Properties

Label 2205.2.a.v.1.1
Level $2205$
Weight $2$
Character 2205.1
Self dual yes
Analytic conductor $17.607$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2205 = 3^{2} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2205.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.6070136457\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 245)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2205.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.41421 q^{2} +1.00000 q^{5} +2.82843 q^{8} +O(q^{10})\) \(q-1.41421 q^{2} +1.00000 q^{5} +2.82843 q^{8} -1.41421 q^{10} +5.82843 q^{11} -1.58579 q^{13} -4.00000 q^{16} -5.24264 q^{17} -6.00000 q^{19} -8.24264 q^{22} -4.58579 q^{23} +1.00000 q^{25} +2.24264 q^{26} -2.65685 q^{29} -1.75736 q^{31} +7.41421 q^{34} -6.24264 q^{37} +8.48528 q^{38} +2.82843 q^{40} +2.24264 q^{41} +2.00000 q^{43} +6.48528 q^{46} +1.24264 q^{47} -1.41421 q^{50} +4.24264 q^{53} +5.82843 q^{55} +3.75736 q^{58} +6.24264 q^{59} -2.82843 q^{61} +2.48528 q^{62} +8.00000 q^{64} -1.58579 q^{65} +0.242641 q^{67} +8.82843 q^{71} -8.48528 q^{73} +8.82843 q^{74} -15.4853 q^{79} -4.00000 q^{80} -3.17157 q^{82} -5.24264 q^{85} -2.82843 q^{86} +16.4853 q^{88} -8.00000 q^{89} -1.75736 q^{94} -6.00000 q^{95} -4.75736 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 2q^{5} + O(q^{10}) \) \( 2q + 2q^{5} + 6q^{11} - 6q^{13} - 8q^{16} - 2q^{17} - 12q^{19} - 8q^{22} - 12q^{23} + 2q^{25} - 4q^{26} + 6q^{29} - 12q^{31} + 12q^{34} - 4q^{37} - 4q^{41} + 4q^{43} - 4q^{46} - 6q^{47} + 6q^{55} + 16q^{58} + 4q^{59} - 12q^{62} + 16q^{64} - 6q^{65} - 8q^{67} + 12q^{71} + 12q^{74} - 14q^{79} - 8q^{80} - 12q^{82} - 2q^{85} + 16q^{88} - 16q^{89} - 12q^{94} - 12q^{95} - 18q^{97} + O(q^{100}) \)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.41421 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(3\) 0 0
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 2.82843 1.00000
\(9\) 0 0
\(10\) −1.41421 −0.447214
\(11\) 5.82843 1.75734 0.878668 0.477432i \(-0.158432\pi\)
0.878668 + 0.477432i \(0.158432\pi\)
\(12\) 0 0
\(13\) −1.58579 −0.439818 −0.219909 0.975520i \(-0.570576\pi\)
−0.219909 + 0.975520i \(0.570576\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −5.24264 −1.27153 −0.635764 0.771884i \(-0.719315\pi\)
−0.635764 + 0.771884i \(0.719315\pi\)
\(18\) 0 0
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −8.24264 −1.75734
\(23\) −4.58579 −0.956203 −0.478101 0.878305i \(-0.658675\pi\)
−0.478101 + 0.878305i \(0.658675\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.24264 0.439818
\(27\) 0 0
\(28\) 0 0
\(29\) −2.65685 −0.493365 −0.246683 0.969096i \(-0.579341\pi\)
−0.246683 + 0.969096i \(0.579341\pi\)
\(30\) 0 0
\(31\) −1.75736 −0.315631 −0.157816 0.987469i \(-0.550445\pi\)
−0.157816 + 0.987469i \(0.550445\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 7.41421 1.27153
\(35\) 0 0
\(36\) 0 0
\(37\) −6.24264 −1.02628 −0.513142 0.858304i \(-0.671519\pi\)
−0.513142 + 0.858304i \(0.671519\pi\)
\(38\) 8.48528 1.37649
\(39\) 0 0
\(40\) 2.82843 0.447214
\(41\) 2.24264 0.350242 0.175121 0.984547i \(-0.443968\pi\)
0.175121 + 0.984547i \(0.443968\pi\)
\(42\) 0 0
\(43\) 2.00000 0.304997 0.152499 0.988304i \(-0.451268\pi\)
0.152499 + 0.988304i \(0.451268\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.48528 0.956203
\(47\) 1.24264 0.181258 0.0906289 0.995885i \(-0.471112\pi\)
0.0906289 + 0.995885i \(0.471112\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.41421 −0.200000
\(51\) 0 0
\(52\) 0 0
\(53\) 4.24264 0.582772 0.291386 0.956606i \(-0.405884\pi\)
0.291386 + 0.956606i \(0.405884\pi\)
\(54\) 0 0
\(55\) 5.82843 0.785905
\(56\) 0 0
\(57\) 0 0
\(58\) 3.75736 0.493365
\(59\) 6.24264 0.812723 0.406361 0.913712i \(-0.366797\pi\)
0.406361 + 0.913712i \(0.366797\pi\)
\(60\) 0 0
\(61\) −2.82843 −0.362143 −0.181071 0.983470i \(-0.557957\pi\)
−0.181071 + 0.983470i \(0.557957\pi\)
\(62\) 2.48528 0.315631
\(63\) 0 0
\(64\) 8.00000 1.00000
\(65\) −1.58579 −0.196693
\(66\) 0 0
\(67\) 0.242641 0.0296433 0.0148216 0.999890i \(-0.495282\pi\)
0.0148216 + 0.999890i \(0.495282\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.82843 1.04774 0.523871 0.851798i \(-0.324487\pi\)
0.523871 + 0.851798i \(0.324487\pi\)
