Properties

Label 7728.2.a.cd.1.1
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: 4.4.24197.1
Defining polynomial: \(x^{4} - 6 x^{2} - x + 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
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 \(-0.700017\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.15706 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.15706 q^{5} -1.00000 q^{7} +1.00000 q^{9} -3.66704 q^{11} +2.34710 q^{13} -1.15706 q^{15} +4.80996 q^{17} -7.06707 q^{19} -1.00000 q^{21} +1.00000 q^{23} -3.66122 q^{25} +1.00000 q^{27} -6.52411 q^{29} -2.80996 q^{31} -3.66704 q^{33} +1.15706 q^{35} +5.40993 q^{37} +2.34710 q^{39} +0.647082 q^{41} +4.76709 q^{43} -1.15706 q^{45} -4.85708 q^{47} +1.00000 q^{49} +4.80996 q^{51} +7.63405 q^{53} +4.24297 q^{55} -7.06707 q^{57} +10.2142 q^{59} +9.91001 q^{61} -1.00000 q^{63} -2.71573 q^{65} -6.07114 q^{67} +1.00000 q^{69} +3.65290 q^{71} +11.8183 q^{73} -3.66122 q^{75} +3.66704 q^{77} +9.12408 q^{79} +1.00000 q^{81} +15.8582 q^{83} -5.56541 q^{85} -6.52411 q^{87} +4.66122 q^{89} -2.34710 q^{91} -2.80996 q^{93} +8.17701 q^{95} -4.90419 q^{97} -3.66704 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9} + O(q^{10}) \) \( 4 q + 4 q^{3} + 5 q^{5} - 4 q^{7} + 4 q^{9} + 5 q^{11} + 7 q^{13} + 5 q^{15} + 12 q^{17} - 3 q^{19} - 4 q^{21} + 4 q^{23} + 7 q^{25} + 4 q^{27} + 6 q^{29} - 4 q^{31} + 5 q^{33} - 5 q^{35} + 20 q^{37} + 7 q^{39} + 3 q^{41} - 9 q^{43} + 5 q^{45} - 7 q^{47} + 4 q^{49} + 12 q^{51} - 6 q^{53} + 21 q^{55} - 3 q^{57} + 2 q^{59} + 24 q^{61} - 4 q^{63} - 14 q^{65} - q^{67} + 4 q^{69} + 17 q^{71} + 16 q^{73} + 7 q^{75} - 5 q^{77} + 10 q^{79} + 4 q^{81} - 8 q^{83} + 17 q^{85} + 6 q^{87} - 3 q^{89} - 7 q^{91} - 4 q^{93} + 3 q^{95} - 2 q^{97} + 5 q^{99} + O(q^{100}) \)

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
\(5\) −1.15706 −0.517452 −0.258726 0.965951i \(-0.583303\pi\)
−0.258726 + 0.965951i \(0.583303\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.66704 −1.10565 −0.552826 0.833296i \(-0.686451\pi\)
−0.552826 + 0.833296i \(0.686451\pi\)
\(12\) 0 0
\(13\) 2.34710 0.650968 0.325484 0.945548i \(-0.394473\pi\)
0.325484 + 0.945548i \(0.394473\pi\)
\(14\) 0 0
\(15\) −1.15706 −0.298751
\(16\) 0 0
\(17\) 4.80996 1.16659 0.583293 0.812262i \(-0.301764\pi\)
0.583293 + 0.812262i \(0.301764\pi\)
\(18\) 0 0
\(19\) −7.06707 −1.62130 −0.810648 0.585533i \(-0.800885\pi\)
−0.810648 + 0.585533i \(0.800885\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −3.66122 −0.732243
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −6.52411 −1.21150 −0.605748 0.795656i \(-0.707126\pi\)
−0.605748 + 0.795656i \(0.707126\pi\)
\(30\) 0 0
\(31\) −2.80996 −0.504684 −0.252342 0.967638i \(-0.581201\pi\)
−0.252342 + 0.967638i \(0.581201\pi\)
\(32\) 0 0
\(33\) −3.66704 −0.638349
\(34\) 0 0
\(35\) 1.15706 0.195579
\(36\) 0 0
\(37\) 5.40993 0.889387 0.444693 0.895683i \(-0.353313\pi\)
0.444693 + 0.895683i \(0.353313\pi\)
\(38\) 0 0
\(39\) 2.34710 0.375837
\(40\) 0 0
\(41\) 0.647082 0.101057 0.0505286 0.998723i \(-0.483909\pi\)
0.0505286 + 0.998723i \(0.483909\pi\)
\(42\) 0 0
\(43\) 4.76709 0.726974 0.363487 0.931599i \(-0.381586\pi\)
0.363487 + 0.931599i \(0.381586\pi\)
\(44\) 0 0
\(45\) −1.15706 −0.172484
\(46\) 0 0
\(47\) −4.85708 −0.708477 −0.354239 0.935155i \(-0.615260\pi\)
−0.354239 + 0.935155i \(0.615260\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 4.80996 0.673529
\(52\) 0 0
\(53\) 7.63405 1.04862 0.524309 0.851528i \(-0.324324\pi\)
0.524309 + 0.851528i \(0.324324\pi\)
\(54\) 0 0
\(55\) 4.24297 0.572123
\(56\) 0 0
\(57\) −7.06707 −0.936056
\(58\) 0 0
\(59\) 10.2142 1.32978 0.664890 0.746941i \(-0.268478\pi\)
0.664890 + 0.746941i \(0.268478\pi\)
\(60\) 0 0
\(61\) 9.91001 1.26885 0.634423 0.772986i \(-0.281238\pi\)
0.634423 + 0.772986i \(0.281238\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) −2.71573 −0.336845
\(66\) 0 0
\(67\) −6.07114 −0.741708 −0.370854 0.928691i \(-0.620935\pi\)
−0.370854 + 0.928691i \(0.620935\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) 3.65290 0.433520 0.216760 0.976225i \(-0.430451\pi\)
0.216760 + 0.976225i \(0.430451\pi\)
