Properties

Label 2352.4.a.cj.1.1
Level $2352$
Weight $4$
Character 2352.1
Self dual yes
Analytic conductor $138.772$
Analytic rank $0$
Dimension $3$
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] = [2352,4,Mod(1,2352)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2352, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2352.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2352 = 2^{4} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 2352.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: \(138.772492334\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.58461.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 65x - 126 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 168)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.16736\) of defining polynomial
Character \(\chi\) \(=\) 2352.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -10.1566 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -10.1566 q^{5} +9.00000 q^{9} +37.8005 q^{11} +74.0881 q^{13} -30.4699 q^{15} +44.6694 q^{17} +139.367 q^{19} -57.8200 q^{23} -21.8428 q^{25} +27.0000 q^{27} +39.5039 q^{29} -254.837 q^{31} +113.401 q^{33} +18.1048 q^{37} +222.264 q^{39} +420.819 q^{41} +382.346 q^{43} -91.4097 q^{45} -484.901 q^{47} +134.008 q^{51} +36.0875 q^{53} -383.925 q^{55} +418.102 q^{57} -145.831 q^{59} -78.1359 q^{61} -752.486 q^{65} +273.053 q^{67} -173.460 q^{69} -717.642 q^{71} -1108.28 q^{73} -65.5283 q^{75} +1085.66 q^{79} +81.0000 q^{81} +404.777 q^{83} -453.691 q^{85} +118.512 q^{87} -1354.42 q^{89} -764.510 q^{93} -1415.50 q^{95} +681.847 q^{97} +340.204 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 9 q^{3} + 11 q^{5} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 9 q^{3} + 11 q^{5} + 27 q^{9} + 19 q^{11} + 22 q^{13} + 33 q^{15} + 104 q^{17} + 202 q^{19} + 280 q^{23} + 186 q^{25} + 81 q^{27} - 73 q^{29} - 131 q^{31} + 57 q^{33} - 326 q^{37} + 66 q^{39} + 516 q^{41} - 36 q^{43} + 99 q^{45} - 126 q^{47} + 312 q^{51} - 385 q^{53} - 611 q^{55} + 606 q^{57} + 285 q^{59} + 34 q^{61} - 920 q^{65} + 100 q^{67} + 840 q^{69} - 34 q^{71} + 108 q^{73} + 558 q^{75} + 2463 q^{79} + 243 q^{81} - 115 q^{83} - 500 q^{85} - 219 q^{87} + 110 q^{89} - 393 q^{93} + 2064 q^{95} + 2941 q^{97} + 171 q^{99}+O(q^{100}) \) 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
\(3\) 3.00000 0.577350
\(4\) 0 0
\(5\) −10.1566 −0.908437 −0.454218 0.890890i \(-0.650081\pi\)
−0.454218 + 0.890890i \(0.650081\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 37.8005 1.03612 0.518058 0.855346i \(-0.326655\pi\)
0.518058 + 0.855346i \(0.326655\pi\)
\(12\) 0 0
\(13\) 74.0881 1.58064 0.790321 0.612693i \(-0.209914\pi\)
0.790321 + 0.612693i \(0.209914\pi\)
\(14\) 0 0
\(15\) −30.4699 −0.524486
\(16\) 0 0
\(17\) 44.6694 0.637290 0.318645 0.947874i \(-0.396772\pi\)
0.318645 + 0.947874i \(0.396772\pi\)
\(18\) 0 0
\(19\) 139.367 1.68279 0.841397 0.540418i \(-0.181734\pi\)
0.841397 + 0.540418i \(0.181734\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −57.8200 −0.524187 −0.262094 0.965042i \(-0.584413\pi\)
−0.262094 + 0.965042i \(0.584413\pi\)
\(24\) 0 0
\(25\) −21.8428 −0.174742
\(26\) 0 0
\(27\) 27.0000 0.192450
\(28\) 0 0
\(29\) 39.5039 0.252955 0.126477 0.991969i \(-0.459633\pi\)
0.126477 + 0.991969i \(0.459633\pi\)
\(30\) 0 0
\(31\) −254.837 −1.47645 −0.738226 0.674553i \(-0.764336\pi\)
−0.738226 + 0.674553i \(0.764336\pi\)
\(32\) 0 0
\(33\) 113.401 0.598201
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 18.1048 0.0804435 0.0402217 0.999191i \(-0.487194\pi\)
0.0402217 + 0.999191i \(0.487194\pi\)
\(38\) 0 0
\(39\) 222.264 0.912584
\(40\) 0 0
\(41\) 420.819 1.60295 0.801475 0.598028i \(-0.204049\pi\)
0.801475 + 0.598028i \(0.204049\pi\)
\(42\) 0 0
\(43\) 382.346 1.35598 0.677991 0.735070i \(-0.262851\pi\)
0.677991 + 0.735070i \(0.262851\pi\)
\(44\) 0 0
\(45\) −91.4097 −0.302812
\(46\) 0 0
\(47\) −484.901 −1.50490 −0.752448 0.658651i \(-0.771127\pi\)
−0.752448 + 0.658651i \(0.771127\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 134.008 0.367940
\(52\) 0 0
\(53\) 36.0875 0.0935283 0.0467642 0.998906i \(-0.485109\pi\)
0.0467642 + 0.998906i \(0.485109\pi\)
\(54\) 0 0
\(55\) −383.925 −0.941245
\(56\) 0 0
