Properties

Label 4842.2.a.n.1.4
Level $4842$
Weight $2$
Character 4842.1
Self dual yes
Analytic conductor $38.664$
Analytic rank $1$
Dimension $7$
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] = [4842,2,Mod(1,4842)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4842, 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("4842.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4842 = 2 \cdot 3^{2} \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4842.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: \(38.6635646587\)
Analytic rank: \(1\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 538)
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(1.14366\) of defining polynomial
Character \(\chi\) \(=\) 4842.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.725043 q^{5} +1.29978 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -0.725043 q^{5} +1.29978 q^{7} -1.00000 q^{8} +0.725043 q^{10} +3.71687 q^{11} +2.30856 q^{13} -1.29978 q^{14} +1.00000 q^{16} -6.32982 q^{17} +0.0212539 q^{19} -0.725043 q^{20} -3.71687 q^{22} +1.30511 q^{23} -4.47431 q^{25} -2.30856 q^{26} +1.29978 q^{28} +2.06303 q^{29} -5.82316 q^{31} -1.00000 q^{32} +6.32982 q^{34} -0.942394 q^{35} -7.25272 q^{37} -0.0212539 q^{38} +0.725043 q^{40} -8.77454 q^{41} +4.52327 q^{43} +3.71687 q^{44} -1.30511 q^{46} -3.85502 q^{47} -5.31058 q^{49} +4.47431 q^{50} +2.30856 q^{52} +6.86923 q^{53} -2.69489 q^{55} -1.29978 q^{56} -2.06303 q^{58} -7.84518 q^{59} +3.84958 q^{61} +5.82316 q^{62} +1.00000 q^{64} -1.67381 q^{65} -9.31541 q^{67} -6.32982 q^{68} +0.942394 q^{70} +0.643097 q^{71} -2.97920 q^{73} +7.25272 q^{74} +0.0212539 q^{76} +4.83110 q^{77} +0.418245 q^{79} -0.725043 q^{80} +8.77454 q^{82} +15.6322 q^{83} +4.58939 q^{85} -4.52327 q^{86} -3.71687 q^{88} -10.3879 q^{89} +3.00062 q^{91} +1.30511 q^{92} +3.85502 q^{94} -0.0154100 q^{95} +5.07659 q^{97} +5.31058 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q - 7 q^{2} + 7 q^{4} - 7 q^{5} + 6 q^{7} - 7 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 7 q - 7 q^{2} + 7 q^{4} - 7 q^{5} + 6 q^{7} - 7 q^{8} + 7 q^{10} + 3 q^{11} - 9 q^{13} - 6 q^{14} + 7 q^{16} - 8 q^{17} - 11 q^{19} - 7 q^{20} - 3 q^{22} - 12 q^{23} + 22 q^{25} + 9 q^{26} + 6 q^{28} + 5 q^{29} + 14 q^{31} - 7 q^{32} + 8 q^{34} + 4 q^{35} + 13 q^{37} + 11 q^{38} + 7 q^{40} - 12 q^{41} - 11 q^{43} + 3 q^{44} + 12 q^{46} - 2 q^{47} + 15 q^{49} - 22 q^{50} - 9 q^{52} - 19 q^{53} - 40 q^{55} - 6 q^{56} - 5 q^{58} + 9 q^{59} - 3 q^{61} - 14 q^{62} + 7 q^{64} + 10 q^{65} - 33 q^{67} - 8 q^{68} - 4 q^{70} - 28 q^{71} - 14 q^{73} - 13 q^{74} - 11 q^{76} - 10 q^{77} + 2 q^{79} - 7 q^{80} + 12 q^{82} + 7 q^{83} - 16 q^{85} + 11 q^{86} - 3 q^{88} - 18 q^{89} - 26 q^{91} - 12 q^{92} + 2 q^{94} + 34 q^{95} - 4 q^{97} - 15 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −0.725043 −0.324249 −0.162125 0.986770i \(-0.551835\pi\)
−0.162125 + 0.986770i \(0.551835\pi\)
\(6\) 0 0
\(7\) 1.29978 0.491269 0.245635 0.969362i \(-0.421004\pi\)
0.245635 + 0.969362i \(0.421004\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 0.725043 0.229279
\(11\) 3.71687 1.12068 0.560340 0.828263i \(-0.310671\pi\)
0.560340 + 0.828263i \(0.310671\pi\)
\(12\) 0 0
\(13\) 2.30856 0.640280 0.320140 0.947370i \(-0.396270\pi\)
0.320140 + 0.947370i \(0.396270\pi\)
\(14\) −1.29978 −0.347380
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.32982 −1.53521 −0.767603 0.640925i \(-0.778551\pi\)
−0.767603 + 0.640925i \(0.778551\pi\)
\(18\) 0 0
\(19\) 0.0212539 0.00487598 0.00243799 0.999997i \(-0.499224\pi\)
0.00243799 + 0.999997i \(0.499224\pi\)
\(20\) −0.725043 −0.162125
\(21\) 0 0
\(22\) −3.71687 −0.792440
\(23\) 1.30511 0.272134 0.136067 0.990700i \(-0.456554\pi\)
0.136067 + 0.990700i \(0.456554\pi\)
\(24\) 0 0
\(25\) −4.47431 −0.894863
\(26\) −2.30856 −0.452747
\(27\) 0 0
\(28\) 1.29978 0.245635
\(29\) 2.06303 0.383095 0.191548 0.981483i \(-0.438649\pi\)
0.191548 + 0.981483i \(0.438649\pi\)
\(30\) 0 0
\(31\) −5.82316 −1.04587 −0.522935 0.852372i \(-0.675163\pi\)
−0.522935 + 0.852372i \(0.675163\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 6.32982 1.08555
\(35\) −0.942394 −0.159294
\(36\) 0 0
\(37\) −7.25272 −1.19234 −0.596170 0.802858i \(-0.703312\pi\)
−0.596170 + 0.802858i \(0.703312\pi\)
\(38\) −0.0212539 −0.00344784
\(39\) 0 0
\(40\) 0.725043 0.114639
\(41\) −8.77454 −1.37035 −0.685176 0.728377i \(-0.740275\pi\)
−0.685176 + 0.728377i \(0.740275\pi\)
\(42\) 0 0
\(43\) 4.52327 0.689792 0.344896 0.938641i \(-0.387914\pi\)
0.344896 + 0.938641i \(0.387914\pi\)
\(44\) 3.71687 0.560340
\(45\) 0 0
\(46\) −1.30511 −0.192428
