Properties

Label 1728.3.e.u.1025.4
Level $1728$
Weight $3$
Character 1728.1025
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{10}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1025.4
Root \(0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1728.1025
Dual form 1728.3.e.u.1025.6

$q$-expansion

\(f(q)\) \(=\) \(q-3.78194i q^{5} +0.953512 q^{7} +O(q^{10})\) \(q-3.78194i q^{5} +0.953512 q^{7} -11.6969i q^{11} +8.69694 q^{13} -4.73545i q^{17} +29.2699 q^{19} +2.69694i q^{23} +10.6969 q^{25} +21.7060i q^{29} -22.5953 q^{31} -3.60612i q^{35} -24.0908 q^{37} +19.7990i q^{41} +49.0047 q^{43} -59.3939i q^{47} -48.0908 q^{49} -48.1796i q^{53} -44.2371 q^{55} -12.6061i q^{59} +4.00000 q^{61} -32.8913i q^{65} +111.358 q^{67} -56.6969i q^{71} +35.6969 q^{73} -11.1532i q^{77} -144.442 q^{79} -101.697i q^{83} -17.9092 q^{85} -19.0060i q^{89} +8.29263 q^{91} -110.697i q^{95} +27.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 48 q^{13} - 32 q^{25} + 160 q^{37} - 32 q^{49} + 32 q^{61} + 168 q^{73} - 496 q^{85} + 216 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(5\) − 3.78194i − 0.756388i −0.925726 0.378194i \(-0.876545\pi\)
0.925726 0.378194i \(-0.123455\pi\)
\(6\) 0 0
\(7\) 0.953512 0.136216 0.0681080 0.997678i \(-0.478304\pi\)
0.0681080 + 0.997678i \(0.478304\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 11.6969i − 1.06336i −0.846946 0.531679i \(-0.821561\pi\)
0.846946 0.531679i \(-0.178439\pi\)
\(12\) 0 0
\(13\) 8.69694 0.668995 0.334498 0.942397i \(-0.391433\pi\)
0.334498 + 0.942397i \(0.391433\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 4.73545i − 0.278556i −0.990253 0.139278i \(-0.955522\pi\)
0.990253 0.139278i \(-0.0444781\pi\)
\(18\) 0 0
\(19\) 29.2699 1.54052 0.770260 0.637730i \(-0.220126\pi\)
0.770260 + 0.637730i \(0.220126\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.69694i 0.117258i 0.998280 + 0.0586291i \(0.0186729\pi\)
−0.998280 + 0.0586291i \(0.981327\pi\)
\(24\) 0 0
\(25\) 10.6969 0.427878
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 21.7060i 0.748483i 0.927331 + 0.374242i \(0.122097\pi\)
−0.927331 + 0.374242i \(0.877903\pi\)
\(30\) 0 0
\(31\) −22.5953 −0.728881 −0.364440 0.931227i \(-0.618740\pi\)
−0.364440 + 0.931227i \(0.618740\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 3.60612i − 0.103032i
\(36\) 0 0
\(37\) −24.0908 −0.651103 −0.325552 0.945524i \(-0.605550\pi\)
−0.325552 + 0.945524i \(0.605550\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 19.7990i 0.482902i 0.970413 + 0.241451i \(0.0776233\pi\)
−0.970413 + 0.241451i \(0.922377\pi\)
\(42\) 0 0
\(43\) 49.0047 1.13964 0.569822 0.821768i \(-0.307012\pi\)
0.569822 + 0.821768i \(0.307012\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 59.3939i − 1.26370i −0.775091 0.631850i \(-0.782296\pi\)
0.775091 0.631850i \(-0.217704\pi\)
\(48\) 0 0
\(49\) −48.0908 −0.981445
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 48.1796i − 0.909049i −0.890734 0.454524i \(-0.849809\pi\)
0.890734 0.454524i \(-0.150191\pi\)
\(54\) 0 0
\(55\) −44.2371 −0.804311
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 12.6061i − 0.213663i −0.994277 0.106832i \(-0.965929\pi\)
0.994277 0.106832i \(-0.0340706\pi\)
\(60\) 0 0
\(61\) 4.00000 0.0655738 0.0327869 0.999462i \(-0.489562\pi\)
0.0327869 + 0.999462i \(0.489562\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 32.8913i − 0.506020i
\(66\) 0 0
\(67\) 111.358 1.66207 0.831034 0.556222i \(-0.187750\pi\)
0.831034 + 0.556222i \(0.187750\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 56.6969i − 0.798548i −0.916832 0.399274i \(-0.869262\pi\)
0.916832 0.399274i \(-0.130738\pi\)
\(72\) 0 0
\(73\) 35.6969 0.488999 0.244500 0.969649i \(-0.421376\pi\)
0.244500 + 0.969649i \(0.421376\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 11.1532i − 0.144846i
\(78\) 0 0
\(79\) −144.442 −1.82839 −0.914193 0.405280i \(-0.867174\pi\)
−0.914193 + 0.405280i \(0.867174\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 101.697i − 1.22526i −0.790368 0.612632i \(-0.790111\pi\)
