Properties

Label 1968.2.a.w.1.1
Level $1968$
Weight $2$
Character 1968.1
Self dual yes
Analytic conductor $15.715$
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] = [1968,2,Mod(1,1968)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1968, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1968.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1968 = 2^{4} \cdot 3 \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1968.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: \(15.7145591178\)
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: \(\Z[a_1, \ldots, a_{11}]\)
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 1.1
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 1968.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.10278 q^{5} -2.52444 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.10278 q^{5} -2.52444 q^{7} +1.00000 q^{9} -0.813607 q^{11} +5.10278 q^{13} -1.10278 q^{15} +3.39194 q^{17} -3.10278 q^{19} -2.52444 q^{21} +0.897225 q^{23} -3.78389 q^{25} +1.00000 q^{27} +4.44082 q^{29} +8.96526 q^{31} -0.813607 q^{33} +2.78389 q^{35} +2.08362 q^{37} +5.10278 q^{39} +1.00000 q^{41} -9.07306 q^{43} -1.10278 q^{45} +0.235269 q^{47} -0.627213 q^{49} +3.39194 q^{51} +13.8328 q^{53} +0.897225 q^{55} -3.10278 q^{57} -1.04888 q^{59} +1.91638 q^{61} -2.52444 q^{63} -5.62721 q^{65} +10.0383 q^{67} +0.897225 q^{69} +12.8136 q^{71} +4.75971 q^{73} -3.78389 q^{75} +2.05390 q^{77} +15.8328 q^{79} +1.00000 q^{81} +7.68111 q^{83} -3.74055 q^{85} +4.44082 q^{87} +13.2544 q^{89} -12.8816 q^{91} +8.96526 q^{93} +3.42166 q^{95} +2.72999 q^{97} -0.813607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} + 4 q^{5} - 2 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} + 4 q^{5} - 2 q^{7} + 3 q^{9} + 4 q^{11} + 8 q^{13} + 4 q^{15} + 2 q^{17} - 2 q^{19} - 2 q^{21} + 10 q^{23} + 5 q^{25} + 3 q^{27} - 6 q^{29} + 2 q^{31} + 4 q^{33} - 8 q^{35} + 20 q^{37} + 8 q^{39} + 3 q^{41} - 10 q^{43} + 4 q^{45} - 4 q^{47} + 11 q^{49} + 2 q^{51} + 14 q^{53} + 10 q^{55} - 2 q^{57} + 8 q^{59} - 8 q^{61} - 2 q^{63} - 4 q^{65} - 12 q^{67} + 10 q^{69} + 32 q^{71} + 4 q^{73} + 5 q^{75} + 10 q^{77} + 20 q^{79} + 3 q^{81} + 14 q^{83} - 22 q^{85} - 6 q^{87} + 14 q^{89} + 2 q^{93} + 12 q^{95} - 12 q^{97} + 4 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.10278 −0.493176 −0.246588 0.969120i \(-0.579309\pi\)
−0.246588 + 0.969120i \(0.579309\pi\)
\(6\) 0 0
\(7\) −2.52444 −0.954148 −0.477074 0.878863i \(-0.658303\pi\)
−0.477074 + 0.878863i \(0.658303\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −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 \(0.249765\pi\)
\(14\) 0 0
\(15\) −1.10278 −0.284735
\(16\) 0 0
\(17\) 3.39194 0.822667 0.411334 0.911485i \(-0.365063\pi\)
0.411334 + 0.911485i \(0.365063\pi\)
\(18\) 0 0
\(19\) −3.10278 −0.711825 −0.355913 0.934519i \(-0.615830\pi\)
−0.355913 + 0.934519i \(0.615830\pi\)
\(20\) 0 0
\(21\) −2.52444 −0.550878
\(22\) 0 0
\(23\) 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
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 4.44082 0.824639 0.412320 0.911039i \(-0.364719\pi\)
0.412320 + 0.911039i \(0.364719\pi\)
\(30\) 0 0
\(31\) 8.96526 1.61021 0.805104 0.593134i \(-0.202110\pi\)
0.805104 + 0.593134i \(0.202110\pi\)
\(32\) 0 0
\(33\) −0.813607 −0.141631
\(34\) 0 0
\(35\) 2.78389 0.470563
\(36\) 0 0
\(37\) 2.08362 0.342545 0.171272 0.985224i \(-0.445212\pi\)
0.171272 + 0.985224i \(0.445212\pi\)
\(38\) 0 0
\(39\) 5.10278 0.817098
\(40\) 0 0
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −9.07306 −1.38363 −0.691814 0.722076i \(-0.743188\pi\)
−0.691814 + 0.722076i \(0.743188\pi\)
\(44\) 0 0
\(45\) −1.10278 −0.164392
\(46\) 0 0
\(47\) 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
\(50\) 0 0
\(51\) 3.39194 0.474967
