Properties

Label 1728.3.g.j.703.1
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 703.1
Root \(0.500000 + 0.564882i\) of defining polynomial
Character \(\chi\) \(=\) 1728.703
Dual form 1728.3.g.j.703.2

$q$-expansion

\(f(q)\) \(=\) \(q-6.42091 q^{5} -13.8102i q^{7} +O(q^{10})\) \(q-6.42091 q^{5} -13.8102i q^{7} -13.0350i q^{11} -7.16454 q^{13} +31.5918 q^{17} -16.4875i q^{19} -16.9778i q^{23} +16.2281 q^{25} +42.4562 q^{29} -29.6836i q^{31} +88.6741i q^{35} -39.3037 q^{37} +39.8856 q^{41} -16.3291i q^{43} -57.8888i q^{47} -141.722 q^{49} -46.4110 q^{53} +83.6964i q^{55} +14.2179i q^{59} +63.7479 q^{61} +46.0028 q^{65} +32.5634i q^{67} -22.4738i q^{71} +24.9252 q^{73} -180.016 q^{77} +61.9501i q^{79} +44.7901i q^{83} -202.848 q^{85} -1.95333 q^{89} +98.9437i q^{91} +105.865i q^{95} -44.5813 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 8 q^{5}+O(q^{10}) \) 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
\(5\) −6.42091 −1.28418 −0.642091 0.766628i \(-0.721933\pi\)
−0.642091 + 0.766628i \(0.721933\pi\)
\(6\) 0 0
\(7\) − 13.8102i − 1.97289i −0.164102 0.986443i \(-0.552473\pi\)
0.164102 0.986443i \(-0.447527\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 13.0350i − 1.18500i −0.805572 0.592498i \(-0.798142\pi\)
0.805572 0.592498i \(-0.201858\pi\)
\(12\) 0 0
\(13\) −7.16454 −0.551118 −0.275559 0.961284i \(-0.588863\pi\)
−0.275559 + 0.961284i \(0.588863\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 31.5918 1.85834 0.929171 0.369651i \(-0.120523\pi\)
0.929171 + 0.369651i \(0.120523\pi\)
\(18\) 0 0
\(19\) − 16.4875i − 0.867763i −0.900970 0.433881i \(-0.857144\pi\)
0.900970 0.433881i \(-0.142856\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 16.9778i − 0.738163i −0.929397 0.369082i \(-0.879672\pi\)
0.929397 0.369082i \(-0.120328\pi\)
\(24\) 0 0
\(25\) 16.2281 0.649124
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 42.4562 1.46401 0.732004 0.681301i \(-0.238585\pi\)
0.732004 + 0.681301i \(0.238585\pi\)
\(30\) 0 0
\(31\) − 29.6836i − 0.957537i −0.877941 0.478769i \(-0.841083\pi\)
0.877941 0.478769i \(-0.158917\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 88.6741i 2.53355i
\(36\) 0 0
\(37\) −39.3037 −1.06226 −0.531131 0.847289i \(-0.678233\pi\)
−0.531131 + 0.847289i \(0.678233\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 39.8856 0.972819 0.486409 0.873731i \(-0.338306\pi\)
0.486409 + 0.873731i \(0.338306\pi\)
\(42\) 0 0
\(43\) − 16.3291i − 0.379746i −0.981809 0.189873i \(-0.939192\pi\)
0.981809 0.189873i \(-0.0608076\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 57.8888i − 1.23168i −0.787872 0.615839i \(-0.788817\pi\)
0.787872 0.615839i \(-0.211183\pi\)
\(48\) 0 0
\(49\) −141.722 −2.89228
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −46.4110 −0.875680 −0.437840 0.899053i \(-0.644256\pi\)
−0.437840 + 0.899053i \(0.644256\pi\)
\(54\) 0 0
\(55\) 83.6964i 1.52175i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 14.2179i 0.240981i 0.992714 + 0.120491i \(0.0384468\pi\)
−0.992714 + 0.120491i \(0.961553\pi\)
\(60\) 0 0
\(61\) 63.7479 1.04505 0.522524 0.852625i \(-0.324991\pi\)
0.522524 + 0.852625i \(0.324991\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 46.0028 0.707736
\(66\) 0 0
\(67\) 32.5634i 0.486022i 0.970024 + 0.243011i \(0.0781350\pi\)
−0.970024 + 0.243011i \(0.921865\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 22.4738i − 0.316532i −0.987397 0.158266i \(-0.949410\pi\)
0.987397 0.158266i \(-0.0505904\pi\)
\(72\) 0 0
\(73\) 24.9252 0.341441 0.170721 0.985319i \(-0.445390\pi\)
0.170721 + 0.985319i \(0.445390\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −180.016 −2.33786
\(78\) 0 0
\(79\) 61.9501i 0.784178i 0.919927 + 0.392089i \(0.128247\pi\)
−0.919927 + 0.392089i \(0.871753\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 44.7901i 0.539640i 0.962911 + 0.269820i \(0.0869642\pi\)
−0.962911 + 0.269820i \(0.913036\pi\)
\(84\) 0 0
\(85\) −202.848 −2.38645
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.95333 −0.0219475 −0.0109738 0.999940i \(-0.503493\pi\)