\(72\) 0 0
\(73\) 11.8183 1.38322 0.691612 0.722269i \(-0.256901\pi\)
0.691612 + 0.722269i \(0.256901\pi\)
\(74\) 0 0
\(75\) −3.66122 −0.422761
\(76\) 0 0
\(77\) 3.66704 0.417897
\(78\) 0 0
\(79\) 9.12408 1.02654 0.513269 0.858228i \(-0.328434\pi\)
0.513269 + 0.858228i \(0.328434\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 15.8582 1.74066 0.870331 0.492468i \(-0.163905\pi\)
0.870331 + 0.492468i \(0.163905\pi\)
\(84\) 0 0
\(85\) −5.56541 −0.603653
\(86\) 0 0
\(87\) −6.52411 −0.699458
\(88\) 0 0
\(89\) 4.66122 0.494088 0.247044 0.969004i \(-0.420541\pi\)
0.247044 + 0.969004i \(0.420541\pi\)
\(90\) 0 0
\(91\) −2.34710 −0.246043
\(92\) 0 0
\(93\) −2.80996 −0.291379
\(94\) 0 0
\(95\) 8.17701 0.838944
\(96\) 0 0
\(97\) −4.90419 −0.497945 −0.248973 0.968511i \(-0.580093\pi\)
−0.248973 + 0.968511i \(0.580093\pi\)
\(98\) 0 0
\(99\) −3.66704 −0.368551
\(100\) 0 0
\(101\) −5.80589 −0.577707 −0.288854 0.957373i \(-0.593274\pi\)
−0.288854 + 0.957373i \(0.593274\pi\)
\(102\) 0 0
\(103\) −17.1169 −1.68658 −0.843288 0.537463i \(-0.819383\pi\)
−0.843288 + 0.537463i \(0.819383\pi\)
\(104\) 0 0
\(105\) 1.15706 0.112917
\(106\) 0 0
\(107\) 8.34710 0.806944 0.403472 0.914992i \(-0.367803\pi\)
0.403472 + 0.914992i \(0.367803\pi\)
\(108\) 0 0
\(109\) 16.3771 1.56864 0.784321 0.620355i \(-0.213011\pi\)
0.784321 + 0.620355i \(0.213011\pi\)
\(110\) 0 0
\(111\) 5.40993 0.513488
\(112\) 0 0
\(113\) −14.1770 −1.33366 −0.666831 0.745209i \(-0.732350\pi\)
−0.666831 + 0.745209i \(0.732350\pi\)
\(114\) 0 0
\(115\) −1.15706 −0.107896
\(116\) 0 0
\(117\) 2.34710 0.216989
\(118\) 0 0
\(119\) −4.80996 −0.440928
\(120\) 0 0
\(121\) 2.44715 0.222468
\(122\) 0 0
\(123\) 0.647082 0.0583454
\(124\) 0 0
\(125\) 10.0215 0.896353
\(126\) 0 0
\(127\) −13.1184 −1.16407 −0.582036 0.813163i \(-0.697744\pi\)
−0.582036 + 0.813163i \(0.697744\pi\)
\(128\) 0 0
\(129\) 4.76709 0.419718
\(130\) 0 0
\(131\) 1.28696 0.112442 0.0562209 0.998418i \(-0.482095\pi\)
0.0562209 + 0.998418i \(0.482095\pi\)
\(132\) 0 0
\(133\) 7.06707 0.612793
\(134\) 0 0
\(135\) −1.15706 −0.0995837
\(136\) 0 0
\(137\) −11.3152 −0.966725 −0.483362 0.875420i \(-0.660585\pi\)
−0.483362 + 0.875420i \(0.660585\pi\)
\(138\) 0 0
\(139\) −8.93717 −0.758041 −0.379020 0.925388i \(-0.623739\pi\)
−0.379020 + 0.925388i \(0.623739\pi\)
\(140\) 0 0
\(141\) −4.85708 −0.409040
\(142\) 0 0
\(143\) −8.60689 −0.719745
\(144\) 0 0
\(145\) 7.54878 0.626892
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 0 0
\(149\) 1.67693 0.137379 0.0686897 0.997638i \(-0.478118\pi\)
0.0686897 + 0.997638i \(0.478118\pi\)
\(150\) 0 0
\(151\) 19.0110 1.54709 0.773547 0.633739i \(-0.218481\pi\)
0.773547 + 0.633739i \(0.218481\pi\)
\(152\) 0 0
\(153\) 4.80996 0.388862
\(154\) 0 0
\(155\) 3.25129 0.261150
\(156\) 0 0
\(157\) 12.5712 1.00329 0.501647 0.865073i \(-0.332728\pi\)
0.501647 + 0.865073i \(0.332728\pi\)
\(158\) 0 0
\(159\) 7.63405 0.605420
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 15.6740 1.22768 0.613840 0.789431i \(-0.289624\pi\)
0.613840 + 0.789431i \(0.289624\pi\)
\(164\) 0 0
\(165\) 4.24297 0.330315
\(166\) 0 0
\(167\) −4.96120 −0.383909 −0.191955 0.981404i \(-0.561483\pi\)
−0.191955 + 0.981404i \(0.561483\pi\)
\(168\) 0 0
\(169\) −7.49113 −0.576241
\(170\) 0 0
\(171\) −7.06707 −0.540432
\(172\) 0 0
\(173\) 12.9042 0.981087 0.490544 0.871417i \(-0.336798\pi\)
0.490544 + 0.871417i \(0.336798\pi\)
\(174\) 0 0
\(175\) 3.66122 0.276762
\(176\) 0 0
\(177\) 10.2142 0.767749
\(178\) 0 0
\(179\) 0.262762 0.0196398 0.00981989 0.999952i \(-0.496874\pi\)
0.00981989 + 0.999952i \(0.496874\pi\)
\(180\) 0 0
\(181\) −12.1440 −0.902659 −0.451329 0.892357i \(-0.649050\pi\)
−0.451329 + 0.892357i \(0.649050\pi\)
\(182\) 0 0
\(183\) 9.91001 0.732569
\(184\) 0 0
\(185\) −6.25960 −0.460215
\(186\) 0 0
\(187\) −17.6383 −1.28984
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) −2.68524 −0.194297 −0.0971487 0.995270i \(-0.530972\pi\)
−0.0971487 + 0.995270i \(0.530972\pi\)
\(192\) 0 0
\(193\) −18.7252 −1.34787 −0.673933 0.738793i \(-0.735396\pi\)
−0.673933 + 0.738793i \(0.735396\pi\)
\(194\) 0 0
\(195\) −2.71573 −0.194477
\(196\) 0 0
