Properties

Label 960.4.f.e.769.1
Level $960$
Weight $4$
Character 960.769
Analytic conductor $56.642$
Analytic rank $0$
Dimension $2$
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] = [960,4,Mod(769,960)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(960, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 1]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("960.769");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 960 = 2^{6} \cdot 3 \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 960.f (of order \(2\), degree \(1\), 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: \(56.6418336055\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.1
Root \(-1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 960.769
Dual form 960.4.f.e.769.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.00000i q^{3} +(2.00000 - 11.0000i) q^{5} -10.0000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q-3.00000i q^{3} +(2.00000 - 11.0000i) q^{5} -10.0000i q^{7} -9.00000 q^{9} -14.0000 q^{11} -82.0000i q^{13} +(-33.0000 - 6.00000i) q^{15} +18.0000i q^{17} +136.000 q^{19} -30.0000 q^{21} -140.000i q^{23} +(-117.000 - 44.0000i) q^{25} +27.0000i q^{27} +112.000 q^{29} -72.0000 q^{31} +42.0000i q^{33} +(-110.000 - 20.0000i) q^{35} -26.0000i q^{37} -246.000 q^{39} -446.000 q^{41} -396.000i q^{43} +(-18.0000 + 99.0000i) q^{45} +144.000i q^{47} +243.000 q^{49} +54.0000 q^{51} +158.000i q^{53} +(-28.0000 + 154.000i) q^{55} -408.000i q^{57} +342.000 q^{59} -314.000 q^{61} +90.0000i q^{63} +(-902.000 - 164.000i) q^{65} -152.000i q^{67} -420.000 q^{69} +932.000 q^{71} +548.000i q^{73} +(-132.000 + 351.000i) q^{75} +140.000i q^{77} -512.000 q^{79} +81.0000 q^{81} -284.000i q^{83} +(198.000 + 36.0000i) q^{85} -336.000i q^{87} +810.000 q^{89} -820.000 q^{91} +216.000i q^{93} +(272.000 - 1496.00i) q^{95} +1304.00i q^{97} +126.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 4 q^{5} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 4 q^{5} - 18 q^{9} - 28 q^{11} - 66 q^{15} + 272 q^{19} - 60 q^{21} - 234 q^{25} + 224 q^{29} - 144 q^{31} - 220 q^{35} - 492 q^{39} - 892 q^{41} - 36 q^{45} + 486 q^{49} + 108 q^{51} - 56 q^{55} + 684 q^{59} - 628 q^{61} - 1804 q^{65} - 840 q^{69} + 1864 q^{71} - 264 q^{75} - 1024 q^{79} + 162 q^{81} + 396 q^{85} + 1620 q^{89} - 1640 q^{91} + 544 q^{95} + 252 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/960\mathbb{Z}\right)^\times\).

\(n\) \(511\) \(577\) \(641\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000i 0.577350i
\(4\) 0 0
\(5\) 2.00000 11.0000i 0.178885 0.983870i
\(6\) 0 0
\(7\) 10.0000i 0.539949i −0.962867 0.269975i \(-0.912985\pi\)
0.962867 0.269975i \(-0.0870153\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −14.0000 −0.383742 −0.191871 0.981420i \(-0.561455\pi\)
−0.191871 + 0.981420i \(0.561455\pi\)
\(12\) 0 0
\(13\) 82.0000i 1.74944i −0.484629 0.874720i \(-0.661046\pi\)
0.484629 0.874720i \(-0.338954\pi\)
\(14\) 0 0
\(15\) −33.0000 6.00000i −0.568038 0.103280i
\(16\) 0 0
\(17\) 18.0000i 0.256802i 0.991722 + 0.128401i \(0.0409845\pi\)
−0.991722 + 0.128401i \(0.959015\pi\)
\(18\) 0 0
\(19\) 136.000 1.64213 0.821067 0.570832i \(-0.193379\pi\)
0.821067 + 0.570832i \(0.193379\pi\)
\(20\) 0 0
\(21\) −30.0000 −0.311740
\(22\) 0 0
\(23\) 140.000i 1.26922i −0.772833 0.634609i \(-0.781161\pi\)
0.772833 0.634609i \(-0.218839\pi\)
\(24\) 0 0
\(25\) −117.000 44.0000i −0.936000 0.352000i
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 112.000 0.717168 0.358584 0.933497i \(-0.383260\pi\)
0.358584 + 0.933497i \(0.383260\pi\)
\(30\) 0 0
\(31\) −72.0000 −0.417148 −0.208574 0.978007i \(-0.566882\pi\)
−0.208574 + 0.978007i \(0.566882\pi\)
\(32\) 0 0
\(33\) 42.0000i 0.221553i
\(34\) 0 0
\(35\) −110.000 20.0000i −0.531240 0.0965891i
\(36\) 0 0
\(37\) 26.0000i 0.115524i −0.998330 0.0577618i \(-0.981604\pi\)
0.998330 0.0577618i \(-0.0183964\pi\)
\(38\) 0 0
\(39\) −246.000 −1.01004
\(40\) 0 0
\(41\) −446.000 −1.69887 −0.849433 0.527697i \(-0.823056\pi\)
−0.849433 + 0.527697i \(0.823056\pi\)
\(42\) 0 0
\(43\) 396.000i 1.40441i −0.711977 0.702203i \(-0.752200\pi\)
0.711977 0.702203i \(-0.247800\pi\)
\(44\) 0 0
\(45\) −18.0000 + 99.0000i −0.0596285 + 0.327957i
\(46\) 0 0
\(47\) 144.000i 0.446906i 0.974715 + 0.223453i \(0.0717328\pi\)
−0.974715 + 0.223453i \(0.928267\pi\)
\(48\) 0 0
\(49\) 243.000 0.708455
\(50\) 0 0
\(51\) 54.0000 0.148265
\(52\) 0 0
\(53\) 158.000i 0.409490i 0.978815 + 0.204745i \(0.0656365\pi\)
−0.978815 + 0.204745i \(0.934363\pi\)
\(54\) 0 0
\(55\) −28.0000 + 154.000i −0.0686458 + 0.377552i
\(56\) 0 0
\(57\) 408.000i 0.948086i
\(58\) 0 0
\(59\) 342.000 0.754654 0.377327 0.926080i \(-0.376843\pi\)
