Properties

Label 8752.2.a.s.1.16
Level $8752$
Weight $2$
Character 8752.1
Self dual yes
Analytic conductor $69.885$
Analytic rank $1$
Dimension $18$
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] = [8752,2,Mod(1,8752)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8752, 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("8752.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8752 = 2^{4} \cdot 547 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8752.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: \(69.8850718490\)
Analytic rank: \(1\)
Dimension: \(18\)
Coefficient field: \(\mathbb{Q}[x]/(x^{18} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} - 137 x^{10} - 7703 x^{9} + 2068 x^{8} + 11068 x^{7} - 4274 x^{6} - 9021 x^{5} + \cdots + 328 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 547)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(-2.50138\) of defining polynomial
Character \(\chi\) \(=\) 8752.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.08733 q^{3} -3.57921 q^{5} -1.44216 q^{7} +6.53160 q^{9} +O(q^{10})\) \(q+3.08733 q^{3} -3.57921 q^{5} -1.44216 q^{7} +6.53160 q^{9} +5.34528 q^{11} -5.39279 q^{13} -11.0502 q^{15} -5.31413 q^{17} -2.56610 q^{19} -4.45242 q^{21} +6.63074 q^{23} +7.81075 q^{25} +10.9032 q^{27} +5.19544 q^{29} -4.77081 q^{31} +16.5026 q^{33} +5.16179 q^{35} -3.44033 q^{37} -16.6493 q^{39} +8.38934 q^{41} -5.49566 q^{43} -23.3780 q^{45} +0.427891 q^{47} -4.92018 q^{49} -16.4065 q^{51} -3.73075 q^{53} -19.1319 q^{55} -7.92240 q^{57} -2.87686 q^{59} -3.71286 q^{61} -9.41961 q^{63} +19.3019 q^{65} -6.57367 q^{67} +20.4713 q^{69} +11.1354 q^{71} -0.468824 q^{73} +24.1144 q^{75} -7.70874 q^{77} -5.80618 q^{79} +14.0670 q^{81} -0.338379 q^{83} +19.0204 q^{85} +16.0400 q^{87} +1.79829 q^{89} +7.77726 q^{91} -14.7291 q^{93} +9.18462 q^{95} -11.5278 q^{97} +34.9132 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 18 q + 10 q^{3} - 27 q^{5} + 11 q^{7} + 14 q^{9} - 2 q^{11} - 25 q^{13} - 9 q^{15} - 30 q^{17} - 4 q^{19} - 16 q^{21} + 26 q^{23} + 31 q^{25} + 37 q^{27} - 18 q^{29} + 5 q^{31} - 10 q^{33} + 9 q^{35} - 18 q^{37} - 7 q^{39} - 17 q^{41} - 8 q^{43} - 44 q^{45} + 52 q^{47} + 29 q^{49} - 19 q^{51} - 60 q^{53} - 11 q^{55} + 4 q^{57} + 8 q^{59} - 26 q^{61} + q^{63} - 6 q^{65} - 12 q^{67} - 38 q^{69} + q^{71} - 2 q^{73} + 17 q^{75} - 73 q^{77} - 18 q^{79} + 18 q^{81} + 43 q^{83} + 51 q^{85} - 3 q^{87} - 28 q^{89} + q^{91} - 60 q^{93} + 18 q^{95} - 34 q^{97} - 15 q^{99}+O(q^{100}) \) 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
\(3\) 3.08733 1.78247 0.891235 0.453541i \(-0.149840\pi\)
0.891235 + 0.453541i \(0.149840\pi\)
\(4\) 0 0
\(5\) −3.57921 −1.60067 −0.800336 0.599552i \(-0.795346\pi\)
−0.800336 + 0.599552i \(0.795346\pi\)
\(6\) 0 0
\(7\) −1.44216 −0.545085 −0.272542 0.962144i \(-0.587865\pi\)
−0.272542 + 0.962144i \(0.587865\pi\)
\(8\) 0 0
\(9\) 6.53160 2.17720
\(10\) 0 0
\(11\) 5.34528 1.61166 0.805831 0.592145i \(-0.201719\pi\)
0.805831 + 0.592145i \(0.201719\pi\)
\(12\) 0 0
\(13\) −5.39279 −1.49569 −0.747845 0.663873i \(-0.768911\pi\)
−0.747845 + 0.663873i \(0.768911\pi\)
\(14\) 0 0
\(15\) −11.0502 −2.85315
\(16\) 0 0
\(17\) −5.31413 −1.28887 −0.644433 0.764661i \(-0.722907\pi\)
−0.644433 + 0.764661i \(0.722907\pi\)
\(18\) 0 0
\(19\) −2.56610 −0.588704 −0.294352 0.955697i \(-0.595104\pi\)
−0.294352 + 0.955697i \(0.595104\pi\)
\(20\) 0 0
\(21\) −4.45242 −0.971598
\(22\) 0 0
\(23\) 6.63074 1.38260 0.691302 0.722566i \(-0.257037\pi\)
0.691302 + 0.722566i \(0.257037\pi\)
\(24\) 0 0
\(25\) 7.81075 1.56215
\(26\) 0 0
\(27\) 10.9032 2.09832
\(28\) 0 0
\(29\) 5.19544 0.964768 0.482384 0.875960i \(-0.339771\pi\)
0.482384 + 0.875960i \(0.339771\pi\)
\(30\) 0 0
\(31\) −4.77081 −0.856864 −0.428432 0.903574i \(-0.640934\pi\)
−0.428432 + 0.903574i \(0.640934\pi\)
\(32\) 0 0
\(33\) 16.5026 2.87274
\(34\) 0 0
\(35\) 5.16179 0.872502
\(36\) 0 0
\(37\) −3.44033 −0.565586 −0.282793 0.959181i \(-0.591261\pi\)
−0.282793 + 0.959181i \(0.591261\pi\)
\(38\) 0 0
\(39\) −16.6493 −2.66602
\(40\) 0 0
\(41\) 8.38934 1.31019 0.655097 0.755545i \(-0.272628\pi\)
0.655097 + 0.755545i \(0.272628\pi\)
\(42\) 0 0
\(43\) −5.49566 −0.838080 −0.419040 0.907968i \(-0.637633\pi\)
−0.419040 + 0.907968i \(0.637633\pi\)
