Properties

Label 1536.2.a.f.1.1
Level $1536$
Weight $2$
Character 1536.1
Self dual yes
Analytic conductor $12.265$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1536,2,Mod(1,1536)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1536, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1536.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1536 = 2^{9} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1536.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(12.2650217505\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 1536.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} +0.585786 q^{5} -1.41421 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} +0.585786 q^{5} -1.41421 q^{7} +1.00000 q^{9} -0.828427 q^{11} +4.82843 q^{13} -0.585786 q^{15} -0.828427 q^{17} +2.82843 q^{19} +1.41421 q^{21} -6.82843 q^{23} -4.65685 q^{25} -1.00000 q^{27} +4.58579 q^{29} +7.07107 q^{31} +0.828427 q^{33} -0.828427 q^{35} +0.343146 q^{37} -4.82843 q^{39} +6.48528 q^{41} -1.17157 q^{43} +0.585786 q^{45} +4.48528 q^{47} -5.00000 q^{49} +0.828427 q^{51} +10.2426 q^{53} -0.485281 q^{55} -2.82843 q^{57} +9.65685 q^{59} +11.6569 q^{61} -1.41421 q^{63} +2.82843 q^{65} -5.65685 q^{67} +6.82843 q^{69} -8.48528 q^{71} +11.3137 q^{73} +4.65685 q^{75} +1.17157 q^{77} +14.5858 q^{79} +1.00000 q^{81} +3.17157 q^{83} -0.485281 q^{85} -4.58579 q^{87} -17.3137 q^{89} -6.82843 q^{91} -7.07107 q^{93} +1.65685 q^{95} +3.65685 q^{97} -0.828427 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{3} + 4 q^{5} + 2 q^{9} + 4 q^{11} + 4 q^{13} - 4 q^{15} + 4 q^{17} - 8 q^{23} + 2 q^{25} - 2 q^{27} + 12 q^{29} - 4 q^{33} + 4 q^{35} + 12 q^{37} - 4 q^{39} - 4 q^{41} - 8 q^{43} + 4 q^{45} - 8 q^{47} - 10 q^{49} - 4 q^{51} + 12 q^{53} + 16 q^{55} + 8 q^{59} + 12 q^{61} + 8 q^{69} - 2 q^{75} + 8 q^{77} + 32 q^{79} + 2 q^{81} + 12 q^{83} + 16 q^{85} - 12 q^{87} - 12 q^{89} - 8 q^{91} - 8 q^{95} - 4 q^{97} + 4 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) 0.585786 0.261972 0.130986 0.991384i \(-0.458186\pi\)
0.130986 + 0.991384i \(0.458186\pi\)
\(6\) 0 0
\(7\) −1.41421 −0.534522 −0.267261 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.828427 −0.249780 −0.124890 0.992171i \(-0.539858\pi\)
−0.124890 + 0.992171i \(0.539858\pi\)
\(12\) 0 0
\(13\) 4.82843 1.33916 0.669582 0.742738i \(-0.266473\pi\)
0.669582 + 0.742738i \(0.266473\pi\)
\(14\) 0 0
\(15\) −0.585786 −0.151249
\(16\) 0 0
\(17\) −0.828427 −0.200923 −0.100462 0.994941i \(-0.532032\pi\)
−0.100462 + 0.994941i \(0.532032\pi\)
\(18\) 0 0
\(19\) 2.82843 0.648886 0.324443 0.945905i \(-0.394823\pi\)
0.324443 + 0.945905i \(0.394823\pi\)
\(20\) 0 0
\(21\) 1.41421 0.308607
\(22\) 0 0
\(23\) −6.82843 −1.42383 −0.711913 0.702268i \(-0.752171\pi\)
−0.711913 + 0.702268i \(0.752171\pi\)
\(24\) 0 0
\(25\) −4.65685 −0.931371
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 4.58579 0.851559 0.425780 0.904827i \(-0.360000\pi\)
0.425780 + 0.904827i \(0.360000\pi\)
\(30\) 0 0
\(31\) 7.07107 1.27000 0.635001 0.772512i \(-0.281000\pi\)
0.635001 + 0.772512i \(0.281000\pi\)
\(32\) 0 0
\(33\) 0.828427 0.144211
\(34\) 0 0
\(35\) −0.828427 −0.140030
\(36\) 0 0
\(37\) 0.343146 0.0564128 0.0282064 0.999602i \(-0.491020\pi\)
0.0282064 + 0.999602i \(0.491020\pi\)
\(38\) 0 0
\(39\) −4.82843 −0.773167
\(40\) 0 0
\(41\) 6.48528 1.01283 0.506415 0.862290i \(-0.330970\pi\)
0.506415 + 0.862290i \(0.330970\pi\)
\(42\) 0 0
\(43\) −1.17157 −0.178663 −0.0893316 0.996002i \(-0.528473\pi\)
−0.0893316 + 0.996002i \(0.528473\pi\)
\(44\) 0 0
\(45\) 0.585786 0.0873239
\(46\) 0 0
\(47\) 4.48528 0.654246 0.327123 0.944982i \(-0.393921\pi\)
0.327123 + 0.944982i \(0.393921\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) 0.828427 0.116003
\(52\) 0 0
\(53\) 10.2426 1.40693 0.703467 0.710727i \(-0.251634\pi\)
0.703467 + 0.710727i \(0.251634\pi\)
\(54\) 0 0
\(55\) −0.485281 −0.0654353
\(56\) 0 0
\(57\) −2.82843 −0.374634
\(58\) 0 0
\(59\) 9.65685 1.25722 0.628608 0.777723i \(-0.283625\pi\)
0.628608 + 0.777723i \(0.283625\pi\)
\(60\) 0 0
\(61\) 11.6569 1.49251 0.746254 0.665662i \(-0.231851\pi\)
0.746254 + 0.665662i \(0.231851\pi\)
\(62\) 0 0
\(63\) −1.41421 −0.178174
\(64\) 0 0
\(65\) 2.82843 0.350823
