Properties

Label 836.2.a.e.1.2
Level $836$
Weight $2$
Character 836.1
Self dual yes
Analytic conductor $6.675$
Analytic rank $0$
Dimension $6$
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] = [836,2,Mod(1,836)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(836, 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("836.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 836 = 2^{2} \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 836.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: \(6.67549360898\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.744786576.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 17x^{4} + 13x^{3} + 69x^{2} - 21x - 30 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.05861\) of defining polynomial
Character \(\chi\) \(=\) 836.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.05861 q^{3} +0.680036 q^{5} +1.59123 q^{7} +1.23787 q^{9} +O(q^{10})\) \(q-2.05861 q^{3} +0.680036 q^{5} +1.59123 q^{7} +1.23787 q^{9} -1.00000 q^{11} -0.546391 q^{13} -1.39993 q^{15} +6.61704 q^{17} -1.00000 q^{19} -3.27573 q^{21} -6.44615 q^{23} -4.53755 q^{25} +3.62754 q^{27} +7.32229 q^{29} +3.39993 q^{31} +2.05861 q^{33} +1.08210 q^{35} +8.83065 q^{37} +1.12481 q^{39} +3.18632 q^{41} +10.4060 q^{43} +0.841795 q^{45} -7.25697 q^{47} -4.46798 q^{49} -13.6219 q^{51} +7.47729 q^{53} -0.680036 q^{55} +2.05861 q^{57} +11.1288 q^{59} +8.75715 q^{61} +1.96974 q^{63} -0.371565 q^{65} +11.1326 q^{67} +13.2701 q^{69} -7.88100 q^{71} +3.00155 q^{73} +9.34104 q^{75} -1.59123 q^{77} +8.73426 q^{79} -11.1813 q^{81} -7.76611 q^{83} +4.49983 q^{85} -15.0737 q^{87} +15.7031 q^{89} -0.869435 q^{91} -6.99912 q^{93} -0.680036 q^{95} +9.99329 q^{97} -1.23787 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{3} + 5 q^{5} - 2 q^{7} + 17 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{3} + 5 q^{5} - 2 q^{7} + 17 q^{9} - 6 q^{11} + 8 q^{13} + 9 q^{15} - 2 q^{17} - 6 q^{19} + 18 q^{21} + 5 q^{23} + 13 q^{25} + 7 q^{27} + 10 q^{29} + 3 q^{31} - q^{33} - 4 q^{35} + 7 q^{37} - 10 q^{39} + 6 q^{41} + 16 q^{43} + 42 q^{45} + 12 q^{49} + 8 q^{51} + 20 q^{53} - 5 q^{55} - q^{57} + 15 q^{59} + 24 q^{61} - 20 q^{63} - 28 q^{65} + 25 q^{67} - 33 q^{69} - 9 q^{71} - 26 q^{73} + 28 q^{75} + 2 q^{77} - 16 q^{79} + 58 q^{81} - 2 q^{83} + 12 q^{85} - 36 q^{87} + 7 q^{89} + 8 q^{91} - 55 q^{93} - 5 q^{95} + 37 q^{97} - 17 q^{99}+O(q^{100}) \) 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
\(3\) −2.05861 −1.18854 −0.594269 0.804266i \(-0.702559\pi\)
−0.594269 + 0.804266i \(0.702559\pi\)
\(4\) 0 0
\(5\) 0.680036 0.304121 0.152061 0.988371i \(-0.451409\pi\)
0.152061 + 0.988371i \(0.451409\pi\)
\(6\) 0 0
\(7\) 1.59123 0.601429 0.300715 0.953714i \(-0.402775\pi\)
0.300715 + 0.953714i \(0.402775\pi\)
\(8\) 0 0
\(9\) 1.23787 0.412623
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −0.546391 −0.151542 −0.0757708 0.997125i \(-0.524142\pi\)
−0.0757708 + 0.997125i \(0.524142\pi\)
\(14\) 0 0
\(15\) −1.39993 −0.361460
\(16\) 0 0
\(17\) 6.61704 1.60487 0.802434 0.596740i \(-0.203538\pi\)
0.802434 + 0.596740i \(0.203538\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −3.27573 −0.714822
\(22\) 0 0
\(23\) −6.44615 −1.34411 −0.672057 0.740499i \(-0.734589\pi\)
−0.672057 + 0.740499i \(0.734589\pi\)
\(24\) 0 0
\(25\) −4.53755 −0.907510
\(26\) 0 0
\(27\) 3.62754 0.698120
\(28\) 0 0
\(29\) 7.32229 1.35971 0.679857 0.733344i \(-0.262042\pi\)
0.679857 + 0.733344i \(0.262042\pi\)
\(30\) 0 0
\(31\) 3.39993 0.610645 0.305323 0.952249i \(-0.401236\pi\)
0.305323 + 0.952249i \(0.401236\pi\)
\(32\) 0 0
\(33\) 2.05861 0.358358
\(34\) 0 0
\(35\) 1.08210 0.182907
\(36\) 0 0
\(37\) 8.83065 1.45175 0.725875 0.687826i \(-0.241435\pi\)
0.725875 + 0.687826i \(0.241435\pi\)
\(38\) 0 0
\(39\) 1.12481 0.180113
\(40\) 0 0
\(41\) 3.18632 0.497619 0.248810 0.968552i \(-0.419961\pi\)
0.248810 + 0.968552i \(0.419961\pi\)
\(42\) 0 0
\(43\) 10.4060 1.58691 0.793453 0.608632i \(-0.208281\pi\)
0.793453 + 0.608632i \(0.208281\pi\)
\(44\) 0 0
\(45\) 0.841795 0.125487
\(46\) 0 0
\(47\) −7.25697 −1.05854 −0.529269 0.848454i \(-0.677534\pi\)
−0.529269 + 0.848454i \(0.677534\pi\)
\(48\) 0 0
\(49\) −4.46798 −0.638283
\(50\) 0 0
\(51\) −13.6219 −1.90745
\(52\) 0 0
