Properties

Label 2200.2.a.o.1.1
Level $2200$
Weight $2$
Character 2200.1
Self dual yes
Analytic conductor $17.567$
Analytic rank $1$
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] = [2200,2,Mod(1,2200)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2200, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2200.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2200.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: \(17.5670884447\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 88)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) of defining polynomial
Character \(\chi\) \(=\) 2200.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.56155 q^{3} +5.12311 q^{7} +3.56155 q^{9} +O(q^{10})\) \(q-2.56155 q^{3} +5.12311 q^{7} +3.56155 q^{9} -1.00000 q^{11} -3.12311 q^{13} -2.00000 q^{17} -4.00000 q^{19} -13.1231 q^{21} -6.56155 q^{23} -1.43845 q^{27} +3.12311 q^{29} -1.43845 q^{31} +2.56155 q^{33} +3.43845 q^{37} +8.00000 q^{39} +7.12311 q^{41} -1.12311 q^{43} -8.00000 q^{47} +19.2462 q^{49} +5.12311 q^{51} +4.24621 q^{53} +10.2462 q^{57} -12.8078 q^{59} -7.12311 q^{61} +18.2462 q^{63} -5.43845 q^{67} +16.8078 q^{69} +3.68466 q^{71} +3.12311 q^{73} -5.12311 q^{77} -2.87689 q^{79} -7.00000 q^{81} -9.12311 q^{83} -8.00000 q^{87} -9.68466 q^{89} -16.0000 q^{91} +3.68466 q^{93} -11.4384 q^{97} -3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 4 q^{17} - 8 q^{19} - 18 q^{21} - 9 q^{23} - 7 q^{27} - 2 q^{29} - 7 q^{31} + q^{33} + 11 q^{37} + 16 q^{39} + 6 q^{41} + 6 q^{43} - 16 q^{47} + 22 q^{49} + 2 q^{51} - 8 q^{53} + 4 q^{57} - 5 q^{59} - 6 q^{61} + 20 q^{63} - 15 q^{67} + 13 q^{69} - 5 q^{71} - 2 q^{73} - 2 q^{77} - 14 q^{79} - 14 q^{81} - 10 q^{83} - 16 q^{87} - 7 q^{89} - 32 q^{91} - 5 q^{93} - 27 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(3\) −2.56155 −1.47891 −0.739457 0.673204i \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 5.12311 1.93635 0.968176 0.250270i \(-0.0805195\pi\)
0.968176 + 0.250270i \(0.0805195\pi\)
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.12311 −0.866194 −0.433097 0.901347i \(-0.642579\pi\)
−0.433097 + 0.901347i \(0.642579\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.00000 −0.485071 −0.242536 0.970143i \(-0.577979\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −13.1231 −2.86370
\(22\) 0 0
\(23\) −6.56155 −1.36818 −0.684089 0.729398i \(-0.739800\pi\)
−0.684089 + 0.729398i \(0.739800\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) 3.12311 0.579946 0.289973 0.957035i \(-0.406354\pi\)
0.289973 + 0.957035i \(0.406354\pi\)
\(30\) 0 0
\(31\) −1.43845 −0.258353 −0.129176 0.991622i \(-0.541233\pi\)
−0.129176 + 0.991622i \(0.541233\pi\)
\(32\) 0 0
\(33\) 2.56155 0.445909
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.43845 0.565277 0.282639 0.959226i \(-0.408790\pi\)
0.282639 + 0.959226i \(0.408790\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) 7.12311 1.11244 0.556221 0.831034i \(-0.312251\pi\)
0.556221 + 0.831034i \(0.312251\pi\)
\(42\) 0 0
\(43\) −1.12311 −0.171272 −0.0856360 0.996326i \(-0.527292\pi\)
−0.0856360 + 0.996326i \(0.527292\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −8.00000 −1.16692 −0.583460 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 19.2462 2.74946
\(50\) 0 0
\(51\) 5.12311 0.717378
\(52\) 0 0
\(53\) 4.24621 0.583262 0.291631 0.956531i \(-0.405802\pi\)
0.291631 + 0.956531i \(0.405802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 10.2462 1.35714
\(58\) 0 0
\(59\) −12.8078 −1.66743 −0.833714 0.552196i \(-0.813790\pi\)
−0.833714 + 0.552196i \(0.813790\pi\)
\(60\) 0 0
\(61\) −7.12311 −0.912020 −0.456010 0.889975i \(-0.650722\pi\)
−0.456010 + 0.889975i \(0.650722\pi\)
\(62\) 0 0
\(63\) 18.2462 2.29881
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −5.43845 −0.664412 −0.332206 0.943207i \(-0.607793\pi\)
−0.332206 + 0.943207i \(0.607793\pi\)
\(68\) 0 0
\(69\) 16.8078 2.02342
\(70\) 0 0
\(71\) 3.68466 0.437289 0.218644 0.975805i \(-0.429837\pi\)
0.218644 + 0.975805i \(0.429837\pi\)
\(72\) 0 0
\(73\) 3.12311 0.365532 0.182766 0.983156i \(-0.441495\pi\)
0.182766 + 0.983156i \(0.441495\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −5.12311 −0.583832
