Properties

Label 8001.2.a.n.1.12
Level $8001$
Weight $2$
Character 8001.1
Self dual yes
Analytic conductor $63.888$
Analytic rank $0$
Dimension $12$
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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 5 x^{11} - 3 x^{10} + 41 x^{9} - 11 x^{8} - 123 x^{7} + 44 x^{6} + 159 x^{5} - 39 x^{4} - 71 x^{3} + 16 x^{2} + 7 x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 889)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(-1.77927\) 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)
 
\(f(q)\) \(=\) \(q+2.77927 q^{2} +5.72435 q^{4} +1.30046 q^{5} +1.00000 q^{7} +10.3510 q^{8} +O(q^{10})\) \(q+2.77927 q^{2} +5.72435 q^{4} +1.30046 q^{5} +1.00000 q^{7} +10.3510 q^{8} +3.61432 q^{10} -1.22388 q^{11} +1.99365 q^{13} +2.77927 q^{14} +17.3194 q^{16} -3.84365 q^{17} -7.61093 q^{19} +7.44427 q^{20} -3.40150 q^{22} +5.59005 q^{23} -3.30881 q^{25} +5.54090 q^{26} +5.72435 q^{28} +1.90776 q^{29} +4.73151 q^{31} +27.4335 q^{32} -10.6826 q^{34} +1.30046 q^{35} +7.21595 q^{37} -21.1528 q^{38} +13.4610 q^{40} +6.26895 q^{41} +1.73579 q^{43} -7.00593 q^{44} +15.5363 q^{46} +3.16118 q^{47} +1.00000 q^{49} -9.19608 q^{50} +11.4123 q^{52} +0.926242 q^{53} -1.59161 q^{55} +10.3510 q^{56} +5.30219 q^{58} +4.37513 q^{59} -8.05486 q^{61} +13.1502 q^{62} +41.6062 q^{64} +2.59266 q^{65} -1.47559 q^{67} -22.0024 q^{68} +3.61432 q^{70} +8.77195 q^{71} -1.71314 q^{73} +20.0551 q^{74} -43.5676 q^{76} -1.22388 q^{77} +5.14825 q^{79} +22.5232 q^{80} +17.4231 q^{82} -6.39753 q^{83} -4.99851 q^{85} +4.82423 q^{86} -12.6684 q^{88} +6.15316 q^{89} +1.99365 q^{91} +31.9994 q^{92} +8.78579 q^{94} -9.89768 q^{95} -9.84875 q^{97} +2.77927 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 7 q^{2} + 9 q^{4} + 7 q^{5} + 12 q^{7} + 15 q^{8} - 2 q^{10} + 22 q^{11} + 7 q^{14} + 7 q^{16} + 6 q^{17} - 7 q^{19} + 8 q^{20} + 13 q^{22} + 29 q^{23} + 3 q^{25} + 9 q^{28} + 22 q^{29} - 16 q^{31} + 27 q^{32} - 5 q^{34} + 7 q^{35} - 4 q^{37} - 2 q^{38} + 16 q^{40} + 21 q^{41} + 11 q^{43} + 11 q^{44} + 31 q^{47} + 12 q^{49} + 21 q^{50} + 3 q^{52} + 38 q^{53} - 11 q^{55} + 15 q^{56} + 20 q^{58} + 15 q^{59} - 3 q^{61} + 4 q^{62} + 29 q^{64} + 32 q^{65} - q^{67} - 17 q^{68} - 2 q^{70} + 57 q^{71} - 7 q^{73} + 42 q^{74} - 44 q^{76} + 22 q^{77} - 18 q^{79} - q^{80} + 56 q^{82} + 21 q^{83} - 5 q^{85} + 32 q^{86} - 10 q^{88} - 6 q^{89} + 15 q^{92} + 35 q^{94} + 57 q^{95} + 4 q^{97} + 7 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.77927 1.96524 0.982621 0.185625i \(-0.0594311\pi\)
0.982621 + 0.185625i \(0.0594311\pi\)
\(3\) 0 0
\(4\) 5.72435 2.86217
\(5\) 1.30046 0.581582 0.290791 0.956787i \(-0.406082\pi\)
0.290791 + 0.956787i \(0.406082\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 10.3510 3.65962
\(9\) 0 0
\(10\) 3.61432 1.14295
\(11\) −1.22388 −0.369015 −0.184507 0.982831i \(-0.559069\pi\)
−0.184507 + 0.982831i \(0.559069\pi\)
\(12\) 0 0
\(13\) 1.99365 0.552939 0.276470 0.961023i \(-0.410835\pi\)
0.276470 + 0.961023i \(0.410835\pi\)
\(14\) 2.77927 0.742791
\(15\) 0 0
\(16\) 17.3194 4.32986
\(17\) −3.84365 −0.932223 −0.466112 0.884726i \(-0.654345\pi\)
−0.466112 + 0.884726i \(0.654345\pi\)
\(18\) 0 0
\(19\) −7.61093 −1.74607 −0.873033 0.487661i \(-0.837850\pi\)
−0.873033 + 0.487661i \(0.837850\pi\)
\(20\) 7.44427 1.66459
\(21\) 0 0
\(22\) −3.40150 −0.725203
\(23\) 5.59005 1.16561 0.582803 0.812613i \(-0.301956\pi\)
0.582803 + 0.812613i \(0.301956\pi\)
\(24\) 0 0
\(25\) −3.30881 −0.661762
\(26\) 5.54090 1.08666
\(27\) 0 0
\(28\) 5.72435 1.08180
\(29\) 1.90776 0.354263 0.177131 0.984187i \(-0.443318\pi\)
0.177131 + 0.984187i \(0.443318\pi\)
\(30\) 0 0
\(31\) 4.73151 0.849805 0.424903 0.905239i \(-0.360308\pi\)
0.424903 + 0.905239i \(0.360308\pi\)
\(32\) 27.4335 4.84960
\(33\) 0 0
\(34\) −10.6826 −1.83204
\(35\) 1.30046 0.219817
\(36\) 0 0
\(37\) 7.21595 1.18630 0.593148 0.805094i \(-0.297885\pi\)
0.593148 + 0.805094i \(0.297885\pi\)
\(38\) −21.1528 −3.43144
\(39\) 0 0
\(40\) 13.4610 2.12837
\(41\) 6.26895 0.979046 0.489523 0.871990i \(-0.337171\pi\)
0.489523 + 0.871990i \(0.337171\pi\)
\(42\) 0 0
\(43\) 1.73579 0.264705 0.132353 0.991203i \(-0.457747\pi\)
0.132353 + 0.991203i \(0.457747\pi\)
\(44\) −7.00593 −1.05618
\(45\) 0 0
\(46\) 15.5363 2.29070
\(47\) 3.16118 0.461106 0.230553 0.973060i \(-0.425946\pi\)
0.230553 + 0.973060i \(0.425946\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −9.19608 −1.30052
\(51\) 0 0
\(52\) 11.4123 1.58261
\(53\) 0.926242 0.127229 0.0636146 0.997975i \(-0.479737\pi\)
