Properties

Label 2368.2.a.bg.1.2
Level $2368$
Weight $2$
Character 2368.1
Self dual yes
Analytic conductor $18.909$
Analytic rank $1$
Dimension $4$
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] = [2368,2,Mod(1,2368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2368.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2368 = 2^{6} \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2368.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: \(18.9085751986\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.48389.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 2x^{3} - 8x^{2} + 3x + 10 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 296)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.26726\) of defining polynomial
Character \(\chi\) \(=\) 2368.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.26726 q^{3} +2.21602 q^{5} +3.07555 q^{7} +2.14048 q^{9} +O(q^{10})\) \(q-2.26726 q^{3} +2.21602 q^{5} +3.07555 q^{7} +2.14048 q^{9} -1.14048 q^{11} -2.80828 q^{13} -5.02431 q^{15} +2.00000 q^{17} -3.89752 q^{19} -6.97307 q^{21} -7.62376 q^{23} -0.0892395 q^{25} +1.94876 q^{27} +3.34281 q^{29} -10.3671 q^{31} +2.58576 q^{33} +6.81548 q^{35} -1.00000 q^{37} +6.36711 q^{39} +2.54822 q^{41} -8.43205 q^{43} +4.74335 q^{45} -7.17802 q^{47} +2.45898 q^{49} -4.53452 q^{51} -5.61007 q^{53} -2.52733 q^{55} +8.83671 q^{57} +6.17848 q^{59} +10.6481 q^{61} +6.58314 q^{63} -6.22322 q^{65} +0.318501 q^{67} +17.2851 q^{69} +6.42555 q^{71} +16.0046 q^{73} +0.202329 q^{75} -3.50759 q^{77} -8.69281 q^{79} -10.8398 q^{81} +9.76116 q^{83} +4.43205 q^{85} -7.57902 q^{87} -8.25357 q^{89} -8.63700 q^{91} +23.5050 q^{93} -8.63700 q^{95} -13.2201 q^{97} -2.44117 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2 q^{3} - 5 q^{5} - q^{7} + 8 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2 q^{3} - 5 q^{5} - q^{7} + 8 q^{9} - 4 q^{11} - 5 q^{13} + 8 q^{17} - 2 q^{19} - q^{21} - 9 q^{23} + 7 q^{25} + q^{27} - 7 q^{29} - q^{31} + 3 q^{33} + 12 q^{35} - 4 q^{37} - 15 q^{39} + 2 q^{41} - 6 q^{43} - 29 q^{47} + 9 q^{49} - 4 q^{51} + 5 q^{53} - 5 q^{55} - 32 q^{57} + 10 q^{59} + q^{61} - 28 q^{63} - 2 q^{65} + q^{67} + 27 q^{69} - 17 q^{71} + 8 q^{73} - 19 q^{75} + 27 q^{77} + 15 q^{79} - 8 q^{81} - 15 q^{83} - 10 q^{85} - 17 q^{87} - 20 q^{89} - 34 q^{91} + 6 q^{93} - 34 q^{95} + 2 q^{97} - 44 q^{99}+O(q^{100}) \) 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
\(3\) −2.26726 −1.30900 −0.654502 0.756060i \(-0.727122\pi\)
−0.654502 + 0.756060i \(0.727122\pi\)
\(4\) 0 0
\(5\) 2.21602 0.991036 0.495518 0.868598i \(-0.334978\pi\)
0.495518 + 0.868598i \(0.334978\pi\)
\(6\) 0 0
\(7\) 3.07555 1.16245 0.581223 0.813744i \(-0.302574\pi\)
0.581223 + 0.813744i \(0.302574\pi\)
\(8\) 0 0
\(9\) 2.14048 0.713493
\(10\) 0 0
\(11\) −1.14048 −0.343867 −0.171934 0.985109i \(-0.555001\pi\)
−0.171934 + 0.985109i \(0.555001\pi\)
\(12\) 0 0
\(13\) −2.80828 −0.778878 −0.389439 0.921052i \(-0.627331\pi\)
−0.389439 + 0.921052i \(0.627331\pi\)
\(14\) 0 0
\(15\) −5.02431 −1.29727
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −3.89752 −0.894153 −0.447077 0.894496i \(-0.647535\pi\)
−0.447077 + 0.894496i \(0.647535\pi\)
\(20\) 0 0
\(21\) −6.97307 −1.52165
\(22\) 0 0
\(23\) −7.62376 −1.58966 −0.794832 0.606829i \(-0.792441\pi\)
−0.794832 + 0.606829i \(0.792441\pi\)
\(24\) 0 0
\(25\) −0.0892395 −0.0178479
\(26\) 0 0
\(27\) 1.94876 0.375039
\(28\) 0 0
\(29\) 3.34281 0.620744 0.310372 0.950615i \(-0.399546\pi\)
0.310372 + 0.950615i \(0.399546\pi\)
\(30\) 0 0
\(31\) −10.3671 −1.86199 −0.930994 0.365034i \(-0.881057\pi\)
−0.930994 + 0.365034i \(0.881057\pi\)
\(32\) 0 0
\(33\) 2.58576 0.450124
\(34\) 0 0
\(35\) 6.81548 1.15203
\(36\) 0 0
\(37\) −1.00000 −0.164399
\(38\) 0 0
\(39\) 6.36711 1.01955
\(40\) 0 0
\(41\) 2.54822 0.397965 0.198982 0.980003i \(-0.436236\pi\)
0.198982 + 0.980003i \(0.436236\pi\)
\(42\) 0 0
\(43\) −8.43205 −1.28588 −0.642938 0.765919i \(-0.722285\pi\)
−0.642938 + 0.765919i \(0.722285\pi\)
\(44\) 0 0
\(45\) 4.74335 0.707097
\(46\) 0 0
\(47\) −7.17802 −1.04702 −0.523511 0.852019i \(-0.675378\pi\)
−0.523511 + 0.852019i \(0.675378\pi\)
\(48\) 0 0
\(49\) 2.45898 0.351283
\(50\) 0 0
\(51\) −4.53452 −0.634960
\(52\) 0 0
\(53\) −5.61007 −0.770602 −0.385301 0.922791i \(-0.625902\pi\)
