Properties

Label 369.2.a.e.1.3
Level $369$
Weight $2$
Character 369.1
Self dual yes
Analytic conductor $2.946$
Analytic rank $0$
Dimension $3$
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] = [369,2,Mod(1,369)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(369, 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("369.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 369 = 3^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 369.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: \(2.94647983459\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 123)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 369.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.81361 q^{2} +1.28917 q^{4} +1.10278 q^{5} +2.52444 q^{7} -1.28917 q^{8} +O(q^{10})\) \(q+1.81361 q^{2} +1.28917 q^{4} +1.10278 q^{5} +2.52444 q^{7} -1.28917 q^{8} +2.00000 q^{10} -0.813607 q^{11} +5.10278 q^{13} +4.57834 q^{14} -4.91638 q^{16} -3.39194 q^{17} +3.10278 q^{19} +1.42166 q^{20} -1.47556 q^{22} +0.897225 q^{23} -3.78389 q^{25} +9.25443 q^{26} +3.25443 q^{28} -4.44082 q^{29} -8.96526 q^{31} -6.33804 q^{32} -6.15165 q^{34} +2.78389 q^{35} +2.08362 q^{37} +5.62721 q^{38} -1.42166 q^{40} -1.00000 q^{41} +9.07306 q^{43} -1.04888 q^{44} +1.62721 q^{46} +0.235269 q^{47} -0.627213 q^{49} -6.86248 q^{50} +6.57834 q^{52} -13.8328 q^{53} -0.897225 q^{55} -3.25443 q^{56} -8.05390 q^{58} -1.04888 q^{59} +1.91638 q^{61} -16.2594 q^{62} -1.66196 q^{64} +5.62721 q^{65} -10.0383 q^{67} -4.37279 q^{68} +5.04888 q^{70} +12.8136 q^{71} +4.75971 q^{73} +3.77886 q^{74} +4.00000 q^{76} -2.05390 q^{77} -15.8328 q^{79} -5.42166 q^{80} -1.81361 q^{82} +7.68111 q^{83} -3.74055 q^{85} +16.4550 q^{86} +1.04888 q^{88} -13.2544 q^{89} +12.8816 q^{91} +1.15667 q^{92} +0.426686 q^{94} +3.42166 q^{95} +2.72999 q^{97} -1.13752 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 3 q^{4} - 4 q^{5} + 2 q^{7} - 3 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 3 q^{4} - 4 q^{5} + 2 q^{7} - 3 q^{8} + 6 q^{10} + 4 q^{11} + 8 q^{13} + 12 q^{14} - q^{16} - 2 q^{17} + 2 q^{19} + 6 q^{20} - 10 q^{22} + 10 q^{23} + 5 q^{25} + 2 q^{26} - 16 q^{28} + 6 q^{29} - 2 q^{31} - 7 q^{32} - 8 q^{35} + 20 q^{37} + 4 q^{38} - 6 q^{40} - 3 q^{41} + 10 q^{43} + 8 q^{44} - 8 q^{46} - 4 q^{47} + 11 q^{49} - 3 q^{50} + 18 q^{52} - 14 q^{53} - 10 q^{55} + 16 q^{56} - 28 q^{58} + 8 q^{59} - 8 q^{61} - 38 q^{62} - 17 q^{64} + 4 q^{65} + 12 q^{67} - 26 q^{68} + 4 q^{70} + 32 q^{71} + 4 q^{73} - 20 q^{74} + 12 q^{76} - 10 q^{77} - 20 q^{79} - 18 q^{80} + q^{82} + 14 q^{83} - 22 q^{85} - 6 q^{86} - 8 q^{88} - 14 q^{89} + 18 q^{94} + 12 q^{95} - 12 q^{97} - 21 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.81361 1.28241 0.641207 0.767368i \(-0.278434\pi\)
0.641207 + 0.767368i \(0.278434\pi\)
\(3\) 0 0
\(4\) 1.28917 0.644584
\(5\) 1.10278 0.493176 0.246588 0.969120i \(-0.420691\pi\)
0.246588 + 0.969120i \(0.420691\pi\)
\(6\) 0 0
\(7\) 2.52444 0.954148 0.477074 0.878863i \(-0.341697\pi\)
0.477074 + 0.878863i \(0.341697\pi\)
\(8\) −1.28917 −0.455790
\(9\) 0 0
\(10\) 2.00000 0.632456
\(11\) −0.813607 −0.245312 −0.122656 0.992449i \(-0.539141\pi\)
−0.122656 + 0.992449i \(0.539141\pi\)
\(12\) 0 0
\(13\) 5.10278 1.41526 0.707628 0.706586i \(-0.249765\pi\)
0.707628 + 0.706586i \(0.249765\pi\)
\(14\) 4.57834 1.22361
\(15\) 0 0
\(16\) −4.91638 −1.22910
\(17\) −3.39194 −0.822667 −0.411334 0.911485i \(-0.634937\pi\)
−0.411334 + 0.911485i \(0.634937\pi\)
\(18\) 0 0
\(19\) 3.10278 0.711825 0.355913 0.934519i \(-0.384170\pi\)
0.355913 + 0.934519i \(0.384170\pi\)
\(20\) 1.42166 0.317893
\(21\) 0 0
\(22\) −1.47556 −0.314591
\(23\) 0.897225 0.187084 0.0935422 0.995615i \(-0.470181\pi\)
0.0935422 + 0.995615i \(0.470181\pi\)
\(24\) 0 0
\(25\) −3.78389 −0.756777
\(26\) 9.25443 1.81494
\(27\) 0 0
\(28\) 3.25443 0.615029
\(29\) −4.44082 −0.824639 −0.412320 0.911039i \(-0.635281\pi\)
−0.412320 + 0.911039i \(0.635281\pi\)
\(30\) 0 0
\(31\) −8.96526 −1.61021 −0.805104 0.593134i \(-0.797890\pi\)
−0.805104 + 0.593134i \(0.797890\pi\)
\(32\) −6.33804 −1.12042
\(33\) 0 0
\(34\) −6.15165 −1.05500
\(35\) 2.78389 0.470563
\(36\) 0 0
\(37\) 2.08362 0.342545 0.171272 0.985224i \(-0.445212\pi\)
0.171272 + 0.985224i \(0.445212\pi\)
\(38\) 5.62721 0.912854
\(39\) 0 0
\(40\) −1.42166 −0.224785
\(41\) −1.00000 −0.156174
\(42\) 0 0
\(43\) 9.07306 1.38363 0.691814 0.722076i \(-0.256812\pi\)
0.691814 + 0.722076i \(0.256812\pi\)
\(44\) −1.04888 −0.158124
\(45\) 0 0
\(46\) 1.62721 0.239919
\(47\) 0.235269 0.0343176 0.0171588 0.999853i \(-0.494538\pi\)
0.0171588 + 0.999853i \(0.494538\pi\)
\(48\) 0 0
\(49\) −0.627213 −0.0896019
