Properties

Label 1575.4.a.t.1.1
Level $1575$
Weight $4$
Character 1575.1
Self dual yes
Analytic conductor $92.928$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1575 = 3^{2} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1575.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(92.9280082590\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{19}) \)
Defining polynomial: \(x^{2} - 19\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 63)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.35890\) of defining polynomial
Character \(\chi\) \(=\) 1575.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.35890 q^{2} +11.0000 q^{4} +7.00000 q^{7} -13.0767 q^{8} +O(q^{10})\) \(q-4.35890 q^{2} +11.0000 q^{4} +7.00000 q^{7} -13.0767 q^{8} -43.5890 q^{11} -82.0000 q^{13} -30.5123 q^{14} -31.0000 q^{16} +78.4602 q^{17} -20.0000 q^{19} +190.000 q^{22} +130.767 q^{23} +357.430 q^{26} +77.0000 q^{28} +244.098 q^{29} +156.000 q^{31} +239.739 q^{32} -342.000 q^{34} -186.000 q^{37} +87.1780 q^{38} -165.638 q^{41} -164.000 q^{43} -479.479 q^{44} -570.000 q^{46} -470.761 q^{47} +49.0000 q^{49} -902.000 q^{52} +156.920 q^{53} -91.5369 q^{56} -1064.00 q^{58} +156.920 q^{59} +790.000 q^{61} -679.988 q^{62} -797.000 q^{64} +44.0000 q^{67} +863.062 q^{68} +444.608 q^{71} -126.000 q^{73} +810.755 q^{74} -220.000 q^{76} -305.123 q^{77} -712.000 q^{79} +722.000 q^{82} +1464.59 q^{83} +714.859 q^{86} +570.000 q^{88} -1455.87 q^{89} -574.000 q^{91} +1438.44 q^{92} +2052.00 q^{94} -798.000 q^{97} -213.586 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 22q^{4} + 14q^{7} + O(q^{10}) \) \( 2q + 22q^{4} + 14q^{7} - 164q^{13} - 62q^{16} - 40q^{19} + 380q^{22} + 154q^{28} + 312q^{31} - 684q^{34} - 372q^{37} - 328q^{43} - 1140q^{46} + 98q^{49} - 1804q^{52} - 2128q^{58} + 1580q^{61} - 1594q^{64} + 88q^{67} - 252q^{73} - 440q^{76} - 1424q^{79} + 1444q^{82} + 1140q^{88} - 1148q^{91} + 4104q^{94} - 1596q^{97} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.35890 −1.54110 −0.770552 0.637377i \(-0.780019\pi\)
−0.770552 + 0.637377i \(0.780019\pi\)
\(3\) 0 0
\(4\) 11.0000 1.37500
\(5\) 0 0
\(6\) 0 0
\(7\) 7.00000 0.377964
\(8\) −13.0767 −0.577914
\(9\) 0 0
\(10\) 0 0
\(11\) −43.5890 −1.19478 −0.597390 0.801951i \(-0.703795\pi\)
−0.597390 + 0.801951i \(0.703795\pi\)
\(12\) 0 0
\(13\) −82.0000 −1.74944 −0.874720 0.484629i \(-0.838954\pi\)
−0.874720 + 0.484629i \(0.838954\pi\)
\(14\) −30.5123 −0.582482
\(15\) 0 0
\(16\) −31.0000 −0.484375
\(17\) 78.4602 1.11938 0.559688 0.828703i \(-0.310921\pi\)
0.559688 + 0.828703i \(0.310921\pi\)
\(18\) 0 0
\(19\) −20.0000 −0.241490 −0.120745 0.992684i \(-0.538528\pi\)
−0.120745 + 0.992684i \(0.538528\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 190.000 1.84128
\(23\) 130.767 1.18551 0.592756 0.805382i \(-0.298040\pi\)
0.592756 + 0.805382i \(0.298040\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 357.430 2.69607
\(27\) 0 0
\(28\) 77.0000 0.519701
\(29\) 244.098 1.56303 0.781516 0.623885i \(-0.214447\pi\)
0.781516 + 0.623885i \(0.214447\pi\)
\(30\) 0 0
\(31\) 156.000 0.903820 0.451910 0.892063i \(-0.350743\pi\)
0.451910 + 0.892063i \(0.350743\pi\)
\(32\) 239.739 1.32439
\(33\) 0 0
\(34\) −342.000 −1.72507
\(35\) 0 0
\(36\) 0 0
\(37\) −186.000 −0.826438 −0.413219 0.910632i \(-0.635596\pi\)
−0.413219 + 0.910632i \(0.635596\pi\)
\(38\) 87.1780 0.372161
\(39\) 0 0
\(40\) 0 0
\(41\) −165.638 −0.630935 −0.315467 0.948936i \(-0.602161\pi\)
−0.315467 + 0.948936i \(0.602161\pi\)
\(42\) 0 0
\(43\) −164.000 −0.581622 −0.290811 0.956780i \(-0.593925\pi\)
−0.290811 + 0.956780i \(0.593925\pi\)
\(44\) −479.479 −1.64282
\(45\) 0 0
\(46\) −570.000 −1.82700
\(47\) −470.761 −1.46101 −0.730506 0.682906i \(-0.760716\pi\)
−0.730506 + 0.682906i \(0.760716\pi\)
\(48\) 0 0
\(49\) 49.0000 0.142857
\(50\) 0 0
\(51\) 0 0
\(52\) −902.000 −2.40548
\(53\) 156.920 0.406692 0.203346 0.979107i \(-0.434818\pi\)
0.203346 + 0.979107i \(0.434818\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −91.5369 −0.218431
\(57\) 0 0
\(58\) −1064.00 −2.40879
\(59\) 156.920 0.346259 0.173130 0.984899i \(-0.444612\pi\)
0.173130 + 0.984899i \(0.444612\pi\)
\(60\) 0 0
\(61\) 790.000 1.65818 0.829091 0.559113i \(-0.188858\pi\)
0.829091 + 0.559113i \(0.188858\pi\)
\(62\) −679.988 −1.39288
\(63\) 0 0
\(64\) −797.000 −1.55664
\(65\) 0 0
\(66\) 0 0
\(67\) 44.0000 0.0802307 0.0401153 0.999195i \(-0.487227\pi\)
