Properties

Label 6048.2.c.e.3025.8
Level 6048
Weight 2
Character 6048.3025
Analytic conductor 48.294
Analytic rank 0
Dimension 20
CM no
Inner twists 4

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Newspace parameters

Level: \( N \) = \( 6048 = 2^{5} \cdot 3^{3} \cdot 7 \)
Weight: \( k \) = \( 2 \)
Character orbit: \([\chi]\) = 6048.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} + \cdots)\)
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{8} \)
Twist minimal: no (minimal twist has level 1512)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3025.8
Root \(-0.328272 - 1.37559i\)
Character \(\chi\) = 6048.3025
Dual form 6048.2.c.e.3025.13

$q$-expansion

\(f(q)\) \(=\) \(q-0.512447i q^{5} +1.00000 q^{7} +O(q^{10})\) \(q-0.512447i q^{5} +1.00000 q^{7} -1.82890i q^{11} +1.80627i q^{13} +8.11456 q^{17} -3.43946i q^{19} +3.65626 q^{23} +4.73740 q^{25} +7.98990i q^{29} +1.56895 q^{31} -0.512447i q^{35} -8.44370i q^{37} -2.30828 q^{41} +10.7778i q^{43} -11.3833 q^{47} +1.00000 q^{49} +8.87795i q^{53} -0.937212 q^{55} -2.50234i q^{59} -5.31310i q^{61} +0.925616 q^{65} -6.44648i q^{67} -9.10975 q^{71} +9.24419 q^{73} -1.82890i q^{77} -3.64300 q^{79} -4.53105i q^{83} -4.15828i q^{85} -11.7359 q^{89} +1.80627i q^{91} -1.76254 q^{95} +15.2323 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 20q^{7} + O(q^{10}) \) \( 20q + 20q^{7} - 28q^{25} - 36q^{31} + 20q^{49} - 48q^{55} - 64q^{79} + 56q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/6048\mathbb{Z}\right)^\times\).

\(n\) \(2593\) \(3781\) \(3809\) \(4159\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) − 0.512447i − 0.229173i −0.993413 0.114587i \(-0.963446\pi\)
0.993413 0.114587i \(-0.0365543\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 1.82890i − 0.551433i −0.961239 0.275716i \(-0.911085\pi\)
0.961239 0.275716i \(-0.0889151\pi\)
\(12\) 0 0
\(13\) 1.80627i 0.500968i 0.968121 + 0.250484i \(0.0805898\pi\)
−0.968121 + 0.250484i \(0.919410\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 8.11456 1.96807 0.984035 0.177977i \(-0.0569552\pi\)
0.984035 + 0.177977i \(0.0569552\pi\)
\(18\) 0 0
\(19\) − 3.43946i − 0.789066i −0.918882 0.394533i \(-0.870906\pi\)
0.918882 0.394533i \(-0.129094\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 3.65626 0.762384 0.381192 0.924496i \(-0.375514\pi\)
0.381192 + 0.924496i \(0.375514\pi\)
\(24\) 0 0
\(25\) 4.73740 0.947480
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 7.98990i 1.48369i 0.670573 + 0.741843i \(0.266048\pi\)
−0.670573 + 0.741843i \(0.733952\pi\)
\(30\) 0 0
\(31\) 1.56895 0.281792 0.140896 0.990024i \(-0.455002\pi\)
0.140896 + 0.990024i \(0.455002\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 0.512447i − 0.0866193i
\(36\) 0 0
\(37\) − 8.44370i − 1.38814i −0.719909 0.694068i \(-0.755817\pi\)
0.719909 0.694068i \(-0.244183\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.30828 −0.360493 −0.180247 0.983621i \(-0.557690\pi\)
−0.180247 + 0.983621i \(0.557690\pi\)
\(42\) 0 0
\(43\) 10.7778i 1.64360i 0.569773 + 0.821802i \(0.307031\pi\)
−0.569773 + 0.821802i \(0.692969\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −11.3833 −1.66043 −0.830216 0.557442i \(-0.811783\pi\)
−0.830216 + 0.557442i \(0.811783\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.87795i 1.21948i 0.792601 + 0.609740i \(0.208726\pi\)
−0.792601 + 0.609740i \(0.791274\pi\)
\(54\) 0 0
\(55\) −0.937212 −0.126374
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 2.50234i − 0.325778i −0.986644 0.162889i \(-0.947919\pi\)
0.986644 0.162889i \(-0.0520812\pi\)
\(60\) 0 0
\(61\) − 5.31310i − 0.680272i −0.940376 0.340136i \(-0.889527\pi\)
0.940376 0.340136i \(-0.110473\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.925616 0.114809
\(66\) 0 0
\(67\) − 6.44648i − 0.787562i −0.919204 0.393781i \(-0.871167\pi\)
0.919204 0.393781i \(-0.128833\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.10975 −1.08113 −0.540564 0.841303i \(-0.681789\pi\)
−0.540564 + 0.841303i \(0.681789\pi\)
\(72\) 0 0
\(73\) 9.24419 1.08195 0.540975 0.841039i \(-0.318055\pi\)
