Properties

Label 6003.2.a.r.1.2
Level $6003$
Weight $2$
Character 6003.1
Self dual yes
Analytic conductor $47.934$
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] = [6003,2,Mod(1,6003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6003, 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("6003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6003.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: \(47.9341963334\)
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: \( x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + 5778 x^{8} - 5124 x^{7} - 9405 x^{6} + 8288 x^{5} + 6405 x^{4} - 6032 x^{3} + \cdots - 192 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 2001)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.50224\) of defining polynomial
Character \(\chi\) \(=\) 6003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.50224 q^{2} +4.26120 q^{4} -0.499377 q^{5} +0.377265 q^{7} -5.65806 q^{8} +O(q^{10})\) \(q-2.50224 q^{2} +4.26120 q^{4} -0.499377 q^{5} +0.377265 q^{7} -5.65806 q^{8} +1.24956 q^{10} -0.280990 q^{11} +2.81091 q^{13} -0.944007 q^{14} +5.63543 q^{16} -3.47193 q^{17} -3.13228 q^{19} -2.12795 q^{20} +0.703104 q^{22} +1.00000 q^{23} -4.75062 q^{25} -7.03357 q^{26} +1.60760 q^{28} +1.00000 q^{29} -8.49306 q^{31} -2.78506 q^{32} +8.68760 q^{34} -0.188397 q^{35} -3.30358 q^{37} +7.83770 q^{38} +2.82551 q^{40} +3.07528 q^{41} +6.93400 q^{43} -1.19735 q^{44} -2.50224 q^{46} -1.62857 q^{47} -6.85767 q^{49} +11.8872 q^{50} +11.9779 q^{52} -5.80717 q^{53} +0.140320 q^{55} -2.13459 q^{56} -2.50224 q^{58} -4.19192 q^{59} +14.5885 q^{61} +21.2517 q^{62} -4.30197 q^{64} -1.40370 q^{65} +6.61416 q^{67} -14.7946 q^{68} +0.471415 q^{70} +8.99710 q^{71} -3.65881 q^{73} +8.26635 q^{74} -13.3473 q^{76} -0.106008 q^{77} +10.1301 q^{79} -2.81420 q^{80} -7.69509 q^{82} -14.4012 q^{83} +1.73380 q^{85} -17.3505 q^{86} +1.58986 q^{88} +15.5648 q^{89} +1.06046 q^{91} +4.26120 q^{92} +4.07507 q^{94} +1.56419 q^{95} +0.680933 q^{97} +17.1595 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.50224 −1.76935 −0.884675 0.466208i \(-0.845620\pi\)
−0.884675 + 0.466208i \(0.845620\pi\)
\(3\) 0 0
\(4\) 4.26120 2.13060
\(5\) −0.499377 −0.223328 −0.111664 0.993746i \(-0.535618\pi\)
−0.111664 + 0.993746i \(0.535618\pi\)
\(6\) 0 0
\(7\) 0.377265 0.142593 0.0712963 0.997455i \(-0.477286\pi\)
0.0712963 + 0.997455i \(0.477286\pi\)
\(8\) −5.65806 −2.00043
\(9\) 0 0
\(10\) 1.24956 0.395146
\(11\) −0.280990 −0.0847216 −0.0423608 0.999102i \(-0.513488\pi\)
−0.0423608 + 0.999102i \(0.513488\pi\)
\(12\) 0 0
\(13\) 2.81091 0.779606 0.389803 0.920898i \(-0.372543\pi\)
0.389803 + 0.920898i \(0.372543\pi\)
\(14\) −0.944007 −0.252296
\(15\) 0 0
\(16\) 5.63543 1.40886
\(17\) −3.47193 −0.842067 −0.421033 0.907045i \(-0.638333\pi\)
−0.421033 + 0.907045i \(0.638333\pi\)
\(18\) 0 0
\(19\) −3.13228 −0.718593 −0.359297 0.933223i \(-0.616983\pi\)
−0.359297 + 0.933223i \(0.616983\pi\)
\(20\) −2.12795 −0.475823
\(21\) 0 0
\(22\) 0.703104 0.149902
\(23\) 1.00000 0.208514
\(24\) 0 0
\(25\) −4.75062 −0.950124
\(26\) −7.03357 −1.37940
\(27\) 0 0
\(28\) 1.60760 0.303808
\(29\) 1.00000 0.185695
\(30\) 0 0
\(31\) −8.49306 −1.52540 −0.762700 0.646753i \(-0.776127\pi\)
−0.762700 + 0.646753i \(0.776127\pi\)
\(32\) −2.78506 −0.492334
\(33\) 0 0
\(34\) 8.68760 1.48991
\(35\) −0.188397 −0.0318450
\(36\) 0 0
\(37\) −3.30358 −0.543106 −0.271553 0.962424i \(-0.587537\pi\)
−0.271553 + 0.962424i \(0.587537\pi\)
\(38\) 7.83770 1.27144
\(39\) 0 0
\(40\) 2.82551 0.446752
\(41\) 3.07528 0.480278 0.240139 0.970739i \(-0.422807\pi\)
0.240139 + 0.970739i \(0.422807\pi\)
\(42\) 0 0
\(43\) 6.93400 1.05743 0.528713 0.848801i \(-0.322675\pi\)
0.528713 + 0.848801i \(0.322675\pi\)
\(44\) −1.19735 −0.180508
\(45\) 0 0
\(46\) −2.50224 −0.368935
\(47\) −1.62857 −0.237551 −0.118776 0.992921i \(-0.537897\pi\)
−0.118776 + 0.992921i \(0.537897\pi\)
\(48\) 0 0
\(49\) −6.85767 −0.979667
\(50\) 11.8872 1.68110
\(51\) 0 0
\(52\) 11.9779 1.66103
\(53\) −5.80717 −0.797676 −0.398838 0.917021i \(-0.630586\pi\)
−0.398838 + 0.917021i \(0.630586\pi\)
\(54\) 0 0
\(55\) 0.140320 0.0189207
\(56\) −2.13459 −0.285246
\(57\) 0 0
\(58\) −2.50224 −0.328560
\(59\) −4.19192 −0.545742 −0.272871 0.962051i \(-0.587973\pi\)
−0.272871 + 0.962051i \(0.587973\pi\)
\(60\) 0 0
\(61\) 14.5885 1.86786 0.933932 0.357451i \(-0.116354\pi\)
0.933932 + 0.357451i \(0.116354\pi\)
\(62\) 21.2517 2.69897
\(63\) 0 0
\(64\) −4.30197 −0.537746
\(65\) −1.40370 −0.174108
