Properties

Label 1352.4.a.n.1.4
Level $1352$
Weight $4$
Character 1352.1
Self dual yes
Analytic conductor $79.771$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1352,4,Mod(1,1352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1352.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1352 = 2^{3} \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1352.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: \(79.7705823278\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 3x^{5} - 123x^{4} + 243x^{3} + 3138x^{2} - 3264x - 4096 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 104)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(-0.753179\) of defining polynomial
Character \(\chi\) \(=\) 1352.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.753179 q^{3} -7.26675 q^{5} -2.85313 q^{7} -26.4327 q^{9} +O(q^{10})\) \(q+0.753179 q^{3} -7.26675 q^{5} -2.85313 q^{7} -26.4327 q^{9} +58.0893 q^{11} -5.47317 q^{15} +73.0634 q^{17} -142.815 q^{19} -2.14891 q^{21} +48.8276 q^{23} -72.1943 q^{25} -40.2444 q^{27} +38.8864 q^{29} +153.866 q^{31} +43.7516 q^{33} +20.7330 q^{35} -356.246 q^{37} +3.88294 q^{41} +12.1821 q^{43} +192.080 q^{45} +258.312 q^{47} -334.860 q^{49} +55.0298 q^{51} -302.642 q^{53} -422.120 q^{55} -107.565 q^{57} +110.686 q^{59} -415.136 q^{61} +75.4159 q^{63} +933.920 q^{67} +36.7759 q^{69} +661.899 q^{71} +436.800 q^{73} -54.3753 q^{75} -165.736 q^{77} -318.158 q^{79} +683.372 q^{81} +595.656 q^{83} -530.934 q^{85} +29.2884 q^{87} +1146.68 q^{89} +115.889 q^{93} +1037.80 q^{95} -669.840 q^{97} -1535.46 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 3 q^{3} + 9 q^{5} + q^{7} + 93 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 3 q^{3} + 9 q^{5} + q^{7} + 93 q^{9} + 11 q^{11} + 168 q^{15} + 50 q^{17} + 211 q^{19} - 83 q^{21} - 103 q^{23} + 393 q^{25} - 243 q^{27} - 48 q^{29} + 190 q^{31} - 133 q^{33} - 226 q^{35} + 476 q^{37} + 10 q^{41} + 13 q^{43} - 57 q^{45} + 122 q^{47} - 31 q^{49} - 745 q^{51} - 483 q^{53} + 1510 q^{55} + 1177 q^{57} - 731 q^{59} - 704 q^{61} - 518 q^{63} + 901 q^{67} + 3479 q^{69} - 673 q^{71} + 1551 q^{73} - 2739 q^{75} + 133 q^{77} - 840 q^{79} + 3018 q^{81} + 2016 q^{83} - 1373 q^{85} - 1485 q^{87} + 295 q^{89} + 3220 q^{93} + 3128 q^{95} + 2575 q^{97} + 458 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0.753179 0.144949 0.0724747 0.997370i \(-0.476910\pi\)
0.0724747 + 0.997370i \(0.476910\pi\)
\(4\) 0 0
\(5\) −7.26675 −0.649958 −0.324979 0.945721i \(-0.605357\pi\)
−0.324979 + 0.945721i \(0.605357\pi\)
\(6\) 0 0
\(7\) −2.85313 −0.154054 −0.0770271 0.997029i \(-0.524543\pi\)
−0.0770271 + 0.997029i \(0.524543\pi\)
\(8\) 0 0
\(9\) −26.4327 −0.978990
\(10\) 0 0
\(11\) 58.0893 1.59223 0.796117 0.605143i \(-0.206884\pi\)
0.796117 + 0.605143i \(0.206884\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 0 0
\(15\) −5.47317 −0.0942110
\(16\) 0 0
\(17\) 73.0634 1.04238 0.521190 0.853440i \(-0.325488\pi\)
0.521190 + 0.853440i \(0.325488\pi\)
\(18\) 0 0
\(19\) −142.815 −1.72442 −0.862212 0.506548i \(-0.830921\pi\)
−0.862212 + 0.506548i \(0.830921\pi\)
\(20\) 0 0
\(21\) −2.14891 −0.0223301
\(22\) 0 0
\(23\) 48.8276 0.442663 0.221332 0.975199i \(-0.428960\pi\)
0.221332 + 0.975199i \(0.428960\pi\)
\(24\) 0 0
\(25\) −72.1943 −0.577555
\(26\) 0 0
\(27\) −40.2444 −0.286853
\(28\) 0 0
\(29\) 38.8864 0.249001 0.124500 0.992220i \(-0.460267\pi\)
0.124500 + 0.992220i \(0.460267\pi\)
\(30\) 0 0
\(31\) 153.866 0.891459 0.445730 0.895168i \(-0.352944\pi\)
0.445730 + 0.895168i \(0.352944\pi\)
\(32\) 0 0
\(33\) 43.7516 0.230793
\(34\) 0 0
\(35\) 20.7330 0.100129
\(36\) 0 0
\(37\) −356.246 −1.58288 −0.791439 0.611249i \(-0.790668\pi\)
−0.791439 + 0.611249i \(0.790668\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 3.88294 0.0147906 0.00739528 0.999973i \(-0.497646\pi\)
0.00739528 + 0.999973i \(0.497646\pi\)
\(42\) 0 0
\(43\) 12.1821 0.0432034 0.0216017 0.999767i \(-0.493123\pi\)
0.0216017 + 0.999767i \(0.493123\pi\)
\(44\) 0 0
\(45\) 192.080 0.636302
\(46\) 0 0
\(47\) 258.312 0.801675 0.400837 0.916149i \(-0.368719\pi\)
0.400837 + 0.916149i \(0.368719\pi\)
\(48\) 0 0
\(49\) −334.860 −0.976267
\(50\) 0 0
\(51\) 55.0298 0.151092
\(52\) 0 0
\(53\) −302.642 −0.784361 −0.392180 0.919888i \(-0.628279\pi\)
