Properties

Label 572.2.p.a.309.9
Level $572$
Weight $2$
Character 572.309
Analytic conductor $4.567$
Analytic rank $0$
Dimension $24$
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] = [572,2,Mod(309,572)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(572, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("572.309");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 572 = 2^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 572.p (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: \(4.56744299562\)
Analytic rank: \(0\)
Dimension: \(24\)
Relative dimension: \(12\) over \(\Q(\zeta_{6})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 309.9
Character \(\chi\) \(=\) 572.309
Dual form 572.2.p.a.485.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.716368 + 1.24079i) q^{3} -2.80197i q^{5} +(4.48758 + 2.59090i) q^{7} +(0.473633 - 0.820357i) q^{9} +O(q^{10})\) \(q+(0.716368 + 1.24079i) q^{3} -2.80197i q^{5} +(4.48758 + 2.59090i) q^{7} +(0.473633 - 0.820357i) q^{9} +(-0.866025 + 0.500000i) q^{11} +(-1.99912 - 3.00059i) q^{13} +(3.47664 - 2.00724i) q^{15} +(1.08476 - 1.87887i) q^{17} +(-1.69474 - 0.978461i) q^{19} +7.42416i q^{21} +(2.85806 + 4.95030i) q^{23} -2.85102 q^{25} +5.65539 q^{27} +(-2.78406 - 4.82213i) q^{29} +4.69366i q^{31} +(-1.24079 - 0.716368i) q^{33} +(7.25963 - 12.5740i) q^{35} +(4.30742 - 2.48689i) q^{37} +(2.29098 - 4.63000i) q^{39} +(3.64181 - 2.10260i) q^{41} +(-5.38594 + 9.32871i) q^{43} +(-2.29861 - 1.32710i) q^{45} +4.57836i q^{47} +(9.92556 + 17.1916i) q^{49} +3.10836 q^{51} +1.56839 q^{53} +(1.40098 + 2.42658i) q^{55} -2.80375i q^{57} +(-11.3947 - 6.57871i) q^{59} +(-6.29341 + 10.9005i) q^{61} +(4.25093 - 2.45428i) q^{63} +(-8.40755 + 5.60146i) q^{65} +(6.44028 - 3.71830i) q^{67} +(-4.09484 + 7.09247i) q^{69} +(-7.51196 - 4.33703i) q^{71} +2.25882i q^{73} +(-2.04238 - 3.53751i) q^{75} -5.18181 q^{77} -6.15778 q^{79} +(2.63044 + 4.55606i) q^{81} -6.08702i q^{83} +(-5.26452 - 3.03947i) q^{85} +(3.98882 - 6.90884i) q^{87} +(2.80742 - 1.62086i) q^{89} +(-1.19696 - 18.6449i) q^{91} +(-5.82383 + 3.36239i) q^{93} +(-2.74162 + 4.74862i) q^{95} +(-3.75096 - 2.16562i) q^{97} +0.947266i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - 2 q^{3} + 6 q^{7} - 14 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - 2 q^{3} + 6 q^{7} - 14 q^{9} - 2 q^{13} - 6 q^{19} + 10 q^{23} - 40 q^{25} - 8 q^{27} - 8 q^{29} + 8 q^{35} + 18 q^{37} + 36 q^{41} + 10 q^{43} - 30 q^{45} + 14 q^{49} + 44 q^{51} + 16 q^{53} - 24 q^{59} + 6 q^{61} - 6 q^{63} - 24 q^{65} - 54 q^{67} + 10 q^{69} + 18 q^{71} + 6 q^{75} - 16 q^{77} - 32 q^{79} - 4 q^{81} + 52 q^{87} - 18 q^{89} - 18 q^{91} + 30 q^{93} - 12 q^{95} + 42 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/572\mathbb{Z}\right)^\times\).

\(n\) \(287\) \(353\) \(365\)
\(\chi(n)\) \(1\) \(e\left(\frac{5}{6}\right)\) \(1\)

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
\(3\) 0.716368 + 1.24079i 0.413595 + 0.716368i 0.995280 0.0970463i \(-0.0309395\pi\)
−0.581684 + 0.813415i \(0.697606\pi\)
\(4\) 0 0
\(5\) 2.80197i 1.25308i −0.779390 0.626539i \(-0.784471\pi\)
0.779390 0.626539i \(-0.215529\pi\)
\(6\) 0 0
\(7\) 4.48758 + 2.59090i 1.69614 + 0.979269i 0.949353 + 0.314211i \(0.101740\pi\)
0.746791 + 0.665058i \(0.231593\pi\)
\(8\) 0 0
\(9\) 0.473633 0.820357i 0.157878 0.273452i
\(10\) 0 0
\(11\) −0.866025 + 0.500000i −0.261116 + 0.150756i
\(12\) 0 0
\(13\) −1.99912 3.00059i −0.554456 0.832213i
\(14\) 0 0
\(15\) 3.47664 2.00724i 0.897665 0.518267i
\(16\) 0 0
\(17\) 1.08476 1.87887i 0.263094 0.455692i −0.703969 0.710231i \(-0.748590\pi\)
0.967062 + 0.254539i \(0.0819237\pi\)
\(18\) 0 0
\(19\) −1.69474 0.978461i −0.388801 0.224474i 0.292840 0.956162i \(-0.405400\pi\)
−0.681640 + 0.731687i \(0.738733\pi\)
\(20\) 0 0
\(21\) 7.42416i 1.62009i
\(22\) 0 0
\(23\) 2.85806 + 4.95030i 0.595946 + 1.03221i 0.993413 + 0.114592i \(0.0365561\pi\)
−0.397467 + 0.917617i \(0.630111\pi\)
\(24\) 0 0
\(25\) −2.85102 −0.570205
\(26\) 0 0
\(27\) 5.65539 1.08838
\(28\) 0 0
\(29\) −2.78406 4.82213i −0.516986 0.895446i −0.999805 0.0197262i \(-0.993721\pi\)
0.482819 0.875720i \(-0.339613\pi\)
\(30\) 0 0
\(31\) 4.69366i 0.843007i 0.906827 + 0.421503i \(0.138497\pi\)
−0.906827 + 0.421503i \(0.861503\pi\)
\(32\) 0 0
\(33\) −1.24079 0.716368i −0.215993 0.124704i
\(34\) 0 0
\(35\) 7.25963 12.5740i 1.22710 2.12540i
\(36\) 0 0
\(37\) 4.30742 2.48689i 0.708135 0.408842i −0.102235 0.994760i \(-0.532599\pi\)
0.810370 + 0.585918i \(0.199266\pi\)
\(38\) 0 0
\(39\) 2.29098 4.63000i 0.366851 0.741394i
\(40\) 0 0
\(41\) 3.64181 2.10260i 0.568755 0.328371i −0.187897 0.982189i \(-0.560167\pi\)
0.756652 + 0.653818i \(0.226834\pi\)
\(42\) 0 0
\(43\) −5.38594 + 9.32871i −0.821348 + 1.42262i 0.0833315 + 0.996522i \(0.473444\pi\)
−0.904679 + 0.426094i \(0.859889\pi\)
\(44\) 0 0
\(45\) −2.29861 1.32710i −0.342657 0.197833i
\(46\) 0 0
\(47\) 4.57836i 0.667822i 0.942605 + 0.333911i \(0.108368\pi\)
−0.942605 + 0.333911i \(0.891632\pi\)
\(48\) 0 0
\(49\) 9.92556 + 17.1916i 1.41794 + 2.45594i
\(50\) 0 0
\(51\) 3.10836 0.435258
\(52\) 0 0
\(53\) 1.56839 0.215435 0.107717 0.994182i \(-0.465646\pi\)
0.107717 + 0.994182i \(0.465646\pi\)
\(54\) 0 0
\(55\) 1.40098 + 2.42658i 0.188909 + 0.327199i