0.0636146 + 0.997975i \(0.479737\pi\)
\(54\) 0 0
\(55\) −1.59161 −0.214612
\(56\) 10.3510 1.38321
\(57\) 0 0
\(58\) 5.30219 0.696211
\(59\) 4.37513 0.569593 0.284796 0.958588i \(-0.408074\pi\)
0.284796 + 0.958588i \(0.408074\pi\)
\(60\) 0 0
\(61\) −8.05486 −1.03132 −0.515660 0.856793i \(-0.672453\pi\)
−0.515660 + 0.856793i \(0.672453\pi\)
\(62\) 13.1502 1.67007
\(63\) 0 0
\(64\) 41.6062 5.20078
\(65\) 2.59266 0.321580
\(66\) 0 0
\(67\) −1.47559 −0.180273 −0.0901363 0.995929i \(-0.528730\pi\)
−0.0901363 + 0.995929i \(0.528730\pi\)
\(68\) −22.0024 −2.66818
\(69\) 0 0
\(70\) 3.61432 0.431994
\(71\) 8.77195 1.04104 0.520520 0.853850i \(-0.325738\pi\)
0.520520 + 0.853850i \(0.325738\pi\)
\(72\) 0 0
\(73\) −1.71314 −0.200508 −0.100254 0.994962i \(-0.531966\pi\)
−0.100254 + 0.994962i \(0.531966\pi\)
\(74\) 20.0551 2.33136
\(75\) 0 0
\(76\) −43.5676 −4.99754
\(77\) −1.22388 −0.139474
\(78\) 0 0
\(79\) 5.14825 0.579223 0.289612 0.957144i \(-0.406474\pi\)
0.289612 + 0.957144i \(0.406474\pi\)
\(80\) 22.5232 2.51817
\(81\) 0 0
\(82\) 17.4231 1.92406
\(83\) −6.39753 −0.702220 −0.351110 0.936334i \(-0.614196\pi\)
−0.351110 + 0.936334i \(0.614196\pi\)
\(84\) 0 0
\(85\) −4.99851 −0.542164
\(86\) 4.82423 0.520210
\(87\) 0 0
\(88\) −12.6684 −1.35045
\(89\) 6.15316 0.652234 0.326117 0.945329i \(-0.394260\pi\)
0.326117 + 0.945329i \(0.394260\pi\)
\(90\) 0 0
\(91\) 1.99365 0.208991
\(92\) 31.9994 3.33617
\(93\) 0 0
\(94\) 8.78579 0.906185
\(95\) −9.89768 −1.01548
\(96\) 0 0
\(97\) −9.84875 −0.999989 −0.499995 0.866029i \(-0.666665\pi\)
−0.499995 + 0.866029i \(0.666665\pi\)
\(98\) 2.77927 0.280749
\(99\) 0 0
\(100\) −18.9408 −1.89408
\(101\) 17.3913 1.73050 0.865250 0.501340i \(-0.167159\pi\)
0.865250 + 0.501340i \(0.167159\pi\)
\(102\) 0 0
\(103\) −6.13951 −0.604943 −0.302472 0.953158i \(-0.597812\pi\)
−0.302472 + 0.953158i \(0.597812\pi\)
\(104\) 20.6362 2.02355
\(105\) 0 0
\(106\) 2.57428 0.250036
\(107\) 10.7901 1.04312 0.521559 0.853215i \(-0.325350\pi\)
0.521559 + 0.853215i \(0.325350\pi\)
\(108\) 0 0
\(109\) −14.4321 −1.38234 −0.691170 0.722692i \(-0.742905\pi\)
−0.691170 + 0.722692i \(0.742905\pi\)
\(110\) −4.42351 −0.421765
\(111\) 0 0
\(112\) 17.3194 1.63653
\(113\) −19.2485 −1.81075 −0.905375 0.424614i \(-0.860410\pi\)
−0.905375 + 0.424614i \(0.860410\pi\)
\(114\) 0 0
\(115\) 7.26962 0.677896
\(116\) 10.9207 1.01396
\(117\) 0 0
\(118\) 12.1597 1.11939
\(119\) −3.84365 −0.352347
\(120\) 0 0
\(121\) −9.50211 −0.863828
\(122\) −22.3866 −2.02679
\(123\) 0 0
\(124\) 27.0848 2.43229
\(125\) −10.8053 −0.966451
\(126\) 0 0
\(127\) 1.00000 0.0887357
\(128\) 60.7679 5.37118
\(129\) 0 0
\(130\) 7.20570 0.631981
\(131\) −10.2834 −0.898468 −0.449234 0.893414i \(-0.648303\pi\)
−0.449234 + 0.893414i \(0.648303\pi\)
\(132\) 0 0
\(133\) −7.61093 −0.659951
\(134\) −4.10108 −0.354279
\(135\) 0 0
\(136\) −39.7855 −3.41158
\(137\) 14.7634 1.26132 0.630661 0.776058i \(-0.282784\pi\)
0.630661 + 0.776058i \(0.282784\pi\)
\(138\) 0 0
\(139\) −19.4741 −1.65177 −0.825885 0.563838i \(-0.809324\pi\)
−0.825885 + 0.563838i \(0.809324\pi\)
\(140\) 7.44427 0.629155
\(141\) 0 0
\(142\) 24.3796 2.04589
\(143\) −2.44000 −0.204043
\(144\) 0 0
\(145\) 2.48096 0.206033
\(146\) −4.76128 −0.394047
\(147\) 0 0
\(148\) 41.3066 3.39538
\(149\) 7.55089 0.618593 0.309296 0.950966i \(-0.399906\pi\)
0.309296 + 0.950966i \(0.399906\pi\)
\(150\) 0 0
\(151\) −23.1272 −1.88206 −0.941032 0.338318i \(-0.890142\pi\)
−0.941032 + 0.338318i \(0.890142\pi\)
\(152\) −78.7804 −6.38994
\(153\) 0 0
\(154\) −3.40150 −0.274101
\(155\) 6.15313 0.494231
\(156\) 0 0
\(157\) 16.3630 1.30591 0.652954 0.757398i \(-0.273530\pi\)
0.652954 + 0.757398i \(0.273530\pi\)
\(158\) 14.3084 1.13831
\(159\) 0 0
\(160\) 35.6761 2.82044
\(161\) 5.59005 0.440558
\(162\) 0 0
\(163\) −19.4504 −1.52347 −0.761737 0.647886i \(-0.775653\pi\)
−0.761737 + 0.647886i \(0.775653\pi\)
\(164\) 35.8857 2.80220
\(165\) 0 0
\(166\) −17.7805 −1.38003
\(167\) −21.8043 −1.68726 −0.843632 0.536922i \(-0.819587\pi\)
−0.843632 + 0.536922i \(0.819587\pi\)
\(168\) 0 0
\(169\) −9.02536 −0.694258
\(170\) −13.8922 −1.06548
\(171\) 0 0
\(172\) 9.93625 0.757632
\(173\) −10.1178 −0.769239 −0.384620 0.923075i \(-0.625667\pi\)
−0.384620 + 0.923075i \(0.625667\pi\)
\(174\) 0 0
\(175\) −3.30881 −0.250123
\(176\) −21.1970 −1.59778
\(177\) 0 0
\(178\) 17.1013 1.28180
\(179\) 5.76419 0.430836 0.215418 0.976522i \(-0.430889\pi\)
0.215418 + 0.976522i \(0.430889\pi\)
\(180\) 0 0
\(181\) 10.4234 0.774763 0.387382 0.921919i \(-0.373380\pi\)
