Properties

Label 4950.2.a.bw.1.1
Level $4950$
Weight $2$
Character 4950.1
Self dual yes
Analytic conductor $39.526$
Analytic rank $1$
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] = [4950,2,Mod(1,4950)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4950, 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("4950.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4950 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4950.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: \(39.5259490005\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 110)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(3.37228\) of defining polynomial
Character \(\chi\) \(=\) 4950.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.37228 q^{7} -1.00000 q^{8} +1.00000 q^{11} -2.00000 q^{13} +3.37228 q^{14} +1.00000 q^{16} +1.37228 q^{17} +0.627719 q^{19} -1.00000 q^{22} +2.74456 q^{23} +2.00000 q^{26} -3.37228 q^{28} -1.37228 q^{29} +3.37228 q^{31} -1.00000 q^{32} -1.37228 q^{34} -9.37228 q^{37} -0.627719 q^{38} +11.4891 q^{41} +4.00000 q^{43} +1.00000 q^{44} -2.74456 q^{46} +2.74456 q^{47} +4.37228 q^{49} -2.00000 q^{52} -4.11684 q^{53} +3.37228 q^{56} +1.37228 q^{58} +2.74456 q^{59} -5.37228 q^{61} -3.37228 q^{62} +1.00000 q^{64} -8.00000 q^{67} +1.37228 q^{68} -10.1168 q^{71} +15.4891 q^{73} +9.37228 q^{74} +0.627719 q^{76} -3.37228 q^{77} -1.25544 q^{79} -11.4891 q^{82} -2.74456 q^{83} -4.00000 q^{86} -1.00000 q^{88} +1.37228 q^{89} +6.74456 q^{91} +2.74456 q^{92} -2.74456 q^{94} +12.7446 q^{97} -4.37228 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} + 2 q^{4} - q^{7} - 2 q^{8} + 2 q^{11} - 4 q^{13} + q^{14} + 2 q^{16} - 3 q^{17} + 7 q^{19} - 2 q^{22} - 6 q^{23} + 4 q^{26} - q^{28} + 3 q^{29} + q^{31} - 2 q^{32} + 3 q^{34} - 13 q^{37}+ \cdots - 3 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) −3.37228 −1.27460 −0.637301 0.770615i \(-0.719949\pi\)
−0.637301 + 0.770615i \(0.719949\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 3.37228 0.901280
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 1.37228 0.332827 0.166414 0.986056i \(-0.446781\pi\)
0.166414 + 0.986056i \(0.446781\pi\)
\(18\) 0 0
\(19\) 0.627719 0.144009 0.0720043 0.997404i \(-0.477060\pi\)
0.0720043 + 0.997404i \(0.477060\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) 2.74456 0.572281 0.286140 0.958188i \(-0.407628\pi\)
0.286140 + 0.958188i \(0.407628\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) −3.37228 −0.637301
\(29\) −1.37228 −0.254826 −0.127413 0.991850i \(-0.540667\pi\)
−0.127413 + 0.991850i \(0.540667\pi\)
\(30\) 0 0
\(31\) 3.37228 0.605680 0.302840 0.953041i \(-0.402065\pi\)
0.302840 + 0.953041i \(0.402065\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) −1.37228 −0.235344
\(35\) 0 0
\(36\) 0 0
\(37\) −9.37228 −1.54079 −0.770397 0.637565i \(-0.779942\pi\)
−0.770397 + 0.637565i \(0.779942\pi\)
\(38\) −0.627719 −0.101829
\(39\) 0 0
\(40\) 0 0
\(41\) 11.4891 1.79430 0.897150 0.441726i \(-0.145634\pi\)
0.897150 + 0.441726i \(0.145634\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) −2.74456 −0.404664
\(47\) 2.74456 0.400336 0.200168 0.979762i \(-0.435851\pi\)
0.200168 + 0.979762i \(0.435851\pi\)
\(48\) 0 0
\(49\) 4.37228 0.624612
\(50\) 0 0
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −4.11684 −0.565492 −0.282746 0.959195i \(-0.591245\pi\)
−0.282746 + 0.959195i \(0.591245\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 3.37228 0.450640
\(57\) 0 0
\(58\) 1.37228 0.180189
\(59\) 2.74456 0.357312 0.178656 0.983912i \(-0.442825\pi\)
0.178656 + 0.983912i \(0.442825\pi\)
\(60\) 0 0
\(61\) −5.37228 −0.687850 −0.343925 0.938997i \(-0.611757\pi\)
−0.343925 + 0.938997i \(0.611757\pi\)
\(62\) −3.37228 −0.428280
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 1.37228 0.166414
\(69\) 0 0
\(70\) 0 0
\(71\) −10.1168 −1.20065 −0.600324 0.799757i \(-0.704962\pi\)
−0.600324 + 0.799757i \(0.704962\pi\)
\(72\) 0 0
\(73\) 15.4891 1.81286 0.906432 0.422351i \(-0.138795\pi\)
0.906432 + 0.422351i \(0.138795\pi\)
\(74\) 9.37228 1.08951
\(75\) 0 0
\(76\) 0.627719 0.0720043
\(77\) −3.37228 −0.384307
\(78\) 0 0
\(79\) −1.25544 −0.141248 −0.0706239 0.997503i \(-0.522499\pi\)
