Properties

Label 1336.2.a.d.1.11
Level $1336$
Weight $2$
Character 1336.1
Self dual yes
Analytic conductor $10.668$
Analytic rank $0$
Dimension $12$
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] = [1336,2,Mod(1,1336)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1336, 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("1336.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1336 = 2^{3} \cdot 167 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1336.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: \(10.6680137100\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{11} - 25 x^{10} + 20 x^{9} + 224 x^{8} - 135 x^{7} - 865 x^{6} + 330 x^{5} + 1341 x^{4} - 127 x^{3} - 652 x^{2} - 48 x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(-2.49668\) of defining polynomial
Character \(\chi\) \(=\) 1336.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.49668 q^{3} +0.494819 q^{5} +2.18883 q^{7} +3.23342 q^{9} +O(q^{10})\) \(q+2.49668 q^{3} +0.494819 q^{5} +2.18883 q^{7} +3.23342 q^{9} +3.30196 q^{11} +3.85717 q^{13} +1.23541 q^{15} -2.84942 q^{17} -0.107129 q^{19} +5.46481 q^{21} -0.225506 q^{23} -4.75515 q^{25} +0.582788 q^{27} -7.21858 q^{29} -1.87493 q^{31} +8.24395 q^{33} +1.08307 q^{35} -5.20486 q^{37} +9.63013 q^{39} +6.59154 q^{41} -6.49233 q^{43} +1.59996 q^{45} +9.83971 q^{47} -2.20903 q^{49} -7.11410 q^{51} +3.68672 q^{53} +1.63387 q^{55} -0.267468 q^{57} +5.17220 q^{59} +5.59021 q^{61} +7.07741 q^{63} +1.90860 q^{65} -4.59776 q^{67} -0.563018 q^{69} +5.33280 q^{71} +2.39626 q^{73} -11.8721 q^{75} +7.22743 q^{77} -6.02964 q^{79} -8.24524 q^{81} +4.72036 q^{83} -1.40995 q^{85} -18.0225 q^{87} +0.842438 q^{89} +8.44268 q^{91} -4.68111 q^{93} -0.0530097 q^{95} +14.3558 q^{97} +10.6766 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - q^{3} + 8 q^{5} - 4 q^{7} + 15 q^{9} + 6 q^{11} + 13 q^{13} + 4 q^{15} + 10 q^{17} - q^{19} + 13 q^{21} + 3 q^{23} + 18 q^{25} - 10 q^{27} + 29 q^{29} - 3 q^{31} + 3 q^{35} + 41 q^{37} + 10 q^{39} + 20 q^{41} - q^{43} + 42 q^{45} + 5 q^{47} + 20 q^{49} + 14 q^{51} + 39 q^{53} + 3 q^{55} + 3 q^{57} + 8 q^{59} + 30 q^{61} - 2 q^{63} + 21 q^{65} - 9 q^{67} + 33 q^{69} + 29 q^{71} + 12 q^{73} + q^{75} + 19 q^{77} - 2 q^{79} + 24 q^{81} - 5 q^{83} + 44 q^{85} - 2 q^{87} + 7 q^{89} - 4 q^{91} + 37 q^{93} + 18 q^{95} - 14 q^{97} + 5 q^{99}+O(q^{100}) \) 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
\(3\) 2.49668 1.44146 0.720730 0.693216i \(-0.243807\pi\)
0.720730 + 0.693216i \(0.243807\pi\)
\(4\) 0 0
\(5\) 0.494819 0.221290 0.110645 0.993860i \(-0.464708\pi\)
0.110645 + 0.993860i \(0.464708\pi\)
\(6\) 0 0
\(7\) 2.18883 0.827299 0.413650 0.910436i \(-0.364254\pi\)
0.413650 + 0.910436i \(0.364254\pi\)
\(8\) 0 0
\(9\) 3.23342 1.07781
\(10\) 0 0
\(11\) 3.30196 0.995579 0.497790 0.867298i \(-0.334145\pi\)
0.497790 + 0.867298i \(0.334145\pi\)
\(12\) 0 0
\(13\) 3.85717 1.06979 0.534893 0.844920i \(-0.320352\pi\)
0.534893 + 0.844920i \(0.320352\pi\)
\(14\) 0 0
\(15\) 1.23541 0.318980
\(16\) 0 0
\(17\) −2.84942 −0.691086 −0.345543 0.938403i \(-0.612305\pi\)
−0.345543 + 0.938403i \(0.612305\pi\)
\(18\) 0 0
\(19\) −0.107129 −0.0245772 −0.0122886 0.999924i \(-0.503912\pi\)
−0.0122886 + 0.999924i \(0.503912\pi\)
\(20\) 0 0
\(21\) 5.46481 1.19252
\(22\) 0 0
\(23\) −0.225506 −0.0470213 −0.0235107 0.999724i \(-0.507484\pi\)
−0.0235107 + 0.999724i \(0.507484\pi\)
\(24\) 0 0
\(25\) −4.75515 −0.951031
\(26\) 0 0
\(27\) 0.582788 0.112158
\(28\) 0 0
\(29\) −7.21858 −1.34046 −0.670229 0.742155i \(-0.733804\pi\)
−0.670229 + 0.742155i \(0.733804\pi\)
\(30\) 0 0
\(31\) −1.87493 −0.336747 −0.168374 0.985723i \(-0.553852\pi\)
−0.168374 + 0.985723i \(0.553852\pi\)
\(32\) 0 0
\(33\) 8.24395 1.43509
\(34\) 0 0
\(35\) 1.08307 0.183073
\(36\) 0 0
\(37\) −5.20486 −0.855674 −0.427837 0.903856i \(-0.640724\pi\)
−0.427837 + 0.903856i \(0.640724\pi\)
\(38\) 0 0
\(39\) 9.63013 1.54206
\(40\) 0 0
\(41\) 6.59154 1.02943 0.514713 0.857363i \(-0.327898\pi\)
0.514713 + 0.857363i \(0.327898\pi\)
\(42\) 0 0
\(43\) −6.49233 −0.990071 −0.495036 0.868873i \(-0.664845\pi\)
−0.495036 + 0.868873i \(0.664845\pi\)
\(44\) 0 0
\(45\) 1.59996 0.238508
\(46\) 0 0
\(47\) 9.83971 1.43527 0.717635 0.696419i \(-0.245225\pi\)
