Properties

Label 1728.3.q.g.1601.1
Level $1728$
Weight $3$
Character 1728.1601
Analytic conductor $47.085$
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] = [1728,3,Mod(449,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 5]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.q (of order \(6\), degree \(2\), not 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: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{2} \)
Twist minimal: no (minimal twist has level 36)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 1601.1
Root \(1.68614 - 0.396143i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1601
Dual form 1728.3.q.g.449.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-2.05842 + 1.18843i) q^{5} +(4.05842 - 7.02939i) q^{7} +O(q^{10})\) \(q+(-2.05842 + 1.18843i) q^{5} +(4.05842 - 7.02939i) q^{7} +(-17.6168 - 10.1711i) q^{11} +(3.05842 + 5.29734i) q^{13} -17.9653i q^{17} -9.11684 q^{19} +(-29.0584 + 16.7769i) q^{23} +(-9.67527 + 16.7581i) q^{25} +(14.4090 + 8.31901i) q^{29} +(11.1753 + 19.3561i) q^{31} +19.2926i q^{35} +50.4674 q^{37} +(-29.9674 + 17.3017i) q^{41} +(11.5000 - 19.9186i) q^{43} +(33.1753 + 19.1537i) q^{47} +(-8.44158 - 14.6212i) q^{49} +19.0149i q^{53} +48.3505 q^{55} +(-2.96738 + 1.71322i) q^{59} +(-23.1753 + 40.1407i) q^{61} +(-12.5910 - 7.26944i) q^{65} +(-3.14947 - 5.45504i) q^{67} +35.9306i q^{71} +47.3505 q^{73} +(-142.993 + 82.5571i) q^{77} +(42.2921 - 73.2521i) q^{79} +(-33.1753 - 19.1537i) q^{83} +(21.3505 + 36.9802i) q^{85} +143.723i q^{89} +49.6495 q^{91} +(18.7663 - 10.8347i) q^{95} +(-40.3832 + 69.9457i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 9 q^{5} - q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 9 q^{5} - q^{7} - 36 q^{11} - 5 q^{13} - 2 q^{19} - 99 q^{23} + 13 q^{25} - 63 q^{29} - 7 q^{31} + 64 q^{37} + 18 q^{41} + 46 q^{43} + 81 q^{47} - 51 q^{49} + 90 q^{55} + 126 q^{59} - 41 q^{61} - 171 q^{65} - 116 q^{67} + 86 q^{73} - 279 q^{77} + 83 q^{79} - 81 q^{83} - 18 q^{85} + 302 q^{91} + 144 q^{95} - 196 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(5\) −2.05842 + 1.18843i −0.411684 + 0.237686i −0.691513 0.722364i \(-0.743056\pi\)
0.279829 + 0.960050i \(0.409722\pi\)
\(6\) 0 0
\(7\) 4.05842 7.02939i 0.579775 1.00420i −0.415730 0.909488i \(-0.636474\pi\)
0.995505 0.0947110i \(-0.0301927\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −17.6168 10.1711i −1.60153 0.924645i −0.991181 0.132513i \(-0.957695\pi\)
−0.610350 0.792132i \(-0.708971\pi\)
\(12\) 0 0
\(13\) 3.05842 + 5.29734i 0.235263 + 0.407488i 0.959349 0.282222i \(-0.0910714\pi\)
−0.724086 + 0.689710i \(0.757738\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 17.9653i 1.05678i −0.849001 0.528392i \(-0.822795\pi\)
0.849001 0.528392i \(-0.177205\pi\)
\(18\) 0 0
\(19\) −9.11684 −0.479834 −0.239917 0.970793i \(-0.577120\pi\)
−0.239917 + 0.970793i \(0.577120\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −29.0584 + 16.7769i −1.26341 + 0.729430i −0.973733 0.227695i \(-0.926881\pi\)
−0.289677 + 0.957124i \(0.593548\pi\)
\(24\) 0 0
\(25\) −9.67527 + 16.7581i −0.387011 + 0.670322i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 14.4090 + 8.31901i 0.496860 + 0.286863i 0.727416 0.686197i \(-0.240721\pi\)
−0.230556 + 0.973059i \(0.574054\pi\)
\(30\) 0 0
\(31\) 11.1753 + 19.3561i 0.360492 + 0.624391i 0.988042 0.154185i \(-0.0492753\pi\)
−0.627549 + 0.778577i \(0.715942\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 19.2926i 0.551217i
\(36\) 0 0
\(37\) 50.4674 1.36398 0.681992 0.731360i \(-0.261114\pi\)
0.681992 + 0.731360i \(0.261114\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −29.9674 + 17.3017i −0.730912 + 0.421992i −0.818756 0.574142i \(-0.805336\pi\)
0.0878440 + 0.996134i \(0.472002\pi\)
\(42\) 0 0
\(43\) 11.5000 19.9186i 0.267442 0.463223i −0.700759 0.713398i \(-0.747155\pi\)
0.968200 + 0.250176i \(0.0804883\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 33.1753 + 19.1537i 0.705857 + 0.407527i 0.809525 0.587085i \(-0.199725\pi\)
−0.103668 + 0.994612i \(0.533058\pi\)
\(48\) 0 0
\(49\) −8.44158 14.6212i −0.172277 0.298393i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 19.0149i 0.358771i 0.983779 + 0.179386i \(0.0574110\pi\)
−0.983779 + 0.179386i \(0.942589\pi\)
\(54\) 0 0
\(55\) 48.3505 0.879101
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −2.96738 + 1.71322i −0.0502945 + 0.0290375i −0.524936 0.851141i \(-0.675911\pi\)
0.474642 + 0.880179i \(0.342578\pi\)
\(60\) 0 0
\(61\) −23.1753 + 40.1407i −0.379922 + 0.658045i −0.991051 0.133487i \(-0.957383\pi\)
0.611128 + 0.791532i \(0.290716\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −12.5910 7.26944i −0.193708 0.111838i
\(66\) 0 0
\(67\) −3.14947 5.45504i −0.0470070 0.0814185i 0.841565 0.540157i \(-0.181635\pi\)
−0.888572 + 0.458738i \(0.848302\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 35.9306i 0.506065i 0.967458 + 0.253033i \(0.0814280\pi\)
