Properties

Label 3800.2.a.bc.1.4
Level $3800$
Weight $2$
Character 3800.1
Self dual yes
Analytic conductor $30.343$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3800 = 2^{3} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3800.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(30.3431527681\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 2 x^{5} - 10 x^{4} + 16 x^{3} + 15 x^{2} - 14 x - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.848258\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.848258 q^{3} +1.74484 q^{7} -2.28046 q^{9} +O(q^{10})\) \(q+0.848258 q^{3} +1.74484 q^{7} -2.28046 q^{9} +5.92425 q^{11} +6.78582 q^{13} +1.86024 q^{17} -1.00000 q^{19} +1.48007 q^{21} +5.94357 q^{23} -4.47919 q^{27} +3.29977 q^{29} -5.75242 q^{31} +5.02529 q^{33} -4.36379 q^{37} +5.75613 q^{39} +7.12500 q^{41} -6.98455 q^{43} -4.02529 q^{47} -3.95555 q^{49} +1.57796 q^{51} -9.19015 q^{53} -0.848258 q^{57} +2.51553 q^{59} -2.49621 q^{61} -3.97903 q^{63} +6.90485 q^{67} +5.04168 q^{69} +1.27288 q^{71} +12.1217 q^{73} +10.3368 q^{77} -13.8376 q^{79} +3.04187 q^{81} -4.94955 q^{83} +2.79906 q^{87} +15.6067 q^{89} +11.8401 q^{91} -4.87953 q^{93} -15.7764 q^{97} -13.5100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{3} + 2 q^{7} + 6 q^{9} + O(q^{10}) \) \( 6 q + 2 q^{3} + 2 q^{7} + 6 q^{9} + 3 q^{11} - q^{13} - 14 q^{17} - 6 q^{19} + 15 q^{21} + 12 q^{23} + 8 q^{27} + 9 q^{29} + 5 q^{31} + 2 q^{33} - 8 q^{37} + 12 q^{39} + 3 q^{41} + 15 q^{43} + 4 q^{47} + 22 q^{49} + 33 q^{51} + 13 q^{53} - 2 q^{57} + 9 q^{61} + 21 q^{63} - 3 q^{67} - 11 q^{69} + 19 q^{71} + 3 q^{73} - 36 q^{77} - 16 q^{79} + 26 q^{81} + 31 q^{83} - 25 q^{87} + 14 q^{89} + 42 q^{91} + 39 q^{93} - 11 q^{97} - 6 q^{99} + O(q^{100}) \)

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
\(3\) 0.848258 0.489742 0.244871 0.969556i \(-0.421254\pi\)
0.244871 + 0.969556i \(0.421254\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 1.74484 0.659486 0.329743 0.944071i \(-0.393038\pi\)
0.329743 + 0.944071i \(0.393038\pi\)
\(8\) 0 0
\(9\) −2.28046 −0.760153
\(10\) 0 0
\(11\) 5.92425 1.78623 0.893115 0.449829i \(-0.148515\pi\)
0.893115 + 0.449829i \(0.148515\pi\)
\(12\) 0 0
\(13\) 6.78582 1.88205 0.941024 0.338339i \(-0.109865\pi\)
0.941024 + 0.338339i \(0.109865\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.86024 0.451174 0.225587 0.974223i \(-0.427570\pi\)
0.225587 + 0.974223i \(0.427570\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) 1.48007 0.322978
\(22\) 0 0
\(23\) 5.94357 1.23932 0.619660 0.784871i \(-0.287271\pi\)
0.619660 + 0.784871i \(0.287271\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −4.47919 −0.862021
\(28\) 0 0
\(29\) 3.29977 0.612752 0.306376 0.951911i \(-0.400883\pi\)
0.306376 + 0.951911i \(0.400883\pi\)
\(30\) 0 0
\(31\) −5.75242 −1.03316 −0.516582 0.856238i \(-0.672796\pi\)
−0.516582 + 0.856238i \(0.672796\pi\)
\(32\) 0 0
\(33\) 5.02529 0.874791
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.36379 −0.717402 −0.358701 0.933453i \(-0.616780\pi\)
−0.358701 + 0.933453i \(0.616780\pi\)
\(38\) 0 0
\(39\) 5.75613 0.921718
\(40\) 0 0
\(41\) 7.12500 1.11274 0.556369 0.830935i \(-0.312194\pi\)
0.556369 + 0.830935i \(0.312194\pi\)
\(42\) 0 0
\(43\) −6.98455 −1.06513 −0.532567 0.846388i \(-0.678773\pi\)
−0.532567 + 0.846388i \(0.678773\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −4.02529 −0.587149 −0.293575 0.955936i \(-0.594845\pi\)
−0.293575 + 0.955936i \(0.594845\pi\)
\(48\) 0 0
\(49\) −3.95555 −0.565078
\(50\) 0 0
\(51\) 1.57796 0.220959
\(52\) 0 0
\(53\) −9.19015 −1.26236 −0.631182 0.775635i \(-0.717430\pi\)
−0.631182 + 0.775635i \(0.717430\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.848258 −0.112354
\(58\) 0 0
\(59\) 2.51553 0.327494 0.163747 0.986502i \(-0.447642\pi\)
0.163747 + 0.986502i \(0.447642\pi\)
\(60\) 0 0
\(61\) −2.49621 −0.319607 −0.159804 0.987149i \(-0.551086\pi\)
−0.159804 + 0.987149i \(0.551086\pi\)
\(62\) 0 0
\(63\) −3.97903 −0.501310
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.90485 0.843561 0.421781 0.906698i \(-0.361405\pi\)
