Properties

Label 6021.2.a.l.1.6
Level $6021$
Weight $2$
Character 6021.1
Self dual yes
Analytic conductor $48.078$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6021,2,Mod(1,6021)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6021, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("6021.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6021 = 3^{3} \cdot 223 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6021.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: \(48.0779270570\)
Analytic rank: \(0\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 20x^{8} + 139x^{6} - 384x^{4} + 331x^{2} - 63 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(0.513295\) of defining polynomial
Character \(\chi\) \(=\) 6021.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.513295 q^{2} -1.73653 q^{4} +3.39171 q^{5} -2.33725 q^{7} -1.91794 q^{8} +O(q^{10})\) \(q+0.513295 q^{2} -1.73653 q^{4} +3.39171 q^{5} -2.33725 q^{7} -1.91794 q^{8} +1.74095 q^{10} -6.00138 q^{11} -0.529519 q^{13} -1.19970 q^{14} +2.48859 q^{16} -3.47145 q^{17} +1.60072 q^{19} -5.88979 q^{20} -3.08048 q^{22} +4.78346 q^{23} +6.50367 q^{25} -0.271799 q^{26} +4.05870 q^{28} -0.564435 q^{29} -3.16166 q^{31} +5.11326 q^{32} -1.78188 q^{34} -7.92726 q^{35} -6.70734 q^{37} +0.821642 q^{38} -6.50509 q^{40} -1.61963 q^{41} +7.24461 q^{43} +10.4216 q^{44} +2.45533 q^{46} +3.63167 q^{47} -1.53727 q^{49} +3.33830 q^{50} +0.919525 q^{52} +0.415335 q^{53} -20.3549 q^{55} +4.48271 q^{56} -0.289722 q^{58} +8.45889 q^{59} +4.30107 q^{61} -1.62286 q^{62} -2.35256 q^{64} -1.79597 q^{65} +7.14056 q^{67} +6.02827 q^{68} -4.06902 q^{70} +9.02792 q^{71} +4.46165 q^{73} -3.44285 q^{74} -2.77970 q^{76} +14.0267 q^{77} +2.33392 q^{79} +8.44056 q^{80} -0.831346 q^{82} +1.10162 q^{83} -11.7741 q^{85} +3.71862 q^{86} +11.5103 q^{88} -7.97816 q^{89} +1.23762 q^{91} -8.30661 q^{92} +1.86412 q^{94} +5.42917 q^{95} -9.18700 q^{97} -0.789071 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 20 q^{4} + 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 20 q^{4} + 2 q^{7} - 10 q^{10} + 2 q^{13} + 44 q^{16} + 28 q^{19} - 42 q^{22} + 22 q^{25} + 40 q^{28} - 18 q^{31} + 36 q^{34} + 20 q^{37} - 4 q^{40} + 2 q^{43} - 30 q^{46} - 32 q^{49} - 2 q^{52} + 52 q^{55} + 84 q^{58} + 40 q^{61} + 64 q^{64} + 18 q^{67} + 18 q^{70} + 32 q^{73} + 104 q^{76} - 16 q^{79} - 94 q^{82} - 40 q^{85} - 32 q^{88} + 14 q^{91} - 56 q^{94} - 18 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.513295 0.362954 0.181477 0.983395i \(-0.441912\pi\)
0.181477 + 0.983395i \(0.441912\pi\)
\(3\) 0 0
\(4\) −1.73653 −0.868264
\(5\) 3.39171 1.51682 0.758408 0.651780i \(-0.225977\pi\)
0.758408 + 0.651780i \(0.225977\pi\)
\(6\) 0 0
\(7\) −2.33725 −0.883397 −0.441699 0.897164i \(-0.645624\pi\)
−0.441699 + 0.897164i \(0.645624\pi\)
\(8\) −1.91794 −0.678094
\(9\) 0 0
\(10\) 1.74095 0.550535
\(11\) −6.00138 −1.80948 −0.904741 0.425961i \(-0.859936\pi\)
−0.904741 + 0.425961i \(0.859936\pi\)
\(12\) 0 0
\(13\) −0.529519 −0.146862 −0.0734311 0.997300i \(-0.523395\pi\)
−0.0734311 + 0.997300i \(0.523395\pi\)
\(14\) −1.19970 −0.320633
\(15\) 0 0
\(16\) 2.48859 0.622147
\(17\) −3.47145 −0.841950 −0.420975 0.907072i \(-0.638312\pi\)
−0.420975 + 0.907072i \(0.638312\pi\)
\(18\) 0 0
\(19\) 1.60072 0.367231 0.183615 0.982998i \(-0.441220\pi\)
0.183615 + 0.982998i \(0.441220\pi\)
\(20\) −5.88979 −1.31700
\(21\) 0 0
\(22\) −3.08048 −0.656759
\(23\) 4.78346 0.997420 0.498710 0.866769i \(-0.333807\pi\)
0.498710 + 0.866769i \(0.333807\pi\)
\(24\) 0 0
\(25\) 6.50367 1.30073
\(26\) −0.271799 −0.0533043
\(27\) 0 0
\(28\) 4.05870 0.767022
\(29\) −0.564435 −0.104813 −0.0524065 0.998626i \(-0.516689\pi\)
−0.0524065 + 0.998626i \(0.516689\pi\)
\(30\) 0 0
\(31\) −3.16166 −0.567851 −0.283926 0.958846i \(-0.591637\pi\)
−0.283926 + 0.958846i \(0.591637\pi\)
\(32\) 5.11326 0.903905
\(33\) 0 0
\(34\) −1.78188 −0.305589
\(35\) −7.92726 −1.33995
\(36\) 0 0
\(37\) −6.70734 −1.10268 −0.551340 0.834280i \(-0.685883\pi\)
−0.551340 + 0.834280i \(0.685883\pi\)
\(38\) 0.821642 0.133288
\(39\) 0 0
\(40\) −6.50509 −1.02855
\(41\) −1.61963 −0.252943 −0.126472 0.991970i \(-0.540365\pi\)
−0.126472 + 0.991970i \(0.540365\pi\)
\(42\) 0 0
\(43\) 7.24461 1.10479 0.552396 0.833582i \(-0.313713\pi\)
0.552396 + 0.833582i \(0.313713\pi\)
\(44\) 10.4216 1.57111
\(45\) 0 0
\(46\) 2.45533 0.362018
\(47\) 3.63167 0.529733 0.264866 0.964285i \(-0.414672\pi\)
0.264866 + 0.964285i \(0.414672\pi\)
\(48\) 0 0
\(49\) −1.53727 −0.219609
\(50\) 3.33830 0.472107
\(51\) 0 0
\(52\) 0.919525 0.127515
