Properties

Label 1380.2.a.c.1.1
Level $1380$
Weight $2$
Character 1380.1
Self dual yes
Analytic conductor $11.019$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1380 = 2^{2} \cdot 3 \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1380.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(11.0193554789\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1380.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -4.00000 q^{13} -1.00000 q^{15} -3.00000 q^{17} -4.00000 q^{19} -1.00000 q^{21} -1.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -3.00000 q^{29} -7.00000 q^{31} +1.00000 q^{35} +11.0000 q^{37} -4.00000 q^{39} -9.00000 q^{41} -4.00000 q^{43} -1.00000 q^{45} +6.00000 q^{47} -6.00000 q^{49} -3.00000 q^{51} +9.00000 q^{53} -4.00000 q^{57} +3.00000 q^{59} -10.0000 q^{61} -1.00000 q^{63} +4.00000 q^{65} -13.0000 q^{67} -1.00000 q^{69} +9.00000 q^{71} -16.0000 q^{73} +1.00000 q^{75} +8.00000 q^{79} +1.00000 q^{81} -15.0000 q^{83} +3.00000 q^{85} -3.00000 q^{87} +4.00000 q^{91} -7.00000 q^{93} +4.00000 q^{95} +2.00000 q^{97} +O(q^{100})\)

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
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) −1.00000 −0.377964 −0.188982 0.981981i \(-0.560519\pi\)
−0.188982 + 0.981981i \(0.560519\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −3.00000 −0.727607 −0.363803 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −3.00000 −0.557086 −0.278543 0.960424i \(-0.589851\pi\)
−0.278543 + 0.960424i \(0.589851\pi\)
\(30\) 0 0
\(31\) −7.00000 −1.25724 −0.628619 0.777714i \(-0.716379\pi\)
−0.628619 + 0.777714i \(0.716379\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.00000 0.169031
\(36\) 0 0
\(37\) 11.0000 1.80839 0.904194 0.427121i \(-0.140472\pi\)
0.904194 + 0.427121i \(0.140472\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −6.00000 −0.857143
\(50\) 0 0
\(51\) −3.00000 −0.420084
\(52\) 0 0
\(53\) 9.00000 1.23625 0.618123 0.786082i \(-0.287894\pi\)
0.618123 + 0.786082i \(0.287894\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) 3.00000 0.390567 0.195283 0.980747i \(-0.437437\pi\)
0.195283 + 0.980747i \(0.437437\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) 0 0
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) −13.0000 −1.58820 −0.794101 0.607785i \(-0.792058\pi\)
−0.794101 + 0.607785i \(0.792058\pi\)
\(68\) 0 0
\(69\) −1.00000 −0.120386
\(70\) 0 0
\(71\) 9.00000 1.06810 0.534052 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) −16.0000 −1.87266 −0.936329 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −15.0000 −1.64646 −0.823232 0.567705i \(-0.807831\pi\)
−0.823232 + 0.567705i \(0.807831\pi\)
\(84\) 0 0
\(85\) 3.00000 0.325396
\(86\) 0 0
\(87\) −3.00000 −0.321634
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 0 0
\(93\) −7.00000 −0.725866
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.00000 −0.298511 −0.149256 0.988799i \(-0.547688\pi\)
−0.149256 + 0.988799i \(0.547688\pi\)
\(102\) 0 0
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) 0 0
\(105\) 1.00000 0.0975900
\(106\) 0 0
\(107\) −3.00000 −0.290021 −0.145010 0.989430i \(-0.546322\pi\)
−0.145010 + 0.989430i \(0.546322\pi\)
\(108\) 0 0
\(109\) −4.00000 −0.383131 −0.191565 0.981480i \(-0.561356\pi\)
−0.191565 + 0.981480i \(0.561356\pi\)
\(110\) 0 0
\(111\) 11.0000 1.04407
\(112\) 0 0
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) 1.00000 0.0932505
