Properties

Label 1089.4.a.u.1.2
Level $1089$
Weight $4$
Character 1089.1
Self dual yes
Analytic conductor $64.253$
Analytic rank $0$
Dimension $2$
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] = [1089,4,Mod(1,1089)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1089, 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("1089.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1089 = 3^{2} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1089.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: \(64.2530799963\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 33)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(5.42443\) of defining polynomial
Character \(\chi\) \(=\) 1089.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.42443 q^{2} +21.4244 q^{4} +16.8489 q^{5} +7.69772 q^{7} +72.8199 q^{8} +O(q^{10})\) \(q+5.42443 q^{2} +21.4244 q^{4} +16.8489 q^{5} +7.69772 q^{7} +72.8199 q^{8} +91.3954 q^{10} -24.8489 q^{13} +41.7557 q^{14} +223.611 q^{16} -15.9420 q^{17} -15.1511 q^{19} +360.977 q^{20} -17.7557 q^{23} +158.884 q^{25} -134.791 q^{26} +164.919 q^{28} -128.547 q^{29} +219.395 q^{31} +630.402 q^{32} -86.4763 q^{34} +129.698 q^{35} +92.0703 q^{37} -82.1863 q^{38} +1226.93 q^{40} -459.942 q^{41} -64.9648 q^{43} -96.3146 q^{46} -497.408 q^{47} -283.745 q^{49} +861.855 q^{50} -532.373 q^{52} +526.919 q^{53} +560.547 q^{56} -697.292 q^{58} +578.443 q^{59} +221.569 q^{61} +1190.09 q^{62} +1630.68 q^{64} -418.675 q^{65} -860.745 q^{67} -341.548 q^{68} +703.536 q^{70} -580.919 q^{71} -510.116 q^{73} +499.429 q^{74} -324.605 q^{76} -1035.12 q^{79} +3767.59 q^{80} -2494.92 q^{82} +606.211 q^{83} -268.605 q^{85} -352.397 q^{86} +23.4411 q^{89} -191.279 q^{91} -380.406 q^{92} -2698.15 q^{94} -255.279 q^{95} +719.490 q^{97} -1539.16 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 33 q^{4} + 14 q^{5} - 24 q^{7} + 57 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 33 q^{4} + 14 q^{5} - 24 q^{7} + 57 q^{8} + 104 q^{10} - 30 q^{13} + 182 q^{14} + 201 q^{16} + 106 q^{17} - 50 q^{19} + 328 q^{20} - 134 q^{23} + 42 q^{25} - 112 q^{26} - 202 q^{28} - 198 q^{29} + 360 q^{31} + 857 q^{32} - 626 q^{34} + 220 q^{35} - 328 q^{37} + 72 q^{38} + 1272 q^{40} - 782 q^{41} - 386 q^{43} + 418 q^{46} - 266 q^{47} + 378 q^{49} + 1379 q^{50} - 592 q^{52} + 522 q^{53} + 1062 q^{56} - 390 q^{58} + 172 q^{59} + 778 q^{61} + 568 q^{62} + 809 q^{64} - 404 q^{65} - 776 q^{67} + 1070 q^{68} + 304 q^{70} - 630 q^{71} - 1296 q^{73} + 2358 q^{74} - 728 q^{76} - 652 q^{79} + 3832 q^{80} - 1070 q^{82} - 324 q^{83} - 616 q^{85} + 1068 q^{86} + 756 q^{89} - 28 q^{91} - 1726 q^{92} - 3722 q^{94} - 156 q^{95} - 452 q^{97} - 4467 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.42443 1.91783 0.958913 0.283702i \(-0.0915625\pi\)
0.958913 + 0.283702i \(0.0915625\pi\)
\(3\) 0 0
\(4\) 21.4244 2.67805
\(5\) 16.8489 1.50701 0.753504 0.657444i \(-0.228362\pi\)
0.753504 + 0.657444i \(0.228362\pi\)
\(6\) 0 0
\(7\) 7.69772 0.415638 0.207819 0.978167i \(-0.433364\pi\)
0.207819 + 0.978167i \(0.433364\pi\)
\(8\) 72.8199 3.21821
\(9\) 0 0
\(10\) 91.3954 2.89018
\(11\) 0 0
\(12\) 0 0
\(13\) −24.8489 −0.530141 −0.265071 0.964229i \(-0.585395\pi\)
−0.265071 + 0.964229i \(0.585395\pi\)
\(14\) 41.7557 0.797120
\(15\) 0 0
\(16\) 223.611 3.49392
\(17\) −15.9420 −0.227441 −0.113721 0.993513i \(-0.536277\pi\)
−0.113721 + 0.993513i \(0.536277\pi\)
\(18\) 0 0
\(19\) −15.1511 −0.182943 −0.0914713 0.995808i \(-0.529157\pi\)
−0.0914713 + 0.995808i \(0.529157\pi\)
\(20\) 360.977 4.03585
\(21\) 0 0
\(22\) 0 0
\(23\) −17.7557 −0.160971 −0.0804853 0.996756i \(-0.525647\pi\)
−0.0804853 + 0.996756i \(0.525647\pi\)
\(24\) 0 0
\(25\) 158.884 1.27107
\(26\) −134.791 −1.01672
\(27\) 0 0
\(28\) 164.919 1.11310
\(29\) −128.547 −0.823121 −0.411560 0.911383i \(-0.635016\pi\)
−0.411560 + 0.911383i \(0.635016\pi\)
\(30\) 0 0
\(31\) 219.395 1.27112 0.635558 0.772053i \(-0.280770\pi\)
0.635558 + 0.772053i \(0.280770\pi\)
\(32\) 630.402 3.48251
\(33\) 0 0
\(34\) −86.4763 −0.436193
\(35\) 129.698 0.626369
\(36\) 0 0
\(37\) 92.0703 0.409088 0.204544 0.978857i \(-0.434429\pi\)
0.204544 + 0.978857i \(0.434429\pi\)
\(38\) −82.1863 −0.350852
\(39\) 0 0
\(40\) 1226.93 4.84987
\(41\) −459.942 −1.75197 −0.875986 0.482336i \(-0.839788\pi\)
−0.875986 + 0.482336i \(0.839788\pi\)
\(42\) 0 0
\(43\) −64.9648 −0.230396 −0.115198 0.993343i \(-0.536750\pi\)
−0.115198 + 0.993343i \(0.536750\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −96.3146 −0.308713
\(47\) −497.408 −1.54371 −0.771855 0.635799i \(-0.780671\pi\)
