Properties

Label 225.4.a.c.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 25)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 225.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -7.00000 q^{4} +6.00000 q^{7} +15.0000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -7.00000 q^{4} +6.00000 q^{7} +15.0000 q^{8} +43.0000 q^{11} -28.0000 q^{13} -6.00000 q^{14} +41.0000 q^{16} -91.0000 q^{17} -35.0000 q^{19} -43.0000 q^{22} -162.000 q^{23} +28.0000 q^{26} -42.0000 q^{28} -160.000 q^{29} +42.0000 q^{31} -161.000 q^{32} +91.0000 q^{34} -314.000 q^{37} +35.0000 q^{38} +203.000 q^{41} +92.0000 q^{43} -301.000 q^{44} +162.000 q^{46} -196.000 q^{47} -307.000 q^{49} +196.000 q^{52} -82.0000 q^{53} +90.0000 q^{56} +160.000 q^{58} +280.000 q^{59} -518.000 q^{61} -42.0000 q^{62} -167.000 q^{64} +141.000 q^{67} +637.000 q^{68} -412.000 q^{71} -763.000 q^{73} +314.000 q^{74} +245.000 q^{76} +258.000 q^{77} +510.000 q^{79} -203.000 q^{82} -777.000 q^{83} -92.0000 q^{86} +645.000 q^{88} +945.000 q^{89} -168.000 q^{91} +1134.00 q^{92} +196.000 q^{94} +1246.00 q^{97} +307.000 q^{98} +O(q^{100})\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.353553 −0.176777 0.984251i \(-0.556567\pi\)
−0.176777 + 0.984251i \(0.556567\pi\)
\(3\) 0 0
\(4\) −7.00000 −0.875000
\(5\) 0 0
\(6\) 0 0
\(7\) 6.00000 0.323970 0.161985 0.986793i \(-0.448210\pi\)
0.161985 + 0.986793i \(0.448210\pi\)
\(8\) 15.0000 0.662913
\(9\) 0 0
\(10\) 0 0
\(11\) 43.0000 1.17864 0.589318 0.807901i \(-0.299397\pi\)
0.589318 + 0.807901i \(0.299397\pi\)
\(12\) 0 0
\(13\) −28.0000 −0.597369 −0.298685 0.954352i \(-0.596548\pi\)
−0.298685 + 0.954352i \(0.596548\pi\)
\(14\) −6.00000 −0.114541
\(15\) 0 0
\(16\) 41.0000 0.640625
\(17\) −91.0000 −1.29828 −0.649139 0.760669i \(-0.724871\pi\)
−0.649139 + 0.760669i \(0.724871\pi\)
\(18\) 0 0
\(19\) −35.0000 −0.422608 −0.211304 0.977420i \(-0.567771\pi\)
−0.211304 + 0.977420i \(0.567771\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −43.0000 −0.416710
\(23\) −162.000 −1.46867 −0.734333 0.678789i \(-0.762505\pi\)
−0.734333 + 0.678789i \(0.762505\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 28.0000 0.211202
\(27\) 0 0
\(28\) −42.0000 −0.283473
\(29\) −160.000 −1.02453 −0.512263 0.858829i \(-0.671193\pi\)
−0.512263 + 0.858829i \(0.671193\pi\)
\(30\) 0 0
\(31\) 42.0000 0.243336 0.121668 0.992571i \(-0.461176\pi\)
0.121668 + 0.992571i \(0.461176\pi\)
\(32\) −161.000 −0.889408
\(33\) 0 0
\(34\) 91.0000 0.459011
\(35\) 0 0
\(36\) 0 0
\(37\) −314.000 −1.39517 −0.697585 0.716502i \(-0.745742\pi\)
−0.697585 + 0.716502i \(0.745742\pi\)
\(38\) 35.0000 0.149414
\(39\) 0 0
\(40\) 0 0
\(41\) 203.000 0.773251 0.386625 0.922237i \(-0.373641\pi\)
0.386625 + 0.922237i \(0.373641\pi\)
\(42\) 0 0
\(43\) 92.0000 0.326276 0.163138 0.986603i \(-0.447838\pi\)
0.163138 + 0.986603i \(0.447838\pi\)
\(44\) −301.000 −1.03131
\(45\) 0 0
\(46\) 162.000 0.519252
\(47\) −196.000 −0.608288 −0.304144 0.952626i \(-0.598370\pi\)
−0.304144 + 0.952626i \(0.598370\pi\)
\(48\) 0 0
\(49\) −307.000 −0.895044
\(50\) 0 0
\(51\) 0 0
\(52\) 196.000 0.522698
\(53\) −82.0000 −0.212520 −0.106260 0.994338i \(-0.533888\pi\)
−0.106260 + 0.994338i \(0.533888\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 90.0000 0.214763
\(57\) 0 0
\(58\) 160.000 0.362225
\(59\) 280.000 0.617846 0.308923 0.951087i \(-0.400032\pi\)
0.308923 + 0.951087i \(0.400032\pi\)
\(60\) 0 0
\(61\) −518.000 −1.08726 −0.543632 0.839324i \(-0.682951\pi\)
−0.543632 + 0.839324i \(0.682951\pi\)
\(62\) −42.0000 −0.0860323
\(63\) 0 0
\(64\) −167.000 −0.326172
\(65\) 0 0
\(66\) 0 0
\(67\) 141.000 0.257103 0.128551 0.991703i \(-0.458967\pi\)
0.128551 + 0.991703i \(0.458967\pi\)
\(68\) 637.000 1.13599
\(69\) 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
\(73\) −763.000 −1.22332 −0.611660 0.791121i \(-0.709498\pi\)
−0.611660 + 0.791121i \(0.709498\pi\)
\(74\) 314.000 0.493267
\(75\) 0 0
\(76\) 245.000 0.369782
\(77\) 258.000 0.381842
\(78\) 0 0
\(79\) 510.000 0.726323 0.363161 0.931726i \(-0.381697\pi\)
0.363161 + 0.931726i \(0.381697\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −203.000 −0.273385
\(83\) −777.000 −1.02755 −0.513776 0.857924i \(-0.671754\pi\)
−0.513776 + 0.857924i \(0.671754\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −92.0000 −0.115356
\(87\) 0 0
\(88\) 645.000 0.781332
