Properties

Label 3920.2.a.bw.1.2
Level $3920$
Weight $2$
Character 3920.1
Self dual yes
Analytic conductor $31.301$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3920 = 2^{4} \cdot 5 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3920.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(31.3013575923\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
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 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 3920.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421 q^{3} -1.00000 q^{5} +2.82843 q^{9} +O(q^{10})\) \(q+2.41421 q^{3} -1.00000 q^{5} +2.82843 q^{9} +5.82843 q^{11} -1.58579 q^{13} -2.41421 q^{15} +5.24264 q^{17} +6.00000 q^{19} -4.58579 q^{23} +1.00000 q^{25} -0.414214 q^{27} +2.65685 q^{29} +1.75736 q^{31} +14.0711 q^{33} -6.24264 q^{37} -3.82843 q^{39} -2.24264 q^{41} -2.00000 q^{43} -2.82843 q^{45} +1.24264 q^{47} +12.6569 q^{51} -4.24264 q^{53} -5.82843 q^{55} +14.4853 q^{57} +6.24264 q^{59} -2.82843 q^{61} +1.58579 q^{65} -0.242641 q^{67} -11.0711 q^{69} +8.82843 q^{71} -8.48528 q^{73} +2.41421 q^{75} +15.4853 q^{79} -9.48528 q^{81} -5.24264 q^{85} +6.41421 q^{87} +8.00000 q^{89} +4.24264 q^{93} -6.00000 q^{95} -4.75736 q^{97} +16.4853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{3} - 2 q^{5} + O(q^{10}) \) \( 2 q + 2 q^{3} - 2 q^{5} + 6 q^{11} - 6 q^{13} - 2 q^{15} + 2 q^{17} + 12 q^{19} - 12 q^{23} + 2 q^{25} + 2 q^{27} - 6 q^{29} + 12 q^{31} + 14 q^{33} - 4 q^{37} - 2 q^{39} + 4 q^{41} - 4 q^{43} - 6 q^{47} + 14 q^{51} - 6 q^{55} + 12 q^{57} + 4 q^{59} + 6 q^{65} + 8 q^{67} - 8 q^{69} + 12 q^{71} + 2 q^{75} + 14 q^{79} - 2 q^{81} - 2 q^{85} + 10 q^{87} + 16 q^{89} - 12 q^{95} - 18 q^{97} + 16 q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.41421 1.39385 0.696923 0.717146i \(-0.254552\pi\)
0.696923 + 0.717146i \(0.254552\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 2.82843 0.942809
\(10\) 0 0
\(11\) 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
\(15\) −2.41421 −0.623347
\(16\) 0 0
\(17\) 5.24264 1.27153 0.635764 0.771884i \(-0.280685\pi\)
0.635764 + 0.771884i \(0.280685\pi\)
\(18\) 0 0
\(19\) 6.00000 1.37649 0.688247 0.725476i \(-0.258380\pi\)
0.688247 + 0.725476i \(0.258380\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −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
\(26\) 0 0
\(27\) −0.414214 −0.0797154
\(28\) 0 0
\(29\) 2.65685 0.493365 0.246683 0.969096i \(-0.420659\pi\)
0.246683 + 0.969096i \(0.420659\pi\)
\(30\) 0 0
\(31\) 1.75736 0.315631 0.157816 0.987469i \(-0.449555\pi\)
0.157816 + 0.987469i \(0.449555\pi\)
\(32\) 0 0
\(33\) 14.0711 2.44946
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −6.24264 −1.02628 −0.513142 0.858304i \(-0.671519\pi\)
−0.513142 + 0.858304i \(0.671519\pi\)
\(38\) 0 0
\(39\) −3.82843 −0.613039
\(40\) 0 0
\(41\) −2.24264 −0.350242 −0.175121 0.984547i \(-0.556032\pi\)
−0.175121 + 0.984547i \(0.556032\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 0 0
\(45\) −2.82843 −0.421637
\(46\) 0 0
\(47\) 1.24264 0.181258 0.0906289 0.995885i \(-0.471112\pi\)
0.0906289 + 0.995885i \(0.471112\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 12.6569 1.77231
\(52\) 0 0
\(53\) −4.24264 −0.582772 −0.291386 0.956606i \(-0.594116\pi\)
−0.291386 + 0.956606i \(0.594116\pi\)
\(54\) 0 0
\(55\) −5.82843 −0.785905
\(56\) 0 0
\(57\) 14.4853 1.91862
\(58\) 0 0
\(59\) 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 \(0.557957\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.58579 0.196693
\(66\) 0 0