0.387382 + 0.921919i \(0.373380\pi\)
\(182\) 5.54090 0.410718
\(183\) 0 0
\(184\) 57.8624 4.26568
\(185\) 9.38404 0.689928
\(186\) 0 0
\(187\) 4.70418 0.344004
\(188\) 18.0957 1.31977
\(189\) 0 0
\(190\) −27.5083 −1.99567
\(191\) −18.9663 −1.37235 −0.686176 0.727436i \(-0.740712\pi\)
−0.686176 + 0.727436i \(0.740712\pi\)
\(192\) 0 0
\(193\) −12.5961 −0.906688 −0.453344 0.891336i \(-0.649769\pi\)
−0.453344 + 0.891336i \(0.649769\pi\)
\(194\) −27.3723 −1.96522
\(195\) 0 0
\(196\) 5.72435 0.408882
\(197\) −4.25793 −0.303365 −0.151682 0.988429i \(-0.548469\pi\)
−0.151682 + 0.988429i \(0.548469\pi\)
\(198\) 0 0
\(199\) 11.0155 0.780867 0.390433 0.920631i \(-0.372325\pi\)
0.390433 + 0.920631i \(0.372325\pi\)
\(200\) −34.2494 −2.42180
\(201\) 0 0
\(202\) 48.3352 3.40085
\(203\) 1.90776 0.133899
\(204\) 0 0
\(205\) 8.15250 0.569396
\(206\) −17.0633 −1.18886
\(207\) 0 0
\(208\) 34.5289 2.39415
\(209\) 9.31489 0.644324
\(210\) 0 0
\(211\) 19.1891 1.32103 0.660515 0.750813i \(-0.270338\pi\)
0.660515 + 0.750813i \(0.270338\pi\)
\(212\) 5.30213 0.364152
\(213\) 0 0
\(214\) 29.9886 2.04998
\(215\) 2.25732 0.153948
\(216\) 0 0
\(217\) 4.73151 0.321196
\(218\) −40.1106 −2.71663
\(219\) 0 0
\(220\) −9.11091 −0.614258
\(221\) −7.66290 −0.515463
\(222\) 0 0
\(223\) −14.8093 −0.991704 −0.495852 0.868407i \(-0.665144\pi\)
−0.495852 + 0.868407i \(0.665144\pi\)
\(224\) 27.4335 1.83298
\(225\) 0 0
\(226\) −53.4969 −3.55856
\(227\) 1.08703 0.0721487 0.0360743 0.999349i \(-0.488515\pi\)
0.0360743 + 0.999349i \(0.488515\pi\)
\(228\) 0 0
\(229\) −29.8776 −1.97437 −0.987185 0.159580i \(-0.948986\pi\)
−0.987185 + 0.159580i \(0.948986\pi\)
\(230\) 20.2043 1.33223
\(231\) 0 0
\(232\) 19.7472 1.29647
\(233\) −12.5209 −0.820274 −0.410137 0.912024i \(-0.634519\pi\)
−0.410137 + 0.912024i \(0.634519\pi\)
\(234\) 0 0
\(235\) 4.11098 0.268171
\(236\) 25.0447 1.63027
\(237\) 0 0
\(238\) −10.6826 −0.692447
\(239\) 13.5100 0.873891 0.436946 0.899488i \(-0.356060\pi\)
0.436946 + 0.899488i \(0.356060\pi\)
\(240\) 0 0
\(241\) 3.97228 0.255877 0.127939 0.991782i \(-0.459164\pi\)
0.127939 + 0.991782i \(0.459164\pi\)
\(242\) −26.4089 −1.69763
\(243\) 0 0
\(244\) −46.1088 −2.95181
\(245\) 1.30046 0.0830832
\(246\) 0 0
\(247\) −15.1735 −0.965469
\(248\) 48.9757 3.10996
\(249\) 0 0
\(250\) −30.0307 −1.89931
\(251\) −16.6044 −1.04806 −0.524029 0.851701i \(-0.675572\pi\)
−0.524029 + 0.851701i \(0.675572\pi\)
\(252\) 0 0
\(253\) −6.84157 −0.430126
\(254\) 2.77927 0.174387
\(255\) 0 0
\(256\) 85.6781 5.35488
\(257\) −17.9690 −1.12088 −0.560438 0.828197i \(-0.689367\pi\)
−0.560438 + 0.828197i \(0.689367\pi\)
\(258\) 0 0
\(259\) 7.21595 0.448378
\(260\) 14.8413 0.920416
\(261\) 0 0
\(262\) −28.5805 −1.76571
\(263\) 7.18373 0.442968 0.221484 0.975164i \(-0.428910\pi\)
0.221484 + 0.975164i \(0.428910\pi\)
\(264\) 0 0
\(265\) 1.20454 0.0739942
\(266\) −21.1528 −1.29696
\(267\) 0 0
\(268\) −8.44681 −0.515971
\(269\) −15.3889 −0.938280 −0.469140 0.883124i \(-0.655436\pi\)
−0.469140 + 0.883124i \(0.655436\pi\)
\(270\) 0 0
\(271\) −24.1037 −1.46419 −0.732096 0.681201i \(-0.761458\pi\)
−0.732096 + 0.681201i \(0.761458\pi\)
\(272\) −66.5700 −4.03640
\(273\) 0 0
\(274\) 41.0315 2.47880
\(275\) 4.04960 0.244200
\(276\) 0 0
\(277\) 27.5990 1.65826 0.829132 0.559053i \(-0.188835\pi\)
0.829132 + 0.559053i \(0.188835\pi\)
\(278\) −54.1238 −3.24613
\(279\) 0 0
\(280\) 13.4610 0.804448
\(281\) −5.36018 −0.319762 −0.159881 0.987136i \(-0.551111\pi\)
−0.159881 + 0.987136i \(0.551111\pi\)
\(282\) 0 0
\(283\) −6.46356 −0.384219 −0.192109 0.981374i \(-0.561533\pi\)
−0.192109 + 0.981374i \(0.561533\pi\)
\(284\) 50.2137 2.97963
\(285\) 0 0
\(286\) −6.78141 −0.400993
\(287\) 6.26895 0.370045
\(288\) 0 0
\(289\) −2.22632 −0.130960
\(290\) 6.89527 0.404904
\(291\) 0 0
\(292\) −9.80662 −0.573889
\(293\) 7.99366 0.466995 0.233497 0.972357i \(-0.424983\pi\)
0.233497 + 0.972357i \(0.424983\pi\)
\(294\) 0 0
\(295\) 5.68966 0.331265
\(296\) 74.6921 4.34139
\(297\) 0 0
\(298\) 20.9860 1.21568
\(299\) 11.1446 0.644510
\(300\) 0 0
\(301\) 1.73579 0.100049
\(302\) −64.2767 −3.69871
\(303\) 0 0
\(304\) −131.817 −7.56022
\(305\) −10.4750 −0.599797
\(306\) 0 0
\(307\) 11.0044 0.628053 0.314026 0.949414i \(-0.398322\pi\)
0.314026 + 0.949414i \(0.398322\pi\)
\(308\) −7.00593 −0.399200
\(309\) 0 0
\(310\) 17.1012 0.971284
\(311\) −7.25990 −0.411671 −0.205836 0.978587i \(-0.565991\pi\)
−0.205836 + 0.978587i \(0.565991\pi\)
\(312\) 0 0
\(313\) 21.5487 1.21800 0.609001 0.793169i \(-0.291571\pi\)
