Properties

Label 3381.2.a.v.1.1
Level $3381$
Weight $2$
Character 3381.1
Self dual yes
Analytic conductor $26.997$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3381,2,Mod(1,3381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3381, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3381.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3381 = 3 \cdot 7^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3381.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: \(26.9974209234\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 6x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.36147\) of defining polynomial
Character \(\chi\) \(=\) 3381.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.36147 q^{2} -1.00000 q^{3} +3.57653 q^{4} -3.36147 q^{5} +2.36147 q^{6} -3.72294 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.36147 q^{2} -1.00000 q^{3} +3.57653 q^{4} -3.36147 q^{5} +2.36147 q^{6} -3.72294 q^{8} +1.00000 q^{9} +7.93800 q^{10} +5.93800 q^{11} -3.57653 q^{12} -1.42347 q^{13} +3.36147 q^{15} +1.63853 q^{16} -2.78493 q^{17} -2.36147 q^{18} +5.93800 q^{19} -12.0224 q^{20} -14.0224 q^{22} -1.00000 q^{23} +3.72294 q^{24} +6.29947 q^{25} +3.36147 q^{26} -1.00000 q^{27} -6.66094 q^{29} -7.93800 q^{30} +9.93800 q^{31} +3.57653 q^{32} -5.93800 q^{33} +6.57653 q^{34} +3.57653 q^{36} +10.6609 q^{37} -14.0224 q^{38} +1.42347 q^{39} +12.5145 q^{40} +9.09107 q^{41} +3.29947 q^{43} +21.2375 q^{44} -3.36147 q^{45} +2.36147 q^{46} -9.15307 q^{47} -1.63853 q^{48} -14.8760 q^{50} +2.78493 q^{51} -5.09107 q^{52} +3.36147 q^{53} +2.36147 q^{54} -19.9604 q^{55} -5.93800 q^{57} +15.7296 q^{58} -0.208399 q^{59} +12.0224 q^{60} -5.29947 q^{61} -23.4683 q^{62} -11.7229 q^{64} +4.78493 q^{65} +14.0224 q^{66} -10.9313 q^{67} -9.96041 q^{68} +1.00000 q^{69} +3.00667 q^{71} -3.72294 q^{72} +1.50787 q^{73} -25.1755 q^{74} -6.29947 q^{75} +21.2375 q^{76} -3.36147 q^{78} -4.66094 q^{79} -5.50787 q^{80} +1.00000 q^{81} -21.4683 q^{82} -4.78493 q^{83} +9.36147 q^{85} -7.79160 q^{86} +6.66094 q^{87} -22.1068 q^{88} -6.57653 q^{89} +7.93800 q^{90} -3.57653 q^{92} -9.93800 q^{93} +21.6147 q^{94} -19.9604 q^{95} -3.57653 q^{96} +9.50787 q^{97} +5.93800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{5} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 6 q^{4} - 3 q^{5} + 3 q^{8} + 3 q^{9} + 12 q^{10} + 6 q^{11} - 6 q^{12} - 9 q^{13} + 3 q^{15} + 12 q^{16} - 6 q^{17} + 6 q^{19} - 3 q^{20} - 9 q^{22} - 3 q^{23} - 3 q^{24} + 3 q^{26} - 3 q^{27} + 6 q^{29} - 12 q^{30} + 18 q^{31} + 6 q^{32} - 6 q^{33} + 15 q^{34} + 6 q^{36} + 6 q^{37} - 9 q^{38} + 9 q^{39} + 21 q^{40} + 6 q^{41} - 9 q^{43} + 33 q^{44} - 3 q^{45} - 18 q^{47} - 12 q^{48} - 21 q^{50} + 6 q^{51} + 6 q^{52} + 3 q^{53} - 15 q^{55} - 6 q^{57} + 33 q^{58} - 3 q^{59} + 3 q^{60} + 3 q^{61} - 9 q^{62} - 21 q^{64} + 12 q^{65} + 9 q^{66} - 21 q^{67} + 15 q^{68} + 3 q^{69} + 9 q^{71} + 3 q^{72} - 12 q^{73} - 33 q^{74} + 33 q^{76} - 3 q^{78} + 12 q^{79} + 3 q^{81} - 3 q^{82} - 12 q^{83} + 21 q^{85} - 21 q^{86} - 6 q^{87} - 12 q^{88} - 15 q^{89} + 12 q^{90} - 6 q^{92} - 18 q^{93} - 6 q^{94} - 15 q^{95} - 6 q^{96} + 12 q^{97} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.36147 −1.66981 −0.834905 0.550394i \(-0.814478\pi\)
−0.834905 + 0.550394i \(0.814478\pi\)
\(3\) −1.00000 −0.577350
\(4\) 3.57653 1.78827
\(5\) −3.36147 −1.50329 −0.751647 0.659565i \(-0.770740\pi\)
−0.751647 + 0.659565i \(0.770740\pi\)
\(6\) 2.36147 0.964066
\(7\) 0 0
\(8\) −3.72294 −1.31626
\(9\) 1.00000 0.333333
\(10\) 7.93800 2.51022
\(11\) 5.93800 1.79038 0.895188 0.445689i \(-0.147041\pi\)
0.895188 + 0.445689i \(0.147041\pi\)
\(12\) −3.57653 −1.03246
\(13\) −1.42347 −0.394798 −0.197399 0.980323i \(-0.563249\pi\)
−0.197399 + 0.980323i \(0.563249\pi\)
\(14\) 0 0
\(15\) 3.36147 0.867928
\(16\) 1.63853 0.409633
\(17\) −2.78493 −0.675446 −0.337723 0.941246i \(-0.609657\pi\)
−0.337723 + 0.941246i \(0.609657\pi\)
\(18\) −2.36147 −0.556604
\(19\) 5.93800 1.36227 0.681136 0.732157i \(-0.261486\pi\)
0.681136 + 0.732157i \(0.261486\pi\)
\(20\) −12.0224 −2.68829
\(21\) 0 0
\(22\) −14.0224 −2.98959
\(23\) −1.00000 −0.208514
\(24\) 3.72294 0.759941
\(25\) 6.29947 1.25989
\(26\) 3.36147 0.659238
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) −6.66094 −1.23691 −0.618453 0.785822i \(-0.712240\pi\)
−0.618453 + 0.785822i \(0.712240\pi\)
\(30\) −7.93800 −1.44927
\(31\) 9.93800 1.78492 0.892459 0.451128i \(-0.148978\pi\)
0.892459 + 0.451128i \(0.148978\pi\)
\(32\) 3.57653 0.632248
\(33\) −5.93800 −1.03367
\(34\) 6.57653 1.12787
\(35\) 0 0
\(36\) 3.57653 0.596089
\(37\) 10.6609 1.75265 0.876324 0.481722i \(-0.159989\pi\)
0.876324 + 0.481722i \(0.159989\pi\)
\(38\) −14.0224 −2.27474
\(39\) 1.42347 0.227937
\(40\) 12.5145 1.97872
\(41\) 9.09107 1.41979 0.709894 0.704309i \(-0.248743\pi\)
0.709894 + 0.704309i \(0.248743\pi\)
\(42\) 0 0
\(43\) 3.29947 0.503165 0.251582 0.967836i \(-0.419049\pi\)
0.251582 + 0.967836i \(0.419049\pi\)
