Properties

Label 4608.2.f.l.2303.3
Level $4608$
Weight $2$
Character 4608.2303
Analytic conductor $36.795$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4608,2,Mod(2303,4608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4608.2303");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.f (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: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2303.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 4608.2303
Dual form 4608.2.f.l.2303.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.82843 q^{5} -1.41421i q^{7} +O(q^{10})\) \(q+2.82843 q^{5} -1.41421i q^{7} -2.00000i q^{13} -1.41421i q^{17} +2.82843 q^{19} +6.00000 q^{23} +3.00000 q^{25} +5.65685 q^{29} +9.89949i q^{31} -4.00000i q^{35} -2.00000i q^{37} -4.24264i q^{41} -8.48528 q^{43} -10.0000 q^{47} +5.00000 q^{49} +5.65685 q^{53} +12.0000i q^{59} -10.0000i q^{61} -5.65685i q^{65} +5.65685 q^{67} +2.00000 q^{71} +10.0000 q^{73} -12.7279i q^{79} -4.00000i q^{83} -4.00000i q^{85} -9.89949i q^{89} -2.82843 q^{91} +8.00000 q^{95} +12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 24 q^{23} + 12 q^{25} - 40 q^{47} + 20 q^{49} + 8 q^{71} + 40 q^{73} + 32 q^{95} + 48 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(3583\) \(4097\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(5\) 2.82843 1.26491 0.632456 0.774597i \(-0.282047\pi\)
0.632456 + 0.774597i \(0.282047\pi\)
\(6\) 0 0
\(7\) − 1.41421i − 0.534522i −0.963624 0.267261i \(-0.913881\pi\)
0.963624 0.267261i \(-0.0861187\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) − 2.00000i − 0.554700i −0.960769 0.277350i \(-0.910544\pi\)
0.960769 0.277350i \(-0.0894562\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 1.41421i − 0.342997i −0.985184 0.171499i \(-0.945139\pi\)
0.985184 0.171499i \(-0.0548609\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 3.00000 0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.65685 1.05045 0.525226 0.850963i \(-0.323981\pi\)
0.525226 + 0.850963i \(0.323981\pi\)
\(30\) 0 0
\(31\) 9.89949i 1.77800i 0.457905 + 0.889001i \(0.348600\pi\)
−0.457905 + 0.889001i \(0.651400\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 4.00000i − 0.676123i
\(36\) 0 0
\(37\) − 2.00000i − 0.328798i −0.986394 0.164399i \(-0.947432\pi\)
0.986394 0.164399i \(-0.0525685\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 4.24264i − 0.662589i −0.943527 0.331295i \(-0.892515\pi\)
0.943527 0.331295i \(-0.107485\pi\)
\(42\) 0 0
\(43\) −8.48528 −1.29399 −0.646997 0.762493i \(-0.723975\pi\)
−0.646997 + 0.762493i \(0.723975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −10.0000 −1.45865 −0.729325 0.684167i \(-0.760166\pi\)
−0.729325 + 0.684167i \(0.760166\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 5.65685 0.777029 0.388514 0.921443i \(-0.372988\pi\)
0.388514 + 0.921443i \(0.372988\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.0000i 1.56227i 0.624364 + 0.781133i \(0.285358\pi\)
−0.624364 + 0.781133i \(0.714642\pi\)
\(60\) 0 0
\(61\) − 10.0000i − 1.28037i −0.768221 0.640184i \(-0.778858\pi\)
0.768221 0.640184i \(-0.221142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 5.65685i − 0.701646i
\(66\) 0 0
\(67\) 5.65685 0.691095 0.345547 0.938401i \(-0.387693\pi\)
0.345547 + 0.938401i \(0.387693\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 12.7279i − 1.43200i −0.698099 0.716002i \(-0.745970\pi\)
0.698099 0.716002i \(-0.254030\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 4.00000i − 0.439057i −0.975606 0.219529i \(-0.929548\pi\)