0.609001 + 0.793169i \(0.291571\pi\)
\(314\) 45.4771 2.56642
\(315\) 0 0
\(316\) 29.4704 1.65784
\(317\) −19.3937 −1.08926 −0.544628 0.838677i \(-0.683329\pi\)
−0.544628 + 0.838677i \(0.683329\pi\)
\(318\) 0 0
\(319\) −2.33488 −0.130728
\(320\) 54.1071 3.02468
\(321\) 0 0
\(322\) 15.5363 0.865803
\(323\) 29.2538 1.62772
\(324\) 0 0
\(325\) −6.59661 −0.365914
\(326\) −54.0580 −2.99399
\(327\) 0 0
\(328\) 64.8897 3.58293
\(329\) 3.16118 0.174282
\(330\) 0 0
\(331\) 31.1563 1.71250 0.856252 0.516559i \(-0.172787\pi\)
0.856252 + 0.516559i \(0.172787\pi\)
\(332\) −36.6216 −2.00987
\(333\) 0 0
\(334\) −60.5999 −3.31588
\(335\) −1.91895 −0.104843
\(336\) 0 0
\(337\) −29.6388 −1.61453 −0.807264 0.590190i \(-0.799053\pi\)
−0.807264 + 0.590190i \(0.799053\pi\)
\(338\) −25.0839 −1.36438
\(339\) 0 0
\(340\) −28.6132 −1.55177
\(341\) −5.79082 −0.313591
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 17.9671 0.968720
\(345\) 0 0
\(346\) −28.1200 −1.51174
\(347\) 2.89297 0.155303 0.0776514 0.996981i \(-0.475258\pi\)
0.0776514 + 0.996981i \(0.475258\pi\)
\(348\) 0 0
\(349\) 8.62684 0.461784 0.230892 0.972979i \(-0.425836\pi\)
0.230892 + 0.972979i \(0.425836\pi\)
\(350\) −9.19608 −0.491551
\(351\) 0 0
\(352\) −33.5754 −1.78957
\(353\) −29.3126 −1.56015 −0.780077 0.625684i \(-0.784820\pi\)
−0.780077 + 0.625684i \(0.784820\pi\)
\(354\) 0 0
\(355\) 11.4075 0.605450
\(356\) 35.2228 1.86681
\(357\) 0 0
\(358\) 16.0203 0.846697
\(359\) 29.4378 1.55367 0.776833 0.629707i \(-0.216825\pi\)
0.776833 + 0.629707i \(0.216825\pi\)
\(360\) 0 0
\(361\) 38.9262 2.04875
\(362\) 28.9694 1.52260
\(363\) 0 0
\(364\) 11.4123 0.598169
\(365\) −2.22787 −0.116612
\(366\) 0 0
\(367\) 0.0234964 0.00122650 0.000613251 1.00000i \(-0.499805\pi\)
0.000613251 1.00000i \(0.499805\pi\)
\(368\) 96.8166 5.04691
\(369\) 0 0
\(370\) 26.0808 1.35588
\(371\) 0.926242 0.0480881
\(372\) 0 0
\(373\) 0.413267 0.0213982 0.0106991 0.999943i \(-0.496594\pi\)
0.0106991 + 0.999943i \(0.496594\pi\)
\(374\) 13.0742 0.676051
\(375\) 0 0
\(376\) 32.7213 1.68747
\(377\) 3.80341 0.195886
\(378\) 0 0
\(379\) −12.7696 −0.655928 −0.327964 0.944690i \(-0.606362\pi\)
−0.327964 + 0.944690i \(0.606362\pi\)
\(380\) −56.6578 −2.90648
\(381\) 0 0
\(382\) −52.7124 −2.69700
\(383\) 21.2538 1.08602 0.543009 0.839727i \(-0.317285\pi\)
0.543009 + 0.839727i \(0.317285\pi\)
\(384\) 0 0
\(385\) −1.59161 −0.0811159
\(386\) −35.0080 −1.78186
\(387\) 0 0
\(388\) −56.3777 −2.86214
\(389\) 0.127498 0.00646441 0.00323220 0.999995i \(-0.498971\pi\)
0.00323220 + 0.999995i \(0.498971\pi\)
\(390\) 0 0
\(391\) −21.4862 −1.08661
\(392\) 10.3510 0.522803
\(393\) 0 0
\(394\) −11.8339 −0.596185
\(395\) 6.69508 0.336866
\(396\) 0 0
\(397\) −6.42114 −0.322268 −0.161134 0.986933i \(-0.551515\pi\)
−0.161134 + 0.986933i \(0.551515\pi\)
\(398\) 30.6150 1.53459
\(399\) 0 0
\(400\) −57.3068 −2.86534
\(401\) −31.3010 −1.56310 −0.781549 0.623843i \(-0.785570\pi\)
−0.781549 + 0.623843i \(0.785570\pi\)
\(402\) 0 0
\(403\) 9.43299 0.469891
\(404\) 99.5539 4.95299
\(405\) 0 0
\(406\) 5.30219 0.263143
\(407\) −8.83149 −0.437760
\(408\) 0 0
\(409\) 32.1486 1.58965 0.794824 0.606841i \(-0.207563\pi\)
0.794824 + 0.606841i \(0.207563\pi\)
\(410\) 22.6580 1.11900
\(411\) 0 0
\(412\) −35.1447 −1.73145
\(413\) 4.37513 0.215286
\(414\) 0 0
\(415\) −8.31971 −0.408398
\(416\) 54.6928 2.68154
\(417\) 0 0
\(418\) 25.8886 1.26625
\(419\) −30.9045 −1.50978 −0.754892 0.655849i \(-0.772311\pi\)
−0.754892 + 0.655849i \(0.772311\pi\)
\(420\) 0 0
\(421\) −9.57947 −0.466875 −0.233437 0.972372i \(-0.574997\pi\)
−0.233437 + 0.972372i \(0.574997\pi\)
\(422\) 53.3316 2.59614
\(423\) 0 0
\(424\) 9.58750 0.465610
\(425\) 12.7179 0.616910
\(426\) 0 0
\(427\) −8.05486 −0.389802
\(428\) 61.7663 2.98559
\(429\) 0 0
\(430\) 6.27370 0.302545
\(431\) 31.9913 1.54097 0.770483 0.637461i \(-0.220015\pi\)
0.770483 + 0.637461i \(0.220015\pi\)
\(432\) 0 0
\(433\) 18.2429 0.876697 0.438349 0.898805i \(-0.355564\pi\)
0.438349 + 0.898805i \(0.355564\pi\)
\(434\) 13.1502 0.631228
\(435\) 0 0
\(436\) −82.6141 −3.95650
\(437\) −42.5455 −2.03523
\(438\) 0 0
\(439\) 36.4213 1.73829 0.869146 0.494556i \(-0.164669\pi\)
0.869146 + 0.494556i \(0.164669\pi\)
\(440\) −16.4747 −0.785399
\(441\) 0 0
\(442\) −21.2973 −1.01301
\(443\) 20.3418 0.966468 0.483234 0.875491i \(-0.339462\pi\)
0.483234 + 0.875491i \(0.339462\pi\)
\(444\) 0 0
\(445\) 8.00193 0.379328
\(446\) −41.1590 −1.94894
\(447\) 0 0
\(448\) 41.6062 1.96571
\(449\) 8.60191 0.405949 0.202975 0.979184i \(-0.434939\pi\)
