Properties

Label 1296.3.e.d.161.1
Level $1296$
Weight $3$
Character 1296.161
Analytic conductor $35.313$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1296 = 2^{4} \cdot 3^{4} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1296.e (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.3134422611\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{3})\)
Defining polynomial: \( x^{4} + 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 3^{3} \)
Twist minimal: no (minimal twist has level 162)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.1
Root \(-1.93185i\) of defining polynomial
Character \(\chi\) \(=\) 1296.161
Dual form 1296.3.e.d.161.4

$q$-expansion

\(f(q)\) \(=\) \(q-5.79555i q^{5} +8.39230 q^{7} +O(q^{10})\) \(q-5.79555i q^{5} +8.39230 q^{7} +14.6969i q^{11} -21.1962 q^{13} -7.76457i q^{17} -24.3923 q^{19} -14.6969i q^{23} -8.58846 q^{25} -35.4940i q^{29} -8.00000 q^{31} -48.6381i q^{35} -60.5692 q^{37} +33.6365i q^{41} -9.17691 q^{43} -16.9706i q^{47} +21.4308 q^{49} -25.7605i q^{53} +85.1769 q^{55} +61.6706i q^{59} -13.0000 q^{61} +122.843i q^{65} -21.1769 q^{67} -101.214i q^{71} +40.4115 q^{73} +123.341i q^{77} -98.7461 q^{79} +103.488i q^{83} -45.0000 q^{85} -134.130i q^{89} -177.885 q^{91} +141.367i q^{95} -75.1384 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{7} - 64 q^{13} - 56 q^{19} + 28 q^{25} - 32 q^{31} - 76 q^{37} + 88 q^{43} + 252 q^{49} + 216 q^{55} - 52 q^{61} + 40 q^{67} + 224 q^{73} - 104 q^{79} - 180 q^{85} - 88 q^{91} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1296\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(1135\) \(1217\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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
\(5\) − 5.79555i − 1.15911i −0.814933 0.579555i \(-0.803226\pi\)
0.814933 0.579555i \(-0.196774\pi\)
\(6\) 0 0
\(7\) 8.39230 1.19890 0.599450 0.800412i \(-0.295386\pi\)
0.599450 + 0.800412i \(0.295386\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 14.6969i 1.33609i 0.744123 + 0.668043i \(0.232868\pi\)
−0.744123 + 0.668043i \(0.767132\pi\)
\(12\) 0 0
\(13\) −21.1962 −1.63047 −0.815237 0.579128i \(-0.803393\pi\)
−0.815237 + 0.579128i \(0.803393\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 7.76457i − 0.456739i −0.973574 0.228370i \(-0.926660\pi\)
0.973574 0.228370i \(-0.0733395\pi\)
\(18\) 0 0
\(19\) −24.3923 −1.28381 −0.641903 0.766786i \(-0.721855\pi\)
−0.641903 + 0.766786i \(0.721855\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 14.6969i − 0.638997i −0.947587 0.319499i \(-0.896486\pi\)
0.947587 0.319499i \(-0.103514\pi\)
\(24\) 0 0
\(25\) −8.58846 −0.343538
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 35.4940i − 1.22393i −0.790884 0.611966i \(-0.790379\pi\)
0.790884 0.611966i \(-0.209621\pi\)
\(30\) 0 0
\(31\) −8.00000 −0.258065 −0.129032 0.991640i \(-0.541187\pi\)
−0.129032 + 0.991640i \(0.541187\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 48.6381i − 1.38966i
\(36\) 0 0
\(37\) −60.5692 −1.63701 −0.818503 0.574502i \(-0.805196\pi\)
−0.818503 + 0.574502i \(0.805196\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 33.6365i 0.820403i 0.911995 + 0.410201i \(0.134542\pi\)
−0.911995 + 0.410201i \(0.865458\pi\)
\(42\) 0 0
\(43\) −9.17691 −0.213417 −0.106708 0.994290i \(-0.534031\pi\)
−0.106708 + 0.994290i \(0.534031\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 16.9706i − 0.361076i −0.983568 0.180538i \(-0.942216\pi\)
0.983568 0.180538i \(-0.0577838\pi\)
\(48\) 0 0
\(49\) 21.4308 0.437363
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 25.7605i − 0.486046i −0.970020 0.243023i \(-0.921861\pi\)
0.970020 0.243023i \(-0.0781391\pi\)
\(54\) 0 0
\(55\) 85.1769 1.54867
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 61.6706i 1.04526i 0.852558 + 0.522632i \(0.175050\pi\)
−0.852558 + 0.522632i \(0.824950\pi\)
\(60\) 0 0
\(61\) −13.0000 −0.213115 −0.106557 0.994307i \(-0.533983\pi\)
−0.106557 + 0.994307i \(0.533983\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 122.843i 1.88990i
\(66\) 0 0
\(67\) −21.1769 −0.316073 −0.158037 0.987433i \(-0.550516\pi\)
