Properties

Label 1521.2.a.j.1.1
Level $1521$
Weight $2$
Character 1521.1
Self dual yes
Analytic conductor $12.145$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1521 = 3^{2} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1521.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(12.1452461474\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 117)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 1521.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.73205 q^{8} +O(q^{10})\) \(q-1.73205 q^{2} +1.00000 q^{4} -2.00000 q^{7} +1.73205 q^{8} +3.46410 q^{11} +3.46410 q^{14} -5.00000 q^{16} -6.92820 q^{17} -2.00000 q^{19} -6.00000 q^{22} +6.92820 q^{23} -5.00000 q^{25} -2.00000 q^{28} +6.92820 q^{29} -2.00000 q^{31} +5.19615 q^{32} +12.0000 q^{34} -2.00000 q^{37} +3.46410 q^{38} -6.92820 q^{41} +8.00000 q^{43} +3.46410 q^{44} -12.0000 q^{46} +10.3923 q^{47} -3.00000 q^{49} +8.66025 q^{50} -3.46410 q^{56} -12.0000 q^{58} -3.46410 q^{59} -10.0000 q^{61} +3.46410 q^{62} +1.00000 q^{64} -14.0000 q^{67} -6.92820 q^{68} -3.46410 q^{71} +10.0000 q^{73} +3.46410 q^{74} -2.00000 q^{76} -6.92820 q^{77} -4.00000 q^{79} +12.0000 q^{82} -10.3923 q^{83} -13.8564 q^{86} +6.00000 q^{88} -6.92820 q^{89} +6.92820 q^{92} -18.0000 q^{94} +10.0000 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} - 4 q^{7} - 10 q^{16} - 4 q^{19} - 12 q^{22} - 10 q^{25} - 4 q^{28} - 4 q^{31} + 24 q^{34} - 4 q^{37} + 16 q^{43} - 24 q^{46} - 6 q^{49} - 24 q^{58} - 20 q^{61} + 2 q^{64} - 28 q^{67} + 20 q^{73} - 4 q^{76} - 8 q^{79} + 24 q^{82} + 12 q^{88} - 36 q^{94} + 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.73205 0.612372
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410 1.04447 0.522233 0.852803i \(-0.325099\pi\)
0.522233 + 0.852803i \(0.325099\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 3.46410 0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) −6.92820 −1.68034 −0.840168 0.542326i \(-0.817544\pi\)
−0.840168 + 0.542326i \(0.817544\pi\)
\(18\) 0 0
\(19\) −2.00000 −0.458831 −0.229416 0.973329i \(-0.573682\pi\)
−0.229416 + 0.973329i \(0.573682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 6.92820 1.44463 0.722315 0.691564i \(-0.243078\pi\)
0.722315 + 0.691564i \(0.243078\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −2.00000 −0.377964
\(29\) 6.92820 1.28654 0.643268 0.765641i \(-0.277578\pi\)
0.643268 + 0.765641i \(0.277578\pi\)
\(30\) 0 0
\(31\) −2.00000 −0.359211 −0.179605 0.983739i \(-0.557482\pi\)
−0.179605 + 0.983739i \(0.557482\pi\)
\(32\) 5.19615 0.918559
\(33\) 0 0
\(34\) 12.0000 2.05798
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 −0.328798 −0.164399 0.986394i \(-0.552568\pi\)
−0.164399 + 0.986394i \(0.552568\pi\)
\(38\) 3.46410 0.561951
\(39\) 0 0
\(40\) 0 0
\(41\) −6.92820 −1.08200 −0.541002 0.841021i \(-0.681955\pi\)
−0.541002 + 0.841021i \(0.681955\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 3.46410 0.522233
\(45\) 0 0
\(46\) −12.0000 −1.76930
\(47\) 10.3923 1.51587 0.757937 0.652328i \(-0.226208\pi\)
0.757937 + 0.652328i \(0.226208\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 8.66025 1.22474
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.46410 −0.462910
\(57\) 0 0
\(58\) −12.0000 −1.57568
\(59\) −3.46410 −0.450988 −0.225494 0.974245i \(-0.572400\pi\)
−0.225494 + 0.974245i \(0.572400\pi\)
\(60\) 0 0
\(61\) −10.0000 −1.28037 −0.640184 0.768221i \(-0.721142\pi\)
−0.640184 + 0.768221i \(0.721142\pi\)
\(62\) 3.46410 0.439941
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) −14.0000 −1.71037 −0.855186 0.518321i \(-0.826557\pi\)
−0.855186 + 0.518321i \(0.826557\pi\)
\(68\) −6.92820 −0.840168
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 0 0
\(73\) 10.0000 1.17041 0.585206 0.810885i \(-0.301014\pi\)
0.585206 + 0.810885i \(0.301014\pi\)
\(74\) 3.46410 0.402694
\(75\) 0 0
\(76\) −2.00000 −0.229416
\(77\) −6.92820 −0.789542
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 12.0000 1.32518
\(83\) −10.3923 −1.14070 −0.570352 0.821401i \(-0.693193\pi\)