\(78\) 0 0
\(79\) −2.87689 −0.323676 −0.161838 0.986817i \(-0.551742\pi\)
−0.161838 + 0.986817i \(0.551742\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −9.12311 −1.00139 −0.500695 0.865624i \(-0.666922\pi\)
−0.500695 + 0.865624i \(0.666922\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −8.00000 −0.857690
\(88\) 0 0
\(89\) −9.68466 −1.02657 −0.513286 0.858218i \(-0.671572\pi\)
−0.513286 + 0.858218i \(0.671572\pi\)
\(90\) 0 0
\(91\) −16.0000 −1.67726
\(92\) 0 0
\(93\) 3.68466 0.382081
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −11.4384 −1.16140 −0.580699 0.814118i \(-0.697221\pi\)
−0.580699 + 0.814118i \(0.697221\pi\)
\(98\) 0 0
\(99\) −3.56155 −0.357950
\(100\) 0 0
\(101\) −2.00000 −0.199007 −0.0995037 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −11.3693 −1.09911 −0.549557 0.835456i \(-0.685203\pi\)
−0.549557 + 0.835456i \(0.685203\pi\)
\(108\) 0 0
\(109\) −4.24621 −0.406713 −0.203357 0.979105i \(-0.565185\pi\)
−0.203357 + 0.979105i \(0.565185\pi\)
\(110\) 0 0
\(111\) −8.80776 −0.835996
\(112\) 0 0
\(113\) 4.56155 0.429115 0.214557 0.976711i \(-0.431169\pi\)
0.214557 + 0.976711i \(0.431169\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −11.1231 −1.02833
\(118\) 0 0
\(119\) −10.2462 −0.939269
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −18.2462 −1.64521
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 10.2462 0.909204 0.454602 0.890695i \(-0.349781\pi\)
0.454602 + 0.890695i \(0.349781\pi\)
\(128\) 0 0
\(129\) 2.87689 0.253296
\(130\) 0 0
\(131\) −11.3693 −0.993342 −0.496671 0.867939i \(-0.665444\pi\)
−0.496671 + 0.867939i \(0.665444\pi\)
\(132\) 0 0
\(133\) −20.4924 −1.77692
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 12.5616 1.07321 0.536603 0.843835i \(-0.319707\pi\)
0.536603 + 0.843835i \(0.319707\pi\)
\(138\) 0 0
\(139\) 6.87689 0.583291 0.291645 0.956527i \(-0.405797\pi\)
0.291645 + 0.956527i \(0.405797\pi\)
\(140\) 0 0
\(141\) 20.4924 1.72577
\(142\) 0 0
\(143\) 3.12311 0.261167
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −49.3002 −4.06621
\(148\) 0 0
\(149\) −12.2462 −1.00325 −0.501624 0.865086i \(-0.667264\pi\)
−0.501624 + 0.865086i \(0.667264\pi\)
\(150\) 0 0
\(151\) 15.3693 1.25074 0.625369 0.780329i \(-0.284949\pi\)
0.625369 + 0.780329i \(0.284949\pi\)
\(152\) 0 0
\(153\) −7.12311 −0.575869
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 8.56155 0.683286 0.341643 0.939830i \(-0.389017\pi\)
0.341643 + 0.939830i \(0.389017\pi\)
\(158\) 0 0
\(159\) −10.8769 −0.862594
\(160\) 0 0
\(161\) −33.6155 −2.64927
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −8.00000 −0.619059 −0.309529 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −3.24621 −0.249709
\(170\) 0 0
\(171\) −14.2462 −1.08944
\(172\) 0 0
\(173\) 4.24621 0.322833 0.161417 0.986886i \(-0.448394\pi\)
0.161417 + 0.986886i \(0.448394\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 32.8078 2.46598
\(178\) 0 0
\(179\) −10.5616 −0.789408 −0.394704 0.918808i \(-0.629153\pi\)
−0.394704 + 0.918808i \(0.629153\pi\)
\(180\) 0 0
\(181\) −13.6847 −1.01717 −0.508586 0.861011i \(-0.669832\pi\)
−0.508586 + 0.861011i \(0.669832\pi\)
\(182\) 0 0
\(183\) 18.2462 1.34880
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.00000 0.146254
\(188\) 0 0
\(189\) −7.36932 −0.536039
\(190\) 0 0
\(191\) −14.5616 −1.05364 −0.526818 0.849978i \(-0.676615\pi\)
−0.526818 + 0.849978i \(0.676615\pi\)
\(192\) 0 0
\(193\) 0.876894 0.0631202 0.0315601 0.999502i \(-0.489952\pi\)
0.0315601 + 0.999502i \(0.489952\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −18.4924 −1.31753 −0.658765 0.752349i \(-0.728921\pi\)
−0.658765 + 0.752349i \(0.728921\pi\)
\(198\) 0 0
\(199\) 20.4924 1.45267 0.726335 0.687341i \(-0.241222\pi\)
0.726335 + 0.687341i \(0.241222\pi\)
\(200\) 0 0
\(201\) 13.9309 0.982608
\(202\) 0 0
\(203\) 16.0000 1.12298
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −23.3693 −1.62428
\(208\) 0 0
\(209\) 4.00000 0.276686
\(210\) 0 0
\(211\) −24.4924 −1.68613 −0.843064 0.537813i \(-0.819251\pi\)
−0.843064 + 0.537813i \(0.819251\pi\)
\(212\) 0 0
\(213\) −9.43845 −0.646712
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −7.36932 −0.500262