0.202975 + 0.979184i \(0.434939\pi\)
\(450\) 0 0
\(451\) −7.67247 −0.361282
\(452\) −110.185 −5.18268
\(453\) 0 0
\(454\) 3.02115 0.141790
\(455\) 2.59266 0.121546
\(456\) 0 0
\(457\) 25.0041 1.16964 0.584821 0.811163i \(-0.301165\pi\)
0.584821 + 0.811163i \(0.301165\pi\)
\(458\) −83.0381 −3.88011
\(459\) 0 0
\(460\) 41.6138 1.94026
\(461\) −27.0304 −1.25893 −0.629465 0.777029i \(-0.716726\pi\)
−0.629465 + 0.777029i \(0.716726\pi\)
\(462\) 0 0
\(463\) 15.2177 0.707228 0.353614 0.935391i \(-0.384953\pi\)
0.353614 + 0.935391i \(0.384953\pi\)
\(464\) 33.0414 1.53391
\(465\) 0 0
\(466\) −34.7991 −1.61204
\(467\) 0.539310 0.0249563 0.0124781 0.999922i \(-0.496028\pi\)
0.0124781 + 0.999922i \(0.496028\pi\)
\(468\) 0 0
\(469\) −1.47559 −0.0681366
\(470\) 11.4255 0.527021
\(471\) 0 0
\(472\) 45.2868 2.08449
\(473\) −2.12440 −0.0976801
\(474\) 0 0
\(475\) 25.1831 1.15548
\(476\) −22.0024 −1.00848
\(477\) 0 0
\(478\) 37.5480 1.71741
\(479\) 22.5610 1.03084 0.515420 0.856938i \(-0.327636\pi\)
0.515420 + 0.856938i \(0.327636\pi\)
\(480\) 0 0
\(481\) 14.3861 0.655949
\(482\) 11.0400 0.502860
\(483\) 0 0
\(484\) −54.3934 −2.47243
\(485\) −12.8079 −0.581576
\(486\) 0 0
\(487\) 30.8681 1.39877 0.699383 0.714747i \(-0.253458\pi\)
0.699383 + 0.714747i \(0.253458\pi\)
\(488\) −83.3756 −3.77424
\(489\) 0 0
\(490\) 3.61432 0.163278
\(491\) −7.14793 −0.322582 −0.161291 0.986907i \(-0.551566\pi\)
−0.161291 + 0.986907i \(0.551566\pi\)
\(492\) 0 0
\(493\) −7.33278 −0.330252
\(494\) −42.1713 −1.89738
\(495\) 0 0
\(496\) 81.9472 3.67954
\(497\) 8.77195 0.393476
\(498\) 0 0
\(499\) 6.21761 0.278338 0.139169 0.990269i \(-0.455557\pi\)
0.139169 + 0.990269i \(0.455557\pi\)
\(500\) −61.8530 −2.76615
\(501\) 0 0
\(502\) −46.1480 −2.05969
\(503\) 33.1941 1.48005 0.740026 0.672579i \(-0.234813\pi\)
0.740026 + 0.672579i \(0.234813\pi\)
\(504\) 0 0
\(505\) 22.6167 1.00643
\(506\) −19.0146 −0.845301
\(507\) 0 0
\(508\) 5.72435 0.253977
\(509\) −39.3796 −1.74547 −0.872736 0.488193i \(-0.837656\pi\)
−0.872736 + 0.488193i \(0.837656\pi\)
\(510\) 0 0
\(511\) −1.71314 −0.0757849
\(512\) 116.587 5.15246
\(513\) 0 0
\(514\) −49.9407 −2.20279
\(515\) −7.98416 −0.351824
\(516\) 0 0
\(517\) −3.86892 −0.170155
\(518\) 20.0551 0.881170
\(519\) 0 0
\(520\) 26.8365 1.17686
\(521\) 13.6202 0.596714 0.298357 0.954454i \(-0.403561\pi\)
0.298357 + 0.954454i \(0.403561\pi\)
\(522\) 0 0
\(523\) −38.5952 −1.68765 −0.843826 0.536617i \(-0.819702\pi\)
−0.843826 + 0.536617i \(0.819702\pi\)
\(524\) −58.8660 −2.57157
\(525\) 0 0
\(526\) 19.9655 0.870539
\(527\) −18.1863 −0.792208
\(528\) 0 0
\(529\) 8.24869 0.358639
\(530\) 3.34774 0.145416
\(531\) 0 0
\(532\) −43.5676 −1.88889
\(533\) 12.4981 0.541353
\(534\) 0 0
\(535\) 14.0321 0.606659
\(536\) −15.2738 −0.659729
\(537\) 0 0
\(538\) −42.7700 −1.84395
\(539\) −1.22388 −0.0527164
\(540\) 0 0
\(541\) 7.36149 0.316495 0.158248 0.987399i \(-0.449416\pi\)
0.158248 + 0.987399i \(0.449416\pi\)
\(542\) −66.9906 −2.87749
\(543\) 0 0
\(544\) −105.445 −4.52091
\(545\) −18.7683 −0.803945
\(546\) 0 0
\(547\) −30.7632 −1.31534 −0.657669 0.753307i \(-0.728458\pi\)
−0.657669 + 0.753307i \(0.728458\pi\)
\(548\) 84.5108 3.61012
\(549\) 0 0
\(550\) 11.2549 0.479912
\(551\) −14.5198 −0.618566
\(552\) 0 0
\(553\) 5.14825 0.218926
\(554\) 76.7051 3.25889
\(555\) 0 0
\(556\) −111.476 −4.72765
\(557\) −3.54294 −0.150119 −0.0750595 0.997179i \(-0.523915\pi\)
−0.0750595 + 0.997179i \(0.523915\pi\)
\(558\) 0 0
\(559\) 3.46056 0.146366
\(560\) 22.5232 0.951779
\(561\) 0 0
\(562\) −14.8974 −0.628409
\(563\) 4.73790 0.199679 0.0998394 0.995004i \(-0.468167\pi\)
0.0998394 + 0.995004i \(0.468167\pi\)
\(564\) 0 0
\(565\) −25.0319 −1.05310
\(566\) −17.9640 −0.755083
\(567\) 0 0
\(568\) 90.7982 3.80981
\(569\) −0.229871 −0.00963669 −0.00481834 0.999988i \(-0.501534\pi\)
−0.00481834 + 0.999988i \(0.501534\pi\)
\(570\) 0 0
\(571\) −7.08679 −0.296573 −0.148286 0.988944i \(-0.547376\pi\)
−0.148286 + 0.988944i \(0.547376\pi\)
\(572\) −13.9674 −0.584006
\(573\) 0 0
\(574\) 17.4231 0.727227
\(575\) −18.4964 −0.771354
\(576\) 0 0
\(577\) 44.3215 1.84513 0.922563 0.385846i \(-0.126090\pi\)
0.922563 + 0.385846i \(0.126090\pi\)
\(578\) −6.18755 −0.257368
\(579\) 0 0
\(580\) 14.2019 0.589701
\(581\) −6.39753 −0.265414
\(582\) 0 0
\(583\) −1.13361 −0.0469494
\(584\) −17.7327 −0.733783
\(585\) 0 0
\(586\) 22.2166 0.917758
\(587\) −41.7718 −1.72411 −0.862053 0.506817i \(-0.830822\pi\)
−0.862053 + 0.506817i \(0.830822\pi\)
\(588\) 0 0