−0.570352 + 0.821401i \(0.693193\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −13.8564 −1.49417
\(87\) 0 0
\(88\) 6.00000 0.639602
\(89\) −6.92820 −0.734388 −0.367194 0.930144i \(-0.619682\pi\)
−0.367194 + 0.930144i \(0.619682\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 6.92820 0.722315
\(93\) 0 0
\(94\) −18.0000 −1.85656
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000 1.01535 0.507673 0.861550i \(-0.330506\pi\)
0.507673 + 0.861550i \(0.330506\pi\)
\(98\) 5.19615 0.524891
\(99\) 0 0
\(100\) −5.00000 −0.500000
\(101\) −13.8564 −1.37876 −0.689382 0.724398i \(-0.742118\pi\)
−0.689382 + 0.724398i \(0.742118\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.92820 0.669775 0.334887 0.942258i \(-0.391302\pi\)
0.334887 + 0.942258i \(0.391302\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 10.0000 0.944911
\(113\) −6.92820 −0.651751 −0.325875 0.945413i \(-0.605659\pi\)
−0.325875 + 0.945413i \(0.605659\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 6.92820 0.643268
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) 13.8564 1.27021
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 17.3205 1.56813
\(123\) 0 0
\(124\) −2.00000 −0.179605
\(125\) 0 0
\(126\) 0 0
\(127\) −16.0000 −1.41977 −0.709885 0.704317i \(-0.751253\pi\)
−0.709885 + 0.704317i \(0.751253\pi\)
\(128\) −12.1244 −1.07165
\(129\) 0 0
\(130\) 0 0
\(131\) −20.7846 −1.81596 −0.907980 0.419014i \(-0.862376\pi\)
−0.907980 + 0.419014i \(0.862376\pi\)
\(132\) 0 0
\(133\) 4.00000 0.346844
\(134\) 24.2487 2.09477
\(135\) 0 0
\(136\) −12.0000 −1.02899
\(137\) 6.92820 0.591916 0.295958 0.955201i \(-0.404361\pi\)
0.295958 + 0.955201i \(0.404361\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) −17.3205 −1.43346
\(147\) 0 0
\(148\) −2.00000 −0.164399
\(149\) −13.8564 −1.13516 −0.567581 0.823318i \(-0.692120\pi\)
−0.567581 + 0.823318i \(0.692120\pi\)
\(150\) 0 0
\(151\) −14.0000 −1.13930 −0.569652 0.821886i \(-0.692922\pi\)
−0.569652 + 0.821886i \(0.692922\pi\)
\(152\) −3.46410 −0.280976
\(153\) 0 0
\(154\) 12.0000 0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) 2.00000 0.159617 0.0798087 0.996810i \(-0.474569\pi\)
0.0798087 + 0.996810i \(0.474569\pi\)
\(158\) 6.92820 0.551178
\(159\) 0 0
\(160\) 0 0
\(161\) −13.8564 −1.09204
\(162\) 0 0
\(163\) −14.0000 −1.09656 −0.548282 0.836293i \(-0.684718\pi\)
−0.548282 + 0.836293i \(0.684718\pi\)
\(164\) −6.92820 −0.541002
\(165\) 0 0
\(166\) 18.0000 1.39707
\(167\) −17.3205 −1.34030 −0.670151 0.742225i \(-0.733770\pi\)
−0.670151 + 0.742225i \(0.733770\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 13.8564 1.05348 0.526742 0.850026i \(-0.323414\pi\)
0.526742 + 0.850026i \(0.323414\pi\)
\(174\) 0 0
\(175\) 10.0000 0.755929
\(176\) −17.3205 −1.30558
\(177\) 0 0
\(178\) 12.0000 0.899438
\(179\) −13.8564 −1.03568 −0.517838 0.855479i \(-0.673263\pi\)
−0.517838 + 0.855479i \(0.673263\pi\)
\(180\) 0 0
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 12.0000 0.884652
\(185\) 0 0
\(186\) 0 0
\(187\) −24.0000 −1.75505
\(188\) 10.3923 0.757937
\(189\) 0 0
\(190\) 0 0
\(191\) 13.8564 1.00261 0.501307 0.865269i \(-0.332853\pi\)
0.501307 + 0.865269i \(0.332853\pi\)
\(192\) 0 0
\(193\) −26.0000 −1.87152 −0.935760 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) −17.3205 −1.24354
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) 13.8564 0.987228 0.493614 0.869681i \(-0.335676\pi\)
0.493614 + 0.869681i \(0.335676\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −8.66025 −0.612372
\(201\) 0 0
\(202\) 24.0000 1.68863
\(203\) −13.8564 −0.972529
\(204\) 0 0
\(205\) 0 0
\(206\) 6.92820 0.482711
\(207\) 0 0
\(208\) 0 0
\(209\) −6.92820 −0.479234
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) 0 0
\(216\) 0 0
\(217\) 4.00000 0.271538
\(218\) −17.3205 −1.17309
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −10.3923 −0.694365
\(225\) 0 0
\(226\) 12.0000 0.798228
\(227\) −3.46410 −0.229920 −0.114960 0.993370i \(-0.536674\pi\)
−0.114960 + 0.993370i \(0.536674\pi\)
\(228\) 0 0
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 12.0000 0.787839