\(66\) 0 0
\(67\) 6.61416 0.808048 0.404024 0.914748i \(-0.367611\pi\)
0.404024 + 0.914748i \(0.367611\pi\)
\(68\) −14.7946 −1.79411
\(69\) 0 0
\(70\) 0.471415 0.0563449
\(71\) 8.99710 1.06776 0.533880 0.845561i \(-0.320734\pi\)
0.533880 + 0.845561i \(0.320734\pi\)
\(72\) 0 0
\(73\) −3.65881 −0.428231 −0.214115 0.976808i \(-0.568687\pi\)
−0.214115 + 0.976808i \(0.568687\pi\)
\(74\) 8.26635 0.960944
\(75\) 0 0
\(76\) −13.3473 −1.53104
\(77\) −0.106008 −0.0120807
\(78\) 0 0
\(79\) 10.1301 1.13973 0.569865 0.821738i \(-0.306995\pi\)
0.569865 + 0.821738i \(0.306995\pi\)
\(80\) −2.81420 −0.314638
\(81\) 0 0
\(82\) −7.69509 −0.849780
\(83\) −14.4012 −1.58073 −0.790367 0.612633i \(-0.790110\pi\)
−0.790367 + 0.612633i \(0.790110\pi\)
\(84\) 0 0
\(85\) 1.73380 0.188057
\(86\) −17.3505 −1.87096
\(87\) 0 0
\(88\) 1.58986 0.169480
\(89\) 15.5648 1.64987 0.824933 0.565230i \(-0.191213\pi\)
0.824933 + 0.565230i \(0.191213\pi\)
\(90\) 0 0
\(91\) 1.06046 0.111166
\(92\) 4.26120 0.444261
\(93\) 0 0
\(94\) 4.07507 0.420311
\(95\) 1.56419 0.160482
\(96\) 0 0
\(97\) 0.680933 0.0691383 0.0345691 0.999402i \(-0.488994\pi\)
0.0345691 + 0.999402i \(0.488994\pi\)
\(98\) 17.1595 1.73337
\(99\) 0 0
\(100\) −20.2434 −2.02434
\(101\) 3.18551 0.316970 0.158485 0.987361i \(-0.449339\pi\)
0.158485 + 0.987361i \(0.449339\pi\)
\(102\) 0 0
\(103\) −10.4798 −1.03261 −0.516305 0.856405i \(-0.672693\pi\)
−0.516305 + 0.856405i \(0.672693\pi\)
\(104\) −15.9043 −1.55955
\(105\) 0 0
\(106\) 14.5309 1.41137
\(107\) −6.58541 −0.636636 −0.318318 0.947984i \(-0.603118\pi\)
−0.318318 + 0.947984i \(0.603118\pi\)
\(108\) 0 0
\(109\) 0.159462 0.0152737 0.00763685 0.999971i \(-0.497569\pi\)
0.00763685 + 0.999971i \(0.497569\pi\)
\(110\) −0.351114 −0.0334774
\(111\) 0 0
\(112\) 2.12605 0.200893
\(113\) 15.4677 1.45508 0.727538 0.686067i \(-0.240664\pi\)
0.727538 + 0.686067i \(0.240664\pi\)
\(114\) 0 0
\(115\) −0.499377 −0.0465672
\(116\) 4.26120 0.395643
\(117\) 0 0
\(118\) 10.4892 0.965609
\(119\) −1.30984 −0.120073
\(120\) 0 0
\(121\) −10.9210 −0.992822
\(122\) −36.5039 −3.30491
\(123\) 0 0
\(124\) −36.1906 −3.25002
\(125\) 4.86924 0.435518
\(126\) 0 0
\(127\) 10.3000 0.913977 0.456989 0.889473i \(-0.348928\pi\)
0.456989 + 0.889473i \(0.348928\pi\)
\(128\) 16.3347 1.44379
\(129\) 0 0
\(130\) 3.51240 0.308058
\(131\) 5.86940 0.512812 0.256406 0.966569i \(-0.417462\pi\)
0.256406 + 0.966569i \(0.417462\pi\)
\(132\) 0 0
\(133\) −1.18170 −0.102466
\(134\) −16.5502 −1.42972
\(135\) 0 0
\(136\) 19.6444 1.68449
\(137\) 8.10245 0.692239 0.346119 0.938190i \(-0.387499\pi\)
0.346119 + 0.938190i \(0.387499\pi\)
\(138\) 0 0
\(139\) −0.897029 −0.0760850 −0.0380425 0.999276i \(-0.512112\pi\)
−0.0380425 + 0.999276i \(0.512112\pi\)
\(140\) −0.802799 −0.0678489
\(141\) 0 0
\(142\) −22.5129 −1.88924
\(143\) −0.789837 −0.0660495
\(144\) 0 0
\(145\) −0.499377 −0.0414710
\(146\) 9.15521 0.757690
\(147\) 0 0
\(148\) −14.0772 −1.15714
\(149\) 8.58730 0.703499 0.351749 0.936094i \(-0.385587\pi\)
0.351749 + 0.936094i \(0.385587\pi\)
\(150\) 0 0
\(151\) −19.7857 −1.61013 −0.805067 0.593183i \(-0.797871\pi\)
−0.805067 + 0.593183i \(0.797871\pi\)
\(152\) 17.7226 1.43749
\(153\) 0 0
\(154\) 0.265256 0.0213750
\(155\) 4.24124 0.340665
\(156\) 0 0
\(157\) −1.76040 −0.140495 −0.0702476 0.997530i \(-0.522379\pi\)
−0.0702476 + 0.997530i \(0.522379\pi\)
\(158\) −25.3480 −2.01658
\(159\) 0 0
\(160\) 1.39080 0.109952
\(161\) 0.377265 0.0297326
\(162\) 0 0
\(163\) 16.4964 1.29210 0.646048 0.763296i \(-0.276420\pi\)
0.646048 + 0.763296i \(0.276420\pi\)
\(164\) 13.1044 1.02328
\(165\) 0 0
\(166\) 36.0352 2.79687
\(167\) 21.1152 1.63394 0.816970 0.576680i \(-0.195652\pi\)
0.816970 + 0.576680i \(0.195652\pi\)
\(168\) 0 0
\(169\) −5.09878 −0.392214
\(170\) −4.33839 −0.332739
\(171\) 0 0
\(172\) 29.5472 2.25295
\(173\) 1.30202 0.0989910 0.0494955 0.998774i \(-0.484239\pi\)
0.0494955 + 0.998774i \(0.484239\pi\)
\(174\) 0 0
\(175\) −1.79224 −0.135481
\(176\) −1.58350 −0.119361
\(177\) 0 0
\(178\) −38.9469 −2.91919
\(179\) −5.51022 −0.411853 −0.205927 0.978567i \(-0.566021\pi\)
−0.205927 + 0.978567i \(0.566021\pi\)
\(180\) 0 0
\(181\) 3.29654 0.245030 0.122515 0.992467i \(-0.460904\pi\)
0.122515 + 0.992467i \(0.460904\pi\)
\(182\) −2.65352 −0.196692
\(183\) 0 0
\(184\) −5.65806 −0.417118
\(185\) 1.64973 0.121291
\(186\) 0 0
\(187\) 0.975578 0.0713413
\(188\) −6.93966 −0.506127
\(189\) 0 0
\(190\) −3.91397 −0.283949
\(191\) 22.1285 1.60116 0.800581 0.599225i \(-0.204524\pi\)