\(44\) 21.2375 3.20167
\(45\) −3.36147 −0.501098
\(46\) 2.36147 0.348180
\(47\) −9.15307 −1.33511 −0.667556 0.744559i \(-0.732660\pi\)
−0.667556 + 0.744559i \(0.732660\pi\)
\(48\) −1.63853 −0.236502
\(49\) 0 0
\(50\) −14.8760 −2.10379
\(51\) 2.78493 0.389969
\(52\) −5.09107 −0.706005
\(53\) 3.36147 0.461733 0.230867 0.972985i \(-0.425844\pi\)
0.230867 + 0.972985i \(0.425844\pi\)
\(54\) 2.36147 0.321355
\(55\) −19.9604 −2.69146
\(56\) 0 0
\(57\) −5.93800 −0.786508
\(58\) 15.7296 2.06540
\(59\) −0.208399 −0.0271313 −0.0135656 0.999908i \(-0.504318\pi\)
−0.0135656 + 0.999908i \(0.504318\pi\)
\(60\) 12.0224 1.55209
\(61\) −5.29947 −0.678528 −0.339264 0.940691i \(-0.610178\pi\)
−0.339264 + 0.940691i \(0.610178\pi\)
\(62\) −23.4683 −2.98048
\(63\) 0 0
\(64\) −11.7229 −1.46537
\(65\) 4.78493 0.593498
\(66\) 14.0224 1.72604
\(67\) −10.9313 −1.33548 −0.667738 0.744397i \(-0.732737\pi\)
−0.667738 + 0.744397i \(0.732737\pi\)
\(68\) −9.96041 −1.20788
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 3.00667 0.356826 0.178413 0.983956i \(-0.442904\pi\)
0.178413 + 0.983956i \(0.442904\pi\)
\(72\) −3.72294 −0.438752
\(73\) 1.50787 0.176483 0.0882415 0.996099i \(-0.471875\pi\)
0.0882415 + 0.996099i \(0.471875\pi\)
\(74\) −25.1755 −2.92659
\(75\) −6.29947 −0.727400
\(76\) 21.2375 2.43611
\(77\) 0 0
\(78\) −3.36147 −0.380611
\(79\) −4.66094 −0.524397 −0.262198 0.965014i \(-0.584447\pi\)
−0.262198 + 0.965014i \(0.584447\pi\)
\(80\) −5.50787 −0.615799
\(81\) 1.00000 0.111111
\(82\) −21.4683 −2.37078
\(83\) −4.78493 −0.525215 −0.262607 0.964903i \(-0.584582\pi\)
−0.262607 + 0.964903i \(0.584582\pi\)
\(84\) 0 0
\(85\) 9.36147 1.01539
\(86\) −7.79160 −0.840190
\(87\) 6.66094 0.714128
\(88\) −22.1068 −2.35659
\(89\) −6.57653 −0.697111 −0.348556 0.937288i \(-0.613328\pi\)
−0.348556 + 0.937288i \(0.613328\pi\)
\(90\) 7.93800 0.836739
\(91\) 0 0
\(92\) −3.57653 −0.372880
\(93\) −9.93800 −1.03052
\(94\) 21.6147 2.22938
\(95\) −19.9604 −2.04790
\(96\) −3.57653 −0.365029
\(97\) 9.50787 0.965378 0.482689 0.875792i \(-0.339660\pi\)
0.482689 + 0.875792i \(0.339660\pi\)
\(98\) 0 0
\(99\) 5.93800 0.596792
\(100\) 22.5303 2.25303
\(101\) −5.66761 −0.563948 −0.281974 0.959422i \(-0.590989\pi\)
−0.281974 + 0.959422i \(0.590989\pi\)
\(102\) −6.57653 −0.651174
\(103\) −13.4459 −1.32486 −0.662431 0.749123i \(-0.730475\pi\)
−0.662431 + 0.749123i \(0.730475\pi\)
\(104\) 5.29947 0.519656
\(105\) 0 0
\(106\) −7.93800 −0.771007
\(107\) −11.2995 −1.09236 −0.546181 0.837667i \(-0.683919\pi\)
−0.546181 + 0.837667i \(0.683919\pi\)
\(108\) −3.57653 −0.344152
\(109\) 6.08441 0.582780 0.291390 0.956604i \(-0.405882\pi\)
0.291390 + 0.956604i \(0.405882\pi\)
\(110\) 47.1359 4.49423
\(111\) −10.6609 −1.01189
\(112\) 0 0
\(113\) 3.06866 0.288676 0.144338 0.989528i \(-0.453895\pi\)
0.144338 + 0.989528i \(0.453895\pi\)
\(114\) 14.0224 1.31332
\(115\) 3.36147 0.313459
\(116\) −23.8231 −2.21192
\(117\) −1.42347 −0.131599
\(118\) 0.492128 0.0453041
\(119\) 0 0
\(120\) −12.5145 −1.14242
\(121\) 24.2599 2.20544
\(122\) 12.5145 1.13301
\(123\) −9.09107 −0.819714
\(124\) 35.5436 3.19191
\(125\) −4.36814 −0.390698
\(126\) 0 0
\(127\) −21.8984 −1.94317 −0.971585 0.236690i \(-0.923937\pi\)
−0.971585 + 0.236690i \(0.923937\pi\)
\(128\) 20.5303 1.81464
\(129\) −3.29947 −0.290502
\(130\) −11.2995 −0.991029
\(131\) 5.93800 0.518806 0.259403 0.965769i \(-0.416474\pi\)
0.259403 + 0.965769i \(0.416474\pi\)
\(132\) −21.2375 −1.84848
\(133\) 0 0
\(134\) 25.8140 2.22999
\(135\) 3.36147 0.289309
\(136\) 10.3681 0.889060
\(137\) −1.33906 −0.114404 −0.0572018 0.998363i \(-0.518218\pi\)
−0.0572018 + 0.998363i \(0.518218\pi\)
\(138\) −2.36147 −0.201022
\(139\) 17.3615 1.47258 0.736290 0.676666i \(-0.236576\pi\)
0.736290 + 0.676666i \(0.236576\pi\)
\(140\) 0 0
\(141\) 9.15307 0.770828
\(142\) −7.10015 −0.595831
\(143\) −8.45254 −0.706837
\(144\) 1.63853 0.136544
\(145\) 22.3905 1.85943
\(146\) −3.56079 −0.294693
\(147\) 0 0
\(148\) 38.1292 3.13420
\(149\) −7.02908 −0.575844 −0.287922 0.957654i \(-0.592964\pi\)
−0.287922 + 0.957654i \(0.592964\pi\)
\(150\) 14.8760 1.21462
\(151\) 6.59894 0.537014 0.268507 0.963278i \(-0.413470\pi\)
0.268507 + 0.963278i \(0.413470\pi\)
\(152\) −22.1068 −1.79310
\(153\) −2.78493 −0.225149
\(154\) 0 0
\(155\) −33.4063 −2.68326
\(156\) 5.09107 0.407612
\(157\) 19.7387 1.57532 0.787659 0.616111i \(-0.211293\pi\)
0.787659 + 0.616111i \(0.211293\pi\)
\(158\) 11.0067 0.875643
\(159\) −3.36147 −0.266582
\(160\) −12.0224 −0.950455
\(161\) 0 0
\(162\) −2.36147 −0.185535
\(163\) 2.93134 0.229600 0.114800 0.993389i \(-0.463377\pi\)
0.114800 + 0.993389i \(0.463377\pi\)
\(164\) 32.5145 2.53896
\(165\) 19.9604 1.55392
\(166\) 11.2995 0.877009
\(167\) 11.2599 0.871316 0.435658 0.900112i \(-0.356516\pi\)
0.435658 + 0.900112i \(0.356516\pi\)
\(168\) 0 0
\(169\) −10.9737 −0.844134
\(170\) −22.1068 −1.69552
\(171\) 5.93800 0.454090
\(172\) 11.8007 0.899793
