Properties

Label 3072.2.a.b.1.1
Level $3072$
Weight $2$
Character 3072.1
Self dual yes
Analytic conductor $24.530$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3072 = 2^{10} \cdot 3 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3072.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(24.5300435009\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1536)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3072.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.41421 q^{5} -2.82843 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.41421 q^{5} -2.82843 q^{7} +1.00000 q^{9} +4.24264 q^{13} +1.41421 q^{15} +4.00000 q^{17} -8.00000 q^{19} +2.82843 q^{21} +5.65685 q^{23} -3.00000 q^{25} -1.00000 q^{27} +1.41421 q^{29} +2.82843 q^{31} +4.00000 q^{35} +4.24264 q^{37} -4.24264 q^{39} -4.00000 q^{41} -1.41421 q^{45} +11.3137 q^{47} +1.00000 q^{49} -4.00000 q^{51} -7.07107 q^{53} +8.00000 q^{57} -12.0000 q^{59} +1.41421 q^{61} -2.82843 q^{63} -6.00000 q^{65} +4.00000 q^{67} -5.65685 q^{69} -11.3137 q^{71} +14.0000 q^{73} +3.00000 q^{75} +8.48528 q^{79} +1.00000 q^{81} -16.0000 q^{83} -5.65685 q^{85} -1.41421 q^{87} +6.00000 q^{89} -12.0000 q^{91} -2.82843 q^{93} +11.3137 q^{95} +16.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{9} + 8q^{17} - 16q^{19} - 6q^{25} - 2q^{27} + 8q^{35} - 8q^{41} + 2q^{49} - 8q^{51} + 16q^{57} - 24q^{59} - 12q^{65} + 8q^{67} + 28q^{73} + 6q^{75} + 2q^{81} - 32q^{83} + 12q^{89} - 24q^{91} + 32q^{97} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.41421 −0.632456 −0.316228 0.948683i \(-0.602416\pi\)
−0.316228 + 0.948683i \(0.602416\pi\)
\(6\) 0 0
\(7\) −2.82843 −1.06904 −0.534522 0.845154i \(-0.679509\pi\)
−0.534522 + 0.845154i \(0.679509\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 0 0
\(15\) 1.41421 0.365148
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) −8.00000 −1.83533 −0.917663 0.397360i \(-0.869927\pi\)
−0.917663 + 0.397360i \(0.869927\pi\)
\(20\) 0 0
\(21\) 2.82843 0.617213
\(22\) 0 0
\(23\) 5.65685 1.17954 0.589768 0.807573i \(-0.299219\pi\)
0.589768 + 0.807573i \(0.299219\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 1.41421 0.262613 0.131306 0.991342i \(-0.458083\pi\)
0.131306 + 0.991342i \(0.458083\pi\)
\(30\) 0 0
\(31\) 2.82843 0.508001 0.254000 0.967204i \(-0.418254\pi\)
0.254000 + 0.967204i \(0.418254\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 4.24264 0.697486 0.348743 0.937218i \(-0.386609\pi\)
0.348743 + 0.937218i \(0.386609\pi\)
\(38\) 0 0
\(39\) −4.24264 −0.679366
\(40\) 0 0
\(41\) −4.00000 −0.624695 −0.312348 0.949968i \(-0.601115\pi\)
−0.312348 + 0.949968i \(0.601115\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −1.41421 −0.210819
\(46\) 0 0
\(47\) 11.3137 1.65027 0.825137 0.564933i \(-0.191098\pi\)
0.825137 + 0.564933i \(0.191098\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −4.00000 −0.560112
\(52\) 0 0
\(53\) −7.07107 −0.971286 −0.485643 0.874157i \(-0.661414\pi\)
−0.485643 + 0.874157i \(0.661414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 8.00000 1.05963
\(58\) 0 0
\(59\) −12.0000 −1.56227 −0.781133 0.624364i \(-0.785358\pi\)
−0.781133 + 0.624364i \(0.785358\pi\)
\(60\) 0 0
\(61\) 1.41421 0.181071 0.0905357 0.995893i \(-0.471142\pi\)
0.0905357 + 0.995893i \(0.471142\pi\)
\(62\) 0 0
\(63\) −2.82843 −0.356348
\(64\) 0 0
\(65\) −6.00000 −0.744208
\(66\) 0 0
\(67\) 4.00000 0.488678 0.244339 0.969690i \(-0.421429\pi\)
0.244339 + 0.969690i \(0.421429\pi\)
\(68\) 0 0
\(69\) −5.65685 −0.681005
\(70\) 0 0
\(71\) −11.3137 −1.34269 −0.671345 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) 3.00000 0.346410
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.48528 0.954669 0.477334 0.878722i \(-0.341603\pi\)
0.477334 + 0.878722i \(0.341603\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −16.0000 −1.75623 −0.878114 0.478451i \(-0.841198\pi\)
−0.878114 + 0.478451i \(0.841198\pi\)