−0.392180 + 0.919888i \(0.628279\pi\)
\(54\) 0 0
\(55\) −422.120 −1.03489
\(56\) 0 0
\(57\) −107.565 −0.249954
\(58\) 0 0
\(59\) 110.686 0.244238 0.122119 0.992515i \(-0.461031\pi\)
0.122119 + 0.992515i \(0.461031\pi\)
\(60\) 0 0
\(61\) −415.136 −0.871355 −0.435678 0.900103i \(-0.643491\pi\)
−0.435678 + 0.900103i \(0.643491\pi\)
\(62\) 0 0
\(63\) 75.4159 0.150818
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 933.920 1.70293 0.851466 0.524409i \(-0.175714\pi\)
0.851466 + 0.524409i \(0.175714\pi\)
\(68\) 0 0
\(69\) 36.7759 0.0641638
\(70\) 0 0
\(71\) 661.899 1.10638 0.553190 0.833055i \(-0.313410\pi\)
0.553190 + 0.833055i \(0.313410\pi\)
\(72\) 0 0
\(73\) 436.800 0.700323 0.350162 0.936689i \(-0.386127\pi\)
0.350162 + 0.936689i \(0.386127\pi\)
\(74\) 0 0
\(75\) −54.3753 −0.0837162
\(76\) 0 0
\(77\) −165.736 −0.245290
\(78\) 0 0
\(79\) −318.158 −0.453109 −0.226554 0.973999i \(-0.572746\pi\)
−0.226554 + 0.973999i \(0.572746\pi\)
\(80\) 0 0
\(81\) 683.372 0.937410
\(82\) 0 0
\(83\) 595.656 0.787732 0.393866 0.919168i \(-0.371137\pi\)
0.393866 + 0.919168i \(0.371137\pi\)
\(84\) 0 0
\(85\) −530.934 −0.677504
\(86\) 0 0
\(87\) 29.2884 0.0360925
\(88\) 0 0
\(89\) 1146.68 1.36571 0.682853 0.730556i \(-0.260739\pi\)
0.682853 + 0.730556i \(0.260739\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 115.889 0.129216
\(94\) 0 0
\(95\) 1037.80 1.12080
\(96\) 0 0
\(97\) −669.840 −0.701154 −0.350577 0.936534i \(-0.614015\pi\)
−0.350577 + 0.936534i \(0.614015\pi\)
\(98\) 0 0
\(99\) −1535.46 −1.55878
\(100\) 0 0
\(101\) 1433.86 1.41261 0.706307 0.707906i \(-0.250360\pi\)
0.706307 + 0.707906i \(0.250360\pi\)
\(102\) 0 0
\(103\) 794.312 0.759863 0.379931 0.925015i \(-0.375948\pi\)
0.379931 + 0.925015i \(0.375948\pi\)
\(104\) 0 0
\(105\) 15.6156 0.0145136
\(106\) 0 0
\(107\) 1564.54 1.41355 0.706776 0.707438i \(-0.250149\pi\)
0.706776 + 0.707438i \(0.250149\pi\)
\(108\) 0 0
\(109\) 2083.84 1.83115 0.915575 0.402147i \(-0.131736\pi\)
0.915575 + 0.402147i \(0.131736\pi\)
\(110\) 0 0
\(111\) −268.317 −0.229437
\(112\) 0 0
\(113\) 198.368 0.165140 0.0825701 0.996585i \(-0.473687\pi\)
0.0825701 + 0.996585i \(0.473687\pi\)
\(114\) 0 0
\(115\) −354.818 −0.287712
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −208.459 −0.160583
\(120\) 0 0
\(121\) 2043.36 1.53521
\(122\) 0 0
\(123\) 2.92455 0.00214388
\(124\) 0 0
\(125\) 1432.96 1.02534
\(126\) 0 0
\(127\) −2167.88 −1.51471 −0.757356 0.653003i \(-0.773509\pi\)
−0.757356 + 0.653003i \(0.773509\pi\)
\(128\) 0 0
\(129\) 9.17527 0.00626231
\(130\) 0 0
\(131\) −800.420 −0.533840 −0.266920 0.963719i \(-0.586006\pi\)
−0.266920 + 0.963719i \(0.586006\pi\)
\(132\) 0 0
\(133\) 407.470 0.265655
\(134\) 0 0
\(135\) 292.446 0.186443
\(136\) 0 0
\(137\) −1135.55 −0.708149 −0.354074 0.935217i \(-0.615204\pi\)
−0.354074 + 0.935217i \(0.615204\pi\)
\(138\) 0 0
\(139\) −2006.92 −1.22464 −0.612318 0.790611i \(-0.709763\pi\)
−0.612318 + 0.790611i \(0.709763\pi\)
\(140\) 0 0
\(141\) 194.555 0.116202
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −282.578 −0.161840
\(146\) 0 0
\(147\) −252.209 −0.141509
\(148\) 0 0
\(149\) 2587.13 1.42246 0.711229 0.702960i \(-0.248139\pi\)
0.711229 + 0.702960i \(0.248139\pi\)
\(150\) 0 0
\(151\) 1582.79 0.853016 0.426508 0.904484i \(-0.359744\pi\)
0.426508 + 0.904484i \(0.359744\pi\)
\(152\) 0 0
\(153\) −1931.26 −1.02048
\(154\) 0 0
\(155\) −1118.11 −0.579411
\(156\) 0 0
\(157\) −270.338 −0.137423 −0.0687113 0.997637i \(-0.521889\pi\)
−0.0687113 + 0.997637i \(0.521889\pi\)
\(158\) 0 0
\(159\) −227.944 −0.113693
\(160\) 0 0
\(161\) −139.311 −0.0681942
\(162\) 0 0
\(163\) −142.203 −0.0683325 −0.0341662 0.999416i \(-0.510878\pi\)
−0.0341662 + 0.999416i \(0.510878\pi\)
\(164\) 0 0
\(165\) −317.932 −0.150006
\(166\) 0 0
\(167\) 242.913 0.112558 0.0562789 0.998415i \(-0.482076\pi\)
0.0562789 + 0.998415i \(0.482076\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) 0 0
\(171\) 3774.99 1.68819
\(172\) 0 0
\(173\) 3472.16 1.52592 0.762958 0.646448i \(-0.223746\pi\)
0.762958 + 0.646448i \(0.223746\pi\)
\(174\) 0 0
\(175\) 205.979 0.0889748
\(176\) 0 0
\(177\) 83.3661 0.0354022
\(178\) 0 0
\(179\) −834.865 −0.348607 −0.174304 0.984692i \(-0.555767\pi\)
−0.174304 + 0.984692i \(0.555767\pi\)
