Properties

Label 8100.2.d.k.649.3
Level $8100$
Weight $2$
Character 8100.649
Analytic conductor $64.679$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [8100,2,Mod(649,8100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8100.649");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8100 = 2^{2} \cdot 3^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8100.d (of order \(2\), degree \(1\), not 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: \(64.6788256372\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 649.3
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 8100.649
Dual form 8100.2.d.k.649.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.85410i q^{7} +O(q^{10})\) \(q+2.85410i q^{7} +1.85410 q^{11} +0.854102i q^{13} -1.14590i q^{17} -2.00000 q^{19} -4.85410i q^{23} -3.70820 q^{29} +2.70820 q^{31} +5.85410i q^{37} -11.5623 q^{41} +0.854102i q^{43} +6.70820i q^{47} -1.14590 q^{49} +4.85410i q^{53} -1.14590 q^{59} +0.854102 q^{61} +7.00000i q^{67} +9.00000 q^{71} +2.70820i q^{73} +5.29180i q^{77} -11.7082 q^{79} +6.70820i q^{83} +12.0000 q^{89} -2.43769 q^{91} +10.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 6 q^{11} - 8 q^{19} + 12 q^{29} - 16 q^{31} - 6 q^{41} - 18 q^{49} - 18 q^{59} - 10 q^{61} + 36 q^{71} - 20 q^{79} + 48 q^{89} - 50 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/8100\mathbb{Z}\right)^\times\).

\(n\) \(4051\) \(6401\) \(7777\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.85410i 1.07875i 0.842066 + 0.539375i \(0.181339\pi\)
−0.842066 + 0.539375i \(0.818661\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.85410 0.559033 0.279516 0.960141i \(-0.409826\pi\)
0.279516 + 0.960141i \(0.409826\pi\)
\(12\) 0 0
\(13\) 0.854102i 0.236885i 0.992961 + 0.118443i \(0.0377902\pi\)
−0.992961 + 0.118443i \(0.962210\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.14590i − 0.277921i −0.990298 0.138961i \(-0.955624\pi\)
0.990298 0.138961i \(-0.0443761\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.85410i − 1.01215i −0.862489 0.506075i \(-0.831096\pi\)
0.862489 0.506075i \(-0.168904\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −3.70820 −0.688596 −0.344298 0.938860i \(-0.611883\pi\)
−0.344298 + 0.938860i \(0.611883\pi\)
\(30\) 0 0
\(31\) 2.70820 0.486408 0.243204 0.969975i \(-0.421802\pi\)
0.243204 + 0.969975i \(0.421802\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 5.85410i 0.962408i 0.876609 + 0.481204i \(0.159800\pi\)
−0.876609 + 0.481204i \(0.840200\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −11.5623 −1.80573 −0.902864 0.429925i \(-0.858540\pi\)
−0.902864 + 0.429925i \(0.858540\pi\)
\(42\) 0 0
\(43\) 0.854102i 0.130249i 0.997877 + 0.0651247i \(0.0207445\pi\)
−0.997877 + 0.0651247i \(0.979255\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.70820i 0.978492i 0.872146 + 0.489246i \(0.162728\pi\)
−0.872146 + 0.489246i \(0.837272\pi\)
\(48\) 0 0
\(49\) −1.14590 −0.163700
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 4.85410i 0.666762i 0.942792 + 0.333381i \(0.108190\pi\)
−0.942792 + 0.333381i \(0.891810\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −1.14590 −0.149183 −0.0745916 0.997214i \(-0.523765\pi\)
−0.0745916 + 0.997214i \(0.523765\pi\)
\(60\) 0 0
\(61\) 0.854102 0.109357 0.0546783 0.998504i \(-0.482587\pi\)
0.0546783 + 0.998504i \(0.482587\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 7.00000i 0.855186i 0.903971 + 0.427593i \(0.140638\pi\)
−0.903971 + 0.427593i \(0.859362\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) 2.70820i 0.316971i 0.987361 + 0.158486i \(0.0506612\pi\)
−0.987361 + 0.158486i \(0.949339\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.29180i 0.603056i
