Properties

Label 105.4.a.a.1.1
Level 105
Weight 4
Character 105.1
Self dual yes
Analytic conductor 6.195
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 105 = 3 \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 105.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(6.19520055060\)
Analytic rank: \(0\)
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
Root \(0\) of \(x\)
Character \(\chi\) \(=\) 105.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{3} -8.00000 q^{4} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q-3.00000 q^{3} -8.00000 q^{4} +5.00000 q^{5} +7.00000 q^{7} +9.00000 q^{9} +42.0000 q^{11} +24.0000 q^{12} +20.0000 q^{13} -15.0000 q^{15} +64.0000 q^{16} +66.0000 q^{17} +38.0000 q^{19} -40.0000 q^{20} -21.0000 q^{21} +12.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -56.0000 q^{28} -258.000 q^{29} +146.000 q^{31} -126.000 q^{33} +35.0000 q^{35} -72.0000 q^{36} +434.000 q^{37} -60.0000 q^{39} -282.000 q^{41} +20.0000 q^{43} -336.000 q^{44} +45.0000 q^{45} -72.0000 q^{47} -192.000 q^{48} +49.0000 q^{49} -198.000 q^{51} -160.000 q^{52} +336.000 q^{53} +210.000 q^{55} -114.000 q^{57} -360.000 q^{59} +120.000 q^{60} -682.000 q^{61} +63.0000 q^{63} -512.000 q^{64} +100.000 q^{65} +812.000 q^{67} -528.000 q^{68} -36.0000 q^{69} +810.000 q^{71} -124.000 q^{73} -75.0000 q^{75} -304.000 q^{76} +294.000 q^{77} +1136.00 q^{79} +320.000 q^{80} +81.0000 q^{81} +156.000 q^{83} +168.000 q^{84} +330.000 q^{85} +774.000 q^{87} -1038.00 q^{89} +140.000 q^{91} -96.0000 q^{92} -438.000 q^{93} +190.000 q^{95} +1208.00 q^{97} +378.000 q^{99} +O(q^{100})\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) −3.00000 −0.577350
\(4\) −8.00000 −1.00000
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 42.0000 1.15123 0.575613 0.817723i \(-0.304764\pi\)
0.575613 + 0.817723i \(0.304764\pi\)
\(12\) 24.0000 0.577350
\(13\) 20.0000 0.426692 0.213346 0.976977i \(-0.431564\pi\)
0.213346 + 0.976977i \(0.431564\pi\)
\(14\) 0 0
\(15\) −15.0000 −0.258199
\(16\) 64.0000 1.00000
\(17\) 66.0000 0.941609 0.470804 0.882238i \(-0.343964\pi\)
0.470804 + 0.882238i \(0.343964\pi\)
\(18\) 0 0
\(19\) 38.0000 0.458831 0.229416 0.973329i \(-0.426318\pi\)
0.229416 + 0.973329i \(0.426318\pi\)
\(20\) −40.0000 −0.447214
\(21\) −21.0000 −0.218218
\(22\) 0 0
\(23\) 12.0000 0.108790 0.0543951 0.998519i \(-0.482677\pi\)
0.0543951 + 0.998519i \(0.482677\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 0 0
\(27\) −27.0000 −0.192450
\(28\) −56.0000 −0.377964
\(29\) −258.000 −1.65205 −0.826024 0.563635i \(-0.809403\pi\)
−0.826024 + 0.563635i \(0.809403\pi\)
\(30\) 0 0
\(31\) 146.000 0.845883 0.422942 0.906157i \(-0.360998\pi\)
0.422942 + 0.906157i \(0.360998\pi\)
\(32\) 0 0
\(33\) −126.000 −0.664660
\(34\) 0 0
\(35\) 35.0000 0.169031
\(36\) −72.0000 −0.333333
\(37\) 434.000 1.92836 0.964178 0.265257i \(-0.0854567\pi\)
0.964178 + 0.265257i \(0.0854567\pi\)
\(38\) 0 0
\(39\) −60.0000 −0.246351
\(40\) 0 0
\(41\) −282.000 −1.07417 −0.537085 0.843528i \(-0.680475\pi\)
−0.537085 + 0.843528i \(0.680475\pi\)
\(42\) 0 0
\(43\) 20.0000 0.0709296 0.0354648 0.999371i \(-0.488709\pi\)
0.0354648 + 0.999371i \(0.488709\pi\)
\(44\) −336.000 −1.15123
\(45\) 45.0000 0.149071
\(46\) 0 0
\(47\) −72.0000 −0.223453 −0.111726 0.993739i \(-0.535638\pi\)
−0.111726 + 0.993739i \(0.535638\pi\)
\(48\) −192.000 −0.577350
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) −198.000 −0.543638
\(52\) −160.000 −0.426692
\(53\) 336.000 0.870814 0.435407 0.900234i \(-0.356604\pi\)
0.435407 + 0.900234i \(0.356604\pi\)
\(54\) 0 0
\(55\) 210.000 0.514844
\(56\) 0 0
\(57\) −114.000 −0.264906
\(58\) 0 0
\(59\) −360.000 −0.794373 −0.397187 0.917738i \(-0.630013\pi\)
−0.397187 + 0.917738i \(0.630013\pi\)
\(60\) 120.000 0.258199
\(61\) −682.000 −1.43149 −0.715747 0.698360i \(-0.753914\pi\)
−0.715747 + 0.698360i \(0.753914\pi\)
\(62\) 0 0
\(63\) 63.0000 0.125988
\(64\) −512.000 −1.00000
\(65\) 100.000 0.190823
\(66\) 0 0
\(67\) 812.000 1.48062 0.740310 0.672265i \(-0.234679\pi\)
0.740310 + 0.672265i \(0.234679\pi\)
\(68\) −528.000 −0.941609
\(69\) −36.0000 −0.0628100
\(70\) 0 0
\(71\) 810.000 1.35393 0.676967 0.736013i \(-0.263294\pi\)
0.676967 + 0.736013i \(0.263294\pi\)
\(72\) 0 0
\(73\) −124.000 −0.198810 −0.0994048 0.995047i \(-0.531694\pi\)
−0.0994048 + 0.995047i \(0.531694\pi\)
\(74\) 0 0
\(75\) −75.0000 −0.115470
\(76\) −304.000 −0.458831
\(77\) 294.000 0.435122
\(78\) 0 0
\(79\) 1136.00 1.61785 0.808924 0.587913i \(-0.200050\pi\)