0.800581 + 0.599225i \(0.204524\pi\)
\(192\) 0 0
\(193\) 18.7430 1.34915 0.674574 0.738207i \(-0.264327\pi\)
0.674574 + 0.738207i \(0.264327\pi\)
\(194\) −1.70386 −0.122330
\(195\) 0 0
\(196\) −29.2219 −2.08728
\(197\) −16.3147 −1.16237 −0.581186 0.813771i \(-0.697411\pi\)
−0.581186 + 0.813771i \(0.697411\pi\)
\(198\) 0 0
\(199\) 12.9140 0.915453 0.457726 0.889093i \(-0.348664\pi\)
0.457726 + 0.889093i \(0.348664\pi\)
\(200\) 26.8793 1.90066
\(201\) 0 0
\(202\) −7.97091 −0.560831
\(203\) 0.377265 0.0264788
\(204\) 0 0
\(205\) −1.53572 −0.107260
\(206\) 26.2231 1.82705
\(207\) 0 0
\(208\) 15.8407 1.09835
\(209\) 0.880138 0.0608804
\(210\) 0 0
\(211\) 14.4592 0.995414 0.497707 0.867345i \(-0.334176\pi\)
0.497707 + 0.867345i \(0.334176\pi\)
\(212\) −24.7455 −1.69953
\(213\) 0 0
\(214\) 16.4783 1.12643
\(215\) −3.46268 −0.236153
\(216\) 0 0
\(217\) −3.20413 −0.217511
\(218\) −0.399012 −0.0270245
\(219\) 0 0
\(220\) 0.597931 0.0403125
\(221\) −9.75929 −0.656481
\(222\) 0 0
\(223\) 7.78586 0.521380 0.260690 0.965423i \(-0.416050\pi\)
0.260690 + 0.965423i \(0.416050\pi\)
\(224\) −1.05071 −0.0702032
\(225\) 0 0
\(226\) −38.7038 −2.57454
\(227\) 19.0309 1.26313 0.631563 0.775325i \(-0.282414\pi\)
0.631563 + 0.775325i \(0.282414\pi\)
\(228\) 0 0
\(229\) −29.2950 −1.93587 −0.967935 0.251201i \(-0.919175\pi\)
−0.967935 + 0.251201i \(0.919175\pi\)
\(230\) 1.24956 0.0823936
\(231\) 0 0
\(232\) −5.65806 −0.371470
\(233\) 13.0823 0.857053 0.428526 0.903529i \(-0.359033\pi\)
0.428526 + 0.903529i \(0.359033\pi\)
\(234\) 0 0
\(235\) 0.813270 0.0530519
\(236\) −17.8626 −1.16276
\(237\) 0 0
\(238\) 3.27753 0.212450
\(239\) −25.0289 −1.61899 −0.809494 0.587128i \(-0.800258\pi\)
−0.809494 + 0.587128i \(0.800258\pi\)
\(240\) 0 0
\(241\) −0.924120 −0.0595278 −0.0297639 0.999557i \(-0.509476\pi\)
−0.0297639 + 0.999557i \(0.509476\pi\)
\(242\) 27.3271 1.75665
\(243\) 0 0
\(244\) 62.1645 3.97967
\(245\) 3.42456 0.218787
\(246\) 0 0
\(247\) −8.80455 −0.560220
\(248\) 48.0543 3.05145
\(249\) 0 0
\(250\) −12.1840 −0.770584
\(251\) 0.948617 0.0598762 0.0299381 0.999552i \(-0.490469\pi\)
0.0299381 + 0.999552i \(0.490469\pi\)
\(252\) 0 0
\(253\) −0.280990 −0.0176657
\(254\) −25.7731 −1.61715
\(255\) 0 0
\(256\) −32.2693 −2.01683
\(257\) −11.2748 −0.703302 −0.351651 0.936131i \(-0.614380\pi\)
−0.351651 + 0.936131i \(0.614380\pi\)
\(258\) 0 0
\(259\) −1.24633 −0.0774429
\(260\) −5.98147 −0.370955
\(261\) 0 0
\(262\) −14.6866 −0.907343
\(263\) −18.0974 −1.11593 −0.557965 0.829864i \(-0.688418\pi\)
−0.557965 + 0.829864i \(0.688418\pi\)
\(264\) 0 0
\(265\) 2.89997 0.178144
\(266\) 2.95689 0.181298
\(267\) 0 0
\(268\) 28.1842 1.72163
\(269\) 16.1467 0.984479 0.492240 0.870460i \(-0.336178\pi\)
0.492240 + 0.870460i \(0.336178\pi\)
\(270\) 0 0
\(271\) 12.1429 0.737631 0.368815 0.929503i \(-0.379763\pi\)
0.368815 + 0.929503i \(0.379763\pi\)
\(272\) −19.5658 −1.18635
\(273\) 0 0
\(274\) −20.2743 −1.22481
\(275\) 1.33488 0.0804961
\(276\) 0 0
\(277\) 3.34216 0.200811 0.100405 0.994947i \(-0.467986\pi\)
0.100405 + 0.994947i \(0.467986\pi\)
\(278\) 2.24458 0.134621
\(279\) 0 0
\(280\) 1.06596 0.0637036
\(281\) 0.0599960 0.00357906 0.00178953 0.999998i \(-0.499430\pi\)
0.00178953 + 0.999998i \(0.499430\pi\)
\(282\) 0 0
\(283\) 14.3954 0.855718 0.427859 0.903845i \(-0.359268\pi\)
0.427859 + 0.903845i \(0.359268\pi\)
\(284\) 38.3384 2.27497
\(285\) 0 0
\(286\) 1.97636 0.116865
\(287\) 1.16019 0.0684841
\(288\) 0 0
\(289\) −4.94570 −0.290923
\(290\) 1.24956 0.0733767
\(291\) 0 0
\(292\) −15.5909 −0.912389
\(293\) −11.9152 −0.696091 −0.348046 0.937478i \(-0.613155\pi\)
−0.348046 + 0.937478i \(0.613155\pi\)
\(294\) 0 0
\(295\) 2.09335 0.121880
\(296\) 18.6919 1.08644
\(297\) 0 0
\(298\) −21.4875 −1.24474
\(299\) 2.81091 0.162559
\(300\) 0 0
\(301\) 2.61595 0.150781
\(302\) 49.5085 2.84889
\(303\) 0 0
\(304\) −17.6517 −1.01240
\(305\) −7.28515 −0.417147
\(306\) 0 0
\(307\) 3.76417 0.214832 0.107416 0.994214i \(-0.465742\pi\)
0.107416 + 0.994214i \(0.465742\pi\)
\(308\) −0.451720 −0.0257391
\(309\) 0 0
\(310\) −10.6126 −0.602755
\(311\) −21.5328 −1.22101 −0.610506 0.792011i \(-0.709034\pi\)
−0.610506 + 0.792011i \(0.709034\pi\)
\(312\) 0 0
\(313\) 8.37433 0.473345 0.236673 0.971589i \(-0.423943\pi\)
0.236673 + 0.971589i \(0.423943\pi\)
\(314\) 4.40494 0.248585
\(315\) 0 0
\(316\) 43.1666 2.42831
\(317\) −29.4201 −1.65240 −0.826200 0.563377i \(-0.809502\pi\)
−0.826200 + 0.563377i \(0.809502\pi\)
\(318\) 0 0