−0.967458 + 0.253033i \(0.918572\pi\)
\(72\) 0 0
\(73\) 47.3505 0.648637 0.324319 0.945948i \(-0.394865\pi\)
0.324319 + 0.945948i \(0.394865\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −142.993 + 82.5571i −1.85705 + 1.07217i
\(78\) 0 0
\(79\) 42.2921 73.2521i 0.535343 0.927242i −0.463803 0.885938i \(-0.653516\pi\)
0.999147 0.0413035i \(-0.0131510\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −33.1753 19.1537i −0.399702 0.230768i 0.286653 0.958034i \(-0.407457\pi\)
−0.686355 + 0.727266i \(0.740791\pi\)
\(84\) 0 0
\(85\) 21.3505 + 36.9802i 0.251183 + 0.435061i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 143.723i 1.61486i 0.589963 + 0.807430i \(0.299142\pi\)
−0.589963 + 0.807430i \(0.700858\pi\)
\(90\) 0 0
\(91\) 49.6495 0.545599
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 18.7663 10.8347i 0.197540 0.114050i
\(96\) 0 0
\(97\) −40.3832 + 69.9457i −0.416321 + 0.721089i −0.995566 0.0940641i \(-0.970014\pi\)
0.579245 + 0.815154i \(0.303347\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 105.942 + 61.1654i 1.04893 + 0.605598i 0.922349 0.386359i \(-0.126267\pi\)
0.126578 + 0.991957i \(0.459601\pi\)
\(102\) 0 0
\(103\) −36.8247 63.7823i −0.357522 0.619246i 0.630024 0.776575i \(-0.283045\pi\)
−0.987546 + 0.157330i \(0.949712\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 72.9108i 0.681410i 0.940170 + 0.340705i \(0.110666\pi\)
−0.940170 + 0.340705i \(0.889334\pi\)
\(108\) 0 0
\(109\) −31.2989 −0.287146 −0.143573 0.989640i \(-0.545859\pi\)
−0.143573 + 0.989640i \(0.545859\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −16.2269 + 9.36858i −0.143601 + 0.0829078i −0.570079 0.821590i \(-0.693087\pi\)
0.426478 + 0.904498i \(0.359754\pi\)
\(114\) 0 0
\(115\) 39.8763 69.0678i 0.346751 0.600590i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −126.285 72.9108i −1.06122 0.612696i
\(120\) 0 0
\(121\) 146.402 + 253.576i 1.20993 + 2.09567i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 105.415i 0.843320i
\(126\) 0 0
\(127\) −126.103 −0.992939 −0.496469 0.868054i \(-0.665370\pi\)
−0.496469 + 0.868054i \(0.665370\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 140.694 81.2299i 1.07400 0.620075i 0.144730 0.989471i \(-0.453769\pi\)
0.929272 + 0.369396i \(0.120435\pi\)
\(132\) 0 0
\(133\) −37.0000 + 64.0859i −0.278195 + 0.481849i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −90.3832 52.1827i −0.659731 0.380896i 0.132443 0.991191i \(-0.457718\pi\)
−0.792174 + 0.610295i \(0.791051\pi\)
\(138\) 0 0
\(139\) 30.6168 + 53.0299i 0.220265 + 0.381510i 0.954888 0.296965i \(-0.0959744\pi\)
−0.734623 + 0.678475i \(0.762641\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 124.430i 0.870139i
\(144\) 0 0
\(145\) −39.5463 −0.272733
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −128.344 + 74.0993i −0.861367 + 0.497311i −0.864470 0.502685i \(-0.832346\pi\)
0.00310272 + 0.999995i \(0.499012\pi\)
\(150\) 0 0
\(151\) −127.526 + 220.881i −0.844542 + 1.46279i 0.0414769 + 0.999139i \(0.486794\pi\)
−0.886019 + 0.463650i \(0.846540\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −46.0068 26.5621i −0.296818 0.171368i
\(156\) 0 0
\(157\) 146.227 + 253.272i 0.931381 + 1.61320i 0.780963 + 0.624577i \(0.214729\pi\)
0.150418 + 0.988622i \(0.451938\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 272.351i 1.69162i
\(162\) 0 0
\(163\) −93.5326 −0.573820 −0.286910 0.957958i \(-0.592628\pi\)
−0.286910 + 0.957958i \(0.592628\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 97.2269 56.1340i 0.582197 0.336131i −0.179809 0.983702i \(-0.557548\pi\)
0.762006 + 0.647570i \(0.224215\pi\)
\(168\) 0 0
\(169\) 65.7921 113.955i 0.389302 0.674292i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −205.227 118.488i −1.18628 0.684900i −0.228823 0.973468i \(-0.573488\pi\)
−0.957460 + 0.288568i \(0.906821\pi\)
\(174\) 0 0
\(175\) 78.5326 + 136.022i 0.448758 + 0.777271i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 234.599i 1.31061i 0.755366 + 0.655304i \(0.227459\pi\)
−0.755366 + 0.655304i \(0.772541\pi\)
\(180\) 0 0
\(181\) −221.636 −1.22451 −0.612254 0.790661i \(-0.709737\pi\)
−0.612254 + 0.790661i \(0.709737\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −103.883 + 59.9770i −0.561531 + 0.324200i
\(186\) 0 0
\(187\) −182.727 + 316.492i −0.977149 + 1.69247i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 130.162 + 75.1488i 0.681474 + 0.393449i 0.800410 0.599452i \(-0.204615\pi\)
−0.118936 + 0.992902i \(0.537948\pi\)
\(192\) 0 0
\(193\) 24.5000 + 42.4352i 0.126943 + 0.219872i 0.922491 0.386019i \(-0.126150\pi\)
−0.795548 + 0.605891i \(0.792817\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 276.827i 1.40521i 0.711579 + 0.702606i \(0.247980\pi\)
−0.711579 + 0.702606i \(0.752020\pi\)
\(198\) 0 0
\(199\) −198.935 −0.999672 −0.499836 0.866120i \(-0.666606\pi\)