\(50\) −6.86248 −0.970502
\(51\) 0 0
\(52\) 6.57834 0.912251
\(53\) −13.8328 −1.90008 −0.950038 0.312134i \(-0.898956\pi\)
−0.950038 + 0.312134i \(0.898956\pi\)
\(54\) 0 0
\(55\) −0.897225 −0.120982
\(56\) −3.25443 −0.434891
\(57\) 0 0
\(58\) −8.05390 −1.05753
\(59\) −1.04888 −0.136552 −0.0682760 0.997666i \(-0.521750\pi\)
−0.0682760 + 0.997666i \(0.521750\pi\)
\(60\) 0 0
\(61\) 1.91638 0.245368 0.122684 0.992446i \(-0.460850\pi\)
0.122684 + 0.992446i \(0.460850\pi\)
\(62\) −16.2594 −2.06495
\(63\) 0 0
\(64\) −1.66196 −0.207744
\(65\) 5.62721 0.697970
\(66\) 0 0
\(67\) −10.0383 −1.22638 −0.613188 0.789937i \(-0.710113\pi\)
−0.613188 + 0.789937i \(0.710113\pi\)
\(68\) −4.37279 −0.530278
\(69\) 0 0
\(70\) 5.04888 0.603456
\(71\) 12.8136 1.52070 0.760348 0.649516i \(-0.225029\pi\)
0.760348 + 0.649516i \(0.225029\pi\)
\(72\) 0 0
\(73\) 4.75971 0.557082 0.278541 0.960424i \(-0.410149\pi\)
0.278541 + 0.960424i \(0.410149\pi\)
\(74\) 3.77886 0.439284
\(75\) 0 0
\(76\) 4.00000 0.458831
\(77\) −2.05390 −0.234064
\(78\) 0 0
\(79\) −15.8328 −1.78133 −0.890663 0.454665i \(-0.849759\pi\)
−0.890663 + 0.454665i \(0.849759\pi\)
\(80\) −5.42166 −0.606160
\(81\) 0 0
\(82\) −1.81361 −0.200279
\(83\) 7.68111 0.843112 0.421556 0.906802i \(-0.361484\pi\)
0.421556 + 0.906802i \(0.361484\pi\)
\(84\) 0 0
\(85\) −3.74055 −0.405720
\(86\) 16.4550 1.77438
\(87\) 0 0
\(88\) 1.04888 0.111811
\(89\) −13.2544 −1.40497 −0.702483 0.711700i \(-0.747925\pi\)
−0.702483 + 0.711700i \(0.747925\pi\)
\(90\) 0 0
\(91\) 12.8816 1.35036
\(92\) 1.15667 0.120592
\(93\) 0 0
\(94\) 0.426686 0.0440093
\(95\) 3.42166 0.351055
\(96\) 0 0
\(97\) 2.72999 0.277188 0.138594 0.990349i \(-0.455742\pi\)
0.138594 + 0.990349i \(0.455742\pi\)
\(98\) −1.13752 −0.114907
\(99\) 0 0
\(100\) −4.87807 −0.487807
\(101\) 11.8625 1.18036 0.590181 0.807271i \(-0.299057\pi\)
0.590181 + 0.807271i \(0.299057\pi\)
\(102\) 0 0
\(103\) −0.494719 −0.0487461 −0.0243730 0.999703i \(-0.507759\pi\)
−0.0243730 + 0.999703i \(0.507759\pi\)
\(104\) −6.57834 −0.645059
\(105\) 0 0
\(106\) −25.0872 −2.43668
\(107\) 15.0872 1.45853 0.729267 0.684229i \(-0.239861\pi\)
0.729267 + 0.684229i \(0.239861\pi\)
\(108\) 0 0
\(109\) 5.10278 0.488757 0.244379 0.969680i \(-0.421416\pi\)
0.244379 + 0.969680i \(0.421416\pi\)
\(110\) −1.62721 −0.155149
\(111\) 0 0
\(112\) −12.4111 −1.17274
\(113\) 17.5139 1.64757 0.823783 0.566905i \(-0.191859\pi\)
0.823783 + 0.566905i \(0.191859\pi\)
\(114\) 0 0
\(115\) 0.989437 0.0922655
\(116\) −5.72496 −0.531550
\(117\) 0 0
\(118\) −1.90225 −0.175116
\(119\) −8.56275 −0.784946
\(120\) 0 0
\(121\) −10.3380 −0.939822
\(122\) 3.47556 0.314663
\(123\) 0 0
\(124\) −11.5577 −1.03791
\(125\) −9.68665 −0.866400
\(126\) 0 0
\(127\) 12.8816 1.14306 0.571530 0.820581i \(-0.306350\pi\)
0.571530 + 0.820581i \(0.306350\pi\)
\(128\) 9.66196 0.854004
\(129\) 0 0
\(130\) 10.2056 0.895086
\(131\) 6.35720 0.555431 0.277716 0.960663i \(-0.410423\pi\)
0.277716 + 0.960663i \(0.410423\pi\)
\(132\) 0 0
\(133\) 7.83276 0.679187
\(134\) −18.2056 −1.57272
\(135\) 0 0
\(136\) 4.37279 0.374963
\(137\) 17.4897 1.49425 0.747123 0.664686i \(-0.231435\pi\)
0.747123 + 0.664686i \(0.231435\pi\)
\(138\) 0 0
\(139\) 15.7250 1.33377 0.666887 0.745159i \(-0.267626\pi\)
0.666887 + 0.745159i \(0.267626\pi\)
\(140\) 3.58890 0.303317
\(141\) 0 0
\(142\) 23.2388 1.95016
\(143\) −4.15165 −0.347178
\(144\) 0 0
\(145\) −4.89722 −0.406692
\(146\) 8.63224 0.714409
\(147\) 0 0
\(148\) 2.68614 0.220799
\(149\) −11.5678 −0.947669 −0.473835 0.880614i \(-0.657131\pi\)
−0.473835 + 0.880614i \(0.657131\pi\)
\(150\) 0 0
\(151\) 19.2544 1.56690 0.783451 0.621453i \(-0.213457\pi\)
0.783451 + 0.621453i \(0.213457\pi\)
\(152\) −4.00000 −0.324443
\(153\) 0 0
\(154\) −3.72496 −0.300166
\(155\) −9.88666 −0.794116
\(156\) 0 0
\(157\) −20.9200 −1.66959 −0.834797 0.550558i \(-0.814415\pi\)
−0.834797 + 0.550558i \(0.814415\pi\)
\(158\) −28.7144 −2.28440
\(159\) 0 0
\(160\) −6.98944 −0.552564
\(161\) 2.26499 0.178506
\(162\) 0 0
\(163\) −12.7980 −1.00242 −0.501209 0.865326i \(-0.667111\pi\)
−0.501209 + 0.865326i \(0.667111\pi\)
\(164\) −1.28917 −0.100667
\(165\) 0 0
\(166\) 13.9305 1.08122
\(167\) −9.62721 −0.744976 −0.372488 0.928037i \(-0.621495\pi\)
−0.372488 + 0.928037i \(0.621495\pi\)
\(168\) 0 0
\(169\) 13.0383 1.00295
\(170\) −6.78389 −0.520300
\(171\) 0 0
\(172\) 11.6967 0.891865
\(173\) −11.9461 −0.908245 −0.454123 0.890939i \(-0.650047\pi\)
−0.454123 + 0.890939i \(0.650047\pi\)
\(174\) 0 0
\(175\) −9.55219 −0.722078
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −24.0383 −1.80175
\(179\) 2.13752 0.159766 0.0798828 0.996804i \(-0.474545\pi\)
