Properties

Label 7728.2.a.ch.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $6$
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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
Defining polynomial: \(x^{6} - 3 x^{5} - 11 x^{4} + 23 x^{3} + 9 x^{2} - 23 x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 3864)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(1.35141\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.60318 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.60318 q^{5} +1.00000 q^{7} +1.00000 q^{9} -3.69569 q^{11} -5.98292 q^{13} -2.60318 q^{15} -5.23227 q^{17} -3.69569 q^{19} +1.00000 q^{21} +1.00000 q^{23} +1.77655 q^{25} +1.00000 q^{27} -4.36825 q^{29} -4.00295 q^{31} -3.69569 q^{33} -2.60318 q^{35} -8.22079 q^{37} -5.98292 q^{39} +6.55101 q^{41} +8.60613 q^{43} -2.60318 q^{45} +2.68126 q^{47} +1.00000 q^{49} -5.23227 q^{51} +11.3133 q^{53} +9.62055 q^{55} -3.69569 q^{57} +14.6671 q^{59} +4.08103 q^{61} +1.00000 q^{63} +15.5746 q^{65} -3.79098 q^{67} +1.00000 q^{69} +1.77377 q^{71} +1.52720 q^{73} +1.77655 q^{75} -3.69569 q^{77} -1.97409 q^{79} +1.00000 q^{81} +7.08481 q^{83} +13.6206 q^{85} -4.36825 q^{87} +12.9801 q^{89} -5.98292 q^{91} -4.00295 q^{93} +9.62055 q^{95} +1.66060 q^{97} -3.69569 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 6q^{3} + 2q^{5} + 6q^{7} + 6q^{9} + O(q^{10}) \) \( 6q + 6q^{3} + 2q^{5} + 6q^{7} + 6q^{9} + 3q^{11} + 2q^{15} + 6q^{17} + 3q^{19} + 6q^{21} + 6q^{23} + 10q^{25} + 6q^{27} + 5q^{29} + 4q^{31} + 3q^{33} + 2q^{35} + q^{37} + 12q^{41} + 6q^{43} + 2q^{45} + 6q^{47} + 6q^{49} + 6q^{51} + 10q^{53} - 3q^{55} + 3q^{57} + 14q^{59} + 4q^{61} + 6q^{63} + 27q^{65} - 7q^{67} + 6q^{69} - 7q^{71} + 10q^{73} + 10q^{75} + 3q^{77} - 14q^{79} + 6q^{81} - 14q^{83} + 21q^{85} + 5q^{87} + 25q^{89} + 4q^{93} - 3q^{95} + 11q^{97} + 3q^{99} + O(q^{100}) \)

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
\(5\) −2.60318 −1.16418 −0.582089 0.813125i \(-0.697764\pi\)
−0.582089 + 0.813125i \(0.697764\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.69569 −1.11429 −0.557146 0.830414i \(-0.688104\pi\)
−0.557146 + 0.830414i \(0.688104\pi\)
\(12\) 0 0
\(13\) −5.98292 −1.65936 −0.829681 0.558238i \(-0.811478\pi\)
−0.829681 + 0.558238i \(0.811478\pi\)
\(14\) 0 0
\(15\) −2.60318 −0.672139
\(16\) 0 0
\(17\) −5.23227 −1.26901 −0.634506 0.772918i \(-0.718797\pi\)
−0.634506 + 0.772918i \(0.718797\pi\)
\(18\) 0 0
\(19\) −3.69569 −0.847850 −0.423925 0.905697i \(-0.639348\pi\)
−0.423925 + 0.905697i \(0.639348\pi\)
\(20\) 0 0
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) 1.77655 0.355311
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.36825 −0.811164 −0.405582 0.914059i \(-0.632931\pi\)
−0.405582 + 0.914059i \(0.632931\pi\)
\(30\) 0 0
\(31\) −4.00295 −0.718950 −0.359475 0.933155i \(-0.617044\pi\)
−0.359475 + 0.933155i \(0.617044\pi\)
\(32\) 0 0
\(33\) −3.69569 −0.643337
\(34\) 0 0
\(35\) −2.60318 −0.440018
\(36\) 0 0
\(37\) −8.22079 −1.35149 −0.675745 0.737136i \(-0.736178\pi\)
−0.675745 + 0.737136i \(0.736178\pi\)
\(38\) 0 0
\(39\) −5.98292 −0.958033
\(40\) 0 0
\(41\) 6.55101 1.02310 0.511548 0.859255i \(-0.329072\pi\)
0.511548 + 0.859255i \(0.329072\pi\)
\(42\) 0 0
\(43\) 8.60613 1.31242 0.656211 0.754577i \(-0.272158\pi\)
0.656211 + 0.754577i \(0.272158\pi\)
\(44\) 0 0
\(45\) −2.60318 −0.388059
\(46\) 0 0
\(47\) 2.68126 0.391102 0.195551 0.980693i \(-0.437350\pi\)
0.195551 + 0.980693i \(0.437350\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −5.23227 −0.732665
\(52\) 0 0
\(53\) 11.3133 1.55400 0.777001 0.629500i \(-0.216740\pi\)
0.777001 + 0.629500i \(0.216740\pi\)
\(54\) 0 0
\(55\) 9.62055 1.29724
\(56\) 0 0
\(57\) −3.69569 −0.489506
\(58\) 0 0
\(59\) 14.6671 1.90950 0.954749 0.297414i \(-0.0961243\pi\)
0.954749 + 0.297414i \(0.0961243\pi\)
\(60\) 0 0
\(61\) 4.08103 0.522522 0.261261 0.965268i \(-0.415862\pi\)
0.261261 + 0.965268i \(0.415862\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) 15.5746 1.93179
\(66\) 0 0
\(67\) −3.79098 −0.463142 −0.231571 0.972818i \(-0.574387\pi\)
−0.231571 + 0.972818i \(0.574387\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 1.77377 0.210508 0.105254 0.994445i \(-0.466434\pi\)
