Properties

Label 1728.3.g.j.703.6
Level $1728$
Weight $3$
Character 1728.703
Analytic conductor $47.085$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Newform invariants

Self dual: no
Analytic conductor: \(47.0845896815\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.56070144.2
Defining polynomial: \( x^{8} - 4x^{7} + 16x^{6} - 34x^{5} + 63x^{4} - 74x^{3} + 70x^{2} - 38x + 13 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 864)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 703.6
Root \(0.500000 - 1.56488i\) of defining polynomial
Character \(\chi\) \(=\) 1728.703
Dual form 1728.3.g.j.703.5

$q$-expansion

\(f(q)\) \(=\) \(q+0.956810 q^{5} +6.34610i q^{7} +O(q^{10})\) \(q+0.956810 q^{5} +6.34610i q^{7} +14.4991i q^{11} -1.76367 q^{13} -1.34309 q^{17} +13.0234i q^{19} -4.19916i q^{23} -24.0845 q^{25} -38.1690 q^{29} -0.172759i q^{31} +6.07202i q^{35} -15.1937 q^{37} +69.3965 q^{41} -5.52734i q^{43} -66.7137i q^{47} +8.72695 q^{49} -31.6556 q^{53} +13.8729i q^{55} +12.2410i q^{59} -92.1017 q^{61} -1.68750 q^{65} -26.4583i q^{67} -85.4493i q^{71} -103.920 q^{73} -92.0126 q^{77} -68.3424i q^{79} +137.277i q^{83} -1.28509 q^{85} +67.3405 q^{89} -11.1924i q^{91} +12.4609i q^{95} -95.6956 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 8 q^{5} - 8 q^{13} + 24 q^{17} + 24 q^{25} + 128 q^{29} - 24 q^{37} + 160 q^{41} - 144 q^{49} + 48 q^{53} + 136 q^{61} - 280 q^{65} + 72 q^{73} - 520 q^{77} - 96 q^{85} - 168 q^{89} + 104 q^{97}+O(q^{100}) \) 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
\(5\) 0.956810 0.191362 0.0956810 0.995412i \(-0.469497\pi\)
0.0956810 + 0.995412i \(0.469497\pi\)
\(6\) 0 0
\(7\) 6.34610i 0.906586i 0.891361 + 0.453293i \(0.149751\pi\)
−0.891361 + 0.453293i \(0.850249\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 14.4991i 1.31810i 0.752100 + 0.659049i \(0.229041\pi\)
−0.752100 + 0.659049i \(0.770959\pi\)
\(12\) 0 0
\(13\) −1.76367 −0.135667 −0.0678334 0.997697i \(-0.521609\pi\)
−0.0678334 + 0.997697i \(0.521609\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.34309 −0.0790056 −0.0395028 0.999219i \(-0.512577\pi\)
−0.0395028 + 0.999219i \(0.512577\pi\)
\(18\) 0 0
\(19\) 13.0234i 0.685442i 0.939437 + 0.342721i \(0.111349\pi\)
−0.939437 + 0.342721i \(0.888651\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.19916i − 0.182572i −0.995825 0.0912861i \(-0.970902\pi\)
0.995825 0.0912861i \(-0.0290978\pi\)
\(24\) 0 0
\(25\) −24.0845 −0.963381
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −38.1690 −1.31617 −0.658087 0.752942i \(-0.728634\pi\)
−0.658087 + 0.752942i \(0.728634\pi\)
\(30\) 0 0
\(31\) − 0.172759i − 0.00557288i −0.999996 0.00278644i \(-0.999113\pi\)
0.999996 0.00278644i \(-0.000886953\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 6.07202i 0.173486i
\(36\) 0 0
\(37\) −15.1937 −0.410641 −0.205320 0.978695i \(-0.565824\pi\)
−0.205320 + 0.978695i \(0.565824\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 69.3965 1.69260 0.846298 0.532709i \(-0.178826\pi\)
0.846298 + 0.532709i \(0.178826\pi\)
\(42\) 0 0
\(43\) − 5.52734i − 0.128543i −0.997932 0.0642713i \(-0.979528\pi\)
0.997932 0.0642713i \(-0.0204723\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 66.7137i − 1.41944i −0.704484 0.709720i \(-0.748821\pi\)
0.704484 0.709720i \(-0.251179\pi\)
\(48\) 0 0
\(49\) 8.72695 0.178101
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −31.6556 −0.597275 −0.298638 0.954367i \(-0.596532\pi\)
−0.298638 + 0.954367i \(0.596532\pi\)
\(54\) 0 0
\(55\) 13.8729i 0.252234i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.2410i 0.207475i 0.994605 + 0.103738i \(0.0330802\pi\)
−0.994605 + 0.103738i \(0.966920\pi\)
\(60\) 0 0
\(61\) −92.1017 −1.50986 −0.754932 0.655803i \(-0.772330\pi\)
−0.754932 + 0.655803i \(0.772330\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −1.68750 −0.0259615
\(66\) 0 0
\(67\) − 26.4583i − 0.394900i −0.980313 0.197450i \(-0.936734\pi\)
0.980313 0.197450i \(-0.0632661\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 85.4493i − 1.20351i −0.798680 0.601755i \(-0.794468\pi\)
0.798680 0.601755i \(-0.205532\pi\)
