Properties

Label 2736.2.bm.j.1855.1
Level $2736$
Weight $2$
Character 2736.1855
Analytic conductor $21.847$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2736,2,Mod(559,2736)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2736, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2736.559");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2736 = 2^{4} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2736.bm (of order \(6\), degree \(2\), minimal)

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: no
Analytic conductor: \(21.8470699930\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 912)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1855.1
Root \(-1.22474 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2736.1855
Dual form 2736.2.bm.j.559.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.00000 - 1.73205i) q^{5} -4.56048i q^{7} +O(q^{10})\) \(q+(-1.00000 - 1.73205i) q^{5} -4.56048i q^{7} -5.65685i q^{11} +(3.94949 + 2.28024i) q^{13} +(-1.00000 - 1.73205i) q^{17} +(4.00000 - 1.73205i) q^{19} +(4.89898 + 2.82843i) q^{23} +(0.500000 - 0.866025i) q^{25} +(1.89898 + 1.09638i) q^{29} +7.89898 q^{31} +(-7.89898 + 4.56048i) q^{35} -1.09638i q^{37} +(4.50000 - 2.59808i) q^{43} +(-7.89898 - 4.56048i) q^{47} -13.7980 q^{49} +(-10.8990 - 6.29253i) q^{53} +(-9.79796 + 5.65685i) q^{55} +(5.89898 + 10.2173i) q^{59} +(4.94949 - 8.57277i) q^{61} -9.12096i q^{65} +(-4.39898 + 7.61926i) q^{67} +(-2.00000 - 3.46410i) q^{71} +(3.39898 + 5.88721i) q^{73} -25.7980 q^{77} +(-0.0505103 - 0.0874863i) q^{79} +3.46410i q^{83} +(-2.00000 + 3.46410i) q^{85} +(1.89898 + 1.09638i) q^{89} +(10.3990 - 18.0116i) q^{91} +(-7.00000 - 5.19615i) q^{95} +(3.79796 - 2.19275i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} + 6 q^{13} - 4 q^{17} + 16 q^{19} + 2 q^{25} - 12 q^{29} + 12 q^{31} - 12 q^{35} + 18 q^{43} - 12 q^{47} - 16 q^{49} - 24 q^{53} + 4 q^{59} + 10 q^{61} + 2 q^{67} - 8 q^{71} - 6 q^{73} - 64 q^{77} - 10 q^{79} - 8 q^{85} - 12 q^{89} + 22 q^{91} - 28 q^{95} - 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2736\mathbb{Z}\right)^\times\).

\(n\) \(1009\) \(1217\) \(1711\) \(2053\)
\(\chi(n)\) \(e\left(\frac{5}{6}\right)\) \(1\) \(-1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) −1.00000 1.73205i −0.447214 0.774597i 0.550990 0.834512i \(-0.314250\pi\)
−0.998203 + 0.0599153i \(0.980917\pi\)
\(6\) 0 0
\(7\) 4.56048i 1.72370i −0.507164 0.861849i \(-0.669306\pi\)
0.507164 0.861849i \(-0.330694\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685i 1.70561i −0.522233 0.852803i \(-0.674901\pi\)
0.522233 0.852803i \(-0.325099\pi\)
\(12\) 0 0
\(13\) 3.94949 + 2.28024i 1.09539 + 0.632425i 0.935007 0.354630i \(-0.115393\pi\)
0.160385 + 0.987055i \(0.448727\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.00000 1.73205i −0.242536 0.420084i 0.718900 0.695113i \(-0.244646\pi\)
−0.961436 + 0.275029i \(0.911312\pi\)
\(18\) 0 0
\(19\) 4.00000 1.73205i 0.917663 0.397360i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.89898 + 2.82843i 1.02151 + 0.589768i 0.914540 0.404495i \(-0.132553\pi\)
0.106967 + 0.994263i \(0.465886\pi\)
\(24\) 0 0
\(25\) 0.500000 0.866025i 0.100000 0.173205i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.89898 + 1.09638i 0.352632 + 0.203592i 0.665844 0.746091i \(-0.268072\pi\)
−0.313212 + 0.949683i \(0.601405\pi\)
\(30\) 0 0
\(31\) 7.89898 1.41870 0.709349 0.704857i \(-0.248989\pi\)
0.709349 + 0.704857i \(0.248989\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.89898 + 4.56048i −1.33517 + 0.770861i
\(36\) 0 0
\(37\) 1.09638i 0.180243i −0.995931 0.0901216i \(-0.971274\pi\)
0.995931 0.0901216i \(-0.0287256\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(42\) 0 0
\(43\) 4.50000 2.59808i 0.686244 0.396203i −0.115960 0.993254i \(-0.536994\pi\)
0.802203 + 0.597051i \(0.203661\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.89898 4.56048i −1.15218 0.665214i −0.202766 0.979227i \(-0.564993\pi\)
−0.949419 + 0.314013i \(0.898326\pi\)
\(48\) 0 0
\(49\) −13.7980 −1.97114
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −10.8990 6.29253i −1.49709 0.864345i −0.497096 0.867696i \(-0.665600\pi\)
−0.999994 + 0.00335049i \(0.998934\pi\)
\(54\) 0 0
\(55\) −9.79796 + 5.65685i −1.32116 + 0.762770i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.89898 + 10.2173i 0.767982 + 1.33018i 0.938656 + 0.344856i \(0.112072\pi\)
−0.170674 + 0.985328i \(0.554594\pi\)
\(60\) 0 0
\(61\) 4.94949 8.57277i 0.633717 1.09763i −0.353068 0.935598i \(-0.614862\pi\)
0.986785 0.162033i \(-0.0518050\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 9.12096i 1.13132i