\(589\) −36.0112 −1.48382
\(590\) 15.8131 0.651016
\(591\) 0 0
\(592\) 124.976 5.13649
\(593\) 19.4400 0.798304 0.399152 0.916885i \(-0.369305\pi\)
0.399152 + 0.916885i \(0.369305\pi\)
\(594\) 0 0
\(595\) −4.99851 −0.204919
\(596\) 43.2239 1.77052
\(597\) 0 0
\(598\) 30.9739 1.26662
\(599\) −0.285281 −0.0116563 −0.00582813 0.999983i \(-0.501855\pi\)
−0.00582813 + 0.999983i \(0.501855\pi\)
\(600\) 0 0
\(601\) −7.41769 −0.302574 −0.151287 0.988490i \(-0.548342\pi\)
−0.151287 + 0.988490i \(0.548342\pi\)
\(602\) 4.82423 0.196621
\(603\) 0 0
\(604\) −132.388 −5.38679
\(605\) −12.3571 −0.502387
\(606\) 0 0
\(607\) −27.5980 −1.12017 −0.560084 0.828436i \(-0.689231\pi\)
−0.560084 + 0.828436i \(0.689231\pi\)
\(608\) −208.794 −8.46773
\(609\) 0 0
\(610\) −29.1129 −1.17875
\(611\) 6.30230 0.254964
\(612\) 0 0
\(613\) −33.7794 −1.36434 −0.682169 0.731194i \(-0.738963\pi\)
−0.682169 + 0.731194i \(0.738963\pi\)
\(614\) 30.5841 1.23428
\(615\) 0 0
\(616\) −12.6684 −0.510423
\(617\) 31.6964 1.27605 0.638024 0.770017i \(-0.279752\pi\)
0.638024 + 0.770017i \(0.279752\pi\)
\(618\) 0 0
\(619\) −1.48718 −0.0597747 −0.0298874 0.999553i \(-0.509515\pi\)
−0.0298874 + 0.999553i \(0.509515\pi\)
\(620\) 35.2227 1.41458
\(621\) 0 0
\(622\) −20.1772 −0.809033
\(623\) 6.15316 0.246521
\(624\) 0 0
\(625\) 2.49229 0.0996914
\(626\) 59.8895 2.39367
\(627\) 0 0
\(628\) 93.6673 3.73773
\(629\) −27.7356 −1.10589
\(630\) 0 0
\(631\) 6.35865 0.253134 0.126567 0.991958i \(-0.459604\pi\)
0.126567 + 0.991958i \(0.459604\pi\)
\(632\) 53.2894 2.11974
\(633\) 0 0
\(634\) −53.9002 −2.14065
\(635\) 1.30046 0.0516071
\(636\) 0 0
\(637\) 1.99365 0.0789913
\(638\) −6.48926 −0.256912
\(639\) 0 0
\(640\) 79.0261 3.12378
\(641\) −17.6050 −0.695357 −0.347678 0.937614i \(-0.613030\pi\)
−0.347678 + 0.937614i \(0.613030\pi\)
\(642\) 0 0
\(643\) 35.1486 1.38612 0.693062 0.720878i \(-0.256261\pi\)
0.693062 + 0.720878i \(0.256261\pi\)
\(644\) 31.9994 1.26095
\(645\) 0 0
\(646\) 81.3042 3.19887
\(647\) 21.8303 0.858239 0.429120 0.903248i \(-0.358824\pi\)
0.429120 + 0.903248i \(0.358824\pi\)
\(648\) 0 0
\(649\) −5.35464 −0.210188
\(650\) −18.3338 −0.719110
\(651\) 0 0
\(652\) −111.341 −4.36045
\(653\) 23.8766 0.934364 0.467182 0.884161i \(-0.345269\pi\)
0.467182 + 0.884161i \(0.345269\pi\)
\(654\) 0 0
\(655\) −13.3732 −0.522533
\(656\) 108.575 4.23913
\(657\) 0 0
\(658\) 8.78579 0.342506
\(659\) 12.8713 0.501393 0.250697 0.968066i \(-0.419340\pi\)
0.250697 + 0.968066i \(0.419340\pi\)
\(660\) 0 0
\(661\) 0.331054 0.0128765 0.00643826 0.999979i \(-0.497951\pi\)
0.00643826 + 0.999979i \(0.497951\pi\)
\(662\) 86.5917 3.36548
\(663\) 0 0
\(664\) −66.2206 −2.56986
\(665\) −9.89768 −0.383816
\(666\) 0 0
\(667\) 10.6645 0.412931
\(668\) −124.815 −4.82924
\(669\) 0 0
\(670\) −5.33327 −0.206042
\(671\) 9.85821 0.380572
\(672\) 0 0
\(673\) 46.1771 1.78000 0.889998 0.455965i \(-0.150706\pi\)
0.889998 + 0.455965i \(0.150706\pi\)
\(674\) −82.3743 −3.17294
\(675\) 0 0
\(676\) −51.6643 −1.98709
\(677\) 15.5181 0.596408 0.298204 0.954502i \(-0.403612\pi\)
0.298204 + 0.954502i \(0.403612\pi\)
\(678\) 0 0
\(679\) −9.84875 −0.377960
\(680\) −51.7394 −1.98411
\(681\) 0 0
\(682\) −16.0943 −0.616281
\(683\) 30.6461 1.17264 0.586319 0.810080i \(-0.300576\pi\)
0.586319 + 0.810080i \(0.300576\pi\)
\(684\) 0 0
\(685\) 19.1992 0.733563
\(686\) 2.77927 0.106113
\(687\) 0 0
\(688\) 30.0629 1.14614
\(689\) 1.84660 0.0703500
\(690\) 0 0
\(691\) 39.6963 1.51012 0.755060 0.655656i \(-0.227608\pi\)
0.755060 + 0.655656i \(0.227608\pi\)
\(692\) −57.9176 −2.20170
\(693\) 0 0
\(694\) 8.04035 0.305208
\(695\) −25.3252 −0.960640
\(696\) 0 0
\(697\) −24.0957 −0.912689
\(698\) 23.9763 0.907517
\(699\) 0 0
\(700\) −18.9408 −0.715894
\(701\) 27.9108 1.05417 0.527087 0.849811i \(-0.323284\pi\)
0.527087 + 0.849811i \(0.323284\pi\)
\(702\) 0 0
\(703\) −54.9201 −2.07135
\(704\) −50.9211 −1.91916
\(705\) 0 0
\(706\) −81.4677 −3.06608
\(707\) 17.3913 0.654068
\(708\) 0 0
\(709\) −29.9459 −1.12464 −0.562320 0.826920i \(-0.690091\pi\)
−0.562320 + 0.826920i \(0.690091\pi\)
\(710\) 31.7047 1.18985
\(711\) 0 0
\(712\) 63.6912 2.38693
\(713\) 26.4494 0.990538
\(714\) 0 0
\(715\) −3.17311 −0.118668
\(716\) 32.9962 1.23313
\(717\) 0 0
\(718\) 81.8155 3.05333
\(719\) 10.8582 0.404944 0.202472 0.979288i \(-0.435103\pi\)
0.202472 + 0.979288i \(0.435103\pi\)
\(720\) 0 0
\(721\) −6.13951 −0.228647
\(722\) 108.186 4.02628
\(723\) 0 0
\(724\) 59.6670 2.21751
\(725\) −6.31242 −0.234438
\(726\) 0 0
\(727\) −29.9620 −1.11123 −0.555614 0.831440i \(-0.687517\pi\)