\(67\) −0.242641 −0.0296433 −0.0148216 0.999890i \(-0.504718\pi\)
−0.0148216 + 0.999890i \(0.504718\pi\)
\(68\) 0 0
\(69\) −11.0711 −1.33280
\(70\) 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 \(0.665405\pi\)
\(74\) 0 0
\(75\) 2.41421 0.278769
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 15.4853 1.74223 0.871115 0.491079i \(-0.163397\pi\)
0.871115 + 0.491079i \(0.163397\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −5.24264 −0.568644
\(86\) 0 0
\(87\) 6.41421 0.687676
\(88\) 0 0
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 4.24264 0.439941
\(94\) 0 0
\(95\) −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
\(99\) 16.4853 1.65683
\(100\) 0 0
\(101\) 14.4853 1.44134 0.720670 0.693279i \(-0.243834\pi\)
0.720670 + 0.693279i \(0.243834\pi\)
\(102\) 0 0
\(103\) 10.7574 1.05995 0.529977 0.848012i \(-0.322201\pi\)
0.529977 + 0.848012i \(0.322201\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −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 \(0.423031\pi\)
\(110\) 0 0
\(111\) −15.0711 −1.43048
\(112\) 0 0
\(113\) 1.07107 0.100758 0.0503788 0.998730i \(-0.483957\pi\)
0.0503788 + 0.998730i \(0.483957\pi\)
\(114\) 0 0
\(115\) 4.58579 0.427627
\(116\) 0 0
\(117\) −4.48528 −0.414664
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 22.9706 2.08823
\(122\) 0 0
\(123\) −5.41421 −0.488183
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0.242641 0.0215309 0.0107654 0.999942i \(-0.496573\pi\)
0.0107654 + 0.999942i \(0.496573\pi\)
\(128\) 0 0
\(129\) −4.82843 −0.425119
\(130\) 0 0
\(131\) 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 0
\(135\) 0.414214 0.0356498
\(136\) 0 0
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) 16.2426 1.37768 0.688841 0.724912i \(-0.258120\pi\)
0.688841 + 0.724912i \(0.258120\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) 0 0
\(143\) −9.24264 −0.772908
\(144\) 0 0
\(145\) −2.65685 −0.220640
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −14.8284 −1.21479 −0.607396 0.794399i \(-0.707786\pi\)
−0.607396 + 0.794399i \(0.707786\pi\)
\(150\) 0 0
\(151\) −9.48528 −0.771901 −0.385951 0.922519i \(-0.626126\pi\)
−0.385951 + 0.922519i \(0.626126\pi\)
\(152\) 0 0
\(153\) 14.8284 1.19881
\(154\) 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 \(0.187686\pi\)
\(158\) 0 0
\(159\) −10.2426 −0.812294
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.75736 0.137647 0.0688235 0.997629i \(-0.478075\pi\)
0.0688235 + 0.997629i \(0.478075\pi\)
\(164\) 0 0
\(165\) −14.0711 −1.09543
\(166\) 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
\(170\) 0 0
\(171\) 16.9706 1.29777
\(172\) 0 0
\(173\) −1.24264 −0.0944762 −0.0472381 0.998884i \(-0.515042\pi\)
−0.0472381 + 0.998884i \(0.515042\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 15.0711 1.13281
\(178\) 0 0
\(179\) −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
\(183\) −6.82843 −0.504772
\(184\) 0 0
\(185\) 6.24264 0.458968
\(186\) 0 0
\(187\) 30.5563 2.23450
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 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 \(0.695330\pi\)
\(194\) 0 0
\(195\) 3.82843 0.274159
\(196\) 0 0
\(197\) 10.5858 0.754206 0.377103 0.926171i \(-0.376920\pi\)
0.377103 + 0.926171i \(0.376920\pi\)
\(198\) 0 0
\(199\) −4.58579 −0.325078 −0.162539 0.986702i \(-0.551968\pi\)
−0.162539 + 0.986702i \(0.551968\pi\)
\(200\) 0 0
\(201\) −0.585786 −0.0413182
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2.24264 0.156633
\(206\) 0 0
\(207\) −12.9706 −0.901516
\(208\) 0 0
\(209\) 34.9706 2.41896
\(210\) 0 0
