Properties

Label 116.4.a.a.1.1
Level $116$
Weight $4$
Character 116.1
Self dual yes
Analytic conductor $6.844$
Analytic rank $1$
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] = [116,4,Mod(1,116)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(116, 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("116.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 116 = 2^{2} \cdot 29 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 116.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: \(6.84422156067\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{13}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.30278\) of defining polynomial
Character \(\chi\) \(=\) 116.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.60555 q^{3} +2.21110 q^{5} +4.42221 q^{7} -14.0000 q^{9} +O(q^{10})\) \(q-3.60555 q^{3} +2.21110 q^{5} +4.42221 q^{7} -14.0000 q^{9} -12.3944 q^{11} -48.6333 q^{13} -7.97224 q^{15} -101.322 q^{17} -59.2666 q^{19} -15.9445 q^{21} +18.0555 q^{23} -120.111 q^{25} +147.828 q^{27} -29.0000 q^{29} +20.8167 q^{31} +44.6888 q^{33} +9.77795 q^{35} +101.066 q^{37} +175.350 q^{39} +40.7889 q^{41} +152.450 q^{43} -30.9554 q^{45} +121.328 q^{47} -323.444 q^{49} +365.322 q^{51} +177.700 q^{53} -27.4054 q^{55} +213.689 q^{57} +109.611 q^{59} +61.7662 q^{61} -61.9109 q^{63} -107.533 q^{65} -471.532 q^{67} -65.1001 q^{69} +546.522 q^{71} +169.733 q^{73} +433.066 q^{75} -54.8108 q^{77} +184.916 q^{79} -155.000 q^{81} +210.500 q^{83} -224.034 q^{85} +104.561 q^{87} -1393.32 q^{89} -215.066 q^{91} -75.0555 q^{93} -131.045 q^{95} -1099.43 q^{97} +173.522 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 10 q^{5} - 20 q^{7} - 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 10 q^{5} - 20 q^{7} - 28 q^{9} - 32 q^{11} - 54 q^{13} - 52 q^{15} - 44 q^{17} - 32 q^{19} - 104 q^{21} - 36 q^{23} - 96 q^{25} - 58 q^{29} + 20 q^{31} - 26 q^{33} + 308 q^{35} - 144 q^{37} + 156 q^{39} + 96 q^{41} + 240 q^{43} + 140 q^{45} + 596 q^{47} - 70 q^{49} + 572 q^{51} - 34 q^{53} + 212 q^{55} + 312 q^{57} + 724 q^{59} - 612 q^{61} + 280 q^{63} - 42 q^{65} + 528 q^{67} - 260 q^{69} - 104 q^{71} - 872 q^{73} + 520 q^{75} + 424 q^{77} - 820 q^{79} - 310 q^{81} - 228 q^{83} - 924 q^{85} - 32 q^{89} - 84 q^{91} - 78 q^{93} - 464 q^{95} - 1896 q^{97} + 448 q^{99}+O(q^{100}) \) 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
\(3\) −3.60555 −0.693889 −0.346944 0.937886i \(-0.612781\pi\)
−0.346944 + 0.937886i \(0.612781\pi\)
\(4\) 0 0
\(5\) 2.21110 0.197767 0.0988835 0.995099i \(-0.468473\pi\)
0.0988835 + 0.995099i \(0.468473\pi\)
\(6\) 0 0
\(7\) 4.42221 0.238777 0.119388 0.992848i \(-0.461907\pi\)
0.119388 + 0.992848i \(0.461907\pi\)
\(8\) 0 0
\(9\) −14.0000 −0.518519
\(10\) 0 0
\(11\) −12.3944 −0.339733 −0.169867 0.985467i \(-0.554334\pi\)
−0.169867 + 0.985467i \(0.554334\pi\)
\(12\) 0 0
\(13\) −48.6333 −1.03757 −0.518787 0.854904i \(-0.673616\pi\)
−0.518787 + 0.854904i \(0.673616\pi\)
\(14\) 0 0
\(15\) −7.97224 −0.137228
\(16\) 0 0
\(17\) −101.322 −1.44554 −0.722771 0.691087i \(-0.757132\pi\)
−0.722771 + 0.691087i \(0.757132\pi\)
\(18\) 0 0
\(19\) −59.2666 −0.715615 −0.357808 0.933795i \(-0.616476\pi\)
−0.357808 + 0.933795i \(0.616476\pi\)
\(20\) 0 0
\(21\) −15.9445 −0.165684
\(22\) 0 0
\(23\) 18.0555 0.163688 0.0818442 0.996645i \(-0.473919\pi\)
0.0818442 + 0.996645i \(0.473919\pi\)
\(24\) 0 0
\(25\) −120.111 −0.960888
\(26\) 0 0
\(27\) 147.828 1.05368
\(28\) 0 0
\(29\) −29.0000 −0.185695
\(30\) 0 0
\(31\) 20.8167 0.120606 0.0603029 0.998180i \(-0.480793\pi\)
0.0603029 + 0.998180i \(0.480793\pi\)
\(32\) 0 0
\(33\) 44.6888 0.235737
\(34\) 0 0
\(35\) 9.77795 0.0472221
\(36\) 0 0
\(37\) 101.066 0.449060 0.224530 0.974467i \(-0.427915\pi\)
0.224530 + 0.974467i \(0.427915\pi\)
\(38\) 0 0
\(39\) 175.350 0.719960
\(40\) 0 0
\(41\) 40.7889 0.155370 0.0776848 0.996978i \(-0.475247\pi\)
0.0776848 + 0.996978i \(0.475247\pi\)
\(42\) 0 0
\(43\) 152.450 0.540660 0.270330 0.962768i \(-0.412867\pi\)
0.270330 + 0.962768i \(0.412867\pi\)
\(44\) 0 0
\(45\) −30.9554 −0.102546
\(46\) 0 0
\(47\) 121.328 0.376543 0.188271 0.982117i \(-0.439712\pi\)
0.188271 + 0.982117i \(0.439712\pi\)
\(48\) 0 0
\(49\) −323.444 −0.942986
\(50\) 0 0
\(51\) 365.322 1.00305
\(52\) 0 0
