Properties

Label 5.4.a.a.1.1
Level $5$
Weight $4$
Character 5.1
Self dual yes
Analytic conductor $0.295$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

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

Newform invariants

Self dual: yes
Analytic conductor: \(0.295009550029\)
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
Character \(\chi\) \(=\) 5.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} -5.00000 q^{5} -8.00000 q^{6} +6.00000 q^{7} -23.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} -5.00000 q^{5} -8.00000 q^{6} +6.00000 q^{7} -23.0000 q^{9} +20.0000 q^{10} +32.0000 q^{11} +16.0000 q^{12} -38.0000 q^{13} -24.0000 q^{14} -10.0000 q^{15} -64.0000 q^{16} +26.0000 q^{17} +92.0000 q^{18} +100.000 q^{19} -40.0000 q^{20} +12.0000 q^{21} -128.000 q^{22} -78.0000 q^{23} +25.0000 q^{25} +152.000 q^{26} -100.000 q^{27} +48.0000 q^{28} -50.0000 q^{29} +40.0000 q^{30} -108.000 q^{31} +256.000 q^{32} +64.0000 q^{33} -104.000 q^{34} -30.0000 q^{35} -184.000 q^{36} +266.000 q^{37} -400.000 q^{38} -76.0000 q^{39} +22.0000 q^{41} -48.0000 q^{42} +442.000 q^{43} +256.000 q^{44} +115.000 q^{45} +312.000 q^{46} -514.000 q^{47} -128.000 q^{48} -307.000 q^{49} -100.000 q^{50} +52.0000 q^{51} -304.000 q^{52} +2.00000 q^{53} +400.000 q^{54} -160.000 q^{55} +200.000 q^{57} +200.000 q^{58} +500.000 q^{59} -80.0000 q^{60} -518.000 q^{61} +432.000 q^{62} -138.000 q^{63} -512.000 q^{64} +190.000 q^{65} -256.000 q^{66} +126.000 q^{67} +208.000 q^{68} -156.000 q^{69} +120.000 q^{70} +412.000 q^{71} -878.000 q^{73} -1064.00 q^{74} +50.0000 q^{75} +800.000 q^{76} +192.000 q^{77} +304.000 q^{78} +600.000 q^{79} +320.000 q^{80} +421.000 q^{81} -88.0000 q^{82} +282.000 q^{83} +96.0000 q^{84} -130.000 q^{85} -1768.00 q^{86} -100.000 q^{87} -150.000 q^{89} -460.000 q^{90} -228.000 q^{91} -624.000 q^{92} -216.000 q^{93} +2056.00 q^{94} -500.000 q^{95} +512.000 q^{96} +386.000 q^{97} +1228.00 q^{98} -736.000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 2.00000 0.384900 0.192450 0.981307i \(-0.438357\pi\)
0.192450 + 0.981307i \(0.438357\pi\)
\(4\) 8.00000 1.00000
\(5\) −5.00000 −0.447214
\(6\) −8.00000 −0.544331
\(7\) 6.00000 0.323970 0.161985 0.986793i \(-0.448210\pi\)
0.161985 + 0.986793i \(0.448210\pi\)
\(8\) 0 0
\(9\) −23.0000 −0.851852
\(10\) 20.0000 0.632456
\(11\) 32.0000 0.877124 0.438562 0.898701i \(-0.355488\pi\)
0.438562 + 0.898701i \(0.355488\pi\)
\(12\) 16.0000 0.384900
\(13\) −38.0000 −0.810716 −0.405358 0.914158i \(-0.632853\pi\)
−0.405358 + 0.914158i \(0.632853\pi\)
\(14\) −24.0000 −0.458162
\(15\) −10.0000 −0.172133
\(16\) −64.0000 −1.00000
\(17\) 26.0000 0.370937 0.185468 0.982650i \(-0.440620\pi\)
0.185468 + 0.982650i \(0.440620\pi\)
\(18\) 92.0000 1.20470
\(19\) 100.000 1.20745 0.603726 0.797192i \(-0.293682\pi\)
0.603726 + 0.797192i \(0.293682\pi\)
\(20\) −40.0000 −0.447214
\(21\) 12.0000 0.124696
\(22\) −128.000 −1.24044
\(23\) −78.0000 −0.707136 −0.353568 0.935409i \(-0.615032\pi\)
−0.353568 + 0.935409i \(0.615032\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 152.000 1.14653
\(27\) −100.000 −0.712778
\(28\) 48.0000 0.323970
\(29\) −50.0000 −0.320164 −0.160082 0.987104i \(-0.551176\pi\)
−0.160082 + 0.987104i \(0.551176\pi\)
\(30\) 40.0000 0.243432
\(31\) −108.000 −0.625722 −0.312861 0.949799i \(-0.601287\pi\)
−0.312861 + 0.949799i \(0.601287\pi\)
\(32\) 256.000 1.41421
\(33\) 64.0000 0.337605
\(34\) −104.000 −0.524584
\(35\) −30.0000 −0.144884
\(36\) −184.000 −0.851852
\(37\) 266.000 1.18190 0.590948 0.806710i \(-0.298754\pi\)
0.590948 + 0.806710i \(0.298754\pi\)
\(38\) −400.000 −1.70759
\(39\) −76.0000 −0.312045
\(40\) 0 0
\(41\) 22.0000 0.0838006 0.0419003 0.999122i \(-0.486659\pi\)
0.0419003 + 0.999122i \(0.486659\pi\)
\(42\) −48.0000 −0.176347
\(43\) 442.000 1.56754 0.783772 0.621049i \(-0.213293\pi\)
0.783772 + 0.621049i \(0.213293\pi\)
\(44\) 256.000 0.877124
\(45\) 115.000 0.380960
\(46\) 312.000 1.00004
\(47\) −514.000 −1.59520 −0.797602 0.603184i \(-0.793899\pi\)
−0.797602 + 0.603184i \(0.793899\pi\)
\(48\) −128.000 −0.384900
\(49\) −307.000 −0.895044
\(50\) −100.000 −0.282843
\(51\) 52.0000 0.142774
\(52\) −304.000 −0.810716
\(53\) 2.00000 0.00518342 0.00259171 0.999997i \(-0.499175\pi\)
0.00259171 + 0.999997i \(0.499175\pi\)
\(54\) 400.000 1.00802
\(55\) −160.000 −0.392262
\(56\) 0 0
\(57\) 200.000 0.464748
\(58\) 200.000 0.452781
\(59\) 500.000 1.10330 0.551648 0.834077i \(-0.313999\pi\)
0.551648 + 0.834077i \(0.313999\pi\)
\(60\) −80.0000 −0.172133
\(61\) −518.000 −1.08726 −0.543632 0.839324i \(-0.682951\pi\)
−0.543632 + 0.839324i \(0.682951\pi\)
\(62\) 432.000 0.884904
\(63\) −138.000 −0.275974
\(64\) −512.000 −1.00000
\(65\) 190.000 0.362563
\(66\) −256.000 −0.477446
\(67\) 126.000 0.229751 0.114876 0.993380i \(-0.463353\pi\)
0.114876 + 0.993380i \(0.463353\pi\)
\(68\) 208.000 0.370937
\(69\) −156.000 −0.272177
\(70\) 120.000 0.204896
\(71\) 412.000 0.688668 0.344334 0.938847i \(-0.388105\pi\)
0.344334 + 0.938847i \(0.388105\pi\)
\(72\) 0 0
\(73\) −878.000 −1.40770 −0.703850 0.710348i \(-0.748537\pi\)
−0.703850 + 0.710348i \(0.748537\pi\)
\(74\) −1064.00 −1.67145
\(75\) 50.0000 0.0769800
\(76\) 800.000 1.20745
