Properties

Label 225.4.a.b.1.1
Level $225$
Weight $4$
Character 225.1
Self dual yes
Analytic conductor $13.275$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(13.2754297513\)
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 \(\chi\) \(=\) 225.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +8.00000 q^{4} -6.00000 q^{7} +O(q^{10})\) \(q-4.00000 q^{2} +8.00000 q^{4} -6.00000 q^{7} -32.0000 q^{11} +38.0000 q^{13} +24.0000 q^{14} -64.0000 q^{16} +26.0000 q^{17} +100.000 q^{19} +128.000 q^{22} -78.0000 q^{23} -152.000 q^{26} -48.0000 q^{28} +50.0000 q^{29} -108.000 q^{31} +256.000 q^{32} -104.000 q^{34} -266.000 q^{37} -400.000 q^{38} -22.0000 q^{41} -442.000 q^{43} -256.000 q^{44} +312.000 q^{46} -514.000 q^{47} -307.000 q^{49} +304.000 q^{52} +2.00000 q^{53} -200.000 q^{58} -500.000 q^{59} -518.000 q^{61} +432.000 q^{62} -512.000 q^{64} -126.000 q^{67} +208.000 q^{68} -412.000 q^{71} +878.000 q^{73} +1064.00 q^{74} +800.000 q^{76} +192.000 q^{77} +600.000 q^{79} +88.0000 q^{82} +282.000 q^{83} +1768.00 q^{86} +150.000 q^{89} -228.000 q^{91} -624.000 q^{92} +2056.00 q^{94} -386.000 q^{97} +1228.00 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 8.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) −6.00000 −0.323970 −0.161985 0.986793i \(-0.551790\pi\)
−0.161985 + 0.986793i \(0.551790\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −32.0000 −0.877124 −0.438562 0.898701i \(-0.644512\pi\)
−0.438562 + 0.898701i \(0.644512\pi\)
\(12\) 0 0
\(13\) 38.0000 0.810716 0.405358 0.914158i \(-0.367147\pi\)
0.405358 + 0.914158i \(0.367147\pi\)
\(14\) 24.0000 0.458162
\(15\) 0 0
\(16\) −64.0000 −1.00000
\(17\) 26.0000 0.370937 0.185468 0.982650i \(-0.440620\pi\)
0.185468 + 0.982650i \(0.440620\pi\)
\(18\) 0 0
\(19\) 100.000 1.20745 0.603726 0.797192i \(-0.293682\pi\)
0.603726 + 0.797192i \(0.293682\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 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
\(25\) 0 0
\(26\) −152.000 −1.14653
\(27\) 0 0
\(28\) −48.0000 −0.323970
\(29\) 50.0000 0.320164 0.160082 0.987104i \(-0.448824\pi\)
0.160082 + 0.987104i \(0.448824\pi\)
\(30\) 0 0
\(31\) −108.000 −0.625722 −0.312861 0.949799i \(-0.601287\pi\)
−0.312861 + 0.949799i \(0.601287\pi\)
\(32\) 256.000 1.41421
\(33\) 0 0
\(34\) −104.000 −0.524584
\(35\) 0 0
\(36\) 0 0
\(37\) −266.000 −1.18190 −0.590948 0.806710i \(-0.701246\pi\)
−0.590948 + 0.806710i \(0.701246\pi\)
\(38\) −400.000 −1.70759
\(39\) 0 0
\(40\) 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 \(0.793899\pi\)
\(48\) 0 0
\(49\) −307.000 −0.895044
\(50\) 0 0
\(51\) 0 0
\(52\) 304.000 0.810716
\(53\) 2.00000 0.00518342 0.00259171 0.999997i \(-0.499175\pi\)
0.00259171 + 0.999997i \(0.499175\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −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 \(-0.682951\pi\)
−0.543632 + 0.839324i \(0.682951\pi\)
\(62\) 432.000 0.884904
\(63\) 0 0
\(64\) −512.000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −126.000 −0.229751 −0.114876 0.993380i \(-0.536647\pi\)
−0.114876 + 0.993380i \(0.536647\pi\)
\(68\) 208.000 0.370937
\(69\) 0 0
\(70\) 0 0
\(71\) −412.000 −0.688668 −0.344334 0.938847i \(-0.611895\pi\)
−0.344334 + 0.938847i \(0.611895\pi\)
\(72\) 0 0
\(73\) 878.000 1.40770 0.703850 0.710348i \(-0.251463\pi\)
0.703850 + 0.710348i \(0.251463\pi\)
\(74\) 1064.00 1.67145
\(75\) 0 0
\(76\) 800.000 1.20745
\(77\) 192.000 0.284161
\(78\) 0 0
\(79\) 600.000 0.854497 0.427249 0.904134i \(-0.359483\pi\)
0.427249 + 0.904134i \(0.359483\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 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