\(84\) 0 0
\(85\) −5.65685 −0.613572
\(86\) 0 0
\(87\) −1.41421 −0.151620
\(88\) 0 0
\(89\) 6.00000 0.635999 0.317999 0.948091i \(-0.396989\pi\)
0.317999 + 0.948091i \(0.396989\pi\)
\(90\) 0 0
\(91\) −12.0000 −1.25794
\(92\) 0 0
\(93\) −2.82843 −0.293294
\(94\) 0 0
\(95\) 11.3137 1.16076
\(96\) 0 0
\(97\) 16.0000 1.62455 0.812277 0.583272i \(-0.198228\pi\)
0.812277 + 0.583272i \(0.198228\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −15.5563 −1.54791 −0.773957 0.633238i \(-0.781726\pi\)
−0.773957 + 0.633238i \(0.781726\pi\)
\(102\) 0 0
\(103\) −19.7990 −1.95085 −0.975426 0.220326i \(-0.929288\pi\)
−0.975426 + 0.220326i \(0.929288\pi\)
\(104\) 0 0
\(105\) −4.00000 −0.390360
\(106\) 0 0
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) −4.24264 −0.406371 −0.203186 0.979140i \(-0.565129\pi\)
−0.203186 + 0.979140i \(0.565129\pi\)
\(110\) 0 0
\(111\) −4.24264 −0.402694
\(112\) 0 0
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 0 0
\(117\) 4.24264 0.392232
\(118\) 0 0
\(119\) −11.3137 −1.03713
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 4.00000 0.360668
\(124\) 0 0
\(125\) 11.3137 1.01193
\(126\) 0 0
\(127\) −8.48528 −0.752947 −0.376473 0.926427i \(-0.622863\pi\)
−0.376473 + 0.926427i \(0.622863\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 0 0
\(133\) 22.6274 1.96205
\(134\) 0 0
\(135\) 1.41421 0.121716
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −11.3137 −0.952786
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −2.00000 −0.166091
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 12.7279 1.04271 0.521356 0.853339i \(-0.325426\pi\)
0.521356 + 0.853339i \(0.325426\pi\)
\(150\) 0 0
\(151\) 2.82843 0.230174 0.115087 0.993355i \(-0.463285\pi\)
0.115087 + 0.993355i \(0.463285\pi\)
\(152\) 0 0
\(153\) 4.00000 0.323381
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −21.2132 −1.69300 −0.846499 0.532390i \(-0.821294\pi\)
−0.846499 + 0.532390i \(0.821294\pi\)
\(158\) 0 0
\(159\) 7.07107 0.560772
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −11.3137 −0.875481 −0.437741 0.899101i \(-0.644221\pi\)
−0.437741 + 0.899101i \(0.644221\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) −8.00000 −0.611775
\(172\) 0 0
\(173\) 7.07107 0.537603 0.268802 0.963196i \(-0.413372\pi\)
0.268802 + 0.963196i \(0.413372\pi\)
\(174\) 0 0
\(175\) 8.48528 0.641427
\(176\) 0 0
\(177\) 12.0000 0.901975
\(178\) 0 0
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) −12.7279 −0.946059 −0.473029 0.881047i \(-0.656840\pi\)
−0.473029 + 0.881047i \(0.656840\pi\)
\(182\) 0 0
\(183\) −1.41421 −0.104542
\(184\) 0 0
\(185\) −6.00000 −0.441129
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 2.82843 0.205738
\(190\) 0 0
\(191\) −5.65685 −0.409316 −0.204658 0.978834i \(-0.565608\pi\)
−0.204658 + 0.978834i \(0.565608\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 6.00000 0.429669
\(196\) 0 0
\(197\) −7.07107 −0.503793 −0.251896 0.967754i \(-0.581054\pi\)
−0.251896 + 0.967754i \(0.581054\pi\)
\(198\) 0 0
\(199\) 8.48528 0.601506 0.300753 0.953702i \(-0.402762\pi\)
0.300753 + 0.953702i \(0.402762\pi\)
\(200\) 0 0
\(201\) −4.00000 −0.282138
\(202\) 0 0
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 5.65685 0.395092
\(206\) 0 0
\(207\) 5.65685 0.393179
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 11.3137 0.775203
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −8.00000 −0.543075
\(218\) 0 0
\(219\) −14.0000 −0.946032
\(220\) 0 0
\(221\) 16.9706 1.14156
\(222\) 0 0
\(223\) 25.4558 1.70465 0.852325 0.523013i \(-0.175192\pi\)
0.852325 + 0.523013i \(0.175192\pi\)
\(224\) 0 0
\(225\) −3.00000 −0.200000
\(226\) 0 0
\(227\) −8.00000 −0.530979 −0.265489 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 0 0
\(229\) 24.0416 1.58872 0.794358 0.607450i \(-0.207808\pi\)
0.794358 + 0.607450i \(0.207808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −10.0000 −0.655122 −0.327561 0.944830i \(-0.606227\pi\)