\(66\) 0 0
\(67\) −4.39898 + 7.61926i −0.537421 + 0.930840i 0.461621 + 0.887077i \(0.347268\pi\)
−0.999042 + 0.0437630i \(0.986065\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −2.00000 3.46410i −0.237356 0.411113i 0.722599 0.691268i \(-0.242948\pi\)
−0.959955 + 0.280155i \(0.909614\pi\)
\(72\) 0 0
\(73\) 3.39898 + 5.88721i 0.397820 + 0.689045i 0.993457 0.114209i \(-0.0364334\pi\)
−0.595636 + 0.803254i \(0.703100\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −25.7980 −2.93995
\(78\) 0 0
\(79\) −0.0505103 0.0874863i −0.00568285 0.00984298i 0.863170 0.504913i \(-0.168476\pi\)
−0.868853 + 0.495070i \(0.835142\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 3.46410i 0.380235i 0.981761 + 0.190117i \(0.0608868\pi\)
−0.981761 + 0.190117i \(0.939113\pi\)
\(84\) 0 0
\(85\) −2.00000 + 3.46410i −0.216930 + 0.375735i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.89898 + 1.09638i 0.201291 + 0.116216i 0.597258 0.802049i \(-0.296257\pi\)
−0.395966 + 0.918265i \(0.629590\pi\)
\(90\) 0 0
\(91\) 10.3990 18.0116i 1.09011 1.88812i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −7.00000 5.19615i −0.718185 0.533114i
\(96\) 0 0
\(97\) 3.79796 2.19275i 0.385624 0.222640i −0.294638 0.955609i \(-0.595199\pi\)
0.680262 + 0.732969i \(0.261866\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −6.00000 + 10.3923i −0.597022 + 1.03407i 0.396236 + 0.918149i \(0.370316\pi\)
−0.993258 + 0.115924i \(0.963017\pi\)
\(102\) 0 0
\(103\) −2.10102 −0.207020 −0.103510 0.994628i \(-0.533007\pi\)
−0.103510 + 0.994628i \(0.533007\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 8.00000 0.773389 0.386695 0.922208i \(-0.373617\pi\)
0.386695 + 0.922208i \(0.373617\pi\)
\(108\) 0 0
\(109\) −14.6969 + 8.48528i −1.40771 + 0.812743i −0.995167 0.0981950i \(-0.968693\pi\)
−0.412544 + 0.910938i \(0.635360\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 4.73545i 0.445474i −0.974879 0.222737i \(-0.928501\pi\)
0.974879 0.222737i \(-0.0714991\pi\)
\(114\) 0 0
\(115\) 11.3137i 1.05501i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −7.89898 + 4.56048i −0.724098 + 0.418058i
\(120\) 0 0
\(121\) −21.0000 −1.90909
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) 2.89898 5.02118i 0.257243 0.445558i −0.708259 0.705952i \(-0.750519\pi\)
0.965502 + 0.260395i \(0.0838526\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 6.92820i 1.04844 0.605320i 0.126231 0.992001i \(-0.459712\pi\)
0.922214 + 0.386681i \(0.126379\pi\)
\(132\) 0 0
\(133\) −7.89898 18.2419i −0.684928 1.58177i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −6.00000 + 10.3923i −0.512615 + 0.887875i 0.487278 + 0.873247i \(0.337990\pi\)
−0.999893 + 0.0146279i \(0.995344\pi\)
\(138\) 0 0
\(139\) 3.39898 + 1.96240i 0.288298 + 0.166449i 0.637174 0.770720i \(-0.280103\pi\)
−0.348876 + 0.937169i \(0.613437\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 12.8990 22.3417i 1.07867 1.86831i
\(144\) 0 0
\(145\) 4.38551i 0.364196i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 6.89898 + 11.9494i 0.565186 + 0.978932i 0.997032 + 0.0769842i \(0.0245291\pi\)
−0.431846 + 0.901947i \(0.642138\pi\)
\(150\) 0 0
\(151\) −21.7980 −1.77389 −0.886946 0.461872i \(-0.847178\pi\)
−0.886946 + 0.461872i \(0.847178\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −7.89898 13.6814i −0.634461 1.09892i
\(156\) 0 0
\(157\) 7.84847 + 13.5939i 0.626376 + 1.08492i 0.988273 + 0.152697i \(0.0487959\pi\)
−0.361897 + 0.932218i \(0.617871\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 12.8990 22.3417i 1.01658 1.76077i
\(162\) 0 0
\(163\) 15.5885i 1.22098i 0.792023 + 0.610491i \(0.209028\pi\)
−0.792023 + 0.610491i \(0.790972\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 3.00000 5.19615i 0.232147 0.402090i −0.726293 0.687386i \(-0.758758\pi\)
0.958440 + 0.285295i \(0.0920916\pi\)
\(168\) 0 0
\(169\) 3.89898 + 6.75323i 0.299921 + 0.519479i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.89898 1.09638i 0.144377 0.0833559i −0.426072 0.904689i \(-0.640103\pi\)
0.570448 + 0.821334i \(0.306769\pi\)
\(174\) 0 0
\(175\) −3.94949 2.28024i −0.298553 0.172370i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 25.7980 1.92823 0.964115 0.265485i \(-0.0855321\pi\)
0.964115 + 0.265485i \(0.0855321\pi\)
\(180\) 0 0
\(181\) 14.6969 + 8.48528i 1.09241 + 0.630706i 0.934218 0.356702i \(-0.116099\pi\)
0.158196 + 0.987408i \(0.449432\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.89898 + 1.09638i −0.139616 + 0.0806072i
\(186\) 0 0
\(187\) −9.79796 + 5.65685i −0.716498 + 0.413670i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 24.2487i 1.75458i −0.479965 0.877288i \(-0.659351\pi\)
