Properties

Label 6864.2.a.br.1.1
Level $6864$
Weight $2$
Character 6864.1
Self dual yes
Analytic conductor $54.809$
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] = [6864,2,Mod(1,6864)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6864, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 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("6864.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6864 = 2^{4} \cdot 3 \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6864.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: \(54.8093159474\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.564.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 5x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1716)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.51414\) of defining polynomial
Character \(\chi\) \(=\) 6864.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{3} -1.51414 q^{5} +1.32088 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{3} -1.51414 q^{5} +1.32088 q^{7} +1.00000 q^{9} -1.00000 q^{11} -1.00000 q^{13} +1.51414 q^{15} -4.83502 q^{17} -5.32088 q^{19} -1.32088 q^{21} -8.73566 q^{23} -2.70739 q^{25} -1.00000 q^{27} +10.4485 q^{29} -0.485863 q^{31} +1.00000 q^{33} -2.00000 q^{35} +2.64177 q^{37} +1.00000 q^{39} +10.7357 q^{41} +0.193252 q^{43} -1.51414 q^{45} -5.02827 q^{47} -5.25526 q^{49} +4.83502 q^{51} -7.67004 q^{53} +1.51414 q^{55} +5.32088 q^{57} +7.02827 q^{59} +6.25526 q^{61} +1.32088 q^{63} +1.51414 q^{65} -9.18418 q^{67} +8.73566 q^{69} -14.6983 q^{71} +10.9909 q^{73} +2.70739 q^{75} -1.32088 q^{77} +16.5051 q^{79} +1.00000 q^{81} -0.386505 q^{83} +7.32088 q^{85} -10.4485 q^{87} -2.92892 q^{89} -1.32088 q^{91} +0.485863 q^{93} +8.05655 q^{95} -6.64177 q^{97} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 3 q^{3} + 2 q^{5} - 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 3 q^{3} + 2 q^{5} - 4 q^{7} + 3 q^{9} - 3 q^{11} - 3 q^{13} - 2 q^{15} - 8 q^{19} + 4 q^{21} - 8 q^{23} - 3 q^{25} - 3 q^{27} + 14 q^{29} - 8 q^{31} + 3 q^{33} - 6 q^{35} - 8 q^{37} + 3 q^{39} + 14 q^{41} + 2 q^{43} + 2 q^{45} - 2 q^{47} + 3 q^{49} + 6 q^{53} - 2 q^{55} + 8 q^{57} + 8 q^{59} - 4 q^{63} - 2 q^{65} + 8 q^{67} + 8 q^{69} - 2 q^{71} - 4 q^{73} + 3 q^{75} + 4 q^{77} + 6 q^{79} + 3 q^{81} - 4 q^{83} + 14 q^{85} - 14 q^{87} + 8 q^{89} + 4 q^{91} + 8 q^{93} - 2 q^{95} - 4 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(3\) −1.00000 −0.577350
\(4\) 0 0
\(5\) −1.51414 −0.677143 −0.338571 0.940941i \(-0.609944\pi\)
−0.338571 + 0.940941i \(0.609944\pi\)
\(6\) 0 0
\(7\) 1.32088 0.499247 0.249624 0.968343i \(-0.419693\pi\)
0.249624 + 0.968343i \(0.419693\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350
\(14\) 0 0
\(15\) 1.51414 0.390948
\(16\) 0 0
\(17\) −4.83502 −1.17266 −0.586332 0.810071i \(-0.699429\pi\)
−0.586332 + 0.810071i \(0.699429\pi\)
\(18\) 0 0
\(19\) −5.32088 −1.22069 −0.610347 0.792134i \(-0.708970\pi\)
−0.610347 + 0.792134i \(0.708970\pi\)
\(20\) 0 0
\(21\) −1.32088 −0.288241
\(22\) 0 0
\(23\) −8.73566 −1.82151 −0.910756 0.412945i \(-0.864500\pi\)
−0.910756 + 0.412945i \(0.864500\pi\)
\(24\) 0 0
\(25\) −2.70739 −0.541478
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) 0 0
\(29\) 10.4485 1.94024 0.970120 0.242624i \(-0.0780081\pi\)
0.970120 + 0.242624i \(0.0780081\pi\)
\(30\) 0 0
\(31\) −0.485863 −0.0872636 −0.0436318 0.999048i \(-0.513893\pi\)
−0.0436318 + 0.999048i \(0.513893\pi\)
\(32\) 0 0
\(33\) 1.00000 0.174078
\(34\) 0 0
\(35\) −2.00000 −0.338062
\(36\) 0 0
\(37\) 2.64177 0.434304 0.217152 0.976138i \(-0.430323\pi\)
0.217152 + 0.976138i \(0.430323\pi\)
\(38\) 0 0
\(39\) 1.00000 0.160128
\(40\) 0 0
\(41\) 10.7357 1.67663 0.838314 0.545187i \(-0.183541\pi\)
0.838314 + 0.545187i \(0.183541\pi\)
\(42\) 0 0
\(43\) 0.193252 0.0294707 0.0147354 0.999891i \(-0.495309\pi\)
0.0147354 + 0.999891i \(0.495309\pi\)
\(44\) 0 0
\(45\) −1.51414 −0.225714
\(46\) 0 0
\(47\) −5.02827 −0.733449 −0.366725 0.930330i \(-0.619521\pi\)
−0.366725 + 0.930330i \(0.619521\pi\)
\(48\) 0 0
\(49\) −5.25526 −0.750752
\(50\) 0 0
\(51\) 4.83502 0.677038
\(52\) 0 0
\(53\) −7.67004 −1.05356 −0.526781 0.850001i \(-0.676601\pi\)
−0.526781 + 0.850001i \(0.676601\pi\)
\(54\) 0 0