−0.385301 + 0.922791i \(0.625902\pi\)
\(54\) 0 0
\(55\) −2.52733 −0.340785
\(56\) 0 0
\(57\) 8.83671 1.17045
\(58\) 0 0
\(59\) 6.17848 0.804369 0.402185 0.915559i \(-0.368251\pi\)
0.402185 + 0.915559i \(0.368251\pi\)
\(60\) 0 0
\(61\) 10.6481 1.36335 0.681673 0.731657i \(-0.261253\pi\)
0.681673 + 0.731657i \(0.261253\pi\)
\(62\) 0 0
\(63\) 6.58314 0.829397
\(64\) 0 0
\(65\) −6.22322 −0.771896
\(66\) 0 0
\(67\) 0.318501 0.0389111 0.0194555 0.999811i \(-0.493807\pi\)
0.0194555 + 0.999811i \(0.493807\pi\)
\(68\) 0 0
\(69\) 17.2851 2.08088
\(70\) 0 0
\(71\) 6.42555 0.762573 0.381286 0.924457i \(-0.375481\pi\)
0.381286 + 0.924457i \(0.375481\pi\)
\(72\) 0 0
\(73\) 16.0046 1.87319 0.936597 0.350409i \(-0.113957\pi\)
0.936597 + 0.350409i \(0.113957\pi\)
\(74\) 0 0
\(75\) 0.202329 0.0233630
\(76\) 0 0
\(77\) −3.50759 −0.399727
\(78\) 0 0
\(79\) −8.69281 −0.978018 −0.489009 0.872279i \(-0.662641\pi\)
−0.489009 + 0.872279i \(0.662641\pi\)
\(80\) 0 0
\(81\) −10.8398 −1.20442
\(82\) 0 0
\(83\) 9.76116 1.07143 0.535713 0.844400i \(-0.320043\pi\)
0.535713 + 0.844400i \(0.320043\pi\)
\(84\) 0 0
\(85\) 4.43205 0.480723
\(86\) 0 0
\(87\) −7.57902 −0.812556
\(88\) 0 0
\(89\) −8.25357 −0.874876 −0.437438 0.899248i \(-0.644114\pi\)
−0.437438 + 0.899248i \(0.644114\pi\)
\(90\) 0 0
\(91\) −8.63700 −0.905404
\(92\) 0 0
\(93\) 23.5050 2.43735
\(94\) 0 0
\(95\) −8.63700 −0.886138
\(96\) 0 0
\(97\) −13.2201 −1.34230 −0.671151 0.741321i \(-0.734200\pi\)
−0.671151 + 0.741321i \(0.734200\pi\)
\(98\) 0 0
\(99\) −2.44117 −0.245347
\(100\) 0 0
\(101\) −16.3231 −1.62421 −0.812103 0.583514i \(-0.801677\pi\)
−0.812103 + 0.583514i \(0.801677\pi\)
\(102\) 0 0
\(103\) 9.93507 0.978931 0.489466 0.872023i \(-0.337192\pi\)
0.489466 + 0.872023i \(0.337192\pi\)
\(104\) 0 0
\(105\) −15.4525 −1.50801
\(106\) 0 0
\(107\) −5.34281 −0.516509 −0.258254 0.966077i \(-0.583147\pi\)
−0.258254 + 0.966077i \(0.583147\pi\)
\(108\) 0 0
\(109\) −5.79505 −0.555065 −0.277532 0.960716i \(-0.589517\pi\)
−0.277532 + 0.960716i \(0.589517\pi\)
\(110\) 0 0
\(111\) 2.26726 0.215199
\(112\) 0 0
\(113\) 11.0690 1.04129 0.520644 0.853774i \(-0.325692\pi\)
0.520644 + 0.853774i \(0.325692\pi\)
\(114\) 0 0
\(115\) −16.8944 −1.57541
\(116\) 0 0
\(117\) −6.01107 −0.555724
\(118\) 0 0
\(119\) 6.15109 0.563870
\(120\) 0 0
\(121\) −9.69931 −0.881755
\(122\) 0 0
\(123\) −5.77748 −0.520938
\(124\) 0 0
\(125\) −11.2779 −1.00872
\(126\) 0 0
\(127\) −2.43854 −0.216386 −0.108193 0.994130i \(-0.534506\pi\)
−0.108193 + 0.994130i \(0.534506\pi\)
\(128\) 0 0
\(129\) 19.1177 1.68322
\(130\) 0 0
\(131\) −12.3296 −1.07724 −0.538620 0.842549i \(-0.681054\pi\)
−0.538620 + 0.842549i \(0.681054\pi\)
\(132\) 0 0
\(133\) −11.9870 −1.03941
\(134\) 0 0
\(135\) 4.31850 0.371677
\(136\) 0 0
\(137\) −13.5516 −1.15779 −0.578897 0.815401i \(-0.696517\pi\)
−0.578897 + 0.815401i \(0.696517\pi\)
\(138\) 0 0
\(139\) 6.20690 0.526463 0.263231 0.964733i \(-0.415212\pi\)
0.263231 + 0.964733i \(0.415212\pi\)
\(140\) 0 0
\(141\) 16.2745 1.37056
\(142\) 0 0
\(143\) 3.20279 0.267830
\(144\) 0 0
\(145\) 7.40774 0.615179
\(146\) 0 0
\(147\) −5.57515 −0.459831
\(148\) 0 0
\(149\) −2.26006 −0.185152 −0.0925758 0.995706i \(-0.529510\pi\)
−0.0925758 + 0.995706i \(0.529510\pi\)
\(150\) 0 0
\(151\) 17.5011 1.42422 0.712110 0.702068i \(-0.247740\pi\)
0.712110 + 0.702068i \(0.247740\pi\)
\(152\) 0 0
\(153\) 4.28096 0.346095
\(154\) 0 0
\(155\) −22.9738 −1.84530
\(156\) 0 0
\(157\) −11.9661 −0.955000 −0.477500 0.878632i \(-0.658457\pi\)
−0.477500 + 0.878632i \(0.658457\pi\)
\(158\) 0 0
\(159\) 12.7195 1.00872
\(160\) 0 0
\(161\) −23.4472 −1.84790
\(162\) 0 0
\(163\) −1.18452 −0.0927787 −0.0463893 0.998923i \(-0.514771\pi\)
−0.0463893 + 0.998923i \(0.514771\pi\)
\(164\) 0 0
\(165\) 5.73011 0.446089
\(166\) 0 0
\(167\) −20.2160 −1.56436 −0.782181 0.623051i \(-0.785893\pi\)
−0.782181 + 0.623051i \(0.785893\pi\)
\(168\) 0 0
\(169\) −5.11355 −0.393350
\(170\) 0 0
\(171\) −8.34256 −0.637972
\(172\) 0 0
\(173\) −22.6791 −1.72426 −0.862131 0.506686i \(-0.830870\pi\)
−0.862131 + 0.506686i \(0.830870\pi\)
\(174\) 0 0
\(175\) −0.274460 −0.0207472
\(176\) 0 0
\(177\) −14.0082 −1.05292
\(178\) 0 0
\(179\) 14.5831 1.08999 0.544997 0.838438i \(-0.316531\pi\)
