Properties

Label 4608.2.c.d.4607.1
Level $4608$
Weight $2$
Character 4608.4607
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.c (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4607.1
Root \(-1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 4608.4607
Dual form 4608.2.c.d.4607.2

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843i q^{5} +1.41421i q^{7} +O(q^{10})\) \(q-2.82843i q^{5} +1.41421i q^{7} -2.00000 q^{13} +1.41421i q^{17} -2.82843i q^{19} +6.00000 q^{23} -3.00000 q^{25} +5.65685i q^{29} +9.89949i q^{31} +4.00000 q^{35} +2.00000 q^{37} -4.24264i q^{41} -8.48528i q^{43} +10.0000 q^{47} +5.00000 q^{49} -5.65685i q^{53} +12.0000 q^{59} -10.0000 q^{61} +5.65685i q^{65} -5.65685i q^{67} +2.00000 q^{71} -10.0000 q^{73} -12.7279i q^{79} +4.00000 q^{83} +4.00000 q^{85} -9.89949i q^{89} -2.82843i q^{91} -8.00000 q^{95} +12.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 4 q^{13} + 12 q^{23} - 6 q^{25} + 8 q^{35} + 4 q^{37} + 20 q^{47} + 10 q^{49} + 24 q^{59} - 20 q^{61} + 4 q^{71} - 20 q^{73} + 8 q^{83} + 8 q^{85} - 16 q^{95} + 24 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) 1.41421i 0.534522i 0.963624 + 0.267261i \(0.0861187\pi\)
−0.963624 + 0.267261i \(0.913881\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.41421i 0.342997i 0.985184 + 0.171499i \(0.0548609\pi\)
−0.985184 + 0.171499i \(0.945139\pi\)
\(18\) 0 0
\(19\) − 2.82843i − 0.648886i −0.945905 0.324443i \(-0.894823\pi\)
0.945905 0.324443i \(-0.105177\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) −3.00000 −0.600000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.65685i 1.05045i 0.850963 + 0.525226i \(0.176019\pi\)
−0.850963 + 0.525226i \(0.823981\pi\)
\(30\) 0 0
\(31\) 9.89949i 1.77800i 0.457905 + 0.889001i \(0.348600\pi\)
−0.457905 + 0.889001i \(0.651400\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 2.00000 0.328798 0.164399 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 4.24264i − 0.662589i −0.943527 0.331295i \(-0.892515\pi\)
0.943527 0.331295i \(-0.107485\pi\)
\(42\) 0 0
\(43\) − 8.48528i − 1.29399i −0.762493 0.646997i \(-0.776025\pi\)
0.762493 0.646997i \(-0.223975\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 10.0000 1.45865 0.729325 0.684167i \(-0.239834\pi\)
0.729325 + 0.684167i \(0.239834\pi\)
\(48\) 0 0
\(49\) 5.00000 0.714286
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 5.65685i − 0.777029i −0.921443 0.388514i \(-0.872988\pi\)
0.921443 0.388514i \(-0.127012\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 12.0000 1.56227 0.781133 0.624364i \(-0.214642\pi\)
0.781133 + 0.624364i \(0.214642\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.65685i 0.701646i
\(66\) 0 0
\(67\) − 5.65685i − 0.691095i −0.938401 0.345547i \(-0.887693\pi\)
0.938401 0.345547i \(-0.112307\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 2.00000 0.237356 0.118678 0.992933i \(-0.462134\pi\)
0.118678 + 0.992933i \(0.462134\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) − 12.7279i − 1.43200i −0.698099 0.716002i \(-0.745970\pi\)
0.698099 0.716002i \(-0.254030\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 4.00000 0.433861
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 9.89949i − 1.04934i −0.851304 0.524672i \(-0.824188\pi\)
0.851304 0.524672i \(-0.175812\pi\)
\(90\) 0 0
\(91\) − 2.82843i − 0.296500i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −8.00000 −0.820783
\(96\) 0 0
\(97\) 12.0000 1.21842 0.609208 0.793011i \(-0.291488\pi\)
