Properties

Label 8001.2.a.s.1.16
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $16$
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] = [8001,2,Mod(1,8001)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8001, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("8001.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8001 = 3^{2} \cdot 7 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8001.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: \(63.8883066572\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 4 x^{15} - 18 x^{14} + 83 x^{13} + 112 x^{12} - 668 x^{11} - 235 x^{10} + 2648 x^{9} - 298 x^{8} - 5422 x^{7} + 2075 x^{6} + 5385 x^{5} - 3163 x^{4} - 1882 x^{3} + \cdots - 20 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 2667)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.16
Root \(-2.57436\) of defining polynomial
Character \(\chi\) \(=\) 8001.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.57436 q^{2} +4.62732 q^{4} -2.78465 q^{5} -1.00000 q^{7} +6.76367 q^{8} +O(q^{10})\) \(q+2.57436 q^{2} +4.62732 q^{4} -2.78465 q^{5} -1.00000 q^{7} +6.76367 q^{8} -7.16870 q^{10} -1.44556 q^{11} +4.74033 q^{13} -2.57436 q^{14} +8.15746 q^{16} -4.74437 q^{17} +5.68628 q^{19} -12.8855 q^{20} -3.72140 q^{22} -0.885825 q^{23} +2.75429 q^{25} +12.2033 q^{26} -4.62732 q^{28} +1.81353 q^{29} +4.74962 q^{31} +7.47290 q^{32} -12.2137 q^{34} +2.78465 q^{35} +5.22583 q^{37} +14.6385 q^{38} -18.8345 q^{40} +5.00751 q^{41} -6.34876 q^{43} -6.68909 q^{44} -2.28043 q^{46} +13.1360 q^{47} +1.00000 q^{49} +7.09054 q^{50} +21.9350 q^{52} -8.73749 q^{53} +4.02539 q^{55} -6.76367 q^{56} +4.66867 q^{58} +7.36544 q^{59} +2.00342 q^{61} +12.2272 q^{62} +2.92299 q^{64} -13.2002 q^{65} -4.76572 q^{67} -21.9537 q^{68} +7.16870 q^{70} +13.1659 q^{71} +12.7431 q^{73} +13.4532 q^{74} +26.3123 q^{76} +1.44556 q^{77} +3.77816 q^{79} -22.7157 q^{80} +12.8911 q^{82} -13.2837 q^{83} +13.2114 q^{85} -16.3440 q^{86} -9.77731 q^{88} +2.64036 q^{89} -4.74033 q^{91} -4.09900 q^{92} +33.8168 q^{94} -15.8343 q^{95} +14.9175 q^{97} +2.57436 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 4 q^{2} + 20 q^{4} - 5 q^{5} - 16 q^{7} - 15 q^{8} - 4 q^{10} - q^{11} + 20 q^{13} + 4 q^{14} + 32 q^{16} - 3 q^{17} + 13 q^{19} - 17 q^{20} + 13 q^{22} - 5 q^{23} + 17 q^{25} + 2 q^{26} - 20 q^{28} - 22 q^{29} + 26 q^{31} - 54 q^{32} - 6 q^{34} + 5 q^{35} + 30 q^{37} - 5 q^{38} + 13 q^{40} - q^{41} + 31 q^{43} - 22 q^{44} - 2 q^{46} + q^{47} + 16 q^{49} - 5 q^{50} + 31 q^{52} - 24 q^{53} + 8 q^{55} + 15 q^{56} + 13 q^{58} + 17 q^{59} + 32 q^{61} + 5 q^{62} + 61 q^{64} + 3 q^{65} + 16 q^{67} + 10 q^{68} + 4 q^{70} + 10 q^{71} + 23 q^{73} - q^{74} + 18 q^{76} + q^{77} + 48 q^{79} - 38 q^{80} + 12 q^{82} - 9 q^{83} + 22 q^{85} + 4 q^{86} + 27 q^{88} - 17 q^{89} - 20 q^{91} - 16 q^{92} + 13 q^{94} - 22 q^{95} + 17 q^{97} - 4 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.57436 1.82035 0.910173 0.414228i \(-0.135948\pi\)
0.910173 + 0.414228i \(0.135948\pi\)
\(3\) 0 0
\(4\) 4.62732 2.31366
\(5\) −2.78465 −1.24533 −0.622667 0.782487i \(-0.713951\pi\)
−0.622667 + 0.782487i \(0.713951\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) 6.76367 2.39132
\(9\) 0 0
\(10\) −7.16870 −2.26694
\(11\) −1.44556 −0.435854 −0.217927 0.975965i \(-0.569929\pi\)
−0.217927 + 0.975965i \(0.569929\pi\)
\(12\) 0 0
\(13\) 4.74033 1.31473 0.657365 0.753572i \(-0.271671\pi\)
0.657365 + 0.753572i \(0.271671\pi\)
\(14\) −2.57436 −0.688026
\(15\) 0 0
\(16\) 8.15746 2.03937
\(17\) −4.74437 −1.15068 −0.575340 0.817914i \(-0.695130\pi\)
−0.575340 + 0.817914i \(0.695130\pi\)
\(18\) 0 0
\(19\) 5.68628 1.30452 0.652261 0.757994i \(-0.273820\pi\)
0.652261 + 0.757994i \(0.273820\pi\)
\(20\) −12.8855 −2.88128
\(21\) 0 0
\(22\) −3.72140 −0.793405
\(23\) −0.885825 −0.184707 −0.0923537 0.995726i \(-0.529439\pi\)
−0.0923537 + 0.995726i \(0.529439\pi\)
\(24\) 0 0
\(25\) 2.75429 0.550858
\(26\) 12.2033 2.39326
\(27\) 0 0
\(28\) −4.62732 −0.874482
\(29\) 1.81353 0.336764 0.168382 0.985722i \(-0.446146\pi\)
0.168382 + 0.985722i \(0.446146\pi\)
\(30\) 0 0
\(31\) 4.74962 0.853057 0.426528 0.904474i \(-0.359736\pi\)
0.426528 + 0.904474i \(0.359736\pi\)
\(32\) 7.47290 1.32103
\(33\) 0 0
\(34\) −12.2137 −2.09464
\(35\) 2.78465 0.470692
\(36\) 0 0
\(37\) 5.22583 0.859122 0.429561 0.903038i \(-0.358668\pi\)
0.429561 + 0.903038i \(0.358668\pi\)
\(38\) 14.6385 2.37468
\(39\) 0 0
\(40\) −18.8345 −2.97799
\(41\) 5.00751 0.782042 0.391021 0.920382i \(-0.372122\pi\)
0.391021 + 0.920382i \(0.372122\pi\)
\(42\) 0 0