\(52\) 0 0
\(53\) 13.8328 1.90008 0.950038 0.312134i \(-0.101044\pi\)
0.950038 + 0.312134i \(0.101044\pi\)
\(54\) 0 0
\(55\) 0.897225 0.120982
\(56\) 0 0
\(57\) −3.10278 −0.410973
\(58\) 0 0
\(59\) −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 \(0.460850\pi\)
\(62\) 0 0
\(63\) −2.52444 −0.318049
\(64\) 0 0
\(65\) −5.62721 −0.697970
\(66\) 0 0
\(67\) 10.0383 1.22638 0.613188 0.789937i \(-0.289887\pi\)
0.613188 + 0.789937i \(0.289887\pi\)
\(68\) 0 0
\(69\) 0.897225 0.108013
\(70\) 0 0
\(71\) 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 \(0.410149\pi\)
\(74\) 0 0
\(75\) −3.78389 −0.436926
\(76\) 0 0
\(77\) 2.05390 0.234064
\(78\) 0 0
\(79\) 15.8328 1.78133 0.890663 0.454665i \(-0.150241\pi\)
0.890663 + 0.454665i \(0.150241\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 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
\(86\) 0 0
\(87\) 4.44082 0.476106
\(88\) 0 0
\(89\) 13.2544 1.40497 0.702483 0.711700i \(-0.252075\pi\)
0.702483 + 0.711700i \(0.252075\pi\)
\(90\) 0 0
\(91\) −12.8816 −1.35036
\(92\) 0 0
\(93\) 8.96526 0.929654
\(94\) 0 0
\(95\) 3.42166 0.351055
\(96\) 0 0
\(97\) 2.72999 0.277188 0.138594 0.990349i \(-0.455742\pi\)
0.138594 + 0.990349i \(0.455742\pi\)
\(98\) 0 0
\(99\) −0.813607 −0.0817705
\(100\) 0 0
\(101\) −11.8625 −1.18036 −0.590181 0.807271i \(-0.700943\pi\)
−0.590181 + 0.807271i \(0.700943\pi\)
\(102\) 0 0
\(103\) 0.494719 0.0487461 0.0243730 0.999703i \(-0.492241\pi\)
0.0243730 + 0.999703i \(0.492241\pi\)
\(104\) 0 0
\(105\) 2.78389 0.271680
\(106\) 0 0
\(107\) 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 \(0.421416\pi\)
\(110\) 0 0
\(111\) 2.08362 0.197768
\(112\) 0 0
\(113\) −17.5139 −1.64757 −0.823783 0.566905i \(-0.808141\pi\)
−0.823783 + 0.566905i \(0.808141\pi\)
\(114\) 0 0
\(115\) −0.989437 −0.0922655
\(116\) 0 0
\(117\) 5.10278 0.471752
\(118\) 0 0
\(119\) −8.56275 −0.784946
\(120\) 0 0
\(121\) −10.3380 −0.939822
\(122\) 0 0
\(123\) 1.00000 0.0901670
\(124\) 0 0
\(125\) 9.68665 0.866400
\(126\) 0 0
\(127\) −12.8816 −1.14306 −0.571530 0.820581i \(-0.693650\pi\)
−0.571530 + 0.820581i \(0.693650\pi\)
\(128\) 0 0
\(129\) −9.07306 −0.798838
\(130\) 0 0
\(131\) 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
\(134\) 0 0
\(135\) −1.10278 −0.0949118
\(136\) 0 0
\(137\) −17.4897 −1.49425 −0.747123 0.664686i \(-0.768565\pi\)
−0.747123 + 0.664686i \(0.768565\pi\)
\(138\) 0 0
\(139\) −15.7250 −1.33377 −0.666887 0.745159i \(-0.732374\pi\)
−0.666887 + 0.745159i \(0.732374\pi\)
\(140\) 0 0
\(141\) 0.235269 0.0198133
\(142\) 0 0
\(143\) −4.15165 −0.347178
\(144\) 0 0
\(145\) −4.89722 −0.406692
\(146\) 0 0
\(147\) −0.627213 −0.0517317
\(148\) 0 0
\(149\) 11.5678 0.947669 0.473835 0.880614i \(-0.342869\pi\)
0.473835 + 0.880614i \(0.342869\pi\)
\(150\) 0 0
\(151\) −19.2544 −1.56690 −0.783451 0.621453i \(-0.786543\pi\)
−0.783451 + 0.621453i \(0.786543\pi\)
\(152\) 0 0
\(153\) 3.39194 0.274222
\(154\) 0 0
\(155\) −9.88666 −0.794116
\(156\) 0 0
\(157\) −20.9200 −1.66959 −0.834797 0.550558i \(-0.814415\pi\)
−0.834797 + 0.550558i \(0.814415\pi\)
\(158\) 0 0
\(159\) 13.8328 1.09701
\(160\) 0 0
\(161\) −2.26499 −0.178506
\(162\) 0 0
\(163\) 12.7980 1.00242 0.501209 0.865326i \(-0.332889\pi\)
0.501209 + 0.865326i \(0.332889\pi\)
\(164\) 0 0
\(165\) 0.897225 0.0698489
\(166\) 0 0
\(167\) −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
\(170\) 0 0
\(171\) −3.10278 −0.237275
\(172\) 0 0
\(173\) 11.9461 0.908245 0.454123 0.890939i \(-0.349953\pi\)
0.454123 + 0.890939i \(0.349953\pi\)
\(174\) 0 0
\(175\) 9.55219 0.722078
\(176\) 0 0
\(177\) −1.04888 −0.0788383
\(178\) 0 0
\(179\) 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 \(0.521698\pi\)
\(182\) 0 0
\(183\) 1.91638 0.141663
\(184\) 0 0
\(185\) −2.29776 −0.168935