−0.771855 + 0.635799i \(0.780671\pi\)
\(48\) 0 0
\(49\) −283.745 −0.827245
\(50\) 861.855 2.43769
\(51\) 0 0
\(52\) −532.373 −1.41975
\(53\) 526.919 1.36562 0.682811 0.730596i \(-0.260757\pi\)
0.682811 + 0.730596i \(0.260757\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 560.547 1.33761
\(57\) 0 0
\(58\) −697.292 −1.57860
\(59\) 578.443 1.27639 0.638194 0.769876i \(-0.279682\pi\)
0.638194 + 0.769876i \(0.279682\pi\)
\(60\) 0 0
\(61\) 221.569 0.465067 0.232533 0.972588i \(-0.425299\pi\)
0.232533 + 0.972588i \(0.425299\pi\)
\(62\) 1190.09 2.43778
\(63\) 0 0
\(64\) 1630.68 3.18493
\(65\) −418.675 −0.798927
\(66\) 0 0
\(67\) −860.745 −1.56950 −0.784752 0.619810i \(-0.787210\pi\)
−0.784752 + 0.619810i \(0.787210\pi\)
\(68\) −341.548 −0.609100
\(69\) 0 0
\(70\) 703.536 1.20127
\(71\) −580.919 −0.971020 −0.485510 0.874231i \(-0.661366\pi\)
−0.485510 + 0.874231i \(0.661366\pi\)
\(72\) 0 0
\(73\) −510.116 −0.817871 −0.408935 0.912563i \(-0.634100\pi\)
−0.408935 + 0.912563i \(0.634100\pi\)
\(74\) 499.429 0.784560
\(75\) 0 0
\(76\) −324.605 −0.489930
\(77\) 0 0
\(78\) 0 0
\(79\) −1035.12 −1.47418 −0.737088 0.675797i \(-0.763800\pi\)
−0.737088 + 0.675797i \(0.763800\pi\)
\(80\) 3767.59 5.26536
\(81\) 0 0
\(82\) −2494.92 −3.35998
\(83\) 606.211 0.801690 0.400845 0.916146i \(-0.368717\pi\)
0.400845 + 0.916146i \(0.368717\pi\)
\(84\) 0 0
\(85\) −268.605 −0.342756
\(86\) −352.397 −0.441860
\(87\) 0 0
\(88\) 0 0
\(89\) 23.4411 0.0279186 0.0139593 0.999903i \(-0.495556\pi\)
0.0139593 + 0.999903i \(0.495556\pi\)
\(90\) 0 0
\(91\) −191.279 −0.220347
\(92\) −380.406 −0.431088
\(93\) 0 0
\(94\) −2698.15 −2.96057
\(95\) −255.279 −0.275696
\(96\) 0 0
\(97\) 719.490 0.753126 0.376563 0.926391i \(-0.377106\pi\)
0.376563 + 0.926391i \(0.377106\pi\)
\(98\) −1539.16 −1.58651
\(99\) 0 0
\(100\) 3404.00 3.40400
\(101\) 1871.27 1.84355 0.921774 0.387727i \(-0.126740\pi\)
0.921774 + 0.387727i \(0.126740\pi\)
\(102\) 0 0
\(103\) −428.745 −0.410151 −0.205075 0.978746i \(-0.565744\pi\)
−0.205075 + 0.978746i \(0.565744\pi\)
\(104\) −1809.49 −1.70611
\(105\) 0 0
\(106\) 2858.24 2.61902
\(107\) 1148.02 1.03723 0.518616 0.855008i \(-0.326448\pi\)
0.518616 + 0.855008i \(0.326448\pi\)
\(108\) 0 0
\(109\) 1828.32 1.60662 0.803308 0.595564i \(-0.203071\pi\)
0.803308 + 0.595564i \(0.203071\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1721.29 1.45220
\(113\) −1126.40 −0.937722 −0.468861 0.883272i \(-0.655335\pi\)
−0.468861 + 0.883272i \(0.655335\pi\)
\(114\) 0 0
\(115\) −299.163 −0.242584
\(116\) −2754.04 −2.20436
\(117\) 0 0
\(118\) 3137.72 2.44789
\(119\) −122.717 −0.0945332
\(120\) 0 0
\(121\) 0 0
\(122\) 1201.89 0.891916
\(123\) 0 0
\(124\) 4700.42 3.40412
\(125\) 570.907 0.408508
\(126\) 0 0
\(127\) −661.304 −0.462057 −0.231029 0.972947i \(-0.574209\pi\)
−0.231029 + 0.972947i \(0.574209\pi\)
\(128\) 3802.31 2.62562
\(129\) 0 0
\(130\) −2271.07 −1.53220
\(131\) 622.186 0.414967 0.207483 0.978239i \(-0.433473\pi\)
0.207483 + 0.978239i \(0.433473\pi\)
\(132\) 0 0
\(133\) −116.629 −0.0760378
\(134\) −4669.05 −3.01003
\(135\) 0 0
\(136\) −1160.89 −0.731955
\(137\) 1872.84 1.16794 0.583969 0.811776i \(-0.301499\pi\)
0.583969 + 0.811776i \(0.301499\pi\)
\(138\) 0 0
\(139\) 954.058 0.582174 0.291087 0.956697i \(-0.405983\pi\)
0.291087 + 0.956697i \(0.405983\pi\)
\(140\) 2778.70 1.67745
\(141\) 0 0
\(142\) −3151.15 −1.86225
\(143\) 0 0
\(144\) 0 0
\(145\) −2165.86 −1.24045
\(146\) −2767.09 −1.56853
\(147\) 0 0
\(148\) 1972.55 1.09556
\(149\) −2047.01 −1.12549 −0.562745 0.826631i \(-0.690255\pi\)
−0.562745 + 0.826631i \(0.690255\pi\)
\(150\) 0 0
\(151\) −475.863 −0.256458 −0.128229 0.991745i \(-0.540929\pi\)
−0.128229 + 0.991745i \(0.540929\pi\)
\(152\) −1103.30 −0.588749
\(153\) 0 0
\(154\) 0 0
\(155\) 3696.56 1.91558
\(156\) 0 0
\(157\) −647.466 −0.329130 −0.164565 0.986366i \(-0.552622\pi\)
−0.164565 + 0.986366i \(0.552622\pi\)
\(158\) −5614.92 −2.82721
\(159\) 0 0
\(160\) 10621.5 5.24817
\(161\) −136.678 −0.0669054
\(162\) 0 0
\(163\) 1093.23 0.525329 0.262665 0.964887i \(-0.415399\pi\)
0.262665 + 0.964887i \(0.415399\pi\)
\(164\) −9853.99 −4.69188
\(165\) 0 0
\(166\) 3288.35 1.53750
\(167\) 1123.25 0.520479 0.260240 0.965544i \(-0.416198\pi\)
0.260240 + 0.965544i \(0.416198\pi\)
\(168\) 0 0
\(169\) −1579.53 −0.718951
\(170\) −1457.03 −0.657346
\(171\) 0 0
\(172\) −1391.83 −0.617014
\(173\) 46.0123 0.0202211 0.0101106 0.999949i \(-0.496782\pi\)
0.0101106 + 0.999949i \(0.496782\pi\)
\(174\) 0 0
\(175\) 1223.04 0.528305
\(176\) 0 0
\(177\) 0 0
\(178\) 127.155 0.0535430