−0.327561 + 0.944830i \(0.606227\pi\)
\(234\) 0 0
\(235\) −16.0000 −1.04372
\(236\) 0 0
\(237\) −8.48528 −0.551178
\(238\) 0 0
\(239\) −5.65685 −0.365911 −0.182956 0.983121i \(-0.558567\pi\)
−0.182956 + 0.983121i \(0.558567\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.41421 −0.0903508
\(246\) 0 0
\(247\) −33.9411 −2.15962
\(248\) 0 0
\(249\) 16.0000 1.01396
\(250\) 0 0
\(251\) 8.00000 0.504956 0.252478 0.967603i \(-0.418755\pi\)
0.252478 + 0.967603i \(0.418755\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 5.65685 0.354246
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) −12.0000 −0.745644
\(260\) 0 0
\(261\) 1.41421 0.0875376
\(262\) 0 0
\(263\) 28.2843 1.74408 0.872041 0.489432i \(-0.162796\pi\)
0.872041 + 0.489432i \(0.162796\pi\)
\(264\) 0 0
\(265\) 10.0000 0.614295
\(266\) 0 0
\(267\) −6.00000 −0.367194
\(268\) 0 0
\(269\) 21.2132 1.29339 0.646696 0.762748i \(-0.276150\pi\)
0.646696 + 0.762748i \(0.276150\pi\)
\(270\) 0 0
\(271\) −8.48528 −0.515444 −0.257722 0.966219i \(-0.582972\pi\)
−0.257722 + 0.966219i \(0.582972\pi\)
\(272\) 0 0
\(273\) 12.0000 0.726273
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.41421 0.0849719 0.0424859 0.999097i \(-0.486472\pi\)
0.0424859 + 0.999097i \(0.486472\pi\)
\(278\) 0 0
\(279\) 2.82843 0.169334
\(280\) 0 0
\(281\) −22.0000 −1.31241 −0.656205 0.754583i \(-0.727839\pi\)
−0.656205 + 0.754583i \(0.727839\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 0 0
\(285\) −11.3137 −0.670166
\(286\) 0 0
\(287\) 11.3137 0.667827
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) −16.0000 −0.937937
\(292\) 0 0
\(293\) −18.3848 −1.07405 −0.537025 0.843566i \(-0.680452\pi\)
−0.537025 + 0.843566i \(0.680452\pi\)
\(294\) 0 0
\(295\) 16.9706 0.988064
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 24.0000 1.38796
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 15.5563 0.893689
\(304\) 0 0
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −4.00000 −0.228292 −0.114146 0.993464i \(-0.536413\pi\)
−0.114146 + 0.993464i \(0.536413\pi\)
\(308\) 0 0
\(309\) 19.7990 1.12633
\(310\) 0 0
\(311\) −28.2843 −1.60385 −0.801927 0.597422i \(-0.796192\pi\)
−0.801927 + 0.597422i \(0.796192\pi\)
\(312\) 0 0
\(313\) −16.0000 −0.904373 −0.452187 0.891923i \(-0.649356\pi\)
−0.452187 + 0.891923i \(0.649356\pi\)
\(314\) 0 0
\(315\) 4.00000 0.225374
\(316\) 0 0
\(317\) −21.2132 −1.19145 −0.595726 0.803188i \(-0.703136\pi\)
−0.595726 + 0.803188i \(0.703136\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) −32.0000 −1.78053
\(324\) 0 0
\(325\) −12.7279 −0.706018
\(326\) 0 0
\(327\) 4.24264 0.234619
\(328\) 0 0
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) 4.00000 0.219860 0.109930 0.993939i \(-0.464937\pi\)
0.109930 + 0.993939i \(0.464937\pi\)
\(332\) 0 0
\(333\) 4.24264 0.232495
\(334\) 0 0
\(335\) −5.65685 −0.309067
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) 18.0000 0.977626
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16.9706 0.916324
\(344\) 0 0
\(345\) 8.00000 0.430706
\(346\) 0 0
\(347\) 32.0000 1.71785 0.858925 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) 0 0
\(349\) −12.7279 −0.681310 −0.340655 0.940188i \(-0.610649\pi\)
−0.340655 + 0.940188i \(0.610649\pi\)
\(350\) 0 0
\(351\) −4.24264 −0.226455
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 16.0000 0.849192
\(356\) 0 0
\(357\) 11.3137 0.598785
\(358\) 0 0
\(359\) −11.3137 −0.597115 −0.298557 0.954392i \(-0.596505\pi\)
−0.298557 + 0.954392i \(0.596505\pi\)
\(360\) 0 0
\(361\) 45.0000 2.36842
\(362\) 0 0
\(363\) 11.0000 0.577350
\(364\) 0 0
\(365\) −19.7990 −1.03633
\(366\) 0 0
\(367\) 2.82843 0.147643 0.0738213 0.997271i \(-0.476481\pi\)
0.0738213 + 0.997271i \(0.476481\pi\)
\(368\) 0 0
\(369\) −4.00000 −0.208232
\(370\) 0 0
\(371\) 20.0000 1.03835
\(372\) 0 0
\(373\) −4.24264 −0.219676 −0.109838 0.993950i \(-0.535033\pi\)