0.421781 + 0.906698i \(0.361405\pi\)
\(68\) 0 0
\(69\) 5.04168 0.606947
\(70\) 0 0
\(71\) 1.27288 0.151063 0.0755314 0.997143i \(-0.475935\pi\)
0.0755314 + 0.997143i \(0.475935\pi\)
\(72\) 0 0
\(73\) 12.1217 1.41874 0.709371 0.704836i \(-0.248979\pi\)
0.709371 + 0.704836i \(0.248979\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 10.3368 1.17799
\(78\) 0 0
\(79\) −13.8376 −1.55685 −0.778424 0.627739i \(-0.783980\pi\)
−0.778424 + 0.627739i \(0.783980\pi\)
\(80\) 0 0
\(81\) 3.04187 0.337985
\(82\) 0 0
\(83\) −4.94955 −0.543283 −0.271642 0.962398i \(-0.587567\pi\)
−0.271642 + 0.962398i \(0.587567\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.79906 0.300091
\(88\) 0 0
\(89\) 15.6067 1.65430 0.827151 0.561980i \(-0.189960\pi\)
0.827151 + 0.561980i \(0.189960\pi\)
\(90\) 0 0
\(91\) 11.8401 1.24118
\(92\) 0 0
\(93\) −4.87953 −0.505984
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −15.7764 −1.60185 −0.800924 0.598766i \(-0.795658\pi\)
−0.800924 + 0.598766i \(0.795658\pi\)
\(98\) 0 0
\(99\) −13.5100 −1.35781
\(100\) 0 0
\(101\) −6.81483 −0.678101 −0.339050 0.940768i \(-0.610106\pi\)
−0.339050 + 0.940768i \(0.610106\pi\)
\(102\) 0 0
\(103\) 13.3177 1.31224 0.656118 0.754659i \(-0.272197\pi\)
0.656118 + 0.754659i \(0.272197\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.69935 0.357629 0.178815 0.983883i \(-0.442774\pi\)
0.178815 + 0.983883i \(0.442774\pi\)
\(108\) 0 0
\(109\) 12.7753 1.22366 0.611828 0.790991i \(-0.290434\pi\)
0.611828 + 0.790991i \(0.290434\pi\)
\(110\) 0 0
\(111\) −3.70162 −0.351342
\(112\) 0 0
\(113\) −8.81845 −0.829570 −0.414785 0.909919i \(-0.636143\pi\)
−0.414785 + 0.909919i \(0.636143\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −15.4748 −1.43064
\(118\) 0 0
\(119\) 3.24581 0.297543
\(120\) 0 0
\(121\) 24.0968 2.19062
\(122\) 0 0
\(123\) 6.04384 0.544955
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 4.39650 0.390127 0.195063 0.980791i \(-0.437509\pi\)
0.195063 + 0.980791i \(0.437509\pi\)
\(128\) 0 0
\(129\) −5.92470 −0.521641
\(130\) 0 0
\(131\) 21.0076 1.83544 0.917719 0.397229i \(-0.130028\pi\)
0.917719 + 0.397229i \(0.130028\pi\)
\(132\) 0 0
\(133\) −1.74484 −0.151296
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −11.3766 −0.971973 −0.485986 0.873966i \(-0.661540\pi\)
−0.485986 + 0.873966i \(0.661540\pi\)
\(138\) 0 0
\(139\) 5.18664 0.439925 0.219962 0.975508i \(-0.429407\pi\)
0.219962 + 0.975508i \(0.429407\pi\)
\(140\) 0 0
\(141\) −3.41449 −0.287552
\(142\) 0 0
\(143\) 40.2009 3.36177
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.35533 −0.276743
\(148\) 0 0
\(149\) 0.474871 0.0389030 0.0194515 0.999811i \(-0.493808\pi\)
0.0194515 + 0.999811i \(0.493808\pi\)
\(150\) 0 0
\(151\) 16.6164 1.35222 0.676110 0.736800i \(-0.263664\pi\)
0.676110 + 0.736800i \(0.263664\pi\)
\(152\) 0 0
\(153\) −4.24220 −0.342962
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.83889 0.545803 0.272901 0.962042i \(-0.412017\pi\)
0.272901 + 0.962042i \(0.412017\pi\)
\(158\) 0 0
\(159\) −7.79561 −0.618232
\(160\) 0 0
\(161\) 10.3705 0.817314
\(162\) 0 0
\(163\) 2.42620 0.190035 0.0950173 0.995476i \(-0.469709\pi\)
0.0950173 + 0.995476i \(0.469709\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.02699 0.0794705 0.0397353 0.999210i \(-0.487349\pi\)
0.0397353 + 0.999210i \(0.487349\pi\)
\(168\) 0 0
\(169\) 33.0474 2.54211
\(170\) 0 0
\(171\) 2.28046 0.174391
\(172\) 0 0
\(173\) −13.2443 −1.00694 −0.503472 0.864012i \(-0.667944\pi\)
−0.503472 + 0.864012i \(0.667944\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 2.13382 0.160387
\(178\) 0 0
\(179\) −18.8025 −1.40536 −0.702681 0.711505i \(-0.748014\pi\)
−0.702681 + 0.711505i \(0.748014\pi\)
\(180\) 0 0
\(181\) −8.01073 −0.595433 −0.297716 0.954654i \(-0.596225\pi\)
−0.297716 + 0.954654i \(0.596225\pi\)
\(182\) 0 0
\(183\) −2.11743 −0.156525
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 11.0205 0.805901
\(188\) 0 0
\(189\) −7.81545 −0.568490
\(190\) 0 0
\(191\) 23.3198 1.68736 0.843682 0.536844i \(-0.180384\pi\)
0.843682 + 0.536844i \(0.180384\pi\)
\(192\) 0 0