\(53\) 177.700 0.460546 0.230273 0.973126i \(-0.426038\pi\)
0.230273 + 0.973126i \(0.426038\pi\)
\(54\) 0 0
\(55\) −27.4054 −0.0671881
\(56\) 0 0
\(57\) 213.689 0.496557
\(58\) 0 0
\(59\) 109.611 0.241868 0.120934 0.992661i \(-0.461411\pi\)
0.120934 + 0.992661i \(0.461411\pi\)
\(60\) 0 0
\(61\) 61.7662 0.129645 0.0648226 0.997897i \(-0.479352\pi\)
0.0648226 + 0.997897i \(0.479352\pi\)
\(62\) 0 0
\(63\) −61.9109 −0.123810
\(64\) 0 0
\(65\) −107.533 −0.205198
\(66\) 0 0
\(67\) −471.532 −0.859804 −0.429902 0.902876i \(-0.641452\pi\)
−0.429902 + 0.902876i \(0.641452\pi\)
\(68\) 0 0
\(69\) −65.1001 −0.113582
\(70\) 0 0
\(71\) 546.522 0.913524 0.456762 0.889589i \(-0.349009\pi\)
0.456762 + 0.889589i \(0.349009\pi\)
\(72\) 0 0
\(73\) 169.733 0.272133 0.136066 0.990700i \(-0.456554\pi\)
0.136066 + 0.990700i \(0.456554\pi\)
\(74\) 0 0
\(75\) 433.066 0.666749
\(76\) 0 0
\(77\) −54.8108 −0.0811204
\(78\) 0 0
\(79\) 184.916 0.263350 0.131675 0.991293i \(-0.457964\pi\)
0.131675 + 0.991293i \(0.457964\pi\)
\(80\) 0 0
\(81\) −155.000 −0.212620
\(82\) 0 0
\(83\) 210.500 0.278378 0.139189 0.990266i \(-0.455551\pi\)
0.139189 + 0.990266i \(0.455551\pi\)
\(84\) 0 0
\(85\) −224.034 −0.285881
\(86\) 0 0
\(87\) 104.561 0.128852
\(88\) 0 0
\(89\) −1393.32 −1.65946 −0.829729 0.558167i \(-0.811505\pi\)
−0.829729 + 0.558167i \(0.811505\pi\)
\(90\) 0 0
\(91\) −215.066 −0.247748
\(92\) 0 0
\(93\) −75.0555 −0.0836870
\(94\) 0 0
\(95\) −131.045 −0.141525
\(96\) 0 0
\(97\) −1099.43 −1.15083 −0.575415 0.817862i \(-0.695159\pi\)
−0.575415 + 0.817862i \(0.695159\pi\)
\(98\) 0 0
\(99\) 173.522 0.176158
\(100\) 0 0
\(101\) 322.799 0.318017 0.159008 0.987277i \(-0.449170\pi\)
0.159008 + 0.987277i \(0.449170\pi\)
\(102\) 0 0
\(103\) −1741.61 −1.66608 −0.833038 0.553215i \(-0.813401\pi\)
−0.833038 + 0.553215i \(0.813401\pi\)
\(104\) 0 0
\(105\) −35.2549 −0.0327669
\(106\) 0 0
\(107\) 560.244 0.506176 0.253088 0.967443i \(-0.418554\pi\)
0.253088 + 0.967443i \(0.418554\pi\)
\(108\) 0 0
\(109\) −987.700 −0.867931 −0.433966 0.900929i \(-0.642886\pi\)
−0.433966 + 0.900929i \(0.642886\pi\)
\(110\) 0 0
\(111\) −364.400 −0.311598
\(112\) 0 0
\(113\) 493.532 0.410864 0.205432 0.978671i \(-0.434140\pi\)
0.205432 + 0.978671i \(0.434140\pi\)
\(114\) 0 0
\(115\) 39.9226 0.0323722
\(116\) 0 0
\(117\) 680.866 0.538001
\(118\) 0 0
\(119\) −448.067 −0.345162
\(120\) 0 0
\(121\) −1177.38 −0.884581
\(122\) 0 0
\(123\) −147.066 −0.107809
\(124\) 0 0
\(125\) −541.966 −0.387799
\(126\) 0 0
\(127\) −663.822 −0.463816 −0.231908 0.972738i \(-0.574497\pi\)
−0.231908 + 0.972738i \(0.574497\pi\)
\(128\) 0 0
\(129\) −549.666 −0.375158
\(130\) 0 0
\(131\) 1456.67 0.971523 0.485761 0.874091i \(-0.338542\pi\)
0.485761 + 0.874091i \(0.338542\pi\)
\(132\) 0 0
\(133\) −262.089 −0.170872
\(134\) 0 0
\(135\) 326.862 0.208384
\(136\) 0 0
\(137\) 2670.15 1.66516 0.832579 0.553907i \(-0.186864\pi\)
0.832579 + 0.553907i \(0.186864\pi\)
\(138\) 0 0
\(139\) 958.476 0.584870 0.292435 0.956285i \(-0.405535\pi\)
0.292435 + 0.956285i \(0.405535\pi\)
\(140\) 0 0
\(141\) −437.454 −0.261279
\(142\) 0 0
\(143\) 602.783 0.352498
\(144\) 0 0
\(145\) −64.1220 −0.0367244
\(146\) 0 0
\(147\) 1166.19 0.654327
\(148\) 0 0
\(149\) 490.435 0.269651 0.134825 0.990869i \(-0.456953\pi\)
0.134825 + 0.990869i \(0.456953\pi\)
\(150\) 0 0
\(151\) −1999.24 −1.07746 −0.538728 0.842480i \(-0.681095\pi\)
−0.538728 + 0.842480i \(0.681095\pi\)
\(152\) 0 0
\(153\) 1418.51 0.749541
\(154\) 0 0
\(155\) 46.0278 0.0238519
\(156\) 0 0
\(157\) 1830.68 0.930598 0.465299 0.885154i \(-0.345947\pi\)
0.465299 + 0.885154i \(0.345947\pi\)
\(158\) 0 0
\(159\) −640.706 −0.319568
\(160\) 0 0
\(161\) 79.8452 0.0390850
\(162\) 0 0
\(163\) 2884.29 1.38598 0.692991 0.720946i \(-0.256292\pi\)
0.692991 + 0.720946i \(0.256292\pi\)
\(164\) 0 0
\(165\) 98.8116 0.0466210
\(166\) 0 0
\(167\) 1658.79 0.768628 0.384314 0.923203i \(-0.374438\pi\)
0.384314 + 0.923203i \(0.374438\pi\)
\(168\) 0 0
\(169\) 168.199 0.0765583
\(170\) 0 0
\(171\) 829.733 0.371060
\(172\) 0 0
\(173\) 2925.53 1.28569 0.642844 0.765997i \(-0.277754\pi\)
0.642844 + 0.765997i \(0.277754\pi\)
\(174\) 0 0
\(175\) −531.156 −0.229438
\(176\) 0 0
\(177\) −395.210 −0.167829
\(178\) 0 0
\(179\) −4190.23 −1.74968 −0.874839 0.484413i \(-0.839033\pi\)
