Properties

Label 9801.2.a.cp.1.10
Level $9801$
Weight $2$
Character 9801.1
Self dual yes
Analytic conductor $78.261$
Analytic rank $0$
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: \(0\)
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 1.10
Root \(0.490494\) 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)
 
\(f(q)\) \(=\) \(q+0.490494 q^{2} -1.75942 q^{4} -1.70818 q^{5} +3.41052 q^{7} -1.84397 q^{8} -0.837851 q^{10} -3.99792 q^{13} +1.67284 q^{14} +2.61437 q^{16} +3.30458 q^{17} -4.41765 q^{19} +3.00539 q^{20} -7.49201 q^{23} -2.08213 q^{25} -1.96096 q^{26} -6.00053 q^{28} -1.69129 q^{29} -3.89852 q^{31} +4.97028 q^{32} +1.62087 q^{34} -5.82578 q^{35} +3.06553 q^{37} -2.16683 q^{38} +3.14983 q^{40} -0.390875 q^{41} +9.61268 q^{43} -3.67479 q^{46} +1.08026 q^{47} +4.63167 q^{49} -1.02127 q^{50} +7.03401 q^{52} +12.5638 q^{53} -6.28890 q^{56} -0.829567 q^{58} +3.65654 q^{59} -3.85318 q^{61} -1.91220 q^{62} -2.79086 q^{64} +6.82916 q^{65} +3.10120 q^{67} -5.81412 q^{68} -2.85751 q^{70} -7.01715 q^{71} -15.0426 q^{73} +1.50362 q^{74} +7.77248 q^{76} -3.99913 q^{79} -4.46581 q^{80} -0.191722 q^{82} +0.124966 q^{83} -5.64480 q^{85} +4.71496 q^{86} +7.93327 q^{89} -13.6350 q^{91} +13.1816 q^{92} +0.529860 q^{94} +7.54613 q^{95} +0.343160 q^{97} +2.27180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 2 q^{2} + 12 q^{4} + q^{5} + q^{7} + 6 q^{8} - 2 q^{10} + 3 q^{13} - 8 q^{16} + 20 q^{17} - 3 q^{19} + 5 q^{20} + 10 q^{23} + 7 q^{25} - 2 q^{26} + 19 q^{28} + 21 q^{29} + 6 q^{31} + 9 q^{32} - 4 q^{34}+ \cdots + 82 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.490494 0.346832 0.173416 0.984849i \(-0.444520\pi\)
0.173416 + 0.984849i \(0.444520\pi\)
\(3\) 0 0
\(4\) −1.75942 −0.879708
\(5\) −1.70818 −0.763920 −0.381960 0.924179i \(-0.624751\pi\)
−0.381960 + 0.924179i \(0.624751\pi\)
\(6\) 0 0
\(7\) 3.41052 1.28906 0.644528 0.764581i \(-0.277054\pi\)
0.644528 + 0.764581i \(0.277054\pi\)
\(8\) −1.84397 −0.651942
\(9\) 0 0
\(10\) −0.837851 −0.264952
\(11\) 0 0
\(12\) 0 0
\(13\) −3.99792 −1.10882 −0.554412 0.832242i \(-0.687057\pi\)
−0.554412 + 0.832242i \(0.687057\pi\)
\(14\) 1.67284 0.447086
\(15\) 0 0
\(16\) 2.61437 0.653594
\(17\) 3.30458 0.801477 0.400739 0.916192i \(-0.368754\pi\)
0.400739 + 0.916192i \(0.368754\pi\)
\(18\) 0 0
\(19\) −4.41765 −1.01348 −0.506739 0.862099i \(-0.669149\pi\)
−0.506739 + 0.862099i \(0.669149\pi\)
\(20\) 3.00539 0.672026
\(21\) 0 0
\(22\) 0 0
\(23\) −7.49201 −1.56219 −0.781096 0.624411i \(-0.785339\pi\)
−0.781096 + 0.624411i \(0.785339\pi\)
\(24\) 0 0
\(25\) −2.08213 −0.416426
\(26\) −1.96096 −0.384575
\(27\) 0 0
\(28\) −6.00053 −1.13399
\(29\) −1.69129 −0.314064 −0.157032 0.987593i \(-0.550193\pi\)
−0.157032 + 0.987593i \(0.550193\pi\)
\(30\) 0 0
\(31\) −3.89852 −0.700194 −0.350097 0.936713i \(-0.613851\pi\)
−0.350097 + 0.936713i \(0.613851\pi\)
\(32\) 4.97028 0.878629
\(33\) 0 0
\(34\) 1.62087 0.277978
\(35\) −5.82578 −0.984736
\(36\) 0 0
\(37\) 3.06553 0.503970 0.251985 0.967731i \(-0.418917\pi\)
0.251985 + 0.967731i \(0.418917\pi\)
\(38\) −2.16683 −0.351506
\(39\) 0 0
\(40\) 3.14983 0.498032
\(41\) −0.390875 −0.0610444 −0.0305222 0.999534i \(-0.509717\pi\)
−0.0305222 + 0.999534i \(0.509717\pi\)
\(42\) 0 0
\(43\) 9.61268 1.46592 0.732960 0.680272i \(-0.238138\pi\)
0.732960 + 0.680272i \(0.238138\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −3.67479 −0.541818
\(47\) 1.08026 0.157572 0.0787859 0.996892i \(-0.474896\pi\)
0.0787859 + 0.996892i \(0.474896\pi\)
\(48\) 0 0
\(49\) 4.63167 0.661666
\(50\) −1.02127 −0.144430
\(51\) 0 0
\(52\) 7.03401 0.975442
\(53\) 12.5638 1.72577 0.862884 0.505402i \(-0.168656\pi\)
0.862884 + 0.505402i \(0.168656\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −6.28890 −0.840390
\(57\) 0 0
\(58\) −0.829567 −0.108927
\(59\) 3.65654 0.476041 0.238021 0.971260i \(-0.423501\pi\)
0.238021 + 0.971260i \(0.423501\pi\)
\(60\) 0 0
\(61\) −3.85318 −0.493349 −0.246674 0.969098i \(-0.579338\pi\)
−0.246674 + 0.969098i \(0.579338\pi\)
\(62\) −1.91220 −0.242849
\(63\) 0 0
\(64\) −2.79086 −0.348857
\(65\) 6.82916 0.847053
\(66\) 0 0
\(67\) 3.10120 0.378872 0.189436 0.981893i \(-0.439334\pi\)
0.189436 + 0.981893i \(0.439334\pi\)
\(68\) −5.81412 −0.705066
\(69\) 0 0
\(70\) −2.85751 −0.341538
\(71\) −7.01715 −0.832783 −0.416391 0.909186i \(-0.636705\pi\)
−0.416391 + 0.909186i \(0.636705\pi\)
\(72\) 0 0
\(73\) −15.0426 −1.76060 −0.880302 0.474413i \(-0.842660\pi\)
−0.880302 + 0.474413i \(0.842660\pi\)
\(74\) 1.50362 0.174793