\(87\) 0 0
\(88\) 0 0
\(89\) 150.000 0.178651 0.0893257 0.996002i \(-0.471529\pi\)
0.0893257 + 0.996002i \(0.471529\pi\)
\(90\) 0 0
\(91\) −228.000 −0.262647
\(92\) −624.000 −0.707136
\(93\) 0 0
\(94\) 2056.00 2.25596
\(95\) 0 0
\(96\) 0 0
\(97\) −386.000 −0.404045 −0.202022 0.979381i \(-0.564751\pi\)
−0.202022 + 0.979381i \(0.564751\pi\)
\(98\) 1228.00 1.26578
\(99\) 0 0
\(100\) 0 0
\(101\) −702.000 −0.691600 −0.345800 0.938308i \(-0.612392\pi\)
−0.345800 + 0.938308i \(0.612392\pi\)
\(102\) 0 0
\(103\) 598.000 0.572065 0.286032 0.958220i \(-0.407663\pi\)
0.286032 + 0.958220i \(0.407663\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −8.00000 −0.00733046
\(107\) −1194.00 −1.07877 −0.539385 0.842059i \(-0.681343\pi\)
−0.539385 + 0.842059i \(0.681343\pi\)
\(108\) 0 0
\(109\) −550.000 −0.483307 −0.241653 0.970363i \(-0.577690\pi\)
−0.241653 + 0.970363i \(0.577690\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 384.000 0.323970
\(113\) 1562.00 1.30036 0.650180 0.759781i \(-0.274694\pi\)
0.650180 + 0.759781i \(0.274694\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 400.000 0.320164
\(117\) 0 0
\(118\) 2000.00 1.56030
\(119\) −156.000 −0.120172
\(120\) 0 0
\(121\) −307.000 −0.230654
\(122\) 2072.00 1.53762
\(123\) 0 0
\(124\) −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
\(129\) 0 0
\(130\) 0 0
\(131\) 2208.00 1.47262 0.736312 0.676642i \(-0.236565\pi\)
0.736312 + 0.676642i \(0.236565\pi\)
\(132\) 0 0
\(133\) −600.000 −0.391177
\(134\) 504.000 0.324918
\(135\) 0 0
\(136\) 0 0
\(137\) −2334.00 −1.45553 −0.727763 0.685829i \(-0.759440\pi\)
−0.727763 + 0.685829i \(0.759440\pi\)
\(138\) 0 0
\(139\) −700.000 −0.427146 −0.213573 0.976927i \(-0.568510\pi\)
−0.213573 + 0.976927i \(0.568510\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1648.00 0.973923
\(143\) −1216.00 −0.711098
\(144\) 0 0
\(145\) 0 0
\(146\) −3512.00 −1.99079
\(147\) 0 0
\(148\) −2128.00 −1.18190
\(149\) −2050.00 −1.12713 −0.563566 0.826071i \(-0.690571\pi\)
−0.563566 + 0.826071i \(0.690571\pi\)
\(150\) 0 0
\(151\) 1852.00 0.998103 0.499052 0.866572i \(-0.333682\pi\)
0.499052 + 0.866572i \(0.333682\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −768.000 −0.401865
\(155\) 0 0
\(156\) 0 0
\(157\) 2494.00 1.26779 0.633894 0.773420i \(-0.281455\pi\)
0.633894 + 0.773420i \(0.281455\pi\)
\(158\) −2400.00 −1.20844
\(159\) 0 0
\(160\) 0 0
\(161\) 468.000 0.229090
\(162\) 0 0
\(163\) −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
\(171\) 0 0
\(172\) −3536.00 −1.56754
\(173\) −78.0000 −0.0342788 −0.0171394 0.999853i \(-0.505456\pi\)
−0.0171394 + 0.999853i \(0.505456\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2048.00 0.877124
\(177\) 0 0
\(178\) −600.000 −0.252651
\(179\) 1300.00 0.542830 0.271415 0.962462i \(-0.412508\pi\)
0.271415 + 0.962462i \(0.412508\pi\)
\(180\) 0 0
\(181\) 1742.00 0.715369 0.357685 0.933842i \(-0.383566\pi\)
0.357685 + 0.933842i \(0.383566\pi\)
\(182\) 912.000 0.371439
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −832.000 −0.325358
\(188\) −4112.00 −1.59520
\(189\) 0 0
\(190\) 0 0
\(191\) −3772.00 −1.42897 −0.714483 0.699653i \(-0.753338\pi\)
−0.714483 + 0.699653i \(0.753338\pi\)
\(192\) 0 0
\(193\) 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
\(196\) −2456.00 −0.895044
\(197\) −2214.00 −0.800716 −0.400358 0.916359i \(-0.631114\pi\)
−0.400358 + 0.916359i \(0.631114\pi\)
\(198\) 0 0
\(199\) −2600.00 −0.926176 −0.463088 0.886312i \(-0.653259\pi\)
−0.463088 + 0.886312i \(0.653259\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 2808.00 0.978070
