Properties

Label 880.2.a.m.1.1
Level $880$
Weight $2$
Character 880.1
Self dual yes
Analytic conductor $7.027$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 880 = 2^{4} \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 880.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.02683537787\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 55)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 880.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.82843 q^{3} -1.00000 q^{5} +2.00000 q^{7} +5.00000 q^{9} +O(q^{10})\) \(q-2.82843 q^{3} -1.00000 q^{5} +2.00000 q^{7} +5.00000 q^{9} -1.00000 q^{11} -6.82843 q^{13} +2.82843 q^{15} +1.17157 q^{17} -5.65685 q^{21} -2.82843 q^{23} +1.00000 q^{25} -5.65685 q^{27} +7.65685 q^{29} +2.82843 q^{33} -2.00000 q^{35} +3.65685 q^{37} +19.3137 q^{39} +6.00000 q^{41} +6.00000 q^{43} -5.00000 q^{45} +2.82843 q^{47} -3.00000 q^{49} -3.31371 q^{51} +0.343146 q^{53} +1.00000 q^{55} +9.65685 q^{59} +13.3137 q^{61} +10.0000 q^{63} +6.82843 q^{65} +4.48528 q^{67} +8.00000 q^{69} +11.3137 q^{71} -6.82843 q^{73} -2.82843 q^{75} -2.00000 q^{77} -4.00000 q^{79} +1.00000 q^{81} +6.00000 q^{83} -1.17157 q^{85} -21.6569 q^{87} +9.31371 q^{89} -13.6569 q^{91} -7.65685 q^{97} -5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{5} + 4 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{5} + 4 q^{7} + 10 q^{9} - 2 q^{11} - 8 q^{13} + 8 q^{17} + 2 q^{25} + 4 q^{29} - 4 q^{35} - 4 q^{37} + 16 q^{39} + 12 q^{41} + 12 q^{43} - 10 q^{45} - 6 q^{49} + 16 q^{51} + 12 q^{53} + 2 q^{55} + 8 q^{59} + 4 q^{61} + 20 q^{63} + 8 q^{65} - 8 q^{67} + 16 q^{69} - 8 q^{73} - 4 q^{77} - 8 q^{79} + 2 q^{81} + 12 q^{83} - 8 q^{85} - 32 q^{87} - 4 q^{89} - 16 q^{91} - 4 q^{97} - 10 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −2.82843 −1.63299 −0.816497 0.577350i \(-0.804087\pi\)
−0.816497 + 0.577350i \(0.804087\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 5.00000 1.66667
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −6.82843 −1.89386 −0.946932 0.321433i \(-0.895836\pi\)
−0.946932 + 0.321433i \(0.895836\pi\)
\(14\) 0 0
\(15\) 2.82843 0.730297
\(16\) 0 0
\(17\) 1.17157 0.284148 0.142074 0.989856i \(-0.454623\pi\)
0.142074 + 0.989856i \(0.454623\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) −5.65685 −1.23443
\(22\) 0 0
\(23\) −2.82843 −0.589768 −0.294884 0.955533i \(-0.595281\pi\)
−0.294884 + 0.955533i \(0.595281\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −5.65685 −1.08866
\(28\) 0 0
\(29\) 7.65685 1.42184 0.710921 0.703272i \(-0.248278\pi\)
0.710921 + 0.703272i \(0.248278\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 2.82843 0.492366
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 3.65685 0.601183 0.300592 0.953753i \(-0.402816\pi\)
0.300592 + 0.953753i \(0.402816\pi\)
\(38\) 0 0
\(39\) 19.3137 3.09267
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 6.00000 0.914991 0.457496 0.889212i \(-0.348747\pi\)
0.457496 + 0.889212i \(0.348747\pi\)
\(44\) 0 0
\(45\) −5.00000 −0.745356
\(46\) 0 0
\(47\) 2.82843 0.412568 0.206284 0.978492i \(-0.433863\pi\)
0.206284 + 0.978492i \(0.433863\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −3.31371 −0.464012
\(52\) 0 0
\(53\) 0.343146 0.0471347 0.0235673 0.999722i \(-0.492498\pi\)
0.0235673 + 0.999722i \(0.492498\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.65685 1.25722 0.628608 0.777723i \(-0.283625\pi\)
0.628608 + 0.777723i \(0.283625\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0 0
\(63\) 10.0000 1.25988
\(64\) 0 0
\(65\) 6.82843 0.846962
\(66\) 0 0
\(67\) 4.48528 0.547964 0.273982 0.961735i \(-0.411659\pi\)
0.273982 + 0.961735i \(0.411659\pi\)
\(68\) 0 0
\(69\) 8.00000 0.963087
\(70\) 0 0
\(71\) 11.3137 1.34269 0.671345 0.741145i \(-0.265717\pi\)
0.671345 + 0.741145i \(0.265717\pi\)
\(72\) 0 0
\(73\) −6.82843 −0.799207 −0.399603 0.916688i \(-0.630852\pi\)
−0.399603 + 0.916688i \(0.630852\pi\)
\(74\) 0 0
\(75\) −2.82843 −0.326599
\(76\) 0 0
\(77\) −2.00000 −0.227921
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −1.17157 −0.127075
\(86\) 0 0
