Properties

Label 7728.2.a.x.1.2
Level $7728$
Weight $2$
Character 7728.1
Self dual yes
Analytic conductor $61.708$
Analytic rank $1$
Dimension $2$
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: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
Defining polynomial: \(x^{2} - x - 3\)
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 1.2
Root \(-1.30278\) of defining polynomial
Character \(\chi\) \(=\) 7728.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -0.697224 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -0.697224 q^{5} +1.00000 q^{7} +1.00000 q^{9} +5.00000 q^{11} +2.30278 q^{13} +0.697224 q^{15} -5.60555 q^{17} +1.60555 q^{19} -1.00000 q^{21} -1.00000 q^{23} -4.51388 q^{25} -1.00000 q^{27} +6.21110 q^{29} -3.00000 q^{31} -5.00000 q^{33} -0.697224 q^{35} -9.00000 q^{37} -2.30278 q^{39} -12.2111 q^{41} -5.51388 q^{43} -0.697224 q^{45} +8.60555 q^{47} +1.00000 q^{49} +5.60555 q^{51} -12.5139 q^{53} -3.48612 q^{55} -1.60555 q^{57} -3.90833 q^{59} -1.09167 q^{61} +1.00000 q^{63} -1.60555 q^{65} +11.9083 q^{67} +1.00000 q^{69} -0.908327 q^{71} -2.21110 q^{73} +4.51388 q^{75} +5.00000 q^{77} +1.00000 q^{79} +1.00000 q^{81} -5.60555 q^{83} +3.90833 q^{85} -6.21110 q^{87} +10.9083 q^{89} +2.30278 q^{91} +3.00000 q^{93} -1.11943 q^{95} -17.6056 q^{97} +5.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 5q^{5} + 2q^{7} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 5q^{5} + 2q^{7} + 2q^{9} + 10q^{11} + q^{13} + 5q^{15} - 4q^{17} - 4q^{19} - 2q^{21} - 2q^{23} + 9q^{25} - 2q^{27} - 2q^{29} - 6q^{31} - 10q^{33} - 5q^{35} - 18q^{37} - q^{39} - 10q^{41} + 7q^{43} - 5q^{45} + 10q^{47} + 2q^{49} + 4q^{51} - 7q^{53} - 25q^{55} + 4q^{57} + 3q^{59} - 13q^{61} + 2q^{63} + 4q^{65} + 13q^{67} + 2q^{69} + 9q^{71} + 10q^{73} - 9q^{75} + 10q^{77} + 2q^{79} + 2q^{81} - 4q^{83} - 3q^{85} + 2q^{87} + 11q^{89} + q^{91} + 6q^{93} + 23q^{95} - 28q^{97} + 10q^{99} + O(q^{100}) \)

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
\(5\) −0.697224 −0.311808 −0.155904 0.987772i \(-0.549829\pi\)
−0.155904 + 0.987772i \(0.549829\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 5.00000 1.50756 0.753778 0.657129i \(-0.228229\pi\)
0.753778 + 0.657129i \(0.228229\pi\)
\(12\) 0 0
\(13\) 2.30278 0.638675 0.319338 0.947641i \(-0.396540\pi\)
0.319338 + 0.947641i \(0.396540\pi\)
\(14\) 0 0
\(15\) 0.697224 0.180023
\(16\) 0 0
\(17\) −5.60555 −1.35955 −0.679773 0.733423i \(-0.737922\pi\)
−0.679773 + 0.733423i \(0.737922\pi\)
\(18\) 0 0
\(19\) 1.60555 0.368339 0.184169 0.982895i \(-0.441041\pi\)
0.184169 + 0.982895i \(0.441041\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.51388 −0.902776
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 6.21110 1.15337 0.576686 0.816966i \(-0.304345\pi\)
0.576686 + 0.816966i \(0.304345\pi\)
\(30\) 0 0
\(31\) −3.00000 −0.538816 −0.269408 0.963026i \(-0.586828\pi\)
−0.269408 + 0.963026i \(0.586828\pi\)
\(32\) 0 0
\(33\) −5.00000 −0.870388
\(34\) 0 0
\(35\) −0.697224 −0.117852
\(36\) 0 0
\(37\) −9.00000 −1.47959 −0.739795 0.672832i \(-0.765078\pi\)
−0.739795 + 0.672832i \(0.765078\pi\)
\(38\) 0 0
\(39\) −2.30278 −0.368739
\(40\) 0 0
\(41\) −12.2111 −1.90705 −0.953527 0.301308i \(-0.902577\pi\)
−0.953527 + 0.301308i \(0.902577\pi\)
\(42\) 0 0
\(43\) −5.51388 −0.840859 −0.420429 0.907325i \(-0.638121\pi\)
−0.420429 + 0.907325i \(0.638121\pi\)
\(44\) 0 0
\(45\) −0.697224 −0.103936
\(46\) 0 0
\(47\) 8.60555 1.25525 0.627624 0.778516i \(-0.284027\pi\)
0.627624 + 0.778516i \(0.284027\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 5.60555 0.784934
\(52\) 0 0
\(53\) −12.5139 −1.71891 −0.859457 0.511209i \(-0.829198\pi\)
−0.859457 + 0.511209i \(0.829198\pi\)
\(54\) 0 0
\(55\) −3.48612 −0.470069
\(56\) 0 0
\(57\) −1.60555 −0.212660
\(58\) 0 0
\(59\) −3.90833 −0.508821 −0.254410 0.967096i \(-0.581881\pi\)
−0.254410 + 0.967096i \(0.581881\pi\)
\(60\) 0 0
\(61\) −1.09167 −0.139774 −0.0698872 0.997555i \(-0.522264\pi\)
−0.0698872 + 0.997555i \(0.522264\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) 0 0
\(65\) −1.60555 −0.199144
\(66\) 0 0
\(67\) 11.9083 1.45483 0.727417 0.686196i \(-0.240721\pi\)
0.727417 + 0.686196i \(0.240721\pi\)