\(197\) 19.3953 1.38186 0.690930 0.722922i \(-0.257201\pi\)
0.690930 + 0.722922i \(0.257201\pi\)
\(198\) 0 0
\(199\) −1.10005 −0.0779805 −0.0389902 0.999240i \(-0.512414\pi\)
−0.0389902 + 0.999240i \(0.512414\pi\)
\(200\) 0 0
\(201\) −6.07114 −0.428225
\(202\) 0 0
\(203\) 6.52411 0.457903
\(204\) 0 0
\(205\) −0.748712 −0.0522923
\(206\) 0 0
\(207\) 1.00000 0.0695048
\(208\) 0 0
\(209\) 25.9152 1.79259
\(210\) 0 0
\(211\) −25.5235 −1.75711 −0.878554 0.477643i \(-0.841491\pi\)
−0.878554 + 0.477643i \(0.841491\pi\)
\(212\) 0 0
\(213\) 3.65290 0.250293
\(214\) 0 0
\(215\) −5.51580 −0.376174
\(216\) 0 0
\(217\) 2.80996 0.190753
\(218\) 0 0
\(219\) 11.8183 0.798605
\(220\) 0 0
\(221\) 11.2894 0.759411
\(222\) 0 0
\(223\) 17.1772 1.15027 0.575134 0.818059i \(-0.304950\pi\)
0.575134 + 0.818059i \(0.304950\pi\)
\(224\) 0 0
\(225\) −3.66122 −0.244081
\(226\) 0 0
\(227\) 26.0334 1.72790 0.863950 0.503577i \(-0.167983\pi\)
0.863950 + 0.503577i \(0.167983\pi\)
\(228\) 0 0
\(229\) −7.01430 −0.463518 −0.231759 0.972773i \(-0.574448\pi\)
−0.231759 + 0.972773i \(0.574448\pi\)
\(230\) 0 0
\(231\) 3.66704 0.241273
\(232\) 0 0
\(233\) 0.671109 0.0439658 0.0219829 0.999758i \(-0.493002\pi\)
0.0219829 + 0.999758i \(0.493002\pi\)
\(234\) 0 0
\(235\) 5.61992 0.366603
\(236\) 0 0
\(237\) 9.12408 0.592673
\(238\) 0 0
\(239\) 26.5571 1.71784 0.858918 0.512114i \(-0.171137\pi\)
0.858918 + 0.512114i \(0.171137\pi\)
\(240\) 0 0
\(241\) −4.97284 −0.320329 −0.160164 0.987090i \(-0.551202\pi\)
−0.160164 + 0.987090i \(0.551202\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −1.15706 −0.0739218
\(246\) 0 0
\(247\) −16.5871 −1.05541
\(248\) 0 0
\(249\) 15.8582 1.00497
\(250\) 0 0
\(251\) 14.4281 0.910696 0.455348 0.890314i \(-0.349515\pi\)
0.455348 + 0.890314i \(0.349515\pi\)
\(252\) 0 0
\(253\) −3.66704 −0.230545
\(254\) 0 0
\(255\) −5.56541 −0.348519
\(256\) 0 0
\(257\) 5.44889 0.339893 0.169946 0.985453i \(-0.445641\pi\)
0.169946 + 0.985453i \(0.445641\pi\)
\(258\) 0 0
\(259\) −5.40993 −0.336157
\(260\) 0 0
\(261\) −6.52411 −0.403832
\(262\) 0 0
\(263\) −15.8128 −0.975060 −0.487530 0.873106i \(-0.662102\pi\)
−0.487530 + 0.873106i \(0.662102\pi\)
\(264\) 0 0
\(265\) −8.83305 −0.542610
\(266\) 0 0
\(267\) 4.66122 0.285262
\(268\) 0 0
\(269\) 11.9011 0.725620 0.362810 0.931863i \(-0.381817\pi\)
0.362810 + 0.931863i \(0.381817\pi\)
\(270\) 0 0
\(271\) 16.6664 1.01241 0.506206 0.862413i \(-0.331048\pi\)
0.506206 + 0.862413i \(0.331048\pi\)
\(272\) 0 0
\(273\) −2.34710 −0.142053
\(274\) 0 0
\(275\) 13.4258 0.809607
\(276\) 0 0
\(277\) 20.1299 1.20949 0.604744 0.796420i \(-0.293275\pi\)
0.604744 + 0.796420i \(0.293275\pi\)
\(278\) 0 0
\(279\) −2.80996 −0.168228
\(280\) 0 0
\(281\) 19.4022 1.15744 0.578720 0.815526i \(-0.303552\pi\)
0.578720 + 0.815526i \(0.303552\pi\)
\(282\) 0 0
\(283\) 10.8313 0.643854 0.321927 0.946764i \(-0.395669\pi\)
0.321927 + 0.946764i \(0.395669\pi\)
\(284\) 0 0
\(285\) 8.17701 0.484364
\(286\) 0 0
\(287\) −0.647082 −0.0381960
\(288\) 0 0
\(289\) 6.13572 0.360925
\(290\) 0 0
\(291\) −4.90419 −0.287489
\(292\) 0 0
\(293\) 8.89430 0.519610 0.259805 0.965661i \(-0.416342\pi\)
0.259805 + 0.965661i \(0.416342\pi\)
\(294\) 0 0
\(295\) −11.8185 −0.688098
\(296\) 0 0
\(297\) −3.66704 −0.212783
\(298\) 0 0
\(299\) 2.34710 0.135736
\(300\) 0 0
\(301\) −4.76709 −0.274770
\(302\) 0 0
\(303\) −5.80589 −0.333539
\(304\) 0 0
\(305\) −11.4665 −0.656568
\(306\) 0 0
\(307\) −19.7442 −1.12686 −0.563429 0.826164i \(-0.690518\pi\)
−0.563429 + 0.826164i \(0.690518\pi\)
\(308\) 0 0
\(309\) −17.1169 −0.973745
\(310\) 0 0
\(311\) 28.5898 1.62118 0.810589 0.585615i \(-0.199147\pi\)
0.810589 + 0.585615i \(0.199147\pi\)
\(312\) 0 0
\(313\) 12.6570 0.715415 0.357707 0.933834i \(-0.383559\pi\)
0.357707 + 0.933834i \(0.383559\pi\)
\(314\) 0 0
\(315\) 1.15706 0.0651929
\(316\) 0 0
\(317\) 4.83130 0.271353 0.135676 0.990753i \(-0.456679\pi\)
0.135676 + 0.990753i \(0.456679\pi\)
\(318\) 0 0
\(319\) 23.9241 1.33949
\(320\) 0 0
\(321\) 8.34710 0.465890
\(322\) 0 0
\(323\) −33.9923 −1.89138
\(324\) 0 0
\(325\) −8.59323 −0.476667
\(326\) 0 0
\(327\) 16.3771 0.905656
\(328\) 0 0
