Properties

Label 3600.3.j.i.1999.3
Level $3600$
Weight $3$
Character 3600.1999
Analytic conductor $98.093$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3600,3,Mod(1999,3600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3600.1999");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3600 = 2^{4} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 3600.j (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: \(98.0928951697\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{12})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{7} \)
Twist minimal: no (minimal twist has level 48)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1999.3
Root \(-0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 3600.1999
Dual form 3600.3.j.i.1999.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+6.92820 q^{7} +O(q^{10})\) \(q+6.92820 q^{7} -20.7846i q^{11} +14.0000i q^{13} +6.00000i q^{17} -6.92820i q^{19} +30.0000 q^{29} -20.7846i q^{31} +26.0000i q^{37} +54.0000 q^{41} +20.7846 q^{43} -41.5692 q^{47} -1.00000 q^{49} -18.0000i q^{53} -20.7846i q^{59} -70.0000 q^{61} +117.779 q^{67} -83.1384i q^{71} -82.0000i q^{73} -144.000i q^{77} +76.2102i q^{79} +20.7846 q^{83} +114.000 q^{89} +96.9948i q^{91} +34.0000i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 120 q^{29} + 216 q^{41} - 4 q^{49} - 280 q^{61} + 456 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3600\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(2801\) \(3151\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(-1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 6.92820 0.989743 0.494872 0.868966i \(-0.335215\pi\)
0.494872 + 0.868966i \(0.335215\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 20.7846i − 1.88951i −0.327777 0.944755i \(-0.606300\pi\)
0.327777 0.944755i \(-0.393700\pi\)
\(12\) 0 0
\(13\) 14.0000i 1.07692i 0.842650 + 0.538462i \(0.180994\pi\)
−0.842650 + 0.538462i \(0.819006\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.00000i 0.352941i 0.984306 + 0.176471i \(0.0564680\pi\)
−0.984306 + 0.176471i \(0.943532\pi\)
\(18\) 0 0
\(19\) − 6.92820i − 0.364642i −0.983239 0.182321i \(-0.941639\pi\)
0.983239 0.182321i \(-0.0583610\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 30.0000 1.03448 0.517241 0.855840i \(-0.326959\pi\)
0.517241 + 0.855840i \(0.326959\pi\)
\(30\) 0 0
\(31\) − 20.7846i − 0.670471i −0.942134 0.335236i \(-0.891184\pi\)
0.942134 0.335236i \(-0.108816\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 26.0000i 0.702703i 0.936244 + 0.351351i \(0.114278\pi\)
−0.936244 + 0.351351i \(0.885722\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 54.0000 1.31707 0.658537 0.752549i \(-0.271176\pi\)
0.658537 + 0.752549i \(0.271176\pi\)
\(42\) 0 0
\(43\) 20.7846 0.483363 0.241682 0.970356i \(-0.422301\pi\)
0.241682 + 0.970356i \(0.422301\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −41.5692 −0.884451 −0.442226 0.896904i \(-0.645811\pi\)
−0.442226 + 0.896904i \(0.645811\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.0204082
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 18.0000i − 0.339623i −0.985477 0.169811i \(-0.945684\pi\)
0.985477 0.169811i \(-0.0543158\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 20.7846i − 0.352282i −0.984365 0.176141i \(-0.943639\pi\)
0.984365 0.176141i \(-0.0563614\pi\)
\(60\) 0 0
\(61\) −70.0000 −1.14754 −0.573770 0.819016i \(-0.694520\pi\)
−0.573770 + 0.819016i \(0.694520\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 117.779 1.75790 0.878951 0.476912i \(-0.158244\pi\)
0.878951 + 0.476912i \(0.158244\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 83.1384i − 1.17096i −0.810685 0.585482i \(-0.800905\pi\)
0.810685 0.585482i \(-0.199095\pi\)
\(72\) 0 0
\(73\) − 82.0000i − 1.12329i −0.827379 0.561644i \(-0.810169\pi\)
0.827379 0.561644i \(-0.189831\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 144.000i − 1.87013i