\(173\) −1.09107 −0.0829527 −0.0414764 0.999139i \(-0.513206\pi\)
−0.0414764 + 0.999139i \(0.513206\pi\)
\(174\) −15.7296 −1.19246
\(175\) 0 0
\(176\) 9.72960 0.733397
\(177\) 0.208399 0.0156643
\(178\) 15.5303 1.16404
\(179\) −13.2375 −0.989415 −0.494708 0.869059i \(-0.664725\pi\)
−0.494708 + 0.869059i \(0.664725\pi\)
\(180\) −12.0224 −0.896098
\(181\) −18.2441 −1.35608 −0.678038 0.735027i \(-0.737170\pi\)
−0.678038 + 0.735027i \(0.737170\pi\)
\(182\) 0 0
\(183\) 5.29947 0.391748
\(184\) 3.72294 0.274459
\(185\) −35.8364 −2.63475
\(186\) 23.4683 1.72078
\(187\) −16.5369 −1.20930
\(188\) −32.7363 −2.38754
\(189\) 0 0
\(190\) 47.1359 3.41960
\(191\) 1.40106 0.101377 0.0506884 0.998715i \(-0.483858\pi\)
0.0506884 + 0.998715i \(0.483858\pi\)
\(192\) 11.7229 0.846030
\(193\) 16.5989 1.19482 0.597409 0.801937i \(-0.296197\pi\)
0.597409 + 0.801937i \(0.296197\pi\)
\(194\) −22.4525 −1.61200
\(195\) −4.78493 −0.342656
\(196\) 0 0
\(197\) −2.86934 −0.204432 −0.102216 0.994762i \(-0.532593\pi\)
−0.102216 + 0.994762i \(0.532593\pi\)
\(198\) −14.0224 −0.996529
\(199\) −18.5594 −1.31564 −0.657819 0.753176i \(-0.728521\pi\)
−0.657819 + 0.753176i \(0.728521\pi\)
\(200\) −23.4525 −1.65835
\(201\) 10.9313 0.771037
\(202\) 13.3839 0.941686
\(203\) 0 0
\(204\) 9.96041 0.697368
\(205\) −30.5594 −2.13436
\(206\) 31.7520 2.21227
\(207\) −1.00000 −0.0695048
\(208\) −2.33239 −0.161722
\(209\) 35.2599 2.43898
\(210\) 0 0
\(211\) −9.81401 −0.675624 −0.337812 0.941214i \(-0.609687\pi\)
−0.337812 + 0.941214i \(0.609687\pi\)
\(212\) 12.0224 0.825702
\(213\) −3.00667 −0.206013
\(214\) 26.6834 1.82404
\(215\) −11.0911 −0.756405
\(216\) 3.72294 0.253314
\(217\) 0 0
\(218\) −14.3681 −0.973133
\(219\) −1.50787 −0.101893
\(220\) −71.3891 −4.81305
\(221\) 3.96426 0.266665
\(222\) 25.1755 1.68967
\(223\) 15.6676 1.04918 0.524590 0.851355i \(-0.324219\pi\)
0.524590 + 0.851355i \(0.324219\pi\)
\(224\) 0 0
\(225\) 6.29947 0.419965
\(226\) −7.24655 −0.482033
\(227\) 7.96041 0.528351 0.264176 0.964475i \(-0.414900\pi\)
0.264176 + 0.964475i \(0.414900\pi\)
\(228\) −21.2375 −1.40649
\(229\) 14.9447 0.987572 0.493786 0.869584i \(-0.335613\pi\)
0.493786 + 0.869584i \(0.335613\pi\)
\(230\) −7.93800 −0.523416
\(231\) 0 0
\(232\) 24.7983 1.62809
\(233\) 9.83642 0.644405 0.322203 0.946671i \(-0.395577\pi\)
0.322203 + 0.946671i \(0.395577\pi\)
\(234\) 3.36147 0.219746
\(235\) 30.7678 2.00707
\(236\) −0.745347 −0.0485180
\(237\) 4.66094 0.302761
\(238\) 0 0
\(239\) 18.8073 1.21655 0.608273 0.793728i \(-0.291863\pi\)
0.608273 + 0.793728i \(0.291863\pi\)
\(240\) 5.50787 0.355532
\(241\) 10.2441 0.659883 0.329942 0.944001i \(-0.392971\pi\)
0.329942 + 0.944001i \(0.392971\pi\)
\(242\) −57.2890 −3.68267
\(243\) −1.00000 −0.0641500
\(244\) −18.9537 −1.21339
\(245\) 0 0
\(246\) 21.4683 1.36877
\(247\) −8.45254 −0.537822
\(248\) −36.9986 −2.34941
\(249\) 4.78493 0.303233
\(250\) 10.3152 0.652392
\(251\) 12.4168 0.783741 0.391871 0.920020i \(-0.371828\pi\)
0.391871 + 0.920020i \(0.371828\pi\)
\(252\) 0 0
\(253\) −5.93800 −0.373319
\(254\) 51.7124 3.24473
\(255\) −9.36147 −0.586238
\(256\) −25.0357 −1.56473
\(257\) 5.58320 0.348271 0.174135 0.984722i \(-0.444287\pi\)
0.174135 + 0.984722i \(0.444287\pi\)
\(258\) 7.79160 0.485084
\(259\) 0 0
\(260\) 17.1135 1.06133
\(261\) −6.66094 −0.412302
\(262\) −14.0224 −0.866307
\(263\) 19.6767 1.21332 0.606658 0.794963i \(-0.292510\pi\)
0.606658 + 0.794963i \(0.292510\pi\)
\(264\) 22.1068 1.36058
\(265\) −11.2995 −0.694121
\(266\) 0 0
\(267\) 6.57653 0.402477
\(268\) −39.0963 −2.38819
\(269\) 0.882674 0.0538176 0.0269088 0.999638i \(-0.491434\pi\)
0.0269088 + 0.999638i \(0.491434\pi\)
\(270\) −7.93800 −0.483092
\(271\) 0.784934 0.0476813 0.0238407 0.999716i \(-0.492411\pi\)
0.0238407 + 0.999716i \(0.492411\pi\)
\(272\) −4.56320 −0.276685
\(273\) 0 0
\(274\) 3.16215 0.191032
\(275\) 37.4063 2.25568
\(276\) 3.57653 0.215282
\(277\) −21.3443 −1.28245 −0.641227 0.767351i \(-0.721574\pi\)
−0.641227 + 0.767351i \(0.721574\pi\)
\(278\) −40.9986 −2.45893
\(279\) 9.93800 0.594973
\(280\) 0 0
\(281\) 21.4592 1.28015 0.640075 0.768313i \(-0.278903\pi\)
0.640075 + 0.768313i \(0.278903\pi\)
\(282\) −21.6147 −1.28714
\(283\) 13.6543 0.811662 0.405831 0.913948i \(-0.366982\pi\)
0.405831 + 0.913948i \(0.366982\pi\)
\(284\) 10.7534 0.638100
\(285\) 19.9604 1.18235
\(286\) 19.9604 1.18028
\(287\) 0 0
\(288\) 3.57653 0.210749
\(289\) −9.24414 −0.543773
\(290\) −52.8746 −3.10490
\(291\) −9.50787 −0.557361
\(292\) 5.39296 0.315599
\(293\) 3.32188 0.194066 0.0970332 0.995281i \(-0.469065\pi\)
0.0970332 + 0.995281i \(0.469065\pi\)
\(294\) 0 0
\(295\) 0.700528 0.0407863
\(296\) −39.6900 −2.30694
\(297\) −5.93800 −0.344558
\(298\) 16.5989 0.961551
\(299\) 1.42347 0.0823211
\(300\) −22.5303 −1.30079
\(301\) 0 0
\(302\) −15.5832 −0.896712
\(303\) 5.66761 0.325596
\(304\) 9.72960 0.558031
\(305\) 17.8140 1.02003
\(306\) 6.57653 0.375955
\(307\) 2.06200 0.117684 0.0588422 0.998267i \(-0.481259\pi\)