\(180\) 0 0
\(181\) 269.143 0.110526 0.0552632 0.998472i \(-0.482400\pi\)
0.0552632 + 0.998472i \(0.482400\pi\)
\(182\) 0 0
\(183\) −312.671 −0.126302
\(184\) 0 0
\(185\) 2588.75 1.02880
\(186\) 0 0
\(187\) 4244.20 1.65971
\(188\) 0 0
\(189\) 114.822 0.0441910
\(190\) 0 0
\(191\) −1320.03 −0.500072 −0.250036 0.968237i \(-0.580442\pi\)
−0.250036 + 0.968237i \(0.580442\pi\)
\(192\) 0 0
\(193\) −1299.38 −0.484617 −0.242309 0.970199i \(-0.577905\pi\)
−0.242309 + 0.970199i \(0.577905\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2470.61 0.893521 0.446761 0.894653i \(-0.352578\pi\)
0.446761 + 0.894653i \(0.352578\pi\)
\(198\) 0 0
\(199\) −1401.88 −0.499380 −0.249690 0.968326i \(-0.580329\pi\)
−0.249690 + 0.968326i \(0.580329\pi\)
\(200\) 0 0
\(201\) 703.409 0.246839
\(202\) 0 0
\(203\) −110.948 −0.0383596
\(204\) 0 0
\(205\) −28.2163 −0.00961325
\(206\) 0 0
\(207\) −1290.65 −0.433363
\(208\) 0 0
\(209\) −8296.03 −2.74569
\(210\) 0 0
\(211\) 2691.00 0.877991 0.438996 0.898489i \(-0.355334\pi\)
0.438996 + 0.898489i \(0.355334\pi\)
\(212\) 0 0
\(213\) 498.528 0.160369
\(214\) 0 0
\(215\) −88.5240 −0.0280804
\(216\) 0 0
\(217\) −439.000 −0.137333
\(218\) 0 0
\(219\) 328.989 0.101511
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 5439.78 1.63352 0.816759 0.576979i \(-0.195769\pi\)
0.816759 + 0.576979i \(0.195769\pi\)
\(224\) 0 0
\(225\) 1908.29 0.565420
\(226\) 0 0
\(227\) −4040.78 −1.18148 −0.590740 0.806862i \(-0.701164\pi\)
−0.590740 + 0.806862i \(0.701164\pi\)
\(228\) 0 0
\(229\) 1072.07 0.309365 0.154682 0.987964i \(-0.450565\pi\)
0.154682 + 0.987964i \(0.450565\pi\)
\(230\) 0 0
\(231\) −124.829 −0.0355547
\(232\) 0 0
\(233\) 4062.66 1.14229 0.571145 0.820849i \(-0.306499\pi\)
0.571145 + 0.820849i \(0.306499\pi\)
\(234\) 0 0
\(235\) −1877.09 −0.521055
\(236\) 0 0
\(237\) −239.630 −0.0656778
\(238\) 0 0
\(239\) 4371.13 1.18303 0.591516 0.806293i \(-0.298530\pi\)
0.591516 + 0.806293i \(0.298530\pi\)
\(240\) 0 0
\(241\) −3447.22 −0.921390 −0.460695 0.887558i \(-0.652400\pi\)
−0.460695 + 0.887558i \(0.652400\pi\)
\(242\) 0 0
\(243\) 1601.30 0.422730
\(244\) 0 0
\(245\) 2433.34 0.634533
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 448.636 0.114181
\(250\) 0 0
\(251\) 6733.95 1.69340 0.846700 0.532071i \(-0.178586\pi\)
0.846700 + 0.532071i \(0.178586\pi\)
\(252\) 0 0
\(253\) 2836.36 0.704823
\(254\) 0 0
\(255\) −399.888 −0.0982038
\(256\) 0 0
\(257\) 3000.95 0.728381 0.364190 0.931325i \(-0.381346\pi\)
0.364190 + 0.931325i \(0.381346\pi\)
\(258\) 0 0
\(259\) 1016.41 0.243849
\(260\) 0 0
\(261\) −1027.87 −0.243769
\(262\) 0 0
\(263\) −3161.43 −0.741224 −0.370612 0.928788i \(-0.620852\pi\)
−0.370612 + 0.928788i \(0.620852\pi\)
\(264\) 0 0
\(265\) 2199.23 0.509802
\(266\) 0 0
\(267\) 863.656 0.197958
\(268\) 0 0
\(269\) −4342.67 −0.984303 −0.492151 0.870510i \(-0.663789\pi\)
−0.492151 + 0.870510i \(0.663789\pi\)
\(270\) 0 0
\(271\) 5871.90 1.31621 0.658104 0.752927i \(-0.271359\pi\)
0.658104 + 0.752927i \(0.271359\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −4193.72 −0.919602
\(276\) 0 0
\(277\) 5414.31 1.17442 0.587210 0.809435i \(-0.300226\pi\)
0.587210 + 0.809435i \(0.300226\pi\)
\(278\) 0 0
\(279\) −4067.11 −0.872729
\(280\) 0 0
\(281\) 6679.68 1.41806 0.709032 0.705176i \(-0.249132\pi\)
0.709032 + 0.705176i \(0.249132\pi\)
\(282\) 0 0
\(283\) −8244.76 −1.73180 −0.865901 0.500215i \(-0.833254\pi\)
−0.865901 + 0.500215i \(0.833254\pi\)
\(284\) 0 0
\(285\) 781.651 0.162460
\(286\) 0 0
\(287\) −11.0785 −0.00227855
\(288\) 0 0
\(289\) 425.259 0.0865579
\(290\) 0 0
\(291\) −504.510 −0.101632
\(292\) 0 0
\(293\) −539.918 −0.107653 −0.0538265 0.998550i \(-0.517142\pi\)
−0.0538265 + 0.998550i \(0.517142\pi\)
\(294\) 0 0
\(295\) −804.325 −0.158744
\(296\) 0 0
\(297\) −2337.77 −0.456738
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −34.7569 −0.00665567
\(302\) 0 0
\(303\) 1079.95 0.204758
\(304\) 0 0
\(305\) 3016.69 0.566344
\(306\) 0 0
\(307\) −4032.04 −0.749578 −0.374789 0.927110i \(-0.622285\pi\)
−0.374789 + 0.927110i \(0.622285\pi\)
\(308\) 0 0
\(309\) 598.259 0.110142
\(310\) 0 0
\(311\) −4146.61 −0.756054 −0.378027 0.925794i \(-0.623397\pi\)
−0.378027 + 0.925794i \(0.623397\pi\)
\(312\) 0 0