0.0401153 + 0.999195i \(0.487227\pi\)
\(68\) 863.062 1.53914
\(69\) 0 0
\(70\) 0 0
\(71\) 444.608 0.743172 0.371586 0.928398i \(-0.378814\pi\)
0.371586 + 0.928398i \(0.378814\pi\)
\(72\) 0 0
\(73\) −126.000 −0.202016 −0.101008 0.994886i \(-0.532207\pi\)
−0.101008 + 0.994886i \(0.532207\pi\)
\(74\) 810.755 1.27363
\(75\) 0 0
\(76\) −220.000 −0.332049
\(77\) −305.123 −0.451584
\(78\) 0 0
\(79\) −712.000 −1.01400 −0.507002 0.861945i \(-0.669246\pi\)
−0.507002 + 0.861945i \(0.669246\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 722.000 0.972336
\(83\) 1464.59 1.93686 0.968432 0.249280i \(-0.0801938\pi\)
0.968432 + 0.249280i \(0.0801938\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 714.859 0.896340
\(87\) 0 0
\(88\) 570.000 0.690480
\(89\) −1455.87 −1.73396 −0.866978 0.498346i \(-0.833941\pi\)
−0.866978 + 0.498346i \(0.833941\pi\)
\(90\) 0 0
\(91\) −574.000 −0.661226
\(92\) 1438.44 1.63008
\(93\) 0 0
\(94\) 2052.00 2.25157
\(95\) 0 0
\(96\) 0 0
\(97\) −798.000 −0.835305 −0.417653 0.908607i \(-0.637147\pi\)
−0.417653 + 0.908607i \(0.637147\pi\)
\(98\) −213.586 −0.220158
\(99\) 0 0
\(100\) 0 0
\(101\) 409.737 0.403666 0.201833 0.979420i \(-0.435310\pi\)
0.201833 + 0.979420i \(0.435310\pi\)
\(102\) 0 0
\(103\) 916.000 0.876273 0.438137 0.898908i \(-0.355639\pi\)
0.438137 + 0.898908i \(0.355639\pi\)
\(104\) 1072.29 1.01103
\(105\) 0 0
\(106\) −684.000 −0.626754
\(107\) −897.933 −0.811275 −0.405638 0.914034i \(-0.632951\pi\)
−0.405638 + 0.914034i \(0.632951\pi\)
\(108\) 0 0
\(109\) −342.000 −0.300529 −0.150264 0.988646i \(-0.548013\pi\)
−0.150264 + 0.988646i \(0.548013\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −217.000 −0.183077
\(113\) 488.197 0.406422 0.203211 0.979135i \(-0.434862\pi\)
0.203211 + 0.979135i \(0.434862\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 2685.08 2.14917
\(117\) 0 0
\(118\) −684.000 −0.533621
\(119\) 549.221 0.423084
\(120\) 0 0
\(121\) 569.000 0.427498
\(122\) −3443.53 −2.55543
\(123\) 0 0
\(124\) 1716.00 1.24275
\(125\) 0 0
\(126\) 0 0
\(127\) −456.000 −0.318610 −0.159305 0.987229i \(-0.550925\pi\)
−0.159305 + 0.987229i \(0.550925\pi\)
\(128\) 1556.13 1.07456
\(129\) 0 0
\(130\) 0 0
\(131\) 1499.46 1.00007 0.500033 0.866007i \(-0.333321\pi\)
0.500033 + 0.866007i \(0.333321\pi\)
\(132\) 0 0
\(133\) −140.000 −0.0912747
\(134\) −191.792 −0.123644
\(135\) 0 0
\(136\) −1026.00 −0.646903
\(137\) 889.215 0.554531 0.277266 0.960793i \(-0.410572\pi\)
0.277266 + 0.960793i \(0.410572\pi\)
\(138\) 0 0
\(139\) −768.000 −0.468640 −0.234320 0.972160i \(-0.575286\pi\)
−0.234320 + 0.972160i \(0.575286\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −1938.00 −1.14531
\(143\) 3574.30 2.09019
\(144\) 0 0
\(145\) 0 0
\(146\) 549.221 0.311328
\(147\) 0 0
\(148\) −2046.00 −1.13635
\(149\) 993.829 0.546427 0.273214 0.961953i \(-0.411913\pi\)
0.273214 + 0.961953i \(0.411913\pi\)
\(150\) 0 0
\(151\) −3496.00 −1.88411 −0.942054 0.335460i \(-0.891108\pi\)
−0.942054 + 0.335460i \(0.891108\pi\)
\(152\) 261.534 0.139561
\(153\) 0 0
\(154\) 1330.00 0.695938
\(155\) 0 0
\(156\) 0 0
\(157\) 506.000 0.257218 0.128609 0.991695i \(-0.458949\pi\)
0.128609 + 0.991695i \(0.458949\pi\)
\(158\) 3103.54 1.56268
\(159\) 0 0
\(160\) 0 0
\(161\) 915.369 0.448082
\(162\) 0 0
\(163\) 2564.00 1.23207 0.616037 0.787717i \(-0.288737\pi\)
0.616037 + 0.787717i \(0.288737\pi\)
\(164\) −1822.02 −0.867536
\(165\) 0 0
\(166\) −6384.00 −2.98491
\(167\) 645.117 0.298926 0.149463 0.988767i \(-0.452245\pi\)
0.149463 + 0.988767i \(0.452245\pi\)
\(168\) 0 0
\(169\) 4527.00 2.06054
\(170\) 0 0
\(171\) 0 0
\(172\) −1804.00 −0.799731
\(173\) −3861.98 −1.69723 −0.848616 0.529009i \(-0.822564\pi\)
−0.848616 + 0.529009i \(0.822564\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1351.26 0.578721
\(177\) 0 0
\(178\) 6346.00 2.67221
\(179\) −270.252 −0.112847 −0.0564234 0.998407i \(-0.517970\pi\)
−0.0564234 + 0.998407i \(0.517970\pi\)
\(180\) 0 0
\(181\) 418.000 0.171656 0.0858279 0.996310i \(-0.472646\pi\)
0.0858279 + 0.996310i \(0.472646\pi\)
\(182\) 2502.01 1.01902
\(183\) 0 0
\(184\) −1710.00 −0.685124
\(185\) 0 0
\(186\) 0 0
\(187\) −3420.00 −1.33741
\(188\) −5178.37 −2.00889
\(189\) 0 0
\(190\) 0 0
\(191\) 1525.61 0.577956 0.288978 0.957336i \(-0.406685\pi\)
0.288978 + 0.957336i \(0.406685\pi\)
\(192\) 0 0
\(193\) −1358.00 −0.506482 −0.253241 0.967403i \(-0.581497\pi\)
−0.253241 + 0.967403i \(0.581497\pi\)
\(194\) 3478.40 1.28729