0.0588422 + 0.998267i \(0.481259\pi\)
\(308\) 0 0
\(309\) 13.4459 0.764909
\(310\) 78.8879 4.48053
\(311\) 8.45254 0.479300 0.239650 0.970859i \(-0.422967\pi\)
0.239650 + 0.970859i \(0.422967\pi\)
\(312\) −5.29947 −0.300024
\(313\) −10.7363 −0.606850 −0.303425 0.952855i \(-0.598130\pi\)
−0.303425 + 0.952855i \(0.598130\pi\)
\(314\) −46.6123 −2.63048
\(315\) 0 0
\(316\) −16.6700 −0.937762
\(317\) 16.8073 0.943994 0.471997 0.881600i \(-0.343533\pi\)
0.471997 + 0.881600i \(0.343533\pi\)
\(318\) 7.93800 0.445141
\(319\) −39.5527 −2.21453
\(320\) 39.4063 2.20288
\(321\) 11.2995 0.630675
\(322\) 0 0
\(323\) −16.5369 −0.920140
\(324\) 3.57653 0.198696
\(325\) −8.96708 −0.497404
\(326\) −6.92226 −0.383389
\(327\) −6.08441 −0.336468
\(328\) −33.8455 −1.86880
\(329\) 0 0
\(330\) −47.1359 −2.59475
\(331\) −21.5699 −1.18559 −0.592794 0.805354i \(-0.701975\pi\)
−0.592794 + 0.805354i \(0.701975\pi\)
\(332\) −17.1135 −0.939224
\(333\) 10.6609 0.584216
\(334\) −26.5899 −1.45493
\(335\) 36.7453 2.00761
\(336\) 0 0
\(337\) 5.42347 0.295435 0.147717 0.989030i \(-0.452807\pi\)
0.147717 + 0.989030i \(0.452807\pi\)
\(338\) 25.9142 1.40954
\(339\) −3.06866 −0.166667
\(340\) 33.4816 1.81580
\(341\) 59.0119 3.19567
\(342\) −14.0224 −0.758245
\(343\) 0 0
\(344\) −12.2837 −0.662294
\(345\) −3.36147 −0.180975
\(346\) 2.57653 0.138515
\(347\) 25.9828 1.39483 0.697416 0.716667i \(-0.254333\pi\)
0.697416 + 0.716667i \(0.254333\pi\)
\(348\) 23.8231 1.27705
\(349\) 2.50120 0.133886 0.0669432 0.997757i \(-0.478675\pi\)
0.0669432 + 0.997757i \(0.478675\pi\)
\(350\) 0 0
\(351\) 1.42347 0.0759790
\(352\) 21.2375 1.13196
\(353\) −19.5212 −1.03901 −0.519504 0.854468i \(-0.673883\pi\)
−0.519504 + 0.854468i \(0.673883\pi\)
\(354\) −0.492128 −0.0261563
\(355\) −10.1068 −0.536414
\(356\) −23.5212 −1.24662
\(357\) 0 0
\(358\) 31.2599 1.65214
\(359\) −37.3615 −1.97186 −0.985931 0.167150i \(-0.946544\pi\)
−0.985931 + 0.167150i \(0.946544\pi\)
\(360\) 12.5145 0.659574
\(361\) 16.2599 0.855783
\(362\) 43.0830 2.26439
\(363\) −24.2599 −1.27331
\(364\) 0 0
\(365\) −5.06866 −0.265306
\(366\) −12.5145 −0.654145
\(367\) −0.452542 −0.0236225 −0.0118112 0.999930i \(-0.503760\pi\)
−0.0118112 + 0.999930i \(0.503760\pi\)
\(368\) −1.63853 −0.0854143
\(369\) 9.09107 0.473262
\(370\) 84.6266 4.39953
\(371\) 0 0
\(372\) −35.5436 −1.84285
\(373\) −13.0911 −0.677830 −0.338915 0.940817i \(-0.610060\pi\)
−0.338915 + 0.940817i \(0.610060\pi\)
\(374\) 39.0515 2.01930
\(375\) 4.36814 0.225570
\(376\) 34.0763 1.75735
\(377\) 9.48162 0.488328
\(378\) 0 0
\(379\) 26.0315 1.33715 0.668574 0.743646i \(-0.266905\pi\)
0.668574 + 0.743646i \(0.266905\pi\)
\(380\) −71.3891 −3.66218
\(381\) 21.8984 1.12189
\(382\) −3.30855 −0.169280
\(383\) 3.09107 0.157946 0.0789732 0.996877i \(-0.474836\pi\)
0.0789732 + 0.996877i \(0.474836\pi\)
\(384\) −20.5303 −1.04768
\(385\) 0 0
\(386\) −39.1979 −1.99512
\(387\) 3.29947 0.167722
\(388\) 34.0052 1.72635
\(389\) −4.77160 −0.241930 −0.120965 0.992657i \(-0.538599\pi\)
−0.120965 + 0.992657i \(0.538599\pi\)
\(390\) 11.2995 0.572171
\(391\) 2.78493 0.140840
\(392\) 0 0
\(393\) −5.93800 −0.299533
\(394\) 6.77586 0.341363
\(395\) 15.6676 0.788323
\(396\) 21.2375 1.06722
\(397\) 9.92083 0.497912 0.248956 0.968515i \(-0.419913\pi\)
0.248956 + 0.968515i \(0.419913\pi\)
\(398\) 43.8273 2.19687
\(399\) 0 0
\(400\) 10.3219 0.516094
\(401\) 16.5503 0.826482 0.413241 0.910622i \(-0.364397\pi\)
0.413241 + 0.910622i \(0.364397\pi\)
\(402\) −25.8140 −1.28749
\(403\) −14.1464 −0.704683
\(404\) −20.2704 −1.00849
\(405\) −3.36147 −0.167033
\(406\) 0 0
\(407\) 63.3047 3.13790
\(408\) −10.3681 −0.513299
\(409\) −12.3548 −0.610906 −0.305453 0.952207i \(-0.598808\pi\)
−0.305453 + 0.952207i \(0.598808\pi\)
\(410\) 72.1650 3.56397
\(411\) 1.33906 0.0660509
\(412\) −48.0896 −2.36921
\(413\) 0 0
\(414\) 2.36147 0.116060
\(415\) 16.0844 0.789552
\(416\) −5.09107 −0.249610
\(417\) −17.3615 −0.850195
\(418\) −83.2651 −4.07263
\(419\) 37.8231 1.84778 0.923889 0.382660i \(-0.124992\pi\)
0.923889 + 0.382660i \(0.124992\pi\)
\(420\) 0 0
\(421\) 21.6189 1.05364 0.526821 0.849976i \(-0.323384\pi\)
0.526821 + 0.849976i \(0.323384\pi\)
\(422\) 23.1755 1.12816
\(423\) −9.15307 −0.445037
\(424\) −12.5145 −0.607760
\(425\) −17.5436 −0.850990
\(426\) 7.10015 0.344003
\(427\) 0 0
\(428\) −40.4130 −1.95343
\(429\) 8.45254 0.408093
\(430\) 26.1912 1.26305
\(431\) 39.6371 1.90925 0.954626 0.297808i \(-0.0962554\pi\)
0.954626 + 0.297808i \(0.0962554\pi\)
\(432\) −1.63853 −0.0788339
\(433\) 25.4592 1.22349 0.611746 0.791054i \(-0.290468\pi\)
0.611746 + 0.791054i \(0.290468\pi\)
\(434\) 0 0
\(435\) −22.3905 −1.07354
\(436\) 21.7611 1.04217
\(437\) −5.93800 −0.284053
\(438\) 3.56079 0.170141
\(439\) 25.6900 1.22612 0.613059 0.790037i \(-0.289939\pi\)
0.613059 + 0.790037i \(0.289939\pi\)
\(440\) 74.3114 3.54266
\(441\) 0 0
\(442\) −9.36147 −0.445280
\(443\) 1.44588 0.0686956 0.0343478 0.999410i \(-0.489065\pi\)