0.717635 + 0.696419i \(0.245225\pi\)
\(48\) 0 0
\(49\) −2.20903 −0.315576
\(50\) 0 0
\(51\) −7.11410 −0.996173
\(52\) 0 0
\(53\) 3.68672 0.506409 0.253205 0.967413i \(-0.418515\pi\)
0.253205 + 0.967413i \(0.418515\pi\)
\(54\) 0 0
\(55\) 1.63387 0.220311
\(56\) 0 0
\(57\) −0.267468 −0.0354270
\(58\) 0 0
\(59\) 5.17220 0.673364 0.336682 0.941618i \(-0.390695\pi\)
0.336682 + 0.941618i \(0.390695\pi\)
\(60\) 0 0
\(61\) 5.59021 0.715753 0.357877 0.933769i \(-0.383501\pi\)
0.357877 + 0.933769i \(0.383501\pi\)
\(62\) 0 0
\(63\) 7.07741 0.891670
\(64\) 0 0
\(65\) 1.90860 0.236733
\(66\) 0 0
\(67\) −4.59776 −0.561705 −0.280853 0.959751i \(-0.590617\pi\)
−0.280853 + 0.959751i \(0.590617\pi\)
\(68\) 0 0
\(69\) −0.563018 −0.0677794
\(70\) 0 0
\(71\) 5.33280 0.632887 0.316444 0.948611i \(-0.397511\pi\)
0.316444 + 0.948611i \(0.397511\pi\)
\(72\) 0 0
\(73\) 2.39626 0.280461 0.140230 0.990119i \(-0.455216\pi\)
0.140230 + 0.990119i \(0.455216\pi\)
\(74\) 0 0
\(75\) −11.8721 −1.37087
\(76\) 0 0
\(77\) 7.22743 0.823642
\(78\) 0 0
\(79\) −6.02964 −0.678388 −0.339194 0.940717i \(-0.610154\pi\)
−0.339194 + 0.940717i \(0.610154\pi\)
\(80\) 0 0
\(81\) −8.24524 −0.916138
\(82\) 0 0
\(83\) 4.72036 0.518127 0.259064 0.965860i \(-0.416586\pi\)
0.259064 + 0.965860i \(0.416586\pi\)
\(84\) 0 0
\(85\) −1.40995 −0.152930
\(86\) 0 0
\(87\) −18.0225 −1.93222
\(88\) 0 0
\(89\) 0.842438 0.0892982 0.0446491 0.999003i \(-0.485783\pi\)
0.0446491 + 0.999003i \(0.485783\pi\)
\(90\) 0 0
\(91\) 8.44268 0.885034
\(92\) 0 0
\(93\) −4.68111 −0.485408
\(94\) 0 0
\(95\) −0.0530097 −0.00543868
\(96\) 0 0
\(97\) 14.3558 1.45761 0.728806 0.684720i \(-0.240076\pi\)
0.728806 + 0.684720i \(0.240076\pi\)
\(98\) 0 0
\(99\) 10.6766 1.07304
\(100\) 0 0
\(101\) 14.0608 1.39911 0.699553 0.714581i \(-0.253383\pi\)
0.699553 + 0.714581i \(0.253383\pi\)
\(102\) 0 0
\(103\) 6.48486 0.638972 0.319486 0.947591i \(-0.396490\pi\)
0.319486 + 0.947591i \(0.396490\pi\)
\(104\) 0 0
\(105\) 2.70409 0.263892
\(106\) 0 0
\(107\) −20.2063 −1.95342 −0.976709 0.214570i \(-0.931165\pi\)
−0.976709 + 0.214570i \(0.931165\pi\)
\(108\) 0 0
\(109\) −0.396911 −0.0380172 −0.0190086 0.999819i \(-0.506051\pi\)
−0.0190086 + 0.999819i \(0.506051\pi\)
\(110\) 0 0
\(111\) −12.9949 −1.23342
\(112\) 0 0
\(113\) −16.7683 −1.57743 −0.788716 0.614757i \(-0.789254\pi\)
−0.788716 + 0.614757i \(0.789254\pi\)
\(114\) 0 0
\(115\) −0.111585 −0.0104053
\(116\) 0 0
\(117\) 12.4719 1.15302
\(118\) 0 0
\(119\) −6.23689 −0.571735
\(120\) 0 0
\(121\) −0.0970461 −0.00882237
\(122\) 0 0
\(123\) 16.4570 1.48388
\(124\) 0 0
\(125\) −4.82703 −0.431743
\(126\) 0 0
\(127\) −13.0545 −1.15840 −0.579199 0.815186i \(-0.696635\pi\)
−0.579199 + 0.815186i \(0.696635\pi\)
\(128\) 0 0
\(129\) −16.2093 −1.42715
\(130\) 0 0
\(131\) −17.6292 −1.54027 −0.770136 0.637879i \(-0.779812\pi\)
−0.770136 + 0.637879i \(0.779812\pi\)
\(132\) 0 0
\(133\) −0.234488 −0.0203327
\(134\) 0 0
\(135\) 0.288374 0.0248193
\(136\) 0 0
\(137\) −3.36166 −0.287206 −0.143603 0.989635i \(-0.545869\pi\)
−0.143603 + 0.989635i \(0.545869\pi\)
\(138\) 0 0
\(139\) −15.3887 −1.30525 −0.652627 0.757679i \(-0.726333\pi\)
−0.652627 + 0.757679i \(0.726333\pi\)
\(140\) 0 0
\(141\) 24.5666 2.06888
\(142\) 0 0
\(143\) 12.7362 1.06506
\(144\) 0 0
\(145\) −3.57189 −0.296629
\(146\) 0 0
\(147\) −5.51526 −0.454891
\(148\) 0 0
\(149\) 9.19953 0.753655 0.376827 0.926284i \(-0.377015\pi\)
0.376827 + 0.926284i \(0.377015\pi\)
\(150\) 0 0
\(151\) −2.85480 −0.232320 −0.116160 0.993231i \(-0.537059\pi\)
−0.116160 + 0.993231i \(0.537059\pi\)
\(152\) 0 0
\(153\) −9.21338 −0.744858
\(154\) 0 0
\(155\) −0.927751 −0.0745188
\(156\) 0 0
\(157\) −0.468866 −0.0374196 −0.0187098 0.999825i \(-0.505956\pi\)
−0.0187098 + 0.999825i \(0.505956\pi\)
\(158\) 0 0
\(159\) 9.20456 0.729969
\(160\) 0 0
\(161\) −0.493594 −0.0389007
\(162\) 0 0
\(163\) −6.65346 −0.521139 −0.260570 0.965455i \(-0.583910\pi\)
−0.260570 + 0.965455i \(0.583910\pi\)
\(164\) 0 0
\(165\) 4.07926 0.317570
\(166\) 0 0
\(167\) −1.00000 −0.0773823
\(168\) 0 0
\(169\) 1.87777 0.144444
\(170\) 0 0
\(171\) −0.346395 −0.0264895
\(172\) 0 0
\(173\) 14.4806 1.10094 0.550470 0.834855i \(-0.314448\pi\)
0.550470 + 0.834855i \(0.314448\pi\)
\(174\) 0 0
\(175\) −10.4082 −0.786787
\(176\) 0 0