\(87\) −21.6569 −2.32186
\(88\) 0 0
\(89\) 9.31371 0.987251 0.493626 0.869675i \(-0.335671\pi\)
0.493626 + 0.869675i \(0.335671\pi\)
\(90\) 0 0
\(91\) −13.6569 −1.43163
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −7.65685 −0.777436 −0.388718 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 0 0
\(101\) −13.3137 −1.32476 −0.662382 0.749166i \(-0.730454\pi\)
−0.662382 + 0.749166i \(0.730454\pi\)
\(102\) 0 0
\(103\) −1.17157 −0.115439 −0.0577193 0.998333i \(-0.518383\pi\)
−0.0577193 + 0.998333i \(0.518383\pi\)
\(104\) 0 0
\(105\) 5.65685 0.552052
\(106\) 0 0
\(107\) 3.65685 0.353521 0.176761 0.984254i \(-0.443438\pi\)
0.176761 + 0.984254i \(0.443438\pi\)
\(108\) 0 0
\(109\) 3.65685 0.350263 0.175132 0.984545i \(-0.443965\pi\)
0.175132 + 0.984545i \(0.443965\pi\)
\(110\) 0 0
\(111\) −10.3431 −0.981728
\(112\) 0 0
\(113\) 8.34315 0.784857 0.392429 0.919782i \(-0.371635\pi\)
0.392429 + 0.919782i \(0.371635\pi\)
\(114\) 0 0
\(115\) 2.82843 0.263752
\(116\) 0 0
\(117\) −34.1421 −3.15644
\(118\) 0 0
\(119\) 2.34315 0.214796
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −16.9706 −1.53018
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −15.6569 −1.38932 −0.694661 0.719338i \(-0.744445\pi\)
−0.694661 + 0.719338i \(0.744445\pi\)
\(128\) 0 0
\(129\) −16.9706 −1.49417
\(130\) 0 0
\(131\) −11.3137 −0.988483 −0.494242 0.869325i \(-0.664554\pi\)
−0.494242 + 0.869325i \(0.664554\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 5.65685 0.486864
\(136\) 0 0
\(137\) 22.9706 1.96251 0.981254 0.192720i \(-0.0617309\pi\)
0.981254 + 0.192720i \(0.0617309\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) 6.82843 0.571022
\(144\) 0 0
\(145\) −7.65685 −0.635867
\(146\) 0 0
\(147\) 8.48528 0.699854
\(148\) 0 0
\(149\) 11.6569 0.954967 0.477483 0.878641i \(-0.341549\pi\)
0.477483 + 0.878641i \(0.341549\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) 5.85786 0.473580
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −14.0000 −1.11732 −0.558661 0.829396i \(-0.688685\pi\)
−0.558661 + 0.829396i \(0.688685\pi\)
\(158\) 0 0
\(159\) −0.970563 −0.0769706
\(160\) 0 0
\(161\) −5.65685 −0.445823
\(162\) 0 0
\(163\) 0.485281 0.0380102 0.0190051 0.999819i \(-0.493950\pi\)
0.0190051 + 0.999819i \(0.493950\pi\)
\(164\) 0 0
\(165\) −2.82843 −0.220193
\(166\) 0 0
\(167\) −10.9706 −0.848928 −0.424464 0.905445i \(-0.639537\pi\)
−0.424464 + 0.905445i \(0.639537\pi\)
\(168\) 0 0
\(169\) 33.6274 2.58672
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 6.14214 0.466978 0.233489 0.972359i \(-0.424986\pi\)
0.233489 + 0.972359i \(0.424986\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) 0 0
\(177\) −27.3137 −2.05302
\(178\) 0 0
\(179\) 1.65685 0.123839 0.0619196 0.998081i \(-0.480278\pi\)
0.0619196 + 0.998081i \(0.480278\pi\)
\(180\) 0 0
\(181\) −1.31371 −0.0976472 −0.0488236 0.998807i \(-0.515547\pi\)
−0.0488236 + 0.998807i \(0.515547\pi\)
\(182\) 0 0
\(183\) −37.6569 −2.78367
\(184\) 0 0
\(185\) −3.65685 −0.268857
\(186\) 0 0
\(187\) −1.17157 −0.0856739
\(188\) 0 0
\(189\) −11.3137 −0.822951
\(190\) 0 0
\(191\) 19.3137 1.39749 0.698745 0.715370i \(-0.253742\pi\)
0.698745 + 0.715370i \(0.253742\pi\)
\(192\) 0 0
\(193\) −6.82843 −0.491521 −0.245760 0.969331i \(-0.579038\pi\)
−0.245760 + 0.969331i \(0.579038\pi\)
\(194\) 0 0
\(195\) −19.3137 −1.38308
\(196\) 0 0
\(197\) −5.17157 −0.368459 −0.184230 0.982883i \(-0.558979\pi\)
−0.184230 + 0.982883i \(0.558979\pi\)
\(198\) 0 0
\(199\) −21.6569 −1.53521 −0.767607 0.640921i \(-0.778553\pi\)
−0.767607 + 0.640921i \(0.778553\pi\)
\(200\) 0 0
\(201\) −12.6863 −0.894822
\(202\) 0 0
\(203\) 15.3137 1.07481
\(204\) 0 0
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) −14.1421 −0.982946
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 16.0000 1.10149 0.550743 0.834675i \(-0.314345\pi\)
0.550743 + 0.834675i \(0.314345\pi\)
\(212\) 0 0
\(213\) −32.0000 −2.19260
\(214\) 0 0
\(215\) −6.00000 −0.409197
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 19.3137 1.30510
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) 5.17157 0.346314 0.173157 0.984894i \(-0.444603\pi\)