0.975606 0.219529i \(-0.0704519\pi\)
\(84\) 0 0
\(85\) − 4.00000i − 0.433861i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 9.89949i − 1.04934i −0.851304 0.524672i \(-0.824188\pi\)
0.851304 0.524672i \(-0.175812\pi\)
\(90\) 0 0
\(91\) −2.82843 −0.296500
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 8.00000 0.820783
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) − 7.07107i − 0.696733i −0.937358 0.348367i \(-0.886736\pi\)
0.937358 0.348367i \(-0.113264\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 8.00000i − 0.773389i −0.922208 0.386695i \(-0.873617\pi\)
0.922208 0.386695i \(-0.126383\pi\)
\(108\) 0 0
\(109\) 10.0000i 0.957826i 0.877862 + 0.478913i \(0.158969\pi\)
−0.877862 + 0.478913i \(0.841031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 9.89949i − 0.931266i −0.884978 0.465633i \(-0.845827\pi\)
0.884978 0.465633i \(-0.154173\pi\)
\(114\) 0 0
\(115\) 16.9706 1.58251
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 4.24264i 0.376473i 0.982124 + 0.188237i \(0.0602772\pi\)
−0.982124 + 0.188237i \(0.939723\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 16.0000i − 1.39793i −0.715158 0.698963i \(-0.753645\pi\)
0.715158 0.698963i \(-0.246355\pi\)
\(132\) 0 0
\(133\) − 4.00000i − 0.346844i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1.41421i − 0.120824i −0.998174 0.0604122i \(-0.980758\pi\)
0.998174 0.0604122i \(-0.0192415\pi\)
\(138\) 0 0
\(139\) −22.6274 −1.91923 −0.959616 0.281312i \(-0.909230\pi\)
−0.959616 + 0.281312i \(0.909230\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 16.0000 1.32873
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −19.7990 −1.62200 −0.810998 0.585049i \(-0.801075\pi\)
−0.810998 + 0.585049i \(0.801075\pi\)
\(150\) 0 0
\(151\) 12.7279i 1.03578i 0.855446 + 0.517892i \(0.173283\pi\)
−0.855446 + 0.517892i \(0.826717\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 28.0000i 2.24901i
\(156\) 0 0
\(157\) 22.0000i 1.75579i 0.478852 + 0.877896i \(0.341053\pi\)
−0.478852 + 0.877896i \(0.658947\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 8.48528i − 0.668734i
\(162\) 0 0
\(163\) 8.48528 0.664619 0.332309 0.943170i \(-0.392172\pi\)
0.332309 + 0.943170i \(0.392172\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) 9.00000 0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) 0 0
\(175\) − 4.24264i − 0.320713i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 4.00000i 0.298974i 0.988764 + 0.149487i \(0.0477622\pi\)
−0.988764 + 0.149487i \(0.952238\pi\)
\(180\) 0 0
\(181\) 18.0000i 1.33793i 0.743294 + 0.668965i \(0.233262\pi\)
−0.743294 + 0.668965i \(0.766738\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 5.65685i − 0.415900i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −11.3137 −0.806068 −0.403034 0.915185i \(-0.632044\pi\)
−0.403034 + 0.915185i \(0.632044\pi\)
\(198\) 0 0
\(199\) − 1.41421i − 0.100251i −0.998743 0.0501255i \(-0.984038\pi\)
0.998743 0.0501255i \(-0.0159621\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 8.00000i − 0.561490i
\(204\) 0 0
\(205\) − 12.0000i − 0.838116i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.9706 1.16830 0.584151 0.811645i \(-0.301428\pi\)
0.584151 + 0.811645i \(0.301428\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −24.0000 −1.63679
\(216\) 0 0
\(217\) 14.0000 0.950382
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −2.82843 −0.190261
\(222\) 0 0
\(223\) − 21.2132i − 1.42054i −0.703929 0.710271i \(-0.748573\pi\)
