Properties

Label 4400.2.b.q.4049.4
Level $4400$
Weight $2$
Character 4400.4049
Analytic conductor $35.134$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 4400 = 2^{4} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4400.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(35.1341768894\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.4
Root \(-0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 4400.4049
Dual form 4400.2.b.q.4049.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.82843i q^{3} +2.00000i q^{7} -5.00000 q^{9} +O(q^{10})\) \(q+2.82843i q^{3} +2.00000i q^{7} -5.00000 q^{9} -1.00000 q^{11} +6.82843i q^{13} +1.17157i q^{17} -5.65685 q^{21} +2.82843i q^{23} -5.65685i q^{27} -7.65685 q^{29} -2.82843i q^{33} +3.65685i q^{37} -19.3137 q^{39} +6.00000 q^{41} -6.00000i q^{43} +2.82843i q^{47} +3.00000 q^{49} -3.31371 q^{51} -0.343146i q^{53} -9.65685 q^{59} +13.3137 q^{61} -10.0000i q^{63} +4.48528i q^{67} -8.00000 q^{69} +11.3137 q^{71} +6.82843i q^{73} -2.00000i q^{77} +4.00000 q^{79} +1.00000 q^{81} -6.00000i q^{83} -21.6569i q^{87} -9.31371 q^{89} -13.6569 q^{91} -7.65685i q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 20 q^{9} + O(q^{10}) \) \( 4 q - 20 q^{9} - 4 q^{11} - 8 q^{29} - 32 q^{39} + 24 q^{41} + 12 q^{49} + 32 q^{51} - 16 q^{59} + 8 q^{61} - 32 q^{69} + 16 q^{79} + 4 q^{81} + 8 q^{89} - 32 q^{91} + 20 q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4400\mathbb{Z}\right)^\times\).

\(n\) \(177\) \(1201\) \(2751\) \(3301\)
\(\chi(n)\) \(-1\) \(1\) \(1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 2.82843i 1.63299i 0.577350 + 0.816497i \(0.304087\pi\)
−0.577350 + 0.816497i \(0.695913\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) −5.00000 −1.66667
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 6.82843i 1.89386i 0.321433 + 0.946932i \(0.395836\pi\)
−0.321433 + 0.946932i \(0.604164\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.17157i 0.284148i 0.989856 + 0.142074i \(0.0453771\pi\)
−0.989856 + 0.142074i \(0.954623\pi\)
\(18\) 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
\(23\) 2.82843i 0.589768i 0.955533 + 0.294884i \(0.0952810\pi\)
−0.955533 + 0.294884i \(0.904719\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.65685i − 1.08866i
\(28\) 0 0
\(29\) −7.65685 −1.42184 −0.710921 0.703272i \(-0.751722\pi\)
−0.710921 + 0.703272i \(0.751722\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) − 2.82843i − 0.492366i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 3.65685i 0.601183i 0.953753 + 0.300592i \(0.0971841\pi\)
−0.953753 + 0.300592i \(0.902816\pi\)
\(38\) 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
\(43\) − 6.00000i − 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.82843i 0.412568i 0.978492 + 0.206284i \(0.0661372\pi\)
−0.978492 + 0.206284i \(0.933863\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) −3.31371 −0.464012
\(52\) 0 0
\(53\) − 0.343146i − 0.0471347i −0.999722 0.0235673i \(-0.992498\pi\)
0.999722 0.0235673i \(-0.00750241\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −9.65685 −1.25722 −0.628608 0.777723i \(-0.716375\pi\)
−0.628608 + 0.777723i \(0.716375\pi\)
\(60\) 0 0
\(61\) 13.3137 1.70465 0.852323 0.523016i \(-0.175193\pi\)
0.852323 + 0.523016i \(0.175193\pi\)
\(62\) 0 0
\(63\) − 10.0000i − 1.25988i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.48528i 0.547964i 0.961735 + 0.273982i \(0.0883409\pi\)
−0.961735 + 0.273982i \(0.911659\pi\)
\(68\) 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
\(73\) 6.82843i 0.799207i 0.916688 + 0.399603i \(0.130852\pi\)
−0.916688 + 0.399603i \(0.869148\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.00000i − 0.227921i