−0.0109738 + 0.999940i \(0.503493\pi\)
\(90\) 0 0
\(91\) 98.9437i 1.08729i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 105.865i 1.11437i
\(96\) 0 0
\(97\) −44.5813 −0.459601 −0.229800 0.973238i \(-0.573807\pi\)
−0.229800 + 0.973238i \(0.573807\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −138.393 −1.37022 −0.685112 0.728438i \(-0.740247\pi\)
−0.685112 + 0.728438i \(0.740247\pi\)
\(102\) 0 0
\(103\) 19.5698i 0.189998i 0.995477 + 0.0949991i \(0.0302848\pi\)
−0.995477 + 0.0949991i \(0.969715\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 39.6245i − 0.370323i −0.982708 0.185161i \(-0.940719\pi\)
0.982708 0.185161i \(-0.0592808\pi\)
\(108\) 0 0
\(109\) −45.2012 −0.414690 −0.207345 0.978268i \(-0.566482\pi\)
−0.207345 + 0.978268i \(0.566482\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 111.845 0.989776 0.494888 0.868957i \(-0.335209\pi\)
0.494888 + 0.868957i \(0.335209\pi\)
\(114\) 0 0
\(115\) 109.013i 0.947936i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 436.289i − 3.66630i
\(120\) 0 0
\(121\) −48.9103 −0.404218
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 56.3235 0.450588
\(126\) 0 0
\(127\) − 110.064i − 0.866642i −0.901240 0.433321i \(-0.857342\pi\)
0.901240 0.433321i \(-0.142658\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 14.3672i 0.109673i 0.998495 + 0.0548367i \(0.0174638\pi\)
−0.998495 + 0.0548367i \(0.982536\pi\)
\(132\) 0 0
\(133\) −227.696 −1.71200
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −170.048 −1.24123 −0.620613 0.784117i \(-0.713116\pi\)
−0.620613 + 0.784117i \(0.713116\pi\)
\(138\) 0 0
\(139\) − 213.171i − 1.53360i −0.641884 0.766802i \(-0.721847\pi\)
0.641884 0.766802i \(-0.278153\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 93.3895i 0.653073i
\(144\) 0 0
\(145\) −272.608 −1.88005
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 84.5143 0.567210 0.283605 0.958941i \(-0.408470\pi\)
0.283605 + 0.958941i \(0.408470\pi\)
\(150\) 0 0
\(151\) − 180.407i − 1.19475i −0.801962 0.597376i \(-0.796210\pi\)
0.801962 0.597376i \(-0.203790\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 190.596i 1.22965i
\(156\) 0 0
\(157\) −49.7071 −0.316605 −0.158303 0.987391i \(-0.550602\pi\)
−0.158303 + 0.987391i \(0.550602\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −234.466 −1.45631
\(162\) 0 0
\(163\) 49.3718i 0.302894i 0.988465 + 0.151447i \(0.0483933\pi\)
−0.988465 + 0.151447i \(0.951607\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 161.617i 0.967766i 0.875133 + 0.483883i \(0.160774\pi\)
−0.875133 + 0.483883i \(0.839226\pi\)
\(168\) 0 0
\(169\) −117.669 −0.696269
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 63.6058 0.367664 0.183832 0.982958i \(-0.441150\pi\)
0.183832 + 0.982958i \(0.441150\pi\)
\(174\) 0 0
\(175\) − 224.114i − 1.28065i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 203.780i 1.13844i 0.822186 + 0.569219i \(0.192754\pi\)
−0.822186 + 0.569219i \(0.807246\pi\)
\(180\) 0 0
\(181\) −23.1886 −0.128114 −0.0640570 0.997946i \(-0.520404\pi\)
−0.0640570 + 0.997946i \(0.520404\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 252.366 1.36414
\(186\) 0 0
\(187\) − 411.798i − 2.20213i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 247.017i 1.29328i 0.762793 + 0.646642i \(0.223827\pi\)
−0.762793 + 0.646642i \(0.776173\pi\)
\(192\) 0 0
\(193\) 126.813 0.657063 0.328531 0.944493i \(-0.393446\pi\)
0.328531 + 0.944493i \(0.393446\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −47.1824 −0.239505 −0.119752 0.992804i \(-0.538210\pi\)
−0.119752 + 0.992804i \(0.538210\pi\)
\(198\) 0 0
\(199\) 298.835i 1.50168i 0.660482 + 0.750842i \(0.270352\pi\)
−0.660482 + 0.750842i \(0.729648\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 586.329i − 2.88832i
\(204\) 0 0
\(205\) −256.102 −1.24928
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −214.914 −1.02830
\(210\) 0 0
\(211\) − 270.034i − 1.27978i −0.768466 0.639891i \(-0.778980\pi\)