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 0 0
\(69\) 6.82843 0.822046
\(70\) 0 0
\(71\) −8.48528 −1.00702 −0.503509 0.863990i \(-0.667958\pi\)
−0.503509 + 0.863990i \(0.667958\pi\)
\(72\) 0 0
\(73\) 11.3137 1.32417 0.662085 0.749429i \(-0.269672\pi\)
0.662085 + 0.749429i \(0.269672\pi\)
\(74\) 0 0
\(75\) 4.65685 0.537727
\(76\) 0 0
\(77\) 1.17157 0.133513
\(78\) 0 0
\(79\) 14.5858 1.64103 0.820515 0.571626i \(-0.193687\pi\)
0.820515 + 0.571626i \(0.193687\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 3.17157 0.348125 0.174063 0.984735i \(-0.444310\pi\)
0.174063 + 0.984735i \(0.444310\pi\)
\(84\) 0 0
\(85\) −0.485281 −0.0526362
\(86\) 0 0
\(87\) −4.58579 −0.491648
\(88\) 0 0
\(89\) −17.3137 −1.83525 −0.917625 0.397448i \(-0.869896\pi\)
−0.917625 + 0.397448i \(0.869896\pi\)
\(90\) 0 0
\(91\) −6.82843 −0.715814
\(92\) 0 0
\(93\) −7.07107 −0.733236
\(94\) 0 0
\(95\) 1.65685 0.169990
\(96\) 0 0
\(97\) 3.65685 0.371297 0.185649 0.982616i \(-0.440561\pi\)
0.185649 + 0.982616i \(0.440561\pi\)
\(98\) 0 0
\(99\) −0.828427 −0.0832601
\(100\) 0 0
\(101\) 14.7279 1.46548 0.732742 0.680507i \(-0.238240\pi\)
0.732742 + 0.680507i \(0.238240\pi\)
\(102\) 0 0
\(103\) 6.58579 0.648917 0.324458 0.945900i \(-0.394818\pi\)
0.324458 + 0.945900i \(0.394818\pi\)
\(104\) 0 0
\(105\) 0.828427 0.0808462
\(106\) 0 0
\(107\) −15.3137 −1.48043 −0.740216 0.672369i \(-0.765277\pi\)
−0.740216 + 0.672369i \(0.765277\pi\)
\(108\) 0 0
\(109\) 12.8284 1.22874 0.614370 0.789018i \(-0.289410\pi\)
0.614370 + 0.789018i \(0.289410\pi\)
\(110\) 0 0
\(111\) −0.343146 −0.0325700
\(112\) 0 0
\(113\) −7.65685 −0.720296 −0.360148 0.932895i \(-0.617274\pi\)
−0.360148 + 0.932895i \(0.617274\pi\)
\(114\) 0 0
\(115\) −4.00000 −0.373002
\(116\) 0 0
\(117\) 4.82843 0.446388
\(118\) 0 0
\(119\) 1.17157 0.107398
\(120\) 0 0
\(121\) −10.3137 −0.937610
\(122\) 0 0
\(123\) −6.48528 −0.584758
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 17.8995 1.58832 0.794162 0.607707i \(-0.207910\pi\)
0.794162 + 0.607707i \(0.207910\pi\)
\(128\) 0 0
\(129\) 1.17157 0.103151
\(130\) 0 0
\(131\) 7.31371 0.639002 0.319501 0.947586i \(-0.396485\pi\)
0.319501 + 0.947586i \(0.396485\pi\)
\(132\) 0 0
\(133\) −4.00000 −0.346844
\(134\) 0 0
\(135\) −0.585786 −0.0504165
\(136\) 0 0
\(137\) 14.4853 1.23756 0.618781 0.785564i \(-0.287627\pi\)
0.618781 + 0.785564i \(0.287627\pi\)
\(138\) 0 0
\(139\) 16.0000 1.35710 0.678551 0.734553i \(-0.262608\pi\)
0.678551 + 0.734553i \(0.262608\pi\)
\(140\) 0 0
\(141\) −4.48528 −0.377729
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) 2.68629 0.223084
\(146\) 0 0
\(147\) 5.00000 0.412393
\(148\) 0 0
\(149\) 0.585786 0.0479895 0.0239947 0.999712i \(-0.492362\pi\)
0.0239947 + 0.999712i \(0.492362\pi\)
\(150\) 0 0
\(151\) 9.41421 0.766118 0.383059 0.923724i \(-0.374871\pi\)
0.383059 + 0.923724i \(0.374871\pi\)
\(152\) 0 0
\(153\) −0.828427 −0.0669744
\(154\) 0 0
\(155\) 4.14214 0.332704
\(156\) 0 0
\(157\) 5.31371 0.424080 0.212040 0.977261i \(-0.431989\pi\)
0.212040 + 0.977261i \(0.431989\pi\)
\(158\) 0 0
\(159\) −10.2426 −0.812294
\(160\) 0 0
\(161\) 9.65685 0.761067
\(162\) 0 0
\(163\) 4.48528 0.351314 0.175657 0.984451i \(-0.443795\pi\)
0.175657 + 0.984451i \(0.443795\pi\)
\(164\) 0 0
\(165\) 0.485281 0.0377791
\(166\) 0 0
\(167\) −12.0000 −0.928588 −0.464294 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) 10.3137 0.793362
\(170\) 0 0
\(171\) 2.82843 0.216295
\(172\) 0 0
\(173\) −11.8995 −0.904702 −0.452351 0.891840i \(-0.649415\pi\)
−0.452351 + 0.891840i \(0.649415\pi\)
\(174\) 0 0
\(175\) 6.58579 0.497839
\(176\) 0 0
\(177\) −9.65685 −0.725854
\(178\) 0 0
\(179\) −25.6569 −1.91768 −0.958842 0.283941i \(-0.908358\pi\)
−0.958842 + 0.283941i \(0.908358\pi\)
\(180\) 0 0
\(181\) −16.1421 −1.19984 −0.599918 0.800062i \(-0.704800\pi\)
−0.599918 + 0.800062i \(0.704800\pi\)
\(182\) 0 0
\(183\) −11.6569 −0.861699
\(184\) 0 0
\(185\) 0.201010 0.0147786
\(186\) 0 0
\(187\) 0.686292 0.0501866
\(188\) 0 0
\(189\) 1.41421 0.102869
\(190\) 0 0
\(191\) −7.31371 −0.529201 −0.264601 0.964358i \(-0.585240\pi\)
−0.264601 + 0.964358i \(0.585240\pi\)
\(192\) 0 0
\(193\) 11.3137 0.814379 0.407189 0.913344i \(-0.366509\pi\)
0.407189 + 0.913344i \(0.366509\pi\)