\(329\) 4.85708 0.267779
\(330\) 0 0
\(331\) 6.84544 0.376259 0.188130 0.982144i \(-0.439758\pi\)
0.188130 + 0.982144i \(0.439758\pi\)
\(332\) 0 0
\(333\) 5.40993 0.296462
\(334\) 0 0
\(335\) 7.02467 0.383799
\(336\) 0 0
\(337\) −8.17527 −0.445335 −0.222668 0.974894i \(-0.571476\pi\)
−0.222668 + 0.974894i \(0.571476\pi\)
\(338\) 0 0
\(339\) −14.1770 −0.769990
\(340\) 0 0
\(341\) 10.3042 0.558005
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −1.15706 −0.0622939
\(346\) 0 0
\(347\) −21.2699 −1.14183 −0.570913 0.821011i \(-0.693411\pi\)
−0.570913 + 0.821011i \(0.693411\pi\)
\(348\) 0 0
\(349\) 1.96527 0.105199 0.0525993 0.998616i \(-0.483249\pi\)
0.0525993 + 0.998616i \(0.483249\pi\)
\(350\) 0 0
\(351\) 2.34710 0.125279
\(352\) 0 0
\(353\) 10.3124 0.548872 0.274436 0.961605i \(-0.411509\pi\)
0.274436 + 0.961605i \(0.411509\pi\)
\(354\) 0 0
\(355\) −4.22662 −0.224326
\(356\) 0 0
\(357\) −4.80996 −0.254570
\(358\) 0 0
\(359\) 26.3856 1.39258 0.696289 0.717761i \(-0.254833\pi\)
0.696289 + 0.717761i \(0.254833\pi\)
\(360\) 0 0
\(361\) 30.9435 1.62860
\(362\) 0 0
\(363\) 2.44715 0.128442
\(364\) 0 0
\(365\) −13.6744 −0.715753
\(366\) 0 0
\(367\) 33.5667 1.75217 0.876083 0.482160i \(-0.160148\pi\)
0.876083 + 0.482160i \(0.160148\pi\)
\(368\) 0 0
\(369\) 0.647082 0.0336857
\(370\) 0 0
\(371\) −7.63405 −0.396340
\(372\) 0 0
\(373\) −17.0698 −0.883838 −0.441919 0.897055i \(-0.645702\pi\)
−0.441919 + 0.897055i \(0.645702\pi\)
\(374\) 0 0
\(375\) 10.0215 0.517510
\(376\) 0 0
\(377\) −15.3127 −0.788646
\(378\) 0 0
\(379\) 15.7541 0.809232 0.404616 0.914487i \(-0.367405\pi\)
0.404616 + 0.914487i \(0.367405\pi\)
\(380\) 0 0
\(381\) −13.1184 −0.672077
\(382\) 0 0
\(383\) −3.85818 −0.197144 −0.0985719 0.995130i \(-0.531427\pi\)
−0.0985719 + 0.995130i \(0.531427\pi\)
\(384\) 0 0
\(385\) −4.24297 −0.216242
\(386\) 0 0
\(387\) 4.76709 0.242325
\(388\) 0 0
\(389\) 35.0666 1.77795 0.888973 0.457959i \(-0.151419\pi\)
0.888973 + 0.457959i \(0.151419\pi\)
\(390\) 0 0
\(391\) 4.80996 0.243250
\(392\) 0 0
\(393\) 1.28696 0.0649183
\(394\) 0 0
\(395\) −10.5571 −0.531185
\(396\) 0 0
\(397\) −15.3422 −0.770004 −0.385002 0.922916i \(-0.625799\pi\)
−0.385002 + 0.922916i \(0.625799\pi\)
\(398\) 0 0
\(399\) 7.06707 0.353796
\(400\) 0 0
\(401\) −9.24705 −0.461776 −0.230888 0.972980i \(-0.574163\pi\)
−0.230888 + 0.972980i \(0.574163\pi\)
\(402\) 0 0
\(403\) −6.59525 −0.328533
\(404\) 0 0
\(405\) −1.15706 −0.0574947
\(406\) 0 0
\(407\) −19.8384 −0.983353
\(408\) 0 0
\(409\) −16.6498 −0.823278 −0.411639 0.911347i \(-0.635044\pi\)
−0.411639 + 0.911347i \(0.635044\pi\)
\(410\) 0 0
\(411\) −11.3152 −0.558139
\(412\) 0 0
\(413\) −10.2142 −0.502610
\(414\) 0 0
\(415\) −18.3488 −0.900709
\(416\) 0 0
\(417\) −8.93717 −0.437655
\(418\) 0 0
\(419\) 6.86306 0.335282 0.167641 0.985848i \(-0.446385\pi\)
0.167641 + 0.985848i \(0.446385\pi\)
\(420\) 0 0
\(421\) −13.1754 −0.642131 −0.321066 0.947057i \(-0.604041\pi\)
−0.321066 + 0.947057i \(0.604041\pi\)
\(422\) 0 0
\(423\) −4.85708 −0.236159
\(424\) 0 0
\(425\) −17.6103 −0.854225
\(426\) 0 0
\(427\) −9.91001 −0.479579
\(428\) 0 0
\(429\) −8.60689 −0.415545
\(430\) 0 0
\(431\) 26.0837 1.25641 0.628204 0.778048i \(-0.283790\pi\)
0.628204 + 0.778048i \(0.283790\pi\)
\(432\) 0 0
\(433\) −39.8676 −1.91591 −0.957957 0.286911i \(-0.907372\pi\)
−0.957957 + 0.286911i \(0.907372\pi\)
\(434\) 0 0
\(435\) 7.54878 0.361936
\(436\) 0 0
\(437\) −7.06707 −0.338064
\(438\) 0 0
\(439\) 17.9160 0.855084 0.427542 0.903996i \(-0.359380\pi\)
0.427542 + 0.903996i \(0.359380\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 3.02810 0.143869 0.0719347 0.997409i \(-0.477083\pi\)
0.0719347 + 0.997409i \(0.477083\pi\)
\(444\) 0 0
\(445\) −5.39330 −0.255667
\(446\) 0 0
\(447\) 1.67693 0.0793160
\(448\) 0 0
\(449\) −30.0253 −1.41698 −0.708491 0.705720i \(-0.750624\pi\)
−0.708491 + 0.705720i \(0.750624\pi\)
\(450\) 0 0
\(451\) −2.37287 −0.111734
\(452\) 0 0
\(453\) 19.0110 0.893215
\(454\) 0 0
\(455\) 2.71573 0.127315
\(456\) 0 0
\(457\) 12.2169 0.571483 0.285742 0.958307i \(-0.407760\pi\)
0.285742 + 0.958307i \(0.407760\pi\)
\(458\) 0 0
\(459\) 4.80996 0.224510
\(460\) 0 0