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) − 6.00000i − 0.658586i −0.944228 0.329293i \(-0.893190\pi\)
0.944228 0.329293i \(-0.106810\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 21.6569i − 2.32186i
\(88\) 0 0
\(89\) −9.31371 −0.987251 −0.493626 0.869675i \(-0.664329\pi\)
−0.493626 + 0.869675i \(0.664329\pi\)
\(90\) 0 0
\(91\) −13.6569 −1.43163
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 7.65685i − 0.777436i −0.921357 0.388718i \(-0.872918\pi\)
0.921357 0.388718i \(-0.127082\pi\)
\(98\) 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
\(103\) 1.17157i 0.115439i 0.998333 + 0.0577193i \(0.0183828\pi\)
−0.998333 + 0.0577193i \(0.981617\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.65685i 0.353521i 0.984254 + 0.176761i \(0.0565619\pi\)
−0.984254 + 0.176761i \(0.943438\pi\)
\(108\) 0 0
\(109\) −3.65685 −0.350263 −0.175132 0.984545i \(-0.556035\pi\)
−0.175132 + 0.984545i \(0.556035\pi\)
\(110\) 0 0
\(111\) −10.3431 −0.981728
\(112\) 0 0
\(113\) − 8.34315i − 0.784857i −0.919782 0.392429i \(-0.871635\pi\)
0.919782 0.392429i \(-0.128365\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 34.1421i − 3.15644i
\(118\) 0 0
\(119\) −2.34315 −0.214796
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 16.9706i 1.53018i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 15.6569i − 1.38932i −0.719338 0.694661i \(-0.755555\pi\)
0.719338 0.694661i \(-0.244445\pi\)
\(128\) 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
\(135\) 0 0
\(136\) 0 0
\(137\) 22.9706i 1.96251i 0.192720 + 0.981254i \(0.438269\pi\)
−0.192720 + 0.981254i \(0.561731\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −8.00000 −0.673722
\(142\) 0 0
\(143\) − 6.82843i − 0.571022i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 8.48528i 0.699854i
\(148\) 0 0
\(149\) −11.6569 −0.954967 −0.477483 0.878641i \(-0.658451\pi\)
−0.477483 + 0.878641i \(0.658451\pi\)
\(150\) 0 0
\(151\) 12.0000 0.976546 0.488273 0.872691i \(-0.337627\pi\)
0.488273 + 0.872691i \(0.337627\pi\)
\(152\) 0 0
\(153\) − 5.85786i − 0.473580i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 14.0000i − 1.11732i −0.829396 0.558661i \(-0.811315\pi\)
0.829396 0.558661i \(-0.188685\pi\)
\(158\) 0 0
\(159\) 0.970563 0.0769706
\(160\) 0 0
\(161\) −5.65685 −0.445823
\(162\) 0 0
\(163\) − 0.485281i − 0.0380102i −0.999819 0.0190051i \(-0.993950\pi\)
0.999819 0.0190051i \(-0.00604987\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 10.9706i − 0.848928i −0.905445 0.424464i \(-0.860463\pi\)
0.905445 0.424464i \(-0.139537\pi\)
\(168\) 0 0
\(169\) −33.6274 −2.58672
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 6.14214i − 0.466978i −0.972359 0.233489i \(-0.924986\pi\)
0.972359 0.233489i \(-0.0750143\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 27.3137i − 2.05302i
\(178\) 0 0
\(179\) −1.65685 −0.123839 −0.0619196 0.998081i \(-0.519722\pi\)
−0.0619196 + 0.998081i \(0.519722\pi\)
\(180\) 0 0
\(181\) −1.31371 −0.0976472 −0.0488236 0.998807i \(-0.515547\pi\)
−0.0488236 + 0.998807i \(0.515547\pi\)
\(182\) 0 0
\(183\) 37.6569i 2.78367i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 1.17157i − 0.0856739i
\(188\) 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
\(193\) 6.82843i 0.491521i 0.969331 + 0.245760i \(0.0790377\pi\)
−0.969331 + 0.245760i \(0.920962\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 5.17157i − 0.368459i −0.982883 0.184230i \(-0.941021\pi\)
0.982883 0.184230i \(-0.0589790\pi\)
\(198\) 0 0
\(199\) 21.6569 1.53521 0.767607 0.640921i \(-0.221447\pi\)
0.767607 + 0.640921i \(0.221447\pi\)
\(200\) 0 0
\(201\) −12.6863 −0.894822
\(202\) 0 0
\(203\) − 15.3137i − 1.07481i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 14.1421i − 0.982946i
\(208\) 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
