Properties

Label 4608.2.d.b.2305.1
Level $4608$
Weight $2$
Character 4608.2305
Analytic conductor $36.795$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \(x^{2} + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2305.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 4608.2305
Dual form 4608.2.d.b.2305.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843i q^{5} +4.00000 q^{7} +O(q^{10})\) \(q-2.82843i q^{5} +4.00000 q^{7} +1.41421i q^{11} +2.82843i q^{13} +4.00000 q^{17} -7.07107i q^{19} -4.00000 q^{23} -3.00000 q^{25} -8.48528i q^{29} -8.00000 q^{31} -11.3137i q^{35} -2.82843i q^{37} +2.00000 q^{41} -4.24264i q^{43} +9.00000 q^{49} +2.82843i q^{53} +4.00000 q^{55} +4.24264i q^{59} -8.48528i q^{61} +8.00000 q^{65} +4.24264i q^{67} +4.00000 q^{71} +4.00000 q^{73} +5.65685i q^{77} +8.00000 q^{79} +9.89949i q^{83} -11.3137i q^{85} +12.0000 q^{89} +11.3137i q^{91} -20.0000 q^{95} -4.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{7} + O(q^{10}) \) \( 2 q + 8 q^{7} + 8 q^{17} - 8 q^{23} - 6 q^{25} - 16 q^{31} + 4 q^{41} + 18 q^{49} + 8 q^{55} + 16 q^{65} + 8 q^{71} + 8 q^{73} + 16 q^{79} + 24 q^{89} - 40 q^{95} - 8 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(3583\) \(4097\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(5\) − 2.82843i − 1.26491i −0.774597 0.632456i \(-0.782047\pi\)
0.774597 0.632456i \(-0.217953\pi\)
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.41421i 0.426401i 0.977008 + 0.213201i \(0.0683888\pi\)
−0.977008 + 0.213201i \(0.931611\pi\)
\(12\) 0 0
\(13\) 2.82843i 0.784465i 0.919866 + 0.392232i \(0.128297\pi\)
−0.919866 + 0.392232i \(0.871703\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 4.00000 0.970143 0.485071 0.874475i \(-0.338794\pi\)
0.485071 + 0.874475i \(0.338794\pi\)
\(18\) 0 0
\(19\) − 7.07107i − 1.62221i −0.584898 0.811107i \(-0.698865\pi\)
0.584898 0.811107i \(-0.301135\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 8.48528i − 1.57568i −0.615882 0.787839i \(-0.711200\pi\)
0.615882 0.787839i \(-0.288800\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 11.3137i − 1.91237i
\(36\) 0 0
\(37\) − 2.82843i − 0.464991i −0.972598 0.232495i \(-0.925311\pi\)
0.972598 0.232495i \(-0.0746890\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 2.00000 0.312348 0.156174 0.987730i \(-0.450084\pi\)
0.156174 + 0.987730i \(0.450084\pi\)
\(42\) 0 0
\(43\) − 4.24264i − 0.646997i −0.946229 0.323498i \(-0.895141\pi\)
0.946229 0.323498i \(-0.104859\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.82843i 0.388514i 0.980951 + 0.194257i \(0.0622296\pi\)
−0.980951 + 0.194257i \(0.937770\pi\)
\(54\) 0 0
\(55\) 4.00000 0.539360
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.24264i 0.552345i 0.961108 + 0.276172i \(0.0890661\pi\)
−0.961108 + 0.276172i \(0.910934\pi\)
\(60\) 0 0
\(61\) − 8.48528i − 1.08643i −0.839594 0.543214i \(-0.817207\pi\)
0.839594 0.543214i \(-0.182793\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 8.00000 0.992278
\(66\) 0 0
\(67\) 4.24264i 0.518321i 0.965834 + 0.259161i \(0.0834459\pi\)
−0.965834 + 0.259161i \(0.916554\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 5.65685i 0.644658i
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 9.89949i 1.08661i 0.839535 + 0.543305i \(0.182827\pi\)
−0.839535 + 0.543305i \(0.817173\pi\)
\(84\) 0 0
\(85\) − 11.3137i − 1.22714i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 12.0000 1.27200 0.635999 0.771690i \(-0.280588\pi\)
0.635999 + 0.771690i \(0.280588\pi\)
\(90\) 0 0
\(91\) 11.3137i 1.18600i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −20.0000 −2.05196
