Properties

Label 9801.2.a.bc.1.1
Level $9801$
Weight $2$
Character 9801.1
Self dual yes
Analytic conductor $78.261$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
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 1.1
Root \(-0.618034\) 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)
 
\(f(q)\) \(=\) \(q+0.381966 q^{2} -1.85410 q^{4} -1.23607 q^{5} +1.00000 q^{7} -1.47214 q^{8} -0.472136 q^{10} +6.47214 q^{13} +0.381966 q^{14} +3.14590 q^{16} -4.85410 q^{17} -1.00000 q^{19} +2.29180 q^{20} +4.61803 q^{23} -3.47214 q^{25} +2.47214 q^{26} -1.85410 q^{28} +4.85410 q^{29} +0.618034 q^{31} +4.14590 q^{32} -1.85410 q^{34} -1.23607 q^{35} -5.09017 q^{37} -0.381966 q^{38} +1.81966 q^{40} +2.52786 q^{41} -1.85410 q^{43} +1.76393 q^{46} +11.9443 q^{47} -6.00000 q^{49} -1.32624 q^{50} -12.0000 q^{52} +4.09017 q^{53} -1.47214 q^{56} +1.85410 q^{58} +1.61803 q^{59} -10.8541 q^{61} +0.236068 q^{62} -4.70820 q^{64} -8.00000 q^{65} +6.00000 q^{67} +9.00000 q^{68} -0.472136 q^{70} -2.90983 q^{71} +0.145898 q^{73} -1.94427 q^{74} +1.85410 q^{76} -10.5623 q^{79} -3.88854 q^{80} +0.965558 q^{82} +9.76393 q^{83} +6.00000 q^{85} -0.708204 q^{86} -6.76393 q^{89} +6.47214 q^{91} -8.56231 q^{92} +4.56231 q^{94} +1.23607 q^{95} +6.00000 q^{97} -2.29180 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 3 q^{2} + 3 q^{4} + 2 q^{5} + 2 q^{7} + 6 q^{8} + 8 q^{10} + 4 q^{13} + 3 q^{14} + 13 q^{16} - 3 q^{17} - 2 q^{19} + 18 q^{20} + 7 q^{23} + 2 q^{25} - 4 q^{26} + 3 q^{28} + 3 q^{29} - q^{31} + 15 q^{32}+ \cdots - 18 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.381966 0.270091 0.135045 0.990839i \(-0.456882\pi\)
0.135045 + 0.990839i \(0.456882\pi\)
\(3\) 0 0
\(4\) −1.85410 −0.927051
\(5\) −1.23607 −0.552786 −0.276393 0.961045i \(-0.589139\pi\)
−0.276393 + 0.961045i \(0.589139\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964 0.188982 0.981981i \(-0.439481\pi\)
0.188982 + 0.981981i \(0.439481\pi\)
\(8\) −1.47214 −0.520479
\(9\) 0 0
\(10\) −0.472136 −0.149302
\(11\) 0 0
\(12\) 0 0
\(13\) 6.47214 1.79505 0.897524 0.440966i \(-0.145364\pi\)
0.897524 + 0.440966i \(0.145364\pi\)
\(14\) 0.381966 0.102085
\(15\) 0 0
\(16\) 3.14590 0.786475
\(17\) −4.85410 −1.17729 −0.588646 0.808391i \(-0.700339\pi\)
−0.588646 + 0.808391i \(0.700339\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 2.29180 0.512461
\(21\) 0 0
\(22\) 0 0
\(23\) 4.61803 0.962927 0.481463 0.876466i \(-0.340105\pi\)
0.481463 + 0.876466i \(0.340105\pi\)
\(24\) 0 0
\(25\) −3.47214 −0.694427
\(26\) 2.47214 0.484826
\(27\) 0 0
\(28\) −1.85410 −0.350392
\(29\) 4.85410 0.901384 0.450692 0.892679i \(-0.351177\pi\)
0.450692 + 0.892679i \(0.351177\pi\)
\(30\) 0 0
\(31\) 0.618034 0.111002 0.0555011 0.998459i \(-0.482324\pi\)
0.0555011 + 0.998459i \(0.482324\pi\)
\(32\) 4.14590 0.732898
\(33\) 0 0
\(34\) −1.85410 −0.317976
\(35\) −1.23607 −0.208934
\(36\) 0 0
\(37\) −5.09017 −0.836819 −0.418409 0.908259i \(-0.637412\pi\)
−0.418409 + 0.908259i \(0.637412\pi\)
\(38\) −0.381966 −0.0619631
\(39\) 0 0
\(40\) 1.81966 0.287714
\(41\) 2.52786 0.394786 0.197393 0.980324i \(-0.436752\pi\)
0.197393 + 0.980324i \(0.436752\pi\)
\(42\) 0 0
\(43\) −1.85410 −0.282748 −0.141374 0.989956i \(-0.545152\pi\)
−0.141374 + 0.989956i \(0.545152\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 1.76393 0.260078
\(47\) 11.9443 1.74225 0.871126 0.491060i \(-0.163391\pi\)
0.871126 + 0.491060i \(0.163391\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) −1.32624 −0.187558
\(51\) 0 0
\(52\) −12.0000 −1.66410
\(53\) 4.09017 0.561828 0.280914 0.959733i \(-0.409362\pi\)
0.280914 + 0.959733i \(0.409362\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.47214 −0.196722
\(57\) 0 0
\(58\) 1.85410 0.243456
\(59\) 1.61803 0.210650 0.105325 0.994438i \(-0.466412\pi\)
0.105325 + 0.994438i \(0.466412\pi\)
\(60\) 0 0
\(61\) −10.8541 −1.38973 −0.694863 0.719142i \(-0.744535\pi\)
−0.694863 + 0.719142i \(0.744535\pi\)
\(62\) 0.236068 0.0299807
\(63\) 0 0
\(64\) −4.70820 −0.588525
\(65\) −8.00000 −0.992278
\(66\) 0 0
\(67\) 6.00000 0.733017 0.366508 0.930415i \(-0.380553\pi\)
0.366508 + 0.930415i \(0.380553\pi\)
\(68\) 9.00000 1.09141
\(69\) 0 0
\(70\) −0.472136 −0.0564310
\(71\) −2.90983 −0.345333 −0.172667 0.984980i \(-0.555238\pi\)
−0.172667 + 0.984980i \(0.555238\pi\)
\(72\) 0 0
\(73\) 0.145898 0.0170761 0.00853804 0.999964i \(-0.497282\pi\)
0.00853804 + 0.999964i \(0.497282\pi\)
\(74\) −1.94427 −0.226017
\(75\) 0 0
\(76\) 1.85410 0.212680
\(77\) 0 0
\(78\) 0 0
