Properties

Label 48.3.g.a.31.1
Level $48$
Weight $3$
Character 48.31
Analytic conductor $1.308$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 48 = 2^{4} \cdot 3 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 48.g (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(1.30790526893\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Defining polynomial: \(x^{2} - x + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 31.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 48.31
Dual form 48.3.g.a.31.2

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{3} +6.00000 q^{5} -6.92820i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-1.73205i q^{3} +6.00000 q^{5} -6.92820i q^{7} -3.00000 q^{9} +20.7846i q^{11} -14.0000 q^{13} -10.3923i q^{15} -6.00000 q^{17} +6.92820i q^{19} -12.0000 q^{21} +11.0000 q^{25} +5.19615i q^{27} +30.0000 q^{29} -20.7846i q^{31} +36.0000 q^{33} -41.5692i q^{35} +26.0000 q^{37} +24.2487i q^{39} -54.0000 q^{41} +20.7846i q^{43} -18.0000 q^{45} -41.5692i q^{47} +1.00000 q^{49} +10.3923i q^{51} -18.0000 q^{53} +124.708i q^{55} +12.0000 q^{57} -20.7846i q^{59} -70.0000 q^{61} +20.7846i q^{63} -84.0000 q^{65} -117.779i q^{67} +83.1384i q^{71} +82.0000 q^{73} -19.0526i q^{75} +144.000 q^{77} -76.2102i q^{79} +9.00000 q^{81} -20.7846i q^{83} -36.0000 q^{85} -51.9615i q^{87} +114.000 q^{89} +96.9948i q^{91} -36.0000 q^{93} +41.5692i q^{95} +34.0000 q^{97} -62.3538i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 12 q^{5} - 6 q^{9} + O(q^{10}) \) \( 2 q + 12 q^{5} - 6 q^{9} - 28 q^{13} - 12 q^{17} - 24 q^{21} + 22 q^{25} + 60 q^{29} + 72 q^{33} + 52 q^{37} - 108 q^{41} - 36 q^{45} + 2 q^{49} - 36 q^{53} + 24 q^{57} - 140 q^{61} - 168 q^{65} + 164 q^{73} + 288 q^{77} + 18 q^{81} - 72 q^{85} + 228 q^{89} - 72 q^{93} + 68 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/48\mathbb{Z}\right)^\times\).

\(n\) \(17\) \(31\) \(37\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) − 1.73205i − 0.577350i
\(4\) 0 0
\(5\) 6.00000 1.20000 0.600000 0.800000i \(-0.295167\pi\)
0.600000 + 0.800000i \(0.295167\pi\)
\(6\) 0 0
\(7\) − 6.92820i − 0.989743i −0.868966 0.494872i \(-0.835215\pi\)
0.868966 0.494872i \(-0.164785\pi\)
\(8\) 0 0
\(9\) −3.00000 −0.333333
\(10\) 0 0
\(11\) 20.7846i 1.88951i 0.327777 + 0.944755i \(0.393700\pi\)
−0.327777 + 0.944755i \(0.606300\pi\)
\(12\) 0 0
\(13\) −14.0000 −1.07692 −0.538462 0.842650i \(-0.680994\pi\)
−0.538462 + 0.842650i \(0.680994\pi\)
\(14\) 0 0
\(15\) − 10.3923i − 0.692820i
\(16\) 0 0
\(17\) −6.00000 −0.352941 −0.176471 0.984306i \(-0.556468\pi\)
−0.176471 + 0.984306i \(0.556468\pi\)
\(18\) 0 0
\(19\) 6.92820i 0.364642i 0.983239 + 0.182321i \(0.0583610\pi\)
−0.983239 + 0.182321i \(0.941639\pi\)
\(20\) 0 0
\(21\) −12.0000 −0.571429
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 11.0000 0.440000
\(26\) 0 0
\(27\) 5.19615i 0.192450i
\(28\) 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
\(33\) 36.0000 1.09091
\(34\) 0 0
\(35\) − 41.5692i − 1.18769i
\(36\) 0 0
\(37\) 26.0000 0.702703 0.351351 0.936244i \(-0.385722\pi\)
0.351351 + 0.936244i \(0.385722\pi\)
\(38\) 0 0
\(39\) 24.2487i 0.621762i
\(40\) 0 0
\(41\) −54.0000 −1.31707 −0.658537 0.752549i \(-0.728824\pi\)
−0.658537 + 0.752549i \(0.728824\pi\)
\(42\) 0 0
\(43\) 20.7846i 0.483363i 0.970356 + 0.241682i \(0.0776989\pi\)
−0.970356 + 0.241682i \(0.922301\pi\)
\(44\) 0 0
\(45\) −18.0000 −0.400000
\(46\) 0 0
\(47\) − 41.5692i − 0.884451i −0.896904 0.442226i \(-0.854189\pi\)
0.896904 0.442226i \(-0.145811\pi\)
\(48\) 0 0
\(49\) 1.00000 0.0204082
\(50\) 0 0
\(51\) 10.3923i 0.203771i
\(52\) 0 0
\(53\) −18.0000 −0.339623 −0.169811 0.985477i \(-0.554316\pi\)
−0.169811 + 0.985477i \(0.554316\pi\)
\(54\) 0 0
\(55\) 124.708i 2.26741i
\(56\) 0 0
\(57\) 12.0000 0.210526
\(58\) 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
\(63\) 20.7846i 0.329914i
\(64\) 0 0
\(65\) −84.0000 −1.29231
\(66\) 0 0
\(67\) − 117.779i − 1.75790i −0.476912 0.878951i \(-0.658244\pi\)
0.476912 0.878951i \(-0.341756\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 83.1384i 1.17096i 0.810685 + 0.585482i \(0.199095\pi\)