0.173157 + 0.984894i \(0.444603\pi\)
\(224\) 0 0
\(225\) 5.00000 0.333333
\(226\) 0 0
\(227\) −2.68629 −0.178295 −0.0891477 0.996018i \(-0.528414\pi\)
−0.0891477 + 0.996018i \(0.528414\pi\)
\(228\) 0 0
\(229\) −21.3137 −1.40845 −0.704225 0.709977i \(-0.748705\pi\)
−0.704225 + 0.709977i \(0.748705\pi\)
\(230\) 0 0
\(231\) 5.65685 0.372194
\(232\) 0 0
\(233\) 22.1421 1.45058 0.725290 0.688444i \(-0.241706\pi\)
0.725290 + 0.688444i \(0.241706\pi\)
\(234\) 0 0
\(235\) −2.82843 −0.184506
\(236\) 0 0
\(237\) 11.3137 0.734904
\(238\) 0 0
\(239\) 0.686292 0.0443925 0.0221963 0.999754i \(-0.492934\pi\)
0.0221963 + 0.999754i \(0.492934\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) 14.1421 0.907218
\(244\) 0 0
\(245\) 3.00000 0.191663
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −16.9706 −1.07547
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) 2.82843 0.177822
\(254\) 0 0
\(255\) 3.31371 0.207512
\(256\) 0 0
\(257\) 13.3137 0.830486 0.415243 0.909710i \(-0.363696\pi\)
0.415243 + 0.909710i \(0.363696\pi\)
\(258\) 0 0
\(259\) 7.31371 0.454452
\(260\) 0 0
\(261\) 38.2843 2.36974
\(262\) 0 0
\(263\) −22.9706 −1.41643 −0.708213 0.705999i \(-0.750498\pi\)
−0.708213 + 0.705999i \(0.750498\pi\)
\(264\) 0 0
\(265\) −0.343146 −0.0210793
\(266\) 0 0
\(267\) −26.3431 −1.61217
\(268\) 0 0
\(269\) −5.31371 −0.323983 −0.161991 0.986792i \(-0.551792\pi\)
−0.161991 + 0.986792i \(0.551792\pi\)
\(270\) 0 0
\(271\) 15.3137 0.930242 0.465121 0.885247i \(-0.346011\pi\)
0.465121 + 0.885247i \(0.346011\pi\)
\(272\) 0 0
\(273\) 38.6274 2.33784
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 1.17157 0.0703930 0.0351965 0.999380i \(-0.488794\pi\)
0.0351965 + 0.999380i \(0.488794\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.31371 −0.316989 −0.158495 0.987360i \(-0.550664\pi\)
−0.158495 + 0.987360i \(0.550664\pi\)
\(282\) 0 0
\(283\) 12.6274 0.750622 0.375311 0.926899i \(-0.377536\pi\)
0.375311 + 0.926899i \(0.377536\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000 0.708338
\(288\) 0 0
\(289\) −15.6274 −0.919260
\(290\) 0 0
\(291\) 21.6569 1.26955
\(292\) 0 0
\(293\) −14.8284 −0.866286 −0.433143 0.901325i \(-0.642595\pi\)
−0.433143 + 0.901325i \(0.642595\pi\)
\(294\) 0 0
\(295\) −9.65685 −0.562244
\(296\) 0 0
\(297\) 5.65685 0.328244
\(298\) 0 0
\(299\) 19.3137 1.11694
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) 37.6569 2.16333
\(304\) 0 0
\(305\) −13.3137 −0.762341
\(306\) 0 0
\(307\) 27.6569 1.57846 0.789230 0.614098i \(-0.210480\pi\)
0.789230 + 0.614098i \(0.210480\pi\)
\(308\) 0 0
\(309\) 3.31371 0.188510
\(310\) 0 0
\(311\) −27.3137 −1.54882 −0.774409 0.632685i \(-0.781953\pi\)
−0.774409 + 0.632685i \(0.781953\pi\)
\(312\) 0 0
\(313\) 21.3137 1.20472 0.602361 0.798224i \(-0.294227\pi\)
0.602361 + 0.798224i \(0.294227\pi\)
\(314\) 0 0
\(315\) −10.0000 −0.563436
\(316\) 0 0
\(317\) 21.3137 1.19710 0.598549 0.801087i \(-0.295744\pi\)
0.598549 + 0.801087i \(0.295744\pi\)
\(318\) 0 0
\(319\) −7.65685 −0.428702
\(320\) 0 0
\(321\) −10.3431 −0.577298
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −6.82843 −0.378773
\(326\) 0 0
\(327\) −10.3431 −0.571977
\(328\) 0 0
\(329\) 5.65685 0.311872
\(330\) 0 0
\(331\) −15.3137 −0.841718 −0.420859 0.907126i \(-0.638271\pi\)
−0.420859 + 0.907126i \(0.638271\pi\)
\(332\) 0 0
\(333\) 18.2843 1.00197
\(334\) 0 0
\(335\) −4.48528 −0.245057
\(336\) 0 0
\(337\) −3.51472 −0.191459 −0.0957295 0.995407i \(-0.530518\pi\)
−0.0957295 + 0.995407i \(0.530518\pi\)
\(338\) 0 0
\(339\) −23.5980 −1.28167
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) −8.00000 −0.430706
\(346\) 0 0
\(347\) 22.9706 1.23312 0.616562 0.787306i \(-0.288525\pi\)
0.616562 + 0.787306i \(0.288525\pi\)
\(348\) 0 0
\(349\) −6.97056 −0.373126 −0.186563 0.982443i \(-0.559735\pi\)
−0.186563 + 0.982443i \(0.559735\pi\)
\(350\) 0 0
\(351\) 38.6274 2.06178
\(352\) 0 0
\(353\) −1.31371 −0.0699216 −0.0349608 0.999389i \(-0.511131\pi\)
−0.0349608 + 0.999389i \(0.511131\pi\)
\(354\) 0 0
\(355\) −11.3137 −0.600469
\(356\) 0 0
