Properties

Label 9801.2.a.cn.1.15
Level $9801$
Weight $2$
Character 9801.1
Self dual yes
Analytic conductor $78.261$
Analytic rank $1$
Dimension $18$
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] = [9801,2,Mod(1,9801)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9801, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9801.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9801 = 3^{4} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9801.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: \(78.2613790211\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 2 x^{17} - 22 x^{16} + 42 x^{15} + 198 x^{14} - 357 x^{13} - 944 x^{12} + 1579 x^{11} + \cdots - 55 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 99)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.15
Root \(-1.54614\) of defining polynomial
Character \(\chi\) \(=\) 9801.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.54614 q^{2} +0.390538 q^{4} -0.592032 q^{5} -0.721951 q^{7} -2.48845 q^{8} -0.915362 q^{10} +1.57496 q^{13} -1.11623 q^{14} -4.62856 q^{16} +4.59200 q^{17} -2.50860 q^{19} -0.231211 q^{20} +4.45200 q^{23} -4.64950 q^{25} +2.43511 q^{26} -0.281949 q^{28} +6.97862 q^{29} -9.12824 q^{31} -2.17948 q^{32} +7.09986 q^{34} +0.427418 q^{35} +2.89170 q^{37} -3.87864 q^{38} +1.47324 q^{40} -1.15412 q^{41} +4.21448 q^{43} +6.88340 q^{46} +0.227555 q^{47} -6.47879 q^{49} -7.18876 q^{50} +0.615084 q^{52} +5.70359 q^{53} +1.79654 q^{56} +10.7899 q^{58} -7.13355 q^{59} -4.55896 q^{61} -14.1135 q^{62} +5.88733 q^{64} -0.932429 q^{65} -8.09142 q^{67} +1.79335 q^{68} +0.660846 q^{70} +12.3094 q^{71} -15.4833 q^{73} +4.47096 q^{74} -0.979704 q^{76} +14.6196 q^{79} +2.74025 q^{80} -1.78443 q^{82} -6.95300 q^{83} -2.71861 q^{85} +6.51616 q^{86} -12.4803 q^{89} -1.13705 q^{91} +1.73868 q^{92} +0.351832 q^{94} +1.48517 q^{95} +14.1805 q^{97} -10.0171 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q - 2 q^{2} + 12 q^{4} + q^{5} - q^{7} - 6 q^{8} + 2 q^{10} - 3 q^{13} - 8 q^{16} - 20 q^{17} + 3 q^{19} + 5 q^{20} + 10 q^{23} + 7 q^{25} - 2 q^{26} - 19 q^{28} - 21 q^{29} + 6 q^{31} - 9 q^{32} - 4 q^{34}+ \cdots - 82 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.54614 1.09328 0.546642 0.837367i \(-0.315906\pi\)
0.546642 + 0.837367i \(0.315906\pi\)
\(3\) 0 0
\(4\) 0.390538 0.195269
\(5\) −0.592032 −0.264765 −0.132382 0.991199i \(-0.542263\pi\)
−0.132382 + 0.991199i \(0.542263\pi\)
\(6\) 0 0
\(7\) −0.721951 −0.272872 −0.136436 0.990649i \(-0.543565\pi\)
−0.136436 + 0.990649i \(0.543565\pi\)
\(8\) −2.48845 −0.879799
\(9\) 0 0
\(10\) −0.915362 −0.289463
\(11\) 0 0
\(12\) 0 0
\(13\) 1.57496 0.436817 0.218408 0.975857i \(-0.429914\pi\)
0.218408 + 0.975857i \(0.429914\pi\)
\(14\) −1.11623 −0.298326
\(15\) 0 0
\(16\) −4.62856 −1.15714
\(17\) 4.59200 1.11372 0.556862 0.830605i \(-0.312005\pi\)
0.556862 + 0.830605i \(0.312005\pi\)
\(18\) 0 0
\(19\) −2.50860 −0.575512 −0.287756 0.957704i \(-0.592909\pi\)
−0.287756 + 0.957704i \(0.592909\pi\)
\(20\) −0.231211 −0.0517004
\(21\) 0 0
\(22\) 0 0
\(23\) 4.45200 0.928306 0.464153 0.885755i \(-0.346359\pi\)
0.464153 + 0.885755i \(0.346359\pi\)
\(24\) 0 0
\(25\) −4.64950 −0.929900
\(26\) 2.43511 0.477564
\(27\) 0 0
\(28\) −0.281949 −0.0532834
\(29\) 6.97862 1.29590 0.647949 0.761684i \(-0.275627\pi\)
0.647949 + 0.761684i \(0.275627\pi\)
\(30\) 0 0
\(31\) −9.12824 −1.63948 −0.819740 0.572735i \(-0.805882\pi\)
−0.819740 + 0.572735i \(0.805882\pi\)
\(32\) −2.17948 −0.385282
\(33\) 0 0
\(34\) 7.09986 1.21762
\(35\) 0.427418 0.0722468
\(36\) 0 0
\(37\) 2.89170 0.475392 0.237696 0.971340i \(-0.423608\pi\)
0.237696 + 0.971340i \(0.423608\pi\)
\(38\) −3.87864 −0.629198
\(39\) 0 0
\(40\) 1.47324 0.232940
\(41\) −1.15412 −0.180243 −0.0901216 0.995931i \(-0.528726\pi\)
−0.0901216 + 0.995931i \(0.528726\pi\)
\(42\) 0 0
\(43\) 4.21448 0.642702 0.321351 0.946960i \(-0.395863\pi\)
0.321351 + 0.946960i \(0.395863\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.88340 1.01490
\(47\) 0.227555 0.0331924 0.0165962 0.999862i \(-0.494717\pi\)
0.0165962 + 0.999862i \(0.494717\pi\)
\(48\) 0 0
\(49\) −6.47879 −0.925541
\(50\) −7.18876 −1.01664
\(51\) 0 0
\(52\) 0.615084 0.0852967
\(53\) 5.70359 0.783449 0.391724 0.920083i \(-0.371879\pi\)
0.391724 + 0.920083i \(0.371879\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 1.79654 0.240072
\(57\) 0 0
\(58\) 10.7899 1.41678
\(59\) −7.13355 −0.928710 −0.464355 0.885649i \(-0.653714\pi\)
−0.464355 + 0.885649i \(0.653714\pi\)
\(60\) 0 0
\(61\) −4.55896 −0.583715 −0.291857 0.956462i \(-0.594273\pi\)
−0.291857 + 0.956462i \(0.594273\pi\)