\(89\) 945.000 1.12550 0.562752 0.826626i \(-0.309743\pi\)
0.562752 + 0.826626i \(0.309743\pi\)
\(90\) 0 0
\(91\) −168.000 −0.193530
\(92\) 1134.00 1.28508
\(93\) 0 0
\(94\) 196.000 0.215062
\(95\) 0 0
\(96\) 0 0
\(97\) 1246.00 1.30425 0.652124 0.758112i \(-0.273878\pi\)
0.652124 + 0.758112i \(0.273878\pi\)
\(98\) 307.000 0.316446
\(99\) 0 0
\(100\) 0 0
\(101\) −1302.00 −1.28271 −0.641356 0.767244i \(-0.721628\pi\)
−0.641356 + 0.767244i \(0.721628\pi\)
\(102\) 0 0
\(103\) 532.000 0.508927 0.254464 0.967082i \(-0.418101\pi\)
0.254464 + 0.967082i \(0.418101\pi\)
\(104\) −420.000 −0.396004
\(105\) 0 0
\(106\) 82.0000 0.0751372
\(107\) 1269.00 1.14653 0.573266 0.819370i \(-0.305676\pi\)
0.573266 + 0.819370i \(0.305676\pi\)
\(108\) 0 0
\(109\) 1070.00 0.940251 0.470126 0.882599i \(-0.344209\pi\)
0.470126 + 0.882599i \(0.344209\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 246.000 0.207543
\(113\) 503.000 0.418746 0.209373 0.977836i \(-0.432858\pi\)
0.209373 + 0.977836i \(0.432858\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1120.00 0.896460
\(117\) 0 0
\(118\) −280.000 −0.218441
\(119\) −546.000 −0.420603
\(120\) 0 0
\(121\) 518.000 0.389181
\(122\) 518.000 0.384406
\(123\) 0 0
\(124\) −294.000 −0.212919
\(125\) 0 0
\(126\) 0 0
\(127\) −874.000 −0.610669 −0.305334 0.952245i \(-0.598768\pi\)
−0.305334 + 0.952245i \(0.598768\pi\)
\(128\) 1455.00 1.00473
\(129\) 0 0
\(130\) 0 0
\(131\) −1092.00 −0.728309 −0.364155 0.931339i \(-0.618642\pi\)
−0.364155 + 0.931339i \(0.618642\pi\)
\(132\) 0 0
\(133\) −210.000 −0.136912
\(134\) −141.000 −0.0908996
\(135\) 0 0
\(136\) −1365.00 −0.860645
\(137\) −411.000 −0.256307 −0.128154 0.991754i \(-0.540905\pi\)
−0.128154 + 0.991754i \(0.540905\pi\)
\(138\) 0 0
\(139\) −595.000 −0.363074 −0.181537 0.983384i \(-0.558107\pi\)
−0.181537 + 0.983384i \(0.558107\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 412.000 0.243481
\(143\) −1204.00 −0.704081
\(144\) 0 0
\(145\) 0 0
\(146\) 763.000 0.432509
\(147\) 0 0
\(148\) 2198.00 1.22077
\(149\) 3200.00 1.75942 0.879712 0.475507i \(-0.157735\pi\)
0.879712 + 0.475507i \(0.157735\pi\)
\(150\) 0 0
\(151\) 202.000 0.108864 0.0544322 0.998517i \(-0.482665\pi\)
0.0544322 + 0.998517i \(0.482665\pi\)
\(152\) −525.000 −0.280152
\(153\) 0 0
\(154\) −258.000 −0.135002
\(155\) 0 0
\(156\) 0 0
\(157\) 406.000 0.206384 0.103192 0.994661i \(-0.467094\pi\)
0.103192 + 0.994661i \(0.467094\pi\)
\(158\) −510.000 −0.256794
\(159\) 0 0
\(160\) 0 0
\(161\) −972.000 −0.475803
\(162\) 0 0
\(163\) −3803.00 −1.82745 −0.913724 0.406336i \(-0.866806\pi\)
−0.913724 + 0.406336i \(0.866806\pi\)
\(164\) −1421.00 −0.676594
\(165\) 0 0
\(166\) 777.000 0.363295
\(167\) −4116.00 −1.90722 −0.953610 0.301046i \(-0.902664\pi\)
−0.953610 + 0.301046i \(0.902664\pi\)
\(168\) 0 0
\(169\) −1413.00 −0.643150
\(170\) 0 0
\(171\) 0 0
\(172\) −644.000 −0.285492
\(173\) −1512.00 −0.664481 −0.332241 0.943195i \(-0.607805\pi\)
−0.332241 + 0.943195i \(0.607805\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1763.00 0.755063
\(177\) 0 0
\(178\) −945.000 −0.397926
\(179\) −2585.00 −1.07940 −0.539698 0.841859i \(-0.681462\pi\)
−0.539698 + 0.841859i \(0.681462\pi\)
\(180\) 0 0
\(181\) −2758.00 −1.13260 −0.566300 0.824199i \(-0.691626\pi\)
−0.566300 + 0.824199i \(0.691626\pi\)
\(182\) 168.000 0.0684230
\(183\) 0 0
\(184\) −2430.00 −0.973598
\(185\) 0 0
\(186\) 0 0
\(187\) −3913.00 −1.53020
\(188\) 1372.00 0.532252
\(189\) 0 0
\(190\) 0 0
\(191\) 2378.00 0.900869 0.450435 0.892809i \(-0.351269\pi\)
0.450435 + 0.892809i \(0.351269\pi\)
\(192\) 0 0
\(193\) 3067.00 1.14387 0.571937 0.820298i \(-0.306192\pi\)
0.571937 + 0.820298i \(0.306192\pi\)
\(194\) −1246.00 −0.461122
\(195\) 0 0
\(196\) 2149.00 0.783163
\(197\) −2346.00 −0.848455 −0.424227 0.905556i \(-0.639454\pi\)
−0.424227 + 0.905556i \(0.639454\pi\)
\(198\) 0 0
\(199\) 4900.00 1.74549 0.872743 0.488180i \(-0.162339\pi\)
0.872743 + 0.488180i \(0.162339\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 1302.00 0.453507
\(203\) −960.000 −0.331915
\(204\) 0 0
\(205\) 0 0
\(206\) −532.000 −0.179933
\(207\) 0 0
\(208\) −1148.00 −0.382690
\(209\) −1505.00 −0.498101
\(210\) 0 0
\(211\) 4307.00 1.40524 0.702621 0.711564i \(-0.252013\pi\)
0.702621 + 0.711564i \(0.252013\pi\)
\(212\) 574.000 0.185955
\(213\) 0 0
\(214\) −1269.00 −0.405360
