Properties

Label 7728.2.a.bt.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 7728 = 2^{4} \cdot 3 \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7728.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(61.7083906820\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.837.1
Defining polynomial: \(x^{3} - 6 x - 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 483)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.52892\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.52892 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.52892 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.86651 q^{11} +0.604574 q^{13} +1.52892 q^{15} -2.92434 q^{17} +1.86651 q^{19} -1.00000 q^{21} +1.00000 q^{23} -2.66241 q^{25} -1.00000 q^{27} +7.19133 q^{29} +5.86651 q^{31} +1.86651 q^{33} -1.52892 q^{35} -3.19133 q^{37} -0.604574 q^{39} -6.65736 q^{41} +5.66241 q^{43} -1.52892 q^{45} -10.7909 q^{47} +1.00000 q^{49} +2.92434 q^{51} -1.52892 q^{53} +2.85374 q^{55} -1.86651 q^{57} +6.31977 q^{59} -3.66241 q^{61} +1.00000 q^{63} -0.924344 q^{65} -5.37761 q^{67} -1.00000 q^{69} -15.2441 q^{71} +13.9822 q^{73} +2.66241 q^{75} -1.86651 q^{77} -9.19133 q^{79} +1.00000 q^{81} +0.924344 q^{83} +4.47108 q^{85} -7.19133 q^{87} +7.39543 q^{89} +0.604574 q^{91} -5.86651 q^{93} -2.85374 q^{95} +5.98218 q^{97} -1.86651 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9} + O(q^{10}) \) \( 3 q - 3 q^{3} + 3 q^{5} + 3 q^{7} + 3 q^{9} - 6 q^{11} + 9 q^{13} - 3 q^{15} + 6 q^{17} + 6 q^{19} - 3 q^{21} + 3 q^{23} - 3 q^{27} + 6 q^{29} + 18 q^{31} + 6 q^{33} + 3 q^{35} + 6 q^{37} - 9 q^{39} - 6 q^{41} + 9 q^{43} + 3 q^{45} - 18 q^{47} + 3 q^{49} - 6 q^{51} + 3 q^{53} - 15 q^{55} - 6 q^{57} - 3 q^{59} - 3 q^{61} + 3 q^{63} + 12 q^{65} + 21 q^{67} - 3 q^{69} - 9 q^{71} + 12 q^{73} - 6 q^{77} - 12 q^{79} + 3 q^{81} - 12 q^{83} + 21 q^{85} - 6 q^{87} + 15 q^{89} + 9 q^{91} - 18 q^{93} + 15 q^{95} - 12 q^{97} - 6 q^{99} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.52892 −0.683753 −0.341876 0.939745i \(-0.611062\pi\)
−0.341876 + 0.939745i \(0.611062\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.86651 −0.562773 −0.281387 0.959594i \(-0.590794\pi\)
−0.281387 + 0.959594i \(0.590794\pi\)
\(12\) 0 0
\(13\) 0.604574 0.167679 0.0838393 0.996479i \(-0.473282\pi\)
0.0838393 + 0.996479i \(0.473282\pi\)
\(14\) 0 0
\(15\) 1.52892 0.394765
\(16\) 0 0
\(17\) −2.92434 −0.709258 −0.354629 0.935007i \(-0.615393\pi\)
−0.354629 + 0.935007i \(0.615393\pi\)
\(18\) 0 0
\(19\) 1.86651 0.428206 0.214103 0.976811i \(-0.431317\pi\)
0.214103 + 0.976811i \(0.431317\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −2.66241 −0.532482
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 7.19133 1.33540 0.667698 0.744432i \(-0.267280\pi\)
0.667698 + 0.744432i \(0.267280\pi\)
\(30\) 0 0
\(31\) 5.86651 1.05366 0.526828 0.849972i \(-0.323381\pi\)
0.526828 + 0.849972i \(0.323381\pi\)
\(32\) 0 0
\(33\) 1.86651 0.324917
\(34\) 0 0
\(35\) −1.52892 −0.258434
\(36\) 0 0
\(37\) −3.19133 −0.524651 −0.262326 0.964979i \(-0.584489\pi\)
−0.262326 + 0.964979i \(0.584489\pi\)
\(38\) 0 0
\(39\) −0.604574 −0.0968093
\(40\) 0 0
\(41\) −6.65736 −1.03970 −0.519852 0.854256i \(-0.674013\pi\)
−0.519852 + 0.854256i \(0.674013\pi\)
\(42\) 0 0
\(43\) 5.66241 0.863509 0.431755 0.901991i \(-0.357895\pi\)
0.431755 + 0.901991i \(0.357895\pi\)
\(44\) 0 0
\(45\) −1.52892 −0.227918
\(46\) 0 0
\(47\) −10.7909 −1.57401 −0.787004 0.616948i \(-0.788369\pi\)
−0.787004 + 0.616948i \(0.788369\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 2.92434 0.409490
\(52\) 0 0
\(53\) −1.52892 −0.210013 −0.105007 0.994472i \(-0.533486\pi\)
−0.105007 + 0.994472i \(0.533486\pi\)
\(54\) 0 0
\(55\) 2.85374 0.384798
\(56\) 0 0
\(57\) −1.86651 −0.247225
\(58\) 0 0
\(59\) 6.31977 0.822764 0.411382 0.911463i \(-0.365046\pi\)
0.411382 + 0.911463i \(0.365046\pi\)
\(60\) 0 0
\(61\) −3.66241 −0.468924 −0.234462 0.972125i \(-0.575333\pi\)
−0.234462 + 0.972125i \(0.575333\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −0.924344 −0.114651
\(66\) 0 0
\(67\) −5.37761 −0.656979 −0.328490 0.944508i \(-0.606540\pi\)