\(203\) −300.000 −0.103724
\(204\) 0 0
\(205\) 0 0
\(206\) −2392.00 −0.809022
\(207\) 0 0
\(208\) −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
\(213\) 0 0
\(214\) 4776.00 1.52561
\(215\) 0 0
\(216\) 0 0
\(217\) 648.000 0.202715
\(218\) 2200.00 0.683499
\(219\) 0 0
\(220\) 0 0
\(221\) 988.000 0.300724
\(222\) 0 0
\(223\) 6478.00 1.94529 0.972643 0.232303i \(-0.0746262\pi\)
0.972643 + 0.232303i \(0.0746262\pi\)
\(224\) −1536.00 −0.458162
\(225\) 0 0
\(226\) −6248.00 −1.83899
\(227\) 646.000 0.188883 0.0944417 0.995530i \(-0.469893\pi\)
0.0944417 + 0.995530i \(0.469893\pi\)
\(228\) 0 0
\(229\) 3750.00 1.08213 0.541063 0.840982i \(-0.318022\pi\)
0.541063 + 0.840982i \(0.318022\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1482.00 0.416691 0.208346 0.978055i \(-0.433192\pi\)
0.208346 + 0.978055i \(0.433192\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −4000.00 −1.10330
\(237\) 0 0
\(238\) 624.000 0.169949
\(239\) −1400.00 −0.378906 −0.189453 0.981890i \(-0.560671\pi\)
−0.189453 + 0.981890i \(0.560671\pi\)
\(240\) 0 0
\(241\) 3022.00 0.807735 0.403867 0.914817i \(-0.367666\pi\)
0.403867 + 0.914817i \(0.367666\pi\)
\(242\) 1228.00 0.326194
\(243\) 0 0
\(244\) −4144.00 −1.08726
\(245\) 0 0
\(246\) 0 0
\(247\) 3800.00 0.978900
\(248\) 0 0
\(249\) 0 0
\(250\) 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 \(0.417733\pi\)
\(258\) 0 0
\(259\) 1596.00 0.382898
\(260\) 0 0
\(261\) 0 0
\(262\) −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
\(265\) 0 0
\(266\) 2400.00 0.553208
\(267\) 0 0
\(268\) −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 \(-0.663658\pi\)
−0.491793 + 0.870712i \(0.663658\pi\)
\(272\) −1664.00 −0.370937
\(273\) 0 0
\(274\) 9336.00 2.05842
\(275\) 0 0
\(276\) 0 0
\(277\) −546.000 −0.118433 −0.0592165 0.998245i \(-0.518860\pi\)
−0.0592165 + 0.998245i \(0.518860\pi\)
\(278\) 2800.00 0.604075
\(279\) 0 0
\(280\) 0 0
\(281\) 6858.00 1.45592 0.727961 0.685619i \(-0.240468\pi\)
0.727961 + 0.685619i \(0.240468\pi\)
\(282\) 0 0
\(283\) −9282.00 −1.94967 −0.974837 0.222920i \(-0.928441\pi\)
−0.974837 + 0.222920i \(0.928441\pi\)
\(284\) −3296.00 −0.688668
\(285\) 0 0
\(286\) 4864.00 1.00564
\(287\) 132.000 0.0271488
\(288\) 0 0
\(289\) −4237.00 −0.862406
\(290\) 0 0
\(291\) 0 0
\(292\) 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
\(297\) 0 0
\(298\) 8200.00 1.59400
\(299\) −2964.00 −0.573286
\(300\) 0 0
\(301\) 2652.00 0.507836
\(302\) −7408.00 −1.41153
\(303\) 0 0
\(304\) −6400.00 −1.20745
\(305\) 0 0
\(306\) 0 0
\(307\) 2594.00 0.482239 0.241120 0.970495i \(-0.422485\pi\)
0.241120 + 0.970495i \(0.422485\pi\)
\(308\) 1536.00 0.284161
\(309\) 0 0
\(310\) 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 \(-0.545044\pi\)
−0.141037 + 0.990004i \(0.545044\pi\)
\(314\) −9976.00 −1.79292
\(315\) 0 0
\(316\) 4800.00 0.854497
\(317\) 1426.00 0.252657 0.126328 0.991988i \(-0.459681\pi\)
0.126328 + 0.991988i \(0.459681\pi\)
\(318\) 0 0
\(319\) −1600.00 −0.280824
\(320\) 0 0
\(321\) 0 0
\(322\) −1872.00 −0.323983
\(323\) 2600.00 0.447888
\(324\) 0 0
\(325\) 0 0
\(326\) 11048.0 1.87697
\(327\) 0 0
\(328\) 0 0
\(329\) 3084.00 0.516798
\(330\) 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
\(333\) 0 0
\(334\) −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
\(339\) 0 0
\(340\) 0 0
\(341\) 3456.00 0.548835
\(342\) 0 0
\(343\) 3900.00 0.613936
\(344\) 0 0
\(345\) 0 0
\(346\) 312.000 0.0484775
\(347\) −1714.00 −0.265165 −0.132583 0.991172i \(-0.542327\pi\)
−0.132583 + 0.991172i \(0.542327\pi\)
\(348\) 0 0