\(44\) 0 0
\(45\) −23.3780 −3.48498
\(46\) 0 0
\(47\) 0.427891 0.0624143 0.0312072 0.999513i \(-0.490065\pi\)
0.0312072 + 0.999513i \(0.490065\pi\)
\(48\) 0 0
\(49\) −4.92018 −0.702882
\(50\) 0 0
\(51\) −16.4065 −2.29737
\(52\) 0 0
\(53\) −3.73075 −0.512458 −0.256229 0.966616i \(-0.582480\pi\)
−0.256229 + 0.966616i \(0.582480\pi\)
\(54\) 0 0
\(55\) −19.1319 −2.57974
\(56\) 0 0
\(57\) −7.92240 −1.04935
\(58\) 0 0
\(59\) −2.87686 −0.374535 −0.187268 0.982309i \(-0.559963\pi\)
−0.187268 + 0.982309i \(0.559963\pi\)
\(60\) 0 0
\(61\) −3.71286 −0.475383 −0.237691 0.971341i \(-0.576391\pi\)
−0.237691 + 0.971341i \(0.576391\pi\)
\(62\) 0 0
\(63\) −9.41961 −1.18676
\(64\) 0 0
\(65\) 19.3019 2.39411
\(66\) 0 0
\(67\) −6.57367 −0.803101 −0.401551 0.915837i \(-0.631529\pi\)
−0.401551 + 0.915837i \(0.631529\pi\)
\(68\) 0 0
\(69\) 20.4713 2.46445
\(70\) 0 0
\(71\) 11.1354 1.32153 0.660764 0.750594i \(-0.270232\pi\)
0.660764 + 0.750594i \(0.270232\pi\)
\(72\) 0 0
\(73\) −0.468824 −0.0548717 −0.0274359 0.999624i \(-0.508734\pi\)
−0.0274359 + 0.999624i \(0.508734\pi\)
\(74\) 0 0
\(75\) 24.1144 2.78449
\(76\) 0 0
\(77\) −7.70874 −0.878493
\(78\) 0 0
\(79\) −5.80618 −0.653246 −0.326623 0.945155i \(-0.605911\pi\)
−0.326623 + 0.945155i \(0.605911\pi\)
\(80\) 0 0
\(81\) 14.0670 1.56300
\(82\) 0 0
\(83\) −0.338379 −0.0371420 −0.0185710 0.999828i \(-0.505912\pi\)
−0.0185710 + 0.999828i \(0.505912\pi\)
\(84\) 0 0
\(85\) 19.0204 2.06305
\(86\) 0 0
\(87\) 16.0400 1.71967
\(88\) 0 0
\(89\) 1.79829 0.190618 0.0953091 0.995448i \(-0.469616\pi\)
0.0953091 + 0.995448i \(0.469616\pi\)
\(90\) 0 0
\(91\) 7.77726 0.815278
\(92\) 0 0
\(93\) −14.7291 −1.52733
\(94\) 0 0
\(95\) 9.18462 0.942322
\(96\) 0 0
\(97\) −11.5278 −1.17047 −0.585234 0.810864i \(-0.698997\pi\)
−0.585234 + 0.810864i \(0.698997\pi\)
\(98\) 0 0
\(99\) 34.9132 3.50891
\(100\) 0 0
\(101\) −6.15589 −0.612534 −0.306267 0.951946i \(-0.599080\pi\)
−0.306267 + 0.951946i \(0.599080\pi\)
\(102\) 0 0
\(103\) −16.0271 −1.57920 −0.789601 0.613621i \(-0.789712\pi\)
−0.789601 + 0.613621i \(0.789712\pi\)
\(104\) 0 0
\(105\) 15.9362 1.55521
\(106\) 0 0
\(107\) −0.286864 −0.0277322 −0.0138661 0.999904i \(-0.504414\pi\)
−0.0138661 + 0.999904i \(0.504414\pi\)
\(108\) 0 0
\(109\) −15.7934 −1.51273 −0.756366 0.654149i \(-0.773027\pi\)
−0.756366 + 0.654149i \(0.773027\pi\)
\(110\) 0 0
\(111\) −10.6214 −1.00814
\(112\) 0 0
\(113\) −12.9488 −1.21812 −0.609058 0.793126i \(-0.708452\pi\)
−0.609058 + 0.793126i \(0.708452\pi\)
\(114\) 0 0
\(115\) −23.7328 −2.21310
\(116\) 0 0
\(117\) −35.2235 −3.25642
\(118\) 0 0
\(119\) 7.66382 0.702541
\(120\) 0 0
\(121\) 17.5720 1.59746
\(122\) 0 0
\(123\) 25.9006 2.33538
\(124\) 0 0
\(125\) −10.0603 −0.899818
\(126\) 0 0
\(127\) 2.62312 0.232764 0.116382 0.993205i \(-0.462870\pi\)
0.116382 + 0.993205i \(0.462870\pi\)
\(128\) 0 0
\(129\) −16.9669 −1.49385
\(130\) 0 0
\(131\) −1.96563 −0.171738 −0.0858691 0.996306i \(-0.527367\pi\)
−0.0858691 + 0.996306i \(0.527367\pi\)
\(132\) 0 0
\(133\) 3.70073 0.320894
\(134\) 0 0
\(135\) −39.0249 −3.35873
\(136\) 0 0
\(137\) −13.8493 −1.18323 −0.591614 0.806221i \(-0.701509\pi\)
−0.591614 + 0.806221i \(0.701509\pi\)
\(138\) 0 0
\(139\) −3.30984 −0.280737 −0.140368 0.990099i \(-0.544829\pi\)
−0.140368 + 0.990099i \(0.544829\pi\)
\(140\) 0 0
\(141\) 1.32104 0.111252
\(142\) 0 0
\(143\) −28.8260 −2.41055
\(144\) 0 0
\(145\) −18.5956 −1.54428
\(146\) 0 0
\(147\) −15.1902 −1.25287
\(148\) 0 0
\(149\) 1.24371 0.101889 0.0509443 0.998701i \(-0.483777\pi\)
0.0509443 + 0.998701i \(0.483777\pi\)
\(150\) 0 0
\(151\) 19.7224 1.60499 0.802494 0.596660i \(-0.203506\pi\)
0.802494 + 0.596660i \(0.203506\pi\)
\(152\) 0 0
\(153\) −34.7098 −2.80612
\(154\) 0 0
\(155\) 17.0758 1.37156
\(156\) 0 0
\(157\) −11.3800 −0.908221 −0.454110 0.890945i \(-0.650043\pi\)
−0.454110 + 0.890945i \(0.650043\pi\)
\(158\) 0 0
\(159\) −11.5181 −0.913441
\(160\) 0 0
\(161\) −9.56258 −0.753637
\(162\) 0 0
\(163\) −2.36146 −0.184964 −0.0924819 0.995714i \(-0.529480\pi\)
−0.0924819 + 0.995714i \(0.529480\pi\)
\(164\) 0 0
\(165\) −59.0664 −4.59832
\(166\) 0 0
\(167\) −3.39282 −0.262544 −0.131272 0.991346i \(-0.541906\pi\)
−0.131272 + 0.991346i \(0.541906\pi\)
\(168\) 0 0
\(169\) 16.0822 1.23709