0.377327 + 0.926080i \(0.376843\pi\)
\(60\) 0 0
\(61\) −314.000 −0.659075 −0.329538 0.944142i \(-0.606893\pi\)
−0.329538 + 0.944142i \(0.606893\pi\)
\(62\) 0 0
\(63\) 90.0000i 0.179983i
\(64\) 0 0
\(65\) −902.000 164.000i −1.72122 0.312949i
\(66\) 0 0
\(67\) 152.000i 0.277161i −0.990351 0.138580i \(-0.955746\pi\)
0.990351 0.138580i \(-0.0442539\pi\)
\(68\) 0 0
\(69\) −420.000 −0.732783
\(70\) 0 0
\(71\) 932.000 1.55786 0.778930 0.627111i \(-0.215763\pi\)
0.778930 + 0.627111i \(0.215763\pi\)
\(72\) 0 0
\(73\) 548.000i 0.878610i 0.898338 + 0.439305i \(0.144775\pi\)
−0.898338 + 0.439305i \(0.855225\pi\)
\(74\) 0 0
\(75\) −132.000 + 351.000i −0.203227 + 0.540400i
\(76\) 0 0
\(77\) 140.000i 0.207201i
\(78\) 0 0
\(79\) −512.000 −0.729171 −0.364585 0.931170i \(-0.618789\pi\)
−0.364585 + 0.931170i \(0.618789\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 284.000i 0.375579i −0.982209 0.187789i \(-0.939868\pi\)
0.982209 0.187789i \(-0.0601323\pi\)
\(84\) 0 0
\(85\) 198.000 + 36.0000i 0.252660 + 0.0459382i
\(86\) 0 0
\(87\) 336.000i 0.414057i
\(88\) 0 0
\(89\) 810.000 0.964717 0.482359 0.875974i \(-0.339780\pi\)
0.482359 + 0.875974i \(0.339780\pi\)
\(90\) 0 0
\(91\) −820.000 −0.944608
\(92\) 0 0
\(93\) 216.000i 0.240840i
\(94\) 0 0
\(95\) 272.000 1496.00i 0.293754 1.61565i
\(96\) 0 0
\(97\) 1304.00i 1.36496i 0.730904 + 0.682480i \(0.239099\pi\)
−0.730904 + 0.682480i \(0.760901\pi\)
\(98\) 0 0
\(99\) 126.000 0.127914
\(100\) 0 0
\(101\) −936.000 −0.922133 −0.461067 0.887365i \(-0.652533\pi\)
−0.461067 + 0.887365i \(0.652533\pi\)
\(102\) 0 0
\(103\) 1450.00i 1.38711i 0.720402 + 0.693557i \(0.243957\pi\)
−0.720402 + 0.693557i \(0.756043\pi\)
\(104\) 0 0
\(105\) −60.0000 + 330.000i −0.0557657 + 0.306711i
\(106\) 0 0
\(107\) 1292.00i 1.16731i 0.812001 + 0.583656i \(0.198378\pi\)
−0.812001 + 0.583656i \(0.801622\pi\)
\(108\) 0 0
\(109\) −2142.00 −1.88226 −0.941130 0.338044i \(-0.890235\pi\)
−0.941130 + 0.338044i \(0.890235\pi\)
\(110\) 0 0
\(111\) −78.0000 −0.0666976
\(112\) 0 0
\(113\) 1418.00i 1.18048i 0.807228 + 0.590240i \(0.200967\pi\)
−0.807228 + 0.590240i \(0.799033\pi\)
\(114\) 0 0
\(115\) −1540.00 280.000i −1.24875 0.227045i
\(116\) 0 0
\(117\) 738.000i 0.583146i
\(118\) 0 0
\(119\) 180.000 0.138660
\(120\) 0 0
\(121\) −1135.00 −0.852742
\(122\) 0 0
\(123\) 1338.00i 0.980841i
\(124\) 0 0
\(125\) −718.000 + 1199.00i −0.513759 + 0.857935i
\(126\) 0 0
\(127\) 1674.00i 1.16963i −0.811165 0.584817i \(-0.801166\pi\)
0.811165 0.584817i \(-0.198834\pi\)
\(128\) 0 0
\(129\) −1188.00 −0.810834
\(130\) 0 0
\(131\) 1134.00 0.756321 0.378160 0.925740i \(-0.376557\pi\)
0.378160 + 0.925740i \(0.376557\pi\)
\(132\) 0 0
\(133\) 1360.00i 0.886669i
\(134\) 0 0
\(135\) 297.000 + 54.0000i 0.189346 + 0.0344265i
\(136\) 0 0
\(137\) 2866.00i 1.78729i −0.448773 0.893646i \(-0.648139\pi\)
0.448773 0.893646i \(-0.351861\pi\)
\(138\) 0 0
\(139\) 764.000 0.466199 0.233099 0.972453i \(-0.425113\pi\)
0.233099 + 0.972453i \(0.425113\pi\)
\(140\) 0 0
\(141\) 432.000 0.258021
\(142\) 0 0
\(143\) 1148.00i 0.671333i
\(144\) 0 0
\(145\) 224.000 1232.00i 0.128291 0.705600i
\(146\) 0 0
\(147\) 729.000i 0.409027i
\(148\) 0 0
\(149\) −3060.00 −1.68245 −0.841225 0.540686i \(-0.818165\pi\)
−0.841225 + 0.540686i \(0.818165\pi\)
\(150\) 0 0
\(151\) 1304.00 0.702768 0.351384 0.936231i \(-0.385711\pi\)
0.351384 + 0.936231i \(0.385711\pi\)
\(152\) 0 0
\(153\) 162.000i 0.0856008i
\(154\) 0 0
\(155\) −144.000 + 792.000i −0.0746217 + 0.410419i
\(156\) 0 0
\(157\) 1582.00i 0.804187i −0.915599 0.402093i \(-0.868283\pi\)
0.915599 0.402093i \(-0.131717\pi\)
\(158\) 0 0
\(159\) 474.000 0.236419
\(160\) 0 0
\(161\) −1400.00 −0.685313
\(162\) 0 0
\(163\) 3232.00i 1.55307i −0.630077 0.776533i \(-0.716976\pi\)
0.630077 0.776533i \(-0.283024\pi\)
\(164\) 0 0
\(165\) 462.000 + 84.0000i 0.217980 + 0.0396327i
\(166\) 0 0
\(167\) 2988.00i 1.38454i 0.721638 + 0.692271i \(0.243390\pi\)
−0.721638 + 0.692271i \(0.756610\pi\)
\(168\) 0 0
\(169\) −4527.00 −2.06054
\(170\) 0 0
\(171\) −1224.00 −0.547378
\(172\) 0 0
\(173\) 918.000i 0.403435i −0.979444 0.201717i \(-0.935348\pi\)
0.979444 0.201717i \(-0.0646523\pi\)
\(174\) 0 0
\(175\) −440.000 + 1170.00i −0.190062 + 0.505392i
\(176\) 0 0
\(177\) 1026.00i 0.435700i
\(178\) 0 0
\(179\) 1238.00 0.516941 0.258471 0.966019i \(-0.416781\pi\)
0.258471 + 0.966019i \(0.416781\pi\)
\(180\) 0 0
\(181\) −1450.00 −0.595457 −0.297728 0.954651i \(-0.596229\pi\)
−0.297728 + 0.954651i \(0.596229\pi\)
\(182\) 0 0
\(183\) 942.000i 0.380517i
\(184\) 0 0
\(185\) −286.000 52.0000i −0.113660 0.0206655i
\(186\) 0 0
\(187\) 252.000i 0.0985458i
\(188\) 0 0