−0.158037 + 0.987433i \(0.550516\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 101.214i − 1.42555i −0.701392 0.712776i \(-0.747438\pi\)
0.701392 0.712776i \(-0.252562\pi\)
\(72\) 0 0
\(73\) 40.4115 0.553583 0.276791 0.960930i \(-0.410729\pi\)
0.276791 + 0.960930i \(0.410729\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 123.341i 1.60183i
\(78\) 0 0
\(79\) −98.7461 −1.24995 −0.624976 0.780644i \(-0.714891\pi\)
−0.624976 + 0.780644i \(0.714891\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 103.488i 1.24684i 0.781887 + 0.623420i \(0.214257\pi\)
−0.781887 + 0.623420i \(0.785743\pi\)
\(84\) 0 0
\(85\) −45.0000 −0.529412
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 134.130i − 1.50708i −0.657403 0.753539i \(-0.728345\pi\)
0.657403 0.753539i \(-0.271655\pi\)
\(90\) 0 0
\(91\) −177.885 −1.95478
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 141.367i 1.48807i
\(96\) 0 0
\(97\) −75.1384 −0.774623 −0.387312 0.921949i \(-0.626596\pi\)
−0.387312 + 0.921949i \(0.626596\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 29.0893i 0.288013i 0.989577 + 0.144006i \(0.0459986\pi\)
−0.989577 + 0.144006i \(0.954001\pi\)
\(102\) 0 0
\(103\) −96.7077 −0.938909 −0.469455 0.882957i \(-0.655549\pi\)
−0.469455 + 0.882957i \(0.655549\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 177.582i 1.65964i 0.558030 + 0.829821i \(0.311558\pi\)
−0.558030 + 0.829821i \(0.688442\pi\)
\(108\) 0 0
\(109\) 61.9423 0.568278 0.284139 0.958783i \(-0.408292\pi\)
0.284139 + 0.958783i \(0.408292\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 109.811i − 0.971778i −0.874020 0.485889i \(-0.838496\pi\)
0.874020 0.485889i \(-0.161504\pi\)
\(114\) 0 0
\(115\) −85.1769 −0.740669
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 65.1626i − 0.547585i
\(120\) 0 0
\(121\) −95.0000 −0.785124
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 95.1140i − 0.760912i
\(126\) 0 0
\(127\) −141.177 −1.11163 −0.555815 0.831306i \(-0.687594\pi\)
−0.555815 + 0.831306i \(0.687594\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 53.1853i − 0.405995i −0.979179 0.202997i \(-0.934932\pi\)
0.979179 0.202997i \(-0.0650683\pi\)
\(132\) 0 0
\(133\) −204.708 −1.53916
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1.63453i − 0.0119309i −0.999982 0.00596545i \(-0.998101\pi\)
0.999982 0.00596545i \(-0.00189887\pi\)
\(138\) 0 0
\(139\) 19.2154 0.138240 0.0691201 0.997608i \(-0.477981\pi\)
0.0691201 + 0.997608i \(0.477981\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 311.519i − 2.17845i
\(144\) 0 0
\(145\) −205.708 −1.41867
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 94.2818i − 0.632764i −0.948632 0.316382i \(-0.897532\pi\)
0.948632 0.316382i \(-0.102468\pi\)
\(150\) 0 0
\(151\) −32.0000 −0.211921 −0.105960 0.994370i \(-0.533792\pi\)
−0.105960 + 0.994370i \(0.533792\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 46.3644i 0.299125i
\(156\) 0 0
\(157\) 0.292342 0.00186205 0.000931025 1.00000i \(-0.499704\pi\)
0.000931025 1.00000i \(0.499704\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 123.341i − 0.766094i
\(162\) 0 0
\(163\) −28.7846 −0.176593 −0.0882963 0.996094i \(-0.528142\pi\)
−0.0882963 + 0.996094i \(0.528142\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 56.0682i 0.335737i 0.985809 + 0.167869i \(0.0536885\pi\)
−0.985809 + 0.167869i \(0.946312\pi\)
\(168\) 0 0
\(169\) 280.277 1.65844
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 220.952i 1.27718i 0.769548 + 0.638589i \(0.220482\pi\)
−0.769548 + 0.638589i \(0.779518\pi\)
\(174\) 0 0
\(175\) −72.0770 −0.411868
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 248.347i − 1.38741i −0.720258 0.693706i \(-0.755977\pi\)
0.720258 0.693706i \(-0.244023\pi\)
\(180\) 0 0
\(181\) −158.277 −0.874458 −0.437229 0.899350i \(-0.644040\pi\)
−0.437229 + 0.899350i \(0.644040\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 351.032i 1.89747i
\(186\) 0 0
\(187\) 114.115 0.610243
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 34.5503i 0.180892i 0.995901 + 0.0904459i \(0.0288292\pi\)
−0.995901 + 0.0904459i \(0.971171\pi\)
\(192\) 0 0
\(193\) −55.0000 −0.284974 −0.142487 0.989797i \(-0.545510\pi\)