\(53\) 7.47729 1.02708 0.513542 0.858064i \(-0.328333\pi\)
0.513542 + 0.858064i \(0.328333\pi\)
\(54\) 0 0
\(55\) −0.680036 −0.0916960
\(56\) 0 0
\(57\) 2.05861 0.272669
\(58\) 0 0
\(59\) 11.1288 1.44884 0.724422 0.689357i \(-0.242107\pi\)
0.724422 + 0.689357i \(0.242107\pi\)
\(60\) 0 0
\(61\) 8.75715 1.12124 0.560619 0.828074i \(-0.310563\pi\)
0.560619 + 0.828074i \(0.310563\pi\)
\(62\) 0 0
\(63\) 1.96974 0.248164
\(64\) 0 0
\(65\) −0.371565 −0.0460870
\(66\) 0 0
\(67\) 11.1326 1.36007 0.680034 0.733180i \(-0.261965\pi\)
0.680034 + 0.733180i \(0.261965\pi\)
\(68\) 0 0
\(69\) 13.2701 1.59753
\(70\) 0 0
\(71\) −7.88100 −0.935303 −0.467651 0.883913i \(-0.654900\pi\)
−0.467651 + 0.883913i \(0.654900\pi\)
\(72\) 0 0
\(73\) 3.00155 0.351305 0.175652 0.984452i \(-0.443796\pi\)
0.175652 + 0.984452i \(0.443796\pi\)
\(74\) 0 0
\(75\) 9.34104 1.07861
\(76\) 0 0
\(77\) −1.59123 −0.181338
\(78\) 0 0
\(79\) 8.73426 0.982681 0.491341 0.870968i \(-0.336507\pi\)
0.491341 + 0.870968i \(0.336507\pi\)
\(80\) 0 0
\(81\) −11.1813 −1.24237
\(82\) 0 0
\(83\) −7.76611 −0.852441 −0.426221 0.904619i \(-0.640155\pi\)
−0.426221 + 0.904619i \(0.640155\pi\)
\(84\) 0 0
\(85\) 4.49983 0.488075
\(86\) 0 0
\(87\) −15.0737 −1.61607
\(88\) 0 0
\(89\) 15.7031 1.66453 0.832264 0.554380i \(-0.187045\pi\)
0.832264 + 0.554380i \(0.187045\pi\)
\(90\) 0 0
\(91\) −0.869435 −0.0911416
\(92\) 0 0
\(93\) −6.99912 −0.725775
\(94\) 0 0
\(95\) −0.680036 −0.0697702
\(96\) 0 0
\(97\) 9.99329 1.01467 0.507333 0.861750i \(-0.330632\pi\)
0.507333 + 0.861750i \(0.330632\pi\)
\(98\) 0 0
\(99\) −1.23787 −0.124411
\(100\) 0 0
\(101\) 5.13976 0.511425 0.255712 0.966753i \(-0.417690\pi\)
0.255712 + 0.966753i \(0.417690\pi\)
\(102\) 0 0
\(103\) −18.5189 −1.82472 −0.912359 0.409390i \(-0.865741\pi\)
−0.912359 + 0.409390i \(0.865741\pi\)
\(104\) 0 0
\(105\) −2.22761 −0.217393
\(106\) 0 0
\(107\) −20.4593 −1.97787 −0.988936 0.148344i \(-0.952606\pi\)
−0.988936 + 0.148344i \(0.952606\pi\)
\(108\) 0 0
\(109\) 3.07296 0.294336 0.147168 0.989112i \(-0.452984\pi\)
0.147168 + 0.989112i \(0.452984\pi\)
\(110\) 0 0
\(111\) −18.1789 −1.72546
\(112\) 0 0
\(113\) −15.8630 −1.49227 −0.746135 0.665795i \(-0.768093\pi\)
−0.746135 + 0.665795i \(0.768093\pi\)
\(114\) 0 0
\(115\) −4.38361 −0.408774
\(116\) 0 0
\(117\) −0.676360 −0.0625295
\(118\) 0 0
\(119\) 10.5293 0.965215
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −6.55938 −0.591440
\(124\) 0 0
\(125\) −6.48588 −0.580114
\(126\) 0 0
\(127\) 17.1267 1.51975 0.759876 0.650069i \(-0.225260\pi\)
0.759876 + 0.650069i \(0.225260\pi\)
\(128\) 0 0
\(129\) −21.4220 −1.88610
\(130\) 0 0
\(131\) −1.47071 −0.128496 −0.0642482 0.997934i \(-0.520465\pi\)
−0.0642482 + 0.997934i \(0.520465\pi\)
\(132\) 0 0
\(133\) −1.59123 −0.137977
\(134\) 0 0
\(135\) 2.46686 0.212313
\(136\) 0 0
\(137\) −8.59949 −0.734704 −0.367352 0.930082i \(-0.619736\pi\)
−0.367352 + 0.930082i \(0.619736\pi\)
\(138\) 0 0
\(139\) −18.5155 −1.57047 −0.785234 0.619199i \(-0.787457\pi\)
−0.785234 + 0.619199i \(0.787457\pi\)
\(140\) 0 0
\(141\) 14.9393 1.25811
\(142\) 0 0
\(143\) 0.546391 0.0456915
\(144\) 0 0
\(145\) 4.97942 0.413518
\(146\) 0 0
\(147\) 9.19782 0.758623
\(148\) 0 0
\(149\) 11.3272 0.927962 0.463981 0.885845i \(-0.346421\pi\)
0.463981 + 0.885845i \(0.346421\pi\)
\(150\) 0 0
\(151\) 13.1169 1.06744 0.533718 0.845663i \(-0.320794\pi\)
0.533718 + 0.845663i \(0.320794\pi\)
\(152\) 0 0
\(153\) 8.19103 0.662206
\(154\) 0 0
\(155\) 2.31207 0.185710
\(156\) 0 0
\(157\) 1.06454 0.0849597 0.0424799 0.999097i \(-0.486474\pi\)
0.0424799 + 0.999097i \(0.486474\pi\)
\(158\) 0 0
\(159\) −15.3928 −1.22073
\(160\) 0 0
\(161\) −10.2573 −0.808390
\(162\) 0 0
\(163\) −11.4544 −0.897178 −0.448589 0.893738i \(-0.648073\pi\)
−0.448589 + 0.893738i \(0.648073\pi\)
\(164\) 0 0
\(165\) 1.39993 0.108984
\(166\) 0 0
\(167\) −18.9897 −1.46947 −0.734733 0.678357i \(-0.762693\pi\)
−0.734733 + 0.678357i \(0.762693\pi\)
\(168\) 0 0
\(169\) −12.7015 −0.977035
\(170\) 0 0
\(171\) −1.23787 −0.0946622
\(172\) 0 0
\(173\) −9.14468 −0.695257 −0.347629 0.937632i \(-0.613013\pi\)
−0.347629 + 0.937632i \(0.613013\pi\)
\(174\) 0 0
\(175\) −7.22030 −0.545803
\(176\) 0 0
\(177\) −22.9098 −1.72201
\(178\) 0 0
\(179\) −2.19458 −0.164030 −0.0820152 0.996631i \(-0.526136\pi\)