\(77\) 192.000 0.284161
\(78\) 304.000 0.441298
\(79\) 600.000 0.854497 0.427249 0.904134i \(-0.359483\pi\)
0.427249 + 0.904134i \(0.359483\pi\)
\(80\) 320.000 0.447214
\(81\) 421.000 0.577503
\(82\) −88.0000 −0.118512
\(83\) 282.000 0.372934 0.186467 0.982461i \(-0.440296\pi\)
0.186467 + 0.982461i \(0.440296\pi\)
\(84\) 96.0000 0.124696
\(85\) −130.000 −0.165888
\(86\) −1768.00 −2.21684
\(87\) −100.000 −0.123231
\(88\) 0 0
\(89\) −150.000 −0.178651 −0.0893257 0.996002i \(-0.528471\pi\)
−0.0893257 + 0.996002i \(0.528471\pi\)
\(90\) −460.000 −0.538758
\(91\) −228.000 −0.262647
\(92\) −624.000 −0.707136
\(93\) −216.000 −0.240840
\(94\) 2056.00 2.25596
\(95\) −500.000 −0.539989
\(96\) 512.000 0.544331
\(97\) 386.000 0.404045 0.202022 0.979381i \(-0.435249\pi\)
0.202022 + 0.979381i \(0.435249\pi\)
\(98\) 1228.00 1.26578
\(99\) −736.000 −0.747180
\(100\) 200.000 0.200000
\(101\) 702.000 0.691600 0.345800 0.938308i \(-0.387608\pi\)
0.345800 + 0.938308i \(0.387608\pi\)
\(102\) −208.000 −0.201912
\(103\) −598.000 −0.572065 −0.286032 0.958220i \(-0.592337\pi\)
−0.286032 + 0.958220i \(0.592337\pi\)
\(104\) 0 0
\(105\) −60.0000 −0.0557657
\(106\) −8.00000 −0.00733046
\(107\) −1194.00 −1.07877 −0.539385 0.842059i \(-0.681343\pi\)
−0.539385 + 0.842059i \(0.681343\pi\)
\(108\) −800.000 −0.712778
\(109\) −550.000 −0.483307 −0.241653 0.970363i \(-0.577690\pi\)
−0.241653 + 0.970363i \(0.577690\pi\)
\(110\) 640.000 0.554742
\(111\) 532.000 0.454912
\(112\) −384.000 −0.323970
\(113\) 1562.00 1.30036 0.650180 0.759781i \(-0.274694\pi\)
0.650180 + 0.759781i \(0.274694\pi\)
\(114\) −800.000 −0.657253
\(115\) 390.000 0.316241
\(116\) −400.000 −0.320164
\(117\) 874.000 0.690610
\(118\) −2000.00 −1.56030
\(119\) 156.000 0.120172
\(120\) 0 0
\(121\) −307.000 −0.230654
\(122\) 2072.00 1.53762
\(123\) 44.0000 0.0322548
\(124\) −864.000 −0.625722
\(125\) −125.000 −0.0894427
\(126\) 552.000 0.390286
\(127\) 1846.00 1.28981 0.644906 0.764262i \(-0.276897\pi\)
0.644906 + 0.764262i \(0.276897\pi\)
\(128\) 0 0
\(129\) 884.000 0.603348
\(130\) −760.000 −0.512742
\(131\) −2208.00 −1.47262 −0.736312 0.676642i \(-0.763435\pi\)
−0.736312 + 0.676642i \(0.763435\pi\)
\(132\) 512.000 0.337605
\(133\) 600.000 0.391177
\(134\) −504.000 −0.324918
\(135\) 500.000 0.318764
\(136\) 0 0
\(137\) −2334.00 −1.45553 −0.727763 0.685829i \(-0.759440\pi\)
−0.727763 + 0.685829i \(0.759440\pi\)
\(138\) 624.000 0.384916
\(139\) −700.000 −0.427146 −0.213573 0.976927i \(-0.568510\pi\)
−0.213573 + 0.976927i \(0.568510\pi\)
\(140\) −240.000 −0.144884
\(141\) −1028.00 −0.613994
\(142\) −1648.00 −0.973923
\(143\) −1216.00 −0.711098
\(144\) 1472.00 0.851852
\(145\) 250.000 0.143182
\(146\) 3512.00 1.99079
\(147\) −614.000 −0.344502
\(148\) 2128.00 1.18190
\(149\) 2050.00 1.12713 0.563566 0.826071i \(-0.309429\pi\)
0.563566 + 0.826071i \(0.309429\pi\)
\(150\) −200.000 −0.108866
\(151\) 1852.00 0.998103 0.499052 0.866572i \(-0.333682\pi\)
0.499052 + 0.866572i \(0.333682\pi\)
\(152\) 0 0
\(153\) −598.000 −0.315983
\(154\) −768.000 −0.401865
\(155\) 540.000 0.279831
\(156\) −608.000 −0.312045
\(157\) −2494.00 −1.26779 −0.633894 0.773420i \(-0.718545\pi\)
−0.633894 + 0.773420i \(0.718545\pi\)
\(158\) −2400.00 −1.20844
\(159\) 4.00000 0.00199510
\(160\) −1280.00 −0.632456
\(161\) −468.000 −0.229090
\(162\) −1684.00 −0.816713
\(163\) 2762.00 1.32722 0.663609 0.748080i \(-0.269024\pi\)
0.663609 + 0.748080i \(0.269024\pi\)
\(164\) 176.000 0.0838006
\(165\) −320.000 −0.150982
\(166\) −1128.00 −0.527408
\(167\) 3126.00 1.44849 0.724243 0.689545i \(-0.242189\pi\)
0.724243 + 0.689545i \(0.242189\pi\)
\(168\) 0 0
\(169\) −753.000 −0.342740
\(170\) 520.000 0.234601
\(171\) −2300.00 −1.02857
\(172\) 3536.00 1.56754
\(173\) −78.0000 −0.0342788 −0.0171394 0.999853i \(-0.505456\pi\)
−0.0171394 + 0.999853i \(0.505456\pi\)
\(174\) 400.000 0.174275
\(175\) 150.000 0.0647939
\(176\) −2048.00 −0.877124
\(177\) 1000.00 0.424659
\(178\) 600.000 0.252651
\(179\) −1300.00 −0.542830 −0.271415 0.962462i \(-0.587492\pi\)
−0.271415 + 0.962462i \(0.587492\pi\)
\(180\) 920.000 0.380960
\(181\) 1742.00 0.715369 0.357685 0.933842i \(-0.383566\pi\)
0.357685 + 0.933842i \(0.383566\pi\)
\(182\) 912.000 0.371439
\(183\) −1036.00 −0.418488
\(184\) 0 0
\(185\) −1330.00 −0.528560
\(186\) 864.000 0.340600
\(187\) 832.000 0.325358
\(188\) −4112.00 −1.59520
\(189\) −600.000 −0.230918
\(190\) 2000.00 0.763659
\(191\) 3772.00 1.42897 0.714483 0.699653i \(-0.246662\pi\)
0.714483 + 0.699653i \(0.246662\pi\)
\(192\) −1024.00 −0.384900
\(193\) −358.000 −0.133520 −0.0667601 0.997769i \(-0.521266\pi\)
−0.0667601 + 0.997769i \(0.521266\pi\)
\(194\) −1544.00 −0.571406
\(195\) 380.000 0.139551
\(196\) −2456.00 −0.895044
\(197\) −2214.00 −0.800716 −0.400358 0.916359i \(-0.631114\pi\)
−0.400358 + 0.916359i \(0.631114\pi\)
\(198\) 2944.00 1.05667
\(199\) −2600.00 −0.926176 −0.463088 0.886312i \(-0.653259\pi\)
−0.463088 + 0.886312i \(0.653259\pi\)
\(200\) 0 0
\(201\) 252.000 0.0884314
\(202\) −2808.00 −0.978070
\(203\) −300.000 −0.103724
\(204\) 416.000 0.142774
\(205\) −110.000 −0.0374767
\(206\) 2392.00 0.809022
\(207\) 1794.00 0.602375
\(208\) 2432.00 0.810716
\(209\) 3200.00 1.05908