\(213\) 32.0000i 2.19260i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −19.3137 −1.30510
\(220\) 0 0
\(221\) −8.00000 −0.538138
\(222\) 0 0
\(223\) − 5.17157i − 0.346314i −0.984894 0.173157i \(-0.944603\pi\)
0.984894 0.173157i \(-0.0553968\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 2.68629i − 0.178295i −0.996018 0.0891477i \(-0.971586\pi\)
0.996018 0.0891477i \(-0.0284143\pi\)
\(228\) 0 0
\(229\) 21.3137 1.40845 0.704225 0.709977i \(-0.251295\pi\)
0.704225 + 0.709977i \(0.251295\pi\)
\(230\) 0 0
\(231\) 5.65685 0.372194
\(232\) 0 0
\(233\) − 22.1421i − 1.45058i −0.688444 0.725290i \(-0.741706\pi\)
0.688444 0.725290i \(-0.258294\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 11.3137i 0.734904i
\(238\) 0 0
\(239\) −0.686292 −0.0443925 −0.0221963 0.999754i \(-0.507066\pi\)
−0.0221963 + 0.999754i \(0.507066\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 0 0
\(243\) − 14.1421i − 0.907218i
\(244\) 0 0
\(245\) 0 0
\(246\) 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
\(253\) − 2.82843i − 0.177822i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 13.3137i 0.830486i 0.909710 + 0.415243i \(0.136304\pi\)
−0.909710 + 0.415243i \(0.863696\pi\)
\(258\) 0 0
\(259\) −7.31371 −0.454452
\(260\) 0 0
\(261\) 38.2843 2.36974
\(262\) 0 0
\(263\) 22.9706i 1.41643i 0.705999 + 0.708213i \(0.250498\pi\)
−0.705999 + 0.708213i \(0.749502\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 26.3431i − 1.61217i
\(268\) 0 0
\(269\) 5.31371 0.323983 0.161991 0.986792i \(-0.448208\pi\)
0.161991 + 0.986792i \(0.448208\pi\)
\(270\) 0 0
\(271\) 15.3137 0.930242 0.465121 0.885247i \(-0.346011\pi\)
0.465121 + 0.885247i \(0.346011\pi\)
\(272\) 0 0
\(273\) − 38.6274i − 2.33784i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.17157i 0.0703930i 0.999380 + 0.0351965i \(0.0112057\pi\)
−0.999380 + 0.0351965i \(0.988794\pi\)
\(278\) 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
\(283\) − 12.6274i − 0.750622i −0.926899 0.375311i \(-0.877536\pi\)
0.926899 0.375311i \(-0.122464\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.0000i 0.708338i
\(288\) 0 0
\(289\) 15.6274 0.919260
\(290\) 0 0
\(291\) 21.6569 1.26955
\(292\) 0 0
\(293\) 14.8284i 0.866286i 0.901325 + 0.433143i \(0.142595\pi\)
−0.901325 + 0.433143i \(0.857405\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 5.65685i 0.328244i
\(298\) 0 0
\(299\) −19.3137 −1.11694
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) 0 0
\(303\) − 37.6569i − 2.16333i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 27.6569i 1.57846i 0.614098 + 0.789230i \(0.289520\pi\)
−0.614098 + 0.789230i \(0.710480\pi\)
\(308\) 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
\(313\) − 21.3137i − 1.20472i −0.798224 0.602361i \(-0.794227\pi\)
0.798224 0.602361i \(-0.205773\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 21.3137i 1.19710i 0.801087 + 0.598549i \(0.204256\pi\)
−0.801087 + 0.598549i \(0.795744\pi\)
\(318\) 0 0
\(319\) 7.65685 0.428702
\(320\) 0 0
\(321\) −10.3431 −0.577298
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 10.3431i − 0.571977i
\(328\) 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
\(333\) − 18.2843i − 1.00197i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 3.51472i − 0.191459i −0.995407 0.0957295i \(-0.969482\pi\)
0.995407 0.0957295i \(-0.0305184\pi\)
\(338\) 0 0
\(339\) 23.5980 1.28167
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 22.9706i 1.23312i 0.787306 + 0.616562i \(0.211475\pi\)
−0.787306 + 0.616562i \(0.788525\pi\)
\(348\) 0 0
\(349\) 6.97056 0.373126 0.186563 0.982443i \(-0.440265\pi\)
0.186563 + 0.982443i \(0.440265\pi\)
\(350\) 0 0
\(351\) 38.6274 2.06178
\(352\) 0 0