\(218\) 0 0
\(219\) −8.00000 −0.540590
\(220\) 0 0
\(221\) 6.24621 0.420166
\(222\) 0 0
\(223\) −8.80776 −0.589812 −0.294906 0.955526i \(-0.595288\pi\)
−0.294906 + 0.955526i \(0.595288\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −14.8769 −0.987414 −0.493707 0.869628i \(-0.664358\pi\)
−0.493707 + 0.869628i \(0.664358\pi\)
\(228\) 0 0
\(229\) 2.31534 0.153002 0.0765010 0.997070i \(-0.475625\pi\)
0.0765010 + 0.997070i \(0.475625\pi\)
\(230\) 0 0
\(231\) 13.1231 0.863437
\(232\) 0 0
\(233\) −17.3693 −1.13790 −0.568951 0.822371i \(-0.692651\pi\)
−0.568951 + 0.822371i \(0.692651\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 7.36932 0.478689
\(238\) 0 0
\(239\) −13.1231 −0.848863 −0.424432 0.905460i \(-0.639526\pi\)
−0.424432 + 0.905460i \(0.639526\pi\)
\(240\) 0 0
\(241\) 20.8769 1.34480 0.672399 0.740188i \(-0.265264\pi\)
0.672399 + 0.740188i \(0.265264\pi\)
\(242\) 0 0
\(243\) 22.2462 1.42710
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 12.4924 0.794874
\(248\) 0 0
\(249\) 23.3693 1.48097
\(250\) 0 0
\(251\) −2.56155 −0.161684 −0.0808419 0.996727i \(-0.525761\pi\)
−0.0808419 + 0.996727i \(0.525761\pi\)
\(252\) 0 0
\(253\) 6.56155 0.412521
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −28.2462 −1.76195 −0.880975 0.473163i \(-0.843112\pi\)
−0.880975 + 0.473163i \(0.843112\pi\)
\(258\) 0 0
\(259\) 17.6155 1.09458
\(260\) 0 0
\(261\) 11.1231 0.688503
\(262\) 0 0
\(263\) −10.8769 −0.670698 −0.335349 0.942094i \(-0.608854\pi\)
−0.335349 + 0.942094i \(0.608854\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 24.8078 1.51821
\(268\) 0 0
\(269\) 28.7386 1.75223 0.876113 0.482106i \(-0.160128\pi\)
0.876113 + 0.482106i \(0.160128\pi\)
\(270\) 0 0
\(271\) 4.49242 0.272895 0.136448 0.990647i \(-0.456431\pi\)
0.136448 + 0.990647i \(0.456431\pi\)
\(272\) 0 0
\(273\) 40.9848 2.48052
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 18.0000 1.08152 0.540758 0.841178i \(-0.318138\pi\)
0.540758 + 0.841178i \(0.318138\pi\)
\(278\) 0 0
\(279\) −5.12311 −0.306712
\(280\) 0 0
\(281\) −0.246211 −0.0146877 −0.00734387 0.999973i \(-0.502338\pi\)
−0.00734387 + 0.999973i \(0.502338\pi\)
\(282\) 0 0
\(283\) −20.0000 −1.18888 −0.594438 0.804141i \(-0.702626\pi\)
−0.594438 + 0.804141i \(0.702626\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 36.4924 2.15408
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 29.3002 1.71761
\(292\) 0 0
\(293\) −21.3693 −1.24841 −0.624204 0.781261i \(-0.714577\pi\)
−0.624204 + 0.781261i \(0.714577\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.43845 0.0834672
\(298\) 0 0
\(299\) 20.4924 1.18511
\(300\) 0 0
\(301\) −5.75379 −0.331643
\(302\) 0 0
\(303\) 5.12311 0.294315
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −0.492423 −0.0281040 −0.0140520 0.999901i \(-0.504473\pi\)
−0.0140520 + 0.999901i \(0.504473\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 26.2462 1.48829 0.744143 0.668020i \(-0.232858\pi\)
0.744143 + 0.668020i \(0.232858\pi\)
\(312\) 0 0
\(313\) −10.8078 −0.610891 −0.305445 0.952210i \(-0.598805\pi\)
−0.305445 + 0.952210i \(0.598805\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 34.8078 1.95500 0.977499 0.210938i \(-0.0676519\pi\)
0.977499 + 0.210938i \(0.0676519\pi\)
\(318\) 0 0
\(319\) −3.12311 −0.174860
\(320\) 0 0
\(321\) 29.1231 1.62549
\(322\) 0 0
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 10.8769 0.601494
\(328\) 0 0
\(329\) −40.9848 −2.25957
\(330\) 0 0
\(331\) 6.06913 0.333590 0.166795 0.985992i \(-0.446658\pi\)
0.166795 + 0.985992i \(0.446658\pi\)
\(332\) 0 0
\(333\) 12.2462 0.671088
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −32.7386 −1.78339 −0.891694 0.452640i \(-0.850482\pi\)
−0.891694 + 0.452640i \(0.850482\pi\)
\(338\) 0 0
\(339\) −11.6847 −0.634624
\(340\) 0 0
\(341\) 1.43845 0.0778963
\(342\) 0 0
\(343\) 62.7386 3.38757
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 26.7386 1.43541 0.717703 0.696350i \(-0.245194\pi\)
0.717703 + 0.696350i \(0.245194\pi\)
\(348\) 0 0
\(349\) −15.7538 −0.843281 −0.421640 0.906763i \(-0.638546\pi\)
−0.421640 + 0.906763i \(0.638546\pi\)
\(350\) 0 0
\(351\) 4.49242 0.239788
\(352\) 0 0
\(353\) −13.0540 −0.694793 −0.347397 0.937718i \(-0.612934\pi\)