\(193\) 7.55652 0.543930 0.271965 0.962307i \(-0.412327\pi\)
0.271965 + 0.962307i \(0.412327\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −20.5405 −1.46345 −0.731726 0.681599i \(-0.761285\pi\)
−0.731726 + 0.681599i \(0.761285\pi\)
\(198\) 0 0
\(199\) −4.93624 −0.349920 −0.174960 0.984576i \(-0.555980\pi\)
−0.174960 + 0.984576i \(0.555980\pi\)
\(200\) 0 0
\(201\) 5.85709 0.413127
\(202\) 0 0
\(203\) 5.75756 0.404102
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −13.5541 −0.942072
\(208\) 0 0
\(209\) −5.92425 −0.409789
\(210\) 0 0
\(211\) −14.6716 −1.01003 −0.505017 0.863109i \(-0.668514\pi\)
−0.505017 + 0.863109i \(0.668514\pi\)
\(212\) 0 0
\(213\) 1.07973 0.0739818
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −10.0370 −0.681357
\(218\) 0 0
\(219\) 10.2824 0.694817
\(220\) 0 0
\(221\) 12.6233 0.849132
\(222\) 0 0
\(223\) 18.8357 1.26133 0.630667 0.776054i \(-0.282782\pi\)
0.630667 + 0.776054i \(0.282782\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −24.8621 −1.65015 −0.825076 0.565021i \(-0.808868\pi\)
−0.825076 + 0.565021i \(0.808868\pi\)
\(228\) 0 0
\(229\) 20.2743 1.33976 0.669880 0.742469i \(-0.266345\pi\)
0.669880 + 0.742469i \(0.266345\pi\)
\(230\) 0 0
\(231\) 8.76831 0.576913
\(232\) 0 0
\(233\) −26.9597 −1.76619 −0.883095 0.469194i \(-0.844545\pi\)
−0.883095 + 0.469194i \(0.844545\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −11.7378 −0.762453
\(238\) 0 0
\(239\) 12.1013 0.782767 0.391384 0.920228i \(-0.371997\pi\)
0.391384 + 0.920228i \(0.371997\pi\)
\(240\) 0 0
\(241\) −18.2563 −1.17599 −0.587997 0.808863i \(-0.700083\pi\)
−0.587997 + 0.808863i \(0.700083\pi\)
\(242\) 0 0
\(243\) 16.0179 1.02755
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −6.78582 −0.431772
\(248\) 0 0
\(249\) −4.19849 −0.266069
\(250\) 0 0
\(251\) −5.65623 −0.357018 −0.178509 0.983938i \(-0.557127\pi\)
−0.178509 + 0.983938i \(0.557127\pi\)
\(252\) 0 0
\(253\) 35.2112 2.21371
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −8.04957 −0.502119 −0.251059 0.967972i \(-0.580779\pi\)
−0.251059 + 0.967972i \(0.580779\pi\)
\(258\) 0 0
\(259\) −7.61409 −0.473116
\(260\) 0 0
\(261\) −7.52500 −0.465786
\(262\) 0 0
\(263\) −8.24808 −0.508598 −0.254299 0.967126i \(-0.581845\pi\)
−0.254299 + 0.967126i \(0.581845\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 13.2385 0.810181
\(268\) 0 0
\(269\) −8.48032 −0.517054 −0.258527 0.966004i \(-0.583237\pi\)
−0.258527 + 0.966004i \(0.583237\pi\)
\(270\) 0 0
\(271\) 30.2291 1.83629 0.918145 0.396244i \(-0.129687\pi\)
0.918145 + 0.396244i \(0.129687\pi\)
\(272\) 0 0
\(273\) 10.0435 0.607860
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.42364 0.145622 0.0728111 0.997346i \(-0.476803\pi\)
0.0728111 + 0.997346i \(0.476803\pi\)
\(278\) 0 0
\(279\) 13.1181 0.785363
\(280\) 0 0
\(281\) −20.6355 −1.23101 −0.615504 0.788133i \(-0.711048\pi\)
−0.615504 + 0.788133i \(0.711048\pi\)
\(282\) 0 0
\(283\) −30.1197 −1.79043 −0.895215 0.445635i \(-0.852978\pi\)
−0.895215 + 0.445635i \(0.852978\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 12.4320 0.733835
\(288\) 0 0
\(289\) −13.5395 −0.796442
\(290\) 0 0
\(291\) −13.3824 −0.784492
\(292\) 0 0
\(293\) 11.8274 0.690963 0.345481 0.938426i \(-0.387716\pi\)
0.345481 + 0.938426i \(0.387716\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −26.5359 −1.53977
\(298\) 0 0
\(299\) 40.3320 2.33246
\(300\) 0 0
\(301\) −12.1869 −0.702441
\(302\) 0 0
\(303\) −5.78073 −0.332094
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 7.22041 0.412091 0.206045 0.978542i \(-0.433941\pi\)
0.206045 + 0.978542i \(0.433941\pi\)
\(308\) 0 0
\(309\) 11.2969 0.642657
\(310\) 0 0
\(311\) −10.1574 −0.575975 −0.287987 0.957634i \(-0.592986\pi\)
−0.287987 + 0.957634i \(0.592986\pi\)
\(312\) 0 0
\(313\) −6.03525 −0.341132 −0.170566 0.985346i \(-0.554560\pi\)
−0.170566 + 0.985346i \(0.554560\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −30.5978 −1.71854 −0.859271 0.511520i \(-0.829083\pi\)
−0.859271 + 0.511520i \(0.829083\pi\)
\(318\) 0 0
\(319\) 19.5487 1.09452
\(320\) 0 0
\(321\) 3.13800 0.175146
\(322\) 0 0
\(323\) −1.86024 −0.103507
