Properties

Label 27.4.a.a.1.1
Level 27
Weight 4
Character 27.1
Self dual yes
Analytic conductor 1.593
Analytic rank 1
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) = \( 27 = 3^{3} \)
Weight: \( k \) = \( 4 \)
Character orbit: \([\chi]\) = 27.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(1.59305157016\)
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
Root \(0\)
Character \(\chi\) = 27.1

$q$-expansion

\(f(q)\) \(=\) \(q-3.00000 q^{2} +1.00000 q^{4} -15.0000 q^{5} -25.0000 q^{7} +21.0000 q^{8} +O(q^{10})\) \(q-3.00000 q^{2} +1.00000 q^{4} -15.0000 q^{5} -25.0000 q^{7} +21.0000 q^{8} +45.0000 q^{10} +15.0000 q^{11} +20.0000 q^{13} +75.0000 q^{14} -71.0000 q^{16} -72.0000 q^{17} +2.00000 q^{19} -15.0000 q^{20} -45.0000 q^{22} -114.000 q^{23} +100.000 q^{25} -60.0000 q^{26} -25.0000 q^{28} -30.0000 q^{29} +101.000 q^{31} +45.0000 q^{32} +216.000 q^{34} +375.000 q^{35} -430.000 q^{37} -6.00000 q^{38} -315.000 q^{40} +30.0000 q^{41} +110.000 q^{43} +15.0000 q^{44} +342.000 q^{46} +330.000 q^{47} +282.000 q^{49} -300.000 q^{50} +20.0000 q^{52} -621.000 q^{53} -225.000 q^{55} -525.000 q^{56} +90.0000 q^{58} +660.000 q^{59} -376.000 q^{61} -303.000 q^{62} +433.000 q^{64} -300.000 q^{65} -250.000 q^{67} -72.0000 q^{68} -1125.00 q^{70} +360.000 q^{71} +785.000 q^{73} +1290.00 q^{74} +2.00000 q^{76} -375.000 q^{77} +488.000 q^{79} +1065.00 q^{80} -90.0000 q^{82} -489.000 q^{83} +1080.00 q^{85} -330.000 q^{86} +315.000 q^{88} +450.000 q^{89} -500.000 q^{91} -114.000 q^{92} -990.000 q^{94} -30.0000 q^{95} -1105.00 q^{97} -846.000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.00000 −1.06066 −0.530330 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) 0 0
\(4\) 1.00000 0.125000
\(5\) −15.0000 −1.34164 −0.670820 0.741620i \(-0.734058\pi\)
−0.670820 + 0.741620i \(0.734058\pi\)
\(6\) 0 0
\(7\) −25.0000 −1.34987 −0.674937 0.737876i \(-0.735829\pi\)
−0.674937 + 0.737876i \(0.735829\pi\)
\(8\) 21.0000 0.928078
\(9\) 0 0
\(10\) 45.0000 1.42302
\(11\) 15.0000 0.411152 0.205576 0.978641i \(-0.434093\pi\)
0.205576 + 0.978641i \(0.434093\pi\)
\(12\) 0 0
\(13\) 20.0000 0.426692 0.213346 0.976977i \(-0.431564\pi\)
0.213346 + 0.976977i \(0.431564\pi\)
\(14\) 75.0000 1.43176
\(15\) 0 0
\(16\) −71.0000 −1.10938
\(17\) −72.0000 −1.02721 −0.513605 0.858027i \(-0.671690\pi\)
−0.513605 + 0.858027i \(0.671690\pi\)
\(18\) 0 0
\(19\) 2.00000 0.0241490 0.0120745 0.999927i \(-0.496156\pi\)
0.0120745 + 0.999927i \(0.496156\pi\)
\(20\) −15.0000 −0.167705
\(21\) 0 0
\(22\) −45.0000 −0.436092
\(23\) −114.000 −1.03351 −0.516753 0.856134i \(-0.672859\pi\)
−0.516753 + 0.856134i \(0.672859\pi\)
\(24\) 0 0
\(25\) 100.000 0.800000
\(26\) −60.0000 −0.452576
\(27\) 0 0
\(28\) −25.0000 −0.168734
\(29\) −30.0000 −0.192099 −0.0960493 0.995377i \(-0.530621\pi\)
−0.0960493 + 0.995377i \(0.530621\pi\)
\(30\) 0 0
\(31\) 101.000 0.585166 0.292583 0.956240i \(-0.405485\pi\)
0.292583 + 0.956240i \(0.405485\pi\)
\(32\) 45.0000 0.248592
\(33\) 0 0
\(34\) 216.000 1.08952
\(35\) 375.000 1.81104
\(36\) 0 0
\(37\) −430.000 −1.91058 −0.955291 0.295666i \(-0.904458\pi\)
−0.955291 + 0.295666i \(0.904458\pi\)
\(38\) −6.00000 −0.0256139
\(39\) 0 0
\(40\) −315.000 −1.24515
\(41\) 30.0000 0.114273 0.0571367 0.998366i \(-0.481803\pi\)
0.0571367 + 0.998366i \(0.481803\pi\)
\(42\) 0 0
\(43\) 110.000 0.390113 0.195056 0.980792i \(-0.437511\pi\)
0.195056 + 0.980792i \(0.437511\pi\)
\(44\) 15.0000 0.0513940
\(45\) 0 0
\(46\) 342.000 1.09620
\(47\) 330.000 1.02416 0.512079 0.858938i \(-0.328875\pi\)
0.512079 + 0.858938i \(0.328875\pi\)
\(48\) 0 0
\(49\) 282.000 0.822157
\(50\) −300.000 −0.848528
\(51\) 0 0
\(52\) 20.0000 0.0533366
\(53\) −621.000 −1.60945 −0.804726 0.593647i \(-0.797688\pi\)
−0.804726 + 0.593647i \(0.797688\pi\)
\(54\) 0 0
\(55\) −225.000 −0.551618
\(56\) −525.000 −1.25279
\(57\) 0 0
\(58\) 90.0000 0.203751
\(59\) 660.000 1.45635 0.728175 0.685391i \(-0.240369\pi\)
0.728175 + 0.685391i \(0.240369\pi\)
\(60\) 0 0
\(61\) −376.000 −0.789211 −0.394605 0.918851i \(-0.629119\pi\)
−0.394605 + 0.918851i \(0.629119\pi\)
\(62\) −303.000 −0.620662
\(63\) 0 0
\(64\) 433.000 0.845703
\(65\) −300.000 −0.572468
\(66\) 0 0
\(67\) −250.000 −0.455856 −0.227928 0.973678i \(-0.573195\pi\)
−0.227928 + 0.973678i \(0.573195\pi\)
\(68\) −72.0000 −0.128401
\(69\) 0 0
\(70\) −1125.00 −1.92090
\(71\) 360.000 0.601748 0.300874 0.953664i \(-0.402722\pi\)
0.300874 + 0.953664i \(0.402722\pi\)
\(72\) 0 0
\(73\) 785.000 1.25859 0.629297 0.777165i \(-0.283343\pi\)
0.629297 + 0.777165i \(0.283343\pi\)
\(74\) 1290.00 2.02648