\(53\) 0.415335 0.0570507 0.0285253 0.999593i \(-0.490919\pi\)
0.0285253 + 0.999593i \(0.490919\pi\)
\(54\) 0 0
\(55\) −20.3549 −2.74465
\(56\) 4.48271 0.599027
\(57\) 0 0
\(58\) −0.289722 −0.0380423
\(59\) 8.45889 1.10125 0.550627 0.834752i \(-0.314389\pi\)
0.550627 + 0.834752i \(0.314389\pi\)
\(60\) 0 0
\(61\) 4.30107 0.550696 0.275348 0.961345i \(-0.411207\pi\)
0.275348 + 0.961345i \(0.411207\pi\)
\(62\) −1.62286 −0.206104
\(63\) 0 0
\(64\) −2.35256 −0.294071
\(65\) −1.79597 −0.222763
\(66\) 0 0
\(67\) 7.14056 0.872359 0.436179 0.899860i \(-0.356331\pi\)
0.436179 + 0.899860i \(0.356331\pi\)
\(68\) 6.02827 0.731035
\(69\) 0 0
\(70\) −4.06902 −0.486341
\(71\) 9.02792 1.07142 0.535708 0.844403i \(-0.320045\pi\)
0.535708 + 0.844403i \(0.320045\pi\)
\(72\) 0 0
\(73\) 4.46165 0.522196 0.261098 0.965312i \(-0.415915\pi\)
0.261098 + 0.965312i \(0.415915\pi\)
\(74\) −3.44285 −0.400223
\(75\) 0 0
\(76\) −2.77970 −0.318853
\(77\) 14.0267 1.59849
\(78\) 0 0
\(79\) 2.33392 0.262586 0.131293 0.991344i \(-0.458087\pi\)
0.131293 + 0.991344i \(0.458087\pi\)
\(80\) 8.44056 0.943683
\(81\) 0 0
\(82\) −0.831346 −0.0918069
\(83\) 1.10162 0.120919 0.0604595 0.998171i \(-0.480743\pi\)
0.0604595 + 0.998171i \(0.480743\pi\)
\(84\) 0 0
\(85\) −11.7741 −1.27708
\(86\) 3.71862 0.400989
\(87\) 0 0
\(88\) 11.5103 1.22700
\(89\) −7.97816 −0.845684 −0.422842 0.906204i \(-0.638967\pi\)
−0.422842 + 0.906204i \(0.638967\pi\)
\(90\) 0 0
\(91\) 1.23762 0.129738
\(92\) −8.30661 −0.866024
\(93\) 0 0
\(94\) 1.86412 0.192269
\(95\) 5.42917 0.557021
\(96\) 0 0
\(97\) −9.18700 −0.932798 −0.466399 0.884574i \(-0.654449\pi\)
−0.466399 + 0.884574i \(0.654449\pi\)
\(98\) −0.789071 −0.0797082
\(99\) 0 0
\(100\) −11.2938 −1.12938
\(101\) 14.4635 1.43917 0.719586 0.694403i \(-0.244332\pi\)
0.719586 + 0.694403i \(0.244332\pi\)
\(102\) 0 0
\(103\) 13.3653 1.31693 0.658463 0.752613i \(-0.271207\pi\)
0.658463 + 0.752613i \(0.271207\pi\)
\(104\) 1.01559 0.0995864
\(105\) 0 0
\(106\) 0.213189 0.0207068
\(107\) −1.80141 −0.174149 −0.0870745 0.996202i \(-0.527752\pi\)
−0.0870745 + 0.996202i \(0.527752\pi\)
\(108\) 0 0
\(109\) −9.60886 −0.920362 −0.460181 0.887825i \(-0.652216\pi\)
−0.460181 + 0.887825i \(0.652216\pi\)
\(110\) −10.4481 −0.996184
\(111\) 0 0
\(112\) −5.81645 −0.549603
\(113\) 4.85776 0.456980 0.228490 0.973546i \(-0.426621\pi\)
0.228490 + 0.973546i \(0.426621\pi\)
\(114\) 0 0
\(115\) 16.2241 1.51290
\(116\) 0.980157 0.0910053
\(117\) 0 0
\(118\) 4.34190 0.399705
\(119\) 8.11364 0.743776
\(120\) 0 0
\(121\) 25.0165 2.27423
\(122\) 2.20772 0.199877
\(123\) 0 0
\(124\) 5.49031 0.493045
\(125\) 5.09999 0.456157
\(126\) 0 0
\(127\) 1.03893 0.0921902 0.0460951 0.998937i \(-0.485322\pi\)
0.0460951 + 0.998937i \(0.485322\pi\)
\(128\) −11.4341 −1.01064
\(129\) 0 0
\(130\) −0.921864 −0.0808528
\(131\) 13.4193 1.17245 0.586226 0.810148i \(-0.300613\pi\)
0.586226 + 0.810148i \(0.300613\pi\)
\(132\) 0 0
\(133\) −3.74128 −0.324410
\(134\) 3.66521 0.316626
\(135\) 0 0
\(136\) 6.65803 0.570922
\(137\) 22.4434 1.91747 0.958733 0.284307i \(-0.0917635\pi\)
0.958733 + 0.284307i \(0.0917635\pi\)
\(138\) 0 0
\(139\) −0.185572 −0.0157400 −0.00787002 0.999969i \(-0.502505\pi\)
−0.00787002 + 0.999969i \(0.502505\pi\)
\(140\) 13.7659 1.16343
\(141\) 0 0
\(142\) 4.63398 0.388875
\(143\) 3.17784 0.265745
\(144\) 0 0
\(145\) −1.91440 −0.158982
\(146\) 2.29014 0.189533
\(147\) 0 0
\(148\) 11.6475 0.957418
\(149\) −15.7053 −1.28663 −0.643314 0.765602i \(-0.722441\pi\)
−0.643314 + 0.765602i \(0.722441\pi\)
\(150\) 0 0
\(151\) 10.1211 0.823641 0.411820 0.911265i \(-0.364893\pi\)
0.411820 + 0.911265i \(0.364893\pi\)
\(152\) −3.07009 −0.249017
\(153\) 0 0
\(154\) 7.19984 0.580179
\(155\) −10.7234 −0.861326
\(156\) 0 0
\(157\) 2.84167 0.226790 0.113395 0.993550i \(-0.463827\pi\)
0.113395 + 0.993550i \(0.463827\pi\)
\(158\) 1.19799 0.0953069
\(159\) 0 0
\(160\) 17.3427 1.37106
\(161\) −11.1801 −0.881118
\(162\) 0 0
\(163\) 3.03794 0.237950 0.118975 0.992897i \(-0.462039\pi\)
0.118975 + 0.992897i \(0.462039\pi\)
\(164\) 2.81253 0.219622
\(165\) 0 0
\(166\) 0.565458 0.0438881
\(167\) −2.86625 −0.221797 −0.110899 0.993832i \(-0.535373\pi\)
−0.110899 + 0.993832i \(0.535373\pi\)
\(168\) 0 0
\(169\) −12.7196 −0.978432
\(170\) −6.04360 −0.463523
\(171\) 0 0
\(172\) −12.5805 −0.959252
\(173\) −1.37172 −0.104290 −0.0521451 0.998640i \(-0.516606\pi\)
−0.0521451 + 0.998640i \(0.516606\pi\)
\(174\) 0 0
\(175\) −15.2007 −1.14906
\(176\) −14.9349 −1.12576
\(177\) 0 0
\(178\) −4.09515 −0.306944
\(179\) 22.0927 1.65128 0.825642 0.564194i \(-0.190813\pi\)