−0.874839 + 0.484413i \(0.839033\pi\)
\(180\) 0 0
\(181\) −321.969 −0.132220 −0.0661098 0.997812i \(-0.521059\pi\)
−0.0661098 + 0.997812i \(0.521059\pi\)
\(182\) 0 0
\(183\) −222.701 −0.0899593
\(184\) 0 0
\(185\) 223.468 0.0888093
\(186\) 0 0
\(187\) 1255.83 0.491099
\(188\) 0 0
\(189\) 653.724 0.251595
\(190\) 0 0
\(191\) −914.466 −0.346432 −0.173216 0.984884i \(-0.555416\pi\)
−0.173216 + 0.984884i \(0.555416\pi\)
\(192\) 0 0
\(193\) −1055.21 −0.393552 −0.196776 0.980448i \(-0.563047\pi\)
−0.196776 + 0.980448i \(0.563047\pi\)
\(194\) 0 0
\(195\) 387.717 0.142384
\(196\) 0 0
\(197\) 549.869 0.198866 0.0994328 0.995044i \(-0.468297\pi\)
0.0994328 + 0.995044i \(0.468297\pi\)
\(198\) 0 0
\(199\) −4225.60 −1.50525 −0.752625 0.658449i \(-0.771213\pi\)
−0.752625 + 0.658449i \(0.771213\pi\)
\(200\) 0 0
\(201\) 1700.13 0.596608
\(202\) 0 0
\(203\) −128.244 −0.0443397
\(204\) 0 0
\(205\) 90.1884 0.0307270
\(206\) 0 0
\(207\) −252.777 −0.0848755
\(208\) 0 0
\(209\) 734.577 0.243118
\(210\) 0 0
\(211\) −692.728 −0.226016 −0.113008 0.993594i \(-0.536049\pi\)
−0.113008 + 0.993594i \(0.536049\pi\)
\(212\) 0 0
\(213\) −1970.51 −0.633884
\(214\) 0 0
\(215\) 337.082 0.106925
\(216\) 0 0
\(217\) 92.0555 0.0287979
\(218\) 0 0
\(219\) −611.980 −0.188830
\(220\) 0 0
\(221\) 4927.63 1.49986
\(222\) 0 0
\(223\) −3864.52 −1.16048 −0.580241 0.814445i \(-0.697042\pi\)
−0.580241 + 0.814445i \(0.697042\pi\)
\(224\) 0 0
\(225\) 1681.55 0.498238
\(226\) 0 0
\(227\) −3410.81 −0.997283 −0.498642 0.866808i \(-0.666168\pi\)
−0.498642 + 0.866808i \(0.666168\pi\)
\(228\) 0 0
\(229\) −2695.50 −0.777833 −0.388916 0.921273i \(-0.627150\pi\)
−0.388916 + 0.921273i \(0.627150\pi\)
\(230\) 0 0
\(231\) 197.623 0.0562885
\(232\) 0 0
\(233\) 4829.82 1.35799 0.678995 0.734143i \(-0.262416\pi\)
0.678995 + 0.734143i \(0.262416\pi\)
\(234\) 0 0
\(235\) 268.269 0.0744677
\(236\) 0 0
\(237\) −666.724 −0.182736
\(238\) 0 0
\(239\) −4869.12 −1.31781 −0.658906 0.752225i \(-0.728981\pi\)
−0.658906 + 0.752225i \(0.728981\pi\)
\(240\) 0 0
\(241\) −3096.69 −0.827698 −0.413849 0.910345i \(-0.635816\pi\)
−0.413849 + 0.910345i \(0.635816\pi\)
\(242\) 0 0
\(243\) −3432.48 −0.906148
\(244\) 0 0
\(245\) −715.168 −0.186491
\(246\) 0 0
\(247\) 2882.33 0.742503
\(248\) 0 0
\(249\) −758.967 −0.193163
\(250\) 0 0
\(251\) 2208.84 0.555461 0.277730 0.960659i \(-0.410418\pi\)
0.277730 + 0.960659i \(0.410418\pi\)
\(252\) 0 0
\(253\) −223.788 −0.0556104
\(254\) 0 0
\(255\) 807.765 0.198369
\(256\) 0 0
\(257\) 526.955 0.127901 0.0639504 0.997953i \(-0.479630\pi\)
0.0639504 + 0.997953i \(0.479630\pi\)
\(258\) 0 0
\(259\) 446.937 0.107225
\(260\) 0 0
\(261\) 406.000 0.0962865
\(262\) 0 0
\(263\) −4892.69 −1.14713 −0.573567 0.819158i \(-0.694441\pi\)
−0.573567 + 0.819158i \(0.694441\pi\)
\(264\) 0 0
\(265\) 392.912 0.0910808
\(266\) 0 0
\(267\) 5023.69 1.15148
\(268\) 0 0
\(269\) −4744.74 −1.07543 −0.537717 0.843125i \(-0.680713\pi\)
−0.537717 + 0.843125i \(0.680713\pi\)
\(270\) 0 0
\(271\) −8187.57 −1.83527 −0.917637 0.397420i \(-0.869906\pi\)
−0.917637 + 0.397420i \(0.869906\pi\)
\(272\) 0 0
\(273\) 775.433 0.171910
\(274\) 0 0
\(275\) 1488.71 0.326446
\(276\) 0 0
\(277\) −4966.39 −1.07726 −0.538631 0.842542i \(-0.681058\pi\)
−0.538631 + 0.842542i \(0.681058\pi\)
\(278\) 0 0
\(279\) −291.433 −0.0625364
\(280\) 0 0
\(281\) 81.1971 0.0172378 0.00861888 0.999963i \(-0.497256\pi\)
0.00861888 + 0.999963i \(0.497256\pi\)
\(282\) 0 0
\(283\) 213.170 0.0447762 0.0223881 0.999749i \(-0.492873\pi\)
0.0223881 + 0.999749i \(0.492873\pi\)
\(284\) 0 0
\(285\) 472.488 0.0982027
\(286\) 0 0
\(287\) 180.377 0.0370986
\(288\) 0 0
\(289\) 5353.17 1.08959
\(290\) 0 0
\(291\) 3964.06 0.798548
\(292\) 0 0
\(293\) −6346.40 −1.26539 −0.632697 0.774399i \(-0.718052\pi\)
−0.632697 + 0.774399i \(0.718052\pi\)
\(294\) 0 0
\(295\) 242.362 0.0478334
\(296\) 0 0
\(297\) −1832.24 −0.357971
\(298\) 0 0
\(299\) −878.099 −0.169839
\(300\) 0 0
\(301\) 674.165 0.129097
\(302\) 0 0
\(303\) −1163.87 −0.220668
\(304\) 0 0
\(305\) 136.571 0.0256395
\(306\) 0 0
\(307\) 5636.38 1.04784 0.523918 0.851769i \(-0.324470\pi\)
0.523918 + 0.851769i \(0.324470\pi\)
\(308\) 0 0
\(309\) 6279.46 1.15607
\(310\) 0 0
\(311\) −586.998 −0.107028 −0.0535138 0.998567i \(-0.517042\pi\)
−0.0535138 + 0.998567i \(0.517042\pi\)