0.768466 0.639891i \(-0.221020\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 104.848i 0.487663i
\(216\) 0 0
\(217\) −409.937 −1.88911
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −226.341 −1.02417
\(222\) 0 0
\(223\) 171.763i 0.770237i 0.922867 + 0.385118i \(0.125839\pi\)
−0.922867 + 0.385118i \(0.874161\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 146.057i − 0.643424i −0.946838 0.321712i \(-0.895742\pi\)
0.946838 0.321712i \(-0.104258\pi\)
\(228\) 0 0
\(229\) 325.016 1.41928 0.709641 0.704563i \(-0.248857\pi\)
0.709641 + 0.704563i \(0.248857\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 405.093 1.73860 0.869299 0.494286i \(-0.164570\pi\)
0.869299 + 0.494286i \(0.164570\pi\)
\(234\) 0 0
\(235\) 371.699i 1.58170i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 85.3071i 0.356933i 0.983946 + 0.178467i \(0.0571137\pi\)
−0.983946 + 0.178467i \(0.942886\pi\)
\(240\) 0 0
\(241\) −357.900 −1.48506 −0.742532 0.669811i \(-0.766375\pi\)
−0.742532 + 0.669811i \(0.766375\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 909.983 3.71422
\(246\) 0 0
\(247\) 118.125i 0.478240i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 88.8097i − 0.353823i −0.984227 0.176912i \(-0.943389\pi\)
0.984227 0.176912i \(-0.0566107\pi\)
\(252\) 0 0
\(253\) −221.304 −0.874721
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 237.843 0.925460 0.462730 0.886499i \(-0.346870\pi\)
0.462730 + 0.886499i \(0.346870\pi\)
\(258\) 0 0
\(259\) 542.793i 2.09572i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 309.702i 1.17757i 0.808289 + 0.588786i \(0.200394\pi\)
−0.808289 + 0.588786i \(0.799606\pi\)
\(264\) 0 0
\(265\) 298.001 1.12453
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 490.940 1.82505 0.912527 0.409016i \(-0.134128\pi\)
0.912527 + 0.409016i \(0.134128\pi\)
\(270\) 0 0
\(271\) − 0.493459i − 0.00182088i −1.00000 0.000910442i \(-0.999710\pi\)
1.00000 0.000910442i \(-0.000289803\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 211.533i − 0.769210i
\(276\) 0 0
\(277\) 10.8497 0.0391686 0.0195843 0.999808i \(-0.493766\pi\)
0.0195843 + 0.999808i \(0.493766\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −351.417 −1.25059 −0.625297 0.780387i \(-0.715022\pi\)
−0.625297 + 0.780387i \(0.715022\pi\)
\(282\) 0 0
\(283\) 62.5357i 0.220974i 0.993878 + 0.110487i \(0.0352411\pi\)
−0.993878 + 0.110487i \(0.964759\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 550.828i − 1.91926i
\(288\) 0 0
\(289\) 709.042 2.45343
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −74.3035 −0.253596 −0.126798 0.991929i \(-0.540470\pi\)
−0.126798 + 0.991929i \(0.540470\pi\)
\(294\) 0 0
\(295\) − 91.2919i − 0.309464i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 121.638i 0.406815i
\(300\) 0 0
\(301\) −225.508 −0.749195
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −409.320 −1.34203
\(306\) 0 0
\(307\) 460.555i 1.50018i 0.661336 + 0.750089i \(0.269990\pi\)
−0.661336 + 0.750089i \(0.730010\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 433.687i 1.39449i 0.716832 + 0.697246i \(0.245591\pi\)
−0.716832 + 0.697246i \(0.754409\pi\)
\(312\) 0 0
\(313\) −434.686 −1.38877 −0.694387 0.719602i \(-0.744324\pi\)
−0.694387 + 0.719602i \(0.744324\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 212.120 0.669149 0.334574 0.942369i \(-0.391407\pi\)
0.334574 + 0.942369i \(0.391407\pi\)
\(318\) 0 0
\(319\) − 553.415i − 1.73484i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 520.870i − 1.61260i
\(324\) 0 0
\(325\) −116.267 −0.357744
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −799.457 −2.42996
\(330\) 0 0
\(331\) − 578.614i − 1.74808i −0.485854 0.874040i \(-0.661491\pi\)
0.485854 0.874040i \(-0.338509\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) − 209.087i − 0.624140i
\(336\) 0 0
\(337\) −214.736 −0.637198 −0.318599 0.947890i \(-0.603212\pi\)
−0.318599 + 0.947890i \(0.603212\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −386.925 −1.13468
\(342\) 0 0