0.0798828 + 0.996804i \(0.474545\pi\)
\(180\) 0 0
\(181\) −1.83276 −0.136228 −0.0681141 0.997678i \(-0.521698\pi\)
−0.0681141 + 0.997678i \(0.521698\pi\)
\(182\) 23.3622 1.73172
\(183\) 0 0
\(184\) −1.15667 −0.0852712
\(185\) 2.29776 0.168935
\(186\) 0 0
\(187\) 2.75971 0.201810
\(188\) 0.303302 0.0221206
\(189\) 0 0
\(190\) 6.20555 0.450198
\(191\) 18.6167 1.34705 0.673527 0.739163i \(-0.264779\pi\)
0.673527 + 0.739163i \(0.264779\pi\)
\(192\) 0 0
\(193\) 15.6116 1.12375 0.561875 0.827222i \(-0.310080\pi\)
0.561875 + 0.827222i \(0.310080\pi\)
\(194\) 4.95112 0.355470
\(195\) 0 0
\(196\) −0.808583 −0.0577559
\(197\) 5.21057 0.371238 0.185619 0.982622i \(-0.440571\pi\)
0.185619 + 0.982622i \(0.440571\pi\)
\(198\) 0 0
\(199\) 2.05390 0.145597 0.0727985 0.997347i \(-0.476807\pi\)
0.0727985 + 0.997347i \(0.476807\pi\)
\(200\) 4.87807 0.344932
\(201\) 0 0
\(202\) 21.5139 1.51371
\(203\) −11.2106 −0.786828
\(204\) 0 0
\(205\) −1.10278 −0.0770212
\(206\) −0.897225 −0.0625126
\(207\) 0 0
\(208\) −25.0872 −1.73948
\(209\) −2.52444 −0.174619
\(210\) 0 0
\(211\) −9.04888 −0.622950 −0.311475 0.950254i \(-0.600823\pi\)
−0.311475 + 0.950254i \(0.600823\pi\)
\(212\) −17.8328 −1.22476
\(213\) 0 0
\(214\) 27.3622 1.87044
\(215\) 10.0055 0.682372
\(216\) 0 0
\(217\) −22.6322 −1.53638
\(218\) 9.25443 0.626789
\(219\) 0 0
\(220\) −1.15667 −0.0779830
\(221\) −17.3083 −1.16428
\(222\) 0 0
\(223\) 24.8222 1.66222 0.831109 0.556110i \(-0.187707\pi\)
0.831109 + 0.556110i \(0.187707\pi\)
\(224\) −16.0000 −1.06904
\(225\) 0 0
\(226\) 31.7633 2.11286
\(227\) −16.6464 −1.10486 −0.552429 0.833560i \(-0.686299\pi\)
−0.552429 + 0.833560i \(0.686299\pi\)
\(228\) 0 0
\(229\) −15.7789 −1.04270 −0.521348 0.853344i \(-0.674571\pi\)
−0.521348 + 0.853344i \(0.674571\pi\)
\(230\) 1.79445 0.118323
\(231\) 0 0
\(232\) 5.72496 0.375862
\(233\) 24.9200 1.63256 0.816280 0.577656i \(-0.196033\pi\)
0.816280 + 0.577656i \(0.196033\pi\)
\(234\) 0 0
\(235\) 0.259449 0.0169246
\(236\) −1.35218 −0.0880193
\(237\) 0 0
\(238\) −15.5295 −1.00663
\(239\) 2.95112 0.190892 0.0954462 0.995435i \(-0.469572\pi\)
0.0954462 + 0.995435i \(0.469572\pi\)
\(240\) 0 0
\(241\) −16.5925 −1.06881 −0.534407 0.845227i \(-0.679465\pi\)
−0.534407 + 0.845227i \(0.679465\pi\)
\(242\) −18.7491 −1.20524
\(243\) 0 0
\(244\) 2.47054 0.158160
\(245\) −0.691675 −0.0441895
\(246\) 0 0
\(247\) 15.8328 1.00741
\(248\) 11.5577 0.733916
\(249\) 0 0
\(250\) −17.5678 −1.11108
\(251\) 20.1361 1.27098 0.635489 0.772110i \(-0.280799\pi\)
0.635489 + 0.772110i \(0.280799\pi\)
\(252\) 0 0
\(253\) −0.729988 −0.0458940
\(254\) 23.3622 1.46588
\(255\) 0 0
\(256\) 20.8469 1.30293
\(257\) −25.9008 −1.61565 −0.807824 0.589424i \(-0.799355\pi\)
−0.807824 + 0.589424i \(0.799355\pi\)
\(258\) 0 0
\(259\) 5.25997 0.326838
\(260\) 7.25443 0.449900
\(261\) 0 0
\(262\) 11.5295 0.712292
\(263\) 22.4408 1.38376 0.691880 0.722012i \(-0.256783\pi\)
0.691880 + 0.722012i \(0.256783\pi\)
\(264\) 0 0
\(265\) −15.2544 −0.937072
\(266\) 14.2056 0.870998
\(267\) 0 0
\(268\) −12.9411 −0.790502
\(269\) −7.62721 −0.465039 −0.232520 0.972592i \(-0.574697\pi\)
−0.232520 + 0.972592i \(0.574697\pi\)
\(270\) 0 0
\(271\) 19.9164 1.20983 0.604917 0.796289i \(-0.293206\pi\)
0.604917 + 0.796289i \(0.293206\pi\)
\(272\) 16.6761 1.01114
\(273\) 0 0
\(274\) 31.7194 1.91624
\(275\) 3.07860 0.185646
\(276\) 0 0
\(277\) −17.6413 −1.05997 −0.529983 0.848008i \(-0.677802\pi\)
−0.529983 + 0.848008i \(0.677802\pi\)
\(278\) 28.5189 1.71045
\(279\) 0 0
\(280\) −3.58890 −0.214478
\(281\) 0.0297193 0.00177291 0.000886453 1.00000i \(-0.499718\pi\)
0.000886453 1.00000i \(0.499718\pi\)
\(282\) 0 0
\(283\) −10.1814 −0.605220 −0.302610 0.953115i \(-0.597858\pi\)
−0.302610 + 0.953115i \(0.597858\pi\)
\(284\) 16.5189 0.980216
\(285\) 0 0
\(286\) −7.52946 −0.445226
\(287\) −2.52444 −0.149013
\(288\) 0 0
\(289\) −5.49472 −0.323219
\(290\) −8.88164 −0.521548
\(291\) 0 0
\(292\) 6.13607 0.359086
\(293\) −3.14808 −0.183913 −0.0919564 0.995763i \(-0.529312\pi\)
−0.0919564 + 0.995763i \(0.529312\pi\)
\(294\) 0 0
\(295\) −1.15667 −0.0673442
\(296\) −2.68614 −0.156128
\(297\) 0 0
\(298\) −20.9794 −1.21530
\(299\) 4.57834 0.264772
\(300\) 0 0
\(301\) 22.9044 1.32019
\(302\) 34.9200 2.00942
\(303\) 0 0
\(304\) −15.2544 −0.874901
\(305\) 2.11334 0.121009
\(306\) 0 0
\(307\) −12.7980 −0.730422 −0.365211 0.930925i \(-0.619003\pi\)
−0.365211 + 0.930925i \(0.619003\pi\)
\(308\) −2.64782 −0.150874
\(309\) 0 0
\(310\) −17.9305 −1.01838
\(311\) −26.6167 −1.50929 −0.754646 0.656132i \(-0.772191\pi\)
−0.754646 + 0.656132i \(0.772191\pi\)
\(312\) 0 0
\(313\) −3.00502 −0.169854 −0.0849270 0.996387i \(-0.527066\pi\)