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 10.8368 0.599276
\(328\) 0 0
\(329\) −7.02348 −0.387217
\(330\) 0 0
\(331\) 19.1478 1.05246 0.526228 0.850343i \(-0.323606\pi\)
0.526228 + 0.850343i \(0.323606\pi\)
\(332\) 0 0
\(333\) 9.95143 0.545335
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 10.8090 0.588804 0.294402 0.955682i \(-0.404880\pi\)
0.294402 + 0.955682i \(0.404880\pi\)
\(338\) 0 0
\(339\) −7.48032 −0.406275
\(340\) 0 0
\(341\) −34.0788 −1.84547
\(342\) 0 0
\(343\) −19.1156 −1.03215
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.0847 1.02452 0.512261 0.858830i \(-0.328808\pi\)
0.512261 + 0.858830i \(0.328808\pi\)
\(348\) 0 0
\(349\) 5.18421 0.277504 0.138752 0.990327i \(-0.455691\pi\)
0.138752 + 0.990327i \(0.455691\pi\)
\(350\) 0 0
\(351\) −30.3950 −1.62236
\(352\) 0 0
\(353\) −15.6398 −0.832423 −0.416212 0.909268i \(-0.636642\pi\)
−0.416212 + 0.909268i \(0.636642\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 2.75329 0.145719
\(358\) 0 0
\(359\) −17.4768 −0.922392 −0.461196 0.887298i \(-0.652580\pi\)
−0.461196 + 0.887298i \(0.652580\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 20.4403 1.07284
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −2.99560 −0.156369 −0.0781845 0.996939i \(-0.524912\pi\)
−0.0781845 + 0.996939i \(0.524912\pi\)
\(368\) 0 0
\(369\) −16.2483 −0.845852
\(370\) 0 0
\(371\) −16.0353 −0.832511
\(372\) 0 0
\(373\) −3.76504 −0.194946 −0.0974731 0.995238i \(-0.531076\pi\)
−0.0974731 + 0.995238i \(0.531076\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 22.3917 1.15323
\(378\) 0 0
\(379\) −10.5503 −0.541933 −0.270967 0.962589i \(-0.587343\pi\)
−0.270967 + 0.962589i \(0.587343\pi\)
\(380\) 0 0
\(381\) 3.72937 0.191061
\(382\) 0 0
\(383\) −6.25532 −0.319632 −0.159816 0.987147i \(-0.551090\pi\)
−0.159816 + 0.987147i \(0.551090\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 15.9280 0.809665
\(388\) 0 0
\(389\) 12.8669 0.652379 0.326190 0.945304i \(-0.394235\pi\)
0.326190 + 0.945304i \(0.394235\pi\)
\(390\) 0 0
\(391\) 11.0565 0.559149
\(392\) 0 0
\(393\) 17.8198 0.898891
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 28.4649 1.42861 0.714306 0.699834i \(-0.246743\pi\)
0.714306 + 0.699834i \(0.246743\pi\)
\(398\) 0 0
\(399\) −1.48007 −0.0740962
\(400\) 0 0
\(401\) 17.2837 0.863106 0.431553 0.902088i \(-0.357966\pi\)
0.431553 + 0.902088i \(0.357966\pi\)
\(402\) 0 0
\(403\) −39.0349 −1.94447
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −25.8522 −1.28144
\(408\) 0 0
\(409\) −28.3677 −1.40269 −0.701346 0.712821i \(-0.747417\pi\)
−0.701346 + 0.712821i \(0.747417\pi\)
\(410\) 0 0
\(411\) −9.65033 −0.476016
\(412\) 0 0
\(413\) 4.38918 0.215978
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.39961 0.215450
\(418\) 0 0
\(419\) −10.0821 −0.492542 −0.246271 0.969201i \(-0.579205\pi\)
−0.246271 + 0.969201i \(0.579205\pi\)
\(420\) 0 0
\(421\) 20.3802 0.993269 0.496635 0.867960i \(-0.334569\pi\)
0.496635 + 0.867960i \(0.334569\pi\)
\(422\) 0 0
\(423\) 9.17952 0.446323
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −4.35548 −0.210777
\(428\) 0 0
\(429\) 34.1008 1.64640
\(430\) 0 0
\(431\) 18.1090 0.872280 0.436140 0.899879i \(-0.356345\pi\)
0.436140 + 0.899879i \(0.356345\pi\)
\(432\) 0 0
\(433\) −33.8792 −1.62813 −0.814066 0.580772i \(-0.802751\pi\)
−0.814066 + 0.580772i \(0.802751\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −5.94357 −0.284319
\(438\) 0 0
\(439\) 21.7440 1.03779 0.518893 0.854839i \(-0.326344\pi\)
0.518893 + 0.854839i \(0.326344\pi\)
\(440\) 0 0
\(441\) 9.02047 0.429546
\(442\) 0 0
\(443\) 20.6461 0.980924 0.490462 0.871463i \(-0.336828\pi\)
0.490462 + 0.871463i \(0.336828\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0.402813 0.0190524
\(448\) 0 0
\(449\) 16.4288 0.775322 0.387661 0.921802i \(-0.373283\pi\)
0.387661 + 0.921802i \(0.373283\pi\)
\(450\) 0 0
\(451\) 42.2103 1.98761
\(452\) 0 0
\(453\) 14.0950 0.662239
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −19.5543 −0.914711 −0.457355 0.889284i \(-0.651203\pi\)
−0.457355 + 0.889284i \(0.651203\pi\)
\(458\) 0 0
