Properties

Label 960.4.f.o.769.3
Level $960$
Weight $4$
Character 960.769
Analytic conductor $56.642$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{129})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 65x^{2} + 1024 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 120)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 769.3
Root \(-5.17891i\) of defining polynomial
Character \(\chi\) \(=\) 960.769
Dual form 960.4.f.o.769.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000i q^{3} +(-11.1789 - 0.178908i) q^{5} +33.0735i q^{7} -9.00000 q^{9} +48.3578 q^{11} -60.3578i q^{13} +(0.536725 - 33.5367i) q^{15} -17.7891i q^{17} +130.863 q^{19} -99.2204 q^{21} +70.8625i q^{23} +(124.936 + 4.00000i) q^{25} -27.0000i q^{27} +104.505 q^{29} +210.441 q^{31} +145.073i q^{33} +(5.91712 - 369.725i) q^{35} -300.945i q^{37} +181.073 q^{39} +240.147 q^{41} +108.000i q^{43} +(100.610 + 1.61018i) q^{45} +278.991i q^{47} -750.853 q^{49} +53.3673 q^{51} +328.358i q^{53} +(-540.588 - 8.65162i) q^{55} +392.588i q^{57} -889.533 q^{59} +241.450 q^{61} -297.661i q^{63} +(-10.7985 + 674.735i) q^{65} +103.834i q^{67} -212.588 q^{69} +277.597 q^{71} -274.403i q^{73} +(-12.0000 + 374.808i) q^{75} +1599.36i q^{77} +366.991 q^{79} +81.0000 q^{81} -57.7251i q^{83} +(-3.18262 + 198.863i) q^{85} +313.514i q^{87} +203.175 q^{89} +1996.24 q^{91} +631.322i q^{93} +(-1462.90 - 23.4124i) q^{95} +1283.45i q^{97} -435.220 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 22 q^{5} - 36 q^{9} + 148 q^{11} - 66 q^{15} + 160 q^{19} + 12 q^{21} + 100 q^{29} + 24 q^{31} + 796 q^{35} + 588 q^{39} + 688 q^{41} + 198 q^{45} - 3276 q^{49} - 468 q^{51} - 1072 q^{55} - 1332 q^{59}+ \cdots - 1332 q^{99}+O(q^{100}) \) 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
\(5\) −11.1789 0.178908i −0.999872 0.0160020i
\(6\) 0 0
\(7\) 33.0735i 1.78580i 0.450257 + 0.892899i \(0.351333\pi\)
−0.450257 + 0.892899i \(0.648667\pi\)
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 48.3578 1.32549 0.662747 0.748844i \(-0.269391\pi\)
0.662747 + 0.748844i \(0.269391\pi\)
\(12\) 0 0
\(13\) 60.3578i 1.28771i −0.765147 0.643856i \(-0.777334\pi\)
0.765147 0.643856i \(-0.222666\pi\)
\(14\) 0 0
\(15\) 0.536725 33.5367i 0.00923879 0.577276i
\(16\) 0 0
\(17\) 17.7891i 0.253793i −0.991916 0.126897i \(-0.959498\pi\)
0.991916 0.126897i \(-0.0405017\pi\)
\(18\) 0 0
\(19\) 130.863 1.58010 0.790051 0.613042i \(-0.210054\pi\)
0.790051 + 0.613042i \(0.210054\pi\)
\(20\) 0 0
\(21\) −99.2204 −1.03103
\(22\) 0 0
\(23\) 70.8625i 0.642429i 0.947007 + 0.321214i \(0.104091\pi\)
−0.947007 + 0.321214i \(0.895909\pi\)
\(24\) 0 0
\(25\) 124.936 + 4.00000i 0.999488 + 0.0320000i
\(26\) 0 0
\(27\) 27.0000i 0.192450i
\(28\) 0 0
\(29\) 104.505 0.669174 0.334587 0.942365i \(-0.391403\pi\)
0.334587 + 0.942365i \(0.391403\pi\)
\(30\) 0 0
\(31\) 210.441 1.21923 0.609617 0.792696i \(-0.291323\pi\)
0.609617 + 0.792696i \(0.291323\pi\)
\(32\) 0 0
\(33\) 145.073i 0.765274i
\(34\) 0 0
\(35\) 5.91712 369.725i 0.0285764 1.78557i
\(36\) 0 0
\(37\) 300.945i 1.33717i −0.743638 0.668583i \(-0.766901\pi\)
0.743638 0.668583i \(-0.233099\pi\)
\(38\) 0 0
\(39\) 181.073 0.743460
\(40\) 0 0
\(41\) 240.147 0.914747 0.457374 0.889275i \(-0.348790\pi\)
0.457374 + 0.889275i \(0.348790\pi\)
\(42\) 0 0
\(43\) 108.000i 0.383020i 0.981491 + 0.191510i \(0.0613384\pi\)
−0.981491 + 0.191510i \(0.938662\pi\)
\(44\) 0 0
\(45\) 100.610 + 1.61018i 0.333291 + 0.00533402i
\(46\) 0 0
\(47\) 278.991i 0.865850i 0.901430 + 0.432925i \(0.142519\pi\)
−0.901430 + 0.432925i \(0.857481\pi\)
\(48\) 0 0
\(49\) −750.853 −2.18908
\(50\) 0 0
\(51\) 53.3673 0.146528
\(52\) 0 0
\(53\) 328.358i 0.851008i 0.904957 + 0.425504i \(0.139903\pi\)
−0.904957 + 0.425504i \(0.860097\pi\)
\(54\) 0 0
\(55\) −540.588 8.65162i −1.32532 0.0212106i
\(56\) 0 0
\(57\) 392.588i 0.912272i
\(58\) 0 0
\(59\) −889.533 −1.96284 −0.981418 0.191882i \(-0.938541\pi\)
−0.981418 + 0.191882i \(0.938541\pi\)
\(60\) 0 0
\(61\) 241.450 0.506795 0.253398 0.967362i \(-0.418452\pi\)
0.253398 + 0.967362i \(0.418452\pi\)
\(62\) 0 0
\(63\) 297.661i 0.595266i
\(64\) 0 0
\(65\) −10.7985 + 674.735i −0.0206060 + 1.28755i
\(66\) 0 0
\(67\) 103.834i 0.189334i 0.995509 + 0.0946669i \(0.0301786\pi\)
−0.995509 + 0.0946669i \(0.969821\pi\)
\(68\) 0 0
\(69\) −212.588 −0.370906
\(70\) 0 0
\(71\) 277.597 0.464010 0.232005 0.972715i \(-0.425471\pi\)
0.232005 + 0.972715i \(0.425471\pi\)
\(72\) 0 0
\(73\) 274.403i 0.439951i −0.975505 0.219976i \(-0.929402\pi\)
0.975505 0.219976i \(-0.0705978\pi\)
\(74\) 0 0
\(75\) −12.0000 + 374.808i −0.0184752 + 0.577055i
