Properties

Label 1728.3.h.h.161.3
Level $1728$
Weight $3$
Character 1728.161
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1728.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.12960000.1
Defining polynomial: \( x^{8} - 3x^{6} + 8x^{4} - 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 161.3
Root \(1.40126 - 0.809017i\) of defining polynomial
Character \(\chi\) \(=\) 1728.161
Dual form 1728.3.h.h.161.4

$q$-expansion

\(f(q)\) \(=\) \(q-7.74597 q^{5} +8.66025 q^{7} +O(q^{10})\) \(q-7.74597 q^{5} +8.66025 q^{7} -13.4164 q^{11} -5.19615i q^{13} -13.4164i q^{17} +23.0000i q^{19} +7.74597i q^{23} +35.0000 q^{25} -30.9839 q^{29} +6.92820 q^{31} -67.0820 q^{35} +29.4449i q^{37} -80.4984i q^{41} -38.0000i q^{43} +54.2218i q^{47} +26.0000 q^{49} +77.4597 q^{53} +103.923 q^{55} +93.9149 q^{59} +60.6218i q^{61} +40.2492i q^{65} -107.000i q^{67} +15.4919i q^{71} -97.0000 q^{73} -116.190 q^{77} -67.5500 q^{79} +103.923i q^{85} +174.413i q^{89} -45.0000i q^{91} -178.157i q^{95} +109.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 280 q^{25} + 208 q^{49} - 776 q^{73} + 872 q^{97}+O(q^{100}) \) 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
\(5\) −7.74597 −1.54919 −0.774597 0.632456i \(-0.782047\pi\)
−0.774597 + 0.632456i \(0.782047\pi\)
\(6\) 0 0
\(7\) 8.66025 1.23718 0.618590 0.785714i \(-0.287704\pi\)
0.618590 + 0.785714i \(0.287704\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −13.4164 −1.21967 −0.609837 0.792527i \(-0.708765\pi\)
−0.609837 + 0.792527i \(0.708765\pi\)
\(12\) 0 0
\(13\) − 5.19615i − 0.399704i −0.979826 0.199852i \(-0.935954\pi\)
0.979826 0.199852i \(-0.0640461\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 13.4164i − 0.789200i −0.918853 0.394600i \(-0.870883\pi\)
0.918853 0.394600i \(-0.129117\pi\)
\(18\) 0 0
\(19\) 23.0000i 1.21053i 0.796025 + 0.605263i \(0.206932\pi\)
−0.796025 + 0.605263i \(0.793068\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.74597i 0.336781i 0.985720 + 0.168391i \(0.0538570\pi\)
−0.985720 + 0.168391i \(0.946143\pi\)
\(24\) 0 0
\(25\) 35.0000 1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −30.9839 −1.06841 −0.534205 0.845355i \(-0.679389\pi\)
−0.534205 + 0.845355i \(0.679389\pi\)
\(30\) 0 0
\(31\) 6.92820 0.223490 0.111745 0.993737i \(-0.464356\pi\)
0.111745 + 0.993737i \(0.464356\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −67.0820 −1.91663
\(36\) 0 0
\(37\) 29.4449i 0.795807i 0.917427 + 0.397904i \(0.130262\pi\)
−0.917427 + 0.397904i \(0.869738\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 80.4984i − 1.96338i −0.190494 0.981688i \(-0.561009\pi\)
0.190494 0.981688i \(-0.438991\pi\)
\(42\) 0 0
\(43\) − 38.0000i − 0.883721i −0.897084 0.441860i \(-0.854319\pi\)
0.897084 0.441860i \(-0.145681\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 54.2218i 1.15365i 0.816866 + 0.576827i \(0.195709\pi\)
−0.816866 + 0.576827i \(0.804291\pi\)
\(48\) 0 0
\(49\) 26.0000 0.530612
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 77.4597 1.46150 0.730752 0.682643i \(-0.239170\pi\)
0.730752 + 0.682643i \(0.239170\pi\)
\(54\) 0 0
\(55\) 103.923 1.88951
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 93.9149 1.59178 0.795889 0.605443i \(-0.207004\pi\)
0.795889 + 0.605443i \(0.207004\pi\)
\(60\) 0 0
\(61\) 60.6218i 0.993800i 0.867808 + 0.496900i \(0.165528\pi\)
−0.867808 + 0.496900i \(0.834472\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 40.2492i 0.619219i
\(66\) 0 0
\(67\) − 107.000i − 1.59701i −0.601985 0.798507i \(-0.705623\pi\)
0.601985 0.798507i \(-0.294377\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 15.4919i 0.218196i 0.994031 + 0.109098i \(0.0347963\pi\)
−0.994031 + 0.109098i \(0.965204\pi\)
\(72\) 0 0
\(73\) −97.0000 −1.32877 −0.664384 0.747392i \(-0.731306\pi\)
−0.664384 + 0.747392i \(0.731306\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −116.190 −1.50895
\(78\) 0 0
\(79\) −67.5500 −0.855063 −0.427532 0.904000i \(-0.640617\pi\)