0.544997 + 0.838438i \(0.316531\pi\)
\(180\) 0 0
\(181\) −24.7277 −1.83800 −0.918999 0.394260i \(-0.871001\pi\)
−0.918999 + 0.394260i \(0.871001\pi\)
\(182\) 0 0
\(183\) −24.1420 −1.78463
\(184\) 0 0
\(185\) −2.21602 −0.162925
\(186\) 0 0
\(187\) −2.28096 −0.166800
\(188\) 0 0
\(189\) 5.99350 0.435963
\(190\) 0 0
\(191\) −14.7241 −1.06540 −0.532698 0.846305i \(-0.678822\pi\)
−0.532698 + 0.846305i \(0.678822\pi\)
\(192\) 0 0
\(193\) 18.1997 1.31004 0.655022 0.755610i \(-0.272659\pi\)
0.655022 + 0.755610i \(0.272659\pi\)
\(194\) 0 0
\(195\) 14.1097 1.01041
\(196\) 0 0
\(197\) −8.52803 −0.607597 −0.303798 0.952736i \(-0.598255\pi\)
−0.303798 + 0.952736i \(0.598255\pi\)
\(198\) 0 0
\(199\) 22.7130 1.61008 0.805041 0.593219i \(-0.202143\pi\)
0.805041 + 0.593219i \(0.202143\pi\)
\(200\) 0 0
\(201\) −0.722125 −0.0509348
\(202\) 0 0
\(203\) 10.2810 0.721582
\(204\) 0 0
\(205\) 5.64691 0.394397
\(206\) 0 0
\(207\) −16.3185 −1.13421
\(208\) 0 0
\(209\) 4.44504 0.307470
\(210\) 0 0
\(211\) 6.22252 0.428376 0.214188 0.976792i \(-0.431289\pi\)
0.214188 + 0.976792i \(0.431289\pi\)
\(212\) 0 0
\(213\) −14.5684 −0.998211
\(214\) 0 0
\(215\) −18.6856 −1.27435
\(216\) 0 0
\(217\) −31.8845 −2.16446
\(218\) 0 0
\(219\) −36.2866 −2.45202
\(220\) 0 0
\(221\) −5.61657 −0.377811
\(222\) 0 0
\(223\) −12.6040 −0.844028 −0.422014 0.906589i \(-0.638677\pi\)
−0.422014 + 0.906589i \(0.638677\pi\)
\(224\) 0 0
\(225\) −0.191015 −0.0127344
\(226\) 0 0
\(227\) 26.3652 1.74992 0.874960 0.484196i \(-0.160888\pi\)
0.874960 + 0.484196i \(0.160888\pi\)
\(228\) 0 0
\(229\) 12.2957 0.812522 0.406261 0.913757i \(-0.366832\pi\)
0.406261 + 0.913757i \(0.366832\pi\)
\(230\) 0 0
\(231\) 7.95263 0.523245
\(232\) 0 0
\(233\) 14.1439 0.926597 0.463299 0.886202i \(-0.346666\pi\)
0.463299 + 0.886202i \(0.346666\pi\)
\(234\) 0 0
\(235\) −15.9067 −1.03764
\(236\) 0 0
\(237\) 19.7089 1.28023
\(238\) 0 0
\(239\) 13.7171 0.887287 0.443643 0.896203i \(-0.353686\pi\)
0.443643 + 0.896203i \(0.353686\pi\)
\(240\) 0 0
\(241\) 28.9127 1.86243 0.931216 0.364469i \(-0.118749\pi\)
0.931216 + 0.364469i \(0.118749\pi\)
\(242\) 0 0
\(243\) 18.7304 1.20155
\(244\) 0 0
\(245\) 5.44916 0.348134
\(246\) 0 0
\(247\) 10.9453 0.696436
\(248\) 0 0
\(249\) −22.1311 −1.40250
\(250\) 0 0
\(251\) −21.3712 −1.34894 −0.674470 0.738302i \(-0.735628\pi\)
−0.674470 + 0.738302i \(0.735628\pi\)
\(252\) 0 0
\(253\) 8.69474 0.546633
\(254\) 0 0
\(255\) −10.0486 −0.629269
\(256\) 0 0
\(257\) 3.64395 0.227304 0.113652 0.993521i \(-0.463745\pi\)
0.113652 + 0.993521i \(0.463745\pi\)
\(258\) 0 0
\(259\) −3.07555 −0.191105
\(260\) 0 0
\(261\) 7.15521 0.442896
\(262\) 0 0
\(263\) −18.2062 −1.12264 −0.561321 0.827598i \(-0.689707\pi\)
−0.561321 + 0.827598i \(0.689707\pi\)
\(264\) 0 0
\(265\) −12.4320 −0.763695
\(266\) 0 0
\(267\) 18.7130 1.14522
\(268\) 0 0
\(269\) −8.04861 −0.490733 −0.245366 0.969430i \(-0.578908\pi\)
−0.245366 + 0.969430i \(0.578908\pi\)
\(270\) 0 0
\(271\) −13.8159 −0.839258 −0.419629 0.907696i \(-0.637840\pi\)
−0.419629 + 0.907696i \(0.637840\pi\)
\(272\) 0 0
\(273\) 19.5823 1.18518
\(274\) 0 0
\(275\) 0.101776 0.00613731
\(276\) 0 0
\(277\) 2.21602 0.133148 0.0665740 0.997781i \(-0.478793\pi\)
0.0665740 + 0.997781i \(0.478793\pi\)
\(278\) 0 0
\(279\) −22.1906 −1.32852
\(280\) 0 0
\(281\) −2.10248 −0.125423 −0.0627116 0.998032i \(-0.519975\pi\)
−0.0627116 + 0.998032i \(0.519975\pi\)
\(282\) 0 0
\(283\) −6.26796 −0.372592 −0.186296 0.982494i \(-0.559648\pi\)
−0.186296 + 0.982494i \(0.559648\pi\)
\(284\) 0 0
\(285\) 19.5823 1.15996
\(286\) 0 0
\(287\) 7.83716 0.462613
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 29.9735 1.75708
\(292\) 0 0
\(293\) 10.3166 0.602701 0.301350 0.953513i \(-0.402563\pi\)
0.301350 + 0.953513i \(0.402563\pi\)
\(294\) 0 0
\(295\) 13.6917 0.797159
\(296\) 0 0
\(297\) −2.22252 −0.128964
\(298\) 0 0
\(299\) 21.4097 1.23815
\(300\) 0 0
\(301\) −25.9331 −1.49476
\(302\) 0 0
\(303\) 37.0087 2.12609
\(304\) 0 0
\(305\) 23.5964 1.35112
\(306\) 0 0
\(307\) −11.1040 −0.633737 −0.316868 0.948469i \(-0.602631\pi\)
−0.316868 + 0.948469i \(0.602631\pi\)
\(308\) 0 0
\(309\) −22.5254 −1.28143
\(310\) 0 0
\(311\) 10.3094 0.584591 0.292296 0.956328i \(-0.405581\pi\)
0.292296 + 0.956328i \(0.405581\pi\)