\(79\) −10.5623 −1.18835 −0.594176 0.804335i \(-0.702522\pi\)
−0.594176 + 0.804335i \(0.702522\pi\)
\(80\) −3.88854 −0.434752
\(81\) 0 0
\(82\) 0.965558 0.106628
\(83\) 9.76393 1.07173 0.535865 0.844303i \(-0.319985\pi\)
0.535865 + 0.844303i \(0.319985\pi\)
\(84\) 0 0
\(85\) 6.00000 0.650791
\(86\) −0.708204 −0.0763676
\(87\) 0 0
\(88\) 0 0
\(89\) −6.76393 −0.716975 −0.358488 0.933534i \(-0.616707\pi\)
−0.358488 + 0.933534i \(0.616707\pi\)
\(90\) 0 0
\(91\) 6.47214 0.678464
\(92\) −8.56231 −0.892682
\(93\) 0 0
\(94\) 4.56231 0.470566
\(95\) 1.23607 0.126818
\(96\) 0 0
\(97\) 6.00000 0.609208 0.304604 0.952479i \(-0.401476\pi\)
0.304604 + 0.952479i \(0.401476\pi\)
\(98\) −2.29180 −0.231506
\(99\) 0 0
\(100\) 6.43769 0.643769
\(101\) 8.47214 0.843009 0.421505 0.906826i \(-0.361502\pi\)
0.421505 + 0.906826i \(0.361502\pi\)
\(102\) 0 0
\(103\) −10.2361 −1.00859 −0.504295 0.863532i \(-0.668248\pi\)
−0.504295 + 0.863532i \(0.668248\pi\)
\(104\) −9.52786 −0.934284
\(105\) 0 0
\(106\) 1.56231 0.151745
\(107\) −14.0344 −1.35676 −0.678380 0.734711i \(-0.737318\pi\)
−0.678380 + 0.734711i \(0.737318\pi\)
\(108\) 0 0
\(109\) −7.14590 −0.684453 −0.342226 0.939618i \(-0.611181\pi\)
−0.342226 + 0.939618i \(0.611181\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.14590 0.297259
\(113\) 17.2361 1.62143 0.810716 0.585439i \(-0.199078\pi\)
0.810716 + 0.585439i \(0.199078\pi\)
\(114\) 0 0
\(115\) −5.70820 −0.532293
\(116\) −9.00000 −0.835629
\(117\) 0 0
\(118\) 0.618034 0.0568946
\(119\) −4.85410 −0.444975
\(120\) 0 0
\(121\) 0 0
\(122\) −4.14590 −0.375352
\(123\) 0 0
\(124\) −1.14590 −0.102905
\(125\) 10.4721 0.936656
\(126\) 0 0
\(127\) 10.5623 0.937253 0.468627 0.883396i \(-0.344749\pi\)
0.468627 + 0.883396i \(0.344749\pi\)
\(128\) −10.0902 −0.891853
\(129\) 0 0
\(130\) −3.05573 −0.268005
\(131\) −21.2361 −1.85540 −0.927702 0.373322i \(-0.878219\pi\)
−0.927702 + 0.373322i \(0.878219\pi\)
\(132\) 0 0
\(133\) −1.00000 −0.0867110
\(134\) 2.29180 0.197981
\(135\) 0 0
\(136\) 7.14590 0.612756
\(137\) −8.61803 −0.736288 −0.368144 0.929769i \(-0.620007\pi\)
−0.368144 + 0.929769i \(0.620007\pi\)
\(138\) 0 0
\(139\) −9.70820 −0.823439 −0.411720 0.911311i \(-0.635072\pi\)
−0.411720 + 0.911311i \(0.635072\pi\)
\(140\) 2.29180 0.193692
\(141\) 0 0
\(142\) −1.11146 −0.0932713
\(143\) 0 0
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 0.0557281 0.00461209
\(147\) 0 0
\(148\) 9.43769 0.775774
\(149\) 15.1803 1.24362 0.621811 0.783167i \(-0.286397\pi\)
0.621811 + 0.783167i \(0.286397\pi\)
\(150\) 0 0
\(151\) −4.32624 −0.352064 −0.176032 0.984384i \(-0.556326\pi\)
−0.176032 + 0.984384i \(0.556326\pi\)
\(152\) 1.47214 0.119406
\(153\) 0 0
\(154\) 0 0
\(155\) −0.763932 −0.0613605
\(156\) 0 0
\(157\) 7.14590 0.570305 0.285152 0.958482i \(-0.407956\pi\)
0.285152 + 0.958482i \(0.407956\pi\)
\(158\) −4.03444 −0.320963
\(159\) 0 0
\(160\) −5.12461 −0.405136
\(161\) 4.61803 0.363952
\(162\) 0 0
\(163\) 20.7082 1.62199 0.810996 0.585052i \(-0.198926\pi\)
0.810996 + 0.585052i \(0.198926\pi\)
\(164\) −4.68692 −0.365987
\(165\) 0 0
\(166\) 3.72949 0.289465
\(167\) 5.76393 0.446026 0.223013 0.974815i \(-0.428411\pi\)
0.223013 + 0.974815i \(0.428411\pi\)
\(168\) 0 0
\(169\) 28.8885 2.22220
\(170\) 2.29180 0.175773
\(171\) 0 0
\(172\) 3.43769 0.262122
\(173\) 23.0902 1.75551 0.877757 0.479107i \(-0.159039\pi\)
0.877757 + 0.479107i \(0.159039\pi\)
\(174\) 0 0
\(175\) −3.47214 −0.262469
\(176\) 0 0
\(177\) 0 0
\(178\) −2.58359 −0.193648
\(179\) −5.05573 −0.377883 −0.188941 0.981988i \(-0.560506\pi\)
−0.188941 + 0.981988i \(0.560506\pi\)
\(180\) 0 0
\(181\) −9.90983 −0.736592 −0.368296 0.929709i \(-0.620059\pi\)
−0.368296 + 0.929709i \(0.620059\pi\)
\(182\) 2.47214 0.183247
\(183\) 0 0
\(184\) −6.79837 −0.501183
\(185\) 6.29180 0.462582
\(186\) 0 0
\(187\) 0 0
\(188\) −22.1459 −1.61516
\(189\) 0 0
\(190\) 0.472136 0.0342523
\(191\) 5.94427 0.430112 0.215056 0.976602i \(-0.431007\pi\)
0.215056 + 0.976602i \(0.431007\pi\)
\(192\) 0 0
\(193\) 11.0000 0.791797 0.395899 0.918294i \(-0.370433\pi\)
0.395899 + 0.918294i \(0.370433\pi\)
\(194\) 2.29180 0.164541
\(195\) 0 0
\(196\) 11.1246 0.794615
\(197\) −8.23607 −0.586796 −0.293398 0.955990i \(-0.594786\pi\)
−0.293398 + 0.955990i \(0.594786\pi\)
\(198\) 0 0
\(199\) −7.14590 −0.506559 −0.253280 0.967393i \(-0.581509\pi\)
−0.253280 + 0.967393i \(0.581509\pi\)
\(200\) 5.11146 0.361435
\(201\) 0 0
\(202\) 3.23607 0.227689
\(203\) 4.85410 0.340691
\(204\) 0 0
\(205\) −3.12461 −0.218232