\(215\) 0 0
\(216\) 0 0
\(217\) 252.000 0.0788335
\(218\) −1070.00 −0.332429
\(219\) 0 0
\(220\) 0 0
\(221\) 2548.00 0.775552
\(222\) 0 0
\(223\) 2212.00 0.664244 0.332122 0.943236i \(-0.392235\pi\)
0.332122 + 0.943236i \(0.392235\pi\)
\(224\) −966.000 −0.288141
\(225\) 0 0
\(226\) −503.000 −0.148049
\(227\) −476.000 −0.139177 −0.0695886 0.997576i \(-0.522169\pi\)
−0.0695886 + 0.997576i \(0.522169\pi\)
\(228\) 0 0
\(229\) −2940.00 −0.848387 −0.424194 0.905572i \(-0.639442\pi\)
−0.424194 + 0.905572i \(0.639442\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2400.00 −0.679171
\(233\) −1002.00 −0.281730 −0.140865 0.990029i \(-0.544988\pi\)
−0.140865 + 0.990029i \(0.544988\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −1960.00 −0.540615
\(237\) 0 0
\(238\) 546.000 0.148706
\(239\) −2480.00 −0.671204 −0.335602 0.942004i \(-0.608940\pi\)
−0.335602 + 0.942004i \(0.608940\pi\)
\(240\) 0 0
\(241\) 1897.00 0.507039 0.253520 0.967330i \(-0.418412\pi\)
0.253520 + 0.967330i \(0.418412\pi\)
\(242\) −518.000 −0.137596
\(243\) 0 0
\(244\) 3626.00 0.951356
\(245\) 0 0
\(246\) 0 0
\(247\) 980.000 0.252453
\(248\) 630.000 0.161311
\(249\) 0 0
\(250\) 0 0
\(251\) 2373.00 0.596743 0.298371 0.954450i \(-0.403557\pi\)
0.298371 + 0.954450i \(0.403557\pi\)
\(252\) 0 0
\(253\) −6966.00 −1.73102
\(254\) 874.000 0.215904
\(255\) 0 0
\(256\) −119.000 −0.0290527
\(257\) 4494.00 1.09077 0.545385 0.838185i \(-0.316383\pi\)
0.545385 + 0.838185i \(0.316383\pi\)
\(258\) 0 0
\(259\) −1884.00 −0.451993
\(260\) 0 0
\(261\) 0 0
\(262\) 1092.00 0.257496
\(263\) −722.000 −0.169279 −0.0846396 0.996412i \(-0.526974\pi\)
−0.0846396 + 0.996412i \(0.526974\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 210.000 0.0484057
\(267\) 0 0
\(268\) −987.000 −0.224965
\(269\) 6160.00 1.39621 0.698107 0.715993i \(-0.254026\pi\)
0.698107 + 0.715993i \(0.254026\pi\)
\(270\) 0 0
\(271\) −7238.00 −1.62243 −0.811213 0.584751i \(-0.801192\pi\)
−0.811213 + 0.584751i \(0.801192\pi\)
\(272\) −3731.00 −0.831710
\(273\) 0 0
\(274\) 411.000 0.0906183
\(275\) 0 0
\(276\) 0 0
\(277\) 1776.00 0.385233 0.192616 0.981274i \(-0.438303\pi\)
0.192616 + 0.981274i \(0.438303\pi\)
\(278\) 595.000 0.128366
\(279\) 0 0
\(280\) 0 0
\(281\) −4542.00 −0.964246 −0.482123 0.876104i \(-0.660134\pi\)
−0.482123 + 0.876104i \(0.660134\pi\)
\(282\) 0 0
\(283\) 7077.00 1.48652 0.743258 0.669005i \(-0.233280\pi\)
0.743258 + 0.669005i \(0.233280\pi\)
\(284\) 2884.00 0.602584
\(285\) 0 0
\(286\) 1204.00 0.248930
\(287\) 1218.00 0.250510
\(288\) 0 0
\(289\) 3368.00 0.685528
\(290\) 0 0
\(291\) 0 0
\(292\) 5341.00 1.07041
\(293\) 4158.00 0.829054 0.414527 0.910037i \(-0.363947\pi\)
0.414527 + 0.910037i \(0.363947\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −4710.00 −0.924876
\(297\) 0 0
\(298\) −3200.00 −0.622050
\(299\) 4536.00 0.877337
\(300\) 0 0
\(301\) 552.000 0.105703
\(302\) −202.000 −0.0384894
\(303\) 0 0
\(304\) −1435.00 −0.270733
\(305\) 0 0
\(306\) 0 0
\(307\) −2569.00 −0.477591 −0.238796 0.971070i \(-0.576753\pi\)
−0.238796 + 0.971070i \(0.576753\pi\)
\(308\) −1806.00 −0.334112
\(309\) 0 0
\(310\) 0 0
\(311\) −2982.00 −0.543710 −0.271855 0.962338i \(-0.587637\pi\)
−0.271855 + 0.962338i \(0.587637\pi\)
\(312\) 0 0
\(313\) 2422.00 0.437379 0.218689 0.975795i \(-0.429822\pi\)
0.218689 + 0.975795i \(0.429822\pi\)
\(314\) −406.000 −0.0729679
\(315\) 0 0
\(316\) −3570.00 −0.635532
\(317\) 9484.00 1.68036 0.840181 0.542307i \(-0.182449\pi\)
0.840181 + 0.542307i \(0.182449\pi\)
\(318\) 0 0
\(319\) −6880.00 −1.20754
\(320\) 0 0
\(321\) 0 0
\(322\) 972.000 0.168222
\(323\) 3185.00 0.548663
\(324\) 0 0
\(325\) 0 0
\(326\) 3803.00 0.646100
\(327\) 0 0
\(328\) 3045.00 0.512598
\(329\) −1176.00 −0.197067
\(330\) 0 0
\(331\) −183.000 −0.0303885 −0.0151942 0.999885i \(-0.504837\pi\)
−0.0151942 + 0.999885i \(0.504837\pi\)
\(332\) 5439.00 0.899108
\(333\) 0 0
\(334\) 4116.00 0.674304
\(335\) 0 0
\(336\) 0 0
\(337\) 2861.00 0.462459 0.231229 0.972899i \(-0.425725\pi\)
0.231229 + 0.972899i \(0.425725\pi\)
\(338\) 1413.00 0.227388
\(339\) 0 0
\(340\) 0 0
\(341\) 1806.00 0.286805
\(342\) 0 0
\(343\) −3900.00 −0.613936
\(344\) 1380.00 0.216292
\(345\) 0 0
\(346\) 1512.00 0.234930
\(347\) 629.000 0.0973098 0.0486549 0.998816i \(-0.484507\pi\)
0.0486549 + 0.998816i \(0.484507\pi\)
\(348\) 0 0