−0.427532 + 0.904000i \(0.640617\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 103.923i 1.22262i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 174.413i 1.95970i 0.199735 + 0.979850i \(0.435992\pi\)
−0.199735 + 0.979850i \(0.564008\pi\)
\(90\) 0 0
\(91\) − 45.0000i − 0.494505i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) − 178.157i − 1.87534i
\(96\) 0 0
\(97\) 109.000 1.12371 0.561856 0.827235i \(-0.310088\pi\)
0.561856 + 0.827235i \(0.310088\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 170.411 1.68724 0.843620 0.536940i \(-0.180420\pi\)
0.843620 + 0.536940i \(0.180420\pi\)
\(102\) 0 0
\(103\) 185.329 1.79931 0.899657 0.436596i \(-0.143816\pi\)
0.899657 + 0.436596i \(0.143816\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 13.4164 0.125387 0.0626935 0.998033i \(-0.480031\pi\)
0.0626935 + 0.998033i \(0.480031\pi\)
\(108\) 0 0
\(109\) 200.918i 1.84328i 0.388042 + 0.921642i \(0.373152\pi\)
−0.388042 + 0.921642i \(0.626848\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 13.4164i − 0.118729i −0.998236 0.0593646i \(-0.981093\pi\)
0.998236 0.0593646i \(-0.0189075\pi\)
\(114\) 0 0
\(115\) − 60.0000i − 0.521739i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 116.190i − 0.976382i
\(120\) 0 0
\(121\) 59.0000 0.487603
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −77.4597 −0.619677
\(126\) 0 0
\(127\) 138.564 1.09106 0.545528 0.838093i \(-0.316329\pi\)
0.545528 + 0.838093i \(0.316329\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 214.663 1.63865 0.819323 0.573333i \(-0.194350\pi\)
0.819323 + 0.573333i \(0.194350\pi\)
\(132\) 0 0
\(133\) 199.186i 1.49764i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 120.748i 0.881370i 0.897662 + 0.440685i \(0.145264\pi\)
−0.897662 + 0.440685i \(0.854736\pi\)
\(138\) 0 0
\(139\) − 49.0000i − 0.352518i −0.984344 0.176259i \(-0.943600\pi\)
0.984344 0.176259i \(-0.0563996\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 69.7137i 0.487508i
\(144\) 0 0
\(145\) 240.000 1.65517
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 139.427 0.935754 0.467877 0.883793i \(-0.345019\pi\)
0.467877 + 0.883793i \(0.345019\pi\)
\(150\) 0 0
\(151\) 164.545 1.08970 0.544850 0.838533i \(-0.316586\pi\)
0.544850 + 0.838533i \(0.316586\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −53.6656 −0.346230
\(156\) 0 0
\(157\) − 159.349i − 1.01496i −0.861664 0.507480i \(-0.830577\pi\)
0.861664 0.507480i \(-0.169423\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 67.0820i 0.416659i
\(162\) 0 0
\(163\) − 97.0000i − 0.595092i −0.954707 0.297546i \(-0.903832\pi\)
0.954707 0.297546i \(-0.0961682\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 38.7298i 0.231915i 0.993254 + 0.115958i \(0.0369937\pi\)
−0.993254 + 0.115958i \(0.963006\pi\)
\(168\) 0 0
\(169\) 142.000 0.840237
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −77.4597 −0.447744 −0.223872 0.974619i \(-0.571870\pi\)
−0.223872 + 0.974619i \(0.571870\pi\)
\(174\) 0 0
\(175\) 303.109 1.73205
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 26.8328 0.149904 0.0749520 0.997187i \(-0.476120\pi\)
0.0749520 + 0.997187i \(0.476120\pi\)
\(180\) 0 0
\(181\) − 174.937i − 0.966503i −0.875481 0.483252i \(-0.839456\pi\)
0.875481 0.483252i \(-0.160544\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 228.079i − 1.23286i
\(186\) 0 0
\(187\) 180.000i 0.962567i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 178.157i 0.932760i 0.884584 + 0.466380i \(0.154442\pi\)
−0.884584 + 0.466380i \(0.845558\pi\)
\(192\) 0 0
\(193\) 59.0000 0.305699 0.152850 0.988249i \(-0.451155\pi\)
0.152850 + 0.988249i \(0.451155\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 23.2379 0.117959 0.0589794 0.998259i \(-0.481215\pi\)
0.0589794 + 0.998259i \(0.481215\pi\)
\(198\) 0 0
\(199\) 88.3346 0.443892 0.221946 0.975059i \(-0.428759\pi\)
0.221946 + 0.975059i \(0.428759\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −268.328 −1.32181
\(204\) 0 0
\(205\) 623.538i 3.04165i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 308.577i − 1.47645i