\(75\) 0 0
\(76\) 2.00000 0.00301863
\(77\) −375.000 −0.555003
\(78\) 0 0
\(79\) 488.000 0.694991 0.347496 0.937682i \(-0.387032\pi\)
0.347496 + 0.937682i \(0.387032\pi\)
\(80\) 1065.00 1.48838
\(81\) 0 0
\(82\) −90.0000 −0.121205
\(83\) −489.000 −0.646683 −0.323342 0.946282i \(-0.604806\pi\)
−0.323342 + 0.946282i \(0.604806\pi\)
\(84\) 0 0
\(85\) 1080.00 1.37815
\(86\) −330.000 −0.413777
\(87\) 0 0
\(88\) 315.000 0.381581
\(89\) 450.000 0.535954 0.267977 0.963425i \(-0.413645\pi\)
0.267977 + 0.963425i \(0.413645\pi\)
\(90\) 0 0
\(91\) −500.000 −0.575981
\(92\) −114.000 −0.129188
\(93\) 0 0
\(94\) −990.000 −1.08628
\(95\) −30.0000 −0.0323993
\(96\) 0 0
\(97\) −1105.00 −1.15666 −0.578329 0.815804i \(-0.696295\pi\)
−0.578329 + 0.815804i \(0.696295\pi\)
\(98\) −846.000 −0.872030
\(99\) 0 0
\(100\) 100.000 0.100000
\(101\) −1425.00 −1.40389 −0.701945 0.712232i \(-0.747685\pi\)
−0.701945 + 0.712232i \(0.747685\pi\)
\(102\) 0 0
\(103\) −1060.00 −1.01403 −0.507014 0.861938i \(-0.669251\pi\)
−0.507014 + 0.861938i \(0.669251\pi\)
\(104\) 420.000 0.396004
\(105\) 0 0
\(106\) 1863.00 1.70708
\(107\) −1485.00 −1.34169 −0.670843 0.741600i \(-0.734067\pi\)
−0.670843 + 0.741600i \(0.734067\pi\)
\(108\) 0 0
\(109\) −862.000 −0.757474 −0.378737 0.925504i \(-0.623641\pi\)
−0.378737 + 0.925504i \(0.623641\pi\)
\(110\) 675.000 0.585079
\(111\) 0 0
\(112\) 1775.00 1.49752
\(113\) −690.000 −0.574422 −0.287211 0.957867i \(-0.592728\pi\)
−0.287211 + 0.957867i \(0.592728\pi\)
\(114\) 0 0
\(115\) 1710.00 1.38659
\(116\) −30.0000 −0.0240123
\(117\) 0 0
\(118\) −1980.00 −1.54469
\(119\) 1800.00 1.38660
\(120\) 0 0
\(121\) −1106.00 −0.830954
\(122\) 1128.00 0.837085
\(123\) 0 0
\(124\) 101.000 0.0731457
\(125\) 375.000 0.268328
\(126\) 0 0
\(127\) 1865.00 1.30309 0.651543 0.758611i \(-0.274122\pi\)
0.651543 + 0.758611i \(0.274122\pi\)
\(128\) −1659.00 −1.14560
\(129\) 0 0
\(130\) 900.000 0.607194
\(131\) 1155.00 0.770327 0.385163 0.922848i \(-0.374145\pi\)
0.385163 + 0.922848i \(0.374145\pi\)
\(132\) 0 0
\(133\) −50.0000 −0.0325981
\(134\) 750.000 0.483508
\(135\) 0 0
\(136\) −1512.00 −0.953330
\(137\) 2778.00 1.73241 0.866206 0.499686i \(-0.166551\pi\)
0.866206 + 0.499686i \(0.166551\pi\)
\(138\) 0 0
\(139\) −1924.00 −1.17404 −0.587020 0.809572i \(-0.699699\pi\)
−0.587020 + 0.809572i \(0.699699\pi\)
\(140\) 375.000 0.226381
\(141\) 0 0
\(142\) −1080.00 −0.638251
\(143\) 300.000 0.175435
\(144\) 0 0
\(145\) 450.000 0.257727
\(146\) −2355.00 −1.33494
\(147\) 0 0
\(148\) −430.000 −0.238823
\(149\) −1455.00 −0.799988 −0.399994 0.916518i \(-0.630988\pi\)
−0.399994 + 0.916518i \(0.630988\pi\)
\(150\) 0 0
\(151\) −727.000 −0.391804 −0.195902 0.980623i \(-0.562763\pi\)
−0.195902 + 0.980623i \(0.562763\pi\)
\(152\) 42.0000 0.0224122
\(153\) 0 0
\(154\) 1125.00 0.588669
\(155\) −1515.00 −0.785082
\(156\) 0 0
\(157\) 3260.00 1.65717 0.828587 0.559860i \(-0.189145\pi\)
0.828587 + 0.559860i \(0.189145\pi\)
\(158\) −1464.00 −0.737149
\(159\) 0 0
\(160\) −675.000 −0.333521
\(161\) 2850.00 1.39510
\(162\) 0 0
\(163\) 2540.00 1.22054 0.610270 0.792193i \(-0.291061\pi\)
0.610270 + 0.792193i \(0.291061\pi\)
\(164\) 30.0000 0.0142842
\(165\) 0 0
\(166\) 1467.00 0.685911
\(167\) −3498.00 −1.62086 −0.810429 0.585837i \(-0.800766\pi\)
−0.810429 + 0.585837i \(0.800766\pi\)
\(168\) 0 0
\(169\) −1797.00 −0.817934
\(170\) −3240.00 −1.46175
\(171\) 0 0
\(172\) 110.000 0.0487641
\(173\) 1149.00 0.504953 0.252476 0.967603i \(-0.418755\pi\)
0.252476 + 0.967603i \(0.418755\pi\)
\(174\) 0 0
\(175\) −2500.00 −1.07990
\(176\) −1065.00 −0.456122
\(177\) 0 0
\(178\) −1350.00 −0.568465
\(179\) −315.000 −0.131532 −0.0657659 0.997835i \(-0.520949\pi\)
−0.0657659 + 0.997835i \(0.520949\pi\)
\(180\) 0 0
\(181\) 1136.00 0.466509 0.233255 0.972416i \(-0.425062\pi\)
0.233255 + 0.972416i \(0.425062\pi\)
\(182\) 1500.00 0.610920
\(183\) 0 0
\(184\) −2394.00 −0.959174
\(185\) 6450.00 2.56332
\(186\) 0 0
\(187\) −1080.00 −0.422339
\(188\) 330.000 0.128020
\(189\) 0 0
\(190\) 90.0000 0.0343647
\(191\) −2460.00 −0.931934 −0.465967 0.884802i \(-0.654293\pi\)
−0.465967 + 0.884802i \(0.654293\pi\)
\(192\) 0 0
\(193\) 965.000 0.359908 0.179954 0.983675i \(-0.442405\pi\)
0.179954 + 0.983675i \(0.442405\pi\)
\(194\) 3315.00 1.22682
\(195\) 0 0
\(196\) 282.000 0.102770
\(197\) −2493.00 −0.901619 −0.450809 0.892620i \(-0.648865\pi\)
−0.450809 + 0.892620i \(0.648865\pi\)
\(198\) 0 0
\(199\) −511.000 −0.182029 −0.0910146 0.995850i \(-0.529011\pi\)
−0.0910146 + 0.995850i \(0.529011\pi\)
\(200\) 2100.00 0.742462
\(201\) 0 0
