Properties

Label 1728.3.o.d.127.1
Level $1728$
Weight $3$
Character 1728.127
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,3,Mod(127,1728)]
 
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([3, 0, 4]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.127");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.o (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: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.121550625.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} - 4x^{6} - 9x^{5} + 23x^{4} + 18x^{3} - 16x^{2} + 8x + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{6} \)
Twist minimal: no (minimal twist has level 144)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 127.1
Root \(2.25820 - 0.369600i\) of defining polynomial
Character \(\chi\) \(=\) 1728.127
Dual form 1728.3.o.d.1279.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-3.31174 - 5.73610i) q^{5} +(-8.46808 - 4.88905i) q^{7} +O(q^{10})\) \(q+(-3.31174 - 5.73610i) q^{5} +(-8.46808 - 4.88905i) q^{7} +(10.0623 + 5.80948i) q^{11} +(8.93521 + 15.4762i) q^{13} -2.37652 q^{17} +14.0337i q^{19} +(35.9635 - 20.7636i) q^{23} +(-9.43521 + 16.3423i) q^{25} +(-5.68826 + 9.85236i) q^{29} +(-18.0334 + 10.4116i) q^{31} +64.7650i q^{35} -35.7409 q^{37} +(2.62957 + 4.55456i) q^{41} +(54.4939 + 31.4621i) q^{43} +(-4.28568 - 2.47434i) q^{47} +(23.3056 + 40.3666i) q^{49} +75.7409 q^{53} -76.9578i q^{55} +(50.3115 - 29.0474i) q^{59} +(-5.93521 + 10.2801i) q^{61} +(59.1822 - 102.507i) q^{65} +(20.6216 - 11.9059i) q^{67} -46.4758i q^{71} -104.352 q^{73} +(-56.8056 - 98.3903i) q^{77} +(-18.0334 - 10.4116i) q^{79} +(-24.4103 - 14.0933i) q^{83} +(7.87043 + 13.6320i) q^{85} -73.0122 q^{89} -174.739i q^{91} +(80.4984 - 46.4758i) q^{95} +(71.1113 - 123.168i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 6 q^{5} + 10 q^{13} - 60 q^{17} - 14 q^{25} - 66 q^{29} - 40 q^{37} + 144 q^{41} + 2 q^{49} + 360 q^{53} + 14 q^{61} + 330 q^{65} - 220 q^{73} - 270 q^{77} - 60 q^{85} - 912 q^{89} + 200 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −3.31174 5.73610i −0.662348 1.14722i −0.979997 0.199011i \(-0.936227\pi\)
0.317650 0.948208i \(-0.397106\pi\)
\(6\) 0 0
\(7\) −8.46808 4.88905i −1.20973 0.698436i −0.247027 0.969009i \(-0.579454\pi\)
−0.962700 + 0.270573i \(0.912787\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 10.0623 + 5.80948i 0.914755 + 0.528134i 0.881958 0.471328i \(-0.156225\pi\)
0.0327970 + 0.999462i \(0.489559\pi\)
\(12\) 0 0
\(13\) 8.93521 + 15.4762i 0.687324 + 1.19048i 0.972700 + 0.232065i \(0.0745481\pi\)
−0.285376 + 0.958416i \(0.592119\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.37652 −0.139796 −0.0698978 0.997554i \(-0.522267\pi\)
−0.0698978 + 0.997554i \(0.522267\pi\)
\(18\) 0 0
\(19\) 14.0337i 0.738614i 0.929308 + 0.369307i \(0.120405\pi\)
−0.929308 + 0.369307i \(0.879595\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 35.9635 20.7636i 1.56363 0.902763i 0.566747 0.823892i \(-0.308202\pi\)
0.996885 0.0788718i \(-0.0251318\pi\)
\(24\) 0 0
\(25\) −9.43521 + 16.3423i −0.377409 + 0.653691i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −5.68826 + 9.85236i −0.196147 + 0.339737i −0.947276 0.320419i \(-0.896176\pi\)
0.751129 + 0.660156i \(0.229510\pi\)
\(30\) 0 0
\(31\) −18.0334 + 10.4116i −0.581723 + 0.335858i −0.761818 0.647791i \(-0.775693\pi\)
0.180095 + 0.983649i \(0.442360\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 64.7650i 1.85043i
\(36\) 0 0
\(37\) −35.7409 −0.965969 −0.482984 0.875629i \(-0.660447\pi\)
−0.482984 + 0.875629i \(0.660447\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.62957 + 4.55456i 0.0641359 + 0.111087i 0.896310 0.443427i \(-0.146238\pi\)
−0.832174 + 0.554514i \(0.812904\pi\)
\(42\) 0 0
\(43\) 54.4939 + 31.4621i 1.26730 + 0.731676i 0.974476 0.224490i \(-0.0720716\pi\)
0.292824 + 0.956166i \(0.405405\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.28568 2.47434i −0.0911848 0.0526456i 0.453714 0.891147i \(-0.350099\pi\)
−0.544899 + 0.838502i \(0.683432\pi\)
\(48\) 0 0
\(49\) 23.3056 + 40.3666i 0.475625 + 0.823807i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 75.7409 1.42907 0.714536 0.699598i \(-0.246638\pi\)
0.714536 + 0.699598i \(0.246638\pi\)
\(54\) 0 0
\(55\) 76.9578i 1.39923i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 50.3115 29.0474i 0.852738 0.492328i −0.00883587 0.999961i \(-0.502813\pi\)
0.861574 + 0.507633i \(0.169479\pi\)
\(60\) 0 0
\(61\) −5.93521 + 10.2801i −0.0972986 + 0.168526i −0.910566 0.413365i \(-0.864354\pi\)
0.813267 + 0.581891i \(0.197687\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 59.1822 102.507i 0.910495 1.57702i
\(66\) 0 0
\(67\) 20.6216 11.9059i 0.307785 0.177700i −0.338150 0.941092i \(-0.609801\pi\)
