Properties

Label 6075.2.a.bn.1.1
Level $6075$
Weight $2$
Character 6075.1
Self dual yes
Analytic conductor $48.509$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6075,2,Mod(1,6075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6075.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6075 = 3^{5} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6075.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: \(48.5091192279\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 243)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.44949\) of defining polynomial
Character \(\chi\) \(=\) 6075.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.44949 q^{2} +4.00000 q^{4} -2.00000 q^{7} -4.89898 q^{8} +O(q^{10})\) \(q-2.44949 q^{2} +4.00000 q^{4} -2.00000 q^{7} -4.89898 q^{8} +2.44949 q^{11} +1.00000 q^{13} +4.89898 q^{14} +4.00000 q^{16} +7.34847 q^{17} -1.00000 q^{19} -6.00000 q^{22} +2.44949 q^{23} -2.44949 q^{26} -8.00000 q^{28} -4.89898 q^{29} -1.00000 q^{31} -18.0000 q^{34} -8.00000 q^{37} +2.44949 q^{38} +4.89898 q^{41} -11.0000 q^{43} +9.79796 q^{44} -6.00000 q^{46} -9.79796 q^{47} -3.00000 q^{49} +4.00000 q^{52} -7.34847 q^{53} +9.79796 q^{56} +12.0000 q^{58} -2.44949 q^{59} +5.00000 q^{61} +2.44949 q^{62} -8.00000 q^{64} +7.00000 q^{67} +29.3939 q^{68} +7.34847 q^{71} -11.0000 q^{73} +19.5959 q^{74} -4.00000 q^{76} -4.89898 q^{77} -7.00000 q^{79} -12.0000 q^{82} +12.2474 q^{83} +26.9444 q^{86} -12.0000 q^{88} -2.00000 q^{91} +9.79796 q^{92} +24.0000 q^{94} +7.00000 q^{97} +7.34847 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 8 q^{4} - 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 8 q^{4} - 4 q^{7} + 2 q^{13} + 8 q^{16} - 2 q^{19} - 12 q^{22} - 16 q^{28} - 2 q^{31} - 36 q^{34} - 16 q^{37} - 22 q^{43} - 12 q^{46} - 6 q^{49} + 8 q^{52} + 24 q^{58} + 10 q^{61} - 16 q^{64} + 14 q^{67} - 22 q^{73} - 8 q^{76} - 14 q^{79} - 24 q^{82} - 24 q^{88} - 4 q^{91} + 48 q^{94} + 14 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.44949 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 0 0
\(4\) 4.00000 2.00000
\(5\) 0 0
\(6\) 0 0
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) −4.89898 −1.73205
\(9\) 0 0
\(10\) 0 0
\(11\) 2.44949 0.738549 0.369274 0.929320i \(-0.379606\pi\)
0.369274 + 0.929320i \(0.379606\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350 0.138675 0.990338i \(-0.455716\pi\)
0.138675 + 0.990338i \(0.455716\pi\)
\(14\) 4.89898 1.30931
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 7.34847 1.78227 0.891133 0.453743i \(-0.149911\pi\)
0.891133 + 0.453743i \(0.149911\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.00000 −1.27920
\(23\) 2.44949 0.510754 0.255377 0.966842i \(-0.417800\pi\)
0.255377 + 0.966842i \(0.417800\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.44949 −0.480384
\(27\) 0 0
\(28\) −8.00000 −1.51186
\(29\) −4.89898 −0.909718 −0.454859 0.890564i \(-0.650310\pi\)
−0.454859 + 0.890564i \(0.650310\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605 −0.0898027 0.995960i \(-0.528624\pi\)
−0.0898027 + 0.995960i \(0.528624\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) −18.0000 −3.08697
\(35\) 0 0
\(36\) 0 0
\(37\) −8.00000 −1.31519 −0.657596 0.753371i \(-0.728427\pi\)
−0.657596 + 0.753371i \(0.728427\pi\)
\(38\) 2.44949 0.397360
\(39\) 0 0
\(40\) 0 0
\(41\) 4.89898 0.765092 0.382546 0.923936i \(-0.375047\pi\)
0.382546 + 0.923936i \(0.375047\pi\)
\(42\) 0 0
\(43\) −11.0000 −1.67748 −0.838742 0.544529i \(-0.816708\pi\)
−0.838742 + 0.544529i \(0.816708\pi\)
\(44\) 9.79796 1.47710
\(45\) 0 0
\(46\) −6.00000 −0.884652
\(47\) −9.79796 −1.42918 −0.714590 0.699544i \(-0.753387\pi\)
−0.714590 + 0.699544i \(0.753387\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 4.00000 0.554700
\(53\) −7.34847 −1.00939 −0.504695 0.863298i \(-0.668395\pi\)
−0.504695 + 0.863298i \(0.668395\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 9.79796 1.30931
\(57\) 0 0
\(58\) 12.0000 1.57568
\(59\) −2.44949 −0.318896 −0.159448 0.987206i \(-0.550971\pi\)
−0.159448 + 0.987206i \(0.550971\pi\)
\(60\) 0 0
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 2.44949 0.311086
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 7.00000 0.855186 0.427593 0.903971i \(-0.359362\pi\)
0.427593 + 0.903971i \(0.359362\pi\)
\(68\) 29.3939 3.56453
\(69\) 0 0
\(70\) 0 0
\(71\) 7.34847 0.872103 0.436051 0.899922i \(-0.356377\pi\)
