Properties

Label 1225.4.a.k.1.1
Level $1225$
Weight $4$
Character 1225.1
Self dual yes
Analytic conductor $72.277$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(72.2773397570\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 5)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 1225.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} +8.00000 q^{6} -23.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} +2.00000 q^{3} +8.00000 q^{4} +8.00000 q^{6} -23.0000 q^{9} +32.0000 q^{11} +16.0000 q^{12} -38.0000 q^{13} -64.0000 q^{16} +26.0000 q^{17} -92.0000 q^{18} -100.000 q^{19} +128.000 q^{22} +78.0000 q^{23} -152.000 q^{26} -100.000 q^{27} -50.0000 q^{29} +108.000 q^{31} -256.000 q^{32} +64.0000 q^{33} +104.000 q^{34} -184.000 q^{36} -266.000 q^{37} -400.000 q^{38} -76.0000 q^{39} -22.0000 q^{41} -442.000 q^{43} +256.000 q^{44} +312.000 q^{46} -514.000 q^{47} -128.000 q^{48} +52.0000 q^{51} -304.000 q^{52} -2.00000 q^{53} -400.000 q^{54} -200.000 q^{57} -200.000 q^{58} -500.000 q^{59} +518.000 q^{61} +432.000 q^{62} -512.000 q^{64} +256.000 q^{66} -126.000 q^{67} +208.000 q^{68} +156.000 q^{69} +412.000 q^{71} -878.000 q^{73} -1064.00 q^{74} -800.000 q^{76} -304.000 q^{78} +600.000 q^{79} +421.000 q^{81} -88.0000 q^{82} +282.000 q^{83} -1768.00 q^{86} -100.000 q^{87} +150.000 q^{89} +624.000 q^{92} +216.000 q^{93} -2056.00 q^{94} -512.000 q^{96} +386.000 q^{97} -736.000 q^{99} +O(q^{100})\)

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 \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 2.00000 0.384900 0.192450 0.981307i \(-0.438357\pi\)
0.192450 + 0.981307i \(0.438357\pi\)
\(4\) 8.00000 1.00000
\(5\) 0 0
\(6\) 8.00000 0.544331
\(7\) 0 0
\(8\) 0 0
\(9\) −23.0000 −0.851852
\(10\) 0 0
\(11\) 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 \(0.632853\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −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 \(-0.706318\pi\)
−0.603726 + 0.797192i \(0.706318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 128.000 1.24044
\(23\) 78.0000 0.707136 0.353568 0.935409i \(-0.384968\pi\)
0.353568 + 0.935409i \(0.384968\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −152.000 −1.14653
\(27\) −100.000 −0.712778
\(28\) 0 0
\(29\) −50.0000 −0.320164 −0.160082 0.987104i \(-0.551176\pi\)
−0.160082 + 0.987104i \(0.551176\pi\)
\(30\) 0 0
\(31\) 108.000 0.625722 0.312861 0.949799i \(-0.398713\pi\)
0.312861 + 0.949799i \(0.398713\pi\)
\(32\) −256.000 −1.41421
\(33\) 64.0000 0.337605
\(34\) 104.000 0.524584
\(35\) 0 0
\(36\) −184.000 −0.851852
\(37\) −266.000 −1.18190 −0.590948 0.806710i \(-0.701246\pi\)
−0.590948 + 0.806710i \(0.701246\pi\)
\(38\) −400.000 −1.70759
\(39\) −76.0000 −0.312045
\(40\) 0 0
\(41\) −22.0000 −0.0838006 −0.0419003 0.999122i \(-0.513341\pi\)
−0.0419003 + 0.999122i \(0.513341\pi\)
\(42\) 0 0
\(43\) −442.000 −1.56754 −0.783772 0.621049i \(-0.786707\pi\)
−0.783772 + 0.621049i \(0.786707\pi\)
\(44\) 256.000 0.877124
\(45\) 0 0
\(46\) 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
\(49\) 0 0
\(50\) 0 0
\(51\) 52.0000 0.142774
\(52\) −304.000 −0.810716
\(53\) −2.00000 −0.00518342 −0.00259171 0.999997i \(-0.500825\pi\)
−0.00259171 + 0.999997i \(0.500825\pi\)
\(54\) −400.000 −1.00802
\(55\) 0 0
\(56\) 0 0
\(57\) −200.000 −0.464748
\(58\) −200.000 −0.452781
\(59\) −500.000 −1.10330 −0.551648 0.834077i \(-0.686001\pi\)
−0.551648 + 0.834077i \(0.686001\pi\)
\(60\) 0 0
\(61\) 518.000 1.08726 0.543632 0.839324i \(-0.317049\pi\)
0.543632 + 0.839324i \(0.317049\pi\)
\(62\) 432.000 0.884904
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) 256.000 0.477446
\(67\) −126.000 −0.229751 −0.114876 0.993380i \(-0.536647\pi\)
−0.114876 + 0.993380i \(0.536647\pi\)
\(68\) 208.000 0.370937
\(69\) 156.000 0.272177
\(70\) 0 0