0.479965 0.877288i \(-0.340649\pi\)
\(192\) 0 0
\(193\) −10.5000 + 6.06218i −0.755807 + 0.436365i −0.827788 0.561041i \(-0.810401\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.00000 0.142494 0.0712470 0.997459i \(-0.477302\pi\)
0.0712470 + 0.997459i \(0.477302\pi\)
\(198\) 0 0
\(199\) −11.0505 6.38002i −0.783350 0.452267i 0.0542663 0.998527i \(-0.482718\pi\)
−0.837616 + 0.546259i \(0.816051\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 5.00000 8.66025i 0.350931 0.607831i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −9.79796 22.6274i −0.677739 1.56517i
\(210\) 0 0
\(211\) 10.3990 + 18.0116i 0.715895 + 1.23997i 0.962613 + 0.270880i \(0.0873145\pi\)
−0.246718 + 0.969087i \(0.579352\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −9.00000 5.19615i −0.613795 0.354375i
\(216\) 0 0
\(217\) 36.0231i 2.44541i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9.12096i 0.613542i
\(222\) 0 0
\(223\) −2.94949 5.10867i −0.197512 0.342102i 0.750209 0.661201i \(-0.229953\pi\)
−0.947721 + 0.319099i \(0.896620\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 15.7980 1.04855 0.524274 0.851550i \(-0.324337\pi\)
0.524274 + 0.851550i \(0.324337\pi\)
\(228\) 0 0
\(229\) −7.69694 −0.508628 −0.254314 0.967122i \(-0.581850\pi\)
−0.254314 + 0.967122i \(0.581850\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 9.79796 + 16.9706i 0.641886 + 1.11178i 0.985011 + 0.172489i \(0.0551809\pi\)
−0.343126 + 0.939289i \(0.611486\pi\)
\(234\) 0 0
\(235\) 18.2419i 1.18997i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 21.7060i 1.40405i 0.712155 + 0.702023i \(0.247719\pi\)
−0.712155 + 0.702023i \(0.752281\pi\)
\(240\) 0 0
\(241\) −14.2980 8.25493i −0.921013 0.531747i −0.0370546 0.999313i \(-0.511798\pi\)
−0.883958 + 0.467566i \(0.845131\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 13.7980 + 23.8988i 0.881519 + 1.52684i
\(246\) 0 0
\(247\) 19.7474 + 2.28024i 1.25650 + 0.145088i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −0.797959 0.460702i −0.0503667 0.0290792i 0.474605 0.880199i \(-0.342591\pi\)
−0.524972 + 0.851120i \(0.675924\pi\)
\(252\) 0 0
\(253\) 16.0000 27.7128i 1.00591 1.74229i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 10.8990 + 6.29253i 0.679860 + 0.392517i 0.799802 0.600264i \(-0.204938\pi\)
−0.119943 + 0.992781i \(0.538271\pi\)
\(258\) 0 0
\(259\) −5.00000 −0.310685
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −15.7980 + 9.12096i −0.974144 + 0.562422i −0.900497 0.434862i \(-0.856797\pi\)
−0.0736468 + 0.997284i \(0.523464\pi\)
\(264\) 0 0
\(265\) 25.1701i 1.54619i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −18.7980 + 10.8530i −1.14613 + 0.661719i −0.947942 0.318444i \(-0.896840\pi\)
−0.198190 + 0.980164i \(0.563506\pi\)
\(270\) 0 0
\(271\) −6.79796 + 3.92480i −0.412947 + 0.238415i −0.692055 0.721845i \(-0.743294\pi\)
0.279108 + 0.960260i \(0.409961\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4.89898 2.82843i −0.295420 0.170561i
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 25.8990 + 14.9528i 1.54500 + 0.892008i 0.998511 + 0.0545441i \(0.0173705\pi\)
0.546492 + 0.837464i \(0.315963\pi\)
\(282\) 0 0
\(283\) 6.79796 3.92480i 0.404097 0.233305i −0.284153 0.958779i \(-0.591713\pi\)
0.688250 + 0.725473i \(0.258379\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.50000 11.2583i 0.382353 0.662255i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 27.3629i 1.59856i −0.600962 0.799278i \(-0.705216\pi\)
0.600962 0.799278i \(-0.294784\pi\)
\(294\) 0 0
\(295\) 11.7980 20.4347i 0.686904 1.18975i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 12.8990 + 22.3417i 0.745967 + 1.29205i
\(300\) 0 0
\(301\) −11.8485 20.5222i −0.682934 1.18288i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −19.7980 −1.13363
\(306\) 0 0
\(307\) −3.79796 6.57826i −0.216761 0.375441i 0.737055 0.675833i \(-0.236216\pi\)
−0.953816 + 0.300392i \(0.902883\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 12.2351i 0.693790i −0.937904 0.346895i \(-0.887236\pi\)
0.937904 0.346895i \(-0.112764\pi\)
\(312\) 0 0
\(313\) −10.7980 + 18.7026i −0.610337 + 1.05713i 0.380847 + 0.924638i \(0.375633\pi\)
−0.991184 + 0.132496i \(0.957701\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.79796 3.92480i −0.381811 0.220439i 0.296795 0.954941i \(-0.404082\pi\)
−0.678606 + 0.734502i \(0.737416\pi\)
\(318\) 0 0
\(319\) 6.20204 10.7423i 0.347248 0.601451i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −7.00000 5.19615i −0.389490 0.289122i
\(324\) 0 0
\(325\) 3.94949 2.28024i 0.219078 0.126485i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −20.7980 + 36.0231i −1.14663 + 1.98602i