\(343\) 1280.51i 3.73326i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 280.616i 0.808692i 0.914606 + 0.404346i \(0.132501\pi\)
−0.914606 + 0.404346i \(0.867499\pi\)
\(348\) 0 0
\(349\) −232.447 −0.666036 −0.333018 0.942920i \(-0.608067\pi\)
−0.333018 + 0.942920i \(0.608067\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −97.0279 −0.274866 −0.137433 0.990511i \(-0.543885\pi\)
−0.137433 + 0.990511i \(0.543885\pi\)
\(354\) 0 0
\(355\) 144.302i 0.406485i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) − 37.7190i − 0.105067i −0.998619 0.0525334i \(-0.983270\pi\)
0.998619 0.0525334i \(-0.0167296\pi\)
\(360\) 0 0
\(361\) 89.1625 0.246988
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −160.043 −0.438473
\(366\) 0 0
\(367\) − 303.506i − 0.826991i −0.910506 0.413496i \(-0.864308\pi\)
0.910506 0.413496i \(-0.135692\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 640.946i 1.72762i
\(372\) 0 0
\(373\) 576.215 1.54481 0.772406 0.635129i \(-0.219053\pi\)
0.772406 + 0.635129i \(0.219053\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −304.179 −0.806841
\(378\) 0 0
\(379\) − 36.0808i − 0.0952001i −0.998866 0.0476000i \(-0.984843\pi\)
0.998866 0.0476000i \(-0.0151573\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 521.053i 1.36045i 0.733003 + 0.680225i \(0.238118\pi\)
−0.733003 + 0.680225i \(0.761882\pi\)
\(384\) 0 0
\(385\) 1155.86 3.00224
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −472.918 −1.21573 −0.607864 0.794041i \(-0.707973\pi\)
−0.607864 + 0.794041i \(0.707973\pi\)
\(390\) 0 0
\(391\) − 536.358i − 1.37176i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 397.776i − 1.00703i
\(396\) 0 0
\(397\) 209.784 0.528423 0.264212 0.964465i \(-0.414888\pi\)
0.264212 + 0.964465i \(0.414888\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 664.907 1.65812 0.829062 0.559157i \(-0.188875\pi\)
0.829062 + 0.559157i \(0.188875\pi\)
\(402\) 0 0
\(403\) 212.670i 0.527716i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 512.323i 1.25878i
\(408\) 0 0
\(409\) −573.373 −1.40189 −0.700945 0.713215i \(-0.747238\pi\)
−0.700945 + 0.713215i \(0.747238\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 196.352 0.475429
\(414\) 0 0
\(415\) − 287.594i − 0.692996i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 672.621i − 1.60530i −0.596450 0.802651i \(-0.703422\pi\)
0.596450 0.802651i \(-0.296578\pi\)
\(420\) 0 0
\(421\) 659.810 1.56724 0.783622 0.621238i \(-0.213370\pi\)
0.783622 + 0.621238i \(0.213370\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 512.675 1.20629
\(426\) 0 0
\(427\) − 880.372i − 2.06176i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 360.903i 0.837361i 0.908134 + 0.418681i \(0.137507\pi\)
−0.908134 + 0.418681i \(0.862493\pi\)
\(432\) 0 0
\(433\) −319.790 −0.738546 −0.369273 0.929321i \(-0.620393\pi\)
−0.369273 + 0.929321i \(0.620393\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −279.921 −0.640551
\(438\) 0 0
\(439\) 80.6635i 0.183744i 0.995771 + 0.0918719i \(0.0292850\pi\)
−0.995771 + 0.0918719i \(0.970715\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 551.362i 1.24461i 0.782775 + 0.622305i \(0.213803\pi\)
−0.782775 + 0.622305i \(0.786197\pi\)
\(444\) 0 0
\(445\) 12.5422 0.0281847
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −43.0735 −0.0959321 −0.0479661 0.998849i \(-0.515274\pi\)
−0.0479661 + 0.998849i \(0.515274\pi\)
\(450\) 0 0
\(451\) − 519.907i − 1.15279i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) − 635.309i − 1.39628i
\(456\) 0 0
\(457\) 205.936 0.450626 0.225313 0.974286i \(-0.427659\pi\)
0.225313 + 0.974286i \(0.427659\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −650.729 −1.41156 −0.705780 0.708431i \(-0.749403\pi\)
−0.705780 + 0.708431i \(0.749403\pi\)
\(462\) 0 0
\(463\) − 333.871i − 0.721103i −0.932739 0.360552i \(-0.882588\pi\)
0.932739 0.360552i \(-0.117412\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 359.357i − 0.769502i −0.923020 0.384751i \(-0.874287\pi\)
0.923020 0.384751i \(-0.125713\pi\)
\(468\) 0 0
\(469\) 449.708 0.958865
