Properties

Label 7569.2.a.bl.1.6
Level $7569$
Weight $2$
Character 7569.1
Self dual yes
Analytic conductor $60.439$
Analytic rank $0$
Dimension $9$
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] = [7569,2,Mod(1,7569)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7569, 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("7569.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7569 = 3^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7569.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: \(60.4387692899\)
Analytic rank: \(0\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - x^{8} - 14x^{7} + 9x^{6} + 70x^{5} - 23x^{4} - 141x^{3} + 14x^{2} + 84x - 7 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 87)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.915678\) of defining polynomial
Character \(\chi\) \(=\) 7569.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.915678 q^{2} -1.16153 q^{4} -0.285007 q^{5} +5.03918 q^{7} -2.89495 q^{8} +O(q^{10})\) \(q+0.915678 q^{2} -1.16153 q^{4} -0.285007 q^{5} +5.03918 q^{7} -2.89495 q^{8} -0.260975 q^{10} +4.18236 q^{11} +2.27527 q^{13} +4.61427 q^{14} -0.327776 q^{16} +3.40834 q^{17} +0.852557 q^{19} +0.331045 q^{20} +3.82970 q^{22} +7.24173 q^{23} -4.91877 q^{25} +2.08342 q^{26} -5.85318 q^{28} -3.27177 q^{31} +5.48976 q^{32} +3.12094 q^{34} -1.43620 q^{35} +0.380705 q^{37} +0.780668 q^{38} +0.825080 q^{40} +5.56008 q^{41} +7.90289 q^{43} -4.85795 q^{44} +6.63110 q^{46} -6.53585 q^{47} +18.3934 q^{49} -4.50401 q^{50} -2.64281 q^{52} -2.89257 q^{53} -1.19200 q^{55} -14.5882 q^{56} +4.68212 q^{59} -6.93927 q^{61} -2.99589 q^{62} +5.68240 q^{64} -0.648469 q^{65} -2.84060 q^{67} -3.95890 q^{68} -1.31510 q^{70} +11.4943 q^{71} -11.3234 q^{73} +0.348603 q^{74} -0.990273 q^{76} +21.0757 q^{77} -6.76944 q^{79} +0.0934183 q^{80} +5.09125 q^{82} -6.27986 q^{83} -0.971400 q^{85} +7.23650 q^{86} -12.1077 q^{88} -5.70609 q^{89} +11.4655 q^{91} -8.41151 q^{92} -5.98474 q^{94} -0.242985 q^{95} +0.502800 q^{97} +16.8424 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 9 q + q^{2} + 11 q^{4} + 5 q^{7} + 12 q^{8} - 4 q^{10} + 3 q^{11} + 5 q^{13} + 15 q^{14} - 5 q^{16} + 16 q^{17} + q^{19} - 8 q^{20} + 24 q^{22} + 10 q^{23} + 9 q^{25} + 12 q^{26} - 24 q^{28} + 4 q^{31} + 25 q^{32} + 24 q^{34} + 44 q^{35} - 25 q^{37} + 10 q^{38} - 5 q^{40} + 34 q^{41} - 12 q^{43} - 23 q^{44} - 6 q^{46} + 8 q^{47} + 26 q^{49} + 27 q^{50} - 23 q^{52} + 32 q^{53} + 5 q^{55} - 14 q^{56} - 10 q^{59} - 51 q^{61} - 8 q^{62} - 8 q^{64} + 11 q^{65} + 7 q^{67} + 11 q^{68} + 14 q^{70} - 7 q^{71} - 17 q^{73} + 62 q^{74} + 6 q^{76} + 64 q^{77} - 13 q^{79} - 54 q^{80} + 37 q^{82} + 31 q^{83} - 42 q^{85} + 70 q^{86} - 29 q^{88} - 32 q^{89} + 45 q^{91} - 9 q^{92} + 38 q^{94} - 20 q^{95} - 16 q^{97} - 12 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.915678 0.647482 0.323741 0.946146i \(-0.395059\pi\)
0.323741 + 0.946146i \(0.395059\pi\)
\(3\) 0 0
\(4\) −1.16153 −0.580766
\(5\) −0.285007 −0.127459 −0.0637295 0.997967i \(-0.520299\pi\)
−0.0637295 + 0.997967i \(0.520299\pi\)
\(6\) 0 0
\(7\) 5.03918 1.90463 0.952316 0.305113i \(-0.0986941\pi\)
0.952316 + 0.305113i \(0.0986941\pi\)
\(8\) −2.89495 −1.02352
\(9\) 0 0
\(10\) −0.260975 −0.0825275
\(11\) 4.18236 1.26103 0.630514 0.776178i \(-0.282844\pi\)
0.630514 + 0.776178i \(0.282844\pi\)
\(12\) 0 0
\(13\) 2.27527 0.631047 0.315524 0.948918i \(-0.397820\pi\)
0.315524 + 0.948918i \(0.397820\pi\)
\(14\) 4.61427 1.23322
\(15\) 0 0
\(16\) −0.327776 −0.0819439
\(17\) 3.40834 0.826644 0.413322 0.910585i \(-0.364368\pi\)
0.413322 + 0.910585i \(0.364368\pi\)
\(18\) 0 0
\(19\) 0.852557 0.195590 0.0977950 0.995207i \(-0.468821\pi\)
0.0977950 + 0.995207i \(0.468821\pi\)
\(20\) 0.331045 0.0740239
\(21\) 0 0
\(22\) 3.82970 0.816494
\(23\) 7.24173 1.51001 0.755003 0.655721i \(-0.227635\pi\)
0.755003 + 0.655721i \(0.227635\pi\)
\(24\) 0 0
\(25\) −4.91877 −0.983754
\(26\) 2.08342 0.408592
\(27\) 0 0
\(28\) −5.85318 −1.10615
\(29\) 0 0
\(30\) 0 0
\(31\) −3.27177 −0.587627 −0.293813 0.955863i \(-0.594924\pi\)
−0.293813 + 0.955863i \(0.594924\pi\)
\(32\) 5.48976 0.970461
\(33\) 0 0
\(34\) 3.12094 0.535237
\(35\) −1.43620 −0.242763
\(36\) 0 0
\(37\) 0.380705 0.0625875 0.0312937 0.999510i \(-0.490037\pi\)
0.0312937 + 0.999510i \(0.490037\pi\)
\(38\) 0.780668 0.126641
\(39\) 0 0
\(40\) 0.825080 0.130457
\(41\) 5.56008 0.868339 0.434169 0.900831i \(-0.357042\pi\)
0.434169 + 0.900831i \(0.357042\pi\)
\(42\) 0 0
\(43\) 7.90289 1.20518 0.602589 0.798051i \(-0.294136\pi\)
0.602589 + 0.798051i \(0.294136\pi\)
\(44\) −4.85795 −0.732363
\(45\) 0 0