\(78\) 0 0
\(79\) 76.2102i 0.964687i 0.875982 + 0.482343i \(0.160214\pi\)
−0.875982 + 0.482343i \(0.839786\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 20.7846 0.250417 0.125208 0.992130i \(-0.460040\pi\)
0.125208 + 0.992130i \(0.460040\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 114.000 1.28090 0.640449 0.768000i \(-0.278748\pi\)
0.640449 + 0.768000i \(0.278748\pi\)
\(90\) 0 0
\(91\) 96.9948i 1.06588i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 34.0000i 0.350515i 0.984523 + 0.175258i \(0.0560759\pi\)
−0.984523 + 0.175258i \(0.943924\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.0000 0.178218 0.0891089 0.996022i \(-0.471598\pi\)
0.0891089 + 0.996022i \(0.471598\pi\)
\(102\) 0 0
\(103\) −131.636 −1.27802 −0.639009 0.769199i \(-0.720655\pi\)
−0.639009 + 0.769199i \(0.720655\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 145.492 1.35974 0.679870 0.733332i \(-0.262036\pi\)
0.679870 + 0.733332i \(0.262036\pi\)
\(108\) 0 0
\(109\) −34.0000 −0.311927 −0.155963 0.987763i \(-0.549848\pi\)
−0.155963 + 0.987763i \(0.549848\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 78.0000i − 0.690265i −0.938554 0.345133i \(-0.887834\pi\)
0.938554 0.345133i \(-0.112166\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 41.5692i 0.349321i
\(120\) 0 0
\(121\) −311.000 −2.57025
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −103.923 −0.818292 −0.409146 0.912469i \(-0.634173\pi\)
−0.409146 + 0.912469i \(0.634173\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 103.923i − 0.793306i −0.917969 0.396653i \(-0.870172\pi\)
0.917969 0.396653i \(-0.129828\pi\)
\(132\) 0 0
\(133\) − 48.0000i − 0.360902i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 186.000i − 1.35766i −0.734294 0.678832i \(-0.762486\pi\)
0.734294 0.678832i \(-0.237514\pi\)
\(138\) 0 0
\(139\) 48.4974i 0.348902i 0.984666 + 0.174451i \(0.0558151\pi\)
−0.984666 + 0.174451i \(0.944185\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 290.985 2.03486
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −186.000 −1.24832 −0.624161 0.781296i \(-0.714559\pi\)
−0.624161 + 0.781296i \(0.714559\pi\)
\(150\) 0 0
\(151\) − 34.6410i − 0.229411i −0.993400 0.114705i \(-0.963408\pi\)
0.993400 0.114705i \(-0.0365924\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 170.000i 1.08280i 0.840764 + 0.541401i \(0.182106\pi\)
−0.840764 + 0.541401i \(0.817894\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 284.056 1.74268 0.871338 0.490683i \(-0.163253\pi\)
0.871338 + 0.490683i \(0.163253\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −207.846 −1.24459 −0.622294 0.782784i \(-0.713799\pi\)
−0.622294 + 0.782784i \(0.713799\pi\)
\(168\) 0 0
\(169\) −27.0000 −0.159763
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 42.0000i − 0.242775i −0.992605 0.121387i \(-0.961266\pi\)
0.992605 0.121387i \(-0.0387343\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 145.492i − 0.812806i −0.913694 0.406403i \(-0.866783\pi\)
0.913694 0.406403i \(-0.133217\pi\)
\(180\) 0 0
\(181\) 82.0000 0.453039 0.226519 0.974007i \(-0.427265\pi\)
0.226519 + 0.974007i \(0.427265\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 124.708 0.666886
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 332.554i − 1.74112i −0.492063 0.870560i \(-0.663757\pi\)
0.492063 0.870560i \(-0.336243\pi\)
\(192\) 0 0
\(193\) 94.0000i 0.487047i 0.969895 + 0.243523i \(0.0783032\pi\)
−0.969895 + 0.243523i \(0.921697\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 258.000i 1.30964i 0.755783 + 0.654822i \(0.227257\pi\)
−0.755783 + 0.654822i \(0.772743\pi\)
\(198\) 0 0
\(199\) − 117.779i − 0.591857i −0.955210 0.295928i \(-0.904371\pi\)
0.955210 0.295928i \(-0.0956289\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 207.846 1.02387
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −144.000 −0.688995
\(210\) 0 0
\(211\) − 90.0666i − 0.426856i −0.976959 0.213428i \(-0.931537\pi\)