\(210\) 0 0
\(211\) 1.00000i 0.00473934i 0.999997 + 0.00236967i \(0.000754290\pi\)
−0.999997 + 0.00236967i \(0.999246\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 294.347i 1.36905i
\(216\) 0 0
\(217\) 60.0000 0.276498
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −69.7137 −0.315447
\(222\) 0 0
\(223\) −48.4974 −0.217477 −0.108739 0.994070i \(-0.534681\pi\)
−0.108739 + 0.994070i \(0.534681\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −80.4984 −0.354619 −0.177309 0.984155i \(-0.556739\pi\)
−0.177309 + 0.984155i \(0.556739\pi\)
\(228\) 0 0
\(229\) 235.559i 1.02864i 0.857598 + 0.514321i \(0.171956\pi\)
−0.857598 + 0.514321i \(0.828044\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 295.161i − 1.26679i −0.773831 0.633393i \(-0.781662\pi\)
0.773831 0.633393i \(-0.218338\pi\)
\(234\) 0 0
\(235\) − 420.000i − 1.78723i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 154.919i 0.648198i 0.946023 + 0.324099i \(0.105061\pi\)
−0.946023 + 0.324099i \(0.894939\pi\)
\(240\) 0 0
\(241\) 299.000 1.24066 0.620332 0.784339i \(-0.286998\pi\)
0.620332 + 0.784339i \(0.286998\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −201.395 −0.822021
\(246\) 0 0
\(247\) 119.512 0.483852
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −134.164 −0.534518 −0.267259 0.963625i \(-0.586118\pi\)
−0.267259 + 0.963625i \(0.586118\pi\)
\(252\) 0 0
\(253\) − 103.923i − 0.410763i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 348.827i 1.35730i 0.734461 + 0.678651i \(0.237435\pi\)
−0.734461 + 0.678651i \(0.762565\pi\)
\(258\) 0 0
\(259\) 255.000i 0.984556i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 464.758i − 1.76714i −0.468298 0.883570i \(-0.655133\pi\)
0.468298 0.883570i \(-0.344867\pi\)
\(264\) 0 0
\(265\) −600.000 −2.26415
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −162.665 −0.604704 −0.302352 0.953196i \(-0.597772\pi\)
−0.302352 + 0.953196i \(0.597772\pi\)
\(270\) 0 0
\(271\) 226.899 0.837264 0.418632 0.908156i \(-0.362510\pi\)
0.418632 + 0.908156i \(0.362510\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −469.574 −1.70754
\(276\) 0 0
\(277\) 297.913i 1.07550i 0.843105 + 0.537749i \(0.180725\pi\)
−0.843105 + 0.537749i \(0.819275\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 53.6656i − 0.190981i −0.995430 0.0954904i \(-0.969558\pi\)
0.995430 0.0954904i \(-0.0304419\pi\)
\(282\) 0 0
\(283\) − 442.000i − 1.56184i −0.624633 0.780919i \(-0.714751\pi\)
0.624633 0.780919i \(-0.285249\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 697.137i − 2.42905i
\(288\) 0 0
\(289\) 109.000 0.377163
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −85.2056 −0.290804 −0.145402 0.989373i \(-0.546448\pi\)
−0.145402 + 0.989373i \(0.546448\pi\)
\(294\) 0 0
\(295\) −727.461 −2.46597
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 40.2492 0.134613
\(300\) 0 0
\(301\) − 329.090i − 1.09332i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 469.574i − 1.53959i
\(306\) 0 0
\(307\) − 302.000i − 0.983713i −0.870676 0.491857i \(-0.836318\pi\)
0.870676 0.491857i \(-0.163682\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 596.439i 1.91781i 0.283724 + 0.958906i \(0.408430\pi\)
−0.283724 + 0.958906i \(0.591570\pi\)
\(312\) 0 0
\(313\) −563.000 −1.79872 −0.899361 0.437207i \(-0.855968\pi\)
−0.899361 + 0.437207i \(0.855968\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 415.692 1.30311
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 308.577 0.955348
\(324\) 0 0
\(325\) − 181.865i − 0.559586i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 469.574i 1.42728i
\(330\) 0 0
\(331\) 359.000i 1.08459i 0.840187 + 0.542296i \(0.182445\pi\)
−0.840187 + 0.542296i \(0.817555\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 828.818i 2.47408i
\(336\) 0 0
\(337\) 253.000 0.750742 0.375371 0.926875i \(-0.377515\pi\)
0.375371 + 0.926875i \(0.377515\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −92.9516 −0.272585
\(342\) 0 0
\(343\) −199.186 −0.580717