\(189\) 270.000 0.103913
\(190\) 0 0
\(191\) −4860.00 −1.84114 −0.920569 0.390581i \(-0.872274\pi\)
−0.920569 + 0.390581i \(0.872274\pi\)
\(192\) 0 0
\(193\) 1412.00i 0.526622i 0.964711 + 0.263311i \(0.0848145\pi\)
−0.964711 + 0.263311i \(0.915186\pi\)
\(194\) 0 0
\(195\) −492.000 + 2706.00i −0.180681 + 0.993747i
\(196\) 0 0
\(197\) 1170.00i 0.423142i 0.977363 + 0.211571i \(0.0678580\pi\)
−0.977363 + 0.211571i \(0.932142\pi\)
\(198\) 0 0
\(199\) 2080.00 0.740941 0.370471 0.928844i \(-0.379196\pi\)
0.370471 + 0.928844i \(0.379196\pi\)
\(200\) 0 0
\(201\) −456.000 −0.160019
\(202\) 0 0
\(203\) 1120.00i 0.387234i
\(204\) 0 0
\(205\) −892.000 + 4906.00i −0.303902 + 1.67146i
\(206\) 0 0
\(207\) 1260.00i 0.423073i
\(208\) 0 0
\(209\) −1904.00 −0.630155
\(210\) 0 0
\(211\) 2692.00 0.878317 0.439159 0.898410i \(-0.355277\pi\)
0.439159 + 0.898410i \(0.355277\pi\)
\(212\) 0 0
\(213\) 2796.00i 0.899431i
\(214\) 0 0
\(215\) −4356.00 792.000i −1.38175 0.251228i
\(216\) 0 0
\(217\) 720.000i 0.225239i
\(218\) 0 0
\(219\) 1644.00 0.507266
\(220\) 0 0
\(221\) 1476.00 0.449260
\(222\) 0 0
\(223\) 846.000i 0.254046i −0.991900 0.127023i \(-0.959458\pi\)
0.991900 0.127023i \(-0.0405423\pi\)
\(224\) 0 0
\(225\) 1053.00 + 396.000i 0.312000 + 0.117333i
\(226\) 0 0
\(227\) 2884.00i 0.843250i 0.906770 + 0.421625i \(0.138540\pi\)
−0.906770 + 0.421625i \(0.861460\pi\)
\(228\) 0 0
\(229\) 3150.00 0.908986 0.454493 0.890750i \(-0.349820\pi\)
0.454493 + 0.890750i \(0.349820\pi\)
\(230\) 0 0
\(231\) 420.000 0.119628
\(232\) 0 0
\(233\) 4014.00i 1.12861i 0.825567 + 0.564304i \(0.190856\pi\)
−0.825567 + 0.564304i \(0.809144\pi\)
\(234\) 0 0
\(235\) 1584.00 + 288.000i 0.439697 + 0.0799449i
\(236\) 0 0
\(237\) 1536.00i 0.420987i
\(238\) 0 0
\(239\) 4900.00 1.32617 0.663085 0.748544i \(-0.269247\pi\)
0.663085 + 0.748544i \(0.269247\pi\)
\(240\) 0 0
\(241\) −2314.00 −0.618497 −0.309249 0.950981i \(-0.600078\pi\)
−0.309249 + 0.950981i \(0.600078\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 486.000 2673.00i 0.126732 0.697027i
\(246\) 0 0
\(247\) 11152.0i 2.87281i
\(248\) 0 0
\(249\) −852.000 −0.216841
\(250\) 0 0
\(251\) −2002.00 −0.503447 −0.251723 0.967799i \(-0.580997\pi\)
−0.251723 + 0.967799i \(0.580997\pi\)
\(252\) 0 0
\(253\) 1960.00i 0.487052i
\(254\) 0 0
\(255\) 108.000 594.000i 0.0265224 0.145873i
\(256\) 0 0
\(257\) 450.000i 0.109223i −0.998508 0.0546113i \(-0.982608\pi\)
0.998508 0.0546113i \(-0.0173920\pi\)
\(258\) 0 0
\(259\) −260.000 −0.0623769
\(260\) 0 0
\(261\) −1008.00 −0.239056
\(262\) 0 0
\(263\) 180.000i 0.0422026i 0.999777 + 0.0211013i \(0.00671725\pi\)
−0.999777 + 0.0211013i \(0.993283\pi\)
\(264\) 0 0
\(265\) 1738.00 + 316.000i 0.402885 + 0.0732518i
\(266\) 0 0
\(267\) 2430.00i 0.556980i
\(268\) 0 0
\(269\) 2448.00 0.554859 0.277430 0.960746i \(-0.410517\pi\)
0.277430 + 0.960746i \(0.410517\pi\)
\(270\) 0 0
\(271\) −6776.00 −1.51887 −0.759433 0.650586i \(-0.774524\pi\)
−0.759433 + 0.650586i \(0.774524\pi\)
\(272\) 0 0
\(273\) 2460.00i 0.545370i
\(274\) 0 0
\(275\) 1638.00 + 616.000i 0.359182 + 0.135077i
\(276\) 0 0
\(277\) 6426.00i 1.39387i −0.717136 0.696933i \(-0.754547\pi\)
0.717136 0.696933i \(-0.245453\pi\)
\(278\) 0 0
\(279\) 648.000 0.139049
\(280\) 0 0
\(281\) 2718.00 0.577019 0.288509 0.957477i \(-0.406840\pi\)
0.288509 + 0.957477i \(0.406840\pi\)
\(282\) 0 0
\(283\) 6048.00i 1.27038i −0.772358 0.635188i \(-0.780923\pi\)
0.772358 0.635188i \(-0.219077\pi\)
\(284\) 0 0
\(285\) −4488.00 816.000i −0.932794 0.169599i
\(286\) 0 0
\(287\) 4460.00i 0.917301i
\(288\) 0 0
\(289\) 4589.00 0.934053
\(290\) 0 0
\(291\) 3912.00 0.788060
\(292\) 0 0
\(293\) 1246.00i 0.248437i 0.992255 + 0.124219i \(0.0396424\pi\)
−0.992255 + 0.124219i \(0.960358\pi\)
\(294\) 0 0
\(295\) 684.000 3762.00i 0.134997 0.742482i
\(296\) 0 0
\(297\) 378.000i 0.0738511i
\(298\) 0 0
\(299\) −11480.0 −2.22042
\(300\) 0 0
\(301\) −3960.00 −0.758308
\(302\) 0 0
\(303\) 2808.00i 0.532394i
\(304\) 0 0
\(305\) −628.000 + 3454.00i −0.117899 + 0.648444i
\(306\) 0 0
\(307\) 1244.00i 0.231267i 0.993292 + 0.115633i \(0.0368897\pi\)
−0.993292 + 0.115633i \(0.963110\pi\)
\(308\) 0 0
\(309\) 4350.00 0.800851
\(310\) 0 0
\(311\) 6372.00 1.16181 0.580905 0.813971i \(-0.302699\pi\)
0.580905 + 0.813971i \(0.302699\pi\)
\(312\) 0 0
\(313\) 5500.00i 0.993222i −0.867973 0.496611i \(-0.834578\pi\)
0.867973 0.496611i \(-0.165422\pi\)
\(314\) 0 0
\(315\) 990.000 + 180.000i 0.177080 + 0.0321964i
\(316\) 0 0
\(317\) 378.000i 0.0669735i −0.999439 0.0334867i \(-0.989339\pi\)
0.999439 0.0334867i \(-0.0106612\pi\)
\(318\) 0 0
\(319\) −1568.00 −0.275207
\(320\) 0 0
\(321\) 3876.00 0.673948
\(322\) 0 0
\(323\) 2448.00i 0.421704i
