Properties

Label 4096.2.a.f.1.3
Level $4096$
Weight $2$
Character 4096.1
Self dual yes
Analytic conductor $32.707$
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] = [4096,2,Mod(1,4096)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4096, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("4096.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4096 = 2^{12} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4096.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: \(32.7067246679\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{16})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 32)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(0.765367\) of defining polynomial
Character \(\chi\) \(=\) 4096.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.765367 q^{3} -2.93015 q^{5} -1.41421 q^{7} -2.41421 q^{9} +O(q^{10})\) \(q+0.765367 q^{3} -2.93015 q^{5} -1.41421 q^{7} -2.41421 q^{9} -4.46088 q^{11} -0.765367 q^{13} -2.24264 q^{15} -2.82843 q^{17} -4.01254 q^{19} -1.08239 q^{21} +8.24264 q^{23} +3.58579 q^{25} -4.14386 q^{27} +3.37849 q^{29} -4.00000 q^{31} -3.41421 q^{33} +4.14386 q^{35} +0.765367 q^{37} -0.585786 q^{39} +0.242641 q^{41} -5.09494 q^{43} +7.07401 q^{45} +0.343146 q^{47} -5.00000 q^{49} -2.16478 q^{51} +1.21371 q^{53} +13.0711 q^{55} -3.07107 q^{57} -4.90923 q^{59} +1.84776 q^{61} +3.41421 q^{63} +2.24264 q^{65} +5.99162 q^{67} +6.30864 q^{69} +8.24264 q^{71} +9.89949 q^{73} +2.74444 q^{75} +6.30864 q^{77} +6.00000 q^{79} +4.07107 q^{81} -4.90923 q^{83} +8.28772 q^{85} +2.58579 q^{87} -12.2426 q^{89} +1.08239 q^{91} -3.06147 q^{93} +11.7574 q^{95} +18.4853 q^{97} +10.7695 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{9} + 8 q^{15} + 16 q^{23} + 20 q^{25} - 16 q^{31} - 8 q^{33} - 8 q^{39} - 16 q^{41} + 24 q^{47} - 20 q^{49} + 24 q^{55} + 16 q^{57} + 8 q^{63} - 8 q^{65} + 16 q^{71} + 24 q^{79} - 12 q^{81} + 16 q^{87} - 32 q^{89} + 64 q^{95} + 40 q^{97}+O(q^{100}) \) 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
\(3\) 0.765367 0.441885 0.220942 0.975287i \(-0.429087\pi\)
0.220942 + 0.975287i \(0.429087\pi\)
\(4\) 0 0
\(5\) −2.93015 −1.31040 −0.655202 0.755454i \(-0.727416\pi\)
−0.655202 + 0.755454i \(0.727416\pi\)
\(6\) 0 0
\(7\) −1.41421 −0.534522 −0.267261 0.963624i \(-0.586119\pi\)
−0.267261 + 0.963624i \(0.586119\pi\)
\(8\) 0 0
\(9\) −2.41421 −0.804738
\(10\) 0 0
\(11\) −4.46088 −1.34501 −0.672504 0.740094i \(-0.734781\pi\)
−0.672504 + 0.740094i \(0.734781\pi\)
\(12\) 0 0
\(13\) −0.765367 −0.212275 −0.106137 0.994351i \(-0.533848\pi\)
−0.106137 + 0.994351i \(0.533848\pi\)
\(14\) 0 0
\(15\) −2.24264 −0.579047
\(16\) 0 0
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) −4.01254 −0.920541 −0.460270 0.887779i \(-0.652248\pi\)
−0.460270 + 0.887779i \(0.652248\pi\)
\(20\) 0 0
\(21\) −1.08239 −0.236197
\(22\) 0 0
\(23\) 8.24264 1.71871 0.859355 0.511380i \(-0.170866\pi\)
0.859355 + 0.511380i \(0.170866\pi\)
\(24\) 0 0
\(25\) 3.58579 0.717157
\(26\) 0 0
\(27\) −4.14386 −0.797486
\(28\) 0 0
\(29\) 3.37849 0.627370 0.313685 0.949527i \(-0.398436\pi\)
0.313685 + 0.949527i \(0.398436\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 0 0
\(33\) −3.41421 −0.594338
\(34\) 0 0
\(35\) 4.14386 0.700440
\(36\) 0 0
\(37\) 0.765367 0.125826 0.0629128 0.998019i \(-0.479961\pi\)
0.0629128 + 0.998019i \(0.479961\pi\)
\(38\) 0 0
\(39\) −0.585786 −0.0938009
\(40\) 0 0
\(41\) 0.242641 0.0378941 0.0189471 0.999820i \(-0.493969\pi\)
0.0189471 + 0.999820i \(0.493969\pi\)
\(42\) 0 0
\(43\) −5.09494 −0.776970 −0.388485 0.921455i \(-0.627002\pi\)
−0.388485 + 0.921455i \(0.627002\pi\)
\(44\) 0 0
\(45\) 7.07401 1.05453
\(46\) 0 0
\(47\) 0.343146 0.0500530 0.0250265 0.999687i \(-0.492033\pi\)
0.0250265 + 0.999687i \(0.492033\pi\)
\(48\) 0 0
\(49\) −5.00000 −0.714286
\(50\) 0 0
\(51\) −2.16478 −0.303130
\(52\) 0 0
\(53\) 1.21371 0.166716 0.0833578 0.996520i \(-0.473436\pi\)
0.0833578 + 0.996520i \(0.473436\pi\)
\(54\) 0 0
\(55\) 13.0711 1.76250
\(56\) 0 0
\(57\) −3.07107 −0.406773
\(58\) 0 0
\(59\) −4.90923 −0.639127 −0.319563 0.947565i \(-0.603536\pi\)
−0.319563 + 0.947565i \(0.603536\pi\)
\(60\) 0 0
\(61\) 1.84776 0.236581 0.118291 0.992979i \(-0.462259\pi\)
0.118291 + 0.992979i \(0.462259\pi\)
\(62\) 0 0
\(63\) 3.41421 0.430150
\(64\) 0 0
\(65\) 2.24264 0.278165
\(66\) 0 0
\(67\) 5.99162 0.731993 0.365996 0.930616i \(-0.380728\pi\)
