Properties

Label 1323.4.a.bb.1.3
Level $1323$
Weight $4$
Character 1323.1
Self dual yes
Analytic conductor $78.060$
Analytic rank $1$
Dimension $6$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1323,4,Mod(1,1323)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1323, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1323.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1323 = 3^{3} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1323.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(78.0595269376\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: 6.6.346909504.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 12x^{4} + 2x^{3} + 39x^{2} + 25x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{4}\cdot 3^{4} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.673097\) of defining polynomial
Character \(\chi\) \(=\) 1323.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.57109 q^{2} -5.53167 q^{4} +6.97204 q^{5} +21.2595 q^{8} +O(q^{10})\) \(q-1.57109 q^{2} -5.53167 q^{4} +6.97204 q^{5} +21.2595 q^{8} -10.9537 q^{10} +32.5591 q^{11} -19.3425 q^{13} +10.8528 q^{16} +22.2459 q^{17} -155.215 q^{19} -38.5671 q^{20} -51.1534 q^{22} +76.9512 q^{23} -76.3906 q^{25} +30.3888 q^{26} -122.079 q^{29} +164.774 q^{31} -187.127 q^{32} -34.9504 q^{34} +66.2329 q^{37} +243.858 q^{38} +148.222 q^{40} -231.844 q^{41} -210.361 q^{43} -180.106 q^{44} -120.897 q^{46} +206.307 q^{47} +120.017 q^{50} +106.996 q^{52} +419.641 q^{53} +227.004 q^{55} +191.797 q^{58} +300.800 q^{59} +24.2225 q^{61} -258.875 q^{62} +207.171 q^{64} -134.857 q^{65} -274.603 q^{67} -123.057 q^{68} +336.736 q^{71} +693.601 q^{73} -104.058 q^{74} +858.601 q^{76} -584.526 q^{79} +75.6661 q^{80} +364.248 q^{82} -1209.75 q^{83} +155.100 q^{85} +330.497 q^{86} +692.191 q^{88} -1258.68 q^{89} -425.669 q^{92} -324.127 q^{94} -1082.17 q^{95} +1085.60 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{4} - 24 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 18 q^{4} - 24 q^{5} - 30 q^{16} - 42 q^{17} - 12 q^{20} + 132 q^{22} + 222 q^{25} - 366 q^{26} - 312 q^{37} - 336 q^{38} - 360 q^{41} + 654 q^{43} + 774 q^{46} - 1812 q^{47} - 378 q^{58} + 6 q^{59} - 2058 q^{62} + 66 q^{64} + 42 q^{67} - 2910 q^{68} + 1956 q^{79} + 2868 q^{80} - 2892 q^{83} - 1944 q^{85} - 2532 q^{88} - 1518 q^{89}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.57109 −0.555465 −0.277732 0.960659i \(-0.589583\pi\)
−0.277732 + 0.960659i \(0.589583\pi\)
\(3\) 0 0
\(4\) −5.53167 −0.691459
\(5\) 6.97204 0.623599 0.311799 0.950148i \(-0.399068\pi\)
0.311799 + 0.950148i \(0.399068\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 21.2595 0.939546
\(9\) 0 0
\(10\) −10.9537 −0.346387
\(11\) 32.5591 0.892450 0.446225 0.894921i \(-0.352768\pi\)
0.446225 + 0.894921i \(0.352768\pi\)
\(12\) 0 0
\(13\) −19.3425 −0.412664 −0.206332 0.978482i \(-0.566153\pi\)
−0.206332 + 0.978482i \(0.566153\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 10.8528 0.169575
\(17\) 22.2459 0.317379 0.158689 0.987329i \(-0.449273\pi\)
0.158689 + 0.987329i \(0.449273\pi\)
\(18\) 0 0
\(19\) −155.215 −1.87415 −0.937075 0.349127i \(-0.886478\pi\)
−0.937075 + 0.349127i \(0.886478\pi\)
\(20\) −38.5671 −0.431193
\(21\) 0 0
\(22\) −51.1534 −0.495724
\(23\) 76.9512 0.697628 0.348814 0.937192i \(-0.386585\pi\)
0.348814 + 0.937192i \(0.386585\pi\)
\(24\) 0 0
\(25\) −76.3906 −0.611125
\(26\) 30.3888 0.229220
\(27\) 0 0
\(28\) 0 0
\(29\) −122.079 −0.781707 −0.390854 0.920453i \(-0.627820\pi\)
−0.390854 + 0.920453i \(0.627820\pi\)
\(30\) 0 0
\(31\) 164.774 0.954653 0.477326 0.878726i \(-0.341606\pi\)
0.477326 + 0.878726i \(0.341606\pi\)
\(32\) −187.127 −1.03374
\(33\) 0 0
\(34\) −34.9504 −0.176293
\(35\) 0 0
\(36\) 0 0
\(37\) 66.2329 0.294287 0.147143 0.989115i \(-0.452992\pi\)
0.147143 + 0.989115i \(0.452992\pi\)
\(38\) 243.858 1.04102
\(39\) 0 0
\(40\) 148.222 0.585899
\(41\) −231.844 −0.883121 −0.441560 0.897232i \(-0.645575\pi\)
−0.441560 + 0.897232i \(0.645575\pi\)
\(42\) 0 0
\(43\) −210.361 −0.746042 −0.373021 0.927823i \(-0.621678\pi\)
−0.373021 + 0.927823i \(0.621678\pi\)
\(44\) −180.106 −0.617093
\(45\) 0 0
\(46\) −120.897 −0.387508
\(47\) 206.307 0.640276 0.320138 0.947371i \(-0.396271\pi\)
0.320138 + 0.947371i \(0.396271\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 120.017 0.339458
\(51\) 0 0
\(52\) 106.996 0.285340
\(53\) 419.641 1.08759 0.543794 0.839219i \(-0.316987\pi\)
0.543794 + 0.839219i \(0.316987\pi\)