0.105254 + 0.994445i \(0.466434\pi\)
\(72\) 0 0
\(73\) 1.52720 0.178745 0.0893724 0.995998i \(-0.471514\pi\)
0.0893724 + 0.995998i \(0.471514\pi\)
\(74\) 0 0
\(75\) 1.77655 0.205139
\(76\) 0 0
\(77\) −3.69569 −0.421163
\(78\) 0 0
\(79\) −1.97409 −0.222102 −0.111051 0.993815i \(-0.535422\pi\)
−0.111051 + 0.993815i \(0.535422\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.08481 0.777659 0.388829 0.921310i \(-0.372880\pi\)
0.388829 + 0.921310i \(0.372880\pi\)
\(84\) 0 0
\(85\) 13.6206 1.47736
\(86\) 0 0
\(87\) −4.36825 −0.468326
\(88\) 0 0
\(89\) 12.9801 1.37589 0.687946 0.725762i \(-0.258513\pi\)
0.687946 + 0.725762i \(0.258513\pi\)
\(90\) 0 0
\(91\) −5.98292 −0.627180
\(92\) 0 0
\(93\) −4.00295 −0.415086
\(94\) 0 0
\(95\) 9.62055 0.987048
\(96\) 0 0
\(97\) 1.66060 0.168609 0.0843043 0.996440i \(-0.473133\pi\)
0.0843043 + 0.996440i \(0.473133\pi\)
\(98\) 0 0
\(99\) −3.69569 −0.371431
\(100\) 0 0
\(101\) −3.46076 −0.344359 −0.172179 0.985066i \(-0.555081\pi\)
−0.172179 + 0.985066i \(0.555081\pi\)
\(102\) 0 0
\(103\) 18.6471 1.83735 0.918676 0.395011i \(-0.129259\pi\)
0.918676 + 0.395011i \(0.129259\pi\)
\(104\) 0 0
\(105\) −2.60318 −0.254044
\(106\) 0 0
\(107\) −16.8097 −1.62506 −0.812528 0.582922i \(-0.801909\pi\)
−0.812528 + 0.582922i \(0.801909\pi\)
\(108\) 0 0
\(109\) −8.58022 −0.821836 −0.410918 0.911672i \(-0.634792\pi\)
−0.410918 + 0.911672i \(0.634792\pi\)
\(110\) 0 0
\(111\) −8.22079 −0.780283
\(112\) 0 0
\(113\) 10.6119 0.998282 0.499141 0.866521i \(-0.333649\pi\)
0.499141 + 0.866521i \(0.333649\pi\)
\(114\) 0 0
\(115\) −2.60318 −0.242748
\(116\) 0 0
\(117\) −5.98292 −0.553121
\(118\) 0 0
\(119\) −5.23227 −0.479642
\(120\) 0 0
\(121\) 2.65813 0.241648
\(122\) 0 0
\(123\) 6.55101 0.590685
\(124\) 0 0
\(125\) 8.39122 0.750533
\(126\) 0 0
\(127\) −12.1427 −1.07749 −0.538746 0.842469i \(-0.681102\pi\)
−0.538746 + 0.842469i \(0.681102\pi\)
\(128\) 0 0
\(129\) 8.60613 0.757727
\(130\) 0 0
\(131\) 7.54823 0.659492 0.329746 0.944070i \(-0.393037\pi\)
0.329746 + 0.944070i \(0.393037\pi\)
\(132\) 0 0
\(133\) −3.69569 −0.320457
\(134\) 0 0
\(135\) −2.60318 −0.224046
\(136\) 0 0
\(137\) 6.65535 0.568605 0.284303 0.958735i \(-0.408238\pi\)
0.284303 + 0.958735i \(0.408238\pi\)
\(138\) 0 0
\(139\) −11.4114 −0.967903 −0.483952 0.875095i \(-0.660799\pi\)
−0.483952 + 0.875095i \(0.660799\pi\)
\(140\) 0 0
\(141\) 2.68126 0.225803
\(142\) 0 0
\(143\) 22.1110 1.84902
\(144\) 0 0
\(145\) 11.3714 0.944339
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 16.7718 1.37400 0.687000 0.726658i \(-0.258927\pi\)
0.687000 + 0.726658i \(0.258927\pi\)
\(150\) 0 0
\(151\) −5.95065 −0.484257 −0.242129 0.970244i \(-0.577846\pi\)
−0.242129 + 0.970244i \(0.577846\pi\)
\(152\) 0 0
\(153\) −5.23227 −0.423004
\(154\) 0 0
\(155\) 10.4204 0.836986
\(156\) 0 0
\(157\) 0.765909 0.0611262 0.0305631 0.999533i \(-0.490270\pi\)
0.0305631 + 0.999533i \(0.490270\pi\)
\(158\) 0 0
\(159\) 11.3133 0.897203
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) −12.5339 −0.981733 −0.490866 0.871235i \(-0.663320\pi\)
−0.490866 + 0.871235i \(0.663320\pi\)
\(164\) 0 0
\(165\) 9.62055 0.748959
\(166\) 0 0
\(167\) −6.18570 −0.478664 −0.239332 0.970938i \(-0.576928\pi\)
−0.239332 + 0.970938i \(0.576928\pi\)
\(168\) 0 0
\(169\) 22.7953 1.75348
\(170\) 0 0
\(171\) −3.69569 −0.282617
\(172\) 0 0
\(173\) −7.41729 −0.563926 −0.281963 0.959425i \(-0.590986\pi\)
−0.281963 + 0.959425i \(0.590986\pi\)
\(174\) 0 0
\(175\) 1.77655 0.134295
\(176\) 0 0
\(177\) 14.6671 1.10245
\(178\) 0 0
\(179\) −21.1053 −1.57748 −0.788742 0.614725i \(-0.789267\pi\)
−0.788742 + 0.614725i \(0.789267\pi\)
\(180\) 0 0
\(181\) 2.30657 0.171446 0.0857231 0.996319i \(-0.472680\pi\)
0.0857231 + 0.996319i \(0.472680\pi\)
\(182\) 0 0
\(183\) 4.08103 0.301678
\(184\) 0 0
\(185\) 21.4002 1.57337
\(186\) 0 0
\(187\) 19.3369 1.41405
\(188\) 0 0
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) −11.0754 −0.801390 −0.400695 0.916212i \(-0.631231\pi\)
−0.400695 + 0.916212i \(0.631231\pi\)
\(192\) 0 0
\(193\) −20.8092 −1.49788 −0.748938 0.662640i \(-0.769436\pi\)