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −212.849 −0.449998
\(474\) 0 0
\(475\) − 267.561i − 0.563286i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) − 635.502i − 1.32673i −0.748297 0.663363i \(-0.769128\pi\)
0.748297 0.663363i \(-0.230872\pi\)
\(480\) 0 0
\(481\) 281.593 0.585432
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 286.252 0.590211
\(486\) 0 0
\(487\) 233.959i 0.480409i 0.970722 + 0.240205i \(0.0772146\pi\)
−0.970722 + 0.240205i \(0.922785\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 494.173i 1.00646i 0.864152 + 0.503231i \(0.167855\pi\)
−0.864152 + 0.503231i \(0.832145\pi\)
\(492\) 0 0
\(493\) 1341.27 2.72063
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −310.368 −0.624482
\(498\) 0 0
\(499\) − 468.260i − 0.938396i −0.883093 0.469198i \(-0.844543\pi\)
0.883093 0.469198i \(-0.155457\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 59.2449i − 0.117783i −0.998264 0.0588916i \(-0.981243\pi\)
0.998264 0.0588916i \(-0.0187566\pi\)
\(504\) 0 0
\(505\) 888.607 1.75962
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 161.243 0.316784 0.158392 0.987376i \(-0.449369\pi\)
0.158392 + 0.987376i \(0.449369\pi\)
\(510\) 0 0
\(511\) − 344.222i − 0.673625i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 125.656i − 0.243992i
\(516\) 0 0
\(517\) −754.579 −1.45953
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 350.872 0.673458 0.336729 0.941602i \(-0.390679\pi\)
0.336729 + 0.941602i \(0.390679\pi\)
\(522\) 0 0
\(523\) 729.342i 1.39454i 0.716811 + 0.697268i \(0.245601\pi\)
−0.716811 + 0.697268i \(0.754399\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 937.760i − 1.77943i
\(528\) 0 0
\(529\) 240.756 0.455115
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −285.762 −0.536138
\(534\) 0 0
\(535\) 254.426i 0.475562i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 1847.34i 3.42734i
\(540\) 0 0
\(541\) 365.728 0.676021 0.338011 0.941142i \(-0.390246\pi\)
0.338011 + 0.941142i \(0.390246\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 290.233 0.532537
\(546\) 0 0
\(547\) 468.636i 0.856738i 0.903604 + 0.428369i \(0.140912\pi\)
−0.903604 + 0.428369i \(0.859088\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) − 699.997i − 1.27041i
\(552\) 0 0
\(553\) 855.543 1.54709
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −288.006 −0.517066 −0.258533 0.966002i \(-0.583239\pi\)
−0.258533 + 0.966002i \(0.583239\pi\)
\(558\) 0 0
\(559\) 116.990i 0.209285i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 158.644i 0.281783i 0.990025 + 0.140892i \(0.0449969\pi\)
−0.990025 + 0.140892i \(0.955003\pi\)
\(564\) 0 0
\(565\) −718.145 −1.27105
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 438.309 0.770315 0.385157 0.922851i \(-0.374147\pi\)
0.385157 + 0.922851i \(0.374147\pi\)
\(570\) 0 0
\(571\) − 683.282i − 1.19664i −0.801257 0.598321i \(-0.795835\pi\)
0.801257 0.598321i \(-0.204165\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 275.517i − 0.479160i
\(576\) 0 0
\(577\) −166.370 −0.288337 −0.144168 0.989553i \(-0.546051\pi\)
−0.144168 + 0.989553i \(0.546051\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 618.561 1.06465
\(582\) 0 0
\(583\) 604.966i 1.03768i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 1095.17i − 1.86570i −0.360265 0.932850i \(-0.617314\pi\)
0.360265 0.932850i \(-0.382686\pi\)
\(588\) 0 0
\(589\) −489.409 −0.830915
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 131.905 0.222437 0.111218 0.993796i \(-0.464525\pi\)
0.111218 + 0.993796i \(0.464525\pi\)
\(594\) 0 0
\(595\) 2801.38i 4.70819i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 228.590i 0.381619i 0.981627 + 0.190809i \(0.0611112\pi\)
−0.981627 + 0.190809i \(0.938889\pi\)
\(600\) 0 0
\(601\) 945.987 1.57402 0.787010 0.616940i \(-0.211628\pi\)
0.787010 + 0.616940i \(0.211628\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 314.049 0.519089
\(606\) 0 0
\(607\) − 539.641i − 0.889029i −0.895772 0.444514i \(-0.853376\pi\)