\(43\) −6.34876 −0.968177 −0.484088 0.875019i \(-0.660849\pi\)
−0.484088 + 0.875019i \(0.660849\pi\)
\(44\) −6.68909 −1.00842
\(45\) 0 0
\(46\) −2.28043 −0.336231
\(47\) 13.1360 1.91609 0.958043 0.286624i \(-0.0925331\pi\)
0.958043 + 0.286624i \(0.0925331\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 7.09054 1.00275
\(51\) 0 0
\(52\) 21.9350 3.04184
\(53\) −8.73749 −1.20019 −0.600093 0.799930i \(-0.704870\pi\)
−0.600093 + 0.799930i \(0.704870\pi\)
\(54\) 0 0
\(55\) 4.02539 0.542784
\(56\) −6.76367 −0.903833
\(57\) 0 0
\(58\) 4.66867 0.613027
\(59\) 7.36544 0.958898 0.479449 0.877570i \(-0.340836\pi\)
0.479449 + 0.877570i \(0.340836\pi\)
\(60\) 0 0
\(61\) 2.00342 0.256511 0.128256 0.991741i \(-0.459062\pi\)
0.128256 + 0.991741i \(0.459062\pi\)
\(62\) 12.2272 1.55286
\(63\) 0 0
\(64\) 2.92299 0.365374
\(65\) −13.2002 −1.63728
\(66\) 0 0
\(67\) −4.76572 −0.582226 −0.291113 0.956689i \(-0.594025\pi\)
−0.291113 + 0.956689i \(0.594025\pi\)
\(68\) −21.9537 −2.66228
\(69\) 0 0
\(70\) 7.16870 0.856823
\(71\) 13.1659 1.56250 0.781251 0.624217i \(-0.214582\pi\)
0.781251 + 0.624217i \(0.214582\pi\)
\(72\) 0 0
\(73\) 12.7431 1.49147 0.745736 0.666242i \(-0.232098\pi\)
0.745736 + 0.666242i \(0.232098\pi\)
\(74\) 13.4532 1.56390
\(75\) 0 0
\(76\) 26.3123 3.01822
\(77\) 1.44556 0.164737
\(78\) 0 0
\(79\) 3.77816 0.425076 0.212538 0.977153i \(-0.431827\pi\)
0.212538 + 0.977153i \(0.431827\pi\)
\(80\) −22.7157 −2.53969
\(81\) 0 0
\(82\) 12.8911 1.42359
\(83\) −13.2837 −1.45807 −0.729036 0.684475i \(-0.760031\pi\)
−0.729036 + 0.684475i \(0.760031\pi\)
\(84\) 0 0
\(85\) 13.2114 1.43298
\(86\) −16.3440 −1.76242
\(87\) 0 0
\(88\) −9.77731 −1.04226
\(89\) 2.64036 0.279877 0.139939 0.990160i \(-0.455309\pi\)
0.139939 + 0.990160i \(0.455309\pi\)
\(90\) 0 0
\(91\) −4.74033 −0.496921
\(92\) −4.09900 −0.427350
\(93\) 0 0
\(94\) 33.8168 3.48794
\(95\) −15.8343 −1.62457
\(96\) 0 0
\(97\) 14.9175 1.51464 0.757322 0.653041i \(-0.226507\pi\)
0.757322 + 0.653041i \(0.226507\pi\)
\(98\) 2.57436 0.260049
\(99\) 0 0
\(100\) 12.7450 1.27450
\(101\) 4.76405 0.474041 0.237020 0.971505i \(-0.423829\pi\)
0.237020 + 0.971505i \(0.423829\pi\)
\(102\) 0 0
\(103\) 3.52975 0.347797 0.173898 0.984764i \(-0.444364\pi\)
0.173898 + 0.984764i \(0.444364\pi\)
\(104\) 32.0620 3.14394
\(105\) 0 0
\(106\) −22.4934 −2.18476
\(107\) 4.73776 0.458017 0.229008 0.973424i \(-0.426452\pi\)
0.229008 + 0.973424i \(0.426452\pi\)
\(108\) 0 0
\(109\) −17.8152 −1.70638 −0.853191 0.521598i \(-0.825336\pi\)
−0.853191 + 0.521598i \(0.825336\pi\)
\(110\) 10.3628 0.988054
\(111\) 0 0
\(112\) −8.15746 −0.770808
\(113\) 16.8253 1.58279 0.791394 0.611307i \(-0.209356\pi\)
0.791394 + 0.611307i \(0.209356\pi\)
\(114\) 0 0
\(115\) 2.46672 0.230022
\(116\) 8.39178 0.779157
\(117\) 0 0
\(118\) 18.9613 1.74553
\(119\) 4.74437 0.434916
\(120\) 0 0
\(121\) −8.91035 −0.810032
\(122\) 5.15752 0.466940
\(123\) 0 0
\(124\) 21.9780 1.97368
\(125\) 6.25352 0.559332
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) −7.42097 −0.655927
\(129\) 0 0
\(130\) −33.9820 −2.98042
\(131\) 3.37663 0.295017 0.147509 0.989061i \(-0.452875\pi\)
0.147509 + 0.989061i \(0.452875\pi\)
\(132\) 0 0
\(133\) −5.68628 −0.493063
\(134\) −12.2687 −1.05985
\(135\) 0 0
\(136\) −32.0894 −2.75164
\(137\) 22.4118 1.91477 0.957386 0.288811i \(-0.0932598\pi\)
0.957386 + 0.288811i \(0.0932598\pi\)
\(138\) 0 0
\(139\) 8.35723 0.708851 0.354426 0.935084i \(-0.384676\pi\)
0.354426 + 0.935084i \(0.384676\pi\)
\(140\) 12.8855 1.08902
\(141\) 0 0
\(142\) 33.8937 2.84430
\(143\) −6.85244 −0.573030
\(144\) 0 0
\(145\) −5.05005 −0.419384
\(146\) 32.8054 2.71499
\(147\) 0 0
\(148\) 24.1816 1.98772
\(149\) −15.1765 −1.24331 −0.621654 0.783292i \(-0.713539\pi\)
−0.621654 + 0.783292i \(0.713539\pi\)
\(150\) 0 0
\(151\) −7.08540 −0.576602 −0.288301 0.957540i \(-0.593090\pi\)
−0.288301 + 0.957540i \(0.593090\pi\)
\(152\) 38.4601 3.11953
\(153\) 0 0
\(154\) 3.72140 0.299879
\(155\) −13.2260 −1.06234
\(156\) 0 0
\(157\) −11.8656 −0.946975 −0.473488 0.880800i \(-0.657005\pi\)
−0.473488 + 0.880800i \(0.657005\pi\)
\(158\) 9.72633 0.773785
\(159\) 0 0
\(160\) −20.8094 −1.64513
\(161\) 0.885825 0.0698128
\(162\) 0 0
\(163\) 2.21744 0.173683 0.0868416 0.996222i \(-0.472323\pi\)
0.0868416 + 0.996222i \(0.472323\pi\)
\(164\) 23.1714 1.80938
\(165\) 0 0
\(166\) −34.1969 −2.65420
\(167\) −5.09063 −0.393925 −0.196962 0.980411i \(-0.563108\pi\)
−0.196962 + 0.980411i \(0.563108\pi\)
\(168\) 0 0
\(169\) 9.47071 0.728516
\(170\) 34.0110 2.60852
\(171\) 0 0
\(172\) −29.3777 −2.24003