\(46\) 6.63110 0.977702
\(47\) −6.53585 −0.953352 −0.476676 0.879079i \(-0.658158\pi\)
−0.476676 + 0.879079i \(0.658158\pi\)
\(48\) 0 0
\(49\) 18.3934 2.62763
\(50\) −4.50401 −0.636964
\(51\) 0 0
\(52\) −2.64281 −0.366491
\(53\) −2.89257 −0.397325 −0.198663 0.980068i \(-0.563660\pi\)
−0.198663 + 0.980068i \(0.563660\pi\)
\(54\) 0 0
\(55\) −1.19200 −0.160729
\(56\) −14.5882 −1.94943
\(57\) 0 0
\(58\) 0 0
\(59\) 4.68212 0.609560 0.304780 0.952423i \(-0.401417\pi\)
0.304780 + 0.952423i \(0.401417\pi\)
\(60\) 0 0
\(61\) −6.93927 −0.888483 −0.444241 0.895907i \(-0.646527\pi\)
−0.444241 + 0.895907i \(0.646527\pi\)
\(62\) −2.99589 −0.380478
\(63\) 0 0
\(64\) 5.68240 0.710301
\(65\) −0.648469 −0.0804327
\(66\) 0 0
\(67\) −2.84060 −0.347035 −0.173518 0.984831i \(-0.555513\pi\)
−0.173518 + 0.984831i \(0.555513\pi\)
\(68\) −3.95890 −0.480087
\(69\) 0 0
\(70\) −1.31510 −0.157184
\(71\) 11.4943 1.36412 0.682061 0.731296i \(-0.261084\pi\)
0.682061 + 0.731296i \(0.261084\pi\)
\(72\) 0 0
\(73\) −11.3234 −1.32531 −0.662653 0.748926i \(-0.730570\pi\)
−0.662653 + 0.748926i \(0.730570\pi\)
\(74\) 0.348603 0.0405243
\(75\) 0 0
\(76\) −0.990273 −0.113592
\(77\) 21.0757 2.40180
\(78\) 0 0
\(79\) −6.76944 −0.761621 −0.380811 0.924653i \(-0.624355\pi\)
−0.380811 + 0.924653i \(0.624355\pi\)
\(80\) 0.0934183 0.0104445
\(81\) 0 0
\(82\) 5.09125 0.562234
\(83\) −6.27986 −0.689304 −0.344652 0.938731i \(-0.612003\pi\)
−0.344652 + 0.938731i \(0.612003\pi\)
\(84\) 0 0
\(85\) −0.971400 −0.105363
\(86\) 7.23650 0.780332
\(87\) 0 0
\(88\) −12.1077 −1.29069
\(89\) −5.70609 −0.604844 −0.302422 0.953174i \(-0.597795\pi\)
−0.302422 + 0.953174i \(0.597795\pi\)
\(90\) 0 0
\(91\) 11.4655 1.20191
\(92\) −8.41151 −0.876961
\(93\) 0 0
\(94\) −5.98474 −0.617278
\(95\) −0.242985 −0.0249297
\(96\) 0 0
\(97\) 0.502800 0.0510516 0.0255258 0.999674i \(-0.491874\pi\)
0.0255258 + 0.999674i \(0.491874\pi\)
\(98\) 16.8424 1.70134
\(99\) 0 0
\(100\) 5.71331 0.571331
\(101\) 4.50787 0.448550 0.224275 0.974526i \(-0.427999\pi\)
0.224275 + 0.974526i \(0.427999\pi\)
\(102\) 0 0
\(103\) 8.12388 0.800469 0.400235 0.916413i \(-0.368929\pi\)
0.400235 + 0.916413i \(0.368929\pi\)
\(104\) −6.58680 −0.645889
\(105\) 0 0
\(106\) −2.64866 −0.257261
\(107\) −9.51920 −0.920256 −0.460128 0.887853i \(-0.652196\pi\)
−0.460128 + 0.887853i \(0.652196\pi\)
\(108\) 0 0
\(109\) −7.43141 −0.711800 −0.355900 0.934524i \(-0.615826\pi\)
−0.355900 + 0.934524i \(0.615826\pi\)
\(110\) −1.09149 −0.104069
\(111\) 0 0
\(112\) −1.65172 −0.156073
\(113\) −4.23804 −0.398682 −0.199341 0.979930i \(-0.563880\pi\)
−0.199341 + 0.979930i \(0.563880\pi\)
\(114\) 0 0
\(115\) −2.06394 −0.192464
\(116\) 0 0
\(117\) 0 0
\(118\) 4.28732 0.394680
\(119\) 17.1752 1.57445
\(120\) 0 0
\(121\) 6.49212 0.590193
\(122\) −6.35414 −0.575277
\(123\) 0 0
\(124\) 3.80027 0.341274
\(125\) 2.82692 0.252847
\(126\) 0 0
\(127\) −16.9993 −1.50845 −0.754224 0.656618i \(-0.771987\pi\)
−0.754224 + 0.656618i \(0.771987\pi\)
\(128\) −5.77626 −0.510554
\(129\) 0 0
\(130\) −0.593789 −0.0520787
\(131\) 12.6952 1.10918 0.554591 0.832123i \(-0.312875\pi\)
0.554591 + 0.832123i \(0.312875\pi\)
\(132\) 0 0
\(133\) 4.29619 0.372527
\(134\) −2.60108 −0.224699
\(135\) 0 0
\(136\) −9.86696 −0.846085
\(137\) −9.74328 −0.832425 −0.416212 0.909267i \(-0.636643\pi\)
−0.416212 + 0.909267i \(0.636643\pi\)
\(138\) 0 0
\(139\) −12.4834 −1.05883 −0.529416 0.848362i \(-0.677589\pi\)
−0.529416 + 0.848362i \(0.677589\pi\)
\(140\) 1.66820 0.140988
\(141\) 0 0
\(142\) 10.5251 0.883245
\(143\) 9.51601 0.795769
\(144\) 0 0
\(145\) 0 0
\(146\) −10.3686 −0.858113
\(147\) 0 0
\(148\) −0.442201 −0.0363487
\(149\) 22.6007 1.85152 0.925760 0.378111i \(-0.123426\pi\)
0.925760 + 0.378111i \(0.123426\pi\)
\(150\) 0 0
\(151\) −6.63564 −0.540001 −0.270000 0.962860i \(-0.587024\pi\)
−0.270000 + 0.962860i \(0.587024\pi\)
\(152\) −2.46811 −0.200190
\(153\) 0 0
\(154\) 19.2985 1.55512
\(155\) 0.932477 0.0748983
\(156\) 0 0
\(157\) −13.3793 −1.06778 −0.533890 0.845554i \(-0.679270\pi\)
−0.533890 + 0.845554i \(0.679270\pi\)
\(158\) −6.19863 −0.493136
\(159\) 0 0
\(160\) −1.56462 −0.123694
\(161\) 36.4924 2.87601
\(162\) 0 0
\(163\) 10.1562 0.795497 0.397748 0.917495i \(-0.369792\pi\)
0.397748 + 0.917495i \(0.369792\pi\)
\(164\) −6.45822 −0.504302
\(165\) 0 0
\(166\) −5.75033 −0.446312
\(167\) −24.5296 −1.89816 −0.949078 0.315042i \(-0.897982\pi\)
−0.949078 + 0.315042i \(0.897982\pi\)
\(168\) 0 0
\(169\) −7.82313 −0.601779
\(170\) −0.889490 −0.0682208
\(171\) 0 0
\(172\) −9.17946 −0.699927
\(173\) 22.0884 1.67935 0.839674 0.543091i \(-0.182746\pi\)
0.839674 + 0.543091i \(0.182746\pi\)
