Properties

Label 1815.2.a.g.1.2
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.73205 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.73205 q^{7} +1.00000 q^{9} +2.00000 q^{12} +1.00000 q^{15} +4.00000 q^{16} -3.46410 q^{17} -5.19615 q^{19} +2.00000 q^{20} -1.73205 q^{21} +6.00000 q^{23} +1.00000 q^{25} -1.00000 q^{27} -3.46410 q^{28} +6.92820 q^{29} +1.00000 q^{31} -1.73205 q^{35} -2.00000 q^{36} -5.00000 q^{37} +3.46410 q^{41} +10.3923 q^{43} -1.00000 q^{45} -12.0000 q^{47} -4.00000 q^{48} -4.00000 q^{49} +3.46410 q^{51} +6.00000 q^{53} +5.19615 q^{57} -2.00000 q^{60} -12.1244 q^{61} +1.73205 q^{63} -8.00000 q^{64} -5.00000 q^{67} +6.92820 q^{68} -6.00000 q^{69} -6.00000 q^{71} -1.73205 q^{73} -1.00000 q^{75} +10.3923 q^{76} -15.5885 q^{79} -4.00000 q^{80} +1.00000 q^{81} -6.92820 q^{83} +3.46410 q^{84} +3.46410 q^{85} -6.92820 q^{87} -6.00000 q^{89} -12.0000 q^{92} -1.00000 q^{93} +5.19615 q^{95} -13.0000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} - 4q^{4} - 2q^{5} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} - 4q^{4} - 2q^{5} + 2q^{9} + 4q^{12} + 2q^{15} + 8q^{16} + 4q^{20} + 12q^{23} + 2q^{25} - 2q^{27} + 2q^{31} - 4q^{36} - 10q^{37} - 2q^{45} - 24q^{47} - 8q^{48} - 8q^{49} + 12q^{53} - 4q^{60} - 16q^{64} - 10q^{67} - 12q^{69} - 12q^{71} - 2q^{75} - 8q^{80} + 2q^{81} - 12q^{89} - 24q^{92} - 2q^{93} - 26q^{97} + O(q^{100}) \)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −1.00000 −0.577350
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.73205 0.654654 0.327327 0.944911i \(-0.393852\pi\)
0.327327 + 0.944911i \(0.393852\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 0 0
\(12\) 2.00000 0.577350
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 1.00000 0.258199
\(16\) 4.00000 1.00000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) −5.19615 −1.19208 −0.596040 0.802955i \(-0.703260\pi\)
−0.596040 + 0.802955i \(0.703260\pi\)
\(20\) 2.00000 0.447214
\(21\) −1.73205 −0.377964
\(22\) 0 0
\(23\) 6.00000 1.25109 0.625543 0.780189i \(-0.284877\pi\)
0.625543 + 0.780189i \(0.284877\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −3.46410 −0.654654
\(29\) 6.92820 1.28654 0.643268 0.765641i \(-0.277578\pi\)
0.643268 + 0.765641i \(0.277578\pi\)
\(30\) 0 0
\(31\) 1.00000 0.179605 0.0898027 0.995960i \(-0.471376\pi\)
0.0898027 + 0.995960i \(0.471376\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.73205 −0.292770
\(36\) −2.00000 −0.333333
\(37\) −5.00000 −0.821995 −0.410997 0.911636i \(-0.634819\pi\)
−0.410997 + 0.911636i \(0.634819\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 0 0
\(43\) 10.3923 1.58481 0.792406 0.609994i \(-0.208828\pi\)
0.792406 + 0.609994i \(0.208828\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −12.0000 −1.75038 −0.875190 0.483779i \(-0.839264\pi\)
−0.875190 + 0.483779i \(0.839264\pi\)
\(48\) −4.00000 −0.577350
\(49\) −4.00000 −0.571429
\(50\) 0 0
\(51\) 3.46410 0.485071
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 5.19615 0.688247
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) −2.00000 −0.258199
\(61\) −12.1244 −1.55236 −0.776182 0.630509i \(-0.782846\pi\)
−0.776182 + 0.630509i \(0.782846\pi\)
\(62\) 0 0
\(63\) 1.73205 0.218218
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.00000 −0.610847 −0.305424 0.952217i \(-0.598798\pi\)
−0.305424 + 0.952217i \(0.598798\pi\)
\(68\) 6.92820 0.840168
\(69\) −6.00000 −0.722315
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −1.73205 −0.202721 −0.101361 0.994850i \(-0.532320\pi\)
−0.101361 + 0.994850i \(0.532320\pi\)
\(74\) 0 0
\(75\) −1.00000 −0.115470
\(76\) 10.3923 1.19208
\(77\) 0 0
\(78\) 0 0
\(79\) −15.5885 −1.75384 −0.876919 0.480638i \(-0.840405\pi\)
−0.876919 + 0.480638i \(0.840405\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −6.92820 −0.760469 −0.380235 0.924890i \(-0.624157\pi\)
−0.380235 + 0.924890i \(0.624157\pi\)
\(84\) 3.46410 0.377964
\(85\) 3.46410 0.375735
\(86\) 0 0
\(87\) −6.92820 −0.742781