\(75\) 0 0
\(76\) 7.77248 0.891565
\(77\) 0 0
\(78\) 0 0
\(79\) −3.99913 −0.449937 −0.224969 0.974366i \(-0.572228\pi\)
−0.224969 + 0.974366i \(0.572228\pi\)
\(80\) −4.46581 −0.499293
\(81\) 0 0
\(82\) −0.191722 −0.0211721
\(83\) 0.124966 0.0137168 0.00685839 0.999976i \(-0.497817\pi\)
0.00685839 + 0.999976i \(0.497817\pi\)
\(84\) 0 0
\(85\) −5.64480 −0.612265
\(86\) 4.71496 0.508428
\(87\) 0 0
\(88\) 0 0
\(89\) 7.93327 0.840925 0.420462 0.907310i \(-0.361868\pi\)
0.420462 + 0.907310i \(0.361868\pi\)
\(90\) 0 0
\(91\) −13.6350 −1.42934
\(92\) 13.1816 1.37427
\(93\) 0 0
\(94\) 0.529860 0.0546509
\(95\) 7.54613 0.774217
\(96\) 0 0
\(97\) 0.343160 0.0348426 0.0174213 0.999848i \(-0.494454\pi\)
0.0174213 + 0.999848i \(0.494454\pi\)
\(98\) 2.27180 0.229487
\(99\) 0 0
\(100\) 3.66333 0.366333
\(101\) −7.11558 −0.708027 −0.354014 0.935240i \(-0.615183\pi\)
−0.354014 + 0.935240i \(0.615183\pi\)
\(102\) 0 0
\(103\) −1.65429 −0.163002 −0.0815009 0.996673i \(-0.525971\pi\)
−0.0815009 + 0.996673i \(0.525971\pi\)
\(104\) 7.37205 0.722889
\(105\) 0 0
\(106\) 6.16246 0.598551
\(107\) −3.30683 −0.319683 −0.159842 0.987143i \(-0.551098\pi\)
−0.159842 + 0.987143i \(0.551098\pi\)
\(108\) 0 0
\(109\) −5.50709 −0.527484 −0.263742 0.964593i \(-0.584957\pi\)
−0.263742 + 0.964593i \(0.584957\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 8.91638 0.842519
\(113\) 16.4092 1.54365 0.771825 0.635835i \(-0.219344\pi\)
0.771825 + 0.635835i \(0.219344\pi\)
\(114\) 0 0
\(115\) 12.7977 1.19339
\(116\) 2.97568 0.276285
\(117\) 0 0
\(118\) 1.79351 0.165106
\(119\) 11.2703 1.03315
\(120\) 0 0
\(121\) 0 0
\(122\) −1.88996 −0.171109
\(123\) 0 0
\(124\) 6.85911 0.615966
\(125\) 12.0975 1.08204
\(126\) 0 0
\(127\) 10.7949 0.957894 0.478947 0.877844i \(-0.341019\pi\)
0.478947 + 0.877844i \(0.341019\pi\)
\(128\) −11.3095 −0.999624
\(129\) 0 0
\(130\) 3.34966 0.293785
\(131\) 21.1557 1.84838 0.924189 0.381935i \(-0.124742\pi\)
0.924189 + 0.381935i \(0.124742\pi\)
\(132\) 0 0
\(133\) −15.0665 −1.30643
\(134\) 1.52112 0.131405
\(135\) 0 0
\(136\) −6.09354 −0.522517
\(137\) −3.20052 −0.273439 −0.136720 0.990610i \(-0.543656\pi\)
−0.136720 + 0.990610i \(0.543656\pi\)
\(138\) 0 0
\(139\) −14.9936 −1.27174 −0.635872 0.771794i \(-0.719360\pi\)
−0.635872 + 0.771794i \(0.719360\pi\)
\(140\) 10.2500 0.866280
\(141\) 0 0
\(142\) −3.44187 −0.288835
\(143\) 0 0
\(144\) 0 0
\(145\) 2.88902 0.239920
\(146\) −7.37831 −0.610633
\(147\) 0 0
\(148\) −5.39354 −0.443346
\(149\) −4.05112 −0.331881 −0.165940 0.986136i \(-0.553066\pi\)
−0.165940 + 0.986136i \(0.553066\pi\)
\(150\) 0 0
\(151\) −7.79399 −0.634266 −0.317133 0.948381i \(-0.602720\pi\)
−0.317133 + 0.948381i \(0.602720\pi\)
\(152\) 8.14602 0.660729
\(153\) 0 0
\(154\) 0 0
\(155\) 6.65936 0.534892
\(156\) 0 0
\(157\) 0.309152 0.0246730 0.0123365 0.999924i \(-0.496073\pi\)
0.0123365 + 0.999924i \(0.496073\pi\)
\(158\) −1.96155 −0.156052
\(159\) 0 0
\(160\) −8.49011 −0.671202
\(161\) −25.5517 −2.01375
\(162\) 0 0
\(163\) 10.9941 0.861125 0.430563 0.902561i \(-0.358315\pi\)
0.430563 + 0.902561i \(0.358315\pi\)
\(164\) 0.687711 0.0537012
\(165\) 0 0
\(166\) 0.0612950 0.00475741
\(167\) 17.5739 1.35991 0.679953 0.733255i \(-0.262000\pi\)
0.679953 + 0.733255i \(0.262000\pi\)
\(168\) 0 0
\(169\) 2.98339 0.229492
\(170\) −2.76874 −0.212353
\(171\) 0 0
\(172\) −16.9127 −1.28958
\(173\) −12.3657 −0.940150 −0.470075 0.882627i \(-0.655773\pi\)
−0.470075 + 0.882627i \(0.655773\pi\)
\(174\) 0 0
\(175\) −7.10116 −0.536797
\(176\) 0 0
\(177\) 0 0
\(178\) 3.89122 0.291659
\(179\) −21.8823 −1.63556 −0.817780 0.575531i \(-0.804795\pi\)
−0.817780 + 0.575531i \(0.804795\pi\)
\(180\) 0 0
\(181\) −4.38018 −0.325576 −0.162788 0.986661i \(-0.552049\pi\)
−0.162788 + 0.986661i \(0.552049\pi\)
\(182\) −6.68789 −0.495739
\(183\) 0 0
\(184\) 13.8150 1.01846
\(185\) −5.23647 −0.384993
\(186\) 0 0
\(187\) 0 0
\(188\) −1.90062 −0.138617
\(189\) 0 0
\(190\) 3.70133 0.268523
\(191\) −12.5147 −0.905534 −0.452767 0.891629i \(-0.649563\pi\)
−0.452767 + 0.891629i \(0.649563\pi\)
\(192\) 0 0
\(193\) −12.5657 −0.904500 −0.452250 0.891891i \(-0.649378\pi\)
−0.452250 + 0.891891i \(0.649378\pi\)
\(194\) 0.168318 0.0120845
\(195\) 0 0
\(196\) −8.14902 −0.582073
\(197\) 10.3453 0.737075 0.368538 0.929613i \(-0.379859\pi\)
0.368538 + 0.929613i \(0.379859\pi\)
\(198\) 0 0
\(199\) 26.4773 1.87693 0.938464 0.345376i \(-0.112249\pi\)
0.938464 + 0.345376i \(0.112249\pi\)
\(200\) 3.83939 0.271486
\(201\) 0 0
\(202\) −3.49015 −0.245566
\(203\) −5.76818 −0.404847