0.976959 0.213428i \(-0.0684629\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) − 144.000i − 0.663594i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −84.0000 −0.380090
\(222\) 0 0
\(223\) 353.338 1.58448 0.792238 0.610212i \(-0.208916\pi\)
0.792238 + 0.610212i \(0.208916\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 145.492 0.640935 0.320468 0.947259i \(-0.396160\pi\)
0.320468 + 0.947259i \(0.396160\pi\)
\(228\) 0 0
\(229\) −226.000 −0.986900 −0.493450 0.869774i \(-0.664264\pi\)
−0.493450 + 0.869774i \(0.664264\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 114.000i 0.489270i 0.969615 + 0.244635i \(0.0786682\pi\)
−0.969615 + 0.244635i \(0.921332\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 332.554i 1.39144i 0.718314 + 0.695719i \(0.244914\pi\)
−0.718314 + 0.695719i \(0.755086\pi\)
\(240\) 0 0
\(241\) 178.000 0.738589 0.369295 0.929312i \(-0.379599\pi\)
0.369295 + 0.929312i \(0.379599\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 96.9948 0.392692
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 103.923i − 0.414036i −0.978337 0.207018i \(-0.933624\pi\)
0.978337 0.207018i \(-0.0663759\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 258.000i − 1.00389i −0.864899 0.501946i \(-0.832618\pi\)
0.864899 0.501946i \(-0.167382\pi\)
\(258\) 0 0
\(259\) 180.133i 0.695495i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 374.123 1.42252 0.711260 0.702929i \(-0.248125\pi\)
0.711260 + 0.702929i \(0.248125\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 510.000 1.89591 0.947955 0.318403i \(-0.103147\pi\)
0.947955 + 0.318403i \(0.103147\pi\)
\(270\) 0 0
\(271\) − 450.333i − 1.66175i −0.556462 0.830873i \(-0.687842\pi\)
0.556462 0.830873i \(-0.312158\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 14.0000i − 0.0505415i −0.999681 0.0252708i \(-0.991955\pi\)
0.999681 0.0252708i \(-0.00804479\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −354.000 −1.25979 −0.629893 0.776682i \(-0.716901\pi\)
−0.629893 + 0.776682i \(0.716901\pi\)
\(282\) 0 0
\(283\) 145.492 0.514107 0.257053 0.966397i \(-0.417248\pi\)
0.257053 + 0.966397i \(0.417248\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 374.123 1.30356
\(288\) 0 0
\(289\) 253.000 0.875433
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 498.000i − 1.69966i −0.527058 0.849829i \(-0.676705\pi\)
0.527058 0.849829i \(-0.323295\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 144.000 0.478405
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −187.061 −0.609321 −0.304660 0.952461i \(-0.598543\pi\)
−0.304660 + 0.952461i \(0.598543\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 41.5692i 0.133663i 0.997764 + 0.0668315i \(0.0212890\pi\)
−0.997764 + 0.0668315i \(0.978711\pi\)
\(312\) 0 0
\(313\) − 290.000i − 0.926518i −0.886223 0.463259i \(-0.846680\pi\)
0.886223 0.463259i \(-0.153320\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 210.000i 0.662461i 0.943550 + 0.331230i \(0.107464\pi\)
−0.943550 + 0.331230i \(0.892536\pi\)
\(318\) 0 0
\(319\) − 623.538i − 1.95467i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 41.5692 0.128697
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −288.000 −0.875380
\(330\) 0 0
\(331\) 200.918i 0.607003i 0.952831 + 0.303501i \(0.0981557\pi\)
−0.952831 + 0.303501i \(0.901844\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 302.000i − 0.896142i −0.893998 0.448071i \(-0.852111\pi\)
0.893998 0.448071i \(-0.147889\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −432.000 −1.26686
\(342\) 0 0
\(343\) −346.410 −1.00994
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −62.3538 −0.179694 −0.0898470 0.995956i \(-0.528638\pi\)
−0.0898470 + 0.995956i \(0.528638\pi\)
\(348\) 0 0
\(349\) 358.000 1.02579 0.512894 0.858452i \(-0.328573\pi\)
0.512894 + 0.858452i \(0.328573\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 558.000i − 1.58074i −0.612632 0.790368i \(-0.709889\pi\)