\(57\) 418.102 0.971561
\(58\) 0 0
\(59\) −145.831 −0.321790 −0.160895 0.986972i \(-0.551438\pi\)
−0.160895 + 0.986972i \(0.551438\pi\)
\(60\) 0 0
\(61\) −78.1359 −0.164005 −0.0820023 0.996632i \(-0.526131\pi\)
−0.0820023 + 0.996632i \(0.526131\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −752.486 −1.43591
\(66\) 0 0
\(67\) 273.053 0.497891 0.248946 0.968517i \(-0.419916\pi\)
0.248946 + 0.968517i \(0.419916\pi\)
\(68\) 0 0
\(69\) −173.460 −0.302640
\(70\) 0 0
\(71\) −717.642 −1.19956 −0.599778 0.800167i \(-0.704744\pi\)
−0.599778 + 0.800167i \(0.704744\pi\)
\(72\) 0 0
\(73\) −1108.28 −1.77691 −0.888453 0.458968i \(-0.848219\pi\)
−0.888453 + 0.458968i \(0.848219\pi\)
\(74\) 0 0
\(75\) −65.5283 −0.100887
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1085.66 1.54615 0.773076 0.634314i \(-0.218717\pi\)
0.773076 + 0.634314i \(0.218717\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 0 0
\(83\) 404.777 0.535301 0.267651 0.963516i \(-0.413753\pi\)
0.267651 + 0.963516i \(0.413753\pi\)
\(84\) 0 0
\(85\) −453.691 −0.578938
\(86\) 0 0
\(87\) 118.512 0.146043
\(88\) 0 0
\(89\) −1354.42 −1.61312 −0.806561 0.591151i \(-0.798674\pi\)
−0.806561 + 0.591151i \(0.798674\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −764.510 −0.852430
\(94\) 0 0
\(95\) −1415.50 −1.52871
\(96\) 0 0
\(97\) 681.847 0.713723 0.356861 0.934157i \(-0.383847\pi\)
0.356861 + 0.934157i \(0.383847\pi\)
\(98\) 0 0
\(99\) 340.204 0.345372
\(100\) 0 0
\(101\) 798.533 0.786703 0.393352 0.919388i \(-0.371316\pi\)
0.393352 + 0.919388i \(0.371316\pi\)
\(102\) 0 0
\(103\) 453.726 0.434048 0.217024 0.976166i \(-0.430365\pi\)
0.217024 + 0.976166i \(0.430365\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 911.741 0.823751 0.411875 0.911240i \(-0.364874\pi\)
0.411875 + 0.911240i \(0.364874\pi\)
\(108\) 0 0
\(109\) 22.8441 0.0200740 0.0100370 0.999950i \(-0.496805\pi\)
0.0100370 + 0.999950i \(0.496805\pi\)
\(110\) 0 0
\(111\) 54.3144 0.0464441
\(112\) 0 0
\(113\) 462.542 0.385065 0.192532 0.981291i \(-0.438330\pi\)
0.192532 + 0.981291i \(0.438330\pi\)
\(114\) 0 0
\(115\) 587.257 0.476191
\(116\) 0 0
\(117\) 666.793 0.526880
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 97.8742 0.0735344
\(122\) 0 0
\(123\) 1262.46 0.925464
\(124\) 0 0
\(125\) 1491.43 1.06718
\(126\) 0 0
\(127\) 1792.58 1.25249 0.626244 0.779627i \(-0.284592\pi\)
0.626244 + 0.779627i \(0.284592\pi\)
\(128\) 0 0
\(129\) 1147.04 0.782877
\(130\) 0 0
\(131\) 580.446 0.387128 0.193564 0.981088i \(-0.437995\pi\)
0.193564 + 0.981088i \(0.437995\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) −274.229 −0.174829
\(136\) 0 0
\(137\) −2850.82 −1.77783 −0.888913 0.458077i \(-0.848538\pi\)
−0.888913 + 0.458077i \(0.848538\pi\)
\(138\) 0 0
\(139\) −187.777 −0.114583 −0.0572915 0.998357i \(-0.518246\pi\)
−0.0572915 + 0.998357i \(0.518246\pi\)
\(140\) 0 0
\(141\) −1454.70 −0.868852
\(142\) 0 0
\(143\) 2800.56 1.63773
\(144\) 0 0
\(145\) −401.226 −0.229793
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 497.684 0.273637 0.136818 0.990596i \(-0.456312\pi\)
0.136818 + 0.990596i \(0.456312\pi\)
\(150\) 0 0
\(151\) 968.646 0.522035 0.261018 0.965334i \(-0.415942\pi\)
0.261018 + 0.965334i \(0.415942\pi\)
\(152\) 0 0
\(153\) 402.025 0.212430
\(154\) 0 0
\(155\) 2588.28 1.34126
\(156\) 0 0
\(157\) −176.120 −0.0895281 −0.0447640 0.998998i \(-0.514254\pi\)
−0.0447640 + 0.998998i \(0.514254\pi\)
\(158\) 0 0
\(159\) 108.263 0.0539986
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 18.7766 0.00902267 0.00451133 0.999990i \(-0.498564\pi\)
0.00451133 + 0.999990i \(0.498564\pi\)
\(164\) 0 0
\(165\) −1151.78 −0.543428
\(166\) 0 0
\(167\) −1325.80 −0.614334 −0.307167 0.951656i \(-0.599381\pi\)
−0.307167 + 0.951656i \(0.599381\pi\)
\(168\) 0 0
\(169\) 3292.05 1.49843
\(170\) 0 0
\(171\) 1254.31 0.560931
\(172\) 0 0
\(173\) −3701.94 −1.62690 −0.813449 0.581636i \(-0.802413\pi\)
−0.813449 + 0.581636i \(0.802413\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −437.494 −0.185785
\(178\) 0 0
\(179\) 4363.41 1.82199 0.910996 0.412416i \(-0.135315\pi\)
0.910996 + 0.412416i \(0.135315\pi\)
\(180\) 0 0
\(181\) −130.885 −0.0537494 −0.0268747 0.999639i \(-0.508556\pi\)
−0.0268747 + 0.999639i \(0.508556\pi\)