−0.555614 + 0.831440i \(0.687517\pi\)
\(728\) 20.6362 0.764829
\(729\) 0 0
\(730\) −6.19185 −0.229171
\(731\) −6.67177 −0.246764
\(732\) 0 0
\(733\) −6.30193 −0.232767 −0.116384 0.993204i \(-0.537130\pi\)
−0.116384 + 0.993204i \(0.537130\pi\)
\(734\) 0.0653029 0.00241037
\(735\) 0 0
\(736\) 153.355 5.65273
\(737\) 1.80596 0.0665232
\(738\) 0 0
\(739\) 20.3572 0.748851 0.374425 0.927257i \(-0.377840\pi\)
0.374425 + 0.927257i \(0.377840\pi\)
\(740\) 53.7175 1.97469
\(741\) 0 0
\(742\) 2.57428 0.0945047
\(743\) −22.8445 −0.838082 −0.419041 0.907967i \(-0.637634\pi\)
−0.419041 + 0.907967i \(0.637634\pi\)
\(744\) 0 0
\(745\) 9.81961 0.359763
\(746\) 1.14858 0.0420526
\(747\) 0 0
\(748\) 26.9284 0.984599
\(749\) 10.7901 0.394262
\(750\) 0 0
\(751\) −22.4377 −0.818763 −0.409382 0.912363i \(-0.634256\pi\)
−0.409382 + 0.912363i \(0.634256\pi\)
\(752\) 54.7499 1.99653
\(753\) 0 0
\(754\) 10.5707 0.384963
\(755\) −30.0759 −1.09457
\(756\) 0 0
\(757\) −39.2552 −1.42675 −0.713377 0.700781i \(-0.752835\pi\)
−0.713377 + 0.700781i \(0.752835\pi\)
\(758\) −35.4900 −1.28906
\(759\) 0 0
\(760\) −102.451 −3.71627
\(761\) 33.9943 1.23229 0.616147 0.787631i \(-0.288693\pi\)
0.616147 + 0.787631i \(0.288693\pi\)
\(762\) 0 0
\(763\) −14.4321 −0.522476
\(764\) −108.570 −3.92791
\(765\) 0 0
\(766\) 59.0701 2.13429
\(767\) 8.72247 0.314950
\(768\) 0 0
\(769\) −43.2905 −1.56109 −0.780547 0.625097i \(-0.785059\pi\)
−0.780547 + 0.625097i \(0.785059\pi\)
\(770\) −4.42351 −0.159412
\(771\) 0 0
\(772\) −72.1045 −2.59510
\(773\) −49.0758 −1.76513 −0.882567 0.470186i \(-0.844187\pi\)
−0.882567 + 0.470186i \(0.844187\pi\)
\(774\) 0 0
\(775\) −15.6557 −0.562369
\(776\) −101.944 −3.65958
\(777\) 0 0
\(778\) 0.354352 0.0127041
\(779\) −47.7125 −1.70948
\(780\) 0 0
\(781\) −10.7358 −0.384159
\(782\) −59.7161 −2.13544
\(783\) 0 0
\(784\) 17.3194 0.618552
\(785\) 21.2793 0.759492
\(786\) 0 0
\(787\) 23.0838 0.822847 0.411424 0.911444i \(-0.365032\pi\)
0.411424 + 0.911444i \(0.365032\pi\)
\(788\) −24.3738 −0.868282
\(789\) 0 0
\(790\) 18.6074 0.662023
\(791\) −19.2485 −0.684399
\(792\) 0 0
\(793\) −16.0586 −0.570257
\(794\) −17.8461 −0.633334
\(795\) 0 0
\(796\) 63.0564 2.23498
\(797\) 39.3361 1.39336 0.696679 0.717383i \(-0.254660\pi\)
0.696679 + 0.717383i \(0.254660\pi\)
\(798\) 0 0
\(799\) −12.1505 −0.429854
\(800\) −90.7722 −3.20928
\(801\) 0 0
\(802\) −86.9940 −3.07187
\(803\) 2.09669 0.0739904
\(804\) 0 0
\(805\) 7.26962 0.256221
\(806\) 26.2168 0.923448
\(807\) 0 0
\(808\) 180.017 6.33297
\(809\) 14.6267 0.514248 0.257124 0.966378i \(-0.417225\pi\)
0.257124 + 0.966378i \(0.417225\pi\)
\(810\) 0 0
\(811\) −26.4080 −0.927311 −0.463656 0.886015i \(-0.653463\pi\)
−0.463656 + 0.886015i \(0.653463\pi\)
\(812\) 10.9207 0.383241
\(813\) 0 0
\(814\) −24.5451 −0.860305
\(815\) −25.2944 −0.886026
\(816\) 0 0
\(817\) −13.2110 −0.462193
\(818\) 89.3497 3.12404
\(819\) 0 0
\(820\) 46.6678 1.62971
\(821\) 8.01626 0.279769 0.139885 0.990168i \(-0.455327\pi\)
0.139885 + 0.990168i \(0.455327\pi\)
\(822\) 0 0
\(823\) −29.1365 −1.01564 −0.507818 0.861465i \(-0.669548\pi\)
−0.507818 + 0.861465i \(0.669548\pi\)
\(824\) −63.5498 −2.21386
\(825\) 0 0
\(826\) 12.1597 0.423089
\(827\) 11.0859 0.385495 0.192748 0.981248i \(-0.438260\pi\)
0.192748 + 0.981248i \(0.438260\pi\)
\(828\) 0 0
\(829\) 6.10638 0.212083 0.106042 0.994362i \(-0.466182\pi\)
0.106042 + 0.994362i \(0.466182\pi\)
\(830\) −23.1227 −0.802601
\(831\) 0 0
\(832\) 82.9483 2.87571
\(833\) −3.84365 −0.133175
\(834\) 0 0
\(835\) −28.3555 −0.981283
\(836\) 53.3216 1.84417
\(837\) 0 0
\(838\) −85.8920 −2.96709
\(839\) 36.9264 1.27484 0.637421 0.770516i \(-0.280001\pi\)
0.637421 + 0.770516i \(0.280001\pi\)
\(840\) 0 0
\(841\) −25.3604 −0.874498
\(842\) −26.6239 −0.917521
\(843\) 0 0
\(844\) 109.845 3.78101
\(845\) −11.7371 −0.403768
\(846\) 0 0
\(847\) −9.50211 −0.326496
\(848\) 16.0420 0.550885
\(849\) 0 0
\(850\) 35.3466 1.21238
\(851\) 40.3376 1.38275
\(852\) 0 0
\(853\) 5.10893 0.174926 0.0874632 0.996168i \(-0.472124\pi\)
0.0874632 + 0.996168i \(0.472124\pi\)
\(854\) −22.3866 −0.766055
\(855\) 0 0
\(856\) 111.688 3.81742
\(857\) 0.0840423 0.00287083 0.00143541 0.999999i \(-0.499543\pi\)
0.00143541 + 0.999999i \(0.499543\pi\)
\(858\) 0 0
\(859\) 36.9612 1.26110 0.630550 0.776149i \(-0.282829\pi\)
0.630550 + 0.776149i \(0.282829\pi\)
\(860\) 12.9217 0.440625
\(861\) 0 0
\(862\) 88.9124 3.02837
\(863\) 9.69124 0.329894 0.164947 0.986302i \(-0.447255\pi\)
0.164947 + 0.986302i \(0.447255\pi\)