\(353\) 1.31371i 0.0699216i 0.999389 + 0.0349608i \(0.0111306\pi\)
−0.999389 + 0.0349608i \(0.988869\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 6.62742i − 0.350760i
\(358\) 0 0
\(359\) 23.3137 1.23045 0.615225 0.788351i \(-0.289065\pi\)
0.615225 + 0.788351i \(0.289065\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 2.82843i 0.148454i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 8.48528i − 0.442928i −0.975169 0.221464i \(-0.928916\pi\)
0.975169 0.221464i \(-0.0710835\pi\)
\(368\) 0 0
\(369\) −30.0000 −1.56174
\(370\) 0 0
\(371\) 0.686292 0.0356305
\(372\) 0 0
\(373\) 3.79899i 0.196704i 0.995152 + 0.0983521i \(0.0313571\pi\)
−0.995152 + 0.0983521i \(0.968643\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 52.2843i − 2.69278i
\(378\) 0 0
\(379\) 22.3431 1.14769 0.573845 0.818964i \(-0.305451\pi\)
0.573845 + 0.818964i \(0.305451\pi\)
\(380\) 0 0
\(381\) 44.2843 2.26875
\(382\) 0 0
\(383\) − 34.1421i − 1.74458i −0.488987 0.872291i \(-0.662634\pi\)
0.488987 0.872291i \(-0.337366\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 30.0000i 1.52499i
\(388\) 0 0
\(389\) 24.6274 1.24866 0.624330 0.781161i \(-0.285372\pi\)
0.624330 + 0.781161i \(0.285372\pi\)
\(390\) 0 0
\(391\) −3.31371 −0.167581
\(392\) 0 0
\(393\) − 32.0000i − 1.61419i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 13.3137i 0.668196i 0.942538 + 0.334098i \(0.108432\pi\)
−0.942538 + 0.334098i \(0.891568\pi\)
\(398\) 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
\(405\) 0 0
\(406\) 0 0
\(407\) − 3.65685i − 0.181264i
\(408\) 0 0
\(409\) −34.9706 −1.72918 −0.864592 0.502475i \(-0.832423\pi\)
−0.864592 + 0.502475i \(0.832423\pi\)
\(410\) 0 0
\(411\) −64.9706 −3.20476
\(412\) 0 0
\(413\) − 19.3137i − 0.950365i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 11.3137i − 0.554035i
\(418\) 0 0
\(419\) −14.3431 −0.700709 −0.350354 0.936617i \(-0.613939\pi\)
−0.350354 + 0.936617i \(0.613939\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 0 0
\(423\) − 14.1421i − 0.687614i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 26.6274i 1.28859i
\(428\) 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
\(433\) − 3.65685i − 0.175737i −0.996132 0.0878686i \(-0.971994\pi\)
0.996132 0.0878686i \(-0.0280056\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −16.0000 −0.763638 −0.381819 0.924237i \(-0.624702\pi\)
−0.381819 + 0.924237i \(0.624702\pi\)
\(440\) 0 0
\(441\) −15.0000 −0.714286
\(442\) 0 0
\(443\) − 21.1716i − 1.00589i −0.864318 0.502946i \(-0.832249\pi\)
0.864318 0.502946i \(-0.167751\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 32.9706i − 1.55945i
\(448\) 0 0
\(449\) 16.6274 0.784696 0.392348 0.919817i \(-0.371663\pi\)
0.392348 + 0.919817i \(0.371663\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) 0 0
\(453\) 33.9411i 1.59469i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 16.4853i − 0.771149i −0.922677 0.385574i \(-0.874003\pi\)
0.922677 0.385574i \(-0.125997\pi\)
\(458\) 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
\(463\) 22.1421i 1.02903i 0.857481 + 0.514516i \(0.172028\pi\)
−0.857481 + 0.514516i \(0.827972\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 9.17157i 0.424410i 0.977225 + 0.212205i \(0.0680644\pi\)
−0.977225 + 0.212205i \(0.931936\pi\)
\(468\) 0 0
\(469\) −8.97056 −0.414222
\(470\) 0 0
\(471\) 39.5980 1.82458
\(472\) 0 0
\(473\) 6.00000i 0.275880i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.71573i 0.0785578i
\(478\) 0 0
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) −24.9706 −1.13856
\(482\) 0 0
\(483\) − 16.0000i − 0.728025i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 7.51472i 0.340524i 0.985399 + 0.170262i \(0.0544615\pi\)
−0.985399 + 0.170262i \(0.945539\pi\)