\(177\) 12.9134 0.970627
\(178\) 0 0
\(179\) −7.70746 −0.576083 −0.288041 0.957618i \(-0.593004\pi\)
−0.288041 + 0.957618i \(0.593004\pi\)
\(180\) 0 0
\(181\) 9.84835 0.732022 0.366011 0.930610i \(-0.380723\pi\)
0.366011 + 0.930610i \(0.380723\pi\)
\(182\) 0 0
\(183\) 13.9570 1.03173
\(184\) 0 0
\(185\) −2.57546 −0.189352
\(186\) 0 0
\(187\) −9.40868 −0.688031
\(188\) 0 0
\(189\) 1.27562 0.0927878
\(190\) 0 0
\(191\) 5.36643 0.388301 0.194151 0.980972i \(-0.437805\pi\)
0.194151 + 0.980972i \(0.437805\pi\)
\(192\) 0 0
\(193\) 6.36645 0.458267 0.229134 0.973395i \(-0.426411\pi\)
0.229134 + 0.973395i \(0.426411\pi\)
\(194\) 0 0
\(195\) 4.76517 0.341241
\(196\) 0 0
\(197\) 15.4195 1.09860 0.549298 0.835626i \(-0.314895\pi\)
0.549298 + 0.835626i \(0.314895\pi\)
\(198\) 0 0
\(199\) 10.2605 0.727346 0.363673 0.931527i \(-0.381523\pi\)
0.363673 + 0.931527i \(0.381523\pi\)
\(200\) 0 0
\(201\) −11.4791 −0.809676
\(202\) 0 0
\(203\) −15.8002 −1.10896
\(204\) 0 0
\(205\) 3.26162 0.227801
\(206\) 0 0
\(207\) −0.729158 −0.0506800
\(208\) 0 0
\(209\) −0.353737 −0.0244685
\(210\) 0 0
\(211\) −6.55220 −0.451072 −0.225536 0.974235i \(-0.572413\pi\)
−0.225536 + 0.974235i \(0.572413\pi\)
\(212\) 0 0
\(213\) 13.3143 0.912282
\(214\) 0 0
\(215\) −3.21253 −0.219093
\(216\) 0 0
\(217\) −4.10390 −0.278591
\(218\) 0 0
\(219\) 5.98270 0.404273
\(220\) 0 0
\(221\) −10.9907 −0.739314
\(222\) 0 0
\(223\) −23.7596 −1.59106 −0.795529 0.605916i \(-0.792807\pi\)
−0.795529 + 0.605916i \(0.792807\pi\)
\(224\) 0 0
\(225\) −15.3754 −1.02503
\(226\) 0 0
\(227\) −5.95489 −0.395240 −0.197620 0.980279i \(-0.563321\pi\)
−0.197620 + 0.980279i \(0.563321\pi\)
\(228\) 0 0
\(229\) −12.9016 −0.852563 −0.426282 0.904590i \(-0.640177\pi\)
−0.426282 + 0.904590i \(0.640177\pi\)
\(230\) 0 0
\(231\) 18.0446 1.18725
\(232\) 0 0
\(233\) 11.3615 0.744314 0.372157 0.928170i \(-0.378618\pi\)
0.372157 + 0.928170i \(0.378618\pi\)
\(234\) 0 0
\(235\) 4.86888 0.317610
\(236\) 0 0
\(237\) −15.0541 −0.977869
\(238\) 0 0
\(239\) 9.86306 0.637989 0.318994 0.947757i \(-0.396655\pi\)
0.318994 + 0.947757i \(0.396655\pi\)
\(240\) 0 0
\(241\) 20.9764 1.35121 0.675604 0.737265i \(-0.263883\pi\)
0.675604 + 0.737265i \(0.263883\pi\)
\(242\) 0 0
\(243\) −22.3341 −1.43273
\(244\) 0 0
\(245\) −1.09307 −0.0698338
\(246\) 0 0
\(247\) −0.413216 −0.0262923
\(248\) 0 0
\(249\) 11.7853 0.746860
\(250\) 0 0
\(251\) 13.9866 0.882826 0.441413 0.897304i \(-0.354477\pi\)
0.441413 + 0.897304i \(0.354477\pi\)
\(252\) 0 0
\(253\) −0.744613 −0.0468134
\(254\) 0 0
\(255\) −3.52019 −0.220443
\(256\) 0 0
\(257\) 10.8077 0.674167 0.337083 0.941475i \(-0.390560\pi\)
0.337083 + 0.941475i \(0.390560\pi\)
\(258\) 0 0
\(259\) −11.3926 −0.707899
\(260\) 0 0
\(261\) −23.3407 −1.44476
\(262\) 0 0
\(263\) −17.3432 −1.06943 −0.534713 0.845034i \(-0.679580\pi\)
−0.534713 + 0.845034i \(0.679580\pi\)
\(264\) 0 0
\(265\) 1.82426 0.112063
\(266\) 0 0
\(267\) 2.10330 0.128720
\(268\) 0 0
\(269\) −19.6873 −1.20036 −0.600179 0.799865i \(-0.704904\pi\)
−0.600179 + 0.799865i \(0.704904\pi\)
\(270\) 0 0
\(271\) −4.06608 −0.246997 −0.123499 0.992345i \(-0.539411\pi\)
−0.123499 + 0.992345i \(0.539411\pi\)
\(272\) 0 0
\(273\) 21.0787 1.27574
\(274\) 0 0
\(275\) −15.7013 −0.946826
\(276\) 0 0
\(277\) −7.61121 −0.457313 −0.228657 0.973507i \(-0.573433\pi\)
−0.228657 + 0.973507i \(0.573433\pi\)
\(278\) 0 0
\(279\) −6.06245 −0.362949
\(280\) 0 0
\(281\) −9.85439 −0.587864 −0.293932 0.955826i \(-0.594964\pi\)
−0.293932 + 0.955826i \(0.594964\pi\)
\(282\) 0 0
\(283\) −13.7241 −0.815815 −0.407907 0.913023i \(-0.633741\pi\)
−0.407907 + 0.913023i \(0.633741\pi\)
\(284\) 0 0
\(285\) −0.132348 −0.00783964
\(286\) 0 0
\(287\) 14.4277 0.851643
\(288\) 0 0
\(289\) −8.88081 −0.522400
\(290\) 0 0
\(291\) 35.8419 2.10109
\(292\) 0 0
\(293\) 19.5483 1.14203 0.571013 0.820941i \(-0.306551\pi\)
0.571013 + 0.820941i \(0.306551\pi\)
\(294\) 0 0
\(295\) 2.55930 0.149008
\(296\) 0 0
\(297\) 1.92434 0.111662
\(298\) 0 0
\(299\) −0.869816 −0.0503028
\(300\) 0 0
\(301\) −14.2106 −0.819085
\(302\) 0 0
\(303\) 35.1054 2.01675
\(304\) 0 0
\(305\) 2.76614 0.158389
\(306\) 0 0
\(307\) 16.0316 0.914970 0.457485 0.889217i \(-0.348750\pi\)
0.457485 + 0.889217i \(0.348750\pi\)
\(308\) 0 0
\(309\) 16.1906 0.921054