\(357\) −6.62742 −0.350760
\(358\) 0 0
\(359\) −23.3137 −1.23045 −0.615225 0.788351i \(-0.710935\pi\)
−0.615225 + 0.788351i \(0.710935\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) −2.82843 −0.148454
\(364\) 0 0
\(365\) 6.82843 0.357416
\(366\) 0 0
\(367\) −8.48528 −0.442928 −0.221464 0.975169i \(-0.571084\pi\)
−0.221464 + 0.975169i \(0.571084\pi\)
\(368\) 0 0
\(369\) 30.0000 1.56174
\(370\) 0 0
\(371\) 0.686292 0.0356305
\(372\) 0 0
\(373\) −3.79899 −0.196704 −0.0983521 0.995152i \(-0.531357\pi\)
−0.0983521 + 0.995152i \(0.531357\pi\)
\(374\) 0 0
\(375\) 2.82843 0.146059
\(376\) 0 0
\(377\) −52.2843 −2.69278
\(378\) 0 0
\(379\) −22.3431 −1.14769 −0.573845 0.818964i \(-0.694549\pi\)
−0.573845 + 0.818964i \(0.694549\pi\)
\(380\) 0 0
\(381\) 44.2843 2.26875
\(382\) 0 0
\(383\) 34.1421 1.74458 0.872291 0.488987i \(-0.162634\pi\)
0.872291 + 0.488987i \(0.162634\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 0 0
\(387\) 30.0000 1.52499
\(388\) 0 0
\(389\) −24.6274 −1.24866 −0.624330 0.781161i \(-0.714628\pi\)
−0.624330 + 0.781161i \(0.714628\pi\)
\(390\) 0 0
\(391\) −3.31371 −0.167581
\(392\) 0 0
\(393\) 32.0000 1.61419
\(394\) 0 0
\(395\) 4.00000 0.201262
\(396\) 0 0
\(397\) 13.3137 0.668196 0.334098 0.942538i \(-0.391568\pi\)
0.334098 + 0.942538i \(0.391568\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 17.3137 0.864605 0.432303 0.901729i \(-0.357701\pi\)
0.432303 + 0.901729i \(0.357701\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −3.65685 −0.181264
\(408\) 0 0
\(409\) 34.9706 1.72918 0.864592 0.502475i \(-0.167577\pi\)
0.864592 + 0.502475i \(0.167577\pi\)
\(410\) 0 0
\(411\) −64.9706 −3.20476
\(412\) 0 0
\(413\) 19.3137 0.950365
\(414\) 0 0
\(415\) −6.00000 −0.294528
\(416\) 0 0
\(417\) −11.3137 −0.554035
\(418\) 0 0
\(419\) 14.3431 0.700709 0.350354 0.936617i \(-0.386061\pi\)
0.350354 + 0.936617i \(0.386061\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) 14.1421 0.687614
\(424\) 0 0
\(425\) 1.17157 0.0568296
\(426\) 0 0
\(427\) 26.6274 1.28859
\(428\) 0 0
\(429\) −19.3137 −0.932475
\(430\) 0 0
\(431\) −11.3137 −0.544962 −0.272481 0.962161i \(-0.587844\pi\)
−0.272481 + 0.962161i \(0.587844\pi\)
\(432\) 0 0
\(433\) 3.65685 0.175737 0.0878686 0.996132i \(-0.471994\pi\)
0.0878686 + 0.996132i \(0.471994\pi\)
\(434\) 0 0
\(435\) 21.6569 1.03837
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 0 0
\(443\) 21.1716 1.00589 0.502946 0.864318i \(-0.332249\pi\)
0.502946 + 0.864318i \(0.332249\pi\)
\(444\) 0 0
\(445\) −9.31371 −0.441512
\(446\) 0 0
\(447\) −32.9706 −1.55945
\(448\) 0 0
\(449\) −16.6274 −0.784696 −0.392348 0.919817i \(-0.628337\pi\)
−0.392348 + 0.919817i \(0.628337\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 0 0
\(453\) −33.9411 −1.59469
\(454\) 0 0
\(455\) 13.6569 0.640243
\(456\) 0 0
\(457\) −16.4853 −0.771149 −0.385574 0.922677i \(-0.625997\pi\)
−0.385574 + 0.922677i \(0.625997\pi\)
\(458\) 0 0
\(459\) −6.62742 −0.309341
\(460\) 0 0
\(461\) −32.6274 −1.51961 −0.759805 0.650151i \(-0.774706\pi\)
−0.759805 + 0.650151i \(0.774706\pi\)
\(462\) 0 0
\(463\) −22.1421 −1.02903 −0.514516 0.857481i \(-0.672028\pi\)
−0.514516 + 0.857481i \(0.672028\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.17157 0.424410 0.212205 0.977225i \(-0.431936\pi\)
0.212205 + 0.977225i \(0.431936\pi\)
\(468\) 0 0
\(469\) 8.97056 0.414222
\(470\) 0 0
\(471\) 39.5980 1.82458
\(472\) 0 0
\(473\) −6.00000 −0.275880
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.71573 0.0785578
\(478\) 0 0
\(479\) 36.0000 1.64488 0.822441 0.568850i \(-0.192612\pi\)
0.822441 + 0.568850i \(0.192612\pi\)
\(480\) 0 0
\(481\) −24.9706 −1.13856
\(482\) 0 0
\(483\) 16.0000 0.728025
\(484\) 0 0
\(485\) 7.65685 0.347680
\(486\) 0 0
\(487\) 7.51472 0.340524 0.170262 0.985399i \(-0.445539\pi\)
0.170262 + 0.985399i \(0.445539\pi\)
\(488\) 0 0
\(489\) −1.37258 −0.0620703
\(490\) 0 0
\(491\) 23.3137 1.05213 0.526066 0.850443i \(-0.323666\pi\)
0.526066 + 0.850443i \(0.323666\pi\)
\(492\) 0 0
\(493\) 8.97056 0.404014
\(494\) 0 0
\(495\) 5.00000 0.224733
\(496\) 0 0
\(497\) 22.6274 1.01498
