Properties

Label 1323.4.a.bb.1.6
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 1.6
Root \(-1.72645\) 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)
 
\(f(q)\) \(=\) \(q+4.77246 q^{2} +14.7764 q^{4} +1.85905 q^{5} +32.3399 q^{8} +O(q^{10})\) \(q+4.77246 q^{2} +14.7764 q^{4} +1.85905 q^{5} +32.3399 q^{8} +8.87225 q^{10} -13.8420 q^{11} -74.4220 q^{13} +36.1299 q^{16} -89.3762 q^{17} -50.0484 q^{19} +27.4700 q^{20} -66.0605 q^{22} +33.2906 q^{23} -121.544 q^{25} -355.176 q^{26} +18.7311 q^{29} -52.1912 q^{31} -86.2907 q^{32} -426.544 q^{34} -152.186 q^{37} -238.854 q^{38} +60.1215 q^{40} +240.210 q^{41} +297.771 q^{43} -204.535 q^{44} +158.878 q^{46} -520.533 q^{47} -580.063 q^{50} -1099.69 q^{52} +566.733 q^{53} -25.7331 q^{55} +89.3936 q^{58} -622.856 q^{59} +714.318 q^{61} -249.080 q^{62} -700.858 q^{64} -138.354 q^{65} -45.3820 q^{67} -1320.66 q^{68} +866.000 q^{71} -966.243 q^{73} -726.303 q^{74} -739.533 q^{76} +1015.70 q^{79} +67.1673 q^{80} +1146.39 q^{82} -380.649 q^{83} -166.155 q^{85} +1421.10 q^{86} -447.650 q^{88} -104.077 q^{89} +491.913 q^{92} -2484.22 q^{94} -93.0426 q^{95} -765.266 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 18 q^{4} - 24 q^{5}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.77246 1.68732 0.843659 0.536879i \(-0.180397\pi\)
0.843659 + 0.536879i \(0.180397\pi\)
\(3\) 0 0
\(4\) 14.7764 1.84704
\(5\) 1.85905 0.166279 0.0831393 0.996538i \(-0.473505\pi\)
0.0831393 + 0.996538i \(0.473505\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 32.3399 1.42923
\(9\) 0 0
\(10\) 8.87225 0.280565
\(11\) −13.8420 −0.379412 −0.189706 0.981841i \(-0.560753\pi\)
−0.189706 + 0.981841i \(0.560753\pi\)
\(12\) 0 0
\(13\) −74.4220 −1.58777 −0.793883 0.608071i \(-0.791944\pi\)
−0.793883 + 0.608071i \(0.791944\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 36.1299 0.564529
\(17\) −89.3762 −1.27511 −0.637557 0.770404i \(-0.720055\pi\)
−0.637557 + 0.770404i \(0.720055\pi\)
\(18\) 0 0
\(19\) −50.0484 −0.604310 −0.302155 0.953259i \(-0.597706\pi\)
−0.302155 + 0.953259i \(0.597706\pi\)
\(20\) 27.4700 0.307124
\(21\) 0 0
\(22\) −66.0605 −0.640189
\(23\) 33.2906 0.301807 0.150903 0.988548i \(-0.451782\pi\)
0.150903 + 0.988548i \(0.451782\pi\)
\(24\) 0 0
\(25\) −121.544 −0.972351
\(26\) −355.176 −2.67907
\(27\) 0 0
\(28\) 0 0
\(29\) 18.7311 0.119941 0.0599705 0.998200i \(-0.480899\pi\)
0.0599705 + 0.998200i \(0.480899\pi\)
\(30\) 0 0
\(31\) −52.1912 −0.302381 −0.151191 0.988505i \(-0.548311\pi\)
−0.151191 + 0.988505i \(0.548311\pi\)
\(32\) −86.2907 −0.476694
\(33\) 0 0
\(34\) −426.544 −2.15152
\(35\) 0 0
\(36\) 0 0
\(37\) −152.186 −0.676197 −0.338099 0.941111i \(-0.609784\pi\)
−0.338099 + 0.941111i \(0.609784\pi\)
\(38\) −238.854 −1.01966
\(39\) 0 0
\(40\) 60.1215 0.237651
\(41\) 240.210 0.914987 0.457493 0.889213i \(-0.348747\pi\)
0.457493 + 0.889213i \(0.348747\pi\)
\(42\) 0 0
\(43\) 297.771 1.05604 0.528020 0.849232i \(-0.322935\pi\)
0.528020 + 0.849232i \(0.322935\pi\)
\(44\) −204.535 −0.700791
\(45\) 0 0
\(46\) 158.878 0.509245
\(47\) −520.533 −1.61548 −0.807739 0.589540i \(-0.799309\pi\)
−0.807739 + 0.589540i \(0.799309\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −580.063 −1.64067
\(51\) 0 0
\(52\) −1099.69 −2.93267
\(53\) 566.733 1.46881 0.734404 0.678713i \(-0.237462\pi\)
0.734404 + 0.678713i \(0.237462\pi\)
\(54\) 0 0
\(55\) −25.7331 −0.0630881
\(56\) 0 0
\(57\) 0 0
\(58\) 89.3936 0.202379
\(59\) −622.856 −1.37439 −0.687194 0.726474i \(-0.741158\pi\)
−0.687194 + 0.726474i \(0.741158\pi\)
\(60\) 0 0
\(61\) 714.318 1.49933 0.749664 0.661818i \(-0.230215\pi\)
0.749664 + 0.661818i \(0.230215\pi\)
\(62\) −249.080 −0.510214
\(63\) 0 0
\(64\) −700.858 −1.36886
\(65\) −138.354 −0.264012
\(66\) 0 0
\(67\) −45.3820 −0.0827506 −0.0413753 0.999144i \(-0.513174\pi\)
−0.0413753 + 0.999144i \(0.513174\pi\)
\(68\) −1320.66 −2.35519
\(69\) 0 0
\(70\) 0 0
\(71\) 866.000 1.44754 0.723770 0.690042i \(-0.242408\pi\)
0.723770 + 0.690042i \(0.242408\pi\)
\(72\) 0 0
\(73\) −966.243 −1.54918 −0.774590 0.632463i \(-0.782044\pi\)
−0.774590 + 0.632463i \(0.782044\pi\)
\(74\) −726.303 −1.14096
\(75\) 0 0
\(76\) −739.533 −1.11619
\(77\) 0 0
\(78\) 0 0
\(79\) 1015.70 1.44653 0.723263 0.690573i \(-0.242641\pi\)
0.723263 + 0.690573i \(0.242641\pi\)
\(80\) 67.1673 0.0938692