\(211\) −9.00000 −0.619586 −0.309793 0.950804i \(-0.600260\pi\)
−0.309793 + 0.950804i \(0.600260\pi\)
\(212\) 0 0
\(213\) 21.3137 1.46039
\(214\) 0 0
\(215\) 2.00000 0.136399
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −20.4853 −1.38427
\(220\) 0 0
\(221\) −8.31371 −0.559241
\(222\) 0 0
\(223\) 18.2132 1.21965 0.609823 0.792538i \(-0.291240\pi\)
0.609823 + 0.792538i \(0.291240\pi\)
\(224\) 0 0
\(225\) 2.82843 0.188562
\(226\) 0 0
\(227\) −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 \(0.0387390\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.8284 −0.971443 −0.485721 0.874114i \(-0.661443\pi\)
−0.485721 + 0.874114i \(0.661443\pi\)
\(234\) 0 0
\(235\) −1.24264 −0.0810609
\(236\) 0 0
\(237\) 37.3848 2.42840
\(238\) 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 \(0.793282\pi\)
\(242\) 0 0
\(243\) −21.6569 −1.38929
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −9.51472 −0.605407
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −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 0
\(255\) −12.6569 −0.792603
\(256\) 0 0
\(257\) 26.4853 1.65211 0.826053 0.563592i \(-0.190581\pi\)
0.826053 + 0.563592i \(0.190581\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 7.51472 0.465149
\(262\) 0 0
\(263\) −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
\(267\) 19.3137 1.18198
\(268\) 0 0
\(269\) 7.75736 0.472975 0.236487 0.971635i \(-0.424004\pi\)
0.236487 + 0.971635i \(0.424004\pi\)
\(270\) 0 0
\(271\) −23.3137 −1.41621 −0.708103 0.706109i \(-0.750449\pi\)
−0.708103 + 0.706109i \(0.750449\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.82843 0.351467
\(276\) 0 0
\(277\) −27.2132 −1.63508 −0.817541 0.575870i \(-0.804664\pi\)
−0.817541 + 0.575870i \(0.804664\pi\)
\(278\) 0 0
\(279\) 4.97056 0.297580
\(280\) 0 0
\(281\) −20.3137 −1.21181 −0.605907 0.795535i \(-0.707190\pi\)
−0.605907 + 0.795535i \(0.707190\pi\)
\(282\) 0 0
\(283\) −6.55635 −0.389735 −0.194867 0.980830i \(-0.562428\pi\)
−0.194867 + 0.980830i \(0.562428\pi\)
\(284\) 0 0
\(285\) −14.4853 −0.858034
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 10.4853 0.616781
\(290\) 0 0
\(291\) −11.4853 −0.673279
\(292\) 0 0
\(293\) −0.272078 −0.0158950 −0.00794748 0.999968i \(-0.502530\pi\)
−0.00794748 + 0.999968i \(0.502530\pi\)
\(294\) 0 0
\(295\) −6.24264 −0.363461
\(296\) 0 0
\(297\) −2.41421 −0.140087
\(298\) 0 0
\(299\) 7.27208 0.420555
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 34.9706 2.00901
\(304\) 0 0
\(305\) 2.82843 0.161955
\(306\) 0 0
\(307\) 11.1005 0.633539 0.316770 0.948503i \(-0.397402\pi\)
0.316770 + 0.948503i \(0.397402\pi\)
\(308\) 0 0
\(309\) 25.9706 1.47741
\(310\) 0 0
\(311\) −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 \(0.739895\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.6569 −0.654714 −0.327357 0.944901i \(-0.606158\pi\)
−0.327357 + 0.944901i \(0.606158\pi\)
\(318\) 0 0
\(319\) 15.4853 0.867009
\(320\) 0 0
\(321\) −34.9706 −1.95187
\(322\) 0 0
\(323\) 31.4558 1.75025
\(324\) 0 0
\(325\) −1.58579 −0.0879636
\(326\) 0 0
\(327\) 12.0711 0.667532
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −23.4558 −1.28925 −0.644625 0.764499i \(-0.722986\pi\)
−0.644625 + 0.764499i \(0.722986\pi\)
\(332\) 0 0
\(333\) −17.6569 −0.967590
\(334\) 0 0
\(335\) 0.242641 0.0132569
\(336\) 0 0
\(337\) 13.7574 0.749411 0.374706 0.927144i \(-0.377744\pi\)
0.374706 + 0.927144i \(0.377744\pi\)
\(338\) 0 0
\(339\) 2.58579 0.140441
\(340\) 0 0
\(341\) 10.2426 0.554670
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 11.0711 0.596046
\(346\) 0 0