\(312\) 0 0
\(313\) −1249.95 −0.225723 −0.112862 0.993611i \(-0.536002\pi\)
−0.112862 + 0.993611i \(0.536002\pi\)
\(314\) 0 0
\(315\) −136.891 −0.0244856
\(316\) 0 0
\(317\) 4821.77 0.854314 0.427157 0.904177i \(-0.359515\pi\)
0.427157 + 0.904177i \(0.359515\pi\)
\(318\) 0 0
\(319\) 359.439 0.0630869
\(320\) 0 0
\(321\) −2019.99 −0.351230
\(322\) 0 0
\(323\) 6005.02 1.03445
\(324\) 0 0
\(325\) 5841.40 0.996992
\(326\) 0 0
\(327\) 3561.20 0.602247
\(328\) 0 0
\(329\) 536.537 0.0899096
\(330\) 0 0
\(331\) 8287.49 1.37620 0.688100 0.725616i \(-0.258445\pi\)
0.688100 + 0.725616i \(0.258445\pi\)
\(332\) 0 0
\(333\) −1414.93 −0.232846
\(334\) 0 0
\(335\) −1042.61 −0.170041
\(336\) 0 0
\(337\) 9910.59 1.60197 0.800985 0.598684i \(-0.204309\pi\)
0.800985 + 0.598684i \(0.204309\pi\)
\(338\) 0 0
\(339\) −1779.46 −0.285094
\(340\) 0 0
\(341\) −258.011 −0.0409738
\(342\) 0 0
\(343\) −2947.15 −0.463940
\(344\) 0 0
\(345\) −143.943 −0.0224627
\(346\) 0 0
\(347\) 9826.54 1.52022 0.760110 0.649794i \(-0.225145\pi\)
0.760110 + 0.649794i \(0.225145\pi\)
\(348\) 0 0
\(349\) −7438.96 −1.14097 −0.570485 0.821308i \(-0.693245\pi\)
−0.570485 + 0.821308i \(0.693245\pi\)
\(350\) 0 0
\(351\) −7189.35 −1.09327
\(352\) 0 0
\(353\) −9385.04 −1.41506 −0.707529 0.706684i \(-0.750190\pi\)
−0.707529 + 0.706684i \(0.750190\pi\)
\(354\) 0 0
\(355\) 1208.42 0.180665
\(356\) 0 0
\(357\) 1615.53 0.239504
\(358\) 0 0
\(359\) 6673.24 0.981059 0.490529 0.871425i \(-0.336803\pi\)
0.490529 + 0.871425i \(0.336803\pi\)
\(360\) 0 0
\(361\) −3346.47 −0.487894
\(362\) 0 0
\(363\) 4245.10 0.613801
\(364\) 0 0
\(365\) 375.296 0.0538189
\(366\) 0 0
\(367\) 12384.4 1.76147 0.880733 0.473613i \(-0.157051\pi\)
0.880733 + 0.473613i \(0.157051\pi\)
\(368\) 0 0
\(369\) −571.045 −0.0805620
\(370\) 0 0
\(371\) 785.825 0.109968
\(372\) 0 0
\(373\) −5167.77 −0.717364 −0.358682 0.933460i \(-0.616774\pi\)
−0.358682 + 0.933460i \(0.616774\pi\)
\(374\) 0 0
\(375\) 1954.08 0.269089
\(376\) 0 0
\(377\) 1410.37 0.192673
\(378\) 0 0
\(379\) 7398.92 1.00279 0.501395 0.865219i \(-0.332820\pi\)
0.501395 + 0.865219i \(0.332820\pi\)
\(380\) 0 0
\(381\) 2393.44 0.321837
\(382\) 0 0
\(383\) −3262.55 −0.435270 −0.217635 0.976030i \(-0.569834\pi\)
−0.217635 + 0.976030i \(0.569834\pi\)
\(384\) 0 0
\(385\) −121.192 −0.0160429
\(386\) 0 0
\(387\) −2134.30 −0.280342
\(388\) 0 0
\(389\) −5819.82 −0.758552 −0.379276 0.925284i \(-0.623827\pi\)
−0.379276 + 0.925284i \(0.623827\pi\)
\(390\) 0 0
\(391\) −1829.42 −0.236619
\(392\) 0 0
\(393\) −5252.08 −0.674129
\(394\) 0 0
\(395\) 408.868 0.0520820
\(396\) 0 0
\(397\) −7678.61 −0.970726 −0.485363 0.874313i \(-0.661313\pi\)
−0.485363 + 0.874313i \(0.661313\pi\)
\(398\) 0 0
\(399\) 944.976 0.118566
\(400\) 0 0
\(401\) −1125.32 −0.140139 −0.0700693 0.997542i \(-0.522322\pi\)
−0.0700693 + 0.997542i \(0.522322\pi\)
\(402\) 0 0
\(403\) −1012.38 −0.125137
\(404\) 0 0
\(405\) −342.721 −0.0420492
\(406\) 0 0
\(407\) −1252.66 −0.152561
\(408\) 0 0
\(409\) −11937.5 −1.44321 −0.721605 0.692305i \(-0.756595\pi\)
−0.721605 + 0.692305i \(0.756595\pi\)
\(410\) 0 0
\(411\) −9627.37 −1.15543
\(412\) 0 0
\(413\) 484.724 0.0577523
\(414\) 0 0
\(415\) 465.436 0.0550539
\(416\) 0 0
\(417\) −3455.84 −0.405834
\(418\) 0 0
\(419\) −16082.3 −1.87511 −0.937554 0.347840i \(-0.886915\pi\)
−0.937554 + 0.347840i \(0.886915\pi\)
\(420\) 0 0
\(421\) 6896.09 0.798326 0.399163 0.916880i \(-0.369301\pi\)
0.399163 + 0.916880i \(0.369301\pi\)
\(422\) 0 0
\(423\) −1698.59 −0.195244
\(424\) 0 0
\(425\) 12169.9 1.38900
\(426\) 0 0
\(427\) 273.143 0.0309562
\(428\) 0 0
\(429\) −2173.37 −0.244595
\(430\) 0 0
\(431\) −9632.03 −1.07647 −0.538235 0.842795i \(-0.680909\pi\)
−0.538235 + 0.842795i \(0.680909\pi\)
\(432\) 0 0
\(433\) 6657.14 0.738850 0.369425 0.929261i \(-0.379555\pi\)
0.369425 + 0.929261i \(0.379555\pi\)
\(434\) 0 0
\(435\) 231.195 0.0254827
\(436\) 0 0
\(437\) −1070.09 −0.117138
\(438\) 0 0
\(439\) 16147.8 1.75556 0.877781 0.479061i \(-0.159023\pi\)
0.877781 + 0.479061i \(0.159023\pi\)
\(440\) 0 0
\(441\) 4528.22 0.488956
\(442\) 0 0
\(443\) −6307.18 −0.676440 −0.338220 0.941067i \(-0.609825\pi\)
−0.338220 + 0.941067i \(0.609825\pi\)
\(444\) 0 0
\(445\) −3080.77 −0.328186
\(446\) 0 0
\(447\) −1768.29 −0.187108
\(448\) 0 0