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.46410 −0.225494
\(237\) 0 0
\(238\) −24.0000 −1.55569
\(239\) 10.3923 0.672222 0.336111 0.941822i \(-0.390888\pi\)
0.336111 + 0.941822i \(0.390888\pi\)
\(240\) 0 0
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) −1.73205 −0.111340
\(243\) 0 0
\(244\) −10.0000 −0.640184
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) −3.46410 −0.219971
\(249\) 0 0
\(250\) 0 0
\(251\) −6.92820 −0.437304 −0.218652 0.975803i \(-0.570166\pi\)
−0.218652 + 0.975803i \(0.570166\pi\)
\(252\) 0 0
\(253\) 24.0000 1.50887
\(254\) 27.7128 1.73886
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 27.7128 1.72868 0.864339 0.502910i \(-0.167737\pi\)
0.864339 + 0.502910i \(0.167737\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 0 0
\(261\) 0 0
\(262\) 36.0000 2.22409
\(263\) 6.92820 0.427211 0.213606 0.976920i \(-0.431479\pi\)
0.213606 + 0.976920i \(0.431479\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −6.92820 −0.424795
\(267\) 0 0
\(268\) −14.0000 −0.855186
\(269\) −6.92820 −0.422420 −0.211210 0.977441i \(-0.567740\pi\)
−0.211210 + 0.977441i \(0.567740\pi\)
\(270\) 0 0
\(271\) 22.0000 1.33640 0.668202 0.743980i \(-0.267064\pi\)
0.668202 + 0.743980i \(0.267064\pi\)
\(272\) 34.6410 2.10042
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) −17.3205 −1.04447
\(276\) 0 0
\(277\) 2.00000 0.120168 0.0600842 0.998193i \(-0.480863\pi\)
0.0600842 + 0.998193i \(0.480863\pi\)
\(278\) 6.92820 0.415526
\(279\) 0 0
\(280\) 0 0
\(281\) 20.7846 1.23991 0.619953 0.784639i \(-0.287152\pi\)
0.619953 + 0.784639i \(0.287152\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −3.46410 −0.205557
\(285\) 0 0
\(286\) 0 0
\(287\) 13.8564 0.817918
\(288\) 0 0
\(289\) 31.0000 1.82353
\(290\) 0 0
\(291\) 0 0
\(292\) 10.0000 0.585206
\(293\) −13.8564 −0.809500 −0.404750 0.914427i \(-0.632641\pi\)
−0.404750 + 0.914427i \(0.632641\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −3.46410 −0.201347
\(297\) 0 0
\(298\) 24.0000 1.39028
\(299\) 0 0
\(300\) 0 0
\(301\) −16.0000 −0.922225
\(302\) 24.2487 1.39536
\(303\) 0 0
\(304\) 10.0000 0.573539
\(305\) 0 0
\(306\) 0 0
\(307\) −26.0000 −1.48390 −0.741949 0.670456i \(-0.766098\pi\)
−0.741949 + 0.670456i \(0.766098\pi\)
\(308\) −6.92820 −0.394771
\(309\) 0 0
\(310\) 0 0
\(311\) −20.7846 −1.17859 −0.589294 0.807919i \(-0.700594\pi\)
−0.589294 + 0.807919i \(0.700594\pi\)
\(312\) 0 0
\(313\) 14.0000 0.791327 0.395663 0.918396i \(-0.370515\pi\)
0.395663 + 0.918396i \(0.370515\pi\)
\(314\) −3.46410 −0.195491
\(315\) 0 0
\(316\) −4.00000 −0.225018
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 24.0000 1.34374
\(320\) 0 0
\(321\) 0 0
\(322\) 24.0000 1.33747
\(323\) 13.8564 0.770991
\(324\) 0 0
\(325\) 0 0
\(326\) 24.2487 1.34301
\(327\) 0 0
\(328\) −12.0000 −0.662589
\(329\) −20.7846 −1.14589
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −10.3923 −0.570352
\(333\) 0 0
\(334\) 30.0000 1.64153
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.108947 0.0544735 0.998515i \(-0.482652\pi\)
0.0544735 + 0.998515i \(0.482652\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −6.92820 −0.375183
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 13.8564 0.747087
\(345\) 0 0
\(346\) −24.0000 −1.29025
\(347\) −27.7128 −1.48770 −0.743851 0.668346i \(-0.767003\pi\)
−0.743851 + 0.668346i \(0.767003\pi\)
\(348\) 0 0
\(349\) −26.0000 −1.39175 −0.695874 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) −17.3205 −0.925820
\(351\) 0 0
\(352\) 18.0000 0.959403
\(353\) −6.92820 −0.368751 −0.184376 0.982856i \(-0.559026\pi\)
−0.184376 + 0.982856i \(0.559026\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6.92820 −0.367194
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) 10.3923 0.548485 0.274242 0.961661i \(-0.411573\pi\)
0.274242 + 0.961661i \(0.411573\pi\)
\(360\) 0 0
\(361\) −15.0000 −0.789474
\(362\) 17.3205 0.910346
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 8.00000 0.417597 0.208798 0.977959i \(-0.433045\pi\)
0.208798 + 0.977959i \(0.433045\pi\)