−0.109838 + 0.993950i \(0.535033\pi\)
\(374\) 0 0
\(375\) −11.3137 −0.584237
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) 16.0000 0.821865 0.410932 0.911666i \(-0.365203\pi\)
0.410932 + 0.911666i \(0.365203\pi\)
\(380\) 0 0
\(381\) 8.48528 0.434714
\(382\) 0 0
\(383\) −28.2843 −1.44526 −0.722629 0.691236i \(-0.757067\pi\)
−0.722629 + 0.691236i \(0.757067\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −9.89949 −0.501924 −0.250962 0.967997i \(-0.580747\pi\)
−0.250962 + 0.967997i \(0.580747\pi\)
\(390\) 0 0
\(391\) 22.6274 1.14432
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) −12.0000 −0.603786
\(396\) 0 0
\(397\) −21.2132 −1.06466 −0.532330 0.846537i \(-0.678683\pi\)
−0.532330 + 0.846537i \(0.678683\pi\)
\(398\) 0 0
\(399\) −22.6274 −1.13279
\(400\) 0 0
\(401\) −20.0000 −0.998752 −0.499376 0.866385i \(-0.666437\pi\)
−0.499376 + 0.866385i \(0.666437\pi\)
\(402\) 0 0
\(403\) 12.0000 0.597763
\(404\) 0 0
\(405\) −1.41421 −0.0702728
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 32.0000 1.58230 0.791149 0.611623i \(-0.209483\pi\)
0.791149 + 0.611623i \(0.209483\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 0 0
\(413\) 33.9411 1.67013
\(414\) 0 0
\(415\) 22.6274 1.11074
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −8.00000 −0.390826 −0.195413 0.980721i \(-0.562605\pi\)
−0.195413 + 0.980721i \(0.562605\pi\)
\(420\) 0 0
\(421\) 12.7279 0.620321 0.310160 0.950684i \(-0.399617\pi\)
0.310160 + 0.950684i \(0.399617\pi\)
\(422\) 0 0
\(423\) 11.3137 0.550091
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) −4.00000 −0.193574
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −11.3137 −0.544962 −0.272481 0.962161i \(-0.587844\pi\)
−0.272481 + 0.962161i \(0.587844\pi\)
\(432\) 0 0
\(433\) 8.00000 0.384455 0.192228 0.981350i \(-0.438429\pi\)
0.192228 + 0.981350i \(0.438429\pi\)
\(434\) 0 0
\(435\) 2.00000 0.0958927
\(436\) 0 0
\(437\) −45.2548 −2.16483
\(438\) 0 0
\(439\) −2.82843 −0.134993 −0.0674967 0.997719i \(-0.521501\pi\)
−0.0674967 + 0.997719i \(0.521501\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 0 0
\(445\) −8.48528 −0.402241
\(446\) 0 0
\(447\) −12.7279 −0.602010
\(448\) 0 0
\(449\) −12.0000 −0.566315 −0.283158 0.959073i \(-0.591382\pi\)
−0.283158 + 0.959073i \(0.591382\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.82843 −0.132891
\(454\) 0 0
\(455\) 16.9706 0.795592
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) 0 0
\(459\) −4.00000 −0.186704
\(460\) 0 0
\(461\) 15.5563 0.724531 0.362266 0.932075i \(-0.382003\pi\)
0.362266 + 0.932075i \(0.382003\pi\)
\(462\) 0 0
\(463\) −31.1127 −1.44593 −0.722965 0.690885i \(-0.757221\pi\)
−0.722965 + 0.690885i \(0.757221\pi\)
\(464\) 0 0
\(465\) 4.00000 0.185496
\(466\) 0 0
\(467\) 8.00000 0.370196 0.185098 0.982720i \(-0.440740\pi\)
0.185098 + 0.982720i \(0.440740\pi\)
\(468\) 0 0
\(469\) −11.3137 −0.522419
\(470\) 0 0
\(471\) 21.2132 0.977453
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 24.0000 1.10120
\(476\) 0 0
\(477\) −7.07107 −0.323762
\(478\) 0 0
\(479\) 28.2843 1.29234 0.646171 0.763193i \(-0.276369\pi\)
0.646171 + 0.763193i \(0.276369\pi\)
\(480\) 0 0
\(481\) 18.0000 0.820729
\(482\) 0 0
\(483\) 16.0000 0.728025
\(484\) 0 0
\(485\) −22.6274 −1.02746
\(486\) 0 0
\(487\) 2.82843 0.128168 0.0640841 0.997944i \(-0.479587\pi\)
0.0640841 + 0.997944i \(0.479587\pi\)
\(488\) 0 0
\(489\) 16.0000 0.723545
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) 5.65685 0.254772
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 32.0000 1.43540
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) 11.3137 0.505459
\(502\) 0 0
\(503\) 11.3137 0.504453 0.252227 0.967668i \(-0.418837\pi\)
0.252227 + 0.967668i \(0.418837\pi\)
\(504\) 0 0
\(505\) 22.0000 0.978987
\(506\) 0 0
\(507\) −5.00000 −0.222058
\(508\) 0 0
\(509\) 32.5269 1.44173 0.720865 0.693075i \(-0.243745\pi\)