\(319\) −0.280990 −0.0157324
\(320\) 2.14830 0.120094
\(321\) 0 0
\(322\) −0.944007 −0.0526074
\(323\) 10.8750 0.605104
\(324\) 0 0
\(325\) −13.3536 −0.740723
\(326\) −41.2779 −2.28617
\(327\) 0 0
\(328\) −17.4001 −0.960762
\(329\) −0.614402 −0.0338731
\(330\) 0 0
\(331\) −8.34926 −0.458917 −0.229458 0.973318i \(-0.573695\pi\)
−0.229458 + 0.973318i \(0.573695\pi\)
\(332\) −61.3663 −3.36791
\(333\) 0 0
\(334\) −52.8352 −2.89101
\(335\) −3.30296 −0.180460
\(336\) 0 0
\(337\) 22.0675 1.20209 0.601047 0.799214i \(-0.294751\pi\)
0.601047 + 0.799214i \(0.294751\pi\)
\(338\) 12.7584 0.693964
\(339\) 0 0
\(340\) 7.38808 0.400675
\(341\) 2.38647 0.129234
\(342\) 0 0
\(343\) −5.22801 −0.282286
\(344\) −39.2330 −2.11530
\(345\) 0 0
\(346\) −3.25797 −0.175150
\(347\) −32.3908 −1.73883 −0.869414 0.494084i \(-0.835504\pi\)
−0.869414 + 0.494084i \(0.835504\pi\)
\(348\) 0 0
\(349\) −16.8925 −0.904233 −0.452117 0.891959i \(-0.649331\pi\)
−0.452117 + 0.891959i \(0.649331\pi\)
\(350\) 4.48462 0.239713
\(351\) 0 0
\(352\) 0.782574 0.0417113
\(353\) 31.9878 1.70254 0.851269 0.524730i \(-0.175834\pi\)
0.851269 + 0.524730i \(0.175834\pi\)
\(354\) 0 0
\(355\) −4.49295 −0.238461
\(356\) 66.3248 3.51521
\(357\) 0 0
\(358\) 13.7879 0.728713
\(359\) −5.58652 −0.294845 −0.147423 0.989074i \(-0.547098\pi\)
−0.147423 + 0.989074i \(0.547098\pi\)
\(360\) 0 0
\(361\) −9.18885 −0.483624
\(362\) −8.24872 −0.433543
\(363\) 0 0
\(364\) 4.51882 0.236851
\(365\) 1.82712 0.0956360
\(366\) 0 0
\(367\) 2.70892 0.141404 0.0707022 0.997497i \(-0.477476\pi\)
0.0707022 + 0.997497i \(0.477476\pi\)
\(368\) 5.63543 0.293767
\(369\) 0 0
\(370\) −4.12803 −0.214606
\(371\) −2.19084 −0.113743
\(372\) 0 0
\(373\) 25.1027 1.29977 0.649883 0.760034i \(-0.274818\pi\)
0.649883 + 0.760034i \(0.274818\pi\)
\(374\) −2.44113 −0.126228
\(375\) 0 0
\(376\) 9.21455 0.475204
\(377\) 2.81091 0.144769
\(378\) 0 0
\(379\) −25.8338 −1.32700 −0.663498 0.748178i \(-0.730929\pi\)
−0.663498 + 0.748178i \(0.730929\pi\)
\(380\) 6.66531 0.341923
\(381\) 0 0
\(382\) −55.3708 −2.83302
\(383\) −18.0940 −0.924561 −0.462281 0.886734i \(-0.652969\pi\)
−0.462281 + 0.886734i \(0.652969\pi\)
\(384\) 0 0
\(385\) 0.0529378 0.00269796
\(386\) −46.8994 −2.38711
\(387\) 0 0
\(388\) 2.90159 0.147306
\(389\) 0.751401 0.0380976 0.0190488 0.999819i \(-0.493936\pi\)
0.0190488 + 0.999819i \(0.493936\pi\)
\(390\) 0 0
\(391\) −3.47193 −0.175583
\(392\) 38.8011 1.95975
\(393\) 0 0
\(394\) 40.8232 2.05664
\(395\) −5.05876 −0.254534
\(396\) 0 0
\(397\) 36.7706 1.84546 0.922731 0.385445i \(-0.125952\pi\)
0.922731 + 0.385445i \(0.125952\pi\)
\(398\) −32.3140 −1.61976
\(399\) 0 0
\(400\) −26.7718 −1.33859
\(401\) −27.6609 −1.38132 −0.690659 0.723181i \(-0.742679\pi\)
−0.690659 + 0.723181i \(0.742679\pi\)
\(402\) 0 0
\(403\) −23.8732 −1.18921
\(404\) 13.5741 0.675337
\(405\) 0 0
\(406\) −0.944007 −0.0468503
\(407\) 0.928273 0.0460128
\(408\) 0 0
\(409\) 16.6783 0.824691 0.412345 0.911028i \(-0.364710\pi\)
0.412345 + 0.911028i \(0.364710\pi\)
\(410\) 3.84275 0.189780
\(411\) 0 0
\(412\) −44.6567 −2.20008
\(413\) −1.58147 −0.0778188
\(414\) 0 0
\(415\) 7.19162 0.353023
\(416\) −7.82856 −0.383827
\(417\) 0 0
\(418\) −2.20232 −0.107719
\(419\) −4.12698 −0.201616 −0.100808 0.994906i \(-0.532143\pi\)
−0.100808 + 0.994906i \(0.532143\pi\)
\(420\) 0 0
\(421\) 10.9526 0.533795 0.266897 0.963725i \(-0.414002\pi\)
0.266897 + 0.963725i \(0.414002\pi\)
\(422\) −36.1804 −1.76124
\(423\) 0 0
\(424\) 32.8573 1.59569
\(425\) 16.4938 0.800068
\(426\) 0 0
\(427\) 5.50372 0.266344
\(428\) −28.0618 −1.35642
\(429\) 0 0
\(430\) 8.66446 0.417837
\(431\) 20.7355 0.998793 0.499396 0.866374i \(-0.333555\pi\)
0.499396 + 0.866374i \(0.333555\pi\)
\(432\) 0 0
\(433\) 23.2333 1.11652 0.558261 0.829665i \(-0.311469\pi\)
0.558261 + 0.829665i \(0.311469\pi\)
\(434\) 8.01751 0.384853
\(435\) 0 0
\(436\) 0.679500 0.0325422
\(437\) −3.13228 −0.149837
\(438\) 0 0
\(439\) 8.90645 0.425082 0.212541 0.977152i \(-0.431826\pi\)
0.212541 + 0.977152i \(0.431826\pi\)
\(440\) −0.793939 −0.0378496
\(441\) 0 0
\(442\) 24.4201 1.16154
\(443\) 6.58936 0.313070 0.156535 0.987672i \(-0.449968\pi\)
0.156535 + 0.987672i \(0.449968\pi\)
\(444\) 0 0
\(445\) −7.77271 −0.368462
\(446\) −19.4821 −0.922504
\(447\) 0 0
\(448\) −1.62298 −0.0766786
\(449\) 2.64648 0.124895 0.0624477 0.998048i \(-0.480109\pi\)
0.0624477 + 0.998048i \(0.480109\pi\)
\(450\) 0 0
\(451\) −0.864123 −0.0406899
\(452\) 65.9109 3.10019
\(453\) 0 0
\(454\) −47.6199 −2.23491