0.365996 + 0.930616i \(0.380728\pi\)
\(68\) 0 0
\(69\) 6.30864 0.759471
\(70\) 0 0
\(71\) 8.24264 0.978221 0.489111 0.872222i \(-0.337321\pi\)
0.489111 + 0.872222i \(0.337321\pi\)
\(72\) 0 0
\(73\) 9.89949 1.15865 0.579324 0.815097i \(-0.303317\pi\)
0.579324 + 0.815097i \(0.303317\pi\)
\(74\) 0 0
\(75\) 2.74444 0.316901
\(76\) 0 0
\(77\) 6.30864 0.718937
\(78\) 0 0
\(79\) 6.00000 0.675053 0.337526 0.941316i \(-0.390410\pi\)
0.337526 + 0.941316i \(0.390410\pi\)
\(80\) 0 0
\(81\) 4.07107 0.452341
\(82\) 0 0
\(83\) −4.90923 −0.538858 −0.269429 0.963020i \(-0.586835\pi\)
−0.269429 + 0.963020i \(0.586835\pi\)
\(84\) 0 0
\(85\) 8.28772 0.898929
\(86\) 0 0
\(87\) 2.58579 0.277225
\(88\) 0 0
\(89\) −12.2426 −1.29772 −0.648859 0.760909i \(-0.724753\pi\)
−0.648859 + 0.760909i \(0.724753\pi\)
\(90\) 0 0
\(91\) 1.08239 0.113466
\(92\) 0 0
\(93\) −3.06147 −0.317459
\(94\) 0 0
\(95\) 11.7574 1.20628
\(96\) 0 0
\(97\) 18.4853 1.87690 0.938448 0.345421i \(-0.112264\pi\)
0.938448 + 0.345421i \(0.112264\pi\)
\(98\) 0 0
\(99\) 10.7695 1.08238
\(100\) 0 0
\(101\) −3.56420 −0.354651 −0.177326 0.984152i \(-0.556745\pi\)
−0.177326 + 0.984152i \(0.556745\pi\)
\(102\) 0 0
\(103\) 13.4142 1.32174 0.660871 0.750500i \(-0.270187\pi\)
0.660871 + 0.750500i \(0.270187\pi\)
\(104\) 0 0
\(105\) 3.17157 0.309514
\(106\) 0 0
\(107\) 4.46088 0.431250 0.215625 0.976476i \(-0.430821\pi\)
0.215625 + 0.976476i \(0.430821\pi\)
\(108\) 0 0
\(109\) −14.9134 −1.42844 −0.714222 0.699919i \(-0.753219\pi\)
−0.714222 + 0.699919i \(0.753219\pi\)
\(110\) 0 0
\(111\) 0.585786 0.0556004
\(112\) 0 0
\(113\) 6.34315 0.596713 0.298356 0.954455i \(-0.403562\pi\)
0.298356 + 0.954455i \(0.403562\pi\)
\(114\) 0 0
\(115\) −24.1522 −2.25220
\(116\) 0 0
\(117\) 1.84776 0.170825
\(118\) 0 0
\(119\) 4.00000 0.366679
\(120\) 0 0
\(121\) 8.89949 0.809045
\(122\) 0 0
\(123\) 0.185709 0.0167448
\(124\) 0 0
\(125\) 4.14386 0.370638
\(126\) 0 0
\(127\) −12.9706 −1.15095 −0.575476 0.817819i \(-0.695183\pi\)
−0.575476 + 0.817819i \(0.695183\pi\)
\(128\) 0 0
\(129\) −3.89949 −0.343331
\(130\) 0 0
\(131\) 17.7122 1.54752 0.773762 0.633476i \(-0.218373\pi\)
0.773762 + 0.633476i \(0.218373\pi\)
\(132\) 0 0
\(133\) 5.67459 0.492050
\(134\) 0 0
\(135\) 12.1421 1.04503
\(136\) 0 0
\(137\) −12.2426 −1.04596 −0.522980 0.852345i \(-0.675180\pi\)
−0.522980 + 0.852345i \(0.675180\pi\)
\(138\) 0 0
\(139\) 14.2793 1.21116 0.605579 0.795785i \(-0.292942\pi\)
0.605579 + 0.795785i \(0.292942\pi\)
\(140\) 0 0
\(141\) 0.262632 0.0221176
\(142\) 0 0
\(143\) 3.41421 0.285511
\(144\) 0 0
\(145\) −9.89949 −0.822108
\(146\) 0 0
\(147\) −3.82683 −0.315632
\(148\) 0 0
\(149\) −16.8925 −1.38388 −0.691942 0.721953i \(-0.743245\pi\)
−0.691942 + 0.721953i \(0.743245\pi\)
\(150\) 0 0
\(151\) −2.10051 −0.170937 −0.0854683 0.996341i \(-0.527239\pi\)
−0.0854683 + 0.996341i \(0.527239\pi\)
\(152\) 0 0
\(153\) 6.82843 0.552046
\(154\) 0 0
\(155\) 11.7206 0.941422
\(156\) 0 0
\(157\) −1.84776 −0.147467 −0.0737336 0.997278i \(-0.523491\pi\)
−0.0737336 + 0.997278i \(0.523491\pi\)
\(158\) 0 0
\(159\) 0.928932 0.0736691
\(160\) 0 0
\(161\) −11.6569 −0.918689
\(162\) 0 0
\(163\) 0.502734 0.0393772 0.0196886 0.999806i \(-0.493733\pi\)
0.0196886 + 0.999806i \(0.493733\pi\)
\(164\) 0 0
\(165\) 10.0042 0.778823
\(166\) 0 0
\(167\) 20.7279 1.60397 0.801987 0.597341i \(-0.203776\pi\)
0.801987 + 0.597341i \(0.203776\pi\)
\(168\) 0 0
\(169\) −12.4142 −0.954940
\(170\) 0 0
\(171\) 9.68714 0.740794
\(172\) 0 0
\(173\) 8.15640 0.620120 0.310060 0.950717i \(-0.399651\pi\)
0.310060 + 0.950717i \(0.399651\pi\)
\(174\) 0 0
\(175\) −5.07107 −0.383337
\(176\) 0 0
\(177\) −3.75736 −0.282420
\(178\) 0 0
\(179\) −4.27518 −0.319542 −0.159771 0.987154i \(-0.551076\pi\)
−0.159771 + 0.987154i \(0.551076\pi\)
\(180\) 0 0
\(181\) −17.5265 −1.30274 −0.651368 0.758762i \(-0.725804\pi\)
−0.651368 + 0.758762i \(0.725804\pi\)
\(182\) 0 0
\(183\) 1.41421 0.104542
\(184\) 0 0
\(185\) −2.24264 −0.164882
\(186\) 0 0
\(187\) 12.6173 0.922667
\(188\) 0 0
\(189\) 5.86030 0.426274
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) 0 0
\(193\) −1.51472 −0.109032 −0.0545159 0.998513i \(-0.517362\pi\)
−0.0545159 + 0.998513i \(0.517362\pi\)
\(194\) 0 0
\(195\) 1.71644 0.122917
\(196\) 0 0
\(197\) −12.1146 −0.863126 −0.431563 0.902083i \(-0.642038\pi\)
−0.431563 + 0.902083i \(0.642038\pi\)
\(198\) 0 0
\(199\) 22.5858 1.60106 0.800532 0.599290i \(-0.204550\pi\)