\(179\) 831.975 0.347401 0.173700 0.984799i \(-0.444428\pi\)
0.173700 + 0.984799i \(0.444428\pi\)
\(180\) 0 0
\(181\) −1810.63 −0.743553 −0.371776 0.928322i \(-0.621251\pi\)
−0.371776 + 0.928322i \(0.621251\pi\)
\(182\) −1037.58 −0.422586
\(183\) 0 0
\(184\) −1292.97 −0.518037
\(185\) 1551.28 0.616499
\(186\) 0 0
\(187\) 0 0
\(188\) −10656.7 −4.13414
\(189\) 0 0
\(190\) −1384.75 −0.528737
\(191\) −458.898 −0.173847 −0.0869233 0.996215i \(-0.527704\pi\)
−0.0869233 + 0.996215i \(0.527704\pi\)
\(192\) 0 0
\(193\) −1778.91 −0.663465 −0.331733 0.943373i \(-0.607633\pi\)
−0.331733 + 0.943373i \(0.607633\pi\)
\(194\) 3902.82 1.44436
\(195\) 0 0
\(196\) −6079.08 −2.21541
\(197\) 5304.53 1.91844 0.959218 0.282666i \(-0.0912188\pi\)
0.959218 + 0.282666i \(0.0912188\pi\)
\(198\) 0 0
\(199\) −5138.40 −1.83041 −0.915205 0.402989i \(-0.867971\pi\)
−0.915205 + 0.402989i \(0.867971\pi\)
\(200\) 11569.9 4.09058
\(201\) 0 0
\(202\) 10150.6 3.53560
\(203\) −989.515 −0.342120
\(204\) 0 0
\(205\) −7749.50 −2.64024
\(206\) −2325.70 −0.786597
\(207\) 0 0
\(208\) −5556.47 −1.85227
\(209\) 0 0
\(210\) 0 0
\(211\) 4262.36 1.39068 0.695339 0.718682i \(-0.255254\pi\)
0.695339 + 0.718682i \(0.255254\pi\)
\(212\) 11288.9 3.65721
\(213\) 0 0
\(214\) 6227.38 1.98923
\(215\) −1094.58 −0.347209
\(216\) 0 0
\(217\) 1688.84 0.528323
\(218\) 9917.58 3.08121
\(219\) 0 0
\(220\) 0 0
\(221\) 396.141 0.120576
\(222\) 0 0
\(223\) −1377.80 −0.413740 −0.206870 0.978368i \(-0.566328\pi\)
−0.206870 + 0.978368i \(0.566328\pi\)
\(224\) 4852.65 1.44746
\(225\) 0 0
\(226\) −6110.06 −1.79839
\(227\) −1227.28 −0.358843 −0.179422 0.983772i \(-0.557423\pi\)
−0.179422 + 0.983772i \(0.557423\pi\)
\(228\) 0 0
\(229\) 3890.28 1.12261 0.561304 0.827610i \(-0.310300\pi\)
0.561304 + 0.827610i \(0.310300\pi\)
\(230\) −1622.79 −0.465233
\(231\) 0 0
\(232\) −9360.74 −2.64898
\(233\) −3218.14 −0.904837 −0.452419 0.891806i \(-0.649439\pi\)
−0.452419 + 0.891806i \(0.649439\pi\)
\(234\) 0 0
\(235\) −8380.75 −2.32638
\(236\) 12392.8 3.41823
\(237\) 0 0
\(238\) −665.670 −0.181298
\(239\) −428.098 −0.115864 −0.0579318 0.998321i \(-0.518451\pi\)
−0.0579318 + 0.998321i \(0.518451\pi\)
\(240\) 0 0
\(241\) −1231.16 −0.329070 −0.164535 0.986371i \(-0.552612\pi\)
−0.164535 + 0.986371i \(0.552612\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 4747.00 1.24547
\(245\) −4780.78 −1.24667
\(246\) 0 0
\(247\) 376.489 0.0969854
\(248\) 15976.3 4.09072
\(249\) 0 0
\(250\) 3096.84 0.783446
\(251\) −2838.22 −0.713732 −0.356866 0.934156i \(-0.616155\pi\)
−0.356866 + 0.934156i \(0.616155\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −3587.20 −0.886145
\(255\) 0 0
\(256\) 7579.90 1.85056
\(257\) −342.007 −0.0830110 −0.0415055 0.999138i \(-0.513215\pi\)
−0.0415055 + 0.999138i \(0.513215\pi\)
\(258\) 0 0
\(259\) 708.731 0.170032
\(260\) −8969.87 −2.13957
\(261\) 0 0
\(262\) 3375.01 0.795834
\(263\) 5895.00 1.38213 0.691067 0.722791i \(-0.257141\pi\)
0.691067 + 0.722791i \(0.257141\pi\)
\(264\) 0 0
\(265\) 8877.99 2.05800
\(266\) −632.647 −0.145827
\(267\) 0 0
\(268\) −18441.0 −4.20322
\(269\) 2496.18 0.565779 0.282890 0.959152i \(-0.408707\pi\)
0.282890 + 0.959152i \(0.408707\pi\)
\(270\) 0 0
\(271\) 2249.68 0.504274 0.252137 0.967692i \(-0.418867\pi\)
0.252137 + 0.967692i \(0.418867\pi\)
\(272\) −3564.80 −0.794662
\(273\) 0 0
\(274\) 10159.1 2.23990
\(275\) 0 0
\(276\) 0 0
\(277\) −4082.59 −0.885556 −0.442778 0.896631i \(-0.646007\pi\)
−0.442778 + 0.896631i \(0.646007\pi\)
\(278\) 5175.22 1.11651
\(279\) 0 0
\(280\) 9444.57 2.01579
\(281\) −1033.79 −0.219468 −0.109734 0.993961i \(-0.535000\pi\)
−0.109734 + 0.993961i \(0.535000\pi\)
\(282\) 0 0
\(283\) 7809.14 1.64030 0.820150 0.572148i \(-0.193890\pi\)
0.820150 + 0.572148i \(0.193890\pi\)
\(284\) −12445.9 −2.60044
\(285\) 0 0
\(286\) 0 0
\(287\) −3540.50 −0.728186
\(288\) 0 0
\(289\) −4658.85 −0.948270
\(290\) −11748.6 −2.37896
\(291\) 0 0
\(292\) −10928.9 −2.19030
\(293\) −1949.19 −0.388645 −0.194323 0.980938i \(-0.562251\pi\)
−0.194323 + 0.980938i \(0.562251\pi\)
\(294\) 0 0
\(295\) 9746.10 1.92353
\(296\) 6704.55 1.31653
\(297\) 0 0
\(298\) −11103.9 −2.15849
\(299\) 441.209 0.0853371
\(300\) 0 0
\(301\) −500.081 −0.0957614
\(302\) −2581.28 −0.491842
\(303\) 0 0
\(304\) −3387.96 −0.639187
\(305\) 3733.19 0.700859
\(306\) 0 0
\(307\) −2364.09 −0.439497 −0.219748 0.975557i \(-0.570524\pi\)
−0.219748 + 0.975557i \(0.570524\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 20051.7 3.67375
\(311\) 1989.17 0.362686 0.181343 0.983420i \(-0.441956\pi\)
0.181343 + 0.983420i \(0.441956\pi\)