\(174\) 0 0
\(175\) −24.7866 −1.87369
\(176\) −1.37087 −0.103334
\(177\) 0 0
\(178\) −5.22494 −0.391626
\(179\) −0.735055 −0.0549406 −0.0274703 0.999623i \(-0.508745\pi\)
−0.0274703 + 0.999623i \(0.508745\pi\)
\(180\) 0 0
\(181\) 13.6765 1.01656 0.508281 0.861191i \(-0.330281\pi\)
0.508281 + 0.861191i \(0.330281\pi\)
\(182\) 10.4987 0.778218
\(183\) 0 0
\(184\) −20.9644 −1.54552
\(185\) −0.108503 −0.00797733
\(186\) 0 0
\(187\) 14.2549 1.04242
\(188\) 7.59160 0.553675
\(189\) 0 0
\(190\) −0.222496 −0.0161415
\(191\) −1.16101 −0.0840075 −0.0420038 0.999117i \(-0.513374\pi\)
−0.0420038 + 0.999117i \(0.513374\pi\)
\(192\) 0 0
\(193\) 20.6010 1.48289 0.741446 0.671013i \(-0.234141\pi\)
0.741446 + 0.671013i \(0.234141\pi\)
\(194\) 0.460403 0.0330550
\(195\) 0 0
\(196\) −21.3645 −1.52604
\(197\) −5.15819 −0.367506 −0.183753 0.982972i \(-0.558825\pi\)
−0.183753 + 0.982972i \(0.558825\pi\)
\(198\) 0 0
\(199\) −2.80583 −0.198900 −0.0994500 0.995043i \(-0.531708\pi\)
−0.0994500 + 0.995043i \(0.531708\pi\)
\(200\) 14.2396 1.00689
\(201\) 0 0
\(202\) 4.12776 0.290428
\(203\) 0 0
\(204\) 0 0
\(205\) −1.58466 −0.110678
\(206\) 7.43886 0.518290
\(207\) 0 0
\(208\) −0.745779 −0.0517105
\(209\) 3.56570 0.246645
\(210\) 0 0
\(211\) −4.84875 −0.333802 −0.166901 0.985974i \(-0.553376\pi\)
−0.166901 + 0.985974i \(0.553376\pi\)
\(212\) 3.35982 0.230753
\(213\) 0 0
\(214\) −8.71653 −0.595850
\(215\) −2.25238 −0.153611
\(216\) 0 0
\(217\) −16.4870 −1.11921
\(218\) −6.80478 −0.460878
\(219\) 0 0
\(220\) 1.38455 0.0933463
\(221\) 7.75490 0.521651
\(222\) 0 0
\(223\) 10.3956 0.696141 0.348070 0.937468i \(-0.386837\pi\)
0.348070 + 0.937468i \(0.386837\pi\)
\(224\) 27.6639 1.84837
\(225\) 0 0
\(226\) −3.88069 −0.258139
\(227\) 27.9845 1.85739 0.928697 0.370838i \(-0.120930\pi\)
0.928697 + 0.370838i \(0.120930\pi\)
\(228\) 0 0
\(229\) −3.49621 −0.231036 −0.115518 0.993305i \(-0.536853\pi\)
−0.115518 + 0.993305i \(0.536853\pi\)
\(230\) −1.88991 −0.124617
\(231\) 0 0
\(232\) 0 0
\(233\) 3.35269 0.219642 0.109821 0.993951i \(-0.464972\pi\)
0.109821 + 0.993951i \(0.464972\pi\)
\(234\) 0 0
\(235\) 1.86276 0.121513
\(236\) −5.43844 −0.354012
\(237\) 0 0
\(238\) 15.7270 1.01943
\(239\) 18.3933 1.18977 0.594883 0.803812i \(-0.297198\pi\)
0.594883 + 0.803812i \(0.297198\pi\)
\(240\) 0 0
\(241\) 5.62516 0.362348 0.181174 0.983451i \(-0.442010\pi\)
0.181174 + 0.983451i \(0.442010\pi\)
\(242\) 5.94470 0.382140
\(243\) 0 0
\(244\) 8.06019 0.516001
\(245\) −5.24224 −0.334914
\(246\) 0 0
\(247\) 1.93980 0.123427
\(248\) 9.47160 0.601447
\(249\) 0 0
\(250\) 2.58855 0.163714
\(251\) 26.8112 1.69231 0.846153 0.532939i \(-0.178913\pi\)
0.846153 + 0.532939i \(0.178913\pi\)
\(252\) 0 0
\(253\) 30.2875 1.90416
\(254\) −15.5659 −0.976693
\(255\) 0 0
\(256\) −16.6540 −1.04088
\(257\) 3.70597 0.231172 0.115586 0.993297i \(-0.463125\pi\)
0.115586 + 0.993297i \(0.463125\pi\)
\(258\) 0 0
\(259\) 1.91844 0.119206
\(260\) 0.753218 0.0467126
\(261\) 0 0
\(262\) 11.6247 0.718176
\(263\) −17.6520 −1.08847 −0.544235 0.838933i \(-0.683180\pi\)
−0.544235 + 0.838933i \(0.683180\pi\)
\(264\) 0 0
\(265\) 0.824403 0.0506427
\(266\) 3.93393 0.241205
\(267\) 0 0
\(268\) 3.29946 0.201546
\(269\) 14.3360 0.874082 0.437041 0.899442i \(-0.356027\pi\)
0.437041 + 0.899442i \(0.356027\pi\)
\(270\) 0 0
\(271\) 30.1902 1.83392 0.916961 0.398977i \(-0.130635\pi\)
0.916961 + 0.398977i \(0.130635\pi\)
\(272\) −1.11717 −0.0677384
\(273\) 0 0
\(274\) −8.92171 −0.538980
\(275\) −20.5721 −1.24054
\(276\) 0 0
\(277\) 8.44567 0.507451 0.253726 0.967276i \(-0.418344\pi\)
0.253726 + 0.967276i \(0.418344\pi\)
\(278\) −11.4308 −0.685575
\(279\) 0 0
\(280\) 4.15773 0.248472
\(281\) 12.6756 0.756163 0.378082 0.925772i \(-0.376584\pi\)
0.378082 + 0.925772i \(0.376584\pi\)
\(282\) 0 0
\(283\) −24.6429 −1.46487 −0.732434 0.680839i \(-0.761616\pi\)
−0.732434 + 0.680839i \(0.761616\pi\)
\(284\) −13.3510 −0.792236
\(285\) 0 0
\(286\) 8.71361 0.515246
\(287\) 28.0183 1.65387
\(288\) 0 0
\(289\) −5.38323 −0.316661
\(290\) 0 0
\(291\) 0 0
\(292\) 13.1525 0.769694
\(293\) 4.60722 0.269157 0.134578 0.990903i \(-0.457032\pi\)
0.134578 + 0.990903i \(0.457032\pi\)
\(294\) 0 0
\(295\) −1.33444 −0.0776939
\(296\) −1.10212 −0.0640594
\(297\) 0 0
\(298\) 20.6950 1.19883
\(299\) 16.4769 0.952885
\(300\) 0 0
\(301\) 39.8241 2.29542
\(302\) −6.07611 −0.349641
\(303\) 0 0
\(304\) −0.279447 −0.0160274
\(305\) 1.97774 0.113245
\(306\) 0 0
\(307\) 2.80588 0.160140 0.0800700 0.996789i \(-0.474486\pi\)
0.0800700 + 0.996789i \(0.474486\pi\)
\(308\) −24.4801 −1.39488
\(309\) 0 0
\(310\) 0.853849 0.0484954