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.82843i 0.281439i 0.990050 + 0.140720i \(0.0449416\pi\)
−0.990050 + 0.140720i \(0.955058\pi\)
\(102\) 0 0
\(103\) 12.0000 1.18240 0.591198 0.806527i \(-0.298655\pi\)
0.591198 + 0.806527i \(0.298655\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 9.89949i − 0.957020i −0.878082 0.478510i \(-0.841177\pi\)
0.878082 0.478510i \(-0.158823\pi\)
\(108\) 0 0
\(109\) − 14.1421i − 1.35457i −0.735720 0.677285i \(-0.763156\pi\)
0.735720 0.677285i \(-0.236844\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −14.0000 −1.31701 −0.658505 0.752577i \(-0.728811\pi\)
−0.658505 + 0.752577i \(0.728811\pi\)
\(114\) 0 0
\(115\) 11.3137i 1.05501i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 16.0000 1.46672
\(120\) 0 0
\(121\) 9.00000 0.818182
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 5.65685i − 0.505964i
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 7.07107i 0.617802i 0.951094 + 0.308901i \(0.0999612\pi\)
−0.951094 + 0.308901i \(0.900039\pi\)
\(132\) 0 0
\(133\) − 28.2843i − 2.45256i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −18.0000 −1.53784 −0.768922 0.639343i \(-0.779207\pi\)
−0.768922 + 0.639343i \(0.779207\pi\)
\(138\) 0 0
\(139\) − 4.24264i − 0.359856i −0.983680 0.179928i \(-0.942414\pi\)
0.983680 0.179928i \(-0.0575865\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −4.00000 −0.334497
\(144\) 0 0
\(145\) −24.0000 −1.99309
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 2.82843i − 0.231714i −0.993266 0.115857i \(-0.963039\pi\)
0.993266 0.115857i \(-0.0369614\pi\)
\(150\) 0 0
\(151\) 4.00000 0.325515 0.162758 0.986666i \(-0.447961\pi\)
0.162758 + 0.986666i \(0.447961\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 22.6274i 1.81748i
\(156\) 0 0
\(157\) − 8.48528i − 0.677199i −0.940931 0.338600i \(-0.890047\pi\)
0.940931 0.338600i \(-0.109953\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −16.0000 −1.26098
\(162\) 0 0
\(163\) − 9.89949i − 0.775388i −0.921788 0.387694i \(-0.873272\pi\)
0.921788 0.387694i \(-0.126728\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 14.1421i − 1.07521i −0.843198 0.537603i \(-0.819330\pi\)
0.843198 0.537603i \(-0.180670\pi\)
\(174\) 0 0
\(175\) −12.0000 −0.907115
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 1.41421i − 0.105703i −0.998602 0.0528516i \(-0.983169\pi\)
0.998602 0.0528516i \(-0.0168310\pi\)
\(180\) 0 0
\(181\) − 8.48528i − 0.630706i −0.948974 0.315353i \(-0.897877\pi\)
0.948974 0.315353i \(-0.102123\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −8.00000 −0.588172
\(186\) 0 0
\(187\) 5.65685i 0.413670i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 8.00000 0.578860 0.289430 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 14.1421i − 1.00759i −0.863825 0.503793i \(-0.831938\pi\)
0.863825 0.503793i \(-0.168062\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 33.9411i − 2.38220i
\(204\) 0 0
\(205\) − 5.65685i − 0.395092i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 10.0000 0.691714
\(210\) 0 0
\(211\) 24.0416i 1.65509i 0.561396 + 0.827547i \(0.310264\pi\)
−0.561396 + 0.827547i \(0.689736\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −12.0000 −0.818393
\(216\) 0 0
\(217\) −32.0000 −2.17230
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 11.3137i 0.761042i
\(222\) 0 0
\(223\) −24.0000 −1.60716 −0.803579 0.595198i \(-0.797074\pi\)
−0.803579 + 0.595198i \(0.797074\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.2132i 1.40797i 0.710215 + 0.703985i \(0.248598\pi\)
−0.710215 + 0.703985i \(0.751402\pi\)
\(228\) 0 0