0.790368 0.612632i \(-0.209889\pi\)
\(84\) 0 0
\(85\) −17.9092 −0.210696
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 19.0060i − 0.213551i −0.994283 0.106775i \(-0.965947\pi\)
0.994283 0.106775i \(-0.0340526\pi\)
\(90\) 0 0
\(91\) 8.29263 0.0911278
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 110.697i − 1.16523i
\(96\) 0 0
\(97\) 27.0000 0.278351 0.139175 0.990268i \(-0.455555\pi\)
0.139175 + 0.990268i \(0.455555\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 189.665i 1.87787i 0.344091 + 0.938936i \(0.388187\pi\)
−0.344091 + 0.938936i \(0.611813\pi\)
\(102\) 0 0
\(103\) −74.4605 −0.722918 −0.361459 0.932388i \(-0.617721\pi\)
−0.361459 + 0.932388i \(0.617721\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 138.576i − 1.29510i −0.762024 0.647549i \(-0.775794\pi\)
0.762024 0.647549i \(-0.224206\pi\)
\(108\) 0 0
\(109\) 114.879 1.05393 0.526966 0.849886i \(-0.323330\pi\)
0.526966 + 0.849886i \(0.323330\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 157.663i 1.39525i 0.716463 + 0.697625i \(0.245760\pi\)
−0.716463 + 0.697625i \(0.754240\pi\)
\(114\) 0 0
\(115\) 10.1997 0.0886927
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 4.51531i − 0.0379438i
\(120\) 0 0
\(121\) −15.8184 −0.130730
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 135.004i − 1.08003i
\(126\) 0 0
\(127\) −125.661 −0.989458 −0.494729 0.869047i \(-0.664733\pi\)
−0.494729 + 0.869047i \(0.664733\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 223.182i − 1.70368i −0.523805 0.851838i \(-0.675488\pi\)
0.523805 0.851838i \(-0.324512\pi\)
\(132\) 0 0
\(133\) 27.9092 0.209843
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 162.142i − 1.18352i −0.806116 0.591758i \(-0.798434\pi\)
0.806116 0.591758i \(-0.201566\pi\)
\(138\) 0 0
\(139\) −69.9819 −0.503467 −0.251734 0.967797i \(-0.581001\pi\)
−0.251734 + 0.967797i \(0.581001\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 101.728i − 0.711381i
\(144\) 0 0
\(145\) 82.0908 0.566144
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 216.171i − 1.45081i −0.688322 0.725405i \(-0.741652\pi\)
0.688322 0.725405i \(-0.258348\pi\)
\(150\) 0 0
\(151\) 92.5772 0.613094 0.306547 0.951855i \(-0.400826\pi\)
0.306547 + 0.951855i \(0.400826\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 85.4541i 0.551317i
\(156\) 0 0
\(157\) −292.545 −1.86334 −0.931672 0.363301i \(-0.881649\pi\)
−0.931672 + 0.363301i \(0.881649\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 2.57156i 0.0159724i
\(162\) 0 0
\(163\) 112.601 0.690804 0.345402 0.938455i \(-0.387743\pi\)
0.345402 + 0.938455i \(0.387743\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 106.182i − 0.635818i −0.948121 0.317909i \(-0.897019\pi\)
0.948121 0.317909i \(-0.102981\pi\)
\(168\) 0 0
\(169\) −93.3633 −0.552445
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 224.271i 1.29636i 0.761486 + 0.648182i \(0.224470\pi\)
−0.761486 + 0.648182i \(0.775530\pi\)
\(174\) 0 0
\(175\) 10.1997 0.0582838
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 113.424i − 0.633656i −0.948483 0.316828i \(-0.897382\pi\)
0.948483 0.316828i \(-0.102618\pi\)
\(180\) 0 0
\(181\) 160.182 0.884981 0.442491 0.896773i \(-0.354095\pi\)
0.442491 + 0.896773i \(0.354095\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 91.1100i 0.492486i
\(186\) 0 0
\(187\) −55.3903 −0.296205
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 200.697i − 1.05077i −0.850865 0.525385i \(-0.823921\pi\)
0.850865 0.525385i \(-0.176079\pi\)
\(192\) 0 0
\(193\) −152.909 −0.792276 −0.396138 0.918191i \(-0.629650\pi\)
−0.396138 + 0.918191i \(0.629650\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 15.2883i 0.0776056i 0.999247 + 0.0388028i \(0.0123544\pi\)
−0.999247 + 0.0388028i \(0.987646\pi\)
\(198\) 0 0
\(199\) 231.299 1.16230 0.581152 0.813795i \(-0.302602\pi\)
0.581152 + 0.813795i \(0.302602\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 20.6969i 0.101955i
\(204\) 0 0
\(205\) 74.8786 0.365261
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 342.368i − 1.63812i