\(488\) 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
\(493\) − 8.97056i − 0.404014i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 22.6274i 1.01498i
\(498\) 0 0
\(499\) −1.65685 −0.0741710 −0.0370855 0.999312i \(-0.511807\pi\)
−0.0370855 + 0.999312i \(0.511807\pi\)
\(500\) 0 0
\(501\) 31.0294 1.38629
\(502\) 0 0
\(503\) − 28.6274i − 1.27643i −0.769857 0.638217i \(-0.779672\pi\)
0.769857 0.638217i \(-0.220328\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 95.1127i − 4.22410i
\(508\) 0 0
\(509\) −9.31371 −0.412823 −0.206411 0.978465i \(-0.566179\pi\)
−0.206411 + 0.978465i \(0.566179\pi\)
\(510\) 0 0
\(511\) −13.6569 −0.604144
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.82843i − 0.124394i
\(518\) 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
\(523\) 37.5980i 1.64404i 0.569455 + 0.822022i \(0.307154\pi\)
−0.569455 + 0.822022i \(0.692846\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 48.2843 2.09536
\(532\) 0 0
\(533\) 40.9706i 1.77463i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 4.68629i − 0.202228i
\(538\) 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
\(543\) − 3.71573i − 0.159457i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 34.0000i 1.45374i 0.686778 + 0.726868i \(0.259025\pi\)
−0.686778 + 0.726868i \(0.740975\pi\)
\(548\) 0 0
\(549\) −66.5685 −2.84108
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 8.00000i 0.340195i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 38.1421i 1.61613i 0.589090 + 0.808067i \(0.299486\pi\)
−0.589090 + 0.808067i \(0.700514\pi\)
\(558\) 0 0
\(559\) 40.9706 1.73287
\(560\) 0 0
\(561\) 3.31371 0.139905
\(562\) 0 0
\(563\) 11.6569i 0.491278i 0.969361 + 0.245639i \(0.0789977\pi\)
−0.969361 + 0.245639i \(0.921002\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.00000i 0.0839921i
\(568\) 0 0
\(569\) −20.3431 −0.852829 −0.426415 0.904528i \(-0.640224\pi\)
−0.426415 + 0.904528i \(0.640224\pi\)
\(570\) 0 0
\(571\) −45.9411 −1.92258 −0.961288 0.275545i \(-0.911142\pi\)
−0.961288 + 0.275545i \(0.911142\pi\)
\(572\) 0 0
\(573\) 54.6274i 2.28209i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 6.97056i 0.290188i 0.989418 + 0.145094i \(0.0463485\pi\)
−0.989418 + 0.145094i \(0.953651\pi\)
\(578\) 0 0
\(579\) −19.3137 −0.802650
\(580\) 0 0
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 0.343146i 0.0142116i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 26.1421i − 1.07900i −0.841985 0.539501i \(-0.818613\pi\)
0.841985 0.539501i \(-0.181387\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 14.6274 0.601692
\(592\) 0 0
\(593\) − 20.4853i − 0.841230i −0.907239 0.420615i \(-0.861814\pi\)
0.907239 0.420615i \(-0.138186\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 61.2548i 2.50699i
\(598\) 0 0
\(599\) 5.65685 0.231133 0.115566 0.993300i \(-0.463132\pi\)
0.115566 + 0.993300i \(0.463132\pi\)
\(600\) 0 0
\(601\) −43.9411 −1.79240 −0.896198 0.443654i \(-0.853682\pi\)
−0.896198 + 0.443654i \(0.853682\pi\)
\(602\) 0 0
\(603\) − 22.4264i − 0.913274i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 18.2843i 0.742136i 0.928606 + 0.371068i \(0.121008\pi\)
−0.928606 + 0.371068i \(0.878992\pi\)
\(608\) 0 0
\(609\) 43.3137 1.75516
\(610\) 0 0
\(611\) −19.3137 −0.781349
\(612\) 0 0
\(613\) − 25.4558i − 1.02815i −0.857745 0.514076i \(-0.828135\pi\)
0.857745 0.514076i \(-0.171865\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11.6569i 0.469287i 0.972081 + 0.234644i \(0.0753923\pi\)
−0.972081 + 0.234644i \(0.924608\pi\)
\(618\) 0 0
\(619\) −25.6569 −1.03124 −0.515618 0.856819i \(-0.672438\pi\)
−0.515618 + 0.856819i \(0.672438\pi\)
\(620\) 0 0
\(621\) 16.0000 0.642058
\(622\) 0 0
\(623\) − 18.6274i − 0.746292i
\(624\) 0 0
\(625\) 0 0