0.612632 0.790368i \(-0.290111\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 83.1384i − 0.231583i −0.993274 0.115792i \(-0.963059\pi\)
0.993274 0.115792i \(-0.0369405\pi\)
\(360\) 0 0
\(361\) 313.000 0.867036
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 214.774 0.585216 0.292608 0.956232i \(-0.405477\pi\)
0.292608 + 0.956232i \(0.405477\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 124.708i − 0.336139i
\(372\) 0 0
\(373\) − 554.000i − 1.48525i −0.669705 0.742627i \(-0.733579\pi\)
0.669705 0.742627i \(-0.266421\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 420.000i 1.11406i
\(378\) 0 0
\(379\) 533.472i 1.40758i 0.710410 + 0.703788i \(0.248510\pi\)
−0.710410 + 0.703788i \(0.751490\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 498.831 1.30243 0.651215 0.758893i \(-0.274260\pi\)
0.651215 + 0.758893i \(0.274260\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 198.000 0.508997 0.254499 0.967073i \(-0.418090\pi\)
0.254499 + 0.967073i \(0.418090\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 646.000i − 1.62720i −0.581422 0.813602i \(-0.697504\pi\)
0.581422 0.813602i \(-0.302496\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −330.000 −0.822943 −0.411471 0.911423i \(-0.634985\pi\)
−0.411471 + 0.911423i \(0.634985\pi\)
\(402\) 0 0
\(403\) 290.985 0.722046
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 540.400 1.32776
\(408\) 0 0
\(409\) −130.000 −0.317848 −0.158924 0.987291i \(-0.550803\pi\)
−0.158924 + 0.987291i \(0.550803\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 144.000i − 0.348668i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 353.338i − 0.843290i −0.906761 0.421645i \(-0.861453\pi\)
0.906761 0.421645i \(-0.138547\pi\)
\(420\) 0 0
\(421\) −398.000 −0.945368 −0.472684 0.881232i \(-0.656715\pi\)
−0.472684 + 0.881232i \(0.656715\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −484.974 −1.13577
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 124.708i 0.289345i 0.989480 + 0.144672i \(0.0462128\pi\)
−0.989480 + 0.144672i \(0.953787\pi\)
\(432\) 0 0
\(433\) 142.000i 0.327945i 0.986465 + 0.163972i \(0.0524308\pi\)
−0.986465 + 0.163972i \(0.947569\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) − 561.184i − 1.27832i −0.769072 0.639162i \(-0.779281\pi\)
0.769072 0.639162i \(-0.220719\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −436.477 −0.985275 −0.492637 0.870235i \(-0.663967\pi\)
−0.492637 + 0.870235i \(0.663967\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −198.000 −0.440980 −0.220490 0.975389i \(-0.570766\pi\)
−0.220490 + 0.975389i \(0.570766\pi\)
\(450\) 0 0
\(451\) − 1122.37i − 2.48862i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 446.000i − 0.975930i −0.872863 0.487965i \(-0.837739\pi\)
0.872863 0.487965i \(-0.162261\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −342.000 −0.741866 −0.370933 0.928660i \(-0.620962\pi\)
−0.370933 + 0.928660i \(0.620962\pi\)
\(462\) 0 0
\(463\) 159.349 0.344166 0.172083 0.985082i \(-0.444950\pi\)
0.172083 + 0.985082i \(0.444950\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 394.908 0.845627 0.422813 0.906217i \(-0.361043\pi\)
0.422813 + 0.906217i \(0.361043\pi\)
\(468\) 0 0
\(469\) 816.000 1.73987
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 432.000i − 0.913319i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 789.815i 1.64888i 0.565947 + 0.824442i \(0.308511\pi\)
−0.565947 + 0.824442i \(0.691489\pi\)
\(480\) 0 0
\(481\) −364.000 −0.756757
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 6.92820 0.0142263 0.00711315 0.999975i \(-0.497736\pi\)
0.00711315 + 0.999975i \(0.497736\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 644.323i − 1.31227i −0.754645 0.656133i \(-0.772191\pi\)
0.754645 0.656133i \(-0.227809\pi\)
\(492\) 0 0
\(493\) 180.000i 0.365112i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 576.000i − 1.15895i