\(347\) 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
\(351\) 0.656854 0.0350603
\(352\) 0 0
\(353\) −36.2132 −1.92743 −0.963717 0.266925i \(-0.913992\pi\)
−0.963717 + 0.266925i \(0.913992\pi\)
\(354\) 0 0
\(355\) −8.82843 −0.468564
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −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
\(362\) 0 0
\(363\) 55.4558 2.91068
\(364\) 0 0
\(365\) 8.48528 0.444140
\(366\) 0 0
\(367\) 29.8701 1.55920 0.779602 0.626275i \(-0.215421\pi\)
0.779602 + 0.626275i \(0.215421\pi\)
\(368\) 0 0
\(369\) −6.34315 −0.330211
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.485281 0.0251269 0.0125635 0.999921i \(-0.496001\pi\)
0.0125635 + 0.999921i \(0.496001\pi\)
\(374\) 0 0
\(375\) −2.41421 −0.124669
\(376\) 0 0
\(377\) −4.21320 −0.216991
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 0 0
\(381\) 0.585786 0.0300107
\(382\) 0 0
\(383\) −12.4853 −0.637968 −0.318984 0.947760i \(-0.603342\pi\)
−0.318984 + 0.947760i \(0.603342\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −5.65685 −0.287554
\(388\) 0 0
\(389\) −35.1421 −1.78178 −0.890889 0.454222i \(-0.849917\pi\)
−0.890889 + 0.454222i \(0.849917\pi\)
\(390\) 0 0
\(391\) −24.0416 −1.21584
\(392\) 0 0
\(393\) 9.07107 0.457575
\(394\) 0 0
\(395\) −15.4853 −0.779149
\(396\) 0 0
\(397\) −4.41421 −0.221543 −0.110772 0.993846i \(-0.535332\pi\)
−0.110772 + 0.993846i \(0.535332\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 6.17157 0.308194 0.154097 0.988056i \(-0.450753\pi\)
0.154097 + 0.988056i \(0.450753\pi\)
\(402\) 0 0
\(403\) −2.78680 −0.138820
\(404\) 0 0
\(405\) 9.48528 0.471327
\(406\) 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 \(0.383416\pi\)
\(410\) 0 0
\(411\) 28.9706 1.42901
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 39.2132 1.92028
\(418\) 0 0
\(419\) −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 \(0.653228\pi\)
\(422\) 0 0
\(423\) 3.51472 0.170891
\(424\) 0 0
\(425\) 5.24264 0.254305
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −22.3137 −1.07732
\(430\) 0 0
\(431\) −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
\(435\) −6.41421 −0.307538
\(436\) 0 0
\(437\) −27.5147 −1.31621
\(438\) 0 0
\(439\) 6.38478 0.304729 0.152364 0.988324i \(-0.451311\pi\)
0.152364 + 0.988324i \(0.451311\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −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
\(446\) 0 0
\(447\) −35.7990 −1.69323
\(448\) 0 0
\(449\) −29.8284 −1.40769 −0.703845 0.710353i \(-0.748535\pi\)
−0.703845 + 0.710353i \(0.748535\pi\)
\(450\) 0 0
\(451\) −13.0711 −0.615493
\(452\) 0 0
\(453\) −22.8995 −1.07591
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −20.2426 −0.946911 −0.473455 0.880818i \(-0.656993\pi\)
−0.473455 + 0.880818i \(0.656993\pi\)
\(458\) 0 0
\(459\) −2.17157 −0.101360
\(460\) 0 0
\(461\) −36.9706 −1.72189 −0.860945 0.508697i \(-0.830127\pi\)
−0.860945 + 0.508697i \(0.830127\pi\)
\(462\) 0 0
\(463\) −29.4558 −1.36893 −0.684465 0.729046i \(-0.739964\pi\)
−0.684465 + 0.729046i \(0.739964\pi\)
\(464\) 0 0
\(465\) −4.24264 −0.196748
\(466\) 0 0
\(467\) 19.7279 0.912899 0.456450 0.889749i \(-0.349121\pi\)
0.456450 + 0.889749i \(0.349121\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 50.2843 2.31698
\(472\) 0 0
\(473\) −11.6569 −0.535983
\(474\) 0 0
\(475\) 6.00000 0.275299
\(476\) 0 0
\(477\) −12.0000 −0.549442
\(478\) 0 0
\(479\) 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
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.75736 0.216021
\(486\) 0 0
\(487\) 31.6985 1.43640 0.718198 0.695839i \(-0.244967\pi\)