\(194\) 0 0
\(195\) −2.82843 −0.202548
\(196\) 0 0
\(197\) −3.41421 −0.243253 −0.121626 0.992576i \(-0.538811\pi\)
−0.121626 + 0.992576i \(0.538811\pi\)
\(198\) 0 0
\(199\) 20.2426 1.43496 0.717481 0.696578i \(-0.245295\pi\)
0.717481 + 0.696578i \(0.245295\pi\)
\(200\) 0 0
\(201\) 5.65685 0.399004
\(202\) 0 0
\(203\) −6.48528 −0.455178
\(204\) 0 0
\(205\) 3.79899 0.265333
\(206\) 0 0
\(207\) −6.82843 −0.474608
\(208\) 0 0
\(209\) −2.34315 −0.162079
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) 8.48528 0.581402
\(214\) 0 0
\(215\) −0.686292 −0.0468047
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) 0 0
\(219\) −11.3137 −0.764510
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −7.55635 −0.506011 −0.253005 0.967465i \(-0.581419\pi\)
−0.253005 + 0.967465i \(0.581419\pi\)
\(224\) 0 0
\(225\) −4.65685 −0.310457
\(226\) 0 0
\(227\) 16.1421 1.07139 0.535696 0.844411i \(-0.320049\pi\)
0.535696 + 0.844411i \(0.320049\pi\)
\(228\) 0 0
\(229\) −21.7990 −1.44052 −0.720259 0.693705i \(-0.755977\pi\)
−0.720259 + 0.693705i \(0.755977\pi\)
\(230\) 0 0
\(231\) −1.17157 −0.0770838
\(232\) 0 0
\(233\) −2.00000 −0.131024 −0.0655122 0.997852i \(-0.520868\pi\)
−0.0655122 + 0.997852i \(0.520868\pi\)
\(234\) 0 0
\(235\) 2.62742 0.171394
\(236\) 0 0
\(237\) −14.5858 −0.947449
\(238\) 0 0
\(239\) −23.3137 −1.50804 −0.754019 0.656852i \(-0.771887\pi\)
−0.754019 + 0.656852i \(0.771887\pi\)
\(240\) 0 0
\(241\) −9.65685 −0.622053 −0.311026 0.950401i \(-0.600673\pi\)
−0.311026 + 0.950401i \(0.600673\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −2.92893 −0.187123
\(246\) 0 0
\(247\) 13.6569 0.868965
\(248\) 0 0
\(249\) −3.17157 −0.200990
\(250\) 0 0
\(251\) 21.7990 1.37594 0.687970 0.725739i \(-0.258502\pi\)
0.687970 + 0.725739i \(0.258502\pi\)
\(252\) 0 0
\(253\) 5.65685 0.355643
\(254\) 0 0
\(255\) 0.485281 0.0303895
\(256\) 0 0
\(257\) −8.34315 −0.520431 −0.260216 0.965551i \(-0.583794\pi\)
−0.260216 + 0.965551i \(0.583794\pi\)
\(258\) 0 0
\(259\) −0.485281 −0.0301539
\(260\) 0 0
\(261\) 4.58579 0.283853
\(262\) 0 0
\(263\) 8.97056 0.553149 0.276574 0.960993i \(-0.410801\pi\)
0.276574 + 0.960993i \(0.410801\pi\)
\(264\) 0 0
\(265\) 6.00000 0.368577
\(266\) 0 0
\(267\) 17.3137 1.05958
\(268\) 0 0
\(269\) −2.24264 −0.136736 −0.0683681 0.997660i \(-0.521779\pi\)
−0.0683681 + 0.997660i \(0.521779\pi\)
\(270\) 0 0
\(271\) −12.2426 −0.743687 −0.371844 0.928295i \(-0.621274\pi\)
−0.371844 + 0.928295i \(0.621274\pi\)
\(272\) 0 0
\(273\) 6.82843 0.413275
\(274\) 0 0
\(275\) 3.85786 0.232638
\(276\) 0 0
\(277\) −20.8284 −1.25146 −0.625729 0.780040i \(-0.715199\pi\)
−0.625729 + 0.780040i \(0.715199\pi\)
\(278\) 0 0
\(279\) 7.07107 0.423334
\(280\) 0 0
\(281\) −12.3431 −0.736330 −0.368165 0.929760i \(-0.620014\pi\)
−0.368165 + 0.929760i \(0.620014\pi\)
\(282\) 0 0
\(283\) −20.9706 −1.24657 −0.623285 0.781995i \(-0.714202\pi\)
−0.623285 + 0.781995i \(0.714202\pi\)
\(284\) 0 0
\(285\) −1.65685 −0.0981436
\(286\) 0 0
\(287\) −9.17157 −0.541381
\(288\) 0 0
\(289\) −16.3137 −0.959630
\(290\) 0 0
\(291\) −3.65685 −0.214369
\(292\) 0 0
\(293\) −17.0711 −0.997302 −0.498651 0.866803i \(-0.666171\pi\)
−0.498651 + 0.866803i \(0.666171\pi\)
\(294\) 0 0
\(295\) 5.65685 0.329355
\(296\) 0 0
\(297\) 0.828427 0.0480702
\(298\) 0 0
\(299\) −32.9706 −1.90674
\(300\) 0 0
\(301\) 1.65685 0.0954995
\(302\) 0 0
\(303\) −14.7279 −0.846097
\(304\) 0 0
\(305\) 6.82843 0.390995
\(306\) 0 0
\(307\) −29.6569 −1.69261 −0.846303 0.532702i \(-0.821177\pi\)
−0.846303 + 0.532702i \(0.821177\pi\)
\(308\) 0 0
\(309\) −6.58579 −0.374652
\(310\) 0 0
\(311\) −4.68629 −0.265735 −0.132868 0.991134i \(-0.542419\pi\)
−0.132868 + 0.991134i \(0.542419\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) −0.828427 −0.0466766
\(316\) 0 0
\(317\) −31.8995 −1.79165 −0.895827 0.444403i \(-0.853416\pi\)
−0.895827 + 0.444403i \(0.853416\pi\)
\(318\) 0 0
\(319\) −3.79899 −0.212703
\(320\) 0 0
\(321\) 15.3137 0.854728
\(322\) 0 0
\(323\) −2.34315 −0.130376
\(324\) 0 0
\(325\) −22.4853 −1.24726
\(326\) 0 0
\(327\) −12.8284 −0.709414
\(328\) 0 0
\(329\) −6.34315 −0.349709
\(330\) 0 0
\(331\) 13.6569 0.750649 0.375324 0.926894i \(-0.377531\pi\)
0.375324 + 0.926894i \(0.377531\pi\)