\(47\) −3.85502 −0.562313 −0.281156 0.959662i \(-0.590718\pi\)
−0.281156 + 0.959662i \(0.590718\pi\)
\(48\) 0 0
\(49\) −5.31058 −0.758654
\(50\) 4.47431 0.632763
\(51\) 0 0
\(52\) 2.30856 0.320140
\(53\) 6.86923 0.943562 0.471781 0.881716i \(-0.343611\pi\)
0.471781 + 0.881716i \(0.343611\pi\)
\(54\) 0 0
\(55\) −2.69489 −0.363379
\(56\) −1.29978 −0.173690
\(57\) 0 0
\(58\) −2.06303 −0.270889
\(59\) −7.84518 −1.02136 −0.510678 0.859772i \(-0.670605\pi\)
−0.510678 + 0.859772i \(0.670605\pi\)
\(60\) 0 0
\(61\) 3.84958 0.492888 0.246444 0.969157i \(-0.420738\pi\)
0.246444 + 0.969157i \(0.420738\pi\)
\(62\) 5.82316 0.739542
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) −1.67381 −0.207610
\(66\) 0 0
\(67\) −9.31541 −1.13806 −0.569029 0.822317i \(-0.692681\pi\)
−0.569029 + 0.822317i \(0.692681\pi\)
\(68\) −6.32982 −0.767603
\(69\) 0 0
\(70\) 0.942394 0.112638
\(71\) 0.643097 0.0763215 0.0381608 0.999272i \(-0.487850\pi\)
0.0381608 + 0.999272i \(0.487850\pi\)
\(72\) 0 0
\(73\) −2.97920 −0.348689 −0.174345 0.984685i \(-0.555781\pi\)
−0.174345 + 0.984685i \(0.555781\pi\)
\(74\) 7.25272 0.843112
\(75\) 0 0
\(76\) 0.0212539 0.00243799
\(77\) 4.83110 0.550555
\(78\) 0 0
\(79\) 0.418245 0.0470563 0.0235281 0.999723i \(-0.492510\pi\)
0.0235281 + 0.999723i \(0.492510\pi\)
\(80\) −0.725043 −0.0810623
\(81\) 0 0
\(82\) 8.77454 0.968985
\(83\) 15.6322 1.71585 0.857927 0.513771i \(-0.171752\pi\)
0.857927 + 0.513771i \(0.171752\pi\)
\(84\) 0 0
\(85\) 4.58939 0.497789
\(86\) −4.52327 −0.487757
\(87\) 0 0
\(88\) −3.71687 −0.396220
\(89\) −10.3879 −1.10111 −0.550556 0.834798i \(-0.685584\pi\)
−0.550556 + 0.834798i \(0.685584\pi\)
\(90\) 0 0
\(91\) 3.00062 0.314550
\(92\) 1.30511 0.136067
\(93\) 0 0
\(94\) 3.85502 0.397615
\(95\) −0.0154100 −0.00158103
\(96\) 0 0
\(97\) 5.07659 0.515450 0.257725 0.966218i \(-0.417027\pi\)
0.257725 + 0.966218i \(0.417027\pi\)
\(98\) 5.31058 0.536450
\(99\) 0 0
\(100\) −4.47431 −0.447431
\(101\) −12.5321 −1.24699 −0.623495 0.781827i \(-0.714288\pi\)
−0.623495 + 0.781827i \(0.714288\pi\)
\(102\) 0 0
\(103\) 5.43374 0.535403 0.267701 0.963502i \(-0.413736\pi\)
0.267701 + 0.963502i \(0.413736\pi\)
\(104\) −2.30856 −0.226373
\(105\) 0 0
\(106\) −6.86923 −0.667199
\(107\) 3.45500 0.334007 0.167004 0.985956i \(-0.446591\pi\)
0.167004 + 0.985956i \(0.446591\pi\)
\(108\) 0 0
\(109\) −6.80801 −0.652089 −0.326044 0.945354i \(-0.605716\pi\)
−0.326044 + 0.945354i \(0.605716\pi\)
\(110\) 2.69489 0.256948
\(111\) 0 0
\(112\) 1.29978 0.122817
\(113\) −0.407582 −0.0383421 −0.0191710 0.999816i \(-0.506103\pi\)
−0.0191710 + 0.999816i \(0.506103\pi\)
\(114\) 0 0
\(115\) −0.946259 −0.0882391
\(116\) 2.06303 0.191548
\(117\) 0 0
\(118\) 7.84518 0.722207
\(119\) −8.22735 −0.754200
\(120\) 0 0
\(121\) 2.81514 0.255922
\(122\) −3.84958 −0.348524
\(123\) 0 0
\(124\) −5.82316 −0.522935
\(125\) 6.86928 0.614407
\(126\) 0 0
\(127\) −9.52589 −0.845286 −0.422643 0.906296i \(-0.638898\pi\)
−0.422643 + 0.906296i \(0.638898\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) 1.67381 0.146803
\(131\) 21.9325 1.91625 0.958127 0.286342i \(-0.0924394\pi\)
0.958127 + 0.286342i \(0.0924394\pi\)
\(132\) 0 0
\(133\) 0.0276253 0.00239542
\(134\) 9.31541 0.804729
\(135\) 0 0
\(136\) 6.32982 0.542777
\(137\) 2.48849 0.212606 0.106303 0.994334i \(-0.466099\pi\)
0.106303 + 0.994334i \(0.466099\pi\)
\(138\) 0 0
\(139\) 21.6244 1.83416 0.917078 0.398708i \(-0.130541\pi\)
0.917078 + 0.398708i \(0.130541\pi\)
\(140\) −0.942394 −0.0796468
\(141\) 0 0
\(142\) −0.643097 −0.0539675
\(143\) 8.58064 0.717549
\(144\) 0 0
\(145\) −1.49579 −0.124218
\(146\) 2.97920 0.246561
\(147\) 0 0
\(148\) −7.25272 −0.596170
\(149\) −20.9106 −1.71306 −0.856532 0.516093i \(-0.827386\pi\)
−0.856532 + 0.516093i \(0.827386\pi\)
\(150\) 0 0
\(151\) −2.07209 −0.168624 −0.0843120 0.996439i \(-0.526869\pi\)
−0.0843120 + 0.996439i \(0.526869\pi\)
\(152\) −0.0212539 −0.00172392
\(153\) 0 0
\(154\) −4.83110 −0.389301
\(155\) 4.22204 0.339123
\(156\) 0 0
\(157\) −2.60975 −0.208281 −0.104140 0.994563i \(-0.533209\pi\)
−0.104140 + 0.994563i \(0.533209\pi\)
\(158\) −0.418245 −0.0332738
\(159\) 0 0
\(160\) 0.725043 0.0573197
\(161\) 1.69635 0.133691
\(162\) 0 0
\(163\) −6.63147 −0.519417 −0.259708 0.965687i \(-0.583626\pi\)
−0.259708 + 0.965687i \(0.583626\pi\)
\(164\) −8.77454 −0.685176
\(165\) 0 0
\(166\) −15.6322 −1.21329
\(167\) −22.6369 −1.75170 −0.875850 0.482584i \(-0.839698\pi\)
−0.875850 + 0.482584i \(0.839698\pi\)
\(168\) 0 0
\(169\) −7.67053 −0.590041
\(170\) −4.58939 −0.351990
\(171\) 0 0
\(172\) 4.52327 0.344896
\(173\) −12.2456 −0.931018 −0.465509 0.885043i \(-0.654129\pi\)