\(54\) 0 0
\(55\) 227.004 0.556530
\(56\) 0 0
\(57\) 0 0
\(58\) 191.797 0.434211
\(59\) 300.800 0.663742 0.331871 0.943325i \(-0.392320\pi\)
0.331871 + 0.943325i \(0.392320\pi\)
\(60\) 0 0
\(61\) 24.2225 0.0508423 0.0254211 0.999677i \(-0.491907\pi\)
0.0254211 + 0.999677i \(0.491907\pi\)
\(62\) −258.875 −0.530276
\(63\) 0 0
\(64\) 207.171 0.404630
\(65\) −134.857 −0.257337
\(66\) 0 0
\(67\) −274.603 −0.500718 −0.250359 0.968153i \(-0.580549\pi\)
−0.250359 + 0.968153i \(0.580549\pi\)
\(68\) −123.057 −0.219454
\(69\) 0 0
\(70\) 0 0
\(71\) 336.736 0.562863 0.281431 0.959581i \(-0.409191\pi\)
0.281431 + 0.959581i \(0.409191\pi\)
\(72\) 0 0
\(73\) 693.601 1.11205 0.556027 0.831164i \(-0.312325\pi\)
0.556027 + 0.831164i \(0.312325\pi\)
\(74\) −104.058 −0.163466
\(75\) 0 0
\(76\) 858.601 1.29590
\(77\) 0 0
\(78\) 0 0
\(79\) −584.526 −0.832459 −0.416230 0.909260i \(-0.636649\pi\)
−0.416230 + 0.909260i \(0.636649\pi\)
\(80\) 75.6661 0.105747
\(81\) 0 0
\(82\) 364.248 0.490542
\(83\) −1209.75 −1.59984 −0.799922 0.600104i \(-0.795126\pi\)
−0.799922 + 0.600104i \(0.795126\pi\)
\(84\) 0 0
\(85\) 155.100 0.197917
\(86\) 330.497 0.414400
\(87\) 0 0
\(88\) 692.191 0.838497
\(89\) −1258.68 −1.49910 −0.749550 0.661948i \(-0.769730\pi\)
−0.749550 + 0.661948i \(0.769730\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −425.669 −0.482381
\(93\) 0 0
\(94\) −324.127 −0.355651
\(95\) −1082.17 −1.16872
\(96\) 0 0
\(97\) 1085.60 1.13636 0.568178 0.822906i \(-0.307649\pi\)
0.568178 + 0.822906i \(0.307649\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 422.568 0.422568
\(101\) 844.120 0.831615 0.415807 0.909453i \(-0.363499\pi\)
0.415807 + 0.909453i \(0.363499\pi\)
\(102\) 0 0
\(103\) −1290.47 −1.23450 −0.617250 0.786767i \(-0.711753\pi\)
−0.617250 + 0.786767i \(0.711753\pi\)
\(104\) −411.211 −0.387717
\(105\) 0 0
\(106\) −659.295 −0.604117
\(107\) 212.075 0.191608 0.0958039 0.995400i \(-0.469458\pi\)
0.0958039 + 0.995400i \(0.469458\pi\)
\(108\) 0 0
\(109\) 524.864 0.461219 0.230609 0.973046i \(-0.425928\pi\)
0.230609 + 0.973046i \(0.425928\pi\)
\(110\) −356.644 −0.309133
\(111\) 0 0
\(112\) 0 0
\(113\) −590.941 −0.491956 −0.245978 0.969275i \(-0.579109\pi\)
−0.245978 + 0.969275i \(0.579109\pi\)
\(114\) 0 0
\(115\) 536.507 0.435040
\(116\) 675.302 0.540519
\(117\) 0 0
\(118\) −472.584 −0.368685
\(119\) 0 0
\(120\) 0 0
\(121\) −270.903 −0.203533
\(122\) −38.0558 −0.0282411
\(123\) 0 0
\(124\) −911.474 −0.660103
\(125\) −1404.10 −1.00470
\(126\) 0 0
\(127\) 2374.15 1.65883 0.829415 0.558633i \(-0.188674\pi\)
0.829415 + 0.558633i \(0.188674\pi\)
\(128\) 1171.53 0.808981
\(129\) 0 0
\(130\) 211.872 0.142941
\(131\) −2661.28 −1.77494 −0.887469 0.460868i \(-0.847538\pi\)
−0.887469 + 0.460868i \(0.847538\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 431.426 0.278131
\(135\) 0 0
\(136\) 472.938 0.298192
\(137\) 1405.23 0.876327 0.438164 0.898895i \(-0.355629\pi\)
0.438164 + 0.898895i \(0.355629\pi\)
\(138\) 0 0
\(139\) −858.529 −0.523881 −0.261941 0.965084i \(-0.584362\pi\)
−0.261941 + 0.965084i \(0.584362\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −529.044 −0.312650
\(143\) −629.774 −0.368282
\(144\) 0 0
\(145\) −851.141 −0.487472
\(146\) −1089.71 −0.617706
\(147\) 0 0
\(148\) −366.378 −0.203487
\(149\) −3114.11 −1.71220 −0.856101 0.516808i \(-0.827120\pi\)
−0.856101 + 0.516808i \(0.827120\pi\)
\(150\) 0 0
\(151\) −26.7635 −0.0144237 −0.00721186 0.999974i \(-0.502296\pi\)
−0.00721186 + 0.999974i \(0.502296\pi\)
\(152\) −3299.80 −1.76085
\(153\) 0 0
\(154\) 0 0
\(155\) 1148.81 0.595320
\(156\) 0 0
\(157\) 2362.94 1.20117 0.600583 0.799562i \(-0.294935\pi\)
0.600583 + 0.799562i \(0.294935\pi\)
\(158\) 918.343 0.462402
\(159\) 0 0
\(160\) −1304.66 −0.644638
\(161\) 0 0
\(162\) 0 0
\(163\) −284.500 −0.136710 −0.0683550 0.997661i \(-0.521775\pi\)
−0.0683550 + 0.997661i \(0.521775\pi\)
\(164\) 1282.49 0.610642
\(165\) 0 0
\(166\) 1900.62 0.888657
\(167\) −3642.66 −1.68789 −0.843946 0.536429i \(-0.819773\pi\)
−0.843946 + 0.536429i \(0.819773\pi\)
\(168\) 0 0
\(169\) −1822.87 −0.829708
\(170\) −243.676 −0.109936
\(171\) 0 0
\(172\) 1163.65 0.515857
\(173\) −2613.40 −1.14852 −0.574258 0.818674i \(-0.694709\pi\)
−0.574258 + 0.818674i \(0.694709\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 353.357 0.151337
\(177\) 0 0
\(178\) 1977.50 0.832697
\(179\) −2757.44 −1.15140 −0.575700 0.817661i \(-0.695270\pi\)
−0.575700 + 0.817661i \(0.695270\pi\)