−0.748938 + 0.662640i \(0.769436\pi\)
\(194\) 0 0
\(195\) 15.5746 1.11532
\(196\) 0 0
\(197\) 7.99412 0.569557 0.284779 0.958593i \(-0.408080\pi\)
0.284779 + 0.958593i \(0.408080\pi\)
\(198\) 0 0
\(199\) 19.9881 1.41692 0.708460 0.705751i \(-0.249390\pi\)
0.708460 + 0.705751i \(0.249390\pi\)
\(200\) 0 0
\(201\) −3.79098 −0.267395
\(202\) 0 0
\(203\) −4.36825 −0.306591
\(204\) 0 0
\(205\) −17.0535 −1.19107
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 13.6581 0.944753
\(210\) 0 0
\(211\) −19.0383 −1.31065 −0.655326 0.755346i \(-0.727469\pi\)
−0.655326 + 0.755346i \(0.727469\pi\)
\(212\) 0 0
\(213\) 1.77377 0.121537
\(214\) 0 0
\(215\) −22.4033 −1.52789
\(216\) 0 0
\(217\) −4.00295 −0.271738
\(218\) 0 0
\(219\) 1.52720 0.103198
\(220\) 0 0
\(221\) 31.3042 2.10575
\(222\) 0 0
\(223\) 11.0349 0.738952 0.369476 0.929240i \(-0.379537\pi\)
0.369476 + 0.929240i \(0.379537\pi\)
\(224\) 0 0
\(225\) 1.77655 0.118437
\(226\) 0 0
\(227\) −4.18514 −0.277778 −0.138889 0.990308i \(-0.544353\pi\)
−0.138889 + 0.990308i \(0.544353\pi\)
\(228\) 0 0
\(229\) 7.15714 0.472957 0.236478 0.971637i \(-0.424007\pi\)
0.236478 + 0.971637i \(0.424007\pi\)
\(230\) 0 0
\(231\) −3.69569 −0.243159
\(232\) 0 0
\(233\) 29.1064 1.90683 0.953413 0.301667i \(-0.0975432\pi\)
0.953413 + 0.301667i \(0.0975432\pi\)
\(234\) 0 0
\(235\) −6.97982 −0.455313
\(236\) 0 0
\(237\) −1.97409 −0.128231
\(238\) 0 0
\(239\) −27.6094 −1.78590 −0.892950 0.450155i \(-0.851369\pi\)
−0.892950 + 0.450155i \(0.851369\pi\)
\(240\) 0 0
\(241\) 13.9424 0.898108 0.449054 0.893505i \(-0.351761\pi\)
0.449054 + 0.893505i \(0.351761\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −2.60318 −0.166311
\(246\) 0 0
\(247\) 22.1110 1.40689
\(248\) 0 0
\(249\) 7.08481 0.448982
\(250\) 0 0
\(251\) 3.89996 0.246163 0.123082 0.992397i \(-0.460722\pi\)
0.123082 + 0.992397i \(0.460722\pi\)
\(252\) 0 0
\(253\) −3.69569 −0.232346
\(254\) 0 0
\(255\) 13.6206 0.852952
\(256\) 0 0
\(257\) 16.0968 1.00409 0.502044 0.864842i \(-0.332581\pi\)
0.502044 + 0.864842i \(0.332581\pi\)
\(258\) 0 0
\(259\) −8.22079 −0.510815
\(260\) 0 0
\(261\) −4.36825 −0.270388
\(262\) 0 0
\(263\) −25.2875 −1.55929 −0.779647 0.626219i \(-0.784602\pi\)
−0.779647 + 0.626219i \(0.784602\pi\)
\(264\) 0 0
\(265\) −29.4506 −1.80913
\(266\) 0 0
\(267\) 12.9801 0.794371
\(268\) 0 0
\(269\) −9.45589 −0.576535 −0.288268 0.957550i \(-0.593079\pi\)
−0.288268 + 0.957550i \(0.593079\pi\)
\(270\) 0 0
\(271\) −0.941543 −0.0571947 −0.0285973 0.999591i \(-0.509104\pi\)
−0.0285973 + 0.999591i \(0.509104\pi\)
\(272\) 0 0
\(273\) −5.98292 −0.362103
\(274\) 0 0
\(275\) −6.56559 −0.395920
\(276\) 0 0
\(277\) −17.8705 −1.07374 −0.536868 0.843666i \(-0.680393\pi\)
−0.536868 + 0.843666i \(0.680393\pi\)
\(278\) 0 0
\(279\) −4.00295 −0.239650
\(280\) 0 0
\(281\) 7.79086 0.464764 0.232382 0.972625i \(-0.425348\pi\)
0.232382 + 0.972625i \(0.425348\pi\)
\(282\) 0 0
\(283\) 17.7680 1.05620 0.528099 0.849183i \(-0.322905\pi\)
0.528099 + 0.849183i \(0.322905\pi\)
\(284\) 0 0
\(285\) 9.62055 0.569872
\(286\) 0 0
\(287\) 6.55101 0.386694
\(288\) 0 0
\(289\) 10.3767 0.610393
\(290\) 0 0
\(291\) 1.66060 0.0973463
\(292\) 0 0
\(293\) 0.0142620 0.000833194 0 0.000416597 1.00000i \(-0.499867\pi\)
0.000416597 1.00000i \(0.499867\pi\)
\(294\) 0 0
\(295\) −38.1812 −2.22299
\(296\) 0 0
\(297\) −3.69569 −0.214446
\(298\) 0 0
\(299\) −5.98292 −0.346001
\(300\) 0 0
\(301\) 8.60613 0.496049
\(302\) 0 0
\(303\) −3.46076 −0.198816
\(304\) 0 0
\(305\) −10.6237 −0.608309
\(306\) 0 0
\(307\) −23.1566 −1.32162 −0.660808 0.750555i \(-0.729786\pi\)
−0.660808 + 0.750555i \(0.729786\pi\)
\(308\) 0 0
\(309\) 18.6471 1.06080
\(310\) 0 0
\(311\) −5.20295 −0.295032 −0.147516 0.989060i \(-0.547128\pi\)
−0.147516 + 0.989060i \(0.547128\pi\)
\(312\) 0 0
\(313\) 4.56663 0.258121 0.129061 0.991637i \(-0.458804\pi\)
0.129061 + 0.991637i \(0.458804\pi\)
\(314\) 0 0
\(315\) −2.60318 −0.146673
\(316\) 0 0
\(317\) −24.7700 −1.39122 −0.695611 0.718419i \(-0.744866\pi\)
−0.695611 + 0.718419i \(0.744866\pi\)
\(318\) 0 0
\(319\) 16.1437 0.903874
\(320\) 0 0
\(321\) −16.8097 −0.938227
\(322\) 0 0