0.808924 + 0.587913i \(0.200050\pi\)
\(80\) 320.000 0.447214
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 156.000 0.206304 0.103152 0.994666i \(-0.467107\pi\)
0.103152 + 0.994666i \(0.467107\pi\)
\(84\) 168.000 0.218218
\(85\) 330.000 0.421100
\(86\) 0 0
\(87\) 774.000 0.953810
\(88\) 0 0
\(89\) −1038.00 −1.23627 −0.618134 0.786073i \(-0.712111\pi\)
−0.618134 + 0.786073i \(0.712111\pi\)
\(90\) 0 0
\(91\) 140.000 0.161275
\(92\) −96.0000 −0.108790
\(93\) −438.000 −0.488371
\(94\) 0 0
\(95\) 190.000 0.205196
\(96\) 0 0
\(97\) 1208.00 1.26447 0.632236 0.774776i \(-0.282137\pi\)
0.632236 + 0.774776i \(0.282137\pi\)
\(98\) 0 0
\(99\) 378.000 0.383742
\(100\) −200.000 −0.200000
\(101\) 546.000 0.537911 0.268956 0.963153i \(-0.413322\pi\)
0.268956 + 0.963153i \(0.413322\pi\)
\(102\) 0 0
\(103\) −520.000 −0.497448 −0.248724 0.968574i \(-0.580011\pi\)
−0.248724 + 0.968574i \(0.580011\pi\)
\(104\) 0 0
\(105\) −105.000 −0.0975900
\(106\) 0 0
\(107\) 1212.00 1.09503 0.547516 0.836795i \(-0.315573\pi\)
0.547516 + 0.836795i \(0.315573\pi\)
\(108\) 216.000 0.192450
\(109\) −1078.00 −0.947281 −0.473641 0.880718i \(-0.657060\pi\)
−0.473641 + 0.880718i \(0.657060\pi\)
\(110\) 0 0
\(111\) −1302.00 −1.11334
\(112\) 448.000 0.377964
\(113\) −1452.00 −1.20878 −0.604392 0.796687i \(-0.706584\pi\)
−0.604392 + 0.796687i \(0.706584\pi\)
\(114\) 0 0
\(115\) 60.0000 0.0486524
\(116\) 2064.00 1.65205
\(117\) 180.000 0.142231
\(118\) 0 0
\(119\) 462.000 0.355895
\(120\) 0 0
\(121\) 433.000 0.325319
\(122\) 0 0
\(123\) 846.000 0.620173
\(124\) −1168.00 −0.845883
\(125\) 125.000 0.0894427
\(126\) 0 0
\(127\) −1312.00 −0.916702 −0.458351 0.888771i \(-0.651560\pi\)
−0.458351 + 0.888771i \(0.651560\pi\)
\(128\) 0 0
\(129\) −60.0000 −0.0409512
\(130\) 0 0
\(131\) −1356.00 −0.904384 −0.452192 0.891921i \(-0.649358\pi\)
−0.452192 + 0.891921i \(0.649358\pi\)
\(132\) 1008.00 0.664660
\(133\) 266.000 0.173422
\(134\) 0 0
\(135\) −135.000 −0.0860663
\(136\) 0 0
\(137\) −984.000 −0.613641 −0.306820 0.951767i \(-0.599265\pi\)
−0.306820 + 0.951767i \(0.599265\pi\)
\(138\) 0 0
\(139\) −394.000 −0.240422 −0.120211 0.992748i \(-0.538357\pi\)
−0.120211 + 0.992748i \(0.538357\pi\)
\(140\) −280.000 −0.169031
\(141\) 216.000 0.129011
\(142\) 0 0
\(143\) 840.000 0.491219
\(144\) 576.000 0.333333
\(145\) −1290.00 −0.738818
\(146\) 0 0
\(147\) −147.000 −0.0824786
\(148\) −3472.00 −1.92836
\(149\) −1014.00 −0.557518 −0.278759 0.960361i \(-0.589923\pi\)
−0.278759 + 0.960361i \(0.589923\pi\)
\(150\) 0 0
\(151\) −1996.00 −1.07571 −0.537855 0.843037i \(-0.680765\pi\)
−0.537855 + 0.843037i \(0.680765\pi\)
\(152\) 0 0
\(153\) 594.000 0.313870
\(154\) 0 0
\(155\) 730.000 0.378290
\(156\) 480.000 0.246351
\(157\) −2392.00 −1.21594 −0.607969 0.793960i \(-0.708016\pi\)
−0.607969 + 0.793960i \(0.708016\pi\)
\(158\) 0 0
\(159\) −1008.00 −0.502765
\(160\) 0 0
\(161\) 84.0000 0.0411188
\(162\) 0 0
\(163\) 2036.00 0.978355 0.489177 0.872184i \(-0.337297\pi\)
0.489177 + 0.872184i \(0.337297\pi\)
\(164\) 2256.00 1.07417
\(165\) −630.000 −0.297245
\(166\) 0 0
\(167\) −3936.00 −1.82381 −0.911907 0.410398i \(-0.865390\pi\)
−0.911907 + 0.410398i \(0.865390\pi\)
\(168\) 0 0
\(169\) −1797.00 −0.817934
\(170\) 0 0
\(171\) 342.000 0.152944
\(172\) −160.000 −0.0709296
\(173\) 378.000 0.166120 0.0830601 0.996545i \(-0.473531\pi\)
0.0830601 + 0.996545i \(0.473531\pi\)
\(174\) 0 0
\(175\) 175.000 0.0755929
\(176\) 2688.00 1.15123
\(177\) 1080.00 0.458631
\(178\) 0 0
\(179\) −222.000 −0.0926987 −0.0463493 0.998925i \(-0.514759\pi\)
−0.0463493 + 0.998925i \(0.514759\pi\)
\(180\) −360.000 −0.149071
\(181\) −2590.00 −1.06361 −0.531804 0.846867i \(-0.678486\pi\)
−0.531804 + 0.846867i \(0.678486\pi\)
\(182\) 0 0
\(183\) 2046.00 0.826474
\(184\) 0 0
\(185\) 2170.00 0.862387
\(186\) 0 0
\(187\) 2772.00 1.08400
\(188\) 576.000 0.223453
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) 2214.00 0.838740 0.419370 0.907815i \(-0.362251\pi\)
0.419370 + 0.907815i \(0.362251\pi\)
\(192\) 1536.00 0.577350
\(193\) 4178.00 1.55823 0.779117 0.626879i \(-0.215668\pi\)
0.779117 + 0.626879i \(0.215668\pi\)
\(194\) 0 0
\(195\) −300.000 −0.110172
\(196\) −392.000 −0.142857
\(197\) −3060.00 −1.10668 −0.553340 0.832955i \(-0.686647\pi\)
−0.553340 + 0.832955i \(0.686647\pi\)
\(198\) 0 0
\(199\) 2666.00 0.949687 0.474844 0.880070i \(-0.342505\pi\)
0.474844 + 0.880070i \(0.342505\pi\)
\(200\) 0 0
\(201\) −2436.00 −0.854837
\(202\) 0 0