\(88\) 0 0
\(89\) −6.00000 −0.635999 −0.317999 0.948091i \(-0.603011\pi\)
−0.317999 + 0.948091i \(0.603011\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −12.0000 −1.25109
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 5.19615 0.533114
\(96\) 0 0
\(97\) −13.0000 −1.31995 −0.659975 0.751288i \(-0.729433\pi\)
−0.659975 + 0.751288i \(0.729433\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −2.00000 −0.200000
\(101\) −13.8564 −1.37876 −0.689382 0.724398i \(-0.742118\pi\)
−0.689382 + 0.724398i \(0.742118\pi\)
\(102\) 0 0
\(103\) −13.0000 −1.28093 −0.640464 0.767988i \(-0.721258\pi\)
−0.640464 + 0.767988i \(0.721258\pi\)
\(104\) 0 0
\(105\) 1.73205 0.169031
\(106\) 0 0
\(107\) −6.92820 −0.669775 −0.334887 0.942258i \(-0.608698\pi\)
−0.334887 + 0.942258i \(0.608698\pi\)
\(108\) 2.00000 0.192450
\(109\) 12.1244 1.16130 0.580651 0.814152i \(-0.302798\pi\)
0.580651 + 0.814152i \(0.302798\pi\)
\(110\) 0 0
\(111\) 5.00000 0.474579
\(112\) 6.92820 0.654654
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) −6.00000 −0.559503
\(116\) −13.8564 −1.28654
\(117\) 0 0
\(118\) 0 0
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) −3.46410 −0.312348
\(124\) −2.00000 −0.179605
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 1.73205 0.153695 0.0768473 0.997043i \(-0.475515\pi\)
0.0768473 + 0.997043i \(0.475515\pi\)
\(128\) 0 0
\(129\) −10.3923 −0.914991
\(130\) 0 0
\(131\) 3.46410 0.302660 0.151330 0.988483i \(-0.451644\pi\)
0.151330 + 0.988483i \(0.451644\pi\)
\(132\) 0 0
\(133\) −9.00000 −0.780399
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 3.46410 0.293821 0.146911 0.989150i \(-0.453067\pi\)
0.146911 + 0.989150i \(0.453067\pi\)
\(140\) 3.46410 0.292770
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) 0 0
\(144\) 4.00000 0.333333
\(145\) −6.92820 −0.575356
\(146\) 0 0
\(147\) 4.00000 0.329914
\(148\) 10.0000 0.821995
\(149\) 24.2487 1.98653 0.993266 0.115857i \(-0.0369614\pi\)
0.993266 + 0.115857i \(0.0369614\pi\)
\(150\) 0 0
\(151\) 17.3205 1.40952 0.704761 0.709444i \(-0.251054\pi\)
0.704761 + 0.709444i \(0.251054\pi\)
\(152\) 0 0
\(153\) −3.46410 −0.280056
\(154\) 0 0
\(155\) −1.00000 −0.0803219
\(156\) 0 0
\(157\) −5.00000 −0.399043 −0.199522 0.979893i \(-0.563939\pi\)
−0.199522 + 0.979893i \(0.563939\pi\)
\(158\) 0 0
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 10.3923 0.819028
\(162\) 0 0
\(163\) −13.0000 −1.01824 −0.509119 0.860696i \(-0.670029\pi\)
−0.509119 + 0.860696i \(0.670029\pi\)
\(164\) −6.92820 −0.541002
\(165\) 0 0
\(166\) 0 0
\(167\) 3.46410 0.268060 0.134030 0.990977i \(-0.457208\pi\)
0.134030 + 0.990977i \(0.457208\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) −5.19615 −0.397360
\(172\) −20.7846 −1.58481
\(173\) −10.3923 −0.790112 −0.395056 0.918657i \(-0.629275\pi\)
−0.395056 + 0.918657i \(0.629275\pi\)
\(174\) 0 0
\(175\) 1.73205 0.130931
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 2.00000 0.149071
\(181\) −7.00000 −0.520306 −0.260153 0.965567i \(-0.583773\pi\)
−0.260153 + 0.965567i \(0.583773\pi\)
\(182\) 0 0
\(183\) 12.1244 0.896258
\(184\) 0 0
\(185\) 5.00000 0.367607
\(186\) 0 0
\(187\) 0 0
\(188\) 24.0000 1.75038
\(189\) −1.73205 −0.125988
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) 8.00000 0.577350
\(193\) 1.73205 0.124676 0.0623379 0.998055i \(-0.480144\pi\)
0.0623379 + 0.998055i \(0.480144\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 8.00000 0.571429
\(197\) −6.92820 −0.493614 −0.246807 0.969065i \(-0.579381\pi\)
−0.246807 + 0.969065i \(0.579381\pi\)
\(198\) 0 0
\(199\) −11.0000 −0.779769 −0.389885 0.920864i \(-0.627485\pi\)
−0.389885 + 0.920864i \(0.627485\pi\)
\(200\) 0 0
\(201\) 5.00000 0.352673
\(202\) 0 0
\(203\) 12.0000 0.842235
\(204\) −6.92820 −0.485071
\(205\) −3.46410 −0.241943
\(206\) 0 0
\(207\) 6.00000 0.417029
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −19.0526 −1.31163 −0.655816 0.754921i \(-0.727675\pi\)
−0.655816 + 0.754921i \(0.727675\pi\)