0.720865 + 0.693075i \(0.243745\pi\)
\(510\) 0 0
\(511\) −39.5980 −1.75171
\(512\) 0 0
\(513\) 8.00000 0.353209
\(514\) 0 0
\(515\) 28.0000 1.23383
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −7.07107 −0.310385
\(520\) 0 0
\(521\) 12.0000 0.525730 0.262865 0.964833i \(-0.415333\pi\)
0.262865 + 0.964833i \(0.415333\pi\)
\(522\) 0 0
\(523\) 8.00000 0.349816 0.174908 0.984585i \(-0.444037\pi\)
0.174908 + 0.984585i \(0.444037\pi\)
\(524\) 0 0
\(525\) −8.48528 −0.370328
\(526\) 0 0
\(527\) 11.3137 0.492833
\(528\) 0 0
\(529\) 9.00000 0.391304
\(530\) 0 0
\(531\) −12.0000 −0.520756
\(532\) 0 0
\(533\) −16.9706 −0.735077
\(534\) 0 0
\(535\) 16.9706 0.733701
\(536\) 0 0
\(537\) −4.00000 −0.172613
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −4.24264 −0.182405 −0.0912027 0.995832i \(-0.529071\pi\)
−0.0912027 + 0.995832i \(0.529071\pi\)
\(542\) 0 0
\(543\) 12.7279 0.546207
\(544\) 0 0
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) −8.00000 −0.342055 −0.171028 0.985266i \(-0.554709\pi\)
−0.171028 + 0.985266i \(0.554709\pi\)
\(548\) 0 0
\(549\) 1.41421 0.0603572
\(550\) 0 0
\(551\) −11.3137 −0.481980
\(552\) 0 0
\(553\) −24.0000 −1.02058
\(554\) 0 0
\(555\) 6.00000 0.254686
\(556\) 0 0
\(557\) 29.6985 1.25837 0.629183 0.777258i \(-0.283390\pi\)
0.629183 + 0.777258i \(0.283390\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −8.00000 −0.337160 −0.168580 0.985688i \(-0.553918\pi\)
−0.168580 + 0.985688i \(0.553918\pi\)
\(564\) 0 0
\(565\) 25.4558 1.07094
\(566\) 0 0
\(567\) −2.82843 −0.118783
\(568\) 0 0
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 5.65685 0.236318
\(574\) 0 0
\(575\) −16.9706 −0.707721
\(576\) 0 0
\(577\) 22.0000 0.915872 0.457936 0.888985i \(-0.348589\pi\)
0.457936 + 0.888985i \(0.348589\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 45.2548 1.87749
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −6.00000 −0.248069
\(586\) 0 0
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 0 0
\(589\) −22.6274 −0.932346
\(590\) 0 0
\(591\) 7.07107 0.290865
\(592\) 0 0
\(593\) −34.0000 −1.39621 −0.698106 0.715994i \(-0.745974\pi\)
−0.698106 + 0.715994i \(0.745974\pi\)
\(594\) 0 0
\(595\) 16.0000 0.655936
\(596\) 0 0
\(597\) −8.48528 −0.347279
\(598\) 0 0
\(599\) 39.5980 1.61793 0.808965 0.587857i \(-0.200028\pi\)
0.808965 + 0.587857i \(0.200028\pi\)
\(600\) 0 0
\(601\) 14.0000 0.571072 0.285536 0.958368i \(-0.407828\pi\)
0.285536 + 0.958368i \(0.407828\pi\)
\(602\) 0 0
\(603\) 4.00000 0.162893
\(604\) 0 0
\(605\) 15.5563 0.632456
\(606\) 0 0
\(607\) −42.4264 −1.72203 −0.861017 0.508576i \(-0.830172\pi\)
−0.861017 + 0.508576i \(0.830172\pi\)
\(608\) 0 0
\(609\) 4.00000 0.162088
\(610\) 0 0
\(611\) 48.0000 1.94187
\(612\) 0 0
\(613\) −15.5563 −0.628315 −0.314158 0.949371i \(-0.601722\pi\)
−0.314158 + 0.949371i \(0.601722\pi\)
\(614\) 0 0
\(615\) −5.65685 −0.228106
\(616\) 0 0
\(617\) −6.00000 −0.241551 −0.120775 0.992680i \(-0.538538\pi\)
−0.120775 + 0.992680i \(0.538538\pi\)
\(618\) 0 0
\(619\) 44.0000 1.76851 0.884255 0.467005i \(-0.154667\pi\)
0.884255 + 0.467005i \(0.154667\pi\)
\(620\) 0 0
\(621\) −5.65685 −0.227002
\(622\) 0 0
\(623\) −16.9706 −0.679911
\(624\) 0 0
\(625\) −1.00000 −0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 16.9706 0.676661
\(630\) 0 0
\(631\) −8.48528 −0.337794 −0.168897 0.985634i \(-0.554020\pi\)
−0.168897 + 0.985634i \(0.554020\pi\)
\(632\) 0 0
\(633\) −12.0000 −0.476957
\(634\) 0 0
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) 4.24264 0.168100
\(638\) 0 0
\(639\) −11.3137 −0.447563
\(640\) 0 0
\(641\) −12.0000 −0.473972 −0.236986 0.971513i \(-0.576159\pi\)
−0.236986 + 0.971513i \(0.576159\pi\)
\(642\) 0 0
\(643\) 16.0000 0.630978 0.315489 0.948929i \(-0.397831\pi\)
0.315489 + 0.948929i \(0.397831\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 39.5980 1.55676 0.778379 0.627795i \(-0.216042\pi\)