\(626\) 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
\(633\) 45.2548i 1.79872i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 20.4853i 0.811656i
\(638\) 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
\(643\) 49.4558i 1.95035i 0.221440 + 0.975174i \(0.428924\pi\)
−0.221440 + 0.975174i \(0.571076\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 35.1127i 1.38042i 0.723608 + 0.690211i \(0.242482\pi\)
−0.723608 + 0.690211i \(0.757518\pi\)
\(648\) 0 0
\(649\) 9.65685 0.379065
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 0.343146i − 0.0134283i −0.999977 0.00671417i \(-0.997863\pi\)
0.999977 0.00671417i \(-0.00213720\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 34.1421i − 1.33201i
\(658\) 0 0
\(659\) −21.9411 −0.854705 −0.427352 0.904085i \(-0.640554\pi\)
−0.427352 + 0.904085i \(0.640554\pi\)
\(660\) 0 0
\(661\) −0.627417 −0.0244037 −0.0122018 0.999926i \(-0.503884\pi\)
−0.0122018 + 0.999926i \(0.503884\pi\)
\(662\) 0 0
\(663\) − 22.6274i − 0.878776i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 21.6569i − 0.838557i
\(668\) 0 0
\(669\) 14.6274 0.565529
\(670\) 0 0
\(671\) −13.3137 −0.513970
\(672\) 0 0
\(673\) − 4.48528i − 0.172895i −0.996256 0.0864474i \(-0.972449\pi\)
0.996256 0.0864474i \(-0.0275515\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.1716i 0.659957i 0.943988 + 0.329979i \(0.107042\pi\)
−0.943988 + 0.329979i \(0.892958\pi\)
\(678\) 0 0
\(679\) 15.3137 0.587686
\(680\) 0 0
\(681\) 7.59798 0.291155
\(682\) 0 0
\(683\) 31.7990i 1.21675i 0.793648 + 0.608377i \(0.208179\pi\)
−0.793648 + 0.608377i \(0.791821\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 60.2843i 2.29999i
\(688\) 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
\(693\) 10.0000i 0.379869i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 7.02944i 0.266259i
\(698\) 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
\(705\) 0 0
\(706\) 0 0
\(707\) − 26.6274i − 1.00143i
\(708\) 0 0
\(709\) 20.6274 0.774679 0.387339 0.921937i \(-0.373394\pi\)
0.387339 + 0.921937i \(0.373394\pi\)
\(710\) 0 0
\(711\) −20.0000 −0.750059
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 1.94113i − 0.0724927i
\(718\) 0 0
\(719\) 29.6569 1.10601 0.553007 0.833177i \(-0.313480\pi\)
0.553007 + 0.833177i \(0.313480\pi\)
\(720\) 0 0
\(721\) −2.34315 −0.0872633
\(722\) 0 0
\(723\) 16.9706i 0.631142i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 36.4853i 1.35316i 0.736367 + 0.676582i \(0.236540\pi\)
−0.736367 + 0.676582i \(0.763460\pi\)
\(728\) 0 0
\(729\) 43.0000 1.59259
\(730\) 0 0
\(731\) 7.02944 0.259993
\(732\) 0 0
\(733\) − 33.4558i − 1.23572i −0.786288 0.617860i \(-0.788000\pi\)
0.786288 0.617860i \(-0.212000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 4.48528i − 0.165217i
\(738\) 0 0
\(739\) −37.9411 −1.39569 −0.697843 0.716250i \(-0.745857\pi\)
−0.697843 + 0.716250i \(0.745857\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 29.5980i 1.08584i 0.839783 + 0.542922i \(0.182682\pi\)
−0.839783 + 0.542922i \(0.817318\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 30.0000i 1.09764i
\(748\) 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
\(753\) − 33.9411i − 1.23688i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 9.31371i − 0.338512i −0.985572 0.169256i \(-0.945863\pi\)
0.985572 0.169256i \(-0.0541365\pi\)
\(758\) 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
\(763\) − 7.31371i − 0.264774i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 65.9411i − 2.38100i
\(768\) 0 0
\(769\) −14.9706 −0.539852 −0.269926 0.962881i \(-0.586999\pi\)
−0.269926 + 0.962881i \(0.586999\pi\)
\(770\) 0 0
\(771\) −37.6569 −1.35618
\(772\) 0 0
\(773\) 30.2843i 1.08925i 0.838680 + 0.544625i \(0.183328\pi\)