\(202\) 4275.00 1.48905
\(203\) 750.000 0.259309
\(204\) 0 0
\(205\) −450.000 −0.153314
\(206\) 3180.00 1.07554
\(207\) 0 0
\(208\) −1420.00 −0.473362
\(209\) 30.0000 0.00992892
\(210\) 0 0
\(211\) −2086.00 −0.680598 −0.340299 0.940317i \(-0.610528\pi\)
−0.340299 + 0.940317i \(0.610528\pi\)
\(212\) −621.000 −0.201181
\(213\) 0 0
\(214\) 4455.00 1.42307
\(215\) −1650.00 −0.523391
\(216\) 0 0
\(217\) −2525.00 −0.789899
\(218\) 2586.00 0.803422
\(219\) 0 0
\(220\) −225.000 −0.0689523
\(221\) −1440.00 −0.438303
\(222\) 0 0
\(223\) 5240.00 1.57353 0.786763 0.617255i \(-0.211755\pi\)
0.786763 + 0.617255i \(0.211755\pi\)
\(224\) −1125.00 −0.335568
\(225\) 0 0
\(226\) 2070.00 0.609267
\(227\) −2388.00 −0.698225 −0.349113 0.937081i \(-0.613517\pi\)
−0.349113 + 0.937081i \(0.613517\pi\)
\(228\) 0 0
\(229\) 182.000 0.0525192 0.0262596 0.999655i \(-0.491640\pi\)
0.0262596 + 0.999655i \(0.491640\pi\)
\(230\) −5130.00 −1.47071
\(231\) 0 0
\(232\) −630.000 −0.178282
\(233\) −450.000 −0.126526 −0.0632628 0.997997i \(-0.520151\pi\)
−0.0632628 + 0.997997i \(0.520151\pi\)
\(234\) 0 0
\(235\) −4950.00 −1.37405
\(236\) 660.000 0.182044
\(237\) 0 0
\(238\) −5400.00 −1.47071
\(239\) −5190.00 −1.40466 −0.702329 0.711853i \(-0.747856\pi\)
−0.702329 + 0.711853i \(0.747856\pi\)
\(240\) 0 0
\(241\) −2266.00 −0.605668 −0.302834 0.953043i \(-0.597933\pi\)
−0.302834 + 0.953043i \(0.597933\pi\)
\(242\) 3318.00 0.881360
\(243\) 0 0
\(244\) −376.000 −0.0986514
\(245\) −4230.00 −1.10304
\(246\) 0 0
\(247\) 40.0000 0.0103042
\(248\) 2121.00 0.543079
\(249\) 0 0
\(250\) −1125.00 −0.284605
\(251\) 2880.00 0.724239 0.362119 0.932132i \(-0.382053\pi\)
0.362119 + 0.932132i \(0.382053\pi\)
\(252\) 0 0
\(253\) −1710.00 −0.424928
\(254\) −5595.00 −1.38213
\(255\) 0 0
\(256\) 1513.00 0.369385
\(257\) 4188.00 1.01650 0.508250 0.861210i \(-0.330293\pi\)
0.508250 + 0.861210i \(0.330293\pi\)
\(258\) 0 0
\(259\) 10750.0 2.57904
\(260\) −300.000 −0.0715585
\(261\) 0 0
\(262\) −3465.00 −0.817055
\(263\) 3030.00 0.710410 0.355205 0.934788i \(-0.384411\pi\)
0.355205 + 0.934788i \(0.384411\pi\)
\(264\) 0 0
\(265\) 9315.00 2.15931
\(266\) 150.000 0.0345755
\(267\) 0 0
\(268\) −250.000 −0.0569820
\(269\) −3510.00 −0.795571 −0.397785 0.917479i \(-0.630221\pi\)
−0.397785 + 0.917479i \(0.630221\pi\)
\(270\) 0 0
\(271\) 2999.00 0.672237 0.336119 0.941820i \(-0.390886\pi\)
0.336119 + 0.941820i \(0.390886\pi\)
\(272\) 5112.00 1.13956
\(273\) 0 0
\(274\) −8334.00 −1.83750
\(275\) 1500.00 0.328921
\(276\) 0 0
\(277\) −7720.00 −1.67455 −0.837274 0.546783i \(-0.815852\pi\)
−0.837274 + 0.546783i \(0.815852\pi\)
\(278\) 5772.00 1.24526
\(279\) 0 0
\(280\) 7875.00 1.68079
\(281\) 7440.00 1.57948 0.789739 0.613443i \(-0.210216\pi\)
0.789739 + 0.613443i \(0.210216\pi\)
\(282\) 0 0
\(283\) 830.000 0.174341 0.0871703 0.996193i \(-0.472218\pi\)
0.0871703 + 0.996193i \(0.472218\pi\)
\(284\) 360.000 0.0752186
\(285\) 0 0
\(286\) −900.000 −0.186077
\(287\) −750.000 −0.154255
\(288\) 0 0
\(289\) 271.000 0.0551598
\(290\) −1350.00 −0.273361
\(291\) 0 0
\(292\) 785.000 0.157324
\(293\) −546.000 −0.108866 −0.0544329 0.998517i \(-0.517335\pi\)
−0.0544329 + 0.998517i \(0.517335\pi\)
\(294\) 0 0
\(295\) −9900.00 −1.95390
\(296\) −9030.00 −1.77317
\(297\) 0 0
\(298\) 4365.00 0.848516
\(299\) −2280.00 −0.440989
\(300\) 0 0
\(301\) −2750.00 −0.526603
\(302\) 2181.00 0.415571
\(303\) 0 0
\(304\) −142.000 −0.0267903
\(305\) 5640.00 1.05884
\(306\) 0 0
\(307\) −5560.00 −1.03364 −0.516818 0.856096i \(-0.672883\pi\)
−0.516818 + 0.856096i \(0.672883\pi\)
\(308\) −375.000 −0.0693754
\(309\) 0 0
\(310\) 4545.00 0.832705
\(311\) 8670.00 1.58081 0.790403 0.612587i \(-0.209871\pi\)
0.790403 + 0.612587i \(0.209871\pi\)
\(312\) 0 0
\(313\) 4565.00 0.824374 0.412187 0.911099i \(-0.364765\pi\)
0.412187 + 0.911099i \(0.364765\pi\)
\(314\) −9780.00 −1.75770
\(315\) 0 0
\(316\) 488.000 0.0868739
\(317\) −4233.00 −0.749997 −0.374998 0.927025i \(-0.622357\pi\)
−0.374998 + 0.927025i \(0.622357\pi\)
\(318\) 0 0
\(319\) −450.000 −0.0789817
\(320\) −6495.00 −1.13463
\(321\) 0 0
\(322\) −8550.00 −1.47973
\(323\) −144.000 −0.0248061
\(324\) 0 0
\(325\) 2000.00 0.341354
\(326\) −7620.00 −1.29458
\(327\) 0 0
\(328\) 630.000 0.106055
\(329\) −8250.00 −1.38248
\(330\) 0 0
\(331\) 542.000 0.0900031 0.0450015 0.998987i \(-0.485671\pi\)
0.0450015 + 0.998987i \(0.485671\pi\)
\(332\) −489.000 −0.0808354
\(333\) 0 0
\(334\) 10494.0 1.71918
\(335\) 3750.00 0.611595
\(336\) 0 0
\(337\) 5690.00 0.919745 0.459872 0.887985i \(-0.347895\pi\)
0.459872 + 0.887985i \(0.347895\pi\)
\(338\) 5391.00 0.867550