\(182\) 0 0
\(183\) −234.408 −0.0946881
\(184\) 0 0
\(185\) −183.884 −0.0730778
\(186\) 0 0
\(187\) 1688.53 0.660306
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4592.05 1.73963 0.869814 0.493379i \(-0.164239\pi\)
0.869814 + 0.493379i \(0.164239\pi\)
\(192\) 0 0
\(193\) 843.489 0.314589 0.157294 0.987552i \(-0.449723\pi\)
0.157294 + 0.987552i \(0.449723\pi\)
\(194\) 0 0
\(195\) −2257.46 −0.829025
\(196\) 0 0
\(197\) 1881.79 0.680570 0.340285 0.940322i \(-0.389476\pi\)
0.340285 + 0.940322i \(0.389476\pi\)
\(198\) 0 0
\(199\) 1546.11 0.550759 0.275380 0.961336i \(-0.411196\pi\)
0.275380 + 0.961336i \(0.411196\pi\)
\(200\) 0 0
\(201\) 819.159 0.287458
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −4274.11 −1.45618
\(206\) 0 0
\(207\) −520.380 −0.174729
\(208\) 0 0
\(209\) 5268.15 1.74357
\(210\) 0 0
\(211\) 1520.42 0.496068 0.248034 0.968751i \(-0.420216\pi\)
0.248034 + 0.968751i \(0.420216\pi\)
\(212\) 0 0
\(213\) −2152.93 −0.692564
\(214\) 0 0
\(215\) −3883.35 −1.23182
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −3324.83 −1.02590
\(220\) 0 0
\(221\) 3309.47 1.00733
\(222\) 0 0
\(223\) 4735.42 1.42201 0.711003 0.703189i \(-0.248241\pi\)
0.711003 + 0.703189i \(0.248241\pi\)
\(224\) 0 0
\(225\) −196.585 −0.0582474
\(226\) 0 0
\(227\) 2285.64 0.668296 0.334148 0.942521i \(-0.391551\pi\)
0.334148 + 0.942521i \(0.391551\pi\)
\(228\) 0 0
\(229\) 479.209 0.138284 0.0691420 0.997607i \(-0.477974\pi\)
0.0691420 + 0.997607i \(0.477974\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 5152.71 1.44878 0.724389 0.689392i \(-0.242122\pi\)
0.724389 + 0.689392i \(0.242122\pi\)
\(234\) 0 0
\(235\) 4924.97 1.36710
\(236\) 0 0
\(237\) 3256.97 0.892671
\(238\) 0 0
\(239\) −2378.94 −0.643852 −0.321926 0.946765i \(-0.604330\pi\)
−0.321926 + 0.946765i \(0.604330\pi\)
\(240\) 0 0
\(241\) −22.2202 −0.00593914 −0.00296957 0.999996i \(-0.500945\pi\)
−0.00296957 + 0.999996i \(0.500945\pi\)
\(242\) 0 0
\(243\) 243.000 0.0641500
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 10325.5 2.65989
\(248\) 0 0
\(249\) 1214.33 0.309056
\(250\) 0 0
\(251\) −4845.08 −1.21840 −0.609201 0.793016i \(-0.708510\pi\)
−0.609201 + 0.793016i \(0.708510\pi\)
\(252\) 0 0
\(253\) −2185.62 −0.543118
\(254\) 0 0
\(255\) −1361.07 −0.334250
\(256\) 0 0
\(257\) 4714.05 1.14418 0.572090 0.820191i \(-0.306133\pi\)
0.572090 + 0.820191i \(0.306133\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 355.535 0.0843182
\(262\) 0 0
\(263\) 3481.06 0.816165 0.408082 0.912945i \(-0.366198\pi\)
0.408082 + 0.912945i \(0.366198\pi\)
\(264\) 0 0
\(265\) −366.528 −0.0849646
\(266\) 0 0
\(267\) −4063.25 −0.931336
\(268\) 0 0
\(269\) 297.092 0.0673384 0.0336692 0.999433i \(-0.489281\pi\)
0.0336692 + 0.999433i \(0.489281\pi\)
\(270\) 0 0
\(271\) −812.504 −0.182126 −0.0910629 0.995845i \(-0.529026\pi\)
−0.0910629 + 0.995845i \(0.529026\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −825.667 −0.181053
\(276\) 0 0
\(277\) 7209.01 1.56371 0.781855 0.623460i \(-0.214274\pi\)
0.781855 + 0.623460i \(0.214274\pi\)
\(278\) 0 0
\(279\) −2293.53 −0.492151
\(280\) 0 0
\(281\) −7861.53 −1.66897 −0.834483 0.551034i \(-0.814233\pi\)
−0.834483 + 0.551034i \(0.814233\pi\)
\(282\) 0 0
\(283\) −2051.70 −0.430957 −0.215479 0.976509i \(-0.569131\pi\)
−0.215479 + 0.976509i \(0.569131\pi\)
\(284\) 0 0
\(285\) −4246.51 −0.882602
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2917.64 −0.593861
\(290\) 0 0
\(291\) 2045.54 0.412068
\(292\) 0 0
\(293\) −6247.59 −1.24569 −0.622847 0.782344i \(-0.714024\pi\)
−0.622847 + 0.782344i \(0.714024\pi\)
\(294\) 0 0
\(295\) 1481.15 0.292326
\(296\) 0 0
\(297\) 1020.61 0.199400
\(298\) 0 0
\(299\) −4283.77 −0.828552
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 2395.60 0.454203
\(304\) 0 0
\(305\) 793.598 0.148988
\(306\) 0 0
\(307\) 9872.98 1.83544 0.917721 0.397225i \(-0.130027\pi\)
0.917721 + 0.397225i \(0.130027\pi\)
\(308\) 0 0
\(309\) 1361.18 0.250598
\(310\) 0 0
\(311\) −1584.47 −0.288898 −0.144449 0.989512i \(-0.546141\pi\)
−0.144449 + 0.989512i \(0.546141\pi\)
\(312\) 0 0
\(313\) −7403.93 −1.33704 −0.668522 0.743693i \(-0.733073\pi\)
−0.668522 + 0.743693i \(0.733073\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −10223.8 −1.81143 −0.905716 0.423884i \(-0.860666\pi\)
