Properties

Label 1840.2.a.v.1.3
Level $1840$
Weight $2$
Character 1840.1
Self dual yes
Analytic conductor $14.692$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.13955077.1
Defining polynomial: \(x^{5} - 14 x^{3} - x^{2} + 32 x + 16\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 920)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.568386\) of defining polynomial
Character \(\chi\) \(=\) 1840.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.568386 q^{3} -1.00000 q^{5} -4.73770 q^{7} -2.67694 q^{9} +O(q^{10})\) \(q+0.568386 q^{3} -1.00000 q^{5} -4.73770 q^{7} -2.67694 q^{9} +0.360532 q^{11} +5.26123 q^{13} -0.568386 q^{15} +0.370852 q^{17} +4.60586 q^{19} -2.69284 q^{21} +1.00000 q^{23} +1.00000 q^{25} -3.22669 q^{27} +0.939238 q^{29} -9.66662 q^{31} +0.204921 q^{33} +4.73770 q^{35} +3.26862 q^{37} +2.99041 q^{39} +5.29977 q^{41} +2.67694 q^{45} +1.25491 q^{47} +15.4458 q^{49} +0.210787 q^{51} +10.9278 q^{53} -0.360532 q^{55} +2.61790 q^{57} +9.66955 q^{59} +9.71441 q^{61} +12.6825 q^{63} -5.26123 q^{65} +7.07001 q^{67} +0.568386 q^{69} -11.3747 q^{71} +0.745086 q^{73} +0.568386 q^{75} -1.70809 q^{77} -0.415709 q^{79} +6.19680 q^{81} +9.26862 q^{83} -0.370852 q^{85} +0.533850 q^{87} +12.6122 q^{89} -24.9261 q^{91} -5.49437 q^{93} -4.60586 q^{95} +14.0404 q^{97} -0.965121 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 5 q^{5} + 2 q^{7} + 13 q^{9} + O(q^{10}) \) \( 5 q - 5 q^{5} + 2 q^{7} + 13 q^{9} + q^{11} + 4 q^{13} + 4 q^{17} - 7 q^{19} + 6 q^{21} + 5 q^{23} + 5 q^{25} - 3 q^{27} + 4 q^{29} - 19 q^{31} + 17 q^{33} - 2 q^{35} + 15 q^{37} - 19 q^{39} + 25 q^{41} - 13 q^{45} + 11 q^{47} + 25 q^{49} - 19 q^{51} + 3 q^{53} - q^{55} + 48 q^{57} + q^{59} - 5 q^{61} + 41 q^{63} - 4 q^{65} - 9 q^{67} - q^{71} - q^{73} + 18 q^{77} + 2 q^{79} + 57 q^{81} + 45 q^{83} - 4 q^{85} + 9 q^{87} + 6 q^{89} - 11 q^{91} - 39 q^{93} + 7 q^{95} + 25 q^{97} + 65 q^{99} + O(q^{100}) \)

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
\(3\) 0.568386 0.328158 0.164079 0.986447i \(-0.447535\pi\)
0.164079 + 0.986447i \(0.447535\pi\)
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −4.73770 −1.79068 −0.895341 0.445381i \(-0.853068\pi\)
−0.895341 + 0.445381i \(0.853068\pi\)
\(8\) 0 0
\(9\) −2.67694 −0.892312
\(10\) 0 0
\(11\) 0.360532 0.108704 0.0543522 0.998522i \(-0.482691\pi\)
0.0543522 + 0.998522i \(0.482691\pi\)
\(12\) 0 0
\(13\) 5.26123 1.45920 0.729601 0.683873i \(-0.239706\pi\)
0.729601 + 0.683873i \(0.239706\pi\)
\(14\) 0 0
\(15\) −0.568386 −0.146757
\(16\) 0 0
\(17\) 0.370852 0.0899449 0.0449724 0.998988i \(-0.485680\pi\)
0.0449724 + 0.998988i \(0.485680\pi\)
\(18\) 0 0
\(19\) 4.60586 1.05666 0.528328 0.849040i \(-0.322819\pi\)
0.528328 + 0.849040i \(0.322819\pi\)
\(20\) 0 0
\(21\) −2.69284 −0.587626
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −3.22669 −0.620977
\(28\) 0 0
\(29\) 0.939238 0.174412 0.0872061 0.996190i \(-0.472206\pi\)
0.0872061 + 0.996190i \(0.472206\pi\)
\(30\) 0 0
\(31\) −9.66662 −1.73618 −0.868088 0.496411i \(-0.834651\pi\)
−0.868088 + 0.496411i \(0.834651\pi\)
\(32\) 0 0
\(33\) 0.204921 0.0356722
\(34\) 0 0
\(35\) 4.73770 0.800817
\(36\) 0 0
\(37\) 3.26862 0.537357 0.268679 0.963230i \(-0.413413\pi\)
0.268679 + 0.963230i \(0.413413\pi\)
\(38\) 0 0
\(39\) 2.99041 0.478849
\(40\) 0 0
\(41\) 5.29977 0.827685 0.413843 0.910348i \(-0.364186\pi\)
0.413843 + 0.910348i \(0.364186\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 2.67694 0.399054
\(46\) 0 0
\(47\) 1.25491 0.183048 0.0915240 0.995803i \(-0.470826\pi\)
0.0915240 + 0.995803i \(0.470826\pi\)
\(48\) 0 0
\(49\) 15.4458 2.20654
\(50\) 0 0
\(51\) 0.210787 0.0295161
\(52\) 0 0
\(53\) 10.9278 1.50106 0.750528 0.660839i \(-0.229799\pi\)
0.750528 + 0.660839i \(0.229799\pi\)
\(54\) 0 0
\(55\) −0.360532 −0.0486141
\(56\) 0 0
\(57\) 2.61790 0.346750
\(58\) 0 0
\(59\) 9.66955 1.25887 0.629434 0.777054i \(-0.283287\pi\)
0.629434 + 0.777054i \(0.283287\pi\)
\(60\) 0 0
\(61\) 9.71441 1.24380 0.621901 0.783096i \(-0.286361\pi\)
0.621901 + 0.783096i \(0.286361\pi\)
\(62\) 0 0
\(63\) 12.6825 1.59785
\(64\) 0 0
\(65\) −5.26123 −0.652575
\(66\) 0 0
\(67\) 7.07001 0.863739 0.431870 0.901936i \(-0.357854\pi\)