\(339\) 0 0
\(340\) 1080.00 0.172268
\(341\) 1515.00 0.240592
\(342\) 0 0
\(343\) 1525.00 0.240065
\(344\) 2310.00 0.362055
\(345\) 0 0
\(346\) −3447.00 −0.535583
\(347\) −5055.00 −0.782036 −0.391018 0.920383i \(-0.627877\pi\)
−0.391018 + 0.920383i \(0.627877\pi\)
\(348\) 0 0
\(349\) 1622.00 0.248778 0.124389 0.992234i \(-0.460303\pi\)
0.124389 + 0.992234i \(0.460303\pi\)
\(350\) 7500.00 1.14541
\(351\) 0 0
\(352\) 675.000 0.102209
\(353\) −30.0000 −0.00452334 −0.00226167 0.999997i \(-0.500720\pi\)
−0.00226167 + 0.999997i \(0.500720\pi\)
\(354\) 0 0
\(355\) −5400.00 −0.807330
\(356\) 450.000 0.0669942
\(357\) 0 0
\(358\) 945.000 0.139511
\(359\) 7470.00 1.09819 0.549097 0.835759i \(-0.314972\pi\)
0.549097 + 0.835759i \(0.314972\pi\)
\(360\) 0 0
\(361\) −6855.00 −0.999417
\(362\) −3408.00 −0.494808
\(363\) 0 0
\(364\) −500.000 −0.0719976
\(365\) −11775.0 −1.68858
\(366\) 0 0
\(367\) −1375.00 −0.195571 −0.0977853 0.995208i \(-0.531176\pi\)
−0.0977853 + 0.995208i \(0.531176\pi\)
\(368\) 8094.00 1.14655
\(369\) 0 0
\(370\) −19350.0 −2.71881
\(371\) 15525.0 2.17255
\(372\) 0 0
\(373\) −4840.00 −0.671865 −0.335933 0.941886i \(-0.609051\pi\)
−0.335933 + 0.941886i \(0.609051\pi\)
\(374\) 3240.00 0.447958
\(375\) 0 0
\(376\) 6930.00 0.950499
\(377\) −600.000 −0.0819670
\(378\) 0 0
\(379\) 1892.00 0.256426 0.128213 0.991747i \(-0.459076\pi\)
0.128213 + 0.991747i \(0.459076\pi\)
\(380\) −30.0000 −0.00404991
\(381\) 0 0
\(382\) 7380.00 0.988465
\(383\) 10704.0 1.42806 0.714032 0.700113i \(-0.246867\pi\)
0.714032 + 0.700113i \(0.246867\pi\)
\(384\) 0 0
\(385\) 5625.00 0.744614
\(386\) −2895.00 −0.381740
\(387\) 0 0
\(388\) −1105.00 −0.144582
\(389\) −7815.00 −1.01860 −0.509301 0.860588i \(-0.670096\pi\)
−0.509301 + 0.860588i \(0.670096\pi\)
\(390\) 0 0
\(391\) 8208.00 1.06163
\(392\) 5922.00 0.763026
\(393\) 0 0
\(394\) 7479.00 0.956311
\(395\) −7320.00 −0.932428
\(396\) 0 0
\(397\) 4700.00 0.594172 0.297086 0.954851i \(-0.403985\pi\)
0.297086 + 0.954851i \(0.403985\pi\)
\(398\) 1533.00 0.193071
\(399\) 0 0
\(400\) −7100.00 −0.887500
\(401\) 2100.00 0.261519 0.130759 0.991414i \(-0.458258\pi\)
0.130759 + 0.991414i \(0.458258\pi\)
\(402\) 0 0
\(403\) 2020.00 0.249686
\(404\) −1425.00 −0.175486
\(405\) 0 0
\(406\) −2250.00 −0.275038
\(407\) −6450.00 −0.785540
\(408\) 0 0
\(409\) −10753.0 −1.30000 −0.650002 0.759933i \(-0.725232\pi\)
−0.650002 + 0.759933i \(0.725232\pi\)
\(410\) 1350.00 0.162614
\(411\) 0 0
\(412\) −1060.00 −0.126754
\(413\) −16500.0 −1.96589
\(414\) 0 0
\(415\) 7335.00 0.867617
\(416\) 900.000 0.106072
\(417\) 0 0
\(418\) −90.0000 −0.0105312
\(419\) −2940.00 −0.342789 −0.171394 0.985203i \(-0.554827\pi\)
−0.171394 + 0.985203i \(0.554827\pi\)
\(420\) 0 0
\(421\) 8696.00 1.00669 0.503346 0.864085i \(-0.332102\pi\)
0.503346 + 0.864085i \(0.332102\pi\)
\(422\) 6258.00 0.721883
\(423\) 0 0
\(424\) −13041.0 −1.49370
\(425\) −7200.00 −0.821768
\(426\) 0 0
\(427\) 9400.00 1.06533
\(428\) −1485.00 −0.167711
\(429\) 0 0
\(430\) 4950.00 0.555140
\(431\) −8370.00 −0.935426 −0.467713 0.883880i \(-0.654922\pi\)
−0.467713 + 0.883880i \(0.654922\pi\)
\(432\) 0 0
\(433\) −5155.00 −0.572133 −0.286066 0.958210i \(-0.592348\pi\)
−0.286066 + 0.958210i \(0.592348\pi\)
\(434\) 7575.00 0.837815
\(435\) 0 0
\(436\) −862.000 −0.0946842
\(437\) −228.000 −0.0249582
\(438\) 0 0
\(439\) −10987.0 −1.19449 −0.597245 0.802059i \(-0.703738\pi\)
−0.597245 + 0.802059i \(0.703738\pi\)
\(440\) −4725.00 −0.511944
\(441\) 0 0
\(442\) 4320.00 0.464890
\(443\) −1956.00 −0.209780 −0.104890 0.994484i \(-0.533449\pi\)
−0.104890 + 0.994484i \(0.533449\pi\)
\(444\) 0 0
\(445\) −6750.00 −0.719058
\(446\) −15720.0 −1.66898
\(447\) 0 0
\(448\) −10825.0 −1.14159
\(449\) 8730.00 0.917582 0.458791 0.888544i \(-0.348283\pi\)
0.458791 + 0.888544i \(0.348283\pi\)
\(450\) 0 0
\(451\) 450.000 0.0469838
\(452\) −690.000 −0.0718028
\(453\) 0 0
\(454\) 7164.00 0.740580
\(455\) 7500.00 0.772759
\(456\) 0 0
\(457\) −8665.00 −0.886940 −0.443470 0.896289i \(-0.646253\pi\)
−0.443470 + 0.896289i \(0.646253\pi\)
\(458\) −546.000 −0.0557050
\(459\) 0 0
\(460\) 1710.00 0.173324
\(461\) 9825.00 0.992616 0.496308 0.868147i \(-0.334689\pi\)
0.496308 + 0.868147i \(0.334689\pi\)
\(462\) 0 0
\(463\) −5245.00 −0.526470 −0.263235 0.964732i \(-0.584790\pi\)
−0.263235 + 0.964732i \(0.584790\pi\)
\(464\) 2130.00 0.213109
\(465\) 0 0
\(466\) 1350.00 0.134201
\(467\) 11007.0 1.09067 0.545335 0.838218i \(-0.316402\pi\)
0.545335 + 0.838218i \(0.316402\pi\)
\(468\) 0 0
\(469\) 6250.00 0.615348
\(470\) 14850.0 1.45740
\(471\) 0 0