\(180\) 0 0
\(181\) 2204.50 0.905298 0.452649 0.891689i \(-0.350479\pi\)
0.452649 + 0.891689i \(0.350479\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1635.94 0.655453
\(185\) 461.778 0.183517
\(186\) 0 0
\(187\) 724.309 0.283244
\(188\) −1141.22 −0.442725
\(189\) 0 0
\(190\) 1700.19 0.649181
\(191\) 3590.90 1.36036 0.680179 0.733046i \(-0.261902\pi\)
0.680179 + 0.733046i \(0.261902\pi\)
\(192\) 0 0
\(193\) −3333.70 −1.24334 −0.621671 0.783278i \(-0.713546\pi\)
−0.621671 + 0.783278i \(0.713546\pi\)
\(194\) −1705.58 −0.631205
\(195\) 0 0
\(196\) 0 0
\(197\) −3125.50 −1.13037 −0.565185 0.824964i \(-0.691195\pi\)
−0.565185 + 0.824964i \(0.691195\pi\)
\(198\) 0 0
\(199\) −4384.56 −1.56188 −0.780938 0.624608i \(-0.785259\pi\)
−0.780938 + 0.624608i \(0.785259\pi\)
\(200\) −1624.03 −0.574180
\(201\) 0 0
\(202\) −1326.19 −0.461933
\(203\) 0 0
\(204\) 0 0
\(205\) −1616.43 −0.550713
\(206\) 2027.44 0.685721
\(207\) 0 0
\(208\) −209.920 −0.0699774
\(209\) −5053.68 −1.67259
\(210\) 0 0
\(211\) −2333.73 −0.761423 −0.380712 0.924694i \(-0.624321\pi\)
−0.380712 + 0.924694i \(0.624321\pi\)
\(212\) −2321.32 −0.752023
\(213\) 0 0
\(214\) −333.189 −0.106431
\(215\) −1466.65 −0.465231
\(216\) 0 0
\(217\) 0 0
\(218\) −824.609 −0.256191
\(219\) 0 0
\(220\) −1255.71 −0.384818
\(221\) −430.292 −0.130971
\(222\) 0 0
\(223\) 226.956 0.0681528 0.0340764 0.999419i \(-0.489151\pi\)
0.0340764 + 0.999419i \(0.489151\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 928.422 0.273264
\(227\) 1298.84 0.379767 0.189884 0.981807i \(-0.439189\pi\)
0.189884 + 0.981807i \(0.439189\pi\)
\(228\) 0 0
\(229\) 445.975 0.128694 0.0643469 0.997928i \(-0.479504\pi\)
0.0643469 + 0.997928i \(0.479504\pi\)
\(230\) −842.902 −0.241649
\(231\) 0 0
\(232\) −2595.34 −0.734450
\(233\) 3599.79 1.01215 0.506073 0.862490i \(-0.331097\pi\)
0.506073 + 0.862490i \(0.331097\pi\)
\(234\) 0 0
\(235\) 1438.38 0.399275
\(236\) −1663.93 −0.458951
\(237\) 0 0
\(238\) 0 0
\(239\) 5201.62 1.40780 0.703902 0.710298i \(-0.251440\pi\)
0.703902 + 0.710298i \(0.251440\pi\)
\(240\) 0 0
\(241\) −5929.56 −1.58488 −0.792440 0.609949i \(-0.791190\pi\)
−0.792440 + 0.609949i \(0.791190\pi\)
\(242\) 425.613 0.113055
\(243\) 0 0
\(244\) −133.991 −0.0351553
\(245\) 0 0
\(246\) 0 0
\(247\) 3002.25 0.773395
\(248\) 3503.01 0.896940
\(249\) 0 0
\(250\) 2205.98 0.558073
\(251\) −5101.02 −1.28276 −0.641381 0.767222i \(-0.721638\pi\)
−0.641381 + 0.767222i \(0.721638\pi\)
\(252\) 0 0
\(253\) 2505.47 0.622598
\(254\) −3730.00 −0.921421
\(255\) 0 0
\(256\) −3497.94 −0.853990
\(257\) −1441.05 −0.349768 −0.174884 0.984589i \(-0.555955\pi\)
−0.174884 + 0.984589i \(0.555955\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 745.982 0.177938
\(261\) 0 0
\(262\) 4181.11 0.985915
\(263\) 7287.63 1.70865 0.854325 0.519740i \(-0.173971\pi\)
0.854325 + 0.519740i \(0.173971\pi\)
\(264\) 0 0
\(265\) 2925.76 0.678218
\(266\) 0 0
\(267\) 0 0
\(268\) 1519.01 0.346226
\(269\) 1964.67 0.445310 0.222655 0.974897i \(-0.428528\pi\)
0.222655 + 0.974897i \(0.428528\pi\)
\(270\) 0 0
\(271\) 2304.66 0.516599 0.258299 0.966065i \(-0.416838\pi\)
0.258299 + 0.966065i \(0.416838\pi\)
\(272\) 241.430 0.0538194
\(273\) 0 0
\(274\) −2207.74 −0.486769
\(275\) −2487.21 −0.545398
\(276\) 0 0
\(277\) −1685.97 −0.365704 −0.182852 0.983140i \(-0.558533\pi\)
−0.182852 + 0.983140i \(0.558533\pi\)
\(278\) 1348.83 0.290997
\(279\) 0 0
\(280\) 0 0
\(281\) −2841.43 −0.603223 −0.301612 0.953431i \(-0.597525\pi\)
−0.301612 + 0.953431i \(0.597525\pi\)
\(282\) 0 0
\(283\) −7307.50 −1.53493 −0.767466 0.641090i \(-0.778483\pi\)
−0.767466 + 0.641090i \(0.778483\pi\)
\(284\) −1862.72 −0.389197
\(285\) 0 0
\(286\) 989.432 0.204568
\(287\) 0 0
\(288\) 0 0
\(289\) −4418.12 −0.899271
\(290\) 1337.22 0.270773
\(291\) 0 0
\(292\) −3836.78 −0.768940
\(293\) −6778.11 −1.35147 −0.675736 0.737143i \(-0.736174\pi\)
−0.675736 + 0.737143i \(0.736174\pi\)
\(294\) 0 0
\(295\) 2097.19 0.413909
\(296\) 1408.08 0.276496
\(297\) 0 0
\(298\) 4892.56 0.951068
\(299\) −1488.43 −0.287886
\(300\) 0 0
\(301\) 0 0
\(302\) 42.0479 0.00801187
\(303\) 0 0
\(304\) −1684.52 −0.317809
\(305\) 168.881 0.0317052
\(306\) 0 0
\(307\) −8157.51 −1.51653 −0.758263 0.651949i \(-0.773952\pi\)
−0.758263 + 0.651949i \(0.773952\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1804.88 −0.330679
\(311\) 2545.26 0.464078 0.232039 0.972706i \(-0.425460\pi\)
0.232039 + 0.972706i \(0.425460\pi\)
\(312\) 0 0