\(186\) 0 0
\(187\) −2.75971 −0.201810
\(188\) 0 0
\(189\) −2.52444 −0.183626
\(190\) 0 0
\(191\) 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 \(0.310080\pi\)
\(194\) 0 0
\(195\) −5.62721 −0.402973
\(196\) 0 0
\(197\) −5.21057 −0.371238 −0.185619 0.982622i \(-0.559429\pi\)
−0.185619 + 0.982622i \(0.559429\pi\)
\(198\) 0 0
\(199\) −2.05390 −0.145597 −0.0727985 0.997347i \(-0.523193\pi\)
−0.0727985 + 0.997347i \(0.523193\pi\)
\(200\) 0 0
\(201\) 10.0383 0.708048
\(202\) 0 0
\(203\) −11.2106 −0.786828
\(204\) 0 0
\(205\) −1.10278 −0.0770212
\(206\) 0 0
\(207\) 0.897225 0.0623614
\(208\) 0 0
\(209\) 2.52444 0.174619
\(210\) 0 0
\(211\) 9.04888 0.622950 0.311475 0.950254i \(-0.399177\pi\)
0.311475 + 0.950254i \(0.399177\pi\)
\(212\) 0 0
\(213\) 12.8136 0.877974
\(214\) 0 0
\(215\) 10.0055 0.682372
\(216\) 0 0
\(217\) −22.6322 −1.53638
\(218\) 0 0
\(219\) 4.75971 0.321631
\(220\) 0 0
\(221\) 17.3083 1.16428
\(222\) 0 0
\(223\) −24.8222 −1.66222 −0.831109 0.556110i \(-0.812293\pi\)
−0.831109 + 0.556110i \(0.812293\pi\)
\(224\) 0 0
\(225\) −3.78389 −0.252259
\(226\) 0 0
\(227\) −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 \(0.674571\pi\)
\(230\) 0 0
\(231\) 2.05390 0.135137
\(232\) 0 0
\(233\) −24.9200 −1.63256 −0.816280 0.577656i \(-0.803967\pi\)
−0.816280 + 0.577656i \(0.803967\pi\)
\(234\) 0 0
\(235\) −0.259449 −0.0169246
\(236\) 0 0
\(237\) 15.8328 1.02845
\(238\) 0 0
\(239\) 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 \(0.679465\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0.691675 0.0441895
\(246\) 0 0
\(247\) −15.8328 −1.00741
\(248\) 0 0
\(249\) 7.68111 0.486771
\(250\) 0 0
\(251\) 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
\(254\) 0 0
\(255\) −3.74055 −0.234242
\(256\) 0 0
\(257\) 25.9008 1.61565 0.807824 0.589424i \(-0.200645\pi\)
0.807824 + 0.589424i \(0.200645\pi\)
\(258\) 0 0
\(259\) −5.25997 −0.326838
\(260\) 0 0
\(261\) 4.44082 0.274880
\(262\) 0 0
\(263\) 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
\(266\) 0 0
\(267\) 13.2544 0.811158
\(268\) 0 0
\(269\) 7.62721 0.465039 0.232520 0.972592i \(-0.425303\pi\)
0.232520 + 0.972592i \(0.425303\pi\)
\(270\) 0 0
\(271\) −19.9164 −1.20983 −0.604917 0.796289i \(-0.706794\pi\)
−0.604917 + 0.796289i \(0.706794\pi\)
\(272\) 0 0
\(273\) −12.8816 −0.779632
\(274\) 0 0
\(275\) 3.07860 0.185646
\(276\) 0 0
\(277\) −17.6413 −1.05997 −0.529983 0.848008i \(-0.677802\pi\)
−0.529983 + 0.848008i \(0.677802\pi\)
\(278\) 0 0
\(279\) 8.96526 0.536736
\(280\) 0 0
\(281\) −0.0297193 −0.00177291 −0.000886453 1.00000i \(-0.500282\pi\)
−0.000886453 1.00000i \(0.500282\pi\)
\(282\) 0 0
\(283\) 10.1814 0.605220 0.302610 0.953115i \(-0.402142\pi\)
0.302610 + 0.953115i \(0.402142\pi\)
\(284\) 0 0
\(285\) 3.42166 0.202682
\(286\) 0 0
\(287\) −2.52444 −0.149013
\(288\) 0 0
\(289\) −5.49472 −0.323219
\(290\) 0 0
\(291\) 2.72999 0.160035
\(292\) 0 0
\(293\) 3.14808 0.183913 0.0919564 0.995763i \(-0.470688\pi\)
0.0919564 + 0.995763i \(0.470688\pi\)
\(294\) 0 0
\(295\) 1.15667 0.0673442
\(296\) 0 0
\(297\) −0.813607 −0.0472102
\(298\) 0 0
\(299\) 4.57834 0.264772
\(300\) 0 0
\(301\) 22.9044 1.32019
\(302\) 0 0
\(303\) −11.8625 −0.681482
\(304\) 0 0
\(305\) −2.11334 −0.121009
\(306\) 0 0
\(307\) 12.7980 0.730422 0.365211 0.930925i \(-0.380997\pi\)
0.365211 + 0.930925i \(0.380997\pi\)
\(308\) 0 0
\(309\) 0.494719 0.0281436
\(310\) 0 0
\(311\) −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 \(0.527066\pi\)
\(314\) 0 0
\(315\) 2.78389 0.156854
\(316\) 0 0
\(317\) −4.33302 −0.243367 −0.121683 0.992569i \(-0.538829\pi\)
−0.121683 + 0.992569i \(0.538829\pi\)
\(318\) 0 0
\(319\) −3.61308 −0.202294
\(320\) 0 0
\(321\) 15.0872 0.842085
\(322\) 0 0
\(323\) −10.5244 −0.585595
\(324\) 0 0
\(325\) −19.3083 −1.07103