\(324\) 0 0
\(325\) −3608.00 + 9594.00i −0.615803 + 1.63747i
\(326\) 0 0
\(327\) 6426.00i 1.08672i
\(328\) 0 0
\(329\) 1440.00 0.241306
\(330\) 0 0
\(331\) −11888.0 −1.97409 −0.987045 0.160446i \(-0.948707\pi\)
−0.987045 + 0.160446i \(0.948707\pi\)
\(332\) 0 0
\(333\) 234.000i 0.0385079i
\(334\) 0 0
\(335\) −1672.00 304.000i −0.272690 0.0495800i
\(336\) 0 0
\(337\) 9116.00i 1.47353i −0.676148 0.736766i \(-0.736352\pi\)
0.676148 0.736766i \(-0.263648\pi\)
\(338\) 0 0
\(339\) 4254.00 0.681550
\(340\) 0 0
\(341\) 1008.00 0.160077
\(342\) 0 0
\(343\) 5860.00i 0.922479i
\(344\) 0 0
\(345\) −840.000 + 4620.00i −0.131084 + 0.720964i
\(346\) 0 0
\(347\) 4676.00i 0.723403i −0.932294 0.361701i \(-0.882196\pi\)
0.932294 0.361701i \(-0.117804\pi\)
\(348\) 0 0
\(349\) 11906.0 1.82611 0.913057 0.407833i \(-0.133715\pi\)
0.913057 + 0.407833i \(0.133715\pi\)
\(350\) 0 0
\(351\) 2214.00 0.336680
\(352\) 0 0
\(353\) 2142.00i 0.322966i 0.986875 + 0.161483i \(0.0516278\pi\)
−0.986875 + 0.161483i \(0.948372\pi\)
\(354\) 0 0
\(355\) 1864.00 10252.0i 0.278678 1.53273i
\(356\) 0 0
\(357\) 540.000i 0.0800555i
\(358\) 0 0
\(359\) 9824.00 1.44426 0.722132 0.691755i \(-0.243162\pi\)
0.722132 + 0.691755i \(0.243162\pi\)
\(360\) 0 0
\(361\) 11637.0 1.69660
\(362\) 0 0
\(363\) 3405.00i 0.492331i
\(364\) 0 0
\(365\) 6028.00 + 1096.00i 0.864438 + 0.157171i
\(366\) 0 0
\(367\) 5354.00i 0.761516i −0.924675 0.380758i \(-0.875663\pi\)
0.924675 0.380758i \(-0.124337\pi\)
\(368\) 0 0
\(369\) 4014.00 0.566289
\(370\) 0 0
\(371\) 1580.00 0.221104
\(372\) 0 0
\(373\) 7694.00i 1.06804i 0.845471 + 0.534022i \(0.179320\pi\)
−0.845471 + 0.534022i \(0.820680\pi\)
\(374\) 0 0
\(375\) 3597.00 + 2154.00i 0.495329 + 0.296619i
\(376\) 0 0
\(377\) 9184.00i 1.25464i
\(378\) 0 0
\(379\) 6004.00 0.813733 0.406866 0.913488i \(-0.366621\pi\)
0.406866 + 0.913488i \(0.366621\pi\)
\(380\) 0 0
\(381\) −5022.00 −0.675288
\(382\) 0 0
\(383\) 9432.00i 1.25836i 0.777259 + 0.629181i \(0.216610\pi\)
−0.777259 + 0.629181i \(0.783390\pi\)
\(384\) 0 0
\(385\) 1540.00 + 280.000i 0.203859 + 0.0370653i
\(386\) 0 0
\(387\) 3564.00i 0.468135i
\(388\) 0 0
\(389\) −6156.00 −0.802369 −0.401185 0.915997i \(-0.631401\pi\)
−0.401185 + 0.915997i \(0.631401\pi\)
\(390\) 0 0
\(391\) 2520.00 0.325938
\(392\) 0 0
\(393\) 3402.00i 0.436662i
\(394\) 0 0
\(395\) −1024.00 + 5632.00i −0.130438 + 0.717409i
\(396\) 0 0
\(397\) 7866.00i 0.994416i −0.867631 0.497208i \(-0.834359\pi\)
0.867631 0.497208i \(-0.165641\pi\)
\(398\) 0 0
\(399\) −4080.00 −0.511918
\(400\) 0 0
\(401\) −2074.00 −0.258281 −0.129140 0.991626i \(-0.541222\pi\)
−0.129140 + 0.991626i \(0.541222\pi\)
\(402\) 0 0
\(403\) 5904.00i 0.729775i
\(404\) 0 0
\(405\) 162.000 891.000i 0.0198762 0.109319i
\(406\) 0 0
\(407\) 364.000i 0.0443312i
\(408\) 0 0
\(409\) 2746.00 0.331983 0.165991 0.986127i \(-0.446918\pi\)
0.165991 + 0.986127i \(0.446918\pi\)
\(410\) 0 0
\(411\) −8598.00 −1.03189
\(412\) 0 0
\(413\) 3420.00i 0.407475i
\(414\) 0 0
\(415\) −3124.00 568.000i −0.369521 0.0671856i
\(416\) 0 0
\(417\) 2292.00i 0.269160i
\(418\) 0 0
\(419\) 6426.00 0.749238 0.374619 0.927179i \(-0.377774\pi\)
0.374619 + 0.927179i \(0.377774\pi\)
\(420\) 0 0
\(421\) 10610.0 1.22827 0.614133 0.789203i \(-0.289506\pi\)
0.614133 + 0.789203i \(0.289506\pi\)
\(422\) 0 0
\(423\) 1296.00i 0.148969i
\(424\) 0 0
\(425\) 792.000 2106.00i 0.0903945 0.240367i
\(426\) 0 0
\(427\) 3140.00i 0.355867i
\(428\) 0 0
\(429\) 3444.00 0.387594
\(430\) 0 0
\(431\) −8384.00 −0.936991 −0.468495 0.883466i \(-0.655204\pi\)
−0.468495 + 0.883466i \(0.655204\pi\)
\(432\) 0 0
\(433\) 1980.00i 0.219752i −0.993945 0.109876i \(-0.964955\pi\)
0.993945 0.109876i \(-0.0350454\pi\)
\(434\) 0 0
\(435\) −3696.00 672.000i −0.407378 0.0740688i
\(436\) 0 0
\(437\) 19040.0i 2.08423i
\(438\) 0 0
\(439\) 864.000 0.0939327 0.0469664 0.998896i \(-0.485045\pi\)
0.0469664 + 0.998896i \(0.485045\pi\)
\(440\) 0 0
\(441\) −2187.00 −0.236152
\(442\) 0 0
\(443\) 13212.0i 1.41698i 0.705722 + 0.708489i \(0.250623\pi\)
−0.705722 + 0.708489i \(0.749377\pi\)
\(444\) 0 0
\(445\) 1620.00 8910.00i 0.172574 0.949156i
\(446\) 0 0
\(447\) 9180.00i 0.971363i
\(448\) 0 0
\(449\) 16290.0 1.71219 0.856094 0.516820i \(-0.172884\pi\)
0.856094 + 0.516820i \(0.172884\pi\)
\(450\) 0 0
\(451\) 6244.00 0.651926
\(452\) 0 0
\(453\) 3912.00i 0.405743i
\(454\) 0 0
\(455\) −1640.00 + 9020.00i −0.168977 + 0.929372i
\(456\) 0 0
\(457\) 6336.00i 0.648546i 0.945964 + 0.324273i \(0.105120\pi\)
−0.945964 + 0.324273i \(0.894880\pi\)
\(458\) 0 0
\(459\) −486.000 −0.0494217
\(460\) 0 0
\(461\) 13716.0 1.38572 0.692861 0.721071i \(-0.256350\pi\)
0.692861 + 0.721071i \(0.256350\pi\)