\(498\) 0 0
\(499\) 810.600i 1.62445i 0.583345 + 0.812224i \(0.301743\pi\)
−0.583345 + 0.812224i \(0.698257\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 332.554 0.661141 0.330570 0.943781i \(-0.392759\pi\)
0.330570 + 0.943781i \(0.392759\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −306.000 −0.601179 −0.300589 0.953754i \(-0.597183\pi\)
−0.300589 + 0.953754i \(0.597183\pi\)
\(510\) 0 0
\(511\) − 568.113i − 1.11177i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 864.000i 1.67118i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −522.000 −1.00192 −0.500960 0.865471i \(-0.667020\pi\)
−0.500960 + 0.865471i \(0.667020\pi\)
\(522\) 0 0
\(523\) 48.4974 0.0927293 0.0463646 0.998925i \(-0.485236\pi\)
0.0463646 + 0.998925i \(0.485236\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 124.708 0.236637
\(528\) 0 0
\(529\) −529.000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 756.000i 1.41839i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 20.7846i 0.0385614i
\(540\) 0 0
\(541\) 802.000 1.48244 0.741220 0.671262i \(-0.234248\pi\)
0.741220 + 0.671262i \(0.234248\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −34.6410 −0.0633291 −0.0316645 0.999499i \(-0.510081\pi\)
−0.0316645 + 0.999499i \(0.510081\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 207.846i − 0.377216i
\(552\) 0 0
\(553\) 528.000i 0.954792i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 474.000i 0.850987i 0.904961 + 0.425494i \(0.139900\pi\)
−0.904961 + 0.425494i \(0.860100\pi\)
\(558\) 0 0
\(559\) 290.985i 0.520545i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 685.892 1.21828 0.609140 0.793062i \(-0.291515\pi\)
0.609140 + 0.793062i \(0.291515\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −150.000 −0.263620 −0.131810 0.991275i \(-0.542079\pi\)
−0.131810 + 0.991275i \(0.542079\pi\)
\(570\) 0 0
\(571\) 672.036i 1.17695i 0.808517 + 0.588473i \(0.200271\pi\)
−0.808517 + 0.588473i \(0.799729\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 46.0000i − 0.0797227i −0.999205 0.0398614i \(-0.987308\pi\)
0.999205 0.0398614i \(-0.0126916\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 144.000 0.247849
\(582\) 0 0
\(583\) −374.123 −0.641720
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −353.338 −0.601939 −0.300970 0.953634i \(-0.597310\pi\)
−0.300970 + 0.953634i \(0.597310\pi\)
\(588\) 0 0
\(589\) −144.000 −0.244482
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 114.000i 0.192243i 0.995370 + 0.0961214i \(0.0306437\pi\)
−0.995370 + 0.0961214i \(0.969356\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 249.415i − 0.416386i −0.978088 0.208193i \(-0.933242\pi\)
0.978088 0.208193i \(-0.0667582\pi\)
\(600\) 0 0
\(601\) 626.000 1.04160 0.520799 0.853680i \(-0.325634\pi\)
0.520799 + 0.853680i \(0.325634\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 672.036 1.10714 0.553571 0.832802i \(-0.313265\pi\)
0.553571 + 0.832802i \(0.313265\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 581.969i − 0.952486i
\(612\) 0 0
\(613\) 694.000i 1.13214i 0.824358 + 0.566069i \(0.191536\pi\)
−0.824358 + 0.566069i \(0.808464\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 30.0000i 0.0486224i 0.999704 + 0.0243112i \(0.00773925\pi\)
−0.999704 + 0.0243112i \(0.992261\pi\)
\(618\) 0 0
\(619\) − 339.482i − 0.548436i −0.961668 0.274218i \(-0.911581\pi\)
0.961668 0.274218i \(-0.0884190\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 789.815 1.26776
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −156.000 −0.248013
\(630\) 0 0
\(631\) 464.190i 0.735641i 0.929897 + 0.367821i \(0.119896\pi\)
−0.929897 + 0.367821i \(0.880104\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 14.0000i − 0.0219780i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 390.000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(642\) 0 0