\(313\) −2777.18 −0.501519 −0.250760 0.968049i \(-0.580680\pi\)
−0.250760 + 0.968049i \(0.580680\pi\)
\(314\) −3712.39 −0.667205
\(315\) 0 0
\(316\) 3233.40 0.575611
\(317\) 4832.35 0.856188 0.428094 0.903734i \(-0.359185\pi\)
0.428094 + 0.903734i \(0.359185\pi\)
\(318\) 0 0
\(319\) −3974.79 −0.697635
\(320\) 1444.40 0.252327
\(321\) 0 0
\(322\) 0 0
\(323\) −3452.91 −0.594815
\(324\) 0 0
\(325\) 1477.58 0.252189
\(326\) 446.975 0.0759376
\(327\) 0 0
\(328\) −4928.89 −0.829732
\(329\) 0 0
\(330\) 0 0
\(331\) 2609.24 0.433284 0.216642 0.976251i \(-0.430490\pi\)
0.216642 + 0.976251i \(0.430490\pi\)
\(332\) 6691.93 1.10623
\(333\) 0 0
\(334\) 5722.96 0.937564
\(335\) −1914.54 −0.312247
\(336\) 0 0
\(337\) −8134.50 −1.31488 −0.657440 0.753507i \(-0.728360\pi\)
−0.657440 + 0.753507i \(0.728360\pi\)
\(338\) 2863.89 0.460874
\(339\) 0 0
\(340\) −857.961 −0.136851
\(341\) 5364.89 0.851980
\(342\) 0 0
\(343\) 0 0
\(344\) −4472.17 −0.700940
\(345\) 0 0
\(346\) 4105.89 0.637960
\(347\) −10500.5 −1.62449 −0.812246 0.583314i \(-0.801756\pi\)
−0.812246 + 0.583314i \(0.801756\pi\)
\(348\) 0 0
\(349\) −10622.4 −1.62924 −0.814618 0.579998i \(-0.803053\pi\)
−0.814618 + 0.579998i \(0.803053\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −6092.68 −0.922560
\(353\) −7575.12 −1.14216 −0.571081 0.820894i \(-0.693476\pi\)
−0.571081 + 0.820894i \(0.693476\pi\)
\(354\) 0 0
\(355\) 2347.74 0.351000
\(356\) 6962.61 1.03657
\(357\) 0 0
\(358\) 4332.18 0.639562
\(359\) −9131.59 −1.34247 −0.671235 0.741244i \(-0.734236\pi\)
−0.671235 + 0.741244i \(0.734236\pi\)
\(360\) 0 0
\(361\) 17232.8 2.51244
\(362\) −3463.47 −0.502861
\(363\) 0 0
\(364\) 0 0
\(365\) 4835.82 0.693475
\(366\) 0 0
\(367\) 3582.52 0.509554 0.254777 0.967000i \(-0.417998\pi\)
0.254777 + 0.967000i \(0.417998\pi\)
\(368\) 835.135 0.118300
\(369\) 0 0
\(370\) −725.496 −0.101937
\(371\) 0 0
\(372\) 0 0
\(373\) 5192.26 0.720765 0.360382 0.932805i \(-0.382646\pi\)
0.360382 + 0.932805i \(0.382646\pi\)
\(374\) −1137.96 −0.157332
\(375\) 0 0
\(376\) 4385.99 0.601569
\(377\) 2361.31 0.322583
\(378\) 0 0
\(379\) 10012.2 1.35696 0.678482 0.734617i \(-0.262638\pi\)
0.678482 + 0.734617i \(0.262638\pi\)
\(380\) 5986.20 0.808120
\(381\) 0 0
\(382\) −5641.63 −0.755631
\(383\) 3347.99 0.446670 0.223335 0.974742i \(-0.428306\pi\)
0.223335 + 0.974742i \(0.428306\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 5237.55 0.690633
\(387\) 0 0
\(388\) −6005.21 −0.785743
\(389\) −510.839 −0.0665824 −0.0332912 0.999446i \(-0.510599\pi\)
−0.0332912 + 0.999446i \(0.510599\pi\)
\(390\) 0 0
\(391\) 1711.85 0.221412
\(392\) 0 0
\(393\) 0 0
\(394\) 4910.45 0.627880
\(395\) −4075.34 −0.519120
\(396\) 0 0
\(397\) −11828.7 −1.49538 −0.747691 0.664046i \(-0.768838\pi\)
−0.747691 + 0.664046i \(0.768838\pi\)
\(398\) 6888.55 0.867567
\(399\) 0 0
\(400\) −829.051 −0.103631
\(401\) −7881.42 −0.981494 −0.490747 0.871302i \(-0.663276\pi\)
−0.490747 + 0.871302i \(0.663276\pi\)
\(402\) 0 0
\(403\) −3187.13 −0.393951
\(404\) −4669.40 −0.575028
\(405\) 0 0
\(406\) 0 0
\(407\) 2156.48 0.262636
\(408\) 0 0
\(409\) −1579.66 −0.190976 −0.0954881 0.995431i \(-0.530441\pi\)
−0.0954881 + 0.995431i \(0.530441\pi\)
\(410\) 2539.55 0.305901
\(411\) 0 0
\(412\) 7138.44 0.853606
\(413\) 0 0
\(414\) 0 0
\(415\) −8434.42 −0.997661
\(416\) 3619.49 0.426587
\(417\) 0 0
\(418\) 7939.79 0.929062
\(419\) −433.887 −0.0505889 −0.0252944 0.999680i \(-0.508052\pi\)
−0.0252944 + 0.999680i \(0.508052\pi\)
\(420\) 0 0
\(421\) 813.228 0.0941432 0.0470716 0.998892i \(-0.485011\pi\)
0.0470716 + 0.998892i \(0.485011\pi\)
\(422\) 3666.50 0.422944
\(423\) 0 0
\(424\) 8921.36 1.02184
\(425\) −1699.38 −0.193958
\(426\) 0 0
\(427\) 0 0
\(428\) −1173.13 −0.132489
\(429\) 0 0
\(430\) 2304.24 0.258419
\(431\) 12624.4 1.41090 0.705450 0.708759i \(-0.250745\pi\)
0.705450 + 0.708759i \(0.250745\pi\)
\(432\) 0 0
\(433\) 8037.33 0.892031 0.446016 0.895025i \(-0.352843\pi\)
0.446016 + 0.895025i \(0.352843\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2903.38 −0.318914
\(437\) −11944.0 −1.30746
\(438\) 0 0
\(439\) −6820.72 −0.741538 −0.370769 0.928725i \(-0.620906\pi\)
−0.370769 + 0.928725i \(0.620906\pi\)
\(440\) 4825.98 0.522886
\(441\) 0 0
\(442\) 676.027 0.0727496
\(443\) −678.956 −0.0728175 −0.0364088 0.999337i \(-0.511592\pi\)
−0.0364088 + 0.999337i \(0.511592\pi\)
\(444\) 0 0
\(445\) −8775.58 −0.934837
\(446\) −356.568 −0.0378565
\(447\) 0 0
\(448\) 0 0