0.895772 0.444514i \(-0.146624\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 414.747i 0.678800i
\(612\) 0 0
\(613\) 305.962 0.499123 0.249561 0.968359i \(-0.419714\pi\)
0.249561 + 0.968359i \(0.419714\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 409.095 0.663038 0.331519 0.943449i \(-0.392439\pi\)
0.331519 + 0.943449i \(0.392439\pi\)
\(618\) 0 0
\(619\) − 453.938i − 0.733341i −0.930351 0.366670i \(-0.880498\pi\)
0.930351 0.366670i \(-0.119502\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 26.9759i 0.0433000i
\(624\) 0 0
\(625\) −767.351 −1.22776
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −1241.68 −1.97405
\(630\) 0 0
\(631\) 393.383i 0.623428i 0.950176 + 0.311714i \(0.100903\pi\)
−0.950176 + 0.311714i \(0.899097\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 706.709i 1.11293i
\(636\) 0 0
\(637\) 1015.37 1.59399
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 982.724 1.53311 0.766555 0.642179i \(-0.221969\pi\)
0.766555 + 0.642179i \(0.221969\pi\)
\(642\) 0 0
\(643\) 1078.21i 1.67684i 0.545025 + 0.838420i \(0.316520\pi\)
−0.545025 + 0.838420i \(0.683480\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 38.4217i 0.0593844i 0.999559 + 0.0296922i \(0.00945271\pi\)
−0.999559 + 0.0296922i \(0.990547\pi\)
\(648\) 0 0
\(649\) 185.330 0.285562
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 317.953 0.486911 0.243456 0.969912i \(-0.421719\pi\)
0.243456 + 0.969912i \(0.421719\pi\)
\(654\) 0 0
\(655\) − 92.2507i − 0.140841i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 792.921i − 1.20322i −0.798790 0.601609i \(-0.794526\pi\)
0.798790 0.601609i \(-0.205474\pi\)
\(660\) 0 0
\(661\) 1029.55 1.55757 0.778786 0.627290i \(-0.215836\pi\)
0.778786 + 0.627290i \(0.215836\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 1462.01 2.19852
\(666\) 0 0
\(667\) − 720.811i − 1.08068i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) − 830.952i − 1.23838i
\(672\) 0 0
\(673\) −6.46119 −0.00960059 −0.00480029 0.999988i \(-0.501528\pi\)
−0.00480029 + 0.999988i \(0.501528\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −21.6650 −0.0320015 −0.0160008 0.999872i \(-0.505093\pi\)
−0.0160008 + 0.999872i \(0.505093\pi\)
\(678\) 0 0
\(679\) 615.676i 0.906740i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 503.008i − 0.736469i −0.929733 0.368234i \(-0.879962\pi\)
0.929733 0.368234i \(-0.120038\pi\)
\(684\) 0 0
\(685\) 1091.86 1.59396
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 332.514 0.482603
\(690\) 0 0
\(691\) − 514.362i − 0.744374i −0.928158 0.372187i \(-0.878608\pi\)
0.928158 0.372187i \(-0.121392\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1368.75i 1.96943i
\(696\) 0 0
\(697\) 1260.06 1.80783
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −457.404 −0.652502 −0.326251 0.945283i \(-0.605786\pi\)
−0.326251 + 0.945283i \(0.605786\pi\)
\(702\) 0 0
\(703\) 648.020i 0.921792i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1911.23i 2.70330i
\(708\) 0 0
\(709\) −390.207 −0.550363 −0.275182 0.961392i \(-0.588738\pi\)
−0.275182 + 0.961392i \(0.588738\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −503.962 −0.706819
\(714\) 0 0
\(715\) − 599.646i − 0.838665i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 979.508i 1.36232i 0.732134 + 0.681160i \(0.238524\pi\)
−0.732134 + 0.681160i \(0.761476\pi\)
\(720\) 0 0
\(721\) 270.263 0.374845
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 688.984 0.950323
\(726\) 0 0
\(727\) 553.802i 0.761764i 0.924624 + 0.380882i \(0.124380\pi\)
−0.924624 + 0.380882i \(0.875620\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 515.865i − 0.705697i
\(732\) 0 0
\(733\) −250.392 −0.341599 −0.170800 0.985306i \(-0.554635\pi\)
−0.170800 + 0.985306i \(0.554635\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 424.463 0.575934
\(738\) 0 0
\(739\) − 403.959i − 0.546630i −0.961925 0.273315i \(-0.911880\pi\)
0.961925 0.273315i \(-0.0881201\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 142.091i 0.191240i 0.995418 + 0.0956201i \(0.0304834\pi\)