\(455\) −0.529568 −0.0248265
\(456\) 0 0
\(457\) 28.1702 1.31775 0.658873 0.752255i \(-0.271034\pi\)
0.658873 + 0.752255i \(0.271034\pi\)
\(458\) 73.3032 3.42523
\(459\) 0 0
\(460\) −2.12795 −0.0992160
\(461\) 9.37170 0.436484 0.218242 0.975895i \(-0.429968\pi\)
0.218242 + 0.975895i \(0.429968\pi\)
\(462\) 0 0
\(463\) −9.05249 −0.420705 −0.210353 0.977626i \(-0.567461\pi\)
−0.210353 + 0.977626i \(0.567461\pi\)
\(464\) 5.63543 0.261618
\(465\) 0 0
\(466\) −32.7351 −1.51643
\(467\) 24.7217 1.14398 0.571992 0.820259i \(-0.306171\pi\)
0.571992 + 0.820259i \(0.306171\pi\)
\(468\) 0 0
\(469\) 2.49529 0.115222
\(470\) −2.03500 −0.0938674
\(471\) 0 0
\(472\) 23.7182 1.09172
\(473\) −1.94838 −0.0895868
\(474\) 0 0
\(475\) 14.8803 0.682753
\(476\) −5.58148 −0.255827
\(477\) 0 0
\(478\) 62.6284 2.86456
\(479\) −6.57206 −0.300285 −0.150143 0.988664i \(-0.547973\pi\)
−0.150143 + 0.988664i \(0.547973\pi\)
\(480\) 0 0
\(481\) −9.28608 −0.423409
\(482\) 2.31237 0.105326
\(483\) 0 0
\(484\) −46.5368 −2.11531
\(485\) −0.340042 −0.0154405
\(486\) 0 0
\(487\) −1.76562 −0.0800077 −0.0400038 0.999200i \(-0.512737\pi\)
−0.0400038 + 0.999200i \(0.512737\pi\)
\(488\) −82.5426 −3.73653
\(489\) 0 0
\(490\) −8.56908 −0.387111
\(491\) −0.943575 −0.0425829 −0.0212915 0.999773i \(-0.506778\pi\)
−0.0212915 + 0.999773i \(0.506778\pi\)
\(492\) 0 0
\(493\) −3.47193 −0.156368
\(494\) 22.0311 0.991225
\(495\) 0 0
\(496\) −47.8621 −2.14907
\(497\) 3.39429 0.152255
\(498\) 0 0
\(499\) −17.7953 −0.796626 −0.398313 0.917250i \(-0.630404\pi\)
−0.398313 + 0.917250i \(0.630404\pi\)
\(500\) 20.7488 0.927914
\(501\) 0 0
\(502\) −2.37367 −0.105942
\(503\) 0.846449 0.0377413 0.0188706 0.999822i \(-0.493993\pi\)
0.0188706 + 0.999822i \(0.493993\pi\)
\(504\) 0 0
\(505\) −1.59077 −0.0707884
\(506\) 0.703104 0.0312568
\(507\) 0 0
\(508\) 43.8904 1.94732
\(509\) 15.7969 0.700184 0.350092 0.936715i \(-0.386150\pi\)
0.350092 + 0.936715i \(0.386150\pi\)
\(510\) 0 0
\(511\) −1.38034 −0.0610626
\(512\) 48.0762 2.12469
\(513\) 0 0
\(514\) 28.2122 1.24439
\(515\) 5.23340 0.230611
\(516\) 0 0
\(517\) 0.457611 0.0201257
\(518\) 3.11860 0.137024
\(519\) 0 0
\(520\) 7.94225 0.348291
\(521\) 31.9344 1.39907 0.699536 0.714597i \(-0.253390\pi\)
0.699536 + 0.714597i \(0.253390\pi\)
\(522\) 0 0
\(523\) 6.79011 0.296911 0.148455 0.988919i \(-0.452570\pi\)
0.148455 + 0.988919i \(0.452570\pi\)
\(524\) 25.0107 1.09260
\(525\) 0 0
\(526\) 45.2839 1.97447
\(527\) 29.4873 1.28449
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) −7.25641 −0.315198
\(531\) 0 0
\(532\) −5.03545 −0.218314
\(533\) 8.64434 0.374428
\(534\) 0 0
\(535\) 3.28860 0.142179
\(536\) −37.4233 −1.61644
\(537\) 0 0
\(538\) −40.4028 −1.74189
\(539\) 1.92694 0.0829990
\(540\) 0 0
\(541\) 14.3565 0.617236 0.308618 0.951186i \(-0.400134\pi\)
0.308618 + 0.951186i \(0.400134\pi\)
\(542\) −30.3845 −1.30513
\(543\) 0 0
\(544\) 9.66954 0.414578
\(545\) −0.0796317 −0.00341105
\(546\) 0 0
\(547\) 28.5383 1.22021 0.610104 0.792321i \(-0.291128\pi\)
0.610104 + 0.792321i \(0.291128\pi\)
\(548\) 34.5261 1.47488
\(549\) 0 0
\(550\) −3.34018 −0.142426
\(551\) −3.13228 −0.133439
\(552\) 0 0
\(553\) 3.82175 0.162517
\(554\) −8.36288 −0.355305
\(555\) 0 0
\(556\) −3.82242 −0.162107
\(557\) −21.9328 −0.929324 −0.464662 0.885488i \(-0.653824\pi\)
−0.464662 + 0.885488i \(0.653824\pi\)
\(558\) 0 0
\(559\) 19.4909 0.824375
\(560\) −1.06170 −0.0448650
\(561\) 0 0
\(562\) −0.150124 −0.00633261
\(563\) 15.4453 0.650941 0.325471 0.945552i \(-0.394477\pi\)
0.325471 + 0.945552i \(0.394477\pi\)
\(564\) 0 0
\(565\) −7.72421 −0.324960
\(566\) −36.0208 −1.51407
\(567\) 0 0
\(568\) −50.9062 −2.13598
\(569\) −21.7539 −0.911970 −0.455985 0.889987i \(-0.650713\pi\)
−0.455985 + 0.889987i \(0.650713\pi\)
\(570\) 0 0
\(571\) −15.6666 −0.655626 −0.327813 0.944743i \(-0.606312\pi\)
−0.327813 + 0.944743i \(0.606312\pi\)
\(572\) −3.36566 −0.140725
\(573\) 0 0
\(574\) −2.90308 −0.121172
\(575\) −4.75062 −0.198115
\(576\) 0 0
\(577\) 22.7042 0.945186 0.472593 0.881281i \(-0.343318\pi\)
0.472593 + 0.881281i \(0.343318\pi\)
\(578\) 12.3753 0.514745
\(579\) 0 0
\(580\) −2.12795 −0.0883581
\(581\) −5.43306 −0.225401
\(582\) 0 0
\(583\) 1.63176 0.0675804
\(584\) 20.7018 0.856645
\(585\) 0 0
\(586\) 29.8146 1.23163
\(587\) −0.863910 −0.0356574 −0.0178287 0.999841i \(-0.505675\pi\)
−0.0178287 + 0.999841i \(0.505675\pi\)
\(588\) 0 0
\(589\) 26.6026 1.09614
\(590\) −5.23806 −0.215648
\(591\) 0 0
\(592\) −18.6171 −0.765158