\(170\) 0 0
\(171\) −16.7607 −1.28173
\(172\) 0 0
\(173\) −24.1775 −1.83819 −0.919093 0.394042i \(-0.871077\pi\)
−0.919093 + 0.394042i \(0.871077\pi\)
\(174\) 0 0
\(175\) −11.2643 −0.851505
\(176\) 0 0
\(177\) −8.88182 −0.667598
\(178\) 0 0
\(179\) 9.73639 0.727732 0.363866 0.931451i \(-0.381457\pi\)
0.363866 + 0.931451i \(0.381457\pi\)
\(180\) 0 0
\(181\) −8.08741 −0.601133 −0.300566 0.953761i \(-0.597176\pi\)
−0.300566 + 0.953761i \(0.597176\pi\)
\(182\) 0 0
\(183\) −11.4628 −0.847355
\(184\) 0 0
\(185\) 12.3137 0.905318
\(186\) 0 0
\(187\) −28.4055 −2.07722
\(188\) 0 0
\(189\) −15.7242 −1.14377
\(190\) 0 0
\(191\) −23.4764 −1.69869 −0.849347 0.527835i \(-0.823004\pi\)
−0.849347 + 0.527835i \(0.823004\pi\)
\(192\) 0 0
\(193\) 0.846561 0.0609368 0.0304684 0.999536i \(-0.490300\pi\)
0.0304684 + 0.999536i \(0.490300\pi\)
\(194\) 0 0
\(195\) 59.5914 4.26743
\(196\) 0 0
\(197\) 24.5035 1.74580 0.872900 0.487900i \(-0.162237\pi\)
0.872900 + 0.487900i \(0.162237\pi\)
\(198\) 0 0
\(199\) 6.60963 0.468544 0.234272 0.972171i \(-0.424729\pi\)
0.234272 + 0.972171i \(0.424729\pi\)
\(200\) 0 0
\(201\) −20.2951 −1.43150
\(202\) 0 0
\(203\) −7.49265 −0.525881
\(204\) 0 0
\(205\) −30.0272 −2.09719
\(206\) 0 0
\(207\) 43.3093 3.01021
\(208\) 0 0
\(209\) −13.7165 −0.948792
\(210\) 0 0
\(211\) −1.49983 −0.103253 −0.0516264 0.998666i \(-0.516441\pi\)
−0.0516264 + 0.998666i \(0.516441\pi\)
\(212\) 0 0
\(213\) 34.3786 2.35558
\(214\) 0 0
\(215\) 19.6701 1.34149
\(216\) 0 0
\(217\) 6.88027 0.467063
\(218\) 0 0
\(219\) −1.44741 −0.0978072
\(220\) 0 0
\(221\) 28.6580 1.92774
\(222\) 0 0
\(223\) −22.0562 −1.47699 −0.738497 0.674257i \(-0.764464\pi\)
−0.738497 + 0.674257i \(0.764464\pi\)
\(224\) 0 0
\(225\) 51.0167 3.40111
\(226\) 0 0
\(227\) −9.69637 −0.643571 −0.321785 0.946813i \(-0.604283\pi\)
−0.321785 + 0.946813i \(0.604283\pi\)
\(228\) 0 0
\(229\) 12.6579 0.836460 0.418230 0.908341i \(-0.362651\pi\)
0.418230 + 0.908341i \(0.362651\pi\)
\(230\) 0 0
\(231\) −23.7994 −1.56589
\(232\) 0 0
\(233\) 12.4477 0.815473 0.407736 0.913100i \(-0.366318\pi\)
0.407736 + 0.913100i \(0.366318\pi\)
\(234\) 0 0
\(235\) −1.53151 −0.0999049
\(236\) 0 0
\(237\) −17.9256 −1.16439
\(238\) 0 0
\(239\) −1.95816 −0.126663 −0.0633314 0.997993i \(-0.520173\pi\)
−0.0633314 + 0.997993i \(0.520173\pi\)
\(240\) 0 0
\(241\) −17.4587 −1.12462 −0.562308 0.826928i \(-0.690086\pi\)
−0.562308 + 0.826928i \(0.690086\pi\)
\(242\) 0 0
\(243\) 10.7199 0.687679
\(244\) 0 0
\(245\) 17.6104 1.12508
\(246\) 0 0
\(247\) 13.8384 0.880519
\(248\) 0 0
\(249\) −1.04469 −0.0662044
\(250\) 0 0
\(251\) −3.61597 −0.228238 −0.114119 0.993467i \(-0.536405\pi\)
−0.114119 + 0.993467i \(0.536405\pi\)
\(252\) 0 0
\(253\) 35.4432 2.22829
\(254\) 0 0
\(255\) 58.7222 3.67733
\(256\) 0 0
\(257\) −10.0934 −0.629606 −0.314803 0.949157i \(-0.601939\pi\)
−0.314803 + 0.949157i \(0.601939\pi\)
\(258\) 0 0
\(259\) 4.96150 0.308293
\(260\) 0 0
\(261\) 33.9345 2.10049
\(262\) 0 0
\(263\) 5.71252 0.352249 0.176124 0.984368i \(-0.443644\pi\)
0.176124 + 0.984368i \(0.443644\pi\)
\(264\) 0 0
\(265\) 13.3531 0.820277
\(266\) 0 0
\(267\) 5.55191 0.339771
\(268\) 0 0
\(269\) −11.2319 −0.684820 −0.342410 0.939551i \(-0.611243\pi\)
−0.342410 + 0.939551i \(0.611243\pi\)
\(270\) 0 0
\(271\) 25.7123 1.56191 0.780954 0.624588i \(-0.214733\pi\)
0.780954 + 0.624588i \(0.214733\pi\)
\(272\) 0 0
\(273\) 24.0110 1.45321
\(274\) 0 0
\(275\) 41.7507 2.51766
\(276\) 0 0
\(277\) 10.7925 0.648461 0.324231 0.945978i \(-0.394895\pi\)
0.324231 + 0.945978i \(0.394895\pi\)
\(278\) 0 0
\(279\) −31.1611 −1.86556
\(280\) 0 0
\(281\) 0.316572 0.0188851 0.00944256 0.999955i \(-0.496994\pi\)
0.00944256 + 0.999955i \(0.496994\pi\)
\(282\) 0 0
\(283\) 30.9827 1.84173 0.920864 0.389883i \(-0.127485\pi\)
0.920864 + 0.389883i \(0.127485\pi\)
\(284\) 0 0
\(285\) 28.3559 1.67966
\(286\) 0 0
\(287\) −12.0988 −0.714167
\(288\) 0 0
\(289\) 11.2400 0.661175
\(290\) 0 0
\(291\) −35.5900 −2.08632
\(292\) 0 0
\(293\) −21.7659 −1.27158 −0.635789 0.771863i \(-0.719325\pi\)
−0.635789 + 0.771863i \(0.719325\pi\)
\(294\) 0 0
\(295\) 10.2969 0.599508
\(296\) 0 0
\(297\) 58.2807 3.38179
\(298\) 0 0
\(299\) −35.7582 −2.06795
\(300\) 0 0
\(301\) 7.92561 0.456825
\(302\) 0 0
\(303\) −19.0053 −1.09182