\(332\) 0 0
\(333\) 0.343146 0.0188043
\(334\) 0 0
\(335\) −3.31371 −0.181047
\(336\) 0 0
\(337\) 3.31371 0.180509 0.0902546 0.995919i \(-0.471232\pi\)
0.0902546 + 0.995919i \(0.471232\pi\)
\(338\) 0 0
\(339\) 7.65685 0.415863
\(340\) 0 0
\(341\) −5.85786 −0.317221
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 4.00000 0.215353
\(346\) 0 0
\(347\) 17.7990 0.955500 0.477750 0.878496i \(-0.341453\pi\)
0.477750 + 0.878496i \(0.341453\pi\)
\(348\) 0 0
\(349\) 1.31371 0.0703212 0.0351606 0.999382i \(-0.488806\pi\)
0.0351606 + 0.999382i \(0.488806\pi\)
\(350\) 0 0
\(351\) −4.82843 −0.257722
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) −4.97056 −0.263810
\(356\) 0 0
\(357\) −1.17157 −0.0620062
\(358\) 0 0
\(359\) 24.4853 1.29228 0.646142 0.763217i \(-0.276381\pi\)
0.646142 + 0.763217i \(0.276381\pi\)
\(360\) 0 0
\(361\) −11.0000 −0.578947
\(362\) 0 0
\(363\) 10.3137 0.541329
\(364\) 0 0
\(365\) 6.62742 0.346895
\(366\) 0 0
\(367\) 15.0711 0.786703 0.393352 0.919388i \(-0.371315\pi\)
0.393352 + 0.919388i \(0.371315\pi\)
\(368\) 0 0
\(369\) 6.48528 0.337610
\(370\) 0 0
\(371\) −14.4853 −0.752038
\(372\) 0 0
\(373\) −0.343146 −0.0177674 −0.00888371 0.999961i \(-0.502828\pi\)
−0.00888371 + 0.999961i \(0.502828\pi\)
\(374\) 0 0
\(375\) 5.65685 0.292119
\(376\) 0 0
\(377\) 22.1421 1.14038
\(378\) 0 0
\(379\) −10.8284 −0.556219 −0.278109 0.960549i \(-0.589708\pi\)
−0.278109 + 0.960549i \(0.589708\pi\)
\(380\) 0 0
\(381\) −17.8995 −0.917019
\(382\) 0 0
\(383\) −33.6569 −1.71978 −0.859892 0.510475i \(-0.829470\pi\)
−0.859892 + 0.510475i \(0.829470\pi\)
\(384\) 0 0
\(385\) 0.686292 0.0349767
\(386\) 0 0
\(387\) −1.17157 −0.0595544
\(388\) 0 0
\(389\) 13.0711 0.662729 0.331365 0.943503i \(-0.392491\pi\)
0.331365 + 0.943503i \(0.392491\pi\)
\(390\) 0 0
\(391\) 5.65685 0.286079
\(392\) 0 0
\(393\) −7.31371 −0.368928
\(394\) 0 0
\(395\) 8.54416 0.429903
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) 1.51472 0.0756414 0.0378207 0.999285i \(-0.487958\pi\)
0.0378207 + 0.999285i \(0.487958\pi\)
\(402\) 0 0
\(403\) 34.1421 1.70074
\(404\) 0 0
\(405\) 0.585786 0.0291080
\(406\) 0 0
\(407\) −0.284271 −0.0140908
\(408\) 0 0
\(409\) 20.3431 1.00590 0.502952 0.864314i \(-0.332247\pi\)
0.502952 + 0.864314i \(0.332247\pi\)
\(410\) 0 0
\(411\) −14.4853 −0.714506
\(412\) 0 0
\(413\) −13.6569 −0.672010
\(414\) 0 0
\(415\) 1.85786 0.0911990
\(416\) 0 0
\(417\) −16.0000 −0.783523
\(418\) 0 0
\(419\) 4.82843 0.235884 0.117942 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(420\) 0 0
\(421\) 5.51472 0.268771 0.134385 0.990929i \(-0.457094\pi\)
0.134385 + 0.990929i \(0.457094\pi\)
\(422\) 0 0
\(423\) 4.48528 0.218082
\(424\) 0 0
\(425\) 3.85786 0.187134
\(426\) 0 0
\(427\) −16.4853 −0.797779
\(428\) 0 0
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 5.85786 0.282163 0.141082 0.989998i \(-0.454942\pi\)
0.141082 + 0.989998i \(0.454942\pi\)
\(432\) 0 0
\(433\) −6.00000 −0.288342 −0.144171 0.989553i \(-0.546051\pi\)
−0.144171 + 0.989553i \(0.546051\pi\)
\(434\) 0 0
\(435\) −2.68629 −0.128798
\(436\) 0 0
\(437\) −19.3137 −0.923900
\(438\) 0 0
\(439\) 21.2132 1.01245 0.506225 0.862401i \(-0.331040\pi\)
0.506225 + 0.862401i \(0.331040\pi\)
\(440\) 0 0
\(441\) −5.00000 −0.238095
\(442\) 0 0
\(443\) 30.4853 1.44840 0.724200 0.689590i \(-0.242209\pi\)
0.724200 + 0.689590i \(0.242209\pi\)
\(444\) 0 0
\(445\) −10.1421 −0.480783
\(446\) 0 0
\(447\) −0.585786 −0.0277067
\(448\) 0 0
\(449\) −25.7990 −1.21753 −0.608765 0.793351i \(-0.708335\pi\)
−0.608765 + 0.793351i \(0.708335\pi\)
\(450\) 0 0
\(451\) −5.37258 −0.252985
\(452\) 0 0
\(453\) −9.41421 −0.442318
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) −22.2843 −1.04241 −0.521207 0.853430i \(-0.674518\pi\)
−0.521207 + 0.853430i \(0.674518\pi\)
\(458\) 0 0
\(459\) 0.828427 0.0386677
\(460\) 0 0
\(461\) −23.2132 −1.08115 −0.540573 0.841297i \(-0.681793\pi\)
−0.540573 + 0.841297i \(0.681793\pi\)
\(462\) 0 0
\(463\) −3.75736 −0.174619 −0.0873096 0.996181i \(-0.527827\pi\)
−0.0873096 + 0.996181i \(0.527827\pi\)
\(464\) 0 0
\(465\) −4.14214 −0.192087
\(466\) 0 0
\(467\) 20.8284 0.963825 0.481912 0.876219i \(-0.339942\pi\)