\(326\) 0 0
\(327\) 5.10278 0.282184
\(328\) 0 0
\(329\) −0.593923 −0.0327440
\(330\) 0 0
\(331\) −5.53500 −0.304231 −0.152116 0.988363i \(-0.548609\pi\)
−0.152116 + 0.988363i \(0.548609\pi\)
\(332\) 0 0
\(333\) 2.08362 0.114182
\(334\) 0 0
\(335\) −11.0700 −0.604819
\(336\) 0 0
\(337\) −3.71083 −0.202142 −0.101071 0.994879i \(-0.532227\pi\)
−0.101071 + 0.994879i \(0.532227\pi\)
\(338\) 0 0
\(339\) −17.5139 −0.951223
\(340\) 0 0
\(341\) −7.29419 −0.395003
\(342\) 0 0
\(343\) 19.2544 1.03964
\(344\) 0 0
\(345\) −0.989437 −0.0532695
\(346\) 0 0
\(347\) −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 \(0.370377\pi\)
\(350\) 0 0
\(351\) 5.10278 0.272366
\(352\) 0 0
\(353\) 26.8222 1.42760 0.713801 0.700349i \(-0.246972\pi\)
0.713801 + 0.700349i \(0.246972\pi\)
\(354\) 0 0
\(355\) −14.1305 −0.749970
\(356\) 0 0
\(357\) −8.56275 −0.453189
\(358\) 0 0
\(359\) −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
\(362\) 0 0
\(363\) −10.3380 −0.542607
\(364\) 0 0
\(365\) −5.24889 −0.274739
\(366\) 0 0
\(367\) −23.1708 −1.20951 −0.604753 0.796413i \(-0.706728\pi\)
−0.604753 + 0.796413i \(0.706728\pi\)
\(368\) 0 0
\(369\) 1.00000 0.0520579
\(370\) 0 0
\(371\) −34.9200 −1.81295
\(372\) 0 0
\(373\) 35.2091 1.82306 0.911530 0.411235i \(-0.134902\pi\)
0.911530 + 0.411235i \(0.134902\pi\)
\(374\) 0 0
\(375\) 9.68665 0.500217
\(376\) 0 0
\(377\) 22.6605 1.16708
\(378\) 0 0
\(379\) 26.0383 1.33750 0.668749 0.743488i \(-0.266830\pi\)
0.668749 + 0.743488i \(0.266830\pi\)
\(380\) 0 0
\(381\) −12.8816 −0.659946
\(382\) 0 0
\(383\) −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
\(386\) 0 0
\(387\) −9.07306 −0.461209
\(388\) 0 0
\(389\) 4.46500 0.226384 0.113192 0.993573i \(-0.463892\pi\)
0.113192 + 0.993573i \(0.463892\pi\)
\(390\) 0 0
\(391\) 3.04334 0.153908
\(392\) 0 0
\(393\) 6.35720 0.320678
\(394\) 0 0
\(395\) −17.4600 −0.878507
\(396\) 0 0
\(397\) −10.5628 −0.530129 −0.265065 0.964231i \(-0.585393\pi\)
−0.265065 + 0.964231i \(0.585393\pi\)
\(398\) 0 0
\(399\) 7.83276 0.392129
\(400\) 0 0
\(401\) 13.0872 0.653543 0.326772 0.945103i \(-0.394039\pi\)
0.326772 + 0.945103i \(0.394039\pi\)
\(402\) 0 0
\(403\) 45.7477 2.27885
\(404\) 0 0
\(405\) −1.10278 −0.0547973
\(406\) 0 0
\(407\) −1.69525 −0.0840302
\(408\) 0 0
\(409\) −18.5542 −0.917444 −0.458722 0.888580i \(-0.651693\pi\)
−0.458722 + 0.888580i \(0.651693\pi\)
\(410\) 0 0
\(411\) −17.4897 −0.862703
\(412\) 0 0
\(413\) 2.64782 0.130291
\(414\) 0 0
\(415\) −8.47054 −0.415802
\(416\) 0 0
\(417\) −15.7250 −0.770055
\(418\) 0 0
\(419\) 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 \(0.567810\pi\)
\(422\) 0 0
\(423\) 0.235269 0.0114392
\(424\) 0 0
\(425\) −12.8347 −0.622576
\(426\) 0 0
\(427\) −4.83779 −0.234117
\(428\) 0 0
\(429\) −4.15165 −0.200444
\(430\) 0 0
\(431\) −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 \(0.585341\pi\)
\(434\) 0 0
\(435\) −4.89722 −0.234804
\(436\) 0 0
\(437\) −2.78389 −0.133171
\(438\) 0 0
\(439\) 1.32391 0.0631868 0.0315934 0.999501i \(-0.489942\pi\)
0.0315934 + 0.999501i \(0.489942\pi\)
\(440\) 0 0
\(441\) −0.627213 −0.0298673
\(442\) 0 0
\(443\) −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
\(446\) 0 0
\(447\) 11.5678 0.547137
\(448\) 0 0
\(449\) −32.7839 −1.54717 −0.773584 0.633694i \(-0.781538\pi\)
−0.773584 + 0.633694i \(0.781538\pi\)
\(450\) 0 0
\(451\) −0.813607 −0.0383112
\(452\) 0 0
\(453\) −19.2544 −0.904652
\(454\) 0 0
\(455\) 14.2056 0.665966
\(456\) 0 0
\(457\) −14.1672 −0.662715 −0.331358 0.943505i \(-0.607507\pi\)
−0.331358 + 0.943505i \(0.607507\pi\)
\(458\) 0 0
\(459\) 3.39194 0.158322
\(460\) 0 0
\(461\) −30.0766 −1.40081 −0.700404 0.713747i \(-0.746997\pi\)
−0.700404 + 0.713747i \(0.746997\pi\)
\(462\) 0 0
\(463\) 23.8766 1.10964 0.554820 0.831970i \(-0.312787\pi\)