\(55\) 1.51414 0.204166
\(56\) 0 0
\(57\) 5.32088 0.704768
\(58\) 0 0
\(59\) 7.02827 0.915003 0.457502 0.889209i \(-0.348744\pi\)
0.457502 + 0.889209i \(0.348744\pi\)
\(60\) 0 0
\(61\) 6.25526 0.800905 0.400452 0.916318i \(-0.368853\pi\)
0.400452 + 0.916318i \(0.368853\pi\)
\(62\) 0 0
\(63\) 1.32088 0.166416
\(64\) 0 0
\(65\) 1.51414 0.187806
\(66\) 0 0
\(67\) −9.18418 −1.12203 −0.561013 0.827807i \(-0.689588\pi\)
−0.561013 + 0.827807i \(0.689588\pi\)
\(68\) 0 0
\(69\) 8.73566 1.05165
\(70\) 0 0
\(71\) −14.6983 −1.74437 −0.872185 0.489177i \(-0.837297\pi\)
−0.872185 + 0.489177i \(0.837297\pi\)
\(72\) 0 0
\(73\) 10.9909 1.28639 0.643195 0.765702i \(-0.277608\pi\)
0.643195 + 0.765702i \(0.277608\pi\)
\(74\) 0 0
\(75\) 2.70739 0.312622
\(76\) 0 0
\(77\) −1.32088 −0.150529
\(78\) 0 0
\(79\) 16.5051 1.85696 0.928482 0.371376i \(-0.121114\pi\)
0.928482 + 0.371376i \(0.121114\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −0.386505 −0.0424244 −0.0212122 0.999775i \(-0.506753\pi\)
−0.0212122 + 0.999775i \(0.506753\pi\)
\(84\) 0 0
\(85\) 7.32088 0.794061
\(86\) 0 0
\(87\) −10.4485 −1.12020
\(88\) 0 0
\(89\) −2.92892 −0.310464 −0.155232 0.987878i \(-0.549613\pi\)
−0.155232 + 0.987878i \(0.549613\pi\)
\(90\) 0 0
\(91\) −1.32088 −0.138466
\(92\) 0 0
\(93\) 0.485863 0.0503817
\(94\) 0 0
\(95\) 8.05655 0.826584
\(96\) 0 0
\(97\) −6.64177 −0.674369 −0.337185 0.941439i \(-0.609475\pi\)
−0.337185 + 0.941439i \(0.609475\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) 0 0
\(101\) 15.4768 1.54000 0.769999 0.638045i \(-0.220257\pi\)
0.769999 + 0.638045i \(0.220257\pi\)
\(102\) 0 0
\(103\) 3.22699 0.317965 0.158982 0.987281i \(-0.449179\pi\)
0.158982 + 0.987281i \(0.449179\pi\)
\(104\) 0 0
\(105\) 2.00000 0.195180
\(106\) 0 0
\(107\) 2.25526 0.218025 0.109012 0.994040i \(-0.465231\pi\)
0.109012 + 0.994040i \(0.465231\pi\)
\(108\) 0 0
\(109\) 13.1222 1.25688 0.628438 0.777860i \(-0.283695\pi\)
0.628438 + 0.777860i \(0.283695\pi\)
\(110\) 0 0
\(111\) −2.64177 −0.250746
\(112\) 0 0
\(113\) −8.25526 −0.776590 −0.388295 0.921535i \(-0.626936\pi\)
−0.388295 + 0.921535i \(0.626936\pi\)
\(114\) 0 0
\(115\) 13.2270 1.23342
\(116\) 0 0
\(117\) −1.00000 −0.0924500
\(118\) 0 0
\(119\) −6.38650 −0.585450
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −10.7357 −0.968002
\(124\) 0 0
\(125\) 11.6700 1.04380
\(126\) 0 0
\(127\) 9.22153 0.818278 0.409139 0.912472i \(-0.365829\pi\)
0.409139 + 0.912472i \(0.365829\pi\)
\(128\) 0 0
\(129\) −0.193252 −0.0170149
\(130\) 0 0
\(131\) −19.9253 −1.74088 −0.870441 0.492273i \(-0.836166\pi\)
−0.870441 + 0.492273i \(0.836166\pi\)
\(132\) 0 0
\(133\) −7.02827 −0.609429
\(134\) 0 0
\(135\) 1.51414 0.130316
\(136\) 0 0
\(137\) −2.92892 −0.250234 −0.125117 0.992142i \(-0.539931\pi\)
−0.125117 + 0.992142i \(0.539931\pi\)
\(138\) 0 0
\(139\) 1.80675 0.153246 0.0766232 0.997060i \(-0.475586\pi\)
0.0766232 + 0.997060i \(0.475586\pi\)
\(140\) 0 0
\(141\) 5.02827 0.423457
\(142\) 0 0
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −15.8205 −1.31382
\(146\) 0 0
\(147\) 5.25526 0.433447
\(148\) 0 0
\(149\) −10.1504 −0.831557 −0.415778 0.909466i \(-0.636491\pi\)
−0.415778 + 0.909466i \(0.636491\pi\)
\(150\) 0 0
\(151\) 7.96265 0.647992 0.323996 0.946059i \(-0.394974\pi\)
0.323996 + 0.946059i \(0.394974\pi\)
\(152\) 0 0
\(153\) −4.83502 −0.390888
\(154\) 0 0
\(155\) 0.735663 0.0590899
\(156\) 0 0
\(157\) −18.3492 −1.46442 −0.732211 0.681078i \(-0.761512\pi\)
−0.732211 + 0.681078i \(0.761512\pi\)
\(158\) 0 0
\(159\) 7.67004 0.608274
\(160\) 0 0
\(161\) −11.5388 −0.909385
\(162\) 0 0
\(163\) 21.8259 1.70954 0.854770 0.519007i \(-0.173698\pi\)
0.854770 + 0.519007i \(0.173698\pi\)
\(164\) 0 0
\(165\) −1.51414 −0.117875
\(166\) 0 0
\(167\) −21.9819 −1.70101 −0.850503 0.525969i \(-0.823703\pi\)
−0.850503 + 0.525969i \(0.823703\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 0 0
\(171\) −5.32088 −0.406898
\(172\) 0 0
\(173\) −2.50506 −0.190457 −0.0952283 0.995455i \(-0.530358\pi\)
−0.0952283 + 0.995455i \(0.530358\pi\)
\(174\) 0 0
\(175\) −3.57615 −0.270331
\(176\) 0 0
\(177\) −7.02827 −0.528277
\(178\) 0 0
\(179\) 13.7074 1.02454 0.512269 0.858825i \(-0.328805\pi\)
0.512269 + 0.858825i \(0.328805\pi\)
\(180\) 0 0