\(81\) 0 0
\(82\) 1146.39 1.54387
\(83\) −380.649 −0.503393 −0.251696 0.967806i \(-0.580988\pi\)
−0.251696 + 0.967806i \(0.580988\pi\)
\(84\) 0 0
\(85\) −166.155 −0.212024
\(86\) 1421.10 1.78187
\(87\) 0 0
\(88\) −447.650 −0.542269
\(89\) −104.077 −0.123957 −0.0619784 0.998077i \(-0.519741\pi\)
−0.0619784 + 0.998077i \(0.519741\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 491.913 0.557451
\(93\) 0 0
\(94\) −2484.22 −2.72583
\(95\) −93.0426 −0.100484
\(96\) 0 0
\(97\) −765.266 −0.801041 −0.400521 0.916288i \(-0.631171\pi\)
−0.400521 + 0.916288i \(0.631171\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −1795.98 −1.79598
\(101\) 1074.14 1.05822 0.529112 0.848552i \(-0.322525\pi\)
0.529112 + 0.848552i \(0.322525\pi\)
\(102\) 0 0
\(103\) 1201.47 1.14936 0.574680 0.818379i \(-0.305127\pi\)
0.574680 + 0.818379i \(0.305127\pi\)
\(104\) −2406.80 −2.26929
\(105\) 0 0
\(106\) 2704.71 2.47835
\(107\) −568.948 −0.514040 −0.257020 0.966406i \(-0.582741\pi\)
−0.257020 + 0.966406i \(0.582741\pi\)
\(108\) 0 0
\(109\) 1360.73 1.19573 0.597866 0.801596i \(-0.296016\pi\)
0.597866 + 0.801596i \(0.296016\pi\)
\(110\) −122.810 −0.106450
\(111\) 0 0
\(112\) 0 0
\(113\) −984.122 −0.819278 −0.409639 0.912248i \(-0.634345\pi\)
−0.409639 + 0.912248i \(0.634345\pi\)
\(114\) 0 0
\(115\) 61.8889 0.0501841
\(116\) 276.778 0.221536
\(117\) 0 0
\(118\) −2972.55 −2.31903
\(119\) 0 0
\(120\) 0 0
\(121\) −1139.40 −0.856047
\(122\) 3409.05 2.52985
\(123\) 0 0
\(124\) −771.196 −0.558512
\(125\) −458.338 −0.327960
\(126\) 0 0
\(127\) −2061.86 −1.44063 −0.720317 0.693645i \(-0.756004\pi\)
−0.720317 + 0.693645i \(0.756004\pi\)
\(128\) −2654.49 −1.83301
\(129\) 0 0
\(130\) −660.291 −0.445472
\(131\) 1985.16 1.32400 0.662000 0.749504i \(-0.269708\pi\)
0.662000 + 0.749504i \(0.269708\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −216.583 −0.139627
\(135\) 0 0
\(136\) −2890.42 −1.82244
\(137\) −1248.15 −0.778369 −0.389184 0.921160i \(-0.627243\pi\)
−0.389184 + 0.921160i \(0.627243\pi\)
\(138\) 0 0
\(139\) −2569.86 −1.56815 −0.784074 0.620668i \(-0.786862\pi\)
−0.784074 + 0.620668i \(0.786862\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4132.95 2.44246
\(143\) 1030.15 0.602417
\(144\) 0 0
\(145\) 34.8222 0.0199436
\(146\) −4611.36 −2.61396
\(147\) 0 0
\(148\) −2248.76 −1.24897
\(149\) 2508.80 1.37939 0.689695 0.724100i \(-0.257745\pi\)
0.689695 + 0.724100i \(0.257745\pi\)
\(150\) 0 0
\(151\) −2435.86 −1.31276 −0.656382 0.754429i \(-0.727914\pi\)
−0.656382 + 0.754429i \(0.727914\pi\)
\(152\) −1618.56 −0.863701
\(153\) 0 0
\(154\) 0 0
\(155\) −97.0262 −0.0502796
\(156\) 0 0
\(157\) 1375.28 0.699103 0.349552 0.936917i \(-0.386334\pi\)
0.349552 + 0.936917i \(0.386334\pi\)
\(158\) 4847.40 2.44075
\(159\) 0 0
\(160\) −160.419 −0.0792640
\(161\) 0 0
\(162\) 0 0
\(163\) −2567.21 −1.23362 −0.616808 0.787114i \(-0.711574\pi\)
−0.616808 + 0.787114i \(0.711574\pi\)
\(164\) 3549.42 1.69002
\(165\) 0 0
\(166\) −1816.63 −0.849384
\(167\) −2041.57 −0.945998 −0.472999 0.881063i \(-0.656829\pi\)
−0.472999 + 0.881063i \(0.656829\pi\)
\(168\) 0 0
\(169\) 3341.64 1.52100
\(170\) −792.968 −0.357752
\(171\) 0 0
\(172\) 4399.97 1.95055
\(173\) 1618.02 0.711072 0.355536 0.934663i \(-0.384298\pi\)
0.355536 + 0.934663i \(0.384298\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −500.111 −0.214189
\(177\) 0 0
\(178\) −496.704 −0.209155
\(179\) −3205.14 −1.33834 −0.669171 0.743108i \(-0.733351\pi\)
−0.669171 + 0.743108i \(0.733351\pi\)
\(180\) 0 0
\(181\) −671.747 −0.275859 −0.137930 0.990442i \(-0.544045\pi\)
−0.137930 + 0.990442i \(0.544045\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1076.61 0.431353
\(185\) −282.922 −0.112437
\(186\) 0 0
\(187\) 1237.15 0.483793
\(188\) −7691.57 −2.98386
\(189\) 0 0
\(190\) −444.042 −0.169548
\(191\) −3344.15 −1.26688 −0.633441 0.773791i \(-0.718358\pi\)
−0.633441 + 0.773791i \(0.718358\pi\)
\(192\) 0 0
\(193\) 296.246 0.110488 0.0552442 0.998473i \(-0.482406\pi\)
0.0552442 + 0.998473i \(0.482406\pi\)
\(194\) −3652.20 −1.35161
\(195\) 0 0
\(196\) 0 0
\(197\) −1908.71 −0.690303 −0.345152 0.938547i \(-0.612173\pi\)
−0.345152 + 0.938547i \(0.612173\pi\)
\(198\) 0 0
\(199\) 2038.09 0.726013 0.363006 0.931787i \(-0.381750\pi\)
0.363006 + 0.931787i \(0.381750\pi\)
\(200\) −3930.72 −1.38972