0.436051 + 0.899922i \(0.356377\pi\)
\(72\) 0 0
\(73\) −11.0000 −1.28745 −0.643726 0.765256i \(-0.722612\pi\)
−0.643726 + 0.765256i \(0.722612\pi\)
\(74\) 19.5959 2.27798
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) −4.89898 −0.558291
\(78\) 0 0
\(79\) −7.00000 −0.787562 −0.393781 0.919204i \(-0.628833\pi\)
−0.393781 + 0.919204i \(0.628833\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −12.0000 −1.32518
\(83\) 12.2474 1.34433 0.672166 0.740400i \(-0.265364\pi\)
0.672166 + 0.740400i \(0.265364\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 26.9444 2.90549
\(87\) 0 0
\(88\) −12.0000 −1.27920
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) −2.00000 −0.209657
\(92\) 9.79796 1.02151
\(93\) 0 0
\(94\) 24.0000 2.47541
\(95\) 0 0
\(96\) 0 0
\(97\) 7.00000 0.710742 0.355371 0.934725i \(-0.384354\pi\)
0.355371 + 0.934725i \(0.384354\pi\)
\(98\) 7.34847 0.742307
\(99\) 0 0
\(100\) 0 0
\(101\) −4.89898 −0.487467 −0.243733 0.969842i \(-0.578372\pi\)
−0.243733 + 0.969842i \(0.578372\pi\)
\(102\) 0 0
\(103\) 7.00000 0.689730 0.344865 0.938652i \(-0.387925\pi\)
0.344865 + 0.938652i \(0.387925\pi\)
\(104\) −4.89898 −0.480384
\(105\) 0 0
\(106\) 18.0000 1.74831
\(107\) −14.6969 −1.42081 −0.710403 0.703795i \(-0.751487\pi\)
−0.710403 + 0.703795i \(0.751487\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −8.00000 −0.755929
\(113\) 9.79796 0.921714 0.460857 0.887474i \(-0.347542\pi\)
0.460857 + 0.887474i \(0.347542\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −19.5959 −1.81944
\(117\) 0 0
\(118\) 6.00000 0.552345
\(119\) −14.6969 −1.34727
\(120\) 0 0
\(121\) −5.00000 −0.454545
\(122\) −12.2474 −1.10883
\(123\) 0 0
\(124\) −4.00000 −0.359211
\(125\) 0 0
\(126\) 0 0
\(127\) 19.0000 1.68598 0.842989 0.537931i \(-0.180794\pi\)
0.842989 + 0.537931i \(0.180794\pi\)
\(128\) 19.5959 1.73205
\(129\) 0 0
\(130\) 0 0
\(131\) 12.2474 1.07006 0.535032 0.844832i \(-0.320299\pi\)
0.535032 + 0.844832i \(0.320299\pi\)
\(132\) 0 0
\(133\) 2.00000 0.173422
\(134\) −17.1464 −1.48123
\(135\) 0 0
\(136\) −36.0000 −3.08697
\(137\) −9.79796 −0.837096 −0.418548 0.908195i \(-0.637461\pi\)
−0.418548 + 0.908195i \(0.637461\pi\)
\(138\) 0 0
\(139\) −10.0000 −0.848189 −0.424094 0.905618i \(-0.639408\pi\)
−0.424094 + 0.905618i \(0.639408\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −18.0000 −1.51053
\(143\) 2.44949 0.204837
\(144\) 0 0
\(145\) 0 0
\(146\) 26.9444 2.22993
\(147\) 0 0
\(148\) −32.0000 −2.63038
\(149\) 12.2474 1.00335 0.501675 0.865056i \(-0.332717\pi\)
0.501675 + 0.865056i \(0.332717\pi\)
\(150\) 0 0
\(151\) 5.00000 0.406894 0.203447 0.979086i \(-0.434786\pi\)
0.203447 + 0.979086i \(0.434786\pi\)
\(152\) 4.89898 0.397360
\(153\) 0 0
\(154\) 12.0000 0.966988
\(155\) 0 0
\(156\) 0 0
\(157\) −17.0000 −1.35675 −0.678374 0.734717i \(-0.737315\pi\)
−0.678374 + 0.734717i \(0.737315\pi\)
\(158\) 17.1464 1.36410
\(159\) 0 0
\(160\) 0 0
\(161\) −4.89898 −0.386094
\(162\) 0 0
\(163\) 10.0000 0.783260 0.391630 0.920123i \(-0.371911\pi\)
0.391630 + 0.920123i \(0.371911\pi\)
\(164\) 19.5959 1.53018
\(165\) 0 0
\(166\) −30.0000 −2.32845
\(167\) −4.89898 −0.379094 −0.189547 0.981872i \(-0.560702\pi\)
−0.189547 + 0.981872i \(0.560702\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 0 0
\(171\) 0 0
\(172\) −44.0000 −3.35497
\(173\) −9.79796 −0.744925 −0.372463 0.928047i \(-0.621486\pi\)
−0.372463 + 0.928047i \(0.621486\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 9.79796 0.738549
\(177\) 0 0
\(178\) 0 0
\(179\) −14.6969 −1.09850 −0.549250 0.835658i \(-0.685087\pi\)
−0.549250 + 0.835658i \(0.685087\pi\)
\(180\) 0 0
\(181\) 8.00000 0.594635 0.297318 0.954779i \(-0.403908\pi\)
0.297318 + 0.954779i \(0.403908\pi\)
\(182\) 4.89898 0.363137
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) 0 0
\(187\) 18.0000 1.31629
\(188\) −39.1918 −2.85836
\(189\) 0 0
\(190\) 0 0
\(191\) 9.79796 0.708955 0.354478 0.935064i \(-0.384659\pi\)
0.354478 + 0.935064i \(0.384659\pi\)
\(192\) 0 0
\(193\) −11.0000 −0.791797 −0.395899 0.918294i \(-0.629567\pi\)
−0.395899 + 0.918294i \(0.629567\pi\)
\(194\) −17.1464 −1.23104
\(195\) 0 0
\(196\) −12.0000 −0.857143
\(197\) −14.6969 −1.04711 −0.523557 0.851991i \(-0.675395\pi\)
−0.523557 + 0.851991i \(0.675395\pi\)
\(198\) 0 0
\(199\) −1.00000 −0.0708881 −0.0354441 0.999372i \(-0.511285\pi\)
