Properties

Label 8001.2.a.t.1.11
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: \( x^{16} - 2 x^{15} - 20 x^{14} + 38 x^{13} + 155 x^{12} - 275 x^{11} - 593 x^{10} + 957 x^{9} + 1177 x^{8} - 1655 x^{7} - 1150 x^{6} + 1279 x^{5} + 474 x^{4} - 280 x^{3} - 83 x^{2} + \cdots + 1 \) 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 1.11
Root \(1.24549\) 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)
 
\(f(q)\) \(=\) \(q+1.24549 q^{2} -0.448744 q^{4} +2.43368 q^{5} -1.00000 q^{7} -3.04990 q^{8} +O(q^{10})\) \(q+1.24549 q^{2} -0.448744 q^{4} +2.43368 q^{5} -1.00000 q^{7} -3.04990 q^{8} +3.03114 q^{10} +4.22568 q^{11} -5.67912 q^{13} -1.24549 q^{14} -2.90114 q^{16} +1.72874 q^{17} -2.50366 q^{19} -1.09210 q^{20} +5.26306 q^{22} +3.83361 q^{23} +0.922820 q^{25} -7.07331 q^{26} +0.448744 q^{28} -5.96884 q^{29} -5.47757 q^{31} +2.48644 q^{32} +2.15314 q^{34} -2.43368 q^{35} +6.12052 q^{37} -3.11830 q^{38} -7.42249 q^{40} +4.31444 q^{41} +4.93422 q^{43} -1.89625 q^{44} +4.77474 q^{46} +10.9899 q^{47} +1.00000 q^{49} +1.14937 q^{50} +2.54847 q^{52} +4.00048 q^{53} +10.2840 q^{55} +3.04990 q^{56} -7.43416 q^{58} +14.3016 q^{59} +3.21888 q^{61} -6.82229 q^{62} +8.89913 q^{64} -13.8212 q^{65} +7.39502 q^{67} -0.775763 q^{68} -3.03114 q^{70} -7.63071 q^{71} +7.53588 q^{73} +7.62307 q^{74} +1.12350 q^{76} -4.22568 q^{77} -15.4421 q^{79} -7.06046 q^{80} +5.37361 q^{82} +12.7925 q^{83} +4.20722 q^{85} +6.14555 q^{86} -12.8879 q^{88} -3.88584 q^{89} +5.67912 q^{91} -1.72031 q^{92} +13.6879 q^{94} -6.09313 q^{95} +4.24345 q^{97} +1.24549 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 2 q^{2} + 12 q^{4} + 9 q^{5} - 16 q^{7} + 6 q^{8} - 2 q^{10} + 22 q^{11} - 4 q^{13} - 2 q^{14} + 12 q^{16} + 18 q^{17} - 15 q^{19} + 40 q^{20} - 11 q^{22} + 5 q^{23} + 15 q^{25} + 24 q^{26} - 12 q^{28} + 12 q^{29} - 32 q^{31} + 9 q^{32} - 14 q^{34} - 9 q^{35} - 2 q^{37} - 3 q^{38} - 14 q^{40} + 45 q^{41} - 3 q^{43} + 54 q^{44} + 49 q^{47} + 16 q^{49} + 6 q^{50} + 38 q^{52} - 16 q^{53} + 7 q^{55} - 6 q^{56} + 16 q^{58} + 35 q^{59} - 11 q^{61} - 17 q^{62} - 2 q^{64} - 14 q^{65} + 17 q^{67} + 71 q^{68} + 2 q^{70} + 81 q^{71} - 15 q^{73} - 13 q^{74} + 14 q^{76} - 22 q^{77} - 34 q^{79} + 33 q^{80} - 14 q^{82} + 39 q^{83} - 17 q^{85} - 36 q^{86} + 61 q^{88} + 32 q^{89} + 4 q^{91} - 37 q^{92} + 13 q^{94} + 33 q^{95} - 4 q^{97} + 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.24549 0.880698 0.440349 0.897827i \(-0.354855\pi\)
0.440349 + 0.897827i \(0.354855\pi\)
\(3\) 0 0
\(4\) −0.448744 −0.224372
\(5\) 2.43368 1.08838 0.544188 0.838963i \(-0.316838\pi\)
0.544188 + 0.838963i \(0.316838\pi\)
\(6\) 0 0
\(7\) −1.00000 −0.377964
\(8\) −3.04990 −1.07830
\(9\) 0 0
\(10\) 3.03114 0.958531
\(11\) 4.22568 1.27409 0.637046 0.770826i \(-0.280156\pi\)
0.637046 + 0.770826i \(0.280156\pi\)
\(12\) 0 0
\(13\) −5.67912 −1.57510 −0.787552 0.616248i \(-0.788652\pi\)
−0.787552 + 0.616248i \(0.788652\pi\)
\(14\) −1.24549 −0.332872
\(15\) 0 0
\(16\) −2.90114 −0.725285
\(17\) 1.72874 0.419282 0.209641 0.977778i \(-0.432771\pi\)
0.209641 + 0.977778i \(0.432771\pi\)
\(18\) 0 0
\(19\) −2.50366 −0.574380 −0.287190 0.957874i \(-0.592721\pi\)
−0.287190 + 0.957874i \(0.592721\pi\)
\(20\) −1.09210 −0.244201
\(21\) 0 0
\(22\) 5.26306 1.12209
\(23\) 3.83361 0.799364 0.399682 0.916654i \(-0.369121\pi\)
0.399682 + 0.916654i \(0.369121\pi\)
\(24\) 0 0
\(25\) 0.922820 0.184564
\(26\) −7.07331 −1.38719
\(27\) 0 0
\(28\) 0.448744 0.0848046
\(29\) −5.96884 −1.10839 −0.554193 0.832388i \(-0.686973\pi\)
−0.554193 + 0.832388i \(0.686973\pi\)
\(30\) 0 0
\(31\) −5.47757 −0.983801 −0.491901 0.870651i \(-0.663698\pi\)
−0.491901 + 0.870651i \(0.663698\pi\)
\(32\) 2.48644 0.439544
\(33\) 0 0
\(34\) 2.15314 0.369260
\(35\) −2.43368 −0.411368
\(36\) 0 0
\(37\) 6.12052 1.00621 0.503103 0.864226i \(-0.332191\pi\)
0.503103 + 0.864226i \(0.332191\pi\)
\(38\) −3.11830 −0.505855
\(39\) 0 0
\(40\) −7.42249 −1.17360
\(41\) 4.31444 0.673802 0.336901 0.941540i \(-0.390621\pi\)
0.336901 + 0.941540i \(0.390621\pi\)
\(42\) 0 0
\(43\) 4.93422 0.752462 0.376231 0.926526i \(-0.377220\pi\)
0.376231 + 0.926526i \(0.377220\pi\)
\(44\) −1.89625 −0.285870
\(45\) 0 0
\(46\) 4.77474 0.703998
\(47\) 10.9899 1.60304 0.801521 0.597966i \(-0.204024\pi\)
0.801521 + 0.597966i \(0.204024\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 1.14937 0.162545
\(51\) 0 0
\(52\) 2.54847 0.353409
\(53\) 4.00048 0.549508 0.274754 0.961514i \(-0.411404\pi\)
0.274754 + 0.961514i \(0.411404\pi\)
\(54\) 0 0
\(55\) 10.2840 1.38669