\(461\) 33.7511 1.57194 0.785972 0.618261i \(-0.212163\pi\)
0.785972 + 0.618261i \(0.212163\pi\)
\(462\) 0 0
\(463\) 21.9187 1.01865 0.509324 0.860575i \(-0.329896\pi\)
0.509324 + 0.860575i \(0.329896\pi\)
\(464\) 0 0
\(465\) 3.25129 0.150775
\(466\) 0 0
\(467\) 15.6499 0.724193 0.362096 0.932141i \(-0.382061\pi\)
0.362096 + 0.932141i \(0.382061\pi\)
\(468\) 0 0
\(469\) 6.07114 0.280339
\(470\) 0 0
\(471\) 12.5712 0.579251
\(472\) 0 0
\(473\) −17.4811 −0.803780
\(474\) 0 0
\(475\) 25.8741 1.18718
\(476\) 0 0
\(477\) 7.63405 0.349539
\(478\) 0 0
\(479\) −6.20999 −0.283742 −0.141871 0.989885i \(-0.545312\pi\)
−0.141871 + 0.989885i \(0.545312\pi\)
\(480\) 0 0
\(481\) 12.6976 0.578962
\(482\) 0 0
\(483\) −1.00000 −0.0455016
\(484\) 0 0
\(485\) 5.67444 0.257663
\(486\) 0 0
\(487\) 7.82642 0.354649 0.177325 0.984152i \(-0.443256\pi\)
0.177325 + 0.984152i \(0.443256\pi\)
\(488\) 0 0
\(489\) 15.6740 0.708801
\(490\) 0 0
\(491\) 32.0550 1.44662 0.723311 0.690523i \(-0.242620\pi\)
0.723311 + 0.690523i \(0.242620\pi\)
\(492\) 0 0
\(493\) −31.3807 −1.41332
\(494\) 0 0
\(495\) 4.24297 0.190708
\(496\) 0 0
\(497\) −3.65290 −0.163855
\(498\) 0 0
\(499\) −42.3592 −1.89626 −0.948129 0.317885i \(-0.897027\pi\)
−0.948129 + 0.317885i \(0.897027\pi\)
\(500\) 0 0
\(501\) −4.96120 −0.221650
\(502\) 0 0
\(503\) −11.6053 −0.517455 −0.258728 0.965950i \(-0.583303\pi\)
−0.258728 + 0.965950i \(0.583303\pi\)
\(504\) 0 0
\(505\) 6.71775 0.298936
\(506\) 0 0
\(507\) −7.49113 −0.332693
\(508\) 0 0
\(509\) −12.1512 −0.538594 −0.269297 0.963057i \(-0.586791\pi\)
−0.269297 + 0.963057i \(0.586791\pi\)
\(510\) 0 0
\(511\) −11.8183 −0.522810
\(512\) 0 0
\(513\) −7.06707 −0.312019
\(514\) 0 0
\(515\) 19.8052 0.872722
\(516\) 0 0
\(517\) 17.8111 0.783330
\(518\) 0 0
\(519\) 12.9042 0.566431
\(520\) 0 0
\(521\) −15.3884 −0.674178 −0.337089 0.941473i \(-0.609442\pi\)
−0.337089 + 0.941473i \(0.609442\pi\)
\(522\) 0 0
\(523\) 14.3536 0.627637 0.313818 0.949483i \(-0.398392\pi\)
0.313818 + 0.949483i \(0.398392\pi\)
\(524\) 0 0
\(525\) 3.66122 0.159789
\(526\) 0 0
\(527\) −13.5158 −0.588757
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 10.2142 0.443260
\(532\) 0 0
\(533\) 1.51876 0.0657850
\(534\) 0 0
\(535\) −9.65808 −0.417555
\(536\) 0 0
\(537\) 0.262762 0.0113390
\(538\) 0 0
\(539\) −3.66704 −0.157950
\(540\) 0 0
\(541\) 31.1451 1.33903 0.669517 0.742797i \(-0.266501\pi\)
0.669517 + 0.742797i \(0.266501\pi\)
\(542\) 0 0
\(543\) −12.1440 −0.521150
\(544\) 0 0
\(545\) −18.9493 −0.811698
\(546\) 0 0
\(547\) −12.6630 −0.541429 −0.270715 0.962660i \(-0.587260\pi\)
−0.270715 + 0.962660i \(0.587260\pi\)
\(548\) 0 0
\(549\) 9.91001 0.422949
\(550\) 0 0
\(551\) 46.1063 1.96420
\(552\) 0 0
\(553\) −9.12408 −0.387995
\(554\) 0 0
\(555\) −6.25960 −0.265705
\(556\) 0 0
\(557\) −30.7740 −1.30394 −0.651968 0.758246i \(-0.726057\pi\)
−0.651968 + 0.758246i \(0.726057\pi\)
\(558\) 0 0
\(559\) 11.1888 0.473237
\(560\) 0 0
\(561\) −17.6383 −0.744689
\(562\) 0 0
\(563\) −35.4352 −1.49342 −0.746708 0.665152i \(-0.768367\pi\)
−0.746708 + 0.665152i \(0.768367\pi\)
\(564\) 0 0
\(565\) 16.4036 0.690106
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 2.15939 0.0905262 0.0452631 0.998975i \(-0.485587\pi\)
0.0452631 + 0.998975i \(0.485587\pi\)
\(570\) 0 0
\(571\) 26.4182 1.10557 0.552784 0.833324i \(-0.313565\pi\)
0.552784 + 0.833324i \(0.313565\pi\)
\(572\) 0 0
\(573\) −2.68524 −0.112178
\(574\) 0 0
\(575\) −3.66122 −0.152683
\(576\) 0 0
\(577\) −12.9885 −0.540719 −0.270360 0.962759i \(-0.587143\pi\)
−0.270360 + 0.962759i \(0.587143\pi\)
\(578\) 0 0
\(579\) −18.7252 −0.778191
\(580\) 0 0
\(581\) −15.8582 −0.657908
\(582\) 0 0
\(583\) −27.9943 −1.15941
\(584\) 0 0
\(585\) −2.71573 −0.112282
\(586\) 0 0
\(587\) −42.1437 −1.73946 −0.869729 0.493529i \(-0.835707\pi\)
−0.869729 + 0.493529i \(0.835707\pi\)
\(588\) 0 0
\(589\) 19.8582 0.818242
\(590\) 0 0
\(591\) 19.3953 0.797817
\(592\) 0 0
\(593\) 10.2777 0.422055 0.211027 0.977480i \(-0.432319\pi\)
0.211027 + 0.977480i \(0.432319\pi\)
\(594\) 0 0
\(595\) 5.56541 0.228159
\(596\) 0 0
\(597\) −1.10005 −0.0450220
\(598\) 0 0