−0.0354441 + 0.999372i \(0.511285\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 12.0000 0.844317
\(203\) 9.79796 0.687682
\(204\) 0 0
\(205\) 0 0
\(206\) −17.1464 −1.19465
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) −2.44949 −0.169435
\(210\) 0 0
\(211\) −1.00000 −0.0688428 −0.0344214 0.999407i \(-0.510959\pi\)
−0.0344214 + 0.999407i \(0.510959\pi\)
\(212\) −29.3939 −2.01878
\(213\) 0 0
\(214\) 36.0000 2.46091
\(215\) 0 0
\(216\) 0 0
\(217\) 2.00000 0.135769
\(218\) 2.44949 0.165900
\(219\) 0 0
\(220\) 0 0
\(221\) 7.34847 0.494312
\(222\) 0 0
\(223\) 7.00000 0.468755 0.234377 0.972146i \(-0.424695\pi\)
0.234377 + 0.972146i \(0.424695\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −24.0000 −1.59646
\(227\) −9.79796 −0.650313 −0.325157 0.945660i \(-0.605417\pi\)
−0.325157 + 0.945660i \(0.605417\pi\)
\(228\) 0 0
\(229\) −1.00000 −0.0660819 −0.0330409 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 24.0000 1.57568
\(233\) 7.34847 0.481414 0.240707 0.970598i \(-0.422621\pi\)
0.240707 + 0.970598i \(0.422621\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −9.79796 −0.637793
\(237\) 0 0
\(238\) 36.0000 2.33353
\(239\) −2.44949 −0.158444 −0.0792222 0.996857i \(-0.525244\pi\)
−0.0792222 + 0.996857i \(0.525244\pi\)
\(240\) 0 0
\(241\) −16.0000 −1.03065 −0.515325 0.856995i \(-0.672329\pi\)
−0.515325 + 0.856995i \(0.672329\pi\)
\(242\) 12.2474 0.787296
\(243\) 0 0
\(244\) 20.0000 1.28037
\(245\) 0 0
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 4.89898 0.311086
\(249\) 0 0
\(250\) 0 0
\(251\) −7.34847 −0.463831 −0.231916 0.972736i \(-0.574499\pi\)
−0.231916 + 0.972736i \(0.574499\pi\)
\(252\) 0 0
\(253\) 6.00000 0.377217
\(254\) −46.5403 −2.92020
\(255\) 0 0
\(256\) −32.0000 −2.00000
\(257\) 17.1464 1.06956 0.534782 0.844990i \(-0.320394\pi\)
0.534782 + 0.844990i \(0.320394\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 0 0
\(262\) −30.0000 −1.85341
\(263\) 26.9444 1.66146 0.830731 0.556674i \(-0.187923\pi\)
0.830731 + 0.556674i \(0.187923\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.89898 −0.300376
\(267\) 0 0
\(268\) 28.0000 1.71037
\(269\) −22.0454 −1.34413 −0.672066 0.740491i \(-0.734593\pi\)
−0.672066 + 0.740491i \(0.734593\pi\)
\(270\) 0 0
\(271\) −7.00000 −0.425220 −0.212610 0.977137i \(-0.568196\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) 29.3939 1.78227
\(273\) 0 0
\(274\) 24.0000 1.44989
\(275\) 0 0
\(276\) 0 0
\(277\) −11.0000 −0.660926 −0.330463 0.943819i \(-0.607205\pi\)
−0.330463 + 0.943819i \(0.607205\pi\)
\(278\) 24.4949 1.46911
\(279\) 0 0
\(280\) 0 0
\(281\) −12.2474 −0.730622 −0.365311 0.930886i \(-0.619037\pi\)
−0.365311 + 0.930886i \(0.619037\pi\)
\(282\) 0 0
\(283\) −17.0000 −1.01055 −0.505273 0.862960i \(-0.668608\pi\)
−0.505273 + 0.862960i \(0.668608\pi\)
\(284\) 29.3939 1.74421
\(285\) 0 0
\(286\) −6.00000 −0.354787
\(287\) −9.79796 −0.578355
\(288\) 0 0
\(289\) 37.0000 2.17647
\(290\) 0 0
\(291\) 0 0
\(292\) −44.0000 −2.57491
\(293\) −4.89898 −0.286201 −0.143101 0.989708i \(-0.545707\pi\)
−0.143101 + 0.989708i \(0.545707\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 39.1918 2.27798
\(297\) 0 0
\(298\) −30.0000 −1.73785
\(299\) 2.44949 0.141658
\(300\) 0 0
\(301\) 22.0000 1.26806
\(302\) −12.2474 −0.704761
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 0 0
\(306\) 0 0
\(307\) −2.00000 −0.114146 −0.0570730 0.998370i \(-0.518177\pi\)
−0.0570730 + 0.998370i \(0.518177\pi\)
\(308\) −19.5959 −1.11658
\(309\) 0 0
\(310\) 0 0
\(311\) −24.4949 −1.38898 −0.694489 0.719503i \(-0.744370\pi\)
−0.694489 + 0.719503i \(0.744370\pi\)
\(312\) 0 0
\(313\) 16.0000 0.904373 0.452187 0.891923i \(-0.350644\pi\)
0.452187 + 0.891923i \(0.350644\pi\)
\(314\) 41.6413 2.34996
\(315\) 0 0
\(316\) −28.0000 −1.57512
\(317\) −9.79796 −0.550308 −0.275154 0.961400i \(-0.588729\pi\)
−0.275154 + 0.961400i \(0.588729\pi\)
\(318\) 0 0
\(319\) −12.0000 −0.671871
\(320\) 0 0
\(321\) 0 0
\(322\) 12.0000 0.668734
\(323\) −7.34847 −0.408880
\(324\) 0 0
\(325\) 0 0
\(326\) −24.4949 −1.35665
\(327\) 0 0
\(328\) −24.0000 −1.32518
\(329\) 19.5959 1.08036
\(330\) 0 0
\(331\) −7.00000 −0.384755 −0.192377 0.981321i \(-0.561620\pi\)
−0.192377 + 0.981321i \(0.561620\pi\)
\(332\) 48.9898 2.68866
\(333\) 0 0
\(334\) 12.0000 0.656611
\(335\) 0 0
\(336\) 0 0
\(337\) 28.0000 1.52526 0.762629 0.646837i \(-0.223908\pi\)