0.718198 + 0.695839i \(0.244967\pi\)
\(488\) 0 0
\(489\) 4.24264 0.191859
\(490\) 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
\(494\) 0 0
\(495\) −16.4853 −0.740958
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −3.00000 −0.134298 −0.0671492 0.997743i \(-0.521390\pi\)
−0.0671492 + 0.997743i \(0.521390\pi\)
\(500\) 0 0
\(501\) 22.3137 0.996903
\(502\) 0 0
\(503\) −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
\(506\) 0 0
\(507\) −25.3137 −1.12422
\(508\) 0 0
\(509\) −17.2132 −0.762962 −0.381481 0.924377i \(-0.624586\pi\)
−0.381481 + 0.924377i \(0.624586\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −2.48528 −0.109728
\(514\) 0 0
\(515\) −10.7574 −0.474026
\(516\) 0 0
\(517\) 7.24264 0.318531
\(518\) 0 0
\(519\) −3.00000 −0.131685
\(520\) 0 0
\(521\) 18.9706 0.831115 0.415558 0.909567i \(-0.363586\pi\)
0.415558 + 0.909567i \(0.363586\pi\)
\(522\) 0 0
\(523\) 15.5147 0.678411 0.339206 0.940712i \(-0.389842\pi\)
0.339206 + 0.940712i \(0.389842\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.21320 0.401333
\(528\) 0 0
\(529\) −1.97056 −0.0856766
\(530\) 0 0
\(531\) 17.6569 0.766242
\(532\) 0 0
\(533\) 3.55635 0.154043
\(534\) 0 0
\(535\) 14.4853 0.626253
\(536\) 0 0
\(537\) −6.00000 −0.258919
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 11.9706 0.514655 0.257327 0.966324i \(-0.417158\pi\)
0.257327 + 0.966324i \(0.417158\pi\)
\(542\) 0 0
\(543\) −16.2426 −0.697038
\(544\) 0 0
\(545\) −5.00000 −0.214176
\(546\) 0 0
\(547\) 7.51472 0.321306 0.160653 0.987011i \(-0.448640\pi\)
0.160653 + 0.987011i \(0.448640\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 15.9411 0.679115
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 15.0711 0.639731
\(556\) 0 0
\(557\) 31.7990 1.34737 0.673683 0.739020i \(-0.264711\pi\)
0.673683 + 0.739020i \(0.264711\pi\)
\(558\) 0 0
\(559\) 3.17157 0.134143
\(560\) 0 0
\(561\) 73.7696 3.11455
\(562\) 0 0
\(563\) 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
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.14214 −0.0898030 −0.0449015 0.998991i \(-0.514297\pi\)
−0.0449015 + 0.998991i \(0.514297\pi\)
\(570\) 0 0
\(571\) 34.4853 1.44316 0.721582 0.692329i \(-0.243415\pi\)
0.721582 + 0.692329i \(0.243415\pi\)
\(572\) 0 0
\(573\) 48.2132 2.01414
\(574\) 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 \(0.435097\pi\)
\(578\) 0 0
\(579\) −38.6274 −1.60530
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −24.7279 −1.02413
\(584\) 0 0
\(585\) 4.48528 0.185444
\(586\) 0 0
\(587\) 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
\(590\) 0 0
\(591\) 25.5563 1.05125
\(592\) 0 0
\(593\) −10.7574 −0.441752 −0.220876 0.975302i \(-0.570892\pi\)
−0.220876 + 0.975302i \(0.570892\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −11.0711 −0.453109
\(598\) 0 0
\(599\) 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
\(603\) −0.686292 −0.0279480
\(604\) 0 0
\(605\) −22.9706 −0.933886
\(606\) 0 0
\(607\) −24.8995 −1.01064 −0.505320 0.862932i \(-0.668625\pi\)
−0.505320 + 0.862932i \(0.668625\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.97056 −0.0797204
\(612\) 0 0
\(613\) 43.9411 1.77477 0.887383 0.461034i \(-0.152521\pi\)
0.887383 + 0.461034i \(0.152521\pi\)
\(614\) 0 0
\(615\) 5.41421 0.218322
\(616\) 0 0
\(617\) 7.41421 0.298485 0.149242 0.988801i \(-0.452316\pi\)
0.149242 + 0.988801i \(0.452316\pi\)
\(618\) 0 0
\(619\) 31.0711 1.24885 0.624426 0.781084i \(-0.285333\pi\)
0.624426 + 0.781084i \(0.285333\pi\)
\(620\) 0 0
\(621\) 1.89949 0.0762241
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 84.4264 3.37167
