Properties

Label 4000.1.cq.a.1377.1
Level $4000$
Weight $1$
Character 4000.1377
Analytic conductor $1.996$
Analytic rank $0$
Dimension $40$
Projective image $D_{100}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4000,1,Mod(33,4000)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4000, base_ring=CyclotomicField(100))
 
chi = DirichletCharacter(H, H._module([0, 0, 83]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4000.33");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4000 = 2^{5} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 4000.cq (of order \(100\), degree \(40\), minimal)

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: no
Analytic conductor: \(1.99626005053\)
Analytic rank: \(0\)
Dimension: \(40\)
Coefficient field: \(\Q(\zeta_{100})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{40} - x^{30} + x^{20} - x^{10} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{100}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{100} - \cdots)\)

Embedding invariants

Embedding label 1377.1
Root \(-0.904827 + 0.425779i\) of defining polynomial
Character \(\chi\) \(=\) 4000.1377
Dual form 4000.1.cq.a.1313.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.844328 - 0.535827i) q^{5} +(-0.904827 - 0.425779i) q^{9} +O(q^{10})\) \(q+(-0.844328 - 0.535827i) q^{5} +(-0.904827 - 0.425779i) q^{9} +(-0.344863 - 0.957895i) q^{13} +(-1.24443 + 1.41153i) q^{17} +(0.425779 + 0.904827i) q^{25} +(-0.0922765 + 0.233064i) q^{29} +(1.01385 + 1.30704i) q^{37} +(0.374023 + 1.96070i) q^{41} +(0.535827 + 0.844328i) q^{45} +(-0.587785 + 0.809017i) q^{49} +(-0.404329 + 0.683684i) q^{53} +(0.316423 - 1.65875i) q^{61} +(-0.222088 + 0.993564i) q^{65} +(-1.91206 + 0.555506i) q^{73} +(0.637424 + 0.770513i) q^{81} +(1.80704 - 0.524995i) q^{85} +(0.659566 - 1.19975i) q^{89} +(-0.221601 - 0.512091i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 40 q+O(q^{10}) \) Copy content Toggle raw display \( 40 q + 10 q^{85} + 10 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4000\mathbb{Z}\right)^\times\).

\(n\) \(1377\) \(2501\) \(2751\)
\(\chi(n)\) \(e\left(\frac{1}{100}\right)\) \(1\) \(1\)

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
\(3\) 0 0 0.218143 0.975917i \(-0.430000\pi\)
−0.218143 + 0.975917i \(0.570000\pi\)
\(4\) 0 0
\(5\) −0.844328 0.535827i −0.844328 0.535827i
\(6\) 0 0
\(7\) 0 0 −0.453990 0.891007i \(-0.650000\pi\)
0.453990 + 0.891007i \(0.350000\pi\)
\(8\) 0 0
\(9\) −0.904827 0.425779i −0.904827 0.425779i
\(10\) 0 0
\(11\) 0 0 −0.684547 0.728969i \(-0.740000\pi\)
0.684547 + 0.728969i \(0.260000\pi\)
\(12\) 0 0
\(13\) −0.344863 0.957895i −0.344863 0.957895i −0.982287 0.187381i \(-0.940000\pi\)
0.637424 0.770513i \(-0.280000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.24443 + 1.41153i −1.24443 + 1.41153i −0.368125 + 0.929776i \(0.620000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(18\) 0 0
\(19\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.827081 0.562083i \(-0.190000\pi\)
−0.827081 + 0.562083i \(0.810000\pi\)
\(24\) 0 0
\(25\) 0.425779 + 0.904827i 0.425779 + 0.904827i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −0.0922765 + 0.233064i −0.0922765 + 0.233064i −0.968583 0.248690i \(-0.920000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(30\) 0 0
\(31\) 0 0 0.998027 0.0627905i \(-0.0200000\pi\)
−0.998027 + 0.0627905i \(0.980000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 1.01385 + 1.30704i 1.01385 + 1.30704i 0.951057 + 0.309017i \(0.100000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.374023 + 1.96070i 0.374023 + 1.96070i 0.248690 + 0.968583i \(0.420000\pi\)
0.125333 + 0.992115i \(0.460000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.156434 0.987688i \(-0.550000\pi\)
0.156434 + 0.987688i \(0.450000\pi\)
\(44\) 0 0
\(45\) 0.535827 + 0.844328i 0.535827 + 0.844328i
\(46\) 0 0
\(47\) 0 0 −0.0941083 0.995562i \(-0.530000\pi\)