−0.0820152 + 0.996631i \(0.526136\pi\)
\(180\) 0 0
\(181\) −10.6906 −0.794622 −0.397311 0.917684i \(-0.630057\pi\)
−0.397311 + 0.917684i \(0.630057\pi\)
\(182\) 0 0
\(183\) −18.0275 −1.33263
\(184\) 0 0
\(185\) 6.00516 0.441508
\(186\) 0 0
\(187\) −6.61704 −0.483886
\(188\) 0 0
\(189\) 5.77226 0.419870
\(190\) 0 0
\(191\) 18.8704 1.36541 0.682707 0.730692i \(-0.260802\pi\)
0.682707 + 0.730692i \(0.260802\pi\)
\(192\) 0 0
\(193\) 5.89542 0.424361 0.212181 0.977230i \(-0.431943\pi\)
0.212181 + 0.977230i \(0.431943\pi\)
\(194\) 0 0
\(195\) 0.764908 0.0547762
\(196\) 0 0
\(197\) −16.1688 −1.15198 −0.575991 0.817456i \(-0.695384\pi\)
−0.575991 + 0.817456i \(0.695384\pi\)
\(198\) 0 0
\(199\) 20.6510 1.46391 0.731955 0.681353i \(-0.238608\pi\)
0.731955 + 0.681353i \(0.238608\pi\)
\(200\) 0 0
\(201\) −22.9177 −1.61649
\(202\) 0 0
\(203\) 11.6515 0.817773
\(204\) 0 0
\(205\) 2.16681 0.151337
\(206\) 0 0
\(207\) −7.97948 −0.554612
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) 0.700318 0.0482119 0.0241059 0.999709i \(-0.492326\pi\)
0.0241059 + 0.999709i \(0.492326\pi\)
\(212\) 0 0
\(213\) 16.2239 1.11164
\(214\) 0 0
\(215\) 7.07648 0.482612
\(216\) 0 0
\(217\) 5.41008 0.367260
\(218\) 0 0
\(219\) −6.17902 −0.417539
\(220\) 0 0
\(221\) −3.61549 −0.243204
\(222\) 0 0
\(223\) 3.32157 0.222429 0.111214 0.993796i \(-0.464526\pi\)
0.111214 + 0.993796i \(0.464526\pi\)
\(224\) 0 0
\(225\) −5.61689 −0.374460
\(226\) 0 0
\(227\) 16.0767 1.06704 0.533522 0.845786i \(-0.320868\pi\)
0.533522 + 0.845786i \(0.320868\pi\)
\(228\) 0 0
\(229\) 3.00688 0.198700 0.0993502 0.995053i \(-0.468324\pi\)
0.0993502 + 0.995053i \(0.468324\pi\)
\(230\) 0 0
\(231\) 3.27573 0.215527
\(232\) 0 0
\(233\) 29.3076 1.92001 0.960004 0.279987i \(-0.0903303\pi\)
0.960004 + 0.279987i \(0.0903303\pi\)
\(234\) 0 0
\(235\) −4.93500 −0.321924
\(236\) 0 0
\(237\) −17.9804 −1.16795
\(238\) 0 0
\(239\) −19.8088 −1.28133 −0.640663 0.767822i \(-0.721340\pi\)
−0.640663 + 0.767822i \(0.721340\pi\)
\(240\) 0 0
\(241\) −19.6586 −1.26632 −0.633162 0.774019i \(-0.718243\pi\)
−0.633162 + 0.774019i \(0.718243\pi\)
\(242\) 0 0
\(243\) 12.1353 0.778478
\(244\) 0 0
\(245\) −3.03839 −0.194115
\(246\) 0 0
\(247\) 0.546391 0.0347660
\(248\) 0 0
\(249\) 15.9874 1.01316
\(250\) 0 0
\(251\) 15.8005 0.997319 0.498660 0.866798i \(-0.333826\pi\)
0.498660 + 0.866798i \(0.333826\pi\)
\(252\) 0 0
\(253\) 6.44615 0.405266
\(254\) 0 0
\(255\) −9.26338 −0.580096
\(256\) 0 0
\(257\) 3.15958 0.197089 0.0985446 0.995133i \(-0.468581\pi\)
0.0985446 + 0.995133i \(0.468581\pi\)
\(258\) 0 0
\(259\) 14.0516 0.873125
\(260\) 0 0
\(261\) 9.06403 0.561050
\(262\) 0 0
\(263\) 1.02188 0.0630116 0.0315058 0.999504i \(-0.489970\pi\)
0.0315058 + 0.999504i \(0.489970\pi\)
\(264\) 0 0
\(265\) 5.08482 0.312358
\(266\) 0 0
\(267\) −32.3266 −1.97835
\(268\) 0 0
\(269\) −19.3680 −1.18089 −0.590445 0.807078i \(-0.701048\pi\)
−0.590445 + 0.807078i \(0.701048\pi\)
\(270\) 0 0
\(271\) 16.9326 1.02858 0.514290 0.857616i \(-0.328055\pi\)
0.514290 + 0.857616i \(0.328055\pi\)
\(272\) 0 0
\(273\) 1.78983 0.108325
\(274\) 0 0
\(275\) 4.53755 0.273625
\(276\) 0 0
\(277\) 16.1370 0.969581 0.484790 0.874630i \(-0.338896\pi\)
0.484790 + 0.874630i \(0.338896\pi\)
\(278\) 0 0
\(279\) 4.20866 0.251966
\(280\) 0 0
\(281\) −21.9217 −1.30774 −0.653868 0.756609i \(-0.726855\pi\)
−0.653868 + 0.756609i \(0.726855\pi\)
\(282\) 0 0
\(283\) 16.4286 0.976578 0.488289 0.872682i \(-0.337621\pi\)
0.488289 + 0.872682i \(0.337621\pi\)
\(284\) 0 0
\(285\) 1.39993 0.0829246
\(286\) 0 0
\(287\) 5.07018 0.299283
\(288\) 0 0
\(289\) 26.7853 1.57560
\(290\) 0 0
\(291\) −20.5723 −1.20597
\(292\) 0 0
\(293\) −29.9404 −1.74914 −0.874569 0.484902i \(-0.838855\pi\)
−0.874569 + 0.484902i \(0.838855\pi\)
\(294\) 0 0
\(295\) 7.56797 0.440624
\(296\) 0 0
\(297\) −3.62754 −0.210491
\(298\) 0 0
\(299\) 3.52212 0.203689
\(300\) 0 0
\(301\) 16.5584 0.954412
\(302\) 0 0
\(303\) −10.5807 −0.607848
\(304\) 0 0
\(305\) 5.95517 0.340992
\(306\) 0 0
\(307\) −6.84067 −0.390418 −0.195209 0.980762i \(-0.562538\pi\)
−0.195209 + 0.980762i \(0.562538\pi\)
\(308\) 0 0
\(309\) 38.1231 2.16875
\(310\) 0 0
\(311\) 6.20500 0.351853 0.175927 0.984403i \(-0.443708\pi\)
0.175927 + 0.984403i \(0.443708\pi\)