\(449\) 9068.89 0.953201 0.476601 0.879120i \(-0.341869\pi\)
0.476601 + 0.879120i \(0.341869\pi\)
\(450\) 0 0
\(451\) −7548.64 −0.788141
\(452\) 3268.89 0.340168
\(453\) 0 0
\(454\) −2040.60 −0.210947
\(455\) 0 0
\(456\) 0 0
\(457\) 4566.21 0.467392 0.233696 0.972310i \(-0.424918\pi\)
0.233696 + 0.972310i \(0.424918\pi\)
\(458\) −700.668 −0.0714848
\(459\) 0 0
\(460\) −2967.78 −0.300812
\(461\) 7604.79 0.768308 0.384154 0.923269i \(-0.374493\pi\)
0.384154 + 0.923269i \(0.374493\pi\)
\(462\) 0 0
\(463\) −10608.6 −1.06484 −0.532422 0.846479i \(-0.678718\pi\)
−0.532422 + 0.846479i \(0.678718\pi\)
\(464\) −1324.90 −0.132558
\(465\) 0 0
\(466\) −5655.60 −0.562212
\(467\) 13597.3 1.34734 0.673668 0.739034i \(-0.264718\pi\)
0.673668 + 0.739034i \(0.264718\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −2259.83 −0.221783
\(471\) 0 0
\(472\) 6394.85 0.623616
\(473\) −6849.18 −0.665805
\(474\) 0 0
\(475\) 11857.0 1.14534
\(476\) 0 0
\(477\) 0 0
\(478\) −8172.22 −0.781985
\(479\) −11950.4 −1.13993 −0.569965 0.821669i \(-0.693043\pi\)
−0.569965 + 0.821669i \(0.693043\pi\)
\(480\) 0 0
\(481\) −1281.11 −0.121442
\(482\) 9315.87 0.880345
\(483\) 0 0
\(484\) 1498.54 0.140735
\(485\) 7568.88 0.708629
\(486\) 0 0
\(487\) 16336.6 1.52009 0.760044 0.649872i \(-0.225178\pi\)
0.760044 + 0.649872i \(0.225178\pi\)
\(488\) 514.959 0.0477686
\(489\) 0 0
\(490\) 0 0
\(491\) −21307.1 −1.95841 −0.979204 0.202880i \(-0.934970\pi\)
−0.979204 + 0.202880i \(0.934970\pi\)
\(492\) 0 0
\(493\) −2715.77 −0.248097
\(494\) −4716.81 −0.429594
\(495\) 0 0
\(496\) 1788.25 0.161885
\(497\) 0 0
\(498\) 0 0
\(499\) 16006.8 1.43600 0.718000 0.696043i \(-0.245058\pi\)
0.718000 + 0.696043i \(0.245058\pi\)
\(500\) 7767.04 0.694706
\(501\) 0 0
\(502\) 8014.16 0.712529
\(503\) 14633.8 1.29719 0.648597 0.761132i \(-0.275356\pi\)
0.648597 + 0.761132i \(0.275356\pi\)
\(504\) 0 0
\(505\) 5885.24 0.518594
\(506\) −3936.31 −0.345831
\(507\) 0 0
\(508\) −13133.0 −1.14701
\(509\) 3992.22 0.347646 0.173823 0.984777i \(-0.444388\pi\)
0.173823 + 0.984777i \(0.444388\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −3876.64 −0.334619
\(513\) 0 0
\(514\) 2264.03 0.194284
\(515\) −8997.19 −0.769832
\(516\) 0 0
\(517\) 6717.18 0.571415
\(518\) 0 0
\(519\) 0 0
\(520\) −2866.98 −0.241780
\(521\) −21464.7 −1.80496 −0.902480 0.430731i \(-0.858256\pi\)
−0.902480 + 0.430731i \(0.858256\pi\)
\(522\) 0 0
\(523\) 5217.23 0.436201 0.218101 0.975926i \(-0.430014\pi\)
0.218101 + 0.975926i \(0.430014\pi\)
\(524\) 14721.3 1.22730
\(525\) 0 0
\(526\) −11449.5 −0.949094
\(527\) 3665.55 0.302986
\(528\) 0 0
\(529\) −6245.51 −0.513316
\(530\) −4596.63 −0.376726
\(531\) 0 0
\(532\) 0 0
\(533\) 4484.44 0.364432
\(534\) 0 0
\(535\) 1478.59 0.119486
\(536\) −5837.92 −0.470447
\(537\) 0 0
\(538\) −3086.68 −0.247354
\(539\) 0 0
\(540\) 0 0
\(541\) −9563.03 −0.759976 −0.379988 0.924991i \(-0.624072\pi\)
−0.379988 + 0.924991i \(0.624072\pi\)
\(542\) −3620.83 −0.286952
\(543\) 0 0
\(544\) −4162.81 −0.328086
\(545\) 3659.37 0.287615
\(546\) 0 0
\(547\) 2206.54 0.172477 0.0862383 0.996275i \(-0.472515\pi\)
0.0862383 + 0.996275i \(0.472515\pi\)
\(548\) −7773.27 −0.605944
\(549\) 0 0
\(550\) 3907.64 0.302949
\(551\) 18948.6 1.46504
\(552\) 0 0
\(553\) 0 0
\(554\) 2648.81 0.203136
\(555\) 0 0
\(556\) 4749.10 0.362242
\(557\) 4723.55 0.359324 0.179662 0.983728i \(-0.442500\pi\)
0.179662 + 0.983728i \(0.442500\pi\)
\(558\) 0 0
\(559\) 4068.91 0.307865
\(560\) 0 0
\(561\) 0 0
\(562\) 4464.15 0.335069
\(563\) −14140.9 −1.05856 −0.529280 0.848447i \(-0.677538\pi\)
−0.529280 + 0.848447i \(0.677538\pi\)
\(564\) 0 0
\(565\) −4120.07 −0.306783
\(566\) 11480.7 0.852600
\(567\) 0 0
\(568\) 7158.85 0.528835
\(569\) 14817.9 1.09174 0.545868 0.837871i \(-0.316200\pi\)
0.545868 + 0.837871i \(0.316200\pi\)
\(570\) 0 0
\(571\) −20877.9 −1.53015 −0.765073 0.643944i \(-0.777297\pi\)
−0.765073 + 0.643944i \(0.777297\pi\)
\(572\) 3483.70 0.254652
\(573\) 0 0
\(574\) 0 0
\(575\) −5878.35 −0.426338
\(576\) 0 0
\(577\) 25816.5 1.86266 0.931330 0.364176i \(-0.118649\pi\)
0.931330 + 0.364176i \(0.118649\pi\)
\(578\) 6941.27 0.499513
\(579\) 0 0
\(580\) 4708.23 0.337067
\(581\) 0 0
\(582\) 0 0
\(583\) 13663.2 0.970618
\(584\) 14745.6 1.04483
\(585\) 0 0
\(586\) 10649.0 0.750695
\(587\) −1727.78 −0.121487 −0.0607435 0.998153i \(-0.519347\pi\)
−0.0607435 + 0.998153i \(0.519347\pi\)
\(588\) 0 0
\(589\) −25575.4 −1.78916
\(590\) −3294.88 −0.229912
\(591\) 0 0