\(368\) −34.6410 −1.80579
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 26.0000 1.34623 0.673114 0.739538i \(-0.264956\pi\)
0.673114 + 0.739538i \(0.264956\pi\)
\(374\) 41.5692 2.14949
\(375\) 0 0
\(376\) 18.0000 0.928279
\(377\) 0 0
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −24.0000 −1.22795
\(383\) 17.3205 0.885037 0.442518 0.896759i \(-0.354085\pi\)
0.442518 + 0.896759i \(0.354085\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 45.0333 2.29214
\(387\) 0 0
\(388\) 10.0000 0.507673
\(389\) 20.7846 1.05382 0.526911 0.849921i \(-0.323350\pi\)
0.526911 + 0.849921i \(0.323350\pi\)
\(390\) 0 0
\(391\) −48.0000 −2.42746
\(392\) −5.19615 −0.262445
\(393\) 0 0
\(394\) −24.0000 −1.20910
\(395\) 0 0
\(396\) 0 0
\(397\) −2.00000 −0.100377 −0.0501886 0.998740i \(-0.515982\pi\)
−0.0501886 + 0.998740i \(0.515982\pi\)
\(398\) 27.7128 1.38912
\(399\) 0 0
\(400\) 25.0000 1.25000
\(401\) 34.6410 1.72989 0.864945 0.501867i \(-0.167353\pi\)
0.864945 + 0.501867i \(0.167353\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −13.8564 −0.689382
\(405\) 0 0
\(406\) 24.0000 1.19110
\(407\) −6.92820 −0.343418
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.00000 −0.197066
\(413\) 6.92820 0.340915
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 12.0000 0.586939
\(419\) 6.92820 0.338465 0.169232 0.985576i \(-0.445871\pi\)
0.169232 + 0.985576i \(0.445871\pi\)
\(420\) 0 0
\(421\) 10.0000 0.487370 0.243685 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) −13.8564 −0.674519
\(423\) 0 0
\(424\) 0 0
\(425\) 34.6410 1.68034
\(426\) 0 0
\(427\) 20.0000 0.967868
\(428\) 6.92820 0.334887
\(429\) 0 0
\(430\) 0 0
\(431\) −38.1051 −1.83546 −0.917729 0.397206i \(-0.869980\pi\)
−0.917729 + 0.397206i \(0.869980\pi\)
\(432\) 0 0
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) −6.92820 −0.332564
\(435\) 0 0
\(436\) 10.0000 0.478913
\(437\) −13.8564 −0.662842
\(438\) 0 0
\(439\) −28.0000 −1.33637 −0.668184 0.743996i \(-0.732928\pi\)
−0.668184 + 0.743996i \(0.732928\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 20.7846 0.987507 0.493753 0.869602i \(-0.335625\pi\)
0.493753 + 0.869602i \(0.335625\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 3.46410 0.164030
\(447\) 0 0
\(448\) −2.00000 −0.0944911
\(449\) 6.92820 0.326962 0.163481 0.986546i \(-0.447728\pi\)
0.163481 + 0.986546i \(0.447728\pi\)
\(450\) 0 0
\(451\) −24.0000 −1.13012
\(452\) −6.92820 −0.325875
\(453\) 0 0
\(454\) 6.00000 0.281594
\(455\) 0 0
\(456\) 0 0
\(457\) 10.0000 0.467780 0.233890 0.972263i \(-0.424854\pi\)
0.233890 + 0.972263i \(0.424854\pi\)
\(458\) 24.2487 1.13307
\(459\) 0 0
\(460\) 0 0
\(461\) 27.7128 1.29071 0.645357 0.763881i \(-0.276709\pi\)
0.645357 + 0.763881i \(0.276709\pi\)
\(462\) 0 0
\(463\) −2.00000 −0.0929479 −0.0464739 0.998920i \(-0.514798\pi\)
−0.0464739 + 0.998920i \(0.514798\pi\)
\(464\) −34.6410 −1.60817
\(465\) 0 0
\(466\) 0 0
\(467\) 20.7846 0.961797 0.480899 0.876776i \(-0.340311\pi\)
0.480899 + 0.876776i \(0.340311\pi\)
\(468\) 0 0
\(469\) 28.0000 1.29292
\(470\) 0 0
\(471\) 0 0
\(472\) −6.00000 −0.276172
\(473\) 27.7128 1.27424
\(474\) 0 0
\(475\) 10.0000 0.458831
\(476\) 13.8564 0.635107
\(477\) 0 0
\(478\) −18.0000 −0.823301
\(479\) 3.46410 0.158279 0.0791394 0.996864i \(-0.474783\pi\)
0.0791394 + 0.996864i \(0.474783\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −17.3205 −0.788928
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) −2.00000 −0.0906287 −0.0453143 0.998973i \(-0.514429\pi\)
−0.0453143 + 0.998973i \(0.514429\pi\)
\(488\) −17.3205 −0.784063
\(489\) 0 0
\(490\) 0 0
\(491\) 6.92820 0.312665 0.156333 0.987704i \(-0.450033\pi\)
0.156333 + 0.987704i \(0.450033\pi\)
\(492\) 0 0
\(493\) −48.0000 −2.16181
\(494\) 0 0
\(495\) 0 0
\(496\) 10.0000 0.449013
\(497\) 6.92820 0.310772
\(498\) 0 0
\(499\) −14.0000 −0.626726 −0.313363 0.949633i \(-0.601456\pi\)
−0.313363 + 0.949633i \(0.601456\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 12.0000 0.535586
\(503\) −6.92820 −0.308913 −0.154457 0.988000i \(-0.549363\pi\)