\(56\) 3.04990 0.407560
\(57\) 0 0
\(58\) −7.43416 −0.976153
\(59\) 14.3016 1.86191 0.930955 0.365133i \(-0.118977\pi\)
0.930955 + 0.365133i \(0.118977\pi\)
\(60\) 0 0
\(61\) 3.21888 0.412136 0.206068 0.978538i \(-0.433933\pi\)
0.206068 + 0.978538i \(0.433933\pi\)
\(62\) −6.82229 −0.866431
\(63\) 0 0
\(64\) 8.89913 1.11239
\(65\) −13.8212 −1.71431
\(66\) 0 0
\(67\) 7.39502 0.903445 0.451723 0.892158i \(-0.350810\pi\)
0.451723 + 0.892158i \(0.350810\pi\)
\(68\) −0.775763 −0.0940750
\(69\) 0 0
\(70\) −3.03114 −0.362291
\(71\) −7.63071 −0.905598 −0.452799 0.891612i \(-0.649575\pi\)
−0.452799 + 0.891612i \(0.649575\pi\)
\(72\) 0 0
\(73\) 7.53588 0.882008 0.441004 0.897505i \(-0.354622\pi\)
0.441004 + 0.897505i \(0.354622\pi\)
\(74\) 7.62307 0.886164
\(75\) 0 0
\(76\) 1.12350 0.128875
\(77\) −4.22568 −0.481561
\(78\) 0 0
\(79\) −15.4421 −1.73737 −0.868685 0.495365i \(-0.835034\pi\)
−0.868685 + 0.495365i \(0.835034\pi\)
\(80\) −7.06046 −0.789384
\(81\) 0 0
\(82\) 5.37361 0.593416
\(83\) 12.7925 1.40415 0.702077 0.712101i \(-0.252256\pi\)
0.702077 + 0.712101i \(0.252256\pi\)
\(84\) 0 0
\(85\) 4.20722 0.456337
\(86\) 6.14555 0.662691
\(87\) 0 0
\(88\) −12.8879 −1.37385
\(89\) −3.88584 −0.411898 −0.205949 0.978563i \(-0.566028\pi\)
−0.205949 + 0.978563i \(0.566028\pi\)
\(90\) 0 0
\(91\) 5.67912 0.595333
\(92\) −1.72031 −0.179355
\(93\) 0 0
\(94\) 13.6879 1.41180
\(95\) −6.09313 −0.625142
\(96\) 0 0
\(97\) 4.24345 0.430857 0.215429 0.976520i \(-0.430885\pi\)
0.215429 + 0.976520i \(0.430885\pi\)
\(98\) 1.24549 0.125814
\(99\) 0 0
\(100\) −0.414110 −0.0414110
\(101\) 17.4200 1.73335 0.866677 0.498870i \(-0.166251\pi\)
0.866677 + 0.498870i \(0.166251\pi\)
\(102\) 0 0
\(103\) −4.42750 −0.436254 −0.218127 0.975920i \(-0.569995\pi\)
−0.218127 + 0.975920i \(0.569995\pi\)
\(104\) 17.3207 1.69844
\(105\) 0 0
\(106\) 4.98258 0.483951
\(107\) 5.71893 0.552869 0.276435 0.961033i \(-0.410847\pi\)
0.276435 + 0.961033i \(0.410847\pi\)
\(108\) 0 0
\(109\) 14.5703 1.39558 0.697790 0.716302i \(-0.254167\pi\)
0.697790 + 0.716302i \(0.254167\pi\)
\(110\) 12.8086 1.22126
\(111\) 0 0
\(112\) 2.90114 0.274132
\(113\) −5.91261 −0.556212 −0.278106 0.960550i \(-0.589707\pi\)
−0.278106 + 0.960550i \(0.589707\pi\)
\(114\) 0 0
\(115\) 9.32981 0.870009
\(116\) 2.67848 0.248691
\(117\) 0 0
\(118\) 17.8126 1.63978
\(119\) −1.72874 −0.158474
\(120\) 0 0
\(121\) 6.85640 0.623309
\(122\) 4.00910 0.362967
\(123\) 0 0
\(124\) 2.45803 0.220737
\(125\) −9.92257 −0.887502
\(126\) 0 0
\(127\) −1.00000 −0.0887357
\(128\) 6.11094 0.540136
\(129\) 0 0
\(130\) −17.2142 −1.50979
\(131\) −19.7982 −1.72978 −0.864890 0.501962i \(-0.832612\pi\)
−0.864890 + 0.501962i \(0.832612\pi\)
\(132\) 0 0
\(133\) 2.50366 0.217095
\(134\) 9.21045 0.795662
\(135\) 0 0
\(136\) −5.27249 −0.452112
\(137\) −8.97609 −0.766879 −0.383439 0.923566i \(-0.625261\pi\)
−0.383439 + 0.923566i \(0.625261\pi\)
\(138\) 0 0
\(139\) 5.86647 0.497587 0.248794 0.968557i \(-0.419966\pi\)
0.248794 + 0.968557i \(0.419966\pi\)
\(140\) 1.09210 0.0922993
\(141\) 0 0
\(142\) −9.50400 −0.797558
\(143\) −23.9982 −2.00683
\(144\) 0 0
\(145\) −14.5263 −1.20634
\(146\) 9.38590 0.776782
\(147\) 0 0
\(148\) −2.74654 −0.225764
\(149\) 21.5978 1.76936 0.884679 0.466201i \(-0.154378\pi\)
0.884679 + 0.466201i \(0.154378\pi\)
\(150\) 0 0
\(151\) −10.8490 −0.882877 −0.441439 0.897291i \(-0.645532\pi\)
−0.441439 + 0.897291i \(0.645532\pi\)
\(152\) 7.63591 0.619354
\(153\) 0 0
\(154\) −5.26306 −0.424110
\(155\) −13.3307 −1.07075
\(156\) 0 0
\(157\) −3.59825 −0.287171 −0.143586 0.989638i \(-0.545863\pi\)
−0.143586 + 0.989638i \(0.545863\pi\)
\(158\) −19.2330 −1.53010
\(159\) 0 0
\(160\) 6.05120 0.478390
\(161\) −3.83361 −0.302131
\(162\) 0 0
\(163\) −5.95370 −0.466330 −0.233165 0.972437i \(-0.574908\pi\)
−0.233165 + 0.972437i \(0.574908\pi\)
\(164\) −1.93608 −0.151182
\(165\) 0 0
\(166\) 15.9329 1.23663
\(167\) −15.7608 −1.21961 −0.609805 0.792552i \(-0.708752\pi\)
−0.609805 + 0.792552i \(0.708752\pi\)
\(168\) 0 0
\(169\) 19.2524 1.48095
\(170\) 5.24006 0.401895
\(171\) 0 0
\(172\) −2.21420 −0.168831
\(173\) 17.0465 1.29602 0.648010 0.761632i \(-0.275602\pi\)
0.648010 + 0.761632i \(0.275602\pi\)
\(174\) 0 0
\(175\) −0.922820 −0.0697587
\(176\) −12.2593 −0.924080
\(177\) 0 0
\(178\) −4.83979 −0.362758
\(179\) −6.68379 −0.499570 −0.249785 0.968301i \(-0.580360\pi\)
−0.249785 + 0.968301i \(0.580360\pi\)
\(180\) 0 0
\(181\) 13.1820 0.979813 0.489906 0.871775i \(-0.337031\pi\)
0.489906 + 0.871775i \(0.337031\pi\)
\(182\) 7.07331 0.524309
