Properties

Label 441.4.a.q.1.2
Level $441$
Weight $4$
Character 441.1
Self dual yes
Analytic conductor $26.020$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.0198423125\)
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 1.2
Root \(4.35890\) of defining polynomial
Character \(\chi\) \(=\) 441.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.35890 q^{2} +11.0000 q^{4} -8.71780 q^{5} +13.0767 q^{8} +O(q^{10})\) \(q+4.35890 q^{2} +11.0000 q^{4} -8.71780 q^{5} +13.0767 q^{8} -38.0000 q^{10} -43.5890 q^{11} -82.0000 q^{13} -31.0000 q^{16} +78.4602 q^{17} +20.0000 q^{19} -95.8958 q^{20} -190.000 q^{22} -130.767 q^{23} -49.0000 q^{25} -357.430 q^{26} +244.098 q^{29} -156.000 q^{31} -239.739 q^{32} +342.000 q^{34} +186.000 q^{37} +87.1780 q^{38} -114.000 q^{40} +165.638 q^{41} +164.000 q^{43} -479.479 q^{44} -570.000 q^{46} -470.761 q^{47} -213.586 q^{50} -902.000 q^{52} -156.920 q^{53} +380.000 q^{55} +1064.00 q^{58} -156.920 q^{59} -790.000 q^{61} -679.988 q^{62} -797.000 q^{64} +714.859 q^{65} -44.0000 q^{67} +863.062 q^{68} +444.608 q^{71} -126.000 q^{73} +810.755 q^{74} +220.000 q^{76} -712.000 q^{79} +270.252 q^{80} +722.000 q^{82} +1464.59 q^{83} -684.000 q^{85} +714.859 q^{86} -570.000 q^{88} +1455.87 q^{89} -1438.44 q^{92} -2052.00 q^{94} -174.356 q^{95} -798.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 22q^{4} + O(q^{10}) \) \( 2q + 22q^{4} - 76q^{10} - 164q^{13} - 62q^{16} + 40q^{19} - 380q^{22} - 98q^{25} - 312q^{31} + 684q^{34} + 372q^{37} - 228q^{40} + 328q^{43} - 1140q^{46} - 1804q^{52} + 760q^{55} + 2128q^{58} - 1580q^{61} - 1594q^{64} - 88q^{67} - 252q^{73} + 440q^{76} - 1424q^{79} + 1444q^{82} - 1368q^{85} - 1140q^{88} - 4104q^{94} - 1596q^{97} + O(q^{100}) \)

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 \(-0.219981\pi\)
0.770552 + 0.637377i \(0.219981\pi\)
\(3\) 0 0
\(4\) 11.0000 1.37500
\(5\) −8.71780 −0.779744 −0.389872 0.920869i \(-0.627481\pi\)
−0.389872 + 0.920869i \(0.627481\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 13.0767 0.577914
\(9\) 0 0
\(10\) −38.0000 −1.20167
\(11\) −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 \(0.838954\pi\)
\(14\) 0 0
\(15\) 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 \(-0.461472\pi\)
0.120745 + 0.992684i \(0.461472\pi\)
\(20\) −95.8958 −1.07215
\(21\) 0 0
\(22\) −190.000 −1.84128
\(23\) −130.767 −1.18551 −0.592756 0.805382i \(-0.701960\pi\)
−0.592756 + 0.805382i \(0.701960\pi\)
\(24\) 0 0
\(25\) −49.0000 −0.392000
\(26\) −357.430 −2.69607
\(27\) 0 0
\(28\) 0 0
\(29\) 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 \(-0.649257\pi\)
−0.451910 + 0.892063i \(0.649257\pi\)
\(32\) −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 \(-0.364404\pi\)
0.413219 + 0.910632i \(0.364404\pi\)
\(38\) 87.1780 0.372161
\(39\) 0 0
\(40\) −114.000 −0.450625
\(41\) 165.638 0.630935 0.315467 0.948936i \(-0.397839\pi\)
0.315467 + 0.948936i \(0.397839\pi\)
\(42\) 0 0
\(43\) 164.000 0.581622 0.290811 0.956780i \(-0.406075\pi\)
0.290811 + 0.956780i \(0.406075\pi\)
\(44\) −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
\(49\) 0 0
\(50\) −213.586 −0.604113
\(51\) 0 0
\(52\) −902.000 −2.40548