\(312\) 0 0
\(313\) 2.95218 0.166867 0.0834334 0.996513i \(-0.473411\pi\)
0.0834334 + 0.996513i \(0.473411\pi\)
\(314\) 0 0
\(315\) 14.5884 0.821963
\(316\) 0 0
\(317\) 14.1867 0.796805 0.398403 0.917211i \(-0.369565\pi\)
0.398403 + 0.917211i \(0.369565\pi\)
\(318\) 0 0
\(319\) −3.81240 −0.213453
\(320\) 0 0
\(321\) 12.1135 0.676112
\(322\) 0 0
\(323\) −7.79505 −0.433728
\(324\) 0 0
\(325\) 0.250610 0.0139013
\(326\) 0 0
\(327\) 13.1389 0.726582
\(328\) 0 0
\(329\) −22.0763 −1.21711
\(330\) 0 0
\(331\) 26.4533 1.45400 0.727002 0.686636i \(-0.240913\pi\)
0.727002 + 0.686636i \(0.240913\pi\)
\(332\) 0 0
\(333\) −2.14048 −0.117297
\(334\) 0 0
\(335\) 0.705806 0.0385623
\(336\) 0 0
\(337\) 2.57857 0.140463 0.0702317 0.997531i \(-0.477626\pi\)
0.0702317 + 0.997531i \(0.477626\pi\)
\(338\) 0 0
\(339\) −25.0964 −1.36305
\(340\) 0 0
\(341\) 11.8235 0.640277
\(342\) 0 0
\(343\) −13.9661 −0.754099
\(344\) 0 0
\(345\) 38.3041 2.06222
\(346\) 0 0
\(347\) 10.2810 0.551911 0.275955 0.961170i \(-0.411006\pi\)
0.275955 + 0.961170i \(0.411006\pi\)
\(348\) 0 0
\(349\) 23.9187 1.28034 0.640171 0.768233i \(-0.278864\pi\)
0.640171 + 0.768233i \(0.278864\pi\)
\(350\) 0 0
\(351\) −5.47267 −0.292110
\(352\) 0 0
\(353\) 21.9596 1.16879 0.584396 0.811468i \(-0.301331\pi\)
0.584396 + 0.811468i \(0.301331\pi\)
\(354\) 0 0
\(355\) 14.2392 0.755737
\(356\) 0 0
\(357\) −13.9461 −0.738108
\(358\) 0 0
\(359\) −2.64350 −0.139518 −0.0697592 0.997564i \(-0.522223\pi\)
−0.0697592 + 0.997564i \(0.522223\pi\)
\(360\) 0 0
\(361\) −3.80932 −0.200490
\(362\) 0 0
\(363\) 21.9909 1.15422
\(364\) 0 0
\(365\) 35.4665 1.85640
\(366\) 0 0
\(367\) −13.9187 −0.726553 −0.363276 0.931681i \(-0.618342\pi\)
−0.363276 + 0.931681i \(0.618342\pi\)
\(368\) 0 0
\(369\) 5.45441 0.283945
\(370\) 0 0
\(371\) −17.2540 −0.895784
\(372\) 0 0
\(373\) 15.2136 0.787733 0.393866 0.919168i \(-0.371137\pi\)
0.393866 + 0.919168i \(0.371137\pi\)
\(374\) 0 0
\(375\) 25.5699 1.32042
\(376\) 0 0
\(377\) −9.38755 −0.483483
\(378\) 0 0
\(379\) −1.17082 −0.0601412 −0.0300706 0.999548i \(-0.509573\pi\)
−0.0300706 + 0.999548i \(0.509573\pi\)
\(380\) 0 0
\(381\) 5.52882 0.283250
\(382\) 0 0
\(383\) −17.6310 −0.900900 −0.450450 0.892802i \(-0.648736\pi\)
−0.450450 + 0.892802i \(0.648736\pi\)
\(384\) 0 0
\(385\) −7.77291 −0.396144
\(386\) 0 0
\(387\) −18.0486 −0.917463
\(388\) 0 0
\(389\) −27.6176 −1.40027 −0.700134 0.714012i \(-0.746876\pi\)
−0.700134 + 0.714012i \(0.746876\pi\)
\(390\) 0 0
\(391\) −15.2475 −0.771101
\(392\) 0 0
\(393\) 27.9544 1.41011
\(394\) 0 0
\(395\) −19.2635 −0.969251
\(396\) 0 0
\(397\) 5.85541 0.293874 0.146937 0.989146i \(-0.453058\pi\)
0.146937 + 0.989146i \(0.453058\pi\)
\(398\) 0 0
\(399\) 27.1777 1.36059
\(400\) 0 0
\(401\) −0.485911 −0.0242653 −0.0121326 0.999926i \(-0.503862\pi\)
−0.0121326 + 0.999926i \(0.503862\pi\)
\(402\) 0 0
\(403\) 29.1138 1.45026
\(404\) 0 0
\(405\) −24.0212 −1.19362
\(406\) 0 0
\(407\) 1.14048 0.0565314
\(408\) 0 0
\(409\) 27.0417 1.33712 0.668562 0.743656i \(-0.266910\pi\)
0.668562 + 0.743656i \(0.266910\pi\)
\(410\) 0 0
\(411\) 30.7251 1.51556
\(412\) 0 0
\(413\) 19.0022 0.935037
\(414\) 0 0
\(415\) 21.6310 1.06182
\(416\) 0 0
\(417\) −14.0727 −0.689142
\(418\) 0 0
\(419\) −20.5704 −1.00493 −0.502464 0.864598i \(-0.667573\pi\)
−0.502464 + 0.864598i \(0.667573\pi\)
\(420\) 0 0
\(421\) 11.6594 0.568244 0.284122 0.958788i \(-0.408298\pi\)
0.284122 + 0.958788i \(0.408298\pi\)
\(422\) 0 0
\(423\) −15.3644 −0.747043
\(424\) 0 0
\(425\) −0.178479 −0.00865751
\(426\) 0 0
\(427\) 32.7486 1.58482
\(428\) 0 0
\(429\) −7.26156 −0.350591
\(430\) 0 0
\(431\) 13.0690 0.629514 0.314757 0.949172i \(-0.398077\pi\)
0.314757 + 0.949172i \(0.398077\pi\)
\(432\) 0 0
\(433\) −26.6454 −1.28050 −0.640249 0.768167i \(-0.721169\pi\)
−0.640249 + 0.768167i \(0.721169\pi\)
\(434\) 0 0
\(435\) −16.7953 −0.805273
\(436\) 0 0
\(437\) 29.7138 1.42140
\(438\) 0 0
\(439\) 15.5964 0.744374 0.372187 0.928158i \(-0.378608\pi\)
0.372187 + 0.928158i \(0.378608\pi\)
\(440\) 0 0
\(441\) 5.26339 0.250638
\(442\) 0 0
\(443\) −30.6546 −1.45644 −0.728221 0.685342i \(-0.759653\pi\)
−0.728221 + 0.685342i \(0.759653\pi\)
\(444\) 0 0
\(445\) −18.2901 −0.867034
\(446\) 0 0
\(447\) 5.12416 0.242364
\(448\) 0 0