0.778379 + 0.627795i \(0.216042\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 8.00000 0.313545
\(652\) 0 0
\(653\) 4.24264 0.166027 0.0830137 0.996548i \(-0.473545\pi\)
0.0830137 + 0.996548i \(0.473545\pi\)
\(654\) 0 0
\(655\) 5.65685 0.221032
\(656\) 0 0
\(657\) 14.0000 0.546192
\(658\) 0 0
\(659\) 20.0000 0.779089 0.389545 0.921008i \(-0.372632\pi\)
0.389545 + 0.921008i \(0.372632\pi\)
\(660\) 0 0
\(661\) 26.8701 1.04512 0.522562 0.852601i \(-0.324976\pi\)
0.522562 + 0.852601i \(0.324976\pi\)
\(662\) 0 0
\(663\) −16.9706 −0.659082
\(664\) 0 0
\(665\) −32.0000 −1.24091
\(666\) 0 0
\(667\) 8.00000 0.309761
\(668\) 0 0
\(669\) −25.4558 −0.984180
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −8.00000 −0.308377 −0.154189 0.988041i \(-0.549276\pi\)
−0.154189 + 0.988041i \(0.549276\pi\)
\(674\) 0 0
\(675\) 3.00000 0.115470
\(676\) 0 0
\(677\) −32.5269 −1.25011 −0.625055 0.780580i \(-0.714924\pi\)
−0.625055 + 0.780580i \(0.714924\pi\)
\(678\) 0 0
\(679\) −45.2548 −1.73672
\(680\) 0 0
\(681\) 8.00000 0.306561
\(682\) 0 0
\(683\) 32.0000 1.22445 0.612223 0.790685i \(-0.290275\pi\)
0.612223 + 0.790685i \(0.290275\pi\)
\(684\) 0 0
\(685\) 16.9706 0.648412
\(686\) 0 0
\(687\) −24.0416 −0.917245
\(688\) 0 0
\(689\) −30.0000 −1.14291
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.65685 0.214577
\(696\) 0 0
\(697\) −16.0000 −0.606043
\(698\) 0 0
\(699\) 10.0000 0.378235
\(700\) 0 0
\(701\) −46.6690 −1.76267 −0.881333 0.472496i \(-0.843353\pi\)
−0.881333 + 0.472496i \(0.843353\pi\)
\(702\) 0 0
\(703\) −33.9411 −1.28011
\(704\) 0 0
\(705\) 16.0000 0.602595
\(706\) 0 0
\(707\) 44.0000 1.65479
\(708\) 0 0
\(709\) −35.3553 −1.32780 −0.663899 0.747822i \(-0.731099\pi\)
−0.663899 + 0.747822i \(0.731099\pi\)
\(710\) 0 0
\(711\) 8.48528 0.318223
\(712\) 0 0
\(713\) 16.0000 0.599205
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 5.65685 0.211259
\(718\) 0 0
\(719\) −16.9706 −0.632895 −0.316448 0.948610i \(-0.602490\pi\)
−0.316448 + 0.948610i \(0.602490\pi\)
\(720\) 0 0
\(721\) 56.0000 2.08555
\(722\) 0 0
\(723\) −8.00000 −0.297523
\(724\) 0 0
\(725\) −4.24264 −0.157568
\(726\) 0 0
\(727\) 42.4264 1.57351 0.786754 0.617266i \(-0.211760\pi\)
0.786754 + 0.617266i \(0.211760\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 4.24264 0.156706 0.0783528 0.996926i \(-0.475034\pi\)
0.0783528 + 0.996926i \(0.475034\pi\)
\(734\) 0 0
\(735\) 1.41421 0.0521641
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) 0 0
\(741\) 33.9411 1.24686
\(742\) 0 0
\(743\) 11.3137 0.415060 0.207530 0.978229i \(-0.433458\pi\)
0.207530 + 0.978229i \(0.433458\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 0 0
\(747\) −16.0000 −0.585409
\(748\) 0 0
\(749\) 33.9411 1.24018
\(750\) 0 0
\(751\) −8.48528 −0.309632 −0.154816 0.987943i \(-0.549479\pi\)
−0.154816 + 0.987943i \(0.549479\pi\)
\(752\) 0 0
\(753\) −8.00000 −0.291536
\(754\) 0 0
\(755\) −4.00000 −0.145575
\(756\) 0 0
\(757\) 21.2132 0.771007 0.385503 0.922706i \(-0.374028\pi\)
0.385503 + 0.922706i \(0.374028\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 36.0000 1.30500 0.652499 0.757789i \(-0.273720\pi\)
0.652499 + 0.757789i \(0.273720\pi\)
\(762\) 0 0
\(763\) 12.0000 0.434429
\(764\) 0 0
\(765\) −5.65685 −0.204524
\(766\) 0 0
\(767\) −50.9117 −1.83831
\(768\) 0 0
\(769\) −32.0000 −1.15395 −0.576975 0.816762i \(-0.695767\pi\)
−0.576975 + 0.816762i \(0.695767\pi\)
\(770\) 0 0
\(771\) −2.00000 −0.0720282
\(772\) 0 0
\(773\) 18.3848 0.661254 0.330627 0.943761i \(-0.392740\pi\)
0.330627 + 0.943761i \(0.392740\pi\)
\(774\) 0 0
\(775\) −8.48528 −0.304800
\(776\) 0 0
\(777\) 12.0000 0.430498
\(778\) 0 0
\(779\) 32.0000 1.14652
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.41421 −0.0505399
\(784\) 0 0
\(785\) 30.0000 1.07075
\(786\) 0 0
\(787\) −32.0000 −1.14068 −0.570338 0.821410i \(-0.693188\pi\)
−0.570338 + 0.821410i \(0.693188\pi\)
\(788\) 0 0
\(789\) −28.2843 −1.00695