\(62\) −14.1135 −1.79242
\(63\) 0 0
\(64\) 5.88733 0.735917
\(65\) −0.932429 −0.115654
\(66\) 0 0
\(67\) −8.09142 −0.988524 −0.494262 0.869313i \(-0.664562\pi\)
−0.494262 + 0.869313i \(0.664562\pi\)
\(68\) 1.79335 0.217476
\(69\) 0 0
\(70\) 0.660846 0.0789862
\(71\) 12.3094 1.46086 0.730429 0.682989i \(-0.239320\pi\)
0.730429 + 0.682989i \(0.239320\pi\)
\(72\) 0 0
\(73\) −15.4833 −1.81219 −0.906093 0.423078i \(-0.860950\pi\)
−0.906093 + 0.423078i \(0.860950\pi\)
\(74\) 4.47096 0.519738
\(75\) 0 0
\(76\) −0.979704 −0.112380
\(77\) 0 0
\(78\) 0 0
\(79\) 14.6196 1.64483 0.822414 0.568889i \(-0.192627\pi\)
0.822414 + 0.568889i \(0.192627\pi\)
\(80\) 2.74025 0.306370
\(81\) 0 0
\(82\) −1.78443 −0.197057
\(83\) −6.95300 −0.763191 −0.381596 0.924329i \(-0.624625\pi\)
−0.381596 + 0.924329i \(0.624625\pi\)
\(84\) 0 0
\(85\) −2.71861 −0.294875
\(86\) 6.51616 0.702656
\(87\) 0 0
\(88\) 0 0
\(89\) −12.4803 −1.32291 −0.661453 0.749986i \(-0.730060\pi\)
−0.661453 + 0.749986i \(0.730060\pi\)
\(90\) 0 0
\(91\) −1.13705 −0.119195
\(92\) 1.73868 0.181269
\(93\) 0 0
\(94\) 0.351832 0.0362887
\(95\) 1.48517 0.152375
\(96\) 0 0
\(97\) 14.1805 1.43981 0.719904 0.694073i \(-0.244186\pi\)
0.719904 + 0.694073i \(0.244186\pi\)
\(98\) −10.0171 −1.01188
\(99\) 0 0
\(100\) −1.81581 −0.181581
\(101\) −14.3685 −1.42972 −0.714861 0.699266i \(-0.753510\pi\)
−0.714861 + 0.699266i \(0.753510\pi\)
\(102\) 0 0
\(103\) −4.42885 −0.436388 −0.218194 0.975905i \(-0.570017\pi\)
−0.218194 + 0.975905i \(0.570017\pi\)
\(104\) −3.91922 −0.384311
\(105\) 0 0
\(106\) 8.81853 0.856532
\(107\) −15.7444 −1.52207 −0.761035 0.648711i \(-0.775308\pi\)
−0.761035 + 0.648711i \(0.775308\pi\)
\(108\) 0 0
\(109\) 13.6970 1.31194 0.655969 0.754788i \(-0.272260\pi\)
0.655969 + 0.754788i \(0.272260\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.34159 0.315750
\(113\) −8.03013 −0.755411 −0.377706 0.925926i \(-0.623287\pi\)
−0.377706 + 0.925926i \(0.623287\pi\)
\(114\) 0 0
\(115\) −2.63573 −0.245783
\(116\) 2.72542 0.253049
\(117\) 0 0
\(118\) −11.0294 −1.01534
\(119\) −3.31520 −0.303904
\(120\) 0 0
\(121\) 0 0
\(122\) −7.04877 −0.638166
\(123\) 0 0
\(124\) −3.56493 −0.320140
\(125\) 5.71281 0.510969
\(126\) 0 0
\(127\) 11.9963 1.06450 0.532250 0.846587i \(-0.321347\pi\)
0.532250 + 0.846587i \(0.321347\pi\)
\(128\) 13.4616 1.18985
\(129\) 0 0
\(130\) −1.44166 −0.126442
\(131\) −6.89109 −0.602077 −0.301039 0.953612i \(-0.597333\pi\)
−0.301039 + 0.953612i \(0.597333\pi\)
\(132\) 0 0
\(133\) 1.81109 0.157041
\(134\) −12.5104 −1.08074
\(135\) 0 0
\(136\) −11.4270 −0.979854
\(137\) −9.24655 −0.789986 −0.394993 0.918684i \(-0.629253\pi\)
−0.394993 + 0.918684i \(0.629253\pi\)
\(138\) 0 0
\(139\) −21.6113 −1.83305 −0.916523 0.399982i \(-0.869016\pi\)
−0.916523 + 0.399982i \(0.869016\pi\)
\(140\) 0.166923 0.0141076
\(141\) 0 0
\(142\) 19.0320 1.59713
\(143\) 0 0
\(144\) 0 0
\(145\) −4.13157 −0.343108
\(146\) −23.9393 −1.98123
\(147\) 0 0
\(148\) 1.12932 0.0928293
\(149\) −1.14643 −0.0939194 −0.0469597 0.998897i \(-0.514953\pi\)
−0.0469597 + 0.998897i \(0.514953\pi\)
\(150\) 0 0
\(151\) −0.880640 −0.0716655 −0.0358327 0.999358i \(-0.511408\pi\)
−0.0358327 + 0.999358i \(0.511408\pi\)
\(152\) 6.24252 0.506335
\(153\) 0 0
\(154\) 0 0
\(155\) 5.40421 0.434077
\(156\) 0 0
\(157\) −9.78621 −0.781025 −0.390512 0.920598i \(-0.627702\pi\)
−0.390512 + 0.920598i \(0.627702\pi\)
\(158\) 22.6038 1.79826
\(159\) 0 0
\(160\) 1.29032 0.102009
\(161\) −3.21412 −0.253308
\(162\) 0 0
\(163\) 4.62976 0.362631 0.181315 0.983425i \(-0.441965\pi\)
0.181315 + 0.983425i \(0.441965\pi\)
\(164\) −0.450728 −0.0351959
\(165\) 0 0
\(166\) −10.7503 −0.834384
\(167\) −16.9874 −1.31452 −0.657262 0.753663i \(-0.728285\pi\)
−0.657262 + 0.753663i \(0.728285\pi\)
\(168\) 0 0
\(169\) −10.5195 −0.809191
\(170\) −4.20335 −0.322382
\(171\) 0 0
\(172\) 1.64592 0.125500
\(173\) −0.591056 −0.0449371 −0.0224686 0.999748i \(-0.507153\pi\)
−0.0224686 + 0.999748i \(0.507153\pi\)
\(174\) 0 0
\(175\) 3.35671 0.253743
\(176\) 0 0
\(177\) 0 0
\(178\) −19.2962 −1.44631
\(179\) 9.64093 0.720597 0.360298 0.932837i \(-0.382675\pi\)
0.360298 + 0.932837i \(0.382675\pi\)
\(180\) 0 0
\(181\) −4.76187 −0.353947 −0.176973 0.984216i \(-0.556631\pi\)
−0.176973 + 0.984216i \(0.556631\pi\)
\(182\) −1.75803 −0.130314
\(183\) 0 0
\(184\) −11.0786 −0.816723
\(185\) −1.71198 −0.125867
\(186\) 0 0
\(187\) 0 0
\(188\) 0.0888691 0.00648144
\(189\) 0 0
\(190\) 2.29628 0.166590
\(191\) 5.19802 0.376116 0.188058 0.982158i \(-0.439781\pi\)
0.188058 + 0.982158i \(0.439781\pi\)
\(192\) 0 0