\(349\) 1150.00 0.176384 0.0881921 0.996103i \(-0.471891\pi\)
0.0881921 + 0.996103i \(0.471891\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −8192.00 −1.24044
\(353\) −4398.00 −0.663122 −0.331561 0.943434i \(-0.607575\pi\)
−0.331561 + 0.943434i \(0.607575\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1200.00 0.178651
\(357\) 0 0
\(358\) −5200.00 −0.767677
\(359\) −1800.00 −0.264625 −0.132312 0.991208i \(-0.542240\pi\)
−0.132312 + 0.991208i \(0.542240\pi\)
\(360\) 0 0
\(361\) 3141.00 0.457938
\(362\) −6968.00 −1.01168
\(363\) 0 0
\(364\) −1824.00 −0.262647
\(365\) 0 0
\(366\) 0 0
\(367\) 5874.00 0.835478 0.417739 0.908567i \(-0.362823\pi\)
0.417739 + 0.908567i \(0.362823\pi\)
\(368\) 4992.00 0.707136
\(369\) 0 0
\(370\) 0 0
\(371\) −12.0000 −0.00167927
\(372\) 0 0
\(373\) 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
\(381\) 0 0
\(382\) 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
\(387\) 0 0
\(388\) −3088.00 −0.404045
\(389\) 1950.00 0.254162 0.127081 0.991892i \(-0.459439\pi\)
0.127081 + 0.991892i \(0.459439\pi\)
\(390\) 0 0
\(391\) −2028.00 −0.262303
\(392\) 0 0
\(393\) 0 0
\(394\) 8856.00 1.13238
\(395\) 0 0
\(396\) 0 0
\(397\) −13786.0 −1.74282 −0.871410 0.490555i \(-0.836794\pi\)
−0.871410 + 0.490555i \(0.836794\pi\)
\(398\) 10400.0 1.30981
\(399\) 0 0
\(400\) 0 0
\(401\) −6402.00 −0.797258 −0.398629 0.917112i \(-0.630514\pi\)
−0.398629 + 0.917112i \(0.630514\pi\)
\(402\) 0 0
\(403\) −4104.00 −0.507282
\(404\) −5616.00 −0.691600
\(405\) 0 0
\(406\) 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 \(0.264575\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 4784.00 0.572065
\(413\) 3000.00 0.357434
\(414\) 0 0
\(415\) 0 0
\(416\) 9728.00 1.14653
\(417\) 0 0
\(418\) 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
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 3108.00 0.352240
\(428\) −9552.00 −1.07877
\(429\) 0 0
\(430\) 0 0
\(431\) −7692.00 −0.859653 −0.429827 0.902911i \(-0.641425\pi\)
−0.429827 + 0.902911i \(0.641425\pi\)
\(432\) 0 0
\(433\) 1118.00 0.124082 0.0620412 0.998074i \(-0.480239\pi\)
0.0620412 + 0.998074i \(0.480239\pi\)
\(434\) −2592.00 −0.286682
\(435\) 0 0
\(436\) −4400.00 −0.483307
\(437\) −7800.00 −0.853832
\(438\) 0 0
\(439\) −2600.00 −0.282668 −0.141334 0.989962i \(-0.545139\pi\)
−0.141334 + 0.989962i \(0.545139\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −3952.00 −0.425288
\(443\) −11958.0 −1.28249 −0.641243 0.767337i \(-0.721581\pi\)
−0.641243 + 0.767337i \(0.721581\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −25912.0 −2.75105
\(447\) 0 0
\(448\) 3072.00 0.323970
\(449\) 17050.0 1.79207 0.896035 0.443984i \(-0.146435\pi\)
0.896035 + 0.443984i \(0.146435\pi\)
\(450\) 0 0
\(451\) 704.000 0.0735035
\(452\) 12496.0 1.30036
\(453\) 0 0
\(454\) −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
\(459\) 0 0
\(460\) 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 \(0.395199\pi\)
\(468\) 0 0
\(469\) 756.000 0.0744325
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 14144.0 1.37493
\(474\) 0 0
\(475\) 0 0
\(476\) −1248.00 −0.120172
\(477\) 0 0
\(478\) 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
\(486\) 0 0
\(487\) −1166.00 −0.108494 −0.0542469 0.998528i \(-0.517276\pi\)
−0.0542469 + 0.998528i \(0.517276\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −7072.00 −0.650010 −0.325005 0.945712i \(-0.605366\pi\)
−0.325005 + 0.945712i \(0.605366\pi\)
\(492\) 0 0
\(493\) 1300.00 0.118761
\(494\) −15200.0 −1.38437
\(495\) 0 0
\(496\) 6912.00 0.625722
\(497\) 2472.00 0.223107
\(498\) 0 0