\(643\) 810.600 1.26065 0.630326 0.776330i \(-0.282921\pi\)
0.630326 + 0.776330i \(0.282921\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −581.969 −0.899489 −0.449744 0.893157i \(-0.648485\pi\)
−0.449744 + 0.893157i \(0.648485\pi\)
\(648\) 0 0
\(649\) −432.000 −0.665639
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 774.000i 1.18530i 0.805461 + 0.592649i \(0.201918\pi\)
−0.805461 + 0.592649i \(0.798082\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 228.631i − 0.346936i −0.984840 0.173468i \(-0.944503\pi\)
0.984840 0.173468i \(-0.0554973\pi\)
\(660\) 0 0
\(661\) −454.000 −0.686838 −0.343419 0.939182i \(-0.611585\pi\)
−0.343419 + 0.939182i \(0.611585\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1454.92i 2.16829i
\(672\) 0 0
\(673\) − 434.000i − 0.644874i −0.946591 0.322437i \(-0.895498\pi\)
0.946591 0.322437i \(-0.104502\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 234.000i 0.345643i 0.984953 + 0.172821i \(0.0552883\pi\)
−0.984953 + 0.172821i \(0.944712\pi\)
\(678\) 0 0
\(679\) 235.559i 0.346920i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −270.200 −0.395608 −0.197804 0.980242i \(-0.563381\pi\)
−0.197804 + 0.980242i \(0.563381\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 252.000 0.365747
\(690\) 0 0
\(691\) − 20.7846i − 0.0300790i −0.999887 0.0150395i \(-0.995213\pi\)
0.999887 0.0150395i \(-0.00478741\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 324.000i 0.464849i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1074.00 1.53210 0.766049 0.642783i \(-0.222220\pi\)
0.766049 + 0.642783i \(0.222220\pi\)
\(702\) 0 0
\(703\) 180.133 0.256235
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 124.708 0.176390
\(708\) 0 0
\(709\) −898.000 −1.26657 −0.633286 0.773918i \(-0.718294\pi\)
−0.633286 + 0.773918i \(0.718294\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 956.092i 1.32975i 0.746954 + 0.664876i \(0.231516\pi\)
−0.746954 + 0.664876i \(0.768484\pi\)
\(720\) 0 0
\(721\) −912.000 −1.26491
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 810.600 1.11499 0.557496 0.830179i \(-0.311762\pi\)
0.557496 + 0.830179i \(0.311762\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 124.708i 0.170599i
\(732\) 0 0
\(733\) − 370.000i − 0.504775i −0.967626 0.252387i \(-0.918784\pi\)
0.967626 0.252387i \(-0.0812157\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 2448.00i − 3.32157i
\(738\) 0 0
\(739\) − 852.169i − 1.15314i −0.817048 0.576569i \(-0.804391\pi\)
0.817048 0.576569i \(-0.195609\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1371.78 −1.84628 −0.923139 0.384467i \(-0.874385\pi\)
−0.923139 + 0.384467i \(0.874385\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1008.00 1.34579
\(750\) 0 0
\(751\) − 76.2102i − 0.101478i −0.998712 0.0507392i \(-0.983842\pi\)
0.998712 0.0507392i \(-0.0161577\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 514.000i 0.678996i 0.940607 + 0.339498i \(0.110257\pi\)
−0.940607 + 0.339498i \(0.889743\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 966.000 1.26938 0.634691 0.772766i \(-0.281127\pi\)
0.634691 + 0.772766i \(0.281127\pi\)
\(762\) 0 0
\(763\) −235.559 −0.308727
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 290.985 0.379380
\(768\) 0 0
\(769\) 958.000 1.24577 0.622887 0.782312i \(-0.285960\pi\)
0.622887 + 0.782312i \(0.285960\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 546.000i − 0.706339i −0.935559 0.353169i \(-0.885104\pi\)
0.935559 0.353169i \(-0.114896\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 374.123i − 0.480261i
\(780\) 0 0
\(781\) −1728.00 −2.21255
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 242.487 0.308116 0.154058 0.988062i \(-0.450766\pi\)
0.154058 + 0.988062i \(0.450766\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 540.400i − 0.683186i
\(792\) 0 0
\(793\) − 980.000i − 1.23581i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1338.00i 1.67880i 0.543518 + 0.839398i \(0.317092\pi\)