\(323\) 19.3369 1.07593
\(324\) 0 0
\(325\) −10.6290 −0.589589
\(326\) 0 0
\(327\) −8.58022 −0.474487
\(328\) 0 0
\(329\) 2.68126 0.147823
\(330\) 0 0
\(331\) −29.3320 −1.61223 −0.806116 0.591758i \(-0.798434\pi\)
−0.806116 + 0.591758i \(0.798434\pi\)
\(332\) 0 0
\(333\) −8.22079 −0.450497
\(334\) 0 0
\(335\) 9.86861 0.539180
\(336\) 0 0
\(337\) −18.5910 −1.01271 −0.506357 0.862324i \(-0.669008\pi\)
−0.506357 + 0.862324i \(0.669008\pi\)
\(338\) 0 0
\(339\) 10.6119 0.576358
\(340\) 0 0
\(341\) 14.7937 0.801121
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −2.60318 −0.140151
\(346\) 0 0
\(347\) 25.7773 1.38380 0.691899 0.721995i \(-0.256775\pi\)
0.691899 + 0.721995i \(0.256775\pi\)
\(348\) 0 0
\(349\) −20.7869 −1.11269 −0.556347 0.830950i \(-0.687798\pi\)
−0.556347 + 0.830950i \(0.687798\pi\)
\(350\) 0 0
\(351\) −5.98292 −0.319344
\(352\) 0 0
\(353\) 21.3198 1.13474 0.567368 0.823464i \(-0.307962\pi\)
0.567368 + 0.823464i \(0.307962\pi\)
\(354\) 0 0
\(355\) −4.61745 −0.245069
\(356\) 0 0
\(357\) −5.23227 −0.276921
\(358\) 0 0
\(359\) 2.09577 0.110610 0.0553051 0.998469i \(-0.482387\pi\)
0.0553051 + 0.998469i \(0.482387\pi\)
\(360\) 0 0
\(361\) −5.34187 −0.281151
\(362\) 0 0
\(363\) 2.65813 0.139516
\(364\) 0 0
\(365\) −3.97557 −0.208091
\(366\) 0 0
\(367\) 9.05801 0.472824 0.236412 0.971653i \(-0.424028\pi\)
0.236412 + 0.971653i \(0.424028\pi\)
\(368\) 0 0
\(369\) 6.55101 0.341032
\(370\) 0 0
\(371\) 11.3133 0.587357
\(372\) 0 0
\(373\) −26.3958 −1.36672 −0.683360 0.730081i \(-0.739482\pi\)
−0.683360 + 0.730081i \(0.739482\pi\)
\(374\) 0 0
\(375\) 8.39122 0.433321
\(376\) 0 0
\(377\) 26.1349 1.34602
\(378\) 0 0
\(379\) 33.1640 1.70352 0.851761 0.523931i \(-0.175535\pi\)
0.851761 + 0.523931i \(0.175535\pi\)
\(380\) 0 0
\(381\) −12.1427 −0.622090
\(382\) 0 0
\(383\) −19.5507 −0.998996 −0.499498 0.866315i \(-0.666482\pi\)
−0.499498 + 0.866315i \(0.666482\pi\)
\(384\) 0 0
\(385\) 9.62055 0.490309
\(386\) 0 0
\(387\) 8.60613 0.437474
\(388\) 0 0
\(389\) −37.9492 −1.92410 −0.962050 0.272872i \(-0.912027\pi\)
−0.962050 + 0.272872i \(0.912027\pi\)
\(390\) 0 0
\(391\) −5.23227 −0.264607
\(392\) 0 0
\(393\) 7.54823 0.380758
\(394\) 0 0
\(395\) 5.13891 0.258567
\(396\) 0 0
\(397\) 18.4074 0.923842 0.461921 0.886921i \(-0.347160\pi\)
0.461921 + 0.886921i \(0.347160\pi\)
\(398\) 0 0
\(399\) −3.69569 −0.185016
\(400\) 0 0
\(401\) 26.2788 1.31230 0.656151 0.754629i \(-0.272183\pi\)
0.656151 + 0.754629i \(0.272183\pi\)
\(402\) 0 0
\(403\) 23.9493 1.19300
\(404\) 0 0
\(405\) −2.60318 −0.129353
\(406\) 0 0
\(407\) 30.3815 1.50596
\(408\) 0 0
\(409\) −13.3366 −0.659453 −0.329727 0.944076i \(-0.606957\pi\)
−0.329727 + 0.944076i \(0.606957\pi\)
\(410\) 0 0
\(411\) 6.65535 0.328284
\(412\) 0 0
\(413\) 14.6671 0.721722
\(414\) 0 0
\(415\) −18.4430 −0.905334
\(416\) 0 0
\(417\) −11.4114 −0.558819
\(418\) 0 0
\(419\) 15.9845 0.780895 0.390447 0.920625i \(-0.372320\pi\)
0.390447 + 0.920625i \(0.372320\pi\)
\(420\) 0 0
\(421\) 26.2583 1.27975 0.639875 0.768479i \(-0.278986\pi\)
0.639875 + 0.768479i \(0.278986\pi\)
\(422\) 0 0
\(423\) 2.68126 0.130367
\(424\) 0 0
\(425\) −9.29541 −0.450894
\(426\) 0 0
\(427\) 4.08103 0.197495
\(428\) 0 0
\(429\) 22.1110 1.06753
\(430\) 0 0
\(431\) 31.7663 1.53013 0.765065 0.643953i \(-0.222707\pi\)
0.765065 + 0.643953i \(0.222707\pi\)
\(432\) 0 0
\(433\) 23.4136 1.12518 0.562592 0.826735i \(-0.309804\pi\)
0.562592 + 0.826735i \(0.309804\pi\)
\(434\) 0 0
\(435\) 11.3714 0.545215
\(436\) 0 0
\(437\) −3.69569 −0.176789
\(438\) 0 0
\(439\) 16.2881 0.777389 0.388695 0.921367i \(-0.372926\pi\)
0.388695 + 0.921367i \(0.372926\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 1.05460 0.0501055 0.0250528 0.999686i \(-0.492025\pi\)
0.0250528 + 0.999686i \(0.492025\pi\)
\(444\) 0 0
\(445\) −33.7896 −1.60178
\(446\) 0 0
\(447\) 16.7718 0.793279
\(448\) 0 0
\(449\) 29.9276 1.41237 0.706185 0.708028i \(-0.250415\pi\)
0.706185 + 0.708028i \(0.250415\pi\)
\(450\) 0 0
\(451\) −24.2105 −1.14003
\(452\) 0 0
\(453\) −5.95065 −0.279586
\(454\) 0 0
\(455\) 15.5746 0.730149
\(456\) 0 0
\(457\) 17.4004 0.813957 0.406979 0.913438i \(-0.366582\pi\)