0.645935 + 0.763393i \(0.276468\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 46.4758i 0.654589i −0.944922 0.327294i \(-0.893863\pi\)
0.944922 0.327294i \(-0.106137\pi\)
\(72\) 0 0
\(73\) −104.352 −1.42948 −0.714741 0.699390i \(-0.753455\pi\)
−0.714741 + 0.699390i \(0.753455\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −56.8056 98.3903i −0.737736 1.27780i
\(78\) 0 0
\(79\) −18.0334 10.4116i −0.228271 0.131792i 0.381503 0.924368i \(-0.375407\pi\)
−0.609774 + 0.792575i \(0.708740\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −24.4103 14.0933i −0.294100 0.169799i 0.345689 0.938349i \(-0.387645\pi\)
−0.639789 + 0.768550i \(0.720978\pi\)
\(84\) 0 0
\(85\) 7.87043 + 13.6320i 0.0925932 + 0.160376i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −73.0122 −0.820362 −0.410181 0.912004i \(-0.634534\pi\)
−0.410181 + 0.912004i \(0.634534\pi\)
\(90\) 0 0
\(91\) 174.739i 1.92021i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 80.4984 46.4758i 0.847352 0.489219i
\(96\) 0 0
\(97\) 71.1113 123.168i 0.733106 1.26978i −0.222443 0.974946i \(-0.571403\pi\)
0.955549 0.294831i \(-0.0952634\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.6761 23.6876i 0.135407 0.234531i −0.790346 0.612661i \(-0.790099\pi\)
0.925753 + 0.378130i \(0.123433\pi\)
\(102\) 0 0
\(103\) 127.021 73.3358i 1.23322 0.711998i 0.265517 0.964106i \(-0.414457\pi\)
0.967699 + 0.252108i \(0.0811240\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 192.573i 1.79975i −0.436146 0.899876i \(-0.643657\pi\)
0.436146 0.899876i \(-0.356343\pi\)
\(108\) 0 0
\(109\) −0.259148 −0.00237750 −0.00118875 0.999999i \(-0.500378\pi\)
−0.00118875 + 0.999999i \(0.500378\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 99.4291 + 172.216i 0.879904 + 1.52404i 0.851446 + 0.524443i \(0.175726\pi\)
0.0284580 + 0.999595i \(0.490940\pi\)
\(114\) 0 0
\(115\) −238.204 137.527i −2.07134 1.19589i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 20.1246 + 11.6190i 0.169114 + 0.0976382i
\(120\) 0 0
\(121\) 7.00000 + 12.1244i 0.0578512 + 0.100201i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −40.5991 −0.324793
\(126\) 0 0
\(127\) 72.2477i 0.568879i −0.958694 0.284440i \(-0.908192\pi\)
0.958694 0.284440i \(-0.0918075\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 159.693 92.1988i 1.21903 0.703808i 0.254320 0.967120i \(-0.418148\pi\)
0.964711 + 0.263312i \(0.0848149\pi\)
\(132\) 0 0
\(133\) 68.6113 118.838i 0.515874 0.893520i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 69.8765 121.030i 0.510048 0.883428i −0.489885 0.871787i \(-0.662961\pi\)
0.999932 0.0116411i \(-0.00370557\pi\)
\(138\) 0 0
\(139\) 151.722 87.5967i 1.09153 0.630192i 0.157543 0.987512i \(-0.449643\pi\)
0.933982 + 0.357320i \(0.116309\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 207.636i 1.45200i
\(144\) 0 0
\(145\) 75.3521 0.519670
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 95.5343 + 165.470i 0.641170 + 1.11054i 0.985172 + 0.171570i \(0.0548839\pi\)
−0.344002 + 0.938969i \(0.611783\pi\)
\(150\) 0 0
\(151\) 6.27360 + 3.62206i 0.0415470 + 0.0239872i 0.520630 0.853783i \(-0.325697\pi\)
−0.479083 + 0.877770i \(0.659031\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 119.444 + 68.9609i 0.770606 + 0.444909i
\(156\) 0 0
\(157\) 36.0648 + 62.4660i 0.229712 + 0.397873i 0.957723 0.287693i \(-0.0928882\pi\)
−0.728011 + 0.685566i \(0.759555\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −406.056 −2.52209
\(162\) 0 0
\(163\) 112.269i 0.688769i −0.938829 0.344384i \(-0.888088\pi\)
0.938829 0.344384i \(-0.111912\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −15.8389 + 9.14461i −0.0948439 + 0.0547581i −0.546672 0.837347i \(-0.684105\pi\)
0.451828 + 0.892105i \(0.350772\pi\)
\(168\) 0 0
\(169\) −75.1761 + 130.209i −0.444829 + 0.770466i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −146.652 + 254.008i −0.847698 + 1.46826i 0.0355603 + 0.999368i \(0.488678\pi\)
−0.883258 + 0.468888i \(0.844655\pi\)
\(174\) 0 0
\(175\) 159.796 92.2585i 0.913122 0.527191i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 92.9516i 0.519283i −0.965705 0.259641i \(-0.916396\pi\)
0.965705 0.259641i \(-0.0836043\pi\)
\(180\) 0 0
\(181\) −128.445 −0.709642 −0.354821 0.934934i \(-0.615458\pi\)
−0.354821 + 0.934934i \(0.615458\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 118.364 + 205.013i 0.639807 + 1.10818i
\(186\) 0 0
\(187\) −23.9133 13.8064i −0.127879 0.0738308i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 260.316 + 150.294i 1.36291 + 0.786877i 0.990011 0.140994i \(-0.0450297\pi\)
0.372901 + 0.927871i \(0.378363\pi\)
\(192\) 0 0
\(193\) 19.6296 + 33.9994i 0.101708 + 0.176163i 0.912388 0.409326i \(-0.134236\pi\)
−0.810681 + 0.585489i \(0.800903\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −94.7530 −0.480980 −0.240490 0.970652i \(-0.577308\pi\)