\(53\) −156.920 −0.406692 −0.203346 0.979107i \(-0.565182\pi\)
−0.203346 + 0.979107i \(0.565182\pi\)
\(54\) 0 0
\(55\) 380.000 0.931622
\(56\) 0 0
\(57\) 0 0
\(58\) 1064.00 2.40879
\(59\) −156.920 −0.346259 −0.173130 0.984899i \(-0.555388\pi\)
−0.173130 + 0.984899i \(0.555388\pi\)
\(60\) 0 0
\(61\) −790.000 −1.65818 −0.829091 0.559113i \(-0.811142\pi\)
−0.829091 + 0.559113i \(0.811142\pi\)
\(62\) −679.988 −1.39288
\(63\) 0 0
\(64\) −797.000 −1.55664
\(65\) 714.859 1.36411
\(66\) 0 0
\(67\) −44.0000 −0.0802307 −0.0401153 0.999195i \(-0.512773\pi\)
−0.0401153 + 0.999195i \(0.512773\pi\)
\(68\) 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
\(77\) 0 0
\(78\) 0 0
\(79\) −712.000 −1.01400 −0.507002 0.861945i \(-0.669246\pi\)
−0.507002 + 0.861945i \(0.669246\pi\)
\(80\) 270.252 0.377688
\(81\) 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
\(85\) −684.000 −0.872826
\(86\) 714.859 0.896340
\(87\) 0 0
\(88\) −570.000 −0.690480
\(89\) 1455.87 1.73396 0.866978 0.498346i \(-0.166059\pi\)
0.866978 + 0.498346i \(0.166059\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −1438.44 −1.63008
\(93\) 0 0
\(94\) −2052.00 −2.25157
\(95\) −174.356 −0.188300
\(96\) 0 0
\(97\) −798.000 −0.835305 −0.417653 0.908607i \(-0.637147\pi\)
−0.417653 + 0.908607i \(0.637147\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −539.000 −0.539000
\(101\) −409.737 −0.403666 −0.201833 0.979420i \(-0.564690\pi\)
−0.201833 + 0.979420i \(0.564690\pi\)
\(102\) 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 \(-0.367049\pi\)
0.405638 + 0.914034i \(0.367049\pi\)
\(108\) 0 0
\(109\) −342.000 −0.300529 −0.150264 0.988646i \(-0.548013\pi\)
−0.150264 + 0.988646i \(0.548013\pi\)
\(110\) 1656.38 1.43573
\(111\) 0 0
\(112\) 0 0
\(113\) −488.197 −0.406422 −0.203211 0.979135i \(-0.565138\pi\)
−0.203211 + 0.979135i \(0.565138\pi\)
\(114\) 0 0
\(115\) 1140.00 0.924396
\(116\) 2685.08 2.14917
\(117\) 0 0
\(118\) −684.000 −0.533621
\(119\) 0 0
\(120\) 0 0
\(121\) 569.000 0.427498
\(122\) −3443.53 −2.55543
\(123\) 0 0
\(124\) −1716.00 −1.24275
\(125\) 1516.90 1.08540
\(126\) 0 0
\(127\) 456.000 0.318610 0.159305 0.987229i \(-0.449075\pi\)
0.159305 + 0.987229i \(0.449075\pi\)
\(128\) −1556.13 −1.07456
\(129\) 0 0
\(130\) 3116.00 2.10224
\(131\) −1499.46 −1.00007 −0.500033 0.866007i \(-0.666679\pi\)
−0.500033 + 0.866007i \(0.666679\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −191.792 −0.123644
\(135\) 0 0
\(136\) 1026.00 0.646903
\(137\) −889.215 −0.554531 −0.277266 0.960793i \(-0.589428\pi\)
−0.277266 + 0.960793i \(0.589428\pi\)
\(138\) 0 0
\(139\) 768.000 0.468640 0.234320 0.972160i \(-0.424714\pi\)
0.234320 + 0.972160i \(0.424714\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1938.00 1.14531
\(143\) 3574.30 2.09019
\(144\) 0 0
\(145\) −2128.00 −1.21876
\(146\) −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
\(154\) 0 0
\(155\) 1359.98 0.704748
\(156\) 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
\(160\) 2090.00 1.03268
\(161\) 0 0
\(162\) 0 0
\(163\) −2564.00 −1.23207 −0.616037 0.787717i \(-0.711263\pi\)
−0.616037 + 0.787717i \(0.711263\pi\)
\(164\) 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