\(72\) 0 0
\(73\) −103.920 −1.42356 −0.711781 0.702401i \(-0.752111\pi\)
−0.711781 + 0.702401i \(0.752111\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −92.0126 −1.19497
\(78\) 0 0
\(79\) − 68.3424i − 0.865093i −0.901612 0.432547i \(-0.857615\pi\)
0.901612 0.432547i \(-0.142385\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 137.277i 1.65393i 0.562251 + 0.826967i \(0.309936\pi\)
−0.562251 + 0.826967i \(0.690064\pi\)
\(84\) 0 0
\(85\) −1.28509 −0.0151187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 67.3405 0.756635 0.378317 0.925676i \(-0.376503\pi\)
0.378317 + 0.925676i \(0.376503\pi\)
\(90\) 0 0
\(91\) − 11.1924i − 0.122994i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 12.4609i 0.131168i
\(96\) 0 0
\(97\) −95.6956 −0.986553 −0.493276 0.869873i \(-0.664201\pi\)
−0.493276 + 0.869873i \(0.664201\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 73.9722 0.732398 0.366199 0.930537i \(-0.380659\pi\)
0.366199 + 0.930537i \(0.380659\pi\)
\(102\) 0 0
\(103\) − 59.6083i − 0.578721i −0.957220 0.289361i \(-0.906557\pi\)
0.957220 0.289361i \(-0.0934427\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 137.299i 1.28317i 0.767053 + 0.641583i \(0.221722\pi\)
−0.767053 + 0.641583i \(0.778278\pi\)
\(108\) 0 0
\(109\) −122.932 −1.12782 −0.563909 0.825837i \(-0.690703\pi\)
−0.563909 + 0.825837i \(0.690703\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 122.117 1.08068 0.540340 0.841447i \(-0.318296\pi\)
0.540340 + 0.841447i \(0.318296\pi\)
\(114\) 0 0
\(115\) − 4.01780i − 0.0349374i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 8.52342i − 0.0716254i
\(120\) 0 0
\(121\) −89.2230 −0.737380
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −46.9646 −0.375716
\(126\) 0 0
\(127\) 91.4995i 0.720469i 0.932862 + 0.360234i \(0.117303\pi\)
−0.932862 + 0.360234i \(0.882697\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 117.372i − 0.895972i −0.894040 0.447986i \(-0.852141\pi\)
0.894040 0.447986i \(-0.147859\pi\)
\(132\) 0 0
\(133\) −82.6478 −0.621412
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −56.4879 −0.412320 −0.206160 0.978518i \(-0.566097\pi\)
−0.206160 + 0.978518i \(0.566097\pi\)
\(138\) 0 0
\(139\) − 105.929i − 0.762080i −0.924559 0.381040i \(-0.875566\pi\)
0.924559 0.381040i \(-0.124434\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) − 25.5715i − 0.178822i
\(144\) 0 0
\(145\) −36.5205 −0.251866
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −290.160 −1.94739 −0.973693 0.227865i \(-0.926826\pi\)
−0.973693 + 0.227865i \(0.926826\pi\)
\(150\) 0 0
\(151\) 188.866i 1.25077i 0.780316 + 0.625385i \(0.215058\pi\)
−0.780316 + 0.625385i \(0.784942\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 0.165298i − 0.00106644i
\(156\) 0 0
\(157\) −213.852 −1.36211 −0.681057 0.732231i \(-0.738479\pi\)
−0.681057 + 0.732231i \(0.738479\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 26.6483 0.165518
\(162\) 0 0
\(163\) 304.944i 1.87082i 0.353566 + 0.935410i \(0.384969\pi\)
−0.353566 + 0.935410i \(0.615031\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 166.488i 0.996935i 0.866908 + 0.498467i \(0.166104\pi\)
−0.866908 + 0.498467i \(0.833896\pi\)
\(168\) 0 0
\(169\) −165.889 −0.981595
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 313.389 1.81150 0.905749 0.423815i \(-0.139310\pi\)
0.905749 + 0.423815i \(0.139310\pi\)
\(174\) 0 0
\(175\) − 152.843i − 0.873388i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 144.278i − 0.806021i −0.915195 0.403011i \(-0.867964\pi\)
0.915195 0.403011i \(-0.132036\pi\)
\(180\) 0 0
\(181\) −120.016 −0.663074 −0.331537 0.943442i \(-0.607567\pi\)
−0.331537 + 0.943442i \(0.607567\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −14.5375 −0.0785810
\(186\) 0 0
\(187\) − 19.4736i − 0.104137i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 257.420i − 1.34775i −0.738846 0.673874i \(-0.764629\pi\)
0.738846 0.673874i \(-0.235371\pi\)
\(192\) 0 0
\(193\) −289.234 −1.49862 −0.749310 0.662220i \(-0.769615\pi\)
−0.749310 + 0.662220i \(0.769615\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 196.282 0.996357 0.498179 0.867074i \(-0.334002\pi\)