\(210\) 0 0
\(211\) −185.819 −0.880659 −0.440329 0.897836i \(-0.645138\pi\)
−0.440329 + 0.897836i \(0.645138\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 185.333i − 0.862012i
\(216\) 0 0
\(217\) −21.5449 −0.0992852
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 41.1839i − 0.186353i
\(222\) 0 0
\(223\) 272.964 1.22405 0.612027 0.790837i \(-0.290354\pi\)
0.612027 + 0.790837i \(0.290354\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 307.757i − 1.35576i −0.735173 0.677879i \(-0.762899\pi\)
0.735173 0.677879i \(-0.237101\pi\)
\(228\) 0 0
\(229\) −200.424 −0.875216 −0.437608 0.899166i \(-0.644174\pi\)
−0.437608 + 0.899166i \(0.644174\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 381.687i 1.63814i 0.573693 + 0.819070i \(0.305510\pi\)
−0.573693 + 0.819070i \(0.694490\pi\)
\(234\) 0 0
\(235\) −224.624 −0.955847
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 254.697i − 1.06568i −0.846217 0.532839i \(-0.821125\pi\)
0.846217 0.532839i \(-0.178875\pi\)
\(240\) 0 0
\(241\) 283.757 1.17742 0.588708 0.808346i \(-0.299637\pi\)
0.588708 + 0.808346i \(0.299637\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 181.877i 0.742353i
\(246\) 0 0
\(247\) 254.558 1.03060
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 54.0000i 0.215139i 0.994198 + 0.107570i \(0.0343069\pi\)
−0.994198 + 0.107570i \(0.965693\pi\)
\(252\) 0 0
\(253\) 31.5459 0.124687
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 202.725i 0.788815i 0.918936 + 0.394407i \(0.129050\pi\)
−0.918936 + 0.394407i \(0.870950\pi\)
\(258\) 0 0
\(259\) −22.9709 −0.0886906
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 162.879i 0.619310i 0.950849 + 0.309655i \(0.100214\pi\)
−0.950849 + 0.309655i \(0.899786\pi\)
\(264\) 0 0
\(265\) −182.212 −0.687593
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 119.715i − 0.445038i −0.974928 0.222519i \(-0.928572\pi\)
0.974928 0.222519i \(-0.0714280\pi\)
\(270\) 0 0
\(271\) 337.601 1.24576 0.622879 0.782318i \(-0.285963\pi\)
0.622879 + 0.782318i \(0.285963\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 125.121i − 0.454987i
\(276\) 0 0
\(277\) 268.182 0.968165 0.484082 0.875022i \(-0.339154\pi\)
0.484082 + 0.875022i \(0.339154\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 272.322i − 0.969117i −0.874759 0.484559i \(-0.838980\pi\)
0.874759 0.484559i \(-0.161020\pi\)
\(282\) 0 0
\(283\) 67.4104 0.238199 0.119100 0.992882i \(-0.461999\pi\)
0.119100 + 0.992882i \(0.461999\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 18.8786i 0.0657790i
\(288\) 0 0
\(289\) 266.576 0.922407
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 378.345i − 1.29128i −0.763642 0.645639i \(-0.776591\pi\)
0.763642 0.645639i \(-0.223409\pi\)
\(294\) 0 0
\(295\) −47.6756 −0.161612
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 23.4551i 0.0784452i
\(300\) 0 0
\(301\) 46.7265 0.155238
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 15.1278i − 0.0495992i
\(306\) 0 0
\(307\) −425.699 −1.38664 −0.693321 0.720629i \(-0.743853\pi\)
−0.693321 + 0.720629i \(0.743853\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 556.999i 1.79099i 0.445068 + 0.895497i \(0.353179\pi\)
−0.445068 + 0.895497i \(0.646821\pi\)
\(312\) 0 0
\(313\) 169.636 0.541967 0.270984 0.962584i \(-0.412651\pi\)
0.270984 + 0.962584i \(0.412651\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 225.128i 0.710183i 0.934832 + 0.355091i \(0.115550\pi\)
−0.934832 + 0.355091i \(0.884450\pi\)
\(318\) 0 0
\(319\) 253.894 0.795906
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 138.606i − 0.429121i
\(324\) 0 0
\(325\) 93.0306 0.286248
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 56.6328i − 0.172136i
\(330\) 0 0
\(331\) −315.005 −0.951677 −0.475839 0.879533i \(-0.657855\pi\)
−0.475839 + 0.879533i \(0.657855\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 421.151i − 1.25717i
\(336\) 0 0
\(337\) −642.545 −1.90666 −0.953331 0.301928i \(-0.902370\pi\)
−0.953331 + 0.301928i \(0.902370\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 264.296i 0.775061i