\(229\) 25.4558i 1.68217i 0.540903 + 0.841085i \(0.318082\pi\)
−0.540903 + 0.841085i \(0.681918\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 12.0000 0.786146 0.393073 0.919507i \(-0.371412\pi\)
0.393073 + 0.919507i \(0.371412\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −8.00000 −0.517477 −0.258738 0.965947i \(-0.583307\pi\)
−0.258738 + 0.965947i \(0.583307\pi\)
\(240\) 0 0
\(241\) 12.0000 0.772988 0.386494 0.922292i \(-0.373686\pi\)
0.386494 + 0.922292i \(0.373686\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 25.4558i − 1.62631i
\(246\) 0 0
\(247\) 20.0000 1.27257
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 4.24264i 0.267793i 0.990995 + 0.133897i \(0.0427490\pi\)
−0.990995 + 0.133897i \(0.957251\pi\)
\(252\) 0 0
\(253\) − 5.65685i − 0.355643i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.00000 0.124757 0.0623783 0.998053i \(-0.480131\pi\)
0.0623783 + 0.998053i \(0.480131\pi\)
\(258\) 0 0
\(259\) − 11.3137i − 0.703000i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −28.0000 −1.72655 −0.863277 0.504730i \(-0.831592\pi\)
−0.863277 + 0.504730i \(0.831592\pi\)
\(264\) 0 0
\(265\) 8.00000 0.491436
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 19.7990i − 1.20717i −0.797300 0.603583i \(-0.793739\pi\)
0.797300 0.603583i \(-0.206261\pi\)
\(270\) 0 0
\(271\) −32.0000 −1.94386 −0.971931 0.235267i \(-0.924404\pi\)
−0.971931 + 0.235267i \(0.924404\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 4.24264i − 0.255841i
\(276\) 0 0
\(277\) 8.48528i 0.509831i 0.966963 + 0.254916i \(0.0820477\pi\)
−0.966963 + 0.254916i \(0.917952\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −20.0000 −1.19310 −0.596550 0.802576i \(-0.703462\pi\)
−0.596550 + 0.802576i \(0.703462\pi\)
\(282\) 0 0
\(283\) − 24.0416i − 1.42913i −0.699571 0.714563i \(-0.746625\pi\)
0.699571 0.714563i \(-0.253375\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 8.00000 0.472225
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 8.48528i 0.495715i 0.968796 + 0.247858i \(0.0797265\pi\)
−0.968796 + 0.247858i \(0.920273\pi\)
\(294\) 0 0
\(295\) 12.0000 0.698667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) − 11.3137i − 0.654289i
\(300\) 0 0
\(301\) − 16.9706i − 0.978167i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −24.0000 −1.37424
\(306\) 0 0
\(307\) − 7.07107i − 0.403567i −0.979430 0.201784i \(-0.935326\pi\)
0.979430 0.201784i \(-0.0646738\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.0000 1.13410 0.567048 0.823685i \(-0.308085\pi\)
0.567048 + 0.823685i \(0.308085\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 2.82843i 0.158860i 0.996840 + 0.0794301i \(0.0253101\pi\)
−0.996840 + 0.0794301i \(0.974690\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 28.2843i − 1.57378i
\(324\) 0 0
\(325\) − 8.48528i − 0.470679i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) − 12.7279i − 0.699590i −0.936826 0.349795i \(-0.886251\pi\)
0.936826 0.349795i \(-0.113749\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) 30.0000 1.63420 0.817102 0.576493i \(-0.195579\pi\)
0.817102 + 0.576493i \(0.195579\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 11.3137i − 0.612672i
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.5563i 0.835109i 0.908652 + 0.417554i \(0.137113\pi\)
−0.908652 + 0.417554i \(0.862887\pi\)
\(348\) 0 0
\(349\) 19.7990i 1.05982i 0.848055 + 0.529908i \(0.177773\pi\)
−0.848055 + 0.529908i \(0.822227\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 14.0000 0.745145 0.372572 0.928003i \(-0.378476\pi\)
0.372572 + 0.928003i \(0.378476\pi\)
\(354\) 0 0