−0.810685 + 0.585482i \(0.800905\pi\)
\(72\) 0 0
\(73\) 82.0000 1.12329 0.561644 0.827379i \(-0.310169\pi\)
0.561644 + 0.827379i \(0.310169\pi\)
\(74\) 0 0
\(75\) − 19.0526i − 0.254034i
\(76\) 0 0
\(77\) 144.000 1.87013
\(78\) 0 0
\(79\) − 76.2102i − 0.964687i −0.875982 0.482343i \(-0.839786\pi\)
0.875982 0.482343i \(-0.160214\pi\)
\(80\) 0 0
\(81\) 9.00000 0.111111
\(82\) 0 0
\(83\) − 20.7846i − 0.250417i −0.992130 0.125208i \(-0.960040\pi\)
0.992130 0.125208i \(-0.0399600\pi\)
\(84\) 0 0
\(85\) −36.0000 −0.423529
\(86\) 0 0
\(87\) − 51.9615i − 0.597259i
\(88\) 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
\(93\) −36.0000 −0.387097
\(94\) 0 0
\(95\) 41.5692i 0.437571i
\(96\) 0 0
\(97\) 34.0000 0.350515 0.175258 0.984523i \(-0.443924\pi\)
0.175258 + 0.984523i \(0.443924\pi\)
\(98\) 0 0
\(99\) − 62.3538i − 0.629837i
\(100\) 0 0
\(101\) −18.0000 −0.178218 −0.0891089 0.996022i \(-0.528402\pi\)
−0.0891089 + 0.996022i \(0.528402\pi\)
\(102\) 0 0
\(103\) − 131.636i − 1.27802i −0.769199 0.639009i \(-0.779345\pi\)
0.769199 0.639009i \(-0.220655\pi\)
\(104\) 0 0
\(105\) −72.0000 −0.685714
\(106\) 0 0
\(107\) 145.492i 1.35974i 0.733332 + 0.679870i \(0.237964\pi\)
−0.733332 + 0.679870i \(0.762036\pi\)
\(108\) 0 0
\(109\) 34.0000 0.311927 0.155963 0.987763i \(-0.450152\pi\)
0.155963 + 0.987763i \(0.450152\pi\)
\(110\) 0 0
\(111\) − 45.0333i − 0.405706i
\(112\) 0 0
\(113\) −78.0000 −0.690265 −0.345133 0.938554i \(-0.612166\pi\)
−0.345133 + 0.938554i \(0.612166\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 42.0000 0.358974
\(118\) 0 0
\(119\) 41.5692i 0.349321i
\(120\) 0 0
\(121\) −311.000 −2.57025
\(122\) 0 0
\(123\) 93.5307i 0.760413i
\(124\) 0 0
\(125\) −84.0000 −0.672000
\(126\) 0 0
\(127\) 103.923i 0.818292i 0.912469 + 0.409146i \(0.134173\pi\)
−0.912469 + 0.409146i \(0.865827\pi\)
\(128\) 0 0
\(129\) 36.0000 0.279070
\(130\) 0 0
\(131\) 103.923i 0.793306i 0.917969 + 0.396653i \(0.129828\pi\)
−0.917969 + 0.396653i \(0.870172\pi\)
\(132\) 0 0
\(133\) 48.0000 0.360902
\(134\) 0 0
\(135\) 31.1769i 0.230940i
\(136\) 0 0
\(137\) 186.000 1.35766 0.678832 0.734294i \(-0.262486\pi\)
0.678832 + 0.734294i \(0.262486\pi\)
\(138\) 0 0
\(139\) − 48.4974i − 0.348902i −0.984666 0.174451i \(-0.944185\pi\)
0.984666 0.174451i \(-0.0558151\pi\)
\(140\) 0 0
\(141\) −72.0000 −0.510638
\(142\) 0 0
\(143\) − 290.985i − 2.03486i
\(144\) 0 0
\(145\) 180.000 1.24138
\(146\) 0 0
\(147\) − 1.73205i − 0.0117827i
\(148\) 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
\(153\) 18.0000 0.117647
\(154\) 0 0
\(155\) − 124.708i − 0.804566i
\(156\) 0 0
\(157\) 170.000 1.08280 0.541401 0.840764i \(-0.317894\pi\)
0.541401 + 0.840764i \(0.317894\pi\)
\(158\) 0 0
\(159\) 31.1769i 0.196081i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 284.056i 1.74268i 0.490683 + 0.871338i \(0.336747\pi\)
−0.490683 + 0.871338i \(0.663253\pi\)
\(164\) 0 0
\(165\) 216.000 1.30909
\(166\) 0 0
\(167\) − 207.846i − 1.24459i −0.782784 0.622294i \(-0.786201\pi\)
0.782784 0.622294i \(-0.213799\pi\)
\(168\) 0 0
\(169\) 27.0000 0.159763
\(170\) 0 0
\(171\) − 20.7846i − 0.121547i
\(172\) 0 0
\(173\) −42.0000 −0.242775 −0.121387 0.992605i \(-0.538734\pi\)
−0.121387 + 0.992605i \(0.538734\pi\)
\(174\) 0 0
\(175\) − 76.2102i − 0.435487i
\(176\) 0 0
\(177\) −36.0000 −0.203390
\(178\) 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
\(183\) 121.244i 0.662533i
\(184\) 0 0
\(185\) 156.000 0.843243
\(186\) 0 0
\(187\) − 124.708i − 0.666886i
\(188\) 0 0
\(189\) 36.0000 0.190476
\(190\) 0 0
\(191\) 332.554i 1.74112i 0.492063 + 0.870560i \(0.336243\pi\)
−0.492063 + 0.870560i \(0.663757\pi\)
\(192\) 0 0
\(193\) −94.0000 −0.487047 −0.243523 0.969895i \(-0.578303\pi\)
−0.243523 + 0.969895i \(0.578303\pi\)
\(194\) 0 0
\(195\) 145.492i 0.746114i
\(196\) 0 0
\(197\) −258.000 −1.30964 −0.654822 0.755783i \(-0.727257\pi\)
−0.654822 + 0.755783i \(0.727257\pi\)
\(198\) 0 0
\(199\) 117.779i 0.591857i 0.955210 + 0.295928i \(0.0956289\pi\)
−0.955210 + 0.295928i \(0.904371\pi\)