\(342\) 0 0
\(343\) −92.5772 −0.269904
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 301.515i 0.868920i 0.900691 + 0.434460i \(0.143061\pi\)
−0.900691 + 0.434460i \(0.856939\pi\)
\(348\) 0 0
\(349\) −180.454 −0.517060 −0.258530 0.966003i \(-0.583238\pi\)
−0.258530 + 0.966003i \(0.583238\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 331.825i − 0.940014i −0.882663 0.470007i \(-0.844251\pi\)
0.882663 0.470007i \(-0.155749\pi\)
\(354\) 0 0
\(355\) −214.424 −0.604012
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 566.908i 1.57913i 0.613666 + 0.789566i \(0.289694\pi\)
−0.613666 + 0.789566i \(0.710306\pi\)
\(360\) 0 0
\(361\) 495.727 1.37320
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 135.004i − 0.369873i
\(366\) 0 0
\(367\) 219.192 0.597253 0.298627 0.954370i \(-0.403471\pi\)
0.298627 + 0.954370i \(0.403471\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 45.9398i − 0.123827i
\(372\) 0 0
\(373\) −606.454 −1.62588 −0.812941 0.582346i \(-0.802135\pi\)
−0.812941 + 0.582346i \(0.802135\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 188.776i 0.500732i
\(378\) 0 0
\(379\) 124.708 0.329044 0.164522 0.986373i \(-0.447392\pi\)
0.164522 + 0.986373i \(0.447392\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 58.4541i − 0.152622i −0.997084 0.0763108i \(-0.975686\pi\)
0.997084 0.0763108i \(-0.0243141\pi\)
\(384\) 0 0
\(385\) −42.1806 −0.109560
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 121.526i 0.312406i 0.987725 + 0.156203i \(0.0499255\pi\)
−0.987725 + 0.156203i \(0.950075\pi\)
\(390\) 0 0
\(391\) 12.7712 0.0326630
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 546.272i 1.38297i
\(396\) 0 0
\(397\) 330.091 0.831463 0.415732 0.909487i \(-0.363526\pi\)
0.415732 + 0.909487i \(0.363526\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 506.202i 1.26235i 0.775641 + 0.631174i \(0.217427\pi\)
−0.775641 + 0.631174i \(0.782573\pi\)
\(402\) 0 0
\(403\) −196.510 −0.487618
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 281.789i 0.692356i
\(408\) 0 0
\(409\) −521.302 −1.27458 −0.637289 0.770625i \(-0.719944\pi\)
−0.637289 + 0.770625i \(0.719944\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 12.0201i − 0.0291043i
\(414\) 0 0
\(415\) −384.612 −0.926775
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 133.151i 0.317783i 0.987296 + 0.158891i \(0.0507920\pi\)
−0.987296 + 0.158891i \(0.949208\pi\)
\(420\) 0 0
\(421\) 129.031 0.306486 0.153243 0.988189i \(-0.451028\pi\)
0.153243 + 0.988189i \(0.451028\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 50.6548i − 0.119188i
\(426\) 0 0
\(427\) 3.81405 0.00893219
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 509.333i − 1.18175i −0.806764 0.590873i \(-0.798783\pi\)
0.806764 0.590873i \(-0.201217\pi\)
\(432\) 0 0
\(433\) −5.54489 −0.0128058 −0.00640288 0.999980i \(-0.502038\pi\)
−0.00640288 + 0.999980i \(0.502038\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 78.9391i 0.180639i
\(438\) 0 0
\(439\) −337.514 −0.768825 −0.384412 0.923162i \(-0.625596\pi\)
−0.384412 + 0.923162i \(0.625596\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 475.212i 1.07271i 0.843991 + 0.536357i \(0.180200\pi\)
−0.843991 + 0.536357i \(0.819800\pi\)
\(444\) 0 0
\(445\) −71.8796 −0.161527
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 251.858i 0.560932i 0.959864 + 0.280466i \(0.0904890\pi\)
−0.959864 + 0.280466i \(0.909511\pi\)
\(450\) 0 0
\(451\) 231.588 0.513498
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 31.3622i − 0.0689280i
\(456\) 0 0
\(457\) 42.5755 0.0931630 0.0465815 0.998914i \(-0.485167\pi\)
0.0465815 + 0.998914i \(0.485167\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 552.346i − 1.19815i −0.800694 0.599074i \(-0.795536\pi\)
0.800694 0.599074i \(-0.204464\pi\)
\(462\) 0 0
\(463\) −524.084 −1.13193 −0.565966 0.824429i \(-0.691496\pi\)
−0.565966 + 0.824429i \(0.691496\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 130.423i − 0.279279i −0.990202 0.139640i \(-0.955406\pi\)