\(312\) 0 0
\(313\) −29.7265 −1.68024 −0.840122 0.542398i \(-0.817517\pi\)
−0.840122 + 0.542398i \(0.817517\pi\)
\(314\) 0 0
\(315\) 1.33949 0.0754718
\(316\) 0 0
\(317\) −9.41590 −0.528850 −0.264425 0.964406i \(-0.585182\pi\)
−0.264425 + 0.964406i \(0.585182\pi\)
\(318\) 0 0
\(319\) −7.32229 −0.409969
\(320\) 0 0
\(321\) 42.1176 2.35078
\(322\) 0 0
\(323\) −6.61704 −0.368182
\(324\) 0 0
\(325\) 2.47928 0.137526
\(326\) 0 0
\(327\) −6.32601 −0.349829
\(328\) 0 0
\(329\) −11.5475 −0.636636
\(330\) 0 0
\(331\) 17.7803 0.977294 0.488647 0.872482i \(-0.337491\pi\)
0.488647 + 0.872482i \(0.337491\pi\)
\(332\) 0 0
\(333\) 10.9312 0.599026
\(334\) 0 0
\(335\) 7.57059 0.413626
\(336\) 0 0
\(337\) −7.23368 −0.394044 −0.197022 0.980399i \(-0.563127\pi\)
−0.197022 + 0.980399i \(0.563127\pi\)
\(338\) 0 0
\(339\) 32.6558 1.77362
\(340\) 0 0
\(341\) −3.39993 −0.184116
\(342\) 0 0
\(343\) −18.2482 −0.985311
\(344\) 0 0
\(345\) 9.02414 0.485843
\(346\) 0 0
\(347\) −7.65820 −0.411114 −0.205557 0.978645i \(-0.565901\pi\)
−0.205557 + 0.978645i \(0.565901\pi\)
\(348\) 0 0
\(349\) 25.5749 1.36899 0.684497 0.729015i \(-0.260022\pi\)
0.684497 + 0.729015i \(0.260022\pi\)
\(350\) 0 0
\(351\) −1.98205 −0.105794
\(352\) 0 0
\(353\) −14.3449 −0.763504 −0.381752 0.924265i \(-0.624679\pi\)
−0.381752 + 0.924265i \(0.624679\pi\)
\(354\) 0 0
\(355\) −5.35936 −0.284446
\(356\) 0 0
\(357\) −21.6756 −1.14720
\(358\) 0 0
\(359\) −8.99828 −0.474911 −0.237455 0.971398i \(-0.576313\pi\)
−0.237455 + 0.971398i \(0.576313\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −2.05861 −0.108049
\(364\) 0 0
\(365\) 2.04116 0.106839
\(366\) 0 0
\(367\) 5.83681 0.304679 0.152339 0.988328i \(-0.451319\pi\)
0.152339 + 0.988328i \(0.451319\pi\)
\(368\) 0 0
\(369\) 3.94425 0.205329
\(370\) 0 0
\(371\) 11.8981 0.617719
\(372\) 0 0
\(373\) −2.76634 −0.143235 −0.0716177 0.997432i \(-0.522816\pi\)
−0.0716177 + 0.997432i \(0.522816\pi\)
\(374\) 0 0
\(375\) 13.3519 0.689488
\(376\) 0 0
\(377\) −4.00083 −0.206053
\(378\) 0 0
\(379\) 37.9993 1.95189 0.975946 0.218011i \(-0.0699569\pi\)
0.975946 + 0.218011i \(0.0699569\pi\)
\(380\) 0 0
\(381\) −35.2572 −1.80628
\(382\) 0 0
\(383\) 0.183768 0.00939013 0.00469507 0.999989i \(-0.498506\pi\)
0.00469507 + 0.999989i \(0.498506\pi\)
\(384\) 0 0
\(385\) −1.08210 −0.0551487
\(386\) 0 0
\(387\) 12.8813 0.654794
\(388\) 0 0
\(389\) −1.95683 −0.0992154 −0.0496077 0.998769i \(-0.515797\pi\)
−0.0496077 + 0.998769i \(0.515797\pi\)
\(390\) 0 0
\(391\) −42.6544 −2.15713
\(392\) 0 0
\(393\) 3.02761 0.152723
\(394\) 0 0
\(395\) 5.93961 0.298854
\(396\) 0 0
\(397\) −30.3125 −1.52134 −0.760669 0.649139i \(-0.775129\pi\)
−0.760669 + 0.649139i \(0.775129\pi\)
\(398\) 0 0
\(399\) 3.27573 0.163991
\(400\) 0 0
\(401\) 10.0474 0.501741 0.250871 0.968021i \(-0.419283\pi\)
0.250871 + 0.968021i \(0.419283\pi\)
\(402\) 0 0
\(403\) −1.85769 −0.0925381
\(404\) 0 0
\(405\) −7.60368 −0.377830
\(406\) 0 0
\(407\) −8.83065 −0.437719
\(408\) 0 0
\(409\) 18.8432 0.931738 0.465869 0.884854i \(-0.345742\pi\)
0.465869 + 0.884854i \(0.345742\pi\)
\(410\) 0 0
\(411\) 17.7030 0.873224
\(412\) 0 0
\(413\) 17.7085 0.871378
\(414\) 0 0
\(415\) −5.28123 −0.259246
\(416\) 0 0
\(417\) 38.1163 1.86656
\(418\) 0 0
\(419\) 9.05528 0.442379 0.221190 0.975231i \(-0.429006\pi\)
0.221190 + 0.975231i \(0.429006\pi\)
\(420\) 0 0
\(421\) 15.4995 0.755399 0.377699 0.925928i \(-0.376715\pi\)
0.377699 + 0.925928i \(0.376715\pi\)
\(422\) 0 0
\(423\) −8.98318 −0.436777
\(424\) 0 0
\(425\) −30.0252 −1.45644
\(426\) 0 0
\(427\) 13.9347 0.674345
\(428\) 0 0
\(429\) −1.12481 −0.0543061
\(430\) 0 0
\(431\) −0.871260 −0.0419671 −0.0209836 0.999780i \(-0.506680\pi\)
−0.0209836 + 0.999780i \(0.506680\pi\)
\(432\) 0 0
\(433\) −17.0358 −0.818688 −0.409344 0.912380i \(-0.634242\pi\)
−0.409344 + 0.912380i \(0.634242\pi\)
\(434\) 0 0
\(435\) −10.2507 −0.491482
\(436\) 0 0
\(437\) 6.44615 0.308361
\(438\) 0 0
\(439\) −30.5207 −1.45668 −0.728338 0.685218i \(-0.759707\pi\)
−0.728338 + 0.685218i \(0.759707\pi\)
\(440\) 0 0
\(441\) −5.53077 −0.263370
\(442\) 0 0
\(443\) 29.2482 1.38962 0.694811 0.719192i \(-0.255488\pi\)
0.694811 + 0.719192i \(0.255488\pi\)
\(444\) 0 0
\(445\) 10.6787 0.506218
\(446\) 0 0
\(447\) −23.3183 −1.10292