−0.240490 + 0.970652i \(0.577308\pi\)
\(198\) 0 0
\(199\) 111.360i 0.559598i 0.960059 + 0.279799i \(0.0902679\pi\)
−0.960059 + 0.279799i \(0.909732\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 96.3374 55.6204i 0.474568 0.273992i
\(204\) 0 0
\(205\) 17.4169 30.1670i 0.0849606 0.147156i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −81.5282 + 141.211i −0.390087 + 0.675651i
\(210\) 0 0
\(211\) −102.714 + 59.3021i −0.486797 + 0.281053i −0.723245 0.690592i \(-0.757350\pi\)
0.236447 + 0.971644i \(0.424017\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 416.777i 1.93850i
\(216\) 0 0
\(217\) 203.611 0.938301
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −21.2348 36.7797i −0.0960849 0.166424i
\(222\) 0 0
\(223\) 21.0153 + 12.1332i 0.0942389 + 0.0544089i 0.546379 0.837538i \(-0.316006\pi\)
−0.452140 + 0.891947i \(0.649339\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.0623 + 5.80948i 0.0443273 + 0.0255924i 0.522000 0.852946i \(-0.325186\pi\)
−0.477673 + 0.878538i \(0.658519\pi\)
\(228\) 0 0
\(229\) −9.06479 15.7007i −0.0395842 0.0685619i 0.845555 0.533889i \(-0.179270\pi\)
−0.885139 + 0.465327i \(0.845937\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 219.785 0.943283 0.471642 0.881790i \(-0.343662\pi\)
0.471642 + 0.881790i \(0.343662\pi\)
\(234\) 0 0
\(235\) 32.7775i 0.139479i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 139.568 80.5799i 0.583968 0.337154i −0.178740 0.983896i \(-0.557202\pi\)
0.762709 + 0.646742i \(0.223869\pi\)
\(240\) 0 0
\(241\) 84.5930 146.519i 0.351008 0.607964i −0.635418 0.772168i \(-0.719172\pi\)
0.986426 + 0.164204i \(0.0525056\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 154.364 267.367i 0.630058 1.09129i
\(246\) 0 0
\(247\) −217.188 + 125.394i −0.879305 + 0.507667i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 219.255i 0.873524i 0.899577 + 0.436762i \(0.143875\pi\)
−0.899577 + 0.436762i \(0.856125\pi\)
\(252\) 0 0
\(253\) 482.502 1.90712
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −88.7348 153.693i −0.345271 0.598028i 0.640132 0.768265i \(-0.278880\pi\)
−0.985403 + 0.170238i \(0.945546\pi\)
\(258\) 0 0
\(259\) 302.657 + 174.739i 1.16856 + 0.674667i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 59.0700 + 34.1041i 0.224601 + 0.129673i 0.608079 0.793877i \(-0.291940\pi\)
−0.383478 + 0.923550i \(0.625274\pi\)
\(264\) 0 0
\(265\) −250.834 434.457i −0.946543 1.63946i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −77.7652 −0.289090 −0.144545 0.989498i \(-0.546172\pi\)
−0.144545 + 0.989498i \(0.546172\pi\)
\(270\) 0 0
\(271\) 447.259i 1.65040i −0.564840 0.825201i \(-0.691062\pi\)
0.564840 0.825201i \(-0.308938\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −189.880 + 109.627i −0.690473 + 0.398645i
\(276\) 0 0
\(277\) −84.0282 + 145.541i −0.303351 + 0.525419i −0.976893 0.213730i \(-0.931439\pi\)
0.673542 + 0.739149i \(0.264772\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 103.170 178.696i 0.367153 0.635927i −0.621966 0.783044i \(-0.713666\pi\)
0.989119 + 0.147117i \(0.0469992\pi\)
\(282\) 0 0
\(283\) 172.653 99.6815i 0.610083 0.352231i −0.162915 0.986640i \(-0.552090\pi\)
0.772998 + 0.634409i \(0.218756\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 51.4245i 0.179179i
\(288\) 0 0
\(289\) −283.352 −0.980457
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 28.5953 + 49.5285i 0.0975948 + 0.169039i 0.910689 0.413094i \(-0.135552\pi\)
−0.813094 + 0.582133i \(0.802218\pi\)
\(294\) 0 0
\(295\) −333.237 192.395i −1.12962 0.652185i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 642.684 + 371.054i 2.14944 + 1.24098i
\(300\) 0 0
\(301\) −307.639 532.847i −1.02206 1.77026i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 78.6235 0.257782
\(306\) 0 0
\(307\) 276.775i 0.901549i 0.892638 + 0.450774i \(0.148852\pi\)
−0.892638 + 0.450774i \(0.851148\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −21.4284 + 12.3717i −0.0689017 + 0.0397804i −0.534055 0.845450i \(-0.679333\pi\)
0.465153 + 0.885230i \(0.345999\pi\)
\(312\) 0 0
\(313\) 32.4070 56.1306i 0.103537 0.179331i −0.809603 0.586978i \(-0.800317\pi\)
0.913139 + 0.407647i \(0.133651\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 227.300 393.694i 0.717033 1.24194i −0.245137 0.969488i \(-0.578833\pi\)
0.962170 0.272450i \(-0.0878338\pi\)
\(318\) 0 0
\(319\) −114.474 + 66.0916i −0.358853 + 0.207184i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 33.3513i 0.103255i
\(324\) 0 0
\(325\) −337.223 −1.03761
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 24.1944 + 41.9059i 0.0735391 + 0.127373i
\(330\) 0 0
\(331\) 236.009 + 136.260i 0.713019 + 0.411661i 0.812178 0.583410i \(-0.198282\pi\)
−0.0991592 + 0.995072i \(0.531615\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −136.587 78.8583i −0.407721 0.235398i