\(116\) 0 0
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) −9.00000 −0.811503
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 14.0000 1.24230 0.621150 0.783692i \(-0.286666\pi\)
0.621150 + 0.783692i \(0.286666\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −13.0000 −1.10265 −0.551323 0.834292i \(-0.685877\pi\)
−0.551323 + 0.834292i \(0.685877\pi\)
\(140\) 0 0
\(141\) 6.00000 0.505291
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 3.00000 0.249136
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) 0 0
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −16.0000 −1.30206 −0.651031 0.759051i \(-0.725663\pi\)
−0.651031 + 0.759051i \(0.725663\pi\)
\(152\) 0 0
\(153\) −3.00000 −0.242536
\(154\) 0 0
\(155\) 7.00000 0.562254
\(156\) 0 0
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 0 0
\(159\) 9.00000 0.713746
\(160\) 0 0
\(161\) 1.00000 0.0788110
\(162\) 0 0
\(163\) 2.00000 0.156652 0.0783260 0.996928i \(-0.475042\pi\)
0.0783260 + 0.996928i \(0.475042\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 18.0000 1.39288 0.696441 0.717614i \(-0.254766\pi\)
0.696441 + 0.717614i \(0.254766\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) −24.0000 −1.82469 −0.912343 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) 0 0
\(177\) 3.00000 0.225494
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −16.0000 −1.18927 −0.594635 0.803996i \(-0.702704\pi\)
−0.594635 + 0.803996i \(0.702704\pi\)
\(182\) 0 0
\(183\) −10.0000 −0.739221
\(184\) 0 0
\(185\) −11.0000 −0.808736
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.00000 −0.0727393
\(190\) 0 0
\(191\) 6.00000 0.434145 0.217072 0.976156i \(-0.430349\pi\)
0.217072 + 0.976156i \(0.430349\pi\)
\(192\) 0 0
\(193\) −4.00000 −0.287926 −0.143963 0.989583i \(-0.545985\pi\)
−0.143963 + 0.989583i \(0.545985\pi\)
\(194\) 0 0
\(195\) 4.00000 0.286446
\(196\) 0 0
\(197\) −6.00000 −0.427482 −0.213741 0.976890i \(-0.568565\pi\)
−0.213741 + 0.976890i \(0.568565\pi\)
\(198\) 0 0
\(199\) 2.00000 0.141776 0.0708881 0.997484i \(-0.477417\pi\)
0.0708881 + 0.997484i \(0.477417\pi\)
\(200\) 0 0
\(201\) −13.0000 −0.916949
\(202\) 0 0
\(203\) 3.00000 0.210559
\(204\) 0 0
\(205\) 9.00000 0.628587
\(206\) 0 0
\(207\) −1.00000 −0.0695048
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 17.0000 1.17033 0.585164 0.810915i \(-0.301030\pi\)
0.585164 + 0.810915i \(0.301030\pi\)
\(212\) 0 0
\(213\) 9.00000 0.616670
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 7.00000 0.475191
\(218\) 0 0
\(219\) −16.0000 −1.08118
\(220\) 0 0
\(221\) 12.0000 0.807207
\(222\) 0 0
\(223\) −10.0000 −0.669650 −0.334825 0.942280i \(-0.608677\pi\)
−0.334825 + 0.942280i \(0.608677\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 12.0000 0.796468 0.398234 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 0 0
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −9.00000 −0.582162 −0.291081 0.956698i \(-0.594015\pi\)
−0.291081 + 0.956698i \(0.594015\pi\)
\(240\) 0 0
\(241\) 8.00000 0.515325 0.257663 0.966235i \(-0.417048\pi\)
0.257663 + 0.966235i \(0.417048\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 0 0
\(249\) −15.0000 −0.950586
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 3.00000 0.187867
\(256\) 0 0
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −11.0000 −0.683507
\(260\) 0 0
\(261\) −3.00000 −0.185695
\(262\) 0 0
\(263\) 27.0000 1.66489 0.832446 0.554107i \(-0.186940\pi\)