0.703929 0.710271i \(-0.251427\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 20.0000i − 1.32745i −0.747978 0.663723i \(-0.768975\pi\)
0.747978 0.663723i \(-0.231025\pi\)
\(228\) 0 0
\(229\) 22.0000i 1.45380i 0.686743 + 0.726900i \(0.259040\pi\)
−0.686743 + 0.726900i \(0.740960\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 26.8701i − 1.76032i −0.474681 0.880158i \(-0.657437\pi\)
0.474681 0.880158i \(-0.342563\pi\)
\(234\) 0 0
\(235\) −28.2843 −1.84506
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 16.0000 1.03495 0.517477 0.855697i \(-0.326871\pi\)
0.517477 + 0.855697i \(0.326871\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 14.1421 0.903508
\(246\) 0 0
\(247\) − 5.65685i − 0.359937i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000i 1.51487i 0.652913 + 0.757433i \(0.273547\pi\)
−0.652913 + 0.757433i \(0.726453\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 9.89949i − 0.617514i −0.951141 0.308757i \(-0.900087\pi\)
0.951141 0.308757i \(-0.0999129\pi\)
\(258\) 0 0
\(259\) −2.82843 −0.175750
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) 16.0000 0.982872
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −11.3137 −0.689809 −0.344904 0.938638i \(-0.612089\pi\)
−0.344904 + 0.938638i \(0.612089\pi\)
\(270\) 0 0
\(271\) 12.7279i 0.773166i 0.922255 + 0.386583i \(0.126345\pi\)
−0.922255 + 0.386583i \(0.873655\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 30.0000i − 1.80253i −0.433273 0.901263i \(-0.642641\pi\)
0.433273 0.901263i \(-0.357359\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 1.41421i − 0.0843649i −0.999110 0.0421825i \(-0.986569\pi\)
0.999110 0.0421825i \(-0.0134311\pi\)
\(282\) 0 0
\(283\) 11.3137 0.672530 0.336265 0.941767i \(-0.390836\pi\)
0.336265 + 0.941767i \(0.390836\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00000 −0.354169
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5.65685 −0.330477 −0.165238 0.986254i \(-0.552839\pi\)
−0.165238 + 0.986254i \(0.552839\pi\)
\(294\) 0 0
\(295\) 33.9411i 1.97613i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 12.0000i − 0.693978i
\(300\) 0 0
\(301\) 12.0000i 0.691669i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 28.2843i − 1.61955i
\(306\) 0 0
\(307\) 11.3137 0.645707 0.322854 0.946449i \(-0.395358\pi\)
0.322854 + 0.946449i \(0.395358\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 8.00000 0.452187 0.226093 0.974106i \(-0.427405\pi\)
0.226093 + 0.974106i \(0.427405\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −22.6274 −1.27088 −0.635441 0.772149i \(-0.719182\pi\)
−0.635441 + 0.772149i \(0.719182\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 4.00000i − 0.222566i
\(324\) 0 0
\(325\) − 6.00000i − 0.332820i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 14.1421i 0.779681i
\(330\) 0 0
\(331\) −22.6274 −1.24372 −0.621858 0.783130i \(-0.713622\pi\)
−0.621858 + 0.783130i \(0.713622\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 16.0000 0.874173
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 16.9706i − 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.0000i 1.50312i 0.659665 + 0.751559i \(0.270698\pi\)
−0.659665 + 0.751559i \(0.729302\pi\)
\(348\) 0 0
\(349\) − 10.0000i − 0.535288i −0.963518 0.267644i \(-0.913755\pi\)
0.963518 0.267644i \(-0.0862451\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 12.7279i 0.677439i 0.940887 + 0.338719i \(0.109994\pi\)
−0.940887 + 0.338719i \(0.890006\pi\)
\(354\) 0 0
\(355\) 5.65685 0.300235