−0.499836 + 0.866120i \(0.666606\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 116.955 67.5241i 0.576134 0.332631i
\(204\) 0 0
\(205\) 41.1237 71.2283i 0.200603 0.347455i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 160.610 + 92.7282i 0.768469 + 0.443676i
\(210\) 0 0
\(211\) 47.0068 + 81.4182i 0.222781 + 0.385868i 0.955651 0.294500i \(-0.0951531\pi\)
−0.732870 + 0.680368i \(0.761820\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 54.6678i 0.254269i
\(216\) 0 0
\(217\) 181.416 0.836017
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 95.1684 54.9455i 0.430626 0.248622i
\(222\) 0 0
\(223\) 77.8763 134.886i 0.349221 0.604869i −0.636890 0.770955i \(-0.719780\pi\)
0.986111 + 0.166086i \(0.0531128\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −138.448 79.9332i −0.609905 0.352129i 0.163023 0.986622i \(-0.447875\pi\)
−0.772928 + 0.634493i \(0.781209\pi\)
\(228\) 0 0
\(229\) −19.1237 33.1232i −0.0835095 0.144643i 0.821246 0.570575i \(-0.193280\pi\)
−0.904755 + 0.425932i \(0.859946\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 157.490i 0.675921i −0.941160 0.337960i \(-0.890263\pi\)
0.941160 0.337960i \(-0.109737\pi\)
\(234\) 0 0
\(235\) −91.0516 −0.387454
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 62.4742 36.0695i 0.261398 0.150918i −0.363574 0.931565i \(-0.618444\pi\)
0.624972 + 0.780647i \(0.285110\pi\)
\(240\) 0 0
\(241\) −113.370 + 196.362i −0.470413 + 0.814779i −0.999427 0.0338337i \(-0.989228\pi\)
0.529015 + 0.848613i \(0.322562\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 34.7527 + 20.0645i 0.141848 + 0.0818957i
\(246\) 0 0
\(247\) −27.8832 48.2950i −0.112887 0.195526i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 222.931i 0.888171i −0.895985 0.444085i \(-0.853529\pi\)
0.895985 0.444085i \(-0.146471\pi\)
\(252\) 0 0
\(253\) 682.557 2.69785
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −92.2011 + 53.2323i −0.358759 + 0.207130i −0.668536 0.743680i \(-0.733079\pi\)
0.309777 + 0.950809i \(0.399746\pi\)
\(258\) 0 0
\(259\) 204.818 354.755i 0.790803 1.36971i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −155.344 89.6877i −0.590660 0.341018i 0.174698 0.984622i \(-0.444105\pi\)
−0.765359 + 0.643604i \(0.777438\pi\)
\(264\) 0 0
\(265\) −22.5979 39.1407i −0.0852750 0.147701i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 416.351i 1.54777i 0.633324 + 0.773887i \(0.281690\pi\)
−0.633324 + 0.773887i \(0.718310\pi\)
\(270\) 0 0
\(271\) 396.907 1.46460 0.732302 0.680980i \(-0.238446\pi\)
0.732302 + 0.680980i \(0.238446\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 340.895 196.816i 1.23962 0.715695i
\(276\) 0 0
\(277\) −57.7731 + 100.066i −0.208567 + 0.361249i −0.951263 0.308379i \(-0.900213\pi\)
0.742696 + 0.669629i \(0.233547\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −422.564 243.967i −1.50379 0.868211i −0.999990 0.00438786i \(-0.998603\pi\)
−0.503795 0.863823i \(-0.668063\pi\)
\(282\) 0 0
\(283\) −169.825 294.145i −0.600087 1.03938i −0.992807 0.119724i \(-0.961799\pi\)
0.392720 0.919658i \(-0.371534\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 280.870i 0.978641i
\(288\) 0 0
\(289\) −33.7527 −0.116791
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 122.409 70.6728i 0.417778 0.241204i −0.276348 0.961058i \(-0.589124\pi\)
0.694126 + 0.719853i \(0.255791\pi\)
\(294\) 0 0
\(295\) 4.07207 7.05304i 0.0138036 0.0239086i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −177.746 102.622i −0.594468 0.343216i
\(300\) 0 0
\(301\) −93.3437 161.676i −0.310112 0.537130i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 110.169i 0.361209i
\(306\) 0 0
\(307\) 120.649 0.392995 0.196498 0.980504i \(-0.437043\pi\)
0.196498 + 0.980504i \(0.437043\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 119.254 68.8514i 0.383454 0.221387i −0.295866 0.955229i \(-0.595608\pi\)
0.679320 + 0.733842i \(0.262275\pi\)
\(312\) 0 0
\(313\) −129.266 + 223.896i −0.412991 + 0.715322i −0.995215 0.0977064i \(-0.968849\pi\)
0.582224 + 0.813029i \(0.302183\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −16.7079 9.64630i −0.0527063 0.0304300i 0.473415 0.880839i \(-0.343021\pi\)
−0.526122 + 0.850409i \(0.676354\pi\)
\(318\) 0 0
\(319\) −169.227 293.110i −0.530492 0.918839i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 163.787i 0.507081i
\(324\) 0 0
\(325\) −118.364 −0.364197
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 269.278 155.468i 0.818476 0.472547i
\(330\) 0 0
\(331\) −98.3953 + 170.426i −0.297267 + 0.514881i −0.975510 0.219957i \(-0.929408\pi\)
0.678243 + 0.734838i \(0.262742\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.9659 + 7.48585i 0.0387041 + 0.0223458i
\(336\) 0 0
\(337\) −158.720 274.911i −0.470979 0.815760i 0.528470 0.848952i \(-0.322766\pi\)
−0.999449 + 0.0331921i \(0.989433\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 454.659i 1.33331i
\(342\) 0 0
\(343\) 260.687 0.760022