\(56\) 0 0
\(57\) 2.80375i 0.371366i
\(58\) 0 0
\(59\) −11.3947 6.57871i −1.48346 0.856475i −0.483634 0.875270i \(-0.660683\pi\)
−0.999823 + 0.0187956i \(0.994017\pi\)
\(60\) 0 0
\(61\) −6.29341 + 10.9005i −0.805788 + 1.39567i 0.109969 + 0.993935i \(0.464925\pi\)
−0.915758 + 0.401731i \(0.868409\pi\)
\(62\) 0 0
\(63\) 4.25093 2.45428i 0.535567 0.309210i
\(64\) 0 0
\(65\) −8.40755 + 5.60146i −1.04283 + 0.694776i
\(66\) 0 0
\(67\) 6.44028 3.71830i 0.786806 0.454262i −0.0520311 0.998645i \(-0.516569\pi\)
0.838837 + 0.544383i \(0.183236\pi\)
\(68\) 0 0
\(69\) −4.09484 + 7.09247i −0.492961 + 0.853834i
\(70\) 0 0
\(71\) −7.51196 4.33703i −0.891505 0.514711i −0.0170707 0.999854i \(-0.505434\pi\)
−0.874435 + 0.485143i \(0.838767\pi\)
\(72\) 0 0
\(73\) 2.25882i 0.264375i 0.991225 + 0.132187i \(0.0422001\pi\)
−0.991225 + 0.132187i \(0.957800\pi\)
\(74\) 0 0
\(75\) −2.04238 3.53751i −0.235834 0.408477i
\(76\) 0 0
\(77\) −5.18181 −0.590522
\(78\) 0 0
\(79\) −6.15778 −0.692805 −0.346402 0.938086i \(-0.612597\pi\)
−0.346402 + 0.938086i \(0.612597\pi\)
\(80\) 0 0
\(81\) 2.63044 + 4.55606i 0.292272 + 0.506229i
\(82\) 0 0
\(83\) 6.08702i 0.668138i −0.942549 0.334069i \(-0.891578\pi\)
0.942549 0.334069i \(-0.108422\pi\)
\(84\) 0 0
\(85\) −5.26452 3.03947i −0.571018 0.329677i
\(86\) 0 0
\(87\) 3.98882 6.90884i 0.427646 0.740705i
\(88\) 0 0
\(89\) 2.80742 1.62086i 0.297586 0.171811i −0.343772 0.939053i \(-0.611705\pi\)
0.641358 + 0.767242i \(0.278371\pi\)
\(90\) 0 0
\(91\) −1.19696 18.6449i −0.125476 1.95452i
\(92\) 0 0
\(93\) −5.82383 + 3.36239i −0.603903 + 0.348664i
\(94\) 0 0
\(95\) −2.74162 + 4.74862i −0.281284 + 0.487198i
\(96\) 0 0
\(97\) −3.75096 2.16562i −0.380853 0.219885i 0.297337 0.954773i \(-0.403902\pi\)
−0.678189 + 0.734887i \(0.737235\pi\)
\(98\) 0 0
\(99\) 0.947266i 0.0952038i
\(100\) 0 0
\(101\) −4.96564 8.60075i −0.494100 0.855806i 0.505877 0.862606i \(-0.331169\pi\)
−0.999977 + 0.00679942i \(0.997836\pi\)
\(102\) 0 0
\(103\) −18.4194 −1.81492 −0.907459 0.420140i \(-0.861981\pi\)
−0.907459 + 0.420140i \(0.861981\pi\)
\(104\) 0 0
\(105\) 20.8023 2.03009
\(106\) 0 0
\(107\) 4.06092 + 7.03373i 0.392584 + 0.679976i 0.992790 0.119870i \(-0.0382478\pi\)
−0.600205 + 0.799846i \(0.704914\pi\)
\(108\) 0 0
\(109\) 11.3837i 1.09036i 0.838319 + 0.545180i \(0.183539\pi\)
−0.838319 + 0.545180i \(0.816461\pi\)
\(110\) 0 0
\(111\) 6.17139 + 3.56306i 0.585763 + 0.338190i
\(112\) 0 0
\(113\) 6.34636 10.9922i 0.597016 1.03406i −0.396243 0.918146i \(-0.629686\pi\)
0.993259 0.115916i \(-0.0369803\pi\)
\(114\) 0 0
\(115\) 13.8706 8.00818i 1.29344 0.746767i
\(116\) 0 0
\(117\) −3.40840 + 0.218812i −0.315107 + 0.0202291i
\(118\) 0 0
\(119\) 9.73592 5.62104i 0.892491 0.515280i
\(120\) 0 0
\(121\) 0.500000 0.866025i 0.0454545 0.0787296i
\(122\) 0 0
\(123\) 5.21775 + 3.01247i 0.470469 + 0.271625i
\(124\) 0 0
\(125\) 6.02136i 0.538567i
\(126\) 0 0
\(127\) −9.23859 16.0017i −0.819792 1.41992i −0.905835 0.423630i \(-0.860756\pi\)
0.0860433 0.996291i \(-0.472578\pi\)
\(128\) 0 0
\(129\) −15.4333 −1.35882
\(130\) 0 0
\(131\) −17.3887 −1.51925 −0.759627 0.650359i \(-0.774618\pi\)
−0.759627 + 0.650359i \(0.774618\pi\)
\(132\) 0 0
\(133\) −5.07020 8.78184i −0.439642 0.761482i
\(134\) 0 0
\(135\) 15.8462i 1.36383i
\(136\) 0 0
\(137\) 7.14707 + 4.12636i 0.610615 + 0.352539i 0.773206 0.634155i \(-0.218652\pi\)
−0.162591 + 0.986694i \(0.551985\pi\)
\(138\) 0 0
\(139\) −5.24005 + 9.07604i −0.444456 + 0.769820i −0.998014 0.0629905i \(-0.979936\pi\)
0.553558 + 0.832810i \(0.313270\pi\)
\(140\) 0 0
\(141\) −5.68076 + 3.27979i −0.478406 + 0.276208i
\(142\) 0 0
\(143\) 3.23158 + 1.59903i 0.270238 + 0.133717i
\(144\) 0 0
\(145\) −13.5114 + 7.80083i −1.12206 + 0.647824i
\(146\) 0 0
\(147\) −14.2207 + 24.6310i −1.17290 + 2.03153i
\(148\) 0 0
\(149\) −3.22926 1.86441i −0.264551 0.152739i 0.361858 0.932233i \(-0.382143\pi\)
−0.626409 + 0.779495i \(0.715476\pi\)
\(150\) 0 0
\(151\) 8.82121i 0.717860i −0.933364 0.358930i \(-0.883142\pi\)
0.933364 0.358930i \(-0.116858\pi\)
\(152\) 0 0
\(153\) −1.02756 1.77979i −0.0830733 0.143887i
\(154\) 0 0
\(155\) 13.1515 1.05635
\(156\) 0 0
\(157\) 0.769573 0.0614186 0.0307093 0.999528i \(-0.490223\pi\)
0.0307093 + 0.999528i \(0.490223\pi\)
\(158\) 0 0
\(159\) 1.12354 + 1.94603i 0.0891028 + 0.154331i
\(160\) 0 0
\(161\) 29.6198i 2.33437i
\(162\) 0 0
\(163\) −0.774673 0.447257i −0.0606770 0.0350319i 0.469355 0.883010i \(-0.344487\pi\)
−0.530032 + 0.847978i \(0.677820\pi\)
\(164\) 0 0
\(165\) −2.00724 + 3.47664i −0.156263 + 0.270656i
\(166\) 0 0
\(167\) −6.69289 + 3.86414i −0.517912 + 0.299016i −0.736080 0.676895i \(-0.763325\pi\)
0.218168 + 0.975911i \(0.429992\pi\)
\(168\) 0 0
\(169\) −5.00706 + 11.9971i −0.385158 + 0.922851i
\(170\) 0 0
\(171\) −1.60537 + 0.926863i −0.122766 + 0.0708790i
\(172\) 0 0
\(173\) −11.8369 + 20.5022i −0.899945 + 1.55875i −0.0723835 + 0.997377i \(0.523061\pi\)
−0.827562 + 0.561374i \(0.810273\pi\)
\(174\) 0 0
\(175\) −12.7942 7.38673i −0.967150 0.558384i
\(176\) 0 0
\(177\) 18.8511i 1.41694i
\(178\) 0 0
\(179\) 11.5758 + 20.0498i 0.865213 + 1.49859i 0.866836 + 0.498594i \(0.166150\pi\)
−0.00162312 + 0.999999i \(0.500517\pi\)
\(180\) 0 0
\(181\) 3.19953 0.237820 0.118910 0.992905i \(-0.462060\pi\)
