Properties

Label 836.2.a.d.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 $1$

Related objects

Downloads

Learn more

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: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 12x^{4} + 28x^{3} + 16x^{2} - 60x + 27 \) 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 \(2.18868\) 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)
 
\(f(q)\) \(=\) \(q-2.18868 q^{3} -3.62873 q^{5} -4.31127 q^{7} +1.79031 q^{9} +O(q^{10})\) \(q-2.18868 q^{3} -3.62873 q^{5} -4.31127 q^{7} +1.79031 q^{9} +1.00000 q^{11} -6.10370 q^{13} +7.94212 q^{15} -0.694815 q^{17} +1.00000 q^{19} +9.43598 q^{21} -7.79640 q^{23} +8.16767 q^{25} +2.64762 q^{27} +6.02709 q^{29} -6.26529 q^{31} -2.18868 q^{33} +15.6444 q^{35} +10.1139 q^{37} +13.3590 q^{39} -2.04380 q^{41} -2.16767 q^{43} -6.49655 q^{45} -10.2074 q^{47} +11.5870 q^{49} +1.52073 q^{51} -1.95798 q^{53} -3.62873 q^{55} -2.18868 q^{57} +5.89831 q^{59} +6.84803 q^{61} -7.71852 q^{63} +22.1487 q^{65} -8.35725 q^{67} +17.0638 q^{69} +0.566034 q^{71} +7.29948 q^{73} -17.8764 q^{75} -4.31127 q^{77} -13.9103 q^{79} -11.1657 q^{81} -7.05772 q^{83} +2.52129 q^{85} -13.1914 q^{87} -14.3590 q^{89} +26.3147 q^{91} +13.7127 q^{93} -3.62873 q^{95} -2.89842 q^{97} +1.79031 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 2 q^{3} + 2 q^{7} + 10 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 2 q^{3} + 2 q^{7} + 10 q^{9} + 6 q^{11} + 2 q^{13} - 12 q^{15} + 12 q^{17} + 6 q^{19} + 2 q^{21} - 2 q^{23} + 26 q^{25} + 16 q^{27} + 4 q^{29} - 20 q^{31} - 2 q^{33} + 20 q^{35} + 22 q^{37} + 8 q^{39} - 2 q^{41} + 10 q^{43} - 6 q^{45} + 16 q^{47} + 48 q^{49} + 36 q^{51} + 12 q^{53} - 2 q^{57} - 14 q^{59} + 12 q^{61} - 4 q^{63} + 10 q^{65} - 12 q^{67} + 30 q^{69} - 30 q^{71} + 24 q^{73} - 50 q^{75} + 2 q^{77} + 10 q^{81} - 14 q^{83} + 12 q^{85} - 30 q^{87} - 14 q^{89} + 20 q^{91} + 14 q^{93} - 46 q^{97} + 10 q^{99}+O(q^{100}) \) 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
\(3\) −2.18868 −1.26363 −0.631817 0.775118i \(-0.717691\pi\)
−0.631817 + 0.775118i \(0.717691\pi\)
\(4\) 0 0
\(5\) −3.62873 −1.62282 −0.811408 0.584480i \(-0.801299\pi\)
−0.811408 + 0.584480i \(0.801299\pi\)
\(6\) 0 0
\(7\) −4.31127 −1.62951 −0.814753 0.579808i \(-0.803128\pi\)
−0.814753 + 0.579808i \(0.803128\pi\)
\(8\) 0 0
\(9\) 1.79031 0.596771
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −6.10370 −1.69286 −0.846431 0.532498i \(-0.821253\pi\)
−0.846431 + 0.532498i \(0.821253\pi\)
\(14\) 0 0
\(15\) 7.94212 2.05065
\(16\) 0 0
\(17\) −0.694815 −0.168517 −0.0842587 0.996444i \(-0.526852\pi\)
−0.0842587 + 0.996444i \(0.526852\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 9.43598 2.05910
\(22\) 0 0
\(23\) −7.79640 −1.62566 −0.812830 0.582500i \(-0.802074\pi\)
−0.812830 + 0.582500i \(0.802074\pi\)
\(24\) 0 0
\(25\) 8.16767 1.63353
\(26\) 0 0
\(27\) 2.64762 0.509534
\(28\) 0 0
\(29\) 6.02709 1.11920 0.559602 0.828762i \(-0.310954\pi\)
0.559602 + 0.828762i \(0.310954\pi\)
\(30\) 0 0
\(31\) −6.26529 −1.12528 −0.562639 0.826703i \(-0.690214\pi\)
−0.562639 + 0.826703i \(0.690214\pi\)
\(32\) 0 0
\(33\) −2.18868 −0.381000
\(34\) 0 0
\(35\) 15.6444 2.64439
\(36\) 0 0
\(37\) 10.1139 1.66271 0.831354 0.555744i \(-0.187566\pi\)
0.831354 + 0.555744i \(0.187566\pi\)
\(38\) 0 0
\(39\) 13.3590 2.13916
\(40\) 0 0
\(41\) −2.04380 −0.319189 −0.159594 0.987183i \(-0.551019\pi\)
−0.159594 + 0.987183i \(0.551019\pi\)
\(42\) 0 0
\(43\) −2.16767 −0.330566 −0.165283 0.986246i \(-0.552854\pi\)
−0.165283 + 0.986246i \(0.552854\pi\)
\(44\) 0 0
\(45\) −6.49655 −0.968449
\(46\) 0 0
\(47\) −10.2074 −1.48890 −0.744451 0.667677i \(-0.767289\pi\)
−0.744451 + 0.667677i \(0.767289\pi\)
\(48\) 0 0
\(49\) 11.5870 1.65529
\(50\) 0 0
\(51\) 1.52073 0.212944
\(52\) 0 0
\(53\) −1.95798 −0.268949 −0.134475 0.990917i \(-0.542935\pi\)
−0.134475 + 0.990917i \(0.542935\pi\)
\(54\) 0 0
\(55\) −3.62873 −0.489298
\(56\) 0 0
\(57\) −2.18868 −0.289897
\(58\) 0 0
\(59\) 5.89831 0.767895 0.383948 0.923355i \(-0.374564\pi\)
0.383948 + 0.923355i \(0.374564\pi\)
\(60\) 0 0
\(61\) 6.84803 0.876800 0.438400 0.898780i \(-0.355545\pi\)
0.438400 + 0.898780i \(0.355545\pi\)