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −537.407 + 310.272i −1.54872 + 0.894157i −0.550485 + 0.834845i \(0.685557\pi\)
−0.998240 + 0.0593116i \(0.981109\pi\)
\(348\) 0 0
\(349\) −189.512 + 328.245i −0.543015 + 0.940529i 0.455714 + 0.890126i \(0.349384\pi\)
−0.998729 + 0.0504030i \(0.983949\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 213.514 + 123.272i 0.604855 + 0.349213i 0.770949 0.636897i \(-0.219782\pi\)
−0.166094 + 0.986110i \(0.553116\pi\)
\(354\) 0 0
\(355\) −42.7011 73.9604i −0.120285 0.208339i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 572.791i 1.59552i 0.602976 + 0.797759i \(0.293981\pi\)
−0.602976 + 0.797759i \(0.706019\pi\)
\(360\) 0 0
\(361\) −277.883 −0.769759
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −97.4674 + 56.2728i −0.267034 + 0.154172i
\(366\) 0 0
\(367\) 93.9279 162.688i 0.255934 0.443291i −0.709214 0.704993i \(-0.750950\pi\)
0.965149 + 0.261701i \(0.0842836\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 133.663 + 77.1704i 0.360278 + 0.208007i
\(372\) 0 0
\(373\) 75.0584 + 130.005i 0.201229 + 0.348539i 0.948925 0.315503i \(-0.102173\pi\)
−0.747696 + 0.664042i \(0.768840\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 101.772i 0.269953i
\(378\) 0 0
\(379\) 26.6222 0.0702432 0.0351216 0.999383i \(-0.488818\pi\)
0.0351216 + 0.999383i \(0.488818\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 444.966 256.901i 1.16179 0.670760i 0.210058 0.977689i \(-0.432635\pi\)
0.951733 + 0.306929i \(0.0993013\pi\)
\(384\) 0 0
\(385\) 196.227 339.875i 0.509680 0.882792i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −22.1616 12.7950i −0.0569707 0.0328921i 0.471244 0.882003i \(-0.343805\pi\)
−0.528215 + 0.849111i \(0.677138\pi\)
\(390\) 0 0
\(391\) 301.402 + 522.044i 0.770849 + 1.33515i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 201.045i 0.508975i
\(396\) 0 0
\(397\) −388.804 −0.979356 −0.489678 0.871903i \(-0.662886\pi\)
−0.489678 + 0.871903i \(0.662886\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 34.0842 19.6785i 0.0849981 0.0490736i −0.456899 0.889519i \(-0.651040\pi\)
0.541897 + 0.840445i \(0.317706\pi\)
\(402\) 0 0
\(403\) −68.3574 + 118.398i −0.169621 + 0.293793i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −889.076 513.308i −2.18446 1.26120i
\(408\) 0 0
\(409\) −86.7200 150.204i −0.212029 0.367246i 0.740320 0.672255i \(-0.234674\pi\)
−0.952350 + 0.305009i \(0.901341\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 27.8118i 0.0673409i
\(414\) 0 0
\(415\) 91.0516 0.219401
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 115.031 66.4132i 0.274537 0.158504i −0.356411 0.934329i \(-0.616000\pi\)
0.630948 + 0.775825i \(0.282666\pi\)
\(420\) 0 0
\(421\) 317.447 549.834i 0.754031 1.30602i −0.191824 0.981429i \(-0.561440\pi\)
0.945855 0.324590i \(-0.105226\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 301.064 + 173.819i 0.708385 + 0.408986i
\(426\) 0 0
\(427\) 188.110 + 325.816i 0.440539 + 0.763035i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 602.424i 1.39774i 0.715251 + 0.698868i \(0.246313\pi\)
−0.715251 + 0.698868i \(0.753687\pi\)
\(432\) 0 0
\(433\) 266.155 0.614676 0.307338 0.951600i \(-0.400562\pi\)
0.307338 + 0.951600i \(0.400562\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 264.921 152.952i 0.606227 0.350005i
\(438\) 0 0
\(439\) −250.330 + 433.584i −0.570228 + 0.987664i 0.426314 + 0.904575i \(0.359812\pi\)
−0.996542 + 0.0830886i \(0.973522\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −261.098 150.745i −0.589386 0.340282i 0.175469 0.984485i \(-0.443856\pi\)
−0.764855 + 0.644203i \(0.777189\pi\)
\(444\) 0 0
\(445\) −170.804 295.842i −0.383830 0.664813i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 565.321i 1.25907i −0.776973 0.629534i \(-0.783246\pi\)
0.776973 0.629534i \(-0.216754\pi\)
\(450\) 0 0
\(451\) 703.907 1.56077
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −102.200 + 59.0049i −0.224614 + 0.129681i
\(456\) 0 0
\(457\) −26.1495 + 45.2922i −0.0572198 + 0.0991077i −0.893216 0.449627i \(-0.851557\pi\)
0.835997 + 0.548735i \(0.184890\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 166.357 + 96.0465i 0.360862 + 0.208344i 0.669459 0.742849i \(-0.266526\pi\)
−0.308597 + 0.951193i \(0.599859\pi\)
\(462\) 0 0
\(463\) −283.110 490.361i −0.611469 1.05909i −0.990993 0.133913i \(-0.957246\pi\)
0.379524 0.925182i \(-0.376088\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 174.405i 0.373459i 0.982411 + 0.186729i \(0.0597888\pi\)
−0.982411 + 0.186729i \(0.940211\pi\)
\(468\) 0 0
\(469\) −51.1275 −0.109014
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −405.187 + 233.935i −0.856633 + 0.494577i
\(474\) 0 0
\(475\) 88.2079 152.781i 0.185701 0.321643i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −473.784 273.539i −0.989110 0.571063i −0.0841020 0.996457i \(-0.526802\pi\)
−0.905008 + 0.425394i \(0.860135\pi\)