\(472\) 13860.0 1.35161
\(473\) 1650.00 0.160396
\(474\) 0 0
\(475\) 200.000 0.0193192
\(476\) 1800.00 0.173325
\(477\) 0 0
\(478\) 15570.0 1.48986
\(479\) −16950.0 −1.61684 −0.808419 0.588608i \(-0.799676\pi\)
−0.808419 + 0.588608i \(0.799676\pi\)
\(480\) 0 0
\(481\) −8600.00 −0.815231
\(482\) 6798.00 0.642408
\(483\) 0 0
\(484\) −1106.00 −0.103869
\(485\) 16575.0 1.55182
\(486\) 0 0
\(487\) 10640.0 0.990030 0.495015 0.868885i \(-0.335163\pi\)
0.495015 + 0.868885i \(0.335163\pi\)
\(488\) −7896.00 −0.732449
\(489\) 0 0
\(490\) 12690.0 1.16995
\(491\) −1635.00 −0.150278 −0.0751390 0.997173i \(-0.523940\pi\)
−0.0751390 + 0.997173i \(0.523940\pi\)
\(492\) 0 0
\(493\) 2160.00 0.197326
\(494\) −120.000 −0.0109293
\(495\) 0 0
\(496\) −7171.00 −0.649168
\(497\) −9000.00 −0.812284
\(498\) 0 0
\(499\) −15802.0 −1.41762 −0.708812 0.705397i \(-0.750769\pi\)
−0.708812 + 0.705397i \(0.750769\pi\)
\(500\) 375.000 0.0335410
\(501\) 0 0
\(502\) −8640.00 −0.768171
\(503\) 7866.00 0.697272 0.348636 0.937258i \(-0.386645\pi\)
0.348636 + 0.937258i \(0.386645\pi\)
\(504\) 0 0
\(505\) 21375.0 1.88351
\(506\) 5130.00 0.450704
\(507\) 0 0
\(508\) 1865.00 0.162886
\(509\) 11955.0 1.04105 0.520527 0.853845i \(-0.325736\pi\)
0.520527 + 0.853845i \(0.325736\pi\)
\(510\) 0 0
\(511\) −19625.0 −1.69894
\(512\) 8733.00 0.753804
\(513\) 0 0
\(514\) −12564.0 −1.07816
\(515\) 15900.0 1.36046
\(516\) 0 0
\(517\) 4950.00 0.421085
\(518\) −32250.0 −2.73549
\(519\) 0 0
\(520\) −6300.00 −0.531295
\(521\) −19260.0 −1.61957 −0.809785 0.586727i \(-0.800416\pi\)
−0.809785 + 0.586727i \(0.800416\pi\)
\(522\) 0 0
\(523\) −18520.0 −1.54842 −0.774209 0.632930i \(-0.781852\pi\)
−0.774209 + 0.632930i \(0.781852\pi\)
\(524\) 1155.00 0.0962909
\(525\) 0 0
\(526\) −9090.00 −0.753503
\(527\) −7272.00 −0.601088
\(528\) 0 0
\(529\) 829.000 0.0681351
\(530\) −27945.0 −2.29029
\(531\) 0 0
\(532\) −50.0000 −0.00407476
\(533\) 600.000 0.0487596
\(534\) 0 0
\(535\) 22275.0 1.80006
\(536\) −5250.00 −0.423070
\(537\) 0 0
\(538\) 10530.0 0.843830
\(539\) 4230.00 0.338032
\(540\) 0 0
\(541\) 8372.00 0.665324 0.332662 0.943046i \(-0.392053\pi\)
0.332662 + 0.943046i \(0.392053\pi\)
\(542\) −8997.00 −0.713015
\(543\) 0 0
\(544\) −3240.00 −0.255356
\(545\) 12930.0 1.01626
\(546\) 0 0
\(547\) 17120.0 1.33821 0.669103 0.743170i \(-0.266679\pi\)
0.669103 + 0.743170i \(0.266679\pi\)
\(548\) 2778.00 0.216552
\(549\) 0 0
\(550\) −4500.00 −0.348874
\(551\) −60.0000 −0.00463899
\(552\) 0 0
\(553\) −12200.0 −0.938150
\(554\) 23160.0 1.77613
\(555\) 0 0
\(556\) −1924.00 −0.146755
\(557\) −10575.0 −0.804447 −0.402224 0.915541i \(-0.631763\pi\)
−0.402224 + 0.915541i \(0.631763\pi\)
\(558\) 0 0
\(559\) 2200.00 0.166458
\(560\) −26625.0 −2.00913
\(561\) 0 0
\(562\) −22320.0 −1.67529
\(563\) −10455.0 −0.782639 −0.391319 0.920255i \(-0.627981\pi\)
−0.391319 + 0.920255i \(0.627981\pi\)
\(564\) 0 0
\(565\) 10350.0 0.770669
\(566\) −2490.00 −0.184916
\(567\) 0 0
\(568\) 7560.00 0.558469
\(569\) 24540.0 1.80803 0.904016 0.427498i \(-0.140605\pi\)
0.904016 + 0.427498i \(0.140605\pi\)
\(570\) 0 0
\(571\) 24644.0 1.80616 0.903082 0.429469i \(-0.141299\pi\)
0.903082 + 0.429469i \(0.141299\pi\)
\(572\) 300.000 0.0219294
\(573\) 0 0
\(574\) 2250.00 0.163612
\(575\) −11400.0 −0.826805
\(576\) 0 0
\(577\) −9610.00 −0.693361 −0.346681 0.937983i \(-0.612691\pi\)
−0.346681 + 0.937983i \(0.612691\pi\)
\(578\) −813.000 −0.0585058
\(579\) 0 0
\(580\) 450.000 0.0322159
\(581\) 12225.0 0.872941
\(582\) 0 0
\(583\) −9315.00 −0.661729
\(584\) 16485.0 1.16807
\(585\) 0 0
\(586\) 1638.00 0.115470
\(587\) −4017.00 −0.282452 −0.141226 0.989977i \(-0.545104\pi\)
−0.141226 + 0.989977i \(0.545104\pi\)
\(588\) 0 0
\(589\) 202.000 0.0141312
\(590\) 29700.0 2.07242
\(591\) 0 0
\(592\) 30530.0 2.11955
\(593\) −594.000 −0.0411343 −0.0205672 0.999788i \(-0.506547\pi\)
−0.0205672 + 0.999788i \(0.506547\pi\)
\(594\) 0 0
\(595\) −27000.0 −1.86032
\(596\) −1455.00 −0.0999985
\(597\) 0 0
\(598\) 6840.00 0.467740
\(599\) −8790.00 −0.599582 −0.299791 0.954005i \(-0.596917\pi\)
−0.299791 + 0.954005i \(0.596917\pi\)
\(600\) 0 0
\(601\) 9371.00 0.636025 0.318013 0.948087i \(-0.396985\pi\)
0.318013 + 0.948087i \(0.396985\pi\)
\(602\) 8250.00 0.558546
\(603\) 0 0
\(604\) −727.000 −0.0489755
\(605\) 16590.0 1.11484
\(606\) 0 0
\(607\) −14560.0 −0.973595 −0.486798 0.873515i \(-0.661835\pi\)
−0.486798 + 0.873515i \(0.661835\pi\)
\(608\) 90.0000 0.00600326
\(609\) 0 0
\(610\) −16920.0 −1.12307
\(611\) 6600.00 0.437001
\(612\) 0 0
\(613\) −18250.0 −1.20246 −0.601232 0.799074i \(-0.705323\pi\)
