Properties

Label 2352.4.a.bd.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more about

Newspace parameters

Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(138.772492334\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 2352.1

$q$-expansion

\(f(q)\) \(=\) \(q+3.00000 q^{3} +3.00000 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +3.00000 q^{5} +9.00000 q^{9} +15.0000 q^{11} +64.0000 q^{13} +9.00000 q^{15} -84.0000 q^{17} -16.0000 q^{19} +84.0000 q^{23} -116.000 q^{25} +27.0000 q^{27} -297.000 q^{29} -253.000 q^{31} +45.0000 q^{33} -316.000 q^{37} +192.000 q^{39} -360.000 q^{41} -26.0000 q^{43} +27.0000 q^{45} -30.0000 q^{47} -252.000 q^{51} +363.000 q^{53} +45.0000 q^{55} -48.0000 q^{57} -15.0000 q^{59} +118.000 q^{61} +192.000 q^{65} +370.000 q^{67} +252.000 q^{69} +342.000 q^{71} -362.000 q^{73} -348.000 q^{75} -467.000 q^{79} +81.0000 q^{81} +477.000 q^{83} -252.000 q^{85} -891.000 q^{87} -906.000 q^{89} -759.000 q^{93} -48.0000 q^{95} -503.000 q^{97} +135.000 q^{99} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) 3.00000 0.268328 0.134164 0.990959i \(-0.457165\pi\)
0.134164 + 0.990959i \(0.457165\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 15.0000 0.411152 0.205576 0.978641i \(-0.434093\pi\)
0.205576 + 0.978641i \(0.434093\pi\)
\(12\) 0 0
\(13\) 64.0000 1.36542 0.682708 0.730691i \(-0.260802\pi\)
0.682708 + 0.730691i \(0.260802\pi\)
\(14\) 0 0
\(15\) 9.00000 0.154919
\(16\) 0 0
\(17\) −84.0000 −1.19841 −0.599206 0.800595i \(-0.704517\pi\)
−0.599206 + 0.800595i \(0.704517\pi\)
\(18\) 0 0
\(19\) −16.0000 −0.193192 −0.0965961 0.995324i \(-0.530796\pi\)
−0.0965961 + 0.995324i \(0.530796\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 84.0000 0.761531 0.380765 0.924672i \(-0.375661\pi\)
0.380765 + 0.924672i \(0.375661\pi\)
\(24\) 0 0
\(25\) −116.000 −0.928000
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) −297.000 −1.90178 −0.950888 0.309535i \(-0.899827\pi\)
−0.950888 + 0.309535i \(0.899827\pi\)
\(30\) 0 0
\(31\) −253.000 −1.46581 −0.732906 0.680330i \(-0.761836\pi\)
−0.732906 + 0.680330i \(0.761836\pi\)
\(32\) 0 0
\(33\) 45.0000 0.237379
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −316.000 −1.40406 −0.702028 0.712149i \(-0.747722\pi\)
−0.702028 + 0.712149i \(0.747722\pi\)
\(38\) 0 0
\(39\) 192.000 0.788323
\(40\) 0 0
\(41\) −360.000 −1.37128 −0.685641 0.727940i \(-0.740478\pi\)
−0.685641 + 0.727940i \(0.740478\pi\)
\(42\) 0 0
\(43\) −26.0000 −0.0922084 −0.0461042 0.998937i \(-0.514681\pi\)
−0.0461042 + 0.998937i \(0.514681\pi\)
\(44\) 0 0
\(45\) 27.0000 0.0894427
\(46\) 0 0
\(47\) −30.0000 −0.0931053 −0.0465527 0.998916i \(-0.514824\pi\)
−0.0465527 + 0.998916i \(0.514824\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −252.000 −0.691903
\(52\) 0 0
\(53\) 363.000 0.940790 0.470395 0.882456i \(-0.344111\pi\)
0.470395 + 0.882456i \(0.344111\pi\)
\(54\) 0 0
\(55\) 45.0000 0.110324
\(56\) 0 0
\(57\) −48.0000 −0.111540
\(58\) 0 0
\(59\) −15.0000 −0.0330989 −0.0165494 0.999863i \(-0.505268\pi\)
−0.0165494 + 0.999863i \(0.505268\pi\)
\(60\) 0 0
\(61\) 118.000 0.247678 0.123839 0.992302i \(-0.460479\pi\)
0.123839 + 0.992302i \(0.460479\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 192.000 0.366380
\(66\) 0 0
\(67\) 370.000 0.674667 0.337334 0.941385i \(-0.390475\pi\)
0.337334 + 0.941385i \(0.390475\pi\)
\(68\) 0 0
\(69\) 252.000 0.439670
\(70\) 0 0
\(71\) 342.000 0.571661 0.285831 0.958280i \(-0.407731\pi\)
0.285831 + 0.958280i \(0.407731\pi\)
\(72\) 0 0
\(73\) −362.000 −0.580396 −0.290198 0.956967i \(-0.593721\pi\)
−0.290198 + 0.956967i \(0.593721\pi\)
\(74\) 0 0
\(75\) −348.000 −0.535781
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −467.000 −0.665084 −0.332542 0.943089i \(-0.607906\pi\)
−0.332542 + 0.943089i \(0.607906\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 477.000 0.630814 0.315407 0.948957i \(-0.397859\pi\)
0.315407 + 0.948957i \(0.397859\pi\)
\(84\) 0 0
\(85\) −252.000 −0.321568
\(86\) 0 0
\(87\) −891.000 −1.09799
\(88\) 0 0
\(89\) −906.000 −1.07905 −0.539527 0.841968i \(-0.681397\pi\)