\(480\) 0 0
\(481\) 154.351 + 267.343i 0.320895 + 0.555807i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 191.970i 0.395815i
\(486\) 0 0
\(487\) −769.945 −1.58100 −0.790498 0.612464i \(-0.790178\pi\)
−0.790498 + 0.612464i \(0.790178\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −154.916 + 89.4407i −0.315511 + 0.182160i −0.649390 0.760456i \(-0.724976\pi\)
0.333879 + 0.942616i \(0.391642\pi\)
\(492\) 0 0
\(493\) 149.454 258.861i 0.303152 0.525074i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 252.571 + 145.822i 0.508190 + 0.293404i
\(498\) 0 0
\(499\) −192.655 333.688i −0.386082 0.668713i 0.605837 0.795589i \(-0.292838\pi\)
−0.991919 + 0.126876i \(0.959505\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 67.6630i 0.134519i −0.997736 0.0672594i \(-0.978574\pi\)
0.997736 0.0672594i \(-0.0214255\pi\)
\(504\) 0 0
\(505\) −290.763 −0.575769
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 523.292 302.123i 1.02808 0.593562i 0.111645 0.993748i \(-0.464388\pi\)
0.916434 + 0.400187i \(0.131055\pi\)
\(510\) 0 0
\(511\) 192.168 332.846i 0.376063 0.651361i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 151.602 + 87.5273i 0.294372 + 0.169956i
\(516\) 0 0
\(517\) −389.629 674.857i −0.753634 1.30533i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 273.678i 0.525294i −0.964892 0.262647i \(-0.915405\pi\)
0.964892 0.262647i \(-0.0845954\pi\)
\(522\) 0 0
\(523\) −687.402 −1.31434 −0.657172 0.753740i \(-0.728248\pi\)
−0.657172 + 0.753740i \(0.728248\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 347.739 200.767i 0.659846 0.380962i
\(528\) 0 0
\(529\) 298.428 516.892i 0.564136 0.977112i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −183.306 105.832i −0.343913 0.198558i
\(534\) 0 0
\(535\) −86.6495 150.081i −0.161962 0.280526i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 343.440i 0.637180i
\(540\) 0 0
\(541\) −664.543 −1.22836 −0.614180 0.789166i \(-0.710513\pi\)
−0.614180 + 0.789166i \(0.710513\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 64.4264 37.1966i 0.118214 0.0682507i
\(546\) 0 0
\(547\) 259.603 449.646i 0.474594 0.822022i −0.524982 0.851113i \(-0.675928\pi\)
0.999577 + 0.0290914i \(0.00926138\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −131.364 75.8431i −0.238410 0.137646i
\(552\) 0 0
\(553\) −343.278 594.576i −0.620757 1.07518i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 422.648i 0.758794i −0.925234 0.379397i \(-0.876131\pi\)
0.925234 0.379397i \(-0.123869\pi\)
\(558\) 0 0
\(559\) 140.687 0.251677
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 798.799 461.187i 1.41883 0.819159i 0.422630 0.906302i \(-0.361107\pi\)
0.996196 + 0.0871428i \(0.0277737\pi\)
\(564\) 0 0
\(565\) 22.2678 38.5690i 0.0394121 0.0682637i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 914.445 + 527.955i 1.60711 + 0.927865i 0.990013 + 0.140978i \(0.0450247\pi\)
0.617097 + 0.786887i \(0.288309\pi\)
\(570\) 0 0
\(571\) 401.524 + 695.460i 0.703195 + 1.21797i 0.967339 + 0.253486i \(0.0815772\pi\)
−0.264144 + 0.964483i \(0.585089\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 649.283i 1.12919i
\(576\) 0 0
\(577\) −96.6495 −0.167503 −0.0837517 0.996487i \(-0.526690\pi\)
−0.0837517 + 0.996487i \(0.526690\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −269.278 + 155.468i −0.463474 + 0.267587i
\(582\) 0 0
\(583\) 193.402 334.982i 0.331736 0.574584i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −870.497 502.582i −1.48296 0.856187i −0.483146 0.875540i \(-0.660506\pi\)
−0.999813 + 0.0193528i \(0.993839\pi\)
\(588\) 0 0
\(589\) −101.883 176.467i −0.172976 0.299604i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 752.444i 1.26888i −0.772973 0.634439i \(-0.781231\pi\)
0.772973 0.634439i \(-0.218769\pi\)
\(594\) 0 0
\(595\) 346.598 0.582517
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 24.0857 13.9059i 0.0402099 0.0232152i −0.479760 0.877400i \(-0.659276\pi\)
0.519970 + 0.854184i \(0.325943\pi\)
\(600\) 0 0
\(601\) 475.356 823.340i 0.790942 1.36995i −0.134443 0.990921i \(-0.542925\pi\)
0.925385 0.379030i \(-0.123742\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −602.715 347.978i −0.996223 0.575169i
\(606\) 0 0
\(607\) 161.306 + 279.390i 0.265743 + 0.460280i 0.967758 0.251882i \(-0.0810495\pi\)
−0.702015 + 0.712162i \(0.747716\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 234.321i 0.383504i
\(612\) 0 0
\(613\) −138.206 −0.225459 −0.112730 0.993626i \(-0.535959\pi\)
−0.112730 + 0.993626i \(0.535959\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 682.084 393.802i 1.10548 0.638252i 0.167829 0.985816i \(-0.446324\pi\)
0.937656 + 0.347564i \(0.112991\pi\)
\(618\) 0 0
\(619\) −121.747 + 210.873i −0.196684 + 0.340667i −0.947451 0.319900i \(-0.896351\pi\)
0.750767 + 0.660567i \(0.229684\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1010.28 + 583.287i 1.62164 + 0.936255i