0.800532 + 0.599290i \(0.204550\pi\)
\(200\) 0 0
\(201\) 4.58579 0.323456
\(202\) 0 0
\(203\) −4.77791 −0.335344
\(204\) 0 0
\(205\) −0.710974 −0.0496566
\(206\) 0 0
\(207\) −19.8995 −1.38311
\(208\) 0 0
\(209\) 17.8995 1.23813
\(210\) 0 0
\(211\) −19.6913 −1.35560 −0.677802 0.735244i \(-0.737068\pi\)
−0.677802 + 0.735244i \(0.737068\pi\)
\(212\) 0 0
\(213\) 6.30864 0.432261
\(214\) 0 0
\(215\) 14.9289 1.01814
\(216\) 0 0
\(217\) 5.65685 0.384012
\(218\) 0 0
\(219\) 7.57675 0.511989
\(220\) 0 0
\(221\) 2.16478 0.145619
\(222\) 0 0
\(223\) −20.9706 −1.40429 −0.702146 0.712033i \(-0.747775\pi\)
−0.702146 + 0.712033i \(0.747775\pi\)
\(224\) 0 0
\(225\) −8.65685 −0.577124
\(226\) 0 0
\(227\) 20.1396 1.33671 0.668357 0.743840i \(-0.266998\pi\)
0.668357 + 0.743840i \(0.266998\pi\)
\(228\) 0 0
\(229\) −24.0978 −1.59243 −0.796213 0.605016i \(-0.793167\pi\)
−0.796213 + 0.605016i \(0.793167\pi\)
\(230\) 0 0
\(231\) 4.82843 0.317687
\(232\) 0 0
\(233\) 3.75736 0.246153 0.123076 0.992397i \(-0.460724\pi\)
0.123076 + 0.992397i \(0.460724\pi\)
\(234\) 0 0
\(235\) −1.00547 −0.0655896
\(236\) 0 0
\(237\) 4.59220 0.298296
\(238\) 0 0
\(239\) 5.31371 0.343715 0.171858 0.985122i \(-0.445023\pi\)
0.171858 + 0.985122i \(0.445023\pi\)
\(240\) 0 0
\(241\) 8.48528 0.546585 0.273293 0.961931i \(-0.411887\pi\)
0.273293 + 0.961931i \(0.411887\pi\)
\(242\) 0 0
\(243\) 15.5474 0.997369
\(244\) 0 0
\(245\) 14.6508 0.936002
\(246\) 0 0
\(247\) 3.07107 0.195407
\(248\) 0 0
\(249\) −3.75736 −0.238113
\(250\) 0 0
\(251\) 17.2639 1.08969 0.544843 0.838538i \(-0.316589\pi\)
0.544843 + 0.838538i \(0.316589\pi\)
\(252\) 0 0
\(253\) −36.7695 −2.31168
\(254\) 0 0
\(255\) 6.34315 0.397223
\(256\) 0 0
\(257\) 6.00000 0.374270 0.187135 0.982334i \(-0.440080\pi\)
0.187135 + 0.982334i \(0.440080\pi\)
\(258\) 0 0
\(259\) −1.08239 −0.0672566
\(260\) 0 0
\(261\) −8.15640 −0.504869
\(262\) 0 0
\(263\) 8.24264 0.508263 0.254131 0.967170i \(-0.418210\pi\)
0.254131 + 0.967170i \(0.418210\pi\)
\(264\) 0 0
\(265\) −3.55635 −0.218465
\(266\) 0 0
\(267\) −9.37011 −0.573442
\(268\) 0 0
\(269\) −23.8352 −1.45326 −0.726628 0.687031i \(-0.758913\pi\)
−0.726628 + 0.687031i \(0.758913\pi\)
\(270\) 0 0
\(271\) 18.0000 1.09342 0.546711 0.837321i \(-0.315880\pi\)
0.546711 + 0.837321i \(0.315880\pi\)
\(272\) 0 0
\(273\) 0.828427 0.0501387
\(274\) 0 0
\(275\) −15.9958 −0.964582
\(276\) 0 0
\(277\) 1.84776 0.111021 0.0555105 0.998458i \(-0.482321\pi\)
0.0555105 + 0.998458i \(0.482321\pi\)
\(278\) 0 0
\(279\) 9.65685 0.578141
\(280\) 0 0
\(281\) 16.7279 0.997904 0.498952 0.866630i \(-0.333718\pi\)
0.498952 + 0.866630i \(0.333718\pi\)
\(282\) 0 0
\(283\) −15.0991 −0.897548 −0.448774 0.893645i \(-0.648139\pi\)
−0.448774 + 0.893645i \(0.648139\pi\)
\(284\) 0 0
\(285\) 8.99869 0.533037
\(286\) 0 0
\(287\) −0.343146 −0.0202553
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 14.1480 0.829372
\(292\) 0 0
\(293\) 25.1033 1.46655 0.733274 0.679933i \(-0.237991\pi\)
0.733274 + 0.679933i \(0.237991\pi\)
\(294\) 0 0
\(295\) 14.3848 0.837514
\(296\) 0 0
\(297\) 18.4853 1.07262
\(298\) 0 0
\(299\) −6.30864 −0.364838
\(300\) 0 0
\(301\) 7.20533 0.415308
\(302\) 0 0
\(303\) −2.72792 −0.156715
\(304\) 0 0
\(305\) −5.41421 −0.310017
\(306\) 0 0
\(307\) −18.1606 −1.03648 −0.518239 0.855236i \(-0.673412\pi\)
−0.518239 + 0.855236i \(0.673412\pi\)
\(308\) 0 0
\(309\) 10.2668 0.584058
\(310\) 0 0
\(311\) 3.75736 0.213060 0.106530 0.994309i \(-0.466026\pi\)
0.106530 + 0.994309i \(0.466026\pi\)
\(312\) 0 0
\(313\) −10.5858 −0.598344 −0.299172 0.954199i \(-0.596710\pi\)
−0.299172 + 0.954199i \(0.596710\pi\)
\(314\) 0 0
\(315\) −10.0042 −0.563671
\(316\) 0 0
\(317\) 18.7946 1.05561 0.527805 0.849365i \(-0.323015\pi\)
0.527805 + 0.849365i \(0.323015\pi\)
\(318\) 0 0
\(319\) −15.0711 −0.843818
\(320\) 0 0
\(321\) 3.41421 0.190563
\(322\) 0 0
\(323\) 11.3492 0.631486
\(324\) 0 0
\(325\) −2.74444 −0.152234
\(326\) 0 0
\(327\) −11.4142 −0.631207
\(328\) 0 0
\(329\) −0.485281 −0.0267544
\(330\) 0 0
\(331\) −1.39942 −0.0769189 −0.0384595 0.999260i \(-0.512245\pi\)
−0.0384595 + 0.999260i \(0.512245\pi\)
\(332\) 0 0
\(333\) −1.84776 −0.101257
\(334\) 0 0
\(335\) −17.5563 −0.959206
\(336\) 0 0
\(337\) −16.9706 −0.924445 −0.462223 0.886764i \(-0.652948\pi\)
−0.462223 + 0.886764i \(0.652948\pi\)
\(338\) 0 0
\(339\) 4.85483 0.263678
\(340\) 0 0