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 28.2843 1.48047
\(366\) 0 0
\(367\) 21.2132i 1.10732i 0.832743 + 0.553660i \(0.186769\pi\)
−0.832743 + 0.553660i \(0.813231\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 8.00000i − 0.415339i
\(372\) 0 0
\(373\) 30.0000i 1.55334i 0.629907 + 0.776671i \(0.283093\pi\)
−0.629907 + 0.776671i \(0.716907\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 11.3137i − 0.582686i
\(378\) 0 0
\(379\) 14.1421 0.726433 0.363216 0.931705i \(-0.381679\pi\)
0.363216 + 0.931705i \(0.381679\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −32.0000 −1.63512 −0.817562 0.575841i \(-0.804675\pi\)
−0.817562 + 0.575841i \(0.804675\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 8.48528 0.430221 0.215110 0.976590i \(-0.430989\pi\)
0.215110 + 0.976590i \(0.430989\pi\)
\(390\) 0 0
\(391\) − 8.48528i − 0.429119i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 36.0000i − 1.81136i
\(396\) 0 0
\(397\) − 22.0000i − 1.10415i −0.833795 0.552074i \(-0.813837\pi\)
0.833795 0.552074i \(-0.186163\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 26.8701i 1.34183i 0.741536 + 0.670913i \(0.234098\pi\)
−0.741536 + 0.670913i \(0.765902\pi\)
\(402\) 0 0
\(403\) 19.7990 0.986258
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −4.00000 −0.197787 −0.0988936 0.995098i \(-0.531530\pi\)
−0.0988936 + 0.995098i \(0.531530\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 16.9706 0.835067
\(414\) 0 0
\(415\) − 11.3137i − 0.555368i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 12.0000i − 0.586238i −0.956076 0.293119i \(-0.905307\pi\)
0.956076 0.293119i \(-0.0946933\pi\)
\(420\) 0 0
\(421\) − 22.0000i − 1.07221i −0.844150 0.536107i \(-0.819894\pi\)
0.844150 0.536107i \(-0.180106\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 4.24264i − 0.205798i
\(426\) 0 0
\(427\) −14.1421 −0.684386
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −14.0000 −0.674356 −0.337178 0.941441i \(-0.609472\pi\)
−0.337178 + 0.941441i \(0.609472\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 16.9706 0.811812
\(438\) 0 0
\(439\) 1.41421i 0.0674967i 0.999430 + 0.0337484i \(0.0107445\pi\)
−0.999430 + 0.0337484i \(0.989256\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 28.0000i 1.33032i 0.746701 + 0.665160i \(0.231637\pi\)
−0.746701 + 0.665160i \(0.768363\pi\)
\(444\) 0 0
\(445\) − 28.0000i − 1.32733i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 32.5269i − 1.53504i −0.641025 0.767520i \(-0.721491\pi\)
0.641025 0.767520i \(-0.278509\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 20.0000 0.935561 0.467780 0.883845i \(-0.345054\pi\)
0.467780 + 0.883845i \(0.345054\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.48528 0.395199 0.197599 0.980283i \(-0.436685\pi\)
0.197599 + 0.980283i \(0.436685\pi\)
\(462\) 0 0
\(463\) 29.6985i 1.38021i 0.723711 + 0.690103i \(0.242435\pi\)
−0.723711 + 0.690103i \(0.757565\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 36.0000i 1.66588i 0.553362 + 0.832941i \(0.313345\pi\)
−0.553362 + 0.832941i \(0.686655\pi\)
\(468\) 0 0
\(469\) − 8.00000i − 0.369406i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 8.48528 0.389331
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −26.0000 −1.18797 −0.593985 0.804476i \(-0.702446\pi\)
−0.593985 + 0.804476i \(0.702446\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 33.9411 1.54119
\(486\) 0 0
\(487\) − 15.5563i − 0.704925i −0.935826 0.352463i \(-0.885344\pi\)