0.762629 + 0.646837i \(0.223908\pi\)
\(338\) 29.3939 1.59882
\(339\) 0 0
\(340\) 0 0
\(341\) −2.44949 −0.132647
\(342\) 0 0
\(343\) 20.0000 1.07990
\(344\) 53.8888 2.90549
\(345\) 0 0
\(346\) 24.0000 1.29025
\(347\) 24.4949 1.31495 0.657477 0.753474i \(-0.271623\pi\)
0.657477 + 0.753474i \(0.271623\pi\)
\(348\) 0 0
\(349\) 20.0000 1.07058 0.535288 0.844670i \(-0.320203\pi\)
0.535288 + 0.844670i \(0.320203\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2.44949 −0.130373 −0.0651866 0.997873i \(-0.520764\pi\)
−0.0651866 + 0.997873i \(0.520764\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 36.0000 1.90266
\(359\) 29.3939 1.55135 0.775675 0.631133i \(-0.217410\pi\)
0.775675 + 0.631133i \(0.217410\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) −19.5959 −1.02994
\(363\) 0 0
\(364\) −8.00000 −0.419314
\(365\) 0 0
\(366\) 0 0
\(367\) −5.00000 −0.260998 −0.130499 0.991448i \(-0.541658\pi\)
−0.130499 + 0.991448i \(0.541658\pi\)
\(368\) 9.79796 0.510754
\(369\) 0 0
\(370\) 0 0
\(371\) 14.6969 0.763027
\(372\) 0 0
\(373\) −35.0000 −1.81223 −0.906116 0.423030i \(-0.860966\pi\)
−0.906116 + 0.423030i \(0.860966\pi\)
\(374\) −44.0908 −2.27988
\(375\) 0 0
\(376\) 48.0000 2.47541
\(377\) −4.89898 −0.252310
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −24.0000 −1.22795
\(383\) −34.2929 −1.75228 −0.876142 0.482054i \(-0.839891\pi\)
−0.876142 + 0.482054i \(0.839891\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 26.9444 1.37143
\(387\) 0 0
\(388\) 28.0000 1.42148
\(389\) −26.9444 −1.36613 −0.683067 0.730355i \(-0.739354\pi\)
−0.683067 + 0.730355i \(0.739354\pi\)
\(390\) 0 0
\(391\) 18.0000 0.910299
\(392\) 14.6969 0.742307
\(393\) 0 0
\(394\) 36.0000 1.81365
\(395\) 0 0
\(396\) 0 0
\(397\) 1.00000 0.0501886 0.0250943 0.999685i \(-0.492011\pi\)
0.0250943 + 0.999685i \(0.492011\pi\)
\(398\) 2.44949 0.122782
\(399\) 0 0
\(400\) 0 0
\(401\) 34.2929 1.71250 0.856252 0.516559i \(-0.172787\pi\)
0.856252 + 0.516559i \(0.172787\pi\)
\(402\) 0 0
\(403\) −1.00000 −0.0498135
\(404\) −19.5959 −0.974933
\(405\) 0 0
\(406\) −24.0000 −1.19110
\(407\) −19.5959 −0.971334
\(408\) 0 0
\(409\) −28.0000 −1.38451 −0.692255 0.721653i \(-0.743383\pi\)
−0.692255 + 0.721653i \(0.743383\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 28.0000 1.37946
\(413\) 4.89898 0.241063
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 6.00000 0.293470
\(419\) 34.2929 1.67532 0.837658 0.546195i \(-0.183924\pi\)
0.837658 + 0.546195i \(0.183924\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 2.44949 0.119239
\(423\) 0 0
\(424\) 36.0000 1.74831
\(425\) 0 0
\(426\) 0 0
\(427\) −10.0000 −0.483934
\(428\) −58.7878 −2.84161
\(429\) 0 0
\(430\) 0 0
\(431\) −7.34847 −0.353963 −0.176982 0.984214i \(-0.556633\pi\)
−0.176982 + 0.984214i \(0.556633\pi\)
\(432\) 0 0
\(433\) −17.0000 −0.816968 −0.408484 0.912766i \(-0.633942\pi\)
−0.408484 + 0.912766i \(0.633942\pi\)
\(434\) −4.89898 −0.235159
\(435\) 0 0
\(436\) −4.00000 −0.191565
\(437\) −2.44949 −0.117175
\(438\) 0 0
\(439\) 14.0000 0.668184 0.334092 0.942541i \(-0.391570\pi\)
0.334092 + 0.942541i \(0.391570\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −18.0000 −0.856173
\(443\) 12.2474 0.581894 0.290947 0.956739i \(-0.406030\pi\)
0.290947 + 0.956739i \(0.406030\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −17.1464 −0.811907
\(447\) 0 0
\(448\) 16.0000 0.755929
\(449\) 22.0454 1.04039 0.520194 0.854048i \(-0.325860\pi\)
0.520194 + 0.854048i \(0.325860\pi\)
\(450\) 0 0
\(451\) 12.0000 0.565058
\(452\) 39.1918 1.84343
\(453\) 0 0
\(454\) 24.0000 1.12638
\(455\) 0 0
\(456\) 0 0
\(457\) −29.0000 −1.35656 −0.678281 0.734802i \(-0.737275\pi\)
−0.678281 + 0.734802i \(0.737275\pi\)
\(458\) 2.44949 0.114457
\(459\) 0 0
\(460\) 0 0
\(461\) −26.9444 −1.25493 −0.627463 0.778647i \(-0.715906\pi\)
−0.627463 + 0.778647i \(0.715906\pi\)
\(462\) 0 0
\(463\) 19.0000 0.883005 0.441502 0.897260i \(-0.354446\pi\)
0.441502 + 0.897260i \(0.354446\pi\)
\(464\) −19.5959 −0.909718
\(465\) 0 0
\(466\) −18.0000 −0.833834
\(467\) −14.6969 −0.680093 −0.340047 0.940409i \(-0.610443\pi\)
−0.340047 + 0.940409i \(0.610443\pi\)
\(468\) 0 0
\(469\) −14.0000 −0.646460
\(470\) 0 0
\(471\) 0 0
\(472\) 12.0000 0.552345
\(473\) −26.9444 −1.23890
\(474\) 0 0
\(475\) 0 0
\(476\) −58.7878 −2.69453
\(477\) 0 0
\(478\) 6.00000 0.274434
\(479\) −26.9444 −1.23112 −0.615560 0.788090i \(-0.711070\pi\)