0.431870 + 0.901936i \(0.357854\pi\)
\(68\) 0 0
\(69\) 0.568386 0.0684257
\(70\) 0 0
\(71\) −11.3747 −1.34993 −0.674965 0.737850i \(-0.735841\pi\)
−0.674965 + 0.737850i \(0.735841\pi\)
\(72\) 0 0
\(73\) 0.745086 0.0872057 0.0436029 0.999049i \(-0.486116\pi\)
0.0436029 + 0.999049i \(0.486116\pi\)
\(74\) 0 0
\(75\) 0.568386 0.0656316
\(76\) 0 0
\(77\) −1.70809 −0.194655
\(78\) 0 0
\(79\) −0.415709 −0.0467709 −0.0233854 0.999727i \(-0.507444\pi\)
−0.0233854 + 0.999727i \(0.507444\pi\)
\(80\) 0 0
\(81\) 6.19680 0.688534
\(82\) 0 0
\(83\) 9.26862 1.01736 0.508681 0.860955i \(-0.330133\pi\)
0.508681 + 0.860955i \(0.330133\pi\)
\(84\) 0 0
\(85\) −0.370852 −0.0402246
\(86\) 0 0
\(87\) 0.533850 0.0572347
\(88\) 0 0
\(89\) 12.6122 1.33689 0.668444 0.743763i \(-0.266961\pi\)
0.668444 + 0.743763i \(0.266961\pi\)
\(90\) 0 0
\(91\) −24.9261 −2.61297
\(92\) 0 0
\(93\) −5.49437 −0.569740
\(94\) 0 0
\(95\) −4.60586 −0.472551
\(96\) 0 0
\(97\) 14.0404 1.42559 0.712793 0.701374i \(-0.247430\pi\)
0.712793 + 0.701374i \(0.247430\pi\)
\(98\) 0 0
\(99\) −0.965121 −0.0969983
\(100\) 0 0
\(101\) −12.7440 −1.26808 −0.634038 0.773302i \(-0.718604\pi\)
−0.634038 + 0.773302i \(0.718604\pi\)
\(102\) 0 0
\(103\) 3.31347 0.326486 0.163243 0.986586i \(-0.447805\pi\)
0.163243 + 0.986586i \(0.447805\pi\)
\(104\) 0 0
\(105\) 2.69284 0.262795
\(106\) 0 0
\(107\) 8.13184 0.786135 0.393067 0.919510i \(-0.371414\pi\)
0.393067 + 0.919510i \(0.371414\pi\)
\(108\) 0 0
\(109\) −5.79508 −0.555068 −0.277534 0.960716i \(-0.589517\pi\)
−0.277534 + 0.960716i \(0.589517\pi\)
\(110\) 0 0
\(111\) 1.85784 0.176338
\(112\) 0 0
\(113\) −7.60724 −0.715629 −0.357815 0.933793i \(-0.616478\pi\)
−0.357815 + 0.933793i \(0.616478\pi\)
\(114\) 0 0
\(115\) −1.00000 −0.0932505
\(116\) 0 0
\(117\) −14.0840 −1.30206
\(118\) 0 0
\(119\) −1.75699 −0.161063
\(120\) 0 0
\(121\) −10.8700 −0.988183
\(122\) 0 0
\(123\) 3.01232 0.271611
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 4.13077 0.366547 0.183273 0.983062i \(-0.441331\pi\)
0.183273 + 0.983062i \(0.441331\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 17.7582 1.55154 0.775770 0.631016i \(-0.217362\pi\)
0.775770 + 0.631016i \(0.217362\pi\)
\(132\) 0 0
\(133\) −21.8212 −1.89213
\(134\) 0 0
\(135\) 3.22669 0.277709
\(136\) 0 0
\(137\) 6.86954 0.586905 0.293452 0.955974i \(-0.405196\pi\)
0.293452 + 0.955974i \(0.405196\pi\)
\(138\) 0 0
\(139\) −20.6635 −1.75266 −0.876330 0.481712i \(-0.840015\pi\)
−0.876330 + 0.481712i \(0.840015\pi\)
\(140\) 0 0
\(141\) 0.713276 0.0600686
\(142\) 0 0
\(143\) 1.89684 0.158622
\(144\) 0 0
\(145\) −0.939238 −0.0779995
\(146\) 0 0
\(147\) 8.77917 0.724094
\(148\) 0 0
\(149\) −16.4763 −1.34979 −0.674896 0.737912i \(-0.735812\pi\)
−0.674896 + 0.737912i \(0.735812\pi\)
\(150\) 0 0
\(151\) 8.89224 0.723640 0.361820 0.932248i \(-0.382156\pi\)
0.361820 + 0.932248i \(0.382156\pi\)
\(152\) 0 0
\(153\) −0.992748 −0.0802589
\(154\) 0 0
\(155\) 9.66662 0.776441
\(156\) 0 0
\(157\) −6.62249 −0.528532 −0.264266 0.964450i \(-0.585130\pi\)
−0.264266 + 0.964450i \(0.585130\pi\)
\(158\) 0 0
\(159\) 6.21124 0.492583
\(160\) 0 0
\(161\) −4.73770 −0.373383
\(162\) 0 0
\(163\) 16.0779 1.25932 0.629658 0.776872i \(-0.283195\pi\)
0.629658 + 0.776872i \(0.283195\pi\)
\(164\) 0 0
\(165\) −0.204921 −0.0159531
\(166\) 0 0
\(167\) −13.8763 −1.07378 −0.536891 0.843651i \(-0.680402\pi\)
−0.536891 + 0.843651i \(0.680402\pi\)
\(168\) 0 0
\(169\) 14.6805 1.12927
\(170\) 0 0
\(171\) −12.3296 −0.942867
\(172\) 0 0
\(173\) −2.05518 −0.156252 −0.0781261 0.996943i \(-0.524894\pi\)
−0.0781261 + 0.996943i \(0.524894\pi\)
\(174\) 0 0
\(175\) −4.73770 −0.358136
\(176\) 0 0
\(177\) 5.49604 0.413108
\(178\) 0 0
\(179\) −17.6478 −1.31906 −0.659531 0.751678i \(-0.729245\pi\)
−0.659531 + 0.751678i \(0.729245\pi\)
\(180\) 0 0
\(181\) 2.85477 0.212193 0.106097 0.994356i \(-0.466165\pi\)
0.106097 + 0.994356i \(0.466165\pi\)
\(182\) 0 0
\(183\) 5.52153 0.408164
\(184\) 0 0
\(185\) −3.26862 −0.240313
\(186\) 0 0
\(187\) 0.133704 0.00977741
\(188\) 0 0
\(189\) 15.2871 1.11197
\(190\) 0 0
\(191\) −1.13677 −0.0822540 −0.0411270 0.999154i \(-0.513095\pi\)
−0.0411270 + 0.999154i \(0.513095\pi\)
\(192\) 0 0
\(193\) −26.4694 −1.90531 −0.952654 0.304055i \(-0.901659\pi\)