\(312\) 0 0
\(313\) −3878.67 −0.700433 −0.350216 0.936669i \(-0.613892\pi\)
−0.350216 + 0.936669i \(0.613892\pi\)
\(314\) −3512.13 −0.631214
\(315\) 0 0
\(316\) −22176.8 −3.94792
\(317\) −2913.73 −0.516251 −0.258126 0.966111i \(-0.583105\pi\)
−0.258126 + 0.966111i \(0.583105\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 27475.1 4.79971
\(321\) 0 0
\(322\) −741.402 −0.128313
\(323\) 241.540 0.0416087
\(324\) 0 0
\(325\) −3948.09 −0.673847
\(326\) 5930.17 1.00749
\(327\) 0 0
\(328\) −33492.9 −5.63822
\(329\) −3828.90 −0.641624
\(330\) 0 0
\(331\) 8104.46 1.34580 0.672902 0.739731i \(-0.265047\pi\)
0.672902 + 0.739731i \(0.265047\pi\)
\(332\) 12987.7 2.14697
\(333\) 0 0
\(334\) 6093.02 0.998189
\(335\) −14502.6 −2.36525
\(336\) 0 0
\(337\) −5919.19 −0.956792 −0.478396 0.878144i \(-0.658782\pi\)
−0.478396 + 0.878144i \(0.658782\pi\)
\(338\) −8568.07 −1.37882
\(339\) 0 0
\(340\) −5754.70 −0.917919
\(341\) 0 0
\(342\) 0 0
\(343\) −4824.51 −0.759472
\(344\) −4730.73 −0.741465
\(345\) 0 0
\(346\) 249.590 0.0387805
\(347\) −8540.59 −1.32128 −0.660638 0.750705i \(-0.729714\pi\)
−0.660638 + 0.750705i \(0.729714\pi\)
\(348\) 0 0
\(349\) −937.337 −0.143767 −0.0718833 0.997413i \(-0.522901\pi\)
−0.0718833 + 0.997413i \(0.522901\pi\)
\(350\) 6634.31 1.01320
\(351\) 0 0
\(352\) 0 0
\(353\) 211.118 0.0318319 0.0159160 0.999873i \(-0.494934\pi\)
0.0159160 + 0.999873i \(0.494934\pi\)
\(354\) 0 0
\(355\) −9787.82 −1.46333
\(356\) 502.213 0.0747675
\(357\) 0 0
\(358\) 4512.99 0.666254
\(359\) −1376.31 −0.202337 −0.101169 0.994869i \(-0.532258\pi\)
−0.101169 + 0.994869i \(0.532258\pi\)
\(360\) 0 0
\(361\) −6629.44 −0.966532
\(362\) −9821.63 −1.42600
\(363\) 0 0
\(364\) −4098.05 −0.590100
\(365\) −8594.87 −1.23254
\(366\) 0 0
\(367\) 1030.45 0.146564 0.0732821 0.997311i \(-0.476653\pi\)
0.0732821 + 0.997311i \(0.476653\pi\)
\(368\) −3970.37 −0.562418
\(369\) 0 0
\(370\) 8414.81 1.18234
\(371\) 4056.07 0.567603
\(372\) 0 0
\(373\) −9365.39 −1.30006 −0.650029 0.759909i \(-0.725243\pi\)
−0.650029 + 0.759909i \(0.725243\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −36221.2 −4.96799
\(377\) 3194.24 0.436370
\(378\) 0 0
\(379\) −7120.23 −0.965017 −0.482509 0.875891i \(-0.660274\pi\)
−0.482509 + 0.875891i \(0.660274\pi\)
\(380\) −5469.22 −0.738329
\(381\) 0 0
\(382\) −2489.26 −0.333407
\(383\) 1163.56 0.155235 0.0776176 0.996983i \(-0.475269\pi\)
0.0776176 + 0.996983i \(0.475269\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −9649.57 −1.27241
\(387\) 0 0
\(388\) 15414.7 2.01691
\(389\) 10958.9 1.42838 0.714188 0.699954i \(-0.246796\pi\)
0.714188 + 0.699954i \(0.246796\pi\)
\(390\) 0 0
\(391\) 283.062 0.0366114
\(392\) −20662.3 −2.66225
\(393\) 0 0
\(394\) 28774.0 3.67923
\(395\) −17440.6 −2.22159
\(396\) 0 0
\(397\) −2172.09 −0.274595 −0.137298 0.990530i \(-0.543842\pi\)
−0.137298 + 0.990530i \(0.543842\pi\)
\(398\) −27872.9 −3.51041
\(399\) 0 0
\(400\) 35528.2 4.44102
\(401\) −7830.71 −0.975180 −0.487590 0.873073i \(-0.662124\pi\)
−0.487590 + 0.873073i \(0.662124\pi\)
\(402\) 0 0
\(403\) −5451.73 −0.673870
\(404\) 40090.9 4.93712
\(405\) 0 0
\(406\) −5367.55 −0.656126
\(407\) 0 0
\(408\) 0 0
\(409\) 10731.2 1.29736 0.648682 0.761060i \(-0.275321\pi\)
0.648682 + 0.761060i \(0.275321\pi\)
\(410\) −42036.6 −5.06351
\(411\) 0 0
\(412\) −9185.62 −1.09841
\(413\) 4452.69 0.530515
\(414\) 0 0
\(415\) 10214.0 1.20815
\(416\) −15664.8 −1.84622
\(417\) 0 0
\(418\) 0 0
\(419\) 7315.88 0.852994 0.426497 0.904489i \(-0.359748\pi\)
0.426497 + 0.904489i \(0.359748\pi\)
\(420\) 0 0
\(421\) −12495.7 −1.44657 −0.723284 0.690551i \(-0.757368\pi\)
−0.723284 + 0.690551i \(0.757368\pi\)
\(422\) 23120.9 2.66708
\(423\) 0 0
\(424\) 38370.2 4.39486
\(425\) −2532.93 −0.289094
\(426\) 0 0
\(427\) 1705.58 0.193299
\(428\) 24595.8 2.77776
\(429\) 0 0
\(430\) −5937.49 −0.665887
\(431\) 6075.01 0.678939 0.339470 0.940617i \(-0.389752\pi\)
0.339470 + 0.940617i \(0.389752\pi\)
\(432\) 0 0
\(433\) 5641.79 0.626160 0.313080 0.949727i \(-0.398639\pi\)
0.313080 + 0.949727i \(0.398639\pi\)
\(434\) 9161.01 1.01323
\(435\) 0 0
\(436\) 39170.7 4.30260
\(437\) 269.019 0.0294484
\(438\) 0 0
\(439\) −10897.0 −1.18470 −0.592351 0.805680i \(-0.701800\pi\)
−0.592351 + 0.805680i \(0.701800\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2148.84 0.231244
\(443\) 7720.83 0.828054 0.414027 0.910265i \(-0.364122\pi\)
0.414027 + 0.910265i \(0.364122\pi\)
\(444\) 0 0
\(445\) 394.956 0.0420735
\(446\) −7473.76 −0.793481
\(447\) 0 0
\(448\) 12552.5 1.32378
\(449\) 7473.86 0.785553 0.392776 0.919634i \(-0.371515\pi\)