\(181\) 25.6892 1.90947 0.954733 0.297463i \(-0.0961406\pi\)
0.954733 + 0.297463i \(0.0961406\pi\)
\(182\) 0 0
\(183\) −6.25526 −0.462402
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 4.83502 0.353572
\(188\) 0 0
\(189\) −1.32088 −0.0960802
\(190\) 0 0
\(191\) −18.9536 −1.37143 −0.685716 0.727869i \(-0.740511\pi\)
−0.685716 + 0.727869i \(0.740511\pi\)
\(192\) 0 0
\(193\) −0.934380 −0.0672581 −0.0336291 0.999434i \(-0.510706\pi\)
−0.0336291 + 0.999434i \(0.510706\pi\)
\(194\) 0 0
\(195\) −1.51414 −0.108430
\(196\) 0 0
\(197\) −4.99093 −0.355589 −0.177794 0.984068i \(-0.556896\pi\)
−0.177794 + 0.984068i \(0.556896\pi\)
\(198\) 0 0
\(199\) 4.00000 0.283552 0.141776 0.989899i \(-0.454719\pi\)
0.141776 + 0.989899i \(0.454719\pi\)
\(200\) 0 0
\(201\) 9.18418 0.647802
\(202\) 0 0
\(203\) 13.8013 0.968660
\(204\) 0 0
\(205\) −16.2553 −1.13532
\(206\) 0 0
\(207\) −8.73566 −0.607171
\(208\) 0 0
\(209\) 5.32088 0.368053
\(210\) 0 0
\(211\) 14.6363 1.00760 0.503802 0.863819i \(-0.331934\pi\)
0.503802 + 0.863819i \(0.331934\pi\)
\(212\) 0 0
\(213\) 14.6983 1.00711
\(214\) 0 0
\(215\) −0.292611 −0.0199559
\(216\) 0 0
\(217\) −0.641769 −0.0435661
\(218\) 0 0
\(219\) −10.9909 −0.742698
\(220\) 0 0
\(221\) 4.83502 0.325239
\(222\) 0 0
\(223\) −24.9108 −1.66815 −0.834074 0.551652i \(-0.813998\pi\)
−0.834074 + 0.551652i \(0.813998\pi\)
\(224\) 0 0
\(225\) −2.70739 −0.180493
\(226\) 0 0
\(227\) 8.60442 0.571096 0.285548 0.958364i \(-0.407824\pi\)
0.285548 + 0.958364i \(0.407824\pi\)
\(228\) 0 0
\(229\) −26.3684 −1.74247 −0.871235 0.490866i \(-0.836680\pi\)
−0.871235 + 0.490866i \(0.836680\pi\)
\(230\) 0 0
\(231\) 1.32088 0.0869078
\(232\) 0 0
\(233\) −5.75020 −0.376708 −0.188354 0.982101i \(-0.560315\pi\)
−0.188354 + 0.982101i \(0.560315\pi\)
\(234\) 0 0
\(235\) 7.61350 0.496650
\(236\) 0 0
\(237\) −16.5051 −1.07212
\(238\) 0 0
\(239\) −15.3209 −0.991026 −0.495513 0.868601i \(-0.665020\pi\)
−0.495513 + 0.868601i \(0.665020\pi\)
\(240\) 0 0
\(241\) 22.3118 1.43723 0.718615 0.695408i \(-0.244776\pi\)
0.718615 + 0.695408i \(0.244776\pi\)
\(242\) 0 0
\(243\) −1.00000 −0.0641500
\(244\) 0 0
\(245\) 7.95719 0.508366
\(246\) 0 0
\(247\) 5.32088 0.338560
\(248\) 0 0
\(249\) 0.386505 0.0244938
\(250\) 0 0
\(251\) 18.4431 1.16412 0.582058 0.813148i \(-0.302248\pi\)
0.582058 + 0.813148i \(0.302248\pi\)
\(252\) 0 0
\(253\) 8.73566 0.549206
\(254\) 0 0
\(255\) −7.32088 −0.458452
\(256\) 0 0
\(257\) 11.4713 0.715562 0.357781 0.933806i \(-0.383533\pi\)
0.357781 + 0.933806i \(0.383533\pi\)
\(258\) 0 0
\(259\) 3.48947 0.216825
\(260\) 0 0
\(261\) 10.4485 0.646747
\(262\) 0 0
\(263\) 8.31181 0.512528 0.256264 0.966607i \(-0.417508\pi\)
0.256264 + 0.966607i \(0.417508\pi\)
\(264\) 0 0
\(265\) 11.6135 0.713411
\(266\) 0 0
\(267\) 2.92892 0.179247
\(268\) 0 0
\(269\) 20.4431 1.24643 0.623217 0.782049i \(-0.285825\pi\)
0.623217 + 0.782049i \(0.285825\pi\)
\(270\) 0 0
\(271\) 18.2926 1.11120 0.555598 0.831451i \(-0.312489\pi\)
0.555598 + 0.831451i \(0.312489\pi\)
\(272\) 0 0
\(273\) 1.32088 0.0799436
\(274\) 0 0
\(275\) 2.70739 0.163262
\(276\) 0 0
\(277\) 21.5279 1.29348 0.646742 0.762709i \(-0.276131\pi\)
0.646742 + 0.762709i \(0.276131\pi\)
\(278\) 0 0
\(279\) −0.485863 −0.0290879
\(280\) 0 0
\(281\) −9.57615 −0.571265 −0.285633 0.958339i \(-0.592204\pi\)
−0.285633 + 0.958339i \(0.592204\pi\)
\(282\) 0 0
\(283\) 2.38104 0.141538 0.0707691 0.997493i \(-0.477455\pi\)
0.0707691 + 0.997493i \(0.477455\pi\)
\(284\) 0 0
\(285\) −8.05655 −0.477229
\(286\) 0 0
\(287\) 14.1806 0.837053
\(288\) 0 0
\(289\) 6.37743 0.375143
\(290\) 0 0
\(291\) 6.64177 0.389347
\(292\) 0 0
\(293\) 16.8031 0.981650 0.490825 0.871258i \(-0.336695\pi\)
0.490825 + 0.871258i \(0.336695\pi\)
\(294\) 0 0
\(295\) −10.6418 −0.619588
\(296\) 0 0
\(297\) 1.00000 0.0580259
\(298\) 0 0
\(299\) 8.73566 0.505196
\(300\) 0 0
\(301\) 0.255264 0.0147132
\(302\) 0 0
\(303\) −15.4768 −0.889118
\(304\) 0 0
\(305\) −9.47133 −0.542327
\(306\) 0 0
\(307\) 30.6044 1.74669 0.873343 0.487105i \(-0.161947\pi\)
0.873343 + 0.487105i \(0.161947\pi\)
\(308\) 0 0
\(309\) −3.22699 −0.183577
\(310\) 0 0
\(311\) 34.0192 1.92905 0.964526 0.263988i \(-0.0850379\pi\)
0.964526 + 0.263988i \(0.0850379\pi\)
\(312\) 0 0
\(313\) 12.0192 0.679365 0.339683 0.940540i \(-0.389680\pi\)