−0.615560 + 0.788090i \(0.711070\pi\)
\(480\) 0 0
\(481\) −8.00000 −0.364769
\(482\) 39.1918 1.78514
\(483\) 0 0
\(484\) −20.0000 −0.909091
\(485\) 0 0
\(486\) 0 0
\(487\) −35.0000 −1.58600 −0.793001 0.609221i \(-0.791482\pi\)
−0.793001 + 0.609221i \(0.791482\pi\)
\(488\) −24.4949 −1.10883
\(489\) 0 0
\(490\) 0 0
\(491\) −39.1918 −1.76870 −0.884351 0.466822i \(-0.845399\pi\)
−0.884351 + 0.466822i \(0.845399\pi\)
\(492\) 0 0
\(493\) −36.0000 −1.62136
\(494\) 2.44949 0.110208
\(495\) 0 0
\(496\) −4.00000 −0.179605
\(497\) −14.6969 −0.659248
\(498\) 0 0
\(499\) 2.00000 0.0895323 0.0447661 0.998997i \(-0.485746\pi\)
0.0447661 + 0.998997i \(0.485746\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 18.0000 0.803379
\(503\) 14.6969 0.655304 0.327652 0.944798i \(-0.393743\pi\)
0.327652 + 0.944798i \(0.393743\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −14.6969 −0.653359
\(507\) 0 0
\(508\) 76.0000 3.37195
\(509\) −9.79796 −0.434287 −0.217143 0.976140i \(-0.569674\pi\)
−0.217143 + 0.976140i \(0.569674\pi\)
\(510\) 0 0
\(511\) 22.0000 0.973223
\(512\) 39.1918 1.73205
\(513\) 0 0
\(514\) −42.0000 −1.85254
\(515\) 0 0
\(516\) 0 0
\(517\) −24.0000 −1.05552
\(518\) −39.1918 −1.72199
\(519\) 0 0
\(520\) 0 0
\(521\) 22.0454 0.965827 0.482913 0.875668i \(-0.339579\pi\)
0.482913 + 0.875668i \(0.339579\pi\)
\(522\) 0 0
\(523\) 25.0000 1.09317 0.546587 0.837402i \(-0.315927\pi\)
0.546587 + 0.837402i \(0.315927\pi\)
\(524\) 48.9898 2.14013
\(525\) 0 0
\(526\) −66.0000 −2.87774
\(527\) −7.34847 −0.320104
\(528\) 0 0
\(529\) −17.0000 −0.739130
\(530\) 0 0
\(531\) 0 0
\(532\) 8.00000 0.346844
\(533\) 4.89898 0.212198
\(534\) 0 0
\(535\) 0 0
\(536\) −34.2929 −1.48123
\(537\) 0 0
\(538\) 54.0000 2.32811
\(539\) −7.34847 −0.316521
\(540\) 0 0
\(541\) −28.0000 −1.20381 −0.601907 0.798566i \(-0.705592\pi\)
−0.601907 + 0.798566i \(0.705592\pi\)
\(542\) 17.1464 0.736502
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 13.0000 0.555840 0.277920 0.960604i \(-0.410355\pi\)
0.277920 + 0.960604i \(0.410355\pi\)
\(548\) −39.1918 −1.67419
\(549\) 0 0
\(550\) 0 0
\(551\) 4.89898 0.208704
\(552\) 0 0
\(553\) 14.0000 0.595341
\(554\) 26.9444 1.14476
\(555\) 0 0
\(556\) −40.0000 −1.69638
\(557\) −7.34847 −0.311365 −0.155682 0.987807i \(-0.549758\pi\)
−0.155682 + 0.987807i \(0.549758\pi\)
\(558\) 0 0
\(559\) −11.0000 −0.465250
\(560\) 0 0
\(561\) 0 0
\(562\) 30.0000 1.26547
\(563\) −12.2474 −0.516168 −0.258084 0.966122i \(-0.583091\pi\)
−0.258084 + 0.966122i \(0.583091\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 41.6413 1.75032
\(567\) 0 0
\(568\) −36.0000 −1.51053
\(569\) −12.2474 −0.513440 −0.256720 0.966486i \(-0.582642\pi\)
−0.256720 + 0.966486i \(0.582642\pi\)
\(570\) 0 0
\(571\) −10.0000 −0.418487 −0.209243 0.977864i \(-0.567100\pi\)
−0.209243 + 0.977864i \(0.567100\pi\)
\(572\) 9.79796 0.409673
\(573\) 0 0
\(574\) 24.0000 1.00174
\(575\) 0 0
\(576\) 0 0
\(577\) 25.0000 1.04076 0.520382 0.853934i \(-0.325790\pi\)
0.520382 + 0.853934i \(0.325790\pi\)
\(578\) −90.6311 −3.76976
\(579\) 0 0
\(580\) 0 0
\(581\) −24.4949 −1.01622
\(582\) 0 0
\(583\) −18.0000 −0.745484
\(584\) 53.8888 2.22993
\(585\) 0 0
\(586\) 12.0000 0.495715
\(587\) −2.44949 −0.101101 −0.0505506 0.998721i \(-0.516098\pi\)
−0.0505506 + 0.998721i \(0.516098\pi\)
\(588\) 0 0
\(589\) 1.00000 0.0412043
\(590\) 0 0
\(591\) 0 0
\(592\) −32.0000 −1.31519
\(593\) −7.34847 −0.301765 −0.150883 0.988552i \(-0.548212\pi\)
−0.150883 + 0.988552i \(0.548212\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 48.9898 2.00670
\(597\) 0 0
\(598\) −6.00000 −0.245358
\(599\) −39.1918 −1.60134 −0.800668 0.599109i \(-0.795522\pi\)
−0.800668 + 0.599109i \(0.795522\pi\)
\(600\) 0 0
\(601\) −7.00000 −0.285536 −0.142768 0.989756i \(-0.545600\pi\)
−0.142768 + 0.989756i \(0.545600\pi\)
\(602\) −53.8888 −2.19634
\(603\) 0 0
\(604\) 20.0000 0.813788
\(605\) 0 0
\(606\) 0 0
\(607\) −44.0000 −1.78590 −0.892952 0.450151i \(-0.851370\pi\)
−0.892952 + 0.450151i \(0.851370\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −9.79796 −0.396383
\(612\) 0 0
\(613\) −11.0000 −0.444286 −0.222143 0.975014i \(-0.571305\pi\)
−0.222143 + 0.975014i \(0.571305\pi\)
\(614\) 4.89898 0.197707
\(615\) 0 0
\(616\) 24.0000 0.966988
\(617\) 24.4949 0.986127 0.493064 0.869993i \(-0.335877\pi\)
0.493064 + 0.869993i \(0.335877\pi\)
\(618\) 0 0