\(203\) −1806.00 −0.624416
\(204\) 1584.00 0.543638
\(205\) −1410.00 −0.480384
\(206\) 0 0
\(207\) 108.000 0.0362634
\(208\) 1280.00 0.426692
\(209\) 1596.00 0.528218
\(210\) 0 0
\(211\) −1348.00 −0.439811 −0.219906 0.975521i \(-0.570575\pi\)
−0.219906 + 0.975521i \(0.570575\pi\)
\(212\) −2688.00 −0.870814
\(213\) −2430.00 −0.781694
\(214\) 0 0
\(215\) 100.000 0.0317207
\(216\) 0 0
\(217\) 1022.00 0.319714
\(218\) 0 0
\(219\) 372.000 0.114783
\(220\) −1680.00 −0.514844
\(221\) 1320.00 0.401777
\(222\) 0 0
\(223\) 3188.00 0.957329 0.478664 0.877998i \(-0.341121\pi\)
0.478664 + 0.877998i \(0.341121\pi\)
\(224\) 0 0
\(225\) 225.000 0.0666667
\(226\) 0 0
\(227\) −3396.00 −0.992953 −0.496477 0.868050i \(-0.665373\pi\)
−0.496477 + 0.868050i \(0.665373\pi\)
\(228\) 912.000 0.264906
\(229\) 5294.00 1.52767 0.763837 0.645409i \(-0.223313\pi\)
0.763837 + 0.645409i \(0.223313\pi\)
\(230\) 0 0
\(231\) −882.000 −0.251218
\(232\) 0 0
\(233\) 852.000 0.239555 0.119778 0.992801i \(-0.461782\pi\)
0.119778 + 0.992801i \(0.461782\pi\)
\(234\) 0 0
\(235\) −360.000 −0.0999311
\(236\) 2880.00 0.794373
\(237\) −3408.00 −0.934065
\(238\) 0 0
\(239\) 4866.00 1.31697 0.658484 0.752595i \(-0.271198\pi\)
0.658484 + 0.752595i \(0.271198\pi\)
\(240\) −960.000 −0.258199
\(241\) −2050.00 −0.547934 −0.273967 0.961739i \(-0.588336\pi\)
−0.273967 + 0.961739i \(0.588336\pi\)
\(242\) 0 0
\(243\) −243.000 −0.0641500
\(244\) 5456.00 1.43149
\(245\) 245.000 0.0638877
\(246\) 0 0
\(247\) 760.000 0.195780
\(248\) 0 0
\(249\) −468.000 −0.119110
\(250\) 0 0
\(251\) −1152.00 −0.289696 −0.144848 0.989454i \(-0.546269\pi\)
−0.144848 + 0.989454i \(0.546269\pi\)
\(252\) −504.000 −0.125988
\(253\) 504.000 0.125242
\(254\) 0 0
\(255\) −990.000 −0.243122
\(256\) 4096.00 1.00000
\(257\) −6450.00 −1.56553 −0.782763 0.622321i \(-0.786190\pi\)
−0.782763 + 0.622321i \(0.786190\pi\)
\(258\) 0 0
\(259\) 3038.00 0.728850
\(260\) −800.000 −0.190823
\(261\) −2322.00 −0.550683
\(262\) 0 0
\(263\) 1968.00 0.461415 0.230707 0.973023i \(-0.425896\pi\)
0.230707 + 0.973023i \(0.425896\pi\)
\(264\) 0 0
\(265\) 1680.00 0.389440
\(266\) 0 0
\(267\) 3114.00 0.713759
\(268\) −6496.00 −1.48062
\(269\) −3894.00 −0.882607 −0.441304 0.897358i \(-0.645484\pi\)
−0.441304 + 0.897358i \(0.645484\pi\)
\(270\) 0 0
\(271\) 7094.00 1.59015 0.795073 0.606513i \(-0.207432\pi\)
0.795073 + 0.606513i \(0.207432\pi\)
\(272\) 4224.00 0.941609
\(273\) −420.000 −0.0931119
\(274\) 0 0
\(275\) 1050.00 0.230245
\(276\) 288.000 0.0628100
\(277\) −3310.00 −0.717973 −0.358987 0.933343i \(-0.616878\pi\)
−0.358987 + 0.933343i \(0.616878\pi\)
\(278\) 0 0
\(279\) 1314.00 0.281961
\(280\) 0 0
\(281\) 7158.00 1.51961 0.759805 0.650151i \(-0.225294\pi\)
0.759805 + 0.650151i \(0.225294\pi\)
\(282\) 0 0
\(283\) −5164.00 −1.08469 −0.542346 0.840155i \(-0.682464\pi\)
−0.542346 + 0.840155i \(0.682464\pi\)
\(284\) −6480.00 −1.35393
\(285\) −570.000 −0.118470
\(286\) 0 0
\(287\) −1974.00 −0.405998
\(288\) 0 0
\(289\) −557.000 −0.113373
\(290\) 0 0
\(291\) −3624.00 −0.730043
\(292\) 992.000 0.198810
\(293\) −8598.00 −1.71434 −0.857168 0.515037i \(-0.827778\pi\)
−0.857168 + 0.515037i \(0.827778\pi\)
\(294\) 0 0
\(295\) −1800.00 −0.355254
\(296\) 0 0
\(297\) −1134.00 −0.221553
\(298\) 0 0
\(299\) 240.000 0.0464199
\(300\) 600.000 0.115470
\(301\) 140.000 0.0268089
\(302\) 0 0
\(303\) −1638.00 −0.310563
\(304\) 2432.00 0.458831
\(305\) −3410.00 −0.640184
\(306\) 0 0
\(307\) −448.000 −0.0832857 −0.0416429 0.999133i \(-0.513259\pi\)
−0.0416429 + 0.999133i \(0.513259\pi\)
\(308\) −2352.00 −0.435122
\(309\) 1560.00 0.287202
\(310\) 0 0
\(311\) −5832.00 −1.06335 −0.531676 0.846948i \(-0.678438\pi\)
−0.531676 + 0.846948i \(0.678438\pi\)
\(312\) 0 0
\(313\) 9848.00 1.77841 0.889204 0.457510i \(-0.151259\pi\)
0.889204 + 0.457510i \(0.151259\pi\)
\(314\) 0 0
\(315\) 315.000 0.0563436
\(316\) −9088.00 −1.61785
\(317\) −5616.00 −0.995035 −0.497517 0.867454i \(-0.665755\pi\)
−0.497517 + 0.867454i \(0.665755\pi\)
\(318\) 0 0
\(319\) −10836.0 −1.90188
\(320\) −2560.00 −0.447214
\(321\) −3636.00 −0.632217
\(322\) 0 0
\(323\) 2508.00 0.432040
\(324\) −648.000 −0.111111
\(325\) 500.000 0.0853385
\(326\) 0 0
\(327\) 3234.00 0.546913
\(328\) 0 0
\(329\) −504.000 −0.0844572
\(330\) 0 0
\(331\) 452.000 0.0750579 0.0375290 0.999296i \(-0.488051\pi\)
0.0375290 + 0.999296i \(0.488051\pi\)
\(332\) −1248.00 −0.206304
\(333\) 3906.00 0.642785
\(334\) 0 0
\(335\) 4060.00 0.662154
\(336\) −1344.00 −0.218218