\(624\) 0 0
\(625\) −116.603 201.963i −0.186565 0.323140i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 906.662i 1.44143i
\(630\) 0 0
\(631\) 111.924 0.177376 0.0886879 0.996059i \(-0.471733\pi\)
0.0886879 + 0.996059i \(0.471733\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 259.574 149.865i 0.408777 0.236008i
\(636\) 0 0
\(637\) 51.6358 89.4359i 0.0810609 0.140402i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −632.095 364.940i −0.986107 0.569329i −0.0819990 0.996632i \(-0.526130\pi\)
−0.904108 + 0.427303i \(0.859464\pi\)
\(642\) 0 0
\(643\) −288.500 499.697i −0.448678 0.777133i 0.549622 0.835413i \(-0.314772\pi\)
−0.998300 + 0.0582801i \(0.981438\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 129.029i 0.199426i −0.995016 0.0997130i \(-0.968208\pi\)
0.995016 0.0997130i \(-0.0317925\pi\)
\(648\) 0 0
\(649\) 69.7011 0.107398
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1026.62 592.717i 1.57215 0.907682i 0.576247 0.817275i \(-0.304517\pi\)
0.995905 0.0904070i \(-0.0288168\pi\)
\(654\) 0 0
\(655\) −193.072 + 334.411i −0.294767 + 0.510551i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 947.808 + 547.217i 1.43825 + 0.830375i 0.997728 0.0673658i \(-0.0214595\pi\)
0.440524 + 0.897741i \(0.354793\pi\)
\(660\) 0 0
\(661\) 604.876 + 1047.68i 0.915093 + 1.58499i 0.806765 + 0.590872i \(0.201216\pi\)
0.108327 + 0.994115i \(0.465451\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 175.888i 0.264493i
\(666\) 0 0
\(667\) −558.269 −0.836984
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 816.550 471.435i 1.21692 0.702586i
\(672\) 0 0
\(673\) −508.615 + 880.948i −0.755743 + 1.30899i 0.189260 + 0.981927i \(0.439391\pi\)
−0.945004 + 0.327059i \(0.893942\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 689.890 + 398.308i 1.01904 + 0.588343i 0.913826 0.406107i \(-0.133114\pi\)
0.105214 + 0.994450i \(0.466447\pi\)
\(678\) 0 0
\(679\) 327.784 + 567.738i 0.482745 + 0.836139i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 400.485i 0.586361i 0.956057 + 0.293181i \(0.0947138\pi\)
−0.956057 + 0.293181i \(0.905286\pi\)
\(684\) 0 0
\(685\) 248.062 0.362135
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −100.728 + 58.1556i −0.146195 + 0.0844057i
\(690\) 0 0
\(691\) −216.423 + 374.855i −0.313202 + 0.542482i −0.979054 0.203602i \(-0.934735\pi\)
0.665852 + 0.746084i \(0.268068\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −126.045 72.7720i −0.181359 0.104708i
\(696\) 0 0
\(697\) 310.830 + 538.373i 0.445954 + 0.772415i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 65.4412i 0.0933541i 0.998910 + 0.0466770i \(0.0148632\pi\)
−0.998910 + 0.0466770i \(0.985137\pi\)
\(702\) 0 0
\(703\) −460.103 −0.654485
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 859.911 496.470i 1.21628 0.702221i
\(708\) 0 0
\(709\) 100.461 174.003i 0.141693 0.245420i −0.786441 0.617665i \(-0.788079\pi\)
0.928134 + 0.372245i \(0.121412\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −649.471 374.972i −0.910899 0.525908i
\(714\) 0 0
\(715\) 147.876 + 256.129i 0.206820 + 0.358223i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1062.98i 1.47841i 0.673478 + 0.739207i \(0.264800\pi\)
−0.673478 + 0.739207i \(0.735200\pi\)
\(720\) 0 0
\(721\) −597.801 −0.829128
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −278.821 + 160.977i −0.384581 + 0.222038i
\(726\) 0 0
\(727\) −495.629 + 858.455i −0.681746 + 1.18082i 0.292702 + 0.956204i \(0.405446\pi\)
−0.974448 + 0.224614i \(0.927888\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −357.844 206.601i −0.489526 0.282628i
\(732\) 0 0
\(733\) −590.134 1022.14i −0.805095 1.39446i −0.916227 0.400659i \(-0.868781\pi\)
0.111133 0.993806i \(-0.464552\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 128.134i 0.173859i
\(738\) 0 0
\(739\) −599.351 −0.811029 −0.405515 0.914089i \(-0.632908\pi\)
−0.405515 + 0.914089i \(0.632908\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −287.083 + 165.747i −0.386383 + 0.223078i −0.680592 0.732663i \(-0.738277\pi\)
0.294209 + 0.955741i \(0.404944\pi\)
\(744\) 0 0
\(745\) 176.124 305.055i 0.236408 0.409470i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 512.519 + 295.903i 0.684271 + 0.395064i
\(750\) 0 0
\(751\) −76.0448 131.713i −0.101258 0.175384i 0.810945 0.585122i \(-0.198953\pi\)
−0.912203 + 0.409738i \(0.865620\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 606.222i 0.802943i
\(756\) 0 0
\(757\) 1179.61 1.55827 0.779134 0.626858i \(-0.215659\pi\)
0.779134 + 0.626858i \(0.215659\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1162.58 671.214i 1.52770 0.882016i 0.528239 0.849096i \(-0.322853\pi\)
0.999458 0.0329205i \(-0.0104808\pi\)
\(762\) 0 0
\(763\) −127.024 + 220.013i −0.166480 + 0.288352i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −18.1510 10.4795i −0.0236649 0.0136629i
\(768\) 0 0