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 53.6656 0.154656 0.0773280 0.997006i \(-0.475361\pi\)
0.0773280 + 0.997006i \(0.475361\pi\)
\(348\) 0 0
\(349\) 71.0141i 0.203479i 0.994811 + 0.101739i \(0.0324408\pi\)
−0.994811 + 0.101739i \(0.967559\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 375.659i 1.06419i 0.846684 + 0.532095i \(0.178595\pi\)
−0.846684 + 0.532095i \(0.821405\pi\)
\(354\) 0 0
\(355\) − 120.000i − 0.338028i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 379.552i − 1.05725i −0.848856 0.528624i \(-0.822708\pi\)
0.848856 0.528624i \(-0.177292\pi\)
\(360\) 0 0
\(361\) −168.000 −0.465374
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 751.359 2.05852
\(366\) 0 0
\(367\) −334.286 −0.910861 −0.455430 0.890271i \(-0.650515\pi\)
−0.455430 + 0.890271i \(0.650515\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 670.820 1.80814
\(372\) 0 0
\(373\) − 233.827i − 0.626882i −0.949608 0.313441i \(-0.898518\pi\)
0.949608 0.313441i \(-0.101482\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 160.997i 0.427047i
\(378\) 0 0
\(379\) 349.000i 0.920844i 0.887700 + 0.460422i \(0.152302\pi\)
−0.887700 + 0.460422i \(0.847698\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 635.169i − 1.65841i −0.558948 0.829203i \(-0.688795\pi\)
0.558948 0.829203i \(-0.311205\pi\)
\(384\) 0 0
\(385\) 900.000 2.33766
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −224.633 −0.577463 −0.288731 0.957410i \(-0.593233\pi\)
−0.288731 + 0.957410i \(0.593233\pi\)
\(390\) 0 0
\(391\) 103.923 0.265788
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 523.240 1.32466
\(396\) 0 0
\(397\) 90.0666i 0.226868i 0.993546 + 0.113434i \(0.0361851\pi\)
−0.993546 + 0.113434i \(0.963815\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 429.325i 1.07064i 0.844651 + 0.535318i \(0.179808\pi\)
−0.844651 + 0.535318i \(0.820192\pi\)
\(402\) 0 0
\(403\) − 36.0000i − 0.0893300i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 395.044i − 0.970625i
\(408\) 0 0
\(409\) −469.000 −1.14670 −0.573350 0.819311i \(-0.694356\pi\)
−0.573350 + 0.819311i \(0.694356\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 813.327 1.96931
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −308.577 −0.736462 −0.368231 0.929734i \(-0.620036\pi\)
−0.368231 + 0.929734i \(0.620036\pi\)
\(420\) 0 0
\(421\) 154.153i 0.366158i 0.983098 + 0.183079i \(0.0586064\pi\)
−0.983098 + 0.183079i \(0.941394\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 469.574i − 1.10488i
\(426\) 0 0
\(427\) 525.000i 1.22951i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 317.585i 0.736855i 0.929656 + 0.368428i \(0.120104\pi\)
−0.929656 + 0.368428i \(0.879896\pi\)
\(432\) 0 0
\(433\) −398.000 −0.919169 −0.459584 0.888134i \(-0.652002\pi\)
−0.459584 + 0.888134i \(0.652002\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −178.157 −0.407682
\(438\) 0 0
\(439\) −769.031 −1.75178 −0.875889 0.482513i \(-0.839724\pi\)
−0.875889 + 0.482513i \(0.839724\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 563.489 1.27198 0.635992 0.771695i \(-0.280591\pi\)
0.635992 + 0.771695i \(0.280591\pi\)
\(444\) 0 0
\(445\) − 1351.00i − 3.03595i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 40.2492i 0.0896419i 0.998995 + 0.0448210i \(0.0142717\pi\)
−0.998995 + 0.0448210i \(0.985728\pi\)
\(450\) 0 0
\(451\) 1080.00i 2.39468i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 348.569i 0.766085i
\(456\) 0 0
\(457\) 326.000 0.713348 0.356674 0.934229i \(-0.383911\pi\)
0.356674 + 0.934229i \(0.383911\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 766.851 1.66345 0.831725 0.555187i \(-0.187353\pi\)
0.831725 + 0.555187i \(0.187353\pi\)
\(462\) 0 0
\(463\) −150.688 −0.325461 −0.162730 0.986671i \(-0.552030\pi\)
−0.162730 + 0.986671i \(0.552030\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −120.748 −0.258560 −0.129280 0.991608i \(-0.541267\pi\)
−0.129280 + 0.991608i \(0.541267\pi\)
\(468\) 0 0
\(469\) − 926.647i − 1.97579i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 509.823i 1.07785i