−0.543518 + 0.839398i \(0.682908\pi\)
\(798\) 0 0
\(799\) − 249.415i − 0.312159i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −1704.34 −2.12246
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −966.000 −1.19407 −0.597033 0.802216i \(-0.703654\pi\)
−0.597033 + 0.802216i \(0.703654\pi\)
\(810\) 0 0
\(811\) 1517.28i 1.87087i 0.353497 + 0.935436i \(0.384992\pi\)
−0.353497 + 0.935436i \(0.615008\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 144.000i − 0.176255i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −222.000 −0.270402 −0.135201 0.990818i \(-0.543168\pi\)
−0.135201 + 0.990818i \(0.543168\pi\)
\(822\) 0 0
\(823\) −1281.72 −1.55737 −0.778686 0.627413i \(-0.784114\pi\)
−0.778686 + 0.627413i \(0.784114\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1434.14 −1.73415 −0.867073 0.498182i \(-0.834001\pi\)
−0.867073 + 0.498182i \(0.834001\pi\)
\(828\) 0 0
\(829\) −226.000 −0.272618 −0.136309 0.990666i \(-0.543524\pi\)
−0.136309 + 0.990666i \(0.543524\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 6.00000i − 0.00720288i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 498.831i − 0.594554i −0.954791 0.297277i \(-0.903922\pi\)
0.954791 0.297277i \(-0.0960784\pi\)
\(840\) 0 0
\(841\) 59.0000 0.0701546
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −2154.67 −2.54389
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 70.0000i 0.0820633i 0.999158 + 0.0410317i \(0.0130644\pi\)
−0.999158 + 0.0410317i \(0.986936\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 42.0000i − 0.0490082i −0.999700 0.0245041i \(-0.992199\pi\)
0.999700 0.0245041i \(-0.00780067\pi\)
\(858\) 0 0
\(859\) 921.451i 1.07270i 0.843995 + 0.536351i \(0.180198\pi\)
−0.843995 + 0.536351i \(0.819802\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −166.277 −0.192673 −0.0963365 0.995349i \(-0.530713\pi\)
−0.0963365 + 0.995349i \(0.530713\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1584.00 1.82278
\(870\) 0 0
\(871\) 1648.91i 1.89313i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 166.000i − 0.189282i −0.995511 0.0946408i \(-0.969830\pi\)
0.995511 0.0946408i \(-0.0301703\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 702.000 0.796822 0.398411 0.917207i \(-0.369562\pi\)
0.398411 + 0.917207i \(0.369562\pi\)
\(882\) 0 0
\(883\) −630.466 −0.714005 −0.357003 0.934103i \(-0.616201\pi\)
−0.357003 + 0.934103i \(0.616201\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −1288.65 −1.45281 −0.726407 0.687265i \(-0.758811\pi\)
−0.726407 + 0.687265i \(0.758811\pi\)
\(888\) 0 0
\(889\) −720.000 −0.809899
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 288.000i 0.322508i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 623.538i − 0.693591i
\(900\) 0 0
\(901\) 108.000 0.119867
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1447.99 1.59647 0.798233 0.602349i \(-0.205768\pi\)
0.798233 + 0.602349i \(0.205768\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 332.554i − 0.365043i −0.983202 0.182521i \(-0.941574\pi\)
0.983202 0.182521i \(-0.0584258\pi\)
\(912\) 0 0
\(913\) − 432.000i − 0.473165i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 720.000i − 0.785169i
\(918\) 0 0
\(919\) 561.184i 0.610647i 0.952249 + 0.305323i \(0.0987646\pi\)
−0.952249 + 0.305323i \(0.901235\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1163.94 1.26104
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −438.000 −0.471475 −0.235737 0.971817i \(-0.575751\pi\)
−0.235737 + 0.971817i \(0.575751\pi\)
\(930\) 0 0
\(931\) 6.92820i 0.00744168i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1826.00i 1.94877i 0.224881 + 0.974386i \(0.427801\pi\)
−0.224881 + 0.974386i \(0.572199\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 330.000 0.350691 0.175345 0.984507i \(-0.443896\pi\)
0.175345 + 0.984507i \(0.443896\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 187.061 0.197531 0.0987653 0.995111i \(-0.468511\pi\)