0.498179 + 0.867074i \(0.334002\pi\)
\(198\) 0 0
\(199\) 192.265i 0.966155i 0.875578 + 0.483078i \(0.160481\pi\)
−0.875578 + 0.483078i \(0.839519\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 242.225i − 1.19322i
\(204\) 0 0
\(205\) 66.3992 0.323899
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −188.827 −0.903479
\(210\) 0 0
\(211\) 20.0620i 0.0950804i 0.998869 + 0.0475402i \(0.0151382\pi\)
−0.998869 + 0.0475402i \(0.984862\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 5.28861i − 0.0245982i
\(216\) 0 0
\(217\) 1.09635 0.00505230
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 2.36877 0.0107184
\(222\) 0 0
\(223\) − 319.896i − 1.43451i −0.696810 0.717256i \(-0.745398\pi\)
0.696810 0.717256i \(-0.254602\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 202.001i 0.889872i 0.895562 + 0.444936i \(0.146774\pi\)
−0.895562 + 0.444936i \(0.853226\pi\)
\(228\) 0 0
\(229\) −196.154 −0.856568 −0.428284 0.903644i \(-0.640882\pi\)
−0.428284 + 0.903644i \(0.640882\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −395.370 −1.69687 −0.848434 0.529301i \(-0.822454\pi\)
−0.848434 + 0.529301i \(0.822454\pi\)
\(234\) 0 0
\(235\) − 63.8324i − 0.271627i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 449.180i 1.87941i 0.341980 + 0.939707i \(0.388902\pi\)
−0.341980 + 0.939707i \(0.611098\pi\)
\(240\) 0 0
\(241\) 187.767 0.779117 0.389558 0.921002i \(-0.372628\pi\)
0.389558 + 0.921002i \(0.372628\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 8.35004 0.0340818
\(246\) 0 0
\(247\) − 22.9689i − 0.0929917i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 57.1800i − 0.227809i −0.993492 0.113904i \(-0.963664\pi\)
0.993492 0.113904i \(-0.0363358\pi\)
\(252\) 0 0
\(253\) 60.8839 0.240648
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 112.952 0.439501 0.219750 0.975556i \(-0.429476\pi\)
0.219750 + 0.975556i \(0.429476\pi\)
\(258\) 0 0
\(259\) − 96.4208i − 0.372281i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 48.0573i 0.182727i 0.995818 + 0.0913636i \(0.0291226\pi\)
−0.995818 + 0.0913636i \(0.970877\pi\)
\(264\) 0 0
\(265\) −30.2884 −0.114296
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 196.360 0.729964 0.364982 0.931015i \(-0.381075\pi\)
0.364982 + 0.931015i \(0.381075\pi\)
\(270\) 0 0
\(271\) 242.829i 0.896050i 0.894021 + 0.448025i \(0.147872\pi\)
−0.894021 + 0.448025i \(0.852128\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 349.203i − 1.26983i
\(276\) 0 0
\(277\) −228.132 −0.823580 −0.411790 0.911279i \(-0.635096\pi\)
−0.411790 + 0.911279i \(0.635096\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −313.999 −1.11743 −0.558716 0.829359i \(-0.688706\pi\)
−0.558716 + 0.829359i \(0.688706\pi\)
\(282\) 0 0
\(283\) − 522.669i − 1.84689i −0.383735 0.923443i \(-0.625362\pi\)
0.383735 0.923443i \(-0.374638\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 440.397i 1.53449i
\(288\) 0 0
\(289\) −287.196 −0.993758
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 173.891 0.593483 0.296742 0.954958i \(-0.404100\pi\)
0.296742 + 0.954958i \(0.404100\pi\)
\(294\) 0 0
\(295\) 11.7124i 0.0397029i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 7.40593i 0.0247690i
\(300\) 0 0
\(301\) 35.0771 0.116535
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −88.1239 −0.288931
\(306\) 0 0
\(307\) 465.568i 1.51651i 0.651959 + 0.758254i \(0.273947\pi\)
−0.651959 + 0.758254i \(0.726053\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 238.054i 0.765448i 0.923863 + 0.382724i \(0.125014\pi\)
−0.923863 + 0.382724i \(0.874986\pi\)
\(312\) 0 0
\(313\) 99.4041 0.317585 0.158793 0.987312i \(-0.449240\pi\)
0.158793 + 0.987312i \(0.449240\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 87.2285 0.275169 0.137584 0.990490i \(-0.456066\pi\)
0.137584 + 0.990490i \(0.456066\pi\)
\(318\) 0 0
\(319\) − 553.415i − 1.73484i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 17.4916i − 0.0541537i
\(324\) 0 0
\(325\) 42.4771 0.130699
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 423.372 1.28685
\(330\) 0 0