\(313\) 4998.17 0.902597 0.451299 0.892373i \(-0.350961\pi\)
0.451299 + 0.892373i \(0.350961\pi\)
\(314\) 0 0
\(315\) −548.028 −0.0980251
\(316\) 0 0
\(317\) 65.9112 0.0116780 0.00583902 0.999983i \(-0.498141\pi\)
0.00583902 + 0.999983i \(0.498141\pi\)
\(318\) 0 0
\(319\) 2258.88 0.396467
\(320\) 0 0
\(321\) 1178.38 0.204893
\(322\) 0 0
\(323\) −10434.6 −1.79751
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1569.50 0.265424
\(328\) 0 0
\(329\) −736.997 −0.123501
\(330\) 0 0
\(331\) 4073.09 0.676367 0.338184 0.941080i \(-0.390188\pi\)
0.338184 + 0.941080i \(0.390188\pi\)
\(332\) 0 0
\(333\) 9416.55 1.54962
\(334\) 0 0
\(335\) −6786.57 −1.10683
\(336\) 0 0
\(337\) −6440.94 −1.04113 −0.520564 0.853822i \(-0.674278\pi\)
−0.520564 + 0.853822i \(0.674278\pi\)
\(338\) 0 0
\(339\) 149.406 0.0239370
\(340\) 0 0
\(341\) 8937.99 1.41941
\(342\) 0 0
\(343\) 1934.02 0.304452
\(344\) 0 0
\(345\) −267.241 −0.0417037
\(346\) 0 0
\(347\) −8303.40 −1.28458 −0.642291 0.766461i \(-0.722016\pi\)
−0.642291 + 0.766461i \(0.722016\pi\)
\(348\) 0 0
\(349\) −650.474 −0.0997681 −0.0498841 0.998755i \(-0.515885\pi\)
−0.0498841 + 0.998755i \(0.515885\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3042.67 −0.458768 −0.229384 0.973336i \(-0.573671\pi\)
−0.229384 + 0.973336i \(0.573671\pi\)
\(354\) 0 0
\(355\) −4809.86 −0.719100
\(356\) 0 0
\(357\) −157.007 −0.0232764
\(358\) 0 0
\(359\) −13014.1 −1.91325 −0.956627 0.291317i \(-0.905907\pi\)
−0.956627 + 0.291317i \(0.905907\pi\)
\(360\) 0 0
\(361\) 13537.2 1.97364
\(362\) 0 0
\(363\) 1539.02 0.222528
\(364\) 0 0
\(365\) −3174.12 −0.455181
\(366\) 0 0
\(367\) 8850.83 1.25888 0.629441 0.777048i \(-0.283284\pi\)
0.629441 + 0.777048i \(0.283284\pi\)
\(368\) 0 0
\(369\) −102.637 −0.0144798
\(370\) 0 0
\(371\) 863.476 0.120834
\(372\) 0 0
\(373\) 10671.7 1.48139 0.740694 0.671843i \(-0.234497\pi\)
0.740694 + 0.671843i \(0.234497\pi\)
\(374\) 0 0
\(375\) 1079.28 0.148623
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 7451.00 1.00985 0.504923 0.863164i \(-0.331521\pi\)
0.504923 + 0.863164i \(0.331521\pi\)
\(380\) 0 0
\(381\) −1632.80 −0.219556
\(382\) 0 0
\(383\) 641.269 0.0855543 0.0427772 0.999085i \(-0.486379\pi\)
0.0427772 + 0.999085i \(0.486379\pi\)
\(384\) 0 0
\(385\) 1204.36 0.159428
\(386\) 0 0
\(387\) −322.005 −0.0422957
\(388\) 0 0
\(389\) 4522.54 0.589465 0.294732 0.955580i \(-0.404769\pi\)
0.294732 + 0.955580i \(0.404769\pi\)
\(390\) 0 0
\(391\) 3567.51 0.461424
\(392\) 0 0
\(393\) −602.859 −0.0773797
\(394\) 0 0
\(395\) 2311.98 0.294502
\(396\) 0 0
\(397\) −7380.01 −0.932978 −0.466489 0.884527i \(-0.654481\pi\)
−0.466489 + 0.884527i \(0.654481\pi\)
\(398\) 0 0
\(399\) 306.898 0.0385065
\(400\) 0 0
\(401\) 9770.19 1.21671 0.608354 0.793666i \(-0.291830\pi\)
0.608354 + 0.793666i \(0.291830\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −4965.90 −0.609277
\(406\) 0 0
\(407\) −20694.1 −2.52031
\(408\) 0 0
\(409\) 7925.84 0.958209 0.479104 0.877758i \(-0.340962\pi\)
0.479104 + 0.877758i \(0.340962\pi\)
\(410\) 0 0
\(411\) −855.271 −0.102646
\(412\) 0 0
\(413\) −315.800 −0.0376259
\(414\) 0 0
\(415\) −4328.48 −0.511992
\(416\) 0 0
\(417\) −1511.57 −0.177510
\(418\) 0 0
\(419\) −15583.2 −1.81692 −0.908462 0.417967i \(-0.862743\pi\)
−0.908462 + 0.417967i \(0.862743\pi\)
\(420\) 0 0
\(421\) −15726.7 −1.82060 −0.910298 0.413954i \(-0.864148\pi\)
−0.910298 + 0.413954i \(0.864148\pi\)
\(422\) 0 0
\(423\) −6827.89 −0.784831
\(424\) 0 0
\(425\) −5274.76 −0.602032
\(426\) 0 0
\(427\) 1184.43 0.134236
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 9844.23 1.10019 0.550093 0.835104i \(-0.314592\pi\)
0.550093 + 0.835104i \(0.314592\pi\)
\(432\) 0 0
\(433\) 12317.4 1.36706 0.683531 0.729921i \(-0.260443\pi\)
0.683531 + 0.729921i \(0.260443\pi\)
\(434\) 0 0
\(435\) −212.832 −0.0234586
\(436\) 0 0
\(437\) −6973.32 −0.763339
\(438\) 0 0
\(439\) 9182.20 0.998274 0.499137 0.866523i \(-0.333650\pi\)
0.499137 + 0.866523i \(0.333650\pi\)
\(440\) 0 0
\(441\) 8851.25 0.955756
\(442\) 0 0
\(443\) 8288.47 0.888933 0.444466 0.895795i \(-0.353393\pi\)
0.444466 + 0.895795i \(0.353393\pi\)
\(444\) 0 0
\(445\) −8332.64 −0.887652
\(446\) 0 0
\(447\) 1948.58 0.206184
\(448\) 0 0
\(449\) 10295.4 1.08212 0.541058 0.840986i \(-0.318024\pi\)