\(170\) −2981.49 −1.34512
\(171\) 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 \(-0.527354\pi\)
−0.0858279 + 0.996310i \(0.527354\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1710.00 −0.685124
\(185\) −1621.51 −0.644410
\(186\) 0 0
\(187\) −3420.00 −1.33741
\(188\) −5178.37 −2.00889
\(189\) 0 0
\(190\) −760.000 −0.290191
\(191\) 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 \(-0.418503\pi\)
0.253241 + 0.967403i \(0.418503\pi\)
\(194\) −3478.40 −1.28729
\(195\) 0 0
\(196\) 0 0
\(197\) 3748.65 1.35574 0.677869 0.735183i \(-0.262904\pi\)
0.677869 + 0.735183i \(0.262904\pi\)
\(198\) 0 0
\(199\) 1056.00 0.376170 0.188085 0.982153i \(-0.439772\pi\)
0.188085 + 0.982153i \(0.439772\pi\)
\(200\) −640.758 −0.226542
\(201\) 0 0
\(202\) −1786.00 −0.622092
\(203\) 0 0
\(204\) 0 0
\(205\) −1444.00 −0.491967
\(206\) 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
\(215\) −1429.72 −0.453516
\(216\) 0 0
\(217\) 0 0
\(218\) −1490.74 −0.463146
\(219\) 0 0
\(220\) 4180.00 1.28098
\(221\) −6433.73 −1.95828
\(222\) 0 0
\(223\) −5368.00 −1.61196 −0.805982 0.591940i \(-0.798362\pi\)
−0.805982 + 0.591940i \(0.798362\pi\)
\(224\) 0 0
\(225\) 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 \(-0.602140\pi\)
−0.315404 + 0.948958i \(0.602140\pi\)
\(230\) 4969.14 1.42459
\(231\) 0 0
\(232\) 3192.00 0.903298
\(233\) −4132.24 −1.16185 −0.580927 0.813956i \(-0.697310\pi\)
−0.580927 + 0.813956i \(0.697310\pi\)
\(234\) 0 0
\(235\) 4104.00 1.13921
\(236\) −1726.12 −0.476106
\(237\) 0 0
\(238\) 0 0
\(239\) −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 \(-0.554979\pi\)
−0.171864 + 0.985121i \(0.554979\pi\)
\(242\) 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
\(250\) 6612.00 1.67272
\(251\) 1795.87 0.451610 0.225805 0.974173i \(-0.427499\pi\)
0.225805 + 0.974173i \(0.427499\pi\)
\(252\) 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
\(259\) 0 0
\(260\) 7863.45 1.87566
\(261\) 0 0
\(262\) −6536.00 −1.54120
\(263\) 5344.01 1.25295 0.626475 0.779442i \(-0.284497\pi\)
0.626475 + 0.779442i \(0.284497\pi\)
\(264\) 0 0
\(265\) 1368.00 0.317115
\(266\) 0 0
\(267\) 0 0
\(268\) −484.000 −0.110317
\(269\) 4001.47 0.906966 0.453483 0.891265i \(-0.350181\pi\)
0.453483 + 0.891265i \(0.350181\pi\)
\(270\) 0 0
\(271\) −2788.00 −0.624941 −0.312470 0.949928i \(-0.601156\pi\)
−0.312470 + 0.949928i \(0.601156\pi\)
\(272\) −2432.27 −0.542198
\(273\) 0 0
\(274\) −3876.00 −0.854590
\(275\) 2135.86 0.468354
\(276\) 0 0
\(277\) −4562.00 −0.989545 −0.494773 0.869022i \(-0.664749\pi\)
−0.494773 + 0.869022i \(0.664749\pi\)
\(278\) 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
\(287\) 0 0
\(288\) 0 0
\(289\) 1243.00 0.253002
\(290\) −9275.74 −1.87824
\(291\) 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
\(295\) 1368.00 0.269993
\(296\) 2432.27 0.477610
\(297\) 0 0
\(298\) 4332.00 0.842101
\(299\) 10722.9 2.07398
\(300\) 0 0
\(301\) 0 0
\(302\) −15238.7 −2.90361
\(303\) 0 0
\(304\) −620.000 −0.116972
\(305\) 6887.06 1.29296
\(306\) 0 0
\(307\) −532.000 −0.0989018 −0.0494509 0.998777i \(-0.515747\pi\)
−0.0494509 + 0.998777i \(0.515747\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 5928.00 1.08609