0.406979 + 0.913438i \(0.366582\pi\)
\(458\) 0 0
\(459\) −5.23227 −0.244222
\(460\) 0 0
\(461\) −2.07921 −0.0968384 −0.0484192 0.998827i \(-0.515418\pi\)
−0.0484192 + 0.998827i \(0.515418\pi\)
\(462\) 0 0
\(463\) −1.63990 −0.0762127 −0.0381063 0.999274i \(-0.512133\pi\)
−0.0381063 + 0.999274i \(0.512133\pi\)
\(464\) 0 0
\(465\) 10.4204 0.483234
\(466\) 0 0
\(467\) 26.7952 1.23994 0.619968 0.784627i \(-0.287146\pi\)
0.619968 + 0.784627i \(0.287146\pi\)
\(468\) 0 0
\(469\) −3.79098 −0.175051
\(470\) 0 0
\(471\) 0.765909 0.0352912
\(472\) 0 0
\(473\) −31.8056 −1.46242
\(474\) 0 0
\(475\) −6.56559 −0.301250
\(476\) 0 0
\(477\) 11.3133 0.518000
\(478\) 0 0
\(479\) −0.159110 −0.00726991 −0.00363496 0.999993i \(-0.501157\pi\)
−0.00363496 + 0.999993i \(0.501157\pi\)
\(480\) 0 0
\(481\) 49.1843 2.24261
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −4.32285 −0.196291
\(486\) 0 0
\(487\) −7.32562 −0.331956 −0.165978 0.986129i \(-0.553078\pi\)
−0.165978 + 0.986129i \(0.553078\pi\)
\(488\) 0 0
\(489\) −12.5339 −0.566804
\(490\) 0 0
\(491\) 21.9361 0.989961 0.494981 0.868904i \(-0.335175\pi\)
0.494981 + 0.868904i \(0.335175\pi\)
\(492\) 0 0
\(493\) 22.8559 1.02938
\(494\) 0 0
\(495\) 9.62055 0.432412
\(496\) 0 0
\(497\) 1.77377 0.0795646
\(498\) 0 0
\(499\) 25.9564 1.16197 0.580985 0.813915i \(-0.302668\pi\)
0.580985 + 0.813915i \(0.302668\pi\)
\(500\) 0 0
\(501\) −6.18570 −0.276357
\(502\) 0 0
\(503\) −41.9105 −1.86870 −0.934348 0.356361i \(-0.884017\pi\)
−0.934348 + 0.356361i \(0.884017\pi\)
\(504\) 0 0
\(505\) 9.00899 0.400895
\(506\) 0 0
\(507\) 22.7953 1.01237
\(508\) 0 0
\(509\) 2.07380 0.0919198 0.0459599 0.998943i \(-0.485365\pi\)
0.0459599 + 0.998943i \(0.485365\pi\)
\(510\) 0 0
\(511\) 1.52720 0.0675592
\(512\) 0 0
\(513\) −3.69569 −0.163169
\(514\) 0 0
\(515\) −48.5418 −2.13901
\(516\) 0 0
\(517\) −9.90912 −0.435803
\(518\) 0 0
\(519\) −7.41729 −0.325583
\(520\) 0 0
\(521\) −31.1781 −1.36594 −0.682968 0.730448i \(-0.739311\pi\)
−0.682968 + 0.730448i \(0.739311\pi\)
\(522\) 0 0
\(523\) −10.3193 −0.451234 −0.225617 0.974216i \(-0.572440\pi\)
−0.225617 + 0.974216i \(0.572440\pi\)
\(524\) 0 0
\(525\) 1.77655 0.0775351
\(526\) 0 0
\(527\) 20.9445 0.912357
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 14.6671 0.636499
\(532\) 0 0
\(533\) −39.1941 −1.69769
\(534\) 0 0
\(535\) 43.7587 1.89185
\(536\) 0 0
\(537\) −21.1053 −0.910760
\(538\) 0 0
\(539\) −3.69569 −0.159185
\(540\) 0 0
\(541\) 18.4970 0.795249 0.397625 0.917548i \(-0.369835\pi\)
0.397625 + 0.917548i \(0.369835\pi\)
\(542\) 0 0
\(543\) 2.30657 0.0989845
\(544\) 0 0
\(545\) 22.3359 0.956763
\(546\) 0 0
\(547\) 28.3400 1.21173 0.605865 0.795568i \(-0.292827\pi\)
0.605865 + 0.795568i \(0.292827\pi\)
\(548\) 0 0
\(549\) 4.08103 0.174174
\(550\) 0 0
\(551\) 16.1437 0.687745
\(552\) 0 0
\(553\) −1.97409 −0.0839469
\(554\) 0 0
\(555\) 21.4002 0.908388
\(556\) 0 0
\(557\) 39.0883 1.65623 0.828113 0.560561i \(-0.189415\pi\)
0.828113 + 0.560561i \(0.189415\pi\)
\(558\) 0 0
\(559\) −51.4897 −2.17778
\(560\) 0 0
\(561\) 19.3369 0.816403
\(562\) 0 0
\(563\) −8.17133 −0.344380 −0.172190 0.985064i \(-0.555084\pi\)
−0.172190 + 0.985064i \(0.555084\pi\)
\(564\) 0 0
\(565\) −27.6247 −1.16218
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −13.6914 −0.573973 −0.286987 0.957935i \(-0.592654\pi\)
−0.286987 + 0.957935i \(0.592654\pi\)
\(570\) 0 0
\(571\) −4.35839 −0.182393 −0.0911965 0.995833i \(-0.529069\pi\)
−0.0911965 + 0.995833i \(0.529069\pi\)
\(572\) 0 0
\(573\) −11.0754 −0.462683
\(574\) 0 0
\(575\) 1.77655 0.0740874
\(576\) 0 0
\(577\) 40.6108 1.69065 0.845325 0.534253i \(-0.179407\pi\)
0.845325 + 0.534253i \(0.179407\pi\)
\(578\) 0 0
\(579\) −20.8092 −0.864799
\(580\) 0 0
\(581\) 7.08481 0.293927
\(582\) 0 0
\(583\) −41.8105 −1.73161
\(584\) 0 0
\(585\) 15.5746 0.643931
\(586\) 0 0
\(587\) 14.2002 0.586105 0.293053 0.956096i \(-0.405329\pi\)
0.293053 + 0.956096i \(0.405329\pi\)
\(588\) 0 0
\(589\) 14.7937 0.609562
\(590\) 0 0
\(591\) 7.99412 0.328834
\(592\) 0 0
\(593\) 8.16321 0.335223 0.167612 0.985853i \(-0.446395\pi\)
0.167612 + 0.985853i \(0.446395\pi\)
\(594\) 0 0
\(595\) 13.6206 0.558388