0.990202 0.139640i \(-0.0445944\pi\)
\(468\) 0 0
\(469\) 106.182 0.226400
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 573.205i − 1.21185i
\(474\) 0 0
\(475\) 313.098 0.659154
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 572.302i 1.19479i 0.801949 + 0.597393i \(0.203797\pi\)
−0.801949 + 0.597393i \(0.796203\pi\)
\(480\) 0 0
\(481\) −209.516 −0.435585
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 102.112i − 0.210541i
\(486\) 0 0
\(487\) −141.957 −0.291494 −0.145747 0.989322i \(-0.546559\pi\)
−0.145747 + 0.989322i \(0.546559\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 464.394i 0.945812i 0.881113 + 0.472906i \(0.156795\pi\)
−0.881113 + 0.472906i \(0.843205\pi\)
\(492\) 0 0
\(493\) 102.788 0.208494
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 54.0612i − 0.108775i
\(498\) 0 0
\(499\) −144.356 −0.289290 −0.144645 0.989484i \(-0.546204\pi\)
−0.144645 + 0.989484i \(0.546204\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 733.485i 1.45822i 0.684396 + 0.729110i \(0.260066\pi\)
−0.684396 + 0.729110i \(0.739934\pi\)
\(504\) 0 0
\(505\) 717.302 1.42040
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 300.680i 0.590727i 0.955385 + 0.295364i \(0.0954408\pi\)
−0.955385 + 0.295364i \(0.904559\pi\)
\(510\) 0 0
\(511\) 34.0374 0.0666095
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 281.605i 0.546806i
\(516\) 0 0
\(517\) −694.727 −1.34377
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 375.988i − 0.721666i −0.932630 0.360833i \(-0.882492\pi\)
0.932630 0.360833i \(-0.117508\pi\)
\(522\) 0 0
\(523\) 665.001 1.27151 0.635757 0.771890i \(-0.280688\pi\)
0.635757 + 0.771890i \(0.280688\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 106.999i 0.203034i
\(528\) 0 0
\(529\) 521.727 0.986251
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 172.191i 0.323059i
\(534\) 0 0
\(535\) −524.084 −0.979596
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 562.515i 1.04363i
\(540\) 0 0
\(541\) 818.302 1.51257 0.756287 0.654241i \(-0.227012\pi\)
0.756287 + 0.654241i \(0.227012\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 434.464i − 0.797181i
\(546\) 0 0
\(547\) 625.532 1.14357 0.571784 0.820404i \(-0.306251\pi\)
0.571784 + 0.820404i \(0.306251\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 635.333i 1.15305i
\(552\) 0 0
\(553\) −137.728 −0.249055
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 27.9087i 0.0501054i 0.999686 + 0.0250527i \(0.00797536\pi\)
−0.999686 + 0.0250527i \(0.992025\pi\)
\(558\) 0 0
\(559\) 426.191 0.762416
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 461.574i − 0.819848i −0.912120 0.409924i \(-0.865555\pi\)
0.912120 0.409924i \(-0.134445\pi\)
\(564\) 0 0
\(565\) 596.272 1.05535
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 983.092i 1.72775i 0.503703 + 0.863877i \(0.331971\pi\)
−0.503703 + 0.863877i \(0.668029\pi\)
\(570\) 0 0
\(571\) 692.856 1.21341 0.606704 0.794928i \(-0.292491\pi\)
0.606704 + 0.794928i \(0.292491\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 28.8490i 0.0501721i
\(576\) 0 0
\(577\) −154.545 −0.267842 −0.133921 0.990992i \(-0.542757\pi\)
−0.133921 + 0.990992i \(0.542757\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 96.9692i − 0.166901i
\(582\) 0 0
\(583\) −563.554 −0.966644
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 867.545i 1.47793i 0.673744 + 0.738965i \(0.264685\pi\)
−0.673744 + 0.738965i \(0.735315\pi\)
\(588\) 0 0
\(589\) −661.362 −1.12286
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 343.675i − 0.579553i −0.957094 0.289776i \(-0.906419\pi\)
0.957094 0.289776i \(-0.0935809\pi\)
\(594\) 0 0
\(595\) −17.0766 −0.0287002
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 908.969i 1.51748i 0.651395 + 0.758739i \(0.274184\pi\)
−0.651395 + 0.758739i \(0.725816\pi\)
\(600\) 0 0
\(601\) −419.817 −0.698531 −0.349266 0.937024i \(-0.613569\pi\)
−0.349266 + 0.937024i \(0.613569\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 59.8241i 0.0988828i
\(606\) 0 0
\(607\) 131.671 0.216921 0.108461 0.994101i \(-0.465408\pi\)