\(68\) 0 0
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −0.908327 −0.107799 −0.0538993 0.998546i \(-0.517165\pi\)
−0.0538993 + 0.998546i \(0.517165\pi\)
\(72\) 0 0
\(73\) −2.21110 −0.258790 −0.129395 0.991593i \(-0.541304\pi\)
−0.129395 + 0.991593i \(0.541304\pi\)
\(74\) 0 0
\(75\) 4.51388 0.521218
\(76\) 0 0
\(77\) 5.00000 0.569803
\(78\) 0 0
\(79\) 1.00000 0.112509 0.0562544 0.998416i \(-0.482084\pi\)
0.0562544 + 0.998416i \(0.482084\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.60555 −0.615289 −0.307645 0.951501i \(-0.599541\pi\)
−0.307645 + 0.951501i \(0.599541\pi\)
\(84\) 0 0
\(85\) 3.90833 0.423918
\(86\) 0 0
\(87\) −6.21110 −0.665900
\(88\) 0 0
\(89\) 10.9083 1.15628 0.578140 0.815937i \(-0.303779\pi\)
0.578140 + 0.815937i \(0.303779\pi\)
\(90\) 0 0
\(91\) 2.30278 0.241396
\(92\) 0 0
\(93\) 3.00000 0.311086
\(94\) 0 0
\(95\) −1.11943 −0.114851
\(96\) 0 0
\(97\) −17.6056 −1.78757 −0.893786 0.448493i \(-0.851961\pi\)
−0.893786 + 0.448493i \(0.851961\pi\)
\(98\) 0 0
\(99\) 5.00000 0.502519
\(100\) 0 0
\(101\) 9.30278 0.925661 0.462830 0.886447i \(-0.346834\pi\)
0.462830 + 0.886447i \(0.346834\pi\)
\(102\) 0 0
\(103\) 4.00000 0.394132 0.197066 0.980390i \(-0.436859\pi\)
0.197066 + 0.980390i \(0.436859\pi\)
\(104\) 0 0
\(105\) 0.697224 0.0680421
\(106\) 0 0
\(107\) −7.51388 −0.726394 −0.363197 0.931712i \(-0.618315\pi\)
−0.363197 + 0.931712i \(0.618315\pi\)
\(108\) 0 0
\(109\) −3.90833 −0.374350 −0.187175 0.982327i \(-0.559933\pi\)
−0.187175 + 0.982327i \(0.559933\pi\)
\(110\) 0 0
\(111\) 9.00000 0.854242
\(112\) 0 0
\(113\) 5.30278 0.498843 0.249422 0.968395i \(-0.419760\pi\)
0.249422 + 0.968395i \(0.419760\pi\)
\(114\) 0 0
\(115\) 0.697224 0.0650165
\(116\) 0 0
\(117\) 2.30278 0.212892
\(118\) 0 0
\(119\) −5.60555 −0.513860
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) 12.2111 1.10104
\(124\) 0 0
\(125\) 6.63331 0.593301
\(126\) 0 0
\(127\) −1.69722 −0.150604 −0.0753022 0.997161i \(-0.523992\pi\)
−0.0753022 + 0.997161i \(0.523992\pi\)
\(128\) 0 0
\(129\) 5.51388 0.485470
\(130\) 0 0
\(131\) −10.3944 −0.908167 −0.454084 0.890959i \(-0.650033\pi\)
−0.454084 + 0.890959i \(0.650033\pi\)
\(132\) 0 0
\(133\) 1.60555 0.139219
\(134\) 0 0
\(135\) 0.697224 0.0600075
\(136\) 0 0
\(137\) 16.8167 1.43674 0.718372 0.695659i \(-0.244888\pi\)
0.718372 + 0.695659i \(0.244888\pi\)
\(138\) 0 0
\(139\) −15.9083 −1.34933 −0.674663 0.738126i \(-0.735711\pi\)
−0.674663 + 0.738126i \(0.735711\pi\)
\(140\) 0 0
\(141\) −8.60555 −0.724718
\(142\) 0 0
\(143\) 11.5139 0.962839
\(144\) 0 0
\(145\) −4.33053 −0.359631
\(146\) 0 0
\(147\) −1.00000 −0.0824786
\(148\) 0 0
\(149\) 8.60555 0.704994 0.352497 0.935813i \(-0.385333\pi\)
0.352497 + 0.935813i \(0.385333\pi\)
\(150\) 0 0
\(151\) −16.6056 −1.35134 −0.675670 0.737204i \(-0.736146\pi\)
−0.675670 + 0.737204i \(0.736146\pi\)
\(152\) 0 0
\(153\) −5.60555 −0.453182
\(154\) 0 0
\(155\) 2.09167 0.168007
\(156\) 0 0
\(157\) −3.81665 −0.304602 −0.152301 0.988334i \(-0.548668\pi\)
−0.152301 + 0.988334i \(0.548668\pi\)
\(158\) 0 0
\(159\) 12.5139 0.992415
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) 13.7250 1.07502 0.537512 0.843256i \(-0.319364\pi\)
0.537512 + 0.843256i \(0.319364\pi\)
\(164\) 0 0
\(165\) 3.48612 0.271394
\(166\) 0 0
\(167\) −2.81665 −0.217959 −0.108980 0.994044i \(-0.534758\pi\)
−0.108980 + 0.994044i \(0.534758\pi\)
\(168\) 0 0
\(169\) −7.69722 −0.592094
\(170\) 0 0
\(171\) 1.60555 0.122780
\(172\) 0 0
\(173\) −18.2111 −1.38456 −0.692282 0.721627i \(-0.743395\pi\)
−0.692282 + 0.721627i \(0.743395\pi\)
\(174\) 0 0
\(175\) −4.51388 −0.341217
\(176\) 0 0
\(177\) 3.90833 0.293768
\(178\) 0 0
\(179\) 2.69722 0.201600 0.100800 0.994907i \(-0.467860\pi\)
0.100800 + 0.994907i \(0.467860\pi\)
\(180\) 0 0
\(181\) −6.81665 −0.506678 −0.253339 0.967378i \(-0.581529\pi\)
−0.253339 + 0.967378i \(0.581529\pi\)
\(182\) 0 0
\(183\) 1.09167 0.0806988
\(184\) 0 0
\(185\) 6.27502 0.461349
\(186\) 0 0
\(187\) −28.0278 −2.04959
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 4.60555 0.333246 0.166623 0.986021i \(-0.446714\pi\)
0.166623 + 0.986021i \(0.446714\pi\)
\(192\) 0 0
\(193\) 9.02776 0.649832 0.324916 0.945743i \(-0.394664\pi\)