\(355\) − 11.3137i − 0.600469i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 36.0000 1.90001 0.950004 0.312239i \(-0.101079\pi\)
0.950004 + 0.312239i \(0.101079\pi\)
\(360\) 0 0
\(361\) −31.0000 −1.63158
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 11.3137i − 0.592187i
\(366\) 0 0
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 11.3137i 0.587378i
\(372\) 0 0
\(373\) − 36.7696i − 1.90386i −0.306323 0.951928i \(-0.599099\pi\)
0.306323 0.951928i \(-0.400901\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 24.0000 1.23606
\(378\) 0 0
\(379\) 29.6985i 1.52551i 0.646688 + 0.762754i \(0.276153\pi\)
−0.646688 + 0.762754i \(0.723847\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 16.0000 0.817562 0.408781 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 16.0000 0.815436
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 19.7990i − 1.00385i −0.864912 0.501924i \(-0.832626\pi\)
0.864912 0.501924i \(-0.167374\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 22.6274i − 1.13851i
\(396\) 0 0
\(397\) − 2.82843i − 0.141955i −0.997478 0.0709773i \(-0.977388\pi\)
0.997478 0.0709773i \(-0.0226118\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 36.0000 1.79775 0.898877 0.438201i \(-0.144384\pi\)
0.898877 + 0.438201i \(0.144384\pi\)
\(402\) 0 0
\(403\) − 22.6274i − 1.12715i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 16.9706i 0.835067i
\(414\) 0 0
\(415\) 28.0000 1.37447
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) − 26.8701i − 1.31269i −0.754462 0.656344i \(-0.772102\pi\)
0.754462 0.656344i \(-0.227898\pi\)
\(420\) 0 0
\(421\) − 8.48528i − 0.413547i −0.978389 0.206774i \(-0.933704\pi\)
0.978389 0.206774i \(-0.0662964\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −12.0000 −0.582086
\(426\) 0 0
\(427\) − 33.9411i − 1.64253i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −24.0000 −1.15604 −0.578020 0.816023i \(-0.696174\pi\)
−0.578020 + 0.816023i \(0.696174\pi\)
\(432\) 0 0
\(433\) −20.0000 −0.961139 −0.480569 0.876957i \(-0.659570\pi\)
−0.480569 + 0.876957i \(0.659570\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 28.2843i 1.35302i
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 35.3553i 1.67978i 0.542754 + 0.839891i \(0.317381\pi\)
−0.542754 + 0.839891i \(0.682619\pi\)
\(444\) 0 0
\(445\) − 33.9411i − 1.60896i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 20.0000 0.943858 0.471929 0.881636i \(-0.343558\pi\)
0.471929 + 0.881636i \(0.343558\pi\)
\(450\) 0 0
\(451\) 2.82843i 0.133185i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 32.0000 1.50018
\(456\) 0 0
\(457\) −2.00000 −0.0935561 −0.0467780 0.998905i \(-0.514895\pi\)
−0.0467780 + 0.998905i \(0.514895\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 14.1421i 0.658665i 0.944214 + 0.329332i \(0.106824\pi\)
−0.944214 + 0.329332i \(0.893176\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.3848i 0.850746i 0.905018 + 0.425373i \(0.139857\pi\)
−0.905018 + 0.425373i \(0.860143\pi\)
\(468\) 0 0
\(469\) 16.9706i 0.783628i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 6.00000 0.275880
\(474\) 0 0
\(475\) 21.2132i 0.973329i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 8.00000 0.364769
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 11.3137i 0.513729i
\(486\) 0 0
\(487\) −12.0000 −0.543772 −0.271886 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.5563i 0.702048i 0.936366 + 0.351024i \(0.114166\pi\)
−0.936366 + 0.351024i \(0.885834\pi\)
\(492\) 0 0
\(493\) − 33.9411i − 1.52863i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 16.0000 0.717698
\(498\) 0 0
\(499\) 12.7279i 0.569780i 0.958560 + 0.284890i \(0.0919571\pi\)