\(499\) 100.000 0.00897117 0.00448559 0.999990i \(-0.498572\pi\)
0.00448559 + 0.999990i \(0.498572\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −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
\(507\) 0 0
\(508\) −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
\(511\) −5268.00 −0.456052
\(512\) −16384.0 −1.41421
\(513\) 0 0
\(514\) −8424.00 −0.722892
\(515\) 0 0
\(516\) 0 0
\(517\) 16448.0 1.39919
\(518\) −6384.00 −0.541500
\(519\) 0 0
\(520\) 0 0
\(521\) 3638.00 0.305919 0.152959 0.988232i \(-0.451120\pi\)
0.152959 + 0.988232i \(0.451120\pi\)
\(522\) 0 0
\(523\) 2078.00 0.173737 0.0868686 0.996220i \(-0.472314\pi\)
0.0868686 + 0.996220i \(0.472314\pi\)
\(524\) 17664.0 1.47262
\(525\) 0 0
\(526\) 14552.0 1.20627
\(527\) −2808.00 −0.232103
\(528\) 0 0
\(529\) −6083.00 −0.499959
\(530\) 0 0
\(531\) 0 0
\(532\) −4800.00 −0.391177
\(533\) −836.000 −0.0679384
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −26200.0 −2.09956
\(539\) 9824.00 0.785064
\(540\) 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
\(543\) 0 0
\(544\) 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
\(549\) 0 0
\(550\) 0 0
\(551\) 5000.00 0.386583
\(552\) 0 0
\(553\) −3600.00 −0.276831
\(554\) 2184.00 0.167490
\(555\) 0 0
\(556\) −5600.00 −0.427146
\(557\) 11706.0 0.890483 0.445242 0.895410i \(-0.353118\pi\)
0.445242 + 0.895410i \(0.353118\pi\)
\(558\) 0 0
\(559\) −16796.0 −1.27083
\(560\) 0 0
\(561\) 0 0
\(562\) −27432.0 −2.05898
\(563\) −25038.0 −1.87429 −0.937146 0.348939i \(-0.886542\pi\)
−0.937146 + 0.348939i \(0.886542\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 37128.0 2.75725
\(567\) 0 0
\(568\) 0 0
\(569\) −17550.0 −1.29303 −0.646515 0.762901i \(-0.723774\pi\)
−0.646515 + 0.762901i \(0.723774\pi\)
\(570\) 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
\(573\) 0 0
\(574\) −528.000 −0.0383942
\(575\) 0 0
\(576\) 0 0
\(577\) 13654.0 0.985136 0.492568 0.870274i \(-0.336058\pi\)
0.492568 + 0.870274i \(0.336058\pi\)
\(578\) 16948.0 1.21963
\(579\) 0 0
\(580\) 0 0
\(581\) −1692.00 −0.120819
\(582\) 0 0
\(583\) −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
\(591\) 0 0
\(592\) 17024.0 1.18190
\(593\) 17842.0 1.23555 0.617777 0.786354i \(-0.288034\pi\)
0.617777 + 0.786354i \(0.288034\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −16400.0 −1.12713
\(597\) 0 0
\(598\) 11856.0 0.810749
\(599\) 17600.0 1.20053 0.600264 0.799802i \(-0.295062\pi\)
0.600264 + 0.799802i \(0.295062\pi\)
\(600\) 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
\(603\) 0 0
\(604\) 14816.0 0.998103
\(605\) 0 0
\(606\) 0 0
\(607\) 3794.00 0.253696 0.126848 0.991922i \(-0.459514\pi\)
0.126848 + 0.991922i \(0.459514\pi\)
\(608\) 25600.0 1.70759
\(609\) 0 0
\(610\) 0 0
\(611\) −19532.0 −1.29326
\(612\) 0 0
\(613\) 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 \(-0.623249\pi\)
−0.377595 + 0.925971i \(0.623249\pi\)
\(618\) 0 0
\(619\) 8300.00 0.538942 0.269471 0.963008i \(-0.413151\pi\)
0.269471 + 0.963008i \(0.413151\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 29328.0 1.89059
\(623\) −900.000 −0.0578776
\(624\) 0 0
\(625\) 0 0
\(626\) 6248.00 0.398914
\(627\) 0 0
\(628\) 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
\(633\) 0 0
\(634\) −5704.00 −0.357310
\(635\) 0 0
\(636\) 0 0
\(637\) −11666.0 −0.725626
\(638\) 6400.00 0.397145
\(639\) 0 0
\(640\) 0 0
\(641\) 27378.0 1.68700 0.843499 0.537130i \(-0.180492\pi\)
0.843499 + 0.537130i \(0.180492\pi\)
\(642\) 0 0
\(643\) −1842.00 −0.112973 −0.0564863 0.998403i \(-0.517990\pi\)
−0.0564863 + 0.998403i \(0.517990\pi\)
\(644\) 3744.00 0.229090