0.554820 + 0.831970i \(0.312787\pi\)
\(464\) 0 0
\(465\) −9.88666 −0.458483
\(466\) 0 0
\(467\) 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 0
\(471\) −20.9200 −0.963941
\(472\) 0 0
\(473\) 7.38190 0.339420
\(474\) 0 0
\(475\) 11.7406 0.538693
\(476\) 0 0
\(477\) 13.8328 0.633359
\(478\) 0 0
\(479\) 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
\(482\) 0 0
\(483\) −2.26499 −0.103061
\(484\) 0 0
\(485\) −3.01056 −0.136703
\(486\) 0 0
\(487\) −20.6025 −0.933589 −0.466795 0.884366i \(-0.654591\pi\)
−0.466795 + 0.884366i \(0.654591\pi\)
\(488\) 0 0
\(489\) 12.7980 0.578746
\(490\) 0 0
\(491\) −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
\(494\) 0 0
\(495\) 0.897225 0.0403273
\(496\) 0 0
\(497\) −32.3472 −1.45097
\(498\) 0 0
\(499\) 21.4600 0.960680 0.480340 0.877082i \(-0.340513\pi\)
0.480340 + 0.877082i \(0.340513\pi\)
\(500\) 0 0
\(501\) −9.62721 −0.430112
\(502\) 0 0
\(503\) 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
\(506\) 0 0
\(507\) 13.0383 0.579052
\(508\) 0 0
\(509\) 22.7542 1.00856 0.504280 0.863540i \(-0.331758\pi\)
0.504280 + 0.863540i \(0.331758\pi\)
\(510\) 0 0
\(511\) −12.0156 −0.531538
\(512\) 0 0
\(513\) −3.10278 −0.136991
\(514\) 0 0
\(515\) −0.545563 −0.0240404
\(516\) 0 0
\(517\) −0.191417 −0.00841850
\(518\) 0 0
\(519\) 11.9461 0.524376
\(520\) 0 0
\(521\) −38.7230 −1.69649 −0.848243 0.529608i \(-0.822339\pi\)
−0.848243 + 0.529608i \(0.822339\pi\)
\(522\) 0 0
\(523\) −6.56829 −0.287211 −0.143606 0.989635i \(-0.545870\pi\)
−0.143606 + 0.989635i \(0.545870\pi\)
\(524\) 0 0
\(525\) 9.55219 0.416892
\(526\) 0 0
\(527\) 30.4096 1.32467
\(528\) 0 0
\(529\) −22.1950 −0.964999
\(530\) 0 0
\(531\) −1.04888 −0.0455173
\(532\) 0 0
\(533\) 5.10278 0.221026
\(534\) 0 0
\(535\) −16.6378 −0.719314
\(536\) 0 0
\(537\) 2.13752 0.0922407
\(538\) 0 0
\(539\) 0.510305 0.0219804
\(540\) 0 0
\(541\) 42.0766 1.80902 0.904508 0.426457i \(-0.140239\pi\)
0.904508 + 0.426457i \(0.140239\pi\)
\(542\) 0 0
\(543\) −1.83276 −0.0786514
\(544\) 0 0
\(545\) −5.62721 −0.241043
\(546\) 0 0
\(547\) −19.9688 −0.853805 −0.426903 0.904298i \(-0.640395\pi\)
−0.426903 + 0.904298i \(0.640395\pi\)
\(548\) 0 0
\(549\) 1.91638 0.0817892
\(550\) 0 0
\(551\) −13.7789 −0.586999
\(552\) 0 0
\(553\) −39.9688 −1.69965
\(554\) 0 0
\(555\) −2.29776 −0.0975346
\(556\) 0 0
\(557\) 13.5491 0.574095 0.287048 0.957916i \(-0.407326\pi\)
0.287048 + 0.957916i \(0.407326\pi\)
\(558\) 0 0
\(559\) −46.2978 −1.95819
\(560\) 0 0
\(561\) −2.75971 −0.116515
\(562\) 0 0
\(563\) −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
\(566\) 0 0
\(567\) −2.52444 −0.106016
\(568\) 0 0
\(569\) −21.4544 −0.899417 −0.449708 0.893175i \(-0.648472\pi\)
−0.449708 + 0.893175i \(0.648472\pi\)
\(570\) 0 0
\(571\) −20.9411 −0.876357 −0.438178 0.898888i \(-0.644376\pi\)
−0.438178 + 0.898888i \(0.644376\pi\)
\(572\) 0 0
\(573\) 18.6167 0.777722
\(574\) 0 0
\(575\) −3.39500 −0.141581
\(576\) 0 0
\(577\) 18.4705 0.768939 0.384469 0.923138i \(-0.374384\pi\)
0.384469 + 0.923138i \(0.374384\pi\)
\(578\) 0 0
\(579\) 15.6116 0.648797
\(580\) 0 0
\(581\) −19.3905 −0.804453
\(582\) 0 0
\(583\) −11.2544 −0.466111
\(584\) 0 0
\(585\) −5.62721 −0.232657
\(586\) 0 0
\(587\) 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
\(590\) 0 0
\(591\) −5.21057 −0.214334
\(592\) 0 0
\(593\) −27.2841 −1.12043 −0.560213 0.828349i \(-0.689281\pi\)
−0.560213 + 0.828349i \(0.689281\pi\)
\(594\) 0 0
\(595\) 9.44279 0.387117
\(596\) 0 0
\(597\) −2.05390 −0.0840605
\(598\) 0 0
\(599\) 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 \(0.603585\pi\)
\(602\) 0 0
\(603\) 10.0383 0.408792
\(604\) 0 0
\(605\) 11.4005 0.463498
\(606\) 0 0
\(607\) 1.93051 0.0783572 0.0391786 0.999232i \(-0.487526\pi\)
0.0391786 + 0.999232i \(0.487526\pi\)