−0.328490 + 0.944508i \(0.606540\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) −15.2441 −1.80914 −0.904572 0.426321i \(-0.859809\pi\)
−0.904572 + 0.426321i \(0.859809\pi\)
\(72\) 0 0
\(73\) 13.9822 1.63649 0.818245 0.574869i \(-0.194947\pi\)
0.818245 + 0.574869i \(0.194947\pi\)
\(74\) 0 0
\(75\) 2.66241 0.307429
\(76\) 0 0
\(77\) −1.86651 −0.212708
\(78\) 0 0
\(79\) −9.19133 −1.03411 −0.517053 0.855954i \(-0.672971\pi\)
−0.517053 + 0.855954i \(0.672971\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 0.924344 0.101460 0.0507300 0.998712i \(-0.483845\pi\)
0.0507300 + 0.998712i \(0.483845\pi\)
\(84\) 0 0
\(85\) 4.47108 0.484957
\(86\) 0 0
\(87\) −7.19133 −0.770991
\(88\) 0 0
\(89\) 7.39543 0.783914 0.391957 0.919984i \(-0.371798\pi\)
0.391957 + 0.919984i \(0.371798\pi\)
\(90\) 0 0
\(91\) 0.604574 0.0633766
\(92\) 0 0
\(93\) −5.86651 −0.608329
\(94\) 0 0
\(95\) −2.85374 −0.292787
\(96\) 0 0
\(97\) 5.98218 0.607398 0.303699 0.952768i \(-0.401778\pi\)
0.303699 + 0.952768i \(0.401778\pi\)
\(98\) 0 0
\(99\) −1.86651 −0.187591
\(100\) 0 0
\(101\) 4.05279 0.403267 0.201634 0.979461i \(-0.435375\pi\)
0.201634 + 0.979461i \(0.435375\pi\)
\(102\) 0 0
\(103\) 6.11567 0.602595 0.301298 0.953530i \(-0.402580\pi\)
0.301298 + 0.953530i \(0.402580\pi\)
\(104\) 0 0
\(105\) 1.52892 0.149207
\(106\) 0 0
\(107\) 2.33759 0.225983 0.112992 0.993596i \(-0.463957\pi\)
0.112992 + 0.993596i \(0.463957\pi\)
\(108\) 0 0
\(109\) −8.58675 −0.822462 −0.411231 0.911531i \(-0.634901\pi\)
−0.411231 + 0.911531i \(0.634901\pi\)
\(110\) 0 0
\(111\) 3.19133 0.302907
\(112\) 0 0
\(113\) 19.3776 1.82289 0.911446 0.411420i \(-0.134967\pi\)
0.911446 + 0.411420i \(0.134967\pi\)
\(114\) 0 0
\(115\) −1.52892 −0.142572
\(116\) 0 0
\(117\) 0.604574 0.0558929
\(118\) 0 0
\(119\) −2.92434 −0.268074
\(120\) 0 0
\(121\) −7.51615 −0.683286
\(122\) 0 0
\(123\) 6.65736 0.600274
\(124\) 0 0
\(125\) 11.7152 1.04784
\(126\) 0 0
\(127\) −4.98723 −0.442545 −0.221273 0.975212i \(-0.571021\pi\)
−0.221273 + 0.975212i \(0.571021\pi\)
\(128\) 0 0
\(129\) −5.66241 −0.498547
\(130\) 0 0
\(131\) 1.86651 0.163078 0.0815388 0.996670i \(-0.474017\pi\)
0.0815388 + 0.996670i \(0.474017\pi\)
\(132\) 0 0
\(133\) 1.86651 0.161847
\(134\) 0 0
\(135\) 1.52892 0.131588
\(136\) 0 0
\(137\) −15.1913 −1.29788 −0.648941 0.760838i \(-0.724788\pi\)
−0.648941 + 0.760838i \(0.724788\pi\)
\(138\) 0 0
\(139\) 12.4711 1.05778 0.528892 0.848689i \(-0.322608\pi\)
0.528892 + 0.848689i \(0.322608\pi\)
\(140\) 0 0
\(141\) 10.7909 0.908754
\(142\) 0 0
\(143\) −1.12844 −0.0943651
\(144\) 0 0
\(145\) −10.9950 −0.913081
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) −0.523868 −0.0429170 −0.0214585 0.999770i \(-0.506831\pi\)
−0.0214585 + 0.999770i \(0.506831\pi\)
\(150\) 0 0
\(151\) 11.3248 0.921601 0.460800 0.887504i \(-0.347562\pi\)
0.460800 + 0.887504i \(0.347562\pi\)
\(152\) 0 0
\(153\) −2.92434 −0.236419
\(154\) 0 0
\(155\) −8.96941 −0.720440
\(156\) 0 0
\(157\) 21.0222 1.67775 0.838877 0.544321i \(-0.183213\pi\)
0.838877 + 0.544321i \(0.183213\pi\)
\(158\) 0 0
\(159\) 1.52892 0.121251
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 13.3776 1.04781 0.523907 0.851775i \(-0.324474\pi\)
0.523907 + 0.851775i \(0.324474\pi\)
\(164\) 0 0
\(165\) −2.85374 −0.222163
\(166\) 0 0
\(167\) −20.5161 −1.58759 −0.793794 0.608187i \(-0.791897\pi\)
−0.793794 + 0.608187i \(0.791897\pi\)
\(168\) 0 0
\(169\) −12.6345 −0.971884
\(170\) 0 0
\(171\) 1.86651 0.142735
\(172\) 0 0
\(173\) −1.34264 −0.102079 −0.0510395 0.998697i \(-0.516253\pi\)
−0.0510395 + 0.998697i \(0.516253\pi\)
\(174\) 0 0
\(175\) −2.66241 −0.201259
\(176\) 0 0
\(177\) −6.31977 −0.475023
\(178\) 0 0
\(179\) 0.204098 0.0152550 0.00762751 0.999971i \(-0.497572\pi\)
0.00762751 + 0.999971i \(0.497572\pi\)
\(180\) 0 0
\(181\) 17.4482 1.29692 0.648458 0.761251i \(-0.275414\pi\)
0.648458 + 0.761251i \(0.275414\pi\)
\(182\) 0 0
\(183\) 3.66241 0.270733
\(184\) 0 0
\(185\) 4.87928 0.358732
\(186\) 0 0
\(187\) 5.45831 0.399151
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −19.3248 −1.39829 −0.699147 0.714978i \(-0.746437\pi\)
−0.699147 + 0.714978i \(0.746437\pi\)
\(192\) 0 0
\(193\) −1.32482 −0.0953626 −0.0476813 0.998863i \(-0.515183\pi\)