\(337\) −2302.00 −0.372101 −0.186050 0.982540i \(-0.559569\pi\)
−0.186050 + 0.982540i \(0.559569\pi\)
\(338\) 0 0
\(339\) 4356.00 0.697892
\(340\) −2640.00 −0.421100
\(341\) 6132.00 0.973802
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 0 0
\(345\) −180.000 −0.0280895
\(346\) 0 0
\(347\) 1584.00 0.245054 0.122527 0.992465i \(-0.460900\pi\)
0.122527 + 0.992465i \(0.460900\pi\)
\(348\) −6192.00 −0.953810
\(349\) 8174.00 1.25371 0.626854 0.779137i \(-0.284342\pi\)
0.626854 + 0.779137i \(0.284342\pi\)
\(350\) 0 0
\(351\) −540.000 −0.0821170
\(352\) 0 0
\(353\) 8610.00 1.29820 0.649099 0.760704i \(-0.275146\pi\)
0.649099 + 0.760704i \(0.275146\pi\)
\(354\) 0 0
\(355\) 4050.00 0.605498
\(356\) 8304.00 1.23627
\(357\) −1386.00 −0.205476
\(358\) 0 0
\(359\) −2154.00 −0.316668 −0.158334 0.987386i \(-0.550612\pi\)
−0.158334 + 0.987386i \(0.550612\pi\)
\(360\) 0 0
\(361\) −5415.00 −0.789474
\(362\) 0 0
\(363\) −1299.00 −0.187823
\(364\) −1120.00 −0.161275
\(365\) −620.000 −0.0889104
\(366\) 0 0
\(367\) 6644.00 0.944997 0.472499 0.881331i \(-0.343352\pi\)
0.472499 + 0.881331i \(0.343352\pi\)
\(368\) 768.000 0.108790
\(369\) −2538.00 −0.358057
\(370\) 0 0
\(371\) 2352.00 0.329137
\(372\) 3504.00 0.488371
\(373\) 7958.00 1.10469 0.552345 0.833615i \(-0.313733\pi\)
0.552345 + 0.833615i \(0.313733\pi\)
\(374\) 0 0
\(375\) −375.000 −0.0516398
\(376\) 0 0
\(377\) −5160.00 −0.704917
\(378\) 0 0
\(379\) 3440.00 0.466229 0.233115 0.972449i \(-0.425108\pi\)
0.233115 + 0.972449i \(0.425108\pi\)
\(380\) −1520.00 −0.205196
\(381\) 3936.00 0.529258
\(382\) 0 0
\(383\) 12936.0 1.72585 0.862923 0.505336i \(-0.168631\pi\)
0.862923 + 0.505336i \(0.168631\pi\)
\(384\) 0 0
\(385\) 1470.00 0.194593
\(386\) 0 0
\(387\) 180.000 0.0236432
\(388\) −9664.00 −1.26447
\(389\) −14862.0 −1.93710 −0.968552 0.248812i \(-0.919960\pi\)
−0.968552 + 0.248812i \(0.919960\pi\)
\(390\) 0 0
\(391\) 792.000 0.102438
\(392\) 0 0
\(393\) 4068.00 0.522146
\(394\) 0 0
\(395\) 5680.00 0.723524
\(396\) −3024.00 −0.383742
\(397\) 10460.0 1.32235 0.661174 0.750232i \(-0.270058\pi\)
0.661174 + 0.750232i \(0.270058\pi\)
\(398\) 0 0
\(399\) −798.000 −0.100125
\(400\) 1600.00 0.200000
\(401\) −9150.00 −1.13947 −0.569737 0.821827i \(-0.692955\pi\)
−0.569737 + 0.821827i \(0.692955\pi\)
\(402\) 0 0
\(403\) 2920.00 0.360932
\(404\) −4368.00 −0.537911
\(405\) 405.000 0.0496904
\(406\) 0 0
\(407\) 18228.0 2.21997
\(408\) 0 0
\(409\) −4894.00 −0.591669 −0.295835 0.955239i \(-0.595598\pi\)
−0.295835 + 0.955239i \(0.595598\pi\)
\(410\) 0 0
\(411\) 2952.00 0.354286
\(412\) 4160.00 0.497448
\(413\) −2520.00 −0.300245
\(414\) 0 0
\(415\) 780.000 0.0922619
\(416\) 0 0
\(417\) 1182.00 0.138808
\(418\) 0 0
\(419\) −1668.00 −0.194480 −0.0972400 0.995261i \(-0.531001\pi\)
−0.0972400 + 0.995261i \(0.531001\pi\)
\(420\) 840.000 0.0975900
\(421\) −12418.0 −1.43757 −0.718784 0.695233i \(-0.755301\pi\)
−0.718784 + 0.695233i \(0.755301\pi\)
\(422\) 0 0
\(423\) −648.000 −0.0744843
\(424\) 0 0
\(425\) 1650.00 0.188322
\(426\) 0 0
\(427\) −4774.00 −0.541054
\(428\) −9696.00 −1.09503
\(429\) −2520.00 −0.283605
\(430\) 0 0
\(431\) 15186.0 1.69718 0.848589 0.529052i \(-0.177452\pi\)
0.848589 + 0.529052i \(0.177452\pi\)
\(432\) −1728.00 −0.192450
\(433\) −5704.00 −0.633064 −0.316532 0.948582i \(-0.602518\pi\)
−0.316532 + 0.948582i \(0.602518\pi\)
\(434\) 0 0
\(435\) 3870.00 0.426557
\(436\) 8624.00 0.947281
\(437\) 456.000 0.0499163
\(438\) 0 0
\(439\) −17206.0 −1.87061 −0.935305 0.353843i \(-0.884875\pi\)
−0.935305 + 0.353843i \(0.884875\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) 0 0
\(443\) 3456.00 0.370654 0.185327 0.982677i \(-0.440666\pi\)
0.185327 + 0.982677i \(0.440666\pi\)
\(444\) 10416.0 1.11334
\(445\) −5190.00 −0.552875
\(446\) 0 0
\(447\) 3042.00 0.321883
\(448\) −3584.00 −0.377964
\(449\) 16074.0 1.68949 0.844743 0.535173i \(-0.179753\pi\)
0.844743 + 0.535173i \(0.179753\pi\)
\(450\) 0 0
\(451\) −11844.0 −1.23661
\(452\) 11616.0 1.20878
\(453\) 5988.00 0.621061
\(454\) 0 0
\(455\) 700.000 0.0721242
\(456\) 0 0
\(457\) 7526.00 0.770353 0.385177 0.922843i \(-0.374141\pi\)
0.385177 + 0.922843i \(0.374141\pi\)
\(458\) 0 0
\(459\) −1782.00 −0.181213
\(460\) −480.000 −0.0486524
\(461\) −2274.00 −0.229741 −0.114871 0.993380i \(-0.536645\pi\)
−0.114871 + 0.993380i \(0.536645\pi\)
\(462\) 0 0
\(463\) −10024.0 −1.00617 −0.503083 0.864238i \(-0.667801\pi\)
−0.503083 + 0.864238i \(0.667801\pi\)
\(464\) −16512.0 −1.65205