0.324916 + 0.945743i \(0.394664\pi\)
\(194\) 0 0
\(195\) 1.60555 0.114976
\(196\) 0 0
\(197\) −7.09167 −0.505261 −0.252630 0.967563i \(-0.581296\pi\)
−0.252630 + 0.967563i \(0.581296\pi\)
\(198\) 0 0
\(199\) −12.5139 −0.887085 −0.443543 0.896253i \(-0.646279\pi\)
−0.443543 + 0.896253i \(0.646279\pi\)
\(200\) 0 0
\(201\) −11.9083 −0.839949
\(202\) 0 0
\(203\) 6.21110 0.435934
\(204\) 0 0
\(205\) 8.51388 0.594635
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 8.02776 0.555292
\(210\) 0 0
\(211\) 27.4222 1.88782 0.943911 0.330199i \(-0.107116\pi\)
0.943911 + 0.330199i \(0.107116\pi\)
\(212\) 0 0
\(213\) 0.908327 0.0622375
\(214\) 0 0
\(215\) 3.84441 0.262187
\(216\) 0 0
\(217\) −3.00000 −0.203653
\(218\) 0 0
\(219\) 2.21110 0.149412
\(220\) 0 0
\(221\) −12.9083 −0.868308
\(222\) 0 0
\(223\) −19.9083 −1.33316 −0.666580 0.745433i \(-0.732243\pi\)
−0.666580 + 0.745433i \(0.732243\pi\)
\(224\) 0 0
\(225\) −4.51388 −0.300925
\(226\) 0 0
\(227\) 20.3305 1.34938 0.674692 0.738099i \(-0.264276\pi\)
0.674692 + 0.738099i \(0.264276\pi\)
\(228\) 0 0
\(229\) 2.51388 0.166122 0.0830609 0.996544i \(-0.473530\pi\)
0.0830609 + 0.996544i \(0.473530\pi\)
\(230\) 0 0
\(231\) −5.00000 −0.328976
\(232\) 0 0
\(233\) −14.3305 −0.938824 −0.469412 0.882979i \(-0.655534\pi\)
−0.469412 + 0.882979i \(0.655534\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) −1.00000 −0.0649570
\(238\) 0 0
\(239\) 14.0917 0.911515 0.455757 0.890104i \(-0.349369\pi\)
0.455757 + 0.890104i \(0.349369\pi\)
\(240\) 0 0
\(241\) 12.0278 0.774776 0.387388 0.921917i \(-0.373377\pi\)
0.387388 + 0.921917i \(0.373377\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) −0.697224 −0.0445440
\(246\) 0 0
\(247\) 3.69722 0.235249
\(248\) 0 0
\(249\) 5.60555 0.355237
\(250\) 0 0
\(251\) −23.8167 −1.50329 −0.751647 0.659566i \(-0.770740\pi\)
−0.751647 + 0.659566i \(0.770740\pi\)
\(252\) 0 0
\(253\) −5.00000 −0.314347
\(254\) 0 0
\(255\) −3.90833 −0.244749
\(256\) 0 0
\(257\) 27.0278 1.68595 0.842973 0.537957i \(-0.180804\pi\)
0.842973 + 0.537957i \(0.180804\pi\)
\(258\) 0 0
\(259\) −9.00000 −0.559233
\(260\) 0 0
\(261\) 6.21110 0.384458
\(262\) 0 0
\(263\) −10.3944 −0.640949 −0.320475 0.947257i \(-0.603842\pi\)
−0.320475 + 0.947257i \(0.603842\pi\)
\(264\) 0 0
\(265\) 8.72498 0.535971
\(266\) 0 0
\(267\) −10.9083 −0.667579
\(268\) 0 0
\(269\) 0.908327 0.0553817 0.0276908 0.999617i \(-0.491185\pi\)
0.0276908 + 0.999617i \(0.491185\pi\)
\(270\) 0 0
\(271\) −3.60555 −0.219022 −0.109511 0.993986i \(-0.534928\pi\)
−0.109511 + 0.993986i \(0.534928\pi\)
\(272\) 0 0
\(273\) −2.30278 −0.139370
\(274\) 0 0
\(275\) −22.5694 −1.36099
\(276\) 0 0
\(277\) −12.3028 −0.739202 −0.369601 0.929191i \(-0.620506\pi\)
−0.369601 + 0.929191i \(0.620506\pi\)
\(278\) 0 0
\(279\) −3.00000 −0.179605
\(280\) 0 0
\(281\) 27.8167 1.65940 0.829701 0.558208i \(-0.188511\pi\)
0.829701 + 0.558208i \(0.188511\pi\)
\(282\) 0 0
\(283\) −17.3028 −1.02854 −0.514272 0.857627i \(-0.671938\pi\)
−0.514272 + 0.857627i \(0.671938\pi\)
\(284\) 0 0
\(285\) 1.11943 0.0663093
\(286\) 0 0
\(287\) −12.2111 −0.720799
\(288\) 0 0
\(289\) 14.4222 0.848365
\(290\) 0 0
\(291\) 17.6056 1.03206
\(292\) 0 0
\(293\) −24.8444 −1.45143 −0.725713 0.687998i \(-0.758490\pi\)
−0.725713 + 0.687998i \(0.758490\pi\)
\(294\) 0 0
\(295\) 2.72498 0.158655
\(296\) 0 0
\(297\) −5.00000 −0.290129
\(298\) 0 0
\(299\) −2.30278 −0.133173
\(300\) 0 0
\(301\) −5.51388 −0.317815
\(302\) 0 0
\(303\) −9.30278 −0.534430
\(304\) 0 0
\(305\) 0.761141 0.0435828
\(306\) 0 0
\(307\) −18.2111 −1.03936 −0.519681 0.854360i \(-0.673949\pi\)
−0.519681 + 0.854360i \(0.673949\pi\)
\(308\) 0 0
\(309\) −4.00000 −0.227552
\(310\) 0 0
\(311\) 9.51388 0.539483 0.269741 0.962933i \(-0.413062\pi\)
0.269741 + 0.962933i \(0.413062\pi\)
\(312\) 0 0
\(313\) 1.21110 0.0684556 0.0342278 0.999414i \(-0.489103\pi\)
0.0342278 + 0.999414i \(0.489103\pi\)
\(314\) 0 0
\(315\) −0.697224 −0.0392841
\(316\) 0 0
\(317\) −30.5139 −1.71383 −0.856915 0.515458i \(-0.827622\pi\)
−0.856915 + 0.515458i \(0.827622\pi\)
\(318\) 0 0
\(319\) 31.0555 1.73877
\(320\) 0 0
\(321\) 7.51388 0.419384
\(322\) 0 0
\(323\) −9.00000 −0.500773