−0.838680 + 0.544625i \(0.816672\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 20.6863i − 0.742117i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) −11.3137 −0.404836
\(782\) 0 0
\(783\) 43.3137i 1.54791i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 18.9706i − 0.676228i −0.941105 0.338114i \(-0.890211\pi\)
0.941105 0.338114i \(-0.109789\pi\)
\(788\) 0 0
\(789\) −64.9706 −2.31301
\(790\) 0 0
\(791\) 16.6863 0.593296
\(792\) 0 0
\(793\) 90.9117i 3.22837i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 12.6274i − 0.447286i −0.974671 0.223643i \(-0.928205\pi\)
0.974671 0.223643i \(-0.0717950\pi\)
\(798\) 0 0
\(799\) −3.31371 −0.117231
\(800\) 0 0
\(801\) 46.5685 1.64542
\(802\) 0 0
\(803\) − 6.82843i − 0.240970i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 15.0294i 0.529061i
\(808\) 0 0
\(809\) −22.9706 −0.807602 −0.403801 0.914847i \(-0.632311\pi\)
−0.403801 + 0.914847i \(0.632311\pi\)
\(810\) 0 0
\(811\) 13.9411 0.489539 0.244770 0.969581i \(-0.421288\pi\)
0.244770 + 0.969581i \(0.421288\pi\)
\(812\) 0 0
\(813\) 43.3137i 1.51908i
\(814\) 0 0
\(815\) 0 0
\(816\) 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
\(823\) 36.4853i 1.27180i 0.771773 + 0.635898i \(0.219370\pi\)
−0.771773 + 0.635898i \(0.780630\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 34.2843i 1.19218i 0.802917 + 0.596090i \(0.203280\pi\)
−0.802917 + 0.596090i \(0.796720\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) −3.31371 −0.114951
\(832\) 0 0
\(833\) 3.51472i 0.121778i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 37.6569 1.30006 0.650029 0.759909i \(-0.274757\pi\)
0.650029 + 0.759909i \(0.274757\pi\)
\(840\) 0 0
\(841\) 29.6274 1.02164
\(842\) 0 0
\(843\) − 15.0294i − 0.517641i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) 0 0
\(849\) 35.7157 1.22576
\(850\) 0 0
\(851\) −10.3431 −0.354558
\(852\) 0 0
\(853\) 32.4853i 1.11227i 0.831090 + 0.556137i \(0.187717\pi\)
−0.831090 + 0.556137i \(0.812283\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 48.7696i − 1.66594i −0.553321 0.832968i \(-0.686640\pi\)
0.553321 0.832968i \(-0.313360\pi\)
\(858\) 0 0
\(859\) −32.2843 −1.10153 −0.550763 0.834662i \(-0.685663\pi\)
−0.550763 + 0.834662i \(0.685663\pi\)
\(860\) 0 0
\(861\) −33.9411 −1.15671
\(862\) 0 0
\(863\) − 14.8284i − 0.504766i −0.967627 0.252383i \(-0.918786\pi\)
0.967627 0.252383i \(-0.0812142\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 44.2010i 1.50115i
\(868\) 0 0
\(869\) −4.00000 −0.135691
\(870\) 0 0
\(871\) −30.6274 −1.03777
\(872\) 0 0
\(873\) 38.2843i 1.29573i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.45584i 0.0491604i 0.999698 + 0.0245802i \(0.00782490\pi\)
−0.999698 + 0.0245802i \(0.992175\pi\)
\(878\) 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
\(883\) 42.8284i 1.44129i 0.693304 + 0.720646i \(0.256155\pi\)
−0.693304 + 0.720646i \(0.743845\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 18.2843i 0.613926i 0.951721 + 0.306963i \(0.0993127\pi\)
−0.951721 + 0.306963i \(0.900687\pi\)
\(888\) 0 0
\(889\) 31.3137 1.05023
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 54.6274i − 1.82396i
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0.402020 0.0133932
\(902\) 0 0
\(903\) 33.9411i 1.12949i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 44.4853i 1.47711i 0.674193 + 0.738555i \(0.264491\pi\)
−0.674193 + 0.738555i \(0.735509\pi\)
\(908\) 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
\(913\) 6.00000i 0.198571i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 22.6274i − 0.747223i
\(918\) 0 0
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) −78.2254 −2.57761
\(922\) 0 0
\(923\) 77.2548i 2.54287i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 5.85786i − 0.192398i