\(311\) −6538.35 −1.19214 −0.596070 0.802932i \(-0.703272\pi\)
−0.596070 + 0.802932i \(0.703272\pi\)
\(312\) 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 \(-0.513279\pi\)
−0.0417044 + 0.999130i \(0.513279\pi\)
\(318\) 0 0
\(319\) −10640.0 −1.86748
\(320\) 6948.08 1.21378
\(321\) 0 0
\(322\) 0 0
\(323\) 1569.20 0.270318
\(324\) 0 0
\(325\) 4018.00 0.685780
\(326\) −11176.2 −1.89875
\(327\) 0 0
\(328\) 2166.00 0.364626
\(329\) 0 0
\(330\) 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
\(335\) 383.583 0.0625594
\(336\) 0 0
\(337\) 238.000 0.0384709 0.0192354 0.999815i \(-0.493877\pi\)
0.0192354 + 0.999815i \(0.493877\pi\)
\(338\) 19732.7 3.17550
\(339\) 0 0
\(340\) −7524.00 −1.20014
\(341\) 6799.88 1.07987
\(342\) 0 0
\(343\) 0 0
\(344\) 2144.58 0.336128
\(345\) 0 0
\(346\) −16834.0 −2.61561
\(347\) 7052.70 1.09109 0.545546 0.838081i \(-0.316322\pi\)
0.545546 + 0.838081i \(0.316322\pi\)
\(348\) 0 0
\(349\) 10850.0 1.66415 0.832073 0.554666i \(-0.187154\pi\)
0.832073 + 0.554666i \(0.187154\pi\)
\(350\) 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
\(355\) −3876.00 −0.579484
\(356\) 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
\(364\) 0 0
\(365\) 1098.44 0.157521
\(366\) 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
\(370\) −7068.00 −0.993102
\(371\) 0 0
\(372\) 0 0
\(373\) −10450.0 −1.45062 −0.725309 0.688423i \(-0.758303\pi\)
−0.725309 + 0.688423i \(0.758303\pi\)
\(374\) −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 \(0.516314\pi\)
\(380\) −1917.92 −0.258913
\(381\) 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
\(392\) 0 0
\(393\) 0 0
\(394\) 16340.0 2.08933
\(395\) 6207.07 0.790663
\(396\) 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
\(400\) 1519.00 0.189875
\(401\) −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
\(406\) 0 0
\(407\) −8107.55 −0.987411
\(408\) 0 0
\(409\) 1330.00 0.160793 0.0803964 0.996763i \(-0.474381\pi\)
0.0803964 + 0.996763i \(0.474381\pi\)
\(410\) −6294.25 −0.758173
\(411\) 0 0
\(412\) 10076.0 1.20488
\(413\) 0 0
\(414\) 0 0
\(415\) −12768.0 −1.51026
\(416\) 19658.6 2.31693
\(417\) 0 0
\(418\) −3800.00 −0.444651
\(419\) −10409.1 −1.21364 −0.606820 0.794839i \(-0.707555\pi\)
−0.606820 + 0.794839i \(0.707555\pi\)
\(420\) 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
\(425\) −3844.55 −0.438795
\(426\) 0 0
\(427\) 0 0
\(428\) 9877.27 1.11550
\(429\) 0 0
\(430\) −6232.00 −0.698916
\(431\) −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 \(0.603748\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3762.00 −0.413227
\(437\) −2615.34 −0.286290
\(438\) 0 0
\(439\) 1872.00 0.203521 0.101760 0.994809i \(-0.467552\pi\)
0.101760 + 0.994809i \(0.467552\pi\)
\(440\) 4969.14 0.538397
\(441\) 0 0
\(442\) −28044.0 −3.01791
\(443\) −11115.2 −1.19210 −0.596048 0.802949i \(-0.703263\pi\)
−0.596048 + 0.802949i \(0.703263\pi\)
\(444\) 0 0
\(445\) −12692.0 −1.35204
\(446\) −23398.6 −2.48420
\(447\) 0 0
\(448\) 0 0
\(449\) −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 \(-0.217770\pi\)
0.774959 + 0.632011i \(0.217770\pi\)
\(458\) −9528.55 −0.972140
\(459\) 0 0
\(460\) 12540.0 1.27104