\(311\) −15.7203 −0.891418 −0.445709 0.895178i \(-0.647048\pi\)
−0.445709 + 0.895178i \(0.647048\pi\)
\(312\) 0 0
\(313\) 18.2248 1.03013 0.515063 0.857152i \(-0.327768\pi\)
0.515063 + 0.857152i \(0.327768\pi\)
\(314\) −12.2511 −0.691369
\(315\) 0 0
\(316\) 7.86293 0.442324
\(317\) −1.87997 −0.105589 −0.0527947 0.998605i \(-0.516813\pi\)
−0.0527947 + 0.998605i \(0.516813\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −1.61952 −0.0905342
\(321\) 0 0
\(322\) 33.4153 1.86216
\(323\) 2.90580 0.161683
\(324\) 0 0
\(325\) −11.1916 −0.620796
\(326\) 9.29984 0.515070
\(327\) 0 0
\(328\) −16.0961 −0.888761
\(329\) −32.9353 −1.81578
\(330\) 0 0
\(331\) 6.14086 0.337532 0.168766 0.985656i \(-0.446022\pi\)
0.168766 + 0.985656i \(0.446022\pi\)
\(332\) 7.29426 0.400324
\(333\) 0 0
\(334\) −22.4612 −1.22902
\(335\) 0.809592 0.0442327
\(336\) 0 0
\(337\) −7.25198 −0.395040 −0.197520 0.980299i \(-0.563289\pi\)
−0.197520 + 0.980299i \(0.563289\pi\)
\(338\) −7.16347 −0.389641
\(339\) 0 0
\(340\) 1.12831 0.0611914
\(341\) −13.6837 −0.741014
\(342\) 0 0
\(343\) 57.4133 3.10003
\(344\) −22.8784 −1.23352
\(345\) 0 0
\(346\) 20.2259 1.08735
\(347\) 1.50956 0.0810374 0.0405187 0.999179i \(-0.487099\pi\)
0.0405187 + 0.999179i \(0.487099\pi\)
\(348\) 0 0
\(349\) 14.5718 0.780011 0.390006 0.920813i \(-0.372473\pi\)
0.390006 + 0.920813i \(0.372473\pi\)
\(350\) −22.6965 −1.21318
\(351\) 0 0
\(352\) 22.9601 1.22378
\(353\) 21.8048 1.16055 0.580276 0.814420i \(-0.302945\pi\)
0.580276 + 0.814420i \(0.302945\pi\)
\(354\) 0 0
\(355\) −3.27595 −0.173870
\(356\) 6.62781 0.351273
\(357\) 0 0
\(358\) −0.673074 −0.0355731
\(359\) −8.45420 −0.446196 −0.223098 0.974796i \(-0.571617\pi\)
−0.223098 + 0.974796i \(0.571617\pi\)
\(360\) 0 0
\(361\) −18.2731 −0.961745
\(362\) 12.5232 0.658207
\(363\) 0 0
\(364\) −13.3176 −0.698031
\(365\) 3.22725 0.168922
\(366\) 0 0
\(367\) −6.88980 −0.359645 −0.179822 0.983699i \(-0.557552\pi\)
−0.179822 + 0.983699i \(0.557552\pi\)
\(368\) −2.37366 −0.123736
\(369\) 0 0
\(370\) −0.0993543 −0.00516518
\(371\) −14.5762 −0.756758
\(372\) 0 0
\(373\) −19.2607 −0.997281 −0.498641 0.866809i \(-0.666167\pi\)
−0.498641 + 0.866809i \(0.666167\pi\)
\(374\) 13.0529 0.674949
\(375\) 0 0
\(376\) 18.9209 0.975773
\(377\) 0 0
\(378\) 0 0
\(379\) −34.9742 −1.79650 −0.898251 0.439483i \(-0.855162\pi\)
−0.898251 + 0.439483i \(0.855162\pi\)
\(380\) 0.282235 0.0144783
\(381\) 0 0
\(382\) −1.06311 −0.0543934
\(383\) 20.4809 1.04652 0.523262 0.852172i \(-0.324715\pi\)
0.523262 + 0.852172i \(0.324715\pi\)
\(384\) 0 0
\(385\) −6.00671 −0.306130
\(386\) 18.8639 0.960146
\(387\) 0 0
\(388\) −0.584019 −0.0296491
\(389\) 7.13745 0.361883 0.180942 0.983494i \(-0.442086\pi\)
0.180942 + 0.983494i \(0.442086\pi\)
\(390\) 0 0
\(391\) 24.6823 1.24824
\(392\) −53.2479 −2.68942
\(393\) 0 0
\(394\) −4.72325 −0.237954
\(395\) 1.92934 0.0970755
\(396\) 0 0
\(397\) −26.3181 −1.32087 −0.660435 0.750884i \(-0.729628\pi\)
−0.660435 + 0.750884i \(0.729628\pi\)
\(398\) −2.56924 −0.128784
\(399\) 0 0
\(400\) 1.61225 0.0806126
\(401\) 5.05597 0.252483 0.126242 0.992000i \(-0.459709\pi\)
0.126242 + 0.992000i \(0.459709\pi\)
\(402\) 0 0
\(403\) −7.44417 −0.370820
\(404\) −5.23604 −0.260503
\(405\) 0 0
\(406\) 0 0
\(407\) 1.59224 0.0789246
\(408\) 0 0
\(409\) 6.45406 0.319133 0.159566 0.987187i \(-0.448990\pi\)
0.159566 + 0.987187i \(0.448990\pi\)
\(410\) −1.45104 −0.0716618
\(411\) 0 0
\(412\) −9.43615 −0.464886
\(413\) 23.5941 1.16099
\(414\) 0 0
\(415\) 1.78980 0.0878580
\(416\) 12.4907 0.612407
\(417\) 0 0
\(418\) 3.26503 0.159698
\(419\) −1.23163 −0.0601693 −0.0300846 0.999547i \(-0.509578\pi\)
−0.0300846 + 0.999547i \(0.509578\pi\)
\(420\) 0 0
\(421\) 8.84808 0.431229 0.215614 0.976479i \(-0.430825\pi\)
0.215614 + 0.976479i \(0.430825\pi\)
\(422\) −4.43990 −0.216131
\(423\) 0 0
\(424\) 8.37384 0.406670
\(425\) −16.7648 −0.813214
\(426\) 0 0
\(427\) −34.9683 −1.69223
\(428\) 11.0569 0.534454
\(429\) 0 0
\(430\) −2.06245 −0.0994603
\(431\) 9.86743 0.475297 0.237649 0.971351i \(-0.423623\pi\)
0.237649 + 0.971351i \(0.423623\pi\)
\(432\) 0 0
\(433\) −33.0899 −1.59020 −0.795099 0.606480i \(-0.792581\pi\)
−0.795099 + 0.606480i \(0.792581\pi\)
\(434\) −15.0968 −0.724671
\(435\) 0 0
\(436\) 8.63182 0.413389
\(437\) 6.17399 0.295342
\(438\) 0 0
\(439\) −10.7748 −0.514253 −0.257126 0.966378i \(-0.582776\pi\)
−0.257126 + 0.966378i \(0.582776\pi\)
\(440\) 3.45078 0.164510
\(441\) 0 0
\(442\) 7.10100 0.337760
\(443\) −34.0393 −1.61726 −0.808629 0.588319i \(-0.799790\pi\)
−0.808629 + 0.588319i \(0.799790\pi\)
\(444\) 0 0
\(445\) 1.62627 0.0770928
\(446\) 9.51903 0.450739