\(324\) 0 0
\(325\) −10.3944 −0.576580
\(326\) 0 0
\(327\) 3.90833 0.216131
\(328\) 0 0
\(329\) 8.60555 0.474439
\(330\) 0 0
\(331\) −17.6333 −0.969214 −0.484607 0.874732i \(-0.661037\pi\)
−0.484607 + 0.874732i \(0.661037\pi\)
\(332\) 0 0
\(333\) −9.00000 −0.493197
\(334\) 0 0
\(335\) −8.30278 −0.453629
\(336\) 0 0
\(337\) −15.6972 −0.855082 −0.427541 0.903996i \(-0.640620\pi\)
−0.427541 + 0.903996i \(0.640620\pi\)
\(338\) 0 0
\(339\) −5.30278 −0.288007
\(340\) 0 0
\(341\) −15.0000 −0.812296
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) −0.697224 −0.0375373
\(346\) 0 0
\(347\) −14.6333 −0.785557 −0.392779 0.919633i \(-0.628486\pi\)
−0.392779 + 0.919633i \(0.628486\pi\)
\(348\) 0 0
\(349\) −10.0917 −0.540195 −0.270097 0.962833i \(-0.587056\pi\)
−0.270097 + 0.962833i \(0.587056\pi\)
\(350\) 0 0
\(351\) −2.30278 −0.122913
\(352\) 0 0
\(353\) −11.0000 −0.585471 −0.292735 0.956193i \(-0.594566\pi\)
−0.292735 + 0.956193i \(0.594566\pi\)
\(354\) 0 0
\(355\) 0.633308 0.0336125
\(356\) 0 0
\(357\) 5.60555 0.296677
\(358\) 0 0
\(359\) 35.5416 1.87582 0.937908 0.346884i \(-0.112760\pi\)
0.937908 + 0.346884i \(0.112760\pi\)
\(360\) 0 0
\(361\) −16.4222 −0.864327
\(362\) 0 0
\(363\) −14.0000 −0.734809
\(364\) 0 0
\(365\) 1.54163 0.0806928
\(366\) 0 0
\(367\) −21.5139 −1.12302 −0.561508 0.827472i \(-0.689778\pi\)
−0.561508 + 0.827472i \(0.689778\pi\)
\(368\) 0 0
\(369\) −12.2111 −0.635685
\(370\) 0 0
\(371\) −12.5139 −0.649688
\(372\) 0 0
\(373\) 16.2111 0.839379 0.419690 0.907668i \(-0.362139\pi\)
0.419690 + 0.907668i \(0.362139\pi\)
\(374\) 0 0
\(375\) −6.63331 −0.342543
\(376\) 0 0
\(377\) 14.3028 0.736630
\(378\) 0 0
\(379\) 22.4222 1.15175 0.575876 0.817537i \(-0.304661\pi\)
0.575876 + 0.817537i \(0.304661\pi\)
\(380\) 0 0
\(381\) 1.69722 0.0869514
\(382\) 0 0
\(383\) −4.81665 −0.246120 −0.123060 0.992399i \(-0.539271\pi\)
−0.123060 + 0.992399i \(0.539271\pi\)
\(384\) 0 0
\(385\) −3.48612 −0.177669
\(386\) 0 0
\(387\) −5.51388 −0.280286
\(388\) 0 0
\(389\) −36.6333 −1.85738 −0.928691 0.370854i \(-0.879065\pi\)
−0.928691 + 0.370854i \(0.879065\pi\)
\(390\) 0 0
\(391\) 5.60555 0.283485
\(392\) 0 0
\(393\) 10.3944 0.524331
\(394\) 0 0
\(395\) −0.697224 −0.0350812
\(396\) 0 0
\(397\) −27.3944 −1.37489 −0.687444 0.726237i \(-0.741267\pi\)
−0.687444 + 0.726237i \(0.741267\pi\)
\(398\) 0 0
\(399\) −1.60555 −0.0803781
\(400\) 0 0
\(401\) −13.4222 −0.670273 −0.335136 0.942170i \(-0.608782\pi\)
−0.335136 + 0.942170i \(0.608782\pi\)
\(402\) 0 0
\(403\) −6.90833 −0.344128
\(404\) 0 0
\(405\) −0.697224 −0.0346454
\(406\) 0 0
\(407\) −45.0000 −2.23057
\(408\) 0 0
\(409\) 24.0278 1.18810 0.594048 0.804430i \(-0.297529\pi\)
0.594048 + 0.804430i \(0.297529\pi\)
\(410\) 0 0
\(411\) −16.8167 −0.829504
\(412\) 0 0
\(413\) −3.90833 −0.192316
\(414\) 0 0
\(415\) 3.90833 0.191852
\(416\) 0 0
\(417\) 15.9083 0.779034
\(418\) 0 0
\(419\) −15.1194 −0.738632 −0.369316 0.929304i \(-0.620408\pi\)
−0.369316 + 0.929304i \(0.620408\pi\)
\(420\) 0 0
\(421\) 9.69722 0.472614 0.236307 0.971678i \(-0.424063\pi\)
0.236307 + 0.971678i \(0.424063\pi\)
\(422\) 0 0
\(423\) 8.60555 0.418416
\(424\) 0 0
\(425\) 25.3028 1.22736
\(426\) 0 0
\(427\) −1.09167 −0.0528298
\(428\) 0 0
\(429\) −11.5139 −0.555895
\(430\) 0 0
\(431\) 8.72498 0.420268 0.210134 0.977673i \(-0.432610\pi\)
0.210134 + 0.977673i \(0.432610\pi\)
\(432\) 0 0
\(433\) −21.0278 −1.01053 −0.505265 0.862964i \(-0.668605\pi\)
−0.505265 + 0.862964i \(0.668605\pi\)
\(434\) 0 0
\(435\) 4.33053 0.207633
\(436\) 0 0
\(437\) −1.60555 −0.0768039
\(438\) 0 0
\(439\) −20.2111 −0.964623 −0.482312 0.876000i \(-0.660203\pi\)
−0.482312 + 0.876000i \(0.660203\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 32.8444 1.56049 0.780243 0.625477i \(-0.215096\pi\)
0.780243 + 0.625477i \(0.215096\pi\)
\(444\) 0 0
\(445\) −7.60555 −0.360538
\(446\) 0 0
\(447\) −8.60555 −0.407029
\(448\) 0 0
\(449\) −35.7250 −1.68597 −0.842983 0.537940i \(-0.819203\pi\)
−0.842983 + 0.537940i \(0.819203\pi\)
\(450\) 0 0
\(451\) −61.0555 −2.87499
\(452\) 0 0
\(453\) 16.6056 0.780197
\(454\) 0 0
\(455\) −1.60555 −0.0752694
\(456\) 0 0
\(457\) −10.8806 −0.508972 −0.254486 0.967077i \(-0.581906\pi\)