\(336\) 0 0
\(337\) 246.945 + 427.721i 0.732775 + 1.26920i 0.955693 + 0.294366i \(0.0951085\pi\)
−0.222918 + 0.974837i \(0.571558\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −241.944 −0.709512
\(342\) 0 0
\(343\) 23.3572i 0.0680967i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −450.196 + 259.921i −1.29740 + 0.749051i −0.979954 0.199226i \(-0.936157\pi\)
−0.317442 + 0.948278i \(0.602824\pi\)
\(348\) 0 0
\(349\) −120.639 + 208.954i −0.345672 + 0.598721i −0.985476 0.169817i \(-0.945682\pi\)
0.639804 + 0.768538i \(0.279016\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 100.463 174.008i 0.284599 0.492940i −0.687913 0.725793i \(-0.741473\pi\)
0.972512 + 0.232853i \(0.0748063\pi\)
\(354\) 0 0
\(355\) −266.590 + 153.916i −0.750957 + 0.433565i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 159.654i 0.444719i −0.974965 0.222360i \(-0.928624\pi\)
0.974965 0.222360i \(-0.0713759\pi\)
\(360\) 0 0
\(361\) 164.056 0.454450
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 345.587 + 598.574i 0.946813 + 1.63993i
\(366\) 0 0
\(367\) 320.690 + 185.150i 0.873815 + 0.504497i 0.868614 0.495489i \(-0.165011\pi\)
0.00520064 + 0.999986i \(0.498345\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −641.380 370.301i −1.72879 0.998116i
\(372\) 0 0
\(373\) 163.194 + 282.661i 0.437518 + 0.757804i 0.997497 0.0707027i \(-0.0225242\pi\)
−0.559979 + 0.828507i \(0.689191\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −203.303 −0.539266
\(378\) 0 0
\(379\) 348.114i 0.918506i −0.888305 0.459253i \(-0.848117\pi\)
0.888305 0.459253i \(-0.151883\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 202.550 116.942i 0.528851 0.305332i −0.211697 0.977335i \(-0.567899\pi\)
0.740548 + 0.672003i \(0.234566\pi\)
\(384\) 0 0
\(385\) −376.251 + 651.685i −0.977275 + 1.69269i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 218.806 378.982i 0.562482 0.974248i −0.434797 0.900529i \(-0.643180\pi\)
0.997279 0.0737194i \(-0.0234869\pi\)
\(390\) 0 0
\(391\) −85.4682 + 49.3451i −0.218589 + 0.126202i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 137.922i 0.349169i
\(396\) 0 0
\(397\) 324.259 0.816774 0.408387 0.912809i \(-0.366091\pi\)
0.408387 + 0.912809i \(0.366091\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −197.087 341.364i −0.491488 0.851283i 0.508463 0.861084i \(-0.330214\pi\)
−0.999952 + 0.00980052i \(0.996880\pi\)
\(402\) 0 0
\(403\) −322.265 186.060i −0.799664 0.461686i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −359.635 207.636i −0.883625 0.510161i
\(408\) 0 0
\(409\) 14.1479 + 24.5048i 0.0345914 + 0.0599140i 0.882803 0.469744i \(-0.155654\pi\)
−0.848211 + 0.529658i \(0.822320\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −568.056 −1.37544
\(414\) 0 0
\(415\) 186.693i 0.449863i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 280.441 161.913i 0.669310 0.386426i −0.126505 0.991966i \(-0.540376\pi\)
0.795815 + 0.605540i \(0.207043\pi\)
\(420\) 0 0
\(421\) −215.676 + 373.562i −0.512295 + 0.887320i 0.487604 + 0.873065i \(0.337871\pi\)
−0.999898 + 0.0142554i \(0.995462\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 22.4230 38.8378i 0.0527600 0.0913831i
\(426\) 0 0
\(427\) 100.520 58.0351i 0.235409 0.135914i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 345.557i 0.801757i 0.916131 + 0.400879i \(0.131295\pi\)
−0.916131 + 0.400879i \(0.868705\pi\)
\(432\) 0 0
\(433\) 369.907 0.854289 0.427144 0.904183i \(-0.359520\pi\)
0.427144 + 0.904183i \(0.359520\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 291.389 + 504.700i 0.666793 + 1.15492i
\(438\) 0 0
\(439\) −358.332 206.883i −0.816245 0.471259i 0.0328748 0.999459i \(-0.489534\pi\)
−0.849120 + 0.528200i \(0.822867\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −467.339 269.818i −1.05494 0.609070i −0.130912 0.991394i \(-0.541791\pi\)
−0.924029 + 0.382323i \(0.875124\pi\)
\(444\) 0 0
\(445\) 241.797 + 418.805i 0.543365 + 0.941135i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 325.056 0.723956 0.361978 0.932187i \(-0.382101\pi\)
0.361978 + 0.932187i \(0.382101\pi\)
\(450\) 0 0
\(451\) 61.1058i 0.135490i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −1002.32 + 578.689i −2.20290 + 1.27184i
\(456\) 0 0
\(457\) −309.075 + 535.333i −0.676312 + 1.17141i 0.299771 + 0.954011i \(0.403090\pi\)
−0.976083 + 0.217396i \(0.930244\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 424.992 736.107i 0.921891 1.59676i 0.125404 0.992106i \(-0.459977\pi\)
0.796487 0.604656i \(-0.206689\pi\)
\(462\) 0 0
\(463\) 706.453 407.871i 1.52582 0.880930i 0.526285 0.850309i \(-0.323585\pi\)
0.999531 0.0306215i \(-0.00974865\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 378.909i 0.811368i −0.914013 0.405684i \(-0.867033\pi\)
0.914013 0.405684i \(-0.132967\pi\)