\(447\) 0 0
\(448\) 28.6347 1.35286
\(449\) 21.8826 1.03270 0.516351 0.856377i \(-0.327290\pi\)
0.516351 + 0.856377i \(0.327290\pi\)
\(450\) 0 0
\(451\) 23.2543 1.09500
\(452\) 4.92263 0.231541
\(453\) 0 0
\(454\) 25.6248 1.20263
\(455\) −3.26775 −0.153195
\(456\) 0 0
\(457\) 29.0963 1.36107 0.680535 0.732716i \(-0.261748\pi\)
0.680535 + 0.732716i \(0.261748\pi\)
\(458\) −3.20140 −0.149592
\(459\) 0 0
\(460\) 2.39734 0.111777
\(461\) −18.7521 −0.873373 −0.436686 0.899614i \(-0.643848\pi\)
−0.436686 + 0.899614i \(0.643848\pi\)
\(462\) 0 0
\(463\) 22.8658 1.06266 0.531332 0.847164i \(-0.321692\pi\)
0.531332 + 0.847164i \(0.321692\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 3.06998 0.142214
\(467\) −19.6329 −0.908500 −0.454250 0.890874i \(-0.650093\pi\)
−0.454250 + 0.890874i \(0.650093\pi\)
\(468\) 0 0
\(469\) −14.3143 −0.660974
\(470\) 1.70569 0.0786777
\(471\) 0 0
\(472\) −13.5545 −0.623896
\(473\) 33.0527 1.51976
\(474\) 0 0
\(475\) −4.19353 −0.192413
\(476\) −19.9496 −0.914389
\(477\) 0 0
\(478\) 16.8424 0.770353
\(479\) −22.5224 −1.02908 −0.514538 0.857468i \(-0.672037\pi\)
−0.514538 + 0.857468i \(0.672037\pi\)
\(480\) 0 0
\(481\) 0.866207 0.0394957
\(482\) 5.15084 0.234614
\(483\) 0 0
\(484\) −7.54081 −0.342764
\(485\) −0.143301 −0.00650698
\(486\) 0 0
\(487\) 1.72031 0.0779548 0.0389774 0.999240i \(-0.487590\pi\)
0.0389774 + 0.999240i \(0.487590\pi\)
\(488\) 20.0888 0.909378
\(489\) 0 0
\(490\) −4.80021 −0.216851
\(491\) −29.4366 −1.32845 −0.664227 0.747531i \(-0.731239\pi\)
−0.664227 + 0.747531i \(0.731239\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 1.77623 0.0799165
\(495\) 0 0
\(496\) 1.07241 0.0481524
\(497\) 57.9219 2.59815
\(498\) 0 0
\(499\) −13.8879 −0.621709 −0.310855 0.950458i \(-0.600615\pi\)
−0.310855 + 0.950458i \(0.600615\pi\)
\(500\) −3.28356 −0.146845
\(501\) 0 0
\(502\) 24.5504 1.09574
\(503\) 26.5111 1.18207 0.591035 0.806646i \(-0.298719\pi\)
0.591035 + 0.806646i \(0.298719\pi\)
\(504\) 0 0
\(505\) −1.28477 −0.0571717
\(506\) 27.7336 1.23291
\(507\) 0 0
\(508\) 19.7453 0.876055
\(509\) 5.77185 0.255833 0.127916 0.991785i \(-0.459171\pi\)
0.127916 + 0.991785i \(0.459171\pi\)
\(510\) 0 0
\(511\) −57.0608 −2.52422
\(512\) −3.69719 −0.163394
\(513\) 0 0
\(514\) 3.39348 0.149680
\(515\) −2.31536 −0.102027
\(516\) 0 0
\(517\) −27.3353 −1.20220
\(518\) 1.75668 0.0771839
\(519\) 0 0
\(520\) 1.87728 0.0823243
\(521\) −23.7796 −1.04180 −0.520901 0.853617i \(-0.674404\pi\)
−0.520901 + 0.853617i \(0.674404\pi\)
\(522\) 0 0
\(523\) 24.9044 1.08899 0.544497 0.838763i \(-0.316720\pi\)
0.544497 + 0.838763i \(0.316720\pi\)
\(524\) −14.7459 −0.644176
\(525\) 0 0
\(526\) −16.1636 −0.704765
\(527\) −11.1513 −0.485758
\(528\) 0 0
\(529\) 29.4427 1.28012
\(530\) 0.754888 0.0327902
\(531\) 0 0
\(532\) −4.99017 −0.216351
\(533\) 12.6507 0.547963
\(534\) 0 0
\(535\) 2.71304 0.117295
\(536\) 8.22340 0.355197
\(537\) 0 0
\(538\) 13.1272 0.565953
\(539\) 76.9277 3.31351
\(540\) 0 0
\(541\) −20.8777 −0.897604 −0.448802 0.893631i \(-0.648149\pi\)
−0.448802 + 0.893631i \(0.648149\pi\)
\(542\) 27.6445 1.18743
\(543\) 0 0
\(544\) 18.7110 0.802226
\(545\) 2.11800 0.0907253
\(546\) 0 0
\(547\) 0.189502 0.00810253 0.00405126 0.999992i \(-0.498710\pi\)
0.00405126 + 0.999992i \(0.498710\pi\)
\(548\) 11.3171 0.483444
\(549\) 0 0
\(550\) −18.8374 −0.803229
\(551\) 0 0
\(552\) 0 0
\(553\) −34.1124 −1.45061
\(554\) 7.73352 0.328566
\(555\) 0 0
\(556\) 14.4999 0.614934
\(557\) 24.6305 1.04363 0.521814 0.853059i \(-0.325255\pi\)
0.521814 + 0.853059i \(0.325255\pi\)
\(558\) 0 0
\(559\) 17.9812 0.760525
\(560\) 0.470752 0.0198929
\(561\) 0 0
\(562\) 11.6068 0.489602
\(563\) 24.8717 1.04822 0.524109 0.851651i \(-0.324398\pi\)
0.524109 + 0.851651i \(0.324398\pi\)
\(564\) 0 0
\(565\) 1.20787 0.0508156
\(566\) −22.5650 −0.948476
\(567\) 0 0
\(568\) −33.2754 −1.39620
\(569\) 14.2830 0.598775 0.299387 0.954132i \(-0.403218\pi\)
0.299387 + 0.954132i \(0.403218\pi\)
\(570\) 0 0
\(571\) 10.5361 0.440921 0.220460 0.975396i \(-0.429244\pi\)
0.220460 + 0.975396i \(0.429244\pi\)
\(572\) −11.0532 −0.462156
\(573\) 0 0
\(574\) 25.6557 1.07085
\(575\) −35.6204 −1.48547
\(576\) 0 0
\(577\) 8.44369 0.351515 0.175758 0.984433i \(-0.443762\pi\)
0.175758 + 0.984433i \(0.443762\pi\)
\(578\) −4.92931 −0.205032
\(579\) 0 0
\(580\) 0 0
\(581\) −31.6454 −1.31287
\(582\) 0 0
\(583\) −12.0978 −0.501038
\(584\) 32.7807 1.35648
\(585\) 0 0
\(586\) 4.21874 0.174274
\(587\) −9.35632 −0.386177 −0.193088 0.981181i \(-0.561850\pi\)
−0.193088 + 0.981181i \(0.561850\pi\)
\(588\) 0 0
\(589\) −2.78937 −0.114934
\(590\) −1.22192 −0.0503055