\(330\) 0 0
\(331\) −20.7980 −1.14316 −0.571580 0.820547i \(-0.693669\pi\)
−0.571580 + 0.820547i \(0.693669\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 17.5959 0.961368
\(336\) 0 0
\(337\) 7.19694 4.15515i 0.392042 0.226346i −0.291002 0.956722i \(-0.593989\pi\)
0.683045 + 0.730377i \(0.260655\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 44.6834i 2.41974i
\(342\) 0 0
\(343\) 31.0019i 1.67395i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.89898 + 4.56048i −0.424039 + 0.244819i −0.696804 0.717262i \(-0.745395\pi\)
0.272765 + 0.962081i \(0.412062\pi\)
\(348\) 0 0
\(349\) 7.69694 0.412008 0.206004 0.978551i \(-0.433954\pi\)
0.206004 + 0.978551i \(0.433954\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −15.7980 −0.840841 −0.420420 0.907329i \(-0.638117\pi\)
−0.420420 + 0.907329i \(0.638117\pi\)
\(354\) 0 0
\(355\) −4.00000 + 6.92820i −0.212298 + 0.367711i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −9.79796 + 5.65685i −0.517116 + 0.298557i −0.735754 0.677249i \(-0.763172\pi\)
0.218638 + 0.975806i \(0.429839\pi\)
\(360\) 0 0
\(361\) 13.0000 13.8564i 0.684211 0.729285i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6.79796 11.7744i 0.355821 0.616301i
\(366\) 0 0
\(367\) −8.84847 5.10867i −0.461886 0.266670i 0.250951 0.968000i \(-0.419257\pi\)
−0.712837 + 0.701330i \(0.752590\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −28.6969 + 49.7046i −1.48987 + 2.58053i
\(372\) 0 0
\(373\) 22.0560i 1.14201i 0.820945 + 0.571007i \(0.193447\pi\)
−0.820945 + 0.571007i \(0.806553\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.00000 + 8.66025i 0.257513 + 0.446026i
\(378\) 0 0
\(379\) 32.5959 1.67434 0.837170 0.546943i \(-0.184209\pi\)
0.837170 + 0.546943i \(0.184209\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 2.89898 + 5.02118i 0.148131 + 0.256570i 0.930537 0.366199i \(-0.119341\pi\)
−0.782406 + 0.622769i \(0.786008\pi\)
\(384\) 0 0
\(385\) 25.7980 + 44.6834i 1.31479 + 2.27728i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −10.6969 + 18.5276i −0.542356 + 0.939389i 0.456412 + 0.889769i \(0.349134\pi\)
−0.998768 + 0.0496201i \(0.984199\pi\)
\(390\) 0 0
\(391\) 11.3137i 0.572159i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.101021 + 0.174973i −0.00508289 + 0.00880383i
\(396\) 0 0
\(397\) 7.94949 + 13.7689i 0.398973 + 0.691042i 0.993600 0.112960i \(-0.0360332\pi\)
−0.594626 + 0.804002i \(0.702700\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 13.8990 8.02458i 0.694082 0.400728i −0.111057 0.993814i \(-0.535424\pi\)
0.805139 + 0.593086i \(0.202090\pi\)
\(402\) 0 0
\(403\) 31.1969 + 18.0116i 1.55403 + 0.897220i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −6.20204 −0.307424
\(408\) 0 0
\(409\) −6.00000 3.46410i −0.296681 0.171289i 0.344270 0.938871i \(-0.388126\pi\)
−0.640951 + 0.767582i \(0.721460\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 46.5959 26.9022i 2.29284 1.32377i
\(414\) 0 0
\(415\) 6.00000 3.46410i 0.294528 0.170046i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 21.3561i 1.04331i −0.853156 0.521656i \(-0.825314\pi\)
0.853156 0.521656i \(-0.174686\pi\)
\(420\) 0 0
\(421\) −4.89898 + 2.82843i −0.238762 + 0.137849i −0.614607 0.788833i \(-0.710686\pi\)
0.375846 + 0.926682i \(0.377352\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −2.00000 −0.0970143
\(426\) 0 0
\(427\) −39.0959 22.5720i −1.89198 1.09234i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.89898 + 3.28913i −0.0914706 + 0.158432i −0.908130 0.418688i \(-0.862490\pi\)
0.816660 + 0.577120i \(0.195823\pi\)
\(432\) 0 0
\(433\) −19.1969 11.0834i −0.922546 0.532632i −0.0380996 0.999274i \(-0.512130\pi\)
−0.884446 + 0.466642i \(0.845464\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 24.4949 + 2.82843i 1.17175 + 0.135302i
\(438\) 0 0
\(439\) 3.84847 + 6.66574i 0.183677 + 0.318139i 0.943130 0.332424i \(-0.107866\pi\)
−0.759453 + 0.650563i \(0.774533\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.797959 + 0.460702i 0.0379122 + 0.0218886i 0.518836 0.854874i \(-0.326365\pi\)
−0.480924 + 0.876762i \(0.659699\pi\)
\(444\) 0 0
\(445\) 4.38551i 0.207893i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 6.00680i 0.283478i −0.989904 0.141739i \(-0.954731\pi\)
0.989904 0.141739i \(-0.0452694\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −41.5959 −1.95005
\(456\) 0 0
\(457\) 1.20204 0.0562291 0.0281146 0.999605i \(-0.491050\pi\)
0.0281146 + 0.999605i \(0.491050\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.79796 + 15.2385i 0.409762 + 0.709728i 0.994863 0.101232i \(-0.0322785\pi\)
−0.585101 + 0.810960i \(0.698945\pi\)