\(195\) 0 0
\(196\) 539.000 0.196429
\(197\) −3748.65 −1.35574 −0.677869 0.735183i \(-0.737096\pi\)
−0.677869 + 0.735183i \(0.737096\pi\)
\(198\) 0 0
\(199\) −1056.00 −0.376170 −0.188085 0.982153i \(-0.560228\pi\)
−0.188085 + 0.982153i \(0.560228\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −1786.00 −0.622092
\(203\) 1708.69 0.590771
\(204\) 0 0
\(205\) 0 0
\(206\) −3992.75 −1.35043
\(207\) 0 0
\(208\) 2542.00 0.847385
\(209\) 871.780 0.288528
\(210\) 0 0
\(211\) −3620.00 −1.18110 −0.590548 0.807003i \(-0.701088\pi\)
−0.590548 + 0.807003i \(0.701088\pi\)
\(212\) 1726.12 0.559201
\(213\) 0 0
\(214\) 3914.00 1.25026
\(215\) 0 0
\(216\) 0 0
\(217\) 1092.00 0.341612
\(218\) 1490.74 0.463146
\(219\) 0 0
\(220\) 0 0
\(221\) −6433.73 −1.95828
\(222\) 0 0
\(223\) −5368.00 −1.61196 −0.805982 0.591940i \(-0.798362\pi\)
−0.805982 + 0.591940i \(0.798362\pi\)
\(224\) 1678.18 0.500571
\(225\) 0 0
\(226\) −2128.00 −0.626338
\(227\) −1621.51 −0.474112 −0.237056 0.971496i \(-0.576182\pi\)
−0.237056 + 0.971496i \(0.576182\pi\)
\(228\) 0 0
\(229\) 2186.00 0.630808 0.315404 0.948958i \(-0.397860\pi\)
0.315404 + 0.948958i \(0.397860\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −3192.00 −0.903298
\(233\) 4132.24 1.16185 0.580927 0.813956i \(-0.302690\pi\)
0.580927 + 0.813956i \(0.302690\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1726.12 0.476106
\(237\) 0 0
\(238\) −2394.00 −0.652017
\(239\) −4838.38 −1.30949 −0.654746 0.755849i \(-0.727224\pi\)
−0.654746 + 0.755849i \(0.727224\pi\)
\(240\) 0 0
\(241\) 1286.00 0.343728 0.171864 0.985121i \(-0.445021\pi\)
0.171864 + 0.985121i \(0.445021\pi\)
\(242\) −2480.21 −0.658819
\(243\) 0 0
\(244\) 8690.00 2.28000
\(245\) 0 0
\(246\) 0 0
\(247\) 1640.00 0.422472
\(248\) −2039.96 −0.522330
\(249\) 0 0
\(250\) 0 0
\(251\) −1795.87 −0.451610 −0.225805 0.974173i \(-0.572501\pi\)
−0.225805 + 0.974173i \(0.572501\pi\)
\(252\) 0 0
\(253\) −5700.00 −1.41643
\(254\) 1987.66 0.491011
\(255\) 0 0
\(256\) −407.000 −0.0993652
\(257\) 1944.07 0.471859 0.235929 0.971770i \(-0.424187\pi\)
0.235929 + 0.971770i \(0.424187\pi\)
\(258\) 0 0
\(259\) −1302.00 −0.312364
\(260\) 0 0
\(261\) 0 0
\(262\) −6536.00 −1.54120
\(263\) −5344.01 −1.25295 −0.626475 0.779442i \(-0.715503\pi\)
−0.626475 + 0.779442i \(0.715503\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 610.246 0.140664
\(267\) 0 0
\(268\) 484.000 0.110317
\(269\) −4001.47 −0.906966 −0.453483 0.891265i \(-0.649819\pi\)
−0.453483 + 0.891265i \(0.649819\pi\)
\(270\) 0 0
\(271\) 2788.00 0.624941 0.312470 0.949928i \(-0.398844\pi\)
0.312470 + 0.949928i \(0.398844\pi\)
\(272\) −2432.27 −0.542198
\(273\) 0 0
\(274\) −3876.00 −0.854590
\(275\) 0 0
\(276\) 0 0
\(277\) 4562.00 0.989545 0.494773 0.869022i \(-0.335251\pi\)
0.494773 + 0.869022i \(0.335251\pi\)
\(278\) 3347.63 0.722222
\(279\) 0 0
\(280\) 0 0
\(281\) −1551.77 −0.329433 −0.164717 0.986341i \(-0.552671\pi\)
−0.164717 + 0.986341i \(0.552671\pi\)
\(282\) 0 0
\(283\) 6788.00 1.42581 0.712906 0.701260i \(-0.247379\pi\)
0.712906 + 0.701260i \(0.247379\pi\)
\(284\) 4890.68 1.02186
\(285\) 0 0
\(286\) −15580.0 −3.22121
\(287\) −1159.47 −0.238471
\(288\) 0 0
\(289\) 1243.00 0.253002
\(290\) 0 0
\(291\) 0 0
\(292\) −1386.00 −0.277772
\(293\) −1142.03 −0.227707 −0.113854 0.993498i \(-0.536319\pi\)
−0.113854 + 0.993498i \(0.536319\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 2432.27 0.477610
\(297\) 0 0
\(298\) −4332.00 −0.842101
\(299\) −10722.9 −2.07398
\(300\) 0 0
\(301\) −1148.00 −0.219833
\(302\) 15238.7 2.90361
\(303\) 0 0
\(304\) 620.000 0.116972
\(305\) 0 0
\(306\) 0 0
\(307\) −532.000 −0.0989018 −0.0494509 0.998777i \(-0.515747\pi\)
−0.0494509 + 0.998777i \(0.515747\pi\)
\(308\) −3356.35 −0.620928
\(309\) 0 0
\(310\) 0 0
\(311\) 6538.35 1.19214 0.596070 0.802932i \(-0.296728\pi\)
0.596070 + 0.802932i \(0.296728\pi\)
\(312\) 0 0
\(313\) −4994.00 −0.901845 −0.450923 0.892563i \(-0.648905\pi\)
−0.450923 + 0.892563i \(0.648905\pi\)
\(314\) −2205.60 −0.396399
\(315\) 0 0
\(316\) −7832.00 −1.39425
\(317\) 470.761 0.0834088 0.0417044 0.999130i \(-0.486721\pi\)
0.0417044 + 0.999130i \(0.486721\pi\)
\(318\) 0 0
\(319\) −10640.0 −1.86748
\(320\) 0 0
\(321\) 0 0
\(322\) −3990.00 −0.690540
\(323\) −1569.20 −0.270318
\(324\) 0 0
\(325\) 0 0
\(326\) −11176.2 −1.89875
\(327\) 0 0
\(328\) 2166.00 0.364626
\(329\) −3295.33 −0.552211
\(330\) 0 0
\(331\) 2588.00 0.429756 0.214878 0.976641i \(-0.431065\pi\)