−0.601232 + 0.799074i \(0.705323\pi\)
\(614\) 16680.0 1.09634
\(615\) 0 0
\(616\) −7875.00 −0.515086
\(617\) −19662.0 −1.28292 −0.641461 0.767156i \(-0.721671\pi\)
−0.641461 + 0.767156i \(0.721671\pi\)
\(618\) 0 0
\(619\) 12044.0 0.782050 0.391025 0.920380i \(-0.372121\pi\)
0.391025 + 0.920380i \(0.372121\pi\)
\(620\) −1515.00 −0.0981353
\(621\) 0 0
\(622\) −26010.0 −1.67670
\(623\) −11250.0 −0.723470
\(624\) 0 0
\(625\) −18125.0 −1.16000
\(626\) −13695.0 −0.874381
\(627\) 0 0
\(628\) 3260.00 0.207147
\(629\) 30960.0 1.96257
\(630\) 0 0
\(631\) 14879.0 0.938706 0.469353 0.883011i \(-0.344487\pi\)
0.469353 + 0.883011i \(0.344487\pi\)
\(632\) 10248.0 0.645006
\(633\) 0 0
\(634\) 12699.0 0.795492
\(635\) −27975.0 −1.74827
\(636\) 0 0
\(637\) 5640.00 0.350808
\(638\) 1350.00 0.0837727
\(639\) 0 0
\(640\) 24885.0 1.53698
\(641\) −8850.00 −0.545326 −0.272663 0.962110i \(-0.587904\pi\)
−0.272663 + 0.962110i \(0.587904\pi\)
\(642\) 0 0
\(643\) 18380.0 1.12727 0.563636 0.826023i \(-0.309402\pi\)
0.563636 + 0.826023i \(0.309402\pi\)
\(644\) 2850.00 0.174388
\(645\) 0 0
\(646\) 432.000 0.0263109
\(647\) −3888.00 −0.236249 −0.118124 0.992999i \(-0.537688\pi\)
−0.118124 + 0.992999i \(0.537688\pi\)
\(648\) 0 0
\(649\) 9900.00 0.598781
\(650\) −6000.00 −0.362061
\(651\) 0 0
\(652\) 2540.00 0.152568
\(653\) 6789.00 0.406852 0.203426 0.979090i \(-0.434792\pi\)
0.203426 + 0.979090i \(0.434792\pi\)
\(654\) 0 0
\(655\) −17325.0 −1.03350
\(656\) −2130.00 −0.126772
\(657\) 0 0
\(658\) 24750.0 1.46635
\(659\) −28335.0 −1.67492 −0.837462 0.546496i \(-0.815962\pi\)
−0.837462 + 0.546496i \(0.815962\pi\)
\(660\) 0 0
\(661\) −6082.00 −0.357886 −0.178943 0.983859i \(-0.557268\pi\)
−0.178943 + 0.983859i \(0.557268\pi\)
\(662\) −1626.00 −0.0954627
\(663\) 0 0
\(664\) −10269.0 −0.600172
\(665\) 750.000 0.0437350
\(666\) 0 0
\(667\) 3420.00 0.198535
\(668\) −3498.00 −0.202607
\(669\) 0 0
\(670\) −11250.0 −0.648695
\(671\) −5640.00 −0.324486
\(672\) 0 0
\(673\) 9965.00 0.570762 0.285381 0.958414i \(-0.407880\pi\)
0.285381 + 0.958414i \(0.407880\pi\)
\(674\) −17070.0 −0.975537
\(675\) 0 0
\(676\) −1797.00 −0.102242
\(677\) −8130.00 −0.461538 −0.230769 0.973009i \(-0.574124\pi\)
−0.230769 + 0.973009i \(0.574124\pi\)
\(678\) 0 0
\(679\) 27625.0 1.56134
\(680\) 22680.0 1.27903
\(681\) 0 0
\(682\) −4545.00 −0.255186
\(683\) −33516.0 −1.87768 −0.938839 0.344356i \(-0.888097\pi\)
−0.938839 + 0.344356i \(0.888097\pi\)
\(684\) 0 0
\(685\) −41670.0 −2.32428
\(686\) −4575.00 −0.254627
\(687\) 0 0
\(688\) −7810.00 −0.432781
\(689\) −12420.0 −0.686741
\(690\) 0 0
\(691\) −22084.0 −1.21580 −0.607898 0.794015i \(-0.707987\pi\)
−0.607898 + 0.794015i \(0.707987\pi\)
\(692\) 1149.00 0.0631191
\(693\) 0 0
\(694\) 15165.0 0.829475
\(695\) 28860.0 1.57514
\(696\) 0 0
\(697\) −2160.00 −0.117383
\(698\) −4866.00 −0.263869
\(699\) 0 0
\(700\) −2500.00 −0.134987
\(701\) 10395.0 0.560077 0.280038 0.959989i \(-0.409653\pi\)
0.280038 + 0.959989i \(0.409653\pi\)
\(702\) 0 0
\(703\) −860.000 −0.0461387
\(704\) 6495.00 0.347712
\(705\) 0 0
\(706\) 90.0000 0.00479773
\(707\) 35625.0 1.89507
\(708\) 0 0
\(709\) −4804.00 −0.254468 −0.127234 0.991873i \(-0.540610\pi\)
−0.127234 + 0.991873i \(0.540610\pi\)
\(710\) 16200.0 0.856303
\(711\) 0 0
\(712\) 9450.00 0.497407
\(713\) −11514.0 −0.604772
\(714\) 0 0
\(715\) −4500.00 −0.235371
\(716\) −315.000 −0.0164415
\(717\) 0 0
\(718\) −22410.0 −1.16481
\(719\) −10980.0 −0.569520 −0.284760 0.958599i \(-0.591914\pi\)
−0.284760 + 0.958599i \(0.591914\pi\)
\(720\) 0 0
\(721\) 26500.0 1.36881
\(722\) 20565.0 1.06004
\(723\) 0 0
\(724\) 1136.00 0.0583137
\(725\) −3000.00 −0.153679
\(726\) 0 0
\(727\) −25945.0 −1.32359 −0.661793 0.749687i \(-0.730204\pi\)
−0.661793 + 0.749687i \(0.730204\pi\)
\(728\) −10500.0 −0.534555
\(729\) 0 0
\(730\) 35325.0 1.79101
\(731\) −7920.00 −0.400727
\(732\) 0 0
\(733\) 18650.0 0.939773 0.469886 0.882727i \(-0.344295\pi\)
0.469886 + 0.882727i \(0.344295\pi\)
\(734\) 4125.00 0.207434
\(735\) 0 0
\(736\) −5130.00 −0.256922
\(737\) −3750.00 −0.187426
\(738\) 0 0
\(739\) −5128.00 −0.255259 −0.127630 0.991822i \(-0.540737\pi\)
−0.127630 + 0.991822i \(0.540737\pi\)
\(740\) 6450.00 0.320414
\(741\) 0 0
\(742\) −46575.0 −2.30434
\(743\) 32700.0 1.61460 0.807299 0.590142i \(-0.200928\pi\)
0.807299 + 0.590142i \(0.200928\pi\)
\(744\) 0 0
\(745\) 21825.0 1.07330
\(746\) 14520.0 0.712621
\(747\) 0 0
\(748\) −1080.00 −0.0527924
\(749\) 37125.0 1.81111
\(750\) 0 0
\(751\) 21161.0 1.02820 0.514098 0.857731i \(-0.328127\pi\)
0.514098 + 0.857731i \(0.328127\pi\)