0.481912 + 0.876219i \(0.339942\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) −5.31371 −0.244843
\(472\) 0 0
\(473\) 0.970563 0.0446265
\(474\) 0 0
\(475\) −13.1716 −0.604353
\(476\) 0 0
\(477\) 10.2426 0.468978
\(478\) 0 0
\(479\) 21.1716 0.967354 0.483677 0.875247i \(-0.339301\pi\)
0.483677 + 0.875247i \(0.339301\pi\)
\(480\) 0 0
\(481\) 1.65685 0.0755461
\(482\) 0 0
\(483\) −9.65685 −0.439402
\(484\) 0 0
\(485\) 2.14214 0.0972694
\(486\) 0 0
\(487\) −4.72792 −0.214243 −0.107121 0.994246i \(-0.534163\pi\)
−0.107121 + 0.994246i \(0.534163\pi\)
\(488\) 0 0
\(489\) −4.48528 −0.202831
\(490\) 0 0
\(491\) −40.2843 −1.81800 −0.909002 0.416792i \(-0.863154\pi\)
−0.909002 + 0.416792i \(0.863154\pi\)
\(492\) 0 0
\(493\) −3.79899 −0.171098
\(494\) 0 0
\(495\) −0.485281 −0.0218118
\(496\) 0 0
\(497\) 12.0000 0.538274
\(498\) 0 0
\(499\) 16.2843 0.728984 0.364492 0.931207i \(-0.381243\pi\)
0.364492 + 0.931207i \(0.381243\pi\)
\(500\) 0 0
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 11.7990 0.526091 0.263045 0.964783i \(-0.415273\pi\)
0.263045 + 0.964783i \(0.415273\pi\)
\(504\) 0 0
\(505\) 8.62742 0.383915
\(506\) 0 0
\(507\) −10.3137 −0.458048
\(508\) 0 0
\(509\) 40.8701 1.81153 0.905767 0.423777i \(-0.139296\pi\)
0.905767 + 0.423777i \(0.139296\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 0 0
\(513\) −2.82843 −0.124878
\(514\) 0 0
\(515\) 3.85786 0.169998
\(516\) 0 0
\(517\) −3.71573 −0.163418
\(518\) 0 0
\(519\) 11.8995 0.522330
\(520\) 0 0
\(521\) −12.1421 −0.531957 −0.265978 0.963979i \(-0.585695\pi\)
−0.265978 + 0.963979i \(0.585695\pi\)
\(522\) 0 0
\(523\) 41.4558 1.81274 0.906369 0.422488i \(-0.138843\pi\)
0.906369 + 0.422488i \(0.138843\pi\)
\(524\) 0 0
\(525\) −6.58579 −0.287427
\(526\) 0 0
\(527\) −5.85786 −0.255173
\(528\) 0 0
\(529\) 23.6274 1.02728
\(530\) 0 0
\(531\) 9.65685 0.419072
\(532\) 0 0
\(533\) 31.3137 1.35635
\(534\) 0 0
\(535\) −8.97056 −0.387831
\(536\) 0 0
\(537\) 25.6569 1.10717
\(538\) 0 0
\(539\) 4.14214 0.178414
\(540\) 0 0
\(541\) 0.828427 0.0356169 0.0178084 0.999841i \(-0.494331\pi\)
0.0178084 + 0.999841i \(0.494331\pi\)
\(542\) 0 0
\(543\) 16.1421 0.692725
\(544\) 0 0
\(545\) 7.51472 0.321895
\(546\) 0 0
\(547\) −18.1421 −0.775702 −0.387851 0.921722i \(-0.626782\pi\)
−0.387851 + 0.921722i \(0.626782\pi\)
\(548\) 0 0
\(549\) 11.6569 0.497502
\(550\) 0 0
\(551\) 12.9706 0.552565
\(552\) 0 0
\(553\) −20.6274 −0.877167
\(554\) 0 0
\(555\) −0.201010 −0.00853240
\(556\) 0 0
\(557\) −10.7279 −0.454557 −0.227278 0.973830i \(-0.572983\pi\)
−0.227278 + 0.973830i \(0.572983\pi\)
\(558\) 0 0
\(559\) −5.65685 −0.239259
\(560\) 0 0
\(561\) −0.686292 −0.0289752
\(562\) 0 0
\(563\) −32.8284 −1.38355 −0.691777 0.722112i \(-0.743172\pi\)
−0.691777 + 0.722112i \(0.743172\pi\)
\(564\) 0 0
\(565\) −4.48528 −0.188697
\(566\) 0 0
\(567\) −1.41421 −0.0593914
\(568\) 0 0
\(569\) 37.1127 1.55585 0.777923 0.628360i \(-0.216274\pi\)
0.777923 + 0.628360i \(0.216274\pi\)
\(570\) 0 0
\(571\) 36.0000 1.50655 0.753277 0.657704i \(-0.228472\pi\)
0.753277 + 0.657704i \(0.228472\pi\)
\(572\) 0 0
\(573\) 7.31371 0.305535
\(574\) 0 0
\(575\) 31.7990 1.32611
\(576\) 0 0
\(577\) 8.68629 0.361615 0.180808 0.983518i \(-0.442129\pi\)
0.180808 + 0.983518i \(0.442129\pi\)
\(578\) 0 0
\(579\) −11.3137 −0.470182
\(580\) 0 0
\(581\) −4.48528 −0.186081
\(582\) 0 0
\(583\) −8.48528 −0.351424
\(584\) 0 0
\(585\) 2.82843 0.116941
\(586\) 0 0
\(587\) −12.9706 −0.535352 −0.267676 0.963509i \(-0.586256\pi\)
−0.267676 + 0.963509i \(0.586256\pi\)
\(588\) 0 0
\(589\) 20.0000 0.824086
\(590\) 0 0
\(591\) 3.41421 0.140442
\(592\) 0 0
\(593\) −31.6569 −1.29999 −0.649996 0.759938i \(-0.725229\pi\)
−0.649996 + 0.759938i \(0.725229\pi\)
\(594\) 0 0
\(595\) 0.686292 0.0281352
\(596\) 0 0
\(597\) −20.2426 −0.828476
\(598\) 0 0
\(599\) −3.51472 −0.143608 −0.0718038 0.997419i \(-0.522876\pi\)
−0.0718038 + 0.997419i \(0.522876\pi\)
\(600\) 0 0
\(601\) 26.6274 1.08615 0.543077 0.839683i \(-0.317259\pi\)
0.543077 + 0.839683i \(0.317259\pi\)
\(602\) 0 0
\(603\) −5.65685 −0.230365
\(604\) 0 0
\(605\) −6.04163 −0.245627
\(606\) 0 0
\(607\) 9.89949 0.401808 0.200904 0.979611i \(-0.435612\pi\)
0.200904 + 0.979611i \(0.435612\pi\)
\(608\) 0 0
\(609\) 6.48528 0.262797
\(610\) 0 0