0.214878 + 0.976641i \(0.431065\pi\)
\(332\) 16110.5 2.66319
\(333\) 0 0
\(334\) −2812.00 −0.460676
\(335\) 0 0
\(336\) 0 0
\(337\) −238.000 −0.0384709 −0.0192354 0.999815i \(-0.506123\pi\)
−0.0192354 + 0.999815i \(0.506123\pi\)
\(338\) −19732.7 −3.17550
\(339\) 0 0
\(340\) 0 0
\(341\) −6799.88 −1.07987
\(342\) 0 0
\(343\) 343.000 0.0539949
\(344\) 2144.58 0.336128
\(345\) 0 0
\(346\) 16834.0 2.61561
\(347\) −7052.70 −1.09109 −0.545546 0.838081i \(-0.683678\pi\)
−0.545546 + 0.838081i \(0.683678\pi\)
\(348\) 0 0
\(349\) −10850.0 −1.66415 −0.832073 0.554666i \(-0.812846\pi\)
−0.832073 + 0.554666i \(0.812846\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −10450.0 −1.58235
\(353\) −5291.70 −0.797872 −0.398936 0.916979i \(-0.630621\pi\)
−0.398936 + 0.916979i \(0.630621\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −16014.6 −2.38419
\(357\) 0 0
\(358\) 1178.00 0.173908
\(359\) −4820.94 −0.708745 −0.354373 0.935104i \(-0.615306\pi\)
−0.354373 + 0.935104i \(0.615306\pi\)
\(360\) 0 0
\(361\) −6459.00 −0.941682
\(362\) −1822.02 −0.264539
\(363\) 0 0
\(364\) −6314.00 −0.909186
\(365\) 0 0
\(366\) 0 0
\(367\) −11712.0 −1.66583 −0.832917 0.553397i \(-0.813331\pi\)
−0.832917 + 0.553397i \(0.813331\pi\)
\(368\) −4053.78 −0.574233
\(369\) 0 0
\(370\) 0 0
\(371\) 1098.44 0.153715
\(372\) 0 0
\(373\) 10450.0 1.45062 0.725309 0.688423i \(-0.241697\pi\)
0.725309 + 0.688423i \(0.241697\pi\)
\(374\) 14907.4 2.06108
\(375\) 0 0
\(376\) 6156.00 0.844339
\(377\) −20016.1 −2.73443
\(378\) 0 0
\(379\) −756.000 −0.102462 −0.0512310 0.998687i \(-0.516314\pi\)
−0.0512310 + 0.998687i \(0.516314\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −6650.00 −0.890690
\(383\) −6381.43 −0.851373 −0.425686 0.904871i \(-0.639967\pi\)
−0.425686 + 0.904871i \(0.639967\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5919.38 0.780541
\(387\) 0 0
\(388\) −8778.00 −1.14854
\(389\) −418.454 −0.0545411 −0.0272705 0.999628i \(-0.508682\pi\)
−0.0272705 + 0.999628i \(0.508682\pi\)
\(390\) 0 0
\(391\) 10260.0 1.32703
\(392\) −640.758 −0.0825591
\(393\) 0 0
\(394\) 16340.0 2.08933
\(395\) 0 0
\(396\) 0 0
\(397\) 5802.00 0.733486 0.366743 0.930322i \(-0.380473\pi\)
0.366743 + 0.930322i \(0.380473\pi\)
\(398\) 4603.00 0.579717
\(399\) 0 0
\(400\) 0 0
\(401\) −4132.24 −0.514599 −0.257299 0.966332i \(-0.582833\pi\)
−0.257299 + 0.966332i \(0.582833\pi\)
\(402\) 0 0
\(403\) −12792.0 −1.58118
\(404\) 4507.10 0.555041
\(405\) 0 0
\(406\) −7448.00 −0.910439
\(407\) 8107.55 0.987411
\(408\) 0 0
\(409\) −1330.00 −0.160793 −0.0803964 0.996763i \(-0.525619\pi\)
−0.0803964 + 0.996763i \(0.525619\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 10076.0 1.20488
\(413\) 1098.44 0.130874
\(414\) 0 0
\(415\) 0 0
\(416\) −19658.6 −2.31693
\(417\) 0 0
\(418\) −3800.00 −0.444651
\(419\) 10409.1 1.21364 0.606820 0.794839i \(-0.292445\pi\)
0.606820 + 0.794839i \(0.292445\pi\)
\(420\) 0 0
\(421\) −12274.0 −1.42090 −0.710449 0.703749i \(-0.751508\pi\)
−0.710449 + 0.703749i \(0.751508\pi\)
\(422\) 15779.2 1.82019
\(423\) 0 0
\(424\) −2052.00 −0.235033
\(425\) 0 0
\(426\) 0 0
\(427\) 5530.00 0.626734
\(428\) −9877.27 −1.11550
\(429\) 0 0
\(430\) 0 0
\(431\) −4681.46 −0.523197 −0.261598 0.965177i \(-0.584250\pi\)
−0.261598 + 0.965177i \(0.584250\pi\)
\(432\) 0 0
\(433\) −5770.00 −0.640389 −0.320195 0.947352i \(-0.603748\pi\)
−0.320195 + 0.947352i \(0.603748\pi\)
\(434\) −4759.92 −0.526459
\(435\) 0 0
\(436\) −3762.00 −0.413227
\(437\) −2615.34 −0.286290
\(438\) 0 0
\(439\) −1872.00 −0.203521 −0.101760 0.994809i \(-0.532448\pi\)
−0.101760 + 0.994809i \(0.532448\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 28044.0 3.01791
\(443\) 11115.2 1.19210 0.596048 0.802949i \(-0.296737\pi\)
0.596048 + 0.802949i \(0.296737\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 23398.6 2.48420
\(447\) 0 0
\(448\) −5579.00 −0.588355
\(449\) −7636.79 −0.802678 −0.401339 0.915930i \(-0.631455\pi\)
−0.401339 + 0.915930i \(0.631455\pi\)
\(450\) 0 0
\(451\) 7220.00 0.753828
\(452\) 5370.16 0.558830
\(453\) 0 0
\(454\) 7068.00 0.730656
\(455\) 0 0
\(456\) 0 0
\(457\) −15142.0 −1.54992 −0.774959 0.632011i \(-0.782230\pi\)
−0.774959 + 0.632011i \(0.782230\pi\)
\(458\) −9528.55 −0.972140
\(459\) 0 0
\(460\) 0 0
\(461\) −13190.0 −1.33258 −0.666292 0.745691i \(-0.732119\pi\)
−0.666292 + 0.745691i \(0.732119\pi\)
\(462\) 0 0
\(463\) −9328.00 −0.936304 −0.468152 0.883648i \(-0.655080\pi\)