\(462\) 0 0
\(463\) 14626.0i 1.46809i −0.679098 0.734047i \(-0.737629\pi\)
0.679098 0.734047i \(-0.262371\pi\)
\(464\) 0 0
\(465\) 2376.00 + 432.000i 0.236956 + 0.0430828i
\(466\) 0 0
\(467\) 5796.00i 0.574319i −0.957883 0.287159i \(-0.907289\pi\)
0.957883 0.287159i \(-0.0927109\pi\)
\(468\) 0 0
\(469\) −1520.00 −0.149653
\(470\) 0 0
\(471\) −4746.00 −0.464298
\(472\) 0 0
\(473\) 5544.00i 0.538929i
\(474\) 0 0
\(475\) −15912.0 5984.00i −1.53704 0.578031i
\(476\) 0 0
\(477\) 1422.00i 0.136497i
\(478\) 0 0
\(479\) −8348.00 −0.796305 −0.398152 0.917319i \(-0.630348\pi\)
−0.398152 + 0.917319i \(0.630348\pi\)
\(480\) 0 0
\(481\) −2132.00 −0.202102
\(482\) 0 0
\(483\) 4200.00i 0.395666i
\(484\) 0 0
\(485\) 14344.0 + 2608.00i 1.34294 + 0.244172i
\(486\) 0 0
\(487\) 1898.00i 0.176605i 0.996094 + 0.0883025i \(0.0281442\pi\)
−0.996094 + 0.0883025i \(0.971856\pi\)
\(488\) 0 0
\(489\) −9696.00 −0.896663
\(490\) 0 0
\(491\) 16654.0 1.53072 0.765361 0.643601i \(-0.222560\pi\)
0.765361 + 0.643601i \(0.222560\pi\)
\(492\) 0 0
\(493\) 2016.00i 0.184171i
\(494\) 0 0
\(495\) 252.000 1386.00i 0.0228819 0.125851i
\(496\) 0 0
\(497\) 9320.00i 0.841165i
\(498\) 0 0
\(499\) 10600.0 0.950944 0.475472 0.879731i \(-0.342277\pi\)
0.475472 + 0.879731i \(0.342277\pi\)
\(500\) 0 0
\(501\) 8964.00 0.799365
\(502\) 0 0
\(503\) 10048.0i 0.890692i −0.895359 0.445346i \(-0.853081\pi\)
0.895359 0.445346i \(-0.146919\pi\)
\(504\) 0 0
\(505\) −1872.00 + 10296.0i −0.164956 + 0.907259i
\(506\) 0 0
\(507\) 13581.0i 1.18965i
\(508\) 0 0
\(509\) −15088.0 −1.31388 −0.656939 0.753944i \(-0.728149\pi\)
−0.656939 + 0.753944i \(0.728149\pi\)
\(510\) 0 0
\(511\) 5480.00 0.474405
\(512\) 0 0
\(513\) 3672.00i 0.316029i
\(514\) 0 0
\(515\) 15950.0 + 2900.00i 1.36474 + 0.248135i
\(516\) 0 0
\(517\) 2016.00i 0.171496i
\(518\) 0 0
\(519\) −2754.00 −0.232923
\(520\) 0 0
\(521\) 8582.00 0.721659 0.360829 0.932632i \(-0.382494\pi\)
0.360829 + 0.932632i \(0.382494\pi\)
\(522\) 0 0
\(523\) 16928.0i 1.41532i −0.706556 0.707658i \(-0.749752\pi\)
0.706556 0.707658i \(-0.250248\pi\)
\(524\) 0 0
\(525\) 3510.00 + 1320.00i 0.291788 + 0.109732i
\(526\) 0 0
\(527\) 1296.00i 0.107125i
\(528\) 0 0
\(529\) −7433.00 −0.610915
\(530\) 0 0
\(531\) −3078.00 −0.251551
\(532\) 0 0
\(533\) 36572.0i 2.97206i
\(534\) 0 0
\(535\) 14212.0 + 2584.00i 1.14848 + 0.208815i
\(536\) 0 0
\(537\) 3714.00i 0.298456i
\(538\) 0 0
\(539\) −3402.00 −0.271864
\(540\) 0 0
\(541\) 11150.0 0.886092 0.443046 0.896499i \(-0.353898\pi\)
0.443046 + 0.896499i \(0.353898\pi\)
\(542\) 0 0
\(543\) 4350.00i 0.343787i
\(544\) 0 0
\(545\) −4284.00 + 23562.0i −0.336709 + 1.85190i
\(546\) 0 0
\(547\) 19628.0i 1.53425i −0.641500 0.767123i \(-0.721688\pi\)
0.641500 0.767123i \(-0.278312\pi\)
\(548\) 0 0
\(549\) 2826.00 0.219692
\(550\) 0 0
\(551\) 15232.0 1.17769
\(552\) 0 0
\(553\) 5120.00i 0.393715i
\(554\) 0 0
\(555\) −156.000 + 858.000i −0.0119312 + 0.0656218i
\(556\) 0 0
\(557\) 8694.00i 0.661358i −0.943743 0.330679i \(-0.892722\pi\)
0.943743 0.330679i \(-0.107278\pi\)
\(558\) 0 0
\(559\) −32472.0 −2.45692
\(560\) 0 0
\(561\) −756.000 −0.0568954
\(562\) 0 0
\(563\) 828.000i 0.0619823i 0.999520 + 0.0309912i \(0.00986637\pi\)
−0.999520 + 0.0309912i \(0.990134\pi\)
\(564\) 0 0
\(565\) 15598.0 + 2836.00i 1.16144 + 0.211171i
\(566\) 0 0
\(567\) 810.000i 0.0599944i
\(568\) 0 0
\(569\) 17514.0 1.29038 0.645189 0.764023i \(-0.276779\pi\)
0.645189 + 0.764023i \(0.276779\pi\)
\(570\) 0 0
\(571\) −5552.00 −0.406907 −0.203454 0.979085i \(-0.565217\pi\)
−0.203454 + 0.979085i \(0.565217\pi\)
\(572\) 0 0
\(573\) 14580.0i 1.06298i
\(574\) 0 0
\(575\) −6160.00 + 16380.0i −0.446765 + 1.18799i
\(576\) 0 0
\(577\) 12896.0i 0.930446i −0.885193 0.465223i \(-0.845974\pi\)
0.885193 0.465223i \(-0.154026\pi\)
\(578\) 0 0
\(579\) 4236.00 0.304045
\(580\) 0 0
\(581\) −2840.00 −0.202794
\(582\) 0 0
\(583\) 2212.00i 0.157138i
\(584\) 0 0
\(585\) 8118.00 + 1476.00i 0.573740 + 0.104316i
\(586\) 0 0
\(587\) 1588.00i 0.111659i −0.998440 0.0558295i \(-0.982220\pi\)
0.998440 0.0558295i \(-0.0177803\pi\)
\(588\) 0 0
\(589\) −9792.00 −0.685012
\(590\) 0 0
\(591\) 3510.00 0.244301
\(592\) 0 0
\(593\) 22262.0i 1.54164i −0.637055 0.770819i \(-0.719848\pi\)
0.637055 0.770819i \(-0.280152\pi\)
\(594\) 0 0
\(595\) 360.000 1980.00i 0.0248043 0.136424i
\(596\) 0 0
\(597\) 6240.00i 0.427783i
\(598\) 0 0
\(599\) 4640.00 0.316503 0.158251 0.987399i \(-0.449414\pi\)
0.158251 + 0.987399i \(0.449414\pi\)
\(600\) 0 0
\(601\) −2574.00 −0.174702 −0.0873508 0.996178i \(-0.527840\pi\)
−0.0873508 + 0.996178i \(0.527840\pi\)
\(602\) 0 0
\(603\) 1368.00i 0.0923868i
\(604\) 0 0