\(459\) −8.33237 −0.388922
\(460\) 0 0
\(461\) 15.4065 0.717554 0.358777 0.933423i \(-0.383194\pi\)
0.358777 + 0.933423i \(0.383194\pi\)
\(462\) 0 0
\(463\) 4.44151 0.206414 0.103207 0.994660i \(-0.467090\pi\)
0.103207 + 0.994660i \(0.467090\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.1240 0.561030 0.280515 0.959850i \(-0.409495\pi\)
0.280515 + 0.959850i \(0.409495\pi\)
\(468\) 0 0
\(469\) 12.0478 0.556317
\(470\) 0 0
\(471\) 5.80114 0.267303
\(472\) 0 0
\(473\) −41.3783 −1.90257
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 20.9577 0.959589
\(478\) 0 0
\(479\) −42.5961 −1.94626 −0.973132 0.230249i \(-0.926046\pi\)
−0.973132 + 0.230249i \(0.926046\pi\)
\(480\) 0 0
\(481\) −29.6119 −1.35019
\(482\) 0 0
\(483\) 8.79690 0.400273
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 19.3424 0.876489 0.438245 0.898856i \(-0.355600\pi\)
0.438245 + 0.898856i \(0.355600\pi\)
\(488\) 0 0
\(489\) 2.05804 0.0930679
\(490\) 0 0
\(491\) 12.2941 0.554823 0.277412 0.960751i \(-0.410523\pi\)
0.277412 + 0.960751i \(0.410523\pi\)
\(492\) 0 0
\(493\) 6.13837 0.276458
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.22096 0.0996238
\(498\) 0 0
\(499\) 1.75085 0.0783789 0.0391894 0.999232i \(-0.487522\pi\)
0.0391894 + 0.999232i \(0.487522\pi\)
\(500\) 0 0
\(501\) 0.871149 0.0389201
\(502\) 0 0
\(503\) 2.13252 0.0950843 0.0475422 0.998869i \(-0.484861\pi\)
0.0475422 + 0.998869i \(0.484861\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 28.0327 1.24498
\(508\) 0 0
\(509\) −35.0078 −1.55169 −0.775846 0.630923i \(-0.782676\pi\)
−0.775846 + 0.630923i \(0.782676\pi\)
\(510\) 0 0
\(511\) 21.1504 0.935640
\(512\) 0 0
\(513\) 4.47919 0.197761
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −23.8469 −1.04878
\(518\) 0 0
\(519\) −11.2346 −0.493143
\(520\) 0 0
\(521\) −15.5581 −0.681613 −0.340807 0.940133i \(-0.610700\pi\)
−0.340807 + 0.940133i \(0.610700\pi\)
\(522\) 0 0
\(523\) 9.91872 0.433715 0.216858 0.976203i \(-0.430419\pi\)
0.216858 + 0.976203i \(0.430419\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.7009 −0.466137
\(528\) 0 0
\(529\) 12.3260 0.535913
\(530\) 0 0
\(531\) −5.73656 −0.248945
\(532\) 0 0
\(533\) 48.3490 2.09423
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −15.9493 −0.688264
\(538\) 0 0
\(539\) −23.4337 −1.00936
\(540\) 0 0
\(541\) 41.7150 1.79347 0.896734 0.442569i \(-0.145933\pi\)
0.896734 + 0.442569i \(0.145933\pi\)
\(542\) 0 0
\(543\) −6.79516 −0.291608
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −18.9230 −0.809089 −0.404545 0.914518i \(-0.632570\pi\)
−0.404545 + 0.914518i \(0.632570\pi\)
\(548\) 0 0
\(549\) 5.69251 0.242951
\(550\) 0 0
\(551\) −3.29977 −0.140575
\(552\) 0 0
\(553\) −24.1443 −1.02672
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.02768 −0.0859157 −0.0429579 0.999077i \(-0.513678\pi\)
−0.0429579 + 0.999077i \(0.513678\pi\)
\(558\) 0 0
\(559\) −47.3960 −2.00464
\(560\) 0 0
\(561\) 9.34825 0.394684
\(562\) 0 0
\(563\) 20.7497 0.874497 0.437248 0.899341i \(-0.355953\pi\)
0.437248 + 0.899341i \(0.355953\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 5.30756 0.222897
\(568\) 0 0
\(569\) 18.2161 0.763659 0.381830 0.924233i \(-0.375294\pi\)
0.381830 + 0.924233i \(0.375294\pi\)
\(570\) 0 0
\(571\) −35.0512 −1.46685 −0.733424 0.679771i \(-0.762079\pi\)
−0.733424 + 0.679771i \(0.762079\pi\)
\(572\) 0 0
\(573\) 19.7812 0.826372
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 21.6181 0.899973 0.449987 0.893035i \(-0.351429\pi\)
0.449987 + 0.893035i \(0.351429\pi\)
\(578\) 0 0
\(579\) 6.40987 0.266385
\(580\) 0 0
\(581\) −8.63615 −0.358288
\(582\) 0 0
\(583\) −54.4448 −2.25487
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20.3305 0.839130 0.419565 0.907725i \(-0.362183\pi\)
0.419565 + 0.907725i \(0.362183\pi\)
\(588\) 0 0
\(589\) 5.75242 0.237024
\(590\) 0 0
\(591\) −17.4237 −0.716714
\(592\) 0 0
\(593\) −2.11878 −0.0870077 −0.0435039 0.999053i \(-0.513852\pi\)
−0.0435039 + 0.999053i \(0.513852\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −4.18720 −0.171371
\(598\) 0 0
\(599\) −4.08633 −0.166963 −0.0834814 0.996509i \(-0.526604\pi\)