\(449\) −11001.7 −1.15636 −0.578178 0.815911i \(-0.696236\pi\)
−0.578178 + 0.815911i \(0.696236\pi\)
\(450\) 0 0
\(451\) −505.556 −0.0527843
\(452\) 0 0
\(453\) 7208.37 0.747635
\(454\) 0 0
\(455\) −475.534 −0.0489964
\(456\) 0 0
\(457\) 1258.34 0.128802 0.0644011 0.997924i \(-0.479486\pi\)
0.0644011 + 0.997924i \(0.479486\pi\)
\(458\) 0 0
\(459\) −14978.2 −1.52314
\(460\) 0 0
\(461\) −7461.56 −0.753838 −0.376919 0.926246i \(-0.623016\pi\)
−0.376919 + 0.926246i \(0.623016\pi\)
\(462\) 0 0
\(463\) 7744.02 0.777311 0.388655 0.921383i \(-0.372940\pi\)
0.388655 + 0.921383i \(0.372940\pi\)
\(464\) 0 0
\(465\) −165.955 −0.0165505
\(466\) 0 0
\(467\) −244.138 −0.0241913 −0.0120957 0.999927i \(-0.503850\pi\)
−0.0120957 + 0.999927i \(0.503850\pi\)
\(468\) 0 0
\(469\) −2085.21 −0.205301
\(470\) 0 0
\(471\) −6600.60 −0.645731
\(472\) 0 0
\(473\) −1889.53 −0.183680
\(474\) 0 0
\(475\) 7118.57 0.687626
\(476\) 0 0
\(477\) −2487.80 −0.238802
\(478\) 0 0
\(479\) −4381.25 −0.417922 −0.208961 0.977924i \(-0.567008\pi\)
−0.208961 + 0.977924i \(0.567008\pi\)
\(480\) 0 0
\(481\) −4915.20 −0.465933
\(482\) 0 0
\(483\) −287.886 −0.0271206
\(484\) 0 0
\(485\) −2430.96 −0.227596
\(486\) 0 0
\(487\) 15726.4 1.46331 0.731654 0.681676i \(-0.238749\pi\)
0.731654 + 0.681676i \(0.238749\pi\)
\(488\) 0 0
\(489\) −10399.5 −0.961718
\(490\) 0 0
\(491\) 17159.0 1.57714 0.788568 0.614947i \(-0.210823\pi\)
0.788568 + 0.614947i \(0.210823\pi\)
\(492\) 0 0
\(493\) 2938.34 0.268431
\(494\) 0 0
\(495\) 383.676 0.0348383
\(496\) 0 0
\(497\) 2416.83 0.218128
\(498\) 0 0
\(499\) 21508.3 1.92955 0.964774 0.263081i \(-0.0847386\pi\)
0.964774 + 0.263081i \(0.0847386\pi\)
\(500\) 0 0
\(501\) −5980.84 −0.533342
\(502\) 0 0
\(503\) −22195.2 −1.96746 −0.983731 0.179648i \(-0.942504\pi\)
−0.983731 + 0.179648i \(0.942504\pi\)
\(504\) 0 0
\(505\) 713.742 0.0628933
\(506\) 0 0
\(507\) −606.449 −0.0531229
\(508\) 0 0
\(509\) 11767.9 1.02476 0.512381 0.858758i \(-0.328763\pi\)
0.512381 + 0.858758i \(0.328763\pi\)
\(510\) 0 0
\(511\) 750.592 0.0649790
\(512\) 0 0
\(513\) −8761.24 −0.754032
\(514\) 0 0
\(515\) −3850.88 −0.329495
\(516\) 0 0
\(517\) −1503.79 −0.127924
\(518\) 0 0
\(519\) −10548.2 −0.892125
\(520\) 0 0
\(521\) 8418.16 0.707881 0.353941 0.935268i \(-0.384842\pi\)
0.353941 + 0.935268i \(0.384842\pi\)
\(522\) 0 0
\(523\) 3414.16 0.285451 0.142726 0.989762i \(-0.454413\pi\)
0.142726 + 0.989762i \(0.454413\pi\)
\(524\) 0 0
\(525\) 1915.11 0.159204
\(526\) 0 0
\(527\) −2109.19 −0.174341
\(528\) 0 0
\(529\) −11841.0 −0.973206
\(530\) 0 0
\(531\) −1534.56 −0.125413
\(532\) 0 0
\(533\) −1983.70 −0.161207
\(534\) 0 0
\(535\) 1238.76 0.100105
\(536\) 0 0
\(537\) 15108.1 1.21408
\(538\) 0 0
\(539\) 4008.91 0.320364
\(540\) 0 0
\(541\) 15393.9 1.22335 0.611677 0.791108i \(-0.290495\pi\)
0.611677 + 0.791108i \(0.290495\pi\)
\(542\) 0 0
\(543\) 1160.87 0.0917457
\(544\) 0 0
\(545\) −2183.91 −0.171648
\(546\) 0 0
\(547\) 9547.99 0.746330 0.373165 0.927765i \(-0.378273\pi\)
0.373165 + 0.927765i \(0.378273\pi\)
\(548\) 0 0
\(549\) −864.727 −0.0672234
\(550\) 0 0
\(551\) 1718.73 0.132886
\(552\) 0 0
\(553\) 817.736 0.0628819
\(554\) 0 0
\(555\) −805.726 −0.0616238
\(556\) 0 0
\(557\) 19667.1 1.49609 0.748044 0.663649i \(-0.230993\pi\)
0.748044 + 0.663649i \(0.230993\pi\)
\(558\) 0 0
\(559\) −7414.15 −0.560975
\(560\) 0 0
\(561\) −4527.97 −0.340768
\(562\) 0 0
\(563\) 13151.8 0.984517 0.492258 0.870449i \(-0.336172\pi\)
0.492258 + 0.870449i \(0.336172\pi\)
\(564\) 0 0
\(565\) 1091.25 0.0812553
\(566\) 0 0
\(567\) −685.442 −0.0507687
\(568\) 0 0
\(569\) 17071.3 1.25776 0.628880 0.777502i \(-0.283514\pi\)
0.628880 + 0.777502i \(0.283514\pi\)
\(570\) 0 0
\(571\) −10724.9 −0.786033 −0.393017 0.919531i \(-0.628568\pi\)
−0.393017 + 0.919531i \(0.628568\pi\)
\(572\) 0 0
\(573\) 3297.15 0.240385
\(574\) 0 0
\(575\) −2168.67 −0.157286
\(576\) 0 0
\(577\) −10700.0 −0.772008 −0.386004 0.922497i \(-0.626145\pi\)
−0.386004 + 0.922497i \(0.626145\pi\)
\(578\) 0 0
\(579\) 3804.61 0.273081
\(580\) 0 0
\(581\) 930.872 0.0664700
\(582\) 0 0
\(583\) −2202.49 −0.156463
\(584\) 0 0
\(585\) 1505.47 0.106399
\(586\) 0 0
\(587\) −10143.6 −0.713237 −0.356618 0.934250i \(-0.616070\pi\)
−0.356618 + 0.934250i \(0.616070\pi\)
\(588\) 0 0
\(589\) −1233.73 −0.0863074