\(592\) 718.811 0.0499036
\(593\) 14580.1 1.00966 0.504832 0.863217i \(-0.331554\pi\)
0.504832 + 0.863217i \(0.331554\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 17226.3 1.18392
\(597\) 0 0
\(598\) 2338.45 0.159911
\(599\) 27691.5 1.88889 0.944444 0.328673i \(-0.106602\pi\)
0.944444 + 0.328673i \(0.106602\pi\)
\(600\) 0 0
\(601\) −783.554 −0.0531811 −0.0265906 0.999646i \(-0.508465\pi\)
−0.0265906 + 0.999646i \(0.508465\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 148.047 0.00997342
\(605\) −1888.74 −0.126923
\(606\) 0 0
\(607\) −23576.1 −1.57648 −0.788241 0.615366i \(-0.789008\pi\)
−0.788241 + 0.615366i \(0.789008\pi\)
\(608\) 29044.9 1.93738
\(609\) 0 0
\(610\) −265.327 −0.0176111
\(611\) −3990.49 −0.264219
\(612\) 0 0
\(613\) 25158.3 1.65764 0.828820 0.559515i \(-0.189012\pi\)
0.828820 + 0.559515i \(0.189012\pi\)
\(614\) 12816.2 0.842377
\(615\) 0 0
\(616\) 0 0
\(617\) 12873.3 0.839968 0.419984 0.907532i \(-0.362036\pi\)
0.419984 + 0.907532i \(0.362036\pi\)
\(618\) 0 0
\(619\) 14591.6 0.947476 0.473738 0.880666i \(-0.342904\pi\)
0.473738 + 0.880666i \(0.342904\pi\)
\(620\) −6354.84 −0.411640
\(621\) 0 0
\(622\) −3998.83 −0.257779
\(623\) 0 0
\(624\) 0 0
\(625\) −240.648 −0.0154015
\(626\) 4363.20 0.278576
\(627\) 0 0
\(628\) −13071.0 −0.830557
\(629\) 1473.41 0.0934003
\(630\) 0 0
\(631\) 20163.1 1.27208 0.636039 0.771657i \(-0.280572\pi\)
0.636039 + 0.771657i \(0.280572\pi\)
\(632\) −12426.7 −0.782133
\(633\) 0 0
\(634\) −7592.05 −0.475582
\(635\) 16552.6 1.03444
\(636\) 0 0
\(637\) 0 0
\(638\) 6244.76 0.387511
\(639\) 0 0
\(640\) 8167.95 0.504479
\(641\) −10687.8 −0.658572 −0.329286 0.944230i \(-0.606808\pi\)
−0.329286 + 0.944230i \(0.606808\pi\)
\(642\) 0 0
\(643\) −9248.53 −0.567226 −0.283613 0.958939i \(-0.591533\pi\)
−0.283613 + 0.958939i \(0.591533\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 5424.84 0.330399
\(647\) −14521.5 −0.882380 −0.441190 0.897414i \(-0.645444\pi\)
−0.441190 + 0.897414i \(0.645444\pi\)
\(648\) 0 0
\(649\) 9793.78 0.592357
\(650\) −2321.42 −0.140082
\(651\) 0 0
\(652\) 1573.76 0.0945294
\(653\) −5894.47 −0.353244 −0.176622 0.984279i \(-0.556517\pi\)
−0.176622 + 0.984279i \(0.556517\pi\)
\(654\) 0 0
\(655\) −18554.5 −1.10685
\(656\) −2516.15 −0.149755
\(657\) 0 0
\(658\) 0 0
\(659\) −8398.63 −0.496455 −0.248228 0.968702i \(-0.579848\pi\)
−0.248228 + 0.968702i \(0.579848\pi\)
\(660\) 0 0
\(661\) 3513.82 0.206765 0.103383 0.994642i \(-0.467033\pi\)
0.103383 + 0.994642i \(0.467033\pi\)
\(662\) −4099.36 −0.240674
\(663\) 0 0
\(664\) −25718.6 −1.50313
\(665\) 0 0
\(666\) 0 0
\(667\) −9394.13 −0.545341
\(668\) 20150.0 1.16711
\(669\) 0 0
\(670\) 3007.92 0.173442
\(671\) 788.665 0.0453742
\(672\) 0 0
\(673\) −14499.1 −0.830457 −0.415229 0.909717i \(-0.636298\pi\)
−0.415229 + 0.909717i \(0.636298\pi\)
\(674\) 12780.0 0.730369
\(675\) 0 0
\(676\) 10083.5 0.573709
\(677\) 26741.4 1.51810 0.759051 0.651031i \(-0.225663\pi\)
0.759051 + 0.651031i \(0.225663\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 3297.34 0.185952
\(681\) 0 0
\(682\) −8428.73 −0.473245
\(683\) −25949.3 −1.45377 −0.726883 0.686761i \(-0.759032\pi\)
−0.726883 + 0.686761i \(0.759032\pi\)
\(684\) 0 0
\(685\) 9797.32 0.546476
\(686\) 0 0
\(687\) 0 0
\(688\) −2283.01 −0.126510
\(689\) −8116.90 −0.448809
\(690\) 0 0
\(691\) 3583.56 0.197286 0.0986432 0.995123i \(-0.468550\pi\)
0.0986432 + 0.995123i \(0.468550\pi\)
\(692\) 14456.5 0.794152
\(693\) 0 0
\(694\) 16497.3 0.902348
\(695\) −5985.70 −0.326691
\(696\) 0 0
\(697\) −5157.59 −0.280284
\(698\) 16688.7 0.904983
\(699\) 0 0
\(700\) 0 0
\(701\) −15096.1 −0.813367 −0.406683 0.913569i \(-0.633315\pi\)
−0.406683 + 0.913569i \(0.633315\pi\)
\(702\) 0 0
\(703\) −10280.4 −0.551538
\(704\) 6745.30 0.361112
\(705\) 0 0
\(706\) 11901.2 0.634430
\(707\) 0 0
\(708\) 0 0
\(709\) −29170.5 −1.54517 −0.772583 0.634914i \(-0.781036\pi\)
−0.772583 + 0.634914i \(0.781036\pi\)
\(710\) −3688.52 −0.194968
\(711\) 0 0
\(712\) −26758.9 −1.40847
\(713\) 12679.5 0.665992
\(714\) 0 0
\(715\) −4390.81 −0.229660
\(716\) 15253.2 0.796146
\(717\) 0 0
\(718\) 14346.6 0.745695
\(719\) 9145.94 0.474390 0.237195 0.971462i \(-0.423772\pi\)
0.237195 + 0.971462i \(0.423772\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −27074.4 −1.39557
\(723\) 0 0
\(724\) −12194.6 −0.625977
\(725\) 9325.70 0.477721
\(726\) 0 0
\(727\) 35509.8 1.81154 0.905768 0.423773i \(-0.139295\pi\)