0.0987653 + 0.995111i \(0.468511\pi\)
\(948\) 0 0
\(949\) 1148.00 1.20969
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 1110.00i − 1.16474i −0.812923 0.582371i \(-0.802125\pi\)
0.812923 0.582371i \(-0.197875\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 1288.65i − 1.34374i
\(960\) 0 0
\(961\) 529.000 0.550468
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −755.174 −0.780945 −0.390473 0.920615i \(-0.627688\pi\)
−0.390473 + 0.920615i \(0.627688\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 394.908i 0.406702i 0.979106 + 0.203351i \(0.0651832\pi\)
−0.979106 + 0.203351i \(0.934817\pi\)
\(972\) 0 0
\(973\) 336.000i 0.345324i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 918.000i 0.939611i 0.882770 + 0.469806i \(0.155676\pi\)
−0.882770 + 0.469806i \(0.844324\pi\)
\(978\) 0 0
\(979\) − 2369.45i − 2.42027i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −41.5692 −0.0422881 −0.0211441 0.999776i \(-0.506731\pi\)
−0.0211441 + 0.999776i \(0.506731\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) − 48.4974i − 0.0489379i −0.999701 0.0244689i \(-0.992211\pi\)
0.999701 0.0244689i \(-0.00778948\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 554.000i 0.555667i 0.960629 + 0.277834i \(0.0896164\pi\)
−0.960629 + 0.277834i \(0.910384\pi\)
\(998\) 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 3600.3.j.i.1999.3 4
3.2 odd 2 1200.3.j.a.799.2 4
4.3 odd 2 inner 3600.3.j.i.1999.2 4
5.2 odd 4 3600.3.e.t.3151.2 2
5.3 odd 4 144.3.g.b.127.1 2
5.4 even 2 inner 3600.3.j.i.1999.1 4
12.11 even 2 1200.3.j.a.799.3 4
15.2 even 4 1200.3.e.h.751.2 2
15.8 even 4 48.3.g.a.31.1 2
15.14 odd 2 1200.3.j.a.799.4 4
20.3 even 4 144.3.g.b.127.2 2
20.7 even 4 3600.3.e.t.3151.1 2
20.19 odd 2 inner 3600.3.j.i.1999.4 4
40.3 even 4 576.3.g.i.127.2 2
40.13 odd 4 576.3.g.i.127.1 2
45.13 odd 12 1296.3.o.p.703.1 2
45.23 even 12 1296.3.o.c.703.1 2
45.38 even 12 1296.3.o.a.271.1 2
45.43 odd 12 1296.3.o.n.271.1 2
60.23 odd 4 48.3.g.a.31.2 yes 2
60.47 odd 4 1200.3.e.h.751.1 2
60.59 even 2 1200.3.j.a.799.1 4
80.3 even 4 2304.3.b.n.127.3 4
80.13 odd 4 2304.3.b.n.127.4 4
80.43 even 4 2304.3.b.n.127.1 4
80.53 odd 4 2304.3.b.n.127.2 4
105.83 odd 4 2352.3.m.a.1471.2 2
120.53 even 4 192.3.g.a.127.2 2
120.83 odd 4 192.3.g.a.127.1 2
180.23 odd 12 1296.3.o.a.703.1 2
180.43 even 12 1296.3.o.p.271.1 2
180.83 odd 12 1296.3.o.c.271.1 2
180.103 even 12 1296.3.o.n.703.1 2
240.53 even 4 768.3.b.b.127.2 4
240.83 odd 4 768.3.b.b.127.1 4
240.173 even 4 768.3.b.b.127.3 4
240.203 odd 4 768.3.b.b.127.4 4
420.83 even 4 2352.3.m.a.1471.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.3.g.a.31.1 2 15.8 even 4
48.3.g.a.31.2 yes 2 60.23 odd 4
144.3.g.b.127.1 2 5.3 odd 4
144.3.g.b.127.2 2 20.3 even 4
192.3.g.a.127.1 2 120.83 odd 4
192.3.g.a.127.2 2 120.53 even 4
576.3.g.i.127.1 2 40.13 odd 4
576.3.g.i.127.2 2 40.3 even 4
768.3.b.b.127.1 4 240.83 odd 4
768.3.b.b.127.2 4 240.53 even 4
768.3.b.b.127.3 4 240.173 even 4
768.3.b.b.127.4 4 240.203 odd 4
1200.3.e.h.751.1 2 60.47 odd 4
1200.3.e.h.751.2 2 15.2 even 4
1200.3.j.a.799.1 4 60.59 even 2
1200.3.j.a.799.2 4 3.2 odd 2
1200.3.j.a.799.3 4 12.11 even 2
1200.3.j.a.799.4 4 15.14 odd 2
1296.3.o.a.271.1 2 45.38 even 12
1296.3.o.a.703.1 2 180.23 odd 12
1296.3.o.c.271.1 2 180.83 odd 12
1296.3.o.c.703.1 2 45.23 even 12
1296.3.o.n.271.1 2 45.43 odd 12
1296.3.o.n.703.1 2 180.103 even 12
1296.3.o.p.271.1 2 180.43 even 12
1296.3.o.p.703.1 2 45.13 odd 12
2304.3.b.n.127.1 4 80.43 even 4
2304.3.b.n.127.2 4 80.53 odd 4
2304.3.b.n.127.3 4 80.3 even 4
2304.3.b.n.127.4 4 80.13 odd 4
2352.3.m.a.1471.1 2 420.83 even 4
2352.3.m.a.1471.2 2 105.83 odd 4
3600.3.e.t.3151.1 2 20.7 even 4
3600.3.e.t.3151.2 2 5.2 odd 4
3600.3.j.i.1999.1 4 5.4 even 2 inner
3600.3.j.i.1999.2 4 4.3 odd 2 inner
3600.3.j.i.1999.3 4 1.1 even 1 trivial
3600.3.j.i.1999.4 4 20.19 odd 2 inner