−0.142487 + 0.989797i \(0.545510\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 35.7170i − 0.181305i −0.995883 0.0906524i \(-0.971105\pi\)
0.995883 0.0906524i \(-0.0288952\pi\)
\(198\) 0 0
\(199\) 375.138 1.88512 0.942559 0.334040i \(-0.108412\pi\)
0.942559 + 0.334040i \(0.108412\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 297.877i − 1.46737i
\(204\) 0 0
\(205\) 194.942 0.950938
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 358.492i − 1.71527i
\(210\) 0 0
\(211\) −307.454 −1.45713 −0.728563 0.684978i \(-0.759812\pi\)
−0.728563 + 0.684978i \(0.759812\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 53.1853i 0.247374i
\(216\) 0 0
\(217\) −67.1384 −0.309394
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 164.579i 0.744702i
\(222\) 0 0
\(223\) 338.592 1.51835 0.759175 0.650886i \(-0.225602\pi\)
0.759175 + 0.650886i \(0.225602\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 123.950i 0.546037i 0.962009 + 0.273019i \(0.0880220\pi\)
−0.962009 + 0.273019i \(0.911978\pi\)
\(228\) 0 0
\(229\) 74.0577 0.323396 0.161698 0.986840i \(-0.448303\pi\)
0.161698 + 0.986840i \(0.448303\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 273.223i 1.17263i 0.810082 + 0.586316i \(0.199422\pi\)
−0.810082 + 0.586316i \(0.800578\pi\)
\(234\) 0 0
\(235\) −98.3538 −0.418527
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 452.885i 1.89492i 0.319877 + 0.947459i \(0.396359\pi\)
−0.319877 + 0.947459i \(0.603641\pi\)
\(240\) 0 0
\(241\) −382.688 −1.58792 −0.793959 0.607971i \(-0.791984\pi\)
−0.793959 + 0.607971i \(0.791984\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 124.203i − 0.506952i
\(246\) 0 0
\(247\) 517.023 2.09321
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 73.9307i 0.294544i 0.989096 + 0.147272i \(0.0470493\pi\)
−0.989096 + 0.147272i \(0.952951\pi\)
\(252\) 0 0
\(253\) 216.000 0.853755
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 159.809i − 0.621824i −0.950439 0.310912i \(-0.899366\pi\)
0.950439 0.310912i \(-0.100634\pi\)
\(258\) 0 0
\(259\) −508.315 −1.96261
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 259.106i − 0.985193i −0.870258 0.492596i \(-0.836048\pi\)
0.870258 0.492596i \(-0.163952\pi\)
\(264\) 0 0
\(265\) −149.296 −0.563382
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 24.8168i 0.0922556i 0.998936 + 0.0461278i \(0.0146881\pi\)
−0.998936 + 0.0461278i \(0.985312\pi\)
\(270\) 0 0
\(271\) −98.1154 −0.362050 −0.181025 0.983479i \(-0.557941\pi\)
−0.181025 + 0.983479i \(0.557941\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 126.224i − 0.458996i
\(276\) 0 0
\(277\) −1.41532 −0.00510945 −0.00255472 0.999997i \(-0.500813\pi\)
−0.00255472 + 0.999997i \(0.500813\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 100.716i − 0.358421i −0.983811 0.179211i \(-0.942646\pi\)
0.983811 0.179211i \(-0.0573544\pi\)
\(282\) 0 0
\(283\) 47.2923 0.167111 0.0835554 0.996503i \(-0.473372\pi\)
0.0835554 + 0.996503i \(0.473372\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 282.288i 0.983582i
\(288\) 0 0
\(289\) 228.711 0.791389
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 333.616i − 1.13862i −0.822123 0.569310i \(-0.807210\pi\)
0.822123 0.569310i \(-0.192790\pi\)
\(294\) 0 0
\(295\) 357.415 1.21158
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 311.519i 1.04187i
\(300\) 0 0
\(301\) −77.0155 −0.255865
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 75.3422i 0.247024i
\(306\) 0 0
\(307\) −10.3538 −0.0337258 −0.0168629 0.999858i \(-0.505368\pi\)
−0.0168629 + 0.999858i \(0.505368\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 9.54047i 0.0306768i 0.999882 + 0.0153384i \(0.00488255\pi\)
−0.999882 + 0.0153384i \(0.995117\pi\)
\(312\) 0 0
\(313\) −479.277 −1.53124 −0.765618 0.643295i \(-0.777567\pi\)
−0.765618 + 0.643295i \(0.777567\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 179.662i 0.566758i 0.959008 + 0.283379i \(0.0914554\pi\)
−0.959008 + 0.283379i \(0.908545\pi\)
\(318\) 0 0
\(319\) 521.654 1.63528
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 189.396i 0.586365i
\(324\) 0 0
\(325\) 182.042 0.560130
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 142.422i − 0.432894i