−0.0476813 + 0.998863i \(0.515183\pi\)
\(194\) 0 0
\(195\) 0.924344 0.0661936
\(196\) 0 0
\(197\) 17.5111 1.24761 0.623807 0.781578i \(-0.285585\pi\)
0.623807 + 0.781578i \(0.285585\pi\)
\(198\) 0 0
\(199\) 22.1786 1.57220 0.786098 0.618102i \(-0.212098\pi\)
0.786098 + 0.618102i \(0.212098\pi\)
\(200\) 0 0
\(201\) 5.37761 0.379307
\(202\) 0 0
\(203\) 7.19133 0.504732
\(204\) 0 0
\(205\) 10.1786 0.710901
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) −3.48385 −0.240983
\(210\) 0 0
\(211\) −2.40048 −0.165256 −0.0826278 0.996580i \(-0.526331\pi\)
−0.0826278 + 0.996580i \(0.526331\pi\)
\(212\) 0 0
\(213\) 15.2441 1.04451
\(214\) 0 0
\(215\) −8.65736 −0.590427
\(216\) 0 0
\(217\) 5.86651 0.398245
\(218\) 0 0
\(219\) −13.9822 −0.944828
\(220\) 0 0
\(221\) −1.76798 −0.118927
\(222\) 0 0
\(223\) 14.0528 0.941044 0.470522 0.882388i \(-0.344066\pi\)
0.470522 + 0.882388i \(0.344066\pi\)
\(224\) 0 0
\(225\) −2.66241 −0.177494
\(226\) 0 0
\(227\) −14.8537 −0.985877 −0.492939 0.870064i \(-0.664077\pi\)
−0.492939 + 0.870064i \(0.664077\pi\)
\(228\) 0 0
\(229\) −23.1106 −1.52719 −0.763596 0.645694i \(-0.776568\pi\)
−0.763596 + 0.645694i \(0.776568\pi\)
\(230\) 0 0
\(231\) 1.86651 0.122807
\(232\) 0 0
\(233\) −21.1207 −1.38366 −0.691832 0.722058i \(-0.743196\pi\)
−0.691832 + 0.722058i \(0.743196\pi\)
\(234\) 0 0
\(235\) 16.4983 1.07623
\(236\) 0 0
\(237\) 9.19133 0.597041
\(238\) 0 0
\(239\) 5.64459 0.365118 0.182559 0.983195i \(-0.441562\pi\)
0.182559 + 0.983195i \(0.441562\pi\)
\(240\) 0 0
\(241\) −9.44821 −0.608613 −0.304306 0.952574i \(-0.598425\pi\)
−0.304306 + 0.952574i \(0.598425\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −1.52892 −0.0976790
\(246\) 0 0
\(247\) 1.12844 0.0718011
\(248\) 0 0
\(249\) −0.924344 −0.0585779
\(250\) 0 0
\(251\) −0.639540 −0.0403674 −0.0201837 0.999796i \(-0.506425\pi\)
−0.0201837 + 0.999796i \(0.506425\pi\)
\(252\) 0 0
\(253\) −1.86651 −0.117346
\(254\) 0 0
\(255\) −4.47108 −0.279990
\(256\) 0 0
\(257\) −18.6395 −1.16270 −0.581351 0.813653i \(-0.697476\pi\)
−0.581351 + 0.813653i \(0.697476\pi\)
\(258\) 0 0
\(259\) −3.19133 −0.198299
\(260\) 0 0
\(261\) 7.19133 0.445132
\(262\) 0 0
\(263\) 25.1557 1.55117 0.775583 0.631246i \(-0.217456\pi\)
0.775583 + 0.631246i \(0.217456\pi\)
\(264\) 0 0
\(265\) 2.33759 0.143597
\(266\) 0 0
\(267\) −7.39543 −0.452593
\(268\) 0 0
\(269\) −4.97713 −0.303461 −0.151730 0.988422i \(-0.548485\pi\)
−0.151730 + 0.988422i \(0.548485\pi\)
\(270\) 0 0
\(271\) −4.92434 −0.299133 −0.149566 0.988752i \(-0.547788\pi\)
−0.149566 + 0.988752i \(0.547788\pi\)
\(272\) 0 0
\(273\) −0.604574 −0.0365905
\(274\) 0 0
\(275\) 4.96941 0.299667
\(276\) 0 0
\(277\) 25.1029 1.50829 0.754144 0.656710i \(-0.228052\pi\)
0.754144 + 0.656710i \(0.228052\pi\)
\(278\) 0 0
\(279\) 5.86651 0.351219
\(280\) 0 0
\(281\) 26.3726 1.57325 0.786627 0.617428i \(-0.211825\pi\)
0.786627 + 0.617428i \(0.211825\pi\)
\(282\) 0 0
\(283\) −12.4354 −0.739210 −0.369605 0.929189i \(-0.620507\pi\)
−0.369605 + 0.929189i \(0.620507\pi\)
\(284\) 0 0
\(285\) 2.85374 0.169041
\(286\) 0 0
\(287\) −6.65736 −0.392972
\(288\) 0 0
\(289\) −8.44821 −0.496954
\(290\) 0 0
\(291\) −5.98218 −0.350682
\(292\) 0 0
\(293\) 24.3827 1.42445 0.712225 0.701951i \(-0.247688\pi\)
0.712225 + 0.701951i \(0.247688\pi\)
\(294\) 0 0
\(295\) −9.66241 −0.562567
\(296\) 0 0
\(297\) 1.86651 0.108306
\(298\) 0 0
\(299\) 0.604574 0.0349634
\(300\) 0 0
\(301\) 5.66241 0.326376
\(302\) 0 0
\(303\) −4.05279 −0.232826
\(304\) 0 0
\(305\) 5.59952 0.320628
\(306\) 0 0
\(307\) 6.13349 0.350057 0.175028 0.984563i \(-0.443998\pi\)
0.175028 + 0.984563i \(0.443998\pi\)
\(308\) 0 0
\(309\) −6.11567 −0.347908
\(310\) 0 0
\(311\) 1.12844 0.0639881 0.0319940 0.999488i \(-0.489814\pi\)
0.0319940 + 0.999488i \(0.489814\pi\)
\(312\) 0 0
\(313\) 25.4304 1.43741 0.718705 0.695315i \(-0.244735\pi\)
0.718705 + 0.695315i \(0.244735\pi\)
\(314\) 0 0
\(315\) −1.52892 −0.0861448
\(316\) 0 0
\(317\) −7.64459 −0.429363 −0.214681 0.976684i \(-0.568871\pi\)
−0.214681 + 0.976684i \(0.568871\pi\)
\(318\) 0 0
\(319\) −13.4227 −0.751525
\(320\) 0 0
\(321\) −2.33759 −0.130472
\(322\) 0 0
\(323\) −5.45831 −0.303709