\(200\) 0 0
\(201\) −204.000 −1.01493
\(202\) 0 0
\(203\) − 207.846i − 1.02387i
\(204\) 0 0
\(205\) −324.000 −1.58049
\(206\) 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
\(213\) 144.000 0.676056
\(214\) 0 0
\(215\) 124.708i 0.580036i
\(216\) 0 0
\(217\) −144.000 −0.663594
\(218\) 0 0
\(219\) − 142.028i − 0.648530i
\(220\) 0 0
\(221\) 84.0000 0.380090
\(222\) 0 0
\(223\) 353.338i 1.58448i 0.610212 + 0.792238i \(0.291084\pi\)
−0.610212 + 0.792238i \(0.708916\pi\)
\(224\) 0 0
\(225\) −33.0000 −0.146667
\(226\) 0 0
\(227\) 145.492i 0.640935i 0.947259 + 0.320468i \(0.103840\pi\)
−0.947259 + 0.320468i \(0.896160\pi\)
\(228\) 0 0
\(229\) 226.000 0.986900 0.493450 0.869774i \(-0.335736\pi\)
0.493450 + 0.869774i \(0.335736\pi\)
\(230\) 0 0
\(231\) − 249.415i − 1.07972i
\(232\) 0 0
\(233\) 114.000 0.489270 0.244635 0.969615i \(-0.421332\pi\)
0.244635 + 0.969615i \(0.421332\pi\)
\(234\) 0 0
\(235\) − 249.415i − 1.06134i
\(236\) 0 0
\(237\) −132.000 −0.556962
\(238\) 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
\(243\) − 15.5885i − 0.0641500i
\(244\) 0 0
\(245\) 6.00000 0.0244898
\(246\) 0 0
\(247\) − 96.9948i − 0.392692i
\(248\) 0 0
\(249\) −36.0000 −0.144578
\(250\) 0 0
\(251\) 103.923i 0.414036i 0.978337 + 0.207018i \(0.0663759\pi\)
−0.978337 + 0.207018i \(0.933624\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 62.3538i 0.244525i
\(256\) 0 0
\(257\) 258.000 1.00389 0.501946 0.864899i \(-0.332618\pi\)
0.501946 + 0.864899i \(0.332618\pi\)
\(258\) 0 0
\(259\) − 180.133i − 0.695495i
\(260\) 0 0
\(261\) −90.0000 −0.344828
\(262\) 0 0
\(263\) − 374.123i − 1.42252i −0.702929 0.711260i \(-0.748125\pi\)
0.702929 0.711260i \(-0.251875\pi\)
\(264\) 0 0
\(265\) −108.000 −0.407547
\(266\) 0 0
\(267\) − 197.454i − 0.739527i
\(268\) 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
\(273\) 168.000 0.615385
\(274\) 0 0
\(275\) 228.631i 0.831384i
\(276\) 0 0
\(277\) −14.0000 −0.0505415 −0.0252708 0.999681i \(-0.508045\pi\)
−0.0252708 + 0.999681i \(0.508045\pi\)
\(278\) 0 0
\(279\) 62.3538i 0.223490i
\(280\) 0 0
\(281\) 354.000 1.25979 0.629893 0.776682i \(-0.283099\pi\)
0.629893 + 0.776682i \(0.283099\pi\)
\(282\) 0 0
\(283\) 145.492i 0.514107i 0.966397 + 0.257053i \(0.0827517\pi\)
−0.966397 + 0.257053i \(0.917248\pi\)
\(284\) 0 0
\(285\) 72.0000 0.252632
\(286\) 0 0
\(287\) 374.123i 1.30356i
\(288\) 0 0
\(289\) −253.000 −0.875433
\(290\) 0 0
\(291\) − 58.8897i − 0.202370i
\(292\) 0 0
\(293\) −498.000 −1.69966 −0.849829 0.527058i \(-0.823295\pi\)
−0.849829 + 0.527058i \(0.823295\pi\)
\(294\) 0 0
\(295\) − 124.708i − 0.422738i
\(296\) 0 0
\(297\) −108.000 −0.363636
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 144.000 0.478405
\(302\) 0 0
\(303\) 31.1769i 0.102894i
\(304\) 0 0
\(305\) −420.000 −1.37705
\(306\) 0 0
\(307\) 187.061i 0.609321i 0.952461 + 0.304660i \(0.0985430\pi\)
−0.952461 + 0.304660i \(0.901457\pi\)
\(308\) 0 0
\(309\) −228.000 −0.737864
\(310\) 0 0
\(311\) − 41.5692i − 0.133663i −0.997764 0.0668315i \(-0.978711\pi\)
0.997764 0.0668315i \(-0.0212890\pi\)
\(312\) 0 0
\(313\) 290.000 0.926518 0.463259 0.886223i \(-0.346680\pi\)
0.463259 + 0.886223i \(0.346680\pi\)
\(314\) 0 0
\(315\) 124.708i 0.395897i
\(316\) 0 0
\(317\) −210.000 −0.662461 −0.331230 0.943550i \(-0.607464\pi\)
−0.331230 + 0.943550i \(0.607464\pi\)
\(318\) 0 0
\(319\) 623.538i 1.95467i
\(320\) 0 0
\(321\) 252.000 0.785047
\(322\) 0 0
\(323\) − 41.5692i − 0.128697i
\(324\) 0 0
\(325\) −154.000 −0.473846
\(326\) 0 0
\(327\) − 58.8897i − 0.180091i
\(328\) 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
\(333\) −78.0000 −0.234234
\(334\) 0 0
\(335\) − 706.677i − 2.10948i
\(336\) 0 0
\(337\) −302.000 −0.896142 −0.448071 0.893998i \(-0.647889\pi\)
−0.448071 + 0.893998i \(0.647889\pi\)
\(338\) 0 0
\(339\) 135.100i 0.398525i
\(340\) 0 0
\(341\) 432.000 1.26686
\(342\) 0 0
\(343\) − 346.410i − 1.00994i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 62.3538i − 0.179694i −0.995956 0.0898470i \(-0.971362\pi\)
0.995956 0.0898470i \(-0.0286378\pi\)
\(348\) 0 0