\(461\) 13190.0 1.33258 0.666292 0.745691i \(-0.267881\pi\)
0.666292 + 0.745691i \(0.267881\pi\)
\(462\) 0 0
\(463\) 9328.00 0.936304 0.468152 0.883648i \(-0.344920\pi\)
0.468152 + 0.883648i \(0.344920\pi\)
\(464\) −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
\(469\) 0 0
\(470\) 17888.9 1.75565
\(471\) 0 0
\(472\) −2052.00 −0.200108
\(473\) −7148.59 −0.694911
\(474\) 0 0
\(475\) −980.000 −0.0946642
\(476\) 0 0
\(477\) 0 0
\(478\) −21090.0 −2.01806
\(479\) −3574.30 −0.340947 −0.170474 0.985362i \(-0.554530\pi\)
−0.170474 + 0.985362i \(0.554530\pi\)
\(480\) 0 0
\(481\) −15252.0 −1.44580
\(482\) −5605.54 −0.529721
\(483\) 0 0
\(484\) 6259.00 0.587810
\(485\) 6956.80 0.651324
\(486\) 0 0
\(487\) 8968.00 0.834454 0.417227 0.908802i \(-0.363002\pi\)
0.417227 + 0.908802i \(0.363002\pi\)
\(488\) −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
\(497\) 0 0
\(498\) 0 0
\(499\) 3940.00 0.353464 0.176732 0.984259i \(-0.443447\pi\)
0.176732 + 0.984259i \(0.443447\pi\)
\(500\) 16685.9 1.49243
\(501\) 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
\(505\) 3572.00 0.314756
\(506\) 24845.7 2.18286
\(507\) 0 0
\(508\) 5016.00 0.438089
\(509\) −9772.65 −0.851012 −0.425506 0.904956i \(-0.639904\pi\)
−0.425506 + 0.904956i \(0.639904\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 10674.9 0.921426
\(513\) 0 0
\(514\) 8474.00 0.727183
\(515\) −7985.50 −0.683269
\(516\) 0 0
\(517\) 20520.0 1.74559
\(518\) 0 0
\(519\) 0 0
\(520\) 9348.00 0.788340
\(521\) 7401.41 0.622383 0.311192 0.950347i \(-0.399272\pi\)
0.311192 + 0.950347i \(0.399272\pi\)
\(522\) 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
\(530\) 5962.97 0.488708
\(531\) 0 0
\(532\) 0 0
\(533\) −13582.3 −1.10378
\(534\) 0 0
\(535\) −7828.00 −0.632587
\(536\) −575.375 −0.0463664
\(537\) 0 0
\(538\) 17442.0 1.39773
\(539\) 0 0
\(540\) 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
\(545\) 2981.49 0.234336
\(546\) 0 0
\(547\) 11140.0 0.870771 0.435386 0.900244i \(-0.356612\pi\)
0.435386 + 0.900244i \(0.356612\pi\)
\(548\) −9781.37 −0.762481
\(549\) 0 0
\(550\) 9310.00 0.721781
\(551\) 4881.97 0.377457
\(552\) 0 0
\(553\) 0 0
\(554\) −19885.3 −1.52499
\(555\) 0 0
\(556\) 8448.00 0.644380
\(557\) 22788.3 1.73352 0.866761 0.498723i \(-0.166197\pi\)
0.866761 + 0.498723i \(0.166197\pi\)
\(558\) 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
\(565\) 4256.00 0.316905
\(566\) 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
\(574\) 0 0
\(575\) 6407.58 0.464721
\(576\) 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
\(580\) −23408.0 −1.67580
\(581\) 0 0
\(582\) 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
\(590\) 5962.97 0.416088
\(591\) 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 \(-0.381041\pi\)
0.365082 + 0.930976i \(0.381041\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −38456.0 −2.59065
\(605\) −4960.43 −0.333339
\(606\) 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
\(610\) 30020.0 1.99258
\(611\) 38602.4 2.55595
\(612\) 0 0
\(613\) −5714.00 −0.376487 −0.188243 0.982122i \(-0.560279\pi\)
−0.188243 + 0.982122i \(0.560279\pi\)
\(614\) −2318.93 −0.152418
\(615\) 0 0
\(616\) 0 0