\(591\) 0 0
\(592\) −0.124786 −0.00512866
\(593\) 5.06904 0.208161 0.104080 0.994569i \(-0.466810\pi\)
0.104080 + 0.994569i \(0.466810\pi\)
\(594\) 0 0
\(595\) −4.89506 −0.200678
\(596\) −26.2514 −1.07530
\(597\) 0 0
\(598\) 15.0876 0.616976
\(599\) −23.2449 −0.949763 −0.474881 0.880050i \(-0.657509\pi\)
−0.474881 + 0.880050i \(0.657509\pi\)
\(600\) 0 0
\(601\) −6.06774 −0.247508 −0.123754 0.992313i \(-0.539493\pi\)
−0.123754 + 0.992313i \(0.539493\pi\)
\(602\) 36.4661 1.48625
\(603\) 0 0
\(604\) 7.70751 0.313614
\(605\) −1.85030 −0.0752254
\(606\) 0 0
\(607\) −39.3130 −1.59566 −0.797832 0.602880i \(-0.794020\pi\)
−0.797832 + 0.602880i \(0.794020\pi\)
\(608\) 4.68033 0.189813
\(609\) 0 0
\(610\) 1.81097 0.0733242
\(611\) −14.8708 −0.601610
\(612\) 0 0
\(613\) 24.6911 0.997266 0.498633 0.866813i \(-0.333836\pi\)
0.498633 + 0.866813i \(0.333836\pi\)
\(614\) 2.56928 0.103688
\(615\) 0 0
\(616\) −61.0130 −2.45828
\(617\) −18.9571 −0.763183 −0.381592 0.924331i \(-0.624624\pi\)
−0.381592 + 0.924331i \(0.624624\pi\)
\(618\) 0 0
\(619\) −28.6122 −1.15002 −0.575010 0.818147i \(-0.695002\pi\)
−0.575010 + 0.818147i \(0.695002\pi\)
\(620\) −1.08310 −0.0434984
\(621\) 0 0
\(622\) −14.3948 −0.577178
\(623\) −28.7540 −1.15201
\(624\) 0 0
\(625\) 23.7882 0.951527
\(626\) 16.6881 0.666989
\(627\) 0 0
\(628\) 15.5404 0.620131
\(629\) 1.29757 0.0517375
\(630\) 0 0
\(631\) 21.0131 0.836518 0.418259 0.908328i \(-0.362640\pi\)
0.418259 + 0.908328i \(0.362640\pi\)
\(632\) 19.5972 0.779534
\(633\) 0 0
\(634\) −1.72144 −0.0683673
\(635\) 4.84493 0.192265
\(636\) 0 0
\(637\) 41.8500 1.65816
\(638\) 0 0
\(639\) 0 0
\(640\) 1.64627 0.0650747
\(641\) −6.06906 −0.239713 −0.119857 0.992791i \(-0.538244\pi\)
−0.119857 + 0.992791i \(0.538244\pi\)
\(642\) 0 0
\(643\) −45.2441 −1.78425 −0.892126 0.451787i \(-0.850787\pi\)
−0.892126 + 0.451787i \(0.850787\pi\)
\(644\) −42.3872 −1.67029
\(645\) 0 0
\(646\) 2.66078 0.104687
\(647\) 11.5929 0.455762 0.227881 0.973689i \(-0.426820\pi\)
0.227881 + 0.973689i \(0.426820\pi\)
\(648\) 0 0
\(649\) 19.5823 0.768673
\(650\) −10.2479 −0.401954
\(651\) 0 0
\(652\) −11.7968 −0.461998
\(653\) −12.6420 −0.494720 −0.247360 0.968924i \(-0.579563\pi\)
−0.247360 + 0.968924i \(0.579563\pi\)
\(654\) 0 0
\(655\) −3.61821 −0.141375
\(656\) −1.82246 −0.0711551
\(657\) 0 0
\(658\) −30.1582 −1.17569
\(659\) −46.2995 −1.80357 −0.901786 0.432183i \(-0.857744\pi\)
−0.901786 + 0.432183i \(0.857744\pi\)
\(660\) 0 0
\(661\) −3.17597 −0.123531 −0.0617655 0.998091i \(-0.519673\pi\)
−0.0617655 + 0.998091i \(0.519673\pi\)
\(662\) 5.62306 0.218546
\(663\) 0 0
\(664\) 18.1799 0.705515
\(665\) −1.22444 −0.0474819
\(666\) 0 0
\(667\) 0 0
\(668\) 28.4919 1.10239
\(669\) 0 0
\(670\) 0.741326 0.0286399
\(671\) −29.0225 −1.12040
\(672\) 0 0
\(673\) 11.3207 0.436383 0.218191 0.975906i \(-0.429984\pi\)
0.218191 + 0.975906i \(0.429984\pi\)
\(674\) −6.64048 −0.255782
\(675\) 0 0
\(676\) 9.08682 0.349493
\(677\) −4.44930 −0.171000 −0.0855002 0.996338i \(-0.527249\pi\)
−0.0855002 + 0.996338i \(0.527249\pi\)
\(678\) 0 0
\(679\) 2.53370 0.0972345
\(680\) 2.81215 0.107841
\(681\) 0 0
\(682\) −12.5299 −0.479794
\(683\) −25.0062 −0.956835 −0.478418 0.878132i \(-0.658789\pi\)
−0.478418 + 0.878132i \(0.658789\pi\)
\(684\) 0 0
\(685\) 2.77690 0.106100
\(686\) 52.5721 2.00721
\(687\) 0 0
\(688\) −2.59037 −0.0987570
\(689\) −6.58139 −0.250731
\(690\) 0 0
\(691\) −9.03178 −0.343585 −0.171793 0.985133i \(-0.554956\pi\)
−0.171793 + 0.985133i \(0.554956\pi\)
\(692\) −25.6564 −0.975309
\(693\) 0 0
\(694\) 1.38227 0.0524703
\(695\) 3.55787 0.134958
\(696\) 0 0
\(697\) 18.9506 0.717807
\(698\) 13.3431 0.505044
\(699\) 0 0
\(700\) 28.7904 1.08818
\(701\) 26.8957 1.01584 0.507919 0.861405i \(-0.330415\pi\)
0.507919 + 0.861405i \(0.330415\pi\)
\(702\) 0 0
\(703\) 0.324572 0.0122415
\(704\) 23.7659 0.895709
\(705\) 0 0
\(706\) 19.9662 0.751437
\(707\) 22.7160 0.854322
\(708\) 0 0
\(709\) −35.3956 −1.32931 −0.664655 0.747151i \(-0.731421\pi\)
−0.664655 + 0.747151i \(0.731421\pi\)
\(710\) −2.99972 −0.112577
\(711\) 0 0
\(712\) 16.5188 0.619069
\(713\) −23.6933 −0.887320
\(714\) 0 0
\(715\) −2.71213 −0.101428
\(716\) 0.853791 0.0319077
\(717\) 0 0
\(718\) −7.74133 −0.288904
\(719\) −10.5778 −0.394486 −0.197243 0.980355i \(-0.563199\pi\)
−0.197243 + 0.980355i \(0.563199\pi\)
\(720\) 0 0
\(721\) 40.9377 1.52460
\(722\) −16.7323 −0.622713
\(723\) 0 0
\(724\) −15.8857 −0.590386
\(725\) 0 0
\(726\) 0 0
\(727\) 45.8396 1.70010 0.850048 0.526705i \(-0.176573\pi\)
0.850048 + 0.526705i \(0.176573\pi\)
\(728\) −33.1921 −1.23018
\(729\) 0 0
\(730\) 2.95513 0.109374