\(76\) 0 0
\(77\) 1599.36i 2.36706i
\(78\) 0 0
\(79\) 366.991 0.522654 0.261327 0.965250i \(-0.415840\pi\)
0.261327 + 0.965250i \(0.415840\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 57.7251i 0.0763391i −0.999271 0.0381696i \(-0.987847\pi\)
0.999271 0.0381696i \(-0.0121527\pi\)
\(84\) 0 0
\(85\) −3.18262 + 198.863i −0.00406121 + 0.253761i
\(86\) 0 0
\(87\) 313.514i 0.386348i
\(88\) 0 0
\(89\) 203.175 0.241983 0.120992 0.992654i \(-0.461393\pi\)
0.120992 + 0.992654i \(0.461393\pi\)
\(90\) 0 0
\(91\) 1996.24 2.29959
\(92\) 0 0
\(93\) 631.322i 0.703925i
\(94\) 0 0
\(95\) −1462.90 23.4124i −1.57990 0.0252849i
\(96\) 0 0
\(97\) 1283.45i 1.34345i 0.740801 + 0.671725i \(0.234446\pi\)
−0.740801 + 0.671725i \(0.765554\pi\)
\(98\) 0 0
\(99\) −435.220 −0.441831
\(100\) 0 0
\(101\) −886.908 −0.873768 −0.436884 0.899518i \(-0.643918\pi\)
−0.436884 + 0.899518i \(0.643918\pi\)
\(102\) 0 0
\(103\) 783.055i 0.749094i −0.927208 0.374547i \(-0.877798\pi\)
0.927208 0.374547i \(-0.122202\pi\)
\(104\) 0 0
\(105\) 1109.18 + 17.7513i 1.03090 + 0.0164986i
\(106\) 0 0
\(107\) 1301.51i 1.17590i 0.808897 + 0.587950i \(0.200065\pi\)
−0.808897 + 0.587950i \(0.799935\pi\)
\(108\) 0 0
\(109\) 1161.91 1.02102 0.510508 0.859873i \(-0.329457\pi\)
0.510508 + 0.859873i \(0.329457\pi\)
\(110\) 0 0
\(111\) 902.836 0.772013
\(112\) 0 0
\(113\) 507.808i 0.422748i −0.977405 0.211374i \(-0.932206\pi\)
0.977405 0.211374i \(-0.0677938\pi\)
\(114\) 0 0
\(115\) 12.6779 792.166i 0.0102802 0.642346i
\(116\) 0 0
\(117\) 543.220i 0.429237i
\(118\) 0 0
\(119\) 588.346 0.453224
\(120\) 0 0
\(121\) 1007.48 0.756933
\(122\) 0 0
\(123\) 720.441i 0.528130i
\(124\) 0 0
\(125\) −1395.93 67.0677i −0.998848 0.0479898i
\(126\) 0 0
\(127\) 796.064i 0.556215i −0.960550 0.278107i \(-0.910293\pi\)
0.960550 0.278107i \(-0.0897071\pi\)
\(128\) 0 0
\(129\) −324.000 −0.221137
\(130\) 0 0
\(131\) −91.4764 −0.0610102 −0.0305051 0.999535i \(-0.509712\pi\)
−0.0305051 + 0.999535i \(0.509712\pi\)
\(132\) 0 0
\(133\) 4328.08i 2.82174i
\(134\) 0 0
\(135\) −4.83053 + 301.831i −0.00307960 + 0.192425i
\(136\) 0 0
\(137\) 2273.28i 1.41766i 0.705380 + 0.708829i \(0.250776\pi\)
−0.705380 + 0.708829i \(0.749224\pi\)
\(138\) 0 0
\(139\) 738.735 0.450782 0.225391 0.974268i \(-0.427634\pi\)
0.225391 + 0.974268i \(0.427634\pi\)
\(140\) 0 0
\(141\) −836.972 −0.499899
\(142\) 0 0
\(143\) 2918.77i 1.70685i
\(144\) 0 0
\(145\) −1168.25 18.6968i −0.669088 0.0107082i
\(146\) 0 0
\(147\) 2252.56i 1.26386i
\(148\) 0 0
\(149\) 1507.15 0.828661 0.414330 0.910127i \(-0.364016\pi\)
0.414330 + 0.910127i \(0.364016\pi\)
\(150\) 0 0
\(151\) −154.365 −0.0831925 −0.0415962 0.999135i \(-0.513244\pi\)
−0.0415962 + 0.999135i \(0.513244\pi\)
\(152\) 0 0
\(153\) 160.102i 0.0845978i
\(154\) 0 0
\(155\) −2352.50 37.6496i −1.21908 0.0195102i
\(156\) 0 0
\(157\) 315.514i 0.160387i 0.996779 + 0.0801935i \(0.0255538\pi\)
−0.996779 + 0.0801935i \(0.974446\pi\)
\(158\) 0 0
\(159\) −985.073 −0.491330
\(160\) 0 0
\(161\) −2343.67 −1.14725
\(162\) 0 0
\(163\) 1457.85i 0.700539i 0.936649 + 0.350270i \(0.113910\pi\)
−0.936649 + 0.350270i \(0.886090\pi\)
\(164\) 0 0
\(165\) 25.9549 1621.76i 0.0122460 0.765176i
\(166\) 0 0
\(167\) 2198.61i 1.01876i 0.860541 + 0.509381i \(0.170126\pi\)
−0.860541 + 0.509381i \(0.829874\pi\)
\(168\) 0 0
\(169\) −1446.07 −0.658200
\(170\) 0 0
\(171\) −1177.76 −0.526700
\(172\) 0 0
\(173\) 2030.49i 0.892343i −0.894948 0.446171i \(-0.852787\pi\)
0.894948 0.446171i \(-0.147213\pi\)
\(174\) 0 0
\(175\) −132.294 + 4132.06i −0.0571456 + 1.78488i
\(176\) 0 0
\(177\) 2668.60i 1.13324i
\(178\) 0 0
\(179\) −427.220 −0.178391 −0.0891954 0.996014i \(-0.528430\pi\)
−0.0891954 + 0.996014i \(0.528430\pi\)
\(180\) 0 0
\(181\) 3779.97 1.55228 0.776140 0.630561i \(-0.217175\pi\)
0.776140 + 0.630561i \(0.217175\pi\)
\(182\) 0 0
\(183\) 724.350i 0.292598i
\(184\) 0 0
\(185\) −53.8416 + 3364.24i −0.0213974 + 1.33699i
\(186\) 0 0
\(187\) 860.241i 0.336401i
\(188\) 0 0
\(189\) 892.983 0.343677
\(190\) 0 0
\(191\) −1565.60 −0.593103 −0.296551 0.955017i \(-0.595837\pi\)
−0.296551 + 0.955017i \(0.595837\pi\)
\(192\) 0 0
\(193\) 2642.63i 0.985599i 0.870143 + 0.492800i \(0.164026\pi\)
−0.870143 + 0.492800i \(0.835974\pi\)
\(194\) 0 0
\(195\) −2024.20 32.3956i −0.743365 0.0118969i
\(196\) 0 0
\(197\) 98.4522i 0.0356062i −0.999842 0.0178031i \(-0.994333\pi\)
0.999842 0.0178031i \(-0.00566721\pi\)
\(198\) 0 0
\(199\) 1131.62 0.403106 0.201553 0.979478i \(-0.435401\pi\)
0.201553 + 0.979478i \(0.435401\pi\)
\(200\) 0 0
\(201\) −311.503 −0.109312