\(331\) − 25.0394i − 0.0756476i −0.999284 0.0378238i \(-0.987957\pi\)
0.999284 0.0378238i \(-0.0120426\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 25.3156i − 0.0755689i
\(336\) 0 0
\(337\) −35.5515 −0.105494 −0.0527471 0.998608i \(-0.516798\pi\)
−0.0527471 + 0.998608i \(0.516798\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 2.50485 0.00734560
\(342\) 0 0
\(343\) 366.341i 1.06805i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 277.688i − 0.800254i −0.916460 0.400127i \(-0.868966\pi\)
0.916460 0.400127i \(-0.131034\pi\)
\(348\) 0 0
\(349\) 382.657 1.09644 0.548219 0.836335i \(-0.315306\pi\)
0.548219 + 0.836335i \(0.315306\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −444.310 −1.25867 −0.629335 0.777134i \(-0.716673\pi\)
−0.629335 + 0.777134i \(0.716673\pi\)
\(354\) 0 0
\(355\) − 81.7587i − 0.230306i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 10.9605i − 0.0305306i −0.999883 0.0152653i \(-0.995141\pi\)
0.999883 0.0152653i \(-0.00485929\pi\)
\(360\) 0 0
\(361\) 191.391 0.530170
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −99.4318 −0.272416
\(366\) 0 0
\(367\) − 166.081i − 0.452538i −0.974065 0.226269i \(-0.927347\pi\)
0.974065 0.226269i \(-0.0726528\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 200.890i − 0.541482i
\(372\) 0 0
\(373\) 240.406 0.644519 0.322259 0.946651i \(-0.395558\pi\)
0.322259 + 0.946651i \(0.395558\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 67.3175 0.178561
\(378\) 0 0
\(379\) 224.504i 0.592359i 0.955132 + 0.296179i \(0.0957127\pi\)
−0.955132 + 0.296179i \(0.904287\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 126.068i − 0.329160i −0.986364 0.164580i \(-0.947373\pi\)
0.986364 0.164580i \(-0.0526268\pi\)
\(384\) 0 0
\(385\) −88.0386 −0.228672
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 435.033 1.11834 0.559169 0.829054i \(-0.311120\pi\)
0.559169 + 0.829054i \(0.311120\pi\)
\(390\) 0 0
\(391\) 5.63987i 0.0144242i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 65.3907i − 0.165546i
\(396\) 0 0
\(397\) 34.0620 0.0857985 0.0428993 0.999079i \(-0.486341\pi\)
0.0428993 + 0.999079i \(0.486341\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −152.431 −0.380126 −0.190063 0.981772i \(-0.560869\pi\)
−0.190063 + 0.981772i \(0.560869\pi\)
\(402\) 0 0
\(403\) 0.304690i 0 0.000756055i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 220.294i − 0.541264i
\(408\) 0 0
\(409\) 714.512 1.74697 0.873486 0.486849i \(-0.161854\pi\)
0.873486 + 0.486849i \(0.161854\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −77.6830 −0.188094
\(414\) 0 0
\(415\) 131.348i 0.316500i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 131.654i 0.314211i 0.987582 + 0.157106i \(0.0502163\pi\)
−0.987582 + 0.157106i \(0.949784\pi\)
\(420\) 0 0
\(421\) 55.5081 0.131848 0.0659241 0.997825i \(-0.479000\pi\)
0.0659241 + 0.997825i \(0.479000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 32.3478 0.0761124
\(426\) 0 0
\(427\) − 584.487i − 1.36882i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) − 743.495i − 1.72505i −0.506018 0.862523i \(-0.668883\pi\)
0.506018 0.862523i \(-0.331117\pi\)
\(432\) 0 0
\(433\) 526.775 1.21657 0.608285 0.793719i \(-0.291858\pi\)
0.608285 + 0.793719i \(0.291858\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 54.6873 0.125143
\(438\) 0 0
\(439\) 684.577i 1.55940i 0.626152 + 0.779701i \(0.284629\pi\)
−0.626152 + 0.779701i \(0.715371\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 59.9868i 0.135410i 0.997705 + 0.0677052i \(0.0215677\pi\)
−0.997705 + 0.0677052i \(0.978432\pi\)
\(444\) 0 0
\(445\) 64.4321 0.144791
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 78.3940 0.174597 0.0872985 0.996182i \(-0.472177\pi\)
0.0872985 + 0.996182i \(0.472177\pi\)
\(450\) 0 0
\(451\) 1006.18i 2.23101i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 10.7090i − 0.0235363i
\(456\) 0 0
\(457\) −64.6748 −0.141520 −0.0707602 0.997493i \(-0.522543\pi\)
−0.0707602 + 0.997493i \(0.522543\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −321.909 −0.698285 −0.349143 0.937070i \(-0.613527\pi\)