−0.0706239 + 0.997503i \(0.522499\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −11.4891 −1.26876
\(83\) −2.74456 −0.301255 −0.150627 0.988591i \(-0.548129\pi\)
−0.150627 + 0.988591i \(0.548129\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 1.37228 0.145462 0.0727308 0.997352i \(-0.476829\pi\)
0.0727308 + 0.997352i \(0.476829\pi\)
\(90\) 0 0
\(91\) 6.74456 0.707022
\(92\) 2.74456 0.286140
\(93\) 0 0
\(94\) −2.74456 −0.283080
\(95\) 0 0
\(96\) 0 0
\(97\) 12.7446 1.29401 0.647007 0.762484i \(-0.276020\pi\)
0.647007 + 0.762484i \(0.276020\pi\)
\(98\) −4.37228 −0.441667
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 9.48913 0.934991 0.467496 0.883995i \(-0.345156\pi\)
0.467496 + 0.883995i \(0.345156\pi\)
\(104\) 2.00000 0.196116
\(105\) 0 0
\(106\) 4.11684 0.399863
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −15.4891 −1.48359 −0.741795 0.670627i \(-0.766025\pi\)
−0.741795 + 0.670627i \(0.766025\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.37228 −0.318651
\(113\) 3.25544 0.306246 0.153123 0.988207i \(-0.451067\pi\)
0.153123 + 0.988207i \(0.451067\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.37228 −0.127413
\(117\) 0 0
\(118\) −2.74456 −0.252657
\(119\) −4.62772 −0.424222
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 5.37228 0.486383
\(123\) 0 0
\(124\) 3.37228 0.302840
\(125\) 0 0
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 0 0
\(131\) −22.1168 −1.93236 −0.966179 0.257873i \(-0.916978\pi\)
−0.966179 + 0.257873i \(0.916978\pi\)
\(132\) 0 0
\(133\) −2.11684 −0.183554
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) −1.37228 −0.117672
\(137\) 8.74456 0.747098 0.373549 0.927610i \(-0.378141\pi\)
0.373549 + 0.927610i \(0.378141\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 10.1168 0.848987
\(143\) −2.00000 −0.167248
\(144\) 0 0
\(145\) 0 0
\(146\) −15.4891 −1.28189
\(147\) 0 0
\(148\) −9.37228 −0.770397
\(149\) −21.6060 −1.77003 −0.885015 0.465563i \(-0.845852\pi\)
−0.885015 + 0.465563i \(0.845852\pi\)
\(150\) 0 0
\(151\) −12.2337 −0.995563 −0.497782 0.867302i \(-0.665852\pi\)
−0.497782 + 0.867302i \(0.665852\pi\)
\(152\) −0.627719 −0.0509147
\(153\) 0 0
\(154\) 3.37228 0.271746
\(155\) 0 0
\(156\) 0 0
\(157\) −9.37228 −0.747989 −0.373995 0.927431i \(-0.622012\pi\)
−0.373995 + 0.927431i \(0.622012\pi\)
\(158\) 1.25544 0.0998772
\(159\) 0 0
\(160\) 0 0
\(161\) −9.25544 −0.729431
\(162\) 0 0
\(163\) 5.88316 0.460804 0.230402 0.973095i \(-0.425996\pi\)
0.230402 + 0.973095i \(0.425996\pi\)
\(164\) 11.4891 0.897150
\(165\) 0 0
\(166\) 2.74456 0.213019
\(167\) −4.62772 −0.358104 −0.179052 0.983840i \(-0.557303\pi\)
−0.179052 + 0.983840i \(0.557303\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) −6.00000 −0.456172 −0.228086 0.973641i \(-0.573247\pi\)
−0.228086 + 0.973641i \(0.573247\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −1.37228 −0.102857
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) −6.74456 −0.499940
\(183\) 0 0
\(184\) −2.74456 −0.202332
\(185\) 0 0
\(186\) 0 0
\(187\) 1.37228 0.100351
\(188\) 2.74456 0.200168
\(189\) 0 0
\(190\) 0 0
\(191\) 5.48913 0.397179 0.198590 0.980083i \(-0.436364\pi\)
0.198590 + 0.980083i \(0.436364\pi\)
\(192\) 0 0
\(193\) −14.8614 −1.06975 −0.534874 0.844932i \(-0.679641\pi\)
−0.534874 + 0.844932i \(0.679641\pi\)
\(194\) −12.7446 −0.915006
\(195\) 0 0
\(196\) 4.37228 0.312306
\(197\) −20.7446 −1.47799 −0.738994 0.673712i \(-0.764699\pi\)
−0.738994 + 0.673712i \(0.764699\pi\)
\(198\) 0 0
\(199\) 18.1168 1.28427 0.642135 0.766592i \(-0.278049\pi\)
0.642135 + 0.766592i \(0.278049\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 4.62772 0.324802
\(204\) 0 0
\(205\) 0 0
\(206\) −9.48913 −0.661139
\(207\) 0 0
\(208\) −2.00000 −0.138675
\(209\) 0.627719 0.0434202
\(210\) 0 0
\(211\) 6.11684 0.421101 0.210550 0.977583i \(-0.432474\pi\)
0.210550 + 0.977583i \(0.432474\pi\)
\(212\) −4.11684 −0.282746
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) −11.3723 −0.772001
\(218\) 15.4891 1.04906
\(219\) 0 0
\(220\) 0 0
\(221\) −2.74456 −0.184619
\(222\) 0 0
\(223\) 18.7446 1.25523 0.627614 0.778524i \(-0.284031\pi\)
0.627614 + 0.778524i \(0.284031\pi\)
\(224\) 3.37228 0.225320
\(225\) 0 0
\(226\) −3.25544 −0.216548