\(202\) 0 0
\(203\) 3456.33i 1.19501i
\(204\) 0 0
\(205\) −2684.58 42.9643i −0.914630 0.0146378i
\(206\) 0 0
\(207\) 637.763i 0.214143i
\(208\) 0 0
\(209\) 6328.23 2.09441
\(210\) 0 0
\(211\) −1902.85 −0.620841 −0.310420 0.950599i \(-0.600470\pi\)
−0.310420 + 0.950599i \(0.600470\pi\)
\(212\) 0 0
\(213\) 832.791i 0.267896i
\(214\) 0 0
\(215\) 19.3221 1207.32i 0.00612910 0.382971i
\(216\) 0 0
\(217\) 6960.00i 2.17731i
\(218\) 0 0
\(219\) 823.209 0.254006
\(220\) 0 0
\(221\) −1073.71 −0.326813
\(222\) 0 0
\(223\) 4855.59i 1.45809i 0.684465 + 0.729046i \(0.260036\pi\)
−0.684465 + 0.729046i \(0.739964\pi\)
\(224\) 0 0
\(225\) −1124.42 36.0000i −0.333163 0.0106667i
\(226\) 0 0
\(227\) 6536.19i 1.91111i −0.294812 0.955555i \(-0.595257\pi\)
0.294812 0.955555i \(-0.404743\pi\)
\(228\) 0 0
\(229\) −5510.35 −1.59011 −0.795053 0.606539i \(-0.792557\pi\)
−0.795053 + 0.606539i \(0.792557\pi\)
\(230\) 0 0
\(231\) −4798.08 −1.36663
\(232\) 0 0
\(233\) 5915.65i 1.66329i −0.555306 0.831646i \(-0.687399\pi\)
0.555306 0.831646i \(-0.312601\pi\)
\(234\) 0 0
\(235\) 49.9137 3118.81i 0.0138554 0.865739i
\(236\) 0 0
\(237\) 1100.97i 0.301754i
\(238\) 0 0
\(239\) 2263.25 0.612543 0.306272 0.951944i \(-0.400918\pi\)
0.306272 + 0.951944i \(0.400918\pi\)
\(240\) 0 0
\(241\) 772.493 0.206476 0.103238 0.994657i \(-0.467080\pi\)
0.103238 + 0.994657i \(0.467080\pi\)
\(242\) 0 0
\(243\) 243.000i 0.0641500i
\(244\) 0 0
\(245\) 8393.72 + 134.334i 2.18880 + 0.0350297i
\(246\) 0 0
\(247\) 7898.58i 2.03471i
\(248\) 0 0
\(249\) 173.175 0.0440744
\(250\) 0 0
\(251\) 596.192 0.149926 0.0749628 0.997186i \(-0.476116\pi\)
0.0749628 + 0.997186i \(0.476116\pi\)
\(252\) 0 0
\(253\) 3426.76i 0.851535i
\(254\) 0 0
\(255\) −596.588 9.54785i −0.146509 0.00234474i
\(256\) 0 0
\(257\) 4139.19i 1.00465i 0.864678 + 0.502326i \(0.167522\pi\)
−0.864678 + 0.502326i \(0.832478\pi\)
\(258\) 0 0
\(259\) 9953.30 2.38791
\(260\) 0 0
\(261\) −940.542 −0.223058
\(262\) 0 0
\(263\) 1611.34i 0.377792i −0.981997 0.188896i \(-0.939509\pi\)
0.981997 0.188896i \(-0.0604909\pi\)
\(264\) 0 0
\(265\) 58.7460 3670.68i 0.0136179 0.850899i
\(266\) 0 0
\(267\) 609.526i 0.139709i
\(268\) 0 0
\(269\) −7031.33 −1.59371 −0.796855 0.604171i \(-0.793504\pi\)
−0.796855 + 0.604171i \(0.793504\pi\)
\(270\) 0 0
\(271\) 5441.27 1.21968 0.609840 0.792524i \(-0.291234\pi\)
0.609840 + 0.792524i \(0.291234\pi\)
\(272\) 0 0
\(273\) 5988.72i 1.32767i
\(274\) 0 0
\(275\) 6041.63 + 193.431i 1.32481 + 0.0424158i
\(276\) 0 0
\(277\) 1080.43i 0.234355i −0.993111 0.117178i \(-0.962615\pi\)
0.993111 0.117178i \(-0.0373847\pi\)
\(278\) 0 0
\(279\) −1893.97 −0.406411
\(280\) 0 0
\(281\) −1602.99 −0.340308 −0.170154 0.985417i \(-0.554427\pi\)
−0.170154 + 0.985417i \(0.554427\pi\)
\(282\) 0 0
\(283\) 334.810i 0.0703265i −0.999382 0.0351632i \(-0.988805\pi\)
0.999382 0.0351632i \(-0.0111951\pi\)
\(284\) 0 0
\(285\) 70.2372 4388.70i 0.0145982 0.912155i
\(286\) 0 0
\(287\) 7942.49i 1.63355i
\(288\) 0 0
\(289\) 4596.55 0.935589
\(290\) 0 0
\(291\) −3850.35 −0.775641
\(292\) 0 0
\(293\) 539.250i 0.107520i 0.998554 + 0.0537599i \(0.0171206\pi\)
−0.998554 + 0.0537599i \(0.982879\pi\)
\(294\) 0 0
\(295\) 9944.01 + 159.145i 1.96258 + 0.0314094i
\(296\) 0 0
\(297\) 1305.66i 0.255091i
\(298\) 0 0
\(299\) 4277.11 0.827263
\(300\) 0 0
\(301\) −3571.93 −0.683996
\(302\) 0 0
\(303\) 2660.72i 0.504470i
\(304\) 0 0
\(305\) −2699.15 43.1974i −0.506731 0.00810977i
\(306\) 0 0
\(307\) 8477.58i 1.57603i 0.615656 + 0.788015i \(0.288891\pi\)
−0.615656 + 0.788015i \(0.711109\pi\)
\(308\) 0 0
\(309\) 2349.16 0.432490
\(310\) 0 0
\(311\) −3646.92 −0.664945 −0.332473 0.943113i \(-0.607883\pi\)
−0.332473 + 0.943113i \(0.607883\pi\)
\(312\) 0 0
\(313\) 7537.05i 1.36108i −0.732709 0.680542i \(-0.761745\pi\)
0.732709 0.680542i \(-0.238255\pi\)
\(314\) 0 0
\(315\) −53.2540 + 3327.53i −0.00952548 + 0.595190i
\(316\) 0 0
\(317\) 10721.2i 1.89956i 0.312916 + 0.949781i \(0.398694\pi\)
−0.312916 + 0.949781i \(0.601306\pi\)
\(318\) 0 0
\(319\) 5053.62 0.886986
\(320\) 0 0
\(321\) −3904.52 −0.678907
\(322\) 0 0
\(323\) 2327.92i 0.401019i
\(324\) 0 0
\(325\) 241.431 7540.86i 0.0412068 1.28705i
\(326\) 0 0
\(327\) 3485.73i 0.589484i
\(328\) 0 0
\(329\) −9227.18 −1.54623
\(330\) 0 0
\(331\) −4405.14 −0.731505 −0.365753 0.930712i \(-0.619188\pi\)
−0.365753 + 0.930712i \(0.619188\pi\)
\(332\) 0 0
\(333\) 2708.51i 0.445722i
\(334\) 0 0
\(335\) 18.5768 1160.75i 0.00302973 0.189310i
\(336\) 0 0
\(337\) 186.117i 0.0300844i −0.999887 0.0150422i \(-0.995212\pi\)
0.999887 0.0150422i \(-0.00478827\pi\)
\(338\) 0 0