\(596\) 0 0
\(597\) 19.9881 0.818059
\(598\) 0 0
\(599\) 10.8584 0.443662 0.221831 0.975085i \(-0.428797\pi\)
0.221831 + 0.975085i \(0.428797\pi\)
\(600\) 0 0
\(601\) 30.4217 1.24093 0.620463 0.784236i \(-0.286945\pi\)
0.620463 + 0.784236i \(0.286945\pi\)
\(602\) 0 0
\(603\) −3.79098 −0.154381
\(604\) 0 0
\(605\) −6.91960 −0.281322
\(606\) 0 0
\(607\) −5.32132 −0.215986 −0.107993 0.994152i \(-0.534442\pi\)
−0.107993 + 0.994152i \(0.534442\pi\)
\(608\) 0 0
\(609\) −4.36825 −0.177011
\(610\) 0 0
\(611\) −16.0418 −0.648981
\(612\) 0 0
\(613\) −29.4731 −1.19041 −0.595203 0.803575i \(-0.702928\pi\)
−0.595203 + 0.803575i \(0.702928\pi\)
\(614\) 0 0
\(615\) −17.0535 −0.687662
\(616\) 0 0
\(617\) 46.4590 1.87037 0.935185 0.354159i \(-0.115233\pi\)
0.935185 + 0.354159i \(0.115233\pi\)
\(618\) 0 0
\(619\) −7.26772 −0.292114 −0.146057 0.989276i \(-0.546658\pi\)
−0.146057 + 0.989276i \(0.546658\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 12.9801 0.520038
\(624\) 0 0
\(625\) −30.7266 −1.22906
\(626\) 0 0
\(627\) 13.6581 0.545453
\(628\) 0 0
\(629\) 43.0134 1.71506
\(630\) 0 0
\(631\) −19.6913 −0.783897 −0.391948 0.919987i \(-0.628199\pi\)
−0.391948 + 0.919987i \(0.628199\pi\)
\(632\) 0 0
\(633\) −19.0383 −0.756705
\(634\) 0 0
\(635\) 31.6097 1.25439
\(636\) 0 0
\(637\) −5.98292 −0.237052
\(638\) 0 0
\(639\) 1.77377 0.0701694
\(640\) 0 0
\(641\) −35.9126 −1.41846 −0.709232 0.704975i \(-0.750958\pi\)
−0.709232 + 0.704975i \(0.750958\pi\)
\(642\) 0 0
\(643\) 15.2297 0.600600 0.300300 0.953845i \(-0.402913\pi\)
0.300300 + 0.953845i \(0.402913\pi\)
\(644\) 0 0
\(645\) −22.4033 −0.882129
\(646\) 0 0
\(647\) −33.5755 −1.31999 −0.659995 0.751270i \(-0.729442\pi\)
−0.659995 + 0.751270i \(0.729442\pi\)
\(648\) 0 0
\(649\) −54.2052 −2.12774
\(650\) 0 0
\(651\) −4.00295 −0.156888
\(652\) 0 0
\(653\) 2.85635 0.111778 0.0558889 0.998437i \(-0.482201\pi\)
0.0558889 + 0.998437i \(0.482201\pi\)
\(654\) 0 0
\(655\) −19.6494 −0.767766
\(656\) 0 0
\(657\) 1.52720 0.0595816
\(658\) 0 0
\(659\) −8.63887 −0.336523 −0.168261 0.985742i \(-0.553815\pi\)
−0.168261 + 0.985742i \(0.553815\pi\)
\(660\) 0 0
\(661\) −44.2990 −1.72303 −0.861516 0.507731i \(-0.830484\pi\)
−0.861516 + 0.507731i \(0.830484\pi\)
\(662\) 0 0
\(663\) 31.3042 1.21576
\(664\) 0 0
\(665\) 9.62055 0.373069
\(666\) 0 0
\(667\) −4.36825 −0.169139
\(668\) 0 0
\(669\) 11.0349 0.426634
\(670\) 0 0
\(671\) −15.0822 −0.582243
\(672\) 0 0
\(673\) 16.2449 0.626194 0.313097 0.949721i \(-0.398633\pi\)
0.313097 + 0.949721i \(0.398633\pi\)
\(674\) 0 0
\(675\) 1.77655 0.0683796
\(676\) 0 0
\(677\) −0.830668 −0.0319251 −0.0159626 0.999873i \(-0.505081\pi\)
−0.0159626 + 0.999873i \(0.505081\pi\)
\(678\) 0 0
\(679\) 1.66060 0.0637281
\(680\) 0 0
\(681\) −4.18514 −0.160375
\(682\) 0 0
\(683\) −29.4028 −1.12507 −0.562534 0.826774i \(-0.690174\pi\)
−0.562534 + 0.826774i \(0.690174\pi\)
\(684\) 0 0
\(685\) −17.3251 −0.661958
\(686\) 0 0
\(687\) 7.15714 0.273062
\(688\) 0 0
\(689\) −67.6865 −2.57865
\(690\) 0 0
\(691\) 38.7775 1.47516 0.737582 0.675257i \(-0.235967\pi\)
0.737582 + 0.675257i \(0.235967\pi\)
\(692\) 0 0
\(693\) −3.69569 −0.140388
\(694\) 0 0
\(695\) 29.7060 1.12681
\(696\) 0 0
\(697\) −34.2767 −1.29832
\(698\) 0 0
\(699\) 29.1064 1.10091
\(700\) 0 0
\(701\) 14.8376 0.560407 0.280203 0.959941i \(-0.409598\pi\)
0.280203 + 0.959941i \(0.409598\pi\)
\(702\) 0 0
\(703\) 30.3815 1.14586
\(704\) 0 0
\(705\) −6.97982 −0.262875
\(706\) 0 0
\(707\) −3.46076 −0.130155
\(708\) 0 0
\(709\) −6.79230 −0.255090 −0.127545 0.991833i \(-0.540710\pi\)
−0.127545 + 0.991833i \(0.540710\pi\)
\(710\) 0 0
\(711\) −1.97409 −0.0740342
\(712\) 0 0
\(713\) −4.00295 −0.149911
\(714\) 0 0
\(715\) −57.5590 −2.15258
\(716\) 0 0
\(717\) −27.6094 −1.03109
\(718\) 0 0
\(719\) −18.4230 −0.687063 −0.343532 0.939141i \(-0.611623\pi\)
−0.343532 + 0.939141i \(0.611623\pi\)
\(720\) 0 0
\(721\) 18.6471 0.694454
\(722\) 0 0
\(723\) 13.9424 0.518523
\(724\) 0 0
\(725\) −7.76043 −0.288215
\(726\) 0 0
\(727\) −24.9481 −0.925275 −0.462637 0.886548i \(-0.653097\pi\)
−0.462637 + 0.886548i \(0.653097\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −45.0296 −1.66548