\(212\) −12.0000 −0.824163
\(213\) 6.00000 0.411113
\(214\) 0 0
\(215\) −10.3923 −0.708749
\(216\) 0 0
\(217\) 1.73205 0.117579
\(218\) 0 0
\(219\) 1.73205 0.117041
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −7.00000 −0.468755 −0.234377 0.972146i \(-0.575305\pi\)
−0.234377 + 0.972146i \(0.575305\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 27.7128 1.83936 0.919682 0.392664i \(-0.128446\pi\)
0.919682 + 0.392664i \(0.128446\pi\)
\(228\) −10.3923 −0.688247
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 13.8564 0.907763 0.453882 0.891062i \(-0.350039\pi\)
0.453882 + 0.891062i \(0.350039\pi\)
\(234\) 0 0
\(235\) 12.0000 0.782794
\(236\) 0 0
\(237\) 15.5885 1.01258
\(238\) 0 0
\(239\) −20.7846 −1.34444 −0.672222 0.740349i \(-0.734660\pi\)
−0.672222 + 0.740349i \(0.734660\pi\)
\(240\) 4.00000 0.258199
\(241\) 6.92820 0.446285 0.223142 0.974786i \(-0.428369\pi\)
0.223142 + 0.974786i \(0.428369\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 24.2487 1.55236
\(245\) 4.00000 0.255551
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 6.92820 0.439057
\(250\) 0 0
\(251\) 30.0000 1.89358 0.946792 0.321847i \(-0.104304\pi\)
0.946792 + 0.321847i \(0.104304\pi\)
\(252\) −3.46410 −0.218218
\(253\) 0 0
\(254\) 0 0
\(255\) −3.46410 −0.216930
\(256\) 16.0000 1.00000
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) −8.66025 −0.538122
\(260\) 0 0
\(261\) 6.92820 0.428845
\(262\) 0 0
\(263\) −10.3923 −0.640817 −0.320408 0.947279i \(-0.603820\pi\)
−0.320408 + 0.947279i \(0.603820\pi\)
\(264\) 0 0
\(265\) −6.00000 −0.368577
\(266\) 0 0
\(267\) 6.00000 0.367194
\(268\) 10.0000 0.610847
\(269\) 24.0000 1.46331 0.731653 0.681677i \(-0.238749\pi\)
0.731653 + 0.681677i \(0.238749\pi\)
\(270\) 0 0
\(271\) −10.3923 −0.631288 −0.315644 0.948878i \(-0.602220\pi\)
−0.315644 + 0.948878i \(0.602220\pi\)
\(272\) −13.8564 −0.840168
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 12.0000 0.722315
\(277\) 25.9808 1.56103 0.780516 0.625135i \(-0.214956\pi\)
0.780516 + 0.625135i \(0.214956\pi\)
\(278\) 0 0
\(279\) 1.00000 0.0598684
\(280\) 0 0
\(281\) 31.1769 1.85986 0.929929 0.367738i \(-0.119868\pi\)
0.929929 + 0.367738i \(0.119868\pi\)
\(282\) 0 0
\(283\) 15.5885 0.926638 0.463319 0.886192i \(-0.346658\pi\)
0.463319 + 0.886192i \(0.346658\pi\)
\(284\) 12.0000 0.712069
\(285\) −5.19615 −0.307794
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) 0 0
\(291\) 13.0000 0.762073
\(292\) 3.46410 0.202721
\(293\) −31.1769 −1.82137 −0.910687 0.413096i \(-0.864447\pi\)
−0.910687 + 0.413096i \(0.864447\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 2.00000 0.115470
\(301\) 18.0000 1.03750
\(302\) 0 0
\(303\) 13.8564 0.796030
\(304\) −20.7846 −1.19208
\(305\) 12.1244 0.694239
\(306\) 0 0
\(307\) −8.66025 −0.494267 −0.247133 0.968981i \(-0.579489\pi\)
−0.247133 + 0.968981i \(0.579489\pi\)
\(308\) 0 0
\(309\) 13.0000 0.739544
\(310\) 0 0
\(311\) 18.0000 1.02069 0.510343 0.859971i \(-0.329518\pi\)
0.510343 + 0.859971i \(0.329518\pi\)
\(312\) 0 0
\(313\) −26.0000 −1.46961 −0.734803 0.678280i \(-0.762726\pi\)
−0.734803 + 0.678280i \(0.762726\pi\)
\(314\) 0 0
\(315\) −1.73205 −0.0975900
\(316\) 31.1769 1.75384
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 8.00000 0.447214
\(321\) 6.92820 0.386695
\(322\) 0 0
\(323\) 18.0000 1.00155
\(324\) −2.00000 −0.111111
\(325\) 0 0
\(326\) 0 0
\(327\) −12.1244 −0.670478
\(328\) 0 0
\(329\) −20.7846 −1.14589
\(330\) 0 0
\(331\) −19.0000 −1.04433 −0.522167 0.852843i \(-0.674876\pi\)
−0.522167 + 0.852843i \(0.674876\pi\)
\(332\) 13.8564 0.760469
\(333\) −5.00000 −0.273998
\(334\) 0 0
\(335\) 5.00000 0.273179
\(336\) −6.92820 −0.377964
\(337\) −29.4449 −1.60396 −0.801982 0.597348i \(-0.796221\pi\)
−0.801982 + 0.597348i \(0.796221\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −6.92820 −0.375735
\(341\) 0 0
\(342\) 0 0
\(343\) −19.0526 −1.02874
\(344\) 0 0
\(345\) 6.00000 0.323029
\(346\) 0 0