\(227\) −2.74456 −0.182163 −0.0910815 0.995843i \(-0.529032\pi\)
−0.0910815 + 0.995843i \(0.529032\pi\)
\(228\) 0 0
\(229\) −10.0000 −0.660819 −0.330409 0.943838i \(-0.607187\pi\)
−0.330409 + 0.943838i \(0.607187\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.37228 0.0900947
\(233\) 1.37228 0.0899011 0.0449506 0.998989i \(-0.485687\pi\)
0.0449506 + 0.998989i \(0.485687\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2.74456 0.178656
\(237\) 0 0
\(238\) 4.62772 0.299970
\(239\) 14.7446 0.953746 0.476873 0.878972i \(-0.341770\pi\)
0.476873 + 0.878972i \(0.341770\pi\)
\(240\) 0 0
\(241\) −22.0000 −1.41714 −0.708572 0.705638i \(-0.750660\pi\)
−0.708572 + 0.705638i \(0.750660\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) −5.37228 −0.343925
\(245\) 0 0
\(246\) 0 0
\(247\) −1.25544 −0.0798816
\(248\) −3.37228 −0.214140
\(249\) 0 0
\(250\) 0 0
\(251\) −2.74456 −0.173235 −0.0866176 0.996242i \(-0.527606\pi\)
−0.0866176 + 0.996242i \(0.527606\pi\)
\(252\) 0 0
\(253\) 2.74456 0.172549
\(254\) 8.00000 0.501965
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 18.0000 1.12281 0.561405 0.827541i \(-0.310261\pi\)
0.561405 + 0.827541i \(0.310261\pi\)
\(258\) 0 0
\(259\) 31.6060 1.96390
\(260\) 0 0
\(261\) 0 0
\(262\) 22.1168 1.36638
\(263\) −24.8614 −1.53302 −0.766510 0.642232i \(-0.778008\pi\)
−0.766510 + 0.642232i \(0.778008\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 2.11684 0.129792
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) 8.74456 0.533165 0.266583 0.963812i \(-0.414105\pi\)
0.266583 + 0.963812i \(0.414105\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) 1.37228 0.0832068
\(273\) 0 0
\(274\) −8.74456 −0.528278
\(275\) 0 0
\(276\) 0 0
\(277\) 12.7446 0.765747 0.382873 0.923801i \(-0.374935\pi\)
0.382873 + 0.923801i \(0.374935\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −5.25544 −0.312403 −0.156202 0.987725i \(-0.549925\pi\)
−0.156202 + 0.987725i \(0.549925\pi\)
\(284\) −10.1168 −0.600324
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) −38.7446 −2.28702
\(288\) 0 0
\(289\) −15.1168 −0.889226
\(290\) 0 0
\(291\) 0 0
\(292\) 15.4891 0.906432
\(293\) 23.4891 1.37225 0.686125 0.727484i \(-0.259310\pi\)
0.686125 + 0.727484i \(0.259310\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 9.37228 0.544753
\(297\) 0 0
\(298\) 21.6060 1.25160
\(299\) −5.48913 −0.317444
\(300\) 0 0
\(301\) −13.4891 −0.777500
\(302\) 12.2337 0.703970
\(303\) 0 0
\(304\) 0.627719 0.0360021
\(305\) 0 0
\(306\) 0 0
\(307\) −5.25544 −0.299944 −0.149972 0.988690i \(-0.547918\pi\)
−0.149972 + 0.988690i \(0.547918\pi\)
\(308\) −3.37228 −0.192154
\(309\) 0 0
\(310\) 0 0
\(311\) −19.3723 −1.09850 −0.549251 0.835658i \(-0.685087\pi\)
−0.549251 + 0.835658i \(0.685087\pi\)
\(312\) 0 0
\(313\) 22.0000 1.24351 0.621757 0.783210i \(-0.286419\pi\)
0.621757 + 0.783210i \(0.286419\pi\)
\(314\) 9.37228 0.528908
\(315\) 0 0
\(316\) −1.25544 −0.0706239
\(317\) −24.3505 −1.36766 −0.683831 0.729640i \(-0.739687\pi\)
−0.683831 + 0.729640i \(0.739687\pi\)
\(318\) 0 0
\(319\) −1.37228 −0.0768330
\(320\) 0 0
\(321\) 0 0
\(322\) 9.25544 0.515785
\(323\) 0.861407 0.0479299
\(324\) 0 0
\(325\) 0 0
\(326\) −5.88316 −0.325838
\(327\) 0 0
\(328\) −11.4891 −0.634381
\(329\) −9.25544 −0.510269
\(330\) 0 0
\(331\) 30.9783 1.70272 0.851359 0.524583i \(-0.175779\pi\)
0.851359 + 0.524583i \(0.175779\pi\)
\(332\) −2.74456 −0.150627
\(333\) 0 0
\(334\) 4.62772 0.253217
\(335\) 0 0
\(336\) 0 0
\(337\) −24.1168 −1.31373 −0.656864 0.754009i \(-0.728117\pi\)
−0.656864 + 0.754009i \(0.728117\pi\)
\(338\) 9.00000 0.489535
\(339\) 0 0
\(340\) 0 0
\(341\) 3.37228 0.182619
\(342\) 0 0
\(343\) 8.86141 0.478471
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) 6.00000 0.322562
\(347\) −32.2337 −1.73040 −0.865198 0.501431i \(-0.832807\pi\)
−0.865198 + 0.501431i \(0.832807\pi\)
\(348\) 0 0
\(349\) 19.4891 1.04323 0.521614 0.853181i \(-0.325330\pi\)
0.521614 + 0.853181i \(0.325330\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) −0.510875 −0.0271911 −0.0135956 0.999908i \(-0.504328\pi\)
−0.0135956 + 0.999908i \(0.504328\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.37228 0.0727308
\(357\) 0 0
\(358\) −12.0000 −0.634220