\(193\) −3.26805 −0.235239 −0.117620 0.993059i \(-0.537526\pi\)
−0.117620 + 0.993059i \(0.537526\pi\)
\(194\) 21.9249 1.57412
\(195\) 0 0
\(196\) −2.53021 −0.180730
\(197\) −22.4626 −1.60039 −0.800197 0.599737i \(-0.795272\pi\)
−0.800197 + 0.599737i \(0.795272\pi\)
\(198\) 0 0
\(199\) −20.8291 −1.47654 −0.738269 0.674507i \(-0.764356\pi\)
−0.738269 + 0.674507i \(0.764356\pi\)
\(200\) 11.5700 0.818125
\(201\) 0 0
\(202\) −22.2157 −1.56309
\(203\) −5.03822 −0.353614
\(204\) 0 0
\(205\) 0.683276 0.0477220
\(206\) −6.84761 −0.477096
\(207\) 0 0
\(208\) −7.28981 −0.505457
\(209\) 0 0
\(210\) 0 0
\(211\) −3.39202 −0.233517 −0.116758 0.993160i \(-0.537250\pi\)
−0.116758 + 0.993160i \(0.537250\pi\)
\(212\) 2.22747 0.152983
\(213\) 0 0
\(214\) −24.3430 −1.66405
\(215\) −2.49511 −0.170165
\(216\) 0 0
\(217\) 6.59014 0.447368
\(218\) 21.1775 1.43432
\(219\) 0 0
\(220\) 0 0
\(221\) 7.23224 0.486493
\(222\) 0 0
\(223\) −11.0264 −0.738382 −0.369191 0.929354i \(-0.620365\pi\)
−0.369191 + 0.929354i \(0.620365\pi\)
\(224\) 1.57348 0.105133
\(225\) 0 0
\(226\) −12.4157 −0.825879
\(227\) −5.96689 −0.396036 −0.198018 0.980198i \(-0.563451\pi\)
−0.198018 + 0.980198i \(0.563451\pi\)
\(228\) 0 0
\(229\) 24.7507 1.63557 0.817785 0.575523i \(-0.195202\pi\)
0.817785 + 0.575523i \(0.195202\pi\)
\(230\) −4.07519 −0.268710
\(231\) 0 0
\(232\) −17.3659 −1.14013
\(233\) 4.18842 0.274392 0.137196 0.990544i \(-0.456191\pi\)
0.137196 + 0.990544i \(0.456191\pi\)
\(234\) 0 0
\(235\) −0.134720 −0.00878817
\(236\) −2.78592 −0.181348
\(237\) 0 0
\(238\) −5.12575 −0.332253
\(239\) −7.37284 −0.476910 −0.238455 0.971154i \(-0.576641\pi\)
−0.238455 + 0.971154i \(0.576641\pi\)
\(240\) 0 0
\(241\) 8.82924 0.568741 0.284371 0.958714i \(-0.408215\pi\)
0.284371 + 0.958714i \(0.408215\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −1.78045 −0.113981
\(245\) 3.83565 0.245051
\(246\) 0 0
\(247\) −3.95096 −0.251393
\(248\) 22.7152 1.44241
\(249\) 0 0
\(250\) 8.83279 0.558634
\(251\) −20.5731 −1.29856 −0.649280 0.760550i \(-0.724930\pi\)
−0.649280 + 0.760550i \(0.724930\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 18.5479 1.16380
\(255\) 0 0
\(256\) 9.03879 0.564924
\(257\) 3.38239 0.210987 0.105494 0.994420i \(-0.466358\pi\)
0.105494 + 0.994420i \(0.466358\pi\)
\(258\) 0 0
\(259\) −2.08766 −0.129721
\(260\) −0.364149 −0.0225836
\(261\) 0 0
\(262\) −10.6546 −0.658241
\(263\) −14.2822 −0.880681 −0.440341 0.897831i \(-0.645142\pi\)
−0.440341 + 0.897831i \(0.645142\pi\)
\(264\) 0 0
\(265\) −3.37671 −0.207430
\(266\) 2.80019 0.171690
\(267\) 0 0
\(268\) −3.16001 −0.193028
\(269\) −6.72041 −0.409751 −0.204875 0.978788i \(-0.565679\pi\)
−0.204875 + 0.978788i \(0.565679\pi\)
\(270\) 0 0
\(271\) 3.98035 0.241789 0.120895 0.992665i \(-0.461424\pi\)
0.120895 + 0.992665i \(0.461424\pi\)
\(272\) −21.2543 −1.28873
\(273\) 0 0
\(274\) −14.2964 −0.863679
\(275\) 0 0
\(276\) 0 0
\(277\) 13.1320 0.789026 0.394513 0.918890i \(-0.370913\pi\)
0.394513 + 0.918890i \(0.370913\pi\)
\(278\) −33.4140 −2.00404
\(279\) 0 0
\(280\) −1.06361 −0.0635627
\(281\) −2.96787 −0.177048 −0.0885241 0.996074i \(-0.528215\pi\)
−0.0885241 + 0.996074i \(0.528215\pi\)
\(282\) 0 0
\(283\) −8.09278 −0.481066 −0.240533 0.970641i \(-0.577322\pi\)
−0.240533 + 0.970641i \(0.577322\pi\)
\(284\) 4.80729 0.285260
\(285\) 0 0
\(286\) 0 0
\(287\) 0.833217 0.0491833
\(288\) 0 0
\(289\) 4.08649 0.240382
\(290\) −6.38797 −0.375114
\(291\) 0 0
\(292\) −6.04683 −0.353864
\(293\) −15.7156 −0.918114 −0.459057 0.888407i \(-0.651813\pi\)
−0.459057 + 0.888407i \(0.651813\pi\)
\(294\) 0 0
\(295\) 4.22329 0.245890
\(296\) −7.19584 −0.418249
\(297\) 0 0
\(298\) −1.77254 −0.102681
\(299\) 7.01174 0.405499
\(300\) 0 0
\(301\) −3.04265 −0.175375
\(302\) −1.36159 −0.0783507
\(303\) 0 0
\(304\) 11.6112 0.665948
\(305\) 2.69905 0.154547
\(306\) 0 0
\(307\) 3.51315 0.200506 0.100253 0.994962i \(-0.468035\pi\)
0.100253 + 0.994962i \(0.468035\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 8.35565 0.474569
\(311\) −26.3344 −1.49329 −0.746644 0.665224i \(-0.768336\pi\)
−0.746644 + 0.665224i \(0.768336\pi\)
\(312\) 0 0
\(313\) −2.46114 −0.139112 −0.0695560 0.997578i \(-0.522158\pi\)
−0.0695560 + 0.997578i \(0.522158\pi\)
\(314\) −15.1308 −0.853882
\(315\) 0 0
\(316\) 5.70949 0.321184
\(317\) 26.5071 1.48879 0.744395 0.667740i \(-0.232738\pi\)
0.744395 + 0.667740i \(0.232738\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.48549 −0.194845
\(321\) 0 0
\(322\) −4.96947 −0.276938
\(323\) −11.5195 −0.640962
\(324\) 0 0
\(325\) −7.32279 −0.406196
\(326\) 7.15824 0.396458
\(327\) 0 0
\(328\) 2.87197 0.158578
\(329\) −0.164284 −0.00905726