\(593\) 31.8728 1.30886 0.654429 0.756123i \(-0.272909\pi\)
0.654429 + 0.756123i \(0.272909\pi\)
\(594\) 0 0
\(595\) 0.654103 0.0268156
\(596\) 36.5922 1.49887
\(597\) 0 0
\(598\) −7.03357 −0.287624
\(599\) −20.4372 −0.835042 −0.417521 0.908667i \(-0.637101\pi\)
−0.417521 + 0.908667i \(0.637101\pi\)
\(600\) 0 0
\(601\) −5.55415 −0.226559 −0.113279 0.993563i \(-0.536135\pi\)
−0.113279 + 0.993563i \(0.536135\pi\)
\(602\) −6.54574 −0.266785
\(603\) 0 0
\(604\) −84.3107 −3.43055
\(605\) 5.45372 0.221725
\(606\) 0 0
\(607\) 8.04486 0.326531 0.163265 0.986582i \(-0.447797\pi\)
0.163265 + 0.986582i \(0.447797\pi\)
\(608\) 8.72358 0.353788
\(609\) 0 0
\(610\) 18.2292 0.738079
\(611\) −4.57776 −0.185196
\(612\) 0 0
\(613\) 2.92270 0.118047 0.0590234 0.998257i \(-0.481201\pi\)
0.0590234 + 0.998257i \(0.481201\pi\)
\(614\) −9.41884 −0.380114
\(615\) 0 0
\(616\) 0.599798 0.0241665
\(617\) 32.7544 1.31864 0.659322 0.751861i \(-0.270843\pi\)
0.659322 + 0.751861i \(0.270843\pi\)
\(618\) 0 0
\(619\) 7.77841 0.312641 0.156320 0.987706i \(-0.450037\pi\)
0.156320 + 0.987706i \(0.450037\pi\)
\(620\) 18.0728 0.725820
\(621\) 0 0
\(622\) 53.8802 2.16040
\(623\) 5.87205 0.235259
\(624\) 0 0
\(625\) 21.3215 0.852861
\(626\) −20.9546 −0.837513
\(627\) 0 0
\(628\) −7.50142 −0.299339
\(629\) 11.4698 0.457331
\(630\) 0 0
\(631\) 45.0815 1.79467 0.897333 0.441354i \(-0.145502\pi\)
0.897333 + 0.441354i \(0.145502\pi\)
\(632\) −57.3170 −2.27995
\(633\) 0 0
\(634\) 73.6162 2.92367
\(635\) −5.14359 −0.204117
\(636\) 0 0
\(637\) −19.2763 −0.763755
\(638\) 0.703104 0.0278362
\(639\) 0 0
\(640\) −8.15716 −0.322440
\(641\) −25.1043 −0.991561 −0.495780 0.868448i \(-0.665118\pi\)
−0.495780 + 0.868448i \(0.665118\pi\)
\(642\) 0 0
\(643\) 37.8245 1.49165 0.745826 0.666140i \(-0.232055\pi\)
0.745826 + 0.666140i \(0.232055\pi\)
\(644\) 1.60760 0.0633483
\(645\) 0 0
\(646\) −27.2120 −1.07064
\(647\) −6.99373 −0.274952 −0.137476 0.990505i \(-0.543899\pi\)
−0.137476 + 0.990505i \(0.543899\pi\)
\(648\) 0 0
\(649\) 1.17789 0.0462362
\(650\) 33.4138 1.31060
\(651\) 0 0
\(652\) 70.2944 2.75294
\(653\) 34.0332 1.33182 0.665911 0.746031i \(-0.268043\pi\)
0.665911 + 0.746031i \(0.268043\pi\)
\(654\) 0 0
\(655\) −2.93104 −0.114525
\(656\) 17.3305 0.676643
\(657\) 0 0
\(658\) 1.53738 0.0599333
\(659\) −27.9925 −1.09043 −0.545217 0.838295i \(-0.683553\pi\)
−0.545217 + 0.838295i \(0.683553\pi\)
\(660\) 0 0
\(661\) −27.9800 −1.08829 −0.544147 0.838990i \(-0.683147\pi\)
−0.544147 + 0.838990i \(0.683147\pi\)
\(662\) 20.8918 0.811984
\(663\) 0 0
\(664\) 81.4828 3.16215
\(665\) 0.590113 0.0228836
\(666\) 0 0
\(667\) 1.00000 0.0387202
\(668\) 89.9759 3.48127
\(669\) 0 0
\(670\) 8.26479 0.319297
\(671\) −4.09922 −0.158249
\(672\) 0 0
\(673\) 49.9531 1.92555 0.962776 0.270300i \(-0.0871228\pi\)
0.962776 + 0.270300i \(0.0871228\pi\)
\(674\) −55.2182 −2.12692
\(675\) 0 0
\(676\) −21.7269 −0.835651
\(677\) −13.7220 −0.527380 −0.263690 0.964607i \(-0.584940\pi\)
−0.263690 + 0.964607i \(0.584940\pi\)
\(678\) 0 0
\(679\) 0.256892 0.00985861
\(680\) −9.80997 −0.376195
\(681\) 0 0
\(682\) −5.97151 −0.228661
\(683\) 51.7531 1.98028 0.990138 0.140095i \(-0.0447408\pi\)
0.990138 + 0.140095i \(0.0447408\pi\)
\(684\) 0 0
\(685\) −4.04618 −0.154596
\(686\) 13.0817 0.499463
\(687\) 0 0
\(688\) 39.0761 1.48976
\(689\) −16.3234 −0.621873
\(690\) 0 0
\(691\) −24.2514 −0.922566 −0.461283 0.887253i \(-0.652611\pi\)
−0.461283 + 0.887253i \(0.652611\pi\)
\(692\) 5.54818 0.210910
\(693\) 0 0
\(694\) 81.0495 3.07660
\(695\) 0.447956 0.0169919
\(696\) 0 0
\(697\) −10.6772 −0.404426
\(698\) 42.2690 1.59991
\(699\) 0 0
\(700\) −7.63710 −0.288655
\(701\) 3.36677 0.127161 0.0635807 0.997977i \(-0.479748\pi\)
0.0635807 + 0.997977i \(0.479748\pi\)
\(702\) 0 0
\(703\) 10.3477 0.390272
\(704\) 1.20881 0.0455587
\(705\) 0 0
\(706\) −80.0410 −3.01238
\(707\) 1.20178 0.0451976
\(708\) 0 0
\(709\) 15.4206 0.579133 0.289566 0.957158i \(-0.406489\pi\)
0.289566 + 0.957158i \(0.406489\pi\)
\(710\) 11.2424 0.421921
\(711\) 0 0
\(712\) −88.0667 −3.30044
\(713\) −8.49306 −0.318068
\(714\) 0 0
\(715\) 0.394427 0.0147507
\(716\) −23.4802 −0.877495
\(717\) 0 0
\(718\) 13.9788 0.521685
\(719\) 29.3650 1.09513 0.547565 0.836763i \(-0.315555\pi\)
0.547565 + 0.836763i \(0.315555\pi\)
\(720\) 0 0
\(721\) −3.95368 −0.147243
\(722\) 22.9927 0.855700
\(723\) 0 0
\(724\) 14.0472 0.522060
\(725\) −4.75062 −0.176434
\(726\) 0 0
\(727\) 19.1116 0.708811 0.354405 0.935092i \(-0.384683\pi\)