0.609208 + 0.793011i \(0.291488\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 7.07107i 0.696733i 0.937358 + 0.348367i \(0.113264\pi\)
−0.937358 + 0.348367i \(0.886736\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −8.00000 −0.773389 −0.386695 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 9.89949i 0.931266i 0.884978 + 0.465633i \(0.154173\pi\)
−0.884978 + 0.465633i \(0.845827\pi\)
\(114\) 0 0
\(115\) − 16.9706i − 1.58251i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 5.65685i − 0.505964i
\(126\) 0 0
\(127\) 4.24264i 0.376473i 0.982124 + 0.188237i \(0.0602772\pi\)
−0.982124 + 0.188237i \(0.939723\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 16.0000 1.39793 0.698963 0.715158i \(-0.253645\pi\)
0.698963 + 0.715158i \(0.253645\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 1.41421i − 0.120824i −0.998174 0.0604122i \(-0.980758\pi\)
0.998174 0.0604122i \(-0.0192415\pi\)
\(138\) 0 0
\(139\) − 22.6274i − 1.91923i −0.281312 0.959616i \(-0.590770\pi\)
0.281312 0.959616i \(-0.409230\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 16.0000 1.32873
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 19.7990i 1.62200i 0.585049 + 0.810998i \(0.301075\pi\)
−0.585049 + 0.810998i \(0.698925\pi\)
\(150\) 0 0
\(151\) − 12.7279i − 1.03578i −0.855446 0.517892i \(-0.826717\pi\)
0.855446 0.517892i \(-0.173283\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 28.0000 2.24901
\(156\) 0 0
\(157\) 22.0000 1.75579 0.877896 0.478852i \(-0.158947\pi\)
0.877896 + 0.478852i \(0.158947\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 8.48528i 0.668734i
\(162\) 0 0
\(163\) − 8.48528i − 0.664619i −0.943170 0.332309i \(-0.892172\pi\)
0.943170 0.332309i \(-0.107828\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −16.0000 −1.23812 −0.619059 0.785345i \(-0.712486\pi\)
−0.619059 + 0.785345i \(0.712486\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 2.82843i − 0.215041i −0.994203 0.107521i \(-0.965709\pi\)
0.994203 0.107521i \(-0.0342912\pi\)
\(174\) 0 0
\(175\) − 4.24264i − 0.320713i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −4.00000 −0.298974 −0.149487 0.988764i \(-0.547762\pi\)
−0.149487 + 0.988764i \(0.547762\pi\)
\(180\) 0 0
\(181\) −18.0000 −1.33793 −0.668965 0.743294i \(-0.733262\pi\)
−0.668965 + 0.743294i \(0.733262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 5.65685i − 0.415900i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.00000 −0.578860 −0.289430 0.957199i \(-0.593466\pi\)
−0.289430 + 0.957199i \(0.593466\pi\)
\(192\) 0 0
\(193\) 8.00000 0.575853 0.287926 0.957653i \(-0.407034\pi\)
0.287926 + 0.957653i \(0.407034\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 11.3137i 0.806068i 0.915185 + 0.403034i \(0.132044\pi\)
−0.915185 + 0.403034i \(0.867956\pi\)
\(198\) 0 0
\(199\) 1.41421i 0.100251i 0.998743 + 0.0501255i \(0.0159621\pi\)
−0.998743 + 0.0501255i \(0.984038\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −8.00000 −0.561490
\(204\) 0 0
\(205\) −12.0000 −0.838116
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) − 16.9706i − 1.16830i −0.811645 0.584151i \(-0.801428\pi\)
0.811645 0.584151i \(-0.198572\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −24.0000 −1.63679
\(216\) 0 0
\(217\) −14.0000 −0.950382
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 2.82843i − 0.190261i
\(222\) 0 0
\(223\) − 21.2132i − 1.42054i −0.703929 0.710271i \(-0.748573\pi\)
0.703929 0.710271i \(-0.251427\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.0000 1.32745 0.663723 0.747978i \(-0.268975\pi\)