0.540975 + 0.841039i \(0.318055\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 1.82890i − 0.208422i
\(78\) 0 0
\(79\) −3.64300 −0.409870 −0.204935 0.978776i \(-0.565698\pi\)
−0.204935 + 0.978776i \(0.565698\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 4.53105i − 0.497348i −0.968587 0.248674i \(-0.920005\pi\)
0.968587 0.248674i \(-0.0799948\pi\)
\(84\) 0 0
\(85\) − 4.15828i − 0.451029i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −11.7359 −1.24401 −0.622003 0.783015i \(-0.713681\pi\)
−0.622003 + 0.783015i \(0.713681\pi\)
\(90\) 0 0
\(91\) 1.80627i 0.189348i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −1.76254 −0.180833
\(96\) 0 0
\(97\) 15.2323 1.54661 0.773303 0.634037i \(-0.218603\pi\)
0.773303 + 0.634037i \(0.218603\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 8.13524i 0.809487i 0.914430 + 0.404744i \(0.132639\pi\)
−0.914430 + 0.404744i \(0.867361\pi\)
\(102\) 0 0
\(103\) 15.9816 1.57471 0.787356 0.616498i \(-0.211449\pi\)
0.787356 + 0.616498i \(0.211449\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 12.1695i − 1.17647i −0.808690 0.588235i \(-0.799823\pi\)
0.808690 0.588235i \(-0.200177\pi\)
\(108\) 0 0
\(109\) 10.8452i 1.03878i 0.854537 + 0.519391i \(0.173841\pi\)
−0.854537 + 0.519391i \(0.826159\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 3.07563 0.289331 0.144665 0.989481i \(-0.453789\pi\)
0.144665 + 0.989481i \(0.453789\pi\)
\(114\) 0 0
\(115\) − 1.87364i − 0.174718i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 8.11456 0.743860
\(120\) 0 0
\(121\) 7.65514 0.695922
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 4.98990i − 0.446310i
\(126\) 0 0
\(127\) −6.67524 −0.592332 −0.296166 0.955137i \(-0.595708\pi\)
−0.296166 + 0.955137i \(0.595708\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 1.26995i − 0.110956i −0.998460 0.0554778i \(-0.982332\pi\)
0.998460 0.0554778i \(-0.0176682\pi\)
\(132\) 0 0
\(133\) − 3.43946i − 0.298239i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.8507 1.69596 0.847980 0.530029i \(-0.177819\pi\)
0.847980 + 0.530029i \(0.177819\pi\)
\(138\) 0 0
\(139\) 13.9758i 1.18541i 0.805419 + 0.592706i \(0.201941\pi\)
−0.805419 + 0.592706i \(0.798059\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 3.30348 0.276251
\(144\) 0 0
\(145\) 4.09440 0.340021
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 11.8561i − 0.971286i −0.874157 0.485643i \(-0.838585\pi\)
0.874157 0.485643i \(-0.161415\pi\)
\(150\) 0 0
\(151\) 13.4949 1.09820 0.549100 0.835757i \(-0.314971\pi\)
0.549100 + 0.835757i \(0.314971\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 0.804003i − 0.0645791i
\(156\) 0 0
\(157\) − 9.88593i − 0.788983i −0.918900 0.394492i \(-0.870921\pi\)
0.918900 0.394492i \(-0.129079\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.65626 0.288154
\(162\) 0 0
\(163\) 4.43706i 0.347537i 0.984786 + 0.173769i \(0.0555945\pi\)
−0.984786 + 0.173769i \(0.944405\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.38391 0.416619 0.208310 0.978063i \(-0.433204\pi\)
0.208310 + 0.978063i \(0.433204\pi\)
\(168\) 0 0
\(169\) 9.73740 0.749031
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 17.9611i 1.36556i 0.730624 + 0.682780i \(0.239229\pi\)
−0.730624 + 0.682780i \(0.760771\pi\)
\(174\) 0 0
\(175\) 4.73740 0.358114
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 16.0482i − 1.19950i −0.800188 0.599749i \(-0.795267\pi\)
0.800188 0.599749i \(-0.204733\pi\)
\(180\) 0 0
\(181\) − 5.91861i − 0.439927i −0.975508 0.219964i \(-0.929406\pi\)
0.975508 0.219964i \(-0.0705938\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.32695 −0.318123
\(186\) 0 0
\(187\) − 14.8407i − 1.08526i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.0697837 −0.00504937 −0.00252469 0.999997i \(-0.500804\pi\)
−0.00252469 + 0.999997i \(0.500804\pi\)
\(192\) 0 0
\(193\) −8.24356 −0.593384 −0.296692 0.954973i \(-0.595884\pi\)
−0.296692 + 0.954973i \(0.595884\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 1.91295i − 0.136292i −0.997675 0.0681459i \(-0.978292\pi\)
0.997675 0.0681459i \(-0.0217083\pi\)
\(198\) 0 0
