Properties

Label 6048.2.h.d.2591.1
Level 6048
Weight 2
Character 6048.2591
Analytic conductor 48.294
Analytic rank 0
Dimension 8
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.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.2935231425\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2591.1
Root \(-0.965926 - 0.258819i\)
Character \(\chi\) = 6048.2591
Dual form 6048.2.h.d.2591.8

$q$-expansion

\(f(q)\) \(=\) \(q-3.14626i q^{5} -1.00000i q^{7} +O(q^{10})\) \(q-3.14626i q^{5} -1.00000i q^{7} +2.04989 q^{11} +2.44949 q^{13} -3.14626i q^{17} -4.44949i q^{19} +2.82843 q^{23} -4.89898 q^{25} -0.778539i q^{29} -0.449490i q^{31} -3.14626 q^{35} -7.00000 q^{37} -2.51059i q^{41} -4.34847i q^{43} +7.38891 q^{47} -1.00000 q^{49} -6.29253i q^{53} -6.44949i q^{55} +5.83183 q^{59} -5.34847 q^{61} -7.70674i q^{65} -8.00000i q^{67} +12.8990 q^{73} -2.04989i q^{77} +13.4495i q^{79} +6.75323 q^{83} -9.89898 q^{85} +14.1421i q^{89} -2.44949i q^{91} -13.9993 q^{95} -12.2474 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 56q^{37} - 8q^{49} + 16q^{61} + 64q^{73} - 40q^{85} + O(q^{100}) \)

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
\(5\) − 3.14626i − 1.40705i −0.710669 0.703526i \(-0.751608\pi\)
0.710669 0.703526i \(-0.248392\pi\)
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.04989 0.618065 0.309032 0.951052i \(-0.399995\pi\)
0.309032 + 0.951052i \(0.399995\pi\)
\(12\) 0 0
\(13\) 2.44949 0.679366 0.339683 0.940540i \(-0.389680\pi\)
0.339683 + 0.940540i \(0.389680\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 3.14626i − 0.763081i −0.924352 0.381541i \(-0.875394\pi\)
0.924352 0.381541i \(-0.124606\pi\)
\(18\) 0 0
\(19\) − 4.44949i − 1.02078i −0.859942 0.510391i \(-0.829501\pi\)
0.859942 0.510391i \(-0.170499\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 2.82843 0.589768 0.294884 0.955533i \(-0.404719\pi\)
0.294884 + 0.955533i \(0.404719\pi\)
\(24\) 0 0
\(25\) −4.89898 −0.979796
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) − 0.778539i − 0.144571i −0.997384 0.0722855i \(-0.976971\pi\)
0.997384 0.0722855i \(-0.0230293\pi\)
\(30\) 0 0
\(31\) − 0.449490i − 0.0807307i −0.999185 0.0403654i \(-0.987148\pi\)
0.999185 0.0403654i \(-0.0128522\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −3.14626 −0.531816
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 2.51059i − 0.392088i −0.980595 0.196044i \(-0.937190\pi\)
0.980595 0.196044i \(-0.0628096\pi\)
\(42\) 0 0
\(43\) − 4.34847i − 0.663135i −0.943431 0.331568i \(-0.892422\pi\)
0.943431 0.331568i \(-0.107578\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.38891 1.07778 0.538891 0.842375i \(-0.318843\pi\)
0.538891 + 0.842375i \(0.318843\pi\)
\(48\) 0 0
\(49\) −1.00000 −0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 6.29253i − 0.864345i −0.901791 0.432173i \(-0.857747\pi\)
0.901791 0.432173i \(-0.142253\pi\)
\(54\) 0 0
\(55\) − 6.44949i − 0.869649i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.83183 0.759239 0.379620 0.925143i \(-0.376055\pi\)
0.379620 + 0.925143i \(0.376055\pi\)
\(60\) 0 0
\(61\) −5.34847 −0.684801 −0.342401 0.939554i \(-0.611240\pi\)
−0.342401 + 0.939554i \(0.611240\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) − 7.70674i − 0.955904i
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 12.8990 1.50971 0.754856 0.655891i \(-0.227707\pi\)
0.754856 + 0.655891i \(0.227707\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.04989i − 0.233606i
\(78\) 0 0
\(79\) 13.4495i 1.51319i 0.653886 + 0.756593i \(0.273137\pi\)
−0.653886 + 0.756593i \(0.726863\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 6.75323 0.741263 0.370632 0.928780i \(-0.379141\pi\)
0.370632 + 0.928780i \(0.379141\pi\)
\(84\) 0 0
\(85\) −9.89898 −1.07370
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 14.1421i 1.49906i 0.661968 + 0.749532i \(0.269721\pi\)
−0.661968 + 0.749532i \(0.730279\pi\)
\(90\) 0 0