\(617\) 6747.58 0.440271 0.220135 0.975469i \(-0.429350\pi\)
0.220135 + 0.975469i \(0.429350\pi\)
\(618\) 0 0
\(619\) 1880.00 0.122074 0.0610368 0.998136i \(-0.480559\pi\)
0.0610368 + 0.998136i \(0.480559\pi\)
\(620\) 14959.7 0.969028
\(621\) 0 0
\(622\) −28500.0 −1.83721
\(623\) 0 0
\(624\) 0 0
\(625\) −7099.00 −0.454336
\(626\) −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
\(635\) −3975.32 −0.248434
\(636\) 0 0
\(637\) 0 0
\(638\) −46378.7 −2.87798
\(639\) 0 0
\(640\) 13566.0 0.837880
\(641\) 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 \(0.774871\pi\)
\(644\) 0 0
\(645\) 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
\(650\) 17514.1 1.05686
\(651\) 0 0
\(652\) −28204.0 −1.69410
\(653\) 3243.02 0.194348 0.0971740 0.995267i \(-0.469020\pi\)
0.0971740 + 0.995267i \(0.469020\pi\)
\(654\) 0 0
\(655\) 13072.0 0.779794
\(656\) −5134.78 −0.305609
\(657\) 0 0
\(658\) 0 0
\(659\) 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 \(-0.610765\pi\)
−0.340998 + 0.940064i \(0.610765\pi\)
\(662\) 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
\(670\) 1672.00 0.0964104
\(671\) 34435.3 1.98116
\(672\) 0 0
\(673\) 23062.0 1.32091 0.660457 0.750864i \(-0.270363\pi\)
0.660457 + 0.750864i \(0.270363\pi\)
\(674\) 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
\(679\) 0 0
\(680\) −8944.46 −0.504418
\(681\) 0 0
\(682\) 29640.0 1.66419
\(683\) −24715.0 −1.38461 −0.692307 0.721603i \(-0.743406\pi\)
−0.692307 + 0.721603i \(0.743406\pi\)
\(684\) 0 0
\(685\) 7752.00 0.432392
\(686\) 0 0
\(687\) 0 0
\(688\) −5084.00 −0.281723
\(689\) 12867.5 0.711483
\(690\) 0 0
\(691\) 10600.0 0.583564 0.291782 0.956485i \(-0.405752\pi\)
0.291782 + 0.956485i \(0.405752\pi\)
\(692\) −42481.8 −2.33369
\(693\) 0 0
\(694\) 30742.0 1.68148
\(695\) −6695.27 −0.365419
\(696\) 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
\(707\) 0 0
\(708\) 0 0
\(709\) −13710.0 −0.726220 −0.363110 0.931746i \(-0.618285\pi\)
−0.363110 + 0.931746i \(0.618285\pi\)
\(710\) −16895.1 −0.893044
\(711\) 0 0
\(712\) 19038.0 1.00208
\(713\) 20399.6 1.07149
\(714\) 0 0
\(715\) −31160.0 −1.62982
\(716\) −2972.77 −0.155164
\(717\) 0 0
\(718\) −21014.0 −1.09225
\(719\) 2510.73 0.130228 0.0651142 0.997878i \(-0.479259\pi\)
0.0651142 + 0.997878i \(0.479259\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −28154.1 −1.45123
\(723\) 0 0
\(724\) −4598.00 −0.236027
\(725\) −11960.8 −0.612708
\(726\) 0 0
\(727\) −620.000 −0.0316293 −0.0158147 0.999875i \(-0.505034\pi\)
−0.0158147 + 0.999875i \(0.505034\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 4788.00 0.242756
\(731\) 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 \(0.599235\pi\)
\(740\) −17836.6 −0.886063
\(741\) 0 0
\(742\) 0 0
\(743\) 29736.4 1.46827 0.734134 0.679005i \(-0.237588\pi\)
0.734134 + 0.679005i \(0.237588\pi\)
\(744\) 0 0
\(745\) −8664.00 −0.426073
\(746\) −45550.5 −2.23555
\(747\) 0 0
\(748\) −37620.0 −1.83894
\(749\) 0 0
\(750\) 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
\(755\) 30477.4 1.46912
\(756\) 0 0
\(757\) 15986.0 0.767531 0.383766 0.923431i \(-0.374627\pi\)
0.383766 + 0.923431i \(0.374627\pi\)
\(758\) −3295.33 −0.157905
\(759\) 0 0
