Properties

Label 7600.2.a.by.1.1
Level $7600$
Weight $2$
Character 7600.1
Self dual yes
Analytic conductor $60.686$
Analytic rank $0$
Dimension $3$
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] = [7600,2,Mod(1,7600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("7600.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7600 = 2^{4} \cdot 5^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7600.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: \(60.6863055362\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.257.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1900)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.19869\) of defining polynomial
Character \(\chi\) \(=\) 7600.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.19869 q^{3} -1.19869 q^{7} -1.56314 q^{9} +O(q^{10})\) \(q-1.19869 q^{3} -1.19869 q^{7} -1.56314 q^{9} +5.86718 q^{11} -0.364448 q^{13} +1.19869 q^{17} +1.00000 q^{19} +1.43686 q^{21} +8.23163 q^{23} +5.46980 q^{27} -7.86718 q^{29} -7.30404 q^{31} -7.03293 q^{33} +7.13828 q^{37} +0.436861 q^{39} +2.43686 q^{41} +7.39738 q^{43} +13.7014 q^{47} -5.56314 q^{49} -1.43686 q^{51} -7.39738 q^{53} -1.19869 q^{57} -12.8606 q^{59} -1.30404 q^{61} +1.87372 q^{63} -11.9330 q^{67} -9.86718 q^{69} -2.12628 q^{71} +2.50273 q^{73} -7.03293 q^{77} +7.74090 q^{79} -1.86718 q^{81} +3.02093 q^{83} +9.43032 q^{87} -5.68942 q^{89} +0.436861 q^{91} +8.75529 q^{93} +1.09334 q^{97} -9.17122 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 2 q^{3} + 2 q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 2 q^{3} + 2 q^{7} + q^{9} + q^{11} - q^{13} - 2 q^{17} + 3 q^{19} + 10 q^{21} + 8 q^{23} + 11 q^{27} - 7 q^{29} - 11 q^{31} - 10 q^{33} + 5 q^{37} + 7 q^{39} + 13 q^{41} + 11 q^{43} + 19 q^{47} - 11 q^{49} - 10 q^{51} - 11 q^{53} + 2 q^{57} + 6 q^{59} + 7 q^{61} + 17 q^{63} + 3 q^{67} - 13 q^{69} + 5 q^{71} - 9 q^{73} - 10 q^{77} + 18 q^{79} + 11 q^{81} + 3 q^{83} + 6 q^{87} + 7 q^{91} - 13 q^{93} + 3 q^{97}+O(q^{100}) \) 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 0
\(3\) −1.19869 −0.692065 −0.346032 0.938223i \(-0.612471\pi\)
−0.346032 + 0.938223i \(0.612471\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −1.19869 −0.453063 −0.226531 0.974004i \(-0.572739\pi\)
−0.226531 + 0.974004i \(0.572739\pi\)
\(8\) 0 0
\(9\) −1.56314 −0.521046
\(10\) 0 0
\(11\) 5.86718 1.76902 0.884510 0.466521i \(-0.154493\pi\)
0.884510 + 0.466521i \(0.154493\pi\)
\(12\) 0 0
\(13\) −0.364448 −0.101080 −0.0505399 0.998722i \(-0.516094\pi\)
−0.0505399 + 0.998722i \(0.516094\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.19869 0.290725 0.145363 0.989378i \(-0.453565\pi\)
0.145363 + 0.989378i \(0.453565\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 0 0
\(21\) 1.43686 0.313549
\(22\) 0 0
\(23\) 8.23163 1.71641 0.858206 0.513305i \(-0.171579\pi\)
0.858206 + 0.513305i \(0.171579\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 5.46980 1.05266
\(28\) 0 0