−0.468152 + 0.883648i \(0.655080\pi\)
\(464\) −7567.05 −0.757094
\(465\) 0 0
\(466\) −18012.0 −1.79054
\(467\) −3399.94 −0.336896 −0.168448 0.985711i \(-0.553876\pi\)
−0.168448 + 0.985711i \(0.553876\pi\)
\(468\) 0 0
\(469\) 308.000 0.0303243
\(470\) 0 0
\(471\) 0 0
\(472\) −2052.00 −0.200108
\(473\) 7148.59 0.694911
\(474\) 0 0
\(475\) 0 0
\(476\) 6041.43 0.581741
\(477\) 0 0
\(478\) 21090.0 2.01806
\(479\) 3574.30 0.340947 0.170474 0.985362i \(-0.445470\pi\)
0.170474 + 0.985362i \(0.445470\pi\)
\(480\) 0 0
\(481\) 15252.0 1.44580
\(482\) −5605.54 −0.529721
\(483\) 0 0
\(484\) 6259.00 0.587810
\(485\) 0 0
\(486\) 0 0
\(487\) −8968.00 −0.834454 −0.417227 0.908802i \(-0.636998\pi\)
−0.417227 + 0.908802i \(0.636998\pi\)
\(488\) −10330.6 −0.958287
\(489\) 0 0
\(490\) 0 0
\(491\) 5169.65 0.475159 0.237580 0.971368i \(-0.423646\pi\)
0.237580 + 0.971368i \(0.423646\pi\)
\(492\) 0 0
\(493\) 19152.0 1.74962
\(494\) −7148.59 −0.651074
\(495\) 0 0
\(496\) −4836.00 −0.437788
\(497\) 3112.25 0.280893
\(498\) 0 0
\(499\) 3940.00 0.353464 0.176732 0.984259i \(-0.443447\pi\)
0.176732 + 0.984259i \(0.443447\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 7828.00 0.695978
\(503\) −10252.1 −0.908787 −0.454394 0.890801i \(-0.650144\pi\)
−0.454394 + 0.890801i \(0.650144\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 24845.7 2.18286
\(507\) 0 0
\(508\) −5016.00 −0.438089
\(509\) 9772.65 0.851012 0.425506 0.904956i \(-0.360096\pi\)
0.425506 + 0.904956i \(0.360096\pi\)
\(510\) 0 0
\(511\) −882.000 −0.0763550
\(512\) −10674.9 −0.921426
\(513\) 0 0
\(514\) −8474.00 −0.727183
\(515\) 0 0
\(516\) 0 0
\(517\) 20520.0 1.74559
\(518\) 5675.29 0.481386
\(519\) 0 0
\(520\) 0 0
\(521\) −7401.41 −0.622383 −0.311192 0.950347i \(-0.600728\pi\)
−0.311192 + 0.950347i \(0.600728\pi\)
\(522\) 0 0
\(523\) 2768.00 0.231427 0.115713 0.993283i \(-0.463085\pi\)
0.115713 + 0.993283i \(0.463085\pi\)
\(524\) 16494.1 1.37509
\(525\) 0 0
\(526\) 23294.0 1.93093
\(527\) 12239.8 1.01171
\(528\) 0 0
\(529\) 4933.00 0.405441
\(530\) 0 0
\(531\) 0 0
\(532\) −1540.00 −0.125503
\(533\) 13582.3 1.10378
\(534\) 0 0
\(535\) 0 0
\(536\) −575.375 −0.0463664
\(537\) 0 0
\(538\) 17442.0 1.39773
\(539\) −2135.86 −0.170683
\(540\) 0 0
\(541\) 16310.0 1.29616 0.648079 0.761573i \(-0.275573\pi\)
0.648079 + 0.761573i \(0.275573\pi\)
\(542\) −12152.6 −0.963098
\(543\) 0 0
\(544\) 18810.0 1.48249
\(545\) 0 0
\(546\) 0 0
\(547\) −11140.0 −0.870771 −0.435386 0.900244i \(-0.643388\pi\)
−0.435386 + 0.900244i \(0.643388\pi\)
\(548\) 9781.37 0.762481
\(549\) 0 0
\(550\) 0 0
\(551\) −4881.97 −0.377457
\(552\) 0 0
\(553\) −4984.00 −0.383257
\(554\) −19885.3 −1.52499
\(555\) 0 0
\(556\) −8448.00 −0.644380
\(557\) −22788.3 −1.73352 −0.866761 0.498723i \(-0.833803\pi\)
−0.866761 + 0.498723i \(0.833803\pi\)
\(558\) 0 0
\(559\) 13448.0 1.01751
\(560\) 0 0
\(561\) 0 0
\(562\) 6764.00 0.507691
\(563\) 11524.9 0.862732 0.431366 0.902177i \(-0.358032\pi\)
0.431366 + 0.902177i \(0.358032\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −29588.2 −2.19732
\(567\) 0 0
\(568\) −5814.00 −0.429489
\(569\) −1691.25 −0.124606 −0.0623032 0.998057i \(-0.519845\pi\)
−0.0623032 + 0.998057i \(0.519845\pi\)
\(570\) 0 0
\(571\) 11228.0 0.822902 0.411451 0.911432i \(-0.365022\pi\)
0.411451 + 0.911432i \(0.365022\pi\)
\(572\) 39317.3 2.87402
\(573\) 0 0
\(574\) 5054.00 0.367509
\(575\) 0 0
\(576\) 0 0
\(577\) −2050.00 −0.147907 −0.0739537 0.997262i \(-0.523562\pi\)
−0.0739537 + 0.997262i \(0.523562\pi\)
\(578\) −5418.11 −0.389903
\(579\) 0 0
\(580\) 0 0
\(581\) 10252.1 0.732065
\(582\) 0 0
\(583\) −6840.00 −0.485907
\(584\) 1647.66 0.116748
\(585\) 0 0
\(586\) 4978.00 0.350920
\(587\) −18394.6 −1.29340 −0.646699 0.762745i \(-0.723851\pi\)
−0.646699 + 0.762745i \(0.723851\pi\)
\(588\) 0 0
\(589\) −3120.00 −0.218264
\(590\) 0 0
\(591\) 0 0
\(592\) 5766.00 0.400306
\(593\) 12632.1 0.874769 0.437384 0.899275i \(-0.355905\pi\)
0.437384 + 0.899275i \(0.355905\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 10932.1 0.751337
\(597\) 0 0
\(598\) 46740.0 3.19622
\(599\) −9598.30 −0.654717 −0.327359 0.944900i \(-0.606159\pi\)
−0.327359 + 0.944900i \(0.606159\pi\)
\(600\) 0 0
\(601\) −10758.0 −0.730163 −0.365082 0.930976i \(-0.618959\pi\)
−0.365082 + 0.930976i \(0.618959\pi\)
\(602\) 5004.02 0.338785
\(603\) 0 0
\(604\) −38456.0 −2.59065
\(605\) 0 0
\(606\) 0 0