\(310\) 0 0
\(311\) 11.8931 0.674398 0.337199 0.941433i \(-0.390521\pi\)
0.337199 + 0.941433i \(0.390521\pi\)
\(312\) 0 0
\(313\) −10.4125 −0.588551 −0.294276 0.955721i \(-0.595078\pi\)
−0.294276 + 0.955721i \(0.595078\pi\)
\(314\) 0 0
\(315\) 3.50204 0.197317
\(316\) 0 0
\(317\) 22.2739 1.25103 0.625514 0.780213i \(-0.284889\pi\)
0.625514 + 0.780213i \(0.284889\pi\)
\(318\) 0 0
\(319\) −23.8355 −1.33453
\(320\) 0 0
\(321\) −50.4487 −2.81577
\(322\) 0 0
\(323\) 0.305257 0.0169849
\(324\) 0 0
\(325\) −18.3414 −1.01740
\(326\) 0 0
\(327\) −0.990961 −0.0548003
\(328\) 0 0
\(329\) 21.5374 1.18740
\(330\) 0 0
\(331\) −4.31964 −0.237429 −0.118714 0.992928i \(-0.537877\pi\)
−0.118714 + 0.992928i \(0.537877\pi\)
\(332\) 0 0
\(333\) −16.8295 −0.922253
\(334\) 0 0
\(335\) −2.27506 −0.124300
\(336\) 0 0
\(337\) 0.451358 0.0245870 0.0122935 0.999924i \(-0.496087\pi\)
0.0122935 + 0.999924i \(0.496087\pi\)
\(338\) 0 0
\(339\) −41.8652 −2.27381
\(340\) 0 0
\(341\) −6.19095 −0.335259
\(342\) 0 0
\(343\) −20.1570 −1.08838
\(344\) 0 0
\(345\) −0.278592 −0.0149989
\(346\) 0 0
\(347\) 1.11053 0.0596163 0.0298081 0.999556i \(-0.490510\pi\)
0.0298081 + 0.999556i \(0.490510\pi\)
\(348\) 0 0
\(349\) −2.44374 −0.130810 −0.0654052 0.997859i \(-0.520834\pi\)
−0.0654052 + 0.997859i \(0.520834\pi\)
\(350\) 0 0
\(351\) 2.24791 0.119985
\(352\) 0 0
\(353\) 12.9606 0.689825 0.344912 0.938635i \(-0.387909\pi\)
0.344912 + 0.938635i \(0.387909\pi\)
\(354\) 0 0
\(355\) 2.63877 0.140051
\(356\) 0 0
\(357\) −15.5715 −0.824133
\(358\) 0 0
\(359\) −18.9552 −1.00042 −0.500209 0.865905i \(-0.666743\pi\)
−0.500209 + 0.865905i \(0.666743\pi\)
\(360\) 0 0
\(361\) −18.9885 −0.999396
\(362\) 0 0
\(363\) −0.242293 −0.0127171
\(364\) 0 0
\(365\) 1.18571 0.0620631
\(366\) 0 0
\(367\) 15.2063 0.793763 0.396882 0.917870i \(-0.370092\pi\)
0.396882 + 0.917870i \(0.370092\pi\)
\(368\) 0 0
\(369\) 21.3133 1.10952
\(370\) 0 0
\(371\) 8.06959 0.418952
\(372\) 0 0
\(373\) 25.6863 1.32999 0.664993 0.746849i \(-0.268434\pi\)
0.664993 + 0.746849i \(0.268434\pi\)
\(374\) 0 0
\(375\) −12.0516 −0.622341
\(376\) 0 0
\(377\) −27.8433 −1.43400
\(378\) 0 0
\(379\) 17.5271 0.900308 0.450154 0.892951i \(-0.351369\pi\)
0.450154 + 0.892951i \(0.351369\pi\)
\(380\) 0 0
\(381\) −32.5929 −1.66979
\(382\) 0 0
\(383\) −10.7028 −0.546886 −0.273443 0.961888i \(-0.588162\pi\)
−0.273443 + 0.961888i \(0.588162\pi\)
\(384\) 0 0
\(385\) 3.57627 0.182263
\(386\) 0 0
\(387\) −20.9925 −1.06711
\(388\) 0 0
\(389\) −18.9576 −0.961188 −0.480594 0.876943i \(-0.659579\pi\)
−0.480594 + 0.876943i \(0.659579\pi\)
\(390\) 0 0
\(391\) 0.642562 0.0324958
\(392\) 0 0
\(393\) −44.0146 −2.22024
\(394\) 0 0
\(395\) −2.98358 −0.150120
\(396\) 0 0
\(397\) −4.52120 −0.226912 −0.113456 0.993543i \(-0.536192\pi\)
−0.113456 + 0.993543i \(0.536192\pi\)
\(398\) 0 0
\(399\) −0.585442 −0.0293087
\(400\) 0 0
\(401\) 16.9485 0.846365 0.423183 0.906044i \(-0.360913\pi\)
0.423183 + 0.906044i \(0.360913\pi\)
\(402\) 0 0
\(403\) −7.23193 −0.360248
\(404\) 0 0
\(405\) −4.07990 −0.202732
\(406\) 0 0
\(407\) −17.1863 −0.851892
\(408\) 0 0
\(409\) −12.0728 −0.596964 −0.298482 0.954415i \(-0.596480\pi\)
−0.298482 + 0.954415i \(0.596480\pi\)
\(410\) 0 0
\(411\) −8.39300 −0.413996
\(412\) 0 0
\(413\) 11.3211 0.557073
\(414\) 0 0
\(415\) 2.33573 0.114656
\(416\) 0 0
\(417\) −38.4208 −1.88147
\(418\) 0 0
\(419\) −2.78051 −0.135837 −0.0679183 0.997691i \(-0.521636\pi\)
−0.0679183 + 0.997691i \(0.521636\pi\)
\(420\) 0 0
\(421\) 32.6855 1.59299 0.796496 0.604644i \(-0.206685\pi\)
0.796496 + 0.604644i \(0.206685\pi\)
\(422\) 0 0
\(423\) 31.8160 1.54695
\(424\) 0 0
\(425\) 13.5494 0.657244
\(426\) 0 0
\(427\) 12.2360 0.592142
\(428\) 0 0
\(429\) 31.7983 1.53524
\(430\) 0 0
\(431\) −4.02096 −0.193683 −0.0968415 0.995300i \(-0.530874\pi\)
−0.0968415 + 0.995300i \(0.530874\pi\)
\(432\) 0 0
\(433\) −30.1701 −1.44988 −0.724942 0.688810i \(-0.758133\pi\)
−0.724942 + 0.688810i \(0.758133\pi\)
\(434\) 0 0
\(435\) −8.91788 −0.427580
\(436\) 0 0
\(437\) 0.0241584 0.00115565
\(438\) 0 0
\(439\) −12.1555 −0.580150 −0.290075 0.957004i \(-0.593680\pi\)
−0.290075 + 0.957004i \(0.593680\pi\)
\(440\) 0 0
\(441\) −7.14275 −0.340131
\(442\) 0 0
\(443\) 14.4039 0.684350 0.342175 0.939636i \(-0.388836\pi\)