\(330\) 0 0
\(331\) 8.10397 0.445434 0.222717 0.974883i \(-0.428507\pi\)
0.222717 + 0.974883i \(0.428507\pi\)
\(332\) −2.71541 −0.149028
\(333\) 0 0
\(334\) −26.2648 −1.43715
\(335\) 4.79038 0.261726
\(336\) 0 0
\(337\) −24.7206 −1.34662 −0.673309 0.739361i \(-0.735128\pi\)
−0.673309 + 0.739361i \(0.735128\pi\)
\(338\) −16.2646 −0.884676
\(339\) 0 0
\(340\) −1.06172 −0.0575799
\(341\) 0 0
\(342\) 0 0
\(343\) 9.73102 0.525426
\(344\) −10.4875 −0.565449
\(345\) 0 0
\(346\) −0.913853 −0.0491290
\(347\) −2.06964 −0.111104 −0.0555522 0.998456i \(-0.517692\pi\)
−0.0555522 + 0.998456i \(0.517692\pi\)
\(348\) 0 0
\(349\) 5.21427 0.279113 0.139557 0.990214i \(-0.455432\pi\)
0.139557 + 0.990214i \(0.455432\pi\)
\(350\) 5.18993 0.277413
\(351\) 0 0
\(352\) 0 0
\(353\) −21.6725 −1.15351 −0.576756 0.816917i \(-0.695681\pi\)
−0.576756 + 0.816917i \(0.695681\pi\)
\(354\) 0 0
\(355\) −7.28756 −0.386784
\(356\) −4.87402 −0.258323
\(357\) 0 0
\(358\) 14.9062 0.787817
\(359\) 8.75285 0.461958 0.230979 0.972959i \(-0.425807\pi\)
0.230979 + 0.972959i \(0.425807\pi\)
\(360\) 0 0
\(361\) −12.7069 −0.668786
\(362\) −7.36250 −0.386964
\(363\) 0 0
\(364\) −0.444060 −0.0232751
\(365\) 9.16662 0.479803
\(366\) 0 0
\(367\) −6.76623 −0.353194 −0.176597 0.984283i \(-0.556509\pi\)
−0.176597 + 0.984283i \(0.556509\pi\)
\(368\) −20.6063 −1.07418
\(369\) 0 0
\(370\) −2.64695 −0.137608
\(371\) −4.11771 −0.213781
\(372\) 0 0
\(373\) 21.8933 1.13359 0.566797 0.823857i \(-0.308182\pi\)
0.566797 + 0.823857i \(0.308182\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −0.566260 −0.0292026
\(377\) 10.9911 0.566069
\(378\) 0 0
\(379\) −33.8236 −1.73740 −0.868701 0.495337i \(-0.835045\pi\)
−0.868701 + 0.495337i \(0.835045\pi\)
\(380\) 0.580016 0.0297542
\(381\) 0 0
\(382\) 8.03686 0.411201
\(383\) 18.3739 0.938862 0.469431 0.882969i \(-0.344459\pi\)
0.469431 + 0.882969i \(0.344459\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −5.05284 −0.257183
\(387\) 0 0
\(388\) 5.53801 0.281150
\(389\) 30.8065 1.56195 0.780977 0.624560i \(-0.214722\pi\)
0.780977 + 0.624560i \(0.214722\pi\)
\(390\) 0 0
\(391\) 20.4436 1.03388
\(392\) 16.1221 0.814290
\(393\) 0 0
\(394\) −34.7302 −1.74968
\(395\) −8.65524 −0.435493
\(396\) 0 0
\(397\) −8.29578 −0.416353 −0.208177 0.978091i \(-0.566753\pi\)
−0.208177 + 0.978091i \(0.566753\pi\)
\(398\) −32.2047 −1.61427
\(399\) 0 0
\(400\) 21.5205 1.07602
\(401\) 3.93362 0.196436 0.0982179 0.995165i \(-0.468686\pi\)
0.0982179 + 0.995165i \(0.468686\pi\)
\(402\) 0 0
\(403\) −14.3767 −0.716152
\(404\) −5.61146 −0.279181
\(405\) 0 0
\(406\) −7.78977 −0.386600
\(407\) 0 0
\(408\) 0 0
\(409\) 2.50425 0.123827 0.0619137 0.998082i \(-0.480280\pi\)
0.0619137 + 0.998082i \(0.480280\pi\)
\(410\) 1.05644 0.0521737
\(411\) 0 0
\(412\) −1.72964 −0.0852130
\(413\) 5.15007 0.253418
\(414\) 0 0
\(415\) 4.11640 0.202066
\(416\) −3.43261 −0.168298
\(417\) 0 0
\(418\) 0 0
\(419\) −10.7394 −0.524654 −0.262327 0.964979i \(-0.584490\pi\)
−0.262327 + 0.964979i \(0.584490\pi\)
\(420\) 0 0
\(421\) 22.1875 1.08135 0.540675 0.841232i \(-0.318169\pi\)
0.540675 + 0.841232i \(0.318169\pi\)
\(422\) −5.24453 −0.255300
\(423\) 0 0
\(424\) −14.1931 −0.689278
\(425\) −21.3505 −1.03565
\(426\) 0 0
\(427\) 3.29134 0.159279
\(428\) −6.14879 −0.297213
\(429\) 0 0
\(430\) −3.85778 −0.186038
\(431\) −18.4582 −0.889100 −0.444550 0.895754i \(-0.646636\pi\)
−0.444550 + 0.895754i \(0.646636\pi\)
\(432\) 0 0
\(433\) 15.1166 0.726459 0.363230 0.931700i \(-0.381674\pi\)
0.363230 + 0.931700i \(0.381674\pi\)
\(434\) 10.1893 0.489100
\(435\) 0 0
\(436\) 5.34921 0.256181
\(437\) −11.1683 −0.534252
\(438\) 0 0
\(439\) −23.9553 −1.14333 −0.571663 0.820488i \(-0.693702\pi\)
−0.571663 + 0.820488i \(0.693702\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 11.1820 0.531875
\(443\) −25.4977 −1.21143 −0.605716 0.795681i \(-0.707113\pi\)
−0.605716 + 0.795681i \(0.707113\pi\)
\(444\) 0 0
\(445\) 7.38872 0.350259
\(446\) −17.0483 −0.807261
\(447\) 0 0
\(448\) −4.25036 −0.200811
\(449\) 22.4824 1.06101 0.530505 0.847682i \(-0.322002\pi\)
0.530505 + 0.847682i \(0.322002\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −3.13607 −0.147508
\(453\) 0 0
\(454\) −9.22562 −0.432980
\(455\) 0.673168 0.0315586
\(456\) 0 0
\(457\) −21.6896 −1.01460 −0.507298 0.861771i \(-0.669356\pi\)
−0.507298 + 0.861771i \(0.669356\pi\)
\(458\) 38.2679 1.78814
\(459\) 0 0
\(460\) −1.02935 −0.0479938
\(461\) −12.1036 −0.563722 −0.281861 0.959455i \(-0.590952\pi\)
−0.281861 + 0.959455i \(0.590952\pi\)
\(462\) 0 0
\(463\) 20.0308 0.930910 0.465455 0.885071i \(-0.345891\pi\)