\(78\) 0 0
\(79\) −11.7082 −1.31728 −0.658638 0.752460i \(-0.728867\pi\)
−0.658638 + 0.752460i \(0.728867\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.70820i 0.736321i 0.929762 + 0.368161i \(0.120012\pi\)
−0.929762 + 0.368161i \(0.879988\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) −2.43769 −0.255540
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 4.85410 0.483001 0.241501 0.970401i \(-0.422360\pi\)
0.241501 + 0.970401i \(0.422360\pi\)
\(102\) 0 0
\(103\) 0.145898i 0.0143758i 0.999974 + 0.00718788i \(0.00228799\pi\)
−0.999974 + 0.00718788i \(0.997712\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 18.7082i − 1.80859i −0.426908 0.904295i \(-0.640397\pi\)
0.426908 0.904295i \(-0.359603\pi\)
\(108\) 0 0
\(109\) −3.85410 −0.369156 −0.184578 0.982818i \(-0.559092\pi\)
−0.184578 + 0.982818i \(0.559092\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 1.85410i − 0.174419i −0.996190 0.0872096i \(-0.972205\pi\)
0.996190 0.0872096i \(-0.0277950\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.27051 0.299807
\(120\) 0 0
\(121\) −7.56231 −0.687482
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 7.00000i 0.621150i 0.950549 + 0.310575i \(0.100522\pi\)
−0.950549 + 0.310575i \(0.899478\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.29180 −0.200235 −0.100118 0.994976i \(-0.531922\pi\)
−0.100118 + 0.994976i \(0.531922\pi\)
\(132\) 0 0
\(133\) − 5.70820i − 0.494964i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 2.56231i − 0.218913i −0.993992 0.109456i \(-0.965089\pi\)
0.993992 0.109456i \(-0.0349110\pi\)
\(138\) 0 0
\(139\) −17.2705 −1.46487 −0.732433 0.680839i \(-0.761615\pi\)
−0.732433 + 0.680839i \(0.761615\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.58359i 0.132427i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −15.0000 −1.22885 −0.614424 0.788976i \(-0.710612\pi\)
−0.614424 + 0.788976i \(0.710612\pi\)
\(150\) 0 0
\(151\) −8.14590 −0.662904 −0.331452 0.943472i \(-0.607538\pi\)
−0.331452 + 0.943472i \(0.607538\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 16.7082i 1.33346i 0.745299 + 0.666730i \(0.232307\pi\)
−0.745299 + 0.666730i \(0.767693\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 13.8541 1.09186
\(162\) 0 0
\(163\) − 14.4164i − 1.12918i −0.825371 0.564590i \(-0.809034\pi\)
0.825371 0.564590i \(-0.190966\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 22.8541i 1.76850i 0.467011 + 0.884252i \(0.345331\pi\)
−0.467011 + 0.884252i \(0.654669\pi\)
\(168\) 0 0
\(169\) 12.2705 0.943885
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 15.7082i − 1.19427i −0.802140 0.597136i \(-0.796305\pi\)
0.802140 0.597136i \(-0.203695\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.29180 0.395527 0.197764 0.980250i \(-0.436632\pi\)
0.197764 + 0.980250i \(0.436632\pi\)
\(180\) 0 0
\(181\) −14.4164 −1.07156 −0.535782 0.844357i \(-0.679983\pi\)
−0.535782 + 0.844357i \(0.679983\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 2.12461i − 0.155367i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.4164 −1.40492 −0.702461 0.711722i \(-0.747915\pi\)
−0.702461 + 0.711722i \(0.747915\pi\)
\(192\) 0 0
\(193\) − 16.7082i − 1.20268i −0.798992 0.601341i \(-0.794633\pi\)
0.798992 0.601341i \(-0.205367\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 25.8541i − 1.84203i −0.389529 0.921014i \(-0.627362\pi\)
0.389529 0.921014i \(-0.372638\pi\)
\(198\) 0 0
\(199\) −15.8541 −1.12387 −0.561934 0.827182i \(-0.689942\pi\)
−0.561934 + 0.827182i \(0.689942\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 10.5836i − 0.742823i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3.70820 −0.256502