\(462\) 0 0
\(463\) 13.6814i 0.635830i −0.948119 0.317915i \(-0.897017\pi\)
0.948119 0.317915i \(-0.102983\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 39.0265i 1.80593i 0.429712 + 0.902966i \(0.358615\pi\)
−0.429712 + 0.902966i \(0.641385\pi\)
\(468\) 0 0
\(469\) 34.7474 + 20.0614i 1.60449 + 0.926352i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −14.6969 25.4558i −0.675766 1.17046i
\(474\) 0 0
\(475\) 0.500000 4.33013i 0.0229416 0.198680i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 2.20204 + 1.27135i 0.100614 + 0.0580894i 0.549463 0.835518i \(-0.314832\pi\)
−0.448849 + 0.893608i \(0.648166\pi\)
\(480\) 0 0
\(481\) 2.50000 4.33013i 0.113990 0.197437i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.59592 4.38551i −0.344913 0.199136i
\(486\) 0 0
\(487\) 17.7980 0.806503 0.403251 0.915089i \(-0.367880\pi\)
0.403251 + 0.915089i \(0.367880\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2.20204 + 1.27135i −0.0993767 + 0.0573752i −0.548865 0.835911i \(-0.684940\pi\)
0.449488 + 0.893286i \(0.351606\pi\)
\(492\) 0 0
\(493\) 4.38551i 0.197513i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −15.7980 + 9.12096i −0.708635 + 0.409131i
\(498\) 0 0
\(499\) −31.1969 + 18.0116i −1.39657 + 0.806308i −0.994031 0.109096i \(-0.965204\pi\)
−0.402536 + 0.915404i \(0.631871\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 5.69694 + 3.28913i 0.254014 + 0.146655i 0.621601 0.783334i \(-0.286483\pi\)
−0.367587 + 0.929989i \(0.619816\pi\)
\(504\) 0 0
\(505\) 24.0000 1.06799
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 22.1010 + 12.7600i 0.979611 + 0.565578i 0.902153 0.431417i \(-0.141986\pi\)
0.0774580 + 0.996996i \(0.475320\pi\)
\(510\) 0 0
\(511\) 26.8485 15.5010i 1.18771 0.685723i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2.10102 + 3.63907i 0.0925820 + 0.160357i
\(516\) 0 0
\(517\) −25.7980 + 44.6834i −1.13459 + 1.96517i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.5851i 0.551361i 0.961249 + 0.275681i \(0.0889032\pi\)
−0.961249 + 0.275681i \(0.911097\pi\)
\(522\) 0 0
\(523\) −6.29796 + 10.9084i −0.275391 + 0.476990i −0.970234 0.242171i \(-0.922140\pi\)
0.694843 + 0.719161i \(0.255474\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −7.89898 13.6814i −0.344085 0.595973i
\(528\) 0 0
\(529\) 4.50000 + 7.79423i 0.195652 + 0.338880i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −8.00000 13.8564i −0.345870 0.599065i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 78.0530i 3.36198i
\(540\) 0 0
\(541\) 15.9495 27.6253i 0.685722 1.18771i −0.287487 0.957784i \(-0.592820\pi\)
0.973209 0.229921i \(-0.0738468\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 29.3939 + 16.9706i 1.25910 + 0.726939i
\(546\) 0 0
\(547\) 19.5000 33.7750i 0.833760 1.44411i −0.0612764 0.998121i \(-0.519517\pi\)
0.895036 0.445993i \(-0.147150\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 9.49490 + 1.09638i 0.404496 + 0.0467072i
\(552\) 0 0
\(553\) −0.398979 + 0.230351i −0.0169663 + 0.00979552i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −12.0000 + 20.7846i −0.508456 + 0.880672i 0.491496 + 0.870880i \(0.336450\pi\)
−0.999952 + 0.00979220i \(0.996883\pi\)
\(558\) 0 0
\(559\) 23.6969 1.00227
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 4.40408 0.185610 0.0928050 0.995684i \(-0.470417\pi\)
0.0928050 + 0.995684i \(0.470417\pi\)
\(564\) 0 0
\(565\) −8.20204 + 4.73545i −0.345062 + 0.199222i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 25.1701i 1.05519i −0.849497 0.527593i \(-0.823095\pi\)
0.849497 0.527593i \(-0.176905\pi\)
\(570\) 0 0
\(571\) 21.2453i 0.889089i −0.895757 0.444544i \(-0.853366\pi\)
0.895757 0.444544i \(-0.146634\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 4.89898 2.82843i 0.204302 0.117954i
\(576\) 0 0
\(577\) −41.5959 −1.73166 −0.865830 0.500338i \(-0.833209\pi\)
−0.865830 + 0.500338i \(0.833209\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 15.7980 0.655410
\(582\) 0 0
\(583\) −35.5959 + 61.6539i −1.47423 + 2.55345i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 13.8990 8.02458i 0.573672 0.331210i −0.184942 0.982749i \(-0.559210\pi\)
0.758615 + 0.651540i \(0.225877\pi\)
\(588\) 0 0
\(589\) 31.5959 13.6814i 1.30189 0.563734i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −7.89898 + 13.6814i −0.324372 + 0.561829i −0.981385 0.192050i \(-0.938486\pi\)
0.657013 + 0.753879i \(0.271820\pi\)
\(594\) 0 0
\(595\) 15.7980 + 9.12096i 0.647653 + 0.373923i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −11.1010 + 19.2275i −0.453575 + 0.785616i −0.998605 0.0528011i \(-0.983185\pi\)
0.545030 + 0.838417i \(0.316518\pi\)