−0.0849270 + 0.996387i \(0.527066\pi\)
\(314\) −37.9406 −2.14111
\(315\) 0 0
\(316\) −20.4111 −1.14821
\(317\) 4.33302 0.243367 0.121683 0.992569i \(-0.461171\pi\)
0.121683 + 0.992569i \(0.461171\pi\)
\(318\) 0 0
\(319\) 3.61308 0.202294
\(320\) −1.83276 −0.102455
\(321\) 0 0
\(322\) 4.10780 0.228919
\(323\) −10.5244 −0.585595
\(324\) 0 0
\(325\) −19.3083 −1.07103
\(326\) −23.2106 −1.28551
\(327\) 0 0
\(328\) 1.28917 0.0711824
\(329\) 0.593923 0.0327440
\(330\) 0 0
\(331\) 5.53500 0.304231 0.152116 0.988363i \(-0.451391\pi\)
0.152116 + 0.988363i \(0.451391\pi\)
\(332\) 9.90225 0.543456
\(333\) 0 0
\(334\) −17.4600 −0.955367
\(335\) −11.0700 −0.604819
\(336\) 0 0
\(337\) −3.71083 −0.202142 −0.101071 0.994879i \(-0.532227\pi\)
−0.101071 + 0.994879i \(0.532227\pi\)
\(338\) 23.6464 1.28619
\(339\) 0 0
\(340\) −4.82220 −0.261521
\(341\) 7.29419 0.395003
\(342\) 0 0
\(343\) −19.2544 −1.03964
\(344\) −11.6967 −0.630644
\(345\) 0 0
\(346\) −21.6655 −1.16475
\(347\) −3.07860 −0.165268 −0.0826338 0.996580i \(-0.526333\pi\)
−0.0826338 + 0.996580i \(0.526333\pi\)
\(348\) 0 0
\(349\) 14.7980 0.792120 0.396060 0.918225i \(-0.370377\pi\)
0.396060 + 0.918225i \(0.370377\pi\)
\(350\) −17.3239 −0.926002
\(351\) 0 0
\(352\) 5.15667 0.274852
\(353\) −26.8222 −1.42760 −0.713801 0.700349i \(-0.753028\pi\)
−0.713801 + 0.700349i \(0.753028\pi\)
\(354\) 0 0
\(355\) 14.1305 0.749970
\(356\) −17.0872 −0.905619
\(357\) 0 0
\(358\) 3.87662 0.204886
\(359\) −2.35720 −0.124408 −0.0622042 0.998063i \(-0.519813\pi\)
−0.0622042 + 0.998063i \(0.519813\pi\)
\(360\) 0 0
\(361\) −9.37279 −0.493305
\(362\) −3.32391 −0.174701
\(363\) 0 0
\(364\) 16.6066 0.870423
\(365\) 5.24889 0.274739
\(366\) 0 0
\(367\) 23.1708 1.20951 0.604753 0.796413i \(-0.293272\pi\)
0.604753 + 0.796413i \(0.293272\pi\)
\(368\) −4.41110 −0.229944
\(369\) 0 0
\(370\) 4.16724 0.216644
\(371\) −34.9200 −1.81295
\(372\) 0 0
\(373\) 35.2091 1.82306 0.911530 0.411235i \(-0.134902\pi\)
0.911530 + 0.411235i \(0.134902\pi\)
\(374\) 5.00502 0.258804
\(375\) 0 0
\(376\) −0.303302 −0.0156416
\(377\) −22.6605 −1.16708
\(378\) 0 0
\(379\) −26.0383 −1.33750 −0.668749 0.743488i \(-0.733170\pi\)
−0.668749 + 0.743488i \(0.733170\pi\)
\(380\) 4.41110 0.226285
\(381\) 0 0
\(382\) 33.7633 1.72748
\(383\) −15.3819 −0.785978 −0.392989 0.919543i \(-0.628559\pi\)
−0.392989 + 0.919543i \(0.628559\pi\)
\(384\) 0 0
\(385\) −2.26499 −0.115435
\(386\) 28.3133 1.44111
\(387\) 0 0
\(388\) 3.51941 0.178671
\(389\) −4.46500 −0.226384 −0.113192 0.993573i \(-0.536108\pi\)
−0.113192 + 0.993573i \(0.536108\pi\)
\(390\) 0 0
\(391\) −3.04334 −0.153908
\(392\) 0.808583 0.0408396
\(393\) 0 0
\(394\) 9.44993 0.476081
\(395\) −17.4600 −0.878507
\(396\) 0 0
\(397\) −10.5628 −0.530129 −0.265065 0.964231i \(-0.585393\pi\)
−0.265065 + 0.964231i \(0.585393\pi\)
\(398\) 3.72496 0.186716
\(399\) 0 0
\(400\) 18.6030 0.930152
\(401\) −13.0872 −0.653543 −0.326772 0.945103i \(-0.605961\pi\)
−0.326772 + 0.945103i \(0.605961\pi\)
\(402\) 0 0
\(403\) −45.7477 −2.27885
\(404\) 15.2927 0.760842
\(405\) 0 0
\(406\) −20.3316 −1.00904
\(407\) −1.69525 −0.0840302
\(408\) 0 0
\(409\) −18.5542 −0.917444 −0.458722 0.888580i \(-0.651693\pi\)
−0.458722 + 0.888580i \(0.651693\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) −0.637776 −0.0314210
\(413\) −2.64782 −0.130291
\(414\) 0 0
\(415\) 8.47054 0.415802
\(416\) −32.3416 −1.58568
\(417\) 0 0
\(418\) −4.57834 −0.223934
\(419\) 25.6116 1.25121 0.625605 0.780140i \(-0.284852\pi\)
0.625605 + 0.780140i \(0.284852\pi\)
\(420\) 0 0
\(421\) −8.67609 −0.422847 −0.211423 0.977395i \(-0.567810\pi\)
−0.211423 + 0.977395i \(0.567810\pi\)
\(422\) −16.4111 −0.798880
\(423\) 0 0
\(424\) 17.8328 0.866036
\(425\) 12.8347 0.622576
\(426\) 0 0
\(427\) 4.83779 0.234117
\(428\) 19.4499 0.940148
\(429\) 0 0
\(430\) 18.1461 0.875083
\(431\) −7.10278 −0.342129 −0.171064 0.985260i \(-0.554721\pi\)
−0.171064 + 0.985260i \(0.554721\pi\)
\(432\) 0 0
\(433\) −11.0247 −0.529813 −0.264907 0.964274i \(-0.585341\pi\)
−0.264907 + 0.964274i \(0.585341\pi\)
\(434\) −41.0460 −1.97027
\(435\) 0 0
\(436\) 6.57834 0.315045
\(437\) 2.78389 0.133171
\(438\) 0 0
\(439\) −1.32391 −0.0631868 −0.0315934 0.999501i \(-0.510058\pi\)
−0.0315934 + 0.999501i \(0.510058\pi\)
\(440\) 1.15667 0.0551423
\(441\) 0 0
\(442\) −31.3905 −1.49309
\(443\) −15.5889 −0.740651 −0.370325 0.928902i \(-0.620754\pi\)
−0.370325 + 0.928902i \(0.620754\pi\)
\(444\) 0 0
\(445\) −14.6167 −0.692896
\(446\) 45.0177 2.13165
\(447\) 0 0
\(448\) −4.19550 −0.198219
\(449\) 32.7839 1.54717 0.773584 0.633694i \(-0.218462\pi\)
0.773584 + 0.633694i \(0.218462\pi\)