0.465455 + 0.885071i \(0.345891\pi\)
\(464\) −32.3009 −1.49953
\(465\) 0 0
\(466\) 6.47586 0.299988
\(467\) −6.06750 −0.280770 −0.140385 0.990097i \(-0.544834\pi\)
−0.140385 + 0.990097i \(0.544834\pi\)
\(468\) 0 0
\(469\) 5.84161 0.269740
\(470\) −0.208296 −0.00960796
\(471\) 0 0
\(472\) 17.7515 0.817078
\(473\) 0 0
\(474\) 0 0
\(475\) 11.6637 0.535169
\(476\) −1.29471 −0.0593430
\(477\) 0 0
\(478\) −11.3994 −0.521397
\(479\) 36.2728 1.65735 0.828673 0.559732i \(-0.189096\pi\)
0.828673 + 0.559732i \(0.189096\pi\)
\(480\) 0 0
\(481\) 4.55432 0.207659
\(482\) 13.6512 0.621795
\(483\) 0 0
\(484\) 0 0
\(485\) −8.39529 −0.381211
\(486\) 0 0
\(487\) −8.68493 −0.393552 −0.196776 0.980449i \(-0.563047\pi\)
−0.196776 + 0.980449i \(0.563047\pi\)
\(488\) 11.3447 0.513552
\(489\) 0 0
\(490\) 5.93044 0.267910
\(491\) −33.1973 −1.49817 −0.749087 0.662472i \(-0.769508\pi\)
−0.749087 + 0.662472i \(0.769508\pi\)
\(492\) 0 0
\(493\) 32.0458 1.44327
\(494\) −6.10872 −0.274844
\(495\) 0 0
\(496\) 42.2506 1.89711
\(497\) −8.88678 −0.398627
\(498\) 0 0
\(499\) 20.8769 0.934578 0.467289 0.884105i \(-0.345231\pi\)
0.467289 + 0.884105i \(0.345231\pi\)
\(500\) 2.23107 0.0997765
\(501\) 0 0
\(502\) −31.8087 −1.41969
\(503\) 23.3189 1.03974 0.519869 0.854246i \(-0.325981\pi\)
0.519869 + 0.854246i \(0.325981\pi\)
\(504\) 0 0
\(505\) 8.50663 0.378540
\(506\) 0 0
\(507\) 0 0
\(508\) 4.68502 0.207864
\(509\) −6.48121 −0.287275 −0.143637 0.989630i \(-0.545880\pi\)
−0.143637 + 0.989630i \(0.545880\pi\)
\(510\) 0 0
\(511\) 11.1782 0.494494
\(512\) −12.9480 −0.572225
\(513\) 0 0
\(514\) 5.22963 0.230669
\(515\) 2.62202 0.115540
\(516\) 0 0
\(517\) 0 0
\(518\) −3.22781 −0.141822
\(519\) 0 0
\(520\) 2.32030 0.101752
\(521\) 33.2103 1.45497 0.727484 0.686124i \(-0.240689\pi\)
0.727484 + 0.686124i \(0.240689\pi\)
\(522\) 0 0
\(523\) 40.1386 1.75514 0.877569 0.479451i \(-0.159164\pi\)
0.877569 + 0.479451i \(0.159164\pi\)
\(524\) −2.69123 −0.117567
\(525\) 0 0
\(526\) −22.0823 −0.962834
\(527\) −41.9169 −1.82593
\(528\) 0 0
\(529\) −3.17970 −0.138248
\(530\) −5.22085 −0.226779
\(531\) 0 0
\(532\) 0.707298 0.0306652
\(533\) −1.81770 −0.0787332
\(534\) 0 0
\(535\) 9.32119 0.402990
\(536\) 20.1351 0.869703
\(537\) 0 0
\(538\) −10.3907 −0.447974
\(539\) 0 0
\(540\) 0 0
\(541\) −24.6485 −1.05972 −0.529860 0.848085i \(-0.677756\pi\)
−0.529860 + 0.848085i \(0.677756\pi\)
\(542\) 6.15417 0.264344
\(543\) 0 0
\(544\) −10.0082 −0.429098
\(545\) −8.10908 −0.347355
\(546\) 0 0
\(547\) −25.7204 −1.09972 −0.549861 0.835256i \(-0.685319\pi\)
−0.549861 + 0.835256i \(0.685319\pi\)
\(548\) −3.61113 −0.154260
\(549\) 0 0
\(550\) 0 0
\(551\) −17.5066 −0.745805
\(552\) 0 0
\(553\) −10.5546 −0.448827
\(554\) 20.3039 0.862629
\(555\) 0 0
\(556\) −8.44003 −0.357937
\(557\) 11.6815 0.494962 0.247481 0.968893i \(-0.420397\pi\)
0.247481 + 0.968893i \(0.420397\pi\)
\(558\) 0 0
\(559\) 6.63766 0.280743
\(560\) −1.97833 −0.0835996
\(561\) 0 0
\(562\) −4.58873 −0.193564
\(563\) 8.12210 0.342306 0.171153 0.985244i \(-0.445251\pi\)
0.171153 + 0.985244i \(0.445251\pi\)
\(564\) 0 0
\(565\) 4.75409 0.200006
\(566\) −12.5125 −0.525941
\(567\) 0 0
\(568\) −30.6313 −1.28526
\(569\) −18.0066 −0.754877 −0.377438 0.926035i \(-0.623195\pi\)
−0.377438 + 0.926035i \(0.623195\pi\)
\(570\) 0 0
\(571\) 2.60645 0.109076 0.0545382 0.998512i \(-0.482631\pi\)
0.0545382 + 0.998512i \(0.482631\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 1.28827 0.0537712
\(575\) −20.6996 −0.863231
\(576\) 0 0
\(577\) 35.1888 1.46493 0.732465 0.680805i \(-0.238370\pi\)
0.732465 + 0.680805i \(0.238370\pi\)
\(578\) 6.31827 0.262805
\(579\) 0 0
\(580\) −1.61353 −0.0669984
\(581\) 5.01972 0.208253
\(582\) 0 0
\(583\) 0 0
\(584\) 38.5294 1.59436
\(585\) 0 0
\(586\) −24.2984 −1.00376
\(587\) −23.5524 −0.972112 −0.486056 0.873928i \(-0.661565\pi\)
−0.486056 + 0.873928i \(0.661565\pi\)
\(588\) 0 0
\(589\) 22.8991 0.943541
\(590\) 6.52979 0.268827
\(591\) 0 0
\(592\) −13.3844 −0.550095
\(593\) 13.1828 0.541354 0.270677 0.962670i \(-0.412752\pi\)
0.270677 + 0.962670i \(0.412752\pi\)
\(594\) 0 0
\(595\) 1.96270 0.0804630
\(596\) −0.447726 −0.0183396
\(597\) 0 0
\(598\) 10.8411 0.443326
\(599\) −34.2164 −1.39804 −0.699022 0.715100i \(-0.746381\pi\)
−0.699022 + 0.715100i \(0.746381\pi\)
\(600\) 0 0
\(601\) −29.0285 −1.18410 −0.592049 0.805902i \(-0.701681\pi\)
−0.592049 + 0.805902i \(0.701681\pi\)
\(602\) −4.70435 −0.191735
\(603\) 0 0
\(604\) −0.343924 −0.0139941
\(605\) 0 0
\(606\) 0 0
\(607\) 3.15635 0.128112 0.0640562 0.997946i \(-0.479596\pi\)
0.0640562 + 0.997946i \(0.479596\pi\)