\(607\) 21352.0 1.42776 0.713881 0.700268i \(-0.246936\pi\)
0.713881 + 0.700268i \(0.246936\pi\)
\(608\) −4794.79 −0.319826
\(609\) 0 0
\(610\) 0 0
\(611\) 38602.4 2.55595
\(612\) 0 0
\(613\) 5714.00 0.376487 0.188243 0.982122i \(-0.439721\pi\)
0.188243 + 0.982122i \(0.439721\pi\)
\(614\) 2318.93 0.152418
\(615\) 0 0
\(616\) 3990.00 0.260977
\(617\) −6747.58 −0.440271 −0.220135 0.975469i \(-0.570650\pi\)
−0.220135 + 0.975469i \(0.570650\pi\)
\(618\) 0 0
\(619\) −1880.00 −0.122074 −0.0610368 0.998136i \(-0.519441\pi\)
−0.0610368 + 0.998136i \(0.519441\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −28500.0 −1.83721
\(623\) −10191.1 −0.655374
\(624\) 0 0
\(625\) 0 0
\(626\) 21768.3 1.38984
\(627\) 0 0
\(628\) 5566.00 0.353674
\(629\) −14593.6 −0.925095
\(630\) 0 0
\(631\) −28888.0 −1.82252 −0.911262 0.411826i \(-0.864891\pi\)
−0.911262 + 0.411826i \(0.864891\pi\)
\(632\) 9310.61 0.586006
\(633\) 0 0
\(634\) −2052.00 −0.128542
\(635\) 0 0
\(636\) 0 0
\(637\) −4018.00 −0.249920
\(638\) 46378.7 2.87798
\(639\) 0 0
\(640\) 0 0
\(641\) 25996.5 1.60187 0.800935 0.598751i \(-0.204336\pi\)
0.800935 + 0.598751i \(0.204336\pi\)
\(642\) 0 0
\(643\) −24788.0 −1.52029 −0.760143 0.649756i \(-0.774871\pi\)
−0.760143 + 0.649756i \(0.774871\pi\)
\(644\) 10069.1 0.616112
\(645\) 0 0
\(646\) 6840.00 0.416589
\(647\) −28472.3 −1.73008 −0.865041 0.501702i \(-0.832708\pi\)
−0.865041 + 0.501702i \(0.832708\pi\)
\(648\) 0 0
\(649\) −6840.00 −0.413703
\(650\) 0 0
\(651\) 0 0
\(652\) 28204.0 1.69410
\(653\) −3243.02 −0.194348 −0.0971740 0.995267i \(-0.530980\pi\)
−0.0971740 + 0.995267i \(0.530980\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 5134.78 0.305609
\(657\) 0 0
\(658\) 14364.0 0.851014
\(659\) 1176.90 0.0695685 0.0347842 0.999395i \(-0.488926\pi\)
0.0347842 + 0.999395i \(0.488926\pi\)
\(660\) 0 0
\(661\) 11590.0 0.681995 0.340998 0.940064i \(-0.389235\pi\)
0.340998 + 0.940064i \(0.389235\pi\)
\(662\) −11280.8 −0.662299
\(663\) 0 0
\(664\) −19152.0 −1.11934
\(665\) 0 0
\(666\) 0 0
\(667\) 31920.0 1.85299
\(668\) 7096.29 0.411023
\(669\) 0 0
\(670\) 0 0
\(671\) −34435.3 −1.98116
\(672\) 0 0
\(673\) −23062.0 −1.32091 −0.660457 0.750864i \(-0.729637\pi\)
−0.660457 + 0.750864i \(0.729637\pi\)
\(674\) 1037.42 0.0592876
\(675\) 0 0
\(676\) 49797.0 2.83324
\(677\) 22884.2 1.29913 0.649566 0.760305i \(-0.274951\pi\)
0.649566 + 0.760305i \(0.274951\pi\)
\(678\) 0 0
\(679\) −5586.00 −0.315716
\(680\) 0 0
\(681\) 0 0
\(682\) 29640.0 1.66419
\(683\) 24715.0 1.38461 0.692307 0.721603i \(-0.256594\pi\)
0.692307 + 0.721603i \(0.256594\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −1495.10 −0.0832118
\(687\) 0 0
\(688\) 5084.00 0.281723
\(689\) −12867.5 −0.711483
\(690\) 0 0
\(691\) −10600.0 −0.583564 −0.291782 0.956485i \(-0.594248\pi\)
−0.291782 + 0.956485i \(0.594248\pi\)
\(692\) −42481.8 −2.33369
\(693\) 0 0
\(694\) 30742.0 1.68148
\(695\) 0 0
\(696\) 0 0
\(697\) −12996.0 −0.706253
\(698\) 47294.1 2.56462
\(699\) 0 0
\(700\) 0 0
\(701\) 12449.0 0.670746 0.335373 0.942085i \(-0.391138\pi\)
0.335373 + 0.942085i \(0.391138\pi\)
\(702\) 0 0
\(703\) 3720.00 0.199577
\(704\) 34740.4 1.85984
\(705\) 0 0
\(706\) 23066.0 1.22960
\(707\) 2868.16 0.152572
\(708\) 0 0
\(709\) −13710.0 −0.726220 −0.363110 0.931746i \(-0.618285\pi\)
−0.363110 + 0.931746i \(0.618285\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 19038.0 1.00208
\(713\) 20399.6 1.07149
\(714\) 0 0
\(715\) 0 0
\(716\) −2972.77 −0.155164
\(717\) 0 0
\(718\) 21014.0 1.09225
\(719\) −2510.73 −0.130228 −0.0651142 0.997878i \(-0.520741\pi\)
−0.0651142 + 0.997878i \(0.520741\pi\)
\(720\) 0 0
\(721\) 6412.00 0.331200
\(722\) 28154.1 1.45123
\(723\) 0 0
\(724\) 4598.00 0.236027
\(725\) 0 0
\(726\) 0 0
\(727\) −620.000 −0.0316293 −0.0158147 0.999875i \(-0.505034\pi\)
−0.0158147 + 0.999875i \(0.505034\pi\)
\(728\) 7506.02 0.382132
\(729\) 0 0
\(730\) 0 0
\(731\) −12867.5 −0.651054
\(732\) 0 0
\(733\) 20214.0 1.01858 0.509291 0.860594i \(-0.329908\pi\)
0.509291 + 0.860594i \(0.329908\pi\)
\(734\) 51051.4 2.56722
\(735\) 0 0
\(736\) 31350.0 1.57008
\(737\) −1917.92 −0.0958580
\(738\) 0 0
\(739\) −12324.0 −0.613458 −0.306729 0.951797i \(-0.599235\pi\)
−0.306729 + 0.951797i \(0.599235\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −4788.00 −0.236891
\(743\) −29736.4 −1.46827 −0.734134 0.679005i \(-0.762412\pi\)
−0.734134 + 0.679005i \(0.762412\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −45550.5 −2.23555