\(29\) −7.86718 −1.46090 −0.730449 0.682967i \(-0.760689\pi\)
−0.730449 + 0.682967i \(0.760689\pi\)
\(30\) 0 0
\(31\) −7.30404 −1.31184 −0.655922 0.754829i \(-0.727720\pi\)
−0.655922 + 0.754829i \(0.727720\pi\)
\(32\) 0 0
\(33\) −7.03293 −1.22428
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.13828 1.17353 0.586763 0.809759i \(-0.300402\pi\)
0.586763 + 0.809759i \(0.300402\pi\)
\(38\) 0 0
\(39\) 0.436861 0.0699537
\(40\) 0 0
\(41\) 2.43686 0.380574 0.190287 0.981729i \(-0.439058\pi\)
0.190287 + 0.981729i \(0.439058\pi\)
\(42\) 0 0
\(43\) 7.39738 1.12809 0.564045 0.825744i \(-0.309244\pi\)
0.564045 + 0.825744i \(0.309244\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 13.7014 1.99856 0.999279 0.0379717i \(-0.0120897\pi\)
0.999279 + 0.0379717i \(0.0120897\pi\)
\(48\) 0 0
\(49\) −5.56314 −0.794734
\(50\) 0 0
\(51\) −1.43686 −0.201201
\(52\) 0 0
\(53\) −7.39738 −1.01611 −0.508054 0.861325i \(-0.669635\pi\)
−0.508054 + 0.861325i \(0.669635\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.19869 −0.158771
\(58\) 0 0
\(59\) −12.8606 −1.67431 −0.837156 0.546964i \(-0.815783\pi\)
−0.837156 + 0.546964i \(0.815783\pi\)
\(60\) 0 0
\(61\) −1.30404 −0.166965 −0.0834825 0.996509i \(-0.526604\pi\)
−0.0834825 + 0.996509i \(0.526604\pi\)
\(62\) 0 0
\(63\) 1.87372 0.236067
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −11.9330 −1.45785 −0.728927 0.684592i \(-0.759981\pi\)
−0.728927 + 0.684592i \(0.759981\pi\)
\(68\) 0 0
\(69\) −9.86718 −1.18787
\(70\) 0 0
\(71\) −2.12628 −0.252343 −0.126171 0.992008i \(-0.540269\pi\)
−0.126171 + 0.992008i \(0.540269\pi\)
\(72\) 0 0
\(73\) 2.50273 0.292922 0.146461 0.989216i \(-0.453212\pi\)
0.146461 + 0.989216i \(0.453212\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −7.03293 −0.801477
\(78\) 0 0
\(79\) 7.74090 0.870919 0.435460 0.900208i \(-0.356586\pi\)
0.435460 + 0.900208i \(0.356586\pi\)
\(80\) 0 0
\(81\) −1.86718 −0.207464
\(82\) 0 0
\(83\) 3.02093 0.331590 0.165795 0.986160i \(-0.446981\pi\)
0.165795 + 0.986160i \(0.446981\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 9.43032 1.01104
\(88\) 0 0
\(89\) −5.68942 −0.603077 −0.301539 0.953454i \(-0.597500\pi\)
−0.301539 + 0.953454i \(0.597500\pi\)
\(90\) 0 0
\(91\) 0.436861 0.0457954
\(92\) 0 0
\(93\) 8.75529 0.907881
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.09334 0.111012 0.0555061 0.998458i \(-0.482323\pi\)
0.0555061 + 0.998458i \(0.482323\pi\)
\(98\) 0 0
\(99\) −9.17122 −0.921742
\(100\) 0 0
\(101\) 10.8672 1.08132 0.540662 0.841240i \(-0.318174\pi\)
0.540662 + 0.841240i \(0.318174\pi\)
\(102\) 0 0
\(103\) 9.74090 0.959799 0.479900 0.877323i \(-0.340673\pi\)
0.479900 + 0.877323i \(0.340673\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −16.3634 −1.58191 −0.790953 0.611877i \(-0.790415\pi\)
−0.790953 + 0.611877i \(0.790415\pi\)
\(108\) 0 0
\(109\) 9.29749 0.890538 0.445269 0.895397i \(-0.353108\pi\)
0.445269 + 0.895397i \(0.353108\pi\)
\(110\) 0 0
\(111\) −8.55660 −0.812156
\(112\) 0 0