0.935826 0.352463i \(-0.114656\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 12.0000i − 0.541552i −0.962642 0.270776i \(-0.912720\pi\)
0.962642 0.270776i \(-0.0872803\pi\)
\(492\) 0 0
\(493\) − 8.00000i − 0.360302i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 2.82843i − 0.126872i
\(498\) 0 0
\(499\) −16.9706 −0.759707 −0.379853 0.925047i \(-0.624026\pi\)
−0.379853 + 0.925047i \(0.624026\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.9706 0.752207 0.376103 0.926578i \(-0.377264\pi\)
0.376103 + 0.926578i \(0.377264\pi\)
\(510\) 0 0
\(511\) − 14.1421i − 0.625611i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 20.0000i − 0.881305i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.7279i 0.557620i 0.960346 + 0.278810i \(0.0899400\pi\)
−0.960346 + 0.278810i \(0.910060\pi\)
\(522\) 0 0
\(523\) −8.48528 −0.371035 −0.185518 0.982641i \(-0.559396\pi\)
−0.185518 + 0.982641i \(0.559396\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 14.0000 0.609850
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −8.48528 −0.367538
\(534\) 0 0
\(535\) − 22.6274i − 0.978269i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000i 1.63375i 0.576816 + 0.816874i \(0.304295\pi\)
−0.576816 + 0.816874i \(0.695705\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 28.2843i 1.21157i
\(546\) 0 0
\(547\) −36.7696 −1.57215 −0.786076 0.618130i \(-0.787891\pi\)
−0.786076 + 0.618130i \(0.787891\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) −18.0000 −0.765438
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.82843 0.119844 0.0599222 0.998203i \(-0.480915\pi\)
0.0599222 + 0.998203i \(0.480915\pi\)
\(558\) 0 0
\(559\) 16.9706i 0.717778i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 8.00000i 0.337160i 0.985688 + 0.168580i \(0.0539181\pi\)
−0.985688 + 0.168580i \(0.946082\pi\)
\(564\) 0 0
\(565\) − 28.0000i − 1.17797i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.5563i 0.652156i 0.945343 + 0.326078i \(0.105727\pi\)
−0.945343 + 0.326078i \(0.894273\pi\)
\(570\) 0 0
\(571\) 16.9706 0.710196 0.355098 0.934829i \(-0.384448\pi\)
0.355098 + 0.934829i \(0.384448\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 18.0000 0.750652
\(576\) 0 0
\(577\) −20.0000 −0.832611 −0.416305 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5.65685 −0.234686
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 12.0000i − 0.495293i −0.968850 0.247647i \(-0.920343\pi\)
0.968850 0.247647i \(-0.0796572\pi\)
\(588\) 0 0
\(589\) 28.0000i 1.15372i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 1.41421i − 0.0580748i −0.999578 0.0290374i \(-0.990756\pi\)
0.999578 0.0290374i \(-0.00924419\pi\)
\(594\) 0 0
\(595\) −5.65685 −0.231908
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 34.0000 1.38920 0.694601 0.719395i \(-0.255581\pi\)
0.694601 + 0.719395i \(0.255581\pi\)
\(600\) 0 0
\(601\) −20.0000 −0.815817 −0.407909 0.913023i \(-0.633742\pi\)
−0.407909 + 0.913023i \(0.633742\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 31.1127 1.26491
\(606\) 0 0
\(607\) 15.5563i 0.631413i 0.948857 + 0.315706i \(0.102241\pi\)
−0.948857 + 0.315706i \(0.897759\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.0000i 0.809113i
\(612\) 0 0
\(613\) 6.00000i 0.242338i 0.992632 + 0.121169i \(0.0386643\pi\)
−0.992632 + 0.121169i \(0.961336\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 24.0416i − 0.967880i −0.875101 0.483940i \(-0.839205\pi\)
0.875101 0.483940i \(-0.160795\pi\)
\(618\) 0 0
\(619\) 16.9706 0.682105 0.341052 0.940044i \(-0.389217\pi\)