−0.347397 + 0.937718i \(0.612934\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 26.2462 1.38910
\(358\) 0 0
\(359\) −28.4924 −1.50377 −0.751886 0.659293i \(-0.770856\pi\)
−0.751886 + 0.659293i \(0.770856\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −2.56155 −0.134447
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 5.93087 0.309589 0.154794 0.987947i \(-0.450528\pi\)
0.154794 + 0.987947i \(0.450528\pi\)
\(368\) 0 0
\(369\) 25.3693 1.32067
\(370\) 0 0
\(371\) 21.7538 1.12940
\(372\) 0 0
\(373\) −8.24621 −0.426973 −0.213486 0.976946i \(-0.568482\pi\)
−0.213486 + 0.976946i \(0.568482\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −9.75379 −0.502346
\(378\) 0 0
\(379\) 20.8078 1.06882 0.534412 0.845224i \(-0.320533\pi\)
0.534412 + 0.845224i \(0.320533\pi\)
\(380\) 0 0
\(381\) −26.2462 −1.34463
\(382\) 0 0
\(383\) −35.0540 −1.79117 −0.895587 0.444886i \(-0.853244\pi\)
−0.895587 + 0.444886i \(0.853244\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −0.561553 −0.0284719 −0.0142359 0.999899i \(-0.504532\pi\)
−0.0142359 + 0.999899i \(0.504532\pi\)
\(390\) 0 0
\(391\) 13.1231 0.663664
\(392\) 0 0
\(393\) 29.1231 1.46907
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 22.4924 1.12886 0.564431 0.825480i \(-0.309096\pi\)
0.564431 + 0.825480i \(0.309096\pi\)
\(398\) 0 0
\(399\) 52.4924 2.62791
\(400\) 0 0
\(401\) −2.49242 −0.124466 −0.0622328 0.998062i \(-0.519822\pi\)
−0.0622328 + 0.998062i \(0.519822\pi\)
\(402\) 0 0
\(403\) 4.49242 0.223784
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −3.43845 −0.170437
\(408\) 0 0
\(409\) −10.4924 −0.518817 −0.259408 0.965768i \(-0.583528\pi\)
−0.259408 + 0.965768i \(0.583528\pi\)
\(410\) 0 0
\(411\) −32.1771 −1.58718
\(412\) 0 0
\(413\) −65.6155 −3.22873
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −17.6155 −0.862636
\(418\) 0 0
\(419\) 0.492423 0.0240564 0.0120282 0.999928i \(-0.496171\pi\)
0.0120282 + 0.999928i \(0.496171\pi\)
\(420\) 0 0
\(421\) −30.4924 −1.48611 −0.743055 0.669230i \(-0.766624\pi\)
−0.743055 + 0.669230i \(0.766624\pi\)
\(422\) 0 0
\(423\) −28.4924 −1.38535
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −36.4924 −1.76599
\(428\) 0 0
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 18.8769 0.909268 0.454634 0.890678i \(-0.349770\pi\)
0.454634 + 0.890678i \(0.349770\pi\)
\(432\) 0 0
\(433\) 10.3153 0.495724 0.247862 0.968795i \(-0.420272\pi\)
0.247862 + 0.968795i \(0.420272\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 26.2462 1.25553
\(438\) 0 0
\(439\) −28.4924 −1.35987 −0.679935 0.733273i \(-0.737992\pi\)
−0.679935 + 0.733273i \(0.737992\pi\)
\(440\) 0 0
\(441\) 68.5464 3.26411
\(442\) 0 0
\(443\) −23.6847 −1.12529 −0.562646 0.826698i \(-0.690217\pi\)
−0.562646 + 0.826698i \(0.690217\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 31.3693 1.48372
\(448\) 0 0
\(449\) 33.5464 1.58315 0.791576 0.611070i \(-0.209261\pi\)
0.791576 + 0.611070i \(0.209261\pi\)
\(450\) 0 0
\(451\) −7.12311 −0.335414
\(452\) 0 0
\(453\) −39.3693 −1.84973
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 33.8617 1.58399 0.791993 0.610531i \(-0.209044\pi\)
0.791993 + 0.610531i \(0.209044\pi\)
\(458\) 0 0
\(459\) 2.87689 0.134282
\(460\) 0 0
\(461\) −7.12311 −0.331756 −0.165878 0.986146i \(-0.553046\pi\)
−0.165878 + 0.986146i \(0.553046\pi\)
\(462\) 0 0
\(463\) −27.6847 −1.28662 −0.643308 0.765608i \(-0.722438\pi\)
−0.643308 + 0.765608i \(0.722438\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 25.3002 1.17075 0.585377 0.810762i \(-0.300947\pi\)
0.585377 + 0.810762i \(0.300947\pi\)
\(468\) 0 0
\(469\) −27.8617 −1.28654
\(470\) 0 0
\(471\) −21.9309 −1.01052
\(472\) 0 0
\(473\) 1.12311 0.0516405
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 15.1231 0.692439
\(478\) 0 0
\(479\) −16.0000 −0.731059 −0.365529 0.930800i \(-0.619112\pi\)
−0.365529 + 0.930800i \(0.619112\pi\)
\(480\) 0 0
\(481\) −10.7386 −0.489640
\(482\) 0 0
\(483\) 86.1080 3.91805
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 13.9309 0.631268 0.315634 0.948881i \(-0.397783\pi\)
0.315634 + 0.948881i \(0.397783\pi\)
\(488\) 0 0
\(489\) −10.2462 −0.463350
\(490\) 0 0