\(645\) 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 \(-0.399106\pi\)
0.311686 + 0.950185i \(0.399106\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1408.00 0.0838006
\(657\) 0 0
\(658\) −12336.0 −0.730862
\(659\) −7100.00 −0.419692 −0.209846 0.977734i \(-0.567296\pi\)
−0.209846 + 0.977734i \(0.567296\pi\)
\(660\) 0 0
\(661\) −7118.00 −0.418847 −0.209424 0.977825i \(-0.567159\pi\)
−0.209424 + 0.977825i \(0.567159\pi\)
\(662\) 16032.0 0.941241
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3900.00 −0.226400
\(668\) 25008.0 1.44849
\(669\) 0 0
\(670\) 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 \(0.825268\pi\)
\(678\) 0 0
\(679\) 2316.00 0.130898
\(680\) 0 0
\(681\) 0 0
\(682\) −13824.0 −0.776171
\(683\) −4518.00 −0.253113 −0.126557 0.991959i \(-0.540393\pi\)
−0.126557 + 0.991959i \(0.540393\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −15600.0 −0.868237
\(687\) 0 0
\(688\) 28288.0 1.56754
\(689\) 76.0000 0.00420228
\(690\) 0 0
\(691\) 29272.0 1.61152 0.805759 0.592243i \(-0.201758\pi\)
0.805759 + 0.592243i \(0.201758\pi\)
\(692\) −624.000 −0.0342788
\(693\) 0 0
\(694\) 6856.00 0.375000
\(695\) 0 0
\(696\) 0 0
\(697\) −572.000 −0.0310847
\(698\) −4600.00 −0.249445
\(699\) 0 0
\(700\) 0 0
\(701\) 5798.00 0.312393 0.156196 0.987726i \(-0.450077\pi\)
0.156196 + 0.987726i \(0.450077\pi\)
\(702\) 0 0
\(703\) −26600.0 −1.42708
\(704\) 16384.0 0.877124
\(705\) 0 0
\(706\) 17592.0 0.937796
\(707\) 4212.00 0.224057
\(708\) 0 0
\(709\) 8950.00 0.474082 0.237041 0.971500i \(-0.423822\pi\)
0.237041 + 0.971500i \(0.423822\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 8424.00 0.442470
\(714\) 0 0
\(715\) 0 0
\(716\) 10400.0 0.542830
\(717\) 0 0
\(718\) 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
\(721\) −3588.00 −0.185332
\(722\) −12564.0 −0.647623
\(723\) 0 0
\(724\) 13936.0 0.715369
\(725\) 0 0
\(726\) 0 0
\(727\) 8554.00 0.436383 0.218191 0.975906i \(-0.429984\pi\)
0.218191 + 0.975906i \(0.429984\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −11492.0 −0.581460
\(732\) 0 0
\(733\) −2882.00 −0.145224 −0.0726119 0.997360i \(-0.523133\pi\)
−0.0726119 + 0.997360i \(0.523133\pi\)
\(734\) −23496.0 −1.18154
\(735\) 0 0
\(736\) −19968.0 −1.00004
\(737\) 4032.00 0.201521
\(738\) 0 0
\(739\) 18700.0 0.930840 0.465420 0.885090i \(-0.345903\pi\)
0.465420 + 0.885090i \(0.345903\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 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
\(745\) 0 0
\(746\) −8312.00 −0.407941
\(747\) 0 0
\(748\) −6656.00 −0.325358
\(749\) 7164.00 0.349488
\(750\) 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
\(753\) 0 0
\(754\) −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
\(759\) 0 0
\(760\) 0 0
\(761\) 4518.00 0.215213 0.107607 0.994194i \(-0.465681\pi\)
0.107607 + 0.994194i \(0.465681\pi\)
\(762\) 0 0
\(763\) 3300.00 0.156577
\(764\) −30176.0 −1.42897
\(765\) 0 0
\(766\) 30072.0 1.41847
\(767\) −19000.0 −0.894459
\(768\) 0 0
\(769\) −39550.0 −1.85463 −0.927314 0.374283i \(-0.877889\pi\)
−0.927314 + 0.374283i \(0.877889\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2864.00 0.133520
\(773\) 22122.0 1.02933 0.514666 0.857391i \(-0.327916\pi\)
0.514666 + 0.857391i \(0.327916\pi\)
\(774\) 0 0
\(775\) 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
\(783\) 0 0
\(784\) 19648.0 0.895044
\(785\) 0 0
\(786\) 0 0
\(787\) 16634.0 0.753416 0.376708 0.926332i \(-0.377056\pi\)
0.376708 + 0.926332i \(0.377056\pi\)
\(788\) −17712.0 −0.800716
\(789\) 0 0
\(790\) 0 0
\(791\) −9372.00 −0.421277