\(608\) 5.46745 0.221735
\(609\) 0 0
\(610\) 4.17310 0.168964
\(611\) 0.358392 0.0144990
\(612\) 0 0
\(613\) −33.9410 −1.37086 −0.685432 0.728136i \(-0.740387\pi\)
−0.685432 + 0.728136i \(0.740387\pi\)
\(614\) 5.43181 0.219210
\(615\) 0 0
\(616\) 0 0
\(617\) 21.6077 0.869895 0.434948 0.900456i \(-0.356767\pi\)
0.434948 + 0.900456i \(0.356767\pi\)
\(618\) 0 0
\(619\) 20.5126 0.824471 0.412235 0.911077i \(-0.364748\pi\)
0.412235 + 0.911077i \(0.364748\pi\)
\(620\) 2.11055 0.0847617
\(621\) 0 0
\(622\) −40.7166 −1.63259
\(623\) 9.01014 0.360984
\(624\) 0 0
\(625\) 19.8653 0.794613
\(626\) −3.80526 −0.152089
\(627\) 0 0
\(628\) −3.82189 −0.152510
\(629\) 13.2787 0.529456
\(630\) 0 0
\(631\) 29.8645 1.18889 0.594443 0.804138i \(-0.297372\pi\)
0.594443 + 0.804138i \(0.297372\pi\)
\(632\) −36.3800 −1.44712
\(633\) 0 0
\(634\) 40.9837 1.62767
\(635\) −7.10220 −0.281842
\(636\) 0 0
\(637\) −10.2039 −0.404292
\(638\) 0 0
\(639\) 0 0
\(640\) −7.96969 −0.315030
\(641\) 38.1201 1.50565 0.752826 0.658220i \(-0.228690\pi\)
0.752826 + 0.658220i \(0.228690\pi\)
\(642\) 0 0
\(643\) −16.8251 −0.663519 −0.331759 0.943364i \(-0.607642\pi\)
−0.331759 + 0.943364i \(0.607642\pi\)
\(644\) −1.25524 −0.0494633
\(645\) 0 0
\(646\) −17.8107 −0.700753
\(647\) 20.9745 0.824592 0.412296 0.911050i \(-0.364727\pi\)
0.412296 + 0.911050i \(0.364727\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −11.3220 −0.444087
\(651\) 0 0
\(652\) 1.80810 0.0708106
\(653\) −19.7665 −0.773522 −0.386761 0.922180i \(-0.626406\pi\)
−0.386761 + 0.922180i \(0.626406\pi\)
\(654\) 0 0
\(655\) 4.07975 0.159409
\(656\) 5.34191 0.208566
\(657\) 0 0
\(658\) −0.254005 −0.00990215
\(659\) −11.8055 −0.459879 −0.229939 0.973205i \(-0.573853\pi\)
−0.229939 + 0.973205i \(0.573853\pi\)
\(660\) 0 0
\(661\) 17.8076 0.692637 0.346318 0.938117i \(-0.387432\pi\)
0.346318 + 0.938117i \(0.387432\pi\)
\(662\) 12.5298 0.486986
\(663\) 0 0
\(664\) 17.3022 0.671455
\(665\) −1.07222 −0.0415789
\(666\) 0 0
\(667\) 31.0688 1.20299
\(668\) −6.63422 −0.256686
\(669\) 0 0
\(670\) 7.40658 0.286141
\(671\) 0 0
\(672\) 0 0
\(673\) −19.7053 −0.759583 −0.379791 0.925072i \(-0.624004\pi\)
−0.379791 + 0.925072i \(0.624004\pi\)
\(674\) −38.2215 −1.47224
\(675\) 0 0
\(676\) −4.10826 −0.158010
\(677\) −12.4139 −0.477105 −0.238553 0.971130i \(-0.576673\pi\)
−0.238553 + 0.971130i \(0.576673\pi\)
\(678\) 0 0
\(679\) −10.2376 −0.392883
\(680\) 6.76512 0.259431
\(681\) 0 0
\(682\) 0 0
\(683\) −2.02837 −0.0776135 −0.0388068 0.999247i \(-0.512356\pi\)
−0.0388068 + 0.999247i \(0.512356\pi\)
\(684\) 0 0
\(685\) 5.47425 0.209160
\(686\) 15.0455 0.574439
\(687\) 0 0
\(688\) −19.5070 −0.743696
\(689\) 8.98296 0.342223
\(690\) 0 0
\(691\) 2.34287 0.0891270 0.0445635 0.999007i \(-0.485810\pi\)
0.0445635 + 0.999007i \(0.485810\pi\)
\(692\) −0.230830 −0.00877483
\(693\) 0 0
\(694\) −3.19995 −0.121469
\(695\) 12.7946 0.485326
\(696\) 0 0
\(697\) −5.29972 −0.200741
\(698\) 8.06197 0.305150
\(699\) 0 0
\(700\) 1.31092 0.0495482
\(701\) −4.43368 −0.167458 −0.0837289 0.996489i \(-0.526683\pi\)
−0.0837289 + 0.996489i \(0.526683\pi\)
\(702\) 0 0
\(703\) −7.25411 −0.273594
\(704\) 0 0
\(705\) 0 0
\(706\) −33.5087 −1.26111
\(707\) 10.3734 0.390131
\(708\) 0 0
\(709\) 29.8825 1.12226 0.561131 0.827727i \(-0.310366\pi\)
0.561131 + 0.827727i \(0.310366\pi\)
\(710\) −11.2676 −0.422864
\(711\) 0 0
\(712\) 31.0565 1.16389
\(713\) −40.6389 −1.52194
\(714\) 0 0
\(715\) 0 0
\(716\) 3.76515 0.140710
\(717\) 0 0
\(718\) 13.5331 0.505051
\(719\) 21.6321 0.806742 0.403371 0.915037i \(-0.367838\pi\)
0.403371 + 0.915037i \(0.367838\pi\)
\(720\) 0 0
\(721\) 3.19741 0.119078
\(722\) −19.6466 −0.731172
\(723\) 0 0
\(724\) −1.85969 −0.0691149
\(725\) −32.4471 −1.20505
\(726\) 0 0
\(727\) −12.4422 −0.461455 −0.230727 0.973018i \(-0.574111\pi\)
−0.230727 + 0.973018i \(0.574111\pi\)
\(728\) 2.82948 0.104868
\(729\) 0 0
\(730\) 14.1729 0.524561
\(731\) 19.3529 0.715793
\(732\) 0 0
\(733\) 52.3522 1.93367 0.966836 0.255396i \(-0.0822060\pi\)
0.966836 + 0.255396i \(0.0822060\pi\)
\(734\) −10.4615 −0.386141
\(735\) 0 0
\(736\) −9.70306 −0.357660
\(737\) 0 0
\(738\) 0 0
\(739\) 46.6760 1.71701 0.858503 0.512809i \(-0.171395\pi\)
0.858503 + 0.512809i \(0.171395\pi\)
\(740\) −0.668592 −0.0245779
\(741\) 0 0
\(742\) −6.36655 −0.233723
\(743\) −21.8719 −0.802401 −0.401201 0.915990i \(-0.631407\pi\)
−0.401201 + 0.915990i \(0.631407\pi\)
\(744\) 0 0
\(745\) 0.678725 0.0248665
\(746\) 33.8501 1.23934
\(747\) 0 0
\(748\) 0 0
\(749\) 11.3667 0.415330
\(750\) 0 0
\(751\) 39.5921 1.44474 0.722369 0.691508i \(-0.243053\pi\)
0.722369 + 0.691508i \(0.243053\pi\)