\(210\) 0 0
\(211\) −22.7082 −1.56330 −0.781649 0.623719i \(-0.785621\pi\)
−0.781649 + 0.623719i \(0.785621\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 7.72949i 0.524712i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.978714 0.0658354
\(222\) 0 0
\(223\) − 3.29180i − 0.220435i −0.993907 0.110217i \(-0.964845\pi\)
0.993907 0.110217i \(-0.0351547\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 3.00000i − 0.199117i −0.995032 0.0995585i \(-0.968257\pi\)
0.995032 0.0995585i \(-0.0317430\pi\)
\(228\) 0 0
\(229\) 8.41641 0.556172 0.278086 0.960556i \(-0.410300\pi\)
0.278086 + 0.960556i \(0.410300\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.1246i 1.31841i 0.751965 + 0.659204i \(0.229106\pi\)
−0.751965 + 0.659204i \(0.770894\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −13.4164 −0.867835 −0.433918 0.900953i \(-0.642869\pi\)
−0.433918 + 0.900953i \(0.642869\pi\)
\(240\) 0 0
\(241\) −17.4164 −1.12189 −0.560945 0.827853i \(-0.689562\pi\)
−0.560945 + 0.827853i \(0.689562\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 1.70820i − 0.108690i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 10.8541 0.685105 0.342552 0.939499i \(-0.388709\pi\)
0.342552 + 0.939499i \(0.388709\pi\)
\(252\) 0 0
\(253\) − 9.00000i − 0.565825i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 25.4164i 1.58543i 0.609591 + 0.792716i \(0.291334\pi\)
−0.609591 + 0.792716i \(0.708666\pi\)
\(258\) 0 0
\(259\) −16.7082 −1.03820
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 2.29180i 0.141318i 0.997501 + 0.0706591i \(0.0225102\pi\)
−0.997501 + 0.0706591i \(0.977490\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 27.2705 1.66271 0.831356 0.555740i \(-0.187565\pi\)
0.831356 + 0.555740i \(0.187565\pi\)
\(270\) 0 0
\(271\) −8.41641 −0.511260 −0.255630 0.966775i \(-0.582283\pi\)
−0.255630 + 0.966775i \(0.582283\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 23.4164i 1.40696i 0.710717 + 0.703478i \(0.248371\pi\)
−0.710717 + 0.703478i \(0.751629\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.70820 −0.400178 −0.200089 0.979778i \(-0.564123\pi\)
−0.200089 + 0.979778i \(0.564123\pi\)
\(282\) 0 0
\(283\) − 11.8541i − 0.704653i −0.935877 0.352327i \(-0.885391\pi\)
0.935877 0.352327i \(-0.114609\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 33.0000i − 1.94793i
\(288\) 0 0
\(289\) 15.6869 0.922760
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 20.1246i 1.17569i 0.808973 + 0.587846i \(0.200024\pi\)
−0.808973 + 0.587846i \(0.799976\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.14590 0.239763
\(300\) 0 0
\(301\) −2.43769 −0.140506
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 10.0000i 0.570730i 0.958419 + 0.285365i \(0.0921148\pi\)
−0.958419 + 0.285365i \(0.907885\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0.708204 0.0401586 0.0200793 0.999798i \(-0.493608\pi\)
0.0200793 + 0.999798i \(0.493608\pi\)
\(312\) 0 0
\(313\) − 20.8541i − 1.17874i −0.807862 0.589372i \(-0.799375\pi\)
0.807862 0.589372i \(-0.200625\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.58359i 0.0889434i 0.999011 + 0.0444717i \(0.0141605\pi\)
−0.999011 + 0.0444717i \(0.985840\pi\)
\(318\) 0 0
\(319\) −6.87539 −0.384948
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 2.29180i 0.127519i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −19.1459 −1.05555
\(330\) 0 0
\(331\) −17.4164 −0.957292 −0.478646 0.878008i \(-0.658872\pi\)
−0.478646 + 0.878008i \(0.658872\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.2918i 0.996418i 0.867057 + 0.498209i \(0.166009\pi\)
−0.867057 + 0.498209i \(0.833991\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.02129 0.271918