\(204\) 0 0
\(205\) 0.667683 0.0466330
\(206\) −0.811418 −0.0565342
\(207\) 0 0
\(208\) −10.4521 −0.724721
\(209\) 0 0
\(210\) 0 0
\(211\) −7.28390 −0.501444 −0.250722 0.968059i \(-0.580668\pi\)
−0.250722 + 0.968059i \(0.580668\pi\)
\(212\) −22.1049 −1.51817
\(213\) 0 0
\(214\) −1.62198 −0.110876
\(215\) −16.4202 −1.11985
\(216\) 0 0
\(217\) −13.2960 −0.902590
\(218\) −2.70120 −0.182948
\(219\) 0 0
\(220\) 0 0
\(221\) −13.2114 −0.888698
\(222\) 0 0
\(223\) 23.5101 1.57435 0.787175 0.616730i \(-0.211543\pi\)
0.787175 + 0.616730i \(0.211543\pi\)
\(224\) 16.9512 1.13260
\(225\) 0 0
\(226\) 8.04863 0.535387
\(227\) 11.9222 0.791303 0.395652 0.918401i \(-0.370519\pi\)
0.395652 + 0.918401i \(0.370519\pi\)
\(228\) 0 0
\(229\) 3.05757 0.202050 0.101025 0.994884i \(-0.467788\pi\)
0.101025 + 0.994884i \(0.467788\pi\)
\(230\) 6.27719 0.413905
\(231\) 0 0
\(232\) 3.11869 0.204752
\(233\) 24.5314 1.60710 0.803552 0.595235i \(-0.202941\pi\)
0.803552 + 0.595235i \(0.202941\pi\)
\(234\) 0 0
\(235\) −1.84527 −0.120372
\(236\) −6.43338 −0.418777
\(237\) 0 0
\(238\) 5.52803 0.358329
\(239\) 6.95063 0.449599 0.224799 0.974405i \(-0.427827\pi\)
0.224799 + 0.974405i \(0.427827\pi\)
\(240\) 0 0
\(241\) −7.77059 −0.500547 −0.250274 0.968175i \(-0.580521\pi\)
−0.250274 + 0.968175i \(0.580521\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 6.77934 0.434003
\(245\) −7.91170 −0.505460
\(246\) 0 0
\(247\) 17.6614 1.12377
\(248\) 7.18875 0.456486
\(249\) 0 0
\(250\) 5.93377 0.375284
\(251\) −0.327168 −0.0206506 −0.0103253 0.999947i \(-0.503287\pi\)
−0.0103253 + 0.999947i \(0.503287\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 5.29484 0.332228
\(255\) 0 0
\(256\) 0.0344986 0.00215616
\(257\) 13.7292 0.856402 0.428201 0.903684i \(-0.359148\pi\)
0.428201 + 0.903684i \(0.359148\pi\)
\(258\) 0 0
\(259\) 10.4551 0.649645
\(260\) −12.0153 −0.745159
\(261\) 0 0
\(262\) 10.3767 0.641076
\(263\) 16.8542 1.03927 0.519637 0.854387i \(-0.326067\pi\)
0.519637 + 0.854387i \(0.326067\pi\)
\(264\) 0 0
\(265\) −21.4612 −1.31835
\(266\) −7.39003 −0.453112
\(267\) 0 0
\(268\) −5.45630 −0.333297
\(269\) −12.3974 −0.755882 −0.377941 0.925830i \(-0.623368\pi\)
−0.377941 + 0.925830i \(0.623368\pi\)
\(270\) 0 0
\(271\) 19.2695 1.17054 0.585269 0.810839i \(-0.300989\pi\)
0.585269 + 0.810839i \(0.300989\pi\)
\(272\) 8.63940 0.523840
\(273\) 0 0
\(274\) −1.56984 −0.0948374
\(275\) 0 0
\(276\) 0 0
\(277\) 10.3511 0.621936 0.310968 0.950420i \(-0.399347\pi\)
0.310968 + 0.950420i \(0.399347\pi\)
\(278\) −7.35429 −0.441081
\(279\) 0 0
\(280\) 10.7426 0.641991
\(281\) 16.4358 0.980478 0.490239 0.871588i \(-0.336910\pi\)
0.490239 + 0.871588i \(0.336910\pi\)
\(282\) 0 0
\(283\) −1.26495 −0.0751937 −0.0375968 0.999293i \(-0.511970\pi\)
−0.0375968 + 0.999293i \(0.511970\pi\)
\(284\) 12.3461 0.732605
\(285\) 0 0
\(286\) 0 0
\(287\) −1.33309 −0.0786896
\(288\) 0 0
\(289\) −6.07978 −0.357634
\(290\) 1.41705 0.0832119
\(291\) 0 0
\(292\) 26.4662 1.54882
\(293\) 24.0874 1.40720 0.703601 0.710596i \(-0.251574\pi\)
0.703601 + 0.710596i \(0.251574\pi\)
\(294\) 0 0
\(295\) −6.24602 −0.363658
\(296\) −5.65274 −0.328559
\(297\) 0 0
\(298\) −1.98705 −0.115107
\(299\) 29.9525 1.73220
\(300\) 0 0
\(301\) 32.7843 1.88965
\(302\) −3.82290 −0.219983
\(303\) 0 0
\(304\) −11.5494 −0.662403
\(305\) 6.58191 0.376879
\(306\) 0 0
\(307\) 16.2949 0.930001 0.465001 0.885310i \(-0.346054\pi\)
0.465001 + 0.885310i \(0.346054\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 3.26637 0.185518
\(311\) −21.3713 −1.21185 −0.605926 0.795521i \(-0.707197\pi\)
−0.605926 + 0.795521i \(0.707197\pi\)
\(312\) 0 0
\(313\) 3.11613 0.176134 0.0880669 0.996115i \(-0.471931\pi\)
0.0880669 + 0.996115i \(0.471931\pi\)
\(314\) 0.151637 0.00855738
\(315\) 0 0
\(316\) 7.03613 0.395813
\(317\) 18.2648 1.02586 0.512928 0.858432i \(-0.328561\pi\)
0.512928 + 0.858432i \(0.328561\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 4.76728 0.266499
\(321\) 0 0
\(322\) −12.5329 −0.698434
\(323\) −14.5985 −0.812280
\(324\) 0 0
\(325\) 8.32420 0.461744
\(326\) 5.39254 0.298665
\(327\) 0 0
\(328\) 0.720761 0.0397974
\(329\) 3.68425 0.203119
\(330\) 0 0
\(331\) 3.76616 0.207007 0.103503 0.994629i \(-0.466995\pi\)
0.103503 + 0.994629i \(0.466995\pi\)
\(332\) −0.219867 −0.0120668
\(333\) 0 0
\(334\) 8.61988 0.471659
\(335\) −5.29740 −0.289428
\(336\) 0 0
\(337\) 5.96618 0.324998 0.162499 0.986709i \(-0.448045\pi\)
0.162499 + 0.986709i \(0.448045\pi\)
\(338\) 1.46334 0.0795950
\(339\) 0 0