0.832446 + 0.554107i \(0.186940\pi\)
\(264\) 0 0
\(265\) −9.00000 −0.552866
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −21.0000 −1.28039 −0.640196 0.768211i \(-0.721147\pi\)
−0.640196 + 0.768211i \(0.721147\pi\)
\(270\) 0 0
\(271\) −13.0000 −0.789694 −0.394847 0.918747i \(-0.629202\pi\)
−0.394847 + 0.918747i \(0.629202\pi\)
\(272\) 0 0
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 0 0
\(279\) −7.00000 −0.419079
\(280\) 0 0
\(281\) −12.0000 −0.715860 −0.357930 0.933748i \(-0.616517\pi\)
−0.357930 + 0.933748i \(0.616517\pi\)
\(282\) 0 0
\(283\) 5.00000 0.297219 0.148610 0.988896i \(-0.452520\pi\)
0.148610 + 0.988896i \(0.452520\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) 9.00000 0.531253
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) 0 0
\(293\) −9.00000 −0.525786 −0.262893 0.964825i \(-0.584677\pi\)
−0.262893 + 0.964825i \(0.584677\pi\)
\(294\) 0 0
\(295\) −3.00000 −0.174667
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.00000 0.231326
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 0 0
\(303\) −3.00000 −0.172345
\(304\) 0 0
\(305\) 10.0000 0.572598
\(306\) 0 0
\(307\) 20.0000 1.14146 0.570730 0.821138i \(-0.306660\pi\)
0.570730 + 0.821138i \(0.306660\pi\)
\(308\) 0 0
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 24.0000 1.36092 0.680458 0.732787i \(-0.261781\pi\)
0.680458 + 0.732787i \(0.261781\pi\)
\(312\) 0 0
\(313\) 17.0000 0.960897 0.480448 0.877023i \(-0.340474\pi\)
0.480448 + 0.877023i \(0.340474\pi\)
\(314\) 0 0
\(315\) 1.00000 0.0563436
\(316\) 0 0
\(317\) 18.0000 1.01098 0.505490 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) −3.00000 −0.167444
\(322\) 0 0
\(323\) 12.0000 0.667698
\(324\) 0 0
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) −4.00000 −0.221201
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 0 0
\(333\) 11.0000 0.602796
\(334\) 0 0
\(335\) 13.0000 0.710266
\(336\) 0 0
\(337\) −22.0000 −1.19842 −0.599208 0.800593i \(-0.704518\pi\)
−0.599208 + 0.800593i \(0.704518\pi\)
\(338\) 0 0
\(339\) 9.00000 0.488813
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.0000 0.701934
\(344\) 0 0
\(345\) 1.00000 0.0538382
\(346\) 0 0
\(347\) 24.0000 1.28839 0.644194 0.764862i \(-0.277193\pi\)
0.644194 + 0.764862i \(0.277193\pi\)
\(348\) 0 0
\(349\) 35.0000 1.87351 0.936754 0.349990i \(-0.113815\pi\)
0.936754 + 0.349990i \(0.113815\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) 6.00000 0.319348 0.159674 0.987170i \(-0.448956\pi\)
0.159674 + 0.987170i \(0.448956\pi\)
\(354\) 0 0
\(355\) −9.00000 −0.477670
\(356\) 0 0
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) 30.0000 1.58334 0.791670 0.610949i \(-0.209212\pi\)
0.791670 + 0.610949i \(0.209212\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −11.0000 −0.577350
\(364\) 0 0
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) 29.0000 1.51379 0.756894 0.653538i \(-0.226716\pi\)
0.756894 + 0.653538i \(0.226716\pi\)
\(368\) 0 0
\(369\) −9.00000 −0.468521
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 0 0
\(373\) −22.0000 −1.13912 −0.569558 0.821951i \(-0.692886\pi\)
−0.569558 + 0.821951i \(0.692886\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −16.0000 −0.821865 −0.410932 0.911666i \(-0.634797\pi\)
−0.410932 + 0.911666i \(0.634797\pi\)
\(380\) 0 0
\(381\) 14.0000 0.717242
\(382\) 0 0
\(383\) −21.0000 −1.07305 −0.536525 0.843884i \(-0.680263\pi\)