0.0941083 + 0.995562i \(0.470000\pi\)
\(48\) 0 0
\(49\) −0.587785 + 0.809017i −0.587785 + 0.809017i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −0.404329 + 0.683684i −0.404329 + 0.683684i −0.992115 0.125333i \(-0.960000\pi\)
0.587785 + 0.809017i \(0.300000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(60\) 0 0
\(61\) 0.316423 1.65875i 0.316423 1.65875i −0.368125 0.929776i \(-0.620000\pi\)
0.684547 0.728969i \(-0.260000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −0.222088 + 0.993564i −0.222088 + 0.993564i
\(66\) 0 0
\(67\) 0 0 0.397148 0.917755i \(-0.370000\pi\)
−0.397148 + 0.917755i \(0.630000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.770513 0.637424i \(-0.780000\pi\)
0.770513 + 0.637424i \(0.220000\pi\)
\(72\) 0 0
\(73\) −1.91206 + 0.555506i −1.91206 + 0.555506i −0.929776 + 0.368125i \(0.880000\pi\)
−0.982287 + 0.187381i \(0.940000\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(80\) 0 0
\(81\) 0.637424 + 0.770513i 0.637424 + 0.770513i
\(82\) 0 0
\(83\) 0 0 0.975917 0.218143i \(-0.0700000\pi\)
−0.975917 + 0.218143i \(0.930000\pi\)
\(84\) 0 0
\(85\) 1.80704 0.524995i 1.80704 0.524995i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0.659566 1.19975i 0.659566 1.19975i −0.309017 0.951057i \(-0.600000\pi\)
0.968583 0.248690i \(-0.0800000\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.221601 0.512091i −0.221601 0.512091i 0.770513 0.637424i \(-0.220000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −0.541587 + 1.66683i −0.541587 + 1.66683i 0.187381 + 0.982287i \(0.440000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(102\) 0 0
\(103\) 0 0 0.750111 0.661312i \(-0.230000\pi\)
−0.750111 + 0.661312i \(0.770000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(108\) 0 0
\(109\) 0.159781 1.26480i 0.159781 1.26480i −0.684547 0.728969i \(-0.740000\pi\)
0.844328 0.535827i \(-0.180000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.557707 + 1.91964i −0.557707 + 1.91964i −0.248690 + 0.968583i \(0.580000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.0958101 + 1.01356i −0.0958101 + 1.01356i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.0627905 + 0.998027i −0.0627905 + 0.998027i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0.125333 0.992115i 0.125333 0.992115i
\(126\) 0 0
\(127\) 0 0 0.0314108 0.999507i \(-0.490000\pi\)
−0.0314108 + 0.999507i \(0.510000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.248690 0.968583i \(-0.420000\pi\)
−0.248690 + 0.968583i \(0.580000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −0.418963 0.121720i −0.418963 0.121720i 0.0627905 0.998027i \(-0.480000\pi\)
−0.481754 + 0.876307i \(0.660000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0.202793 0.147338i 0.202793 0.147338i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.76854 0.574633i −1.76854 0.574633i −0.770513 0.637424i \(-0.780000\pi\)
−0.998027 + 0.0627905i \(0.980000\pi\)
\(150\) 0 0
\(151\) 0 0 0.951057 0.309017i \(-0.100000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(152\) 0 0
\(153\) 1.72700 0.747338i 1.72700 0.747338i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.0620481 + 0.00982745i 0.0620481 + 0.00982745i 0.187381 0.982287i \(-0.440000\pi\)
−0.125333 + 0.992115i \(0.540000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 0.562083 0.827081i \(-0.310000\pi\)
−0.562083 + 0.827081i \(0.690000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.218143 0.975917i \(-0.570000\pi\)
0.218143 + 0.975917i \(0.430000\pi\)
\(168\) 0 0
\(169\) −0.0281181 + 0.0232613i −0.0281181 + 0.0232613i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.80704 0.650576i −1.80704 0.650576i −0.998027 0.0627905i \(-0.980000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(180\) 0 0
\(181\) −1.35556 + 0.536702i −1.35556 + 0.536702i −0.929776 0.368125i \(-0.880000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.155670 1.64682i −0.155670 1.64682i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.481754 0.876307i \(-0.660000\pi\)