−0.254486 + 0.967077i \(0.581906\pi\)
\(458\) 0 0
\(459\) 5.60555 0.261645
\(460\) 0 0
\(461\) 7.72498 0.359788 0.179894 0.983686i \(-0.442424\pi\)
0.179894 + 0.983686i \(0.442424\pi\)
\(462\) 0 0
\(463\) 18.8167 0.874484 0.437242 0.899344i \(-0.355955\pi\)
0.437242 + 0.899344i \(0.355955\pi\)
\(464\) 0 0
\(465\) −2.09167 −0.0969990
\(466\) 0 0
\(467\) −23.6056 −1.09233 −0.546167 0.837676i \(-0.683914\pi\)
−0.546167 + 0.837676i \(0.683914\pi\)
\(468\) 0 0
\(469\) 11.9083 0.549875
\(470\) 0 0
\(471\) 3.81665 0.175862
\(472\) 0 0
\(473\) −27.5694 −1.26764
\(474\) 0 0
\(475\) −7.24726 −0.332527
\(476\) 0 0
\(477\) −12.5139 −0.572971
\(478\) 0 0
\(479\) 18.2111 0.832087 0.416043 0.909345i \(-0.363416\pi\)
0.416043 + 0.909345i \(0.363416\pi\)
\(480\) 0 0
\(481\) −20.7250 −0.944978
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) 12.2750 0.557380
\(486\) 0 0
\(487\) 0.816654 0.0370061 0.0185031 0.999829i \(-0.494110\pi\)
0.0185031 + 0.999829i \(0.494110\pi\)
\(488\) 0 0
\(489\) −13.7250 −0.620665
\(490\) 0 0
\(491\) 17.9361 0.809444 0.404722 0.914440i \(-0.367368\pi\)
0.404722 + 0.914440i \(0.367368\pi\)
\(492\) 0 0
\(493\) −34.8167 −1.56806
\(494\) 0 0
\(495\) −3.48612 −0.156690
\(496\) 0 0
\(497\) −0.908327 −0.0407440
\(498\) 0 0
\(499\) −31.9083 −1.42841 −0.714206 0.699935i \(-0.753212\pi\)
−0.714206 + 0.699935i \(0.753212\pi\)
\(500\) 0 0
\(501\) 2.81665 0.125839
\(502\) 0 0
\(503\) −17.7250 −0.790318 −0.395159 0.918613i \(-0.629310\pi\)
−0.395159 + 0.918613i \(0.629310\pi\)
\(504\) 0 0
\(505\) −6.48612 −0.288629
\(506\) 0 0
\(507\) 7.69722 0.341846
\(508\) 0 0
\(509\) −33.4500 −1.48264 −0.741322 0.671150i \(-0.765801\pi\)
−0.741322 + 0.671150i \(0.765801\pi\)
\(510\) 0 0
\(511\) −2.21110 −0.0978134
\(512\) 0 0
\(513\) −1.60555 −0.0708868
\(514\) 0 0
\(515\) −2.78890 −0.122894
\(516\) 0 0
\(517\) 43.0278 1.89236
\(518\) 0 0
\(519\) 18.2111 0.799379
\(520\) 0 0
\(521\) 21.6333 0.947772 0.473886 0.880586i \(-0.342851\pi\)
0.473886 + 0.880586i \(0.342851\pi\)
\(522\) 0 0
\(523\) 0.422205 0.0184617 0.00923087 0.999957i \(-0.497062\pi\)
0.00923087 + 0.999957i \(0.497062\pi\)
\(524\) 0 0
\(525\) 4.51388 0.197002
\(526\) 0 0
\(527\) 16.8167 0.732545
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −3.90833 −0.169607
\(532\) 0 0
\(533\) −28.1194 −1.21799
\(534\) 0 0
\(535\) 5.23886 0.226496
\(536\) 0 0
\(537\) −2.69722 −0.116394
\(538\) 0 0
\(539\) 5.00000 0.215365
\(540\) 0 0
\(541\) −5.81665 −0.250077 −0.125039 0.992152i \(-0.539906\pi\)
−0.125039 + 0.992152i \(0.539906\pi\)
\(542\) 0 0
\(543\) 6.81665 0.292531
\(544\) 0 0
\(545\) 2.72498 0.116725
\(546\) 0 0
\(547\) 7.72498 0.330296 0.165148 0.986269i \(-0.447190\pi\)
0.165148 + 0.986269i \(0.447190\pi\)
\(548\) 0 0
\(549\) −1.09167 −0.0465915
\(550\) 0 0
\(551\) 9.97224 0.424832
\(552\) 0 0
\(553\) 1.00000 0.0425243
\(554\) 0 0
\(555\) −6.27502 −0.266360
\(556\) 0 0
\(557\) 5.02776 0.213033 0.106516 0.994311i \(-0.466030\pi\)
0.106516 + 0.994311i \(0.466030\pi\)
\(558\) 0 0
\(559\) −12.6972 −0.537035
\(560\) 0 0
\(561\) 28.0278 1.18333
\(562\) 0 0
\(563\) 43.3305 1.82616 0.913082 0.407776i \(-0.133695\pi\)
0.913082 + 0.407776i \(0.133695\pi\)
\(564\) 0 0
\(565\) −3.69722 −0.155543
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) 10.7889 0.452294 0.226147 0.974093i \(-0.427387\pi\)
0.226147 + 0.974093i \(0.427387\pi\)
\(570\) 0 0
\(571\) −10.2111 −0.427321 −0.213661 0.976908i \(-0.568539\pi\)
−0.213661 + 0.976908i \(0.568539\pi\)
\(572\) 0 0
\(573\) −4.60555 −0.192400
\(574\) 0 0
\(575\) 4.51388 0.188242
\(576\) 0 0
\(577\) −36.4222 −1.51628 −0.758138 0.652094i \(-0.773891\pi\)
−0.758138 + 0.652094i \(0.773891\pi\)
\(578\) 0 0
\(579\) −9.02776 −0.375181
\(580\) 0 0
\(581\) −5.60555 −0.232557
\(582\) 0 0
\(583\) −62.5694 −2.59136
\(584\) 0 0
\(585\) −1.60555 −0.0663814
\(586\) 0 0
\(587\) 39.1472 1.61578 0.807889 0.589335i \(-0.200610\pi\)
0.807889 + 0.589335i \(0.200610\pi\)
\(588\) 0 0
\(589\) −4.81665 −0.198467
\(590\) 0 0
\(591\) 7.09167 0.291712
\(592\) 0 0
\(593\) 37.0278 1.52055 0.760274 0.649603i \(-0.225065\pi\)
0.760274 + 0.649603i \(0.225065\pi\)
\(594\) 0 0
\(595\) 3.90833 0.160226