−0.536525 + 0.843884i \(0.680263\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −4.00000 −0.203331
\(388\) 0 0
\(389\) −30.0000 −1.52106 −0.760530 0.649303i \(-0.775061\pi\)
−0.760530 + 0.649303i \(0.775061\pi\)
\(390\) 0 0
\(391\) 3.00000 0.151717
\(392\) 0 0
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) −16.0000 −0.803017 −0.401508 0.915855i \(-0.631514\pi\)
−0.401508 + 0.915855i \(0.631514\pi\)
\(398\) 0 0
\(399\) 4.00000 0.200250
\(400\) 0 0
\(401\) −18.0000 −0.898877 −0.449439 0.893311i \(-0.648376\pi\)
−0.449439 + 0.893311i \(0.648376\pi\)
\(402\) 0 0
\(403\) 28.0000 1.39478
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −31.0000 −1.53285 −0.766426 0.642333i \(-0.777967\pi\)
−0.766426 + 0.642333i \(0.777967\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 0 0
\(413\) −3.00000 −0.147620
\(414\) 0 0
\(415\) 15.0000 0.736321
\(416\) 0 0
\(417\) −13.0000 −0.636613
\(418\) 0 0
\(419\) −36.0000 −1.75872 −0.879358 0.476162i \(-0.842028\pi\)
−0.879358 + 0.476162i \(0.842028\pi\)
\(420\) 0 0
\(421\) 8.00000 0.389896 0.194948 0.980814i \(-0.437546\pi\)
0.194948 + 0.980814i \(0.437546\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 10.0000 0.483934
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) 11.0000 0.528626 0.264313 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) 3.00000 0.143839
\(436\) 0 0
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 0 0
\(441\) −6.00000 −0.285714
\(442\) 0 0
\(443\) 6.00000 0.285069 0.142534 0.989790i \(-0.454475\pi\)
0.142534 + 0.989790i \(0.454475\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 0 0
\(449\) 21.0000 0.991051 0.495526 0.868593i \(-0.334975\pi\)
0.495526 + 0.868593i \(0.334975\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) −4.00000 −0.187523
\(456\) 0 0
\(457\) 29.0000 1.35656 0.678281 0.734802i \(-0.262725\pi\)
0.678281 + 0.734802i \(0.262725\pi\)
\(458\) 0 0
\(459\) −3.00000 −0.140028
\(460\) 0 0
\(461\) −30.0000 −1.39724 −0.698620 0.715493i \(-0.746202\pi\)
−0.698620 + 0.715493i \(0.746202\pi\)
\(462\) 0 0
\(463\) −16.0000 −0.743583 −0.371792 0.928316i \(-0.621256\pi\)
−0.371792 + 0.928316i \(0.621256\pi\)
\(464\) 0 0
\(465\) 7.00000 0.324617
\(466\) 0 0
\(467\) 21.0000 0.971764 0.485882 0.874024i \(-0.338498\pi\)
0.485882 + 0.874024i \(0.338498\pi\)
\(468\) 0 0
\(469\) 13.0000 0.600284
\(470\) 0 0
\(471\) 17.0000 0.783319
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) 9.00000 0.412082
\(478\) 0 0
\(479\) 6.00000 0.274147 0.137073 0.990561i \(-0.456230\pi\)
0.137073 + 0.990561i \(0.456230\pi\)
\(480\) 0 0
\(481\) −44.0000 −2.00623
\(482\) 0 0
\(483\) 1.00000 0.0455016
\(484\) 0 0
\(485\) −2.00000 −0.0908153
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) 0 0
\(489\) 2.00000 0.0904431
\(490\) 0 0
\(491\) 21.0000 0.947717 0.473858 0.880601i \(-0.342861\pi\)
0.473858 + 0.880601i \(0.342861\pi\)
\(492\) 0 0
\(493\) 9.00000 0.405340
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −9.00000 −0.403705
\(498\) 0 0
\(499\) −13.0000 −0.581960 −0.290980 0.956729i \(-0.593981\pi\)
−0.290980 + 0.956729i \(0.593981\pi\)
\(500\) 0 0
\(501\) 18.0000 0.804181
\(502\) 0 0
\(503\) −9.00000 −0.401290 −0.200645 0.979664i \(-0.564304\pi\)
−0.200645 + 0.979664i \(0.564304\pi\)
\(504\) 0 0
\(505\) 3.00000 0.133498
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 0 0