\(474\) 0 0
\(475\) 805.000i 1.69474i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 511.234i − 1.06729i −0.845707 0.533647i \(-0.820821\pi\)
0.845707 0.533647i \(-0.179179\pi\)
\(480\) 0 0
\(481\) 153.000 0.318087
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −844.310 −1.74085
\(486\) 0 0
\(487\) 472.850 0.970944 0.485472 0.874252i \(-0.338648\pi\)
0.485472 + 0.874252i \(0.338648\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 415.909 0.847064 0.423532 0.905881i \(-0.360790\pi\)
0.423532 + 0.905881i \(0.360790\pi\)
\(492\) 0 0
\(493\) 415.692i 0.843189i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 134.164i 0.269948i
\(498\) 0 0
\(499\) − 506.000i − 1.01403i −0.861938 0.507014i \(-0.830749\pi\)
0.861938 0.507014i \(-0.169251\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 286.601i 0.569783i 0.958560 + 0.284891i \(0.0919575\pi\)
−0.958560 + 0.284891i \(0.908042\pi\)
\(504\) 0 0
\(505\) −1320.00 −2.61386
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −178.157 −0.350014 −0.175007 0.984567i \(-0.555995\pi\)
−0.175007 + 0.984567i \(0.555995\pi\)
\(510\) 0 0
\(511\) −840.045 −1.64392
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1435.56 −2.78749
\(516\) 0 0
\(517\) − 727.461i − 1.40708i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 254.912i 0.489274i 0.969615 + 0.244637i \(0.0786688\pi\)
−0.969615 + 0.244637i \(0.921331\pi\)
\(522\) 0 0
\(523\) − 839.000i − 1.60421i −0.597185 0.802103i \(-0.703714\pi\)
0.597185 0.802103i \(-0.296286\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 92.9516i − 0.176379i
\(528\) 0 0
\(529\) 469.000 0.886578
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −418.282 −0.784770
\(534\) 0 0
\(535\) −103.923 −0.194249
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −348.827 −0.647174
\(540\) 0 0
\(541\) − 510.955i − 0.944464i −0.881474 0.472232i \(-0.843448\pi\)
0.881474 0.472232i \(-0.156552\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 1556.30i − 2.85560i
\(546\) 0 0
\(547\) − 13.0000i − 0.0237660i −0.999929 0.0118830i \(-0.996217\pi\)
0.999929 0.0118830i \(-0.00378256\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 712.629i − 1.29334i
\(552\) 0 0
\(553\) −585.000 −1.05787
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −720.375 −1.29331 −0.646656 0.762782i \(-0.723833\pi\)
−0.646656 + 0.762782i \(0.723833\pi\)
\(558\) 0 0
\(559\) −197.454 −0.353227
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 348.827 0.619585 0.309793 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(564\) 0 0
\(565\) 103.923i 0.183935i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 415.909i 0.730947i 0.930822 + 0.365473i \(0.119093\pi\)
−0.930822 + 0.365473i \(0.880907\pi\)
\(570\) 0 0
\(571\) 287.000i 0.502627i 0.967906 + 0.251313i \(0.0808625\pi\)
−0.967906 + 0.251313i \(0.919137\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 271.109i 0.471494i
\(576\) 0 0
\(577\) 169.000 0.292894 0.146447 0.989218i \(-0.453216\pi\)
0.146447 + 0.989218i \(0.453216\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1039.23 −1.78256
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −442.741 −0.754244 −0.377122 0.926164i \(-0.623086\pi\)
−0.377122 + 0.926164i \(0.623086\pi\)
\(588\) 0 0
\(589\) 159.349i 0.270541i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 751.319i 1.26698i 0.773751 + 0.633490i \(0.218378\pi\)
−0.773751 + 0.633490i \(0.781622\pi\)
\(594\) 0 0
\(595\) 900.000i 1.51261i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 123.935i 0.206904i 0.994634 + 0.103452i \(0.0329888\pi\)
−0.994634 + 0.103452i \(0.967011\pi\)
\(600\) 0 0
\(601\) −346.000 −0.575707 −0.287854 0.957674i \(-0.592942\pi\)
−0.287854 + 0.957674i \(0.592942\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −457.012 −0.755392
\(606\) 0 0
\(607\) −698.016 −1.14994 −0.574972 0.818173i \(-0.694987\pi\)
−0.574972 + 0.818173i \(0.694987\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 281.745 0.461120