\(605\) −2270.00 + 12485.0i −0.152543 + 0.838987i
\(606\) 0 0
\(607\) 11170.0i 0.746913i 0.927648 + 0.373457i \(0.121828\pi\)
−0.927648 + 0.373457i \(0.878172\pi\)
\(608\) 0 0
\(609\) −3360.00 −0.223570
\(610\) 0 0
\(611\) 11808.0 0.781834
\(612\) 0 0
\(613\) 3646.00i 0.240229i −0.992760 0.120115i \(-0.961674\pi\)
0.992760 0.120115i \(-0.0383262\pi\)
\(614\) 0 0
\(615\) 14718.0 + 2676.00i 0.965020 + 0.175458i
\(616\) 0 0
\(617\) 7646.00i 0.498892i −0.968389 0.249446i \(-0.919751\pi\)
0.968389 0.249446i \(-0.0802485\pi\)
\(618\) 0 0
\(619\) −26668.0 −1.73163 −0.865814 0.500366i \(-0.833199\pi\)
−0.865814 + 0.500366i \(0.833199\pi\)
\(620\) 0 0
\(621\) 3780.00 0.244261
\(622\) 0 0
\(623\) 8100.00i 0.520898i
\(624\) 0 0
\(625\) 11753.0 + 10296.0i 0.752192 + 0.658944i
\(626\) 0 0
\(627\) 5712.00i 0.363820i
\(628\) 0 0
\(629\) 468.000 0.0296667
\(630\) 0 0
\(631\) −7712.00 −0.486545 −0.243272 0.969958i \(-0.578221\pi\)
−0.243272 + 0.969958i \(0.578221\pi\)
\(632\) 0 0
\(633\) 8076.00i 0.507097i
\(634\) 0 0
\(635\) −18414.0 3348.00i −1.15077 0.209230i
\(636\) 0 0
\(637\) 19926.0i 1.23940i
\(638\) 0 0
\(639\) −8388.00 −0.519287
\(640\) 0 0
\(641\) −22302.0 −1.37422 −0.687111 0.726553i \(-0.741121\pi\)
−0.687111 + 0.726553i \(0.741121\pi\)
\(642\) 0 0
\(643\) 2232.00i 0.136892i −0.997655 0.0684459i \(-0.978196\pi\)
0.997655 0.0684459i \(-0.0218041\pi\)
\(644\) 0 0
\(645\) −2376.00 + 13068.0i −0.145046 + 0.797755i
\(646\) 0 0
\(647\) 9464.00i 0.575067i −0.957771 0.287533i \(-0.907165\pi\)
0.957771 0.287533i \(-0.0928352\pi\)
\(648\) 0 0
\(649\) −4788.00 −0.289592
\(650\) 0 0
\(651\) 2160.00 0.130042
\(652\) 0 0
\(653\) 4878.00i 0.292329i −0.989260 0.146165i \(-0.953307\pi\)
0.989260 0.146165i \(-0.0466929\pi\)
\(654\) 0 0
\(655\) 2268.00 12474.0i 0.135295 0.744121i
\(656\) 0 0
\(657\) 4932.00i 0.292870i
\(658\) 0 0
\(659\) −10206.0 −0.603292 −0.301646 0.953420i \(-0.597536\pi\)
−0.301646 + 0.953420i \(0.597536\pi\)
\(660\) 0 0
\(661\) −20906.0 −1.23018 −0.615090 0.788457i \(-0.710880\pi\)
−0.615090 + 0.788457i \(0.710880\pi\)
\(662\) 0 0
\(663\) 4428.00i 0.259380i
\(664\) 0 0
\(665\) −14960.0 2720.00i −0.872367 0.158612i
\(666\) 0 0
\(667\) 15680.0i 0.910243i
\(668\) 0 0
\(669\) −2538.00 −0.146674
\(670\) 0 0
\(671\) 4396.00 0.252915
\(672\) 0 0
\(673\) 6812.00i 0.390168i 0.980787 + 0.195084i \(0.0624980\pi\)
−0.980787 + 0.195084i \(0.937502\pi\)
\(674\) 0 0
\(675\) 1188.00 3159.00i 0.0677424 0.180133i
\(676\) 0 0
\(677\) 10026.0i 0.569174i −0.958650 0.284587i \(-0.908144\pi\)
0.958650 0.284587i \(-0.0918564\pi\)
\(678\) 0 0
\(679\) 13040.0 0.737009
\(680\) 0 0
\(681\) 8652.00 0.486851
\(682\) 0 0
\(683\) 21236.0i 1.18971i 0.803832 + 0.594856i \(0.202791\pi\)
−0.803832 + 0.594856i \(0.797209\pi\)
\(684\) 0 0
\(685\) −31526.0 5732.00i −1.75846 0.319720i
\(686\) 0 0
\(687\) 9450.00i 0.524803i
\(688\) 0 0
\(689\) 12956.0 0.716378
\(690\) 0 0
\(691\) −11520.0 −0.634213 −0.317107 0.948390i \(-0.602711\pi\)
−0.317107 + 0.948390i \(0.602711\pi\)
\(692\) 0 0
\(693\) 1260.00i 0.0690670i
\(694\) 0 0
\(695\) 1528.00 8404.00i 0.0833962 0.458679i
\(696\) 0 0
\(697\) 8028.00i 0.436273i
\(698\) 0 0
\(699\) 12042.0 0.651603
\(700\) 0 0
\(701\) 400.000 0.0215518 0.0107759 0.999942i \(-0.496570\pi\)
0.0107759 + 0.999942i \(0.496570\pi\)
\(702\) 0 0
\(703\) 3536.00i 0.189705i
\(704\) 0 0
\(705\) 864.000 4752.00i 0.0461562 0.253859i
\(706\) 0 0
\(707\) 9360.00i 0.497905i
\(708\) 0 0
\(709\) −5930.00 −0.314113 −0.157056 0.987590i \(-0.550200\pi\)
−0.157056 + 0.987590i \(0.550200\pi\)
\(710\) 0 0
\(711\) 4608.00 0.243057
\(712\) 0 0
\(713\) 10080.0i 0.529452i
\(714\) 0 0
\(715\) 12628.0 + 2296.00i 0.660504 + 0.120092i
\(716\) 0 0
\(717\) 14700.0i 0.765665i
\(718\) 0 0
\(719\) 7160.00 0.371381 0.185691 0.982608i \(-0.440548\pi\)
0.185691 + 0.982608i \(0.440548\pi\)
\(720\) 0 0
\(721\) 14500.0 0.748971
\(722\) 0 0
\(723\) 6942.00i 0.357090i
\(724\) 0 0
\(725\) −13104.0 4928.00i −0.671269 0.252443i
\(726\) 0 0
\(727\) 8874.00i 0.452708i 0.974045 + 0.226354i \(0.0726806\pi\)
−0.974045 + 0.226354i \(0.927319\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 7128.00 0.360655
\(732\) 0 0
\(733\) 5562.00i 0.280269i 0.990132 + 0.140134i \(0.0447535\pi\)
−0.990132 + 0.140134i \(0.955247\pi\)
\(734\) 0 0
\(735\) −8019.00 1458.00i −0.402429 0.0731689i
\(736\) 0 0
\(737\) 2128.00i 0.106358i
\(738\) 0 0
\(739\) 12096.0 0.602109 0.301055 0.953607i \(-0.402661\pi\)
0.301055 + 0.953607i \(0.402661\pi\)
\(740\) 0 0
\(741\) −33456.0 −1.65862
\(742\) 0 0
\(743\) 21312.0i 1.05230i 0.850391 + 0.526152i \(0.176366\pi\)
−0.850391 + 0.526152i \(0.823634\pi\)
\(744\) 0 0
\(745\) −6120.00 + 33660.0i −0.300966 + 1.65531i
