Properties

Label 378.2.a.b.1.1
Level $378$
Weight $2$
Character 378.1
Self dual yes
Analytic conductor $3.018$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 378 = 2 \cdot 3^{3} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 378.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.01834519640\)
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 \(\chi\) \(=\) 378.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -3.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +3.00000 q^{10} +3.00000 q^{11} -4.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -6.00000 q^{17} -7.00000 q^{19} -3.00000 q^{20} -3.00000 q^{22} -3.00000 q^{23} +4.00000 q^{25} +4.00000 q^{26} +1.00000 q^{28} +5.00000 q^{31} -1.00000 q^{32} +6.00000 q^{34} -3.00000 q^{35} -7.00000 q^{37} +7.00000 q^{38} +3.00000 q^{40} -9.00000 q^{41} -10.0000 q^{43} +3.00000 q^{44} +3.00000 q^{46} +6.00000 q^{47} +1.00000 q^{49} -4.00000 q^{50} -4.00000 q^{52} +12.0000 q^{53} -9.00000 q^{55} -1.00000 q^{56} -6.00000 q^{59} +8.00000 q^{61} -5.00000 q^{62} +1.00000 q^{64} +12.0000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +3.00000 q^{70} +9.00000 q^{71} +2.00000 q^{73} +7.00000 q^{74} -7.00000 q^{76} +3.00000 q^{77} -10.0000 q^{79} -3.00000 q^{80} +9.00000 q^{82} +18.0000 q^{85} +10.0000 q^{86} -3.00000 q^{88} +15.0000 q^{89} -4.00000 q^{91} -3.00000 q^{92} -6.00000 q^{94} +21.0000 q^{95} +8.00000 q^{97} -1.00000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −3.00000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 3.00000 0.948683
\(11\) 3.00000 0.904534 0.452267 0.891883i \(-0.350615\pi\)
0.452267 + 0.891883i \(0.350615\pi\)
\(12\) 0 0
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.00000 −1.45521 −0.727607 0.685994i \(-0.759367\pi\)
−0.727607 + 0.685994i \(0.759367\pi\)
\(18\) 0 0
\(19\) −7.00000 −1.60591 −0.802955 0.596040i \(-0.796740\pi\)
−0.802955 + 0.596040i \(0.796740\pi\)
\(20\) −3.00000 −0.670820
\(21\) 0 0
\(22\) −3.00000 −0.639602
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 4.00000 0.784465
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 5.00000 0.898027 0.449013 0.893525i \(-0.351776\pi\)
0.449013 + 0.893525i \(0.351776\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) −3.00000 −0.507093
\(36\) 0 0
\(37\) −7.00000 −1.15079 −0.575396 0.817875i \(-0.695152\pi\)
−0.575396 + 0.817875i \(0.695152\pi\)
\(38\) 7.00000 1.13555
\(39\) 0 0
\(40\) 3.00000 0.474342
\(41\) −9.00000 −1.40556 −0.702782 0.711405i \(-0.748059\pi\)
−0.702782 + 0.711405i \(0.748059\pi\)
\(42\) 0 0
\(43\) −10.0000 −1.52499 −0.762493 0.646997i \(-0.776025\pi\)
−0.762493 + 0.646997i \(0.776025\pi\)
\(44\) 3.00000 0.452267
\(45\) 0 0
\(46\) 3.00000 0.442326
\(47\) 6.00000 0.875190 0.437595 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −4.00000 −0.565685
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 12.0000 1.64833 0.824163 0.566352i \(-0.191646\pi\)
0.824163 + 0.566352i \(0.191646\pi\)
\(54\) 0 0
\(55\) −9.00000 −1.21356
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) 0 0
\(59\) −6.00000 −0.781133 −0.390567 0.920575i \(-0.627721\pi\)
−0.390567 + 0.920575i \(0.627721\pi\)
\(60\) 0 0
\(61\) 8.00000 1.02430 0.512148 0.858898i \(-0.328850\pi\)
0.512148 + 0.858898i \(0.328850\pi\)
\(62\) −5.00000 −0.635001
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 12.0000 1.48842
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 3.00000 0.358569
\(71\) 9.00000 1.06810 0.534052 0.845452i \(-0.320669\pi\)
0.534052 + 0.845452i \(0.320669\pi\)
\(72\) 0 0
\(73\) 2.00000 0.234082 0.117041 0.993127i \(-0.462659\pi\)
0.117041 + 0.993127i \(0.462659\pi\)
\(74\) 7.00000 0.813733
\(75\) 0 0
\(76\) −7.00000 −0.802955
\(77\) 3.00000 0.341882
\(78\) 0 0
\(79\) −10.0000 −1.12509 −0.562544 0.826767i \(-0.690177\pi\)
−0.562544 + 0.826767i \(0.690177\pi\)
\(80\) −3.00000 −0.335410
\(81\) 0 0
\(82\) 9.00000 0.993884
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 18.0000 1.95237
\(86\) 10.0000 1.07833
\(87\) 0 0
\(88\) −3.00000 −0.319801