\(210\) 240.000 0.0788646
\(211\) −1168.00 −0.381083 −0.190541 0.981679i \(-0.561024\pi\)
−0.190541 + 0.981679i \(0.561024\pi\)
\(212\) 16.0000 0.00518342
\(213\) 824.000 0.265068
\(214\) 4776.00 1.52561
\(215\) −2210.00 −0.701027
\(216\) 0 0
\(217\) −648.000 −0.202715
\(218\) 2200.00 0.683499
\(219\) −1756.00 −0.541824
\(220\) −1280.00 −0.392262
\(221\) −988.000 −0.300724
\(222\) −2128.00 −0.643342
\(223\) −6478.00 −1.94529 −0.972643 0.232303i \(-0.925374\pi\)
−0.972643 + 0.232303i \(0.925374\pi\)
\(224\) 1536.00 0.458162
\(225\) −575.000 −0.170370
\(226\) −6248.00 −1.83899
\(227\) 646.000 0.188883 0.0944417 0.995530i \(-0.469893\pi\)
0.0944417 + 0.995530i \(0.469893\pi\)
\(228\) 1600.00 0.464748
\(229\) 3750.00 1.08213 0.541063 0.840982i \(-0.318022\pi\)
0.541063 + 0.840982i \(0.318022\pi\)
\(230\) −1560.00 −0.447232
\(231\) 384.000 0.109374
\(232\) 0 0
\(233\) 1482.00 0.416691 0.208346 0.978055i \(-0.433192\pi\)
0.208346 + 0.978055i \(0.433192\pi\)
\(234\) −3496.00 −0.976670
\(235\) 2570.00 0.713397
\(236\) 4000.00 1.10330
\(237\) 1200.00 0.328896
\(238\) −624.000 −0.169949
\(239\) 1400.00 0.378906 0.189453 0.981890i \(-0.439329\pi\)
0.189453 + 0.981890i \(0.439329\pi\)
\(240\) 640.000 0.172133
\(241\) 3022.00 0.807735 0.403867 0.914817i \(-0.367666\pi\)
0.403867 + 0.914817i \(0.367666\pi\)
\(242\) 1228.00 0.326194
\(243\) 3542.00 0.935059
\(244\) −4144.00 −1.08726
\(245\) 1535.00 0.400276
\(246\) −176.000 −0.0456152
\(247\) −3800.00 −0.978900
\(248\) 0 0
\(249\) 564.000 0.143542
\(250\) 500.000 0.126491
\(251\) −1248.00 −0.313837 −0.156918 0.987612i \(-0.550156\pi\)
−0.156918 + 0.987612i \(0.550156\pi\)
\(252\) −1104.00 −0.275974
\(253\) −2496.00 −0.620246
\(254\) −7384.00 −1.82407
\(255\) −260.000 −0.0638503
\(256\) 4096.00 1.00000
\(257\) 2106.00 0.511162 0.255581 0.966788i \(-0.417733\pi\)
0.255581 + 0.966788i \(0.417733\pi\)
\(258\) −3536.00 −0.853263
\(259\) 1596.00 0.382898
\(260\) 1520.00 0.362563
\(261\) 1150.00 0.272733
\(262\) 8832.00 2.08261
\(263\) −3638.00 −0.852961 −0.426480 0.904497i \(-0.640247\pi\)
−0.426480 + 0.904497i \(0.640247\pi\)
\(264\) 0 0
\(265\) −10.0000 −0.00231809
\(266\) −2400.00 −0.553208
\(267\) −300.000 −0.0687629
\(268\) 1008.00 0.229751
\(269\) −6550.00 −1.48461 −0.742306 0.670061i \(-0.766268\pi\)
−0.742306 + 0.670061i \(0.766268\pi\)
\(270\) −2000.00 −0.450800
\(271\) −4388.00 −0.983587 −0.491793 0.870712i \(-0.663658\pi\)
−0.491793 + 0.870712i \(0.663658\pi\)
\(272\) −1664.00 −0.370937
\(273\) −456.000 −0.101093
\(274\) 9336.00 2.05842
\(275\) 800.000 0.175425
\(276\) −1248.00 −0.272177
\(277\) 546.000 0.118433 0.0592165 0.998245i \(-0.481140\pi\)
0.0592165 + 0.998245i \(0.481140\pi\)
\(278\) 2800.00 0.604075
\(279\) 2484.00 0.533022
\(280\) 0 0
\(281\) −6858.00 −1.45592 −0.727961 0.685619i \(-0.759532\pi\)
−0.727961 + 0.685619i \(0.759532\pi\)
\(282\) 4112.00 0.868319
\(283\) 9282.00 1.94967 0.974837 0.222920i \(-0.0715588\pi\)
0.974837 + 0.222920i \(0.0715588\pi\)
\(284\) 3296.00 0.688668
\(285\) −1000.00 −0.207842
\(286\) 4864.00 1.00564
\(287\) 132.000 0.0271488
\(288\) −5888.00 −1.20470
\(289\) −4237.00 −0.862406
\(290\) −1000.00 −0.202490
\(291\) 772.000 0.155517
\(292\) −7024.00 −1.40770
\(293\) 4842.00 0.965436 0.482718 0.875776i \(-0.339650\pi\)
0.482718 + 0.875776i \(0.339650\pi\)
\(294\) 2456.00 0.487200
\(295\) −2500.00 −0.493409
\(296\) 0 0
\(297\) −3200.00 −0.625195
\(298\) −8200.00 −1.59400
\(299\) 2964.00 0.573286
\(300\) 400.000 0.0769800
\(301\) 2652.00 0.507836
\(302\) −7408.00 −1.41153
\(303\) 1404.00 0.266197
\(304\) −6400.00 −1.20745
\(305\) 2590.00 0.486239
\(306\) 2392.00 0.446868
\(307\) −2594.00 −0.482239 −0.241120 0.970495i \(-0.577515\pi\)
−0.241120 + 0.970495i \(0.577515\pi\)
\(308\) 1536.00 0.284161
\(309\) −1196.00 −0.220188
\(310\) −2160.00 −0.395741
\(311\) 7332.00 1.33685 0.668424 0.743781i \(-0.266969\pi\)
0.668424 + 0.743781i \(0.266969\pi\)
\(312\) 0 0
\(313\) 1562.00 0.282075 0.141037 0.990004i \(-0.454956\pi\)
0.141037 + 0.990004i \(0.454956\pi\)
\(314\) 9976.00 1.79292
\(315\) 690.000 0.123419
\(316\) 4800.00 0.854497
\(317\) 1426.00 0.252657 0.126328 0.991988i \(-0.459681\pi\)
0.126328 + 0.991988i \(0.459681\pi\)
\(318\) −16.0000 −0.00282150
\(319\) −1600.00 −0.280824
\(320\) 2560.00 0.447214
\(321\) −2388.00 −0.415219
\(322\) 1872.00 0.323983
\(323\) 2600.00 0.447888
\(324\) 3368.00 0.577503
\(325\) −950.000 −0.162143
\(326\) −11048.0 −1.87697
\(327\) −1100.00 −0.186025
\(328\) 0 0
\(329\) −3084.00 −0.516798
\(330\) 1280.00 0.213520
\(331\) −4008.00 −0.665558 −0.332779 0.943005i \(-0.607986\pi\)
−0.332779 + 0.943005i \(0.607986\pi\)
\(332\) 2256.00 0.372934
\(333\) −6118.00 −1.00680
\(334\) −12504.0 −2.04847
\(335\) −630.000 −0.102748
\(336\) −768.000 −0.124696
\(337\) 8866.00 1.43312 0.716561 0.697525i \(-0.245715\pi\)
0.716561 + 0.697525i \(0.245715\pi\)
\(338\) 3012.00 0.484708
\(339\) 3124.00 0.500509
\(340\) −1040.00 −0.165888
\(341\) −3456.00 −0.548835
\(342\) 9200.00 1.45462
\(343\) −3900.00 −0.613936
\(344\) 0 0
\(345\) 780.000 0.121721
\(346\) 312.000 0.0484775
\(347\) −1714.00 −0.265165 −0.132583 0.991172i \(-0.542327\pi\)
−0.132583 + 0.991172i \(0.542327\pi\)