0.392776 + 0.919634i \(0.371515\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −24132.4 −2.51127
\(453\) 0 0
\(454\) −6657.29 −0.688198
\(455\) −3222.84 −0.332064
\(456\) 0 0
\(457\) −11140.5 −1.14033 −0.570167 0.821529i \(-0.693121\pi\)
−0.570167 + 0.821529i \(0.693121\pi\)
\(458\) 21102.6 2.15296
\(459\) 0 0
\(460\) −6409.41 −0.649652
\(461\) 14328.8 1.44763 0.723817 0.689992i \(-0.242386\pi\)
0.723817 + 0.689992i \(0.242386\pi\)
\(462\) 0 0
\(463\) 11760.7 1.18049 0.590246 0.807223i \(-0.299031\pi\)
0.590246 + 0.807223i \(0.299031\pi\)
\(464\) −28744.4 −2.87592
\(465\) 0 0
\(466\) −17456.5 −1.73532
\(467\) −11854.9 −1.17469 −0.587343 0.809338i \(-0.699826\pi\)
−0.587343 + 0.809338i \(0.699826\pi\)
\(468\) 0 0
\(469\) −6625.77 −0.652345
\(470\) −45460.8 −4.46160
\(471\) 0 0
\(472\) 42122.1 4.10769
\(473\) 0 0
\(474\) 0 0
\(475\) −2407.27 −0.232533
\(476\) −2629.14 −0.253165
\(477\) 0 0
\(478\) −2322.19 −0.222206
\(479\) −1324.68 −0.126359 −0.0631796 0.998002i \(-0.520124\pi\)
−0.0631796 + 0.998002i \(0.520124\pi\)
\(480\) 0 0
\(481\) −2287.84 −0.216874
\(482\) −6678.34 −0.631100
\(483\) 0 0
\(484\) 0 0
\(485\) 12122.6 1.13497
\(486\) 0 0
\(487\) 18636.4 1.73408 0.867040 0.498239i \(-0.166020\pi\)
0.867040 + 0.498239i \(0.166020\pi\)
\(488\) 16134.7 1.49668
\(489\) 0 0
\(490\) −25933.0 −2.39089
\(491\) 124.552 0.0114480 0.00572398 0.999984i \(-0.498178\pi\)
0.00572398 + 0.999984i \(0.498178\pi\)
\(492\) 0 0
\(493\) 2049.29 0.187212
\(494\) 2042.24 0.186001
\(495\) 0 0
\(496\) 49059.2 4.44117
\(497\) −4471.75 −0.403592
\(498\) 0 0
\(499\) 10230.2 0.917768 0.458884 0.888496i \(-0.348249\pi\)
0.458884 + 0.888496i \(0.348249\pi\)
\(500\) 12231.4 1.09401
\(501\) 0 0
\(502\) −15395.7 −1.36881
\(503\) 5150.81 0.456587 0.228294 0.973592i \(-0.426685\pi\)
0.228294 + 0.973592i \(0.426685\pi\)
\(504\) 0 0
\(505\) 31528.8 2.77824
\(506\) 0 0
\(507\) 0 0
\(508\) −14168.1 −1.23741
\(509\) 22.7715 0.00198296 0.000991481 1.00000i \(-0.499684\pi\)
0.000991481 1.00000i \(0.499684\pi\)
\(510\) 0 0
\(511\) −3926.73 −0.339938
\(512\) 10698.1 0.923429
\(513\) 0 0
\(514\) −1855.19 −0.159201
\(515\) −7223.87 −0.618100
\(516\) 0 0
\(517\) 0 0
\(518\) 3844.46 0.326093
\(519\) 0 0
\(520\) −30487.8 −2.57112
\(521\) −21521.7 −1.80976 −0.904879 0.425669i \(-0.860039\pi\)
−0.904879 + 0.425669i \(0.860039\pi\)
\(522\) 0 0
\(523\) −2923.36 −0.244416 −0.122208 0.992504i \(-0.538998\pi\)
−0.122208 + 0.992504i \(0.538998\pi\)
\(524\) 13330.0 1.11130
\(525\) 0 0
\(526\) 31977.0 2.65069
\(527\) −3497.60 −0.289104
\(528\) 0 0
\(529\) −11851.7 −0.974088
\(530\) 48158.0 3.94689
\(531\) 0 0
\(532\) −2498.71 −0.203633
\(533\) 11429.0 0.928792
\(534\) 0 0
\(535\) 19342.9 1.56312
\(536\) −62679.3 −5.05100
\(537\) 0 0
\(538\) 13540.3 1.08507
\(539\) 0 0
\(540\) 0 0
\(541\) −21272.8 −1.69056 −0.845278 0.534327i \(-0.820565\pi\)
−0.845278 + 0.534327i \(0.820565\pi\)
\(542\) 12203.2 0.967108
\(543\) 0 0
\(544\) −10049.9 −0.792067
\(545\) 30805.1 2.42118
\(546\) 0 0
\(547\) 18730.5 1.46409 0.732046 0.681256i \(-0.238566\pi\)
0.732046 + 0.681256i \(0.238566\pi\)
\(548\) 40124.5 3.12780
\(549\) 0 0
\(550\) 0 0
\(551\) 1947.63 0.150584
\(552\) 0 0
\(553\) −7968.04 −0.612723
\(554\) −22145.7 −1.69834
\(555\) 0 0
\(556\) 20440.1 1.55909
\(557\) −18885.0 −1.43659 −0.718297 0.695736i \(-0.755078\pi\)
−0.718297 + 0.695736i \(0.755078\pi\)
\(558\) 0 0
\(559\) 1614.30 0.122143
\(560\) 29001.8 2.18848
\(561\) 0 0
\(562\) −5607.71 −0.420902
\(563\) 10285.1 0.769922 0.384961 0.922933i \(-0.374215\pi\)
0.384961 + 0.922933i \(0.374215\pi\)
\(564\) 0 0
\(565\) −18978.5 −1.41315
\(566\) 42360.1 3.14581
\(567\) 0 0
\(568\) −42302.5 −3.12495
\(569\) 18008.8 1.32683 0.663415 0.748251i \(-0.269106\pi\)
0.663415 + 0.748251i \(0.269106\pi\)
\(570\) 0 0
\(571\) 7010.79 0.513822 0.256911 0.966435i \(-0.417295\pi\)
0.256911 + 0.966435i \(0.417295\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −19205.2 −1.39653
\(575\) −2821.10 −0.204605
\(576\) 0 0
\(577\) −16398.9 −1.18318 −0.591589 0.806240i \(-0.701499\pi\)
−0.591589 + 0.806240i \(0.701499\pi\)
\(578\) −25271.6 −1.81862
\(579\) 0 0
\(580\) −46402.4 −3.32199
\(581\) 4666.44 0.333213
\(582\) 0 0
\(583\) 0 0
\(584\) −37146.6 −2.63208
\(585\) 0 0
\(586\) −10573.3 −0.745354
\(587\) −12823.5 −0.901671 −0.450836 0.892607i \(-0.648874\pi\)
−0.450836 + 0.892607i \(0.648874\pi\)
\(588\) 0 0
\(589\) −3324.09 −0.232541
\(590\) 52867.0 3.68899
\(591\) 0 0
\(592\) 20587.9 1.42932
\(593\) 16899.5 1.17029 0.585144 0.810929i \(-0.301038\pi\)
0.585144 + 0.810929i \(0.301038\pi\)