\(449\) 18.8641 0.890252 0.445126 0.895468i \(-0.353159\pi\)
0.445126 + 0.895468i \(0.353159\pi\)
\(450\) 0 0
\(451\) −2.90619 −0.136847
\(452\) 0 0
\(453\) −39.6796 −1.86431
\(454\) 0 0
\(455\) −19.1398 −0.897288
\(456\) 0 0
\(457\) −29.8571 −1.39666 −0.698329 0.715777i \(-0.746073\pi\)
−0.698329 + 0.715777i \(0.746073\pi\)
\(458\) 0 0
\(459\) 3.89752 0.181921
\(460\) 0 0
\(461\) −30.1390 −1.40371 −0.701857 0.712318i \(-0.747645\pi\)
−0.701857 + 0.712318i \(0.747645\pi\)
\(462\) 0 0
\(463\) −20.1857 −0.938108 −0.469054 0.883169i \(-0.655405\pi\)
−0.469054 + 0.883169i \(0.655405\pi\)
\(464\) 0 0
\(465\) 52.0876 2.41550
\(466\) 0 0
\(467\) −31.2201 −1.44470 −0.722348 0.691530i \(-0.756937\pi\)
−0.722348 + 0.691530i \(0.756937\pi\)
\(468\) 0 0
\(469\) 0.979564 0.0452321
\(470\) 0 0
\(471\) 27.1303 1.25010
\(472\) 0 0
\(473\) 9.61657 0.442170
\(474\) 0 0
\(475\) 0.347813 0.0159588
\(476\) 0 0
\(477\) −12.0082 −0.549819
\(478\) 0 0
\(479\) 36.7900 1.68098 0.840490 0.541827i \(-0.182267\pi\)
0.840490 + 0.541827i \(0.182267\pi\)
\(480\) 0 0
\(481\) 2.80828 0.128047
\(482\) 0 0
\(483\) 53.1610 2.41891
\(484\) 0 0
\(485\) −29.2961 −1.33027
\(486\) 0 0
\(487\) 17.9461 0.813217 0.406609 0.913602i \(-0.366711\pi\)
0.406609 + 0.913602i \(0.366711\pi\)
\(488\) 0 0
\(489\) 2.68562 0.121448
\(490\) 0 0
\(491\) 29.3428 1.32422 0.662111 0.749406i \(-0.269661\pi\)
0.662111 + 0.749406i \(0.269661\pi\)
\(492\) 0 0
\(493\) 6.68562 0.301105
\(494\) 0 0
\(495\) −5.40969 −0.243147
\(496\) 0 0
\(497\) 19.7621 0.886450
\(498\) 0 0
\(499\) 6.17848 0.276587 0.138293 0.990391i \(-0.455838\pi\)
0.138293 + 0.990391i \(0.455838\pi\)
\(500\) 0 0
\(501\) 45.8350 2.04776
\(502\) 0 0
\(503\) 6.14917 0.274178 0.137089 0.990559i \(-0.456225\pi\)
0.137089 + 0.990559i \(0.456225\pi\)
\(504\) 0 0
\(505\) −36.1723 −1.60965
\(506\) 0 0
\(507\) 11.5938 0.514897
\(508\) 0 0
\(509\) −27.6183 −1.22416 −0.612080 0.790796i \(-0.709667\pi\)
−0.612080 + 0.790796i \(0.709667\pi\)
\(510\) 0 0
\(511\) 49.2228 2.17749
\(512\) 0 0
\(513\) −7.59534 −0.335343
\(514\) 0 0
\(515\) 22.0163 0.970156
\(516\) 0 0
\(517\) 8.18638 0.360037
\(518\) 0 0
\(519\) 51.4195 2.25707
\(520\) 0 0
\(521\) −10.1720 −0.445643 −0.222821 0.974859i \(-0.571527\pi\)
−0.222821 + 0.974859i \(0.571527\pi\)
\(522\) 0 0
\(523\) −9.41161 −0.411541 −0.205771 0.978600i \(-0.565970\pi\)
−0.205771 + 0.978600i \(0.565970\pi\)
\(524\) 0 0
\(525\) 0.622273 0.0271582
\(526\) 0 0
\(527\) −20.7342 −0.903197
\(528\) 0 0
\(529\) 35.1218 1.52703
\(530\) 0 0
\(531\) 13.2249 0.573912
\(532\) 0 0
\(533\) −7.15612 −0.309966
\(534\) 0 0
\(535\) −11.8398 −0.511879
\(536\) 0 0
\(537\) −33.0638 −1.42681
\(538\) 0 0
\(539\) −2.80441 −0.120795
\(540\) 0 0
\(541\) 2.04371 0.0878659 0.0439329 0.999034i \(-0.486011\pi\)
0.0439329 + 0.999034i \(0.486011\pi\)
\(542\) 0 0
\(543\) 56.0643 2.40595
\(544\) 0 0
\(545\) −12.8420 −0.550089
\(546\) 0 0
\(547\) −22.9271 −0.980292 −0.490146 0.871640i \(-0.663057\pi\)
−0.490146 + 0.871640i \(0.663057\pi\)
\(548\) 0 0
\(549\) 22.7920 0.972737
\(550\) 0 0
\(551\) −13.0287 −0.555040
\(552\) 0 0
\(553\) −26.7351 −1.13689
\(554\) 0 0
\(555\) 5.02431 0.213270
\(556\) 0 0
\(557\) 33.6633 1.42636 0.713179 0.700982i \(-0.247255\pi\)
0.713179 + 0.700982i \(0.247255\pi\)
\(558\) 0 0
\(559\) 23.6796 1.00154
\(560\) 0 0
\(561\) 5.17153 0.218342
\(562\) 0 0
\(563\) 10.0486 0.423499 0.211749 0.977324i \(-0.432084\pi\)
0.211749 + 0.977324i \(0.432084\pi\)
\(564\) 0 0
\(565\) 24.5293 1.03195
\(566\) 0 0
\(567\) −33.3383 −1.40008
\(568\) 0 0
\(569\) 33.9947 1.42513 0.712567 0.701604i \(-0.247532\pi\)
0.712567 + 0.701604i \(0.247532\pi\)
\(570\) 0 0
\(571\) 6.71755 0.281121 0.140560 0.990072i \(-0.455110\pi\)
0.140560 + 0.990072i \(0.455110\pi\)
\(572\) 0 0
\(573\) 33.3833 1.39461
\(574\) 0 0
\(575\) 0.680341 0.0283722
\(576\) 0 0
\(577\) −32.5701 −1.35591 −0.677956 0.735102i \(-0.737134\pi\)
−0.677956 + 0.735102i \(0.737134\pi\)
\(578\) 0 0
\(579\) −41.2635 −1.71485
\(580\) 0 0
\(581\) 30.0209 1.24548
\(582\) 0 0
\(583\) 6.39816 0.264985
\(584\) 0 0
\(585\) −13.3207 −0.550742
\(586\) 0 0
\(587\) −30.6948 −1.26691 −0.633454 0.773780i \(-0.718363\pi\)
−0.633454 + 0.773780i \(0.718363\pi\)
\(588\) 0 0
\(589\) 40.4061 1.66490
\(590\) 0 0
\(591\) 19.3353 0.795347