\(341\) 17.8435 0.966282
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) −18.4853 −0.995214
\(346\) 0 0
\(347\) −4.27518 −0.229503 −0.114752 0.993394i \(-0.536607\pi\)
−0.114752 + 0.993394i \(0.536607\pi\)
\(348\) 0 0
\(349\) −26.7109 −1.42980 −0.714901 0.699225i \(-0.753528\pi\)
−0.714901 + 0.699225i \(0.753528\pi\)
\(350\) 0 0
\(351\) 3.17157 0.169286
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) −24.1522 −1.28186
\(356\) 0 0
\(357\) 3.06147 0.162030
\(358\) 0 0
\(359\) −25.2132 −1.33070 −0.665351 0.746531i \(-0.731718\pi\)
−0.665351 + 0.746531i \(0.731718\pi\)
\(360\) 0 0
\(361\) −2.89949 −0.152605
\(362\) 0 0
\(363\) 6.81138 0.357505
\(364\) 0 0
\(365\) −29.0070 −1.51830
\(366\) 0 0
\(367\) 6.00000 0.313197 0.156599 0.987662i \(-0.449947\pi\)
0.156599 + 0.987662i \(0.449947\pi\)
\(368\) 0 0
\(369\) −0.585786 −0.0304948
\(370\) 0 0
\(371\) −1.71644 −0.0891133
\(372\) 0 0
\(373\) 11.1409 0.576856 0.288428 0.957502i \(-0.406867\pi\)
0.288428 + 0.957502i \(0.406867\pi\)
\(374\) 0 0
\(375\) 3.17157 0.163779
\(376\) 0 0
\(377\) −2.58579 −0.133175
\(378\) 0 0
\(379\) −35.7415 −1.83592 −0.917958 0.396677i \(-0.870163\pi\)
−0.917958 + 0.396677i \(0.870163\pi\)
\(380\) 0 0
\(381\) −9.92724 −0.508588
\(382\) 0 0
\(383\) 16.9706 0.867155 0.433578 0.901116i \(-0.357251\pi\)
0.433578 + 0.901116i \(0.357251\pi\)
\(384\) 0 0
\(385\) −18.4853 −0.942097
\(386\) 0 0
\(387\) 12.3003 0.625257
\(388\) 0 0
\(389\) 21.9330 1.11205 0.556024 0.831166i \(-0.312326\pi\)
0.556024 + 0.831166i \(0.312326\pi\)
\(390\) 0 0
\(391\) −23.3137 −1.17902
\(392\) 0 0
\(393\) 13.5563 0.683827
\(394\) 0 0
\(395\) −17.5809 −0.884591
\(396\) 0 0
\(397\) 24.0978 1.20943 0.604717 0.796441i \(-0.293286\pi\)
0.604717 + 0.796441i \(0.293286\pi\)
\(398\) 0 0
\(399\) 4.34315 0.217429
\(400\) 0 0
\(401\) −2.82843 −0.141245 −0.0706225 0.997503i \(-0.522499\pi\)
−0.0706225 + 0.997503i \(0.522499\pi\)
\(402\) 0 0
\(403\) 3.06147 0.152503
\(404\) 0 0
\(405\) −11.9288 −0.592749
\(406\) 0 0
\(407\) −3.41421 −0.169236
\(408\) 0 0
\(409\) −30.3848 −1.50243 −0.751215 0.660057i \(-0.770532\pi\)
−0.751215 + 0.660057i \(0.770532\pi\)
\(410\) 0 0
\(411\) −9.37011 −0.462194
\(412\) 0 0
\(413\) 6.94269 0.341628
\(414\) 0 0
\(415\) 14.3848 0.706121
\(416\) 0 0
\(417\) 10.9289 0.535192
\(418\) 0 0
\(419\) 13.6453 0.666616 0.333308 0.942818i \(-0.391835\pi\)
0.333308 + 0.942818i \(0.391835\pi\)
\(420\) 0 0
\(421\) 16.4441 0.801437 0.400719 0.916201i \(-0.368760\pi\)
0.400719 + 0.916201i \(0.368760\pi\)
\(422\) 0 0
\(423\) −0.828427 −0.0402795
\(424\) 0 0
\(425\) −10.1421 −0.491966
\(426\) 0 0
\(427\) −2.61313 −0.126458
\(428\) 0 0
\(429\) 2.61313 0.126163
\(430\) 0 0
\(431\) 12.3431 0.594548 0.297274 0.954792i \(-0.403922\pi\)
0.297274 + 0.954792i \(0.403922\pi\)
\(432\) 0 0
\(433\) 15.5147 0.745590 0.372795 0.927914i \(-0.378400\pi\)
0.372795 + 0.927914i \(0.378400\pi\)
\(434\) 0 0
\(435\) −7.57675 −0.363277
\(436\) 0 0
\(437\) −33.0740 −1.58214
\(438\) 0 0
\(439\) 24.0416 1.14744 0.573722 0.819050i \(-0.305499\pi\)
0.573722 + 0.819050i \(0.305499\pi\)
\(440\) 0 0
\(441\) 12.0711 0.574813
\(442\) 0 0
\(443\) 1.58513 0.0753116 0.0376558 0.999291i \(-0.488011\pi\)
0.0376558 + 0.999291i \(0.488011\pi\)
\(444\) 0 0
\(445\) 35.8728 1.70053
\(446\) 0 0
\(447\) −12.9289 −0.611518
\(448\) 0 0
\(449\) −19.4558 −0.918178 −0.459089 0.888390i \(-0.651824\pi\)
−0.459089 + 0.888390i \(0.651824\pi\)
\(450\) 0 0
\(451\) −1.08239 −0.0509679
\(452\) 0 0
\(453\) −1.60766 −0.0755343
\(454\) 0 0
\(455\) −3.17157 −0.148686
\(456\) 0 0
\(457\) −10.5858 −0.495182 −0.247591 0.968865i \(-0.579639\pi\)
−0.247591 + 0.968865i \(0.579639\pi\)
\(458\) 0 0
\(459\) 11.7206 0.547071
\(460\) 0 0
\(461\) −1.66205 −0.0774094 −0.0387047 0.999251i \(-0.512323\pi\)
−0.0387047 + 0.999251i \(0.512323\pi\)
\(462\) 0 0
\(463\) 22.9706 1.06753 0.533766 0.845632i \(-0.320776\pi\)
0.533766 + 0.845632i \(0.320776\pi\)
\(464\) 0 0
\(465\) 8.97056 0.416000
\(466\) 0 0
\(467\) 23.7582 1.09940 0.549700 0.835362i \(-0.314742\pi\)
0.549700 + 0.835362i \(0.314742\pi\)
\(468\) 0 0
\(469\) −8.47343 −0.391267
\(470\) 0 0
\(471\) −1.41421 −0.0651635
\(472\) 0 0
\(473\) 22.7279 1.04503
\(474\) 0 0
\(475\) −14.3881 −0.660172
\(476\) 0 0
\(477\) −2.93015 −0.134162
\(478\) 0 0
\(479\) 28.9706 1.32370 0.661849 0.749637i \(-0.269772\pi\)
0.661849 + 0.749637i \(0.269772\pi\)
\(480\) 0 0