\(769\) −548.512 950.051i −0.713280 1.23544i −0.963619 0.267279i \(-0.913876\pi\)
0.250339 0.968158i \(-0.419458\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1181.39i 1.52832i 0.645028 + 0.764159i \(0.276846\pi\)
−0.645028 + 0.764159i \(0.723154\pi\)
\(774\) 0 0
\(775\) −432.495 −0.558058
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 273.208 157.737i 0.350716 0.202486i
\(780\) 0 0
\(781\) 365.454 632.984i 0.467931 0.810479i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −601.993 347.561i −0.766870 0.442753i
\(786\) 0 0
\(787\) −18.0311 31.2308i −0.0229112 0.0396834i 0.854342 0.519710i \(-0.173960\pi\)
−0.877254 + 0.480027i \(0.840627\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 152.087i 0.192271i
\(792\) 0 0
\(793\) −283.519 −0.357527
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −115.618 + 66.7523i −0.145067 + 0.0837544i −0.570777 0.821105i \(-0.693358\pi\)
0.425710 + 0.904860i \(0.360024\pi\)
\(798\) 0 0
\(799\) 344.103 596.004i 0.430667 0.745938i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −834.167 481.607i −1.03881 0.599759i
\(804\) 0 0
\(805\) −323.670 560.613i −0.402074 0.696413i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1053.66i 1.30242i 0.758898 + 0.651209i \(0.225738\pi\)
−0.758898 + 0.651209i \(0.774262\pi\)
\(810\) 0 0
\(811\) −434.464 −0.535714 −0.267857 0.963459i \(-0.586316\pi\)
−0.267857 + 0.963459i \(0.586316\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 192.530 111.157i 0.236233 0.136389i
\(816\) 0 0
\(817\) −104.844 + 181.595i −0.128328 + 0.222270i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 252.436 + 145.744i 0.307474 + 0.177520i 0.645796 0.763510i \(-0.276526\pi\)
−0.338322 + 0.941031i \(0.609859\pi\)
\(822\) 0 0
\(823\) −168.409 291.693i −0.204628 0.354426i 0.745386 0.666633i \(-0.232265\pi\)
−0.950014 + 0.312207i \(0.898932\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1029.27i 1.24458i −0.782785 0.622292i \(-0.786202\pi\)
0.782785 0.622292i \(-0.213798\pi\)
\(828\) 0 0
\(829\) −790.674 −0.953768 −0.476884 0.878966i \(-0.658234\pi\)
−0.476884 + 0.878966i \(0.658234\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −262.675 + 151.656i −0.315336 + 0.182060i
\(834\) 0 0
\(835\) −133.423 + 231.095i −0.159788 + 0.276760i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −655.031 378.182i −0.780728 0.450754i 0.0559600 0.998433i \(-0.482178\pi\)
−0.836688 + 0.547679i \(0.815511\pi\)
\(840\) 0 0
\(841\) −282.088 488.591i −0.335420 0.580964i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 312.757i 0.370127i
\(846\) 0 0
\(847\) 2376.65 2.80596
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1466.50 + 846.686i −1.72327 + 0.994930i
\(852\) 0 0
\(853\) −599.175 + 1037.80i −0.702433 + 1.21665i 0.265177 + 0.964200i \(0.414570\pi\)
−0.967610 + 0.252450i \(0.918764\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −605.629 349.660i −0.706685 0.408005i 0.103147 0.994666i \(-0.467109\pi\)
−0.809832 + 0.586661i \(0.800442\pi\)
\(858\) 0 0
\(859\) −278.734 482.781i −0.324486 0.562027i 0.656922 0.753959i \(-0.271858\pi\)
−0.981408 + 0.191932i \(0.938525\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 99.3954i 0.115174i −0.998340 0.0575871i \(-0.981659\pi\)
0.998340 0.0575871i \(-0.0183407\pi\)
\(864\) 0 0
\(865\) 563.258 0.651165
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1490.11 + 860.314i −1.71474 + 0.990004i
\(870\) 0 0
\(871\) 19.2648 33.3676i 0.0221180 0.0383096i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −741.004 427.819i −0.846861 0.488936i
\(876\) 0 0
\(877\) 401.292 + 695.058i 0.457574 + 0.792541i 0.998832 0.0483154i \(-0.0153853\pi\)
−0.541258 + 0.840856i \(0.682052\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 524.266i 0.595080i 0.954709 + 0.297540i \(0.0961662\pi\)
−0.954709 + 0.297540i \(0.903834\pi\)
\(882\) 0 0
\(883\) 993.894 1.12559 0.562794 0.826597i \(-0.309727\pi\)
0.562794 + 0.826597i \(0.309727\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −616.643 + 356.019i −0.695200 + 0.401374i −0.805557 0.592518i \(-0.798134\pi\)
0.110357 + 0.993892i \(0.464801\pi\)
\(888\) 0 0
\(889\) −511.780 + 886.429i −0.575681 + 0.997108i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −302.454 174.622i −0.338694 0.195545i
\(894\) 0 0
\(895\) −278.804 482.903i −0.311513 0.539557i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 371.869i 0.413647i
\(900\) 0 0
\(901\) 341.609 0.379144
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 456.220 263.399i 0.504111 0.291048i
\(906\) 0 0
\(907\) −374.473 + 648.606i −0.412870 + 0.715111i −0.995202 0.0978396i \(-0.968807\pi\)
0.582333 + 0.812951i \(0.302140\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 3.87633 + 2.23800i 0.00425503 + 0.00245664i 0.502126 0.864794i \(-0.332551\pi\)
−0.497871 + 0.867251i \(0.665885\pi\)
\(912\) 0 0
\(913\) 389.629 + 674.857i 0.426757 + 0.739165i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1318.66i 1.43802i