0.339683 + 0.940540i \(0.389680\pi\)
\(314\) 0 0
\(315\) −2.00000 −0.112687
\(316\) 0 0
\(317\) 15.6382 0.878327 0.439163 0.898407i \(-0.355275\pi\)
0.439163 + 0.898407i \(0.355275\pi\)
\(318\) 0 0
\(319\) −10.4485 −0.585005
\(320\) 0 0
\(321\) −2.25526 −0.125877
\(322\) 0 0
\(323\) 25.7266 1.43147
\(324\) 0 0
\(325\) 2.70739 0.150179
\(326\) 0 0
\(327\) −13.1222 −0.725658
\(328\) 0 0
\(329\) −6.64177 −0.366173
\(330\) 0 0
\(331\) 17.1276 0.941420 0.470710 0.882288i \(-0.343998\pi\)
0.470710 + 0.882288i \(0.343998\pi\)
\(332\) 0 0
\(333\) 2.64177 0.144768
\(334\) 0 0
\(335\) 13.9061 0.759772
\(336\) 0 0
\(337\) −21.8013 −1.18759 −0.593796 0.804616i \(-0.702371\pi\)
−0.593796 + 0.804616i \(0.702371\pi\)
\(338\) 0 0
\(339\) 8.25526 0.448364
\(340\) 0 0
\(341\) 0.485863 0.0263110
\(342\) 0 0
\(343\) −16.1878 −0.874058
\(344\) 0 0
\(345\) −13.2270 −0.712117
\(346\) 0 0
\(347\) 16.6418 0.893377 0.446688 0.894690i \(-0.352603\pi\)
0.446688 + 0.894690i \(0.352603\pi\)
\(348\) 0 0
\(349\) 35.1979 1.88410 0.942051 0.335471i \(-0.108895\pi\)
0.942051 + 0.335471i \(0.108895\pi\)
\(350\) 0 0
\(351\) 1.00000 0.0533761
\(352\) 0 0
\(353\) 28.7411 1.52974 0.764868 0.644187i \(-0.222804\pi\)
0.764868 + 0.644187i \(0.222804\pi\)
\(354\) 0 0
\(355\) 22.2553 1.18119
\(356\) 0 0
\(357\) 6.38650 0.338010
\(358\) 0 0
\(359\) 5.26434 0.277841 0.138921 0.990304i \(-0.455637\pi\)
0.138921 + 0.990304i \(0.455637\pi\)
\(360\) 0 0
\(361\) 9.31181 0.490095
\(362\) 0 0
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) −16.6418 −0.871070
\(366\) 0 0
\(367\) 35.7266 1.86491 0.932456 0.361282i \(-0.117661\pi\)
0.932456 + 0.361282i \(0.117661\pi\)
\(368\) 0 0
\(369\) 10.7357 0.558876
\(370\) 0 0
\(371\) −10.1312 −0.525988
\(372\) 0 0
\(373\) −20.3684 −1.05463 −0.527317 0.849669i \(-0.676802\pi\)
−0.527317 + 0.849669i \(0.676802\pi\)
\(374\) 0 0
\(375\) −11.6700 −0.602638
\(376\) 0 0
\(377\) −10.4485 −0.538126
\(378\) 0 0
\(379\) 15.4576 0.794003 0.397002 0.917818i \(-0.370051\pi\)
0.397002 + 0.917818i \(0.370051\pi\)
\(380\) 0 0
\(381\) −9.22153 −0.472433
\(382\) 0 0
\(383\) −19.7266 −1.00798 −0.503991 0.863709i \(-0.668135\pi\)
−0.503991 + 0.863709i \(0.668135\pi\)
\(384\) 0 0
\(385\) 2.00000 0.101929
\(386\) 0 0
\(387\) 0.193252 0.00982357
\(388\) 0 0
\(389\) −18.3118 −0.928446 −0.464223 0.885718i \(-0.653666\pi\)
−0.464223 + 0.885718i \(0.653666\pi\)
\(390\) 0 0
\(391\) 42.2371 2.13602
\(392\) 0 0
\(393\) 19.9253 1.00510
\(394\) 0 0
\(395\) −24.9909 −1.25743
\(396\) 0 0
\(397\) −21.9819 −1.10324 −0.551619 0.834096i \(-0.685990\pi\)
−0.551619 + 0.834096i \(0.685990\pi\)
\(398\) 0 0
\(399\) 7.02827 0.351854
\(400\) 0 0
\(401\) −19.9572 −0.996614 −0.498307 0.867001i \(-0.666045\pi\)
−0.498307 + 0.867001i \(0.666045\pi\)
\(402\) 0 0
\(403\) 0.485863 0.0242026
\(404\) 0 0
\(405\) −1.51414 −0.0752381
\(406\) 0 0
\(407\) −2.64177 −0.130948
\(408\) 0 0
\(409\) 18.9909 0.939041 0.469520 0.882922i \(-0.344427\pi\)
0.469520 + 0.882922i \(0.344427\pi\)
\(410\) 0 0
\(411\) 2.92892 0.144473
\(412\) 0 0
\(413\) 9.28354 0.456813
\(414\) 0 0
\(415\) 0.585221 0.0287274
\(416\) 0 0
\(417\) −1.80675 −0.0884768
\(418\) 0 0
\(419\) −30.5188 −1.49094 −0.745471 0.666539i \(-0.767775\pi\)
−0.745471 + 0.666539i \(0.767775\pi\)
\(420\) 0 0
\(421\) −13.9253 −0.678678 −0.339339 0.940664i \(-0.610203\pi\)
−0.339339 + 0.940664i \(0.610203\pi\)
\(422\) 0 0
\(423\) −5.02827 −0.244483
\(424\) 0 0
\(425\) 13.0903 0.634972
\(426\) 0 0
\(427\) 8.26248 0.399850
\(428\) 0 0
\(429\) −1.00000 −0.0482805
\(430\) 0 0
\(431\) −4.58522 −0.220862 −0.110431 0.993884i \(-0.535223\pi\)
−0.110431 + 0.993884i \(0.535223\pi\)
\(432\) 0 0
\(433\) −28.9427 −1.39090 −0.695448 0.718577i \(-0.744794\pi\)
−0.695448 + 0.718577i \(0.744794\pi\)
\(434\) 0 0
\(435\) 15.8205 0.758534
\(436\) 0 0
\(437\) 46.4815 2.22351
\(438\) 0 0
\(439\) −22.7494 −1.08577 −0.542885 0.839807i \(-0.682668\pi\)
−0.542885 + 0.839807i \(0.682668\pi\)
\(440\) 0 0
\(441\) −5.25526 −0.250251
\(442\) 0 0
\(443\) 4.77301 0.226773 0.113386 0.993551i \(-0.463830\pi\)
0.113386 + 0.993551i \(0.463830\pi\)
\(444\) 0 0
\(445\) 4.43478 0.210229
\(446\) 0 0
\(447\) 10.1504 0.480099
\(448\) 0 0
\(449\) 7.95719 0.375523 0.187761 0.982215i \(-0.439877\pi\)