\(89\) 15.0000 1.59000 0.794998 0.606612i \(-0.207472\pi\)
0.794998 + 0.606612i \(0.207472\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) −3.00000 −0.312772
\(93\) 0 0
\(94\) −6.00000 −0.618853
\(95\) 21.0000 2.15455
\(96\) 0 0
\(97\) 8.00000 0.812277 0.406138 0.913812i \(-0.366875\pi\)
0.406138 + 0.913812i \(0.366875\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 4.00000 0.400000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 0 0
\(103\) 5.00000 0.492665 0.246332 0.969185i \(-0.420775\pi\)
0.246332 + 0.969185i \(0.420775\pi\)
\(104\) 4.00000 0.392232
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) −12.0000 −1.16008 −0.580042 0.814587i \(-0.696964\pi\)
−0.580042 + 0.814587i \(0.696964\pi\)
\(108\) 0 0
\(109\) 11.0000 1.05361 0.526804 0.849987i \(-0.323390\pi\)
0.526804 + 0.849987i \(0.323390\pi\)
\(110\) 9.00000 0.858116
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −18.0000 −1.69330 −0.846649 0.532152i \(-0.821383\pi\)
−0.846649 + 0.532152i \(0.821383\pi\)
\(114\) 0 0
\(115\) 9.00000 0.839254
\(116\) 0 0
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) −6.00000 −0.550019
\(120\) 0 0
\(121\) −2.00000 −0.181818
\(122\) −8.00000 −0.724286
\(123\) 0 0
\(124\) 5.00000 0.449013
\(125\) 3.00000 0.268328
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −12.0000 −1.05247
\(131\) 6.00000 0.524222 0.262111 0.965038i \(-0.415581\pi\)
0.262111 + 0.965038i \(0.415581\pi\)
\(132\) 0 0
\(133\) −7.00000 −0.606977
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 6.00000 0.514496
\(137\) 12.0000 1.02523 0.512615 0.858619i \(-0.328677\pi\)
0.512615 + 0.858619i \(0.328677\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) −3.00000 −0.253546
\(141\) 0 0
\(142\) −9.00000 −0.755263
\(143\) −12.0000 −1.00349
\(144\) 0 0
\(145\) 0 0
\(146\) −2.00000 −0.165521
\(147\) 0 0
\(148\) −7.00000 −0.575396
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) −10.0000 −0.813788 −0.406894 0.913475i \(-0.633388\pi\)
−0.406894 + 0.913475i \(0.633388\pi\)
\(152\) 7.00000 0.567775
\(153\) 0 0
\(154\) −3.00000 −0.241747
\(155\) −15.0000 −1.20483
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) 10.0000 0.795557
\(159\) 0 0
\(160\) 3.00000 0.237171
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −9.00000 −0.702782
\(165\) 0 0
\(166\) 0 0
\(167\) −18.0000 −1.39288 −0.696441 0.717614i \(-0.745234\pi\)
−0.696441 + 0.717614i \(0.745234\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) −18.0000 −1.38054
\(171\) 0 0
\(172\) −10.0000 −0.762493
\(173\) −21.0000 −1.59660 −0.798300 0.602260i \(-0.794267\pi\)
−0.798300 + 0.602260i \(0.794267\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) −15.0000 −1.12430
\(179\) −24.0000 −1.79384 −0.896922 0.442189i \(-0.854202\pi\)
−0.896922 + 0.442189i \(0.854202\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 4.00000 0.296500
\(183\) 0 0
\(184\) 3.00000 0.221163
\(185\) 21.0000 1.54395
\(186\) 0 0
\(187\) −18.0000 −1.31629
\(188\) 6.00000 0.437595
\(189\) 0 0
\(190\) −21.0000 −1.52350
\(191\) 15.0000 1.08536 0.542681 0.839939i \(-0.317409\pi\)
0.542681 + 0.839939i \(0.317409\pi\)
\(192\) 0 0
\(193\) 14.0000 1.00774 0.503871 0.863779i \(-0.331909\pi\)
0.503871 + 0.863779i \(0.331909\pi\)
\(194\) −8.00000 −0.574367
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −18.0000 −1.28245 −0.641223 0.767354i \(-0.721573\pi\)
−0.641223 + 0.767354i \(0.721573\pi\)
\(198\) 0 0
\(199\) −7.00000 −0.496217 −0.248108 0.968732i \(-0.579809\pi\)
−0.248108 + 0.968732i \(0.579809\pi\)
\(200\) −4.00000 −0.282843
\(201\) 0 0
\(202\) 6.00000 0.422159
\(203\) 0 0
\(204\) 0 0
\(205\) 27.0000 1.88576
\(206\) −5.00000 −0.348367
\(207\) 0 0
\(208\) −4.00000 −0.277350
\(209\) −21.0000 −1.45260
\(210\) 0 0
\(211\) 14.0000 0.963800 0.481900 0.876226i \(-0.339947\pi\)
0.481900 + 0.876226i \(0.339947\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) 12.0000 0.820303
\(215\) 30.0000 2.04598
\(216\) 0 0
\(217\) 5.00000 0.339422
\(218\) −11.0000 −0.745014
\(219\) 0 0
\(220\) −9.00000 −0.606780
\(221\) 24.0000 1.61441
\(222\) 0 0