0.342175 + 0.939636i \(0.388836\pi\)
\(444\) 0 0
\(445\) 0.416854 0.0197608
\(446\) 0 0
\(447\) 22.9683 1.08636
\(448\) 0 0
\(449\) −28.7784 −1.35814 −0.679069 0.734074i \(-0.737616\pi\)
−0.679069 + 0.734074i \(0.737616\pi\)
\(450\) 0 0
\(451\) 21.7650 1.02487
\(452\) 0 0
\(453\) −7.12753 −0.334880
\(454\) 0 0
\(455\) 4.17760 0.195849
\(456\) 0 0
\(457\) −6.06627 −0.283768 −0.141884 0.989883i \(-0.545316\pi\)
−0.141884 + 0.989883i \(0.545316\pi\)
\(458\) 0 0
\(459\) −1.66061 −0.0775105
\(460\) 0 0
\(461\) −20.3042 −0.945660 −0.472830 0.881154i \(-0.656768\pi\)
−0.472830 + 0.881154i \(0.656768\pi\)
\(462\) 0 0
\(463\) −10.6608 −0.495451 −0.247725 0.968830i \(-0.579683\pi\)
−0.247725 + 0.968830i \(0.579683\pi\)
\(464\) 0 0
\(465\) −2.31630 −0.107416
\(466\) 0 0
\(467\) 19.1698 0.887072 0.443536 0.896257i \(-0.353724\pi\)
0.443536 + 0.896257i \(0.353724\pi\)
\(468\) 0 0
\(469\) −10.0637 −0.464698
\(470\) 0 0
\(471\) −1.17061 −0.0539389
\(472\) 0 0
\(473\) −21.4374 −0.985694
\(474\) 0 0
\(475\) 0.509417 0.0233737
\(476\) 0 0
\(477\) 11.9207 0.545812
\(478\) 0 0
\(479\) 27.8451 1.27228 0.636138 0.771575i \(-0.280531\pi\)
0.636138 + 0.771575i \(0.280531\pi\)
\(480\) 0 0
\(481\) −20.0761 −0.915389
\(482\) 0 0
\(483\) −1.23235 −0.0560738
\(484\) 0 0
\(485\) 7.10353 0.322555
\(486\) 0 0
\(487\) −1.32856 −0.0602027 −0.0301014 0.999547i \(-0.509583\pi\)
−0.0301014 + 0.999547i \(0.509583\pi\)
\(488\) 0 0
\(489\) −16.6116 −0.751201
\(490\) 0 0
\(491\) 29.1475 1.31541 0.657704 0.753277i \(-0.271528\pi\)
0.657704 + 0.753277i \(0.271528\pi\)
\(492\) 0 0
\(493\) 20.5688 0.926371
\(494\) 0 0
\(495\) 5.28301 0.237453
\(496\) 0 0
\(497\) 11.6726 0.523587
\(498\) 0 0
\(499\) 0.303970 0.0136076 0.00680379 0.999977i \(-0.497834\pi\)
0.00680379 + 0.999977i \(0.497834\pi\)
\(500\) 0 0
\(501\) −2.49668 −0.111544
\(502\) 0 0
\(503\) 19.9185 0.888121 0.444061 0.895997i \(-0.353537\pi\)
0.444061 + 0.895997i \(0.353537\pi\)
\(504\) 0 0
\(505\) 6.95756 0.309608
\(506\) 0 0
\(507\) 4.68819 0.208210
\(508\) 0 0
\(509\) −6.68611 −0.296357 −0.148178 0.988961i \(-0.547341\pi\)
−0.148178 + 0.988961i \(0.547341\pi\)
\(510\) 0 0
\(511\) 5.24500 0.232025
\(512\) 0 0
\(513\) −0.0624337 −0.00275652
\(514\) 0 0
\(515\) 3.20883 0.141398
\(516\) 0 0
\(517\) 32.4904 1.42892
\(518\) 0 0
\(519\) 36.1535 1.58696
\(520\) 0 0
\(521\) −33.2937 −1.45862 −0.729312 0.684181i \(-0.760160\pi\)
−0.729312 + 0.684181i \(0.760160\pi\)
\(522\) 0 0
\(523\) −12.2397 −0.535205 −0.267602 0.963529i \(-0.586231\pi\)
−0.267602 + 0.963529i \(0.586231\pi\)
\(524\) 0 0
\(525\) −25.9860 −1.13412
\(526\) 0 0
\(527\) 5.34246 0.232721
\(528\) 0 0
\(529\) −22.9491 −0.997789
\(530\) 0 0
\(531\) 16.7239 0.725757
\(532\) 0 0
\(533\) 25.4247 1.10127
\(534\) 0 0
\(535\) −9.99846 −0.432271
\(536\) 0 0
\(537\) −19.2431 −0.830401
\(538\) 0 0
\(539\) −7.29415 −0.314181
\(540\) 0 0
\(541\) −4.88900 −0.210195 −0.105097 0.994462i \(-0.533515\pi\)
−0.105097 + 0.994462i \(0.533515\pi\)
\(542\) 0 0
\(543\) 24.5882 1.05518
\(544\) 0 0
\(545\) −0.196399 −0.00841281
\(546\) 0 0
\(547\) 8.59128 0.367337 0.183668 0.982988i \(-0.441203\pi\)
0.183668 + 0.982988i \(0.441203\pi\)
\(548\) 0 0
\(549\) 18.0755 0.771445
\(550\) 0 0
\(551\) 0.773323 0.0329447
\(552\) 0 0
\(553\) −13.1978 −0.561229
\(554\) 0 0
\(555\) −6.43012 −0.272943
\(556\) 0 0
\(557\) −21.8867 −0.927371 −0.463686 0.886000i \(-0.653473\pi\)
−0.463686 + 0.886000i \(0.653473\pi\)
\(558\) 0 0
\(559\) −25.0420 −1.05917
\(560\) 0 0
\(561\) −23.4905 −0.991769
\(562\) 0 0
\(563\) 41.5539 1.75129 0.875644 0.482958i \(-0.160438\pi\)
0.875644 + 0.482958i \(0.160438\pi\)
\(564\) 0 0
\(565\) −8.29729 −0.349070
\(566\) 0 0
\(567\) −18.0474 −0.757920
\(568\) 0 0
\(569\) 45.4970 1.90733 0.953667 0.300865i \(-0.0972754\pi\)
0.953667 + 0.300865i \(0.0972754\pi\)
\(570\) 0 0
\(571\) 13.8624 0.580124 0.290062 0.957008i \(-0.406324\pi\)
0.290062 + 0.957008i \(0.406324\pi\)
\(572\) 0 0
\(573\) 13.3983 0.559721
\(574\) 0 0
\(575\) 1.07232 0.0447187
\(576\) 0 0
\(577\) −3.35521 −0.139679 −0.0698395 0.997558i \(-0.522249\pi\)
−0.0698395 + 0.997558i \(0.522249\pi\)
\(578\) 0 0
\(579\) 15.8950 0.660574
\(580\) 0 0
\(581\) 10.3321 0.428646
\(582\) 0 0
\(583\) 12.1734 0.504171
\(584\) 0 0
\(585\) 6.17132 0.255153