\(590\) 0 0
\(591\) −1982.58 −0.137991
\(592\) 0 0
\(593\) −4547.22 −0.314894 −0.157447 0.987527i \(-0.550326\pi\)
−0.157447 + 0.987527i \(0.550326\pi\)
\(594\) 0 0
\(595\) −990.723 −0.0682616
\(596\) 0 0
\(597\) 15235.6 1.04448
\(598\) 0 0
\(599\) 26519.1 1.80892 0.904458 0.426563i \(-0.140276\pi\)
0.904458 + 0.426563i \(0.140276\pi\)
\(600\) 0 0
\(601\) 4927.20 0.334417 0.167209 0.985922i \(-0.446525\pi\)
0.167209 + 0.985922i \(0.446525\pi\)
\(602\) 0 0
\(603\) 6601.45 0.445824
\(604\) 0 0
\(605\) −2603.30 −0.174941
\(606\) 0 0
\(607\) 16406.6 1.09707 0.548535 0.836127i \(-0.315186\pi\)
0.548535 + 0.836127i \(0.315186\pi\)
\(608\) 0 0
\(609\) 462.390 0.0307668
\(610\) 0 0
\(611\) −5900.58 −0.390691
\(612\) 0 0
\(613\) −4914.16 −0.323786 −0.161893 0.986808i \(-0.551760\pi\)
−0.161893 + 0.986808i \(0.551760\pi\)
\(614\) 0 0
\(615\) −325.179 −0.0213211
\(616\) 0 0
\(617\) −186.272 −0.0121540 −0.00607702 0.999982i \(-0.501934\pi\)
−0.00607702 + 0.999982i \(0.501934\pi\)
\(618\) 0 0
\(619\) 408.125 0.0265007 0.0132503 0.999912i \(-0.495782\pi\)
0.0132503 + 0.999912i \(0.495782\pi\)
\(620\) 0 0
\(621\) 2669.10 0.172476
\(622\) 0 0
\(623\) −6161.55 −0.396240
\(624\) 0 0
\(625\) 13815.5 0.884194
\(626\) 0 0
\(627\) −2648.56 −0.168697
\(628\) 0 0
\(629\) −10240.3 −0.649136
\(630\) 0 0
\(631\) 16001.4 1.00952 0.504760 0.863260i \(-0.331581\pi\)
0.504760 + 0.863260i \(0.331581\pi\)
\(632\) 0 0
\(633\) 2497.66 0.156830
\(634\) 0 0
\(635\) −1467.78 −0.0917275
\(636\) 0 0
\(637\) 15730.2 0.978417
\(638\) 0 0
\(639\) −7651.30 −0.473679
\(640\) 0 0
\(641\) −16188.4 −0.997510 −0.498755 0.866743i \(-0.666209\pi\)
−0.498755 + 0.866743i \(0.666209\pi\)
\(642\) 0 0
\(643\) −24550.3 −1.50571 −0.752854 0.658187i \(-0.771323\pi\)
−0.752854 + 0.658187i \(0.771323\pi\)
\(644\) 0 0
\(645\) −1215.37 −0.0741939
\(646\) 0 0
\(647\) 4566.00 0.277447 0.138723 0.990331i \(-0.455700\pi\)
0.138723 + 0.990331i \(0.455700\pi\)
\(648\) 0 0
\(649\) −1358.57 −0.0821705
\(650\) 0 0
\(651\) −331.911 −0.0199825
\(652\) 0 0
\(653\) 12985.7 0.778209 0.389105 0.921194i \(-0.372784\pi\)
0.389105 + 0.921194i \(0.372784\pi\)
\(654\) 0 0
\(655\) 3220.84 0.192135
\(656\) 0 0
\(657\) −2376.26 −0.141106
\(658\) 0 0
\(659\) −22004.9 −1.30074 −0.650370 0.759618i \(-0.725386\pi\)
−0.650370 + 0.759618i \(0.725386\pi\)
\(660\) 0 0
\(661\) −10592.0 −0.623267 −0.311634 0.950202i \(-0.600876\pi\)
−0.311634 + 0.950202i \(0.600876\pi\)
\(662\) 0 0
\(663\) −17766.8 −1.04073
\(664\) 0 0
\(665\) −579.506 −0.0337929
\(666\) 0 0
\(667\) −523.610 −0.0303962
\(668\) 0 0
\(669\) 13933.7 0.805246
\(670\) 0 0
\(671\) −765.558 −0.0440448
\(672\) 0 0
\(673\) −131.859 −0.00755242 −0.00377621 0.999993i \(-0.501202\pi\)
−0.00377621 + 0.999993i \(0.501202\pi\)
\(674\) 0 0
\(675\) −17755.7 −1.01247
\(676\) 0 0
\(677\) −15415.4 −0.875130 −0.437565 0.899187i \(-0.644159\pi\)
−0.437565 + 0.899187i \(0.644159\pi\)
\(678\) 0 0
\(679\) −4861.92 −0.274791
\(680\) 0 0
\(681\) 12297.8 0.692003
\(682\) 0 0
\(683\) −26452.5 −1.48196 −0.740978 0.671530i \(-0.765638\pi\)
−0.740978 + 0.671530i \(0.765638\pi\)
\(684\) 0 0
\(685\) 5903.98 0.329313
\(686\) 0 0
\(687\) 9718.76 0.539729
\(688\) 0 0
\(689\) −8642.13 −0.477850
\(690\) 0 0
\(691\) −20331.6 −1.11932 −0.559661 0.828721i \(-0.689069\pi\)
−0.559661 + 0.828721i \(0.689069\pi\)
\(692\) 0 0
\(693\) 767.351 0.0420624
\(694\) 0 0
\(695\) 2119.29 0.115668
\(696\) 0 0
\(697\) −4132.82 −0.224593
\(698\) 0 0
\(699\) −17414.2 −0.942294
\(700\) 0 0
\(701\) −13325.5 −0.717973 −0.358987 0.933343i \(-0.616878\pi\)
−0.358987 + 0.933343i \(0.616878\pi\)
\(702\) 0 0
\(703\) −5989.87 −0.321354
\(704\) 0 0
\(705\) −967.256 −0.0516723
\(706\) 0 0
\(707\) 1427.48 0.0759350
\(708\) 0 0
\(709\) −33177.3 −1.75741 −0.878703 0.477370i \(-0.841590\pi\)
−0.878703 + 0.477370i \(0.841590\pi\)
\(710\) 0 0
\(711\) −2588.82 −0.136552
\(712\) 0 0
\(713\) 375.855 0.0197418
\(714\) 0 0
\(715\) 1332.82 0.0697125
\(716\) 0 0
\(717\) 17555.9 0.914415
\(718\) 0 0
\(719\) 12840.3 0.666013 0.333007 0.942924i \(-0.391937\pi\)
0.333007 + 0.942924i \(0.391937\pi\)
\(720\) 0 0
\(721\) −7701.76 −0.397820
\(722\) 0 0
\(723\) 11165.3 0.574330
\(724\) 0 0
\(725\) 3483.22 0.178432
\(726\) 0 0
\(727\) −23615.7 −1.20475 −0.602377 0.798212i \(-0.705780\pi\)