\(747\) 0 0
\(748\) −37620.0 −1.83894
\(749\) −6285.53 −0.306633
\(750\) 0 0
\(751\) 19336.0 0.939522 0.469761 0.882794i \(-0.344340\pi\)
0.469761 + 0.882794i \(0.344340\pi\)
\(752\) 14593.6 0.707678
\(753\) 0 0
\(754\) 87248.0 4.21404
\(755\) 0 0
\(756\) 0 0
\(757\) −15986.0 −0.767531 −0.383766 0.923431i \(-0.625373\pi\)
−0.383766 + 0.923431i \(0.625373\pi\)
\(758\) 3295.33 0.157905
\(759\) 0 0
\(760\) 0 0
\(761\) 37007.1 1.76282 0.881409 0.472354i \(-0.156596\pi\)
0.881409 + 0.472354i \(0.156596\pi\)
\(762\) 0 0
\(763\) −2394.00 −0.113589
\(764\) 16781.8 0.794690
\(765\) 0 0
\(766\) 27816.0 1.31205
\(767\) −12867.5 −0.605759
\(768\) 0 0
\(769\) −36070.0 −1.69144 −0.845720 0.533627i \(-0.820829\pi\)
−0.845720 + 0.533627i \(0.820829\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −14938.0 −0.696412
\(773\) 531.786 0.0247439 0.0123719 0.999923i \(-0.496062\pi\)
0.0123719 + 0.999923i \(0.496062\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 10435.2 0.482735
\(777\) 0 0
\(778\) 1824.00 0.0840534
\(779\) 3312.76 0.152365
\(780\) 0 0
\(781\) −19380.0 −0.887927
\(782\) −44722.3 −2.04510
\(783\) 0 0
\(784\) −1519.00 −0.0691964
\(785\) 0 0
\(786\) 0 0
\(787\) 1136.00 0.0514537 0.0257268 0.999669i \(-0.491810\pi\)
0.0257268 + 0.999669i \(0.491810\pi\)
\(788\) −41235.2 −1.86414
\(789\) 0 0
\(790\) 0 0
\(791\) 3417.38 0.153613
\(792\) 0 0
\(793\) −64780.0 −2.90089
\(794\) −25290.3 −1.13038
\(795\) 0 0
\(796\) −11616.0 −0.517234
\(797\) −18054.6 −0.802416 −0.401208 0.915987i \(-0.631409\pi\)
−0.401208 + 0.915987i \(0.631409\pi\)
\(798\) 0 0
\(799\) −36936.0 −1.63542
\(800\) 0 0
\(801\) 0 0
\(802\) 18012.0 0.793050
\(803\) 5492.21 0.241365
\(804\) 0 0
\(805\) 0 0
\(806\) 55759.0 2.43676
\(807\) 0 0
\(808\) −5358.00 −0.233284
\(809\) −38707.0 −1.68216 −0.841079 0.540912i \(-0.818079\pi\)
−0.841079 + 0.540912i \(0.818079\pi\)
\(810\) 0 0
\(811\) 17936.0 0.776595 0.388297 0.921534i \(-0.373063\pi\)
0.388297 + 0.921534i \(0.373063\pi\)
\(812\) 18795.6 0.812309
\(813\) 0 0
\(814\) −35340.0 −1.52170
\(815\) 0 0
\(816\) 0 0
\(817\) 3280.00 0.140456
\(818\) 5797.34 0.247798
\(819\) 0 0
\(820\) 0 0
\(821\) 24915.5 1.05914 0.529571 0.848266i \(-0.322353\pi\)
0.529571 + 0.848266i \(0.322353\pi\)
\(822\) 0 0
\(823\) −23424.0 −0.992113 −0.496057 0.868290i \(-0.665219\pi\)
−0.496057 + 0.868290i \(0.665219\pi\)
\(824\) −11978.3 −0.506411
\(825\) 0 0
\(826\) −4788.00 −0.201690
\(827\) 26650.3 1.12058 0.560291 0.828296i \(-0.310689\pi\)
0.560291 + 0.828296i \(0.310689\pi\)
\(828\) 0 0
\(829\) −26254.0 −1.09993 −0.549963 0.835189i \(-0.685358\pi\)
−0.549963 + 0.835189i \(0.685358\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 65354.0 2.72325
\(833\) 3844.55 0.159911
\(834\) 0 0
\(835\) 0 0
\(836\) 9589.58 0.396725
\(837\) 0 0
\(838\) −45372.0 −1.87035
\(839\) −6189.64 −0.254696 −0.127348 0.991858i \(-0.540647\pi\)
−0.127348 + 0.991858i \(0.540647\pi\)
\(840\) 0 0
\(841\) 35195.0 1.44307
\(842\) 53501.1 2.18975
\(843\) 0 0
\(844\) −39820.0 −1.62401
\(845\) 0 0
\(846\) 0 0
\(847\) 3983.00 0.161579
\(848\) −4864.53 −0.196991
\(849\) 0 0
\(850\) 0 0
\(851\) −24322.7 −0.979753
\(852\) 0 0
\(853\) 45322.0 1.81922 0.909611 0.415462i \(-0.136380\pi\)
0.909611 + 0.415462i \(0.136380\pi\)
\(854\) −24104.7 −0.965862
\(855\) 0 0
\(856\) 11742.0 0.468847
\(857\) −8691.64 −0.346442 −0.173221 0.984883i \(-0.555418\pi\)
−0.173221 + 0.984883i \(0.555418\pi\)
\(858\) 0 0
\(859\) −43252.0 −1.71797 −0.858987 0.511998i \(-0.828906\pi\)
−0.858987 + 0.511998i \(0.828906\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 20406.0 0.806301
\(863\) 29318.0 1.15642 0.578212 0.815886i \(-0.303750\pi\)
0.578212 + 0.815886i \(0.303750\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 25150.8 0.986906
\(867\) 0 0
\(868\) 12012.0 0.469716
\(869\) 31035.4 1.21151
\(870\) 0 0
\(871\) −3608.00 −0.140359
\(872\) 4472.23 0.173680
\(873\) 0 0
\(874\) 11400.0 0.441202
\(875\) 0 0
\(876\) 0 0
\(877\) −47110.0 −1.81390 −0.906951 0.421237i \(-0.861596\pi\)
−0.906951 + 0.421237i \(0.861596\pi\)
\(878\) 8159.86 0.313647
\(879\) 0 0
\(880\) 0 0
\(881\) −42133.1 −1.61124 −0.805619 0.592434i \(-0.798167\pi\)
−0.805619 + 0.592434i \(0.798167\pi\)
\(882\) 0 0
\(883\) 22732.0 0.866356 0.433178 0.901308i \(-0.357392\pi\)
0.433178 + 0.901308i \(0.357392\pi\)
\(884\) −70771.1 −2.69263
\(885\) 0 0
\(886\) −48450.0 −1.83714