−0.958560 + 0.284890i \(0.908043\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −20.0000 −0.891756 −0.445878 0.895094i \(-0.647108\pi\)
−0.445878 + 0.895094i \(0.647108\pi\)
\(504\) 0 0
\(505\) 8.00000 0.355995
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 2.82843i − 0.125368i −0.998033 0.0626839i \(-0.980034\pi\)
0.998033 0.0626839i \(-0.0199660\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 33.9411i − 1.49562i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 14.0000 0.613351 0.306676 0.951814i \(-0.400783\pi\)
0.306676 + 0.951814i \(0.400783\pi\)
\(522\) 0 0
\(523\) 9.89949i 0.432875i 0.976297 + 0.216437i \(0.0694437\pi\)
−0.976297 + 0.216437i \(0.930556\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −32.0000 −1.39394
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 5.65685i 0.245026i
\(534\) 0 0
\(535\) −28.0000 −1.21055
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 12.7279i 0.548230i
\(540\) 0 0
\(541\) − 8.48528i − 0.364811i −0.983223 0.182405i \(-0.941612\pi\)
0.983223 0.182405i \(-0.0583883\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −40.0000 −1.71341
\(546\) 0 0
\(547\) 26.8701i 1.14888i 0.818546 + 0.574440i \(0.194780\pi\)
−0.818546 + 0.574440i \(0.805220\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −60.0000 −2.55609
\(552\) 0 0
\(553\) 32.0000 1.36078
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 31.1127i 1.31829i 0.752017 + 0.659144i \(0.229081\pi\)
−0.752017 + 0.659144i \(0.770919\pi\)
\(558\) 0 0
\(559\) 12.0000 0.507546
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 7.07107i 0.298010i 0.988836 + 0.149005i \(0.0476070\pi\)
−0.988836 + 0.149005i \(0.952393\pi\)
\(564\) 0 0
\(565\) 39.5980i 1.66590i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 30.0000 1.25767 0.628833 0.777541i \(-0.283533\pi\)
0.628833 + 0.777541i \(0.283533\pi\)
\(570\) 0 0
\(571\) 32.5269i 1.36121i 0.732651 + 0.680604i \(0.238283\pi\)
−0.732651 + 0.680604i \(0.761717\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 12.0000 0.500435
\(576\) 0 0
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 39.5980i 1.64280i
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.41421i 0.0583708i 0.999574 + 0.0291854i \(0.00929133\pi\)
−0.999574 + 0.0291854i \(0.990709\pi\)
\(588\) 0 0
\(589\) 56.5685i 2.33087i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −18.0000 −0.739171 −0.369586 0.929197i \(-0.620500\pi\)
−0.369586 + 0.929197i \(0.620500\pi\)
\(594\) 0 0
\(595\) − 45.2548i − 1.85527i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 36.0000 1.47092 0.735460 0.677568i \(-0.236966\pi\)
0.735460 + 0.677568i \(0.236966\pi\)
\(600\) 0 0
\(601\) 4.00000 0.163163 0.0815817 0.996667i \(-0.474003\pi\)
0.0815817 + 0.996667i \(0.474003\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 25.4558i − 1.03493i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) − 8.48528i − 0.342717i −0.985209 0.171359i \(-0.945184\pi\)
0.985209 0.171359i \(-0.0548157\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 28.0000 1.12724 0.563619 0.826035i \(-0.309409\pi\)
0.563619 + 0.826035i \(0.309409\pi\)
\(618\) 0 0
\(619\) − 1.41421i − 0.0568420i −0.999596 0.0284210i \(-0.990952\pi\)
0.999596 0.0284210i \(-0.00904791\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 48.0000 1.92308
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 11.3137i − 0.451107i
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 22.6274i 0.897942i
\(636\) 0 0
\(637\) 25.4558i 1.00860i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 4.00000 0.157991 0.0789953 0.996875i \(-0.474829\pi\)