0.905768 + 0.423773i \(0.139295\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −7597.51 −0.385201
\(731\) −4679.69 −0.236778
\(732\) 0 0
\(733\) 6476.19 0.326335 0.163168 0.986598i \(-0.447829\pi\)
0.163168 + 0.986598i \(0.447829\pi\)
\(734\) −5628.47 −0.283039
\(735\) 0 0
\(736\) −14399.6 −0.721165
\(737\) −8940.83 −0.446866
\(738\) 0 0
\(739\) 31421.5 1.56408 0.782042 0.623226i \(-0.214178\pi\)
0.782042 + 0.623226i \(0.214178\pi\)
\(740\) −2554.41 −0.126894
\(741\) 0 0
\(742\) 0 0
\(743\) 12845.7 0.634268 0.317134 0.948381i \(-0.397279\pi\)
0.317134 + 0.948381i \(0.397279\pi\)
\(744\) 0 0
\(745\) −21711.7 −1.06773
\(746\) −8157.52 −0.400359
\(747\) 0 0
\(748\) −4006.64 −0.195852
\(749\) 0 0
\(750\) 0 0
\(751\) 24892.6 1.20951 0.604756 0.796411i \(-0.293271\pi\)
0.604756 + 0.796411i \(0.293271\pi\)
\(752\) 2239.01 0.108575
\(753\) 0 0
\(754\) −3709.83 −0.179183
\(755\) −186.596 −0.00899461
\(756\) 0 0
\(757\) 34261.1 1.64497 0.822483 0.568789i \(-0.192588\pi\)
0.822483 + 0.568789i \(0.192588\pi\)
\(758\) −15730.0 −0.753746
\(759\) 0 0
\(760\) −23006.4 −1.09806
\(761\) 8054.98 0.383696 0.191848 0.981425i \(-0.438552\pi\)
0.191848 + 0.981425i \(0.438552\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −19863.7 −0.940632
\(765\) 0 0
\(766\) −5260.00 −0.248109
\(767\) −5818.21 −0.273903
\(768\) 0 0
\(769\) −1915.85 −0.0898403 −0.0449202 0.998991i \(-0.514303\pi\)
−0.0449202 + 0.998991i \(0.514303\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 18441.0 0.859721
\(773\) −15431.6 −0.718027 −0.359014 0.933332i \(-0.616887\pi\)
−0.359014 + 0.933332i \(0.616887\pi\)
\(774\) 0 0
\(775\) −12587.2 −0.583412
\(776\) 23079.4 1.06766
\(777\) 0 0
\(778\) 802.574 0.0369842
\(779\) 35985.8 1.65510
\(780\) 0 0
\(781\) 10963.8 0.502327
\(782\) −2689.48 −0.122987
\(783\) 0 0
\(784\) 0 0
\(785\) 16474.5 0.749046
\(786\) 0 0
\(787\) 18487.3 0.837358 0.418679 0.908134i \(-0.362493\pi\)
0.418679 + 0.908134i \(0.362493\pi\)
\(788\) 17289.3 0.781604
\(789\) 0 0
\(790\) 6402.73 0.288353
\(791\) 0 0
\(792\) 0 0
\(793\) −468.524 −0.0209808
\(794\) 18584.0 0.830632
\(795\) 0 0
\(796\) 24254.0 1.07997
\(797\) −7017.35 −0.311879 −0.155939 0.987767i \(-0.549840\pi\)
−0.155939 + 0.987767i \(0.549840\pi\)
\(798\) 0 0
\(799\) 4589.50 0.203210
\(800\) 14294.7 0.631743
\(801\) 0 0
\(802\) 12382.4 0.545185
\(803\) 22583.1 0.992452
\(804\) 0 0
\(805\) 0 0
\(806\) 5007.27 0.218826
\(807\) 0 0
\(808\) 17945.6 0.781340
\(809\) 13231.2 0.575012 0.287506 0.957779i \(-0.407174\pi\)
0.287506 + 0.957779i \(0.407174\pi\)
\(810\) 0 0
\(811\) −8930.40 −0.386669 −0.193335 0.981133i \(-0.561930\pi\)
−0.193335 + 0.981133i \(0.561930\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) −3388.03 −0.145885
\(815\) −1983.54 −0.0852521
\(816\) 0 0
\(817\) 32651.3 1.39819
\(818\) 2481.79 0.106081
\(819\) 0 0
\(820\) 8941.54 0.380795
\(821\) −35165.3 −1.49486 −0.747429 0.664341i \(-0.768712\pi\)
−0.747429 + 0.664341i \(0.768712\pi\)
\(822\) 0 0
\(823\) −9236.94 −0.391227 −0.195613 0.980681i \(-0.562670\pi\)
−0.195613 + 0.980681i \(0.562670\pi\)
\(824\) −27434.7 −1.15987
\(825\) 0 0
\(826\) 0 0
\(827\) 22176.8 0.932483 0.466241 0.884658i \(-0.345608\pi\)
0.466241 + 0.884658i \(0.345608\pi\)
\(828\) 0 0
\(829\) −36563.5 −1.53185 −0.765925 0.642930i \(-0.777719\pi\)
−0.765925 + 0.642930i \(0.777719\pi\)
\(830\) 13251.2 0.554165
\(831\) 0 0
\(832\) −4007.19 −0.166976
\(833\) 0 0
\(834\) 0 0
\(835\) −25396.8 −1.05257
\(836\) 27955.3 1.15652
\(837\) 0 0
\(838\) 681.675 0.0281003
\(839\) −44231.4 −1.82007 −0.910035 0.414531i \(-0.863945\pi\)
−0.910035 + 0.414531i \(0.863945\pi\)
\(840\) 0 0
\(841\) −9485.70 −0.388933
\(842\) −1277.65 −0.0522932
\(843\) 0 0
\(844\) 12909.4 0.526493
\(845\) −12709.1 −0.517405
\(846\) 0 0
\(847\) 0 0
\(848\) 4554.28 0.184427
\(849\) 0 0
\(850\) 2669.88 0.107737
\(851\) 5096.70 0.205303
\(852\) 0 0
\(853\) −19451.0 −0.780762 −0.390381 0.920653i \(-0.627657\pi\)
−0.390381 + 0.920653i \(0.627657\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 4508.60 0.180024
\(857\) −26386.1 −1.05173 −0.525865 0.850568i \(-0.676258\pi\)
−0.525865 + 0.850568i \(0.676258\pi\)
\(858\) 0 0
\(859\) 27227.7 1.08149 0.540744 0.841187i \(-0.318143\pi\)
0.540744 + 0.841187i \(0.318143\pi\)
\(860\) 8113.02 0.321688
\(861\) 0 0
\(862\) −19834.2 −0.783705
\(863\) −20901.0 −0.824425 −0.412212 0.911088i \(-0.635244\pi\)
−0.412212 + 0.911088i \(0.635244\pi\)
\(864\) 0 0
\(865\) −18220.8 −0.716213
\(866\) −12627.4 −0.495492