\(465\) −2190.00 −0.218406
\(466\) 0 0
\(467\) −2460.00 −0.243759 −0.121879 0.992545i \(-0.538892\pi\)
−0.121879 + 0.992545i \(0.538892\pi\)
\(468\) −1440.00 −0.142231
\(469\) 5684.00 0.559622
\(470\) 0 0
\(471\) 7176.00 0.702023
\(472\) 0 0
\(473\) 840.000 0.0816559
\(474\) 0 0
\(475\) 950.000 0.0917663
\(476\) −3696.00 −0.355895
\(477\) 3024.00 0.290271
\(478\) 0 0
\(479\) 19320.0 1.84291 0.921454 0.388486i \(-0.127002\pi\)
0.921454 + 0.388486i \(0.127002\pi\)
\(480\) 0 0
\(481\) 8680.00 0.822815
\(482\) 0 0
\(483\) −252.000 −0.0237400
\(484\) −3464.00 −0.325319
\(485\) 6040.00 0.565489
\(486\) 0 0
\(487\) −12544.0 −1.16719 −0.583596 0.812044i \(-0.698355\pi\)
−0.583596 + 0.812044i \(0.698355\pi\)
\(488\) 0 0
\(489\) −6108.00 −0.564853
\(490\) 0 0
\(491\) −15510.0 −1.42557 −0.712787 0.701381i \(-0.752567\pi\)
−0.712787 + 0.701381i \(0.752567\pi\)
\(492\) −6768.00 −0.620173
\(493\) −17028.0 −1.55558
\(494\) 0 0
\(495\) 1890.00 0.171615
\(496\) 9344.00 0.845883
\(497\) 5670.00 0.511739
\(498\) 0 0
\(499\) −14344.0 −1.28682 −0.643412 0.765520i \(-0.722482\pi\)
−0.643412 + 0.765520i \(0.722482\pi\)
\(500\) −1000.00 −0.0894427
\(501\) 11808.0 1.05298
\(502\) 0 0
\(503\) −21384.0 −1.89556 −0.947779 0.318929i \(-0.896677\pi\)
−0.947779 + 0.318929i \(0.896677\pi\)
\(504\) 0 0
\(505\) 2730.00 0.240561
\(506\) 0 0
\(507\) 5391.00 0.472234
\(508\) 10496.0 0.916702
\(509\) −7134.00 −0.621236 −0.310618 0.950535i \(-0.600536\pi\)
−0.310618 + 0.950535i \(0.600536\pi\)
\(510\) 0 0
\(511\) −868.000 −0.0751430
\(512\) 0 0
\(513\) −1026.00 −0.0883022
\(514\) 0 0
\(515\) −2600.00 −0.222465
\(516\) 480.000 0.0409512
\(517\) −3024.00 −0.257244
\(518\) 0 0
\(519\) −1134.00 −0.0959096
\(520\) 0 0
\(521\) −19122.0 −1.60797 −0.803983 0.594653i \(-0.797290\pi\)
−0.803983 + 0.594653i \(0.797290\pi\)
\(522\) 0 0
\(523\) −15640.0 −1.30763 −0.653814 0.756655i \(-0.726832\pi\)
−0.653814 + 0.756655i \(0.726832\pi\)
\(524\) 10848.0 0.904384
\(525\) −525.000 −0.0436436
\(526\) 0 0
\(527\) 9636.00 0.796491
\(528\) −8064.00 −0.664660
\(529\) −12023.0 −0.988165
\(530\) 0 0
\(531\) −3240.00 −0.264791
\(532\) −2128.00 −0.173422
\(533\) −5640.00 −0.458341
\(534\) 0 0
\(535\) 6060.00 0.489713
\(536\) 0 0
\(537\) 666.000 0.0535196
\(538\) 0 0
\(539\) 2058.00 0.164461
\(540\) 1080.00 0.0860663
\(541\) 2846.00 0.226172 0.113086 0.993585i \(-0.463926\pi\)
0.113086 + 0.993585i \(0.463926\pi\)
\(542\) 0 0
\(543\) 7770.00 0.614075
\(544\) 0 0
\(545\) −5390.00 −0.423637
\(546\) 0 0
\(547\) −4444.00 −0.347371 −0.173685 0.984801i \(-0.555568\pi\)
−0.173685 + 0.984801i \(0.555568\pi\)
\(548\) 7872.00 0.613641
\(549\) −6138.00 −0.477165
\(550\) 0 0
\(551\) −9804.00 −0.758012
\(552\) 0 0
\(553\) 7952.00 0.611489
\(554\) 0 0
\(555\) −6510.00 −0.497899
\(556\) 3152.00 0.240422
\(557\) 18552.0 1.41126 0.705631 0.708579i \(-0.250663\pi\)
0.705631 + 0.708579i \(0.250663\pi\)
\(558\) 0 0
\(559\) 400.000 0.0302651
\(560\) 2240.00 0.169031
\(561\) −8316.00 −0.625850
\(562\) 0 0
\(563\) −16452.0 −1.23156 −0.615781 0.787918i \(-0.711159\pi\)
−0.615781 + 0.787918i \(0.711159\pi\)
\(564\) −1728.00 −0.129011
\(565\) −7260.00 −0.540585
\(566\) 0 0
\(567\) 567.000 0.0419961
\(568\) 0 0
\(569\) 7722.00 0.568933 0.284467 0.958686i \(-0.408183\pi\)
0.284467 + 0.958686i \(0.408183\pi\)
\(570\) 0 0
\(571\) 2576.00 0.188796 0.0943978 0.995535i \(-0.469907\pi\)
0.0943978 + 0.995535i \(0.469907\pi\)
\(572\) −6720.00 −0.491219
\(573\) −6642.00 −0.484247
\(574\) 0 0
\(575\) 300.000 0.0217580
\(576\) −4608.00 −0.333333
\(577\) −2464.00 −0.177778 −0.0888888 0.996042i \(-0.528332\pi\)
−0.0888888 + 0.996042i \(0.528332\pi\)
\(578\) 0 0
\(579\) −12534.0 −0.899646
\(580\) 10320.0 0.738818
\(581\) 1092.00 0.0779755
\(582\) 0 0
\(583\) 14112.0 1.00250
\(584\) 0 0
\(585\) 900.000 0.0636076
\(586\) 0 0
\(587\) 1452.00 0.102096 0.0510481 0.998696i \(-0.483744\pi\)
0.0510481 + 0.998696i \(0.483744\pi\)
\(588\) 1176.00 0.0824786
\(589\) 5548.00 0.388118
\(590\) 0 0
\(591\) 9180.00 0.638942
\(592\) 27776.0 1.92836
\(593\) 10698.0 0.740833 0.370417 0.928866i \(-0.379215\pi\)
0.370417 + 0.928866i \(0.379215\pi\)
\(594\) 0 0
\(595\) 2310.00 0.159161
\(596\) 8112.00 0.557518
\(597\) −7998.00 −0.548302
\(598\) 0 0
\(599\) −8730.00 −0.595489 −0.297745 0.954646i \(-0.596234\pi\)
−0.297745 + 0.954646i \(0.596234\pi\)
\(600\) 0 0
\(601\) 1910.00 0.129635 0.0648174 0.997897i \(-0.479354\pi\)
0.0648174 + 0.997897i \(0.479354\pi\)
\(602\) 0 0
\(603\) 7308.00 0.493540