0.481754 + 0.876307i \(0.340000\pi\)
\(192\) 0 0
\(193\) −1.38015 + 1.38015i −1.38015 + 1.38015i −0.535827 + 0.844328i \(0.680000\pi\)
−0.844328 + 0.535827i \(0.820000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.29130 + 0.763675i 1.29130 + 0.763675i 0.982287 0.187381i \(-0.0600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(198\) 0 0
\(199\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0.734796 1.85588i 0.734796 1.85588i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.125333 0.992115i \(-0.540000\pi\)
0.125333 + 0.992115i \(0.460000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.78126 + 0.705249i 1.78126 + 0.705249i
\(222\) 0 0
\(223\) 0 0 −0.790155 0.612907i \(-0.790000\pi\)
0.790155 + 0.612907i \(0.210000\pi\)
\(224\) 0 0
\(225\) 1.00000i 1.00000i
\(226\) 0 0
\(227\) 0 0 −0.562083 0.827081i \(-0.690000\pi\)
0.562083 + 0.827081i \(0.310000\pi\)
\(228\) 0 0
\(229\) 0.183098 + 0.713118i 0.183098 + 0.713118i 0.992115 + 0.125333i \(0.0400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.992115 + 0.874667i 0.992115 + 0.874667i 0.992115 0.125333i \(-0.0400000\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(240\) 0 0
\(241\) 0.836475 1.77760i 0.836475 1.77760i 0.248690 0.968583i \(-0.420000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0.929776 0.368125i 0.929776 0.368125i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −0.167702 + 0.0854486i −0.167702 + 0.0854486i −0.535827 0.844328i \(-0.680000\pi\)
0.368125 + 0.929776i \(0.380000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0.182728 0.171593i 0.182728 0.171593i
\(262\) 0 0
\(263\) 0 0 0.940881 0.338738i \(-0.110000\pi\)
−0.940881 + 0.338738i \(0.890000\pi\)
\(264\) 0 0
\(265\) 0.707723 0.360603i 0.707723 0.360603i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.904827 + 0.574221i 0.904827 + 0.574221i 0.904827 0.425779i \(-0.140000\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 −0.248690 0.968583i \(-0.580000\pi\)
0.248690 + 0.968583i \(0.420000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.968583 0.751310i −0.968583 0.751310i 1.00000i \(-0.5\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.0462295 0.734796i −0.0462295 0.734796i −0.951057 0.309017i \(-0.900000\pi\)
0.904827 0.425779i \(-0.140000\pi\)
\(282\) 0 0
\(283\) 0 0 −0.509041 0.860742i \(-0.670000\pi\)
0.509041 + 0.860742i \(0.330000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.318475 2.52099i −0.318475 2.52099i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.56085 + 0.247215i −1.56085 + 0.247215i −0.876307 0.481754i \(-0.840000\pi\)
−0.684547 + 0.728969i \(0.740000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.15596 + 1.23098i −1.15596 + 1.23098i
\(306\) 0 0
\(307\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.982287 0.187381i \(-0.940000\pi\)
0.982287 + 0.187381i \(0.0600000\pi\)
\(312\) 0 0
\(313\) 0.907533 0.0285204i 0.907533 0.0285204i 0.425779 0.904827i \(-0.360000\pi\)
0.481754 + 0.876307i \(0.340000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −0.287137 0.124255i −0.287137 0.124255i 0.248690 0.968583i \(-0.420000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0.719893 0.719893i 0.719893 0.719893i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.770513 0.637424i \(-0.220000\pi\)
−0.770513 + 0.637424i \(0.780000\pi\)
\(332\) 0 0
\(333\) −0.360844 1.61432i −0.360844 1.61432i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −1.11033 + 1.63380i −1.11033 + 1.63380i −0.425779 + 0.904827i \(0.640000\pi\)
−0.684547 + 0.728969i \(0.740000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.917755 0.397148i \(-0.130000\pi\)
−0.917755 + 0.397148i \(0.870000\pi\)
\(348\) 0 0
\(349\) 1.86842 0.607087i 1.86842 0.607087i 0.876307 0.481754i \(-0.160000\pi\)