\(592\) 0 0
\(593\) −44.7258 −1.83667 −0.918334 0.395805i \(-0.870465\pi\)
−0.918334 + 0.395805i \(0.870465\pi\)
\(594\) 0 0
\(595\) 13.6310 0.558815
\(596\) 0 0
\(597\) −51.4963 −2.10760
\(598\) 0 0
\(599\) 4.23268 0.172942 0.0864712 0.996254i \(-0.472441\pi\)
0.0864712 + 0.996254i \(0.472441\pi\)
\(600\) 0 0
\(601\) 7.01028 0.285955 0.142978 0.989726i \(-0.454332\pi\)
0.142978 + 0.989726i \(0.454332\pi\)
\(602\) 0 0
\(603\) 0.681744 0.0277628
\(604\) 0 0
\(605\) −21.4939 −0.873851
\(606\) 0 0
\(607\) 23.4188 0.950540 0.475270 0.879840i \(-0.342350\pi\)
0.475270 + 0.879840i \(0.342350\pi\)
\(608\) 0 0
\(609\) −23.3096 −0.944554
\(610\) 0 0
\(611\) 20.1579 0.815502
\(612\) 0 0
\(613\) 7.78855 0.314577 0.157288 0.987553i \(-0.449725\pi\)
0.157288 + 0.987553i \(0.449725\pi\)
\(614\) 0 0
\(615\) −12.8030 −0.516268
\(616\) 0 0
\(617\) −25.4388 −1.02413 −0.512064 0.858948i \(-0.671119\pi\)
−0.512064 + 0.858948i \(0.671119\pi\)
\(618\) 0 0
\(619\) 21.0948 0.847873 0.423937 0.905692i \(-0.360648\pi\)
0.423937 + 0.905692i \(0.360648\pi\)
\(620\) 0 0
\(621\) −14.8569 −0.596187
\(622\) 0 0
\(623\) −25.3842 −1.01700
\(624\) 0 0
\(625\) −24.5458 −0.981834
\(626\) 0 0
\(627\) −10.0781 −0.402479
\(628\) 0 0
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) 3.60937 0.143687 0.0718433 0.997416i \(-0.477112\pi\)
0.0718433 + 0.997416i \(0.477112\pi\)
\(632\) 0 0
\(633\) −14.1081 −0.560746
\(634\) 0 0
\(635\) −5.40387 −0.214446
\(636\) 0 0
\(637\) −6.90551 −0.273606
\(638\) 0 0
\(639\) 13.7538 0.544090
\(640\) 0 0
\(641\) −36.8109 −1.45394 −0.726972 0.686667i \(-0.759073\pi\)
−0.726972 + 0.686667i \(0.759073\pi\)
\(642\) 0 0
\(643\) 28.3296 1.11721 0.558605 0.829434i \(-0.311337\pi\)
0.558605 + 0.829434i \(0.311337\pi\)
\(644\) 0 0
\(645\) 42.3652 1.66813
\(646\) 0 0
\(647\) −0.581191 −0.0228490 −0.0114245 0.999935i \(-0.503637\pi\)
−0.0114245 + 0.999935i \(0.503637\pi\)
\(648\) 0 0
\(649\) −7.04642 −0.276596
\(650\) 0 0
\(651\) 72.2906 2.83329
\(652\) 0 0
\(653\) 3.71276 0.145291 0.0726457 0.997358i \(-0.476856\pi\)
0.0726457 + 0.997358i \(0.476856\pi\)
\(654\) 0 0
\(655\) −27.3226 −1.06758
\(656\) 0 0
\(657\) 34.2574 1.33651
\(658\) 0 0
\(659\) −15.5269 −0.604841 −0.302420 0.953175i \(-0.597795\pi\)
−0.302420 + 0.953175i \(0.597795\pi\)
\(660\) 0 0
\(661\) −17.2760 −0.671957 −0.335978 0.941870i \(-0.609067\pi\)
−0.335978 + 0.941870i \(0.609067\pi\)
\(662\) 0 0
\(663\) 12.7342 0.494556
\(664\) 0 0
\(665\) −26.5635 −1.03009
\(666\) 0 0
\(667\) −25.4848 −0.986774
\(668\) 0 0
\(669\) 28.5766 1.10484
\(670\) 0 0
\(671\) −12.1439 −0.468810
\(672\) 0 0
\(673\) −8.68491 −0.334779 −0.167389 0.985891i \(-0.553534\pi\)
−0.167389 + 0.985891i \(0.553534\pi\)
\(674\) 0 0
\(675\) −0.173907 −0.00669367
\(676\) 0 0
\(677\) −35.7279 −1.37313 −0.686567 0.727067i \(-0.740883\pi\)
−0.686567 + 0.727067i \(0.740883\pi\)
\(678\) 0 0
\(679\) −40.6591 −1.56035
\(680\) 0 0
\(681\) −59.7768 −2.29065
\(682\) 0 0
\(683\) 8.80932 0.337079 0.168540 0.985695i \(-0.446095\pi\)
0.168540 + 0.985695i \(0.446095\pi\)
\(684\) 0 0
\(685\) −30.0307 −1.14742
\(686\) 0 0
\(687\) −27.8775 −1.06359
\(688\) 0 0
\(689\) 15.7547 0.600205
\(690\) 0 0
\(691\) −6.36904 −0.242290 −0.121145 0.992635i \(-0.538657\pi\)
−0.121145 + 0.992635i \(0.538657\pi\)
\(692\) 0 0
\(693\) −7.50793 −0.285203
\(694\) 0 0
\(695\) 13.7546 0.521743
\(696\) 0 0
\(697\) 5.09644 0.193041
\(698\) 0 0
\(699\) −32.0679 −1.21292
\(700\) 0 0
\(701\) −11.6199 −0.438877 −0.219439 0.975626i \(-0.570423\pi\)
−0.219439 + 0.975626i \(0.570423\pi\)
\(702\) 0 0
\(703\) 3.89752 0.146998
\(704\) 0 0
\(705\) 36.0646 1.35827
\(706\) 0 0
\(707\) −50.2024 −1.88805
\(708\) 0 0
\(709\) 20.2372 0.760026 0.380013 0.924981i \(-0.375920\pi\)
0.380013 + 0.924981i \(0.375920\pi\)
\(710\) 0 0
\(711\) −18.6068 −0.697809
\(712\) 0 0
\(713\) 79.0364 2.95994
\(714\) 0 0
\(715\) 7.09745 0.265430
\(716\) 0 0
\(717\) −31.1003 −1.16146
\(718\) 0 0
\(719\) −10.8432 −0.404383 −0.202192 0.979346i \(-0.564806\pi\)
−0.202192 + 0.979346i \(0.564806\pi\)
\(720\) 0 0
\(721\) 30.5557 1.13796
\(722\) 0 0
\(723\) −65.5527 −2.43793
\(724\) 0 0
\(725\) −0.298311 −0.0110790
\(726\) 0 0
\(727\) −35.3481 −1.31099 −0.655494 0.755200i \(-0.727540\pi\)
−0.655494 + 0.755200i \(0.727540\pi\)
