Properties

Label 3800.2.a.ba.1.3
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: \( x^{6} - 2x^{5} - 10x^{4} + 16x^{3} + 15x^{2} - 14x - 3 \) Copy content Toggle raw display
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 1.3
Root \(0.848258\) of defining polynomial
Character \(\chi\) \(=\) 3800.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.848258 q^{3} -1.74484 q^{7} -2.28046 q^{9} +O(q^{10})\) \(q-0.848258 q^{3} -1.74484 q^{7} -2.28046 q^{9} +5.92425 q^{11} -6.78582 q^{13} -1.86024 q^{17} -1.00000 q^{19} +1.48007 q^{21} -5.94357 q^{23} +4.47919 q^{27} +3.29977 q^{29} -5.75242 q^{31} -5.02529 q^{33} +4.36379 q^{37} +5.75613 q^{39} +7.12500 q^{41} +6.98455 q^{43} +4.02529 q^{47} -3.95555 q^{49} +1.57796 q^{51} +9.19015 q^{53} +0.848258 q^{57} +2.51553 q^{59} -2.49621 q^{61} +3.97903 q^{63} -6.90485 q^{67} +5.04168 q^{69} +1.27288 q^{71} -12.1217 q^{73} -10.3368 q^{77} -13.8376 q^{79} +3.04187 q^{81} +4.94955 q^{83} -2.79906 q^{87} +15.6067 q^{89} +11.8401 q^{91} +4.87953 q^{93} +15.7764 q^{97} -13.5100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} - 2 q^{7} + 6 q^{9} + 3 q^{11} + q^{13} + 14 q^{17} - 6 q^{19} + 15 q^{21} - 12 q^{23} - 8 q^{27} + 9 q^{29} + 5 q^{31} - 2 q^{33} + 8 q^{37} + 12 q^{39} + 3 q^{41} - 15 q^{43} - 4 q^{47} + 22 q^{49} + 33 q^{51} - 13 q^{53} + 2 q^{57} + 9 q^{61} - 21 q^{63} + 3 q^{67} - 11 q^{69} + 19 q^{71} - 3 q^{73} + 36 q^{77} - 16 q^{79} + 26 q^{81} - 31 q^{83} + 25 q^{87} + 14 q^{89} + 42 q^{91} - 39 q^{93} + 11 q^{97} - 6 q^{99}+O(q^{100}) \) 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
\(3\) −0.848258 −0.489742 −0.244871 0.969556i \(-0.578746\pi\)
−0.244871 + 0.969556i \(0.578746\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.74484 −0.659486 −0.329743 0.944071i \(-0.606962\pi\)
−0.329743 + 0.944071i \(0.606962\pi\)
\(8\) 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 \(-0.890135\pi\)
−0.941024 + 0.338339i \(0.890135\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.86024 −0.451174 −0.225587 0.974223i \(-0.572430\pi\)
−0.225587 + 0.974223i \(0.572430\pi\)
\(18\) 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 \(-0.712729\pi\)
−0.619660 + 0.784871i \(0.712729\pi\)
\(24\) 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 \(-0.383220\pi\)
0.358701 + 0.933453i \(0.383220\pi\)
\(38\) 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 \(-0.321227\pi\)
0.532567 + 0.846388i \(0.321227\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.02529 0.587149 0.293575 0.955936i \(-0.405155\pi\)
0.293575 + 0.955936i \(0.405155\pi\)
\(48\) 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 \(-0.282570\pi\)
0.631182 + 0.775635i \(0.282570\pi\)
\(54\) 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 \(-0.638595\pi\)
−0.421781 + 0.906698i \(0.638595\pi\)
\(68\) 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 \(-0.751021\pi\)
−0.709371 + 0.704836i \(0.751021\pi\)
\(74\) 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 \(-0.412433\pi\)
0.271642 + 0.962398i \(0.412433\pi\)
\(84\) 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 \(-0.204342\pi\)
0.800924 + 0.598766i \(0.204342\pi\)
\(98\) 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 \(-0.727803\pi\)
−0.656118 + 0.754659i \(0.727803\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −3.69935 −0.357629 −0.178815 0.983883i \(-0.557226\pi\)
−0.178815 + 0.983883i \(0.557226\pi\)
\(108\) 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 \(-0.363857\pi\)
0.414785 + 0.909919i \(0.363857\pi\)
\(114\) 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 \(-0.562491\pi\)
−0.195063 + 0.980791i \(0.562491\pi\)
\(128\) 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 \(-0.338460\pi\)
0.485986 + 0.873966i \(0.338460\pi\)
\(138\) 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 \(-0.587983\pi\)
−0.272901 + 0.962042i \(0.587983\pi\)
\(158\) 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 \(-0.530291\pi\)
−0.0950173 + 0.995476i \(0.530291\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.02699 −0.0794705 −0.0397353 0.999210i \(-0.512651\pi\)
−0.0397353 + 0.999210i \(0.512651\pi\)
\(168\) 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 \(-0.332056\pi\)