0.825642 + 0.564194i \(0.190813\pi\)
\(180\) 0 0
\(181\) 9.30691 0.691777 0.345889 0.938276i \(-0.387577\pi\)
0.345889 + 0.938276i \(0.387577\pi\)
\(182\) 0.635263 0.0470888
\(183\) 0 0
\(184\) −9.17439 −0.676345
\(185\) −22.7493 −1.67256
\(186\) 0 0
\(187\) 20.8335 1.52349
\(188\) −6.30649 −0.459948
\(189\) 0 0
\(190\) 2.78677 0.202173
\(191\) −3.18205 −0.230245 −0.115123 0.993351i \(-0.536726\pi\)
−0.115123 + 0.993351i \(0.536726\pi\)
\(192\) 0 0
\(193\) 14.7508 1.06178 0.530892 0.847439i \(-0.321857\pi\)
0.530892 + 0.847439i \(0.321857\pi\)
\(194\) −4.71564 −0.338563
\(195\) 0 0
\(196\) 2.66951 0.190679
\(197\) 4.18236 0.297981 0.148991 0.988839i \(-0.452398\pi\)
0.148991 + 0.988839i \(0.452398\pi\)
\(198\) 0 0
\(199\) 8.83782 0.626497 0.313248 0.949671i \(-0.398583\pi\)
0.313248 + 0.949671i \(0.398583\pi\)
\(200\) −12.4736 −0.882020
\(201\) 0 0
\(202\) 7.42404 0.522354
\(203\) 1.31923 0.0925915
\(204\) 0 0
\(205\) −5.49330 −0.383669
\(206\) 6.86036 0.477984
\(207\) 0 0
\(208\) −1.31775 −0.0913698
\(209\) −9.60653 −0.664497
\(210\) 0 0
\(211\) −2.97482 −0.204795 −0.102397 0.994744i \(-0.532651\pi\)
−0.102397 + 0.994744i \(0.532651\pi\)
\(212\) −0.721241 −0.0495351
\(213\) 0 0
\(214\) −0.924656 −0.0632082
\(215\) 24.5716 1.67577
\(216\) 0 0
\(217\) 7.38959 0.501638
\(218\) −4.93218 −0.334049
\(219\) 0 0
\(220\) 35.3469 2.38308
\(221\) 1.83820 0.123651
\(222\) 0 0
\(223\) −1.00000 −0.0669650
\(224\) −11.9510 −0.798507
\(225\) 0 0
\(226\) 2.49347 0.165863
\(227\) −7.61794 −0.505621 −0.252810 0.967516i \(-0.581355\pi\)
−0.252810 + 0.967516i \(0.581355\pi\)
\(228\) 0 0
\(229\) −8.83725 −0.583982 −0.291991 0.956421i \(-0.594318\pi\)
−0.291991 + 0.956421i \(0.594318\pi\)
\(230\) 8.32774 0.549115
\(231\) 0 0
\(232\) 1.08255 0.0710731
\(233\) −13.3252 −0.872964 −0.436482 0.899713i \(-0.643776\pi\)
−0.436482 + 0.899713i \(0.643776\pi\)
\(234\) 0 0
\(235\) 12.3175 0.803508
\(236\) −14.6891 −0.956179
\(237\) 0 0
\(238\) 4.16469 0.269957
\(239\) −14.0347 −0.907830 −0.453915 0.891045i \(-0.649973\pi\)
−0.453915 + 0.891045i \(0.649973\pi\)
\(240\) 0 0
\(241\) 23.0863 1.48712 0.743561 0.668668i \(-0.233135\pi\)
0.743561 + 0.668668i \(0.233135\pi\)
\(242\) 12.8408 0.825441
\(243\) 0 0
\(244\) −7.46894 −0.478150
\(245\) −5.21395 −0.333107
\(246\) 0 0
\(247\) −0.847612 −0.0539323
\(248\) 6.06388 0.385057
\(249\) 0 0
\(250\) 2.61780 0.165564
\(251\) −17.9910 −1.13558 −0.567792 0.823172i \(-0.692202\pi\)
−0.567792 + 0.823172i \(0.692202\pi\)
\(252\) 0 0
\(253\) −28.7073 −1.80481
\(254\) 0.533278 0.0334608
\(255\) 0 0
\(256\) −1.16393 −0.0727454
\(257\) −2.89339 −0.180485 −0.0902424 0.995920i \(-0.528764\pi\)
−0.0902424 + 0.995920i \(0.528764\pi\)
\(258\) 0 0
\(259\) 15.6767 0.974105
\(260\) 3.11876 0.193417
\(261\) 0 0
\(262\) 6.88807 0.425546
\(263\) 8.25888 0.509265 0.254632 0.967038i \(-0.418046\pi\)
0.254632 + 0.967038i \(0.418046\pi\)
\(264\) 0 0
\(265\) 1.40869 0.0865354
\(266\) −1.92038 −0.117746
\(267\) 0 0
\(268\) −12.3998 −0.757438
\(269\) −27.4532 −1.67385 −0.836927 0.547315i \(-0.815650\pi\)
−0.836927 + 0.547315i \(0.815650\pi\)
\(270\) 0 0
\(271\) 21.5102 1.30665 0.653325 0.757077i \(-0.273373\pi\)
0.653325 + 0.757077i \(0.273373\pi\)
\(272\) −8.63900 −0.523816
\(273\) 0 0
\(274\) 11.5201 0.695953
\(275\) −39.0309 −2.35365
\(276\) 0 0
\(277\) 12.7081 0.763556 0.381778 0.924254i \(-0.375312\pi\)
0.381778 + 0.924254i \(0.375312\pi\)
\(278\) −0.0952533 −0.00571292
\(279\) 0 0
\(280\) 15.2040 0.908614
\(281\) −3.18529 −0.190018 −0.0950091 0.995476i \(-0.530288\pi\)
−0.0950091 + 0.995476i \(0.530288\pi\)
\(282\) 0 0
\(283\) −19.8124 −1.17773 −0.588864 0.808232i \(-0.700425\pi\)
−0.588864 + 0.808232i \(0.700425\pi\)
\(284\) −15.6772 −0.930273
\(285\) 0 0
\(286\) 1.63117 0.0964531
\(287\) 3.78547 0.223449
\(288\) 0 0
\(289\) −4.94905 −0.291120
\(290\) −0.982650 −0.0577032
\(291\) 0 0
\(292\) −7.74778 −0.453404
\(293\) 16.4181 0.959156 0.479578 0.877499i \(-0.340790\pi\)
0.479578 + 0.877499i \(0.340790\pi\)
\(294\) 0 0
\(295\) 28.6901 1.67040
\(296\) 12.8643 0.747722
\(297\) 0 0
\(298\) −8.06145 −0.466988
\(299\) −2.53293 −0.146483
\(300\) 0 0
\(301\) −16.9325 −0.975971
\(302\) 5.19509 0.298944
\(303\) 0 0
\(304\) 3.98353 0.228471
\(305\) 14.5880 0.835305
\(306\) 0 0
\(307\) 5.61879 0.320681 0.160341 0.987062i \(-0.448741\pi\)
0.160341 + 0.987062i \(0.448741\pi\)
\(308\) −24.3578 −1.38791
\(309\) 0 0
\(310\) −5.50428 −0.312622
\(311\) −12.9632 −0.735077 −0.367539 0.930008i \(-0.619799\pi\)
−0.367539 + 0.930008i \(0.619799\pi\)
\(312\) 0 0