\(72\) 0 0
\(73\) −8.48528 −0.993127 −0.496564 0.868000i \(-0.665405\pi\)
−0.496564 + 0.868000i \(0.665405\pi\)
\(74\) 8.82843 1.02628
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −15.4853 −1.74223 −0.871115 0.491079i \(-0.836603\pi\)
−0.871115 + 0.491079i \(0.836603\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) −3.17157 −0.350242
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −5.24264 −0.568644
\(86\) −2.82843 −0.304997
\(87\) 0 0
\(88\) 16.4853 1.75734
\(89\) −8.00000 −0.847998 −0.423999 0.905663i \(-0.639374\pi\)
−0.423999 + 0.905663i \(0.639374\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −1.75736 −0.181258
\(95\) −6.00000 −0.615587
\(96\) 0 0
\(97\) −4.75736 −0.483037 −0.241518 0.970396i \(-0.577645\pi\)
−0.241518 + 0.970396i \(0.577645\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −14.4853 −1.44134 −0.720670 0.693279i \(-0.756166\pi\)
−0.720670 + 0.693279i \(0.756166\pi\)
\(102\) 0 0
\(103\) −10.7574 −1.05995 −0.529977 0.848012i \(-0.677799\pi\)
−0.529977 + 0.848012i \(0.677799\pi\)
\(104\) −4.48528 −0.439818
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) −14.4853 −1.40035 −0.700173 0.713974i \(-0.746894\pi\)
−0.700173 + 0.713974i \(0.746894\pi\)
\(108\) 0 0
\(109\) 5.00000 0.478913 0.239457 0.970907i \(-0.423031\pi\)
0.239457 + 0.970907i \(0.423031\pi\)
\(110\) −8.24264 −0.785905
\(111\) 0 0
\(112\) 0 0
\(113\) −1.07107 −0.100758 −0.0503788 0.998730i \(-0.516043\pi\)
−0.0503788 + 0.998730i \(0.516043\pi\)
\(114\) 0 0
\(115\) −4.58579 −0.427627
\(116\) 0 0
\(117\) 0 0
\(118\) −8.82843 −0.812723
\(119\) 0 0
\(120\) 0 0
\(121\) 22.9706 2.08823
\(122\) 4.00000 0.362143
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −0.242641 −0.0215309 −0.0107654 0.999942i \(-0.503427\pi\)
−0.0107654 + 0.999942i \(0.503427\pi\)
\(128\) −11.3137 −1.00000
\(129\) 0 0
\(130\) 2.24264 0.196693
\(131\) 3.75736 0.328282 0.164141 0.986437i \(-0.447515\pi\)
0.164141 + 0.986437i \(0.447515\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −0.343146 −0.0296433
\(135\) 0 0
\(136\) −14.8284 −1.27153
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) −16.2426 −1.37768 −0.688841 0.724912i \(-0.741880\pi\)
−0.688841 + 0.724912i \(0.741880\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −12.4853 −1.04774
\(143\) −9.24264 −0.772908
\(144\) 0 0
\(145\) −2.65685 −0.220640
\(146\) 12.0000 0.993127
\(147\) 0 0
\(148\) 0 0
\(149\) 14.8284 1.21479 0.607396 0.794399i \(-0.292214\pi\)
0.607396 + 0.794399i \(0.292214\pi\)
\(150\) 0 0
\(151\) 9.48528 0.771901 0.385951 0.922519i \(-0.373874\pi\)
0.385951 + 0.922519i \(0.373874\pi\)
\(152\) −16.9706 −1.37649
\(153\) 0 0
\(154\) 0 0
\(155\) −1.75736 −0.141154
\(156\) 0 0
\(157\) 20.8284 1.66229 0.831145 0.556056i \(-0.187686\pi\)
0.831145 + 0.556056i \(0.187686\pi\)
\(158\) 21.8995 1.74223
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.75736 −0.137647 −0.0688235 0.997629i \(-0.521925\pi\)
−0.0688235 + 0.997629i \(0.521925\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.24264 0.715217 0.357609 0.933872i \(-0.383592\pi\)
0.357609 + 0.933872i \(0.383592\pi\)
\(168\) 0 0
\(169\) −10.4853 −0.806560
\(170\) 7.41421 0.568644
\(171\) 0 0
\(172\) 0 0
\(173\) 1.24264 0.0944762 0.0472381 0.998884i \(-0.484958\pi\)
0.0472381 + 0.998884i \(0.484958\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −23.3137 −1.75734
\(177\) 0 0
\(178\) 11.3137 0.847998
\(179\) −2.48528 −0.185759 −0.0928793 0.995677i \(-0.529607\pi\)
−0.0928793 + 0.995677i \(0.529607\pi\)
\(180\) 0 0
\(181\) −6.72792 −0.500083 −0.250041 0.968235i \(-0.580444\pi\)
−0.250041 + 0.968235i \(0.580444\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −12.9706 −0.956203
\(185\) −6.24264 −0.458968
\(186\) 0 0
\(187\) −30.5563 −2.23450
\(188\) 0 0
\(189\) 0 0
\(190\) 8.48528 0.615587
\(191\) 19.9706 1.44502 0.722510 0.691361i \(-0.242989\pi\)
0.722510 + 0.691361i \(0.242989\pi\)
\(192\) 0 0
\(193\) −16.0000 −1.15171 −0.575853 0.817554i \(-0.695330\pi\)
−0.575853 + 0.817554i \(0.695330\pi\)
\(194\) 6.72792 0.483037
\(195\) 0 0
\(196\) 0 0
\(197\) −10.5858 −0.754206 −0.377103 0.926171i \(-0.623080\pi\)
−0.377103 + 0.926171i \(0.623080\pi\)
\(198\) 0 0
\(199\) 4.58579 0.325078 0.162539 0.986702i \(-0.448032\pi\)