\(201\) 0 0
\(202\) 5126.27 1.78556
\(203\) 0 0
\(204\) 0 0
\(205\) 446.562 0.152143
\(206\) 5733.95 1.93934
\(207\) 0 0
\(208\) −2688.86 −0.896340
\(209\) 692.772 0.229282
\(210\) 0 0
\(211\) −378.754 −0.123576 −0.0617879 0.998089i \(-0.519680\pi\)
−0.0617879 + 0.998089i \(0.519680\pi\)
\(212\) 8374.25 2.71295
\(213\) 0 0
\(214\) −2715.28 −0.867349
\(215\) 553.572 0.175597
\(216\) 0 0
\(217\) 0 0
\(218\) 6494.05 2.01758
\(219\) 0 0
\(220\) −380.241 −0.116527
\(221\) 6651.56 2.02458
\(222\) 0 0
\(223\) 2180.53 0.654793 0.327397 0.944887i \(-0.393829\pi\)
0.327397 + 0.944887i \(0.393829\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −4696.68 −1.38238
\(227\) −3061.26 −0.895080 −0.447540 0.894264i \(-0.647700\pi\)
−0.447540 + 0.894264i \(0.647700\pi\)
\(228\) 0 0
\(229\) 5611.09 1.61918 0.809588 0.586998i \(-0.199690\pi\)
0.809588 + 0.586998i \(0.199690\pi\)
\(230\) 295.362 0.0846765
\(231\) 0 0
\(232\) 605.763 0.171424
\(233\) 2873.42 0.807915 0.403958 0.914778i \(-0.367634\pi\)
0.403958 + 0.914778i \(0.367634\pi\)
\(234\) 0 0
\(235\) −967.697 −0.268620
\(236\) −9203.54 −2.53856
\(237\) 0 0
\(238\) 0 0
\(239\) 938.880 0.254105 0.127052 0.991896i \(-0.459448\pi\)
0.127052 + 0.991896i \(0.459448\pi\)
\(240\) 0 0
\(241\) 3863.39 1.03262 0.516312 0.856400i \(-0.327304\pi\)
0.516312 + 0.856400i \(0.327304\pi\)
\(242\) −5437.73 −1.44442
\(243\) 0 0
\(244\) 10555.0 2.76933
\(245\) 0 0
\(246\) 0 0
\(247\) 3724.70 0.959503
\(248\) −1687.86 −0.432174
\(249\) 0 0
\(250\) −2187.40 −0.553373
\(251\) 557.651 0.140234 0.0701168 0.997539i \(-0.477663\pi\)
0.0701168 + 0.997539i \(0.477663\pi\)
\(252\) 0 0
\(253\) −460.809 −0.114509
\(254\) −9840.14 −2.43081
\(255\) 0 0
\(256\) −7061.58 −1.72402
\(257\) −5963.84 −1.44753 −0.723763 0.690048i \(-0.757589\pi\)
−0.723763 + 0.690048i \(0.757589\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2044.37 −0.487641
\(261\) 0 0
\(262\) 9474.08 2.23401
\(263\) −3804.77 −0.892062 −0.446031 0.895018i \(-0.647163\pi\)
−0.446031 + 0.895018i \(0.647163\pi\)
\(264\) 0 0
\(265\) 1053.59 0.244231
\(266\) 0 0
\(267\) 0 0
\(268\) −670.580 −0.152844
\(269\) −6288.67 −1.42538 −0.712689 0.701480i \(-0.752523\pi\)
−0.712689 + 0.701480i \(0.752523\pi\)
\(270\) 0 0
\(271\) 1976.24 0.442982 0.221491 0.975162i \(-0.428908\pi\)
0.221491 + 0.975162i \(0.428908\pi\)
\(272\) −3229.15 −0.719839
\(273\) 0 0
\(274\) −5956.74 −1.31336
\(275\) 1682.42 0.368922
\(276\) 0 0
\(277\) 3746.00 0.812546 0.406273 0.913752i \(-0.366828\pi\)
0.406273 + 0.913752i \(0.366828\pi\)
\(278\) −12264.5 −2.64596
\(279\) 0 0
\(280\) 0 0
\(281\) −1764.79 −0.374656 −0.187328 0.982297i \(-0.559983\pi\)
−0.187328 + 0.982297i \(0.559983\pi\)
\(282\) 0 0
\(283\) 8724.12 1.83249 0.916246 0.400616i \(-0.131204\pi\)
0.916246 + 0.400616i \(0.131204\pi\)
\(284\) 12796.3 2.67367
\(285\) 0 0
\(286\) 4916.36 1.01647
\(287\) 0 0
\(288\) 0 0
\(289\) 3075.11 0.625914
\(290\) 166.187 0.0336512
\(291\) 0 0
\(292\) −14277.6 −2.86141
\(293\) −4421.02 −0.881498 −0.440749 0.897630i \(-0.645287\pi\)
−0.440749 + 0.897630i \(0.645287\pi\)
\(294\) 0 0
\(295\) −1157.92 −0.228531
\(296\) −4921.69 −0.966444
\(297\) 0 0
\(298\) 11973.2 2.32747
\(299\) −2477.55 −0.479199
\(300\) 0 0
\(301\) 0 0
\(302\) −11625.0 −2.21505
\(303\) 0 0
\(304\) −1808.24 −0.341151
\(305\) 1327.95 0.249306
\(306\) 0 0
\(307\) 1646.75 0.306140 0.153070 0.988215i \(-0.451084\pi\)
0.153070 + 0.988215i \(0.451084\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −463.053 −0.0848376
\(311\) −1141.93 −0.208210 −0.104105 0.994566i \(-0.533198\pi\)
−0.104105 + 0.994566i \(0.533198\pi\)
\(312\) 0 0
\(313\) 2425.80 0.438064 0.219032 0.975718i \(-0.429710\pi\)
0.219032 + 0.975718i \(0.429710\pi\)
\(314\) 6563.46 1.17961
\(315\) 0 0
\(316\) 15008.4 2.67180
\(317\) −5841.89 −1.03506 −0.517529 0.855666i \(-0.673148\pi\)
−0.517529 + 0.855666i \(0.673148\pi\)
\(318\) 0 0
\(319\) −259.277 −0.0455070
\(320\) −1302.93 −0.227613
\(321\) 0 0
\(322\) 0 0
\(323\) 4473.14 0.770564
\(324\) 0 0
\(325\) 9045.54 1.54387
\(326\) −12251.9 −2.08150
\(327\) 0 0
\(328\) 7768.35 1.30773
\(329\) 0 0
\(330\) 0 0
\(331\) −662.069 −0.109941 −0.0549707 0.998488i \(-0.517507\pi\)
−0.0549707 + 0.998488i \(0.517507\pi\)
\(332\) −5624.60 −0.929789
\(333\) 0 0
\(334\) −9743.32 −1.59620