\(448\) 0 0
\(449\) 2.08177 0.0982448 0.0491224 0.998793i \(-0.484358\pi\)
0.0491224 + 0.998793i \(0.484358\pi\)
\(450\) 0 0
\(451\) −3.18632 −0.150038
\(452\) 0 0
\(453\) −27.0025 −1.26869
\(454\) 0 0
\(455\) −0.591247 −0.0277181
\(456\) 0 0
\(457\) 10.6300 0.497248 0.248624 0.968600i \(-0.420022\pi\)
0.248624 + 0.968600i \(0.420022\pi\)
\(458\) 0 0
\(459\) 24.0036 1.12039
\(460\) 0 0
\(461\) −24.3691 −1.13498 −0.567491 0.823380i \(-0.692086\pi\)
−0.567491 + 0.823380i \(0.692086\pi\)
\(462\) 0 0
\(463\) 28.2627 1.31348 0.656740 0.754117i \(-0.271935\pi\)
0.656740 + 0.754117i \(0.271935\pi\)
\(464\) 0 0
\(465\) −4.75965 −0.220724
\(466\) 0 0
\(467\) −30.9960 −1.43433 −0.717163 0.696905i \(-0.754560\pi\)
−0.717163 + 0.696905i \(0.754560\pi\)
\(468\) 0 0
\(469\) 17.7146 0.817985
\(470\) 0 0
\(471\) −2.19148 −0.100978
\(472\) 0 0
\(473\) −10.4060 −0.478470
\(474\) 0 0
\(475\) 4.53755 0.208197
\(476\) 0 0
\(477\) 9.25590 0.423799
\(478\) 0 0
\(479\) −8.49550 −0.388169 −0.194085 0.980985i \(-0.562174\pi\)
−0.194085 + 0.980985i \(0.562174\pi\)
\(480\) 0 0
\(481\) −4.82499 −0.220001
\(482\) 0 0
\(483\) 21.1158 0.960802
\(484\) 0 0
\(485\) 6.79580 0.308581
\(486\) 0 0
\(487\) −11.4239 −0.517666 −0.258833 0.965922i \(-0.583338\pi\)
−0.258833 + 0.965922i \(0.583338\pi\)
\(488\) 0 0
\(489\) 23.5801 1.06633
\(490\) 0 0
\(491\) −27.2553 −1.23001 −0.615007 0.788522i \(-0.710847\pi\)
−0.615007 + 0.788522i \(0.710847\pi\)
\(492\) 0 0
\(493\) 48.4519 2.18216
\(494\) 0 0
\(495\) −0.841795 −0.0378359
\(496\) 0 0
\(497\) −12.5405 −0.562519
\(498\) 0 0
\(499\) 38.6475 1.73010 0.865051 0.501684i \(-0.167286\pi\)
0.865051 + 0.501684i \(0.167286\pi\)
\(500\) 0 0
\(501\) 39.0923 1.74652
\(502\) 0 0
\(503\) −43.0702 −1.92040 −0.960202 0.279305i \(-0.909896\pi\)
−0.960202 + 0.279305i \(0.909896\pi\)
\(504\) 0 0
\(505\) 3.49522 0.155535
\(506\) 0 0
\(507\) 26.1473 1.16124
\(508\) 0 0
\(509\) 2.52410 0.111879 0.0559393 0.998434i \(-0.482185\pi\)
0.0559393 + 0.998434i \(0.482185\pi\)
\(510\) 0 0
\(511\) 4.77617 0.211285
\(512\) 0 0
\(513\) −3.62754 −0.160160
\(514\) 0 0
\(515\) −12.5935 −0.554936
\(516\) 0 0
\(517\) 7.25697 0.319161
\(518\) 0 0
\(519\) 18.8253 0.826340
\(520\) 0 0
\(521\) −0.528257 −0.0231434 −0.0115717 0.999933i \(-0.503683\pi\)
−0.0115717 + 0.999933i \(0.503683\pi\)
\(522\) 0 0
\(523\) −27.3788 −1.19719 −0.598597 0.801051i \(-0.704275\pi\)
−0.598597 + 0.801051i \(0.704275\pi\)
\(524\) 0 0
\(525\) 14.8638 0.648708
\(526\) 0 0
\(527\) 22.4975 0.980005
\(528\) 0 0
\(529\) 18.5528 0.806643
\(530\) 0 0
\(531\) 13.7760 0.597826
\(532\) 0 0
\(533\) −1.74098 −0.0754100
\(534\) 0 0
\(535\) −13.9130 −0.601513
\(536\) 0 0
\(537\) 4.51778 0.194956
\(538\) 0 0
\(539\) 4.46798 0.192449
\(540\) 0 0
\(541\) −33.1725 −1.42620 −0.713099 0.701063i \(-0.752709\pi\)
−0.713099 + 0.701063i \(0.752709\pi\)
\(542\) 0 0
\(543\) 22.0077 0.944439
\(544\) 0 0
\(545\) 2.08972 0.0895138
\(546\) 0 0
\(547\) 17.2954 0.739499 0.369749 0.929132i \(-0.379444\pi\)
0.369749 + 0.929132i \(0.379444\pi\)
\(548\) 0 0
\(549\) 10.8402 0.462648
\(550\) 0 0
\(551\) −7.32229 −0.311940
\(552\) 0 0
\(553\) 13.8982 0.591013
\(554\) 0 0
\(555\) −12.3623 −0.524749
\(556\) 0 0
\(557\) 9.53222 0.403893 0.201947 0.979397i \(-0.435273\pi\)
0.201947 + 0.979397i \(0.435273\pi\)
\(558\) 0 0
\(559\) −5.68577 −0.240482
\(560\) 0 0
\(561\) 13.6219 0.575117
\(562\) 0 0
\(563\) −6.88207 −0.290045 −0.145022 0.989428i \(-0.546325\pi\)
−0.145022 + 0.989428i \(0.546325\pi\)
\(564\) 0 0
\(565\) −10.7874 −0.453831
\(566\) 0 0
\(567\) −17.7920 −0.747195
\(568\) 0 0
\(569\) −16.0041 −0.670927 −0.335463 0.942053i \(-0.608893\pi\)
−0.335463 + 0.942053i \(0.608893\pi\)
\(570\) 0 0
\(571\) 23.2586 0.973342 0.486671 0.873585i \(-0.338211\pi\)
0.486671 + 0.873585i \(0.338211\pi\)
\(572\) 0 0
\(573\) −38.8468 −1.62285
\(574\) 0 0
\(575\) 29.2497 1.21980
\(576\) 0 0
\(577\) 9.90262 0.412251 0.206126 0.978526i \(-0.433914\pi\)
0.206126 + 0.978526i \(0.433914\pi\)
\(578\) 0 0
\(579\) −12.1364 −0.504370
\(580\) 0 0
\(581\) −12.3577 −0.512683
\(582\) 0 0
\(583\) −7.47729 −0.309678
\(584\) 0 0
\(585\) −0.459949 −0.0190166
\(586\) 0 0
\(587\) −22.0985 −0.912102 −0.456051 0.889954i \(-0.650737\pi\)
−0.456051 + 0.889954i \(0.650737\pi\)
\(588\) 0 0