−0.349143 + 0.937070i \(0.613527\pi\)
\(462\) 0 0
\(463\) − 520.291i − 1.12374i −0.827226 0.561869i \(-0.810083\pi\)
0.827226 0.561869i \(-0.189917\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 454.944i 0.974185i 0.873350 + 0.487093i \(0.161943\pi\)
−0.873350 + 0.487093i \(0.838057\pi\)
\(468\) 0 0
\(469\) 167.907 0.358011
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 80.1412 0.169432
\(474\) 0 0
\(475\) − 313.662i − 0.660341i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 341.743i 0.713451i 0.934209 + 0.356726i \(0.116107\pi\)
−0.934209 + 0.356726i \(0.883893\pi\)
\(480\) 0 0
\(481\) 26.7966 0.0557103
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −91.5625 −0.188789
\(486\) 0 0
\(487\) 683.859i 1.40423i 0.712065 + 0.702113i \(0.247760\pi\)
−0.712065 + 0.702113i \(0.752240\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 419.194i 0.853756i 0.904309 + 0.426878i \(0.140387\pi\)
−0.904309 + 0.426878i \(0.859613\pi\)
\(492\) 0 0
\(493\) 51.2646 0.103985
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 542.270 1.09109
\(498\) 0 0
\(499\) 402.803i 0.807221i 0.914931 + 0.403610i \(0.132245\pi\)
−0.914931 + 0.403610i \(0.867755\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 321.360i 0.638887i 0.947605 + 0.319444i \(0.103496\pi\)
−0.947605 + 0.319444i \(0.896504\pi\)
\(504\) 0 0
\(505\) 70.7773 0.140153
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −366.245 −0.719539 −0.359769 0.933041i \(-0.617145\pi\)
−0.359769 + 0.933041i \(0.617145\pi\)
\(510\) 0 0
\(511\) − 659.488i − 1.29058i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 57.0338i − 0.110745i
\(516\) 0 0
\(517\) 967.287 1.87096
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 660.982 1.26868 0.634340 0.773054i \(-0.281272\pi\)
0.634340 + 0.773054i \(0.281272\pi\)
\(522\) 0 0
\(523\) − 197.073i − 0.376812i −0.982091 0.188406i \(-0.939668\pi\)
0.982091 0.188406i \(-0.0603321\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.232032i 0 0.000440289i
\(528\) 0 0
\(529\) 511.367 0.966667
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −122.392 −0.229629
\(534\) 0 0
\(535\) 131.369i 0.245549i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 126.533i 0.234755i
\(540\) 0 0
\(541\) 777.149 1.43650 0.718252 0.695783i \(-0.244942\pi\)
0.718252 + 0.695783i \(0.244942\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −117.623 −0.215821
\(546\) 0 0
\(547\) 400.931i 0.732963i 0.930425 + 0.366482i \(0.119438\pi\)
−0.930425 + 0.366482i \(0.880562\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 497.090i − 0.902160i
\(552\) 0 0
\(553\) 433.708 0.784282
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −553.074 −0.992951 −0.496476 0.868051i \(-0.665373\pi\)
−0.496476 + 0.868051i \(0.665373\pi\)
\(558\) 0 0
\(559\) 9.74839i 0.0174390i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 548.074i 0.973489i 0.873545 + 0.486744i \(0.161816\pi\)
−0.873545 + 0.486744i \(0.838184\pi\)
\(564\) 0 0
\(565\) 116.843 0.206801
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 627.765 1.10328 0.551639 0.834083i \(-0.314003\pi\)
0.551639 + 0.834083i \(0.314003\pi\)
\(570\) 0 0
\(571\) − 652.428i − 1.14261i −0.820739 0.571303i \(-0.806438\pi\)
0.820739 0.571303i \(-0.193562\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 101.135i 0.175887i
\(576\) 0 0
\(577\) 300.791 0.521301 0.260651 0.965433i \(-0.416063\pi\)
0.260651 + 0.965433i \(0.416063\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −871.171 −1.49943
\(582\) 0 0
\(583\) − 458.977i − 0.787267i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 513.527i 0.874833i 0.899259 + 0.437417i \(0.144107\pi\)
−0.899259 + 0.437417i \(0.855893\pi\)
\(588\) 0 0
\(589\) 2.24991 0.00381989
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −114.700 −0.193423 −0.0967116 0.995312i \(-0.530832\pi\)
−0.0967116 + 0.995312i \(0.530832\pi\)
\(594\) 0 0
\(595\) − 8.15529i − 0.0137064i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 926.005i − 1.54592i −0.634456 0.772959i \(-0.718776\pi\)
0.634456 0.772959i \(-0.281224\pi\)
\(600\) 0 0