\(335\) −84.3674 −0.0137597
\(336\) 0 0
\(337\) −840.027 −0.135784 −0.0678920 0.997693i \(-0.521627\pi\)
−0.0678920 + 0.997693i \(0.521627\pi\)
\(338\) 15947.8 2.56641
\(339\) 0 0
\(340\) −2455.17 −0.391618
\(341\) 722.433 0.114727
\(342\) 0 0
\(343\) 0 0
\(344\) 9629.89 1.50933
\(345\) 0 0
\(346\) 7721.91 1.19980
\(347\) 508.292 0.0786356 0.0393178 0.999227i \(-0.487482\pi\)
0.0393178 + 0.999227i \(0.487482\pi\)
\(348\) 0 0
\(349\) −8514.36 −1.30591 −0.652956 0.757396i \(-0.726471\pi\)
−0.652956 + 0.757396i \(0.726471\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1194.44 0.180863
\(353\) −3862.36 −0.582359 −0.291179 0.956668i \(-0.594048\pi\)
−0.291179 + 0.956668i \(0.594048\pi\)
\(354\) 0 0
\(355\) 1609.94 0.240695
\(356\) −1537.88 −0.228954
\(357\) 0 0
\(358\) −15296.4 −2.25821
\(359\) −3611.48 −0.530938 −0.265469 0.964119i \(-0.585527\pi\)
−0.265469 + 0.964119i \(0.585527\pi\)
\(360\) 0 0
\(361\) −4354.16 −0.634809
\(362\) −3205.88 −0.465463
\(363\) 0 0
\(364\) 0 0
\(365\) −1796.30 −0.257596
\(366\) 0 0
\(367\) 12974.0 1.84534 0.922668 0.385595i \(-0.126004\pi\)
0.922668 + 0.385595i \(0.126004\pi\)
\(368\) 1202.78 0.170379
\(369\) 0 0
\(370\) −1350.24 −0.189717
\(371\) 0 0
\(372\) 0 0
\(373\) −2794.20 −0.387877 −0.193938 0.981014i \(-0.562126\pi\)
−0.193938 + 0.981014i \(0.562126\pi\)
\(374\) 5904.24 0.816313
\(375\) 0 0
\(376\) −16834.0 −2.30890
\(377\) −1394.01 −0.190438
\(378\) 0 0
\(379\) 10793.6 1.46287 0.731437 0.681909i \(-0.238850\pi\)
0.731437 + 0.681909i \(0.238850\pi\)
\(380\) −1374.83 −0.185598
\(381\) 0 0
\(382\) −15959.8 −2.13763
\(383\) 165.603 0.0220938 0.0110469 0.999939i \(-0.496484\pi\)
0.0110469 + 0.999939i \(0.496484\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 1413.82 0.186429
\(387\) 0 0
\(388\) −11307.8 −1.47956
\(389\) 2454.47 0.319914 0.159957 0.987124i \(-0.448864\pi\)
0.159957 + 0.987124i \(0.448864\pi\)
\(390\) 0 0
\(391\) −2975.38 −0.384838
\(392\) 0 0
\(393\) 0 0
\(394\) −9109.23 −1.16476
\(395\) 1888.25 0.240526
\(396\) 0 0
\(397\) 6958.83 0.879733 0.439866 0.898063i \(-0.355026\pi\)
0.439866 + 0.898063i \(0.355026\pi\)
\(398\) 9726.71 1.22501
\(399\) 0 0
\(400\) −4391.37 −0.548921
\(401\) 12870.7 1.60283 0.801414 0.598110i \(-0.204081\pi\)
0.801414 + 0.598110i \(0.204081\pi\)
\(402\) 0 0
\(403\) 3884.18 0.480111
\(404\) 15871.8 1.95459
\(405\) 0 0
\(406\) 0 0
\(407\) 2106.57 0.256557
\(408\) 0 0
\(409\) −2705.89 −0.327134 −0.163567 0.986532i \(-0.552300\pi\)
−0.163567 + 0.986532i \(0.552300\pi\)
\(410\) 2131.20 0.256713
\(411\) 0 0
\(412\) 17753.3 2.12292
\(413\) 0 0
\(414\) 0 0
\(415\) −707.645 −0.0837035
\(416\) 6421.93 0.756878
\(417\) 0 0
\(418\) 3306.22 0.386873
\(419\) 2691.53 0.313819 0.156909 0.987613i \(-0.449847\pi\)
0.156909 + 0.987613i \(0.449847\pi\)
\(420\) 0 0
\(421\) −12789.3 −1.48055 −0.740274 0.672305i \(-0.765304\pi\)
−0.740274 + 0.672305i \(0.765304\pi\)
\(422\) −1807.59 −0.208512
\(423\) 0 0
\(424\) 18328.1 2.09927
\(425\) 10863.1 1.23986
\(426\) 0 0
\(427\) 0 0
\(428\) −8406.98 −0.949455
\(429\) 0 0
\(430\) 2641.90 0.296288
\(431\) 6144.20 0.686672 0.343336 0.939213i \(-0.388443\pi\)
0.343336 + 0.939213i \(0.388443\pi\)
\(432\) 0 0
\(433\) −3505.49 −0.389061 −0.194530 0.980897i \(-0.562318\pi\)
−0.194530 + 0.980897i \(0.562318\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 20106.7 2.20857
\(437\) −1666.14 −0.182385
\(438\) 0 0
\(439\) 15385.2 1.67265 0.836325 0.548234i \(-0.184700\pi\)
0.836325 + 0.548234i \(0.184700\pi\)
\(440\) −832.204 −0.0901677
\(441\) 0 0
\(442\) 31744.3 3.41611
\(443\) −12656.4 −1.35739 −0.678694 0.734422i \(-0.737454\pi\)
−0.678694 + 0.734422i \(0.737454\pi\)
\(444\) 0 0
\(445\) −193.485 −0.0206114
\(446\) 10406.5 1.10485
\(447\) 0 0
\(448\) 0 0
\(449\) 18500.2 1.94449 0.972247 0.233959i \(-0.0751681\pi\)
0.972247 + 0.233959i \(0.0751681\pi\)
\(450\) 0 0
\(451\) −3324.99 −0.347157
\(452\) −14541.7 −1.51324
\(453\) 0 0
\(454\) −14609.8 −1.51029
\(455\) 0 0
\(456\) 0 0
\(457\) −16243.7 −1.66269 −0.831343 0.555759i \(-0.812428\pi\)
−0.831343 + 0.555759i \(0.812428\pi\)
\(458\) 26778.7 2.73207
\(459\) 0 0
\(460\) 914.492 0.0926922
\(461\) −14402.7 −1.45510 −0.727550 0.686054i \(-0.759341\pi\)
−0.727550 + 0.686054i \(0.759341\pi\)
\(462\) 0 0
\(463\) −1610.62 −0.161667 −0.0808335 0.996728i \(-0.525758\pi\)