\(731\) 26.9357 0.996253
\(732\) 0 0
\(733\) −51.9120 −1.91741 −0.958707 0.284394i \(-0.908208\pi\)
−0.958707 + 0.284394i \(0.908208\pi\)
\(734\) −6.30884 −0.232864
\(735\) 0 0
\(736\) 39.7554 1.46540
\(737\) −11.8804 −0.437621
\(738\) 0 0
\(739\) −16.2033 −0.596046 −0.298023 0.954559i \(-0.596327\pi\)
−0.298023 + 0.954559i \(0.596327\pi\)
\(740\) 0.126030 0.00463297
\(741\) 0 0
\(742\) −13.3471 −0.489988
\(743\) −36.1219 −1.32518 −0.662591 0.748981i \(-0.730543\pi\)
−0.662591 + 0.748981i \(0.730543\pi\)
\(744\) 0 0
\(745\) −6.44135 −0.235993
\(746\) −17.6366 −0.645722
\(747\) 0 0
\(748\) −16.5575 −0.605403
\(749\) −47.9690 −1.75275
\(750\) 0 0
\(751\) 41.6701 1.52056 0.760282 0.649593i \(-0.225061\pi\)
0.760282 + 0.649593i \(0.225061\pi\)
\(752\) 2.14229 0.0781213
\(753\) 0 0
\(754\) 0 0
\(755\) 1.89120 0.0688279
\(756\) 0 0
\(757\) 7.81689 0.284110 0.142055 0.989859i \(-0.454629\pi\)
0.142055 + 0.989859i \(0.454629\pi\)
\(758\) −32.0251 −1.16320
\(759\) 0 0
\(760\) 0.703428 0.0255160
\(761\) 21.6344 0.784245 0.392123 0.919913i \(-0.371741\pi\)
0.392123 + 0.919913i \(0.371741\pi\)
\(762\) 0 0
\(763\) −37.4482 −1.35572
\(764\) 1.34855 0.0487887
\(765\) 0 0
\(766\) 18.7539 0.677606
\(767\) 10.6531 0.384661
\(768\) 0 0
\(769\) 4.41685 0.159276 0.0796379 0.996824i \(-0.474624\pi\)
0.0796379 + 0.996824i \(0.474624\pi\)
\(770\) −5.50022 −0.198214
\(771\) 0 0
\(772\) −23.9287 −0.861213
\(773\) −3.01602 −0.108479 −0.0542394 0.998528i \(-0.517273\pi\)
−0.0542394 + 0.998528i \(0.517273\pi\)
\(774\) 0 0
\(775\) 16.0931 0.578081
\(776\) −1.45558 −0.0522523
\(777\) 0 0
\(778\) 6.53561 0.234313
\(779\) 4.74029 0.169838
\(780\) 0 0
\(781\) 48.0733 1.72020
\(782\) 22.6010 0.808211
\(783\) 0 0
\(784\) −6.02890 −0.215318
\(785\) 3.81318 0.136098
\(786\) 0 0
\(787\) −27.7911 −0.990646 −0.495323 0.868709i \(-0.664950\pi\)
−0.495323 + 0.868709i \(0.664950\pi\)
\(788\) 5.99141 0.213435
\(789\) 0 0
\(790\) 1.76665 0.0628547
\(791\) −21.3563 −0.759342
\(792\) 0 0
\(793\) −15.7887 −0.560675
\(794\) −24.0989 −0.855240
\(795\) 0 0
\(796\) 3.25906 0.115514
\(797\) −28.1611 −0.997519 −0.498759 0.866741i \(-0.666211\pi\)
−0.498759 + 0.866741i \(0.666211\pi\)
\(798\) 0 0
\(799\) −22.2764 −0.788082
\(800\) −27.0029 −0.954695
\(801\) 0 0
\(802\) 4.62965 0.163479
\(803\) −47.3586 −1.67125
\(804\) 0 0
\(805\) −10.4006 −0.366573
\(806\) −6.81647 −0.240100
\(807\) 0 0
\(808\) −13.0500 −0.459099
\(809\) 24.8636 0.874159 0.437080 0.899423i \(-0.356013\pi\)
0.437080 + 0.899423i \(0.356013\pi\)
\(810\) 0 0
\(811\) −17.5911 −0.617705 −0.308853 0.951110i \(-0.599945\pi\)
−0.308853 + 0.951110i \(0.599945\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1.45798 0.0511023
\(815\) −2.89459 −0.101393
\(816\) 0 0
\(817\) 6.73766 0.235721
\(818\) 5.90984 0.206633
\(819\) 0 0
\(820\) 1.84064 0.0642778
\(821\) 14.3123 0.499503 0.249752 0.968310i \(-0.419651\pi\)
0.249752 + 0.968310i \(0.419651\pi\)
\(822\) 0 0
\(823\) 15.1918 0.529551 0.264776 0.964310i \(-0.414702\pi\)
0.264776 + 0.964310i \(0.414702\pi\)
\(824\) −23.5182 −0.819295
\(825\) 0 0
\(826\) 21.6046 0.751720
\(827\) 35.3653 1.22977 0.614887 0.788615i \(-0.289202\pi\)
0.614887 + 0.788615i \(0.289202\pi\)
\(828\) 0 0
\(829\) 19.3932 0.673555 0.336777 0.941584i \(-0.390663\pi\)
0.336777 + 0.941584i \(0.390663\pi\)
\(830\) 1.63888 0.0568865
\(831\) 0 0
\(832\) 12.9290 0.448233
\(833\) 62.6909 2.17211
\(834\) 0 0
\(835\) 6.99110 0.241937
\(836\) −4.14168 −0.143243
\(837\) 0 0
\(838\) −1.12778 −0.0389585
\(839\) 35.7024 1.23258 0.616292 0.787518i \(-0.288634\pi\)
0.616292 + 0.787518i \(0.288634\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 8.10199 0.279213
\(843\) 0 0
\(844\) 5.63199 0.193861
\(845\) 2.22965 0.0767022
\(846\) 0 0
\(847\) 32.7150 1.12410
\(848\) 0.948114 0.0325584
\(849\) 0 0
\(850\) −15.3512 −0.526542
\(851\) 2.75696 0.0945074
\(852\) 0 0
\(853\) −2.67243 −0.0915024 −0.0457512 0.998953i \(-0.514568\pi\)
−0.0457512 + 0.998953i \(0.514568\pi\)
\(854\) −32.0197 −1.09569
\(855\) 0 0
\(856\) 27.5576 0.941899
\(857\) 32.1387 1.09784 0.548918 0.835876i \(-0.315040\pi\)
0.548918 + 0.835876i \(0.315040\pi\)
\(858\) 0 0
\(859\) −11.7709 −0.401617 −0.200808 0.979631i \(-0.564357\pi\)
−0.200808 + 0.979631i \(0.564357\pi\)
\(860\) 2.61621 0.0892120
\(861\) 0 0
\(862\) 9.03539 0.307747
\(863\) −31.9118 −1.08629 −0.543146 0.839638i \(-0.682767\pi\)
−0.543146 + 0.839638i \(0.682767\pi\)
\(864\) 0 0
\(865\) −6.29534 −0.214048
\(866\) −30.2997 −1.02963
\(867\) 0 0
\(868\) 19.1502 0.650002
\(869\) −28.3122 −0.960426
\(870\) 0 0
\(871\) −6.46315 −0.218996
\(872\) 21.5135 0.728540
\(873\) 0 0