\(589\) −3.39993 −0.140092
\(590\) 0 0
\(591\) 33.2853 1.36917
\(592\) 0 0
\(593\) 15.4666 0.635138 0.317569 0.948235i \(-0.397133\pi\)
0.317569 + 0.948235i \(0.397133\pi\)
\(594\) 0 0
\(595\) 7.16027 0.293543
\(596\) 0 0
\(597\) −42.5123 −1.73991
\(598\) 0 0
\(599\) −3.25975 −0.133190 −0.0665950 0.997780i \(-0.521214\pi\)
−0.0665950 + 0.997780i \(0.521214\pi\)
\(600\) 0 0
\(601\) 8.34590 0.340436 0.170218 0.985406i \(-0.445553\pi\)
0.170218 + 0.985406i \(0.445553\pi\)
\(602\) 0 0
\(603\) 13.7807 0.561195
\(604\) 0 0
\(605\) 0.680036 0.0276474
\(606\) 0 0
\(607\) −6.57007 −0.266671 −0.133335 0.991071i \(-0.542569\pi\)
−0.133335 + 0.991071i \(0.542569\pi\)
\(608\) 0 0
\(609\) −23.9858 −0.971954
\(610\) 0 0
\(611\) 3.96514 0.160413
\(612\) 0 0
\(613\) −15.5765 −0.629128 −0.314564 0.949236i \(-0.601858\pi\)
−0.314564 + 0.949236i \(0.601858\pi\)
\(614\) 0 0
\(615\) −4.46062 −0.179869
\(616\) 0 0
\(617\) −18.1508 −0.730725 −0.365363 0.930865i \(-0.619055\pi\)
−0.365363 + 0.930865i \(0.619055\pi\)
\(618\) 0 0
\(619\) 6.94456 0.279125 0.139563 0.990213i \(-0.455430\pi\)
0.139563 + 0.990213i \(0.455430\pi\)
\(620\) 0 0
\(621\) −23.3836 −0.938353
\(622\) 0 0
\(623\) 24.9873 1.00110
\(624\) 0 0
\(625\) 18.2771 0.731085
\(626\) 0 0
\(627\) −2.05861 −0.0822129
\(628\) 0 0
\(629\) 58.4328 2.32987
\(630\) 0 0
\(631\) −0.103144 −0.00410608 −0.00205304 0.999998i \(-0.500654\pi\)
−0.00205304 + 0.999998i \(0.500654\pi\)
\(632\) 0 0
\(633\) −1.44168 −0.0573016
\(634\) 0 0
\(635\) 11.6468 0.462189
\(636\) 0 0
\(637\) 2.44126 0.0967264
\(638\) 0 0
\(639\) −9.75565 −0.385927
\(640\) 0 0
\(641\) 30.4033 1.20086 0.600429 0.799678i \(-0.294996\pi\)
0.600429 + 0.799678i \(0.294996\pi\)
\(642\) 0 0
\(643\) 34.7980 1.37230 0.686150 0.727460i \(-0.259299\pi\)
0.686150 + 0.727460i \(0.259299\pi\)
\(644\) 0 0
\(645\) −14.5677 −0.573603
\(646\) 0 0
\(647\) −25.6437 −1.00816 −0.504078 0.863658i \(-0.668168\pi\)
−0.504078 + 0.863658i \(0.668168\pi\)
\(648\) 0 0
\(649\) −11.1288 −0.436843
\(650\) 0 0
\(651\) −11.1372 −0.436502
\(652\) 0 0
\(653\) 42.5402 1.66473 0.832363 0.554231i \(-0.186987\pi\)
0.832363 + 0.554231i \(0.186987\pi\)
\(654\) 0 0
\(655\) −1.00013 −0.0390785
\(656\) 0 0
\(657\) 3.71553 0.144956
\(658\) 0 0
\(659\) 3.33064 0.129743 0.0648717 0.997894i \(-0.479336\pi\)
0.0648717 + 0.997894i \(0.479336\pi\)
\(660\) 0 0
\(661\) 8.14466 0.316791 0.158395 0.987376i \(-0.449368\pi\)
0.158395 + 0.987376i \(0.449368\pi\)
\(662\) 0 0
\(663\) 7.44289 0.289058
\(664\) 0 0
\(665\) −1.08210 −0.0419619
\(666\) 0 0
\(667\) −47.2005 −1.82761
\(668\) 0 0
\(669\) −6.83781 −0.264365
\(670\) 0 0
\(671\) −8.75715 −0.338066
\(672\) 0 0
\(673\) 3.62445 0.139712 0.0698561 0.997557i \(-0.477746\pi\)
0.0698561 + 0.997557i \(0.477746\pi\)
\(674\) 0 0
\(675\) −16.4601 −0.633551
\(676\) 0 0
\(677\) −4.21027 −0.161814 −0.0809070 0.996722i \(-0.525782\pi\)
−0.0809070 + 0.996722i \(0.525782\pi\)
\(678\) 0 0
\(679\) 15.9017 0.610249
\(680\) 0 0
\(681\) −33.0955 −1.26822
\(682\) 0 0
\(683\) 39.3883 1.50715 0.753576 0.657361i \(-0.228327\pi\)
0.753576 + 0.657361i \(0.228327\pi\)
\(684\) 0 0
\(685\) −5.84796 −0.223439
\(686\) 0 0
\(687\) −6.18999 −0.236163
\(688\) 0 0
\(689\) −4.08552 −0.155646
\(690\) 0 0
\(691\) −37.5947 −1.43017 −0.715085 0.699038i \(-0.753612\pi\)
−0.715085 + 0.699038i \(0.753612\pi\)
\(692\) 0 0
\(693\) −1.96974 −0.0748241
\(694\) 0 0
\(695\) −12.5912 −0.477613
\(696\) 0 0
\(697\) 21.0840 0.798614
\(698\) 0 0
\(699\) −60.3330 −2.28200
\(700\) 0 0
\(701\) −18.7978 −0.709981 −0.354991 0.934870i \(-0.615516\pi\)
−0.354991 + 0.934870i \(0.615516\pi\)
\(702\) 0 0
\(703\) −8.83065 −0.333054
\(704\) 0 0
\(705\) 10.1592 0.382619
\(706\) 0 0
\(707\) 8.17855 0.307586
\(708\) 0 0
\(709\) 10.4624 0.392926 0.196463 0.980511i \(-0.437055\pi\)
0.196463 + 0.980511i \(0.437055\pi\)
\(710\) 0 0
\(711\) 10.8119 0.405477
\(712\) 0 0
\(713\) −21.9164 −0.820777
\(714\) 0 0
\(715\) 0.371565 0.0138958
\(716\) 0 0
\(717\) 40.7786 1.52290
\(718\) 0 0
\(719\) 14.2534 0.531561 0.265780 0.964034i \(-0.414370\pi\)
0.265780 + 0.964034i \(0.414370\pi\)
\(720\) 0 0
\(721\) −29.4678 −1.09744
\(722\) 0 0
\(723\) 40.4694 1.50507
\(724\) 0 0
\(725\) −33.2253 −1.23396
\(726\) 0 0
\(727\) −25.0455 −0.928887 −0.464443 0.885603i \(-0.653746\pi\)