0.503472 + 0.864012i \(0.332056\pi\)
\(174\) 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 \(-0.587673\pi\)
−0.271965 + 0.962307i \(0.587673\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.5405 1.46345 0.731726 0.681599i \(-0.238715\pi\)
0.731726 + 0.681599i \(0.238715\pi\)
\(198\) 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 \(-0.717218\pi\)
−0.630667 + 0.776054i \(0.717218\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 24.8621 1.65015 0.825076 0.565021i \(-0.191132\pi\)
0.825076 + 0.565021i \(0.191132\pi\)
\(228\) 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 \(-0.155455\pi\)
0.883095 + 0.469194i \(0.155455\pi\)
\(234\) 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 \(-0.419221\pi\)
0.251059 + 0.967972i \(0.419221\pi\)
\(258\) 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 \(-0.418155\pi\)
0.254299 + 0.967126i \(0.418155\pi\)
\(264\) 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 \(-0.523197\pi\)
−0.0728111 + 0.997346i \(0.523197\pi\)
\(278\) 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 \(-0.147022\pi\)
0.895215 + 0.445635i \(0.147022\pi\)
\(284\) 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 \(-0.612284\pi\)
−0.345481 + 0.938426i \(0.612284\pi\)
\(294\) 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 \(-0.566059\pi\)
−0.206045 + 0.978542i \(0.566059\pi\)
\(308\) 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 \(-0.445440\pi\)
0.170566 + 0.985346i \(0.445440\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 30.5978 1.71854 0.859271 0.511520i \(-0.170917\pi\)
0.859271 + 0.511520i \(0.170917\pi\)
\(318\) 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 \(-0.595120\pi\)
−0.294402 + 0.955682i \(0.595120\pi\)
\(338\) 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 \(-0.671192\pi\)
−0.512261 + 0.858830i \(0.671192\pi\)
\(348\) 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 \(-0.363358\pi\)
0.416212 + 0.909268i \(0.363358\pi\)
\(354\) 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 \(-0.475088\pi\)
0.0781845 + 0.996939i \(0.475088\pi\)
\(368\) 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 \(-0.468924\pi\)
0.0974731 + 0.995238i \(0.468924\pi\)
\(374\) 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 \(-0.448910\pi\)
0.159816 + 0.987147i \(0.448910\pi\)
\(384\) 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 \(-0.753257\pi\)
−0.714306 + 0.699834i \(0.753257\pi\)
\(398\) 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 \(-0.197249\pi\)
0.814066 + 0.580772i \(0.197249\pi\)
\(434\) 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 \(-0.663172\pi\)
−0.490462 + 0.871463i \(0.663172\pi\)
\(444\) 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 \(-0.348797\pi\)
0.457355 + 0.889284i \(0.348797\pi\)
\(458\) 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 \(-0.532910\pi\)
−0.103207 + 0.994660i \(0.532910\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −12.1240 −0.561030 −0.280515 0.959850i \(-0.590505\pi\)
−0.280515 + 0.959850i \(0.590505\pi\)
\(468\) 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 \(-0.644400\pi\)
−0.438245 + 0.898856i \(0.644400\pi\)
\(488\) 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 \(-0.515139\pi\)
−0.0475422 + 0.998869i \(0.515139\pi\)
\(504\) 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 \(-0.569581\pi\)
−0.216858 + 0.976203i \(0.569581\pi\)
\(524\) 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 \(-0.367430\pi\)
0.404545 + 0.914518i \(0.367430\pi\)
\(548\) 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 \(-0.486322\pi\)
0.0429579 + 0.999077i \(0.486322\pi\)
\(558\) 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 \(-0.644047\pi\)
−0.437248 + 0.899341i \(0.644047\pi\)
\(564\) 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 \(-0.648571\pi\)
−0.449987 + 0.893035i \(0.648571\pi\)
\(578\) 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 \(-0.637817\pi\)
−0.419565 + 0.907725i \(0.637817\pi\)
\(588\) 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 \(-0.486148\pi\)
0.0435039 + 0.999053i \(0.486148\pi\)
\(594\) 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 \(-0.658938\pi\)
−0.478826 + 0.877910i \(0.658938\pi\)