\(199\) 3.70579 0.262696 0.131348 0.991336i \(-0.458069\pi\)
0.131348 + 0.991336i \(0.458069\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 7.98990i 0.560781i
\(204\) 0 0
\(205\) 1.18287i 0.0826153i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −6.29041 −0.435117
\(210\) 0 0
\(211\) − 25.6006i − 1.76242i −0.472724 0.881210i \(-0.656729\pi\)
0.472724 0.881210i \(-0.343271\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 5.52307 0.376670
\(216\) 0 0
\(217\) 1.56895 0.106507
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 14.6571i 0.985941i
\(222\) 0 0
\(223\) 14.2639 0.955182 0.477591 0.878582i \(-0.341510\pi\)
0.477591 + 0.878582i \(0.341510\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 2.58553i 0.171608i 0.996312 + 0.0858039i \(0.0273458\pi\)
−0.996312 + 0.0858039i \(0.972654\pi\)
\(228\) 0 0
\(229\) 27.5800i 1.82254i 0.411813 + 0.911268i \(0.364896\pi\)
−0.411813 + 0.911268i \(0.635104\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5.61196 0.367652 0.183826 0.982959i \(-0.441152\pi\)
0.183826 + 0.982959i \(0.441152\pi\)
\(234\) 0 0
\(235\) 5.83336i 0.380526i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.69275 −0.109495 −0.0547476 0.998500i \(-0.517435\pi\)
−0.0547476 + 0.998500i \(0.517435\pi\)
\(240\) 0 0
\(241\) −22.6246 −1.45738 −0.728689 0.684845i \(-0.759870\pi\)
−0.728689 + 0.684845i \(0.759870\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 0.512447i − 0.0327390i
\(246\) 0 0
\(247\) 6.21258 0.395297
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 18.1391i 1.14493i 0.819930 + 0.572464i \(0.194012\pi\)
−0.819930 + 0.572464i \(0.805988\pi\)
\(252\) 0 0
\(253\) − 6.68693i − 0.420403i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8.62562 −0.538051 −0.269026 0.963133i \(-0.586702\pi\)
−0.269026 + 0.963133i \(0.586702\pi\)
\(258\) 0 0
\(259\) − 8.44370i − 0.524666i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2.79262 −0.172201 −0.0861003 0.996286i \(-0.527441\pi\)
−0.0861003 + 0.996286i \(0.527441\pi\)
\(264\) 0 0
\(265\) 4.54948 0.279472
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 17.8391i − 1.08767i −0.839192 0.543836i \(-0.816971\pi\)
0.839192 0.543836i \(-0.183029\pi\)
\(270\) 0 0
\(271\) 22.5304 1.36863 0.684313 0.729188i \(-0.260102\pi\)
0.684313 + 0.729188i \(0.260102\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 8.66421i − 0.522472i
\(276\) 0 0
\(277\) − 2.18249i − 0.131133i −0.997848 0.0655666i \(-0.979115\pi\)
0.997848 0.0655666i \(-0.0208855\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.54152 0.449890 0.224945 0.974372i \(-0.427780\pi\)
0.224945 + 0.974372i \(0.427780\pi\)
\(282\) 0 0
\(283\) − 11.5743i − 0.688021i −0.938966 0.344011i \(-0.888214\pi\)
0.938966 0.344011i \(-0.111786\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.30828 −0.136254
\(288\) 0 0
\(289\) 48.8460 2.87330
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 8.12045i − 0.474402i −0.971461 0.237201i \(-0.923770\pi\)
0.971461 0.237201i \(-0.0762300\pi\)
\(294\) 0 0
\(295\) −1.28232 −0.0746595
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.60419i 0.381930i
\(300\) 0 0
\(301\) 10.7778i 0.621224i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.72268 −0.155900
\(306\) 0 0
\(307\) − 2.73852i − 0.156295i −0.996942 0.0781477i \(-0.975099\pi\)
0.996942 0.0781477i \(-0.0249006\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −17.9488 −1.01778 −0.508891 0.860831i \(-0.669945\pi\)
−0.508891 + 0.860831i \(0.669945\pi\)
\(312\) 0 0
\(313\) 5.95819 0.336777 0.168388 0.985721i \(-0.446144\pi\)
0.168388 + 0.985721i \(0.446144\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 21.6033i 1.21336i 0.794945 + 0.606682i \(0.207500\pi\)
−0.794945 + 0.606682i \(0.792500\pi\)
\(318\) 0 0
\(319\) 14.6127 0.818154
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) − 27.9097i − 1.55294i
\(324\) 0 0
\(325\) 8.55701i 0.474657i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −11.3833 −0.627584
\(330\) 0 0
\(331\) − 12.6066i − 0.692920i −0.938065 0.346460i \(-0.887384\pi\)
