Properties

Label 2320.2.a.k.1.1
Level $2320$
Weight $2$
Character 2320.1
Self dual yes
Analytic conductor $18.525$
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] = [2320,2,Mod(1,2320)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2320, 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("2320.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2320 = 2^{4} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2320.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: \(18.5252932689\)
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: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 145)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2320.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{3} +1.00000 q^{5} -0.828427 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} +1.00000 q^{5} -0.828427 q^{7} +1.00000 q^{9} +4.82843 q^{11} -2.00000 q^{13} +2.00000 q^{15} -2.82843 q^{17} -0.828427 q^{19} -1.65685 q^{21} +8.82843 q^{23} +1.00000 q^{25} -4.00000 q^{27} +1.00000 q^{29} +10.4853 q^{31} +9.65685 q^{33} -0.828427 q^{35} +8.48528 q^{37} -4.00000 q^{39} -6.00000 q^{41} +6.00000 q^{43} +1.00000 q^{45} +0.343146 q^{47} -6.31371 q^{49} -5.65685 q^{51} +7.65685 q^{53} +4.82843 q^{55} -1.65685 q^{57} +7.65685 q^{61} -0.828427 q^{63} -2.00000 q^{65} +10.4853 q^{67} +17.6569 q^{69} -7.31371 q^{71} -8.48528 q^{73} +2.00000 q^{75} -4.00000 q^{77} -14.4853 q^{79} -11.0000 q^{81} -12.8284 q^{83} -2.82843 q^{85} +2.00000 q^{87} +3.65685 q^{89} +1.65685 q^{91} +20.9706 q^{93} -0.828427 q^{95} +4.48528 q^{97} +4.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{3} + 2 q^{5} + 4 q^{7} + 2 q^{9} + 4 q^{11} - 4 q^{13} + 4 q^{15} + 4 q^{19} + 8 q^{21} + 12 q^{23} + 2 q^{25} - 8 q^{27} + 2 q^{29} + 4 q^{31} + 8 q^{33} + 4 q^{35} - 8 q^{39} - 12 q^{41} + 12 q^{43} + 2 q^{45} + 12 q^{47} + 10 q^{49} + 4 q^{53} + 4 q^{55} + 8 q^{57} + 4 q^{61} + 4 q^{63} - 4 q^{65} + 4 q^{67} + 24 q^{69} + 8 q^{71} + 4 q^{75} - 8 q^{77} - 12 q^{79} - 22 q^{81} - 20 q^{83} + 4 q^{87} - 4 q^{89} - 8 q^{91} + 8 q^{93} + 4 q^{95} - 8 q^{97} + 4 q^{99}+O(q^{100}) \) 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
\(3\) 2.00000 1.15470 0.577350 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) −0.828427 −0.313116 −0.156558 0.987669i \(-0.550040\pi\)
−0.156558 + 0.987669i \(0.550040\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 4.82843 1.45583 0.727913 0.685670i \(-0.240491\pi\)
0.727913 + 0.685670i \(0.240491\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 2.00000 0.516398
\(16\) 0 0
\(17\) −2.82843 −0.685994 −0.342997 0.939336i \(-0.611442\pi\)
−0.342997 + 0.939336i \(0.611442\pi\)
\(18\) 0 0
\(19\) −0.828427 −0.190054 −0.0950271 0.995475i \(-0.530294\pi\)
−0.0950271 + 0.995475i \(0.530294\pi\)
\(20\) 0 0
\(21\) −1.65685 −0.361555
\(22\) 0 0
\(23\) 8.82843 1.84085 0.920427 0.390914i \(-0.127841\pi\)
0.920427 + 0.390914i \(0.127841\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −4.00000 −0.769800
\(28\) 0 0
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) 10.4853 1.88321 0.941606 0.336717i \(-0.109316\pi\)
0.941606 + 0.336717i \(0.109316\pi\)
\(32\) 0 0
\(33\) 9.65685 1.68104
\(34\) 0 0
\(35\) −0.828427 −0.140030
\(36\) 0 0
\(37\) 8.48528 1.39497 0.697486 0.716599i \(-0.254302\pi\)
0.697486 + 0.716599i \(0.254302\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) 1.00000 0.149071
\(46\) 0 0
\(47\) 0.343146 0.0500530 0.0250265 0.999687i \(-0.492033\pi\)
0.0250265 + 0.999687i \(0.492033\pi\)
\(48\) 0 0
\(49\) −6.31371 −0.901958
\(50\) 0 0
\(51\) −5.65685 −0.792118
\(52\) 0 0
\(53\) 7.65685 1.05175 0.525875 0.850562i \(-0.323738\pi\)
0.525875 + 0.850562i \(0.323738\pi\)
\(54\) 0 0
\(55\) 4.82843 0.651065
\(56\) 0 0
\(57\) −1.65685 −0.219456
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 7.65685 0.980360 0.490180 0.871621i \(-0.336931\pi\)
0.490180 + 0.871621i \(0.336931\pi\)
\(62\) 0 0
\(63\) −0.828427 −0.104372
\(64\) 0 0
\(65\) −2.00000 −0.248069
\(66\) 0 0
\(67\) 10.4853 1.28098 0.640490 0.767966i \(-0.278731\pi\)
0.640490 + 0.767966i \(0.278731\pi\)
\(68\) 0 0
\(69\) 17.6569 2.12564