\(887\) −21463.2 −0.812474 −0.406237 0.913768i \(-0.633159\pi\)
−0.406237 + 0.913768i \(0.633159\pi\)
\(888\) 0 0
\(889\) −3192.00 −0.120423
\(890\) 0 0
\(891\) 0 0
\(892\) −59048.0 −2.21645
\(893\) 9415.22 0.352820
\(894\) 0 0
\(895\) 0 0
\(896\) 10892.9 0.406145
\(897\) 0 0
\(898\) 33288.0 1.23701
\(899\) 38079.3 1.41270
\(900\) 0 0
\(901\) 12312.0 0.455241
\(902\) −31471.3 −1.16173
\(903\) 0 0
\(904\) −6384.00 −0.234877
\(905\) 0 0
\(906\) 0 0
\(907\) 6916.00 0.253189 0.126594 0.991955i \(-0.459595\pi\)
0.126594 + 0.991955i \(0.459595\pi\)
\(908\) −17836.6 −0.651904
\(909\) 0 0
\(910\) 0 0
\(911\) 38210.1 1.38963 0.694817 0.719186i \(-0.255485\pi\)
0.694817 + 0.719186i \(0.255485\pi\)
\(912\) 0 0
\(913\) −63840.0 −2.31412
\(914\) 66002.4 2.38859
\(915\) 0 0
\(916\) 24046.0 0.867360
\(917\) 10496.2 0.377989
\(918\) 0 0
\(919\) −47632.0 −1.70972 −0.854861 0.518857i \(-0.826358\pi\)
−0.854861 + 0.518857i \(0.826358\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 57494.0 2.05365
\(923\) −36457.8 −1.30013
\(924\) 0 0
\(925\) 0 0
\(926\) 40659.8 1.44294
\(927\) 0 0
\(928\) 58520.0 2.07006
\(929\) −3304.05 −0.116687 −0.0583435 0.998297i \(-0.518582\pi\)
−0.0583435 + 0.998297i \(0.518582\pi\)
\(930\) 0 0
\(931\) −980.000 −0.0344986
\(932\) 45454.6 1.59755
\(933\) 0 0
\(934\) 14820.0 0.519192
\(935\) 0 0
\(936\) 0 0
\(937\) −21858.0 −0.762081 −0.381040 0.924558i \(-0.624434\pi\)
−0.381040 + 0.924558i \(0.624434\pi\)
\(938\) −1342.54 −0.0467330
\(939\) 0 0
\(940\) 0 0
\(941\) −50380.2 −1.74532 −0.872660 0.488328i \(-0.837607\pi\)
−0.872660 + 0.488328i \(0.837607\pi\)
\(942\) 0 0
\(943\) −21660.0 −0.747982
\(944\) −4864.53 −0.167719
\(945\) 0 0
\(946\) −31160.0 −1.07093
\(947\) −31061.5 −1.06585 −0.532927 0.846161i \(-0.678908\pi\)
−0.532927 + 0.846161i \(0.678908\pi\)
\(948\) 0 0
\(949\) 10332.0 0.353415
\(950\) 0 0
\(951\) 0 0
\(952\) −7182.00 −0.244506
\(953\) −22770.9 −0.773999 −0.387000 0.922080i \(-0.626489\pi\)
−0.387000 + 0.922080i \(0.626489\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −53222.2 −1.80055
\(957\) 0 0
\(958\) −15580.0 −0.525435
\(959\) 6224.51 0.209593
\(960\) 0 0
\(961\) −5455.00 −0.183109
\(962\) −66481.9 −2.22813
\(963\) 0 0
\(964\) 14146.0 0.472627
\(965\) 0 0
\(966\) 0 0
\(967\) 36416.0 1.21102 0.605512 0.795836i \(-0.292968\pi\)
0.605512 + 0.795836i \(0.292968\pi\)
\(968\) −7440.64 −0.247057
\(969\) 0 0
\(970\) 0 0
\(971\) −18621.2 −0.615431 −0.307715 0.951478i \(-0.599564\pi\)
−0.307715 + 0.951478i \(0.599564\pi\)
\(972\) 0 0
\(973\) −5376.00 −0.177129
\(974\) 39090.6 1.28598
\(975\) 0 0
\(976\) −24490.0 −0.803182
\(977\) 5806.05 0.190125 0.0950625 0.995471i \(-0.469695\pi\)
0.0950625 + 0.995471i \(0.469695\pi\)
\(978\) 0 0
\(979\) 63460.0 2.07170
\(980\) 0 0
\(981\) 0 0
\(982\) −22534.0 −0.732270
\(983\) 3243.02 0.105225 0.0526126 0.998615i \(-0.483245\pi\)
0.0526126 + 0.998615i \(0.483245\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −83481.6 −2.69635
\(987\) 0 0
\(988\) 18040.0 0.580900
\(989\) −21445.8 −0.689521
\(990\) 0 0
\(991\) 49448.0 1.58503 0.792516 0.609851i \(-0.208771\pi\)
0.792516 + 0.609851i \(0.208771\pi\)
\(992\) 37399.4 1.19701
\(993\) 0 0
\(994\) −13566.0 −0.432885
\(995\) 0 0
\(996\) 0 0
\(997\) −16294.0 −0.517589 −0.258794 0.965932i \(-0.583325\pi\)
−0.258794 + 0.965932i \(0.583325\pi\)
\(998\) −17174.1 −0.544725
\(999\) 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 1575.4.a.t.1.1 2
3.2 odd 2 inner 1575.4.a.t.1.2 2
5.4 even 2 63.4.a.d.1.2 yes 2
15.14 odd 2 63.4.a.d.1.1 2
20.19 odd 2 1008.4.a.be.1.2 2
35.4 even 6 441.4.e.r.226.1 4
35.9 even 6 441.4.e.r.361.1 4
35.19 odd 6 441.4.e.s.361.1 4
35.24 odd 6 441.4.e.s.226.1 4
35.34 odd 2 441.4.a.q.1.2 2
60.59 even 2 1008.4.a.be.1.1 2
105.44 odd 6 441.4.e.r.361.2 4
105.59 even 6 441.4.e.s.226.2 4
105.74 odd 6 441.4.e.r.226.2 4
105.89 even 6 441.4.e.s.361.2 4
105.104 even 2 441.4.a.q.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.a.d.1.1 2 15.14 odd 2
63.4.a.d.1.2 yes 2 5.4 even 2
441.4.a.q.1.1 2 105.104 even 2
441.4.a.q.1.2 2 35.34 odd 2
441.4.e.r.226.1 4 35.4 even 6
441.4.e.r.226.2 4 105.74 odd 6
441.4.e.r.361.1 4 35.9 even 6
441.4.e.r.361.2 4 105.44 odd 6
441.4.e.s.226.1 4 35.24 odd 6
441.4.e.s.226.2 4 105.59 even 6
441.4.e.s.361.1 4 35.19 odd 6
441.4.e.s.361.2 4 105.89 even 6
1008.4.a.be.1.1 2 60.59 even 2
1008.4.a.be.1.2 2 20.19 odd 2
1575.4.a.t.1.1 2 1.1 even 1 trivial
1575.4.a.t.1.2 2 3.2 odd 2 inner