\(62\) 0 0
\(63\) −7.71852 −0.972442
\(64\) 0 0
\(65\) 22.1487 2.74720
\(66\) 0 0
\(67\) −8.35725 −1.02100 −0.510501 0.859877i \(-0.670540\pi\)
−0.510501 + 0.859877i \(0.670540\pi\)
\(68\) 0 0
\(69\) 17.0638 2.05424
\(70\) 0 0
\(71\) 0.566034 0.0671759 0.0335880 0.999436i \(-0.489307\pi\)
0.0335880 + 0.999436i \(0.489307\pi\)
\(72\) 0 0
\(73\) 7.29948 0.854339 0.427170 0.904172i \(-0.359511\pi\)
0.427170 + 0.904172i \(0.359511\pi\)
\(74\) 0 0
\(75\) −17.8764 −2.06419
\(76\) 0 0
\(77\) −4.31127 −0.491315
\(78\) 0 0
\(79\) −13.9103 −1.56503 −0.782513 0.622635i \(-0.786062\pi\)
−0.782513 + 0.622635i \(0.786062\pi\)
\(80\) 0 0
\(81\) −11.1657 −1.24064
\(82\) 0 0
\(83\) −7.05772 −0.774685 −0.387343 0.921936i \(-0.626607\pi\)
−0.387343 + 0.921936i \(0.626607\pi\)
\(84\) 0 0
\(85\) 2.52129 0.273473
\(86\) 0 0
\(87\) −13.1914 −1.41426
\(88\) 0 0
\(89\) −14.3590 −1.52205 −0.761027 0.648720i \(-0.775305\pi\)
−0.761027 + 0.648720i \(0.775305\pi\)
\(90\) 0 0
\(91\) 26.3147 2.75853
\(92\) 0 0
\(93\) 13.7127 1.42194
\(94\) 0 0
\(95\) −3.62873 −0.372300
\(96\) 0 0
\(97\) −2.89842 −0.294290 −0.147145 0.989115i \(-0.547008\pi\)
−0.147145 + 0.989115i \(0.547008\pi\)
\(98\) 0 0
\(99\) 1.79031 0.179933
\(100\) 0 0
\(101\) −7.57491 −0.753732 −0.376866 0.926268i \(-0.622998\pi\)
−0.376866 + 0.926268i \(0.622998\pi\)
\(102\) 0 0
\(103\) 3.29026 0.324199 0.162099 0.986774i \(-0.448173\pi\)
0.162099 + 0.986774i \(0.448173\pi\)
\(104\) 0 0
\(105\) −34.2406 −3.34154
\(106\) 0 0
\(107\) 10.9079 1.05451 0.527255 0.849707i \(-0.323221\pi\)
0.527255 + 0.849707i \(0.323221\pi\)
\(108\) 0 0
\(109\) 5.42741 0.519852 0.259926 0.965629i \(-0.416302\pi\)
0.259926 + 0.965629i \(0.416302\pi\)
\(110\) 0 0
\(111\) −22.1360 −2.10105
\(112\) 0 0
\(113\) 11.9439 1.12359 0.561794 0.827277i \(-0.310111\pi\)
0.561794 + 0.827277i \(0.310111\pi\)
\(114\) 0 0
\(115\) 28.2910 2.63815
\(116\) 0 0
\(117\) −10.9275 −1.01025
\(118\) 0 0
\(119\) 2.99553 0.274600
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 4.47323 0.403337
\(124\) 0 0
\(125\) −11.4946 −1.02811
\(126\) 0 0
\(127\) 7.72813 0.685761 0.342880 0.939379i \(-0.388597\pi\)
0.342880 + 0.939379i \(0.388597\pi\)
\(128\) 0 0
\(129\) 4.74433 0.417715
\(130\) 0 0
\(131\) −10.5909 −0.925335 −0.462668 0.886532i \(-0.653108\pi\)
−0.462668 + 0.886532i \(0.653108\pi\)
\(132\) 0 0
\(133\) −4.31127 −0.373834
\(134\) 0 0
\(135\) −9.60749 −0.826881
\(136\) 0 0
\(137\) −8.74829 −0.747417 −0.373708 0.927546i \(-0.621914\pi\)
−0.373708 + 0.927546i \(0.621914\pi\)
\(138\) 0 0
\(139\) 10.2696 0.871055 0.435527 0.900175i \(-0.356562\pi\)
0.435527 + 0.900175i \(0.356562\pi\)
\(140\) 0 0
\(141\) 22.3407 1.88143
\(142\) 0 0
\(143\) −6.10370 −0.510417
\(144\) 0 0
\(145\) −21.8707 −1.81626
\(146\) 0 0
\(147\) −25.3603 −2.09168
\(148\) 0 0
\(149\) −21.5725 −1.76729 −0.883643 0.468161i \(-0.844917\pi\)
−0.883643 + 0.468161i \(0.844917\pi\)
\(150\) 0 0
\(151\) −0.461397 −0.0375479 −0.0187740 0.999824i \(-0.505976\pi\)
−0.0187740 + 0.999824i \(0.505976\pi\)
\(152\) 0 0
\(153\) −1.24394 −0.100566
\(154\) 0 0
\(155\) 22.7350 1.82612
\(156\) 0 0
\(157\) −4.52449 −0.361093 −0.180547 0.983566i \(-0.557787\pi\)
−0.180547 + 0.983566i \(0.557787\pi\)
\(158\) 0 0
\(159\) 4.28539 0.339853
\(160\) 0 0
\(161\) 33.6124 2.64903
\(162\) 0 0
\(163\) 11.9622 0.936953 0.468477 0.883476i \(-0.344803\pi\)
0.468477 + 0.883476i \(0.344803\pi\)
\(164\) 0 0
\(165\) 7.94212 0.618293
\(166\) 0 0
\(167\) −9.83576 −0.761114 −0.380557 0.924758i \(-0.624268\pi\)
−0.380557 + 0.924758i \(0.624268\pi\)
\(168\) 0 0
\(169\) 24.2552 1.86578
\(170\) 0 0
\(171\) 1.79031 0.136909
\(172\) 0 0
\(173\) −12.7818 −0.971783 −0.485891 0.874019i \(-0.661505\pi\)
−0.485891 + 0.874019i \(0.661505\pi\)
\(174\) 0 0
\(175\) −35.2130 −2.66185
\(176\) 0 0
\(177\) −12.9095 −0.970338
\(178\) 0 0
\(179\) −8.13311 −0.607897 −0.303949 0.952688i \(-0.598305\pi\)
−0.303949 + 0.952688i \(0.598305\pi\)
\(180\) 0 0
\(181\) 18.5543 1.37913 0.689564 0.724225i \(-0.257802\pi\)
0.689564 + 0.724225i \(0.257802\pi\)
\(182\) 0 0
\(183\) −14.9881 −1.10795
\(184\) 0 0
\(185\) −36.7004 −2.69827
\(186\) 0 0
\(187\) −0.694815 −0.0508099
\(188\) 0 0
\(189\) −11.4146 −0.830290