−0.0808335 + 0.996728i \(0.525758\pi\)
\(464\) 676.754 0.0677102
\(465\) 0 0
\(466\) 13713.3 1.36321
\(467\) 14343.3 1.42126 0.710632 0.703564i \(-0.248409\pi\)
0.710632 + 0.703564i \(0.248409\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −4618.29 −0.453247
\(471\) 0 0
\(472\) −20143.1 −1.96432
\(473\) −4121.76 −0.400674
\(474\) 0 0
\(475\) 6083.08 0.587602
\(476\) 0 0
\(477\) 0 0
\(478\) 4480.76 0.428756
\(479\) −6691.97 −0.638338 −0.319169 0.947698i \(-0.603404\pi\)
−0.319169 + 0.947698i \(0.603404\pi\)
\(480\) 0 0
\(481\) 11326.0 1.07364
\(482\) 18437.9 1.74237
\(483\) 0 0
\(484\) −16836.2 −1.58116
\(485\) −1422.67 −0.133196
\(486\) 0 0
\(487\) 479.926 0.0446561 0.0223280 0.999751i \(-0.492892\pi\)
0.0223280 + 0.999751i \(0.492892\pi\)
\(488\) 23101.0 2.14289
\(489\) 0 0
\(490\) 0 0
\(491\) −11907.5 −1.09446 −0.547228 0.836984i \(-0.684317\pi\)
−0.547228 + 0.836984i \(0.684317\pi\)
\(492\) 0 0
\(493\) −1674.12 −0.152938
\(494\) 17776.0 1.61899
\(495\) 0 0
\(496\) −1885.66 −0.170703
\(497\) 0 0
\(498\) 0 0
\(499\) 8818.03 0.791081 0.395540 0.918449i \(-0.370557\pi\)
0.395540 + 0.918449i \(0.370557\pi\)
\(500\) −6772.57 −0.605757
\(501\) 0 0
\(502\) 2661.37 0.236619
\(503\) −17398.9 −1.54231 −0.771154 0.636649i \(-0.780320\pi\)
−0.771154 + 0.636649i \(0.780320\pi\)
\(504\) 0 0
\(505\) 1996.87 0.175960
\(506\) −2199.19 −0.193213
\(507\) 0 0
\(508\) −30466.8 −2.66091
\(509\) 21592.9 1.88033 0.940164 0.340721i \(-0.110671\pi\)
0.940164 + 0.340721i \(0.110671\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −12465.2 −1.07595
\(513\) 0 0
\(514\) −28462.2 −2.44244
\(515\) 2233.59 0.191114
\(516\) 0 0
\(517\) 7205.23 0.612932
\(518\) 0 0
\(519\) 0 0
\(520\) −4474.36 −0.377334
\(521\) −10099.8 −0.849294 −0.424647 0.905359i \(-0.639602\pi\)
−0.424647 + 0.905359i \(0.639602\pi\)
\(522\) 0 0
\(523\) 19888.3 1.66282 0.831410 0.555660i \(-0.187534\pi\)
0.831410 + 0.555660i \(0.187534\pi\)
\(524\) 29333.4 2.44549
\(525\) 0 0
\(526\) −18158.1 −1.50519
\(527\) 4664.66 0.385570
\(528\) 0 0
\(529\) −11058.7 −0.908913
\(530\) 5028.20 0.412096
\(531\) 0 0
\(532\) 0 0
\(533\) −17876.9 −1.45278
\(534\) 0 0
\(535\) −1057.70 −0.0854739
\(536\) −1467.65 −0.118270
\(537\) 0 0
\(538\) −30012.4 −2.40507
\(539\) 0 0
\(540\) 0 0
\(541\) 14042.0 1.11592 0.557962 0.829867i \(-0.311584\pi\)
0.557962 + 0.829867i \(0.311584\pi\)
\(542\) 9431.53 0.747452
\(543\) 0 0
\(544\) 7712.34 0.607838
\(545\) 2529.68 0.198825
\(546\) 0 0
\(547\) −6803.53 −0.531806 −0.265903 0.964000i \(-0.585670\pi\)
−0.265903 + 0.964000i \(0.585670\pi\)
\(548\) −18443.1 −1.43768
\(549\) 0 0
\(550\) 8029.26 0.622489
\(551\) −937.464 −0.0724815
\(552\) 0 0
\(553\) 0 0
\(554\) 17877.6 1.37102
\(555\) 0 0
\(556\) −37973.1 −2.89644
\(557\) −22156.8 −1.68548 −0.842742 0.538317i \(-0.819060\pi\)
−0.842742 + 0.538317i \(0.819060\pi\)
\(558\) 0 0
\(559\) −22160.7 −1.67674
\(560\) 0 0
\(561\) 0 0
\(562\) −8422.38 −0.632165
\(563\) 1316.71 0.0985662 0.0492831 0.998785i \(-0.484306\pi\)
0.0492831 + 0.998785i \(0.484306\pi\)
\(564\) 0 0
\(565\) −1829.53 −0.136229
\(566\) 41635.5 3.09200
\(567\) 0 0
\(568\) 28006.3 2.06887
\(569\) 1242.28 0.0915273 0.0457637 0.998952i \(-0.485428\pi\)
0.0457637 + 0.998952i \(0.485428\pi\)
\(570\) 0 0
\(571\) 13218.4 0.968779 0.484390 0.874852i \(-0.339042\pi\)
0.484390 + 0.874852i \(0.339042\pi\)
\(572\) 15221.9 1.11269
\(573\) 0 0
\(574\) 0 0
\(575\) −4046.26 −0.293462
\(576\) 0 0
\(577\) 16225.0 1.17064 0.585318 0.810803i \(-0.300969\pi\)
0.585318 + 0.810803i \(0.300969\pi\)
\(578\) 14675.9 1.05612
\(579\) 0 0
\(580\) 514.545 0.0368368
\(581\) 0 0
\(582\) 0 0
\(583\) −7844.74 −0.557283
\(584\) −31248.2 −2.21414
\(585\) 0 0
\(586\) −21099.1 −1.48737
\(587\) −12248.0 −0.861208 −0.430604 0.902541i \(-0.641699\pi\)
−0.430604 + 0.902541i \(0.641699\pi\)
\(588\) 0 0
\(589\) 2612.09 0.182732
\(590\) −5526.13 −0.385605
\(591\) 0 0
\(592\) −5498.48 −0.381733
\(593\) −21282.1 −1.47378 −0.736891 0.676012i \(-0.763707\pi\)
−0.736891 + 0.676012i \(0.763707\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 37071.0 2.54779
\(597\) 0 0
\(598\) −11824.0 −0.808561
\(599\) −2441.07 −0.166510 −0.0832551 0.996528i \(-0.526532\pi\)
−0.0832551 + 0.996528i \(0.526532\pi\)
\(600\) 0 0
\(601\) −24463.5 −1.66038 −0.830190 0.557480i \(-0.811768\pi\)