0.938065 0.346460i \(-0.112616\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −3.30348 −0.180488
\(336\) 0 0
\(337\) −28.9818 −1.57874 −0.789370 0.613917i \(-0.789593\pi\)
−0.789370 + 0.613917i \(0.789593\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 2.86945i − 0.155389i
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 12.2675i − 0.658553i −0.944234 0.329277i \(-0.893195\pi\)
0.944234 0.329277i \(-0.106805\pi\)
\(348\) 0 0
\(349\) − 28.3289i − 1.51641i −0.652016 0.758205i \(-0.726077\pi\)
0.652016 0.758205i \(-0.273923\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.40996 −0.0750444 −0.0375222 0.999296i \(-0.511946\pi\)
−0.0375222 + 0.999296i \(0.511946\pi\)
\(354\) 0 0
\(355\) 4.66826i 0.247766i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −17.0581 −0.900290 −0.450145 0.892955i \(-0.648628\pi\)
−0.450145 + 0.892955i \(0.648628\pi\)
\(360\) 0 0
\(361\) 7.17014 0.377376
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 4.73715i − 0.247954i
\(366\) 0 0
\(367\) 25.4558 1.32878 0.664390 0.747386i \(-0.268692\pi\)
0.664390 + 0.747386i \(0.268692\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 8.87795i 0.460920i
\(372\) 0 0
\(373\) − 20.3176i − 1.05201i −0.850483 0.526003i \(-0.823690\pi\)
0.850483 0.526003i \(-0.176310\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −14.4319 −0.743280
\(378\) 0 0
\(379\) 11.8832i 0.610397i 0.952289 + 0.305198i \(0.0987228\pi\)
−0.952289 + 0.305198i \(0.901277\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 18.6261 0.951749 0.475874 0.879513i \(-0.342132\pi\)
0.475874 + 0.879513i \(0.342132\pi\)
\(384\) 0 0
\(385\) −0.937212 −0.0477647
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 20.7636i 1.05276i 0.850251 + 0.526378i \(0.176450\pi\)
−0.850251 + 0.526378i \(0.823550\pi\)
\(390\) 0 0
\(391\) 29.6690 1.50042
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 1.86684i 0.0939311i
\(396\) 0 0
\(397\) − 20.9726i − 1.05259i −0.850303 0.526293i \(-0.823582\pi\)
0.850303 0.526293i \(-0.176418\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.476166 −0.0237786 −0.0118893 0.999929i \(-0.503785\pi\)
−0.0118893 + 0.999929i \(0.503785\pi\)
\(402\) 0 0
\(403\) 2.83394i 0.141169i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −15.4427 −0.765464
\(408\) 0 0
\(409\) −2.08700 −0.103196 −0.0515978 0.998668i \(-0.516431\pi\)
−0.0515978 + 0.998668i \(0.516431\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 2.50234i − 0.123132i
\(414\) 0 0
\(415\) −2.32192 −0.113979
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 15.8104i 0.772388i 0.922418 + 0.386194i \(0.126210\pi\)
−0.922418 + 0.386194i \(0.873790\pi\)
\(420\) 0 0
\(421\) 21.8824i 1.06648i 0.845963 + 0.533241i \(0.179026\pi\)
−0.845963 + 0.533241i \(0.820974\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 38.4419 1.86471
\(426\) 0 0
\(427\) − 5.31310i − 0.257119i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −25.3231 −1.21977 −0.609885 0.792490i \(-0.708784\pi\)
−0.609885 + 0.792490i \(0.708784\pi\)
\(432\) 0 0
\(433\) 30.9197 1.48590 0.742952 0.669344i \(-0.233425\pi\)
0.742952 + 0.669344i \(0.233425\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 12.5756i − 0.601571i
\(438\) 0 0
\(439\) 8.75669 0.417934 0.208967 0.977923i \(-0.432990\pi\)
0.208967 + 0.977923i \(0.432990\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 27.6190i 1.31222i 0.754666 + 0.656109i \(0.227799\pi\)
−0.754666 + 0.656109i \(0.772201\pi\)
\(444\) 0 0
\(445\) 6.01404i 0.285093i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 13.6994 0.646516 0.323258 0.946311i \(-0.395222\pi\)
0.323258 + 0.946311i \(0.395222\pi\)
\(450\) 0 0
\(451\) 4.22161i 0.198788i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0.925616 0.0433935
\(456\) 0 0
\(457\) 5.85452 0.273863 0.136931 0.990581i \(-0.456276\pi\)
0.136931 + 0.990581i \(0.456276\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 2.42544i 0.112964i 0.998404 + 0.0564821i \(0.0179884\pi\)
−0.998404 + 0.0564821i \(0.982012\pi\)
\(462\) 0 0
\(463\) 1.26366 0.0587273 0.0293636 0.999569i \(-0.490652\pi\)