\(190\) 0 0
\(191\) −10.6265 −0.768910 −0.384455 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(192\) 0 0
\(193\) −0.759689 −0.0546836 −0.0273418 0.999626i \(-0.508704\pi\)
−0.0273418 + 0.999626i \(0.508704\pi\)
\(194\) 0 0
\(195\) −48.4763 −3.47146
\(196\) 0 0
\(197\) 1.68678 0.120178 0.0600891 0.998193i \(-0.480862\pi\)
0.0600891 + 0.998193i \(0.480862\pi\)
\(198\) 0 0
\(199\) 24.2153 1.71658 0.858290 0.513166i \(-0.171527\pi\)
0.858290 + 0.513166i \(0.171527\pi\)
\(200\) 0 0
\(201\) 18.2913 1.29017
\(202\) 0 0
\(203\) −25.9844 −1.82375
\(204\) 0 0
\(205\) 7.41641 0.517984
\(206\) 0 0
\(207\) −13.9580 −0.970147
\(208\) 0 0
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −28.4049 −1.95547 −0.977735 0.209841i \(-0.932705\pi\)
−0.977735 + 0.209841i \(0.932705\pi\)
\(212\) 0 0
\(213\) −1.23887 −0.0848858
\(214\) 0 0
\(215\) 7.86588 0.536448
\(216\) 0 0
\(217\) 27.0113 1.83365
\(218\) 0 0
\(219\) −15.9762 −1.07957
\(220\) 0 0
\(221\) 4.24094 0.285277
\(222\) 0 0
\(223\) −8.82043 −0.590660 −0.295330 0.955395i \(-0.595430\pi\)
−0.295330 + 0.955395i \(0.595430\pi\)
\(224\) 0 0
\(225\) 14.6227 0.974845
\(226\) 0 0
\(227\) −17.2796 −1.14689 −0.573443 0.819246i \(-0.694393\pi\)
−0.573443 + 0.819246i \(0.694393\pi\)
\(228\) 0 0
\(229\) 2.15539 0.142432 0.0712162 0.997461i \(-0.477312\pi\)
0.0712162 + 0.997461i \(0.477312\pi\)
\(230\) 0 0
\(231\) 9.43598 0.620842
\(232\) 0 0
\(233\) 11.9733 0.784398 0.392199 0.919880i \(-0.371715\pi\)
0.392199 + 0.919880i \(0.371715\pi\)
\(234\) 0 0
\(235\) 37.0399 2.41622
\(236\) 0 0
\(237\) 30.4451 1.97762
\(238\) 0 0
\(239\) 18.7692 1.21408 0.607038 0.794673i \(-0.292357\pi\)
0.607038 + 0.794673i \(0.292357\pi\)
\(240\) 0 0
\(241\) 14.9468 0.962807 0.481404 0.876499i \(-0.340127\pi\)
0.481404 + 0.876499i \(0.340127\pi\)
\(242\) 0 0
\(243\) 16.4953 1.05817
\(244\) 0 0
\(245\) −42.0462 −2.68624
\(246\) 0 0
\(247\) −6.10370 −0.388369
\(248\) 0 0
\(249\) 15.4471 0.978918
\(250\) 0 0
\(251\) −12.1955 −0.769771 −0.384885 0.922964i \(-0.625759\pi\)
−0.384885 + 0.922964i \(0.625759\pi\)
\(252\) 0 0
\(253\) −7.79640 −0.490155
\(254\) 0 0
\(255\) −5.51830 −0.345569
\(256\) 0 0
\(257\) −6.96669 −0.434570 −0.217285 0.976108i \(-0.569720\pi\)
−0.217285 + 0.976108i \(0.569720\pi\)
\(258\) 0 0
\(259\) −43.6036 −2.70939
\(260\) 0 0
\(261\) 10.7904 0.667908
\(262\) 0 0
\(263\) 25.0396 1.54401 0.772005 0.635617i \(-0.219254\pi\)
0.772005 + 0.635617i \(0.219254\pi\)
\(264\) 0 0
\(265\) 7.10498 0.436455
\(266\) 0 0
\(267\) 31.4273 1.92332
\(268\) 0 0
\(269\) 1.58130 0.0964134 0.0482067 0.998837i \(-0.484649\pi\)
0.0482067 + 0.998837i \(0.484649\pi\)
\(270\) 0 0
\(271\) 5.57867 0.338880 0.169440 0.985540i \(-0.445804\pi\)
0.169440 + 0.985540i \(0.445804\pi\)
\(272\) 0 0
\(273\) −57.5944 −3.48577
\(274\) 0 0
\(275\) 8.16767 0.492529
\(276\) 0 0
\(277\) 30.5385 1.83488 0.917440 0.397874i \(-0.130252\pi\)
0.917440 + 0.397874i \(0.130252\pi\)
\(278\) 0 0
\(279\) −11.2168 −0.671533
\(280\) 0 0
\(281\) −15.0369 −0.897029 −0.448514 0.893776i \(-0.648047\pi\)
−0.448514 + 0.893776i \(0.648047\pi\)
\(282\) 0 0
\(283\) 29.1158 1.73076 0.865378 0.501120i \(-0.167079\pi\)
0.865378 + 0.501120i \(0.167079\pi\)
\(284\) 0 0
\(285\) 7.94212 0.470450
\(286\) 0 0
\(287\) 8.81139 0.520120
\(288\) 0 0
\(289\) −16.5172 −0.971602
\(290\) 0 0
\(291\) 6.34371 0.371875
\(292\) 0 0
\(293\) 5.88816 0.343990 0.171995 0.985098i \(-0.444979\pi\)
0.171995 + 0.985098i \(0.444979\pi\)
\(294\) 0 0
\(295\) −21.4034 −1.24615
\(296\) 0 0
\(297\) 2.64762 0.153630
\(298\) 0 0
\(299\) 47.5869 2.75202
\(300\) 0 0
\(301\) 9.34540 0.538660
\(302\) 0 0
\(303\) 16.5791 0.952442
\(304\) 0 0
\(305\) −24.8496 −1.42289
\(306\) 0 0
\(307\) 19.2140 1.09660 0.548300 0.836282i \(-0.315275\pi\)
0.548300 + 0.836282i \(0.315275\pi\)
\(308\) 0 0
\(309\) −7.20132 −0.409669
\(310\) 0 0
\(311\) 26.7459 1.51662 0.758310 0.651894i \(-0.226025\pi\)
0.758310 + 0.651894i \(0.226025\pi\)
\(312\) 0 0
\(313\) −16.0435 −0.906834 −0.453417 0.891298i \(-0.649795\pi\)
−0.453417 + 0.891298i \(0.649795\pi\)
\(314\) 0 0
\(315\) 28.0084 1.57809
\(316\) 0 0
\(317\) 18.5664 1.04280 0.521398 0.853314i \(-0.325411\pi\)
0.521398 + 0.853314i \(0.325411\pi\)