\(70\) 0 0
\(71\) −7.31371 −0.867978 −0.433989 0.900918i \(-0.642894\pi\)
−0.433989 + 0.900918i \(0.642894\pi\)
\(72\) 0 0
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) 0 0
\(75\) 2.00000 0.230940
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −14.4853 −1.62972 −0.814861 0.579657i \(-0.803187\pi\)
−0.814861 + 0.579657i \(0.803187\pi\)
\(80\) 0 0
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) −12.8284 −1.40810 −0.704051 0.710149i \(-0.748628\pi\)
−0.704051 + 0.710149i \(0.748628\pi\)
\(84\) 0 0
\(85\) −2.82843 −0.306786
\(86\) 0 0
\(87\) 2.00000 0.214423
\(88\) 0 0
\(89\) 3.65685 0.387626 0.193813 0.981039i \(-0.437915\pi\)
0.193813 + 0.981039i \(0.437915\pi\)
\(90\) 0 0
\(91\) 1.65685 0.173686
\(92\) 0 0
\(93\) 20.9706 2.17455
\(94\) 0 0
\(95\) −0.828427 −0.0849948
\(96\) 0 0
\(97\) 4.48528 0.455411 0.227706 0.973730i \(-0.426878\pi\)
0.227706 + 0.973730i \(0.426878\pi\)
\(98\) 0 0
\(99\) 4.82843 0.485275
\(100\) 0 0
\(101\) 4.34315 0.432159 0.216080 0.976376i \(-0.430673\pi\)
0.216080 + 0.976376i \(0.430673\pi\)
\(102\) 0 0
\(103\) 12.1421 1.19640 0.598200 0.801347i \(-0.295883\pi\)
0.598200 + 0.801347i \(0.295883\pi\)
\(104\) 0 0
\(105\) −1.65685 −0.161692
\(106\) 0 0
\(107\) 8.14214 0.787130 0.393565 0.919297i \(-0.371242\pi\)
0.393565 + 0.919297i \(0.371242\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) 16.9706 1.61077
\(112\) 0 0
\(113\) 2.82843 0.266076 0.133038 0.991111i \(-0.457527\pi\)
0.133038 + 0.991111i \(0.457527\pi\)
\(114\) 0 0
\(115\) 8.82843 0.823255
\(116\) 0 0
\(117\) −2.00000 −0.184900
\(118\) 0 0
\(119\) 2.34315 0.214796
\(120\) 0 0
\(121\) 12.3137 1.11943
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −6.00000 −0.532414 −0.266207 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 0 0
\(129\) 12.0000 1.05654
\(130\) 0 0
\(131\) −16.1421 −1.41034 −0.705172 0.709036i \(-0.749130\pi\)
−0.705172 + 0.709036i \(0.749130\pi\)
\(132\) 0 0
\(133\) 0.686292 0.0595090
\(134\) 0 0
\(135\) −4.00000 −0.344265
\(136\) 0 0
\(137\) −10.8284 −0.925135 −0.462567 0.886584i \(-0.653072\pi\)
−0.462567 + 0.886584i \(0.653072\pi\)
\(138\) 0 0
\(139\) −10.3431 −0.877294 −0.438647 0.898659i \(-0.644542\pi\)
−0.438647 + 0.898659i \(0.644542\pi\)
\(140\) 0 0
\(141\) 0.686292 0.0577962
\(142\) 0 0
\(143\) −9.65685 −0.807547
\(144\) 0 0
\(145\) 1.00000 0.0830455
\(146\) 0 0
\(147\) −12.6274 −1.04149
\(148\) 0 0
\(149\) −13.3137 −1.09070 −0.545351 0.838208i \(-0.683604\pi\)
−0.545351 + 0.838208i \(0.683604\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) −2.82843 −0.228665
\(154\) 0 0
\(155\) 10.4853 0.842198
\(156\) 0 0
\(157\) −16.4853 −1.31567 −0.657834 0.753163i \(-0.728527\pi\)
−0.657834 + 0.753163i \(0.728527\pi\)
\(158\) 0 0
\(159\) 15.3137 1.21446
\(160\) 0 0
\(161\) −7.31371 −0.576401
\(162\) 0 0
\(163\) 19.6569 1.53964 0.769822 0.638259i \(-0.220345\pi\)
0.769822 + 0.638259i \(0.220345\pi\)
\(164\) 0 0
\(165\) 9.65685 0.751785
\(166\) 0 0
\(167\) −14.4853 −1.12090 −0.560452 0.828187i \(-0.689373\pi\)
−0.560452 + 0.828187i \(0.689373\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −0.828427 −0.0633514
\(172\) 0 0
\(173\) −5.31371 −0.403994 −0.201997 0.979386i \(-0.564743\pi\)
−0.201997 + 0.979386i \(0.564743\pi\)
\(174\) 0 0
\(175\) −0.828427 −0.0626232
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0.686292 0.0512958 0.0256479 0.999671i \(-0.491835\pi\)
0.0256479 + 0.999671i \(0.491835\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 15.3137 1.13202
\(184\) 0 0
\(185\) 8.48528 0.623850
\(186\) 0 0
\(187\) −13.6569 −0.998688
\(188\) 0 0
\(189\) 3.31371 0.241037
\(190\) 0 0
\(191\) 15.1716 1.09778 0.548888 0.835896i \(-0.315051\pi\)
0.548888 + 0.835896i \(0.315051\pi\)
\(192\) 0 0
\(193\) −12.4853 −0.898710 −0.449355 0.893353i \(-0.648346\pi\)
−0.449355 + 0.893353i \(0.648346\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) 0 0
\(197\) −8.34315 −0.594425 −0.297212 0.954811i \(-0.596057\pi\)
−0.297212 + 0.954811i \(0.596057\pi\)
\(198\) 0 0
\(199\) −12.0000 −0.850657 −0.425329 0.905039i \(-0.639842\pi\)
−0.425329 + 0.905039i \(0.639842\pi\)