\(339\) 1523.42 0.244074
\(340\) 0 0
\(341\) 10176.5 1.61609
\(342\) 0 0
\(343\) 13489.1i 2.12345i
\(344\) 0 0
\(345\) 2376.50 + 38.0337i 0.370859 + 0.00593526i
\(346\) 0 0
\(347\) 5547.38i 0.858211i 0.903254 + 0.429105i \(0.141171\pi\)
−0.903254 + 0.429105i \(0.858829\pi\)
\(348\) 0 0
\(349\) 9078.82 1.39249 0.696244 0.717805i \(-0.254853\pi\)
0.696244 + 0.717805i \(0.254853\pi\)
\(350\) 0 0
\(351\) −1629.66 −0.247820
\(352\) 0 0
\(353\) 10678.1i 1.61002i −0.593262 0.805009i \(-0.702160\pi\)
0.593262 0.805009i \(-0.297840\pi\)
\(354\) 0 0
\(355\) −3103.23 49.6644i −0.463951 0.00742511i
\(356\) 0 0
\(357\) 1765.04i 0.261669i
\(358\) 0 0
\(359\) −265.733 −0.0390665 −0.0195332 0.999809i \(-0.506218\pi\)
−0.0195332 + 0.999809i \(0.506218\pi\)
\(360\) 0 0
\(361\) 10266.0 1.49672
\(362\) 0 0
\(363\) 3022.44i 0.437016i
\(364\) 0 0
\(365\) −49.0930 + 3067.53i −0.00704012 + 0.439895i
\(366\) 0 0
\(367\) 5854.21i 0.832663i 0.909213 + 0.416332i \(0.136684\pi\)
−0.909213 + 0.416332i \(0.863316\pi\)
\(368\) 0 0
\(369\) −2161.32 −0.304916
\(370\) 0 0
\(371\) −10859.9 −1.51973
\(372\) 0 0
\(373\) 10134.5i 1.40682i −0.710787 0.703408i \(-0.751661\pi\)
0.710787 0.703408i \(-0.248339\pi\)
\(374\) 0 0
\(375\) 201.203 4187.80i 0.0277069 0.576685i
\(376\) 0 0
\(377\) 6307.68i 0.861703i
\(378\) 0 0
\(379\) 4235.70 0.574072 0.287036 0.957920i \(-0.407330\pi\)
0.287036 + 0.957920i \(0.407330\pi\)
\(380\) 0 0
\(381\) 2388.19 0.321131
\(382\) 0 0
\(383\) 8100.84i 1.08077i 0.841419 + 0.540383i \(0.181721\pi\)
−0.841419 + 0.540383i \(0.818279\pi\)
\(384\) 0 0
\(385\) 286.139 17879.1i 0.0378779 2.36676i
\(386\) 0 0
\(387\) 972.000i 0.127673i
\(388\) 0 0
\(389\) −13820.2 −1.80131 −0.900656 0.434532i \(-0.856914\pi\)
−0.900656 + 0.434532i \(0.856914\pi\)
\(390\) 0 0
\(391\) 1260.58 0.163044
\(392\) 0 0
\(393\) 274.429i 0.0352242i
\(394\) 0 0
\(395\) −4102.55 65.6577i −0.522587 0.00836353i
\(396\) 0 0
\(397\) 11523.9i 1.45685i −0.685125 0.728425i \(-0.740252\pi\)
0.685125 0.728425i \(-0.259748\pi\)
\(398\) 0 0
\(399\) −12984.2 −1.62913
\(400\) 0 0
\(401\) 700.915 0.0872869 0.0436434 0.999047i \(-0.486103\pi\)
0.0436434 + 0.999047i \(0.486103\pi\)
\(402\) 0 0
\(403\) 12701.7i 1.57002i
\(404\) 0 0
\(405\) −905.492 14.4916i −0.111097 0.00177801i
\(406\) 0 0
\(407\) 14553.1i 1.77240i
\(408\) 0 0
\(409\) −6650.69 −0.804047 −0.402024 0.915629i \(-0.631693\pi\)
−0.402024 + 0.915629i \(0.631693\pi\)
\(410\) 0 0
\(411\) −6819.83 −0.818485
\(412\) 0 0
\(413\) 29419.9i 3.50523i
\(414\) 0 0
\(415\) −10.3275 + 645.303i −0.00122158 + 0.0763294i
\(416\) 0 0
\(417\) 2216.20i 0.260259i
\(418\) 0 0
\(419\) −13844.4 −1.61418 −0.807091 0.590426i \(-0.798960\pi\)
−0.807091 + 0.590426i \(0.798960\pi\)
\(420\) 0 0
\(421\) −12576.3 −1.45590 −0.727949 0.685631i \(-0.759526\pi\)
−0.727949 + 0.685631i \(0.759526\pi\)
\(422\) 0 0
\(423\) 2510.92i 0.288617i
\(424\) 0 0
\(425\) 71.1563 2222.50i 0.00812139 0.253663i
\(426\) 0 0
\(427\) 7985.59i 0.905035i
\(428\) 0 0
\(429\) 8756.32 0.985452
\(430\) 0 0
\(431\) −13440.6 −1.50212 −0.751059 0.660236i \(-0.770456\pi\)
−0.751059 + 0.660236i \(0.770456\pi\)
\(432\) 0 0
\(433\) 3012.87i 0.334387i −0.985924 0.167193i \(-0.946530\pi\)
0.985924 0.167193i \(-0.0534704\pi\)
\(434\) 0 0
\(435\) 56.0903 3504.75i 0.00618235 0.386298i
\(436\) 0 0
\(437\) 9273.25i 1.01510i
\(438\) 0 0
\(439\) 10675.0 1.16057 0.580283 0.814415i \(-0.302942\pi\)
0.580283 + 0.814415i \(0.302942\pi\)
\(440\) 0 0
\(441\) 6757.68 0.729692
\(442\) 0 0
\(443\) 125.868i 0.0134992i −0.999977 0.00674962i \(-0.997852\pi\)
0.999977 0.00674962i \(-0.00214849\pi\)
\(444\) 0 0
\(445\) −2271.28 36.3497i −0.241952 0.00387223i
\(446\) 0 0
\(447\) 4521.45i 0.478428i
\(448\) 0 0
\(449\) 9707.21 1.02029 0.510146 0.860088i \(-0.329591\pi\)
0.510146 + 0.860088i \(0.329591\pi\)
\(450\) 0 0
\(451\) 11613.0 1.21249
\(452\) 0 0
\(453\) 463.096i 0.0480312i
\(454\) 0 0
\(455\) −22315.8 357.144i −2.29930 0.0367982i
\(456\) 0 0
\(457\) 1279.97i 0.131016i −0.997852 0.0655082i \(-0.979133\pi\)
0.997852 0.0655082i \(-0.0208668\pi\)
\(458\) 0 0
\(459\) −480.305 −0.0488425
\(460\) 0 0
\(461\) −3080.48 −0.311220 −0.155610 0.987819i \(-0.549734\pi\)
−0.155610 + 0.987819i \(0.549734\pi\)
\(462\) 0 0
\(463\) 18017.3i 1.80850i 0.427008 + 0.904248i \(0.359568\pi\)
−0.427008 + 0.904248i \(0.640432\pi\)
\(464\) 0 0
\(465\) 112.949 7057.49i 0.0112642 0.703835i
\(466\) 0 0
\(467\) 7236.77i 0.717083i −0.933514 0.358541i \(-0.883274\pi\)
0.933514 0.358541i \(-0.116726\pi\)
\(468\) 0 0
\(469\) −3434.16 −0.338112
\(470\) 0 0
\(471\) −946.542 −0.0925995