\(594\) 0 0
\(595\) −2067.64 −0.142462
\(596\) −43856.1 −3.01412
\(597\) 0 0
\(598\) 2393.31 0.163662
\(599\) 15074.9 1.02829 0.514143 0.857704i \(-0.328110\pi\)
0.514143 + 0.857704i \(0.328110\pi\)
\(600\) 0 0
\(601\) 11418.8 0.775014 0.387507 0.921867i \(-0.373336\pi\)
0.387507 + 0.921867i \(0.373336\pi\)
\(602\) −2712.65 −0.183654
\(603\) 0 0
\(604\) −10195.1 −0.686809
\(605\) 0 0
\(606\) 0 0
\(607\) 17952.8 1.20046 0.600232 0.799826i \(-0.295075\pi\)
0.600232 + 0.799826i \(0.295075\pi\)
\(608\) −9551.30 −0.637100
\(609\) 0 0
\(610\) 20250.4 1.34412
\(611\) 12360.0 0.818384
\(612\) 0 0
\(613\) −12528.9 −0.825507 −0.412753 0.910843i \(-0.635433\pi\)
−0.412753 + 0.910843i \(0.635433\pi\)
\(614\) −12823.8 −0.842878
\(615\) 0 0
\(616\) 0 0
\(617\) 8586.10 0.560232 0.280116 0.959966i \(-0.409627\pi\)
0.280116 + 0.959966i \(0.409627\pi\)
\(618\) 0 0
\(619\) 18415.4 1.19576 0.597882 0.801584i \(-0.296009\pi\)
0.597882 + 0.801584i \(0.296009\pi\)
\(620\) 79196.7 5.13003
\(621\) 0 0
\(622\) 10790.1 0.695569
\(623\) 180.443 0.0116040
\(624\) 0 0
\(625\) −10241.4 −0.655448
\(626\) −21039.6 −1.34331
\(627\) 0 0
\(628\) −13871.6 −0.881427
\(629\) −1467.79 −0.0930436
\(630\) 0 0
\(631\) 2374.38 0.149798 0.0748989 0.997191i \(-0.476137\pi\)
0.0748989 + 0.997191i \(0.476137\pi\)
\(632\) −75377.1 −4.74421
\(633\) 0 0
\(634\) −15805.3 −0.990080
\(635\) −11142.2 −0.696324
\(636\) 0 0
\(637\) 7050.74 0.438557
\(638\) 0 0
\(639\) 0 0
\(640\) 64064.6 3.95684
\(641\) −11086.0 −0.683104 −0.341552 0.939863i \(-0.610953\pi\)
−0.341552 + 0.939863i \(0.610953\pi\)
\(642\) 0 0
\(643\) 19934.1 1.22259 0.611294 0.791403i \(-0.290649\pi\)
0.611294 + 0.791403i \(0.290649\pi\)
\(644\) −2928.26 −0.179176
\(645\) 0 0
\(646\) 1310.21 0.0797983
\(647\) 30634.8 1.86148 0.930739 0.365684i \(-0.119165\pi\)
0.930739 + 0.365684i \(0.119165\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −21416.1 −1.29232
\(651\) 0 0
\(652\) 23421.9 1.40686
\(653\) 9818.07 0.588378 0.294189 0.955747i \(-0.404951\pi\)
0.294189 + 0.955747i \(0.404951\pi\)
\(654\) 0 0
\(655\) 10483.1 0.625358
\(656\) −102848. −6.12125
\(657\) 0 0
\(658\) −20769.6 −1.23052
\(659\) 16478.5 0.974070 0.487035 0.873383i \(-0.338078\pi\)
0.487035 + 0.873383i \(0.338078\pi\)
\(660\) 0 0
\(661\) 2958.12 0.174066 0.0870328 0.996205i \(-0.472262\pi\)
0.0870328 + 0.996205i \(0.472262\pi\)
\(662\) 43962.1 2.58102
\(663\) 0 0
\(664\) 44144.2 2.58001
\(665\) −1965.07 −0.114590
\(666\) 0 0
\(667\) 2282.44 0.132498
\(668\) 24065.1 1.39387
\(669\) 0 0
\(670\) −78668.2 −4.53614
\(671\) 0 0
\(672\) 0 0
\(673\) 29960.3 1.71602 0.858012 0.513630i \(-0.171700\pi\)
0.858012 + 0.513630i \(0.171700\pi\)
\(674\) −32108.2 −1.83496
\(675\) 0 0
\(676\) −33840.6 −1.92539
\(677\) −4514.73 −0.256300 −0.128150 0.991755i \(-0.540904\pi\)
−0.128150 + 0.991755i \(0.540904\pi\)
\(678\) 0 0
\(679\) 5538.43 0.313027
\(680\) −19559.7 −1.10306
\(681\) 0 0
\(682\) 0 0
\(683\) 13555.7 0.759438 0.379719 0.925102i \(-0.376021\pi\)
0.379719 + 0.925102i \(0.376021\pi\)
\(684\) 0 0
\(685\) 31555.2 1.76009
\(686\) −26170.2 −1.45653
\(687\) 0 0
\(688\) −14526.8 −0.804986
\(689\) −13093.3 −0.723972
\(690\) 0 0
\(691\) −11471.3 −0.631535 −0.315768 0.948837i \(-0.602262\pi\)
−0.315768 + 0.948837i \(0.602262\pi\)
\(692\) 985.787 0.0541532
\(693\) 0 0
\(694\) −46327.8 −2.53398
\(695\) 16074.8 0.877340
\(696\) 0 0
\(697\) 7332.40 0.398471
\(698\) −5084.52 −0.275719
\(699\) 0 0
\(700\) 26203.0 1.41483
\(701\) 22229.0 1.19769 0.598843 0.800866i \(-0.295627\pi\)
0.598843 + 0.800866i \(0.295627\pi\)
\(702\) 0 0
\(703\) −1394.97 −0.0748397
\(704\) 0 0
\(705\) 0 0
\(706\) 1145.19 0.0610480
\(707\) 14404.5 0.766248
\(708\) 0 0
\(709\) −15081.2 −0.798851 −0.399426 0.916766i \(-0.630790\pi\)
−0.399426 + 0.916766i \(0.630790\pi\)
\(710\) −53093.4 −2.80642
\(711\) 0 0
\(712\) 1706.98 0.0898480
\(713\) −3895.52 −0.204612
\(714\) 0 0
\(715\) 0 0
\(716\) 17824.6 0.930358
\(717\) 0 0
\(718\) −7465.71 −0.388047
\(719\) 7399.80 0.383819 0.191910 0.981413i \(-0.438532\pi\)
0.191910 + 0.981413i \(0.438532\pi\)
\(720\) 0 0
\(721\) −3300.36 −0.170474
\(722\) −35960.9 −1.85364
\(723\) 0 0
\(724\) −38791.7 −1.99127
\(725\) −20424.0 −1.04625
\(726\) 0 0
\(727\) 1705.77 0.0870202 0.0435101 0.999053i \(-0.486146\pi\)
0.0435101 + 0.999053i \(0.486146\pi\)
\(728\) −13928.9 −0.709122
\(729\) 0 0
\(730\) −46622.3 −2.36379
\(731\) 1035.67 0.0524017
\(732\) 0 0
\(733\) −37122.6 −1.87061 −0.935303 0.353847i \(-0.884873\pi\)
−0.935303 + 0.353847i \(0.884873\pi\)
\(734\) 5589.60 0.281084
\(735\) 0 0