\(752\) −23430.0 −1.13618
\(753\) 0 0
\(754\) 1800.00 0.0869392
\(755\) 10905.0 0.525660
\(756\) 0 0
\(757\) 7130.00 0.342331 0.171165 0.985242i \(-0.445247\pi\)
0.171165 + 0.985242i \(0.445247\pi\)
\(758\) −5676.00 −0.271981
\(759\) 0 0
\(760\) −630.000 −0.0300691
\(761\) 3360.00 0.160052 0.0800262 0.996793i \(-0.474500\pi\)
0.0800262 + 0.996793i \(0.474500\pi\)
\(762\) 0 0
\(763\) 21550.0 1.02249
\(764\) −2460.00 −0.116492
\(765\) 0 0
\(766\) −32112.0 −1.51469
\(767\) 13200.0 0.621414
\(768\) 0 0
\(769\) 33473.0 1.56966 0.784829 0.619712i \(-0.212751\pi\)
0.784829 + 0.619712i \(0.212751\pi\)
\(770\) −16875.0 −0.789783
\(771\) 0 0
\(772\) 965.000 0.0449885
\(773\) 3546.00 0.164995 0.0824973 0.996591i \(-0.473710\pi\)
0.0824973 + 0.996591i \(0.473710\pi\)
\(774\) 0 0
\(775\) 10100.0 0.468133
\(776\) −23205.0 −1.07347
\(777\) 0 0
\(778\) 23445.0 1.08039
\(779\) 60.0000 0.00275959
\(780\) 0 0
\(781\) 5400.00 0.247410
\(782\) −24624.0 −1.12603
\(783\) 0 0
\(784\) −20022.0 −0.912081
\(785\) −48900.0 −2.22333
\(786\) 0 0
\(787\) −31840.0 −1.44215 −0.721076 0.692856i \(-0.756352\pi\)
−0.721076 + 0.692856i \(0.756352\pi\)
\(788\) −2493.00 −0.112702
\(789\) 0 0
\(790\) 21960.0 0.988990
\(791\) 17250.0 0.775397
\(792\) 0 0
\(793\) −7520.00 −0.336750
\(794\) −14100.0 −0.630214
\(795\) 0 0
\(796\) −511.000 −0.0227537
\(797\) 15717.0 0.698525 0.349263 0.937025i \(-0.386432\pi\)
0.349263 + 0.937025i \(0.386432\pi\)
\(798\) 0 0
\(799\) −23760.0 −1.05203
\(800\) 4500.00 0.198874
\(801\) 0 0
\(802\) −6300.00 −0.277382
\(803\) 11775.0 0.517473
\(804\) 0 0
\(805\) −42750.0 −1.87173
\(806\) −6060.00 −0.264832
\(807\) 0 0
\(808\) −29925.0 −1.30292
\(809\) −10530.0 −0.457621 −0.228810 0.973471i \(-0.573484\pi\)
−0.228810 + 0.973471i \(0.573484\pi\)
\(810\) 0 0
\(811\) −26782.0 −1.15961 −0.579805 0.814755i \(-0.696871\pi\)
−0.579805 + 0.814755i \(0.696871\pi\)
\(812\) 750.000 0.0324136
\(813\) 0 0
\(814\) 19350.0 0.833191
\(815\) −38100.0 −1.63753
\(816\) 0 0
\(817\) 220.000 0.00942084
\(818\) 32259.0 1.37886
\(819\) 0 0
\(820\) −450.000 −0.0191642
\(821\) −10110.0 −0.429770 −0.214885 0.976639i \(-0.568938\pi\)
−0.214885 + 0.976639i \(0.568938\pi\)
\(822\) 0 0
\(823\) −12535.0 −0.530914 −0.265457 0.964123i \(-0.585523\pi\)
−0.265457 + 0.964123i \(0.585523\pi\)
\(824\) −22260.0 −0.941097
\(825\) 0 0
\(826\) 49500.0 2.08514
\(827\) −9792.00 −0.411731 −0.205865 0.978580i \(-0.566001\pi\)
−0.205865 + 0.978580i \(0.566001\pi\)
\(828\) 0 0
\(829\) −4534.00 −0.189955 −0.0949773 0.995479i \(-0.530278\pi\)
−0.0949773 + 0.995479i \(0.530278\pi\)
\(830\) −22005.0 −0.920247
\(831\) 0 0
\(832\) 8660.00 0.360855
\(833\) −20304.0 −0.844528
\(834\) 0 0
\(835\) 52470.0 2.17461
\(836\) 30.0000 0.00124111
\(837\) 0 0
\(838\) 8820.00 0.363582
\(839\) 8880.00 0.365401 0.182701 0.983169i \(-0.441516\pi\)
0.182701 + 0.983169i \(0.441516\pi\)
\(840\) 0 0
\(841\) −23489.0 −0.963098
\(842\) −26088.0 −1.06776
\(843\) 0 0
\(844\) −2086.00 −0.0850747
\(845\) 26955.0 1.09737
\(846\) 0 0
\(847\) 27650.0 1.12168
\(848\) 44091.0 1.78548
\(849\) 0 0
\(850\) 21600.0 0.871616
\(851\) 49020.0 1.97460
\(852\) 0 0
\(853\) 2270.00 0.0911176 0.0455588 0.998962i \(-0.485493\pi\)
0.0455588 + 0.998962i \(0.485493\pi\)
\(854\) −28200.0 −1.12996
\(855\) 0 0
\(856\) −31185.0 −1.24519
\(857\) 19608.0 0.781560 0.390780 0.920484i \(-0.372205\pi\)
0.390780 + 0.920484i \(0.372205\pi\)
\(858\) 0 0
\(859\) −952.000 −0.0378135 −0.0189068 0.999821i \(-0.506019\pi\)
−0.0189068 + 0.999821i \(0.506019\pi\)
\(860\) −1650.00 −0.0654239
\(861\) 0 0
\(862\) 25110.0 0.992169
\(863\) 17604.0 0.694377 0.347188 0.937795i \(-0.387136\pi\)
0.347188 + 0.937795i \(0.387136\pi\)
\(864\) 0 0
\(865\) −17235.0 −0.677465
\(866\) 15465.0 0.606838
\(867\) 0 0
\(868\) −2525.00 −0.0987374
\(869\) 7320.00 0.285747
\(870\) 0 0
\(871\) −5000.00 −0.194510
\(872\) −18102.0 −0.702994
\(873\) 0 0
\(874\) 684.000 0.0264721
\(875\) −9375.00 −0.362209
\(876\) 0 0
\(877\) 21890.0 0.842842 0.421421 0.906865i \(-0.361531\pi\)
0.421421 + 0.906865i \(0.361531\pi\)
\(878\) 32961.0 1.26695
\(879\) 0 0
\(880\) 15975.0 0.611951
\(881\) −23940.0 −0.915504 −0.457752 0.889080i \(-0.651345\pi\)
−0.457752 + 0.889080i \(0.651345\pi\)
\(882\) 0 0
\(883\) −34990.0 −1.33353 −0.666765 0.745268i \(-0.732322\pi\)
−0.666765 + 0.745268i \(0.732322\pi\)
\(884\) −1440.00 −0.0547878
\(885\) 0 0
\(886\) 5868.00 0.222505
\(887\) 22188.0 0.839910 0.419955 0.907545i \(-0.362046\pi\)
0.419955 + 0.907545i \(0.362046\pi\)
\(888\) 0 0
\(889\) −46625.0 −1.75900
\(890\) 20250.0 0.762676
\(891\) 0 0