\(586\) 0 0
\(587\) −28.8863 −1.19226 −0.596132 0.802886i \(-0.703297\pi\)
−0.596132 + 0.802886i \(0.703297\pi\)
\(588\) 0 0
\(589\) 0.200860 0.00827630
\(590\) 0 0
\(591\) 38.4977 1.58358
\(592\) 0 0
\(593\) 29.6015 1.21559 0.607795 0.794094i \(-0.292054\pi\)
0.607795 + 0.794094i \(0.292054\pi\)
\(594\) 0 0
\(595\) −3.08613 −0.126519
\(596\) 0 0
\(597\) 25.6172 1.04844
\(598\) 0 0
\(599\) 23.8865 0.975976 0.487988 0.872850i \(-0.337731\pi\)
0.487988 + 0.872850i \(0.337731\pi\)
\(600\) 0 0
\(601\) −9.08030 −0.370393 −0.185197 0.982701i \(-0.559292\pi\)
−0.185197 + 0.982701i \(0.559292\pi\)
\(602\) 0 0
\(603\) −14.8665 −0.605411
\(604\) 0 0
\(605\) −0.0480202 −0.00195230
\(606\) 0 0
\(607\) −22.4683 −0.911959 −0.455979 0.889990i \(-0.650711\pi\)
−0.455979 + 0.889990i \(0.650711\pi\)
\(608\) 0 0
\(609\) −39.4482 −1.59852
\(610\) 0 0
\(611\) 37.9535 1.53543
\(612\) 0 0
\(613\) 41.3738 1.67107 0.835536 0.549435i \(-0.185157\pi\)
0.835536 + 0.549435i \(0.185157\pi\)
\(614\) 0 0
\(615\) 8.14323 0.328367
\(616\) 0 0
\(617\) −5.34719 −0.215270 −0.107635 0.994190i \(-0.534328\pi\)
−0.107635 + 0.994190i \(0.534328\pi\)
\(618\) 0 0
\(619\) −30.7477 −1.23586 −0.617928 0.786235i \(-0.712028\pi\)
−0.617928 + 0.786235i \(0.712028\pi\)
\(620\) 0 0
\(621\) −0.131422 −0.00527379
\(622\) 0 0
\(623\) 1.84395 0.0738763
\(624\) 0 0
\(625\) 21.3873 0.855491
\(626\) 0 0
\(627\) −0.883170 −0.0352704
\(628\) 0 0
\(629\) 14.8308 0.591344
\(630\) 0 0
\(631\) −3.83570 −0.152697 −0.0763483 0.997081i \(-0.524326\pi\)
−0.0763483 + 0.997081i \(0.524326\pi\)
\(632\) 0 0
\(633\) −16.3588 −0.650202
\(634\) 0 0
\(635\) −6.45961 −0.256342
\(636\) 0 0
\(637\) −8.52062 −0.337599
\(638\) 0 0
\(639\) 17.2432 0.682131
\(640\) 0 0
\(641\) 41.3733 1.63415 0.817073 0.576534i \(-0.195595\pi\)
0.817073 + 0.576534i \(0.195595\pi\)
\(642\) 0 0
\(643\) 36.6923 1.44700 0.723501 0.690323i \(-0.242532\pi\)
0.723501 + 0.690323i \(0.242532\pi\)
\(644\) 0 0
\(645\) −8.02066 −0.315813
\(646\) 0 0
\(647\) 4.32189 0.169911 0.0849554 0.996385i \(-0.472925\pi\)
0.0849554 + 0.996385i \(0.472925\pi\)
\(648\) 0 0
\(649\) 17.0784 0.670387
\(650\) 0 0
\(651\) −10.2461 −0.401578
\(652\) 0 0
\(653\) 11.9208 0.466495 0.233248 0.972417i \(-0.425065\pi\)
0.233248 + 0.972417i \(0.425065\pi\)
\(654\) 0 0
\(655\) −8.72327 −0.340846
\(656\) 0 0
\(657\) 7.74812 0.302283
\(658\) 0 0
\(659\) 7.12791 0.277664 0.138832 0.990316i \(-0.455665\pi\)
0.138832 + 0.990316i \(0.455665\pi\)
\(660\) 0 0
\(661\) 41.5190 1.61490 0.807450 0.589936i \(-0.200847\pi\)
0.807450 + 0.589936i \(0.200847\pi\)
\(662\) 0 0
\(663\) −27.4403 −1.06569
\(664\) 0 0
\(665\) −0.116029 −0.00449941
\(666\) 0 0
\(667\) 1.62784 0.0630301
\(668\) 0 0
\(669\) −59.3201 −2.29345
\(670\) 0 0
\(671\) 18.4587 0.712589
\(672\) 0 0
\(673\) 40.2896 1.55305 0.776526 0.630086i \(-0.216980\pi\)
0.776526 + 0.630086i \(0.216980\pi\)
\(674\) 0 0
\(675\) −2.77124 −0.106665
\(676\) 0 0
\(677\) 12.1689 0.467690 0.233845 0.972274i \(-0.424869\pi\)
0.233845 + 0.972274i \(0.424869\pi\)
\(678\) 0 0
\(679\) 31.4224 1.20588
\(680\) 0 0
\(681\) −14.8675 −0.569723
\(682\) 0 0
\(683\) 25.2454 0.965989 0.482995 0.875623i \(-0.339549\pi\)
0.482995 + 0.875623i \(0.339549\pi\)
\(684\) 0 0
\(685\) −1.66341 −0.0635557
\(686\) 0 0
\(687\) −32.2113 −1.22894
\(688\) 0 0
\(689\) 14.2203 0.541750
\(690\) 0 0
\(691\) 24.6993 0.939607 0.469803 0.882771i \(-0.344325\pi\)
0.469803 + 0.882771i \(0.344325\pi\)
\(692\) 0 0
\(693\) 23.3693 0.887728
\(694\) 0 0
\(695\) −7.61463 −0.288839
\(696\) 0 0
\(697\) −18.7821 −0.711422
\(698\) 0 0
\(699\) 28.3660 1.07290
\(700\) 0 0
\(701\) −46.3982 −1.75243 −0.876217 0.481916i \(-0.839941\pi\)
−0.876217 + 0.481916i \(0.839941\pi\)
\(702\) 0 0
\(703\) 0.557594 0.0210301
\(704\) 0 0
\(705\) 12.1560 0.457823
\(706\) 0 0
\(707\) 30.7767 1.15748
\(708\) 0 0
\(709\) 0.362206 0.0136029 0.00680146 0.999977i \(-0.497835\pi\)
0.00680146 + 0.999977i \(0.497835\pi\)
\(710\) 0 0
\(711\) −19.4964 −0.731172
\(712\) 0 0
\(713\) 0.422809 0.0158343
\(714\) 0 0
\(715\) 6.30213 0.235686
\(716\) 0 0
\(717\) 24.6249 0.919635
\(718\) 0 0
\(719\) 47.0968 1.75641 0.878207 0.478280i \(-0.158740\pi\)
0.878207 + 0.478280i \(0.158740\pi\)
\(720\) 0 0
\(721\) 14.1942 0.528621
\(722\) 0 0