\(760\) −2280.00 −0.108821
\(761\) −37007.1 −1.76282 −0.881409 0.472354i \(-0.843404\pi\)
−0.881409 + 0.472354i \(0.843404\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 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 \(-0.179171\pi\)
0.845720 + 0.533627i \(0.179171\pi\)
\(770\) 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
\(775\) 7644.00 0.354298
\(776\) −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
\(784\) 0 0
\(785\) −4411.21 −0.200564
\(786\) 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
\(790\) 27056.0 1.21849
\(791\) 0 0
\(792\) 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
\(800\) 11747.2 0.519159
\(801\) 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 \(-0.626937\pi\)
−0.388297 + 0.921534i \(0.626937\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −35340.0 −1.52170
\(815\) 22352.4 0.960701
\(816\) 0 0
\(817\) 3280.00 0.140456
\(818\) 5797.34 0.247798
\(819\) 0 0
\(820\) −15884.0 −0.676455
\(821\) 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 \(-0.334781\pi\)
0.496057 + 0.868290i \(0.334781\pi\)
\(824\) 11978.3 0.506411
\(825\) 0 0
\(826\) 0 0
\(827\) −26650.3 −1.12058 −0.560291 0.828296i \(-0.689311\pi\)
−0.560291 + 0.828296i \(0.689311\pi\)
\(828\) 0 0
\(829\) 26254.0 1.09993 0.549963 0.835189i \(-0.314642\pi\)
0.549963 + 0.835189i \(0.314642\pi\)
\(830\) −55654.4 −2.32746
\(831\) 0 0
\(832\) 65354.0 2.72325
\(833\) 0 0
\(834\) 0 0
\(835\) −5624.00 −0.233086
\(836\) −9589.58 −0.396725
\(837\) 0 0
\(838\) −45372.0 −1.87035
\(839\) 6189.64 0.254696 0.127348 0.991858i \(-0.459353\pi\)
0.127348 + 0.991858i \(0.459353\pi\)
\(840\) 0 0
\(841\) 35195.0 1.44307
\(842\) −53501.1 −2.18975
\(843\) 0 0
\(844\) −39820.0 −1.62401
\(845\) −39465.5 −1.60669
\(846\) 0 0
\(847\) 0 0
\(848\) 4864.53 0.196991
\(849\) 0 0
\(850\) −16758.0 −0.676229
\(851\) −24322.7 −0.979753
\(852\) 0 0
\(853\) 45322.0 1.81922 0.909611 0.415462i \(-0.136380\pi\)
0.909611 + 0.415462i \(0.136380\pi\)
\(854\) 0 0
\(855\) 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 \(-0.171094\pi\)
0.858987 + 0.511998i \(0.171094\pi\)
\(860\) −15726.9 −0.623585
\(861\) 0 0
\(862\) −20406.0 −0.806301
\(863\) −29318.0 −1.15642 −0.578212 0.815886i \(-0.696250\pi\)
−0.578212 + 0.815886i \(0.696250\pi\)
\(864\) 0 0
\(865\) 33668.0 1.32341
\(866\) −25150.8 −0.986906
\(867\) 0 0
\(868\) 0 0
\(869\) 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 \(-0.138404\pi\)
0.906951 + 0.421237i \(0.138404\pi\)
\(878\) 8159.86 0.313647
\(879\) 0 0
\(880\) −11780.0 −0.451254
\(881\) 42133.1 1.61124 0.805619 0.592434i \(-0.201833\pi\)
0.805619 + 0.592434i \(0.201833\pi\)
\(882\) 0 0
\(883\) −22732.0 −0.866356 −0.433178 0.901308i \(-0.642608\pi\)
−0.433178 + 0.901308i \(0.642608\pi\)
\(884\) −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
\(889\) 0 0
\(890\) −55323.1 −2.08364
\(891\) 0 0
\(892\) −59048.0 −2.21645
\(893\) −9415.22 −0.352820
\(894\) 0 0
\(895\) 2356.00 0.0879915
\(896\) 0 0
\(897\) 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
\(905\) 3644.04 0.133847
\(906\) 0 0
\(907\) −6916.00 −0.253189 −0.126594 0.991955i \(-0.540405\pi\)
−0.126594 + 0.991955i \(0.540405\pi\)