0.341052 + 0.940044i \(0.389217\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −14.0000 −0.560898
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2.82843 −0.112777
\(630\) 0 0
\(631\) − 4.24264i − 0.168897i −0.996428 0.0844484i \(-0.973087\pi\)
0.996428 0.0844484i \(-0.0269128\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.0000i 0.476205i
\(636\) 0 0
\(637\) − 10.0000i − 0.396214i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 18.3848i 0.726155i 0.931759 + 0.363078i \(0.118274\pi\)
−0.931759 + 0.363078i \(0.881726\pi\)
\(642\) 0 0
\(643\) −2.82843 −0.111542 −0.0557711 0.998444i \(-0.517762\pi\)
−0.0557711 + 0.998444i \(0.517762\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22.0000 0.864909 0.432455 0.901656i \(-0.357648\pi\)
0.432455 + 0.901656i \(0.357648\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −42.4264 −1.66027 −0.830137 0.557560i \(-0.811738\pi\)
−0.830137 + 0.557560i \(0.811738\pi\)
\(654\) 0 0
\(655\) − 45.2548i − 1.76825i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 4.00000i − 0.155818i −0.996960 0.0779089i \(-0.975176\pi\)
0.996960 0.0779089i \(-0.0248243\pi\)
\(660\) 0 0
\(661\) − 14.0000i − 0.544537i −0.962221 0.272268i \(-0.912226\pi\)
0.962221 0.272268i \(-0.0877739\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 11.3137i − 0.438727i
\(666\) 0 0
\(667\) 33.9411 1.31421
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −25.4558 −0.978348 −0.489174 0.872186i \(-0.662702\pi\)
−0.489174 + 0.872186i \(0.662702\pi\)
\(678\) 0 0
\(679\) − 16.9706i − 0.651270i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 44.0000i − 1.68361i −0.539779 0.841807i \(-0.681492\pi\)
0.539779 0.841807i \(-0.318508\pi\)
\(684\) 0 0
\(685\) − 4.00000i − 0.152832i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 11.3137i − 0.431018i
\(690\) 0 0
\(691\) 19.7990 0.753189 0.376595 0.926378i \(-0.377095\pi\)
0.376595 + 0.926378i \(0.377095\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −64.0000 −2.42766
\(696\) 0 0
\(697\) −6.00000 −0.227266
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −11.3137 −0.427313 −0.213656 0.976909i \(-0.568537\pi\)
−0.213656 + 0.976909i \(0.568537\pi\)
\(702\) 0 0
\(703\) − 5.65685i − 0.213352i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 14.0000i 0.525781i 0.964826 + 0.262891i \(0.0846758\pi\)
−0.964826 + 0.262891i \(0.915324\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 59.3970i 2.22443i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) −10.0000 −0.372419
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 16.9706 0.630271
\(726\) 0 0
\(727\) 26.8701i 0.996555i 0.867018 + 0.498278i \(0.166034\pi\)
−0.867018 + 0.498278i \(0.833966\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.0000i 0.443836i
\(732\) 0 0
\(733\) − 14.0000i − 0.517102i −0.965998 0.258551i \(-0.916755\pi\)
0.965998 0.258551i \(-0.0832450\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −5.65685 −0.208091 −0.104045 0.994573i \(-0.533179\pi\)
−0.104045 + 0.994573i \(0.533179\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) −56.0000 −2.05168
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −11.3137 −0.413394
\(750\) 0 0
\(751\) − 26.8701i − 0.980502i −0.871581 0.490251i \(-0.836905\pi\)
0.871581 0.490251i \(-0.163095\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 36.0000i 1.31017i
\(756\) 0 0
\(757\) 38.0000i 1.38113i 0.723269 + 0.690567i \(0.242639\pi\)