\(601\) −67.6176 −0.112509 −0.0562543 0.998416i \(-0.517916\pi\)
−0.0562543 + 0.998416i \(0.517916\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −85.3694 −0.141106
\(606\) 0 0
\(607\) − 634.634i − 1.04552i −0.852478 0.522762i \(-0.824902\pi\)
0.852478 0.522762i \(-0.175098\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 117.661i 0.192571i
\(612\) 0 0
\(613\) −115.485 −0.188394 −0.0941969 0.995554i \(-0.530028\pi\)
−0.0941969 + 0.995554i \(0.530028\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −692.513 −1.12239 −0.561193 0.827685i \(-0.689658\pi\)
−0.561193 + 0.827685i \(0.689658\pi\)
\(618\) 0 0
\(619\) − 711.629i − 1.14964i −0.818279 0.574821i \(-0.805072\pi\)
0.818279 0.574821i \(-0.194928\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 427.350i 0.685955i
\(624\) 0 0
\(625\) 557.177 0.891483
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 20.4066 0.0324429
\(630\) 0 0
\(631\) 1036.16i 1.64210i 0.570859 + 0.821048i \(0.306610\pi\)
−0.570859 + 0.821048i \(0.693390\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 87.5477i 0.137870i
\(636\) 0 0
\(637\) −15.3915 −0.0241624
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1024.87 1.59886 0.799431 0.600757i \(-0.205134\pi\)
0.799431 + 0.600757i \(0.205134\pi\)
\(642\) 0 0
\(643\) − 824.393i − 1.28210i −0.767498 0.641052i \(-0.778498\pi\)
0.767498 0.641052i \(-0.221502\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 414.268i − 0.640290i −0.947369 0.320145i \(-0.896268\pi\)
0.947369 0.320145i \(-0.103732\pi\)
\(648\) 0 0
\(649\) −177.484 −0.273473
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 796.975 1.22048 0.610241 0.792216i \(-0.291072\pi\)
0.610241 + 0.792216i \(0.291072\pi\)
\(654\) 0 0
\(655\) − 112.303i − 0.171455i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 53.1521i 0.0806557i 0.999187 + 0.0403279i \(0.0128402\pi\)
−0.999187 + 0.0403279i \(0.987160\pi\)
\(660\) 0 0
\(661\) 649.763 0.983000 0.491500 0.870878i \(-0.336449\pi\)
0.491500 + 0.870878i \(0.336449\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −79.0783 −0.118915
\(666\) 0 0
\(667\) 160.278i 0.240297i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 1335.39i − 1.99015i
\(672\) 0 0
\(673\) 829.302 1.23225 0.616123 0.787650i \(-0.288702\pi\)
0.616123 + 0.787650i \(0.288702\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1166.76 −1.72343 −0.861716 0.507392i \(-0.830610\pi\)
−0.861716 + 0.507392i \(0.830610\pi\)
\(678\) 0 0
\(679\) − 607.294i − 0.894395i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 690.100i − 1.01039i −0.863004 0.505197i \(-0.831420\pi\)
0.863004 0.505197i \(-0.168580\pi\)
\(684\) 0 0
\(685\) −54.0482 −0.0789024
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 55.8300 0.0810304
\(690\) 0 0
\(691\) 819.624i 1.18614i 0.805150 + 0.593071i \(0.202085\pi\)
−0.805150 + 0.593071i \(0.797915\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) − 101.354i − 0.145833i
\(696\) 0 0
\(697\) −93.2060 −0.133725
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 378.889 0.540498 0.270249 0.962791i \(-0.412894\pi\)
0.270249 + 0.962791i \(0.412894\pi\)
\(702\) 0 0
\(703\) − 197.874i − 0.281470i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 469.435i 0.663982i
\(708\) 0 0
\(709\) 296.838 0.418672 0.209336 0.977844i \(-0.432870\pi\)
0.209336 + 0.977844i \(0.432870\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −0.725445 −0.00101745
\(714\) 0 0
\(715\) − 24.4671i − 0.0342197i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 523.144i − 0.727600i −0.931477 0.363800i \(-0.881479\pi\)
0.931477 0.363800i \(-0.118521\pi\)
\(720\) 0 0
\(721\) 378.280 0.524661
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 919.283 1.26798
\(726\) 0 0
\(727\) 636.546i 0.875579i 0.899077 + 0.437790i \(0.144239\pi\)
−0.899077 + 0.437790i \(0.855761\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 7.42373i 0.0101556i
\(732\) 0 0
\(733\) −420.326 −0.573432 −0.286716 0.958016i \(-0.592564\pi\)
−0.286716 + 0.958016i \(0.592564\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 383.621 0.520517
\(738\) 0 0