0.0293636 + 0.999569i \(0.490652\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 27.2456i − 1.26078i −0.776280 0.630388i \(-0.782896\pi\)
0.776280 0.630388i \(-0.217104\pi\)
\(468\) 0 0
\(469\) − 6.44648i − 0.297671i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 19.7115 0.906338
\(474\) 0 0
\(475\) − 16.2941i − 0.747624i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 27.4035 1.25210 0.626050 0.779783i \(-0.284671\pi\)
0.626050 + 0.779783i \(0.284671\pi\)
\(480\) 0 0
\(481\) 15.2516 0.695412
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) − 7.80574i − 0.354440i
\(486\) 0 0
\(487\) −27.0530 −1.22589 −0.612944 0.790126i \(-0.710015\pi\)
−0.612944 + 0.790126i \(0.710015\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 29.2739i 1.32111i 0.750776 + 0.660556i \(0.229680\pi\)
−0.750776 + 0.660556i \(0.770320\pi\)
\(492\) 0 0
\(493\) 64.8345i 2.92000i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.10975 −0.408628
\(498\) 0 0
\(499\) 39.8348i 1.78325i 0.452775 + 0.891625i \(0.350434\pi\)
−0.452775 + 0.891625i \(0.649566\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 36.3079 1.61889 0.809444 0.587197i \(-0.199769\pi\)
0.809444 + 0.587197i \(0.199769\pi\)
\(504\) 0 0
\(505\) 4.16888 0.185513
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 22.2853i − 0.987778i −0.869525 0.493889i \(-0.835575\pi\)
0.869525 0.493889i \(-0.164425\pi\)
\(510\) 0 0
\(511\) 9.24419 0.408939
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 8.18971i − 0.360882i
\(516\) 0 0
\(517\) 20.8190i 0.915617i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −28.8909 −1.26573 −0.632866 0.774261i \(-0.718122\pi\)
−0.632866 + 0.774261i \(0.718122\pi\)
\(522\) 0 0
\(523\) − 22.7284i − 0.993842i −0.867796 0.496921i \(-0.834464\pi\)
0.867796 0.496921i \(-0.165536\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 12.7313 0.554585
\(528\) 0 0
\(529\) −9.63174 −0.418771
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 4.16938i − 0.180596i
\(534\) 0 0
\(535\) −6.23622 −0.269615
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 1.82890i − 0.0787761i
\(540\) 0 0
\(541\) − 12.7291i − 0.547268i −0.961834 0.273634i \(-0.911774\pi\)
0.961834 0.273634i \(-0.0882257\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 5.55759 0.238061
\(546\) 0 0
\(547\) 28.7154i 1.22778i 0.789391 + 0.613890i \(0.210396\pi\)
−0.789391 + 0.613890i \(0.789604\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 27.4809 1.17073
\(552\) 0 0
\(553\) −3.64300 −0.154916
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 7.36295i − 0.311978i −0.987759 0.155989i \(-0.950143\pi\)
0.987759 0.155989i \(-0.0498565\pi\)
\(558\) 0 0
\(559\) −19.4676 −0.823394
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 15.9713i − 0.673110i −0.941664 0.336555i \(-0.890738\pi\)
0.941664 0.336555i \(-0.109262\pi\)
\(564\) 0 0
\(565\) − 1.57610i − 0.0663068i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.9769 −1.13093 −0.565465 0.824772i \(-0.691303\pi\)
−0.565465 + 0.824772i \(0.691303\pi\)
\(570\) 0 0
\(571\) − 35.5632i − 1.48827i −0.668027 0.744137i \(-0.732861\pi\)
0.668027 0.744137i \(-0.267139\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 17.3212 0.722343
\(576\) 0 0
\(577\) −23.4330 −0.975528 −0.487764 0.872976i \(-0.662187\pi\)
−0.487764 + 0.872976i \(0.662187\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 4.53105i − 0.187980i
\(582\) 0 0
\(583\) 16.2369 0.672462
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 37.1113i 1.53175i 0.642991 + 0.765874i \(0.277693\pi\)
−0.642991 + 0.765874i \(0.722307\pi\)
\(588\) 0 0
\(589\) − 5.39633i − 0.222352i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.8317 0.978650 0.489325 0.872102i \(-0.337243\pi\)
0.489325 + 0.872102i \(0.337243\pi\)
\(594\) 0 0
\(595\) − 4.15828i − 0.170473i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −12.0955 −0.494210 −0.247105 0.968989i \(-0.579479\pi\)
−0.247105 + 0.968989i \(0.579479\pi\)
\(600\) 0 0
\(601\) 20.4752 0.835202 0.417601 0.908631i \(-0.362871\pi\)
0.417601 + 0.908631i \(0.362871\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 3.92285i − 0.159487i