\(746\) 0 0
\(747\) 2556.00i 0.125193i
\(748\) 0 0
\(749\) 12920.0 0.630289
\(750\) 0 0
\(751\) 6832.00 0.331962 0.165981 0.986129i \(-0.446921\pi\)
0.165981 + 0.986129i \(0.446921\pi\)
\(752\) 0 0
\(753\) 6006.00i 0.290665i
\(754\) 0 0
\(755\) 2608.00 14344.0i 0.125715 0.691433i
\(756\) 0 0
\(757\) 7174.00i 0.344443i 0.985058 + 0.172222i \(0.0550945\pi\)
−0.985058 + 0.172222i \(0.944905\pi\)
\(758\) 0 0
\(759\) 5880.00 0.281200
\(760\) 0 0
\(761\) −15394.0 −0.733288 −0.366644 0.930361i \(-0.619493\pi\)
−0.366644 + 0.930361i \(0.619493\pi\)
\(762\) 0 0
\(763\) 21420.0i 1.01633i
\(764\) 0 0
\(765\) −1782.00 324.000i −0.0842201 0.0153127i
\(766\) 0 0
\(767\) 28044.0i 1.32022i
\(768\) 0 0
\(769\) −9730.00 −0.456271 −0.228136 0.973629i \(-0.573263\pi\)
−0.228136 + 0.973629i \(0.573263\pi\)
\(770\) 0 0
\(771\) −1350.00 −0.0630597
\(772\) 0 0
\(773\) 24206.0i 1.12630i −0.826355 0.563150i \(-0.809589\pi\)
0.826355 0.563150i \(-0.190411\pi\)
\(774\) 0 0
\(775\) 8424.00 + 3168.00i 0.390450 + 0.146836i
\(776\) 0 0
\(777\) 780.000i 0.0360133i
\(778\) 0 0
\(779\) −60656.0 −2.78976
\(780\) 0 0
\(781\) −13048.0 −0.597816
\(782\) 0 0
\(783\) 3024.00i 0.138019i
\(784\) 0 0
\(785\) −17402.0 3164.00i −0.791215 0.143857i
\(786\) 0 0
\(787\) 31312.0i 1.41824i −0.705089 0.709118i \(-0.749093\pi\)
0.705089 0.709118i \(-0.250907\pi\)
\(788\) 0 0
\(789\) 540.000 0.0243657
\(790\) 0 0
\(791\) 14180.0 0.637399
\(792\) 0 0
\(793\) 25748.0i 1.15301i
\(794\) 0 0
\(795\) 948.000 5214.00i 0.0422919 0.232606i
\(796\) 0 0
\(797\) 10746.0i 0.477595i −0.971069 0.238797i \(-0.923247\pi\)
0.971069 0.238797i \(-0.0767531\pi\)
\(798\) 0 0
\(799\) −2592.00 −0.114766
\(800\) 0 0
\(801\) −7290.00 −0.321572
\(802\) 0 0
\(803\) 7672.00i 0.337159i
\(804\) 0 0
\(805\) −2800.00 + 15400.0i −0.122593 + 0.674259i
\(806\) 0 0
\(807\) 7344.00i 0.320348i
\(808\) 0 0
\(809\) −30546.0 −1.32749 −0.663745 0.747959i \(-0.731034\pi\)
−0.663745 + 0.747959i \(0.731034\pi\)
\(810\) 0 0
\(811\) −2628.00 −0.113787 −0.0568937 0.998380i \(-0.518120\pi\)
−0.0568937 + 0.998380i \(0.518120\pi\)
\(812\) 0 0
\(813\) 20328.0i 0.876918i
\(814\) 0 0
\(815\) −35552.0 6464.00i −1.52802 0.277821i
\(816\) 0 0
\(817\) 53856.0i 2.30622i
\(818\) 0 0
\(819\) 7380.00 0.314869
\(820\) 0 0
\(821\) −26280.0 −1.11715 −0.558574 0.829455i \(-0.688651\pi\)
−0.558574 + 0.829455i \(0.688651\pi\)
\(822\) 0 0
\(823\) 26146.0i 1.10740i −0.832715 0.553701i \(-0.813215\pi\)
0.832715 0.553701i \(-0.186785\pi\)
\(824\) 0 0
\(825\) 1848.00 4914.00i 0.0779868 0.207374i
\(826\) 0 0
\(827\) 29268.0i 1.23065i 0.788273 + 0.615325i \(0.210975\pi\)
−0.788273 + 0.615325i \(0.789025\pi\)
\(828\) 0 0
\(829\) 6202.00 0.259836 0.129918 0.991525i \(-0.458529\pi\)
0.129918 + 0.991525i \(0.458529\pi\)
\(830\) 0 0
\(831\) −19278.0 −0.804749
\(832\) 0 0
\(833\) 4374.00i 0.181933i
\(834\) 0 0
\(835\) 32868.0 + 5976.00i 1.36221 + 0.247674i
\(836\) 0 0
\(837\) 1944.00i 0.0802801i
\(838\) 0 0
\(839\) 4680.00 0.192576 0.0962882 0.995353i \(-0.469303\pi\)
0.0962882 + 0.995353i \(0.469303\pi\)
\(840\) 0 0
\(841\) −11845.0 −0.485670
\(842\) 0 0
\(843\) 8154.00i 0.333142i
\(844\) 0 0
\(845\) −9054.00 + 49797.0i −0.368600 + 2.02730i
\(846\) 0 0
\(847\) 11350.0i 0.460438i
\(848\) 0 0
\(849\) −18144.0 −0.733452
\(850\) 0 0
\(851\) −3640.00 −0.146625
\(852\) 0 0
\(853\) 12122.0i 0.486576i 0.969954 + 0.243288i \(0.0782260\pi\)
−0.969954 + 0.243288i \(0.921774\pi\)
\(854\) 0 0
\(855\) −2448.00 + 13464.0i −0.0979179 + 0.538549i
\(856\) 0 0
\(857\) 26006.0i 1.03658i −0.855205 0.518289i \(-0.826569\pi\)
0.855205 0.518289i \(-0.173431\pi\)
\(858\) 0 0
\(859\) 17684.0 0.702410 0.351205 0.936299i \(-0.385772\pi\)
0.351205 + 0.936299i \(0.385772\pi\)
\(860\) 0 0
\(861\) 13380.0 0.529604
\(862\) 0 0
\(863\) 15084.0i 0.594977i 0.954725 + 0.297489i \(0.0961490\pi\)
−0.954725 + 0.297489i \(0.903851\pi\)
\(864\) 0 0
\(865\) −10098.0 1836.00i −0.396928 0.0721686i
\(866\) 0 0
\(867\) 13767.0i 0.539275i
\(868\) 0 0
\(869\) 7168.00 0.279813
\(870\) 0 0
\(871\) −12464.0 −0.484875
\(872\) 0 0
\(873\) 11736.0i 0.454987i
\(874\) 0 0
\(875\) 11990.0 + 7180.00i 0.463241 + 0.277404i
\(876\) 0 0
\(877\) 17614.0i 0.678201i −0.940750 0.339101i \(-0.889877\pi\)
0.940750 0.339101i \(-0.110123\pi\)
\(878\) 0 0
\(879\) 3738.00 0.143435
\(880\) 0 0
\(881\) −3298.00 −0.126121 −0.0630604 0.998010i \(-0.520086\pi\)
−0.0630604 + 0.998010i \(0.520086\pi\)
\(882\) 0 0
\(883\) 9496.00i 0.361909i −0.983491 0.180955i \(-0.942081\pi\)
0.983491 0.180955i \(-0.0579187\pi\)
\(884\) 0 0
\(885\) −11286.0 2052.00i −0.428672 0.0779404i
\(886\) 0 0
\(887\) 50220.0i 1.90104i 0.310663 + 0.950520i \(0.399449\pi\)