\(342\) 0 0
\(343\) 16.7082i 0.902158i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 35.5623i 1.90908i 0.298075 + 0.954542i \(0.403655\pi\)
−0.298075 + 0.954542i \(0.596345\pi\)
\(348\) 0 0
\(349\) −23.7082 −1.26907 −0.634536 0.772894i \(-0.718809\pi\)
−0.634536 + 0.772894i \(0.718809\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 17.5623i − 0.934747i −0.884060 0.467374i \(-0.845200\pi\)
0.884060 0.467374i \(-0.154800\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 16.1459 0.852148 0.426074 0.904688i \(-0.359896\pi\)
0.426074 + 0.904688i \(0.359896\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 9.56231i 0.499148i 0.968356 + 0.249574i \(0.0802906\pi\)
−0.968356 + 0.249574i \(0.919709\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −13.8541 −0.719269
\(372\) 0 0
\(373\) 16.1246i 0.834901i 0.908700 + 0.417450i \(0.137076\pi\)
−0.908700 + 0.417450i \(0.862924\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 3.16718i − 0.163118i
\(378\) 0 0
\(379\) −20.2705 −1.04123 −0.520613 0.853793i \(-0.674297\pi\)
−0.520613 + 0.853793i \(0.674297\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 30.9787i − 1.58294i −0.611209 0.791469i \(-0.709317\pi\)
0.611209 0.791469i \(-0.290683\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −6.70820 −0.340119 −0.170060 0.985434i \(-0.554396\pi\)
−0.170060 + 0.985434i \(0.554396\pi\)
\(390\) 0 0
\(391\) −5.56231 −0.281298
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 31.5410i 1.58300i 0.611170 + 0.791499i \(0.290699\pi\)
−0.611170 + 0.791499i \(0.709301\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −35.1246 −1.75404 −0.877020 0.480454i \(-0.840472\pi\)
−0.877020 + 0.480454i \(0.840472\pi\)
\(402\) 0 0
\(403\) 2.31308i 0.115223i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 10.8541i 0.538018i
\(408\) 0 0
\(409\) −14.2705 −0.705631 −0.352816 0.935693i \(-0.614776\pi\)
−0.352816 + 0.935693i \(0.614776\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 3.27051i − 0.160931i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −4.14590 −0.202540 −0.101270 0.994859i \(-0.532291\pi\)
−0.101270 + 0.994859i \(0.532291\pi\)
\(420\) 0 0
\(421\) 5.70820 0.278201 0.139100 0.990278i \(-0.455579\pi\)
0.139100 + 0.990278i \(0.455579\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 2.43769i 0.117968i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 21.0000 1.01153 0.505767 0.862670i \(-0.331209\pi\)
0.505767 + 0.862670i \(0.331209\pi\)
\(432\) 0 0
\(433\) − 37.7082i − 1.81214i −0.423127 0.906070i \(-0.639068\pi\)
0.423127 0.906070i \(-0.360932\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 9.70820i 0.464406i
\(438\) 0 0
\(439\) 9.56231 0.456384 0.228192 0.973616i \(-0.426719\pi\)
0.228192 + 0.973616i \(0.426719\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 19.1459i 0.909649i 0.890581 + 0.454825i \(0.150298\pi\)
−0.890581 + 0.454825i \(0.849702\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 7.14590 0.337236 0.168618 0.985681i \(-0.446070\pi\)
0.168618 + 0.985681i \(0.446070\pi\)
\(450\) 0 0
\(451\) −21.4377 −1.00946
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000i 0.467780i 0.972263 + 0.233890i \(0.0751456\pi\)
−0.972263 + 0.233890i \(0.924854\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 23.5623 1.09741 0.548703 0.836017i \(-0.315122\pi\)
0.548703 + 0.836017i \(0.315122\pi\)
\(462\) 0 0
\(463\) 3.14590i 0.146202i 0.997325 + 0.0731011i \(0.0232896\pi\)
−0.997325 + 0.0731011i \(0.976710\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 12.0000i − 0.555294i −0.960683 0.277647i \(-0.910445\pi\)
0.960683 0.277647i \(-0.0895545\pi\)