−0.905716 + 0.423884i \(0.860666\pi\)
\(318\) 0 0
\(319\) 1493.26 0.262090
\(320\) 0 0
\(321\) 2735.22 0.475593
\(322\) 0 0
\(323\) 6225.46 1.07243
\(324\) 0 0
\(325\) −1618.29 −0.276205
\(326\) 0 0
\(327\) 68.5322 0.0115897
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −6847.01 −1.13700 −0.568498 0.822684i \(-0.692475\pi\)
−0.568498 + 0.822684i \(0.692475\pi\)
\(332\) 0 0
\(333\) 162.943 0.0268145
\(334\) 0 0
\(335\) −2773.30 −0.452303
\(336\) 0 0
\(337\) 6735.78 1.08879 0.544394 0.838830i \(-0.316760\pi\)
0.544394 + 0.838830i \(0.316760\pi\)
\(338\) 0 0
\(339\) 1387.63 0.222317
\(340\) 0 0
\(341\) −9632.94 −1.52977
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 1761.77 0.274929
\(346\) 0 0
\(347\) 7776.58 1.20308 0.601540 0.798843i \(-0.294554\pi\)
0.601540 + 0.798843i \(0.294554\pi\)
\(348\) 0 0
\(349\) 3043.15 0.466750 0.233375 0.972387i \(-0.425023\pi\)
0.233375 + 0.972387i \(0.425023\pi\)
\(350\) 0 0
\(351\) 2000.38 0.304195
\(352\) 0 0
\(353\) 2410.49 0.363449 0.181724 0.983350i \(-0.441832\pi\)
0.181724 + 0.983350i \(0.441832\pi\)
\(354\) 0 0
\(355\) 7288.83 1.08972
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −11356.6 −1.66957 −0.834786 0.550574i \(-0.814409\pi\)
−0.834786 + 0.550574i \(0.814409\pi\)
\(360\) 0 0
\(361\) 12564.3 1.83179
\(362\) 0 0
\(363\) 293.623 0.0424551
\(364\) 0 0
\(365\) 11256.4 1.61421
\(366\) 0 0
\(367\) 1860.34 0.264602 0.132301 0.991210i \(-0.457763\pi\)
0.132301 + 0.991210i \(0.457763\pi\)
\(368\) 0 0
\(369\) 3787.37 0.534317
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 3507.96 0.486957 0.243479 0.969906i \(-0.421711\pi\)
0.243479 + 0.969906i \(0.421711\pi\)
\(374\) 0 0
\(375\) 4474.29 0.616136
\(376\) 0 0
\(377\) 2926.77 0.399831
\(378\) 0 0
\(379\) 766.351 0.103865 0.0519324 0.998651i \(-0.483462\pi\)
0.0519324 + 0.998651i \(0.483462\pi\)
\(380\) 0 0
\(381\) 5377.74 0.723124
\(382\) 0 0
\(383\) −13914.7 −1.85642 −0.928212 0.372053i \(-0.878654\pi\)
−0.928212 + 0.372053i \(0.878654\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 3441.12 0.451994
\(388\) 0 0
\(389\) −4598.31 −0.599340 −0.299670 0.954043i \(-0.596877\pi\)
−0.299670 + 0.954043i \(0.596877\pi\)
\(390\) 0 0
\(391\) −2582.79 −0.334059
\(392\) 0 0
\(393\) 1741.34 0.223509
\(394\) 0 0
\(395\) −11026.6 −1.40458
\(396\) 0 0
\(397\) −12735.2 −1.60998 −0.804991 0.593287i \(-0.797830\pi\)
−0.804991 + 0.593287i \(0.797830\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1802.56 0.224477 0.112239 0.993681i \(-0.464198\pi\)
0.112239 + 0.993681i \(0.464198\pi\)
\(402\) 0 0
\(403\) −18880.4 −2.33374
\(404\) 0 0
\(405\) −822.687 −0.100937
\(406\) 0 0
\(407\) 684.369 0.0833487
\(408\) 0 0
\(409\) 8336.30 1.00783 0.503916 0.863753i \(-0.331892\pi\)
0.503916 + 0.863753i \(0.331892\pi\)
\(410\) 0 0
\(411\) −8552.46 −1.02643
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −4111.17 −0.486288
\(416\) 0 0
\(417\) −563.331 −0.0661546
\(418\) 0 0
\(419\) 11940.4 1.39219 0.696093 0.717952i \(-0.254920\pi\)
0.696093 + 0.717952i \(0.254920\pi\)
\(420\) 0 0
\(421\) −3876.28 −0.448737 −0.224368 0.974504i \(-0.572032\pi\)
−0.224368 + 0.974504i \(0.572032\pi\)
\(422\) 0 0
\(423\) −4364.11 −0.501632
\(424\) 0 0
\(425\) −975.705 −0.111362
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 8401.69 0.945542
\(430\) 0 0
\(431\) 6252.60 0.698786 0.349393 0.936976i \(-0.386388\pi\)
0.349393 + 0.936976i \(0.386388\pi\)
\(432\) 0 0
\(433\) −12189.7 −1.35289 −0.676446 0.736493i \(-0.736481\pi\)
−0.676446 + 0.736493i \(0.736481\pi\)
\(434\) 0 0
\(435\) −1203.68 −0.132671
\(436\) 0 0
\(437\) −8058.22 −0.882098
\(438\) 0 0
\(439\) −8281.78 −0.900382 −0.450191 0.892932i \(-0.648644\pi\)
−0.450191 + 0.892932i \(0.648644\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −626.486 −0.0671902 −0.0335951 0.999436i \(-0.510696\pi\)
−0.0335951 + 0.999436i \(0.510696\pi\)
\(444\) 0 0
\(445\) 13756.3 1.46542
\(446\) 0 0
\(447\) 1493.05 0.157984
\(448\) 0 0
\(449\) 12789.8 1.34430 0.672148 0.740416i \(-0.265372\pi\)
0.672148 + 0.740416i \(0.265372\pi\)
\(450\) 0 0
\(451\) 15907.2 1.66084
\(452\) 0 0
\(453\) 2905.94 0.301397