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) −18.6060 −0.979262
\(362\) 10.0000 0.525588
\(363\) 0 0
\(364\) 6.74456 0.353511
\(365\) 0 0
\(366\) 0 0
\(367\) 4.00000 0.208798 0.104399 0.994535i \(-0.466708\pi\)
0.104399 + 0.994535i \(0.466708\pi\)
\(368\) 2.74456 0.143070
\(369\) 0 0
\(370\) 0 0
\(371\) 13.8832 0.720778
\(372\) 0 0
\(373\) −31.4891 −1.63045 −0.815223 0.579148i \(-0.803385\pi\)
−0.815223 + 0.579148i \(0.803385\pi\)
\(374\) −1.37228 −0.0709590
\(375\) 0 0
\(376\) −2.74456 −0.141540
\(377\) 2.74456 0.141352
\(378\) 0 0
\(379\) −0.233688 −0.0120037 −0.00600187 0.999982i \(-0.501910\pi\)
−0.00600187 + 0.999982i \(0.501910\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −5.48913 −0.280848
\(383\) 32.2337 1.64706 0.823532 0.567269i \(-0.192000\pi\)
0.823532 + 0.567269i \(0.192000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 14.8614 0.756426
\(387\) 0 0
\(388\) 12.7446 0.647007
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 3.76631 0.190471
\(392\) −4.37228 −0.220834
\(393\) 0 0
\(394\) 20.7446 1.04510
\(395\) 0 0
\(396\) 0 0
\(397\) −24.9783 −1.25362 −0.626811 0.779171i \(-0.715640\pi\)
−0.626811 + 0.779171i \(0.715640\pi\)
\(398\) −18.1168 −0.908115
\(399\) 0 0
\(400\) 0 0
\(401\) −13.3723 −0.667780 −0.333890 0.942612i \(-0.608361\pi\)
−0.333890 + 0.942612i \(0.608361\pi\)
\(402\) 0 0
\(403\) −6.74456 −0.335971
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) −4.62772 −0.229670
\(407\) −9.37228 −0.464567
\(408\) 0 0
\(409\) −1.76631 −0.0873385 −0.0436693 0.999046i \(-0.513905\pi\)
−0.0436693 + 0.999046i \(0.513905\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 9.48913 0.467496
\(413\) −9.25544 −0.455430
\(414\) 0 0
\(415\) 0 0
\(416\) 2.00000 0.0980581
\(417\) 0 0
\(418\) −0.627719 −0.0307027
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 10.2337 0.498759 0.249380 0.968406i \(-0.419773\pi\)
0.249380 + 0.968406i \(0.419773\pi\)
\(422\) −6.11684 −0.297763
\(423\) 0 0
\(424\) 4.11684 0.199932
\(425\) 0 0
\(426\) 0 0
\(427\) 18.1168 0.876736
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 34.9783 1.68484 0.842422 0.538819i \(-0.181129\pi\)
0.842422 + 0.538819i \(0.181129\pi\)
\(432\) 0 0
\(433\) −27.7228 −1.33227 −0.666137 0.745830i \(-0.732053\pi\)
−0.666137 + 0.745830i \(0.732053\pi\)
\(434\) 11.3723 0.545887
\(435\) 0 0
\(436\) −15.4891 −0.741795
\(437\) 1.72281 0.0824133
\(438\) 0 0
\(439\) 18.9783 0.905782 0.452891 0.891566i \(-0.350393\pi\)
0.452891 + 0.891566i \(0.350393\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.74456 0.130546
\(443\) 29.4891 1.40107 0.700535 0.713618i \(-0.252945\pi\)
0.700535 + 0.713618i \(0.252945\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −18.7446 −0.887581
\(447\) 0 0
\(448\) −3.37228 −0.159325
\(449\) −28.9783 −1.36757 −0.683784 0.729684i \(-0.739667\pi\)
−0.683784 + 0.729684i \(0.739667\pi\)
\(450\) 0 0
\(451\) 11.4891 0.541002
\(452\) 3.25544 0.153123
\(453\) 0 0
\(454\) 2.74456 0.128809
\(455\) 0 0
\(456\) 0 0
\(457\) 16.3505 0.764846 0.382423 0.923987i \(-0.375090\pi\)
0.382423 + 0.923987i \(0.375090\pi\)
\(458\) 10.0000 0.467269
\(459\) 0 0
\(460\) 0 0
\(461\) −16.1168 −0.750636 −0.375318 0.926896i \(-0.622467\pi\)
−0.375318 + 0.926896i \(0.622467\pi\)
\(462\) 0 0
\(463\) 0.233688 0.0108604 0.00543020 0.999985i \(-0.498272\pi\)
0.00543020 + 0.999985i \(0.498272\pi\)
\(464\) −1.37228 −0.0637066
\(465\) 0 0
\(466\) −1.37228 −0.0635697
\(467\) −19.3723 −0.896442 −0.448221 0.893923i \(-0.647942\pi\)
−0.448221 + 0.893923i \(0.647942\pi\)
\(468\) 0 0
\(469\) 26.9783 1.24574
\(470\) 0 0
\(471\) 0 0
\(472\) −2.74456 −0.126329
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 0 0
\(476\) −4.62772 −0.212111
\(477\) 0 0
\(478\) −14.7446 −0.674401
\(479\) −5.48913 −0.250805 −0.125402 0.992106i \(-0.540022\pi\)
−0.125402 + 0.992106i \(0.540022\pi\)
\(480\) 0 0
\(481\) 18.7446 0.854678
\(482\) 22.0000 1.00207
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) −20.0000 −0.906287 −0.453143 0.891438i \(-0.649697\pi\)
−0.453143 + 0.891438i \(0.649697\pi\)
\(488\) 5.37228 0.243192
\(489\) 0 0
\(490\) 0 0
\(491\) 7.37228 0.332706 0.166353 0.986066i \(-0.446801\pi\)
0.166353 + 0.986066i \(0.446801\pi\)
\(492\) 0 0