\(600\) 0 0
\(601\) 38.5658i 1.57313i 0.617506 + 0.786566i \(0.288143\pi\)
−0.617506 + 0.786566i \(0.711857\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 21.0000 + 36.3731i 0.853771 + 1.47878i
\(606\) 0 0
\(607\) −21.8990 −0.888852 −0.444426 0.895816i \(-0.646592\pi\)
−0.444426 + 0.895816i \(0.646592\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −20.7980 36.0231i −0.841395 1.45734i
\(612\) 0 0
\(613\) −2.79796 4.84621i −0.113008 0.195736i 0.803973 0.594665i \(-0.202715\pi\)
−0.916982 + 0.398929i \(0.869382\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.89898 5.02118i 0.116709 0.202145i −0.801753 0.597656i \(-0.796099\pi\)
0.918461 + 0.395511i \(0.129432\pi\)
\(618\) 0 0
\(619\) 1.38211i 0.0555515i −0.999614 0.0277758i \(-0.991158\pi\)
0.999614 0.0277758i \(-0.00884243\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 5.00000 8.66025i 0.200321 0.346966i
\(624\) 0 0
\(625\) 9.50000 + 16.4545i 0.380000 + 0.658179i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1.89898 + 1.09638i −0.0757173 + 0.0437154i
\(630\) 0 0
\(631\) 2.35357 + 1.35884i 0.0936942 + 0.0540944i 0.546115 0.837710i \(-0.316106\pi\)
−0.452421 + 0.891805i \(0.649439\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −11.5959 −0.460170
\(636\) 0 0
\(637\) −54.4949 31.4626i −2.15917 1.24660i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 30.7980 17.7812i 1.21645 0.702316i 0.252290 0.967652i \(-0.418816\pi\)
0.964156 + 0.265336i \(0.0854829\pi\)
\(642\) 0 0
\(643\) 24.0959 13.9118i 0.950250 0.548627i 0.0570916 0.998369i \(-0.481817\pi\)
0.893159 + 0.449742i \(0.148484\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 50.6902i 1.99284i −0.0845621 0.996418i \(-0.526949\pi\)
0.0845621 0.996418i \(-0.473051\pi\)
\(648\) 0 0
\(649\) 57.7980 33.3697i 2.26877 1.30987i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 4.40408 0.172345 0.0861725 0.996280i \(-0.472536\pi\)
0.0861725 + 0.996280i \(0.472536\pi\)
\(654\) 0 0
\(655\) −24.0000 13.8564i −0.937758 0.541415i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 10.6969 18.5276i 0.416694 0.721734i −0.578911 0.815391i \(-0.696522\pi\)
0.995605 + 0.0936563i \(0.0298555\pi\)
\(660\) 0 0
\(661\) 6.49490 + 3.74983i 0.252622 + 0.145852i 0.620964 0.783839i \(-0.286741\pi\)
−0.368342 + 0.929690i \(0.620074\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −23.6969 + 31.9233i −0.918928 + 1.23793i
\(666\) 0 0
\(667\) 6.20204 + 10.7423i 0.240144 + 0.415942i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −48.4949 27.9985i −1.87212 1.08087i
\(672\) 0 0
\(673\) 37.8659i 1.45962i 0.683648 + 0.729812i \(0.260392\pi\)
−0.683648 + 0.729812i \(0.739608\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 25.5201i 0.980816i −0.871493 0.490408i \(-0.836848\pi\)
0.871493 0.490408i \(-0.163152\pi\)
\(678\) 0 0
\(679\) −10.0000 17.3205i −0.383765 0.664700i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −16.2020 −0.619954 −0.309977 0.950744i \(-0.600321\pi\)
−0.309977 + 0.950744i \(0.600321\pi\)
\(684\) 0 0
\(685\) 24.0000 0.916993
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −28.6969 49.7046i −1.09327 1.89359i
\(690\) 0 0
\(691\) 24.2487i 0.922464i 0.887279 + 0.461232i \(0.152592\pi\)
−0.887279 + 0.461232i \(0.847408\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.84961i 0.297753i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 21.0000 + 36.3731i 0.793159 + 1.37379i 0.924002 + 0.382389i \(0.124898\pi\)
−0.130843 + 0.991403i \(0.541768\pi\)
\(702\) 0 0
\(703\) −1.89898 4.38551i −0.0716214 0.165402i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 47.3939 + 27.3629i 1.78243 + 1.02909i
\(708\) 0 0
\(709\) 4.15153 7.19066i 0.155914 0.270051i −0.777477 0.628911i \(-0.783501\pi\)
0.933391 + 0.358860i \(0.116834\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 38.6969 + 22.3417i 1.44921 + 0.836703i
\(714\) 0 0
\(715\) −51.5959 −1.92958
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 22.8990 13.2207i 0.853988 0.493050i −0.00800654 0.999968i \(-0.502549\pi\)
0.861994 + 0.506918i \(0.169215\pi\)
\(720\) 0 0
\(721\) 9.58166i 0.356840i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.89898 1.09638i 0.0705263 0.0407184i
\(726\) 0 0
\(727\) 21.3434 12.3226i 0.791582 0.457020i −0.0489374 0.998802i \(-0.515583\pi\)
0.840519 + 0.541782i \(0.182250\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −9.00000 5.19615i −0.332877 0.192187i
\(732\) 0 0
\(733\) −2.40408 −0.0887968 −0.0443984 0.999014i \(-0.514137\pi\)
−0.0443984 + 0.999014i \(0.514137\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 43.1010 + 24.8844i 1.58765 + 0.916628i
\(738\) 0 0