\(498\) 0 0
\(499\) 1.65685 0.0741710 0.0370855 0.999312i \(-0.488193\pi\)
0.0370855 + 0.999312i \(0.488193\pi\)
\(500\) 0 0
\(501\) 31.0294 1.38629
\(502\) 0 0
\(503\) 28.6274 1.27643 0.638217 0.769857i \(-0.279672\pi\)
0.638217 + 0.769857i \(0.279672\pi\)
\(504\) 0 0
\(505\) 13.3137 0.592452
\(506\) 0 0
\(507\) −95.1127 −4.22410
\(508\) 0 0
\(509\) 9.31371 0.412823 0.206411 0.978465i \(-0.433821\pi\)
0.206411 + 0.978465i \(0.433821\pi\)
\(510\) 0 0
\(511\) −13.6569 −0.604144
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 1.17157 0.0516257
\(516\) 0 0
\(517\) −2.82843 −0.124394
\(518\) 0 0
\(519\) −17.3726 −0.762572
\(520\) 0 0
\(521\) 2.68629 0.117689 0.0588443 0.998267i \(-0.481258\pi\)
0.0588443 + 0.998267i \(0.481258\pi\)
\(522\) 0 0
\(523\) −37.5980 −1.64404 −0.822022 0.569455i \(-0.807154\pi\)
−0.822022 + 0.569455i \(0.807154\pi\)
\(524\) 0 0
\(525\) −5.65685 −0.246885
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −15.0000 −0.652174
\(530\) 0 0
\(531\) 48.2843 2.09536
\(532\) 0 0
\(533\) −40.9706 −1.77463
\(534\) 0 0
\(535\) −3.65685 −0.158100
\(536\) 0 0
\(537\) −4.68629 −0.202228
\(538\) 0 0
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) 0 0
\(543\) 3.71573 0.159457
\(544\) 0 0
\(545\) −3.65685 −0.156642
\(546\) 0 0
\(547\) 34.0000 1.45374 0.726868 0.686778i \(-0.240975\pi\)
0.726868 + 0.686778i \(0.240975\pi\)
\(548\) 0 0
\(549\) 66.5685 2.84108
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) 10.3431 0.439042
\(556\) 0 0
\(557\) 38.1421 1.61613 0.808067 0.589090i \(-0.200514\pi\)
0.808067 + 0.589090i \(0.200514\pi\)
\(558\) 0 0
\(559\) −40.9706 −1.73287
\(560\) 0 0
\(561\) 3.31371 0.139905
\(562\) 0 0
\(563\) −11.6569 −0.491278 −0.245639 0.969361i \(-0.578998\pi\)
−0.245639 + 0.969361i \(0.578998\pi\)
\(564\) 0 0
\(565\) −8.34315 −0.350999
\(566\) 0 0
\(567\) 2.00000 0.0839921
\(568\) 0 0
\(569\) 20.3431 0.852829 0.426415 0.904528i \(-0.359776\pi\)
0.426415 + 0.904528i \(0.359776\pi\)
\(570\) 0 0
\(571\) −45.9411 −1.92258 −0.961288 0.275545i \(-0.911142\pi\)
−0.961288 + 0.275545i \(0.911142\pi\)
\(572\) 0 0
\(573\) −54.6274 −2.28209
\(574\) 0 0
\(575\) −2.82843 −0.117954
\(576\) 0 0
\(577\) 6.97056 0.290188 0.145094 0.989418i \(-0.453651\pi\)
0.145094 + 0.989418i \(0.453651\pi\)
\(578\) 0 0
\(579\) 19.3137 0.802650
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) −0.343146 −0.0142116
\(584\) 0 0
\(585\) 34.1421 1.41160
\(586\) 0 0
\(587\) −26.1421 −1.07900 −0.539501 0.841985i \(-0.681387\pi\)
−0.539501 + 0.841985i \(0.681387\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 14.6274 0.601692
\(592\) 0 0
\(593\) 20.4853 0.841230 0.420615 0.907239i \(-0.361814\pi\)
0.420615 + 0.907239i \(0.361814\pi\)
\(594\) 0 0
\(595\) −2.34315 −0.0960596
\(596\) 0 0
\(597\) 61.2548 2.50699
\(598\) 0 0
\(599\) −5.65685 −0.231133 −0.115566 0.993300i \(-0.536868\pi\)
−0.115566 + 0.993300i \(0.536868\pi\)
\(600\) 0 0
\(601\) −43.9411 −1.79240 −0.896198 0.443654i \(-0.853682\pi\)
−0.896198 + 0.443654i \(0.853682\pi\)
\(602\) 0 0
\(603\) 22.4264 0.913274
\(604\) 0 0
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 18.2843 0.742136 0.371068 0.928606i \(-0.378992\pi\)
0.371068 + 0.928606i \(0.378992\pi\)
\(608\) 0 0
\(609\) −43.3137 −1.75516
\(610\) 0 0
\(611\) −19.3137 −0.781349
\(612\) 0 0
\(613\) 25.4558 1.02815 0.514076 0.857745i \(-0.328135\pi\)
0.514076 + 0.857745i \(0.328135\pi\)
\(614\) 0 0
\(615\) 16.9706 0.684319
\(616\) 0 0
\(617\) 11.6569 0.469287 0.234644 0.972081i \(-0.424608\pi\)
0.234644 + 0.972081i \(0.424608\pi\)
\(618\) 0 0
\(619\) 25.6569 1.03124 0.515618 0.856819i \(-0.327562\pi\)
0.515618 + 0.856819i \(0.327562\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) 0 0
\(623\) 18.6274 0.746292
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 4.28427 0.170825
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) −45.2548 −1.79872
\(634\) 0 0
\(635\) 15.6569 0.621323
\(636\) 0 0
\(637\) 20.4853 0.811656
\(638\) 0 0
\(639\) 56.5685 2.23782
\(640\) 0 0
\(641\) 30.0000 1.18493 0.592464 0.805597i \(-0.298155\pi\)