\(200\) 0 0
\(201\) 20.9706 1.47915
\(202\) 0 0
\(203\) −0.828427 −0.0581442
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) 8.82843 0.613618
\(208\) 0 0
\(209\) −4.00000 −0.276686
\(210\) 0 0
\(211\) −4.82843 −0.332403 −0.166201 0.986092i \(-0.553150\pi\)
−0.166201 + 0.986092i \(0.553150\pi\)
\(212\) 0 0
\(213\) −14.6274 −1.00225
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) −8.68629 −0.589664
\(218\) 0 0
\(219\) −16.9706 −1.14676
\(220\) 0 0
\(221\) 5.65685 0.380521
\(222\) 0 0
\(223\) −21.7990 −1.45977 −0.729884 0.683571i \(-0.760426\pi\)
−0.729884 + 0.683571i \(0.760426\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 8.14214 0.540413 0.270206 0.962802i \(-0.412908\pi\)
0.270206 + 0.962802i \(0.412908\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) −8.00000 −0.526361
\(232\) 0 0
\(233\) 18.0000 1.17922 0.589610 0.807688i \(-0.299282\pi\)
0.589610 + 0.807688i \(0.299282\pi\)
\(234\) 0 0
\(235\) 0.343146 0.0223844
\(236\) 0 0
\(237\) −28.9706 −1.88184
\(238\) 0 0
\(239\) 23.3137 1.50804 0.754019 0.656852i \(-0.228113\pi\)
0.754019 + 0.656852i \(0.228113\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 0 0
\(245\) −6.31371 −0.403368
\(246\) 0 0
\(247\) 1.65685 0.105423
\(248\) 0 0
\(249\) −25.6569 −1.62594
\(250\) 0 0
\(251\) −3.17157 −0.200188 −0.100094 0.994978i \(-0.531914\pi\)
−0.100094 + 0.994978i \(0.531914\pi\)
\(252\) 0 0
\(253\) 42.6274 2.67996
\(254\) 0 0
\(255\) −5.65685 −0.354246
\(256\) 0 0
\(257\) 29.3137 1.82854 0.914269 0.405107i \(-0.132766\pi\)
0.914269 + 0.405107i \(0.132766\pi\)
\(258\) 0 0
\(259\) −7.02944 −0.436788
\(260\) 0 0
\(261\) 1.00000 0.0618984
\(262\) 0 0
\(263\) 8.34315 0.514460 0.257230 0.966350i \(-0.417190\pi\)
0.257230 + 0.966350i \(0.417190\pi\)
\(264\) 0 0
\(265\) 7.65685 0.470357
\(266\) 0 0
\(267\) 7.31371 0.447592
\(268\) 0 0
\(269\) 1.31371 0.0800982 0.0400491 0.999198i \(-0.487249\pi\)
0.0400491 + 0.999198i \(0.487249\pi\)
\(270\) 0 0
\(271\) −29.7990 −1.81016 −0.905080 0.425242i \(-0.860189\pi\)
−0.905080 + 0.425242i \(0.860189\pi\)
\(272\) 0 0
\(273\) 3.31371 0.200555
\(274\) 0 0
\(275\) 4.82843 0.291165
\(276\) 0 0
\(277\) 7.65685 0.460056 0.230028 0.973184i \(-0.426118\pi\)
0.230028 + 0.973184i \(0.426118\pi\)
\(278\) 0 0
\(279\) 10.4853 0.627737
\(280\) 0 0
\(281\) −6.68629 −0.398871 −0.199435 0.979911i \(-0.563911\pi\)
−0.199435 + 0.979911i \(0.563911\pi\)
\(282\) 0 0
\(283\) 0.828427 0.0492449 0.0246224 0.999697i \(-0.492162\pi\)
0.0246224 + 0.999697i \(0.492162\pi\)
\(284\) 0 0
\(285\) −1.65685 −0.0981436
\(286\) 0 0
\(287\) 4.97056 0.293403
\(288\) 0 0
\(289\) −9.00000 −0.529412
\(290\) 0 0
\(291\) 8.97056 0.525864
\(292\) 0 0
\(293\) −8.48528 −0.495715 −0.247858 0.968796i \(-0.579727\pi\)
−0.247858 + 0.968796i \(0.579727\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −19.3137 −1.12070
\(298\) 0 0
\(299\) −17.6569 −1.02112
\(300\) 0 0
\(301\) −4.97056 −0.286498
\(302\) 0 0
\(303\) 8.68629 0.499014
\(304\) 0 0
\(305\) 7.65685 0.438430
\(306\) 0 0
\(307\) −10.9706 −0.626123 −0.313062 0.949733i \(-0.601355\pi\)
−0.313062 + 0.949733i \(0.601355\pi\)
\(308\) 0 0
\(309\) 24.2843 1.38148
\(310\) 0 0
\(311\) 2.48528 0.140927 0.0704637 0.997514i \(-0.477552\pi\)
0.0704637 + 0.997514i \(0.477552\pi\)
\(312\) 0 0
\(313\) −6.00000 −0.339140 −0.169570 0.985518i \(-0.554238\pi\)
−0.169570 + 0.985518i \(0.554238\pi\)
\(314\) 0 0
\(315\) −0.828427 −0.0466766
\(316\) 0 0
\(317\) −2.82843 −0.158860 −0.0794301 0.996840i \(-0.525310\pi\)
−0.0794301 + 0.996840i \(0.525310\pi\)
\(318\) 0 0
\(319\) 4.82843 0.270340
\(320\) 0 0
\(321\) 16.2843 0.908899
\(322\) 0 0
\(323\) 2.34315 0.130376
\(324\) 0 0
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) 4.00000 0.221201
\(328\) 0 0
\(329\) −0.284271 −0.0156724
\(330\) 0 0
\(331\) 17.7990 0.978321 0.489160 0.872194i \(-0.337303\pi\)
0.489160 + 0.872194i \(0.337303\pi\)
\(332\) 0 0
\(333\) 8.48528 0.464991
\(334\) 0 0
\(335\) 10.4853 0.572872
\(336\) 0 0
\(337\) −6.82843 −0.371968 −0.185984 0.982553i \(-0.559547\pi\)
−0.185984 + 0.982553i \(0.559547\pi\)
\(338\) 0 0
\(339\) 5.65685 0.307238