−0.539527 + 0.841968i \(0.681397\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −759.000 −0.846286
\(94\) 0 0
\(95\) −48.0000 −0.0518389
\(96\) 0 0
\(97\) −503.000 −0.526515 −0.263257 0.964726i \(-0.584797\pi\)
−0.263257 + 0.964726i \(0.584797\pi\)
\(98\) 0 0
\(99\) 135.000 0.137051
\(100\) 0 0
\(101\) 1086.00 1.06991 0.534956 0.844880i \(-0.320328\pi\)
0.534956 + 0.844880i \(0.320328\pi\)
\(102\) 0 0
\(103\) 1736.00 1.66071 0.830355 0.557235i \(-0.188138\pi\)
0.830355 + 0.557235i \(0.188138\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1353.00 1.22242 0.611212 0.791467i \(-0.290682\pi\)
0.611212 + 0.791467i \(0.290682\pi\)
\(108\) 0 0
\(109\) −370.000 −0.325134 −0.162567 0.986698i \(-0.551977\pi\)
−0.162567 + 0.986698i \(0.551977\pi\)
\(110\) 0 0
\(111\) −948.000 −0.810632
\(112\) 0 0
\(113\) −648.000 −0.539458 −0.269729 0.962936i \(-0.586934\pi\)
−0.269729 + 0.962936i \(0.586934\pi\)
\(114\) 0 0
\(115\) 252.000 0.204340
\(116\) 0 0
\(117\) 576.000 0.455139
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1106.00 −0.830954
\(122\) 0 0
\(123\) −1080.00 −0.791710
\(124\) 0 0
\(125\) −723.000 −0.517337
\(126\) 0 0
\(127\) −377.000 −0.263412 −0.131706 0.991289i \(-0.542046\pi\)
−0.131706 + 0.991289i \(0.542046\pi\)
\(128\) 0 0
\(129\) −78.0000 −0.0532366
\(130\) 0 0
\(131\) −651.000 −0.434184 −0.217092 0.976151i \(-0.569657\pi\)
−0.217092 + 0.976151i \(0.569657\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 81.0000 0.0516398
\(136\) 0 0
\(137\) −1770.00 −1.10381 −0.551903 0.833909i \(-0.686098\pi\)
−0.551903 + 0.833909i \(0.686098\pi\)
\(138\) 0 0
\(139\) −1558.00 −0.950704 −0.475352 0.879796i \(-0.657679\pi\)
−0.475352 + 0.879796i \(0.657679\pi\)
\(140\) 0 0
\(141\) −90.0000 −0.0537544
\(142\) 0 0
\(143\) 960.000 0.561393
\(144\) 0 0
\(145\) −891.000 −0.510300
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 2454.00 1.34926 0.674629 0.738157i \(-0.264304\pi\)
0.674629 + 0.738157i \(0.264304\pi\)
\(150\) 0 0
\(151\) −1259.00 −0.678516 −0.339258 0.940693i \(-0.610176\pi\)
−0.339258 + 0.940693i \(0.610176\pi\)
\(152\) 0 0
\(153\) −756.000 −0.399470
\(154\) 0 0
\(155\) −759.000 −0.393318
\(156\) 0 0
\(157\) 196.000 0.0996338 0.0498169 0.998758i \(-0.484136\pi\)
0.0498169 + 0.998758i \(0.484136\pi\)
\(158\) 0 0
\(159\) 1089.00 0.543166
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1252.00 0.601621 0.300810 0.953684i \(-0.402743\pi\)
0.300810 + 0.953684i \(0.402743\pi\)
\(164\) 0 0
\(165\) 135.000 0.0636954
\(166\) 0 0
\(167\) −2646.00 −1.22607 −0.613035 0.790056i \(-0.710051\pi\)
−0.613035 + 0.790056i \(0.710051\pi\)
\(168\) 0 0
\(169\) 1899.00 0.864360
\(170\) 0 0
\(171\) −144.000 −0.0643974
\(172\) 0 0
\(173\) 786.000 0.345425 0.172712 0.984972i \(-0.444747\pi\)
0.172712 + 0.984972i \(0.444747\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −45.0000 −0.0191096
\(178\) 0 0
\(179\) −2892.00 −1.20759 −0.603794 0.797140i \(-0.706345\pi\)
−0.603794 + 0.797140i \(0.706345\pi\)
\(180\) 0 0
\(181\) −1352.00 −0.555212 −0.277606 0.960695i \(-0.589541\pi\)
−0.277606 + 0.960695i \(0.589541\pi\)
\(182\) 0 0
\(183\) 354.000 0.142997
\(184\) 0 0
\(185\) −948.000 −0.376748
\(186\) 0 0
\(187\) −1260.00 −0.492729
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −3912.00 −1.48200 −0.741001 0.671504i \(-0.765649\pi\)
−0.741001 + 0.671504i \(0.765649\pi\)
\(192\) 0 0
\(193\) 1493.00 0.556832 0.278416 0.960461i \(-0.410191\pi\)
0.278416 + 0.960461i \(0.410191\pi\)
\(194\) 0 0
\(195\) 576.000 0.211529
\(196\) 0 0
\(197\) −4086.00 −1.47774 −0.738872 0.673846i \(-0.764641\pi\)
−0.738872 + 0.673846i \(0.764641\pi\)
\(198\) 0 0
\(199\) −3556.00 −1.26672 −0.633362 0.773855i \(-0.718326\pi\)
−0.633362 + 0.773855i \(0.718326\pi\)
\(200\) 0 0
\(201\) 1110.00 0.389519
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −1080.00 −0.367954
\(206\) 0 0
\(207\) 756.000 0.253844
\(208\) 0 0
\(209\) −240.000 −0.0794313
\(210\) 0 0
\(211\) −1250.00 −0.407837 −0.203918 0.978988i \(-0.565368\pi\)
−0.203918 + 0.978988i \(0.565368\pi\)
\(212\) 0 0
\(213\) 1026.00 0.330049
\(214\) 0 0
\(215\) −78.0000 −0.0247421