\(113\) −2.96707 −0.279118 −0.139559 0.990214i \(-0.544568\pi\)
−0.139559 + 0.990214i \(0.544568\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.569683 0.0526672
\(118\) 0 0
\(119\) −1.43686 −0.131717
\(120\) 0 0
\(121\) 23.4238 2.12943
\(122\) 0 0
\(123\) −2.92104 −0.263382
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 11.3579 1.00785 0.503926 0.863747i \(-0.331889\pi\)
0.503926 + 0.863747i \(0.331889\pi\)
\(128\) 0 0
\(129\) −8.86718 −0.780711
\(130\) 0 0
\(131\) 1.56314 0.136572 0.0682861 0.997666i \(-0.478247\pi\)
0.0682861 + 0.997666i \(0.478247\pi\)
\(132\) 0 0
\(133\) −1.19869 −0.103940
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −15.8397 −1.35328 −0.676639 0.736315i \(-0.736564\pi\)
−0.676639 + 0.736315i \(0.736564\pi\)
\(138\) 0 0
\(139\) 19.8606 1.68456 0.842278 0.539043i \(-0.181214\pi\)
0.842278 + 0.539043i \(0.181214\pi\)
\(140\) 0 0
\(141\) −16.4238 −1.38313
\(142\) 0 0
\(143\) −2.13828 −0.178812
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 6.66849 0.550007
\(148\) 0 0
\(149\) 2.68942 0.220326 0.110163 0.993914i \(-0.464863\pi\)
0.110163 + 0.993914i \(0.464863\pi\)
\(150\) 0 0
\(151\) −17.1647 −1.39684 −0.698421 0.715688i \(-0.746113\pi\)
−0.698421 + 0.715688i \(0.746113\pi\)
\(152\) 0 0
\(153\) −1.87372 −0.151481
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −16.3634 −1.30594 −0.652969 0.757384i \(-0.726477\pi\)
−0.652969 + 0.757384i \(0.726477\pi\)
\(158\) 0 0
\(159\) 8.86718 0.703213
\(160\) 0 0
\(161\) −9.86718 −0.777643
\(162\) 0 0
\(163\) 8.28549 0.648970 0.324485 0.945891i \(-0.394809\pi\)
0.324485 + 0.945891i \(0.394809\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 7.49727 0.580156 0.290078 0.957003i \(-0.406319\pi\)
0.290078 + 0.957003i \(0.406319\pi\)
\(168\) 0 0
\(169\) −12.8672 −0.989783
\(170\) 0 0
\(171\) −1.56314 −0.119536
\(172\) 0 0
\(173\) −14.7464 −1.12114 −0.560572 0.828105i \(-0.689419\pi\)
−0.560572 + 0.828105i \(0.689419\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 15.4159 1.15873
\(178\) 0 0
\(179\) −13.2526 −0.990543 −0.495271 0.868738i \(-0.664931\pi\)
−0.495271 + 0.868738i \(0.664931\pi\)
\(180\) 0 0
\(181\) 16.7344 1.24385 0.621927 0.783075i \(-0.286350\pi\)
0.621927 + 0.783075i \(0.286350\pi\)
\(182\) 0 0
\(183\) 1.56314 0.115551
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 7.03293 0.514299
\(188\) 0 0
\(189\) −6.55660 −0.476922
\(190\) 0 0
\(191\) 4.13282 0.299041 0.149520 0.988759i \(-0.452227\pi\)
0.149520 + 0.988759i \(0.452227\pi\)
\(192\) 0 0
\(193\) 7.08680 0.510119 0.255060 0.966925i \(-0.417905\pi\)
0.255060 + 0.966925i \(0.417905\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 20.4753 1.45880 0.729401 0.684087i \(-0.239799\pi\)
0.729401 + 0.684087i \(0.239799\pi\)
\(198\) 0 0
\(199\) −8.74090 −0.619626 −0.309813 0.950798i \(-0.600266\pi\)
−0.309813 + 0.950798i \(0.600266\pi\)
\(200\) 0 0
\(201\) 14.3040 1.00893
\(202\) 0 0
\(203\) 9.43032 0.661878
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −12.8672 −0.894331