0.992115 0.125333i \(-0.0400000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.17847 1.33671i −1.17847 1.33671i −0.929776 0.368125i \(-0.880000\pi\)
−0.248690 0.968583i \(-0.580000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(360\) 0 0
\(361\) −0.425779 0.904827i −0.425779 0.904827i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.91206 + 0.555506i 1.91206 + 0.555506i
\(366\) 0 0
\(367\) 0 0 −0.995562 0.0941083i \(-0.970000\pi\)
0.995562 + 0.0941083i \(0.0300000\pi\)
\(368\) 0 0
\(369\) 0.496398 1.93334i 0.496398 1.93334i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.0559744 1.78113i 0.0559744 1.78113i −0.425779 0.904827i \(-0.640000\pi\)
0.481754 0.876307i \(-0.340000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.255073 + 0.00801600i 0.255073 + 0.00801600i
\(378\) 0 0
\(379\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.0941083 0.995562i \(-0.470000\pi\)
−0.0941083 + 0.995562i \(0.530000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −0.113629 + 0.0534698i −0.113629 + 0.0534698i −0.481754 0.876307i \(-0.660000\pi\)
0.368125 + 0.929776i \(0.380000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.48175 + 1.30634i −1.48175 + 1.30634i −0.637424 + 0.770513i \(0.720000\pi\)
−0.844328 + 0.535827i \(0.820000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.331159 + 1.01920i 0.331159 + 1.01920i 0.968583 + 0.248690i \(0.0800000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.125333 0.992115i −0.125333 0.992115i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.256543 0.273190i 0.256543 0.273190i −0.587785 0.809017i \(-0.700000\pi\)
0.844328 + 0.535827i \(0.180000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(420\) 0 0
\(421\) 0.516273 + 0.813516i 0.516273 + 0.813516i 0.998027 0.0627905i \(-0.0200000\pi\)
−0.481754 + 0.876307i \(0.660000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −1.80704 0.524995i −1.80704 0.524995i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.368125 0.929776i \(-0.620000\pi\)
0.368125 + 0.929776i \(0.380000\pi\)
\(432\) 0 0
\(433\) −0.595810 + 1.37684i −0.595810 + 1.37684i 0.309017 + 0.951057i \(0.400000\pi\)
−0.904827 + 0.425779i \(0.860000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(440\) 0 0
\(441\) 0.876307 0.481754i 0.876307 0.481754i
\(442\) 0 0
\(443\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(444\) 0 0
\(445\) −1.19975 + 0.659566i −1.19975 + 0.659566i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.856954 + 1.17950i 0.856954 + 1.17950i 0.982287 + 0.187381i \(0.0600000\pi\)
−0.125333 + 0.992115i \(0.540000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.0489435 0.309017i −0.0489435 0.309017i 0.951057 0.309017i \(-0.100000\pi\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −0.844844 + 0.106729i −0.844844 + 0.106729i −0.535827 0.844328i \(-0.680000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.612907 0.790155i \(-0.710000\pi\)
0.612907 + 0.790155i \(0.290000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.860742 0.509041i \(-0.170000\pi\)
−0.860742 + 0.509041i \(0.830000\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.656947 0.446460i 0.656947 0.446460i
\(478\) 0 0
\(479\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(480\) 0 0
\(481\) 0.902371 1.42191i 0.902371 1.42191i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.0872876 + 0.551113i −0.0872876 + 0.551113i
\(486\) 0 0
\(487\) 0 0 −0.338738 0.940881i \(-0.610000\pi\)
0.338738 + 0.940881i \(0.390000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.904827 0.425779i \(-0.860000\pi\)
0.904827 + 0.425779i \(0.140000\pi\)
\(492\) 0 0
\(493\) −0.214145 0.420283i −0.214145 0.420283i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.218143 0.975917i \(-0.430000\pi\)
−0.218143 + 0.975917i \(0.570000\pi\)
\(504\) 0 0