−0.464443 + 0.885603i \(0.653746\pi\)
\(728\) 0 0
\(729\) 8.56207 0.317114
\(730\) 0 0
\(731\) 68.8572 2.54678
\(732\) 0 0
\(733\) −39.5761 −1.46178 −0.730889 0.682496i \(-0.760894\pi\)
−0.730889 + 0.682496i \(0.760894\pi\)
\(734\) 0 0
\(735\) 6.25485 0.230713
\(736\) 0 0
\(737\) −11.1326 −0.410076
\(738\) 0 0
\(739\) −6.93030 −0.254935 −0.127468 0.991843i \(-0.540685\pi\)
−0.127468 + 0.991843i \(0.540685\pi\)
\(740\) 0 0
\(741\) −1.12481 −0.0413207
\(742\) 0 0
\(743\) 27.3372 1.00290 0.501452 0.865185i \(-0.332799\pi\)
0.501452 + 0.865185i \(0.332799\pi\)
\(744\) 0 0
\(745\) 7.70291 0.282213
\(746\) 0 0
\(747\) −9.61343 −0.351737
\(748\) 0 0
\(749\) −32.5554 −1.18955
\(750\) 0 0
\(751\) −0.430584 −0.0157122 −0.00785611 0.999969i \(-0.502501\pi\)
−0.00785611 + 0.999969i \(0.502501\pi\)
\(752\) 0 0
\(753\) −32.5271 −1.18535
\(754\) 0 0
\(755\) 8.91994 0.324630
\(756\) 0 0
\(757\) −44.5951 −1.62083 −0.810417 0.585853i \(-0.800760\pi\)
−0.810417 + 0.585853i \(0.800760\pi\)
\(758\) 0 0
\(759\) −13.2701 −0.481674
\(760\) 0 0
\(761\) −48.8363 −1.77031 −0.885157 0.465292i \(-0.845949\pi\)
−0.885157 + 0.465292i \(0.845949\pi\)
\(762\) 0 0
\(763\) 4.88979 0.177022
\(764\) 0 0
\(765\) 5.57020 0.201391
\(766\) 0 0
\(767\) −6.08067 −0.219560
\(768\) 0 0
\(769\) −52.3956 −1.88943 −0.944716 0.327889i \(-0.893663\pi\)
−0.944716 + 0.327889i \(0.893663\pi\)
\(770\) 0 0
\(771\) −6.50434 −0.234248
\(772\) 0 0
\(773\) −25.3076 −0.910253 −0.455126 0.890427i \(-0.650406\pi\)
−0.455126 + 0.890427i \(0.650406\pi\)
\(774\) 0 0
\(775\) −15.4273 −0.554167
\(776\) 0 0
\(777\) −28.9268 −1.03774
\(778\) 0 0
\(779\) −3.18632 −0.114162
\(780\) 0 0
\(781\) 7.88100 0.282004
\(782\) 0 0
\(783\) 26.5619 0.949244
\(784\) 0 0
\(785\) 0.723927 0.0258381
\(786\) 0 0
\(787\) −0.354709 −0.0126440 −0.00632201 0.999980i \(-0.502012\pi\)
−0.00632201 + 0.999980i \(0.502012\pi\)
\(788\) 0 0
\(789\) −2.10364 −0.0748917
\(790\) 0 0
\(791\) −25.2418 −0.897495
\(792\) 0 0
\(793\) −4.78483 −0.169914
\(794\) 0 0
\(795\) −10.4677 −0.371250
\(796\) 0 0
\(797\) −51.0269 −1.80747 −0.903733 0.428097i \(-0.859184\pi\)
−0.903733 + 0.428097i \(0.859184\pi\)
\(798\) 0 0
\(799\) −48.0197 −1.69882
\(800\) 0 0
\(801\) 19.4384 0.686822
\(802\) 0 0
\(803\) −3.00155 −0.105922
\(804\) 0 0
\(805\) −6.97534 −0.245849
\(806\) 0 0
\(807\) 39.8712 1.40353
\(808\) 0 0
\(809\) 23.3721 0.821718 0.410859 0.911699i \(-0.365229\pi\)
0.410859 + 0.911699i \(0.365229\pi\)
\(810\) 0 0
\(811\) 15.3578 0.539286 0.269643 0.962960i \(-0.413094\pi\)
0.269643 + 0.962960i \(0.413094\pi\)
\(812\) 0 0
\(813\) −34.8575 −1.22251
\(814\) 0 0
\(815\) −7.78941 −0.272851
\(816\) 0 0
\(817\) −10.4060 −0.364061
\(818\) 0 0
\(819\) −1.07625 −0.0376071
\(820\) 0 0
\(821\) 14.4538 0.504441 0.252221 0.967670i \(-0.418839\pi\)
0.252221 + 0.967670i \(0.418839\pi\)
\(822\) 0 0
\(823\) 14.2522 0.496801 0.248400 0.968657i \(-0.420095\pi\)
0.248400 + 0.968657i \(0.420095\pi\)
\(824\) 0 0
\(825\) −9.34104 −0.325213
\(826\) 0 0
\(827\) 6.68323 0.232399 0.116199 0.993226i \(-0.462929\pi\)
0.116199 + 0.993226i \(0.462929\pi\)
\(828\) 0 0
\(829\) 25.9015 0.899596 0.449798 0.893130i \(-0.351496\pi\)
0.449798 + 0.893130i \(0.351496\pi\)
\(830\) 0 0
\(831\) −33.2199 −1.15238
\(832\) 0 0
\(833\) −29.5648 −1.02436
\(834\) 0 0
\(835\) −12.9137 −0.446896
\(836\) 0 0
\(837\) 12.3334 0.426304
\(838\) 0 0
\(839\) 11.9779 0.413524 0.206762 0.978391i \(-0.433707\pi\)
0.206762 + 0.978391i \(0.433707\pi\)
\(840\) 0 0
\(841\) 24.6159 0.848825
\(842\) 0 0
\(843\) 45.1281 1.55429
\(844\) 0 0
\(845\) −8.63745 −0.297137
\(846\) 0 0
\(847\) 1.59123 0.0546754
\(848\) 0 0
\(849\) −33.8200 −1.16070
\(850\) 0 0
\(851\) −56.9237 −1.95132
\(852\) 0 0
\(853\) −3.22318 −0.110359 −0.0551797 0.998476i \(-0.517573\pi\)
−0.0551797 + 0.998476i \(0.517573\pi\)
\(854\) 0 0
\(855\) −0.841795 −0.0287888
\(856\) 0 0
\(857\) 43.5297 1.48695 0.743473 0.668766i \(-0.233177\pi\)
0.743473 + 0.668766i \(0.233177\pi\)
\(858\) 0 0
\(859\) 18.1218 0.618309 0.309154 0.951012i \(-0.399954\pi\)
0.309154 + 0.951012i \(0.399954\pi\)
\(860\) 0 0
\(861\) −10.4375 −0.355709
\(862\) 0 0
\(863\) 18.8526 0.641749 0.320875 0.947122i \(-0.396023\pi\)
0.320875 + 0.947122i \(0.396023\pi\)
\(864\) 0 0