\(606\) 0 0
\(607\) 24.0329 0.975466 0.487733 0.872993i \(-0.337824\pi\)
0.487733 + 0.872993i \(0.337824\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) − 20.5614i − 0.831824i
\(612\) 0 0
\(613\) − 45.2862i − 1.82909i −0.404480 0.914547i \(-0.632548\pi\)
0.404480 0.914547i \(-0.367452\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −20.4393 −0.822855 −0.411428 0.911442i \(-0.634970\pi\)
−0.411428 + 0.911442i \(0.634970\pi\)
\(618\) 0 0
\(619\) − 15.3328i − 0.616277i −0.951342 0.308138i \(-0.900294\pi\)
0.951342 0.308138i \(-0.0997060\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −11.7359 −0.470190
\(624\) 0 0
\(625\) 21.1299 0.845197
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 68.5169i − 2.73195i
\(630\) 0 0
\(631\) −20.4869 −0.815572 −0.407786 0.913078i \(-0.633699\pi\)
−0.407786 + 0.913078i \(0.633699\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.42070i 0.135747i
\(636\) 0 0
\(637\) 1.80627i 0.0715669i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −42.6868 −1.68603 −0.843013 0.537893i \(-0.819220\pi\)
−0.843013 + 0.537893i \(0.819220\pi\)
\(642\) 0 0
\(643\) − 29.7804i − 1.17442i −0.809434 0.587211i \(-0.800226\pi\)
0.809434 0.587211i \(-0.199774\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −6.63482 −0.260842 −0.130421 0.991459i \(-0.541633\pi\)
−0.130421 + 0.991459i \(0.541633\pi\)
\(648\) 0 0
\(649\) −4.57653 −0.179644
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 4.48147i − 0.175373i −0.996148 0.0876867i \(-0.972053\pi\)
0.996148 0.0876867i \(-0.0279474\pi\)
\(654\) 0 0
\(655\) −0.650779 −0.0254281
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 49.4245i 1.92530i 0.270741 + 0.962652i \(0.412731\pi\)
−0.270741 + 0.962652i \(0.587269\pi\)
\(660\) 0 0
\(661\) − 12.5279i − 0.487278i −0.969866 0.243639i \(-0.921659\pi\)
0.969866 0.243639i \(-0.0783413\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −1.76254 −0.0683483
\(666\) 0 0
\(667\) 29.2132i 1.13114i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9.71710 −0.375125
\(672\) 0 0
\(673\) −26.3343 −1.01511 −0.507556 0.861619i \(-0.669451\pi\)
−0.507556 + 0.861619i \(0.669451\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 14.4868i 0.556774i 0.960469 + 0.278387i \(0.0897998\pi\)
−0.960469 + 0.278387i \(0.910200\pi\)
\(678\) 0 0
\(679\) 15.2323 0.584562
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.0232069i 0 0.000887989i 1.00000 0.000443994i \(0.000141328\pi\)
−1.00000 0.000443994i \(0.999859\pi\)
\(684\) 0 0
\(685\) − 10.1724i − 0.388668i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −16.0360 −0.610921
\(690\) 0 0
\(691\) 17.2719i 0.657054i 0.944495 + 0.328527i \(0.106552\pi\)
−0.944495 + 0.328527i \(0.893448\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 7.16186 0.271665
\(696\) 0 0
\(697\) −18.7307 −0.709476
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 23.9519i − 0.904651i −0.891853 0.452326i \(-0.850595\pi\)
0.891853 0.452326i \(-0.149405\pi\)
\(702\) 0 0
\(703\) −29.0417 −1.09533
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 8.13524i 0.305957i
\(708\) 0 0
\(709\) − 32.1099i − 1.20591i −0.797774 0.602956i \(-0.793989\pi\)
0.797774 0.602956i \(-0.206011\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 5.73649 0.214833
\(714\) 0 0
\(715\) − 1.69286i − 0.0633092i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.4708 −0.726139 −0.363070 0.931762i \(-0.618271\pi\)
−0.363070 + 0.931762i \(0.618271\pi\)
\(720\) 0 0
\(721\) 15.9816 0.595185
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 37.8513i 1.40576i
\(726\) 0 0
\(727\) −18.3066 −0.678954 −0.339477 0.940614i \(-0.610250\pi\)
−0.339477 + 0.940614i \(0.610250\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 87.4573i 3.23473i
\(732\) 0 0
\(733\) − 36.3167i − 1.34139i −0.741735 0.670693i \(-0.765997\pi\)
0.741735 0.670693i \(-0.234003\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −11.7899 −0.434288
\(738\) 0 0
\(739\) 15.6764i 0.576665i 0.957530 + 0.288332i \(0.0931008\pi\)