\(491\) 30.2462 1.36499 0.682496 0.730889i \(-0.260894\pi\)
0.682496 + 0.730889i \(0.260894\pi\)
\(492\) 0 0
\(493\) −6.24621 −0.281315
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 18.8769 0.846744
\(498\) 0 0
\(499\) 36.9848 1.65567 0.827835 0.560972i \(-0.189573\pi\)
0.827835 + 0.560972i \(0.189573\pi\)
\(500\) 0 0
\(501\) 20.4924 0.915534
\(502\) 0 0
\(503\) −9.61553 −0.428735 −0.214368 0.976753i \(-0.568769\pi\)
−0.214368 + 0.976753i \(0.568769\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 8.31534 0.369297
\(508\) 0 0
\(509\) 35.3002 1.56465 0.782327 0.622868i \(-0.214033\pi\)
0.782327 + 0.622868i \(0.214033\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 0 0
\(513\) 5.75379 0.254036
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 8.00000 0.351840
\(518\) 0 0
\(519\) −10.8769 −0.477443
\(520\) 0 0
\(521\) 13.6847 0.599536 0.299768 0.954012i \(-0.403091\pi\)
0.299768 + 0.954012i \(0.403091\pi\)
\(522\) 0 0
\(523\) 12.0000 0.524723 0.262362 0.964970i \(-0.415499\pi\)
0.262362 + 0.964970i \(0.415499\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 2.87689 0.125319
\(528\) 0 0
\(529\) 20.0540 0.871912
\(530\) 0 0
\(531\) −45.6155 −1.97955
\(532\) 0 0
\(533\) −22.2462 −0.963590
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 27.0540 1.16747
\(538\) 0 0
\(539\) −19.2462 −0.828993
\(540\) 0 0
\(541\) 33.8617 1.45583 0.727915 0.685667i \(-0.240490\pi\)
0.727915 + 0.685667i \(0.240490\pi\)
\(542\) 0 0
\(543\) 35.0540 1.50431
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 25.7538 1.10115 0.550576 0.834785i \(-0.314408\pi\)
0.550576 + 0.834785i \(0.314408\pi\)
\(548\) 0 0
\(549\) −25.3693 −1.08274
\(550\) 0 0
\(551\) −12.4924 −0.532195
\(552\) 0 0
\(553\) −14.7386 −0.626750
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 20.2462 0.857860 0.428930 0.903338i \(-0.358891\pi\)
0.428930 + 0.903338i \(0.358891\pi\)
\(558\) 0 0
\(559\) 3.50758 0.148355
\(560\) 0 0
\(561\) −5.12311 −0.216298
\(562\) 0 0
\(563\) 8.49242 0.357913 0.178956 0.983857i \(-0.442728\pi\)
0.178956 + 0.983857i \(0.442728\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −35.8617 −1.50605
\(568\) 0 0
\(569\) −35.1231 −1.47244 −0.736219 0.676744i \(-0.763390\pi\)
−0.736219 + 0.676744i \(0.763390\pi\)
\(570\) 0 0
\(571\) −16.4924 −0.690186 −0.345093 0.938568i \(-0.612153\pi\)
−0.345093 + 0.938568i \(0.612153\pi\)
\(572\) 0 0
\(573\) 37.3002 1.55824
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −11.4384 −0.476189 −0.238094 0.971242i \(-0.576523\pi\)
−0.238094 + 0.971242i \(0.576523\pi\)
\(578\) 0 0
\(579\) −2.24621 −0.0933494
\(580\) 0 0
\(581\) −46.7386 −1.93905
\(582\) 0 0
\(583\) −4.24621 −0.175860
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.49242 −0.350520 −0.175260 0.984522i \(-0.556077\pi\)
−0.175260 + 0.984522i \(0.556077\pi\)
\(588\) 0 0
\(589\) 5.75379 0.237081
\(590\) 0 0
\(591\) 47.3693 1.94851
\(592\) 0 0
\(593\) 21.3693 0.877533 0.438766 0.898601i \(-0.355416\pi\)
0.438766 + 0.898601i \(0.355416\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −52.4924 −2.14837
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) 20.2462 0.825860 0.412930 0.910763i \(-0.364505\pi\)
0.412930 + 0.910763i \(0.364505\pi\)
\(602\) 0 0
\(603\) −19.3693 −0.788780
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −11.8617 −0.481453 −0.240727 0.970593i \(-0.577386\pi\)
−0.240727 + 0.970593i \(0.577386\pi\)
\(608\) 0 0
\(609\) −40.9848 −1.66079
\(610\) 0 0
\(611\) 24.9848 1.01078
\(612\) 0 0
\(613\) 45.8617 1.85234 0.926169 0.377108i \(-0.123082\pi\)
0.926169 + 0.377108i \(0.123082\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −30.4924 −1.22758 −0.613789 0.789470i \(-0.710356\pi\)
−0.613789 + 0.789470i \(0.710356\pi\)
\(618\) 0 0
\(619\) −9.93087 −0.399155 −0.199578 0.979882i \(-0.563957\pi\)
−0.199578 + 0.979882i \(0.563957\pi\)
\(620\) 0 0
\(621\) 9.43845 0.378752
\(622\) 0 0
\(623\) −49.6155 −1.98780
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −10.2462 −0.409194
\(628\) 0 0
\(629\) −6.87689 −0.274200
\(630\) 0 0
\(631\) −4.94602 −0.196898 −0.0984491 0.995142i \(-0.531388\pi\)
−0.0984491 + 0.995142i \(0.531388\pi\)
\(632\) 0 0
\(633\) 62.7386 2.49364