\(318\) 0 0
\(319\) 6.02709 0.337453
\(320\) 0 0
\(321\) −23.8739 −1.33251
\(322\) 0 0
\(323\) −0.694815 −0.0386605
\(324\) 0 0
\(325\) −49.8530 −2.76535
\(326\) 0 0
\(327\) −11.8789 −0.656902
\(328\) 0 0
\(329\) 44.0069 2.42618
\(330\) 0 0
\(331\) −34.6196 −1.90286 −0.951431 0.307861i \(-0.900387\pi\)
−0.951431 + 0.307861i \(0.900387\pi\)
\(332\) 0 0
\(333\) 18.1070 0.992255
\(334\) 0 0
\(335\) 30.3262 1.65690
\(336\) 0 0
\(337\) 2.84625 0.155045 0.0775224 0.996991i \(-0.475299\pi\)
0.0775224 + 0.996991i \(0.475299\pi\)
\(338\) 0 0
\(339\) −26.1414 −1.41980
\(340\) 0 0
\(341\) −6.26529 −0.339284
\(342\) 0 0
\(343\) −19.7760 −1.06780
\(344\) 0 0
\(345\) −61.9199 −3.33366
\(346\) 0 0
\(347\) 10.0153 0.537651 0.268826 0.963189i \(-0.413364\pi\)
0.268826 + 0.963189i \(0.413364\pi\)
\(348\) 0 0
\(349\) −0.712288 −0.0381279 −0.0190640 0.999818i \(-0.506069\pi\)
−0.0190640 + 0.999818i \(0.506069\pi\)
\(350\) 0 0
\(351\) −16.1603 −0.862571
\(352\) 0 0
\(353\) 32.4528 1.72729 0.863645 0.504101i \(-0.168176\pi\)
0.863645 + 0.504101i \(0.168176\pi\)
\(354\) 0 0
\(355\) −2.05398 −0.109014
\(356\) 0 0
\(357\) −6.55626 −0.346994
\(358\) 0 0
\(359\) 19.6749 1.03840 0.519200 0.854653i \(-0.326230\pi\)
0.519200 + 0.854653i \(0.326230\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −2.18868 −0.114876
\(364\) 0 0
\(365\) −26.4878 −1.38644
\(366\) 0 0
\(367\) −31.5887 −1.64891 −0.824457 0.565924i \(-0.808520\pi\)
−0.824457 + 0.565924i \(0.808520\pi\)
\(368\) 0 0
\(369\) −3.65905 −0.190482
\(370\) 0 0
\(371\) 8.44138 0.438255
\(372\) 0 0
\(373\) 20.4557 1.05916 0.529579 0.848260i \(-0.322350\pi\)
0.529579 + 0.848260i \(0.322350\pi\)
\(374\) 0 0
\(375\) 25.1580 1.29915
\(376\) 0 0
\(377\) −36.7876 −1.89466
\(378\) 0 0
\(379\) 6.22508 0.319761 0.159880 0.987136i \(-0.448889\pi\)
0.159880 + 0.987136i \(0.448889\pi\)
\(380\) 0 0
\(381\) −16.9144 −0.866550
\(382\) 0 0
\(383\) −0.592813 −0.0302913 −0.0151457 0.999885i \(-0.504821\pi\)
−0.0151457 + 0.999885i \(0.504821\pi\)
\(384\) 0 0
\(385\) 15.6444 0.797314
\(386\) 0 0
\(387\) −3.88080 −0.197272
\(388\) 0 0
\(389\) 32.2424 1.63475 0.817377 0.576103i \(-0.195427\pi\)
0.817377 + 0.576103i \(0.195427\pi\)
\(390\) 0 0
\(391\) 5.41705 0.273952
\(392\) 0 0
\(393\) 23.1802 1.16929
\(394\) 0 0
\(395\) 50.4765 2.53975
\(396\) 0 0
\(397\) −25.1889 −1.26420 −0.632098 0.774888i \(-0.717806\pi\)
−0.632098 + 0.774888i \(0.717806\pi\)
\(398\) 0 0
\(399\) 9.43598 0.472390
\(400\) 0 0
\(401\) −22.3272 −1.11497 −0.557484 0.830188i \(-0.688233\pi\)
−0.557484 + 0.830188i \(0.688233\pi\)
\(402\) 0 0
\(403\) 38.2414 1.90494
\(404\) 0 0
\(405\) 40.5174 2.01332
\(406\) 0 0
\(407\) 10.1139 0.501325
\(408\) 0 0
\(409\) 24.1005 1.19169 0.595847 0.803098i \(-0.296817\pi\)
0.595847 + 0.803098i \(0.296817\pi\)
\(410\) 0 0
\(411\) 19.1472 0.944461
\(412\) 0 0
\(413\) −25.4292 −1.25129
\(414\) 0 0
\(415\) 25.6105 1.25717
\(416\) 0 0
\(417\) −22.4768 −1.10069
\(418\) 0 0
\(419\) −17.0480 −0.832851 −0.416425 0.909170i \(-0.636717\pi\)
−0.416425 + 0.909170i \(0.636717\pi\)
\(420\) 0 0
\(421\) 4.12963 0.201266 0.100633 0.994924i \(-0.467913\pi\)
0.100633 + 0.994924i \(0.467913\pi\)
\(422\) 0 0
\(423\) −18.2744 −0.888533
\(424\) 0 0
\(425\) −5.67502 −0.275279
\(426\) 0 0
\(427\) −29.5237 −1.42875
\(428\) 0 0
\(429\) 13.3590 0.644980
\(430\) 0 0
\(431\) −5.86783 −0.282643 −0.141322 0.989964i \(-0.545135\pi\)
−0.141322 + 0.989964i \(0.545135\pi\)
\(432\) 0 0
\(433\) 15.5840 0.748920 0.374460 0.927243i \(-0.377828\pi\)
0.374460 + 0.927243i \(0.377828\pi\)
\(434\) 0 0
\(435\) 47.8679 2.29509
\(436\) 0 0
\(437\) −7.79640 −0.372952
\(438\) 0 0
\(439\) 13.1937 0.629699 0.314850 0.949142i \(-0.398046\pi\)
0.314850 + 0.949142i \(0.398046\pi\)
\(440\) 0 0
\(441\) 20.7444 0.987830
\(442\) 0 0
\(443\) −4.14411 −0.196893 −0.0984463 0.995142i \(-0.531387\pi\)
−0.0984463 + 0.995142i \(0.531387\pi\)
\(444\) 0 0
\(445\) 52.1050 2.47002
\(446\) 0 0
\(447\) 47.2152 2.23320
\(448\) 0 0
\(449\) −35.9971 −1.69881 −0.849405 0.527742i \(-0.823039\pi\)
−0.849405 + 0.527742i \(0.823039\pi\)
\(450\) 0 0
\(451\) −2.04380 −0.0962390
\(452\) 0 0
\(453\) 1.00985 0.0474468
\(454\) 0 0