\(91\) − 2.44949i − 0.256776i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −13.9993 −1.43629
\(96\) 0 0
\(97\) −12.2474 −1.24354 −0.621770 0.783200i \(-0.713586\pi\)
−0.621770 + 0.783200i \(0.713586\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 18.2419i 1.81514i 0.419903 + 0.907569i \(0.362064\pi\)
−0.419903 + 0.907569i \(0.637936\pi\)
\(102\) 0 0
\(103\) 12.8990i 1.27097i 0.772111 + 0.635487i \(0.219201\pi\)
−0.772111 + 0.635487i \(0.780799\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 14.1421 1.36717 0.683586 0.729870i \(-0.260419\pi\)
0.683586 + 0.729870i \(0.260419\pi\)
\(108\) 0 0
\(109\) 7.00000 0.670478 0.335239 0.942133i \(-0.391183\pi\)
0.335239 + 0.942133i \(0.391183\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 11.1708i − 1.05086i −0.850835 0.525432i \(-0.823904\pi\)
0.850835 0.525432i \(-0.176096\pi\)
\(114\) 0 0
\(115\) − 8.89898i − 0.829834i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −3.14626 −0.288418
\(120\) 0 0
\(121\) −6.79796 −0.617996
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 0.317837i − 0.0284282i
\(126\) 0 0
\(127\) − 9.24745i − 0.820578i −0.911955 0.410289i \(-0.865428\pi\)
0.911955 0.410289i \(-0.134572\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 4.73545 0.413738 0.206869 0.978369i \(-0.433673\pi\)
0.206869 + 0.978369i \(0.433673\pi\)
\(132\) 0 0
\(133\) −4.44949 −0.385820
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 11.8065i − 1.00870i −0.863500 0.504349i \(-0.831732\pi\)
0.863500 0.504349i \(-0.168268\pi\)
\(138\) 0 0
\(139\) − 12.2474i − 1.03882i −0.854527 0.519408i \(-0.826153\pi\)
0.854527 0.519408i \(-0.173847\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 5.02118 0.419892
\(144\) 0 0
\(145\) −2.44949 −0.203419
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 2.97129i − 0.243418i −0.992566 0.121709i \(-0.961163\pi\)
0.992566 0.121709i \(-0.0388374\pi\)
\(150\) 0 0
\(151\) − 7.24745i − 0.589789i −0.955530 0.294895i \(-0.904715\pi\)
0.955530 0.294895i \(-0.0952845\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.41421 −0.113592
\(156\) 0 0
\(157\) 0.651531 0.0519978 0.0259989 0.999662i \(-0.491723\pi\)
0.0259989 + 0.999662i \(0.491723\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) − 2.82843i − 0.222911i
\(162\) 0 0
\(163\) − 13.2474i − 1.03762i −0.854889 0.518810i \(-0.826375\pi\)
0.854889 0.518810i \(-0.173625\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −21.2453 −1.64401 −0.822006 0.569479i \(-0.807145\pi\)
−0.822006 + 0.569479i \(0.807145\pi\)
\(168\) 0 0
\(169\) −7.00000 −0.538462
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 2.19275i 0.166712i 0.996520 + 0.0833559i \(0.0265638\pi\)
−0.996520 + 0.0833559i \(0.973436\pi\)
\(174\) 0 0
\(175\) 4.89898i 0.370328i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.04989 0.153216 0.0766079 0.997061i \(-0.475591\pi\)
0.0766079 + 0.997061i \(0.475591\pi\)
\(180\) 0 0
\(181\) −4.69694 −0.349121 −0.174560 0.984646i \(-0.555850\pi\)
−0.174560 + 0.984646i \(0.555850\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 22.0239i 1.61923i
\(186\) 0 0
\(187\) − 6.44949i − 0.471633i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −4.73545 −0.342645 −0.171323 0.985215i \(-0.554804\pi\)
−0.171323 + 0.985215i \(0.554804\pi\)
\(192\) 0 0
\(193\) −11.0000 −0.791797 −0.395899 0.918294i \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 7.70674i 0.549083i 0.961575 + 0.274541i \(0.0885260\pi\)
−0.961575 + 0.274541i \(0.911474\pi\)
\(198\) 0 0
\(199\) − 10.0000i − 0.708881i −0.935079 0.354441i \(-0.884671\pi\)
0.935079 0.354441i \(-0.115329\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.778539 −0.0546427
\(204\) 0 0
\(205\) −7.89898 −0.551689
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) − 9.12096i − 0.630910i
\(210\) 0 0
\(211\) − 6.00000i − 0.413057i −0.978441 0.206529i \(-0.933783\pi\)
0.978441 0.206529i \(-0.0662166\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −13.6814 −0.933066