\(611\) 21.6569 0.876143
\(612\) 0 0
\(613\) 40.6274 1.64093 0.820463 0.571700i \(-0.193716\pi\)
0.820463 + 0.571700i \(0.193716\pi\)
\(614\) 0 0
\(615\) −3.79899 −0.153190
\(616\) 0 0
\(617\) −29.3137 −1.18013 −0.590063 0.807357i \(-0.700897\pi\)
−0.590063 + 0.807357i \(0.700897\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 0 0
\(621\) 6.82843 0.274015
\(622\) 0 0
\(623\) 24.4853 0.980982
\(624\) 0 0
\(625\) 19.9706 0.798823
\(626\) 0 0
\(627\) 2.34315 0.0935762
\(628\) 0 0
\(629\) −0.284271 −0.0113346
\(630\) 0 0
\(631\) −44.7279 −1.78059 −0.890295 0.455384i \(-0.849502\pi\)
−0.890295 + 0.455384i \(0.849502\pi\)
\(632\) 0 0
\(633\) −16.0000 −0.635943
\(634\) 0 0
\(635\) 10.4853 0.416096
\(636\) 0 0
\(637\) −24.1421 −0.956546
\(638\) 0 0
\(639\) −8.48528 −0.335673
\(640\) 0 0
\(641\) −28.8284 −1.13865 −0.569327 0.822111i \(-0.692796\pi\)
−0.569327 + 0.822111i \(0.692796\pi\)
\(642\) 0 0
\(643\) −19.1127 −0.753731 −0.376866 0.926268i \(-0.622998\pi\)
−0.376866 + 0.926268i \(0.622998\pi\)
\(644\) 0 0
\(645\) 0.686292 0.0270227
\(646\) 0 0
\(647\) −38.4264 −1.51070 −0.755349 0.655323i \(-0.772533\pi\)
−0.755349 + 0.655323i \(0.772533\pi\)
\(648\) 0 0
\(649\) −8.00000 −0.314027
\(650\) 0 0
\(651\) 10.0000 0.391931
\(652\) 0 0
\(653\) 32.3848 1.26731 0.633657 0.773614i \(-0.281553\pi\)
0.633657 + 0.773614i \(0.281553\pi\)
\(654\) 0 0
\(655\) 4.28427 0.167400
\(656\) 0 0
\(657\) 11.3137 0.441390
\(658\) 0 0
\(659\) 14.3431 0.558730 0.279365 0.960185i \(-0.409876\pi\)
0.279365 + 0.960185i \(0.409876\pi\)
\(660\) 0 0
\(661\) 18.0000 0.700119 0.350059 0.936727i \(-0.386161\pi\)
0.350059 + 0.936727i \(0.386161\pi\)
\(662\) 0 0
\(663\) 4.00000 0.155347
\(664\) 0 0
\(665\) −2.34315 −0.0908633
\(666\) 0 0
\(667\) −31.3137 −1.21247
\(668\) 0 0
\(669\) 7.55635 0.292145
\(670\) 0 0
\(671\) −9.65685 −0.372799
\(672\) 0 0
\(673\) −18.6863 −0.720304 −0.360152 0.932894i \(-0.617275\pi\)
−0.360152 + 0.932894i \(0.617275\pi\)
\(674\) 0 0
\(675\) 4.65685 0.179242
\(676\) 0 0
\(677\) −21.0711 −0.809827 −0.404913 0.914355i \(-0.632698\pi\)
−0.404913 + 0.914355i \(0.632698\pi\)
\(678\) 0 0
\(679\) −5.17157 −0.198467
\(680\) 0 0
\(681\) −16.1421 −0.618568
\(682\) 0 0
\(683\) −15.8579 −0.606784 −0.303392 0.952866i \(-0.598119\pi\)
−0.303392 + 0.952866i \(0.598119\pi\)
\(684\) 0 0
\(685\) 8.48528 0.324206
\(686\) 0 0
\(687\) 21.7990 0.831683
\(688\) 0 0
\(689\) 49.4558 1.88412
\(690\) 0 0
\(691\) −30.8284 −1.17277 −0.586384 0.810033i \(-0.699449\pi\)
−0.586384 + 0.810033i \(0.699449\pi\)
\(692\) 0 0
\(693\) 1.17157 0.0445044
\(694\) 0 0
\(695\) 9.37258 0.355522
\(696\) 0 0
\(697\) −5.37258 −0.203501
\(698\) 0 0
\(699\) 2.00000 0.0756469
\(700\) 0 0
\(701\) 30.7279 1.16058 0.580289 0.814411i \(-0.302940\pi\)
0.580289 + 0.814411i \(0.302940\pi\)
\(702\) 0 0
\(703\) 0.970563 0.0366055
\(704\) 0 0
\(705\) −2.62742 −0.0989542
\(706\) 0 0
\(707\) −20.8284 −0.783334
\(708\) 0 0
\(709\) 35.1716 1.32090 0.660448 0.750872i \(-0.270366\pi\)
0.660448 + 0.750872i \(0.270366\pi\)
\(710\) 0 0
\(711\) 14.5858 0.547010
\(712\) 0 0
\(713\) −48.2843 −1.80826
\(714\) 0 0
\(715\) −2.34315 −0.0876287
\(716\) 0 0
\(717\) 23.3137 0.870666
\(718\) 0 0
\(719\) −8.48528 −0.316448 −0.158224 0.987403i \(-0.550577\pi\)
−0.158224 + 0.987403i \(0.550577\pi\)
\(720\) 0 0
\(721\) −9.31371 −0.346861
\(722\) 0 0
\(723\) 9.65685 0.359142
\(724\) 0 0
\(725\) −21.3553 −0.793117
\(726\) 0 0
\(727\) −40.5269 −1.50306 −0.751530 0.659699i \(-0.770684\pi\)
−0.751530 + 0.659699i \(0.770684\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0.970563 0.0358976
\(732\) 0 0
\(733\) 3.17157 0.117145 0.0585724 0.998283i \(-0.481345\pi\)
0.0585724 + 0.998283i \(0.481345\pi\)
\(734\) 0 0
\(735\) 2.92893 0.108035
\(736\) 0 0
\(737\) 4.68629 0.172622
\(738\) 0 0
\(739\) 31.3137 1.15189 0.575947 0.817487i \(-0.304634\pi\)
0.575947 + 0.817487i \(0.304634\pi\)
\(740\) 0 0
\(741\) −13.6569 −0.501697
\(742\) 0 0
\(743\) −48.2843 −1.77138 −0.885689 0.464279i \(-0.846314\pi\)
−0.885689 + 0.464279i \(0.846314\pi\)
\(744\) 0 0
\(745\) 0.343146 0.0125719
\(746\) 0 0
\(747\) 3.17157 0.116042
\(748\) 0 0
\(749\) 21.6569 0.791324
\(750\) 0 0
\(751\) −17.4142 −0.635454 −0.317727 0.948182i \(-0.602919\pi\)