\(340\) 9.93155 0.538614
\(341\) 0 0
\(342\) 0 0
\(343\) −8.07726 −0.436131
\(344\) −17.7255 −0.955695
\(345\) 0 0
\(346\) −6.06532 −0.326074
\(347\) −24.3493 −1.30714 −0.653570 0.756866i \(-0.726730\pi\)
−0.653570 + 0.756866i \(0.726730\pi\)
\(348\) 0 0
\(349\) 16.7418 0.896166 0.448083 0.893992i \(-0.352107\pi\)
0.448083 + 0.893992i \(0.352107\pi\)
\(350\) −3.48307 −0.186178
\(351\) 0 0
\(352\) 0 0
\(353\) 8.03132 0.427464 0.213732 0.976892i \(-0.431438\pi\)
0.213732 + 0.976892i \(0.431438\pi\)
\(354\) 0 0
\(355\) 11.9865 0.636179
\(356\) −13.9579 −0.739768
\(357\) 0 0
\(358\) −10.7331 −0.567264
\(359\) 28.8250 1.52133 0.760663 0.649147i \(-0.224874\pi\)
0.760663 + 0.649147i \(0.224874\pi\)
\(360\) 0 0
\(361\) 0.515640 0.0271390
\(362\) −2.14845 −0.112920
\(363\) 0 0
\(364\) 23.9896 1.25740
\(365\) 25.6954 1.34496
\(366\) 0 0
\(367\) 22.3434 1.16632 0.583158 0.812359i \(-0.301817\pi\)
0.583158 + 0.812359i \(0.301817\pi\)
\(368\) −19.5869 −1.02104
\(369\) 0 0
\(370\) −2.56845 −0.133528
\(371\) 42.8491 2.22461
\(372\) 0 0
\(373\) 7.07872 0.366522 0.183261 0.983064i \(-0.441335\pi\)
0.183261 + 0.983064i \(0.441335\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −1.99196 −0.102728
\(377\) 6.76164 0.348242
\(378\) 0 0
\(379\) −24.3461 −1.25057 −0.625287 0.780394i \(-0.715018\pi\)
−0.625287 + 0.780394i \(0.715018\pi\)
\(380\) −13.2768 −0.681084
\(381\) 0 0
\(382\) −6.13840 −0.314068
\(383\) 17.6379 0.901254 0.450627 0.892712i \(-0.351200\pi\)
0.450627 + 0.892712i \(0.351200\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.16341 −0.313709
\(387\) 0 0
\(388\) −0.603760 −0.0306513
\(389\) 17.8830 0.906704 0.453352 0.891332i \(-0.350228\pi\)
0.453352 + 0.891332i \(0.350228\pi\)
\(390\) 0 0
\(391\) −24.7579 −1.25206
\(392\) −8.54066 −0.431368
\(393\) 0 0
\(394\) 5.07433 0.255641
\(395\) 6.83122 0.343716
\(396\) 0 0
\(397\) −21.9395 −1.10111 −0.550556 0.834798i \(-0.685584\pi\)
−0.550556 + 0.834798i \(0.685584\pi\)
\(398\) 12.9870 0.650978
\(399\) 0 0
\(400\) −5.44347 −0.272174
\(401\) −5.40756 −0.270041 −0.135020 0.990843i \(-0.543110\pi\)
−0.135020 + 0.990843i \(0.543110\pi\)
\(402\) 0 0
\(403\) 15.5860 0.776392
\(404\) 12.5193 0.622857
\(405\) 0 0
\(406\) −2.82926 −0.140414
\(407\) 0 0
\(408\) 0 0
\(409\) 23.6201 1.16794 0.583970 0.811775i \(-0.301498\pi\)
0.583970 + 0.811775i \(0.301498\pi\)
\(410\) 0.327495 0.0161738
\(411\) 0 0
\(412\) 2.91058 0.143394
\(413\) 12.4707 0.613644
\(414\) 0 0
\(415\) −0.213464 −0.0104785
\(416\) −19.8708 −0.974245
\(417\) 0 0
\(418\) 0 0
\(419\) 18.1313 0.885773 0.442886 0.896578i \(-0.353955\pi\)
0.442886 + 0.896578i \(0.353955\pi\)
\(420\) 0 0
\(421\) −12.8122 −0.624428 −0.312214 0.950012i \(-0.601071\pi\)
−0.312214 + 0.950012i \(0.601071\pi\)
\(422\) −3.57271 −0.173917
\(423\) 0 0
\(424\) −23.1672 −1.12510
\(425\) −6.88056 −0.333756
\(426\) 0 0
\(427\) −13.1413 −0.635954
\(428\) 5.81809 0.281228
\(429\) 0 0
\(430\) −8.05399 −0.388398
\(431\) 6.57916 0.316907 0.158453 0.987366i \(-0.449349\pi\)
0.158453 + 0.987366i \(0.449349\pi\)
\(432\) 0 0
\(433\) 13.9978 0.672690 0.336345 0.941739i \(-0.390809\pi\)
0.336345 + 0.941739i \(0.390809\pi\)
\(434\) −6.52160 −0.313047
\(435\) 0 0
\(436\) 9.68927 0.464032
\(437\) 33.0971 1.58325
\(438\) 0 0
\(439\) 25.1782 1.20169 0.600846 0.799365i \(-0.294831\pi\)
0.600846 + 0.799365i \(0.294831\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −6.48013 −0.308228
\(443\) −26.8225 −1.27437 −0.637187 0.770709i \(-0.719902\pi\)
−0.637187 + 0.770709i \(0.719902\pi\)
\(444\) 0 0
\(445\) −13.5514 −0.642399
\(446\) 11.5315 0.546034
\(447\) 0 0
\(448\) −9.51829 −0.449697
\(449\) −14.1758 −0.668999 −0.334499 0.942396i \(-0.608567\pi\)
−0.334499 + 0.942396i \(0.608567\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −28.8706 −1.35796
\(453\) 0 0
\(454\) 5.84776 0.274449
\(455\) 23.2910 1.09190
\(456\) 0 0
\(457\) 24.6413 1.15267 0.576335 0.817213i \(-0.304482\pi\)
0.576335 + 0.817213i \(0.304482\pi\)
\(458\) 1.49972 0.0700773
\(459\) 0 0
\(460\) −22.5164 −1.04983
\(461\) 26.2454 1.22237 0.611185 0.791488i \(-0.290693\pi\)
0.611185 + 0.791488i \(0.290693\pi\)
\(462\) 0 0
\(463\) 12.6922 0.589855 0.294928 0.955520i \(-0.404704\pi\)
0.294928 + 0.955520i \(0.404704\pi\)
\(464\) −4.42166 −0.205270
\(465\) 0 0
\(466\) 12.0325 0.557394
\(467\) 27.7040 1.28199 0.640994 0.767546i \(-0.278522\pi\)
0.640994 + 0.767546i \(0.278522\pi\)
\(468\) 0 0
\(469\) 10.5767 0.488387
\(470\) −0.905095 −0.0417489
\(471\) 0 0
\(472\) −6.74256 −0.310351
\(473\) 0 0