\(450\) 0 0
\(451\) 0.813607 0.0383112
\(452\) 22.5783 1.06200
\(453\) 0 0
\(454\) −30.1900 −1.41689
\(455\) 14.2056 0.665966
\(456\) 0 0
\(457\) −14.1672 −0.662715 −0.331358 0.943505i \(-0.607507\pi\)
−0.331358 + 0.943505i \(0.607507\pi\)
\(458\) −28.6167 −1.33717
\(459\) 0 0
\(460\) 1.27555 0.0594729
\(461\) 30.0766 1.40081 0.700404 0.713747i \(-0.253003\pi\)
0.700404 + 0.713747i \(0.253003\pi\)
\(462\) 0 0
\(463\) −23.8766 −1.10964 −0.554820 0.831970i \(-0.687213\pi\)
−0.554820 + 0.831970i \(0.687213\pi\)
\(464\) 21.8328 1.01356
\(465\) 0 0
\(466\) 45.1950 2.09362
\(467\) 9.73501 0.450483 0.225241 0.974303i \(-0.427683\pi\)
0.225241 + 0.974303i \(0.427683\pi\)
\(468\) 0 0
\(469\) −25.3411 −1.17014
\(470\) 0.470539 0.0217043
\(471\) 0 0
\(472\) 1.35218 0.0622390
\(473\) −7.38190 −0.339420
\(474\) 0 0
\(475\) −11.7406 −0.538693
\(476\) −11.0388 −0.505964
\(477\) 0 0
\(478\) 5.35218 0.244803
\(479\) 1.17635 0.0537487 0.0268743 0.999639i \(-0.491445\pi\)
0.0268743 + 0.999639i \(0.491445\pi\)
\(480\) 0 0
\(481\) 10.6322 0.484788
\(482\) −30.0922 −1.37066
\(483\) 0 0
\(484\) −13.3275 −0.605795
\(485\) 3.01056 0.136703
\(486\) 0 0
\(487\) 20.6025 0.933589 0.466795 0.884366i \(-0.345409\pi\)
0.466795 + 0.884366i \(0.345409\pi\)
\(488\) −2.47054 −0.111836
\(489\) 0 0
\(490\) −1.25443 −0.0566692
\(491\) −14.1900 −0.640384 −0.320192 0.947353i \(-0.603747\pi\)
−0.320192 + 0.947353i \(0.603747\pi\)
\(492\) 0 0
\(493\) 15.0630 0.678404
\(494\) 28.7144 1.29192
\(495\) 0 0
\(496\) 44.0766 1.97910
\(497\) 32.3472 1.45097
\(498\) 0 0
\(499\) −21.4600 −0.960680 −0.480340 0.877082i \(-0.659487\pi\)
−0.480340 + 0.877082i \(0.659487\pi\)
\(500\) −12.4877 −0.558468
\(501\) 0 0
\(502\) 36.5189 1.62992
\(503\) 15.5491 0.693302 0.346651 0.937994i \(-0.387319\pi\)
0.346651 + 0.937994i \(0.387319\pi\)
\(504\) 0 0
\(505\) 13.0816 0.582126
\(506\) −1.32391 −0.0588550
\(507\) 0 0
\(508\) 16.6066 0.736799
\(509\) −22.7542 −1.00856 −0.504280 0.863540i \(-0.668242\pi\)
−0.504280 + 0.863540i \(0.668242\pi\)
\(510\) 0 0
\(511\) 12.0156 0.531538
\(512\) 18.4842 0.816892
\(513\) 0 0
\(514\) −46.9739 −2.07193
\(515\) −0.545563 −0.0240404
\(516\) 0 0
\(517\) −0.191417 −0.00841850
\(518\) 9.53951 0.419142
\(519\) 0 0
\(520\) −7.25443 −0.318128
\(521\) 38.7230 1.69649 0.848243 0.529608i \(-0.177661\pi\)
0.848243 + 0.529608i \(0.177661\pi\)
\(522\) 0 0
\(523\) 6.56829 0.287211 0.143606 0.989635i \(-0.454130\pi\)
0.143606 + 0.989635i \(0.454130\pi\)
\(524\) 8.19550 0.358022
\(525\) 0 0
\(526\) 40.6988 1.77455
\(527\) 30.4096 1.32467
\(528\) 0 0
\(529\) −22.1950 −0.964999
\(530\) −27.6655 −1.20171
\(531\) 0 0
\(532\) 10.0978 0.437793
\(533\) −5.10278 −0.221026
\(534\) 0 0
\(535\) 16.6378 0.719314
\(536\) 12.9411 0.558969
\(537\) 0 0
\(538\) −13.8328 −0.596373
\(539\) 0.510305 0.0219804
\(540\) 0 0
\(541\) 42.0766 1.80902 0.904508 0.426457i \(-0.140239\pi\)
0.904508 + 0.426457i \(0.140239\pi\)
\(542\) 36.1205 1.55151
\(543\) 0 0
\(544\) 21.4983 0.921732
\(545\) 5.62721 0.241043
\(546\) 0 0
\(547\) 19.9688 0.853805 0.426903 0.904298i \(-0.359605\pi\)
0.426903 + 0.904298i \(0.359605\pi\)
\(548\) 22.5472 0.963167
\(549\) 0 0
\(550\) 5.58336 0.238075
\(551\) −13.7789 −0.586999
\(552\) 0 0
\(553\) −39.9688 −1.69965
\(554\) −31.9945 −1.35931
\(555\) 0 0
\(556\) 20.2721 0.859730
\(557\) −13.5491 −0.574095 −0.287048 0.957916i \(-0.592674\pi\)
−0.287048 + 0.957916i \(0.592674\pi\)
\(558\) 0 0
\(559\) 46.2978 1.95819
\(560\) −13.6867 −0.578367
\(561\) 0 0
\(562\) 0.0538991 0.00227360
\(563\) −9.75468 −0.411111 −0.205555 0.978645i \(-0.565900\pi\)
−0.205555 + 0.978645i \(0.565900\pi\)
\(564\) 0 0
\(565\) 19.3139 0.812540
\(566\) −18.4650 −0.776142
\(567\) 0 0
\(568\) −16.5189 −0.693118
\(569\) 21.4544 0.899417 0.449708 0.893175i \(-0.351528\pi\)
0.449708 + 0.893175i \(0.351528\pi\)
\(570\) 0 0
\(571\) 20.9411 0.876357 0.438178 0.898888i \(-0.355624\pi\)
0.438178 + 0.898888i \(0.355624\pi\)
\(572\) −5.35218 −0.223786
\(573\) 0 0
\(574\) −4.57834 −0.191096
\(575\) −3.39500 −0.141581
\(576\) 0 0
\(577\) 18.4705 0.768939 0.384469 0.923138i \(-0.374384\pi\)
0.384469 + 0.923138i \(0.374384\pi\)
\(578\) −9.96526 −0.414500
\(579\) 0 0
\(580\) −6.31335 −0.262148
\(581\) 19.3905 0.804453
\(582\) 0 0
\(583\) 11.2544 0.466111
\(584\) −6.13607 −0.253912
\(585\) 0 0
\(586\) −5.70938 −0.235852
\(587\) 0.127471 0.00526130 0.00263065 0.999997i \(-0.499163\pi\)
0.00263065 + 0.999997i \(0.499163\pi\)
\(588\) 0 0
\(589\) −27.8172 −1.14619
\(590\) −2.09775 −0.0863631
\(591\) 0 0
\(592\) −10.2439 −0.421020
\(593\) 27.2841 1.12043 0.560213 0.828349i \(-0.310719\pi\)