\(468\) 0 0
\(469\) −19.9787 −0.922531
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 1.58359i 0.0728136i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −16.1459 −0.737725 −0.368862 0.929484i \(-0.620253\pi\)
−0.368862 + 0.929484i \(0.620253\pi\)
\(480\) 0 0
\(481\) −5.00000 −0.227980
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 3.41641i − 0.154812i −0.997000 0.0774061i \(-0.975336\pi\)
0.997000 0.0774061i \(-0.0246638\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −17.8328 −0.804784 −0.402392 0.915468i \(-0.631821\pi\)
−0.402392 + 0.915468i \(0.631821\pi\)
\(492\) 0 0
\(493\) 4.24922i 0.191375i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 25.6869i 1.15222i
\(498\) 0 0
\(499\) −5.43769 −0.243425 −0.121712 0.992565i \(-0.538839\pi\)
−0.121712 + 0.992565i \(0.538839\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 24.2705i 1.08217i 0.840968 + 0.541084i \(0.181986\pi\)
−0.840968 + 0.541084i \(0.818014\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −21.0000 −0.930809 −0.465404 0.885098i \(-0.654091\pi\)
−0.465404 + 0.885098i \(0.654091\pi\)
\(510\) 0 0
\(511\) −7.72949 −0.341933
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 12.4377i 0.547009i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 35.1246 1.53884 0.769419 0.638745i \(-0.220546\pi\)
0.769419 + 0.638745i \(0.220546\pi\)
\(522\) 0 0
\(523\) 18.4164i 0.805293i 0.915355 + 0.402647i \(0.131910\pi\)
−0.915355 + 0.402647i \(0.868090\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 3.10333i − 0.135183i
\(528\) 0 0
\(529\) −0.562306 −0.0244481
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 9.87539i − 0.427751i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.12461 −0.0915135
\(540\) 0 0
\(541\) 40.3951 1.73672 0.868361 0.495933i \(-0.165174\pi\)
0.868361 + 0.495933i \(0.165174\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 38.4164i 1.64257i 0.570520 + 0.821283i \(0.306742\pi\)
−0.570520 + 0.821283i \(0.693258\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.41641 0.315950
\(552\) 0 0
\(553\) − 33.4164i − 1.42101i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 40.2492i − 1.70541i −0.522389 0.852707i \(-0.674959\pi\)
0.522389 0.852707i \(-0.325041\pi\)
\(558\) 0 0
\(559\) −0.729490 −0.0308541
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 36.7082i − 1.54707i −0.633756 0.773533i \(-0.718488\pi\)
0.633756 0.773533i \(-0.281512\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.70820 0.155456 0.0777280 0.996975i \(-0.475233\pi\)
0.0777280 + 0.996975i \(0.475233\pi\)
\(570\) 0 0
\(571\) 10.8328 0.453339 0.226670 0.973972i \(-0.427216\pi\)
0.226670 + 0.973972i \(0.427216\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 0.416408i − 0.0173353i −0.999962 0.00866764i \(-0.997241\pi\)
0.999962 0.00866764i \(-0.00275903\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −19.1459 −0.794306
\(582\) 0 0
\(583\) 9.00000i 0.372742i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 15.0000i − 0.619116i −0.950881 0.309558i \(-0.899819\pi\)
0.950881 0.309558i \(-0.100181\pi\)
\(588\) 0 0
\(589\) −5.41641 −0.223179
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 3.27051i − 0.134304i −0.997743 0.0671519i \(-0.978609\pi\)
0.997743 0.0671519i \(-0.0213912\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 20.5623 0.840153 0.420077 0.907489i \(-0.362003\pi\)
0.420077 + 0.907489i \(0.362003\pi\)
\(600\) 0 0
\(601\) 9.85410 0.401957 0.200979 0.979596i \(-0.435588\pi\)
0.200979 + 0.979596i \(0.435588\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 28.5410i 1.15844i 0.815170 + 0.579222i \(0.196644\pi\)