−0.0834814 + 0.996509i \(0.526604\pi\)
\(600\) 0 0
\(601\) −23.7445 −0.968557 −0.484279 0.874914i \(-0.660918\pi\)
−0.484279 + 0.874914i \(0.660918\pi\)
\(602\) 0 0
\(603\) −15.7462 −0.641236
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 23.5940 0.957653 0.478826 0.877910i \(-0.341062\pi\)
0.478826 + 0.877910i \(0.341062\pi\)
\(608\) 0 0
\(609\) 4.88390 0.197905
\(610\) 0 0
\(611\) −27.3149 −1.10504
\(612\) 0 0
\(613\) −48.4877 −1.95840 −0.979200 0.202897i \(-0.934964\pi\)
−0.979200 + 0.202897i \(0.934964\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −31.1353 −1.25346 −0.626730 0.779236i \(-0.715607\pi\)
−0.626730 + 0.779236i \(0.715607\pi\)
\(618\) 0 0
\(619\) 30.1220 1.21071 0.605353 0.795957i \(-0.293032\pi\)
0.605353 + 0.795957i \(0.293032\pi\)
\(620\) 0 0
\(621\) −26.6224 −1.06832
\(622\) 0 0
\(623\) 27.2310 1.09099
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −5.02529 −0.200691
\(628\) 0 0
\(629\) −8.11769 −0.323673
\(630\) 0 0
\(631\) −6.15070 −0.244855 −0.122428 0.992477i \(-0.539068\pi\)
−0.122428 + 0.992477i \(0.539068\pi\)
\(632\) 0 0
\(633\) −12.4453 −0.494656
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −26.8417 −1.06351
\(638\) 0 0
\(639\) −2.90275 −0.114831
\(640\) 0 0
\(641\) −20.9554 −0.827690 −0.413845 0.910347i \(-0.635814\pi\)
−0.413845 + 0.910347i \(0.635814\pi\)
\(642\) 0 0
\(643\) −24.6309 −0.971349 −0.485675 0.874140i \(-0.661426\pi\)
−0.485675 + 0.874140i \(0.661426\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 16.1496 0.634906 0.317453 0.948274i \(-0.397172\pi\)
0.317453 + 0.948274i \(0.397172\pi\)
\(648\) 0 0
\(649\) 14.9026 0.584979
\(650\) 0 0
\(651\) −8.51398 −0.333689
\(652\) 0 0
\(653\) 33.3511 1.30513 0.652565 0.757733i \(-0.273693\pi\)
0.652565 + 0.757733i \(0.273693\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −27.6431 −1.07846
\(658\) 0 0
\(659\) −11.1727 −0.435226 −0.217613 0.976035i \(-0.569827\pi\)
−0.217613 + 0.976035i \(0.569827\pi\)
\(660\) 0 0
\(661\) −33.5484 −1.30488 −0.652440 0.757840i \(-0.726255\pi\)
−0.652440 + 0.757840i \(0.726255\pi\)
\(662\) 0 0
\(663\) 10.7078 0.415856
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 19.6124 0.759396
\(668\) 0 0
\(669\) 15.9775 0.617728
\(670\) 0 0
\(671\) −14.7882 −0.570892
\(672\) 0 0
\(673\) 4.13528 0.159403 0.0797017 0.996819i \(-0.474603\pi\)
0.0797017 + 0.996819i \(0.474603\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −17.7150 −0.680842 −0.340421 0.940273i \(-0.610570\pi\)
−0.340421 + 0.940273i \(0.610570\pi\)
\(678\) 0 0
\(679\) −27.5272 −1.05640
\(680\) 0 0
\(681\) −21.0894 −0.808149
\(682\) 0 0
\(683\) 13.1004 0.501273 0.250637 0.968081i \(-0.419360\pi\)
0.250637 + 0.968081i \(0.419360\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 17.1978 0.656137
\(688\) 0 0
\(689\) −62.3627 −2.37583
\(690\) 0 0
\(691\) −18.6952 −0.711200 −0.355600 0.934638i \(-0.615724\pi\)
−0.355600 + 0.934638i \(0.615724\pi\)
\(692\) 0 0
\(693\) −23.5728 −0.895455
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 13.2542 0.502039
\(698\) 0 0
\(699\) −22.8688 −0.864977
\(700\) 0 0
\(701\) −33.3160 −1.25833 −0.629164 0.777273i \(-0.716603\pi\)
−0.629164 + 0.777273i \(0.716603\pi\)
\(702\) 0 0
\(703\) 4.36379 0.164583
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −11.8908 −0.447198
\(708\) 0 0
\(709\) 11.1888 0.420206 0.210103 0.977679i \(-0.432620\pi\)
0.210103 + 0.977679i \(0.432620\pi\)
\(710\) 0 0
\(711\) 31.5560 1.18344
\(712\) 0 0
\(713\) −34.1899 −1.28042
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 10.2650 0.383354
\(718\) 0 0
\(719\) 18.3734 0.685212 0.342606 0.939479i \(-0.388690\pi\)
0.342606 + 0.939479i \(0.388690\pi\)
\(720\) 0 0
\(721\) 23.2373 0.865401
\(722\) 0 0
\(723\) −15.4861 −0.575934
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 8.69538 0.322494 0.161247 0.986914i \(-0.448448\pi\)
0.161247 + 0.986914i \(0.448448\pi\)
\(728\) 0 0
\(729\) 4.46167 0.165247
\(730\) 0 0
\(731\) −12.9929 −0.480562
\(732\) 0 0
\(733\) −41.9016 −1.54767 −0.773836 0.633386i \(-0.781665\pi\)