0.118910 + 0.992905i \(0.462060\pi\)
\(182\) 0 0
\(183\) −18.0336 −1.33308
\(184\) 0 0
\(185\) −6.96818 12.0692i −0.512311 0.887348i
\(186\) 0 0
\(187\) 2.16953i 0.158652i
\(188\) 0 0
\(189\) 25.3790 + 14.6526i 1.84605 + 1.06582i
\(190\) 0 0
\(191\) −1.55499 + 2.69332i −0.112515 + 0.194882i −0.916784 0.399384i \(-0.869224\pi\)
0.804269 + 0.594266i \(0.202557\pi\)
\(192\) 0 0
\(193\) 1.04569 0.603729i 0.0752704 0.0434574i −0.461892 0.886936i \(-0.652829\pi\)
0.537163 + 0.843479i \(0.319496\pi\)
\(194\) 0 0
\(195\) −12.9731 6.41926i −0.929025 0.459693i
\(196\) 0 0
\(197\) 3.89615 2.24944i 0.277589 0.160266i −0.354742 0.934964i \(-0.615431\pi\)
0.632331 + 0.774698i \(0.282098\pi\)
\(198\) 0 0
\(199\) −3.88849 + 6.73506i −0.275648 + 0.477436i −0.970298 0.241911i \(-0.922226\pi\)
0.694651 + 0.719347i \(0.255559\pi\)
\(200\) 0 0
\(201\) 9.22723 + 5.32734i 0.650838 + 0.375762i
\(202\) 0 0
\(203\) 28.8529i 2.02507i
\(204\) 0 0
\(205\) −5.89142 10.2042i −0.411474 0.712694i
\(206\) 0 0
\(207\) 5.41468 0.376346
\(208\) 0 0
\(209\) 1.95692 0.135363
\(210\) 0 0
\(211\) 9.33166 + 16.1629i 0.642418 + 1.11270i 0.984891 + 0.173173i \(0.0554019\pi\)
−0.342474 + 0.939527i \(0.611265\pi\)
\(212\) 0 0
\(213\) 12.4276i 0.851528i
\(214\) 0 0
\(215\) 26.1388 + 15.0912i 1.78265 + 1.02921i
\(216\) 0 0
\(217\) −12.1608 + 21.0632i −0.825531 + 1.42986i
\(218\) 0 0
\(219\) −2.80271 + 1.61815i −0.189390 + 0.109344i
\(220\) 0 0
\(221\) −7.80628 + 0.501145i −0.525107 + 0.0337107i
\(222\) 0 0
\(223\) 2.81019 1.62246i 0.188184 0.108648i −0.402948 0.915223i \(-0.632014\pi\)
0.591132 + 0.806575i \(0.298681\pi\)
\(224\) 0 0
\(225\) −1.35034 + 2.33886i −0.0900226 + 0.155924i
\(226\) 0 0
\(227\) 13.5533 + 7.82501i 0.899565 + 0.519364i 0.877059 0.480383i \(-0.159502\pi\)
0.0225060 + 0.999747i \(0.492836\pi\)
\(228\) 0 0
\(229\) 20.9070i 1.38157i −0.723058 0.690787i \(-0.757264\pi\)
0.723058 0.690787i \(-0.242736\pi\)
\(230\) 0 0
\(231\) −3.71208 6.42951i −0.244237 0.423031i
\(232\) 0 0
\(233\) −10.3007 −0.674818 −0.337409 0.941358i \(-0.609551\pi\)
−0.337409 + 0.941358i \(0.609551\pi\)
\(234\) 0 0
\(235\) 12.8284 0.836833
\(236\) 0 0
\(237\) −4.41124 7.64049i −0.286541 0.496303i
\(238\) 0 0
\(239\) 8.22766i 0.532203i −0.963945 0.266101i \(-0.914264\pi\)
0.963945 0.266101i \(-0.0857356\pi\)
\(240\) 0 0
\(241\) −1.41266 0.815602i −0.0909977 0.0525375i 0.453811 0.891098i \(-0.350064\pi\)
−0.544808 + 0.838561i \(0.683398\pi\)
\(242\) 0 0
\(243\) 4.71436 8.16550i 0.302426 0.523817i
\(244\) 0 0
\(245\) 48.1702 27.8111i 3.07748 1.77679i
\(246\) 0 0
\(247\) 0.452035 + 7.04129i 0.0287623 + 0.448026i
\(248\) 0 0
\(249\) 7.55270 4.36055i 0.478633 0.276339i
\(250\) 0 0
\(251\) 2.15846 3.73857i 0.136241 0.235976i −0.789830 0.613326i \(-0.789831\pi\)
0.926071 + 0.377350i \(0.123165\pi\)
\(252\) 0 0
\(253\) −4.95030 2.85806i −0.311223 0.179684i
\(254\) 0 0
\(255\) 8.70953i 0.545412i
\(256\) 0 0
\(257\) −5.45856 9.45451i −0.340496 0.589756i 0.644029 0.765001i \(-0.277262\pi\)
−0.984525 + 0.175245i \(0.943928\pi\)
\(258\) 0 0
\(259\) 25.7732 1.60147
\(260\) 0 0
\(261\) −5.27448 −0.326482
\(262\) 0 0
\(263\) −4.40732 7.63370i −0.271767 0.470714i 0.697547 0.716539i \(-0.254275\pi\)
−0.969314 + 0.245825i \(0.920941\pi\)
\(264\) 0 0
\(265\) 4.39457i 0.269956i
\(266\) 0 0
\(267\) 4.02229 + 2.32227i 0.246160 + 0.142121i
\(268\) 0 0
\(269\) 11.3739 19.7002i 0.693480 1.20114i −0.277210 0.960809i \(-0.589410\pi\)
0.970690 0.240333i \(-0.0772567\pi\)
\(270\) 0 0
\(271\) 0.0882364 0.0509433i 0.00535998 0.00309458i −0.497318 0.867569i \(-0.665682\pi\)
0.502678 + 0.864474i \(0.332348\pi\)
\(272\) 0 0
\(273\) 22.2769 14.8418i 1.34826 0.898265i
\(274\) 0 0
\(275\) 2.46906 1.42551i 0.148890 0.0859616i
\(276\) 0 0
\(277\) 10.2604 17.7715i 0.616486 1.06779i −0.373636 0.927576i \(-0.621889\pi\)
0.990122 0.140210i \(-0.0447777\pi\)
\(278\) 0 0
\(279\) 3.85048 + 2.22307i 0.230522 + 0.133092i
\(280\) 0 0
\(281\) 26.6740i 1.59124i 0.605798 + 0.795619i \(0.292854\pi\)
−0.605798 + 0.795619i \(0.707146\pi\)
\(282\) 0 0
\(283\) 8.70388 + 15.0756i 0.517392 + 0.896149i 0.999796 + 0.0202004i \(0.00643043\pi\)
−0.482404 + 0.875949i \(0.660236\pi\)
\(284\) 0 0
\(285\) −7.85603 −0.465351
\(286\) 0 0
\(287\) 21.7905 1.28625
\(288\) 0 0
\(289\) 6.14657 + 10.6462i 0.361563 + 0.626246i
\(290\) 0 0
\(291\) 6.20552i 0.363774i
\(292\) 0 0
\(293\) 5.29450 + 3.05678i 0.309308 + 0.178579i 0.646617 0.762815i \(-0.276183\pi\)
−0.337309 + 0.941394i \(0.609517\pi\)
\(294\) 0 0
\(295\) −18.4333 + 31.9275i −1.07323 + 1.85889i
\(296\) 0 0
\(297\) −4.89771 + 2.82770i −0.284194 + 0.164080i
\(298\) 0 0
\(299\) 9.14022 18.4721i 0.528592 1.06827i
\(300\) 0 0
\(301\) −48.3396 + 27.9089i −2.78625 + 1.60864i
\(302\) 0 0
\(303\) 7.11446 12.3226i 0.408715 0.707915i
\(304\) 0 0
\(305\) 30.5429 + 17.6339i 1.74888 + 1.00972i
\(306\) 0 0
\(307\) 15.3633i 0.876832i −0.898772 0.438416i \(-0.855540\pi\)
0.898772 0.438416i \(-0.144460\pi\)
\(308\) 0 0
\(309\) −13.1951 22.8546i −0.750642 1.30015i
\(310\) 0 0
\(311\) −15.2427 −0.864337 −0.432169 0.901793i \(-0.642251\pi\)
−0.432169 + 0.901793i \(0.642251\pi\)
\(312\) 0 0
\(313\) 28.4800 1.60978 0.804891 0.593422i \(-0.202224\pi\)
0.804891 + 0.593422i \(0.202224\pi\)
\(314\) 0 0
\(315\) −6.87680 11.9110i −0.387464 0.671107i