\(604\) 15968.0 1.07571
\(605\) 2165.00 0.145487
\(606\) 0 0
\(607\) −5596.00 −0.374192 −0.187096 0.982342i \(-0.559908\pi\)
−0.187096 + 0.982342i \(0.559908\pi\)
\(608\) 0 0
\(609\) 5418.00 0.360506
\(610\) 0 0
\(611\) −1440.00 −0.0953456
\(612\) −4752.00 −0.313870
\(613\) 28586.0 1.88349 0.941744 0.336332i \(-0.109186\pi\)
0.941744 + 0.336332i \(0.109186\pi\)
\(614\) 0 0
\(615\) 4230.00 0.277350
\(616\) 0 0
\(617\) −19236.0 −1.25513 −0.627563 0.778566i \(-0.715947\pi\)
−0.627563 + 0.778566i \(0.715947\pi\)
\(618\) 0 0
\(619\) 6734.00 0.437257 0.218629 0.975808i \(-0.429842\pi\)
0.218629 + 0.975808i \(0.429842\pi\)
\(620\) −5840.00 −0.378290
\(621\) −324.000 −0.0209367
\(622\) 0 0
\(623\) −7266.00 −0.467265
\(624\) −3840.00 −0.246351
\(625\) 625.000 0.0400000
\(626\) 0 0
\(627\) −4788.00 −0.304967
\(628\) 19136.0 1.21594
\(629\) 28644.0 1.81576
\(630\) 0 0
\(631\) 7184.00 0.453234 0.226617 0.973984i \(-0.427233\pi\)
0.226617 + 0.973984i \(0.427233\pi\)
\(632\) 0 0
\(633\) 4044.00 0.253925
\(634\) 0 0
\(635\) −6560.00 −0.409962
\(636\) 8064.00 0.502765
\(637\) 980.000 0.0609561
\(638\) 0 0
\(639\) 7290.00 0.451311
\(640\) 0 0
\(641\) 510.000 0.0314256 0.0157128 0.999877i \(-0.494998\pi\)
0.0157128 + 0.999877i \(0.494998\pi\)
\(642\) 0 0
\(643\) −20752.0 −1.27275 −0.636376 0.771379i \(-0.719567\pi\)
−0.636376 + 0.771379i \(0.719567\pi\)
\(644\) −672.000 −0.0411188
\(645\) −300.000 −0.0183139
\(646\) 0 0
\(647\) 21072.0 1.28041 0.640205 0.768204i \(-0.278849\pi\)
0.640205 + 0.768204i \(0.278849\pi\)
\(648\) 0 0
\(649\) −15120.0 −0.914502
\(650\) 0 0
\(651\) −3066.00 −0.184587
\(652\) −16288.0 −0.978355
\(653\) 2892.00 0.173312 0.0866560 0.996238i \(-0.472382\pi\)
0.0866560 + 0.996238i \(0.472382\pi\)
\(654\) 0 0
\(655\) −6780.00 −0.404453
\(656\) −18048.0 −1.07417
\(657\) −1116.00 −0.0662699
\(658\) 0 0
\(659\) 750.000 0.0443336 0.0221668 0.999754i \(-0.492944\pi\)
0.0221668 + 0.999754i \(0.492944\pi\)
\(660\) 5040.00 0.297245
\(661\) 30062.0 1.76895 0.884475 0.466587i \(-0.154517\pi\)
0.884475 + 0.466587i \(0.154517\pi\)
\(662\) 0 0
\(663\) −3960.00 −0.231966
\(664\) 0 0
\(665\) 1330.00 0.0775567
\(666\) 0 0
\(667\) −3096.00 −0.179727
\(668\) 31488.0 1.82381
\(669\) −9564.00 −0.552714
\(670\) 0 0
\(671\) −28644.0 −1.64797
\(672\) 0 0
\(673\) 15446.0 0.884695 0.442347 0.896844i \(-0.354146\pi\)
0.442347 + 0.896844i \(0.354146\pi\)
\(674\) 0 0
\(675\) −675.000 −0.0384900
\(676\) 14376.0 0.817934
\(677\) −25110.0 −1.42549 −0.712744 0.701424i \(-0.752548\pi\)
−0.712744 + 0.701424i \(0.752548\pi\)
\(678\) 0 0
\(679\) 8456.00 0.477926
\(680\) 0 0
\(681\) 10188.0 0.573282
\(682\) 0 0
\(683\) 7968.00 0.446394 0.223197 0.974773i \(-0.428351\pi\)
0.223197 + 0.974773i \(0.428351\pi\)
\(684\) −2736.00 −0.152944
\(685\) −4920.00 −0.274429
\(686\) 0 0
\(687\) −15882.0 −0.882003
\(688\) 1280.00 0.0709296
\(689\) 6720.00 0.371570
\(690\) 0 0
\(691\) −14398.0 −0.792657 −0.396328 0.918109i \(-0.629716\pi\)
−0.396328 + 0.918109i \(0.629716\pi\)
\(692\) −3024.00 −0.166120
\(693\) 2646.00 0.145041
\(694\) 0 0
\(695\) −1970.00 −0.107520
\(696\) 0 0
\(697\) −18612.0 −1.01145
\(698\) 0 0
\(699\) −2556.00 −0.138307
\(700\) −1400.00 −0.0755929
\(701\) −9234.00 −0.497523 −0.248761 0.968565i \(-0.580023\pi\)
−0.248761 + 0.968565i \(0.580023\pi\)
\(702\) 0 0
\(703\) 16492.0 0.884790
\(704\) −21504.0 −1.15123
\(705\) 1080.00 0.0576953
\(706\) 0 0
\(707\) 3822.00 0.203311
\(708\) −8640.00 −0.458631
\(709\) 8030.00 0.425350 0.212675 0.977123i \(-0.431782\pi\)
0.212675 + 0.977123i \(0.431782\pi\)
\(710\) 0 0
\(711\) 10224.0 0.539283
\(712\) 0 0
\(713\) 1752.00 0.0920237
\(714\) 0 0
\(715\) 4200.00 0.219680
\(716\) 1776.00 0.0926987
\(717\) −14598.0 −0.760352
\(718\) 0 0
\(719\) 27060.0 1.40357 0.701786 0.712388i \(-0.252386\pi\)
0.701786 + 0.712388i \(0.252386\pi\)
\(720\) 2880.00 0.149071
\(721\) −3640.00 −0.188018
\(722\) 0 0
\(723\) 6150.00 0.316350
\(724\) 20720.0 1.06361
\(725\) −6450.00 −0.330410
\(726\) 0 0
\(727\) −3724.00 −0.189980 −0.0949900 0.995478i \(-0.530282\pi\)
−0.0949900 + 0.995478i \(0.530282\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 1320.00 0.0667879
\(732\) −16368.0 −0.826474
\(733\) −5668.00 −0.285610 −0.142805 0.989751i \(-0.545612\pi\)
−0.142805 + 0.989751i \(0.545612\pi\)
\(734\) 0 0
\(735\) −735.000 −0.0368856
\(736\) 0 0
\(737\) 34104.0 1.70453
\(738\) 0 0
\(739\) −16072.0 −0.800024 −0.400012 0.916510i \(-0.630994\pi\)
−0.400012 + 0.916510i \(0.630994\pi\)