0.663723 + 0.747978i \(0.268975\pi\)
\(228\) 0 0
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 26.8701i − 1.76032i −0.474681 0.880158i \(-0.657437\pi\)
0.474681 0.880158i \(-0.342563\pi\)
\(234\) 0 0
\(235\) − 28.2843i − 1.84506i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −16.0000 −1.03495 −0.517477 0.855697i \(-0.673129\pi\)
−0.517477 + 0.855697i \(0.673129\pi\)
\(240\) 0 0
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 14.1421i − 0.903508i
\(246\) 0 0
\(247\) 5.65685i 0.359937i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 24.0000 1.51487 0.757433 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 9.89949i 0.617514i 0.951141 + 0.308757i \(0.0999129\pi\)
−0.951141 + 0.308757i \(0.900087\pi\)
\(258\) 0 0
\(259\) 2.82843i 0.175750i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 8.00000 0.493301 0.246651 0.969104i \(-0.420670\pi\)
0.246651 + 0.969104i \(0.420670\pi\)
\(264\) 0 0
\(265\) −16.0000 −0.982872
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 11.3137i − 0.689809i −0.938638 0.344904i \(-0.887911\pi\)
0.938638 0.344904i \(-0.112089\pi\)
\(270\) 0 0
\(271\) 12.7279i 0.773166i 0.922255 + 0.386583i \(0.126345\pi\)
−0.922255 + 0.386583i \(0.873655\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 30.0000 1.80253 0.901263 0.433273i \(-0.142641\pi\)
0.901263 + 0.433273i \(0.142641\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 1.41421i − 0.0843649i −0.999110 0.0421825i \(-0.986569\pi\)
0.999110 0.0421825i \(-0.0134311\pi\)
\(282\) 0 0
\(283\) 11.3137i 0.672530i 0.941767 + 0.336265i \(0.109164\pi\)
−0.941767 + 0.336265i \(0.890836\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) 15.0000 0.882353
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.65685i 0.330477i 0.986254 + 0.165238i \(0.0528394\pi\)
−0.986254 + 0.165238i \(0.947161\pi\)
\(294\) 0 0
\(295\) − 33.9411i − 1.97613i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −12.0000 −0.693978
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 28.2843i 1.61955i
\(306\) 0 0
\(307\) − 11.3137i − 0.645707i −0.946449 0.322854i \(-0.895358\pi\)
0.946449 0.322854i \(-0.104642\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) −8.00000 −0.452187 −0.226093 0.974106i \(-0.572595\pi\)
−0.226093 + 0.974106i \(0.572595\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 22.6274i − 1.27088i −0.772149 0.635441i \(-0.780818\pi\)
0.772149 0.635441i \(-0.219182\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 4.00000 0.222566
\(324\) 0 0
\(325\) 6.00000 0.332820
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 14.1421i 0.779681i
\(330\) 0 0
\(331\) − 22.6274i − 1.24372i −0.783130 0.621858i \(-0.786378\pi\)
0.783130 0.621858i \(-0.213622\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −16.0000 −0.874173
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 16.9706i 0.916324i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 12.7279i − 0.677439i −0.940887 0.338719i \(-0.890006\pi\)
0.940887 0.338719i \(-0.109994\pi\)
\(354\) 0 0
\(355\) − 5.65685i − 0.300235i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 18.0000 0.950004 0.475002 0.879985i \(-0.342447\pi\)
0.475002 + 0.879985i \(0.342447\pi\)
\(360\) 0 0
\(361\) 11.0000 0.578947
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 28.2843i 1.48047i
\(366\) 0 0
\(367\) 21.2132i 1.10732i 0.832743 + 0.553660i \(0.186769\pi\)
−0.832743 + 0.553660i \(0.813231\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.00000 0.415339