\(619\) −49.0000 −1.96948 −0.984738 0.174042i \(-0.944317\pi\)
−0.984738 + 0.174042i \(0.944317\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 60.0000 2.40578
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −39.1918 −1.56642
\(627\) 0 0
\(628\) −68.0000 −2.71350
\(629\) −58.7878 −2.34402
\(630\) 0 0
\(631\) 44.0000 1.75161 0.875806 0.482663i \(-0.160330\pi\)
0.875806 + 0.482663i \(0.160330\pi\)
\(632\) 34.2929 1.36410
\(633\) 0 0
\(634\) 24.0000 0.953162
\(635\) 0 0
\(636\) 0 0
\(637\) −3.00000 −0.118864
\(638\) 29.3939 1.16371
\(639\) 0 0
\(640\) 0 0
\(641\) 17.1464 0.677243 0.338622 0.940923i \(-0.390039\pi\)
0.338622 + 0.940923i \(0.390039\pi\)
\(642\) 0 0
\(643\) −38.0000 −1.49857 −0.749287 0.662246i \(-0.769604\pi\)
−0.749287 + 0.662246i \(0.769604\pi\)
\(644\) −19.5959 −0.772187
\(645\) 0 0
\(646\) 18.0000 0.708201
\(647\) −36.7423 −1.44449 −0.722245 0.691637i \(-0.756890\pi\)
−0.722245 + 0.691637i \(0.756890\pi\)
\(648\) 0 0
\(649\) −6.00000 −0.235521
\(650\) 0 0
\(651\) 0 0
\(652\) 40.0000 1.56652
\(653\) 9.79796 0.383424 0.191712 0.981451i \(-0.438596\pi\)
0.191712 + 0.981451i \(0.438596\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 19.5959 0.765092
\(657\) 0 0
\(658\) −48.0000 −1.87123
\(659\) −19.5959 −0.763349 −0.381674 0.924297i \(-0.624652\pi\)
−0.381674 + 0.924297i \(0.624652\pi\)
\(660\) 0 0
\(661\) 11.0000 0.427850 0.213925 0.976850i \(-0.431375\pi\)
0.213925 + 0.976850i \(0.431375\pi\)
\(662\) 17.1464 0.666415
\(663\) 0 0
\(664\) −60.0000 −2.32845
\(665\) 0 0
\(666\) 0 0
\(667\) −12.0000 −0.464642
\(668\) −19.5959 −0.758189
\(669\) 0 0
\(670\) 0 0
\(671\) 12.2474 0.472808
\(672\) 0 0
\(673\) −29.0000 −1.11787 −0.558934 0.829212i \(-0.688789\pi\)
−0.558934 + 0.829212i \(0.688789\pi\)
\(674\) −68.5857 −2.64182
\(675\) 0 0
\(676\) −48.0000 −1.84615
\(677\) −46.5403 −1.78869 −0.894345 0.447379i \(-0.852358\pi\)
−0.894345 + 0.447379i \(0.852358\pi\)
\(678\) 0 0
\(679\) −14.0000 −0.537271
\(680\) 0 0
\(681\) 0 0
\(682\) 6.00000 0.229752
\(683\) 22.0454 0.843544 0.421772 0.906702i \(-0.361408\pi\)
0.421772 + 0.906702i \(0.361408\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −48.9898 −1.87044
\(687\) 0 0
\(688\) −44.0000 −1.67748
\(689\) −7.34847 −0.279954
\(690\) 0 0
\(691\) 47.0000 1.78796 0.893982 0.448103i \(-0.147900\pi\)
0.893982 + 0.448103i \(0.147900\pi\)
\(692\) −39.1918 −1.48985
\(693\) 0 0
\(694\) −60.0000 −2.27757
\(695\) 0 0
\(696\) 0 0
\(697\) 36.0000 1.36360
\(698\) −48.9898 −1.85429
\(699\) 0 0
\(700\) 0 0
\(701\) −14.6969 −0.555096 −0.277548 0.960712i \(-0.589522\pi\)
−0.277548 + 0.960712i \(0.589522\pi\)
\(702\) 0 0
\(703\) 8.00000 0.301726
\(704\) −19.5959 −0.738549
\(705\) 0 0
\(706\) 6.00000 0.225813
\(707\) 9.79796 0.368490
\(708\) 0 0
\(709\) −7.00000 −0.262891 −0.131445 0.991323i \(-0.541962\pi\)
−0.131445 + 0.991323i \(0.541962\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −2.44949 −0.0917341
\(714\) 0 0
\(715\) 0 0
\(716\) −58.7878 −2.19700
\(717\) 0 0
\(718\) −72.0000 −2.68702
\(719\) −36.7423 −1.37026 −0.685129 0.728422i \(-0.740254\pi\)
−0.685129 + 0.728422i \(0.740254\pi\)
\(720\) 0 0
\(721\) −14.0000 −0.521387
\(722\) 44.0908 1.64089
\(723\) 0 0
\(724\) 32.0000 1.18927
\(725\) 0 0
\(726\) 0 0
\(727\) −14.0000 −0.519231 −0.259616 0.965712i \(-0.583596\pi\)
−0.259616 + 0.965712i \(0.583596\pi\)
\(728\) 9.79796 0.363137
\(729\) 0 0
\(730\) 0 0
\(731\) −80.8332 −2.98972
\(732\) 0 0
\(733\) −17.0000 −0.627909 −0.313955 0.949438i \(-0.601654\pi\)
−0.313955 + 0.949438i \(0.601654\pi\)
\(734\) 12.2474 0.452062
\(735\) 0 0
\(736\) 0 0
\(737\) 17.1464 0.631597
\(738\) 0 0
\(739\) −1.00000 −0.0367856 −0.0183928 0.999831i \(-0.505855\pi\)
−0.0183928 + 0.999831i \(0.505855\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −36.0000 −1.32160
\(743\) 31.8434 1.16822 0.584110 0.811675i \(-0.301444\pi\)
0.584110 + 0.811675i \(0.301444\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 85.7321 3.13888
\(747\) 0 0
\(748\) 72.0000 2.63258
\(749\) 29.3939 1.07403
\(750\) 0 0
\(751\) 26.0000 0.948753 0.474377 0.880322i \(-0.342673\pi\)
0.474377 + 0.880322i \(0.342673\pi\)
\(752\) −39.1918 −1.42918
\(753\) 0 0
\(754\) 12.0000 0.437014
\(755\) 0 0
\(756\) 0 0
\(757\) 7.00000 0.254419 0.127210 0.991876i \(-0.459398\pi\)
0.127210 + 0.991876i \(0.459398\pi\)
\(758\) −19.5959 −0.711756
\(759\) 0 0
\(760\) 0 0