\(347\) 17.3205 0.929814 0.464907 0.885360i \(-0.346088\pi\)
0.464907 + 0.885360i \(0.346088\pi\)
\(348\) 13.8564 0.742781
\(349\) 5.19615 0.278144 0.139072 0.990282i \(-0.455588\pi\)
0.139072 + 0.990282i \(0.455588\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 12.0000 0.635999
\(357\) 6.00000 0.317554
\(358\) 0 0
\(359\) −24.2487 −1.27980 −0.639899 0.768459i \(-0.721024\pi\)
−0.639899 + 0.768459i \(0.721024\pi\)
\(360\) 0 0
\(361\) 8.00000 0.421053
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1.73205 0.0906597
\(366\) 0 0
\(367\) −28.0000 −1.46159 −0.730794 0.682598i \(-0.760850\pi\)
−0.730794 + 0.682598i \(0.760850\pi\)
\(368\) 24.0000 1.25109
\(369\) 3.46410 0.180334
\(370\) 0 0
\(371\) 10.3923 0.539542
\(372\) 2.00000 0.103695
\(373\) −19.0526 −0.986504 −0.493252 0.869886i \(-0.664192\pi\)
−0.493252 + 0.869886i \(0.664192\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) −10.3923 −0.533114
\(381\) −1.73205 −0.0887357
\(382\) 0 0
\(383\) −6.00000 −0.306586 −0.153293 0.988181i \(-0.548988\pi\)
−0.153293 + 0.988181i \(0.548988\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 10.3923 0.528271
\(388\) 26.0000 1.31995
\(389\) 6.00000 0.304212 0.152106 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) −20.7846 −1.05112
\(392\) 0 0
\(393\) −3.46410 −0.174741
\(394\) 0 0
\(395\) 15.5885 0.784340
\(396\) 0 0
\(397\) −19.0000 −0.953583 −0.476791 0.879017i \(-0.658200\pi\)
−0.476791 + 0.879017i \(0.658200\pi\)
\(398\) 0 0
\(399\) 9.00000 0.450564
\(400\) 4.00000 0.200000
\(401\) −30.0000 −1.49813 −0.749064 0.662497i \(-0.769497\pi\)
−0.749064 + 0.662497i \(0.769497\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 27.7128 1.37876
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 19.0526 0.942088 0.471044 0.882110i \(-0.343877\pi\)
0.471044 + 0.882110i \(0.343877\pi\)
\(410\) 0 0
\(411\) 12.0000 0.591916
\(412\) 26.0000 1.28093
\(413\) 0 0
\(414\) 0 0
\(415\) 6.92820 0.340092
\(416\) 0 0
\(417\) −3.46410 −0.169638
\(418\) 0 0
\(419\) 12.0000 0.586238 0.293119 0.956076i \(-0.405307\pi\)
0.293119 + 0.956076i \(0.405307\pi\)
\(420\) −3.46410 −0.169031
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) −12.0000 −0.583460
\(424\) 0 0
\(425\) −3.46410 −0.168034
\(426\) 0 0
\(427\) −21.0000 −1.01626
\(428\) 13.8564 0.669775
\(429\) 0 0
\(430\) 0 0
\(431\) −38.1051 −1.83546 −0.917729 0.397206i \(-0.869980\pi\)
−0.917729 + 0.397206i \(0.869980\pi\)
\(432\) −4.00000 −0.192450
\(433\) −7.00000 −0.336399 −0.168199 0.985753i \(-0.553795\pi\)
−0.168199 + 0.985753i \(0.553795\pi\)
\(434\) 0 0
\(435\) 6.92820 0.332182
\(436\) −24.2487 −1.16130
\(437\) −31.1769 −1.49139
\(438\) 0 0
\(439\) 15.5885 0.743996 0.371998 0.928233i \(-0.378673\pi\)
0.371998 + 0.928233i \(0.378673\pi\)
\(440\) 0 0
\(441\) −4.00000 −0.190476
\(442\) 0 0
\(443\) 18.0000 0.855206 0.427603 0.903967i \(-0.359358\pi\)
0.427603 + 0.903967i \(0.359358\pi\)
\(444\) −10.0000 −0.474579
\(445\) 6.00000 0.284427
\(446\) 0 0
\(447\) −24.2487 −1.14692
\(448\) −13.8564 −0.654654
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −17.3205 −0.813788
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 34.6410 1.62044 0.810219 0.586127i \(-0.199348\pi\)
0.810219 + 0.586127i \(0.199348\pi\)
\(458\) 0 0
\(459\) 3.46410 0.161690
\(460\) 12.0000 0.559503
\(461\) −17.3205 −0.806696 −0.403348 0.915047i \(-0.632154\pi\)
−0.403348 + 0.915047i \(0.632154\pi\)
\(462\) 0 0
\(463\) −40.0000 −1.85896 −0.929479 0.368875i \(-0.879743\pi\)
−0.929479 + 0.368875i \(0.879743\pi\)
\(464\) 27.7128 1.28654
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) −6.00000 −0.277647 −0.138823 0.990317i \(-0.544332\pi\)
−0.138823 + 0.990317i \(0.544332\pi\)
\(468\) 0 0
\(469\) −8.66025 −0.399893
\(470\) 0 0
\(471\) 5.00000 0.230388
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −5.19615 −0.238416
\(476\) 12.0000 0.550019
\(477\) 6.00000 0.274721
\(478\) 0 0
\(479\) 27.7128 1.26623 0.633115 0.774057i \(-0.281776\pi\)