0.0343478 + 0.999410i \(0.489065\pi\)
\(444\) −38.1292 −1.80953
\(445\) 22.1068 1.04796
\(446\) −36.9986 −1.75193
\(447\) 7.02908 0.332464
\(448\) 0 0
\(449\) 10.2532 0.483879 0.241940 0.970291i \(-0.422216\pi\)
0.241940 + 0.970291i \(0.422216\pi\)
\(450\) −14.8760 −0.701262
\(451\) 53.9828 2.54195
\(452\) 10.9752 0.516229
\(453\) −6.59894 −0.310045
\(454\) −18.7983 −0.882246
\(455\) 0 0
\(456\) 22.1068 1.03525
\(457\) −35.5303 −1.66204 −0.831018 0.556245i \(-0.812242\pi\)
−0.831018 + 0.556245i \(0.812242\pi\)
\(458\) −35.2914 −1.64906
\(459\) 2.78493 0.129990
\(460\) 12.0224 0.560548
\(461\) −19.9471 −0.929028 −0.464514 0.885566i \(-0.653771\pi\)
−0.464514 + 0.885566i \(0.653771\pi\)
\(462\) 0 0
\(463\) 3.92467 0.182395 0.0911974 0.995833i \(-0.470931\pi\)
0.0911974 + 0.995833i \(0.470931\pi\)
\(464\) −10.9142 −0.506677
\(465\) 33.4063 1.54918
\(466\) −23.2284 −1.07603
\(467\) 15.0911 0.698332 0.349166 0.937061i \(-0.386465\pi\)
0.349166 + 0.937061i \(0.386465\pi\)
\(468\) −5.09107 −0.235335
\(469\) 0 0
\(470\) −72.6571 −3.35142
\(471\) −19.7387 −0.909510
\(472\) 0.775858 0.0357117
\(473\) 19.5923 0.900854
\(474\) −11.0067 −0.505553
\(475\) 37.4063 1.71632
\(476\) 0 0
\(477\) 3.36147 0.153911
\(478\) −44.4130 −2.03140
\(479\) −16.1201 −0.736548 −0.368274 0.929717i \(-0.620051\pi\)
−0.368274 + 0.929717i \(0.620051\pi\)
\(480\) 12.0224 0.548745
\(481\) −15.1755 −0.691942
\(482\) −24.1912 −1.10188
\(483\) 0 0
\(484\) 86.7663 3.94392
\(485\) −31.9604 −1.45125
\(486\) 2.36147 0.107118
\(487\) 15.8007 0.715997 0.357999 0.933722i \(-0.383459\pi\)
0.357999 + 0.933722i \(0.383459\pi\)
\(488\) 19.7296 0.893117
\(489\) −2.93134 −0.132560
\(490\) 0 0
\(491\) −1.97759 −0.0892474 −0.0446237 0.999004i \(-0.514209\pi\)
−0.0446237 + 0.999004i \(0.514209\pi\)
\(492\) −32.5145 −1.46587
\(493\) 18.5503 0.835463
\(494\) 19.9604 0.898061
\(495\) −19.9604 −0.897154
\(496\) 16.2837 0.731161
\(497\) 0 0
\(498\) −11.2995 −0.506341
\(499\) 20.5012 0.917760 0.458880 0.888498i \(-0.348251\pi\)
0.458880 + 0.888498i \(0.348251\pi\)
\(500\) −15.6228 −0.698672
\(501\) −11.2599 −0.503055
\(502\) −29.3219 −1.30870
\(503\) −20.1292 −0.897518 −0.448759 0.893653i \(-0.648134\pi\)
−0.448759 + 0.893653i \(0.648134\pi\)
\(504\) 0 0
\(505\) 19.0515 0.847780
\(506\) 14.0224 0.623372
\(507\) 10.9737 0.487361
\(508\) −78.3204 −3.47491
\(509\) −2.44347 −0.108305 −0.0541523 0.998533i \(-0.517246\pi\)
−0.0541523 + 0.998533i \(0.517246\pi\)
\(510\) 22.1068 0.978906
\(511\) 0 0
\(512\) 18.0606 0.798172
\(513\) −5.93800 −0.262169
\(514\) −13.1846 −0.581546
\(515\) 45.1979 1.99166
\(516\) −11.8007 −0.519496
\(517\) −54.3510 −2.39035
\(518\) 0 0
\(519\) 1.09107 0.0478928
\(520\) −17.8140 −0.781196
\(521\) 5.16640 0.226344 0.113172 0.993575i \(-0.463899\pi\)
0.113172 + 0.993575i \(0.463899\pi\)
\(522\) 15.7296 0.688466
\(523\) −34.5198 −1.50944 −0.754722 0.656045i \(-0.772228\pi\)
−0.754722 + 0.656045i \(0.772228\pi\)
\(524\) 21.2375 0.927763
\(525\) 0 0
\(526\) −46.4659 −2.02601
\(527\) −27.6767 −1.20562
\(528\) −9.72960 −0.423427
\(529\) 1.00000 0.0434783
\(530\) 26.6834 1.15905
\(531\) −0.208399 −0.00904376
\(532\) 0 0
\(533\) −12.9408 −0.560529
\(534\) −15.5303 −0.672061
\(535\) 37.9828 1.64214
\(536\) 40.6967 1.75783
\(537\) 13.2375 0.571239
\(538\) −2.08441 −0.0898651
\(539\) 0 0
\(540\) 12.0224 0.517362
\(541\) 43.3667 1.86448 0.932240 0.361840i \(-0.117851\pi\)
0.932240 + 0.361840i \(0.117851\pi\)
\(542\) −1.85360 −0.0796188
\(543\) 18.2441 0.782931
\(544\) −9.96041 −0.427049
\(545\) −20.4525 −0.876091
\(546\) 0 0
\(547\) 16.5279 0.706681 0.353340 0.935495i \(-0.385046\pi\)
0.353340 + 0.935495i \(0.385046\pi\)
\(548\) −4.78919 −0.204584
\(549\) −5.29947 −0.226176
\(550\) −88.3338 −3.76657
\(551\) −39.5527 −1.68500
\(552\) −3.72294 −0.158459
\(553\) 0 0
\(554\) 50.4039 2.14146
\(555\) 35.8364 1.52117
\(556\) 62.0939 2.63337
\(557\) 20.3510 0.862298 0.431149 0.902281i \(-0.358108\pi\)
0.431149 + 0.902281i \(0.358108\pi\)
\(558\) −23.4683 −0.993492
\(559\) −4.69668 −0.198649
\(560\) 0 0
\(561\) 16.5369 0.698190
\(562\) −50.6753 −2.13761
\(563\) 29.1888 1.23016 0.615081 0.788464i \(-0.289123\pi\)
0.615081 + 0.788464i \(0.289123\pi\)
\(564\) 32.7363 1.37845
\(565\) −10.3152 −0.433964
\(566\) −32.2441 −1.35532
\(567\) 0 0
\(568\) −11.1936 −0.469674
\(569\) 16.1821 0.678391 0.339195 0.940716i \(-0.389845\pi\)
0.339195 + 0.940716i \(0.389845\pi\)
\(570\) −47.1359 −1.97431
\(571\) −37.1226 −1.55353 −0.776765 0.629790i \(-0.783141\pi\)
−0.776765 + 0.629790i \(0.783141\pi\)
\(572\) −30.2308 −1.26401
\(573\) −1.40106 −0.0585299
\(574\) 0 0
\(575\) −6.29947 −0.262706
\(576\) −11.7229 −0.488456
\(577\) −41.7520 −1.73816 −0.869080 0.494672i \(-0.835288\pi\)
−0.869080 + 0.494672i \(0.835288\pi\)
\(578\) 21.8298 0.907998
\(579\) −16.5989 −0.689829
\(580\) 80.0806 3.32516
\(581\) 0 0
\(582\) 22.4525 0.930688
\(583\) 19.9604 0.826676
\(584\) −5.61371 −0.232297
\(585\) 4.78493 0.197833
\(586\) −7.84452 −0.324054