\(223\) −1.00000 −0.0669650 −0.0334825 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) −6.00000 −0.398234 −0.199117 0.979976i \(-0.563807\pi\)
−0.199117 + 0.979976i \(0.563807\pi\)
\(228\) 0 0
\(229\) −4.00000 −0.264327 −0.132164 0.991228i \(-0.542192\pi\)
−0.132164 + 0.991228i \(0.542192\pi\)
\(230\) −9.00000 −0.593442
\(231\) 0 0
\(232\) 0 0
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −18.0000 −1.17419
\(236\) −6.00000 −0.390567
\(237\) 0 0
\(238\) 6.00000 0.388922
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 26.0000 1.67481 0.837404 0.546585i \(-0.184072\pi\)
0.837404 + 0.546585i \(0.184072\pi\)
\(242\) 2.00000 0.128565
\(243\) 0 0
\(244\) 8.00000 0.512148
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 28.0000 1.78160
\(248\) −5.00000 −0.317500
\(249\) 0 0
\(250\) −3.00000 −0.189737
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −9.00000 −0.565825
\(254\) −20.0000 −1.25491
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −21.0000 −1.30994 −0.654972 0.755653i \(-0.727320\pi\)
−0.654972 + 0.755653i \(0.727320\pi\)
\(258\) 0 0
\(259\) −7.00000 −0.434959
\(260\) 12.0000 0.744208
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) −15.0000 −0.924940 −0.462470 0.886635i \(-0.653037\pi\)
−0.462470 + 0.886635i \(0.653037\pi\)
\(264\) 0 0
\(265\) −36.0000 −2.21146
\(266\) 7.00000 0.429198
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) −15.0000 −0.914566 −0.457283 0.889321i \(-0.651177\pi\)
−0.457283 + 0.889321i \(0.651177\pi\)
\(270\) 0 0
\(271\) −16.0000 −0.971931 −0.485965 0.873978i \(-0.661532\pi\)
−0.485965 + 0.873978i \(0.661532\pi\)
\(272\) −6.00000 −0.363803
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) 12.0000 0.723627
\(276\) 0 0
\(277\) 17.0000 1.02143 0.510716 0.859750i \(-0.329381\pi\)
0.510716 + 0.859750i \(0.329381\pi\)
\(278\) 4.00000 0.239904
\(279\) 0 0
\(280\) 3.00000 0.179284
\(281\) 18.0000 1.07379 0.536895 0.843649i \(-0.319597\pi\)
0.536895 + 0.843649i \(0.319597\pi\)
\(282\) 0 0
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) 9.00000 0.534052
\(285\) 0 0
\(286\) 12.0000 0.709575
\(287\) −9.00000 −0.531253
\(288\) 0 0
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) 0 0
\(292\) 2.00000 0.117041
\(293\) 18.0000 1.05157 0.525786 0.850617i \(-0.323771\pi\)
0.525786 + 0.850617i \(0.323771\pi\)
\(294\) 0 0
\(295\) 18.0000 1.04800
\(296\) 7.00000 0.406867
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −10.0000 −0.576390
\(302\) 10.0000 0.575435
\(303\) 0 0
\(304\) −7.00000 −0.401478
\(305\) −24.0000 −1.37424
\(306\) 0 0
\(307\) 29.0000 1.65512 0.827559 0.561379i \(-0.189729\pi\)
0.827559 + 0.561379i \(0.189729\pi\)
\(308\) 3.00000 0.170941
\(309\) 0 0
\(310\) 15.0000 0.851943
\(311\) 12.0000 0.680458 0.340229 0.940343i \(-0.389495\pi\)
0.340229 + 0.940343i \(0.389495\pi\)
\(312\) 0 0
\(313\) −28.0000 −1.58265 −0.791327 0.611393i \(-0.790609\pi\)
−0.791327 + 0.611393i \(0.790609\pi\)
\(314\) 4.00000 0.225733
\(315\) 0 0
\(316\) −10.0000 −0.562544
\(317\) −30.0000 −1.68497 −0.842484 0.538721i \(-0.818908\pi\)
−0.842484 + 0.538721i \(0.818908\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.00000 −0.167705
\(321\) 0 0
\(322\) 3.00000 0.167183
\(323\) 42.0000 2.33694
\(324\) 0 0
\(325\) −16.0000 −0.887520
\(326\) 16.0000 0.886158
\(327\) 0 0
\(328\) 9.00000 0.496942
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) −28.0000 −1.53902 −0.769510 0.638635i \(-0.779499\pi\)
−0.769510 + 0.638635i \(0.779499\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 18.0000 0.984916
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) −3.00000 −0.163178
\(339\) 0 0
\(340\) 18.0000 0.976187
\(341\) 15.0000 0.812296
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 10.0000 0.539164
\(345\) 0 0
\(346\) 21.0000 1.12897
\(347\) 3.00000 0.161048 0.0805242 0.996753i \(-0.474341\pi\)
0.0805242 + 0.996753i \(0.474341\pi\)
\(348\) 0 0
\(349\) −10.0000 −0.535288 −0.267644 0.963518i \(-0.586245\pi\)
−0.267644 + 0.963518i \(0.586245\pi\)
\(350\) −4.00000 −0.213809
\(351\) 0 0
\(352\) −3.00000 −0.159901