\(728\) 0 0
\(729\) −9.94727 −0.368417
\(730\) 0 0
\(731\) −16.8641 −0.623741
\(732\) 0 0
\(733\) −27.3253 −1.00928 −0.504641 0.863329i \(-0.668375\pi\)
−0.504641 + 0.863329i \(0.668375\pi\)
\(734\) 0 0
\(735\) −12.3547 −0.455709
\(736\) 0 0
\(737\) −0.363243 −0.0133802
\(738\) 0 0
\(739\) 27.8208 1.02341 0.511703 0.859162i \(-0.329015\pi\)
0.511703 + 0.859162i \(0.329015\pi\)
\(740\) 0 0
\(741\) −24.8160 −0.911638
\(742\) 0 0
\(743\) −37.9335 −1.39164 −0.695822 0.718214i \(-0.744960\pi\)
−0.695822 + 0.718214i \(0.744960\pi\)
\(744\) 0 0
\(745\) −5.00836 −0.183492
\(746\) 0 0
\(747\) 20.8936 0.764455
\(748\) 0 0
\(749\) −16.4320 −0.600414
\(750\) 0 0
\(751\) 47.0299 1.71615 0.858073 0.513528i \(-0.171662\pi\)
0.858073 + 0.513528i \(0.171662\pi\)
\(752\) 0 0
\(753\) 48.4542 1.76577
\(754\) 0 0
\(755\) 38.7828 1.41145
\(756\) 0 0
\(757\) −13.0041 −0.472643 −0.236321 0.971675i \(-0.575942\pi\)
−0.236321 + 0.971675i \(0.575942\pi\)
\(758\) 0 0
\(759\) −19.7132 −0.715546
\(760\) 0 0
\(761\) 48.7562 1.76741 0.883705 0.468045i \(-0.155041\pi\)
0.883705 + 0.468045i \(0.155041\pi\)
\(762\) 0 0
\(763\) −17.8229 −0.645233
\(764\) 0 0
\(765\) 9.48670 0.342992
\(766\) 0 0
\(767\) −17.3509 −0.626505
\(768\) 0 0
\(769\) 25.5641 0.921865 0.460933 0.887435i \(-0.347515\pi\)
0.460933 + 0.887435i \(0.347515\pi\)
\(770\) 0 0
\(771\) −8.26180 −0.297541
\(772\) 0 0
\(773\) −10.6026 −0.381350 −0.190675 0.981653i \(-0.561068\pi\)
−0.190675 + 0.981653i \(0.561068\pi\)
\(774\) 0 0
\(775\) 0.925156 0.0332326
\(776\) 0 0
\(777\) 6.97307 0.250157
\(778\) 0 0
\(779\) −9.93174 −0.355842
\(780\) 0 0
\(781\) −7.32820 −0.262224
\(782\) 0 0
\(783\) 6.51433 0.232803
\(784\) 0 0
\(785\) −26.5172 −0.946439
\(786\) 0 0
\(787\) 40.9813 1.46083 0.730413 0.683006i \(-0.239328\pi\)
0.730413 + 0.683006i \(0.239328\pi\)
\(788\) 0 0
\(789\) 41.2782 1.46954
\(790\) 0 0
\(791\) 34.0434 1.21044
\(792\) 0 0
\(793\) −29.9028 −1.06188
\(794\) 0 0
\(795\) 28.1867 0.999680
\(796\) 0 0
\(797\) 45.0984 1.59747 0.798733 0.601685i \(-0.205504\pi\)
0.798733 + 0.601685i \(0.205504\pi\)
\(798\) 0 0
\(799\) −14.3560 −0.507880
\(800\) 0 0
\(801\) −17.6666 −0.624218
\(802\) 0 0
\(803\) −18.2529 −0.644130
\(804\) 0 0
\(805\) −51.9596 −1.83134
\(806\) 0 0
\(807\) 18.2483 0.642371
\(808\) 0 0
\(809\) 13.4342 0.472323 0.236161 0.971714i \(-0.424111\pi\)
0.236161 + 0.971714i \(0.424111\pi\)
\(810\) 0 0
\(811\) 33.6020 1.17993 0.589963 0.807431i \(-0.299142\pi\)
0.589963 + 0.807431i \(0.299142\pi\)
\(812\) 0 0
\(813\) 31.3244 1.09859
\(814\) 0 0
\(815\) −2.62492 −0.0919470
\(816\) 0 0
\(817\) 32.8641 1.14977
\(818\) 0 0
\(819\) −18.4873 −0.645999
\(820\) 0 0
\(821\) 7.15680 0.249774 0.124887 0.992171i \(-0.460143\pi\)
0.124887 + 0.992171i \(0.460143\pi\)
\(822\) 0 0
\(823\) 17.7282 0.617966 0.308983 0.951068i \(-0.400011\pi\)
0.308983 + 0.951068i \(0.400011\pi\)
\(824\) 0 0
\(825\) −0.230752 −0.00803376
\(826\) 0 0
\(827\) 30.4129 1.05756 0.528780 0.848759i \(-0.322650\pi\)
0.528780 + 0.848759i \(0.322650\pi\)
\(828\) 0 0
\(829\) 32.3315 1.12292 0.561460 0.827504i \(-0.310240\pi\)
0.561460 + 0.827504i \(0.310240\pi\)
\(830\) 0 0
\(831\) −5.02431 −0.174291
\(832\) 0 0
\(833\) 4.91796 0.170397
\(834\) 0 0
\(835\) −44.7992 −1.55034
\(836\) 0 0
\(837\) −20.2030 −0.698319
\(838\) 0 0
\(839\) 3.00604 0.103780 0.0518900 0.998653i \(-0.483475\pi\)
0.0518900 + 0.998653i \(0.483475\pi\)
\(840\) 0 0
\(841\) −17.8256 −0.614677
\(842\) 0 0
\(843\) 4.76687 0.164180
\(844\) 0 0
\(845\) −11.3317 −0.389824
\(846\) 0 0
\(847\) −29.8307 −1.02499
\(848\) 0 0
\(849\) 14.2111 0.487724
\(850\) 0 0
\(851\) 7.62376 0.261339
\(852\) 0 0
\(853\) −32.2920 −1.10566 −0.552829 0.833295i \(-0.686452\pi\)
−0.552829 + 0.833295i \(0.686452\pi\)
\(854\) 0 0
\(855\) −18.4873 −0.632253
\(856\) 0 0
\(857\) 42.8093 1.46234 0.731169 0.682196i \(-0.238975\pi\)
0.731169 + 0.682196i \(0.238975\pi\)
\(858\) 0 0
\(859\) 20.5567 0.701384 0.350692 0.936491i \(-0.385946\pi\)
0.350692 + 0.936491i \(0.385946\pi\)
\(860\) 0 0
\(861\) −17.7689 −0.605563
\(862\) 0 0
\(863\) −30.2353 −1.02922 −0.514611 0.857424i \(-0.672064\pi\)
−0.514611 + 0.857424i \(0.672064\pi\)
\(864\) 0 0
\(865\) −50.2575 −1.70880
\(866\) 0 0
\(867\) 29.4744 1.00100
\(868\) 0 0
\(869\) 9.91396 0.336308