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1086.00 −0.335092
\(220\) 0 0
\(221\) −5376.00 −1.63633
\(222\) 0 0
\(223\) 425.000 0.127624 0.0638119 0.997962i \(-0.479674\pi\)
0.0638119 + 0.997962i \(0.479674\pi\)
\(224\) 0 0
\(225\) −1044.00 −0.309333
\(226\) 0 0
\(227\) 3855.00 1.12716 0.563580 0.826061i \(-0.309424\pi\)
0.563580 + 0.826061i \(0.309424\pi\)
\(228\) 0 0
\(229\) 2188.00 0.631385 0.315692 0.948862i \(-0.397763\pi\)
0.315692 + 0.948862i \(0.397763\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 852.000 0.239555 0.119778 0.992801i \(-0.461782\pi\)
0.119778 + 0.992801i \(0.461782\pi\)
\(234\) 0 0
\(235\) −90.0000 −0.0249828
\(236\) 0 0
\(237\) −1401.00 −0.383986
\(238\) 0 0
\(239\) −5508.00 −1.49072 −0.745362 0.666660i \(-0.767723\pi\)
−0.745362 + 0.666660i \(0.767723\pi\)
\(240\) 0 0
\(241\) −791.000 −0.211422 −0.105711 0.994397i \(-0.533712\pi\)
−0.105711 + 0.994397i \(0.533712\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1024.00 −0.263788
\(248\) 0 0
\(249\) 1431.00 0.364201
\(250\) 0 0
\(251\) 5265.00 1.32400 0.662000 0.749504i \(-0.269708\pi\)
0.662000 + 0.749504i \(0.269708\pi\)
\(252\) 0 0
\(253\) 1260.00 0.313105
\(254\) 0 0
\(255\) −756.000 −0.185657
\(256\) 0 0
\(257\) 6870.00 1.66747 0.833733 0.552168i \(-0.186199\pi\)
0.833733 + 0.552168i \(0.186199\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −2673.00 −0.633925
\(262\) 0 0
\(263\) 222.000 0.0520498 0.0260249 0.999661i \(-0.491715\pi\)
0.0260249 + 0.999661i \(0.491715\pi\)
\(264\) 0 0
\(265\) 1089.00 0.252441
\(266\) 0 0
\(267\) −2718.00 −0.622992
\(268\) 0 0
\(269\) −7851.00 −1.77949 −0.889747 0.456454i \(-0.849119\pi\)
−0.889747 + 0.456454i \(0.849119\pi\)
\(270\) 0 0
\(271\) 5183.00 1.16179 0.580895 0.813979i \(-0.302703\pi\)
0.580895 + 0.813979i \(0.302703\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1740.00 −0.381549
\(276\) 0 0
\(277\) −4960.00 −1.07588 −0.537938 0.842985i \(-0.680796\pi\)
−0.537938 + 0.842985i \(0.680796\pi\)
\(278\) 0 0
\(279\) −2277.00 −0.488604
\(280\) 0 0
\(281\) −774.000 −0.164317 −0.0821583 0.996619i \(-0.526181\pi\)
−0.0821583 + 0.996619i \(0.526181\pi\)
\(282\) 0 0
\(283\) 3698.00 0.776761 0.388380 0.921499i \(-0.373035\pi\)
0.388380 + 0.921499i \(0.373035\pi\)
\(284\) 0 0
\(285\) −144.000 −0.0299292
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2143.00 0.436190
\(290\) 0 0
\(291\) −1509.00 −0.303983
\(292\) 0 0
\(293\) 6273.00 1.25076 0.625380 0.780321i \(-0.284944\pi\)
0.625380 + 0.780321i \(0.284944\pi\)
\(294\) 0 0
\(295\) −45.0000 −0.00888136
\(296\) 0 0
\(297\) 405.000 0.0791262
\(298\) 0 0
\(299\) 5376.00 1.03981
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 3258.00 0.617714
\(304\) 0 0
\(305\) 354.000 0.0664590
\(306\) 0 0
\(307\) −1684.00 −0.313065 −0.156533 0.987673i \(-0.550032\pi\)
−0.156533 + 0.987673i \(0.550032\pi\)
\(308\) 0 0
\(309\) 5208.00 0.958812
\(310\) 0 0
\(311\) −1320.00 −0.240676 −0.120338 0.992733i \(-0.538398\pi\)
−0.120338 + 0.992733i \(0.538398\pi\)
\(312\) 0 0
\(313\) 8503.00 1.53552 0.767760 0.640737i \(-0.221371\pi\)
0.767760 + 0.640737i \(0.221371\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2577.00 −0.456589 −0.228295 0.973592i \(-0.573315\pi\)
−0.228295 + 0.973592i \(0.573315\pi\)
\(318\) 0 0
\(319\) −4455.00 −0.781919
\(320\) 0 0
\(321\) 4059.00 0.705767
\(322\) 0 0
\(323\) 1344.00 0.231524
\(324\) 0 0
\(325\) −7424.00 −1.26711
\(326\) 0 0
\(327\) −1110.00 −0.187716
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 484.000 0.0803717 0.0401859 0.999192i \(-0.487205\pi\)
0.0401859 + 0.999192i \(0.487205\pi\)
\(332\) 0 0
\(333\) −2844.00 −0.468019
\(334\) 0 0
\(335\) 1110.00 0.181032
\(336\) 0 0
\(337\) −8359.00 −1.35117 −0.675584 0.737283i \(-0.736109\pi\)
−0.675584 + 0.737283i \(0.736109\pi\)
\(338\) 0 0
\(339\) −1944.00 −0.311456
\(340\) 0 0
\(341\) −3795.00 −0.602671
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 756.000 0.117976
\(346\) 0 0
\(347\) 1860.00 0.287752 0.143876 0.989596i \(-0.454043\pi\)
0.143876 + 0.989596i \(0.454043\pi\)
\(348\) 0 0
\(349\) 1918.00 0.294178 0.147089 0.989123i \(-0.453010\pi\)