0.541058 + 0.840986i \(0.318024\pi\)
\(450\) 0 0
\(451\) 225.557 0.0235500
\(452\) 0 0
\(453\) 1192.12 0.123644
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −12210.5 −1.24985 −0.624924 0.780685i \(-0.714870\pi\)
−0.624924 + 0.780685i \(0.714870\pi\)
\(458\) 0 0
\(459\) −2940.39 −0.299010
\(460\) 0 0
\(461\) −9114.70 −0.920854 −0.460427 0.887697i \(-0.652304\pi\)
−0.460427 + 0.887697i \(0.652304\pi\)
\(462\) 0 0
\(463\) 0.376664 3.78079e−5 0 1.89040e−5 1.00000i \(-0.499994\pi\)
1.89040e−5 1.00000i \(0.499994\pi\)
\(464\) 0 0
\(465\) −842.137 −0.0839853
\(466\) 0 0
\(467\) 5771.35 0.571877 0.285938 0.958248i \(-0.407695\pi\)
0.285938 + 0.958248i \(0.407695\pi\)
\(468\) 0 0
\(469\) −2664.59 −0.262344
\(470\) 0 0
\(471\) −203.613 −0.0199193
\(472\) 0 0
\(473\) 707.647 0.0687899
\(474\) 0 0
\(475\) 10310.4 0.995949
\(476\) 0 0
\(477\) 7999.66 0.767881
\(478\) 0 0
\(479\) 8609.07 0.821207 0.410604 0.911814i \(-0.365318\pi\)
0.410604 + 0.911814i \(0.365318\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −104.926 −0.00988470
\(484\) 0 0
\(485\) 4867.56 0.455721
\(486\) 0 0
\(487\) −3635.16 −0.338244 −0.169122 0.985595i \(-0.554093\pi\)
−0.169122 + 0.985595i \(0.554093\pi\)
\(488\) 0 0
\(489\) −107.104 −0.00990475
\(490\) 0 0
\(491\) −17387.5 −1.59814 −0.799068 0.601241i \(-0.794673\pi\)
−0.799068 + 0.601241i \(0.794673\pi\)
\(492\) 0 0
\(493\) 2841.17 0.259554
\(494\) 0 0
\(495\) 11157.8 1.01314
\(496\) 0 0
\(497\) −1888.48 −0.170443
\(498\) 0 0
\(499\) −1174.90 −0.105403 −0.0527014 0.998610i \(-0.516783\pi\)
−0.0527014 + 0.998610i \(0.516783\pi\)
\(500\) 0 0
\(501\) 182.957 0.0163152
\(502\) 0 0
\(503\) 21573.0 1.91231 0.956155 0.292862i \(-0.0946077\pi\)
0.956155 + 0.292862i \(0.0946077\pi\)
\(504\) 0 0
\(505\) −10419.5 −0.918140
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −21860.2 −1.90361 −0.951804 0.306708i \(-0.900773\pi\)
−0.951804 + 0.306708i \(0.900773\pi\)
\(510\) 0 0
\(511\) −1246.25 −0.107888
\(512\) 0 0
\(513\) 5747.51 0.494657
\(514\) 0 0
\(515\) −5772.07 −0.493879
\(516\) 0 0
\(517\) 15005.2 1.27645
\(518\) 0 0
\(519\) 2615.16 0.221180
\(520\) 0 0
\(521\) 18858.4 1.58580 0.792899 0.609353i \(-0.208571\pi\)
0.792899 + 0.609353i \(0.208571\pi\)
\(522\) 0 0
\(523\) −14433.7 −1.20677 −0.603386 0.797450i \(-0.706182\pi\)
−0.603386 + 0.797450i \(0.706182\pi\)
\(524\) 0 0
\(525\) 155.139 0.0128968
\(526\) 0 0
\(527\) 11242.0 0.929240
\(528\) 0 0
\(529\) −9782.87 −0.804049
\(530\) 0 0
\(531\) −2925.72 −0.239107
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −11369.1 −0.918749
\(536\) 0 0
\(537\) −628.803 −0.0505304
\(538\) 0 0
\(539\) −19451.8 −1.55445
\(540\) 0 0
\(541\) 9419.76 0.748590 0.374295 0.927310i \(-0.377885\pi\)
0.374295 + 0.927310i \(0.377885\pi\)
\(542\) 0 0
\(543\) 202.713 0.0160207
\(544\) 0 0
\(545\) −15142.7 −1.19017
\(546\) 0 0
\(547\) −12270.8 −0.959162 −0.479581 0.877498i \(-0.659211\pi\)
−0.479581 + 0.877498i \(0.659211\pi\)
\(548\) 0 0
\(549\) 10973.2 0.853048
\(550\) 0 0
\(551\) −5553.57 −0.429383
\(552\) 0 0
\(553\) 907.745 0.0698034
\(554\) 0 0
\(555\) 1949.79 0.149124
\(556\) 0 0
\(557\) −18279.2 −1.39051 −0.695254 0.718765i \(-0.744708\pi\)
−0.695254 + 0.718765i \(0.744708\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 3196.64 0.240575
\(562\) 0 0
\(563\) −23994.1 −1.79615 −0.898073 0.439847i \(-0.855033\pi\)
−0.898073 + 0.439847i \(0.855033\pi\)
\(564\) 0 0
\(565\) −1441.49 −0.107334
\(566\) 0 0
\(567\) −1949.75 −0.144412
\(568\) 0 0
\(569\) 10263.6 0.756191 0.378096 0.925767i \(-0.376579\pi\)
0.378096 + 0.925767i \(0.376579\pi\)
\(570\) 0 0
\(571\) −18173.6 −1.33195 −0.665974 0.745975i \(-0.731984\pi\)
−0.665974 + 0.745975i \(0.731984\pi\)
\(572\) 0 0
\(573\) −994.216 −0.0724851
\(574\) 0 0
\(575\) −3525.07 −0.255662
\(576\) 0 0
\(577\) −21688.8 −1.56484 −0.782422 0.622748i \(-0.786016\pi\)
−0.782422 + 0.622748i \(0.786016\pi\)
\(578\) 0 0
\(579\) −978.663 −0.0702450
\(580\) 0 0
\(581\) −1699.48 −0.121353
\(582\) 0 0
\(583\) −17580.3 −1.24889
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −929.325 −0.0653447 −0.0326724 0.999466i \(-0.510402\pi\)
−0.0326724 + 0.999466i \(0.510402\pi\)
\(588\) 0 0
\(589\) −21974.5 −1.53725
\(590\) 0 0
\(591\) 1860.81 0.129515
\(592\) 0 0