\(792\) 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
\(801\) 0 0
\(802\) 25608.0 1.12749
\(803\) −28096.0 −1.23473
\(804\) 0 0
\(805\) 0 0
\(806\) 16416.0 0.717406
\(807\) 0 0
\(808\) 0 0
\(809\) −3850.00 −0.167316 −0.0836581 0.996495i \(-0.526660\pi\)
−0.0836581 + 0.996495i \(0.526660\pi\)
\(810\) 0 0
\(811\) 10032.0 0.434366 0.217183 0.976131i \(-0.430313\pi\)
0.217183 + 0.976131i \(0.430313\pi\)
\(812\) −2400.00 −0.103724
\(813\) 0 0
\(814\) −34048.0 −1.46607
\(815\) 0 0
\(816\) 0 0
\(817\) −44200.0 −1.89273
\(818\) −44600.0 −1.90636
\(819\) 0 0
\(820\) 0 0
\(821\) −20562.0 −0.874079 −0.437039 0.899442i \(-0.643973\pi\)
−0.437039 + 0.899442i \(0.643973\pi\)
\(822\) 0 0
\(823\) −10322.0 −0.437184 −0.218592 0.975816i \(-0.570146\pi\)
−0.218592 + 0.975816i \(0.570146\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −12000.0 −0.505488
\(827\) 8846.00 0.371954 0.185977 0.982554i \(-0.440455\pi\)
0.185977 + 0.982554i \(0.440455\pi\)
\(828\) 0 0
\(829\) −25350.0 −1.06205 −0.531026 0.847355i \(-0.678194\pi\)
−0.531026 + 0.847355i \(0.678194\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −19456.0 −0.810716
\(833\) −7982.00 −0.332005
\(834\) 0 0
\(835\) 0 0
\(836\) −25600.0 −1.05908
\(837\) 0 0
\(838\) −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
\(843\) 0 0
\(844\) −9344.00 −0.381083
\(845\) 0 0
\(846\) 0 0
\(847\) 1842.00 0.0747248
\(848\) −128.000 −0.00518342
\(849\) 0 0
\(850\) 0 0
\(851\) 20748.0 0.835761
\(852\) 0 0
\(853\) 16998.0 0.682298 0.341149 0.940009i \(-0.389184\pi\)
0.341149 + 0.940009i \(0.389184\pi\)
\(854\) −12432.0 −0.498143
\(855\) 0 0
\(856\) 0 0
\(857\) −26494.0 −1.05603 −0.528015 0.849235i \(-0.677064\pi\)
−0.528015 + 0.849235i \(0.677064\pi\)
\(858\) 0 0
\(859\) −21500.0 −0.853982 −0.426991 0.904256i \(-0.640426\pi\)
−0.426991 + 0.904256i \(0.640426\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 30768.0 1.21573
\(863\) 25762.0 1.01616 0.508082 0.861309i \(-0.330355\pi\)
0.508082 + 0.861309i \(0.330355\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −4472.00 −0.175479
\(867\) 0 0
\(868\) 5184.00 0.202715
\(869\) −19200.0 −0.749500
\(870\) 0 0
\(871\) −4788.00 −0.186263
\(872\) 0 0
\(873\) 0 0
\(874\) 31200.0 1.20750
\(875\) 0 0
\(876\) 0 0
\(877\) −30546.0 −1.17613 −0.588064 0.808814i \(-0.700110\pi\)
−0.588064 + 0.808814i \(0.700110\pi\)
\(878\) 10400.0 0.399753
\(879\) 0 0
\(880\) 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
\(889\) 11076.0 0.417860
\(890\) 0 0
\(891\) 0 0
\(892\) 51824.0 1.94529
\(893\) −51400.0 −1.92613
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −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
\(906\) 0 0
\(907\) 1794.00 0.0656767 0.0328384 0.999461i \(-0.489545\pi\)
0.0328384 + 0.999461i \(0.489545\pi\)
\(908\) 5168.00 0.188883
\(909\) 0 0
\(910\) 0 0
\(911\) −41732.0 −1.51772 −0.758860 0.651254i \(-0.774243\pi\)
−0.758860 + 0.651254i \(0.774243\pi\)
\(912\) 0 0
\(913\) −9024.00 −0.327109
\(914\) −37976.0 −1.37433
\(915\) 0 0
\(916\) 30000.0 1.08213
\(917\) −13248.0 −0.477086
\(918\) 0 0
\(919\) 29200.0 1.04812 0.524058 0.851682i \(-0.324417\pi\)
0.524058 + 0.851682i \(0.324417\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −45672.0 −1.63137
\(923\) −15656.0 −0.558314
\(924\) 0 0
\(925\) 0 0
\(926\) 31848.0 1.13023
\(927\) 0 0
\(928\) 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
\(931\) −30700.0 −1.08072
\(932\) 11856.0 0.416691
\(933\) 0 0
\(934\) −26104.0 −0.914506
\(935\) 0 0
\(936\) 0 0
\(937\) 11334.0 0.395161 0.197580 0.980287i \(-0.436692\pi\)
0.197580 + 0.980287i \(0.436692\pi\)