0.147089 + 0.989123i \(0.453010\pi\)
\(350\) 0 0
\(351\) 1728.00 0.262774
\(352\) 0 0
\(353\) 3048.00 0.459571 0.229786 0.973241i \(-0.426197\pi\)
0.229786 + 0.973241i \(0.426197\pi\)
\(354\) 0 0
\(355\) 1026.00 0.153393
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 30.0000 0.00441042 0.00220521 0.999998i \(-0.499298\pi\)
0.00220521 + 0.999998i \(0.499298\pi\)
\(360\) 0 0
\(361\) −6603.00 −0.962677
\(362\) 0 0
\(363\) −3318.00 −0.479752
\(364\) 0 0
\(365\) −1086.00 −0.155737
\(366\) 0 0
\(367\) −11311.0 −1.60880 −0.804400 0.594088i \(-0.797513\pi\)
−0.804400 + 0.594088i \(0.797513\pi\)
\(368\) 0 0
\(369\) −3240.00 −0.457094
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1208.00 0.167689 0.0838443 0.996479i \(-0.473280\pi\)
0.0838443 + 0.996479i \(0.473280\pi\)
\(374\) 0 0
\(375\) −2169.00 −0.298684
\(376\) 0 0
\(377\) −19008.0 −2.59672
\(378\) 0 0
\(379\) −7640.00 −1.03546 −0.517731 0.855543i \(-0.673223\pi\)
−0.517731 + 0.855543i \(0.673223\pi\)
\(380\) 0 0
\(381\) −1131.00 −0.152081
\(382\) 0 0
\(383\) 12750.0 1.70103 0.850515 0.525951i \(-0.176290\pi\)
0.850515 + 0.525951i \(0.176290\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −234.000 −0.0307361
\(388\) 0 0
\(389\) 3126.00 0.407441 0.203720 0.979029i \(-0.434697\pi\)
0.203720 + 0.979029i \(0.434697\pi\)
\(390\) 0 0
\(391\) −7056.00 −0.912627
\(392\) 0 0
\(393\) −1953.00 −0.250676
\(394\) 0 0
\(395\) −1401.00 −0.178461
\(396\) 0 0
\(397\) 5932.00 0.749921 0.374960 0.927041i \(-0.377656\pi\)
0.374960 + 0.927041i \(0.377656\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1608.00 0.200249 0.100124 0.994975i \(-0.468076\pi\)
0.100124 + 0.994975i \(0.468076\pi\)
\(402\) 0 0
\(403\) −16192.0 −2.00144
\(404\) 0 0
\(405\) 243.000 0.0298142
\(406\) 0 0
\(407\) −4740.00 −0.577280
\(408\) 0 0
\(409\) 4465.00 0.539805 0.269902 0.962888i \(-0.413009\pi\)
0.269902 + 0.962888i \(0.413009\pi\)
\(410\) 0 0
\(411\) −5310.00 −0.637282
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1431.00 0.169265
\(416\) 0 0
\(417\) −4674.00 −0.548889
\(418\) 0 0
\(419\) −1584.00 −0.184686 −0.0923430 0.995727i \(-0.529436\pi\)
−0.0923430 + 0.995727i \(0.529436\pi\)
\(420\) 0 0
\(421\) −1330.00 −0.153967 −0.0769837 0.997032i \(-0.524529\pi\)
−0.0769837 + 0.997032i \(0.524529\pi\)
\(422\) 0 0
\(423\) −270.000 −0.0310351
\(424\) 0 0
\(425\) 9744.00 1.11213
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 2880.00 0.324121
\(430\) 0 0
\(431\) −9588.00 −1.07155 −0.535775 0.844361i \(-0.679980\pi\)
−0.535775 + 0.844361i \(0.679980\pi\)
\(432\) 0 0
\(433\) −494.000 −0.0548271 −0.0274135 0.999624i \(-0.508727\pi\)
−0.0274135 + 0.999624i \(0.508727\pi\)
\(434\) 0 0
\(435\) −2673.00 −0.294622
\(436\) 0 0
\(437\) −1344.00 −0.147122
\(438\) 0 0
\(439\) −16009.0 −1.74047 −0.870237 0.492634i \(-0.836034\pi\)
−0.870237 + 0.492634i \(0.836034\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −7773.00 −0.833649 −0.416824 0.908987i \(-0.636857\pi\)
−0.416824 + 0.908987i \(0.636857\pi\)
\(444\) 0 0
\(445\) −2718.00 −0.289541
\(446\) 0 0
\(447\) 7362.00 0.778995
\(448\) 0 0
\(449\) 864.000 0.0908122 0.0454061 0.998969i \(-0.485542\pi\)
0.0454061 + 0.998969i \(0.485542\pi\)
\(450\) 0 0
\(451\) −5400.00 −0.563805
\(452\) 0 0
\(453\) −3777.00 −0.391742
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2519.00 0.257842 0.128921 0.991655i \(-0.458849\pi\)
0.128921 + 0.991655i \(0.458849\pi\)
\(458\) 0 0
\(459\) −2268.00 −0.230634
\(460\) 0 0
\(461\) 342.000 0.0345521 0.0172761 0.999851i \(-0.494501\pi\)
0.0172761 + 0.999851i \(0.494501\pi\)
\(462\) 0 0
\(463\) 4336.00 0.435229 0.217614 0.976035i \(-0.430172\pi\)
0.217614 + 0.976035i \(0.430172\pi\)
\(464\) 0 0
\(465\) −2277.00 −0.227082
\(466\) 0 0
\(467\) 18636.0 1.84662 0.923310 0.384056i \(-0.125473\pi\)
0.923310 + 0.384056i \(0.125473\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 588.000 0.0575236
\(472\) 0 0
\(473\) −390.000 −0.0379117
\(474\) 0 0
\(475\) 1856.00 0.179282
\(476\) 0 0
\(477\) 3267.00 0.313597
\(478\) 0 0
\(479\) 15078.0 1.43827 0.719135 0.694870i \(-0.244538\pi\)