0.560213 + 0.828349i \(0.310719\pi\)
\(594\) 0 0
\(595\) −9.44279 −0.387117
\(596\) −14.9128 −0.610853
\(597\) 0 0
\(598\) 8.30330 0.339547
\(599\) 42.3260 1.72939 0.864697 0.502293i \(-0.167510\pi\)
0.864697 + 0.502293i \(0.167510\pi\)
\(600\) 0 0
\(601\) −15.6756 −0.639420 −0.319710 0.947515i \(-0.603585\pi\)
−0.319710 + 0.947515i \(0.603585\pi\)
\(602\) 41.5395 1.69302
\(603\) 0 0
\(604\) 24.8222 1.01000
\(605\) −11.4005 −0.463498
\(606\) 0 0
\(607\) −1.93051 −0.0783572 −0.0391786 0.999232i \(-0.512474\pi\)
−0.0391786 + 0.999232i \(0.512474\pi\)
\(608\) −19.6655 −0.797542
\(609\) 0 0
\(610\) 3.83276 0.155184
\(611\) 1.20053 0.0485681
\(612\) 0 0
\(613\) −0.372787 −0.0150567 −0.00752836 0.999972i \(-0.502396\pi\)
−0.00752836 + 0.999972i \(0.502396\pi\)
\(614\) −23.2106 −0.936703
\(615\) 0 0
\(616\) 2.64782 0.106684
\(617\) −31.3083 −1.26043 −0.630213 0.776422i \(-0.717032\pi\)
−0.630213 + 0.776422i \(0.717032\pi\)
\(618\) 0 0
\(619\) 34.3658 1.38128 0.690639 0.723200i \(-0.257329\pi\)
0.690639 + 0.723200i \(0.257329\pi\)
\(620\) −12.7456 −0.511875
\(621\) 0 0
\(622\) −48.2721 −1.93554
\(623\) −33.4600 −1.34055
\(624\) 0 0
\(625\) 8.23724 0.329490
\(626\) −5.44993 −0.217823
\(627\) 0 0
\(628\) −26.9693 −1.07619
\(629\) −7.06752 −0.281800
\(630\) 0 0
\(631\) 7.33804 0.292123 0.146061 0.989276i \(-0.453340\pi\)
0.146061 + 0.989276i \(0.453340\pi\)
\(632\) 20.4111 0.811910
\(633\) 0 0
\(634\) 7.85840 0.312097
\(635\) 14.2056 0.563730
\(636\) 0 0
\(637\) −3.20053 −0.126809
\(638\) 6.55270 0.259424
\(639\) 0 0
\(640\) 10.6550 0.421174
\(641\) 31.5764 1.24719 0.623596 0.781747i \(-0.285671\pi\)
0.623596 + 0.781747i \(0.285671\pi\)
\(642\) 0 0
\(643\) −4.51890 −0.178208 −0.0891040 0.996022i \(-0.528400\pi\)
−0.0891040 + 0.996022i \(0.528400\pi\)
\(644\) 2.91995 0.115062
\(645\) 0 0
\(646\) −19.0872 −0.750975
\(647\) 20.6705 0.812643 0.406322 0.913730i \(-0.366811\pi\)
0.406322 + 0.913730i \(0.366811\pi\)
\(648\) 0 0
\(649\) 0.853372 0.0334978
\(650\) −35.0177 −1.37351
\(651\) 0 0
\(652\) −16.4988 −0.646143
\(653\) −0.381381 −0.0149246 −0.00746229 0.999972i \(-0.502375\pi\)
−0.00746229 + 0.999972i \(0.502375\pi\)
\(654\) 0 0
\(655\) 7.01056 0.273925
\(656\) 4.91638 0.191952
\(657\) 0 0
\(658\) 1.07714 0.0419914
\(659\) −13.4600 −0.524326 −0.262163 0.965024i \(-0.584436\pi\)
−0.262163 + 0.965024i \(0.584436\pi\)
\(660\) 0 0
\(661\) 30.7738 1.19696 0.598482 0.801136i \(-0.295771\pi\)
0.598482 + 0.801136i \(0.295771\pi\)
\(662\) 10.0383 0.390150
\(663\) 0 0
\(664\) −9.90225 −0.384282
\(665\) 8.63778 0.334959
\(666\) 0 0
\(667\) −3.98441 −0.154277
\(668\) −12.4111 −0.480200
\(669\) 0 0
\(670\) −20.0766 −0.775628
\(671\) −1.55918 −0.0601915
\(672\) 0 0
\(673\) 1.48110 0.0570923 0.0285461 0.999592i \(-0.490912\pi\)
0.0285461 + 0.999592i \(0.490912\pi\)
\(674\) −6.72999 −0.259229
\(675\) 0 0
\(676\) 16.8086 0.646484
\(677\) −45.3749 −1.74390 −0.871950 0.489596i \(-0.837144\pi\)
−0.871950 + 0.489596i \(0.837144\pi\)
\(678\) 0 0
\(679\) 6.89169 0.264479
\(680\) 4.82220 0.184923
\(681\) 0 0
\(682\) 13.2288 0.506557
\(683\) 0.0594386 0.00227436 0.00113718 0.999999i \(-0.499638\pi\)
0.00113718 + 0.999999i \(0.499638\pi\)
\(684\) 0 0
\(685\) 19.2872 0.736926
\(686\) −34.9200 −1.33325
\(687\) 0 0
\(688\) −44.6066 −1.70061
\(689\) −70.5855 −2.68909
\(690\) 0 0
\(691\) −18.9355 −0.720342 −0.360171 0.932886i \(-0.617282\pi\)
−0.360171 + 0.932886i \(0.617282\pi\)
\(692\) −15.4005 −0.585441
\(693\) 0 0
\(694\) −5.58336 −0.211941
\(695\) 17.3411 0.657785
\(696\) 0 0
\(697\) 3.39194 0.128479
\(698\) 26.8378 1.01583
\(699\) 0 0
\(700\) −12.3144 −0.465440
\(701\) 0.540024 0.0203964 0.0101982 0.999948i \(-0.496754\pi\)
0.0101982 + 0.999948i \(0.496754\pi\)
\(702\) 0 0
\(703\) 6.46500 0.243832
\(704\) 1.35218 0.0509621
\(705\) 0 0
\(706\) −48.6449 −1.83078
\(707\) 29.9461 1.12624
\(708\) 0 0
\(709\) 28.0867 1.05482 0.527409 0.849612i \(-0.323164\pi\)
0.527409 + 0.849612i \(0.323164\pi\)
\(710\) 25.6272 0.961772
\(711\) 0 0
\(712\) 17.0872 0.640369
\(713\) −8.04385 −0.301245
\(714\) 0 0
\(715\) −4.57834 −0.171220
\(716\) 2.75562 0.102982
\(717\) 0 0
\(718\) −4.27504 −0.159543
\(719\) −9.86248 −0.367809 −0.183904 0.982944i \(-0.558874\pi\)
−0.183904 + 0.982944i \(0.558874\pi\)
\(720\) 0 0
\(721\) −1.24889 −0.0465110
\(722\) −16.9985 −0.632620
\(723\) 0 0
\(724\) −2.36274 −0.0878106
\(725\) 16.8036 0.624069
\(726\) 0 0
\(727\) −30.4650 −1.12988 −0.564942 0.825131i \(-0.691102\pi\)
−0.564942 + 0.825131i \(0.691102\pi\)
\(728\) −16.6066 −0.615482
\(729\) 0 0
\(730\) 9.51941 0.352329
\(731\) −30.7753 −1.13827
\(732\) 0 0
\(733\) 27.5436 1.01735 0.508673 0.860960i \(-0.330136\pi\)