\(790\) 0 0
\(791\) 50.9117 1.81021
\(792\) 0 0
\(793\) 6.00000 0.213066
\(794\) 0 0
\(795\) −10.0000 −0.354663
\(796\) 0 0
\(797\) 49.4975 1.75329 0.876645 0.481137i \(-0.159776\pi\)
0.876645 + 0.481137i \(0.159776\pi\)
\(798\) 0 0
\(799\) 45.2548 1.60100
\(800\) 0 0
\(801\) 6.00000 0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 22.6274 0.797512
\(806\) 0 0
\(807\) −21.2132 −0.746740
\(808\) 0 0
\(809\) 4.00000 0.140633 0.0703163 0.997525i \(-0.477599\pi\)
0.0703163 + 0.997525i \(0.477599\pi\)
\(810\) 0 0
\(811\) 40.0000 1.40459 0.702295 0.711886i \(-0.252159\pi\)
0.702295 + 0.711886i \(0.252159\pi\)
\(812\) 0 0
\(813\) 8.48528 0.297592
\(814\) 0 0
\(815\) 22.6274 0.792604
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −12.0000 −0.419314
\(820\) 0 0
\(821\) −7.07107 −0.246782 −0.123391 0.992358i \(-0.539377\pi\)
−0.123391 + 0.992358i \(0.539377\pi\)
\(822\) 0 0
\(823\) 25.4558 0.887335 0.443667 0.896191i \(-0.353677\pi\)
0.443667 + 0.896191i \(0.353677\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 0 0
\(829\) 52.3259 1.81735 0.908677 0.417500i \(-0.137094\pi\)
0.908677 + 0.417500i \(0.137094\pi\)
\(830\) 0 0
\(831\) −1.41421 −0.0490585
\(832\) 0 0
\(833\) 4.00000 0.138592
\(834\) 0 0
\(835\) 16.0000 0.553703
\(836\) 0 0
\(837\) −2.82843 −0.0977647
\(838\) 0 0
\(839\) −45.2548 −1.56237 −0.781185 0.624299i \(-0.785385\pi\)
−0.781185 + 0.624299i \(0.785385\pi\)
\(840\) 0 0
\(841\) −27.0000 −0.931034
\(842\) 0 0
\(843\) 22.0000 0.757720
\(844\) 0 0
\(845\) −7.07107 −0.243252
\(846\) 0 0
\(847\) 31.1127 1.06904
\(848\) 0 0
\(849\) 4.00000 0.137280
\(850\) 0 0
\(851\) 24.0000 0.822709
\(852\) 0 0
\(853\) −15.5563 −0.532639 −0.266320 0.963885i \(-0.585808\pi\)
−0.266320 + 0.963885i \(0.585808\pi\)
\(854\) 0 0
\(855\) 11.3137 0.386921
\(856\) 0 0
\(857\) −20.0000 −0.683187 −0.341593 0.939848i \(-0.610967\pi\)
−0.341593 + 0.939848i \(0.610967\pi\)
\(858\) 0 0
\(859\) −8.00000 −0.272956 −0.136478 0.990643i \(-0.543578\pi\)
−0.136478 + 0.990643i \(0.543578\pi\)
\(860\) 0 0
\(861\) −11.3137 −0.385570
\(862\) 0 0
\(863\) 5.65685 0.192562 0.0962808 0.995354i \(-0.469305\pi\)
0.0962808 + 0.995354i \(0.469305\pi\)
\(864\) 0 0
\(865\) −10.0000 −0.340010
\(866\) 0 0
\(867\) 1.00000 0.0339618
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 16.9706 0.575026
\(872\) 0 0
\(873\) 16.0000 0.541518
\(874\) 0 0
\(875\) −32.0000 −1.08180
\(876\) 0 0
\(877\) −32.5269 −1.09836 −0.549178 0.835705i \(-0.685059\pi\)
−0.549178 + 0.835705i \(0.685059\pi\)
\(878\) 0 0
\(879\) 18.3848 0.620103
\(880\) 0 0
\(881\) −30.0000 −1.01073 −0.505363 0.862907i \(-0.668641\pi\)
−0.505363 + 0.862907i \(0.668641\pi\)
\(882\) 0 0
\(883\) 24.0000 0.807664 0.403832 0.914833i \(-0.367678\pi\)
0.403832 + 0.914833i \(0.367678\pi\)
\(884\) 0 0
\(885\) −16.9706 −0.570459
\(886\) 0 0
\(887\) 16.9706 0.569816 0.284908 0.958555i \(-0.408037\pi\)
0.284908 + 0.958555i \(0.408037\pi\)
\(888\) 0 0
\(889\) 24.0000 0.804934
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −90.5097 −3.02879
\(894\) 0 0
\(895\) −5.65685 −0.189088
\(896\) 0 0
\(897\) −24.0000 −0.801337
\(898\) 0 0
\(899\) 4.00000 0.133407
\(900\) 0 0
\(901\) −28.2843 −0.942286
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 18.0000 0.598340
\(906\) 0 0
\(907\) −32.0000 −1.06254 −0.531271 0.847202i \(-0.678286\pi\)
−0.531271 + 0.847202i \(0.678286\pi\)
\(908\) 0 0
\(909\) −15.5563 −0.515972
\(910\) 0 0
\(911\) −5.65685 −0.187420 −0.0937100 0.995600i \(-0.529873\pi\)
−0.0937100 + 0.995600i \(0.529873\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 2.00000 0.0661180
\(916\) 0 0
\(917\) 11.3137 0.373612
\(918\) 0 0
\(919\) 2.82843 0.0933012 0.0466506 0.998911i \(-0.485145\pi\)
0.0466506 + 0.998911i \(0.485145\pi\)
\(920\) 0 0
\(921\) 4.00000 0.131804
\(922\) 0 0
\(923\) −48.0000 −1.57994
\(924\) 0 0
\(925\) −12.7279 −0.418491
\(926\) 0 0
\(927\) −19.7990 −0.650284