\(206\) −3.90983 −0.272411
\(207\) 0 0
\(208\) 20.3607 1.41176
\(209\) 0 0
\(210\) 0 0
\(211\) −20.4721 −1.40936 −0.704680 0.709525i \(-0.748909\pi\)
−0.704680 + 0.709525i \(0.748909\pi\)
\(212\) −7.58359 −0.520843
\(213\) 0 0
\(214\) −5.36068 −0.366449
\(215\) 2.29180 0.156299
\(216\) 0 0
\(217\) 0.618034 0.0419549
\(218\) −2.72949 −0.184864
\(219\) 0 0
\(220\) 0 0
\(221\) −31.4164 −2.11330
\(222\) 0 0
\(223\) −15.5066 −1.03840 −0.519199 0.854654i \(-0.673770\pi\)
−0.519199 + 0.854654i \(0.673770\pi\)
\(224\) 4.14590 0.277009
\(225\) 0 0
\(226\) 6.58359 0.437934
\(227\) 16.6180 1.10298 0.551489 0.834182i \(-0.314060\pi\)
0.551489 + 0.834182i \(0.314060\pi\)
\(228\) 0 0
\(229\) 10.1803 0.672736 0.336368 0.941731i \(-0.390801\pi\)
0.336368 + 0.941731i \(0.390801\pi\)
\(230\) −2.18034 −0.143767
\(231\) 0 0
\(232\) −7.14590 −0.469151
\(233\) 11.1246 0.728798 0.364399 0.931243i \(-0.381275\pi\)
0.364399 + 0.931243i \(0.381275\pi\)
\(234\) 0 0
\(235\) −14.7639 −0.963093
\(236\) −3.00000 −0.195283
\(237\) 0 0
\(238\) −1.85410 −0.120184
\(239\) −10.4721 −0.677386 −0.338693 0.940897i \(-0.609985\pi\)
−0.338693 + 0.940897i \(0.609985\pi\)
\(240\) 0 0
\(241\) 26.8328 1.72845 0.864227 0.503102i \(-0.167808\pi\)
0.864227 + 0.503102i \(0.167808\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 20.1246 1.28835
\(245\) 7.41641 0.473817
\(246\) 0 0
\(247\) −6.47214 −0.411812
\(248\) −0.909830 −0.0577743
\(249\) 0 0
\(250\) 4.00000 0.252982
\(251\) 20.5623 1.29788 0.648941 0.760839i \(-0.275212\pi\)
0.648941 + 0.760839i \(0.275212\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 4.03444 0.253143
\(255\) 0 0
\(256\) 5.56231 0.347644
\(257\) −14.9098 −0.930050 −0.465025 0.885298i \(-0.653955\pi\)
−0.465025 + 0.885298i \(0.653955\pi\)
\(258\) 0 0
\(259\) −5.09017 −0.316288
\(260\) 14.8328 0.919892
\(261\) 0 0
\(262\) −8.11146 −0.501127
\(263\) 12.7082 0.783621 0.391811 0.920046i \(-0.371849\pi\)
0.391811 + 0.920046i \(0.371849\pi\)
\(264\) 0 0
\(265\) −5.05573 −0.310571
\(266\) −0.381966 −0.0234198
\(267\) 0 0
\(268\) −11.1246 −0.679544
\(269\) −21.2705 −1.29689 −0.648443 0.761263i \(-0.724579\pi\)
−0.648443 + 0.761263i \(0.724579\pi\)
\(270\) 0 0
\(271\) −10.7639 −0.653862 −0.326931 0.945048i \(-0.606015\pi\)
−0.326931 + 0.945048i \(0.606015\pi\)
\(272\) −15.2705 −0.925911
\(273\) 0 0
\(274\) −3.29180 −0.198865
\(275\) 0 0
\(276\) 0 0
\(277\) 19.2361 1.15578 0.577892 0.816113i \(-0.303876\pi\)
0.577892 + 0.816113i \(0.303876\pi\)
\(278\) −3.70820 −0.222403
\(279\) 0 0
\(280\) 1.81966 0.108745
\(281\) 9.32624 0.556357 0.278178 0.960529i \(-0.410269\pi\)
0.278178 + 0.960529i \(0.410269\pi\)
\(282\) 0 0
\(283\) 17.0000 1.01055 0.505273 0.862960i \(-0.331392\pi\)
0.505273 + 0.862960i \(0.331392\pi\)
\(284\) 5.39512 0.320142
\(285\) 0 0
\(286\) 0 0
\(287\) 2.52786 0.149215
\(288\) 0 0
\(289\) 6.56231 0.386018
\(290\) −2.29180 −0.134579
\(291\) 0 0
\(292\) −0.270510 −0.0158304
\(293\) 29.1803 1.70473 0.852367 0.522944i \(-0.175166\pi\)
0.852367 + 0.522944i \(0.175166\pi\)
\(294\) 0 0
\(295\) −2.00000 −0.116445
\(296\) 7.49342 0.435546
\(297\) 0 0
\(298\) 5.79837 0.335891
\(299\) 29.8885 1.72850
\(300\) 0 0
\(301\) −1.85410 −0.106869
\(302\) −1.65248 −0.0950893
\(303\) 0 0
\(304\) −3.14590 −0.180430
\(305\) 13.4164 0.768221
\(306\) 0 0
\(307\) −7.85410 −0.448257 −0.224129 0.974560i \(-0.571954\pi\)
−0.224129 + 0.974560i \(0.571954\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −0.291796 −0.0165729
\(311\) 15.1803 0.860798 0.430399 0.902639i \(-0.358373\pi\)
0.430399 + 0.902639i \(0.358373\pi\)
\(312\) 0 0
\(313\) 3.47214 0.196257 0.0981284 0.995174i \(-0.468714\pi\)
0.0981284 + 0.995174i \(0.468714\pi\)
\(314\) 2.72949 0.154034
\(315\) 0 0
\(316\) 19.5836 1.10166
\(317\) 20.9443 1.17635 0.588174 0.808735i \(-0.299847\pi\)
0.588174 + 0.808735i \(0.299847\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5.81966 0.325329
\(321\) 0 0
\(322\) 1.76393 0.0983001
\(323\) 4.85410 0.270089
\(324\) 0 0
\(325\) −22.4721 −1.24653
\(326\) 7.90983 0.438085
\(327\) 0 0
\(328\) −3.72136 −0.205478
\(329\) 11.9443 0.658509
\(330\) 0 0
\(331\) 14.4164 0.792397 0.396199 0.918165i \(-0.370329\pi\)
0.396199 + 0.918165i \(0.370329\pi\)
\(332\) −18.1033 −0.993549
\(333\) 0 0
\(334\) 2.20163 0.120468
\(335\) −7.41641 −0.405202
\(336\) 0 0
\(337\) 11.4721 0.624927 0.312464 0.949930i \(-0.398846\pi\)
0.312464 + 0.949930i \(0.398846\pi\)
\(338\) 11.0344 0.600195
\(339\) 0 0
\(340\) −11.1246 −0.603317
\(341\) 0 0