−0.154457 + 0.988000i \(0.549363\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −41.5692 −1.84798
\(507\) 0 0
\(508\) −16.0000 −0.709885
\(509\) 13.8564 0.614174 0.307087 0.951681i \(-0.400646\pi\)
0.307087 + 0.951681i \(0.400646\pi\)
\(510\) 0 0
\(511\) −20.0000 −0.884748
\(512\) −8.66025 −0.382733
\(513\) 0 0
\(514\) −48.0000 −2.11719
\(515\) 0 0
\(516\) 0 0
\(517\) 36.0000 1.58328
\(518\) −6.92820 −0.304408
\(519\) 0 0
\(520\) 0 0
\(521\) 20.7846 0.910590 0.455295 0.890341i \(-0.349534\pi\)
0.455295 + 0.890341i \(0.349534\pi\)
\(522\) 0 0
\(523\) 20.0000 0.874539 0.437269 0.899331i \(-0.355946\pi\)
0.437269 + 0.899331i \(0.355946\pi\)
\(524\) −20.7846 −0.907980
\(525\) 0 0
\(526\) −12.0000 −0.523225
\(527\) 13.8564 0.603595
\(528\) 0 0
\(529\) 25.0000 1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) 4.00000 0.173422
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −24.2487 −1.04738
\(537\) 0 0
\(538\) 12.0000 0.517357
\(539\) −10.3923 −0.447628
\(540\) 0 0
\(541\) 10.0000 0.429934 0.214967 0.976621i \(-0.431036\pi\)
0.214967 + 0.976621i \(0.431036\pi\)
\(542\) −38.1051 −1.63675
\(543\) 0 0
\(544\) −36.0000 −1.54349
\(545\) 0 0
\(546\) 0 0
\(547\) −4.00000 −0.171028 −0.0855138 0.996337i \(-0.527253\pi\)
−0.0855138 + 0.996337i \(0.527253\pi\)
\(548\) 6.92820 0.295958
\(549\) 0 0
\(550\) 30.0000 1.27920
\(551\) −13.8564 −0.590303
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) −3.46410 −0.147176
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −27.7128 −1.17423 −0.587115 0.809504i \(-0.699736\pi\)
−0.587115 + 0.809504i \(0.699736\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −36.0000 −1.51857
\(563\) 13.8564 0.583978 0.291989 0.956422i \(-0.405683\pi\)
0.291989 + 0.956422i \(0.405683\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 6.92820 0.291214
\(567\) 0 0
\(568\) −6.00000 −0.251754
\(569\) −13.8564 −0.580891 −0.290445 0.956892i \(-0.593803\pi\)
−0.290445 + 0.956892i \(0.593803\pi\)
\(570\) 0 0
\(571\) 20.0000 0.836974 0.418487 0.908223i \(-0.362561\pi\)
0.418487 + 0.908223i \(0.362561\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −24.0000 −1.00174
\(575\) −34.6410 −1.44463
\(576\) 0 0
\(577\) −26.0000 −1.08239 −0.541197 0.840896i \(-0.682029\pi\)
−0.541197 + 0.840896i \(0.682029\pi\)
\(578\) −53.6936 −2.23336
\(579\) 0 0
\(580\) 0 0
\(581\) 20.7846 0.862291
\(582\) 0 0
\(583\) 0 0
\(584\) 17.3205 0.716728
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) 45.0333 1.85872 0.929362 0.369170i \(-0.120358\pi\)
0.929362 + 0.369170i \(0.120358\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 0 0
\(592\) 10.0000 0.410997
\(593\) −20.7846 −0.853522 −0.426761 0.904365i \(-0.640345\pi\)
−0.426761 + 0.904365i \(0.640345\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −13.8564 −0.567581
\(597\) 0 0
\(598\) 0 0
\(599\) 20.7846 0.849236 0.424618 0.905373i \(-0.360408\pi\)
0.424618 + 0.905373i \(0.360408\pi\)
\(600\) 0 0
\(601\) −34.0000 −1.38689 −0.693444 0.720510i \(-0.743908\pi\)
−0.693444 + 0.720510i \(0.743908\pi\)
\(602\) 27.7128 1.12949
\(603\) 0 0
\(604\) −14.0000 −0.569652
\(605\) 0 0
\(606\) 0 0
\(607\) −40.0000 −1.62355 −0.811775 0.583970i \(-0.801498\pi\)
−0.811775 + 0.583970i \(0.801498\pi\)
\(608\) −10.3923 −0.421464
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −2.00000 −0.0807792 −0.0403896 0.999184i \(-0.512860\pi\)
−0.0403896 + 0.999184i \(0.512860\pi\)
\(614\) 45.0333 1.81740
\(615\) 0 0
\(616\) −12.0000 −0.483494
\(617\) 6.92820 0.278919 0.139459 0.990228i \(-0.455464\pi\)
0.139459 + 0.990228i \(0.455464\pi\)
\(618\) 0 0
\(619\) −26.0000 −1.04503 −0.522514 0.852631i \(-0.675006\pi\)
−0.522514 + 0.852631i \(0.675006\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 36.0000 1.44347
\(623\) 13.8564 0.555145
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) −24.2487 −0.969173
\(627\) 0 0
\(628\) 2.00000 0.0798087
\(629\) 13.8564 0.552491
\(630\) 0 0
\(631\) −38.0000 −1.51276 −0.756378 0.654135i \(-0.773033\pi\)
−0.756378 + 0.654135i \(0.773033\pi\)
\(632\) −6.92820 −0.275589