0.633115 + 0.774057i \(0.281776\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −10.3923 −0.472866
\(484\) 0 0
\(485\) 13.0000 0.590300
\(486\) 0 0
\(487\) 16.0000 0.725029 0.362515 0.931978i \(-0.381918\pi\)
0.362515 + 0.931978i \(0.381918\pi\)
\(488\) 0 0
\(489\) 13.0000 0.587880
\(490\) 0 0
\(491\) 24.2487 1.09433 0.547165 0.837025i \(-0.315707\pi\)
0.547165 + 0.837025i \(0.315707\pi\)
\(492\) 6.92820 0.312348
\(493\) −24.0000 −1.08091
\(494\) 0 0
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −10.3923 −0.466159
\(498\) 0 0
\(499\) 1.00000 0.0447661 0.0223831 0.999749i \(-0.492875\pi\)
0.0223831 + 0.999749i \(0.492875\pi\)
\(500\) 2.00000 0.0894427
\(501\) −3.46410 −0.154765
\(502\) 0 0
\(503\) 34.6410 1.54457 0.772283 0.635278i \(-0.219115\pi\)
0.772283 + 0.635278i \(0.219115\pi\)
\(504\) 0 0
\(505\) 13.8564 0.616602
\(506\) 0 0
\(507\) 13.0000 0.577350
\(508\) −3.46410 −0.153695
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −3.00000 −0.132712
\(512\) 0 0
\(513\) 5.19615 0.229416
\(514\) 0 0
\(515\) 13.0000 0.572848
\(516\) 20.7846 0.914991
\(517\) 0 0
\(518\) 0 0
\(519\) 10.3923 0.456172
\(520\) 0 0
\(521\) 36.0000 1.57719 0.788594 0.614914i \(-0.210809\pi\)
0.788594 + 0.614914i \(0.210809\pi\)
\(522\) 0 0
\(523\) 29.4449 1.28753 0.643767 0.765222i \(-0.277371\pi\)
0.643767 + 0.765222i \(0.277371\pi\)
\(524\) −6.92820 −0.302660
\(525\) −1.73205 −0.0755929
\(526\) 0 0
\(527\) −3.46410 −0.150899
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 0 0
\(532\) 18.0000 0.780399
\(533\) 0 0
\(534\) 0 0
\(535\) 6.92820 0.299532
\(536\) 0 0
\(537\) −12.0000 −0.517838
\(538\) 0 0
\(539\) 0 0
\(540\) −2.00000 −0.0860663
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 7.00000 0.300399
\(544\) 0 0
\(545\) −12.1244 −0.519350
\(546\) 0 0
\(547\) −31.1769 −1.33303 −0.666514 0.745492i \(-0.732214\pi\)
−0.666514 + 0.745492i \(0.732214\pi\)
\(548\) 24.0000 1.02523
\(549\) −12.1244 −0.517455
\(550\) 0 0
\(551\) −36.0000 −1.53365
\(552\) 0 0
\(553\) −27.0000 −1.14816
\(554\) 0 0
\(555\) −5.00000 −0.212238
\(556\) −6.92820 −0.293821
\(557\) 3.46410 0.146779 0.0733893 0.997303i \(-0.476618\pi\)
0.0733893 + 0.997303i \(0.476618\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −6.92820 −0.292770
\(561\) 0 0
\(562\) 0 0
\(563\) 34.6410 1.45994 0.729972 0.683477i \(-0.239533\pi\)
0.729972 + 0.683477i \(0.239533\pi\)
\(564\) −24.0000 −1.01058
\(565\) 0 0
\(566\) 0 0
\(567\) 1.73205 0.0727393
\(568\) 0 0
\(569\) −20.7846 −0.871336 −0.435668 0.900107i \(-0.643488\pi\)
−0.435668 + 0.900107i \(0.643488\pi\)
\(570\) 0 0
\(571\) 19.0526 0.797325 0.398662 0.917098i \(-0.369475\pi\)
0.398662 + 0.917098i \(0.369475\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 0 0
\(575\) 6.00000 0.250217
\(576\) −8.00000 −0.333333
\(577\) 7.00000 0.291414 0.145707 0.989328i \(-0.453454\pi\)
0.145707 + 0.989328i \(0.453454\pi\)
\(578\) 0 0
\(579\) −1.73205 −0.0719816
\(580\) 13.8564 0.575356
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −8.00000 −0.329914
\(589\) −5.19615 −0.214104
\(590\) 0 0
\(591\) 6.92820 0.284988
\(592\) −20.0000 −0.821995
\(593\) −10.3923 −0.426761 −0.213380 0.976969i \(-0.568447\pi\)
−0.213380 + 0.976969i \(0.568447\pi\)
\(594\) 0 0
\(595\) 6.00000 0.245976
\(596\) −48.4974 −1.98653
\(597\) 11.0000 0.450200
\(598\) 0 0
\(599\) 6.00000 0.245153 0.122577 0.992459i \(-0.460884\pi\)
0.122577 + 0.992459i \(0.460884\pi\)
\(600\) 0 0
\(601\) 22.5167 0.918474 0.459237 0.888314i \(-0.348123\pi\)
0.459237 + 0.888314i \(0.348123\pi\)
\(602\) 0 0
\(603\) −5.00000 −0.203616
\(604\) −34.6410 −1.40952
\(605\) 0 0
\(606\) 0 0
\(607\) −3.46410 −0.140604 −0.0703018 0.997526i \(-0.522396\pi\)
−0.0703018 + 0.997526i \(0.522396\pi\)
\(608\) 0 0
\(609\) −12.0000 −0.486265
\(610\) 0 0
\(611\) 0 0
\(612\) 6.92820 0.280056
\(613\) −29.4449 −1.18927 −0.594633 0.803997i \(-0.702703\pi\)
−0.594633 + 0.803997i \(0.702703\pi\)
\(614\) 0 0
\(615\) 3.46410 0.139686