\(349\) 5950.00 0.912597 0.456298 0.889827i \(-0.349175\pi\)
0.456298 + 0.889827i \(0.349175\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6923.00 −1.04829
\(353\) 11718.0 1.76682 0.883408 0.468604i \(-0.155243\pi\)
0.883408 + 0.468604i \(0.155243\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −6615.00 −0.984815
\(357\) 0 0
\(358\) 2585.00 0.381624
\(359\) −8070.00 −1.18640 −0.593201 0.805054i \(-0.702136\pi\)
−0.593201 + 0.805054i \(0.702136\pi\)
\(360\) 0 0
\(361\) −5634.00 −0.821403
\(362\) 2758.00 0.400434
\(363\) 0 0
\(364\) 1176.00 0.169338
\(365\) 0 0
\(366\) 0 0
\(367\) 8316.00 1.18281 0.591406 0.806374i \(-0.298573\pi\)
0.591406 + 0.806374i \(0.298573\pi\)
\(368\) −6642.00 −0.940865
\(369\) 0 0
\(370\) 0 0
\(371\) −492.000 −0.0688500
\(372\) 0 0
\(373\) 12062.0 1.67439 0.837194 0.546906i \(-0.184195\pi\)
0.837194 + 0.546906i \(0.184195\pi\)
\(374\) 3913.00 0.541006
\(375\) 0 0
\(376\) −2940.00 −0.403242
\(377\) 4480.00 0.612021
\(378\) 0 0
\(379\) 1735.00 0.235148 0.117574 0.993064i \(-0.462488\pi\)
0.117574 + 0.993064i \(0.462488\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2378.00 −0.318505
\(383\) −7602.00 −1.01421 −0.507107 0.861883i \(-0.669285\pi\)
−0.507107 + 0.861883i \(0.669285\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3067.00 −0.404420
\(387\) 0 0
\(388\) −8722.00 −1.14122
\(389\) −3030.00 −0.394928 −0.197464 0.980310i \(-0.563271\pi\)
−0.197464 + 0.980310i \(0.563271\pi\)
\(390\) 0 0
\(391\) 14742.0 1.90674
\(392\) −4605.00 −0.593336
\(393\) 0 0
\(394\) 2346.00 0.299974
\(395\) 0 0
\(396\) 0 0
\(397\) −1204.00 −0.152209 −0.0761046 0.997100i \(-0.524248\pi\)
−0.0761046 + 0.997100i \(0.524248\pi\)
\(398\) −4900.00 −0.617123
\(399\) 0 0
\(400\) 0 0
\(401\) −1077.00 −0.134122 −0.0670609 0.997749i \(-0.521362\pi\)
−0.0670609 + 0.997749i \(0.521362\pi\)
\(402\) 0 0
\(403\) −1176.00 −0.145362
\(404\) 9114.00 1.12237
\(405\) 0 0
\(406\) 960.000 0.117350
\(407\) −13502.0 −1.64440
\(408\) 0 0
\(409\) −3955.00 −0.478147 −0.239074 0.971001i \(-0.576844\pi\)
−0.239074 + 0.971001i \(0.576844\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −3724.00 −0.445311
\(413\) 1680.00 0.200163
\(414\) 0 0
\(415\) 0 0
\(416\) 4508.00 0.531305
\(417\) 0 0
\(418\) 1505.00 0.176105
\(419\) −6265.00 −0.730466 −0.365233 0.930916i \(-0.619011\pi\)
−0.365233 + 0.930916i \(0.619011\pi\)
\(420\) 0 0
\(421\) −3788.00 −0.438517 −0.219259 0.975667i \(-0.570364\pi\)
−0.219259 + 0.975667i \(0.570364\pi\)
\(422\) −4307.00 −0.496828
\(423\) 0 0
\(424\) −1230.00 −0.140882
\(425\) 0 0
\(426\) 0 0
\(427\) −3108.00 −0.352240
\(428\) −8883.00 −1.00321
\(429\) 0 0
\(430\) 0 0
\(431\) 15258.0 1.70523 0.852613 0.522544i \(-0.175017\pi\)
0.852613 + 0.522544i \(0.175017\pi\)
\(432\) 0 0
\(433\) −13573.0 −1.50641 −0.753206 0.657784i \(-0.771494\pi\)
−0.753206 + 0.657784i \(0.771494\pi\)
\(434\) −252.000 −0.0278719
\(435\) 0 0
\(436\) −7490.00 −0.822720
\(437\) 5670.00 0.620670
\(438\) 0 0
\(439\) −8120.00 −0.882794 −0.441397 0.897312i \(-0.645517\pi\)
−0.441397 + 0.897312i \(0.645517\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −2548.00 −0.274199
\(443\) 6183.00 0.663122 0.331561 0.943434i \(-0.392425\pi\)
0.331561 + 0.943434i \(0.392425\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −2212.00 −0.234846
\(447\) 0 0
\(448\) −1002.00 −0.105670
\(449\) 1975.00 0.207586 0.103793 0.994599i \(-0.466902\pi\)
0.103793 + 0.994599i \(0.466902\pi\)
\(450\) 0 0
\(451\) 8729.00 0.911380
\(452\) −3521.00 −0.366402
\(453\) 0 0
\(454\) 476.000 0.0492066
\(455\) 0 0
\(456\) 0 0
\(457\) 11831.0 1.21101 0.605504 0.795842i \(-0.292971\pi\)
0.605504 + 0.795842i \(0.292971\pi\)
\(458\) 2940.00 0.299950
\(459\) 0 0
\(460\) 0 0
\(461\) −1932.00 −0.195189 −0.0975946 0.995226i \(-0.531115\pi\)
−0.0975946 + 0.995226i \(0.531115\pi\)
\(462\) 0 0
\(463\) −9228.00 −0.926267 −0.463133 0.886289i \(-0.653275\pi\)
−0.463133 + 0.886289i \(0.653275\pi\)
\(464\) −6560.00 −0.656337
\(465\) 0 0
\(466\) 1002.00 0.0996068
\(467\) −13916.0 −1.37892 −0.689460 0.724324i \(-0.742152\pi\)
−0.689460 + 0.724324i \(0.742152\pi\)
\(468\) 0 0
\(469\) 846.000 0.0832935
\(470\) 0 0
\(471\) 0 0
\(472\) 4200.00 0.409578
\(473\) 3956.00 0.384560
\(474\) 0 0
\(475\) 0 0
\(476\) 3822.00 0.368027
\(477\) 0 0
\(478\) 2480.00 0.237307
\(479\) −2310.00 −0.220348 −0.110174 0.993912i \(-0.535141\pi\)