\(593\) 8355.69 0.578629 0.289315 0.957234i \(-0.406573\pi\)
0.289315 + 0.957234i \(0.406573\pi\)
\(594\) 0 0
\(595\) 1514.82 0.104372
\(596\) 0 0
\(597\) −1055.87 −0.0723848
\(598\) 0 0
\(599\) 241.352 0.0164631 0.00823154 0.999966i \(-0.497380\pi\)
0.00823154 + 0.999966i \(0.497380\pi\)
\(600\) 0 0
\(601\) −8607.03 −0.584173 −0.292087 0.956392i \(-0.594350\pi\)
−0.292087 + 0.956392i \(0.594350\pi\)
\(602\) 0 0
\(603\) −24686.1 −1.66715
\(604\) 0 0
\(605\) −14848.6 −0.997821
\(606\) 0 0
\(607\) −4892.10 −0.327124 −0.163562 0.986533i \(-0.552298\pi\)
−0.163562 + 0.986533i \(0.552298\pi\)
\(608\) 0 0
\(609\) −83.5635 −0.00556021
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 3875.72 0.255365 0.127682 0.991815i \(-0.459246\pi\)
0.127682 + 0.991815i \(0.459246\pi\)
\(614\) 0 0
\(615\) −21.2520 −0.00139343
\(616\) 0 0
\(617\) −8663.76 −0.565300 −0.282650 0.959223i \(-0.591213\pi\)
−0.282650 + 0.959223i \(0.591213\pi\)
\(618\) 0 0
\(619\) −3269.09 −0.212271 −0.106136 0.994352i \(-0.533848\pi\)
−0.106136 + 0.994352i \(0.533848\pi\)
\(620\) 0 0
\(621\) −1965.04 −0.126979
\(622\) 0 0
\(623\) −3271.62 −0.210393
\(624\) 0 0
\(625\) −1388.69 −0.0888761
\(626\) 0 0
\(627\) −6248.40 −0.397985
\(628\) 0 0
\(629\) −26028.5 −1.64996
\(630\) 0 0
\(631\) 11170.0 0.704710 0.352355 0.935866i \(-0.385381\pi\)
0.352355 + 0.935866i \(0.385381\pi\)
\(632\) 0 0
\(633\) 2026.81 0.127264
\(634\) 0 0
\(635\) 15753.5 0.984499
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −17495.8 −1.08313
\(640\) 0 0
\(641\) −20320.3 −1.25211 −0.626057 0.779778i \(-0.715332\pi\)
−0.626057 + 0.779778i \(0.715332\pi\)
\(642\) 0 0
\(643\) −10873.6 −0.666892 −0.333446 0.942769i \(-0.608212\pi\)
−0.333446 + 0.942769i \(0.608212\pi\)
\(644\) 0 0
\(645\) −66.6744 −0.00407024
\(646\) 0 0
\(647\) 7533.72 0.457776 0.228888 0.973453i \(-0.426491\pi\)
0.228888 + 0.973453i \(0.426491\pi\)
\(648\) 0 0
\(649\) 6429.65 0.388884
\(650\) 0 0
\(651\) −330.646 −0.0199063
\(652\) 0 0
\(653\) −3839.50 −0.230094 −0.115047 0.993360i \(-0.536702\pi\)
−0.115047 + 0.993360i \(0.536702\pi\)
\(654\) 0 0
\(655\) 5816.45 0.346973
\(656\) 0 0
\(657\) −11545.8 −0.685609
\(658\) 0 0
\(659\) −5500.42 −0.325138 −0.162569 0.986697i \(-0.551978\pi\)
−0.162569 + 0.986697i \(0.551978\pi\)
\(660\) 0 0
\(661\) 13859.1 0.815517 0.407759 0.913090i \(-0.366310\pi\)
0.407759 + 0.913090i \(0.366310\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2960.98 −0.172665
\(666\) 0 0
\(667\) 1898.73 0.110223
\(668\) 0 0
\(669\) 4097.13 0.236777
\(670\) 0 0
\(671\) −24114.9 −1.38740
\(672\) 0 0
\(673\) 2069.86 0.118554 0.0592772 0.998242i \(-0.481120\pi\)
0.0592772 + 0.998242i \(0.481120\pi\)
\(674\) 0 0
\(675\) 2905.42 0.165673
\(676\) 0 0
\(677\) 7030.18 0.399102 0.199551 0.979887i \(-0.436052\pi\)
0.199551 + 0.979887i \(0.436052\pi\)
\(678\) 0 0
\(679\) 1911.14 0.108016
\(680\) 0 0
\(681\) −3043.43 −0.171255
\(682\) 0 0
\(683\) 3831.34 0.214644 0.107322 0.994224i \(-0.465772\pi\)
0.107322 + 0.994224i \(0.465772\pi\)
\(684\) 0 0
\(685\) 8251.74 0.460267
\(686\) 0 0
\(687\) 807.463 0.0448422
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 8186.94 0.450718 0.225359 0.974276i \(-0.427645\pi\)
0.225359 + 0.974276i \(0.427645\pi\)
\(692\) 0 0
\(693\) 4380.85 0.240137
\(694\) 0 0
\(695\) 14583.8 0.795962
\(696\) 0 0
\(697\) 283.701 0.0154174
\(698\) 0 0
\(699\) 3059.91 0.165574
\(700\) 0 0
\(701\) −2297.51 −0.123789 −0.0618943 0.998083i \(-0.519714\pi\)
−0.0618943 + 0.998083i \(0.519714\pi\)
\(702\) 0 0
\(703\) 50877.3 2.72955
\(704\) 0 0
\(705\) −1413.79 −0.0755266
\(706\) 0 0
\(707\) −4090.97 −0.217619
\(708\) 0 0
\(709\) 28328.7 1.50057 0.750285 0.661114i \(-0.229916\pi\)
0.750285 + 0.661114i \(0.229916\pi\)
\(710\) 0 0
\(711\) 8409.79 0.443589
\(712\) 0 0
\(713\) 7512.93 0.394616
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3292.24 0.171480
\(718\) 0 0
\(719\) 1170.26 0.0607001 0.0303500 0.999539i \(-0.490338\pi\)
0.0303500 + 0.999539i \(0.490338\pi\)
\(720\) 0 0
\(721\) −2266.27 −0.117060
\(722\) 0 0
\(723\) −2596.38 −0.133555
\(724\) 0 0
\(725\) −2807.38 −0.143812
\(726\) 0 0
\(727\) 30312.6 1.54640 0.773201 0.634162i \(-0.218655\pi\)
0.773201 + 0.634162i \(0.218655\pi\)
\(728\) 0 0
\(729\) −17245.0 −0.876136