\(372\) 0 0
\(373\) −30.0000 −1.55334 −0.776671 0.629907i \(-0.783093\pi\)
−0.776671 + 0.629907i \(0.783093\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 11.3137i − 0.582686i
\(378\) 0 0
\(379\) 14.1421i 0.726433i 0.931705 + 0.363216i \(0.118321\pi\)
−0.931705 + 0.363216i \(0.881679\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 32.0000 1.63512 0.817562 0.575841i \(-0.195325\pi\)
0.817562 + 0.575841i \(0.195325\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 8.48528i − 0.430221i −0.976590 0.215110i \(-0.930989\pi\)
0.976590 0.215110i \(-0.0690111\pi\)
\(390\) 0 0
\(391\) 8.48528i 0.429119i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −36.0000 −1.81136
\(396\) 0 0
\(397\) −22.0000 −1.10415 −0.552074 0.833795i \(-0.686163\pi\)
−0.552074 + 0.833795i \(0.686163\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 26.8701i − 1.34183i −0.741536 0.670913i \(-0.765902\pi\)
0.741536 0.670913i \(-0.234098\pi\)
\(402\) 0 0
\(403\) − 19.7990i − 0.986258i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.00000 0.197787 0.0988936 0.995098i \(-0.468470\pi\)
0.0988936 + 0.995098i \(0.468470\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 16.9706i 0.835067i
\(414\) 0 0
\(415\) − 11.3137i − 0.555368i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) − 4.24264i − 0.205798i
\(426\) 0 0
\(427\) − 14.1421i − 0.684386i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.0000 0.674356 0.337178 0.941441i \(-0.390528\pi\)
0.337178 + 0.941441i \(0.390528\pi\)
\(432\) 0 0
\(433\) −14.0000 −0.672797 −0.336399 0.941720i \(-0.609209\pi\)
−0.336399 + 0.941720i \(0.609209\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 16.9706i − 0.811812i
\(438\) 0 0
\(439\) − 1.41421i − 0.0674967i −0.999430 0.0337484i \(-0.989256\pi\)
0.999430 0.0337484i \(-0.0107445\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 28.0000 1.33032 0.665160 0.746701i \(-0.268363\pi\)
0.665160 + 0.746701i \(0.268363\pi\)
\(444\) 0 0
\(445\) −28.0000 −1.32733
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 32.5269i 1.53504i 0.641025 + 0.767520i \(0.278509\pi\)
−0.641025 + 0.767520i \(0.721491\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −20.0000 −0.935561 −0.467780 0.883845i \(-0.654946\pi\)
−0.467780 + 0.883845i \(0.654946\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.48528i 0.395199i 0.980283 + 0.197599i \(0.0633145\pi\)
−0.980283 + 0.197599i \(0.936685\pi\)
\(462\) 0 0
\(463\) 29.6985i 1.38021i 0.723711 + 0.690103i \(0.242435\pi\)
−0.723711 + 0.690103i \(0.757565\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −36.0000 −1.66588 −0.832941 0.553362i \(-0.813345\pi\)
−0.832941 + 0.553362i \(0.813345\pi\)
\(468\) 0 0
\(469\) 8.00000 0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 8.48528i 0.389331i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 26.0000 1.18797 0.593985 0.804476i \(-0.297554\pi\)
0.593985 + 0.804476i \(0.297554\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 33.9411i − 1.54119i
\(486\) 0 0
\(487\) 15.5563i 0.704925i 0.935826 + 0.352463i \(0.114656\pi\)
−0.935826 + 0.352463i \(0.885344\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −12.0000 −0.541552 −0.270776 0.962642i \(-0.587280\pi\)
−0.270776 + 0.962642i \(0.587280\pi\)
\(492\) 0 0
\(493\) −8.00000 −0.360302
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.82843i 0.126872i
\(498\) 0 0
\(499\) 16.9706i 0.759707i 0.925047 + 0.379853i \(0.124026\pi\)