\(599\) 6.73566 0.275212 0.137606 0.990487i \(-0.456059\pi\)
0.137606 + 0.990487i \(0.456059\pi\)
\(600\) 0 0
\(601\) −45.6995 −1.86412 −0.932062 0.362300i \(-0.881992\pi\)
−0.932062 + 0.362300i \(0.881992\pi\)
\(602\) 0 0
\(603\) −6.07114 −0.247236
\(604\) 0 0
\(605\) −2.83149 −0.115117
\(606\) 0 0
\(607\) −46.6518 −1.89354 −0.946769 0.321914i \(-0.895674\pi\)
−0.946769 + 0.321914i \(0.895674\pi\)
\(608\) 0 0
\(609\) 6.52411 0.264370
\(610\) 0 0
\(611\) −11.4000 −0.461196
\(612\) 0 0
\(613\) 40.2204 1.62448 0.812242 0.583320i \(-0.198247\pi\)
0.812242 + 0.583320i \(0.198247\pi\)
\(614\) 0 0
\(615\) −0.748712 −0.0301910
\(616\) 0 0
\(617\) 11.1211 0.447719 0.223860 0.974621i \(-0.428134\pi\)
0.223860 + 0.974621i \(0.428134\pi\)
\(618\) 0 0
\(619\) 45.1974 1.81664 0.908319 0.418278i \(-0.137366\pi\)
0.908319 + 0.418278i \(0.137366\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) −4.66122 −0.186748
\(624\) 0 0
\(625\) 6.71058 0.268423
\(626\) 0 0
\(627\) 25.9152 1.03495
\(628\) 0 0
\(629\) 26.0215 1.03755
\(630\) 0 0
\(631\) 22.9463 0.913478 0.456739 0.889601i \(-0.349017\pi\)
0.456739 + 0.889601i \(0.349017\pi\)
\(632\) 0 0
\(633\) −25.5235 −1.01447
\(634\) 0 0
\(635\) 15.1788 0.602352
\(636\) 0 0
\(637\) 2.34710 0.0929954
\(638\) 0 0
\(639\) 3.65290 0.144507
\(640\) 0 0
\(641\) −12.5876 −0.497179 −0.248590 0.968609i \(-0.579967\pi\)
−0.248590 + 0.968609i \(0.579967\pi\)
\(642\) 0 0
\(643\) −0.547672 −0.0215981 −0.0107990 0.999942i \(-0.503438\pi\)
−0.0107990 + 0.999942i \(0.503438\pi\)
\(644\) 0 0
\(645\) −5.51580 −0.217184
\(646\) 0 0
\(647\) 16.7393 0.658089 0.329045 0.944314i \(-0.393273\pi\)
0.329045 + 0.944314i \(0.393273\pi\)
\(648\) 0 0
\(649\) −37.4560 −1.47027
\(650\) 0 0
\(651\) 2.80996 0.110131
\(652\) 0 0
\(653\) 2.63120 0.102967 0.0514834 0.998674i \(-0.483605\pi\)
0.0514834 + 0.998674i \(0.483605\pi\)
\(654\) 0 0
\(655\) −1.48908 −0.0581833
\(656\) 0 0
\(657\) 11.8183 0.461075
\(658\) 0 0
\(659\) 28.7983 1.12182 0.560912 0.827876i \(-0.310451\pi\)
0.560912 + 0.827876i \(0.310451\pi\)
\(660\) 0 0
\(661\) 2.76143 0.107407 0.0537036 0.998557i \(-0.482897\pi\)
0.0537036 + 0.998557i \(0.482897\pi\)
\(662\) 0 0
\(663\) 11.2894 0.438446
\(664\) 0 0
\(665\) −8.17701 −0.317091
\(666\) 0 0
\(667\) −6.52411 −0.252615
\(668\) 0 0
\(669\) 17.1772 0.664108
\(670\) 0 0
\(671\) −36.3404 −1.40290
\(672\) 0 0
\(673\) 32.8777 1.26734 0.633670 0.773603i \(-0.281548\pi\)
0.633670 + 0.773603i \(0.281548\pi\)
\(674\) 0 0
\(675\) −3.66122 −0.140920
\(676\) 0 0
\(677\) 34.5014 1.32599 0.662997 0.748622i \(-0.269284\pi\)
0.662997 + 0.748622i \(0.269284\pi\)
\(678\) 0 0
\(679\) 4.90419 0.188206
\(680\) 0 0
\(681\) 26.0334 0.997604
\(682\) 0 0
\(683\) −34.9805 −1.33849 −0.669246 0.743041i \(-0.733383\pi\)
−0.669246 + 0.743041i \(0.733383\pi\)
\(684\) 0 0
\(685\) 13.0924 0.500234
\(686\) 0 0
\(687\) −7.01430 −0.267612
\(688\) 0 0
\(689\) 17.9179 0.682617
\(690\) 0 0
\(691\) −34.0895 −1.29683 −0.648413 0.761289i \(-0.724567\pi\)
−0.648413 + 0.761289i \(0.724567\pi\)
\(692\) 0 0
\(693\) 3.66704 0.139299
\(694\) 0 0
\(695\) 10.3408 0.392250
\(696\) 0 0
\(697\) 3.11244 0.117892
\(698\) 0 0
\(699\) 0.671109 0.0253837
\(700\) 0 0
\(701\) −43.6564 −1.64888 −0.824439 0.565950i \(-0.808509\pi\)
−0.824439 + 0.565950i \(0.808509\pi\)
\(702\) 0 0
\(703\) −38.2323 −1.44196
\(704\) 0 0
\(705\) 5.61992 0.211658
\(706\) 0 0
\(707\) 5.80589 0.218353
\(708\) 0 0
\(709\) 11.8247 0.444087 0.222044 0.975037i \(-0.428727\pi\)
0.222044 + 0.975037i \(0.428727\pi\)
\(710\) 0 0
\(711\) 9.12408 0.342180
\(712\) 0 0
\(713\) −2.80996 −0.105234
\(714\) 0 0
\(715\) 9.95868 0.372433
\(716\) 0 0
\(717\) 26.5571 0.991793
\(718\) 0 0
\(719\) −33.8406 −1.26204 −0.631021 0.775766i \(-0.717364\pi\)
−0.631021 + 0.775766i \(0.717364\pi\)
\(720\) 0 0
\(721\) 17.1169 0.637466
\(722\) 0 0
\(723\) −4.97284 −0.184942
\(724\) 0 0
\(725\) 23.8862 0.887110
\(726\) 0 0
\(727\) −43.3730 −1.60862 −0.804308 0.594212i \(-0.797464\pi\)
−0.804308 + 0.594212i \(0.797464\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 22.9295 0.848078
\(732\) 0 0
\(733\) −35.3627 −1.30615 −0.653075 0.757293i \(-0.726521\pi\)