\(612\) 0 0
\(613\) 497.099i 0.810928i 0.914111 + 0.405464i \(0.132890\pi\)
−0.914111 + 0.405464i \(0.867110\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 818.401i − 1.32642i −0.748434 0.663210i \(-0.769194\pi\)
0.748434 0.663210i \(-0.230806\pi\)
\(618\) 0 0
\(619\) − 277.000i − 0.447496i −0.974647 0.223748i \(-0.928171\pi\)
0.974647 0.223748i \(-0.0718293\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 1510.46i 2.42450i
\(624\) 0 0
\(625\) −275.000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 395.044 0.628051
\(630\) 0 0
\(631\) −278.860 −0.441934 −0.220967 0.975281i \(-0.570921\pi\)
−0.220967 + 0.975281i \(0.570921\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −1073.31 −1.69026
\(636\) 0 0
\(637\) − 135.100i − 0.212088i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 778.152i − 1.21397i −0.794715 0.606983i \(-0.792380\pi\)
0.794715 0.606983i \(-0.207620\pi\)
\(642\) 0 0
\(643\) − 466.000i − 0.724728i −0.932037 0.362364i \(-0.881970\pi\)
0.932037 0.362364i \(-0.118030\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 852.056i 1.31693i 0.752610 + 0.658467i \(0.228795\pi\)
−0.752610 + 0.658467i \(0.771205\pi\)
\(648\) 0 0
\(649\) −1260.00 −1.94145
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −185.903 −0.284691 −0.142345 0.989817i \(-0.545464\pi\)
−0.142345 + 0.989817i \(0.545464\pi\)
\(654\) 0 0
\(655\) −1662.77 −2.53858
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 295.161 0.447892 0.223946 0.974602i \(-0.428106\pi\)
0.223946 + 0.974602i \(0.428106\pi\)
\(660\) 0 0
\(661\) − 1124.10i − 1.70061i −0.526293 0.850303i \(-0.676418\pi\)
0.526293 0.850303i \(-0.323582\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 1542.89i − 2.32013i
\(666\) 0 0
\(667\) − 240.000i − 0.359820i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 813.327i − 1.21211i
\(672\) 0 0
\(673\) 37.0000 0.0549777 0.0274889 0.999622i \(-0.491249\pi\)
0.0274889 + 0.999622i \(0.491249\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 735.867 1.08695 0.543476 0.839425i \(-0.317108\pi\)
0.543476 + 0.839425i \(0.317108\pi\)
\(678\) 0 0
\(679\) 943.968 1.39023
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1153.81 −1.68933 −0.844664 0.535297i \(-0.820200\pi\)
−0.844664 + 0.535297i \(0.820200\pi\)
\(684\) 0 0
\(685\) − 935.307i − 1.36541i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 402.492i − 0.584169i
\(690\) 0 0
\(691\) 662.000i 0.958032i 0.877806 + 0.479016i \(0.159006\pi\)
−0.877806 + 0.479016i \(0.840994\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 379.552i 0.546119i
\(696\) 0 0
\(697\) −1080.00 −1.54950
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 1045.71 1.49173 0.745867 0.666095i \(-0.232035\pi\)
0.745867 + 0.666095i \(0.232035\pi\)
\(702\) 0 0
\(703\) −677.232 −0.963345
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1475.80 2.08742
\(708\) 0 0
\(709\) − 129.904i − 0.183221i −0.995795 0.0916106i \(-0.970799\pi\)
0.995795 0.0916106i \(-0.0292015\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 53.6656i 0.0752674i
\(714\) 0 0
\(715\) − 540.000i − 0.755245i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 92.9516i − 0.129279i −0.997909 0.0646395i \(-0.979410\pi\)
0.997909 0.0646395i \(-0.0205897\pi\)
\(720\) 0 0
\(721\) 1605.00 2.22607
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1084.44 −1.49577
\(726\) 0 0
\(727\) 810.600 1.11499 0.557496 0.830179i \(-0.311762\pi\)
0.557496 + 0.830179i \(0.311762\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −509.823 −0.697433
\(732\) 0 0
\(733\) 658.179i 0.897925i 0.893551 + 0.448963i \(0.148206\pi\)
−0.893551 + 0.448963i \(0.851794\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1435.56i 1.94784i
\(738\) 0 0
\(739\) 82.0000i 0.110961i 0.998460 + 0.0554804i \(0.0176690\pi\)
−0.998460 + 0.0554804i \(0.982331\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 704.883i − 0.948698i −0.880337 0.474349i \(-0.842683\pi\)
0.880337 0.474349i \(-0.157317\pi\)
\(744\) 0 0
\(745\) −1080.00 −1.44966