\(587\) 43.4683 1.79413 0.897064 0.441901i \(-0.145696\pi\)
0.897064 + 0.441901i \(0.145696\pi\)
\(588\) 0 0
\(589\) 59.0119 2.43154
\(590\) −1.65427 −0.0681054
\(591\) 2.86934 0.118029
\(592\) 17.4683 0.717942
\(593\) 4.84693 0.199040 0.0995198 0.995036i \(-0.468269\pi\)
0.0995198 + 0.995036i \(0.468269\pi\)
\(594\) 14.0224 0.575346
\(595\) 0 0
\(596\) −25.1397 −1.02976
\(597\) 18.5594 0.759584
\(598\) −3.36147 −0.137461
\(599\) 2.27040 0.0927659 0.0463829 0.998924i \(-0.485231\pi\)
0.0463829 + 0.998924i \(0.485231\pi\)
\(600\) 23.4525 0.957446
\(601\) 34.7940 1.41928 0.709639 0.704566i \(-0.248858\pi\)
0.709639 + 0.704566i \(0.248858\pi\)
\(602\) 0 0
\(603\) −10.9313 −0.445158
\(604\) 23.6014 0.960325
\(605\) −81.5488 −3.31543
\(606\) −13.3839 −0.543683
\(607\) −18.1912 −0.738359 −0.369179 0.929358i \(-0.620361\pi\)
−0.369179 + 0.929358i \(0.620361\pi\)
\(608\) 21.2375 0.861293
\(609\) 0 0
\(610\) −42.0672 −1.70325
\(611\) 13.0291 0.527100
\(612\) −9.96041 −0.402626
\(613\) 22.2175 0.897355 0.448678 0.893694i \(-0.351895\pi\)
0.448678 + 0.893694i \(0.351895\pi\)
\(614\) −4.86934 −0.196511
\(615\) 30.5594 1.23227
\(616\) 0 0
\(617\) −19.5789 −0.788219 −0.394109 0.919064i \(-0.628947\pi\)
−0.394109 + 0.919064i \(0.628947\pi\)
\(618\) −31.7520 −1.27725
\(619\) −28.4525 −1.14360 −0.571802 0.820392i \(-0.693755\pi\)
−0.571802 + 0.820392i \(0.693755\pi\)
\(620\) −119.479 −4.79838
\(621\) 1.00000 0.0401286
\(622\) −19.9604 −0.800340
\(623\) 0 0
\(624\) 2.33239 0.0933704
\(625\) −16.8140 −0.672560
\(626\) 25.3534 1.01332
\(627\) −35.2599 −1.40814
\(628\) 70.5961 2.81709
\(629\) −29.6900 −1.18382
\(630\) 0 0
\(631\) −42.0276 −1.67309 −0.836547 0.547895i \(-0.815429\pi\)
−0.836547 + 0.547895i \(0.815429\pi\)
\(632\) 17.3524 0.690241
\(633\) 9.81401 0.390072
\(634\) −39.6900 −1.57629
\(635\) 73.6108 2.92116
\(636\) −12.0224 −0.476720
\(637\) 0 0
\(638\) 93.4024 3.69784
\(639\) 3.00667 0.118942
\(640\) −69.0119 −2.72793
\(641\) 4.85601 0.191801 0.0959004 0.995391i \(-0.469427\pi\)
0.0959004 + 0.995391i \(0.469427\pi\)
\(642\) −26.6834 −1.05311
\(643\) 37.6056 1.48302 0.741510 0.670942i \(-0.234110\pi\)
0.741510 + 0.670942i \(0.234110\pi\)
\(644\) 0 0
\(645\) 11.0911 0.436711
\(646\) 39.0515 1.53646
\(647\) 33.7258 1.32590 0.662948 0.748665i \(-0.269305\pi\)
0.662948 + 0.748665i \(0.269305\pi\)
\(648\) −3.72294 −0.146251
\(649\) −1.23748 −0.0485752
\(650\) 21.1755 0.830571
\(651\) 0 0
\(652\) 10.4840 0.410586
\(653\) 30.6700 1.20021 0.600105 0.799921i \(-0.295125\pi\)
0.600105 + 0.799921i \(0.295125\pi\)
\(654\) 14.3681 0.561839
\(655\) −19.9604 −0.779918
\(656\) 14.8960 0.581591
\(657\) 1.50787 0.0588277
\(658\) 0 0
\(659\) −7.50787 −0.292465 −0.146233 0.989250i \(-0.546715\pi\)
−0.146233 + 0.989250i \(0.546715\pi\)
\(660\) 71.3891 2.77882
\(661\) −7.35239 −0.285975 −0.142987 0.989725i \(-0.545671\pi\)
−0.142987 + 0.989725i \(0.545671\pi\)
\(662\) 50.9366 1.97971
\(663\) −3.96426 −0.153959
\(664\) 17.8140 0.691318
\(665\) 0 0
\(666\) −25.1755 −0.975530
\(667\) 6.66094 0.257913
\(668\) 40.2714 1.55815
\(669\) −15.6676 −0.605745
\(670\) −86.7730 −3.35233
\(671\) −31.4683 −1.21482
\(672\) 0 0
\(673\) −25.0119 −0.964138 −0.482069 0.876133i \(-0.660115\pi\)
−0.482069 + 0.876133i \(0.660115\pi\)
\(674\) −12.8073 −0.493320
\(675\) −6.29947 −0.242467
\(676\) −39.2480 −1.50954
\(677\) −10.1597 −0.390470 −0.195235 0.980756i \(-0.562547\pi\)
−0.195235 + 0.980756i \(0.562547\pi\)
\(678\) 7.24655 0.278302
\(679\) 0 0
\(680\) −34.8522 −1.33652
\(681\) −7.96041 −0.305044
\(682\) −139.355 −5.33617
\(683\) −17.9828 −0.688094 −0.344047 0.938953i \(-0.611798\pi\)
−0.344047 + 0.938953i \(0.611798\pi\)
\(684\) 21.2375 0.812035
\(685\) 4.50120 0.171982
\(686\) 0 0
\(687\) −14.9447 −0.570175
\(688\) 5.40629 0.206113
\(689\) −4.78493 −0.182291
\(690\) 7.93800 0.302195
\(691\) −26.7320 −1.01693 −0.508467 0.861082i \(-0.669788\pi\)
−0.508467 + 0.861082i \(0.669788\pi\)
\(692\) −3.90226 −0.148342
\(693\) 0 0
\(694\) −61.3576 −2.32910
\(695\) −58.3600 −2.21372
\(696\) −24.7983 −0.939976
\(697\) −25.3180 −0.958989
\(698\) −5.90652 −0.223565
\(699\) −9.83642 −0.372048
\(700\) 0 0
\(701\) −41.7572 −1.57715 −0.788575 0.614939i \(-0.789181\pi\)
−0.788575 + 0.614939i \(0.789181\pi\)
\(702\) −3.36147 −0.126870
\(703\) 63.3047 2.38758
\(704\) −69.6108 −2.62356
\(705\) −30.7678 −1.15878
\(706\) 46.0987 1.73495
\(707\) 0 0
\(708\) 0.745347 0.0280119
\(709\) −31.1621 −1.17032 −0.585159 0.810918i \(-0.698968\pi\)
−0.585159 + 0.810918i \(0.698968\pi\)
\(710\) 23.8669 0.895710
\(711\) −4.66094 −0.174799
\(712\) 24.4840 0.917578
\(713\) −9.93800 −0.372181
\(714\) 0 0
\(715\) 28.4130 1.06258
\(716\) −47.3443 −1.76934
\(717\) −18.8073 −0.702373
\(718\) 88.2279 3.29264
\(719\) 36.1650 1.34873 0.674363 0.738400i \(-0.264418\pi\)
0.674363 + 0.738400i \(0.264418\pi\)
\(720\) −5.50787 −0.205266
\(721\) 0 0
\(722\) −38.3972 −1.42900
\(723\) −10.2441 −0.380984
\(724\) −65.2508 −2.42503