−0.110174 + 0.993912i \(0.535141\pi\)
\(480\) 0 0
\(481\) 8792.00 0.833432
\(482\) −1897.00 −0.179266
\(483\) 0 0
\(484\) −3626.00 −0.340533
\(485\) 0 0
\(486\) 0 0
\(487\) −17114.0 −1.59242 −0.796211 0.605019i \(-0.793165\pi\)
−0.796211 + 0.605019i \(0.793165\pi\)
\(488\) −7770.00 −0.720761
\(489\) 0 0
\(490\) 0 0
\(491\) 17228.0 1.58348 0.791740 0.610858i \(-0.209175\pi\)
0.791740 + 0.610858i \(0.209175\pi\)
\(492\) 0 0
\(493\) 14560.0 1.33012
\(494\) −980.000 −0.0892556
\(495\) 0 0
\(496\) 1722.00 0.155887
\(497\) −2472.00 −0.223107
\(498\) 0 0
\(499\) −12500.0 −1.12140 −0.560698 0.828020i \(-0.689467\pi\)
−0.560698 + 0.828020i \(0.689467\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −2373.00 −0.210980
\(503\) 868.000 0.0769428 0.0384714 0.999260i \(-0.487751\pi\)
0.0384714 + 0.999260i \(0.487751\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6966.00 0.612009
\(507\) 0 0
\(508\) 6118.00 0.534335
\(509\) −13370.0 −1.16427 −0.582136 0.813091i \(-0.697783\pi\)
−0.582136 + 0.813091i \(0.697783\pi\)
\(510\) 0 0
\(511\) −4578.00 −0.396319
\(512\) −11521.0 −0.994455
\(513\) 0 0
\(514\) −4494.00 −0.385646
\(515\) 0 0
\(516\) 0 0
\(517\) −8428.00 −0.716950
\(518\) 1884.00 0.159803
\(519\) 0 0
\(520\) 0 0
\(521\) −21637.0 −1.81945 −0.909726 0.415210i \(-0.863708\pi\)
−0.909726 + 0.415210i \(0.863708\pi\)
\(522\) 0 0
\(523\) 287.000 0.0239955 0.0119977 0.999928i \(-0.496181\pi\)
0.0119977 + 0.999928i \(0.496181\pi\)
\(524\) 7644.00 0.637270
\(525\) 0 0
\(526\) 722.000 0.0598492
\(527\) −3822.00 −0.315918
\(528\) 0 0
\(529\) 14077.0 1.15698
\(530\) 0 0
\(531\) 0 0
\(532\) 1470.00 0.119798
\(533\) −5684.00 −0.461916
\(534\) 0 0
\(535\) 0 0
\(536\) 2115.00 0.170437
\(537\) 0 0
\(538\) −6160.00 −0.493637
\(539\) −13201.0 −1.05493
\(540\) 0 0
\(541\) −5328.00 −0.423417 −0.211709 0.977333i \(-0.567903\pi\)
−0.211709 + 0.977333i \(0.567903\pi\)
\(542\) 7238.00 0.573614
\(543\) 0 0
\(544\) 14651.0 1.15470
\(545\) 0 0
\(546\) 0 0
\(547\) 71.0000 0.00554980 0.00277490 0.999996i \(-0.499117\pi\)
0.00277490 + 0.999996i \(0.499117\pi\)
\(548\) 2877.00 0.224269
\(549\) 0 0
\(550\) 0 0
\(551\) 5600.00 0.432973
\(552\) 0 0
\(553\) 3060.00 0.235306
\(554\) −1776.00 −0.136200
\(555\) 0 0
\(556\) 4165.00 0.317689
\(557\) 18444.0 1.40305 0.701524 0.712646i \(-0.252503\pi\)
0.701524 + 0.712646i \(0.252503\pi\)
\(558\) 0 0
\(559\) −2576.00 −0.194907
\(560\) 0 0
\(561\) 0 0
\(562\) 4542.00 0.340912
\(563\) −672.000 −0.0503045 −0.0251522 0.999684i \(-0.508007\pi\)
−0.0251522 + 0.999684i \(0.508007\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −7077.00 −0.525563
\(567\) 0 0
\(568\) −6180.00 −0.456526
\(569\) 10935.0 0.805657 0.402829 0.915275i \(-0.368027\pi\)
0.402829 + 0.915275i \(0.368027\pi\)
\(570\) 0 0
\(571\) −13588.0 −0.995867 −0.497934 0.867215i \(-0.665908\pi\)
−0.497934 + 0.867215i \(0.665908\pi\)
\(572\) 8428.00 0.616071
\(573\) 0 0
\(574\) −1218.00 −0.0885685
\(575\) 0 0
\(576\) 0 0
\(577\) 8701.00 0.627777 0.313889 0.949460i \(-0.398368\pi\)
0.313889 + 0.949460i \(0.398368\pi\)
\(578\) −3368.00 −0.242371
\(579\) 0 0
\(580\) 0 0
\(581\) −4662.00 −0.332896
\(582\) 0 0
\(583\) −3526.00 −0.250484
\(584\) −11445.0 −0.810955
\(585\) 0 0
\(586\) −4158.00 −0.293115
\(587\) −11361.0 −0.798839 −0.399420 0.916768i \(-0.630788\pi\)
−0.399420 + 0.916768i \(0.630788\pi\)
\(588\) 0 0
\(589\) −1470.00 −0.102836
\(590\) 0 0
\(591\) 0 0
\(592\) −12874.0 −0.893781
\(593\) −11417.0 −0.790624 −0.395312 0.918547i \(-0.629364\pi\)
−0.395312 + 0.918547i \(0.629364\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −22400.0 −1.53950
\(597\) 0 0
\(598\) −4536.00 −0.310185
\(599\) 21050.0 1.43586 0.717930 0.696116i \(-0.245090\pi\)
0.717930 + 0.696116i \(0.245090\pi\)
\(600\) 0 0
\(601\) 7427.00 0.504083 0.252041 0.967716i \(-0.418898\pi\)
0.252041 + 0.967716i \(0.418898\pi\)
\(602\) −552.000 −0.0373718
\(603\) 0 0
\(604\) −1414.00 −0.0952564
\(605\) 0 0
\(606\) 0 0
\(607\) −4144.00 −0.277100 −0.138550 0.990355i \(-0.544244\pi\)
−0.138550 + 0.990355i \(0.544244\pi\)
\(608\) 5635.00 0.375871
\(609\) 0 0
\(610\) 0 0
\(611\) 5488.00 0.363373
\(612\) 0 0
\(613\) 30122.0 1.98469 0.992346 0.123489i \(-0.0394084\pi\)
0.992346 + 0.123489i \(0.0394084\pi\)
\(614\) 2569.00 0.168854
\(615\) 0 0
\(616\) 3870.00 0.253128