\(493\) −1.88316 −0.0848131
\(494\) 1.25544 0.0564848
\(495\) 0 0
\(496\) 3.37228 0.151420
\(497\) 34.1168 1.53035
\(498\) 0 0
\(499\) −33.4891 −1.49918 −0.749590 0.661903i \(-0.769749\pi\)
−0.749590 + 0.661903i \(0.769749\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 2.74456 0.122496
\(503\) −34.9783 −1.55960 −0.779802 0.626027i \(-0.784680\pi\)
−0.779802 + 0.626027i \(0.784680\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −2.74456 −0.122011
\(507\) 0 0
\(508\) −8.00000 −0.354943
\(509\) −9.76631 −0.432884 −0.216442 0.976295i \(-0.569445\pi\)
−0.216442 + 0.976295i \(0.569445\pi\)
\(510\) 0 0
\(511\) −52.2337 −2.31068
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −18.0000 −0.793946
\(515\) 0 0
\(516\) 0 0
\(517\) 2.74456 0.120706
\(518\) −31.6060 −1.38869
\(519\) 0 0
\(520\) 0 0
\(521\) −12.5109 −0.548111 −0.274056 0.961714i \(-0.588365\pi\)
−0.274056 + 0.961714i \(0.588365\pi\)
\(522\) 0 0
\(523\) −30.9783 −1.35458 −0.677292 0.735714i \(-0.736847\pi\)
−0.677292 + 0.735714i \(0.736847\pi\)
\(524\) −22.1168 −0.966179
\(525\) 0 0
\(526\) 24.8614 1.08401
\(527\) 4.62772 0.201587
\(528\) 0 0
\(529\) −15.4674 −0.672495
\(530\) 0 0
\(531\) 0 0
\(532\) −2.11684 −0.0917768
\(533\) −22.9783 −0.995299
\(534\) 0 0
\(535\) 0 0
\(536\) 8.00000 0.345547
\(537\) 0 0
\(538\) −8.74456 −0.377005
\(539\) 4.37228 0.188327
\(540\) 0 0
\(541\) −20.1168 −0.864891 −0.432445 0.901660i \(-0.642349\pi\)
−0.432445 + 0.901660i \(0.642349\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) −1.37228 −0.0588361
\(545\) 0 0
\(546\) 0 0
\(547\) −20.0000 −0.855138 −0.427569 0.903983i \(-0.640630\pi\)
−0.427569 + 0.903983i \(0.640630\pi\)
\(548\) 8.74456 0.373549
\(549\) 0 0
\(550\) 0 0
\(551\) −0.861407 −0.0366972
\(552\) 0 0
\(553\) 4.23369 0.180035
\(554\) −12.7446 −0.541465
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 4.97825 0.210935 0.105468 0.994423i \(-0.466366\pi\)
0.105468 + 0.994423i \(0.466366\pi\)
\(558\) 0 0
\(559\) −8.00000 −0.338364
\(560\) 0 0
\(561\) 0 0
\(562\) 18.0000 0.759284
\(563\) −8.23369 −0.347009 −0.173504 0.984833i \(-0.555509\pi\)
−0.173504 + 0.984833i \(0.555509\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 5.25544 0.220903
\(567\) 0 0
\(568\) 10.1168 0.424493
\(569\) 15.2554 0.639541 0.319771 0.947495i \(-0.396394\pi\)
0.319771 + 0.947495i \(0.396394\pi\)
\(570\) 0 0
\(571\) 15.3723 0.643310 0.321655 0.946857i \(-0.395761\pi\)
0.321655 + 0.946857i \(0.395761\pi\)
\(572\) −2.00000 −0.0836242
\(573\) 0 0
\(574\) 38.7446 1.61717
\(575\) 0 0
\(576\) 0 0
\(577\) −36.9783 −1.53942 −0.769712 0.638391i \(-0.779600\pi\)
−0.769712 + 0.638391i \(0.779600\pi\)
\(578\) 15.1168 0.628778
\(579\) 0 0
\(580\) 0 0
\(581\) 9.25544 0.383980
\(582\) 0 0
\(583\) −4.11684 −0.170502
\(584\) −15.4891 −0.640945
\(585\) 0 0
\(586\) −23.4891 −0.970327
\(587\) 24.8614 1.02614 0.513070 0.858347i \(-0.328508\pi\)
0.513070 + 0.858347i \(0.328508\pi\)
\(588\) 0 0
\(589\) 2.11684 0.0872230
\(590\) 0 0
\(591\) 0 0
\(592\) −9.37228 −0.385198
\(593\) −12.5109 −0.513760 −0.256880 0.966443i \(-0.582695\pi\)
−0.256880 + 0.966443i \(0.582695\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −21.6060 −0.885015
\(597\) 0 0
\(598\) 5.48913 0.224467
\(599\) 39.6060 1.61826 0.809128 0.587632i \(-0.199940\pi\)
0.809128 + 0.587632i \(0.199940\pi\)
\(600\) 0 0
\(601\) −16.5109 −0.673493 −0.336746 0.941595i \(-0.609326\pi\)
−0.336746 + 0.941595i \(0.609326\pi\)
\(602\) 13.4891 0.549776
\(603\) 0 0
\(604\) −12.2337 −0.497782
\(605\) 0 0
\(606\) 0 0
\(607\) 5.88316 0.238790 0.119395 0.992847i \(-0.461905\pi\)
0.119395 + 0.992847i \(0.461905\pi\)
\(608\) −0.627719 −0.0254574
\(609\) 0 0
\(610\) 0 0
\(611\) −5.48913 −0.222066
\(612\) 0 0
\(613\) −20.5109 −0.828426 −0.414213 0.910180i \(-0.635943\pi\)
−0.414213 + 0.910180i \(0.635943\pi\)
\(614\) 5.25544 0.212092
\(615\) 0 0
\(616\) 3.37228 0.135873
\(617\) −2.23369 −0.0899249 −0.0449624 0.998989i \(-0.514317\pi\)
−0.0449624 + 0.998989i \(0.514317\pi\)
\(618\) 0 0
\(619\) −44.4674 −1.78729 −0.893647 0.448770i \(-0.851862\pi\)
−0.893647 + 0.448770i \(0.851862\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 19.3723 0.776758
\(623\) −4.62772 −0.185406
\(624\) 0 0
\(625\) 0 0