\(349\) −358.000 −1.02579 −0.512894 0.858452i \(-0.671427\pi\)
−0.512894 + 0.858452i \(0.671427\pi\)
\(350\) 0 0
\(351\) − 72.7461i − 0.207254i
\(352\) 0 0
\(353\) −558.000 −1.58074 −0.790368 0.612632i \(-0.790111\pi\)
−0.790368 + 0.612632i \(0.790111\pi\)
\(354\) 0 0
\(355\) 498.831i 1.40516i
\(356\) 0 0
\(357\) 72.0000 0.201681
\(358\) 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
\(363\) 538.668i 1.48393i
\(364\) 0 0
\(365\) 492.000 1.34795
\(366\) 0 0
\(367\) − 214.774i − 0.585216i −0.956232 0.292608i \(-0.905477\pi\)
0.956232 0.292608i \(-0.0945231\pi\)
\(368\) 0 0
\(369\) 162.000 0.439024
\(370\) 0 0
\(371\) 124.708i 0.336139i
\(372\) 0 0
\(373\) 554.000 1.48525 0.742627 0.669705i \(-0.233579\pi\)
0.742627 + 0.669705i \(0.233579\pi\)
\(374\) 0 0
\(375\) 145.492i 0.387979i
\(376\) 0 0
\(377\) −420.000 −1.11406
\(378\) 0 0
\(379\) − 533.472i − 1.40758i −0.710410 0.703788i \(-0.751490\pi\)
0.710410 0.703788i \(-0.248510\pi\)
\(380\) 0 0
\(381\) 180.000 0.472441
\(382\) 0 0
\(383\) − 498.831i − 1.30243i −0.758893 0.651215i \(-0.774260\pi\)
0.758893 0.651215i \(-0.225740\pi\)
\(384\) 0 0
\(385\) 864.000 2.24416
\(386\) 0 0
\(387\) − 62.3538i − 0.161121i
\(388\) 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
\(393\) 180.000 0.458015
\(394\) 0 0
\(395\) − 457.261i − 1.15762i
\(396\) 0 0
\(397\) −646.000 −1.62720 −0.813602 0.581422i \(-0.802496\pi\)
−0.813602 + 0.581422i \(0.802496\pi\)
\(398\) 0 0
\(399\) − 83.1384i − 0.208367i
\(400\) 0 0
\(401\) 330.000 0.822943 0.411471 0.911423i \(-0.365015\pi\)
0.411471 + 0.911423i \(0.365015\pi\)
\(402\) 0 0
\(403\) 290.985i 0.722046i
\(404\) 0 0
\(405\) 54.0000 0.133333
\(406\) 0 0
\(407\) 540.400i 1.32776i
\(408\) 0 0
\(409\) 130.000 0.317848 0.158924 0.987291i \(-0.449197\pi\)
0.158924 + 0.987291i \(0.449197\pi\)
\(410\) 0 0
\(411\) − 322.161i − 0.783848i
\(412\) 0 0
\(413\) −144.000 −0.348668
\(414\) 0 0
\(415\) − 124.708i − 0.300500i
\(416\) 0 0
\(417\) −84.0000 −0.201439
\(418\) 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
\(423\) 124.708i 0.294817i
\(424\) 0 0
\(425\) −66.0000 −0.155294
\(426\) 0 0
\(427\) 484.974i 1.13577i
\(428\) 0 0
\(429\) −504.000 −1.17483
\(430\) 0 0
\(431\) − 124.708i − 0.289345i −0.989480 0.144672i \(-0.953787\pi\)
0.989480 0.144672i \(-0.0462128\pi\)
\(432\) 0 0
\(433\) −142.000 −0.327945 −0.163972 0.986465i \(-0.552431\pi\)
−0.163972 + 0.986465i \(0.552431\pi\)
\(434\) 0 0
\(435\) − 311.769i − 0.716711i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 561.184i 1.27832i 0.769072 + 0.639162i \(0.220719\pi\)
−0.769072 + 0.639162i \(0.779281\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.00680272
\(442\) 0 0
\(443\) 436.477i 0.985275i 0.870235 + 0.492637i \(0.163967\pi\)
−0.870235 + 0.492637i \(0.836033\pi\)
\(444\) 0 0
\(445\) 684.000 1.53708
\(446\) 0 0
\(447\) 322.161i 0.720719i
\(448\) 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
\(453\) −60.0000 −0.132450
\(454\) 0 0
\(455\) 581.969i 1.27905i
\(456\) 0 0
\(457\) −446.000 −0.975930 −0.487965 0.872863i \(-0.662261\pi\)
−0.487965 + 0.872863i \(0.662261\pi\)
\(458\) 0 0
\(459\) − 31.1769i − 0.0679236i
\(460\) 0 0
\(461\) 342.000 0.741866 0.370933 0.928660i \(-0.379038\pi\)
0.370933 + 0.928660i \(0.379038\pi\)
\(462\) 0 0
\(463\) 159.349i 0.344166i 0.985082 + 0.172083i \(0.0550497\pi\)
−0.985082 + 0.172083i \(0.944950\pi\)
\(464\) 0 0
\(465\) −216.000 −0.464516
\(466\) 0 0
\(467\) 394.908i 0.845627i 0.906217 + 0.422813i \(0.138957\pi\)
−0.906217 + 0.422813i \(0.861043\pi\)
\(468\) 0 0
\(469\) −816.000 −1.73987
\(470\) 0 0
\(471\) − 294.449i − 0.625156i
\(472\) 0 0
\(473\) −432.000 −0.913319
\(474\) 0 0
\(475\) 76.2102i 0.160443i
\(476\) 0 0
\(477\) 54.0000 0.113208
\(478\) 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
\(485\) 204.000 0.420619
\(486\) 0 0
\(487\) − 6.92820i − 0.0142263i −0.999975 0.00711315i \(-0.997736\pi\)
0.999975 0.00711315i \(-0.00226420\pi\)
\(488\) 0 0
\(489\) 492.000 1.00613
\(490\) 0 0
\(491\) 644.323i 1.31227i 0.754645 + 0.656133i \(0.227809\pi\)
−0.754645 + 0.656133i \(0.772191\pi\)
\(492\) 0 0