\(216\) 0 0
\(217\) −0.449490 −0.0305134
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) − 7.70674i − 0.518412i
\(222\) 0 0
\(223\) 6.24745i 0.418360i 0.977877 + 0.209180i \(0.0670795\pi\)
−0.977877 + 0.209180i \(0.932921\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 9.75663 0.647570 0.323785 0.946131i \(-0.395045\pi\)
0.323785 + 0.946131i \(0.395045\pi\)
\(228\) 0 0
\(229\) −16.6969 −1.10336 −0.551682 0.834054i \(-0.686014\pi\)
−0.551682 + 0.834054i \(0.686014\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 15.9063i 1.04206i 0.853540 + 0.521028i \(0.174451\pi\)
−0.853540 + 0.521028i \(0.825549\pi\)
\(234\) 0 0
\(235\) − 23.2474i − 1.51650i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −17.4634 −1.12961 −0.564806 0.825224i \(-0.691049\pi\)
−0.564806 + 0.825224i \(0.691049\pi\)
\(240\) 0 0
\(241\) −28.2474 −1.81958 −0.909789 0.415071i \(-0.863757\pi\)
−0.909789 + 0.415071i \(0.863757\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 3.14626i 0.201007i
\(246\) 0 0
\(247\) − 10.8990i − 0.693485i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.7600 0.805406 0.402703 0.915331i \(-0.368071\pi\)
0.402703 + 0.915331i \(0.368071\pi\)
\(252\) 0 0
\(253\) 5.79796 0.364515
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 16.6848i 1.04077i 0.853931 + 0.520386i \(0.174212\pi\)
−0.853931 + 0.520386i \(0.825788\pi\)
\(258\) 0 0
\(259\) 7.00000i 0.434959i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.07107 −0.436021 −0.218010 0.975946i \(-0.569957\pi\)
−0.218010 + 0.975946i \(0.569957\pi\)
\(264\) 0 0
\(265\) −19.7980 −1.21618
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.72452i 0.592915i 0.955046 + 0.296457i \(0.0958052\pi\)
−0.955046 + 0.296457i \(0.904195\pi\)
\(270\) 0 0
\(271\) − 12.6969i − 0.771284i −0.922648 0.385642i \(-0.873980\pi\)
0.922648 0.385642i \(-0.126020\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −10.0424 −0.605577
\(276\) 0 0
\(277\) −10.7980 −0.648786 −0.324393 0.945922i \(-0.605160\pi\)
−0.324393 + 0.945922i \(0.605160\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) − 8.48528i − 0.506189i −0.967442 0.253095i \(-0.918552\pi\)
0.967442 0.253095i \(-0.0814484\pi\)
\(282\) 0 0
\(283\) 7.75255i 0.460841i 0.973091 + 0.230421i \(0.0740102\pi\)
−0.973091 + 0.230421i \(0.925990\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.51059 −0.148195
\(288\) 0 0
\(289\) 7.10102 0.417707
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 10.3602i − 0.605249i −0.953110 0.302625i \(-0.902137\pi\)
0.953110 0.302625i \(-0.0978628\pi\)
\(294\) 0 0
\(295\) − 18.3485i − 1.06829i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 6.92820 0.400668
\(300\) 0 0
\(301\) −4.34847 −0.250642
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 16.8277i 0.963551i
\(306\) 0 0
\(307\) 4.89898i 0.279600i 0.990180 + 0.139800i \(0.0446459\pi\)
−0.990180 + 0.139800i \(0.955354\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −13.0458 −0.739757 −0.369879 0.929080i \(-0.620601\pi\)
−0.369879 + 0.929080i \(0.620601\pi\)
\(312\) 0 0
\(313\) −2.89898 −0.163860 −0.0819300 0.996638i \(-0.526108\pi\)
−0.0819300 + 0.996638i \(0.526108\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.9985i 1.57255i 0.617874 + 0.786277i \(0.287994\pi\)
−0.617874 + 0.786277i \(0.712006\pi\)
\(318\) 0 0
\(319\) − 1.59592i − 0.0893543i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −13.9993 −0.778940
\(324\) 0 0
\(325\) −12.0000 −0.665640
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) − 7.38891i − 0.407364i
\(330\) 0 0
\(331\) 17.2474i 0.948006i 0.880523 + 0.474003i \(0.157191\pi\)
−0.880523 + 0.474003i \(0.842809\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −25.1701 −1.37519
\(336\) 0 0
\(337\) 3.00000 0.163420 0.0817102 0.996656i \(-0.473962\pi\)
0.0817102 + 0.996656i \(0.473962\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 0.921404i − 0.0498968i