\(739\) 18.3990 10.6227i 0.676817 0.390761i −0.121837 0.992550i \(-0.538879\pi\)
0.798655 + 0.601789i \(0.205545\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 10.8990 + 18.8776i 0.399845 + 0.692551i 0.993706 0.112016i \(-0.0357307\pi\)
−0.593862 + 0.804567i \(0.702397\pi\)
\(744\) 0 0
\(745\) 13.7980 23.8988i 0.505518 0.875583i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 36.4838i 1.33309i
\(750\) 0 0
\(751\) 3.74745 6.49077i 0.136746 0.236852i −0.789517 0.613729i \(-0.789669\pi\)
0.926263 + 0.376877i \(0.123002\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 21.7980 + 37.7552i 0.793309 + 1.37405i
\(756\) 0 0
\(757\) 8.74745 + 15.1510i 0.317931 + 0.550673i 0.980056 0.198720i \(-0.0636785\pi\)
−0.662125 + 0.749394i \(0.730345\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.79796 −0.137676 −0.0688380 0.997628i \(-0.521929\pi\)
−0.0688380 + 0.997628i \(0.521929\pi\)
\(762\) 0 0
\(763\) 38.6969 + 67.0251i 1.40092 + 2.42647i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 53.8043i 1.94276i
\(768\) 0 0
\(769\) 23.0959 40.0033i 0.832860 1.44256i −0.0629008 0.998020i \(-0.520035\pi\)
0.895761 0.444536i \(-0.146632\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 35.6969 + 20.6096i 1.28393 + 0.741277i 0.977564 0.210637i \(-0.0675538\pi\)
0.306365 + 0.951914i \(0.400887\pi\)
\(774\) 0 0
\(775\) 3.94949 6.84072i 0.141870 0.245726i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −19.5959 + 11.3137i −0.701197 + 0.404836i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 15.6969 27.1879i 0.560248 0.970378i
\(786\) 0 0
\(787\) 4.59592 0.163827 0.0819134 0.996639i \(-0.473897\pi\)
0.0819134 + 0.996639i \(0.473897\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −21.5959 −0.767862
\(792\) 0 0
\(793\) 39.0959 22.5720i 1.38834 0.801557i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 38.3266i 1.35760i 0.734324 + 0.678799i \(0.237499\pi\)
−0.734324 + 0.678799i \(0.762501\pi\)
\(798\) 0 0
\(799\) 18.2419i 0.645352i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 33.3031 19.2275i 1.17524 0.678525i
\(804\) 0 0
\(805\) −51.5959 −1.81852
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 15.3939 0.541220 0.270610 0.962689i \(-0.412775\pi\)
0.270610 + 0.962689i \(0.412775\pi\)
\(810\) 0 0
\(811\) −4.00000 + 6.92820i −0.140459 + 0.243282i −0.927670 0.373402i \(-0.878191\pi\)
0.787211 + 0.616684i \(0.211524\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 27.0000 15.5885i 0.945769 0.546040i
\(816\) 0 0
\(817\) 13.5000 18.1865i 0.472305 0.636266i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.69694 + 9.86739i −0.198825 + 0.344374i −0.948148 0.317830i \(-0.897046\pi\)
0.749323 + 0.662205i \(0.230379\pi\)
\(822\) 0 0
\(823\) −20.3939 11.7744i −0.710886 0.410430i 0.100503 0.994937i \(-0.467955\pi\)
−0.811389 + 0.584507i \(0.801288\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −3.20204 + 5.54610i −0.111346 + 0.192857i −0.916313 0.400462i \(-0.868849\pi\)
0.804967 + 0.593319i \(0.202183\pi\)
\(828\) 0 0
\(829\) 2.71767i 0.0943886i −0.998886 0.0471943i \(-0.984972\pi\)
0.998886 0.0471943i \(-0.0150280\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 13.7980 + 23.8988i 0.478071 + 0.828043i
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −3.00000 5.19615i −0.103572 0.179391i 0.809582 0.587007i \(-0.199694\pi\)
−0.913154 + 0.407615i \(0.866360\pi\)
\(840\) 0 0
\(841\) −12.0959 20.9507i −0.417101 0.722439i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 7.79796 13.5065i 0.268258 0.464636i
\(846\) 0 0
\(847\) 95.7700i 3.29070i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 3.10102 5.37113i 0.106302 0.184120i
\(852\) 0 0
\(853\) −11.6464 20.1722i −0.398766 0.690683i 0.594808 0.803868i \(-0.297228\pi\)
−0.993574 + 0.113185i \(0.963895\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −7.40408 + 4.27475i −0.252919 + 0.146023i −0.621100 0.783731i \(-0.713314\pi\)
0.368181 + 0.929754i \(0.379981\pi\)
\(858\) 0 0
\(859\) −23.2980 13.4511i −0.794916 0.458945i 0.0467743 0.998905i \(-0.485106\pi\)
−0.841690 + 0.539960i \(0.818439\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11.7980 −0.401607 −0.200804 0.979632i \(-0.564355\pi\)
−0.200804 + 0.979632i \(0.564355\pi\)
\(864\) 0 0
\(865\) −3.79796 2.19275i −0.129134 0.0745558i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −0.494897 + 0.285729i −0.0167882 + 0.00969270i
\(870\) 0 0
\(871\) −34.7474 + 20.0614i −1.17737 + 0.679756i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 54.7257i 1.85007i
\(876\) 0 0
\(877\) −23.5454 + 13.5939i −0.795072 + 0.459035i −0.841745 0.539875i \(-0.818471\pi\)