\(324\) 0 0
\(325\) −1.60962 −0.0892859
\(326\) 0 0
\(327\) 8.58675 0.474849
\(328\) 0 0
\(329\) −10.7909 −0.594919
\(330\) 0 0
\(331\) 10.1513 0.557967 0.278983 0.960296i \(-0.410003\pi\)
0.278983 + 0.960296i \(0.410003\pi\)
\(332\) 0 0
\(333\) −3.19133 −0.174884
\(334\) 0 0
\(335\) 8.22192 0.449211
\(336\) 0 0
\(337\) 4.60457 0.250827 0.125414 0.992105i \(-0.459974\pi\)
0.125414 + 0.992105i \(0.459974\pi\)
\(338\) 0 0
\(339\) −19.3776 −1.05245
\(340\) 0 0
\(341\) −10.9499 −0.592970
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 1.52892 0.0823142
\(346\) 0 0
\(347\) 15.5740 0.836055 0.418028 0.908434i \(-0.362722\pi\)
0.418028 + 0.908434i \(0.362722\pi\)
\(348\) 0 0
\(349\) 25.2263 1.35033 0.675166 0.737666i \(-0.264072\pi\)
0.675166 + 0.737666i \(0.264072\pi\)
\(350\) 0 0
\(351\) −0.604574 −0.0322698
\(352\) 0 0
\(353\) 28.5060 1.51722 0.758612 0.651543i \(-0.225878\pi\)
0.758612 + 0.651543i \(0.225878\pi\)
\(354\) 0 0
\(355\) 23.3070 1.23701
\(356\) 0 0
\(357\) 2.92434 0.154773
\(358\) 0 0
\(359\) 32.4711 1.71376 0.856879 0.515517i \(-0.172400\pi\)
0.856879 + 0.515517i \(0.172400\pi\)
\(360\) 0 0
\(361\) −15.5161 −0.816639
\(362\) 0 0
\(363\) 7.51615 0.394495
\(364\) 0 0
\(365\) −21.3776 −1.11896
\(366\) 0 0
\(367\) 6.87156 0.358692 0.179346 0.983786i \(-0.442602\pi\)
0.179346 + 0.983786i \(0.442602\pi\)
\(368\) 0 0
\(369\) −6.65736 −0.346568
\(370\) 0 0
\(371\) −1.52892 −0.0793775
\(372\) 0 0
\(373\) −10.6574 −0.551817 −0.275909 0.961184i \(-0.588979\pi\)
−0.275909 + 0.961184i \(0.588979\pi\)
\(374\) 0 0
\(375\) −11.7152 −0.604970
\(376\) 0 0
\(377\) 4.34769 0.223917
\(378\) 0 0
\(379\) 35.9287 1.84553 0.922767 0.385358i \(-0.125922\pi\)
0.922767 + 0.385358i \(0.125922\pi\)
\(380\) 0 0
\(381\) 4.98723 0.255504
\(382\) 0 0
\(383\) 0.657360 0.0335895 0.0167948 0.999859i \(-0.494654\pi\)
0.0167948 + 0.999859i \(0.494654\pi\)
\(384\) 0 0
\(385\) 2.85374 0.145440
\(386\) 0 0
\(387\) 5.66241 0.287836
\(388\) 0 0
\(389\) 25.4126 1.28847 0.644234 0.764828i \(-0.277176\pi\)
0.644234 + 0.764828i \(0.277176\pi\)
\(390\) 0 0
\(391\) −2.92434 −0.147890
\(392\) 0 0
\(393\) −1.86651 −0.0941529
\(394\) 0 0
\(395\) 14.0528 0.707072
\(396\) 0 0
\(397\) 35.7075 1.79211 0.896053 0.443946i \(-0.146422\pi\)
0.896053 + 0.443946i \(0.146422\pi\)
\(398\) 0 0
\(399\) −1.86651 −0.0934423
\(400\) 0 0
\(401\) 19.0299 0.950309 0.475154 0.879902i \(-0.342392\pi\)
0.475154 + 0.879902i \(0.342392\pi\)
\(402\) 0 0
\(403\) 3.54674 0.176676
\(404\) 0 0
\(405\) −1.52892 −0.0759725
\(406\) 0 0
\(407\) 5.95664 0.295260
\(408\) 0 0
\(409\) −4.77303 −0.236011 −0.118006 0.993013i \(-0.537650\pi\)
−0.118006 + 0.993013i \(0.537650\pi\)
\(410\) 0 0
\(411\) 15.1913 0.749333
\(412\) 0 0
\(413\) 6.31977 0.310976
\(414\) 0 0
\(415\) −1.41325 −0.0693735
\(416\) 0 0
\(417\) −12.4711 −0.610712
\(418\) 0 0
\(419\) −17.6089 −0.860253 −0.430127 0.902769i \(-0.641531\pi\)
−0.430127 + 0.902769i \(0.641531\pi\)
\(420\) 0 0
\(421\) 40.4075 1.96934 0.984671 0.174422i \(-0.0558056\pi\)
0.984671 + 0.174422i \(0.0558056\pi\)
\(422\) 0 0
\(423\) −10.7909 −0.524669
\(424\) 0 0
\(425\) 7.78580 0.377667
\(426\) 0 0
\(427\) −3.66241 −0.177236
\(428\) 0 0
\(429\) 1.12844 0.0544817
\(430\) 0 0
\(431\) 28.0094 1.34917 0.674583 0.738199i \(-0.264323\pi\)
0.674583 + 0.738199i \(0.264323\pi\)
\(432\) 0 0
\(433\) −30.3726 −1.45961 −0.729806 0.683654i \(-0.760390\pi\)
−0.729806 + 0.683654i \(0.760390\pi\)
\(434\) 0 0
\(435\) 10.9950 0.527168
\(436\) 0 0
\(437\) 1.86651 0.0892872
\(438\) 0 0
\(439\) 5.33254 0.254508 0.127254 0.991870i \(-0.459384\pi\)
0.127254 + 0.991870i \(0.459384\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 18.1157 0.860702 0.430351 0.902662i \(-0.358390\pi\)
0.430351 + 0.902662i \(0.358390\pi\)
\(444\) 0 0
\(445\) −11.3070 −0.536003
\(446\) 0 0
\(447\) 0.523868 0.0247781
\(448\) 0 0
\(449\) −33.7603 −1.59325 −0.796623 0.604477i \(-0.793382\pi\)
−0.796623 + 0.604477i \(0.793382\pi\)
\(450\) 0 0
\(451\) 12.4260 0.585118
\(452\) 0 0
\(453\) −11.3248 −0.532086
\(454\) 0 0
\(455\) −0.924344 −0.0433339
\(456\) 0 0
\(457\) −1.29757 −0.0606980 −0.0303490 0.999539i \(-0.509662\pi\)