\(340\) 0 0
\(341\) 50.6274 2.74163
\(342\) 0 0
\(343\) 11.0294 0.595534
\(344\) 0 0
\(345\) 17.6569 0.950613
\(346\) 0 0
\(347\) −20.1421 −1.08129 −0.540643 0.841252i \(-0.681819\pi\)
−0.540643 + 0.841252i \(0.681819\pi\)
\(348\) 0 0
\(349\) −24.6274 −1.31828 −0.659138 0.752022i \(-0.729079\pi\)
−0.659138 + 0.752022i \(0.729079\pi\)
\(350\) 0 0
\(351\) 8.00000 0.427008
\(352\) 0 0
\(353\) −15.6569 −0.833330 −0.416665 0.909060i \(-0.636801\pi\)
−0.416665 + 0.909060i \(0.636801\pi\)
\(354\) 0 0
\(355\) −7.31371 −0.388171
\(356\) 0 0
\(357\) 4.68629 0.248025
\(358\) 0 0
\(359\) 32.1421 1.69640 0.848199 0.529678i \(-0.177687\pi\)
0.848199 + 0.529678i \(0.177687\pi\)
\(360\) 0 0
\(361\) −18.3137 −0.963879
\(362\) 0 0
\(363\) 24.6274 1.29260
\(364\) 0 0
\(365\) −8.48528 −0.444140
\(366\) 0 0
\(367\) −18.0000 −0.939592 −0.469796 0.882775i \(-0.655673\pi\)
−0.469796 + 0.882775i \(0.655673\pi\)
\(368\) 0 0
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −6.34315 −0.329320
\(372\) 0 0
\(373\) 26.9706 1.39648 0.698241 0.715862i \(-0.253966\pi\)
0.698241 + 0.715862i \(0.253966\pi\)
\(374\) 0 0
\(375\) 2.00000 0.103280
\(376\) 0 0
\(377\) −2.00000 −0.103005
\(378\) 0 0
\(379\) −5.51472 −0.283272 −0.141636 0.989919i \(-0.545236\pi\)
−0.141636 + 0.989919i \(0.545236\pi\)
\(380\) 0 0
\(381\) −12.0000 −0.614779
\(382\) 0 0
\(383\) 14.4853 0.740163 0.370082 0.928999i \(-0.379330\pi\)
0.370082 + 0.928999i \(0.379330\pi\)
\(384\) 0 0
\(385\) −4.00000 −0.203859
\(386\) 0 0
\(387\) 6.00000 0.304997
\(388\) 0 0
\(389\) −6.68629 −0.339008 −0.169504 0.985529i \(-0.554217\pi\)
−0.169504 + 0.985529i \(0.554217\pi\)
\(390\) 0 0
\(391\) −24.9706 −1.26282
\(392\) 0 0
\(393\) −32.2843 −1.62853
\(394\) 0 0
\(395\) −14.4853 −0.728834
\(396\) 0 0
\(397\) −8.34315 −0.418730 −0.209365 0.977838i \(-0.567140\pi\)
−0.209365 + 0.977838i \(0.567140\pi\)
\(398\) 0 0
\(399\) 1.37258 0.0687151
\(400\) 0 0
\(401\) −29.3137 −1.46386 −0.731928 0.681382i \(-0.761379\pi\)
−0.731928 + 0.681382i \(0.761379\pi\)
\(402\) 0 0
\(403\) −20.9706 −1.04462
\(404\) 0 0
\(405\) −11.0000 −0.546594
\(406\) 0 0
\(407\) 40.9706 2.03084
\(408\) 0 0
\(409\) 30.9706 1.53140 0.765698 0.643200i \(-0.222394\pi\)
0.765698 + 0.643200i \(0.222394\pi\)
\(410\) 0 0
\(411\) −21.6569 −1.06825
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −12.8284 −0.629723
\(416\) 0 0
\(417\) −20.6863 −1.01301
\(418\) 0 0
\(419\) 4.97056 0.242828 0.121414 0.992602i \(-0.461257\pi\)
0.121414 + 0.992602i \(0.461257\pi\)
\(420\) 0 0
\(421\) −14.9706 −0.729621 −0.364810 0.931082i \(-0.618866\pi\)
−0.364810 + 0.931082i \(0.618866\pi\)
\(422\) 0 0
\(423\) 0.343146 0.0166843
\(424\) 0 0
\(425\) −2.82843 −0.137199
\(426\) 0 0
\(427\) −6.34315 −0.306966
\(428\) 0 0
\(429\) −19.3137 −0.932475
\(430\) 0 0
\(431\) 19.3137 0.930309 0.465154 0.885230i \(-0.345999\pi\)
0.465154 + 0.885230i \(0.345999\pi\)
\(432\) 0 0
\(433\) −34.8284 −1.67375 −0.836874 0.547396i \(-0.815619\pi\)
−0.836874 + 0.547396i \(0.815619\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) 0 0
\(437\) −7.31371 −0.349862
\(438\) 0 0
\(439\) 21.6569 1.03363 0.516813 0.856099i \(-0.327118\pi\)
0.516813 + 0.856099i \(0.327118\pi\)
\(440\) 0 0
\(441\) −6.31371 −0.300653
\(442\) 0 0
\(443\) −3.65685 −0.173742 −0.0868712 0.996220i \(-0.527687\pi\)
−0.0868712 + 0.996220i \(0.527687\pi\)
\(444\) 0 0
\(445\) 3.65685 0.173352
\(446\) 0 0
\(447\) −26.6274 −1.25943
\(448\) 0 0
\(449\) 0.343146 0.0161940 0.00809702 0.999967i \(-0.497423\pi\)
0.00809702 + 0.999967i \(0.497423\pi\)
\(450\) 0 0
\(451\) −28.9706 −1.36417
\(452\) 0 0
\(453\) 24.0000 1.12762
\(454\) 0 0
\(455\) 1.65685 0.0776745
\(456\) 0 0
\(457\) 8.34315 0.390276 0.195138 0.980776i \(-0.437485\pi\)
0.195138 + 0.980776i \(0.437485\pi\)
\(458\) 0 0
\(459\) 11.3137 0.528079
\(460\) 0 0
\(461\) −24.3431 −1.13377 −0.566887 0.823796i \(-0.691852\pi\)
−0.566887 + 0.823796i \(0.691852\pi\)
\(462\) 0 0
\(463\) 17.7990 0.827189 0.413595 0.910461i \(-0.364273\pi\)
0.413595 + 0.910461i \(0.364273\pi\)
\(464\) 0 0
\(465\) 20.9706 0.972487
\(466\) 0 0
\(467\) 22.9706 1.06295 0.531475 0.847074i \(-0.321638\pi\)