\(208\) 0 0
\(209\) 5.86718 0.405841
\(210\) 0 0
\(211\) 18.7344 1.28973 0.644863 0.764298i \(-0.276914\pi\)
0.644863 + 0.764298i \(0.276914\pi\)
\(212\) 0 0
\(213\) 2.54875 0.174638
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.75529 0.594348
\(218\) 0 0
\(219\) −3.00000 −0.202721
\(220\) 0 0
\(221\) −0.436861 −0.0293864
\(222\) 0 0
\(223\) −4.27110 −0.286014 −0.143007 0.989722i \(-0.545677\pi\)
−0.143007 + 0.989722i \(0.545677\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 21.5631 1.43120 0.715598 0.698512i \(-0.246154\pi\)
0.715598 + 0.698512i \(0.246154\pi\)
\(228\) 0 0
\(229\) 27.4172 1.81178 0.905891 0.423511i \(-0.139203\pi\)
0.905891 + 0.423511i \(0.139203\pi\)
\(230\) 0 0
\(231\) 8.43032 0.554674
\(232\) 0 0
\(233\) 7.86718 0.515396 0.257698 0.966226i \(-0.417036\pi\)
0.257698 + 0.966226i \(0.417036\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −9.27895 −0.602732
\(238\) 0 0
\(239\) −9.30404 −0.601828 −0.300914 0.953651i \(-0.597292\pi\)
−0.300914 + 0.953651i \(0.597292\pi\)
\(240\) 0 0
\(241\) −21.9056 −1.41106 −0.705531 0.708679i \(-0.749291\pi\)
−0.705531 + 0.708679i \(0.749291\pi\)
\(242\) 0 0
\(243\) −14.1712 −0.909084
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −0.364448 −0.0231893
\(248\) 0 0
\(249\) −3.62116 −0.229482
\(250\) 0 0
\(251\) 7.43032 0.468997 0.234499 0.972116i \(-0.424655\pi\)
0.234499 + 0.972116i \(0.424655\pi\)
\(252\) 0 0
\(253\) 48.2964 3.03637
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 5.05387 0.315251 0.157626 0.987499i \(-0.449616\pi\)
0.157626 + 0.987499i \(0.449616\pi\)
\(258\) 0 0
\(259\) −8.55660 −0.531681
\(260\) 0 0
\(261\) 12.2975 0.761196
\(262\) 0 0
\(263\) 17.7673 1.09558 0.547789 0.836617i \(-0.315470\pi\)
0.547789 + 0.836617i \(0.315470\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 6.81986 0.417368
\(268\) 0 0
\(269\) 9.17776 0.559578 0.279789 0.960062i \(-0.409736\pi\)
0.279789 + 0.960062i \(0.409736\pi\)
\(270\) 0 0
\(271\) 20.7278 1.25912 0.629562 0.776950i \(-0.283234\pi\)
0.629562 + 0.776950i \(0.283234\pi\)
\(272\) 0 0
\(273\) −0.523661 −0.0316934
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 24.5895 1.47744 0.738721 0.674012i \(-0.235430\pi\)
0.738721 + 0.674012i \(0.235430\pi\)
\(278\) 0 0
\(279\) 11.4172 0.683532
\(280\) 0 0
\(281\) 3.73436 0.222773 0.111386 0.993777i \(-0.464471\pi\)
0.111386 + 0.993777i \(0.464471\pi\)
\(282\) 0 0
\(283\) 13.0318 0.774663 0.387332 0.921941i \(-0.373397\pi\)
0.387332 + 0.921941i \(0.373397\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.92104 −0.172424
\(288\) 0 0
\(289\) −15.5631 −0.915479
\(290\) 0 0
\(291\) −1.31058 −0.0768277
\(292\) 0 0
\(293\) 19.9001 1.16258 0.581288 0.813698i \(-0.302549\pi\)
0.581288 + 0.813698i \(0.302549\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 32.0923 1.86218
\(298\) 0 0
\(299\) −3.00000 −0.173494
\(300\) 0 0
\(301\) −8.86718 −0.511096
\(302\) 0 0
\(303\) −13.0264 −0.748347