\(867\) 0 0
\(868\) 0 0
\(869\) −19031.6 −0.742928
\(870\) 0 0
\(871\) 5311.50 0.206628
\(872\) 11158.3 0.433336
\(873\) 0 0
\(874\) 18765.1 0.726248
\(875\) 0 0
\(876\) 0 0
\(877\) 12516.3 0.481921 0.240961 0.970535i \(-0.422538\pi\)
0.240961 + 0.970535i \(0.422538\pi\)
\(878\) 10716.0 0.411898
\(879\) 0 0
\(880\) 2463.62 0.0943735
\(881\) 1308.26 0.0500301 0.0250150 0.999687i \(-0.492037\pi\)
0.0250150 + 0.999687i \(0.492037\pi\)
\(882\) 0 0
\(883\) 19537.5 0.744609 0.372305 0.928111i \(-0.378568\pi\)
0.372305 + 0.928111i \(0.378568\pi\)
\(884\) 2380.23 0.0905609
\(885\) 0 0
\(886\) 1066.70 0.0404475
\(887\) 286.574 0.0108480 0.00542402 0.999985i \(-0.498273\pi\)
0.00542402 + 0.999985i \(0.498273\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 13787.2 0.519269
\(891\) 0 0
\(892\) −1255.44 −0.0471249
\(893\) −32022.1 −1.19997
\(894\) 0 0
\(895\) −19225.0 −0.718011
\(896\) 0 0
\(897\) 0 0
\(898\) −14248.1 −0.529470
\(899\) −20115.4 −0.746259
\(900\) 0 0
\(901\) 9335.32 0.345177
\(902\) 11859.6 0.437784
\(903\) 0 0
\(904\) −12563.1 −0.462215
\(905\) 15369.9 0.564543
\(906\) 0 0
\(907\) −7960.25 −0.291418 −0.145709 0.989328i \(-0.546546\pi\)
−0.145709 + 0.989328i \(0.546546\pi\)
\(908\) −7184.77 −0.262594
\(909\) 0 0
\(910\) 0 0
\(911\) 38900.9 1.41476 0.707378 0.706835i \(-0.249878\pi\)
0.707378 + 0.706835i \(0.249878\pi\)
\(912\) 0 0
\(913\) −39388.4 −1.42778
\(914\) −7173.93 −0.259620
\(915\) 0 0
\(916\) −2466.99 −0.0889865
\(917\) 0 0
\(918\) 0 0
\(919\) 1315.54 0.0472205 0.0236103 0.999721i \(-0.492484\pi\)
0.0236103 + 0.999721i \(0.492484\pi\)
\(920\) 11405.9 0.408740
\(921\) 0 0
\(922\) −11947.8 −0.426768
\(923\) −6513.31 −0.232273
\(924\) 0 0
\(925\) −5059.57 −0.179846
\(926\) 16667.1 0.591483
\(927\) 0 0
\(928\) 22844.2 0.808081
\(929\) −46155.4 −1.63004 −0.815021 0.579432i \(-0.803274\pi\)
−0.815021 + 0.579432i \(0.803274\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −19912.9 −0.699858
\(933\) 0 0
\(934\) −21362.5 −0.748397
\(935\) 5049.91 0.176631
\(936\) 0 0
\(937\) −41274.8 −1.43905 −0.719525 0.694466i \(-0.755641\pi\)
−0.719525 + 0.694466i \(0.755641\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −7956.66 −0.276083
\(941\) −29642.6 −1.02691 −0.513454 0.858117i \(-0.671634\pi\)
−0.513454 + 0.858117i \(0.671634\pi\)
\(942\) 0 0
\(943\) −17840.7 −0.616090
\(944\) 3264.52 0.112554
\(945\) 0 0
\(946\) 10760.7 0.369831
\(947\) 24856.8 0.852944 0.426472 0.904501i \(-0.359756\pi\)
0.426472 + 0.904501i \(0.359756\pi\)
\(948\) 0 0
\(949\) −13416.0 −0.458905
\(950\) −18628.4 −0.636196
\(951\) 0 0
\(952\) 0 0
\(953\) −36288.1 −1.23346 −0.616729 0.787175i \(-0.711543\pi\)
−0.616729 + 0.787175i \(0.711543\pi\)
\(954\) 0 0
\(955\) 25035.9 0.848317
\(956\) −28773.7 −0.973438
\(957\) 0 0
\(958\) 18775.1 0.633190
\(959\) 0 0
\(960\) 0 0
\(961\) −2640.61 −0.0886378
\(962\) 2012.74 0.0674566
\(963\) 0 0
\(964\) 32800.4 1.09588
\(965\) −23242.7 −0.775347
\(966\) 0 0
\(967\) −16773.7 −0.557815 −0.278908 0.960318i \(-0.589972\pi\)
−0.278908 + 0.960318i \(0.589972\pi\)
\(968\) −5759.25 −0.191229
\(969\) 0 0
\(970\) −11891.4 −0.393619
\(971\) −4458.85 −0.147365 −0.0736825 0.997282i \(-0.523475\pi\)
−0.0736825 + 0.997282i \(0.523475\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −25666.3 −0.844355
\(975\) 0 0
\(976\) 262.882 0.00862156
\(977\) 4100.18 0.134265 0.0671323 0.997744i \(-0.478615\pi\)
0.0671323 + 0.997744i \(0.478615\pi\)
\(978\) 0 0
\(979\) −40981.6 −1.33787
\(980\) 0 0
\(981\) 0 0
\(982\) 33475.5 1.08783
\(983\) −8704.08 −0.282418 −0.141209 0.989980i \(-0.545099\pi\)
−0.141209 + 0.989980i \(0.545099\pi\)
\(984\) 0 0
\(985\) −21791.1 −0.704897
\(986\) 4266.71 0.137809
\(987\) 0 0
\(988\) −16607.5 −0.534771
\(989\) −16187.6 −0.520460
\(990\) 0 0
\(991\) −20466.1 −0.656030 −0.328015 0.944673i \(-0.606380\pi\)
−0.328015 + 0.944673i \(0.606380\pi\)
\(992\) −30833.6 −0.986861
\(993\) 0 0
\(994\) 0 0
\(995\) −30569.4 −0.973984
\(996\) 0 0
\(997\) 9354.70 0.297158 0.148579 0.988901i \(-0.452530\pi\)
0.148579 + 0.988901i \(0.452530\pi\)
\(998\) −25148.2 −0.797647
\(999\) 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 1323.4.a.bb.1.3 6
3.2 odd 2 1323.4.a.bc.1.4 yes 6
7.6 odd 2 1323.4.a.bc.1.3 yes 6
21.20 even 2 inner 1323.4.a.bb.1.4 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.4.a.bb.1.3 6 1.1 even 1 trivial
1323.4.a.bb.1.4 yes 6 21.20 even 2 inner
1323.4.a.bc.1.3 yes 6 7.6 odd 2
1323.4.a.bc.1.4 yes 6 3.2 odd 2