−0.317727 + 0.948182i \(0.602919\pi\)
\(752\) 0 0
\(753\) −21.7990 −0.794399
\(754\) 0 0
\(755\) 5.51472 0.200701
\(756\) 0 0
\(757\) 26.7696 0.972956 0.486478 0.873693i \(-0.338281\pi\)
0.486478 + 0.873693i \(0.338281\pi\)
\(758\) 0 0
\(759\) −5.65685 −0.205331
\(760\) 0 0
\(761\) 28.8284 1.04503 0.522515 0.852630i \(-0.324994\pi\)
0.522515 + 0.852630i \(0.324994\pi\)
\(762\) 0 0
\(763\) −18.1421 −0.656789
\(764\) 0 0
\(765\) −0.485281 −0.0175454
\(766\) 0 0
\(767\) 46.6274 1.68362
\(768\) 0 0
\(769\) −27.3137 −0.984958 −0.492479 0.870324i \(-0.663909\pi\)
−0.492479 + 0.870324i \(0.663909\pi\)
\(770\) 0 0
\(771\) 8.34315 0.300471
\(772\) 0 0
\(773\) 29.7574 1.07030 0.535149 0.844758i \(-0.320255\pi\)
0.535149 + 0.844758i \(0.320255\pi\)
\(774\) 0 0
\(775\) −32.9289 −1.18284
\(776\) 0 0
\(777\) 0.485281 0.0174094
\(778\) 0 0
\(779\) 18.3431 0.657211
\(780\) 0 0
\(781\) 7.02944 0.251533
\(782\) 0 0
\(783\) −4.58579 −0.163883
\(784\) 0 0
\(785\) 3.11270 0.111097
\(786\) 0 0
\(787\) 18.8284 0.671161 0.335580 0.942012i \(-0.391068\pi\)
0.335580 + 0.942012i \(0.391068\pi\)
\(788\) 0 0
\(789\) −8.97056 −0.319360
\(790\) 0 0
\(791\) 10.8284 0.385015
\(792\) 0 0
\(793\) 56.2843 1.99871
\(794\) 0 0
\(795\) −6.00000 −0.212798
\(796\) 0 0
\(797\) 26.5269 0.939631 0.469816 0.882765i \(-0.344320\pi\)
0.469816 + 0.882765i \(0.344320\pi\)
\(798\) 0 0
\(799\) −3.71573 −0.131453
\(800\) 0 0
\(801\) −17.3137 −0.611750
\(802\) 0 0
\(803\) −9.37258 −0.330751
\(804\) 0 0
\(805\) 5.65685 0.199378
\(806\) 0 0
\(807\) 2.24264 0.0789447
\(808\) 0 0
\(809\) 21.5147 0.756417 0.378209 0.925720i \(-0.376540\pi\)
0.378209 + 0.925720i \(0.376540\pi\)
\(810\) 0 0
\(811\) −16.4853 −0.578877 −0.289438 0.957197i \(-0.593468\pi\)
−0.289438 + 0.957197i \(0.593468\pi\)
\(812\) 0 0
\(813\) 12.2426 0.429368
\(814\) 0 0
\(815\) 2.62742 0.0920344
\(816\) 0 0
\(817\) −3.31371 −0.115932
\(818\) 0 0
\(819\) −6.82843 −0.238605
\(820\) 0 0
\(821\) 35.0122 1.22193 0.610967 0.791656i \(-0.290781\pi\)
0.610967 + 0.791656i \(0.290781\pi\)
\(822\) 0 0
\(823\) −31.0711 −1.08307 −0.541535 0.840678i \(-0.682157\pi\)
−0.541535 + 0.840678i \(0.682157\pi\)
\(824\) 0 0
\(825\) −3.85786 −0.134314
\(826\) 0 0
\(827\) 34.6274 1.20411 0.602057 0.798453i \(-0.294348\pi\)
0.602057 + 0.798453i \(0.294348\pi\)
\(828\) 0 0
\(829\) 25.5147 0.886163 0.443081 0.896481i \(-0.353885\pi\)
0.443081 + 0.896481i \(0.353885\pi\)
\(830\) 0 0
\(831\) 20.8284 0.722530
\(832\) 0 0
\(833\) 4.14214 0.143516
\(834\) 0 0
\(835\) −7.02944 −0.243264
\(836\) 0 0
\(837\) −7.07107 −0.244412
\(838\) 0 0
\(839\) 55.1127 1.90270 0.951351 0.308110i \(-0.0996964\pi\)
0.951351 + 0.308110i \(0.0996964\pi\)
\(840\) 0 0
\(841\) −7.97056 −0.274847
\(842\) 0 0
\(843\) 12.3431 0.425121
\(844\) 0 0
\(845\) 6.04163 0.207838
\(846\) 0 0
\(847\) 14.5858 0.501174
\(848\) 0 0
\(849\) 20.9706 0.719708
\(850\) 0 0
\(851\) −2.34315 −0.0803220
\(852\) 0 0
\(853\) 21.3137 0.729767 0.364884 0.931053i \(-0.381109\pi\)
0.364884 + 0.931053i \(0.381109\pi\)
\(854\) 0 0
\(855\) 1.65685 0.0566632
\(856\) 0 0
\(857\) −10.4853 −0.358170 −0.179085 0.983834i \(-0.557314\pi\)
−0.179085 + 0.983834i \(0.557314\pi\)
\(858\) 0 0
\(859\) 14.8284 0.505939 0.252970 0.967474i \(-0.418593\pi\)
0.252970 + 0.967474i \(0.418593\pi\)
\(860\) 0 0
\(861\) 9.17157 0.312566
\(862\) 0 0
\(863\) −4.00000 −0.136162 −0.0680808 0.997680i \(-0.521688\pi\)
−0.0680808 + 0.997680i \(0.521688\pi\)
\(864\) 0 0
\(865\) −6.97056 −0.237006
\(866\) 0 0
\(867\) 16.3137 0.554043
\(868\) 0 0
\(869\) −12.0833 −0.409897
\(870\) 0 0
\(871\) −27.3137 −0.925490
\(872\) 0 0
\(873\) 3.65685 0.123766
\(874\) 0 0
\(875\) 8.00000 0.270449
\(876\) 0 0
\(877\) −35.9411 −1.21365 −0.606823 0.794837i \(-0.707556\pi\)
−0.606823 + 0.794837i \(0.707556\pi\)
\(878\) 0 0
\(879\) 17.0711 0.575793
\(880\) 0 0
\(881\) 15.6569 0.527493 0.263746 0.964592i \(-0.415042\pi\)
0.263746 + 0.964592i \(0.415042\pi\)
\(882\) 0 0
\(883\) −15.7990 −0.531678 −0.265839 0.964017i \(-0.585649\pi\)
−0.265839 + 0.964017i \(0.585649\pi\)
\(884\) 0 0
\(885\) −5.65685 −0.190153
\(886\) 0 0
\(887\) −52.2843 −1.75553 −0.877767 0.479088i \(-0.840968\pi\)
−0.877767 + 0.479088i \(0.840968\pi\)
\(888\) 0 0
\(889\) −25.3137 −0.848995