\(316\) 0 0
\(317\) 26.0223i 1.46156i 0.682615 + 0.730779i \(0.260843\pi\)
−0.682615 + 0.730779i \(0.739157\pi\)
\(318\) 0 0
\(319\) 4.82213 + 2.78406i 0.269987 + 0.155877i
\(320\) 0 0
\(321\) −5.81823 + 10.0775i −0.324742 + 0.562470i
\(322\) 0 0
\(323\) −3.67680 + 2.12280i −0.204582 + 0.118116i
\(324\) 0 0
\(325\) 5.69953 + 8.55475i 0.316153 + 0.474532i
\(326\) 0 0
\(327\) −14.1247 + 8.15492i −0.781099 + 0.450968i
\(328\) 0 0
\(329\) −11.8621 + 20.5457i −0.653977 + 1.13272i
\(330\) 0 0
\(331\) −5.51765 3.18562i −0.303277 0.175097i 0.340637 0.940195i \(-0.389357\pi\)
−0.643914 + 0.765098i \(0.722691\pi\)
\(332\) 0 0
\(333\) 4.71149i 0.258188i
\(334\) 0 0
\(335\) −10.4186 18.0455i −0.569226 0.985929i
\(336\) 0 0
\(337\) −22.1004 −1.20388 −0.601942 0.798540i \(-0.705606\pi\)
−0.601942 + 0.798540i \(0.705606\pi\)
\(338\) 0 0
\(339\) 18.1853 0.987692
\(340\) 0 0
\(341\) −2.34683 4.06483i −0.127088 0.220123i
\(342\) 0 0
\(343\) 66.5920i 3.59563i
\(344\) 0 0
\(345\) 19.8729 + 11.4736i 1.06992 + 0.617719i
\(346\) 0 0
\(347\) −5.42631 + 9.39864i −0.291299 + 0.504545i −0.974117 0.226043i \(-0.927421\pi\)
0.682818 + 0.730589i \(0.260754\pi\)
\(348\) 0 0
\(349\) 21.1946 12.2367i 1.13452 0.655017i 0.189454 0.981890i \(-0.439328\pi\)
0.945068 + 0.326873i \(0.105995\pi\)
\(350\) 0 0
\(351\) −11.3058 16.9695i −0.603459 0.905765i
\(352\) 0 0
\(353\) 13.1279 7.57938i 0.698726 0.403410i −0.108147 0.994135i \(-0.534492\pi\)
0.806873 + 0.590725i \(0.201158\pi\)
\(354\) 0 0
\(355\) −12.1522 + 21.0483i −0.644973 + 1.11713i
\(356\) 0 0
\(357\) 13.9490 + 8.05347i 0.738260 + 0.426235i
\(358\) 0 0
\(359\) 26.4827i 1.39771i −0.715265 0.698853i \(-0.753694\pi\)
0.715265 0.698853i \(-0.246306\pi\)
\(360\) 0 0
\(361\) −7.58523 13.1380i −0.399223 0.691474i
\(362\) 0 0
\(363\) 1.43274 0.0751992
\(364\) 0 0
\(365\) 6.32914 0.331283
\(366\) 0 0
\(367\) −0.314518 0.544762i −0.0164177 0.0284363i 0.857700 0.514151i \(-0.171893\pi\)
−0.874118 + 0.485714i \(0.838559\pi\)
\(368\) 0 0
\(369\) 3.98344i 0.207370i
\(370\) 0 0
\(371\) 7.03826 + 4.06354i 0.365408 + 0.210969i
\(372\) 0 0
\(373\) −7.00503 + 12.1331i −0.362707 + 0.628227i −0.988405 0.151838i \(-0.951481\pi\)
0.625698 + 0.780065i \(0.284814\pi\)
\(374\) 0 0
\(375\) 7.47122 4.31351i 0.385812 0.222749i
\(376\) 0 0
\(377\) −8.90356 + 17.9938i −0.458557 + 0.926728i
\(378\) 0 0
\(379\) 31.7847 18.3509i 1.63267 0.942623i 0.649406 0.760442i \(-0.275018\pi\)
0.983265 0.182181i \(-0.0583157\pi\)
\(380\) 0 0
\(381\) 13.2365 22.9262i 0.678124 1.17455i
\(382\) 0 0
\(383\) −16.0246 9.25179i −0.818817 0.472744i 0.0311911 0.999513i \(-0.490070\pi\)
−0.850008 + 0.526769i \(0.823403\pi\)
\(384\) 0 0
\(385\) 14.5193i 0.739970i
\(386\) 0 0
\(387\) 5.10192 + 8.83678i 0.259345 + 0.449199i
\(388\) 0 0
\(389\) 7.42893 0.376662 0.188331 0.982106i \(-0.439692\pi\)
0.188331 + 0.982106i \(0.439692\pi\)
\(390\) 0 0
\(391\) 12.4013 0.627159
\(392\) 0 0
\(393\) −12.4567 21.5756i −0.628356 1.08835i
\(394\) 0 0
\(395\) 17.2539i 0.868139i
\(396\) 0 0
\(397\) −4.98398 2.87750i −0.250139 0.144418i 0.369689 0.929156i \(-0.379464\pi\)
−0.619828 + 0.784738i \(0.712798\pi\)
\(398\) 0 0
\(399\) 7.26425 12.5821i 0.363668 0.629891i
\(400\) 0 0
\(401\) 12.0290 6.94494i 0.600699 0.346814i −0.168617 0.985682i \(-0.553930\pi\)
0.769317 + 0.638868i \(0.220597\pi\)
\(402\) 0 0
\(403\) 14.0838 9.38319i 0.701562 0.467410i
\(404\) 0 0
\(405\) 12.7659 7.37042i 0.634345 0.366239i
\(406\) 0 0
\(407\) −2.48689 + 4.30742i −0.123270 + 0.213511i
\(408\) 0 0
\(409\) −11.0622 6.38679i −0.546993 0.315806i 0.200916 0.979609i \(-0.435608\pi\)
−0.747908 + 0.663802i \(0.768942\pi\)
\(410\) 0 0
\(411\) 11.8240i 0.583234i
\(412\) 0 0
\(413\) −34.0896 59.0449i −1.67744 2.90541i
\(414\) 0 0
\(415\) −17.0556 −0.837229
\(416\) 0 0
\(417\) −15.0152 −0.735299
\(418\) 0 0
\(419\) 19.5615 + 33.8815i 0.955642 + 1.65522i 0.732892 + 0.680345i \(0.238170\pi\)
0.222750 + 0.974875i \(0.428496\pi\)
\(420\) 0 0
\(421\) 12.2666i 0.597836i 0.954279 + 0.298918i \(0.0966256\pi\)
−0.954279 + 0.298918i \(0.903374\pi\)
\(422\) 0 0
\(423\) 3.75588 + 2.16846i 0.182617 + 0.105434i
\(424\) 0 0
\(425\) −3.09269 + 5.35669i −0.150017 + 0.259838i
\(426\) 0 0
\(427\) −56.4843 + 32.6112i −2.73347 + 1.57817i
\(428\) 0 0
\(429\) 0.330952 + 5.15519i 0.0159785 + 0.248895i
\(430\) 0 0
\(431\) 30.5866 17.6592i 1.47330 0.850612i 0.473755 0.880657i \(-0.342898\pi\)
0.999549 + 0.0300446i \(0.00956494\pi\)
\(432\) 0 0
\(433\) 3.33151 5.77034i 0.160102 0.277305i −0.774803 0.632203i \(-0.782151\pi\)
0.934905 + 0.354898i \(0.115484\pi\)
\(434\) 0 0
\(435\) −19.3583 11.1765i −0.928161 0.535874i
\(436\) 0 0
\(437\) 11.1860i 0.535098i
\(438\) 0 0
\(439\) 16.7212 + 28.9620i 0.798060 + 1.38228i 0.920878 + 0.389851i \(0.127474\pi\)
−0.122818 + 0.992429i \(0.539193\pi\)
\(440\) 0 0
\(441\) 18.8043 0.895443
\(442\) 0 0
\(443\) −27.6425 −1.31333 −0.656667 0.754181i \(-0.728034\pi\)
−0.656667 + 0.754181i \(0.728034\pi\)
\(444\) 0 0
\(445\) −4.54161 7.86629i −0.215293 0.372898i
\(446\) 0 0
\(447\) 5.34243i 0.252688i
\(448\) 0 0
\(449\) 14.8470 + 8.57190i 0.700672 + 0.404533i 0.807598 0.589734i \(-0.200767\pi\)
−0.106926 + 0.994267i \(0.534101\pi\)
\(450\) 0 0
\(451\) −2.10260 + 3.64181i −0.0990075 + 0.171486i