\(739\) 987.498i 1.33626i 0.744044 + 0.668131i \(0.232905\pi\)
−0.744044 + 0.668131i \(0.767095\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 463.675i 0.624058i 0.950073 + 0.312029i \(0.101009\pi\)
−0.950073 + 0.312029i \(0.898991\pi\)
\(744\) 0 0
\(745\) −277.628 −0.372656
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −871.313 −1.16330
\(750\) 0 0
\(751\) − 326.135i − 0.434268i −0.976142 0.217134i \(-0.930329\pi\)
0.976142 0.217134i \(-0.0696709\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 180.709i 0.239350i
\(756\) 0 0
\(757\) 404.389 0.534199 0.267100 0.963669i \(-0.413935\pi\)
0.267100 + 0.963669i \(0.413935\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −322.469 −0.423744 −0.211872 0.977297i \(-0.567956\pi\)
−0.211872 + 0.977297i \(0.567956\pi\)
\(762\) 0 0
\(763\) − 780.140i − 1.02246i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 21.5891i − 0.0281475i
\(768\) 0 0
\(769\) −520.611 −0.676997 −0.338499 0.940967i \(-0.609919\pi\)
−0.338499 + 0.940967i \(0.609919\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −664.521 −0.859666 −0.429833 0.902909i \(-0.641427\pi\)
−0.429833 + 0.902909i \(0.641427\pi\)
\(774\) 0 0
\(775\) 4.16083i 0.00536881i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 903.777i 1.16018i
\(780\) 0 0
\(781\) 1238.93 1.58634
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −204.616 −0.260657
\(786\) 0 0
\(787\) 711.537i 0.904113i 0.891989 + 0.452057i \(0.149310\pi\)
−0.891989 + 0.452057i \(0.850690\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 774.966i 0.979729i
\(792\) 0 0
\(793\) 162.437 0.204838
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 691.498 0.867626 0.433813 0.901003i \(-0.357168\pi\)
0.433813 + 0.901003i \(0.357168\pi\)
\(798\) 0 0
\(799\) 89.6028i 0.112144i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 1506.74i − 1.87639i
\(804\) 0 0
\(805\) 25.4974 0.0316738
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −646.526 −0.799167 −0.399583 0.916697i \(-0.630845\pi\)
−0.399583 + 0.916697i \(0.630845\pi\)
\(810\) 0 0
\(811\) 391.295i 0.482485i 0.970465 + 0.241243i \(0.0775549\pi\)
−0.970465 + 0.241243i \(0.922445\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 291.773i 0.358004i
\(816\) 0 0
\(817\) 71.9847 0.0881085
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −271.408 −0.330582 −0.165291 0.986245i \(-0.552856\pi\)
−0.165291 + 0.986245i \(0.552856\pi\)
\(822\) 0 0
\(823\) 1135.15i 1.37928i 0.724152 + 0.689641i \(0.242231\pi\)
−0.724152 + 0.689641i \(0.757769\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 266.279i − 0.321982i −0.986956 0.160991i \(-0.948531\pi\)
0.986956 0.160991i \(-0.0514690\pi\)
\(828\) 0 0
\(829\) −723.241 −0.872426 −0.436213 0.899844i \(-0.643681\pi\)
−0.436213 + 0.899844i \(0.643681\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −11.7211 −0.0140710
\(834\) 0 0
\(835\) 159.298i 0.190776i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1162.52i 1.38560i 0.721131 + 0.692799i \(0.243623\pi\)
−0.721131 + 0.692799i \(0.756377\pi\)
\(840\) 0 0
\(841\) 615.875 0.732312
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −158.725 −0.187840
\(846\) 0 0
\(847\) − 566.218i − 0.668498i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 63.8008i 0.0749716i
\(852\) 0 0
\(853\) −770.807 −0.903642 −0.451821 0.892109i \(-0.649225\pi\)
−0.451821 + 0.892109i \(0.649225\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 383.535 0.447532 0.223766 0.974643i \(-0.428165\pi\)
0.223766 + 0.974643i \(0.428165\pi\)
\(858\) 0 0
\(859\) − 338.150i − 0.393655i −0.980438 0.196828i \(-0.936936\pi\)
0.980438 0.196828i \(-0.0630640\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 783.026i 0.907331i 0.891172 + 0.453665i \(0.149884\pi\)
−0.891172 + 0.453665i \(0.850116\pi\)
\(864\) 0 0
\(865\) 299.854 0.346652
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 990.901 1.14028
\(870\) 0 0
\(871\) 46.6637i 0.0535749i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 298.042i − 0.340619i
\(876\) 0 0
\(877\) −1668.78 −1.90283 −0.951414 0.307916i \(-0.900369\pi\)