−0.952654 + 0.304055i \(0.901659\pi\)
\(194\) 0 0
\(195\) −2.99041 −0.214148
\(196\) 0 0
\(197\) 5.44424 0.387886 0.193943 0.981013i \(-0.437872\pi\)
0.193943 + 0.981013i \(0.437872\pi\)
\(198\) 0 0
\(199\) 18.6122 1.31938 0.659691 0.751537i \(-0.270687\pi\)
0.659691 + 0.751537i \(0.270687\pi\)
\(200\) 0 0
\(201\) 4.01850 0.283443
\(202\) 0 0
\(203\) −4.44983 −0.312317
\(204\) 0 0
\(205\) −5.29977 −0.370152
\(206\) 0 0
\(207\) −2.67694 −0.186060
\(208\) 0 0
\(209\) 1.66056 0.114863
\(210\) 0 0
\(211\) −6.10003 −0.419944 −0.209972 0.977707i \(-0.567337\pi\)
−0.209972 + 0.977707i \(0.567337\pi\)
\(212\) 0 0
\(213\) −6.46523 −0.442990
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 45.7975 3.10894
\(218\) 0 0
\(219\) 0.423497 0.0286173
\(220\) 0 0
\(221\) 1.95114 0.131248
\(222\) 0 0
\(223\) 16.4136 1.09913 0.549567 0.835450i \(-0.314793\pi\)
0.549567 + 0.835450i \(0.314793\pi\)
\(224\) 0 0
\(225\) −2.67694 −0.178462
\(226\) 0 0
\(227\) −1.21171 −0.0804240 −0.0402120 0.999191i \(-0.512803\pi\)
−0.0402120 + 0.999191i \(0.512803\pi\)
\(228\) 0 0
\(229\) −22.8293 −1.50860 −0.754300 0.656529i \(-0.772024\pi\)
−0.754300 + 0.656529i \(0.772024\pi\)
\(230\) 0 0
\(231\) −0.970856 −0.0638776
\(232\) 0 0
\(233\) 21.9420 1.43747 0.718735 0.695284i \(-0.244722\pi\)
0.718735 + 0.695284i \(0.244722\pi\)
\(234\) 0 0
\(235\) −1.25491 −0.0818616
\(236\) 0 0
\(237\) −0.236283 −0.0153482
\(238\) 0 0
\(239\) 8.58274 0.555172 0.277586 0.960701i \(-0.410466\pi\)
0.277586 + 0.960701i \(0.410466\pi\)
\(240\) 0 0
\(241\) 15.3857 0.991079 0.495540 0.868585i \(-0.334970\pi\)
0.495540 + 0.868585i \(0.334970\pi\)
\(242\) 0 0
\(243\) 13.2023 0.846925
\(244\) 0 0
\(245\) −15.4458 −0.986795
\(246\) 0 0
\(247\) 24.2325 1.54187
\(248\) 0 0
\(249\) 5.26815 0.333856
\(250\) 0 0
\(251\) 8.85477 0.558908 0.279454 0.960159i \(-0.409847\pi\)
0.279454 + 0.960159i \(0.409847\pi\)
\(252\) 0 0
\(253\) 0.360532 0.0226664
\(254\) 0 0
\(255\) −0.210787 −0.0132000
\(256\) 0 0
\(257\) −22.6008 −1.40980 −0.704899 0.709308i \(-0.749008\pi\)
−0.704899 + 0.709308i \(0.749008\pi\)
\(258\) 0 0
\(259\) −15.4857 −0.962236
\(260\) 0 0
\(261\) −2.51428 −0.155630
\(262\) 0 0
\(263\) 7.73590 0.477016 0.238508 0.971141i \(-0.423342\pi\)
0.238508 + 0.971141i \(0.423342\pi\)
\(264\) 0 0
\(265\) −10.9278 −0.671292
\(266\) 0 0
\(267\) 7.16858 0.438710
\(268\) 0 0
\(269\) −3.87154 −0.236052 −0.118026 0.993011i \(-0.537657\pi\)
−0.118026 + 0.993011i \(0.537657\pi\)
\(270\) 0 0
\(271\) −4.24705 −0.257990 −0.128995 0.991645i \(-0.541175\pi\)
−0.128995 + 0.991645i \(0.541175\pi\)
\(272\) 0 0
\(273\) −14.1677 −0.857466
\(274\) 0 0
\(275\) 0.360532 0.0217409
\(276\) 0 0
\(277\) −7.26754 −0.436664 −0.218332 0.975875i \(-0.570062\pi\)
−0.218332 + 0.975875i \(0.570062\pi\)
\(278\) 0 0
\(279\) 25.8769 1.54921
\(280\) 0 0
\(281\) −13.0131 −0.776297 −0.388148 0.921597i \(-0.626885\pi\)
−0.388148 + 0.921597i \(0.626885\pi\)
\(282\) 0 0
\(283\) −17.8342 −1.06013 −0.530067 0.847956i \(-0.677833\pi\)
−0.530067 + 0.847956i \(0.677833\pi\)
\(284\) 0 0
\(285\) −2.61790 −0.155071
\(286\) 0 0
\(287\) −25.1087 −1.48212
\(288\) 0 0
\(289\) −16.8625 −0.991910
\(290\) 0 0
\(291\) 7.98037 0.467818
\(292\) 0 0
\(293\) 7.32291 0.427809 0.213905 0.976855i \(-0.431382\pi\)
0.213905 + 0.976855i \(0.431382\pi\)
\(294\) 0 0
\(295\) −9.66955 −0.562983
\(296\) 0 0
\(297\) −1.16333 −0.0675030
\(298\) 0 0
\(299\) 5.26123 0.304265
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −7.24352 −0.416129
\(304\) 0 0
\(305\) −9.71441 −0.556245
\(306\) 0 0
\(307\) 29.1127 1.66155 0.830775 0.556608i \(-0.187897\pi\)
0.830775 + 0.556608i \(0.187897\pi\)
\(308\) 0 0
\(309\) 1.88333 0.107139
\(310\) 0 0
\(311\) −0.458912 −0.0260225 −0.0130113 0.999915i \(-0.504142\pi\)
−0.0130113 + 0.999915i \(0.504142\pi\)
\(312\) 0 0
\(313\) −17.8279 −1.00769 −0.503846 0.863794i \(-0.668082\pi\)
−0.503846 + 0.863794i \(0.668082\pi\)
\(314\) 0 0
\(315\) −12.6825 −0.714579
\(316\) 0 0
\(317\) 23.8799 1.34123 0.670614 0.741807i \(-0.266031\pi\)
0.670614 + 0.741807i \(0.266031\pi\)
\(318\) 0 0
\(319\) 0.338625 0.0189594
\(320\) 0 0
\(321\) 4.62203 0.257976
\(322\) 0 0
\(323\) 1.70809 0.0950407
\(324\) 0 0
\(325\) 5.26123 0.291840