0.108461 + 0.994101i \(0.465408\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 516.545i − 0.845409i
\(612\) 0 0
\(613\) 494.727 0.807058 0.403529 0.914967i \(-0.367783\pi\)
0.403529 + 0.914967i \(0.367783\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 500.330i 0.810908i 0.914116 + 0.405454i \(0.132886\pi\)
−0.914116 + 0.405454i \(0.867114\pi\)
\(618\) 0 0
\(619\) 636.974 1.02904 0.514519 0.857479i \(-0.327971\pi\)
0.514519 + 0.857479i \(0.327971\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 18.1225i − 0.0290890i
\(624\) 0 0
\(625\) −243.152 −0.389043
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 114.081i 0.181369i
\(630\) 0 0
\(631\) −365.541 −0.579305 −0.289652 0.957132i \(-0.593540\pi\)
−0.289652 + 0.957132i \(0.593540\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 475.243i 0.748414i
\(636\) 0 0
\(637\) −418.243 −0.656582
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 843.812i 1.31640i 0.752843 + 0.658200i \(0.228682\pi\)
−0.752843 + 0.658200i \(0.771318\pi\)
\(642\) 0 0
\(643\) 1252.22 1.94746 0.973732 0.227696i \(-0.0731192\pi\)
0.973732 + 0.227696i \(0.0731192\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 799.151i 1.23516i 0.786507 + 0.617582i \(0.211888\pi\)
−0.786507 + 0.617582i \(0.788112\pi\)
\(648\) 0 0
\(649\) −147.453 −0.227200
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 798.227i 1.22240i 0.791477 + 0.611199i \(0.209313\pi\)
−0.791477 + 0.611199i \(0.790687\pi\)
\(654\) 0 0
\(655\) −844.059 −1.28864
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 106.151i − 0.161079i −0.996751 0.0805395i \(-0.974336\pi\)
0.996751 0.0805395i \(-0.0256643\pi\)
\(660\) 0 0
\(661\) −772.272 −1.16834 −0.584170 0.811631i \(-0.698580\pi\)
−0.584170 + 0.811631i \(0.698580\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 105.551i − 0.158723i
\(666\) 0 0
\(667\) −58.5398 −0.0877658
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 46.7878i − 0.0697284i
\(672\) 0 0
\(673\) 494.637 0.734973 0.367486 0.930029i \(-0.380218\pi\)
0.367486 + 0.930029i \(0.380218\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 1089.14i − 1.60877i −0.594109 0.804385i \(-0.702495\pi\)
0.594109 0.804385i \(-0.297505\pi\)
\(678\) 0 0
\(679\) 25.7448 0.0379158
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1126.60i 1.64949i 0.565502 + 0.824747i \(0.308682\pi\)
−0.565502 + 0.824747i \(0.691318\pi\)
\(684\) 0 0
\(685\) −613.210 −0.895197
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 419.015i − 0.608149i
\(690\) 0 0
\(691\) −462.019 −0.668624 −0.334312 0.942462i \(-0.608504\pi\)
−0.334312 + 0.942462i \(0.608504\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 264.667i 0.380816i
\(696\) 0 0
\(697\) 93.7571 0.134515
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 189.193i − 0.269891i −0.990853 0.134945i \(-0.956914\pi\)
0.990853 0.134945i \(-0.0430859\pi\)
\(702\) 0 0
\(703\) −705.136 −1.00304
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 180.848i 0.255796i
\(708\) 0 0
\(709\) 926.302 1.30649 0.653245 0.757146i \(-0.273407\pi\)
0.653245 + 0.757146i \(0.273407\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 60.9382i − 0.0854673i
\(714\) 0 0
\(715\) −384.727 −0.538080
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 957.453i − 1.33165i −0.746110 0.665823i \(-0.768081\pi\)
0.746110 0.665823i \(-0.231919\pi\)
\(720\) 0 0
\(721\) −70.9990 −0.0984729
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 232.188i 0.320259i
\(726\) 0 0
\(727\) 979.545 1.34738 0.673690 0.739015i \(-0.264709\pi\)
0.673690 + 0.739015i \(0.264709\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 232.059i − 0.317454i
\(732\) 0 0
\(733\) 452.636 0.617511 0.308756 0.951141i \(-0.400087\pi\)
0.308756 + 0.951141i \(0.400087\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1302.55i − 1.76737i
\(738\) 0 0
\(739\) 460.026 0.622497 0.311249 0.950328i \(-0.399253\pi\)
0.311249 + 0.950328i \(0.399253\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 393.119i − 0.529097i −0.964372 0.264549i \(-0.914777\pi\)