\(468\) 0 0
\(469\) −232.834 −0.496447
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 365.556 + 633.162i 0.772846 + 1.33861i
\(474\) 0 0
\(475\) −229.342 132.411i −0.482825 0.278759i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −297.958 172.026i −0.622041 0.359136i 0.155622 0.987817i \(-0.450262\pi\)
−0.777663 + 0.628681i \(0.783595\pi\)
\(480\) 0 0
\(481\) −319.352 553.134i −0.663934 1.14997i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −942.008 −1.94228
\(486\) 0 0
\(487\) 587.595i 1.20656i 0.797529 + 0.603281i \(0.206140\pi\)
−0.797529 + 0.603281i \(0.793860\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 430.072 248.302i 0.875909 0.505707i 0.00660193 0.999978i \(-0.497899\pi\)
0.869307 + 0.494272i \(0.164565\pi\)
\(492\) 0 0
\(493\) 13.5183 23.4144i 0.0274205 0.0474937i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −227.223 + 393.561i −0.457188 + 0.791873i
\(498\) 0 0
\(499\) 30.1869 17.4284i 0.0604948 0.0349267i −0.469448 0.882960i \(-0.655547\pi\)
0.529942 + 0.848034i \(0.322214\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 630.866i 1.25421i 0.778936 + 0.627104i \(0.215760\pi\)
−0.778936 + 0.627104i \(0.784240\pi\)
\(504\) 0 0
\(505\) −181.166 −0.358745
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −41.0282 71.0629i −0.0806055 0.139613i 0.822905 0.568179i \(-0.192352\pi\)
−0.903510 + 0.428567i \(0.859019\pi\)
\(510\) 0 0
\(511\) 883.663 + 510.183i 1.72928 + 0.998401i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −841.322 485.738i −1.63364 0.943180i
\(516\) 0 0
\(517\) −28.7492 49.7952i −0.0556078 0.0963156i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −562.797 −1.08023 −0.540113 0.841593i \(-0.681618\pi\)
−0.540113 + 0.841593i \(0.681618\pi\)
\(522\) 0 0
\(523\) 909.721i 1.73943i 0.493555 + 0.869714i \(0.335697\pi\)
−0.493555 + 0.869714i \(0.664303\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 42.8568 24.7434i 0.0813223 0.0469514i
\(528\) 0 0
\(529\) 597.751 1035.33i 1.12996 1.95715i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −46.9916 + 81.3918i −0.0881644 + 0.152705i
\(534\) 0 0
\(535\) −1104.62 + 637.753i −2.06471 + 1.19206i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 541.574i 1.00478i
\(540\) 0 0
\(541\) −468.259 −0.865544 −0.432772 0.901503i \(-0.642464\pi\)
−0.432772 + 0.901503i \(0.642464\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.858229 + 1.48650i 0.00157473 + 0.00272752i
\(546\) 0 0
\(547\) −271.063 156.498i −0.495544 0.286102i 0.231328 0.972876i \(-0.425693\pi\)
−0.726872 + 0.686773i \(0.759026\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −138.265 79.8271i −0.250934 0.144877i
\(552\) 0 0
\(553\) 101.806 + 176.333i 0.184097 + 0.318865i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 382.841 0.687328 0.343664 0.939093i \(-0.388332\pi\)
0.343664 + 0.939093i \(0.388332\pi\)
\(558\) 0 0
\(559\) 1124.48i 2.01160i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −518.767 + 299.510i −0.921434 + 0.531990i −0.884092 0.467313i \(-0.845222\pi\)
−0.0373414 + 0.999303i \(0.511889\pi\)
\(564\) 0 0
\(565\) 658.566 1140.67i 1.16560 2.01889i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 422.087 731.076i 0.741805 1.28484i −0.209868 0.977730i \(-0.567303\pi\)
0.951673 0.307114i \(-0.0993632\pi\)
\(570\) 0 0
\(571\) −672.354 + 388.184i −1.17750 + 0.679832i −0.955436 0.295200i \(-0.904614\pi\)
−0.222067 + 0.975031i \(0.571281\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 783.634i 1.36284i
\(576\) 0 0
\(577\) −487.316 −0.844568 −0.422284 0.906464i \(-0.638771\pi\)
−0.422284 + 0.906464i \(0.638771\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 137.806 + 238.686i 0.237187 + 0.410820i
\(582\) 0 0
\(583\) 762.128 + 440.015i 1.30725 + 0.754742i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 487.463 + 281.437i 0.830432 + 0.479450i 0.854001 0.520272i \(-0.174170\pi\)
−0.0235687 + 0.999722i \(0.507503\pi\)
\(588\) 0 0
\(589\) −146.113 253.075i −0.248069 0.429669i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 316.915 0.534426 0.267213 0.963637i \(-0.413897\pi\)
0.267213 + 0.963637i \(0.413897\pi\)
\(594\) 0 0
\(595\) 153.916i 0.258682i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 76.5870 44.2175i 0.127858 0.0738189i −0.434707 0.900572i \(-0.643148\pi\)
0.562565 + 0.826753i \(0.309815\pi\)
\(600\) 0 0
\(601\) 55.9085 96.8364i 0.0930258 0.161125i −0.815757 0.578395i \(-0.803679\pi\)
0.908783 + 0.417269i \(0.137013\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 46.3643 80.3054i 0.0766353 0.132736i
\(606\) 0 0
\(607\) 596.058 344.134i 0.981973 0.566942i 0.0791078 0.996866i \(-0.474793\pi\)
0.902865 + 0.429924i \(0.141460\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 88.4351i 0.144738i
\(612\) 0 0