\(353\) 21.0000 1.11772 0.558859 0.829263i \(-0.311239\pi\)
0.558859 + 0.829263i \(0.311239\pi\)
\(354\) 0 0
\(355\) −27.0000 −1.43301
\(356\) 15.0000 0.794998
\(357\) 0 0
\(358\) 24.0000 1.26844
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) 30.0000 1.57895
\(362\) −2.00000 −0.105118
\(363\) 0 0
\(364\) −4.00000 −0.209657
\(365\) −6.00000 −0.314054
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) −3.00000 −0.156386
\(369\) 0 0
\(370\) −21.0000 −1.09174
\(371\) 12.0000 0.623009
\(372\) 0 0
\(373\) −13.0000 −0.673114 −0.336557 0.941663i \(-0.609263\pi\)
−0.336557 + 0.941663i \(0.609263\pi\)
\(374\) 18.0000 0.930758
\(375\) 0 0
\(376\) −6.00000 −0.309426
\(377\) 0 0
\(378\) 0 0
\(379\) 2.00000 0.102733 0.0513665 0.998680i \(-0.483642\pi\)
0.0513665 + 0.998680i \(0.483642\pi\)
\(380\) 21.0000 1.07728
\(381\) 0 0
\(382\) −15.0000 −0.767467
\(383\) −30.0000 −1.53293 −0.766464 0.642287i \(-0.777986\pi\)
−0.766464 + 0.642287i \(0.777986\pi\)
\(384\) 0 0
\(385\) −9.00000 −0.458682
\(386\) −14.0000 −0.712581
\(387\) 0 0
\(388\) 8.00000 0.406138
\(389\) 12.0000 0.608424 0.304212 0.952604i \(-0.401607\pi\)
0.304212 + 0.952604i \(0.401607\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 18.0000 0.906827
\(395\) 30.0000 1.50946
\(396\) 0 0
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 7.00000 0.350878
\(399\) 0 0
\(400\) 4.00000 0.200000
\(401\) 24.0000 1.19850 0.599251 0.800561i \(-0.295465\pi\)
0.599251 + 0.800561i \(0.295465\pi\)
\(402\) 0 0
\(403\) −20.0000 −0.996271
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 0 0
\(407\) −21.0000 −1.04093
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) −27.0000 −1.33343
\(411\) 0 0
\(412\) 5.00000 0.246332
\(413\) −6.00000 −0.295241
\(414\) 0 0
\(415\) 0 0
\(416\) 4.00000 0.196116
\(417\) 0 0
\(418\) 21.0000 1.02714
\(419\) 18.0000 0.879358 0.439679 0.898155i \(-0.355092\pi\)
0.439679 + 0.898155i \(0.355092\pi\)
\(420\) 0 0
\(421\) 17.0000 0.828529 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) −14.0000 −0.681509
\(423\) 0 0
\(424\) −12.0000 −0.582772
\(425\) −24.0000 −1.16417
\(426\) 0 0
\(427\) 8.00000 0.387147
\(428\) −12.0000 −0.580042
\(429\) 0 0
\(430\) −30.0000 −1.44673
\(431\) 3.00000 0.144505 0.0722525 0.997386i \(-0.476981\pi\)
0.0722525 + 0.997386i \(0.476981\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) −5.00000 −0.240008
\(435\) 0 0
\(436\) 11.0000 0.526804
\(437\) 21.0000 1.00457
\(438\) 0 0
\(439\) 8.00000 0.381819 0.190910 0.981608i \(-0.438856\pi\)
0.190910 + 0.981608i \(0.438856\pi\)
\(440\) 9.00000 0.429058
\(441\) 0 0
\(442\) −24.0000 −1.14156
\(443\) −3.00000 −0.142534 −0.0712672 0.997457i \(-0.522704\pi\)
−0.0712672 + 0.997457i \(0.522704\pi\)
\(444\) 0 0
\(445\) −45.0000 −2.13320
\(446\) 1.00000 0.0473514
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) 18.0000 0.849473 0.424736 0.905317i \(-0.360367\pi\)
0.424736 + 0.905317i \(0.360367\pi\)
\(450\) 0 0
\(451\) −27.0000 −1.27138
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) 6.00000 0.281594
\(455\) 12.0000 0.562569
\(456\) 0 0
\(457\) −19.0000 −0.888783 −0.444391 0.895833i \(-0.646580\pi\)
−0.444391 + 0.895833i \(0.646580\pi\)
\(458\) 4.00000 0.186908
\(459\) 0 0
\(460\) 9.00000 0.419627
\(461\) 27.0000 1.25752 0.628758 0.777601i \(-0.283564\pi\)
0.628758 + 0.777601i \(0.283564\pi\)
\(462\) 0 0
\(463\) 14.0000 0.650635 0.325318 0.945605i \(-0.394529\pi\)
0.325318 + 0.945605i \(0.394529\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −6.00000 −0.277945
\(467\) 6.00000 0.277647 0.138823 0.990317i \(-0.455668\pi\)
0.138823 + 0.990317i \(0.455668\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 18.0000 0.830278
\(471\) 0 0
\(472\) 6.00000 0.276172
\(473\) −30.0000 −1.37940
\(474\) 0 0
\(475\) −28.0000 −1.28473
\(476\) −6.00000 −0.275010
\(477\) 0 0
\(478\) 0 0
\(479\) −24.0000 −1.09659 −0.548294 0.836286i \(-0.684723\pi\)
−0.548294 + 0.836286i \(0.684723\pi\)
\(480\) 0 0
\(481\) 28.0000 1.27669
\(482\) −26.0000 −1.18427
\(483\) 0 0
\(484\) −2.00000 −0.0909091