\(313\) 30.5913 1.72912 0.864561 0.502528i \(-0.167597\pi\)
0.864561 + 0.502528i \(0.167597\pi\)
\(314\) 1.45861 0.0823144
\(315\) 0 0
\(316\) −4.05292 −0.227994
\(317\) −0.145803 −0.00818910 −0.00409455 0.999992i \(-0.501303\pi\)
−0.00409455 + 0.999992i \(0.501303\pi\)
\(318\) 0 0
\(319\) 3.38739 0.189657
\(320\) −7.97921 −0.446051
\(321\) 0 0
\(322\) −5.73871 −0.319806
\(323\) −5.55682 −0.309190
\(324\) 0 0
\(325\) −3.44382 −0.191028
\(326\) 1.55936 0.0863649
\(327\) 0 0
\(328\) 3.10635 0.171519
\(329\) −8.48811 −0.467964
\(330\) 0 0
\(331\) 3.06165 0.168284 0.0841419 0.996454i \(-0.473185\pi\)
0.0841419 + 0.996454i \(0.473185\pi\)
\(332\) −1.91300 −0.104990
\(333\) 0 0
\(334\) −1.47123 −0.0805022
\(335\) 24.2187 1.32321
\(336\) 0 0
\(337\) −19.7483 −1.07576 −0.537879 0.843022i \(-0.680774\pi\)
−0.537879 + 0.843022i \(0.680774\pi\)
\(338\) −6.52891 −0.355126
\(339\) 0 0
\(340\) 20.4461 1.10885
\(341\) 18.9743 1.02752
\(342\) 0 0
\(343\) 19.9537 1.07740
\(344\) −13.8947 −0.749154
\(345\) 0 0
\(346\) −0.704098 −0.0378526
\(347\) 28.1298 1.51009 0.755043 0.655675i \(-0.227616\pi\)
0.755043 + 0.655675i \(0.227616\pi\)
\(348\) 0 0
\(349\) 28.0056 1.49911 0.749553 0.661944i \(-0.230268\pi\)
0.749553 + 0.661944i \(0.230268\pi\)
\(350\) −7.80243 −0.417058
\(351\) 0 0
\(352\) −30.6866 −1.63560
\(353\) −25.9135 −1.37923 −0.689617 0.724174i \(-0.742221\pi\)
−0.689617 + 0.724174i \(0.742221\pi\)
\(354\) 0 0
\(355\) 30.6200 1.62514
\(356\) 13.8543 0.734277
\(357\) 0 0
\(358\) 11.3401 0.599341
\(359\) −7.00993 −0.369970 −0.184985 0.982741i \(-0.559224\pi\)
−0.184985 + 0.982741i \(0.559224\pi\)
\(360\) 0 0
\(361\) −16.4377 −0.865142
\(362\) 4.77719 0.251084
\(363\) 0 0
\(364\) −2.14916 −0.112647
\(365\) 15.1326 0.792076
\(366\) 0 0
\(367\) −21.8727 −1.14174 −0.570872 0.821039i \(-0.693395\pi\)
−0.570872 + 0.821039i \(0.693395\pi\)
\(368\) 11.9041 0.620542
\(369\) 0 0
\(370\) −11.6771 −0.607064
\(371\) −0.970742 −0.0503984
\(372\) 0 0
\(373\) −1.41439 −0.0732345 −0.0366173 0.999329i \(-0.511658\pi\)
−0.0366173 + 0.999329i \(0.511658\pi\)
\(374\) 10.6937 0.552959
\(375\) 0 0
\(376\) −6.96532 −0.359209
\(377\) 0.298879 0.0153931
\(378\) 0 0
\(379\) 29.4270 1.51156 0.755782 0.654823i \(-0.227257\pi\)
0.755782 + 0.654823i \(0.227257\pi\)
\(380\) −9.42791 −0.483642
\(381\) 0 0
\(382\) −1.63333 −0.0835684
\(383\) 8.41983 0.430233 0.215117 0.976588i \(-0.430987\pi\)
0.215117 + 0.976588i \(0.430987\pi\)
\(384\) 0 0
\(385\) 47.5745 2.42462
\(386\) 7.57150 0.385379
\(387\) 0 0
\(388\) 15.9535 0.809915
\(389\) −15.0428 −0.762700 −0.381350 0.924431i \(-0.624541\pi\)
−0.381350 + 0.924431i \(0.624541\pi\)
\(390\) 0 0
\(391\) −16.6055 −0.839778
\(392\) 2.94839 0.148916
\(393\) 0 0
\(394\) 2.14679 0.108154
\(395\) 7.91597 0.398295
\(396\) 0 0
\(397\) 30.2828 1.51985 0.759925 0.650011i \(-0.225236\pi\)
0.759925 + 0.650011i \(0.225236\pi\)
\(398\) 4.53641 0.227390
\(399\) 0 0
\(400\) 16.1849 0.809247
\(401\) −21.9421 −1.09574 −0.547869 0.836564i \(-0.684561\pi\)
−0.547869 + 0.836564i \(0.684561\pi\)
\(402\) 0 0
\(403\) 1.67416 0.0833958
\(404\) −25.1163 −1.24958
\(405\) 0 0
\(406\) 0.677152 0.0336065
\(407\) 40.2533 1.99528
\(408\) 0 0
\(409\) 38.4944 1.90343 0.951713 0.306989i \(-0.0993215\pi\)
0.951713 + 0.306989i \(0.0993215\pi\)
\(410\) −2.81968 −0.139254
\(411\) 0 0
\(412\) −23.2093 −1.14344
\(413\) −19.7705 −0.972844
\(414\) 0 0
\(415\) 3.73639 0.183412
\(416\) −2.70757 −0.132750
\(417\) 0 0
\(418\) −4.93098 −0.241182
\(419\) 9.68878 0.473328 0.236664 0.971592i \(-0.423946\pi\)
0.236664 + 0.971592i \(0.423946\pi\)
\(420\) 0 0
\(421\) −28.0991 −1.36946 −0.684732 0.728795i \(-0.740081\pi\)
−0.684732 + 0.728795i \(0.740081\pi\)
\(422\) −1.52696 −0.0743312
\(423\) 0 0
\(424\) −0.796588 −0.0386857
\(425\) −22.5771 −1.09515
\(426\) 0 0
\(427\) −10.0527 −0.486483
\(428\) 3.12820 0.151207
\(429\) 0 0
\(430\) 12.6125 0.608227
\(431\) −7.94768 −0.382826 −0.191413 0.981510i \(-0.561307\pi\)
−0.191413 + 0.981510i \(0.561307\pi\)
\(432\) 0 0
\(433\) −16.8345 −0.809012 −0.404506 0.914535i \(-0.632557\pi\)
−0.404506 + 0.914535i \(0.632557\pi\)
\(434\) 3.79304 0.182072
\(435\) 0 0
\(436\) 16.6861 0.799118
\(437\) 7.65698 0.366283
\(438\) 0 0
\(439\) −1.63085 −0.0778362 −0.0389181 0.999242i \(-0.512391\pi\)
−0.0389181 + 0.999242i \(0.512391\pi\)
\(440\) 39.0395 1.86113
\(441\) 0 0
\(442\) 0.943538 0.0448795
\(443\) 8.13294 0.386408 0.193204 0.981159i \(-0.438112\pi\)
0.193204 + 0.981159i \(0.438112\pi\)
\(444\) 0 0
\(445\) −27.0596 −1.28275
\(446\) −0.513295 −0.0243052
\(447\) 0 0