\(455\) −95.4889 −4.47659
\(456\) 0 0
\(457\) 7.94870 0.371825 0.185912 0.982566i \(-0.440476\pi\)
0.185912 + 0.982566i \(0.440476\pi\)
\(458\) 0 0
\(459\) −1.83960 −0.0858654
\(460\) 0 0
\(461\) −9.33968 −0.434992 −0.217496 0.976061i \(-0.569789\pi\)
−0.217496 + 0.976061i \(0.569789\pi\)
\(462\) 0 0
\(463\) 13.0328 0.605686 0.302843 0.953040i \(-0.402064\pi\)
0.302843 + 0.953040i \(0.402064\pi\)
\(464\) 0 0
\(465\) −49.7596 −2.30755
\(466\) 0 0
\(467\) 17.1169 0.792076 0.396038 0.918234i \(-0.370385\pi\)
0.396038 + 0.918234i \(0.370385\pi\)
\(468\) 0 0
\(469\) 36.0304 1.66373
\(470\) 0 0
\(471\) 9.90264 0.456290
\(472\) 0 0
\(473\) −2.16767 −0.0996695
\(474\) 0 0
\(475\) 8.16767 0.374758
\(476\) 0 0
\(477\) −3.50539 −0.160501
\(478\) 0 0
\(479\) 39.8917 1.82270 0.911349 0.411635i \(-0.135042\pi\)
0.911349 + 0.411635i \(0.135042\pi\)
\(480\) 0 0
\(481\) −61.7319 −2.81473
\(482\) 0 0
\(483\) −73.5666 −3.34740
\(484\) 0 0
\(485\) 10.5176 0.477578
\(486\) 0 0
\(487\) −21.3504 −0.967478 −0.483739 0.875212i \(-0.660722\pi\)
−0.483739 + 0.875212i \(0.660722\pi\)
\(488\) 0 0
\(489\) −26.1814 −1.18397
\(490\) 0 0
\(491\) −17.0293 −0.768524 −0.384262 0.923224i \(-0.625544\pi\)
−0.384262 + 0.923224i \(0.625544\pi\)
\(492\) 0 0
\(493\) −4.18771 −0.188605
\(494\) 0 0
\(495\) −6.49655 −0.291998
\(496\) 0 0
\(497\) −2.44033 −0.109464
\(498\) 0 0
\(499\) −4.94667 −0.221443 −0.110722 0.993851i \(-0.535316\pi\)
−0.110722 + 0.993851i \(0.535316\pi\)
\(500\) 0 0
\(501\) 21.5273 0.961769
\(502\) 0 0
\(503\) −3.22196 −0.143660 −0.0718301 0.997417i \(-0.522884\pi\)
−0.0718301 + 0.997417i \(0.522884\pi\)
\(504\) 0 0
\(505\) 27.4873 1.22317
\(506\) 0 0
\(507\) −53.0868 −2.35767
\(508\) 0 0
\(509\) 10.1751 0.451003 0.225502 0.974243i \(-0.427598\pi\)
0.225502 + 0.974243i \(0.427598\pi\)
\(510\) 0 0
\(511\) −31.4700 −1.39215
\(512\) 0 0
\(513\) 2.64762 0.116895
\(514\) 0 0
\(515\) −11.9395 −0.526115
\(516\) 0 0
\(517\) −10.2074 −0.448921
\(518\) 0 0
\(519\) 27.9753 1.22798
\(520\) 0 0
\(521\) −33.3792 −1.46237 −0.731186 0.682179i \(-0.761033\pi\)
−0.731186 + 0.682179i \(0.761033\pi\)
\(522\) 0 0
\(523\) −2.17534 −0.0951208 −0.0475604 0.998868i \(-0.515145\pi\)
−0.0475604 + 0.998868i \(0.515145\pi\)
\(524\) 0 0
\(525\) 77.0700 3.36361
\(526\) 0 0
\(527\) 4.35321 0.189629
\(528\) 0 0
\(529\) 37.7838 1.64277
\(530\) 0 0
\(531\) 10.5598 0.458257
\(532\) 0 0
\(533\) 12.4748 0.540342
\(534\) 0 0
\(535\) −39.5819 −1.71127
\(536\) 0 0
\(537\) 17.8008 0.768160
\(538\) 0 0
\(539\) 11.5870 0.499089
\(540\) 0 0
\(541\) 24.6958 1.06176 0.530878 0.847448i \(-0.321862\pi\)
0.530878 + 0.847448i \(0.321862\pi\)
\(542\) 0 0
\(543\) −40.6093 −1.74271
\(544\) 0 0
\(545\) −19.6946 −0.843624
\(546\) 0 0
\(547\) −9.80330 −0.419159 −0.209579 0.977792i \(-0.567209\pi\)
−0.209579 + 0.977792i \(0.567209\pi\)
\(548\) 0 0
\(549\) 12.2601 0.523249
\(550\) 0 0
\(551\) 6.02709 0.256763
\(552\) 0 0
\(553\) 59.9708 2.55022
\(554\) 0 0
\(555\) 80.3254 3.40962
\(556\) 0 0
\(557\) −12.1796 −0.516065 −0.258033 0.966136i \(-0.583074\pi\)
−0.258033 + 0.966136i \(0.583074\pi\)
\(558\) 0 0
\(559\) 13.2308 0.559603
\(560\) 0 0
\(561\) 1.52073 0.0642051
\(562\) 0 0
\(563\) 10.6703 0.449698 0.224849 0.974394i \(-0.427811\pi\)
0.224849 + 0.974394i \(0.427811\pi\)
\(564\) 0 0
\(565\) −43.3412 −1.82338
\(566\) 0 0
\(567\) 48.1384 2.02162
\(568\) 0 0
\(569\) −20.8260 −0.873072 −0.436536 0.899687i \(-0.643795\pi\)
−0.436536 + 0.899687i \(0.643795\pi\)
\(570\) 0 0
\(571\) 21.5755 0.902907 0.451454 0.892295i \(-0.350906\pi\)
0.451454 + 0.892295i \(0.350906\pi\)
\(572\) 0 0
\(573\) 23.2581 0.971621
\(574\) 0 0
\(575\) −63.6784 −2.65557
\(576\) 0 0
\(577\) 15.5212 0.646157 0.323079 0.946372i \(-0.395282\pi\)
0.323079 + 0.946372i \(0.395282\pi\)
\(578\) 0 0
\(579\) 1.66272 0.0691001
\(580\) 0 0
\(581\) 30.4277 1.26235
\(582\) 0 0
\(583\) −1.95798 −0.0810912
\(584\) 0 0
\(585\) 39.6530 1.63945
\(586\) 0 0
\(587\) 4.08383 0.168558 0.0842789 0.996442i \(-0.473141\pi\)
0.0842789 + 0.996442i \(0.473141\pi\)
\(588\) 0 0
\(589\) −6.26529 −0.258157
\(590\) 0 0
\(591\) −3.69182 −0.151861
\(592\) 0 0
\(593\) −45.1039 −1.85220 −0.926098 0.377283i \(-0.876858\pi\)