\(481\) −0.585786 −0.0267096
\(482\) 0 0
\(483\) −8.92177 −0.405955
\(484\) 0 0
\(485\) −54.1647 −2.45949
\(486\) 0 0
\(487\) −15.5563 −0.704925 −0.352463 0.935826i \(-0.614656\pi\)
−0.352463 + 0.935826i \(0.614656\pi\)
\(488\) 0 0
\(489\) 0.384776 0.0174002
\(490\) 0 0
\(491\) −42.5754 −1.92140 −0.960700 0.277589i \(-0.910465\pi\)
−0.960700 + 0.277589i \(0.910465\pi\)
\(492\) 0 0
\(493\) −9.55582 −0.430373
\(494\) 0 0
\(495\) −31.5563 −1.41835
\(496\) 0 0
\(497\) −11.6569 −0.522881
\(498\) 0 0
\(499\) −2.48181 −0.111101 −0.0555505 0.998456i \(-0.517691\pi\)
−0.0555505 + 0.998456i \(0.517691\pi\)
\(500\) 0 0
\(501\) 15.8645 0.708772
\(502\) 0 0
\(503\) 15.7574 0.702586 0.351293 0.936266i \(-0.385742\pi\)
0.351293 + 0.936266i \(0.385742\pi\)
\(504\) 0 0
\(505\) 10.4437 0.464736
\(506\) 0 0
\(507\) −9.50143 −0.421973
\(508\) 0 0
\(509\) −28.2417 −1.25179 −0.625895 0.779908i \(-0.715266\pi\)
−0.625895 + 0.779908i \(0.715266\pi\)
\(510\) 0 0
\(511\) −14.0000 −0.619324
\(512\) 0 0
\(513\) 16.6274 0.734118
\(514\) 0 0
\(515\) −39.3057 −1.73201
\(516\) 0 0
\(517\) −1.53073 −0.0673216
\(518\) 0 0
\(519\) 6.24264 0.274022
\(520\) 0 0
\(521\) 4.72792 0.207134 0.103567 0.994622i \(-0.466974\pi\)
0.103567 + 0.994622i \(0.466974\pi\)
\(522\) 0 0
\(523\) 20.7737 0.908370 0.454185 0.890907i \(-0.349930\pi\)
0.454185 + 0.890907i \(0.349930\pi\)
\(524\) 0 0
\(525\) −3.88123 −0.169391
\(526\) 0 0
\(527\) 11.3137 0.492833
\(528\) 0 0
\(529\) 44.9411 1.95396
\(530\) 0 0
\(531\) 11.8519 0.514330
\(532\) 0 0
\(533\) −0.185709 −0.00804396
\(534\) 0 0
\(535\) −13.0711 −0.565112
\(536\) 0 0
\(537\) −3.27208 −0.141201
\(538\) 0 0
\(539\) 22.3044 0.960720
\(540\) 0 0
\(541\) 29.5098 1.26872 0.634362 0.773036i \(-0.281263\pi\)
0.634362 + 0.773036i \(0.281263\pi\)
\(542\) 0 0
\(543\) −13.4142 −0.575659
\(544\) 0 0
\(545\) 43.6985 1.87184
\(546\) 0 0
\(547\) −18.9803 −0.811540 −0.405770 0.913975i \(-0.632997\pi\)
−0.405770 + 0.913975i \(0.632997\pi\)
\(548\) 0 0
\(549\) −4.46088 −0.190386
\(550\) 0 0
\(551\) −13.5563 −0.577520
\(552\) 0 0
\(553\) −8.48528 −0.360831
\(554\) 0 0
\(555\) −1.71644 −0.0728589
\(556\) 0 0
\(557\) 39.5139 1.67426 0.837129 0.547005i \(-0.184232\pi\)
0.837129 + 0.547005i \(0.184232\pi\)
\(558\) 0 0
\(559\) 3.89949 0.164931
\(560\) 0 0
\(561\) 9.65685 0.407713
\(562\) 0 0
\(563\) 20.5880 0.867680 0.433840 0.900990i \(-0.357158\pi\)
0.433840 + 0.900990i \(0.357158\pi\)
\(564\) 0 0
\(565\) −18.5864 −0.781935
\(566\) 0 0
\(567\) −5.75736 −0.241786
\(568\) 0 0
\(569\) 20.7279 0.868960 0.434480 0.900682i \(-0.356932\pi\)
0.434480 + 0.900682i \(0.356932\pi\)
\(570\) 0 0
\(571\) −7.07401 −0.296038 −0.148019 0.988985i \(-0.547290\pi\)
−0.148019 + 0.988985i \(0.547290\pi\)
\(572\) 0 0
\(573\) 9.18440 0.383684
\(574\) 0 0
\(575\) 29.5563 1.23258
\(576\) 0 0
\(577\) 18.9706 0.789755 0.394877 0.918734i \(-0.370787\pi\)
0.394877 + 0.918734i \(0.370787\pi\)
\(578\) 0 0
\(579\) −1.15932 −0.0481795
\(580\) 0 0
\(581\) 6.94269 0.288032
\(582\) 0 0
\(583\) −5.41421 −0.224234
\(584\) 0 0
\(585\) −5.41421 −0.223850
\(586\) 0 0
\(587\) −13.6453 −0.563201 −0.281601 0.959532i \(-0.590865\pi\)
−0.281601 + 0.959532i \(0.590865\pi\)
\(588\) 0 0
\(589\) 16.0502 0.661336
\(590\) 0 0
\(591\) −9.27208 −0.381402
\(592\) 0 0
\(593\) −28.2843 −1.16150 −0.580748 0.814083i \(-0.697240\pi\)
−0.580748 + 0.814083i \(0.697240\pi\)
\(594\) 0 0
\(595\) −11.7206 −0.480498
\(596\) 0 0
\(597\) 17.2864 0.707486
\(598\) 0 0
\(599\) −37.6985 −1.54032 −0.770159 0.637852i \(-0.779823\pi\)
−0.770159 + 0.637852i \(0.779823\pi\)
\(600\) 0 0
\(601\) 31.0711 1.26742 0.633708 0.773573i \(-0.281532\pi\)
0.633708 + 0.773573i \(0.281532\pi\)
\(602\) 0 0
\(603\) −14.4650 −0.589062
\(604\) 0 0
\(605\) −26.0769 −1.06018
\(606\) 0 0
\(607\) −32.9706 −1.33823 −0.669117 0.743157i \(-0.733327\pi\)
−0.669117 + 0.743157i \(0.733327\pi\)
\(608\) 0 0
\(609\) −3.65685 −0.148183
\(610\) 0 0
\(611\) −0.262632 −0.0106250
\(612\) 0 0
\(613\) 3.45542 0.139563 0.0697815 0.997562i \(-0.477770\pi\)
0.0697815 + 0.997562i \(0.477770\pi\)
\(614\) 0 0
\(615\) −0.544156 −0.0219425
\(616\) 0 0
\(617\) −32.2426 −1.29804 −0.649020 0.760771i \(-0.724821\pi\)
−0.649020 + 0.760771i \(0.724821\pi\)
\(618\) 0 0
\(619\) 23.5725 0.947460 0.473730 0.880670i \(-0.342907\pi\)
0.473730 + 0.880670i \(0.342907\pi\)
\(620\) 0 0
\(621\) −34.1563 −1.37065
\(622\) 0 0
\(623\) 17.3137 0.693659