0.964372 0.264549i \(-0.0852230\pi\)
\(744\) 0 0
\(745\) −817.545 −1.09738
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 132.133i − 0.176413i
\(750\) 0 0
\(751\) 611.229 0.813887 0.406944 0.913453i \(-0.366595\pi\)
0.406944 + 0.913453i \(0.366595\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 350.121i − 0.463737i
\(756\) 0 0
\(757\) −106.182 −0.140266 −0.0701332 0.997538i \(-0.522342\pi\)
−0.0701332 + 0.997538i \(0.522342\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 844.843i − 1.11017i −0.831792 0.555087i \(-0.812685\pi\)
0.831792 0.555087i \(-0.187315\pi\)
\(762\) 0 0
\(763\) 109.538 0.143562
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 109.635i − 0.142940i
\(768\) 0 0
\(769\) −588.728 −0.765575 −0.382788 0.923836i \(-0.625036\pi\)
−0.382788 + 0.923836i \(0.625036\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 9.87858i 0.0127795i 0.999980 + 0.00638976i \(0.00203394\pi\)
−0.999980 + 0.00638976i \(0.997966\pi\)
\(774\) 0 0
\(775\) −241.701 −0.311872
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 579.514i 0.743921i
\(780\) 0 0
\(781\) −663.181 −0.849143
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1106.39i 1.40941i
\(786\) 0 0
\(787\) −118.409 −0.150456 −0.0752279 0.997166i \(-0.523968\pi\)
−0.0752279 + 0.997166i \(0.523968\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 150.334i 0.190055i
\(792\) 0 0
\(793\) 34.7878 0.0438685
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 371.477i 0.466095i 0.972465 + 0.233047i \(0.0748697\pi\)
−0.972465 + 0.233047i \(0.925130\pi\)
\(798\) 0 0
\(799\) −281.257 −0.352011
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 417.545i − 0.519981i
\(804\) 0 0
\(805\) 9.72549 0.0120814
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 64.7746i − 0.0800675i −0.999198 0.0400337i \(-0.987253\pi\)
0.999198 0.0400337i \(-0.0127465\pi\)
\(810\) 0 0
\(811\) −782.832 −0.965268 −0.482634 0.875822i \(-0.660320\pi\)
−0.482634 + 0.875822i \(0.660320\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 425.850i − 0.522515i
\(816\) 0 0
\(817\) 1434.36 1.75564
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 589.748i − 0.718329i −0.933274 0.359164i \(-0.883062\pi\)
0.933274 0.359164i \(-0.116938\pi\)
\(822\) 0 0
\(823\) −492.821 −0.598810 −0.299405 0.954126i \(-0.596788\pi\)
−0.299405 + 0.954126i \(0.596788\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 475.212i 0.574622i 0.957837 + 0.287311i \(0.0927613\pi\)
−0.957837 + 0.287311i \(0.907239\pi\)
\(828\) 0 0
\(829\) 597.637 0.720913 0.360456 0.932776i \(-0.382621\pi\)
0.360456 + 0.932776i \(0.382621\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 227.732i 0.273387i
\(834\) 0 0
\(835\) −401.572 −0.480925
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 290.636i − 0.346407i −0.984886 0.173204i \(-0.944588\pi\)
0.984886 0.173204i \(-0.0554119\pi\)
\(840\) 0 0
\(841\) 369.849 0.439773
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 353.094i 0.417863i
\(846\) 0 0
\(847\) −15.0830 −0.0178076
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 64.9714i − 0.0763472i
\(852\) 0 0
\(853\) 201.273 0.235960 0.117980 0.993016i \(-0.462358\pi\)
0.117980 + 0.993016i \(0.462358\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1009.64i 1.17811i 0.808093 + 0.589055i \(0.200500\pi\)
−0.808093 + 0.589055i \(0.799500\pi\)
\(858\) 0 0
\(859\) −1172.12 −1.36452 −0.682261 0.731109i \(-0.739003\pi\)
−0.682261 + 0.731109i \(0.739003\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 755.755i 0.875730i 0.899041 + 0.437865i \(0.144265\pi\)
−0.899041 + 0.437865i \(0.855735\pi\)
\(864\) 0 0
\(865\) 848.179 0.980553
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 1689.53i 1.94423i
\(870\) 0 0
\(871\) 968.478 1.11192
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 128.728i − 0.147117i
\(876\) 0 0
\(877\) 317.818 0.362393 0.181196 0.983447i \(-0.442003\pi\)
0.181196 + 0.983447i \(0.442003\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 705.755i 0.801084i 0.916278 + 0.400542i \(0.131178\pi\)