\(493\) −180.000 −0.365112
\(494\) 0 0
\(495\) − 374.123i − 0.755804i
\(496\) 0 0
\(497\) 576.000 1.15895
\(498\) 0 0
\(499\) − 810.600i − 1.62445i −0.583345 0.812224i \(-0.698257\pi\)
0.583345 0.812224i \(-0.301743\pi\)
\(500\) 0 0
\(501\) −360.000 −0.718563
\(502\) 0 0
\(503\) − 332.554i − 0.661141i −0.943781 0.330570i \(-0.892759\pi\)
0.943781 0.330570i \(-0.107241\pi\)
\(504\) 0 0
\(505\) −108.000 −0.213861
\(506\) 0 0
\(507\) − 46.7654i − 0.0922394i
\(508\) 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
\(513\) −36.0000 −0.0701754
\(514\) 0 0
\(515\) − 789.815i − 1.53362i
\(516\) 0 0
\(517\) 864.000 1.67118
\(518\) 0 0
\(519\) 72.7461i 0.140166i
\(520\) 0 0
\(521\) 522.000 1.00192 0.500960 0.865471i \(-0.332980\pi\)
0.500960 + 0.865471i \(0.332980\pi\)
\(522\) 0 0
\(523\) 48.4974i 0.0927293i 0.998925 + 0.0463646i \(0.0147636\pi\)
−0.998925 + 0.0463646i \(0.985236\pi\)
\(524\) 0 0
\(525\) −132.000 −0.251429
\(526\) 0 0
\(527\) 124.708i 0.236637i
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 62.3538i 0.117427i
\(532\) 0 0
\(533\) 756.000 1.41839
\(534\) 0 0
\(535\) 872.954i 1.63169i
\(536\) 0 0
\(537\) −252.000 −0.469274
\(538\) 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
\(543\) − 142.028i − 0.261562i
\(544\) 0 0
\(545\) 204.000 0.374312
\(546\) 0 0
\(547\) 34.6410i 0.0633291i 0.999499 + 0.0316645i \(0.0100808\pi\)
−0.999499 + 0.0316645i \(0.989919\pi\)
\(548\) 0 0
\(549\) 210.000 0.382514
\(550\) 0 0
\(551\) 207.846i 0.377216i
\(552\) 0 0
\(553\) −528.000 −0.954792
\(554\) 0 0
\(555\) − 270.200i − 0.486847i
\(556\) 0 0
\(557\) −474.000 −0.850987 −0.425494 0.904961i \(-0.639900\pi\)
−0.425494 + 0.904961i \(0.639900\pi\)
\(558\) 0 0
\(559\) − 290.985i − 0.520545i
\(560\) 0 0
\(561\) −216.000 −0.385027
\(562\) 0 0
\(563\) − 685.892i − 1.21828i −0.793062 0.609140i \(-0.791515\pi\)
0.793062 0.609140i \(-0.208485\pi\)
\(564\) 0 0
\(565\) −468.000 −0.828319
\(566\) 0 0
\(567\) − 62.3538i − 0.109971i
\(568\) 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
\(573\) 576.000 1.00524
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −46.0000 −0.0797227 −0.0398614 0.999205i \(-0.512692\pi\)
−0.0398614 + 0.999205i \(0.512692\pi\)
\(578\) 0 0
\(579\) 162.813i 0.281197i
\(580\) 0 0
\(581\) −144.000 −0.247849
\(582\) 0 0
\(583\) − 374.123i − 0.641720i
\(584\) 0 0
\(585\) 252.000 0.430769
\(586\) 0 0
\(587\) − 353.338i − 0.601939i −0.953634 0.300970i \(-0.902690\pi\)
0.953634 0.300970i \(-0.0973103\pi\)
\(588\) 0 0
\(589\) 144.000 0.244482
\(590\) 0 0
\(591\) 446.869i 0.756124i
\(592\) 0 0
\(593\) 114.000 0.192243 0.0961214 0.995370i \(-0.469356\pi\)
0.0961214 + 0.995370i \(0.469356\pi\)
\(594\) 0 0
\(595\) 249.415i 0.419185i
\(596\) 0 0
\(597\) 204.000 0.341709
\(598\) 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
\(603\) 353.338i 0.585967i
\(604\) 0 0
\(605\) −1866.00 −3.08430
\(606\) 0 0
\(607\) − 672.036i − 1.10714i −0.832802 0.553571i \(-0.813265\pi\)
0.832802 0.553571i \(-0.186735\pi\)
\(608\) 0 0
\(609\) −360.000 −0.591133
\(610\) 0 0
\(611\) 581.969i 0.952486i
\(612\) 0 0
\(613\) −694.000 −1.13214 −0.566069 0.824358i \(-0.691536\pi\)
−0.566069 + 0.824358i \(0.691536\pi\)
\(614\) 0 0
\(615\) 561.184i 0.912495i
\(616\) 0 0
\(617\) −30.0000 −0.0486224 −0.0243112 0.999704i \(-0.507739\pi\)
−0.0243112 + 0.999704i \(0.507739\pi\)
\(618\) 0 0
\(619\) 339.482i 0.548436i 0.961668 + 0.274218i \(0.0884190\pi\)
−0.961668 + 0.274218i \(0.911581\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 789.815i − 1.26776i
\(624\) 0 0
\(625\) −779.000 −1.24640
\(626\) 0 0
\(627\) 249.415i 0.397792i
\(628\) 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
\(633\) −156.000 −0.246445
\(634\) 0 0
\(635\) 623.538i 0.981950i
\(636\) 0 0
\(637\) −14.0000 −0.0219780
\(638\) 0 0
\(639\) − 249.415i − 0.390321i
\(640\) 0 0
\(641\) −390.000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(642\) 0 0
\(643\) 810.600i 1.26065i 0.776330 + 0.630326i \(0.217079\pi\)
−0.776330 + 0.630326i \(0.782921\pi\)
\(644\) 0 0
\(645\) 216.000 0.334884
\(646\) 0 0