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −60.1080 −2.38156
\(638\) 0 0
\(639\) 13.1231 0.519142
\(640\) 0 0
\(641\) 7.30019 0.288340 0.144170 0.989553i \(-0.453949\pi\)
0.144170 + 0.989553i \(0.453949\pi\)
\(642\) 0 0
\(643\) −36.1771 −1.42668 −0.713342 0.700816i \(-0.752820\pi\)
−0.713342 + 0.700816i \(0.752820\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −9.43845 −0.371064 −0.185532 0.982638i \(-0.559401\pi\)
−0.185532 + 0.982638i \(0.559401\pi\)
\(648\) 0 0
\(649\) 12.8078 0.502749
\(650\) 0 0
\(651\) 18.8769 0.739844
\(652\) 0 0
\(653\) 10.1771 0.398260 0.199130 0.979973i \(-0.436188\pi\)
0.199130 + 0.979973i \(0.436188\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 11.1231 0.433954
\(658\) 0 0
\(659\) 29.6155 1.15366 0.576829 0.816865i \(-0.304290\pi\)
0.576829 + 0.816865i \(0.304290\pi\)
\(660\) 0 0
\(661\) 21.1922 0.824282 0.412141 0.911120i \(-0.364781\pi\)
0.412141 + 0.911120i \(0.364781\pi\)
\(662\) 0 0
\(663\) −16.0000 −0.621389
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −20.4924 −0.793470
\(668\) 0 0
\(669\) 22.5616 0.872280
\(670\) 0 0
\(671\) 7.12311 0.274984
\(672\) 0 0
\(673\) 49.2311 1.89772 0.948859 0.315701i \(-0.102239\pi\)
0.948859 + 0.315701i \(0.102239\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −28.7386 −1.10452 −0.552258 0.833673i \(-0.686234\pi\)
−0.552258 + 0.833673i \(0.686234\pi\)
\(678\) 0 0
\(679\) −58.6004 −2.24888
\(680\) 0 0
\(681\) 38.1080 1.46030
\(682\) 0 0
\(683\) 42.7386 1.63535 0.817674 0.575681i \(-0.195263\pi\)
0.817674 + 0.575681i \(0.195263\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −5.93087 −0.226277
\(688\) 0 0
\(689\) −13.2614 −0.505218
\(690\) 0 0
\(691\) −47.0540 −1.79002 −0.895009 0.446049i \(-0.852831\pi\)
−0.895009 + 0.446049i \(0.852831\pi\)
\(692\) 0 0
\(693\) −18.2462 −0.693116
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −14.2462 −0.539614
\(698\) 0 0
\(699\) 44.4924 1.68286
\(700\) 0 0
\(701\) 17.5076 0.661252 0.330626 0.943762i \(-0.392740\pi\)
0.330626 + 0.943762i \(0.392740\pi\)
\(702\) 0 0
\(703\) −13.7538 −0.518734
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −10.2462 −0.385348
\(708\) 0 0
\(709\) 22.8078 0.856564 0.428282 0.903645i \(-0.359119\pi\)
0.428282 + 0.903645i \(0.359119\pi\)
\(710\) 0 0
\(711\) −10.2462 −0.384263
\(712\) 0 0
\(713\) 9.43845 0.353473
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 33.6155 1.25539
\(718\) 0 0
\(719\) −26.4233 −0.985423 −0.492711 0.870193i \(-0.663994\pi\)
−0.492711 + 0.870193i \(0.663994\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −53.4773 −1.98884
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 35.6847 1.32347 0.661735 0.749737i \(-0.269820\pi\)
0.661735 + 0.749737i \(0.269820\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) 2.24621 0.0830791
\(732\) 0 0
\(733\) 7.12311 0.263098 0.131549 0.991310i \(-0.458005\pi\)
0.131549 + 0.991310i \(0.458005\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5.43845 0.200328
\(738\) 0 0
\(739\) −27.3693 −1.00680 −0.503398 0.864054i \(-0.667917\pi\)
−0.503398 + 0.864054i \(0.667917\pi\)
\(740\) 0 0
\(741\) −32.0000 −1.17555
\(742\) 0 0
\(743\) 38.7386 1.42118 0.710591 0.703605i \(-0.248428\pi\)
0.710591 + 0.703605i \(0.248428\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −32.4924 −1.18884
\(748\) 0 0
\(749\) −58.2462 −2.12827
\(750\) 0 0
\(751\) 1.43845 0.0524897 0.0262448 0.999656i \(-0.491645\pi\)
0.0262448 + 0.999656i \(0.491645\pi\)
\(752\) 0 0
\(753\) 6.56155 0.239116
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −32.2462 −1.17201 −0.586004 0.810308i \(-0.699300\pi\)
−0.586004 + 0.810308i \(0.699300\pi\)
\(758\) 0 0
\(759\) −16.8078 −0.610083
\(760\) 0 0
\(761\) −3.12311 −0.113212 −0.0566062 0.998397i \(-0.518028\pi\)
−0.0566062 + 0.998397i \(0.518028\pi\)
\(762\) 0 0
\(763\) −21.7538 −0.787540
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 40.0000 1.44432
\(768\) 0 0
\(769\) −15.6155 −0.563110 −0.281555 0.959545i \(-0.590850\pi\)
−0.281555 + 0.959545i \(0.590850\pi\)
\(770\) 0 0
\(771\) 72.3542 2.60577
\(772\) 0 0
\(773\) 8.73863 0.314307 0.157153 0.987574i \(-0.449768\pi\)
0.157153 + 0.987574i \(0.449768\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −45.1231 −1.61878