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 116.190 0.155126
\(750\) 0 0
\(751\) 465.922 0.620402 0.310201 0.950671i \(-0.399604\pi\)
0.310201 + 0.950671i \(0.399604\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −1274.56 −1.68816
\(756\) 0 0
\(757\) 147.224i 0.194484i 0.995261 + 0.0972420i \(0.0310021\pi\)
−0.995261 + 0.0972420i \(0.968998\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 40.2492i 0.0528899i 0.999650 + 0.0264450i \(0.00841867\pi\)
−0.999650 + 0.0264450i \(0.991581\pi\)
\(762\) 0 0
\(763\) 1740.00i 2.28047i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 487.996i − 0.636240i
\(768\) 0 0
\(769\) 551.000 0.716515 0.358257 0.933623i \(-0.383371\pi\)
0.358257 + 0.933623i \(0.383371\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 588.693 0.761570 0.380785 0.924664i \(-0.375654\pi\)
0.380785 + 0.924664i \(0.375654\pi\)
\(774\) 0 0
\(775\) 242.487 0.312887
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1851.46 2.37672
\(780\) 0 0
\(781\) − 207.846i − 0.266128i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1234.31i 1.57237i
\(786\) 0 0
\(787\) 71.0000i 0.0902160i 0.998982 + 0.0451080i \(0.0143632\pi\)
−0.998982 + 0.0451080i \(0.985637\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 116.190i − 0.146889i
\(792\) 0 0
\(793\) 315.000 0.397226
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −635.169 −0.796950 −0.398475 0.917179i \(-0.630460\pi\)
−0.398475 + 0.917179i \(0.630460\pi\)
\(798\) 0 0
\(799\) 727.461 0.910465
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 1301.39 1.62066
\(804\) 0 0
\(805\) − 519.615i − 0.645485i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1100.15i 1.35988i 0.733266 + 0.679942i \(0.237995\pi\)
−0.733266 + 0.679942i \(0.762005\pi\)
\(810\) 0 0
\(811\) − 974.000i − 1.20099i −0.799630 0.600493i \(-0.794971\pi\)
0.799630 0.600493i \(-0.205029\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 751.359i 0.921913i
\(816\) 0 0
\(817\) 874.000 1.06977
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 689.391 0.839697 0.419848 0.907594i \(-0.362083\pi\)
0.419848 + 0.907594i \(0.362083\pi\)
\(822\) 0 0
\(823\) −410.496 −0.498780 −0.249390 0.968403i \(-0.580230\pi\)
−0.249390 + 0.968403i \(0.580230\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1033.06 1.24917 0.624585 0.780957i \(-0.285268\pi\)
0.624585 + 0.780957i \(0.285268\pi\)
\(828\) 0 0
\(829\) − 348.142i − 0.419954i −0.977706 0.209977i \(-0.932661\pi\)
0.977706 0.209977i \(-0.0673390\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 348.827i − 0.418759i
\(834\) 0 0
\(835\) − 300.000i − 0.359281i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 728.121i 0.867844i 0.900951 + 0.433922i \(0.142871\pi\)
−0.900951 + 0.433922i \(0.857129\pi\)
\(840\) 0 0
\(841\) 119.000 0.141498
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1099.93 −1.30169
\(846\) 0 0
\(847\) 510.955 0.603253
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −228.079 −0.268013
\(852\) 0 0
\(853\) 1103.32i 1.29345i 0.762722 + 0.646727i \(0.223863\pi\)
−0.762722 + 0.646727i \(0.776137\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 26.8328i − 0.0313102i −0.999877 0.0156551i \(-0.995017\pi\)
0.999877 0.0156551i \(-0.00498337\pi\)
\(858\) 0 0
\(859\) 1549.00i 1.80326i 0.432509 + 0.901630i \(0.357629\pi\)
−0.432509 + 0.901630i \(0.642371\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 1309.07i − 1.51688i −0.651742 0.758441i \(-0.725962\pi\)
0.651742 0.758441i \(-0.274038\pi\)
\(864\) 0 0
\(865\) 600.000 0.693642
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 906.278 1.04290
\(870\) 0 0
\(871\) −555.988 −0.638333
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −670.820 −0.766652
\(876\) 0 0
\(877\) 8.66025i 0.00987486i 0.999988 + 0.00493743i \(0.00157164\pi\)
−0.999988 + 0.00493743i \(0.998428\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 389.076i − 0.441630i −0.975316 0.220815i \(-0.929128\pi\)
0.975316 0.220815i \(-0.0708717\pi\)