−0.957530 + 0.288332i \(0.906899\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 49.3578 1.81076 0.905381 0.424599i \(-0.139585\pi\)
0.905381 + 0.424599i \(0.139585\pi\)
\(744\) 0 0
\(745\) −6.07560 −0.222593
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 12.1695i − 0.444664i
\(750\) 0 0
\(751\) −29.7930 −1.08716 −0.543582 0.839356i \(-0.682932\pi\)
−0.543582 + 0.839356i \(0.682932\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 6.91542i − 0.251678i
\(756\) 0 0
\(757\) 13.6736i 0.496974i 0.968635 + 0.248487i \(0.0799333\pi\)
−0.968635 + 0.248487i \(0.920067\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 3.34618 0.121299 0.0606494 0.998159i \(-0.480683\pi\)
0.0606494 + 0.998159i \(0.480683\pi\)
\(762\) 0 0
\(763\) 10.8452i 0.392623i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.51990 0.163204
\(768\) 0 0
\(769\) 10.2544 0.369783 0.184891 0.982759i \(-0.440807\pi\)
0.184891 + 0.982759i \(0.440807\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 27.9261i − 1.00443i −0.864742 0.502217i \(-0.832518\pi\)
0.864742 0.502217i \(-0.167482\pi\)
\(774\) 0 0
\(775\) 7.43274 0.266992
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 7.93924i 0.284453i
\(780\) 0 0
\(781\) 16.6608i 0.596170i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −5.06601 −0.180814
\(786\) 0 0
\(787\) 19.9536i 0.711268i 0.934625 + 0.355634i \(0.115735\pi\)
−0.934625 + 0.355634i \(0.884265\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 3.07563 0.109357
\(792\) 0 0
\(793\) 9.59687 0.340795
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 22.3652i 0.792217i 0.918204 + 0.396108i \(0.129640\pi\)
−0.918204 + 0.396108i \(0.870360\pi\)
\(798\) 0 0
\(799\) −92.3708 −3.26785
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 16.9067i − 0.596623i
\(804\) 0 0
\(805\) − 1.87364i − 0.0660371i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 8.28121 0.291152 0.145576 0.989347i \(-0.453496\pi\)
0.145576 + 0.989347i \(0.453496\pi\)
\(810\) 0 0
\(811\) 20.5177i 0.720474i 0.932861 + 0.360237i \(0.117304\pi\)
−0.932861 + 0.360237i \(0.882696\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 2.27376 0.0796462
\(816\) 0 0
\(817\) 37.0699 1.29691
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 6.88906i 0.240430i 0.992748 + 0.120215i \(0.0383584\pi\)
−0.992748 + 0.120215i \(0.961642\pi\)
\(822\) 0 0
\(823\) 6.14442 0.214181 0.107091 0.994249i \(-0.465847\pi\)
0.107091 + 0.994249i \(0.465847\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 29.6562i 1.03125i 0.856815 + 0.515623i \(0.172440\pi\)
−0.856815 + 0.515623i \(0.827560\pi\)
\(828\) 0 0
\(829\) − 7.16328i − 0.248791i −0.992233 0.124396i \(-0.960301\pi\)
0.992233 0.124396i \(-0.0396992\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 8.11456 0.281153
\(834\) 0 0
\(835\) − 2.75897i − 0.0954780i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 15.6136 0.539041 0.269520 0.962995i \(-0.413135\pi\)
0.269520 + 0.962995i \(0.413135\pi\)
\(840\) 0 0
\(841\) −34.8385 −1.20133
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 4.98990i − 0.171658i
\(846\) 0 0
\(847\) 7.65514 0.263034
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 30.8724i − 1.05829i
\(852\) 0 0
\(853\) 0.960301i 0.0328801i 0.999865 + 0.0164400i \(0.00523327\pi\)
−0.999865 + 0.0164400i \(0.994767\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 17.3062 0.591168 0.295584 0.955317i \(-0.404486\pi\)
0.295584 + 0.955317i \(0.404486\pi\)
\(858\) 0 0
\(859\) 15.5416i 0.530273i 0.964211 + 0.265137i \(0.0854171\pi\)
−0.964211 + 0.265137i \(0.914583\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −30.9082 −1.05213 −0.526063 0.850445i \(-0.676333\pi\)
−0.526063 + 0.850445i \(0.676333\pi\)
\(864\) 0 0
\(865\) 9.20413 0.312950
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 6.66267i 0.226016i
\(870\) 0 0
\(871\) 11.6441 0.394544
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) − 4.98990i − 0.168689i
\(876\) 0 0
\(877\) 30.0443i 1.01452i 0.861792 + 0.507262i \(0.169342\pi\)
−0.861792 + 0.507262i \(0.830658\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −43.7362 −1.47351 −0.736754 0.676160i \(-0.763643\pi\)