\(183\) 0 0
\(184\) −11.6921 −0.861955
\(185\) 14.8954 1.09513
\(186\) 0 0
\(187\) 7.30512 0.534203
\(188\) −4.93165 −0.359678
\(189\) 0 0
\(190\) −7.58896 −0.550561
\(191\) −17.4273 −1.26100 −0.630498 0.776191i \(-0.717149\pi\)
−0.630498 + 0.776191i \(0.717149\pi\)
\(192\) 0 0
\(193\) 8.81879 0.634790 0.317395 0.948293i \(-0.397192\pi\)
0.317395 + 0.948293i \(0.397192\pi\)
\(194\) 5.28520 0.379455
\(195\) 0 0
\(196\) −0.448744 −0.0320531
\(197\) 22.4395 1.59874 0.799372 0.600836i \(-0.205166\pi\)
0.799372 + 0.600836i \(0.205166\pi\)
\(198\) 0 0
\(199\) 6.40127 0.453774 0.226887 0.973921i \(-0.427145\pi\)
0.226887 + 0.973921i \(0.427145\pi\)
\(200\) −2.81451 −0.199016
\(201\) 0 0
\(202\) 21.6965 1.52656
\(203\) 5.96884 0.418931
\(204\) 0 0
\(205\) 10.5000 0.733351
\(206\) −5.51443 −0.384208
\(207\) 0 0
\(208\) 16.4759 1.14240
\(209\) −10.5797 −0.731812
\(210\) 0 0
\(211\) 17.0999 1.17721 0.588604 0.808422i \(-0.299678\pi\)
0.588604 + 0.808422i \(0.299678\pi\)
\(212\) −1.79519 −0.123294
\(213\) 0 0
\(214\) 7.12289 0.486911
\(215\) 12.0083 0.818962
\(216\) 0 0
\(217\) 5.47757 0.371842
\(218\) 18.1472 1.22908
\(219\) 0 0
\(220\) −4.61487 −0.311135
\(221\) −9.81774 −0.660413
\(222\) 0 0
\(223\) −3.55249 −0.237892 −0.118946 0.992901i \(-0.537952\pi\)
−0.118946 + 0.992901i \(0.537952\pi\)
\(224\) −2.48644 −0.166132
\(225\) 0 0
\(226\) −7.36413 −0.489854
\(227\) 0.461961 0.0306614 0.0153307 0.999882i \(-0.495120\pi\)
0.0153307 + 0.999882i \(0.495120\pi\)
\(228\) 0 0
\(229\) 22.8631 1.51084 0.755418 0.655244i \(-0.227434\pi\)
0.755418 + 0.655244i \(0.227434\pi\)
\(230\) 11.6202 0.766215
\(231\) 0 0
\(232\) 18.2044 1.19517
\(233\) −2.25856 −0.147963 −0.0739816 0.997260i \(-0.523571\pi\)
−0.0739816 + 0.997260i \(0.523571\pi\)
\(234\) 0 0
\(235\) 26.7460 1.74471
\(236\) −6.41776 −0.417760
\(237\) 0 0
\(238\) −2.15314 −0.139567
\(239\) 28.5593 1.84735 0.923673 0.383182i \(-0.125172\pi\)
0.923673 + 0.383182i \(0.125172\pi\)
\(240\) 0 0
\(241\) −13.6906 −0.881888 −0.440944 0.897535i \(-0.645356\pi\)
−0.440944 + 0.897535i \(0.645356\pi\)
\(242\) 8.53960 0.548947
\(243\) 0 0
\(244\) −1.44445 −0.0924717
\(245\) 2.43368 0.155482
\(246\) 0 0
\(247\) 14.2186 0.904708
\(248\) 16.7060 1.06083
\(249\) 0 0
\(250\) −12.3585 −0.781620
\(251\) −7.41487 −0.468022 −0.234011 0.972234i \(-0.575185\pi\)
−0.234011 + 0.972234i \(0.575185\pi\)
\(252\) 0 0
\(253\) 16.1996 1.01846
\(254\) −1.24549 −0.0781493
\(255\) 0 0
\(256\) −10.1871 −0.636695
\(257\) −16.3544 −1.02016 −0.510080 0.860127i \(-0.670384\pi\)
−0.510080 + 0.860127i \(0.670384\pi\)
\(258\) 0 0
\(259\) −6.12052 −0.380310
\(260\) 6.20217 0.384642
\(261\) 0 0
\(262\) −24.6586 −1.52341
\(263\) −4.68637 −0.288974 −0.144487 0.989507i \(-0.546153\pi\)
−0.144487 + 0.989507i \(0.546153\pi\)
\(264\) 0 0
\(265\) 9.73591 0.598072
\(266\) 3.11830 0.191195
\(267\) 0 0
\(268\) −3.31847 −0.202708
\(269\) 14.6818 0.895167 0.447584 0.894242i \(-0.352285\pi\)
0.447584 + 0.894242i \(0.352285\pi\)
\(270\) 0 0
\(271\) −10.7458 −0.652764 −0.326382 0.945238i \(-0.605830\pi\)
−0.326382 + 0.945238i \(0.605830\pi\)
\(272\) −5.01533 −0.304099
\(273\) 0 0
\(274\) −11.1797 −0.675388
\(275\) 3.89955 0.235152
\(276\) 0 0
\(277\) 0.856063 0.0514358 0.0257179 0.999669i \(-0.491813\pi\)
0.0257179 + 0.999669i \(0.491813\pi\)
\(278\) 7.30665 0.438224
\(279\) 0 0
\(280\) 7.42249 0.443578
\(281\) −14.3744 −0.857504 −0.428752 0.903422i \(-0.641047\pi\)
−0.428752 + 0.903422i \(0.641047\pi\)
\(282\) 0 0
\(283\) 18.7855 1.11668 0.558341 0.829612i \(-0.311438\pi\)
0.558341 + 0.829612i \(0.311438\pi\)
\(284\) 3.42423 0.203191
\(285\) 0 0
\(286\) −29.8896 −1.76741
\(287\) −4.31444 −0.254673
\(288\) 0 0
\(289\) −14.0114 −0.824203
\(290\) −18.0924 −1.06242
\(291\) 0 0
\(292\) −3.38168 −0.197898
\(293\) 14.6548 0.856145 0.428072 0.903744i \(-0.359193\pi\)
0.428072 + 0.903744i \(0.359193\pi\)
\(294\) 0 0
\(295\) 34.8056 2.02646
\(296\) −18.6669 −1.08499
\(297\) 0 0
\(298\) 26.8999 1.55827
\(299\) −21.7715 −1.25908
\(300\) 0 0
\(301\) −4.93422 −0.284404
\(302\) −13.5123 −0.777548
\(303\) 0 0
\(304\) 7.26348 0.416589
\(305\) 7.83375 0.448559
\(306\) 0 0
\(307\) −11.1328 −0.635385 −0.317692 0.948194i \(-0.602908\pi\)
−0.317692 + 0.948194i \(0.602908\pi\)
\(308\) 1.89625 0.108049
\(309\) 0 0
\(310\) −16.6033 −0.943004
\(311\) 16.8169 0.953602 0.476801 0.879011i \(-0.341796\pi\)
0.476801 + 0.879011i \(0.341796\pi\)
\(312\) 0 0
\(313\) −13.4057 −0.757732 −0.378866 0.925451i \(-0.623686\pi\)
−0.378866 + 0.925451i \(0.623686\pi\)
\(314\) −4.48160 −0.252911