\(474\) 0 0
\(475\) 9.19813 0.422039
\(476\) −19.8292 −0.908870
\(477\) 0 0
\(478\) 3.40924 0.155935
\(479\) 15.6222 0.713797 0.356898 0.934143i \(-0.383834\pi\)
0.356898 + 0.934143i \(0.383834\pi\)
\(480\) 0 0
\(481\) −12.2557 −0.558814
\(482\) −3.81143 −0.173606
\(483\) 0 0
\(484\) 0 0
\(485\) −0.586177 −0.0266169
\(486\) 0 0
\(487\) 26.9270 1.22018 0.610090 0.792332i \(-0.291133\pi\)
0.610090 + 0.792332i \(0.291133\pi\)
\(488\) 7.10514 0.321635
\(489\) 0 0
\(490\) −3.88064 −0.175310
\(491\) −38.9997 −1.76003 −0.880017 0.474942i \(-0.842469\pi\)
−0.880017 + 0.474942i \(0.842469\pi\)
\(492\) 0 0
\(493\) −5.58899 −0.251715
\(494\) 8.66283 0.389759
\(495\) 0 0
\(496\) −10.1922 −0.457642
\(497\) −23.9322 −1.07350
\(498\) 0 0
\(499\) −24.0351 −1.07596 −0.537980 0.842958i \(-0.680812\pi\)
−0.537980 + 0.842958i \(0.680812\pi\)
\(500\) −21.2846 −0.951876
\(501\) 0 0
\(502\) −0.160474 −0.00716229
\(503\) 23.9865 1.06950 0.534752 0.845009i \(-0.320405\pi\)
0.534752 + 0.845009i \(0.320405\pi\)
\(504\) 0 0
\(505\) 12.1547 0.540876
\(506\) 0 0
\(507\) 0 0
\(508\) −18.9927 −0.842667
\(509\) −3.69886 −0.163949 −0.0819746 0.996634i \(-0.526123\pi\)
−0.0819746 + 0.996634i \(0.526123\pi\)
\(510\) 0 0
\(511\) −51.3032 −2.26952
\(512\) 22.6358 1.00037
\(513\) 0 0
\(514\) 6.73407 0.297027
\(515\) 2.82582 0.124520
\(516\) 0 0
\(517\) 0 0
\(518\) 5.12814 0.225318
\(519\) 0 0
\(520\) −12.5928 −0.552230
\(521\) 6.41318 0.280966 0.140483 0.990083i \(-0.455134\pi\)
0.140483 + 0.990083i \(0.455134\pi\)
\(522\) 0 0
\(523\) −2.73856 −0.119749 −0.0598745 0.998206i \(-0.519070\pi\)
−0.0598745 + 0.998206i \(0.519070\pi\)
\(524\) −37.2216 −1.62603
\(525\) 0 0
\(526\) 8.26688 0.360453
\(527\) −12.8829 −0.561190
\(528\) 0 0
\(529\) 33.1302 1.44044
\(530\) −10.5266 −0.457245
\(531\) 0 0
\(532\) 26.5082 1.14928
\(533\) 1.56269 0.0676875
\(534\) 0 0
\(535\) 5.64865 0.244213
\(536\) −5.71852 −0.247003
\(537\) 0 0
\(538\) −6.08085 −0.262164
\(539\) 0 0
\(540\) 0 0
\(541\) 20.9037 0.898719 0.449359 0.893351i \(-0.351652\pi\)
0.449359 + 0.893351i \(0.351652\pi\)
\(542\) 9.45157 0.405980
\(543\) 0 0
\(544\) 16.4247 0.704201
\(545\) 9.40709 0.402956
\(546\) 0 0
\(547\) −41.8764 −1.79050 −0.895252 0.445560i \(-0.853004\pi\)
−0.895252 + 0.445560i \(0.853004\pi\)
\(548\) 5.63105 0.240547
\(549\) 0 0
\(550\) 0 0
\(551\) 7.47152 0.318298
\(552\) 0 0
\(553\) −13.6391 −0.579995
\(554\) 5.07714 0.215707
\(555\) 0 0
\(556\) 26.3801 1.11876
\(557\) 12.5244 0.530675 0.265337 0.964156i \(-0.414517\pi\)
0.265337 + 0.964156i \(0.414517\pi\)
\(558\) 0 0
\(559\) −38.4308 −1.62545
\(560\) −15.2308 −0.643617
\(561\) 0 0
\(562\) 8.06166 0.340061
\(563\) 14.5087 0.611470 0.305735 0.952117i \(-0.401098\pi\)
0.305735 + 0.952117i \(0.401098\pi\)
\(564\) 0 0
\(565\) −28.0299 −1.17923
\(566\) −0.620452 −0.0260796
\(567\) 0 0
\(568\) 12.9394 0.542926
\(569\) 27.9724 1.17266 0.586332 0.810071i \(-0.300571\pi\)
0.586332 + 0.810071i \(0.300571\pi\)
\(570\) 0 0
\(571\) 9.04886 0.378683 0.189341 0.981911i \(-0.439365\pi\)
0.189341 + 0.981911i \(0.439365\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −0.653871 −0.0272921
\(575\) 15.5993 0.650538
\(576\) 0 0
\(577\) 38.3471 1.59641 0.798204 0.602387i \(-0.205783\pi\)
0.798204 + 0.602387i \(0.205783\pi\)
\(578\) −2.98210 −0.124039
\(579\) 0 0
\(580\) −5.08299 −0.211060
\(581\) 0.426199 0.0176817
\(582\) 0 0
\(583\) 0 0
\(584\) 27.7381 1.14781
\(585\) 0 0
\(586\) 11.8147 0.488062
\(587\) 0.722617 0.0298256 0.0149128 0.999889i \(-0.495253\pi\)
0.0149128 + 0.999889i \(0.495253\pi\)
\(588\) 0 0
\(589\) 17.2223 0.709632
\(590\) −3.06364 −0.126128
\(591\) 0 0
\(592\) 8.01444 0.329391
\(593\) 39.6596 1.62863 0.814313 0.580426i \(-0.197114\pi\)
0.814313 + 0.580426i \(0.197114\pi\)
\(594\) 0 0
\(595\) −19.2517 −0.789244
\(596\) 7.12761 0.291958
\(597\) 0 0
\(598\) 14.6915 0.600781
\(599\) 20.3768 0.832573 0.416287 0.909233i \(-0.363331\pi\)
0.416287 + 0.909233i \(0.363331\pi\)
\(600\) 0 0
\(601\) −48.3717 −1.97312 −0.986562 0.163389i \(-0.947757\pi\)
−0.986562 + 0.163389i \(0.947757\pi\)
\(602\) 16.0805 0.655392
\(603\) 0 0
\(604\) 13.7129 0.557968
\(605\) 0 0
\(606\) 0 0
\(607\) −19.7023 −0.799691 −0.399845 0.916583i \(-0.630936\pi\)
−0.399845 + 0.916583i \(0.630936\pi\)
\(608\) −21.9569 −0.890472
\(609\) 0 0
\(610\) 3.22839 0.130714
\(611\) −4.31879 −0.174720
\(612\) 0 0
\(613\) 5.71781 0.230940 0.115470 0.993311i \(-0.463163\pi\)
0.115470 + 0.993311i \(0.463163\pi\)
\(614\) 7.99257 0.322554
\(615\) 0 0
\(616\) 0 0