\(938\) −3024.00 −0.105263
\(939\) 0 0
\(940\) 0 0
\(941\) 31178.0 1.08010 0.540050 0.841633i \(-0.318405\pi\)
0.540050 + 0.841633i \(0.318405\pi\)
\(942\) 0 0
\(943\) 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 \(-0.474381\pi\)
0.0803984 + 0.996763i \(0.474381\pi\)
\(948\) 0 0
\(949\) 33364.0 1.14124
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −598.000 −0.0203265 −0.0101632 0.999948i \(-0.503235\pi\)
−0.0101632 + 0.999948i \(0.503235\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −11200.0 −0.378906
\(957\) 0 0
\(958\) 69600.0 2.34726
\(959\) 14004.0 0.471546
\(960\) 0 0
\(961\) −18127.0 −0.608472
\(962\) 40432.0 1.35507
\(963\) 0 0
\(964\) 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
\(969\) 0 0
\(970\) 0 0
\(971\) −24312.0 −0.803511 −0.401756 0.915747i \(-0.631600\pi\)
−0.401756 + 0.915747i \(0.631600\pi\)
\(972\) 0 0
\(973\) 4200.00 0.138382
\(974\) 4664.00 0.153433
\(975\) 0 0
\(976\) 33152.0 1.08726
\(977\) 40946.0 1.34082 0.670409 0.741992i \(-0.266119\pi\)
0.670409 + 0.741992i \(0.266119\pi\)
\(978\) 0 0
\(979\) −4800.00 −0.156699
\(980\) 0 0
\(981\) 0 0
\(982\) 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
\(993\) 0 0
\(994\) −9888.00 −0.315521
\(995\) 0 0
\(996\) 0 0
\(997\) 31614.0 1.00424 0.502119 0.864798i \(-0.332554\pi\)
0.502119 + 0.864798i \(0.332554\pi\)
\(998\) −400.000 −0.0126872
\(999\) 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 225.4.a.b.1.1 1
3.2 odd 2 25.4.a.c.1.1 1
5.2 odd 4 225.4.b.c.199.1 2
5.3 odd 4 225.4.b.c.199.2 2
5.4 even 2 45.4.a.d.1.1 1
12.11 even 2 400.4.a.m.1.1 1
15.2 even 4 25.4.b.a.24.2 2
15.8 even 4 25.4.b.a.24.1 2
15.14 odd 2 5.4.a.a.1.1 1
20.19 odd 2 720.4.a.u.1.1 1
21.20 even 2 1225.4.a.k.1.1 1
24.5 odd 2 1600.4.a.bi.1.1 1
24.11 even 2 1600.4.a.s.1.1 1
35.34 odd 2 2205.4.a.q.1.1 1
45.4 even 6 405.4.e.c.136.1 2
45.14 odd 6 405.4.e.l.136.1 2
45.29 odd 6 405.4.e.l.271.1 2
45.34 even 6 405.4.e.c.271.1 2
60.23 odd 4 400.4.c.k.49.2 2
60.47 odd 4 400.4.c.k.49.1 2
60.59 even 2 80.4.a.d.1.1 1
105.44 odd 6 245.4.e.f.116.1 2
105.59 even 6 245.4.e.g.226.1 2
105.74 odd 6 245.4.e.f.226.1 2
105.89 even 6 245.4.e.g.116.1 2
105.104 even 2 245.4.a.a.1.1 1
120.29 odd 2 320.4.a.g.1.1 1
120.59 even 2 320.4.a.h.1.1 1
165.164 even 2 605.4.a.d.1.1 1
195.194 odd 2 845.4.a.b.1.1 1
240.29 odd 4 1280.4.d.e.641.2 2
240.59 even 4 1280.4.d.l.641.2 2
240.149 odd 4 1280.4.d.e.641.1 2
240.179 even 4 1280.4.d.l.641.1 2
255.254 odd 2 1445.4.a.a.1.1 1
285.284 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 15.14 odd 2
25.4.a.c.1.1 1 3.2 odd 2
25.4.b.a.24.1 2 15.8 even 4
25.4.b.a.24.2 2 15.2 even 4
45.4.a.d.1.1 1 5.4 even 2
80.4.a.d.1.1 1 60.59 even 2
225.4.a.b.1.1 1 1.1 even 1 trivial
225.4.b.c.199.1 2 5.2 odd 4
225.4.b.c.199.2 2 5.3 odd 4
245.4.a.a.1.1 1 105.104 even 2
245.4.e.f.116.1 2 105.44 odd 6
245.4.e.f.226.1 2 105.74 odd 6
245.4.e.g.116.1 2 105.89 even 6
245.4.e.g.226.1 2 105.59 even 6
320.4.a.g.1.1 1 120.29 odd 2
320.4.a.h.1.1 1 120.59 even 2
400.4.a.m.1.1 1 12.11 even 2
400.4.c.k.49.1 2 60.47 odd 4
400.4.c.k.49.2 2 60.23 odd 4
405.4.e.c.136.1 2 45.4 even 6
405.4.e.c.271.1 2 45.34 even 6
405.4.e.l.136.1 2 45.14 odd 6
405.4.e.l.271.1 2 45.29 odd 6
605.4.a.d.1.1 1 165.164 even 2
720.4.a.u.1.1 1 20.19 odd 2
845.4.a.b.1.1 1 195.194 odd 2
1225.4.a.k.1.1 1 21.20 even 2
1280.4.d.e.641.1 2 240.149 odd 4
1280.4.d.e.641.2 2 240.29 odd 4
1280.4.d.l.641.1 2 240.179 even 4
1280.4.d.l.641.2 2 240.59 even 4
1445.4.a.a.1.1 1 255.254 odd 2
1600.4.a.s.1.1 1 24.11 even 2
1600.4.a.bi.1.1 1 24.5 odd 2
1805.4.a.h.1.1 1 285.284 even 2
2205.4.a.q.1.1 1 35.34 odd 2