\(725\) −41.9604 −1.55837
\(726\) 57.2890 2.12619
\(727\) −44.3338 −1.64425 −0.822124 0.569308i \(-0.807211\pi\)
−0.822124 + 0.569308i \(0.807211\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 11.9695 0.443011
\(731\) −9.18881 −0.339861
\(732\) 18.9537 0.700551
\(733\) 16.5989 0.613096 0.306548 0.951855i \(-0.400826\pi\)
0.306548 + 0.951855i \(0.400826\pi\)
\(734\) 1.06866 0.0394451
\(735\) 0 0
\(736\) −3.57653 −0.131833
\(737\) −64.9103 −2.39100
\(738\) −21.4683 −0.790258
\(739\) 21.6900 0.797880 0.398940 0.916977i \(-0.369378\pi\)
0.398940 + 0.916977i \(0.369378\pi\)
\(740\) −128.170 −4.71163
\(741\) 8.45254 0.310512
\(742\) 0 0
\(743\) −13.1135 −0.481087 −0.240544 0.970638i \(-0.577326\pi\)
−0.240544 + 0.970638i \(0.577326\pi\)
\(744\) 36.9986 1.35643
\(745\) 23.6280 0.865664
\(746\) 30.9142 1.13185
\(747\) −4.78493 −0.175072
\(748\) −59.1450 −2.16255
\(749\) 0 0
\(750\) −10.3152 −0.376658
\(751\) −48.6700 −1.77599 −0.887997 0.459849i \(-0.847904\pi\)
−0.887997 + 0.459849i \(0.847904\pi\)
\(752\) −14.9976 −0.546906
\(753\) −12.4168 −0.452493
\(754\) −22.3905 −0.815416
\(755\) −22.1821 −0.807291
\(756\) 0 0
\(757\) 28.2308 1.02607 0.513033 0.858369i \(-0.328522\pi\)
0.513033 + 0.858369i \(0.328522\pi\)
\(758\) −61.4725 −2.23278
\(759\) 5.93800 0.215536
\(760\) 74.3114 2.69556
\(761\) −25.6014 −0.928048 −0.464024 0.885823i \(-0.653595\pi\)
−0.464024 + 0.885823i \(0.653595\pi\)
\(762\) −51.7124 −1.87334
\(763\) 0 0
\(764\) 5.01092 0.181289
\(765\) 9.36147 0.338465
\(766\) −7.29947 −0.263741
\(767\) 0.296649 0.0107114
\(768\) 25.0357 0.903400
\(769\) 18.8784 0.680773 0.340387 0.940286i \(-0.389442\pi\)
0.340387 + 0.940286i \(0.389442\pi\)
\(770\) 0 0
\(771\) −5.58320 −0.201074
\(772\) 59.3667 2.13665
\(773\) −29.6319 −1.06578 −0.532892 0.846183i \(-0.678895\pi\)
−0.532892 + 0.846183i \(0.678895\pi\)
\(774\) −7.79160 −0.280063
\(775\) 62.6042 2.24881
\(776\) −35.3972 −1.27069
\(777\) 0 0
\(778\) 11.2680 0.403977
\(779\) 53.9828 1.93414
\(780\) −17.1135 −0.612761
\(781\) 17.8536 0.638852
\(782\) −6.57653 −0.235176
\(783\) 6.66094 0.238043
\(784\) 0 0
\(785\) −66.3510 −2.36817
\(786\) 14.0224 0.500163
\(787\) 12.3324 0.439602 0.219801 0.975545i \(-0.429459\pi\)
0.219801 + 0.975545i \(0.429459\pi\)
\(788\) −10.2623 −0.365579
\(789\) −19.6767 −0.700509
\(790\) −36.9986 −1.31635
\(791\) 0 0
\(792\) −22.1068 −0.785532
\(793\) 7.54361 0.267882
\(794\) −23.4277 −0.831419
\(795\) 11.2995 0.400751
\(796\) −66.3782 −2.35271
\(797\) −9.55653 −0.338510 −0.169255 0.985572i \(-0.554136\pi\)
−0.169255 + 0.985572i \(0.554136\pi\)
\(798\) 0 0
\(799\) 25.4907 0.901796
\(800\) 22.5303 0.796566
\(801\) −6.57653 −0.232370
\(802\) −39.0830 −1.38007
\(803\) 8.95375 0.315971
\(804\) 39.0963 1.37882
\(805\) 0 0
\(806\) 33.4063 1.17669
\(807\) −0.882674 −0.0310716
\(808\) 21.1001 0.742301
\(809\) −40.7587 −1.43300 −0.716499 0.697588i \(-0.754257\pi\)
−0.716499 + 0.697588i \(0.754257\pi\)
\(810\) 7.93800 0.278913
\(811\) 6.35480 0.223147 0.111574 0.993756i \(-0.464411\pi\)
0.111574 + 0.993756i \(0.464411\pi\)
\(812\) 0 0
\(813\) −0.784934 −0.0275288
\(814\) −149.492 −5.23969
\(815\) −9.85360 −0.345156
\(816\) 4.56320 0.159744
\(817\) 19.5923 0.685447
\(818\) 29.1755 1.02010
\(819\) 0 0
\(820\) −109.297 −3.81680
\(821\) 32.2270 1.12473 0.562364 0.826890i \(-0.309892\pi\)
0.562364 + 0.826890i \(0.309892\pi\)
\(822\) −3.16215 −0.110293
\(823\) 5.19266 0.181005 0.0905023 0.995896i \(-0.471153\pi\)
0.0905023 + 0.995896i \(0.471153\pi\)
\(824\) 50.0582 1.74386
\(825\) −37.4063 −1.30232
\(826\) 0 0
\(827\) 39.1268 1.36057 0.680286 0.732946i \(-0.261855\pi\)
0.680286 + 0.732946i \(0.261855\pi\)
\(828\) −3.57653 −0.124293
\(829\) 11.2732 0.391535 0.195768 0.980650i \(-0.437280\pi\)
0.195768 + 0.980650i \(0.437280\pi\)
\(830\) −37.9828 −1.31840
\(831\) 21.3443 0.740425
\(832\) 16.6872 0.578524
\(833\) 0 0
\(834\) 40.9986 1.41966
\(835\) −37.8498 −1.30984
\(836\) 126.108 4.36154
\(837\) −9.93800 −0.343508
\(838\) −89.3180 −3.08544
\(839\) −26.1912 −0.904221 −0.452111 0.891962i \(-0.649329\pi\)
−0.452111 + 0.891962i \(0.649329\pi\)
\(840\) 0 0
\(841\) 15.3681 0.529936
\(842\) −51.0525 −1.75938
\(843\) −21.4592 −0.739094
\(844\) −35.1001 −1.20820
\(845\) 36.8879 1.26898
\(846\) 21.6147 0.743128
\(847\) 0 0
\(848\) 5.50787 0.189141
\(849\) −13.6543 −0.468613
\(850\) 41.4287 1.42099
\(851\) −10.6609 −0.365452
\(852\) −10.7534 −0.368407
\(853\) −39.7387 −1.36063 −0.680313 0.732921i \(-0.738156\pi\)
−0.680313 + 0.732921i \(0.738156\pi\)
\(854\) 0 0
\(855\) −19.9604 −0.682632
\(856\) 42.0672 1.43783
\(857\) 36.9366 1.26173 0.630865 0.775893i \(-0.282700\pi\)
0.630865 + 0.775893i \(0.282700\pi\)
\(858\) −19.9604 −0.681437
\(859\) −9.59653 −0.327430 −0.163715 0.986508i \(-0.552348\pi\)
−0.163715 + 0.986508i \(0.552348\pi\)
\(860\) −39.6676 −1.35265
\(861\) 0 0
\(862\) −93.6018 −3.18809
\(863\) 26.2270 0.892776 0.446388 0.894839i \(-0.352710\pi\)