\(342\) 0 0
\(343\) 1.00000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −29.4128 −1.57896 −0.789480 0.613777i \(-0.789650\pi\)
−0.789480 + 0.613777i \(0.789650\pi\)
\(348\) 0 0
\(349\) 33.7980 1.80916 0.904582 0.426300i \(-0.140183\pi\)
0.904582 + 0.426300i \(0.140183\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 20.7525i 1.10454i 0.833664 + 0.552272i \(0.186239\pi\)
−0.833664 + 0.552272i \(0.813761\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 26.0915 1.37706 0.688529 0.725209i \(-0.258257\pi\)
0.688529 + 0.725209i \(0.258257\pi\)
\(360\) 0 0
\(361\) −0.797959 −0.0419978
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) − 40.5836i − 2.12424i
\(366\) 0 0
\(367\) − 16.6969i − 0.871573i −0.900050 0.435787i \(-0.856470\pi\)
0.900050 0.435787i \(-0.143530\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.29253 −0.326692
\(372\) 0 0
\(373\) −35.2929 −1.82739 −0.913697 0.406395i \(-0.866786\pi\)
−0.913697 + 0.406395i \(0.866786\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1.90702i − 0.0982167i
\(378\) 0 0
\(379\) − 11.0454i − 0.567364i −0.958918 0.283682i \(-0.908444\pi\)
0.958918 0.283682i \(-0.0915561\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11.4887 −0.587044 −0.293522 0.955952i \(-0.594827\pi\)
−0.293522 + 0.955952i \(0.594827\pi\)
\(384\) 0 0
\(385\) −6.44949 −0.328696
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 7.07107i 0.358517i 0.983802 + 0.179259i \(0.0573699\pi\)
−0.983802 + 0.179259i \(0.942630\pi\)
\(390\) 0 0
\(391\) − 8.89898i − 0.450041i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 42.3157 2.12913
\(396\) 0 0
\(397\) −24.8990 −1.24964 −0.624822 0.780767i \(-0.714828\pi\)
−0.624822 + 0.780767i \(0.714828\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 22.9774i − 1.14743i −0.819053 0.573717i \(-0.805501\pi\)
0.819053 0.573717i \(-0.194499\pi\)
\(402\) 0 0
\(403\) − 1.10102i − 0.0548457i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −14.3492 −0.711264
\(408\) 0 0
\(409\) 13.7980 0.682265 0.341133 0.940015i \(-0.389189\pi\)
0.341133 + 0.940015i \(0.389189\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 5.83183i − 0.286965i
\(414\) 0 0
\(415\) − 21.2474i − 1.04300i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3.63907 0.177780 0.0888902 0.996041i \(-0.471668\pi\)
0.0888902 + 0.996041i \(0.471668\pi\)
\(420\) 0 0
\(421\) 16.8990 0.823606 0.411803 0.911273i \(-0.364899\pi\)
0.411803 + 0.911273i \(0.364899\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 15.4135i 0.747664i
\(426\) 0 0
\(427\) 5.34847i 0.258831i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −40.2979 −1.94108 −0.970540 0.240940i \(-0.922544\pi\)
−0.970540 + 0.240940i \(0.922544\pi\)
\(432\) 0 0
\(433\) −37.8434 −1.81864 −0.909318 0.416102i \(-0.863396\pi\)
−0.909318 + 0.416102i \(0.863396\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) − 12.5851i − 0.602025i
\(438\) 0 0
\(439\) − 2.00000i − 0.0954548i −0.998860 0.0477274i \(-0.984802\pi\)
0.998860 0.0477274i \(-0.0151979\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 30.6841 1.45785 0.728923 0.684596i \(-0.240021\pi\)
0.728923 + 0.684596i \(0.240021\pi\)
\(444\) 0 0
\(445\) 44.4949 2.10926
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 10.8209i 0.510670i 0.966853 + 0.255335i \(0.0821857\pi\)
−0.966853 + 0.255335i \(0.917814\pi\)
\(450\) 0 0
\(451\) − 5.14643i − 0.242336i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −7.70674 −0.361298
\(456\) 0 0
\(457\) 37.3939 1.74921 0.874606 0.484835i \(-0.161120\pi\)
0.874606 + 0.484835i \(0.161120\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.953512i 0.0444095i 0.999753 + 0.0222047i \(0.00706857\pi\)
−0.999753 + 0.0222047i \(0.992931\pi\)
\(462\) 0 0
\(463\) − 34.8434i − 1.61931i −0.586907 0.809654i \(-0.699655\pi\)
0.586907 0.809654i \(-0.300345\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 26.7272 1.23679 0.618393 0.785869i \(-0.287784\pi\)