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −41.5692 −1.64574
\(639\) 0 0
\(640\) 0 0
\(641\) 34.6410 1.36824 0.684119 0.729370i \(-0.260187\pi\)
0.684119 + 0.729370i \(0.260187\pi\)
\(642\) 0 0
\(643\) −14.0000 −0.552106 −0.276053 0.961142i \(-0.589027\pi\)
−0.276053 + 0.961142i \(0.589027\pi\)
\(644\) −13.8564 −0.546019
\(645\) 0 0
\(646\) −24.0000 −0.944267
\(647\) −13.8564 −0.544752 −0.272376 0.962191i \(-0.587809\pi\)
−0.272376 + 0.962191i \(0.587809\pi\)
\(648\) 0 0
\(649\) −12.0000 −0.471041
\(650\) 0 0
\(651\) 0 0
\(652\) −14.0000 −0.548282
\(653\) −34.6410 −1.35561 −0.677804 0.735243i \(-0.737068\pi\)
−0.677804 + 0.735243i \(0.737068\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 34.6410 1.35250
\(657\) 0 0
\(658\) 36.0000 1.40343
\(659\) 13.8564 0.539769 0.269884 0.962893i \(-0.413014\pi\)
0.269884 + 0.962893i \(0.413014\pi\)
\(660\) 0 0
\(661\) −26.0000 −1.01128 −0.505641 0.862744i \(-0.668744\pi\)
−0.505641 + 0.862744i \(0.668744\pi\)
\(662\) −17.3205 −0.673181
\(663\) 0 0
\(664\) −18.0000 −0.698535
\(665\) 0 0
\(666\) 0 0
\(667\) 48.0000 1.85857
\(668\) −17.3205 −0.670151
\(669\) 0 0
\(670\) 0 0
\(671\) −34.6410 −1.33730
\(672\) 0 0
\(673\) −10.0000 −0.385472 −0.192736 0.981251i \(-0.561736\pi\)
−0.192736 + 0.981251i \(0.561736\pi\)
\(674\) −3.46410 −0.133432
\(675\) 0 0
\(676\) 0 0
\(677\) −41.5692 −1.59763 −0.798817 0.601574i \(-0.794541\pi\)
−0.798817 + 0.601574i \(0.794541\pi\)
\(678\) 0 0
\(679\) −20.0000 −0.767530
\(680\) 0 0
\(681\) 0 0
\(682\) 12.0000 0.459504
\(683\) 17.3205 0.662751 0.331375 0.943499i \(-0.392487\pi\)
0.331375 + 0.943499i \(0.392487\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −34.6410 −1.32260
\(687\) 0 0
\(688\) −40.0000 −1.52499
\(689\) 0 0
\(690\) 0 0
\(691\) 10.0000 0.380418 0.190209 0.981744i \(-0.439083\pi\)
0.190209 + 0.981744i \(0.439083\pi\)
\(692\) 13.8564 0.526742
\(693\) 0 0
\(694\) 48.0000 1.82206
\(695\) 0 0
\(696\) 0 0
\(697\) 48.0000 1.81813
\(698\) 45.0333 1.70454
\(699\) 0 0
\(700\) 10.0000 0.377964
\(701\) −20.7846 −0.785024 −0.392512 0.919747i \(-0.628394\pi\)
−0.392512 + 0.919747i \(0.628394\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 3.46410 0.130558
\(705\) 0 0
\(706\) 12.0000 0.451626
\(707\) 27.7128 1.04225
\(708\) 0 0
\(709\) −38.0000 −1.42712 −0.713560 0.700594i \(-0.752918\pi\)
−0.713560 + 0.700594i \(0.752918\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −12.0000 −0.449719
\(713\) −13.8564 −0.518927
\(714\) 0 0
\(715\) 0 0
\(716\) −13.8564 −0.517838
\(717\) 0 0
\(718\) −18.0000 −0.671754
\(719\) 34.6410 1.29189 0.645946 0.763383i \(-0.276463\pi\)
0.645946 + 0.763383i \(0.276463\pi\)
\(720\) 0 0
\(721\) 8.00000 0.297936
\(722\) 25.9808 0.966904
\(723\) 0 0
\(724\) −10.0000 −0.371647
\(725\) −34.6410 −1.28654
\(726\) 0 0
\(727\) 32.0000 1.18681 0.593407 0.804902i \(-0.297782\pi\)
0.593407 + 0.804902i \(0.297782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −55.4256 −2.04999
\(732\) 0 0
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) −13.8564 −0.511449
\(735\) 0 0
\(736\) 36.0000 1.32698
\(737\) −48.4974 −1.78643
\(738\) 0 0
\(739\) 22.0000 0.809283 0.404642 0.914475i \(-0.367396\pi\)
0.404642 + 0.914475i \(0.367396\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.2487 0.889599 0.444799 0.895630i \(-0.353275\pi\)
0.444799 + 0.895630i \(0.353275\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −45.0333 −1.64879
\(747\) 0 0
\(748\) −24.0000 −0.877527
\(749\) −13.8564 −0.506302
\(750\) 0 0
\(751\) −4.00000 −0.145962 −0.0729810 0.997333i \(-0.523251\pi\)
−0.0729810 + 0.997333i \(0.523251\pi\)
\(752\) −51.9615 −1.89484
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) 3.46410 0.125822
\(759\) 0 0
\(760\) 0 0
\(761\) −34.6410 −1.25574 −0.627868 0.778320i \(-0.716072\pi\)
−0.627868 + 0.778320i \(0.716072\pi\)
\(762\) 0 0
\(763\) −20.0000 −0.724049
\(764\) 13.8564 0.501307
\(765\) 0 0
\(766\) −30.0000 −1.08394
\(767\) 0 0
\(768\) 0 0
\(769\) −50.0000 −1.80305 −0.901523 0.432731i \(-0.857550\pi\)
−0.901523 + 0.432731i \(0.857550\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −26.0000 −0.935760