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −4.00000 −0.160774 −0.0803868 0.996764i \(-0.525616\pi\)
−0.0803868 + 0.996764i \(0.525616\pi\)
\(620\) 2.00000 0.0803219
\(621\) −6.00000 −0.240772
\(622\) 0 0
\(623\) −10.3923 −0.416359
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 0 0
\(628\) 10.0000 0.399043
\(629\) 17.3205 0.690614
\(630\) 0 0
\(631\) 16.0000 0.636950 0.318475 0.947931i \(-0.396829\pi\)
0.318475 + 0.947931i \(0.396829\pi\)
\(632\) 0 0
\(633\) 19.0526 0.757271
\(634\) 0 0
\(635\) −1.73205 −0.0687343
\(636\) 12.0000 0.475831
\(637\) 0 0
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −31.0000 −1.22252 −0.611260 0.791430i \(-0.709337\pi\)
−0.611260 + 0.791430i \(0.709337\pi\)
\(644\) −20.7846 −0.819028
\(645\) 10.3923 0.409197
\(646\) 0 0
\(647\) 12.0000 0.471769 0.235884 0.971781i \(-0.424201\pi\)
0.235884 + 0.971781i \(0.424201\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −1.73205 −0.0678844
\(652\) 26.0000 1.01824
\(653\) −42.0000 −1.64359 −0.821794 0.569785i \(-0.807026\pi\)
−0.821794 + 0.569785i \(0.807026\pi\)
\(654\) 0 0
\(655\) −3.46410 −0.135354
\(656\) 13.8564 0.541002
\(657\) −1.73205 −0.0675737
\(658\) 0 0
\(659\) −10.3923 −0.404827 −0.202413 0.979300i \(-0.564878\pi\)
−0.202413 + 0.979300i \(0.564878\pi\)
\(660\) 0 0
\(661\) 49.0000 1.90588 0.952940 0.303160i \(-0.0980418\pi\)
0.952940 + 0.303160i \(0.0980418\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 9.00000 0.349005
\(666\) 0 0
\(667\) 41.5692 1.60957
\(668\) −6.92820 −0.268060
\(669\) 7.00000 0.270636
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −15.5885 −0.600891 −0.300445 0.953799i \(-0.597135\pi\)
−0.300445 + 0.953799i \(0.597135\pi\)
\(674\) 0 0
\(675\) −1.00000 −0.0384900
\(676\) 26.0000 1.00000
\(677\) 20.7846 0.798817 0.399409 0.916773i \(-0.369215\pi\)
0.399409 + 0.916773i \(0.369215\pi\)
\(678\) 0 0
\(679\) −22.5167 −0.864110
\(680\) 0 0
\(681\) −27.7128 −1.06196
\(682\) 0 0
\(683\) 42.0000 1.60709 0.803543 0.595247i \(-0.202946\pi\)
0.803543 + 0.595247i \(0.202946\pi\)
\(684\) 10.3923 0.397360
\(685\) 12.0000 0.458496
\(686\) 0 0
\(687\) 14.0000 0.534133
\(688\) 41.5692 1.58481
\(689\) 0 0
\(690\) 0 0
\(691\) 35.0000 1.33146 0.665731 0.746191i \(-0.268120\pi\)
0.665731 + 0.746191i \(0.268120\pi\)
\(692\) 20.7846 0.790112
\(693\) 0 0
\(694\) 0 0
\(695\) −3.46410 −0.131401
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) 0 0
\(699\) −13.8564 −0.524097
\(700\) −3.46410 −0.130931
\(701\) 24.2487 0.915861 0.457931 0.888988i \(-0.348591\pi\)
0.457931 + 0.888988i \(0.348591\pi\)
\(702\) 0 0
\(703\) 25.9808 0.979883
\(704\) 0 0
\(705\) −12.0000 −0.451946
\(706\) 0 0
\(707\) −24.0000 −0.902613
\(708\) 0 0
\(709\) 10.0000 0.375558 0.187779 0.982211i \(-0.439871\pi\)
0.187779 + 0.982211i \(0.439871\pi\)
\(710\) 0 0
\(711\) −15.5885 −0.584613
\(712\) 0 0
\(713\) 6.00000 0.224702
\(714\) 0 0
\(715\) 0 0
\(716\) −24.0000 −0.896922
\(717\) 20.7846 0.776215
\(718\) 0 0
\(719\) 12.0000 0.447524 0.223762 0.974644i \(-0.428166\pi\)
0.223762 + 0.974644i \(0.428166\pi\)
\(720\) −4.00000 −0.149071
\(721\) −22.5167 −0.838564
\(722\) 0 0
\(723\) −6.92820 −0.257663
\(724\) 14.0000 0.520306
\(725\) 6.92820 0.257307
\(726\) 0 0
\(727\) 40.0000 1.48352 0.741759 0.670667i \(-0.233992\pi\)
0.741759 + 0.670667i \(0.233992\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −36.0000 −1.33151
\(732\) −24.2487 −0.896258
\(733\) 27.7128 1.02360 0.511798 0.859106i \(-0.328980\pi\)
0.511798 + 0.859106i \(0.328980\pi\)
\(734\) 0 0
\(735\) −4.00000 −0.147542
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −15.5885 −0.573431 −0.286715 0.958016i \(-0.592563\pi\)
−0.286715 + 0.958016i \(0.592563\pi\)
\(740\) −10.0000 −0.367607
\(741\) 0 0
\(742\) 0 0
\(743\) −20.7846 −0.762513 −0.381257 0.924469i \(-0.624509\pi\)
−0.381257 + 0.924469i \(0.624509\pi\)
\(744\) 0 0
\(745\) −24.2487 −0.888404
\(746\) 0 0
\(747\) −6.92820 −0.253490
\(748\) 0 0
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −11.0000 −0.401396 −0.200698 0.979653i \(-0.564321\pi\)