\(617\) −6.10692 −0.245855 −0.122928 0.992416i \(-0.539228\pi\)
−0.122928 + 0.992416i \(0.539228\pi\)
\(618\) 0 0
\(619\) −1.77973 −0.0715335 −0.0357668 0.999360i \(-0.511387\pi\)
−0.0357668 + 0.999360i \(0.511387\pi\)
\(620\) −11.7166 −0.470549
\(621\) 0 0
\(622\) −10.4825 −0.420309
\(623\) 27.0566 1.08400
\(624\) 0 0
\(625\) −10.2541 −0.410163
\(626\) 1.52844 0.0610888
\(627\) 0 0
\(628\) −0.543927 −0.0217050
\(629\) 10.1303 0.403920
\(630\) 0 0
\(631\) −44.8407 −1.78508 −0.892541 0.450966i \(-0.851079\pi\)
−0.892541 + 0.450966i \(0.851079\pi\)
\(632\) 7.37428 0.293333
\(633\) 0 0
\(634\) 8.95880 0.355799
\(635\) −18.4396 −0.731754
\(636\) 0 0
\(637\) −18.5170 −0.733672
\(638\) 0 0
\(639\) 0 0
\(640\) 19.3185 0.763633
\(641\) 27.0096 1.06681 0.533407 0.845859i \(-0.320912\pi\)
0.533407 + 0.845859i \(0.320912\pi\)
\(642\) 0 0
\(643\) −32.5574 −1.28394 −0.641969 0.766731i \(-0.721882\pi\)
−0.641969 + 0.766731i \(0.721882\pi\)
\(644\) 44.9560 1.77152
\(645\) 0 0
\(646\) −7.16046 −0.281724
\(647\) 31.4833 1.23774 0.618868 0.785495i \(-0.287592\pi\)
0.618868 + 0.785495i \(0.287592\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 4.08297 0.160147
\(651\) 0 0
\(652\) −19.3432 −0.757538
\(653\) 4.65977 0.182351 0.0911755 0.995835i \(-0.470938\pi\)
0.0911755 + 0.995835i \(0.470938\pi\)
\(654\) 0 0
\(655\) −36.1376 −1.41201
\(656\) −1.02189 −0.0398982
\(657\) 0 0
\(658\) 1.80710 0.0704481
\(659\) 32.0418 1.24817 0.624086 0.781355i \(-0.285471\pi\)
0.624086 + 0.781355i \(0.285471\pi\)
\(660\) 0 0
\(661\) −40.7054 −1.58325 −0.791627 0.611004i \(-0.790766\pi\)
−0.791627 + 0.611004i \(0.790766\pi\)
\(662\) 1.84728 0.0717965
\(663\) 0 0
\(664\) −0.230433 −0.00894255
\(665\) 25.7362 0.998009
\(666\) 0 0
\(667\) 12.6712 0.490629
\(668\) −30.9197 −1.19632
\(669\) 0 0
\(670\) −2.59834 −0.100383
\(671\) 0 0
\(672\) 0 0
\(673\) −37.7635 −1.45568 −0.727838 0.685749i \(-0.759475\pi\)
−0.727838 + 0.685749i \(0.759475\pi\)
\(674\) 2.92637 0.112720
\(675\) 0 0
\(676\) −5.24903 −0.201886
\(677\) −18.9007 −0.726412 −0.363206 0.931709i \(-0.618318\pi\)
−0.363206 + 0.931709i \(0.618318\pi\)
\(678\) 0 0
\(679\) 1.17035 0.0449140
\(680\) 10.4088 0.399161
\(681\) 0 0
\(682\) 0 0
\(683\) −31.1053 −1.19021 −0.595105 0.803648i \(-0.702890\pi\)
−0.595105 + 0.803648i \(0.702890\pi\)
\(684\) 0 0
\(685\) 5.46706 0.208886
\(686\) −3.96185 −0.151264
\(687\) 0 0
\(688\) 25.1312 0.958116
\(689\) −50.2290 −1.91357
\(690\) 0 0
\(691\) 22.2114 0.844962 0.422481 0.906372i \(-0.361159\pi\)
0.422481 + 0.906372i \(0.361159\pi\)
\(692\) 21.7565 0.827057
\(693\) 0 0
\(694\) −11.9432 −0.453358
\(695\) 25.6118 0.971511
\(696\) 0 0
\(697\) −1.29167 −0.0489257
\(698\) 8.21174 0.310819
\(699\) 0 0
\(700\) 12.4939 0.472224
\(701\) 29.9457 1.13103 0.565517 0.824736i \(-0.308677\pi\)
0.565517 + 0.824736i \(0.308677\pi\)
\(702\) 0 0
\(703\) −13.5424 −0.510763
\(704\) 0 0
\(705\) 0 0
\(706\) 3.93931 0.148258
\(707\) −24.2679 −0.912687
\(708\) 0 0
\(709\) 17.2057 0.646175 0.323087 0.946369i \(-0.395279\pi\)
0.323087 + 0.946369i \(0.395279\pi\)
\(710\) 5.87932 0.220647
\(711\) 0 0
\(712\) −14.6287 −0.548234
\(713\) 29.2077 1.09384
\(714\) 0 0
\(715\) 0 0
\(716\) 38.5001 1.43881
\(717\) 0 0
\(718\) 14.1385 0.527644
\(719\) −4.72508 −0.176216 −0.0881079 0.996111i \(-0.528082\pi\)
−0.0881079 + 0.996111i \(0.528082\pi\)
\(720\) 0 0
\(721\) −5.64199 −0.210119
\(722\) 0.252919 0.00941265
\(723\) 0 0
\(724\) 7.70655 0.286412
\(725\) 3.52148 0.130785
\(726\) 0 0
\(727\) 5.20088 0.192890 0.0964450 0.995338i \(-0.469253\pi\)
0.0964450 + 0.995338i \(0.469253\pi\)
\(728\) 25.1426 0.931845
\(729\) 0 0
\(730\) 12.6035 0.466475
\(731\) 31.7658 1.17490
\(732\) 0 0
\(733\) −49.0632 −1.81219 −0.906095 0.423075i \(-0.860951\pi\)
−0.906095 + 0.423075i \(0.860951\pi\)
\(734\) 10.9593 0.404515
\(735\) 0 0
\(736\) −37.2374 −1.37259
\(737\) 0 0
\(738\) 0 0
\(739\) −15.5942 −0.573641 −0.286821 0.957984i \(-0.592598\pi\)
−0.286821 + 0.957984i \(0.592598\pi\)
\(740\) 9.21312 0.338681
\(741\) 0 0
\(742\) 21.0172 0.771566
\(743\) −11.1344 −0.408483 −0.204242 0.978921i \(-0.565473\pi\)
−0.204242 + 0.978921i \(0.565473\pi\)
\(744\) 0 0
\(745\) 6.92004 0.253530
\(746\) 3.47207 0.127122
\(747\) 0 0
\(748\) 0 0
\(749\) −11.2780 −0.412090
\(750\) 0 0
\(751\) −8.08520 −0.295033 −0.147517 0.989060i \(-0.547128\pi\)
−0.147517 + 0.989060i \(0.547128\pi\)
\(752\) 2.82420 0.102988
\(753\) 0 0
\(754\) 3.31654 0.120781
\(755\) 13.3135 0.484528
\(756\) 0 0
\(757\) 29.8616 1.08534 0.542669 0.839947i \(-0.317414\pi\)