\(865\) −6.21871 −0.211443
\(866\) 0 0
\(867\) −55.1404 −1.87267
\(868\) 0 0
\(869\) −8.73426 −0.296290
\(870\) 0 0
\(871\) −6.08277 −0.206107
\(872\) 0 0
\(873\) 12.3704 0.418674
\(874\) 0 0
\(875\) −10.3205 −0.348898
\(876\) 0 0
\(877\) −48.4436 −1.63582 −0.817911 0.575344i \(-0.804868\pi\)
−0.817911 + 0.575344i \(0.804868\pi\)
\(878\) 0 0
\(879\) 61.6356 2.07892
\(880\) 0 0
\(881\) 38.2825 1.28977 0.644885 0.764280i \(-0.276905\pi\)
0.644885 + 0.764280i \(0.276905\pi\)
\(882\) 0 0
\(883\) 5.98538 0.201424 0.100712 0.994916i \(-0.467888\pi\)
0.100712 + 0.994916i \(0.467888\pi\)
\(884\) 0 0
\(885\) −15.5795 −0.523699
\(886\) 0 0
\(887\) −13.5203 −0.453966 −0.226983 0.973899i \(-0.572886\pi\)
−0.226983 + 0.973899i \(0.572886\pi\)
\(888\) 0 0
\(889\) 27.2526 0.914023
\(890\) 0 0
\(891\) 11.1813 0.374587
\(892\) 0 0
\(893\) 7.25697 0.242845
\(894\) 0 0
\(895\) −1.49239 −0.0498852
\(896\) 0 0
\(897\) −7.25066 −0.242092
\(898\) 0 0
\(899\) 24.8953 0.830303
\(900\) 0 0
\(901\) 49.4775 1.64834
\(902\) 0 0
\(903\) −34.0873 −1.13436
\(904\) 0 0
\(905\) −7.26996 −0.241662
\(906\) 0 0
\(907\) −0.321956 −0.0106904 −0.00534519 0.999986i \(-0.501701\pi\)
−0.00534519 + 0.999986i \(0.501701\pi\)
\(908\) 0 0
\(909\) 6.36234 0.211026
\(910\) 0 0
\(911\) 2.89500 0.0959155 0.0479578 0.998849i \(-0.484729\pi\)
0.0479578 + 0.998849i \(0.484729\pi\)
\(912\) 0 0
\(913\) 7.76611 0.257021
\(914\) 0 0
\(915\) −12.2594 −0.405282
\(916\) 0 0
\(917\) −2.34024 −0.0772815
\(918\) 0 0
\(919\) −22.5001 −0.742211 −0.371105 0.928591i \(-0.621021\pi\)
−0.371105 + 0.928591i \(0.621021\pi\)
\(920\) 0 0
\(921\) 14.0823 0.464026
\(922\) 0 0
\(923\) 4.30611 0.141737
\(924\) 0 0
\(925\) −40.0695 −1.31748
\(926\) 0 0
\(927\) −22.9239 −0.752921
\(928\) 0 0
\(929\) −20.8990 −0.685674 −0.342837 0.939395i \(-0.611388\pi\)
−0.342837 + 0.939395i \(0.611388\pi\)
\(930\) 0 0
\(931\) 4.46798 0.146432
\(932\) 0 0
\(933\) −12.7737 −0.418191
\(934\) 0 0
\(935\) −4.49983 −0.147160
\(936\) 0 0
\(937\) 34.5162 1.12760 0.563798 0.825913i \(-0.309340\pi\)
0.563798 + 0.825913i \(0.309340\pi\)
\(938\) 0 0
\(939\) 61.1953 1.99703
\(940\) 0 0
\(941\) −25.8848 −0.843821 −0.421910 0.906638i \(-0.638640\pi\)
−0.421910 + 0.906638i \(0.638640\pi\)
\(942\) 0 0
\(943\) −20.5395 −0.668857
\(944\) 0 0
\(945\) 3.92534 0.127691
\(946\) 0 0
\(947\) −11.2617 −0.365955 −0.182978 0.983117i \(-0.558574\pi\)
−0.182978 + 0.983117i \(0.558574\pi\)
\(948\) 0 0
\(949\) −1.64002 −0.0532373
\(950\) 0 0
\(951\) 19.3837 0.628558
\(952\) 0 0
\(953\) 34.3302 1.11206 0.556032 0.831161i \(-0.312323\pi\)
0.556032 + 0.831161i \(0.312323\pi\)
\(954\) 0 0
\(955\) 12.8326 0.415252
\(956\) 0 0
\(957\) 15.0737 0.487264
\(958\) 0 0
\(959\) −13.6838 −0.441873
\(960\) 0 0
\(961\) −19.4405 −0.627113
\(962\) 0 0
\(963\) −25.3259 −0.816115
\(964\) 0 0
\(965\) 4.00910 0.129057
\(966\) 0 0
\(967\) 24.4450 0.786100 0.393050 0.919517i \(-0.371420\pi\)
0.393050 + 0.919517i \(0.371420\pi\)
\(968\) 0 0
\(969\) 13.6219 0.437599
\(970\) 0 0
\(971\) 9.32129 0.299134 0.149567 0.988752i \(-0.452212\pi\)
0.149567 + 0.988752i \(0.452212\pi\)
\(972\) 0 0
\(973\) −29.4625 −0.944526
\(974\) 0 0
\(975\) −5.10386 −0.163454
\(976\) 0 0
\(977\) 20.0702 0.642102 0.321051 0.947062i \(-0.395964\pi\)
0.321051 + 0.947062i \(0.395964\pi\)
\(978\) 0 0
\(979\) −15.7031 −0.501874
\(980\) 0 0
\(981\) 3.80392 0.121450
\(982\) 0 0
\(983\) 0.585569 0.0186767 0.00933837 0.999956i \(-0.497027\pi\)
0.00933837 + 0.999956i \(0.497027\pi\)
\(984\) 0 0
\(985\) −10.9954 −0.350342
\(986\) 0 0
\(987\) 23.7718 0.756666
\(988\) 0 0
\(989\) −67.0788 −2.13298
\(990\) 0 0
\(991\) −52.8562 −1.67903 −0.839516 0.543335i \(-0.817161\pi\)
−0.839516 + 0.543335i \(0.817161\pi\)
\(992\) 0 0
\(993\) −36.6027 −1.16155
\(994\) 0 0
\(995\) 14.0434 0.445206
\(996\) 0 0
\(997\) −35.9223 −1.13767 −0.568836 0.822451i \(-0.692606\pi\)
−0.568836 + 0.822451i \(0.692606\pi\)
\(998\) 0 0
\(999\) 32.0335 1.01350
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 836.2.a.e.1.2 6
3.2 odd 2 7524.2.a.q.1.4 6
4.3 odd 2 3344.2.a.w.1.5 6
11.10 odd 2 9196.2.a.n.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
836.2.a.e.1.2 6 1.1 even 1 trivial
3344.2.a.w.1.5 6 4.3 odd 2
7524.2.a.q.1.4 6 3.2 odd 2
9196.2.a.n.1.2 6 11.10 odd 2