0.719135 + 0.694870i \(0.244538\pi\)
\(480\) 0 0
\(481\) −20224.0 −1.91712
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −1509.00 −0.141279
\(486\) 0 0
\(487\) −6221.00 −0.578851 −0.289425 0.957201i \(-0.593464\pi\)
−0.289425 + 0.957201i \(0.593464\pi\)
\(488\) 0 0
\(489\) 3756.00 0.347346
\(490\) 0 0
\(491\) 7371.00 0.677492 0.338746 0.940878i \(-0.389997\pi\)
0.338746 + 0.940878i \(0.389997\pi\)
\(492\) 0 0
\(493\) 24948.0 2.27911
\(494\) 0 0
\(495\) 405.000 0.0367745
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4274.00 −0.383428 −0.191714 0.981451i \(-0.561405\pi\)
−0.191714 + 0.981451i \(0.561405\pi\)
\(500\) 0 0
\(501\) −7938.00 −0.707872
\(502\) 0 0
\(503\) −2520.00 −0.223382 −0.111691 0.993743i \(-0.535627\pi\)
−0.111691 + 0.993743i \(0.535627\pi\)
\(504\) 0 0
\(505\) 3258.00 0.287087
\(506\) 0 0
\(507\) 5697.00 0.499039
\(508\) 0 0
\(509\) 14277.0 1.24326 0.621628 0.783313i \(-0.286472\pi\)
0.621628 + 0.783313i \(0.286472\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −432.000 −0.0371799
\(514\) 0 0
\(515\) 5208.00 0.445615
\(516\) 0 0
\(517\) −450.000 −0.0382804
\(518\) 0 0
\(519\) 2358.00 0.199431
\(520\) 0 0
\(521\) 6306.00 0.530270 0.265135 0.964211i \(-0.414583\pi\)
0.265135 + 0.964211i \(0.414583\pi\)
\(522\) 0 0
\(523\) 8072.00 0.674883 0.337442 0.941346i \(-0.390438\pi\)
0.337442 + 0.941346i \(0.390438\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 21252.0 1.75664
\(528\) 0 0
\(529\) −5111.00 −0.420071
\(530\) 0 0
\(531\) −135.000 −0.0110330
\(532\) 0 0
\(533\) −23040.0 −1.87237
\(534\) 0 0
\(535\) 4059.00 0.328011
\(536\) 0 0
\(537\) −8676.00 −0.697201
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −22858.0 −1.81653 −0.908264 0.418396i \(-0.862592\pi\)
−0.908264 + 0.418396i \(0.862592\pi\)
\(542\) 0 0
\(543\) −4056.00 −0.320552
\(544\) 0 0
\(545\) −1110.00 −0.0872425
\(546\) 0 0
\(547\) 24724.0 1.93258 0.966291 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(548\) 0 0
\(549\) 1062.00 0.0825593
\(550\) 0 0
\(551\) 4752.00 0.367408
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −2844.00 −0.217515
\(556\) 0 0
\(557\) −9843.00 −0.748764 −0.374382 0.927275i \(-0.622145\pi\)
−0.374382 + 0.927275i \(0.622145\pi\)
\(558\) 0 0
\(559\) −1664.00 −0.125903
\(560\) 0 0
\(561\) −3780.00 −0.284477
\(562\) 0 0
\(563\) −13371.0 −1.00092 −0.500462 0.865758i \(-0.666837\pi\)
−0.500462 + 0.865758i \(0.666837\pi\)
\(564\) 0 0
\(565\) −1944.00 −0.144752
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −5232.00 −0.385478 −0.192739 0.981250i \(-0.561737\pi\)
−0.192739 + 0.981250i \(0.561737\pi\)
\(570\) 0 0
\(571\) 14398.0 1.05523 0.527616 0.849483i \(-0.323086\pi\)
0.527616 + 0.849483i \(0.323086\pi\)
\(572\) 0 0
\(573\) −11736.0 −0.855634
\(574\) 0 0
\(575\) −9744.00 −0.706701
\(576\) 0 0
\(577\) −19871.0 −1.43369 −0.716846 0.697231i \(-0.754415\pi\)
−0.716846 + 0.697231i \(0.754415\pi\)
\(578\) 0 0
\(579\) 4479.00 0.321487
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 5445.00 0.386808
\(584\) 0 0
\(585\) 1728.00 0.122127
\(586\) 0 0
\(587\) −16137.0 −1.13466 −0.567330 0.823491i \(-0.692024\pi\)
−0.567330 + 0.823491i \(0.692024\pi\)
\(588\) 0 0
\(589\) 4048.00 0.283183
\(590\) 0 0
\(591\) −12258.0 −0.853176
\(592\) 0 0
\(593\) 21324.0 1.47668 0.738340 0.674428i \(-0.235610\pi\)
0.738340 + 0.674428i \(0.235610\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −10668.0 −0.731344
\(598\) 0 0
\(599\) 8646.00 0.589760 0.294880 0.955534i \(-0.404720\pi\)
0.294880 + 0.955534i \(0.404720\pi\)
\(600\) 0 0
\(601\) −11195.0 −0.759823 −0.379911 0.925023i \(-0.624046\pi\)
−0.379911 + 0.925023i \(0.624046\pi\)
\(602\) 0 0
\(603\) 3330.00 0.224889
\(604\) 0 0
\(605\) −3318.00 −0.222968
\(606\) 0 0
\(607\) −8971.00 −0.599871 −0.299935 0.953959i \(-0.596965\pi\)
−0.299935 + 0.953959i \(0.596965\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1920.00 −0.127127
\(612\) 0 0
\(613\) −12772.0 −0.841527 −0.420764 0.907170i \(-0.638238\pi\)
−0.420764 + 0.907170i \(0.638238\pi\)
\(614\) 0 0
\(615\) −3240.00 −0.212438
\(616\) 0 0
\(617\) 12762.0 0.832705 0.416352 0.909203i \(-0.363308\pi\)