\(624\) 0 0
\(625\) −30.0711 −1.20284
\(626\) 0 0
\(627\) 13.6997 0.547113
\(628\) 0 0
\(629\) −2.16478 −0.0863156
\(630\) 0 0
\(631\) −45.8995 −1.82723 −0.913615 0.406580i \(-0.866721\pi\)
−0.913615 + 0.406580i \(0.866721\pi\)
\(632\) 0 0
\(633\) −15.0711 −0.599021
\(634\) 0 0
\(635\) 38.0057 1.50821
\(636\) 0 0
\(637\) 3.82683 0.151625
\(638\) 0 0
\(639\) −19.8995 −0.787212
\(640\) 0 0
\(641\) 7.45584 0.294488 0.147244 0.989100i \(-0.452960\pi\)
0.147244 + 0.989100i \(0.452960\pi\)
\(642\) 0 0
\(643\) −12.3772 −0.488109 −0.244054 0.969762i \(-0.578478\pi\)
−0.244054 + 0.969762i \(0.578478\pi\)
\(644\) 0 0
\(645\) 11.4261 0.449903
\(646\) 0 0
\(647\) −8.72792 −0.343130 −0.171565 0.985173i \(-0.554882\pi\)
−0.171565 + 0.985173i \(0.554882\pi\)
\(648\) 0 0
\(649\) 21.8995 0.859630
\(650\) 0 0
\(651\) 4.32957 0.169689
\(652\) 0 0
\(653\) −5.46635 −0.213915 −0.106957 0.994264i \(-0.534111\pi\)
−0.106957 + 0.994264i \(0.534111\pi\)
\(654\) 0 0
\(655\) −51.8995 −2.02788
\(656\) 0 0
\(657\) −23.8995 −0.932408
\(658\) 0 0
\(659\) 26.4483 1.03028 0.515139 0.857106i \(-0.327740\pi\)
0.515139 + 0.857106i \(0.327740\pi\)
\(660\) 0 0
\(661\) −45.1885 −1.75763 −0.878815 0.477163i \(-0.841665\pi\)
−0.878815 + 0.477163i \(0.841665\pi\)
\(662\) 0 0
\(663\) 1.65685 0.0643469
\(664\) 0 0
\(665\) −16.6274 −0.644784
\(666\) 0 0
\(667\) 27.8477 1.07827
\(668\) 0 0
\(669\) −16.0502 −0.620536
\(670\) 0 0
\(671\) −8.24264 −0.318204
\(672\) 0 0
\(673\) −22.4853 −0.866744 −0.433372 0.901215i \(-0.642676\pi\)
−0.433372 + 0.901215i \(0.642676\pi\)
\(674\) 0 0
\(675\) −14.8590 −0.571923
\(676\) 0 0
\(677\) 40.7820 1.56738 0.783690 0.621152i \(-0.213335\pi\)
0.783690 + 0.621152i \(0.213335\pi\)
\(678\) 0 0
\(679\) −26.1421 −1.00324
\(680\) 0 0
\(681\) 15.4142 0.590674
\(682\) 0 0
\(683\) 10.9552 0.419191 0.209595 0.977788i \(-0.432785\pi\)
0.209595 + 0.977788i \(0.432785\pi\)
\(684\) 0 0
\(685\) 35.8728 1.37063
\(686\) 0 0
\(687\) −18.4437 −0.703669
\(688\) 0 0
\(689\) −0.928932 −0.0353895
\(690\) 0 0
\(691\) 32.6800 1.24321 0.621603 0.783332i \(-0.286482\pi\)
0.621603 + 0.783332i \(0.286482\pi\)
\(692\) 0 0
\(693\) −15.2304 −0.578556
\(694\) 0 0
\(695\) −41.8406 −1.58711
\(696\) 0 0
\(697\) −0.686292 −0.0259951
\(698\) 0 0
\(699\) 2.87576 0.108771
\(700\) 0 0
\(701\) 3.11586 0.117684 0.0588422 0.998267i \(-0.481259\pi\)
0.0588422 + 0.998267i \(0.481259\pi\)
\(702\) 0 0
\(703\) −3.07107 −0.115828
\(704\) 0 0
\(705\) −0.769553 −0.0289830
\(706\) 0 0
\(707\) 5.04054 0.189569
\(708\) 0 0
\(709\) −22.9385 −0.861473 −0.430736 0.902478i \(-0.641746\pi\)
−0.430736 + 0.902478i \(0.641746\pi\)
\(710\) 0 0
\(711\) −14.4853 −0.543240
\(712\) 0 0
\(713\) −32.9706 −1.23476
\(714\) 0 0
\(715\) −10.0042 −0.374134
\(716\) 0 0
\(717\) 4.06694 0.151883
\(718\) 0 0
\(719\) 35.6569 1.32978 0.664888 0.746943i \(-0.268479\pi\)
0.664888 + 0.746943i \(0.268479\pi\)
\(720\) 0 0
\(721\) −18.9706 −0.706501
\(722\) 0 0
\(723\) 6.49435 0.241528
\(724\) 0 0
\(725\) 12.1146 0.449923
\(726\) 0 0
\(727\) −14.1005 −0.522959 −0.261479 0.965209i \(-0.584210\pi\)
−0.261479 + 0.965209i \(0.584210\pi\)
\(728\) 0 0
\(729\) −0.313708 −0.0116188
\(730\) 0 0
\(731\) 14.4107 0.532997
\(732\) 0 0
\(733\) 36.0041 1.32984 0.664921 0.746914i \(-0.268465\pi\)
0.664921 + 0.746914i \(0.268465\pi\)
\(734\) 0 0
\(735\) 11.2132 0.413605
\(736\) 0 0
\(737\) −26.7279 −0.984536
\(738\) 0 0
\(739\) 0.502734 0.0184934 0.00924669 0.999957i \(-0.497057\pi\)
0.00924669 + 0.999957i \(0.497057\pi\)
\(740\) 0 0
\(741\) 2.35049 0.0863475
\(742\) 0 0
\(743\) 44.7279 1.64091 0.820454 0.571712i \(-0.193721\pi\)
0.820454 + 0.571712i \(0.193721\pi\)
\(744\) 0 0
\(745\) 49.4975 1.81345
\(746\) 0 0
\(747\) 11.8519 0.433639
\(748\) 0 0
\(749\) −6.30864 −0.230513
\(750\) 0 0
\(751\) 10.9706 0.400322 0.200161 0.979763i \(-0.435854\pi\)
0.200161 + 0.979763i \(0.435854\pi\)
\(752\) 0 0
\(753\) 13.2132 0.481516
\(754\) 0 0
\(755\) 6.15480 0.223996
\(756\) 0 0
\(757\) 36.0041 1.30859 0.654296 0.756239i \(-0.272965\pi\)
0.654296 + 0.756239i \(0.272965\pi\)
\(758\) 0 0
\(759\) −28.1421 −1.02149
\(760\) 0 0
\(761\) −42.1838 −1.52916 −0.764580 0.644529i \(-0.777054\pi\)
−0.764580 + 0.644529i \(0.777054\pi\)
\(762\) 0 0
\(763\) 21.0907 0.763535
\(764\) 0 0
\(765\) −20.0083 −0.723402
\(766\) 0 0
\(767\) 3.75736 0.135670
\(768\) 0 0
\(769\) 5.51472 0.198866 0.0994329 0.995044i \(-0.468297\pi\)