\(613\) 1021.67 1.66667 0.833334 0.552770i \(-0.186429\pi\)
0.833334 + 0.552770i \(0.186429\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −381.884 661.443i −0.618937 1.07203i −0.989680 0.143295i \(-0.954230\pi\)
0.370743 0.928735i \(-0.379103\pi\)
\(618\) 0 0
\(619\) 661.382 + 381.849i 1.06847 + 0.616880i 0.927763 0.373171i \(-0.121730\pi\)
0.140706 + 0.990051i \(0.455063\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 618.273 + 356.960i 0.992413 + 0.572970i
\(624\) 0 0
\(625\) 370.334 + 641.437i 0.592534 + 1.02630i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 84.9390 0.135038
\(630\) 0 0
\(631\) 166.586i 0.264002i 0.991250 + 0.132001i \(0.0421403\pi\)
−0.991250 + 0.132001i \(0.957860\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −414.420 + 239.265i −0.652629 + 0.376796i
\(636\) 0 0
\(637\) −416.482 + 721.367i −0.653817 + 1.13245i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 102.136 176.904i 0.159338 0.275982i −0.775292 0.631603i \(-0.782397\pi\)
0.934630 + 0.355621i \(0.115731\pi\)
\(642\) 0 0
\(643\) −826.974 + 477.454i −1.28612 + 0.742541i −0.977960 0.208794i \(-0.933046\pi\)
−0.308159 + 0.951335i \(0.599713\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 424.736i 0.656470i 0.944596 + 0.328235i \(0.106454\pi\)
−0.944596 + 0.328235i \(0.893546\pi\)
\(648\) 0 0
\(649\) 675.000 1.04006
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 307.178 + 532.047i 0.470410 + 0.814774i 0.999427 0.0338373i \(-0.0107728\pi\)
−0.529018 + 0.848611i \(0.677439\pi\)
\(654\) 0 0
\(655\) −1057.72 610.677i −1.61484 0.932331i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 843.930 + 487.243i 1.28062 + 0.739367i 0.976962 0.213413i \(-0.0684580\pi\)
0.303660 + 0.952780i \(0.401791\pi\)
\(660\) 0 0
\(661\) −274.380 475.241i −0.415099 0.718972i 0.580340 0.814374i \(-0.302920\pi\)
−0.995439 + 0.0954020i \(0.969586\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −908.890 −1.36675
\(666\) 0 0
\(667\) 472.434i 0.708297i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −119.444 + 68.9609i −0.178009 + 0.102773i
\(672\) 0 0
\(673\) 186.992 323.879i 0.277848 0.481247i −0.693002 0.720936i \(-0.743712\pi\)
0.970850 + 0.239689i \(0.0770456\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 120.972 209.529i 0.178688 0.309497i −0.762743 0.646701i \(-0.776148\pi\)
0.941431 + 0.337205i \(0.109481\pi\)
\(678\) 0 0
\(679\) −1204.35 + 695.333i −1.77372 + 1.02405i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 524.358i 0.767728i 0.923390 + 0.383864i \(0.125407\pi\)
−0.923390 + 0.383864i \(0.874593\pi\)
\(684\) 0 0
\(685\) −925.651 −1.35132
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 676.761 + 1172.18i 0.982236 + 1.70128i
\(690\) 0 0
\(691\) 306.568 + 176.997i 0.443658 + 0.256146i 0.705148 0.709060i \(-0.250880\pi\)
−0.261490 + 0.965206i \(0.584214\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1004.93 580.195i −1.44594 0.834813i
\(696\) 0 0
\(697\) −6.24925 10.8240i −0.00896592 0.0155294i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −621.287 −0.886286 −0.443143 0.896451i \(-0.646137\pi\)
−0.443143 + 0.896451i \(0.646137\pi\)
\(702\) 0 0
\(703\) 501.575i 0.713478i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −231.620 + 133.726i −0.327610 + 0.189146i
\(708\) 0 0
\(709\) 393.992 682.413i 0.555700 0.962501i −0.442148 0.896942i \(-0.645783\pi\)
0.997849 0.0655594i \(-0.0208832\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −432.364 + 748.876i −0.606400 + 1.05032i
\(714\) 0 0
\(715\) 1191.02 687.635i 1.66576 0.961727i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 1248.82i 1.73689i −0.495785 0.868445i \(-0.665119\pi\)
0.495785 0.868445i \(-0.334881\pi\)
\(720\) 0 0
\(721\) −1434.17 −1.98914
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −107.340 185.918i −0.148055 0.256439i
\(726\) 0 0
\(727\) −634.487 366.321i −0.872746 0.503880i −0.00448649 0.999990i \(-0.501428\pi\)
−0.868260 + 0.496110i \(0.834761\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −129.506 74.7704i −0.177163 0.102285i
\(732\) 0 0
\(733\) −182.361 315.858i −0.248787 0.430911i 0.714403 0.699735i \(-0.246698\pi\)
−0.963189 + 0.268824i \(0.913365\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 276.668 0.375397
\(738\) 0 0
\(739\) 1162.07i 1.57248i 0.617918 + 0.786242i \(0.287976\pi\)
−0.617918 + 0.786242i \(0.712024\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 156.711 90.4773i 0.210917 0.121773i −0.390821 0.920467i \(-0.627809\pi\)
0.601737 + 0.798694i \(0.294475\pi\)
\(744\) 0 0
\(745\) 632.769 1095.99i 0.849354 1.47113i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −941.502 + 1630.73i −1.25701 + 2.17721i
\(750\) 0 0
\(751\) 152.903 88.2786i 0.203599 0.117548i −0.394734 0.918795i \(-0.629163\pi\)