\(882\) 0 0
\(883\) − 61.0000i − 0.0690827i −0.999403 0.0345413i \(-0.989003\pi\)
0.999403 0.0345413i \(-0.0109970\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1208.37i − 1.36231i −0.732138 0.681156i \(-0.761478\pi\)
0.732138 0.681156i \(-0.238522\pi\)
\(888\) 0 0
\(889\) 1200.00 1.34983
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −1247.10 −1.39653
\(894\) 0 0
\(895\) −207.846 −0.232230
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −214.663 −0.238779
\(900\) 0 0
\(901\) − 1039.23i − 1.15342i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1355.06i 1.49730i
\(906\) 0 0
\(907\) 433.000i 0.477398i 0.971094 + 0.238699i \(0.0767209\pi\)
−0.971094 + 0.238699i \(0.923279\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 1595.67i − 1.75156i −0.482712 0.875779i \(-0.660348\pi\)
0.482712 0.875779i \(-0.339652\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1859.03 2.02730
\(918\) 0 0
\(919\) −145.492 −0.158316 −0.0791579 0.996862i \(-0.525223\pi\)
−0.0791579 + 0.996862i \(0.525223\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 80.4984 0.0872139
\(924\) 0 0
\(925\) 1030.57i 1.11413i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 536.656i 0.577671i 0.957379 + 0.288835i \(0.0932681\pi\)
−0.957379 + 0.288835i \(0.906732\pi\)
\(930\) 0 0
\(931\) 598.000i 0.642320i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 1394.27i − 1.49120i
\(936\) 0 0
\(937\) 469.000 0.500534 0.250267 0.968177i \(-0.419482\pi\)
0.250267 + 0.968177i \(0.419482\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −472.504 −0.502130 −0.251065 0.967970i \(-0.580781\pi\)
−0.251065 + 0.967970i \(0.580781\pi\)
\(942\) 0 0
\(943\) 623.538 0.661228
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 576.906 0.609193 0.304596 0.952482i \(-0.401478\pi\)
0.304596 + 0.952482i \(0.401478\pi\)
\(948\) 0 0
\(949\) 504.027i 0.531114i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 845.234i 0.886919i 0.896294 + 0.443459i \(0.146249\pi\)
−0.896294 + 0.443459i \(0.853751\pi\)
\(954\) 0 0
\(955\) − 1380.00i − 1.44503i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 1045.71i 1.09041i
\(960\) 0 0
\(961\) −913.000 −0.950052
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −457.012 −0.473588
\(966\) 0 0
\(967\) 999.393 1.03350 0.516749 0.856137i \(-0.327142\pi\)
0.516749 + 0.856137i \(0.327142\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 469.574 0.483599 0.241799 0.970326i \(-0.422262\pi\)
0.241799 + 0.970326i \(0.422262\pi\)
\(972\) 0 0
\(973\) − 424.352i − 0.436128i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 536.656i 0.549290i 0.961546 + 0.274645i \(0.0885603\pi\)
−0.961546 + 0.274645i \(0.911440\pi\)
\(978\) 0 0
\(979\) − 2340.00i − 2.39019i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1231.61i 1.25291i 0.779458 + 0.626454i \(0.215494\pi\)
−0.779458 + 0.626454i \(0.784506\pi\)
\(984\) 0 0
\(985\) −180.000 −0.182741
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 294.347 0.297621
\(990\) 0 0
\(991\) −43.3013 −0.0436945 −0.0218473 0.999761i \(-0.506955\pi\)
−0.0218473 + 0.999761i \(0.506955\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −684.237 −0.687675
\(996\) 0 0
\(997\) 1558.85i 1.56354i 0.623569 + 0.781768i \(0.285682\pi\)
−0.623569 + 0.781768i \(0.714318\pi\)
\(998\) 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.h.h.161.3 yes 8
3.2 odd 2 inner 1728.3.h.h.161.7 yes 8
4.3 odd 2 inner 1728.3.h.h.161.1 8
8.3 odd 2 inner 1728.3.h.h.161.6 yes 8
8.5 even 2 inner 1728.3.h.h.161.8 yes 8
12.11 even 2 inner 1728.3.h.h.161.5 yes 8
24.5 odd 2 inner 1728.3.h.h.161.4 yes 8
24.11 even 2 inner 1728.3.h.h.161.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1728.3.h.h.161.1 8 4.3 odd 2 inner
1728.3.h.h.161.2 yes 8 24.11 even 2 inner
1728.3.h.h.161.3 yes 8 1.1 even 1 trivial
1728.3.h.h.161.4 yes 8 24.5 odd 2 inner
1728.3.h.h.161.5 yes 8 12.11 even 2 inner
1728.3.h.h.161.6 yes 8 8.3 odd 2 inner
1728.3.h.h.161.7 yes 8 3.2 odd 2 inner
1728.3.h.h.161.8 yes 8 8.5 even 2 inner