\(617\) 11934.0 0.778679 0.389339 0.921094i \(-0.372703\pi\)
0.389339 + 0.921094i \(0.372703\pi\)
\(618\) 0 0
\(619\) 8540.00 0.554526 0.277263 0.960794i \(-0.410573\pi\)
0.277263 + 0.960794i \(0.410573\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 2982.00 0.192230
\(623\) 5670.00 0.364629
\(624\) 0 0
\(625\) 0 0
\(626\) −2422.00 −0.154637
\(627\) 0 0
\(628\) −2842.00 −0.180586
\(629\) 28574.0 1.81132
\(630\) 0 0
\(631\) −3158.00 −0.199236 −0.0996181 0.995026i \(-0.531762\pi\)
−0.0996181 + 0.995026i \(0.531762\pi\)
\(632\) 7650.00 0.481488
\(633\) 0 0
\(634\) −9484.00 −0.594097
\(635\) 0 0
\(636\) 0 0
\(637\) 8596.00 0.534672
\(638\) 6880.00 0.426931
\(639\) 0 0
\(640\) 0 0
\(641\) 4278.00 0.263605 0.131803 0.991276i \(-0.457924\pi\)
0.131803 + 0.991276i \(0.457924\pi\)
\(642\) 0 0
\(643\) −11508.0 −0.705803 −0.352901 0.935661i \(-0.614805\pi\)
−0.352901 + 0.935661i \(0.614805\pi\)
\(644\) 6804.00 0.416328
\(645\) 0 0
\(646\) −3185.00 −0.193982
\(647\) 8204.00 0.498505 0.249252 0.968439i \(-0.419815\pi\)
0.249252 + 0.968439i \(0.419815\pi\)
\(648\) 0 0
\(649\) 12040.0 0.728215
\(650\) 0 0
\(651\) 0 0
\(652\) 26621.0 1.59902
\(653\) 5518.00 0.330683 0.165342 0.986236i \(-0.447127\pi\)
0.165342 + 0.986236i \(0.447127\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 8323.00 0.495364
\(657\) 0 0
\(658\) 1176.00 0.0696736
\(659\) −13295.0 −0.785887 −0.392944 0.919563i \(-0.628543\pi\)
−0.392944 + 0.919563i \(0.628543\pi\)
\(660\) 0 0
\(661\) −9968.00 −0.586551 −0.293276 0.956028i \(-0.594745\pi\)
−0.293276 + 0.956028i \(0.594745\pi\)
\(662\) 183.000 0.0107440
\(663\) 0 0
\(664\) −11655.0 −0.681177
\(665\) 0 0
\(666\) 0 0
\(667\) 25920.0 1.50469
\(668\) 28812.0 1.66882
\(669\) 0 0
\(670\) 0 0
\(671\) −22274.0 −1.28149
\(672\) 0 0
\(673\) −15738.0 −0.901419 −0.450710 0.892671i \(-0.648829\pi\)
−0.450710 + 0.892671i \(0.648829\pi\)
\(674\) −2861.00 −0.163504
\(675\) 0 0
\(676\) 9891.00 0.562756
\(677\) 19824.0 1.12540 0.562702 0.826660i \(-0.309762\pi\)
0.562702 + 0.826660i \(0.309762\pi\)
\(678\) 0 0
\(679\) 7476.00 0.422537
\(680\) 0 0
\(681\) 0 0
\(682\) −1806.00 −0.101401
\(683\) 11073.0 0.620346 0.310173 0.950680i \(-0.399613\pi\)
0.310173 + 0.950680i \(0.399613\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 3900.00 0.217059
\(687\) 0 0
\(688\) 3772.00 0.209021
\(689\) 2296.00 0.126953
\(690\) 0 0
\(691\) −6503.00 −0.358011 −0.179006 0.983848i \(-0.557288\pi\)
−0.179006 + 0.983848i \(0.557288\pi\)
\(692\) 10584.0 0.581421
\(693\) 0 0
\(694\) −629.000 −0.0344042
\(695\) 0 0
\(696\) 0 0
\(697\) −18473.0 −1.00389
\(698\) −5950.00 −0.322652
\(699\) 0 0
\(700\) 0 0
\(701\) 10148.0 0.546768 0.273384 0.961905i \(-0.411857\pi\)
0.273384 + 0.961905i \(0.411857\pi\)
\(702\) 0 0
\(703\) 10990.0 0.589610
\(704\) −7181.00 −0.384438
\(705\) 0 0
\(706\) −11718.0 −0.624664
\(707\) −7812.00 −0.415559
\(708\) 0 0
\(709\) −9980.00 −0.528641 −0.264321 0.964435i \(-0.585148\pi\)
−0.264321 + 0.964435i \(0.585148\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 14175.0 0.746110
\(713\) −6804.00 −0.357380
\(714\) 0 0
\(715\) 0 0
\(716\) 18095.0 0.944472
\(717\) 0 0
\(718\) 8070.00 0.419456
\(719\) 27510.0 1.42691 0.713456 0.700700i \(-0.247129\pi\)
0.713456 + 0.700700i \(0.247129\pi\)
\(720\) 0 0
\(721\) 3192.00 0.164877
\(722\) 5634.00 0.290410
\(723\) 0 0
\(724\) 19306.0 0.991025
\(725\) 0 0
\(726\) 0 0
\(727\) −17024.0 −0.868480 −0.434240 0.900797i \(-0.642983\pi\)
−0.434240 + 0.900797i \(0.642983\pi\)
\(728\) −2520.00 −0.128293
\(729\) 0 0
\(730\) 0 0
\(731\) −8372.00 −0.423597
\(732\) 0 0
\(733\) −34748.0 −1.75095 −0.875475 0.483263i \(-0.839451\pi\)
−0.875475 + 0.483263i \(0.839451\pi\)
\(734\) −8316.00 −0.418187
\(735\) 0 0
\(736\) 26082.0 1.30624
\(737\) 6063.00 0.303030
\(738\) 0 0
\(739\) −12020.0 −0.598326 −0.299163 0.954202i \(-0.596707\pi\)
−0.299163 + 0.954202i \(0.596707\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 492.000 0.0243422
\(743\) −28642.0 −1.41423 −0.707115 0.707098i \(-0.750004\pi\)
−0.707115 + 0.707098i \(0.750004\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −12062.0 −0.591986
\(747\) 0 0
\(748\) 27391.0 1.33892
\(749\) 7614.00 0.371441
\(750\) 0 0
\(751\) 8752.00 0.425253 0.212627 0.977134i \(-0.431798\pi\)
0.212627 + 0.977134i \(0.431798\pi\)
\(752\) −8036.00 −0.389685
\(753\) 0 0
\(754\) −4480.00 −0.216382