\(730\) 0 0
\(731\) 890.062 0.0450344
\(732\) 0 0
\(733\) 27438.1 1.38260 0.691302 0.722565i \(-0.257037\pi\)
0.691302 + 0.722565i \(0.257037\pi\)
\(734\) 0 0
\(735\) 1832.74 0.0919751
\(736\) 0 0
\(737\) 54250.7 2.71147
\(738\) 0 0
\(739\) −17327.6 −0.862526 −0.431263 0.902226i \(-0.641932\pi\)
−0.431263 + 0.902226i \(0.641932\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −17296.0 −0.854007 −0.427004 0.904250i \(-0.640431\pi\)
−0.427004 + 0.904250i \(0.640431\pi\)
\(744\) 0 0
\(745\) −18800.1 −0.924538
\(746\) 0 0
\(747\) −15744.8 −0.771181
\(748\) 0 0
\(749\) −4463.83 −0.217764
\(750\) 0 0
\(751\) −11089.6 −0.538835 −0.269417 0.963024i \(-0.586831\pi\)
−0.269417 + 0.963024i \(0.586831\pi\)
\(752\) 0 0
\(753\) 5071.87 0.245457
\(754\) 0 0
\(755\) −11501.7 −0.554425
\(756\) 0 0
\(757\) −23761.4 −1.14085 −0.570425 0.821350i \(-0.693221\pi\)
−0.570425 + 0.821350i \(0.693221\pi\)
\(758\) 0 0
\(759\) 2136.29 0.102164
\(760\) 0 0
\(761\) 17689.7 0.842641 0.421321 0.906912i \(-0.361567\pi\)
0.421321 + 0.906912i \(0.361567\pi\)
\(762\) 0 0
\(763\) −5945.45 −0.282097
\(764\) 0 0
\(765\) 14034.0 0.663269
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −28732.2 −1.34735 −0.673673 0.739029i \(-0.735284\pi\)
−0.673673 + 0.739029i \(0.735284\pi\)
\(770\) 0 0
\(771\) 2260.25 0.105578
\(772\) 0 0
\(773\) 5188.28 0.241410 0.120705 0.992688i \(-0.461485\pi\)
0.120705 + 0.992688i \(0.461485\pi\)
\(774\) 0 0
\(775\) −11108.3 −0.514866
\(776\) 0 0
\(777\) 765.542 0.0353458
\(778\) 0 0
\(779\) −554.543 −0.0255052
\(780\) 0 0
\(781\) 38449.2 1.76162
\(782\) 0 0
\(783\) −1564.96 −0.0714267
\(784\) 0 0
\(785\) 1964.48 0.0893189
\(786\) 0 0
\(787\) −15807.0 −0.715956 −0.357978 0.933730i \(-0.616534\pi\)
−0.357978 + 0.933730i \(0.616534\pi\)
\(788\) 0 0
\(789\) −2381.12 −0.107440
\(790\) 0 0
\(791\) −565.967 −0.0254406
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 1656.41 0.0738954
\(796\) 0 0
\(797\) 22603.4 1.00458 0.502292 0.864698i \(-0.332490\pi\)
0.502292 + 0.864698i \(0.332490\pi\)
\(798\) 0 0
\(799\) 18873.2 0.835650
\(800\) 0 0
\(801\) −30309.9 −1.33701
\(802\) 0 0
\(803\) 25373.4 1.11508
\(804\) 0 0
\(805\) 1012.34 0.0443233
\(806\) 0 0
\(807\) −3270.81 −0.142674
\(808\) 0 0
\(809\) −9902.20 −0.430337 −0.215169 0.976577i \(-0.569030\pi\)
−0.215169 + 0.976577i \(0.569030\pi\)
\(810\) 0 0
\(811\) 3136.80 0.135818 0.0679088 0.997692i \(-0.478367\pi\)
0.0679088 + 0.997692i \(0.478367\pi\)
\(812\) 0 0
\(813\) 4422.59 0.190784
\(814\) 0 0
\(815\) 1033.35 0.0444133
\(816\) 0 0
\(817\) −1739.78 −0.0745010
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 773.946 0.0329000 0.0164500 0.999865i \(-0.494764\pi\)
0.0164500 + 0.999865i \(0.494764\pi\)
\(822\) 0 0
\(823\) −12381.7 −0.524423 −0.262212 0.965010i \(-0.584452\pi\)
−0.262212 + 0.965010i \(0.584452\pi\)
\(824\) 0 0
\(825\) −3158.62 −0.133296
\(826\) 0 0
\(827\) 39362.3 1.65509 0.827545 0.561399i \(-0.189737\pi\)
0.827545 + 0.561399i \(0.189737\pi\)
\(828\) 0 0
\(829\) 11667.1 0.488800 0.244400 0.969674i \(-0.421409\pi\)
0.244400 + 0.969674i \(0.421409\pi\)
\(830\) 0 0
\(831\) 4077.94 0.170231
\(832\) 0 0
\(833\) −24466.0 −1.01764
\(834\) 0 0
\(835\) −1765.19 −0.0731578
\(836\) 0 0
\(837\) −6192.26 −0.255718
\(838\) 0 0
\(839\) −46018.4 −1.89360 −0.946801 0.321821i \(-0.895705\pi\)
−0.946801 + 0.321821i \(0.895705\pi\)
\(840\) 0 0
\(841\) −22876.8 −0.937999
\(842\) 0 0
\(843\) 5030.99 0.205548
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −5829.97 −0.236505
\(848\) 0 0
\(849\) −6209.78 −0.251024
\(850\) 0 0
\(851\) −17394.6 −0.700682
\(852\) 0 0
\(853\) −6667.50 −0.267633 −0.133817 0.991006i \(-0.542723\pi\)
−0.133817 + 0.991006i \(0.542723\pi\)
\(854\) 0 0
\(855\) −27431.9 −1.09725
\(856\) 0 0
\(857\) 23981.7 0.955890 0.477945 0.878390i \(-0.341382\pi\)
0.477945 + 0.878390i \(0.341382\pi\)
\(858\) 0 0
\(859\) 4760.36 0.189082 0.0945409 0.995521i \(-0.469862\pi\)
0.0945409 + 0.995521i \(0.469862\pi\)
\(860\) 0 0
\(861\) −8.34410 −0.000330274 0
\(862\) 0 0
\(863\) −49009.9 −1.93316 −0.966580 0.256363i \(-0.917476\pi\)
−0.966580 + 0.256363i \(0.917476\pi\)
\(864\) 0 0
\(865\) −25231.3 −0.991781
\(866\) 0 0
\(867\) 320.296 0.0125465
\(868\) 0 0
\(869\) −18481.6 −0.721455