−0.773836 + 0.633386i \(0.781665\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 40.9061 1.50679
\(738\) 0 0
\(739\) 49.4636 1.81955 0.909774 0.415104i \(-0.136255\pi\)
0.909774 + 0.415104i \(0.136255\pi\)
\(740\) 0 0
\(741\) −5.75613 −0.211457
\(742\) 0 0
\(743\) −31.1266 −1.14192 −0.570962 0.820977i \(-0.693430\pi\)
−0.570962 + 0.820977i \(0.693430\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 11.2872 0.412978
\(748\) 0 0
\(749\) 6.45475 0.235852
\(750\) 0 0
\(751\) 6.27888 0.229119 0.114560 0.993416i \(-0.463454\pi\)
0.114560 + 0.993416i \(0.463454\pi\)
\(752\) 0 0
\(753\) −4.79794 −0.174847
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 38.1507 1.38661 0.693304 0.720645i \(-0.256154\pi\)
0.693304 + 0.720645i \(0.256154\pi\)
\(758\) 0 0
\(759\) 29.8682 1.08415
\(760\) 0 0
\(761\) −2.59360 −0.0940180 −0.0470090 0.998894i \(-0.514969\pi\)
−0.0470090 + 0.998894i \(0.514969\pi\)
\(762\) 0 0
\(763\) 22.2909 0.806984
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 17.0699 0.616359
\(768\) 0 0
\(769\) −52.8561 −1.90604 −0.953020 0.302908i \(-0.902043\pi\)
−0.953020 + 0.302908i \(0.902043\pi\)
\(770\) 0 0
\(771\) −6.82811 −0.245908
\(772\) 0 0
\(773\) 21.1684 0.761374 0.380687 0.924704i \(-0.375688\pi\)
0.380687 + 0.924704i \(0.375688\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −6.45871 −0.231705
\(778\) 0 0
\(779\) −7.12500 −0.255280
\(780\) 0 0
\(781\) 7.54085 0.269833
\(782\) 0 0
\(783\) −14.7803 −0.528205
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 48.1097 1.71492 0.857462 0.514547i \(-0.172040\pi\)
0.857462 + 0.514547i \(0.172040\pi\)
\(788\) 0 0
\(789\) −6.99650 −0.249082
\(790\) 0 0
\(791\) −15.3867 −0.547090
\(792\) 0 0
\(793\) −16.9389 −0.601517
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 14.3510 0.508338 0.254169 0.967160i \(-0.418198\pi\)
0.254169 + 0.967160i \(0.418198\pi\)
\(798\) 0 0
\(799\) −7.48801 −0.264907
\(800\) 0 0
\(801\) −35.5903 −1.25752
\(802\) 0 0
\(803\) 71.8122 2.53420
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −7.19350 −0.253223
\(808\) 0 0
\(809\) −28.2263 −0.992384 −0.496192 0.868213i \(-0.665269\pi\)
−0.496192 + 0.868213i \(0.665269\pi\)
\(810\) 0 0
\(811\) −17.7490 −0.623250 −0.311625 0.950205i \(-0.600873\pi\)
−0.311625 + 0.950205i \(0.600873\pi\)
\(812\) 0 0
\(813\) 25.6421 0.899308
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 6.98455 0.244359
\(818\) 0 0
\(819\) −27.0010 −0.943490
\(820\) 0 0
\(821\) −32.2066 −1.12402 −0.562010 0.827130i \(-0.689972\pi\)
−0.562010 + 0.827130i \(0.689972\pi\)
\(822\) 0 0
\(823\) −27.0278 −0.942130 −0.471065 0.882099i \(-0.656130\pi\)
−0.471065 + 0.882099i \(0.656130\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 41.2334 1.43383 0.716913 0.697163i \(-0.245555\pi\)
0.716913 + 0.697163i \(0.245555\pi\)
\(828\) 0 0
\(829\) −22.0919 −0.767285 −0.383642 0.923482i \(-0.625330\pi\)
−0.383642 + 0.923482i \(0.625330\pi\)
\(830\) 0 0
\(831\) 2.05587 0.0713173
\(832\) 0 0
\(833\) −7.35827 −0.254949
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 25.7662 0.890609
\(838\) 0 0
\(839\) 22.1096 0.763310 0.381655 0.924305i \(-0.375354\pi\)
0.381655 + 0.924305i \(0.375354\pi\)
\(840\) 0 0
\(841\) −18.1115 −0.624534
\(842\) 0 0
\(843\) −17.5042 −0.602877
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 42.0449 1.44468
\(848\) 0 0
\(849\) −25.5493 −0.876848
\(850\) 0 0
\(851\) −25.9365 −0.889090
\(852\) 0 0
\(853\) 24.0868 0.824717 0.412359 0.911022i \(-0.364705\pi\)
0.412359 + 0.911022i \(0.364705\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −35.7836 −1.22235 −0.611173 0.791497i \(-0.709302\pi\)
−0.611173 + 0.791497i \(0.709302\pi\)
\(858\) 0 0
\(859\) −16.4286 −0.560538 −0.280269 0.959922i \(-0.590424\pi\)
−0.280269 + 0.959922i \(0.590424\pi\)
\(860\) 0 0
\(861\) 10.5455 0.359390
\(862\) 0 0
\(863\) −16.2841 −0.554319 −0.277159 0.960824i \(-0.589393\pi\)
−0.277159 + 0.960824i \(0.589393\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −11.4850 −0.390051
\(868\) 0 0
\(869\) −81.9772 −2.78089
\(870\) 0 0
\(871\) 46.8551 1.58762
\(872\) 0 0
\(873\) 35.9774 1.21765