0.508673 + 0.860960i \(0.330136\pi\)
\(734\) 42.0227 1.55109
\(735\) 0 0
\(736\) −5.68665 −0.209613
\(737\) 8.16724 0.300844
\(738\) 0 0
\(739\) 13.1013 0.481940 0.240970 0.970533i \(-0.422534\pi\)
0.240970 + 0.970533i \(0.422534\pi\)
\(740\) 2.96220 0.108893
\(741\) 0 0
\(742\) −63.3311 −2.32496
\(743\) −34.7527 −1.27495 −0.637477 0.770470i \(-0.720022\pi\)
−0.637477 + 0.770470i \(0.720022\pi\)
\(744\) 0 0
\(745\) −12.7567 −0.467368
\(746\) 63.8555 2.33792
\(747\) 0 0
\(748\) 3.55773 0.130083
\(749\) 38.0867 1.39166
\(750\) 0 0
\(751\) 32.7738 1.19593 0.597967 0.801521i \(-0.295975\pi\)
0.597967 + 0.801521i \(0.295975\pi\)
\(752\) −1.15667 −0.0421796
\(753\) 0 0
\(754\) −41.0972 −1.49667
\(755\) 21.2333 0.772759
\(756\) 0 0
\(757\) −32.2978 −1.17388 −0.586941 0.809630i \(-0.699668\pi\)
−0.586941 + 0.809630i \(0.699668\pi\)
\(758\) −47.2233 −1.71523
\(759\) 0 0
\(760\) −4.41110 −0.160007
\(761\) −44.4494 −1.61129 −0.805645 0.592399i \(-0.798181\pi\)
−0.805645 + 0.592399i \(0.798181\pi\)
\(762\) 0 0
\(763\) 12.8816 0.466347
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) −27.8967 −1.00795
\(767\) −5.35218 −0.193256
\(768\) 0 0
\(769\) −19.7108 −0.710791 −0.355395 0.934716i \(-0.615654\pi\)
−0.355395 + 0.934716i \(0.615654\pi\)
\(770\) −4.10780 −0.148035
\(771\) 0 0
\(772\) 20.1260 0.724351
\(773\) −43.2530 −1.55570 −0.777851 0.628449i \(-0.783690\pi\)
−0.777851 + 0.628449i \(0.783690\pi\)
\(774\) 0 0
\(775\) 33.9235 1.21857
\(776\) −3.51941 −0.126340
\(777\) 0 0
\(778\) −8.09775 −0.290318
\(779\) −3.10278 −0.111168
\(780\) 0 0
\(781\) −10.4252 −0.373044
\(782\) −5.51941 −0.197374
\(783\) 0 0
\(784\) 3.08362 0.110129
\(785\) −23.0700 −0.823404
\(786\) 0 0
\(787\) −21.4358 −0.764104 −0.382052 0.924141i \(-0.624782\pi\)
−0.382052 + 0.924141i \(0.624782\pi\)
\(788\) 6.71731 0.239294
\(789\) 0 0
\(790\) −31.6655 −1.12661
\(791\) 44.2127 1.57202
\(792\) 0 0
\(793\) 9.77886 0.347258
\(794\) −19.1567 −0.679845
\(795\) 0 0
\(796\) 2.64782 0.0938496
\(797\) 4.40105 0.155893 0.0779467 0.996958i \(-0.475164\pi\)
0.0779467 + 0.996958i \(0.475164\pi\)
\(798\) 0 0
\(799\) −0.798021 −0.0282319
\(800\) 23.9824 0.847907
\(801\) 0 0
\(802\) −23.7350 −0.838112
\(803\) −3.87253 −0.136659
\(804\) 0 0
\(805\) 2.49777 0.0880349
\(806\) −82.9683 −2.92243
\(807\) 0 0
\(808\) −15.2927 −0.537997
\(809\) 26.3799 0.927469 0.463734 0.885974i \(-0.346509\pi\)
0.463734 + 0.885974i \(0.346509\pi\)
\(810\) 0 0
\(811\) 4.96526 0.174354 0.0871769 0.996193i \(-0.472215\pi\)
0.0871769 + 0.996193i \(0.472215\pi\)
\(812\) −14.4523 −0.507177
\(813\) 0 0
\(814\) −3.07451 −0.107761
\(815\) −14.1133 −0.494369
\(816\) 0 0
\(817\) 28.1517 0.984902
\(818\) −33.6499 −1.17654
\(819\) 0 0
\(820\) −1.42166 −0.0496466
\(821\) 15.7789 0.550686 0.275343 0.961346i \(-0.411209\pi\)
0.275343 + 0.961346i \(0.411209\pi\)
\(822\) 0 0
\(823\) −8.35166 −0.291121 −0.145560 0.989349i \(-0.546498\pi\)
−0.145560 + 0.989349i \(0.546498\pi\)
\(824\) 0.637776 0.0222180
\(825\) 0 0
\(826\) −4.80211 −0.167087
\(827\) 18.3925 0.639568 0.319784 0.947490i \(-0.396390\pi\)
0.319784 + 0.947490i \(0.396390\pi\)
\(828\) 0 0
\(829\) −29.4147 −1.02161 −0.510807 0.859695i \(-0.670653\pi\)
−0.510807 + 0.859695i \(0.670653\pi\)
\(830\) 15.3622 0.533231
\(831\) 0 0
\(832\) −8.48059 −0.294011
\(833\) 2.12747 0.0737125
\(834\) 0 0
\(835\) −10.6167 −0.367404
\(836\) −3.25443 −0.112557
\(837\) 0 0
\(838\) 46.4494 1.60457
\(839\) 33.4600 1.15517 0.577583 0.816332i \(-0.303996\pi\)
0.577583 + 0.816332i \(0.303996\pi\)
\(840\) 0 0
\(841\) −9.27912 −0.319970
\(842\) −15.7350 −0.542264
\(843\) 0 0
\(844\) −11.6655 −0.401544
\(845\) 14.3783 0.494629
\(846\) 0 0
\(847\) −26.0978 −0.896729
\(848\) 68.0071 2.33537
\(849\) 0 0
\(850\) 23.2772 0.798400
\(851\) 1.86947 0.0640848
\(852\) 0 0
\(853\) 40.4494 1.38496 0.692481 0.721436i \(-0.256518\pi\)
0.692481 + 0.721436i \(0.256518\pi\)
\(854\) 8.77384 0.300235
\(855\) 0 0
\(856\) −19.4499 −0.664785
\(857\) −12.4494 −0.425264 −0.212632 0.977132i \(-0.568204\pi\)
−0.212632 + 0.977132i \(0.568204\pi\)
\(858\) 0 0
\(859\) −21.6514 −0.738736 −0.369368 0.929283i \(-0.620426\pi\)
−0.369368 + 0.929283i \(0.620426\pi\)
\(860\) 12.8988 0.439846
\(861\) 0 0
\(862\) −12.8816 −0.438750
\(863\) −19.3778 −0.659628 −0.329814 0.944046i \(-0.606986\pi\)
−0.329814 + 0.944046i \(0.606986\pi\)
\(864\) 0 0
\(865\) −13.1739 −0.447925
\(866\) −19.9945 −0.679439
\(867\) 0 0
\(868\) −29.1768 −0.990324
\(869\) 12.8816 0.436980
\(870\) 0 0
\(871\) −51.2233 −1.73563
\(872\) −6.57834 −0.222771
\(873\) 0 0
\(874\) 5.04888 0.170781
\(875\) −24.4534 −0.826674
\(876\) 0 0