\(348\) −800.000 −0.123231
\(349\) 1150.00 0.176384 0.0881921 0.996103i \(-0.471891\pi\)
0.0881921 + 0.996103i \(0.471891\pi\)
\(350\) −600.000 −0.0916324
\(351\) 3800.00 0.577860
\(352\) 8192.00 1.24044
\(353\) −4398.00 −0.663122 −0.331561 0.943434i \(-0.607575\pi\)
−0.331561 + 0.943434i \(0.607575\pi\)
\(354\) −4000.00 −0.600558
\(355\) −2060.00 −0.307982
\(356\) −1200.00 −0.178651
\(357\) 312.000 0.0462543
\(358\) 5200.00 0.767677
\(359\) 1800.00 0.264625 0.132312 0.991208i \(-0.457760\pi\)
0.132312 + 0.991208i \(0.457760\pi\)
\(360\) 0 0
\(361\) 3141.00 0.457938
\(362\) −6968.00 −1.01168
\(363\) −614.000 −0.0887786
\(364\) −1824.00 −0.262647
\(365\) 4390.00 0.629543
\(366\) 4144.00 0.591832
\(367\) −5874.00 −0.835478 −0.417739 0.908567i \(-0.637177\pi\)
−0.417739 + 0.908567i \(0.637177\pi\)
\(368\) 4992.00 0.707136
\(369\) −506.000 −0.0713857
\(370\) 5320.00 0.747496
\(371\) 12.0000 0.00167927
\(372\) −1728.00 −0.240840
\(373\) −2078.00 −0.288458 −0.144229 0.989544i \(-0.546070\pi\)
−0.144229 + 0.989544i \(0.546070\pi\)
\(374\) −3328.00 −0.460125
\(375\) −250.000 −0.0344265
\(376\) 0 0
\(377\) 1900.00 0.259562
\(378\) 2400.00 0.326568
\(379\) 7900.00 1.07070 0.535351 0.844630i \(-0.320179\pi\)
0.535351 + 0.844630i \(0.320179\pi\)
\(380\) −4000.00 −0.539989
\(381\) 3692.00 0.496449
\(382\) −15088.0 −2.02086
\(383\) −7518.00 −1.00301 −0.501504 0.865155i \(-0.667220\pi\)
−0.501504 + 0.865155i \(0.667220\pi\)
\(384\) 0 0
\(385\) −960.000 −0.127081
\(386\) 1432.00 0.188826
\(387\) −10166.0 −1.33531
\(388\) 3088.00 0.404045
\(389\) −1950.00 −0.254162 −0.127081 0.991892i \(-0.540561\pi\)
−0.127081 + 0.991892i \(0.540561\pi\)
\(390\) −1520.00 −0.197354
\(391\) −2028.00 −0.262303
\(392\) 0 0
\(393\) −4416.00 −0.566814
\(394\) 8856.00 1.13238
\(395\) −3000.00 −0.382143
\(396\) −5888.00 −0.747180
\(397\) 13786.0 1.74282 0.871410 0.490555i \(-0.163206\pi\)
0.871410 + 0.490555i \(0.163206\pi\)
\(398\) 10400.0 1.30981
\(399\) 1200.00 0.150564
\(400\) −1600.00 −0.200000
\(401\) 6402.00 0.797258 0.398629 0.917112i \(-0.369486\pi\)
0.398629 + 0.917112i \(0.369486\pi\)
\(402\) −1008.00 −0.125061
\(403\) 4104.00 0.507282
\(404\) 5616.00 0.691600
\(405\) −2105.00 −0.258267
\(406\) 1200.00 0.146687
\(407\) 8512.00 1.03667
\(408\) 0 0
\(409\) 11150.0 1.34800 0.674000 0.738731i \(-0.264575\pi\)
0.674000 + 0.738731i \(0.264575\pi\)
\(410\) 440.000 0.0530001
\(411\) −4668.00 −0.560232
\(412\) −4784.00 −0.572065
\(413\) 3000.00 0.357434
\(414\) −7176.00 −0.851887
\(415\) −1410.00 −0.166781
\(416\) −9728.00 −1.14653
\(417\) −1400.00 −0.164408
\(418\) −12800.0 −1.49777
\(419\) −13700.0 −1.59735 −0.798674 0.601764i \(-0.794465\pi\)
−0.798674 + 0.601764i \(0.794465\pi\)
\(420\) −480.000 −0.0557657
\(421\) −5438.00 −0.629529 −0.314765 0.949170i \(-0.601926\pi\)
−0.314765 + 0.949170i \(0.601926\pi\)
\(422\) 4672.00 0.538932
\(423\) 11822.0 1.35888
\(424\) 0 0
\(425\) 650.000 0.0741874
\(426\) −3296.00 −0.374863
\(427\) −3108.00 −0.352240
\(428\) −9552.00 −1.07877
\(429\) −2432.00 −0.273702
\(430\) 8840.00 0.991402
\(431\) 7692.00 0.859653 0.429827 0.902911i \(-0.358575\pi\)
0.429827 + 0.902911i \(0.358575\pi\)
\(432\) 6400.00 0.712778
\(433\) −1118.00 −0.124082 −0.0620412 0.998074i \(-0.519761\pi\)
−0.0620412 + 0.998074i \(0.519761\pi\)
\(434\) 2592.00 0.286682
\(435\) 500.000 0.0551107
\(436\) −4400.00 −0.483307
\(437\) −7800.00 −0.853832
\(438\) 7024.00 0.766255
\(439\) −2600.00 −0.282668 −0.141334 0.989962i \(-0.545139\pi\)
−0.141334 + 0.989962i \(0.545139\pi\)
\(440\) 0 0
\(441\) 7061.00 0.762445
\(442\) 3952.00 0.425288
\(443\) −11958.0 −1.28249 −0.641243 0.767337i \(-0.721581\pi\)
−0.641243 + 0.767337i \(0.721581\pi\)
\(444\) 4256.00 0.454912
\(445\) 750.000 0.0798953
\(446\) 25912.0 2.75105
\(447\) 4100.00 0.433833
\(448\) −3072.00 −0.323970
\(449\) −17050.0 −1.79207 −0.896035 0.443984i \(-0.853565\pi\)
−0.896035 + 0.443984i \(0.853565\pi\)
\(450\) 2300.00 0.240940
\(451\) 704.000 0.0735035
\(452\) 12496.0 1.30036
\(453\) 3704.00 0.384170
\(454\) −2584.00 −0.267121
\(455\) 1140.00 0.117459
\(456\) 0 0
\(457\) −9494.00 −0.971796 −0.485898 0.874016i \(-0.661507\pi\)
−0.485898 + 0.874016i \(0.661507\pi\)
\(458\) −15000.0 −1.53036
\(459\) −2600.00 −0.264396
\(460\) 3120.00 0.316241
\(461\) −11418.0 −1.15356 −0.576778 0.816901i \(-0.695690\pi\)
−0.576778 + 0.816901i \(0.695690\pi\)
\(462\) −1536.00 −0.154678
\(463\) 7962.00 0.799191 0.399596 0.916692i \(-0.369151\pi\)
0.399596 + 0.916692i \(0.369151\pi\)
\(464\) 3200.00 0.320164
\(465\) 1080.00 0.107707
\(466\) −5928.00 −0.589290
\(467\) 6526.00 0.646654 0.323327 0.946287i \(-0.395199\pi\)
0.323327 + 0.946287i \(0.395199\pi\)
\(468\) 6992.00 0.690610
\(469\) 756.000 0.0744325
\(470\) −10280.0 −1.00890
\(471\) −4988.00 −0.487972
\(472\) 0 0
\(473\) 14144.0 1.37493
\(474\) −4800.00 −0.465129
\(475\) 2500.00 0.241490
\(476\) 1248.00 0.120172
\(477\) −46.0000 −0.00441550
\(478\) −5600.00 −0.535854
\(479\) 17400.0 1.65976 0.829881 0.557940i \(-0.188408\pi\)
0.829881 + 0.557940i \(0.188408\pi\)
\(480\) −2560.00 −0.243432
\(481\) −10108.0 −0.958181
\(482\) −12088.0 −1.14231
\(483\) −936.000 −0.0881770
\(484\) −2456.00 −0.230654
\(485\) −1930.00 −0.180694