−0.0303490 + 0.999539i \(0.509662\pi\)
\(458\) 0 0
\(459\) 2.92434 0.136497
\(460\) 0 0
\(461\) −27.3420 −1.27344 −0.636721 0.771094i \(-0.719710\pi\)
−0.636721 + 0.771094i \(0.719710\pi\)
\(462\) 0 0
\(463\) 24.6217 1.14427 0.572134 0.820160i \(-0.306116\pi\)
0.572134 + 0.820160i \(0.306116\pi\)
\(464\) 0 0
\(465\) 8.96941 0.415946
\(466\) 0 0
\(467\) 12.6574 0.585713 0.292856 0.956156i \(-0.405394\pi\)
0.292856 + 0.956156i \(0.405394\pi\)
\(468\) 0 0
\(469\) −5.37761 −0.248315
\(470\) 0 0
\(471\) −21.0222 −0.968652
\(472\) 0 0
\(473\) −10.5689 −0.485960
\(474\) 0 0
\(475\) −4.96941 −0.228012
\(476\) 0 0
\(477\) −1.52892 −0.0700043
\(478\) 0 0
\(479\) −7.18123 −0.328119 −0.164059 0.986450i \(-0.552459\pi\)
−0.164059 + 0.986450i \(0.552459\pi\)
\(480\) 0 0
\(481\) −1.92939 −0.0879728
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) −9.14626 −0.415310
\(486\) 0 0
\(487\) 20.8887 0.946558 0.473279 0.880913i \(-0.343070\pi\)
0.473279 + 0.880913i \(0.343070\pi\)
\(488\) 0 0
\(489\) −13.3776 −0.604956
\(490\) 0 0
\(491\) 20.7202 0.935092 0.467546 0.883969i \(-0.345138\pi\)
0.467546 + 0.883969i \(0.345138\pi\)
\(492\) 0 0
\(493\) −21.0299 −0.947140
\(494\) 0 0
\(495\) 2.85374 0.128266
\(496\) 0 0
\(497\) −15.2441 −0.683792
\(498\) 0 0
\(499\) 7.22629 0.323493 0.161747 0.986832i \(-0.448287\pi\)
0.161747 + 0.986832i \(0.448287\pi\)
\(500\) 0 0
\(501\) 20.5161 0.916594
\(502\) 0 0
\(503\) 32.0272 1.42802 0.714012 0.700133i \(-0.246876\pi\)
0.714012 + 0.700133i \(0.246876\pi\)
\(504\) 0 0
\(505\) −6.19638 −0.275735
\(506\) 0 0
\(507\) 12.6345 0.561117
\(508\) 0 0
\(509\) 38.3369 1.69925 0.849627 0.527384i \(-0.176827\pi\)
0.849627 + 0.527384i \(0.176827\pi\)
\(510\) 0 0
\(511\) 13.9822 0.618535
\(512\) 0 0
\(513\) −1.86651 −0.0824083
\(514\) 0 0
\(515\) −9.35036 −0.412026
\(516\) 0 0
\(517\) 20.1412 0.885810
\(518\) 0 0
\(519\) 1.34264 0.0589353
\(520\) 0 0
\(521\) −31.2791 −1.37036 −0.685181 0.728373i \(-0.740277\pi\)
−0.685181 + 0.728373i \(0.740277\pi\)
\(522\) 0 0
\(523\) 29.0323 1.26949 0.634747 0.772720i \(-0.281104\pi\)
0.634747 + 0.772720i \(0.281104\pi\)
\(524\) 0 0
\(525\) 2.66241 0.116197
\(526\) 0 0
\(527\) −17.1557 −0.747313
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 6.31977 0.274255
\(532\) 0 0
\(533\) −4.02487 −0.174336
\(534\) 0 0
\(535\) −3.57398 −0.154517
\(536\) 0 0
\(537\) −0.204098 −0.00880749
\(538\) 0 0
\(539\) −1.86651 −0.0803962
\(540\) 0 0
\(541\) −21.8231 −0.938250 −0.469125 0.883132i \(-0.655431\pi\)
−0.469125 + 0.883132i \(0.655431\pi\)
\(542\) 0 0
\(543\) −17.4482 −0.748774
\(544\) 0 0
\(545\) 13.1284 0.562361
\(546\) 0 0
\(547\) −37.7502 −1.61408 −0.807040 0.590497i \(-0.798932\pi\)
−0.807040 + 0.590497i \(0.798932\pi\)
\(548\) 0 0
\(549\) −3.66241 −0.156308
\(550\) 0 0
\(551\) 13.4227 0.571825
\(552\) 0 0
\(553\) −9.19133 −0.390855
\(554\) 0 0
\(555\) −4.87928 −0.207114
\(556\) 0 0
\(557\) −13.8588 −0.587216 −0.293608 0.955926i \(-0.594856\pi\)
−0.293608 + 0.955926i \(0.594856\pi\)
\(558\) 0 0
\(559\) 3.42335 0.144792
\(560\) 0 0
\(561\) −5.45831 −0.230450
\(562\) 0 0
\(563\) 36.5588 1.54077 0.770386 0.637578i \(-0.220064\pi\)
0.770386 + 0.637578i \(0.220064\pi\)
\(564\) 0 0
\(565\) −29.6268 −1.24641
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 11.3147 0.474338 0.237169 0.971468i \(-0.423781\pi\)
0.237169 + 0.971468i \(0.423781\pi\)
\(570\) 0 0
\(571\) −27.2714 −1.14127 −0.570635 0.821204i \(-0.693303\pi\)
−0.570635 + 0.821204i \(0.693303\pi\)
\(572\) 0 0
\(573\) 19.3248 0.807306
\(574\) 0 0
\(575\) −2.66241 −0.111030
\(576\) 0 0
\(577\) 25.4660 1.06016 0.530082 0.847946i \(-0.322161\pi\)
0.530082 + 0.847946i \(0.322161\pi\)
\(578\) 0 0
\(579\) 1.32482 0.0550576
\(580\) 0 0
\(581\) 0.924344 0.0383483
\(582\) 0 0
\(583\) 2.85374 0.118190
\(584\) 0 0
\(585\) −0.924344 −0.0382169
\(586\) 0 0
\(587\) 5.16408 0.213144 0.106572 0.994305i \(-0.466012\pi\)
0.106572 + 0.994305i \(0.466012\pi\)
\(588\) 0 0
\(589\) 10.9499 0.451182
\(590\) 0 0
\(591\) −17.5111 −0.720310
\(592\) 0 0
\(593\) −3.20915 −0.131784 −0.0658920 0.997827i \(-0.520989\pi\)
−0.0658920 + 0.997827i \(0.520989\pi\)
\(594\) 0 0
\(595\) 4.47108 0.183296
\(596\) 0 0
\(597\) −22.1786 −0.907708