\(330\) 0 0
\(331\) −295.454 −0.892610 −0.446305 0.894881i \(-0.647260\pi\)
−0.446305 + 0.894881i \(0.647260\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 122.732i 0.366364i
\(336\) 0 0
\(337\) 489.261 1.45181 0.725907 0.687793i \(-0.241420\pi\)
0.725907 + 0.687793i \(0.241420\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 117.576i − 0.344796i
\(342\) 0 0
\(343\) −231.369 −0.674546
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 666.682i − 1.92127i −0.277807 0.960637i \(-0.589608\pi\)
0.277807 0.960637i \(-0.410392\pi\)
\(348\) 0 0
\(349\) 511.969 1.46696 0.733480 0.679711i \(-0.237895\pi\)
0.733480 + 0.679711i \(0.237895\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 586.681i 1.66199i 0.556283 + 0.830993i \(0.312227\pi\)
−0.556283 + 0.830993i \(0.687773\pi\)
\(354\) 0 0
\(355\) −586.592 −1.65237
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 534.573i 1.48906i 0.667589 + 0.744530i \(0.267326\pi\)
−0.667589 + 0.744530i \(0.732674\pi\)
\(360\) 0 0
\(361\) 233.985 0.648157
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 234.207i − 0.641664i
\(366\) 0 0
\(367\) −15.2923 −0.0416685 −0.0208343 0.999783i \(-0.506632\pi\)
−0.0208343 + 0.999783i \(0.506632\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 216.190i − 0.582721i
\(372\) 0 0
\(373\) 652.985 1.75063 0.875314 0.483554i \(-0.160654\pi\)
0.875314 + 0.483554i \(0.160654\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 752.337i 1.99559i
\(378\) 0 0
\(379\) 655.215 1.72880 0.864400 0.502804i \(-0.167698\pi\)
0.864400 + 0.502804i \(0.167698\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 299.541i 0.782092i 0.920371 + 0.391046i \(0.127887\pi\)
−0.920371 + 0.391046i \(0.872113\pi\)
\(384\) 0 0
\(385\) 714.831 1.85670
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 464.558i − 1.19424i −0.802153 0.597119i \(-0.796312\pi\)
0.802153 0.597119i \(-0.203688\pi\)
\(390\) 0 0
\(391\) −114.115 −0.291855
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 572.289i 1.44883i
\(396\) 0 0
\(397\) −185.708 −0.467777 −0.233889 0.972263i \(-0.575145\pi\)
−0.233889 + 0.972263i \(0.575145\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 62.8592i − 0.156756i −0.996924 0.0783780i \(-0.975026\pi\)
0.996924 0.0783780i \(-0.0249741\pi\)
\(402\) 0 0
\(403\) 169.569 0.420767
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 890.182i − 2.18718i
\(408\) 0 0
\(409\) −327.281 −0.800197 −0.400099 0.916472i \(-0.631024\pi\)
−0.400099 + 0.916472i \(0.631024\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 517.558i 1.25317i
\(414\) 0 0
\(415\) 599.769 1.44523
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 649.875i − 1.55101i −0.631339 0.775507i \(-0.717494\pi\)
0.631339 0.775507i \(-0.282506\pi\)
\(420\) 0 0
\(421\) 3.31913 0.00788391 0.00394196 0.999992i \(-0.498745\pi\)
0.00394196 + 0.999992i \(0.498745\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 66.6857i 0.156908i
\(426\) 0 0
\(427\) −109.100 −0.255503
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 803.502i − 1.86427i −0.362107 0.932136i \(-0.617943\pi\)
0.362107 0.932136i \(-0.382057\pi\)
\(432\) 0 0
\(433\) −93.1230 −0.215065 −0.107532 0.994202i \(-0.534295\pi\)
−0.107532 + 0.994202i \(0.534295\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 358.492i 0.820348i
\(438\) 0 0
\(439\) 472.000 1.07517 0.537585 0.843209i \(-0.319337\pi\)
0.537585 + 0.843209i \(0.319337\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 569.689i − 1.28598i −0.765875 0.642989i \(-0.777694\pi\)
0.765875 0.642989i \(-0.222306\pi\)
\(444\) 0 0
\(445\) −777.358 −1.74687
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 559.115i − 1.24524i −0.782523 0.622622i \(-0.786067\pi\)
0.782523 0.622622i \(-0.213933\pi\)
\(450\) 0 0
\(451\) −494.354 −1.09613
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 1030.94i 2.26580i
\(456\) 0 0
\(457\) 604.296 1.32231 0.661155 0.750249i \(-0.270066\pi\)
0.661155 + 0.750249i \(0.270066\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) − 5.13459i − 0.0111379i −0.999984 0.00556897i \(-0.998227\pi\)
0.999984 0.00556897i \(-0.00177267\pi\)