−0.465509 + 0.885043i \(0.654129\pi\)
\(174\) 0 0
\(175\) −5.81561 −0.439619
\(176\) 3.71687 0.280170
\(177\) 0 0
\(178\) 10.3879 0.778603
\(179\) 18.0181 1.34673 0.673367 0.739308i \(-0.264847\pi\)
0.673367 + 0.739308i \(0.264847\pi\)
\(180\) 0 0
\(181\) 7.77690 0.578052 0.289026 0.957321i \(-0.406669\pi\)
0.289026 + 0.957321i \(0.406669\pi\)
\(182\) −3.00062 −0.222421
\(183\) 0 0
\(184\) −1.30511 −0.0962138
\(185\) 5.25854 0.386615
\(186\) 0 0
\(187\) −23.5271 −1.72047
\(188\) −3.85502 −0.281156
\(189\) 0 0
\(190\) 0.0154100 0.00111796
\(191\) −0.312243 −0.0225931 −0.0112965 0.999936i \(-0.503596\pi\)
−0.0112965 + 0.999936i \(0.503596\pi\)
\(192\) 0 0
\(193\) −0.336949 −0.0242541 −0.0121271 0.999926i \(-0.503860\pi\)
−0.0121271 + 0.999926i \(0.503860\pi\)
\(194\) −5.07659 −0.364478
\(195\) 0 0
\(196\) −5.31058 −0.379327
\(197\) −26.8275 −1.91138 −0.955691 0.294370i \(-0.904890\pi\)
−0.955691 + 0.294370i \(0.904890\pi\)
\(198\) 0 0
\(199\) 4.03560 0.286076 0.143038 0.989717i \(-0.454313\pi\)
0.143038 + 0.989717i \(0.454313\pi\)
\(200\) 4.47431 0.316382
\(201\) 0 0
\(202\) 12.5321 0.881755
\(203\) 2.68148 0.188203
\(204\) 0 0
\(205\) 6.36192 0.444335
\(206\) −5.43374 −0.378587
\(207\) 0 0
\(208\) 2.30856 0.160070
\(209\) 0.0789980 0.00546440
\(210\) 0 0
\(211\) 9.22688 0.635205 0.317602 0.948224i \(-0.397122\pi\)
0.317602 + 0.948224i \(0.397122\pi\)
\(212\) 6.86923 0.471781
\(213\) 0 0
\(214\) −3.45500 −0.236179
\(215\) −3.27956 −0.223664
\(216\) 0 0
\(217\) −7.56881 −0.513804
\(218\) 6.80801 0.461096
\(219\) 0 0
\(220\) −2.69489 −0.181690
\(221\) −14.6128 −0.982963
\(222\) 0 0
\(223\) −9.44842 −0.632713 −0.316357 0.948640i \(-0.602460\pi\)
−0.316357 + 0.948640i \(0.602460\pi\)
\(224\) −1.29978 −0.0868450
\(225\) 0 0
\(226\) 0.407582 0.0271119
\(227\) −15.2061 −1.00926 −0.504632 0.863335i \(-0.668372\pi\)
−0.504632 + 0.863335i \(0.668372\pi\)
\(228\) 0 0
\(229\) 28.6411 1.89266 0.946329 0.323205i \(-0.104760\pi\)
0.946329 + 0.323205i \(0.104760\pi\)
\(230\) 0.946259 0.0623945
\(231\) 0 0
\(232\) −2.06303 −0.135445
\(233\) −3.14875 −0.206281 −0.103141 0.994667i \(-0.532889\pi\)
−0.103141 + 0.994667i \(0.532889\pi\)
\(234\) 0 0
\(235\) 2.79506 0.182329
\(236\) −7.84518 −0.510678
\(237\) 0 0
\(238\) 8.22735 0.533300
\(239\) −12.5059 −0.808940 −0.404470 0.914551i \(-0.632544\pi\)
−0.404470 + 0.914551i \(0.632544\pi\)
\(240\) 0 0
\(241\) 3.73027 0.240287 0.120144 0.992757i \(-0.461664\pi\)
0.120144 + 0.992757i \(0.461664\pi\)
\(242\) −2.81514 −0.180964
\(243\) 0 0
\(244\) 3.84958 0.246444
\(245\) 3.85040 0.245993
\(246\) 0 0
\(247\) 0.0490660 0.00312199
\(248\) 5.82316 0.369771
\(249\) 0 0
\(250\) −6.86928 −0.434452
\(251\) 26.2795 1.65875 0.829374 0.558694i \(-0.188697\pi\)
0.829374 + 0.558694i \(0.188697\pi\)
\(252\) 0 0
\(253\) 4.85092 0.304975
\(254\) 9.52589 0.597708
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −28.2237 −1.76055 −0.880273 0.474468i \(-0.842641\pi\)
−0.880273 + 0.474468i \(0.842641\pi\)
\(258\) 0 0
\(259\) −9.42692 −0.585760
\(260\) −1.67381 −0.103805
\(261\) 0 0
\(262\) −21.9325 −1.35500
\(263\) −25.3865 −1.56540 −0.782701 0.622398i \(-0.786158\pi\)
−0.782701 + 0.622398i \(0.786158\pi\)
\(264\) 0 0
\(265\) −4.98049 −0.305949
\(266\) −0.0276253 −0.00169382
\(267\) 0 0
\(268\) −9.31541 −0.569029
\(269\) 1.00000 0.0609711
\(270\) 0 0
\(271\) −17.9078 −1.08782 −0.543911 0.839143i \(-0.683057\pi\)
−0.543911 + 0.839143i \(0.683057\pi\)
\(272\) −6.32982 −0.383802
\(273\) 0 0
\(274\) −2.48849 −0.150335
\(275\) −16.6304 −1.00285
\(276\) 0 0
\(277\) −6.51988 −0.391741 −0.195871 0.980630i \(-0.562753\pi\)
−0.195871 + 0.980630i \(0.562753\pi\)
\(278\) −21.6244 −1.29694
\(279\) 0 0
\(280\) 0.942394 0.0563188
\(281\) 2.61394 0.155934 0.0779672 0.996956i \(-0.475157\pi\)
0.0779672 + 0.996956i \(0.475157\pi\)
\(282\) 0 0
\(283\) −9.65338 −0.573834 −0.286917 0.957955i \(-0.592630\pi\)
−0.286917 + 0.957955i \(0.592630\pi\)
\(284\) 0.643097 0.0381608
\(285\) 0 0
\(286\) −8.58064 −0.507384
\(287\) −11.4049 −0.673212
\(288\) 0 0
\(289\) 23.0666 1.35686
\(290\) 1.49579 0.0878356
\(291\) 0 0
\(292\) −2.97920 −0.174345
\(293\) −2.04724 −0.119601 −0.0598005 0.998210i \(-0.519046\pi\)
−0.0598005 + 0.998210i \(0.519046\pi\)
\(294\) 0 0
\(295\) 5.68809 0.331173
\(296\) 7.25272 0.421556
\(297\) 0 0
\(298\) 20.9106 1.21132
\(299\) 3.01293 0.174242
\(300\) 0 0
\(301\) 5.87924 0.338874
\(302\) 2.07209 0.119235
\(303\) 0 0
\(304\) 0.0212539 0.00121899
\(305\) −2.79111 −0.159818
\(306\) 0 0
\(307\) −18.6159 −1.06246 −0.531231 0.847227i \(-0.678271\pi\)
−0.531231 + 0.847227i \(0.678271\pi\)