\(452\) 0 0
\(453\) 10.9452 6.31924i 0.514252 0.296904i
\(454\) 0 0
\(455\) −52.2424 + 3.35384i −2.44916 + 0.157231i
\(456\) 0 0
\(457\) 30.5138 17.6172i 1.42738 0.824096i 0.430462 0.902609i \(-0.358351\pi\)
0.996913 + 0.0785130i \(0.0250172\pi\)
\(458\) 0 0
\(459\) 6.13477 10.6257i 0.286346 0.495967i
\(460\) 0 0
\(461\) −24.9831 14.4240i −1.16358 0.671793i −0.211421 0.977395i \(-0.567809\pi\)
−0.952159 + 0.305602i \(0.901142\pi\)
\(462\) 0 0
\(463\) 28.0607i 1.30409i 0.758180 + 0.652046i \(0.226089\pi\)
−0.758180 + 0.652046i \(0.773911\pi\)
\(464\) 0 0
\(465\) 9.42131 + 16.3182i 0.436903 + 0.756738i
\(466\) 0 0
\(467\) −40.8923 −1.89227 −0.946136 0.323769i \(-0.895050\pi\)
−0.946136 + 0.323769i \(0.895050\pi\)
\(468\) 0 0
\(469\) 38.5350 1.77938
\(470\) 0 0
\(471\) 0.551298 + 0.954875i 0.0254025 + 0.0439983i
\(472\) 0 0
\(473\) 10.7719i 0.495291i
\(474\) 0 0
\(475\) 4.83176 + 2.78962i 0.221696 + 0.127996i
\(476\) 0 0
\(477\) 0.742840 1.28664i 0.0340123 0.0589111i
\(478\) 0 0
\(479\) −11.2896 + 6.51805i −0.515835 + 0.297817i −0.735229 0.677819i \(-0.762925\pi\)
0.219394 + 0.975636i \(0.429592\pi\)
\(480\) 0 0
\(481\) −16.0732 7.95320i −0.732873 0.362635i
\(482\) 0 0
\(483\) −36.7518 + 21.2187i −1.67227 + 0.965483i
\(484\) 0 0
\(485\) −6.06800 + 10.5101i −0.275533 + 0.477238i
\(486\) 0 0
\(487\) 11.4433 + 6.60677i 0.518543 + 0.299381i 0.736339 0.676613i \(-0.236553\pi\)
−0.217795 + 0.975995i \(0.569886\pi\)
\(488\) 0 0
\(489\) 1.28160i 0.0579561i
\(490\) 0 0
\(491\) −19.7078 34.1349i −0.889399 1.54048i −0.840587 0.541676i \(-0.817790\pi\)
−0.0488119 0.998808i \(-0.515543\pi\)
\(492\) 0 0
\(493\) −12.0802 −0.544064
\(494\) 0 0
\(495\) 2.65421 0.119298
\(496\) 0 0
\(497\) −22.4737 38.9255i −1.00808 1.74605i
\(498\) 0 0
\(499\) 4.09468i 0.183303i −0.995791 0.0916515i \(-0.970785\pi\)
0.995791 0.0916515i \(-0.0292146\pi\)
\(500\) 0 0
\(501\) −9.58915 5.53630i −0.428412 0.247344i
\(502\) 0 0
\(503\) 4.04613 7.00810i 0.180408 0.312476i −0.761612 0.648034i \(-0.775592\pi\)
0.942019 + 0.335558i \(0.108925\pi\)
\(504\) 0 0
\(505\) −24.0990 + 13.9136i −1.07239 + 0.619146i
\(506\) 0 0
\(507\) −18.4727 + 2.38162i −0.820400 + 0.105772i
\(508\) 0 0
\(509\) 18.8371 10.8756i 0.834941 0.482053i −0.0206004 0.999788i \(-0.506558\pi\)
0.855541 + 0.517734i \(0.173224\pi\)
\(510\) 0 0
\(511\) −5.85239 + 10.1366i −0.258894 + 0.448418i
\(512\) 0 0
\(513\) −9.58444 5.53358i −0.423163 0.244314i
\(514\) 0 0
\(515\) 51.6106i 2.27424i
\(516\) 0 0
\(517\) −2.28918 3.96497i −0.100678 0.174379i
\(518\) 0 0
\(519\) −33.9184 −1.48885
\(520\) 0 0
\(521\) 40.4837 1.77362 0.886812 0.462130i \(-0.152915\pi\)
0.886812 + 0.462130i \(0.152915\pi\)
\(522\) 0 0
\(523\) 4.78756 + 8.29230i 0.209345 + 0.362597i 0.951508 0.307623i \(-0.0995334\pi\)
−0.742163 + 0.670219i \(0.766200\pi\)
\(524\) 0 0
\(525\) 21.1665i 0.923780i
\(526\) 0 0
\(527\) 8.81877 + 5.09152i 0.384152 + 0.221790i
\(528\) 0 0
\(529\) −4.83697 + 8.37789i −0.210303 + 0.364256i
\(530\) 0 0
\(531\) −10.7938 + 6.23179i −0.468410 + 0.270436i
\(532\) 0 0
\(533\) −13.5894 6.72422i −0.588624 0.291258i
\(534\) 0 0
\(535\) 19.7083 11.3786i 0.852063 0.491939i
\(536\) 0 0
\(537\) −16.5850 + 28.7261i −0.715696 + 1.23962i
\(538\) 0 0
\(539\) −17.1916 9.92556i −0.740494 0.427524i
\(540\) 0 0
\(541\) 3.03711i 0.130576i −0.997866 0.0652878i \(-0.979203\pi\)
0.997866 0.0652878i \(-0.0207965\pi\)
\(542\) 0 0
\(543\) 2.29205 + 3.96994i 0.0983611 + 0.170366i
\(544\) 0 0
\(545\) 31.8967 1.36631
\(546\) 0 0
\(547\) 11.8042 0.504711 0.252356 0.967635i \(-0.418795\pi\)
0.252356 + 0.967635i \(0.418795\pi\)
\(548\) 0 0
\(549\) 5.96153 + 10.3257i 0.254432 + 0.440689i
\(550\) 0 0
\(551\) 10.8964i 0.464200i
\(552\) 0 0
\(553\) −27.6335 15.9542i −1.17510 0.678443i
\(554\) 0 0
\(555\) 9.98357 17.2920i 0.423779 0.734006i
\(556\) 0 0
\(557\) −28.1966 + 16.2793i −1.19473 + 0.689777i −0.959375 0.282133i \(-0.908958\pi\)
−0.235353 + 0.971910i \(0.575625\pi\)
\(558\) 0 0
\(559\) 38.7587 2.48822i 1.63932 0.105241i
\(560\) 0 0
\(561\) −2.69192 + 1.55418i −0.113653 + 0.0656176i
\(562\) 0 0
\(563\) −0.809155 + 1.40150i −0.0341018 + 0.0590661i −0.882573 0.470176i \(-0.844190\pi\)
0.848471 + 0.529242i \(0.177524\pi\)
\(564\) 0 0
\(565\) −30.7999 17.7823i −1.29576 0.748107i
\(566\) 0 0
\(567\) 27.2609i 1.14485i
\(568\) 0 0
\(569\) 8.50400 + 14.7294i 0.356506 + 0.617487i 0.987375 0.158403i \(-0.0506345\pi\)
−0.630868 + 0.775890i \(0.717301\pi\)
\(570\) 0 0
\(571\) 13.8556 0.579840 0.289920 0.957051i \(-0.406371\pi\)
0.289920 + 0.957051i \(0.406371\pi\)
\(572\) 0 0
\(573\) −4.45578 −0.186143
\(574\) 0 0
\(575\) −8.14839 14.1134i −0.339811 0.588570i
\(576\) 0 0
\(577\) 4.25714i 0.177227i −0.996066 0.0886134i \(-0.971756\pi\)
0.996066 0.0886134i \(-0.0282436\pi\)
\(578\) 0 0
\(579\) 1.49820 + 0.864985i 0.0622630 + 0.0359475i
\(580\) 0 0
\(581\) 15.7709 27.3160i 0.654287 1.13326i
\(582\) 0 0
\(583\) −1.35826 + 0.784194i −0.0562535 + 0.0324780i
\(584\) 0 0
\(585\) 0.613103 + 9.55023i 0.0253487 + 0.394853i
\(586\) 0 0
\(587\) 0.0717203 0.0414077i 0.00296021 0.00170908i −0.498519 0.866879i \(-0.666123\pi\)
0.501479 + 0.865170i \(0.332789\pi\)
\(588\) 0 0
\(589\) 4.59257 7.95456i 0.189233 0.327762i
\(590\) 0 0
\(591\) 5.58215 + 3.22286i 0.229619 + 0.132571i