\(486\) −14168.0 −1.32237
\(487\) 1166.00 0.108494 0.0542469 0.998528i \(-0.482724\pi\)
0.0542469 + 0.998528i \(0.482724\pi\)
\(488\) 0 0
\(489\) 5524.00 0.510846
\(490\) −6140.00 −0.566075
\(491\) 7072.00 0.650010 0.325005 0.945712i \(-0.394634\pi\)
0.325005 + 0.945712i \(0.394634\pi\)
\(492\) 352.000 0.0322548
\(493\) −1300.00 −0.118761
\(494\) 15200.0 1.38437
\(495\) 3680.00 0.334149
\(496\) 6912.00 0.625722
\(497\) 2472.00 0.223107
\(498\) −2256.00 −0.203000
\(499\) 100.000 0.00897117 0.00448559 0.999990i \(-0.498572\pi\)
0.00448559 + 0.999990i \(0.498572\pi\)
\(500\) −1000.00 −0.0894427
\(501\) 6252.00 0.557522
\(502\) 4992.00 0.443832
\(503\) 2602.00 0.230651 0.115325 0.993328i \(-0.463209\pi\)
0.115325 + 0.993328i \(0.463209\pi\)
\(504\) 0 0
\(505\) −3510.00 −0.309293
\(506\) 9984.00 0.877160
\(507\) −1506.00 −0.131921
\(508\) 14768.0 1.28981
\(509\) 11150.0 0.970953 0.485476 0.874250i \(-0.338646\pi\)
0.485476 + 0.874250i \(0.338646\pi\)
\(510\) 1040.00 0.0902980
\(511\) −5268.00 −0.456052
\(512\) −16384.0 −1.41421
\(513\) −10000.0 −0.860645
\(514\) −8424.00 −0.722892
\(515\) 2990.00 0.255835
\(516\) 7072.00 0.603348
\(517\) −16448.0 −1.39919
\(518\) −6384.00 −0.541500
\(519\) −156.000 −0.0131939
\(520\) 0 0
\(521\) −3638.00 −0.305919 −0.152959 0.988232i \(-0.548880\pi\)
−0.152959 + 0.988232i \(0.548880\pi\)
\(522\) −4600.00 −0.385702
\(523\) −2078.00 −0.173737 −0.0868686 0.996220i \(-0.527686\pi\)
−0.0868686 + 0.996220i \(0.527686\pi\)
\(524\) −17664.0 −1.47262
\(525\) 300.000 0.0249392
\(526\) 14552.0 1.20627
\(527\) −2808.00 −0.232103
\(528\) −4096.00 −0.337605
\(529\) −6083.00 −0.499959
\(530\) 40.0000 0.00327828
\(531\) −11500.0 −0.939845
\(532\) 4800.00 0.391177
\(533\) −836.000 −0.0679384
\(534\) 1200.00 0.0972455
\(535\) 5970.00 0.482440
\(536\) 0 0
\(537\) −2600.00 −0.208935
\(538\) 26200.0 2.09956
\(539\) −9824.00 −0.785064
\(540\) 4000.00 0.318764
\(541\) 5622.00 0.446781 0.223391 0.974729i \(-0.428287\pi\)
0.223391 + 0.974729i \(0.428287\pi\)
\(542\) 17552.0 1.39100
\(543\) 3484.00 0.275346
\(544\) 6656.00 0.524584
\(545\) 2750.00 0.216141
\(546\) 1824.00 0.142967
\(547\) 16486.0 1.28865 0.644324 0.764753i \(-0.277139\pi\)
0.644324 + 0.764753i \(0.277139\pi\)
\(548\) −18672.0 −1.45553
\(549\) 11914.0 0.926188
\(550\) −3200.00 −0.248088
\(551\) −5000.00 −0.386583
\(552\) 0 0
\(553\) 3600.00 0.276831
\(554\) −2184.00 −0.167490
\(555\) −2660.00 −0.203443
\(556\) −5600.00 −0.427146
\(557\) 11706.0 0.890483 0.445242 0.895410i \(-0.353118\pi\)
0.445242 + 0.895410i \(0.353118\pi\)
\(558\) −9936.00 −0.753807
\(559\) −16796.0 −1.27083
\(560\) 1920.00 0.144884
\(561\) 1664.00 0.125230
\(562\) 27432.0 2.05898
\(563\) −25038.0 −1.87429 −0.937146 0.348939i \(-0.886542\pi\)
−0.937146 + 0.348939i \(0.886542\pi\)
\(564\) −8224.00 −0.613994
\(565\) −7810.00 −0.581538
\(566\) −37128.0 −2.75725
\(567\) 2526.00 0.187094
\(568\) 0 0
\(569\) 17550.0 1.29303 0.646515 0.762901i \(-0.276226\pi\)
0.646515 + 0.762901i \(0.276226\pi\)
\(570\) 4000.00 0.293933
\(571\) 10712.0 0.785084 0.392542 0.919734i \(-0.371596\pi\)
0.392542 + 0.919734i \(0.371596\pi\)
\(572\) −9728.00 −0.711098
\(573\) 7544.00 0.550009
\(574\) −528.000 −0.0383942
\(575\) −1950.00 −0.141427
\(576\) 11776.0 0.851852
\(577\) −13654.0 −0.985136 −0.492568 0.870274i \(-0.663942\pi\)
−0.492568 + 0.870274i \(0.663942\pi\)
\(578\) 16948.0 1.21963
\(579\) −716.000 −0.0513920
\(580\) 2000.00 0.143182
\(581\) 1692.00 0.120819
\(582\) −3088.00 −0.219934
\(583\) 64.0000 0.00454650
\(584\) 0 0
\(585\) −4370.00 −0.308850
\(586\) −19368.0 −1.36533
\(587\) 14166.0 0.996071 0.498035 0.867157i \(-0.334055\pi\)
0.498035 + 0.867157i \(0.334055\pi\)
\(588\) −4912.00 −0.344502
\(589\) −10800.0 −0.755528
\(590\) 10000.0 0.697786
\(591\) −4428.00 −0.308196
\(592\) −17024.0 −1.18190
\(593\) 17842.0 1.23555 0.617777 0.786354i \(-0.288034\pi\)
0.617777 + 0.786354i \(0.288034\pi\)
\(594\) 12800.0 0.884159
\(595\) −780.000 −0.0537427
\(596\) 16400.0 1.12713
\(597\) −5200.00 −0.356485
\(598\) −11856.0 −0.810749
\(599\) −17600.0 −1.20053 −0.600264 0.799802i \(-0.704938\pi\)
−0.600264 + 0.799802i \(0.704938\pi\)
\(600\) 0 0
\(601\) 27302.0 1.85303 0.926516 0.376256i \(-0.122789\pi\)
0.926516 + 0.376256i \(0.122789\pi\)
\(602\) −10608.0 −0.718189
\(603\) −2898.00 −0.195714
\(604\) 14816.0 0.998103
\(605\) 1535.00 0.103151
\(606\) −5616.00 −0.376459
\(607\) −3794.00 −0.253696 −0.126848 0.991922i \(-0.540486\pi\)
−0.126848 + 0.991922i \(0.540486\pi\)
\(608\) 25600.0 1.70759
\(609\) −600.000 −0.0399232
\(610\) −10360.0 −0.687646
\(611\) 19532.0 1.29326
\(612\) −4784.00 −0.315983
\(613\) −13238.0 −0.872231 −0.436116 0.899891i \(-0.643646\pi\)
−0.436116 + 0.899891i \(0.643646\pi\)
\(614\) 10376.0 0.681989
\(615\) −220.000 −0.0144248
\(616\) 0 0
\(617\) −11574.0 −0.755189 −0.377595 0.925971i \(-0.623249\pi\)
−0.377595 + 0.925971i \(0.623249\pi\)
\(618\) 4784.00 0.311393
\(619\) 8300.00 0.538942 0.269471 0.963008i \(-0.413151\pi\)
0.269471 + 0.963008i \(0.413151\pi\)
\(620\) 4320.00 0.279831
\(621\) 7800.00 0.504031
\(622\) −29328.0 −1.89059
\(623\) −900.000 −0.0578776
\(624\) 4864.00 0.312045
\(625\) 625.000 0.0400000
\(626\) −6248.00 −0.398914