−0.200698 + 0.979653i \(0.564321\pi\)
\(752\) −48.0000 −1.75038
\(753\) −30.0000 −1.09326
\(754\) 0 0
\(755\) −17.3205 −0.630358
\(756\) 3.46410 0.125988
\(757\) 13.0000 0.472493 0.236247 0.971693i \(-0.424083\pi\)
0.236247 + 0.971693i \(0.424083\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −3.46410 −0.125574 −0.0627868 0.998027i \(-0.519999\pi\)
−0.0627868 + 0.998027i \(0.519999\pi\)
\(762\) 0 0
\(763\) 21.0000 0.760251
\(764\) −48.0000 −1.73658
\(765\) 3.46410 0.125245
\(766\) 0 0
\(767\) 0 0
\(768\) −16.0000 −0.577350
\(769\) 1.73205 0.0624593 0.0312297 0.999512i \(-0.490058\pi\)
0.0312297 + 0.999512i \(0.490058\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) −3.46410 −0.124676
\(773\) −24.0000 −0.863220 −0.431610 0.902060i \(-0.642054\pi\)
−0.431610 + 0.902060i \(0.642054\pi\)
\(774\) 0 0
\(775\) 1.00000 0.0359211
\(776\) 0 0
\(777\) 8.66025 0.310685
\(778\) 0 0
\(779\) −18.0000 −0.644917
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −6.92820 −0.247594
\(784\) −16.0000 −0.571429
\(785\) 5.00000 0.178458
\(786\) 0 0
\(787\) −24.2487 −0.864373 −0.432187 0.901784i \(-0.642258\pi\)
−0.432187 + 0.901784i \(0.642258\pi\)
\(788\) 13.8564 0.493614
\(789\) 10.3923 0.369976
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 6.00000 0.212798
\(796\) 22.0000 0.779769
\(797\) 18.0000 0.637593 0.318796 0.947823i \(-0.396721\pi\)
0.318796 + 0.947823i \(0.396721\pi\)
\(798\) 0 0
\(799\) 41.5692 1.47061
\(800\) 0 0
\(801\) −6.00000 −0.212000
\(802\) 0 0
\(803\) 0 0
\(804\) −10.0000 −0.352673
\(805\) −10.3923 −0.366281
\(806\) 0 0
\(807\) −24.0000 −0.844840
\(808\) 0 0
\(809\) −38.1051 −1.33970 −0.669852 0.742494i \(-0.733643\pi\)
−0.669852 + 0.742494i \(0.733643\pi\)
\(810\) 0 0
\(811\) 29.4449 1.03395 0.516975 0.856001i \(-0.327058\pi\)
0.516975 + 0.856001i \(0.327058\pi\)
\(812\) −24.0000 −0.842235
\(813\) 10.3923 0.364474
\(814\) 0 0
\(815\) 13.0000 0.455370
\(816\) 13.8564 0.485071
\(817\) −54.0000 −1.88922
\(818\) 0 0
\(819\) 0 0
\(820\) 6.92820 0.241943
\(821\) 10.3923 0.362694 0.181347 0.983419i \(-0.441954\pi\)
0.181347 + 0.983419i \(0.441954\pi\)
\(822\) 0 0
\(823\) 23.0000 0.801730 0.400865 0.916137i \(-0.368710\pi\)
0.400865 + 0.916137i \(0.368710\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −41.5692 −1.44550 −0.722752 0.691108i \(-0.757123\pi\)
−0.722752 + 0.691108i \(0.757123\pi\)
\(828\) −12.0000 −0.417029
\(829\) −55.0000 −1.91023 −0.955114 0.296237i \(-0.904268\pi\)
−0.955114 + 0.296237i \(0.904268\pi\)
\(830\) 0 0
\(831\) −25.9808 −0.901263
\(832\) 0 0
\(833\) 13.8564 0.480096
\(834\) 0 0
\(835\) −3.46410 −0.119880
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) 0 0
\(843\) −31.1769 −1.07379
\(844\) 38.1051 1.31163
\(845\) 13.0000 0.447214
\(846\) 0 0
\(847\) 0 0
\(848\) 24.0000 0.824163
\(849\) −15.5885 −0.534994
\(850\) 0 0
\(851\) −30.0000 −1.02839
\(852\) −12.0000 −0.411113
\(853\) −43.3013 −1.48261 −0.741304 0.671170i \(-0.765792\pi\)
−0.741304 + 0.671170i \(0.765792\pi\)
\(854\) 0 0
\(855\) 5.19615 0.177705
\(856\) 0 0
\(857\) −10.3923 −0.354994 −0.177497 0.984121i \(-0.556800\pi\)
−0.177497 + 0.984121i \(0.556800\pi\)
\(858\) 0 0
\(859\) 23.0000 0.784750 0.392375 0.919805i \(-0.371654\pi\)
0.392375 + 0.919805i \(0.371654\pi\)
\(860\) 20.7846 0.708749
\(861\) −6.00000 −0.204479
\(862\) 0 0
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 10.3923 0.353349
\(866\) 0 0
\(867\) 5.00000 0.169809
\(868\) −3.46410 −0.117579
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −13.0000 −0.439983
\(874\) 0 0
\(875\) −1.73205 −0.0585540
\(876\) −3.46410 −0.117041
\(877\) −29.4449 −0.994282 −0.497141 0.867670i \(-0.665617\pi\)
−0.497141 + 0.867670i \(0.665617\pi\)
\(878\) 0 0
\(879\) 31.1769 1.05157
\(880\) 0 0
\(881\) −24.0000 −0.808581 −0.404290 0.914631i \(-0.632481\pi\)
−0.404290 + 0.914631i \(0.632481\pi\)
\(882\) 0 0
\(883\) −11.0000 −0.370179 −0.185090 0.982722i \(-0.559258\pi\)