−0.310663 + 0.950520i \(0.600551\pi\)
\(888\) 0 0
\(889\) −16740.0 −0.631543
\(890\) 0 0
\(891\) −1134.00 −0.0426380
\(892\) 0 0
\(893\) 19584.0i 0.733879i
\(894\) 0 0
\(895\) 2476.00 13618.0i 0.0924732 0.508603i
\(896\) 0 0
\(897\) 34440.0i 1.28196i
\(898\) 0 0
\(899\) −8064.00 −0.299165
\(900\) 0 0
\(901\) −2844.00 −0.105158
\(902\) 0 0
\(903\) 11880.0i 0.437809i
\(904\) 0 0
\(905\) −2900.00 + 15950.0i −0.106519 + 0.585852i
\(906\) 0 0
\(907\) 20636.0i 0.755465i 0.925915 + 0.377733i \(0.123296\pi\)
−0.925915 + 0.377733i \(0.876704\pi\)
\(908\) 0 0
\(909\) 8424.00 0.307378
\(910\) 0 0
\(911\) −13680.0 −0.497518 −0.248759 0.968565i \(-0.580023\pi\)
−0.248759 + 0.968565i \(0.580023\pi\)
\(912\) 0 0
\(913\) 3976.00i 0.144125i
\(914\) 0 0
\(915\) 10362.0 + 1884.00i 0.374379 + 0.0680690i
\(916\) 0 0
\(917\) 11340.0i 0.408375i
\(918\) 0 0
\(919\) 21456.0 0.770150 0.385075 0.922885i \(-0.374176\pi\)
0.385075 + 0.922885i \(0.374176\pi\)
\(920\) 0 0
\(921\) 3732.00 0.133522
\(922\) 0 0
\(923\) 76424.0i 2.72538i
\(924\) 0 0
\(925\) −1144.00 + 3042.00i −0.0406643 + 0.108130i
\(926\) 0 0
\(927\) 13050.0i 0.462371i
\(928\) 0 0
\(929\) −16510.0 −0.583074 −0.291537 0.956560i \(-0.594167\pi\)
−0.291537 + 0.956560i \(0.594167\pi\)
\(930\) 0 0
\(931\) 33048.0 1.16338
\(932\) 0 0
\(933\) 19116.0i 0.670771i
\(934\) 0 0
\(935\) −2772.00 504.000i −0.0969562 0.0176284i
\(936\) 0 0
\(937\) 36296.0i 1.26546i −0.774371 0.632731i \(-0.781934\pi\)
0.774371 0.632731i \(-0.218066\pi\)
\(938\) 0 0
\(939\) −16500.0 −0.573437
\(940\) 0 0
\(941\) −13540.0 −0.469066 −0.234533 0.972108i \(-0.575356\pi\)
−0.234533 + 0.972108i \(0.575356\pi\)
\(942\) 0 0
\(943\) 62440.0i 2.15623i
\(944\) 0 0
\(945\) 540.000 2970.00i 0.0185886 0.102237i
\(946\) 0 0
\(947\) 32940.0i 1.13031i 0.824984 + 0.565156i \(0.191184\pi\)
−0.824984 + 0.565156i \(0.808816\pi\)
\(948\) 0 0
\(949\) 44936.0 1.53708
\(950\) 0 0
\(951\) −1134.00 −0.0386672
\(952\) 0 0
\(953\) 22482.0i 0.764180i 0.924125 + 0.382090i \(0.124796\pi\)
−0.924125 + 0.382090i \(0.875204\pi\)
\(954\) 0 0
\(955\) −9720.00 + 53460.0i −0.329353 + 1.81144i
\(956\) 0 0
\(957\) 4704.00i 0.158891i
\(958\) 0 0
\(959\) −28660.0 −0.965047
\(960\) 0 0
\(961\) −24607.0 −0.825988
\(962\) 0 0
\(963\) 11628.0i 0.389104i
\(964\) 0 0
\(965\) 15532.0 + 2824.00i 0.518127 + 0.0942049i
\(966\) 0 0
\(967\) 9566.00i 0.318120i 0.987269 + 0.159060i \(0.0508463\pi\)
−0.987269 + 0.159060i \(0.949154\pi\)
\(968\) 0 0
\(969\) 7344.00 0.243471
\(970\) 0 0
\(971\) 10062.0 0.332549 0.166274 0.986080i \(-0.446826\pi\)
0.166274 + 0.986080i \(0.446826\pi\)
\(972\) 0 0
\(973\) 7640.00i 0.251724i
\(974\) 0 0
\(975\) 28782.0 + 10824.0i 0.945397 + 0.355534i
\(976\) 0 0
\(977\) 48506.0i 1.58838i 0.607671 + 0.794189i \(0.292104\pi\)
−0.607671 + 0.794189i \(0.707896\pi\)
\(978\) 0 0
\(979\) −11340.0 −0.370202
\(980\) 0 0
\(981\) 19278.0 0.627420
\(982\) 0 0
\(983\) 41144.0i 1.33498i −0.744617 0.667492i \(-0.767368\pi\)
0.744617 0.667492i \(-0.232632\pi\)
\(984\) 0 0
\(985\) 12870.0 + 2340.00i 0.416317 + 0.0756940i
\(986\) 0 0
\(987\) 4320.00i 0.139318i
\(988\) 0 0
\(989\) −55440.0 −1.78250
\(990\) 0 0
\(991\) −16120.0 −0.516719 −0.258360 0.966049i \(-0.583182\pi\)
−0.258360 + 0.966049i \(0.583182\pi\)
\(992\) 0 0
\(993\) 35664.0i 1.13974i
\(994\) 0 0
\(995\) 4160.00 22880.0i 0.132544 0.728990i
\(996\) 0 0
\(997\) 36666.0i 1.16472i 0.812932 + 0.582359i \(0.197870\pi\)
−0.812932 + 0.582359i \(0.802130\pi\)
\(998\) 0 0
\(999\) 702.000 0.0222325
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 960.4.f.e.769.1 2
4.3 odd 2 960.4.f.f.769.2 2
5.4 even 2 inner 960.4.f.e.769.2 2
8.3 odd 2 120.4.f.b.49.1 2
8.5 even 2 240.4.f.c.49.2 2
20.19 odd 2 960.4.f.f.769.1 2
24.5 odd 2 720.4.f.e.289.1 2
24.11 even 2 360.4.f.b.289.1 2
40.3 even 4 600.4.a.p.1.1 1
40.13 odd 4 1200.4.a.e.1.1 1
40.19 odd 2 120.4.f.b.49.2 yes 2
40.27 even 4 600.4.a.c.1.1 1
40.29 even 2 240.4.f.c.49.1 2
40.37 odd 4 1200.4.a.bg.1.1 1
120.29 odd 2 720.4.f.e.289.2 2
120.59 even 2 360.4.f.b.289.2 2
120.83 odd 4 1800.4.a.x.1.1 1
120.107 odd 4 1800.4.a.i.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.f.b.49.1 2 8.3 odd 2
120.4.f.b.49.2 yes 2 40.19 odd 2
240.4.f.c.49.1 2 40.29 even 2
240.4.f.c.49.2 2 8.5 even 2
360.4.f.b.289.1 2 24.11 even 2
360.4.f.b.289.2 2 120.59 even 2
600.4.a.c.1.1 1 40.27 even 4
600.4.a.p.1.1 1 40.3 even 4
720.4.f.e.289.1 2 24.5 odd 2
720.4.f.e.289.2 2 120.29 odd 2
960.4.f.e.769.1 2 1.1 even 1 trivial
960.4.f.e.769.2 2 5.4 even 2 inner
960.4.f.f.769.1 2 20.19 odd 2
960.4.f.f.769.2 2 4.3 odd 2
1200.4.a.e.1.1 1 40.13 odd 4
1200.4.a.bg.1.1 1 40.37 odd 4
1800.4.a.i.1.1 1 120.107 odd 4
1800.4.a.x.1.1 1 120.83 odd 4