−0.925047 + 0.379853i \(0.875974\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −6.00000 −0.267527 −0.133763 0.991013i \(-0.542706\pi\)
−0.133763 + 0.991013i \(0.542706\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 16.9706i 0.752207i 0.926578 + 0.376103i \(0.122736\pi\)
−0.926578 + 0.376103i \(0.877264\pi\)
\(510\) 0 0
\(511\) − 14.1421i − 0.625611i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 20.0000 0.881305
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 12.7279i 0.557620i 0.960346 + 0.278810i \(0.0899400\pi\)
−0.960346 + 0.278810i \(0.910060\pi\)
\(522\) 0 0
\(523\) − 8.48528i − 0.371035i −0.982641 0.185518i \(-0.940604\pi\)
0.982641 0.185518i \(-0.0593962\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −14.0000 −0.609850
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 8.48528i 0.367538i
\(534\) 0 0
\(535\) 22.6274i 0.978269i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 38.0000 1.63375 0.816874 0.576816i \(-0.195705\pi\)
0.816874 + 0.576816i \(0.195705\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 28.2843i − 1.21157i
\(546\) 0 0
\(547\) 36.7696i 1.57215i 0.618130 + 0.786076i \(0.287891\pi\)
−0.618130 + 0.786076i \(0.712109\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 16.0000 0.681623
\(552\) 0 0
\(553\) 18.0000 0.765438
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 2.82843i 0.119844i 0.998203 + 0.0599222i \(0.0190852\pi\)
−0.998203 + 0.0599222i \(0.980915\pi\)
\(558\) 0 0
\(559\) 16.9706i 0.717778i
\(560\) 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
\(565\) 28.0000 1.17797
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 15.5563i 0.652156i 0.945343 + 0.326078i \(0.105727\pi\)
−0.945343 + 0.326078i \(0.894273\pi\)
\(570\) 0 0
\(571\) 16.9706i 0.710196i 0.934829 + 0.355098i \(0.115552\pi\)
−0.934829 + 0.355098i \(0.884448\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −18.0000 −0.750652
\(576\) 0 0
\(577\) −20.0000 −0.832611 −0.416305 0.909225i \(-0.636675\pi\)
−0.416305 + 0.909225i \(0.636675\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 5.65685i 0.234686i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 0 0
\(589\) 28.0000 1.15372
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.41421i 0.0580748i 0.999578 + 0.0290374i \(0.00924419\pi\)
−0.999578 + 0.0290374i \(0.990756\pi\)
\(594\) 0 0
\(595\) 5.65685i 0.231908i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 34.0000 1.38920 0.694601 0.719395i \(-0.255581\pi\)
0.694601 + 0.719395i \(0.255581\pi\)
\(600\) 0 0
\(601\) 20.0000 0.815817 0.407909 0.913023i \(-0.366258\pi\)
0.407909 + 0.913023i \(0.366258\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 31.1127i 1.26491i
\(606\) 0 0
\(607\) 15.5563i 0.631413i 0.948857 + 0.315706i \(0.102241\pi\)
−0.948857 + 0.315706i \(0.897759\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −20.0000 −0.809113
\(612\) 0 0
\(613\) −6.00000 −0.242338 −0.121169 0.992632i \(-0.538664\pi\)
−0.121169 + 0.992632i \(0.538664\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 24.0416i − 0.967880i −0.875101 0.483940i \(-0.839205\pi\)
0.875101 0.483940i \(-0.160795\pi\)
\(618\) 0 0
\(619\) 16.9706i 0.682105i 0.940044 + 0.341052i \(0.110783\pi\)
−0.940044 + 0.341052i \(0.889217\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.0000 0.560898
\(624\) 0 0
\(625\) −31.0000 −1.24000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 2.82843i 0.112777i
\(630\) 0 0
\(631\) 4.24264i 0.168897i 0.996428 + 0.0844484i \(0.0269128\pi\)