\(626\) −22.0000 −0.879297
\(627\) 0 0
\(628\) −9.37228 −0.373995
\(629\) −12.8614 −0.512818
\(630\) 0 0
\(631\) 42.1168 1.67665 0.838323 0.545175i \(-0.183537\pi\)
0.838323 + 0.545175i \(0.183537\pi\)
\(632\) 1.25544 0.0499386
\(633\) 0 0
\(634\) 24.3505 0.967083
\(635\) 0 0
\(636\) 0 0
\(637\) −8.74456 −0.346472
\(638\) 1.37228 0.0543291
\(639\) 0 0
\(640\) 0 0
\(641\) 27.0951 1.07019 0.535096 0.844791i \(-0.320275\pi\)
0.535096 + 0.844791i \(0.320275\pi\)
\(642\) 0 0
\(643\) 5.88316 0.232009 0.116005 0.993249i \(-0.462991\pi\)
0.116005 + 0.993249i \(0.462991\pi\)
\(644\) −9.25544 −0.364715
\(645\) 0 0
\(646\) −0.861407 −0.0338916
\(647\) −37.7228 −1.48304 −0.741518 0.670933i \(-0.765894\pi\)
−0.741518 + 0.670933i \(0.765894\pi\)
\(648\) 0 0
\(649\) 2.74456 0.107734
\(650\) 0 0
\(651\) 0 0
\(652\) 5.88316 0.230402
\(653\) 10.6277 0.415895 0.207947 0.978140i \(-0.433322\pi\)
0.207947 + 0.978140i \(0.433322\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 11.4891 0.448575
\(657\) 0 0
\(658\) 9.25544 0.360815
\(659\) −12.8614 −0.501009 −0.250505 0.968115i \(-0.580597\pi\)
−0.250505 + 0.968115i \(0.580597\pi\)
\(660\) 0 0
\(661\) 8.51087 0.331035 0.165517 0.986207i \(-0.447071\pi\)
0.165517 + 0.986207i \(0.447071\pi\)
\(662\) −30.9783 −1.20400
\(663\) 0 0
\(664\) 2.74456 0.106510
\(665\) 0 0
\(666\) 0 0
\(667\) −3.76631 −0.145832
\(668\) −4.62772 −0.179052
\(669\) 0 0
\(670\) 0 0
\(671\) −5.37228 −0.207395
\(672\) 0 0
\(673\) −14.8614 −0.572865 −0.286433 0.958100i \(-0.592469\pi\)
−0.286433 + 0.958100i \(0.592469\pi\)
\(674\) 24.1168 0.928946
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 3.25544 0.125117 0.0625583 0.998041i \(-0.480074\pi\)
0.0625583 + 0.998041i \(0.480074\pi\)
\(678\) 0 0
\(679\) −42.9783 −1.64935
\(680\) 0 0
\(681\) 0 0
\(682\) −3.37228 −0.129131
\(683\) −28.6277 −1.09541 −0.547705 0.836672i \(-0.684498\pi\)
−0.547705 + 0.836672i \(0.684498\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −8.86141 −0.338330
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) 8.23369 0.313679
\(690\) 0 0
\(691\) 40.2337 1.53056 0.765281 0.643697i \(-0.222600\pi\)
0.765281 + 0.643697i \(0.222600\pi\)
\(692\) −6.00000 −0.228086
\(693\) 0 0
\(694\) 32.2337 1.22357
\(695\) 0 0
\(696\) 0 0
\(697\) 15.7663 0.597192
\(698\) −19.4891 −0.737674
\(699\) 0 0
\(700\) 0 0
\(701\) 37.3723 1.41153 0.705766 0.708445i \(-0.250603\pi\)
0.705766 + 0.708445i \(0.250603\pi\)
\(702\) 0 0
\(703\) −5.88316 −0.221887
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) 0.510875 0.0192270
\(707\) 20.2337 0.760966
\(708\) 0 0
\(709\) −10.0000 −0.375558 −0.187779 0.982211i \(-0.560129\pi\)
−0.187779 + 0.982211i \(0.560129\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.37228 −0.0514284
\(713\) 9.25544 0.346619
\(714\) 0 0
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) 0 0
\(718\) 0 0
\(719\) −13.8832 −0.517754 −0.258877 0.965910i \(-0.583352\pi\)
−0.258877 + 0.965910i \(0.583352\pi\)
\(720\) 0 0
\(721\) −32.0000 −1.19174
\(722\) 18.6060 0.692442
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) 0 0
\(726\) 0 0
\(727\) 24.2337 0.898778 0.449389 0.893336i \(-0.351642\pi\)
0.449389 + 0.893336i \(0.351642\pi\)
\(728\) −6.74456 −0.249970
\(729\) 0 0
\(730\) 0 0
\(731\) 5.48913 0.203023
\(732\) 0 0
\(733\) −46.2337 −1.70768 −0.853840 0.520535i \(-0.825732\pi\)
−0.853840 + 0.520535i \(0.825732\pi\)
\(734\) −4.00000 −0.147643
\(735\) 0 0
\(736\) −2.74456 −0.101166
\(737\) −8.00000 −0.294684
\(738\) 0 0
\(739\) −20.4674 −0.752905 −0.376452 0.926436i \(-0.622856\pi\)
−0.376452 + 0.926436i \(0.622856\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −13.8832 −0.509667
\(743\) −4.62772 −0.169775 −0.0848873 0.996391i \(-0.527053\pi\)
−0.0848873 + 0.996391i \(0.527053\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 31.4891 1.15290
\(747\) 0 0
\(748\) 1.37228 0.0501756
\(749\) 40.4674 1.47865
\(750\) 0 0
\(751\) 8.86141 0.323357 0.161679 0.986843i \(-0.448309\pi\)
0.161679 + 0.986843i \(0.448309\pi\)
\(752\) 2.74456 0.100084
\(753\) 0 0
\(754\) −2.74456 −0.0999511
\(755\) 0 0
\(756\) 0 0
\(757\) 20.9783 0.762467 0.381234 0.924479i \(-0.375499\pi\)
0.381234 + 0.924479i \(0.375499\pi\)
\(758\) 0.233688 0.00848793