−0.830190 + 0.557480i \(0.811768\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −35993.1 −2.42473
\(605\) −2118.20 −0.142342
\(606\) 0 0
\(607\) −13981.9 −0.934937 −0.467468 0.884010i \(-0.654834\pi\)
−0.467468 + 0.884010i \(0.654834\pi\)
\(608\) 4318.71 0.288071
\(609\) 0 0
\(610\) 6337.61 0.420659
\(611\) 38739.1 2.56500
\(612\) 0 0
\(613\) 14610.7 0.962674 0.481337 0.876535i \(-0.340151\pi\)
0.481337 + 0.876535i \(0.340151\pi\)
\(614\) 7859.04 0.516556
\(615\) 0 0
\(616\) 0 0
\(617\) 17304.1 1.12907 0.564536 0.825408i \(-0.309055\pi\)
0.564536 + 0.825408i \(0.309055\pi\)
\(618\) 0 0
\(619\) −20746.3 −1.34712 −0.673558 0.739134i \(-0.735235\pi\)
−0.673558 + 0.739134i \(0.735235\pi\)
\(620\) −1433.69 −0.0928686
\(621\) 0 0
\(622\) −5449.84 −0.351316
\(623\) 0 0
\(624\) 0 0
\(625\) 14340.9 0.917819
\(626\) 11577.0 0.739154
\(627\) 0 0
\(628\) 20321.6 1.29128
\(629\) 13601.9 0.862228
\(630\) 0 0
\(631\) 9117.96 0.575246 0.287623 0.957744i \(-0.407135\pi\)
0.287623 + 0.957744i \(0.407135\pi\)
\(632\) 32847.7 2.06742
\(633\) 0 0
\(634\) −27880.2 −1.74647
\(635\) −3833.11 −0.239547
\(636\) 0 0
\(637\) 0 0
\(638\) −1237.39 −0.0767848
\(639\) 0 0
\(640\) −4934.83 −0.304791
\(641\) 5056.08 0.311549 0.155775 0.987793i \(-0.450213\pi\)
0.155775 + 0.987793i \(0.450213\pi\)
\(642\) 0 0
\(643\) −28901.0 −1.77254 −0.886271 0.463167i \(-0.846713\pi\)
−0.886271 + 0.463167i \(0.846713\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 21347.9 1.30019
\(647\) −15005.3 −0.911777 −0.455889 0.890037i \(-0.650679\pi\)
−0.455889 + 0.890037i \(0.650679\pi\)
\(648\) 0 0
\(649\) 8621.59 0.521459
\(650\) 43169.5 2.60499
\(651\) 0 0
\(652\) −37934.0 −2.27854
\(653\) −27354.3 −1.63929 −0.819644 0.572873i \(-0.805829\pi\)
−0.819644 + 0.572873i \(0.805829\pi\)
\(654\) 0 0
\(655\) 3690.51 0.220153
\(656\) 8678.75 0.516537
\(657\) 0 0
\(658\) 0 0
\(659\) −27099.4 −1.60188 −0.800942 0.598742i \(-0.795668\pi\)
−0.800942 + 0.598742i \(0.795668\pi\)
\(660\) 0 0
\(661\) −6956.66 −0.409354 −0.204677 0.978830i \(-0.565614\pi\)
−0.204677 + 0.978830i \(0.565614\pi\)
\(662\) −3159.69 −0.185506
\(663\) 0 0
\(664\) −12310.1 −0.719466
\(665\) 0 0
\(666\) 0 0
\(667\) 623.570 0.0361990
\(668\) −30167.0 −1.74730
\(669\) 0 0
\(670\) −402.640 −0.0232169
\(671\) −9887.61 −0.568863
\(672\) 0 0
\(673\) 6529.01 0.373960 0.186980 0.982364i \(-0.440130\pi\)
0.186980 + 0.982364i \(0.440130\pi\)
\(674\) −4008.99 −0.229111
\(675\) 0 0
\(676\) 49377.2 2.80935
\(677\) −14813.7 −0.840970 −0.420485 0.907299i \(-0.638140\pi\)
−0.420485 + 0.907299i \(0.638140\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −5373.44 −0.303032
\(681\) 0 0
\(682\) 3447.78 0.193581
\(683\) −4262.59 −0.238805 −0.119402 0.992846i \(-0.538098\pi\)
−0.119402 + 0.992846i \(0.538098\pi\)
\(684\) 0 0
\(685\) −2320.37 −0.129426
\(686\) 0 0
\(687\) 0 0
\(688\) 10758.4 0.596165
\(689\) −42177.4 −2.33212
\(690\) 0 0
\(691\) −1334.48 −0.0734672 −0.0367336 0.999325i \(-0.511695\pi\)
−0.0367336 + 0.999325i \(0.511695\pi\)
\(692\) 23908.4 1.31338
\(693\) 0 0
\(694\) 2425.80 0.132683
\(695\) −4777.50 −0.260749
\(696\) 0 0
\(697\) −21469.0 −1.16671
\(698\) −40634.4 −2.20349
\(699\) 0 0
\(700\) 0 0
\(701\) −5412.90 −0.291644 −0.145822 0.989311i \(-0.546583\pi\)
−0.145822 + 0.989311i \(0.546583\pi\)
\(702\) 0 0
\(703\) 7616.69 0.408633
\(704\) 9701.30 0.519363
\(705\) 0 0
\(706\) −18433.0 −0.982625
\(707\) 0 0
\(708\) 0 0
\(709\) 27830.8 1.47420 0.737101 0.675783i \(-0.236194\pi\)
0.737101 + 0.675783i \(0.236194\pi\)
\(710\) 7683.37 0.406129
\(711\) 0 0
\(712\) −3365.84 −0.177163
\(713\) −1737.47 −0.0912608
\(714\) 0 0
\(715\) 1915.11 0.100169
\(716\) −47360.3 −2.47198
\(717\) 0 0
\(718\) −17235.6 −0.895861
\(719\) 25992.6 1.34821 0.674103 0.738637i \(-0.264530\pi\)
0.674103 + 0.738637i \(0.264530\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −20780.0 −1.07113
\(723\) 0 0
\(724\) −9925.97 −0.509525
\(725\) −2276.66 −0.116625
\(726\) 0 0
\(727\) 22425.6 1.14405 0.572023 0.820238i \(-0.306159\pi\)
0.572023 + 0.820238i \(0.306159\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −8572.75 −0.434646
\(731\) −26613.7 −1.34657
\(732\) 0 0
\(733\) 8151.82 0.410770 0.205385 0.978681i \(-0.434155\pi\)
0.205385 + 0.978681i \(0.434155\pi\)
\(734\) 61918.0 3.11367