\(71\) 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
\(75\) 0 0
\(76\) −800.000 −1.20745
\(77\) 0 0
\(78\) −304.000 −0.441298
\(79\) 600.000 0.854497 0.427249 0.904134i \(-0.359483\pi\)
0.427249 + 0.904134i \(0.359483\pi\)
\(80\) 0 0
\(81\) 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 \(0.440296\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −1768.00 −2.21684
\(87\) −100.000 −0.123231
\(88\) 0 0
\(89\) 150.000 0.178651 0.0893257 0.996002i \(-0.471529\pi\)
0.0893257 + 0.996002i \(0.471529\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 624.000 0.707136
\(93\) 216.000 0.240840
\(94\) −2056.00 −2.25596
\(95\) 0 0
\(96\) −512.000 −0.544331
\(97\) 386.000 0.404045 0.202022 0.979381i \(-0.435249\pi\)
0.202022 + 0.979381i \(0.435249\pi\)
\(98\) 0 0
\(99\) −736.000 −0.747180
\(100\) 0 0
\(101\) −702.000 −0.691600 −0.345800 0.938308i \(-0.612392\pi\)
−0.345800 + 0.938308i \(0.612392\pi\)
\(102\) 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
\(105\) 0 0
\(106\) −8.00000 −0.00733046
\(107\) 1194.00 1.07877 0.539385 0.842059i \(-0.318657\pi\)
0.539385 + 0.842059i \(0.318657\pi\)
\(108\) −800.000 −0.712778
\(109\) −550.000 −0.483307 −0.241653 0.970363i \(-0.577690\pi\)
−0.241653 + 0.970363i \(0.577690\pi\)
\(110\) 0 0
\(111\) −532.000 −0.454912
\(112\) 0 0
\(113\) −1562.00 −1.30036 −0.650180 0.759781i \(-0.725306\pi\)
−0.650180 + 0.759781i \(0.725306\pi\)
\(114\) −800.000 −0.657253
\(115\) 0 0
\(116\) −400.000 −0.320164
\(117\) 874.000 0.690610
\(118\) −2000.00 −1.56030
\(119\) 0 0
\(120\) 0 0
\(121\) −307.000 −0.230654
\(122\) 2072.00 1.53762
\(123\) −44.0000 −0.0322548
\(124\) 864.000 0.625722
\(125\) 0 0
\(126\) 0 0
\(127\) −1846.00 −1.28981 −0.644906 0.764262i \(-0.723103\pi\)
−0.644906 + 0.764262i \(0.723103\pi\)
\(128\) 0 0
\(129\) −884.000 −0.603348
\(130\) 0 0
\(131\) 2208.00 1.47262 0.736312 0.676642i \(-0.236565\pi\)
0.736312 + 0.676642i \(0.236565\pi\)
\(132\) 512.000 0.337605
\(133\) 0 0
\(134\) −504.000 −0.324918
\(135\) 0 0
\(136\) 0 0
\(137\) 2334.00 1.45553 0.727763 0.685829i \(-0.240560\pi\)
0.727763 + 0.685829i \(0.240560\pi\)
\(138\) 624.000 0.384916
\(139\) 700.000 0.427146 0.213573 0.976927i \(-0.431490\pi\)
0.213573 + 0.976927i \(0.431490\pi\)
\(140\) 0 0
\(141\) −1028.00 −0.613994
\(142\) 1648.00 0.973923
\(143\) −1216.00 −0.711098
\(144\) 1472.00 0.851852
\(145\) 0 0
\(146\) −3512.00 −1.99079
\(147\) 0 0
\(148\) −2128.00 −1.18190
\(149\) 2050.00 1.12713 0.563566 0.826071i \(-0.309429\pi\)
0.563566 + 0.826071i \(0.309429\pi\)
\(150\) 0 0
\(151\) 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
\(154\) 0 0
\(155\) 0 0
\(156\) −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
\(160\) 0 0
\(161\) 0 0
\(162\) 1684.00 0.816713
\(163\) −2762.00 −1.32722 −0.663609 0.748080i \(-0.730976\pi\)
−0.663609 + 0.748080i \(0.730976\pi\)
\(164\) −176.000 −0.0838006
\(165\) 0 0
\(166\) 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
\(170\) 0 0
\(171\) 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
\(175\) 0 0
\(176\) −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 \(0.587492\pi\)
\(180\) 0 0
\(181\) −1742.00 −0.715369 −0.357685 0.933842i \(-0.616434\pi\)
−0.357685 + 0.933842i \(0.616434\pi\)
\(182\) 0 0
\(183\) 1036.00 0.418488
\(184\) 0 0
\(185\) 0 0
\(186\) 864.000 0.340600
\(187\) 832.000 0.325358
\(188\) −4112.00 −1.59520
\(189\) 0 0
\(190\) 0 0
\(191\) 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 \(-0.478734\pi\)
0.0667601 + 0.997769i \(0.478734\pi\)
\(194\) 1544.00 0.571406
\(195\) 0 0
\(196\) 0 0
\(197\) 2214.00 0.800716 0.400358 0.916359i \(-0.368886\pi\)
0.400358 + 0.916359i \(0.368886\pi\)
\(198\) −2944.00 −1.05667
\(199\) 2600.00 0.926176 0.463088 0.886312i \(-0.346741\pi\)
0.463088 + 0.886312i \(0.346741\pi\)
\(200\) 0 0
\(201\) −252.000 −0.0884314