−0.815170 + 0.579222i \(0.803356\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −5.72949 −0.231790
\(612\) 0 0
\(613\) 11.7082i 0.472890i 0.971645 + 0.236445i \(0.0759823\pi\)
−0.971645 + 0.236445i \(0.924018\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 20.1246i − 0.810186i −0.914275 0.405093i \(-0.867239\pi\)
0.914275 0.405093i \(-0.132761\pi\)
\(618\) 0 0
\(619\) −33.4164 −1.34312 −0.671559 0.740951i \(-0.734375\pi\)
−0.671559 + 0.740951i \(0.734375\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 34.2492i 1.37217i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 6.70820 0.267474
\(630\) 0 0
\(631\) 0.145898 0.00580811 0.00290405 0.999996i \(-0.499076\pi\)
0.00290405 + 0.999996i \(0.499076\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 0.978714i − 0.0387781i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 11.5623 0.456684 0.228342 0.973581i \(-0.426670\pi\)
0.228342 + 0.973581i \(0.426670\pi\)
\(642\) 0 0
\(643\) 37.5623i 1.48131i 0.671884 + 0.740656i \(0.265485\pi\)
−0.671884 + 0.740656i \(0.734515\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 24.7082i − 0.971380i −0.874131 0.485690i \(-0.838568\pi\)
0.874131 0.485690i \(-0.161432\pi\)
\(648\) 0 0
\(649\) −2.12461 −0.0833983
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.14590i 0.162242i 0.996704 + 0.0811208i \(0.0258499\pi\)
−0.996704 + 0.0811208i \(0.974150\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −18.2705 −0.711718 −0.355859 0.934540i \(-0.615812\pi\)
−0.355859 + 0.934540i \(0.615812\pi\)
\(660\) 0 0
\(661\) 25.8328 1.00478 0.502390 0.864641i \(-0.332454\pi\)
0.502390 + 0.864641i \(0.332454\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 18.0000i 0.696963i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.58359 0.0611339
\(672\) 0 0
\(673\) 2.70820i 0.104394i 0.998637 + 0.0521968i \(0.0166223\pi\)
−0.998637 + 0.0521968i \(0.983378\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 16.4164i 0.630934i 0.948937 + 0.315467i \(0.102161\pi\)
−0.948937 + 0.315467i \(0.897839\pi\)
\(678\) 0 0
\(679\) −28.5410 −1.09530
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 29.3951i 1.12477i 0.826874 + 0.562387i \(0.190117\pi\)
−0.826874 + 0.562387i \(0.809883\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −4.14590 −0.157946
\(690\) 0 0
\(691\) 10.1246 0.385158 0.192579 0.981281i \(-0.438315\pi\)
0.192579 + 0.981281i \(0.438315\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 13.2492i 0.501850i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −15.7082 −0.593291 −0.296645 0.954988i \(-0.595868\pi\)
−0.296645 + 0.954988i \(0.595868\pi\)
\(702\) 0 0
\(703\) − 11.7082i − 0.441583i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 13.8541i 0.521037i
\(708\) 0 0
\(709\) 19.0000 0.713560 0.356780 0.934188i \(-0.383875\pi\)
0.356780 + 0.934188i \(0.383875\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 13.1459i − 0.492318i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 20.5623 0.766845 0.383422 0.923573i \(-0.374745\pi\)
0.383422 + 0.923573i \(0.374745\pi\)
\(720\) 0 0
\(721\) −0.416408 −0.0155078
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 37.5623i − 1.39311i −0.717504 0.696554i \(-0.754715\pi\)
0.717504 0.696554i \(-0.245285\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0.978714 0.0361990
\(732\) 0 0
\(733\) − 36.5623i − 1.35046i −0.737607 0.675230i \(-0.764044\pi\)
0.737607 0.675230i \(-0.235956\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 12.9787i 0.478077i
\(738\) 0 0
\(739\) −5.43769 −0.200029 −0.100014 0.994986i \(-0.531889\pi\)
−0.100014 + 0.994986i \(0.531889\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 0.875388i − 0.0321149i −0.999871 0.0160574i \(-0.994889\pi\)