−0.736754 + 0.676160i \(0.763643\pi\)
\(882\) 0 0
\(883\) 11.9694i 0.402804i 0.979509 + 0.201402i \(0.0645497\pi\)
−0.979509 + 0.201402i \(0.935450\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −50.6980 −1.70227 −0.851136 0.524945i \(-0.824086\pi\)
−0.851136 + 0.524945i \(0.824086\pi\)
\(888\) 0 0
\(889\) −6.67524 −0.223880
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 39.1525i 1.31019i
\(894\) 0 0
\(895\) −8.22384 −0.274893
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.5357i 0.418090i
\(900\) 0 0
\(901\) 72.0406i 2.40002i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −3.03297 −0.100819
\(906\) 0 0
\(907\) 5.94868i 0.197523i 0.995111 + 0.0987614i \(0.0314881\pi\)
−0.995111 + 0.0987614i \(0.968512\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 49.6827 1.64606 0.823031 0.567996i \(-0.192281\pi\)
0.823031 + 0.567996i \(0.192281\pi\)
\(912\) 0 0
\(913\) −8.28683 −0.274254
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 1.26995i − 0.0419373i
\(918\) 0 0
\(919\) −16.5497 −0.545923 −0.272961 0.962025i \(-0.588003\pi\)
−0.272961 + 0.962025i \(0.588003\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) − 16.4546i − 0.541611i
\(924\) 0 0
\(925\) − 40.0012i − 1.31523i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −15.8292 −0.519339 −0.259669 0.965698i \(-0.583614\pi\)
−0.259669 + 0.965698i \(0.583614\pi\)
\(930\) 0 0
\(931\) − 3.43946i − 0.112724i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −7.60506 −0.248712
\(936\) 0 0
\(937\) −14.1814 −0.463286 −0.231643 0.972801i \(-0.574410\pi\)
−0.231643 + 0.972801i \(0.574410\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 59.4006i − 1.93641i −0.250166 0.968203i \(-0.580485\pi\)
0.250166 0.968203i \(-0.419515\pi\)
\(942\) 0 0
\(943\) −8.43969 −0.274834
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 6.81038i 0.221308i 0.993859 + 0.110654i \(0.0352945\pi\)
−0.993859 + 0.110654i \(0.964706\pi\)
\(948\) 0 0
\(949\) 16.6975i 0.542023i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −25.4827 −0.825466 −0.412733 0.910852i \(-0.635426\pi\)
−0.412733 + 0.910852i \(0.635426\pi\)
\(954\) 0 0
\(955\) 0.0357604i 0.00115718i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 19.8507 0.641012
\(960\) 0 0
\(961\) −28.5384 −0.920593
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 4.22439i 0.135988i
\(966\) 0 0
\(967\) −44.8473 −1.44219 −0.721096 0.692835i \(-0.756361\pi\)
−0.721096 + 0.692835i \(0.756361\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) − 23.4868i − 0.753728i −0.926269 0.376864i \(-0.877002\pi\)
0.926269 0.376864i \(-0.122998\pi\)
\(972\) 0 0
\(973\) 13.9758i 0.448044i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 54.0970 1.73072 0.865359 0.501153i \(-0.167091\pi\)
0.865359 + 0.501153i \(0.167091\pi\)
\(978\) 0 0
\(979\) 21.4638i 0.685986i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −15.0476 −0.479943 −0.239972 0.970780i \(-0.577138\pi\)
−0.239972 + 0.970780i \(0.577138\pi\)
\(984\) 0 0
\(985\) −0.980283 −0.0312344
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 39.4066i 1.25306i
\(990\) 0 0
\(991\) −24.0651 −0.764455 −0.382227 0.924068i \(-0.624843\pi\)
−0.382227 + 0.924068i \(0.624843\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 1.89902i − 0.0602030i
\(996\) 0 0
\(997\) − 46.1183i − 1.46058i −0.683136 0.730291i \(-0.739384\pi\)
0.683136 0.730291i \(-0.260616\pi\)
\(998\) 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 6048.2.c.e.3025.8 20
3.2 odd 2 inner 6048.2.c.e.3025.14 20
4.3 odd 2 1512.2.c.e.757.11 yes 20
8.3 odd 2 1512.2.c.e.757.12 yes 20
8.5 even 2 inner 6048.2.c.e.3025.13 20
12.11 even 2 1512.2.c.e.757.10 yes 20
24.5 odd 2 inner 6048.2.c.e.3025.7 20
24.11 even 2 1512.2.c.e.757.9 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1512.2.c.e.757.9 20 24.11 even 2
1512.2.c.e.757.10 yes 20 12.11 even 2
1512.2.c.e.757.11 yes 20 4.3 odd 2
1512.2.c.e.757.12 yes 20 8.3 odd 2
6048.2.c.e.3025.7 20 24.5 odd 2 inner
6048.2.c.e.3025.8 20 1.1 even 1 trivial
6048.2.c.e.3025.13 20 8.5 even 2 inner
6048.2.c.e.3025.14 20 3.2 odd 2 inner