0.542669 + 0.839947i \(0.317414\pi\)
\(758\) −11.9416 −0.433739
\(759\) 0 0
\(760\) −13.9148 −0.504744
\(761\) −34.8926 −1.26486 −0.632428 0.774619i \(-0.717941\pi\)
−0.632428 + 0.774619i \(0.717941\pi\)
\(762\) 0 0
\(763\) −18.7821 −0.679957
\(764\) 22.0186 0.796606
\(765\) 0 0
\(766\) 8.65128 0.312583
\(767\) −14.6186 −0.527846
\(768\) 0 0
\(769\) −5.60723 −0.202202 −0.101101 0.994876i \(-0.532237\pi\)
−0.101101 + 0.994876i \(0.532237\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 22.1083 0.795696
\(773\) 10.8401 0.389893 0.194946 0.980814i \(-0.437547\pi\)
0.194946 + 0.980814i \(0.437547\pi\)
\(774\) 0 0
\(775\) 8.11722 0.291579
\(776\) −0.632776 −0.0227153
\(777\) 0 0
\(778\) 8.77150 0.314473
\(779\) 1.72675 0.0618672
\(780\) 0 0
\(781\) 0 0
\(782\) −12.1436 −0.434255
\(783\) 0 0
\(784\) 12.1089 0.432461
\(785\) −0.528086 −0.0188482
\(786\) 0 0
\(787\) −2.47643 −0.0882752 −0.0441376 0.999025i \(-0.514054\pi\)
−0.0441376 + 0.999025i \(0.514054\pi\)
\(788\) −18.2018 −0.648411
\(789\) 0 0
\(790\) 3.35067 0.119212
\(791\) 55.9640 1.98985
\(792\) 0 0
\(793\) 15.4047 0.547037
\(794\) −10.7612 −0.381900
\(795\) 0 0
\(796\) −46.5846 −1.65115
\(797\) 9.67969 0.342872 0.171436 0.985195i \(-0.445159\pi\)
0.171436 + 0.985195i \(0.445159\pi\)
\(798\) 0 0
\(799\) 3.56980 0.126290
\(800\) −10.3488 −0.365884
\(801\) 0 0
\(802\) −2.65237 −0.0936586
\(803\) 0 0
\(804\) 0 0
\(805\) 43.6468 1.53835
\(806\) 7.64482 0.269277
\(807\) 0 0
\(808\) 13.1209 0.461593
\(809\) 7.86532 0.276530 0.138265 0.990395i \(-0.455847\pi\)
0.138265 + 0.990395i \(0.455847\pi\)
\(810\) 0 0
\(811\) 39.5047 1.38720 0.693599 0.720361i \(-0.256024\pi\)
0.693599 + 0.720361i \(0.256024\pi\)
\(812\) 10.1486 0.356147
\(813\) 0 0
\(814\) 0 0
\(815\) −18.7799 −0.657831
\(816\) 0 0
\(817\) −42.4655 −1.48568
\(818\) 11.5855 0.405079
\(819\) 0 0
\(820\) −1.17473 −0.0410234
\(821\) 3.35362 0.117042 0.0585211 0.998286i \(-0.481362\pi\)
0.0585211 + 0.998286i \(0.481362\pi\)
\(822\) 0 0
\(823\) 16.1018 0.561274 0.280637 0.959814i \(-0.409454\pi\)
0.280637 + 0.959814i \(0.409454\pi\)
\(824\) 3.05046 0.106268
\(825\) 0 0
\(826\) 6.11681 0.212831
\(827\) −15.2905 −0.531702 −0.265851 0.964014i \(-0.585653\pi\)
−0.265851 + 0.964014i \(0.585653\pi\)
\(828\) 0 0
\(829\) −42.1463 −1.46380 −0.731901 0.681411i \(-0.761367\pi\)
−0.731901 + 0.681411i \(0.761367\pi\)
\(830\) −0.104703 −0.00363428
\(831\) 0 0
\(832\) 11.1576 0.386822
\(833\) 15.3057 0.530311
\(834\) 0 0
\(835\) −30.0193 −1.03886
\(836\) 0 0
\(837\) 0 0
\(838\) 8.89330 0.307214
\(839\) −2.64739 −0.0913981 −0.0456990 0.998955i \(-0.514552\pi\)
−0.0456990 + 0.998955i \(0.514552\pi\)
\(840\) 0 0
\(841\) −26.1395 −0.901364
\(842\) −6.28431 −0.216572
\(843\) 0 0
\(844\) 12.8154 0.441125
\(845\) −5.09616 −0.175313
\(846\) 0 0
\(847\) 0 0
\(848\) 32.8464 1.12795
\(849\) 0 0
\(850\) −3.37487 −0.115757
\(851\) −22.9670 −0.787298
\(852\) 0 0
\(853\) 9.27037 0.317411 0.158706 0.987326i \(-0.449268\pi\)
0.158706 + 0.987326i \(0.449268\pi\)
\(854\) −6.44575 −0.220569
\(855\) 0 0
\(856\) 6.09770 0.208415
\(857\) 20.5356 0.701483 0.350742 0.936472i \(-0.385930\pi\)
0.350742 + 0.936472i \(0.385930\pi\)
\(858\) 0 0
\(859\) 36.0278 1.22925 0.614626 0.788819i \(-0.289307\pi\)
0.614626 + 0.788819i \(0.289307\pi\)
\(860\) 28.8899 0.985137
\(861\) 0 0
\(862\) 3.22704 0.109913
\(863\) −4.31646 −0.146934 −0.0734670 0.997298i \(-0.523406\pi\)
−0.0734670 + 0.997298i \(0.523406\pi\)
\(864\) 0 0
\(865\) 21.1229 0.718199
\(866\) 6.86583 0.233310
\(867\) 0 0
\(868\) 23.3932 0.794015
\(869\) 0 0
\(870\) 0 0
\(871\) −12.3984 −0.420102
\(872\) 10.1549 0.343889
\(873\) 0 0
\(874\) 16.2339 0.549121
\(875\) 41.2589 1.39481
\(876\) 0 0
\(877\) 31.1160 1.05071 0.525356 0.850882i \(-0.323932\pi\)
0.525356 + 0.850882i \(0.323932\pi\)
\(878\) 12.3498 0.416785
\(879\) 0 0
\(880\) 0 0
\(881\) −43.9267 −1.47993 −0.739964 0.672647i \(-0.765157\pi\)
−0.739964 + 0.672647i \(0.765157\pi\)
\(882\) 0 0
\(883\) −5.87946 −0.197860 −0.0989298 0.995094i \(-0.531542\pi\)
−0.0989298 + 0.995094i \(0.531542\pi\)
\(884\) 23.2444 0.781794
\(885\) 0 0
\(886\) −13.1563 −0.441993
\(887\) 7.69510 0.258376 0.129188 0.991620i \(-0.458763\pi\)
0.129188 + 0.991620i \(0.458763\pi\)
\(888\) 0 0
\(889\) 36.8163 1.23478
\(890\) −6.64689 −0.222804
\(891\) 0 0
\(892\) −41.3640 −1.38497
\(893\) −4.77220 −0.159696
\(894\) 0 0
\(895\) 37.3788 1.24944
\(896\) −38.5711 −1.28857
\(897\) 0 0
\(898\) −6.95316 −0.232030
\(899\) 6.59352 0.219906
\(900\) 0 0