0.999871 0.0160574i \(-0.00511146\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 53.3951 1.95102
\(750\) 0 0
\(751\) −40.9787 −1.49533 −0.747667 0.664074i \(-0.768826\pi\)
−0.747667 + 0.664074i \(0.768826\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 34.2705i 1.24558i 0.782388 + 0.622791i \(0.214001\pi\)
−0.782388 + 0.622791i \(0.785999\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −34.8541 −1.26346 −0.631730 0.775188i \(-0.717655\pi\)
−0.631730 + 0.775188i \(0.717655\pi\)
\(762\) 0 0
\(763\) − 11.0000i − 0.398227i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 0.978714i − 0.0353393i
\(768\) 0 0
\(769\) 26.1459 0.942845 0.471423 0.881907i \(-0.343741\pi\)
0.471423 + 0.881907i \(0.343741\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 33.5410i − 1.20639i −0.797595 0.603193i \(-0.793895\pi\)
0.797595 0.603193i \(-0.206105\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 23.1246 0.828525
\(780\) 0 0
\(781\) 16.6869 0.597105
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 2.14590i 0.0764930i 0.999268 + 0.0382465i \(0.0121772\pi\)
−0.999268 + 0.0382465i \(0.987823\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.29180 0.188155
\(792\) 0 0
\(793\) 0.729490i 0.0259050i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 9.27051i 0.328378i 0.986429 + 0.164189i \(0.0525007\pi\)
−0.986429 + 0.164189i \(0.947499\pi\)
\(798\) 0 0
\(799\) 7.68692 0.271944
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.02129i 0.177197i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −8.12461 −0.285646 −0.142823 0.989748i \(-0.545618\pi\)
−0.142823 + 0.989748i \(0.545618\pi\)
\(810\) 0 0
\(811\) −16.9787 −0.596203 −0.298102 0.954534i \(-0.596353\pi\)
−0.298102 + 0.954534i \(0.596353\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 1.70820i − 0.0597625i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 4.41641 0.154134 0.0770668 0.997026i \(-0.475445\pi\)
0.0770668 + 0.997026i \(0.475445\pi\)
\(822\) 0 0
\(823\) 17.5410i 0.611442i 0.952121 + 0.305721i \(0.0988974\pi\)
−0.952121 + 0.305721i \(0.901103\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 44.3951i 1.54377i 0.635762 + 0.771885i \(0.280686\pi\)
−0.635762 + 0.771885i \(0.719314\pi\)
\(828\) 0 0
\(829\) 5.41641 0.188120 0.0940598 0.995567i \(-0.470016\pi\)
0.0940598 + 0.995567i \(0.470016\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.31308i 0.0454956i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −53.1246 −1.83407 −0.917033 0.398812i \(-0.869423\pi\)
−0.917033 + 0.398812i \(0.869423\pi\)
\(840\) 0 0
\(841\) −15.2492 −0.525835
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 21.5836i − 0.741621i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 28.4164 0.974102
\(852\) 0 0
\(853\) − 24.1246i − 0.826011i −0.910729 0.413005i \(-0.864479\pi\)
0.910729 0.413005i \(-0.135521\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.14590i 0.141621i 0.997490 + 0.0708106i \(0.0225586\pi\)
−0.997490 + 0.0708106i \(0.977441\pi\)
\(858\) 0 0
\(859\) −9.58359 −0.326988 −0.163494 0.986544i \(-0.552276\pi\)
−0.163494 + 0.986544i \(0.552276\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 15.2705i 0.519814i 0.965634 + 0.259907i \(0.0836919\pi\)
−0.965634 + 0.259907i \(0.916308\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −21.7082 −0.736400
\(870\) 0 0
\(871\) −5.97871 −0.202581
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 14.2705i − 0.481881i −0.970540 0.240940i \(-0.922544\pi\)
0.970540 0.240940i \(-0.0774558\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 26.1246 0.880161 0.440080 0.897958i \(-0.354950\pi\)