\(908\) −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
\(917\) 0 0
\(918\) 0 0
\(919\) −47632.0 −1.70972 −0.854861 0.518857i \(-0.826358\pi\)
−0.854861 + 0.518857i \(0.826358\pi\)
\(920\) 14907.4 0.534221
\(921\) 0 0
\(922\) 57494.0 2.05365
\(923\) −36457.8 −1.30013
\(924\) 0 0
\(925\) −9114.00 −0.323964
\(926\) 40659.8 1.44294
\(927\) 0 0
\(928\) −58520.0 −2.07006
\(929\) 3304.05 0.116687 0.0583435 0.998297i \(-0.481418\pi\)
0.0583435 + 0.998297i \(0.481418\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −45454.6 −1.59755
\(933\) 0 0
\(934\) −14820.0 −0.519192
\(935\) 29814.9 1.04283
\(936\) 0 0
\(937\) −21858.0 −0.762081 −0.381040 0.924558i \(-0.624434\pi\)
−0.381040 + 0.924558i \(0.624434\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 45144.0 1.56642
\(941\) 50380.2 1.74532 0.872660 0.488328i \(-0.162393\pi\)
0.872660 + 0.488328i \(0.162393\pi\)
\(942\) 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 \(-0.321092\pi\)
0.532927 + 0.846161i \(0.321092\pi\)
\(948\) 0 0
\(949\) 10332.0 0.353415
\(950\) −4271.72 −0.145887
\(951\) 0 0
\(952\) 0 0
\(953\) 22770.9 0.773999 0.387000 0.922080i \(-0.373511\pi\)
0.387000 + 0.922080i \(0.373511\pi\)
\(954\) 0 0
\(955\) −13300.0 −0.450657
\(956\) −53222.2 −1.80055
\(957\) 0 0
\(958\) −15580.0 −0.525435
\(959\) 0 0
\(960\) 0 0
\(961\) −5455.00 −0.183109
\(962\) −66481.9 −2.22813
\(963\) 0 0
\(964\) −14146.0 −0.472627
\(965\) −11838.8 −0.394926
\(966\) 0 0
\(967\) −36416.0 −1.21102 −0.605512 0.795836i \(-0.707032\pi\)
−0.605512 + 0.795836i \(0.707032\pi\)
\(968\) 7440.64 0.247057
\(969\) 0 0
\(970\) 30324.0 1.00376
\(971\) 18621.2 0.615431 0.307715 0.951478i \(-0.400436\pi\)
0.307715 + 0.951478i \(0.400436\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 39090.6 1.28598
\(975\) 0 0
\(976\) 24490.0 0.803182
\(977\) −5806.05 −0.190125 −0.0950625 0.995471i \(-0.530305\pi\)
−0.0950625 + 0.995471i \(0.530305\pi\)
\(978\) 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
\(985\) −32680.0 −1.05713
\(986\) 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
\(994\) 0 0
\(995\) −9205.99 −0.293316
\(996\) 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 441.4.a.q.1.2 2
3.2 odd 2 inner 441.4.a.q.1.1 2
7.2 even 3 441.4.e.s.361.1 4
7.3 odd 6 441.4.e.r.226.1 4
7.4 even 3 441.4.e.s.226.1 4
7.5 odd 6 441.4.e.r.361.1 4
7.6 odd 2 63.4.a.d.1.2 yes 2
21.2 odd 6 441.4.e.s.361.2 4
21.5 even 6 441.4.e.r.361.2 4
21.11 odd 6 441.4.e.s.226.2 4
21.17 even 6 441.4.e.r.226.2 4
21.20 even 2 63.4.a.d.1.1 2
28.27 even 2 1008.4.a.be.1.2 2
35.34 odd 2 1575.4.a.t.1.1 2
84.83 odd 2 1008.4.a.be.1.1 2
105.104 even 2 1575.4.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
63.4.a.d.1.1 2 21.20 even 2
63.4.a.d.1.2 yes 2 7.6 odd 2
441.4.a.q.1.1 2 3.2 odd 2 inner
441.4.a.q.1.2 2 1.1 even 1 trivial
441.4.e.r.226.1 4 7.3 odd 6
441.4.e.r.226.2 4 21.17 even 6
441.4.e.r.361.1 4 7.5 odd 6
441.4.e.r.361.2 4 21.5 even 6
441.4.e.s.226.1 4 7.4 even 3
441.4.e.s.226.2 4 21.11 odd 6
441.4.e.s.361.1 4 7.2 even 3
441.4.e.s.361.2 4 21.2 odd 6
1008.4.a.be.1.1 2 84.83 odd 2
1008.4.a.be.1.2 2 28.27 even 2
1575.4.a.t.1.1 2 35.34 odd 2
1575.4.a.t.1.2 2 105.104 even 2