\(304\) 0 0
\(305\) 13.2891 0.760932
\(306\) 0 0
\(307\) −3.11297 −0.177667 −0.0888334 0.996046i \(-0.528314\pi\)
−0.0888334 + 0.996046i \(0.528314\pi\)
\(308\) 0 0
\(309\) −49.4811 −2.81488
\(310\) 0 0
\(311\) −27.3164 −1.54897 −0.774485 0.632592i \(-0.781991\pi\)
−0.774485 + 0.632592i \(0.781991\pi\)
\(312\) 0 0
\(313\) 13.8638 0.783630 0.391815 0.920044i \(-0.371847\pi\)
0.391815 + 0.920044i \(0.371847\pi\)
\(314\) 0 0
\(315\) 33.7148 1.89961
\(316\) 0 0
\(317\) −14.2600 −0.800923 −0.400461 0.916314i \(-0.631150\pi\)
−0.400461 + 0.916314i \(0.631150\pi\)
\(318\) 0 0
\(319\) 27.7711 1.55488
\(320\) 0 0
\(321\) −0.885643 −0.0494318
\(322\) 0 0
\(323\) 13.6366 0.758761
\(324\) 0 0
\(325\) −42.1217 −2.33649
\(326\) 0 0
\(327\) −48.7594 −2.69640
\(328\) 0 0
\(329\) −0.617087 −0.0340211
\(330\) 0 0
\(331\) 14.0211 0.770667 0.385334 0.922777i \(-0.374086\pi\)
0.385334 + 0.922777i \(0.374086\pi\)
\(332\) 0 0
\(333\) −22.4709 −1.23140
\(334\) 0 0
\(335\) 23.5285 1.28550
\(336\) 0 0
\(337\) −2.77663 −0.151253 −0.0756263 0.997136i \(-0.524096\pi\)
−0.0756263 + 0.997136i \(0.524096\pi\)
\(338\) 0 0
\(339\) −39.9771 −2.17126
\(340\) 0 0
\(341\) −25.5013 −1.38098
\(342\) 0 0
\(343\) 17.1908 0.928216
\(344\) 0 0
\(345\) −73.2710 −3.94478
\(346\) 0 0
\(347\) −34.2051 −1.83623 −0.918114 0.396317i \(-0.870288\pi\)
−0.918114 + 0.396317i \(0.870288\pi\)
\(348\) 0 0
\(349\) 31.8576 1.70530 0.852649 0.522485i \(-0.174995\pi\)
0.852649 + 0.522485i \(0.174995\pi\)
\(350\) 0 0
\(351\) −58.7987 −3.13844
\(352\) 0 0
\(353\) 19.7596 1.05170 0.525849 0.850578i \(-0.323748\pi\)
0.525849 + 0.850578i \(0.323748\pi\)
\(354\) 0 0
\(355\) −39.8559 −2.11533
\(356\) 0 0
\(357\) 23.6607 1.25226
\(358\) 0 0
\(359\) −6.12948 −0.323501 −0.161751 0.986832i \(-0.551714\pi\)
−0.161751 + 0.986832i \(0.551714\pi\)
\(360\) 0 0
\(361\) −12.4151 −0.653428
\(362\) 0 0
\(363\) 54.2506 2.84742
\(364\) 0 0
\(365\) 1.67802 0.0878316
\(366\) 0 0
\(367\) 13.4451 0.701828 0.350914 0.936408i \(-0.385871\pi\)
0.350914 + 0.936408i \(0.385871\pi\)
\(368\) 0 0
\(369\) 54.7958 2.85256
\(370\) 0 0
\(371\) 5.38034 0.279333
\(372\) 0 0
\(373\) 12.5335 0.648963 0.324481 0.945892i \(-0.394810\pi\)
0.324481 + 0.945892i \(0.394810\pi\)
\(374\) 0 0
\(375\) −31.0594 −1.60390
\(376\) 0 0
\(377\) −28.0179 −1.44299
\(378\) 0 0
\(379\) 8.91515 0.457940 0.228970 0.973433i \(-0.426464\pi\)
0.228970 + 0.973433i \(0.426464\pi\)
\(380\) 0 0
\(381\) 8.09843 0.414895
\(382\) 0 0
\(383\) 8.41053 0.429758 0.214879 0.976641i \(-0.431064\pi\)
0.214879 + 0.976641i \(0.431064\pi\)
\(384\) 0 0
\(385\) 27.5912 1.40618
\(386\) 0 0
\(387\) −35.8954 −1.82467
\(388\) 0 0
\(389\) −6.85721 −0.347674 −0.173837 0.984774i \(-0.555617\pi\)
−0.173837 + 0.984774i \(0.555617\pi\)
\(390\) 0 0
\(391\) −35.2366 −1.78199
\(392\) 0 0
\(393\) −6.06855 −0.306118
\(394\) 0 0
\(395\) 20.7815 1.04563
\(396\) 0 0
\(397\) −7.82424 −0.392688 −0.196344 0.980535i \(-0.562907\pi\)
−0.196344 + 0.980535i \(0.562907\pi\)
\(398\) 0 0
\(399\) 11.4254 0.571983
\(400\) 0 0
\(401\) −16.2178 −0.809879 −0.404939 0.914344i \(-0.632707\pi\)
−0.404939 + 0.914344i \(0.632707\pi\)
\(402\) 0 0
\(403\) 25.7280 1.28160
\(404\) 0 0
\(405\) −50.3488 −2.50185
\(406\) 0 0
\(407\) −18.3895 −0.911535
\(408\) 0 0
\(409\) 10.0577 0.497322 0.248661 0.968591i \(-0.420009\pi\)
0.248661 + 0.968591i \(0.420009\pi\)
\(410\) 0 0
\(411\) −42.7574 −2.10907
\(412\) 0 0
\(413\) 4.14889 0.204154
\(414\) 0 0
\(415\) 1.21113 0.0594521
\(416\) 0 0
\(417\) −10.2186 −0.500405
\(418\) 0 0
\(419\) −29.3701 −1.43482 −0.717412 0.696649i \(-0.754673\pi\)
−0.717412 + 0.696649i \(0.754673\pi\)
\(420\) 0 0
\(421\) −28.4591 −1.38701 −0.693507 0.720450i \(-0.743935\pi\)
−0.693507 + 0.720450i \(0.743935\pi\)
\(422\) 0 0
\(423\) 2.79481 0.135889
\(424\) 0 0
\(425\) −41.5074 −2.01340
\(426\) 0 0
\(427\) 5.35453 0.259124
\(428\) 0 0
\(429\) −88.9952 −4.29673
\(430\) 0 0
\(431\) 16.8736 0.812772 0.406386 0.913701i \(-0.366789\pi\)
0.406386 + 0.913701i \(0.366789\pi\)
\(432\) 0 0
\(433\) −17.5129 −0.841615 −0.420807 0.907150i \(-0.638253\pi\)
−0.420807 + 0.907150i \(0.638253\pi\)
\(434\) 0 0
\(435\) −57.4106 −2.75263
\(436\) 0 0
\(437\) −17.0151 −0.813945
\(438\) 0 0
\(439\) 1.19663 0.0571120 0.0285560 0.999592i \(-0.490909\pi\)