\(864\) 0 0
\(865\) −13.1577 −0.447376
\(866\) 50.7019 1.72292
\(867\) 0 0
\(868\) 27.0848 0.919319
\(869\) −6.30086 −0.213742
\(870\) 0 0
\(871\) −2.94182 −0.0996797
\(872\) −149.386 −5.05884
\(873\) 0 0
\(874\) −118.245 −3.99971
\(875\) −10.8053 −0.365284
\(876\) 0 0
\(877\) 46.7769 1.57954 0.789772 0.613400i \(-0.210199\pi\)
0.789772 + 0.613400i \(0.210199\pi\)
\(878\) 101.225 3.41616
\(879\) 0 0
\(880\) −27.5658 −0.929242
\(881\) −41.5070 −1.39841 −0.699203 0.714924i \(-0.746461\pi\)
−0.699203 + 0.714924i \(0.746461\pi\)
\(882\) 0 0
\(883\) −26.0657 −0.877181 −0.438590 0.898687i \(-0.644522\pi\)
−0.438590 + 0.898687i \(0.644522\pi\)
\(884\) −43.8651 −1.47534
\(885\) 0 0
\(886\) 56.5353 1.89934
\(887\) −1.81271 −0.0608647 −0.0304324 0.999537i \(-0.509688\pi\)
−0.0304324 + 0.999537i \(0.509688\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) 22.2395 0.745470
\(891\) 0 0
\(892\) −84.7735 −2.83843
\(893\) −24.0595 −0.805122
\(894\) 0 0
\(895\) 7.49609 0.250567
\(896\) 60.7679 2.03011
\(897\) 0 0
\(898\) 23.9070 0.797788
\(899\) 9.02660 0.301054
\(900\) 0 0
\(901\) −3.56016 −0.118606
\(902\) −21.3239 −0.710007
\(903\) 0 0
\(904\) −199.241 −6.62665
\(905\) 13.5552 0.450588
\(906\) 0 0
\(907\) 2.56696 0.0852344 0.0426172 0.999091i \(-0.486430\pi\)
0.0426172 + 0.999091i \(0.486430\pi\)
\(908\) 6.22253 0.206502
\(909\) 0 0
\(910\) 7.20570 0.238867
\(911\) −39.1665 −1.29764 −0.648822 0.760940i \(-0.724738\pi\)
−0.648822 + 0.760940i \(0.724738\pi\)
\(912\) 0 0
\(913\) 7.82982 0.259129
\(914\) 69.4931 2.29863
\(915\) 0 0
\(916\) −171.030 −5.65099
\(917\) −10.2834 −0.339589
\(918\) 0 0
\(919\) 0.749666 0.0247292 0.0123646 0.999924i \(-0.496064\pi\)
0.0123646 + 0.999924i \(0.496064\pi\)
\(920\) 75.2476 2.48084
\(921\) 0 0
\(922\) −75.1248 −2.47410
\(923\) 17.4882 0.575631
\(924\) 0 0
\(925\) −23.8762 −0.785045
\(926\) 42.2942 1.38987
\(927\) 0 0
\(928\) 52.3366 1.71803
\(929\) −2.66217 −0.0873428 −0.0436714 0.999046i \(-0.513905\pi\)
−0.0436714 + 0.999046i \(0.513905\pi\)
\(930\) 0 0
\(931\) −7.61093 −0.249438
\(932\) −71.6742 −2.34777
\(933\) 0 0
\(934\) 1.49889 0.0490451
\(935\) 6.11759 0.200067
\(936\) 0 0
\(937\) −4.03927 −0.131957 −0.0659787 0.997821i \(-0.521017\pi\)
−0.0659787 + 0.997821i \(0.521017\pi\)
\(938\) −4.10108 −0.133905
\(939\) 0 0
\(940\) 23.5327 0.767552
\(941\) −45.3228 −1.47748 −0.738740 0.673990i \(-0.764579\pi\)
−0.738740 + 0.673990i \(0.764579\pi\)
\(942\) 0 0
\(943\) 35.0438 1.14118
\(944\) 75.7747 2.46626
\(945\) 0 0
\(946\) −5.90429 −0.191965
\(947\) 42.7254 1.38839 0.694195 0.719787i \(-0.255761\pi\)
0.694195 + 0.719787i \(0.255761\pi\)
\(948\) 0 0
\(949\) −3.41541 −0.110869
\(950\) 69.9907 2.27080
\(951\) 0 0
\(952\) −39.7855 −1.28946
\(953\) 16.1878 0.524375 0.262188 0.965017i \(-0.415556\pi\)
0.262188 + 0.965017i \(0.415556\pi\)
\(954\) 0 0
\(955\) −24.6648 −0.798135
\(956\) 77.3361 2.50123
\(957\) 0 0
\(958\) 62.7032 2.02585
\(959\) 14.7634 0.476735
\(960\) 0 0
\(961\) −8.61277 −0.277831
\(962\) 39.9828 1.28910
\(963\) 0 0
\(964\) 22.7387 0.732365
\(965\) −16.3807 −0.527314
\(966\) 0 0
\(967\) −20.4936 −0.659030 −0.329515 0.944150i \(-0.606885\pi\)
−0.329515 + 0.944150i \(0.606885\pi\)
\(968\) −98.3560 −3.16128
\(969\) 0 0
\(970\) −35.5966 −1.14294
\(971\) −18.7588 −0.601997 −0.300999 0.953625i \(-0.597320\pi\)
−0.300999 + 0.953625i \(0.597320\pi\)
\(972\) 0 0
\(973\) −19.4741 −0.624311
\(974\) 85.7907 2.74891
\(975\) 0 0
\(976\) −139.506 −4.46547
\(977\) −27.9229 −0.893331 −0.446666 0.894701i \(-0.647389\pi\)
−0.446666 + 0.894701i \(0.647389\pi\)
\(978\) 0 0
\(979\) −7.53075 −0.240684
\(980\) 7.44427 0.237798
\(981\) 0 0
\(982\) −19.8660 −0.633951
\(983\) −17.0824 −0.544845 −0.272423 0.962178i \(-0.587825\pi\)
−0.272423 + 0.962178i \(0.587825\pi\)
\(984\) 0 0
\(985\) −5.53725 −0.176431
\(986\) −20.3798 −0.649024
\(987\) 0 0
\(988\) −86.8585 −2.76334
\(989\) 9.70315 0.308542
\(990\) 0 0
\(991\) −13.7507 −0.436804 −0.218402 0.975859i \(-0.570084\pi\)
−0.218402 + 0.975859i \(0.570084\pi\)
\(992\) 129.802 4.12122
\(993\) 0 0
\(994\) 24.3796 0.773275
\(995\) 14.3252 0.454138
\(996\) 0 0
\(997\) 9.83250 0.311398 0.155699 0.987805i \(-0.450237\pi\)
0.155699 + 0.987805i \(0.450237\pi\)
\(998\) 17.2804 0.547002
\(999\) 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.n.1.12 12
3.2 odd 2 889.2.a.a.1.1 12
21.20 even 2 6223.2.a.i.1.1 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
889.2.a.a.1.1 12 3.2 odd 2
6223.2.a.i.1.1 12 21.20 even 2
8001.2.a.n.1.12 12 1.1 even 1 trivial