0.416352 + 0.909203i \(0.363308\pi\)
\(618\) 0 0
\(619\) 12842.0 0.833867 0.416933 0.908937i \(-0.363105\pi\)
0.416933 + 0.908937i \(0.363105\pi\)
\(620\) 0 0
\(621\) 2268.00 0.146557
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 12331.0 0.789184
\(626\) 0 0
\(627\) −720.000 −0.0458597
\(628\) 0 0
\(629\) 26544.0 1.68264
\(630\) 0 0
\(631\) −21365.0 −1.34790 −0.673952 0.738775i \(-0.735404\pi\)
−0.673952 + 0.738775i \(0.735404\pi\)
\(632\) 0 0
\(633\) −3750.00 −0.235465
\(634\) 0 0
\(635\) −1131.00 −0.0706809
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 3078.00 0.190554
\(640\) 0 0
\(641\) 8274.00 0.509834 0.254917 0.966963i \(-0.417952\pi\)
0.254917 + 0.966963i \(0.417952\pi\)
\(642\) 0 0
\(643\) 27998.0 1.71716 0.858580 0.512680i \(-0.171347\pi\)
0.858580 + 0.512680i \(0.171347\pi\)
\(644\) 0 0
\(645\) −234.000 −0.0142849
\(646\) 0 0
\(647\) −17466.0 −1.06130 −0.530649 0.847592i \(-0.678052\pi\)
−0.530649 + 0.847592i \(0.678052\pi\)
\(648\) 0 0
\(649\) −225.000 −0.0136087
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2157.00 0.129265 0.0646324 0.997909i \(-0.479413\pi\)
0.0646324 + 0.997909i \(0.479413\pi\)
\(654\) 0 0
\(655\) −1953.00 −0.116504
\(656\) 0 0
\(657\) −3258.00 −0.193465
\(658\) 0 0
\(659\) −19944.0 −1.17892 −0.589460 0.807798i \(-0.700659\pi\)
−0.589460 + 0.807798i \(0.700659\pi\)
\(660\) 0 0
\(661\) −27506.0 −1.61855 −0.809273 0.587432i \(-0.800139\pi\)
−0.809273 + 0.587432i \(0.800139\pi\)
\(662\) 0 0
\(663\) −16128.0 −0.944735
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −24948.0 −1.44826
\(668\) 0 0
\(669\) 1275.00 0.0736836
\(670\) 0 0
\(671\) 1770.00 0.101833
\(672\) 0 0
\(673\) −19123.0 −1.09530 −0.547650 0.836707i \(-0.684478\pi\)
−0.547650 + 0.836707i \(0.684478\pi\)
\(674\) 0 0
\(675\) −3132.00 −0.178594
\(676\) 0 0
\(677\) −13857.0 −0.786658 −0.393329 0.919398i \(-0.628677\pi\)
−0.393329 + 0.919398i \(0.628677\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 11565.0 0.650766
\(682\) 0 0
\(683\) 22245.0 1.24624 0.623120 0.782127i \(-0.285865\pi\)
0.623120 + 0.782127i \(0.285865\pi\)
\(684\) 0 0
\(685\) −5310.00 −0.296182
\(686\) 0 0
\(687\) 6564.00 0.364530
\(688\) 0 0
\(689\) 23232.0 1.28457
\(690\) 0 0
\(691\) −640.000 −0.0352341 −0.0176170 0.999845i \(-0.505608\pi\)
−0.0176170 + 0.999845i \(0.505608\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −4674.00 −0.255101
\(696\) 0 0
\(697\) 30240.0 1.64336
\(698\) 0 0
\(699\) 2556.00 0.138307
\(700\) 0 0
\(701\) −15561.0 −0.838418 −0.419209 0.907890i \(-0.637693\pi\)
−0.419209 + 0.907890i \(0.637693\pi\)
\(702\) 0 0
\(703\) 5056.00 0.271253
\(704\) 0 0
\(705\) −270.000 −0.0144238
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 5534.00 0.293136 0.146568 0.989201i \(-0.453177\pi\)
0.146568 + 0.989201i \(0.453177\pi\)
\(710\) 0 0
\(711\) −4203.00 −0.221695
\(712\) 0 0
\(713\) −21252.0 −1.11626
\(714\) 0 0
\(715\) 2880.00 0.150638
\(716\) 0 0
\(717\) −16524.0 −0.860670
\(718\) 0 0
\(719\) −21846.0 −1.13313 −0.566564 0.824018i \(-0.691727\pi\)
−0.566564 + 0.824018i \(0.691727\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −2373.00 −0.122065
\(724\) 0 0
\(725\) 34452.0 1.76485
\(726\) 0 0
\(727\) −11089.0 −0.565706 −0.282853 0.959163i \(-0.591281\pi\)
−0.282853 + 0.959163i \(0.591281\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 2184.00 0.110504
\(732\) 0 0
\(733\) −11762.0 −0.592687 −0.296343 0.955081i \(-0.595767\pi\)
−0.296343 + 0.955081i \(0.595767\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 5550.00 0.277391
\(738\) 0 0
\(739\) 22726.0 1.13124 0.565622 0.824665i \(-0.308636\pi\)
0.565622 + 0.824665i \(0.308636\pi\)
\(740\) 0 0
\(741\) −3072.00 −0.152298
\(742\) 0 0
\(743\) −6678.00 −0.329734 −0.164867 0.986316i \(-0.552719\pi\)
−0.164867 + 0.986316i \(0.552719\pi\)
\(744\) 0 0
\(745\) 7362.00 0.362044
\(746\) 0 0
\(747\) 4293.00 0.210271
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 19987.0 0.971153 0.485577 0.874194i \(-0.338610\pi\)
0.485577 + 0.874194i \(0.338610\pi\)
\(752\) 0 0
\(753\) 15795.0 0.764411
\(754\) 0 0