−0.926098 + 0.377283i \(0.876858\pi\)
\(594\) 0 0
\(595\) −10.8700 −0.445626
\(596\) 0 0
\(597\) −52.9996 −2.16913
\(598\) 0 0
\(599\) −1.29026 −0.0527186 −0.0263593 0.999653i \(-0.508391\pi\)
−0.0263593 + 0.999653i \(0.508391\pi\)
\(600\) 0 0
\(601\) −29.0011 −1.18298 −0.591489 0.806313i \(-0.701460\pi\)
−0.591489 + 0.806313i \(0.701460\pi\)
\(602\) 0 0
\(603\) −14.9621 −0.609303
\(604\) 0 0
\(605\) −3.62873 −0.147529
\(606\) 0 0
\(607\) −11.0119 −0.446958 −0.223479 0.974709i \(-0.571741\pi\)
−0.223479 + 0.974709i \(0.571741\pi\)
\(608\) 0 0
\(609\) 56.8715 2.30455
\(610\) 0 0
\(611\) 62.3029 2.52051
\(612\) 0 0
\(613\) 1.94010 0.0783600 0.0391800 0.999232i \(-0.487525\pi\)
0.0391800 + 0.999232i \(0.487525\pi\)
\(614\) 0 0
\(615\) −16.2321 −0.654543
\(616\) 0 0
\(617\) 15.1760 0.610964 0.305482 0.952198i \(-0.401182\pi\)
0.305482 + 0.952198i \(0.401182\pi\)
\(618\) 0 0
\(619\) 35.8548 1.44113 0.720564 0.693389i \(-0.243883\pi\)
0.720564 + 0.693389i \(0.243883\pi\)
\(620\) 0 0
\(621\) −20.6419 −0.828330
\(622\) 0 0
\(623\) 61.9057 2.48020
\(624\) 0 0
\(625\) 0.872460 0.0348984
\(626\) 0 0
\(627\) −2.18868 −0.0874074
\(628\) 0 0
\(629\) −7.02726 −0.280195
\(630\) 0 0
\(631\) −1.43092 −0.0569640 −0.0284820 0.999594i \(-0.509067\pi\)
−0.0284820 + 0.999594i \(0.509067\pi\)
\(632\) 0 0
\(633\) 62.1691 2.47100
\(634\) 0 0
\(635\) −28.0433 −1.11286
\(636\) 0 0
\(637\) −70.7239 −2.80218
\(638\) 0 0
\(639\) 1.01338 0.0400886
\(640\) 0 0
\(641\) −27.8685 −1.10074 −0.550369 0.834922i \(-0.685513\pi\)
−0.550369 + 0.834922i \(0.685513\pi\)
\(642\) 0 0
\(643\) −1.90753 −0.0752256 −0.0376128 0.999292i \(-0.511975\pi\)
−0.0376128 + 0.999292i \(0.511975\pi\)
\(644\) 0 0
\(645\) −17.2159 −0.677874
\(646\) 0 0
\(647\) −44.9164 −1.76584 −0.882922 0.469519i \(-0.844427\pi\)
−0.882922 + 0.469519i \(0.844427\pi\)
\(648\) 0 0
\(649\) 5.89831 0.231529
\(650\) 0 0
\(651\) −59.1191 −2.31706
\(652\) 0 0
\(653\) 23.6282 0.924642 0.462321 0.886713i \(-0.347017\pi\)
0.462321 + 0.886713i \(0.347017\pi\)
\(654\) 0 0
\(655\) 38.4317 1.50165
\(656\) 0 0
\(657\) 13.0683 0.509844
\(658\) 0 0
\(659\) −10.7391 −0.418334 −0.209167 0.977880i \(-0.567075\pi\)
−0.209167 + 0.977880i \(0.567075\pi\)
\(660\) 0 0
\(661\) −46.0837 −1.79245 −0.896224 0.443603i \(-0.853700\pi\)
−0.896224 + 0.443603i \(0.853700\pi\)
\(662\) 0 0
\(663\) −9.28206 −0.360485
\(664\) 0 0
\(665\) 15.6444 0.606665
\(666\) 0 0
\(667\) −46.9896 −1.81945
\(668\) 0 0
\(669\) 19.3051 0.746378
\(670\) 0 0
\(671\) 6.84803 0.264365
\(672\) 0 0
\(673\) −22.6538 −0.873242 −0.436621 0.899646i \(-0.643825\pi\)
−0.436621 + 0.899646i \(0.643825\pi\)
\(674\) 0 0
\(675\) 21.6249 0.832341
\(676\) 0 0
\(677\) 24.3654 0.936438 0.468219 0.883613i \(-0.344896\pi\)
0.468219 + 0.883613i \(0.344896\pi\)
\(678\) 0 0
\(679\) 12.4959 0.479547
\(680\) 0 0
\(681\) 37.8194 1.44924
\(682\) 0 0
\(683\) −26.6471 −1.01962 −0.509811 0.860286i \(-0.670285\pi\)
−0.509811 + 0.860286i \(0.670285\pi\)
\(684\) 0 0
\(685\) 31.7452 1.21292
\(686\) 0 0
\(687\) −4.71746 −0.179982
\(688\) 0 0
\(689\) 11.9509 0.455294
\(690\) 0 0
\(691\) 39.6176 1.50713 0.753563 0.657376i \(-0.228334\pi\)
0.753563 + 0.657376i \(0.228334\pi\)
\(692\) 0 0
\(693\) −7.71852 −0.293202
\(694\) 0 0
\(695\) −37.2655 −1.41356
\(696\) 0 0
\(697\) 1.42007 0.0537888
\(698\) 0 0
\(699\) −26.2057 −0.991192
\(700\) 0 0
\(701\) 1.92410 0.0726722 0.0363361 0.999340i \(-0.488431\pi\)
0.0363361 + 0.999340i \(0.488431\pi\)
\(702\) 0 0
\(703\) 10.1139 0.381451
\(704\) 0 0
\(705\) −81.0684 −3.05321
\(706\) 0 0
\(707\) 32.6575 1.22821
\(708\) 0 0
\(709\) −6.91616 −0.259742 −0.129871 0.991531i \(-0.541456\pi\)
−0.129871 + 0.991531i \(0.541456\pi\)
\(710\) 0 0
\(711\) −24.9037 −0.933961
\(712\) 0 0
\(713\) 48.8466 1.82932
\(714\) 0 0
\(715\) 22.1487 0.828313
\(716\) 0 0
\(717\) −41.0797 −1.53415
\(718\) 0 0
\(719\) −12.3035 −0.458843 −0.229422 0.973327i \(-0.573683\pi\)
−0.229422 + 0.973327i \(0.573683\pi\)
\(720\) 0 0
\(721\) −14.1852 −0.528284
\(722\) 0 0
\(723\) −32.7137 −1.21664
\(724\) 0 0
\(725\) 49.2273 1.82826
\(726\) 0 0
\(727\) −27.6098 −1.02399 −0.511995 0.858989i \(-0.671093\pi\)
−0.511995 + 0.858989i \(0.671093\pi\)
\(728\) 0 0