0.618393 + 0.785869i \(0.287784\pi\)
\(468\) 0 0
\(469\) −8.00000 −0.369406
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) − 8.91388i − 0.409860i
\(474\) 0 0
\(475\) 21.7980i 1.00016i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 22.8024 1.04187 0.520934 0.853597i \(-0.325584\pi\)
0.520934 + 0.853597i \(0.325584\pi\)
\(480\) 0 0
\(481\) −17.1464 −0.781810
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 38.5337i 1.74973i
\(486\) 0 0
\(487\) 25.7980i 1.16902i 0.811388 + 0.584509i \(0.198713\pi\)
−0.811388 + 0.584509i \(0.801287\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 10.8851 0.491238 0.245619 0.969366i \(-0.421009\pi\)
0.245619 + 0.969366i \(0.421009\pi\)
\(492\) 0 0
\(493\) −2.44949 −0.110319
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) − 16.5505i − 0.740903i −0.928852 0.370451i \(-0.879203\pi\)
0.928852 0.370451i \(-0.120797\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 16.7956 0.748878 0.374439 0.927251i \(-0.377835\pi\)
0.374439 + 0.927251i \(0.377835\pi\)
\(504\) 0 0
\(505\) 57.3939 2.55399
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 33.9732i − 1.50584i −0.658114 0.752919i \(-0.728645\pi\)
0.658114 0.752919i \(-0.271355\pi\)
\(510\) 0 0
\(511\) − 12.8990i − 0.570617i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 40.5836 1.78833
\(516\) 0 0
\(517\) 15.1464 0.666139
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 18.9097i 0.828449i 0.910175 + 0.414224i \(0.135947\pi\)
−0.910175 + 0.414224i \(0.864053\pi\)
\(522\) 0 0
\(523\) 40.7423i 1.78154i 0.454456 + 0.890769i \(0.349834\pi\)
−0.454456 + 0.890769i \(0.650166\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −1.41421 −0.0616041
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 6.14966i − 0.266372i
\(534\) 0 0
\(535\) − 44.4949i − 1.92368i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −2.04989 −0.0882949
\(540\) 0 0
\(541\) −36.5959 −1.57338 −0.786691 0.617347i \(-0.788207\pi\)
−0.786691 + 0.617347i \(0.788207\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 22.0239i − 0.943398i
\(546\) 0 0
\(547\) − 3.04541i − 0.130212i −0.997878 0.0651061i \(-0.979261\pi\)
0.997878 0.0651061i \(-0.0207386\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3.46410 −0.147576
\(552\) 0 0
\(553\) 13.4495 0.571930
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 16.3991i − 0.694852i −0.937707 0.347426i \(-0.887056\pi\)
0.937707 0.347426i \(-0.112944\pi\)
\(558\) 0 0
\(559\) − 10.6515i − 0.450512i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 36.4838 1.53761 0.768805 0.639483i \(-0.220852\pi\)
0.768805 + 0.639483i \(0.220852\pi\)
\(564\) 0 0
\(565\) −35.1464 −1.47862
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 13.5707i − 0.568912i −0.958689 0.284456i \(-0.908187\pi\)
0.958689 0.284456i \(-0.0918130\pi\)
\(570\) 0 0
\(571\) 7.44949i 0.311751i 0.987777 + 0.155876i \(0.0498199\pi\)
−0.987777 + 0.155876i \(0.950180\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −13.8564 −0.577852
\(576\) 0 0
\(577\) −2.89898 −0.120686 −0.0603430 0.998178i \(-0.519219\pi\)
−0.0603430 + 0.998178i \(0.519219\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 6.75323i − 0.280171i
\(582\) 0 0
\(583\) − 12.8990i − 0.534221i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −8.48528 −0.350225 −0.175113 0.984548i \(-0.556029\pi\)
−0.175113 + 0.984548i \(0.556029\pi\)
\(588\) 0 0
\(589\) −2.00000 −0.0824086
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 35.8803i − 1.47343i −0.676206 0.736713i \(-0.736377\pi\)
0.676206 0.736713i \(-0.263623\pi\)
\(594\) 0 0
\(595\) 9.89898i 0.405819i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 13.3636 0.546022 0.273011 0.962011i \(-0.411980\pi\)
0.273011 + 0.962011i \(0.411980\pi\)
\(600\) 0 0
\(601\) −36.2474 −1.47856 −0.739282 0.673396i \(-0.764835\pi\)
−0.739282 + 0.673396i \(0.764835\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 21.3882i 0.869553i