\(773\) −13.8564 −0.498380 −0.249190 0.968455i \(-0.580164\pi\)
−0.249190 + 0.968455i \(0.580164\pi\)
\(774\) 0 0
\(775\) 10.0000 0.359211
\(776\) 17.3205 0.621770
\(777\) 0 0
\(778\) −36.0000 −1.29066
\(779\) 13.8564 0.496457
\(780\) 0 0
\(781\) −12.0000 −0.429394
\(782\) 83.1384 2.97302
\(783\) 0 0
\(784\) 15.0000 0.535714
\(785\) 0 0
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 13.8564 0.493614
\(789\) 0 0
\(790\) 0 0
\(791\) 13.8564 0.492677
\(792\) 0 0
\(793\) 0 0
\(794\) 3.46410 0.122936
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 34.6410 1.22705 0.613524 0.789676i \(-0.289751\pi\)
0.613524 + 0.789676i \(0.289751\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) −25.9808 −0.918559
\(801\) 0 0
\(802\) −60.0000 −2.11867
\(803\) 34.6410 1.22245
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −24.0000 −0.844317
\(809\) 6.92820 0.243583 0.121791 0.992556i \(-0.461136\pi\)
0.121791 + 0.992556i \(0.461136\pi\)
\(810\) 0 0
\(811\) −26.0000 −0.912983 −0.456492 0.889728i \(-0.650894\pi\)
−0.456492 + 0.889728i \(0.650894\pi\)
\(812\) −13.8564 −0.486265
\(813\) 0 0
\(814\) 12.0000 0.420600
\(815\) 0 0
\(816\) 0 0
\(817\) −16.0000 −0.559769
\(818\) −17.3205 −0.605597
\(819\) 0 0
\(820\) 0 0
\(821\) −27.7128 −0.967184 −0.483592 0.875294i \(-0.660668\pi\)
−0.483592 + 0.875294i \(0.660668\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) −6.92820 −0.241355
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) −10.3923 −0.361376 −0.180688 0.983540i \(-0.557832\pi\)
−0.180688 + 0.983540i \(0.557832\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 20.7846 0.720144
\(834\) 0 0
\(835\) 0 0
\(836\) −6.92820 −0.239617
\(837\) 0 0
\(838\) −12.0000 −0.414533
\(839\) −24.2487 −0.837158 −0.418579 0.908180i \(-0.637472\pi\)
−0.418579 + 0.908180i \(0.637472\pi\)
\(840\) 0 0
\(841\) 19.0000 0.655172
\(842\) −17.3205 −0.596904
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 0 0
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 0 0
\(849\) 0 0
\(850\) −60.0000 −2.05798
\(851\) −13.8564 −0.474991
\(852\) 0 0
\(853\) 46.0000 1.57501 0.787505 0.616308i \(-0.211372\pi\)
0.787505 + 0.616308i \(0.211372\pi\)
\(854\) −34.6410 −1.18539
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −40.0000 −1.36478 −0.682391 0.730987i \(-0.739060\pi\)
−0.682391 + 0.730987i \(0.739060\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 66.0000 2.24797
\(863\) 31.1769 1.06127 0.530637 0.847599i \(-0.321953\pi\)
0.530637 + 0.847599i \(0.321953\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −3.46410 −0.117715
\(867\) 0 0
\(868\) 4.00000 0.135769
\(869\) −13.8564 −0.470046
\(870\) 0 0
\(871\) 0 0
\(872\) 17.3205 0.586546
\(873\) 0 0
\(874\) 24.0000 0.811812
\(875\) 0 0
\(876\) 0 0
\(877\) 22.0000 0.742887 0.371444 0.928456i \(-0.378863\pi\)
0.371444 + 0.928456i \(0.378863\pi\)
\(878\) 48.4974 1.63671
\(879\) 0 0
\(880\) 0 0
\(881\) −13.8564 −0.466834 −0.233417 0.972377i \(-0.574991\pi\)
−0.233417 + 0.972377i \(0.574991\pi\)
\(882\) 0 0
\(883\) 44.0000 1.48072 0.740359 0.672212i \(-0.234656\pi\)
0.740359 + 0.672212i \(0.234656\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −36.0000 −1.20944
\(887\) 27.7128 0.930505 0.465253 0.885178i \(-0.345963\pi\)
0.465253 + 0.885178i \(0.345963\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 0 0
\(892\) −2.00000 −0.0669650
\(893\) −20.7846 −0.695530
\(894\) 0 0
\(895\) 0 0
\(896\) 24.2487 0.810093
\(897\) 0 0
\(898\) −12.0000 −0.400445
\(899\) −13.8564 −0.462137
\(900\) 0 0
\(901\) 0 0
\(902\) 41.5692 1.38410
\(903\) 0 0
\(904\) −12.0000 −0.399114
\(905\) 0 0
\(906\) 0 0
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) −3.46410 −0.114960
\(909\) 0 0
\(910\) 0 0
\(911\) −41.5692 −1.37725 −0.688625 0.725118i \(-0.741785\pi\)
−0.688625 + 0.725118i \(0.741785\pi\)
\(912\) 0 0
\(913\) −36.0000 −1.19143
\(914\) −17.3205 −0.572911
\(915\) 0 0
\(916\) −14.0000 −0.462573
\(917\) 41.5692 1.37274
\(918\) 0 0
\(919\) −4.00000 −0.131948 −0.0659739 0.997821i \(-0.521015\pi\)