\(326\) 0 0
\(327\) −3.29384 −0.182150
\(328\) 0 0
\(329\) −5.94540 −0.327781
\(330\) 0 0
\(331\) −30.0214 −1.65013 −0.825063 0.565041i \(-0.808860\pi\)
−0.825063 + 0.565041i \(0.808860\pi\)
\(332\) 0 0
\(333\) −8.74988 −0.479490
\(334\) 0 0
\(335\) −7.07001 −0.386276
\(336\) 0 0
\(337\) 11.9440 0.650631 0.325316 0.945605i \(-0.394529\pi\)
0.325316 + 0.945605i \(0.394529\pi\)
\(338\) 0 0
\(339\) −4.32385 −0.234839
\(340\) 0 0
\(341\) −3.48512 −0.188730
\(342\) 0 0
\(343\) −40.0136 −2.16053
\(344\) 0 0
\(345\) −0.568386 −0.0306009
\(346\) 0 0
\(347\) 0.873132 0.0468722 0.0234361 0.999725i \(-0.492539\pi\)
0.0234361 + 0.999725i \(0.492539\pi\)
\(348\) 0 0
\(349\) −18.8868 −1.01099 −0.505493 0.862831i \(-0.668689\pi\)
−0.505493 + 0.862831i \(0.668689\pi\)
\(350\) 0 0
\(351\) −16.9764 −0.906131
\(352\) 0 0
\(353\) −5.43875 −0.289475 −0.144738 0.989470i \(-0.546234\pi\)
−0.144738 + 0.989470i \(0.546234\pi\)
\(354\) 0 0
\(355\) 11.3747 0.603707
\(356\) 0 0
\(357\) −0.998646 −0.0528540
\(358\) 0 0
\(359\) 23.4229 1.23622 0.618108 0.786093i \(-0.287899\pi\)
0.618108 + 0.786093i \(0.287899\pi\)
\(360\) 0 0
\(361\) 2.21390 0.116521
\(362\) 0 0
\(363\) −6.17837 −0.324280
\(364\) 0 0
\(365\) −0.745086 −0.0389996
\(366\) 0 0
\(367\) −27.2014 −1.41990 −0.709951 0.704252i \(-0.751283\pi\)
−0.709951 + 0.704252i \(0.751283\pi\)
\(368\) 0 0
\(369\) −14.1872 −0.738554
\(370\) 0 0
\(371\) −51.7728 −2.68791
\(372\) 0 0
\(373\) 20.0431 1.03779 0.518897 0.854837i \(-0.326343\pi\)
0.518897 + 0.854837i \(0.326343\pi\)
\(374\) 0 0
\(375\) −0.568386 −0.0293513
\(376\) 0 0
\(377\) 4.94155 0.254503
\(378\) 0 0
\(379\) 12.8364 0.659362 0.329681 0.944092i \(-0.393059\pi\)
0.329681 + 0.944092i \(0.393059\pi\)
\(380\) 0 0
\(381\) 2.34787 0.120285
\(382\) 0 0
\(383\) 30.5405 1.56055 0.780273 0.625439i \(-0.215080\pi\)
0.780273 + 0.625439i \(0.215080\pi\)
\(384\) 0 0
\(385\) 1.70809 0.0870524
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 32.3392 1.63966 0.819831 0.572605i \(-0.194067\pi\)
0.819831 + 0.572605i \(0.194067\pi\)
\(390\) 0 0
\(391\) 0.370852 0.0187548
\(392\) 0 0
\(393\) 10.0935 0.509150
\(394\) 0 0
\(395\) 0.415709 0.0209166
\(396\) 0 0
\(397\) 28.9147 1.45119 0.725594 0.688123i \(-0.241565\pi\)
0.725594 + 0.688123i \(0.241565\pi\)
\(398\) 0 0
\(399\) −12.4028 −0.620919
\(400\) 0 0
\(401\) 24.2862 1.21279 0.606397 0.795162i \(-0.292614\pi\)
0.606397 + 0.795162i \(0.292614\pi\)
\(402\) 0 0
\(403\) −50.8583 −2.53343
\(404\) 0 0
\(405\) −6.19680 −0.307922
\(406\) 0 0
\(407\) 1.17844 0.0584131
\(408\) 0 0
\(409\) 31.8371 1.57424 0.787122 0.616797i \(-0.211570\pi\)
0.787122 + 0.616797i \(0.211570\pi\)
\(410\) 0 0
\(411\) 3.90455 0.192597
\(412\) 0 0
\(413\) −45.8114 −2.25423
\(414\) 0 0
\(415\) −9.26862 −0.454978
\(416\) 0 0
\(417\) −11.7449 −0.575149
\(418\) 0 0
\(419\) 21.4880 1.04976 0.524879 0.851177i \(-0.324110\pi\)
0.524879 + 0.851177i \(0.324110\pi\)
\(420\) 0 0
\(421\) −13.2930 −0.647859 −0.323930 0.946081i \(-0.605004\pi\)
−0.323930 + 0.946081i \(0.605004\pi\)
\(422\) 0 0
\(423\) −3.35933 −0.163336
\(424\) 0 0
\(425\) 0.370852 0.0179890
\(426\) 0 0
\(427\) −46.0239 −2.22725
\(428\) 0 0
\(429\) 1.07814 0.0520530
\(430\) 0 0
\(431\) 0.534460 0.0257440 0.0128720 0.999917i \(-0.495903\pi\)
0.0128720 + 0.999917i \(0.495903\pi\)
\(432\) 0 0
\(433\) 16.3377 0.785140 0.392570 0.919722i \(-0.371586\pi\)
0.392570 + 0.919722i \(0.371586\pi\)
\(434\) 0 0
\(435\) −0.533850 −0.0255962
\(436\) 0 0
\(437\) 4.60586 0.220328
\(438\) 0 0
\(439\) −28.7573 −1.37251 −0.686255 0.727361i \(-0.740746\pi\)
−0.686255 + 0.727361i \(0.740746\pi\)
\(440\) 0 0
\(441\) −41.3474 −1.96892
\(442\) 0 0
\(443\) 15.6542 0.743751 0.371876 0.928283i \(-0.378715\pi\)
0.371876 + 0.928283i \(0.378715\pi\)
\(444\) 0 0
\(445\) −12.6122 −0.597874
\(446\) 0 0
\(447\) −9.36491 −0.442945
\(448\) 0 0
\(449\) 38.4302 1.81363 0.906817 0.421525i \(-0.138505\pi\)
0.906817 + 0.421525i \(0.138505\pi\)
\(450\) 0 0
\(451\) 1.91074 0.0899730
\(452\) 0 0
\(453\) 5.05422 0.237468
\(454\) 0 0
\(455\) 24.9261 1.16855
\(456\) 0 0
\(457\) −37.2199 −1.74107 −0.870536 0.492104i \(-0.836228\pi\)
−0.870536 + 0.492104i \(0.836228\pi\)
\(458\) 0 0
\(459\) −1.19663 −0.0558537
\(460\) 0 0