0.0994329 + 0.995044i \(0.468297\pi\)
\(770\) 0 0
\(771\) 4.59220 0.165384
\(772\) 0 0
\(773\) −31.5976 −1.13649 −0.568244 0.822860i \(-0.692377\pi\)
−0.568244 + 0.822860i \(0.692377\pi\)
\(774\) 0 0
\(775\) −14.3431 −0.515221
\(776\) 0 0
\(777\) −0.828427 −0.0297197
\(778\) 0 0
\(779\) −0.973606 −0.0348831
\(780\) 0 0
\(781\) −36.7695 −1.31572
\(782\) 0 0
\(783\) −14.0000 −0.500319
\(784\) 0 0
\(785\) 5.41421 0.193242
\(786\) 0 0
\(787\) 2.48181 0.0884670 0.0442335 0.999021i \(-0.485915\pi\)
0.0442335 + 0.999021i \(0.485915\pi\)
\(788\) 0 0
\(789\) 6.30864 0.224594
\(790\) 0 0
\(791\) −8.97056 −0.318956
\(792\) 0 0
\(793\) −1.41421 −0.0502202
\(794\) 0 0
\(795\) −2.72191 −0.0965363
\(796\) 0 0
\(797\) 28.2417 1.00037 0.500185 0.865918i \(-0.333265\pi\)
0.500185 + 0.865918i \(0.333265\pi\)
\(798\) 0 0
\(799\) −0.970563 −0.0343360
\(800\) 0 0
\(801\) 29.5563 1.04432
\(802\) 0 0
\(803\) −44.1605 −1.55839
\(804\) 0 0
\(805\) 34.1563 1.20385
\(806\) 0 0
\(807\) −18.2426 −0.642171
\(808\) 0 0
\(809\) 41.2132 1.44898 0.724490 0.689286i \(-0.242076\pi\)
0.724490 + 0.689286i \(0.242076\pi\)
\(810\) 0 0
\(811\) 45.7456 1.60635 0.803173 0.595746i \(-0.203143\pi\)
0.803173 + 0.595746i \(0.203143\pi\)
\(812\) 0 0
\(813\) 13.7766 0.483167
\(814\) 0 0
\(815\) −1.47309 −0.0516000
\(816\) 0 0
\(817\) 20.4437 0.715233
\(818\) 0 0
\(819\) −2.61313 −0.0913100
\(820\) 0 0
\(821\) 23.3868 0.816206 0.408103 0.912936i \(-0.366191\pi\)
0.408103 + 0.912936i \(0.366191\pi\)
\(822\) 0 0
\(823\) −50.8701 −1.77322 −0.886609 0.462519i \(-0.846946\pi\)
−0.886609 + 0.462519i \(0.846946\pi\)
\(824\) 0 0
\(825\) −12.2426 −0.426234
\(826\) 0 0
\(827\) 42.1270 1.46490 0.732450 0.680820i \(-0.238377\pi\)
0.732450 + 0.680820i \(0.238377\pi\)
\(828\) 0 0
\(829\) 37.0096 1.28540 0.642698 0.766120i \(-0.277815\pi\)
0.642698 + 0.766120i \(0.277815\pi\)
\(830\) 0 0
\(831\) 1.41421 0.0490585
\(832\) 0 0
\(833\) 14.1421 0.489996
\(834\) 0 0
\(835\) −60.7359 −2.10185
\(836\) 0 0
\(837\) 16.5754 0.572931
\(838\) 0 0
\(839\) −13.6985 −0.472924 −0.236462 0.971641i \(-0.575988\pi\)
−0.236462 + 0.971641i \(0.575988\pi\)
\(840\) 0 0
\(841\) −17.5858 −0.606406
\(842\) 0 0
\(843\) 12.8030 0.440959
\(844\) 0 0
\(845\) 36.3755 1.25136
\(846\) 0 0
\(847\) −12.5858 −0.432453
\(848\) 0 0
\(849\) −11.5563 −0.396613
\(850\) 0 0
\(851\) 6.30864 0.216258
\(852\) 0 0
\(853\) 55.3784 1.89612 0.948060 0.318092i \(-0.103042\pi\)
0.948060 + 0.318092i \(0.103042\pi\)
\(854\) 0 0
\(855\) −28.3848 −0.970739
\(856\) 0 0
\(857\) −13.6985 −0.467931 −0.233966 0.972245i \(-0.575170\pi\)
−0.233966 + 0.972245i \(0.575170\pi\)
\(858\) 0 0
\(859\) 4.38396 0.149579 0.0747894 0.997199i \(-0.476172\pi\)
0.0747894 + 0.997199i \(0.476172\pi\)
\(860\) 0 0
\(861\) −0.262632 −0.00895049
\(862\) 0 0
\(863\) 21.9411 0.746885 0.373442 0.927653i \(-0.378177\pi\)
0.373442 + 0.927653i \(0.378177\pi\)
\(864\) 0 0
\(865\) −23.8995 −0.812607
\(866\) 0 0
\(867\) −6.88830 −0.233939
\(868\) 0 0
\(869\) −26.7653 −0.907951
\(870\) 0 0
\(871\) −4.58579 −0.155383
\(872\) 0 0
\(873\) −44.6274 −1.51041
\(874\) 0 0
\(875\) −5.86030 −0.198114
\(876\) 0 0
\(877\) −1.92468 −0.0649919 −0.0324960 0.999472i \(-0.510346\pi\)
−0.0324960 + 0.999472i \(0.510346\pi\)
\(878\) 0 0
\(879\) 19.2132 0.648045
\(880\) 0 0
\(881\) −22.6274 −0.762337 −0.381169 0.924506i \(-0.624478\pi\)
−0.381169 + 0.924506i \(0.624478\pi\)
\(882\) 0 0
\(883\) 53.7389 1.80846 0.904228 0.427049i \(-0.140447\pi\)
0.904228 + 0.427049i \(0.140447\pi\)
\(884\) 0 0
\(885\) 11.0096 0.370085
\(886\) 0 0
\(887\) −3.27208 −0.109866 −0.0549328 0.998490i \(-0.517494\pi\)
−0.0549328 + 0.998490i \(0.517494\pi\)
\(888\) 0 0
\(889\) 18.3431 0.615209
\(890\) 0 0
\(891\) −18.1606 −0.608402
\(892\) 0 0
\(893\) −1.37689 −0.0460758
\(894\) 0 0
\(895\) 12.5269 0.418728
\(896\) 0 0
\(897\) −4.82843 −0.161216
\(898\) 0 0
\(899\) −13.5140 −0.450716
\(900\) 0 0
\(901\) −3.43289 −0.114366
\(902\) 0 0
\(903\) 5.51472 0.183518
\(904\) 0 0
\(905\) 51.3553 1.70711
\(906\) 0 0
\(907\) −17.0782 −0.567071 −0.283536 0.958962i \(-0.591507\pi\)
−0.283536 + 0.958962i \(0.591507\pi\)
\(908\) 0 0
\(909\) 8.60474 0.285401
\(910\) 0 0
\(911\) 33.5980 1.11315 0.556575 0.830797i \(-0.312115\pi\)
0.556575 + 0.830797i \(0.312115\pi\)
\(912\) 0 0
\(913\) 21.8995 0.724767
\(914\) 0 0
\(915\) −4.14386 −0.136992
\(916\) 0 0
\(917\) −25.0489 −0.827186
\(918\) 0 0