\(608\) 0 0
\(609\) −11.2106 −0.454275
\(610\) 0 0
\(611\) 1.20053 0.0485681
\(612\) 0 0
\(613\) −0.372787 −0.0150567 −0.00752836 0.999972i \(-0.502396\pi\)
−0.00752836 + 0.999972i \(0.502396\pi\)
\(614\) 0 0
\(615\) −1.10278 −0.0444682
\(616\) 0 0
\(617\) 31.3083 1.26043 0.630213 0.776422i \(-0.282968\pi\)
0.630213 + 0.776422i \(0.282968\pi\)
\(618\) 0 0
\(619\) −34.3658 −1.38128 −0.690639 0.723200i \(-0.742671\pi\)
−0.690639 + 0.723200i \(0.742671\pi\)
\(620\) 0 0
\(621\) 0.897225 0.0360044
\(622\) 0 0
\(623\) −33.4600 −1.34055
\(624\) 0 0
\(625\) 8.23724 0.329490
\(626\) 0 0
\(627\) 2.52444 0.100816
\(628\) 0 0
\(629\) 7.06752 0.281800
\(630\) 0 0
\(631\) −7.33804 −0.292123 −0.146061 0.989276i \(-0.546660\pi\)
−0.146061 + 0.989276i \(0.546660\pi\)
\(632\) 0 0
\(633\) 9.04888 0.359661
\(634\) 0 0
\(635\) 14.2056 0.563730
\(636\) 0 0
\(637\) −3.20053 −0.126809
\(638\) 0 0
\(639\) 12.8136 0.506898
\(640\) 0 0
\(641\) −31.5764 −1.24719 −0.623596 0.781747i \(-0.714329\pi\)
−0.623596 + 0.781747i \(0.714329\pi\)
\(642\) 0 0
\(643\) 4.51890 0.178208 0.0891040 0.996022i \(-0.471600\pi\)
0.0891040 + 0.996022i \(0.471600\pi\)
\(644\) 0 0
\(645\) 10.0055 0.393968
\(646\) 0 0
\(647\) 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
\(650\) 0 0
\(651\) −22.6322 −0.887027
\(652\) 0 0
\(653\) 0.381381 0.0149246 0.00746229 0.999972i \(-0.497625\pi\)
0.00746229 + 0.999972i \(0.497625\pi\)
\(654\) 0 0
\(655\) −7.01056 −0.273925
\(656\) 0 0
\(657\) 4.75971 0.185694
\(658\) 0 0
\(659\) −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 \(0.295771\pi\)
\(662\) 0 0
\(663\) 17.3083 0.672200
\(664\) 0 0
\(665\) −8.63778 −0.334959
\(666\) 0 0
\(667\) 3.98441 0.154277
\(668\) 0 0
\(669\) −24.8222 −0.959682
\(670\) 0 0
\(671\) −1.55918 −0.0601915
\(672\) 0 0
\(673\) 1.48110 0.0570923 0.0285461 0.999592i \(-0.490912\pi\)
0.0285461 + 0.999592i \(0.490912\pi\)
\(674\) 0 0
\(675\) −3.78389 −0.145642
\(676\) 0 0
\(677\) 45.3749 1.74390 0.871950 0.489596i \(-0.162856\pi\)
0.871950 + 0.489596i \(0.162856\pi\)
\(678\) 0 0
\(679\) −6.89169 −0.264479
\(680\) 0 0
\(681\) −16.6464 −0.637890
\(682\) 0 0
\(683\) 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
\(686\) 0 0
\(687\) −15.7789 −0.602001
\(688\) 0 0
\(689\) 70.5855 2.68909
\(690\) 0 0
\(691\) 18.9355 0.720342 0.360171 0.932886i \(-0.382718\pi\)
0.360171 + 0.932886i \(0.382718\pi\)
\(692\) 0 0
\(693\) 2.05390 0.0780212
\(694\) 0 0
\(695\) 17.3411 0.657785
\(696\) 0 0
\(697\) 3.39194 0.128479
\(698\) 0 0
\(699\) −24.9200 −0.942559
\(700\) 0 0
\(701\) −0.540024 −0.0203964 −0.0101982 0.999948i \(-0.503246\pi\)
−0.0101982 + 0.999948i \(0.503246\pi\)
\(702\) 0 0
\(703\) −6.46500 −0.243832
\(704\) 0 0
\(705\) −0.259449 −0.00977142
\(706\) 0 0
\(707\) 29.9461 1.12624
\(708\) 0 0
\(709\) 28.0867 1.05482 0.527409 0.849612i \(-0.323164\pi\)
0.527409 + 0.849612i \(0.323164\pi\)
\(710\) 0 0
\(711\) 15.8328 0.593775
\(712\) 0 0
\(713\) 8.04385 0.301245
\(714\) 0 0
\(715\) 4.57834 0.171220
\(716\) 0 0
\(717\) 2.95112 0.110212
\(718\) 0 0
\(719\) −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
\(722\) 0 0
\(723\) −16.5925 −0.617081
\(724\) 0 0
\(725\) −16.8036 −0.624069
\(726\) 0 0
\(727\) 30.4650 1.12988 0.564942 0.825131i \(-0.308898\pi\)
0.564942 + 0.825131i \(0.308898\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −30.7753 −1.13827
\(732\) 0 0
\(733\) 27.5436 1.01735 0.508673 0.860960i \(-0.330136\pi\)
0.508673 + 0.860960i \(0.330136\pi\)
\(734\) 0 0
\(735\) 0.691675 0.0255128
\(736\) 0 0
\(737\) −8.16724 −0.300844
\(738\) 0 0
\(739\) −13.1013 −0.481940 −0.240970 0.970533i \(-0.577466\pi\)
−0.240970 + 0.970533i \(0.577466\pi\)
\(740\) 0 0
\(741\) −15.8328 −0.581631
\(742\) 0 0
\(743\) −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