−0.602377 + 0.798212i \(0.705780\pi\)
\(728\) 0 0
\(729\) 16561.0 0.841386
\(730\) 0 0
\(731\) −15446.6 −0.781548
\(732\) 0 0
\(733\) 23648.0 1.19162 0.595812 0.803124i \(-0.296830\pi\)
0.595812 + 0.803124i \(0.296830\pi\)
\(734\) 0 0
\(735\) 2578.58 0.129404
\(736\) 0 0
\(737\) 5844.38 0.292104
\(738\) 0 0
\(739\) −18418.8 −0.916841 −0.458420 0.888735i \(-0.651585\pi\)
−0.458420 + 0.888735i \(0.651585\pi\)
\(740\) 0 0
\(741\) −10392.4 −0.515215
\(742\) 0 0
\(743\) 19005.1 0.938396 0.469198 0.883093i \(-0.344543\pi\)
0.469198 + 0.883093i \(0.344543\pi\)
\(744\) 0 0
\(745\) 1084.40 0.0533280
\(746\) 0 0
\(747\) −2946.99 −0.144344
\(748\) 0 0
\(749\) 2477.51 0.120863
\(750\) 0 0
\(751\) 26136.1 1.26993 0.634967 0.772540i \(-0.281014\pi\)
0.634967 + 0.772540i \(0.281014\pi\)
\(752\) 0 0
\(753\) −7964.08 −0.385428
\(754\) 0 0
\(755\) −4420.53 −0.213085
\(756\) 0 0
\(757\) −19642.3 −0.943081 −0.471540 0.881844i \(-0.656302\pi\)
−0.471540 + 0.881844i \(0.656302\pi\)
\(758\) 0 0
\(759\) 806.880 0.0385874
\(760\) 0 0
\(761\) 36349.8 1.73151 0.865754 0.500469i \(-0.166839\pi\)
0.865754 + 0.500469i \(0.166839\pi\)
\(762\) 0 0
\(763\) −4367.81 −0.207242
\(764\) 0 0
\(765\) 3136.47 0.148234
\(766\) 0 0
\(767\) −5330.77 −0.250955
\(768\) 0 0
\(769\) −4940.72 −0.231686 −0.115843 0.993268i \(-0.536957\pi\)
−0.115843 + 0.993268i \(0.536957\pi\)
\(770\) 0 0
\(771\) −1899.96 −0.0887490
\(772\) 0 0
\(773\) −4543.81 −0.211422 −0.105711 0.994397i \(-0.533712\pi\)
−0.105711 + 0.994397i \(0.533712\pi\)
\(774\) 0 0
\(775\) −2500.31 −0.115889
\(776\) 0 0
\(777\) −1611.45 −0.0744023
\(778\) 0 0
\(779\) −2417.42 −0.111185
\(780\) 0 0
\(781\) −6773.83 −0.310354
\(782\) 0 0
\(783\) −4287.00 −0.195664
\(784\) 0 0
\(785\) 4047.81 0.184042
\(786\) 0 0
\(787\) 18523.9 0.839016 0.419508 0.907752i \(-0.362203\pi\)
0.419508 + 0.907752i \(0.362203\pi\)
\(788\) 0 0
\(789\) 17640.9 0.795984
\(790\) 0 0
\(791\) 2182.50 0.0981047
\(792\) 0 0
\(793\) −3003.90 −0.134516
\(794\) 0 0
\(795\) −1416.67 −0.0631999
\(796\) 0 0
\(797\) 9666.66 0.429624 0.214812 0.976655i \(-0.431086\pi\)
0.214812 + 0.976655i \(0.431086\pi\)
\(798\) 0 0
\(799\) −12293.2 −0.544309
\(800\) 0 0
\(801\) 19506.5 0.860459
\(802\) 0 0
\(803\) −2103.74 −0.0924526
\(804\) 0 0
\(805\) 176.546 0.00772972
\(806\) 0 0
\(807\) 17107.4 0.746232
\(808\) 0 0
\(809\) 8733.19 0.379533 0.189767 0.981829i \(-0.439227\pi\)
0.189767 + 0.981829i \(0.439227\pi\)
\(810\) 0 0
\(811\) 14082.6 0.609750 0.304875 0.952392i \(-0.401385\pi\)
0.304875 + 0.952392i \(0.401385\pi\)
\(812\) 0 0
\(813\) 29520.7 1.27348
\(814\) 0 0
\(815\) 6377.47 0.274102
\(816\) 0 0
\(817\) −9035.19 −0.386905
\(818\) 0 0
\(819\) 3010.93 0.128462
\(820\) 0 0
\(821\) −15741.2 −0.669149 −0.334574 0.942369i \(-0.608593\pi\)
−0.334574 + 0.942369i \(0.608593\pi\)
\(822\) 0 0
\(823\) −2136.76 −0.0905014 −0.0452507 0.998976i \(-0.514409\pi\)
−0.0452507 + 0.998976i \(0.514409\pi\)
\(824\) 0 0
\(825\) −5367.62 −0.226517
\(826\) 0 0
\(827\) 2966.17 0.124721 0.0623603 0.998054i \(-0.480137\pi\)
0.0623603 + 0.998054i \(0.480137\pi\)
\(828\) 0 0
\(829\) −28562.2 −1.19663 −0.598316 0.801260i \(-0.704163\pi\)
−0.598316 + 0.801260i \(0.704163\pi\)
\(830\) 0 0
\(831\) 17906.6 0.747500
\(832\) 0 0
\(833\) 32772.0 1.36313
\(834\) 0 0
\(835\) 3667.75 0.152009
\(836\) 0 0
\(837\) 3077.28 0.127080
\(838\) 0 0
\(839\) −9658.72 −0.397445 −0.198722 0.980056i \(-0.563679\pi\)
−0.198722 + 0.980056i \(0.563679\pi\)
\(840\) 0 0
\(841\) 841.000 0.0344828
\(842\) 0 0
\(843\) −292.760 −0.0119611
\(844\) 0 0
\(845\) 371.904 0.0151407
\(846\) 0 0
\(847\) −5206.61 −0.211217
\(848\) 0 0
\(849\) −768.597 −0.0310697
\(850\) 0 0
\(851\) 1824.81 0.0735060
\(852\) 0 0
\(853\) −5702.59 −0.228901 −0.114451 0.993429i \(-0.536511\pi\)
−0.114451 + 0.993429i \(0.536511\pi\)
\(854\) 0 0
\(855\) 1834.62 0.0733834
\(856\) 0 0
\(857\) −11141.0 −0.444071 −0.222035 0.975039i \(-0.571270\pi\)
−0.222035 + 0.975039i \(0.571270\pi\)
\(858\) 0 0
\(859\) −15527.7 −0.616763 −0.308382 0.951263i \(-0.599787\pi\)
−0.308382 + 0.951263i \(0.599787\pi\)
\(860\) 0 0
\(861\) −650.358 −0.0257423
\(862\) 0 0
\(863\) 19168.3 0.756080 0.378040 0.925789i \(-0.376598\pi\)
0.378040 + 0.925789i \(0.376598\pi\)
\(864\) 0 0
\(865\) 6468.65 0.254267
\(866\) 0 0