−0.996428 + 0.0844484i \(0.973087\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 12.0000 0.476205
\(636\) 0 0
\(637\) −10.0000 −0.396214
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 18.3848i − 0.726155i −0.931759 0.363078i \(-0.881726\pi\)
0.931759 0.363078i \(-0.118274\pi\)
\(642\) 0 0
\(643\) 2.82843i 0.111542i 0.998444 + 0.0557711i \(0.0177617\pi\)
−0.998444 + 0.0557711i \(0.982238\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 22.0000 0.864909 0.432455 0.901656i \(-0.357648\pi\)
0.432455 + 0.901656i \(0.357648\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 42.4264i − 1.66027i −0.557560 0.830137i \(-0.688262\pi\)
0.557560 0.830137i \(-0.311738\pi\)
\(654\) 0 0
\(655\) − 45.2548i − 1.76825i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.00000 0.155818 0.0779089 0.996960i \(-0.475176\pi\)
0.0779089 + 0.996960i \(0.475176\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) − 11.3137i − 0.438727i
\(666\) 0 0
\(667\) 33.9411i 1.31421i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.00000 0.0770943 0.0385472 0.999257i \(-0.487727\pi\)
0.0385472 + 0.999257i \(0.487727\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 25.4558i 0.978348i 0.872186 + 0.489174i \(0.162702\pi\)
−0.872186 + 0.489174i \(0.837298\pi\)
\(678\) 0 0
\(679\) 16.9706i 0.651270i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −44.0000 −1.68361 −0.841807 0.539779i \(-0.818508\pi\)
−0.841807 + 0.539779i \(0.818508\pi\)
\(684\) 0 0
\(685\) −4.00000 −0.152832
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 11.3137i 0.431018i
\(690\) 0 0
\(691\) − 19.7990i − 0.753189i −0.926378 0.376595i \(-0.877095\pi\)
0.926378 0.376595i \(-0.122905\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −64.0000 −2.42766
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 11.3137i − 0.427313i −0.976909 0.213656i \(-0.931463\pi\)
0.976909 0.213656i \(-0.0685373\pi\)
\(702\) 0 0
\(703\) − 5.65685i − 0.213352i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −14.0000 −0.525781 −0.262891 0.964826i \(-0.584676\pi\)
−0.262891 + 0.964826i \(0.584676\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 59.3970i 2.22443i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 30.0000 1.11881 0.559406 0.828894i \(-0.311029\pi\)
0.559406 + 0.828894i \(0.311029\pi\)
\(720\) 0 0
\(721\) −10.0000 −0.372419
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 16.9706i − 0.630271i
\(726\) 0 0
\(727\) − 26.8701i − 0.996555i −0.867018 0.498278i \(-0.833966\pi\)
0.867018 0.498278i \(-0.166034\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 0 0
\(733\) −14.0000 −0.517102 −0.258551 0.965998i \(-0.583245\pi\)
−0.258551 + 0.965998i \(0.583245\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 5.65685i 0.208091i 0.994573 + 0.104045i \(0.0331787\pi\)
−0.994573 + 0.104045i \(0.966821\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −40.0000 −1.46746 −0.733729 0.679442i \(-0.762222\pi\)
−0.733729 + 0.679442i \(0.762222\pi\)
\(744\) 0 0
\(745\) 56.0000 2.05168
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 11.3137i − 0.413394i
\(750\) 0 0
\(751\) − 26.8701i − 0.980502i −0.871581 0.490251i \(-0.836905\pi\)
0.871581 0.490251i \(-0.163095\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −36.0000 −1.31017
\(756\) 0 0
\(757\) −38.0000 −1.38113 −0.690567 0.723269i \(-0.742639\pi\)
−0.690567 + 0.723269i \(0.742639\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 49.4975i 1.79428i 0.441744 + 0.897141i \(0.354360\pi\)