\(736\) −11193.2 −0.560581
\(737\) 0 0
\(738\) 0 0
\(739\) −34256.3 −1.70520 −0.852598 0.522568i \(-0.824974\pi\)
−0.852598 + 0.522568i \(0.824974\pi\)
\(740\) 33235.3 1.65102
\(741\) 0 0
\(742\) 22001.9 1.08856
\(743\) 1567.88 0.0774160 0.0387080 0.999251i \(-0.487676\pi\)
0.0387080 + 0.999251i \(0.487676\pi\)
\(744\) 0 0
\(745\) −34489.8 −1.69612
\(746\) −50801.9 −2.49328
\(747\) 0 0
\(748\) 0 0
\(749\) 8837.17 0.431112
\(750\) 0 0
\(751\) −955.613 −0.0464325 −0.0232163 0.999730i \(-0.507391\pi\)
−0.0232163 + 0.999730i \(0.507391\pi\)
\(752\) −111226. −5.39360
\(753\) 0 0
\(754\) 17326.9 0.836881
\(755\) −8017.75 −0.386484
\(756\) 0 0
\(757\) −14015.4 −0.672918 −0.336459 0.941698i \(-0.609229\pi\)
−0.336459 + 0.941698i \(0.609229\pi\)
\(758\) −38623.2 −1.85073
\(759\) 0 0
\(760\) −18589.4 −0.887249
\(761\) −36271.0 −1.72776 −0.863879 0.503699i \(-0.831972\pi\)
−0.863879 + 0.503699i \(0.831972\pi\)
\(762\) 0 0
\(763\) 14073.9 0.667770
\(764\) −9831.63 −0.465571
\(765\) 0 0
\(766\) 6311.64 0.297714
\(767\) −14373.6 −0.676665
\(768\) 0 0
\(769\) 18163.6 0.851749 0.425874 0.904782i \(-0.359967\pi\)
0.425874 + 0.904782i \(0.359967\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −38112.1 −1.77680
\(773\) 8345.65 0.388321 0.194160 0.980970i \(-0.437802\pi\)
0.194160 + 0.980970i \(0.437802\pi\)
\(774\) 0 0
\(775\) 34858.4 1.61568
\(776\) 52393.2 2.42372
\(777\) 0 0
\(778\) 59445.7 2.73937
\(779\) 6968.65 0.320510
\(780\) 0 0
\(781\) 0 0
\(782\) 1535.45 0.0702142
\(783\) 0 0
\(784\) −63448.5 −2.89033
\(785\) −10909.1 −0.496001
\(786\) 0 0
\(787\) 22996.2 1.04158 0.520791 0.853684i \(-0.325637\pi\)
0.520791 + 0.853684i \(0.325637\pi\)
\(788\) 113647. 5.13768
\(789\) 0 0
\(790\) −94605.0 −4.26063
\(791\) −8670.69 −0.389752
\(792\) 0 0
\(793\) −5505.75 −0.246551
\(794\) −11782.4 −0.526626
\(795\) 0 0
\(796\) −110087. −4.90194
\(797\) 2743.82 0.121946 0.0609730 0.998139i \(-0.480580\pi\)
0.0609730 + 0.998139i \(0.480580\pi\)
\(798\) 0 0
\(799\) 7929.68 0.351104
\(800\) 100161. 4.42652
\(801\) 0 0
\(802\) −42477.1 −1.87022
\(803\) 0 0
\(804\) 0 0
\(805\) −2302.88 −0.100827
\(806\) −29572.5 −1.29237
\(807\) 0 0
\(808\) 136266. 5.93293
\(809\) 41241.7 1.79231 0.896156 0.443738i \(-0.146348\pi\)
0.896156 + 0.443738i \(0.146348\pi\)
\(810\) 0 0
\(811\) −12832.9 −0.555641 −0.277820 0.960633i \(-0.589612\pi\)
−0.277820 + 0.960633i \(0.589612\pi\)
\(812\) −21199.8 −0.916215
\(813\) 0 0
\(814\) 0 0
\(815\) 18419.7 0.791675
\(816\) 0 0
\(817\) 984.292 0.0421493
\(818\) 58210.4 2.48812
\(819\) 0 0
\(820\) −166029. −7.07069
\(821\) 16368.5 0.695817 0.347908 0.937529i \(-0.386892\pi\)
0.347908 + 0.937529i \(0.386892\pi\)
\(822\) 0 0
\(823\) 3869.53 0.163892 0.0819461 0.996637i \(-0.473886\pi\)
0.0819461 + 0.996637i \(0.473886\pi\)
\(824\) −31221.2 −1.31995
\(825\) 0 0
\(826\) 24153.3 1.01743
\(827\) 7388.69 0.310677 0.155339 0.987861i \(-0.450353\pi\)
0.155339 + 0.987861i \(0.450353\pi\)
\(828\) 0 0
\(829\) 23990.1 1.00508 0.502539 0.864554i \(-0.332399\pi\)
0.502539 + 0.864554i \(0.332399\pi\)
\(830\) 55404.9 2.31703
\(831\) 0 0
\(832\) −40520.6 −1.68846
\(833\) 4523.47 0.188150
\(834\) 0 0
\(835\) 18925.6 0.784367
\(836\) 0 0
\(837\) 0 0
\(838\) 39684.5 1.63589
\(839\) 18228.3 0.750074 0.375037 0.927010i \(-0.377630\pi\)
0.375037 + 0.927010i \(0.377630\pi\)
\(840\) 0 0
\(841\) −7864.78 −0.322472
\(842\) −67782.3 −2.77427
\(843\) 0 0
\(844\) 91318.7 3.72431
\(845\) −26613.3 −1.08346
\(846\) 0 0
\(847\) 0 0
\(848\) 117825. 4.77137
\(849\) 0 0
\(850\) −13739.7 −0.554433
\(851\) −1634.77 −0.0658511
\(852\) 0 0
\(853\) 21737.3 0.872534 0.436267 0.899817i \(-0.356300\pi\)
0.436267 + 0.899817i \(0.356300\pi\)
\(854\) 9251.79 0.370714
\(855\) 0 0
\(856\) 83599.0 3.33803
\(857\) −18712.2 −0.745852 −0.372926 0.927861i \(-0.621645\pi\)
−0.372926 + 0.927861i \(0.621645\pi\)
\(858\) 0 0
\(859\) 30527.6 1.21256 0.606279 0.795252i \(-0.292661\pi\)
0.606279 + 0.795252i \(0.292661\pi\)
\(860\) −23450.8 −0.929845
\(861\) 0 0
\(862\) 32953.4 1.30209
\(863\) −10906.4 −0.430196 −0.215098 0.976592i \(-0.569007\pi\)
−0.215098 + 0.976592i \(0.569007\pi\)
\(864\) 0 0
\(865\) 775.255 0.0304734
\(866\) 30603.5 1.20087
\(867\) 0 0
\(868\) 36182.5 1.41488
\(869\) 0 0
\(870\) 0 0
\(871\) 21388.5 0.832058
\(872\) 133138. 5.17043
\(873\) 0 0
\(874\) 1459.28 0.0564768
\(875\) 4394.68 0.169791
\(876\) 0 0
\(877\) −21770.9 −0.838256 −0.419128 0.907927i \(-0.637664\pi\)
−0.419128 + 0.907927i \(0.637664\pi\)
\(878\) −59109.8 −2.27205
\(879\) 0 0
\(880\) 0 0
\(881\) −47206.9 −1.80527 −0.902634 0.430409i \(-0.858369\pi\)