\(596\) 0 0
\(597\) 12.5139 0.512159
\(598\) 0 0
\(599\) 39.5139 1.61449 0.807247 0.590214i \(-0.200957\pi\)
0.807247 + 0.590214i \(0.200957\pi\)
\(600\) 0 0
\(601\) 40.9361 1.66982 0.834909 0.550388i \(-0.185520\pi\)
0.834909 + 0.550388i \(0.185520\pi\)
\(602\) 0 0
\(603\) 11.9083 0.484945
\(604\) 0 0
\(605\) −9.76114 −0.396847
\(606\) 0 0
\(607\) 24.4861 0.993861 0.496931 0.867790i \(-0.334460\pi\)
0.496931 + 0.867790i \(0.334460\pi\)
\(608\) 0 0
\(609\) −6.21110 −0.251687
\(610\) 0 0
\(611\) 19.8167 0.801696
\(612\) 0 0
\(613\) 2.81665 0.113764 0.0568818 0.998381i \(-0.481884\pi\)
0.0568818 + 0.998381i \(0.481884\pi\)
\(614\) 0 0
\(615\) −8.51388 −0.343313
\(616\) 0 0
\(617\) 14.7250 0.592805 0.296403 0.955063i \(-0.404213\pi\)
0.296403 + 0.955063i \(0.404213\pi\)
\(618\) 0 0
\(619\) −9.88057 −0.397134 −0.198567 0.980087i \(-0.563629\pi\)
−0.198567 + 0.980087i \(0.563629\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) 0 0
\(623\) 10.9083 0.437033
\(624\) 0 0
\(625\) 17.9445 0.717779
\(626\) 0 0
\(627\) −8.02776 −0.320598
\(628\) 0 0
\(629\) 50.4500 2.01157
\(630\) 0 0
\(631\) 11.9722 0.476607 0.238304 0.971191i \(-0.423409\pi\)
0.238304 + 0.971191i \(0.423409\pi\)
\(632\) 0 0
\(633\) −27.4222 −1.08993
\(634\) 0 0
\(635\) 1.18335 0.0469597
\(636\) 0 0
\(637\) 2.30278 0.0912393
\(638\) 0 0
\(639\) −0.908327 −0.0359329
\(640\) 0 0
\(641\) −10.5416 −0.416370 −0.208185 0.978090i \(-0.566756\pi\)
−0.208185 + 0.978090i \(0.566756\pi\)
\(642\) 0 0
\(643\) −43.5694 −1.71821 −0.859105 0.511800i \(-0.828979\pi\)
−0.859105 + 0.511800i \(0.828979\pi\)
\(644\) 0 0
\(645\) −3.84441 −0.151374
\(646\) 0 0
\(647\) −16.3305 −0.642019 −0.321010 0.947076i \(-0.604022\pi\)
−0.321010 + 0.947076i \(0.604022\pi\)
\(648\) 0 0
\(649\) −19.5416 −0.767076
\(650\) 0 0
\(651\) 3.00000 0.117579
\(652\) 0 0
\(653\) 30.5139 1.19410 0.597050 0.802204i \(-0.296339\pi\)
0.597050 + 0.802204i \(0.296339\pi\)
\(654\) 0 0
\(655\) 7.24726 0.283174
\(656\) 0 0
\(657\) −2.21110 −0.0862633
\(658\) 0 0
\(659\) −42.6333 −1.66076 −0.830379 0.557199i \(-0.811876\pi\)
−0.830379 + 0.557199i \(0.811876\pi\)
\(660\) 0 0
\(661\) 20.8167 0.809674 0.404837 0.914389i \(-0.367328\pi\)
0.404837 + 0.914389i \(0.367328\pi\)
\(662\) 0 0
\(663\) 12.9083 0.501318
\(664\) 0 0
\(665\) −1.11943 −0.0434096
\(666\) 0 0
\(667\) −6.21110 −0.240495
\(668\) 0 0
\(669\) 19.9083 0.769700
\(670\) 0 0
\(671\) −5.45837 −0.210718
\(672\) 0 0
\(673\) −26.6333 −1.02664 −0.513319 0.858198i \(-0.671584\pi\)
−0.513319 + 0.858198i \(0.671584\pi\)
\(674\) 0 0
\(675\) 4.51388 0.173739
\(676\) 0 0
\(677\) 37.1472 1.42768 0.713841 0.700308i \(-0.246954\pi\)
0.713841 + 0.700308i \(0.246954\pi\)
\(678\) 0 0
\(679\) −17.6056 −0.675639
\(680\) 0 0
\(681\) −20.3305 −0.779068
\(682\) 0 0
\(683\) 1.42221 0.0544192 0.0272096 0.999630i \(-0.491338\pi\)
0.0272096 + 0.999630i \(0.491338\pi\)
\(684\) 0 0
\(685\) −11.7250 −0.447988
\(686\) 0 0
\(687\) −2.51388 −0.0959104
\(688\) 0 0
\(689\) −28.8167 −1.09783
\(690\) 0 0
\(691\) 31.5139 1.19884 0.599422 0.800433i \(-0.295397\pi\)
0.599422 + 0.800433i \(0.295397\pi\)
\(692\) 0 0
\(693\) 5.00000 0.189934
\(694\) 0 0
\(695\) 11.0917 0.420731
\(696\) 0 0
\(697\) 68.4500 2.59273
\(698\) 0 0
\(699\) 14.3305 0.542031
\(700\) 0 0
\(701\) −25.5416 −0.964694 −0.482347 0.875980i \(-0.660216\pi\)
−0.482347 + 0.875980i \(0.660216\pi\)
\(702\) 0 0
\(703\) −14.4500 −0.544991
\(704\) 0 0
\(705\) 6.00000 0.225973
\(706\) 0 0
\(707\) 9.30278 0.349867
\(708\) 0 0
\(709\) −47.3305 −1.77754 −0.888768 0.458358i \(-0.848438\pi\)
−0.888768 + 0.458358i \(0.848438\pi\)
\(710\) 0 0
\(711\) 1.00000 0.0375029
\(712\) 0 0
\(713\) 3.00000 0.112351
\(714\) 0 0
\(715\) −8.02776 −0.300221
\(716\) 0 0
\(717\) −14.0917 −0.526263
\(718\) 0 0
\(719\) −14.2111 −0.529985 −0.264992 0.964251i \(-0.585369\pi\)
−0.264992 + 0.964251i \(0.585369\pi\)
\(720\) 0 0
\(721\) 4.00000 0.148968
\(722\) 0 0
\(723\) −12.0278 −0.447317
\(724\) 0 0
\(725\) −28.0362 −1.04124
\(726\) 0 0
\(727\) 46.8722 1.73839 0.869196 0.494467i \(-0.164637\pi\)
0.869196 + 0.494467i \(0.164637\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.9083 1.14319
\(732\) 0 0