0.446388 + 0.894839i \(0.352710\pi\)
\(864\) −3.57653 −0.121676
\(865\) 3.66761 0.124702
\(866\) −60.1211 −2.04300
\(867\) 9.24414 0.313948
\(868\) 0 0
\(869\) −27.6767 −0.938867
\(870\) 52.8746 1.79262
\(871\) 15.5604 0.527243
\(872\) −22.6519 −0.767089
\(873\) 9.50787 0.321793
\(874\) 14.0224 0.474315
\(875\) 0 0
\(876\) −5.39296 −0.182211
\(877\) 26.1240 0.882145 0.441072 0.897472i \(-0.354598\pi\)
0.441072 + 0.897472i \(0.354598\pi\)
\(878\) −60.6662 −2.04738
\(879\) −3.32188 −0.112044
\(880\) −32.7058 −1.10251
\(881\) 20.8469 0.702351 0.351175 0.936310i \(-0.385782\pi\)
0.351175 + 0.936310i \(0.385782\pi\)
\(882\) 0 0
\(883\) 58.6347 1.97321 0.986607 0.163114i \(-0.0521539\pi\)
0.986607 + 0.163114i \(0.0521539\pi\)
\(884\) 14.1783 0.476868
\(885\) −0.700528 −0.0235480
\(886\) −3.41439 −0.114709
\(887\) −28.1597 −0.945511 −0.472756 0.881194i \(-0.656741\pi\)
−0.472756 + 0.881194i \(0.656741\pi\)
\(888\) 39.6900 1.33191
\(889\) 0 0
\(890\) −52.2046 −1.74990
\(891\) 5.93800 0.198931
\(892\) 56.0357 1.87622
\(893\) −54.3510 −1.81879
\(894\) −16.5989 −0.555152
\(895\) 44.4974 1.48738
\(896\) 0 0
\(897\) −1.42347 −0.0475281
\(898\) −24.2127 −0.807987
\(899\) −66.1965 −2.20778
\(900\) 22.5303 0.751009
\(901\) −9.36147 −0.311876
\(902\) −127.479 −4.24458
\(903\) 0 0
\(904\) −11.4244 −0.379971
\(905\) 61.3271 2.03858
\(906\) 15.5832 0.517717
\(907\) −11.5884 −0.384788 −0.192394 0.981318i \(-0.561625\pi\)
−0.192394 + 0.981318i \(0.561625\pi\)
\(908\) 28.4707 0.944833
\(909\) −5.66761 −0.187983
\(910\) 0 0
\(911\) 14.5989 0.483685 0.241842 0.970316i \(-0.422248\pi\)
0.241842 + 0.970316i \(0.422248\pi\)
\(912\) −9.72960 −0.322179
\(913\) −28.4130 −0.940332
\(914\) 83.9036 2.77529
\(915\) −17.8140 −0.588913
\(916\) 53.4501 1.76604
\(917\) 0 0
\(918\) −6.57653 −0.217058
\(919\) −33.9380 −1.11951 −0.559756 0.828658i \(-0.689105\pi\)
−0.559756 + 0.828658i \(0.689105\pi\)
\(920\) −12.5145 −0.412592
\(921\) −2.06200 −0.0679451
\(922\) 47.1044 1.55130
\(923\) −4.27989 −0.140874
\(924\) 0 0
\(925\) 67.1583 2.20815
\(926\) −9.26799 −0.304565
\(927\) −13.4459 −0.441620
\(928\) −23.8231 −0.782031
\(929\) 5.17932 0.169928 0.0849640 0.996384i \(-0.472922\pi\)
0.0849640 + 0.996384i \(0.472922\pi\)
\(930\) −78.8879 −2.58684
\(931\) 0 0
\(932\) 35.1803 1.15237
\(933\) −8.45254 −0.276724
\(934\) −35.6371 −1.16608
\(935\) 55.5884 1.81794
\(936\) 5.29947 0.173219
\(937\) −28.6924 −0.937341 −0.468670 0.883373i \(-0.655267\pi\)
−0.468670 + 0.883373i \(0.655267\pi\)
\(938\) 0 0
\(939\) 10.7363 0.350365
\(940\) 110.042 3.58917
\(941\) 31.9342 1.04102 0.520512 0.853854i \(-0.325741\pi\)
0.520512 + 0.853854i \(0.325741\pi\)
\(942\) 46.6123 1.51871
\(943\) −9.09107 −0.296046
\(944\) −0.341469 −0.0111139
\(945\) 0 0
\(946\) −46.2666 −1.50426
\(947\) −8.46162 −0.274966 −0.137483 0.990504i \(-0.543901\pi\)
−0.137483 + 0.990504i \(0.543901\pi\)
\(948\) 16.6700 0.541417
\(949\) −2.14640 −0.0696752
\(950\) −88.3338 −2.86593
\(951\) −16.8073 −0.545015
\(952\) 0 0
\(953\) −21.5475 −0.697991 −0.348995 0.937124i \(-0.613477\pi\)
−0.348995 + 0.937124i \(0.613477\pi\)
\(954\) −7.93800 −0.257002
\(955\) −4.70960 −0.152399
\(956\) 67.2651 2.17551
\(957\) 39.5527 1.27856
\(958\) 38.0672 1.22990
\(959\) 0 0
\(960\) −39.4063 −1.27183
\(961\) 67.7639 2.18593
\(962\) 35.8364 1.15541
\(963\) −11.2995 −0.364120
\(964\) 36.6385 1.18005
\(965\) −55.7968 −1.79616
\(966\) 0 0
\(967\) 32.4883 1.04475 0.522376 0.852715i \(-0.325046\pi\)
0.522376 + 0.852715i \(0.325046\pi\)
\(968\) −90.3180 −2.90293
\(969\) 16.5369 0.531243
\(970\) 75.4735 2.42331
\(971\) −17.1621 −0.550759 −0.275380 0.961336i \(-0.588804\pi\)
−0.275380 + 0.961336i \(0.588804\pi\)
\(972\) −3.57653 −0.114717
\(973\) 0 0
\(974\) −37.3128 −1.19558
\(975\) 8.96708 0.287176
\(976\) −8.68335 −0.277947
\(977\) 22.3285 0.714354 0.357177 0.934037i \(-0.383739\pi\)
0.357177 + 0.934037i \(0.383739\pi\)
\(978\) 6.92226 0.221349
\(979\) −39.0515 −1.24809
\(980\) 0 0
\(981\) 6.08441 0.194260
\(982\) 4.67002 0.149026
\(983\) −39.2151 −1.25077 −0.625383 0.780318i \(-0.715057\pi\)
−0.625383 + 0.780318i \(0.715057\pi\)
\(984\) 33.8455 1.07896
\(985\) 9.64520 0.307322
\(986\) −43.8059 −1.39506
\(987\) 0 0
\(988\) −30.2308 −0.961770
\(989\) −3.29947 −0.104917
\(990\) 47.1359 1.49808
\(991\) −36.4974 −1.15938 −0.579688 0.814838i \(-0.696826\pi\)
−0.579688 + 0.814838i \(0.696826\pi\)
\(992\) 35.5436 1.12851
\(993\) 21.5699 0.684499
\(994\) 0 0
\(995\) 62.3867 1.97779
\(996\) 17.1135 0.542261
\(997\) −29.0644 −0.920479 −0.460239 0.887795i \(-0.652236\pi\)
−0.460239 + 0.887795i \(0.652236\pi\)
\(998\) −48.4130 −1.53249
\(999\) −10.6609 −0.337297
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 3381.2.a.v.1.1 3
7.6 odd 2 483.2.a.h.1.1 3
21.20 even 2 1449.2.a.l.1.3 3
28.27 even 2 7728.2.a.bt.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.h.1.1 3 7.6 odd 2
1449.2.a.l.1.3 3 21.20 even 2
3381.2.a.v.1.1 3 1.1 even 1 trivial
7728.2.a.bt.1.3 3 28.27 even 2