−0.902634 + 0.430409i \(0.858369\pi\)
\(882\) 0 0
\(883\) −6059.68 −0.230945 −0.115473 0.993311i \(-0.536838\pi\)
−0.115473 + 0.993311i \(0.536838\pi\)
\(884\) 8487.09 0.322909
\(885\) 0 0
\(886\) 41881.1 1.58806
\(887\) −37130.2 −1.40553 −0.702767 0.711420i \(-0.748052\pi\)
−0.702767 + 0.711420i \(0.748052\pi\)
\(888\) 0 0
\(889\) −5090.53 −0.192048
\(890\) 2142.41 0.0806896
\(891\) 0 0
\(892\) −29518.5 −1.10802
\(893\) 7536.30 0.282410
\(894\) 0 0
\(895\) 14017.8 0.523536
\(896\) 29269.1 1.09131
\(897\) 0 0
\(898\) 40541.4 1.50655
\(899\) −28202.5 −1.04628
\(900\) 0 0
\(901\) −8400.15 −0.310599
\(902\) 0 0
\(903\) 0 0
\(904\) −82024.1 −3.01779
\(905\) −30507.0 −1.12054
\(906\) 0 0
\(907\) 1182.94 0.0433064 0.0216532 0.999766i \(-0.493107\pi\)
0.0216532 + 0.999766i \(0.493107\pi\)
\(908\) −26293.8 −0.961001
\(909\) 0 0
\(910\) −17482.1 −0.636841
\(911\) −37676.4 −1.37022 −0.685112 0.728438i \(-0.740247\pi\)
−0.685112 + 0.728438i \(0.740247\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −60431.1 −2.18696
\(915\) 0 0
\(916\) 83347.1 3.00640
\(917\) 4789.41 0.172476
\(918\) 0 0
\(919\) −8697.82 −0.312203 −0.156101 0.987741i \(-0.549893\pi\)
−0.156101 + 0.987741i \(0.549893\pi\)
\(920\) −21785.0 −0.780686
\(921\) 0 0
\(922\) 77725.7 2.77631
\(923\) 14435.2 0.514778
\(924\) 0 0
\(925\) 14628.5 0.519981
\(926\) 63795.3 2.26398
\(927\) 0 0
\(928\) −81036.0 −2.86653
\(929\) 17247.5 0.609119 0.304559 0.952493i \(-0.401491\pi\)
0.304559 + 0.952493i \(0.401491\pi\)
\(930\) 0 0
\(931\) 4299.06 0.151338
\(932\) −68946.7 −2.42320
\(933\) 0 0
\(934\) −64306.0 −2.25284
\(935\) 0 0
\(936\) 0 0
\(937\) 41812.4 1.45779 0.728896 0.684624i \(-0.240034\pi\)
0.728896 + 0.684624i \(0.240034\pi\)
\(938\) −35941.0 −1.25108
\(939\) 0 0
\(940\) −179553. −6.23018
\(941\) 37655.9 1.30451 0.652257 0.757998i \(-0.273822\pi\)
0.652257 + 0.757998i \(0.273822\pi\)
\(942\) 0 0
\(943\) 8166.60 0.282016
\(944\) 129346. 4.45959
\(945\) 0 0
\(946\) 0 0
\(947\) 21244.4 0.728986 0.364493 0.931206i \(-0.381242\pi\)
0.364493 + 0.931206i \(0.381242\pi\)
\(948\) 0 0
\(949\) 12675.8 0.433587
\(950\) −13058.1 −0.445958
\(951\) 0 0
\(952\) −8936.24 −0.304228
\(953\) 1324.27 0.0450130 0.0225065 0.999747i \(-0.492835\pi\)
0.0225065 + 0.999747i \(0.492835\pi\)
\(954\) 0 0
\(955\) −7731.91 −0.261988
\(956\) −9171.77 −0.310289
\(957\) 0 0
\(958\) −7185.62 −0.242335
\(959\) 14416.6 0.485439
\(960\) 0 0
\(961\) 18343.4 0.615735
\(962\) −12410.2 −0.415927
\(963\) 0 0
\(964\) −26376.9 −0.881268
\(965\) −29972.6 −0.999847
\(966\) 0 0
\(967\) −52267.1 −1.73815 −0.869077 0.494676i \(-0.835287\pi\)
−0.869077 + 0.494676i \(0.835287\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 65758.1 2.17667
\(971\) 52489.8 1.73479 0.867394 0.497622i \(-0.165793\pi\)
0.867394 + 0.497622i \(0.165793\pi\)
\(972\) 0 0
\(973\) 7344.07 0.241973
\(974\) 101092. 3.32566
\(975\) 0 0
\(976\) 49545.3 1.62490
\(977\) −8324.11 −0.272581 −0.136291 0.990669i \(-0.543518\pi\)
−0.136291 + 0.990669i \(0.543518\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −102426. −3.33864
\(981\) 0 0
\(982\) 675.623 0.0219552
\(983\) 44407.1 1.44086 0.720431 0.693527i \(-0.243944\pi\)
0.720431 + 0.693527i \(0.243944\pi\)
\(984\) 0 0
\(985\) 89375.3 2.89110
\(986\) 11116.2 0.359039
\(987\) 0 0
\(988\) 8066.05 0.259732
\(989\) 1153.50 0.0370870
\(990\) 0 0
\(991\) −45124.7 −1.44645 −0.723226 0.690612i \(-0.757341\pi\)
−0.723226 + 0.690612i \(0.757341\pi\)
\(992\) 138307. 4.42667
\(993\) 0 0
\(994\) −24256.7 −0.774020
\(995\) −86576.2 −2.75844
\(996\) 0 0
\(997\) −5480.61 −0.174095 −0.0870474 0.996204i \(-0.527743\pi\)
−0.0870474 + 0.996204i \(0.527743\pi\)
\(998\) 55492.9 1.76012
\(999\) 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 1089.4.a.u.1.2 2
3.2 odd 2 363.4.a.i.1.1 2
11.10 odd 2 99.4.a.f.1.1 2
33.32 even 2 33.4.a.c.1.2 2
44.43 even 2 1584.4.a.bj.1.2 2
55.54 odd 2 2475.4.a.p.1.2 2
132.131 odd 2 528.4.a.p.1.1 2
165.32 odd 4 825.4.c.h.199.4 4
165.98 odd 4 825.4.c.h.199.1 4
165.164 even 2 825.4.a.l.1.1 2
231.230 odd 2 1617.4.a.k.1.2 2
264.131 odd 2 2112.4.a.bg.1.2 2
264.197 even 2 2112.4.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.2 2 33.32 even 2
99.4.a.f.1.1 2 11.10 odd 2
363.4.a.i.1.1 2 3.2 odd 2
528.4.a.p.1.1 2 132.131 odd 2
825.4.a.l.1.1 2 165.164 even 2
825.4.c.h.199.1 4 165.98 odd 4
825.4.c.h.199.4 4 165.32 odd 4
1089.4.a.u.1.2 2 1.1 even 1 trivial
1584.4.a.bj.1.2 2 44.43 even 2
1617.4.a.k.1.2 2 231.230 odd 2
2112.4.a.bg.1.2 2 264.131 odd 2
2112.4.a.bn.1.2 2 264.197 even 2
2475.4.a.p.1.2 2 55.54 odd 2