0.598333 + 0.801247i \(0.295830\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 47.9813i 0.0635514i
\(756\) 0 0
\(757\) −146.665 −0.193745 −0.0968723 0.995297i \(-0.530884\pi\)
−0.0968723 + 0.995297i \(0.530884\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 403.405 + 698.717i 0.530098 + 0.918157i 0.999383 + 0.0351104i \(0.0111783\pi\)
−0.469285 + 0.883047i \(0.655488\pi\)
\(762\) 0 0
\(763\) 2.19448 + 1.26699i 0.00287613 + 0.00166053i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 899.088 + 519.089i 1.17221 + 0.676778i
\(768\) 0 0
\(769\) −275.566 477.295i −0.358344 0.620669i 0.629341 0.777130i \(-0.283325\pi\)
−0.987684 + 0.156460i \(0.949992\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 856.915 1.10856 0.554279 0.832331i \(-0.312994\pi\)
0.554279 + 0.832331i \(0.312994\pi\)
\(774\) 0 0
\(775\) 392.942i 0.507023i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −63.9171 + 36.9025i −0.0820502 + 0.0473717i
\(780\) 0 0
\(781\) 270.000 467.654i 0.345711 0.598788i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 238.874 413.742i 0.304298 0.527060i
\(786\) 0 0
\(787\) −466.532 + 269.352i −0.592798 + 0.342252i −0.766203 0.642598i \(-0.777856\pi\)
0.173405 + 0.984851i \(0.444523\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1944.46i 2.45823i
\(792\) 0 0
\(793\) −212.130 −0.267503
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −674.153 1167.67i −0.845863 1.46508i −0.884869 0.465840i \(-0.845752\pi\)
0.0390058 0.999239i \(-0.487581\pi\)
\(798\) 0 0
\(799\) 10.1850 + 5.88033i 0.0127472 + 0.00735962i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1050.02 606.231i −1.30763 0.754958i
\(804\) 0 0
\(805\) 1344.75 + 2329.18i 1.67050 + 2.89339i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −1163.88 −1.43867 −0.719334 0.694664i \(-0.755553\pi\)
−0.719334 + 0.694664i \(0.755553\pi\)
\(810\) 0 0
\(811\) 627.878i 0.774202i −0.922037 0.387101i \(-0.873476\pi\)
0.922037 0.387101i \(-0.126524\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −643.988 + 371.806i −0.790169 + 0.456204i
\(816\) 0 0
\(817\) −441.528 + 764.749i −0.540426 + 0.936046i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −32.1822 + 55.7411i −0.0391987 + 0.0678942i −0.884959 0.465669i \(-0.845814\pi\)
0.845760 + 0.533563i \(0.179147\pi\)
\(822\) 0 0
\(823\) 817.183 471.801i 0.992932 0.573269i 0.0867825 0.996227i \(-0.472341\pi\)
0.906149 + 0.422958i \(0.139008\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 571.482i 0.691031i 0.938413 + 0.345515i \(0.112296\pi\)
−0.938413 + 0.345515i \(0.887704\pi\)
\(828\) 0 0
\(829\) 519.555 0.626725 0.313362 0.949634i \(-0.398545\pi\)
0.313362 + 0.949634i \(0.398545\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −55.3864 95.9321i −0.0664903 0.115165i
\(834\) 0 0
\(835\) 104.909 + 60.5691i 0.125639 + 0.0725378i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1039.96 600.422i −1.23952 0.715640i −0.270527 0.962712i \(-0.587198\pi\)
−0.968997 + 0.247073i \(0.920531\pi\)
\(840\) 0 0
\(841\) 355.787 + 616.242i 0.423053 + 0.732749i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 995.854 1.17853
\(846\) 0 0
\(847\) 136.893i 0.161622i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −1285.37 + 742.107i −1.51042 + 0.872041i
\(852\) 0 0
\(853\) −299.583 + 518.893i −0.351211 + 0.608315i −0.986462 0.163990i \(-0.947563\pi\)
0.635251 + 0.772306i \(0.280897\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 345.664 598.707i 0.403342 0.698608i −0.590785 0.806829i \(-0.701182\pi\)
0.994127 + 0.108221i \(0.0345153\pi\)
\(858\) 0 0
\(859\) 1215.72 701.896i 1.41527 0.817108i 0.419394 0.907805i \(-0.362243\pi\)
0.995879 + 0.0906967i \(0.0289094\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 253.038i 0.293207i 0.989195 + 0.146604i \(0.0468342\pi\)
−0.989195 + 0.146604i \(0.953166\pi\)
\(864\) 0 0
\(865\) 1942.69 2.24588
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −120.972 209.529i −0.139208 0.241115i
\(870\) 0 0
\(871\) 368.517 + 212.763i 0.423096 + 0.244275i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 343.796 + 198.491i 0.392910 + 0.226847i
\(876\) 0 0
\(877\) −53.9352 93.4185i −0.0614997 0.106521i 0.833636 0.552314i \(-0.186255\pi\)
−0.895136 + 0.445793i \(0.852922\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 119.927 0.136126 0.0680629 0.997681i \(-0.478318\pi\)
0.0680629 + 0.997681i \(0.478318\pi\)
\(882\) 0 0
\(883\) 306.661i 0.347295i −0.984808 0.173647i \(-0.944445\pi\)
0.984808 0.173647i \(-0.0555553\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 680.325 392.786i 0.766996 0.442825i −0.0648060 0.997898i \(-0.520643\pi\)
0.831802 + 0.555073i \(0.187310\pi\)
\(888\) 0 0
\(889\) −353.223 + 611.799i −0.397326 + 0.688188i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 34.7241 60.1438i 0.0388847 0.0673503i