\(202\) −2808.00 −0.978070
\(203\) 0 0
\(204\) 416.000 0.142774
\(205\) 0 0
\(206\) −2392.00 −0.809022
\(207\) −1794.00 −0.602375
\(208\) 2432.00 0.810716
\(209\) −3200.00 −1.05908
\(210\) 0 0
\(211\) −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
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −2200.00 −0.683499
\(219\) −1756.00 −0.541824
\(220\) 0 0
\(221\) −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 \(0.925374\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −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 \(-0.681978\pi\)
−0.541063 + 0.840982i \(0.681978\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1482.00 −0.416691 −0.208346 0.978055i \(-0.566808\pi\)
−0.208346 + 0.978055i \(0.566808\pi\)
\(234\) 3496.00 0.976670
\(235\) 0 0
\(236\) −4000.00 −1.10330
\(237\) 1200.00 0.328896
\(238\) 0 0
\(239\) 1400.00 0.378906 0.189453 0.981890i \(-0.439329\pi\)
0.189453 + 0.981890i \(0.439329\pi\)
\(240\) 0 0
\(241\) −3022.00 −0.807735 −0.403867 0.914817i \(-0.632334\pi\)
−0.403867 + 0.914817i \(0.632334\pi\)
\(242\) −1228.00 −0.326194
\(243\) 3542.00 0.935059
\(244\) 4144.00 1.08726
\(245\) 0 0
\(246\) −176.000 −0.0456152
\(247\) 3800.00 0.978900
\(248\) 0 0
\(249\) 564.000 0.143542
\(250\) 0 0
\(251\) 1248.00 0.313837 0.156918 0.987612i \(-0.449844\pi\)
0.156918 + 0.987612i \(0.449844\pi\)
\(252\) 0 0
\(253\) 2496.00 0.620246
\(254\) −7384.00 −1.82407
\(255\) 0 0
\(256\) 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
\(259\) 0 0
\(260\) 0 0
\(261\) 1150.00 0.272733
\(262\) 8832.00 2.08261
\(263\) 3638.00 0.852961 0.426480 0.904497i \(-0.359753\pi\)
0.426480 + 0.904497i \(0.359753\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 300.000 0.0687629
\(268\) −1008.00 −0.229751
\(269\) 6550.00 1.48461 0.742306 0.670061i \(-0.233732\pi\)
0.742306 + 0.670061i \(0.233732\pi\)
\(270\) 0 0
\(271\) 4388.00 0.983587 0.491793 0.870712i \(-0.336342\pi\)
0.491793 + 0.870712i \(0.336342\pi\)
\(272\) −1664.00 −0.370937
\(273\) 0 0
\(274\) 9336.00 2.05842
\(275\) 0 0
\(276\) 1248.00 0.272177
\(277\) −546.000 −0.118433 −0.0592165 0.998245i \(-0.518860\pi\)
−0.0592165 + 0.998245i \(0.518860\pi\)
\(278\) 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
\(285\) 0 0
\(286\) −4864.00 −1.00564
\(287\) 0 0
\(288\) 5888.00 1.20470
\(289\) −4237.00 −0.862406
\(290\) 0 0
\(291\) 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 \(0.339650\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −3200.00 −0.625195
\(298\) 8200.00 1.59400
\(299\) −2964.00 −0.573286
\(300\) 0 0
\(301\) 0 0
\(302\) 7408.00 1.41153
\(303\) −1404.00 −0.266197
\(304\) 6400.00 1.20745
\(305\) 0 0
\(306\) −2392.00 −0.446868
\(307\) −2594.00 −0.482239 −0.241120 0.970495i \(-0.577515\pi\)
−0.241120 + 0.970495i \(0.577515\pi\)
\(308\) 0 0
\(309\) −1196.00 −0.220188
\(310\) 0 0
\(311\) −7332.00 −1.33685 −0.668424 0.743781i \(-0.733031\pi\)
−0.668424 + 0.743781i \(0.733031\pi\)
\(312\) 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
\(315\) 0 0
\(316\) 4800.00 0.854497
\(317\) −1426.00 −0.252657 −0.126328 0.991988i \(-0.540319\pi\)
−0.126328 + 0.991988i \(0.540319\pi\)
\(318\) −16.0000 −0.00282150
\(319\) −1600.00 −0.280824
\(320\) 0 0
\(321\) 2388.00 0.415219
\(322\) 0 0
\(323\) −2600.00 −0.447888
\(324\) 3368.00 0.577503
\(325\) 0 0
\(326\) −11048.0 −1.87697
\(327\) −1100.00 −0.186025
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −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
\(335\) 0 0
\(336\) 0 0
\(337\) −8866.00 −1.43312 −0.716561 0.697525i \(-0.754285\pi\)
−0.716561 + 0.697525i \(0.754285\pi\)
\(338\) −3012.00 −0.484708
\(339\) −3124.00 −0.500509
\(340\) 0 0
\(341\) 3456.00 0.548835
\(342\) 9200.00 1.45462
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −312.000 −0.0484775