\(173\) −4.99185 −0.379524 −0.189762 0.981830i \(-0.560772\pi\)
−0.189762 + 0.981830i \(0.560772\pi\)
\(174\) 0 0
\(175\) −2.75429 −0.208205
\(176\) −11.7921 −0.888865
\(177\) 0 0
\(178\) 6.79722 0.509473
\(179\) 16.1437 1.20663 0.603317 0.797502i \(-0.293846\pi\)
0.603317 + 0.797502i \(0.293846\pi\)
\(180\) 0 0
\(181\) 19.5581 1.45374 0.726870 0.686775i \(-0.240974\pi\)
0.726870 + 0.686775i \(0.240974\pi\)
\(182\) −12.2033 −0.904569
\(183\) 0 0
\(184\) −5.99143 −0.441694
\(185\) −14.5521 −1.06989
\(186\) 0 0
\(187\) 6.85829 0.501528
\(188\) 60.7846 4.43317
\(189\) 0 0
\(190\) −40.7632 −2.95728
\(191\) −1.86303 −0.134804 −0.0674019 0.997726i \(-0.521471\pi\)
−0.0674019 + 0.997726i \(0.521471\pi\)
\(192\) 0 0
\(193\) 11.9749 0.861974 0.430987 0.902358i \(-0.358165\pi\)
0.430987 + 0.902358i \(0.358165\pi\)
\(194\) 38.4031 2.75718
\(195\) 0 0
\(196\) 4.62732 0.330523
\(197\) 1.66527 0.118646 0.0593228 0.998239i \(-0.481106\pi\)
0.0593228 + 0.998239i \(0.481106\pi\)
\(198\) 0 0
\(199\) −13.1338 −0.931029 −0.465514 0.885040i \(-0.654131\pi\)
−0.465514 + 0.885040i \(0.654131\pi\)
\(200\) 18.6291 1.31728
\(201\) 0 0
\(202\) 12.2644 0.862918
\(203\) −1.81353 −0.127285
\(204\) 0 0
\(205\) −13.9442 −0.973903
\(206\) 9.08685 0.633111
\(207\) 0 0
\(208\) 38.6690 2.68122
\(209\) −8.21988 −0.568581
\(210\) 0 0
\(211\) 3.92135 0.269957 0.134978 0.990849i \(-0.456903\pi\)
0.134978 + 0.990849i \(0.456903\pi\)
\(212\) −40.4312 −2.77683
\(213\) 0 0
\(214\) 12.1967 0.833749
\(215\) 17.6791 1.20570
\(216\) 0 0
\(217\) −4.74962 −0.322425
\(218\) −45.8626 −3.10621
\(219\) 0 0
\(220\) 18.6268 1.25582
\(221\) −22.4899 −1.51283
\(222\) 0 0
\(223\) 4.53320 0.303565 0.151783 0.988414i \(-0.451499\pi\)
0.151783 + 0.988414i \(0.451499\pi\)
\(224\) −7.47290 −0.499304
\(225\) 0 0
\(226\) 43.3142 2.88122
\(227\) −18.1116 −1.20211 −0.601055 0.799208i \(-0.705253\pi\)
−0.601055 + 0.799208i \(0.705253\pi\)
\(228\) 0 0
\(229\) 16.5931 1.09650 0.548250 0.836314i \(-0.315294\pi\)
0.548250 + 0.836314i \(0.315294\pi\)
\(230\) 6.35021 0.418721
\(231\) 0 0
\(232\) 12.2661 0.805309
\(233\) 14.4081 0.943906 0.471953 0.881624i \(-0.343549\pi\)
0.471953 + 0.881624i \(0.343549\pi\)
\(234\) 0 0
\(235\) −36.5793 −2.38617
\(236\) 34.0823 2.21857
\(237\) 0 0
\(238\) 12.2137 0.791698
\(239\) −18.7529 −1.21302 −0.606511 0.795075i \(-0.707431\pi\)
−0.606511 + 0.795075i \(0.707431\pi\)
\(240\) 0 0
\(241\) −19.1134 −1.23120 −0.615601 0.788058i \(-0.711087\pi\)
−0.615601 + 0.788058i \(0.711087\pi\)
\(242\) −22.9384 −1.47454
\(243\) 0 0
\(244\) 9.27046 0.593480
\(245\) −2.78465 −0.177905
\(246\) 0 0
\(247\) 26.9548 1.71510
\(248\) 32.1248 2.03993
\(249\) 0 0
\(250\) 16.0988 1.01818
\(251\) 11.8661 0.748981 0.374491 0.927231i \(-0.377818\pi\)
0.374491 + 0.927231i \(0.377818\pi\)
\(252\) 0 0
\(253\) 1.28052 0.0805054
\(254\) −2.57436 −0.161530
\(255\) 0 0
\(256\) −24.9502 −1.55939
\(257\) −31.6311 −1.97310 −0.986548 0.163470i \(-0.947731\pi\)
−0.986548 + 0.163470i \(0.947731\pi\)
\(258\) 0 0
\(259\) −5.22583 −0.324718
\(260\) −61.0814 −3.78811
\(261\) 0 0
\(262\) 8.69265 0.537034
\(263\) 6.20289 0.382487 0.191243 0.981543i \(-0.438748\pi\)
0.191243 + 0.981543i \(0.438748\pi\)
\(264\) 0 0
\(265\) 24.3309 1.49463
\(266\) −14.6385 −0.897546
\(267\) 0 0
\(268\) −22.0525 −1.34707
\(269\) 16.2838 0.992841 0.496421 0.868082i \(-0.334647\pi\)
0.496421 + 0.868082i \(0.334647\pi\)
\(270\) 0 0
\(271\) −16.3251 −0.991679 −0.495840 0.868414i \(-0.665140\pi\)
−0.495840 + 0.868414i \(0.665140\pi\)
\(272\) −38.7021 −2.34666
\(273\) 0 0
\(274\) 57.6961 3.48555
\(275\) −3.98150 −0.240094
\(276\) 0 0
\(277\) −8.19322 −0.492283 −0.246141 0.969234i \(-0.579163\pi\)
−0.246141 + 0.969234i \(0.579163\pi\)
\(278\) 21.5145 1.29035
\(279\) 0 0
\(280\) 18.8345 1.12557
\(281\) −14.9657 −0.892777 −0.446389 0.894839i \(-0.647290\pi\)
−0.446389 + 0.894839i \(0.647290\pi\)
\(282\) 0 0
\(283\) −0.946807 −0.0562818 −0.0281409 0.999604i \(-0.508959\pi\)
−0.0281409 + 0.999604i \(0.508959\pi\)
\(284\) 60.9228 3.61510
\(285\) 0 0
\(286\) −17.6406 −1.04311
\(287\) −5.00751 −0.295584
\(288\) 0 0
\(289\) 5.50909 0.324064
\(290\) −13.0006 −0.763423
\(291\) 0 0
\(292\) 58.9666 3.45076
\(293\) −23.1429 −1.35202 −0.676012 0.736891i \(-0.736293\pi\)
−0.676012 + 0.736891i \(0.736293\pi\)
\(294\) 0 0
\(295\) −20.5102 −1.19415
\(296\) 35.3458 2.05443
\(297\) 0 0
\(298\) −39.0698 −2.26325
\(299\) −4.19910 −0.242840
\(300\) 0 0
\(301\) 6.34876 0.365936
\(302\) −18.2404 −1.04961
\(303\) 0 0
\(304\) 46.3856 2.66040
\(305\) −5.57882 −0.319443
\(306\) 0 0