\(894\) 0 0
\(895\) −533.179 + 307.831i −0.595731 + 0.343946i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 236.896i 0.263510i
\(900\) 0 0
\(901\) −180.000 −0.199778
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 425.377 + 736.774i 0.470029 + 0.814115i
\(906\) 0 0
\(907\) 1055.14 + 609.182i 1.16332 + 0.671646i 0.952098 0.305792i \(-0.0989211\pi\)
0.211226 + 0.977437i \(0.432254\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 760.824 + 439.262i 0.835152 + 0.482175i 0.855614 0.517615i \(-0.173180\pi\)
−0.0204612 + 0.999791i \(0.506513\pi\)
\(912\) 0 0
\(913\) −163.749 283.622i −0.179353 0.310648i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1803.06 −1.96626
\(918\) 0 0
\(919\) 1036.67i 1.12804i −0.825760 0.564022i \(-0.809253\pi\)
0.825760 0.564022i \(-0.190747\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 719.271 415.271i 0.779275 0.449915i
\(924\) 0 0
\(925\) 337.223 584.087i 0.364565 0.631445i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 124.911 216.352i 0.134457 0.232887i −0.790933 0.611903i \(-0.790404\pi\)
0.925390 + 0.379016i \(0.123738\pi\)
\(930\) 0 0
\(931\) −566.490 + 327.063i −0.608475 + 0.351303i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 182.892i 0.195607i
\(936\) 0 0
\(937\) 190.744 0.203569 0.101784 0.994806i \(-0.467545\pi\)
0.101784 + 0.994806i \(0.467545\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −141.150 244.479i −0.150000 0.259808i 0.781227 0.624247i \(-0.214594\pi\)
−0.931227 + 0.364439i \(0.881261\pi\)
\(942\) 0 0
\(943\) 189.138 + 109.199i 0.200570 + 0.115799i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 573.926 + 331.356i 0.606046 + 0.349901i 0.771416 0.636331i \(-0.219549\pi\)
−0.165370 + 0.986232i \(0.552882\pi\)
\(948\) 0 0
\(949\) −932.409 1614.98i −0.982517 1.70177i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1166.70 −1.22424 −0.612119 0.790765i \(-0.709683\pi\)
−0.612119 + 0.790765i \(0.709683\pi\)
\(954\) 0 0
\(955\) 1990.93i 2.08475i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1183.44 + 683.260i −1.23404 + 0.712471i
\(960\) 0 0
\(961\) −263.697 + 456.737i −0.274399 + 0.475273i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 130.016 225.194i 0.134732 0.233362i
\(966\) 0 0
\(967\) 1010.52 583.422i 1.04500 0.603332i 0.123756 0.992313i \(-0.460506\pi\)
0.921246 + 0.388981i \(0.127173\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1481.20i 1.52544i −0.646728 0.762721i \(-0.723863\pi\)
0.646728 0.762721i \(-0.276137\pi\)
\(972\) 0 0
\(973\) −1713.06 −1.76060
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −592.779 1026.72i −0.606734 1.05089i −0.991775 0.127994i \(-0.959146\pi\)
0.385041 0.922899i \(-0.374187\pi\)
\(978\) 0 0
\(979\) −734.671 424.163i −0.750430 0.433261i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −56.0882 32.3825i −0.0570581 0.0329425i 0.471200 0.882027i \(-0.343821\pi\)
−0.528258 + 0.849084i \(0.677154\pi\)
\(984\) 0 0
\(985\) 313.797 + 543.513i 0.318576 + 0.551790i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2613.06 2.64212
\(990\) 0 0
\(991\) 1394.27i 1.40694i −0.710727 0.703468i \(-0.751634\pi\)
0.710727 0.703468i \(-0.248366\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 638.772 368.795i 0.641982 0.370649i
\(996\) 0 0
\(997\) −487.271 + 843.977i −0.488737 + 0.846517i −0.999916 0.0129572i \(-0.995875\pi\)
0.511179 + 0.859474i \(0.329209\pi\)
\(998\) 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.o.d.127.1 8
3.2 odd 2 576.3.o.e.511.3 8
4.3 odd 2 inner 1728.3.o.d.127.2 8
8.3 odd 2 432.3.o.c.127.4 8
8.5 even 2 432.3.o.c.127.3 8
9.4 even 3 inner 1728.3.o.d.1279.2 8
9.5 odd 6 576.3.o.e.319.2 8
12.11 even 2 576.3.o.e.511.2 8
24.5 odd 2 144.3.o.b.79.2 yes 8
24.11 even 2 144.3.o.b.79.3 yes 8
36.23 even 6 576.3.o.e.319.3 8
36.31 odd 6 inner 1728.3.o.d.1279.1 8
72.5 odd 6 144.3.o.b.31.3 yes 8
72.11 even 6 1296.3.g.h.1135.3 4
72.13 even 6 432.3.o.c.415.4 8
72.29 odd 6 1296.3.g.h.1135.4 4
72.43 odd 6 1296.3.g.d.1135.1 4
72.59 even 6 144.3.o.b.31.2 8
72.61 even 6 1296.3.g.d.1135.2 4
72.67 odd 6 432.3.o.c.415.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
144.3.o.b.31.2 8 72.59 even 6
144.3.o.b.31.3 yes 8 72.5 odd 6
144.3.o.b.79.2 yes 8 24.5 odd 2
144.3.o.b.79.3 yes 8 24.11 even 2
432.3.o.c.127.3 8 8.5 even 2
432.3.o.c.127.4 8 8.3 odd 2
432.3.o.c.415.3 8 72.67 odd 6
432.3.o.c.415.4 8 72.13 even 6
576.3.o.e.319.2 8 9.5 odd 6
576.3.o.e.319.3 8 36.23 even 6
576.3.o.e.511.2 8 12.11 even 2
576.3.o.e.511.3 8 3.2 odd 2
1296.3.g.d.1135.1 4 72.43 odd 6
1296.3.g.d.1135.2 4 72.61 even 6
1296.3.g.h.1135.3 4 72.11 even 6
1296.3.g.h.1135.4 4 72.29 odd 6
1728.3.o.d.127.1 8 1.1 even 1 trivial
1728.3.o.d.127.2 8 4.3 odd 2 inner
1728.3.o.d.1279.1 8 36.31 odd 6 inner
1728.3.o.d.1279.2 8 9.4 even 3 inner