−0.653075 + 0.757293i \(0.726521\pi\)
\(734\) 0 0
\(735\) −1.15706 −0.0426787
\(736\) 0 0
\(737\) 22.2631 0.820072
\(738\) 0 0
\(739\) 20.9639 0.771169 0.385584 0.922673i \(-0.374000\pi\)
0.385584 + 0.922673i \(0.374000\pi\)
\(740\) 0 0
\(741\) −16.5871 −0.609343
\(742\) 0 0
\(743\) 3.66859 0.134587 0.0672937 0.997733i \(-0.478564\pi\)
0.0672937 + 0.997733i \(0.478564\pi\)
\(744\) 0 0
\(745\) −1.94031 −0.0710873
\(746\) 0 0
\(747\) 15.8582 0.580221
\(748\) 0 0
\(749\) −8.34710 −0.304996
\(750\) 0 0
\(751\) −26.4450 −0.964990 −0.482495 0.875899i \(-0.660269\pi\)
−0.482495 + 0.875899i \(0.660269\pi\)
\(752\) 0 0
\(753\) 14.4281 0.525790
\(754\) 0 0
\(755\) −21.9968 −0.800547
\(756\) 0 0
\(757\) 31.6239 1.14939 0.574695 0.818368i \(-0.305121\pi\)
0.574695 + 0.818368i \(0.305121\pi\)
\(758\) 0 0
\(759\) −3.66704 −0.133105
\(760\) 0 0
\(761\) 13.8173 0.500878 0.250439 0.968132i \(-0.419425\pi\)
0.250439 + 0.968132i \(0.419425\pi\)
\(762\) 0 0
\(763\) −16.3771 −0.592891
\(764\) 0 0
\(765\) −5.56541 −0.201218
\(766\) 0 0
\(767\) 23.9738 0.865644
\(768\) 0 0
\(769\) 24.1624 0.871319 0.435659 0.900112i \(-0.356515\pi\)
0.435659 + 0.900112i \(0.356515\pi\)
\(770\) 0 0
\(771\) 5.44889 0.196237
\(772\) 0 0
\(773\) 24.9557 0.897595 0.448798 0.893633i \(-0.351852\pi\)
0.448798 + 0.893633i \(0.351852\pi\)
\(774\) 0 0
\(775\) 10.2879 0.369551
\(776\) 0 0
\(777\) −5.40993 −0.194080
\(778\) 0 0
\(779\) −4.57297 −0.163844
\(780\) 0 0
\(781\) −13.3953 −0.479322
\(782\) 0 0
\(783\) −6.52411 −0.233153
\(784\) 0 0
\(785\) −14.5456 −0.519156
\(786\) 0 0
\(787\) −27.6309 −0.984935 −0.492468 0.870331i \(-0.663905\pi\)
−0.492468 + 0.870331i \(0.663905\pi\)
\(788\) 0 0
\(789\) −15.8128 −0.562951
\(790\) 0 0
\(791\) 14.1770 0.504077
\(792\) 0 0
\(793\) 23.2598 0.825979
\(794\) 0 0
\(795\) −8.83305 −0.313276
\(796\) 0 0
\(797\) −47.8222 −1.69395 −0.846975 0.531632i \(-0.821579\pi\)
−0.846975 + 0.531632i \(0.821579\pi\)
\(798\) 0 0
\(799\) −23.3623 −0.826500
\(800\) 0 0
\(801\) 4.66122 0.164696
\(802\) 0 0
\(803\) −43.3380 −1.52937
\(804\) 0 0
\(805\) 1.15706 0.0407810
\(806\) 0 0
\(807\) 11.9011 0.418937
\(808\) 0 0
\(809\) −53.9350 −1.89625 −0.948127 0.317891i \(-0.897025\pi\)
−0.948127 + 0.317891i \(0.897025\pi\)
\(810\) 0 0
\(811\) −29.8321 −1.04755 −0.523774 0.851857i \(-0.675476\pi\)
−0.523774 + 0.851857i \(0.675476\pi\)
\(812\) 0 0
\(813\) 16.6664 0.584516
\(814\) 0 0
\(815\) −18.1357 −0.635266
\(816\) 0 0
\(817\) −33.6893 −1.17864
\(818\) 0 0
\(819\) −2.34710 −0.0820143
\(820\) 0 0
\(821\) 38.4517 1.34198 0.670988 0.741469i \(-0.265870\pi\)
0.670988 + 0.741469i \(0.265870\pi\)
\(822\) 0 0
\(823\) −22.4705 −0.783274 −0.391637 0.920120i \(-0.628091\pi\)
−0.391637 + 0.920120i \(0.628091\pi\)
\(824\) 0 0
\(825\) 13.4258 0.467427
\(826\) 0 0
\(827\) −15.5282 −0.539968 −0.269984 0.962865i \(-0.587018\pi\)
−0.269984 + 0.962865i \(0.587018\pi\)
\(828\) 0 0
\(829\) −35.6581 −1.23846 −0.619228 0.785211i \(-0.712554\pi\)
−0.619228 + 0.785211i \(0.712554\pi\)
\(830\) 0 0
\(831\) 20.1299 0.698298
\(832\) 0 0
\(833\) 4.80996 0.166655
\(834\) 0 0
\(835\) 5.74040 0.198655
\(836\) 0 0
\(837\) −2.80996 −0.0971264
\(838\) 0 0
\(839\) −16.9411 −0.584873 −0.292436 0.956285i \(-0.594466\pi\)
−0.292436 + 0.956285i \(0.594466\pi\)
\(840\) 0 0
\(841\) 13.5640 0.467725
\(842\) 0 0
\(843\) 19.4022 0.668249
\(844\) 0 0
\(845\) 8.66768 0.298177
\(846\) 0 0
\(847\) −2.44715 −0.0840850
\(848\) 0 0
\(849\) 10.8313 0.371729
\(850\) 0 0
\(851\) 5.40993 0.185450
\(852\) 0 0
\(853\) −2.83149 −0.0969485 −0.0484742 0.998824i \(-0.515436\pi\)
−0.0484742 + 0.998824i \(0.515436\pi\)
\(854\) 0 0
\(855\) 8.17701 0.279648
\(856\) 0 0
\(857\) −3.73125 −0.127457 −0.0637286 0.997967i \(-0.520299\pi\)
−0.0637286 + 0.997967i \(0.520299\pi\)
\(858\) 0 0
\(859\) −3.83133 −0.130723 −0.0653616 0.997862i \(-0.520820\pi\)
−0.0653616 + 0.997862i \(0.520820\pi\)
\(860\) 0 0
\(861\) −0.647082 −0.0220525
\(862\) 0 0
\(863\) 21.3080 0.725333 0.362667 0.931919i \(-0.381866\pi\)
0.362667 + 0.931919i \(0.381866\pi\)
\(864\) 0 0
\(865\) −14.9309 −0.507666
\(866\) 0 0
\(867\) 6.13572 0.208380
\(868\) 0 0
\(869\) −33.4583 −1.13500
\(870\) 0 0