\(755\) 0 0
\(756\) 0 0
\(757\) 10256.0 0.492418 0.246209 0.969217i \(-0.420815\pi\)
0.246209 + 0.969217i \(0.420815\pi\)
\(758\) −1735.00 −0.0831373
\(759\) 0 0
\(760\) 0 0
\(761\) −33957.0 −1.61753 −0.808765 0.588132i \(-0.799864\pi\)
−0.808765 + 0.588132i \(0.799864\pi\)
\(762\) 0 0
\(763\) 6420.00 0.304613
\(764\) −16646.0 −0.788261
\(765\) 0 0
\(766\) 7602.00 0.358579
\(767\) −7840.00 −0.369082
\(768\) 0 0
\(769\) 27965.0 1.31137 0.655685 0.755034i \(-0.272380\pi\)
0.655685 + 0.755034i \(0.272380\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −21469.0 −1.00089
\(773\) −9912.00 −0.461203 −0.230601 0.973048i \(-0.574069\pi\)
−0.230601 + 0.973048i \(0.574069\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 18690.0 0.864603
\(777\) 0 0
\(778\) 3030.00 0.139628
\(779\) −7105.00 −0.326782
\(780\) 0 0
\(781\) −17716.0 −0.811688
\(782\) −14742.0 −0.674134
\(783\) 0 0
\(784\) −12587.0 −0.573387
\(785\) 0 0
\(786\) 0 0
\(787\) −25564.0 −1.15789 −0.578944 0.815367i \(-0.696535\pi\)
−0.578944 + 0.815367i \(0.696535\pi\)
\(788\) 16422.0 0.742398
\(789\) 0 0
\(790\) 0 0
\(791\) 3018.00 0.135661
\(792\) 0 0
\(793\) 14504.0 0.649498
\(794\) 1204.00 0.0538141
\(795\) 0 0
\(796\) −34300.0 −1.52730
\(797\) −12446.0 −0.553149 −0.276575 0.960992i \(-0.589199\pi\)
−0.276575 + 0.960992i \(0.589199\pi\)
\(798\) 0 0
\(799\) 17836.0 0.789728
\(800\) 0 0
\(801\) 0 0
\(802\) 1077.00 0.0474192
\(803\) −32809.0 −1.44185
\(804\) 0 0
\(805\) 0 0
\(806\) 1176.00 0.0513931
\(807\) 0 0
\(808\) −19530.0 −0.850325
\(809\) −33970.0 −1.47629 −0.738147 0.674640i \(-0.764299\pi\)
−0.738147 + 0.674640i \(0.764299\pi\)
\(810\) 0 0
\(811\) 18732.0 0.811060 0.405530 0.914082i \(-0.367087\pi\)
0.405530 + 0.914082i \(0.367087\pi\)
\(812\) 6720.00 0.290426
\(813\) 0 0
\(814\) 13502.0 0.581382
\(815\) 0 0
\(816\) 0 0
\(817\) −3220.00 −0.137887
\(818\) 3955.00 0.169051
\(819\) 0 0
\(820\) 0 0
\(821\) −6162.00 −0.261943 −0.130972 0.991386i \(-0.541810\pi\)
−0.130972 + 0.991386i \(0.541810\pi\)
\(822\) 0 0
\(823\) −25388.0 −1.07530 −0.537649 0.843169i \(-0.680687\pi\)
−0.537649 + 0.843169i \(0.680687\pi\)
\(824\) 7980.00 0.337374
\(825\) 0 0
\(826\) −1680.00 −0.0707684
\(827\) −25201.0 −1.05964 −0.529821 0.848109i \(-0.677741\pi\)
−0.529821 + 0.848109i \(0.677741\pi\)
\(828\) 0 0
\(829\) −19740.0 −0.827019 −0.413509 0.910500i \(-0.635697\pi\)
−0.413509 + 0.910500i \(0.635697\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 4676.00 0.194845
\(833\) 27937.0 1.16202
\(834\) 0 0
\(835\) 0 0
\(836\) 10535.0 0.435838
\(837\) 0 0
\(838\) 6265.00 0.258259
\(839\) −29680.0 −1.22130 −0.610648 0.791902i \(-0.709091\pi\)
−0.610648 + 0.791902i \(0.709091\pi\)
\(840\) 0 0
\(841\) 1211.00 0.0496535
\(842\) 3788.00 0.155039
\(843\) 0 0
\(844\) −30149.0 −1.22959
\(845\) 0 0
\(846\) 0 0
\(847\) 3108.00 0.126083
\(848\) −3362.00 −0.136146
\(849\) 0 0
\(850\) 0 0
\(851\) 50868.0 2.04904
\(852\) 0 0
\(853\) −1218.00 −0.0488904 −0.0244452 0.999701i \(-0.507782\pi\)
−0.0244452 + 0.999701i \(0.507782\pi\)
\(854\) 3108.00 0.124536
\(855\) 0 0
\(856\) 19035.0 0.760050
\(857\) −38731.0 −1.54379 −0.771894 0.635752i \(-0.780690\pi\)
−0.771894 + 0.635752i \(0.780690\pi\)
\(858\) 0 0
\(859\) −23555.0 −0.935607 −0.467803 0.883833i \(-0.654954\pi\)
−0.467803 + 0.883833i \(0.654954\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −15258.0 −0.602888
\(863\) −24872.0 −0.981058 −0.490529 0.871425i \(-0.663196\pi\)
−0.490529 + 0.871425i \(0.663196\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 13573.0 0.532597
\(867\) 0 0
\(868\) −1764.00 −0.0689793
\(869\) 21930.0 0.856069
\(870\) 0 0
\(871\) −3948.00 −0.153585
\(872\) 16050.0 0.623305
\(873\) 0 0
\(874\) −5670.00 −0.219440
\(875\) 0 0
\(876\) 0 0
\(877\) −17124.0 −0.659335 −0.329667 0.944097i \(-0.606937\pi\)
−0.329667 + 0.944097i \(0.606937\pi\)
\(878\) 8120.00 0.312115
\(879\) 0 0
\(880\) 0 0
\(881\) 658.000 0.0251630 0.0125815 0.999921i \(-0.495995\pi\)
0.0125815 + 0.999921i \(0.495995\pi\)
\(882\) 0 0
\(883\) 33727.0 1.28540 0.642698 0.766120i \(-0.277815\pi\)
0.642698 + 0.766120i \(0.277815\pi\)
\(884\) −17836.0 −0.678608
\(885\) 0 0
\(886\) −6183.00 −0.234449
\(887\) −36036.0 −1.36412 −0.682058 0.731298i \(-0.738915\pi\)
−0.682058 + 0.731298i \(0.738915\pi\)
\(888\) 0 0
\(889\) −5244.00 −0.197838
\(890\) 0 0
\(891\) 0 0