\(735\) 0 0
\(736\) −2872.67 −0.143869
\(737\) 628.179 0.0313966
\(738\) 0 0
\(739\) −11002.4 −0.547673 −0.273837 0.961776i \(-0.588293\pi\)
−0.273837 + 0.961776i \(0.588293\pi\)
\(740\) −4180.56 −0.207676
\(741\) 0 0
\(742\) 0 0
\(743\) 31840.9 1.57218 0.786090 0.618112i \(-0.212102\pi\)
0.786090 + 0.618112i \(0.212102\pi\)
\(744\) 0 0
\(745\) 4663.99 0.229363
\(746\) −13335.2 −0.654472
\(747\) 0 0
\(748\) 18280.6 0.893587
\(749\) 0 0
\(750\) 0 0
\(751\) −22379.7 −1.08741 −0.543707 0.839275i \(-0.682980\pi\)
−0.543707 + 0.839275i \(0.682980\pi\)
\(752\) −18806.8 −0.911985
\(753\) 0 0
\(754\) −6652.85 −0.321330
\(755\) −4528.39 −0.218285
\(756\) 0 0
\(757\) −17522.9 −0.841321 −0.420660 0.907218i \(-0.638202\pi\)
−0.420660 + 0.907218i \(0.638202\pi\)
\(758\) 51511.9 2.46834
\(759\) 0 0
\(760\) −3008.99 −0.143615
\(761\) −23910.9 −1.13899 −0.569493 0.821996i \(-0.692860\pi\)
−0.569493 + 0.821996i \(0.692860\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −49414.4 −2.33999
\(765\) 0 0
\(766\) 790.335 0.0372793
\(767\) 46354.2 2.18221
\(768\) 0 0
\(769\) 28549.3 1.33877 0.669386 0.742915i \(-0.266557\pi\)
0.669386 + 0.742915i \(0.266557\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4377.44 0.204077
\(773\) −7608.00 −0.353998 −0.176999 0.984211i \(-0.556639\pi\)
−0.176999 + 0.984211i \(0.556639\pi\)
\(774\) 0 0
\(775\) 6343.53 0.294021
\(776\) −24748.6 −1.14488
\(777\) 0 0
\(778\) 11713.9 0.539797
\(779\) −12022.1 −0.552936
\(780\) 0 0
\(781\) −11987.2 −0.549214
\(782\) −14199.9 −0.649344
\(783\) 0 0
\(784\) 0 0
\(785\) 2556.72 0.116246
\(786\) 0 0
\(787\) 13189.1 0.597381 0.298691 0.954350i \(-0.403450\pi\)
0.298691 + 0.954350i \(0.403450\pi\)
\(788\) −28203.7 −1.27502
\(789\) 0 0
\(790\) 9011.57 0.405845
\(791\) 0 0
\(792\) 0 0
\(793\) −53161.0 −2.38058
\(794\) 33210.7 1.48439
\(795\) 0 0
\(796\) 30115.6 1.34098
\(797\) −21354.5 −0.949080 −0.474540 0.880234i \(-0.657385\pi\)
−0.474540 + 0.880234i \(0.657385\pi\)
\(798\) 0 0
\(799\) 46523.2 2.05992
\(800\) 10488.1 0.463514
\(801\) 0 0
\(802\) 61425.1 2.70448
\(803\) 13374.8 0.587778
\(804\) 0 0
\(805\) 0 0
\(806\) 18537.1 0.810100
\(807\) 0 0
\(808\) 34737.4 1.51245
\(809\) −17159.4 −0.745728 −0.372864 0.927886i \(-0.621624\pi\)
−0.372864 + 0.927886i \(0.621624\pi\)
\(810\) 0 0
\(811\) 28817.4 1.24774 0.623869 0.781529i \(-0.285560\pi\)
0.623869 + 0.781529i \(0.285560\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 10053.5 0.432894
\(815\) −4772.58 −0.205124
\(816\) 0 0
\(817\) −14903.0 −0.638175
\(818\) −12913.7 −0.551979
\(819\) 0 0
\(820\) 6598.57 0.281015
\(821\) −33190.4 −1.41090 −0.705452 0.708758i \(-0.749256\pi\)
−0.705452 + 0.708758i \(0.749256\pi\)
\(822\) 0 0
\(823\) 7007.52 0.296800 0.148400 0.988927i \(-0.452588\pi\)
0.148400 + 0.988927i \(0.452588\pi\)
\(824\) 38855.3 1.64270
\(825\) 0 0
\(826\) 0 0
\(827\) 28707.3 1.20708 0.603538 0.797334i \(-0.293757\pi\)
0.603538 + 0.797334i \(0.293757\pi\)
\(828\) 0 0
\(829\) −27277.5 −1.14281 −0.571403 0.820670i \(-0.693601\pi\)
−0.571403 + 0.820670i \(0.693601\pi\)
\(830\) −3377.21 −0.141234
\(831\) 0 0
\(832\) 52159.3 2.17343
\(833\) 0 0
\(834\) 0 0
\(835\) −3795.39 −0.157299
\(836\) 10236.6 0.423495
\(837\) 0 0
\(838\) 12845.2 0.529512
\(839\) 23773.1 0.978236 0.489118 0.872218i \(-0.337319\pi\)
0.489118 + 0.872218i \(0.337319\pi\)
\(840\) 0 0
\(841\) −24038.1 −0.985614
\(842\) −61036.3 −2.49816
\(843\) 0 0
\(844\) −5596.60 −0.228250
\(845\) 6212.27 0.252910
\(846\) 0 0
\(847\) 0 0
\(848\) 20476.0 0.829185
\(849\) 0 0
\(850\) 51843.9 2.09204
\(851\) −5066.37 −0.204081
\(852\) 0 0
\(853\) 9089.25 0.364842 0.182421 0.983221i \(-0.441607\pi\)
0.182421 + 0.983221i \(0.441607\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −18399.7 −0.734683
\(857\) 12309.3 0.490640 0.245320 0.969442i \(-0.421107\pi\)
0.245320 + 0.969442i \(0.421107\pi\)
\(858\) 0 0
\(859\) −24303.7 −0.965347 −0.482674 0.875800i \(-0.660334\pi\)
−0.482674 + 0.875800i \(0.660334\pi\)
\(860\) 8179.78 0.324335
\(861\) 0 0
\(862\) 29322.9 1.15863
\(863\) 21329.5 0.841326 0.420663 0.907217i \(-0.361797\pi\)
0.420663 + 0.907217i \(0.361797\pi\)
\(864\) 0 0
\(865\) 3007.98 0.118236
\(866\) −16729.8 −0.656469
\(867\) 0 0
\(868\) 0 0
\(869\) −14059.4 −0.548829
\(870\) 0 0
\(871\) 3377.42 0.131388
\(872\) 44006.0 1.70898