\(919\) −12.0416 −0.397217 −0.198608 0.980079i \(-0.563642\pi\)
−0.198608 + 0.980079i \(0.563642\pi\)
\(920\) 0 0
\(921\) −13.8995 −0.458004
\(922\) 0 0
\(923\) −6.30864 −0.207652
\(924\) 0 0
\(925\) 2.74444 0.0902367
\(926\) 0 0
\(927\) −32.3848 −1.06366
\(928\) 0 0
\(929\) 9.51472 0.312168 0.156084 0.987744i \(-0.450113\pi\)
0.156084 + 0.987744i \(0.450113\pi\)
\(930\) 0 0
\(931\) 20.0627 0.657529
\(932\) 0 0
\(933\) 2.87576 0.0941481
\(934\) 0 0
\(935\) −36.9706 −1.20907
\(936\) 0 0
\(937\) −26.8701 −0.877807 −0.438903 0.898534i \(-0.644633\pi\)
−0.438903 + 0.898534i \(0.644633\pi\)
\(938\) 0 0
\(939\) −8.10201 −0.264399
\(940\) 0 0
\(941\) 1.66205 0.0541813 0.0270906 0.999633i \(-0.491376\pi\)
0.0270906 + 0.999633i \(0.491376\pi\)
\(942\) 0 0
\(943\) 2.00000 0.0651290
\(944\) 0 0
\(945\) −17.1716 −0.558591
\(946\) 0 0
\(947\) 24.3923 0.792643 0.396321 0.918112i \(-0.370287\pi\)
0.396321 + 0.918112i \(0.370287\pi\)
\(948\) 0 0
\(949\) −7.57675 −0.245952
\(950\) 0 0
\(951\) 14.3848 0.466458
\(952\) 0 0
\(953\) 20.7279 0.671443 0.335722 0.941961i \(-0.391020\pi\)
0.335722 + 0.941961i \(0.391020\pi\)
\(954\) 0 0
\(955\) −35.1618 −1.13781
\(956\) 0 0
\(957\) −11.5349 −0.372870
\(958\) 0 0
\(959\) 17.3137 0.559089
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) −10.7695 −0.347043
\(964\) 0 0
\(965\) 4.43835 0.142876
\(966\) 0 0
\(967\) 8.52691 0.274207 0.137104 0.990557i \(-0.456221\pi\)
0.137104 + 0.990557i \(0.456221\pi\)
\(968\) 0 0
\(969\) 8.68629 0.279044
\(970\) 0 0
\(971\) 24.2066 0.776826 0.388413 0.921485i \(-0.373023\pi\)
0.388413 + 0.921485i \(0.373023\pi\)
\(972\) 0 0
\(973\) −20.1940 −0.647391
\(974\) 0 0
\(975\) −2.10051 −0.0672700
\(976\) 0 0
\(977\) 14.1421 0.452447 0.226224 0.974075i \(-0.427362\pi\)
0.226224 + 0.974075i \(0.427362\pi\)
\(978\) 0 0
\(979\) 54.6130 1.74544
\(980\) 0 0
\(981\) 36.0041 1.14952
\(982\) 0 0
\(983\) −27.7574 −0.885322 −0.442661 0.896689i \(-0.645966\pi\)
−0.442661 + 0.896689i \(0.645966\pi\)
\(984\) 0 0
\(985\) 35.4975 1.13104
\(986\) 0 0
\(987\) −0.371418 −0.0118224
\(988\) 0 0
\(989\) −41.9957 −1.33539
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) 0 0
\(993\) −1.07107 −0.0339893
\(994\) 0 0
\(995\) −66.1798 −2.09804
\(996\) 0 0
\(997\) −52.7653 −1.67109 −0.835546 0.549420i \(-0.814849\pi\)
−0.835546 + 0.549420i \(0.814849\pi\)
\(998\) 0 0
\(999\) −3.17157 −0.100344
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 4096.2.a.f.1.3 4
4.3 odd 2 4096.2.a.e.1.2 4
8.3 odd 2 4096.2.a.e.1.3 4
8.5 even 2 inner 4096.2.a.f.1.2 4
64.3 odd 16 256.2.g.b.161.1 4
64.5 even 16 512.2.g.c.449.1 4
64.11 odd 16 32.2.g.a.21.1 4
64.13 even 16 512.2.g.c.65.1 4
64.19 odd 16 512.2.g.d.65.1 4
64.21 even 16 256.2.g.a.97.1 4
64.27 odd 16 512.2.g.d.449.1 4
64.29 even 16 128.2.g.a.81.1 4
64.35 odd 16 32.2.g.a.29.1 yes 4
64.37 even 16 512.2.g.b.449.1 4
64.43 odd 16 256.2.g.b.97.1 4
64.45 even 16 512.2.g.b.65.1 4
64.51 odd 16 512.2.g.a.65.1 4
64.53 even 16 128.2.g.a.49.1 4
64.59 odd 16 512.2.g.a.449.1 4
64.61 even 16 256.2.g.a.161.1 4
192.11 even 16 288.2.v.a.181.1 4
192.29 odd 16 1152.2.v.a.721.1 4
192.35 even 16 288.2.v.a.253.1 4
192.53 odd 16 1152.2.v.a.433.1 4
320.99 odd 16 800.2.y.a.701.1 4
320.139 odd 16 800.2.y.a.501.1 4
320.163 even 16 800.2.ba.b.349.1 4
320.203 even 16 800.2.ba.a.149.1 4
320.227 even 16 800.2.ba.a.349.1 4
320.267 even 16 800.2.ba.b.149.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
32.2.g.a.21.1 4 64.11 odd 16
32.2.g.a.29.1 yes 4 64.35 odd 16
128.2.g.a.49.1 4 64.53 even 16
128.2.g.a.81.1 4 64.29 even 16
256.2.g.a.97.1 4 64.21 even 16
256.2.g.a.161.1 4 64.61 even 16
256.2.g.b.97.1 4 64.43 odd 16
256.2.g.b.161.1 4 64.3 odd 16
288.2.v.a.181.1 4 192.11 even 16
288.2.v.a.253.1 4 192.35 even 16
512.2.g.a.65.1 4 64.51 odd 16
512.2.g.a.449.1 4 64.59 odd 16
512.2.g.b.65.1 4 64.45 even 16
512.2.g.b.449.1 4 64.37 even 16
512.2.g.c.65.1 4 64.13 even 16
512.2.g.c.449.1 4 64.5 even 16
512.2.g.d.65.1 4 64.19 odd 16
512.2.g.d.449.1 4 64.27 odd 16
800.2.y.a.501.1 4 320.139 odd 16
800.2.y.a.701.1 4 320.99 odd 16
800.2.ba.a.149.1 4 320.203 even 16
800.2.ba.a.349.1 4 320.227 even 16
800.2.ba.b.149.1 4 320.267 even 16
800.2.ba.b.349.1 4 320.163 even 16
1152.2.v.a.433.1 4 192.53 odd 16
1152.2.v.a.721.1 4 192.29 odd 16
4096.2.a.e.1.2 4 4.3 odd 2
4096.2.a.e.1.3 4 8.3 odd 2
4096.2.a.f.1.2 4 8.5 even 2 inner
4096.2.a.f.1.3 4 1.1 even 1 trivial