\(732\) 0 0
\(733\) 14.1542 0.522799 0.261399 0.965231i \(-0.415816\pi\)
0.261399 + 0.965231i \(0.415816\pi\)
\(734\) 0 0
\(735\) −2.60318 −0.0960198
\(736\) 0 0
\(737\) 14.0103 0.516076
\(738\) 0 0
\(739\) 46.3978 1.70677 0.853385 0.521281i \(-0.174546\pi\)
0.853385 + 0.521281i \(0.174546\pi\)
\(740\) 0 0
\(741\) 22.1110 0.812268
\(742\) 0 0
\(743\) 41.6993 1.52980 0.764899 0.644150i \(-0.222789\pi\)
0.764899 + 0.644150i \(0.222789\pi\)
\(744\) 0 0
\(745\) −43.6600 −1.59958
\(746\) 0 0
\(747\) 7.08481 0.259220
\(748\) 0 0
\(749\) −16.8097 −0.614213
\(750\) 0 0
\(751\) −9.29644 −0.339232 −0.169616 0.985510i \(-0.554253\pi\)
−0.169616 + 0.985510i \(0.554253\pi\)
\(752\) 0 0
\(753\) 3.89996 0.142122
\(754\) 0 0
\(755\) 15.4906 0.563761
\(756\) 0 0
\(757\) −3.81280 −0.138578 −0.0692892 0.997597i \(-0.522073\pi\)
−0.0692892 + 0.997597i \(0.522073\pi\)
\(758\) 0 0
\(759\) −3.69569 −0.134145
\(760\) 0 0
\(761\) −48.5275 −1.75912 −0.879560 0.475788i \(-0.842163\pi\)
−0.879560 + 0.475788i \(0.842163\pi\)
\(762\) 0 0
\(763\) −8.58022 −0.310625
\(764\) 0 0
\(765\) 13.6206 0.492452
\(766\) 0 0
\(767\) −87.7522 −3.16855
\(768\) 0 0
\(769\) 22.1402 0.798398 0.399199 0.916864i \(-0.369288\pi\)
0.399199 + 0.916864i \(0.369288\pi\)
\(770\) 0 0
\(771\) 16.0968 0.579711
\(772\) 0 0
\(773\) −1.18180 −0.0425062 −0.0212531 0.999774i \(-0.506766\pi\)
−0.0212531 + 0.999774i \(0.506766\pi\)
\(774\) 0 0
\(775\) −7.11145 −0.255451
\(776\) 0 0
\(777\) −8.22079 −0.294919
\(778\) 0 0
\(779\) −24.2105 −0.867431
\(780\) 0 0
\(781\) −6.55532 −0.234568
\(782\) 0 0
\(783\) −4.36825 −0.156109
\(784\) 0 0
\(785\) −1.99380 −0.0711617
\(786\) 0 0
\(787\) −39.7670 −1.41754 −0.708770 0.705440i \(-0.750749\pi\)
−0.708770 + 0.705440i \(0.750749\pi\)
\(788\) 0 0
\(789\) −25.2875 −0.900259
\(790\) 0 0
\(791\) 10.6119 0.377315
\(792\) 0 0
\(793\) −24.4164 −0.867053
\(794\) 0 0
\(795\) −29.4506 −1.04450
\(796\) 0 0
\(797\) −6.93837 −0.245770 −0.122885 0.992421i \(-0.539215\pi\)
−0.122885 + 0.992421i \(0.539215\pi\)
\(798\) 0 0
\(799\) −14.0291 −0.496314
\(800\) 0 0
\(801\) 12.9801 0.458631
\(802\) 0 0
\(803\) −5.64405 −0.199174
\(804\) 0 0
\(805\) −2.60318 −0.0917501
\(806\) 0 0
\(807\) −9.45589 −0.332863
\(808\) 0 0
\(809\) 1.71075 0.0601468 0.0300734 0.999548i \(-0.490426\pi\)
0.0300734 + 0.999548i \(0.490426\pi\)
\(810\) 0 0
\(811\) 1.63104 0.0572737 0.0286368 0.999590i \(-0.490883\pi\)
0.0286368 + 0.999590i \(0.490883\pi\)
\(812\) 0 0
\(813\) −0.941543 −0.0330214
\(814\) 0 0
\(815\) 32.6281 1.14291
\(816\) 0 0
\(817\) −31.8056 −1.11274
\(818\) 0 0
\(819\) −5.98292 −0.209060
\(820\) 0 0
\(821\) 18.1679 0.634066 0.317033 0.948415i \(-0.397313\pi\)
0.317033 + 0.948415i \(0.397313\pi\)
\(822\) 0 0
\(823\) 22.2847 0.776797 0.388398 0.921492i \(-0.373028\pi\)
0.388398 + 0.921492i \(0.373028\pi\)
\(824\) 0 0
\(825\) −6.56559 −0.228585
\(826\) 0 0
\(827\) 13.2556 0.460942 0.230471 0.973079i \(-0.425973\pi\)
0.230471 + 0.973079i \(0.425973\pi\)
\(828\) 0 0
\(829\) 12.1676 0.422599 0.211300 0.977421i \(-0.432230\pi\)
0.211300 + 0.977421i \(0.432230\pi\)
\(830\) 0 0
\(831\) −17.8705 −0.619922
\(832\) 0 0
\(833\) −5.23227 −0.181288
\(834\) 0 0
\(835\) 16.1025 0.557250
\(836\) 0 0
\(837\) −4.00295 −0.138362
\(838\) 0 0
\(839\) −1.07369 −0.0370679 −0.0185340 0.999828i \(-0.505900\pi\)
−0.0185340 + 0.999828i \(0.505900\pi\)
\(840\) 0 0
\(841\) −9.91837 −0.342013
\(842\) 0 0
\(843\) 7.79086 0.268331
\(844\) 0 0
\(845\) −59.3403 −2.04137
\(846\) 0 0
\(847\) 2.65813 0.0913345
\(848\) 0 0
\(849\) 17.7680 0.609797
\(850\) 0 0
\(851\) −8.22079 −0.281805
\(852\) 0 0
\(853\) 34.4834 1.18069 0.590344 0.807152i \(-0.298992\pi\)
0.590344 + 0.807152i \(0.298992\pi\)
\(854\) 0 0
\(855\) 9.62055 0.329016
\(856\) 0 0
\(857\) −34.8518 −1.19052 −0.595258 0.803535i \(-0.702950\pi\)
−0.595258 + 0.803535i \(0.702950\pi\)
\(858\) 0 0
\(859\) 23.0040 0.784886 0.392443 0.919776i \(-0.371630\pi\)
0.392443 + 0.919776i \(0.371630\pi\)
\(860\) 0 0
\(861\) 6.55101 0.223258
\(862\) 0 0
\(863\) −8.40516 −0.286115 −0.143057 0.989714i \(-0.545693\pi\)
−0.143057 + 0.989714i \(0.545693\pi\)
\(864\) 0 0