\(592\) 0 0
\(593\) 14.3010i 0.587271i 0.955917 + 0.293636i \(0.0948652\pi\)
−0.955917 + 0.293636i \(0.905135\pi\)
\(594\) 0 0
\(595\) −15.7500 27.2797i −0.645686 1.11836i
\(596\) 0 0
\(597\) −11.1424 −0.456027
\(598\) 0 0
\(599\) 4.97772 0.203384 0.101692 0.994816i \(-0.467574\pi\)
0.101692 + 0.994816i \(0.467574\pi\)
\(600\) 0 0
\(601\) −14.4440 25.0177i −0.589182 1.02049i −0.994340 0.106247i \(-0.966117\pi\)
0.405158 0.914247i \(-0.367217\pi\)
\(602\) 0 0
\(603\) 7.04444i 0.286872i
\(604\) 0 0
\(605\) −2.42658 1.40098i −0.0986543 0.0569581i
\(606\) 0 0
\(607\) 6.36749 11.0288i 0.258448 0.447646i −0.707378 0.706835i \(-0.750122\pi\)
0.965826 + 0.259190i \(0.0834555\pi\)
\(608\) 0 0
\(609\) 35.8003 20.6693i 1.45070 0.837562i
\(610\) 0 0
\(611\) 13.7378 9.15267i 0.555770 0.370277i
\(612\) 0 0
\(613\) −11.6240 + 6.71111i −0.469488 + 0.271059i −0.716026 0.698074i \(-0.754041\pi\)
0.246537 + 0.969133i \(0.420707\pi\)
\(614\) 0 0
\(615\) 8.44085 14.6200i 0.340368 0.589534i
\(616\) 0 0
\(617\) 20.7276 + 11.9671i 0.834462 + 0.481777i 0.855378 0.518004i \(-0.173325\pi\)
−0.0209160 + 0.999781i \(0.506658\pi\)
\(618\) 0 0
\(619\) 23.8252i 0.957614i −0.877920 0.478807i \(-0.841069\pi\)
0.877920 0.478807i \(-0.158931\pi\)
\(620\) 0 0
\(621\) 16.1634 + 27.9959i 0.648616 + 1.12344i
\(622\) 0 0
\(623\) 16.7980 0.672998
\(624\) 0 0
\(625\) −31.1268 −1.24507
\(626\) 0 0
\(627\) 1.40188 + 2.42812i 0.0559856 + 0.0969698i
\(628\) 0 0
\(629\) 10.7908i 0.430255i
\(630\) 0 0
\(631\) −10.0986 5.83041i −0.402018 0.232105i 0.285337 0.958427i \(-0.407895\pi\)
−0.687354 + 0.726322i \(0.741228\pi\)
\(632\) 0 0
\(633\) −13.3698 + 23.1572i −0.531402 + 0.920415i
\(634\) 0 0
\(635\) −44.8363 + 25.8862i −1.77927 + 1.02726i
\(636\) 0 0
\(637\) 31.7425 64.1505i 1.25768 2.54174i
\(638\) 0 0
\(639\) −7.11582 + 4.10832i −0.281498 + 0.162523i
\(640\) 0 0
\(641\) −18.6700 + 32.3373i −0.737419 + 1.27725i 0.216235 + 0.976341i \(0.430622\pi\)
−0.953654 + 0.300906i \(0.902711\pi\)
\(642\) 0 0
\(643\) 15.9097 + 9.18548i 0.627418 + 0.362240i 0.779751 0.626089i \(-0.215345\pi\)
−0.152333 + 0.988329i \(0.548679\pi\)
\(644\) 0 0
\(645\) 43.2435i 1.70271i
\(646\) 0 0
\(647\) 0.697205 + 1.20759i 0.0274099 + 0.0474754i 0.879405 0.476074i \(-0.157941\pi\)
−0.851995 + 0.523550i \(0.824607\pi\)
\(648\) 0 0
\(649\) 13.1574 0.516474
\(650\) 0 0
\(651\) −34.8465 −1.36574
\(652\) 0 0
\(653\) 13.7695 + 23.8495i 0.538844 + 0.933305i 0.998967 + 0.0454496i \(0.0144720\pi\)
−0.460123 + 0.887855i \(0.652195\pi\)
\(654\) 0 0
\(655\) 48.7225i 1.90374i
\(656\) 0 0
\(657\) 1.85304 + 1.06985i 0.0722939 + 0.0417389i
\(658\) 0 0
\(659\) −3.16810 + 5.48731i −0.123412 + 0.213755i −0.921111 0.389300i \(-0.872717\pi\)
0.797699 + 0.603055i \(0.206050\pi\)
\(660\) 0 0
\(661\) 21.7039 12.5307i 0.844184 0.487390i −0.0145005 0.999895i \(-0.504616\pi\)
0.858684 + 0.512505i \(0.171282\pi\)
\(662\) 0 0
\(663\) −6.21398 9.32691i −0.241331 0.362227i
\(664\) 0 0
\(665\) −24.6064 + 14.2065i −0.954196 + 0.550905i
\(666\) 0 0
\(667\) 15.9140 27.5638i 0.616192 1.06728i
\(668\) 0 0
\(669\) 4.02626 + 2.32456i 0.155664 + 0.0898727i
\(670\) 0 0
\(671\) 12.5868i 0.485909i
\(672\) 0 0
\(673\) −4.60132 7.96972i −0.177368 0.307210i 0.763610 0.645677i \(-0.223425\pi\)
−0.940978 + 0.338467i \(0.890092\pi\)
\(674\) 0 0
\(675\) −16.1237 −0.620600
\(676\) 0 0
\(677\) 14.8987 0.572604 0.286302 0.958139i \(-0.407574\pi\)
0.286302 + 0.958139i \(0.407574\pi\)
\(678\) 0 0
\(679\) −11.2218 19.4368i −0.430654 0.745915i
\(680\) 0 0
\(681\) 22.4224i 0.859226i
\(682\) 0 0
\(683\) −17.8390 10.2993i −0.682590 0.394093i 0.118240 0.992985i \(-0.462275\pi\)
−0.800830 + 0.598892i \(0.795608\pi\)
\(684\) 0 0
\(685\) 11.5619 20.0258i 0.441759 0.765148i
\(686\) 0 0
\(687\) 25.9411 14.9771i 0.989716 0.571413i
\(688\) 0 0
\(689\) −3.13539 4.70609i −0.119449 0.179288i
\(690\) 0 0
\(691\) 41.5382 23.9821i 1.58019 0.912321i 0.585355 0.810777i \(-0.300955\pi\)
0.994831 0.101544i \(-0.0323782\pi\)
\(692\) 0 0
\(693\) −2.45428 + 4.25093i −0.0932302 + 0.161479i
\(694\) 0 0
\(695\) 25.4308 + 14.6825i 0.964644 + 0.556938i
\(696\) 0 0
\(697\) 9.12330i 0.345570i
\(698\) 0 0
\(699\) −7.37906 12.7809i −0.279102 0.483418i
\(700\) 0 0
\(701\) −8.08062 −0.305201 −0.152600 0.988288i \(-0.548765\pi\)
−0.152600 + 0.988288i \(0.548765\pi\)
\(702\) 0 0
\(703\) −9.73329 −0.367098
\(704\) 0 0
\(705\) 9.18986 + 15.9173i 0.346110 + 0.599480i
\(706\) 0 0
\(707\) 51.4620i 1.93543i
\(708\) 0 0
\(709\) −4.27537 2.46838i −0.160565 0.0927021i 0.417565 0.908647i \(-0.362884\pi\)
−0.578129 + 0.815945i \(0.696217\pi\)
\(710\) 0 0
\(711\) −2.91653 + 5.05158i −0.109378 + 0.189449i
\(712\) 0 0
\(713\) −23.2350 + 13.4148i −0.870159 + 0.502387i
\(714\) 0 0
\(715\) 4.48042 9.05479i 0.167558 0.338630i
\(716\) 0 0
\(717\) 10.2088 5.89403i 0.381253 0.220117i
\(718\) 0 0
\(719\) 0.0204414 0.0354055i 0.000762335 0.00132040i −0.865644 0.500660i \(-0.833091\pi\)
0.866406 + 0.499340i \(0.166424\pi\)
\(720\) 0 0
\(721\) −82.6585 47.7229i −3.07836 1.77729i
\(722\) 0 0
\(723\) 2.33708i 0.0869171i
\(724\) 0 0
\(725\) 7.93741 + 13.7480i 0.294788 + 0.510588i
\(726\) 0 0
\(727\) −25.3751 −0.941112 −0.470556 0.882370i \(-0.655947\pi\)
−0.470556 + 0.882370i \(0.655947\pi\)