0.354405 + 0.935092i \(0.384683\pi\)
\(728\) −6.00014 −0.222380
\(729\) 0 0
\(730\) −4.57190 −0.169214
\(731\) −24.0744 −0.890423
\(732\) 0 0
\(733\) −35.6961 −1.31846 −0.659232 0.751939i \(-0.729119\pi\)
−0.659232 + 0.751939i \(0.729119\pi\)
\(734\) −6.77836 −0.250194
\(735\) 0 0
\(736\) −2.78506 −0.102659
\(737\) −1.85851 −0.0684591
\(738\) 0 0
\(739\) 17.4540 0.642057 0.321028 0.947070i \(-0.395972\pi\)
0.321028 + 0.947070i \(0.395972\pi\)
\(740\) 7.02985 0.258422
\(741\) 0 0
\(742\) 5.48200 0.201251
\(743\) −22.3796 −0.821028 −0.410514 0.911854i \(-0.634651\pi\)
−0.410514 + 0.911854i \(0.634651\pi\)
\(744\) 0 0
\(745\) −4.28830 −0.157111
\(746\) −62.8129 −2.29974
\(747\) 0 0
\(748\) 4.15713 0.152000
\(749\) −2.48444 −0.0907796
\(750\) 0 0
\(751\) 20.1094 0.733804 0.366902 0.930260i \(-0.380418\pi\)
0.366902 + 0.930260i \(0.380418\pi\)
\(752\) −9.17768 −0.334676
\(753\) 0 0
\(754\) −7.03357 −0.256148
\(755\) 9.88051 0.359589
\(756\) 0 0
\(757\) 37.1745 1.35113 0.675565 0.737300i \(-0.263900\pi\)
0.675565 + 0.737300i \(0.263900\pi\)
\(758\) 64.6424 2.34792
\(759\) 0 0
\(760\) −8.85027 −0.321033
\(761\) 51.3836 1.86265 0.931327 0.364184i \(-0.118652\pi\)
0.931327 + 0.364184i \(0.118652\pi\)
\(762\) 0 0
\(763\) 0.0601594 0.00217792
\(764\) 94.2940 3.41144
\(765\) 0 0
\(766\) 45.2756 1.63587
\(767\) −11.7831 −0.425464
\(768\) 0 0
\(769\) −32.3771 −1.16755 −0.583775 0.811916i \(-0.698425\pi\)
−0.583775 + 0.811916i \(0.698425\pi\)
\(770\) −0.132463 −0.00477363
\(771\) 0 0
\(772\) 79.8675 2.87449
\(773\) 13.8667 0.498751 0.249375 0.968407i \(-0.419775\pi\)
0.249375 + 0.968407i \(0.419775\pi\)
\(774\) 0 0
\(775\) 40.3473 1.44932
\(776\) −3.85276 −0.138306
\(777\) 0 0
\(778\) −1.88019 −0.0674079
\(779\) −9.63262 −0.345125
\(780\) 0 0
\(781\) −2.52809 −0.0904623
\(782\) 8.68760 0.310668
\(783\) 0 0
\(784\) −38.6459 −1.38021
\(785\) 0.879103 0.0313765
\(786\) 0 0
\(787\) 30.4534 1.08555 0.542773 0.839879i \(-0.317374\pi\)
0.542773 + 0.839879i \(0.317374\pi\)
\(788\) −69.5201 −2.47655
\(789\) 0 0
\(790\) 12.6582 0.450360
\(791\) 5.83541 0.207483
\(792\) 0 0
\(793\) 41.0069 1.45620
\(794\) −92.0087 −3.26527
\(795\) 0 0
\(796\) 55.0293 1.95046
\(797\) 24.1771 0.856395 0.428198 0.903685i \(-0.359149\pi\)
0.428198 + 0.903685i \(0.359149\pi\)
\(798\) 0 0
\(799\) 5.65428 0.200034
\(800\) 13.2308 0.467779
\(801\) 0 0
\(802\) 69.2141 2.44404
\(803\) 1.02809 0.0362804
\(804\) 0 0
\(805\) −0.188397 −0.00664014
\(806\) 59.7366 2.10413
\(807\) 0 0
\(808\) −18.0238 −0.634076
\(809\) −28.4412 −0.999941 −0.499970 0.866042i \(-0.666656\pi\)
−0.499970 + 0.866042i \(0.666656\pi\)
\(810\) 0 0
\(811\) −6.10171 −0.214260 −0.107130 0.994245i \(-0.534166\pi\)
−0.107130 + 0.994245i \(0.534166\pi\)
\(812\) 1.60760 0.0564157
\(813\) 0 0
\(814\) −2.32276 −0.0814128
\(815\) −8.23792 −0.288562
\(816\) 0 0
\(817\) −21.7192 −0.759859
\(818\) −41.7332 −1.45917
\(819\) 0 0
\(820\) −6.54403 −0.228527
\(821\) 34.8159 1.21508 0.607542 0.794288i \(-0.292156\pi\)
0.607542 + 0.794288i \(0.292156\pi\)
\(822\) 0 0
\(823\) 28.6531 0.998783 0.499392 0.866376i \(-0.333557\pi\)
0.499392 + 0.866376i \(0.333557\pi\)
\(824\) 59.2957 2.06566
\(825\) 0 0
\(826\) 3.95720 0.137689
\(827\) −35.5993 −1.23791 −0.618954 0.785427i \(-0.712443\pi\)
−0.618954 + 0.785427i \(0.712443\pi\)
\(828\) 0 0
\(829\) −32.0733 −1.11395 −0.556976 0.830529i \(-0.688038\pi\)
−0.556976 + 0.830529i \(0.688038\pi\)
\(830\) −17.9951 −0.624621
\(831\) 0 0
\(832\) −12.0924 −0.419230
\(833\) 23.8094 0.824945
\(834\) 0 0
\(835\) −10.5444 −0.364905
\(836\) 3.75044 0.129712
\(837\) 0 0
\(838\) 10.3267 0.356729
\(839\) 1.42992 0.0493662 0.0246831 0.999695i \(-0.492142\pi\)
0.0246831 + 0.999695i \(0.492142\pi\)
\(840\) 0 0
\(841\) 1.00000 0.0344828
\(842\) −27.4059 −0.944470
\(843\) 0 0
\(844\) 61.6136 2.12083
\(845\) 2.54622 0.0875925
\(846\) 0 0
\(847\) −4.12013 −0.141569
\(848\) −32.7259 −1.12381
\(849\) 0 0
\(850\) −41.2715 −1.41560
\(851\) −3.30358 −0.113245
\(852\) 0 0
\(853\) −1.61657 −0.0553503 −0.0276752 0.999617i \(-0.508810\pi\)
−0.0276752 + 0.999617i \(0.508810\pi\)
\(854\) −13.7716 −0.471255
\(855\) 0 0
\(856\) 37.2607 1.27354
\(857\) 29.6871 1.01409 0.507046 0.861919i \(-0.330737\pi\)
0.507046 + 0.861919i \(0.330737\pi\)
\(858\) 0 0
\(859\) 32.4766 1.10809 0.554044 0.832487i \(-0.313084\pi\)
0.554044 + 0.832487i \(0.313084\pi\)
\(860\) −14.7552 −0.503147
\(861\) 0 0
\(862\) −51.8851 −1.76721
\(863\) 1.28776 0.0438357 0.0219179 0.999760i \(-0.493023\pi\)