\(867\) −19301.1 −0.756057
\(868\) 0 0
\(869\) −2291.93 −0.0894689
\(870\) 0 0
\(871\) 22932.2 0.892110
\(872\) 0 0
\(873\) 15392.1 0.596727
\(874\) 0 0
\(875\) −2396.68 −0.0925973
\(876\) 0 0
\(877\) −33530.4 −1.29104 −0.645519 0.763744i \(-0.723359\pi\)
−0.645519 + 0.763744i \(0.723359\pi\)
\(878\) 0 0
\(879\) 22882.3 0.878043
\(880\) 0 0
\(881\) −1779.67 −0.0680574 −0.0340287 0.999421i \(-0.510834\pi\)
−0.0340287 + 0.999421i \(0.510834\pi\)
\(882\) 0 0
\(883\) 47347.6 1.80450 0.902250 0.431212i \(-0.141914\pi\)
0.902250 + 0.431212i \(0.141914\pi\)
\(884\) 0 0
\(885\) −873.849 −0.0331911
\(886\) 0 0
\(887\) 33076.7 1.25210 0.626048 0.779785i \(-0.284672\pi\)
0.626048 + 0.779785i \(0.284672\pi\)
\(888\) 0 0
\(889\) −2935.56 −0.110748
\(890\) 0 0
\(891\) 1921.14 0.0722341
\(892\) 0 0
\(893\) −7190.70 −0.269460
\(894\) 0 0
\(895\) −9265.03 −0.346029
\(896\) 0 0
\(897\) 3166.03 0.117849
\(898\) 0 0
\(899\) −603.683 −0.0223959
\(900\) 0 0
\(901\) −18004.9 −0.665739
\(902\) 0 0
\(903\) −2430.74 −0.0895790
\(904\) 0 0
\(905\) −711.906 −0.0261487
\(906\) 0 0
\(907\) 2824.54 0.103404 0.0517019 0.998663i \(-0.483535\pi\)
0.0517019 + 0.998663i \(0.483535\pi\)
\(908\) 0 0
\(909\) −4519.19 −0.164898
\(910\) 0 0
\(911\) 41606.9 1.51317 0.756586 0.653895i \(-0.226866\pi\)
0.756586 + 0.653895i \(0.226866\pi\)
\(912\) 0 0
\(913\) −2609.03 −0.0945741
\(914\) 0 0
\(915\) −492.415 −0.0177910
\(916\) 0 0
\(917\) 6441.68 0.231977
\(918\) 0 0
\(919\) 618.058 0.0221848 0.0110924 0.999938i \(-0.496469\pi\)
0.0110924 + 0.999938i \(0.496469\pi\)
\(920\) 0 0
\(921\) −20322.3 −0.727081
\(922\) 0 0
\(923\) −26579.1 −0.947848
\(924\) 0 0
\(925\) −12139.2 −0.431497
\(926\) 0 0
\(927\) 24382.5 0.863892
\(928\) 0 0
\(929\) −8931.48 −0.315428 −0.157714 0.987485i \(-0.550412\pi\)
−0.157714 + 0.987485i \(0.550412\pi\)
\(930\) 0 0
\(931\) 19169.4 0.674815
\(932\) 0 0
\(933\) 2116.45 0.0742652
\(934\) 0 0
\(935\) 2776.77 0.0971232
\(936\) 0 0
\(937\) −37657.4 −1.31293 −0.656464 0.754357i \(-0.727949\pi\)
−0.656464 + 0.754357i \(0.727949\pi\)
\(938\) 0 0
\(939\) 4506.76 0.156627
\(940\) 0 0
\(941\) 43174.4 1.49569 0.747846 0.663872i \(-0.231088\pi\)
0.747846 + 0.663872i \(0.231088\pi\)
\(942\) 0 0
\(943\) 736.464 0.0254322
\(944\) 0 0
\(945\) 1445.45 0.0497572
\(946\) 0 0
\(947\) −36963.1 −1.26836 −0.634181 0.773185i \(-0.718663\pi\)
−0.634181 + 0.773185i \(0.718663\pi\)
\(948\) 0 0
\(949\) −8254.66 −0.282358
\(950\) 0 0
\(951\) −17385.1 −0.592799
\(952\) 0 0
\(953\) 32697.1 1.11140 0.555699 0.831384i \(-0.312451\pi\)
0.555699 + 0.831384i \(0.312451\pi\)
\(954\) 0 0
\(955\) −2021.98 −0.0685127
\(956\) 0 0
\(957\) −1295.98 −0.0437753
\(958\) 0 0
\(959\) 11808.0 0.397601
\(960\) 0 0
\(961\) −29357.7 −0.985454
\(962\) 0 0
\(963\) −7843.42 −0.262462
\(964\) 0 0
\(965\) −2333.17 −0.0778316
\(966\) 0 0
\(967\) 4649.77 0.154629 0.0773146 0.997007i \(-0.475365\pi\)
0.0773146 + 0.997007i \(0.475365\pi\)
\(968\) 0 0
\(969\) −21651.4 −0.717795
\(970\) 0 0
\(971\) 27533.7 0.909987 0.454994 0.890495i \(-0.349642\pi\)
0.454994 + 0.890495i \(0.349642\pi\)
\(972\) 0 0
\(973\) 4238.58 0.139653
\(974\) 0 0
\(975\) −21061.5 −0.691801
\(976\) 0 0
\(977\) −16474.7 −0.539482 −0.269741 0.962933i \(-0.586938\pi\)
−0.269741 + 0.962933i \(0.586938\pi\)
\(978\) 0 0
\(979\) 17269.4 0.563773
\(980\) 0 0
\(981\) 13827.8 0.450038
\(982\) 0 0
\(983\) 26969.9 0.875084 0.437542 0.899198i \(-0.355849\pi\)
0.437542 + 0.899198i \(0.355849\pi\)
\(984\) 0 0
\(985\) 1215.82 0.0393291
\(986\) 0 0
\(987\) −1934.51 −0.0623872
\(988\) 0 0
\(989\) 2752.56 0.0884999
\(990\) 0 0
\(991\) 6331.81 0.202963 0.101482 0.994837i \(-0.467642\pi\)
0.101482 + 0.994837i \(0.467642\pi\)
\(992\) 0 0
\(993\) −29881.0 −0.954929
\(994\) 0 0
\(995\) −9343.23 −0.297689
\(996\) 0 0
\(997\) −53115.7 −1.68725 −0.843626 0.536932i \(-0.819583\pi\)
−0.843626 + 0.536932i \(0.819583\pi\)
\(998\) 0 0
\(999\) 14940.4 0.473167
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 116.4.a.a.1.1 2
3.2 odd 2 1044.4.a.d.1.1 2
4.3 odd 2 464.4.a.d.1.2 2
8.3 odd 2 1856.4.a.k.1.1 2
8.5 even 2 1856.4.a.j.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
116.4.a.a.1.1 2 1.1 even 1 trivial
464.4.a.d.1.2 2 4.3 odd 2
1044.4.a.d.1.1 2 3.2 odd 2
1856.4.a.j.1.2 2 8.5 even 2
1856.4.a.k.1.1 2 8.3 odd 2