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 10708.8 1.09614 0.548069 0.836433i \(-0.315363\pi\)
0.548069 + 0.836433i \(0.315363\pi\)
\(458\) 0 0
\(459\) 1206.08 0.122647
\(460\) 0 0
\(461\) 2549.45 0.257570 0.128785 0.991673i \(-0.458892\pi\)
0.128785 + 0.991673i \(0.458892\pi\)
\(462\) 0 0
\(463\) 13906.1 1.39584 0.697919 0.716177i \(-0.254110\pi\)
0.697919 + 0.716177i \(0.254110\pi\)
\(464\) 0 0
\(465\) 7764.85 0.774379
\(466\) 0 0
\(467\) −5827.33 −0.577423 −0.288712 0.957416i \(-0.593227\pi\)
−0.288712 + 0.957416i \(0.593227\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −528.360 −0.0516891
\(472\) 0 0
\(473\) 14452.9 1.40495
\(474\) 0 0
\(475\) −3044.17 −0.294055
\(476\) 0 0
\(477\) 324.788 0.0311761
\(478\) 0 0
\(479\) −3813.03 −0.363720 −0.181860 0.983324i \(-0.558212\pi\)
−0.181860 + 0.983324i \(0.558212\pi\)
\(480\) 0 0
\(481\) 1341.35 0.127152
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −6925.27 −0.648372
\(486\) 0 0
\(487\) 5783.76 0.538167 0.269083 0.963117i \(-0.413279\pi\)
0.269083 + 0.963117i \(0.413279\pi\)
\(488\) 0 0
\(489\) 56.3297 0.00520924
\(490\) 0 0
\(491\) 10840.9 0.996419 0.498209 0.867057i \(-0.333991\pi\)
0.498209 + 0.867057i \(0.333991\pi\)
\(492\) 0 0
\(493\) 1764.62 0.161206
\(494\) 0 0
\(495\) −3455.33 −0.313748
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12738.1 1.14276 0.571379 0.820686i \(-0.306409\pi\)
0.571379 + 0.820686i \(0.306409\pi\)
\(500\) 0 0
\(501\) −3977.41 −0.354686
\(502\) 0 0
\(503\) −5014.31 −0.444487 −0.222243 0.974991i \(-0.571338\pi\)
−0.222243 + 0.974991i \(0.571338\pi\)
\(504\) 0 0
\(505\) −8110.41 −0.714670
\(506\) 0 0
\(507\) 9876.14 0.865117
\(508\) 0 0
\(509\) 14977.5 1.30425 0.652126 0.758111i \(-0.273877\pi\)
0.652126 + 0.758111i \(0.273877\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 3762.92 0.323854
\(514\) 0 0
\(515\) −4608.33 −0.394306
\(516\) 0 0
\(517\) −18329.5 −1.55925
\(518\) 0 0
\(519\) −11105.8 −0.939290
\(520\) 0 0
\(521\) −3122.30 −0.262553 −0.131277 0.991346i \(-0.541908\pi\)
−0.131277 + 0.991346i \(0.541908\pi\)
\(522\) 0 0
\(523\) −9649.39 −0.806765 −0.403383 0.915031i \(-0.632166\pi\)
−0.403383 + 0.915031i \(0.632166\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −11383.4 −0.940929
\(528\) 0 0
\(529\) −8823.85 −0.725228
\(530\) 0 0
\(531\) −1312.48 −0.107263
\(532\) 0 0
\(533\) 31177.7 2.53369
\(534\) 0 0
\(535\) −9260.22 −0.748326
\(536\) 0 0
\(537\) 13090.2 1.05193
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 15861.6 1.26052 0.630262 0.776382i \(-0.282947\pi\)
0.630262 + 0.776382i \(0.282947\pi\)
\(542\) 0 0
\(543\) −392.656 −0.0310322
\(544\) 0 0
\(545\) −232.019 −0.0182360
\(546\) 0 0
\(547\) −7477.29 −0.584471 −0.292236 0.956346i \(-0.594399\pi\)
−0.292236 + 0.956346i \(0.594399\pi\)
\(548\) 0 0
\(549\) −703.223 −0.0546682
\(550\) 0 0
\(551\) 5505.55 0.425670
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −551.651 −0.0421915
\(556\) 0 0
\(557\) −18170.4 −1.38224 −0.691119 0.722741i \(-0.742882\pi\)
−0.691119 + 0.722741i \(0.742882\pi\)
\(558\) 0 0
\(559\) 28327.3 2.14332
\(560\) 0 0
\(561\) 5065.58 0.381228
\(562\) 0 0
\(563\) 467.613 0.0350045 0.0175023 0.999847i \(-0.494429\pi\)
0.0175023 + 0.999847i \(0.494429\pi\)
\(564\) 0 0
\(565\) −4697.87 −0.349807
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 26449.8 1.94874 0.974368 0.224958i \(-0.0722245\pi\)
0.974368 + 0.224958i \(0.0722245\pi\)
\(570\) 0 0
\(571\) −8770.96 −0.642826 −0.321413 0.946939i \(-0.604158\pi\)
−0.321413 + 0.946939i \(0.604158\pi\)
\(572\) 0 0
\(573\) 13776.1 1.00438
\(574\) 0 0
\(575\) 1262.95 0.0915976
\(576\) 0 0
\(577\) 3036.56 0.219088 0.109544 0.993982i \(-0.465061\pi\)
0.109544 + 0.993982i \(0.465061\pi\)
\(578\) 0 0
\(579\) 2530.47 0.181628
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 1364.12 0.0969061
\(584\) 0 0
\(585\) −6772.37 −0.478638
\(586\) 0 0
\(587\) 1620.33 0.113932 0.0569661 0.998376i \(-0.481857\pi\)
0.0569661 + 0.998376i \(0.481857\pi\)
\(588\) 0 0
\(589\) −35515.9 −2.48456
\(590\) 0 0
\(591\) 5645.38 0.392927
\(592\) 0 0
\(593\) −27146.4 −1.87988 −0.939940 0.341340i \(-0.889119\pi\)
−0.939940 + 0.341340i \(0.889119\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 4638.34 0.317981