\(347\) 1714.00 0.265165 0.132583 0.991172i \(-0.457673\pi\)
0.132583 + 0.991172i \(0.457673\pi\)
\(348\) −800.000 −0.123231
\(349\) −1150.00 −0.176384 −0.0881921 0.996103i \(-0.528109\pi\)
−0.0881921 + 0.996103i \(0.528109\pi\)
\(350\) 0 0
\(351\) 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
\(355\) 0 0
\(356\) 1200.00 0.178651
\(357\) 0 0
\(358\) −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
\(364\) 0 0
\(365\) 0 0
\(366\) 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
\(370\) 0 0
\(371\) 0 0
\(372\) 1728.00 0.240840
\(373\) 2078.00 0.288458 0.144229 0.989544i \(-0.453930\pi\)
0.144229 + 0.989544i \(0.453930\pi\)
\(374\) 3328.00 0.460125
\(375\) 0 0
\(376\) 0 0
\(377\) 1900.00 0.259562
\(378\) 0 0
\(379\) 7900.00 1.07070 0.535351 0.844630i \(-0.320179\pi\)
0.535351 + 0.844630i \(0.320179\pi\)
\(380\) 0 0
\(381\) −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
\(385\) 0 0
\(386\) 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 \(0.540561\pi\)
\(390\) 0 0
\(391\) 2028.00 0.262303
\(392\) 0 0
\(393\) 4416.00 0.566814
\(394\) 8856.00 1.13238
\(395\) 0 0
\(396\) −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
\(399\) 0 0
\(400\) 0 0
\(401\) 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
\(405\) 0 0
\(406\) 0 0
\(407\) −8512.00 −1.03667
\(408\) 0 0
\(409\) −11150.0 −1.34800 −0.674000 0.738731i \(-0.735425\pi\)
−0.674000 + 0.738731i \(0.735425\pi\)
\(410\) 0 0
\(411\) 4668.00 0.560232
\(412\) −4784.00 −0.572065
\(413\) 0 0
\(414\) −7176.00 −0.851887
\(415\) 0 0
\(416\) 9728.00 1.14653
\(417\) 1400.00 0.164408
\(418\) −12800.0 −1.49777
\(419\) 13700.0 1.59735 0.798674 0.601764i \(-0.205535\pi\)
0.798674 + 0.601764i \(0.205535\pi\)
\(420\) 0 0
\(421\) −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
\(425\) 0 0
\(426\) 3296.00 0.374863
\(427\) 0 0
\(428\) 9552.00 1.07877
\(429\) −2432.00 −0.273702
\(430\) 0 0
\(431\) 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 \(0.519761\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −4400.00 −0.483307
\(437\) −7800.00 −0.853832
\(438\) −7024.00 −0.766255
\(439\) 2600.00 0.282668 0.141334 0.989962i \(-0.454861\pi\)
0.141334 + 0.989962i \(0.454861\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3952.00 −0.425288
\(443\) 11958.0 1.28249 0.641243 0.767337i \(-0.278419\pi\)
0.641243 + 0.767337i \(0.278419\pi\)
\(444\) −4256.00 −0.454912
\(445\) 0 0
\(446\) −25912.0 −2.75105
\(447\) 4100.00 0.433833
\(448\) 0 0
\(449\) −17050.0 −1.79207 −0.896035 0.443984i \(-0.853565\pi\)
−0.896035 + 0.443984i \(0.853565\pi\)
\(450\) 0 0
\(451\) −704.000 −0.0735035
\(452\) −12496.0 −1.30036
\(453\) 3704.00 0.384170
\(454\) 2584.00 0.267121
\(455\) 0 0
\(456\) 0 0
\(457\) 9494.00 0.971796 0.485898 0.874016i \(-0.338493\pi\)
0.485898 + 0.874016i \(0.338493\pi\)
\(458\) −15000.0 −1.53036
\(459\) −2600.00 −0.264396
\(460\) 0 0
\(461\) 11418.0 1.15356 0.576778 0.816901i \(-0.304310\pi\)
0.576778 + 0.816901i \(0.304310\pi\)
\(462\) 0 0
\(463\) −7962.00 −0.799191 −0.399596 0.916692i \(-0.630849\pi\)
−0.399596 + 0.916692i \(0.630849\pi\)
\(464\) 3200.00 0.320164
\(465\) 0 0
\(466\) −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
\(469\) 0 0
\(470\) 0 0
\(471\) −4988.00 −0.487972
\(472\) 0 0
\(473\) −14144.0 −1.37493
\(474\) 4800.00 0.465129
\(475\) 0 0
\(476\) 0 0
\(477\) 46.0000 0.00441550
\(478\) 5600.00 0.535854
\(479\) −17400.0 −1.65976 −0.829881 0.557940i \(-0.811592\pi\)
−0.829881 + 0.557940i \(0.811592\pi\)
\(480\) 0 0
\(481\) 10108.0 0.958181
\(482\) −12088.0 −1.14231
\(483\) 0 0
\(484\) −2456.00 −0.230654
\(485\) 0 0
\(486\) 14168.0 1.32237
\(487\) −1166.00 −0.108494 −0.0542469 0.998528i \(-0.517276\pi\)
−0.0542469 + 0.998528i \(0.517276\pi\)
\(488\) 0 0
\(489\) −5524.00 −0.510846
\(490\) 0 0
\(491\) 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