\(509\) −18.0000 −0.797836 −0.398918 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) −8.00000 −0.352522
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) −24.0000 −1.05348
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −28.0000 −1.22435 −0.612177 0.790721i \(-0.709706\pi\)
−0.612177 + 0.790721i \(0.709706\pi\)
\(524\) 0 0
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 21.0000 0.914774
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 3.00000 0.130189
\(532\) 0 0
\(533\) 36.0000 1.55933
\(534\) 0 0
\(535\) 3.00000 0.129701
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −34.0000 −1.46177 −0.730887 0.682498i \(-0.760893\pi\)
−0.730887 + 0.682498i \(0.760893\pi\)
\(542\) 0 0
\(543\) −16.0000 −0.686626
\(544\) 0 0
\(545\) 4.00000 0.171341
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 0 0
\(549\) −10.0000 −0.426790
\(550\) 0 0
\(551\) 12.0000 0.511217
\(552\) 0 0
\(553\) −8.00000 −0.340195
\(554\) 0 0
\(555\) −11.0000 −0.466924
\(556\) 0 0
\(557\) −39.0000 −1.65248 −0.826242 0.563316i \(-0.809525\pi\)
−0.826242 + 0.563316i \(0.809525\pi\)
\(558\) 0 0
\(559\) 16.0000 0.676728
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21.0000 −0.885044 −0.442522 0.896758i \(-0.645916\pi\)
−0.442522 + 0.896758i \(0.645916\pi\)
\(564\) 0 0
\(565\) −9.00000 −0.378633
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) 18.0000 0.754599 0.377300 0.926091i \(-0.376853\pi\)
0.377300 + 0.926091i \(0.376853\pi\)
\(570\) 0 0
\(571\) −22.0000 −0.920671 −0.460336 0.887745i \(-0.652271\pi\)
−0.460336 + 0.887745i \(0.652271\pi\)
\(572\) 0 0
\(573\) 6.00000 0.250654
\(574\) 0 0
\(575\) −1.00000 −0.0417029
\(576\) 0 0
\(577\) 44.0000 1.83174 0.915872 0.401470i \(-0.131501\pi\)
0.915872 + 0.401470i \(0.131501\pi\)
\(578\) 0 0
\(579\) −4.00000 −0.166234
\(580\) 0 0
\(581\) 15.0000 0.622305
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 4.00000 0.165380
\(586\) 0 0
\(587\) −30.0000 −1.23823 −0.619116 0.785299i \(-0.712509\pi\)
−0.619116 + 0.785299i \(0.712509\pi\)
\(588\) 0 0
\(589\) 28.0000 1.15372
\(590\) 0 0
\(591\) −6.00000 −0.246807
\(592\) 0 0
\(593\) 6.00000 0.246390 0.123195 0.992382i \(-0.460686\pi\)
0.123195 + 0.992382i \(0.460686\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 0 0
\(597\) 2.00000 0.0818546
\(598\) 0 0
\(599\) −36.0000 −1.47092 −0.735460 0.677568i \(-0.763034\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 23.0000 0.938190 0.469095 0.883148i \(-0.344580\pi\)
0.469095 + 0.883148i \(0.344580\pi\)
\(602\) 0 0
\(603\) −13.0000 −0.529401
\(604\) 0 0
\(605\) 11.0000 0.447214
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) 3.00000 0.121566
\(610\) 0 0
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 14.0000 0.565455 0.282727 0.959200i \(-0.408761\pi\)
0.282727 + 0.959200i \(0.408761\pi\)
\(614\) 0 0
\(615\) 9.00000 0.362915
\(616\) 0 0
\(617\) −9.00000 −0.362326 −0.181163 0.983453i \(-0.557986\pi\)
−0.181163 + 0.983453i \(0.557986\pi\)
\(618\) 0 0
\(619\) 20.0000 0.803868 0.401934 0.915669i \(-0.368338\pi\)
0.401934 + 0.915669i \(0.368338\pi\)
\(620\) 0 0
\(621\) −1.00000 −0.0401286
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −33.0000 −1.31580
\(630\) 0 0
\(631\) −34.0000 −1.35352 −0.676759 0.736204i \(-0.736616\pi\)
−0.676759 + 0.736204i \(0.736616\pi\)
\(632\) 0 0
\(633\) 17.0000 0.675689
\(634\) 0 0
\(635\) −14.0000 −0.555573
\(636\) 0 0
\(637\) 24.0000 0.950915
\(638\) 0 0
\(639\) 9.00000 0.356034