0.592464 + 0.805597i \(0.298155\pi\)
\(642\) 0 0
\(643\) −49.4558 −1.95035 −0.975174 0.221440i \(-0.928924\pi\)
−0.975174 + 0.221440i \(0.928924\pi\)
\(644\) 0 0
\(645\) 16.9706 0.668215
\(646\) 0 0
\(647\) 35.1127 1.38042 0.690211 0.723608i \(-0.257518\pi\)
0.690211 + 0.723608i \(0.257518\pi\)
\(648\) 0 0
\(649\) −9.65685 −0.379065
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.343146 0.0134283 0.00671417 0.999977i \(-0.497863\pi\)
0.00671417 + 0.999977i \(0.497863\pi\)
\(654\) 0 0
\(655\) 11.3137 0.442063
\(656\) 0 0
\(657\) −34.1421 −1.33201
\(658\) 0 0
\(659\) 21.9411 0.854705 0.427352 0.904085i \(-0.359446\pi\)
0.427352 + 0.904085i \(0.359446\pi\)
\(660\) 0 0
\(661\) −0.627417 −0.0244037 −0.0122018 0.999926i \(-0.503884\pi\)
−0.0122018 + 0.999926i \(0.503884\pi\)
\(662\) 0 0
\(663\) 22.6274 0.878776
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −21.6569 −0.838557
\(668\) 0 0
\(669\) −14.6274 −0.565529
\(670\) 0 0
\(671\) −13.3137 −0.513970
\(672\) 0 0
\(673\) 4.48528 0.172895 0.0864474 0.996256i \(-0.472449\pi\)
0.0864474 + 0.996256i \(0.472449\pi\)
\(674\) 0 0
\(675\) −5.65685 −0.217732
\(676\) 0 0
\(677\) 17.1716 0.659957 0.329979 0.943988i \(-0.392958\pi\)
0.329979 + 0.943988i \(0.392958\pi\)
\(678\) 0 0
\(679\) −15.3137 −0.587686
\(680\) 0 0
\(681\) 7.59798 0.291155
\(682\) 0 0
\(683\) −31.7990 −1.21675 −0.608377 0.793648i \(-0.708179\pi\)
−0.608377 + 0.793648i \(0.708179\pi\)
\(684\) 0 0
\(685\) −22.9706 −0.877660
\(686\) 0 0
\(687\) 60.2843 2.29999
\(688\) 0 0
\(689\) −2.34315 −0.0892667
\(690\) 0 0
\(691\) 16.6863 0.634776 0.317388 0.948296i \(-0.397194\pi\)
0.317388 + 0.948296i \(0.397194\pi\)
\(692\) 0 0
\(693\) −10.0000 −0.379869
\(694\) 0 0
\(695\) −4.00000 −0.151729
\(696\) 0 0
\(697\) 7.02944 0.266259
\(698\) 0 0
\(699\) −62.6274 −2.36879
\(700\) 0 0
\(701\) 32.6274 1.23232 0.616160 0.787621i \(-0.288687\pi\)
0.616160 + 0.787621i \(0.288687\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 8.00000 0.301297
\(706\) 0 0
\(707\) −26.6274 −1.00143
\(708\) 0 0
\(709\) −20.6274 −0.774679 −0.387339 0.921937i \(-0.626606\pi\)
−0.387339 + 0.921937i \(0.626606\pi\)
\(710\) 0 0
\(711\) −20.0000 −0.750059
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −6.82843 −0.255369
\(716\) 0 0
\(717\) −1.94113 −0.0724927
\(718\) 0 0
\(719\) −29.6569 −1.10601 −0.553007 0.833177i \(-0.686520\pi\)
−0.553007 + 0.833177i \(0.686520\pi\)
\(720\) 0 0
\(721\) −2.34315 −0.0872633
\(722\) 0 0
\(723\) −16.9706 −0.631142
\(724\) 0 0
\(725\) 7.65685 0.284368
\(726\) 0 0
\(727\) 36.4853 1.35316 0.676582 0.736367i \(-0.263460\pi\)
0.676582 + 0.736367i \(0.263460\pi\)
\(728\) 0 0
\(729\) −43.0000 −1.59259
\(730\) 0 0
\(731\) 7.02944 0.259993
\(732\) 0 0
\(733\) 33.4558 1.23572 0.617860 0.786288i \(-0.288000\pi\)
0.617860 + 0.786288i \(0.288000\pi\)
\(734\) 0 0
\(735\) −8.48528 −0.312984
\(736\) 0 0
\(737\) −4.48528 −0.165217
\(738\) 0 0
\(739\) 37.9411 1.39569 0.697843 0.716250i \(-0.254143\pi\)
0.697843 + 0.716250i \(0.254143\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −29.5980 −1.08584 −0.542922 0.839783i \(-0.682682\pi\)
−0.542922 + 0.839783i \(0.682682\pi\)
\(744\) 0 0
\(745\) −11.6569 −0.427074
\(746\) 0 0
\(747\) 30.0000 1.09764
\(748\) 0 0
\(749\) 7.31371 0.267237
\(750\) 0 0
\(751\) 16.0000 0.583848 0.291924 0.956441i \(-0.405705\pi\)
0.291924 + 0.956441i \(0.405705\pi\)
\(752\) 0 0
\(753\) 33.9411 1.23688
\(754\) 0 0
\(755\) −12.0000 −0.436725
\(756\) 0 0
\(757\) −9.31371 −0.338512 −0.169256 0.985572i \(-0.554137\pi\)
−0.169256 + 0.985572i \(0.554137\pi\)
\(758\) 0 0
\(759\) −8.00000 −0.290382
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 7.31371 0.264774
\(764\) 0 0
\(765\) −5.85786 −0.211792
\(766\) 0 0
\(767\) −65.9411 −2.38100
\(768\) 0 0
\(769\) 14.9706 0.539852 0.269926 0.962881i \(-0.413001\pi\)
0.269926 + 0.962881i \(0.413001\pi\)
\(770\) 0 0
\(771\) −37.6569 −1.35618
\(772\) 0 0
\(773\) −30.2843 −1.08925 −0.544625 0.838680i \(-0.683328\pi\)
−0.544625 + 0.838680i \(0.683328\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −20.6863 −0.742117