\(505\) 1.35041 1.11716i 1.35041 1.11716i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −1.63742 0.770513i −1.63742 0.770513i −0.637424 0.770513i \(-0.720000\pi\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.80113 0.462452i 1.80113 0.462452i 0.809017 0.587785i \(-0.200000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(522\) 0 0
\(523\) 0 0 0.827081 0.562083i \(-0.190000\pi\)
−0.827081 + 0.562083i \(0.810000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.368125 0.929776i 0.368125 0.929776i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1.74915 1.03445i 1.74915 1.03445i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.371808 1.94908i −0.371808 1.94908i −0.309017 0.951057i \(-0.600000\pi\)
−0.0627905 0.998027i \(-0.520000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −0.812619 + 0.982287i −0.812619 + 0.982287i
\(546\) 0 0
\(547\) 0 0 −0.0941083 0.995562i \(-0.530000\pi\)
0.0941083 + 0.995562i \(0.470000\pi\)
\(548\) 0 0
\(549\) −0.992567 + 1.36615i −0.992567 + 1.36615i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.41352 + 1.41352i 1.41352 + 1.41352i 0.728969 + 0.684547i \(0.240000\pi\)
0.684547 + 0.728969i \(0.260000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.0314108 0.999507i \(-0.510000\pi\)
0.0314108 + 0.999507i \(0.490000\pi\)
\(564\) 0 0
\(565\) 1.49948 1.32197i 1.49948 1.32197i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.313480 0.791759i −0.313480 0.791759i −0.998027 0.0627905i \(-0.980000\pi\)
0.684547 0.728969i \(-0.260000\pi\)
\(570\) 0 0
\(571\) 0 0 −0.770513 0.637424i \(-0.780000\pi\)
0.770513 + 0.637424i \(0.220000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0.105981 0.294372i 0.105981 0.294372i −0.876307 0.481754i \(-0.840000\pi\)
0.982287 + 0.187381i \(0.0600000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0.623990 0.804443i 0.623990 0.804443i
\(586\) 0 0
\(587\) 0 0 −0.827081 0.562083i \(-0.810000\pi\)
0.827081 + 0.562083i \(0.190000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0.191760 1.21072i 0.191760 1.21072i −0.684547 0.728969i \(-0.740000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(600\) 0 0
\(601\) −0.153699 + 0.473036i −0.153699 + 0.473036i −0.998027 0.0627905i \(-0.980000\pi\)
0.844328 + 0.535827i \(0.180000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.587785 0.809017i 0.587785 0.809017i
\(606\) 0 0
\(607\) 0 0 −0.891007 0.453990i \(-0.850000\pi\)
0.891007 + 0.453990i \(0.150000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −0.313633 + 1.07953i −0.313633 + 1.07953i 0.637424 + 0.770513i \(0.280000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.141183 + 1.49356i −0.141183 + 1.49356i 0.587785 + 0.809017i \(0.300000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(618\) 0 0
\(619\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.637424 + 0.770513i −0.637424 + 0.770513i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.10659 0.195450i −3.10659 0.195450i
\(630\) 0 0
\(631\) 0 0 0.248690 0.968583i \(-0.420000\pi\)
−0.248690 + 0.968583i \(0.580000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.977659 + 0.284036i 0.977659 + 0.284036i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.88711 + 0.238398i 1.88711 + 0.238398i 0.982287 0.187381i \(-0.0600000\pi\)
0.904827 + 0.425779i \(0.140000\pi\)
\(642\) 0 0
\(643\) 0 0 0.453990 0.891007i \(-0.350000\pi\)
−0.453990 + 0.891007i \(0.650000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.661312 0.750111i \(-0.730000\pi\)
0.661312 + 0.750111i \(0.270000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.512091 + 0.221601i −0.512091 + 0.221601i −0.637424 0.770513i \(-0.720000\pi\)
0.125333 + 0.992115i \(0.460000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.96661 + 0.311480i 1.96661 + 0.311480i
\(658\) 0 0
\(659\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(660\) 0 0
\(661\) 1.66683 + 0.916350i 1.66683 + 0.916350i 0.982287 + 0.187381i \(0.0600000\pi\)