\(472\) 0 0
\(473\) 5222.64i 0.507690i
\(474\) 0 0
\(475\) 16349.4 + 523.450i 1.57929 + 0.0505632i
\(476\) 0 0
\(477\) 2955.22i 0.283669i
\(478\) 0 0
\(479\) 4932.78 0.470532 0.235266 0.971931i \(-0.424404\pi\)
0.235266 + 0.971931i \(0.424404\pi\)
\(480\) 0 0
\(481\) −18164.4 −1.72188
\(482\) 0 0
\(483\) 7031.01i 0.662364i
\(484\) 0 0
\(485\) 229.620 14347.6i 0.0214979 1.34328i
\(486\) 0 0
\(487\) 5937.91i 0.552510i 0.961084 + 0.276255i \(0.0890934\pi\)
−0.961084 + 0.276255i \(0.910907\pi\)
\(488\) 0 0
\(489\) −4373.56 −0.404456
\(490\) 0 0
\(491\) 15703.5 1.44336 0.721678 0.692229i \(-0.243371\pi\)
0.721678 + 0.692229i \(0.243371\pi\)
\(492\) 0 0
\(493\) 1859.04i 0.169832i
\(494\) 0 0
\(495\) 4865.29 + 77.8646i 0.441775 + 0.00707020i
\(496\) 0 0
\(497\) 9181.09i 0.828628i
\(498\) 0 0
\(499\) 5656.77 0.507478 0.253739 0.967273i \(-0.418340\pi\)
0.253739 + 0.967273i \(0.418340\pi\)
\(500\) 0 0
\(501\) −6595.82 −0.588183
\(502\) 0 0
\(503\) 290.441i 0.0257457i 0.999917 + 0.0128729i \(0.00409768\pi\)
−0.999917 + 0.0128729i \(0.995902\pi\)
\(504\) 0 0
\(505\) 9914.66 + 158.675i 0.873657 + 0.0139821i
\(506\) 0 0
\(507\) 4338.20i 0.380012i
\(508\) 0 0
\(509\) 17330.6 1.50916 0.754582 0.656206i \(-0.227840\pi\)
0.754582 + 0.656206i \(0.227840\pi\)
\(510\) 0 0
\(511\) 9075.45 0.785664
\(512\) 0 0
\(513\) 3533.29i 0.304091i
\(514\) 0 0
\(515\) −140.095 + 8753.70i −0.0119870 + 0.748998i
\(516\) 0 0
\(517\) 13491.4i 1.14768i
\(518\) 0 0
\(519\) 6091.47 0.515194
\(520\) 0 0
\(521\) 6174.36 0.519201 0.259601 0.965716i \(-0.416409\pi\)
0.259601 + 0.965716i \(0.416409\pi\)
\(522\) 0 0
\(523\) 13389.4i 1.11946i 0.828675 + 0.559730i \(0.189095\pi\)
−0.828675 + 0.559730i \(0.810905\pi\)
\(524\) 0 0
\(525\) −12396.2 396.881i −1.03050 0.0329930i
\(526\) 0 0
\(527\) 3743.55i 0.309434i
\(528\) 0 0
\(529\) 7145.50 0.587285
\(530\) 0 0
\(531\) 8005.80 0.654279
\(532\) 0 0
\(533\) 14494.7i 1.17793i
\(534\) 0 0
\(535\) 232.850 14549.4i 0.0188168 1.17575i
\(536\) 0 0
\(537\) 1281.66i 0.102994i
\(538\) 0 0
\(539\) −36309.6 −2.90161
\(540\) 0 0
\(541\) 14355.5 1.14084 0.570418 0.821354i \(-0.306781\pi\)
0.570418 + 0.821354i \(0.306781\pi\)
\(542\) 0 0
\(543\) 11339.9i 0.896209i
\(544\) 0 0
\(545\) −12988.9 207.875i −1.02089 0.0163384i
\(546\) 0 0
\(547\) 21133.9i 1.65195i −0.563704 0.825977i \(-0.690624\pi\)
0.563704 0.825977i \(-0.309376\pi\)
\(548\) 0 0
\(549\) −2173.05 −0.168932
\(550\) 0 0
\(551\) 13675.8 1.05736
\(552\) 0 0
\(553\) 12137.6i 0.933355i
\(554\) 0 0
\(555\) −10092.7 161.525i −0.771914 0.0123538i
\(556\) 0 0
\(557\) 3098.60i 0.235713i −0.993031 0.117856i \(-0.962398\pi\)
0.993031 0.117856i \(-0.0376023\pi\)
\(558\) 0 0
\(559\) 6518.64 0.493219
\(560\) 0 0
\(561\) 2580.72 0.194221
\(562\) 0 0
\(563\) 7908.77i 0.592033i −0.955183 0.296017i \(-0.904342\pi\)
0.955183 0.296017i \(-0.0956584\pi\)
\(564\) 0 0
\(565\) −90.8511 + 5676.74i −0.00676484 + 0.422694i
\(566\) 0 0
\(567\) 2678.95i 0.198422i
\(568\) 0 0
\(569\) 10740.7 0.791345 0.395673 0.918392i \(-0.370512\pi\)
0.395673 + 0.918392i \(0.370512\pi\)
\(570\) 0 0
\(571\) 14701.2 1.07745 0.538725 0.842482i \(-0.318906\pi\)
0.538725 + 0.842482i \(0.318906\pi\)
\(572\) 0 0
\(573\) 4696.79i 0.342428i
\(574\) 0 0
\(575\) −283.450 + 8853.28i −0.0205577 + 0.642100i
\(576\) 0 0
\(577\) 15788.1i 1.13911i 0.821954 + 0.569554i \(0.192884\pi\)
−0.821954 + 0.569554i \(0.807116\pi\)
\(578\) 0 0
\(579\) −7927.89 −0.569036
\(580\) 0 0
\(581\) 1909.17 0.136326
\(582\) 0 0
\(583\) 15878.7i 1.12801i
\(584\) 0 0
\(585\) 97.1867 6072.61i 0.00686867 0.429182i
\(586\) 0 0
\(587\) 14778.5i 1.03913i 0.854430 + 0.519567i \(0.173907\pi\)
−0.854430 + 0.519567i \(0.826093\pi\)
\(588\) 0 0
\(589\) 27538.8 1.92651
\(590\) 0 0
\(591\) 295.356 0.0205573
\(592\) 0 0
\(593\) 11143.9i 0.771713i −0.922559 0.385857i \(-0.873906\pi\)
0.922559 0.385857i \(-0.126094\pi\)
\(594\) 0 0
\(595\) −6577.07 105.260i −0.453166 0.00725251i
\(596\) 0 0
\(597\) 3394.85i 0.232733i
\(598\) 0 0
\(599\) −7707.87 −0.525768 −0.262884 0.964827i \(-0.584674\pi\)
−0.262884 + 0.964827i \(0.584674\pi\)
\(600\) 0 0
\(601\) −13681.4 −0.928580 −0.464290 0.885683i \(-0.653691\pi\)
−0.464290 + 0.885683i \(0.653691\pi\)
\(602\) 0 0
\(603\) 934.508i 0.0631113i
\(604\) 0 0
\(605\) −11262.5 180.246i −0.756837 0.0121125i
\(606\) 0 0
\(607\) 11552.4i 0.772481i −0.922398 0.386241i \(-0.873773\pi\)
0.922398 0.386241i \(-0.126227\pi\)
\(608\) 0 0
\(609\) −10369.0 −0.689939
\(610\) 0 0
\(611\) 16839.3 1.11496
\(612\) 0 0
\(613\) 9904.66i 0.652603i −0.945266 0.326301i \(-0.894198\pi\)