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 17705.7 0.686423
\(874\) 0 0
\(875\) −4088.42 −0.157959
\(876\) 0 0
\(877\) 48161.3 1.85438 0.927189 0.374593i \(-0.122218\pi\)
0.927189 + 0.374593i \(0.122218\pi\)
\(878\) 0 0
\(879\) −406.655 −0.0156042
\(880\) 0 0
\(881\) −25752.2 −0.984804 −0.492402 0.870368i \(-0.663881\pi\)
−0.492402 + 0.870368i \(0.663881\pi\)
\(882\) 0 0
\(883\) −41104.6 −1.56657 −0.783285 0.621663i \(-0.786457\pi\)
−0.783285 + 0.621663i \(0.786457\pi\)
\(884\) 0 0
\(885\) −605.801 −0.0230099
\(886\) 0 0
\(887\) 48960.9 1.85338 0.926689 0.375829i \(-0.122642\pi\)
0.926689 + 0.375829i \(0.122642\pi\)
\(888\) 0 0
\(889\) 6185.24 0.233348
\(890\) 0 0
\(891\) 39696.6 1.49258
\(892\) 0 0
\(893\) −36890.9 −1.38243
\(894\) 0 0
\(895\) 6066.75 0.226580
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 5983.31 0.221974
\(900\) 0 0
\(901\) −22112.1 −0.817603
\(902\) 0 0
\(903\) −26.1782 −0.000964735 0
\(904\) 0 0
\(905\) −1955.80 −0.0718375
\(906\) 0 0
\(907\) 42398.2 1.55216 0.776081 0.630634i \(-0.217205\pi\)
0.776081 + 0.630634i \(0.217205\pi\)
\(908\) 0 0
\(909\) −37900.7 −1.38293
\(910\) 0 0
\(911\) −8150.26 −0.296411 −0.148205 0.988957i \(-0.547350\pi\)
−0.148205 + 0.988957i \(0.547350\pi\)
\(912\) 0 0
\(913\) 34601.2 1.25425
\(914\) 0 0
\(915\) 2272.11 0.0820913
\(916\) 0 0
\(917\) 2283.70 0.0822403
\(918\) 0 0
\(919\) −27015.2 −0.969693 −0.484847 0.874599i \(-0.661125\pi\)
−0.484847 + 0.874599i \(0.661125\pi\)
\(920\) 0 0
\(921\) −3036.85 −0.108651
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 25718.9 0.914198
\(926\) 0 0
\(927\) −20995.8 −0.743898
\(928\) 0 0
\(929\) −18722.9 −0.661227 −0.330613 0.943766i \(-0.607256\pi\)
−0.330613 + 0.943766i \(0.607256\pi\)
\(930\) 0 0
\(931\) 47823.0 1.68350
\(932\) 0 0
\(933\) −3123.14 −0.109590
\(934\) 0 0
\(935\) −30841.5 −1.07874
\(936\) 0 0
\(937\) 15955.0 0.556274 0.278137 0.960541i \(-0.410283\pi\)
0.278137 + 0.960541i \(0.410283\pi\)
\(938\) 0 0
\(939\) 3764.51 0.130831
\(940\) 0 0
\(941\) 9021.18 0.312521 0.156260 0.987716i \(-0.450056\pi\)
0.156260 + 0.987716i \(0.450056\pi\)
\(942\) 0 0
\(943\) 189.594 0.00654724
\(944\) 0 0
\(945\) −834.385 −0.0287223
\(946\) 0 0
\(947\) −12497.9 −0.428857 −0.214428 0.976740i \(-0.568789\pi\)
−0.214428 + 0.976740i \(0.568789\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 49.6429 0.00169272
\(952\) 0 0
\(953\) 36334.4 1.23503 0.617516 0.786558i \(-0.288139\pi\)
0.617516 + 0.786558i \(0.288139\pi\)
\(954\) 0 0
\(955\) 9592.30 0.325026
\(956\) 0 0
\(957\) 1701.34 0.0574677
\(958\) 0 0
\(959\) 3239.86 0.109093
\(960\) 0 0
\(961\) −6116.11 −0.205301
\(962\) 0 0
\(963\) −41355.1 −1.38385
\(964\) 0 0
\(965\) 9442.24 0.314981
\(966\) 0 0
\(967\) 40930.2 1.36114 0.680572 0.732681i \(-0.261731\pi\)
0.680572 + 0.732681i \(0.261731\pi\)
\(968\) 0 0
\(969\) −7859.09 −0.260547
\(970\) 0 0
\(971\) 29443.0 0.973092 0.486546 0.873655i \(-0.338257\pi\)
0.486546 + 0.873655i \(0.338257\pi\)
\(972\) 0 0
\(973\) 5725.99 0.188661
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 26831.2 0.878614 0.439307 0.898337i \(-0.355224\pi\)
0.439307 + 0.898337i \(0.355224\pi\)
\(978\) 0 0
\(979\) 66609.8 2.17452
\(980\) 0 0
\(981\) −55081.5 −1.79268
\(982\) 0 0
\(983\) 3409.34 0.110622 0.0553108 0.998469i \(-0.482385\pi\)
0.0553108 + 0.998469i \(0.482385\pi\)
\(984\) 0 0
\(985\) −17953.3 −0.580751
\(986\) 0 0
\(987\) −555.091 −0.0179015
\(988\) 0 0
\(989\) 594.820 0.0191246
\(990\) 0 0
\(991\) −23214.8 −0.744138 −0.372069 0.928205i \(-0.621352\pi\)
−0.372069 + 0.928205i \(0.621352\pi\)
\(992\) 0 0
\(993\) 3067.77 0.0980390
\(994\) 0 0
\(995\) 10187.1 0.324576
\(996\) 0 0
\(997\) −13047.0 −0.414446 −0.207223 0.978294i \(-0.566443\pi\)
−0.207223 + 0.978294i \(0.566443\pi\)
\(998\) 0 0
\(999\) 14336.9 0.454054
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 1352.4.a.n.1.4 6
13.3 even 3 104.4.i.c.9.3 12
13.9 even 3 104.4.i.c.81.3 yes 12
13.12 even 2 1352.4.a.m.1.4 6
52.3 odd 6 208.4.i.h.113.4 12
52.35 odd 6 208.4.i.h.81.4 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.4.i.c.9.3 12 13.3 even 3
104.4.i.c.81.3 yes 12 13.9 even 3
208.4.i.h.81.4 12 52.35 odd 6
208.4.i.h.113.4 12 52.3 odd 6
1352.4.a.m.1.4 6 13.12 even 2
1352.4.a.n.1.4 6 1.1 even 1 trivial