0.0285560 + 0.999592i \(0.490909\pi\)
\(440\) 0 0
\(441\) −32.1366 −1.53032
\(442\) 0 0
\(443\) 1.77986 0.0845637 0.0422819 0.999106i \(-0.486537\pi\)
0.0422819 + 0.999106i \(0.486537\pi\)
\(444\) 0 0
\(445\) −6.43646 −0.305117
\(446\) 0 0
\(447\) 3.83974 0.181614
\(448\) 0 0
\(449\) 35.7981 1.68942 0.844708 0.535227i \(-0.179774\pi\)
0.844708 + 0.535227i \(0.179774\pi\)
\(450\) 0 0
\(451\) 44.8433 2.11159
\(452\) 0 0
\(453\) 60.8896 2.86084
\(454\) 0 0
\(455\) −27.8364 −1.30499
\(456\) 0 0
\(457\) −9.01813 −0.421850 −0.210925 0.977502i \(-0.567648\pi\)
−0.210925 + 0.977502i \(0.567648\pi\)
\(458\) 0 0
\(459\) −57.9411 −2.70446
\(460\) 0 0
\(461\) −30.8599 −1.43729 −0.718646 0.695377i \(-0.755238\pi\)
−0.718646 + 0.695377i \(0.755238\pi\)
\(462\) 0 0
\(463\) 1.05816 0.0491767 0.0245884 0.999698i \(-0.492172\pi\)
0.0245884 + 0.999698i \(0.492172\pi\)
\(464\) 0 0
\(465\) 52.7185 2.44476
\(466\) 0 0
\(467\) 34.1893 1.58209 0.791047 0.611755i \(-0.209536\pi\)
0.791047 + 0.611755i \(0.209536\pi\)
\(468\) 0 0
\(469\) 9.48027 0.437758
\(470\) 0 0
\(471\) −35.1337 −1.61888
\(472\) 0 0
\(473\) −29.3758 −1.35070
\(474\) 0 0
\(475\) −20.0432 −0.919644
\(476\) 0 0
\(477\) −24.3678 −1.11572
\(478\) 0 0
\(479\) 30.8709 1.41053 0.705265 0.708944i \(-0.250828\pi\)
0.705265 + 0.708944i \(0.250828\pi\)
\(480\) 0 0
\(481\) 18.5530 0.845942
\(482\) 0 0
\(483\) −29.5228 −1.34334
\(484\) 0 0
\(485\) 41.2603 1.87354
\(486\) 0 0
\(487\) 23.2537 1.05373 0.526863 0.849950i \(-0.323368\pi\)
0.526863 + 0.849950i \(0.323368\pi\)
\(488\) 0 0
\(489\) −7.29061 −0.329693
\(490\) 0 0
\(491\) 9.10242 0.410786 0.205393 0.978680i \(-0.434153\pi\)
0.205393 + 0.978680i \(0.434153\pi\)
\(492\) 0 0
\(493\) −27.6092 −1.24346
\(494\) 0 0
\(495\) −124.962 −5.61662
\(496\) 0 0
\(497\) −16.0590 −0.720345
\(498\) 0 0
\(499\) −25.0594 −1.12181 −0.560907 0.827879i \(-0.689548\pi\)
−0.560907 + 0.827879i \(0.689548\pi\)
\(500\) 0 0
\(501\) −10.4748 −0.467978
\(502\) 0 0
\(503\) 7.25454 0.323464 0.161732 0.986835i \(-0.448292\pi\)
0.161732 + 0.986835i \(0.448292\pi\)
\(504\) 0 0
\(505\) 22.0332 0.980467
\(506\) 0 0
\(507\) 49.6509 2.20507
\(508\) 0 0
\(509\) 1.63509 0.0724740 0.0362370 0.999343i \(-0.488463\pi\)
0.0362370 + 0.999343i \(0.488463\pi\)
\(510\) 0 0
\(511\) 0.676119 0.0299098
\(512\) 0 0
\(513\) −27.9788 −1.23529
\(514\) 0 0
\(515\) 57.3645 2.52778
\(516\) 0 0
\(517\) 2.28720 0.100591
\(518\) 0 0
\(519\) −74.6441 −3.27651
\(520\) 0 0
\(521\) 32.4953 1.42365 0.711823 0.702359i \(-0.247870\pi\)
0.711823 + 0.702359i \(0.247870\pi\)
\(522\) 0 0
\(523\) −42.1414 −1.84272 −0.921358 0.388714i \(-0.872919\pi\)
−0.921358 + 0.388714i \(0.872919\pi\)
\(524\) 0 0
\(525\) −34.7767 −1.51778
\(526\) 0 0
\(527\) 25.3527 1.10438
\(528\) 0 0
\(529\) 20.9667 0.911595
\(530\) 0 0
\(531\) −18.7905 −0.815439
\(532\) 0 0
\(533\) −45.2419 −1.95964
\(534\) 0 0
\(535\) 1.02675 0.0443901
\(536\) 0 0
\(537\) 30.0595 1.29716
\(538\) 0 0
\(539\) −26.2997 −1.13281
\(540\) 0 0
\(541\) 43.3831 1.86519 0.932593 0.360930i \(-0.117541\pi\)
0.932593 + 0.360930i \(0.117541\pi\)
\(542\) 0 0
\(543\) −24.9685 −1.07150
\(544\) 0 0
\(545\) 56.5278 2.42139
\(546\) 0 0
\(547\) 1.00000 0.0427569
\(548\) 0 0
\(549\) −24.2509 −1.03500
\(550\) 0 0
\(551\) −13.3320 −0.567963
\(552\) 0 0
\(553\) 8.37344 0.356075
\(554\) 0 0
\(555\) 38.0163 1.61370
\(556\) 0 0
\(557\) −26.5599 −1.12538 −0.562689 0.826669i \(-0.690233\pi\)
−0.562689 + 0.826669i \(0.690233\pi\)
\(558\) 0 0
\(559\) 29.6369 1.25351
\(560\) 0 0
\(561\) −87.6972 −3.70258
\(562\) 0 0
\(563\) 41.6328 1.75461 0.877306 0.479931i \(-0.159338\pi\)
0.877306 + 0.479931i \(0.159338\pi\)
\(564\) 0 0
\(565\) 46.3463 1.94980
\(566\) 0 0
\(567\) −20.2869 −0.851969
\(568\) 0 0
\(569\) 15.6265 0.655097 0.327549 0.944834i \(-0.393777\pi\)
0.327549 + 0.944834i \(0.393777\pi\)
\(570\) 0 0
\(571\) −37.2288 −1.55798 −0.778988 0.627038i \(-0.784267\pi\)
−0.778988 + 0.627038i \(0.784267\pi\)
\(572\) 0 0
\(573\) −72.4794 −3.02787
\(574\) 0 0
\(575\) 51.7911 2.15984
\(576\) 0 0
\(577\) −9.81210 −0.408483 −0.204242 0.978921i \(-0.565473\pi\)
−0.204242 + 0.978921i \(0.565473\pi\)
\(578\) 0 0
\(579\) 2.61361 0.108618
\(580\) 0 0
\(581\) 0.487997 0.0202455
\(582\) 0 0
\(583\) −19.9419 −0.825909