−0.441744 + 0.897141i \(0.645640\pi\)
\(762\) 0 0
\(763\) 14.1421i 0.511980i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −24.0000 −0.866590
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 11.3137i 0.406926i 0.979083 + 0.203463i \(0.0652196\pi\)
−0.979083 + 0.203463i \(0.934780\pi\)
\(774\) 0 0
\(775\) − 29.6985i − 1.06680i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −12.0000 −0.429945
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 62.2254i − 2.22092i
\(786\) 0 0
\(787\) 42.4264i 1.51234i 0.654376 + 0.756169i \(0.272931\pi\)
−0.654376 + 0.756169i \(0.727069\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −14.0000 −0.497783
\(792\) 0 0
\(793\) 20.0000 0.710221
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 14.1421i − 0.500940i −0.968124 0.250470i \(-0.919415\pi\)
0.968124 0.250470i \(-0.0805852\pi\)
\(798\) 0 0
\(799\) 14.1421i 0.500313i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 24.0000 0.845889
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 43.8406i − 1.54135i −0.637226 0.770677i \(-0.719918\pi\)
0.637226 0.770677i \(-0.280082\pi\)
\(810\) 0 0
\(811\) − 8.48528i − 0.297959i −0.988840 0.148979i \(-0.952401\pi\)
0.988840 0.148979i \(-0.0475988\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −24.0000 −0.840683
\(816\) 0 0
\(817\) −24.0000 −0.839654
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 56.5685i 1.97426i 0.159933 + 0.987128i \(0.448872\pi\)
−0.159933 + 0.987128i \(0.551128\pi\)
\(822\) 0 0
\(823\) − 26.8701i − 0.936631i −0.883561 0.468316i \(-0.844861\pi\)
0.883561 0.468316i \(-0.155139\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −48.0000 −1.66912 −0.834562 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 7.07107i 0.244998i
\(834\) 0 0
\(835\) 45.2548i 1.56611i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −18.0000 −0.621429 −0.310715 0.950503i \(-0.600568\pi\)
−0.310715 + 0.950503i \(0.600568\pi\)
\(840\) 0 0
\(841\) −3.00000 −0.103448
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 25.4558i 0.875708i
\(846\) 0 0
\(847\) − 15.5563i − 0.534522i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 0 0
\(853\) −42.0000 −1.43805 −0.719026 0.694983i \(-0.755412\pi\)
−0.719026 + 0.694983i \(0.755412\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 18.3848i 0.628012i 0.949421 + 0.314006i \(0.101671\pi\)
−0.949421 + 0.314006i \(0.898329\pi\)
\(858\) 0 0
\(859\) 31.1127i 1.06155i 0.847512 + 0.530776i \(0.178099\pi\)
−0.847512 + 0.530776i \(0.821901\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −48.0000 −1.63394 −0.816970 0.576681i \(-0.804348\pi\)
−0.816970 + 0.576681i \(0.804348\pi\)
\(864\) 0 0
\(865\) −8.00000 −0.272008
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 11.3137i 0.383350i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 8.00000 0.270449
\(876\) 0 0
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 21.2132i 0.714691i 0.933972 + 0.357345i \(0.116318\pi\)
−0.933972 + 0.357345i \(0.883682\pi\)
\(882\) 0 0
\(883\) 31.1127i 1.04703i 0.852018 + 0.523513i \(0.175379\pi\)
−0.852018 + 0.523513i \(0.824621\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) −6.00000 −0.201234
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 28.2843i − 0.946497i
\(894\) 0 0
\(895\) 11.3137i 0.378176i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −56.0000 −1.86770
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 50.9117i 1.69236i
\(906\) 0 0
\(907\) 42.4264i 1.40875i 0.709830 + 0.704373i \(0.248772\pi\)