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −11.3137 −0.404836
\(782\) 0 0
\(783\) −43.3137 −1.54791
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) −18.9706 −0.676228 −0.338114 0.941105i \(-0.609789\pi\)
−0.338114 + 0.941105i \(0.609789\pi\)
\(788\) 0 0
\(789\) 64.9706 2.31301
\(790\) 0 0
\(791\) 16.6863 0.593296
\(792\) 0 0
\(793\) −90.9117 −3.22837
\(794\) 0 0
\(795\) 0.970563 0.0344223
\(796\) 0 0
\(797\) −12.6274 −0.447286 −0.223643 0.974671i \(-0.571795\pi\)
−0.223643 + 0.974671i \(0.571795\pi\)
\(798\) 0 0
\(799\) 3.31371 0.117231
\(800\) 0 0
\(801\) 46.5685 1.64542
\(802\) 0 0
\(803\) 6.82843 0.240970
\(804\) 0 0
\(805\) 5.65685 0.199378
\(806\) 0 0
\(807\) 15.0294 0.529061
\(808\) 0 0
\(809\) 22.9706 0.807602 0.403801 0.914847i \(-0.367689\pi\)
0.403801 + 0.914847i \(0.367689\pi\)
\(810\) 0 0
\(811\) 13.9411 0.489539 0.244770 0.969581i \(-0.421288\pi\)
0.244770 + 0.969581i \(0.421288\pi\)
\(812\) 0 0
\(813\) −43.3137 −1.51908
\(814\) 0 0
\(815\) −0.485281 −0.0169987
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −68.2843 −2.38605
\(820\) 0 0
\(821\) −18.6863 −0.652156 −0.326078 0.945343i \(-0.605727\pi\)
−0.326078 + 0.945343i \(0.605727\pi\)
\(822\) 0 0
\(823\) −36.4853 −1.27180 −0.635898 0.771773i \(-0.719370\pi\)
−0.635898 + 0.771773i \(0.719370\pi\)
\(824\) 0 0
\(825\) 2.82843 0.0984732
\(826\) 0 0
\(827\) 34.2843 1.19218 0.596090 0.802917i \(-0.296720\pi\)
0.596090 + 0.802917i \(0.296720\pi\)
\(828\) 0 0
\(829\) 18.0000 0.625166 0.312583 0.949890i \(-0.398806\pi\)
0.312583 + 0.949890i \(0.398806\pi\)
\(830\) 0 0
\(831\) −3.31371 −0.114951
\(832\) 0 0
\(833\) −3.51472 −0.121778
\(834\) 0 0
\(835\) 10.9706 0.379652
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −37.6569 −1.30006 −0.650029 0.759909i \(-0.725243\pi\)
−0.650029 + 0.759909i \(0.725243\pi\)
\(840\) 0 0
\(841\) 29.6274 1.02164
\(842\) 0 0
\(843\) 15.0294 0.517641
\(844\) 0 0
\(845\) −33.6274 −1.15682
\(846\) 0 0
\(847\) 2.00000 0.0687208
\(848\) 0 0
\(849\) −35.7157 −1.22576
\(850\) 0 0
\(851\) −10.3431 −0.354558
\(852\) 0 0
\(853\) −32.4853 −1.11227 −0.556137 0.831090i \(-0.687717\pi\)
−0.556137 + 0.831090i \(0.687717\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −48.7696 −1.66594 −0.832968 0.553321i \(-0.813360\pi\)
−0.832968 + 0.553321i \(0.813360\pi\)
\(858\) 0 0
\(859\) 32.2843 1.10153 0.550763 0.834662i \(-0.314337\pi\)
0.550763 + 0.834662i \(0.314337\pi\)
\(860\) 0 0
\(861\) −33.9411 −1.15671
\(862\) 0 0
\(863\) 14.8284 0.504766 0.252383 0.967627i \(-0.418786\pi\)
0.252383 + 0.967627i \(0.418786\pi\)
\(864\) 0 0
\(865\) −6.14214 −0.208839
\(866\) 0 0
\(867\) 44.2010 1.50115
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) −30.6274 −1.03777
\(872\) 0 0
\(873\) −38.2843 −1.29573
\(874\) 0 0
\(875\) −2.00000 −0.0676123
\(876\) 0 0
\(877\) 1.45584 0.0491604 0.0245802 0.999698i \(-0.492175\pi\)
0.0245802 + 0.999698i \(0.492175\pi\)
\(878\) 0 0
\(879\) 41.9411 1.41464
\(880\) 0 0
\(881\) −52.6274 −1.77306 −0.886531 0.462668i \(-0.846892\pi\)
−0.886531 + 0.462668i \(0.846892\pi\)
\(882\) 0 0
\(883\) −42.8284 −1.44129 −0.720646 0.693304i \(-0.756155\pi\)
−0.720646 + 0.693304i \(0.756155\pi\)
\(884\) 0 0
\(885\) 27.3137 0.918140
\(886\) 0 0
\(887\) 18.2843 0.613926 0.306963 0.951721i \(-0.400687\pi\)
0.306963 + 0.951721i \(0.400687\pi\)
\(888\) 0 0
\(889\) −31.3137 −1.05023
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −1.65685 −0.0553825
\(896\) 0 0
\(897\) −54.6274 −1.82396
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0.402020 0.0133932
\(902\) 0 0
\(903\) −33.9411 −1.12949
\(904\) 0 0
\(905\) 1.31371 0.0436691
\(906\) 0 0
\(907\) 44.4853 1.47711 0.738555 0.674193i \(-0.235509\pi\)
0.738555 + 0.674193i \(0.235509\pi\)
\(908\) 0 0
\(909\) −66.5685 −2.20794
\(910\) 0 0
\(911\) −57.9411 −1.91968 −0.959838 0.280556i \(-0.909481\pi\)
−0.959838 + 0.280556i \(0.909481\pi\)
\(912\) 0 0
\(913\) −6.00000 −0.198571
\(914\) 0 0
\(915\) 37.6569 1.24490
\(916\) 0 0
\(917\) −22.6274 −0.747223
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) −78.2254 −2.57761
\(922\) 0 0