\(495\) 0 0
\(496\) −6912.00 −0.625722
\(497\) 0 0
\(498\) 2256.00 0.203000
\(499\) 100.000 0.00897117 0.00448559 0.999990i \(-0.498572\pi\)
0.00448559 + 0.999990i \(0.498572\pi\)
\(500\) 0 0
\(501\) 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
\(505\) 0 0
\(506\) 9984.00 0.877160
\(507\) −1506.00 −0.131921
\(508\) −14768.0 −1.28981
\(509\) −11150.0 −0.970953 −0.485476 0.874250i \(-0.661354\pi\)
−0.485476 + 0.874250i \(0.661354\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 16384.0 1.41421
\(513\) 10000.0 0.860645
\(514\) 8424.00 0.722892
\(515\) 0 0
\(516\) −7072.00 −0.603348
\(517\) −16448.0 −1.39919
\(518\) 0 0
\(519\) −156.000 −0.0131939
\(520\) 0 0
\(521\) 3638.00 0.305919 0.152959 0.988232i \(-0.451120\pi\)
0.152959 + 0.988232i \(0.451120\pi\)
\(522\) 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
\(525\) 0 0
\(526\) 14552.0 1.20627
\(527\) 2808.00 0.232103
\(528\) −4096.00 −0.337605
\(529\) −6083.00 −0.499959
\(530\) 0 0
\(531\) 11500.0 0.939845
\(532\) 0 0
\(533\) 836.000 0.0679384
\(534\) 1200.00 0.0972455
\(535\) 0 0
\(536\) 0 0
\(537\) −2600.00 −0.208935
\(538\) 26200.0 2.09956
\(539\) 0 0
\(540\) 0 0
\(541\) 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
\(545\) 0 0
\(546\) 0 0
\(547\) −16486.0 −1.28865 −0.644324 0.764753i \(-0.722861\pi\)
−0.644324 + 0.764753i \(0.722861\pi\)
\(548\) 18672.0 1.45553
\(549\) −11914.0 −0.926188
\(550\) 0 0
\(551\) 5000.00 0.386583
\(552\) 0 0
\(553\) 0 0
\(554\) −2184.00 −0.167490
\(555\) 0 0
\(556\) 5600.00 0.427146
\(557\) −11706.0 −0.890483 −0.445242 0.895410i \(-0.646882\pi\)
−0.445242 + 0.895410i \(0.646882\pi\)
\(558\) −9936.00 −0.753807
\(559\) 16796.0 1.27083
\(560\) 0 0
\(561\) 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
\(565\) 0 0
\(566\) 37128.0 2.75725
\(567\) 0 0
\(568\) 0 0
\(569\) 17550.0 1.29303 0.646515 0.762901i \(-0.276226\pi\)
0.646515 + 0.762901i \(0.276226\pi\)
\(570\) 0 0
\(571\) 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
\(574\) 0 0
\(575\) 0 0
\(576\) 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
\(580\) 0 0
\(581\) 0 0
\(582\) 3088.00 0.219934
\(583\) −64.0000 −0.00454650
\(584\) 0 0
\(585\) 0 0
\(586\) 19368.0 1.36533
\(587\) 14166.0 0.996071 0.498035 0.867157i \(-0.334055\pi\)
0.498035 + 0.867157i \(0.334055\pi\)
\(588\) 0 0
\(589\) −10800.0 −0.755528
\(590\) 0 0
\(591\) 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
\(595\) 0 0
\(596\) 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 \(-0.877211\pi\)
−0.926516 + 0.376256i \(0.877211\pi\)
\(602\) 0 0
\(603\) 2898.00 0.195714
\(604\) 14816.0 0.998103
\(605\) 0 0
\(606\) −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
\(609\) 0 0
\(610\) 0 0
\(611\) 19532.0 1.29326
\(612\) −4784.00 −0.315983
\(613\) 13238.0 0.872231 0.436116 0.899891i \(-0.356354\pi\)
0.436116 + 0.899891i \(0.356354\pi\)
\(614\) −10376.0 −0.681989
\(615\) 0 0
\(616\) 0 0
\(617\) 11574.0 0.755189 0.377595 0.925971i \(-0.376751\pi\)
0.377595 + 0.925971i \(0.376751\pi\)
\(618\) −4784.00 −0.311393
\(619\) −8300.00 −0.538942 −0.269471 0.963008i \(-0.586849\pi\)
−0.269471 + 0.963008i \(0.586849\pi\)
\(620\) 0 0
\(621\) −7800.00 −0.504031
\(622\) −29328.0 −1.89059
\(623\) 0 0
\(624\) 4864.00 0.312045
\(625\) 0 0
\(626\) 6248.00 0.398914
\(627\) −6400.00 −0.407642
\(628\) −19952.0 −1.26779
\(629\) −6916.00 −0.438409
\(630\) 0 0
\(631\) −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
\(635\) 0 0
\(636\) −32.0000 −0.00199510
\(637\) 0 0
\(638\) −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 \(0.482010\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −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
\(650\) 0 0
\(651\) 0 0
\(652\) −22096.0 −1.32722
\(653\) −10402.0 −0.623372 −0.311686 0.950185i \(-0.600894\pi\)