\(890\) 0 0
\(891\) −0.828427 −0.0277534
\(892\) 0 0
\(893\) 12.6863 0.424531
\(894\) 0 0
\(895\) −15.0294 −0.502379
\(896\) 0 0
\(897\) 32.9706 1.10086
\(898\) 0 0
\(899\) 32.4264 1.08148
\(900\) 0 0
\(901\) −8.48528 −0.282686
\(902\) 0 0
\(903\) −1.65685 −0.0551367
\(904\) 0 0
\(905\) −9.45584 −0.314323
\(906\) 0 0
\(907\) 30.8284 1.02364 0.511821 0.859092i \(-0.328971\pi\)
0.511821 + 0.859092i \(0.328971\pi\)
\(908\) 0 0
\(909\) 14.7279 0.488494
\(910\) 0 0
\(911\) 59.5980 1.97457 0.987285 0.158963i \(-0.0508150\pi\)
0.987285 + 0.158963i \(0.0508150\pi\)
\(912\) 0 0
\(913\) −2.62742 −0.0869548
\(914\) 0 0
\(915\) −6.82843 −0.225741
\(916\) 0 0
\(917\) −10.3431 −0.341561
\(918\) 0 0
\(919\) 15.5563 0.513157 0.256578 0.966523i \(-0.417405\pi\)
0.256578 + 0.966523i \(0.417405\pi\)
\(920\) 0 0
\(921\) 29.6569 0.977227
\(922\) 0 0
\(923\) −40.9706 −1.34856
\(924\) 0 0
\(925\) −1.59798 −0.0525413
\(926\) 0 0
\(927\) 6.58579 0.216306
\(928\) 0 0
\(929\) 19.8579 0.651515 0.325758 0.945453i \(-0.394381\pi\)
0.325758 + 0.945453i \(0.394381\pi\)
\(930\) 0 0
\(931\) −14.1421 −0.463490
\(932\) 0 0
\(933\) 4.68629 0.153422
\(934\) 0 0
\(935\) 0.402020 0.0131475
\(936\) 0 0
\(937\) −8.62742 −0.281845 −0.140923 0.990021i \(-0.545007\pi\)
−0.140923 + 0.990021i \(0.545007\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 30.0416 0.979329 0.489665 0.871911i \(-0.337119\pi\)
0.489665 + 0.871911i \(0.337119\pi\)
\(942\) 0 0
\(943\) −44.2843 −1.44209
\(944\) 0 0
\(945\) 0.828427 0.0269487
\(946\) 0 0
\(947\) −38.3431 −1.24598 −0.622992 0.782228i \(-0.714083\pi\)
−0.622992 + 0.782228i \(0.714083\pi\)
\(948\) 0 0
\(949\) 54.6274 1.77328
\(950\) 0 0
\(951\) 31.8995 1.03441
\(952\) 0 0
\(953\) 34.4853 1.11709 0.558544 0.829475i \(-0.311360\pi\)
0.558544 + 0.829475i \(0.311360\pi\)
\(954\) 0 0
\(955\) −4.28427 −0.138636
\(956\) 0 0
\(957\) 3.79899 0.122804
\(958\) 0 0
\(959\) −20.4853 −0.661504
\(960\) 0 0
\(961\) 19.0000 0.612903
\(962\) 0 0
\(963\) −15.3137 −0.493477
\(964\) 0 0
\(965\) 6.62742 0.213344
\(966\) 0 0
\(967\) −48.5269 −1.56052 −0.780260 0.625455i \(-0.784913\pi\)
−0.780260 + 0.625455i \(0.784913\pi\)
\(968\) 0 0
\(969\) 2.34315 0.0752727
\(970\) 0 0
\(971\) −19.1716 −0.615245 −0.307623 0.951508i \(-0.599533\pi\)
−0.307623 + 0.951508i \(0.599533\pi\)
\(972\) 0 0
\(973\) −22.6274 −0.725402
\(974\) 0 0
\(975\) 22.4853 0.720105
\(976\) 0 0
\(977\) −40.8284 −1.30622 −0.653109 0.757264i \(-0.726535\pi\)
−0.653109 + 0.757264i \(0.726535\pi\)
\(978\) 0 0
\(979\) 14.3431 0.458409
\(980\) 0 0
\(981\) 12.8284 0.409580
\(982\) 0 0
\(983\) 26.3431 0.840216 0.420108 0.907474i \(-0.361992\pi\)
0.420108 + 0.907474i \(0.361992\pi\)
\(984\) 0 0
\(985\) −2.00000 −0.0637253
\(986\) 0 0
\(987\) 6.34315 0.201905
\(988\) 0 0
\(989\) 8.00000 0.254385
\(990\) 0 0
\(991\) −22.1005 −0.702046 −0.351023 0.936367i \(-0.614166\pi\)
−0.351023 + 0.936367i \(0.614166\pi\)
\(992\) 0 0
\(993\) −13.6569 −0.433387
\(994\) 0 0
\(995\) 11.8579 0.375920
\(996\) 0 0
\(997\) 6.97056 0.220760 0.110380 0.993889i \(-0.464793\pi\)
0.110380 + 0.993889i \(0.464793\pi\)
\(998\) 0 0
\(999\) −0.343146 −0.0108567
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 1536.2.a.f.1.1 yes 2
3.2 odd 2 4608.2.a.b.1.2 2
4.3 odd 2 1536.2.a.k.1.1 yes 2
8.3 odd 2 1536.2.a.a.1.2 2
8.5 even 2 1536.2.a.h.1.2 yes 2
12.11 even 2 4608.2.a.d.1.2 2
16.3 odd 4 1536.2.d.e.769.4 4
16.5 even 4 1536.2.d.b.769.3 4
16.11 odd 4 1536.2.d.e.769.1 4
16.13 even 4 1536.2.d.b.769.2 4
24.5 odd 2 4608.2.a.q.1.1 2
24.11 even 2 4608.2.a.o.1.1 2
48.5 odd 4 4608.2.d.f.2305.2 4
48.11 even 4 4608.2.d.h.2305.2 4
48.29 odd 4 4608.2.d.f.2305.3 4
48.35 even 4 4608.2.d.h.2305.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.a.a.1.2 2 8.3 odd 2
1536.2.a.f.1.1 yes 2 1.1 even 1 trivial
1536.2.a.h.1.2 yes 2 8.5 even 2
1536.2.a.k.1.1 yes 2 4.3 odd 2
1536.2.d.b.769.2 4 16.13 even 4
1536.2.d.b.769.3 4 16.5 even 4
1536.2.d.e.769.1 4 16.11 odd 4
1536.2.d.e.769.4 4 16.3 odd 4
4608.2.a.b.1.2 2 3.2 odd 2
4608.2.a.d.1.2 2 12.11 even 2
4608.2.a.o.1.1 2 24.11 even 2
4608.2.a.q.1.1 2 24.5 odd 2
4608.2.d.f.2305.2 4 48.5 odd 4
4608.2.d.f.2305.3 4 48.29 odd 4
4608.2.d.h.2305.2 4 48.11 even 4
4608.2.d.h.2305.3 4 48.35 even 4