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 57.3960 1.93813 0.969063 0.246814i \(-0.0793835\pi\)
0.969063 + 0.246814i \(0.0793835\pi\)
\(878\) 0 0
\(879\) 10.0327 0.338393
\(880\) 0 0
\(881\) 51.6832 1.74125 0.870625 0.491947i \(-0.163715\pi\)
0.870625 + 0.491947i \(0.163715\pi\)
\(882\) 0 0
\(883\) −32.8329 −1.10492 −0.552458 0.833541i \(-0.686310\pi\)
−0.552458 + 0.833541i \(0.686310\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −7.04248 −0.236463 −0.118232 0.992986i \(-0.537723\pi\)
−0.118232 + 0.992986i \(0.537723\pi\)
\(888\) 0 0
\(889\) 7.67118 0.257283
\(890\) 0 0
\(891\) 18.0208 0.603719
\(892\) 0 0
\(893\) 4.02529 0.134701
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 34.2119 1.14230
\(898\) 0 0
\(899\) −18.9817 −0.633074
\(900\) 0 0
\(901\) −17.0959 −0.569546
\(902\) 0 0
\(903\) −10.3376 −0.344015
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.94068 0.0644391 0.0322195 0.999481i \(-0.489742\pi\)
0.0322195 + 0.999481i \(0.489742\pi\)
\(908\) 0 0
\(909\) 15.5409 0.515460
\(910\) 0 0
\(911\) −57.4852 −1.90457 −0.952284 0.305212i \(-0.901273\pi\)
−0.952284 + 0.305212i \(0.901273\pi\)
\(912\) 0 0
\(913\) −29.3224 −0.970429
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 36.6547 1.21045
\(918\) 0 0
\(919\) 49.4475 1.63112 0.815561 0.578672i \(-0.196429\pi\)
0.815561 + 0.578672i \(0.196429\pi\)
\(920\) 0 0
\(921\) 6.12477 0.201818
\(922\) 0 0
\(923\) 8.63753 0.284308
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −30.3705 −0.997500
\(928\) 0 0
\(929\) 43.8092 1.43733 0.718667 0.695354i \(-0.244753\pi\)
0.718667 + 0.695354i \(0.244753\pi\)
\(930\) 0 0
\(931\) 3.95555 0.129638
\(932\) 0 0
\(933\) −8.61612 −0.282079
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −26.4692 −0.864712 −0.432356 0.901703i \(-0.642318\pi\)
−0.432356 + 0.901703i \(0.642318\pi\)
\(938\) 0 0
\(939\) −5.11945 −0.167067
\(940\) 0 0
\(941\) −1.38141 −0.0450327 −0.0225164 0.999746i \(-0.507168\pi\)
−0.0225164 + 0.999746i \(0.507168\pi\)
\(942\) 0 0
\(943\) 42.3479 1.37904
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −1.40723 −0.0457289 −0.0228645 0.999739i \(-0.507279\pi\)
−0.0228645 + 0.999739i \(0.507279\pi\)
\(948\) 0 0
\(949\) 82.2559 2.67014
\(950\) 0 0
\(951\) −25.9548 −0.841642
\(952\) 0 0
\(953\) −22.5899 −0.731760 −0.365880 0.930662i \(-0.619232\pi\)
−0.365880 + 0.930662i \(0.619232\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 16.5823 0.536031
\(958\) 0 0
\(959\) −19.8504 −0.641002
\(960\) 0 0
\(961\) 2.09029 0.0674287
\(962\) 0 0
\(963\) −8.43621 −0.271853
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −33.8866 −1.08972 −0.544860 0.838527i \(-0.683417\pi\)
−0.544860 + 0.838527i \(0.683417\pi\)
\(968\) 0 0
\(969\) −1.57796 −0.0506915
\(970\) 0 0
\(971\) 19.0982 0.612890 0.306445 0.951888i \(-0.400860\pi\)
0.306445 + 0.951888i \(0.400860\pi\)
\(972\) 0 0
\(973\) 9.04983 0.290124
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −35.4149 −1.13302 −0.566511 0.824054i \(-0.691707\pi\)
−0.566511 + 0.824054i \(0.691707\pi\)
\(978\) 0 0
\(979\) 92.4578 2.95496
\(980\) 0 0
\(981\) −29.1336 −0.930166
\(982\) 0 0
\(983\) −37.3537 −1.19140 −0.595700 0.803207i \(-0.703125\pi\)
−0.595700 + 0.803207i \(0.703125\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −5.95772 −0.189636
\(988\) 0 0
\(989\) −41.5132 −1.32004
\(990\) 0 0
\(991\) 21.4720 0.682082 0.341041 0.940048i \(-0.389221\pi\)
0.341041 + 0.940048i \(0.389221\pi\)
\(992\) 0 0
\(993\) 16.2422 0.515432
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 20.3410 0.644204 0.322102 0.946705i \(-0.395611\pi\)
0.322102 + 0.946705i \(0.395611\pi\)
\(998\) 0 0
\(999\) 19.5462 0.618415
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 3800.2.a.bc.1.4 yes 6
4.3 odd 2 7600.2.a.ch.1.3 6
5.2 odd 4 3800.2.d.q.3649.5 12
5.3 odd 4 3800.2.d.q.3649.8 12
5.4 even 2 3800.2.a.ba.1.3 6
20.19 odd 2 7600.2.a.cl.1.4 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.3 6 5.4 even 2
3800.2.a.bc.1.4 yes 6 1.1 even 1 trivial
3800.2.d.q.3649.5 12 5.2 odd 4
3800.2.d.q.3649.8 12 5.3 odd 4
7600.2.a.ch.1.3 6 4.3 odd 2
7600.2.a.cl.1.4 6 20.19 odd 2