Properties

Label 9280.2.a.be.1.2
Level $9280$
Weight $2$
Character 9280.1
Self dual yes
Analytic conductor $74.101$
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] = [9280,2,Mod(1,9280)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9280, 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("9280.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9280 = 2^{6} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9280.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: \(74.1011730757\)
Analytic rank: \(1\)
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 1.2
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 9280.1

$q$-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 \(-0.449960\pi\)
0.156558 + 0.987669i \(0.449960\pi\)
\(8\) 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 \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 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 \(-0.872159\pi\)
−0.920427 + 0.390914i \(0.872159\pi\)
\(24\) 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 \(-0.890684\pi\)
−0.941606 + 0.336717i \(0.890684\pi\)
\(32\) 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 \(-0.745698\pi\)
−0.697486 + 0.716599i \(0.745698\pi\)
\(38\) 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 \(-0.507967\pi\)
−0.0250265 + 0.999687i \(0.507967\pi\)
\(48\) 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 \(-0.676262\pi\)
−0.525875 + 0.850562i \(0.676262\pi\)
\(54\) 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 \(-0.663069\pi\)
−0.490180 + 0.871621i \(0.663069\pi\)
\(62\) 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 \(-0.357106\pi\)
0.433989 + 0.900918i \(0.357106\pi\)
\(72\) 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 \(-0.196813\pi\)
0.814861 + 0.579657i \(0.196813\pi\)
\(80\) 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 \(-0.569327\pi\)
−0.216080 + 0.976376i \(0.569327\pi\)
\(102\) 0 0
\(103\) −12.1421 −1.19640 −0.598200 0.801347i \(-0.704117\pi\)
−0.598200 + 0.801347i \(0.704117\pi\)
\(104\) 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 \(-0.530535\pi\)
−0.0957826 + 0.995402i \(0.530535\pi\)
\(110\) 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 \(-0.414230\pi\)
0.266207 + 0.963916i \(0.414230\pi\)
\(128\) 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 \(-0.316396\pi\)
0.545351 + 0.838208i \(0.316396\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 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 \(-0.271473\pi\)
0.657834 + 0.753163i \(0.271473\pi\)
\(158\) 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 \(-0.310627\pi\)
0.560452 + 0.828187i \(0.310627\pi\)
\(168\) 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 \(-0.435257\pi\)
0.201997 + 0.979386i \(0.435257\pi\)
\(174\) 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 \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) 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 \(-0.684949\pi\)
−0.548888 + 0.835896i \(0.684949\pi\)
\(192\) 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 \(-0.403943\pi\)
0.297212 + 0.954811i \(0.403943\pi\)
\(198\) 0 0
\(199\) 12.0000 0.850657 0.425329 0.905039i \(-0.360158\pi\)
0.425329 + 0.905039i \(0.360158\pi\)
\(200\) 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 \(-0.239574\pi\)
0.729884 + 0.683571i \(0.239574\pi\)
\(224\) 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 \(-0.478950\pi\)
0.0660819 + 0.997814i \(0.478950\pi\)
\(230\) 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 \(-0.771887\pi\)
−0.754019 + 0.656852i \(0.771887\pi\)
\(240\) 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 \(-0.582810\pi\)
−0.257230 + 0.966350i \(0.582810\pi\)
\(264\) 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 \(-0.512751\pi\)
−0.0400491 + 0.999198i \(0.512751\pi\)
\(270\) 0 0
\(271\) 29.7990 1.81016 0.905080 0.425242i \(-0.139811\pi\)
0.905080 + 0.425242i \(0.139811\pi\)
\(272\) 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 \(-0.573882\pi\)
−0.230028 + 0.973184i \(0.573882\pi\)
\(278\) 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 \(-0.420273\pi\)
0.247858 + 0.968796i \(0.420273\pi\)
\(294\) 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 \(-0.522448\pi\)
−0.0704637 + 0.997514i \(0.522448\pi\)
\(312\) 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 \(-0.474690\pi\)
0.0794301 + 0.996840i \(0.474690\pi\)
\(318\) 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 \(-0.270921\pi\)
0.659138 + 0.752022i \(0.270921\pi\)
\(350\) 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 \(-0.822313\pi\)
−0.848199 + 0.529678i \(0.822313\pi\)
\(360\) 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 \(-0.344327\pi\)
0.469796 + 0.882775i \(0.344327\pi\)
\(368\) 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 \(-0.746034\pi\)
−0.698241 + 0.715862i \(0.746034\pi\)
\(374\) 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 \(-0.620670\pi\)
−0.370082 + 0.928999i \(0.620670\pi\)
\(384\) 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 \(-0.445783\pi\)
0.169504 + 0.985529i \(0.445783\pi\)
\(390\) 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 \(-0.432860\pi\)
0.209365 + 0.977838i \(0.432860\pi\)
\(398\) 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 \(-0.381134\pi\)
0.364810 + 0.931082i \(0.381134\pi\)
\(422\) 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 \(-0.654001\pi\)
−0.465154 + 0.885230i \(0.654001\pi\)
\(432\) 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 \(-0.672882\pi\)