\(307\) −16.7783 −0.957588 −0.478794 0.877927i \(-0.658926\pi\)
−0.478794 + 0.877927i \(0.658926\pi\)
\(308\) 6.68909 0.381146
\(309\) 0 0
\(310\) −34.0486 −1.93383
\(311\) −10.2652 −0.582088 −0.291044 0.956710i \(-0.594003\pi\)
−0.291044 + 0.956710i \(0.594003\pi\)
\(312\) 0 0
\(313\) 26.3636 1.49016 0.745079 0.666976i \(-0.232412\pi\)
0.745079 + 0.666976i \(0.232412\pi\)
\(314\) −30.5462 −1.72382
\(315\) 0 0
\(316\) 17.4827 0.983481
\(317\) 6.89867 0.387468 0.193734 0.981054i \(-0.437940\pi\)
0.193734 + 0.981054i \(0.437940\pi\)
\(318\) 0 0
\(319\) −2.62157 −0.146780
\(320\) −8.13952 −0.455013
\(321\) 0 0
\(322\) 2.28043 0.127083
\(323\) −26.9779 −1.50109
\(324\) 0 0
\(325\) 13.0562 0.724230
\(326\) 5.70848 0.316164
\(327\) 0 0
\(328\) 33.8691 1.87011
\(329\) −13.1360 −0.724213
\(330\) 0 0
\(331\) −7.07030 −0.388619 −0.194309 0.980940i \(-0.562247\pi\)
−0.194309 + 0.980940i \(0.562247\pi\)
\(332\) −61.4678 −3.37348
\(333\) 0 0
\(334\) −13.1051 −0.717080
\(335\) 13.2709 0.725066
\(336\) 0 0
\(337\) −12.3787 −0.674312 −0.337156 0.941449i \(-0.609465\pi\)
−0.337156 + 0.941449i \(0.609465\pi\)
\(338\) 24.3810 1.32615
\(339\) 0 0
\(340\) 61.1336 3.31543
\(341\) −6.86587 −0.371808
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −42.9409 −2.31522
\(345\) 0 0
\(346\) −12.8508 −0.690864
\(347\) 15.0504 0.807948 0.403974 0.914770i \(-0.367629\pi\)
0.403974 + 0.914770i \(0.367629\pi\)
\(348\) 0 0
\(349\) 23.7118 1.26926 0.634631 0.772816i \(-0.281152\pi\)
0.634631 + 0.772816i \(0.281152\pi\)
\(350\) −7.09054 −0.379005
\(351\) 0 0
\(352\) −10.8025 −0.575778
\(353\) −22.5750 −1.20154 −0.600772 0.799420i \(-0.705140\pi\)
−0.600772 + 0.799420i \(0.705140\pi\)
\(354\) 0 0
\(355\) −36.6624 −1.94584
\(356\) 12.2178 0.647541
\(357\) 0 0
\(358\) 41.5595 2.19649
\(359\) 30.2864 1.59846 0.799228 0.601029i \(-0.205242\pi\)
0.799228 + 0.601029i \(0.205242\pi\)
\(360\) 0 0
\(361\) 13.3338 0.701780
\(362\) 50.3495 2.64631
\(363\) 0 0
\(364\) −21.9350 −1.14971
\(365\) −35.4852 −1.85738
\(366\) 0 0
\(367\) 2.31654 0.120922 0.0604612 0.998171i \(-0.480743\pi\)
0.0604612 + 0.998171i \(0.480743\pi\)
\(368\) −7.22609 −0.376686
\(369\) 0 0
\(370\) −37.4624 −1.94758
\(371\) 8.73749 0.453628
\(372\) 0 0
\(373\) 26.2206 1.35765 0.678826 0.734299i \(-0.262489\pi\)
0.678826 + 0.734299i \(0.262489\pi\)
\(374\) 17.6557 0.912955
\(375\) 0 0
\(376\) 88.8477 4.58197
\(377\) 8.59672 0.442754
\(378\) 0 0
\(379\) −8.95185 −0.459825 −0.229913 0.973211i \(-0.573844\pi\)
−0.229913 + 0.973211i \(0.573844\pi\)
\(380\) −73.2705 −3.75870
\(381\) 0 0
\(382\) −4.79610 −0.245390
\(383\) −20.6076 −1.05300 −0.526500 0.850175i \(-0.676496\pi\)
−0.526500 + 0.850175i \(0.676496\pi\)
\(384\) 0 0
\(385\) −4.02539 −0.205153
\(386\) 30.8278 1.56909
\(387\) 0 0
\(388\) 69.0282 3.50437
\(389\) 11.0810 0.561828 0.280914 0.959733i \(-0.409362\pi\)
0.280914 + 0.959733i \(0.409362\pi\)
\(390\) 0 0
\(391\) 4.20269 0.212539
\(392\) 6.76367 0.341617
\(393\) 0 0
\(394\) 4.28701 0.215976
\(395\) −10.5209 −0.529362
\(396\) 0 0
\(397\) −25.2409 −1.26680 −0.633402 0.773823i \(-0.718342\pi\)
−0.633402 + 0.773823i \(0.718342\pi\)
\(398\) −33.8111 −1.69480
\(399\) 0 0
\(400\) 22.4680 1.12340
\(401\) 6.42176 0.320687 0.160344 0.987061i \(-0.448740\pi\)
0.160344 + 0.987061i \(0.448740\pi\)
\(402\) 0 0
\(403\) 22.5147 1.12154
\(404\) 22.0448 1.09677
\(405\) 0 0
\(406\) −4.66867 −0.231702
\(407\) −7.55427 −0.374451
\(408\) 0 0
\(409\) 21.8312 1.07948 0.539742 0.841830i \(-0.318522\pi\)
0.539742 + 0.841830i \(0.318522\pi\)
\(410\) −35.8973 −1.77284
\(411\) 0 0
\(412\) 16.3333 0.804684
\(413\) −7.36544 −0.362430
\(414\) 0 0
\(415\) 36.9904 1.81579
\(416\) 35.4240 1.73680
\(417\) 0 0
\(418\) −21.1609 −1.03501
\(419\) −21.5047 −1.05057 −0.525286 0.850926i \(-0.676042\pi\)
−0.525286 + 0.850926i \(0.676042\pi\)
\(420\) 0 0
\(421\) 4.18572 0.204000 0.102000 0.994784i \(-0.467476\pi\)
0.102000 + 0.994784i \(0.467476\pi\)
\(422\) 10.0950 0.491415
\(423\) 0 0
\(424\) −59.0975 −2.87003
\(425\) −13.0674 −0.633862
\(426\) 0 0
\(427\) −2.00342 −0.0969522
\(428\) 21.9231 1.05970
\(429\) 0 0
\(430\) 45.5123 2.19480
\(431\) 14.6079 0.703638 0.351819 0.936068i \(-0.385563\pi\)
0.351819 + 0.936068i \(0.385563\pi\)
\(432\) 0 0
\(433\) −31.6255 −1.51983 −0.759913 0.650025i \(-0.774759\pi\)
−0.759913 + 0.650025i \(0.774759\pi\)
\(434\) −12.2272 −0.586925
\(435\) 0 0
\(436\) −82.4364 −3.94799
\(437\) −5.03705 −0.240955
\(438\) 0 0
\(439\) 6.74432 0.321889 0.160944 0.986963i \(-0.448546\pi\)
0.160944 + 0.986963i \(0.448546\pi\)
\(440\) 27.2264 1.29797
\(441\) 0 0