\(448\) 5.49853 0.259781
\(449\) 8.35560 0.394325 0.197163 0.980371i \(-0.436827\pi\)
0.197163 + 0.980371i \(0.436827\pi\)
\(450\) 0 0
\(451\) 9.71999 0.457697
\(452\) −8.43564 −0.396779
\(453\) 0 0
\(454\) −3.91025 −0.183517
\(455\) 4.19764 0.196788
\(456\) 0 0
\(457\) 22.7900 1.06607 0.533036 0.846093i \(-0.321051\pi\)
0.533036 + 0.846093i \(0.321051\pi\)
\(458\) −4.53612 −0.211959
\(459\) 0 0
\(460\) −28.1736 −1.31360
\(461\) 34.1572 1.59086 0.795430 0.606045i \(-0.207245\pi\)
0.795430 + 0.606045i \(0.207245\pi\)
\(462\) 0 0
\(463\) 3.71082 0.172456 0.0862281 0.996275i \(-0.472519\pi\)
0.0862281 + 0.996275i \(0.472519\pi\)
\(464\) −1.40465 −0.0652091
\(465\) 0 0
\(466\) −6.83976 −0.316846
\(467\) −14.9207 −0.690446 −0.345223 0.938521i \(-0.612197\pi\)
−0.345223 + 0.938521i \(0.612197\pi\)
\(468\) 0 0
\(469\) −16.6893 −0.770639
\(470\) 6.32253 0.291637
\(471\) 0 0
\(472\) −16.2236 −0.746754
\(473\) −43.4776 −1.99910
\(474\) 0 0
\(475\) 10.4106 0.477669
\(476\) −14.0896 −0.645794
\(477\) 0 0
\(478\) −7.20395 −0.329501
\(479\) 29.0217 1.32604 0.663018 0.748603i \(-0.269275\pi\)
0.663018 + 0.748603i \(0.269275\pi\)
\(480\) 0 0
\(481\) 3.55167 0.161942
\(482\) 11.8501 0.539757
\(483\) 0 0
\(484\) −43.4419 −1.97463
\(485\) −31.1596 −1.41488
\(486\) 0 0
\(487\) −16.8875 −0.765246 −0.382623 0.923905i \(-0.624979\pi\)
−0.382623 + 0.923905i \(0.624979\pi\)
\(488\) −8.24920 −0.373424
\(489\) 0 0
\(490\) −2.67630 −0.120903
\(491\) −25.5037 −1.15097 −0.575484 0.817813i \(-0.695186\pi\)
−0.575484 + 0.817813i \(0.695186\pi\)
\(492\) 0 0
\(493\) 1.95941 0.0882473
\(494\) −0.435075 −0.0195750
\(495\) 0 0
\(496\) −7.86807 −0.353287
\(497\) −21.1005 −0.946486
\(498\) 0 0
\(499\) −12.0031 −0.537335 −0.268667 0.963233i \(-0.586583\pi\)
−0.268667 + 0.963233i \(0.586583\pi\)
\(500\) −8.85628 −0.396065
\(501\) 0 0
\(502\) −9.23471 −0.412165
\(503\) 35.0040 1.56075 0.780375 0.625312i \(-0.215028\pi\)
0.780375 + 0.625312i \(0.215028\pi\)
\(504\) 0 0
\(505\) 49.0559 2.18296
\(506\) −14.7353 −0.655065
\(507\) 0 0
\(508\) −1.80413 −0.0800455
\(509\) −35.9815 −1.59485 −0.797426 0.603417i \(-0.793805\pi\)
−0.797426 + 0.603417i \(0.793805\pi\)
\(510\) 0 0
\(511\) −10.4280 −0.461307
\(512\) 22.2707 0.984236
\(513\) 0 0
\(514\) −1.48516 −0.0655077
\(515\) 45.3313 1.99754
\(516\) 0 0
\(517\) −21.7950 −0.958542
\(518\) 8.04679 0.353556
\(519\) 0 0
\(520\) 3.44457 0.151054
\(521\) 21.9220 0.960418 0.480209 0.877154i \(-0.340561\pi\)
0.480209 + 0.877154i \(0.340561\pi\)
\(522\) 0 0
\(523\) 27.8296 1.21690 0.608451 0.793592i \(-0.291791\pi\)
0.608451 + 0.793592i \(0.291791\pi\)
\(524\) −23.3030 −1.01800
\(525\) 0 0
\(526\) 4.23924 0.184840
\(527\) 10.9755 0.478102
\(528\) 0 0
\(529\) −0.118510 −0.00515259
\(530\) 0.723076 0.0314084
\(531\) 0 0
\(532\) 6.49685 0.281674
\(533\) 0.857624 0.0371478
\(534\) 0 0
\(535\) −6.10986 −0.264152
\(536\) −13.6952 −0.591542
\(537\) 0 0
\(538\) −14.0916 −0.607532
\(539\) 9.22571 0.397379
\(540\) 0 0
\(541\) 18.2815 0.785982 0.392991 0.919542i \(-0.371440\pi\)
0.392991 + 0.919542i \(0.371440\pi\)
\(542\) 11.0411 0.474254
\(543\) 0 0
\(544\) −17.7504 −0.761043
\(545\) −32.5904 −1.39602
\(546\) 0 0
\(547\) −30.5937 −1.30809 −0.654047 0.756454i \(-0.726930\pi\)
−0.654047 + 0.756454i \(0.726930\pi\)
\(548\) −38.9735 −1.66487
\(549\) 0 0
\(550\) −20.0344 −0.854269
\(551\) −0.903503 −0.0384905
\(552\) 0 0
\(553\) −5.45495 −0.231968
\(554\) 6.52300 0.277136
\(555\) 0 0
\(556\) 0.322252 0.0136665
\(557\) 41.3981 1.75409 0.877046 0.480406i \(-0.159511\pi\)
0.877046 + 0.480406i \(0.159511\pi\)
\(558\) 0 0
\(559\) −3.83616 −0.162252
\(560\) −19.7277 −0.833647
\(561\) 0 0
\(562\) −1.63499 −0.0689679
\(563\) 32.0399 1.35032 0.675161 0.737670i \(-0.264074\pi\)
0.675161 + 0.737670i \(0.264074\pi\)
\(564\) 0 0
\(565\) 16.4761 0.693155
\(566\) −10.1696 −0.427461
\(567\) 0 0
\(568\) −17.3150 −0.726522
\(569\) 23.8638 1.00042 0.500211 0.865903i \(-0.333256\pi\)
0.500211 + 0.865903i \(0.333256\pi\)
\(570\) 0 0
\(571\) 12.0209 0.503058 0.251529 0.967850i \(-0.419067\pi\)
0.251529 + 0.967850i \(0.419067\pi\)
\(572\) −5.51841 −0.230736
\(573\) 0 0
\(574\) 1.94306 0.0811019
\(575\) 31.1100 1.29738
\(576\) 0 0
\(577\) 1.61611 0.0672795 0.0336398 0.999434i \(-0.489290\pi\)
0.0336398 + 0.999434i \(0.489290\pi\)
\(578\) −2.54032 −0.105663
\(579\) 0 0
\(580\) 3.32441 0.138038
\(581\) −2.57477 −0.106820
\(582\) 0 0
\(583\) −2.49258 −0.103232
\(584\) −8.55718 −0.354098
\(585\) 0 0
\(586\) 8.42733 0.348130
\(587\) −8.53142 −0.352130 −0.176065 0.984379i \(-0.556337\pi\)