0.187761 + 0.982215i \(0.439877\pi\)
\(450\) 0 0
\(451\) −10.7357 −0.505523
\(452\) 0 0
\(453\) −7.96265 −0.374118
\(454\) 0 0
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) 17.5279 0.819919 0.409960 0.912104i \(-0.365543\pi\)
0.409960 + 0.912104i \(0.365543\pi\)
\(458\) 0 0
\(459\) 4.83502 0.225679
\(460\) 0 0
\(461\) −10.6610 −0.496531 −0.248265 0.968692i \(-0.579861\pi\)
−0.248265 + 0.968692i \(0.579861\pi\)
\(462\) 0 0
\(463\) 15.9572 0.741593 0.370797 0.928714i \(-0.379085\pi\)
0.370797 + 0.928714i \(0.379085\pi\)
\(464\) 0 0
\(465\) −0.735663 −0.0341156
\(466\) 0 0
\(467\) −30.9053 −1.43013 −0.715064 0.699059i \(-0.753602\pi\)
−0.715064 + 0.699059i \(0.753602\pi\)
\(468\) 0 0
\(469\) −12.1312 −0.560169
\(470\) 0 0
\(471\) 18.3492 0.845485
\(472\) 0 0
\(473\) −0.193252 −0.00888576
\(474\) 0 0
\(475\) 14.4057 0.660979
\(476\) 0 0
\(477\) −7.67004 −0.351187
\(478\) 0 0
\(479\) 14.5479 0.664709 0.332355 0.943154i \(-0.392157\pi\)
0.332355 + 0.943154i \(0.392157\pi\)
\(480\) 0 0
\(481\) −2.64177 −0.120454
\(482\) 0 0
\(483\) 11.5388 0.525034
\(484\) 0 0
\(485\) 10.0565 0.456644
\(486\) 0 0
\(487\) 36.3437 1.64689 0.823445 0.567396i \(-0.192049\pi\)
0.823445 + 0.567396i \(0.192049\pi\)
\(488\) 0 0
\(489\) −21.8259 −0.987004
\(490\) 0 0
\(491\) 41.3401 1.86565 0.932826 0.360327i \(-0.117335\pi\)
0.932826 + 0.360327i \(0.117335\pi\)
\(492\) 0 0
\(493\) −50.5188 −2.27525
\(494\) 0 0
\(495\) 1.51414 0.0680554
\(496\) 0 0
\(497\) −19.4148 −0.870872
\(498\) 0 0
\(499\) −12.2234 −0.547194 −0.273597 0.961844i \(-0.588213\pi\)
−0.273597 + 0.961844i \(0.588213\pi\)
\(500\) 0 0
\(501\) 21.9819 0.982077
\(502\) 0 0
\(503\) −16.1987 −0.722265 −0.361133 0.932515i \(-0.617610\pi\)
−0.361133 + 0.932515i \(0.617610\pi\)
\(504\) 0 0
\(505\) −23.4340 −1.04280
\(506\) 0 0
\(507\) −1.00000 −0.0444116
\(508\) 0 0
\(509\) −7.24073 −0.320940 −0.160470 0.987041i \(-0.551301\pi\)
−0.160470 + 0.987041i \(0.551301\pi\)
\(510\) 0 0
\(511\) 14.5177 0.642227
\(512\) 0 0
\(513\) 5.32088 0.234923
\(514\) 0 0
\(515\) −4.88611 −0.215308
\(516\) 0 0
\(517\) 5.02827 0.221143
\(518\) 0 0
\(519\) 2.50506 0.109960
\(520\) 0 0
\(521\) −6.65991 −0.291776 −0.145888 0.989301i \(-0.546604\pi\)
−0.145888 + 0.989301i \(0.546604\pi\)
\(522\) 0 0
\(523\) −16.3063 −0.713027 −0.356513 0.934290i \(-0.616035\pi\)
−0.356513 + 0.934290i \(0.616035\pi\)
\(524\) 0 0
\(525\) 3.57615 0.156076
\(526\) 0 0
\(527\) 2.34916 0.102331
\(528\) 0 0
\(529\) 53.3118 2.31790
\(530\) 0 0
\(531\) 7.02827 0.305001
\(532\) 0 0
\(533\) −10.7357 −0.465013
\(534\) 0 0
\(535\) −3.41478 −0.147634
\(536\) 0 0
\(537\) −13.7074 −0.591518
\(538\) 0 0
\(539\) 5.25526 0.226360
\(540\) 0 0
\(541\) 39.2835 1.68893 0.844466 0.535610i \(-0.179918\pi\)
0.844466 + 0.535610i \(0.179918\pi\)
\(542\) 0 0
\(543\) −25.6892 −1.10243
\(544\) 0 0
\(545\) −19.8688 −0.851084
\(546\) 0 0
\(547\) −1.03374 −0.0441994 −0.0220997 0.999756i \(-0.507035\pi\)
−0.0220997 + 0.999756i \(0.507035\pi\)
\(548\) 0 0
\(549\) 6.25526 0.266968
\(550\) 0 0
\(551\) −55.5953 −2.36844
\(552\) 0 0
\(553\) 21.8013 0.927085
\(554\) 0 0
\(555\) 4.00000 0.169791
\(556\) 0 0
\(557\) −31.8314 −1.34874 −0.674370 0.738393i \(-0.735585\pi\)
−0.674370 + 0.738393i \(0.735585\pi\)
\(558\) 0 0
\(559\) −0.193252 −0.00817371
\(560\) 0 0
\(561\) −4.83502 −0.204135
\(562\) 0 0
\(563\) 29.2088 1.23101 0.615503 0.788135i \(-0.288953\pi\)
0.615503 + 0.788135i \(0.288953\pi\)
\(564\) 0 0
\(565\) 12.4996 0.525862
\(566\) 0 0
\(567\) 1.32088 0.0554719
\(568\) 0 0
\(569\) −32.5616 −1.36505 −0.682527 0.730860i \(-0.739119\pi\)
−0.682527 + 0.730860i \(0.739119\pi\)
\(570\) 0 0
\(571\) 1.16498 0.0487528 0.0243764 0.999703i \(-0.492240\pi\)
0.0243764 + 0.999703i \(0.492240\pi\)
\(572\) 0 0
\(573\) 18.9536 0.791797
\(574\) 0 0
\(575\) 23.6508 0.986308
\(576\) 0 0
\(577\) −33.2654 −1.38486 −0.692428 0.721487i \(-0.743459\pi\)
−0.692428 + 0.721487i \(0.743459\pi\)
\(578\) 0 0
\(579\) 0.934380 0.0388315
\(580\) 0 0
\(581\) −0.510528 −0.0211803
\(582\) 0 0
\(583\) 7.67004 0.317661
\(584\) 0 0
\(585\) 1.51414 0.0626019
\(586\) 0 0
\(587\) 20.3118 0.838358 0.419179 0.907904i \(-0.362318\pi\)
0.419179 + 0.907904i \(0.362318\pi\)
\(588\) 0 0