0.945266 0.326301i \(-0.105802\pi\)
\(614\) 0 0
\(615\) 128.893 8053.74i 0.00845116 0.528062i
\(616\) 0 0
\(617\) 20323.5i 1.32608i 0.748582 + 0.663042i \(0.230735\pi\)
−0.748582 + 0.663042i \(0.769265\pi\)
\(618\) 0 0
\(619\) 9223.99 0.598940 0.299470 0.954106i \(-0.403190\pi\)
0.299470 + 0.954106i \(0.403190\pi\)
\(620\) 0 0
\(621\) 1913.29 0.123635
\(622\) 0 0
\(623\) 6719.70i 0.432134i
\(624\) 0 0
\(625\) 15593.0 + 999.488i 0.997952 + 0.0639672i
\(626\) 0 0
\(627\) 18984.7i 1.20921i
\(628\) 0 0
\(629\) −5353.54 −0.339364
\(630\) 0 0
\(631\) 16916.0 1.06722 0.533610 0.845730i \(-0.320835\pi\)
0.533610 + 0.845730i \(0.320835\pi\)
\(632\) 0 0
\(633\) 5708.54i 0.358443i
\(634\) 0 0
\(635\) −142.422 + 8899.13i −0.00890057 + 0.556143i
\(636\) 0 0
\(637\) 45319.9i 2.81890i
\(638\) 0 0
\(639\) −2498.37 −0.154670
\(640\) 0 0
\(641\) −5811.35 −0.358088 −0.179044 0.983841i \(-0.557300\pi\)
−0.179044 + 0.983841i \(0.557300\pi\)
\(642\) 0 0
\(643\) 27931.7i 1.71309i −0.516069 0.856547i \(-0.672605\pi\)
0.516069 0.856547i \(-0.327395\pi\)
\(644\) 0 0
\(645\) 3621.97 + 57.9663i 0.221108 + 0.00353864i
\(646\) 0 0
\(647\) 25437.8i 1.54569i −0.634595 0.772845i \(-0.718833\pi\)
0.634595 0.772845i \(-0.281167\pi\)
\(648\) 0 0
\(649\) −43015.9 −2.60173
\(650\) 0 0
\(651\) −20880.0 −1.25707
\(652\) 0 0
\(653\) 22647.7i 1.35723i 0.734493 + 0.678617i \(0.237420\pi\)
−0.734493 + 0.678617i \(0.762580\pi\)
\(654\) 0 0
\(655\) 1022.61 + 16.3659i 0.0610024 + 0.000976288i
\(656\) 0 0
\(657\) 2469.63i 0.146650i
\(658\) 0 0
\(659\) −25158.5 −1.48716 −0.743579 0.668649i \(-0.766873\pi\)
−0.743579 + 0.668649i \(0.766873\pi\)
\(660\) 0 0
\(661\) −23441.0 −1.37935 −0.689675 0.724119i \(-0.742247\pi\)
−0.689675 + 0.724119i \(0.742247\pi\)
\(662\) 0 0
\(663\) 3221.13i 0.188685i
\(664\) 0 0
\(665\) 774.329 48383.2i 0.0451537 2.82138i
\(666\) 0 0
\(667\) 7405.47i 0.429896i
\(668\) 0 0
\(669\) −14566.8 −0.841829
\(670\) 0 0
\(671\) 11676.0 0.671754
\(672\) 0 0
\(673\) 3145.33i 0.180154i 0.995935 + 0.0900770i \(0.0287113\pi\)
−0.995935 + 0.0900770i \(0.971289\pi\)
\(674\) 0 0
\(675\) 108.000 3373.27i 0.00615840 0.192352i
\(676\) 0 0
\(677\) 10606.2i 0.602109i 0.953607 + 0.301054i \(0.0973386\pi\)
−0.953607 + 0.301054i \(0.902661\pi\)
\(678\) 0 0
\(679\) −42448.1 −2.39913
\(680\) 0 0
\(681\) 19608.6 1.10338
\(682\) 0 0
\(683\) 7825.06i 0.438386i 0.975682 + 0.219193i \(0.0703424\pi\)
−0.975682 + 0.219193i \(0.929658\pi\)
\(684\) 0 0
\(685\) 406.708 25412.8i 0.0226854 1.41748i
\(686\) 0 0
\(687\) 16531.1i 0.918049i
\(688\) 0 0
\(689\) 19819.0 1.09585
\(690\) 0 0
\(691\) 22750.9 1.25251 0.626256 0.779618i \(-0.284587\pi\)
0.626256 + 0.779618i \(0.284587\pi\)
\(692\) 0 0
\(693\) 14394.2i 0.789022i
\(694\) 0 0
\(695\) −8258.25 132.166i −0.450724 0.00721343i
\(696\) 0 0
\(697\) 4271.99i 0.232157i
\(698\) 0 0
\(699\) 17747.0 0.960302
\(700\) 0 0
\(701\) 133.598 0.00719817 0.00359908 0.999994i \(-0.498854\pi\)
0.00359908 + 0.999994i \(0.498854\pi\)
\(702\) 0 0
\(703\) 39382.5i 2.11286i
\(704\) 0 0
\(705\) 9356.43 + 149.741i 0.499835 + 0.00799941i
\(706\) 0 0
\(707\) 29333.1i 1.56037i
\(708\) 0 0
\(709\) 6886.26 0.364766 0.182383 0.983228i \(-0.441619\pi\)
0.182383 + 0.983228i \(0.441619\pi\)
\(710\) 0 0
\(711\) −3302.92 −0.174218
\(712\) 0 0
\(713\) 14912.4i 0.783271i
\(714\) 0 0
\(715\) −522.193 + 32628.7i −0.0273131 + 1.70663i
\(716\) 0 0
\(717\) 6789.76i 0.353652i
\(718\) 0 0
\(719\) −1570.18 −0.0814436 −0.0407218 0.999171i \(-0.512966\pi\)
−0.0407218 + 0.999171i \(0.512966\pi\)
\(720\) 0 0
\(721\) 25898.3 1.33773
\(722\) 0 0
\(723\) 2317.48i 0.119209i
\(724\) 0 0
\(725\) 13056.4 + 418.019i 0.668831 + 0.0214136i
\(726\) 0 0
\(727\) 3399.18i 0.173409i 0.996234 + 0.0867047i \(0.0276337\pi\)
−0.996234 + 0.0867047i \(0.972366\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 1921.22 0.0972078
\(732\) 0 0
\(733\) 14152.1i 0.713125i −0.934272 0.356562i \(-0.883949\pi\)
0.934272 0.356562i \(-0.116051\pi\)
\(734\) 0 0
\(735\) −403.002 + 25181.2i −0.0202244 + 1.26370i
\(736\) 0 0
\(737\) 5021.20i 0.250961i
\(738\) 0 0
\(739\) 14919.5 0.742655 0.371328 0.928502i \(-0.378903\pi\)
0.371328 + 0.928502i \(0.378903\pi\)
\(740\) 0 0
\(741\) 23695.7 1.17474
\(742\) 0 0
\(743\) 7287.29i 0.359818i 0.983683 + 0.179909i \(0.0575803\pi\)
−0.983683 + 0.179909i \(0.942420\pi\)
\(744\) 0 0
\(745\) −16848.3 269.642i −0.828555 0.0132603i
\(746\) 0 0
\(747\) 519.526i 0.0254464i
\(748\) 0 0
\(749\) −43045.3 −2.09992
\(750\) 0 0
\(751\) −18783.4 −0.912670 −0.456335 0.889808i \(-0.650838\pi\)
−0.456335 + 0.889808i \(0.650838\pi\)
\(752\) 0 0