\(315\) 0 0
\(316\) 6.92953 0.389817
\(317\) 18.6801 1.04918 0.524588 0.851356i \(-0.324219\pi\)
0.524588 + 0.851356i \(0.324219\pi\)
\(318\) 0 0
\(319\) −25.2224 −1.41219
\(320\) 21.6577 1.21070
\(321\) 0 0
\(322\) −4.77474 −0.266086
\(323\) −4.32819 −0.240827
\(324\) 0 0
\(325\) −5.24081 −0.290708
\(326\) −7.41530 −0.410696
\(327\) 0 0
\(328\) −13.1586 −0.726562
\(329\) −10.9899 −0.605893
\(330\) 0 0
\(331\) 2.41940 0.132982 0.0664910 0.997787i \(-0.478820\pi\)
0.0664910 + 0.997787i \(0.478820\pi\)
\(332\) −5.74053 −0.315053
\(333\) 0 0
\(334\) −19.6300 −1.07411
\(335\) 17.9971 0.983289
\(336\) 0 0
\(337\) −17.2871 −0.941687 −0.470844 0.882217i \(-0.656050\pi\)
−0.470844 + 0.882217i \(0.656050\pi\)
\(338\) 23.9788 1.30427
\(339\) 0 0
\(340\) −1.88796 −0.102389
\(341\) −23.1465 −1.25345
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) −15.0489 −0.811380
\(345\) 0 0
\(346\) 21.2313 1.14140
\(347\) 20.7673 1.11485 0.557424 0.830228i \(-0.311790\pi\)
0.557424 + 0.830228i \(0.311790\pi\)
\(348\) 0 0
\(349\) 23.6254 1.26464 0.632318 0.774709i \(-0.282103\pi\)
0.632318 + 0.774709i \(0.282103\pi\)
\(350\) −1.14937 −0.0614363
\(351\) 0 0
\(352\) 10.5069 0.560019
\(353\) 15.5597 0.828161 0.414080 0.910240i \(-0.364103\pi\)
0.414080 + 0.910240i \(0.364103\pi\)
\(354\) 0 0
\(355\) −18.5707 −0.985632
\(356\) 1.74375 0.0924184
\(357\) 0 0
\(358\) −8.32462 −0.439970
\(359\) 7.03515 0.371301 0.185651 0.982616i \(-0.440561\pi\)
0.185651 + 0.982616i \(0.440561\pi\)
\(360\) 0 0
\(361\) −12.7317 −0.670088
\(362\) 16.4181 0.862919
\(363\) 0 0
\(364\) −2.54847 −0.133576
\(365\) 18.3400 0.959957
\(366\) 0 0
\(367\) 28.6265 1.49429 0.747145 0.664661i \(-0.231424\pi\)
0.747145 + 0.664661i \(0.231424\pi\)
\(368\) −11.1219 −0.579767
\(369\) 0 0
\(370\) 18.5521 0.964480
\(371\) −4.00048 −0.207695
\(372\) 0 0
\(373\) 3.75654 0.194506 0.0972531 0.995260i \(-0.468994\pi\)
0.0972531 + 0.995260i \(0.468994\pi\)
\(374\) 9.09849 0.470472
\(375\) 0 0
\(376\) −33.5181 −1.72856
\(377\) 33.8978 1.74582
\(378\) 0 0
\(379\) 20.7800 1.06740 0.533699 0.845675i \(-0.320802\pi\)
0.533699 + 0.845675i \(0.320802\pi\)
\(380\) 2.73425 0.140264
\(381\) 0 0
\(382\) −21.7056 −1.11056
\(383\) −17.6927 −0.904054 −0.452027 0.892004i \(-0.649299\pi\)
−0.452027 + 0.892004i \(0.649299\pi\)
\(384\) 0 0
\(385\) −10.2840 −0.524120
\(386\) 10.9838 0.559058
\(387\) 0 0
\(388\) −1.90422 −0.0966722
\(389\) −27.3076 −1.38455 −0.692275 0.721634i \(-0.743391\pi\)
−0.692275 + 0.721634i \(0.743391\pi\)
\(390\) 0 0
\(391\) 6.62733 0.335159
\(392\) −3.04990 −0.154043
\(393\) 0 0
\(394\) 27.9482 1.40801
\(395\) −37.5811 −1.89091
\(396\) 0 0
\(397\) −12.7728 −0.641046 −0.320523 0.947241i \(-0.603859\pi\)
−0.320523 + 0.947241i \(0.603859\pi\)
\(398\) 7.97275 0.399638
\(399\) 0 0
\(400\) −2.67723 −0.133862
\(401\) −10.4514 −0.521919 −0.260960 0.965350i \(-0.584039\pi\)
−0.260960 + 0.965350i \(0.584039\pi\)
\(402\) 0 0
\(403\) 31.1078 1.54959
\(404\) −7.81711 −0.388916
\(405\) 0 0
\(406\) 7.43416 0.368951
\(407\) 25.8634 1.28200
\(408\) 0 0
\(409\) −10.0572 −0.497297 −0.248648 0.968594i \(-0.579986\pi\)
−0.248648 + 0.968594i \(0.579986\pi\)
\(410\) 13.0777 0.645860
\(411\) 0 0
\(412\) 1.98681 0.0978832
\(413\) −14.3016 −0.703736
\(414\) 0 0
\(415\) 31.1328 1.52825
\(416\) −14.1208 −0.692328
\(417\) 0 0
\(418\) −13.1769 −0.644505
\(419\) 4.78230 0.233631 0.116815 0.993154i \(-0.462731\pi\)
0.116815 + 0.993154i \(0.462731\pi\)
\(420\) 0 0
\(421\) −13.2705 −0.646766 −0.323383 0.946268i \(-0.604820\pi\)
−0.323383 + 0.946268i \(0.604820\pi\)
\(422\) 21.2979 1.03676
\(423\) 0 0
\(424\) −12.2011 −0.592536
\(425\) 1.59532 0.0773844
\(426\) 0 0
\(427\) −3.21888 −0.155773
\(428\) −2.56633 −0.124048
\(429\) 0 0
\(430\) 14.9563 0.721258
\(431\) 21.8535 1.05264 0.526322 0.850285i \(-0.323571\pi\)
0.526322 + 0.850285i \(0.323571\pi\)
\(432\) 0 0
\(433\) −28.0481 −1.34791 −0.673953 0.738774i \(-0.735405\pi\)
−0.673953 + 0.738774i \(0.735405\pi\)
\(434\) 6.82229 0.327480
\(435\) 0 0
\(436\) −6.53832 −0.313129
\(437\) −9.59808 −0.459138
\(438\) 0 0
\(439\) −0.864598 −0.0412650 −0.0206325 0.999787i \(-0.506568\pi\)
−0.0206325 + 0.999787i \(0.506568\pi\)
\(440\) −31.3651 −1.49527
\(441\) 0 0
\(442\) −12.2279 −0.581624
\(443\) −17.3136 −0.822592 −0.411296 0.911502i \(-0.634924\pi\)
−0.411296 + 0.911502i \(0.634924\pi\)
\(444\) 0 0
\(445\) −9.45691 −0.448301
\(446\) −4.42460 −0.209511
\(447\) 0 0
\(448\) −8.89913 −0.420444
\(449\) −8.34805 −0.393969 −0.196985 0.980407i \(-0.563115\pi\)
−0.196985 + 0.980407i \(0.563115\pi\)