\(584\) 0 0
\(585\) 126.072 5.21246
\(586\) 0 0
\(587\) −9.72834 −0.401532 −0.200766 0.979639i \(-0.564343\pi\)
−0.200766 + 0.979639i \(0.564343\pi\)
\(588\) 0 0
\(589\) 12.2424 0.504439
\(590\) 0 0
\(591\) 75.6503 3.11184
\(592\) 0 0
\(593\) 28.2791 1.16128 0.580642 0.814159i \(-0.302802\pi\)
0.580642 + 0.814159i \(0.302802\pi\)
\(594\) 0 0
\(595\) −27.4304 −1.12454
\(596\) 0 0
\(597\) 20.4061 0.835166
\(598\) 0 0
\(599\) −32.5093 −1.32830 −0.664148 0.747601i \(-0.731205\pi\)
−0.664148 + 0.747601i \(0.731205\pi\)
\(600\) 0 0
\(601\) 5.09124 0.207676 0.103838 0.994594i \(-0.466888\pi\)
0.103838 + 0.994594i \(0.466888\pi\)
\(602\) 0 0
\(603\) −42.9366 −1.74851
\(604\) 0 0
\(605\) −62.8940 −2.55700
\(606\) 0 0
\(607\) −26.8978 −1.09175 −0.545875 0.837867i \(-0.683803\pi\)
−0.545875 + 0.837867i \(0.683803\pi\)
\(608\) 0 0
\(609\) −23.1323 −0.937367
\(610\) 0 0
\(611\) −2.30753 −0.0933525
\(612\) 0 0
\(613\) −6.32992 −0.255663 −0.127832 0.991796i \(-0.540802\pi\)
−0.127832 + 0.991796i \(0.540802\pi\)
\(614\) 0 0
\(615\) −92.7039 −3.73818
\(616\) 0 0
\(617\) −12.5277 −0.504344 −0.252172 0.967682i \(-0.581145\pi\)
−0.252172 + 0.967682i \(0.581145\pi\)
\(618\) 0 0
\(619\) −30.6964 −1.23379 −0.616897 0.787044i \(-0.711611\pi\)
−0.616897 + 0.787044i \(0.711611\pi\)
\(620\) 0 0
\(621\) 72.2964 2.90115
\(622\) 0 0
\(623\) −2.59342 −0.103903
\(624\) 0 0
\(625\) −3.04591 −0.121837
\(626\) 0 0
\(627\) −42.3474 −1.69119
\(628\) 0 0
\(629\) 18.2824 0.728965
\(630\) 0 0
\(631\) −23.2190 −0.924334 −0.462167 0.886793i \(-0.652928\pi\)
−0.462167 + 0.886793i \(0.652928\pi\)
\(632\) 0 0
\(633\) −4.63048 −0.184045
\(634\) 0 0
\(635\) −9.38870 −0.372579
\(636\) 0 0
\(637\) 26.5335 1.05129
\(638\) 0 0
\(639\) 72.7319 2.87723
\(640\) 0 0
\(641\) 10.2964 0.406681 0.203341 0.979108i \(-0.434820\pi\)
0.203341 + 0.979108i \(0.434820\pi\)
\(642\) 0 0
\(643\) 49.1690 1.93903 0.969517 0.245022i \(-0.0787952\pi\)
0.969517 + 0.245022i \(0.0787952\pi\)
\(644\) 0 0
\(645\) 60.7281 2.39117
\(646\) 0 0
\(647\) 12.1501 0.477668 0.238834 0.971060i \(-0.423235\pi\)
0.238834 + 0.971060i \(0.423235\pi\)
\(648\) 0 0
\(649\) −15.3776 −0.603625
\(650\) 0 0
\(651\) 21.2417 0.832527
\(652\) 0 0
\(653\) 31.9760 1.25132 0.625658 0.780097i \(-0.284830\pi\)
0.625658 + 0.780097i \(0.284830\pi\)
\(654\) 0 0
\(655\) 7.03541 0.274896
\(656\) 0 0
\(657\) −3.06217 −0.119467
\(658\) 0 0
\(659\) 16.5352 0.644120 0.322060 0.946719i \(-0.395625\pi\)
0.322060 + 0.946719i \(0.395625\pi\)
\(660\) 0 0
\(661\) 4.40089 0.171175 0.0855873 0.996331i \(-0.472723\pi\)
0.0855873 + 0.996331i \(0.472723\pi\)
\(662\) 0 0
\(663\) 88.4766 3.43615
\(664\) 0 0
\(665\) −13.2457 −0.513645
\(666\) 0 0
\(667\) 34.4496 1.33389
\(668\) 0 0
\(669\) −68.0948 −2.63270
\(670\) 0 0
\(671\) −19.8463 −0.766156
\(672\) 0 0
\(673\) −28.2504 −1.08897 −0.544487 0.838769i \(-0.683276\pi\)
−0.544487 + 0.838769i \(0.683276\pi\)
\(674\) 0 0
\(675\) 85.1623 3.27790
\(676\) 0 0
\(677\) 15.1494 0.582240 0.291120 0.956687i \(-0.405972\pi\)
0.291120 + 0.956687i \(0.405972\pi\)
\(678\) 0 0
\(679\) 16.6249 0.638004
\(680\) 0 0
\(681\) −29.9359 −1.14715
\(682\) 0 0
\(683\) 0.816285 0.0312343 0.0156171 0.999878i \(-0.495029\pi\)
0.0156171 + 0.999878i \(0.495029\pi\)
\(684\) 0 0
\(685\) 49.5697 1.89396
\(686\) 0 0
\(687\) 39.0792 1.49097
\(688\) 0 0
\(689\) 20.1191 0.766478
\(690\) 0 0
\(691\) −18.8709 −0.717882 −0.358941 0.933360i \(-0.616862\pi\)
−0.358941 + 0.933360i \(0.616862\pi\)
\(692\) 0 0
\(693\) −50.3504 −1.91266
\(694\) 0 0
\(695\) 11.8466 0.449367
\(696\) 0 0
\(697\) −44.5820 −1.68866
\(698\) 0 0
\(699\) 38.4300 1.45356
\(700\) 0 0
\(701\) 29.2476 1.10467 0.552334 0.833623i \(-0.313737\pi\)
0.552334 + 0.833623i \(0.313737\pi\)
\(702\) 0 0
\(703\) 8.82823 0.332963
\(704\) 0 0
\(705\) −4.72828 −0.178077
\(706\) 0 0
\(707\) 8.87778 0.333883
\(708\) 0 0
\(709\) −5.19560 −0.195125 −0.0975625 0.995229i \(-0.531105\pi\)
−0.0975625 + 0.995229i \(0.531105\pi\)
\(710\) 0 0
\(711\) −37.9237 −1.42225
\(712\) 0 0
\(713\) −31.6340 −1.18470
\(714\) 0 0
\(715\) 103.174 3.85850
\(716\) 0 0
\(717\) −6.04548 −0.225773
\(718\) 0 0
\(719\) 25.5079 0.951286 0.475643 0.879638i \(-0.342215\pi\)
0.475643 + 0.879638i \(0.342215\pi\)
\(720\) 0 0
\(721\) 23.1137 0.860799
\(722\) 0 0