−0.311686 + 0.950185i \(0.600894\pi\)
\(654\) −4400.00 −0.263079
\(655\) 0 0
\(656\) 1408.00 0.0838006
\(657\) 20194.0 1.19915
\(658\) 0 0
\(659\) 7100.00 0.419692 0.209846 0.977734i \(-0.432704\pi\)
0.209846 + 0.977734i \(0.432704\pi\)
\(660\) 0 0
\(661\) 7118.00 0.418847 0.209424 0.977825i \(-0.432841\pi\)
0.209424 + 0.977825i \(0.432841\pi\)
\(662\) −16032.0 −0.941241
\(663\) −1976.00 −0.115749
\(664\) 0 0
\(665\) 0 0
\(666\) 24472.0 1.42383
\(667\) −3900.00 −0.226400
\(668\) 25008.0 1.44849
\(669\) −12956.0 −0.748741
\(670\) 0 0
\(671\) 16576.0 0.953665
\(672\) 0 0
\(673\) 31278.0 1.79150 0.895749 0.444560i \(-0.146640\pi\)
0.895749 + 0.444560i \(0.146640\pi\)
\(674\) −35464.0 −2.02674
\(675\) 0 0
\(676\) −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
\(679\) 0 0
\(680\) 0 0
\(681\) 1292.00 0.0727012
\(682\) 13824.0 0.776171
\(683\) 4518.00 0.253113 0.126557 0.991959i \(-0.459607\pi\)
0.126557 + 0.991959i \(0.459607\pi\)
\(684\) 18400.0 1.02857
\(685\) 0 0
\(686\) 0 0
\(687\) −7500.00 −0.416511
\(688\) 28288.0 1.56754
\(689\) 76.0000 0.00420228
\(690\) 0 0
\(691\) −29272.0 −1.61152 −0.805759 0.592243i \(-0.798242\pi\)
−0.805759 + 0.592243i \(0.798242\pi\)
\(692\) −624.000 −0.0342788
\(693\) 0 0
\(694\) 6856.00 0.375000
\(695\) 0 0
\(696\) 0 0
\(697\) −572.000 −0.0310847
\(698\) −4600.00 −0.249445
\(699\) −2964.00 −0.160385
\(700\) 0 0
\(701\) −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
\(705\) 0 0
\(706\) −17592.0 −0.937796
\(707\) 0 0
\(708\) −8000.00 −0.424659
\(709\) 8950.00 0.474082 0.237041 0.971500i \(-0.423822\pi\)
0.237041 + 0.971500i \(0.423822\pi\)
\(710\) 0 0
\(711\) −13800.0 −0.727905
\(712\) 0 0
\(713\) 8424.00 0.442470
\(714\) 0 0
\(715\) 0 0
\(716\) −10400.0 −0.542830
\(717\) 2800.00 0.145841
\(718\) 7200.00 0.374236
\(719\) −7800.00 −0.404577 −0.202289 0.979326i \(-0.564838\pi\)
−0.202289 + 0.979326i \(0.564838\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 12564.0 0.647623
\(723\) −6044.00 −0.310897
\(724\) −13936.0 −0.715369
\(725\) 0 0
\(726\) −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
\(730\) 0 0
\(731\) −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
\(735\) 0 0
\(736\) −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 \(0.345903\pi\)
\(740\) 0 0
\(741\) 7600.00 0.376779
\(742\) 0 0
\(743\) −12242.0 −0.604462 −0.302231 0.953235i \(-0.597731\pi\)
−0.302231 + 0.953235i \(0.597731\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 8312.00 0.407941
\(747\) −6486.00 −0.317685
\(748\) 6656.00 0.325358
\(749\) 0 0
\(750\) 0 0
\(751\) −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
\(755\) 0 0
\(756\) 0 0
\(757\) 7694.00 0.369410 0.184705 0.982794i \(-0.440867\pi\)
0.184705 + 0.982794i \(0.440867\pi\)
\(758\) 31600.0 1.51420
\(759\) 4992.00 0.238733
\(760\) 0 0
\(761\) 4518.00 0.215213 0.107607 0.994194i \(-0.465681\pi\)
0.107607 + 0.994194i \(0.465681\pi\)
\(762\) −14768.0 −0.702084
\(763\) 0 0
\(764\) 30176.0 1.42897
\(765\) 0 0
\(766\) −30072.0 −1.41847
\(767\) 19000.0 0.894459
\(768\) 8192.00 0.384900
\(769\) 39550.0 1.85463 0.927314 0.374283i \(-0.122111\pi\)
0.927314 + 0.374283i \(0.122111\pi\)
\(770\) 0 0
\(771\) 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
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −7800.00 −0.359439
\(779\) 2200.00 0.101185
\(780\) 0 0
\(781\) 13184.0 0.604047
\(782\) 8112.00 0.370952
\(783\) 5000.00 0.228206
\(784\) 0 0
\(785\) 0 0
\(786\) 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
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −19684.0 −0.881462
\(794\) 55144.0 2.46472
\(795\) 0 0
\(796\) 20800.0 0.926176
\(797\) 27586.0 1.22603 0.613015 0.790071i \(-0.289956\pi\)
0.613015 + 0.790071i \(0.289956\pi\)
\(798\) 0 0
\(799\) −13364.0 −0.591720
\(800\) 0 0
\(801\) −3450.00 −0.152184