\(450\) 0 0
\(451\) 18.2315 0.858486
\(452\) 2.65325 0.124798
\(453\) 0 0
\(454\) 0.575369 0.0270034
\(455\) 13.8212 0.647947
\(456\) 0 0
\(457\) 31.6135 1.47882 0.739408 0.673258i \(-0.235106\pi\)
0.739408 + 0.673258i \(0.235106\pi\)
\(458\) 28.4759 1.33059
\(459\) 0 0
\(460\) −4.18669 −0.195205
\(461\) −14.9040 −0.694148 −0.347074 0.937838i \(-0.612825\pi\)
−0.347074 + 0.937838i \(0.612825\pi\)
\(462\) 0 0
\(463\) −6.32576 −0.293983 −0.146992 0.989138i \(-0.546959\pi\)
−0.146992 + 0.989138i \(0.546959\pi\)
\(464\) 17.3165 0.803896
\(465\) 0 0
\(466\) −2.81303 −0.130311
\(467\) 25.5517 1.18239 0.591196 0.806528i \(-0.298656\pi\)
0.591196 + 0.806528i \(0.298656\pi\)
\(468\) 0 0
\(469\) −7.39502 −0.341470
\(470\) 33.3120 1.53657
\(471\) 0 0
\(472\) −43.6184 −2.00770
\(473\) 20.8505 0.958705
\(474\) 0 0
\(475\) −2.31043 −0.106010
\(476\) 0.775763 0.0355570
\(477\) 0 0
\(478\) 35.5704 1.62695
\(479\) 5.61743 0.256667 0.128333 0.991731i \(-0.459037\pi\)
0.128333 + 0.991731i \(0.459037\pi\)
\(480\) 0 0
\(481\) −34.7592 −1.58488
\(482\) −17.0516 −0.776677
\(483\) 0 0
\(484\) −3.07676 −0.139853
\(485\) 10.3272 0.468935
\(486\) 0 0
\(487\) 25.1524 1.13976 0.569881 0.821727i \(-0.306989\pi\)
0.569881 + 0.821727i \(0.306989\pi\)
\(488\) −9.81726 −0.444407
\(489\) 0 0
\(490\) 3.03114 0.136933
\(491\) 30.3800 1.37103 0.685515 0.728058i \(-0.259577\pi\)
0.685515 + 0.728058i \(0.259577\pi\)
\(492\) 0 0
\(493\) −10.3186 −0.464726
\(494\) 17.7092 0.796774
\(495\) 0 0
\(496\) 15.8912 0.713537
\(497\) 7.63071 0.342284
\(498\) 0 0
\(499\) −28.6191 −1.28116 −0.640582 0.767889i \(-0.721307\pi\)
−0.640582 + 0.767889i \(0.721307\pi\)
\(500\) 4.45269 0.199130
\(501\) 0 0
\(502\) −9.23518 −0.412186
\(503\) −13.1761 −0.587493 −0.293746 0.955883i \(-0.594902\pi\)
−0.293746 + 0.955883i \(0.594902\pi\)
\(504\) 0 0
\(505\) 42.3948 1.88654
\(506\) 20.1766 0.896957
\(507\) 0 0
\(508\) 0.448744 0.0199098
\(509\) 31.0674 1.37704 0.688518 0.725219i \(-0.258262\pi\)
0.688518 + 0.725219i \(0.258262\pi\)
\(510\) 0 0
\(511\) −7.53588 −0.333368
\(512\) −24.9099 −1.10087
\(513\) 0 0
\(514\) −20.3693 −0.898452
\(515\) −10.7751 −0.474809
\(516\) 0 0
\(517\) 46.4399 2.04242
\(518\) −7.62307 −0.334939
\(519\) 0 0
\(520\) 42.1532 1.84854
\(521\) −21.6092 −0.946714 −0.473357 0.880871i \(-0.656958\pi\)
−0.473357 + 0.880871i \(0.656958\pi\)
\(522\) 0 0
\(523\) −11.8477 −0.518063 −0.259032 0.965869i \(-0.583403\pi\)
−0.259032 + 0.965869i \(0.583403\pi\)
\(524\) 8.88433 0.388114
\(525\) 0 0
\(526\) −5.83685 −0.254499
\(527\) −9.46932 −0.412490
\(528\) 0 0
\(529\) −8.30341 −0.361018
\(530\) 12.1260 0.526721
\(531\) 0 0
\(532\) −1.12350 −0.0487100
\(533\) −24.5022 −1.06131
\(534\) 0 0
\(535\) 13.9181 0.601730
\(536\) −22.5540 −0.974186
\(537\) 0 0
\(538\) 18.2861 0.788372
\(539\) 4.22568 0.182013
\(540\) 0 0
\(541\) −13.3103 −0.572253 −0.286126 0.958192i \(-0.592368\pi\)
−0.286126 + 0.958192i \(0.592368\pi\)
\(542\) −13.3839 −0.574887
\(543\) 0 0
\(544\) 4.29841 0.184293
\(545\) 35.4595 1.51892
\(546\) 0 0
\(547\) 0.103601 0.00442966 0.00221483 0.999998i \(-0.499295\pi\)
0.00221483 + 0.999998i \(0.499295\pi\)
\(548\) 4.02796 0.172066
\(549\) 0 0
\(550\) 4.85686 0.207097
\(551\) 14.9440 0.636635
\(552\) 0 0
\(553\) 15.4421 0.656664
\(554\) 1.06622 0.0452994
\(555\) 0 0
\(556\) −2.63254 −0.111645
\(557\) −3.49486 −0.148082 −0.0740410 0.997255i \(-0.523590\pi\)
−0.0740410 + 0.997255i \(0.523590\pi\)
\(558\) 0 0
\(559\) −28.0220 −1.18521
\(560\) 7.06046 0.298359
\(561\) 0 0
\(562\) −17.9032 −0.755202
\(563\) 35.0406 1.47679 0.738393 0.674370i \(-0.235585\pi\)
0.738393 + 0.674370i \(0.235585\pi\)
\(564\) 0 0
\(565\) −14.3894 −0.605368
\(566\) 23.3972 0.983459
\(567\) 0 0
\(568\) 23.2729 0.976508
\(569\) −9.51641 −0.398949 −0.199474 0.979903i \(-0.563923\pi\)
−0.199474 + 0.979903i \(0.563923\pi\)
\(570\) 0 0
\(571\) −41.7756 −1.74825 −0.874127 0.485697i \(-0.838566\pi\)
−0.874127 + 0.485697i \(0.838566\pi\)
\(572\) 10.7690 0.450275
\(573\) 0 0
\(574\) −5.37361 −0.224290
\(575\) 3.53774 0.147534
\(576\) 0 0
\(577\) −18.2875 −0.761320 −0.380660 0.924715i \(-0.624303\pi\)
−0.380660 + 0.924715i \(0.624303\pi\)
\(578\) −17.4512 −0.725873
\(579\) 0 0
\(580\) 6.51858 0.270669
\(581\) −12.7925 −0.530720
\(582\) 0 0
\(583\) 16.9048 0.700124
\(584\) −22.9837 −0.951070
\(585\) 0 0
\(586\) 18.2525 0.754005
\(587\) −2.18408 −0.0901465 −0.0450732 0.998984i \(-0.514352\pi\)
−0.0450732 + 0.998984i \(0.514352\pi\)
\(588\) 0 0
\(589\) 13.7140 0.565076
\(590\) 43.3502 1.78470