\(627\) 6400.00 0.407642
\(628\) −19952.0 −1.26779
\(629\) 6916.00 0.438409
\(630\) −2760.00 −0.174541
\(631\) −7508.00 −0.473675 −0.236837 0.971549i \(-0.576111\pi\)
−0.236837 + 0.971549i \(0.576111\pi\)
\(632\) 0 0
\(633\) −2336.00 −0.146679
\(634\) −5704.00 −0.357310
\(635\) −9230.00 −0.576821
\(636\) 32.0000 0.00199510
\(637\) 11666.0 0.725626
\(638\) 6400.00 0.397145
\(639\) −9476.00 −0.586643
\(640\) 0 0
\(641\) −27378.0 −1.68700 −0.843499 0.537130i \(-0.819508\pi\)
−0.843499 + 0.537130i \(0.819508\pi\)
\(642\) 9552.00 0.587208
\(643\) 1842.00 0.112973 0.0564863 0.998403i \(-0.482010\pi\)
0.0564863 + 0.998403i \(0.482010\pi\)
\(644\) −3744.00 −0.229090
\(645\) −4420.00 −0.269825
\(646\) −10400.0 −0.633409
\(647\) −10114.0 −0.614563 −0.307282 0.951619i \(-0.599419\pi\)
−0.307282 + 0.951619i \(0.599419\pi\)
\(648\) 0 0
\(649\) 16000.0 0.967727
\(650\) 3800.00 0.229305
\(651\) −1296.00 −0.0780250
\(652\) 22096.0 1.32722
\(653\) 10402.0 0.623372 0.311686 0.950185i \(-0.399106\pi\)
0.311686 + 0.950185i \(0.399106\pi\)
\(654\) 4400.00 0.263079
\(655\) 11040.0 0.658578
\(656\) −1408.00 −0.0838006
\(657\) 20194.0 1.19915
\(658\) 12336.0 0.730862
\(659\) 7100.00 0.419692 0.209846 0.977734i \(-0.432704\pi\)
0.209846 + 0.977734i \(0.432704\pi\)
\(660\) −2560.00 −0.150982
\(661\) −7118.00 −0.418847 −0.209424 0.977825i \(-0.567159\pi\)
−0.209424 + 0.977825i \(0.567159\pi\)
\(662\) 16032.0 0.941241
\(663\) −1976.00 −0.115749
\(664\) 0 0
\(665\) −3000.00 −0.174940
\(666\) 24472.0 1.42383
\(667\) 3900.00 0.226400
\(668\) 25008.0 1.44849
\(669\) −12956.0 −0.748741
\(670\) 2520.00 0.145308
\(671\) −16576.0 −0.953665
\(672\) 3072.00 0.176347
\(673\) −31278.0 −1.79150 −0.895749 0.444560i \(-0.853360\pi\)
−0.895749 + 0.444560i \(0.853360\pi\)
\(674\) −35464.0 −2.02674
\(675\) −2500.00 −0.142556
\(676\) −6024.00 −0.342740
\(677\) −30054.0 −1.70616 −0.853079 0.521782i \(-0.825268\pi\)
−0.853079 + 0.521782i \(0.825268\pi\)
\(678\) −12496.0 −0.707826
\(679\) 2316.00 0.130898
\(680\) 0 0
\(681\) 1292.00 0.0727012
\(682\) 13824.0 0.776171
\(683\) −4518.00 −0.253113 −0.126557 0.991959i \(-0.540393\pi\)
−0.126557 + 0.991959i \(0.540393\pi\)
\(684\) −18400.0 −1.02857
\(685\) 11670.0 0.650931
\(686\) 15600.0 0.868237
\(687\) 7500.00 0.416511
\(688\) −28288.0 −1.56754
\(689\) −76.0000 −0.00420228
\(690\) −3120.00 −0.172140
\(691\) 29272.0 1.61152 0.805759 0.592243i \(-0.201758\pi\)
0.805759 + 0.592243i \(0.201758\pi\)
\(692\) −624.000 −0.0342788
\(693\) −4416.00 −0.242063
\(694\) 6856.00 0.375000
\(695\) 3500.00 0.191025
\(696\) 0 0
\(697\) 572.000 0.0310847
\(698\) −4600.00 −0.249445
\(699\) 2964.00 0.160385
\(700\) 1200.00 0.0647939
\(701\) −5798.00 −0.312393 −0.156196 0.987726i \(-0.549923\pi\)
−0.156196 + 0.987726i \(0.549923\pi\)
\(702\) −15200.0 −0.817218
\(703\) 26600.0 1.42708
\(704\) −16384.0 −0.877124
\(705\) 5140.00 0.274587
\(706\) 17592.0 0.937796
\(707\) 4212.00 0.224057
\(708\) 8000.00 0.424659
\(709\) 8950.00 0.474082 0.237041 0.971500i \(-0.423822\pi\)
0.237041 + 0.971500i \(0.423822\pi\)
\(710\) 8240.00 0.435552
\(711\) −13800.0 −0.727905
\(712\) 0 0
\(713\) 8424.00 0.442470
\(714\) −1248.00 −0.0654135
\(715\) 6080.00 0.318013
\(716\) −10400.0 −0.542830
\(717\) 2800.00 0.145841
\(718\) −7200.00 −0.374236
\(719\) 7800.00 0.404577 0.202289 0.979326i \(-0.435162\pi\)
0.202289 + 0.979326i \(0.435162\pi\)
\(720\) −7360.00 −0.380960
\(721\) −3588.00 −0.185332
\(722\) −12564.0 −0.647623
\(723\) 6044.00 0.310897
\(724\) 13936.0 0.715369
\(725\) −1250.00 −0.0640329
\(726\) 2456.00 0.125552
\(727\) −8554.00 −0.436383 −0.218191 0.975906i \(-0.570016\pi\)
−0.218191 + 0.975906i \(0.570016\pi\)
\(728\) 0 0
\(729\) −4283.00 −0.217599
\(730\) −17560.0 −0.890308
\(731\) 11492.0 0.581460
\(732\) −8288.00 −0.418488
\(733\) 2882.00 0.145224 0.0726119 0.997360i \(-0.476867\pi\)
0.0726119 + 0.997360i \(0.476867\pi\)
\(734\) 23496.0 1.18154
\(735\) 3070.00 0.154066
\(736\) −19968.0 −1.00004
\(737\) 4032.00 0.201521
\(738\) 2024.00 0.100955
\(739\) 18700.0 0.930840 0.465420 0.885090i \(-0.345903\pi\)
0.465420 + 0.885090i \(0.345903\pi\)
\(740\) −10640.0 −0.528560
\(741\) −7600.00 −0.376779
\(742\) −48.0000 −0.00237485
\(743\) 12242.0 0.604462 0.302231 0.953235i \(-0.402269\pi\)
0.302231 + 0.953235i \(0.402269\pi\)
\(744\) 0 0
\(745\) −10250.0 −0.504068
\(746\) 8312.00 0.407941
\(747\) −6486.00 −0.317685
\(748\) 6656.00 0.325358
\(749\) −7164.00 −0.349488
\(750\) 1000.00 0.0486864
\(751\) −31148.0 −1.51346 −0.756729 0.653729i \(-0.773204\pi\)
−0.756729 + 0.653729i \(0.773204\pi\)
\(752\) 32896.0 1.59520
\(753\) −2496.00 −0.120796
\(754\) −7600.00 −0.367076
\(755\) −9260.00 −0.446365
\(756\) −4800.00 −0.230918
\(757\) −7694.00 −0.369410 −0.184705 0.982794i \(-0.559133\pi\)
−0.184705 + 0.982794i \(0.559133\pi\)
\(758\) −31600.0 −1.51420
\(759\) −4992.00 −0.238733
\(760\) 0 0
\(761\) −4518.00 −0.215213 −0.107607 0.994194i \(-0.534319\pi\)
−0.107607 + 0.994194i \(0.534319\pi\)
\(762\) −14768.0 −0.702084
\(763\) −3300.00 −0.156577
\(764\) 30176.0 1.42897
\(765\) 2990.00 0.141312
\(766\) 30072.0 1.41847
\(767\) −19000.0 −0.894459
\(768\) 8192.00 0.384900
\(769\) −39550.0 −1.85463 −0.927314 0.374283i \(-0.877889\pi\)