0.0219179 + 0.999760i \(0.493023\pi\)
\(864\) 0 0
\(865\) −0.650201 −0.0221075
\(866\) −58.1353 −1.97552
\(867\) 0 0
\(868\) −13.6535 −0.463428
\(869\) −2.84647 −0.0965598
\(870\) 0 0
\(871\) 18.5918 0.629959
\(872\) −0.902247 −0.0305539
\(873\) 0 0
\(874\) 7.83770 0.265114
\(875\) 1.83699 0.0621017
\(876\) 0 0
\(877\) 20.5699 0.694595 0.347297 0.937755i \(-0.387099\pi\)
0.347297 + 0.937755i \(0.387099\pi\)
\(878\) −22.2861 −0.752119
\(879\) 0 0
\(880\) 0.790763 0.0266566
\(881\) −15.6728 −0.528030 −0.264015 0.964519i \(-0.585047\pi\)
−0.264015 + 0.964519i \(0.585047\pi\)
\(882\) 0 0
\(883\) −24.2059 −0.814594 −0.407297 0.913296i \(-0.633529\pi\)
−0.407297 + 0.913296i \(0.633529\pi\)
\(884\) −41.5863 −1.39870
\(885\) 0 0
\(886\) −16.4881 −0.553930
\(887\) 12.4786 0.418990 0.209495 0.977810i \(-0.432818\pi\)
0.209495 + 0.977810i \(0.432818\pi\)
\(888\) 0 0
\(889\) 3.88583 0.130326
\(890\) 19.4492 0.651938
\(891\) 0 0
\(892\) 33.1771 1.11085
\(893\) 5.10113 0.170703
\(894\) 0 0
\(895\) 2.75168 0.0919785
\(896\) 6.16250 0.205875
\(897\) 0 0
\(898\) −6.62214 −0.220984
\(899\) −8.49306 −0.283260
\(900\) 0 0
\(901\) 20.1621 0.671696
\(902\) 2.16224 0.0719948
\(903\) 0 0
\(904\) −87.5171 −2.91078
\(905\) −1.64621 −0.0547220
\(906\) 0 0
\(907\) −20.8227 −0.691407 −0.345704 0.938344i \(-0.612360\pi\)
−0.345704 + 0.938344i \(0.612360\pi\)
\(908\) 81.0945 2.69122
\(909\) 0 0
\(910\) 1.32511 0.0439268
\(911\) 46.6670 1.54615 0.773074 0.634316i \(-0.218718\pi\)
0.773074 + 0.634316i \(0.218718\pi\)
\(912\) 0 0
\(913\) 4.04659 0.133922
\(914\) −70.4885 −2.33155
\(915\) 0 0
\(916\) −124.832 −4.12457
\(917\) 2.21432 0.0731232
\(918\) 0 0
\(919\) −8.68386 −0.286454 −0.143227 0.989690i \(-0.545748\pi\)
−0.143227 + 0.989690i \(0.545748\pi\)
\(920\) 2.82551 0.0931542
\(921\) 0 0
\(922\) −23.4502 −0.772293
\(923\) 25.2900 0.832432
\(924\) 0 0
\(925\) 15.6941 0.516018
\(926\) 22.6515 0.744375
\(927\) 0 0
\(928\) −2.78506 −0.0914241
\(929\) 16.0979 0.528156 0.264078 0.964501i \(-0.414932\pi\)
0.264078 + 0.964501i \(0.414932\pi\)
\(930\) 0 0
\(931\) 21.4801 0.703982
\(932\) 55.7465 1.82604
\(933\) 0 0
\(934\) −61.8596 −2.02411
\(935\) −0.487181 −0.0159325
\(936\) 0 0
\(937\) −40.3985 −1.31976 −0.659881 0.751370i \(-0.729393\pi\)
−0.659881 + 0.751370i \(0.729393\pi\)
\(938\) −6.24381 −0.203868
\(939\) 0 0
\(940\) 3.46551 0.113032
\(941\) 1.57266 0.0512674 0.0256337 0.999671i \(-0.491840\pi\)
0.0256337 + 0.999671i \(0.491840\pi\)
\(942\) 0 0
\(943\) 3.07528 0.100145
\(944\) −23.6233 −0.768873
\(945\) 0 0
\(946\) 4.87532 0.158510
\(947\) 42.5119 1.38145 0.690725 0.723118i \(-0.257292\pi\)
0.690725 + 0.723118i \(0.257292\pi\)
\(948\) 0 0
\(949\) −10.2846 −0.333851
\(950\) −37.2340 −1.20803
\(951\) 0 0
\(952\) 7.41114 0.240197
\(953\) −21.9061 −0.709608 −0.354804 0.934941i \(-0.615452\pi\)
−0.354804 + 0.934941i \(0.615452\pi\)
\(954\) 0 0
\(955\) −11.0505 −0.357585
\(956\) −106.653 −3.44942
\(957\) 0 0
\(958\) 16.4449 0.531310
\(959\) 3.05677 0.0987082
\(960\) 0 0
\(961\) 41.1321 1.32684
\(962\) 23.2360 0.749158
\(963\) 0 0
\(964\) −3.93786 −0.126830
\(965\) −9.35980 −0.301303
\(966\) 0 0
\(967\) 32.3109 1.03905 0.519524 0.854456i \(-0.326109\pi\)
0.519524 + 0.854456i \(0.326109\pi\)
\(968\) 61.7920 1.98607
\(969\) 0 0
\(970\) 0.850867 0.0273197
\(971\) 11.7064 0.375677 0.187839 0.982200i \(-0.439852\pi\)
0.187839 + 0.982200i \(0.439852\pi\)
\(972\) 0 0
\(973\) −0.338418 −0.0108492
\(974\) 4.41799 0.141562
\(975\) 0 0
\(976\) 82.2124 2.63155
\(977\) 5.86810 0.187737 0.0938685 0.995585i \(-0.470077\pi\)
0.0938685 + 0.995585i \(0.470077\pi\)
\(978\) 0 0
\(979\) −4.37355 −0.139779
\(980\) 14.5928 0.466148
\(981\) 0 0
\(982\) 2.36105 0.0753441
\(983\) 18.1817 0.579907 0.289953 0.957041i \(-0.406360\pi\)
0.289953 + 0.957041i \(0.406360\pi\)
\(984\) 0 0
\(985\) 8.14717 0.259591
\(986\) 8.68760 0.276670
\(987\) 0 0
\(988\) −37.5179 −1.19360
\(989\) 6.93400 0.220488
\(990\) 0 0
\(991\) 45.5666 1.44747 0.723736 0.690077i \(-0.242423\pi\)
0.723736 + 0.690077i \(0.242423\pi\)
\(992\) 23.6537 0.751006
\(993\) 0 0
\(994\) −8.49332 −0.269392
\(995\) −6.44898 −0.204446
\(996\) 0 0
\(997\) 23.5989 0.747384 0.373692 0.927553i \(-0.378092\pi\)
0.373692 + 0.927553i \(0.378092\pi\)
\(998\) 44.5280 1.40951
\(999\) 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 6003.2.a.r.1.2 16
3.2 odd 2 2001.2.a.n.1.15 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2001.2.a.n.1.15 16 3.2 odd 2
6003.2.a.r.1.2 16 1.1 even 1 trivial