\(591\) 0 0
\(592\) −17.7565 −0.729787
\(593\) 31.5402 1.29520 0.647600 0.761981i \(-0.275773\pi\)
0.647600 + 0.761981i \(0.275773\pi\)
\(594\) 0 0
\(595\) −4.20722 −0.172479
\(596\) −9.69186 −0.396994
\(597\) 0 0
\(598\) −27.1163 −1.10887
\(599\) 42.0406 1.71773 0.858866 0.512200i \(-0.171169\pi\)
0.858866 + 0.512200i \(0.171169\pi\)
\(600\) 0 0
\(601\) −13.4807 −0.549891 −0.274945 0.961460i \(-0.588660\pi\)
−0.274945 + 0.961460i \(0.588660\pi\)
\(602\) −6.14555 −0.250474
\(603\) 0 0
\(604\) 4.86841 0.198093
\(605\) 16.6863 0.678395
\(606\) 0 0
\(607\) −6.69701 −0.271823 −0.135912 0.990721i \(-0.543396\pi\)
−0.135912 + 0.990721i \(0.543396\pi\)
\(608\) −6.22520 −0.252465
\(609\) 0 0
\(610\) 9.75689 0.395045
\(611\) −62.4130 −2.52496
\(612\) 0 0
\(613\) −36.2623 −1.46462 −0.732311 0.680970i \(-0.761558\pi\)
−0.732311 + 0.680970i \(0.761558\pi\)
\(614\) −13.8659 −0.559582
\(615\) 0 0
\(616\) 12.8879 0.519268
\(617\) −38.6679 −1.55671 −0.778356 0.627823i \(-0.783946\pi\)
−0.778356 + 0.627823i \(0.783946\pi\)
\(618\) 0 0
\(619\) −8.11816 −0.326296 −0.163148 0.986602i \(-0.552165\pi\)
−0.163148 + 0.986602i \(0.552165\pi\)
\(620\) 5.98206 0.240245
\(621\) 0 0
\(622\) 20.9454 0.839835
\(623\) 3.88584 0.155683
\(624\) 0 0
\(625\) −28.7625 −1.15050
\(626\) −16.6967 −0.667333
\(627\) 0 0
\(628\) 1.61469 0.0644332
\(629\) 10.5808 0.421884
\(630\) 0 0
\(631\) −19.4072 −0.772587 −0.386293 0.922376i \(-0.626245\pi\)
−0.386293 + 0.922376i \(0.626245\pi\)
\(632\) 47.0967 1.87341
\(633\) 0 0
\(634\) 23.2659 0.924007
\(635\) −2.43368 −0.0965778
\(636\) 0 0
\(637\) −5.67912 −0.225015
\(638\) −31.4144 −1.24371
\(639\) 0 0
\(640\) 14.8721 0.587871
\(641\) −11.5636 −0.456736 −0.228368 0.973575i \(-0.573339\pi\)
−0.228368 + 0.973575i \(0.573339\pi\)
\(642\) 0 0
\(643\) −5.96147 −0.235098 −0.117549 0.993067i \(-0.537504\pi\)
−0.117549 + 0.993067i \(0.537504\pi\)
\(644\) 1.72031 0.0677897
\(645\) 0 0
\(646\) −5.39074 −0.212096
\(647\) 50.1845 1.97296 0.986478 0.163892i \(-0.0524050\pi\)
0.986478 + 0.163892i \(0.0524050\pi\)
\(648\) 0 0
\(649\) 60.4341 2.37224
\(650\) −6.52740 −0.256026
\(651\) 0 0
\(652\) 2.67169 0.104631
\(653\) −7.04866 −0.275835 −0.137918 0.990444i \(-0.544041\pi\)
−0.137918 + 0.990444i \(0.544041\pi\)
\(654\) 0 0
\(655\) −48.1826 −1.88265
\(656\) −12.5168 −0.488699
\(657\) 0 0
\(658\) −13.6879 −0.533609
\(659\) 48.8016 1.90104 0.950521 0.310660i \(-0.100550\pi\)
0.950521 + 0.310660i \(0.100550\pi\)
\(660\) 0 0
\(661\) −7.03546 −0.273648 −0.136824 0.990595i \(-0.543689\pi\)
−0.136824 + 0.990595i \(0.543689\pi\)
\(662\) 3.01334 0.117117
\(663\) 0 0
\(664\) −39.0156 −1.51410
\(665\) 6.09313 0.236281
\(666\) 0 0
\(667\) −22.8822 −0.886004
\(668\) 7.07257 0.273646
\(669\) 0 0
\(670\) 22.4153 0.865980
\(671\) 13.6020 0.525099
\(672\) 0 0
\(673\) −48.0452 −1.85201 −0.926004 0.377514i \(-0.876779\pi\)
−0.926004 + 0.377514i \(0.876779\pi\)
\(674\) −21.5310 −0.829342
\(675\) 0 0
\(676\) −8.63939 −0.332284
\(677\) −8.31499 −0.319571 −0.159786 0.987152i \(-0.551080\pi\)
−0.159786 + 0.987152i \(0.551080\pi\)
\(678\) 0 0
\(679\) −4.24345 −0.162849
\(680\) −12.8316 −0.492068
\(681\) 0 0
\(682\) −28.8288 −1.10391
\(683\) −5.68344 −0.217471 −0.108735 0.994071i \(-0.534680\pi\)
−0.108735 + 0.994071i \(0.534680\pi\)
\(684\) 0 0
\(685\) −21.8450 −0.834653
\(686\) −1.24549 −0.0475532
\(687\) 0 0
\(688\) −14.3149 −0.545750
\(689\) −22.7192 −0.865533
\(690\) 0 0
\(691\) 6.26192 0.238215 0.119107 0.992881i \(-0.461997\pi\)
0.119107 + 0.992881i \(0.461997\pi\)
\(692\) −7.64950 −0.290790
\(693\) 0 0
\(694\) 25.8656 0.981844
\(695\) 14.2771 0.541562
\(696\) 0 0
\(697\) 7.45856 0.282513
\(698\) 29.4253 1.11376
\(699\) 0 0
\(700\) 0.414110 0.0156519
\(701\) −9.37500 −0.354089 −0.177044 0.984203i \(-0.556654\pi\)
−0.177044 + 0.984203i \(0.556654\pi\)
\(702\) 0 0
\(703\) −15.3237 −0.577945
\(704\) 37.6049 1.41729
\(705\) 0 0
\(706\) 19.3796 0.729359
\(707\) −17.4200 −0.655146
\(708\) 0 0
\(709\) −34.9615 −1.31301 −0.656504 0.754323i \(-0.727965\pi\)
−0.656504 + 0.754323i \(0.727965\pi\)
\(710\) −23.1297 −0.868044
\(711\) 0 0
\(712\) 11.8514 0.444151
\(713\) −20.9989 −0.786415
\(714\) 0 0
\(715\) −58.4039 −2.18418
\(716\) 2.99931 0.112089
\(717\) 0 0
\(718\) 8.76224 0.327004
\(719\) 36.7725 1.37138 0.685691 0.727893i \(-0.259500\pi\)
0.685691 + 0.727893i \(0.259500\pi\)
\(720\) 0 0
\(721\) 4.42750 0.164889
\(722\) −15.8572 −0.590145
\(723\) 0 0
\(724\) −5.91535 −0.219842
\(725\) −5.50817 −0.204568
\(726\) 0 0
\(727\) −9.14372 −0.339122 −0.169561 0.985520i \(-0.554235\pi\)