\(461\) −3.25491 −0.151596 −0.0757982 0.997123i \(-0.524150\pi\)
−0.0757982 + 0.997123i \(0.524150\pi\)
\(462\) 0 0
\(463\) −29.9534 −1.39205 −0.696027 0.718016i \(-0.745050\pi\)
−0.696027 + 0.718016i \(0.745050\pi\)
\(464\) 0 0
\(465\) 5.49437 0.254795
\(466\) 0 0
\(467\) 39.3302 1.81999 0.909993 0.414623i \(-0.136087\pi\)
0.909993 + 0.414623i \(0.136087\pi\)
\(468\) 0 0
\(469\) −33.4956 −1.54668
\(470\) 0 0
\(471\) −3.76413 −0.173442
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.60586 0.211331
\(476\) 0 0
\(477\) −29.2532 −1.33941
\(478\) 0 0
\(479\) −8.18660 −0.374055 −0.187028 0.982355i \(-0.559885\pi\)
−0.187028 + 0.982355i \(0.559885\pi\)
\(480\) 0 0
\(481\) 17.1969 0.784113
\(482\) 0 0
\(483\) −2.69284 −0.122529
\(484\) 0 0
\(485\) −14.0404 −0.637542
\(486\) 0 0
\(487\) 31.9963 1.44989 0.724946 0.688806i \(-0.241865\pi\)
0.724946 + 0.688806i \(0.241865\pi\)
\(488\) 0 0
\(489\) 9.13844 0.413255
\(490\) 0 0
\(491\) 7.12584 0.321585 0.160792 0.986988i \(-0.448595\pi\)
0.160792 + 0.986988i \(0.448595\pi\)
\(492\) 0 0
\(493\) 0.348319 0.0156875
\(494\) 0 0
\(495\) 0.965121 0.0433790
\(496\) 0 0
\(497\) 53.8899 2.41729
\(498\) 0 0
\(499\) −23.4219 −1.04851 −0.524254 0.851562i \(-0.675656\pi\)
−0.524254 + 0.851562i \(0.675656\pi\)
\(500\) 0 0
\(501\) −7.88711 −0.352370
\(502\) 0 0
\(503\) 3.94170 0.175752 0.0878758 0.996131i \(-0.471992\pi\)
0.0878758 + 0.996131i \(0.471992\pi\)
\(504\) 0 0
\(505\) 12.7440 0.567101
\(506\) 0 0
\(507\) 8.34421 0.370579
\(508\) 0 0
\(509\) 38.5237 1.70753 0.853766 0.520656i \(-0.174313\pi\)
0.853766 + 0.520656i \(0.174313\pi\)
\(510\) 0 0
\(511\) −3.52999 −0.156158
\(512\) 0 0
\(513\) −14.8617 −0.656159
\(514\) 0 0
\(515\) −3.31347 −0.146009
\(516\) 0 0
\(517\) 0.452436 0.0198981
\(518\) 0 0
\(519\) −1.16813 −0.0512754
\(520\) 0 0
\(521\) −39.2428 −1.71926 −0.859630 0.510917i \(-0.829306\pi\)
−0.859630 + 0.510917i \(0.829306\pi\)
\(522\) 0 0
\(523\) 39.0190 1.70618 0.853090 0.521763i \(-0.174726\pi\)
0.853090 + 0.521763i \(0.174726\pi\)
\(524\) 0 0
\(525\) −2.69284 −0.117525
\(526\) 0 0
\(527\) −3.58489 −0.156160
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −25.8848 −1.12330
\(532\) 0 0
\(533\) 27.8833 1.20776
\(534\) 0 0
\(535\) −8.13184 −0.351570
\(536\) 0 0
\(537\) −10.0308 −0.432860
\(538\) 0 0
\(539\) 5.56870 0.239861
\(540\) 0 0
\(541\) 2.54061 0.109230 0.0546148 0.998508i \(-0.482607\pi\)
0.0546148 + 0.998508i \(0.482607\pi\)
\(542\) 0 0
\(543\) 1.62261 0.0696329
\(544\) 0 0
\(545\) 5.79508 0.248234
\(546\) 0 0
\(547\) −6.78776 −0.290224 −0.145112 0.989415i \(-0.546354\pi\)
−0.145112 + 0.989415i \(0.546354\pi\)
\(548\) 0 0
\(549\) −26.0049 −1.10986
\(550\) 0 0
\(551\) 4.32600 0.184294
\(552\) 0 0
\(553\) 1.96950 0.0837517
\(554\) 0 0
\(555\) −1.85784 −0.0788607
\(556\) 0 0
\(557\) −24.7086 −1.04694 −0.523468 0.852045i \(-0.675362\pi\)
−0.523468 + 0.852045i \(0.675362\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0.0759955 0.00320853
\(562\) 0 0
\(563\) 28.1423 1.18606 0.593029 0.805181i \(-0.297932\pi\)
0.593029 + 0.805181i \(0.297932\pi\)
\(564\) 0 0
\(565\) 7.60724 0.320039
\(566\) 0 0
\(567\) −29.3586 −1.23294
\(568\) 0 0
\(569\) 41.1938 1.72694 0.863468 0.504404i \(-0.168288\pi\)
0.863468 + 0.504404i \(0.168288\pi\)
\(570\) 0 0
\(571\) −32.4687 −1.35877 −0.679387 0.733780i \(-0.737754\pi\)
−0.679387 + 0.733780i \(0.737754\pi\)
\(572\) 0 0
\(573\) −0.646126 −0.0269923
\(574\) 0 0
\(575\) 1.00000 0.0417029
\(576\) 0 0
\(577\) 6.61770 0.275498 0.137749 0.990467i \(-0.456013\pi\)
0.137749 + 0.990467i \(0.456013\pi\)
\(578\) 0 0
\(579\) −15.0448 −0.625242
\(580\) 0 0
\(581\) −43.9119 −1.82177
\(582\) 0 0
\(583\) 3.93984 0.163171
\(584\) 0 0
\(585\) 14.0840 0.582301
\(586\) 0 0
\(587\) 8.18008 0.337628 0.168814 0.985648i \(-0.446006\pi\)
0.168814 + 0.985648i \(0.446006\pi\)
\(588\) 0 0
\(589\) −44.5230 −1.83454
\(590\) 0 0
\(591\) 3.09443 0.127288
\(592\) 0 0
\(593\) 29.7489 1.22164 0.610821 0.791768i \(-0.290839\pi\)
0.610821 + 0.791768i \(0.290839\pi\)
\(594\) 0 0
\(595\) 1.75699 0.0720294
\(596\) 0 0
\(597\) 10.5789 0.432966
\(598\) 0 0
\(599\) −39.4069 −1.61012 −0.805061 0.593192i \(-0.797868\pi\)
−0.805061 + 0.593192i \(0.797868\pi\)
\(600\) 0 0