\(740\) −17360.0 −0.862387
\(741\) −2280.00 −0.113034
\(742\) 0 0
\(743\) −8256.00 −0.407649 −0.203825 0.979007i \(-0.565337\pi\)
−0.203825 + 0.979007i \(0.565337\pi\)
\(744\) 0 0
\(745\) −5070.00 −0.249329
\(746\) 0 0
\(747\) 1404.00 0.0687680
\(748\) −22176.0 −1.08400
\(749\) 8484.00 0.413883
\(750\) 0 0
\(751\) −6352.00 −0.308639 −0.154319 0.988021i \(-0.549318\pi\)
−0.154319 + 0.988021i \(0.549318\pi\)
\(752\) −4608.00 −0.223453
\(753\) 3456.00 0.167256
\(754\) 0 0
\(755\) −9980.00 −0.481072
\(756\) 1512.00 0.0727393
\(757\) 11558.0 0.554931 0.277465 0.960736i \(-0.410506\pi\)
0.277465 + 0.960736i \(0.410506\pi\)
\(758\) 0 0
\(759\) −1512.00 −0.0723085
\(760\) 0 0
\(761\) 7770.00 0.370121 0.185061 0.982727i \(-0.440752\pi\)
0.185061 + 0.982727i \(0.440752\pi\)
\(762\) 0 0
\(763\) −7546.00 −0.358039
\(764\) −17712.0 −0.838740
\(765\) 2970.00 0.140367
\(766\) 0 0
\(767\) −7200.00 −0.338953
\(768\) −12288.0 −0.577350
\(769\) 22646.0 1.06194 0.530972 0.847389i \(-0.321827\pi\)
0.530972 + 0.847389i \(0.321827\pi\)
\(770\) 0 0
\(771\) 19350.0 0.903856
\(772\) −33424.0 −1.55823
\(773\) −35502.0 −1.65190 −0.825950 0.563744i \(-0.809361\pi\)
−0.825950 + 0.563744i \(0.809361\pi\)
\(774\) 0 0
\(775\) 3650.00 0.169177
\(776\) 0 0
\(777\) −9114.00 −0.420802
\(778\) 0 0
\(779\) −10716.0 −0.492863
\(780\) 2400.00 0.110172
\(781\) 34020.0 1.55868
\(782\) 0 0
\(783\) 6966.00 0.317937
\(784\) 3136.00 0.142857
\(785\) −11960.0 −0.543784
\(786\) 0 0
\(787\) −17080.0 −0.773617 −0.386808 0.922160i \(-0.626422\pi\)
−0.386808 + 0.922160i \(0.626422\pi\)
\(788\) 24480.0 1.10668
\(789\) −5904.00 −0.266398
\(790\) 0 0
\(791\) −10164.0 −0.456878
\(792\) 0 0
\(793\) −13640.0 −0.610808
\(794\) 0 0
\(795\) −5040.00 −0.224843
\(796\) −21328.0 −0.949687
\(797\) 5730.00 0.254664 0.127332 0.991860i \(-0.459359\pi\)
0.127332 + 0.991860i \(0.459359\pi\)
\(798\) 0 0
\(799\) −4752.00 −0.210405
\(800\) 0 0
\(801\) −9342.00 −0.412089
\(802\) 0 0
\(803\) −5208.00 −0.228875
\(804\) 19488.0 0.854837
\(805\) 420.000 0.0183889
\(806\) 0 0
\(807\) 11682.0 0.509574
\(808\) 0 0
\(809\) 2550.00 0.110820 0.0554099 0.998464i \(-0.482353\pi\)
0.0554099 + 0.998464i \(0.482353\pi\)
\(810\) 0 0
\(811\) −27538.0 −1.19234 −0.596171 0.802857i \(-0.703312\pi\)
−0.596171 + 0.802857i \(0.703312\pi\)
\(812\) 14448.0 0.624416
\(813\) −21282.0 −0.918072
\(814\) 0 0
\(815\) 10180.0 0.437534
\(816\) −12672.0 −0.543638
\(817\) 760.000 0.0325447
\(818\) 0 0
\(819\) 1260.00 0.0537582
\(820\) 11280.0 0.480384
\(821\) −19242.0 −0.817966 −0.408983 0.912542i \(-0.634117\pi\)
−0.408983 + 0.912542i \(0.634117\pi\)
\(822\) 0 0
\(823\) −11752.0 −0.497751 −0.248875 0.968536i \(-0.580061\pi\)
−0.248875 + 0.968536i \(0.580061\pi\)
\(824\) 0 0
\(825\) −3150.00 −0.132932
\(826\) 0 0
\(827\) −28692.0 −1.20643 −0.603216 0.797578i \(-0.706114\pi\)
−0.603216 + 0.797578i \(0.706114\pi\)
\(828\) −864.000 −0.0362634
\(829\) 28442.0 1.19159 0.595797 0.803135i \(-0.296836\pi\)
0.595797 + 0.803135i \(0.296836\pi\)
\(830\) 0 0
\(831\) 9930.00 0.414522
\(832\) −10240.0 −0.426692
\(833\) 3234.00 0.134516
\(834\) 0 0
\(835\) −19680.0 −0.815634
\(836\) −12768.0 −0.528218
\(837\) −3942.00 −0.162790
\(838\) 0 0
\(839\) 20172.0 0.830053 0.415027 0.909809i \(-0.363772\pi\)
0.415027 + 0.909809i \(0.363772\pi\)
\(840\) 0 0
\(841\) 42175.0 1.72926
\(842\) 0 0
\(843\) −21474.0 −0.877347
\(844\) 10784.0 0.439811
\(845\) −8985.00 −0.365791
\(846\) 0 0
\(847\) 3031.00 0.122959
\(848\) 21504.0 0.870814
\(849\) 15492.0 0.626247
\(850\) 0 0
\(851\) 5208.00 0.209786
\(852\) 19440.0 0.781694
\(853\) 19820.0 0.795573 0.397787 0.917478i \(-0.369778\pi\)
0.397787 + 0.917478i \(0.369778\pi\)
\(854\) 0 0
\(855\) 1710.00 0.0683986
\(856\) 0 0
\(857\) −10290.0 −0.410151 −0.205076 0.978746i \(-0.565744\pi\)
−0.205076 + 0.978746i \(0.565744\pi\)
\(858\) 0 0
\(859\) −31606.0 −1.25539 −0.627697 0.778458i \(-0.716002\pi\)
−0.627697 + 0.778458i \(0.716002\pi\)
\(860\) −800.000 −0.0317207
\(861\) 5922.00 0.234403
\(862\) 0 0
\(863\) 23172.0 0.914002 0.457001 0.889466i \(-0.348924\pi\)
0.457001 + 0.889466i \(0.348924\pi\)
\(864\) 0 0
\(865\) 1890.00 0.0742912
\(866\) 0 0
\(867\) 1671.00 0.0654558
\(868\) −8176.00 −0.319714
\(869\) 47712.0 1.86251
\(870\) 0 0
\(871\) 16240.0 0.631770
\(872\) 0 0
\(873\) 10872.0 0.421491
\(874\) 0 0
\(875\) 875.000 0.0338062
\(876\) −2976.00 −0.114783
\(877\) −15550.0 −0.598730 −0.299365 0.954139i \(-0.596775\pi\)
−0.299365 + 0.954139i \(0.596775\pi\)
\(878\) 0 0