\(723\) −53.9009 −2.00459
\(724\) 0 0
\(725\) 40.5803 1.50711
\(726\) 0 0
\(727\) −18.2744 −0.677761 −0.338881 0.940829i \(-0.610048\pi\)
−0.338881 + 0.940829i \(0.610048\pi\)
\(728\) 0 0
\(729\) −9.10533 −0.337234
\(730\) 0 0
\(731\) 29.2046 1.08017
\(732\) 0 0
\(733\) 11.4034 0.421195 0.210597 0.977573i \(-0.432459\pi\)
0.210597 + 0.977573i \(0.432459\pi\)
\(734\) 0 0
\(735\) 54.3690 2.00543
\(736\) 0 0
\(737\) −35.1381 −1.29433
\(738\) 0 0
\(739\) 22.2708 0.819246 0.409623 0.912255i \(-0.365660\pi\)
0.409623 + 0.912255i \(0.365660\pi\)
\(740\) 0 0
\(741\) 42.7238 1.56950
\(742\) 0 0
\(743\) −37.4127 −1.37254 −0.686270 0.727347i \(-0.740753\pi\)
−0.686270 + 0.727347i \(0.740753\pi\)
\(744\) 0 0
\(745\) −4.45150 −0.163090
\(746\) 0 0
\(747\) −2.21016 −0.0808655
\(748\) 0 0
\(749\) 0.413703 0.0151164
\(750\) 0 0
\(751\) −20.9303 −0.763758 −0.381879 0.924212i \(-0.624723\pi\)
−0.381879 + 0.924212i \(0.624723\pi\)
\(752\) 0 0
\(753\) −11.1637 −0.406828
\(754\) 0 0
\(755\) −70.5907 −2.56906
\(756\) 0 0
\(757\) 1.08103 0.0392907 0.0196454 0.999807i \(-0.493746\pi\)
0.0196454 + 0.999807i \(0.493746\pi\)
\(758\) 0 0
\(759\) 109.425 3.97186
\(760\) 0 0
\(761\) −38.7659 −1.40526 −0.702631 0.711554i \(-0.747992\pi\)
−0.702631 + 0.711554i \(0.747992\pi\)
\(762\) 0 0
\(763\) 22.7766 0.824567
\(764\) 0 0
\(765\) 124.234 4.49168
\(766\) 0 0
\(767\) 15.5143 0.560189
\(768\) 0 0
\(769\) −48.8853 −1.76285 −0.881424 0.472326i \(-0.843415\pi\)
−0.881424 + 0.472326i \(0.843415\pi\)
\(770\) 0 0
\(771\) −31.1615 −1.12225
\(772\) 0 0
\(773\) 7.06901 0.254255 0.127127 0.991886i \(-0.459424\pi\)
0.127127 + 0.991886i \(0.459424\pi\)
\(774\) 0 0
\(775\) −37.2637 −1.33855
\(776\) 0 0
\(777\) 15.3178 0.549523
\(778\) 0 0
\(779\) −21.5279 −0.771316
\(780\) 0 0
\(781\) 59.5218 2.12986
\(782\) 0 0
\(783\) 56.6470 2.02440
\(784\) 0 0
\(785\) 40.7313 1.45376
\(786\) 0 0
\(787\) −30.3453 −1.08169 −0.540847 0.841121i \(-0.681896\pi\)
−0.540847 + 0.841121i \(0.681896\pi\)
\(788\) 0 0
\(789\) 17.6364 0.627873
\(790\) 0 0
\(791\) 18.6742 0.663977
\(792\) 0 0
\(793\) 20.0226 0.711025
\(794\) 0 0
\(795\) 41.2256 1.46212
\(796\) 0 0
\(797\) −36.9392 −1.30845 −0.654227 0.756299i \(-0.727006\pi\)
−0.654227 + 0.756299i \(0.727006\pi\)
\(798\) 0 0
\(799\) −2.27387 −0.0804437
\(800\) 0 0
\(801\) 11.7457 0.415014
\(802\) 0 0
\(803\) −2.50600 −0.0884347
\(804\) 0 0
\(805\) 34.2265 1.20633
\(806\) 0 0
\(807\) −34.6765 −1.22067
\(808\) 0 0
\(809\) −35.3975 −1.24451 −0.622255 0.782815i \(-0.713783\pi\)
−0.622255 + 0.782815i \(0.713783\pi\)
\(810\) 0 0
\(811\) 26.8628 0.943281 0.471641 0.881791i \(-0.343662\pi\)
0.471641 + 0.881791i \(0.343662\pi\)
\(812\) 0 0
\(813\) 79.3822 2.78406
\(814\) 0 0
\(815\) 8.45217 0.296066
\(816\) 0 0
\(817\) 14.1024 0.493381
\(818\) 0 0
\(819\) 50.7979 1.77502
\(820\) 0 0
\(821\) 33.5285 1.17015 0.585077 0.810978i \(-0.301064\pi\)
0.585077 + 0.810978i \(0.301064\pi\)
\(822\) 0 0
\(823\) 17.8795 0.623240 0.311620 0.950207i \(-0.399128\pi\)
0.311620 + 0.950207i \(0.399128\pi\)
\(824\) 0 0
\(825\) 128.898 4.48765
\(826\) 0 0
\(827\) −23.8898 −0.830729 −0.415364 0.909655i \(-0.636346\pi\)
−0.415364 + 0.909655i \(0.636346\pi\)
\(828\) 0 0
\(829\) 29.2441 1.01569 0.507844 0.861449i \(-0.330443\pi\)
0.507844 + 0.861449i \(0.330443\pi\)
\(830\) 0 0
\(831\) 33.3201 1.15586
\(832\) 0 0
\(833\) 26.1465 0.905921
\(834\) 0 0
\(835\) 12.1436 0.420248
\(836\) 0 0
\(837\) −52.0172 −1.79798
\(838\) 0 0
\(839\) 50.1966 1.73298 0.866489 0.499196i \(-0.166371\pi\)
0.866489 + 0.499196i \(0.166371\pi\)
\(840\) 0 0
\(841\) −2.00744 −0.0692221
\(842\) 0 0
\(843\) 0.977363 0.0336622
\(844\) 0 0
\(845\) −57.5614 −1.98017
\(846\) 0 0
\(847\) −25.3416 −0.870749
\(848\) 0 0
\(849\) 95.6537 3.28283
\(850\) 0 0
\(851\) −22.8119 −0.781982
\(852\) 0 0
\(853\) 48.6698 1.66642 0.833212 0.552954i \(-0.186499\pi\)
0.833212 + 0.552954i \(0.186499\pi\)
\(854\) 0 0
\(855\) 59.9903 2.05162
\(856\) 0 0
\(857\) 21.2395 0.725528 0.362764 0.931881i \(-0.381833\pi\)
0.362764 + 0.931881i \(0.381833\pi\)
\(858\) 0 0
\(859\) −16.2261 −0.553626 −0.276813 0.960924i \(-0.589278\pi\)
−0.276813 + 0.960924i \(0.589278\pi\)
\(860\) 0 0
\(861\) −37.3528 −1.27298
\(862\) 0 0
\(863\) 12.9330 0.440243 0.220122 0.975472i \(-0.429355\pi\)