\(589\) 2.58522 0.106522
\(590\) 0 0
\(591\) 4.99093 0.205299
\(592\) 0 0
\(593\) 42.1998 1.73294 0.866468 0.499232i \(-0.166384\pi\)
0.866468 + 0.499232i \(0.166384\pi\)
\(594\) 0 0
\(595\) 9.67004 0.396433
\(596\) 0 0
\(597\) −4.00000 −0.163709
\(598\) 0 0
\(599\) 7.98907 0.326425 0.163212 0.986591i \(-0.447814\pi\)
0.163212 + 0.986591i \(0.447814\pi\)
\(600\) 0 0
\(601\) 14.8861 0.607217 0.303608 0.952797i \(-0.401809\pi\)
0.303608 + 0.952797i \(0.401809\pi\)
\(602\) 0 0
\(603\) −9.18418 −0.374009
\(604\) 0 0
\(605\) −1.51414 −0.0615584
\(606\) 0 0
\(607\) −17.4949 −0.710098 −0.355049 0.934848i \(-0.615536\pi\)
−0.355049 + 0.934848i \(0.615536\pi\)
\(608\) 0 0
\(609\) −13.8013 −0.559256
\(610\) 0 0
\(611\) 5.02827 0.203422
\(612\) 0 0
\(613\) 19.6508 0.793690 0.396845 0.917886i \(-0.370105\pi\)
0.396845 + 0.917886i \(0.370105\pi\)
\(614\) 0 0
\(615\) 16.2553 0.655476
\(616\) 0 0
\(617\) 21.1276 0.850566 0.425283 0.905060i \(-0.360174\pi\)
0.425283 + 0.905060i \(0.360174\pi\)
\(618\) 0 0
\(619\) −7.18418 −0.288757 −0.144378 0.989523i \(-0.546118\pi\)
−0.144378 + 0.989523i \(0.546118\pi\)
\(620\) 0 0
\(621\) 8.73566 0.350550
\(622\) 0 0
\(623\) −3.86876 −0.154999
\(624\) 0 0
\(625\) −4.13310 −0.165324
\(626\) 0 0
\(627\) −5.32088 −0.212496
\(628\) 0 0
\(629\) −12.7730 −0.509293
\(630\) 0 0
\(631\) −26.9855 −1.07427 −0.537137 0.843495i \(-0.680494\pi\)
−0.537137 + 0.843495i \(0.680494\pi\)
\(632\) 0 0
\(633\) −14.6363 −0.581741
\(634\) 0 0
\(635\) −13.9627 −0.554091
\(636\) 0 0
\(637\) 5.25526 0.208221
\(638\) 0 0
\(639\) −14.6983 −0.581456
\(640\) 0 0
\(641\) 46.5489 1.83857 0.919286 0.393590i \(-0.128767\pi\)
0.919286 + 0.393590i \(0.128767\pi\)
\(642\) 0 0
\(643\) 36.9217 1.45605 0.728025 0.685551i \(-0.240439\pi\)
0.728025 + 0.685551i \(0.240439\pi\)
\(644\) 0 0
\(645\) 0.292611 0.0115215
\(646\) 0 0
\(647\) 17.1595 0.674610 0.337305 0.941395i \(-0.390485\pi\)
0.337305 + 0.941395i \(0.390485\pi\)
\(648\) 0 0
\(649\) −7.02827 −0.275884
\(650\) 0 0
\(651\) 0.641769 0.0251529
\(652\) 0 0
\(653\) −14.5852 −0.570764 −0.285382 0.958414i \(-0.592120\pi\)
−0.285382 + 0.958414i \(0.592120\pi\)
\(654\) 0 0
\(655\) 30.1696 1.17883
\(656\) 0 0
\(657\) 10.9909 0.428797
\(658\) 0 0
\(659\) −43.0848 −1.67835 −0.839173 0.543864i \(-0.816961\pi\)
−0.839173 + 0.543864i \(0.816961\pi\)
\(660\) 0 0
\(661\) −29.8397 −1.16063 −0.580315 0.814392i \(-0.697070\pi\)
−0.580315 + 0.814392i \(0.697070\pi\)
\(662\) 0 0
\(663\) −4.83502 −0.187777
\(664\) 0 0
\(665\) 10.6418 0.412670
\(666\) 0 0
\(667\) −91.2747 −3.53417
\(668\) 0 0
\(669\) 24.9108 0.963106
\(670\) 0 0
\(671\) −6.25526 −0.241482
\(672\) 0 0
\(673\) 49.4713 1.90698 0.953490 0.301425i \(-0.0974624\pi\)
0.953490 + 0.301425i \(0.0974624\pi\)
\(674\) 0 0
\(675\) 2.70739 0.104207
\(676\) 0 0
\(677\) 8.11856 0.312022 0.156011 0.987755i \(-0.450137\pi\)
0.156011 + 0.987755i \(0.450137\pi\)
\(678\) 0 0
\(679\) −8.77301 −0.336677
\(680\) 0 0
\(681\) −8.60442 −0.329722
\(682\) 0 0
\(683\) 45.3785 1.73636 0.868180 0.496250i \(-0.165290\pi\)
0.868180 + 0.496250i \(0.165290\pi\)
\(684\) 0 0
\(685\) 4.43478 0.169444
\(686\) 0 0
\(687\) 26.3684 1.00602
\(688\) 0 0
\(689\) 7.67004 0.292205
\(690\) 0 0
\(691\) 6.67365 0.253878 0.126939 0.991911i \(-0.459485\pi\)
0.126939 + 0.991911i \(0.459485\pi\)
\(692\) 0 0
\(693\) −1.32088 −0.0501763
\(694\) 0 0
\(695\) −2.73566 −0.103770
\(696\) 0 0
\(697\) −51.9072 −1.96612
\(698\) 0 0
\(699\) 5.75020 0.217493
\(700\) 0 0
\(701\) −46.0329 −1.73864 −0.869320 0.494250i \(-0.835443\pi\)
−0.869320 + 0.494250i \(0.835443\pi\)
\(702\) 0 0
\(703\) −14.0565 −0.530153
\(704\) 0 0
\(705\) −7.61350 −0.286741
\(706\) 0 0
\(707\) 20.4431 0.768840
\(708\) 0 0
\(709\) 11.9891 0.450259 0.225130 0.974329i \(-0.427719\pi\)
0.225130 + 0.974329i \(0.427719\pi\)
\(710\) 0 0
\(711\) 16.5051 0.618988
\(712\) 0 0
\(713\) 4.24434 0.158952
\(714\) 0 0
\(715\) −1.51414 −0.0566255
\(716\) 0 0
\(717\) 15.3209 0.572169
\(718\) 0 0
\(719\) 19.7375 0.736085 0.368043 0.929809i \(-0.380028\pi\)
0.368043 + 0.929809i \(0.380028\pi\)
\(720\) 0 0
\(721\) 4.26248 0.158743
\(722\) 0 0
\(723\) −22.3118 −0.829785
\(724\) 0 0
\(725\) −28.2882 −1.05060
\(726\) 0 0
\(727\) −14.8405 −0.550403 −0.275202 0.961387i \(-0.588745\pi\)