\(753\) 1788.58i 0.0865595i
\(754\) 0 0
\(755\) 1725.63 + 27.6172i 0.0831818 + 0.00133125i
\(756\) 0 0
\(757\) 30614.3i 1.46988i 0.678134 + 0.734939i \(0.262789\pi\)
−0.678134 + 0.734939i \(0.737211\pi\)
\(758\) 0 0
\(759\) −10280.3 −0.491634
\(760\) 0 0
\(761\) −15277.2 −0.727723 −0.363861 0.931453i \(-0.618542\pi\)
−0.363861 + 0.931453i \(0.618542\pi\)
\(762\) 0 0
\(763\) 38428.4i 1.82333i
\(764\) 0 0
\(765\) 28.6435 1789.76i 0.00135374 0.0845869i
\(766\) 0 0
\(767\) 53690.3i 2.52757i
\(768\) 0 0
\(769\) 17700.1 0.830016 0.415008 0.909818i \(-0.363779\pi\)
0.415008 + 0.909818i \(0.363779\pi\)
\(770\) 0 0
\(771\) −12417.6 −0.580036
\(772\) 0 0
\(773\) 29362.5i 1.36623i 0.730310 + 0.683116i \(0.239375\pi\)
−0.730310 + 0.683116i \(0.760625\pi\)
\(774\) 0 0
\(775\) 26291.6 + 841.763i 1.21861 + 0.0390155i
\(776\) 0 0
\(777\) 29859.9i 1.37866i
\(778\) 0 0
\(779\) 31426.2 1.44539
\(780\) 0 0
\(781\) 13424.0 0.615042
\(782\) 0 0
\(783\) 2821.63i 0.128783i
\(784\) 0 0
\(785\) 56.4481 3527.10i 0.00256652 0.160367i
\(786\) 0 0
\(787\) 2816.92i 0.127589i −0.997963 0.0637943i \(-0.979680\pi\)
0.997963 0.0637943i \(-0.0203202\pi\)
\(788\) 0 0
\(789\) 4834.01 0.218118
\(790\) 0 0
\(791\) 16795.0 0.754943
\(792\) 0 0
\(793\) 14573.4i 0.652606i
\(794\) 0 0
\(795\) 11012.0 + 176.238i 0.491267 + 0.00786228i
\(796\) 0 0
\(797\) 18433.9i 0.819276i 0.912248 + 0.409638i \(0.134345\pi\)
−0.912248 + 0.409638i \(0.865655\pi\)
\(798\) 0 0
\(799\) 4962.99 0.219747
\(800\) 0 0
\(801\) −1828.58 −0.0806612
\(802\) 0 0
\(803\) 13269.5i 0.583153i
\(804\) 0 0
\(805\) 26199.7 + 419.302i 1.14710 + 0.0183583i
\(806\) 0 0
\(807\) 21094.0i 0.920128i
\(808\) 0 0
\(809\) 13276.6 0.576983 0.288492 0.957482i \(-0.406846\pi\)
0.288492 + 0.957482i \(0.406846\pi\)
\(810\) 0 0
\(811\) −18842.5 −0.815845 −0.407923 0.913016i \(-0.633747\pi\)
−0.407923 + 0.913016i \(0.633747\pi\)
\(812\) 0 0
\(813\) 16323.8i 0.704183i
\(814\) 0 0
\(815\) 260.822 16297.2i 0.0112101 0.700449i
\(816\) 0 0
\(817\) 14133.2i 0.605210i
\(818\) 0 0
\(819\) −17966.2 −0.766531
\(820\) 0 0
\(821\) −5863.62 −0.249259 −0.124629 0.992203i \(-0.539774\pi\)
−0.124629 + 0.992203i \(0.539774\pi\)
\(822\) 0 0
\(823\) 5018.31i 0.212548i 0.994337 + 0.106274i \(0.0338921\pi\)
−0.994337 + 0.106274i \(0.966108\pi\)
\(824\) 0 0
\(825\) −580.294 + 18124.9i −0.0244888 + 0.764882i
\(826\) 0 0
\(827\) 20303.9i 0.853730i 0.904315 + 0.426865i \(0.140382\pi\)
−0.904315 + 0.426865i \(0.859618\pi\)
\(828\) 0 0
\(829\) −38156.2 −1.59857 −0.799287 0.600949i \(-0.794789\pi\)
−0.799287 + 0.600949i \(0.794789\pi\)
\(830\) 0 0
\(831\) 3241.28 0.135305
\(832\) 0 0
\(833\) 13357.0i 0.555573i
\(834\) 0 0
\(835\) 393.349 24578.0i 0.0163023 1.01863i
\(836\) 0 0
\(837\) 5681.90i 0.234642i
\(838\) 0 0
\(839\) 41004.6 1.68729 0.843645 0.536901i \(-0.180405\pi\)
0.843645 + 0.536901i \(0.180405\pi\)
\(840\) 0 0
\(841\) −13467.8 −0.552206
\(842\) 0 0
\(843\) 4808.98i 0.196477i
\(844\) 0 0
\(845\) 16165.4 + 258.713i 0.658116 + 0.0105326i
\(846\) 0 0
\(847\) 33320.8i 1.35173i
\(848\) 0 0
\(849\) 1004.43 0.0406030
\(850\) 0 0
\(851\) 21325.8 0.859033
\(852\) 0 0
\(853\) 22451.8i 0.901214i −0.892722 0.450607i \(-0.851208\pi\)
0.892722 0.450607i \(-0.148792\pi\)
\(854\) 0 0
\(855\) 13166.1 + 210.712i 0.526633 + 0.00842829i
\(856\) 0 0
\(857\) 39858.0i 1.58871i 0.607455 + 0.794354i \(0.292191\pi\)
−0.607455 + 0.794354i \(0.707809\pi\)
\(858\) 0 0
\(859\) −32585.0 −1.29428 −0.647139 0.762372i \(-0.724035\pi\)
−0.647139 + 0.762372i \(0.724035\pi\)
\(860\) 0 0
\(861\) −23827.5 −0.943133
\(862\) 0 0
\(863\) 15593.6i 0.615078i −0.951535 0.307539i \(-0.900495\pi\)
0.951535 0.307539i \(-0.0995055\pi\)
\(864\) 0 0
\(865\) −363.272 + 22698.7i −0.0142793 + 0.892228i
\(866\) 0 0
\(867\) 13789.6i 0.540163i
\(868\) 0 0
\(869\) 17746.9 0.692775
\(870\) 0 0
\(871\) 6267.21 0.243807
\(872\) 0 0
\(873\) 11551.1i 0.447817i
\(874\) 0 0
\(875\) 2218.16 46168.3i 0.0857000 1.78374i
\(876\) 0 0
\(877\) 1804.75i 0.0694892i 0.999396 + 0.0347446i \(0.0110618\pi\)
−0.999396 + 0.0347446i \(0.988938\pi\)
\(878\) 0 0
\(879\) −1617.75 −0.0620766
\(880\) 0 0
\(881\) −14788.0 −0.565515 −0.282758 0.959191i \(-0.591249\pi\)
−0.282758 + 0.959191i \(0.591249\pi\)
\(882\) 0 0
\(883\) 1999.24i 0.0761947i −0.999274 0.0380973i \(-0.987870\pi\)
0.999274 0.0380973i \(-0.0121297\pi\)
\(884\) 0 0
\(885\) −477.435 + 29832.0i −0.0181342 + 1.13310i
\(886\) 0 0
\(887\) 32474.4i 1.22929i −0.788802 0.614647i \(-0.789299\pi\)
0.788802 0.614647i \(-0.210701\pi\)
\(888\) 0 0
\(889\) 26328.6 0.993287
\(890\) 0 0