\(601\) −34.3183 −1.39987 −0.699937 0.714205i \(-0.746788\pi\)
−0.699937 + 0.714205i \(0.746788\pi\)
\(602\) 0 0
\(603\) −18.9260 −0.770725
\(604\) 0 0
\(605\) 10.8700 0.441929
\(606\) 0 0
\(607\) 11.7570 0.477200 0.238600 0.971118i \(-0.423312\pi\)
0.238600 + 0.971118i \(0.423312\pi\)
\(608\) 0 0
\(609\) −2.52922 −0.102489
\(610\) 0 0
\(611\) 6.60239 0.267104
\(612\) 0 0
\(613\) 5.86862 0.237031 0.118516 0.992952i \(-0.462186\pi\)
0.118516 + 0.992952i \(0.462186\pi\)
\(614\) 0 0
\(615\) −3.01232 −0.121468
\(616\) 0 0
\(617\) −8.45842 −0.340523 −0.170262 0.985399i \(-0.554461\pi\)
−0.170262 + 0.985399i \(0.554461\pi\)
\(618\) 0 0
\(619\) −0.851047 −0.0342065 −0.0171032 0.999854i \(-0.505444\pi\)
−0.0171032 + 0.999854i \(0.505444\pi\)
\(620\) 0 0
\(621\) −3.22669 −0.129483
\(622\) 0 0
\(623\) −59.7527 −2.39394
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0.943838 0.0376933
\(628\) 0 0
\(629\) 1.21217 0.0483325
\(630\) 0 0
\(631\) −19.7131 −0.784768 −0.392384 0.919802i \(-0.628350\pi\)
−0.392384 + 0.919802i \(0.628350\pi\)
\(632\) 0 0
\(633\) −3.46717 −0.137808
\(634\) 0 0
\(635\) −4.13077 −0.163925
\(636\) 0 0
\(637\) 81.2638 3.21979
\(638\) 0 0
\(639\) 30.4494 1.20456
\(640\) 0 0
\(641\) −34.3593 −1.35711 −0.678555 0.734550i \(-0.737393\pi\)
−0.678555 + 0.734550i \(0.737393\pi\)
\(642\) 0 0
\(643\) 24.2817 0.957578 0.478789 0.877930i \(-0.341076\pi\)
0.478789 + 0.877930i \(0.341076\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 30.1465 1.18518 0.592590 0.805504i \(-0.298105\pi\)
0.592590 + 0.805504i \(0.298105\pi\)
\(648\) 0 0
\(649\) 3.48618 0.136845
\(650\) 0 0
\(651\) 26.0307 1.02022
\(652\) 0 0
\(653\) −7.17298 −0.280700 −0.140350 0.990102i \(-0.544823\pi\)
−0.140350 + 0.990102i \(0.544823\pi\)
\(654\) 0 0
\(655\) −17.7582 −0.693870
\(656\) 0 0
\(657\) −1.99455 −0.0778148
\(658\) 0 0
\(659\) 12.8521 0.500645 0.250323 0.968162i \(-0.419463\pi\)
0.250323 + 0.968162i \(0.419463\pi\)
\(660\) 0 0
\(661\) 3.72426 0.144857 0.0724285 0.997374i \(-0.476925\pi\)
0.0724285 + 0.997374i \(0.476925\pi\)
\(662\) 0 0
\(663\) 1.10900 0.0430700
\(664\) 0 0
\(665\) 21.8212 0.846188
\(666\) 0 0
\(667\) 0.939238 0.0363675
\(668\) 0 0
\(669\) 9.32924 0.360689
\(670\) 0 0
\(671\) 3.50235 0.135207
\(672\) 0 0
\(673\) 13.5204 0.521175 0.260587 0.965450i \(-0.416084\pi\)
0.260587 + 0.965450i \(0.416084\pi\)
\(674\) 0 0
\(675\) −3.22669 −0.124195
\(676\) 0 0
\(677\) −17.8534 −0.686161 −0.343081 0.939306i \(-0.611470\pi\)
−0.343081 + 0.939306i \(0.611470\pi\)
\(678\) 0 0
\(679\) −66.5192 −2.55277
\(680\) 0 0
\(681\) −0.688719 −0.0263918
\(682\) 0 0
\(683\) −15.8200 −0.605337 −0.302669 0.953096i \(-0.597878\pi\)
−0.302669 + 0.953096i \(0.597878\pi\)
\(684\) 0 0
\(685\) −6.86954 −0.262472
\(686\) 0 0
\(687\) −12.9758 −0.495059
\(688\) 0 0
\(689\) 57.4939 2.19034
\(690\) 0 0
\(691\) −9.45476 −0.359676 −0.179838 0.983696i \(-0.557557\pi\)
−0.179838 + 0.983696i \(0.557557\pi\)
\(692\) 0 0
\(693\) 4.57245 0.173693
\(694\) 0 0
\(695\) 20.6635 0.783813
\(696\) 0 0
\(697\) 1.96543 0.0744460
\(698\) 0 0
\(699\) 12.4715 0.471717
\(700\) 0 0
\(701\) 29.0096 1.09568 0.547838 0.836584i \(-0.315451\pi\)
0.547838 + 0.836584i \(0.315451\pi\)
\(702\) 0 0
\(703\) 15.0548 0.567801
\(704\) 0 0
\(705\) −0.713276 −0.0268635
\(706\) 0 0
\(707\) 60.3773 2.27072
\(708\) 0 0
\(709\) −8.22423 −0.308868 −0.154434 0.988003i \(-0.549355\pi\)
−0.154434 + 0.988003i \(0.549355\pi\)
\(710\) 0 0
\(711\) 1.11283 0.0417342
\(712\) 0 0
\(713\) −9.66662 −0.362018
\(714\) 0 0
\(715\) −1.89684 −0.0709378
\(716\) 0 0
\(717\) 4.87831 0.182184
\(718\) 0 0
\(719\) 12.5468 0.467917 0.233958 0.972247i \(-0.424832\pi\)
0.233958 + 0.972247i \(0.424832\pi\)
\(720\) 0 0
\(721\) −15.6982 −0.584633
\(722\) 0 0
\(723\) 8.74501 0.325230
\(724\) 0 0
\(725\) 0.939238 0.0348824
\(726\) 0 0
\(727\) 29.1952 1.08279 0.541395 0.840768i \(-0.317896\pi\)
0.541395 + 0.840768i \(0.317896\pi\)
\(728\) 0 0
\(729\) −11.0864 −0.410609
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −36.0414 −1.33122 −0.665611 0.746299i \(-0.731829\pi\)
−0.665611 + 0.746299i \(0.731829\pi\)
\(734\) 0 0
\(735\) −8.77917 −0.323825
\(736\) 0 0
\(737\) 2.54896 0.0938923
\(738\) 0 0