\(462\) 0 0
\(463\) 161.492 0.348795 0.174398 0.984675i \(-0.444202\pi\)
0.174398 + 0.984675i \(0.444202\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 503.025i 1.07714i 0.842581 + 0.538570i \(0.181035\pi\)
−0.842581 + 0.538570i \(0.818965\pi\)
\(468\) 0 0
\(469\) −177.723 −0.378941
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 134.873i − 0.285143i
\(474\) 0 0
\(475\) 209.492 0.441036
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 213.796i − 0.446339i −0.974780 0.223170i \(-0.928360\pi\)
0.974780 0.223170i \(-0.0716404\pi\)
\(480\) 0 0
\(481\) 1283.83 2.66909
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 435.469i 0.897874i
\(486\) 0 0
\(487\) −8.63071 −0.0177222 −0.00886110 0.999961i \(-0.502821\pi\)
−0.00886110 + 0.999961i \(0.502821\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 922.459i − 1.87873i −0.342912 0.939367i \(-0.611413\pi\)
0.342912 0.939367i \(-0.388587\pi\)
\(492\) 0 0
\(493\) −275.596 −0.559018
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 849.420i − 1.70909i
\(498\) 0 0
\(499\) 434.592 0.870926 0.435463 0.900207i \(-0.356585\pi\)
0.435463 + 0.900207i \(0.356585\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 144.087i − 0.286454i −0.989690 0.143227i \(-0.954252\pi\)
0.989690 0.143227i \(-0.0457480\pi\)
\(504\) 0 0
\(505\) 168.588 0.333839
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 697.272i − 1.36989i −0.728596 0.684943i \(-0.759827\pi\)
0.728596 0.684943i \(-0.240173\pi\)
\(510\) 0 0
\(511\) 339.146 0.663691
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 560.475i 1.08830i
\(516\) 0 0
\(517\) 249.415 0.482428
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 426.962i − 0.819504i −0.912197 0.409752i \(-0.865615\pi\)
0.912197 0.409752i \(-0.134385\pi\)
\(522\) 0 0
\(523\) −179.762 −0.343712 −0.171856 0.985122i \(-0.554976\pi\)
−0.171856 + 0.985122i \(0.554976\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 62.1166i 0.117868i
\(528\) 0 0
\(529\) 313.000 0.591682
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 712.965i − 1.33764i
\(534\) 0 0
\(535\) 1029.18 1.92371
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 314.967i 0.584354i
\(540\) 0 0
\(541\) 708.734 1.31005 0.655023 0.755609i \(-0.272659\pi\)
0.655023 + 0.755609i \(0.272659\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 358.990i − 0.658697i
\(546\) 0 0
\(547\) 196.231 0.358740 0.179370 0.983782i \(-0.442594\pi\)
0.179370 + 0.983782i \(0.442594\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 865.781i 1.57129i
\(552\) 0 0
\(553\) −828.708 −1.49857
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 353.610i 0.634848i 0.948284 + 0.317424i \(0.102818\pi\)
−0.948284 + 0.317424i \(0.897182\pi\)
\(558\) 0 0
\(559\) 194.515 0.347970
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 373.026i − 0.662568i −0.943531 0.331284i \(-0.892518\pi\)
0.943531 0.331284i \(-0.107482\pi\)
\(564\) 0 0
\(565\) −636.415 −1.12640
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 60.1177i 0.105655i 0.998604 + 0.0528275i \(0.0168233\pi\)
−0.998604 + 0.0528275i \(0.983177\pi\)
\(570\) 0 0
\(571\) −86.1999 −0.150963 −0.0754815 0.997147i \(-0.524049\pi\)
−0.0754815 + 0.997147i \(0.524049\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 126.224i 0.219520i
\(576\) 0 0
\(577\) 709.123 1.22898 0.614491 0.788924i \(-0.289361\pi\)
0.614491 + 0.788924i \(0.289361\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 868.501i 1.49484i
\(582\) 0 0
\(583\) 378.600 0.649399
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 962.285i 1.63933i 0.572845 + 0.819664i \(0.305840\pi\)
−0.572845 + 0.819664i \(0.694160\pi\)
\(588\) 0 0
\(589\) 195.138 0.331305
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 104.350i − 0.175969i −0.996122 0.0879847i \(-0.971957\pi\)
0.996122 0.0879847i \(-0.0280427\pi\)
\(594\) 0 0
\(595\) −377.654 −0.634712
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 286.063i − 0.477567i −0.971073 0.238784i \(-0.923251\pi\)
0.971073 0.238784i \(-0.0767486\pi\)
\(600\) 0 0
\(601\) −280.415 −0.466581 −0.233291 0.972407i \(-0.574949\pi\)