\(802\) 25608.0 1.12749
\(803\) −28096.0 −1.23473
\(804\) −2016.00 −0.0884314
\(805\) 0 0
\(806\) −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 \(0.473340\pi\)
\(810\) 0 0
\(811\) −10032.0 −0.434366 −0.217183 0.976131i \(-0.569687\pi\)
−0.217183 + 0.976131i \(0.569687\pi\)
\(812\) 0 0
\(813\) 8776.00 0.378583
\(814\) −34048.0 −1.46607
\(815\) 0 0
\(816\) −3328.00 −0.142774
\(817\) 44200.0 1.89273
\(818\) −44600.0 −1.90636
\(819\) 0 0
\(820\) 0 0
\(821\) 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 \(-0.570146\pi\)
−0.218592 + 0.975816i \(0.570146\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −8846.00 −0.371954 −0.185977 0.982554i \(-0.559545\pi\)
−0.185977 + 0.982554i \(0.559545\pi\)
\(828\) −14352.0 −0.602375
\(829\) 25350.0 1.06205 0.531026 0.847355i \(-0.321806\pi\)
0.531026 + 0.847355i \(0.321806\pi\)
\(830\) 0 0
\(831\) −1092.00 −0.0455849
\(832\) 19456.0 0.810716
\(833\) 0 0
\(834\) 5600.00 0.232509
\(835\) 0 0
\(836\) −25600.0 −1.05908
\(837\) −10800.0 −0.446001
\(838\) 54800.0 2.25899
\(839\) −46000.0 −1.89284 −0.946422 0.322932i \(-0.895331\pi\)
−0.946422 + 0.322932i \(0.895331\pi\)
\(840\) 0 0
\(841\) −21889.0 −0.897495
\(842\) −21752.0 −0.890289
\(843\) −13716.0 −0.560385
\(844\) −9344.00 −0.381083
\(845\) 0 0
\(846\) 47288.0 1.92174
\(847\) 0 0
\(848\) 128.000 0.00518342
\(849\) 18564.0 0.750430
\(850\) 0 0
\(851\) −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 \(0.610816\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 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 \(-0.359574\pi\)
0.426991 + 0.904256i \(0.359574\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 30768.0 1.21573
\(863\) −25762.0 −1.01616 −0.508082 0.861309i \(-0.669645\pi\)
−0.508082 + 0.861309i \(0.669645\pi\)
\(864\) 25600.0 1.00802
\(865\) 0 0
\(866\) −4472.00 −0.175479
\(867\) −8474.00 −0.331940
\(868\) 0 0
\(869\) 19200.0 0.749500
\(870\) 0 0
\(871\) 4788.00 0.186263
\(872\) 0 0
\(873\) −8878.00 −0.344186
\(874\) −31200.0 −1.20750
\(875\) 0 0
\(876\) −14048.0 −0.541824
\(877\) −30546.0 −1.17613 −0.588064 0.808814i \(-0.700110\pi\)
−0.588064 + 0.808814i \(0.700110\pi\)
\(878\) 10400.0 0.399753
\(879\) 9684.00 0.371596
\(880\) 0 0
\(881\) −32942.0 −1.25976 −0.629878 0.776694i \(-0.716895\pi\)
−0.629878 + 0.776694i \(0.716895\pi\)
\(882\) 0 0
\(883\) 27118.0 1.03351 0.516757 0.856132i \(-0.327139\pi\)
0.516757 + 0.856132i \(0.327139\pi\)
\(884\) −7904.00 −0.300724
\(885\) 0 0
\(886\) 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
\(889\) 0 0
\(890\) 0 0
\(891\) 13472.0 0.506542
\(892\) −51824.0 −1.94529
\(893\) 51400.0 1.92613
\(894\) 16400.0 0.613532
\(895\) 0 0
\(896\) 0 0
\(897\) −5928.00 −0.220658
\(898\) −68200.0 −2.53437
\(899\) −5400.00 −0.200334
\(900\) 0 0
\(901\) −52.0000 −0.00192272
\(902\) −2816.00 −0.103950
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 14816.0 0.543299
\(907\) 1794.00 0.0656767 0.0328384 0.999461i \(-0.489545\pi\)
0.0328384 + 0.999461i \(0.489545\pi\)
\(908\) 5168.00 0.188883
\(909\) 16146.0 0.589141
\(910\) 0 0
\(911\) 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
\(915\) 0 0
\(916\) −30000.0 −1.08213
\(917\) 0 0
\(918\) −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
\(924\) 0 0
\(925\) 0 0
\(926\) −31848.0 −1.13023
\(927\) 13754.0 0.487315
\(928\) 12800.0 0.452781
\(929\) 48650.0 1.71814 0.859071 0.511856i \(-0.171042\pi\)
0.859071 + 0.511856i \(0.171042\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −11856.0 −0.416691
\(933\) −14664.0 −0.514553
\(934\) 26104.0 0.914506
\(935\) 0 0
\(936\) 0 0
\(937\) −11334.0 −0.395161 −0.197580 0.980287i \(-0.563308\pi\)
−0.197580 + 0.980287i \(0.563308\pi\)
\(938\) 0 0
\(939\) 3124.00 0.108571
\(940\) 0 0
\(941\) 31178.0 1.08010 0.540050 0.841633i \(-0.318405\pi\)