\(755\) −3777.00 −0.182065
\(756\) 0 0
\(757\) 314.000 0.0150760 0.00753799 0.999972i \(-0.497601\pi\)
0.00753799 + 0.999972i \(0.497601\pi\)
\(758\) 0 0
\(759\) 3780.00 0.180771
\(760\) 0 0
\(761\) 11496.0 0.547608 0.273804 0.961786i \(-0.411718\pi\)
0.273804 + 0.961786i \(0.411718\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −2268.00 −0.107189
\(766\) 0 0
\(767\) −960.000 −0.0451937
\(768\) 0 0
\(769\) −2765.00 −0.129660 −0.0648299 0.997896i \(-0.520650\pi\)
−0.0648299 + 0.997896i \(0.520650\pi\)
\(770\) 0 0
\(771\) 20610.0 0.962712
\(772\) 0 0
\(773\) 14046.0 0.653557 0.326778 0.945101i \(-0.394037\pi\)
0.326778 + 0.945101i \(0.394037\pi\)
\(774\) 0 0
\(775\) 29348.0 1.36027
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 5760.00 0.264921
\(780\) 0 0
\(781\) 5130.00 0.235039
\(782\) 0 0
\(783\) −8019.00 −0.365997
\(784\) 0 0
\(785\) 588.000 0.0267345
\(786\) 0 0
\(787\) −18514.0 −0.838568 −0.419284 0.907855i \(-0.637719\pi\)
−0.419284 + 0.907855i \(0.637719\pi\)
\(788\) 0 0
\(789\) 666.000 0.0300510
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 7552.00 0.338183
\(794\) 0 0
\(795\) 3267.00 0.145747
\(796\) 0 0
\(797\) 27495.0 1.22199 0.610993 0.791636i \(-0.290770\pi\)
0.610993 + 0.791636i \(0.290770\pi\)
\(798\) 0 0
\(799\) 2520.00 0.111578
\(800\) 0 0
\(801\) −8154.00 −0.359685
\(802\) 0 0
\(803\) −5430.00 −0.238631
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −23553.0 −1.02739
\(808\) 0 0
\(809\) −7944.00 −0.345236 −0.172618 0.984989i \(-0.555223\pi\)
−0.172618 + 0.984989i \(0.555223\pi\)
\(810\) 0 0
\(811\) −28942.0 −1.25313 −0.626567 0.779368i \(-0.715540\pi\)
−0.626567 + 0.779368i \(0.715540\pi\)
\(812\) 0 0
\(813\) 15549.0 0.670759
\(814\) 0 0
\(815\) 3756.00 0.161432
\(816\) 0 0
\(817\) 416.000 0.0178140
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8187.00 0.348025 0.174012 0.984743i \(-0.444327\pi\)
0.174012 + 0.984743i \(0.444327\pi\)
\(822\) 0 0
\(823\) 280.000 0.0118593 0.00592964 0.999982i \(-0.498113\pi\)
0.00592964 + 0.999982i \(0.498113\pi\)
\(824\) 0 0
\(825\) −5220.00 −0.220287
\(826\) 0 0
\(827\) −25317.0 −1.06452 −0.532260 0.846581i \(-0.678657\pi\)
−0.532260 + 0.846581i \(0.678657\pi\)
\(828\) 0 0
\(829\) −15320.0 −0.641840 −0.320920 0.947106i \(-0.603992\pi\)
−0.320920 + 0.947106i \(0.603992\pi\)
\(830\) 0 0
\(831\) −14880.0 −0.621157
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −7938.00 −0.328989
\(836\) 0 0
\(837\) −6831.00 −0.282095
\(838\) 0 0
\(839\) 34092.0 1.40284 0.701422 0.712746i \(-0.252549\pi\)
0.701422 + 0.712746i \(0.252549\pi\)
\(840\) 0 0
\(841\) 63820.0 2.61675
\(842\) 0 0
\(843\) −2322.00 −0.0948682
\(844\) 0 0
\(845\) 5697.00 0.231932
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 11094.0 0.448463
\(850\) 0 0
\(851\) −26544.0 −1.06923
\(852\) 0 0
\(853\) 7378.00 0.296152 0.148076 0.988976i \(-0.452692\pi\)
0.148076 + 0.988976i \(0.452692\pi\)
\(854\) 0 0
\(855\) −432.000 −0.0172796
\(856\) 0 0
\(857\) 15594.0 0.621565 0.310782 0.950481i \(-0.399409\pi\)
0.310782 + 0.950481i \(0.399409\pi\)
\(858\) 0 0
\(859\) −30538.0 −1.21297 −0.606486 0.795094i \(-0.707421\pi\)
−0.606486 + 0.795094i \(0.707421\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 822.000 0.0324232 0.0162116 0.999869i \(-0.494839\pi\)
0.0162116 + 0.999869i \(0.494839\pi\)
\(864\) 0 0
\(865\) 2358.00 0.0926872
\(866\) 0 0
\(867\) 6429.00 0.251834
\(868\) 0 0
\(869\) −7005.00 −0.273450
\(870\) 0 0
\(871\) 23680.0 0.921201
\(872\) 0 0
\(873\) −4527.00 −0.175505
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −41824.0 −1.61037 −0.805186 0.593022i \(-0.797935\pi\)
−0.805186 + 0.593022i \(0.797935\pi\)
\(878\) 0 0
\(879\) 18819.0 0.722126
\(880\) 0 0
\(881\) 46098.0 1.76286 0.881431 0.472313i \(-0.156581\pi\)
0.881431 + 0.472313i \(0.156581\pi\)
\(882\) 0 0
\(883\) −21008.0 −0.800652 −0.400326 0.916373i \(-0.631103\pi\)
−0.400326 + 0.916373i \(0.631103\pi\)
\(884\) 0 0
\(885\) −135.000 −0.00512766
\(886\) 0 0
\(887\) 24036.0 0.909865 0.454932 0.890526i \(-0.349663\pi\)
0.454932 + 0.890526i \(0.349663\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1215.00 0.0456835