−0.951414 + 0.307916i \(0.900369\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1269.86 1.44139 0.720695 0.693252i \(-0.243823\pi\)
0.720695 + 0.693252i \(0.243823\pi\)
\(882\) 0 0
\(883\) − 471.136i − 0.533562i −0.963757 0.266781i \(-0.914040\pi\)
0.963757 0.266781i \(-0.0859601\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 686.179i − 0.773596i −0.922165 0.386798i \(-0.873581\pi\)
0.922165 0.386798i \(-0.126419\pi\)
\(888\) 0 0
\(889\) −580.666 −0.653167
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 868.839 0.972944
\(894\) 0 0
\(895\) − 138.046i − 0.154242i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.59406i 0.00733488i
\(900\) 0 0
\(901\) 42.5165 0.0471881
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −114.833 −0.126887
\(906\) 0 0
\(907\) − 406.300i − 0.447960i −0.974594 0.223980i \(-0.928095\pi\)
0.974594 0.223980i \(-0.0719051\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1340.86i 1.47186i 0.677060 + 0.735928i \(0.263254\pi\)
−0.677060 + 0.735928i \(0.736746\pi\)
\(912\) 0 0
\(913\) −1990.38 −2.18005
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 744.857 0.812276
\(918\) 0 0
\(919\) 1568.33i 1.70656i 0.521452 + 0.853281i \(0.325391\pi\)
−0.521452 + 0.853281i \(0.674609\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 150.704i 0.163276i
\(924\) 0 0
\(925\) 365.933 0.395603
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1252.87 −1.34862 −0.674312 0.738446i \(-0.735560\pi\)
−0.674312 + 0.738446i \(0.735560\pi\)
\(930\) 0 0
\(931\) 113.655i 0.122078i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) − 18.6326i − 0.0199279i
\(936\) 0 0
\(937\) −423.086 −0.451532 −0.225766 0.974182i \(-0.572489\pi\)
−0.225766 + 0.974182i \(0.572489\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 449.433 0.477612 0.238806 0.971067i \(-0.423244\pi\)
0.238806 + 0.971067i \(0.423244\pi\)
\(942\) 0 0
\(943\) − 291.407i − 0.309021i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 723.129i − 0.763599i −0.924245 0.381800i \(-0.875304\pi\)
0.924245 0.381800i \(-0.124696\pi\)
\(948\) 0 0
\(949\) 183.280 0.193130
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −921.421 −0.966863 −0.483432 0.875382i \(-0.660610\pi\)
−0.483432 + 0.875382i \(0.660610\pi\)
\(954\) 0 0
\(955\) − 246.302i − 0.257908i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 358.478i − 0.373804i
\(960\) 0 0
\(961\) 960.970 0.999969
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −276.742 −0.286779
\(966\) 0 0
\(967\) − 317.463i − 0.328297i −0.986436 0.164148i \(-0.947512\pi\)
0.986436 0.164148i \(-0.0524876\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 559.532i 0.576243i 0.957594 + 0.288122i \(0.0930307\pi\)
−0.957594 + 0.288122i \(0.906969\pi\)
\(972\) 0 0
\(973\) 672.237 0.690891
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1742.63 −1.78366 −0.891829 0.452372i \(-0.850578\pi\)
−0.891829 + 0.452372i \(0.850578\pi\)
\(978\) 0 0
\(979\) 976.374i 0.997318i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1071.37i 1.08989i 0.838470 + 0.544947i \(0.183450\pi\)
−0.838470 + 0.544947i \(0.816550\pi\)
\(984\) 0 0
\(985\) 187.805 0.190665
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −23.2102 −0.0234683
\(990\) 0 0
\(991\) − 1167.78i − 1.17839i −0.807992 0.589193i \(-0.799446\pi\)
0.807992 0.589193i \(-0.200554\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 183.961i 0.184885i
\(996\) 0 0
\(997\) −1320.67 −1.32464 −0.662321 0.749220i \(-0.730429\pi\)
−0.662321 + 0.749220i \(0.730429\pi\)
\(998\) 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.g.j.703.6 8
3.2 odd 2 1728.3.g.m.703.4 8
4.3 odd 2 inner 1728.3.g.j.703.5 8
8.3 odd 2 864.3.g.d.703.3 yes 8
8.5 even 2 864.3.g.d.703.4 yes 8
12.11 even 2 1728.3.g.m.703.3 8
24.5 odd 2 864.3.g.b.703.6 yes 8
24.11 even 2 864.3.g.b.703.5 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.g.b.703.5 8 24.11 even 2
864.3.g.b.703.6 yes 8 24.5 odd 2
864.3.g.d.703.3 yes 8 8.3 odd 2
864.3.g.d.703.4 yes 8 8.5 even 2
1728.3.g.j.703.5 8 4.3 odd 2 inner
1728.3.g.j.703.6 8 1.1 even 1 trivial
1728.3.g.m.703.3 8 12.11 even 2
1728.3.g.m.703.4 8 3.2 odd 2