\(606\) 0 0
\(607\) 23.8434i 0.967772i 0.875131 + 0.483886i \(0.160775\pi\)
−0.875131 + 0.483886i \(0.839225\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 18.0990 0.732209
\(612\) 0 0
\(613\) −20.0000 −0.807792 −0.403896 0.914805i \(-0.632344\pi\)
−0.403896 + 0.914805i \(0.632344\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 11.5994i − 0.466976i −0.972360 0.233488i \(-0.924986\pi\)
0.972360 0.233488i \(-0.0750139\pi\)
\(618\) 0 0
\(619\) − 31.1464i − 1.25188i −0.779871 0.625940i \(-0.784715\pi\)
0.779871 0.625940i \(-0.215285\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 14.1421 0.566593
\(624\) 0 0
\(625\) −25.4949 −1.01980
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 22.0239i 0.878148i
\(630\) 0 0
\(631\) 5.44949i 0.216941i 0.994100 + 0.108470i \(0.0345953\pi\)
−0.994100 + 0.108470i \(0.965405\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −29.0949 −1.15460
\(636\) 0 0
\(637\) −2.44949 −0.0970523
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 29.4128i − 1.16173i −0.813998 0.580867i \(-0.802714\pi\)
0.813998 0.580867i \(-0.197286\pi\)
\(642\) 0 0
\(643\) 15.3031i 0.603494i 0.953388 + 0.301747i \(0.0975698\pi\)
−0.953388 + 0.301747i \(0.902430\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 38.0409 1.49554 0.747771 0.663957i \(-0.231124\pi\)
0.747771 + 0.663957i \(0.231124\pi\)
\(648\) 0 0
\(649\) 11.9546 0.469259
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.34242i 0.326464i 0.986588 + 0.163232i \(0.0521919\pi\)
−0.986588 + 0.163232i \(0.947808\pi\)
\(654\) 0 0
\(655\) − 14.8990i − 0.582151i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.90702 0.0742871 0.0371435 0.999310i \(-0.488174\pi\)
0.0371435 + 0.999310i \(0.488174\pi\)
\(660\) 0 0
\(661\) 26.7423 1.04016 0.520078 0.854119i \(-0.325903\pi\)
0.520078 + 0.854119i \(0.325903\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 13.9993i 0.542868i
\(666\) 0 0
\(667\) − 2.20204i − 0.0852634i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −10.9638 −0.423251
\(672\) 0 0
\(673\) 46.2929 1.78446 0.892229 0.451583i \(-0.149140\pi\)
0.892229 + 0.451583i \(0.149140\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.94258i 0.228392i 0.993458 + 0.114196i \(0.0364292\pi\)
−0.993458 + 0.114196i \(0.963571\pi\)
\(678\) 0 0
\(679\) 12.2474i 0.470014i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 26.2986 1.00629 0.503144 0.864203i \(-0.332177\pi\)
0.503144 + 0.864203i \(0.332177\pi\)
\(684\) 0 0
\(685\) −37.1464 −1.41929
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 15.4135i − 0.587207i
\(690\) 0 0
\(691\) − 28.2474i − 1.07458i −0.843396 0.537292i \(-0.819447\pi\)
0.843396 0.537292i \(-0.180553\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −38.5337 −1.46167
\(696\) 0 0
\(697\) −7.89898 −0.299195
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 15.5563i − 0.587555i −0.955874 0.293778i \(-0.905087\pi\)
0.955874 0.293778i \(-0.0949125\pi\)
\(702\) 0 0
\(703\) 31.1464i 1.17471i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 18.2419 0.686058
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 1.27135i − 0.0476124i
\(714\) 0 0
\(715\) − 15.7980i − 0.590810i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 9.23171 0.344285 0.172142 0.985072i \(-0.444931\pi\)
0.172142 + 0.985072i \(0.444931\pi\)
\(720\) 0 0
\(721\) 12.8990 0.480383
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 3.81405i 0.141650i
\(726\) 0 0
\(727\) 42.2474i 1.56687i 0.621473 + 0.783436i \(0.286535\pi\)
−0.621473 + 0.783436i \(0.713465\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −13.6814 −0.506026
\(732\) 0 0
\(733\) 42.8990 1.58451 0.792255 0.610190i \(-0.208907\pi\)
0.792255 + 0.610190i \(0.208907\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 16.3991i − 0.604069i
\(738\) 0 0
\(739\) 39.5959i 1.45656i 0.685280 + 0.728280i \(0.259680\pi\)