0.0789953 + 0.996875i \(0.474829\pi\)
\(642\) 0 0
\(643\) 4.24264i 0.167313i 0.996495 + 0.0836567i \(0.0266599\pi\)
−0.996495 + 0.0836567i \(0.973340\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −12.0000 −0.471769 −0.235884 0.971781i \(-0.575799\pi\)
−0.235884 + 0.971781i \(0.575799\pi\)
\(648\) 0 0
\(649\) −6.00000 −0.235521
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 48.0833i 1.88164i 0.338902 + 0.940822i \(0.389945\pi\)
−0.338902 + 0.940822i \(0.610055\pi\)
\(654\) 0 0
\(655\) 20.0000 0.781465
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 32.5269i 1.26707i 0.773715 + 0.633534i \(0.218396\pi\)
−0.773715 + 0.633534i \(0.781604\pi\)
\(660\) 0 0
\(661\) − 31.1127i − 1.21014i −0.796171 0.605072i \(-0.793144\pi\)
0.796171 0.605072i \(-0.206856\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −80.0000 −3.10227
\(666\) 0 0
\(667\) 33.9411i 1.31421i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12.0000 0.463255
\(672\) 0 0
\(673\) −20.0000 −0.770943 −0.385472 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) − 42.4264i − 1.63058i −0.579053 0.815290i \(-0.696578\pi\)
0.579053 0.815290i \(-0.303422\pi\)
\(678\) 0 0
\(679\) −16.0000 −0.614024
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 29.6985i − 1.13638i −0.822897 0.568190i \(-0.807644\pi\)
0.822897 0.568190i \(-0.192356\pi\)
\(684\) 0 0
\(685\) 50.9117i 1.94524i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −8.00000 −0.304776
\(690\) 0 0
\(691\) 46.6690i 1.77537i 0.460447 + 0.887687i \(0.347689\pi\)
−0.460447 + 0.887687i \(0.652311\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 48.0833i − 1.81608i −0.418884 0.908040i \(-0.637579\pi\)
0.418884 0.908040i \(-0.362421\pi\)
\(702\) 0 0
\(703\) −20.0000 −0.754314
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 11.3137i 0.425496i
\(708\) 0 0
\(709\) 8.48528i 0.318671i 0.987224 + 0.159336i \(0.0509352\pi\)
−0.987224 + 0.159336i \(0.949065\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 11.3137i 0.423109i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 24.0000 0.895049 0.447524 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) 48.0000 1.78761
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 25.4558i 0.945406i
\(726\) 0 0
\(727\) 28.0000 1.03846 0.519231 0.854634i \(-0.326218\pi\)
0.519231 + 0.854634i \(0.326218\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) − 16.9706i − 0.627679i
\(732\) 0 0
\(733\) 19.7990i 0.731292i 0.930754 + 0.365646i \(0.119152\pi\)
−0.930754 + 0.365646i \(0.880848\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −6.00000 −0.221013
\(738\) 0 0
\(739\) 24.0416i 0.884386i 0.896920 + 0.442193i \(0.145799\pi\)
−0.896920 + 0.442193i \(0.854201\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −4.00000 −0.146746 −0.0733729 0.997305i \(-0.523376\pi\)
−0.0733729 + 0.997305i \(0.523376\pi\)
\(744\) 0 0
\(745\) −8.00000 −0.293097
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 39.5980i − 1.44688i
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 11.3137i − 0.411748i
\(756\) 0 0
\(757\) 25.4558i 0.925208i 0.886565 + 0.462604i \(0.153085\pi\)
−0.886565 + 0.462604i \(0.846915\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 18.0000 0.652499 0.326250 0.945284i \(-0.394215\pi\)
0.326250 + 0.945284i \(0.394215\pi\)
\(762\) 0 0
\(763\) − 56.5685i − 2.04792i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) 44.0000 1.58668 0.793340 0.608778i \(-0.208340\pi\)
0.793340 + 0.608778i \(0.208340\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 8.48528i − 0.305194i −0.988288 0.152597i \(-0.951236\pi\)