\(598\) 0 0
\(599\) 0.186278 0.00761112 0.00380556 0.999993i \(-0.498789\pi\)
0.00380556 + 0.999993i \(0.498789\pi\)
\(600\) 0 0
\(601\) 14.1328 0.576490 0.288245 0.957557i \(-0.406928\pi\)
0.288245 + 0.957557i \(0.406928\pi\)
\(602\) 0 0
\(603\) −5.37761 −0.218993
\(604\) 0 0
\(605\) 11.4916 0.467199
\(606\) 0 0
\(607\) 29.8938 1.21335 0.606675 0.794950i \(-0.292503\pi\)
0.606675 + 0.794950i \(0.292503\pi\)
\(608\) 0 0
\(609\) −7.19133 −0.291407
\(610\) 0 0
\(611\) −6.52387 −0.263927
\(612\) 0 0
\(613\) −27.5282 −1.11186 −0.555928 0.831231i \(-0.687637\pi\)
−0.555928 + 0.831231i \(0.687637\pi\)
\(614\) 0 0
\(615\) −10.1786 −0.410439
\(616\) 0 0
\(617\) 35.0572 1.41135 0.705674 0.708537i \(-0.250644\pi\)
0.705674 + 0.708537i \(0.250644\pi\)
\(618\) 0 0
\(619\) −21.1284 −0.849224 −0.424612 0.905375i \(-0.639589\pi\)
−0.424612 + 0.905375i \(0.639589\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 7.39543 0.296291
\(624\) 0 0
\(625\) −4.59952 −0.183981
\(626\) 0 0
\(627\) 3.48385 0.139132
\(628\) 0 0
\(629\) 9.33254 0.372113
\(630\) 0 0
\(631\) −37.0145 −1.47352 −0.736761 0.676153i \(-0.763646\pi\)
−0.736761 + 0.676153i \(0.763646\pi\)
\(632\) 0 0
\(633\) 2.40048 0.0954103
\(634\) 0 0
\(635\) 7.62506 0.302591
\(636\) 0 0
\(637\) 0.604574 0.0239541
\(638\) 0 0
\(639\) −15.2441 −0.603048
\(640\) 0 0
\(641\) −39.9993 −1.57988 −0.789939 0.613185i \(-0.789888\pi\)
−0.789939 + 0.613185i \(0.789888\pi\)
\(642\) 0 0
\(643\) 31.9193 1.25877 0.629387 0.777092i \(-0.283306\pi\)
0.629387 + 0.777092i \(0.283306\pi\)
\(644\) 0 0
\(645\) 8.65736 0.340883
\(646\) 0 0
\(647\) 19.1005 0.750919 0.375460 0.926839i \(-0.377485\pi\)
0.375460 + 0.926839i \(0.377485\pi\)
\(648\) 0 0
\(649\) −11.7959 −0.463030
\(650\) 0 0
\(651\) −5.86651 −0.229927
\(652\) 0 0
\(653\) −26.3998 −1.03310 −0.516552 0.856256i \(-0.672785\pi\)
−0.516552 + 0.856256i \(0.672785\pi\)
\(654\) 0 0
\(655\) −2.85374 −0.111505
\(656\) 0 0
\(657\) 13.9822 0.545497
\(658\) 0 0
\(659\) −7.98218 −0.310942 −0.155471 0.987840i \(-0.549689\pi\)
−0.155471 + 0.987840i \(0.549689\pi\)
\(660\) 0 0
\(661\) 45.6796 1.77673 0.888364 0.459139i \(-0.151842\pi\)
0.888364 + 0.459139i \(0.151842\pi\)
\(662\) 0 0
\(663\) 1.76798 0.0686627
\(664\) 0 0
\(665\) −2.85374 −0.110663
\(666\) 0 0
\(667\) 7.19133 0.278449
\(668\) 0 0
\(669\) −14.0528 −0.543312
\(670\) 0 0
\(671\) 6.83592 0.263898
\(672\) 0 0
\(673\) 23.0501 0.888517 0.444258 0.895899i \(-0.353467\pi\)
0.444258 + 0.895899i \(0.353467\pi\)
\(674\) 0 0
\(675\) 2.66241 0.102476
\(676\) 0 0
\(677\) 24.0350 0.923739 0.461869 0.886948i \(-0.347179\pi\)
0.461869 + 0.886948i \(0.347179\pi\)
\(678\) 0 0
\(679\) 5.98218 0.229575
\(680\) 0 0
\(681\) 14.8537 0.569196
\(682\) 0 0
\(683\) −23.5740 −0.902033 −0.451017 0.892516i \(-0.648939\pi\)
−0.451017 + 0.892516i \(0.648939\pi\)
\(684\) 0 0
\(685\) 23.2263 0.887431
\(686\) 0 0
\(687\) 23.1106 0.881725
\(688\) 0 0
\(689\) −0.924344 −0.0352147
\(690\) 0 0
\(691\) 26.2663 0.999218 0.499609 0.866251i \(-0.333477\pi\)
0.499609 + 0.866251i \(0.333477\pi\)
\(692\) 0 0
\(693\) −1.86651 −0.0709028
\(694\) 0 0
\(695\) −19.0673 −0.723262
\(696\) 0 0
\(697\) 19.4684 0.737419
\(698\) 0 0
\(699\) 21.1207 0.798859
\(700\) 0 0
\(701\) 34.8282 1.31544 0.657721 0.753261i \(-0.271520\pi\)
0.657721 + 0.753261i \(0.271520\pi\)
\(702\) 0 0
\(703\) −5.95664 −0.224659
\(704\) 0 0
\(705\) −16.4983 −0.621363
\(706\) 0 0
\(707\) 4.05279 0.152421
\(708\) 0 0
\(709\) 10.4176 0.391242 0.195621 0.980680i \(-0.437328\pi\)
0.195621 + 0.980680i \(0.437328\pi\)
\(710\) 0 0
\(711\) −9.19133 −0.344702
\(712\) 0 0
\(713\) 5.86651 0.219702
\(714\) 0 0
\(715\) 1.72530 0.0645224
\(716\) 0 0
\(717\) −5.64459 −0.210801
\(718\) 0 0
\(719\) −10.2593 −0.382606 −0.191303 0.981531i \(-0.561271\pi\)
−0.191303 + 0.981531i \(0.561271\pi\)
\(720\) 0 0
\(721\) 6.11567 0.227760
\(722\) 0 0
\(723\) 9.44821 0.351383
\(724\) 0 0
\(725\) −19.1463 −0.711074
\(726\) 0 0
\(727\) 31.4328 1.16578 0.582888 0.812552i \(-0.301922\pi\)
0.582888 + 0.812552i \(0.301922\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −16.5588 −0.612451
\(732\) 0 0
\(733\) 1.32482 0.0489333 0.0244667 0.999701i \(-0.492211\pi\)