\(733\) −26.6056 −0.982698 −0.491349 0.870963i \(-0.663496\pi\)
−0.491349 + 0.870963i \(0.663496\pi\)
\(734\) 0 0
\(735\) 0.697224 0.0257175
\(736\) 0 0
\(737\) 59.5416 2.19324
\(738\) 0 0
\(739\) 43.4222 1.59731 0.798656 0.601788i \(-0.205545\pi\)
0.798656 + 0.601788i \(0.205545\pi\)
\(740\) 0 0
\(741\) −3.69722 −0.135821
\(742\) 0 0
\(743\) −30.5139 −1.11945 −0.559723 0.828680i \(-0.689092\pi\)
−0.559723 + 0.828680i \(0.689092\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) −5.60555 −0.205096
\(748\) 0 0
\(749\) −7.51388 −0.274551
\(750\) 0 0
\(751\) −8.69722 −0.317366 −0.158683 0.987330i \(-0.550725\pi\)
−0.158683 + 0.987330i \(0.550725\pi\)
\(752\) 0 0
\(753\) 23.8167 0.867927
\(754\) 0 0
\(755\) 11.5778 0.421359
\(756\) 0 0
\(757\) −3.36669 −0.122365 −0.0611823 0.998127i \(-0.519487\pi\)
−0.0611823 + 0.998127i \(0.519487\pi\)
\(758\) 0 0
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) 12.0000 0.435000 0.217500 0.976060i \(-0.430210\pi\)
0.217500 + 0.976060i \(0.430210\pi\)
\(762\) 0 0
\(763\) −3.90833 −0.141491
\(764\) 0 0
\(765\) 3.90833 0.141306
\(766\) 0 0
\(767\) −9.00000 −0.324971
\(768\) 0 0
\(769\) −27.4500 −0.989871 −0.494935 0.868930i \(-0.664808\pi\)
−0.494935 + 0.868930i \(0.664808\pi\)
\(770\) 0 0
\(771\) −27.0278 −0.973381
\(772\) 0 0
\(773\) 31.8444 1.14536 0.572682 0.819778i \(-0.305903\pi\)
0.572682 + 0.819778i \(0.305903\pi\)
\(774\) 0 0
\(775\) 13.5416 0.486430
\(776\) 0 0
\(777\) 9.00000 0.322873
\(778\) 0 0
\(779\) −19.6056 −0.702442
\(780\) 0 0
\(781\) −4.54163 −0.162512
\(782\) 0 0
\(783\) −6.21110 −0.221967
\(784\) 0 0
\(785\) 2.66106 0.0949774
\(786\) 0 0
\(787\) −32.1472 −1.14592 −0.572962 0.819582i \(-0.694206\pi\)
−0.572962 + 0.819582i \(0.694206\pi\)
\(788\) 0 0
\(789\) 10.3944 0.370052
\(790\) 0 0
\(791\) 5.30278 0.188545
\(792\) 0 0
\(793\) −2.51388 −0.0892704
\(794\) 0 0
\(795\) −8.72498 −0.309443
\(796\) 0 0
\(797\) −39.0278 −1.38243 −0.691217 0.722647i \(-0.742925\pi\)
−0.691217 + 0.722647i \(0.742925\pi\)
\(798\) 0 0
\(799\) −48.2389 −1.70657
\(800\) 0 0
\(801\) 10.9083 0.385427
\(802\) 0 0
\(803\) −11.0555 −0.390141
\(804\) 0 0
\(805\) 0.697224 0.0245739
\(806\) 0 0
\(807\) −0.908327 −0.0319746
\(808\) 0 0
\(809\) 25.9361 0.911864 0.455932 0.890015i \(-0.349306\pi\)
0.455932 + 0.890015i \(0.349306\pi\)
\(810\) 0 0
\(811\) 10.3944 0.364998 0.182499 0.983206i \(-0.441581\pi\)
0.182499 + 0.983206i \(0.441581\pi\)
\(812\) 0 0
\(813\) 3.60555 0.126452
\(814\) 0 0
\(815\) −9.56939 −0.335201
\(816\) 0 0
\(817\) −8.85281 −0.309721
\(818\) 0 0
\(819\) 2.30278 0.0804655
\(820\) 0 0
\(821\) −19.3944 −0.676871 −0.338435 0.940990i \(-0.609898\pi\)
−0.338435 + 0.940990i \(0.609898\pi\)
\(822\) 0 0
\(823\) −6.51388 −0.227060 −0.113530 0.993535i \(-0.536216\pi\)
−0.113530 + 0.993535i \(0.536216\pi\)
\(824\) 0 0
\(825\) 22.5694 0.785765
\(826\) 0 0
\(827\) 9.90833 0.344546 0.172273 0.985049i \(-0.444889\pi\)
0.172273 + 0.985049i \(0.444889\pi\)
\(828\) 0 0
\(829\) −30.4500 −1.05757 −0.528785 0.848756i \(-0.677352\pi\)
−0.528785 + 0.848756i \(0.677352\pi\)
\(830\) 0 0
\(831\) 12.3028 0.426779
\(832\) 0 0
\(833\) −5.60555 −0.194221
\(834\) 0 0
\(835\) 1.96384 0.0679615
\(836\) 0 0
\(837\) 3.00000 0.103695
\(838\) 0 0
\(839\) −38.3028 −1.32236 −0.661179 0.750228i \(-0.729944\pi\)
−0.661179 + 0.750228i \(0.729944\pi\)
\(840\) 0 0
\(841\) 9.57779 0.330269
\(842\) 0 0
\(843\) −27.8167 −0.958056
\(844\) 0 0
\(845\) 5.36669 0.184620
\(846\) 0 0
\(847\) 14.0000 0.481046
\(848\) 0 0
\(849\) 17.3028 0.593830
\(850\) 0 0
\(851\) 9.00000 0.308516
\(852\) 0 0
\(853\) −58.2389 −1.99406 −0.997030 0.0770106i \(-0.975462\pi\)
−0.997030 + 0.0770106i \(0.975462\pi\)
\(854\) 0 0
\(855\) −1.11943 −0.0382837
\(856\) 0 0
\(857\) −50.2389 −1.71613 −0.858063 0.513544i \(-0.828332\pi\)
−0.858063 + 0.513544i \(0.828332\pi\)
\(858\) 0 0
\(859\) −43.2111 −1.47434 −0.737172 0.675705i \(-0.763839\pi\)
−0.737172 + 0.675705i \(0.763839\pi\)
\(860\) 0 0
\(861\) 12.2111 0.416153
\(862\) 0 0
\(863\) 5.76114 0.196112 0.0980558 0.995181i \(-0.468738\pi\)
0.0980558 + 0.995181i \(0.468738\pi\)
\(864\) 0 0
\(865\) 12.6972 0.431719
\(866\) 0 0
\(867\) −14.4222 −0.489804
\(868\) 0 0