\(739\) −0.728002 −0.0267800 −0.0133900 0.999910i \(-0.504262\pi\)
−0.0133900 + 0.999910i \(0.504262\pi\)
\(740\) 0 0
\(741\) 13.7734 0.505978
\(742\) 0 0
\(743\) 32.8880 1.20655 0.603273 0.797535i \(-0.293863\pi\)
0.603273 + 0.797535i \(0.293863\pi\)
\(744\) 0 0
\(745\) 16.4763 0.603646
\(746\) 0 0
\(747\) −24.8115 −0.907805
\(748\) 0 0
\(749\) −38.5262 −1.40772
\(750\) 0 0
\(751\) −14.3337 −0.523044 −0.261522 0.965197i \(-0.584224\pi\)
−0.261522 + 0.965197i \(0.584224\pi\)
\(752\) 0 0
\(753\) 5.03293 0.183410
\(754\) 0 0
\(755\) −8.89224 −0.323622
\(756\) 0 0
\(757\) −0.0587519 −0.00213537 −0.00106769 0.999999i \(-0.500340\pi\)
−0.00106769 + 0.999999i \(0.500340\pi\)
\(758\) 0 0
\(759\) 0.204921 0.00743817
\(760\) 0 0
\(761\) 45.7135 1.65711 0.828556 0.559906i \(-0.189163\pi\)
0.828556 + 0.559906i \(0.189163\pi\)
\(762\) 0 0
\(763\) 27.4553 0.993950
\(764\) 0 0
\(765\) 0.992748 0.0358929
\(766\) 0 0
\(767\) 50.8737 1.83694
\(768\) 0 0
\(769\) −39.3949 −1.42061 −0.710307 0.703892i \(-0.751444\pi\)
−0.710307 + 0.703892i \(0.751444\pi\)
\(770\) 0 0
\(771\) −12.8460 −0.462636
\(772\) 0 0
\(773\) 21.1799 0.761788 0.380894 0.924619i \(-0.375616\pi\)
0.380894 + 0.924619i \(0.375616\pi\)
\(774\) 0 0
\(775\) −9.66662 −0.347235
\(776\) 0 0
\(777\) −8.80187 −0.315765
\(778\) 0 0
\(779\) 24.4100 0.874578
\(780\) 0 0
\(781\) −4.10094 −0.146743
\(782\) 0 0
\(783\) −3.03063 −0.108306
\(784\) 0 0
\(785\) 6.62249 0.236367
\(786\) 0 0
\(787\) −21.5154 −0.766941 −0.383470 0.923553i \(-0.625271\pi\)
−0.383470 + 0.923553i \(0.625271\pi\)
\(788\) 0 0
\(789\) 4.39698 0.156537
\(790\) 0 0
\(791\) 36.0408 1.28146
\(792\) 0 0
\(793\) 51.1097 1.81496
\(794\) 0 0
\(795\) −6.21124 −0.220290
\(796\) 0 0
\(797\) −54.7227 −1.93838 −0.969189 0.246320i \(-0.920779\pi\)
−0.969189 + 0.246320i \(0.920779\pi\)
\(798\) 0 0
\(799\) 0.465388 0.0164642
\(800\) 0 0
\(801\) −33.7620 −1.19292
\(802\) 0 0
\(803\) 0.268627 0.00947965
\(804\) 0 0
\(805\) 4.73770 0.166982
\(806\) 0 0
\(807\) −2.20053 −0.0774623
\(808\) 0 0
\(809\) −43.5554 −1.53133 −0.765663 0.643242i \(-0.777589\pi\)
−0.765663 + 0.643242i \(0.777589\pi\)
\(810\) 0 0
\(811\) 13.1837 0.462944 0.231472 0.972842i \(-0.425646\pi\)
0.231472 + 0.972842i \(0.425646\pi\)
\(812\) 0 0
\(813\) −2.41397 −0.0846615
\(814\) 0 0
\(815\) −16.0779 −0.563183
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 66.7256 2.33158
\(820\) 0 0
\(821\) −3.63328 −0.126803 −0.0634013 0.997988i \(-0.520195\pi\)
−0.0634013 + 0.997988i \(0.520195\pi\)
\(822\) 0 0
\(823\) 39.5044 1.37704 0.688518 0.725220i \(-0.258262\pi\)
0.688518 + 0.725220i \(0.258262\pi\)
\(824\) 0 0
\(825\) 0.204921 0.00713444
\(826\) 0 0
\(827\) −48.3819 −1.68240 −0.841202 0.540721i \(-0.818152\pi\)
−0.841202 + 0.540721i \(0.818152\pi\)
\(828\) 0 0
\(829\) −44.5864 −1.54855 −0.774275 0.632850i \(-0.781885\pi\)
−0.774275 + 0.632850i \(0.781885\pi\)
\(830\) 0 0
\(831\) −4.13077 −0.143295
\(832\) 0 0
\(833\) 5.72810 0.198467
\(834\) 0 0
\(835\) 13.8763 0.480210
\(836\) 0 0
\(837\) 31.1912 1.07813
\(838\) 0 0
\(839\) 29.2324 1.00921 0.504606 0.863350i \(-0.331638\pi\)
0.504606 + 0.863350i \(0.331638\pi\)
\(840\) 0 0
\(841\) −28.1178 −0.969580
\(842\) 0 0
\(843\) −7.39647 −0.254748
\(844\) 0 0
\(845\) −14.6805 −0.505025
\(846\) 0 0
\(847\) 51.4989 1.76952
\(848\) 0 0
\(849\) −10.1367 −0.347891
\(850\) 0 0
\(851\) 3.26862 0.112047
\(852\) 0 0
\(853\) 50.5599 1.73114 0.865569 0.500790i \(-0.166957\pi\)
0.865569 + 0.500790i \(0.166957\pi\)
\(854\) 0 0
\(855\) 12.3296 0.421663
\(856\) 0 0
\(857\) −21.9568 −0.750030 −0.375015 0.927019i \(-0.622362\pi\)
−0.375015 + 0.927019i \(0.622362\pi\)
\(858\) 0 0
\(859\) −13.7671 −0.469726 −0.234863 0.972029i \(-0.575464\pi\)
−0.234863 + 0.972029i \(0.575464\pi\)
\(860\) 0 0
\(861\) −14.2714 −0.486370
\(862\) 0 0
\(863\) 26.5748 0.904618 0.452309 0.891861i \(-0.350600\pi\)
0.452309 + 0.891861i \(0.350600\pi\)
\(864\) 0 0
\(865\) 2.05518 0.0698781
\(866\) 0 0
\(867\) −9.58439 −0.325503
\(868\) 0 0
\(869\) −0.149876 −0.00508420
\(870\) 0 0
\(871\) 37.1969 1.26037
\(872\) 0 0
\(873\) −37.5853 −1.27207
\(874\) 0 0
\(875\) 4.73770 0.160163
\(876\) 0 0
\(877\) 45.9414 1.55133 0.775665 0.631145i \(-0.217415\pi\)
0.775665 + 0.631145i \(0.217415\pi\)