\(640\) 0 0
\(641\) −24.0000 −0.947943 −0.473972 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) −13.0000 −0.512670 −0.256335 0.966588i \(-0.582515\pi\)
−0.256335 + 0.966588i \(0.582515\pi\)
\(644\) 0 0
\(645\) 4.00000 0.157500
\(646\) 0 0
\(647\) −24.0000 −0.943537 −0.471769 0.881722i \(-0.656384\pi\)
−0.471769 + 0.881722i \(0.656384\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 7.00000 0.274352
\(652\) 0 0
\(653\) −36.0000 −1.40879 −0.704394 0.709809i \(-0.748781\pi\)
−0.704394 + 0.709809i \(0.748781\pi\)
\(654\) 0 0
\(655\) −12.0000 −0.468879
\(656\) 0 0
\(657\) −16.0000 −0.624219
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 0 0
\(663\) 12.0000 0.466041
\(664\) 0 0
\(665\) −4.00000 −0.155113
\(666\) 0 0
\(667\) 3.00000 0.116160
\(668\) 0 0
\(669\) −10.0000 −0.386622
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −22.0000 −0.848038 −0.424019 0.905653i \(-0.639381\pi\)
−0.424019 + 0.905653i \(0.639381\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) −33.0000 −1.26829 −0.634147 0.773213i \(-0.718648\pi\)
−0.634147 + 0.773213i \(0.718648\pi\)
\(678\) 0 0
\(679\) −2.00000 −0.0767530
\(680\) 0 0
\(681\) 12.0000 0.459841
\(682\) 0 0
\(683\) −30.0000 −1.14792 −0.573959 0.818884i \(-0.694593\pi\)
−0.573959 + 0.818884i \(0.694593\pi\)
\(684\) 0 0
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) −4.00000 −0.152610
\(688\) 0 0
\(689\) −36.0000 −1.37149
\(690\) 0 0
\(691\) −40.0000 −1.52167 −0.760836 0.648944i \(-0.775211\pi\)
−0.760836 + 0.648944i \(0.775211\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 13.0000 0.493118
\(696\) 0 0
\(697\) 27.0000 1.02270
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 30.0000 1.13308 0.566542 0.824033i \(-0.308281\pi\)
0.566542 + 0.824033i \(0.308281\pi\)
\(702\) 0 0
\(703\) −44.0000 −1.65949
\(704\) 0 0
\(705\) −6.00000 −0.225973
\(706\) 0 0
\(707\) 3.00000 0.112827
\(708\) 0 0
\(709\) −22.0000 −0.826227 −0.413114 0.910679i \(-0.635559\pi\)
−0.413114 + 0.910679i \(0.635559\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 7.00000 0.262152
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −9.00000 −0.336111
\(718\) 0 0
\(719\) −45.0000 −1.67822 −0.839108 0.543964i \(-0.816923\pi\)
−0.839108 + 0.543964i \(0.816923\pi\)
\(720\) 0 0
\(721\) −8.00000 −0.297936
\(722\) 0 0
\(723\) 8.00000 0.297523
\(724\) 0 0
\(725\) −3.00000 −0.111417
\(726\) 0 0
\(727\) 17.0000 0.630495 0.315248 0.949009i \(-0.397912\pi\)
0.315248 + 0.949009i \(0.397912\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 12.0000 0.443836
\(732\) 0 0
\(733\) −1.00000 −0.0369358 −0.0184679 0.999829i \(-0.505879\pi\)
−0.0184679 + 0.999829i \(0.505879\pi\)
\(734\) 0 0
\(735\) 6.00000 0.221313
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 5.00000 0.183928 0.0919640 0.995762i \(-0.470686\pi\)
0.0919640 + 0.995762i \(0.470686\pi\)
\(740\) 0 0
\(741\) 16.0000 0.587775
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −6.00000 −0.219823
\(746\) 0 0
\(747\) −15.0000 −0.548821
\(748\) 0 0
\(749\) 3.00000 0.109618
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) 0 0
\(753\) 30.0000 1.09326
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 0 0
\(757\) 17.0000 0.617876 0.308938 0.951082i \(-0.400027\pi\)
0.308938 + 0.951082i \(0.400027\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 15.0000 0.543750 0.271875 0.962333i \(-0.412356\pi\)