−0.995418 + 0.0956201i \(0.969517\pi\)
\(744\) 0 0
\(745\) −542.659 −0.728401
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −547.223 −0.730605
\(750\) 0 0
\(751\) − 667.242i − 0.888471i −0.895910 0.444235i \(-0.853475\pi\)
0.895910 0.444235i \(-0.146525\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 1158.38i 1.53428i
\(756\) 0 0
\(757\) −970.866 −1.28252 −0.641259 0.767325i \(-0.721587\pi\)
−0.641259 + 0.767325i \(0.721587\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −295.323 −0.388072 −0.194036 0.980994i \(-0.562158\pi\)
−0.194036 + 0.980994i \(0.562158\pi\)
\(762\) 0 0
\(763\) 624.238i 0.818136i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 101.865i − 0.132809i
\(768\) 0 0
\(769\) −552.240 −0.718128 −0.359064 0.933313i \(-0.616904\pi\)
−0.359064 + 0.933313i \(0.616904\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 797.819 1.03211 0.516053 0.856556i \(-0.327401\pi\)
0.516053 + 0.856556i \(0.327401\pi\)
\(774\) 0 0
\(775\) − 481.709i − 0.621561i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 657.613i − 0.844176i
\(780\) 0 0
\(781\) −292.945 −0.375090
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 319.165 0.406579
\(786\) 0 0
\(787\) − 1140.72i − 1.44946i −0.689033 0.724730i \(-0.741965\pi\)
0.689033 0.724730i \(-0.258035\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 1544.60i − 1.95272i
\(792\) 0 0
\(793\) −456.724 −0.575945
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1453.75 −1.82403 −0.912016 0.410154i \(-0.865475\pi\)
−0.912016 + 0.410154i \(0.865475\pi\)
\(798\) 0 0
\(799\) − 1828.81i − 2.28888i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 324.899i − 0.404607i
\(804\) 0 0
\(805\) 1505.49 1.87017
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −981.664 −1.21343 −0.606714 0.794920i \(-0.707513\pi\)
−0.606714 + 0.794920i \(0.707513\pi\)
\(810\) 0 0
\(811\) − 1090.81i − 1.34502i −0.740087 0.672511i \(-0.765216\pi\)
0.740087 0.672511i \(-0.234784\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 317.012i − 0.388972i
\(816\) 0 0
\(817\) −269.225 −0.329529
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −282.456 −0.344039 −0.172019 0.985094i \(-0.555029\pi\)
−0.172019 + 0.985094i \(0.555029\pi\)
\(822\) 0 0
\(823\) 497.382i 0.604352i 0.953252 + 0.302176i \(0.0977131\pi\)
−0.953252 + 0.302176i \(0.902287\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 1159.09i − 1.40156i −0.713379 0.700778i \(-0.752836\pi\)
0.713379 0.700778i \(-0.247164\pi\)
\(828\) 0 0
\(829\) 806.908 0.973351 0.486675 0.873583i \(-0.338209\pi\)
0.486675 + 0.873583i \(0.338209\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −4477.25 −5.37485
\(834\) 0 0
\(835\) − 1037.73i − 1.24279i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) − 1045.71i − 1.24638i −0.782071 0.623189i \(-0.785837\pi\)
0.782071 0.623189i \(-0.214163\pi\)
\(840\) 0 0
\(841\) 961.530 1.14332
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 755.545 0.894136
\(846\) 0 0
\(847\) 675.462i 0.797476i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 667.289i 0.784123i
\(852\) 0 0
\(853\) −1126.10 −1.32017 −0.660083 0.751193i \(-0.729479\pi\)
−0.660083 + 0.751193i \(0.729479\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −112.853 −0.131684 −0.0658419 0.997830i \(-0.520973\pi\)
−0.0658419 + 0.997830i \(0.520973\pi\)
\(858\) 0 0
\(859\) − 679.360i − 0.790873i −0.918493 0.395437i \(-0.870593\pi\)
0.918493 0.395437i \(-0.129407\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 560.002i 0.648901i 0.945903 + 0.324451i \(0.105179\pi\)
−0.945903 + 0.324451i \(0.894821\pi\)
\(864\) 0 0
\(865\) −408.408 −0.472147
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 807.517 0.929249
\(870\) 0 0
\(871\) − 233.302i − 0.267855i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 777.840i − 0.888960i
\(876\) 0 0
\(877\) 775.431 0.884186 0.442093 0.896969i \(-0.354236\pi\)
0.442093 + 0.896969i \(0.354236\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −1024.25 −1.16260 −0.581298 0.813691i \(-0.697455\pi\)