\(647\) − 581.969i − 0.899489i −0.893157 0.449744i \(-0.851515\pi\)
0.893157 0.449744i \(-0.148485\pi\)
\(648\) 0 0
\(649\) 432.000 0.665639
\(650\) 0 0
\(651\) 249.415i 0.383126i
\(652\) 0 0
\(653\) 774.000 1.18530 0.592649 0.805461i \(-0.298082\pi\)
0.592649 + 0.805461i \(0.298082\pi\)
\(654\) 0 0
\(655\) 623.538i 0.951967i
\(656\) 0 0
\(657\) −246.000 −0.374429
\(658\) 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
\(663\) − 145.492i − 0.219445i
\(664\) 0 0
\(665\) 288.000 0.433083
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 612.000 0.914798
\(670\) 0 0
\(671\) − 1454.92i − 2.16829i
\(672\) 0 0
\(673\) 434.000 0.644874 0.322437 0.946591i \(-0.395498\pi\)
0.322437 + 0.946591i \(0.395498\pi\)
\(674\) 0 0
\(675\) 57.1577i 0.0846780i
\(676\) 0 0
\(677\) −234.000 −0.345643 −0.172821 0.984953i \(-0.555288\pi\)
−0.172821 + 0.984953i \(0.555288\pi\)
\(678\) 0 0
\(679\) − 235.559i − 0.346920i
\(680\) 0 0
\(681\) 252.000 0.370044
\(682\) 0 0
\(683\) 270.200i 0.395608i 0.980242 + 0.197804i \(0.0633809\pi\)
−0.980242 + 0.197804i \(0.936619\pi\)
\(684\) 0 0
\(685\) 1116.00 1.62920
\(686\) 0 0
\(687\) − 391.443i − 0.569787i
\(688\) 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
\(693\) −432.000 −0.623377
\(694\) 0 0
\(695\) − 290.985i − 0.418683i
\(696\) 0 0
\(697\) 324.000 0.464849
\(698\) 0 0
\(699\) − 197.454i − 0.282480i
\(700\) 0 0
\(701\) −1074.00 −1.53210 −0.766049 0.642783i \(-0.777780\pi\)
−0.766049 + 0.642783i \(0.777780\pi\)
\(702\) 0 0
\(703\) 180.133i 0.256235i
\(704\) 0 0
\(705\) −432.000 −0.612766
\(706\) 0 0
\(707\) 124.708i 0.176390i
\(708\) 0 0
\(709\) 898.000 1.26657 0.633286 0.773918i \(-0.281706\pi\)
0.633286 + 0.773918i \(0.281706\pi\)
\(710\) 0 0
\(711\) 228.631i 0.321562i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) − 1745.91i − 2.44183i
\(716\) 0 0
\(717\) 576.000 0.803347
\(718\) 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
\(723\) − 308.305i − 0.426425i
\(724\) 0 0
\(725\) 330.000 0.455172
\(726\) 0 0
\(727\) − 810.600i − 1.11499i −0.830179 0.557496i \(-0.811762\pi\)
0.830179 0.557496i \(-0.188238\pi\)
\(728\) 0 0
\(729\) −27.0000 −0.0370370
\(730\) 0 0
\(731\) − 124.708i − 0.170599i
\(732\) 0 0
\(733\) 370.000 0.504775 0.252387 0.967626i \(-0.418784\pi\)
0.252387 + 0.967626i \(0.418784\pi\)
\(734\) 0 0
\(735\) − 10.3923i − 0.0141392i
\(736\) 0 0
\(737\) 2448.00 3.32157
\(738\) 0 0
\(739\) 852.169i 1.15314i 0.817048 + 0.576569i \(0.195609\pi\)
−0.817048 + 0.576569i \(0.804391\pi\)
\(740\) 0 0
\(741\) −168.000 −0.226721
\(742\) 0 0
\(743\) 1371.78i 1.84628i 0.384467 + 0.923139i \(0.374385\pi\)
−0.384467 + 0.923139i \(0.625615\pi\)
\(744\) 0 0
\(745\) −1116.00 −1.49799
\(746\) 0 0
\(747\) 62.3538i 0.0834723i
\(748\) 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
\(753\) 180.000 0.239044
\(754\) 0 0
\(755\) − 207.846i − 0.275293i
\(756\) 0 0
\(757\) 514.000 0.678996 0.339498 0.940607i \(-0.389743\pi\)
0.339498 + 0.940607i \(0.389743\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −966.000 −1.26938 −0.634691 0.772766i \(-0.718873\pi\)
−0.634691 + 0.772766i \(0.718873\pi\)
\(762\) 0 0
\(763\) − 235.559i − 0.308727i
\(764\) 0 0
\(765\) 108.000 0.141176
\(766\) 0 0
\(767\) 290.985i 0.379380i
\(768\) 0 0
\(769\) −958.000 −1.24577 −0.622887 0.782312i \(-0.714040\pi\)
−0.622887 + 0.782312i \(0.714040\pi\)
\(770\) 0 0
\(771\) − 446.869i − 0.579597i
\(772\) 0 0
\(773\) −546.000 −0.706339 −0.353169 0.935559i \(-0.614896\pi\)
−0.353169 + 0.935559i \(0.614896\pi\)
\(774\) 0 0
\(775\) − 228.631i − 0.295007i
\(776\) 0 0
\(777\) −312.000 −0.401544
\(778\) 0 0
\(779\) − 374.123i − 0.480261i
\(780\) 0 0
\(781\) −1728.00 −2.21255
\(782\) 0 0
\(783\) 155.885i 0.199086i
\(784\) 0 0
\(785\) 1020.00 1.29936
\(786\) 0 0
\(787\) − 242.487i − 0.308116i −0.988062 0.154058i \(-0.950766\pi\)
0.988062 0.154058i \(-0.0492342\pi\)
\(788\) 0 0
\(789\) −648.000 −0.821293
\(790\) 0 0
\(791\) 540.400i 0.683186i
\(792\) 0 0
\(793\) 980.000 1.23581
\(794\) 0 0
\(795\) 187.061i 0.235297i
\(796\) 0 0
\(797\) −1338.00 −1.67880 −0.839398 0.543518i \(-0.817092\pi\)