\(873\) 0 0
\(874\) −7951.58 −0.307742
\(875\) 0 0
\(876\) 0 0
\(877\) 12828.9 0.493958 0.246979 0.969021i \(-0.420562\pi\)
0.246979 + 0.969021i \(0.420562\pi\)
\(878\) 73425.0 2.82229
\(879\) 0 0
\(880\) −929.732 −0.0356151
\(881\) −20750.1 −0.793515 −0.396758 0.917923i \(-0.629865\pi\)
−0.396758 + 0.917923i \(0.629865\pi\)
\(882\) 0 0
\(883\) −41290.6 −1.57366 −0.786830 0.617170i \(-0.788279\pi\)
−0.786830 + 0.617170i \(0.788279\pi\)
\(884\) 98285.8 3.73949
\(885\) 0 0
\(886\) −60402.0 −2.29034
\(887\) 40623.2 1.53776 0.768879 0.639394i \(-0.220815\pi\)
0.768879 + 0.639394i \(0.220815\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −923.398 −0.0347779
\(891\) 0 0
\(892\) 32220.2 1.20943
\(893\) 26051.8 0.976250
\(894\) 0 0
\(895\) −5958.52 −0.222538
\(896\) 0 0
\(897\) 0 0
\(898\) 88291.3 3.28098
\(899\) −977.601 −0.0362679
\(900\) 0 0
\(901\) −50652.5 −1.87290
\(902\) −15868.4 −0.585764
\(903\) 0 0
\(904\) −31826.4 −1.17094
\(905\) −1248.81 −0.0458695
\(906\) 0 0
\(907\) −28905.0 −1.05818 −0.529092 0.848564i \(-0.677468\pi\)
−0.529092 + 0.848564i \(0.677468\pi\)
\(908\) −45234.3 −1.65325
\(909\) 0 0
\(910\) 0 0
\(911\) −49053.8 −1.78400 −0.892001 0.452034i \(-0.850699\pi\)
−0.892001 + 0.452034i \(0.850699\pi\)
\(912\) 0 0
\(913\) 5268.95 0.190993
\(914\) −77522.3 −2.80548
\(915\) 0 0
\(916\) 82911.5 2.99069
\(917\) 0 0
\(918\) 0 0
\(919\) 12730.6 0.456958 0.228479 0.973549i \(-0.426625\pi\)
0.228479 + 0.973549i \(0.426625\pi\)
\(920\) 2001.48 0.0717248
\(921\) 0 0
\(922\) −68736.4 −2.45522
\(923\) −64449.4 −2.29835
\(924\) 0 0
\(925\) 18497.3 0.657501
\(926\) −7686.61 −0.272784
\(927\) 0 0
\(928\) −1616.32 −0.0571751
\(929\) 46317.2 1.63576 0.817879 0.575391i \(-0.195150\pi\)
0.817879 + 0.575391i \(0.195150\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 42458.7 1.49226
\(933\) 0 0
\(934\) 68453.0 2.39813
\(935\) 2299.93 0.0804445
\(936\) 0 0
\(937\) 12831.0 0.447354 0.223677 0.974663i \(-0.428194\pi\)
0.223677 + 0.974663i \(0.428194\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) −14299.0 −0.496152
\(941\) 515.868 0.0178712 0.00893561 0.999960i \(-0.497156\pi\)
0.00893561 + 0.999960i \(0.497156\pi\)
\(942\) 0 0
\(943\) 7996.72 0.276149
\(944\) −22503.7 −0.775882
\(945\) 0 0
\(946\) −19670.9 −0.676065
\(947\) 14341.4 0.492115 0.246057 0.969255i \(-0.420865\pi\)
0.246057 + 0.969255i \(0.420865\pi\)
\(948\) 0 0
\(949\) 71909.8 2.45974
\(950\) 29031.2 0.991471
\(951\) 0 0
\(952\) 0 0
\(953\) 57092.8 1.94063 0.970313 0.241854i \(-0.0777553\pi\)
0.970313 + 0.241854i \(0.0777553\pi\)
\(954\) 0 0
\(955\) −6216.96 −0.210656
\(956\) 13873.2 0.469343
\(957\) 0 0
\(958\) −31937.1 −1.07708
\(959\) 0 0
\(960\) 0 0
\(961\) −27067.1 −0.908566
\(962\) 54052.9 1.81158
\(963\) 0 0
\(964\) 57086.8 1.90730
\(965\) 550.737 0.0183719
\(966\) 0 0
\(967\) −29393.7 −0.977496 −0.488748 0.872425i \(-0.662546\pi\)
−0.488748 + 0.872425i \(0.662546\pi\)
\(968\) −36848.0 −1.22349
\(969\) 0 0
\(970\) −6789.63 −0.224744
\(971\) 4232.54 0.139885 0.0699427 0.997551i \(-0.477718\pi\)
0.0699427 + 0.997551i \(0.477718\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 2290.43 0.0753491
\(975\) 0 0
\(976\) 25808.2 0.846415
\(977\) −45894.1 −1.50285 −0.751425 0.659819i \(-0.770633\pi\)
−0.751425 + 0.659819i \(0.770633\pi\)
\(978\) 0 0
\(979\) 1440.64 0.0470307
\(980\) 0 0
\(981\) 0 0
\(982\) −56828.0 −1.84670
\(983\) 4596.71 0.149148 0.0745739 0.997215i \(-0.476240\pi\)
0.0745739 + 0.997215i \(0.476240\pi\)
\(984\) 0 0
\(985\) −3548.39 −0.114783
\(986\) −7989.67 −0.258056
\(987\) 0 0
\(988\) 55037.5 1.77224
\(989\) 9912.97 0.318720
\(990\) 0 0
\(991\) 15520.8 0.497512 0.248756 0.968566i \(-0.419978\pi\)
0.248756 + 0.968566i \(0.419978\pi\)
\(992\) 4503.62 0.144143
\(993\) 0 0
\(994\) 0 0
\(995\) 3788.92 0.120720
\(996\) 0 0
\(997\) 26478.5 0.841104 0.420552 0.907268i \(-0.361836\pi\)
0.420552 + 0.907268i \(0.361836\pi\)
\(998\) 42083.7 1.33481
\(999\) 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.1.6 yes 6
3.2 odd 2 1323.4.a.bc.1.1 yes 6
7.6 odd 2 1323.4.a.bc.1.6 yes 6
21.20 even 2 inner 1323.4.a.bb.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1323.4.a.bb.1.1 6 21.20 even 2 inner
1323.4.a.bb.1.6 yes 6 1.1 even 1 trivial
1323.4.a.bc.1.1 yes 6 3.2 odd 2
1323.4.a.bc.1.6 yes 6 7.6 odd 2