\(628\) 0 0
\(629\) −32.7279 −1.30495
\(630\) 0 0
\(631\) −8.45584 −0.336622 −0.168311 0.985734i \(-0.553831\pi\)
−0.168311 + 0.985734i \(0.553831\pi\)
\(632\) 0 0
\(633\) −21.7279 −0.863607
\(634\) 0 0
\(635\) −0.242641 −0.00962890
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 24.9706 0.987820
\(640\) 0 0
\(641\) −0.686292 −0.0271069 −0.0135534 0.999908i \(-0.504314\pi\)
−0.0135534 + 0.999908i \(0.504314\pi\)
\(642\) 0 0
\(643\) 27.7279 1.09348 0.546741 0.837302i \(-0.315868\pi\)
0.546741 + 0.837302i \(0.315868\pi\)
\(644\) 0 0
\(645\) 4.82843 0.190119
\(646\) 0 0
\(647\) −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
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 1.02944 0.0402850 0.0201425 0.999797i \(-0.493588\pi\)
0.0201425 + 0.999797i \(0.493588\pi\)
\(654\) 0 0
\(655\) −3.75736 −0.146812
\(656\) 0 0
\(657\) −24.0000 −0.936329
\(658\) 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 \(0.240250\pi\)
\(662\) 0 0
\(663\) −20.0711 −0.779496
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −12.1838 −0.471757
\(668\) 0 0
\(669\) 43.9706 1.70000
\(670\) 0 0
\(671\) −16.4853 −0.636407
\(672\) 0 0
\(673\) 20.4853 0.789650 0.394825 0.918756i \(-0.370805\pi\)
0.394825 + 0.918756i \(0.370805\pi\)
\(674\) 0 0
\(675\) −0.414214 −0.0159431
\(676\) 0 0
\(677\) −1.78680 −0.0686722 −0.0343361 0.999410i \(-0.510932\pi\)
−0.0343361 + 0.999410i \(0.510932\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −23.4853 −0.899958
\(682\) 0 0
\(683\) 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
\(687\) 72.5269 2.76707
\(688\) 0 0
\(689\) 6.72792 0.256313
\(690\) 0 0
\(691\) 9.17157 0.348903 0.174452 0.984666i \(-0.444185\pi\)
0.174452 + 0.984666i \(0.444185\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −16.2426 −0.616118
\(696\) 0 0
\(697\) −11.7574 −0.445342
\(698\) 0 0
\(699\) −35.7990 −1.35404
\(700\) 0 0
\(701\) −4.45584 −0.168295 −0.0841475 0.996453i \(-0.526817\pi\)
−0.0841475 + 0.996453i \(0.526817\pi\)
\(702\) 0 0
\(703\) −37.4558 −1.41267
\(704\) 0 0
\(705\) −3.00000 −0.112987
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 17.0000 0.638448 0.319224 0.947679i \(-0.396578\pi\)
0.319224 + 0.947679i \(0.396578\pi\)
\(710\) 0 0
\(711\) 43.7990 1.64259
\(712\) 0 0
\(713\) −8.05887 −0.301807
\(714\) 0 0
\(715\) 9.24264 0.345655
\(716\) 0 0
\(717\) 1.24264 0.0464073
\(718\) 0 0
\(719\) 17.2132 0.641944 0.320972 0.947089i \(-0.395990\pi\)
0.320972 + 0.947089i \(0.395990\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −59.6985 −2.22021
\(724\) 0 0
\(725\) 2.65685 0.0986731
\(726\) 0 0
\(727\) −29.3137 −1.08719 −0.543593 0.839349i \(-0.682936\pi\)
−0.543593 + 0.839349i \(0.682936\pi\)
\(728\) 0 0
\(729\) −23.8284 −0.882534
\(730\) 0 0
\(731\) −10.4853 −0.387812
\(732\) 0 0
\(733\) 44.6985 1.65098 0.825488 0.564420i \(-0.190900\pi\)
0.825488 + 0.564420i \(0.190900\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1.41421 −0.0520932
\(738\) 0 0
\(739\) 3.97056 0.146060 0.0730298 0.997330i \(-0.476733\pi\)
0.0730298 + 0.997330i \(0.476733\pi\)
\(740\) 0 0
\(741\) −22.9706 −0.843845
\(742\) 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 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −12.5147 −0.456669 −0.228334 0.973583i \(-0.573328\pi\)
−0.228334 + 0.973583i \(0.573328\pi\)
\(752\) 0 0
\(753\) −60.8701 −2.21823
\(754\) 0 0
\(755\) 9.48528 0.345205
\(756\) 0 0
\(757\) −16.4853 −0.599168 −0.299584 0.954070i \(-0.596848\pi\)
−0.299584 + 0.954070i \(0.596848\pi\)
\(758\) 0 0
\(759\) −64.5269 −2.34218