0.540050 + 0.841633i \(0.318405\pi\)
\(942\) −19952.0 −0.690097
\(943\) −1716.00 −0.0592584
\(944\) 32000.0 1.10330
\(945\) 0 0
\(946\) −56576.0 −1.94444
\(947\) −4686.00 −0.160797 −0.0803984 0.996763i \(-0.525619\pi\)
−0.0803984 + 0.996763i \(0.525619\pi\)
\(948\) 9600.00 0.328896
\(949\) 33364.0 1.14124
\(950\) 0 0
\(951\) −2852.00 −0.0972476
\(952\) 0 0
\(953\) 598.000 0.0203265 0.0101632 0.999948i \(-0.496765\pi\)
0.0101632 + 0.999948i \(0.496765\pi\)
\(954\) 184.000 0.00624447
\(955\) 0 0
\(956\) 11200.0 0.378906
\(957\) −3200.00 −0.108089
\(958\) −69600.0 −2.34726
\(959\) 0 0
\(960\) 0 0
\(961\) −18127.0 −0.608472
\(962\) 40432.0 1.35507
\(963\) −27462.0 −0.918952
\(964\) −24176.0 −0.807735
\(965\) 0 0
\(966\) 0 0
\(967\) −41726.0 −1.38761 −0.693804 0.720163i \(-0.744067\pi\)
−0.693804 + 0.720163i \(0.744067\pi\)
\(968\) 0 0
\(969\) −5200.00 −0.172392
\(970\) 0 0
\(971\) −24312.0 −0.803511 −0.401756 0.915747i \(-0.631600\pi\)
−0.401756 + 0.915747i \(0.631600\pi\)
\(972\) 28336.0 0.935059
\(973\) 0 0
\(974\) −4664.00 −0.153433
\(975\) 0 0
\(976\) −33152.0 −1.08726
\(977\) −40946.0 −1.34082 −0.670409 0.741992i \(-0.733881\pi\)
−0.670409 + 0.741992i \(0.733881\pi\)
\(978\) −22096.0 −0.722446
\(979\) 4800.00 0.156699
\(980\) 0 0
\(981\) 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
\(985\) 0 0
\(986\) −5200.00 −0.167953
\(987\) 0 0
\(988\) 30400.0 0.978900
\(989\) −34476.0 −1.10847
\(990\) 0 0
\(991\) 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
\(994\) 0 0
\(995\) 0 0
\(996\) 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 1225.4.a.k.1.1 1
5.4 even 2 245.4.a.a.1.1 1
7.6 odd 2 25.4.a.c.1.1 1
15.14 odd 2 2205.4.a.q.1.1 1
21.20 even 2 225.4.a.b.1.1 1
28.27 even 2 400.4.a.m.1.1 1
35.4 even 6 245.4.e.g.226.1 2
35.9 even 6 245.4.e.g.116.1 2
35.13 even 4 25.4.b.a.24.1 2
35.19 odd 6 245.4.e.f.116.1 2
35.24 odd 6 245.4.e.f.226.1 2
35.27 even 4 25.4.b.a.24.2 2
35.34 odd 2 5.4.a.a.1.1 1
56.13 odd 2 1600.4.a.bi.1.1 1
56.27 even 2 1600.4.a.s.1.1 1
105.62 odd 4 225.4.b.c.199.1 2
105.83 odd 4 225.4.b.c.199.2 2
105.104 even 2 45.4.a.d.1.1 1
140.27 odd 4 400.4.c.k.49.1 2
140.83 odd 4 400.4.c.k.49.2 2
140.139 even 2 80.4.a.d.1.1 1
280.69 odd 2 320.4.a.g.1.1 1
280.139 even 2 320.4.a.h.1.1 1
315.34 odd 6 405.4.e.l.271.1 2
315.104 even 6 405.4.e.c.136.1 2
315.139 odd 6 405.4.e.l.136.1 2
315.209 even 6 405.4.e.c.271.1 2
385.384 even 2 605.4.a.d.1.1 1
420.419 odd 2 720.4.a.u.1.1 1
455.454 odd 2 845.4.a.b.1.1 1
560.69 odd 4 1280.4.d.e.641.1 2
560.139 even 4 1280.4.d.l.641.2 2
560.349 odd 4 1280.4.d.e.641.2 2
560.419 even 4 1280.4.d.l.641.1 2
595.594 odd 2 1445.4.a.a.1.1 1
665.664 even 2 1805.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
5.4.a.a.1.1 1 35.34 odd 2
25.4.a.c.1.1 1 7.6 odd 2
25.4.b.a.24.1 2 35.13 even 4
25.4.b.a.24.2 2 35.27 even 4
45.4.a.d.1.1 1 105.104 even 2
80.4.a.d.1.1 1 140.139 even 2
225.4.a.b.1.1 1 21.20 even 2
225.4.b.c.199.1 2 105.62 odd 4
225.4.b.c.199.2 2 105.83 odd 4
245.4.a.a.1.1 1 5.4 even 2
245.4.e.f.116.1 2 35.19 odd 6
245.4.e.f.226.1 2 35.24 odd 6
245.4.e.g.116.1 2 35.9 even 6
245.4.e.g.226.1 2 35.4 even 6
320.4.a.g.1.1 1 280.69 odd 2
320.4.a.h.1.1 1 280.139 even 2
400.4.a.m.1.1 1 28.27 even 2
400.4.c.k.49.1 2 140.27 odd 4
400.4.c.k.49.2 2 140.83 odd 4
405.4.e.c.136.1 2 315.104 even 6
405.4.e.c.271.1 2 315.209 even 6
405.4.e.l.136.1 2 315.139 odd 6
405.4.e.l.271.1 2 315.34 odd 6
605.4.a.d.1.1 1 385.384 even 2
720.4.a.u.1.1 1 420.419 odd 2
845.4.a.b.1.1 1 455.454 odd 2
1225.4.a.k.1.1 1 1.1 even 1 trivial
1280.4.d.e.641.1 2 560.69 odd 4
1280.4.d.e.641.2 2 560.349 odd 4
1280.4.d.l.641.1 2 560.419 even 4
1280.4.d.l.641.2 2 560.139 even 4
1445.4.a.a.1.1 1 595.594 odd 2
1600.4.a.s.1.1 1 56.27 even 2
1600.4.a.bi.1.1 1 56.13 odd 2
1805.4.a.h.1.1 1 665.664 even 2
2205.4.a.q.1.1 1 15.14 odd 2