\(308\) 4.83110 0.275278
\(309\) 0 0
\(310\) −4.22204 −0.239796
\(311\) 9.88503 0.560529 0.280264 0.959923i \(-0.409578\pi\)
0.280264 + 0.959923i \(0.409578\pi\)
\(312\) 0 0
\(313\) 17.4438 0.985983 0.492991 0.870034i \(-0.335903\pi\)
0.492991 + 0.870034i \(0.335903\pi\)
\(314\) 2.60975 0.147277
\(315\) 0 0
\(316\) 0.418245 0.0235281
\(317\) −2.92828 −0.164468 −0.0822342 0.996613i \(-0.526206\pi\)
−0.0822342 + 0.996613i \(0.526206\pi\)
\(318\) 0 0
\(319\) 7.66802 0.429327
\(320\) −0.725043 −0.0405311
\(321\) 0 0
\(322\) −1.69635 −0.0945338
\(323\) −0.134533 −0.00748563
\(324\) 0 0
\(325\) −10.3292 −0.572963
\(326\) 6.63147 0.367283
\(327\) 0 0
\(328\) 8.77454 0.484493
\(329\) −5.01067 −0.276247
\(330\) 0 0
\(331\) 5.44140 0.299087 0.149543 0.988755i \(-0.452220\pi\)
0.149543 + 0.988755i \(0.452220\pi\)
\(332\) 15.6322 0.857927
\(333\) 0 0
\(334\) 22.6369 1.23864
\(335\) 6.75407 0.369014
\(336\) 0 0
\(337\) −3.73840 −0.203644 −0.101822 0.994803i \(-0.532467\pi\)
−0.101822 + 0.994803i \(0.532467\pi\)
\(338\) 7.67053 0.417222
\(339\) 0 0
\(340\) 4.58939 0.248895
\(341\) −21.6439 −1.17209
\(342\) 0 0
\(343\) −16.0010 −0.863973
\(344\) −4.52327 −0.243878
\(345\) 0 0
\(346\) 12.2456 0.658329
\(347\) −10.8089 −0.580252 −0.290126 0.956988i \(-0.593697\pi\)
−0.290126 + 0.956988i \(0.593697\pi\)
\(348\) 0 0
\(349\) 19.8978 1.06510 0.532552 0.846397i \(-0.321233\pi\)
0.532552 + 0.846397i \(0.321233\pi\)
\(350\) 5.81561 0.310857
\(351\) 0 0
\(352\) −3.71687 −0.198110
\(353\) 11.8505 0.630736 0.315368 0.948969i \(-0.397872\pi\)
0.315368 + 0.948969i \(0.397872\pi\)
\(354\) 0 0
\(355\) −0.466273 −0.0247472
\(356\) −10.3879 −0.550556
\(357\) 0 0
\(358\) −18.0181 −0.952285
\(359\) −28.6632 −1.51279 −0.756393 0.654118i \(-0.773040\pi\)
−0.756393 + 0.654118i \(0.773040\pi\)
\(360\) 0 0
\(361\) −18.9995 −0.999976
\(362\) −7.77690 −0.408745
\(363\) 0 0
\(364\) 3.00062 0.157275
\(365\) 2.16005 0.113062
\(366\) 0 0
\(367\) 13.1083 0.684246 0.342123 0.939655i \(-0.388854\pi\)
0.342123 + 0.939655i \(0.388854\pi\)
\(368\) 1.30511 0.0680335
\(369\) 0 0
\(370\) −5.25854 −0.273378
\(371\) 8.92847 0.463543
\(372\) 0 0
\(373\) −17.2738 −0.894406 −0.447203 0.894432i \(-0.647580\pi\)
−0.447203 + 0.894432i \(0.647580\pi\)
\(374\) 23.5271 1.21656
\(375\) 0 0
\(376\) 3.85502 0.198808
\(377\) 4.76264 0.245288
\(378\) 0 0
\(379\) 6.30803 0.324022 0.162011 0.986789i \(-0.448202\pi\)
0.162011 + 0.986789i \(0.448202\pi\)
\(380\) −0.0154100 −0.000790515 0
\(381\) 0 0
\(382\) 0.312243 0.0159757
\(383\) 28.2196 1.44196 0.720978 0.692958i \(-0.243693\pi\)
0.720978 + 0.692958i \(0.243693\pi\)
\(384\) 0 0
\(385\) −3.50276 −0.178517
\(386\) 0.336949 0.0171502
\(387\) 0 0
\(388\) 5.07659 0.257725
\(389\) 3.15029 0.159726 0.0798629 0.996806i \(-0.474552\pi\)
0.0798629 + 0.996806i \(0.474552\pi\)
\(390\) 0 0
\(391\) −8.26110 −0.417782
\(392\) 5.31058 0.268225
\(393\) 0 0
\(394\) 26.8275 1.35155
\(395\) −0.303246 −0.0152579
\(396\) 0 0
\(397\) 31.2926 1.57053 0.785265 0.619159i \(-0.212527\pi\)
0.785265 + 0.619159i \(0.212527\pi\)
\(398\) −4.03560 −0.202286
\(399\) 0 0
\(400\) −4.47431 −0.223716
\(401\) −0.744072 −0.0371572 −0.0185786 0.999827i \(-0.505914\pi\)
−0.0185786 + 0.999827i \(0.505914\pi\)
\(402\) 0 0
\(403\) −13.4431 −0.669651
\(404\) −12.5321 −0.623495
\(405\) 0 0
\(406\) −2.68148 −0.133080
\(407\) −26.9574 −1.33623
\(408\) 0 0
\(409\) 3.60242 0.178128 0.0890641 0.996026i \(-0.471612\pi\)
0.0890641 + 0.996026i \(0.471612\pi\)
\(410\) −6.36192 −0.314193
\(411\) 0 0
\(412\) 5.43374 0.267701
\(413\) −10.1970 −0.501760
\(414\) 0 0
\(415\) −11.3340 −0.556364
\(416\) −2.30856 −0.113187
\(417\) 0 0
\(418\) −0.0789980 −0.00386392
\(419\) 17.1323 0.836966 0.418483 0.908225i \(-0.362562\pi\)
0.418483 + 0.908225i \(0.362562\pi\)
\(420\) 0 0
\(421\) 24.1651 1.17773 0.588866 0.808230i \(-0.299574\pi\)
0.588866 + 0.808230i \(0.299574\pi\)
\(422\) −9.22688 −0.449158
\(423\) 0 0
\(424\) −6.86923 −0.333600
\(425\) 28.3216 1.37380
\(426\) 0 0
\(427\) 5.00359 0.242141
\(428\) 3.45500 0.167004
\(429\) 0 0
\(430\) 3.27956 0.158155
\(431\) −16.8823 −0.813193 −0.406596 0.913608i \(-0.633284\pi\)
−0.406596 + 0.913608i \(0.633284\pi\)
\(432\) 0 0
\(433\) −13.0644 −0.627834 −0.313917 0.949450i \(-0.601641\pi\)
−0.313917 + 0.949450i \(0.601641\pi\)
\(434\) 7.56881 0.363314
\(435\) 0 0
\(436\) −6.80801 −0.326044
\(437\) 0.0277386 0.00132692
\(438\) 0 0
\(439\) 3.91599 0.186900 0.0934499 0.995624i \(-0.470211\pi\)
0.0934499 + 0.995624i \(0.470211\pi\)
\(440\) 2.69489 0.128474
\(441\) 0 0
\(442\) 14.6128 0.695060
\(443\) −25.2673 −1.20049 −0.600243 0.799818i \(-0.704929\pi\)