0.162539 + 0.986702i \(0.448032\pi\)
\(200\) 2.82843 0.200000
\(201\) 0 0
\(202\) 20.4853 1.44134
\(203\) 0 0
\(204\) 0 0
\(205\) 2.24264 0.156633
\(206\) 15.2132 1.05995
\(207\) 0 0
\(208\) 6.34315 0.439818
\(209\) −34.9706 −2.41896
\(210\) 0 0
\(211\) 9.00000 0.619586 0.309793 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 20.4853 1.40035
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 0 0
\(218\) −7.07107 −0.478913
\(219\) 0 0
\(220\) 0 0
\(221\) 8.31371 0.559241
\(222\) 0 0
\(223\) −18.2132 −1.21965 −0.609823 0.792538i \(-0.708760\pi\)
−0.609823 + 0.792538i \(0.708760\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.51472 0.100758
\(227\) −9.72792 −0.645665 −0.322832 0.946456i \(-0.604635\pi\)
−0.322832 + 0.946456i \(0.604635\pi\)
\(228\) 0 0
\(229\) 30.0416 1.98521 0.992603 0.121402i \(-0.0387390\pi\)
0.992603 + 0.121402i \(0.0387390\pi\)
\(230\) 6.48528 0.427627
\(231\) 0 0
\(232\) −7.51472 −0.493365
\(233\) 14.8284 0.971443 0.485721 0.874114i \(-0.338557\pi\)
0.485721 + 0.874114i \(0.338557\pi\)
\(234\) 0 0
\(235\) 1.24264 0.0810609
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.514719 0.0332944 0.0166472 0.999861i \(-0.494701\pi\)
0.0166472 + 0.999861i \(0.494701\pi\)
\(240\) 0 0
\(241\) −24.7279 −1.59287 −0.796433 0.604727i \(-0.793282\pi\)
−0.796433 + 0.604727i \(0.793282\pi\)
\(242\) −32.4853 −2.08823
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 9.51472 0.605407
\(248\) −4.97056 −0.315631
\(249\) 0 0
\(250\) −1.41421 −0.0894427
\(251\) −25.2132 −1.59144 −0.795722 0.605663i \(-0.792908\pi\)
−0.795722 + 0.605663i \(0.792908\pi\)
\(252\) 0 0
\(253\) −26.7279 −1.68037
\(254\) 0.343146 0.0215309
\(255\) 0 0
\(256\) 0 0
\(257\) −26.4853 −1.65211 −0.826053 0.563592i \(-0.809419\pi\)
−0.826053 + 0.563592i \(0.809419\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −5.31371 −0.328282
\(263\) −28.6274 −1.76524 −0.882621 0.470085i \(-0.844223\pi\)
−0.882621 + 0.470085i \(0.844223\pi\)
\(264\) 0 0
\(265\) 4.24264 0.260623
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −7.75736 −0.472975 −0.236487 0.971635i \(-0.575996\pi\)
−0.236487 + 0.971635i \(0.575996\pi\)
\(270\) 0 0
\(271\) 23.3137 1.41621 0.708103 0.706109i \(-0.249551\pi\)
0.708103 + 0.706109i \(0.249551\pi\)
\(272\) 20.9706 1.27153
\(273\) 0 0
\(274\) 16.9706 1.02523
\(275\) 5.82843 0.351467
\(276\) 0 0
\(277\) −27.2132 −1.63508 −0.817541 0.575870i \(-0.804664\pi\)
−0.817541 + 0.575870i \(0.804664\pi\)
\(278\) 22.9706 1.37768
\(279\) 0 0
\(280\) 0 0
\(281\) 20.3137 1.21181 0.605907 0.795535i \(-0.292810\pi\)
0.605907 + 0.795535i \(0.292810\pi\)
\(282\) 0 0
\(283\) 6.55635 0.389735 0.194867 0.980830i \(-0.437572\pi\)
0.194867 + 0.980830i \(0.437572\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 13.0711 0.772908
\(287\) 0 0
\(288\) 0 0
\(289\) 10.4853 0.616781
\(290\) 3.75736 0.220640
\(291\) 0 0
\(292\) 0 0
\(293\) 0.272078 0.0158950 0.00794748 0.999968i \(-0.497470\pi\)
0.00794748 + 0.999968i \(0.497470\pi\)
\(294\) 0 0
\(295\) 6.24264 0.363461
\(296\) −17.6569 −1.02628
\(297\) 0 0
\(298\) −20.9706 −1.21479
\(299\) 7.27208 0.420555
\(300\) 0 0
\(301\) 0 0
\(302\) −13.4142 −0.771901
\(303\) 0 0
\(304\) 24.0000 1.37649
\(305\) −2.82843 −0.161955
\(306\) 0 0
\(307\) −11.1005 −0.633539 −0.316770 0.948503i \(-0.602598\pi\)
−0.316770 + 0.948503i \(0.602598\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 2.48528 0.141154
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) 0 0
\(313\) −24.2132 −1.36861 −0.684306 0.729195i \(-0.739895\pi\)
−0.684306 + 0.729195i \(0.739895\pi\)
\(314\) −29.4558 −1.66229
\(315\) 0 0
\(316\) 0 0
\(317\) 11.6569 0.654714 0.327357 0.944901i \(-0.393842\pi\)
0.327357 + 0.944901i \(0.393842\pi\)
\(318\) 0 0
\(319\) −15.4853 −0.867009
\(320\) 8.00000 0.447214
\(321\) 0 0
\(322\) 0 0
\(323\) 31.4558 1.75025
\(324\) 0 0
\(325\) −1.58579 −0.0879636
\(326\) 2.48528 0.137647
\(327\) 0 0
\(328\) 6.34315 0.350242
\(329\) 0 0
\(330\) 0 0
\(331\) 23.4558 1.28925 0.644625 0.764499i \(-0.277014\pi\)
0.644625 + 0.764499i \(0.277014\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) −13.0711 −0.715217
\(335\) 0.242641 0.0132569
\(336\) 0 0
\(337\) 13.7574 0.749411 0.374706 0.927144i \(-0.377744\pi\)