\(778\) 0 0
\(779\) −28.4924 −1.02085
\(780\) 0 0
\(781\) −3.68466 −0.131847
\(782\) 0 0
\(783\) −4.49242 −0.160546
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 46.2462 1.64850 0.824250 0.566226i \(-0.191597\pi\)
0.824250 + 0.566226i \(0.191597\pi\)
\(788\) 0 0
\(789\) 27.8617 0.991904
\(790\) 0 0
\(791\) 23.3693 0.830917
\(792\) 0 0
\(793\) 22.2462 0.789986
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 34.8078 1.23295 0.616477 0.787373i \(-0.288559\pi\)
0.616477 + 0.787373i \(0.288559\pi\)
\(798\) 0 0
\(799\) 16.0000 0.566039
\(800\) 0 0
\(801\) −34.4924 −1.21873
\(802\) 0 0
\(803\) −3.12311 −0.110212
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −73.6155 −2.59139
\(808\) 0 0
\(809\) −4.38447 −0.154150 −0.0770749 0.997025i \(-0.524558\pi\)
−0.0770749 + 0.997025i \(0.524558\pi\)
\(810\) 0 0
\(811\) 1.12311 0.0394376 0.0197188 0.999806i \(-0.493723\pi\)
0.0197188 + 0.999806i \(0.493723\pi\)
\(812\) 0 0
\(813\) −11.5076 −0.403588
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4.49242 0.157170
\(818\) 0 0
\(819\) −56.9848 −1.99121
\(820\) 0 0
\(821\) 22.9848 0.802177 0.401088 0.916039i \(-0.368632\pi\)
0.401088 + 0.916039i \(0.368632\pi\)
\(822\) 0 0
\(823\) −15.5464 −0.541913 −0.270957 0.962592i \(-0.587340\pi\)
−0.270957 + 0.962592i \(0.587340\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 10.7386 0.373419 0.186709 0.982415i \(-0.440218\pi\)
0.186709 + 0.982415i \(0.440218\pi\)
\(828\) 0 0
\(829\) 43.9309 1.52578 0.762891 0.646527i \(-0.223779\pi\)
0.762891 + 0.646527i \(0.223779\pi\)
\(830\) 0 0
\(831\) −46.1080 −1.59947
\(832\) 0 0
\(833\) −38.4924 −1.33368
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 2.06913 0.0715196
\(838\) 0 0
\(839\) 0.807764 0.0278871 0.0139436 0.999903i \(-0.495561\pi\)
0.0139436 + 0.999903i \(0.495561\pi\)
\(840\) 0 0
\(841\) −19.2462 −0.663662
\(842\) 0 0
\(843\) 0.630683 0.0217219
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 5.12311 0.176032
\(848\) 0 0
\(849\) 51.2311 1.75825
\(850\) 0 0
\(851\) −22.5616 −0.773400
\(852\) 0 0
\(853\) 13.5076 0.462491 0.231245 0.972895i \(-0.425720\pi\)
0.231245 + 0.972895i \(0.425720\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 44.1080 1.50670 0.753349 0.657620i \(-0.228437\pi\)
0.753349 + 0.657620i \(0.228437\pi\)
\(858\) 0 0
\(859\) −7.05398 −0.240679 −0.120339 0.992733i \(-0.538398\pi\)
−0.120339 + 0.992733i \(0.538398\pi\)
\(860\) 0 0
\(861\) −93.4773 −3.18570
\(862\) 0 0
\(863\) 3.50758 0.119399 0.0596997 0.998216i \(-0.480986\pi\)
0.0596997 + 0.998216i \(0.480986\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 33.3002 1.13093
\(868\) 0 0
\(869\) 2.87689 0.0975920
\(870\) 0 0
\(871\) 16.9848 0.575510
\(872\) 0 0
\(873\) −40.7386 −1.37879
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −30.6307 −1.03432 −0.517162 0.855887i \(-0.673012\pi\)
−0.517162 + 0.855887i \(0.673012\pi\)
\(878\) 0 0
\(879\) 54.7386 1.84629
\(880\) 0 0
\(881\) −19.3002 −0.650240 −0.325120 0.945673i \(-0.605405\pi\)
−0.325120 + 0.945673i \(0.605405\pi\)
\(882\) 0 0
\(883\) 24.4924 0.824236 0.412118 0.911131i \(-0.364789\pi\)
0.412118 + 0.911131i \(0.364789\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.61553 0.322858 0.161429 0.986884i \(-0.448390\pi\)
0.161429 + 0.986884i \(0.448390\pi\)
\(888\) 0 0
\(889\) 52.4924 1.76054
\(890\) 0 0
\(891\) 7.00000 0.234509
\(892\) 0 0
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −52.4924 −1.75267
\(898\) 0 0
\(899\) −4.49242 −0.149831
\(900\) 0 0
\(901\) −8.49242 −0.282924
\(902\) 0 0
\(903\) 14.7386 0.490471
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 16.4924 0.547622 0.273811 0.961784i \(-0.411716\pi\)
0.273811 + 0.961784i \(0.411716\pi\)
\(908\) 0 0
\(909\) −7.12311 −0.236259
\(910\) 0 0
\(911\) 22.7386 0.753365 0.376682 0.926343i \(-0.377065\pi\)
0.376682 + 0.926343i \(0.377065\pi\)
\(912\) 0 0
\(913\) 9.12311 0.301931
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −58.2462 −1.92346
\(918\) 0 0
\(919\) 31.3693 1.03478 0.517389 0.855750i \(-0.326904\pi\)
0.517389 + 0.855750i \(0.326904\pi\)
\(920\) 0 0
\(921\) 1.26137 0.0415634
\(922\) 0 0
\(923\) −11.5076 −0.378777