\(342\) 0 0
\(343\) −13.0000 −0.701934
\(344\) 2.72949 0.147164
\(345\) 0 0
\(346\) 8.81966 0.474148
\(347\) 8.88854 0.477162 0.238581 0.971123i \(-0.423318\pi\)
0.238581 + 0.971123i \(0.423318\pi\)
\(348\) 0 0
\(349\) 4.14590 0.221925 0.110962 0.993825i \(-0.464607\pi\)
0.110962 + 0.993825i \(0.464607\pi\)
\(350\) −1.32624 −0.0708904
\(351\) 0 0
\(352\) 0 0
\(353\) −16.4164 −0.873757 −0.436879 0.899520i \(-0.643916\pi\)
−0.436879 + 0.899520i \(0.643916\pi\)
\(354\) 0 0
\(355\) 3.59675 0.190896
\(356\) 12.5410 0.664673
\(357\) 0 0
\(358\) −1.93112 −0.102063
\(359\) 22.8541 1.20619 0.603097 0.797668i \(-0.293933\pi\)
0.603097 + 0.797668i \(0.293933\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −3.78522 −0.198947
\(363\) 0 0
\(364\) −12.0000 −0.628971
\(365\) −0.180340 −0.00943942
\(366\) 0 0
\(367\) −12.7082 −0.663363 −0.331681 0.943391i \(-0.607616\pi\)
−0.331681 + 0.943391i \(0.607616\pi\)
\(368\) 14.5279 0.757317
\(369\) 0 0
\(370\) 2.40325 0.124939
\(371\) 4.09017 0.212351
\(372\) 0 0
\(373\) 5.65248 0.292674 0.146337 0.989235i \(-0.453252\pi\)
0.146337 + 0.989235i \(0.453252\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −17.5836 −0.906805
\(377\) 31.4164 1.61803
\(378\) 0 0
\(379\) 15.7984 0.811508 0.405754 0.913982i \(-0.367009\pi\)
0.405754 + 0.913982i \(0.367009\pi\)
\(380\) −2.29180 −0.117567
\(381\) 0 0
\(382\) 2.27051 0.116169
\(383\) 19.8541 1.01450 0.507249 0.861800i \(-0.330663\pi\)
0.507249 + 0.861800i \(0.330663\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.20163 0.213857
\(387\) 0 0
\(388\) −11.1246 −0.564767
\(389\) −22.6525 −1.14853 −0.574263 0.818671i \(-0.694711\pi\)
−0.574263 + 0.818671i \(0.694711\pi\)
\(390\) 0 0
\(391\) −22.4164 −1.13365
\(392\) 8.83282 0.446125
\(393\) 0 0
\(394\) −3.14590 −0.158488
\(395\) 13.0557 0.656905
\(396\) 0 0
\(397\) −2.12461 −0.106631 −0.0533156 0.998578i \(-0.516979\pi\)
−0.0533156 + 0.998578i \(0.516979\pi\)
\(398\) −2.72949 −0.136817
\(399\) 0 0
\(400\) −10.9230 −0.546149
\(401\) 14.5623 0.727207 0.363603 0.931554i \(-0.381546\pi\)
0.363603 + 0.931554i \(0.381546\pi\)
\(402\) 0 0
\(403\) 4.00000 0.199254
\(404\) −15.7082 −0.781512
\(405\) 0 0
\(406\) 1.85410 0.0920175
\(407\) 0 0
\(408\) 0 0
\(409\) 33.7426 1.66847 0.834233 0.551412i \(-0.185911\pi\)
0.834233 + 0.551412i \(0.185911\pi\)
\(410\) −1.19350 −0.0589425
\(411\) 0 0
\(412\) 18.9787 0.935014
\(413\) 1.61803 0.0796182
\(414\) 0 0
\(415\) −12.0689 −0.592438
\(416\) 26.8328 1.31559
\(417\) 0 0
\(418\) 0 0
\(419\) 8.09017 0.395231 0.197615 0.980280i \(-0.436680\pi\)
0.197615 + 0.980280i \(0.436680\pi\)
\(420\) 0 0
\(421\) 10.2361 0.498875 0.249438 0.968391i \(-0.419754\pi\)
0.249438 + 0.968391i \(0.419754\pi\)
\(422\) −7.81966 −0.380655
\(423\) 0 0
\(424\) −6.02129 −0.292420
\(425\) 16.8541 0.817544
\(426\) 0 0
\(427\) −10.8541 −0.525267
\(428\) 26.0213 1.25779
\(429\) 0 0
\(430\) 0.875388 0.0422150
\(431\) 34.0689 1.64104 0.820520 0.571618i \(-0.193684\pi\)
0.820520 + 0.571618i \(0.193684\pi\)
\(432\) 0 0
\(433\) 14.7426 0.708486 0.354243 0.935153i \(-0.384739\pi\)
0.354243 + 0.935153i \(0.384739\pi\)
\(434\) 0.236068 0.0113316
\(435\) 0 0
\(436\) 13.2492 0.634523
\(437\) −4.61803 −0.220911
\(438\) 0 0
\(439\) 0.708204 0.0338007 0.0169004 0.999857i \(-0.494620\pi\)
0.0169004 + 0.999857i \(0.494620\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −12.0000 −0.570782
\(443\) −1.23607 −0.0587274 −0.0293637 0.999569i \(-0.509348\pi\)
−0.0293637 + 0.999569i \(0.509348\pi\)
\(444\) 0 0
\(445\) 8.36068 0.396334
\(446\) −5.92299 −0.280462
\(447\) 0 0
\(448\) −4.70820 −0.222442
\(449\) 7.47214 0.352632 0.176316 0.984334i \(-0.443582\pi\)
0.176316 + 0.984334i \(0.443582\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −31.9574 −1.50315
\(453\) 0 0
\(454\) 6.34752 0.297904
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) 1.00000 0.0467780 0.0233890 0.999726i \(-0.492554\pi\)
0.0233890 + 0.999726i \(0.492554\pi\)
\(458\) 3.88854 0.181700
\(459\) 0 0
\(460\) 10.5836 0.493463
\(461\) 16.1459 0.751989 0.375995 0.926622i \(-0.377301\pi\)
0.375995 + 0.926622i \(0.377301\pi\)
\(462\) 0 0
\(463\) 17.0000 0.790057 0.395029 0.918669i \(-0.370735\pi\)
0.395029 + 0.918669i \(0.370735\pi\)
\(464\) 15.2705 0.708916
\(465\) 0 0
\(466\) 4.24922 0.196841
\(467\) −0.763932 −0.0353506 −0.0176753 0.999844i \(-0.505627\pi\)
−0.0176753 + 0.999844i \(0.505627\pi\)
\(468\) 0 0
\(469\) 6.00000 0.277054
\(470\) −5.63932 −0.260122
\(471\) 0 0
\(472\) −2.38197 −0.109639
\(473\) 0 0