\(878\) 0 0
\(879\) 4.16224 0.140389
\(880\) 0 0
\(881\) −23.9054 −0.805392 −0.402696 0.915334i \(-0.631927\pi\)
−0.402696 + 0.915334i \(0.631927\pi\)
\(882\) 0 0
\(883\) −52.4002 −1.76341 −0.881703 0.471804i \(-0.843603\pi\)
−0.881703 + 0.471804i \(0.843603\pi\)
\(884\) 0 0
\(885\) −5.49604 −0.184747
\(886\) 0 0
\(887\) 32.7608 1.10000 0.550000 0.835165i \(-0.314628\pi\)
0.550000 + 0.835165i \(0.314628\pi\)
\(888\) 0 0
\(889\) −19.5703 −0.656368
\(890\) 0 0
\(891\) 2.23415 0.0748467
\(892\) 0 0
\(893\) 5.77995 0.193419
\(894\) 0 0
\(895\) 17.6478 0.589902
\(896\) 0 0
\(897\) 2.99041 0.0998469
\(898\) 0 0
\(899\) −9.07926 −0.302810
\(900\) 0 0
\(901\) 4.05262 0.135012
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −2.85477 −0.0948957
\(906\) 0 0
\(907\) −20.9426 −0.695388 −0.347694 0.937608i \(-0.613035\pi\)
−0.347694 + 0.937608i \(0.613035\pi\)
\(908\) 0 0
\(909\) 34.1149 1.13152
\(910\) 0 0
\(911\) −40.6418 −1.34652 −0.673262 0.739404i \(-0.735107\pi\)
−0.673262 + 0.739404i \(0.735107\pi\)
\(912\) 0 0
\(913\) 3.34163 0.110592
\(914\) 0 0
\(915\) −5.52153 −0.182536
\(916\) 0 0
\(917\) −84.1330 −2.77831
\(918\) 0 0
\(919\) 50.3241 1.66004 0.830019 0.557735i \(-0.188330\pi\)
0.830019 + 0.557735i \(0.188330\pi\)
\(920\) 0 0
\(921\) 16.5473 0.545251
\(922\) 0 0
\(923\) −59.8449 −1.96982
\(924\) 0 0
\(925\) 3.26862 0.107471
\(926\) 0 0
\(927\) −8.86996 −0.291328
\(928\) 0 0
\(929\) 24.1954 0.793825 0.396912 0.917856i \(-0.370082\pi\)
0.396912 + 0.917856i \(0.370082\pi\)
\(930\) 0 0
\(931\) 71.1411 2.33155
\(932\) 0 0
\(933\) −0.260839 −0.00853949
\(934\) 0 0
\(935\) −0.133704 −0.00437259
\(936\) 0 0
\(937\) −37.2988 −1.21850 −0.609250 0.792978i \(-0.708529\pi\)
−0.609250 + 0.792978i \(0.708529\pi\)
\(938\) 0 0
\(939\) −10.1331 −0.330682
\(940\) 0 0
\(941\) 16.6467 0.542667 0.271334 0.962485i \(-0.412535\pi\)
0.271334 + 0.962485i \(0.412535\pi\)
\(942\) 0 0
\(943\) 5.29977 0.172584
\(944\) 0 0
\(945\) −15.2871 −0.497289
\(946\) 0 0
\(947\) 30.1868 0.980938 0.490469 0.871459i \(-0.336825\pi\)
0.490469 + 0.871459i \(0.336825\pi\)
\(948\) 0 0
\(949\) 3.92007 0.127251
\(950\) 0 0
\(951\) 13.5730 0.440134
\(952\) 0 0
\(953\) 34.6992 1.12402 0.562008 0.827132i \(-0.310029\pi\)
0.562008 + 0.827132i \(0.310029\pi\)
\(954\) 0 0
\(955\) 1.13677 0.0367851
\(956\) 0 0
\(957\) 0.192470 0.00622167
\(958\) 0 0
\(959\) −32.5458 −1.05096
\(960\) 0 0
\(961\) 62.4435 2.01431
\(962\) 0 0
\(963\) −21.7684 −0.701478
\(964\) 0 0
\(965\) 26.4694 0.852080
\(966\) 0 0
\(967\) 11.4788 0.369133 0.184566 0.982820i \(-0.440912\pi\)
0.184566 + 0.982820i \(0.440912\pi\)
\(968\) 0 0
\(969\) 0.970856 0.0311884
\(970\) 0 0
\(971\) 46.8441 1.50330 0.751650 0.659563i \(-0.229258\pi\)
0.751650 + 0.659563i \(0.229258\pi\)
\(972\) 0 0
\(973\) 97.8977 3.13846
\(974\) 0 0
\(975\) 2.99041 0.0957698
\(976\) 0 0
\(977\) 39.3737 1.25968 0.629838 0.776727i \(-0.283121\pi\)
0.629838 + 0.776727i \(0.283121\pi\)
\(978\) 0 0
\(979\) 4.54709 0.145326
\(980\) 0 0
\(981\) 15.5131 0.495294
\(982\) 0 0
\(983\) −42.3505 −1.35077 −0.675385 0.737465i \(-0.736023\pi\)
−0.675385 + 0.737465i \(0.736023\pi\)
\(984\) 0 0
\(985\) −5.44424 −0.173468
\(986\) 0 0
\(987\) −3.37929 −0.107564
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 2.00104 0.0635652 0.0317826 0.999495i \(-0.489882\pi\)
0.0317826 + 0.999495i \(0.489882\pi\)
\(992\) 0 0
\(993\) −17.0638 −0.541502
\(994\) 0 0
\(995\) −18.6122 −0.590045
\(996\) 0 0
\(997\) 1.97936 0.0626869 0.0313435 0.999509i \(-0.490021\pi\)
0.0313435 + 0.999509i \(0.490021\pi\)
\(998\) 0 0
\(999\) −10.5468 −0.333687
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 1840.2.a.v.1.3 5
4.3 odd 2 920.2.a.j.1.3 5
5.4 even 2 9200.2.a.cu.1.3 5
8.3 odd 2 7360.2.a.co.1.3 5
8.5 even 2 7360.2.a.cp.1.3 5
12.11 even 2 8280.2.a.bs.1.5 5
20.3 even 4 4600.2.e.u.4049.5 10
20.7 even 4 4600.2.e.u.4049.6 10
20.19 odd 2 4600.2.a.be.1.3 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
920.2.a.j.1.3 5 4.3 odd 2
1840.2.a.v.1.3 5 1.1 even 1 trivial
4600.2.a.be.1.3 5 20.19 odd 2
4600.2.e.u.4049.5 10 20.3 even 4
4600.2.e.u.4049.6 10 20.7 even 4
7360.2.a.co.1.3 5 8.3 odd 2
7360.2.a.cp.1.3 5 8.5 even 2
8280.2.a.bs.1.5 5 12.11 even 2
9200.2.a.cu.1.3 5 5.4 even 2