0.374706 + 0.927144i \(0.377744\pi\)
\(338\) 14.8284 0.806560
\(339\) 0 0
\(340\) 0 0
\(341\) −10.2426 −0.554670
\(342\) 0 0
\(343\) 0 0
\(344\) 5.65685 0.304997
\(345\) 0 0
\(346\) −1.75736 −0.0944762
\(347\) 1.07107 0.0574979 0.0287490 0.999587i \(-0.490848\pi\)
0.0287490 + 0.999587i \(0.490848\pi\)
\(348\) 0 0
\(349\) −22.9706 −1.22959 −0.614793 0.788688i \(-0.710760\pi\)
−0.614793 + 0.788688i \(0.710760\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 36.2132 1.92743 0.963717 0.266925i \(-0.0860078\pi\)
0.963717 + 0.266925i \(0.0860078\pi\)
\(354\) 0 0
\(355\) 8.82843 0.468564
\(356\) 0 0
\(357\) 0 0
\(358\) 3.51472 0.185759
\(359\) −11.3137 −0.597115 −0.298557 0.954392i \(-0.596505\pi\)
−0.298557 + 0.954392i \(0.596505\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 9.51472 0.500083
\(363\) 0 0
\(364\) 0 0
\(365\) −8.48528 −0.444140
\(366\) 0 0
\(367\) −29.8701 −1.55920 −0.779602 0.626275i \(-0.784579\pi\)
−0.779602 + 0.626275i \(0.784579\pi\)
\(368\) 18.3431 0.956203
\(369\) 0 0
\(370\) 8.82843 0.458968
\(371\) 0 0
\(372\) 0 0
\(373\) 0.485281 0.0251269 0.0125635 0.999921i \(-0.496001\pi\)
0.0125635 + 0.999921i \(0.496001\pi\)
\(374\) 43.2132 2.23450
\(375\) 0 0
\(376\) 3.51472 0.181258
\(377\) 4.21320 0.216991
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −28.2426 −1.44502
\(383\) −12.4853 −0.637968 −0.318984 0.947760i \(-0.603342\pi\)
−0.318984 + 0.947760i \(0.603342\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 22.6274 1.15171
\(387\) 0 0
\(388\) 0 0
\(389\) 35.1421 1.78178 0.890889 0.454222i \(-0.150083\pi\)
0.890889 + 0.454222i \(0.150083\pi\)
\(390\) 0 0
\(391\) 24.0416 1.21584
\(392\) 0 0
\(393\) 0 0
\(394\) 14.9706 0.754206
\(395\) −15.4853 −0.779149
\(396\) 0 0
\(397\) −4.41421 −0.221543 −0.110772 0.993846i \(-0.535332\pi\)
−0.110772 + 0.993846i \(0.535332\pi\)
\(398\) −6.48528 −0.325078
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) −6.17157 −0.308194 −0.154097 0.988056i \(-0.549247\pi\)
−0.154097 + 0.988056i \(0.549247\pi\)
\(402\) 0 0
\(403\) 2.78680 0.138820
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −36.3848 −1.80353
\(408\) 0 0
\(409\) 14.4853 0.716251 0.358126 0.933673i \(-0.383416\pi\)
0.358126 + 0.933673i \(0.383416\pi\)
\(410\) −3.17157 −0.156633
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 49.4558 2.41896
\(419\) −6.72792 −0.328681 −0.164340 0.986404i \(-0.552550\pi\)
−0.164340 + 0.986404i \(0.552550\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) −12.7279 −0.619586
\(423\) 0 0
\(424\) 12.0000 0.582772
\(425\) −5.24264 −0.254305
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) −2.82843 −0.136399
\(431\) −10.7990 −0.520169 −0.260085 0.965586i \(-0.583750\pi\)
−0.260085 + 0.965586i \(0.583750\pi\)
\(432\) 0 0
\(433\) 22.9706 1.10389 0.551947 0.833879i \(-0.313885\pi\)
0.551947 + 0.833879i \(0.313885\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 27.5147 1.31621
\(438\) 0 0
\(439\) −6.38478 −0.304729 −0.152364 0.988324i \(-0.548689\pi\)
−0.152364 + 0.988324i \(0.548689\pi\)
\(440\) 16.4853 0.785905
\(441\) 0 0
\(442\) −11.7574 −0.559241
\(443\) −20.8284 −0.989588 −0.494794 0.869010i \(-0.664757\pi\)
−0.494794 + 0.869010i \(0.664757\pi\)
\(444\) 0 0
\(445\) −8.00000 −0.379236
\(446\) 25.7574 1.21965
\(447\) 0 0
\(448\) 0 0
\(449\) 29.8284 1.40769 0.703845 0.710353i \(-0.251465\pi\)
0.703845 + 0.710353i \(0.251465\pi\)
\(450\) 0 0
\(451\) 13.0711 0.615493
\(452\) 0 0
\(453\) 0 0
\(454\) 13.7574 0.645665
\(455\) 0 0
\(456\) 0 0
\(457\) −20.2426 −0.946911 −0.473455 0.880818i \(-0.656993\pi\)
−0.473455 + 0.880818i \(0.656993\pi\)
\(458\) −42.4853 −1.98521
\(459\) 0 0
\(460\) 0 0
\(461\) 36.9706 1.72189 0.860945 0.508697i \(-0.169873\pi\)
0.860945 + 0.508697i \(0.169873\pi\)
\(462\) 0 0
\(463\) 29.4558 1.36893 0.684465 0.729046i \(-0.260036\pi\)
0.684465 + 0.729046i \(0.260036\pi\)
\(464\) 10.6274 0.493365
\(465\) 0 0
\(466\) −20.9706 −0.971443
\(467\) 19.7279 0.912899 0.456450 0.889749i \(-0.349121\pi\)
0.456450 + 0.889749i \(0.349121\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −1.75736 −0.0810609
\(471\) 0 0
\(472\) 17.6569 0.812723
\(473\) 11.6569 0.535983
\(474\) 0 0
\(475\) −6.00000 −0.275299
\(476\) 0 0
\(477\) 0 0
\(478\) −0.727922 −0.0332944