−0.275202 + 0.961387i \(0.588745\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −0.934380 −0.0345593
\(732\) 0 0
\(733\) 38.5935 1.42548 0.712742 0.701427i \(-0.247453\pi\)
0.712742 + 0.701427i \(0.247453\pi\)
\(734\) 0 0
\(735\) −7.95719 −0.293505
\(736\) 0 0
\(737\) 9.18418 0.338304
\(738\) 0 0
\(739\) 30.9053 1.13687 0.568435 0.822728i \(-0.307549\pi\)
0.568435 + 0.822728i \(0.307549\pi\)
\(740\) 0 0
\(741\) −5.32088 −0.195468
\(742\) 0 0
\(743\) −48.3118 −1.77239 −0.886194 0.463314i \(-0.846660\pi\)
−0.886194 + 0.463314i \(0.846660\pi\)
\(744\) 0 0
\(745\) 15.3692 0.563082
\(746\) 0 0
\(747\) −0.386505 −0.0141415
\(748\) 0 0
\(749\) 2.97894 0.108848
\(750\) 0 0
\(751\) −7.61350 −0.277820 −0.138910 0.990305i \(-0.544360\pi\)
−0.138910 + 0.990305i \(0.544360\pi\)
\(752\) 0 0
\(753\) −18.4431 −0.672102
\(754\) 0 0
\(755\) −12.0565 −0.438783
\(756\) 0 0
\(757\) 2.27447 0.0826668 0.0413334 0.999145i \(-0.486839\pi\)
0.0413334 + 0.999145i \(0.486839\pi\)
\(758\) 0 0
\(759\) −8.73566 −0.317084
\(760\) 0 0
\(761\) −3.00907 −0.109079 −0.0545394 0.998512i \(-0.517369\pi\)
−0.0545394 + 0.998512i \(0.517369\pi\)
\(762\) 0 0
\(763\) 17.3329 0.627492
\(764\) 0 0
\(765\) 7.32088 0.264687
\(766\) 0 0
\(767\) −7.02827 −0.253776
\(768\) 0 0
\(769\) −6.11310 −0.220444 −0.110222 0.993907i \(-0.535156\pi\)
−0.110222 + 0.993907i \(0.535156\pi\)
\(770\) 0 0
\(771\) −11.4713 −0.413130
\(772\) 0 0
\(773\) −3.32635 −0.119640 −0.0598202 0.998209i \(-0.519053\pi\)
−0.0598202 + 0.998209i \(0.519053\pi\)
\(774\) 0 0
\(775\) 1.31542 0.0472513
\(776\) 0 0
\(777\) −3.48947 −0.125184
\(778\) 0 0
\(779\) −57.1232 −2.04665
\(780\) 0 0
\(781\) 14.6983 0.525947
\(782\) 0 0
\(783\) −10.4485 −0.373400
\(784\) 0 0
\(785\) 27.7831 0.991623
\(786\) 0 0
\(787\) −5.43398 −0.193700 −0.0968502 0.995299i \(-0.530877\pi\)
−0.0968502 + 0.995299i \(0.530877\pi\)
\(788\) 0 0
\(789\) −8.31181 −0.295908
\(790\) 0 0
\(791\) −10.9043 −0.387711
\(792\) 0 0
\(793\) −6.25526 −0.222131
\(794\) 0 0
\(795\) −11.6135 −0.411888
\(796\) 0 0
\(797\) 24.4540 0.866204 0.433102 0.901345i \(-0.357419\pi\)
0.433102 + 0.901345i \(0.357419\pi\)
\(798\) 0 0
\(799\) 24.3118 0.860090
\(800\) 0 0
\(801\) −2.92892 −0.103488
\(802\) 0 0
\(803\) −10.9909 −0.387861
\(804\) 0 0
\(805\) 17.4713 0.615783
\(806\) 0 0
\(807\) −20.4431 −0.719630
\(808\) 0 0
\(809\) 0.175107 0.00615644 0.00307822 0.999995i \(-0.499020\pi\)
0.00307822 + 0.999995i \(0.499020\pi\)
\(810\) 0 0
\(811\) −42.0192 −1.47549 −0.737747 0.675077i \(-0.764110\pi\)
−0.737747 + 0.675077i \(0.764110\pi\)
\(812\) 0 0
\(813\) −18.2926 −0.641550
\(814\) 0 0
\(815\) −33.0475 −1.15760
\(816\) 0 0
\(817\) −1.02827 −0.0359747
\(818\) 0 0
\(819\) −1.32088 −0.0461554
\(820\) 0 0
\(821\) −25.0293 −0.873530 −0.436765 0.899576i \(-0.643876\pi\)
−0.436765 + 0.899576i \(0.643876\pi\)
\(822\) 0 0
\(823\) −29.2726 −1.02038 −0.510190 0.860062i \(-0.670425\pi\)
−0.510190 + 0.860062i \(0.670425\pi\)
\(824\) 0 0
\(825\) −2.70739 −0.0942592
\(826\) 0 0
\(827\) 10.6610 0.370718 0.185359 0.982671i \(-0.440655\pi\)
0.185359 + 0.982671i \(0.440655\pi\)
\(828\) 0 0
\(829\) −52.2827 −1.81585 −0.907927 0.419128i \(-0.862336\pi\)
−0.907927 + 0.419128i \(0.862336\pi\)
\(830\) 0 0
\(831\) −21.5279 −0.746794
\(832\) 0 0
\(833\) 25.4093 0.880381
\(834\) 0 0
\(835\) 33.2835 1.15182
\(836\) 0 0
\(837\) 0.485863 0.0167939
\(838\) 0 0
\(839\) 30.8680 1.06568 0.532840 0.846216i \(-0.321125\pi\)
0.532840 + 0.846216i \(0.321125\pi\)
\(840\) 0 0
\(841\) 80.1715 2.76453
\(842\) 0 0
\(843\) 9.57615 0.329820
\(844\) 0 0
\(845\) −1.51414 −0.0520879
\(846\) 0 0
\(847\) 1.32088 0.0453861
\(848\) 0 0
\(849\) −2.38104 −0.0817171
\(850\) 0 0
\(851\) −23.0776 −0.791090
\(852\) 0 0
\(853\) 44.4924 1.52339 0.761695 0.647936i \(-0.224367\pi\)
0.761695 + 0.647936i \(0.224367\pi\)
\(854\) 0 0
\(855\) 8.05655 0.275528
\(856\) 0 0
\(857\) 11.3346 0.387183 0.193592 0.981082i \(-0.437986\pi\)
0.193592 + 0.981082i \(0.437986\pi\)
\(858\) 0 0
\(859\) 13.6700 0.466416 0.233208 0.972427i \(-0.425078\pi\)
0.233208 + 0.972427i \(0.425078\pi\)
\(860\) 0 0
\(861\) −14.1806 −0.483273
\(862\) 0 0
\(863\) −16.2734 −0.553953 −0.276977 0.960877i \(-0.589332\pi\)
−0.276977 + 0.960877i \(0.589332\pi\)
\(864\) 0 0