0.684547 + 0.728969i \(0.260000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.48688 + 0.535311i 1.48688 + 0.535311i 0.951057 0.309017i \(-0.100000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −0.551113 1.89694i −0.551113 1.89694i −0.425779 0.904827i \(-0.640000\pi\)
−0.125333 0.992115i \(-0.540000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.917755 0.397148i \(-0.870000\pi\)
0.917755 + 0.397148i \(0.130000\pi\)
\(684\) 0 0
\(685\) 0.288521 + 0.327263i 0.288521 + 0.327263i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.794335 + 0.151528i 0.794335 + 0.151528i
\(690\) 0 0
\(691\) 0 0 −0.481754 0.876307i \(-0.660000\pi\)
0.481754 + 0.876307i \(0.340000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −3.23303 1.91201i −3.23303 1.91201i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −0.402389 0.292352i −0.402389 0.292352i 0.368125 0.929776i \(-0.380000\pi\)
−0.770513 + 0.637424i \(0.780000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0.488570 0.0931997i 0.488570 0.0931997i 0.0627905 0.998027i \(-0.480000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −0.250172 + 0.0157395i −0.250172 + 0.0157395i
\(726\) 0 0
\(727\) 0 0 −0.562083 0.827081i \(-0.690000\pi\)
0.562083 + 0.827081i \(0.310000\pi\)
\(728\) 0 0
\(729\) −0.248690 0.968583i −0.248690 0.968583i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.33671 + 1.17847i 1.33671 + 1.17847i 0.968583 + 0.248690i \(0.0800000\pi\)
0.368125 + 0.929776i \(0.380000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.891007 0.453990i \(-0.150000\pi\)
−0.891007 + 0.453990i \(0.850000\pi\)
\(744\) 0 0
\(745\) 1.18532 + 1.43281i 1.18532 + 1.43281i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0.388734 0.198070i 0.388734 0.198070i −0.248690 0.968583i \(-0.580000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −0.929324 + 0.872693i −0.929324 + 0.872693i −0.992115 0.125333i \(-0.960000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −1.85859 0.294372i −1.85859 0.294372i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.992567 + 0.629902i 0.992567 + 0.629902i 0.929776 0.368125i \(-0.120000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −0.105793 0.155670i −0.105793 0.155670i 0.770513 0.637424i \(-0.220000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.0471231 0.0415446i −0.0471231 0.0415446i
\(786\) 0 0
\(787\) 0 0 0.790155 0.612907i \(-0.210000\pi\)
−0.790155 + 0.612907i \(0.790000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.69803 + 0.268941i −1.69803 + 0.268941i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.31675 + 0.124470i −1.31675 + 0.124470i −0.728969 0.684547i \(-0.760000\pi\)
−0.587785 + 0.809017i \(0.700000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −1.10762 + 0.804733i −1.10762 + 0.804733i
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −0.871808 1.58581i −0.871808 1.58581i −0.809017 0.587785i \(-0.800000\pi\)
−0.0627905 0.998027i \(-0.520000\pi\)
\(810\) 0 0
\(811\) 0 0 −0.982287 0.187381i \(-0.940000\pi\)
0.982287 + 0.187381i \(0.0600000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.21245 + 1.46560i −1.21245 + 1.46560i −0.368125 + 0.929776i \(0.620000\pi\)
−0.844328 + 0.535827i \(0.820000\pi\)
\(822\) 0 0
\(823\) 0 0 −0.278991 0.960294i \(-0.590000\pi\)
0.278991 + 0.960294i \(0.410000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.940881 0.338738i \(-0.890000\pi\)
0.940881 + 0.338738i \(0.110000\pi\)
\(828\) 0 0
\(829\) −1.63560 + 1.03799i −1.63560 + 1.03799i −0.684547 + 0.728969i \(0.740000\pi\)
−0.951057 + 0.309017i \(0.900000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −0.410494 1.83644i −0.410494 1.83644i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(840\) 0 0