−0.685280 + 0.728280i \(0.740320\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −31.2556 −1.14666 −0.573328 0.819326i \(-0.694348\pi\)
−0.573328 + 0.819326i \(0.694348\pi\)
\(744\) 0 0
\(745\) −9.34847 −0.342501
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) − 14.1421i − 0.516742i
\(750\) 0 0
\(751\) 33.3939i 1.21856i 0.792955 + 0.609280i \(0.208541\pi\)
−0.792955 + 0.609280i \(0.791459\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −22.8024 −0.829864
\(756\) 0 0
\(757\) −24.3031 −0.883310 −0.441655 0.897185i \(-0.645608\pi\)
−0.441655 + 0.897185i \(0.645608\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 2.79632i − 0.101366i −0.998715 0.0506832i \(-0.983860\pi\)
0.998715 0.0506832i \(-0.0161399\pi\)
\(762\) 0 0
\(763\) − 7.00000i − 0.253417i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 14.2850 0.515801
\(768\) 0 0
\(769\) −25.1010 −0.905166 −0.452583 0.891722i \(-0.649497\pi\)
−0.452583 + 0.891722i \(0.649497\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 32.0662i − 1.15334i −0.816977 0.576671i \(-0.804352\pi\)
0.816977 0.576671i \(-0.195648\pi\)
\(774\) 0 0
\(775\) 2.20204i 0.0790996i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −11.1708 −0.400237
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 2.04989i − 0.0731636i
\(786\) 0 0
\(787\) 4.49490i 0.160226i 0.996786 + 0.0801129i \(0.0255281\pi\)
−0.996786 + 0.0801129i \(0.974472\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −11.1708 −0.397189
\(792\) 0 0
\(793\) −13.1010 −0.465231
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 36.1981i − 1.28220i −0.767456 0.641101i \(-0.778478\pi\)
0.767456 0.641101i \(-0.221522\pi\)
\(798\) 0 0
\(799\) − 23.2474i − 0.822436i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 26.4415 0.933099
\(804\) 0 0
\(805\) −8.89898 −0.313648
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 21.4203i 0.753097i 0.926397 + 0.376549i \(0.122889\pi\)
−0.926397 + 0.376549i \(0.877111\pi\)
\(810\) 0 0
\(811\) 36.7423i 1.29020i 0.764099 + 0.645099i \(0.223184\pi\)
−0.764099 + 0.645099i \(0.776816\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −41.6800 −1.45999
\(816\) 0 0
\(817\) −19.3485 −0.676917
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 22.0560i 0.769758i 0.922967 + 0.384879i \(0.125757\pi\)
−0.922967 + 0.384879i \(0.874243\pi\)
\(822\) 0 0
\(823\) − 24.3485i − 0.848734i −0.905490 0.424367i \(-0.860497\pi\)
0.905490 0.424367i \(-0.139503\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 36.9766 1.28580 0.642902 0.765949i \(-0.277730\pi\)
0.642902 + 0.765949i \(0.277730\pi\)
\(828\) 0 0
\(829\) 38.0454 1.32137 0.660686 0.750663i \(-0.270266\pi\)
0.660686 + 0.750663i \(0.270266\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 3.14626i 0.109012i
\(834\) 0 0
\(835\) 66.8434i 2.31321i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 23.7238 0.819036 0.409518 0.912302i \(-0.365697\pi\)
0.409518 + 0.912302i \(0.365697\pi\)
\(840\) 0 0
\(841\) 28.3939 0.979099
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 22.0239i 0.757643i
\(846\) 0 0
\(847\) 6.79796i 0.233581i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −19.7990 −0.678701
\(852\) 0 0
\(853\) 37.1010 1.27031 0.635157 0.772383i \(-0.280935\pi\)
0.635157 + 0.772383i \(0.280935\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 42.8085i 1.46231i 0.682212 + 0.731155i \(0.261018\pi\)
−0.682212 + 0.731155i \(0.738982\pi\)
\(858\) 0 0
\(859\) 22.4949i 0.767516i 0.923434 + 0.383758i \(0.125370\pi\)
−0.923434 + 0.383758i \(0.874630\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −26.0915 −0.888166 −0.444083 0.895986i \(-0.646470\pi\)
−0.444083 + 0.895986i \(0.646470\pi\)
\(864\) 0 0
\(865\) 6.89898 0.234572
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 27.5699i 0.935246i
\(870\) 0 0
\(871\) − 19.5959i − 0.663982i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.317837 −0.0107449
\(876\) 0 0