−0.600243 + 0.799818i \(0.704929\pi\)
\(444\) 0 0
\(445\) 7.53165 0.357034
\(446\) 9.44842 0.447396
\(447\) 0 0
\(448\) 1.29978 0.0614087
\(449\) −27.4706 −1.29642 −0.648210 0.761462i \(-0.724482\pi\)
−0.648210 + 0.761462i \(0.724482\pi\)
\(450\) 0 0
\(451\) −32.6138 −1.53573
\(452\) −0.407582 −0.0191710
\(453\) 0 0
\(454\) 15.2061 0.713657
\(455\) −2.17558 −0.101993
\(456\) 0 0
\(457\) −14.5345 −0.679896 −0.339948 0.940444i \(-0.610409\pi\)
−0.339948 + 0.940444i \(0.610409\pi\)
\(458\) −28.6411 −1.33831
\(459\) 0 0
\(460\) −0.946259 −0.0441196
\(461\) 10.4630 0.487312 0.243656 0.969862i \(-0.421653\pi\)
0.243656 + 0.969862i \(0.421653\pi\)
\(462\) 0 0
\(463\) −27.7519 −1.28974 −0.644871 0.764292i \(-0.723089\pi\)
−0.644871 + 0.764292i \(0.723089\pi\)
\(464\) 2.06303 0.0957738
\(465\) 0 0
\(466\) 3.14875 0.145863
\(467\) 12.3322 0.570665 0.285333 0.958429i \(-0.407896\pi\)
0.285333 + 0.958429i \(0.407896\pi\)
\(468\) 0 0
\(469\) −12.1080 −0.559093
\(470\) −2.79506 −0.128926
\(471\) 0 0
\(472\) 7.84518 0.361104
\(473\) 16.8124 0.773036
\(474\) 0 0
\(475\) −0.0950965 −0.00436333
\(476\) −8.22735 −0.377100
\(477\) 0 0
\(478\) 12.5059 0.572007
\(479\) 2.92366 0.133586 0.0667928 0.997767i \(-0.478723\pi\)
0.0667928 + 0.997767i \(0.478723\pi\)
\(480\) 0 0
\(481\) −16.7434 −0.763432
\(482\) −3.73027 −0.169909
\(483\) 0 0
\(484\) 2.81514 0.127961
\(485\) −3.68075 −0.167134
\(486\) 0 0
\(487\) 10.1739 0.461025 0.230513 0.973069i \(-0.425960\pi\)
0.230513 + 0.973069i \(0.425960\pi\)
\(488\) −3.84958 −0.174262
\(489\) 0 0
\(490\) −3.85040 −0.173943
\(491\) 16.8181 0.758989 0.379494 0.925194i \(-0.376098\pi\)
0.379494 + 0.925194i \(0.376098\pi\)
\(492\) 0 0
\(493\) −13.0586 −0.588130
\(494\) −0.0490660 −0.00220758
\(495\) 0 0
\(496\) −5.82316 −0.261468
\(497\) 0.835882 0.0374944
\(498\) 0 0
\(499\) −30.0514 −1.34529 −0.672643 0.739967i \(-0.734841\pi\)
−0.672643 + 0.739967i \(0.734841\pi\)
\(500\) 6.86928 0.307204
\(501\) 0 0
\(502\) −26.2795 −1.17291
\(503\) 6.83648 0.304823 0.152412 0.988317i \(-0.451296\pi\)
0.152412 + 0.988317i \(0.451296\pi\)
\(504\) 0 0
\(505\) 9.08630 0.404335
\(506\) −4.85092 −0.215650
\(507\) 0 0
\(508\) −9.52589 −0.422643
\(509\) 16.9547 0.751502 0.375751 0.926721i \(-0.377385\pi\)
0.375751 + 0.926721i \(0.377385\pi\)
\(510\) 0 0
\(511\) −3.87230 −0.171300
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 28.2237 1.24489
\(515\) −3.93970 −0.173604
\(516\) 0 0
\(517\) −14.3286 −0.630172
\(518\) 9.42692 0.414195
\(519\) 0 0
\(520\) 1.67381 0.0734013
\(521\) −37.1136 −1.62597 −0.812987 0.582282i \(-0.802160\pi\)
−0.812987 + 0.582282i \(0.802160\pi\)
\(522\) 0 0
\(523\) −9.80481 −0.428734 −0.214367 0.976753i \(-0.568769\pi\)
−0.214367 + 0.976753i \(0.568769\pi\)
\(524\) 21.9325 0.958127
\(525\) 0 0
\(526\) 25.3865 1.10691
\(527\) 36.8596 1.60563
\(528\) 0 0
\(529\) −21.2967 −0.925943
\(530\) 4.98049 0.216339
\(531\) 0 0
\(532\) 0.0276253 0.00119771
\(533\) −20.2566 −0.877410
\(534\) 0 0
\(535\) −2.50502 −0.108302
\(536\) 9.31541 0.402365
\(537\) 0 0
\(538\) −1.00000 −0.0431131
\(539\) −19.7388 −0.850208
\(540\) 0 0
\(541\) −15.4647 −0.664878 −0.332439 0.943125i \(-0.607871\pi\)
−0.332439 + 0.943125i \(0.607871\pi\)
\(542\) 17.9078 0.769206
\(543\) 0 0
\(544\) 6.32982 0.271389
\(545\) 4.93610 0.211439
\(546\) 0 0
\(547\) −31.5080 −1.34719 −0.673593 0.739103i \(-0.735250\pi\)
−0.673593 + 0.739103i \(0.735250\pi\)
\(548\) 2.48849 0.106303
\(549\) 0 0
\(550\) 16.6304 0.709125
\(551\) 0.0438474 0.00186796
\(552\) 0 0
\(553\) 0.543625 0.0231173
\(554\) 6.51988 0.277003
\(555\) 0 0
\(556\) 21.6244 0.917078
\(557\) 5.45649 0.231199 0.115599 0.993296i \(-0.463121\pi\)
0.115599 + 0.993296i \(0.463121\pi\)
\(558\) 0 0
\(559\) 10.4423 0.441660
\(560\) −0.942394 −0.0398234
\(561\) 0 0
\(562\) −2.61394 −0.110262
\(563\) 5.91106 0.249121 0.124561 0.992212i \(-0.460248\pi\)
0.124561 + 0.992212i \(0.460248\pi\)
\(564\) 0 0
\(565\) 0.295514 0.0124324
\(566\) 9.65338 0.405762
\(567\) 0 0
\(568\) −0.643097 −0.0269837
\(569\) −31.9474 −1.33930 −0.669652 0.742675i \(-0.733557\pi\)
−0.669652 + 0.742675i \(0.733557\pi\)
\(570\) 0 0
\(571\) −13.7554 −0.575647 −0.287824 0.957683i \(-0.592932\pi\)
−0.287824 + 0.957683i \(0.592932\pi\)
\(572\) 8.58064 0.358774
\(573\) 0 0
\(574\) 11.4049 0.476033
\(575\) −5.83946 −0.243522
\(576\) 0 0
\(577\) −2.93964 −0.122379 −0.0611893 0.998126i \(-0.519489\pi\)
−0.0611893 + 0.998126i \(0.519489\pi\)
\(578\) −23.0666 −0.959444
\(579\) 0 0
\(580\) −1.49579 −0.0621091
\(581\) 20.3183 0.842947
\(582\) 0 0
\(583\) 25.5321 1.05743
\(584\) 2.97920 0.123280
\(585\) 0 0
\(586\) 2.04724 0.0845707