\(746\) 0 0
\(747\) 7.68111 0.281037
\(748\) 0 0
\(749\) −38.0867 −1.39166
\(750\) 0 0
\(751\) −32.7738 −1.19593 −0.597967 0.801521i \(-0.704025\pi\)
−0.597967 + 0.801521i \(0.704025\pi\)
\(752\) 0 0
\(753\) 20.1361 0.733799
\(754\) 0 0
\(755\) 21.2333 0.772759
\(756\) 0 0
\(757\) −32.2978 −1.17388 −0.586941 0.809630i \(-0.699668\pi\)
−0.586941 + 0.809630i \(0.699668\pi\)
\(758\) 0 0
\(759\) −0.729988 −0.0264969
\(760\) 0 0
\(761\) 44.4494 1.61129 0.805645 0.592399i \(-0.201819\pi\)
0.805645 + 0.592399i \(0.201819\pi\)
\(762\) 0 0
\(763\) −12.8816 −0.466347
\(764\) 0 0
\(765\) −3.74055 −0.135240
\(766\) 0 0
\(767\) −5.35218 −0.193256
\(768\) 0 0
\(769\) −19.7108 −0.710791 −0.355395 0.934716i \(-0.615654\pi\)
−0.355395 + 0.934716i \(0.615654\pi\)
\(770\) 0 0
\(771\) 25.9008 0.932794
\(772\) 0 0
\(773\) 43.2530 1.55570 0.777851 0.628449i \(-0.216310\pi\)
0.777851 + 0.628449i \(0.216310\pi\)
\(774\) 0 0
\(775\) −33.9235 −1.21857
\(776\) 0 0
\(777\) −5.25997 −0.188700
\(778\) 0 0
\(779\) −3.10278 −0.111168
\(780\) 0 0
\(781\) −10.4252 −0.373044
\(782\) 0 0
\(783\) 4.44082 0.158702
\(784\) 0 0
\(785\) 23.0700 0.823404
\(786\) 0 0
\(787\) 21.4358 0.764104 0.382052 0.924141i \(-0.375218\pi\)
0.382052 + 0.924141i \(0.375218\pi\)
\(788\) 0 0
\(789\) 22.4408 0.798914
\(790\) 0 0
\(791\) 44.2127 1.57202
\(792\) 0 0
\(793\) 9.77886 0.347258
\(794\) 0 0
\(795\) −15.2544 −0.541019
\(796\) 0 0
\(797\) −4.40105 −0.155893 −0.0779467 0.996958i \(-0.524836\pi\)
−0.0779467 + 0.996958i \(0.524836\pi\)
\(798\) 0 0
\(799\) 0.798021 0.0282319
\(800\) 0 0
\(801\) 13.2544 0.468322
\(802\) 0 0
\(803\) −3.87253 −0.136659
\(804\) 0 0
\(805\) 2.49777 0.0880349
\(806\) 0 0
\(807\) 7.62721 0.268491
\(808\) 0 0
\(809\) −26.3799 −0.927469 −0.463734 0.885974i \(-0.653491\pi\)
−0.463734 + 0.885974i \(0.653491\pi\)
\(810\) 0 0
\(811\) −4.96526 −0.174354 −0.0871769 0.996193i \(-0.527785\pi\)
−0.0871769 + 0.996193i \(0.527785\pi\)
\(812\) 0 0
\(813\) −19.9164 −0.698498
\(814\) 0 0
\(815\) −14.1133 −0.494369
\(816\) 0 0
\(817\) 28.1517 0.984902
\(818\) 0 0
\(819\) −12.8816 −0.450121
\(820\) 0 0
\(821\) −15.7789 −0.550686 −0.275343 0.961346i \(-0.588791\pi\)
−0.275343 + 0.961346i \(0.588791\pi\)
\(822\) 0 0
\(823\) 8.35166 0.291121 0.145560 0.989349i \(-0.453502\pi\)
0.145560 + 0.989349i \(0.453502\pi\)
\(824\) 0 0
\(825\) 3.07860 0.107183
\(826\) 0 0
\(827\) 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 \(0.670653\pi\)
\(830\) 0 0
\(831\) −17.6413 −0.611972
\(832\) 0 0
\(833\) −2.12747 −0.0737125
\(834\) 0 0
\(835\) 10.6167 0.367404
\(836\) 0 0
\(837\) 8.96526 0.309885
\(838\) 0 0
\(839\) 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
\(842\) 0 0
\(843\) −0.0297193 −0.00102359
\(844\) 0 0
\(845\) −14.3783 −0.494629
\(846\) 0 0
\(847\) 26.0978 0.896729
\(848\) 0 0
\(849\) 10.1814 0.349424
\(850\) 0 0
\(851\) 1.86947 0.0640848
\(852\) 0 0
\(853\) 40.4494 1.38496 0.692481 0.721436i \(-0.256518\pi\)
0.692481 + 0.721436i \(0.256518\pi\)
\(854\) 0 0
\(855\) 3.42166 0.117018
\(856\) 0 0
\(857\) 12.4494 0.425264 0.212632 0.977132i \(-0.431796\pi\)
0.212632 + 0.977132i \(0.431796\pi\)
\(858\) 0 0
\(859\) 21.6514 0.738736 0.369368 0.929283i \(-0.379574\pi\)
0.369368 + 0.929283i \(0.379574\pi\)
\(860\) 0 0
\(861\) −2.52444 −0.0860326
\(862\) 0 0
\(863\) −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
\(866\) 0 0
\(867\) −5.49472 −0.186610
\(868\) 0 0
\(869\) −12.8816 −0.436980
\(870\) 0 0
\(871\) 51.2233 1.73563
\(872\) 0 0
\(873\) 2.72999 0.0923961
\(874\) 0 0
\(875\) −24.4534 −0.826674
\(876\) 0 0
\(877\) −3.08413 −0.104144 −0.0520719 0.998643i \(-0.516583\pi\)
−0.0520719 + 0.998643i \(0.516583\pi\)
\(878\) 0 0
\(879\) 3.14808 0.106182
\(880\) 0 0
\(881\) −34.5783 −1.16497 −0.582487 0.812840i \(-0.697920\pi\)