\(485\) −24.0000 −1.08978
\(486\) 0 0
\(487\) 2.00000 0.0906287 0.0453143 0.998973i \(-0.485571\pi\)
0.0453143 + 0.998973i \(0.485571\pi\)
\(488\) −8.00000 −0.362143
\(489\) 0 0
\(490\) 3.00000 0.135526
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −28.0000 −1.25978
\(495\) 0 0
\(496\) 5.00000 0.224507
\(497\) 9.00000 0.403705
\(498\) 0 0
\(499\) −22.0000 −0.984855 −0.492428 0.870353i \(-0.663890\pi\)
−0.492428 + 0.870353i \(0.663890\pi\)
\(500\) 3.00000 0.134164
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 18.0000 0.800989
\(506\) 9.00000 0.400099
\(507\) 0 0
\(508\) 20.0000 0.887357
\(509\) −30.0000 −1.32973 −0.664863 0.746965i \(-0.731510\pi\)
−0.664863 + 0.746965i \(0.731510\pi\)
\(510\) 0 0
\(511\) 2.00000 0.0884748
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 21.0000 0.926270
\(515\) −15.0000 −0.660979
\(516\) 0 0
\(517\) 18.0000 0.791639
\(518\) 7.00000 0.307562
\(519\) 0 0
\(520\) −12.0000 −0.526235
\(521\) −15.0000 −0.657162 −0.328581 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 0 0
\(523\) 29.0000 1.26808 0.634041 0.773300i \(-0.281395\pi\)
0.634041 + 0.773300i \(0.281395\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) 15.0000 0.654031
\(527\) −30.0000 −1.30682
\(528\) 0 0
\(529\) −14.0000 −0.608696
\(530\) 36.0000 1.56374
\(531\) 0 0
\(532\) −7.00000 −0.303488
\(533\) 36.0000 1.55933
\(534\) 0 0
\(535\) 36.0000 1.55642
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) 15.0000 0.646696
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) −7.00000 −0.300954 −0.150477 0.988614i \(-0.548081\pi\)
−0.150477 + 0.988614i \(0.548081\pi\)
\(542\) 16.0000 0.687259
\(543\) 0 0
\(544\) 6.00000 0.257248
\(545\) −33.0000 −1.41356
\(546\) 0 0
\(547\) −28.0000 −1.19719 −0.598597 0.801050i \(-0.704275\pi\)
−0.598597 + 0.801050i \(0.704275\pi\)
\(548\) 12.0000 0.512615
\(549\) 0 0
\(550\) −12.0000 −0.511682
\(551\) 0 0
\(552\) 0 0
\(553\) −10.0000 −0.425243
\(554\) −17.0000 −0.722261
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) −42.0000 −1.77960 −0.889799 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 0 0
\(559\) 40.0000 1.69182
\(560\) −3.00000 −0.126773
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) −24.0000 −1.01148 −0.505740 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 54.0000 2.27180
\(566\) 4.00000 0.168133
\(567\) 0 0
\(568\) −9.00000 −0.377632
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) 14.0000 0.585882 0.292941 0.956131i \(-0.405366\pi\)
0.292941 + 0.956131i \(0.405366\pi\)
\(572\) −12.0000 −0.501745
\(573\) 0 0
\(574\) 9.00000 0.375653
\(575\) −12.0000 −0.500435
\(576\) 0 0
\(577\) 2.00000 0.0832611 0.0416305 0.999133i \(-0.486745\pi\)
0.0416305 + 0.999133i \(0.486745\pi\)
\(578\) −19.0000 −0.790296
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 36.0000 1.49097
\(584\) −2.00000 −0.0827606
\(585\) 0 0
\(586\) −18.0000 −0.743573
\(587\) 18.0000 0.742940 0.371470 0.928445i \(-0.378854\pi\)
0.371470 + 0.928445i \(0.378854\pi\)
\(588\) 0 0
\(589\) −35.0000 −1.44215
\(590\) −18.0000 −0.741048
\(591\) 0 0
\(592\) −7.00000 −0.287698
\(593\) 33.0000 1.35515 0.677574 0.735455i \(-0.263031\pi\)
0.677574 + 0.735455i \(0.263031\pi\)
\(594\) 0 0
\(595\) 18.0000 0.737928
\(596\) 6.00000 0.245770
\(597\) 0 0
\(598\) −12.0000 −0.490716
\(599\) −33.0000 −1.34834 −0.674172 0.738575i \(-0.735499\pi\)
−0.674172 + 0.738575i \(0.735499\pi\)
\(600\) 0 0
\(601\) −28.0000 −1.14214 −0.571072 0.820900i \(-0.693472\pi\)
−0.571072 + 0.820900i \(0.693472\pi\)
\(602\) 10.0000 0.407570
\(603\) 0 0
\(604\) −10.0000 −0.406894
\(605\) 6.00000 0.243935
\(606\) 0 0
\(607\) −4.00000 −0.162355 −0.0811775 0.996700i \(-0.525868\pi\)
−0.0811775 + 0.996700i \(0.525868\pi\)
\(608\) 7.00000 0.283887
\(609\) 0 0
\(610\) 24.0000 0.971732
\(611\) −24.0000 −0.970936
\(612\) 0 0
\(613\) 29.0000 1.17130 0.585649 0.810564i \(-0.300840\pi\)
0.585649 + 0.810564i \(0.300840\pi\)
\(614\) −29.0000 −1.17034
\(615\) 0 0
\(616\) −3.00000 −0.120873
\(617\) 18.0000 0.724653 0.362326 0.932051i \(-0.381983\pi\)