\(869\) 5.00000 0.169613
\(870\) 0 0
\(871\) 27.4222 0.929166
\(872\) 0 0
\(873\) −17.6056 −0.595858
\(874\) 0 0
\(875\) 6.63331 0.224247
\(876\) 0 0
\(877\) −14.7889 −0.499386 −0.249693 0.968325i \(-0.580330\pi\)
−0.249693 + 0.968325i \(0.580330\pi\)
\(878\) 0 0
\(879\) 24.8444 0.837981
\(880\) 0 0
\(881\) −43.8167 −1.47622 −0.738110 0.674680i \(-0.764282\pi\)
−0.738110 + 0.674680i \(0.764282\pi\)
\(882\) 0 0
\(883\) 18.4861 0.622108 0.311054 0.950392i \(-0.399318\pi\)
0.311054 + 0.950392i \(0.399318\pi\)
\(884\) 0 0
\(885\) −2.72498 −0.0915992
\(886\) 0 0
\(887\) 11.9361 0.400774 0.200387 0.979717i \(-0.435780\pi\)
0.200387 + 0.979717i \(0.435780\pi\)
\(888\) 0 0
\(889\) −1.69722 −0.0569231
\(890\) 0 0
\(891\) 5.00000 0.167506
\(892\) 0 0
\(893\) 13.8167 0.462357
\(894\) 0 0
\(895\) −1.88057 −0.0628605
\(896\) 0 0
\(897\) 2.30278 0.0768874
\(898\) 0 0
\(899\) −18.6333 −0.621456
\(900\) 0 0
\(901\) 70.1472 2.33694
\(902\) 0 0
\(903\) 5.51388 0.183490
\(904\) 0 0
\(905\) 4.75274 0.157986
\(906\) 0 0
\(907\) 8.14719 0.270523 0.135261 0.990810i \(-0.456813\pi\)
0.135261 + 0.990810i \(0.456813\pi\)
\(908\) 0 0
\(909\) 9.30278 0.308554
\(910\) 0 0
\(911\) 24.2389 0.803069 0.401535 0.915844i \(-0.368477\pi\)
0.401535 + 0.915844i \(0.368477\pi\)
\(912\) 0 0
\(913\) −28.0278 −0.927583
\(914\) 0 0
\(915\) −0.761141 −0.0251625
\(916\) 0 0
\(917\) −10.3944 −0.343255
\(918\) 0 0
\(919\) 10.3944 0.342881 0.171441 0.985194i \(-0.445158\pi\)
0.171441 + 0.985194i \(0.445158\pi\)
\(920\) 0 0
\(921\) 18.2111 0.600076
\(922\) 0 0
\(923\) −2.09167 −0.0688483
\(924\) 0 0
\(925\) 40.6249 1.33574
\(926\) 0 0
\(927\) 4.00000 0.131377
\(928\) 0 0
\(929\) 40.3305 1.32320 0.661601 0.749856i \(-0.269877\pi\)
0.661601 + 0.749856i \(0.269877\pi\)
\(930\) 0 0
\(931\) 1.60555 0.0526198
\(932\) 0 0
\(933\) −9.51388 −0.311470
\(934\) 0 0
\(935\) 19.5416 0.639080
\(936\) 0 0
\(937\) −4.57779 −0.149550 −0.0747750 0.997200i \(-0.523824\pi\)
−0.0747750 + 0.997200i \(0.523824\pi\)
\(938\) 0 0
\(939\) −1.21110 −0.0395228
\(940\) 0 0
\(941\) −0.844410 −0.0275270 −0.0137635 0.999905i \(-0.504381\pi\)
−0.0137635 + 0.999905i \(0.504381\pi\)
\(942\) 0 0
\(943\) 12.2111 0.397648
\(944\) 0 0
\(945\) 0.697224 0.0226807
\(946\) 0 0
\(947\) 13.6333 0.443023 0.221511 0.975158i \(-0.428901\pi\)
0.221511 + 0.975158i \(0.428901\pi\)
\(948\) 0 0
\(949\) −5.09167 −0.165283
\(950\) 0 0
\(951\) 30.5139 0.989480
\(952\) 0 0
\(953\) −51.3305 −1.66276 −0.831380 0.555705i \(-0.812448\pi\)
−0.831380 + 0.555705i \(0.812448\pi\)
\(954\) 0 0
\(955\) −3.21110 −0.103909
\(956\) 0 0
\(957\) −31.0555 −1.00388
\(958\) 0 0
\(959\) 16.8167 0.543038
\(960\) 0 0
\(961\) −22.0000 −0.709677
\(962\) 0 0
\(963\) −7.51388 −0.242131
\(964\) 0 0
\(965\) −6.29437 −0.202623
\(966\) 0 0
\(967\) −41.2666 −1.32704 −0.663522 0.748156i \(-0.730939\pi\)
−0.663522 + 0.748156i \(0.730939\pi\)
\(968\) 0 0
\(969\) 9.00000 0.289122
\(970\) 0 0
\(971\) 35.0917 1.12615 0.563073 0.826407i \(-0.309619\pi\)
0.563073 + 0.826407i \(0.309619\pi\)
\(972\) 0 0
\(973\) −15.9083 −0.509998
\(974\) 0 0
\(975\) 10.3944 0.332889
\(976\) 0 0
\(977\) 31.1472 0.996487 0.498243 0.867037i \(-0.333979\pi\)
0.498243 + 0.867037i \(0.333979\pi\)
\(978\) 0 0
\(979\) 54.5416 1.74316
\(980\) 0 0
\(981\) −3.90833 −0.124783
\(982\) 0 0
\(983\) 22.8167 0.727738 0.363869 0.931450i \(-0.381456\pi\)
0.363869 + 0.931450i \(0.381456\pi\)
\(984\) 0 0
\(985\) 4.94449 0.157544
\(986\) 0 0
\(987\) −8.60555 −0.273918
\(988\) 0 0
\(989\) 5.51388 0.175331
\(990\) 0 0
\(991\) −35.5139 −1.12814 −0.564068 0.825729i \(-0.690764\pi\)
−0.564068 + 0.825729i \(0.690764\pi\)
\(992\) 0 0
\(993\) 17.6333 0.559576
\(994\) 0 0
\(995\) 8.72498 0.276600
\(996\) 0 0
\(997\) −0.0277564 −0.000879053 0 −0.000439527 1.00000i \(-0.500140\pi\)
−0.000439527 1.00000i \(0.500140\pi\)
\(998\) 0 0
\(999\) 9.00000 0.284747
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.x.1.2 2
4.3 odd 2 483.2.a.f.1.1 2
12.11 even 2 1449.2.a.j.1.2 2
28.27 even 2 3381.2.a.p.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
483.2.a.f.1.1 2 4.3 odd 2
1449.2.a.j.1.2 2 12.11 even 2
3381.2.a.p.1.1 2 28.27 even 2
7728.2.a.x.1.2 2 1.1 even 1 trivial