−0.176065 + 0.984379i \(0.556337\pi\)
\(588\) 0 0
\(589\) −5.06094 −0.208532
\(590\) 14.7265 0.606279
\(591\) 0 0
\(592\) −16.6918 −0.686029
\(593\) −6.80889 −0.279608 −0.139804 0.990179i \(-0.544647\pi\)
−0.139804 + 0.990179i \(0.544647\pi\)
\(594\) 0 0
\(595\) 27.5191 1.12817
\(596\) 27.2727 1.11713
\(597\) 0 0
\(598\) −1.30014 −0.0531667
\(599\) −3.33538 −0.136280 −0.0681400 0.997676i \(-0.521706\pi\)
−0.0681400 + 0.997676i \(0.521706\pi\)
\(600\) 0 0
\(601\) 44.4736 1.81412 0.907059 0.421004i \(-0.138322\pi\)
0.907059 + 0.421004i \(0.138322\pi\)
\(602\) −8.69135 −0.354233
\(603\) 0 0
\(604\) −17.5755 −0.715138
\(605\) 84.8486 3.44959
\(606\) 0 0
\(607\) −5.74043 −0.232997 −0.116498 0.993191i \(-0.537167\pi\)
−0.116498 + 0.993191i \(0.537167\pi\)
\(608\) 8.18490 0.331942
\(609\) 0 0
\(610\) 7.48793 0.303177
\(611\) −1.92304 −0.0777977
\(612\) 0 0
\(613\) −36.3585 −1.46851 −0.734253 0.678875i \(-0.762468\pi\)
−0.734253 + 0.678875i \(0.762468\pi\)
\(614\) 2.88410 0.116393
\(615\) 0 0
\(616\) −26.9024 −1.08393
\(617\) −14.2699 −0.574486 −0.287243 0.957858i \(-0.592739\pi\)
−0.287243 + 0.957858i \(0.592739\pi\)
\(618\) 0 0
\(619\) 33.1753 1.33343 0.666713 0.745314i \(-0.267701\pi\)
0.666713 + 0.745314i \(0.267701\pi\)
\(620\) 18.6215 0.747859
\(621\) 0 0
\(622\) −6.65396 −0.266799
\(623\) 18.6470 0.747074
\(624\) 0 0
\(625\) −15.2207 −0.608826
\(626\) 15.7024 0.627592
\(627\) 0 0
\(628\) −4.93464 −0.196914
\(629\) 23.2842 0.928402
\(630\) 0 0
\(631\) 21.7317 0.865124 0.432562 0.901604i \(-0.357610\pi\)
0.432562 + 0.901604i \(0.357610\pi\)
\(632\) −4.47632 −0.178058
\(633\) 0 0
\(634\) −0.0748398 −0.00297227
\(635\) 3.52375 0.139836
\(636\) 0 0
\(637\) 0.814012 0.0322523
\(638\) 1.73873 0.0688369
\(639\) 0 0
\(640\) −38.7810 −1.53296
\(641\) 23.6805 0.935322 0.467661 0.883908i \(-0.345097\pi\)
0.467661 + 0.883908i \(0.345097\pi\)
\(642\) 0 0
\(643\) 24.6117 0.970591 0.485296 0.874350i \(-0.338712\pi\)
0.485296 + 0.874350i \(0.338712\pi\)
\(644\) 19.4146 0.765043
\(645\) 0 0
\(646\) −2.85229 −0.112222
\(647\) 0.956553 0.0376060 0.0188030 0.999823i \(-0.494014\pi\)
0.0188030 + 0.999823i \(0.494014\pi\)
\(648\) 0 0
\(649\) −50.7649 −1.99270
\(650\) −1.76769 −0.0693346
\(651\) 0 0
\(652\) −5.27547 −0.206603
\(653\) −20.9835 −0.821150 −0.410575 0.911827i \(-0.634672\pi\)
−0.410575 + 0.911827i \(0.634672\pi\)
\(654\) 0 0
\(655\) 45.5144 1.77839
\(656\) −4.03058 −0.157368
\(657\) 0 0
\(658\) −4.35690 −0.169850
\(659\) 10.8250 0.421681 0.210841 0.977520i \(-0.432380\pi\)
0.210841 + 0.977520i \(0.432380\pi\)
\(660\) 0 0
\(661\) 0.544481 0.0211779 0.0105889 0.999944i \(-0.496629\pi\)
0.0105889 + 0.999944i \(0.496629\pi\)
\(662\) 1.57153 0.0610793
\(663\) 0 0
\(664\) −2.11285 −0.0819945
\(665\) −12.6893 −0.492071
\(666\) 0 0
\(667\) −2.69995 −0.104543
\(668\) 4.97733 0.192579
\(669\) 0 0
\(670\) 12.4313 0.480264
\(671\) −25.8124 −0.996475
\(672\) 0 0
\(673\) 16.5504 0.637971 0.318985 0.947760i \(-0.396658\pi\)
0.318985 + 0.947760i \(0.396658\pi\)
\(674\) −10.1367 −0.390451
\(675\) 0 0
\(676\) 22.0880 0.849537
\(677\) 0.0522237 0.00200712 0.00100356 0.999999i \(-0.499681\pi\)
0.00100356 + 0.999999i \(0.499681\pi\)
\(678\) 0 0
\(679\) 21.4723 0.824031
\(680\) 22.5821 0.865983
\(681\) 0 0
\(682\) 9.73942 0.372942
\(683\) 5.32927 0.203919 0.101959 0.994789i \(-0.467489\pi\)
0.101959 + 0.994789i \(0.467489\pi\)
\(684\) 0 0
\(685\) 76.1213 2.90845
\(686\) 10.2421 0.391047
\(687\) 0 0
\(688\) 18.0288 0.687343
\(689\) −0.219928 −0.00837858
\(690\) 0 0
\(691\) 11.4960 0.437327 0.218663 0.975800i \(-0.429830\pi\)
0.218663 + 0.975800i \(0.429830\pi\)
\(692\) 2.38204 0.0905514
\(693\) 0 0
\(694\) 14.4389 0.548093
\(695\) −0.629407 −0.0238748
\(696\) 0 0
\(697\) 5.62245 0.212966
\(698\) 14.3751 0.544107
\(699\) 0 0
\(700\) 26.3964 0.997691
\(701\) −15.6919 −0.592673 −0.296337 0.955084i \(-0.595765\pi\)
−0.296337 + 0.955084i \(0.595765\pi\)
\(702\) 0 0
\(703\) −10.7366 −0.404938
\(704\) 14.1186 0.532116
\(705\) 0 0
\(706\) −13.3012 −0.500599
\(707\) −33.8048 −1.27136
\(708\) 0 0
\(709\) 38.1458 1.43259 0.716297 0.697795i \(-0.245836\pi\)
0.716297 + 0.697795i \(0.245836\pi\)
\(710\) 15.7171 0.589852
\(711\) 0 0
\(712\) 15.3016 0.573453
\(713\) −15.1237 −0.566386
\(714\) 0 0
\(715\) 10.7783 0.403086
\(716\) −38.3646 −1.43375
\(717\) 0 0
\(718\) −3.59816 −0.134282
\(719\) −15.4177 −0.574983 −0.287491 0.957783i \(-0.592821\pi\)
−0.287491 + 0.957783i \(0.592821\pi\)
\(720\) 0 0
\(721\) −31.2381 −1.16337
\(722\) −8.43738 −0.314007
\(723\) 0 0
\(724\) −16.1617 −0.600646
\(725\) −3.67090 −0.136334