\(879\) 25794.0 0.989772
\(880\) 13440.0 0.514844
\(881\) 28530.0 1.09103 0.545517 0.838100i \(-0.316334\pi\)
0.545517 + 0.838100i \(0.316334\pi\)
\(882\) 0 0
\(883\) −28780.0 −1.09686 −0.548428 0.836198i \(-0.684774\pi\)
−0.548428 + 0.836198i \(0.684774\pi\)
\(884\) −10560.0 −0.401777
\(885\) 5400.00 0.205106
\(886\) 0 0
\(887\) 22872.0 0.865802 0.432901 0.901441i \(-0.357490\pi\)
0.432901 + 0.901441i \(0.357490\pi\)
\(888\) 0 0
\(889\) −9184.00 −0.346481
\(890\) 0 0
\(891\) 3402.00 0.127914
\(892\) −25504.0 −0.957329
\(893\) −2736.00 −0.102527
\(894\) 0 0
\(895\) −1110.00 −0.0414561
\(896\) 0 0
\(897\) −720.000 −0.0268006
\(898\) 0 0
\(899\) −37668.0 −1.39744
\(900\) −1800.00 −0.0666667
\(901\) 22176.0 0.819966
\(902\) 0 0
\(903\) −420.000 −0.0154781
\(904\) 0 0
\(905\) −12950.0 −0.475660
\(906\) 0 0
\(907\) −10708.0 −0.392010 −0.196005 0.980603i \(-0.562797\pi\)
−0.196005 + 0.980603i \(0.562797\pi\)
\(908\) 27168.0 0.992953
\(909\) 4914.00 0.179304
\(910\) 0 0
\(911\) 1326.00 0.0482243 0.0241122 0.999709i \(-0.492324\pi\)
0.0241122 + 0.999709i \(0.492324\pi\)
\(912\) −7296.00 −0.264906
\(913\) 6552.00 0.237502
\(914\) 0 0
\(915\) 10230.0 0.369610
\(916\) −42352.0 −1.52767
\(917\) −9492.00 −0.341825
\(918\) 0 0
\(919\) −13696.0 −0.491610 −0.245805 0.969319i \(-0.579052\pi\)
−0.245805 + 0.969319i \(0.579052\pi\)
\(920\) 0 0
\(921\) 1344.00 0.0480850
\(922\) 0 0
\(923\) 16200.0 0.577713
\(924\) 7056.00 0.251218
\(925\) 10850.0 0.385671
\(926\) 0 0
\(927\) −4680.00 −0.165816
\(928\) 0 0
\(929\) 42354.0 1.49579 0.747895 0.663817i \(-0.231064\pi\)
0.747895 + 0.663817i \(0.231064\pi\)
\(930\) 0 0
\(931\) 1862.00 0.0655474
\(932\) −6816.00 −0.239555
\(933\) 17496.0 0.613926
\(934\) 0 0
\(935\) 13860.0 0.484781
\(936\) 0 0
\(937\) 6644.00 0.231644 0.115822 0.993270i \(-0.463050\pi\)
0.115822 + 0.993270i \(0.463050\pi\)
\(938\) 0 0
\(939\) −29544.0 −1.02676
\(940\) 2880.00 0.0999311
\(941\) 1350.00 0.0467681 0.0233840 0.999727i \(-0.492556\pi\)
0.0233840 + 0.999727i \(0.492556\pi\)
\(942\) 0 0
\(943\) −3384.00 −0.116859
\(944\) −23040.0 −0.794373
\(945\) −945.000 −0.0325300
\(946\) 0 0
\(947\) 49320.0 1.69238 0.846190 0.532881i \(-0.178891\pi\)
0.846190 + 0.532881i \(0.178891\pi\)
\(948\) 27264.0 0.934065
\(949\) −2480.00 −0.0848306
\(950\) 0 0
\(951\) 16848.0 0.574484
\(952\) 0 0
\(953\) 5940.00 0.201905 0.100953 0.994891i \(-0.467811\pi\)
0.100953 + 0.994891i \(0.467811\pi\)
\(954\) 0 0
\(955\) 11070.0 0.375096
\(956\) −38928.0 −1.31697
\(957\) 32508.0 1.09805
\(958\) 0 0
\(959\) −6888.00 −0.231934
\(960\) 7680.00 0.258199
\(961\) −8475.00 −0.284482
\(962\) 0 0
\(963\) 10908.0 0.365011
\(964\) 16400.0 0.547934
\(965\) 20890.0 0.696863
\(966\) 0 0
\(967\) 47216.0 1.57018 0.785090 0.619382i \(-0.212617\pi\)
0.785090 + 0.619382i \(0.212617\pi\)
\(968\) 0 0
\(969\) −7524.00 −0.249438
\(970\) 0 0
\(971\) 12552.0 0.414843 0.207422 0.978252i \(-0.433493\pi\)
0.207422 + 0.978252i \(0.433493\pi\)
\(972\) 1944.00 0.0641500
\(973\) −2758.00 −0.0908709
\(974\) 0 0
\(975\) −1500.00 −0.0492702
\(976\) −43648.0 −1.43149
\(977\) 46908.0 1.53605 0.768025 0.640420i \(-0.221240\pi\)
0.768025 + 0.640420i \(0.221240\pi\)
\(978\) 0 0
\(979\) −43596.0 −1.42322
\(980\) −1960.00 −0.0638877
\(981\) −9702.00 −0.315760
\(982\) 0 0
\(983\) −46128.0 −1.49670 −0.748349 0.663305i \(-0.769153\pi\)
−0.748349 + 0.663305i \(0.769153\pi\)
\(984\) 0 0
\(985\) −15300.0 −0.494922
\(986\) 0 0
\(987\) 1512.00 0.0487614
\(988\) −6080.00 −0.195780
\(989\) 240.000 0.00771644
\(990\) 0 0
\(991\) −12184.0 −0.390552 −0.195276 0.980748i \(-0.562560\pi\)
−0.195276 + 0.980748i \(0.562560\pi\)
\(992\) 0 0
\(993\) −1356.00 −0.0433347
\(994\) 0 0
\(995\) 13330.0 0.424713
\(996\) 3744.00 0.119110
\(997\) −5164.00 −0.164038 −0.0820188 0.996631i \(-0.526137\pi\)
−0.0820188 + 0.996631i \(0.526137\pi\)
\(998\) 0 0
\(999\) −11718.0 −0.371112
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 105.4.a.a.1.1 1
3.2 odd 2 315.4.a.d.1.1 1
4.3 odd 2 1680.4.a.s.1.1 1
5.2 odd 4 525.4.d.f.274.2 2
5.3 odd 4 525.4.d.f.274.1 2
5.4 even 2 525.4.a.e.1.1 1
7.6 odd 2 735.4.a.c.1.1 1
15.14 odd 2 1575.4.a.f.1.1 1
21.20 even 2 2205.4.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
105.4.a.a.1.1 1 1.1 even 1 trivial
315.4.a.d.1.1 1 3.2 odd 2
525.4.a.e.1.1 1 5.4 even 2
525.4.d.f.274.1 2 5.3 odd 4
525.4.d.f.274.2 2 5.2 odd 4
735.4.a.c.1.1 1 7.6 odd 2
1575.4.a.f.1.1 1 15.14 odd 2
1680.4.a.s.1.1 1 4.3 odd 2
2205.4.a.o.1.1 1 21.20 even 2