0.271875 + 0.962333i \(0.412356\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 0 0
\(765\) 3.00000 0.108465
\(766\) 0 0
\(767\) −12.0000 −0.433295
\(768\) 0 0
\(769\) −22.0000 −0.793340 −0.396670 0.917961i \(-0.629834\pi\)
−0.396670 + 0.917961i \(0.629834\pi\)
\(770\) 0 0
\(771\) −18.0000 −0.648254
\(772\) 0 0
\(773\) 42.0000 1.51064 0.755318 0.655359i \(-0.227483\pi\)
0.755318 + 0.655359i \(0.227483\pi\)
\(774\) 0 0
\(775\) −7.00000 −0.251447
\(776\) 0 0
\(777\) −11.0000 −0.394623
\(778\) 0 0
\(779\) 36.0000 1.28983
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −3.00000 −0.107211
\(784\) 0 0
\(785\) −17.0000 −0.606756
\(786\) 0 0
\(787\) 11.0000 0.392108 0.196054 0.980593i \(-0.437187\pi\)
0.196054 + 0.980593i \(0.437187\pi\)
\(788\) 0 0
\(789\) 27.0000 0.961225
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) 0 0
\(793\) 40.0000 1.42044
\(794\) 0 0
\(795\) −9.00000 −0.319197
\(796\) 0 0
\(797\) −33.0000 −1.16892 −0.584460 0.811423i \(-0.698694\pi\)
−0.584460 + 0.811423i \(0.698694\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −1.00000 −0.0352454
\(806\) 0 0
\(807\) −21.0000 −0.739235
\(808\) 0 0
\(809\) −39.0000 −1.37117 −0.685583 0.727994i \(-0.740453\pi\)
−0.685583 + 0.727994i \(0.740453\pi\)
\(810\) 0 0
\(811\) 41.0000 1.43970 0.719852 0.694127i \(-0.244209\pi\)
0.719852 + 0.694127i \(0.244209\pi\)
\(812\) 0 0
\(813\) −13.0000 −0.455930
\(814\) 0 0
\(815\) −2.00000 −0.0700569
\(816\) 0 0
\(817\) 16.0000 0.559769
\(818\) 0 0
\(819\) 4.00000 0.139771
\(820\) 0 0
\(821\) −42.0000 −1.46581 −0.732905 0.680331i \(-0.761836\pi\)
−0.732905 + 0.680331i \(0.761836\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −45.0000 −1.56480 −0.782402 0.622774i \(-0.786006\pi\)
−0.782402 + 0.622774i \(0.786006\pi\)
\(828\) 0 0
\(829\) −7.00000 −0.243120 −0.121560 0.992584i \(-0.538790\pi\)
−0.121560 + 0.992584i \(0.538790\pi\)
\(830\) 0 0
\(831\) 2.00000 0.0693792
\(832\) 0 0
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) −18.0000 −0.622916
\(836\) 0 0
\(837\) −7.00000 −0.241955
\(838\) 0 0
\(839\) 30.0000 1.03572 0.517858 0.855467i \(-0.326730\pi\)
0.517858 + 0.855467i \(0.326730\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −12.0000 −0.413302
\(844\) 0 0
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 11.0000 0.377964
\(848\) 0 0
\(849\) 5.00000 0.171600
\(850\) 0 0
\(851\) −11.0000 −0.377075
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 6.00000 0.204956 0.102478 0.994735i \(-0.467323\pi\)
0.102478 + 0.994735i \(0.467323\pi\)
\(858\) 0 0
\(859\) 5.00000 0.170598 0.0852989 0.996355i \(-0.472815\pi\)
0.0852989 + 0.996355i \(0.472815\pi\)
\(860\) 0 0
\(861\) 9.00000 0.306719
\(862\) 0 0
\(863\) −6.00000 −0.204242 −0.102121 0.994772i \(-0.532563\pi\)
−0.102121 + 0.994772i \(0.532563\pi\)
\(864\) 0 0
\(865\) 24.0000 0.816024
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 52.0000 1.76195
\(872\) 0 0
\(873\) 2.00000 0.0676897
\(874\) 0 0
\(875\) 1.00000 0.0338062
\(876\) 0 0
\(877\) −22.0000 −0.742887 −0.371444 0.928456i \(-0.621137\pi\)
−0.371444 + 0.928456i \(0.621137\pi\)
\(878\) 0 0
\(879\) −9.00000 −0.303562
\(880\) 0 0
\(881\) 18.0000 0.606435 0.303218 0.952921i \(-0.401939\pi\)
0.303218 + 0.952921i \(0.401939\pi\)
\(882\) 0 0
\(883\) 14.0000 0.471138 0.235569 0.971858i \(-0.424305\pi\)
0.235569 + 0.971858i \(0.424305\pi\)
\(884\) 0 0
\(885\) −3.00000 −0.100844
\(886\) 0 0
\(887\) 6.00000 0.201460 0.100730 0.994914i \(-0.467882\pi\)