\(761\) −2.44949 −0.0887939 −0.0443970 0.999014i \(-0.514137\pi\)
−0.0443970 + 0.999014i \(0.514137\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) 39.1918 1.41791
\(765\) 0 0
\(766\) 84.0000 3.03504
\(767\) −2.44949 −0.0884459
\(768\) 0 0
\(769\) −37.0000 −1.33425 −0.667127 0.744944i \(-0.732476\pi\)
−0.667127 + 0.744944i \(0.732476\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −44.0000 −1.58359
\(773\) 44.0908 1.58584 0.792918 0.609328i \(-0.208561\pi\)
0.792918 + 0.609328i \(0.208561\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −34.2929 −1.23104
\(777\) 0 0
\(778\) 66.0000 2.36621
\(779\) −4.89898 −0.175524
\(780\) 0 0
\(781\) 18.0000 0.644091
\(782\) −44.0908 −1.57668
\(783\) 0 0
\(784\) −12.0000 −0.428571
\(785\) 0 0
\(786\) 0 0
\(787\) 25.0000 0.891154 0.445577 0.895244i \(-0.352999\pi\)
0.445577 + 0.895244i \(0.352999\pi\)
\(788\) −58.7878 −2.09423
\(789\) 0 0
\(790\) 0 0
\(791\) −19.5959 −0.696751
\(792\) 0 0
\(793\) 5.00000 0.177555
\(794\) −2.44949 −0.0869291
\(795\) 0 0
\(796\) −4.00000 −0.141776
\(797\) −41.6413 −1.47501 −0.737506 0.675341i \(-0.763997\pi\)
−0.737506 + 0.675341i \(0.763997\pi\)
\(798\) 0 0
\(799\) −72.0000 −2.54718
\(800\) 0 0
\(801\) 0 0
\(802\) −84.0000 −2.96614
\(803\) −26.9444 −0.950847
\(804\) 0 0
\(805\) 0 0
\(806\) 2.44949 0.0862796
\(807\) 0 0
\(808\) 24.0000 0.844317
\(809\) 22.0454 0.775075 0.387538 0.921854i \(-0.373326\pi\)
0.387538 + 0.921854i \(0.373326\pi\)
\(810\) 0 0
\(811\) 35.0000 1.22902 0.614508 0.788911i \(-0.289355\pi\)
0.614508 + 0.788911i \(0.289355\pi\)
\(812\) 39.1918 1.37536
\(813\) 0 0
\(814\) 48.0000 1.68240
\(815\) 0 0
\(816\) 0 0
\(817\) 11.0000 0.384841
\(818\) 68.5857 2.39804
\(819\) 0 0
\(820\) 0 0
\(821\) 39.1918 1.36780 0.683902 0.729574i \(-0.260281\pi\)
0.683902 + 0.729574i \(0.260281\pi\)
\(822\) 0 0
\(823\) −35.0000 −1.22002 −0.610012 0.792392i \(-0.708835\pi\)
−0.610012 + 0.792392i \(0.708835\pi\)
\(824\) −34.2929 −1.19465
\(825\) 0 0
\(826\) −12.0000 −0.417533
\(827\) 22.0454 0.766594 0.383297 0.923625i \(-0.374789\pi\)
0.383297 + 0.923625i \(0.374789\pi\)
\(828\) 0 0
\(829\) −37.0000 −1.28506 −0.642532 0.766259i \(-0.722116\pi\)
−0.642532 + 0.766259i \(0.722116\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −8.00000 −0.277350
\(833\) −22.0454 −0.763828
\(834\) 0 0
\(835\) 0 0
\(836\) −9.79796 −0.338869
\(837\) 0 0
\(838\) −84.0000 −2.90173
\(839\) −4.89898 −0.169132 −0.0845658 0.996418i \(-0.526950\pi\)
−0.0845658 + 0.996418i \(0.526950\pi\)
\(840\) 0 0
\(841\) −5.00000 −0.172414
\(842\) −4.89898 −0.168830
\(843\) 0 0
\(844\) −4.00000 −0.137686
\(845\) 0 0
\(846\) 0 0
\(847\) 10.0000 0.343604
\(848\) −29.3939 −1.00939
\(849\) 0 0
\(850\) 0 0
\(851\) −19.5959 −0.671739
\(852\) 0 0
\(853\) 13.0000 0.445112 0.222556 0.974920i \(-0.428560\pi\)
0.222556 + 0.974920i \(0.428560\pi\)
\(854\) 24.4949 0.838198
\(855\) 0 0
\(856\) 72.0000 2.46091
\(857\) −24.4949 −0.836730 −0.418365 0.908279i \(-0.637397\pi\)
−0.418365 + 0.908279i \(0.637397\pi\)
\(858\) 0 0
\(859\) 26.0000 0.887109 0.443554 0.896248i \(-0.353717\pi\)
0.443554 + 0.896248i \(0.353717\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 18.0000 0.613082
\(863\) 7.34847 0.250145 0.125072 0.992148i \(-0.460084\pi\)
0.125072 + 0.992148i \(0.460084\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 41.6413 1.41503
\(867\) 0 0
\(868\) 8.00000 0.271538
\(869\) −17.1464 −0.581653
\(870\) 0 0
\(871\) 7.00000 0.237186
\(872\) 4.89898 0.165900
\(873\) 0 0
\(874\) 6.00000 0.202953
\(875\) 0 0
\(876\) 0 0
\(877\) −8.00000 −0.270141 −0.135070 0.990836i \(-0.543126\pi\)
−0.135070 + 0.990836i \(0.543126\pi\)
\(878\) −34.2929 −1.15733
\(879\) 0 0
\(880\) 0 0
\(881\) 36.7423 1.23788 0.618941 0.785438i \(-0.287562\pi\)
0.618941 + 0.785438i \(0.287562\pi\)
\(882\) 0 0
\(883\) −17.0000 −0.572096 −0.286048 0.958215i \(-0.592342\pi\)
−0.286048 + 0.958215i \(0.592342\pi\)
\(884\) 29.3939 0.988623
\(885\) 0 0
\(886\) −30.0000 −1.00787
\(887\) 17.1464 0.575721 0.287860 0.957672i \(-0.407056\pi\)
0.287860 + 0.957672i \(0.407056\pi\)
\(888\) 0 0
\(889\) −38.0000 −1.27448
\(890\) 0 0
\(891\) 0 0
\(892\) 28.0000 0.937509
\(893\) 9.79796 0.327876
\(894\) 0 0
\(895\) 0 0
\(896\) −39.1918 −1.30931
\(897\) 0 0
\(898\) −54.0000 −1.80200
\(899\) 4.89898 0.163390
\(900\) 0 0
\(901\) −54.0000 −1.79900
\(902\) −29.3939 −0.978709
\(903\) 0 0