0.988288 0.152597i \(-0.0487637\pi\)
\(774\) 0 0
\(775\) 24.0000 0.862105
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 14.1421i − 0.506695i
\(780\) 0 0
\(781\) 5.65685i 0.202418i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −24.0000 −0.856597
\(786\) 0 0
\(787\) 46.6690i 1.66357i 0.555097 + 0.831786i \(0.312681\pi\)
−0.555097 + 0.831786i \(0.687319\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −56.0000 −1.99113
\(792\) 0 0
\(793\) 24.0000 0.852265
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.7990i 0.701316i 0.936504 + 0.350658i \(0.114042\pi\)
−0.936504 + 0.350658i \(0.885958\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 5.65685i 0.199626i
\(804\) 0 0
\(805\) 45.2548i 1.59502i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) − 24.0416i − 0.844216i −0.906546 0.422108i \(-0.861290\pi\)
0.906546 0.422108i \(-0.138710\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −28.0000 −0.980797
\(816\) 0 0
\(817\) −30.0000 −1.04957
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 8.48528i − 0.296138i −0.988977 0.148069i \(-0.952694\pi\)
0.988977 0.148069i \(-0.0473058\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 21.2132i − 0.737655i −0.929498 0.368828i \(-0.879759\pi\)
0.929498 0.368828i \(-0.120241\pi\)
\(828\) 0 0
\(829\) 25.4558i 0.884118i 0.896986 + 0.442059i \(0.145752\pi\)
−0.896986 + 0.442059i \(0.854248\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 36.0000 1.24733
\(834\) 0 0
\(835\) − 33.9411i − 1.17458i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −20.0000 −0.690477 −0.345238 0.938515i \(-0.612202\pi\)
−0.345238 + 0.938515i \(0.612202\pi\)
\(840\) 0 0
\(841\) −43.0000 −1.48276
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 14.1421i − 0.486504i
\(846\) 0 0
\(847\) 36.0000 1.23697
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 11.3137i 0.387829i
\(852\) 0 0
\(853\) 42.4264i 1.45265i 0.687350 + 0.726326i \(0.258774\pi\)
−0.687350 + 0.726326i \(0.741226\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −34.0000 −1.16142 −0.580709 0.814111i \(-0.697225\pi\)
−0.580709 + 0.814111i \(0.697225\pi\)
\(858\) 0 0
\(859\) − 38.1838i − 1.30281i −0.758729 0.651407i \(-0.774179\pi\)
0.758729 0.651407i \(-0.225821\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) −40.0000 −1.36004
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.3137i 0.383791i
\(870\) 0 0
\(871\) −12.0000 −0.406604
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 22.6274i − 0.764946i
\(876\) 0 0
\(877\) 8.48528i 0.286528i 0.989685 + 0.143264i \(0.0457597\pi\)
−0.989685 + 0.143264i \(0.954240\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 34.0000 1.14549 0.572745 0.819734i \(-0.305879\pi\)
0.572745 + 0.819734i \(0.305879\pi\)
\(882\) 0 0
\(883\) − 41.0122i − 1.38017i −0.723728 0.690085i \(-0.757573\pi\)
0.723728 0.690085i \(-0.242427\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −28.0000 −0.940148 −0.470074 0.882627i \(-0.655773\pi\)
−0.470074 + 0.882627i \(0.655773\pi\)
\(888\) 0 0
\(889\) −32.0000 −1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 67.8823i 2.26400i
\(900\) 0 0
\(901\) 11.3137i 0.376914i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −24.0000 −0.797787
\(906\) 0 0
\(907\) 7.07107i 0.234791i 0.993085 + 0.117395i \(0.0374545\pi\)
−0.993085 + 0.117395i \(0.962545\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 48.0000 1.59031 0.795155 0.606406i \(-0.207389\pi\)
0.795155 + 0.606406i \(0.207389\pi\)
\(912\) 0 0
\(913\) −14.0000 −0.463332
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 28.2843i 0.934029i
\(918\) 0 0