\(598\) 0 0
\(599\) −19211.0 −1.31042 −0.655209 0.755448i \(-0.727420\pi\)
−0.655209 + 0.755448i \(0.727420\pi\)
\(600\) 0 0
\(601\) −24463.9 −1.66041 −0.830204 0.557460i \(-0.811776\pi\)
−0.830204 + 0.557460i \(0.811776\pi\)
\(602\) 0 0
\(603\) 2457.48 0.165964
\(604\) 0 0
\(605\) −994.073 −0.0668013
\(606\) 0 0
\(607\) 20270.4 1.35544 0.677718 0.735322i \(-0.262969\pi\)
0.677718 + 0.735322i \(0.262969\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −35925.4 −2.37870
\(612\) 0 0
\(613\) 15584.8 1.02686 0.513429 0.858132i \(-0.328375\pi\)
0.513429 + 0.858132i \(0.328375\pi\)
\(614\) 0 0
\(615\) −12822.3 −0.840725
\(616\) 0 0
\(617\) 2276.13 0.148515 0.0742573 0.997239i \(-0.476341\pi\)
0.0742573 + 0.997239i \(0.476341\pi\)
\(618\) 0 0
\(619\) 14493.2 0.941081 0.470540 0.882378i \(-0.344059\pi\)
0.470540 + 0.882378i \(0.344059\pi\)
\(620\) 0 0
\(621\) −1561.14 −0.100880
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −12417.5 −0.794723
\(626\) 0 0
\(627\) 15804.4 1.00665
\(628\) 0 0
\(629\) 808.731 0.0512658
\(630\) 0 0
\(631\) −8517.07 −0.537336 −0.268668 0.963233i \(-0.586583\pi\)
−0.268668 + 0.963233i \(0.586583\pi\)
\(632\) 0 0
\(633\) 4561.27 0.286405
\(634\) 0 0
\(635\) −18206.6 −1.13781
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −6458.78 −0.399852
\(640\) 0 0
\(641\) 21914.3 1.35033 0.675167 0.737665i \(-0.264072\pi\)
0.675167 + 0.737665i \(0.264072\pi\)
\(642\) 0 0
\(643\) 26799.5 1.64365 0.821826 0.569739i \(-0.192956\pi\)
0.821826 + 0.569739i \(0.192956\pi\)
\(644\) 0 0
\(645\) −11650.1 −0.711194
\(646\) 0 0
\(647\) −32176.4 −1.95515 −0.977577 0.210578i \(-0.932465\pi\)
−0.977577 + 0.210578i \(0.932465\pi\)
\(648\) 0 0
\(649\) −5512.49 −0.333411
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −270.878 −0.0162332 −0.00811660 0.999967i \(-0.502584\pi\)
−0.00811660 + 0.999967i \(0.502584\pi\)
\(654\) 0 0
\(655\) −5895.38 −0.351682
\(656\) 0 0
\(657\) −9974.50 −0.592302
\(658\) 0 0
\(659\) 913.255 0.0539838 0.0269919 0.999636i \(-0.491407\pi\)
0.0269919 + 0.999636i \(0.491407\pi\)
\(660\) 0 0
\(661\) 17642.8 1.03816 0.519080 0.854726i \(-0.326275\pi\)
0.519080 + 0.854726i \(0.326275\pi\)
\(662\) 0 0
\(663\) 9928.42 0.581581
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2284.11 −0.132596
\(668\) 0 0
\(669\) 14206.3 0.820995
\(670\) 0 0
\(671\) −2953.57 −0.169928
\(672\) 0 0
\(673\) 9839.88 0.563595 0.281798 0.959474i \(-0.409069\pi\)
0.281798 + 0.959474i \(0.409069\pi\)
\(674\) 0 0
\(675\) −589.755 −0.0336292
\(676\) 0 0
\(677\) 27471.3 1.55954 0.779769 0.626067i \(-0.215336\pi\)
0.779769 + 0.626067i \(0.215336\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 6856.92 0.385841
\(682\) 0 0
\(683\) 6363.61 0.356511 0.178255 0.983984i \(-0.442955\pi\)
0.178255 + 0.983984i \(0.442955\pi\)
\(684\) 0 0
\(685\) 28954.7 1.61504
\(686\) 0 0
\(687\) 1437.63 0.0798383
\(688\) 0 0
\(689\) 2673.65 0.147835
\(690\) 0 0
\(691\) 16512.5 0.909068 0.454534 0.890729i \(-0.349806\pi\)
0.454534 + 0.890729i \(0.349806\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 1907.18 0.104091
\(696\) 0 0
\(697\) 18797.8 1.02154
\(698\) 0 0
\(699\) 15458.1 0.836452
\(700\) 0 0
\(701\) −32633.2 −1.75826 −0.879128 0.476585i \(-0.841874\pi\)
−0.879128 + 0.476585i \(0.841874\pi\)
\(702\) 0 0
\(703\) 2523.22 0.135370
\(704\) 0 0
\(705\) 14774.9 0.789298
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −22641.5 −1.19932 −0.599660 0.800255i \(-0.704698\pi\)
−0.599660 + 0.800255i \(0.704698\pi\)
\(710\) 0 0
\(711\) 9770.92 0.515384
\(712\) 0 0
\(713\) 14734.7 0.773937
\(714\) 0 0
\(715\) −28444.3 −1.48777
\(716\) 0 0
\(717\) −7136.81 −0.371728
\(718\) 0 0
\(719\) 28786.0 1.49310 0.746549 0.665330i \(-0.231709\pi\)
0.746549 + 0.665330i \(0.231709\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −66.6607 −0.00342896
\(724\) 0 0
\(725\) −862.875 −0.0442019
\(726\) 0 0
\(727\) 15302.1 0.780639 0.390319 0.920679i \(-0.372365\pi\)
0.390319 + 0.920679i \(0.372365\pi\)
\(728\) 0 0
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 17079.2 0.864155
\(732\) 0 0
\(733\) 16451.3 0.828983 0.414491 0.910053i \(-0.363960\pi\)
0.414491 + 0.910053i \(0.363960\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 10321.5 0.515873