−0.169561 + 0.985520i \(0.554235\pi\)
\(728\) −17.3207 −0.641949
\(729\) 0 0
\(730\) 22.8423 0.845432
\(731\) 8.53000 0.315494
\(732\) 0 0
\(733\) 29.1847 1.07796 0.538981 0.842318i \(-0.318809\pi\)
0.538981 + 0.842318i \(0.318809\pi\)
\(734\) 35.6541 1.31602
\(735\) 0 0
\(736\) 9.53204 0.351356
\(737\) 31.2490 1.15107
\(738\) 0 0
\(739\) −0.392476 −0.0144375 −0.00721873 0.999974i \(-0.502298\pi\)
−0.00721873 + 0.999974i \(0.502298\pi\)
\(740\) −6.68422 −0.245717
\(741\) 0 0
\(742\) −4.98258 −0.182916
\(743\) 1.15399 0.0423359 0.0211680 0.999776i \(-0.493262\pi\)
0.0211680 + 0.999776i \(0.493262\pi\)
\(744\) 0 0
\(745\) 52.5622 1.92573
\(746\) 4.67875 0.171301
\(747\) 0 0
\(748\) −3.27813 −0.119860
\(749\) −5.71893 −0.208965
\(750\) 0 0
\(751\) 48.0457 1.75321 0.876607 0.481208i \(-0.159802\pi\)
0.876607 + 0.481208i \(0.159802\pi\)
\(752\) −31.8833 −1.16266
\(753\) 0 0
\(754\) 42.2195 1.53754
\(755\) −26.4030 −0.960903
\(756\) 0 0
\(757\) 40.4145 1.46889 0.734445 0.678668i \(-0.237443\pi\)
0.734445 + 0.678668i \(0.237443\pi\)
\(758\) 25.8814 0.940054
\(759\) 0 0
\(760\) 18.5834 0.674091
\(761\) 26.8626 0.973769 0.486884 0.873466i \(-0.338133\pi\)
0.486884 + 0.873466i \(0.338133\pi\)
\(762\) 0 0
\(763\) −14.5703 −0.527480
\(764\) 7.82040 0.282932
\(765\) 0 0
\(766\) −22.0362 −0.796199
\(767\) −81.2205 −2.93270
\(768\) 0 0
\(769\) −36.9143 −1.33117 −0.665583 0.746324i \(-0.731817\pi\)
−0.665583 + 0.746324i \(0.731817\pi\)
\(770\) −12.8086 −0.461591
\(771\) 0 0
\(772\) −3.95737 −0.142429
\(773\) −35.0570 −1.26091 −0.630457 0.776224i \(-0.717132\pi\)
−0.630457 + 0.776224i \(0.717132\pi\)
\(774\) 0 0
\(775\) −5.05482 −0.181574
\(776\) −12.9421 −0.464594
\(777\) 0 0
\(778\) −34.0114 −1.21937
\(779\) −10.8019 −0.387018
\(780\) 0 0
\(781\) −32.2450 −1.15382
\(782\) 8.25431 0.295173
\(783\) 0 0
\(784\) −2.90114 −0.103612
\(785\) −8.75700 −0.312551
\(786\) 0 0
\(787\) 4.03988 0.144006 0.0720031 0.997404i \(-0.477061\pi\)
0.0720031 + 0.997404i \(0.477061\pi\)
\(788\) −10.0696 −0.358713
\(789\) 0 0
\(790\) −46.8071 −1.66532
\(791\) 5.91261 0.210228
\(792\) 0 0
\(793\) −18.2804 −0.649157
\(794\) −15.9084 −0.564568
\(795\) 0 0
\(796\) −2.87253 −0.101814
\(797\) −21.3487 −0.756208 −0.378104 0.925763i \(-0.623424\pi\)
−0.378104 + 0.925763i \(0.623424\pi\)
\(798\) 0 0
\(799\) 18.9987 0.672127
\(800\) 2.29454 0.0811241
\(801\) 0 0
\(802\) −13.0172 −0.459653
\(803\) 31.8442 1.12376
\(804\) 0 0
\(805\) −9.32981 −0.328832
\(806\) 38.7446 1.36472
\(807\) 0 0
\(808\) −53.1292 −1.86908
\(809\) 26.7700 0.941182 0.470591 0.882352i \(-0.344041\pi\)
0.470591 + 0.882352i \(0.344041\pi\)
\(810\) 0 0
\(811\) 22.1435 0.777563 0.388781 0.921330i \(-0.372896\pi\)
0.388781 + 0.921330i \(0.372896\pi\)
\(812\) −2.67848 −0.0939962
\(813\) 0 0
\(814\) 32.2127 1.12905
\(815\) −14.4894 −0.507543
\(816\) 0 0
\(817\) −12.3536 −0.432199
\(818\) −12.5262 −0.437968
\(819\) 0 0
\(820\) −4.71180 −0.164543
\(821\) −9.39267 −0.327806 −0.163903 0.986476i \(-0.552408\pi\)
−0.163903 + 0.986476i \(0.552408\pi\)
\(822\) 0 0
\(823\) −0.914372 −0.0318730 −0.0159365 0.999873i \(-0.505073\pi\)
−0.0159365 + 0.999873i \(0.505073\pi\)
\(824\) 13.5034 0.470414
\(825\) 0 0
\(826\) −17.8126 −0.619779
\(827\) −39.0657 −1.35845 −0.679224 0.733931i \(-0.737683\pi\)
−0.679224 + 0.733931i \(0.737683\pi\)
\(828\) 0 0
\(829\) 35.1117 1.21948 0.609741 0.792601i \(-0.291274\pi\)
0.609741 + 0.792601i \(0.291274\pi\)
\(830\) 38.7757 1.34592
\(831\) 0 0
\(832\) −50.5392 −1.75213
\(833\) 1.72874 0.0598974
\(834\) 0 0
\(835\) −38.3569 −1.32740
\(836\) 4.74757 0.164198
\(837\) 0 0
\(838\) 5.95632 0.205758
\(839\) 38.3771 1.32492 0.662462 0.749096i \(-0.269512\pi\)
0.662462 + 0.749096i \(0.269512\pi\)
\(840\) 0 0
\(841\) 6.62708 0.228520
\(842\) −16.5284 −0.569605
\(843\) 0 0
\(844\) −7.67348 −0.264132
\(845\) 46.8543 1.61184
\(846\) 0 0
\(847\) −6.85640 −0.235589
\(848\) −11.6060 −0.398550
\(849\) 0 0
\(850\) 1.98696 0.0681522
\(851\) 23.4637 0.804325
\(852\) 0 0
\(853\) 46.6670 1.59785 0.798924 0.601432i \(-0.205403\pi\)
0.798924 + 0.601432i \(0.205403\pi\)
\(854\) −4.00910 −0.137189
\(855\) 0 0
\(856\) −17.4421 −0.596160
\(857\) −13.5444 −0.462670 −0.231335 0.972874i \(-0.574309\pi\)
−0.231335 + 0.972874i \(0.574309\pi\)
\(858\) 0 0
\(859\) −12.2410 −0.417658 −0.208829 0.977952i \(-0.566965\pi\)
−0.208829 + 0.977952i \(0.566965\pi\)
\(860\) −5.38867 −0.183752
\(861\) 0 0
\(862\) 27.2184 0.927061
\(863\) −31.6870 −1.07864 −0.539318 0.842102i \(-0.681318\pi\)
−0.539318 + 0.842102i \(0.681318\pi\)