0.531475 + 0.847074i \(0.321638\pi\)
\(468\) 0 0
\(469\) −8.68629 −0.401096
\(470\) 0 0
\(471\) −32.9706 −1.51920
\(472\) 0 0
\(473\) 28.9706 1.33207
\(474\) 0 0
\(475\) −0.828427 −0.0380108
\(476\) 0 0
\(477\) 7.65685 0.350583
\(478\) 0 0
\(479\) −12.8284 −0.586146 −0.293073 0.956090i \(-0.594678\pi\)
−0.293073 + 0.956090i \(0.594678\pi\)
\(480\) 0 0
\(481\) −16.9706 −0.773791
\(482\) 0 0
\(483\) −14.6274 −0.665571
\(484\) 0 0
\(485\) 4.48528 0.203666
\(486\) 0 0
\(487\) 29.7990 1.35032 0.675161 0.737671i \(-0.264074\pi\)
0.675161 + 0.737671i \(0.264074\pi\)
\(488\) 0 0
\(489\) 39.3137 1.77783
\(490\) 0 0
\(491\) −43.4558 −1.96113 −0.980567 0.196183i \(-0.937145\pi\)
−0.980567 + 0.196183i \(0.937145\pi\)
\(492\) 0 0
\(493\) −2.82843 −0.127386
\(494\) 0 0
\(495\) 4.82843 0.217022
\(496\) 0 0
\(497\) 6.05887 0.271778
\(498\) 0 0
\(499\) −36.0000 −1.61158 −0.805791 0.592200i \(-0.798259\pi\)
−0.805791 + 0.592200i \(0.798259\pi\)
\(500\) 0 0
\(501\) −28.9706 −1.29431
\(502\) 0 0
\(503\) 30.0000 1.33763 0.668817 0.743427i \(-0.266801\pi\)
0.668817 + 0.743427i \(0.266801\pi\)
\(504\) 0 0
\(505\) 4.34315 0.193267
\(506\) 0 0
\(507\) −18.0000 −0.799408
\(508\) 0 0
\(509\) −44.6274 −1.97808 −0.989038 0.147663i \(-0.952825\pi\)
−0.989038 + 0.147663i \(0.952825\pi\)
\(510\) 0 0
\(511\) 7.02944 0.310964
\(512\) 0 0
\(513\) 3.31371 0.146304
\(514\) 0 0
\(515\) 12.1421 0.535046
\(516\) 0 0
\(517\) 1.65685 0.0728684
\(518\) 0 0
\(519\) −10.6274 −0.466492
\(520\) 0 0
\(521\) 1.31371 0.0575546 0.0287773 0.999586i \(-0.490839\pi\)
0.0287773 + 0.999586i \(0.490839\pi\)
\(522\) 0 0
\(523\) 14.4853 0.633397 0.316699 0.948526i \(-0.397426\pi\)
0.316699 + 0.948526i \(0.397426\pi\)
\(524\) 0 0
\(525\) −1.65685 −0.0723110
\(526\) 0 0
\(527\) −29.6569 −1.29187
\(528\) 0 0
\(529\) 54.9411 2.38874
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 12.0000 0.519778
\(534\) 0 0
\(535\) 8.14214 0.352015
\(536\) 0 0
\(537\) 1.37258 0.0592313
\(538\) 0 0
\(539\) −30.4853 −1.31309
\(540\) 0 0
\(541\) −38.9706 −1.67548 −0.837738 0.546073i \(-0.816122\pi\)
−0.837738 + 0.546073i \(0.816122\pi\)
\(542\) 0 0
\(543\) −12.0000 −0.514969
\(544\) 0 0
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −14.4853 −0.619346 −0.309673 0.950843i \(-0.600220\pi\)
−0.309673 + 0.950843i \(0.600220\pi\)
\(548\) 0 0
\(549\) 7.65685 0.326787
\(550\) 0 0
\(551\) −0.828427 −0.0352922
\(552\) 0 0
\(553\) 12.0000 0.510292
\(554\) 0 0
\(555\) 16.9706 0.720360
\(556\) 0 0
\(557\) 39.9411 1.69236 0.846180 0.532897i \(-0.178897\pi\)
0.846180 + 0.532897i \(0.178897\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) 0 0
\(561\) −27.3137 −1.15319
\(562\) 0 0
\(563\) 3.65685 0.154118 0.0770590 0.997027i \(-0.475447\pi\)
0.0770590 + 0.997027i \(0.475447\pi\)
\(564\) 0 0
\(565\) 2.82843 0.118993
\(566\) 0 0
\(567\) 9.11270 0.382697
\(568\) 0 0
\(569\) 16.3431 0.685140 0.342570 0.939492i \(-0.388703\pi\)
0.342570 + 0.939492i \(0.388703\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 0 0
\(573\) 30.3431 1.26760
\(574\) 0 0
\(575\) 8.82843 0.368171
\(576\) 0 0
\(577\) −15.7990 −0.657721 −0.328860 0.944379i \(-0.606665\pi\)
−0.328860 + 0.944379i \(0.606665\pi\)
\(578\) 0 0
\(579\) −24.9706 −1.03774
\(580\) 0 0
\(581\) 10.6274 0.440900
\(582\) 0 0
\(583\) 36.9706 1.53116
\(584\) 0 0
\(585\) −2.00000 −0.0826898
\(586\) 0 0
\(587\) −9.79899 −0.404448 −0.202224 0.979339i \(-0.564817\pi\)
−0.202224 + 0.979339i \(0.564817\pi\)
\(588\) 0 0
\(589\) −8.68629 −0.357912
\(590\) 0 0
\(591\) −16.6863 −0.686382
\(592\) 0 0
\(593\) 3.65685 0.150169 0.0750845 0.997177i \(-0.476077\pi\)
0.0750845 + 0.997177i \(0.476077\pi\)
\(594\) 0 0
\(595\) 2.34315 0.0960596
\(596\) 0 0
\(597\) −24.0000 −0.982255
\(598\) 0 0
\(599\) −1.79899 −0.0735047 −0.0367524 0.999324i \(-0.511701\pi\)
−0.0367524 + 0.999324i \(0.511701\pi\)
\(600\) 0 0
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 10.4853 0.426994
\(604\) 0 0
\(605\) 12.3137 0.500623
\(606\) 0 0
\(607\) −42.9706 −1.74412 −0.872061 0.489398i \(-0.837217\pi\)
−0.872061 + 0.489398i \(0.837217\pi\)
\(608\) 0 0
\(609\) −1.65685 −0.0671391