\(474\) 0 0
\(475\) 3.47214 0.159313
\(476\) 9.00000 0.412514
\(477\) 0 0
\(478\) −4.00000 −0.182956
\(479\) 11.7639 0.537508 0.268754 0.963209i \(-0.413388\pi\)
0.268754 + 0.963209i \(0.413388\pi\)
\(480\) 0 0
\(481\) −32.9443 −1.50213
\(482\) 10.2492 0.466839
\(483\) 0 0
\(484\) 0 0
\(485\) −7.41641 −0.336762
\(486\) 0 0
\(487\) −12.9787 −0.588122 −0.294061 0.955787i \(-0.595007\pi\)
−0.294061 + 0.955787i \(0.595007\pi\)
\(488\) 15.9787 0.723322
\(489\) 0 0
\(490\) 2.83282 0.127974
\(491\) −7.32624 −0.330628 −0.165314 0.986241i \(-0.552864\pi\)
−0.165314 + 0.986241i \(0.552864\pi\)
\(492\) 0 0
\(493\) −23.5623 −1.06119
\(494\) −2.47214 −0.111227
\(495\) 0 0
\(496\) 1.94427 0.0873004
\(497\) −2.90983 −0.130524
\(498\) 0 0
\(499\) 38.4164 1.71975 0.859877 0.510501i \(-0.170540\pi\)
0.859877 + 0.510501i \(0.170540\pi\)
\(500\) −19.4164 −0.868328
\(501\) 0 0
\(502\) 7.85410 0.350546
\(503\) 9.52786 0.424826 0.212413 0.977180i \(-0.431868\pi\)
0.212413 + 0.977180i \(0.431868\pi\)
\(504\) 0 0
\(505\) −10.4721 −0.466004
\(506\) 0 0
\(507\) 0 0
\(508\) −19.5836 −0.868881
\(509\) −27.1803 −1.20475 −0.602374 0.798214i \(-0.705778\pi\)
−0.602374 + 0.798214i \(0.705778\pi\)
\(510\) 0 0
\(511\) 0.145898 0.00645415
\(512\) 22.3050 0.985749
\(513\) 0 0
\(514\) −5.69505 −0.251198
\(515\) 12.6525 0.557535
\(516\) 0 0
\(517\) 0 0
\(518\) −1.94427 −0.0854264
\(519\) 0 0
\(520\) 11.7771 0.516459
\(521\) −40.3607 −1.76823 −0.884117 0.467266i \(-0.845239\pi\)
−0.884117 + 0.467266i \(0.845239\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 39.3738 1.72005
\(525\) 0 0
\(526\) 4.85410 0.211649
\(527\) −3.00000 −0.130682
\(528\) 0 0
\(529\) −1.67376 −0.0727723
\(530\) −1.93112 −0.0838823
\(531\) 0 0
\(532\) 1.85410 0.0803855
\(533\) 16.3607 0.708660
\(534\) 0 0
\(535\) 17.3475 0.749999
\(536\) −8.83282 −0.381520
\(537\) 0 0
\(538\) −8.12461 −0.350277
\(539\) 0 0
\(540\) 0 0
\(541\) 11.9098 0.512044 0.256022 0.966671i \(-0.417588\pi\)
0.256022 + 0.966671i \(0.417588\pi\)
\(542\) −4.11146 −0.176602
\(543\) 0 0
\(544\) −20.1246 −0.862836
\(545\) 8.83282 0.378356
\(546\) 0 0
\(547\) 32.3262 1.38217 0.691085 0.722773i \(-0.257133\pi\)
0.691085 + 0.722773i \(0.257133\pi\)
\(548\) 15.9787 0.682577
\(549\) 0 0
\(550\) 0 0
\(551\) −4.85410 −0.206792
\(552\) 0 0
\(553\) −10.5623 −0.449155
\(554\) 7.34752 0.312166
\(555\) 0 0
\(556\) 18.0000 0.763370
\(557\) 0.652476 0.0276463 0.0138231 0.999904i \(-0.495600\pi\)
0.0138231 + 0.999904i \(0.495600\pi\)
\(558\) 0 0
\(559\) −12.0000 −0.507546
\(560\) −3.88854 −0.164321
\(561\) 0 0
\(562\) 3.56231 0.150267
\(563\) 20.3050 0.855752 0.427876 0.903838i \(-0.359262\pi\)
0.427876 + 0.903838i \(0.359262\pi\)
\(564\) 0 0
\(565\) −21.3050 −0.896306
\(566\) 6.49342 0.272939
\(567\) 0 0
\(568\) 4.28367 0.179739
\(569\) −38.7984 −1.62651 −0.813256 0.581906i \(-0.802307\pi\)
−0.813256 + 0.581906i \(0.802307\pi\)
\(570\) 0 0
\(571\) 18.5279 0.775367 0.387683 0.921793i \(-0.373275\pi\)
0.387683 + 0.921793i \(0.373275\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0.965558 0.0403016
\(575\) −16.0344 −0.668682
\(576\) 0 0
\(577\) 35.4164 1.47440 0.737202 0.675672i \(-0.236147\pi\)
0.737202 + 0.675672i \(0.236147\pi\)
\(578\) 2.50658 0.104260
\(579\) 0 0
\(580\) 11.1246 0.461924
\(581\) 9.76393 0.405076
\(582\) 0 0
\(583\) 0 0
\(584\) −0.214782 −0.00888773
\(585\) 0 0
\(586\) 11.1459 0.460433
\(587\) 13.4164 0.553754 0.276877 0.960905i \(-0.410700\pi\)
0.276877 + 0.960905i \(0.410700\pi\)
\(588\) 0 0
\(589\) −0.618034 −0.0254656
\(590\) −0.763932 −0.0314506
\(591\) 0 0
\(592\) −16.0132 −0.658137
\(593\) 10.9443 0.449427 0.224714 0.974425i \(-0.427855\pi\)
0.224714 + 0.974425i \(0.427855\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) −28.1459 −1.15290
\(597\) 0 0
\(598\) 11.4164 0.466852
\(599\) −13.3607 −0.545903 −0.272951 0.962028i \(-0.588000\pi\)
−0.272951 + 0.962028i \(0.588000\pi\)
\(600\) 0 0
\(601\) 17.4721 0.712703 0.356352 0.934352i \(-0.384021\pi\)
0.356352 + 0.934352i \(0.384021\pi\)
\(602\) −0.708204 −0.0288642
\(603\) 0 0
\(604\) 8.02129 0.326382
\(605\) 0 0
\(606\) 0 0
\(607\) 3.58359 0.145454 0.0727268 0.997352i \(-0.476830\pi\)
0.0727268 + 0.997352i \(0.476830\pi\)
\(608\) −4.14590 −0.168138
\(609\) 0 0
\(610\) 5.12461 0.207489
\(611\) 77.3050 3.12742
\(612\) 0 0
\(613\) −19.8541 −0.801900 −0.400950 0.916100i \(-0.631320\pi\)
−0.400950 + 0.916100i \(0.631320\pi\)
\(614\) −3.00000 −0.121070
\(615\) 0 0
\(616\) 0 0
\(617\) −27.1591 −1.09338 −0.546691 0.837334i \(-0.684113\pi\)