−0.233291 + 0.972407i \(0.574949\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 550.578i 0.910046i
\(606\) 0 0
\(607\) −737.731 −1.21537 −0.607686 0.794177i \(-0.707902\pi\)
−0.607686 + 0.794177i \(0.707902\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 359.711i 0.588724i
\(612\) 0 0
\(613\) −679.415 −1.10834 −0.554172 0.832402i \(-0.686965\pi\)
−0.554172 + 0.832402i \(0.686965\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 472.649i − 0.766044i −0.923739 0.383022i \(-0.874883\pi\)
0.923739 0.383022i \(-0.125117\pi\)
\(618\) 0 0
\(619\) −886.354 −1.43191 −0.715956 0.698145i \(-0.754009\pi\)
−0.715956 + 0.698145i \(0.754009\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) − 1125.66i − 1.80684i
\(624\) 0 0
\(625\) −765.950 −1.22552
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 470.294i 0.747685i
\(630\) 0 0
\(631\) 729.108 1.15548 0.577740 0.816221i \(-0.303935\pi\)
0.577740 + 0.816221i \(0.303935\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 818.199i 1.28850i
\(636\) 0 0
\(637\) −454.250 −0.713108
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 870.195i 1.35756i 0.734342 + 0.678780i \(0.237491\pi\)
−0.734342 + 0.678780i \(0.762509\pi\)
\(642\) 0 0
\(643\) 618.123 0.961311 0.480656 0.876910i \(-0.340399\pi\)
0.480656 + 0.876910i \(0.340399\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 425.439i − 0.657556i −0.944407 0.328778i \(-0.893363\pi\)
0.944407 0.328778i \(-0.106637\pi\)
\(648\) 0 0
\(649\) −906.369 −1.39656
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 188.808i 0.289140i 0.989495 + 0.144570i \(0.0461799\pi\)
−0.989495 + 0.144570i \(0.953820\pi\)
\(654\) 0 0
\(655\) −308.238 −0.470593
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 660.470i 1.00223i 0.865380 + 0.501116i \(0.167077\pi\)
−0.865380 + 0.501116i \(0.832923\pi\)
\(660\) 0 0
\(661\) −589.831 −0.892331 −0.446165 0.894951i \(-0.647211\pi\)
−0.446165 + 0.894951i \(0.647211\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1186.39i 1.78405i
\(666\) 0 0
\(667\) −521.654 −0.782090
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 191.060i − 0.284739i
\(672\) 0 0
\(673\) −391.431 −0.581621 −0.290810 0.956781i \(-0.593925\pi\)
−0.290810 + 0.956781i \(0.593925\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 693.661i − 1.02461i −0.858804 0.512305i \(-0.828792\pi\)
0.858804 0.512305i \(-0.171208\pi\)
\(678\) 0 0
\(679\) −630.585 −0.928696
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 678.170i − 0.992928i −0.868057 0.496464i \(-0.834632\pi\)
0.868057 0.496464i \(-0.165368\pi\)
\(684\) 0 0
\(685\) −9.47303 −0.0138292
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 546.022i 0.792485i
\(690\) 0 0
\(691\) −675.777 −0.977969 −0.488985 0.872292i \(-0.662633\pi\)
−0.488985 + 0.872292i \(0.662633\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 111.364i − 0.160236i
\(696\) 0 0
\(697\) 261.173 0.374710
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 797.260i − 1.13732i −0.822573 0.568659i \(-0.807462\pi\)
0.822573 0.568659i \(-0.192538\pi\)
\(702\) 0 0
\(703\) 1477.42 2.10160
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 244.126i 0.345298i
\(708\) 0 0
\(709\) 173.504 0.244716 0.122358 0.992486i \(-0.460954\pi\)
0.122358 + 0.992486i \(0.460954\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 117.576i 0.164903i
\(714\) 0 0
\(715\) −1805.42 −2.52507
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 188.177i 0.261721i 0.991401 + 0.130860i \(0.0417740\pi\)
−0.991401 + 0.130860i \(0.958226\pi\)
\(720\) 0 0
\(721\) −811.600 −1.12566
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 304.839i 0.420468i
\(726\) 0 0
\(727\) −711.777 −0.979060 −0.489530 0.871986i \(-0.662832\pi\)
−0.489530 + 0.871986i \(0.662832\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 71.2548i 0.0974758i
\(732\) 0 0
\(733\) 1226.83 1.67371 0.836856 0.547423i \(-0.184391\pi\)
0.836856 + 0.547423i \(0.184391\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 311.236i − 0.422301i
\(738\) 0 0
\(739\) −741.892 −1.00391 −0.501957 0.864893i \(-0.667386\pi\)
−0.501957 + 0.864893i \(0.667386\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 781.984i 1.05247i 0.850340 + 0.526234i \(0.176396\pi\)