0.220122 + 0.975472i \(0.429355\pi\)
\(864\) 0 0
\(865\) 86.5366 2.94233
\(866\) 0 0
\(867\) 34.7015 1.17853
\(868\) 0 0
\(869\) −31.0357 −1.05281
\(870\) 0 0
\(871\) 35.4504 1.20119
\(872\) 0 0
\(873\) −75.2948 −2.54834
\(874\) 0 0
\(875\) 14.5085 0.490477
\(876\) 0 0
\(877\) −4.82463 −0.162916 −0.0814581 0.996677i \(-0.525958\pi\)
−0.0814581 + 0.996677i \(0.525958\pi\)
\(878\) 0 0
\(879\) −67.1985 −2.26655
\(880\) 0 0
\(881\) 31.5509 1.06298 0.531489 0.847065i \(-0.321633\pi\)
0.531489 + 0.847065i \(0.321633\pi\)
\(882\) 0 0
\(883\) −40.4305 −1.36059 −0.680297 0.732937i \(-0.738149\pi\)
−0.680297 + 0.732937i \(0.738149\pi\)
\(884\) 0 0
\(885\) 31.7899 1.06861
\(886\) 0 0
\(887\) −30.6420 −1.02886 −0.514428 0.857534i \(-0.671996\pi\)
−0.514428 + 0.857534i \(0.671996\pi\)
\(888\) 0 0
\(889\) −3.78295 −0.126876
\(890\) 0 0
\(891\) 75.1921 2.51903
\(892\) 0 0
\(893\) −1.09801 −0.0367436
\(894\) 0 0
\(895\) −34.8486 −1.16486
\(896\) 0 0
\(897\) −110.397 −3.68606
\(898\) 0 0
\(899\) −24.7865 −0.826675
\(900\) 0 0
\(901\) 19.8257 0.660490
\(902\) 0 0
\(903\) 24.4690 0.814277
\(904\) 0 0
\(905\) 28.9466 0.962216
\(906\) 0 0
\(907\) −31.9314 −1.06026 −0.530132 0.847915i \(-0.677858\pi\)
−0.530132 + 0.847915i \(0.677858\pi\)
\(908\) 0 0
\(909\) −40.2079 −1.33361
\(910\) 0 0
\(911\) 0.929435 0.0307936 0.0153968 0.999881i \(-0.495099\pi\)
0.0153968 + 0.999881i \(0.495099\pi\)
\(912\) 0 0
\(913\) −1.80873 −0.0598603
\(914\) 0 0
\(915\) 41.0278 1.35634
\(916\) 0 0
\(917\) 2.83475 0.0936119
\(918\) 0 0
\(919\) 58.0653 1.91540 0.957698 0.287775i \(-0.0929155\pi\)
0.957698 + 0.287775i \(0.0929155\pi\)
\(920\) 0 0
\(921\) −9.61078 −0.316686
\(922\) 0 0
\(923\) −60.0508 −1.97660
\(924\) 0 0
\(925\) −26.8716 −0.883531
\(926\) 0 0
\(927\) −104.683 −3.43824
\(928\) 0 0
\(929\) −43.9547 −1.44211 −0.721054 0.692879i \(-0.756342\pi\)
−0.721054 + 0.692879i \(0.756342\pi\)
\(930\) 0 0
\(931\) 12.6257 0.413790
\(932\) 0 0
\(933\) −84.3347 −2.76099
\(934\) 0 0
\(935\) 101.669 3.32494
\(936\) 0 0
\(937\) −40.6735 −1.32875 −0.664373 0.747401i \(-0.731301\pi\)
−0.664373 + 0.747401i \(0.731301\pi\)
\(938\) 0 0
\(939\) 42.8022 1.39680
\(940\) 0 0
\(941\) 32.6479 1.06429 0.532146 0.846653i \(-0.321386\pi\)
0.532146 + 0.846653i \(0.321386\pi\)
\(942\) 0 0
\(943\) 55.6275 1.81148
\(944\) 0 0
\(945\) 56.2801 1.83079
\(946\) 0 0
\(947\) −23.2833 −0.756605 −0.378303 0.925682i \(-0.623492\pi\)
−0.378303 + 0.925682i \(0.623492\pi\)
\(948\) 0 0
\(949\) 2.52827 0.0820711
\(950\) 0 0
\(951\) −44.0254 −1.42762
\(952\) 0 0
\(953\) 9.41717 0.305052 0.152526 0.988299i \(-0.451259\pi\)
0.152526 + 0.988299i \(0.451259\pi\)
\(954\) 0 0
\(955\) 84.0270 2.71905
\(956\) 0 0
\(957\) 85.7384 2.77153
\(958\) 0 0
\(959\) 19.9729 0.644960
\(960\) 0 0
\(961\) −8.23933 −0.265785
\(962\) 0 0
\(963\) −1.87368 −0.0603785
\(964\) 0 0
\(965\) −3.03002 −0.0975398
\(966\) 0 0
\(967\) 14.0391 0.451467 0.225734 0.974189i \(-0.427522\pi\)
0.225734 + 0.974189i \(0.427522\pi\)
\(968\) 0 0
\(969\) 42.1007 1.35247
\(970\) 0 0
\(971\) −16.1363 −0.517838 −0.258919 0.965899i \(-0.583366\pi\)
−0.258919 + 0.965899i \(0.583366\pi\)
\(972\) 0 0
\(973\) 4.77331 0.153025
\(974\) 0 0
\(975\) −130.044 −4.16473
\(976\) 0 0
\(977\) 9.00176 0.287992 0.143996 0.989578i \(-0.454005\pi\)
0.143996 + 0.989578i \(0.454005\pi\)
\(978\) 0 0
\(979\) 9.61236 0.307212
\(980\) 0 0
\(981\) −103.156 −3.29352
\(982\) 0 0
\(983\) 50.8826 1.62290 0.811451 0.584421i \(-0.198678\pi\)
0.811451 + 0.584421i \(0.198678\pi\)
\(984\) 0 0
\(985\) −87.7031 −2.79445
\(986\) 0 0
\(987\) −1.90515 −0.0606416
\(988\) 0 0
\(989\) −36.4403 −1.15873
\(990\) 0 0
\(991\) 20.2975 0.644772 0.322386 0.946608i \(-0.395515\pi\)
0.322386 + 0.946608i \(0.395515\pi\)
\(992\) 0 0
\(993\) 43.2876 1.37369
\(994\) 0 0
\(995\) −23.6573 −0.749985
\(996\) 0 0
\(997\) −45.1716 −1.43060 −0.715300 0.698817i \(-0.753710\pi\)
−0.715300 + 0.698817i \(0.753710\pi\)
\(998\) 0 0
\(999\) −37.5106 −1.18678
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 8752.2.a.s.1.16 18
4.3 odd 2 547.2.a.b.1.18 18
12.11 even 2 4923.2.a.l.1.1 18
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
547.2.a.b.1.18 18 4.3 odd 2
4923.2.a.l.1.1 18 12.11 even 2
8752.2.a.s.1.16 18 1.1 even 1 trivial