0.362326 + 0.932051i \(0.381983\pi\)
\(618\) 0 0
\(619\) 35.0000 1.40677 0.703384 0.710810i \(-0.251671\pi\)
0.703384 + 0.710810i \(0.251671\pi\)
\(620\) −15.0000 −0.602414
\(621\) 0 0
\(622\) −12.0000 −0.481156
\(623\) 15.0000 0.600962
\(624\) 0 0
\(625\) −29.0000 −1.16000
\(626\) 28.0000 1.11911
\(627\) 0 0
\(628\) −4.00000 −0.159617
\(629\) 42.0000 1.67465
\(630\) 0 0
\(631\) 38.0000 1.51276 0.756378 0.654135i \(-0.226967\pi\)
0.756378 + 0.654135i \(0.226967\pi\)
\(632\) 10.0000 0.397779
\(633\) 0 0
\(634\) 30.0000 1.19145
\(635\) −60.0000 −2.38103
\(636\) 0 0
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) 0 0
\(640\) 3.00000 0.118585
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) 5.00000 0.197181 0.0985904 0.995128i \(-0.468567\pi\)
0.0985904 + 0.995128i \(0.468567\pi\)
\(644\) −3.00000 −0.118217
\(645\) 0 0
\(646\) −42.0000 −1.65247
\(647\) −6.00000 −0.235884 −0.117942 0.993020i \(-0.537630\pi\)
−0.117942 + 0.993020i \(0.537630\pi\)
\(648\) 0 0
\(649\) −18.0000 −0.706562
\(650\) 16.0000 0.627572
\(651\) 0 0
\(652\) −16.0000 −0.626608
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) 0 0
\(655\) −18.0000 −0.703318
\(656\) −9.00000 −0.351391
\(657\) 0 0
\(658\) −6.00000 −0.233904
\(659\) −9.00000 −0.350590 −0.175295 0.984516i \(-0.556088\pi\)
−0.175295 + 0.984516i \(0.556088\pi\)
\(660\) 0 0
\(661\) −40.0000 −1.55582 −0.777910 0.628376i \(-0.783720\pi\)
−0.777910 + 0.628376i \(0.783720\pi\)
\(662\) 28.0000 1.08825
\(663\) 0 0
\(664\) 0 0
\(665\) 21.0000 0.814345
\(666\) 0 0
\(667\) 0 0
\(668\) −18.0000 −0.696441
\(669\) 0 0
\(670\) −12.0000 −0.463600
\(671\) 24.0000 0.926510
\(672\) 0 0
\(673\) −46.0000 −1.77317 −0.886585 0.462566i \(-0.846929\pi\)
−0.886585 + 0.462566i \(0.846929\pi\)
\(674\) 13.0000 0.500741
\(675\) 0 0
\(676\) 3.00000 0.115385
\(677\) 15.0000 0.576497 0.288248 0.957556i \(-0.406927\pi\)
0.288248 + 0.957556i \(0.406927\pi\)
\(678\) 0 0
\(679\) 8.00000 0.307012
\(680\) −18.0000 −0.690268
\(681\) 0 0
\(682\) −15.0000 −0.574380
\(683\) −15.0000 −0.573959 −0.286980 0.957937i \(-0.592651\pi\)
−0.286980 + 0.957937i \(0.592651\pi\)
\(684\) 0 0
\(685\) −36.0000 −1.37549
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) −10.0000 −0.381246
\(689\) −48.0000 −1.82865
\(690\) 0 0
\(691\) −28.0000 −1.06517 −0.532585 0.846376i \(-0.678779\pi\)
−0.532585 + 0.846376i \(0.678779\pi\)
\(692\) −21.0000 −0.798300
\(693\) 0 0
\(694\) −3.00000 −0.113878
\(695\) 12.0000 0.455186
\(696\) 0 0
\(697\) 54.0000 2.04540
\(698\) 10.0000 0.378506
\(699\) 0 0
\(700\) 4.00000 0.151186
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 49.0000 1.84807
\(704\) 3.00000 0.113067
\(705\) 0 0
\(706\) −21.0000 −0.790345
\(707\) −6.00000 −0.225653
\(708\) 0 0
\(709\) 35.0000 1.31445 0.657226 0.753693i \(-0.271730\pi\)
0.657226 + 0.753693i \(0.271730\pi\)
\(710\) 27.0000 1.01329
\(711\) 0 0
\(712\) −15.0000 −0.562149
\(713\) −15.0000 −0.561754
\(714\) 0 0
\(715\) 36.0000 1.34632
\(716\) −24.0000 −0.896922
\(717\) 0 0
\(718\) 12.0000 0.447836
\(719\) −30.0000 −1.11881 −0.559406 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 5.00000 0.186210
\(722\) −30.0000 −1.11648
\(723\) 0 0
\(724\) 2.00000 0.0743294
\(725\) 0 0
\(726\) 0 0
\(727\) 8.00000 0.296704 0.148352 0.988935i \(-0.452603\pi\)
0.148352 + 0.988935i \(0.452603\pi\)
\(728\) 4.00000 0.148250
\(729\) 0 0
\(730\) 6.00000 0.222070
\(731\) 60.0000 2.21918
\(732\) 0 0
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) −17.0000 −0.627481
\(735\) 0 0
\(736\) 3.00000 0.110581
\(737\) −12.0000 −0.442026
\(738\) 0 0
\(739\) −34.0000 −1.25071 −0.625355 0.780340i \(-0.715046\pi\)
−0.625355 + 0.780340i \(0.715046\pi\)
\(740\) 21.0000 0.771975
\(741\) 0 0
\(742\) −12.0000 −0.440534
\(743\) 9.00000 0.330178 0.165089 0.986279i \(-0.447209\pi\)
0.165089 + 0.986279i \(0.447209\pi\)
\(744\) 0 0
\(745\) −18.0000 −0.659469
\(746\) 13.0000 0.475964
\(747\) 0 0
\(748\) −18.0000 −0.658145
\(749\) −12.0000 −0.438470
\(750\) 0 0
\(751\) −22.0000 −0.802791 −0.401396 0.915905i \(-0.631475\pi\)
−0.401396 + 0.915905i \(0.631475\pi\)