\(723\) 52.3714 1.94771
\(724\) 0 0
\(725\) 34.3255 1.27482
\(726\) 0 0
\(727\) −33.9450 −1.25895 −0.629475 0.777021i \(-0.716730\pi\)
−0.629475 + 0.777021i \(0.716730\pi\)
\(728\) 0 0
\(729\) −31.0255 −1.14909
\(730\) 0 0
\(731\) 18.4994 0.684224
\(732\) 0 0
\(733\) 24.7995 0.915989 0.457994 0.888955i \(-0.348568\pi\)
0.457994 + 0.888955i \(0.348568\pi\)
\(734\) 0 0
\(735\) −2.72905 −0.100663
\(736\) 0 0
\(737\) −15.1816 −0.559222
\(738\) 0 0
\(739\) 50.4197 1.85472 0.927360 0.374170i \(-0.122073\pi\)
0.927360 + 0.374170i \(0.122073\pi\)
\(740\) 0 0
\(741\) −1.03167 −0.0378994
\(742\) 0 0
\(743\) −29.4622 −1.08086 −0.540432 0.841387i \(-0.681739\pi\)
−0.540432 + 0.841387i \(0.681739\pi\)
\(744\) 0 0
\(745\) 4.55210 0.166776
\(746\) 0 0
\(747\) 15.2629 0.558442
\(748\) 0 0
\(749\) −44.2281 −1.61606
\(750\) 0 0
\(751\) −37.2575 −1.35954 −0.679772 0.733423i \(-0.737921\pi\)
−0.679772 + 0.733423i \(0.737921\pi\)
\(752\) 0 0
\(753\) 34.9201 1.27256
\(754\) 0 0
\(755\) −1.41261 −0.0514101
\(756\) 0 0
\(757\) −29.3744 −1.06763 −0.533815 0.845601i \(-0.679242\pi\)
−0.533815 + 0.845601i \(0.679242\pi\)
\(758\) 0 0
\(759\) −1.85906 −0.0674797
\(760\) 0 0
\(761\) −14.1050 −0.511305 −0.255652 0.966769i \(-0.582290\pi\)
−0.255652 + 0.966769i \(0.582290\pi\)
\(762\) 0 0
\(763\) −0.868770 −0.0314516
\(764\) 0 0
\(765\) −4.55896 −0.164829
\(766\) 0 0
\(767\) 19.9501 0.720356
\(768\) 0 0
\(769\) −0.563122 −0.0203067 −0.0101534 0.999948i \(-0.503232\pi\)
−0.0101534 + 0.999948i \(0.503232\pi\)
\(770\) 0 0
\(771\) 26.9834 0.971785
\(772\) 0 0
\(773\) 32.4287 1.16638 0.583189 0.812336i \(-0.301805\pi\)
0.583189 + 0.812336i \(0.301805\pi\)
\(774\) 0 0
\(775\) 8.91558 0.320257
\(776\) 0 0
\(777\) −28.4436 −1.02041
\(778\) 0 0
\(779\) −0.706148 −0.0253004
\(780\) 0 0
\(781\) 17.6087 0.630089
\(782\) 0 0
\(783\) −4.20690 −0.150342
\(784\) 0 0
\(785\) −0.232004 −0.00828057
\(786\) 0 0
\(787\) 43.3905 1.54671 0.773353 0.633976i \(-0.218578\pi\)
0.773353 + 0.633976i \(0.218578\pi\)
\(788\) 0 0
\(789\) −43.3004 −1.54154
\(790\) 0 0
\(791\) −36.7030 −1.30501
\(792\) 0 0
\(793\) 21.5624 0.765703
\(794\) 0 0
\(795\) 4.55459 0.161535
\(796\) 0 0
\(797\) 17.1876 0.608817 0.304408 0.952542i \(-0.401541\pi\)
0.304408 + 0.952542i \(0.401541\pi\)
\(798\) 0 0
\(799\) −28.0375 −0.991895
\(800\) 0 0
\(801\) 2.72396 0.0962463
\(802\) 0 0
\(803\) 7.91236 0.279221
\(804\) 0 0
\(805\) −0.244240 −0.00860832
\(806\) 0 0
\(807\) −49.1531 −1.73027
\(808\) 0 0
\(809\) −12.9798 −0.456344 −0.228172 0.973621i \(-0.573275\pi\)
−0.228172 + 0.973621i \(0.573275\pi\)
\(810\) 0 0
\(811\) −19.0137 −0.667660 −0.333830 0.942633i \(-0.608341\pi\)
−0.333830 + 0.942633i \(0.608341\pi\)
\(812\) 0 0
\(813\) −10.1517 −0.356036
\(814\) 0 0
\(815\) −3.29226 −0.115323
\(816\) 0 0
\(817\) 0.695520 0.0243332
\(818\) 0 0
\(819\) 27.2988 0.953896
\(820\) 0 0
\(821\) 38.4281 1.34115 0.670574 0.741842i \(-0.266048\pi\)
0.670574 + 0.741842i \(0.266048\pi\)
\(822\) 0 0
\(823\) −36.1962 −1.26172 −0.630860 0.775897i \(-0.717298\pi\)
−0.630860 + 0.775897i \(0.717298\pi\)
\(824\) 0 0
\(825\) −39.2013 −1.36481
\(826\) 0 0
\(827\) −33.8350 −1.17656 −0.588279 0.808658i \(-0.700194\pi\)
−0.588279 + 0.808658i \(0.700194\pi\)
\(828\) 0 0
\(829\) 32.5410 1.13020 0.565099 0.825023i \(-0.308838\pi\)
0.565099 + 0.825023i \(0.308838\pi\)
\(830\) 0 0
\(831\) −19.0028 −0.659199
\(832\) 0 0
\(833\) 6.29447 0.218090
\(834\) 0 0
\(835\) −0.494819 −0.0171239
\(836\) 0 0
\(837\) −1.09269 −0.0377688
\(838\) 0 0
\(839\) −28.6028 −0.987478 −0.493739 0.869610i \(-0.664370\pi\)
−0.493739 + 0.869610i \(0.664370\pi\)
\(840\) 0 0
\(841\) 23.1080 0.796826
\(842\) 0 0
\(843\) −24.6033 −0.847382
\(844\) 0 0
\(845\) 0.929154 0.0319639
\(846\) 0 0
\(847\) −0.212417 −0.00729874
\(848\) 0 0
\(849\) −34.2648 −1.17596
\(850\) 0 0
\(851\) 1.17373 0.0402349
\(852\) 0 0
\(853\) 29.4616 1.00874 0.504372 0.863486i \(-0.331724\pi\)
0.504372 + 0.863486i \(0.331724\pi\)
\(854\) 0 0
\(855\) −0.171403 −0.00586185
\(856\) 0 0
\(857\) −2.23304 −0.0762793 −0.0381397 0.999272i \(-0.512143\pi\)
−0.0381397 + 0.999272i \(0.512143\pi\)
\(858\) 0 0
\(859\) −29.7098 −1.01369 −0.506843 0.862038i \(-0.669188\pi\)
−0.506843 + 0.862038i \(0.669188\pi\)
\(860\) 0 0
\(861\) 36.0215 1.22761
\(862\) 0 0