\(892\) 5240.00 0.196691
\(893\) 660.000 0.0247324
\(894\) 0 0
\(895\) 4725.00 0.176469
\(896\) 41475.0 1.54641
\(897\) 0 0
\(898\) −26190.0 −0.973242
\(899\) −3030.00 −0.112410
\(900\) 0 0
\(901\) 44712.0 1.65324
\(902\) −1350.00 −0.0498338
\(903\) 0 0
\(904\) −14490.0 −0.533109
\(905\) −17040.0 −0.625888
\(906\) 0 0
\(907\) 37370.0 1.36808 0.684041 0.729444i \(-0.260221\pi\)
0.684041 + 0.729444i \(0.260221\pi\)
\(908\) −2388.00 −0.0872782
\(909\) 0 0
\(910\) −22500.0 −0.819635
\(911\) −40710.0 −1.48055 −0.740276 0.672303i \(-0.765305\pi\)
−0.740276 + 0.672303i \(0.765305\pi\)
\(912\) 0 0
\(913\) −7335.00 −0.265885
\(914\) 25995.0 0.940742
\(915\) 0 0
\(916\) 182.000 0.00656490
\(917\) −28875.0 −1.03984
\(918\) 0 0
\(919\) 20981.0 0.753100 0.376550 0.926396i \(-0.377110\pi\)
0.376550 + 0.926396i \(0.377110\pi\)
\(920\) 35910.0 1.28687
\(921\) 0 0
\(922\) −29475.0 −1.05283
\(923\) 7200.00 0.256762
\(924\) 0 0
\(925\) −43000.0 −1.52847
\(926\) 15735.0 0.558406
\(927\) 0 0
\(928\) −1350.00 −0.0477542
\(929\) −20100.0 −0.709860 −0.354930 0.934893i \(-0.615495\pi\)
−0.354930 + 0.934893i \(0.615495\pi\)
\(930\) 0 0
\(931\) 564.000 0.0198543
\(932\) −450.000 −0.0158157
\(933\) 0 0
\(934\) −33021.0 −1.15683
\(935\) 16200.0 0.566627
\(936\) 0 0
\(937\) 15635.0 0.545115 0.272558 0.962139i \(-0.412130\pi\)
0.272558 + 0.962139i \(0.412130\pi\)
\(938\) −18750.0 −0.652675
\(939\) 0 0
\(940\) −4950.00 −0.171757
\(941\) −23955.0 −0.829873 −0.414937 0.909850i \(-0.636196\pi\)
−0.414937 + 0.909850i \(0.636196\pi\)
\(942\) 0 0
\(943\) −3420.00 −0.118102
\(944\) −46860.0 −1.61564
\(945\) 0 0
\(946\) −4950.00 −0.170125
\(947\) 36393.0 1.24880 0.624400 0.781105i \(-0.285344\pi\)
0.624400 + 0.781105i \(0.285344\pi\)
\(948\) 0 0
\(949\) 15700.0 0.537032
\(950\) −600.000 −0.0204911
\(951\) 0 0
\(952\) 37800.0 1.28688
\(953\) −43020.0 −1.46228 −0.731141 0.682227i \(-0.761012\pi\)
−0.731141 + 0.682227i \(0.761012\pi\)
\(954\) 0 0
\(955\) 36900.0 1.25032
\(956\) −5190.00 −0.175582
\(957\) 0 0
\(958\) 50850.0 1.71492
\(959\) −69450.0 −2.33854
\(960\) 0 0
\(961\) −19590.0 −0.657581
\(962\) 25800.0 0.864683
\(963\) 0 0
\(964\) −2266.00 −0.0757084
\(965\) −14475.0 −0.482867
\(966\) 0 0
\(967\) −43585.0 −1.44943 −0.724715 0.689049i \(-0.758029\pi\)
−0.724715 + 0.689049i \(0.758029\pi\)
\(968\) −23226.0 −0.771190
\(969\) 0 0
\(970\) −49725.0 −1.64595
\(971\) 43335.0 1.43222 0.716110 0.697987i \(-0.245921\pi\)
0.716110 + 0.697987i \(0.245921\pi\)
\(972\) 0 0
\(973\) 48100.0 1.58480
\(974\) −31920.0 −1.05008
\(975\) 0 0
\(976\) 26696.0 0.875531
\(977\) −30390.0 −0.995151 −0.497575 0.867421i \(-0.665776\pi\)
−0.497575 + 0.867421i \(0.665776\pi\)
\(978\) 0 0
\(979\) 6750.00 0.220358
\(980\) −4230.00 −0.137880
\(981\) 0 0
\(982\) 4905.00 0.159394
\(983\) 59226.0 1.92168 0.960842 0.277096i \(-0.0893719\pi\)
0.960842 + 0.277096i \(0.0893719\pi\)
\(984\) 0 0
\(985\) 37395.0 1.20965
\(986\) −6480.00 −0.209295
\(987\) 0 0
\(988\) 40.0000 0.00128803
\(989\) −12540.0 −0.403184
\(990\) 0 0
\(991\) 8399.00 0.269226 0.134613 0.990898i \(-0.457021\pi\)
0.134613 + 0.990898i \(0.457021\pi\)
\(992\) 4545.00 0.145468
\(993\) 0 0
\(994\) 27000.0 0.861557
\(995\) 7665.00 0.244218
\(996\) 0 0
\(997\) 13340.0 0.423753 0.211877 0.977296i \(-0.432042\pi\)
0.211877 + 0.977296i \(0.432042\pi\)
\(998\) 47406.0 1.50362
\(999\) 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 27.4.a.a.1.1 1
3.2 odd 2 27.4.a.b.1.1 yes 1
4.3 odd 2 432.4.a.a.1.1 1
5.2 odd 4 675.4.b.b.649.1 2
5.3 odd 4 675.4.b.b.649.2 2
5.4 even 2 675.4.a.j.1.1 1
7.6 odd 2 1323.4.a.d.1.1 1
8.3 odd 2 1728.4.a.bd.1.1 1
8.5 even 2 1728.4.a.bc.1.1 1
9.2 odd 6 81.4.c.a.28.1 2
9.4 even 3 81.4.c.c.55.1 2
9.5 odd 6 81.4.c.a.55.1 2
9.7 even 3 81.4.c.c.28.1 2
12.11 even 2 432.4.a.n.1.1 1
15.2 even 4 675.4.b.a.649.2 2
15.8 even 4 675.4.b.a.649.1 2
15.14 odd 2 675.4.a.a.1.1 1
21.20 even 2 1323.4.a.k.1.1 1
24.5 odd 2 1728.4.a.c.1.1 1
24.11 even 2 1728.4.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.4.a.a.1.1 1 1.1 even 1 trivial
27.4.a.b.1.1 yes 1 3.2 odd 2
81.4.c.a.28.1 2 9.2 odd 6
81.4.c.a.55.1 2 9.5 odd 6
81.4.c.c.28.1 2 9.7 even 3
81.4.c.c.55.1 2 9.4 even 3
432.4.a.a.1.1 1 4.3 odd 2
432.4.a.n.1.1 1 12.11 even 2
675.4.a.a.1.1 1 15.14 odd 2
675.4.a.j.1.1 1 5.4 even 2
675.4.b.a.649.1 2 15.8 even 4
675.4.b.a.649.2 2 15.2 even 4
675.4.b.b.649.1 2 5.2 odd 4
675.4.b.b.649.2 2 5.3 odd 4
1323.4.a.d.1.1 1 7.6 odd 2
1323.4.a.k.1.1 1 21.20 even 2
1728.4.a.c.1.1 1 24.5 odd 2
1728.4.a.d.1.1 1 24.11 even 2
1728.4.a.bc.1.1 1 8.5 even 2
1728.4.a.bd.1.1 1 8.3 odd 2