\(479\) 20.2426 0.924910 0.462455 0.886643i \(-0.346969\pi\)
0.462455 + 0.886643i \(0.346969\pi\)
\(480\) 0 0
\(481\) 9.89949 0.451378
\(482\) 34.9706 1.59287
\(483\) 0 0
\(484\) 0 0
\(485\) −4.75736 −0.216021
\(486\) 0 0
\(487\) −31.6985 −1.43640 −0.718198 0.695839i \(-0.755033\pi\)
−0.718198 + 0.695839i \(0.755033\pi\)
\(488\) −8.00000 −0.362143
\(489\) 0 0
\(490\) 0 0
\(491\) −19.2843 −0.870287 −0.435143 0.900361i \(-0.643302\pi\)
−0.435143 + 0.900361i \(0.643302\pi\)
\(492\) 0 0
\(493\) 13.9289 0.627328
\(494\) −13.4558 −0.605407
\(495\) 0 0
\(496\) 7.02944 0.315631
\(497\) 0 0
\(498\) 0 0
\(499\) 3.00000 0.134298 0.0671492 0.997743i \(-0.478610\pi\)
0.0671492 + 0.997743i \(0.478610\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 35.6569 1.59144
\(503\) −32.7574 −1.46058 −0.730289 0.683138i \(-0.760615\pi\)
−0.730289 + 0.683138i \(0.760615\pi\)
\(504\) 0 0
\(505\) −14.4853 −0.644587
\(506\) 37.7990 1.68037
\(507\) 0 0
\(508\) 0 0
\(509\) 17.2132 0.762962 0.381481 0.924377i \(-0.375414\pi\)
0.381481 + 0.924377i \(0.375414\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274 1.00000
\(513\) 0 0
\(514\) 37.4558 1.65211
\(515\) −10.7574 −0.474026
\(516\) 0 0
\(517\) 7.24264 0.318531
\(518\) 0 0
\(519\) 0 0
\(520\) −4.48528 −0.196693
\(521\) −18.9706 −0.831115 −0.415558 0.909567i \(-0.636414\pi\)
−0.415558 + 0.909567i \(0.636414\pi\)
\(522\) 0 0
\(523\) −15.5147 −0.678411 −0.339206 0.940712i \(-0.610158\pi\)
−0.339206 + 0.940712i \(0.610158\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 40.4853 1.76524
\(527\) 9.21320 0.401333
\(528\) 0 0
\(529\) −1.97056 −0.0856766
\(530\) −6.00000 −0.260623
\(531\) 0 0
\(532\) 0 0
\(533\) −3.55635 −0.154043
\(534\) 0 0
\(535\) −14.4853 −0.626253
\(536\) 0.686292 0.0296433
\(537\) 0 0
\(538\) 10.9706 0.472975
\(539\) 0 0
\(540\) 0 0
\(541\) 11.9706 0.514655 0.257327 0.966324i \(-0.417158\pi\)
0.257327 + 0.966324i \(0.417158\pi\)
\(542\) −32.9706 −1.41621
\(543\) 0 0
\(544\) 0 0
\(545\) 5.00000 0.214176
\(546\) 0 0
\(547\) −7.51472 −0.321306 −0.160653 0.987011i \(-0.551360\pi\)
−0.160653 + 0.987011i \(0.551360\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −8.24264 −0.351467
\(551\) 15.9411 0.679115
\(552\) 0 0
\(553\) 0 0
\(554\) 38.4853 1.63508
\(555\) 0 0
\(556\) 0 0
\(557\) −31.7990 −1.34737 −0.673683 0.739020i \(-0.735289\pi\)
−0.673683 + 0.739020i \(0.735289\pi\)
\(558\) 0 0
\(559\) −3.17157 −0.134143
\(560\) 0 0
\(561\) 0 0
\(562\) −28.7279 −1.21181
\(563\) 35.9411 1.51474 0.757369 0.652987i \(-0.226484\pi\)
0.757369 + 0.652987i \(0.226484\pi\)
\(564\) 0 0
\(565\) −1.07107 −0.0450602
\(566\) −9.27208 −0.389735
\(567\) 0 0
\(568\) 24.9706 1.04774
\(569\) 2.14214 0.0898030 0.0449015 0.998991i \(-0.485703\pi\)
0.0449015 + 0.998991i \(0.485703\pi\)
\(570\) 0 0
\(571\) −34.4853 −1.44316 −0.721582 0.692329i \(-0.756585\pi\)
−0.721582 + 0.692329i \(0.756585\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.58579 −0.191241
\(576\) 0 0
\(577\) 9.72792 0.404979 0.202489 0.979284i \(-0.435097\pi\)
0.202489 + 0.979284i \(0.435097\pi\)
\(578\) −14.8284 −0.616781
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 24.7279 1.02413
\(584\) −24.0000 −0.993127
\(585\) 0 0
\(586\) −0.384776 −0.0158950
\(587\) 13.4558 0.555382 0.277691 0.960670i \(-0.410431\pi\)
0.277691 + 0.960670i \(0.410431\pi\)
\(588\) 0 0
\(589\) 10.5442 0.434464
\(590\) −8.82843 −0.363461
\(591\) 0 0
\(592\) 24.9706 1.02628
\(593\) 10.7574 0.441752 0.220876 0.975302i \(-0.429108\pi\)
0.220876 + 0.975302i \(0.429108\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) −10.2843 −0.420555
\(599\) 12.1716 0.497317 0.248658 0.968591i \(-0.420010\pi\)
0.248658 + 0.968591i \(0.420010\pi\)
\(600\) 0 0
\(601\) −22.9706 −0.936989 −0.468494 0.883466i \(-0.655203\pi\)
−0.468494 + 0.883466i \(0.655203\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 22.9706 0.933886
\(606\) 0 0
\(607\) 24.8995 1.01064 0.505320 0.862932i \(-0.331375\pi\)
0.505320 + 0.862932i \(0.331375\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 4.00000 0.161955
\(611\) −1.97056 −0.0797204
\(612\) 0 0
\(613\) 43.9411 1.77477 0.887383 0.461034i \(-0.152521\pi\)
0.887383 + 0.461034i \(0.152521\pi\)
\(614\) 15.6985 0.633539
\(615\) 0 0