\(728\) 0 0
\(729\) 29.2915 1.08487
\(730\) 0 0
\(731\) 11.6849 + 20.2389i 0.432183 + 0.748563i
\(732\) 0 0
\(733\) 33.7943i 1.24822i 0.781336 + 0.624111i \(0.214539\pi\)
−0.781336 + 0.624111i \(0.785461\pi\)
\(734\) 0 0
\(735\) 69.0153 + 39.8460i 2.54567 + 1.46974i
\(736\) 0 0
\(737\) −3.71830 + 6.44028i −0.136965 + 0.237231i
\(738\) 0 0
\(739\) −37.5447 + 21.6764i −1.38110 + 0.797380i −0.992290 0.123936i \(-0.960448\pi\)
−0.388813 + 0.921317i \(0.627115\pi\)
\(740\) 0 0
\(741\) −8.41291 + 5.60503i −0.309056 + 0.205906i
\(742\) 0 0
\(743\) 11.6616 6.73281i 0.427822 0.247003i −0.270597 0.962693i \(-0.587221\pi\)
0.698418 + 0.715690i \(0.253888\pi\)
\(744\) 0 0
\(745\) −5.22403 + 9.04828i −0.191394 + 0.331503i
\(746\) 0 0
\(747\) −4.99353 2.88302i −0.182704 0.105484i
\(748\) 0 0
\(749\) 42.0858i 1.53778i
\(750\) 0 0
\(751\) 9.89892 + 17.1454i 0.361217 + 0.625646i 0.988161 0.153418i \(-0.0490282\pi\)
−0.626945 + 0.779064i \(0.715695\pi\)
\(752\) 0 0
\(753\) 6.18502 0.225395
\(754\) 0 0
\(755\) −24.7168 −0.899535
\(756\) 0 0
\(757\) −18.5987 32.2138i −0.675980 1.17083i −0.976181 0.216956i \(-0.930387\pi\)
0.300202 0.953876i \(-0.402946\pi\)
\(758\) 0 0
\(759\) 8.18968i 0.297267i
\(760\) 0 0
\(761\) −29.3204 16.9281i −1.06286 0.613645i −0.136641 0.990621i \(-0.543631\pi\)
−0.926223 + 0.376976i \(0.876964\pi\)
\(762\) 0 0
\(763\) −29.4940 + 51.0852i −1.06776 + 1.84941i
\(764\) 0 0
\(765\) −4.98691 + 2.87919i −0.180302 + 0.104097i
\(766\) 0 0
\(767\) 3.03927 + 47.3423i 0.109742 + 1.70943i
\(768\) 0 0
\(769\) 13.7689 7.94946i 0.496518 0.286665i −0.230756 0.973012i \(-0.574120\pi\)
0.727275 + 0.686347i \(0.240787\pi\)
\(770\) 0 0
\(771\) 7.82068 13.5458i 0.281655 0.487841i
\(772\) 0 0
\(773\) −46.0728 26.6002i −1.65712 0.956741i −0.974033 0.226408i \(-0.927302\pi\)
−0.683091 0.730333i \(-0.739365\pi\)
\(774\) 0 0
\(775\) 13.3817i 0.480687i
\(776\) 0 0
\(777\) 18.4631 + 31.9790i 0.662359 + 1.14724i
\(778\) 0 0
\(779\) −8.22925 −0.294843
\(780\) 0 0
\(781\) 8.67406 0.310382
\(782\) 0 0
\(783\) −15.7449 27.2710i −0.562678 0.974586i
\(784\) 0 0
\(785\) 2.15632i 0.0769623i
\(786\) 0 0
\(787\) −20.2310 11.6803i −0.721156 0.416359i 0.0940223 0.995570i \(-0.470028\pi\)
−0.815178 + 0.579211i \(0.803361\pi\)
\(788\) 0 0
\(789\) 6.31452 10.9371i 0.224803 0.389370i
\(790\) 0 0
\(791\) 56.9596 32.8856i 2.02525 1.16928i
\(792\) 0 0
\(793\) 45.2892 2.90746i 1.60827 0.103247i
\(794\) 0 0
\(795\) 5.45272 3.14813i 0.193388 0.111653i
\(796\) 0 0
\(797\) −13.6403 + 23.6258i −0.483166 + 0.836868i −0.999813 0.0193307i \(-0.993846\pi\)
0.516647 + 0.856198i \(0.327180\pi\)
\(798\) 0 0
\(799\) 8.60212 + 4.96644i 0.304321 + 0.175700i
\(800\) 0 0
\(801\) 3.07078i 0.108501i
\(802\) 0 0
\(803\) −1.12941 1.95620i −0.0398560 0.0690327i
\(804\) 0 0
\(805\) 82.9937 2.92514
\(806\) 0 0
\(807\) 32.5917 1.14728
\(808\) 0 0
\(809\) 18.7886 + 32.5428i 0.660573 + 1.14415i 0.980465 + 0.196692i \(0.0630198\pi\)
−0.319893 + 0.947454i \(0.603647\pi\)
\(810\) 0 0
\(811\) 10.1264i 0.355586i 0.984068 + 0.177793i \(0.0568957\pi\)
−0.984068 + 0.177793i \(0.943104\pi\)
\(812\) 0 0
\(813\) 0.126419 + 0.0729883i 0.00443372 + 0.00255981i
\(814\) 0 0
\(815\) −1.25320 + 2.17061i −0.0438977 + 0.0760331i
\(816\) 0 0
\(817\) 18.2556 10.5399i 0.638681 0.368743i
\(818\) 0 0
\(819\) −15.8624 7.84890i −0.554276 0.274263i
\(820\) 0 0
\(821\) −25.8787 + 14.9411i −0.903174 + 0.521448i −0.878229 0.478241i \(-0.841274\pi\)
−0.0249452 + 0.999689i \(0.507941\pi\)
\(822\) 0 0
\(823\) −3.11768 + 5.39999i −0.108676 + 0.188232i −0.915234 0.402923i \(-0.867994\pi\)
0.806558 + 0.591155i \(0.201328\pi\)
\(824\) 0 0
\(825\) 3.53751 + 2.04238i 0.123160 + 0.0711066i
\(826\) 0 0
\(827\) 49.0708i 1.70636i −0.521617 0.853180i \(-0.674671\pi\)
0.521617 0.853180i \(-0.325329\pi\)
\(828\) 0 0
\(829\) −2.77989 4.81491i −0.0965496 0.167229i 0.813705 0.581279i \(-0.197447\pi\)
−0.910254 + 0.414050i \(0.864114\pi\)
\(830\) 0 0
\(831\) 29.4008 1.01990
\(832\) 0 0
\(833\) 43.0676 1.49220
\(834\) 0 0
\(835\) 10.8272 + 18.7533i 0.374691 + 0.648984i
\(836\) 0 0
\(837\) 26.5445i 0.917513i
\(838\) 0 0
\(839\) 9.60576 + 5.54589i 0.331628 + 0.191465i 0.656564 0.754271i \(-0.272009\pi\)
−0.324936 + 0.945736i \(0.605343\pi\)
\(840\) 0 0
\(841\) −1.00193 + 1.73539i −0.0345493 + 0.0598412i
\(842\) 0 0
\(843\) −33.0967 + 19.1084i −1.13991 + 0.658128i
\(844\) 0 0
\(845\) 33.6154 + 14.0296i 1.15640 + 0.482633i
\(846\) 0 0
\(847\) 4.48758 2.59090i 0.154195 0.0890245i
\(848\) 0 0
\(849\) −12.4704 + 21.5993i −0.427982 + 0.741286i
\(850\) 0 0
\(851\) 24.6217 + 14.2153i 0.844020 + 0.487295i
\(852\) 0 0
\(853\) 10.8585i 0.371788i −0.982570 0.185894i \(-0.940482\pi\)
0.982570 0.185894i \(-0.0595181\pi\)
\(854\) 0 0
\(855\) 2.59704 + 4.49821i 0.0888169 + 0.153835i
\(856\) 0 0
\(857\) 28.3654 0.968942 0.484471 0.874807i \(-0.339012\pi\)
0.484471 + 0.874807i \(0.339012\pi\)
\(858\) 0 0
\(859\) 18.4665 0.630068 0.315034 0.949080i \(-0.397984\pi\)
0.315034 + 0.949080i \(0.397984\pi\)
\(860\) 0 0
\(861\) 15.6100 + 27.0374i 0.531989 + 0.921432i
\(862\) 0 0
\(863\) 29.0321i 0.988263i −0.869387 0.494131i \(-0.835486\pi\)
0.869387 0.494131i \(-0.164514\pi\)
\(864\) 0 0
\(865\) 57.4464 + 33.1667i 1.95324 + 1.12770i
\(866\) 0 0