−0.723269 + 0.690567i \(0.757361\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 49.4975i 1.79428i 0.441744 + 0.897141i \(0.354360\pi\)
−0.441744 + 0.897141i \(0.645640\pi\)
\(762\) 0 0
\(763\) 14.1421 0.511980
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −11.3137 −0.406926 −0.203463 0.979083i \(-0.565220\pi\)
−0.203463 + 0.979083i \(0.565220\pi\)
\(774\) 0 0
\(775\) 29.6985i 1.06680i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 12.0000i − 0.429945i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 62.2254i 2.22092i
\(786\) 0 0
\(787\) −42.4264 −1.51234 −0.756169 0.654376i \(-0.772931\pi\)
−0.756169 + 0.654376i \(0.772931\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −14.1421 −0.500940 −0.250470 0.968124i \(-0.580585\pi\)
−0.250470 + 0.968124i \(0.580585\pi\)
\(798\) 0 0
\(799\) 14.1421i 0.500313i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) − 24.0000i − 0.845889i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 43.8406i − 1.54135i −0.637226 0.770677i \(-0.719918\pi\)
0.637226 0.770677i \(-0.280082\pi\)
\(810\) 0 0
\(811\) −8.48528 −0.297959 −0.148979 0.988840i \(-0.547599\pi\)
−0.148979 + 0.988840i \(0.547599\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 24.0000 0.840683
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −56.5685 −1.97426 −0.987128 0.159933i \(-0.948872\pi\)
−0.987128 + 0.159933i \(0.948872\pi\)
\(822\) 0 0
\(823\) 26.8701i 0.936631i 0.883561 + 0.468316i \(0.155139\pi\)
−0.883561 + 0.468316i \(0.844861\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 48.0000i − 1.66912i −0.550914 0.834562i \(-0.685721\pi\)
0.550914 0.834562i \(-0.314279\pi\)
\(828\) 0 0
\(829\) 2.00000i 0.0694629i 0.999397 + 0.0347314i \(0.0110576\pi\)
−0.999397 + 0.0347314i \(0.988942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 7.07107i − 0.244998i
\(834\) 0 0
\(835\) −45.2548 −1.56611
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) 3.00000 0.103448
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 25.4558 0.875708
\(846\) 0 0
\(847\) − 15.5563i − 0.534522i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 12.0000i − 0.411355i
\(852\) 0 0
\(853\) 42.0000i 1.43805i 0.694983 + 0.719026i \(0.255412\pi\)
−0.694983 + 0.719026i \(0.744588\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.3848i 0.628012i 0.949421 + 0.314006i \(0.101671\pi\)
−0.949421 + 0.314006i \(0.898329\pi\)
\(858\) 0 0
\(859\) 31.1127 1.06155 0.530776 0.847512i \(-0.321901\pi\)
0.530776 + 0.847512i \(0.321901\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 48.0000 1.63394 0.816970 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) − 11.3137i − 0.383350i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 8.00000i 0.270449i
\(876\) 0 0
\(877\) 22.0000i 0.742887i 0.928456 + 0.371444i \(0.121137\pi\)
−0.928456 + 0.371444i \(0.878863\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 21.2132i − 0.714691i −0.933972 0.357345i \(-0.883682\pi\)
0.933972 0.357345i \(-0.116318\pi\)
\(882\) 0 0
\(883\) −31.1127 −1.04703 −0.523513 0.852018i \(-0.675379\pi\)
−0.523513 + 0.852018i \(0.675379\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) 6.00000 0.201234
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −28.2843 −0.946497
\(894\) 0 0
\(895\) 11.3137i 0.378176i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 56.0000i 1.86770i
\(900\) 0 0
\(901\) − 8.00000i − 0.266519i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 50.9117i 1.69236i
\(906\) 0 0
\(907\) 42.4264 1.40875 0.704373 0.709830i \(-0.251228\pi\)