\(892\) 0 0
\(893\) 480.000 0.0179872
\(894\) 0 0
\(895\) −8676.00 −0.324030
\(896\) 0 0
\(897\) 16128.0 0.600332
\(898\) 0 0
\(899\) 75141.0 2.78764
\(900\) 0 0
\(901\) −30492.0 −1.12745
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4056.00 −0.148979
\(906\) 0 0
\(907\) −13292.0 −0.486608 −0.243304 0.969950i \(-0.578231\pi\)
−0.243304 + 0.969950i \(0.578231\pi\)
\(908\) 0 0
\(909\) 9774.00 0.356637
\(910\) 0 0
\(911\) 9306.00 0.338443 0.169221 0.985578i \(-0.445875\pi\)
0.169221 + 0.985578i \(0.445875\pi\)
\(912\) 0 0
\(913\) 7155.00 0.259360
\(914\) 0 0
\(915\) 1062.00 0.0383701
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −16496.0 −0.592114 −0.296057 0.955170i \(-0.595672\pi\)
−0.296057 + 0.955170i \(0.595672\pi\)
\(920\) 0 0
\(921\) −5052.00 −0.180748
\(922\) 0 0
\(923\) 21888.0 0.780555
\(924\) 0 0
\(925\) 36656.0 1.30296
\(926\) 0 0
\(927\) 15624.0 0.553570
\(928\) 0 0
\(929\) −14154.0 −0.499868 −0.249934 0.968263i \(-0.580409\pi\)
−0.249934 + 0.968263i \(0.580409\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −3960.00 −0.138955
\(934\) 0 0
\(935\) −3780.00 −0.132213
\(936\) 0 0
\(937\) 3781.00 0.131825 0.0659124 0.997825i \(-0.479004\pi\)
0.0659124 + 0.997825i \(0.479004\pi\)
\(938\) 0 0
\(939\) 25509.0 0.886533
\(940\) 0 0
\(941\) 25863.0 0.895972 0.447986 0.894041i \(-0.352141\pi\)
0.447986 + 0.894041i \(0.352141\pi\)
\(942\) 0 0
\(943\) −30240.0 −1.04427
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 42384.0 1.45438 0.727188 0.686438i \(-0.240827\pi\)
0.727188 + 0.686438i \(0.240827\pi\)
\(948\) 0 0
\(949\) −23168.0 −0.792482
\(950\) 0 0
\(951\) −7731.00 −0.263612
\(952\) 0 0
\(953\) 10530.0 0.357923 0.178961 0.983856i \(-0.442726\pi\)
0.178961 + 0.983856i \(0.442726\pi\)
\(954\) 0 0
\(955\) −11736.0 −0.397663
\(956\) 0 0
\(957\) −13365.0 −0.451441
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 34218.0 1.14860
\(962\) 0 0
\(963\) 12177.0 0.407475
\(964\) 0 0
\(965\) 4479.00 0.149414
\(966\) 0 0
\(967\) 38341.0 1.27504 0.637520 0.770434i \(-0.279960\pi\)
0.637520 + 0.770434i \(0.279960\pi\)
\(968\) 0 0
\(969\) 4032.00 0.133670
\(970\) 0 0
\(971\) 1923.00 0.0635551 0.0317776 0.999495i \(-0.489883\pi\)
0.0317776 + 0.999495i \(0.489883\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −22272.0 −0.731564
\(976\) 0 0
\(977\) 57090.0 1.86947 0.934734 0.355347i \(-0.115637\pi\)
0.934734 + 0.355347i \(0.115637\pi\)
\(978\) 0 0
\(979\) −13590.0 −0.443655
\(980\) 0 0
\(981\) −3330.00 −0.108378
\(982\) 0 0
\(983\) 5484.00 0.177937 0.0889687 0.996034i \(-0.471643\pi\)
0.0889687 + 0.996034i \(0.471643\pi\)
\(984\) 0 0
\(985\) −12258.0 −0.396520
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2184.00 −0.0702196
\(990\) 0 0
\(991\) 22465.0 0.720105 0.360053 0.932932i \(-0.382759\pi\)
0.360053 + 0.932932i \(0.382759\pi\)
\(992\) 0 0
\(993\) 1452.00 0.0464026
\(994\) 0 0
\(995\) −10668.0 −0.339898
\(996\) 0 0
\(997\) −29366.0 −0.932829 −0.466415 0.884566i \(-0.654454\pi\)
−0.466415 + 0.884566i \(0.654454\pi\)
\(998\) 0 0
\(999\) −8532.00 −0.270211
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 2352.4.a.bd.1.1 1
4.3 odd 2 147.4.a.a.1.1 1
7.3 odd 6 336.4.q.e.289.1 2
7.5 odd 6 336.4.q.e.193.1 2
7.6 odd 2 2352.4.a.i.1.1 1
12.11 even 2 441.4.a.k.1.1 1
28.3 even 6 21.4.e.a.16.1 yes 2
28.11 odd 6 147.4.e.h.79.1 2
28.19 even 6 21.4.e.a.4.1 2
28.23 odd 6 147.4.e.h.67.1 2
28.27 even 2 147.4.a.b.1.1 1
84.11 even 6 441.4.e.c.226.1 2
84.23 even 6 441.4.e.c.361.1 2
84.47 odd 6 63.4.e.a.46.1 2
84.59 odd 6 63.4.e.a.37.1 2
84.83 odd 2 441.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.e.a.4.1 2 28.19 even 6
21.4.e.a.16.1 yes 2 28.3 even 6
63.4.e.a.37.1 2 84.59 odd 6
63.4.e.a.46.1 2 84.47 odd 6
147.4.a.a.1.1 1 4.3 odd 2
147.4.a.b.1.1 1 28.27 even 2
147.4.e.h.67.1 2 28.23 odd 6
147.4.e.h.79.1 2 28.11 odd 6
336.4.q.e.193.1 2 7.5 odd 6
336.4.q.e.289.1 2 7.3 odd 6
441.4.a.k.1.1 1 12.11 even 2
441.4.a.l.1.1 1 84.83 odd 2
441.4.e.c.226.1 2 84.11 even 6
441.4.e.c.361.1 2 84.23 even 6
2352.4.a.i.1.1 1 7.6 odd 2
2352.4.a.bd.1.1 1 1.1 even 1 trivial