\(442\) −57.8970 −2.75388
\(443\) 18.9280 0.899294 0.449647 0.893206i \(-0.351550\pi\)
0.449647 + 0.893206i \(0.351550\pi\)
\(444\) 0 0
\(445\) −7.35248 −0.348541
\(446\) 11.6701 0.552594
\(447\) 0 0
\(448\) −2.92299 −0.138098
\(449\) −10.7611 −0.507846 −0.253923 0.967224i \(-0.581721\pi\)
−0.253923 + 0.967224i \(0.581721\pi\)
\(450\) 0 0
\(451\) −7.23867 −0.340856
\(452\) 77.8559 3.66203
\(453\) 0 0
\(454\) −46.6258 −2.18825
\(455\) 13.2002 0.618833
\(456\) 0 0
\(457\) −6.44982 −0.301710 −0.150855 0.988556i \(-0.548203\pi\)
−0.150855 + 0.988556i \(0.548203\pi\)
\(458\) 42.7165 1.99601
\(459\) 0 0
\(460\) 11.4143 0.532194
\(461\) −5.25564 −0.244779 −0.122390 0.992482i \(-0.539056\pi\)
−0.122390 + 0.992482i \(0.539056\pi\)
\(462\) 0 0
\(463\) −23.1048 −1.07377 −0.536886 0.843655i \(-0.680399\pi\)
−0.536886 + 0.843655i \(0.680399\pi\)
\(464\) 14.7938 0.686785
\(465\) 0 0
\(466\) 37.0916 1.71824
\(467\) −8.23695 −0.381160 −0.190580 0.981672i \(-0.561037\pi\)
−0.190580 + 0.981672i \(0.561037\pi\)
\(468\) 0 0
\(469\) 4.76572 0.220061
\(470\) −94.1682 −4.34365
\(471\) 0 0
\(472\) 49.8174 2.29303
\(473\) 9.17753 0.421983
\(474\) 0 0
\(475\) 15.6617 0.718607
\(476\) 21.9537 1.00625
\(477\) 0 0
\(478\) −48.2766 −2.20812
\(479\) −32.9121 −1.50379 −0.751897 0.659281i \(-0.770861\pi\)
−0.751897 + 0.659281i \(0.770861\pi\)
\(480\) 0 0
\(481\) 24.7722 1.12951
\(482\) −49.2048 −2.24122
\(483\) 0 0
\(484\) −41.2310 −1.87414
\(485\) −41.5401 −1.88624
\(486\) 0 0
\(487\) 0.480422 0.0217700 0.0108850 0.999941i \(-0.496535\pi\)
0.0108850 + 0.999941i \(0.496535\pi\)
\(488\) 13.5505 0.613400
\(489\) 0 0
\(490\) −7.16870 −0.323849
\(491\) 15.2161 0.686694 0.343347 0.939209i \(-0.388439\pi\)
0.343347 + 0.939209i \(0.388439\pi\)
\(492\) 0 0
\(493\) −8.60406 −0.387507
\(494\) 69.3914 3.12207
\(495\) 0 0
\(496\) 38.7448 1.73969
\(497\) −13.1659 −0.590571
\(498\) 0 0
\(499\) −5.10495 −0.228529 −0.114264 0.993450i \(-0.536451\pi\)
−0.114264 + 0.993450i \(0.536451\pi\)
\(500\) 28.9370 1.29410
\(501\) 0 0
\(502\) 30.5476 1.36341
\(503\) −17.3084 −0.771741 −0.385871 0.922553i \(-0.626099\pi\)
−0.385871 + 0.922553i \(0.626099\pi\)
\(504\) 0 0
\(505\) −13.2662 −0.590339
\(506\) 3.29651 0.146548
\(507\) 0 0
\(508\) −4.62732 −0.205304
\(509\) 32.5988 1.44492 0.722458 0.691415i \(-0.243012\pi\)
0.722458 + 0.691415i \(0.243012\pi\)
\(510\) 0 0
\(511\) −12.7431 −0.563723
\(512\) −49.3889 −2.18270
\(513\) 0 0
\(514\) −81.4299 −3.59172
\(515\) −9.82913 −0.433123
\(516\) 0 0
\(517\) −18.9890 −0.835133
\(518\) −13.4532 −0.591098
\(519\) 0 0
\(520\) −89.2816 −3.91526
\(521\) 20.6553 0.904925 0.452463 0.891783i \(-0.350546\pi\)
0.452463 + 0.891783i \(0.350546\pi\)
\(522\) 0 0
\(523\) 42.8597 1.87412 0.937061 0.349166i \(-0.113535\pi\)
0.937061 + 0.349166i \(0.113535\pi\)
\(524\) 15.6247 0.682570
\(525\) 0 0
\(526\) 15.9685 0.696258
\(527\) −22.5340 −0.981595
\(528\) 0 0
\(529\) −22.2153 −0.965883
\(530\) 62.6364 2.72075
\(531\) 0 0
\(532\) −26.3123 −1.14078
\(533\) 23.7372 1.02817
\(534\) 0 0
\(535\) −13.1930 −0.570384
\(536\) −32.2338 −1.39229
\(537\) 0 0
\(538\) 41.9204 1.80732
\(539\) −1.44556 −0.0622648
\(540\) 0 0
\(541\) −17.7567 −0.763419 −0.381709 0.924282i \(-0.624664\pi\)
−0.381709 + 0.924282i \(0.624664\pi\)
\(542\) −42.0267 −1.80520
\(543\) 0 0
\(544\) −35.4542 −1.52009
\(545\) 49.6090 2.12502
\(546\) 0 0
\(547\) −6.99775 −0.299202 −0.149601 0.988746i \(-0.547799\pi\)
−0.149601 + 0.988746i \(0.547799\pi\)
\(548\) 103.707 4.43013
\(549\) 0 0
\(550\) −10.2498 −0.437054
\(551\) 10.3122 0.439316
\(552\) 0 0
\(553\) −3.77816 −0.160664
\(554\) −21.0923 −0.896125
\(555\) 0 0
\(556\) 38.6716 1.64004
\(557\) −31.1241 −1.31877 −0.659384 0.751806i \(-0.729183\pi\)
−0.659384 + 0.751806i \(0.729183\pi\)
\(558\) 0 0
\(559\) −30.0952 −1.27289
\(560\) 22.7157 0.959914
\(561\) 0 0
\(562\) −38.5270 −1.62516
\(563\) 40.6243 1.71211 0.856055 0.516884i \(-0.172908\pi\)
0.856055 + 0.516884i \(0.172908\pi\)
\(564\) 0 0
\(565\) −46.8525 −1.97110
\(566\) −2.43742 −0.102452
\(567\) 0 0
\(568\) 89.0497 3.73644
\(569\) −45.2818 −1.89831 −0.949155 0.314809i \(-0.898060\pi\)
−0.949155 + 0.314809i \(0.898060\pi\)
\(570\) 0 0
\(571\) 13.5280 0.566131 0.283065 0.959101i \(-0.408649\pi\)
0.283065 + 0.959101i \(0.408649\pi\)
\(572\) −31.7085 −1.32580
\(573\) 0 0
\(574\) −12.8911 −0.538065
\(575\) −2.43982 −0.101748
\(576\) 0 0
\(577\) 25.8904 1.07783 0.538916 0.842360i \(-0.318834\pi\)
0.538916 + 0.842360i \(0.318834\pi\)
\(578\) 14.1824 0.589909
\(579\) 0 0
\(580\) −23.3682 −0.970311
\(581\) 13.2837 0.551100
\(582\) 0 0