\(901\) 41.5180 1.38316
\(902\) 0 0
\(903\) 0 0
\(904\) −30.2581 −1.00637
\(905\) 7.48212 0.248714
\(906\) 0 0
\(907\) −18.9906 −0.630571 −0.315285 0.948997i \(-0.602100\pi\)
−0.315285 + 0.948997i \(0.602100\pi\)
\(908\) −20.9761 −0.696116
\(909\) 0 0
\(910\) 11.4241 0.378705
\(911\) −50.0353 −1.65774 −0.828871 0.559439i \(-0.811017\pi\)
−0.828871 + 0.559439i \(0.811017\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 12.0864 0.399783
\(915\) 0 0
\(916\) −5.37954 −0.177745
\(917\) 72.1519 2.38266
\(918\) 0 0
\(919\) 3.07419 0.101408 0.0507042 0.998714i \(-0.483853\pi\)
0.0507042 + 0.998714i \(0.483853\pi\)
\(920\) −23.5985 −0.778021
\(921\) 0 0
\(922\) 12.8732 0.423956
\(923\) 28.0540 0.923410
\(924\) 0 0
\(925\) −6.38283 −0.209866
\(926\) 6.22544 0.204581
\(927\) 0 0
\(928\) −8.40617 −0.275946
\(929\) 0.905574 0.0297109 0.0148555 0.999890i \(-0.495271\pi\)
0.0148555 + 0.999890i \(0.495271\pi\)
\(930\) 0 0
\(931\) −20.4611 −0.670585
\(932\) −43.1609 −1.41378
\(933\) 0 0
\(934\) 13.5887 0.444634
\(935\) 0 0
\(936\) 0 0
\(937\) −22.0542 −0.720481 −0.360240 0.932860i \(-0.617305\pi\)
−0.360240 + 0.932860i \(0.617305\pi\)
\(938\) 5.18781 0.169388
\(939\) 0 0
\(940\) 3.24660 0.105892
\(941\) −5.46092 −0.178021 −0.0890105 0.996031i \(-0.528370\pi\)
−0.0890105 + 0.996031i \(0.528370\pi\)
\(942\) 0 0
\(943\) 2.92844 0.0953630
\(944\) 9.55957 0.311138
\(945\) 0 0
\(946\) 0 0
\(947\) −32.4406 −1.05418 −0.527089 0.849810i \(-0.676717\pi\)
−0.527089 + 0.849810i \(0.676717\pi\)
\(948\) 0 0
\(949\) 60.1392 1.95220
\(950\) 4.51163 0.146377
\(951\) 0 0
\(952\) −20.7822 −0.673554
\(953\) 53.8088 1.74304 0.871518 0.490363i \(-0.163136\pi\)
0.871518 + 0.490363i \(0.163136\pi\)
\(954\) 0 0
\(955\) 21.3774 0.691756
\(956\) −12.2290 −0.395516
\(957\) 0 0
\(958\) 7.66260 0.247567
\(959\) −10.9155 −0.352479
\(960\) 0 0
\(961\) −15.8016 −0.509728
\(962\) −6.01137 −0.193814
\(963\) 0 0
\(964\) 13.6717 0.440335
\(965\) 21.4645 0.690966
\(966\) 0 0
\(967\) −28.8151 −0.926630 −0.463315 0.886194i \(-0.653340\pi\)
−0.463315 + 0.886194i \(0.653340\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −0.287516 −0.00923160
\(971\) −5.39405 −0.173103 −0.0865517 0.996247i \(-0.527585\pi\)
−0.0865517 + 0.996247i \(0.527585\pi\)
\(972\) 0 0
\(973\) −51.1362 −1.63935
\(974\) 13.2075 0.423197
\(975\) 0 0
\(976\) −10.0736 −0.322450
\(977\) 3.31236 0.105972 0.0529859 0.998595i \(-0.483126\pi\)
0.0529859 + 0.998595i \(0.483126\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 13.9200 0.444657
\(981\) 0 0
\(982\) −19.1291 −0.610435
\(983\) −33.8721 −1.08035 −0.540176 0.841552i \(-0.681643\pi\)
−0.540176 + 0.841552i \(0.681643\pi\)
\(984\) 0 0
\(985\) −17.6717 −0.563067
\(986\) −2.74137 −0.0873029
\(987\) 0 0
\(988\) −31.0738 −0.988589
\(989\) −72.0183 −2.29005
\(990\) 0 0
\(991\) 8.66640 0.275297 0.137649 0.990481i \(-0.456046\pi\)
0.137649 + 0.990481i \(0.456046\pi\)
\(992\) −19.3767 −0.615211
\(993\) 0 0
\(994\) −11.7386 −0.372325
\(995\) −45.2280 −1.43382
\(996\) 0 0
\(997\) −30.2654 −0.958515 −0.479258 0.877674i \(-0.659094\pi\)
−0.479258 + 0.877674i \(0.659094\pi\)
\(998\) −11.7891 −0.373177
\(999\) 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.cp.1.10 18
3.2 odd 2 9801.2.a.cm.1.9 18
9.2 odd 6 1089.2.e.p.364.10 36
9.5 odd 6 1089.2.e.p.727.10 36
11.5 even 5 891.2.f.e.487.5 36
11.9 even 5 891.2.f.e.730.5 36
11.10 odd 2 9801.2.a.cn.1.9 18
33.5 odd 10 891.2.f.f.487.5 36
33.20 odd 10 891.2.f.f.730.5 36
33.32 even 2 9801.2.a.co.1.10 18
99.5 odd 30 99.2.m.b.25.5 yes 72
99.16 even 15 297.2.n.b.91.5 72
99.20 odd 30 99.2.m.b.4.5 72
99.31 even 15 297.2.n.b.235.5 72
99.32 even 6 1089.2.e.o.727.9 36
99.38 odd 30 99.2.m.b.58.5 yes 72
99.49 even 15 297.2.n.b.289.5 72
99.65 even 6 1089.2.e.o.364.9 36
99.86 odd 30 99.2.m.b.70.5 yes 72
99.97 even 15 297.2.n.b.37.5 72
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.m.b.4.5 72 99.20 odd 30
99.2.m.b.25.5 yes 72 99.5 odd 30
99.2.m.b.58.5 yes 72 99.38 odd 30
99.2.m.b.70.5 yes 72 99.86 odd 30
297.2.n.b.37.5 72 99.97 even 15
297.2.n.b.91.5 72 99.16 even 15
297.2.n.b.235.5 72 99.31 even 15
297.2.n.b.289.5 72 99.49 even 15
891.2.f.e.487.5 36 11.5 even 5
891.2.f.e.730.5 36 11.9 even 5
891.2.f.f.487.5 36 33.5 odd 10
891.2.f.f.730.5 36 33.20 odd 10
1089.2.e.o.364.9 36 99.65 even 6
1089.2.e.o.727.9 36 99.32 even 6
1089.2.e.p.364.10 36 9.2 odd 6
1089.2.e.p.727.10 36 9.5 odd 6
9801.2.a.cm.1.9 18 3.2 odd 2
9801.2.a.cn.1.9 18 11.10 odd 2
9801.2.a.co.1.10 18 33.32 even 2
9801.2.a.cp.1.10 18 1.1 even 1 trivial