−0.916278 + 0.400542i \(0.868822\pi\)
\(882\) 0 0
\(883\) 505.389 0.572355 0.286177 0.958177i \(-0.407615\pi\)
0.286177 + 0.958177i \(0.407615\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1581.00i 1.78241i 0.453602 + 0.891205i \(0.350139\pi\)
−0.453602 + 0.891205i \(0.649861\pi\)
\(888\) 0 0
\(889\) −119.819 −0.134780
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 1738.45i − 1.94676i
\(894\) 0 0
\(895\) −428.964 −0.479290
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 490.454i − 0.545555i
\(900\) 0 0
\(901\) −228.152 −0.253221
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 605.797i − 0.669389i
\(906\) 0 0
\(907\) −773.875 −0.853225 −0.426612 0.904435i \(-0.640293\pi\)
−0.426612 + 0.904435i \(0.640293\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1785.51i − 1.95995i −0.199121 0.979975i \(-0.563809\pi\)
0.199121 0.979975i \(-0.436191\pi\)
\(912\) 0 0
\(913\) −1189.54 −1.30289
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 212.806i − 0.232068i
\(918\) 0 0
\(919\) 992.402 1.07987 0.539936 0.841706i \(-0.318448\pi\)
0.539936 + 0.841706i \(0.318448\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 493.090i − 0.534225i
\(924\) 0 0
\(925\) −257.698 −0.278592
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 887.568i 0.955401i 0.878523 + 0.477701i \(0.158530\pi\)
−0.878523 + 0.477701i \(0.841470\pi\)
\(930\) 0 0
\(931\) −1407.61 −1.51194
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 209.483i 0.224046i
\(936\) 0 0
\(937\) 493.000 0.526147 0.263074 0.964776i \(-0.415264\pi\)
0.263074 + 0.964776i \(0.415264\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 379.770i − 0.403581i −0.979429 0.201791i \(-0.935324\pi\)
0.979429 0.201791i \(-0.0646761\pi\)
\(942\) 0 0
\(943\) −53.3967 −0.0566242
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 414.092i − 0.437267i −0.975807 0.218633i \(-0.929840\pi\)
0.975807 0.218633i \(-0.0701599\pi\)
\(948\) 0 0
\(949\) 310.454 0.327138
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 1254.36i − 1.31623i −0.752919 0.658113i \(-0.771355\pi\)
0.752919 0.658113i \(-0.228645\pi\)
\(954\) 0 0
\(955\) −759.024 −0.794789
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 154.604i − 0.161214i
\(960\) 0 0
\(961\) −450.452 −0.468733
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 578.293i 0.599268i
\(966\) 0 0
\(967\) 522.755 0.540595 0.270297 0.962777i \(-0.412878\pi\)
0.270297 + 0.962777i \(0.412878\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 1044.66i − 1.07586i −0.842988 0.537932i \(-0.819206\pi\)
0.842988 0.537932i \(-0.180794\pi\)
\(972\) 0 0
\(973\) −66.7286 −0.0685803
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 1375.71i − 1.40809i −0.710154 0.704047i \(-0.751375\pi\)
0.710154 0.704047i \(-0.248625\pi\)
\(978\) 0 0
\(979\) −222.312 −0.227081
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 451.757i 0.459570i 0.973241 + 0.229785i \(0.0738023\pi\)
−0.973241 + 0.229785i \(0.926198\pi\)
\(984\) 0 0
\(985\) 57.8194 0.0586999
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 132.163i 0.133633i
\(990\) 0 0
\(991\) 1811.29 1.82774 0.913872 0.406002i \(-0.133077\pi\)
0.913872 + 0.406002i \(0.133077\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 874.757i − 0.879153i
\(996\) 0 0
\(997\) 377.546 0.378682 0.189341 0.981911i \(-0.439365\pi\)
0.189341 + 0.981911i \(0.439365\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 1728.3.e.u.1025.4 8
3.2 odd 2 inner 1728.3.e.u.1025.6 8
4.3 odd 2 inner 1728.3.e.u.1025.3 8
8.3 odd 2 864.3.e.f.161.5 yes 8
8.5 even 2 864.3.e.f.161.6 yes 8
12.11 even 2 inner 1728.3.e.u.1025.5 8
24.5 odd 2 864.3.e.f.161.4 yes 8
24.11 even 2 864.3.e.f.161.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.e.f.161.3 8 24.11 even 2
864.3.e.f.161.4 yes 8 24.5 odd 2
864.3.e.f.161.5 yes 8 8.3 odd 2
864.3.e.f.161.6 yes 8 8.5 even 2
1728.3.e.u.1025.3 8 4.3 odd 2 inner
1728.3.e.u.1025.4 8 1.1 even 1 trivial
1728.3.e.u.1025.5 8 12.11 even 2 inner
1728.3.e.u.1025.6 8 3.2 odd 2 inner