\(871\) −14.2496 −0.482828
\(872\) 0 0
\(873\) −4.90419 −0.165982
\(874\) 0 0
\(875\) −10.0215 −0.338790
\(876\) 0 0
\(877\) −34.4456 −1.16314 −0.581572 0.813495i \(-0.697562\pi\)
−0.581572 + 0.813495i \(0.697562\pi\)
\(878\) 0 0
\(879\) 8.89430 0.299997
\(880\) 0 0
\(881\) 5.70600 0.192240 0.0961201 0.995370i \(-0.469357\pi\)
0.0961201 + 0.995370i \(0.469357\pi\)
\(882\) 0 0
\(883\) −5.89756 −0.198469 −0.0992344 0.995064i \(-0.531639\pi\)
−0.0992344 + 0.995064i \(0.531639\pi\)
\(884\) 0 0
\(885\) −11.8185 −0.397273
\(886\) 0 0
\(887\) 41.8153 1.40402 0.702010 0.712167i \(-0.252286\pi\)
0.702010 + 0.712167i \(0.252286\pi\)
\(888\) 0 0
\(889\) 13.1184 0.439978
\(890\) 0 0
\(891\) −3.66704 −0.122850
\(892\) 0 0
\(893\) 34.3253 1.14865
\(894\) 0 0
\(895\) −0.304031 −0.0101626
\(896\) 0 0
\(897\) 2.34710 0.0783673
\(898\) 0 0
\(899\) 18.3325 0.611423
\(900\) 0 0
\(901\) 36.7195 1.22330
\(902\) 0 0
\(903\) −4.76709 −0.158639
\(904\) 0 0
\(905\) 14.0514 0.467083
\(906\) 0 0
\(907\) 39.9593 1.32683 0.663414 0.748253i \(-0.269107\pi\)
0.663414 + 0.748253i \(0.269107\pi\)
\(908\) 0 0
\(909\) −5.80589 −0.192569
\(910\) 0 0
\(911\) 25.1107 0.831954 0.415977 0.909375i \(-0.363440\pi\)
0.415977 + 0.909375i \(0.363440\pi\)
\(912\) 0 0
\(913\) −58.1525 −1.92457
\(914\) 0 0
\(915\) −11.4665 −0.379070
\(916\) 0 0
\(917\) −1.28696 −0.0424990
\(918\) 0 0
\(919\) −37.6701 −1.24262 −0.621310 0.783565i \(-0.713399\pi\)
−0.621310 + 0.783565i \(0.713399\pi\)
\(920\) 0 0
\(921\) −19.7442 −0.650592
\(922\) 0 0
\(923\) 8.57372 0.282207
\(924\) 0 0
\(925\) −19.8069 −0.651247
\(926\) 0 0
\(927\) −17.1169 −0.562192
\(928\) 0 0
\(929\) −15.5425 −0.509933 −0.254967 0.966950i \(-0.582064\pi\)
−0.254967 + 0.966950i \(0.582064\pi\)
\(930\) 0 0
\(931\) −7.06707 −0.231614
\(932\) 0 0
\(933\) 28.5898 0.935988
\(934\) 0 0
\(935\) 20.4085 0.667431
\(936\) 0 0
\(937\) 18.2576 0.596449 0.298224 0.954496i \(-0.403606\pi\)
0.298224 + 0.954496i \(0.403606\pi\)
\(938\) 0 0
\(939\) 12.6570 0.413045
\(940\) 0 0
\(941\) −0.0776022 −0.00252976 −0.00126488 0.999999i \(-0.500403\pi\)
−0.00126488 + 0.999999i \(0.500403\pi\)
\(942\) 0 0
\(943\) 0.647082 0.0210719
\(944\) 0 0
\(945\) 1.15706 0.0376391
\(946\) 0 0
\(947\) −10.0364 −0.326140 −0.163070 0.986615i \(-0.552140\pi\)
−0.163070 + 0.986615i \(0.552140\pi\)
\(948\) 0 0
\(949\) 27.7387 0.900435
\(950\) 0 0
\(951\) 4.83130 0.156666
\(952\) 0 0
\(953\) 51.2265 1.65939 0.829695 0.558218i \(-0.188515\pi\)
0.829695 + 0.558218i \(0.188515\pi\)
\(954\) 0 0
\(955\) 3.10698 0.100540
\(956\) 0 0
\(957\) 23.9241 0.773358
\(958\) 0 0
\(959\) 11.3152 0.365388
\(960\) 0 0
\(961\) −23.1041 −0.745294
\(962\) 0 0
\(963\) 8.34710 0.268981
\(964\) 0 0
\(965\) 21.6661 0.697456
\(966\) 0 0
\(967\) −12.2315 −0.393339 −0.196670 0.980470i \(-0.563013\pi\)
−0.196670 + 0.980470i \(0.563013\pi\)
\(968\) 0 0
\(969\) −33.9923 −1.09199
\(970\) 0 0
\(971\) −33.5753 −1.07748 −0.538741 0.842471i \(-0.681100\pi\)
−0.538741 + 0.842471i \(0.681100\pi\)
\(972\) 0 0
\(973\) 8.93717 0.286513
\(974\) 0 0
\(975\) −8.59323 −0.275204
\(976\) 0 0
\(977\) 23.7149 0.758707 0.379353 0.925252i \(-0.376146\pi\)
0.379353 + 0.925252i \(0.376146\pi\)
\(978\) 0 0
\(979\) −17.0928 −0.546290
\(980\) 0 0
\(981\) 16.3771 0.522881
\(982\) 0 0
\(983\) 1.47810 0.0471441 0.0235721 0.999722i \(-0.492496\pi\)
0.0235721 + 0.999722i \(0.492496\pi\)
\(984\) 0 0
\(985\) −22.4415 −0.715046
\(986\) 0 0
\(987\) 4.85708 0.154602
\(988\) 0 0
\(989\) 4.76709 0.151584
\(990\) 0 0
\(991\) −20.5203 −0.651850 −0.325925 0.945396i \(-0.605676\pi\)
−0.325925 + 0.945396i \(0.605676\pi\)
\(992\) 0 0
\(993\) 6.84544 0.217233
\(994\) 0 0
\(995\) 1.27282 0.0403512
\(996\) 0 0
\(997\) −22.7211 −0.719583 −0.359791 0.933033i \(-0.617152\pi\)
−0.359791 + 0.933033i \(0.617152\pi\)
\(998\) 0 0
\(999\) 5.40993 0.171163
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.cd.1.1 4
4.3 odd 2 483.2.a.i.1.3 4
12.11 even 2 1449.2.a.p.1.2 4
28.27 even 2 3381.2.a.w.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.i.1.3 4 4.3 odd 2
1449.2.a.p.1.2 4 12.11 even 2
3381.2.a.w.1.3 4 28.27 even 2
7728.2.a.cd.1.1 4 1.1 even 1 trivial