−0.850340 + 0.526234i \(0.823604\pi\)
\(744\) 0 0
\(745\) −546.415 −0.733443
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 1490.32i 1.98975i
\(750\) 0 0
\(751\) 1033.33 1.37594 0.687970 0.725739i \(-0.258502\pi\)
0.687970 + 0.725739i \(0.258502\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 185.458i 0.245639i
\(756\) 0 0
\(757\) 473.877 0.625993 0.312997 0.949754i \(-0.398667\pi\)
0.312997 + 0.949754i \(0.398667\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 940.455i 1.23581i 0.786251 + 0.617907i \(0.212019\pi\)
−0.786251 + 0.617907i \(0.787981\pi\)
\(762\) 0 0
\(763\) 519.839 0.681309
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 1307.18i − 1.70428i
\(768\) 0 0
\(769\) 980.815 1.27544 0.637721 0.770267i \(-0.279877\pi\)
0.637721 + 0.770267i \(0.279877\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 1042.20i − 1.34825i −0.738618 0.674124i \(-0.764521\pi\)
0.738618 0.674124i \(-0.235479\pi\)
\(774\) 0 0
\(775\) 68.7077 0.0886550
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 820.472i − 1.05324i
\(780\) 0 0
\(781\) 1487.54 1.90466
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 1.69428i − 0.00215832i
\(786\) 0 0
\(787\) 144.700 0.183863 0.0919315 0.995765i \(-0.470696\pi\)
0.0919315 + 0.995765i \(0.470696\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 921.567i − 1.16507i
\(792\) 0 0
\(793\) 275.550 0.347478
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 155.729i − 0.195394i −0.995216 0.0976972i \(-0.968852\pi\)
0.995216 0.0976972i \(-0.0311477\pi\)
\(798\) 0 0
\(799\) −131.769 −0.164918
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 593.926i 0.739634i
\(804\) 0 0
\(805\) −714.831 −0.887988
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 1216.92i − 1.50422i −0.659035 0.752112i \(-0.729035\pi\)
0.659035 0.752112i \(-0.270965\pi\)
\(810\) 0 0
\(811\) −1243.31 −1.53305 −0.766527 0.642212i \(-0.778017\pi\)
−0.766527 + 0.642212i \(0.778017\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 166.823i 0.204691i
\(816\) 0 0
\(817\) 223.846 0.273985
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.2854i 0.0161820i 0.999967 + 0.00809098i \(0.00257547\pi\)
−0.999967 + 0.00809098i \(0.997425\pi\)
\(822\) 0 0
\(823\) 214.200 0.260267 0.130134 0.991496i \(-0.458459\pi\)
0.130134 + 0.991496i \(0.458459\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1221.27i − 1.47675i −0.674391 0.738374i \(-0.735594\pi\)
0.674391 0.738374i \(-0.264406\pi\)
\(828\) 0 0
\(829\) −358.431 −0.432365 −0.216183 0.976353i \(-0.569361\pi\)
−0.216183 + 0.976353i \(0.569361\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 166.401i − 0.199761i
\(834\) 0 0
\(835\) 324.946 0.389157
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 327.597i 0.390461i 0.980757 + 0.195231i \(0.0625456\pi\)
−0.980757 + 0.195231i \(0.937454\pi\)
\(840\) 0 0
\(841\) −418.827 −0.498011
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 1624.36i − 1.92232i
\(846\) 0 0
\(847\) −797.269 −0.941286
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 890.182i 1.04604i
\(852\) 0 0
\(853\) −52.8306 −0.0619351 −0.0309675 0.999520i \(-0.509859\pi\)
−0.0309675 + 0.999520i \(0.509859\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 1248.42i − 1.45673i −0.685187 0.728367i \(-0.740280\pi\)
0.685187 0.728367i \(-0.259720\pi\)
\(858\) 0 0
\(859\) 446.739 0.520068 0.260034 0.965599i \(-0.416266\pi\)
0.260034 + 0.965599i \(0.416266\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 635.297i − 0.736150i −0.929796 0.368075i \(-0.880017\pi\)
0.929796 0.368075i \(-0.119983\pi\)
\(864\) 0 0
\(865\) 1280.54 1.48039
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 1451.27i − 1.67004i
\(870\) 0 0
\(871\) 448.869 0.515349
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 798.226i − 0.912258i
\(876\) 0 0
\(877\) −788.692 −0.899307 −0.449653 0.893203i \(-0.648453\pi\)
−0.449653 + 0.893203i \(0.648453\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 558.506i − 0.633945i −0.948435 0.316972i \(-0.897334\pi\)
0.948435 0.316972i \(-0.102666\pi\)
\(882\) 0 0
\(883\) −313.338 −0.354857 −0.177428 0.984134i \(-0.556778\pi\)