\(892\) −15484.0 −0.581214
\(893\) 6860.00 0.257067
\(894\) 0 0
\(895\) 0 0
\(896\) 8730.00 0.325501
\(897\) 0 0
\(898\) −1975.00 −0.0733927
\(899\) −6720.00 −0.249304
\(900\) 0 0
\(901\) 7462.00 0.275910
\(902\) −8729.00 −0.322222
\(903\) 0 0
\(904\) 7545.00 0.277592
\(905\) 0 0
\(906\) 0 0
\(907\) 39156.0 1.43347 0.716733 0.697348i \(-0.245637\pi\)
0.716733 + 0.697348i \(0.245637\pi\)
\(908\) 3332.00 0.121780
\(909\) 0 0
\(910\) 0 0
\(911\) −43532.0 −1.58318 −0.791591 0.611051i \(-0.790747\pi\)
−0.791591 + 0.611051i \(0.790747\pi\)
\(912\) 0 0
\(913\) −33411.0 −1.21111
\(914\) −11831.0 −0.428156
\(915\) 0 0
\(916\) 20580.0 0.742339
\(917\) −6552.00 −0.235950
\(918\) 0 0
\(919\) −28610.0 −1.02694 −0.513469 0.858108i \(-0.671640\pi\)
−0.513469 + 0.858108i \(0.671640\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1932.00 0.0690098
\(923\) 11536.0 0.411389
\(924\) 0 0
\(925\) 0 0
\(926\) 9228.00 0.327485
\(927\) 0 0
\(928\) 25760.0 0.911221
\(929\) 24290.0 0.857835 0.428918 0.903344i \(-0.358895\pi\)
0.428918 + 0.903344i \(0.358895\pi\)
\(930\) 0 0
\(931\) 10745.0 0.378253
\(932\) 7014.00 0.246514
\(933\) 0 0
\(934\) 13916.0 0.487522
\(935\) 0 0
\(936\) 0 0
\(937\) 34461.0 1.20149 0.600743 0.799442i \(-0.294872\pi\)
0.600743 + 0.799442i \(0.294872\pi\)
\(938\) −846.000 −0.0294487
\(939\) 0 0
\(940\) 0 0
\(941\) 40628.0 1.40748 0.703738 0.710460i \(-0.251513\pi\)
0.703738 + 0.710460i \(0.251513\pi\)
\(942\) 0 0
\(943\) −32886.0 −1.13565
\(944\) 11480.0 0.395807
\(945\) 0 0
\(946\) −3956.00 −0.135963
\(947\) 20904.0 0.717306 0.358653 0.933471i \(-0.383236\pi\)
0.358653 + 0.933471i \(0.383236\pi\)
\(948\) 0 0
\(949\) 21364.0 0.730774
\(950\) 0 0
\(951\) 0 0
\(952\) −8190.00 −0.278823
\(953\) −1807.00 −0.0614213 −0.0307106 0.999528i \(-0.509777\pi\)
−0.0307106 + 0.999528i \(0.509777\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 17360.0 0.587304
\(957\) 0 0
\(958\) 2310.00 0.0779047
\(959\) −2466.00 −0.0830358
\(960\) 0 0
\(961\) −28027.0 −0.940787
\(962\) −8792.00 −0.294663
\(963\) 0 0
\(964\) −13279.0 −0.443660
\(965\) 0 0
\(966\) 0 0
\(967\) −57584.0 −1.91497 −0.957485 0.288482i \(-0.906849\pi\)
−0.957485 + 0.288482i \(0.906849\pi\)
\(968\) 7770.00 0.257993
\(969\) 0 0
\(970\) 0 0
\(971\) −27237.0 −0.900182 −0.450091 0.892983i \(-0.648608\pi\)
−0.450091 + 0.892983i \(0.648608\pi\)
\(972\) 0 0
\(973\) −3570.00 −0.117625
\(974\) 17114.0 0.563006
\(975\) 0 0
\(976\) −21238.0 −0.696528
\(977\) 13649.0 0.446950 0.223475 0.974710i \(-0.428260\pi\)
0.223475 + 0.974710i \(0.428260\pi\)
\(978\) 0 0
\(979\) 40635.0 1.32656
\(980\) 0 0
\(981\) 0 0
\(982\) −17228.0 −0.559845
\(983\) −16002.0 −0.519211 −0.259606 0.965715i \(-0.583593\pi\)
−0.259606 + 0.965715i \(0.583593\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −14560.0 −0.470269
\(987\) 0 0
\(988\) −6860.00 −0.220896
\(989\) −14904.0 −0.479191
\(990\) 0 0
\(991\) 37022.0 1.18672 0.593362 0.804936i \(-0.297800\pi\)
0.593362 + 0.804936i \(0.297800\pi\)
\(992\) −6762.00 −0.216425
\(993\) 0 0
\(994\) 2472.00 0.0788804
\(995\) 0 0
\(996\) 0 0
\(997\) 18396.0 0.584360 0.292180 0.956363i \(-0.405619\pi\)
0.292180 + 0.956363i \(0.405619\pi\)
\(998\) 12500.0 0.396474
\(999\) 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.c.1.1 1
3.2 odd 2 25.4.a.b.1.1 yes 1
5.2 odd 4 225.4.b.f.199.1 2
5.3 odd 4 225.4.b.f.199.2 2
5.4 even 2 225.4.a.e.1.1 1
12.11 even 2 400.4.a.c.1.1 1
15.2 even 4 25.4.b.b.24.2 2
15.8 even 4 25.4.b.b.24.1 2
15.14 odd 2 25.4.a.a.1.1 1
21.20 even 2 1225.4.a.i.1.1 1
24.5 odd 2 1600.4.a.i.1.1 1
24.11 even 2 1600.4.a.bs.1.1 1
60.23 odd 4 400.4.c.e.49.1 2
60.47 odd 4 400.4.c.e.49.2 2
60.59 even 2 400.4.a.s.1.1 1
105.104 even 2 1225.4.a.h.1.1 1
120.29 odd 2 1600.4.a.bt.1.1 1
120.59 even 2 1600.4.a.h.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
25.4.a.a.1.1 1 15.14 odd 2
25.4.a.b.1.1 yes 1 3.2 odd 2
25.4.b.b.24.1 2 15.8 even 4
25.4.b.b.24.2 2 15.2 even 4
225.4.a.c.1.1 1 1.1 even 1 trivial
225.4.a.e.1.1 1 5.4 even 2
225.4.b.f.199.1 2 5.2 odd 4
225.4.b.f.199.2 2 5.3 odd 4
400.4.a.c.1.1 1 12.11 even 2
400.4.a.s.1.1 1 60.59 even 2
400.4.c.e.49.1 2 60.23 odd 4
400.4.c.e.49.2 2 60.47 odd 4
1225.4.a.h.1.1 1 105.104 even 2
1225.4.a.i.1.1 1 21.20 even 2
1600.4.a.h.1.1 1 120.59 even 2
1600.4.a.i.1.1 1 24.5 odd 2
1600.4.a.bs.1.1 1 24.11 even 2
1600.4.a.bt.1.1 1 120.29 odd 2