0.0244667 + 0.999701i \(0.492211\pi\)
\(734\) 0 0
\(735\) 1.52892 0.0563950
\(736\) 0 0
\(737\) 10.0373 0.369730
\(738\) 0 0
\(739\) −1.33254 −0.0490183 −0.0245091 0.999700i \(-0.507802\pi\)
−0.0245091 + 0.999700i \(0.507802\pi\)
\(740\) 0 0
\(741\) −1.12844 −0.0414544
\(742\) 0 0
\(743\) −8.06289 −0.295799 −0.147899 0.989002i \(-0.547251\pi\)
−0.147899 + 0.989002i \(0.547251\pi\)
\(744\) 0 0
\(745\) 0.800952 0.0293446
\(746\) 0 0
\(747\) 0.924344 0.0338200
\(748\) 0 0
\(749\) 2.33759 0.0854137
\(750\) 0 0
\(751\) −8.39980 −0.306513 −0.153257 0.988186i \(-0.548976\pi\)
−0.153257 + 0.988186i \(0.548976\pi\)
\(752\) 0 0
\(753\) 0.639540 0.0233061
\(754\) 0 0
\(755\) −17.3147 −0.630147
\(756\) 0 0
\(757\) 2.95998 0.107582 0.0537912 0.998552i \(-0.482869\pi\)
0.0537912 + 0.998552i \(0.482869\pi\)
\(758\) 0 0
\(759\) 1.86651 0.0677500
\(760\) 0 0
\(761\) −47.7774 −1.73193 −0.865965 0.500105i \(-0.833295\pi\)
−0.865965 + 0.500105i \(0.833295\pi\)
\(762\) 0 0
\(763\) −8.58675 −0.310861
\(764\) 0 0
\(765\) 4.47108 0.161652
\(766\) 0 0
\(767\) 3.82077 0.137960
\(768\) 0 0
\(769\) 44.7196 1.61263 0.806315 0.591487i \(-0.201459\pi\)
0.806315 + 0.591487i \(0.201459\pi\)
\(770\) 0 0
\(771\) 18.6395 0.671287
\(772\) 0 0
\(773\) 22.2848 0.801529 0.400764 0.916181i \(-0.368745\pi\)
0.400764 + 0.916181i \(0.368745\pi\)
\(774\) 0 0
\(775\) −15.6190 −0.561053
\(776\) 0 0
\(777\) 3.19133 0.114488
\(778\) 0 0
\(779\) −12.4260 −0.445208
\(780\) 0 0
\(781\) 28.4533 1.01814
\(782\) 0 0
\(783\) −7.19133 −0.256997
\(784\) 0 0
\(785\) −32.1412 −1.14717
\(786\) 0 0
\(787\) 13.9472 0.497164 0.248582 0.968611i \(-0.420035\pi\)
0.248582 + 0.968611i \(0.420035\pi\)
\(788\) 0 0
\(789\) −25.1557 −0.895566
\(790\) 0 0
\(791\) 19.3776 0.688988
\(792\) 0 0
\(793\) −2.21420 −0.0786285
\(794\) 0 0
\(795\) −2.33759 −0.0829058
\(796\) 0 0
\(797\) −26.3369 −0.932901 −0.466451 0.884547i \(-0.654468\pi\)
−0.466451 + 0.884547i \(0.654468\pi\)
\(798\) 0 0
\(799\) 31.5562 1.11638
\(800\) 0 0
\(801\) 7.39543 0.261305
\(802\) 0 0
\(803\) −26.0979 −0.920973
\(804\) 0 0
\(805\) −1.52892 −0.0538873
\(806\) 0 0
\(807\) 4.97713 0.175203
\(808\) 0 0
\(809\) −36.7101 −1.29066 −0.645330 0.763904i \(-0.723280\pi\)
−0.645330 + 0.763904i \(0.723280\pi\)
\(810\) 0 0
\(811\) −10.7730 −0.378292 −0.189146 0.981949i \(-0.560572\pi\)
−0.189146 + 0.981949i \(0.560572\pi\)
\(812\) 0 0
\(813\) 4.92434 0.172704
\(814\) 0 0
\(815\) −20.4533 −0.716447
\(816\) 0 0
\(817\) 10.5689 0.369760
\(818\) 0 0
\(819\) 0.604574 0.0211255
\(820\) 0 0
\(821\) −10.1258 −0.353392 −0.176696 0.984265i \(-0.556541\pi\)
−0.176696 + 0.984265i \(0.556541\pi\)
\(822\) 0 0
\(823\) −29.6446 −1.03335 −0.516673 0.856183i \(-0.672830\pi\)
−0.516673 + 0.856183i \(0.672830\pi\)
\(824\) 0 0
\(825\) −4.96941 −0.173013
\(826\) 0 0
\(827\) −42.4253 −1.47527 −0.737637 0.675198i \(-0.764058\pi\)
−0.737637 + 0.675198i \(0.764058\pi\)
\(828\) 0 0
\(829\) −3.97208 −0.137956 −0.0689780 0.997618i \(-0.521974\pi\)
−0.0689780 + 0.997618i \(0.521974\pi\)
\(830\) 0 0
\(831\) −25.1029 −0.870810
\(832\) 0 0
\(833\) −2.92434 −0.101323
\(834\) 0 0
\(835\) 31.3675 1.08552
\(836\) 0 0
\(837\) −5.86651 −0.202776
\(838\) 0 0
\(839\) 21.8938 0.755856 0.377928 0.925835i \(-0.376637\pi\)
0.377928 + 0.925835i \(0.376637\pi\)
\(840\) 0 0
\(841\) 22.7152 0.783283
\(842\) 0 0
\(843\) −26.3726 −0.908319
\(844\) 0 0
\(845\) 19.3171 0.664528
\(846\) 0 0
\(847\) −7.51615 −0.258258
\(848\) 0 0
\(849\) 12.4354 0.426783
\(850\) 0 0
\(851\) −3.19133 −0.109397
\(852\) 0 0
\(853\) −1.02220 −0.0349993 −0.0174997 0.999847i \(-0.505571\pi\)
−0.0174997 + 0.999847i \(0.505571\pi\)
\(854\) 0 0
\(855\) −2.85374 −0.0975958
\(856\) 0 0
\(857\) 39.6718 1.35516 0.677582 0.735447i \(-0.263028\pi\)
0.677582 + 0.735447i \(0.263028\pi\)
\(858\) 0 0
\(859\) −47.1278 −1.60798 −0.803989 0.594644i \(-0.797293\pi\)
−0.803989 + 0.594644i \(0.797293\pi\)
\(860\) 0 0
\(861\) 6.65736 0.226882
\(862\) 0 0
\(863\) 16.1258 0.548928 0.274464 0.961597i \(-0.411500\pi\)
0.274464 + 0.961597i \(0.411500\pi\)
\(864\) 0 0
\(865\) 2.05279 0.0697968
\(866\) 0 0
\(867\) 8.44821 0.286916
\(868\) 0 0
\(869\) 17.1557 0.581967