\(864\) 0 0
\(865\) 41.4858 1.41056
\(866\) −34.9338 −1.18710
\(867\) 0 0
\(868\) −2.45803 −0.0834308
\(869\) −65.2533 −2.21357
\(870\) 0 0
\(871\) −41.9972 −1.42302
\(872\) −44.4379 −1.50486
\(873\) 0 0
\(874\) −11.9544 −0.404362
\(875\) 9.92257 0.335444
\(876\) 0 0
\(877\) 34.1129 1.15191 0.575955 0.817481i \(-0.304630\pi\)
0.575955 + 0.817481i \(0.304630\pi\)
\(878\) −1.07685 −0.0363420
\(879\) 0 0
\(880\) −29.8353 −1.00575
\(881\) 23.3863 0.787905 0.393952 0.919131i \(-0.371107\pi\)
0.393952 + 0.919131i \(0.371107\pi\)
\(882\) 0 0
\(883\) 6.39758 0.215296 0.107648 0.994189i \(-0.465668\pi\)
0.107648 + 0.994189i \(0.465668\pi\)
\(884\) 4.40565 0.148178
\(885\) 0 0
\(886\) −21.5640 −0.724455
\(887\) −16.3462 −0.548851 −0.274425 0.961608i \(-0.588488\pi\)
−0.274425 + 0.961608i \(0.588488\pi\)
\(888\) 0 0
\(889\) 1.00000 0.0335389
\(890\) −11.7785 −0.394817
\(891\) 0 0
\(892\) 1.59416 0.0533763
\(893\) −27.5150 −0.920755
\(894\) 0 0
\(895\) −16.2662 −0.543720
\(896\) −6.11094 −0.204152
\(897\) 0 0
\(898\) −10.3975 −0.346968
\(899\) 32.6948 1.09043
\(900\) 0 0
\(901\) 6.91581 0.230399
\(902\) 22.7072 0.756066
\(903\) 0 0
\(904\) 18.0329 0.599764
\(905\) 32.0809 1.06641
\(906\) 0 0
\(907\) −12.0928 −0.401534 −0.200767 0.979639i \(-0.564343\pi\)
−0.200767 + 0.979639i \(0.564343\pi\)
\(908\) −0.207302 −0.00687955
\(909\) 0 0
\(910\) 17.2142 0.570645
\(911\) 6.38071 0.211402 0.105701 0.994398i \(-0.466291\pi\)
0.105701 + 0.994398i \(0.466291\pi\)
\(912\) 0 0
\(913\) 54.0568 1.78902
\(914\) 39.3744 1.30239
\(915\) 0 0
\(916\) −10.2597 −0.338989
\(917\) 19.7982 0.653795
\(918\) 0 0
\(919\) −10.8878 −0.359156 −0.179578 0.983744i \(-0.557473\pi\)
−0.179578 + 0.983744i \(0.557473\pi\)
\(920\) −28.4549 −0.938132
\(921\) 0 0
\(922\) −18.5628 −0.611334
\(923\) 43.3357 1.42641
\(924\) 0 0
\(925\) 5.64814 0.185710
\(926\) −7.87870 −0.258910
\(927\) 0 0
\(928\) −14.8412 −0.487185
\(929\) 15.7838 0.517851 0.258925 0.965897i \(-0.416632\pi\)
0.258925 + 0.965897i \(0.416632\pi\)
\(930\) 0 0
\(931\) −2.50366 −0.0820543
\(932\) 1.01351 0.0331988
\(933\) 0 0
\(934\) 31.8245 1.04133
\(935\) 17.7784 0.581415
\(936\) 0 0
\(937\) −28.4718 −0.930134 −0.465067 0.885275i \(-0.653970\pi\)
−0.465067 + 0.885275i \(0.653970\pi\)
\(938\) −9.21045 −0.300732
\(939\) 0 0
\(940\) −12.0021 −0.391465
\(941\) −47.9062 −1.56170 −0.780849 0.624720i \(-0.785213\pi\)
−0.780849 + 0.624720i \(0.785213\pi\)
\(942\) 0 0
\(943\) 16.5399 0.538613
\(944\) −41.4910 −1.35042
\(945\) 0 0
\(946\) 25.9691 0.844329
\(947\) −13.1474 −0.427233 −0.213617 0.976918i \(-0.568524\pi\)
−0.213617 + 0.976918i \(0.568524\pi\)
\(948\) 0 0
\(949\) −42.7972 −1.38925
\(950\) −2.87763 −0.0933626
\(951\) 0 0
\(952\) 5.27249 0.170882
\(953\) −4.75915 −0.154164 −0.0770820 0.997025i \(-0.524560\pi\)
−0.0770820 + 0.997025i \(0.524560\pi\)
\(954\) 0 0
\(955\) −42.4126 −1.37244
\(956\) −12.8158 −0.414492
\(957\) 0 0
\(958\) 6.99648 0.226046
\(959\) 8.97609 0.289853
\(960\) 0 0
\(961\) −0.996189 −0.0321351
\(962\) −43.2923 −1.39580
\(963\) 0 0
\(964\) 6.14356 0.197871
\(965\) 21.4621 0.690891
\(966\) 0 0
\(967\) −14.6017 −0.469558 −0.234779 0.972049i \(-0.575437\pi\)
−0.234779 + 0.972049i \(0.575437\pi\)
\(968\) −20.9113 −0.672115
\(969\) 0 0
\(970\) 12.8625 0.412990
\(971\) −2.56019 −0.0821603 −0.0410801 0.999156i \(-0.513080\pi\)
−0.0410801 + 0.999156i \(0.513080\pi\)
\(972\) 0 0
\(973\) −5.86647 −0.188070
\(974\) 31.3271 1.00379
\(975\) 0 0
\(976\) −9.33844 −0.298916
\(977\) −20.9064 −0.668856 −0.334428 0.942421i \(-0.608543\pi\)
−0.334428 + 0.942421i \(0.608543\pi\)
\(978\) 0 0
\(979\) −16.4203 −0.524796
\(980\) −1.09210 −0.0348859
\(981\) 0 0
\(982\) 37.8381 1.20746
\(983\) 0.0609831 0.00194506 0.000972530 1.00000i \(-0.499690\pi\)
0.000972530 1.00000i \(0.499690\pi\)
\(984\) 0 0
\(985\) 54.6105 1.74004
\(986\) −12.8518 −0.409283
\(987\) 0 0
\(988\) −6.38051 −0.202991
\(989\) 18.9159 0.601491
\(990\) 0 0
\(991\) 6.39312 0.203084 0.101542 0.994831i \(-0.467622\pi\)
0.101542 + 0.994831i \(0.467622\pi\)
\(992\) −13.6196 −0.432424
\(993\) 0 0
\(994\) 9.50400 0.301449
\(995\) 15.5787 0.493877
\(996\) 0 0
\(997\) 33.1091 1.04858 0.524288 0.851541i \(-0.324332\pi\)
0.524288 + 0.851541i \(0.324332\pi\)
\(998\) −35.6449 −1.12832
\(999\) 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.t.1.11 16
3.2 odd 2 889.2.a.c.1.6 16
21.20 even 2 6223.2.a.k.1.6 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
889.2.a.c.1.6 16 3.2 odd 2
6223.2.a.k.1.6 16 21.20 even 2
8001.2.a.t.1.11 16 1.1 even 1 trivial