\(877\) −3.08413 −0.104144 −0.0520719 0.998643i \(-0.516583\pi\)
−0.0520719 + 0.998643i \(0.516583\pi\)
\(878\) −2.40105 −0.0810316
\(879\) 0 0
\(880\) 4.41110 0.148698
\(881\) 34.5783 1.16497 0.582487 0.812840i \(-0.302080\pi\)
0.582487 + 0.812840i \(0.302080\pi\)
\(882\) 0 0
\(883\) −9.64280 −0.324506 −0.162253 0.986749i \(-0.551876\pi\)
−0.162253 + 0.986749i \(0.551876\pi\)
\(884\) −22.3133 −0.750479
\(885\) 0 0
\(886\) −28.2721 −0.949821
\(887\) −32.2041 −1.08131 −0.540654 0.841245i \(-0.681823\pi\)
−0.540654 + 0.841245i \(0.681823\pi\)
\(888\) 0 0
\(889\) 32.5189 1.09065
\(890\) −26.5089 −0.888579
\(891\) 0 0
\(892\) 32.0000 1.07144
\(893\) 0.729988 0.0244281
\(894\) 0 0
\(895\) 2.35720 0.0787925
\(896\) 24.3910 0.814846
\(897\) 0 0
\(898\) 59.4571 1.98411
\(899\) 39.8131 1.32784
\(900\) 0 0
\(901\) 46.9200 1.56313
\(902\) 1.47556 0.0491308
\(903\) 0 0
\(904\) −22.5783 −0.750944
\(905\) −2.02113 −0.0671845
\(906\) 0 0
\(907\) −17.8227 −0.591794 −0.295897 0.955220i \(-0.595618\pi\)
−0.295897 + 0.955220i \(0.595618\pi\)
\(908\) −21.4600 −0.712174
\(909\) 0 0
\(910\) 25.7633 0.854044
\(911\) 20.7894 0.688784 0.344392 0.938826i \(-0.388085\pi\)
0.344392 + 0.938826i \(0.388085\pi\)
\(912\) 0 0
\(913\) −6.24940 −0.206825
\(914\) −25.6938 −0.849875
\(915\) 0 0
\(916\) −20.3416 −0.672106
\(917\) 16.0484 0.529964
\(918\) 0 0
\(919\) 33.8555 1.11679 0.558395 0.829575i \(-0.311417\pi\)
0.558395 + 0.829575i \(0.311417\pi\)
\(920\) −1.27555 −0.0420537
\(921\) 0 0
\(922\) 54.5472 1.79642
\(923\) 65.3850 2.15217
\(924\) 0 0
\(925\) −7.88418 −0.259230
\(926\) −43.3028 −1.42302
\(927\) 0 0
\(928\) 28.1461 0.923941
\(929\) 4.10635 0.134725 0.0673624 0.997729i \(-0.478542\pi\)
0.0673624 + 0.997729i \(0.478542\pi\)
\(930\) 0 0
\(931\) −1.94610 −0.0637809
\(932\) 32.1260 1.05232
\(933\) 0 0
\(934\) 17.6555 0.577705
\(935\) 3.04334 0.0995277
\(936\) 0 0
\(937\) 13.3028 0.434583 0.217292 0.976107i \(-0.430278\pi\)
0.217292 + 0.976107i \(0.430278\pi\)
\(938\) −45.9588 −1.50061
\(939\) 0 0
\(940\) 0.334474 0.0109093
\(941\) 11.7633 0.383472 0.191736 0.981447i \(-0.438588\pi\)
0.191736 + 0.981447i \(0.438588\pi\)
\(942\) 0 0
\(943\) −0.897225 −0.0292177
\(944\) 5.15667 0.167835
\(945\) 0 0
\(946\) −13.3879 −0.435277
\(947\) −1.40054 −0.0455114 −0.0227557 0.999741i \(-0.507244\pi\)
−0.0227557 + 0.999741i \(0.507244\pi\)
\(948\) 0 0
\(949\) 24.2877 0.788413
\(950\) −21.2927 −0.690828
\(951\) 0 0
\(952\) 11.0388 0.357771
\(953\) −33.7577 −1.09352 −0.546760 0.837289i \(-0.684139\pi\)
−0.546760 + 0.837289i \(0.684139\pi\)
\(954\) 0 0
\(955\) 20.5300 0.664334
\(956\) 3.80450 0.123046
\(957\) 0 0
\(958\) 2.13343 0.0689280
\(959\) 44.1517 1.42573
\(960\) 0 0
\(961\) 49.3758 1.59277
\(962\) 19.2827 0.621699
\(963\) 0 0
\(964\) −21.3905 −0.688941
\(965\) 17.2161 0.554206
\(966\) 0 0
\(967\) −10.4806 −0.337033 −0.168516 0.985699i \(-0.553898\pi\)
−0.168516 + 0.985699i \(0.553898\pi\)
\(968\) 13.3275 0.428361
\(969\) 0 0
\(970\) 5.45998 0.175309
\(971\) −57.6358 −1.84962 −0.924811 0.380428i \(-0.875777\pi\)
−0.924811 + 0.380428i \(0.875777\pi\)
\(972\) 0 0
\(973\) 39.6967 1.27262
\(974\) 37.3649 1.19725
\(975\) 0 0
\(976\) −9.42166 −0.301580
\(977\) −41.9008 −1.34053 −0.670263 0.742124i \(-0.733819\pi\)
−0.670263 + 0.742124i \(0.733819\pi\)
\(978\) 0 0
\(979\) 10.7839 0.344655
\(980\) −0.891685 −0.0284838
\(981\) 0 0
\(982\) −25.7350 −0.821237
\(983\) 18.4056 0.587046 0.293523 0.955952i \(-0.405172\pi\)
0.293523 + 0.955952i \(0.405172\pi\)
\(984\) 0 0
\(985\) 5.74609 0.183086
\(986\) 27.3184 0.869994
\(987\) 0 0
\(988\) 20.4111 0.649364
\(989\) 8.14057 0.258855
\(990\) 0 0
\(991\) 26.0666 0.828032 0.414016 0.910270i \(-0.364126\pi\)
0.414016 + 0.910270i \(0.364126\pi\)
\(992\) 56.8222 1.80411
\(993\) 0 0
\(994\) 58.6650 1.86074
\(995\) 2.26499 0.0718050
\(996\) 0 0
\(997\) −29.8483 −0.945307 −0.472653 0.881248i \(-0.656704\pi\)
−0.472653 + 0.881248i \(0.656704\pi\)
\(998\) −38.9200 −1.23199
\(999\) 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 369.2.a.e.1.3 3
3.2 odd 2 123.2.a.d.1.1 3
4.3 odd 2 5904.2.a.bd.1.3 3
5.4 even 2 9225.2.a.bx.1.1 3
12.11 even 2 1968.2.a.w.1.1 3
15.14 odd 2 3075.2.a.t.1.3 3
21.20 even 2 6027.2.a.s.1.1 3
24.5 odd 2 7872.2.a.bx.1.3 3
24.11 even 2 7872.2.a.bs.1.3 3
123.122 odd 2 5043.2.a.n.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
123.2.a.d.1.1 3 3.2 odd 2
369.2.a.e.1.3 3 1.1 even 1 trivial
1968.2.a.w.1.1 3 12.11 even 2
3075.2.a.t.1.3 3 15.14 odd 2
5043.2.a.n.1.1 3 123.122 odd 2
5904.2.a.bd.1.3 3 4.3 odd 2
6027.2.a.s.1.1 3 21.20 even 2
7872.2.a.bs.1.3 3 24.11 even 2
7872.2.a.bx.1.3 3 24.5 odd 2
9225.2.a.bx.1.1 3 5.4 even 2