\(841\) 0.683165 + 0.641534i 0.683165 + 0.641534i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0.0362049 0.00457374i 0.0362049 0.00457374i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.21384 1.37684i −1.21384 1.37684i −0.904827 0.425779i \(-0.860000\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.896802 1.76007i 0.896802 1.76007i 0.309017 0.951057i \(-0.400000\pi\)
0.587785 0.809017i \(-0.300000\pi\)
\(858\) 0 0
\(859\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.960294 0.278991i \(-0.910000\pi\)
0.960294 + 0.278991i \(0.0900000\pi\)
\(864\) 0 0
\(865\) 1.17714 + 1.51756i 1.17714 + 1.51756i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −0.0175266 + 0.557707i −0.0175266 + 0.557707i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.99014 + 0.0625427i 1.99014 + 0.0625427i 0.998027 0.0627905i \(-0.0200000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1.84235 0.473036i −1.84235 0.473036i −0.844328 0.535827i \(-0.820000\pi\)
−0.998027 + 0.0627905i \(0.980000\pi\)
\(882\) 0 0
\(883\) 0 0 0.0941083 0.995562i \(-0.470000\pi\)
−0.0941083 + 0.995562i \(0.530000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.278991 0.960294i \(-0.410000\pi\)
−0.278991 + 0.960294i \(0.590000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −0.461880 1.42152i −0.461880 1.42152i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.43211 + 0.273190i 1.43211 + 0.273190i
\(906\) 0 0
\(907\) 0 0 0.156434 0.987688i \(-0.450000\pi\)
−0.156434 + 0.987688i \(0.550000\pi\)
\(908\) 0 0
\(909\) 1.19975 1.27760i 1.19975 1.27760i
\(910\) 0 0
\(911\) 0 0 0.481754 0.876307i \(-0.340000\pi\)
−0.481754 + 0.876307i \(0.660000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −0.750973 + 1.47387i −0.750973 + 1.47387i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.35041 1.11716i −1.35041 1.11716i −0.982287 0.187381i \(-0.940000\pi\)
−0.368125 0.929776i \(-0.620000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −0.0353109 1.12361i −0.0353109 1.12361i −0.844328 0.535827i \(-0.820000\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1.73879 + 0.955910i −1.73879 + 0.955910i −0.809017 + 0.587785i \(0.800000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.509041 0.860742i \(-0.330000\pi\)
−0.509041 + 0.860742i \(0.670000\pi\)
\(948\) 0 0
\(949\) 1.19152 + 1.63998i 1.19152 + 1.63998i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.0637561 0.674469i −0.0637561 0.674469i −0.968583 0.248690i \(-0.920000\pi\)
0.904827 0.425779i \(-0.140000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.992115 0.125333i 0.992115 0.125333i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.90483 0.425779i 1.90483 0.425779i
\(966\) 0 0
\(967\) 0 0 0.860742 0.509041i \(-0.170000\pi\)
−0.860742 + 0.509041i \(0.830000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.368125 0.929776i \(-0.380000\pi\)
−0.368125 + 0.929776i \(0.620000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.155670 + 0.105793i −0.155670 + 0.105793i −0.637424 0.770513i \(-0.720000\pi\)
0.481754 + 0.876307i \(0.340000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −0.683098 + 1.07639i −0.683098 + 1.07639i
\(982\) 0 0
\(983\) 0 0 0.661312 0.750111i \(-0.270000\pi\)
−0.661312 + 0.750111i \(0.730000\pi\)
\(984\) 0 0
\(985\) −0.681087 1.33671i −0.681087 1.33671i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.904827 0.425779i \(-0.860000\pi\)
0.904827 + 0.425779i \(0.140000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −0.0682502 + 0.305334i −0.0682502 + 0.305334i −0.998027 0.0627905i \(-0.980000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(998\) 0 0
\(999\) 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 4000.1.cq.a.1377.1 yes 40
4.3 odd 2 CM 4000.1.cq.a.1377.1 yes 40
125.63 odd 100 inner 4000.1.cq.a.1313.1 40
500.63 even 100 inner 4000.1.cq.a.1313.1 40
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4000.1.cq.a.1313.1 40 125.63 odd 100 inner
4000.1.cq.a.1313.1 40 500.63 even 100 inner
4000.1.cq.a.1377.1 yes 40 1.1 even 1 trivial
4000.1.cq.a.1377.1 yes 40 4.3 odd 2 CM