\(726\) 0 0
\(727\) −31.3341 −1.16212 −0.581058 0.813862i \(-0.697361\pi\)
−0.581058 + 0.813862i \(0.697361\pi\)
\(728\) −2.37368 −0.0879744
\(729\) 0 0
\(730\) 7.76748 0.287487
\(731\) −25.1493 −0.930180
\(732\) 0 0
\(733\) −16.2871 −0.601576 −0.300788 0.953691i \(-0.597250\pi\)
−0.300788 + 0.953691i \(0.597250\pi\)
\(734\) −11.2271 −0.414401
\(735\) 0 0
\(736\) 24.4591 0.901574
\(737\) −42.8532 −1.57852
\(738\) 0 0
\(739\) −27.9825 −1.02935 −0.514677 0.857384i \(-0.672088\pi\)
−0.514677 + 0.857384i \(0.672088\pi\)
\(740\) 39.5049 1.45223
\(741\) 0 0
\(742\) −0.498277 −0.0182923
\(743\) 19.8961 0.729916 0.364958 0.931024i \(-0.381083\pi\)
0.364958 + 0.931024i \(0.381083\pi\)
\(744\) 0 0
\(745\) −53.2678 −1.95158
\(746\) −0.726001 −0.0265808
\(747\) 0 0
\(748\) −36.1779 −1.32279
\(749\) 4.21035 0.153843
\(750\) 0 0
\(751\) 40.2940 1.47035 0.735175 0.677878i \(-0.237100\pi\)
0.735175 + 0.677878i \(0.237100\pi\)
\(752\) 9.03772 0.329572
\(753\) 0 0
\(754\) 0.153413 0.00558698
\(755\) 34.3277 1.24931
\(756\) 0 0
\(757\) −47.8159 −1.73790 −0.868949 0.494901i \(-0.835204\pi\)
−0.868949 + 0.494901i \(0.835204\pi\)
\(758\) 15.1047 0.548629
\(759\) 0 0
\(760\) −10.4128 −0.377713
\(761\) −14.3712 −0.520956 −0.260478 0.965480i \(-0.583880\pi\)
−0.260478 + 0.965480i \(0.583880\pi\)
\(762\) 0 0
\(763\) 22.4583 0.813045
\(764\) 5.52572 0.199914
\(765\) 0 0
\(766\) 4.32186 0.156155
\(767\) −4.47914 −0.161732
\(768\) 0 0
\(769\) 6.40950 0.231132 0.115566 0.993300i \(-0.463132\pi\)
0.115566 + 0.993300i \(0.463132\pi\)
\(770\) 24.4197 0.880026
\(771\) 0 0
\(772\) −25.6152 −0.921910
\(773\) 15.5035 0.557624 0.278812 0.960346i \(-0.410059\pi\)
0.278812 + 0.960346i \(0.410059\pi\)
\(774\) 0 0
\(775\) −20.5624 −0.738623
\(776\) 17.6201 0.632525
\(777\) 0 0
\(778\) −7.72139 −0.276825
\(779\) −2.59257 −0.0928885
\(780\) 0 0
\(781\) −54.1799 −1.93871
\(782\) −8.52354 −0.304801
\(783\) 0 0
\(784\) −3.82562 −0.136629
\(785\) 9.63810 0.343999
\(786\) 0 0
\(787\) −55.2404 −1.96911 −0.984554 0.175082i \(-0.943981\pi\)
−0.984554 + 0.175082i \(0.943981\pi\)
\(788\) −7.26279 −0.258726
\(789\) 0 0
\(790\) 4.06322 0.144563
\(791\) −11.3538 −0.403695
\(792\) 0 0
\(793\) −2.27750 −0.0808764
\(794\) 15.5440 0.551636
\(795\) 0 0
\(796\) −15.3471 −0.543965
\(797\) 13.1924 0.467299 0.233650 0.972321i \(-0.424933\pi\)
0.233650 + 0.972321i \(0.424933\pi\)
\(798\) 0 0
\(799\) −12.6071 −0.446008
\(800\) 33.2549 1.17574
\(801\) 0 0
\(802\) −11.2628 −0.397703
\(803\) −26.7760 −0.944905
\(804\) 0 0
\(805\) −37.9197 −1.33650
\(806\) 0.859338 0.0302689
\(807\) 0 0
\(808\) −27.7401 −0.975895
\(809\) −14.0518 −0.494035 −0.247017 0.969011i \(-0.579450\pi\)
−0.247017 + 0.969011i \(0.579450\pi\)
\(810\) 0 0
\(811\) −54.7981 −1.92422 −0.962111 0.272657i \(-0.912097\pi\)
−0.962111 + 0.272657i \(0.912097\pi\)
\(812\) −2.29087 −0.0803939
\(813\) 0 0
\(814\) 20.6618 0.724196
\(815\) 10.3038 0.360926
\(816\) 0 0
\(817\) 11.5966 0.405714
\(818\) 19.7590 0.690857
\(819\) 0 0
\(820\) 9.53927 0.333126
\(821\) 9.96717 0.347857 0.173928 0.984758i \(-0.444354\pi\)
0.173928 + 0.984758i \(0.444354\pi\)
\(822\) 0 0
\(823\) 47.2593 1.64735 0.823677 0.567059i \(-0.191919\pi\)
0.823677 + 0.567059i \(0.191919\pi\)
\(824\) −25.6339 −0.893000
\(825\) 0 0
\(826\) −10.1481 −0.353098
\(827\) −2.83027 −0.0984181 −0.0492090 0.998789i \(-0.515670\pi\)
−0.0492090 + 0.998789i \(0.515670\pi\)
\(828\) 0 0
\(829\) 0.814307 0.0282820 0.0141410 0.999900i \(-0.495499\pi\)
0.0141410 + 0.999900i \(0.495499\pi\)
\(830\) 1.91787 0.0665702
\(831\) 0 0
\(832\) 1.24573 0.0431878
\(833\) 5.33654 0.184900
\(834\) 0 0
\(835\) −9.72148 −0.336426
\(836\) 16.6820 0.576959
\(837\) 0 0
\(838\) 4.97320 0.171796
\(839\) −32.6993 −1.12891 −0.564453 0.825465i \(-0.690913\pi\)
−0.564453 + 0.825465i \(0.690913\pi\)
\(840\) 0 0
\(841\) −28.6814 −0.989014
\(842\) −14.4231 −0.497053
\(843\) 0 0
\(844\) 5.16585 0.177816
\(845\) −43.1412 −1.48410
\(846\) 0 0
\(847\) −58.4698 −2.00905
\(848\) 1.03360 0.0354939
\(849\) 0 0
\(850\) −11.5887 −0.397490
\(851\) −32.0843 −1.09984
\(852\) 0 0
\(853\) 11.6427 0.398638 0.199319 0.979935i \(-0.436127\pi\)
0.199319 + 0.979935i \(0.436127\pi\)
\(854\) −5.15999 −0.176571
\(855\) 0 0
\(856\) 3.45500 0.118090
\(857\) −53.6791 −1.83364 −0.916822 0.399295i \(-0.869255\pi\)
−0.916822 + 0.399295i \(0.869255\pi\)
\(858\) 0 0
\(859\) 50.9481 1.73833 0.869164 0.494524i \(-0.164658\pi\)
0.869164 + 0.494524i \(0.164658\pi\)
\(860\) −42.6693 −1.45501
\(861\) 0 0
\(862\) −4.07950 −0.138948
\(863\) −28.8475 −0.981979 −0.490990 0.871165i \(-0.663365\pi\)