\(610\) 0 0
\(611\) −0.686292 −0.0277644
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 0 0
\(615\) −12.0000 −0.483887
\(616\) 0 0
\(617\) −14.8284 −0.596970 −0.298485 0.954414i \(-0.596481\pi\)
−0.298485 + 0.954414i \(0.596481\pi\)
\(618\) 0 0
\(619\) −29.7990 −1.19772 −0.598861 0.800853i \(-0.704380\pi\)
−0.598861 + 0.800853i \(0.704380\pi\)
\(620\) 0 0
\(621\) −35.3137 −1.41709
\(622\) 0 0
\(623\) −3.02944 −0.121372
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −8.00000 −0.319489
\(628\) 0 0
\(629\) −24.0000 −0.956943
\(630\) 0 0
\(631\) −3.02944 −0.120600 −0.0603000 0.998180i \(-0.519206\pi\)
−0.0603000 + 0.998180i \(0.519206\pi\)
\(632\) 0 0
\(633\) −9.65685 −0.383825
\(634\) 0 0
\(635\) −6.00000 −0.238103
\(636\) 0 0
\(637\) 12.6274 0.500316
\(638\) 0 0
\(639\) −7.31371 −0.289326
\(640\) 0 0
\(641\) −44.6274 −1.76268 −0.881338 0.472485i \(-0.843357\pi\)
−0.881338 + 0.472485i \(0.843357\pi\)
\(642\) 0 0
\(643\) 31.4558 1.24050 0.620249 0.784405i \(-0.287032\pi\)
0.620249 + 0.784405i \(0.287032\pi\)
\(644\) 0 0
\(645\) 12.0000 0.472500
\(646\) 0 0
\(647\) −21.1127 −0.830026 −0.415013 0.909816i \(-0.636223\pi\)
−0.415013 + 0.909816i \(0.636223\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −17.3726 −0.680885
\(652\) 0 0
\(653\) −22.8284 −0.893345 −0.446673 0.894697i \(-0.647391\pi\)
−0.446673 + 0.894697i \(0.647391\pi\)
\(654\) 0 0
\(655\) −16.1421 −0.630725
\(656\) 0 0
\(657\) −8.48528 −0.331042
\(658\) 0 0
\(659\) 37.7990 1.47244 0.736220 0.676743i \(-0.236609\pi\)
0.736220 + 0.676743i \(0.236609\pi\)
\(660\) 0 0
\(661\) 26.0000 1.01128 0.505641 0.862744i \(-0.331256\pi\)
0.505641 + 0.862744i \(0.331256\pi\)
\(662\) 0 0
\(663\) 11.3137 0.439388
\(664\) 0 0
\(665\) 0.686292 0.0266132
\(666\) 0 0
\(667\) 8.82843 0.341838
\(668\) 0 0
\(669\) −43.5980 −1.68560
\(670\) 0 0
\(671\) 36.9706 1.42723
\(672\) 0 0
\(673\) −10.9706 −0.422884 −0.211442 0.977391i \(-0.567816\pi\)
−0.211442 + 0.977391i \(0.567816\pi\)
\(674\) 0 0
\(675\) −4.00000 −0.153960
\(676\) 0 0
\(677\) 36.7696 1.41317 0.706584 0.707629i \(-0.250235\pi\)
0.706584 + 0.707629i \(0.250235\pi\)
\(678\) 0 0
\(679\) −3.71573 −0.142597
\(680\) 0 0
\(681\) 16.2843 0.624015
\(682\) 0 0
\(683\) −40.1421 −1.53600 −0.767998 0.640452i \(-0.778747\pi\)
−0.767998 + 0.640452i \(0.778747\pi\)
\(684\) 0 0
\(685\) −10.8284 −0.413733
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) 0 0
\(689\) −15.3137 −0.583406
\(690\) 0 0
\(691\) 11.0294 0.419580 0.209790 0.977747i \(-0.432722\pi\)
0.209790 + 0.977747i \(0.432722\pi\)
\(692\) 0 0
\(693\) −4.00000 −0.151947
\(694\) 0 0
\(695\) −10.3431 −0.392338
\(696\) 0 0
\(697\) 16.9706 0.642806
\(698\) 0 0
\(699\) 36.0000 1.36165
\(700\) 0 0
\(701\) 29.3137 1.10716 0.553582 0.832795i \(-0.313261\pi\)
0.553582 + 0.832795i \(0.313261\pi\)
\(702\) 0 0
\(703\) −7.02944 −0.265120
\(704\) 0 0
\(705\) 0.686292 0.0258472
\(706\) 0 0
\(707\) −3.59798 −0.135316
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) −14.4853 −0.543240
\(712\) 0 0
\(713\) 92.5685 3.46672
\(714\) 0 0
\(715\) −9.65685 −0.361146
\(716\) 0 0
\(717\) 46.6274 1.74133
\(718\) 0 0
\(719\) −10.6274 −0.396336 −0.198168 0.980168i \(-0.563499\pi\)
−0.198168 + 0.980168i \(0.563499\pi\)
\(720\) 0 0
\(721\) −10.0589 −0.374612
\(722\) 0 0
\(723\) 20.0000 0.743808
\(724\) 0 0
\(725\) 1.00000 0.0371391
\(726\) 0 0
\(727\) −43.9411 −1.62969 −0.814843 0.579682i \(-0.803177\pi\)
−0.814843 + 0.579682i \(0.803177\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) −16.9706 −0.627679
\(732\) 0 0
\(733\) −17.1716 −0.634247 −0.317123 0.948384i \(-0.602717\pi\)
−0.317123 + 0.948384i \(0.602717\pi\)
\(734\) 0 0
\(735\) −12.6274 −0.465769
\(736\) 0 0
\(737\) 50.6274 1.86488
\(738\) 0 0
\(739\) 2.48528 0.0914226 0.0457113 0.998955i \(-0.485445\pi\)
0.0457113 + 0.998955i \(0.485445\pi\)
\(740\) 0 0
\(741\) 3.31371 0.121732
\(742\) 0 0
\(743\) 7.37258 0.270474 0.135237 0.990813i \(-0.456820\pi\)
0.135237 + 0.990813i \(0.456820\pi\)
\(744\) 0 0
\(745\) −13.3137 −0.487777
\(746\) 0 0
\(747\) −12.8284 −0.469368
\(748\) 0 0
\(749\) −6.74517 −0.246463