\(863\) 48.6497 1.65605 0.828027 0.560688i \(-0.189463\pi\)
0.828027 + 0.560688i \(0.189463\pi\)
\(864\) 0 0
\(865\) 7.16528 0.243627
\(866\) 0 0
\(867\) −22.1726 −0.753020
\(868\) 0 0
\(869\) −19.9096 −0.675389
\(870\) 0 0
\(871\) −17.7343 −0.600905
\(872\) 0 0
\(873\) 46.4184 1.57103
\(874\) 0 0
\(875\) −10.5655 −0.357181
\(876\) 0 0
\(877\) 55.2990 1.86732 0.933658 0.358167i \(-0.116598\pi\)
0.933658 + 0.358167i \(0.116598\pi\)
\(878\) 0 0
\(879\) 48.8060 1.64619
\(880\) 0 0
\(881\) 5.19113 0.174893 0.0874467 0.996169i \(-0.472129\pi\)
0.0874467 + 0.996169i \(0.472129\pi\)
\(882\) 0 0
\(883\) −51.0328 −1.71739 −0.858696 0.512486i \(-0.828725\pi\)
−0.858696 + 0.512486i \(0.828725\pi\)
\(884\) 0 0
\(885\) 6.38977 0.214790
\(886\) 0 0
\(887\) 18.0401 0.605728 0.302864 0.953034i \(-0.402057\pi\)
0.302864 + 0.953034i \(0.402057\pi\)
\(888\) 0 0
\(889\) −28.5740 −0.958342
\(890\) 0 0
\(891\) −27.2255 −0.912087
\(892\) 0 0
\(893\) −1.05412 −0.0352749
\(894\) 0 0
\(895\) −3.81380 −0.127481
\(896\) 0 0
\(897\) −2.17166 −0.0725095
\(898\) 0 0
\(899\) 13.5343 0.451396
\(900\) 0 0
\(901\) −10.5050 −0.349972
\(902\) 0 0
\(903\) −35.4793 −1.18068
\(904\) 0 0
\(905\) 4.87315 0.161989
\(906\) 0 0
\(907\) −47.1331 −1.56503 −0.782515 0.622632i \(-0.786063\pi\)
−0.782515 + 0.622632i \(0.786063\pi\)
\(908\) 0 0
\(909\) 45.4646 1.50797
\(910\) 0 0
\(911\) −45.8008 −1.51745 −0.758724 0.651412i \(-0.774177\pi\)
−0.758724 + 0.651412i \(0.774177\pi\)
\(912\) 0 0
\(913\) 15.5865 0.515837
\(914\) 0 0
\(915\) 6.90618 0.228311
\(916\) 0 0
\(917\) −38.5873 −1.27427
\(918\) 0 0
\(919\) 9.37906 0.309387 0.154693 0.987963i \(-0.450561\pi\)
0.154693 + 0.987963i \(0.450561\pi\)
\(920\) 0 0
\(921\) 40.0257 1.31889
\(922\) 0 0
\(923\) 20.5695 0.677054
\(924\) 0 0
\(925\) 24.7499 0.813773
\(926\) 0 0
\(927\) 20.9683 0.688690
\(928\) 0 0
\(929\) 51.8937 1.70258 0.851288 0.524699i \(-0.175822\pi\)
0.851288 + 0.524699i \(0.175822\pi\)
\(930\) 0 0
\(931\) 0.236653 0.00775597
\(932\) 0 0
\(933\) 29.6934 0.972118
\(934\) 0 0
\(935\) −4.65559 −0.152254
\(936\) 0 0
\(937\) 11.5263 0.376549 0.188274 0.982116i \(-0.439711\pi\)
0.188274 + 0.982116i \(0.439711\pi\)
\(938\) 0 0
\(939\) −25.9968 −0.848373
\(940\) 0 0
\(941\) −23.2254 −0.757126 −0.378563 0.925576i \(-0.623582\pi\)
−0.378563 + 0.925576i \(0.623582\pi\)
\(942\) 0 0
\(943\) −1.48643 −0.0484050
\(944\) 0 0
\(945\) 0.631202 0.0205330
\(946\) 0 0
\(947\) 2.30372 0.0748610 0.0374305 0.999299i \(-0.488083\pi\)
0.0374305 + 0.999299i \(0.488083\pi\)
\(948\) 0 0
\(949\) 9.24278 0.300033
\(950\) 0 0
\(951\) 55.6109 1.80331
\(952\) 0 0
\(953\) −14.0992 −0.456718 −0.228359 0.973577i \(-0.573336\pi\)
−0.228359 + 0.973577i \(0.573336\pi\)
\(954\) 0 0
\(955\) 2.65541 0.0859271
\(956\) 0 0
\(957\) −59.5097 −1.92367
\(958\) 0 0
\(959\) −7.35809 −0.237605
\(960\) 0 0
\(961\) −27.4846 −0.886601
\(962\) 0 0
\(963\) −65.3356 −2.10541
\(964\) 0 0
\(965\) 3.15024 0.101410
\(966\) 0 0
\(967\) −49.5491 −1.59339 −0.796696 0.604380i \(-0.793421\pi\)
−0.796696 + 0.604380i \(0.793421\pi\)
\(968\) 0 0
\(969\) 0.762129 0.0244831
\(970\) 0 0
\(971\) 9.32257 0.299175 0.149588 0.988748i \(-0.452205\pi\)
0.149588 + 0.988748i \(0.452205\pi\)
\(972\) 0 0
\(973\) −33.6833 −1.07984
\(974\) 0 0
\(975\) −45.7928 −1.46654
\(976\) 0 0
\(977\) −33.7294 −1.07910 −0.539550 0.841954i \(-0.681406\pi\)
−0.539550 + 0.841954i \(0.681406\pi\)
\(978\) 0 0
\(979\) 2.78170 0.0889034
\(980\) 0 0
\(981\) −1.28338 −0.0409752
\(982\) 0 0
\(983\) 8.58538 0.273831 0.136915 0.990583i \(-0.456281\pi\)
0.136915 + 0.990583i \(0.456281\pi\)
\(984\) 0 0
\(985\) 7.62988 0.243108
\(986\) 0 0
\(987\) 53.7722 1.71159
\(988\) 0 0
\(989\) 1.46406 0.0465545
\(990\) 0 0
\(991\) −51.4087 −1.63305 −0.816525 0.577310i \(-0.804102\pi\)
−0.816525 + 0.577310i \(0.804102\pi\)
\(992\) 0 0
\(993\) −10.7848 −0.342244
\(994\) 0 0
\(995\) 5.07708 0.160954
\(996\) 0 0
\(997\) −5.24495 −0.166109 −0.0830546 0.996545i \(-0.526468\pi\)
−0.0830546 + 0.996545i \(0.526468\pi\)
\(998\) 0 0
\(999\) −3.03333 −0.0959703
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 1336.2.a.d.1.11 12
4.3 odd 2 2672.2.a.p.1.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1336.2.a.d.1.11 12 1.1 even 1 trivial
2672.2.a.p.1.2 12 4.3 odd 2