\(865\) 19.3086 0.656511
\(866\) 0 0
\(867\) 10.3767 0.352410
\(868\) 0 0
\(869\) 7.29563 0.247487
\(870\) 0 0
\(871\) 22.6811 0.768520
\(872\) 0 0
\(873\) 1.66060 0.0562029
\(874\) 0 0
\(875\) 8.39122 0.283675
\(876\) 0 0
\(877\) −10.3832 −0.350615 −0.175307 0.984514i \(-0.556092\pi\)
−0.175307 + 0.984514i \(0.556092\pi\)
\(878\) 0 0
\(879\) 0.0142620 0.000481045 0
\(880\) 0 0
\(881\) 40.9118 1.37835 0.689176 0.724594i \(-0.257973\pi\)
0.689176 + 0.724594i \(0.257973\pi\)
\(882\) 0 0
\(883\) 18.1694 0.611449 0.305725 0.952120i \(-0.401101\pi\)
0.305725 + 0.952120i \(0.401101\pi\)
\(884\) 0 0
\(885\) −38.1812 −1.28345
\(886\) 0 0
\(887\) −1.58708 −0.0532889 −0.0266444 0.999645i \(-0.508482\pi\)
−0.0266444 + 0.999645i \(0.508482\pi\)
\(888\) 0 0
\(889\) −12.1427 −0.407253
\(890\) 0 0
\(891\) −3.69569 −0.123810
\(892\) 0 0
\(893\) −9.90912 −0.331596
\(894\) 0 0
\(895\) 54.9409 1.83647
\(896\) 0 0
\(897\) −5.98292 −0.199764
\(898\) 0 0
\(899\) 17.4859 0.583187
\(900\) 0 0
\(901\) −59.1943 −1.97205
\(902\) 0 0
\(903\) 8.60613 0.286394
\(904\) 0 0
\(905\) −6.00442 −0.199594
\(906\) 0 0
\(907\) 7.51319 0.249471 0.124736 0.992190i \(-0.460192\pi\)
0.124736 + 0.992190i \(0.460192\pi\)
\(908\) 0 0
\(909\) −3.46076 −0.114786
\(910\) 0 0
\(911\) 47.2129 1.56423 0.782117 0.623131i \(-0.214140\pi\)
0.782117 + 0.623131i \(0.214140\pi\)
\(912\) 0 0
\(913\) −26.1833 −0.866540
\(914\) 0 0
\(915\) −10.6237 −0.351207
\(916\) 0 0
\(917\) 7.54823 0.249264
\(918\) 0 0
\(919\) 26.4159 0.871381 0.435690 0.900097i \(-0.356504\pi\)
0.435690 + 0.900097i \(0.356504\pi\)
\(920\) 0 0
\(921\) −23.1566 −0.763035
\(922\) 0 0
\(923\) −10.6123 −0.349309
\(924\) 0 0
\(925\) −14.6047 −0.480199
\(926\) 0 0
\(927\) 18.6471 0.612451
\(928\) 0 0
\(929\) 11.8816 0.389823 0.194911 0.980821i \(-0.437558\pi\)
0.194911 + 0.980821i \(0.437558\pi\)
\(930\) 0 0
\(931\) −3.69569 −0.121121
\(932\) 0 0
\(933\) −5.20295 −0.170337
\(934\) 0 0
\(935\) −50.3374 −1.64621
\(936\) 0 0
\(937\) 52.3277 1.70947 0.854736 0.519063i \(-0.173719\pi\)
0.854736 + 0.519063i \(0.173719\pi\)
\(938\) 0 0
\(939\) 4.56663 0.149026
\(940\) 0 0
\(941\) −6.10300 −0.198952 −0.0994761 0.995040i \(-0.531717\pi\)
−0.0994761 + 0.995040i \(0.531717\pi\)
\(942\) 0 0
\(943\) 6.55101 0.213330
\(944\) 0 0
\(945\) −2.60318 −0.0846815
\(946\) 0 0
\(947\) −1.77999 −0.0578417 −0.0289209 0.999582i \(-0.509207\pi\)
−0.0289209 + 0.999582i \(0.509207\pi\)
\(948\) 0 0
\(949\) −9.13709 −0.296602
\(950\) 0 0
\(951\) −24.7700 −0.803222
\(952\) 0 0
\(953\) 30.0788 0.974347 0.487174 0.873305i \(-0.338028\pi\)
0.487174 + 0.873305i \(0.338028\pi\)
\(954\) 0 0
\(955\) 28.8313 0.932960
\(956\) 0 0
\(957\) 16.1437 0.521852
\(958\) 0 0
\(959\) 6.65535 0.214913
\(960\) 0 0
\(961\) −14.9764 −0.483111
\(962\) 0 0
\(963\) −16.8097 −0.541685
\(964\) 0 0
\(965\) 54.1700 1.74379
\(966\) 0 0
\(967\) 62.0094 1.99409 0.997044 0.0768279i \(-0.0244792\pi\)
0.997044 + 0.0768279i \(0.0244792\pi\)
\(968\) 0 0
\(969\) 19.3369 0.621190
\(970\) 0 0
\(971\) −10.3406 −0.331847 −0.165923 0.986139i \(-0.553060\pi\)
−0.165923 + 0.986139i \(0.553060\pi\)
\(972\) 0 0
\(973\) −11.4114 −0.365833
\(974\) 0 0
\(975\) −10.6290 −0.340399
\(976\) 0 0
\(977\) −32.4038 −1.03669 −0.518345 0.855172i \(-0.673452\pi\)
−0.518345 + 0.855172i \(0.673452\pi\)
\(978\) 0 0
\(979\) −47.9706 −1.53315
\(980\) 0 0
\(981\) −8.58022 −0.273945
\(982\) 0 0
\(983\) 51.4249 1.64020 0.820099 0.572221i \(-0.193918\pi\)
0.820099 + 0.572221i \(0.193918\pi\)
\(984\) 0 0
\(985\) −20.8101 −0.663066
\(986\) 0 0
\(987\) 2.68126 0.0853456
\(988\) 0 0
\(989\) 8.60613 0.273659
\(990\) 0 0
\(991\) 6.14387 0.195166 0.0975832 0.995227i \(-0.468889\pi\)
0.0975832 + 0.995227i \(0.468889\pi\)
\(992\) 0 0
\(993\) −29.3320 −0.930822
\(994\) 0 0
\(995\) −52.0327 −1.64955
\(996\) 0 0
\(997\) 32.8573 1.04060 0.520300 0.853984i \(-0.325820\pi\)
0.520300 + 0.853984i \(0.325820\pi\)
\(998\) 0 0
\(999\) −8.22079 −0.260094
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.ch.1.1 6
4.3 odd 2 3864.2.a.w.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3864.2.a.w.1.1 6 4.3 odd 2
7728.2.a.ch.1.1 6 1.1 even 1 trivial