−0.185090 + 0.982722i \(0.559258\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −17.3205 −0.581566 −0.290783 0.956789i \(-0.593916\pi\)
−0.290783 + 0.956789i \(0.593916\pi\)
\(888\) 0 0
\(889\) 3.00000 0.100617
\(890\) 0 0
\(891\) 0 0
\(892\) 14.0000 0.468755
\(893\) 62.3538 2.08659
\(894\) 0 0
\(895\) −12.0000 −0.401116
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 6.92820 0.231069
\(900\) −2.00000 −0.0666667
\(901\) −20.7846 −0.692436
\(902\) 0 0
\(903\) −18.0000 −0.599002
\(904\) 0 0
\(905\) 7.00000 0.232688
\(906\) 0 0
\(907\) −1.00000 −0.0332045 −0.0166022 0.999862i \(-0.505285\pi\)
−0.0166022 + 0.999862i \(0.505285\pi\)
\(908\) −55.4256 −1.83936
\(909\) −13.8564 −0.459588
\(910\) 0 0
\(911\) −12.0000 −0.397578 −0.198789 0.980042i \(-0.563701\pi\)
−0.198789 + 0.980042i \(0.563701\pi\)
\(912\) 20.7846 0.688247
\(913\) 0 0
\(914\) 0 0
\(915\) −12.1244 −0.400819
\(916\) 28.0000 0.925146
\(917\) 6.00000 0.198137
\(918\) 0 0
\(919\) 15.5885 0.514216 0.257108 0.966383i \(-0.417230\pi\)
0.257108 + 0.966383i \(0.417230\pi\)
\(920\) 0 0
\(921\) 8.66025 0.285365
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −5.00000 −0.164399
\(926\) 0 0
\(927\) −13.0000 −0.426976
\(928\) 0 0
\(929\) −30.0000 −0.984268 −0.492134 0.870519i \(-0.663783\pi\)
−0.492134 + 0.870519i \(0.663783\pi\)
\(930\) 0 0
\(931\) 20.7846 0.681188
\(932\) −27.7128 −0.907763
\(933\) −18.0000 −0.589294
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 29.4449 0.961922 0.480961 0.876742i \(-0.340288\pi\)
0.480961 + 0.876742i \(0.340288\pi\)
\(938\) 0 0
\(939\) 26.0000 0.848478
\(940\) −24.0000 −0.782794
\(941\) 24.2487 0.790485 0.395243 0.918577i \(-0.370660\pi\)
0.395243 + 0.918577i \(0.370660\pi\)
\(942\) 0 0
\(943\) 20.7846 0.676840
\(944\) 0 0
\(945\) 1.73205 0.0563436
\(946\) 0 0
\(947\) −36.0000 −1.16984 −0.584921 0.811090i \(-0.698875\pi\)
−0.584921 + 0.811090i \(0.698875\pi\)
\(948\) −31.1769 −1.01258
\(949\) 0 0
\(950\) 0 0
\(951\) 30.0000 0.972817
\(952\) 0 0
\(953\) −17.3205 −0.561066 −0.280533 0.959844i \(-0.590511\pi\)
−0.280533 + 0.959844i \(0.590511\pi\)
\(954\) 0 0
\(955\) −24.0000 −0.776622
\(956\) 41.5692 1.34444
\(957\) 0 0
\(958\) 0 0
\(959\) −20.7846 −0.671170
\(960\) −8.00000 −0.258199
\(961\) −30.0000 −0.967742
\(962\) 0 0
\(963\) −6.92820 −0.223258
\(964\) −13.8564 −0.446285
\(965\) −1.73205 −0.0557567
\(966\) 0 0
\(967\) 43.3013 1.39247 0.696237 0.717812i \(-0.254856\pi\)
0.696237 + 0.717812i \(0.254856\pi\)
\(968\) 0 0
\(969\) −18.0000 −0.578243
\(970\) 0 0
\(971\) 18.0000 0.577647 0.288824 0.957382i \(-0.406736\pi\)
0.288824 + 0.957382i \(0.406736\pi\)
\(972\) 2.00000 0.0641500
\(973\) 6.00000 0.192351
\(974\) 0 0
\(975\) 0 0
\(976\) −48.4974 −1.55236
\(977\) −30.0000 −0.959785 −0.479893 0.877327i \(-0.659324\pi\)
−0.479893 + 0.877327i \(0.659324\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −8.00000 −0.255551
\(981\) 12.1244 0.387101
\(982\) 0 0
\(983\) −30.0000 −0.956851 −0.478426 0.878128i \(-0.658792\pi\)
−0.478426 + 0.878128i \(0.658792\pi\)
\(984\) 0 0
\(985\) 6.92820 0.220751
\(986\) 0 0
\(987\) 20.7846 0.661581
\(988\) 0 0
\(989\) 62.3538 1.98274
\(990\) 0 0
\(991\) 16.0000 0.508257 0.254128 0.967170i \(-0.418211\pi\)
0.254128 + 0.967170i \(0.418211\pi\)
\(992\) 0 0
\(993\) 19.0000 0.602947
\(994\) 0 0
\(995\) 11.0000 0.348723
\(996\) −13.8564 −0.439057
\(997\) 8.66025 0.274273 0.137136 0.990552i \(-0.456210\pi\)
0.137136 + 0.990552i \(0.456210\pi\)
\(998\) 0 0
\(999\) 5.00000 0.158193
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 1815.2.a.g.1.2 yes 2
3.2 odd 2 5445.2.a.q.1.2 2
5.4 even 2 9075.2.a.bl.1.1 2
11.10 odd 2 inner 1815.2.a.g.1.1 2
33.32 even 2 5445.2.a.q.1.1 2
55.54 odd 2 9075.2.a.bl.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1815.2.a.g.1.1 2 11.10 odd 2 inner
1815.2.a.g.1.2 yes 2 1.1 even 1 trivial
5445.2.a.q.1.1 2 33.32 even 2
5445.2.a.q.1.2 2 3.2 odd 2
9075.2.a.bl.1.1 2 5.4 even 2
9075.2.a.bl.1.2 2 55.54 odd 2