\(608\) 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 \(-0.0650356\pi\)
0.979200 + 0.202897i \(0.0650356\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 31.1353 1.25346 0.626730 0.779236i \(-0.284393\pi\)
0.626730 + 0.779236i \(0.284393\pi\)
\(618\) 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 \(-0.338574\pi\)
0.485675 + 0.874140i \(0.338574\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −16.1496 −0.634906 −0.317453 0.948274i \(-0.602828\pi\)
−0.317453 + 0.948274i \(0.602828\pi\)
\(648\) 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 \(-0.726307\pi\)
−0.652565 + 0.757733i \(0.726307\pi\)
\(654\) 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 \(-0.525397\pi\)
−0.0797017 + 0.996819i \(0.525397\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.7150 0.680842 0.340421 0.940273i \(-0.389430\pi\)
0.340421 + 0.940273i \(0.389430\pi\)
\(678\) 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 \(-0.580640\pi\)
−0.250637 + 0.968081i \(0.580640\pi\)
\(684\) 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 \(-0.551552\pi\)
−0.161247 + 0.986914i \(0.551552\pi\)
\(728\) 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 \(-0.218335\pi\)
0.773836 + 0.633386i \(0.218335\pi\)
\(734\) 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 \(-0.306570\pi\)
0.570962 + 0.820977i \(0.306570\pi\)
\(744\) 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 \(-0.743846\pi\)
−0.693304 + 0.720645i \(0.743846\pi\)
\(758\) 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 \(-0.624312\pi\)
−0.380687 + 0.924704i \(0.624312\pi\)
\(774\) 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 \(-0.827960\pi\)
−0.857462 + 0.514547i \(0.827960\pi\)
\(788\) 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 \(-0.581802\pi\)
−0.254169 + 0.967160i \(0.581802\pi\)
\(798\) 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 \(-0.343870\pi\)
0.471065 + 0.882099i \(0.343870\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −41.2334 −1.43383 −0.716913 0.697163i \(-0.754445\pi\)
−0.716913 + 0.697163i \(0.754445\pi\)
\(828\) 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 \(-0.635295\pi\)
−0.412359 + 0.911022i \(0.635295\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 35.7836 1.22235 0.611173 0.791497i \(-0.290698\pi\)
0.611173 + 0.791497i \(0.290698\pi\)
\(858\) 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 \(-0.410607\pi\)
0.277159 + 0.960824i \(0.410607\pi\)
\(864\) 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 \(-0.920616\pi\)
−0.969063 + 0.246814i \(0.920616\pi\)
\(878\) 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 \(-0.313690\pi\)
0.552458 + 0.833541i \(0.313690\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 7.04248 0.236463 0.118232 0.992986i \(-0.462277\pi\)
0.118232 + 0.992986i \(0.462277\pi\)
\(888\) 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 \(-0.510258\pi\)
−0.0322195 + 0.999481i \(0.510258\pi\)
\(908\) 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 \(-0.357682\pi\)
0.432356 + 0.901703i \(0.357682\pi\)
\(938\) 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 \(-0.492721\pi\)
0.0228645 + 0.999739i \(0.492721\pi\)
\(948\) 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 \(-0.380768\pi\)
0.365880 + 0.930662i \(0.380768\pi\)
\(954\) 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 \(-0.316583\pi\)
0.544860 + 0.838527i \(0.316583\pi\)
\(968\) 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 \(-0.308293\pi\)
0.566511 + 0.824054i \(0.308293\pi\)
\(978\) 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 \(-0.296875\pi\)
0.595700 + 0.803207i \(0.296875\pi\)
\(984\) 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 \(-0.604389\pi\)
−0.322102 + 0.946705i \(0.604389\pi\)
\(998\) 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.ba.1.3 6
4.3 odd 2 7600.2.a.cl.1.4 6
5.2 odd 4 3800.2.d.q.3649.8 12
5.3 odd 4 3800.2.d.q.3649.5 12
5.4 even 2 3800.2.a.bc.1.4 yes 6
20.19 odd 2 7600.2.a.ch.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3800.2.a.ba.1.3 6 1.1 even 1 trivial
3800.2.a.bc.1.4 yes 6 5.4 even 2
3800.2.d.q.3649.5 12 5.3 odd 4
3800.2.d.q.3649.8 12 5.2 odd 4
7600.2.a.ch.1.3 6 20.19 odd 2
7600.2.a.cl.1.4 6 4.3 odd 2