\(891\) 3916.98 0.147277
\(892\) 0 0
\(893\) 36509.4i 1.36813i
\(894\) 0 0
\(895\) 4775.86 + 76.4333i 0.178368 + 0.00285462i
\(896\) 0 0
\(897\) 12831.3i 0.477620i
\(898\) 0 0
\(899\) 21992.0 0.815880
\(900\) 0 0
\(901\) 5841.18 0.215980
\(902\) 0 0
\(903\) 10715.8i 0.394905i
\(904\) 0 0
\(905\) −42255.9 676.268i −1.55208 0.0248397i
\(906\) 0 0
\(907\) 27597.9i 1.01033i −0.863022 0.505167i \(-0.831431\pi\)
0.863022 0.505167i \(-0.168569\pi\)
\(908\) 0 0
\(909\) 7982.17 0.291256
\(910\) 0 0
\(911\) 19861.4 0.722325 0.361163 0.932503i \(-0.382380\pi\)
0.361163 + 0.932503i \(0.382380\pi\)
\(912\) 0 0
\(913\) 2791.46i 0.101187i
\(914\) 0 0
\(915\) 129.592 8097.45i 0.00468218 0.292561i
\(916\) 0 0
\(917\) 3025.44i 0.108952i
\(918\) 0 0
\(919\) −20886.0 −0.749691 −0.374845 0.927087i \(-0.622304\pi\)
−0.374845 + 0.927087i \(0.622304\pi\)
\(920\) 0 0
\(921\) −25432.7 −0.909921
\(922\) 0 0
\(923\) 16755.2i 0.597511i
\(924\) 0 0
\(925\) 1203.78 37598.9i 0.0427893 1.33648i
\(926\) 0 0
\(927\) 7047.49i 0.249698i
\(928\) 0 0
\(929\) −37461.2 −1.32299 −0.661497 0.749948i \(-0.730078\pi\)
−0.661497 + 0.749948i \(0.730078\pi\)
\(930\) 0 0
\(931\) −98258.5 −3.45896
\(932\) 0 0
\(933\) 10940.8i 0.383906i
\(934\) 0 0
\(935\) −153.904 + 9616.56i −0.00538311 + 0.336358i
\(936\) 0 0
\(937\) 27149.0i 0.946552i 0.880914 + 0.473276i \(0.156929\pi\)
−0.880914 + 0.473276i \(0.843071\pi\)
\(938\) 0 0
\(939\) 22611.1 0.785822
\(940\) 0 0
\(941\) 43068.3 1.49202 0.746008 0.665937i \(-0.231968\pi\)
0.746008 + 0.665937i \(0.231968\pi\)
\(942\) 0 0
\(943\) 17017.4i 0.587660i
\(944\) 0 0
\(945\) −9982.58 159.762i −0.343633 0.00549954i
\(946\) 0 0
\(947\) 30021.1i 1.03015i 0.857145 + 0.515076i \(0.172236\pi\)
−0.857145 + 0.515076i \(0.827764\pi\)
\(948\) 0 0
\(949\) −16562.4 −0.566530
\(950\) 0 0
\(951\) −32163.5 −1.09671
\(952\) 0 0
\(953\) 53820.4i 1.82939i 0.404140 + 0.914697i \(0.367571\pi\)
−0.404140 + 0.914697i \(0.632429\pi\)
\(954\) 0 0
\(955\) 17501.7 + 280.098i 0.593027 + 0.00949086i
\(956\) 0 0
\(957\) 15160.9i 0.512101i
\(958\) 0 0
\(959\) −75185.1 −2.53165
\(960\) 0 0
\(961\) 14494.3 0.486532
\(962\) 0 0
\(963\) 11713.6i 0.391967i
\(964\) 0 0
\(965\) 472.788 29541.7i 0.0157716 0.985473i
\(966\) 0 0
\(967\) 32210.2i 1.07116i 0.844485 + 0.535580i \(0.179907\pi\)
−0.844485 + 0.535580i \(0.820093\pi\)
\(968\) 0 0
\(969\) 6983.77 0.231529
\(970\) 0 0
\(971\) −21616.1 −0.714411 −0.357206 0.934026i \(-0.616270\pi\)
−0.357206 + 0.934026i \(0.616270\pi\)
\(972\) 0 0
\(973\) 24432.5i 0.805005i
\(974\) 0 0
\(975\) 22622.6 + 724.294i 0.743080 + 0.0237907i
\(976\) 0 0
\(977\) 46002.8i 1.50641i 0.657787 + 0.753204i \(0.271493\pi\)
−0.657787 + 0.753204i \(0.728507\pi\)
\(978\) 0 0
\(979\) 9825.11 0.320748
\(980\) 0 0
\(981\) −10457.2 −0.340339
\(982\) 0 0
\(983\) 3926.43i 0.127400i 0.997969 + 0.0636998i \(0.0202900\pi\)
−0.997969 + 0.0636998i \(0.979710\pi\)
\(984\) 0 0
\(985\) −17.6139 + 1100.59i −0.000569772 + 0.0356017i
\(986\) 0 0
\(987\) 27681.5i 0.892719i
\(988\) 0 0
\(989\) −7653.15 −0.246063
\(990\) 0 0
\(991\) −55802.3 −1.78872 −0.894359 0.447351i \(-0.852368\pi\)
−0.894359 + 0.447351i \(0.852368\pi\)
\(992\) 0 0
\(993\) 13215.4i 0.422335i
\(994\) 0 0
\(995\) −12650.2 202.456i −0.403055 0.00645052i
\(996\) 0 0
\(997\) 43880.8i 1.39390i 0.717119 + 0.696951i \(0.245460\pi\)
−0.717119 + 0.696951i \(0.754540\pi\)
\(998\) 0 0
\(999\) −8125.53 −0.257338
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.o.769.3 4
4.3 odd 2 960.4.f.n.769.1 4
5.4 even 2 inner 960.4.f.o.769.1 4
8.3 odd 2 120.4.f.d.49.4 yes 4
8.5 even 2 240.4.f.g.49.2 4
20.19 odd 2 960.4.f.n.769.3 4
24.5 odd 2 720.4.f.i.289.1 4
24.11 even 2 360.4.f.d.289.1 4
40.3 even 4 600.4.a.t.1.1 2
40.13 odd 4 1200.4.a.bq.1.2 2
40.19 odd 2 120.4.f.d.49.2 4
40.27 even 4 600.4.a.v.1.2 2
40.29 even 2 240.4.f.g.49.4 4
40.37 odd 4 1200.4.a.bo.1.1 2
120.29 odd 2 720.4.f.i.289.2 4
120.59 even 2 360.4.f.d.289.2 4
120.83 odd 4 1800.4.a.bn.1.1 2
120.107 odd 4 1800.4.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
120.4.f.d.49.2 4 40.19 odd 2
120.4.f.d.49.4 yes 4 8.3 odd 2
240.4.f.g.49.2 4 8.5 even 2
240.4.f.g.49.4 4 40.29 even 2
360.4.f.d.289.1 4 24.11 even 2
360.4.f.d.289.2 4 120.59 even 2
600.4.a.t.1.1 2 40.3 even 4
600.4.a.v.1.2 2 40.27 even 4
720.4.f.i.289.1 4 24.5 odd 2
720.4.f.i.289.2 4 120.29 odd 2
960.4.f.n.769.1 4 4.3 odd 2
960.4.f.n.769.3 4 20.19 odd 2
960.4.f.o.769.1 4 5.4 even 2 inner
960.4.f.o.769.3 4 1.1 even 1 trivial
1200.4.a.bo.1.1 2 40.37 odd 4
1200.4.a.bq.1.2 2 40.13 odd 4
1800.4.a.bl.1.2 2 120.107 odd 4
1800.4.a.bn.1.1 2 120.83 odd 4