\(760\) 0 0
\(761\) −12.7279 −0.461387 −0.230693 0.973026i \(-0.574099\pi\)
−0.230693 + 0.973026i \(0.574099\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −14.8284 −0.536123
\(766\) 0 0
\(767\) −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
\(771\) 63.9411 2.30278
\(772\) 0 0
\(773\) −36.2132 −1.30250 −0.651249 0.758864i \(-0.725755\pi\)
−0.651249 + 0.758864i \(0.725755\pi\)
\(774\) 0 0
\(775\) 1.75736 0.0631262
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −13.4558 −0.482106
\(780\) 0 0
\(781\) 51.4558 1.84123
\(782\) 0 0
\(783\) −1.10051 −0.0393288
\(784\) 0 0
\(785\) −20.8284 −0.743398
\(786\) 0 0
\(787\) 45.7279 1.63002 0.815012 0.579444i \(-0.196730\pi\)
0.815012 + 0.579444i \(0.196730\pi\)
\(788\) 0 0
\(789\) −69.1127 −2.46048
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 4.48528 0.159277
\(794\) 0 0
\(795\) 10.2426 0.363269
\(796\) 0 0
\(797\) −55.1838 −1.95471 −0.977355 0.211608i \(-0.932130\pi\)
−0.977355 + 0.211608i \(0.932130\pi\)
\(798\) 0 0
\(799\) 6.51472 0.230474
\(800\) 0 0
\(801\) 22.6274 0.799500
\(802\) 0 0
\(803\) −49.4558 −1.74526
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 18.7279 0.659254
\(808\) 0 0
\(809\) 30.5980 1.07577 0.537884 0.843019i \(-0.319224\pi\)
0.537884 + 0.843019i \(0.319224\pi\)
\(810\) 0 0
\(811\) −23.3553 −0.820117 −0.410058 0.912059i \(-0.634492\pi\)
−0.410058 + 0.912059i \(0.634492\pi\)
\(812\) 0 0
\(813\) −56.2843 −1.97398
\(814\) 0 0
\(815\) −1.75736 −0.0615576
\(816\) 0 0
\(817\) −12.0000 −0.419827
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −6.51472 −0.227365 −0.113683 0.993517i \(-0.536265\pi\)
−0.113683 + 0.993517i \(0.536265\pi\)
\(822\) 0 0
\(823\) 8.72792 0.304236 0.152118 0.988362i \(-0.451391\pi\)
0.152118 + 0.988362i \(0.451391\pi\)
\(824\) 0 0
\(825\) 14.0711 0.489892
\(826\) 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
\(831\) −65.6985 −2.27906
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −9.24264 −0.319855
\(836\) 0 0
\(837\) −0.727922 −0.0251607
\(838\) 0 0
\(839\) −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
\(842\) 0 0
\(843\) −49.0416 −1.68908
\(844\) 0 0
\(845\) 10.4853 0.360705
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −15.8284 −0.543230
\(850\) 0 0
\(851\) 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
\(855\) −16.9706 −0.580381
\(856\) 0 0
\(857\) 5.51472 0.188379 0.0941896 0.995554i \(-0.469974\pi\)
0.0941896 + 0.995554i \(0.469974\pi\)
\(858\) 0 0
\(859\) 48.7696 1.66400 0.831998 0.554779i \(-0.187197\pi\)
0.831998 + 0.554779i \(0.187197\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 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
\(866\) 0 0
\(867\) 25.3137 0.859699
\(868\) 0 0
\(869\) 90.2548 3.06169
\(870\) 0 0
\(871\) 0.384776 0.0130376
\(872\) 0 0
\(873\) −13.4558 −0.455411
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 46.9706 1.58608 0.793042 0.609167i \(-0.208496\pi\)
0.793042 + 0.609167i \(0.208496\pi\)
\(878\) 0 0
\(879\) −0.656854 −0.0221551
\(880\) 0 0
\(881\) 19.0294 0.641118 0.320559 0.947229i \(-0.396129\pi\)
0.320559 + 0.947229i \(0.396129\pi\)
\(882\) 0 0
\(883\) −48.4853 −1.63166 −0.815830 0.578292i \(-0.803719\pi\)
−0.815830 + 0.578292i \(0.803719\pi\)
\(884\) 0 0
\(885\) −15.0711 −0.506608
\(886\) 0 0
\(887\) −30.9706 −1.03989 −0.519945 0.854200i \(-0.674048\pi\)
−0.519945 + 0.854200i \(0.674048\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −55.2843 −1.85209
\(892\) 0 0
\(893\) 7.45584 0.249500
\(894\) 0 0
\(895\) 2.48528 0.0830738
\(896\) 0 0
\(897\) 17.5563 0.586189