\(752\) −1.05325 −0.0384082
\(753\) 0 0
\(754\) 16.9937 0.618874
\(755\) 0.521367 0.0189745
\(756\) 0 0
\(757\) 23.1457 0.841246 0.420623 0.907235i \(-0.361811\pi\)
0.420623 + 0.907235i \(0.361811\pi\)
\(758\) −52.2959 −1.89947
\(759\) 0 0
\(760\) −3.69577 −0.134060
\(761\) −14.8929 −0.539866 −0.269933 0.962879i \(-0.587002\pi\)
−0.269933 + 0.962879i \(0.587002\pi\)
\(762\) 0 0
\(763\) −9.88858 −0.357991
\(764\) 2.03003 0.0734438
\(765\) 0 0
\(766\) 28.4085 1.02644
\(767\) −11.2351 −0.405676
\(768\) 0 0
\(769\) −21.6233 −0.779757 −0.389879 0.920866i \(-0.627483\pi\)
−0.389879 + 0.920866i \(0.627483\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.27630 −0.0459349
\(773\) 6.07292 0.218428 0.109214 0.994018i \(-0.465167\pi\)
0.109214 + 0.994018i \(0.465167\pi\)
\(774\) 0 0
\(775\) 42.4417 1.52455
\(776\) −35.2874 −1.26674
\(777\) 0 0
\(778\) 47.6311 1.70766
\(779\) 2.89522 0.103732
\(780\) 0 0
\(781\) 0 0
\(782\) 31.6086 1.13032
\(783\) 0 0
\(784\) 29.9874 1.07098
\(785\) 5.79375 0.206788
\(786\) 0 0
\(787\) −51.1554 −1.82349 −0.911746 0.410754i \(-0.865265\pi\)
−0.911746 + 0.410754i \(0.865265\pi\)
\(788\) −8.77250 −0.312507
\(789\) 0 0
\(790\) −13.3822 −0.476117
\(791\) 5.79736 0.206130
\(792\) 0 0
\(793\) −7.18019 −0.254976
\(794\) −12.8264 −0.455192
\(795\) 0 0
\(796\) −8.13457 −0.288322
\(797\) 25.2612 0.894798 0.447399 0.894334i \(-0.352350\pi\)
0.447399 + 0.894334i \(0.352350\pi\)
\(798\) 0 0
\(799\) 1.04494 0.0369672
\(800\) 10.1335 0.358274
\(801\) 0 0
\(802\) 6.08192 0.214760
\(803\) 0 0
\(804\) 0 0
\(805\) 1.90286 0.0670671
\(806\) −22.2283 −0.782957
\(807\) 0 0
\(808\) 35.7554 1.25787
\(809\) 51.3275 1.80458 0.902290 0.431130i \(-0.141885\pi\)
0.902290 + 0.431130i \(0.141885\pi\)
\(810\) 0 0
\(811\) −4.54589 −0.159628 −0.0798138 0.996810i \(-0.525433\pi\)
−0.0798138 + 0.996810i \(0.525433\pi\)
\(812\) −1.96762 −0.0690498
\(813\) 0 0
\(814\) 0 0
\(815\) −2.74097 −0.0960119
\(816\) 0 0
\(817\) −10.5724 −0.369883
\(818\) 3.87192 0.135378
\(819\) 0 0
\(820\) 0.266845 0.00931864
\(821\) 10.8738 0.379498 0.189749 0.981833i \(-0.439233\pi\)
0.189749 + 0.981833i \(0.439233\pi\)
\(822\) 0 0
\(823\) −23.6161 −0.823207 −0.411604 0.911363i \(-0.635031\pi\)
−0.411604 + 0.911363i \(0.635031\pi\)
\(824\) 11.0210 0.383934
\(825\) 0 0
\(826\) 7.96271 0.277058
\(827\) −27.6040 −0.959886 −0.479943 0.877300i \(-0.659343\pi\)
−0.479943 + 0.877300i \(0.659343\pi\)
\(828\) 0 0
\(829\) −16.1009 −0.559208 −0.279604 0.960115i \(-0.590203\pi\)
−0.279604 + 0.960115i \(0.590203\pi\)
\(830\) 6.36452 0.220916
\(831\) 0 0
\(832\) 9.27234 0.321461
\(833\) −29.7506 −1.03080
\(834\) 0 0
\(835\) 10.0571 0.348039
\(836\) 0 0
\(837\) 0 0
\(838\) −16.6046 −0.573596
\(839\) 19.4319 0.670865 0.335432 0.942064i \(-0.391118\pi\)
0.335432 + 0.942064i \(0.391118\pi\)
\(840\) 0 0
\(841\) 19.7012 0.679350
\(842\) 34.3048 1.18222
\(843\) 0 0
\(844\) −1.32471 −0.0455986
\(845\) 6.22787 0.214245
\(846\) 0 0
\(847\) 0 0
\(848\) −26.3994 −0.906559
\(849\) 0 0
\(850\) −33.0108 −1.13226
\(851\) 12.8738 0.441309
\(852\) 0 0
\(853\) 19.7855 0.677443 0.338722 0.940887i \(-0.390006\pi\)
0.338722 + 0.940887i \(0.390006\pi\)
\(854\) 5.08886 0.174137
\(855\) 0 0
\(856\) 39.1791 1.33912
\(857\) −4.34065 −0.148274 −0.0741369 0.997248i \(-0.523620\pi\)
−0.0741369 + 0.997248i \(0.523620\pi\)
\(858\) 0 0
\(859\) −5.17935 −0.176717 −0.0883586 0.996089i \(-0.528162\pi\)
−0.0883586 + 0.996089i \(0.528162\pi\)
\(860\) −0.974434 −0.0332279
\(861\) 0 0
\(862\) −28.5389 −0.972039
\(863\) 2.69392 0.0917021 0.0458510 0.998948i \(-0.485400\pi\)
0.0458510 + 0.998948i \(0.485400\pi\)
\(864\) 0 0
\(865\) 0.349924 0.0118978
\(866\) 23.3724 0.794226
\(867\) 0 0
\(868\) 2.57370 0.0873571
\(869\) 0 0
\(870\) 0 0
\(871\) −12.7437 −0.431804
\(872\) −34.0843 −1.15424
\(873\) 0 0
\(874\) −17.2677 −0.584088
\(875\) −4.12437 −0.139429
\(876\) 0 0
\(877\) 20.8448 0.703878 0.351939 0.936023i \(-0.385522\pi\)
0.351939 + 0.936023i \(0.385522\pi\)
\(878\) −37.0382 −1.24998
\(879\) 0 0
\(880\) 0 0
\(881\) −11.5843 −0.390286 −0.195143 0.980775i \(-0.562517\pi\)
−0.195143 + 0.980775i \(0.562517\pi\)
\(882\) 0 0
\(883\) 15.3973 0.518159 0.259079 0.965856i \(-0.416581\pi\)
0.259079 + 0.965856i \(0.416581\pi\)
\(884\) 2.82447 0.0949970
\(885\) 0 0
\(886\) −39.4229 −1.32444
\(887\) −5.90996 −0.198437 −0.0992185 0.995066i \(-0.531634\pi\)
−0.0992185 + 0.995066i \(0.531634\pi\)
\(888\) 0 0
\(889\) −8.66074 −0.290472
\(890\) 11.4240 0.382932
\(891\) 0 0
\(892\) −4.30623 −0.144183
\(893\) −0.570846 −0.0191026
\(894\) 0 0
\(895\) −5.70774 −0.190789