\(877\) −19.2020 −0.648407 −0.324203 0.945987i \(-0.605096\pi\)
−0.324203 + 0.945987i \(0.605096\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 7.49966i 0.252670i 0.991988 + 0.126335i \(0.0403214\pi\)
−0.991988 + 0.126335i \(0.959679\pi\)
\(882\) 0 0
\(883\) − 37.0454i − 1.24668i −0.781952 0.623339i \(-0.785776\pi\)
0.781952 0.623339i \(-0.214224\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −13.0458 −0.438034 −0.219017 0.975721i \(-0.570285\pi\)
−0.219017 + 0.975721i \(0.570285\pi\)
\(888\) 0 0
\(889\) −9.24745 −0.310149
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) − 32.8769i − 1.10018i
\(894\) 0 0
\(895\) − 6.44949i − 0.215583i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −0.349945 −0.0116713
\(900\) 0 0
\(901\) −19.7980 −0.659566
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 14.7778i 0.491231i
\(906\) 0 0
\(907\) − 55.9444i − 1.85760i −0.370578 0.928801i \(-0.620840\pi\)
0.370578 0.928801i \(-0.379160\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 7.70674 0.255336 0.127668 0.991817i \(-0.459251\pi\)
0.127668 + 0.991817i \(0.459251\pi\)
\(912\) 0 0
\(913\) 13.8434 0.458149
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 4.73545i − 0.156378i
\(918\) 0 0
\(919\) 37.2474i 1.22868i 0.789041 + 0.614340i \(0.210578\pi\)
−0.789041 + 0.614340i \(0.789422\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 34.2929 1.12754
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 13.8885i 0.455667i 0.973700 + 0.227834i \(0.0731643\pi\)
−0.973700 + 0.227834i \(0.926836\pi\)
\(930\) 0 0
\(931\) 4.44949i 0.145826i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −20.2918 −0.663613
\(936\) 0 0
\(937\) 0.651531 0.0212846 0.0106423 0.999943i \(-0.496612\pi\)
0.0106423 + 0.999943i \(0.496612\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 22.3096i − 0.727272i −0.931541 0.363636i \(-0.881535\pi\)
0.931541 0.363636i \(-0.118465\pi\)
\(942\) 0 0
\(943\) − 7.10102i − 0.231241i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.9842 0.941859 0.470929 0.882171i \(-0.343919\pi\)
0.470929 + 0.882171i \(0.343919\pi\)
\(948\) 0 0
\(949\) 31.5959 1.02565
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 2.25697i − 0.0731104i −0.999332 0.0365552i \(-0.988362\pi\)
0.999332 0.0365552i \(-0.0116385\pi\)
\(954\) 0 0
\(955\) 14.8990i 0.482120i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.8065 −0.381252
\(960\) 0 0
\(961\) 30.7980 0.993483
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 34.6089i 1.11410i
\(966\) 0 0
\(967\) 4.89898i 0.157541i 0.996893 + 0.0787703i \(0.0250994\pi\)
−0.996893 + 0.0787703i \(0.974901\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 55.8221 1.79142 0.895708 0.444642i \(-0.146669\pi\)
0.895708 + 0.444642i \(0.146669\pi\)
\(972\) 0 0
\(973\) −12.2474 −0.392635
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 43.7620i − 1.40007i −0.714109 0.700035i \(-0.753168\pi\)
0.714109 0.700035i \(-0.246832\pi\)
\(978\) 0 0
\(979\) 28.9898i 0.926518i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −28.1735 −0.898596 −0.449298 0.893382i \(-0.648326\pi\)
−0.449298 + 0.893382i \(0.648326\pi\)
\(984\) 0 0
\(985\) 24.2474 0.772588
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 12.2993i − 0.391096i
\(990\) 0 0
\(991\) − 14.5505i − 0.462212i −0.972929 0.231106i \(-0.925766\pi\)
0.972929 0.231106i \(-0.0742344\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −31.4626 −0.997433
\(996\) 0 0
\(997\) 32.2474 1.02129 0.510643 0.859793i \(-0.329407\pi\)
0.510643 + 0.859793i \(0.329407\pi\)
\(998\) 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.h.d.2591.1 8
3.2 odd 2 inner 6048.2.h.d.2591.7 yes 8
4.3 odd 2 inner 6048.2.h.d.2591.2 yes 8
12.11 even 2 inner 6048.2.h.d.2591.8 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6048.2.h.d.2591.1 8 1.1 even 1 trivial
6048.2.h.d.2591.2 yes 8 4.3 odd 2 inner
6048.2.h.d.2591.7 yes 8 3.2 odd 2 inner
6048.2.h.d.2591.8 yes 8 12.11 even 2 inner