\(738\) 0 0
\(739\) 3878.19 0.193047 0.0965233 0.995331i \(-0.469228\pi\)
0.0965233 + 0.995331i \(0.469228\pi\)
\(740\) 0 0
\(741\) 30976.4 1.53569
\(742\) 0 0
\(743\) 7702.39 0.380314 0.190157 0.981754i \(-0.439100\pi\)
0.190157 + 0.981754i \(0.439100\pi\)
\(744\) 0 0
\(745\) −5054.79 −0.248582
\(746\) 0 0
\(747\) 3642.99 0.178434
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −7068.66 −0.343461 −0.171730 0.985144i \(-0.554936\pi\)
−0.171730 + 0.985144i \(0.554936\pi\)
\(752\) 0 0
\(753\) −14535.2 −0.703445
\(754\) 0 0
\(755\) −9838.19 −0.474236
\(756\) 0 0
\(757\) 21371.0 1.02608 0.513039 0.858365i \(-0.328520\pi\)
0.513039 + 0.858365i \(0.328520\pi\)
\(758\) 0 0
\(759\) −6556.87 −0.313569
\(760\) 0 0
\(761\) −16307.2 −0.776786 −0.388393 0.921494i \(-0.626970\pi\)
−0.388393 + 0.921494i \(0.626970\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −4083.22 −0.192979
\(766\) 0 0
\(767\) −10804.4 −0.508634
\(768\) 0 0
\(769\) −40553.7 −1.90169 −0.950847 0.309660i \(-0.899785\pi\)
−0.950847 + 0.309660i \(0.899785\pi\)
\(770\) 0 0
\(771\) 14142.2 0.660593
\(772\) 0 0
\(773\) 11220.5 0.522089 0.261045 0.965327i \(-0.415933\pi\)
0.261045 + 0.965327i \(0.415933\pi\)
\(774\) 0 0
\(775\) 5566.34 0.257999
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 58648.5 2.69743
\(780\) 0 0
\(781\) −27127.2 −1.24288
\(782\) 0 0
\(783\) 1066.60 0.0486812
\(784\) 0 0
\(785\) 1788.79 0.0813306
\(786\) 0 0
\(787\) −7836.87 −0.354961 −0.177480 0.984124i \(-0.556795\pi\)
−0.177480 + 0.984124i \(0.556795\pi\)
\(788\) 0 0
\(789\) 10443.2 0.471213
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −5788.94 −0.259232
\(794\) 0 0
\(795\) −1099.58 −0.0490543
\(796\) 0 0
\(797\) 6521.90 0.289859 0.144929 0.989442i \(-0.453704\pi\)
0.144929 + 0.989442i \(0.453704\pi\)
\(798\) 0 0
\(799\) −21660.3 −0.959056
\(800\) 0 0
\(801\) −12189.7 −0.537707
\(802\) 0 0
\(803\) −41893.4 −1.84108
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 891.277 0.0388779
\(808\) 0 0
\(809\) −22997.8 −0.999457 −0.499728 0.866182i \(-0.666567\pi\)
−0.499728 + 0.866182i \(0.666567\pi\)
\(810\) 0 0
\(811\) 5253.94 0.227485 0.113743 0.993510i \(-0.463716\pi\)
0.113743 + 0.993510i \(0.463716\pi\)
\(812\) 0 0
\(813\) −2437.51 −0.105150
\(814\) 0 0
\(815\) −190.707 −0.00819653
\(816\) 0 0
\(817\) 53286.6 2.28184
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −14044.6 −0.597029 −0.298515 0.954405i \(-0.596491\pi\)
−0.298515 + 0.954405i \(0.596491\pi\)
\(822\) 0 0
\(823\) −13677.4 −0.579300 −0.289650 0.957133i \(-0.593539\pi\)
−0.289650 + 0.957133i \(0.593539\pi\)
\(824\) 0 0
\(825\) −2477.00 −0.104531
\(826\) 0 0
\(827\) −3206.14 −0.134811 −0.0674054 0.997726i \(-0.521472\pi\)
−0.0674054 + 0.997726i \(0.521472\pi\)
\(828\) 0 0
\(829\) −30876.6 −1.29359 −0.646796 0.762663i \(-0.723892\pi\)
−0.646796 + 0.762663i \(0.723892\pi\)
\(830\) 0 0
\(831\) 21627.0 0.902808
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 13465.7 0.558083
\(836\) 0 0
\(837\) −6880.59 −0.284143
\(838\) 0 0
\(839\) −11040.6 −0.454306 −0.227153 0.973859i \(-0.572942\pi\)
−0.227153 + 0.973859i \(0.572942\pi\)
\(840\) 0 0
\(841\) −22828.4 −0.936014
\(842\) 0 0
\(843\) −23584.6 −0.963578
\(844\) 0 0
\(845\) −33436.1 −1.36123
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −6155.10 −0.248813
\(850\) 0 0
\(851\) −1046.82 −0.0421674
\(852\) 0 0
\(853\) 12245.3 0.491524 0.245762 0.969330i \(-0.420962\pi\)
0.245762 + 0.969330i \(0.420962\pi\)
\(854\) 0 0
\(855\) −12739.5 −0.509570
\(856\) 0 0
\(857\) −18764.2 −0.747925 −0.373962 0.927444i \(-0.622001\pi\)
−0.373962 + 0.927444i \(0.622001\pi\)
\(858\) 0 0
\(859\) 30566.2 1.21409 0.607046 0.794667i \(-0.292354\pi\)
0.607046 + 0.794667i \(0.292354\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 14027.6 0.553307 0.276654 0.960970i \(-0.410775\pi\)
0.276654 + 0.960970i \(0.410775\pi\)
\(864\) 0 0
\(865\) 37599.3 1.47793
\(866\) 0 0
\(867\) −8752.92 −0.342866
\(868\) 0 0
\(869\) 41038.3 1.60199
\(870\) 0 0
\(871\) 20230.0 0.786988
\(872\) 0 0
\(873\) 6136.62 0.237908
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −33737.9 −1.29903 −0.649514 0.760349i \(-0.725028\pi\)
−0.649514 + 0.760349i \(0.725028\pi\)
\(878\) 0 0