\(616\) 0 0
\(617\) −7.41421 −0.298485 −0.149242 0.988801i \(-0.547684\pi\)
−0.149242 + 0.988801i \(0.547684\pi\)
\(618\) 0 0
\(619\) −31.0711 −1.24885 −0.624426 0.781084i \(-0.714667\pi\)
−0.624426 + 0.781084i \(0.714667\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 14.1421 0.567048
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 34.2426 1.36861
\(627\) 0 0
\(628\) 0 0
\(629\) 32.7279 1.30495
\(630\) 0 0
\(631\) 8.45584 0.336622 0.168311 0.985734i \(-0.446169\pi\)
0.168311 + 0.985734i \(0.446169\pi\)
\(632\) −43.7990 −1.74223
\(633\) 0 0
\(634\) −16.4853 −0.654714
\(635\) −0.242641 −0.00962890
\(636\) 0 0
\(637\) 0 0
\(638\) 21.8995 0.867009
\(639\) 0 0
\(640\) −11.3137 −0.447214
\(641\) 0.686292 0.0271069 0.0135534 0.999908i \(-0.495686\pi\)
0.0135534 + 0.999908i \(0.495686\pi\)
\(642\) 0 0
\(643\) −27.7279 −1.09348 −0.546741 0.837302i \(-0.684132\pi\)
−0.546741 + 0.837302i \(0.684132\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −44.4853 −1.75025
\(647\) −28.4853 −1.11987 −0.559936 0.828536i \(-0.689174\pi\)
−0.559936 + 0.828536i \(0.689174\pi\)
\(648\) 0 0
\(649\) 36.3848 1.42823
\(650\) 2.24264 0.0879636
\(651\) 0 0
\(652\) 0 0
\(653\) −1.02944 −0.0402850 −0.0201425 0.999797i \(-0.506412\pi\)
−0.0201425 + 0.999797i \(0.506412\pi\)
\(654\) 0 0
\(655\) 3.75736 0.146812
\(656\) −8.97056 −0.350242
\(657\) 0 0
\(658\) 0 0
\(659\) 13.9706 0.544216 0.272108 0.962267i \(-0.412279\pi\)
0.272108 + 0.962267i \(0.412279\pi\)
\(660\) 0 0
\(661\) 37.4558 1.45686 0.728432 0.685118i \(-0.240250\pi\)
0.728432 + 0.685118i \(0.240250\pi\)
\(662\) −33.1716 −1.28925
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.1838 0.471757
\(668\) 0 0
\(669\) 0 0
\(670\) −0.343146 −0.0132569
\(671\) −16.4853 −0.636407
\(672\) 0 0
\(673\) 20.4853 0.789650 0.394825 0.918756i \(-0.370805\pi\)
0.394825 + 0.918756i \(0.370805\pi\)
\(674\) −19.4558 −0.749411
\(675\) 0 0
\(676\) 0 0
\(677\) 1.78680 0.0686722 0.0343361 0.999410i \(-0.489068\pi\)
0.0343361 + 0.999410i \(0.489068\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −14.8284 −0.568644
\(681\) 0 0
\(682\) 14.4853 0.554670
\(683\) 7.79899 0.298420 0.149210 0.988806i \(-0.452327\pi\)
0.149210 + 0.988806i \(0.452327\pi\)
\(684\) 0 0
\(685\) −12.0000 −0.458496
\(686\) 0 0
\(687\) 0 0
\(688\) −8.00000 −0.304997
\(689\) −6.72792 −0.256313
\(690\) 0 0
\(691\) −9.17157 −0.348903 −0.174452 0.984666i \(-0.555815\pi\)
−0.174452 + 0.984666i \(0.555815\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −1.51472 −0.0574979
\(695\) −16.2426 −0.616118
\(696\) 0 0
\(697\) −11.7574 −0.445342
\(698\) 32.4853 1.22959
\(699\) 0 0
\(700\) 0 0
\(701\) 4.45584 0.168295 0.0841475 0.996453i \(-0.473183\pi\)
0.0841475 + 0.996453i \(0.473183\pi\)
\(702\) 0 0
\(703\) 37.4558 1.41267
\(704\) 46.6274 1.75734
\(705\) 0 0
\(706\) −51.2132 −1.92743
\(707\) 0 0
\(708\) 0 0
\(709\) 17.0000 0.638448 0.319224 0.947679i \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) −12.4853 −0.468564
\(711\) 0 0
\(712\) −22.6274 −0.847998
\(713\) 8.05887 0.301807
\(714\) 0 0
\(715\) −9.24264 −0.345655
\(716\) 0 0
\(717\) 0 0
\(718\) 16.0000 0.597115
\(719\) 17.2132 0.641944 0.320972 0.947089i \(-0.395990\pi\)
0.320972 + 0.947089i \(0.395990\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −24.0416 −0.894737
\(723\) 0 0
\(724\) 0 0
\(725\) −2.65685 −0.0986731
\(726\) 0 0
\(727\) 29.3137 1.08719 0.543593 0.839349i \(-0.317064\pi\)
0.543593 + 0.839349i \(0.317064\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 12.0000 0.444140
\(731\) −10.4853 −0.387812
\(732\) 0 0
\(733\) 44.6985 1.65098 0.825488 0.564420i \(-0.190900\pi\)
0.825488 + 0.564420i \(0.190900\pi\)
\(734\) 42.2426 1.55920
\(735\) 0 0
\(736\) 0 0
\(737\) 1.41421 0.0520932
\(738\) 0 0
\(739\) −3.97056 −0.146060 −0.0730298 0.997330i \(-0.523267\pi\)
−0.0730298 + 0.997330i \(0.523267\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −36.7279 −1.34742 −0.673708 0.738997i \(-0.735300\pi\)
−0.673708 + 0.738997i \(0.735300\pi\)
\(744\) 0 0
\(745\) 14.8284 0.543272
\(746\) −0.686292 −0.0251269
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 12.5147 0.456669 0.228334 0.973583i \(-0.426672\pi\)
0.228334 + 0.973583i \(0.426672\pi\)
\(752\) −4.97056 −0.181258