\(865\) 3.79301 0.128966
\(866\) 0 0
\(867\) −6.37743 −0.216589
\(868\) 0 0
\(869\) −16.5051 −0.559896
\(870\) 0 0
\(871\) 9.18418 0.311194
\(872\) 0 0
\(873\) −6.64177 −0.224790
\(874\) 0 0
\(875\) 15.4148 0.521115
\(876\) 0 0
\(877\) 5.42571 0.183213 0.0916065 0.995795i \(-0.470800\pi\)
0.0916065 + 0.995795i \(0.470800\pi\)
\(878\) 0 0
\(879\) −16.8031 −0.566756
\(880\) 0 0
\(881\) −1.60257 −0.0539919 −0.0269959 0.999636i \(-0.508594\pi\)
−0.0269959 + 0.999636i \(0.508594\pi\)
\(882\) 0 0
\(883\) 13.2835 0.447027 0.223513 0.974701i \(-0.428247\pi\)
0.223513 + 0.974701i \(0.428247\pi\)
\(884\) 0 0
\(885\) 10.6418 0.357719
\(886\) 0 0
\(887\) 40.8789 1.37258 0.686289 0.727329i \(-0.259238\pi\)
0.686289 + 0.727329i \(0.259238\pi\)
\(888\) 0 0
\(889\) 12.1806 0.408523
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 0 0
\(893\) 26.7549 0.895317
\(894\) 0 0
\(895\) −20.7549 −0.693759
\(896\) 0 0
\(897\) −8.73566 −0.291675
\(898\) 0 0
\(899\) −5.07655 −0.169312
\(900\) 0 0
\(901\) 37.0848 1.23547
\(902\) 0 0
\(903\) −0.255264 −0.00849466
\(904\) 0 0
\(905\) −38.8970 −1.29298
\(906\) 0 0
\(907\) −26.2937 −0.873067 −0.436533 0.899688i \(-0.643794\pi\)
−0.436533 + 0.899688i \(0.643794\pi\)
\(908\) 0 0
\(909\) 15.4768 0.513333
\(910\) 0 0
\(911\) −21.5569 −0.714214 −0.357107 0.934064i \(-0.616237\pi\)
−0.357107 + 0.934064i \(0.616237\pi\)
\(912\) 0 0
\(913\) 0.386505 0.0127914
\(914\) 0 0
\(915\) 9.47133 0.313112
\(916\) 0 0
\(917\) −26.3190 −0.869131
\(918\) 0 0
\(919\) −38.7785 −1.27918 −0.639592 0.768715i \(-0.720897\pi\)
−0.639592 + 0.768715i \(0.720897\pi\)
\(920\) 0 0
\(921\) −30.6044 −1.00845
\(922\) 0 0
\(923\) 14.6983 0.483801
\(924\) 0 0
\(925\) −7.15230 −0.235166
\(926\) 0 0
\(927\) 3.22699 0.105988
\(928\) 0 0
\(929\) 23.2023 0.761244 0.380622 0.924731i \(-0.375710\pi\)
0.380622 + 0.924731i \(0.375710\pi\)
\(930\) 0 0
\(931\) 27.9627 0.916439
\(932\) 0 0
\(933\) −34.0192 −1.11374
\(934\) 0 0
\(935\) −7.32088 −0.239419
\(936\) 0 0
\(937\) 3.60257 0.117691 0.0588454 0.998267i \(-0.481258\pi\)
0.0588454 + 0.998267i \(0.481258\pi\)
\(938\) 0 0
\(939\) −12.0192 −0.392232
\(940\) 0 0
\(941\) 26.3382 0.858602 0.429301 0.903162i \(-0.358760\pi\)
0.429301 + 0.903162i \(0.358760\pi\)
\(942\) 0 0
\(943\) −93.7831 −3.05400
\(944\) 0 0
\(945\) 2.00000 0.0650600
\(946\) 0 0
\(947\) −34.8970 −1.13400 −0.567001 0.823717i \(-0.691896\pi\)
−0.567001 + 0.823717i \(0.691896\pi\)
\(948\) 0 0
\(949\) −10.9909 −0.356781
\(950\) 0 0
\(951\) −15.6382 −0.507102
\(952\) 0 0
\(953\) −16.8532 −0.545928 −0.272964 0.962024i \(-0.588004\pi\)
−0.272964 + 0.962024i \(0.588004\pi\)
\(954\) 0 0
\(955\) 28.6983 0.928656
\(956\) 0 0
\(957\) 10.4485 0.337753
\(958\) 0 0
\(959\) −3.86876 −0.124929
\(960\) 0 0
\(961\) −30.7639 −0.992385
\(962\) 0 0
\(963\) 2.25526 0.0726749
\(964\) 0 0
\(965\) 1.41478 0.0455433
\(966\) 0 0
\(967\) −32.3492 −1.04028 −0.520139 0.854081i \(-0.674120\pi\)
−0.520139 + 0.854081i \(0.674120\pi\)
\(968\) 0 0
\(969\) −25.7266 −0.826457
\(970\) 0 0
\(971\) −2.98000 −0.0956327 −0.0478164 0.998856i \(-0.515226\pi\)
−0.0478164 + 0.998856i \(0.515226\pi\)
\(972\) 0 0
\(973\) 2.38650 0.0765078
\(974\) 0 0
\(975\) −2.70739 −0.0867059
\(976\) 0 0
\(977\) −15.2407 −0.487594 −0.243797 0.969826i \(-0.578393\pi\)
−0.243797 + 0.969826i \(0.578393\pi\)
\(978\) 0 0
\(979\) 2.92892 0.0936086
\(980\) 0 0
\(981\) 13.1222 0.418959
\(982\) 0 0
\(983\) −9.75566 −0.311157 −0.155579 0.987824i \(-0.549724\pi\)
−0.155579 + 0.987824i \(0.549724\pi\)
\(984\) 0 0
\(985\) 7.55695 0.240784
\(986\) 0 0
\(987\) 6.64177 0.211410
\(988\) 0 0
\(989\) −1.68819 −0.0536813
\(990\) 0 0
\(991\) −31.6519 −1.00546 −0.502728 0.864445i \(-0.667670\pi\)
−0.502728 + 0.864445i \(0.667670\pi\)
\(992\) 0 0
\(993\) −17.1276 −0.543529
\(994\) 0 0
\(995\) −6.05655 −0.192005
\(996\) 0 0
\(997\) 11.0848 0.351060 0.175530 0.984474i \(-0.443836\pi\)
0.175530 + 0.984474i \(0.443836\pi\)
\(998\) 0 0
\(999\) −2.64177 −0.0835819
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 6864.2.a.br.1.1 3
4.3 odd 2 1716.2.a.g.1.1 3
12.11 even 2 5148.2.a.m.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1716.2.a.g.1.1 3 4.3 odd 2
5148.2.a.m.1.3 3 12.11 even 2
6864.2.a.br.1.1 3 1.1 even 1 trivial