\(879\) −18742.8 −0.719201
\(880\) 0 0
\(881\) −8512.70 −0.325539 −0.162770 0.986664i \(-0.552043\pi\)
−0.162770 + 0.986664i \(0.552043\pi\)
\(882\) 0 0
\(883\) 27302.9 1.04056 0.520281 0.853995i \(-0.325827\pi\)
0.520281 + 0.853995i \(0.325827\pi\)
\(884\) 0 0
\(885\) 4443.46 0.168774
\(886\) 0 0
\(887\) −25804.0 −0.976789 −0.488395 0.872623i \(-0.662417\pi\)
−0.488395 + 0.872623i \(0.662417\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 3061.84 0.115124
\(892\) 0 0
\(893\) −67579.4 −2.53243
\(894\) 0 0
\(895\) −44317.5 −1.65516
\(896\) 0 0
\(897\) −12851.3 −0.478365
\(898\) 0 0
\(899\) −10067.0 −0.373476
\(900\) 0 0
\(901\) 1612.01 0.0596047
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1329.36 0.0488279
\(906\) 0 0
\(907\) −29536.1 −1.08129 −0.540645 0.841251i \(-0.681820\pi\)
−0.540645 + 0.841251i \(0.681820\pi\)
\(908\) 0 0
\(909\) 7186.80 0.262234
\(910\) 0 0
\(911\) 19543.9 0.710777 0.355389 0.934719i \(-0.384349\pi\)
0.355389 + 0.934719i \(0.384349\pi\)
\(912\) 0 0
\(913\) 15300.7 0.554634
\(914\) 0 0
\(915\) 2380.79 0.0860182
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 8300.26 0.297933 0.148967 0.988842i \(-0.452405\pi\)
0.148967 + 0.988842i \(0.452405\pi\)
\(920\) 0 0
\(921\) 29618.9 1.05969
\(922\) 0 0
\(923\) −53168.7 −1.89607
\(924\) 0 0
\(925\) −395.459 −0.0140569
\(926\) 0 0
\(927\) 4083.54 0.144683
\(928\) 0 0
\(929\) −11835.8 −0.417999 −0.208999 0.977916i \(-0.567021\pi\)
−0.208999 + 0.977916i \(0.567021\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −4753.42 −0.166795
\(934\) 0 0
\(935\) −17149.7 −0.599846
\(936\) 0 0
\(937\) 19702.4 0.686926 0.343463 0.939166i \(-0.388400\pi\)
0.343463 + 0.939166i \(0.388400\pi\)
\(938\) 0 0
\(939\) −22211.8 −0.771942
\(940\) 0 0
\(941\) −32940.6 −1.14116 −0.570581 0.821241i \(-0.693282\pi\)
−0.570581 + 0.821241i \(0.693282\pi\)
\(942\) 0 0
\(943\) −24331.8 −0.840246
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −37692.0 −1.29337 −0.646687 0.762755i \(-0.723846\pi\)
−0.646687 + 0.762755i \(0.723846\pi\)
\(948\) 0 0
\(949\) −82110.2 −2.80865
\(950\) 0 0
\(951\) −30671.3 −1.04583
\(952\) 0 0
\(953\) 2904.61 0.0987298 0.0493649 0.998781i \(-0.484280\pi\)
0.0493649 + 0.998781i \(0.484280\pi\)
\(954\) 0 0
\(955\) −46639.8 −1.58034
\(956\) 0 0
\(957\) 4479.79 0.151318
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 35150.7 1.17991
\(962\) 0 0
\(963\) 8205.67 0.274584
\(964\) 0 0
\(965\) −8567.01 −0.285784
\(966\) 0 0
\(967\) 7954.56 0.264531 0.132266 0.991214i \(-0.457775\pi\)
0.132266 + 0.991214i \(0.457775\pi\)
\(968\) 0 0
\(969\) 18676.4 0.619166
\(970\) 0 0
\(971\) −15318.3 −0.506268 −0.253134 0.967431i \(-0.581461\pi\)
−0.253134 + 0.967431i \(0.581461\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −4854.87 −0.159467
\(976\) 0 0
\(977\) −34372.3 −1.12555 −0.562777 0.826609i \(-0.690267\pi\)
−0.562777 + 0.826609i \(0.690267\pi\)
\(978\) 0 0
\(979\) −51197.5 −1.67138
\(980\) 0 0
\(981\) 205.597 0.00669133
\(982\) 0 0
\(983\) −15356.6 −0.498271 −0.249136 0.968469i \(-0.580146\pi\)
−0.249136 + 0.968469i \(0.580146\pi\)
\(984\) 0 0
\(985\) −19112.7 −0.618255
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −22107.3 −0.710789
\(990\) 0 0
\(991\) 50048.5 1.60428 0.802141 0.597134i \(-0.203694\pi\)
0.802141 + 0.597134i \(0.203694\pi\)
\(992\) 0 0
\(993\) −20541.0 −0.656445
\(994\) 0 0
\(995\) −15703.3 −0.500330
\(996\) 0 0
\(997\) 7695.26 0.244445 0.122222 0.992503i \(-0.460998\pi\)
0.122222 + 0.992503i \(0.460998\pi\)
\(998\) 0 0
\(999\) 488.829 0.0154814
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 2352.4.a.cj.1.1 3
4.3 odd 2 1176.4.a.x.1.1 3
7.3 odd 6 336.4.q.l.289.1 6
7.5 odd 6 336.4.q.l.193.1 6
7.6 odd 2 2352.4.a.ch.1.3 3
28.3 even 6 168.4.q.e.121.1 yes 6
28.19 even 6 168.4.q.e.25.1 6
28.27 even 2 1176.4.a.y.1.3 3
84.47 odd 6 504.4.s.g.361.3 6
84.59 odd 6 504.4.s.g.289.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.4.q.e.25.1 6 28.19 even 6
168.4.q.e.121.1 yes 6 28.3 even 6
336.4.q.l.193.1 6 7.5 odd 6
336.4.q.l.289.1 6 7.3 odd 6
504.4.s.g.289.3 6 84.59 odd 6
504.4.s.g.361.3 6 84.47 odd 6
1176.4.a.x.1.1 3 4.3 odd 2
1176.4.a.y.1.3 3 28.27 even 2
2352.4.a.ch.1.3 3 7.6 odd 2
2352.4.a.cj.1.1 3 1.1 even 1 trivial