\(729\) −2.60577 −0.0965099
\(730\) 0 0
\(731\) 1.50613 0.0557062
\(732\) 0 0
\(733\) −45.0293 −1.66319 −0.831597 0.555380i \(-0.812573\pi\)
−0.831597 + 0.555380i \(0.812573\pi\)
\(734\) 0 0
\(735\) 92.0257 3.39442
\(736\) 0 0
\(737\) −8.35725 −0.307843
\(738\) 0 0
\(739\) 37.8963 1.39404 0.697018 0.717053i \(-0.254510\pi\)
0.697018 + 0.717053i \(0.254510\pi\)
\(740\) 0 0
\(741\) 13.3590 0.490757
\(742\) 0 0
\(743\) 23.3103 0.855172 0.427586 0.903975i \(-0.359364\pi\)
0.427586 + 0.903975i \(0.359364\pi\)
\(744\) 0 0
\(745\) 78.2807 2.86798
\(746\) 0 0
\(747\) −12.6355 −0.462309
\(748\) 0 0
\(749\) −47.0270 −1.71833
\(750\) 0 0
\(751\) −14.9643 −0.546056 −0.273028 0.962006i \(-0.588025\pi\)
−0.273028 + 0.962006i \(0.588025\pi\)
\(752\) 0 0
\(753\) 26.6919 0.972709
\(754\) 0 0
\(755\) 1.67428 0.0609334
\(756\) 0 0
\(757\) −13.3436 −0.484980 −0.242490 0.970154i \(-0.577964\pi\)
−0.242490 + 0.970154i \(0.577964\pi\)
\(758\) 0 0
\(759\) 17.0638 0.619377
\(760\) 0 0
\(761\) −11.6962 −0.423986 −0.211993 0.977271i \(-0.567995\pi\)
−0.211993 + 0.977271i \(0.567995\pi\)
\(762\) 0 0
\(763\) −23.3990 −0.847102
\(764\) 0 0
\(765\) 4.51390 0.163200
\(766\) 0 0
\(767\) −36.0015 −1.29994
\(768\) 0 0
\(769\) 41.2234 1.48655 0.743276 0.668985i \(-0.233271\pi\)
0.743276 + 0.668985i \(0.233271\pi\)
\(770\) 0 0
\(771\) 15.2478 0.549137
\(772\) 0 0
\(773\) −12.9999 −0.467574 −0.233787 0.972288i \(-0.575112\pi\)
−0.233787 + 0.972288i \(0.575112\pi\)
\(774\) 0 0
\(775\) −51.1728 −1.83818
\(776\) 0 0
\(777\) 95.4341 3.42368
\(778\) 0 0
\(779\) −2.04380 −0.0732269
\(780\) 0 0
\(781\) 0.566034 0.0202543
\(782\) 0 0
\(783\) 15.9574 0.570273
\(784\) 0 0
\(785\) 16.4181 0.585988
\(786\) 0 0
\(787\) 4.63820 0.165334 0.0826669 0.996577i \(-0.473656\pi\)
0.0826669 + 0.996577i \(0.473656\pi\)
\(788\) 0 0
\(789\) −54.8037 −1.95106
\(790\) 0 0
\(791\) −51.4934 −1.83089
\(792\) 0 0
\(793\) −41.7983 −1.48430
\(794\) 0 0
\(795\) −15.5505 −0.551520
\(796\) 0 0
\(797\) −11.0271 −0.390600 −0.195300 0.980744i \(-0.562568\pi\)
−0.195300 + 0.980744i \(0.562568\pi\)
\(798\) 0 0
\(799\) 7.09226 0.250906
\(800\) 0 0
\(801\) −25.7072 −0.908318
\(802\) 0 0
\(803\) 7.29948 0.257593
\(804\) 0 0
\(805\) −121.970 −4.29888
\(806\) 0 0
\(807\) −3.46095 −0.121831
\(808\) 0 0
\(809\) −18.8659 −0.663290 −0.331645 0.943404i \(-0.607604\pi\)
−0.331645 + 0.943404i \(0.607604\pi\)
\(810\) 0 0
\(811\) −13.9642 −0.490348 −0.245174 0.969479i \(-0.578845\pi\)
−0.245174 + 0.969479i \(0.578845\pi\)
\(812\) 0 0
\(813\) −12.2099 −0.428221
\(814\) 0 0
\(815\) −43.4076 −1.52050
\(816\) 0 0
\(817\) −2.16767 −0.0758371
\(818\) 0 0
\(819\) 47.1115 1.64621
\(820\) 0 0
\(821\) −30.8485 −1.07662 −0.538310 0.842747i \(-0.680937\pi\)
−0.538310 + 0.842747i \(0.680937\pi\)
\(822\) 0 0
\(823\) −9.48137 −0.330500 −0.165250 0.986252i \(-0.552843\pi\)
−0.165250 + 0.986252i \(0.552843\pi\)
\(824\) 0 0
\(825\) −17.8764 −0.622376
\(826\) 0 0
\(827\) 16.7454 0.582296 0.291148 0.956678i \(-0.405963\pi\)
0.291148 + 0.956678i \(0.405963\pi\)
\(828\) 0 0
\(829\) 1.59130 0.0552682 0.0276341 0.999618i \(-0.491203\pi\)
0.0276341 + 0.999618i \(0.491203\pi\)
\(830\) 0 0
\(831\) −66.8389 −2.31862
\(832\) 0 0
\(833\) −8.05085 −0.278945
\(834\) 0 0
\(835\) 35.6913 1.23515
\(836\) 0 0
\(837\) −16.5881 −0.573368
\(838\) 0 0
\(839\) −5.12222 −0.176839 −0.0884193 0.996083i \(-0.528182\pi\)
−0.0884193 + 0.996083i \(0.528182\pi\)
\(840\) 0 0
\(841\) 7.32587 0.252616
\(842\) 0 0
\(843\) 32.9110 1.13352
\(844\) 0 0
\(845\) −88.0154 −3.02782
\(846\) 0 0
\(847\) −4.31127 −0.148137
\(848\) 0 0
\(849\) −63.7251 −2.18704
\(850\) 0 0
\(851\) −78.8516 −2.70300
\(852\) 0 0
\(853\) −51.2020 −1.75312 −0.876561 0.481290i \(-0.840168\pi\)
−0.876561 + 0.481290i \(0.840168\pi\)
\(854\) 0 0
\(855\) −6.49655 −0.222177
\(856\) 0 0
\(857\) −7.92412 −0.270683 −0.135341 0.990799i \(-0.543213\pi\)
−0.135341 + 0.990799i \(0.543213\pi\)
\(858\) 0 0
\(859\) −39.4668 −1.34659 −0.673296 0.739373i \(-0.735122\pi\)
−0.673296 + 0.739373i \(0.735122\pi\)
\(860\) 0 0
\(861\) −19.2853 −0.657241
\(862\) 0 0
\(863\) −52.1231 −1.77429 −0.887145 0.461491i \(-0.847315\pi\)
−0.887145 + 0.461491i \(0.847315\pi\)
\(864\) 0 0
\(865\) 46.3817 1.57703