\(750\) 0 0
\(751\) 12.1421 0.443073 0.221536 0.975152i \(-0.428893\pi\)
0.221536 + 0.975152i \(0.428893\pi\)
\(752\) 0 0
\(753\) −6.34315 −0.231157
\(754\) 0 0
\(755\) 12.0000 0.436725
\(756\) 0 0
\(757\) 36.4853 1.32608 0.663040 0.748584i \(-0.269266\pi\)
0.663040 + 0.748584i \(0.269266\pi\)
\(758\) 0 0
\(759\) 85.2548 3.09455
\(760\) 0 0
\(761\) −36.6274 −1.32774 −0.663871 0.747847i \(-0.731088\pi\)
−0.663871 + 0.747847i \(0.731088\pi\)
\(762\) 0 0
\(763\) −1.65685 −0.0599822
\(764\) 0 0
\(765\) −2.82843 −0.102262
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −4.34315 −0.156618 −0.0783089 0.996929i \(-0.524952\pi\)
−0.0783089 + 0.996929i \(0.524952\pi\)
\(770\) 0 0
\(771\) 58.6274 2.11141
\(772\) 0 0
\(773\) 8.48528 0.305194 0.152597 0.988288i \(-0.451236\pi\)
0.152597 + 0.988288i \(0.451236\pi\)
\(774\) 0 0
\(775\) 10.4853 0.376642
\(776\) 0 0
\(777\) −14.0589 −0.504359
\(778\) 0 0
\(779\) 4.97056 0.178089
\(780\) 0 0
\(781\) −35.3137 −1.26362
\(782\) 0 0
\(783\) −4.00000 −0.142948
\(784\) 0 0
\(785\) −16.4853 −0.588385
\(786\) 0 0
\(787\) 21.7990 0.777050 0.388525 0.921438i \(-0.372985\pi\)
0.388525 + 0.921438i \(0.372985\pi\)
\(788\) 0 0
\(789\) 16.6863 0.594048
\(790\) 0 0
\(791\) −2.34315 −0.0833127
\(792\) 0 0
\(793\) −15.3137 −0.543806
\(794\) 0 0
\(795\) 15.3137 0.543121
\(796\) 0 0
\(797\) −34.1421 −1.20938 −0.604688 0.796462i \(-0.706702\pi\)
−0.604688 + 0.796462i \(0.706702\pi\)
\(798\) 0 0
\(799\) −0.970563 −0.0343360
\(800\) 0 0
\(801\) 3.65685 0.129209
\(802\) 0 0
\(803\) −40.9706 −1.44582
\(804\) 0 0
\(805\) −7.31371 −0.257774
\(806\) 0 0
\(807\) 2.62742 0.0924895
\(808\) 0 0
\(809\) −14.2843 −0.502208 −0.251104 0.967960i \(-0.580794\pi\)
−0.251104 + 0.967960i \(0.580794\pi\)
\(810\) 0 0
\(811\) −26.3431 −0.925033 −0.462516 0.886611i \(-0.653053\pi\)
−0.462516 + 0.886611i \(0.653053\pi\)
\(812\) 0 0
\(813\) −59.5980 −2.09019
\(814\) 0 0
\(815\) 19.6569 0.688550
\(816\) 0 0
\(817\) −4.97056 −0.173898
\(818\) 0 0
\(819\) 1.65685 0.0578952
\(820\) 0 0
\(821\) −45.3137 −1.58146 −0.790730 0.612165i \(-0.790299\pi\)
−0.790730 + 0.612165i \(0.790299\pi\)
\(822\) 0 0
\(823\) −2.97056 −0.103547 −0.0517737 0.998659i \(-0.516487\pi\)
−0.0517737 + 0.998659i \(0.516487\pi\)
\(824\) 0 0
\(825\) 9.65685 0.336209
\(826\) 0 0
\(827\) 5.31371 0.184776 0.0923879 0.995723i \(-0.470550\pi\)
0.0923879 + 0.995723i \(0.470550\pi\)
\(828\) 0 0
\(829\) −24.6274 −0.855346 −0.427673 0.903934i \(-0.640666\pi\)
−0.427673 + 0.903934i \(0.640666\pi\)
\(830\) 0 0
\(831\) 15.3137 0.531227
\(832\) 0 0
\(833\) 17.8579 0.618738
\(834\) 0 0
\(835\) −14.4853 −0.501284
\(836\) 0 0
\(837\) −41.9411 −1.44970
\(838\) 0 0
\(839\) 14.4853 0.500087 0.250044 0.968235i \(-0.419555\pi\)
0.250044 + 0.968235i \(0.419555\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) 0 0
\(843\) −13.3726 −0.460576
\(844\) 0 0
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −10.2010 −0.350511
\(848\) 0 0
\(849\) 1.65685 0.0568631
\(850\) 0 0
\(851\) 74.9117 2.56794
\(852\) 0 0
\(853\) −11.1127 −0.380492 −0.190246 0.981736i \(-0.560928\pi\)
−0.190246 + 0.981736i \(0.560928\pi\)
\(854\) 0 0
\(855\) −0.828427 −0.0283316
\(856\) 0 0
\(857\) 48.6274 1.66108 0.830540 0.556958i \(-0.188032\pi\)
0.830540 + 0.556958i \(0.188032\pi\)
\(858\) 0 0
\(859\) −28.4264 −0.969896 −0.484948 0.874543i \(-0.661162\pi\)
−0.484948 + 0.874543i \(0.661162\pi\)
\(860\) 0 0
\(861\) 9.94113 0.338793
\(862\) 0 0
\(863\) 7.85786 0.267485 0.133742 0.991016i \(-0.457301\pi\)
0.133742 + 0.991016i \(0.457301\pi\)
\(864\) 0 0
\(865\) −5.31371 −0.180672
\(866\) 0 0
\(867\) −18.0000 −0.611312
\(868\) 0 0
\(869\) −69.9411 −2.37259
\(870\) 0 0
\(871\) −20.9706 −0.710560
\(872\) 0 0
\(873\) 4.48528 0.151804
\(874\) 0 0
\(875\) −0.828427 −0.0280059
\(876\) 0 0
\(877\) −18.2843 −0.617416 −0.308708 0.951157i \(-0.599897\pi\)
−0.308708 + 0.951157i \(0.599897\pi\)
\(878\) 0 0
\(879\) −16.9706 −0.572403
\(880\) 0 0
\(881\) 6.68629 0.225267 0.112633 0.993637i \(-0.464071\pi\)
0.112633 + 0.993637i \(0.464071\pi\)
\(882\) 0 0
\(883\) −2.48528 −0.0836364 −0.0418182 0.999125i \(-0.513315\pi\)