\(759\) 0 0
\(760\) 0 0
\(761\) −4.97825 −0.180461 −0.0902307 0.995921i \(-0.528760\pi\)
−0.0902307 + 0.995921i \(0.528760\pi\)
\(762\) 0 0
\(763\) 52.2337 1.89099
\(764\) 5.48913 0.198590
\(765\) 0 0
\(766\) −32.2337 −1.16465
\(767\) −5.48913 −0.198201
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14.8614 −0.534874
\(773\) −33.6060 −1.20872 −0.604361 0.796710i \(-0.706572\pi\)
−0.604361 + 0.796710i \(0.706572\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −12.7446 −0.457503
\(777\) 0 0
\(778\) 6.00000 0.215110
\(779\) 7.21194 0.258395
\(780\) 0 0
\(781\) −10.1168 −0.362009
\(782\) −3.76631 −0.134683
\(783\) 0 0
\(784\) 4.37228 0.156153
\(785\) 0 0
\(786\) 0 0
\(787\) 44.4674 1.58509 0.792545 0.609813i \(-0.208755\pi\)
0.792545 + 0.609813i \(0.208755\pi\)
\(788\) −20.7446 −0.738994
\(789\) 0 0
\(790\) 0 0
\(791\) −10.9783 −0.390342
\(792\) 0 0
\(793\) 10.7446 0.381551
\(794\) 24.9783 0.886445
\(795\) 0 0
\(796\) 18.1168 0.642135
\(797\) 11.4891 0.406966 0.203483 0.979079i \(-0.434774\pi\)
0.203483 + 0.979079i \(0.434774\pi\)
\(798\) 0 0
\(799\) 3.76631 0.133243
\(800\) 0 0
\(801\) 0 0
\(802\) 13.3723 0.472192
\(803\) 15.4891 0.546599
\(804\) 0 0
\(805\) 0 0
\(806\) 6.74456 0.237567
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) −18.0000 −0.632846 −0.316423 0.948618i \(-0.602482\pi\)
−0.316423 + 0.948618i \(0.602482\pi\)
\(810\) 0 0
\(811\) 44.8614 1.57530 0.787649 0.616125i \(-0.211298\pi\)
0.787649 + 0.616125i \(0.211298\pi\)
\(812\) 4.62772 0.162401
\(813\) 0 0
\(814\) 9.37228 0.328498
\(815\) 0 0
\(816\) 0 0
\(817\) 2.51087 0.0878444
\(818\) 1.76631 0.0617577
\(819\) 0 0
\(820\) 0 0
\(821\) −11.4891 −0.400973 −0.200487 0.979696i \(-0.564252\pi\)
−0.200487 + 0.979696i \(0.564252\pi\)
\(822\) 0 0
\(823\) 28.0000 0.976019 0.488009 0.872838i \(-0.337723\pi\)
0.488009 + 0.872838i \(0.337723\pi\)
\(824\) −9.48913 −0.330569
\(825\) 0 0
\(826\) 9.25544 0.322038
\(827\) 46.9783 1.63359 0.816797 0.576925i \(-0.195748\pi\)
0.816797 + 0.576925i \(0.195748\pi\)
\(828\) 0 0
\(829\) −24.7446 −0.859414 −0.429707 0.902968i \(-0.641383\pi\)
−0.429707 + 0.902968i \(0.641383\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −2.00000 −0.0693375
\(833\) 6.00000 0.207888
\(834\) 0 0
\(835\) 0 0
\(836\) 0.627719 0.0217101
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) 10.9783 0.379011 0.189506 0.981880i \(-0.439311\pi\)
0.189506 + 0.981880i \(0.439311\pi\)
\(840\) 0 0
\(841\) −27.1168 −0.935064
\(842\) −10.2337 −0.352676
\(843\) 0 0
\(844\) 6.11684 0.210550
\(845\) 0 0
\(846\) 0 0
\(847\) −3.37228 −0.115873
\(848\) −4.11684 −0.141373
\(849\) 0 0
\(850\) 0 0
\(851\) −25.7228 −0.881767
\(852\) 0 0
\(853\) 38.4674 1.31710 0.658549 0.752538i \(-0.271171\pi\)
0.658549 + 0.752538i \(0.271171\pi\)
\(854\) −18.1168 −0.619946
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 36.3505 1.24171 0.620855 0.783925i \(-0.286785\pi\)
0.620855 + 0.783925i \(0.286785\pi\)
\(858\) 0 0
\(859\) −42.7446 −1.45843 −0.729213 0.684287i \(-0.760114\pi\)
−0.729213 + 0.684287i \(0.760114\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −34.9783 −1.19136
\(863\) 21.2554 0.723544 0.361772 0.932267i \(-0.382172\pi\)
0.361772 + 0.932267i \(0.382172\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 27.7228 0.942060
\(867\) 0 0
\(868\) −11.3723 −0.386000
\(869\) −1.25544 −0.0425878
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) 15.4891 0.524528
\(873\) 0 0
\(874\) −1.72281 −0.0582750
\(875\) 0 0
\(876\) 0 0
\(877\) −36.9783 −1.24867 −0.624333 0.781158i \(-0.714629\pi\)
−0.624333 + 0.781158i \(0.714629\pi\)
\(878\) −18.9783 −0.640485
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) 0 0
\(883\) −3.37228 −0.113486 −0.0567432 0.998389i \(-0.518072\pi\)
−0.0567432 + 0.998389i \(0.518072\pi\)
\(884\) −2.74456 −0.0923096
\(885\) 0 0
\(886\) −29.4891 −0.990707
\(887\) −10.9783 −0.368614 −0.184307 0.982869i \(-0.559004\pi\)
−0.184307 + 0.982869i \(0.559004\pi\)
\(888\) 0 0
\(889\) 26.9783 0.904821
\(890\) 0 0
\(891\) 0 0
\(892\) 18.7446 0.627614
\(893\) 1.72281 0.0576517
\(894\) 0 0
\(895\) 0 0
\(896\) 3.37228 0.112660
\(897\) 0 0
\(898\) 28.9783 0.967017
\(899\) −4.62772 −0.154343
\(900\) 0 0
\(901\) −5.64947 −0.188211