\(753\) 0 0
\(754\) −5.95837 −0.216991
\(755\) 9.48528 0.345205
\(756\) 0 0
\(757\) −16.4853 −0.599168 −0.299584 0.954070i \(-0.596848\pi\)
−0.299584 + 0.954070i \(0.596848\pi\)
\(758\) −2.82843 −0.102733
\(759\) 0 0
\(760\) −16.9706 −0.615587
\(761\) 12.7279 0.461387 0.230693 0.973026i \(-0.425901\pi\)
0.230693 + 0.973026i \(0.425901\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 17.6569 0.637968
\(767\) −9.89949 −0.357450
\(768\) 0 0
\(769\) 3.17157 0.114370 0.0571849 0.998364i \(-0.481788\pi\)
0.0571849 + 0.998364i \(0.481788\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 36.2132 1.30250 0.651249 0.758864i \(-0.274245\pi\)
0.651249 + 0.758864i \(0.274245\pi\)
\(774\) 0 0
\(775\) −1.75736 −0.0631262
\(776\) −13.4558 −0.483037
\(777\) 0 0
\(778\) −49.6985 −1.78178
\(779\) −13.4558 −0.482106
\(780\) 0 0
\(781\) 51.4558 1.84123
\(782\) −34.0000 −1.21584
\(783\) 0 0
\(784\) 0 0
\(785\) 20.8284 0.743398
\(786\) 0 0
\(787\) −45.7279 −1.63002 −0.815012 0.579444i \(-0.803270\pi\)
−0.815012 + 0.579444i \(0.803270\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 21.8995 0.779149
\(791\) 0 0
\(792\) 0 0
\(793\) 4.48528 0.159277
\(794\) 6.24264 0.221543
\(795\) 0 0
\(796\) 0 0
\(797\) 55.1838 1.95471 0.977355 0.211608i \(-0.0678700\pi\)
0.977355 + 0.211608i \(0.0678700\pi\)
\(798\) 0 0
\(799\) −6.51472 −0.230474
\(800\) 0 0
\(801\) 0 0
\(802\) 8.72792 0.308194
\(803\) −49.4558 −1.74526
\(804\) 0 0
\(805\) 0 0
\(806\) −3.94113 −0.138820
\(807\) 0 0
\(808\) −40.9706 −1.44134
\(809\) −30.5980 −1.07577 −0.537884 0.843019i \(-0.680776\pi\)
−0.537884 + 0.843019i \(0.680776\pi\)
\(810\) 0 0
\(811\) 23.3553 0.820117 0.410058 0.912059i \(-0.365508\pi\)
0.410058 + 0.912059i \(0.365508\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 51.4558 1.80353
\(815\) −1.75736 −0.0615576
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) −20.4853 −0.716251
\(819\) 0 0
\(820\) 0 0
\(821\) 6.51472 0.227365 0.113683 0.993517i \(-0.463735\pi\)
0.113683 + 0.993517i \(0.463735\pi\)
\(822\) 0 0
\(823\) −8.72792 −0.304236 −0.152118 0.988362i \(-0.548609\pi\)
−0.152118 + 0.988362i \(0.548609\pi\)
\(824\) −30.4264 −1.05995
\(825\) 0 0
\(826\) 0 0
\(827\) 6.04163 0.210088 0.105044 0.994468i \(-0.466502\pi\)
0.105044 + 0.994468i \(0.466502\pi\)
\(828\) 0 0
\(829\) −54.0416 −1.87694 −0.938472 0.345356i \(-0.887758\pi\)
−0.938472 + 0.345356i \(0.887758\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −12.6863 −0.439818
\(833\) 0 0
\(834\) 0 0
\(835\) 9.24264 0.319855
\(836\) 0 0
\(837\) 0 0
\(838\) 9.51472 0.328681
\(839\) −48.7279 −1.68227 −0.841137 0.540822i \(-0.818113\pi\)
−0.841137 + 0.540822i \(0.818113\pi\)
\(840\) 0 0
\(841\) −21.9411 −0.756591
\(842\) 26.8701 0.926003
\(843\) 0 0
\(844\) 0 0
\(845\) −10.4853 −0.360705
\(846\) 0 0
\(847\) 0 0
\(848\) −16.9706 −0.582772
\(849\) 0 0
\(850\) 7.41421 0.254305
\(851\) 28.6274 0.981335
\(852\) 0 0
\(853\) 22.9706 0.786497 0.393249 0.919432i \(-0.371351\pi\)
0.393249 + 0.919432i \(0.371351\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −40.9706 −1.40035
\(857\) −5.51472 −0.188379 −0.0941896 0.995554i \(-0.530026\pi\)
−0.0941896 + 0.995554i \(0.530026\pi\)
\(858\) 0 0
\(859\) −48.7696 −1.66400 −0.831998 0.554779i \(-0.812803\pi\)
−0.831998 + 0.554779i \(0.812803\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 15.2721 0.520169
\(863\) 18.3848 0.625825 0.312913 0.949782i \(-0.398695\pi\)
0.312913 + 0.949782i \(0.398695\pi\)
\(864\) 0 0
\(865\) 1.24264 0.0422511
\(866\) −32.4853 −1.10389
\(867\) 0 0
\(868\) 0 0
\(869\) −90.2548 −3.06169
\(870\) 0 0
\(871\) −0.384776 −0.0130376
\(872\) 14.1421 0.478913
\(873\) 0 0
\(874\) −38.9117 −1.31621
\(875\) 0 0
\(876\) 0 0
\(877\) 46.9706 1.58608 0.793042 0.609167i \(-0.208496\pi\)
0.793042 + 0.609167i \(0.208496\pi\)
\(878\) 9.02944 0.304729
\(879\) 0 0
\(880\) −23.3137 −0.785905
\(881\) −19.0294 −0.641118 −0.320559 0.947229i \(-0.603871\pi\)
−0.320559 + 0.947229i \(0.603871\pi\)
\(882\) 0 0
\(883\) 48.4853 1.63166 0.815830 0.578292i \(-0.196281\pi\)
0.815830 + 0.578292i \(0.196281\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 29.4558 0.989588
\(887\) −30.9706 −1.03989 −0.519945 0.854200i \(-0.674048\pi\)