−0.0659739 + 0.997821i \(0.521015\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −48.0000 −1.58080
\(923\) 0 0
\(924\) 0 0
\(925\) 10.0000 0.328798
\(926\) 3.46410 0.113837
\(927\) 0 0
\(928\) 36.0000 1.18176
\(929\) 48.4974 1.59115 0.795574 0.605856i \(-0.207169\pi\)
0.795574 + 0.605856i \(0.207169\pi\)
\(930\) 0 0
\(931\) 6.00000 0.196642
\(932\) 0 0
\(933\) 0 0
\(934\) −36.0000 −1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) 26.0000 0.849383 0.424691 0.905338i \(-0.360383\pi\)
0.424691 + 0.905338i \(0.360383\pi\)
\(938\) −48.4974 −1.58350
\(939\) 0 0
\(940\) 0 0
\(941\) −41.5692 −1.35512 −0.677559 0.735469i \(-0.736962\pi\)
−0.677559 + 0.735469i \(0.736962\pi\)
\(942\) 0 0
\(943\) −48.0000 −1.56310
\(944\) 17.3205 0.563735
\(945\) 0 0
\(946\) −48.0000 −1.56061
\(947\) −17.3205 −0.562841 −0.281420 0.959585i \(-0.590806\pi\)
−0.281420 + 0.959585i \(0.590806\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) −17.3205 −0.561951
\(951\) 0 0
\(952\) 24.0000 0.777844
\(953\) −27.7128 −0.897706 −0.448853 0.893606i \(-0.648167\pi\)
−0.448853 + 0.893606i \(0.648167\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 10.3923 0.336111
\(957\) 0 0
\(958\) −6.00000 −0.193851
\(959\) −13.8564 −0.447447
\(960\) 0 0
\(961\) −27.0000 −0.870968
\(962\) 0 0
\(963\) 0 0
\(964\) 10.0000 0.322078
\(965\) 0 0
\(966\) 0 0
\(967\) −2.00000 −0.0643157 −0.0321578 0.999483i \(-0.510238\pi\)
−0.0321578 + 0.999483i \(0.510238\pi\)
\(968\) 1.73205 0.0556702
\(969\) 0 0
\(970\) 0 0
\(971\) 55.4256 1.77869 0.889346 0.457234i \(-0.151160\pi\)
0.889346 + 0.457234i \(0.151160\pi\)
\(972\) 0 0
\(973\) 8.00000 0.256468
\(974\) 3.46410 0.110997
\(975\) 0 0
\(976\) 50.0000 1.60046
\(977\) −6.92820 −0.221653 −0.110826 0.993840i \(-0.535350\pi\)
−0.110826 + 0.993840i \(0.535350\pi\)
\(978\) 0 0
\(979\) −24.0000 −0.767043
\(980\) 0 0
\(981\) 0 0
\(982\) −12.0000 −0.382935
\(983\) −10.3923 −0.331463 −0.165732 0.986171i \(-0.552999\pi\)
−0.165732 + 0.986171i \(0.552999\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 83.1384 2.64767
\(987\) 0 0
\(988\) 0 0
\(989\) 55.4256 1.76243
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −10.3923 −0.329956
\(993\) 0 0
\(994\) −12.0000 −0.380617
\(995\) 0 0
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 24.2487 0.767580
\(999\) 0 0
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 1521.2.a.j.1.1 2
3.2 odd 2 inner 1521.2.a.j.1.2 2
13.5 odd 4 1521.2.b.i.1351.3 4
13.8 odd 4 1521.2.b.i.1351.2 4
13.12 even 2 117.2.a.b.1.2 yes 2
39.5 even 4 1521.2.b.i.1351.1 4
39.8 even 4 1521.2.b.i.1351.4 4
39.38 odd 2 117.2.a.b.1.1 2
52.51 odd 2 1872.2.a.v.1.2 2
65.12 odd 4 2925.2.c.s.2224.4 4
65.38 odd 4 2925.2.c.s.2224.1 4
65.64 even 2 2925.2.a.y.1.1 2
91.90 odd 2 5733.2.a.t.1.2 2
104.51 odd 2 7488.2.a.cj.1.1 2
104.77 even 2 7488.2.a.cq.1.2 2
117.25 even 6 1053.2.e.i.352.1 4
117.38 odd 6 1053.2.e.i.352.2 4
117.77 odd 6 1053.2.e.i.703.2 4
117.103 even 6 1053.2.e.i.703.1 4
156.155 even 2 1872.2.a.v.1.1 2
195.38 even 4 2925.2.c.s.2224.3 4
195.77 even 4 2925.2.c.s.2224.2 4
195.194 odd 2 2925.2.a.y.1.2 2
273.272 even 2 5733.2.a.t.1.1 2
312.77 odd 2 7488.2.a.cq.1.1 2
312.155 even 2 7488.2.a.cj.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
117.2.a.b.1.1 2 39.38 odd 2
117.2.a.b.1.2 yes 2 13.12 even 2
1053.2.e.i.352.1 4 117.25 even 6
1053.2.e.i.352.2 4 117.38 odd 6
1053.2.e.i.703.1 4 117.103 even 6
1053.2.e.i.703.2 4 117.77 odd 6
1521.2.a.j.1.1 2 1.1 even 1 trivial
1521.2.a.j.1.2 2 3.2 odd 2 inner
1521.2.b.i.1351.1 4 39.5 even 4
1521.2.b.i.1351.2 4 13.8 odd 4
1521.2.b.i.1351.3 4 13.5 odd 4
1521.2.b.i.1351.4 4 39.8 even 4
1872.2.a.v.1.1 2 156.155 even 2
1872.2.a.v.1.2 2 52.51 odd 2
2925.2.a.y.1.1 2 65.64 even 2
2925.2.a.y.1.2 2 195.194 odd 2
2925.2.c.s.2224.1 4 65.38 odd 4
2925.2.c.s.2224.2 4 195.77 even 4
2925.2.c.s.2224.3 4 195.38 even 4
2925.2.c.s.2224.4 4 65.12 odd 4
5733.2.a.t.1.1 2 273.272 even 2
5733.2.a.t.1.2 2 91.90 odd 2
7488.2.a.cj.1.1 2 104.51 odd 2
7488.2.a.cj.1.2 2 312.155 even 2
7488.2.a.cq.1.1 2 312.77 odd 2
7488.2.a.cq.1.2 2 104.77 even 2