0.0466731 + 0.998910i \(0.485138\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −15.7980 −0.532247 −0.266123 0.963939i \(-0.585743\pi\)
−0.266123 + 0.963939i \(0.585743\pi\)
\(882\) 0 0
\(883\) 38.2980 + 22.1113i 1.28883 + 0.744106i 0.978445 0.206506i \(-0.0662092\pi\)
0.310384 + 0.950611i \(0.399543\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −26.5959 + 46.0655i −0.893004 + 1.54673i −0.0567473 + 0.998389i \(0.518073\pi\)
−0.836256 + 0.548339i \(0.815260\pi\)
\(888\) 0 0
\(889\) −22.8990 13.2207i −0.768007 0.443409i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −39.4949 4.56048i −1.32165 0.152611i
\(894\) 0 0
\(895\) −25.7980 44.6834i −0.862331 1.49360i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 15.0000 + 8.66025i 0.500278 + 0.288836i
\(900\) 0 0
\(901\) 25.1701i 0.838538i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 33.9411i 1.12824i
\(906\) 0 0
\(907\) 11.5959 + 20.0847i 0.385036 + 0.666902i 0.991774 0.128000i \(-0.0408556\pi\)
−0.606738 + 0.794902i \(0.707522\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −15.5959 −0.516716 −0.258358 0.966049i \(-0.583181\pi\)
−0.258358 + 0.966049i \(0.583181\pi\)
\(912\) 0 0
\(913\) 19.5959 0.648530
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −31.5959 54.7257i −1.04339 1.80720i
\(918\) 0 0
\(919\) 25.9165i 0.854908i −0.904037 0.427454i \(-0.859411\pi\)
0.904037 0.427454i \(-0.140589\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 18.2419i 0.600440i
\(924\) 0 0
\(925\) −0.949490 0.548188i −0.0312190 0.0180243i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 3.89898 + 6.75323i 0.127921 + 0.221566i 0.922871 0.385109i \(-0.125836\pi\)
−0.794950 + 0.606675i \(0.792503\pi\)
\(930\) 0 0
\(931\) −55.1918 + 23.8988i −1.80884 + 0.783250i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 19.5959 + 11.3137i 0.640855 + 0.369998i
\(936\) 0 0
\(937\) 14.5000 25.1147i 0.473694 0.820463i −0.525852 0.850576i \(-0.676253\pi\)
0.999546 + 0.0301133i \(0.00958681\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −9.79796 5.65685i −0.319404 0.184408i 0.331723 0.943377i \(-0.392370\pi\)
−0.651127 + 0.758969i \(0.725703\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 42.0000 24.2487i 1.36482 0.787977i 0.374556 0.927204i \(-0.377795\pi\)
0.990260 + 0.139227i \(0.0444618\pi\)
\(948\) 0 0
\(949\) 31.0019i 1.00637i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.10102 0.635674i 0.0356656 0.0205915i −0.482061 0.876138i \(-0.660112\pi\)
0.517727 + 0.855546i \(0.326778\pi\)
\(954\) 0 0
\(955\) −42.0000 + 24.2487i −1.35909 + 0.784670i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 47.3939 + 27.3629i 1.53043 + 0.883593i
\(960\) 0 0
\(961\) 31.3939 1.01271
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 21.0000 + 12.1244i 0.676014 + 0.390297i
\(966\) 0 0
\(967\) −21.3434 + 12.3226i −0.686356 + 0.396268i −0.802246 0.596994i \(-0.796362\pi\)
0.115889 + 0.993262i \(0.463028\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −26.5959 46.0655i −0.853504 1.47831i −0.878026 0.478612i \(-0.841140\pi\)
0.0245227 0.999699i \(-0.492193\pi\)
\(972\) 0 0
\(973\) 8.94949 15.5010i 0.286907 0.496938i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 33.0197i 1.05639i 0.849122 + 0.528197i \(0.177132\pi\)
−0.849122 + 0.528197i \(0.822868\pi\)
\(978\) 0 0
\(979\) 6.20204 10.7423i 0.198218 0.343324i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −18.8990 32.7340i −0.602784 1.04405i −0.992398 0.123074i \(-0.960725\pi\)
0.389614 0.920978i \(-0.372609\pi\)
\(984\) 0 0
\(985\) −2.00000 3.46410i −0.0637253 0.110375i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 29.3939 0.934671
\(990\) 0 0
\(991\) 2.84847 + 4.93369i 0.0904846 + 0.156724i 0.907715 0.419587i \(-0.137825\pi\)
−0.817231 + 0.576311i \(0.804492\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 25.5201i 0.809040i
\(996\) 0 0
\(997\) −7.05051 + 12.2118i −0.223292 + 0.386753i −0.955806 0.293999i \(-0.905014\pi\)
0.732514 + 0.680752i \(0.238347\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2736.2.bm.j.1855.1 4
3.2 odd 2 912.2.bb.c.31.1 4
4.3 odd 2 2736.2.bm.i.1855.2 4
12.11 even 2 912.2.bb.d.31.2 yes 4
19.8 odd 6 2736.2.bm.i.559.1 4
57.8 even 6 912.2.bb.d.559.1 yes 4
76.27 even 6 inner 2736.2.bm.j.559.2 4
228.179 odd 6 912.2.bb.c.559.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
912.2.bb.c.31.1 4 3.2 odd 2
912.2.bb.c.559.2 yes 4 228.179 odd 6
912.2.bb.d.31.2 yes 4 12.11 even 2
912.2.bb.d.559.1 yes 4 57.8 even 6
2736.2.bm.i.559.1 4 19.8 odd 6
2736.2.bm.i.1855.2 4 4.3 odd 2
2736.2.bm.j.559.2 4 76.27 even 6 inner
2736.2.bm.j.1855.1 4 1.1 even 1 trivial