−0.516813 + 0.856099i \(0.672882\pi\)
\(440\) 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 \(-0.308148\pi\)
0.566887 + 0.823796i \(0.308148\pi\)
\(462\) 0 0
\(463\) −17.7990 −0.827189 −0.413595 0.910461i \(-0.635727\pi\)
−0.413595 + 0.910461i \(0.635727\pi\)
\(464\) 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 \(-0.405322\pi\)
0.293073 + 0.956090i \(0.405322\pi\)
\(480\) 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 \(-0.735926\pi\)
−0.675161 + 0.737671i \(0.735926\pi\)
\(488\) 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 \(-0.733199\pi\)
−0.668817 + 0.743427i \(0.733199\pi\)
\(504\) 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 \(-0.0471751\pi\)
0.989038 + 0.147663i \(0.0471751\pi\)
\(510\) 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 \(-0.183878\pi\)
0.837738 + 0.546073i \(0.183878\pi\)
\(542\) 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 \(-0.821103\pi\)
−0.846180 + 0.532897i \(0.821103\pi\)
\(558\) 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 \(-0.488299\pi\)
0.0367524 + 0.999324i \(0.488299\pi\)
\(600\) 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 \(-0.162783\pi\)
0.872061 + 0.489398i \(0.162783\pi\)
\(608\) 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 \(-0.487140\pi\)
0.0403896 + 0.999184i \(0.487140\pi\)
\(614\) 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 \(-0.480794\pi\)
0.0603000 + 0.998180i \(0.480794\pi\)
\(632\) 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 \(-0.363777\pi\)
0.415013 + 0.909816i \(0.363777\pi\)
\(648\) 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 \(-0.352609\pi\)
0.446673 + 0.894697i \(0.352609\pi\)
\(654\) 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 \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) 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 \(-0.749765\pi\)
−0.706584 + 0.707629i \(0.749765\pi\)
\(678\) 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 \(-0.686739\pi\)
−0.553582 + 0.832795i \(0.686739\pi\)
\(702\) 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 \(-0.364441\pi\)
0.413114 + 0.910679i \(0.364441\pi\)
\(710\) 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 \(-0.436501\pi\)
0.198168 + 0.980168i \(0.436501\pi\)
\(720\) 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 \(-0.196823\pi\)
0.814843 + 0.579682i \(0.196823\pi\)
\(728\) 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 \(-0.397283\pi\)
0.317123 + 0.948384i \(0.397283\pi\)
\(734\) 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 \(-0.543180\pi\)
−0.135237 + 0.990813i \(0.543180\pi\)
\(744\) 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 \(-0.571107\pi\)
−0.221536 + 0.975152i \(0.571107\pi\)
\(752\) 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 \(-0.730734\pi\)
−0.663040 + 0.748584i \(0.730734\pi\)
\(758\) 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 \(-0.548764\pi\)
−0.152597 + 0.988288i \(0.548764\pi\)
\(774\) 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 \(-0.293298\pi\)
0.604688 + 0.796462i \(0.293298\pi\)
\(798\) 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 \(-0.209701\pi\)
0.790730 + 0.612165i \(0.209701\pi\)
\(822\) 0 0
\(823\) 2.97056 0.103547 0.0517737 0.998659i \(-0.483513\pi\)
0.0517737 + 0.998659i \(0.483513\pi\)
\(824\) 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 \(-0.359334\pi\)
0.427673 + 0.903934i \(0.359334\pi\)
\(830\) 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 \(-0.580445\pi\)
−0.250044 + 0.968235i \(0.580445\pi\)
\(840\) 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 \(-0.439072\pi\)
0.190246 + 0.981736i \(0.439072\pi\)
\(854\) 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 \(-0.542699\pi\)
−0.133742 + 0.991016i \(0.542699\pi\)
\(864\) 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 \(-0.400103\pi\)
0.308708 + 0.951157i \(0.400103\pi\)
\(878\) 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 \(-0.663781\pi\)
−0.492129 + 0.870522i \(0.663781\pi\)
\(888\) 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 \(-0.479643\pi\)
0.0639084 + 0.997956i \(0.479643\pi\)
\(912\) 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 \(-0.702359\pi\)
−0.593765 + 0.804638i \(0.702359\pi\)
\(920\) 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 \(-0.725659\pi\)
−0.651022 + 0.759059i \(0.725659\pi\)
\(942\) 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 \(-0.598665\pi\)
−0.305026 + 0.952344i \(0.598665\pi\)
\(968\) 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 \(-0.431898\pi\)
0.212321 + 0.977200i \(0.431898\pi\)
\(984\) 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 \(-0.190658\pi\)
0.825917 + 0.563791i \(0.190658\pi\)
\(992\) 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 \(-0.505906\pi\)
−0.0185520 + 0.999828i \(0.505906\pi\)
\(998\) 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 9280.2.a.be.1.2 2
4.3 odd 2 9280.2.a.w.1.1 2
8.3 odd 2 2320.2.a.k.1.1 2
8.5 even 2 145.2.a.b.1.1 2
24.5 odd 2 1305.2.a.n.1.2 2
40.13 odd 4 725.2.b.c.349.4 4
40.29 even 2 725.2.a.c.1.2 2
40.37 odd 4 725.2.b.c.349.1 4
56.13 odd 2 7105.2.a.e.1.1 2
120.29 odd 2 6525.2.a.p.1.1 2
232.173 even 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 8.5 even 2
725.2.a.c.1.2 2 40.29 even 2
725.2.b.c.349.1 4 40.37 odd 4
725.2.b.c.349.4 4 40.13 odd 4
1305.2.a.n.1.2 2 24.5 odd 2
2320.2.a.k.1.1 2 8.3 odd 2
4205.2.a.d.1.2 2 232.173 even 2
6525.2.a.p.1.1 2 120.29 odd 2
7105.2.a.e.1.1 2 56.13 odd 2
9280.2.a.w.1.1 2 4.3 odd 2
9280.2.a.be.1.2 2 1.1 even 1 trivial