−0.839398 + 0.543518i \(0.817092\pi\)
\(798\) 0 0
\(799\) 249.415i 0.312159i
\(800\) 0 0
\(801\) −342.000 −0.426966
\(802\) 0 0
\(803\) 1704.34i 2.12246i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 883.346i − 1.09460i
\(808\) 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
\(813\) −780.000 −0.959410
\(814\) 0 0
\(815\) 1704.34i 2.09121i
\(816\) 0 0
\(817\) −144.000 −0.176255
\(818\) 0 0
\(819\) − 290.985i − 0.355292i
\(820\) 0 0
\(821\) 222.000 0.270402 0.135201 0.990818i \(-0.456832\pi\)
0.135201 + 0.990818i \(0.456832\pi\)
\(822\) 0 0
\(823\) − 1281.72i − 1.55737i −0.627413 0.778686i \(-0.715886\pi\)
0.627413 0.778686i \(-0.284114\pi\)
\(824\) 0 0
\(825\) 396.000 0.480000
\(826\) 0 0
\(827\) − 1434.14i − 1.73415i −0.498182 0.867073i \(-0.665999\pi\)
0.498182 0.867073i \(-0.334001\pi\)
\(828\) 0 0
\(829\) 226.000 0.272618 0.136309 0.990666i \(-0.456476\pi\)
0.136309 + 0.990666i \(0.456476\pi\)
\(830\) 0 0
\(831\) 24.2487i 0.0291802i
\(832\) 0 0
\(833\) −6.00000 −0.00720288
\(834\) 0 0
\(835\) − 1247.08i − 1.49350i
\(836\) 0 0
\(837\) 108.000 0.129032
\(838\) 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
\(843\) − 613.146i − 0.727338i
\(844\) 0 0
\(845\) 162.000 0.191716
\(846\) 0 0
\(847\) 2154.67i 2.54389i
\(848\) 0 0
\(849\) 252.000 0.296820
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −70.0000 −0.0820633 −0.0410317 0.999158i \(-0.513064\pi\)
−0.0410317 + 0.999158i \(0.513064\pi\)
\(854\) 0 0
\(855\) − 124.708i − 0.145857i
\(856\) 0 0
\(857\) 42.0000 0.0490082 0.0245041 0.999700i \(-0.492199\pi\)
0.0245041 + 0.999700i \(0.492199\pi\)
\(858\) 0 0
\(859\) − 921.451i − 1.07270i −0.843995 0.536351i \(-0.819802\pi\)
0.843995 0.536351i \(-0.180198\pi\)
\(860\) 0 0
\(861\) 648.000 0.752613
\(862\) 0 0
\(863\) 166.277i 0.192673i 0.995349 + 0.0963365i \(0.0307125\pi\)
−0.995349 + 0.0963365i \(0.969287\pi\)
\(864\) 0 0
\(865\) −252.000 −0.291329
\(866\) 0 0
\(867\) 438.209i 0.505431i
\(868\) 0 0
\(869\) 1584.00 1.82278
\(870\) 0 0
\(871\) 1648.91i 1.89313i
\(872\) 0 0
\(873\) −102.000 −0.116838
\(874\) 0 0
\(875\) 581.969i 0.665108i
\(876\) 0 0
\(877\) −166.000 −0.189282 −0.0946408 0.995511i \(-0.530170\pi\)
−0.0946408 + 0.995511i \(0.530170\pi\)
\(878\) 0 0
\(879\) 862.561i 0.981298i
\(880\) 0 0
\(881\) −702.000 −0.796822 −0.398411 0.917207i \(-0.630438\pi\)
−0.398411 + 0.917207i \(0.630438\pi\)
\(882\) 0 0
\(883\) − 630.466i − 0.714005i −0.934103 0.357003i \(-0.883799\pi\)
0.934103 0.357003i \(-0.116201\pi\)
\(884\) 0 0
\(885\) −216.000 −0.244068
\(886\) 0 0
\(887\) − 1288.65i − 1.45281i −0.687265 0.726407i \(-0.741189\pi\)
0.687265 0.726407i \(-0.258811\pi\)
\(888\) 0 0
\(889\) 720.000 0.809899
\(890\) 0 0
\(891\) 187.061i 0.209946i
\(892\) 0 0
\(893\) 288.000 0.322508
\(894\) 0 0
\(895\) − 872.954i − 0.975367i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 623.538i − 0.693591i
\(900\) 0 0
\(901\) 108.000 0.119867
\(902\) 0 0
\(903\) − 249.415i − 0.276207i
\(904\) 0 0
\(905\) 492.000 0.543646
\(906\) 0 0
\(907\) − 1447.99i − 1.59647i −0.602349 0.798233i \(-0.705768\pi\)
0.602349 0.798233i \(-0.294232\pi\)
\(908\) 0 0
\(909\) 54.0000 0.0594059
\(910\) 0 0
\(911\) 332.554i 0.365043i 0.983202 + 0.182521i \(0.0584258\pi\)
−0.983202 + 0.182521i \(0.941574\pi\)
\(912\) 0 0
\(913\) 432.000 0.473165
\(914\) 0 0
\(915\) 727.461i 0.795040i
\(916\) 0 0
\(917\) 720.000 0.785169
\(918\) 0 0
\(919\) − 561.184i − 0.610647i −0.952249 0.305323i \(-0.901235\pi\)
0.952249 0.305323i \(-0.0987646\pi\)
\(920\) 0 0
\(921\) 324.000 0.351792
\(922\) 0 0
\(923\) − 1163.94i − 1.26104i
\(924\) 0 0
\(925\) 286.000 0.309189
\(926\) 0 0
\(927\) 394.908i 0.426006i
\(928\) 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
\(933\) −72.0000 −0.0771704
\(934\) 0 0
\(935\) − 748.246i − 0.800263i
\(936\) 0 0
\(937\) 1826.00 1.94877 0.974386 0.224881i \(-0.0721992\pi\)
0.974386 + 0.224881i \(0.0721992\pi\)
\(938\) 0 0
\(939\) − 502.295i − 0.534925i
\(940\) 0 0
\(941\) −330.000 −0.350691 −0.175345 0.984507i \(-0.556104\pi\)