−0.546691 + 0.837334i \(0.684113\pi\)
\(618\) 0 0
\(619\) 27.5623 1.10782 0.553911 0.832576i \(-0.313135\pi\)
0.553911 + 0.832576i \(0.313135\pi\)
\(620\) 1.41641 0.0568843
\(621\) 0 0
\(622\) 5.79837 0.232494
\(623\) −6.76393 −0.270991
\(624\) 0 0
\(625\) 4.41641 0.176656
\(626\) 1.32624 0.0530071
\(627\) 0 0
\(628\) −13.2492 −0.528702
\(629\) 24.7082 0.985181
\(630\) 0 0
\(631\) −10.8885 −0.433466 −0.216733 0.976231i \(-0.569540\pi\)
−0.216733 + 0.976231i \(0.569540\pi\)
\(632\) 15.5492 0.618512
\(633\) 0 0
\(634\) 8.00000 0.317721
\(635\) −13.0557 −0.518101
\(636\) 0 0
\(637\) −38.8328 −1.53861
\(638\) 0 0
\(639\) 0 0
\(640\) 12.4721 0.493004
\(641\) −20.4508 −0.807760 −0.403880 0.914812i \(-0.632339\pi\)
−0.403880 + 0.914812i \(0.632339\pi\)
\(642\) 0 0
\(643\) −6.85410 −0.270299 −0.135150 0.990825i \(-0.543152\pi\)
−0.135150 + 0.990825i \(0.543152\pi\)
\(644\) −8.56231 −0.337402
\(645\) 0 0
\(646\) 1.85410 0.0729487
\(647\) −4.18034 −0.164346 −0.0821731 0.996618i \(-0.526186\pi\)
−0.0821731 + 0.996618i \(0.526186\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −8.58359 −0.336676
\(651\) 0 0
\(652\) −38.3951 −1.50367
\(653\) −1.36068 −0.0532475 −0.0266238 0.999646i \(-0.508476\pi\)
−0.0266238 + 0.999646i \(0.508476\pi\)
\(654\) 0 0
\(655\) 26.2492 1.02564
\(656\) 7.95240 0.310489
\(657\) 0 0
\(658\) 4.56231 0.177857
\(659\) −9.97871 −0.388715 −0.194358 0.980931i \(-0.562262\pi\)
−0.194358 + 0.980931i \(0.562262\pi\)
\(660\) 0 0
\(661\) −4.29180 −0.166932 −0.0834658 0.996511i \(-0.526599\pi\)
−0.0834658 + 0.996511i \(0.526599\pi\)
\(662\) 5.50658 0.214019
\(663\) 0 0
\(664\) −14.3738 −0.557813
\(665\) 1.23607 0.0479327
\(666\) 0 0
\(667\) 22.4164 0.867967
\(668\) −10.6869 −0.413489
\(669\) 0 0
\(670\) −2.83282 −0.109441
\(671\) 0 0
\(672\) 0 0
\(673\) 29.5623 1.13954 0.569772 0.821803i \(-0.307032\pi\)
0.569772 + 0.821803i \(0.307032\pi\)
\(674\) 4.38197 0.168787
\(675\) 0 0
\(676\) −53.5623 −2.06009
\(677\) −6.05573 −0.232741 −0.116370 0.993206i \(-0.537126\pi\)
−0.116370 + 0.993206i \(0.537126\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) −8.83282 −0.338723
\(681\) 0 0
\(682\) 0 0
\(683\) −5.52786 −0.211518 −0.105759 0.994392i \(-0.533727\pi\)
−0.105759 + 0.994392i \(0.533727\pi\)
\(684\) 0 0
\(685\) 10.6525 0.407010
\(686\) −4.96556 −0.189586
\(687\) 0 0
\(688\) −5.83282 −0.222374
\(689\) 26.4721 1.00851
\(690\) 0 0
\(691\) 14.5967 0.555286 0.277643 0.960684i \(-0.410447\pi\)
0.277643 + 0.960684i \(0.410447\pi\)
\(692\) −42.8115 −1.62745
\(693\) 0 0
\(694\) 3.39512 0.128877
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) −12.2705 −0.464779
\(698\) 1.58359 0.0599398
\(699\) 0 0
\(700\) 6.43769 0.243322
\(701\) 47.7214 1.80241 0.901205 0.433392i \(-0.142683\pi\)
0.901205 + 0.433392i \(0.142683\pi\)
\(702\) 0 0
\(703\) 5.09017 0.191979
\(704\) 0 0
\(705\) 0 0
\(706\) −6.27051 −0.235994
\(707\) 8.47214 0.318627
\(708\) 0 0
\(709\) −25.7639 −0.967585 −0.483792 0.875183i \(-0.660741\pi\)
−0.483792 + 0.875183i \(0.660741\pi\)
\(710\) 1.37384 0.0515591
\(711\) 0 0
\(712\) 9.95743 0.373170
\(713\) 2.85410 0.106887
\(714\) 0 0
\(715\) 0 0
\(716\) 9.37384 0.350317
\(717\) 0 0
\(718\) 8.72949 0.325782
\(719\) −19.5967 −0.730835 −0.365418 0.930844i \(-0.619074\pi\)
−0.365418 + 0.930844i \(0.619074\pi\)
\(720\) 0 0
\(721\) −10.2361 −0.381211
\(722\) −6.87539 −0.255875
\(723\) 0 0
\(724\) 18.3738 0.682858
\(725\) −16.8541 −0.625946
\(726\) 0 0
\(727\) 18.8328 0.698470 0.349235 0.937035i \(-0.386441\pi\)
0.349235 + 0.937035i \(0.386441\pi\)
\(728\) −9.52786 −0.353126
\(729\) 0 0
\(730\) −0.0688837 −0.00254950
\(731\) 9.00000 0.332877
\(732\) 0 0
\(733\) −48.4721 −1.79036 −0.895180 0.445706i \(-0.852953\pi\)
−0.895180 + 0.445706i \(0.852953\pi\)
\(734\) −4.85410 −0.179168
\(735\) 0 0
\(736\) 19.1459 0.705727
\(737\) 0 0
\(738\) 0 0
\(739\) −42.5410 −1.56490 −0.782448 0.622715i \(-0.786029\pi\)
−0.782448 + 0.622715i \(0.786029\pi\)
\(740\) −11.6656 −0.428837
\(741\) 0 0
\(742\) 1.56231 0.0573541
\(743\) 42.3607 1.55406 0.777031 0.629462i \(-0.216725\pi\)
0.777031 + 0.629462i \(0.216725\pi\)
\(744\) 0 0
\(745\) −18.7639 −0.687457
\(746\) 2.15905 0.0790486
\(747\) 0 0
\(748\) 0 0
\(749\) −14.0344 −0.512807
\(750\) 0 0
\(751\) −31.2492 −1.14030 −0.570150 0.821540i \(-0.693115\pi\)
−0.570150 + 0.821540i \(0.693115\pi\)
\(752\) 37.5755 1.37024
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 5.34752 0.194616
\(756\) 0 0
\(757\) 11.0000 0.399802 0.199901 0.979816i \(-0.435938\pi\)