\(923\) −77.2548 −2.54287
\(924\) 0 0
\(925\) 3.65685 0.120237
\(926\) 0 0
\(927\) −5.85786 −0.192398
\(928\) 0 0
\(929\) −17.3137 −0.568044 −0.284022 0.958818i \(-0.591669\pi\)
−0.284022 + 0.958818i \(0.591669\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 77.2548 2.52921
\(934\) 0 0
\(935\) 1.17157 0.0383145
\(936\) 0 0
\(937\) 49.4558 1.61565 0.807826 0.589421i \(-0.200644\pi\)
0.807826 + 0.589421i \(0.200644\pi\)
\(938\) 0 0
\(939\) −60.2843 −1.96730
\(940\) 0 0
\(941\) −29.3137 −0.955600 −0.477800 0.878469i \(-0.658566\pi\)
−0.477800 + 0.878469i \(0.658566\pi\)
\(942\) 0 0
\(943\) −16.9706 −0.552638
\(944\) 0 0
\(945\) 11.3137 0.368035
\(946\) 0 0
\(947\) −46.8284 −1.52172 −0.760860 0.648916i \(-0.775222\pi\)
−0.760860 + 0.648916i \(0.775222\pi\)
\(948\) 0 0
\(949\) 46.6274 1.51359
\(950\) 0 0
\(951\) −60.2843 −1.95485
\(952\) 0 0
\(953\) 58.8284 1.90564 0.952820 0.303536i \(-0.0981674\pi\)
0.952820 + 0.303536i \(0.0981674\pi\)
\(954\) 0 0
\(955\) −19.3137 −0.624977
\(956\) 0 0
\(957\) 21.6569 0.700067
\(958\) 0 0
\(959\) 45.9411 1.48352
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 18.2843 0.589202
\(964\) 0 0
\(965\) 6.82843 0.219815
\(966\) 0 0
\(967\) −18.9706 −0.610052 −0.305026 0.952344i \(-0.598665\pi\)
−0.305026 + 0.952344i \(0.598665\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 31.3137 1.00490 0.502452 0.864605i \(-0.332431\pi\)
0.502452 + 0.864605i \(0.332431\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 0 0
\(975\) 19.3137 0.618534
\(976\) 0 0
\(977\) 43.6569 1.39671 0.698353 0.715753i \(-0.253916\pi\)
0.698353 + 0.715753i \(0.253916\pi\)
\(978\) 0 0
\(979\) −9.31371 −0.297667
\(980\) 0 0
\(981\) 18.2843 0.583772
\(982\) 0 0
\(983\) 50.1421 1.59929 0.799643 0.600476i \(-0.205022\pi\)
0.799643 + 0.600476i \(0.205022\pi\)
\(984\) 0 0
\(985\) 5.17157 0.164780
\(986\) 0 0
\(987\) −16.0000 −0.509286
\(988\) 0 0
\(989\) −16.9706 −0.539633
\(990\) 0 0
\(991\) −9.94113 −0.315790 −0.157895 0.987456i \(-0.550471\pi\)
−0.157895 + 0.987456i \(0.550471\pi\)
\(992\) 0 0
\(993\) 43.3137 1.37452
\(994\) 0 0
\(995\) 21.6569 0.686568
\(996\) 0 0
\(997\) 9.45584 0.299470 0.149735 0.988726i \(-0.452158\pi\)
0.149735 + 0.988726i \(0.452158\pi\)
\(998\) 0 0
\(999\) −20.6863 −0.654485
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 880.2.a.m.1.1 2
3.2 odd 2 7920.2.a.ch.1.1 2
4.3 odd 2 55.2.a.b.1.1 2
5.2 odd 4 4400.2.b.q.4049.4 4
5.3 odd 4 4400.2.b.q.4049.1 4
5.4 even 2 4400.2.a.bn.1.2 2
8.3 odd 2 3520.2.a.bn.1.1 2
8.5 even 2 3520.2.a.bo.1.2 2
11.10 odd 2 9680.2.a.bn.1.1 2
12.11 even 2 495.2.a.b.1.2 2
20.3 even 4 275.2.b.d.199.3 4
20.7 even 4 275.2.b.d.199.2 4
20.19 odd 2 275.2.a.c.1.2 2
28.27 even 2 2695.2.a.f.1.1 2
44.3 odd 10 605.2.g.f.251.1 8
44.7 even 10 605.2.g.l.511.2 8
44.15 odd 10 605.2.g.f.511.1 8
44.19 even 10 605.2.g.l.251.2 8
44.27 odd 10 605.2.g.f.366.2 8
44.31 odd 10 605.2.g.f.81.2 8
44.35 even 10 605.2.g.l.81.1 8
44.39 even 10 605.2.g.l.366.1 8
44.43 even 2 605.2.a.d.1.2 2
52.51 odd 2 9295.2.a.g.1.2 2
60.23 odd 4 2475.2.c.l.199.2 4
60.47 odd 4 2475.2.c.l.199.3 4
60.59 even 2 2475.2.a.x.1.1 2
132.131 odd 2 5445.2.a.y.1.1 2
220.219 even 2 3025.2.a.o.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.1 2 4.3 odd 2
275.2.a.c.1.2 2 20.19 odd 2
275.2.b.d.199.2 4 20.7 even 4
275.2.b.d.199.3 4 20.3 even 4
495.2.a.b.1.2 2 12.11 even 2
605.2.a.d.1.2 2 44.43 even 2
605.2.g.f.81.2 8 44.31 odd 10
605.2.g.f.251.1 8 44.3 odd 10
605.2.g.f.366.2 8 44.27 odd 10
605.2.g.f.511.1 8 44.15 odd 10
605.2.g.l.81.1 8 44.35 even 10
605.2.g.l.251.2 8 44.19 even 10
605.2.g.l.366.1 8 44.39 even 10
605.2.g.l.511.2 8 44.7 even 10
880.2.a.m.1.1 2 1.1 even 1 trivial
2475.2.a.x.1.1 2 60.59 even 2
2475.2.c.l.199.2 4 60.23 odd 4
2475.2.c.l.199.3 4 60.47 odd 4
2695.2.a.f.1.1 2 28.27 even 2
3025.2.a.o.1.1 2 220.219 even 2
3520.2.a.bn.1.1 2 8.3 odd 2
3520.2.a.bo.1.2 2 8.5 even 2
4400.2.a.bn.1.2 2 5.4 even 2
4400.2.b.q.4049.1 4 5.3 odd 4
4400.2.b.q.4049.4 4 5.2 odd 4
5445.2.a.y.1.1 2 132.131 odd 2
7920.2.a.ch.1.1 2 3.2 odd 2
9295.2.a.g.1.2 2 52.51 odd 2
9680.2.a.bn.1.1 2 11.10 odd 2