\(752\) 6.00000 0.218797
\(753\) 0 0
\(754\) 0 0
\(755\) 30.0000 1.09181
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −2.00000 −0.0726433
\(759\) 0 0
\(760\) −21.0000 −0.761750
\(761\) 42.0000 1.52250 0.761249 0.648459i \(-0.224586\pi\)
0.761249 + 0.648459i \(0.224586\pi\)
\(762\) 0 0
\(763\) 11.0000 0.398227
\(764\) 15.0000 0.542681
\(765\) 0 0
\(766\) 30.0000 1.08394
\(767\) 24.0000 0.866590
\(768\) 0 0
\(769\) −4.00000 −0.144244 −0.0721218 0.997396i \(-0.522977\pi\)
−0.0721218 + 0.997396i \(0.522977\pi\)
\(770\) 9.00000 0.324337
\(771\) 0 0
\(772\) 14.0000 0.503871
\(773\) 39.0000 1.40273 0.701366 0.712801i \(-0.252574\pi\)
0.701366 + 0.712801i \(0.252574\pi\)
\(774\) 0 0
\(775\) 20.0000 0.718421
\(776\) −8.00000 −0.287183
\(777\) 0 0
\(778\) −12.0000 −0.430221
\(779\) 63.0000 2.25721
\(780\) 0 0
\(781\) 27.0000 0.966136
\(782\) −18.0000 −0.643679
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) 12.0000 0.428298
\(786\) 0 0
\(787\) 32.0000 1.14068 0.570338 0.821410i \(-0.306812\pi\)
0.570338 + 0.821410i \(0.306812\pi\)
\(788\) −18.0000 −0.641223
\(789\) 0 0
\(790\) −30.0000 −1.06735
\(791\) −18.0000 −0.640006
\(792\) 0 0
\(793\) −32.0000 −1.13635
\(794\) −2.00000 −0.0709773
\(795\) 0 0
\(796\) −7.00000 −0.248108
\(797\) 27.0000 0.956389 0.478195 0.878254i \(-0.341291\pi\)
0.478195 + 0.878254i \(0.341291\pi\)
\(798\) 0 0
\(799\) −36.0000 −1.27359
\(800\) −4.00000 −0.141421
\(801\) 0 0
\(802\) −24.0000 −0.847469
\(803\) 6.00000 0.211735
\(804\) 0 0
\(805\) 9.00000 0.317208
\(806\) 20.0000 0.704470
\(807\) 0 0
\(808\) 6.00000 0.211079
\(809\) −24.0000 −0.843795 −0.421898 0.906644i \(-0.638636\pi\)
−0.421898 + 0.906644i \(0.638636\pi\)
\(810\) 0 0
\(811\) −25.0000 −0.877869 −0.438934 0.898519i \(-0.644644\pi\)
−0.438934 + 0.898519i \(0.644644\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 21.0000 0.736050
\(815\) 48.0000 1.68137
\(816\) 0 0
\(817\) 70.0000 2.44899
\(818\) 22.0000 0.769212
\(819\) 0 0
\(820\) 27.0000 0.942881
\(821\) −30.0000 −1.04701 −0.523504 0.852023i \(-0.675375\pi\)
−0.523504 + 0.852023i \(0.675375\pi\)
\(822\) 0 0
\(823\) 14.0000 0.488009 0.244005 0.969774i \(-0.421539\pi\)
0.244005 + 0.969774i \(0.421539\pi\)
\(824\) −5.00000 −0.174183
\(825\) 0 0
\(826\) 6.00000 0.208767
\(827\) 9.00000 0.312961 0.156480 0.987681i \(-0.449985\pi\)
0.156480 + 0.987681i \(0.449985\pi\)
\(828\) 0 0
\(829\) −34.0000 −1.18087 −0.590434 0.807086i \(-0.701044\pi\)
−0.590434 + 0.807086i \(0.701044\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.00000 −0.138675
\(833\) −6.00000 −0.207888
\(834\) 0 0
\(835\) 54.0000 1.86875
\(836\) −21.0000 −0.726300
\(837\) 0 0
\(838\) −18.0000 −0.621800
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) −17.0000 −0.585859
\(843\) 0 0
\(844\) 14.0000 0.481900
\(845\) −9.00000 −0.309609
\(846\) 0 0
\(847\) −2.00000 −0.0687208
\(848\) 12.0000 0.412082
\(849\) 0 0
\(850\) 24.0000 0.823193
\(851\) 21.0000 0.719871
\(852\) 0 0
\(853\) 8.00000 0.273915 0.136957 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 12.0000 0.410152
\(857\) 3.00000 0.102478 0.0512390 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) −31.0000 −1.05771 −0.528853 0.848713i \(-0.677378\pi\)
−0.528853 + 0.848713i \(0.677378\pi\)
\(860\) 30.0000 1.02299
\(861\) 0 0
\(862\) −3.00000 −0.102180
\(863\) 24.0000 0.816970 0.408485 0.912765i \(-0.366057\pi\)
0.408485 + 0.912765i \(0.366057\pi\)
\(864\) 0 0
\(865\) 63.0000 2.14206
\(866\) 16.0000 0.543702
\(867\) 0 0
\(868\) 5.00000 0.169711
\(869\) −30.0000 −1.01768
\(870\) 0 0
\(871\) 16.0000 0.542139
\(872\) −11.0000 −0.372507
\(873\) 0 0
\(874\) −21.0000 −0.710336
\(875\) 3.00000 0.101419
\(876\) 0 0
\(877\) 14.0000 0.472746 0.236373 0.971662i \(-0.424041\pi\)
0.236373 + 0.971662i \(0.424041\pi\)
\(878\) −8.00000 −0.269987
\(879\) 0 0
\(880\) −9.00000 −0.303390
\(881\) 9.00000 0.303218 0.151609 0.988441i \(-0.451555\pi\)
0.151609 + 0.988441i \(0.451555\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 24.0000 0.807207
\(885\) 0 0
\(886\) 3.00000 0.100787