\(898\) 0 0
\(899\) 4.66905 0.155721
\(900\) 0 0
\(901\) −22.2426 −0.741010
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 6.72792 0.223644
\(906\) 0 0
\(907\) −32.1838 −1.06864 −0.534322 0.845281i \(-0.679433\pi\)
−0.534322 + 0.845281i \(0.679433\pi\)
\(908\) 0 0
\(909\) 40.9706 1.35891
\(910\) 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
\(914\) 0 0
\(915\) 6.82843 0.225741
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 38.4558 1.26854 0.634271 0.773111i \(-0.281301\pi\)
0.634271 + 0.773111i \(0.281301\pi\)
\(920\) 0 0
\(921\) 26.7990 0.883057
\(922\) 0 0
\(923\) −14.0000 −0.460816
\(924\) 0 0
\(925\) −6.24264 −0.205257
\(926\) 0 0
\(927\) 30.4264 0.999334
\(928\) 0 0
\(929\) −54.7279 −1.79556 −0.897782 0.440439i \(-0.854823\pi\)
−0.897782 + 0.440439i \(0.854823\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −24.1421 −0.790378
\(934\) 0 0
\(935\) −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
\(939\) −58.4558 −1.90763
\(940\) 0 0
\(941\) 4.97056 0.162036 0.0810179 0.996713i \(-0.474183\pi\)
0.0810179 + 0.996713i \(0.474183\pi\)
\(942\) 0 0
\(943\) 10.2843 0.334902
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 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
\(950\) 0 0
\(951\) −28.1421 −0.912571
\(952\) 0 0
\(953\) −29.0122 −0.939797 −0.469899 0.882720i \(-0.655710\pi\)
−0.469899 + 0.882720i \(0.655710\pi\)
\(954\) 0 0
\(955\) −19.9706 −0.646232
\(956\) 0 0
\(957\) 37.3848 1.20848
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −27.9117 −0.900377
\(962\) 0 0
\(963\) −40.9706 −1.32026
\(964\) 0 0
\(965\) 16.0000 0.515058
\(966\) 0 0
\(967\) −24.4264 −0.785500 −0.392750 0.919645i \(-0.628476\pi\)
−0.392750 + 0.919645i \(0.628476\pi\)
\(968\) 0 0
\(969\) 75.9411 2.43958
\(970\) 0 0
\(971\) −42.7279 −1.37120 −0.685602 0.727976i \(-0.740461\pi\)
−0.685602 + 0.727976i \(0.740461\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −3.82843 −0.122608
\(976\) 0 0
\(977\) −30.7696 −0.984405 −0.492203 0.870481i \(-0.663808\pi\)
−0.492203 + 0.870481i \(0.663808\pi\)
\(978\) 0 0
\(979\) 46.6274 1.49022
\(980\) 0 0
\(981\) 14.1421 0.451524
\(982\) 0 0
\(983\) −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
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 9.17157 0.291639
\(990\) 0 0
\(991\) 47.9411 1.52290 0.761450 0.648224i \(-0.224488\pi\)
0.761450 + 0.648224i \(0.224488\pi\)
\(992\) 0 0
\(993\) −56.6274 −1.79702
\(994\) 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 \(0.518801\pi\)
\(998\) 0 0
\(999\) 2.58579 0.0818107
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 3920.2.a.bw.1.2 2
4.3 odd 2 245.2.a.e.1.2 2
7.6 odd 2 3920.2.a.br.1.1 2
12.11 even 2 2205.2.a.v.1.1 2
20.3 even 4 1225.2.b.i.99.1 4
20.7 even 4 1225.2.b.i.99.4 4
20.19 odd 2 1225.2.a.r.1.1 2
28.3 even 6 245.2.e.f.226.1 4
28.11 odd 6 245.2.e.g.226.1 4
28.19 even 6 245.2.e.f.116.1 4
28.23 odd 6 245.2.e.g.116.1 4
28.27 even 2 245.2.a.f.1.2 yes 2
84.83 odd 2 2205.2.a.t.1.1 2
140.27 odd 4 1225.2.b.j.99.3 4
140.83 odd 4 1225.2.b.j.99.2 4
140.139 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 4.3 odd 2
245.2.a.f.1.2 yes 2 28.27 even 2
245.2.e.f.116.1 4 28.19 even 6
245.2.e.f.226.1 4 28.3 even 6
245.2.e.g.116.1 4 28.23 odd 6
245.2.e.g.226.1 4 28.11 odd 6
1225.2.a.p.1.1 2 140.139 even 2
1225.2.a.r.1.1 2 20.19 odd 2
1225.2.b.i.99.1 4 20.3 even 4
1225.2.b.i.99.4 4 20.7 even 4
1225.2.b.j.99.2 4 140.83 odd 4
1225.2.b.j.99.3 4 140.27 odd 4
2205.2.a.t.1.1 2 84.83 odd 2
2205.2.a.v.1.1 2 12.11 even 2
3920.2.a.br.1.1 2 7.6 odd 2
3920.2.a.bw.1.2 2 1.1 even 1 trivial