\(583\) 12.6306 0.523106
\(584\) 86.1904 3.56658
\(585\) 0 0
\(586\) −59.5782 −2.46115
\(587\) −46.0434 −1.90041 −0.950207 0.311621i \(-0.899128\pi\)
−0.950207 + 0.311621i \(0.899128\pi\)
\(588\) 0 0
\(589\) 27.0077 1.11283
\(590\) −52.8006 −2.17377
\(591\) 0 0
\(592\) 42.6295 1.75206
\(593\) −27.1990 −1.11693 −0.558465 0.829528i \(-0.688610\pi\)
−0.558465 + 0.829528i \(0.688610\pi\)
\(594\) 0 0
\(595\) −13.2114 −0.541616
\(596\) −70.2266 −2.87659
\(597\) 0 0
\(598\) −10.8100 −0.442054
\(599\) −26.4774 −1.08184 −0.540918 0.841075i \(-0.681923\pi\)
−0.540918 + 0.841075i \(0.681923\pi\)
\(600\) 0 0
\(601\) 20.4346 0.833546 0.416773 0.909011i \(-0.363161\pi\)
0.416773 + 0.909011i \(0.363161\pi\)
\(602\) 16.3440 0.666131
\(603\) 0 0
\(604\) −32.7864 −1.33406
\(605\) 24.8122 1.00876
\(606\) 0 0
\(607\) 38.3850 1.55800 0.779000 0.627024i \(-0.215727\pi\)
0.779000 + 0.627024i \(0.215727\pi\)
\(608\) 42.4930 1.72332
\(609\) 0 0
\(610\) −14.3619 −0.581496
\(611\) 62.2691 2.51914
\(612\) 0 0
\(613\) −31.1100 −1.25652 −0.628260 0.778003i \(-0.716233\pi\)
−0.628260 + 0.778003i \(0.716233\pi\)
\(614\) −43.1933 −1.74314
\(615\) 0 0
\(616\) 9.77731 0.393939
\(617\) 3.99996 0.161032 0.0805161 0.996753i \(-0.474343\pi\)
0.0805161 + 0.996753i \(0.474343\pi\)
\(618\) 0 0
\(619\) −37.4699 −1.50604 −0.753021 0.657996i \(-0.771404\pi\)
−0.753021 + 0.657996i \(0.771404\pi\)
\(620\) −61.2011 −2.45790
\(621\) 0 0
\(622\) −26.4264 −1.05960
\(623\) −2.64036 −0.105784
\(624\) 0 0
\(625\) −31.1853 −1.24741
\(626\) 67.8694 2.71260
\(627\) 0 0
\(628\) −54.9058 −2.19098
\(629\) −24.7933 −0.988574
\(630\) 0 0
\(631\) −23.9103 −0.951853 −0.475926 0.879485i \(-0.657887\pi\)
−0.475926 + 0.879485i \(0.657887\pi\)
\(632\) 25.5542 1.01649
\(633\) 0 0
\(634\) 17.7596 0.705326
\(635\) 2.78465 0.110506
\(636\) 0 0
\(637\) 4.74033 0.187819
\(638\) −6.74886 −0.267190
\(639\) 0 0
\(640\) 20.6648 0.816849
\(641\) −29.3723 −1.16014 −0.580069 0.814567i \(-0.696974\pi\)
−0.580069 + 0.814567i \(0.696974\pi\)
\(642\) 0 0
\(643\) −3.61735 −0.142654 −0.0713272 0.997453i \(-0.522723\pi\)
−0.0713272 + 0.997453i \(0.522723\pi\)
\(644\) 4.09900 0.161523
\(645\) 0 0
\(646\) −69.4507 −2.73250
\(647\) −12.6711 −0.498152 −0.249076 0.968484i \(-0.580127\pi\)
−0.249076 + 0.968484i \(0.580127\pi\)
\(648\) 0 0
\(649\) −10.6472 −0.417939
\(650\) 33.6115 1.31835
\(651\) 0 0
\(652\) 10.2608 0.401844
\(653\) −16.5935 −0.649355 −0.324678 0.945825i \(-0.605256\pi\)
−0.324678 + 0.945825i \(0.605256\pi\)
\(654\) 0 0
\(655\) −9.40273 −0.367395
\(656\) 40.8486 1.59487
\(657\) 0 0
\(658\) −33.8168 −1.31832
\(659\) 15.9424 0.621028 0.310514 0.950569i \(-0.399499\pi\)
0.310514 + 0.950569i \(0.399499\pi\)
\(660\) 0 0
\(661\) −25.7378 −1.00109 −0.500543 0.865712i \(-0.666866\pi\)
−0.500543 + 0.865712i \(0.666866\pi\)
\(662\) −18.2015 −0.707421
\(663\) 0 0
\(664\) −89.8463 −3.48671
\(665\) 15.8343 0.614029
\(666\) 0 0
\(667\) −1.60647 −0.0622027
\(668\) −23.5560 −0.911408
\(669\) 0 0
\(670\) 34.1640 1.31987
\(671\) −2.89607 −0.111801
\(672\) 0 0
\(673\) −15.7394 −0.606709 −0.303355 0.952878i \(-0.598107\pi\)
−0.303355 + 0.952878i \(0.598107\pi\)
\(674\) −31.8673 −1.22748
\(675\) 0 0
\(676\) 43.8240 1.68554
\(677\) 6.89543 0.265013 0.132506 0.991182i \(-0.457697\pi\)
0.132506 + 0.991182i \(0.457697\pi\)
\(678\) 0 0
\(679\) −14.9175 −0.572482
\(680\) 89.3578 3.42671
\(681\) 0 0
\(682\) −17.6752 −0.676819
\(683\) 10.6634 0.408025 0.204013 0.978968i \(-0.434602\pi\)
0.204013 + 0.978968i \(0.434602\pi\)
\(684\) 0 0
\(685\) −62.4092 −2.38453
\(686\) −2.57436 −0.0982895
\(687\) 0 0
\(688\) −51.7898 −1.97447
\(689\) −41.4186 −1.57792
\(690\) 0 0
\(691\) −10.2270 −0.389052 −0.194526 0.980897i \(-0.562317\pi\)
−0.194526 + 0.980897i \(0.562317\pi\)
\(692\) −23.0989 −0.878089
\(693\) 0 0
\(694\) 38.7451 1.47075
\(695\) −23.2720 −0.882757
\(696\) 0 0
\(697\) −23.7575 −0.899879
\(698\) 61.0426 2.31049
\(699\) 0 0
\(700\) −12.7450 −0.481716
\(701\) 2.61443 0.0987457 0.0493729 0.998780i \(-0.484278\pi\)
0.0493729 + 0.998780i \(0.484278\pi\)
\(702\) 0 0
\(703\) 29.7156 1.12074
\(704\) −4.22537 −0.159250
\(705\) 0 0
\(706\) −58.1161 −2.18723
\(707\) −4.76405 −0.179171
\(708\) 0 0
\(709\) 12.2028 0.458286 0.229143 0.973393i \(-0.426408\pi\)
0.229143 + 0.973393i \(0.426408\pi\)
\(710\) −94.3822 −3.54210
\(711\) 0 0
\(712\) 17.8585 0.669275
\(713\) −4.20733 −0.157566
\(714\) 0 0
\(715\) 19.0817 0.713614
\(716\) 74.7019 2.79174
\(717\) 0 0
\(718\) 77.9681 2.90974
\(719\) 45.8176 1.70871 0.854354 0.519691i \(-0.173953\pi\)
0.854354 + 0.519691i \(0.173953\pi\)