0.199901 + 0.979816i \(0.435938\pi\)
\(758\) 6.03444 0.219181
\(759\) 0 0
\(760\) −1.81966 −0.0660060
\(761\) −6.97871 −0.252978 −0.126489 0.991968i \(-0.540371\pi\)
−0.126489 + 0.991968i \(0.540371\pi\)
\(762\) 0 0
\(763\) −7.14590 −0.258699
\(764\) −11.0213 −0.398736
\(765\) 0 0
\(766\) 7.58359 0.274006
\(767\) 10.4721 0.378127
\(768\) 0 0
\(769\) 15.7984 0.569704 0.284852 0.958572i \(-0.408056\pi\)
0.284852 + 0.958572i \(0.408056\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −20.3951 −0.734036
\(773\) 23.6525 0.850720 0.425360 0.905024i \(-0.360147\pi\)
0.425360 + 0.905024i \(0.360147\pi\)
\(774\) 0 0
\(775\) −2.14590 −0.0770829
\(776\) −8.83282 −0.317080
\(777\) 0 0
\(778\) −8.65248 −0.310206
\(779\) −2.52786 −0.0905701
\(780\) 0 0
\(781\) 0 0
\(782\) −8.56231 −0.306187
\(783\) 0 0
\(784\) −18.8754 −0.674121
\(785\) −8.83282 −0.315257
\(786\) 0 0
\(787\) −36.0000 −1.28326 −0.641631 0.767014i \(-0.721742\pi\)
−0.641631 + 0.767014i \(0.721742\pi\)
\(788\) 15.2705 0.543989
\(789\) 0 0
\(790\) 4.98684 0.177424
\(791\) 17.2361 0.612844
\(792\) 0 0
\(793\) −70.2492 −2.49462
\(794\) −0.811529 −0.0288001
\(795\) 0 0
\(796\) 13.2492 0.469606
\(797\) 24.2705 0.859706 0.429853 0.902899i \(-0.358565\pi\)
0.429853 + 0.902899i \(0.358565\pi\)
\(798\) 0 0
\(799\) −57.9787 −2.05114
\(800\) −14.3951 −0.508944
\(801\) 0 0
\(802\) 5.56231 0.196412
\(803\) 0 0
\(804\) 0 0
\(805\) −5.70820 −0.201188
\(806\) 1.52786 0.0538167
\(807\) 0 0
\(808\) −12.4721 −0.438768
\(809\) 18.0902 0.636017 0.318008 0.948088i \(-0.396986\pi\)
0.318008 + 0.948088i \(0.396986\pi\)
\(810\) 0 0
\(811\) −8.68692 −0.305039 −0.152519 0.988300i \(-0.548739\pi\)
−0.152519 + 0.988300i \(0.548739\pi\)
\(812\) −9.00000 −0.315838
\(813\) 0 0
\(814\) 0 0
\(815\) −25.5967 −0.896615
\(816\) 0 0
\(817\) 1.85410 0.0648668
\(818\) 12.8885 0.450637
\(819\) 0 0
\(820\) 5.79335 0.202313
\(821\) −38.2148 −1.33371 −0.666853 0.745190i \(-0.732359\pi\)
−0.666853 + 0.745190i \(0.732359\pi\)
\(822\) 0 0
\(823\) −42.2918 −1.47420 −0.737100 0.675784i \(-0.763805\pi\)
−0.737100 + 0.675784i \(0.763805\pi\)
\(824\) 15.0689 0.524949
\(825\) 0 0
\(826\) 0.618034 0.0215042
\(827\) 57.0344 1.98328 0.991641 0.129028i \(-0.0411858\pi\)
0.991641 + 0.129028i \(0.0411858\pi\)
\(828\) 0 0
\(829\) 43.5410 1.51224 0.756121 0.654432i \(-0.227092\pi\)
0.756121 + 0.654432i \(0.227092\pi\)
\(830\) −4.60990 −0.160012
\(831\) 0 0
\(832\) −30.4721 −1.05643
\(833\) 29.1246 1.00911
\(834\) 0 0
\(835\) −7.12461 −0.246557
\(836\) 0 0
\(837\) 0 0
\(838\) 3.09017 0.106748
\(839\) −17.2918 −0.596979 −0.298490 0.954413i \(-0.596483\pi\)
−0.298490 + 0.954413i \(0.596483\pi\)
\(840\) 0 0
\(841\) −5.43769 −0.187507
\(842\) 3.90983 0.134742
\(843\) 0 0
\(844\) 37.9574 1.30655
\(845\) −35.7082 −1.22840
\(846\) 0 0
\(847\) 0 0
\(848\) 12.8673 0.441863
\(849\) 0 0
\(850\) 6.43769 0.220811
\(851\) −23.5066 −0.805795
\(852\) 0 0
\(853\) 35.4377 1.21336 0.606682 0.794945i \(-0.292500\pi\)
0.606682 + 0.794945i \(0.292500\pi\)
\(854\) −4.14590 −0.141870
\(855\) 0 0
\(856\) 20.6606 0.706165
\(857\) −25.3607 −0.866304 −0.433152 0.901321i \(-0.642599\pi\)
−0.433152 + 0.901321i \(0.642599\pi\)
\(858\) 0 0
\(859\) −32.5623 −1.11101 −0.555506 0.831513i \(-0.687475\pi\)
−0.555506 + 0.831513i \(0.687475\pi\)
\(860\) −4.24922 −0.144897
\(861\) 0 0
\(862\) 13.0132 0.443230
\(863\) 14.4508 0.491913 0.245956 0.969281i \(-0.420898\pi\)
0.245956 + 0.969281i \(0.420898\pi\)
\(864\) 0 0
\(865\) −28.5410 −0.970424
\(866\) 5.63119 0.191356
\(867\) 0 0
\(868\) −1.14590 −0.0388943
\(869\) 0 0
\(870\) 0 0
\(871\) 38.8328 1.31580
\(872\) 10.5197 0.356243
\(873\) 0 0
\(874\) −1.76393 −0.0596659
\(875\) 10.4721 0.354023
\(876\) 0 0
\(877\) −11.5967 −0.391594 −0.195797 0.980644i \(-0.562729\pi\)
−0.195797 + 0.980644i \(0.562729\pi\)
\(878\) 0.270510 0.00912926
\(879\) 0 0
\(880\) 0 0
\(881\) 1.81966 0.0613059 0.0306530 0.999530i \(-0.490241\pi\)
0.0306530 + 0.999530i \(0.490241\pi\)
\(882\) 0 0
\(883\) −35.6525 −1.19980 −0.599901 0.800074i \(-0.704793\pi\)
−0.599901 + 0.800074i \(0.704793\pi\)
\(884\) 58.2492 1.95913
\(885\) 0 0
\(886\) −0.472136 −0.0158617
\(887\) 4.03444 0.135463 0.0677316 0.997704i \(-0.478424\pi\)
0.0677316 + 0.997704i \(0.478424\pi\)
\(888\) 0 0
\(889\) 10.5623 0.354248
\(890\) 3.19350 0.107046
\(891\) 0 0
\(892\) 28.7508 0.962647
\(893\) −11.9443 −0.399700
\(894\) 0 0
\(895\) 6.24922 0.208889
\(896\) −10.0902 −0.337089
\(897\) 0 0
\(898\) 2.85410 0.0952426