0.100730 + 0.994914i \(0.467882\pi\)
\(888\) 0 0
\(889\) −14.0000 −0.469545
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −24.0000 −0.803129
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 4.00000 0.133556
\(898\) 0 0
\(899\) 21.0000 0.700389
\(900\) 0 0
\(901\) −27.0000 −0.899500
\(902\) 0 0
\(903\) 4.00000 0.133112
\(904\) 0 0
\(905\) 16.0000 0.531858
\(906\) 0 0
\(907\) −13.0000 −0.431658 −0.215829 0.976431i \(-0.569245\pi\)
−0.215829 + 0.976431i \(0.569245\pi\)
\(908\) 0 0
\(909\) −3.00000 −0.0995037
\(910\) 0 0
\(911\) −24.0000 −0.795155 −0.397578 0.917568i \(-0.630149\pi\)
−0.397578 + 0.917568i \(0.630149\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 10.0000 0.330590
\(916\) 0 0
\(917\) −12.0000 −0.396275
\(918\) 0 0
\(919\) −16.0000 −0.527791 −0.263896 0.964551i \(-0.585007\pi\)
−0.263896 + 0.964551i \(0.585007\pi\)
\(920\) 0 0
\(921\) 20.0000 0.659022
\(922\) 0 0
\(923\) −36.0000 −1.18495
\(924\) 0 0
\(925\) 11.0000 0.361678
\(926\) 0 0
\(927\) 8.00000 0.262754
\(928\) 0 0
\(929\) −15.0000 −0.492134 −0.246067 0.969253i \(-0.579138\pi\)
−0.246067 + 0.969253i \(0.579138\pi\)
\(930\) 0 0
\(931\) 24.0000 0.786568
\(932\) 0 0
\(933\) 24.0000 0.785725
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 17.0000 0.554774
\(940\) 0 0
\(941\) 30.0000 0.977972 0.488986 0.872292i \(-0.337367\pi\)
0.488986 + 0.872292i \(0.337367\pi\)
\(942\) 0 0
\(943\) 9.00000 0.293080
\(944\) 0 0
\(945\) 1.00000 0.0325300
\(946\) 0 0
\(947\) −48.0000 −1.55979 −0.779895 0.625910i \(-0.784728\pi\)
−0.779895 + 0.625910i \(0.784728\pi\)
\(948\) 0 0
\(949\) 64.0000 2.07753
\(950\) 0 0
\(951\) 18.0000 0.583690
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −6.00000 −0.194155
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −18.0000 −0.581250
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) 0 0
\(963\) −3.00000 −0.0966736
\(964\) 0 0
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) 26.0000 0.836104 0.418052 0.908423i \(-0.362713\pi\)
0.418052 + 0.908423i \(0.362713\pi\)
\(968\) 0 0
\(969\) 12.0000 0.385496
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 13.0000 0.416761
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 0 0
\(977\) −33.0000 −1.05576 −0.527882 0.849318i \(-0.677014\pi\)
−0.527882 + 0.849318i \(0.677014\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −4.00000 −0.127710
\(982\) 0 0
\(983\) −27.0000 −0.861166 −0.430583 0.902551i \(-0.641692\pi\)
−0.430583 + 0.902551i \(0.641692\pi\)
\(984\) 0 0
\(985\) 6.00000 0.191176
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) 0 0
\(989\) 4.00000 0.127193
\(990\) 0 0
\(991\) 53.0000 1.68360 0.841800 0.539789i \(-0.181496\pi\)
0.841800 + 0.539789i \(0.181496\pi\)
\(992\) 0 0
\(993\) −7.00000 −0.222138
\(994\) 0 0
\(995\) −2.00000 −0.0634043
\(996\) 0 0
\(997\) 8.00000 0.253363 0.126681 0.991943i \(-0.459567\pi\)
0.126681 + 0.991943i \(0.459567\pi\)
\(998\) 0 0
\(999\) 11.0000 0.348025
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 1380.2.a.c.1.1 1
3.2 odd 2 4140.2.a.i.1.1 1
4.3 odd 2 5520.2.a.g.1.1 1
5.2 odd 4 6900.2.f.e.6349.1 2
5.3 odd 4 6900.2.f.e.6349.2 2
5.4 even 2 6900.2.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1380.2.a.c.1.1 1 1.1 even 1 trivial
4140.2.a.i.1.1 1 3.2 odd 2
5520.2.a.g.1.1 1 4.3 odd 2
6900.2.a.b.1.1 1 5.4 even 2
6900.2.f.e.6349.1 2 5.2 odd 4
6900.2.f.e.6349.2 2 5.3 odd 4