\(867\) −8.80642 + 15.2532i −0.299082 + 0.518025i
\(868\) 0 0
\(869\) 5.33280 3.07889i 0.180903 0.104444i
\(870\) 0 0
\(871\) −24.0320 11.8913i −0.814292 0.402922i
\(872\) 0 0
\(873\) −3.55316 + 2.05142i −0.120256 + 0.0694300i
\(874\) 0 0
\(875\) 15.6008 27.0213i 0.527402 0.913487i
\(876\) 0 0
\(877\) 10.9577 + 6.32644i 0.370016 + 0.213629i 0.673465 0.739219i \(-0.264805\pi\)
−0.303450 + 0.952847i \(0.598138\pi\)
\(878\) 0 0
\(879\) 8.75912i 0.295438i
\(880\) 0 0
\(881\) 2.52114 + 4.36674i 0.0849393 + 0.147119i 0.905365 0.424634i \(-0.139597\pi\)
−0.820426 + 0.571753i \(0.806264\pi\)
\(882\) 0 0
\(883\) 53.8609 1.81256 0.906281 0.422675i \(-0.138909\pi\)
0.906281 + 0.422675i \(0.138909\pi\)
\(884\) 0 0
\(885\) −52.8202 −1.77553
\(886\) 0 0
\(887\) 3.45147 + 5.97813i 0.115889 + 0.200726i 0.918135 0.396268i \(-0.129695\pi\)
−0.802246 + 0.596994i \(0.796362\pi\)
\(888\) 0 0
\(889\) 95.7451i 3.21119i
\(890\) 0 0
\(891\) −4.55606 2.63044i −0.152634 0.0881232i
\(892\) 0 0
\(893\) 4.47974 7.75914i 0.149909 0.259650i
\(894\) 0 0
\(895\) 56.1789 32.4349i 1.87785 1.08418i
\(896\) 0 0
\(897\) 29.4677 1.89176i 0.983897 0.0631640i
\(898\) 0 0
\(899\) 22.6334 13.0674i 0.754867 0.435823i
\(900\) 0 0
\(901\) 1.70133 2.94679i 0.0566795 0.0981719i
\(902\) 0 0
\(903\) −69.2579 39.9861i −2.30476 1.33065i
\(904\) 0 0
\(905\) 8.96499i 0.298006i
\(906\) 0 0
\(907\) 2.56474 + 4.44226i 0.0851607 + 0.147503i 0.905460 0.424432i \(-0.139526\pi\)
−0.820299 + 0.571935i \(0.806193\pi\)
\(908\) 0 0
\(909\) −9.40757 −0.312029
\(910\) 0 0
\(911\) −40.9426 −1.35649 −0.678245 0.734836i \(-0.737259\pi\)
−0.678245 + 0.734836i \(0.737259\pi\)
\(912\) 0 0
\(913\) 3.04351 + 5.27152i 0.100726 + 0.174462i
\(914\) 0 0
\(915\) 50.5295i 1.67045i
\(916\) 0 0
\(917\) −78.0329 45.0523i −2.57687 1.48776i
\(918\) 0 0
\(919\) −15.9026 + 27.5441i −0.524579 + 0.908597i 0.475012 + 0.879979i \(0.342444\pi\)
−0.999590 + 0.0286174i \(0.990890\pi\)
\(920\) 0 0
\(921\) 19.0626 11.0058i 0.628134 0.362654i
\(922\) 0 0
\(923\) 2.00365 + 31.2105i 0.0659508 + 1.02731i
\(924\) 0 0
\(925\) −12.2805 + 7.09018i −0.403782 + 0.233124i
\(926\) 0 0
\(927\) −8.72405 + 15.1105i −0.286535 + 0.496294i
\(928\) 0 0
\(929\) 39.4977 + 22.8040i 1.29588 + 0.748175i 0.979689 0.200522i \(-0.0642638\pi\)
0.316187 + 0.948697i \(0.397597\pi\)
\(930\) 0 0
\(931\) 38.8471i 1.27316i
\(932\) 0 0
\(933\) −10.9194 18.9130i −0.357486 0.619184i
\(934\) 0 0
\(935\) 6.07895 0.198803
\(936\) 0 0
\(937\) −8.54956 −0.279302 −0.139651 0.990201i \(-0.544598\pi\)
−0.139651 + 0.990201i \(0.544598\pi\)
\(938\) 0 0
\(939\) 20.4021 + 35.3375i 0.665799 + 1.15320i
\(940\) 0 0
\(941\) 2.66537i 0.0868885i 0.999056 + 0.0434443i \(0.0138331\pi\)
−0.999056 + 0.0434443i \(0.986167\pi\)
\(942\) 0 0
\(943\) 20.8170 + 12.0187i 0.677895 + 0.391383i
\(944\) 0 0
\(945\) 41.0560 71.1112i 1.33555 2.31325i
\(946\) 0 0
\(947\) 44.6122 25.7569i 1.44970 0.836986i 0.451239 0.892403i \(-0.350982\pi\)
0.998463 + 0.0554169i \(0.0176488\pi\)
\(948\) 0 0
\(949\) 6.77779 4.51565i 0.220016 0.146584i
\(950\) 0 0
\(951\) −32.2881 + 18.6415i −1.04701 + 0.604493i
\(952\) 0 0
\(953\) −29.7208 + 51.4780i −0.962752 + 1.66753i −0.247214 + 0.968961i \(0.579515\pi\)
−0.715538 + 0.698574i \(0.753818\pi\)
\(954\) 0 0
\(955\) 7.54659 + 4.35703i 0.244202 + 0.140990i
\(956\) 0 0
\(957\) 7.97764i 0.257880i
\(958\) 0 0
\(959\) 21.3820 + 37.0347i 0.690461 + 1.19591i
\(960\) 0 0
\(961\) 8.96952 0.289339
\(962\) 0 0
\(963\) 7.69355 0.247921
\(964\) 0 0
\(965\) −1.69163 2.92999i −0.0544555 0.0943196i
\(966\) 0 0
\(967\) 50.6572i 1.62903i −0.580144 0.814514i \(-0.697004\pi\)
0.580144 0.814514i \(-0.302996\pi\)
\(968\) 0 0
\(969\) −5.26788 3.04141i −0.169229 0.0977042i
\(970\) 0 0
\(971\) 18.8999 32.7356i 0.606527 1.05054i −0.385281 0.922799i \(-0.625896\pi\)
0.991808 0.127736i \(-0.0407711\pi\)
\(972\) 0 0
\(973\) −47.0303 + 27.1530i −1.50772 + 0.870484i
\(974\) 0 0
\(975\) −6.53165 + 13.2003i −0.209180 + 0.422746i
\(976\) 0 0
\(977\) 20.2681 11.7018i 0.648433 0.374373i −0.139422 0.990233i \(-0.544525\pi\)
0.787856 + 0.615860i \(0.211191\pi\)
\(978\) 0 0
\(979\) −1.62086 + 2.80742i −0.0518030 + 0.0897254i
\(980\) 0 0
\(981\) 9.33869 + 5.39169i 0.298161 + 0.172144i
\(982\) 0 0
\(983\) 30.3104i 0.966752i −0.875413 0.483376i \(-0.839410\pi\)
0.875413 0.483376i \(-0.160590\pi\)
\(984\) 0 0
\(985\) −6.30286 10.9169i −0.200826 0.347841i
\(986\) 0 0
\(987\) −33.9905 −1.08193
\(988\) 0 0
\(989\) −61.5732 −1.95792
\(990\) 0 0
\(991\) 7.25812 + 12.5714i 0.230562 + 0.399345i 0.957974 0.286856i \(-0.0926103\pi\)
−0.727412 + 0.686201i \(0.759277\pi\)
\(992\) 0 0
\(993\) 9.12830i 0.289678i
\(994\) 0 0
\(995\) 18.8714 + 10.8954i 0.598265 + 0.345408i
\(996\) 0 0
\(997\) −0.0246062 + 0.0426191i −0.000779285 + 0.00134976i −0.866415 0.499325i \(-0.833581\pi\)
0.865635 + 0.500675i \(0.166915\pi\)
\(998\) 0 0
\(999\) 24.3601 14.0643i 0.770721 0.444976i
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 572.2.p.a.309.9 24
13.2 odd 12 7436.2.a.v.1.4 12
13.4 even 6 inner 572.2.p.a.485.9 yes 24
13.11 odd 12 7436.2.a.u.1.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
572.2.p.a.309.9 24 1.1 even 1 trivial
572.2.p.a.485.9 yes 24 13.4 even 6 inner
7436.2.a.u.1.4 12 13.11 odd 12
7436.2.a.v.1.4 12 13.2 odd 12