\(587\) −15.2876 −0.630989 −0.315494 0.948927i \(-0.602170\pi\)
−0.315494 + 0.948927i \(0.602170\pi\)
\(588\) 0 0
\(589\) −0.123765 −0.00509964
\(590\) −5.68809 −0.234175
\(591\) 0 0
\(592\) −7.25272 −0.298085
\(593\) 33.5005 1.37570 0.687850 0.725853i \(-0.258555\pi\)
0.687850 + 0.725853i \(0.258555\pi\)
\(594\) 0 0
\(595\) 5.96518 0.244549
\(596\) −20.9106 −0.856532
\(597\) 0 0
\(598\) −3.01293 −0.123208
\(599\) −0.150988 −0.00616920 −0.00308460 0.999995i \(-0.500982\pi\)
−0.00308460 + 0.999995i \(0.500982\pi\)
\(600\) 0 0
\(601\) −34.9546 −1.42583 −0.712915 0.701251i \(-0.752625\pi\)
−0.712915 + 0.701251i \(0.752625\pi\)
\(602\) −5.87924 −0.239620
\(603\) 0 0
\(604\) −2.07209 −0.0843120
\(605\) −2.04110 −0.0829824
\(606\) 0 0
\(607\) 10.5301 0.427404 0.213702 0.976899i \(-0.431448\pi\)
0.213702 + 0.976899i \(0.431448\pi\)
\(608\) −0.0212539 −0.000861959 0
\(609\) 0 0
\(610\) 2.79111 0.113009
\(611\) −8.89957 −0.360038
\(612\) 0 0
\(613\) −20.2271 −0.816965 −0.408483 0.912766i \(-0.633942\pi\)
−0.408483 + 0.912766i \(0.633942\pi\)
\(614\) 18.6159 0.751275
\(615\) 0 0
\(616\) −4.83110 −0.194651
\(617\) −7.82969 −0.315211 −0.157606 0.987502i \(-0.550378\pi\)
−0.157606 + 0.987502i \(0.550378\pi\)
\(618\) 0 0
\(619\) −23.6992 −0.952549 −0.476275 0.879297i \(-0.658013\pi\)
−0.476275 + 0.879297i \(0.658013\pi\)
\(620\) 4.22204 0.169561
\(621\) 0 0
\(622\) −9.88503 −0.396354
\(623\) −13.5019 −0.540942
\(624\) 0 0
\(625\) 17.3910 0.695642
\(626\) −17.4438 −0.697195
\(627\) 0 0
\(628\) −2.60975 −0.104140
\(629\) 45.9084 1.83049
\(630\) 0 0
\(631\) −31.8687 −1.26867 −0.634336 0.773058i \(-0.718726\pi\)
−0.634336 + 0.773058i \(0.718726\pi\)
\(632\) −0.418245 −0.0166369
\(633\) 0 0
\(634\) 2.92828 0.116297
\(635\) 6.90668 0.274083
\(636\) 0 0
\(637\) −12.2598 −0.485752
\(638\) −7.66802 −0.303580
\(639\) 0 0
\(640\) 0.725043 0.0286598
\(641\) −10.3578 −0.409108 −0.204554 0.978855i \(-0.565574\pi\)
−0.204554 + 0.978855i \(0.565574\pi\)
\(642\) 0 0
\(643\) −21.9761 −0.866653 −0.433326 0.901237i \(-0.642660\pi\)
−0.433326 + 0.901237i \(0.642660\pi\)
\(644\) 1.69635 0.0668455
\(645\) 0 0
\(646\) 0.134533 0.00529314
\(647\) −9.10376 −0.357906 −0.178953 0.983858i \(-0.557271\pi\)
−0.178953 + 0.983858i \(0.557271\pi\)
\(648\) 0 0
\(649\) −29.1595 −1.14461
\(650\) 10.3292 0.405146
\(651\) 0 0
\(652\) −6.63147 −0.259708
\(653\) −6.84892 −0.268019 −0.134010 0.990980i \(-0.542785\pi\)
−0.134010 + 0.990980i \(0.542785\pi\)
\(654\) 0 0
\(655\) −15.9020 −0.621344
\(656\) −8.77454 −0.342588
\(657\) 0 0
\(658\) 5.01067 0.195336
\(659\) 10.9102 0.424999 0.212500 0.977161i \(-0.431840\pi\)
0.212500 + 0.977161i \(0.431840\pi\)
\(660\) 0 0
\(661\) −9.26687 −0.360439 −0.180220 0.983626i \(-0.557681\pi\)
−0.180220 + 0.983626i \(0.557681\pi\)
\(662\) −5.44140 −0.211486
\(663\) 0 0
\(664\) −15.6322 −0.606646
\(665\) −0.0200295 −0.000776712 0
\(666\) 0 0
\(667\) 2.69248 0.104253
\(668\) −22.6369 −0.875850
\(669\) 0 0
\(670\) −6.75407 −0.260933
\(671\) 14.3084 0.552369
\(672\) 0 0
\(673\) −15.0272 −0.579255 −0.289627 0.957139i \(-0.593531\pi\)
−0.289627 + 0.957139i \(0.593531\pi\)
\(674\) 3.73840 0.143998
\(675\) 0 0
\(676\) −7.67053 −0.295020
\(677\) 46.3947 1.78309 0.891547 0.452929i \(-0.149621\pi\)
0.891547 + 0.452929i \(0.149621\pi\)
\(678\) 0 0
\(679\) 6.59843 0.253225
\(680\) −4.58939 −0.175995
\(681\) 0 0
\(682\) 21.6439 0.828790
\(683\) 34.2506 1.31056 0.655281 0.755385i \(-0.272550\pi\)
0.655281 + 0.755385i \(0.272550\pi\)
\(684\) 0 0
\(685\) −1.80426 −0.0689373
\(686\) 16.0010 0.610921
\(687\) 0 0
\(688\) 4.52327 0.172448
\(689\) 15.8581 0.604144
\(690\) 0 0
\(691\) −1.19208 −0.0453489 −0.0226744 0.999743i \(-0.507218\pi\)
−0.0226744 + 0.999743i \(0.507218\pi\)
\(692\) −12.2456 −0.465509
\(693\) 0 0
\(694\) 10.8089 0.410300
\(695\) −15.6786 −0.594723
\(696\) 0 0
\(697\) 55.5412 2.10377
\(698\) −19.8978 −0.753142
\(699\) 0 0
\(700\) −5.81561 −0.219809
\(701\) 24.6698 0.931767 0.465883 0.884846i \(-0.345737\pi\)
0.465883 + 0.884846i \(0.345737\pi\)
\(702\) 0 0
\(703\) −0.154149 −0.00581382
\(704\) 3.71687 0.140085
\(705\) 0 0
\(706\) −11.8505 −0.445998
\(707\) −16.2889 −0.612608
\(708\) 0 0
\(709\) 23.3891 0.878398 0.439199 0.898390i \(-0.355262\pi\)
0.439199 + 0.898390i \(0.355262\pi\)
\(710\) 0.466273 0.0174989
\(711\) 0 0
\(712\) 10.3879 0.389302
\(713\) −7.59986 −0.284617
\(714\) 0 0
\(715\) −6.22133 −0.232665
\(716\) 18.0181 0.673367
\(717\) 0 0
\(718\) 28.6632 1.06970
\(719\) 1.52136 0.0567371 0.0283686 0.999598i \(-0.490969\pi\)
0.0283686 + 0.999598i \(0.490969\pi\)
\(720\) 0 0
\(721\) 7.06265 0.263027
\(722\) 18.9995 0.707090