\(870\) 0 0
\(871\) −3.25116 −0.110161
\(872\) 0 0
\(873\) 5.98218 0.202466
\(874\) 0 0
\(875\) 11.7152 0.396046
\(876\) 0 0
\(877\) 34.2670 1.15711 0.578557 0.815642i \(-0.303616\pi\)
0.578557 + 0.815642i \(0.303616\pi\)
\(878\) 0 0
\(879\) −24.3827 −0.822407
\(880\) 0 0
\(881\) −19.2091 −0.647173 −0.323586 0.946199i \(-0.604889\pi\)
−0.323586 + 0.946199i \(0.604889\pi\)
\(882\) 0 0
\(883\) −46.4432 −1.56294 −0.781468 0.623945i \(-0.785529\pi\)
−0.781468 + 0.623945i \(0.785529\pi\)
\(884\) 0 0
\(885\) 9.66241 0.324798
\(886\) 0 0
\(887\) −42.0350 −1.41140 −0.705698 0.708513i \(-0.749366\pi\)
−0.705698 + 0.708513i \(0.749366\pi\)
\(888\) 0 0
\(889\) −4.98723 −0.167266
\(890\) 0 0
\(891\) −1.86651 −0.0625304
\(892\) 0 0
\(893\) −20.1412 −0.674000
\(894\) 0 0
\(895\) −0.312049 −0.0104307
\(896\) 0 0
\(897\) −0.604574 −0.0201861
\(898\) 0 0
\(899\) 42.1880 1.40705
\(900\) 0 0
\(901\) 4.47108 0.148953
\(902\) 0 0
\(903\) −5.66241 −0.188433
\(904\) 0 0
\(905\) −26.6769 −0.886770
\(906\) 0 0
\(907\) −35.6547 −1.18389 −0.591947 0.805977i \(-0.701641\pi\)
−0.591947 + 0.805977i \(0.701641\pi\)
\(908\) 0 0
\(909\) 4.05279 0.134422
\(910\) 0 0
\(911\) 3.32482 0.110156 0.0550781 0.998482i \(-0.482459\pi\)
0.0550781 + 0.998482i \(0.482459\pi\)
\(912\) 0 0
\(913\) −1.72530 −0.0570989
\(914\) 0 0
\(915\) −5.59952 −0.185115
\(916\) 0 0
\(917\) 1.86651 0.0616375
\(918\) 0 0
\(919\) 29.8665 0.985205 0.492603 0.870254i \(-0.336046\pi\)
0.492603 + 0.870254i \(0.336046\pi\)
\(920\) 0 0
\(921\) −6.13349 −0.202105
\(922\) 0 0
\(923\) −9.21619 −0.303355
\(924\) 0 0
\(925\) 8.49662 0.279367
\(926\) 0 0
\(927\) 6.11567 0.200865
\(928\) 0 0
\(929\) −5.15636 −0.169175 −0.0845874 0.996416i \(-0.526957\pi\)
−0.0845874 + 0.996416i \(0.526957\pi\)
\(930\) 0 0
\(931\) 1.86651 0.0611723
\(932\) 0 0
\(933\) −1.12844 −0.0369435
\(934\) 0 0
\(935\) −8.34531 −0.272921
\(936\) 0 0
\(937\) −47.1200 −1.53934 −0.769672 0.638439i \(-0.779580\pi\)
−0.769672 + 0.638439i \(0.779580\pi\)
\(938\) 0 0
\(939\) −25.4304 −0.829889
\(940\) 0 0
\(941\) −10.7808 −0.351442 −0.175721 0.984440i \(-0.556226\pi\)
−0.175721 + 0.984440i \(0.556226\pi\)
\(942\) 0 0
\(943\) −6.65736 −0.216793
\(944\) 0 0
\(945\) 1.52892 0.0497357
\(946\) 0 0
\(947\) −42.0800 −1.36742 −0.683709 0.729755i \(-0.739634\pi\)
−0.683709 + 0.729755i \(0.739634\pi\)
\(948\) 0 0
\(949\) 8.45326 0.274404
\(950\) 0 0
\(951\) 7.64459 0.247893
\(952\) 0 0
\(953\) −28.8716 −0.935241 −0.467621 0.883929i \(-0.654889\pi\)
−0.467621 + 0.883929i \(0.654889\pi\)
\(954\) 0 0
\(955\) 29.5461 0.956088
\(956\) 0 0
\(957\) 13.4227 0.433893
\(958\) 0 0
\(959\) −15.1913 −0.490554
\(960\) 0 0
\(961\) 3.41592 0.110191
\(962\) 0 0
\(963\) 2.33759 0.0753278
\(964\) 0 0
\(965\) 2.02554 0.0652045
\(966\) 0 0
\(967\) −30.8964 −0.993562 −0.496781 0.867876i \(-0.665485\pi\)
−0.496781 + 0.867876i \(0.665485\pi\)
\(968\) 0 0
\(969\) 5.45831 0.175346
\(970\) 0 0
\(971\) 24.4176 0.783599 0.391799 0.920051i \(-0.371853\pi\)
0.391799 + 0.920051i \(0.371853\pi\)
\(972\) 0 0
\(973\) 12.4711 0.399805
\(974\) 0 0
\(975\) 1.60962 0.0515492
\(976\) 0 0
\(977\) 6.86146 0.219518 0.109759 0.993958i \(-0.464992\pi\)
0.109759 + 0.993958i \(0.464992\pi\)
\(978\) 0 0
\(979\) −13.8036 −0.441166
\(980\) 0 0
\(981\) −8.58675 −0.274154
\(982\) 0 0
\(983\) −44.9243 −1.43286 −0.716432 0.697657i \(-0.754226\pi\)
−0.716432 + 0.697657i \(0.754226\pi\)
\(984\) 0 0
\(985\) −26.7730 −0.853060
\(986\) 0 0
\(987\) 10.7909 0.343477
\(988\) 0 0
\(989\) 5.66241 0.180054
\(990\) 0 0
\(991\) −8.31205 −0.264041 −0.132020 0.991247i \(-0.542146\pi\)
−0.132020 + 0.991247i \(0.542146\pi\)
\(992\) 0 0
\(993\) −10.1513 −0.322142
\(994\) 0 0
\(995\) −33.9092 −1.07499
\(996\) 0 0
\(997\) −22.3191 −0.706853 −0.353426 0.935462i \(-0.614984\pi\)
−0.353426 + 0.935462i \(0.614984\pi\)
\(998\) 0 0
\(999\) 3.19133 0.100969
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 7728.2.a.bt.1.1 3
4.3 odd 2 483.2.a.h.1.3 3
12.11 even 2 1449.2.a.l.1.1 3
28.27 even 2 3381.2.a.v.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.h.1.3 3 4.3 odd 2
1449.2.a.l.1.1 3 12.11 even 2
3381.2.a.v.1.3 3 28.27 even 2
7728.2.a.bt.1.1 3 1.1 even 1 trivial