\(720\) 0 0
\(721\) −3.52975 −0.131455
\(722\) 34.3260 1.27748
\(723\) 0 0
\(724\) 90.5015 3.36346
\(725\) 4.99499 0.185509
\(726\) 0 0
\(727\) 6.20108 0.229985 0.114993 0.993366i \(-0.463316\pi\)
0.114993 + 0.993366i \(0.463316\pi\)
\(728\) −32.0620 −1.18830
\(729\) 0 0
\(730\) −91.3517 −3.38108
\(731\) 30.1209 1.11406
\(732\) 0 0
\(733\) −28.0136 −1.03471 −0.517353 0.855772i \(-0.673083\pi\)
−0.517353 + 0.855772i \(0.673083\pi\)
\(734\) 5.96361 0.220121
\(735\) 0 0
\(736\) −6.61968 −0.244005
\(737\) 6.88915 0.253765
\(738\) 0 0
\(739\) 18.5172 0.681166 0.340583 0.940214i \(-0.389376\pi\)
0.340583 + 0.940214i \(0.389376\pi\)
\(740\) −67.3374 −2.47537
\(741\) 0 0
\(742\) 22.4934 0.825760
\(743\) 38.0750 1.39684 0.698419 0.715689i \(-0.253887\pi\)
0.698419 + 0.715689i \(0.253887\pi\)
\(744\) 0 0
\(745\) 42.2613 1.54833
\(746\) 67.5013 2.47140
\(747\) 0 0
\(748\) 31.7355 1.16037
\(749\) −4.73776 −0.173114
\(750\) 0 0
\(751\) 25.6544 0.936144 0.468072 0.883690i \(-0.344949\pi\)
0.468072 + 0.883690i \(0.344949\pi\)
\(752\) 107.157 3.90760
\(753\) 0 0
\(754\) 22.1310 0.805965
\(755\) 19.7304 0.718062
\(756\) 0 0
\(757\) 43.1037 1.56663 0.783315 0.621625i \(-0.213527\pi\)
0.783315 + 0.621625i \(0.213527\pi\)
\(758\) −23.0453 −0.837042
\(759\) 0 0
\(760\) −107.098 −3.88486
\(761\) 41.2866 1.49664 0.748318 0.663340i \(-0.230862\pi\)
0.748318 + 0.663340i \(0.230862\pi\)
\(762\) 0 0
\(763\) 17.8152 0.644952
\(764\) −8.62082 −0.311890
\(765\) 0 0
\(766\) −53.0514 −1.91683
\(767\) 34.9146 1.26069
\(768\) 0 0
\(769\) 40.4188 1.45754 0.728770 0.684759i \(-0.240092\pi\)
0.728770 + 0.684759i \(0.240092\pi\)
\(770\) −10.3628 −0.373449
\(771\) 0 0
\(772\) 55.4118 1.99432
\(773\) −31.2197 −1.12289 −0.561447 0.827513i \(-0.689755\pi\)
−0.561447 + 0.827513i \(0.689755\pi\)
\(774\) 0 0
\(775\) 13.0818 0.469913
\(776\) 100.897 3.62200
\(777\) 0 0
\(778\) 28.5264 1.02272
\(779\) 28.4741 1.02019
\(780\) 0 0
\(781\) −19.0321 −0.681023
\(782\) 10.8192 0.386895
\(783\) 0 0
\(784\) 8.15746 0.291338
\(785\) 33.0415 1.17930
\(786\) 0 0
\(787\) −16.5112 −0.588561 −0.294281 0.955719i \(-0.595080\pi\)
−0.294281 + 0.955719i \(0.595080\pi\)
\(788\) 7.70575 0.274506
\(789\) 0 0
\(790\) −27.0845 −0.963622
\(791\) −16.8253 −0.598237
\(792\) 0 0
\(793\) 9.49686 0.337243
\(794\) −64.9791 −2.30602
\(795\) 0 0
\(796\) −60.7742 −2.15409
\(797\) −21.7364 −0.769943 −0.384972 0.922928i \(-0.625789\pi\)
−0.384972 + 0.922928i \(0.625789\pi\)
\(798\) 0 0
\(799\) −62.3222 −2.20480
\(800\) 20.5825 0.727703
\(801\) 0 0
\(802\) 16.5319 0.583762
\(803\) −18.4210 −0.650063
\(804\) 0 0
\(805\) −2.46672 −0.0869403
\(806\) 57.9610 2.04159
\(807\) 0 0
\(808\) 32.2225 1.13358
\(809\) 24.1216 0.848069 0.424035 0.905646i \(-0.360613\pi\)
0.424035 + 0.905646i \(0.360613\pi\)
\(810\) 0 0
\(811\) 37.2206 1.30699 0.653496 0.756930i \(-0.273302\pi\)
0.653496 + 0.756930i \(0.273302\pi\)
\(812\) −8.39178 −0.294494
\(813\) 0 0
\(814\) −19.4474 −0.681631
\(815\) −6.17480 −0.216294
\(816\) 0 0
\(817\) −36.1008 −1.26301
\(818\) 56.2014 1.96504
\(819\) 0 0
\(820\) −64.5242 −2.25328
\(821\) 25.3553 0.884907 0.442454 0.896791i \(-0.354108\pi\)
0.442454 + 0.896791i \(0.354108\pi\)
\(822\) 0 0
\(823\) −25.1655 −0.877214 −0.438607 0.898679i \(-0.644528\pi\)
−0.438607 + 0.898679i \(0.644528\pi\)
\(824\) 23.8741 0.831693
\(825\) 0 0
\(826\) −18.9613 −0.659747
\(827\) −37.6398 −1.30886 −0.654432 0.756121i \(-0.727092\pi\)
−0.654432 + 0.756121i \(0.727092\pi\)
\(828\) 0 0
\(829\) −10.4051 −0.361385 −0.180693 0.983540i \(-0.557834\pi\)
−0.180693 + 0.983540i \(0.557834\pi\)
\(830\) 95.2266 3.30536
\(831\) 0 0
\(832\) 13.8559 0.480368
\(833\) −4.74437 −0.164383
\(834\) 0 0
\(835\) 14.1756 0.490568
\(836\) −38.0360 −1.31550
\(837\) 0 0
\(838\) −55.3607 −1.91241
\(839\) −2.12792 −0.0734639 −0.0367320 0.999325i \(-0.511695\pi\)
−0.0367320 + 0.999325i \(0.511695\pi\)
\(840\) 0 0
\(841\) −25.7111 −0.886590
\(842\) 10.7755 0.371350
\(843\) 0 0
\(844\) 18.1453 0.624588
\(845\) −26.3726 −0.907246
\(846\) 0 0
\(847\) 8.91035 0.306163
\(848\) −71.2758 −2.44762
\(849\) 0 0
\(850\) −33.6402 −1.15385
\(851\) −4.62918 −0.158686
\(852\) 0 0
\(853\) −26.5959 −0.910625 −0.455312 0.890332i \(-0.650472\pi\)
−0.455312 + 0.890332i \(0.650472\pi\)
\(854\) −5.15752 −0.176487
\(855\) 0 0
\(856\) 32.0446 1.09526
\(857\) 37.6643 1.28659 0.643294 0.765619i \(-0.277567\pi\)
0.643294 + 0.765619i \(0.277567\pi\)
\(858\) 0 0
\(859\) −7.50166 −0.255953 −0.127977 0.991777i \(-0.540848\pi\)
−0.127977 + 0.991777i \(0.540848\pi\)
\(860\) 81.8068 2.78959
\(861\) 0 0