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 13.5076 0.443169 0.221585 0.975141i \(-0.428877\pi\)
0.221585 + 0.975141i \(0.428877\pi\)
\(930\) 0 0
\(931\) −76.9848 −2.52308
\(932\) 0 0
\(933\) −67.2311 −2.20105
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −32.1080 −1.04892 −0.524461 0.851435i \(-0.675733\pi\)
−0.524461 + 0.851435i \(0.675733\pi\)
\(938\) 0 0
\(939\) 27.6847 0.903455
\(940\) 0 0
\(941\) −15.7538 −0.513559 −0.256779 0.966470i \(-0.582661\pi\)
−0.256779 + 0.966470i \(0.582661\pi\)
\(942\) 0 0
\(943\) −46.7386 −1.52202
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0.315342 0.0102472 0.00512361 0.999987i \(-0.498369\pi\)
0.00512361 + 0.999987i \(0.498369\pi\)
\(948\) 0 0
\(949\) −9.75379 −0.316621
\(950\) 0 0
\(951\) −89.1619 −2.89127
\(952\) 0 0
\(953\) 16.2462 0.526266 0.263133 0.964760i \(-0.415244\pi\)
0.263133 + 0.964760i \(0.415244\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 8.00000 0.258603
\(958\) 0 0
\(959\) 64.3542 2.07810
\(960\) 0 0
\(961\) −28.9309 −0.933254
\(962\) 0 0
\(963\) −40.4924 −1.30485
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) 0 0
\(969\) −20.4924 −0.658311
\(970\) 0 0
\(971\) 35.5464 1.14074 0.570369 0.821389i \(-0.306800\pi\)
0.570369 + 0.821389i \(0.306800\pi\)
\(972\) 0 0
\(973\) 35.2311 1.12946
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −33.1922 −1.06191 −0.530957 0.847399i \(-0.678167\pi\)
−0.530957 + 0.847399i \(0.678167\pi\)
\(978\) 0 0
\(979\) 9.68466 0.309523
\(980\) 0 0
\(981\) −15.1231 −0.482844
\(982\) 0 0
\(983\) −2.06913 −0.0659950 −0.0329975 0.999455i \(-0.510505\pi\)
−0.0329975 + 0.999455i \(0.510505\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 104.985 3.34170
\(988\) 0 0
\(989\) 7.36932 0.234331
\(990\) 0 0
\(991\) 28.4924 0.905092 0.452546 0.891741i \(-0.350516\pi\)
0.452546 + 0.891741i \(0.350516\pi\)
\(992\) 0 0
\(993\) −15.5464 −0.493350
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −35.7538 −1.13233 −0.566167 0.824291i \(-0.691574\pi\)
−0.566167 + 0.824291i \(0.691574\pi\)
\(998\) 0 0
\(999\) −4.94602 −0.156485
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 2200.2.a.o.1.1 2
4.3 odd 2 4400.2.a.bp.1.2 2
5.2 odd 4 2200.2.b.g.1849.4 4
5.3 odd 4 2200.2.b.g.1849.1 4
5.4 even 2 88.2.a.b.1.2 2
15.14 odd 2 792.2.a.h.1.2 2
20.3 even 4 4400.2.b.v.4049.4 4
20.7 even 4 4400.2.b.v.4049.1 4
20.19 odd 2 176.2.a.d.1.1 2
35.34 odd 2 4312.2.a.n.1.1 2
40.19 odd 2 704.2.a.p.1.2 2
40.29 even 2 704.2.a.m.1.1 2
55.4 even 10 968.2.i.r.753.1 8
55.9 even 10 968.2.i.r.81.2 8
55.14 even 10 968.2.i.r.9.1 8
55.19 odd 10 968.2.i.q.9.1 8
55.24 odd 10 968.2.i.q.81.2 8
55.29 odd 10 968.2.i.q.753.1 8
55.39 odd 10 968.2.i.q.729.2 8
55.49 even 10 968.2.i.r.729.2 8
55.54 odd 2 968.2.a.j.1.2 2
60.59 even 2 1584.2.a.t.1.2 2
80.19 odd 4 2816.2.c.p.1409.1 4
80.29 even 4 2816.2.c.w.1409.4 4
80.59 odd 4 2816.2.c.p.1409.4 4
80.69 even 4 2816.2.c.w.1409.1 4
120.29 odd 2 6336.2.a.cu.1.1 2
120.59 even 2 6336.2.a.cx.1.1 2
140.139 even 2 8624.2.a.cb.1.2 2
165.164 even 2 8712.2.a.bb.1.2 2
220.219 even 2 1936.2.a.r.1.1 2
440.109 odd 2 7744.2.a.by.1.1 2
440.219 even 2 7744.2.a.cl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
88.2.a.b.1.2 2 5.4 even 2
176.2.a.d.1.1 2 20.19 odd 2
704.2.a.m.1.1 2 40.29 even 2
704.2.a.p.1.2 2 40.19 odd 2
792.2.a.h.1.2 2 15.14 odd 2
968.2.a.j.1.2 2 55.54 odd 2
968.2.i.q.9.1 8 55.19 odd 10
968.2.i.q.81.2 8 55.24 odd 10
968.2.i.q.729.2 8 55.39 odd 10
968.2.i.q.753.1 8 55.29 odd 10
968.2.i.r.9.1 8 55.14 even 10
968.2.i.r.81.2 8 55.9 even 10
968.2.i.r.729.2 8 55.49 even 10
968.2.i.r.753.1 8 55.4 even 10
1584.2.a.t.1.2 2 60.59 even 2
1936.2.a.r.1.1 2 220.219 even 2
2200.2.a.o.1.1 2 1.1 even 1 trivial
2200.2.b.g.1849.1 4 5.3 odd 4
2200.2.b.g.1849.4 4 5.2 odd 4
2816.2.c.p.1409.1 4 80.19 odd 4
2816.2.c.p.1409.4 4 80.59 odd 4
2816.2.c.w.1409.1 4 80.69 even 4
2816.2.c.w.1409.4 4 80.29 even 4
4312.2.a.n.1.1 2 35.34 odd 2
4400.2.a.bp.1.2 2 4.3 odd 2
4400.2.b.v.4049.1 4 20.7 even 4
4400.2.b.v.4049.4 4 20.3 even 4
6336.2.a.cu.1.1 2 120.29 odd 2
6336.2.a.cx.1.1 2 120.59 even 2
7744.2.a.by.1.1 2 440.109 odd 2
7744.2.a.cl.1.2 2 440.219 even 2
8624.2.a.cb.1.2 2 140.139 even 2
8712.2.a.bb.1.2 2 165.164 even 2