\(899\) 3.00000 0.100056
\(900\) 0 0
\(901\) −19.8541 −0.661436
\(902\) 0 0
\(903\) 0 0
\(904\) −25.3738 −0.843921
\(905\) 12.2492 0.407178
\(906\) 0 0
\(907\) −14.5836 −0.484240 −0.242120 0.970246i \(-0.577843\pi\)
−0.242120 + 0.970246i \(0.577843\pi\)
\(908\) −30.8115 −1.02252
\(909\) 0 0
\(910\) −3.05573 −0.101296
\(911\) −19.1803 −0.635473 −0.317737 0.948179i \(-0.602923\pi\)
−0.317737 + 0.948179i \(0.602923\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.381966 0.0126343
\(915\) 0 0
\(916\) −18.8754 −0.623660
\(917\) −21.2361 −0.701277
\(918\) 0 0
\(919\) 26.8673 0.886269 0.443135 0.896455i \(-0.353866\pi\)
0.443135 + 0.896455i \(0.353866\pi\)
\(920\) 8.40325 0.277047
\(921\) 0 0
\(922\) 6.16718 0.203105
\(923\) −18.8328 −0.619890
\(924\) 0 0
\(925\) 17.6738 0.581110
\(926\) 6.49342 0.213387
\(927\) 0 0
\(928\) 20.1246 0.660623
\(929\) −48.2705 −1.58370 −0.791852 0.610713i \(-0.790883\pi\)
−0.791852 + 0.610713i \(0.790883\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) −20.6262 −0.675632
\(933\) 0 0
\(934\) −0.291796 −0.00954786
\(935\) 0 0
\(936\) 0 0
\(937\) 23.0344 0.752502 0.376251 0.926518i \(-0.377213\pi\)
0.376251 + 0.926518i \(0.377213\pi\)
\(938\) 2.29180 0.0748298
\(939\) 0 0
\(940\) 27.3738 0.892836
\(941\) −27.1591 −0.885360 −0.442680 0.896680i \(-0.645972\pi\)
−0.442680 + 0.896680i \(0.645972\pi\)
\(942\) 0 0
\(943\) 11.6738 0.380150
\(944\) 5.09017 0.165671
\(945\) 0 0
\(946\) 0 0
\(947\) 36.7082 1.19286 0.596428 0.802666i \(-0.296586\pi\)
0.596428 + 0.802666i \(0.296586\pi\)
\(948\) 0 0
\(949\) 0.944272 0.0306524
\(950\) 1.32624 0.0430288
\(951\) 0 0
\(952\) 7.14590 0.231600
\(953\) 9.49342 0.307522 0.153761 0.988108i \(-0.450861\pi\)
0.153761 + 0.988108i \(0.450861\pi\)
\(954\) 0 0
\(955\) −7.34752 −0.237760
\(956\) 19.4164 0.627972
\(957\) 0 0
\(958\) 4.49342 0.145176
\(959\) −8.61803 −0.278291
\(960\) 0 0
\(961\) −30.6180 −0.987679
\(962\) −12.5836 −0.405711
\(963\) 0 0
\(964\) −49.7508 −1.60236
\(965\) −13.5967 −0.437695
\(966\) 0 0
\(967\) −52.4853 −1.68781 −0.843907 0.536490i \(-0.819750\pi\)
−0.843907 + 0.536490i \(0.819750\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −2.83282 −0.0909562
\(971\) 9.27051 0.297505 0.148752 0.988874i \(-0.452474\pi\)
0.148752 + 0.988874i \(0.452474\pi\)
\(972\) 0 0
\(973\) −9.70820 −0.311231
\(974\) −4.95743 −0.158846
\(975\) 0 0
\(976\) −34.1459 −1.09298
\(977\) 15.7639 0.504333 0.252166 0.967684i \(-0.418857\pi\)
0.252166 + 0.967684i \(0.418857\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −13.7508 −0.439252
\(981\) 0 0
\(982\) −2.79837 −0.0892997
\(983\) 9.25735 0.295264 0.147632 0.989042i \(-0.452835\pi\)
0.147632 + 0.989042i \(0.452835\pi\)
\(984\) 0 0
\(985\) 10.1803 0.324373
\(986\) −9.00000 −0.286618
\(987\) 0 0
\(988\) 12.0000 0.381771
\(989\) −8.56231 −0.272265
\(990\) 0 0
\(991\) 20.5967 0.654277 0.327139 0.944976i \(-0.393916\pi\)
0.327139 + 0.944976i \(0.393916\pi\)
\(992\) 2.56231 0.0813533
\(993\) 0 0
\(994\) −1.11146 −0.0352532
\(995\) 8.83282 0.280019
\(996\) 0 0
\(997\) −7.38197 −0.233789 −0.116895 0.993144i \(-0.537294\pi\)
−0.116895 + 0.993144i \(0.537294\pi\)
\(998\) 14.6738 0.464490
\(999\) 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.bc.1.1 2
3.2 odd 2 9801.2.a.m.1.2 2
9.4 even 3 1089.2.e.d.727.2 4
9.7 even 3 1089.2.e.d.364.2 4
11.7 odd 10 891.2.f.b.82.1 4
11.8 odd 10 891.2.f.b.163.1 4
11.10 odd 2 9801.2.a.n.1.2 2
33.8 even 10 891.2.f.a.163.1 4
33.29 even 10 891.2.f.a.82.1 4
33.32 even 2 9801.2.a.bb.1.1 2
99.7 odd 30 99.2.m.a.49.1 yes 8
99.29 even 30 297.2.n.a.280.1 8
99.40 odd 30 99.2.m.a.16.1 8
99.41 even 30 297.2.n.a.262.1 8
99.43 odd 6 1089.2.e.g.364.1 4
99.52 odd 30 99.2.m.a.31.1 yes 8
99.74 even 30 297.2.n.a.64.1 8
99.76 odd 6 1089.2.e.g.727.1 4
99.85 odd 30 99.2.m.a.97.1 yes 8
99.95 even 30 297.2.n.a.181.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.m.a.16.1 8 99.40 odd 30
99.2.m.a.31.1 yes 8 99.52 odd 30
99.2.m.a.49.1 yes 8 99.7 odd 30
99.2.m.a.97.1 yes 8 99.85 odd 30
297.2.n.a.64.1 8 99.74 even 30
297.2.n.a.181.1 8 99.95 even 30
297.2.n.a.262.1 8 99.41 even 30
297.2.n.a.280.1 8 99.29 even 30
891.2.f.a.82.1 4 33.29 even 10
891.2.f.a.163.1 4 33.8 even 10
891.2.f.b.82.1 4 11.7 odd 10
891.2.f.b.163.1 4 11.8 odd 10
1089.2.e.d.364.2 4 9.7 even 3
1089.2.e.d.727.2 4 9.4 even 3
1089.2.e.g.364.1 4 99.43 odd 6
1089.2.e.g.727.1 4 99.76 odd 6
9801.2.a.m.1.2 2 3.2 odd 2
9801.2.a.n.1.2 2 11.10 odd 2
9801.2.a.bb.1.1 2 33.32 even 2
9801.2.a.bc.1.1 2 1.1 even 1 trivial