−0.581298 + 0.813691i \(0.697455\pi\)
\(882\) 0 0
\(883\) 1301.28i 1.47370i 0.676057 + 0.736850i \(0.263687\pi\)
−0.676057 + 0.736850i \(0.736313\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 1284.02i − 1.44760i −0.690010 0.723800i \(-0.742394\pi\)
0.690010 0.723800i \(-0.257606\pi\)
\(888\) 0 0
\(889\) −1520.00 −1.70979
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −954.442 −1.06880
\(894\) 0 0
\(895\) − 1308.46i − 1.46196i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 1260.26i − 1.40184i
\(900\) 0 0
\(901\) −1466.21 −1.62731
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 148.892 0.164522
\(906\) 0 0
\(907\) 1042.84i 1.14976i 0.818236 + 0.574882i \(0.194952\pi\)
−0.818236 + 0.574882i \(0.805048\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 857.109i 0.940844i 0.882442 + 0.470422i \(0.155898\pi\)
−0.882442 + 0.470422i \(0.844102\pi\)
\(912\) 0 0
\(913\) 583.838 0.639472
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 198.414 0.216373
\(918\) 0 0
\(919\) 451.721i 0.491536i 0.969329 + 0.245768i \(0.0790401\pi\)
−0.969329 + 0.245768i \(0.920960\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 161.014i 0.174447i
\(924\) 0 0
\(925\) −637.825 −0.689541
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1826.21 1.96578 0.982890 0.184191i \(-0.0589666\pi\)
0.982890 + 0.184191i \(0.0589666\pi\)
\(930\) 0 0
\(931\) 2336.64i 2.50981i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 2644.12i 2.82794i
\(936\) 0 0
\(937\) −264.729 −0.282529 −0.141264 0.989972i \(-0.545117\pi\)
−0.141264 + 0.989972i \(0.545117\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1401.16 1.48901 0.744505 0.667616i \(-0.232685\pi\)
0.744505 + 0.667616i \(0.232685\pi\)
\(942\) 0 0
\(943\) − 677.167i − 0.718099i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 389.952i 0.411776i 0.978576 + 0.205888i \(0.0660082\pi\)
−0.978576 + 0.205888i \(0.933992\pi\)
\(948\) 0 0
\(949\) −178.578 −0.188175
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1291.61 −1.35531 −0.677656 0.735379i \(-0.737004\pi\)
−0.677656 + 0.735379i \(0.737004\pi\)
\(954\) 0 0
\(955\) − 1586.08i − 1.66081i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2348.40i 2.44880i
\(960\) 0 0
\(961\) 79.8811 0.0831229
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −814.256 −0.843789
\(966\) 0 0
\(967\) − 1802.57i − 1.86408i −0.362349 0.932042i \(-0.618025\pi\)
0.362349 0.932042i \(-0.381975\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 473.260i 0.487395i 0.969851 + 0.243697i \(0.0783603\pi\)
−0.969851 + 0.243697i \(0.921640\pi\)
\(972\) 0 0
\(973\) −2943.93 −3.02563
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 656.147 0.671594 0.335797 0.941934i \(-0.390994\pi\)
0.335797 + 0.941934i \(0.390994\pi\)
\(978\) 0 0
\(979\) 25.4616i 0.0260078i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) − 267.000i − 0.271617i −0.990735 0.135809i \(-0.956637\pi\)
0.990735 0.135809i \(-0.0433632\pi\)
\(984\) 0 0
\(985\) 302.954 0.307568
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −277.231 −0.280314
\(990\) 0 0
\(991\) − 549.499i − 0.554489i −0.960799 0.277245i \(-0.910579\pi\)
0.960799 0.277245i \(-0.0894213\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 1918.79i − 1.92844i
\(996\) 0 0
\(997\) −546.254 −0.547898 −0.273949 0.961744i \(-0.588330\pi\)
−0.273949 + 0.961744i \(0.588330\pi\)
\(998\) 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.703.1 8
3.2 odd 2 1728.3.g.m.703.7 8
4.3 odd 2 inner 1728.3.g.j.703.2 8
8.3 odd 2 864.3.g.d.703.8 yes 8
8.5 even 2 864.3.g.d.703.7 yes 8
12.11 even 2 1728.3.g.m.703.8 8
24.5 odd 2 864.3.g.b.703.1 8
24.11 even 2 864.3.g.b.703.2 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
864.3.g.b.703.1 8 24.5 odd 2
864.3.g.b.703.2 yes 8 24.11 even 2
864.3.g.d.703.7 yes 8 8.5 even 2
864.3.g.d.703.8 yes 8 8.3 odd 2
1728.3.g.j.703.1 8 1.1 even 1 trivial
1728.3.g.j.703.2 8 4.3 odd 2 inner
1728.3.g.m.703.7 8 3.2 odd 2
1728.3.g.m.703.8 8 12.11 even 2