\(723\) 0 0
\(724\) 7.77690 0.289026
\(725\) −9.23065 −0.342818
\(726\) 0 0
\(727\) 36.3855 1.34946 0.674732 0.738063i \(-0.264259\pi\)
0.674732 + 0.738063i \(0.264259\pi\)
\(728\) −3.00062 −0.111210
\(729\) 0 0
\(730\) −2.16005 −0.0799470
\(731\) −28.6315 −1.05897
\(732\) 0 0
\(733\) −18.7869 −0.693908 −0.346954 0.937882i \(-0.612784\pi\)
−0.346954 + 0.937882i \(0.612784\pi\)
\(734\) −13.1083 −0.483835
\(735\) 0 0
\(736\) −1.30511 −0.0481069
\(737\) −34.6242 −1.27540
\(738\) 0 0
\(739\) −38.7776 −1.42646 −0.713228 0.700932i \(-0.752767\pi\)
−0.713228 + 0.700932i \(0.752767\pi\)
\(740\) 5.25854 0.193308
\(741\) 0 0
\(742\) −8.92847 −0.327774
\(743\) 6.80190 0.249537 0.124769 0.992186i \(-0.460181\pi\)
0.124769 + 0.992186i \(0.460181\pi\)
\(744\) 0 0
\(745\) 15.1611 0.555460
\(746\) 17.2738 0.632441
\(747\) 0 0
\(748\) −23.5271 −0.860237
\(749\) 4.49073 0.164088
\(750\) 0 0
\(751\) 10.5672 0.385601 0.192801 0.981238i \(-0.438243\pi\)
0.192801 + 0.981238i \(0.438243\pi\)
\(752\) −3.85502 −0.140578
\(753\) 0 0
\(754\) −4.76264 −0.173445
\(755\) 1.50235 0.0546762
\(756\) 0 0
\(757\) −22.6389 −0.822825 −0.411412 0.911449i \(-0.634964\pi\)
−0.411412 + 0.911449i \(0.634964\pi\)
\(758\) −6.30803 −0.229118
\(759\) 0 0
\(760\) 0.0154100 0.000558979 0
\(761\) −1.83584 −0.0665490 −0.0332745 0.999446i \(-0.510594\pi\)
−0.0332745 + 0.999446i \(0.510594\pi\)
\(762\) 0 0
\(763\) −8.84889 −0.320351
\(764\) −0.312243 −0.0112965
\(765\) 0 0
\(766\) −28.2196 −1.01962
\(767\) −18.1111 −0.653954
\(768\) 0 0
\(769\) 50.0137 1.80354 0.901770 0.432215i \(-0.142268\pi\)
0.901770 + 0.432215i \(0.142268\pi\)
\(770\) 3.50276 0.126231
\(771\) 0 0
\(772\) −0.336949 −0.0121271
\(773\) −40.2988 −1.44945 −0.724724 0.689039i \(-0.758033\pi\)
−0.724724 + 0.689039i \(0.758033\pi\)
\(774\) 0 0
\(775\) 26.0546 0.935911
\(776\) −5.07659 −0.182239
\(777\) 0 0
\(778\) −3.15029 −0.112943
\(779\) −0.186493 −0.00668180
\(780\) 0 0
\(781\) 2.39031 0.0855319
\(782\) 8.26110 0.295416
\(783\) 0 0
\(784\) −5.31058 −0.189664
\(785\) 1.89218 0.0675349
\(786\) 0 0
\(787\) 38.2500 1.36347 0.681733 0.731601i \(-0.261227\pi\)
0.681733 + 0.731601i \(0.261227\pi\)
\(788\) −26.8275 −0.955691
\(789\) 0 0
\(790\) 0.303246 0.0107890
\(791\) −0.529765 −0.0188363
\(792\) 0 0
\(793\) 8.88700 0.315586
\(794\) −31.2926 −1.11053
\(795\) 0 0
\(796\) 4.03560 0.143038
\(797\) 33.4247 1.18397 0.591983 0.805951i \(-0.298345\pi\)
0.591983 + 0.805951i \(0.298345\pi\)
\(798\) 0 0
\(799\) 24.4016 0.863266
\(800\) 4.47431 0.158191
\(801\) 0 0
\(802\) 0.744072 0.0262741
\(803\) −11.0733 −0.390769
\(804\) 0 0
\(805\) −1.22993 −0.0433492
\(806\) 13.4431 0.473514
\(807\) 0 0
\(808\) 12.5321 0.440877
\(809\) 39.8841 1.40225 0.701125 0.713038i \(-0.252681\pi\)
0.701125 + 0.713038i \(0.252681\pi\)
\(810\) 0 0
\(811\) 34.7237 1.21931 0.609657 0.792665i \(-0.291307\pi\)
0.609657 + 0.792665i \(0.291307\pi\)
\(812\) 2.68148 0.0941015
\(813\) 0 0
\(814\) 26.9574 0.944858
\(815\) 4.80810 0.168420
\(816\) 0 0
\(817\) 0.0961370 0.00336341
\(818\) −3.60242 −0.125956
\(819\) 0 0
\(820\) 6.36192 0.222168
\(821\) 48.5169 1.69325 0.846626 0.532189i \(-0.178630\pi\)
0.846626 + 0.532189i \(0.178630\pi\)
\(822\) 0 0
\(823\) 19.5287 0.680728 0.340364 0.940294i \(-0.389450\pi\)
0.340364 + 0.940294i \(0.389450\pi\)
\(824\) −5.43374 −0.189293
\(825\) 0 0
\(826\) 10.1970 0.354798
\(827\) −18.5400 −0.644698 −0.322349 0.946621i \(-0.604472\pi\)
−0.322349 + 0.946621i \(0.604472\pi\)
\(828\) 0 0
\(829\) −20.2344 −0.702769 −0.351385 0.936231i \(-0.614289\pi\)
−0.351385 + 0.936231i \(0.614289\pi\)
\(830\) 11.3340 0.393409
\(831\) 0 0
\(832\) 2.30856 0.0800351
\(833\) 33.6150 1.16469
\(834\) 0 0
\(835\) 16.4128 0.567987
\(836\) 0.0789980 0.00273220
\(837\) 0 0
\(838\) −17.1323 −0.591824
\(839\) −46.3931 −1.60167 −0.800833 0.598887i \(-0.795610\pi\)
−0.800833 + 0.598887i \(0.795610\pi\)
\(840\) 0 0
\(841\) −24.7439 −0.853238
\(842\) −24.1651 −0.832783
\(843\) 0 0
\(844\) 9.22688 0.317602
\(845\) 5.56146 0.191320
\(846\) 0 0
\(847\) 3.65905 0.125727
\(848\) 6.86923 0.235890
\(849\) 0 0
\(850\) −28.3216 −0.971422
\(851\) −9.46559 −0.324476
\(852\) 0 0
\(853\) −0.133230 −0.00456172 −0.00228086 0.999997i \(-0.500726\pi\)
−0.00228086 + 0.999997i \(0.500726\pi\)
\(854\) −5.00359 −0.171219
\(855\) 0 0
\(856\) −3.45500 −0.118089
\(857\) 10.2942 0.351642 0.175821 0.984422i \(-0.443742\pi\)
0.175821 + 0.984422i \(0.443742\pi\)
\(858\) 0 0
\(859\) 35.5814 1.21402 0.607011 0.794694i \(-0.292368\pi\)
0.607011 + 0.794694i \(0.292368\pi\)
\(860\) −3.27956 −0.111832
\(861\) 0 0
\(862\) 16.8823 0.575014