\(866\) 0 0
\(867\) 36.1509 1.22775
\(868\) 0 0
\(869\) −13.9103 −0.471873
\(870\) 0 0
\(871\) 51.0102 1.72841
\(872\) 0 0
\(873\) −5.18907 −0.175624
\(874\) 0 0
\(875\) 49.5563 1.67531
\(876\) 0 0
\(877\) 20.6344 0.696773 0.348386 0.937351i \(-0.386730\pi\)
0.348386 + 0.937351i \(0.386730\pi\)
\(878\) 0 0
\(879\) −12.8873 −0.434677
\(880\) 0 0
\(881\) 42.3008 1.42515 0.712574 0.701597i \(-0.247529\pi\)
0.712574 + 0.701597i \(0.247529\pi\)
\(882\) 0 0
\(883\) −47.1397 −1.58638 −0.793188 0.608977i \(-0.791580\pi\)
−0.793188 + 0.608977i \(0.791580\pi\)
\(884\) 0 0
\(885\) 46.8451 1.57468
\(886\) 0 0
\(887\) 5.41860 0.181939 0.0909694 0.995854i \(-0.471003\pi\)
0.0909694 + 0.995854i \(0.471003\pi\)
\(888\) 0 0
\(889\) −33.3180 −1.11745
\(890\) 0 0
\(891\) −11.1657 −0.374066
\(892\) 0 0
\(893\) −10.2074 −0.341578
\(894\) 0 0
\(895\) 29.5129 0.986506
\(896\) 0 0
\(897\) −104.152 −3.47755
\(898\) 0 0
\(899\) −37.7615 −1.25942
\(900\) 0 0
\(901\) 1.36043 0.0453226
\(902\) 0 0
\(903\) −20.4541 −0.680669
\(904\) 0 0
\(905\) −67.3284 −2.23807
\(906\) 0 0
\(907\) 22.1148 0.734309 0.367154 0.930160i \(-0.380332\pi\)
0.367154 + 0.930160i \(0.380332\pi\)
\(908\) 0 0
\(909\) −13.5615 −0.449805
\(910\) 0 0
\(911\) 18.0321 0.597429 0.298715 0.954343i \(-0.403442\pi\)
0.298715 + 0.954343i \(0.403442\pi\)
\(912\) 0 0
\(913\) −7.05772 −0.233576
\(914\) 0 0
\(915\) 54.3879 1.79801
\(916\) 0 0
\(917\) 45.6604 1.50784
\(918\) 0 0
\(919\) −36.9395 −1.21852 −0.609261 0.792970i \(-0.708534\pi\)
−0.609261 + 0.792970i \(0.708534\pi\)
\(920\) 0 0
\(921\) −42.0532 −1.38570
\(922\) 0 0
\(923\) −3.45490 −0.113720
\(924\) 0 0
\(925\) 82.6066 2.71609
\(926\) 0 0
\(927\) 5.89059 0.193472
\(928\) 0 0
\(929\) 10.2792 0.337250 0.168625 0.985680i \(-0.446067\pi\)
0.168625 + 0.985680i \(0.446067\pi\)
\(930\) 0 0
\(931\) 11.5870 0.379750
\(932\) 0 0
\(933\) −58.5382 −1.91645
\(934\) 0 0
\(935\) 2.52129 0.0824551
\(936\) 0 0
\(937\) 1.42527 0.0465615 0.0232807 0.999729i \(-0.492589\pi\)
0.0232807 + 0.999729i \(0.492589\pi\)
\(938\) 0 0
\(939\) 35.1141 1.14591
\(940\) 0 0
\(941\) 32.0967 1.04632 0.523161 0.852234i \(-0.324753\pi\)
0.523161 + 0.852234i \(0.324753\pi\)
\(942\) 0 0
\(943\) 15.9343 0.518892
\(944\) 0 0
\(945\) 41.4205 1.34741
\(946\) 0 0
\(947\) −8.61405 −0.279919 −0.139959 0.990157i \(-0.544697\pi\)
−0.139959 + 0.990157i \(0.544697\pi\)
\(948\) 0 0
\(949\) −44.5538 −1.44628
\(950\) 0 0
\(951\) −40.6360 −1.31771
\(952\) 0 0
\(953\) 6.48103 0.209941 0.104971 0.994475i \(-0.466525\pi\)
0.104971 + 0.994475i \(0.466525\pi\)
\(954\) 0 0
\(955\) 38.5609 1.24780
\(956\) 0 0
\(957\) −13.1914 −0.426416
\(958\) 0 0
\(959\) 37.7162 1.21792
\(960\) 0 0
\(961\) 8.25380 0.266252
\(962\) 0 0
\(963\) 19.5286 0.629300
\(964\) 0 0
\(965\) 2.75671 0.0887415
\(966\) 0 0
\(967\) 55.2454 1.77657 0.888286 0.459292i \(-0.151897\pi\)
0.888286 + 0.459292i \(0.151897\pi\)
\(968\) 0 0
\(969\) 1.52073 0.0488528
\(970\) 0 0
\(971\) 30.1518 0.967618 0.483809 0.875174i \(-0.339253\pi\)
0.483809 + 0.875174i \(0.339253\pi\)
\(972\) 0 0
\(973\) −44.2750 −1.41939
\(974\) 0 0
\(975\) 109.112 3.49439
\(976\) 0 0
\(977\) 22.6608 0.724984 0.362492 0.931987i \(-0.381926\pi\)
0.362492 + 0.931987i \(0.381926\pi\)
\(978\) 0 0
\(979\) −14.3590 −0.458917
\(980\) 0 0
\(981\) 9.71676 0.310232
\(982\) 0 0
\(983\) 17.3274 0.552658 0.276329 0.961063i \(-0.410882\pi\)
0.276329 + 0.961063i \(0.410882\pi\)
\(984\) 0 0
\(985\) −6.12087 −0.195027
\(986\) 0 0
\(987\) −96.3169 −3.06580
\(988\) 0 0
\(989\) 16.9000 0.537389
\(990\) 0 0
\(991\) 27.1931 0.863818 0.431909 0.901917i \(-0.357840\pi\)
0.431909 + 0.901917i \(0.357840\pi\)
\(992\) 0 0
\(993\) 75.7711 2.40452
\(994\) 0 0
\(995\) −87.8709 −2.78569
\(996\) 0 0
\(997\) 21.0570 0.666881 0.333441 0.942771i \(-0.391790\pi\)
0.333441 + 0.942771i \(0.391790\pi\)
\(998\) 0 0
\(999\) 26.7776 0.847207
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.d.1.2 6
3.2 odd 2 7524.2.a.r.1.6 6
4.3 odd 2 3344.2.a.x.1.5 6
11.10 odd 2 9196.2.a.k.1.2 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
836.2.a.d.1.2 6 1.1 even 1 trivial
3344.2.a.x.1.5 6 4.3 odd 2
7524.2.a.r.1.6 6 3.2 odd 2
9196.2.a.k.1.2 6 11.10 odd 2