−0.0418182 + 0.999125i \(0.513315\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 29.3137 0.984258 0.492129 0.870522i \(-0.336219\pi\)
0.492129 + 0.870522i \(0.336219\pi\)
\(888\) 0 0
\(889\) 4.97056 0.166707
\(890\) 0 0
\(891\) −53.1127 −1.77934
\(892\) 0 0
\(893\) −0.284271 −0.00951277
\(894\) 0 0
\(895\) 0.686292 0.0229402
\(896\) 0 0
\(897\) −35.3137 −1.17909
\(898\) 0 0
\(899\) 10.4853 0.349704
\(900\) 0 0
\(901\) −21.6569 −0.721494
\(902\) 0 0
\(903\) −9.94113 −0.330820
\(904\) 0 0
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) 10.0000 0.332045 0.166022 0.986122i \(-0.446908\pi\)
0.166022 + 0.986122i \(0.446908\pi\)
\(908\) 0 0
\(909\) 4.34315 0.144053
\(910\) 0 0
\(911\) −3.85786 −0.127817 −0.0639084 0.997956i \(-0.520357\pi\)
−0.0639084 + 0.997956i \(0.520357\pi\)
\(912\) 0 0
\(913\) −61.9411 −2.04995
\(914\) 0 0
\(915\) 15.3137 0.506256
\(916\) 0 0
\(917\) 13.3726 0.441602
\(918\) 0 0
\(919\) 36.0000 1.18753 0.593765 0.804638i \(-0.297641\pi\)
0.593765 + 0.804638i \(0.297641\pi\)
\(920\) 0 0
\(921\) −21.9411 −0.722985
\(922\) 0 0
\(923\) 14.6274 0.481467
\(924\) 0 0
\(925\) 8.48528 0.278994
\(926\) 0 0
\(927\) 12.1421 0.398800
\(928\) 0 0
\(929\) 40.6274 1.33294 0.666471 0.745531i \(-0.267804\pi\)
0.666471 + 0.745531i \(0.267804\pi\)
\(930\) 0 0
\(931\) 5.23045 0.171421
\(932\) 0 0
\(933\) 4.97056 0.162729
\(934\) 0 0
\(935\) −13.6569 −0.446627
\(936\) 0 0
\(937\) 8.34315 0.272559 0.136279 0.990670i \(-0.456486\pi\)
0.136279 + 0.990670i \(0.456486\pi\)
\(938\) 0 0
\(939\) −12.0000 −0.391605
\(940\) 0 0
\(941\) 39.9411 1.30204 0.651022 0.759059i \(-0.274341\pi\)
0.651022 + 0.759059i \(0.274341\pi\)
\(942\) 0 0
\(943\) −52.9706 −1.72496
\(944\) 0 0
\(945\) 3.31371 0.107795
\(946\) 0 0
\(947\) 56.9117 1.84938 0.924691 0.380719i \(-0.124324\pi\)
0.924691 + 0.380719i \(0.124324\pi\)
\(948\) 0 0
\(949\) 16.9706 0.550888
\(950\) 0 0
\(951\) −5.65685 −0.183436
\(952\) 0 0
\(953\) 6.68629 0.216590 0.108295 0.994119i \(-0.465461\pi\)
0.108295 + 0.994119i \(0.465461\pi\)
\(954\) 0 0
\(955\) 15.1716 0.490941
\(956\) 0 0
\(957\) 9.65685 0.312162
\(958\) 0 0
\(959\) 8.97056 0.289675
\(960\) 0 0
\(961\) 78.9411 2.54649
\(962\) 0 0
\(963\) 8.14214 0.262377
\(964\) 0 0
\(965\) −12.4853 −0.401915
\(966\) 0 0
\(967\) 18.9706 0.610052 0.305026 0.952344i \(-0.401335\pi\)
0.305026 + 0.952344i \(0.401335\pi\)
\(968\) 0 0
\(969\) 4.68629 0.150545
\(970\) 0 0
\(971\) 0.142136 0.00456135 0.00228067 0.999997i \(-0.499274\pi\)
0.00228067 + 0.999997i \(0.499274\pi\)
\(972\) 0 0
\(973\) 8.56854 0.274695
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) −25.3137 −0.809857 −0.404929 0.914348i \(-0.632704\pi\)
−0.404929 + 0.914348i \(0.632704\pi\)
\(978\) 0 0
\(979\) 17.6569 0.564316
\(980\) 0 0
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) −13.3137 −0.424641 −0.212321 0.977200i \(-0.568102\pi\)
−0.212321 + 0.977200i \(0.568102\pi\)
\(984\) 0 0
\(985\) −8.34315 −0.265835
\(986\) 0 0
\(987\) −0.568542 −0.0180969
\(988\) 0 0
\(989\) 52.9706 1.68437
\(990\) 0 0
\(991\) −52.0000 −1.65183 −0.825917 0.563791i \(-0.809342\pi\)
−0.825917 + 0.563791i \(0.809342\pi\)
\(992\) 0 0
\(993\) 35.5980 1.12967
\(994\) 0 0
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) 1.17157 0.0371041 0.0185520 0.999828i \(-0.494094\pi\)
0.0185520 + 0.999828i \(0.494094\pi\)
\(998\) 0 0
\(999\) −33.9411 −1.07385
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 2320.2.a.k.1.1 2
4.3 odd 2 145.2.a.b.1.1 2
8.3 odd 2 9280.2.a.be.1.2 2
8.5 even 2 9280.2.a.w.1.1 2
12.11 even 2 1305.2.a.n.1.2 2
20.3 even 4 725.2.b.c.349.4 4
20.7 even 4 725.2.b.c.349.1 4
20.19 odd 2 725.2.a.c.1.2 2
28.27 even 2 7105.2.a.e.1.1 2
60.59 even 2 6525.2.a.p.1.1 2
116.115 odd 2 4205.2.a.d.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
145.2.a.b.1.1 2 4.3 odd 2
725.2.a.c.1.2 2 20.19 odd 2
725.2.b.c.349.1 4 20.7 even 4
725.2.b.c.349.4 4 20.3 even 4
1305.2.a.n.1.2 2 12.11 even 2
2320.2.a.k.1.1 2 1.1 even 1 trivial
4205.2.a.d.1.2 2 116.115 odd 2
6525.2.a.p.1.1 2 60.59 even 2
7105.2.a.e.1.1 2 28.27 even 2
9280.2.a.w.1.1 2 8.5 even 2
9280.2.a.be.1.2 2 8.3 odd 2