\(863\) −17.9896 −0.612372 −0.306186 0.951972i \(-0.599053\pi\)
−0.306186 + 0.951972i \(0.599053\pi\)
\(864\) 0 0
\(865\) 8.87861 0.301882
\(866\) 13.0644 0.443946
\(867\) 0 0
\(868\) −7.56881 −0.256902
\(869\) 1.55456 0.0527350
\(870\) 0 0
\(871\) −21.5052 −0.728677
\(872\) 6.80801 0.230548
\(873\) 0 0
\(874\) −0.0277386 −0.000938273 0
\(875\) 8.92853 0.301839
\(876\) 0 0
\(877\) 47.6131 1.60778 0.803890 0.594778i \(-0.202760\pi\)
0.803890 + 0.594778i \(0.202760\pi\)
\(878\) −3.91599 −0.132158
\(879\) 0 0
\(880\) −2.69489 −0.0908448
\(881\) 15.7287 0.529914 0.264957 0.964260i \(-0.414642\pi\)
0.264957 + 0.964260i \(0.414642\pi\)
\(882\) 0 0
\(883\) −20.9111 −0.703713 −0.351856 0.936054i \(-0.614449\pi\)
−0.351856 + 0.936054i \(0.614449\pi\)
\(884\) −14.6128 −0.491481
\(885\) 0 0
\(886\) 25.2673 0.848871
\(887\) −3.47543 −0.116694 −0.0583468 0.998296i \(-0.518583\pi\)
−0.0583468 + 0.998296i \(0.518583\pi\)
\(888\) 0 0
\(889\) −12.3815 −0.415263
\(890\) −7.53165 −0.252461
\(891\) 0 0
\(892\) −9.44842 −0.316357
\(893\) −0.0819342 −0.00274182
\(894\) 0 0
\(895\) −13.0639 −0.436677
\(896\) −1.29978 −0.0434225
\(897\) 0 0
\(898\) 27.4706 0.916707
\(899\) −12.0134 −0.400668
\(900\) 0 0
\(901\) −43.4810 −1.44856
\(902\) 32.6138 1.08592
\(903\) 0 0
\(904\) 0.407582 0.0135560
\(905\) −5.63859 −0.187433
\(906\) 0 0
\(907\) 5.99370 0.199018 0.0995088 0.995037i \(-0.468273\pi\)
0.0995088 + 0.995037i \(0.468273\pi\)
\(908\) −15.2061 −0.504632
\(909\) 0 0
\(910\) 2.17558 0.0721196
\(911\) −3.66404 −0.121395 −0.0606976 0.998156i \(-0.519333\pi\)
−0.0606976 + 0.998156i \(0.519333\pi\)
\(912\) 0 0
\(913\) 58.1028 1.92292
\(914\) 14.5345 0.480759
\(915\) 0 0
\(916\) 28.6411 0.946329
\(917\) 28.5074 0.941397
\(918\) 0 0
\(919\) 44.8323 1.47888 0.739441 0.673222i \(-0.235090\pi\)
0.739441 + 0.673222i \(0.235090\pi\)
\(920\) 0.946259 0.0311972
\(921\) 0 0
\(922\) −10.4630 −0.344581
\(923\) 1.48463 0.0488672
\(924\) 0 0
\(925\) 32.4510 1.06698
\(926\) 27.7519 0.911985
\(927\) 0 0
\(928\) −2.06303 −0.0677223
\(929\) 35.1823 1.15429 0.577146 0.816641i \(-0.304166\pi\)
0.577146 + 0.816641i \(0.304166\pi\)
\(930\) 0 0
\(931\) −0.112870 −0.00369918
\(932\) −3.14875 −0.103141
\(933\) 0 0
\(934\) −12.3322 −0.403521
\(935\) 17.0582 0.557862
\(936\) 0 0
\(937\) 37.7329 1.23268 0.616339 0.787481i \(-0.288615\pi\)
0.616339 + 0.787481i \(0.288615\pi\)
\(938\) 12.1080 0.395339
\(939\) 0 0
\(940\) 2.79506 0.0911647
\(941\) 31.0321 1.01162 0.505809 0.862646i \(-0.331194\pi\)
0.505809 + 0.862646i \(0.331194\pi\)
\(942\) 0 0
\(943\) −11.4517 −0.372919
\(944\) −7.84518 −0.255339
\(945\) 0 0
\(946\) −16.8124 −0.546619
\(947\) −19.5388 −0.634924 −0.317462 0.948271i \(-0.602831\pi\)
−0.317462 + 0.948271i \(0.602831\pi\)
\(948\) 0 0
\(949\) −6.87768 −0.223259
\(950\) 0.0950965 0.00308534
\(951\) 0 0
\(952\) 8.22735 0.266650
\(953\) −15.9894 −0.517948 −0.258974 0.965884i \(-0.583384\pi\)
−0.258974 + 0.965884i \(0.583384\pi\)
\(954\) 0 0
\(955\) 0.226389 0.00732579
\(956\) −12.5059 −0.404470
\(957\) 0 0
\(958\) −2.92366 −0.0944592
\(959\) 3.23448 0.104447
\(960\) 0 0
\(961\) 2.90921 0.0938456
\(962\) 16.7434 0.539828
\(963\) 0 0
\(964\) 3.73027 0.120144
\(965\) 0.244302 0.00786437
\(966\) 0 0
\(967\) 38.1787 1.22774 0.613872 0.789406i \(-0.289611\pi\)
0.613872 + 0.789406i \(0.289611\pi\)
\(968\) −2.81514 −0.0904820
\(969\) 0 0
\(970\) 3.68075 0.118182
\(971\) −2.16411 −0.0694497 −0.0347248 0.999397i \(-0.511055\pi\)
−0.0347248 + 0.999397i \(0.511055\pi\)
\(972\) 0 0
\(973\) 28.1069 0.901064
\(974\) −10.1739 −0.325994
\(975\) 0 0
\(976\) 3.84958 0.123222
\(977\) 14.7297 0.471243 0.235622 0.971845i \(-0.424287\pi\)
0.235622 + 0.971845i \(0.424287\pi\)
\(978\) 0 0
\(979\) −38.6104 −1.23399
\(980\) 3.85040 0.122996
\(981\) 0 0
\(982\) −16.8181 −0.536686
\(983\) −36.4217 −1.16167 −0.580836 0.814020i \(-0.697274\pi\)
−0.580836 + 0.814020i \(0.697274\pi\)
\(984\) 0 0
\(985\) 19.4511 0.619764
\(986\) 13.0586 0.415871
\(987\) 0 0
\(988\) 0.0490660 0.00156100
\(989\) 5.90336 0.187716
\(990\) 0 0
\(991\) −24.5429 −0.779632 −0.389816 0.920893i \(-0.627461\pi\)
−0.389816 + 0.920893i \(0.627461\pi\)
\(992\) 5.82316 0.184886
\(993\) 0 0
\(994\) −0.835882 −0.0265126
\(995\) −2.92598 −0.0927598
\(996\) 0 0
\(997\) −52.7413 −1.67033 −0.835167 0.549996i \(-0.814629\pi\)
−0.835167 + 0.549996i \(0.814629\pi\)
\(998\) 30.0514 0.951260
\(999\) 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 4842.2.a.n.1.4 7
3.2 odd 2 538.2.a.e.1.5 7
12.11 even 2 4304.2.a.h.1.3 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.e.1.5 7 3.2 odd 2
4304.2.a.h.1.3 7 12.11 even 2
4842.2.a.n.1.4 7 1.1 even 1 trivial