−0.927314 + 0.374283i \(0.877889\pi\)
\(770\) 3840.00 0.179719
\(771\) 4212.00 0.196746
\(772\) −2864.00 −0.133520
\(773\) 22122.0 1.02933 0.514666 0.857391i \(-0.327916\pi\)
0.514666 + 0.857391i \(0.327916\pi\)
\(774\) 40664.0 1.88842
\(775\) −2700.00 −0.125144
\(776\) 0 0
\(777\) 3192.00 0.147378
\(778\) 7800.00 0.359439
\(779\) 2200.00 0.101185
\(780\) 3040.00 0.139551
\(781\) 13184.0 0.604047
\(782\) 8112.00 0.370952
\(783\) 5000.00 0.228206
\(784\) 19648.0 0.895044
\(785\) 12470.0 0.566972
\(786\) 17664.0 0.801595
\(787\) −16634.0 −0.753416 −0.376708 0.926332i \(-0.622944\pi\)
−0.376708 + 0.926332i \(0.622944\pi\)
\(788\) −17712.0 −0.800716
\(789\) −7276.00 −0.328305
\(790\) 12000.0 0.540431
\(791\) 9372.00 0.421277
\(792\) 0 0
\(793\) 19684.0 0.881462
\(794\) −55144.0 −2.46472
\(795\) −20.0000 −0.000892235 0
\(796\) −20800.0 −0.926176
\(797\) 27586.0 1.22603 0.613015 0.790071i \(-0.289956\pi\)
0.613015 + 0.790071i \(0.289956\pi\)
\(798\) −4800.00 −0.212930
\(799\) −13364.0 −0.591720
\(800\) 6400.00 0.282843
\(801\) 3450.00 0.152184
\(802\) −25608.0 −1.12749
\(803\) −28096.0 −1.23473
\(804\) 2016.00 0.0884314
\(805\) 2340.00 0.102452
\(806\) −16416.0 −0.717406
\(807\) −13100.0 −0.571427
\(808\) 0 0
\(809\) 3850.00 0.167316 0.0836581 0.996495i \(-0.473340\pi\)
0.0836581 + 0.996495i \(0.473340\pi\)
\(810\) 8420.00 0.365245
\(811\) 10032.0 0.434366 0.217183 0.976131i \(-0.430313\pi\)
0.217183 + 0.976131i \(0.430313\pi\)
\(812\) −2400.00 −0.103724
\(813\) −8776.00 −0.378583
\(814\) −34048.0 −1.46607
\(815\) −13810.0 −0.593550
\(816\) −3328.00 −0.142774
\(817\) 44200.0 1.89273
\(818\) −44600.0 −1.90636
\(819\) 5244.00 0.223736
\(820\) −880.000 −0.0374767
\(821\) 20562.0 0.874079 0.437039 0.899442i \(-0.356027\pi\)
0.437039 + 0.899442i \(0.356027\pi\)
\(822\) 18672.0 0.792288
\(823\) 10322.0 0.437184 0.218592 0.975816i \(-0.429854\pi\)
0.218592 + 0.975816i \(0.429854\pi\)
\(824\) 0 0
\(825\) 1600.00 0.0675210
\(826\) −12000.0 −0.505488
\(827\) 8846.00 0.371954 0.185977 0.982554i \(-0.440455\pi\)
0.185977 + 0.982554i \(0.440455\pi\)
\(828\) 14352.0 0.602375
\(829\) −25350.0 −1.06205 −0.531026 0.847355i \(-0.678194\pi\)
−0.531026 + 0.847355i \(0.678194\pi\)
\(830\) 5640.00 0.235864
\(831\) 1092.00 0.0455849
\(832\) 19456.0 0.810716
\(833\) −7982.00 −0.332005
\(834\) 5600.00 0.232509
\(835\) −15630.0 −0.647783
\(836\) 25600.0 1.05908
\(837\) 10800.0 0.446001
\(838\) 54800.0 2.25899
\(839\) 46000.0 1.89284 0.946422 0.322932i \(-0.104669\pi\)
0.946422 + 0.322932i \(0.104669\pi\)
\(840\) 0 0
\(841\) −21889.0 −0.897495
\(842\) 21752.0 0.890289
\(843\) −13716.0 −0.560385
\(844\) −9344.00 −0.381083
\(845\) 3765.00 0.153278
\(846\) −47288.0 −1.92174
\(847\) −1842.00 −0.0747248
\(848\) −128.000 −0.00518342
\(849\) 18564.0 0.750430
\(850\) −2600.00 −0.104917
\(851\) −20748.0 −0.835761
\(852\) 6592.00 0.265068
\(853\) −16998.0 −0.682298 −0.341149 0.940009i \(-0.610816\pi\)
−0.341149 + 0.940009i \(0.610816\pi\)
\(854\) 12432.0 0.498143
\(855\) 11500.0 0.459990
\(856\) 0 0
\(857\) −26494.0 −1.05603 −0.528015 0.849235i \(-0.677064\pi\)
−0.528015 + 0.849235i \(0.677064\pi\)
\(858\) 9728.00 0.387073
\(859\) −21500.0 −0.853982 −0.426991 0.904256i \(-0.640426\pi\)
−0.426991 + 0.904256i \(0.640426\pi\)
\(860\) −17680.0 −0.701027
\(861\) 264.000 0.0104496
\(862\) −30768.0 −1.21573
\(863\) 25762.0 1.01616 0.508082 0.861309i \(-0.330355\pi\)
0.508082 + 0.861309i \(0.330355\pi\)
\(864\) −25600.0 −1.00802
\(865\) 390.000 0.0153299
\(866\) 4472.00 0.175479
\(867\) −8474.00 −0.331940
\(868\) −5184.00 −0.202715
\(869\) 19200.0 0.749500
\(870\) −2000.00 −0.0779383
\(871\) −4788.00 −0.186263
\(872\) 0 0
\(873\) −8878.00 −0.344186
\(874\) 31200.0 1.20750
\(875\) −750.000 −0.0289767
\(876\) −14048.0 −0.541824
\(877\) 30546.0 1.17613 0.588064 0.808814i \(-0.299890\pi\)
0.588064 + 0.808814i \(0.299890\pi\)
\(878\) 10400.0 0.399753
\(879\) 9684.00 0.371596
\(880\) 10240.0 0.392262
\(881\) 32942.0 1.25976 0.629878 0.776694i \(-0.283105\pi\)
0.629878 + 0.776694i \(0.283105\pi\)
\(882\) −28244.0 −1.07826
\(883\) −27118.0 −1.03351 −0.516757 0.856132i \(-0.672861\pi\)
−0.516757 + 0.856132i \(0.672861\pi\)
\(884\) −7904.00 −0.300724
\(885\) −5000.00 −0.189913
\(886\) 47832.0 1.81371
\(887\) −38634.0 −1.46246 −0.731230 0.682131i \(-0.761054\pi\)
−0.731230 + 0.682131i \(0.761054\pi\)
\(888\) 0 0
\(889\) 11076.0 0.417860
\(890\) −3000.00 −0.112989
\(891\) 13472.0 0.506542
\(892\) −51824.0 −1.94529
\(893\) −51400.0 −1.92613
\(894\) −16400.0 −0.613532
\(895\) 6500.00 0.242761
\(896\) 0 0
\(897\) 5928.00 0.220658
\(898\) 68200.0 2.53437
\(899\) 5400.00 0.200334
\(900\) −4600.00 −0.170370
\(901\) 52.0000 0.00192272
\(902\) −2816.00 −0.103950
\(903\) 5304.00 0.195466
\(904\) 0 0
\(905\) −8710.00 −0.319923
\(906\) −14816.0 −0.543299
\(907\) −1794.00 −0.0656767 −0.0328384 0.999461i \(-0.510455\pi\)
−0.0328384 + 0.999461i \(0.510455\pi\)
\(908\) 5168.00 0.188883
\(909\) −16146.0 −0.589141
\(910\) −4560.00 −0.166113
\(911\) 41732.0 1.51772 0.758860 0.651254i \(-0.225757\pi\)
0.758860 + 0.651254i \(0.225757\pi\)
\(912\) −12800.0 −0.464748
\(913\) 9024.00 0.327109
\(914\) 37976.0 1.37433