\(874\) 5.65339 0.191229
\(875\) 14.2454 0.481581
\(876\) 0 0
\(877\) 8.86548 0.299366 0.149683 0.988734i \(-0.452175\pi\)
0.149683 + 0.988734i \(0.452175\pi\)
\(878\) −9.86624 −0.332969
\(879\) 0 0
\(880\) 0.390709 0.0131708
\(881\) −4.16892 −0.140454 −0.0702272 0.997531i \(-0.522372\pi\)
−0.0702272 + 0.997531i \(0.522372\pi\)
\(882\) 0 0
\(883\) 23.5054 0.791018 0.395509 0.918462i \(-0.370568\pi\)
0.395509 + 0.918462i \(0.370568\pi\)
\(884\) −9.00757 −0.302958
\(885\) 0 0
\(886\) −31.1691 −1.04715
\(887\) −20.2426 −0.679680 −0.339840 0.940483i \(-0.610373\pi\)
−0.339840 + 0.940483i \(0.610373\pi\)
\(888\) 0 0
\(889\) −85.6628 −2.87304
\(890\) 1.48914 0.0499162
\(891\) 0 0
\(892\) −12.0748 −0.404295
\(893\) −5.57218 −0.186466
\(894\) 0 0
\(895\) 0.209496 0.00700267
\(896\) −29.1076 −0.972418
\(897\) 0 0
\(898\) 20.0374 0.668657
\(899\) 0 0
\(900\) 0 0
\(901\) −9.85886 −0.328446
\(902\) 21.2934 0.708993
\(903\) 0 0
\(904\) 12.2689 0.408058
\(905\) −3.89789 −0.129570
\(906\) 0 0
\(907\) −2.27974 −0.0756974 −0.0378487 0.999283i \(-0.512050\pi\)
−0.0378487 + 0.999283i \(0.512050\pi\)
\(908\) −32.5049 −1.07871
\(909\) 0 0
\(910\) −2.99221 −0.0991909
\(911\) 18.8815 0.625572 0.312786 0.949824i \(-0.398738\pi\)
0.312786 + 0.949824i \(0.398738\pi\)
\(912\) 0 0
\(913\) −26.2646 −0.869232
\(914\) 26.6429 0.881268
\(915\) 0 0
\(916\) 4.06096 0.134178
\(917\) 63.9733 2.11258
\(918\) 0 0
\(919\) −16.3791 −0.540296 −0.270148 0.962819i \(-0.587073\pi\)
−0.270148 + 0.962819i \(0.587073\pi\)
\(920\) 5.97501 0.196990
\(921\) 0 0
\(922\) −17.1709 −0.565493
\(923\) 26.1527 0.860825
\(924\) 0 0
\(925\) −1.87260 −0.0615707
\(926\) 20.9377 0.688056
\(927\) 0 0
\(928\) 0 0
\(929\) −31.0737 −1.01950 −0.509748 0.860324i \(-0.670261\pi\)
−0.509748 + 0.860324i \(0.670261\pi\)
\(930\) 0 0
\(931\) 15.6814 0.513937
\(932\) −3.89426 −0.127561
\(933\) 0 0
\(934\) −17.9774 −0.588238
\(935\) −4.06274 −0.132866
\(936\) 0 0
\(937\) 26.1466 0.854171 0.427085 0.904211i \(-0.359540\pi\)
0.427085 + 0.904211i \(0.359540\pi\)
\(938\) −13.1073 −0.427969
\(939\) 0 0
\(940\) −2.16366 −0.0705708
\(941\) −40.0522 −1.30566 −0.652832 0.757503i \(-0.726419\pi\)
−0.652832 + 0.757503i \(0.726419\pi\)
\(942\) 0 0
\(943\) 40.2646 1.31120
\(944\) −1.53468 −0.0499497
\(945\) 0 0
\(946\) 30.2656 0.984021
\(947\) −41.8585 −1.36022 −0.680109 0.733111i \(-0.738068\pi\)
−0.680109 + 0.733111i \(0.738068\pi\)
\(948\) 0 0
\(949\) −25.7639 −0.836331
\(950\) −3.83993 −0.124584
\(951\) 0 0
\(952\) −49.7214 −1.61148
\(953\) 30.7325 0.995522 0.497761 0.867314i \(-0.334156\pi\)
0.497761 + 0.867314i \(0.334156\pi\)
\(954\) 0 0
\(955\) 0.330895 0.0107075
\(956\) −21.3645 −0.690976
\(957\) 0 0
\(958\) −20.6233 −0.666309
\(959\) −49.0982 −1.58546
\(960\) 0 0
\(961\) −20.2955 −0.654695
\(962\) 0.793167 0.0255727
\(963\) 0 0
\(964\) −6.53381 −0.210440
\(965\) −5.87142 −0.189008
\(966\) 0 0
\(967\) −11.4758 −0.369037 −0.184519 0.982829i \(-0.559073\pi\)
−0.184519 + 0.982829i \(0.559073\pi\)
\(968\) −18.7944 −0.604074
\(969\) 0 0
\(970\) −0.131218 −0.00421316
\(971\) 12.8983 0.413927 0.206963 0.978349i \(-0.433642\pi\)
0.206963 + 0.978349i \(0.433642\pi\)
\(972\) 0 0
\(973\) −62.9064 −2.01669
\(974\) 1.57525 0.0504744
\(975\) 0 0
\(976\) 2.27452 0.0728057
\(977\) 62.0399 1.98483 0.992415 0.122929i \(-0.0392288\pi\)
0.992415 + 0.122929i \(0.0392288\pi\)
\(978\) 0 0
\(979\) −23.8649 −0.762725
\(980\) 6.08903 0.194507
\(981\) 0 0
\(982\) −26.9544 −0.860150
\(983\) −8.89801 −0.283803 −0.141901 0.989881i \(-0.545322\pi\)
−0.141901 + 0.989881i \(0.545322\pi\)
\(984\) 0 0
\(985\) 1.47012 0.0468420
\(986\) 0 0
\(987\) 0 0
\(988\) −2.25314 −0.0716820
\(989\) 57.2306 1.81983
\(990\) 0 0
\(991\) 2.83908 0.0901863 0.0450931 0.998983i \(-0.485642\pi\)
0.0450931 + 0.998983i \(0.485642\pi\)
\(992\) −17.9612 −0.570269
\(993\) 0 0
\(994\) 53.0378 1.68226
\(995\) 0.799681 0.0253516
\(996\) 0 0
\(997\) −57.0290 −1.80613 −0.903063 0.429508i \(-0.858687\pi\)
−0.903063 + 0.429508i \(0.858687\pi\)
\(998\) −12.7169 −0.402546
\(999\) 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 7569.2.a.bl.1.6 9
3.2 odd 2 2523.2.a.p.1.4 9
29.23 even 7 261.2.k.b.181.2 18
29.24 even 7 261.2.k.b.199.2 18
29.28 even 2 7569.2.a.bk.1.4 9
87.23 odd 14 87.2.g.b.7.2 18
87.53 odd 14 87.2.g.b.25.2 yes 18
87.86 odd 2 2523.2.a.q.1.6 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
87.2.g.b.7.2 18 87.23 odd 14
87.2.g.b.25.2 yes 18 87.53 odd 14
261.2.k.b.181.2 18 29.23 even 7
261.2.k.b.199.2 18 29.24 even 7
2523.2.a.p.1.4 9 3.2 odd 2
2523.2.a.q.1.6 9 87.86 odd 2
7569.2.a.bk.1.4 9 29.28 even 2
7569.2.a.bl.1.6 9 1.1 even 1 trivial