\(918\) 0 0
\(919\) −1592.91 −1.73331 −0.866653 0.498912i \(-0.833733\pi\)
−0.866653 + 0.498912i \(0.833733\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −190.337 + 109.891i −0.206215 + 0.119059i
\(924\) 0 0
\(925\) −488.285 + 845.735i −0.527876 + 0.914308i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 770.784 + 445.012i 0.829692 + 0.479023i 0.853747 0.520688i \(-0.174324\pi\)
−0.0240553 + 0.999711i \(0.507658\pi\)
\(930\) 0 0
\(931\) 76.9605 + 133.300i 0.0826644 + 0.143179i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 868.633i 0.929019i
\(936\) 0 0
\(937\) 443.554 0.473377 0.236688 0.971586i \(-0.423938\pi\)
0.236688 + 0.971586i \(0.423938\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −69.7458 + 40.2678i −0.0741188 + 0.0427925i −0.536601 0.843836i \(-0.680292\pi\)
0.462483 + 0.886628i \(0.346959\pi\)
\(942\) 0 0
\(943\) 580.536 1005.52i 0.615627 1.06630i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 245.861 + 141.948i 0.259621 + 0.149892i 0.624162 0.781295i \(-0.285441\pi\)
−0.364541 + 0.931187i \(0.618774\pi\)
\(948\) 0 0
\(949\) 144.818 + 250.832i 0.152601 + 0.264312i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1123.17i 1.17857i 0.807927 + 0.589283i \(0.200590\pi\)
−0.807927 + 0.589283i \(0.799410\pi\)
\(954\) 0 0
\(955\) −357.237 −0.374070
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −733.626 + 423.559i −0.764991 + 0.441668i
\(960\) 0 0
\(961\) 230.727 399.631i 0.240090 0.415849i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −100.863 58.2331i −0.104521 0.0603452i
\(966\) 0 0
\(967\) 699.536 + 1211.63i 0.723409 + 1.25298i 0.959626 + 0.281281i \(0.0907592\pi\)
−0.236217 + 0.971700i \(0.575907\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1705.41i 1.75634i 0.478345 + 0.878172i \(0.341237\pi\)
−0.478345 + 0.878172i \(0.658763\pi\)
\(972\) 0 0
\(973\) 497.024 0.510816
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1005.50 + 580.524i −1.02917 + 0.594190i −0.916746 0.399470i \(-0.869194\pi\)
−0.112422 + 0.993661i \(0.535861\pi\)
\(978\) 0 0
\(979\) 1461.81 2531.94i 1.49317 2.58625i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1023.46 + 590.895i 1.04116 + 0.601114i 0.920162 0.391539i \(-0.128057\pi\)
0.120999 + 0.992653i \(0.461390\pi\)
\(984\) 0 0
\(985\) −328.989 569.826i −0.333999 0.578504i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 771.737i 0.780320i
\(990\) 0 0
\(991\) −969.527 −0.978332 −0.489166 0.872191i \(-0.662699\pi\)
−0.489166 + 0.872191i \(0.662699\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 409.492 236.420i 0.411549 0.237608i
\(996\) 0 0
\(997\) 390.124 675.714i 0.391298 0.677747i −0.601323 0.799006i \(-0.705360\pi\)
0.992621 + 0.121258i \(0.0386930\pi\)
\(998\) 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 1728.3.q.g.1601.1 4
3.2 odd 2 576.3.q.d.65.1 4
4.3 odd 2 1728.3.q.h.1601.1 4
8.3 odd 2 432.3.q.b.305.2 4
8.5 even 2 108.3.g.a.89.2 4
9.4 even 3 576.3.q.d.257.1 4
9.5 odd 6 inner 1728.3.q.g.449.1 4
12.11 even 2 576.3.q.g.65.2 4
24.5 odd 2 36.3.g.a.29.2 yes 4
24.11 even 2 144.3.q.b.65.1 4
36.23 even 6 1728.3.q.h.449.1 4
36.31 odd 6 576.3.q.g.257.2 4
40.13 odd 4 2700.3.u.b.2249.2 8
40.29 even 2 2700.3.p.b.1601.1 4
40.37 odd 4 2700.3.u.b.2249.3 8
72.5 odd 6 108.3.g.a.17.2 4
72.11 even 6 1296.3.e.e.161.2 4
72.13 even 6 36.3.g.a.5.2 4
72.29 odd 6 324.3.c.b.161.2 4
72.43 odd 6 1296.3.e.e.161.3 4
72.59 even 6 432.3.q.b.17.2 4
72.61 even 6 324.3.c.b.161.3 4
72.67 odd 6 144.3.q.b.113.1 4
120.29 odd 2 900.3.p.a.101.1 4
120.53 even 4 900.3.u.a.749.2 8
120.77 even 4 900.3.u.a.749.3 8
360.13 odd 12 900.3.u.a.149.3 8
360.77 even 12 2700.3.u.b.449.2 8
360.149 odd 6 2700.3.p.b.2501.1 4
360.157 odd 12 900.3.u.a.149.2 8
360.229 even 6 900.3.p.a.401.1 4
360.293 even 12 2700.3.u.b.449.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
36.3.g.a.5.2 4 72.13 even 6
36.3.g.a.29.2 yes 4 24.5 odd 2
108.3.g.a.17.2 4 72.5 odd 6
108.3.g.a.89.2 4 8.5 even 2
144.3.q.b.65.1 4 24.11 even 2
144.3.q.b.113.1 4 72.67 odd 6
324.3.c.b.161.2 4 72.29 odd 6
324.3.c.b.161.3 4 72.61 even 6
432.3.q.b.17.2 4 72.59 even 6
432.3.q.b.305.2 4 8.3 odd 2
576.3.q.d.65.1 4 3.2 odd 2
576.3.q.d.257.1 4 9.4 even 3
576.3.q.g.65.2 4 12.11 even 2
576.3.q.g.257.2 4 36.31 odd 6
900.3.p.a.101.1 4 120.29 odd 2
900.3.p.a.401.1 4 360.229 even 6
900.3.u.a.149.2 8 360.157 odd 12
900.3.u.a.149.3 8 360.13 odd 12
900.3.u.a.749.2 8 120.53 even 4
900.3.u.a.749.3 8 120.77 even 4
1296.3.e.e.161.2 4 72.11 even 6
1296.3.e.e.161.3 4 72.43 odd 6
1728.3.q.g.449.1 4 9.5 odd 6 inner
1728.3.q.g.1601.1 4 1.1 even 1 trivial
1728.3.q.h.449.1 4 36.23 even 6
1728.3.q.h.1601.1 4 4.3 odd 2
2700.3.p.b.1601.1 4 40.29 even 2
2700.3.p.b.2501.1 4 360.149 odd 6
2700.3.u.b.449.2 8 360.77 even 12
2700.3.u.b.449.3 8 360.293 even 12
2700.3.u.b.2249.2 8 40.13 odd 4
2700.3.u.b.2249.3 8 40.37 odd 4