\(928\) 0 0
\(929\) −20.0000 −0.656179 −0.328089 0.944647i \(-0.606405\pi\)
−0.328089 + 0.944647i \(0.606405\pi\)
\(930\) 0 0
\(931\) −8.00000 −0.262189
\(932\) 0 0
\(933\) 28.2843 0.925985
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) 16.0000 0.522140
\(940\) 0 0
\(941\) −52.3259 −1.70578 −0.852888 0.522094i \(-0.825151\pi\)
−0.852888 + 0.522094i \(0.825151\pi\)
\(942\) 0 0
\(943\) −22.6274 −0.736850
\(944\) 0 0
\(945\) −4.00000 −0.130120
\(946\) 0 0
\(947\) −52.0000 −1.68977 −0.844886 0.534946i \(-0.820332\pi\)
−0.844886 + 0.534946i \(0.820332\pi\)
\(948\) 0 0
\(949\) 59.3970 1.92811
\(950\) 0 0
\(951\) 21.2132 0.687885
\(952\) 0 0
\(953\) −36.0000 −1.16615 −0.583077 0.812417i \(-0.698151\pi\)
−0.583077 + 0.812417i \(0.698151\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 33.9411 1.09602
\(960\) 0 0
\(961\) −23.0000 −0.741935
\(962\) 0 0
\(963\) −12.0000 −0.386695
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 42.4264 1.36434 0.682171 0.731193i \(-0.261036\pi\)
0.682171 + 0.731193i \(0.261036\pi\)
\(968\) 0 0
\(969\) 32.0000 1.02799
\(970\) 0 0
\(971\) 24.0000 0.770197 0.385098 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) 11.3137 0.362701
\(974\) 0 0
\(975\) 12.7279 0.407620
\(976\) 0 0
\(977\) −36.0000 −1.15174 −0.575871 0.817541i \(-0.695337\pi\)
−0.575871 + 0.817541i \(0.695337\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −4.24264 −0.135457
\(982\) 0 0
\(983\) 16.9706 0.541277 0.270638 0.962681i \(-0.412765\pi\)
0.270638 + 0.962681i \(0.412765\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 0 0
\(987\) 32.0000 1.01857
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 25.4558 0.808632 0.404316 0.914619i \(-0.367510\pi\)
0.404316 + 0.914619i \(0.367510\pi\)
\(992\) 0 0
\(993\) −4.00000 −0.126936
\(994\) 0 0
\(995\) −12.0000 −0.380426
\(996\) 0 0
\(997\) −7.07107 −0.223943 −0.111971 0.993711i \(-0.535717\pi\)
−0.111971 + 0.993711i \(0.535717\pi\)
\(998\) 0 0
\(999\) −4.24264 −0.134231
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 3072.2.a.b.1.1 2
3.2 odd 2 9216.2.a.h.1.2 2
4.3 odd 2 3072.2.a.h.1.1 2
8.3 odd 2 inner 3072.2.a.b.1.2 2
8.5 even 2 3072.2.a.h.1.2 2
12.11 even 2 9216.2.a.i.1.2 2
16.3 odd 4 3072.2.d.c.1537.4 4
16.5 even 4 3072.2.d.c.1537.3 4
16.11 odd 4 3072.2.d.c.1537.1 4
16.13 even 4 3072.2.d.c.1537.2 4
24.5 odd 2 9216.2.a.i.1.1 2
24.11 even 2 9216.2.a.h.1.1 2
32.3 odd 8 1536.2.j.c.1153.2 yes 4
32.5 even 8 1536.2.j.b.385.2 yes 4
32.11 odd 8 1536.2.j.c.385.2 yes 4
32.13 even 8 1536.2.j.b.1153.2 yes 4
32.19 odd 8 1536.2.j.b.1153.1 yes 4
32.21 even 8 1536.2.j.c.385.1 yes 4
32.27 odd 8 1536.2.j.b.385.1 4
32.29 even 8 1536.2.j.c.1153.1 yes 4
96.5 odd 8 4608.2.k.bb.3457.1 4
96.11 even 8 4608.2.k.y.3457.2 4
96.29 odd 8 4608.2.k.y.1153.2 4
96.35 even 8 4608.2.k.y.1153.1 4
96.53 odd 8 4608.2.k.y.3457.1 4
96.59 even 8 4608.2.k.bb.3457.2 4
96.77 odd 8 4608.2.k.bb.1153.2 4
96.83 even 8 4608.2.k.bb.1153.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1536.2.j.b.385.1 4 32.27 odd 8
1536.2.j.b.385.2 yes 4 32.5 even 8
1536.2.j.b.1153.1 yes 4 32.19 odd 8
1536.2.j.b.1153.2 yes 4 32.13 even 8
1536.2.j.c.385.1 yes 4 32.21 even 8
1536.2.j.c.385.2 yes 4 32.11 odd 8
1536.2.j.c.1153.1 yes 4 32.29 even 8
1536.2.j.c.1153.2 yes 4 32.3 odd 8
3072.2.a.b.1.1 2 1.1 even 1 trivial
3072.2.a.b.1.2 2 8.3 odd 2 inner
3072.2.a.h.1.1 2 4.3 odd 2
3072.2.a.h.1.2 2 8.5 even 2
3072.2.d.c.1537.1 4 16.11 odd 4
3072.2.d.c.1537.2 4 16.13 even 4
3072.2.d.c.1537.3 4 16.5 even 4
3072.2.d.c.1537.4 4 16.3 odd 4
4608.2.k.y.1153.1 4 96.35 even 8
4608.2.k.y.1153.2 4 96.29 odd 8
4608.2.k.y.3457.1 4 96.53 odd 8
4608.2.k.y.3457.2 4 96.11 even 8
4608.2.k.bb.1153.1 4 96.83 even 8
4608.2.k.bb.1153.2 4 96.77 odd 8
4608.2.k.bb.3457.1 4 96.5 odd 8
4608.2.k.bb.3457.2 4 96.59 even 8
9216.2.a.h.1.1 2 24.11 even 2
9216.2.a.h.1.2 2 3.2 odd 2
9216.2.a.i.1.1 2 24.5 odd 2
9216.2.a.i.1.2 2 12.11 even 2