\(928\) 0 0
\(929\) 17.3137 0.568044 0.284022 0.958818i \(-0.408331\pi\)
0.284022 + 0.958818i \(0.408331\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) − 77.2548i − 2.52921i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 49.4558i 1.61565i 0.589421 + 0.807826i \(0.299356\pi\)
−0.589421 + 0.807826i \(0.700644\pi\)
\(938\) 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
\(943\) 16.9706i 0.552638i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 46.8284i − 1.52172i −0.648916 0.760860i \(-0.724778\pi\)
0.648916 0.760860i \(-0.275222\pi\)
\(948\) 0 0
\(949\) −46.6274 −1.51359
\(950\) 0 0
\(951\) −60.2843 −1.95485
\(952\) 0 0
\(953\) − 58.8284i − 1.90564i −0.303536 0.952820i \(-0.598167\pi\)
0.303536 0.952820i \(-0.401833\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 21.6569i 0.700067i
\(958\) 0 0
\(959\) −45.9411 −1.48352
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) − 18.2843i − 0.589202i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 18.9706i − 0.610052i −0.952344 0.305026i \(-0.901335\pi\)
0.952344 0.305026i \(-0.0986652\pi\)
\(968\) 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
\(973\) − 8.00000i − 0.256468i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 43.6569i 1.39671i 0.715753 + 0.698353i \(0.246084\pi\)
−0.715753 + 0.698353i \(0.753916\pi\)
\(978\) 0 0
\(979\) 9.31371 0.297667
\(980\) 0 0
\(981\) 18.2843 0.583772
\(982\) 0 0
\(983\) − 50.1421i − 1.59929i −0.600476 0.799643i \(-0.705022\pi\)
0.600476 0.799643i \(-0.294978\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 16.0000i − 0.509286i
\(988\) 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
\(993\) − 43.3137i − 1.37452i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 9.45584i 0.299470i 0.988726 + 0.149735i \(0.0478420\pi\)
−0.988726 + 0.149735i \(0.952158\pi\)
\(998\) 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 4400.2.b.q.4049.4 4
4.3 odd 2 275.2.b.d.199.2 4
5.2 odd 4 4400.2.a.bn.1.2 2
5.3 odd 4 880.2.a.m.1.1 2
5.4 even 2 inner 4400.2.b.q.4049.1 4
12.11 even 2 2475.2.c.l.199.3 4
15.8 even 4 7920.2.a.ch.1.1 2
20.3 even 4 55.2.a.b.1.1 2
20.7 even 4 275.2.a.c.1.2 2
20.19 odd 2 275.2.b.d.199.3 4
40.3 even 4 3520.2.a.bn.1.1 2
40.13 odd 4 3520.2.a.bo.1.2 2
55.43 even 4 9680.2.a.bn.1.1 2
60.23 odd 4 495.2.a.b.1.2 2
60.47 odd 4 2475.2.a.x.1.1 2
60.59 even 2 2475.2.c.l.199.2 4
140.83 odd 4 2695.2.a.f.1.1 2
220.3 even 20 605.2.g.f.251.1 8
220.43 odd 4 605.2.a.d.1.2 2
220.63 odd 20 605.2.g.l.251.2 8
220.83 odd 20 605.2.g.l.366.1 8
220.87 odd 4 3025.2.a.o.1.1 2
220.103 even 20 605.2.g.f.511.1 8
220.123 odd 20 605.2.g.l.81.1 8
220.163 even 20 605.2.g.f.81.2 8
220.183 odd 20 605.2.g.l.511.2 8
220.203 even 20 605.2.g.f.366.2 8
260.103 even 4 9295.2.a.g.1.2 2
660.263 even 4 5445.2.a.y.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.1 2 20.3 even 4
275.2.a.c.1.2 2 20.7 even 4
275.2.b.d.199.2 4 4.3 odd 2
275.2.b.d.199.3 4 20.19 odd 2
495.2.a.b.1.2 2 60.23 odd 4
605.2.a.d.1.2 2 220.43 odd 4
605.2.g.f.81.2 8 220.163 even 20
605.2.g.f.251.1 8 220.3 even 20
605.2.g.f.366.2 8 220.203 even 20
605.2.g.f.511.1 8 220.103 even 20
605.2.g.l.81.1 8 220.123 odd 20
605.2.g.l.251.2 8 220.63 odd 20
605.2.g.l.366.1 8 220.83 odd 20
605.2.g.l.511.2 8 220.183 odd 20
880.2.a.m.1.1 2 5.3 odd 4
2475.2.a.x.1.1 2 60.47 odd 4
2475.2.c.l.199.2 4 60.59 even 2
2475.2.c.l.199.3 4 12.11 even 2
2695.2.a.f.1.1 2 140.83 odd 4
3025.2.a.o.1.1 2 220.87 odd 4
3520.2.a.bn.1.1 2 40.3 even 4
3520.2.a.bo.1.2 2 40.13 odd 4
4400.2.a.bn.1.2 2 5.2 odd 4
4400.2.b.q.4049.1 4 5.4 even 2 inner
4400.2.b.q.4049.4 4 1.1 even 1 trivial
5445.2.a.y.1.1 2 660.263 even 4
7920.2.a.ch.1.1 2 15.8 even 4
9295.2.a.g.1.2 2 260.103 even 4
9680.2.a.bn.1.1 2 55.43 even 4