\(887\) 24.0000 0.805841 0.402921 0.915235i \(-0.367995\pi\)
0.402921 + 0.915235i \(0.367995\pi\)
\(888\) 0 0
\(889\) 20.0000 0.670778
\(890\) 45.0000 1.50840
\(891\) 0 0
\(892\) −1.00000 −0.0334825
\(893\) −42.0000 −1.40548
\(894\) 0 0
\(895\) 72.0000 2.40669
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) −18.0000 −0.600668
\(899\) 0 0
\(900\) 0 0
\(901\) −72.0000 −2.39867
\(902\) 27.0000 0.899002
\(903\) 0 0
\(904\) 18.0000 0.598671
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) −46.0000 −1.52740 −0.763702 0.645568i \(-0.776621\pi\)
−0.763702 + 0.645568i \(0.776621\pi\)
\(908\) −6.00000 −0.199117
\(909\) 0 0
\(910\) −12.0000 −0.397796
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 19.0000 0.628464
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) 6.00000 0.198137
\(918\) 0 0
\(919\) −52.0000 −1.71532 −0.857661 0.514216i \(-0.828083\pi\)
−0.857661 + 0.514216i \(0.828083\pi\)
\(920\) −9.00000 −0.296721
\(921\) 0 0
\(922\) −27.0000 −0.889198
\(923\) −36.0000 −1.18495
\(924\) 0 0
\(925\) −28.0000 −0.920634
\(926\) −14.0000 −0.460069
\(927\) 0 0
\(928\) 0 0
\(929\) 6.00000 0.196854 0.0984268 0.995144i \(-0.468619\pi\)
0.0984268 + 0.995144i \(0.468619\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) −6.00000 −0.196326
\(935\) 54.0000 1.76599
\(936\) 0 0
\(937\) 20.0000 0.653372 0.326686 0.945133i \(-0.394068\pi\)
0.326686 + 0.945133i \(0.394068\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) −18.0000 −0.587095
\(941\) −15.0000 −0.488986 −0.244493 0.969651i \(-0.578622\pi\)
−0.244493 + 0.969651i \(0.578622\pi\)
\(942\) 0 0
\(943\) 27.0000 0.879241
\(944\) −6.00000 −0.195283
\(945\) 0 0
\(946\) 30.0000 0.975384
\(947\) −39.0000 −1.26733 −0.633665 0.773608i \(-0.718450\pi\)
−0.633665 + 0.773608i \(0.718450\pi\)
\(948\) 0 0
\(949\) −8.00000 −0.259691
\(950\) 28.0000 0.908440
\(951\) 0 0
\(952\) 6.00000 0.194461
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) −45.0000 −1.45617
\(956\) 0 0
\(957\) 0 0
\(958\) 24.0000 0.775405
\(959\) 12.0000 0.387500
\(960\) 0 0
\(961\) −6.00000 −0.193548
\(962\) −28.0000 −0.902756
\(963\) 0 0
\(964\) 26.0000 0.837404
\(965\) −42.0000 −1.35203
\(966\) 0 0
\(967\) −58.0000 −1.86515 −0.932577 0.360971i \(-0.882445\pi\)
−0.932577 + 0.360971i \(0.882445\pi\)
\(968\) 2.00000 0.0642824
\(969\) 0 0
\(970\) 24.0000 0.770594
\(971\) −48.0000 −1.54039 −0.770197 0.637806i \(-0.779842\pi\)
−0.770197 + 0.637806i \(0.779842\pi\)
\(972\) 0 0
\(973\) −4.00000 −0.128234
\(974\) −2.00000 −0.0640841
\(975\) 0 0
\(976\) 8.00000 0.256074
\(977\) 12.0000 0.383914 0.191957 0.981403i \(-0.438517\pi\)
0.191957 + 0.981403i \(0.438517\pi\)
\(978\) 0 0
\(979\) 45.0000 1.43821
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) 9.00000 0.287202
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 0 0
\(985\) 54.0000 1.72058
\(986\) 0 0
\(987\) 0 0
\(988\) 28.0000 0.890799
\(989\) 30.0000 0.953945
\(990\) 0 0
\(991\) −16.0000 −0.508257 −0.254128 0.967170i \(-0.581789\pi\)
−0.254128 + 0.967170i \(0.581789\pi\)
\(992\) −5.00000 −0.158750
\(993\) 0 0
\(994\) −9.00000 −0.285463
\(995\) 21.0000 0.665745
\(996\) 0 0
\(997\) −10.0000 −0.316703 −0.158352 0.987383i \(-0.550618\pi\)
−0.158352 + 0.987383i \(0.550618\pi\)
\(998\) 22.0000 0.696398
\(999\) 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 378.2.a.b.1.1 1
3.2 odd 2 378.2.a.g.1.1 yes 1
4.3 odd 2 3024.2.a.c.1.1 1
5.4 even 2 9450.2.a.cu.1.1 1
7.6 odd 2 2646.2.a.n.1.1 1
9.2 odd 6 1134.2.f.b.757.1 2
9.4 even 3 1134.2.f.o.379.1 2
9.5 odd 6 1134.2.f.b.379.1 2
9.7 even 3 1134.2.f.o.757.1 2
12.11 even 2 3024.2.a.bb.1.1 1
15.14 odd 2 9450.2.a.h.1.1 1
21.20 even 2 2646.2.a.q.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
378.2.a.b.1.1 1 1.1 even 1 trivial
378.2.a.g.1.1 yes 1 3.2 odd 2
1134.2.f.b.379.1 2 9.5 odd 6
1134.2.f.b.757.1 2 9.2 odd 6
1134.2.f.o.379.1 2 9.4 even 3
1134.2.f.o.757.1 2 9.7 even 3
2646.2.a.n.1.1 1 7.6 odd 2
2646.2.a.q.1.1 1 21.20 even 2
3024.2.a.c.1.1 1 4.3 odd 2
3024.2.a.bb.1.1 1 12.11 even 2
9450.2.a.h.1.1 1 15.14 odd 2
9450.2.a.cu.1.1 1 5.4 even 2