\(919\) −44.0000 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 11.3137i 0.372395i
\(924\) 0 0
\(925\) 8.48528i 0.278994i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −60.0000 −1.96854 −0.984268 0.176682i \(-0.943464\pi\)
−0.984268 + 0.176682i \(0.943464\pi\)
\(930\) 0 0
\(931\) − 63.6396i − 2.08570i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 16.0000 0.523256
\(936\) 0 0
\(937\) 20.0000 0.653372 0.326686 0.945133i \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 36.7696i 1.19865i 0.800505 + 0.599327i \(0.204565\pi\)
−0.800505 + 0.599327i \(0.795435\pi\)
\(942\) 0 0
\(943\) −8.00000 −0.260516
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 60.8112i − 1.97610i −0.154140 0.988049i \(-0.549261\pi\)
0.154140 0.988049i \(-0.450739\pi\)
\(948\) 0 0
\(949\) 11.3137i 0.367259i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 30.0000 0.971795 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(954\) 0 0
\(955\) − 22.6274i − 0.732206i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −72.0000 −2.32500
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 11.3137i 0.364201i
\(966\) 0 0
\(967\) 20.0000 0.643157 0.321578 0.946883i \(-0.395787\pi\)
0.321578 + 0.946883i \(0.395787\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 15.5563i 0.499227i 0.968346 + 0.249614i \(0.0803036\pi\)
−0.968346 + 0.249614i \(0.919696\pi\)
\(972\) 0 0
\(973\) − 16.9706i − 0.544051i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −12.0000 −0.383914 −0.191957 0.981403i \(-0.561483\pi\)
−0.191957 + 0.981403i \(0.561483\pi\)
\(978\) 0 0
\(979\) 16.9706i 0.542382i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 36.0000 1.14822 0.574111 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) −40.0000 −1.27451
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 16.9706i 0.539633i
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 11.3137i − 0.358669i
\(996\) 0 0
\(997\) − 14.1421i − 0.447886i −0.974602 0.223943i \(-0.928107\pi\)
0.974602 0.223943i \(-0.0718929\pi\)
\(998\) 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 4608.2.d.b.2305.1 2
3.2 odd 2 512.2.b.b.257.2 2
4.3 odd 2 4608.2.d.a.2305.1 2
8.3 odd 2 4608.2.d.a.2305.2 2
8.5 even 2 inner 4608.2.d.b.2305.2 2
12.11 even 2 512.2.b.a.257.1 2
16.3 odd 4 4608.2.a.m.1.1 2
16.5 even 4 4608.2.a.f.1.2 2
16.11 odd 4 4608.2.a.m.1.2 2
16.13 even 4 4608.2.a.f.1.1 2
24.5 odd 2 512.2.b.b.257.1 2
24.11 even 2 512.2.b.a.257.2 2
48.5 odd 4 512.2.a.c.1.2 yes 2
48.11 even 4 512.2.a.d.1.1 yes 2
48.29 odd 4 512.2.a.c.1.1 2
48.35 even 4 512.2.a.d.1.2 yes 2
96.5 odd 8 1024.2.e.b.257.1 2
96.11 even 8 1024.2.e.c.257.1 2
96.29 odd 8 1024.2.e.b.769.1 2
96.35 even 8 1024.2.e.d.769.1 2
96.53 odd 8 1024.2.e.e.257.1 2
96.59 even 8 1024.2.e.d.257.1 2
96.77 odd 8 1024.2.e.e.769.1 2
96.83 even 8 1024.2.e.c.769.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
512.2.a.c.1.1 2 48.29 odd 4
512.2.a.c.1.2 yes 2 48.5 odd 4
512.2.a.d.1.1 yes 2 48.11 even 4
512.2.a.d.1.2 yes 2 48.35 even 4
512.2.b.a.257.1 2 12.11 even 2
512.2.b.a.257.2 2 24.11 even 2
512.2.b.b.257.1 2 24.5 odd 2
512.2.b.b.257.2 2 3.2 odd 2
1024.2.e.b.257.1 2 96.5 odd 8
1024.2.e.b.769.1 2 96.29 odd 8
1024.2.e.c.257.1 2 96.11 even 8
1024.2.e.c.769.1 2 96.83 even 8
1024.2.e.d.257.1 2 96.59 even 8
1024.2.e.d.769.1 2 96.35 even 8
1024.2.e.e.257.1 2 96.53 odd 8
1024.2.e.e.769.1 2 96.77 odd 8
4608.2.a.f.1.1 2 16.13 even 4
4608.2.a.f.1.2 2 16.5 even 4
4608.2.a.m.1.1 2 16.3 odd 4
4608.2.a.m.1.2 2 16.11 odd 4
4608.2.d.a.2305.1 2 4.3 odd 2
4608.2.d.a.2305.2 2 8.3 odd 2
4608.2.d.b.2305.1 2 1.1 even 1 trivial
4608.2.d.b.2305.2 2 8.5 even 2 inner