\(904\) −48.0000 −1.59646
\(905\) 0 0
\(906\) 0 0
\(907\) 7.00000 0.232431 0.116216 0.993224i \(-0.462924\pi\)
0.116216 + 0.993224i \(0.462924\pi\)
\(908\) −39.1918 −1.30063
\(909\) 0 0
\(910\) 0 0
\(911\) −12.2474 −0.405776 −0.202888 0.979202i \(-0.565033\pi\)
−0.202888 + 0.979202i \(0.565033\pi\)
\(912\) 0 0
\(913\) 30.0000 0.992855
\(914\) 71.0352 2.34964
\(915\) 0 0
\(916\) −4.00000 −0.132164
\(917\) −24.4949 −0.808893
\(918\) 0 0
\(919\) 20.0000 0.659739 0.329870 0.944027i \(-0.392995\pi\)
0.329870 + 0.944027i \(0.392995\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 66.0000 2.17359
\(923\) 7.34847 0.241878
\(924\) 0 0
\(925\) 0 0
\(926\) −46.5403 −1.52941
\(927\) 0 0
\(928\) 0 0
\(929\) −26.9444 −0.884017 −0.442008 0.897011i \(-0.645734\pi\)
−0.442008 + 0.897011i \(0.645734\pi\)
\(930\) 0 0
\(931\) 3.00000 0.0983210
\(932\) 29.3939 0.962828
\(933\) 0 0
\(934\) 36.0000 1.17796
\(935\) 0 0
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) 34.2929 1.11970
\(939\) 0 0
\(940\) 0 0
\(941\) −9.79796 −0.319404 −0.159702 0.987165i \(-0.551053\pi\)
−0.159702 + 0.987165i \(0.551053\pi\)
\(942\) 0 0
\(943\) 12.0000 0.390774
\(944\) −9.79796 −0.318896
\(945\) 0 0
\(946\) 66.0000 2.14585
\(947\) −24.4949 −0.795977 −0.397989 0.917390i \(-0.630292\pi\)
−0.397989 + 0.917390i \(0.630292\pi\)
\(948\) 0 0
\(949\) −11.0000 −0.357075
\(950\) 0 0
\(951\) 0 0
\(952\) 72.0000 2.33353
\(953\) −29.3939 −0.952161 −0.476081 0.879402i \(-0.657943\pi\)
−0.476081 + 0.879402i \(0.657943\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −9.79796 −0.316889
\(957\) 0 0
\(958\) 66.0000 2.13236
\(959\) 19.5959 0.632785
\(960\) 0 0
\(961\) −30.0000 −0.967742
\(962\) 19.5959 0.631798
\(963\) 0 0
\(964\) −64.0000 −2.06130
\(965\) 0 0
\(966\) 0 0
\(967\) 7.00000 0.225105 0.112552 0.993646i \(-0.464097\pi\)
0.112552 + 0.993646i \(0.464097\pi\)
\(968\) 24.4949 0.787296
\(969\) 0 0
\(970\) 0 0
\(971\) 29.3939 0.943294 0.471647 0.881787i \(-0.343660\pi\)
0.471647 + 0.881787i \(0.343660\pi\)
\(972\) 0 0
\(973\) 20.0000 0.641171
\(974\) 85.7321 2.74703
\(975\) 0 0
\(976\) 20.0000 0.640184
\(977\) 24.4949 0.783661 0.391831 0.920037i \(-0.371842\pi\)
0.391831 + 0.920037i \(0.371842\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 96.0000 3.06348
\(983\) 4.89898 0.156253 0.0781266 0.996943i \(-0.475106\pi\)
0.0781266 + 0.996943i \(0.475106\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 88.1816 2.80828
\(987\) 0 0
\(988\) −4.00000 −0.127257
\(989\) −26.9444 −0.856782
\(990\) 0 0
\(991\) −7.00000 −0.222362 −0.111181 0.993800i \(-0.535463\pi\)
−0.111181 + 0.993800i \(0.535463\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 36.0000 1.14185
\(995\) 0 0
\(996\) 0 0
\(997\) −50.0000 −1.58352 −0.791758 0.610835i \(-0.790834\pi\)
−0.791758 + 0.610835i \(0.790834\pi\)
\(998\) −4.89898 −0.155074
\(999\) 0 0
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 6075.2.a.bn.1.1 2
3.2 odd 2 inner 6075.2.a.bn.1.2 2
5.4 even 2 243.2.a.d.1.2 yes 2
15.14 odd 2 243.2.a.d.1.1 2
20.19 odd 2 3888.2.a.z.1.1 2
45.4 even 6 243.2.c.c.163.1 4
45.14 odd 6 243.2.c.c.163.2 4
45.29 odd 6 243.2.c.c.82.2 4
45.34 even 6 243.2.c.c.82.1 4
60.59 even 2 3888.2.a.z.1.2 2
135.4 even 18 729.2.e.p.406.2 12
135.14 odd 18 729.2.e.p.163.2 12
135.29 odd 18 729.2.e.p.568.2 12
135.34 even 18 729.2.e.p.325.2 12
135.49 even 18 729.2.e.p.649.2 12
135.59 odd 18 729.2.e.p.649.1 12
135.74 odd 18 729.2.e.p.325.1 12
135.79 even 18 729.2.e.p.568.1 12
135.94 even 18 729.2.e.p.163.1 12
135.104 odd 18 729.2.e.p.406.1 12
135.119 odd 18 729.2.e.p.82.1 12
135.124 even 18 729.2.e.p.82.2 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
243.2.a.d.1.1 2 15.14 odd 2
243.2.a.d.1.2 yes 2 5.4 even 2
243.2.c.c.82.1 4 45.34 even 6
243.2.c.c.82.2 4 45.29 odd 6
243.2.c.c.163.1 4 45.4 even 6
243.2.c.c.163.2 4 45.14 odd 6
729.2.e.p.82.1 12 135.119 odd 18
729.2.e.p.82.2 12 135.124 even 18
729.2.e.p.163.1 12 135.94 even 18
729.2.e.p.163.2 12 135.14 odd 18
729.2.e.p.325.1 12 135.74 odd 18
729.2.e.p.325.2 12 135.34 even 18
729.2.e.p.406.1 12 135.104 odd 18
729.2.e.p.406.2 12 135.4 even 18
729.2.e.p.568.1 12 135.79 even 18
729.2.e.p.568.2 12 135.29 odd 18
729.2.e.p.649.1 12 135.59 odd 18
729.2.e.p.649.2 12 135.49 even 18
3888.2.a.z.1.1 2 20.19 odd 2
3888.2.a.z.1.2 2 60.59 even 2
6075.2.a.bn.1.1 2 1.1 even 1 trivial
6075.2.a.bn.1.2 2 3.2 odd 2 inner