\(862\) 37.6060 1.28086
\(863\) −4.85207 −0.165167 −0.0825833 0.996584i \(-0.526317\pi\)
−0.0825833 + 0.996584i \(0.526317\pi\)
\(864\) 0 0
\(865\) 13.9006 0.472634
\(866\) −81.4155 −2.76661
\(867\) 0 0
\(868\) −21.9780 −0.745982
\(869\) −5.46156 −0.185271
\(870\) 0 0
\(871\) −22.5911 −0.765470
\(872\) −120.496 −4.08050
\(873\) 0 0
\(874\) −12.9672 −0.438621
\(875\) −6.25352 −0.211407
\(876\) 0 0
\(877\) 4.17291 0.140909 0.0704546 0.997515i \(-0.477555\pi\)
0.0704546 + 0.997515i \(0.477555\pi\)
\(878\) 17.3623 0.585949
\(879\) 0 0
\(880\) 32.8370 1.10693
\(881\) 40.7208 1.37192 0.685959 0.727640i \(-0.259383\pi\)
0.685959 + 0.727640i \(0.259383\pi\)
\(882\) 0 0
\(883\) −25.4037 −0.854904 −0.427452 0.904038i \(-0.640589\pi\)
−0.427452 + 0.904038i \(0.640589\pi\)
\(884\) −104.068 −3.50018
\(885\) 0 0
\(886\) 48.7273 1.63703
\(887\) −11.1346 −0.373862 −0.186931 0.982373i \(-0.559854\pi\)
−0.186931 + 0.982373i \(0.559854\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −18.9279 −0.634465
\(891\) 0 0
\(892\) 20.9766 0.702347
\(893\) 74.6952 2.49958
\(894\) 0 0
\(895\) −44.9545 −1.50266
\(896\) 7.42097 0.247917
\(897\) 0 0
\(898\) −27.7028 −0.924455
\(899\) 8.61357 0.287279
\(900\) 0 0
\(901\) 41.4539 1.38103
\(902\) −18.6349 −0.620475
\(903\) 0 0
\(904\) 113.800 3.78495
\(905\) −54.4624 −1.81039
\(906\) 0 0
\(907\) −35.3986 −1.17539 −0.587696 0.809082i \(-0.699965\pi\)
−0.587696 + 0.809082i \(0.699965\pi\)
\(908\) −83.8082 −2.78127
\(909\) 0 0
\(910\) 33.9820 1.12649
\(911\) −7.20575 −0.238737 −0.119369 0.992850i \(-0.538087\pi\)
−0.119369 + 0.992850i \(0.538087\pi\)
\(912\) 0 0
\(913\) 19.2024 0.635506
\(914\) −16.6042 −0.549217
\(915\) 0 0
\(916\) 76.7814 2.53693
\(917\) −3.37663 −0.111506
\(918\) 0 0
\(919\) −6.33954 −0.209122 −0.104561 0.994518i \(-0.533344\pi\)
−0.104561 + 0.994518i \(0.533344\pi\)
\(920\) 16.6840 0.550057
\(921\) 0 0
\(922\) −13.5299 −0.445583
\(923\) 62.4106 2.05427
\(924\) 0 0
\(925\) 14.3935 0.473255
\(926\) −59.4801 −1.95464
\(927\) 0 0
\(928\) 13.5523 0.444876
\(929\) 34.2525 1.12379 0.561894 0.827209i \(-0.310073\pi\)
0.561894 + 0.827209i \(0.310073\pi\)
\(930\) 0 0
\(931\) 5.68628 0.186360
\(932\) 66.6709 2.18388
\(933\) 0 0
\(934\) −21.2049 −0.693844
\(935\) −19.0980 −0.624570
\(936\) 0 0
\(937\) 27.6294 0.902614 0.451307 0.892369i \(-0.350958\pi\)
0.451307 + 0.892369i \(0.350958\pi\)
\(938\) 12.2687 0.400586
\(939\) 0 0
\(940\) −169.264 −5.52078
\(941\) −14.1355 −0.460804 −0.230402 0.973096i \(-0.574004\pi\)
−0.230402 + 0.973096i \(0.574004\pi\)
\(942\) 0 0
\(943\) −4.43578 −0.144449
\(944\) 60.0833 1.95554
\(945\) 0 0
\(946\) 23.6263 0.768156
\(947\) −40.2007 −1.30635 −0.653173 0.757208i \(-0.726563\pi\)
−0.653173 + 0.757208i \(0.726563\pi\)
\(948\) 0 0
\(949\) 60.4067 1.96088
\(950\) 40.3188 1.30811
\(951\) 0 0
\(952\) 32.0894 1.04002
\(953\) −46.6706 −1.51181 −0.755904 0.654682i \(-0.772802\pi\)
−0.755904 + 0.654682i \(0.772802\pi\)
\(954\) 0 0
\(955\) 5.18788 0.167876
\(956\) −86.7756 −2.80652
\(957\) 0 0
\(958\) −84.7276 −2.73742
\(959\) −22.4118 −0.723716
\(960\) 0 0
\(961\) −8.44113 −0.272295
\(962\) 63.7724 2.05611
\(963\) 0 0
\(964\) −88.4439 −2.84859
\(965\) −33.3460 −1.07345
\(966\) 0 0
\(967\) 20.5147 0.659708 0.329854 0.944032i \(-0.393001\pi\)
0.329854 + 0.944032i \(0.393001\pi\)
\(968\) −60.2666 −1.93704
\(969\) 0 0
\(970\) −106.939 −3.43361
\(971\) 34.6513 1.11201 0.556006 0.831178i \(-0.312333\pi\)
0.556006 + 0.831178i \(0.312333\pi\)
\(972\) 0 0
\(973\) −8.35723 −0.267921
\(974\) 1.23678 0.0396289
\(975\) 0 0
\(976\) 16.3428 0.523121
\(977\) −21.9087 −0.700920 −0.350460 0.936578i \(-0.613975\pi\)
−0.350460 + 0.936578i \(0.613975\pi\)
\(978\) 0 0
\(979\) −3.81680 −0.121986
\(980\) −12.8855 −0.411612
\(981\) 0 0
\(982\) 39.1718 1.25002
\(983\) −37.9579 −1.21067 −0.605334 0.795972i \(-0.706960\pi\)
−0.605334 + 0.795972i \(0.706960\pi\)
\(984\) 0 0
\(985\) −4.63720 −0.147754
\(986\) −22.1499 −0.705398
\(987\) 0 0
\(988\) 124.729 3.96815
\(989\) 5.62389 0.178829
\(990\) 0 0
\(991\) 1.01270 0.0321694 0.0160847 0.999871i \(-0.494880\pi\)
0.0160847 + 0.999871i \(0.494880\pi\)
\(992\) 35.4934 1.12692
\(993\) 0 0
\(994\) −33.8937 −1.07504
\(995\) 36.5730 1.15944
\(996\) 0 0
\(997\) −46.5112 −1.47303 −0.736513 0.676424i \(-0.763529\pi\)
−0.736513 + 0.676424i \(0.763529\pi\)
\(998\) −13.1420 −0.416001
\(999\) 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 8001.2.a.s.1.16 16
3.2 odd 2 2667.2.a.n.1.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2667.2.a.n.1.1 16 3.2 odd 2
8001.2.a.s.1.16 16 1.1 even 1 trivial