−0.519945 + 0.854200i \(0.674048\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 11.3137 0.379236
\(891\) 0 0
\(892\) 0 0
\(893\) −7.45584 −0.249500
\(894\) 0 0
\(895\) −2.48528 −0.0830738
\(896\) 0 0
\(897\) 0 0
\(898\) −42.1838 −1.40769
\(899\) 4.66905 0.155721
\(900\) 0 0
\(901\) −22.2426 −0.741010
\(902\) −18.4853 −0.615493
\(903\) 0 0
\(904\) −3.02944 −0.100758
\(905\) −6.72792 −0.223644
\(906\) 0 0
\(907\) 32.1838 1.06864 0.534322 0.845281i \(-0.320567\pi\)
0.534322 + 0.845281i \(0.320567\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.65685 0.187420 0.0937100 0.995600i \(-0.470127\pi\)
0.0937100 + 0.995600i \(0.470127\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 28.6274 0.946911
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −38.4558 −1.26854 −0.634271 0.773111i \(-0.718699\pi\)
−0.634271 + 0.773111i \(0.718699\pi\)
\(920\) −12.9706 −0.427627
\(921\) 0 0
\(922\) −52.2843 −1.72189
\(923\) −14.0000 −0.460816
\(924\) 0 0
\(925\) −6.24264 −0.205257
\(926\) −41.6569 −1.36893
\(927\) 0 0
\(928\) 0 0
\(929\) 54.7279 1.79556 0.897782 0.440439i \(-0.145177\pi\)
0.897782 + 0.440439i \(0.145177\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −27.8995 −0.912899
\(935\) −30.5563 −0.999299
\(936\) 0 0
\(937\) 17.4437 0.569859 0.284930 0.958548i \(-0.408030\pi\)
0.284930 + 0.958548i \(0.408030\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −4.97056 −0.162036 −0.0810179 0.996713i \(-0.525817\pi\)
−0.0810179 + 0.996713i \(0.525817\pi\)
\(942\) 0 0
\(943\) −10.2843 −0.334902
\(944\) −24.9706 −0.812723
\(945\) 0 0
\(946\) −16.4853 −0.535983
\(947\) 43.7574 1.42192 0.710962 0.703231i \(-0.248260\pi\)
0.710962 + 0.703231i \(0.248260\pi\)
\(948\) 0 0
\(949\) 13.4558 0.436795
\(950\) 8.48528 0.275299
\(951\) 0 0
\(952\) 0 0
\(953\) 29.0122 0.939797 0.469899 0.882720i \(-0.344290\pi\)
0.469899 + 0.882720i \(0.344290\pi\)
\(954\) 0 0
\(955\) 19.9706 0.646232
\(956\) 0 0
\(957\) 0 0
\(958\) −28.6274 −0.924910
\(959\) 0 0
\(960\) 0 0
\(961\) −27.9117 −0.900377
\(962\) −14.0000 −0.451378
\(963\) 0 0
\(964\) 0 0
\(965\) −16.0000 −0.515058
\(966\) 0 0
\(967\) 24.4264 0.785500 0.392750 0.919645i \(-0.371524\pi\)
0.392750 + 0.919645i \(0.371524\pi\)
\(968\) 64.9706 2.08823
\(969\) 0 0
\(970\) 6.72792 0.216021
\(971\) −42.7279 −1.37120 −0.685602 0.727976i \(-0.740461\pi\)
−0.685602 + 0.727976i \(0.740461\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 44.8284 1.43640
\(975\) 0 0
\(976\) 11.3137 0.362143
\(977\) 30.7696 0.984405 0.492203 0.870481i \(-0.336192\pi\)
0.492203 + 0.870481i \(0.336192\pi\)
\(978\) 0 0
\(979\) −46.6274 −1.49022
\(980\) 0 0
\(981\) 0 0
\(982\) 27.2721 0.870287
\(983\) −42.2132 −1.34639 −0.673196 0.739464i \(-0.735079\pi\)
−0.673196 + 0.739464i \(0.735079\pi\)
\(984\) 0 0
\(985\) −10.5858 −0.337291
\(986\) −19.6985 −0.627328
\(987\) 0 0
\(988\) 0 0
\(989\) −9.17157 −0.291639
\(990\) 0 0
\(991\) −47.9411 −1.52290 −0.761450 0.648224i \(-0.775512\pi\)
−0.761450 + 0.648224i \(0.775512\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 4.58579 0.145379
\(996\) 0 0
\(997\) −3.72792 −0.118064 −0.0590322 0.998256i \(-0.518801\pi\)
−0.0590322 + 0.998256i \(0.518801\pi\)
\(998\) −4.24264 −0.134298
\(999\) 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 2205.2.a.v.1.1 2
3.2 odd 2 245.2.a.e.1.2 2
7.6 odd 2 2205.2.a.t.1.1 2
12.11 even 2 3920.2.a.bw.1.2 2
15.2 even 4 1225.2.b.i.99.4 4
15.8 even 4 1225.2.b.i.99.1 4
15.14 odd 2 1225.2.a.r.1.1 2
21.2 odd 6 245.2.e.g.116.1 4
21.5 even 6 245.2.e.f.116.1 4
21.11 odd 6 245.2.e.g.226.1 4
21.17 even 6 245.2.e.f.226.1 4
21.20 even 2 245.2.a.f.1.2 yes 2
84.83 odd 2 3920.2.a.br.1.1 2
105.62 odd 4 1225.2.b.j.99.3 4
105.83 odd 4 1225.2.b.j.99.2 4
105.104 even 2 1225.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.a.e.1.2 2 3.2 odd 2
245.2.a.f.1.2 yes 2 21.20 even 2
245.2.e.f.116.1 4 21.5 even 6
245.2.e.f.226.1 4 21.17 even 6
245.2.e.g.116.1 4 21.2 odd 6
245.2.e.g.226.1 4 21.11 odd 6
1225.2.a.p.1.1 2 105.104 even 2
1225.2.a.r.1.1 2 15.14 odd 2
1225.2.b.i.99.1 4 15.8 even 4
1225.2.b.i.99.4 4 15.2 even 4
1225.2.b.j.99.2 4 105.83 odd 4
1225.2.b.j.99.3 4 105.62 odd 4
2205.2.a.t.1.1 2 7.6 odd 2
2205.2.a.v.1.1 2 1.1 even 1 trivial
3920.2.a.br.1.1 2 84.83 odd 2
3920.2.a.bw.1.2 2 12.11 even 2