\(915\) 5180.00 0.187154
\(916\) 30000.0 1.08213
\(917\) −13248.0 −0.477086
\(918\) 10400.0 0.373912
\(919\) 29200.0 1.04812 0.524058 0.851682i \(-0.324417\pi\)
0.524058 + 0.851682i \(0.324417\pi\)
\(920\) 0 0
\(921\) −5188.00 −0.185614
\(922\) 45672.0 1.63137
\(923\) −15656.0 −0.558314
\(924\) 3072.00 0.109374
\(925\) 6650.00 0.236379
\(926\) −31848.0 −1.13023
\(927\) 13754.0 0.487315
\(928\) −12800.0 −0.452781
\(929\) −48650.0 −1.71814 −0.859071 0.511856i \(-0.828958\pi\)
−0.859071 + 0.511856i \(0.828958\pi\)
\(930\) −4320.00 −0.152321
\(931\) −30700.0 −1.08072
\(932\) 11856.0 0.416691
\(933\) 14664.0 0.514553
\(934\) −26104.0 −0.914506
\(935\) −4160.00 −0.145504
\(936\) 0 0
\(937\) −11334.0 −0.395161 −0.197580 0.980287i \(-0.563308\pi\)
−0.197580 + 0.980287i \(0.563308\pi\)
\(938\) −3024.00 −0.105263
\(939\) 3124.00 0.108571
\(940\) 20560.0 0.713397
\(941\) −31178.0 −1.08010 −0.540050 0.841633i \(-0.681595\pi\)
−0.540050 + 0.841633i \(0.681595\pi\)
\(942\) 19952.0 0.690097
\(943\) −1716.00 −0.0592584
\(944\) −32000.0 −1.10330
\(945\) 3000.00 0.103270
\(946\) −56576.0 −1.94444
\(947\) 4686.00 0.160797 0.0803984 0.996763i \(-0.474381\pi\)
0.0803984 + 0.996763i \(0.474381\pi\)
\(948\) 9600.00 0.328896
\(949\) 33364.0 1.14124
\(950\) −10000.0 −0.341519
\(951\) 2852.00 0.0972476
\(952\) 0 0
\(953\) −598.000 −0.0203265 −0.0101632 0.999948i \(-0.503235\pi\)
−0.0101632 + 0.999948i \(0.503235\pi\)
\(954\) 184.000 0.00624447
\(955\) −18860.0 −0.639053
\(956\) 11200.0 0.378906
\(957\) −3200.00 −0.108089
\(958\) −69600.0 −2.34726
\(959\) −14004.0 −0.471546
\(960\) 5120.00 0.172133
\(961\) −18127.0 −0.608472
\(962\) 40432.0 1.35507
\(963\) 27462.0 0.918952
\(964\) 24176.0 0.807735
\(965\) 1790.00 0.0597121
\(966\) 3744.00 0.124701
\(967\) 41726.0 1.38761 0.693804 0.720163i \(-0.255933\pi\)
0.693804 + 0.720163i \(0.255933\pi\)
\(968\) 0 0
\(969\) 5200.00 0.172392
\(970\) 7720.00 0.255540
\(971\) 24312.0 0.803511 0.401756 0.915747i \(-0.368400\pi\)
0.401756 + 0.915747i \(0.368400\pi\)
\(972\) 28336.0 0.935059
\(973\) −4200.00 −0.138382
\(974\) −4664.00 −0.153433
\(975\) −1900.00 −0.0624089
\(976\) 33152.0 1.08726
\(977\) 40946.0 1.34082 0.670409 0.741992i \(-0.266119\pi\)
0.670409 + 0.741992i \(0.266119\pi\)
\(978\) −22096.0 −0.722446
\(979\) −4800.00 −0.156699
\(980\) 12280.0 0.400276
\(981\) 12650.0 0.411706
\(982\) −28288.0 −0.919253
\(983\) 42282.0 1.37191 0.685954 0.727645i \(-0.259385\pi\)
0.685954 + 0.727645i \(0.259385\pi\)
\(984\) 0 0
\(985\) 11070.0 0.358091
\(986\) 5200.00 0.167953
\(987\) −6168.00 −0.198916
\(988\) −30400.0 −0.978900
\(989\) −34476.0 −1.10847
\(990\) −14720.0 −0.472558
\(991\) 1172.00 0.0375679 0.0187840 0.999824i \(-0.494021\pi\)
0.0187840 + 0.999824i \(0.494021\pi\)
\(992\) −27648.0 −0.884904
\(993\) −8016.00 −0.256173
\(994\) −9888.00 −0.315521
\(995\) 13000.0 0.414199
\(996\) 4512.00 0.143542
\(997\) −31614.0 −1.00424 −0.502119 0.864798i \(-0.667446\pi\)
−0.502119 + 0.864798i \(0.667446\pi\)
\(998\) −400.000 −0.0126872
\(999\) −26600.0 −0.842429
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 5.4.a.a.1.1 1
3.2 odd 2 45.4.a.d.1.1 1
4.3 odd 2 80.4.a.d.1.1 1
5.2 odd 4 25.4.b.a.24.1 2
5.3 odd 4 25.4.b.a.24.2 2
5.4 even 2 25.4.a.c.1.1 1
7.2 even 3 245.4.e.f.116.1 2
7.3 odd 6 245.4.e.g.226.1 2
7.4 even 3 245.4.e.f.226.1 2
7.5 odd 6 245.4.e.g.116.1 2
7.6 odd 2 245.4.a.a.1.1 1
8.3 odd 2 320.4.a.h.1.1 1
8.5 even 2 320.4.a.g.1.1 1
9.2 odd 6 405.4.e.c.271.1 2
9.4 even 3 405.4.e.l.136.1 2
9.5 odd 6 405.4.e.c.136.1 2
9.7 even 3 405.4.e.l.271.1 2
11.10 odd 2 605.4.a.d.1.1 1
12.11 even 2 720.4.a.u.1.1 1
13.12 even 2 845.4.a.b.1.1 1
15.2 even 4 225.4.b.c.199.2 2
15.8 even 4 225.4.b.c.199.1 2
15.14 odd 2 225.4.a.b.1.1 1
16.3 odd 4 1280.4.d.l.641.1 2
16.5 even 4 1280.4.d.e.641.1 2
16.11 odd 4 1280.4.d.l.641.2 2
16.13 even 4 1280.4.d.e.641.2 2
17.16 even 2 1445.4.a.a.1.1 1
19.18 odd 2 1805.4.a.h.1.1 1
20.3 even 4 400.4.c.k.49.1 2
20.7 even 4 400.4.c.k.49.2 2
20.19 odd 2 400.4.a.m.1.1 1
21.20 even 2 2205.4.a.q.1.1 1
35.34 odd 2 1225.4.a.k.1.1 1
40.19 odd 2 1600.4.a.s.1.1 1
40.29 even 2 1600.4.a.bi.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.4.a.a.1.1 1 1.1 even 1 trivial
25.4.a.c.1.1 1 5.4 even 2
25.4.b.a.24.1 2 5.2 odd 4
25.4.b.a.24.2 2 5.3 odd 4
45.4.a.d.1.1 1 3.2 odd 2
80.4.a.d.1.1 1 4.3 odd 2
225.4.a.b.1.1 1 15.14 odd 2
225.4.b.c.199.1 2 15.8 even 4
225.4.b.c.199.2 2 15.2 even 4
245.4.a.a.1.1 1 7.6 odd 2
245.4.e.f.116.1 2 7.2 even 3
245.4.e.f.226.1 2 7.4 even 3
245.4.e.g.116.1 2 7.5 odd 6
245.4.e.g.226.1 2 7.3 odd 6
320.4.a.g.1.1 1 8.5 even 2
320.4.a.h.1.1 1 8.3 odd 2
400.4.a.m.1.1 1 20.19 odd 2
400.4.c.k.49.1 2 20.3 even 4
400.4.c.k.49.2 2 20.7 even 4
405.4.e.c.136.1 2 9.5 odd 6
405.4.e.c.271.1 2 9.2 odd 6
405.4.e.l.136.1 2 9.4 even 3
405.4.e.l.271.1 2 9.7 even 3
605.4.a.d.1.1 1 11.10 odd 2
720.4.a.u.1.1 1 12.11 even 2
845.4.a.b.1.1 1 13.12 even 2
1225.4.a.k.1.1 1 35.34 odd 2
1280.4.d.e.641.1 2 16.5 even 4
1280.4.d.e.641.2 2 16.13 even 4
1280.4.d.l.641.1 2 16.3 odd 4
1280.4.d.l.641.2 2 16.11 odd 4
1445.4.a.a.1.1 1 17.16 even 2
1600.4.a.s.1.1 1 40.19 odd 2
1600.4.a.bi.1.1 1 40.29 even 2
1805.4.a.h.1.1 1 19.18 odd 2
2205.4.a.q.1.1 1 21.20 even 2