−0.177428 + 0.984134i \(0.556778\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 597.701i 0.673845i 0.941532 + 0.336923i \(0.109386\pi\)
−0.941532 + 0.336923i \(0.890614\pi\)
\(888\) 0 0
\(889\) −1184.80 −1.33273
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 413.951i 0.463551i
\(894\) 0 0
\(895\) −1439.31 −1.60816
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 283.952i 0.315854i
\(900\) 0 0
\(901\) −200.019 −0.221997
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 917.302i 1.01359i
\(906\) 0 0
\(907\) −1086.51 −1.19791 −0.598957 0.800781i \(-0.704418\pi\)
−0.598957 + 0.800781i \(0.704418\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 710.283i 0.779674i 0.920884 + 0.389837i \(0.127469\pi\)
−0.920884 + 0.389837i \(0.872531\pi\)
\(912\) 0 0
\(913\) −1520.95 −1.66589
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 446.347i − 0.486747i
\(918\) 0 0
\(919\) −1540.01 −1.67574 −0.837871 0.545868i \(-0.816200\pi\)
−0.837871 + 0.545868i \(0.816200\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2145.35i 2.32432i
\(924\) 0 0
\(925\) 520.196 0.562374
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1582.59i 1.70355i 0.523911 + 0.851773i \(0.324473\pi\)
−0.523911 + 0.851773i \(0.675527\pi\)
\(930\) 0 0
\(931\) −522.746 −0.561489
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 661.362i − 0.707339i
\(936\) 0 0
\(937\) −1747.40 −1.86489 −0.932444 0.361315i \(-0.882328\pi\)
−0.932444 + 0.361315i \(0.882328\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 367.579i 0.390626i 0.980741 + 0.195313i \(0.0625722\pi\)
−0.980741 + 0.195313i \(0.937428\pi\)
\(942\) 0 0
\(943\) 494.354 0.524235
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 190.168i − 0.200811i −0.994947 0.100406i \(-0.967986\pi\)
0.994947 0.100406i \(-0.0320140\pi\)
\(948\) 0 0
\(949\) −856.569 −0.902602
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 861.754i 0.904254i 0.891954 + 0.452127i \(0.149335\pi\)
−0.891954 + 0.452127i \(0.850665\pi\)
\(954\) 0 0
\(955\) 200.238 0.209674
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 13.7175i − 0.0143040i
\(960\) 0 0
\(961\) −897.000 −0.933403
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 318.756i 0.330317i
\(966\) 0 0
\(967\) −1275.31 −1.31884 −0.659418 0.751776i \(-0.729197\pi\)
−0.659418 + 0.751776i \(0.729197\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1238.69i 1.27568i 0.770168 + 0.637841i \(0.220172\pi\)
−0.770168 + 0.637841i \(0.779828\pi\)
\(972\) 0 0
\(973\) 161.261 0.165736
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 18.4936i − 0.0189290i −0.999955 0.00946448i \(-0.996987\pi\)
0.999955 0.00946448i \(-0.00301268\pi\)
\(978\) 0 0
\(979\) 1971.30 2.01359
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 658.643i 0.670033i 0.942212 + 0.335017i \(0.108742\pi\)
−0.942212 + 0.335017i \(0.891258\pi\)
\(984\) 0 0
\(985\) −207.000 −0.210152
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 134.873i 0.136373i
\(990\) 0 0
\(991\) −608.484 −0.614010 −0.307005 0.951708i \(-0.599327\pi\)
−0.307005 + 0.951708i \(0.599327\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 2174.14i − 2.18506i
\(996\) 0 0
\(997\) −617.246 −0.619103 −0.309552 0.950883i \(-0.600179\pi\)
−0.309552 + 0.950883i \(0.600179\pi\)
\(998\) 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 1296.3.e.d.161.1 4
3.2 odd 2 inner 1296.3.e.d.161.4 4
4.3 odd 2 162.3.b.b.161.3 yes 4
9.2 odd 6 1296.3.q.o.1025.1 8
9.4 even 3 1296.3.q.o.593.1 8
9.5 odd 6 1296.3.q.o.593.4 8
9.7 even 3 1296.3.q.o.1025.4 8
12.11 even 2 162.3.b.b.161.2 4
36.7 odd 6 162.3.d.c.53.4 8
36.11 even 6 162.3.d.c.53.1 8
36.23 even 6 162.3.d.c.107.4 8
36.31 odd 6 162.3.d.c.107.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
162.3.b.b.161.2 4 12.11 even 2
162.3.b.b.161.3 yes 4 4.3 odd 2
162.3.d.c.53.1 8 36.11 even 6
162.3.d.c.53.4 8 36.7 odd 6
162.3.d.c.107.1 8 36.31 odd 6
162.3.d.c.107.4 8 36.23 even 6
1296.3.e.d.161.1 4 1.1 even 1 trivial
1296.3.e.d.161.4 4 3.2 odd 2 inner
1296.3.q.o.593.1 8 9.4 even 3
1296.3.q.o.593.4 8 9.5 odd 6
1296.3.q.o.1025.1 8 9.2 odd 6
1296.3.q.o.1025.4 8 9.7 even 3