\(896\) −9.71860 −0.324676
\(897\) 0 0
\(898\) 34.7609 1.15999
\(899\) −63.7025 −2.12460
\(900\) 0 0
\(901\) 26.1909 0.872546
\(902\) 0 0
\(903\) 0 0
\(904\) 19.9826 0.664610
\(905\) 2.81918 0.0937127
\(906\) 0 0
\(907\) −7.39332 −0.245491 −0.122746 0.992438i \(-0.539170\pi\)
−0.122746 + 0.992438i \(0.539170\pi\)
\(908\) −2.33030 −0.0773336
\(909\) 0 0
\(910\) 1.04081 0.0345025
\(911\) 14.1988 0.470427 0.235214 0.971944i \(-0.424421\pi\)
0.235214 + 0.971944i \(0.424421\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −33.5350 −1.10924
\(915\) 0 0
\(916\) 9.66608 0.319376
\(917\) 4.97503 0.164290
\(918\) 0 0
\(919\) 41.6426 1.37366 0.686831 0.726817i \(-0.259001\pi\)
0.686831 + 0.726817i \(0.259001\pi\)
\(920\) 6.55887 0.216239
\(921\) 0 0
\(922\) −18.7139 −0.616308
\(923\) 19.3869 0.638127
\(924\) 0 0
\(925\) −13.4449 −0.442067
\(926\) 30.9703 1.01775
\(927\) 0 0
\(928\) −15.2098 −0.499286
\(929\) 34.1328 1.11986 0.559930 0.828540i \(-0.310828\pi\)
0.559930 + 0.828540i \(0.310828\pi\)
\(930\) 0 0
\(931\) 16.2527 0.532660
\(932\) 1.63574 0.0535803
\(933\) 0 0
\(934\) −9.38118 −0.306962
\(935\) 0 0
\(936\) 0 0
\(937\) −51.6822 −1.68838 −0.844192 0.536041i \(-0.819919\pi\)
−0.844192 + 0.536041i \(0.819919\pi\)
\(938\) 9.03192 0.294903
\(939\) 0 0
\(940\) −0.0526133 −0.00171606
\(941\) 18.0062 0.586985 0.293492 0.955961i \(-0.405182\pi\)
0.293492 + 0.955961i \(0.405182\pi\)
\(942\) 0 0
\(943\) −5.13814 −0.167321
\(944\) 33.0181 1.07465
\(945\) 0 0
\(946\) 0 0
\(947\) −28.6303 −0.930360 −0.465180 0.885216i \(-0.654010\pi\)
−0.465180 + 0.885216i \(0.654010\pi\)
\(948\) 0 0
\(949\) −24.3857 −0.791593
\(950\) 18.0337 0.585091
\(951\) 0 0
\(952\) 8.24970 0.267374
\(953\) 16.6893 0.540620 0.270310 0.962773i \(-0.412874\pi\)
0.270310 + 0.962773i \(0.412874\pi\)
\(954\) 0 0
\(955\) −3.07740 −0.0995822
\(956\) −2.87938 −0.0931257
\(957\) 0 0
\(958\) 56.0827 1.81195
\(959\) 6.67555 0.215565
\(960\) 0 0
\(961\) 52.3248 1.68790
\(962\) 7.04160 0.227030
\(963\) 0 0
\(964\) 3.44815 0.111058
\(965\) 1.93479 0.0622830
\(966\) 0 0
\(967\) −24.3193 −0.782057 −0.391029 0.920379i \(-0.627881\pi\)
−0.391029 + 0.920379i \(0.627881\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −12.9803 −0.416771
\(971\) −7.55191 −0.242352 −0.121176 0.992631i \(-0.538667\pi\)
−0.121176 + 0.992631i \(0.538667\pi\)
\(972\) 0 0
\(973\) 15.6023 0.500186
\(974\) −13.4281 −0.430264
\(975\) 0 0
\(976\) 21.1014 0.675439
\(977\) −2.03229 −0.0650186 −0.0325093 0.999471i \(-0.510350\pi\)
−0.0325093 + 0.999471i \(0.510350\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.49797 0.0478508
\(981\) 0 0
\(982\) −51.3276 −1.63793
\(983\) −33.4396 −1.06656 −0.533279 0.845940i \(-0.679040\pi\)
−0.533279 + 0.845940i \(0.679040\pi\)
\(984\) 0 0
\(985\) 13.2986 0.423728
\(986\) 49.5473 1.57791
\(987\) 0 0
\(988\) −1.54300 −0.0490893
\(989\) 18.7629 0.596624
\(990\) 0 0
\(991\) 40.1009 1.27385 0.636923 0.770927i \(-0.280207\pi\)
0.636923 + 0.770927i \(0.280207\pi\)
\(992\) 19.8949 0.631662
\(993\) 0 0
\(994\) −13.7402 −0.435812
\(995\) 12.3315 0.390935
\(996\) 0 0
\(997\) −16.1301 −0.510845 −0.255423 0.966829i \(-0.582215\pi\)
−0.255423 + 0.966829i \(0.582215\pi\)
\(998\) 32.2785 1.02176
\(999\) 0 0
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 9801.2.a.cn.1.15 18
3.2 odd 2 9801.2.a.co.1.4 18
9.2 odd 6 1089.2.e.o.364.15 36
9.5 odd 6 1089.2.e.o.727.15 36
11.2 odd 10 891.2.f.e.730.8 36
11.6 odd 10 891.2.f.e.487.8 36
11.10 odd 2 9801.2.a.cp.1.4 18
33.2 even 10 891.2.f.f.730.2 36
33.17 even 10 891.2.f.f.487.2 36
33.32 even 2 9801.2.a.cm.1.15 18
99.2 even 30 99.2.m.b.4.2 72
99.13 odd 30 297.2.n.b.235.2 72
99.32 even 6 1089.2.e.p.727.4 36
99.50 even 30 99.2.m.b.25.2 yes 72
99.61 odd 30 297.2.n.b.91.2 72
99.65 even 6 1089.2.e.p.364.4 36
99.68 even 30 99.2.m.b.70.8 yes 72
99.79 odd 30 297.2.n.b.37.8 72
99.83 even 30 99.2.m.b.58.8 yes 72
99.94 odd 30 297.2.n.b.289.8 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.m.b.4.2 72 99.2 even 30
99.2.m.b.25.2 yes 72 99.50 even 30
99.2.m.b.58.8 yes 72 99.83 even 30
99.2.m.b.70.8 yes 72 99.68 even 30
297.2.n.b.37.8 72 99.79 odd 30
297.2.n.b.91.2 72 99.61 odd 30
297.2.n.b.235.2 72 99.13 odd 30
297.2.n.b.289.8 72 99.94 odd 30
891.2.f.e.487.8 36 11.6 odd 10
891.2.f.e.730.8 36 11.2 odd 10
891.2.f.f.487.2 36 33.17 even 10
891.2.f.f.730.2 36 33.2 even 10
1089.2.e.o.364.15 36 9.2 odd 6
1089.2.e.o.727.15 36 9.5 odd 6
1089.2.e.p.364.4 36 99.65 even 6
1089.2.e.p.727.4 36 99.32 even 6
9801.2.a.cm.1.15 18 33.32 even 2
9801.2.a.cn.1.15 18 1.1 even 1 trivial
9801.2.a.co.1.4 18 3.2 odd 2
9801.2.a.cp.1.4 18 11.10 odd 2