\(902\) −11.4891 −0.382546
\(903\) 0 0
\(904\) −3.25544 −0.108274
\(905\) 0 0
\(906\) 0 0
\(907\) 0.394031 0.0130836 0.00654179 0.999979i \(-0.497918\pi\)
0.00654179 + 0.999979i \(0.497918\pi\)
\(908\) −2.74456 −0.0910815
\(909\) 0 0
\(910\) 0 0
\(911\) 8.39403 0.278107 0.139053 0.990285i \(-0.455594\pi\)
0.139053 + 0.990285i \(0.455594\pi\)
\(912\) 0 0
\(913\) −2.74456 −0.0908318
\(914\) −16.3505 −0.540828
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 74.5842 2.46299
\(918\) 0 0
\(919\) 18.9783 0.626035 0.313017 0.949747i \(-0.398660\pi\)
0.313017 + 0.949747i \(0.398660\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 16.1168 0.530780
\(923\) 20.2337 0.666000
\(924\) 0 0
\(925\) 0 0
\(926\) −0.233688 −0.00767946
\(927\) 0 0
\(928\) 1.37228 0.0450473
\(929\) −24.3505 −0.798915 −0.399458 0.916752i \(-0.630801\pi\)
−0.399458 + 0.916752i \(0.630801\pi\)
\(930\) 0 0
\(931\) 2.74456 0.0899494
\(932\) 1.37228 0.0449506
\(933\) 0 0
\(934\) 19.3723 0.633880
\(935\) 0 0
\(936\) 0 0
\(937\) 28.5109 0.931410 0.465705 0.884940i \(-0.345801\pi\)
0.465705 + 0.884940i \(0.345801\pi\)
\(938\) −26.9783 −0.880871
\(939\) 0 0
\(940\) 0 0
\(941\) 15.0951 0.492086 0.246043 0.969259i \(-0.420870\pi\)
0.246043 + 0.969259i \(0.420870\pi\)
\(942\) 0 0
\(943\) 31.5326 1.02684
\(944\) 2.74456 0.0893279
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 48.8614 1.58778 0.793891 0.608060i \(-0.208052\pi\)
0.793891 + 0.608060i \(0.208052\pi\)
\(948\) 0 0
\(949\) −30.9783 −1.00560
\(950\) 0 0
\(951\) 0 0
\(952\) 4.62772 0.149985
\(953\) 40.1168 1.29951 0.649756 0.760143i \(-0.274871\pi\)
0.649756 + 0.760143i \(0.274871\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 14.7446 0.476873
\(957\) 0 0
\(958\) 5.48913 0.177346
\(959\) −29.4891 −0.952254
\(960\) 0 0
\(961\) −19.6277 −0.633152
\(962\) −18.7446 −0.604349
\(963\) 0 0
\(964\) −22.0000 −0.708572
\(965\) 0 0
\(966\) 0 0
\(967\) −47.6060 −1.53090 −0.765452 0.643493i \(-0.777485\pi\)
−0.765452 + 0.643493i \(0.777485\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) 0 0
\(971\) 1.02175 0.0327895 0.0163947 0.999866i \(-0.494781\pi\)
0.0163947 + 0.999866i \(0.494781\pi\)
\(972\) 0 0
\(973\) 13.4891 0.432442
\(974\) 20.0000 0.640841
\(975\) 0 0
\(976\) −5.37228 −0.171963
\(977\) 14.2337 0.455376 0.227688 0.973734i \(-0.426883\pi\)
0.227688 + 0.973734i \(0.426883\pi\)
\(978\) 0 0
\(979\) 1.37228 0.0438583
\(980\) 0 0
\(981\) 0 0
\(982\) −7.37228 −0.235259
\(983\) −13.7228 −0.437690 −0.218845 0.975760i \(-0.570229\pi\)
−0.218845 + 0.975760i \(0.570229\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 1.88316 0.0599719
\(987\) 0 0
\(988\) −1.25544 −0.0399408
\(989\) 10.9783 0.349088
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −3.37228 −0.107070
\(993\) 0 0
\(994\) −34.1168 −1.08212
\(995\) 0 0
\(996\) 0 0
\(997\) −22.2337 −0.704148 −0.352074 0.935972i \(-0.614523\pi\)
−0.352074 + 0.935972i \(0.614523\pi\)
\(998\) 33.4891 1.06008
\(999\) 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 4950.2.a.bw.1.1 2
3.2 odd 2 550.2.a.n.1.2 2
5.2 odd 4 4950.2.c.bc.199.1 4
5.3 odd 4 4950.2.c.bc.199.4 4
5.4 even 2 990.2.a.m.1.2 2
12.11 even 2 4400.2.a.bl.1.1 2
15.2 even 4 550.2.b.f.199.3 4
15.8 even 4 550.2.b.f.199.2 4
15.14 odd 2 110.2.a.d.1.1 2
20.19 odd 2 7920.2.a.bq.1.1 2
33.32 even 2 6050.2.a.cb.1.2 2
60.23 odd 4 4400.2.b.p.4049.1 4
60.47 odd 4 4400.2.b.p.4049.4 4
60.59 even 2 880.2.a.n.1.2 2
105.104 even 2 5390.2.a.bp.1.2 2
120.29 odd 2 3520.2.a.bq.1.2 2
120.59 even 2 3520.2.a.bj.1.1 2
165.164 even 2 1210.2.a.r.1.1 2
660.659 odd 2 9680.2.a.bt.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
110.2.a.d.1.1 2 15.14 odd 2
550.2.a.n.1.2 2 3.2 odd 2
550.2.b.f.199.2 4 15.8 even 4
550.2.b.f.199.3 4 15.2 even 4
880.2.a.n.1.2 2 60.59 even 2
990.2.a.m.1.2 2 5.4 even 2
1210.2.a.r.1.1 2 165.164 even 2
3520.2.a.bj.1.1 2 120.59 even 2
3520.2.a.bq.1.2 2 120.29 odd 2
4400.2.a.bl.1.1 2 12.11 even 2
4400.2.b.p.4049.1 4 60.23 odd 4
4400.2.b.p.4049.4 4 60.47 odd 4
4950.2.a.bw.1.1 2 1.1 even 1 trivial
4950.2.c.bc.199.1 4 5.2 odd 4
4950.2.c.bc.199.4 4 5.3 odd 4
5390.2.a.bp.1.2 2 105.104 even 2
6050.2.a.cb.1.2 2 33.32 even 2
7920.2.a.bq.1.1 2 20.19 odd 2
9680.2.a.bt.1.2 2 660.659 odd 2