\(870\) 0 0
\(871\) −0.894441 −0.0303070
\(872\) 0 0
\(873\) −28.2974 −0.957723
\(874\) 0 0
\(875\) −34.6856 −1.17259
\(876\) 0 0
\(877\) −31.7830 −1.07323 −0.536617 0.843826i \(-0.680298\pi\)
−0.536617 + 0.843826i \(0.680298\pi\)
\(878\) 0 0
\(879\) −23.3904 −0.788938
\(880\) 0 0
\(881\) −9.27159 −0.312368 −0.156184 0.987728i \(-0.549919\pi\)
−0.156184 + 0.987728i \(0.549919\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) −31.0426 −1.04348
\(886\) 0 0
\(887\) 30.8250 1.03500 0.517500 0.855683i \(-0.326863\pi\)
0.517500 + 0.855683i \(0.326863\pi\)
\(888\) 0 0
\(889\) −7.49985 −0.251537
\(890\) 0 0
\(891\) 12.3625 0.414161
\(892\) 0 0
\(893\) 27.9765 0.936198
\(894\) 0 0
\(895\) 32.3166 1.08022
\(896\) 0 0
\(897\) −48.5414 −1.62075
\(898\) 0 0
\(899\) −34.6553 −1.15582
\(900\) 0 0
\(901\) −11.2201 −0.373797
\(902\) 0 0
\(903\) 58.7972 1.95665
\(904\) 0 0
\(905\) −54.7972 −1.82152
\(906\) 0 0
\(907\) 55.9288 1.85709 0.928543 0.371225i \(-0.121062\pi\)
0.928543 + 0.371225i \(0.121062\pi\)
\(908\) 0 0
\(909\) −34.9392 −1.15886
\(910\) 0 0
\(911\) 18.4663 0.611815 0.305907 0.952061i \(-0.401040\pi\)
0.305907 + 0.952061i \(0.401040\pi\)
\(912\) 0 0
\(913\) −11.1324 −0.368428
\(914\) 0 0
\(915\) −53.4992 −1.76863
\(916\) 0 0
\(917\) −37.9202 −1.25223
\(918\) 0 0
\(919\) 5.12383 0.169019 0.0845097 0.996423i \(-0.473068\pi\)
0.0845097 + 0.996423i \(0.473068\pi\)
\(920\) 0 0
\(921\) 25.1756 0.829564
\(922\) 0 0
\(923\) −18.0448 −0.593951
\(924\) 0 0
\(925\) 0.0892395 0.00293418
\(926\) 0 0
\(927\) 21.2658 0.698460
\(928\) 0 0
\(929\) −0.612450 −0.0200938 −0.0100469 0.999950i \(-0.503198\pi\)
−0.0100469 + 0.999950i \(0.503198\pi\)
\(930\) 0 0
\(931\) −9.58393 −0.314101
\(932\) 0 0
\(933\) −23.3741 −0.765233
\(934\) 0 0
\(935\) −5.05465 −0.165305
\(936\) 0 0
\(937\) −3.83662 −0.125337 −0.0626684 0.998034i \(-0.519961\pi\)
−0.0626684 + 0.998034i \(0.519961\pi\)
\(938\) 0 0
\(939\) −6.69336 −0.218429
\(940\) 0 0
\(941\) −3.05465 −0.0995789 −0.0497894 0.998760i \(-0.515855\pi\)
−0.0497894 + 0.998760i \(0.515855\pi\)
\(942\) 0 0
\(943\) −19.4270 −0.632631
\(944\) 0 0
\(945\) 13.2817 0.432055
\(946\) 0 0
\(947\) −38.9604 −1.26604 −0.633022 0.774134i \(-0.718186\pi\)
−0.633022 + 0.774134i \(0.718186\pi\)
\(948\) 0 0
\(949\) −44.9454 −1.45899
\(950\) 0 0
\(951\) −32.1650 −1.04302
\(952\) 0 0
\(953\) 42.2529 1.36871 0.684353 0.729151i \(-0.260085\pi\)
0.684353 + 0.729151i \(0.260085\pi\)
\(954\) 0 0
\(955\) −32.6289 −1.05585
\(956\) 0 0
\(957\) 8.64371 0.279411
\(958\) 0 0
\(959\) −41.6787 −1.34587
\(960\) 0 0
\(961\) 76.4771 2.46700
\(962\) 0 0
\(963\) −11.4362 −0.368525
\(964\) 0 0
\(965\) 40.3310 1.29830
\(966\) 0 0
\(967\) 36.2122 1.16451 0.582253 0.813008i \(-0.302171\pi\)
0.582253 + 0.813008i \(0.302171\pi\)
\(968\) 0 0
\(969\) 17.6734 0.567752
\(970\) 0 0
\(971\) −11.9615 −0.383864 −0.191932 0.981408i \(-0.561475\pi\)
−0.191932 + 0.981408i \(0.561475\pi\)
\(972\) 0 0
\(973\) 19.0896 0.611985
\(974\) 0 0
\(975\) −0.568198 −0.0181969
\(976\) 0 0
\(977\) −38.3440 −1.22673 −0.613366 0.789799i \(-0.710185\pi\)
−0.613366 + 0.789799i \(0.710185\pi\)
\(978\) 0 0
\(979\) 9.41301 0.300841
\(980\) 0 0
\(981\) −12.4042 −0.396035
\(982\) 0 0
\(983\) 31.1742 0.994302 0.497151 0.867664i \(-0.334380\pi\)
0.497151 + 0.867664i \(0.334380\pi\)
\(984\) 0 0
\(985\) −18.8983 −0.602150
\(986\) 0 0
\(987\) 50.0528 1.59320
\(988\) 0 0
\(989\) 64.2839 2.04411
\(990\) 0 0
\(991\) −39.2981 −1.24834 −0.624172 0.781287i \(-0.714564\pi\)
−0.624172 + 0.781287i \(0.714564\pi\)
\(992\) 0 0
\(993\) −59.9765 −1.90330
\(994\) 0 0
\(995\) 50.3326 1.59565
\(996\) 0 0
\(997\) −34.3022 −1.08636 −0.543180 0.839616i \(-0.682780\pi\)
−0.543180 + 0.839616i \(0.682780\pi\)
\(998\) 0 0
\(999\) −1.94876 −0.0616561
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 2368.2.a.bg.1.2 4
4.3 odd 2 2368.2.a.bh.1.3 4
8.3 odd 2 592.2.a.j.1.2 4
8.5 even 2 296.2.a.d.1.3 4
24.5 odd 2 2664.2.a.r.1.4 4
24.11 even 2 5328.2.a.bp.1.4 4
40.29 even 2 7400.2.a.n.1.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
296.2.a.d.1.3 4 8.5 even 2
592.2.a.j.1.2 4 8.3 odd 2
2368.2.a.bg.1.2 4 1.1 even 1 trivial
2368.2.a.bh.1.3 4 4.3 odd 2
2664.2.a.r.1.4 4 24.5 odd 2
5328.2.a.bp.1.4 4 24.11 even 2
7400.2.a.n.1.2 4 40.29 even 2