−0.490990 + 0.871165i \(0.663365\pi\)
\(864\) 0 0
\(865\) −4.65248 −0.158189
\(866\) −8.64104 −0.293634
\(867\) 0 0
\(868\) −12.8322 −0.435554
\(869\) −14.0067 −0.475146
\(870\) 0 0
\(871\) −3.78106 −0.128116
\(872\) 18.4292 0.624093
\(873\) 0 0
\(874\) 3.93029 0.132944
\(875\) −11.9199 −0.402968
\(876\) 0 0
\(877\) 20.6543 0.697445 0.348722 0.937226i \(-0.386616\pi\)
0.348722 + 0.937226i \(0.386616\pi\)
\(878\) −0.837106 −0.0282510
\(879\) 0 0
\(880\) −50.6549 −1.70758
\(881\) −42.9046 −1.44549 −0.722746 0.691114i \(-0.757120\pi\)
−0.722746 + 0.691114i \(0.757120\pi\)
\(882\) 0 0
\(883\) 12.8240 0.431561 0.215781 0.976442i \(-0.430770\pi\)
0.215781 + 0.976442i \(0.430770\pi\)
\(884\) −3.19208 −0.107361
\(885\) 0 0
\(886\) 4.17460 0.140248
\(887\) 23.4296 0.786689 0.393345 0.919391i \(-0.371318\pi\)
0.393345 + 0.919391i \(0.371318\pi\)
\(888\) 0 0
\(889\) −2.42824 −0.0814406
\(890\) −13.8895 −0.465579
\(891\) 0 0
\(892\) 1.73653 0.0581433
\(893\) 5.81328 0.194534
\(894\) 0 0
\(895\) 74.9319 2.50470
\(896\) 26.7243 0.892796
\(897\) 0 0
\(898\) 4.28889 0.143122
\(899\) 1.78455 0.0595182
\(900\) 0 0
\(901\) −1.44181 −0.0480338
\(902\) 4.98922 0.166123
\(903\) 0 0
\(904\) −9.31690 −0.309876
\(905\) 31.5663 1.04930
\(906\) 0 0
\(907\) 59.1109 1.96275 0.981373 0.192113i \(-0.0615340\pi\)
0.981373 + 0.192113i \(0.0615340\pi\)
\(908\) 13.2288 0.439012
\(909\) 0 0
\(910\) 2.15462 0.0714251
\(911\) −12.0703 −0.399907 −0.199954 0.979805i \(-0.564079\pi\)
−0.199954 + 0.979805i \(0.564079\pi\)
\(912\) 0 0
\(913\) −6.61126 −0.218801
\(914\) 11.6980 0.386935
\(915\) 0 0
\(916\) 15.3461 0.507051
\(917\) −31.3643 −1.03574
\(918\) 0 0
\(919\) 18.6172 0.614125 0.307062 0.951689i \(-0.400654\pi\)
0.307062 + 0.951689i \(0.400654\pi\)
\(920\) −31.1168 −1.02589
\(921\) 0 0
\(922\) 17.5327 0.577409
\(923\) −4.78045 −0.157351
\(924\) 0 0
\(925\) −43.6223 −1.43429
\(926\) 1.90474 0.0625937
\(927\) 0 0
\(928\) −2.88610 −0.0947410
\(929\) −17.7819 −0.583406 −0.291703 0.956509i \(-0.594222\pi\)
−0.291703 + 0.956509i \(0.594222\pi\)
\(930\) 0 0
\(931\) −2.46073 −0.0806473
\(932\) 23.1396 0.757963
\(933\) 0 0
\(934\) −7.65870 −0.250600
\(935\) 70.6610 2.31086
\(936\) 0 0
\(937\) −28.2425 −0.922643 −0.461322 0.887233i \(-0.652625\pi\)
−0.461322 + 0.887233i \(0.652625\pi\)
\(938\) −8.56652 −0.279707
\(939\) 0 0
\(940\) −21.3898 −0.697657
\(941\) −56.3982 −1.83853 −0.919265 0.393640i \(-0.871216\pi\)
−0.919265 + 0.393640i \(0.871216\pi\)
\(942\) 0 0
\(943\) −7.74742 −0.252291
\(944\) 21.0507 0.685141
\(945\) 0 0
\(946\) −22.3168 −0.725583
\(947\) −24.1789 −0.785708 −0.392854 0.919601i \(-0.628512\pi\)
−0.392854 + 0.919601i \(0.628512\pi\)
\(948\) 0 0
\(949\) −2.36253 −0.0766909
\(950\) 5.34368 0.173372
\(951\) 0 0
\(952\) −15.5615 −0.504350
\(953\) −25.7051 −0.832670 −0.416335 0.909211i \(-0.636686\pi\)
−0.416335 + 0.909211i \(0.636686\pi\)
\(954\) 0 0
\(955\) −10.7926 −0.349240
\(956\) 24.3717 0.788236
\(957\) 0 0
\(958\) 14.8967 0.481291
\(959\) −52.4557 −1.69388
\(960\) 0 0
\(961\) −21.0039 −0.677545
\(962\) 1.82305 0.0587776
\(963\) 0 0
\(964\) −40.0901 −1.29121
\(965\) 50.0303 1.61053
\(966\) 0 0
\(967\) −14.4596 −0.464991 −0.232495 0.972597i \(-0.574689\pi\)
−0.232495 + 0.972597i \(0.574689\pi\)
\(968\) −47.9802 −1.54214
\(969\) 0 0
\(970\) −15.9941 −0.513538
\(971\) −49.3477 −1.58364 −0.791822 0.610752i \(-0.790867\pi\)
−0.791822 + 0.610752i \(0.790867\pi\)
\(972\) 0 0
\(973\) 0.433729 0.0139047
\(974\) −8.66827 −0.277749
\(975\) 0 0
\(976\) 10.7036 0.342614
\(977\) −35.6884 −1.14177 −0.570886 0.821029i \(-0.693400\pi\)
−0.570886 + 0.821029i \(0.693400\pi\)
\(978\) 0 0
\(979\) 47.8799 1.53025
\(980\) 9.05418 0.289225
\(981\) 0 0
\(982\) −13.0909 −0.417748
\(983\) −9.33759 −0.297823 −0.148911 0.988851i \(-0.547577\pi\)
−0.148911 + 0.988851i \(0.547577\pi\)
\(984\) 0 0
\(985\) 14.1853 0.451983
\(986\) 1.00575 0.0320297
\(987\) 0 0
\(988\) 1.47190 0.0468275
\(989\) 34.6543 1.10194
\(990\) 0 0
\(991\) 33.1586 1.05332 0.526659 0.850077i \(-0.323444\pi\)
0.526659 + 0.850077i \(0.323444\pi\)
\(992\) −16.1664 −0.513284
\(993\) 0 0
\(994\) −10.8308 −0.343531
\(995\) 29.9753 0.950280
\(996\) 0 0
\(997\) 46.2727 1.46547 0.732736 0.680513i \(-0.238243\pi\)
0.732736 + 0.680513i \(0.238243\pi\)
\(998\) −6.16115 −0.195028
\(999\) 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 6021.2.a.l.1.6 yes 10
3.2 odd 2 inner 6021.2.a.l.1.5 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6021.2.a.l.1.5 10 3.2 odd 2 inner
6021.2.a.l.1.6 yes 10 1.1 even 1 trivial