−0.709830 + 0.704373i \(0.751228\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 22.6274i 0.747223i
\(918\) 0 0
\(919\) − 38.1838i − 1.25957i −0.776771 0.629783i \(-0.783144\pi\)
0.776771 0.629783i \(-0.216856\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −4.00000 −0.131662
\(924\) 0 0
\(925\) −6.00000 −0.197279
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 49.4975i − 1.62396i −0.583686 0.811980i \(-0.698390\pi\)
0.583686 0.811980i \(-0.301610\pi\)
\(930\) 0 0
\(931\) − 14.1421i − 0.463490i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 14.0000 0.457360 0.228680 0.973502i \(-0.426559\pi\)
0.228680 + 0.973502i \(0.426559\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 19.7990i − 0.645429i −0.946496 0.322714i \(-0.895405\pi\)
0.946496 0.322714i \(-0.104595\pi\)
\(942\) 0 0
\(943\) − 25.4558i − 0.828956i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −20.0000 −0.649913 −0.324956 0.945729i \(-0.605350\pi\)
−0.324956 + 0.945729i \(0.605350\pi\)
\(948\) 0 0
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 32.5269i − 1.05365i −0.849974 0.526825i \(-0.823382\pi\)
0.849974 0.526825i \(-0.176618\pi\)
\(954\) 0 0
\(955\) 22.6274i 0.732206i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 2.00000 0.0645834
\(960\) 0 0
\(961\) −67.0000 −2.16129
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 22.6274i − 0.728402i
\(966\) 0 0
\(967\) − 7.07107i − 0.227390i −0.993516 0.113695i \(-0.963731\pi\)
0.993516 0.113695i \(-0.0362687\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −20.0000 −0.641831 −0.320915 0.947108i \(-0.603990\pi\)
−0.320915 + 0.947108i \(0.603990\pi\)
\(972\) 0 0
\(973\) 32.0000 1.02587
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 29.6985i − 0.950139i −0.879948 0.475069i \(-0.842423\pi\)
0.879948 0.475069i \(-0.157577\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0000 0.765481 0.382741 0.923856i \(-0.374980\pi\)
0.382741 + 0.923856i \(0.374980\pi\)
\(984\) 0 0
\(985\) 32.0000 1.01960
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 50.9117i − 1.61890i
\(990\) 0 0
\(991\) 41.0122i 1.30280i 0.758737 + 0.651398i \(0.225817\pi\)
−0.758737 + 0.651398i \(0.774183\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.00000 0.126809
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) 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.c.d.4607.1 yes 2
3.2 odd 2 4608.2.c.c.4607.2 yes 2
4.3 odd 2 4608.2.c.c.4607.1 2
8.3 odd 2 4608.2.c.e.4607.2 yes 2
8.5 even 2 4608.2.c.f.4607.2 yes 2
12.11 even 2 inner 4608.2.c.d.4607.2 yes 2
16.3 odd 4 4608.2.f.l.2303.2 4
16.5 even 4 4608.2.f.j.2303.3 4
16.11 odd 4 4608.2.f.l.2303.4 4
16.13 even 4 4608.2.f.j.2303.1 4
24.5 odd 2 4608.2.c.e.4607.1 yes 2
24.11 even 2 4608.2.c.f.4607.1 yes 2
48.5 odd 4 4608.2.f.l.2303.1 4
48.11 even 4 4608.2.f.j.2303.2 4
48.29 odd 4 4608.2.f.l.2303.3 4
48.35 even 4 4608.2.f.j.2303.4 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.c.c.4607.1 2 4.3 odd 2
4608.2.c.c.4607.2 yes 2 3.2 odd 2
4608.2.c.d.4607.1 yes 2 1.1 even 1 trivial
4608.2.c.d.4607.2 yes 2 12.11 even 2 inner
4608.2.c.e.4607.1 yes 2 24.5 odd 2
4608.2.c.e.4607.2 yes 2 8.3 odd 2
4608.2.c.f.4607.1 yes 2 24.11 even 2
4608.2.c.f.4607.2 yes 2 8.5 even 2
4608.2.f.j.2303.1 4 16.13 even 4
4608.2.f.j.2303.2 4 48.11 even 4
4608.2.f.j.2303.3 4 16.5 even 4
4608.2.f.j.2303.4 4 48.35 even 4
4608.2.f.l.2303.1 4 48.5 odd 4
4608.2.f.l.2303.2 4 16.3 odd 4
4608.2.f.l.2303.3 4 48.29 odd 4
4608.2.f.l.2303.4 4 16.11 odd 4