−0.582487 + 0.812840i \(0.697920\pi\)
\(882\) 0 0
\(883\) 9.64280 0.324506 0.162253 0.986749i \(-0.448124\pi\)
0.162253 + 0.986749i \(0.448124\pi\)
\(884\) 0 0
\(885\) 1.15667 0.0388812
\(886\) 0 0
\(887\) −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
\(890\) 0 0
\(891\) −0.813607 −0.0272568
\(892\) 0 0
\(893\) −0.729988 −0.0244281
\(894\) 0 0
\(895\) −2.35720 −0.0787925
\(896\) 0 0
\(897\) 4.57834 0.152866
\(898\) 0 0
\(899\) 39.8131 1.32784
\(900\) 0 0
\(901\) 46.9200 1.56313
\(902\) 0 0
\(903\) 22.9044 0.762210
\(904\) 0 0
\(905\) 2.02113 0.0671845
\(906\) 0 0
\(907\) 17.8227 0.591794 0.295897 0.955220i \(-0.404382\pi\)
0.295897 + 0.955220i \(0.404382\pi\)
\(908\) 0 0
\(909\) −11.8625 −0.393454
\(910\) 0 0
\(911\) 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
\(914\) 0 0
\(915\) −2.11334 −0.0698648
\(916\) 0 0
\(917\) −16.0484 −0.529964
\(918\) 0 0
\(919\) −33.8555 −1.11679 −0.558395 0.829575i \(-0.688583\pi\)
−0.558395 + 0.829575i \(0.688583\pi\)
\(920\) 0 0
\(921\) 12.7980 0.421709
\(922\) 0 0
\(923\) 65.3850 2.15217
\(924\) 0 0
\(925\) −7.88418 −0.259230
\(926\) 0 0
\(927\) 0.494719 0.0162487
\(928\) 0 0
\(929\) −4.10635 −0.134725 −0.0673624 0.997729i \(-0.521458\pi\)
−0.0673624 + 0.997729i \(0.521458\pi\)
\(930\) 0 0
\(931\) 1.94610 0.0637809
\(932\) 0 0
\(933\) −26.6167 −0.871390
\(934\) 0 0
\(935\) 3.04334 0.0995277
\(936\) 0 0
\(937\) 13.3028 0.434583 0.217292 0.976107i \(-0.430278\pi\)
0.217292 + 0.976107i \(0.430278\pi\)
\(938\) 0 0
\(939\) −3.00502 −0.0980652
\(940\) 0 0
\(941\) −11.7633 −0.383472 −0.191736 0.981447i \(-0.561412\pi\)
−0.191736 + 0.981447i \(0.561412\pi\)
\(942\) 0 0
\(943\) 0.897225 0.0292177
\(944\) 0 0
\(945\) 2.78389 0.0905599
\(946\) 0 0
\(947\) −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
\(950\) 0 0
\(951\) −4.33302 −0.140508
\(952\) 0 0
\(953\) 33.7577 1.09352 0.546760 0.837289i \(-0.315861\pi\)
0.546760 + 0.837289i \(0.315861\pi\)
\(954\) 0 0
\(955\) −20.5300 −0.664334
\(956\) 0 0
\(957\) −3.61308 −0.116794
\(958\) 0 0
\(959\) 44.1517 1.42573
\(960\) 0 0
\(961\) 49.3758 1.59277
\(962\) 0 0
\(963\) 15.0872 0.486178
\(964\) 0 0
\(965\) −17.2161 −0.554206
\(966\) 0 0
\(967\) 10.4806 0.337033 0.168516 0.985699i \(-0.446102\pi\)
0.168516 + 0.985699i \(0.446102\pi\)
\(968\) 0 0
\(969\) −10.5244 −0.338094
\(970\) 0 0
\(971\) −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
\(974\) 0 0
\(975\) −19.3083 −0.618361
\(976\) 0 0
\(977\) 41.9008 1.34053 0.670263 0.742124i \(-0.266181\pi\)
0.670263 + 0.742124i \(0.266181\pi\)
\(978\) 0 0
\(979\) −10.7839 −0.344655
\(980\) 0 0
\(981\) 5.10278 0.162919
\(982\) 0 0
\(983\) 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
\(986\) 0 0
\(987\) −0.593923 −0.0189048
\(988\) 0 0
\(989\) −8.14057 −0.258855
\(990\) 0 0
\(991\) −26.0666 −0.828032 −0.414016 0.910270i \(-0.635874\pi\)
−0.414016 + 0.910270i \(0.635874\pi\)
\(992\) 0 0
\(993\) −5.53500 −0.175648
\(994\) 0 0
\(995\) 2.26499 0.0718050
\(996\) 0 0
\(997\) −29.8483 −0.945307 −0.472653 0.881248i \(-0.656704\pi\)
−0.472653 + 0.881248i \(0.656704\pi\)
\(998\) 0 0
\(999\) 2.08362 0.0659228
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 1968.2.a.w.1.1 3
3.2 odd 2 5904.2.a.bd.1.3 3
4.3 odd 2 123.2.a.d.1.1 3
8.3 odd 2 7872.2.a.bx.1.3 3
8.5 even 2 7872.2.a.bs.1.3 3
12.11 even 2 369.2.a.e.1.3 3
20.19 odd 2 3075.2.a.t.1.3 3
28.27 even 2 6027.2.a.s.1.1 3
60.59 even 2 9225.2.a.bx.1.1 3
164.163 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 4.3 odd 2
369.2.a.e.1.3 3 12.11 even 2
1968.2.a.w.1.1 3 1.1 even 1 trivial
3075.2.a.t.1.3 3 20.19 odd 2
5043.2.a.n.1.1 3 164.163 odd 2
5904.2.a.bd.1.3 3 3.2 odd 2
6027.2.a.s.1.1 3 28.27 even 2
7872.2.a.bs.1.3 3 8.5 even 2
7872.2.a.bx.1.3 3 8.3 odd 2
9225.2.a.bx.1.1 3 60.59 even 2