−0.175345 + 0.984507i \(0.556104\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 216.000 0.228571
\(946\) 0 0
\(947\) 187.061i 0.197531i 0.995111 + 0.0987653i \(0.0314893\pi\)
−0.995111 + 0.0987653i \(0.968511\pi\)
\(948\) 0 0
\(949\) −1148.00 −1.20969
\(950\) 0 0
\(951\) 363.731i 0.382472i
\(952\) 0 0
\(953\) −1110.00 −1.16474 −0.582371 0.812923i \(-0.697875\pi\)
−0.582371 + 0.812923i \(0.697875\pi\)
\(954\) 0 0
\(955\) 1995.32i 2.08934i
\(956\) 0 0
\(957\) 1080.00 1.12853
\(958\) 0 0
\(959\) − 1288.65i − 1.34374i
\(960\) 0 0
\(961\) 529.000 0.550468
\(962\) 0 0
\(963\) − 436.477i − 0.453247i
\(964\) 0 0
\(965\) −564.000 −0.584456
\(966\) 0 0
\(967\) 755.174i 0.780945i 0.920615 + 0.390473i \(0.127688\pi\)
−0.920615 + 0.390473i \(0.872312\pi\)
\(968\) 0 0
\(969\) −72.0000 −0.0743034
\(970\) 0 0
\(971\) − 394.908i − 0.406702i −0.979106 0.203351i \(-0.934817\pi\)
0.979106 0.203351i \(-0.0651832\pi\)
\(972\) 0 0
\(973\) −336.000 −0.345324
\(974\) 0 0
\(975\) 266.736i 0.273575i
\(976\) 0 0
\(977\) −918.000 −0.939611 −0.469806 0.882770i \(-0.655676\pi\)
−0.469806 + 0.882770i \(0.655676\pi\)
\(978\) 0 0
\(979\) 2369.45i 2.42027i
\(980\) 0 0
\(981\) −102.000 −0.103976
\(982\) 0 0
\(983\) 41.5692i 0.0422881i 0.999776 + 0.0211441i \(0.00673086\pi\)
−0.999776 + 0.0211441i \(0.993269\pi\)
\(984\) 0 0
\(985\) −1548.00 −1.57157
\(986\) 0 0
\(987\) 498.831i 0.505401i
\(988\) 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
\(993\) 348.000 0.350453
\(994\) 0 0
\(995\) 706.677i 0.710228i
\(996\) 0 0
\(997\) 554.000 0.555667 0.277834 0.960629i \(-0.410384\pi\)
0.277834 + 0.960629i \(0.410384\pi\)
\(998\) 0 0
\(999\) 135.100i 0.135235i
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 48.3.g.a.31.1 2
3.2 odd 2 144.3.g.b.127.1 2
4.3 odd 2 inner 48.3.g.a.31.2 yes 2
5.2 odd 4 1200.3.j.a.799.2 4
5.3 odd 4 1200.3.j.a.799.4 4
5.4 even 2 1200.3.e.h.751.2 2
7.6 odd 2 2352.3.m.a.1471.2 2
8.3 odd 2 192.3.g.a.127.1 2
8.5 even 2 192.3.g.a.127.2 2
9.2 odd 6 1296.3.o.n.271.1 2
9.4 even 3 1296.3.o.c.703.1 2
9.5 odd 6 1296.3.o.p.703.1 2
9.7 even 3 1296.3.o.a.271.1 2
12.11 even 2 144.3.g.b.127.2 2
15.2 even 4 3600.3.j.i.1999.3 4
15.8 even 4 3600.3.j.i.1999.1 4
15.14 odd 2 3600.3.e.t.3151.2 2
16.3 odd 4 768.3.b.b.127.1 4
16.5 even 4 768.3.b.b.127.2 4
16.11 odd 4 768.3.b.b.127.4 4
16.13 even 4 768.3.b.b.127.3 4
20.3 even 4 1200.3.j.a.799.1 4
20.7 even 4 1200.3.j.a.799.3 4
20.19 odd 2 1200.3.e.h.751.1 2
24.5 odd 2 576.3.g.i.127.1 2
24.11 even 2 576.3.g.i.127.2 2
28.27 even 2 2352.3.m.a.1471.1 2
36.7 odd 6 1296.3.o.c.271.1 2
36.11 even 6 1296.3.o.p.271.1 2
36.23 even 6 1296.3.o.n.703.1 2
36.31 odd 6 1296.3.o.a.703.1 2
48.5 odd 4 2304.3.b.n.127.2 4
48.11 even 4 2304.3.b.n.127.1 4
48.29 odd 4 2304.3.b.n.127.4 4
48.35 even 4 2304.3.b.n.127.3 4
60.23 odd 4 3600.3.j.i.1999.4 4
60.47 odd 4 3600.3.j.i.1999.2 4
60.59 even 2 3600.3.e.t.3151.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
48.3.g.a.31.1 2 1.1 even 1 trivial
48.3.g.a.31.2 yes 2 4.3 odd 2 inner
144.3.g.b.127.1 2 3.2 odd 2
144.3.g.b.127.2 2 12.11 even 2
192.3.g.a.127.1 2 8.3 odd 2
192.3.g.a.127.2 2 8.5 even 2
576.3.g.i.127.1 2 24.5 odd 2
576.3.g.i.127.2 2 24.11 even 2
768.3.b.b.127.1 4 16.3 odd 4
768.3.b.b.127.2 4 16.5 even 4
768.3.b.b.127.3 4 16.13 even 4
768.3.b.b.127.4 4 16.11 odd 4
1200.3.e.h.751.1 2 20.19 odd 2
1200.3.e.h.751.2 2 5.4 even 2
1200.3.j.a.799.1 4 20.3 even 4
1200.3.j.a.799.2 4 5.2 odd 4
1200.3.j.a.799.3 4 20.7 even 4
1200.3.j.a.799.4 4 5.3 odd 4
1296.3.o.a.271.1 2 9.7 even 3
1296.3.o.a.703.1 2 36.31 odd 6
1296.3.o.c.271.1 2 36.7 odd 6
1296.3.o.c.703.1 2 9.4 even 3
1296.3.o.n.271.1 2 9.2 odd 6
1296.3.o.n.703.1 2 36.23 even 6
1296.3.o.p.271.1 2 36.11 even 6
1296.3.o.p.703.1 2 9.5 odd 6
2304.3.b.n.127.1 4 48.11 even 4
2304.3.b.n.127.2 4 48.5 odd 4
2304.3.b.n.127.3 4 48.35 even 4
2304.3.b.n.127.4 4 48.29 odd 4
2352.3.m.a.1471.1 2 28.27 even 2
2352.3.m.a.1471.2 2 7.6 odd 2
3600.3.e.t.3151.1 2 60.59 even 2
3600.3.e.t.3151.2 2 15.14 odd 2
3600.3.j.i.1999.1 4 15.8 even 4
3600.3.j.i.1999.2 4 60.47 odd 4
3600.3.j.i.1999.3 4 15.2 even 4
3600.3.j.i.1999.4 4 60.23 odd 4