0.704373 + 0.709830i \(0.251228\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −22.6274 −0.747223
\(918\) 0 0
\(919\) 38.1838i 1.25957i 0.776771 + 0.629783i \(0.216856\pi\)
−0.776771 + 0.629783i \(0.783144\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 4.00000i − 0.131662i
\(924\) 0 0
\(925\) − 6.00000i − 0.197279i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 49.4975i 1.62396i 0.583686 + 0.811980i \(0.301610\pi\)
−0.583686 + 0.811980i \(0.698390\pi\)
\(930\) 0 0
\(931\) 14.1421 0.463490
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −14.0000 −0.457360 −0.228680 0.973502i \(-0.573441\pi\)
−0.228680 + 0.973502i \(0.573441\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −19.7990 −0.645429 −0.322714 0.946496i \(-0.604595\pi\)
−0.322714 + 0.946496i \(0.604595\pi\)
\(942\) 0 0
\(943\) − 25.4558i − 0.828956i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20.0000i 0.649913i 0.945729 + 0.324956i \(0.105350\pi\)
−0.945729 + 0.324956i \(0.894650\pi\)
\(948\) 0 0
\(949\) − 20.0000i − 0.649227i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 32.5269i − 1.05365i −0.849974 0.526825i \(-0.823382\pi\)
0.849974 0.526825i \(-0.176618\pi\)
\(954\) 0 0
\(955\) 22.6274 0.732206
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −2.00000 −0.0645834
\(960\) 0 0
\(961\) −67.0000 −2.16129
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 22.6274 0.728402
\(966\) 0 0
\(967\) 7.07107i 0.227390i 0.993516 + 0.113695i \(0.0362687\pi\)
−0.993516 + 0.113695i \(0.963731\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 20.0000i − 0.641831i −0.947108 0.320915i \(-0.896010\pi\)
0.947108 0.320915i \(-0.103990\pi\)
\(972\) 0 0
\(973\) 32.0000i 1.02587i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 29.6985i 0.950139i 0.879948 + 0.475069i \(0.157577\pi\)
−0.879948 + 0.475069i \(0.842423\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) −32.0000 −1.01960
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −50.9117 −1.61890
\(990\) 0 0
\(991\) 41.0122i 1.30280i 0.758737 + 0.651398i \(0.225817\pi\)
−0.758737 + 0.651398i \(0.774183\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 4.00000i − 0.126809i
\(996\) 0 0
\(997\) 50.0000i 1.58352i 0.610835 + 0.791758i \(0.290834\pi\)
−0.610835 + 0.791758i \(0.709166\pi\)
\(998\) 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 4608.2.f.l.2303.3 4
3.2 odd 2 4608.2.f.j.2303.1 4
4.3 odd 2 4608.2.f.j.2303.4 4
8.3 odd 2 4608.2.f.j.2303.2 4
8.5 even 2 inner 4608.2.f.l.2303.1 4
12.11 even 2 inner 4608.2.f.l.2303.2 4
16.3 odd 4 4608.2.c.f.4607.1 yes 2
16.5 even 4 4608.2.c.c.4607.2 yes 2
16.11 odd 4 4608.2.c.d.4607.2 yes 2
16.13 even 4 4608.2.c.e.4607.1 yes 2
24.5 odd 2 4608.2.f.j.2303.3 4
24.11 even 2 inner 4608.2.f.l.2303.4 4
48.5 odd 4 4608.2.c.d.4607.1 yes 2
48.11 even 4 4608.2.c.c.4607.1 2
48.29 odd 4 4608.2.c.f.4607.2 yes 2
48.35 even 4 4608.2.c.e.4607.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.c.c.4607.1 2 48.11 even 4
4608.2.c.c.4607.2 yes 2 16.5 even 4
4608.2.c.d.4607.1 yes 2 48.5 odd 4
4608.2.c.d.4607.2 yes 2 16.11 odd 4
4608.2.c.e.4607.1 yes 2 16.13 even 4
4608.2.c.e.4607.2 yes 2 48.35 even 4
4608.2.c.f.4607.1 yes 2 16.3 odd 4
4608.2.c.f.4607.2 yes 2 48.29 odd 4
4608.2.f.j.2303.1 4 3.2 odd 2
4608.2.f.j.2303.2 4 8.3 odd 2
4608.2.f.j.2303.3 4 24.5 odd 2
4608.2.f.j.2303.4 4 4.3 odd 2
4608.2.f.l.2303.1 4 8.5 even 2 inner
4608.2.f.l.2303.2 4 12.11 even 2 inner
4608.2.f.l.2303.3 4 1.1 even 1 trivial
4608.2.f.l.2303.4 4 24.11 even 2 inner