0.440080 + 0.897958i \(0.354950\pi\)
\(882\) 0 0
\(883\) 31.1246i 1.04743i 0.851895 + 0.523713i \(0.175454\pi\)
−0.851895 + 0.523713i \(0.824546\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.978714i 0.0328620i 0.999865 + 0.0164310i \(0.00523038\pi\)
−0.999865 + 0.0164310i \(0.994770\pi\)
\(888\) 0 0
\(889\) −19.9787 −0.670065
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 13.4164i − 0.448963i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −10.0426 −0.334939
\(900\) 0 0
\(901\) 5.56231 0.185307
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 13.9787i 0.464156i 0.972697 + 0.232078i \(0.0745524\pi\)
−0.972697 + 0.232078i \(0.925448\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −26.1246 −0.865547 −0.432774 0.901503i \(-0.642465\pi\)
−0.432774 + 0.901503i \(0.642465\pi\)
\(912\) 0 0
\(913\) 12.4377i 0.411628i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 6.54102i − 0.216003i
\(918\) 0 0
\(919\) 24.5623 0.810236 0.405118 0.914264i \(-0.367230\pi\)
0.405118 + 0.914264i \(0.367230\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 7.68692i 0.253018i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 41.2918 1.35474 0.677370 0.735643i \(-0.263120\pi\)
0.677370 + 0.735643i \(0.263120\pi\)
\(930\) 0 0
\(931\) 2.29180 0.0751106
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 43.8328i − 1.43196i −0.698123 0.715978i \(-0.745981\pi\)
0.698123 0.715978i \(-0.254019\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 29.3951 0.958254 0.479127 0.877746i \(-0.340953\pi\)
0.479127 + 0.877746i \(0.340953\pi\)
\(942\) 0 0
\(943\) 56.1246i 1.82767i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 46.4164i − 1.50833i −0.656685 0.754165i \(-0.728042\pi\)
0.656685 0.754165i \(-0.271958\pi\)
\(948\) 0 0
\(949\) −2.31308 −0.0750858
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 3.16718i − 0.102595i −0.998683 0.0512976i \(-0.983664\pi\)
0.998683 0.0512976i \(-0.0163357\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 7.31308 0.236152
\(960\) 0 0
\(961\) −23.6656 −0.763407
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 43.8328i − 1.40957i −0.709422 0.704784i \(-0.751044\pi\)
0.709422 0.704784i \(-0.248956\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −21.9787 −0.705330 −0.352665 0.935750i \(-0.614725\pi\)
−0.352665 + 0.935750i \(0.614725\pi\)
\(972\) 0 0
\(973\) − 49.2918i − 1.58022i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 37.6869i − 1.20571i −0.797850 0.602856i \(-0.794029\pi\)
0.797850 0.602856i \(-0.205971\pi\)
\(978\) 0 0
\(979\) 22.2492 0.711088
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 9.54102i − 0.304311i −0.988357 0.152156i \(-0.951379\pi\)
0.988357 0.152156i \(-0.0486215\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.14590 0.131832
\(990\) 0 0
\(991\) −23.1459 −0.735254 −0.367627 0.929973i \(-0.619830\pi\)
−0.367627 + 0.929973i \(0.619830\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.12461i 0.0989574i 0.998775 + 0.0494787i \(0.0157560\pi\)
−0.998775 + 0.0494787i \(0.984244\pi\)
\(998\) 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 8100.2.d.k.649.3 4
3.2 odd 2 8100.2.d.n.649.3 4
5.2 odd 4 8100.2.a.q.1.1 yes 2
5.3 odd 4 8100.2.a.o.1.2 2
5.4 even 2 inner 8100.2.d.k.649.2 4
15.2 even 4 8100.2.a.r.1.1 yes 2
15.8 even 4 8100.2.a.p.1.2 yes 2
15.14 odd 2 8100.2.d.n.649.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8100.2.a.o.1.2 2 5.3 odd 4
8100.2.a.p.1.2 yes 2 15.8 even 4
8100.2.a.q.1.1 yes 2 5.2 odd 4
8100.2.a.r.1.1 yes 2 15.2 even 4
8100.2.d.k.649.2 4 5.4 even 2 inner
8100.2.d.k.649.3 4 1.1 even 1 trivial
8100.2.d.n.649.2 4 15.14 odd 2
8100.2.d.n.649.3 4 3.2 odd 2