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −8.02093 −0.457779 −0.228889 0.973452i \(-0.573509\pi\)
−0.228889 + 0.973452i \(0.573509\pi\)
\(308\) 0 0
\(309\) −11.6763 −0.664243
\(310\) 0 0
\(311\) 16.1778 0.917357 0.458678 0.888602i \(-0.348323\pi\)
0.458678 + 0.888602i \(0.348323\pi\)
\(312\) 0 0
\(313\) −7.13828 −0.403480 −0.201740 0.979439i \(-0.564660\pi\)
−0.201740 + 0.979439i \(0.564660\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3.96052 0.222445 0.111223 0.993796i \(-0.464523\pi\)
0.111223 + 0.993796i \(0.464523\pi\)
\(318\) 0 0
\(319\) −46.1581 −2.58436
\(320\) 0 0
\(321\) 19.6146 1.09478
\(322\) 0 0
\(323\) 1.19869 0.0666970
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −11.1448 −0.616310
\(328\) 0 0
\(329\) −16.4238 −0.905472
\(330\) 0 0
\(331\) 5.94852 0.326960 0.163480 0.986547i \(-0.447728\pi\)
0.163480 + 0.986547i \(0.447728\pi\)
\(332\) 0 0
\(333\) −11.1581 −0.611462
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 8.90666 0.485176 0.242588 0.970129i \(-0.422004\pi\)
0.242588 + 0.970129i \(0.422004\pi\)
\(338\) 0 0
\(339\) 3.55660 0.193168
\(340\) 0 0
\(341\) −42.8541 −2.32068
\(342\) 0 0
\(343\) 15.0593 0.813127
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 18.5961 0.998290 0.499145 0.866519i \(-0.333648\pi\)
0.499145 + 0.866519i \(0.333648\pi\)
\(348\) 0 0
\(349\) −14.3040 −0.765678 −0.382839 0.923815i \(-0.625054\pi\)
−0.382839 + 0.923815i \(0.625054\pi\)
\(350\) 0 0
\(351\) −1.99346 −0.106403
\(352\) 0 0
\(353\) −12.7673 −0.679534 −0.339767 0.940510i \(-0.610348\pi\)
−0.339767 + 0.940510i \(0.610348\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.72235 0.0911566
\(358\) 0 0
\(359\) −2.69596 −0.142287 −0.0711437 0.997466i \(-0.522665\pi\)
−0.0711437 + 0.997466i \(0.522665\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) −28.0779 −1.47371
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −3.43140 −0.179118 −0.0895589 0.995982i \(-0.528546\pi\)
−0.0895589 + 0.995982i \(0.528546\pi\)
\(368\) 0 0
\(369\) −3.80915 −0.198297
\(370\) 0 0
\(371\) 8.86718 0.460361
\(372\) 0 0
\(373\) 15.5697 0.806168 0.403084 0.915163i \(-0.367938\pi\)
0.403084 + 0.915163i \(0.367938\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.86718 0.147667
\(378\) 0 0
\(379\) 0.164672 0.00845864 0.00422932 0.999991i \(-0.498654\pi\)
0.00422932 + 0.999991i \(0.498654\pi\)
\(380\) 0 0
\(381\) −13.6146 −0.697498
\(382\) 0 0
\(383\) −22.3514 −1.14210 −0.571051 0.820915i \(-0.693464\pi\)
−0.571051 + 0.820915i \(0.693464\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −11.5631 −0.587787
\(388\) 0 0
\(389\) −9.34243 −0.473680 −0.236840 0.971549i \(-0.576112\pi\)
−0.236840 + 0.971549i \(0.576112\pi\)
\(390\) 0 0
\(391\) 9.86718 0.499005
\(392\) 0 0
\(393\) −1.87372 −0.0945167
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.83533 0.242678 0.121339 0.992611i \(-0.461281\pi\)
0.121339 + 0.992611i \(0.461281\pi\)
\(398\) 0 0
\(399\) 1.43686 0.0719330
\(400\) 0 0
\(401\) 37.4622 1.87077 0.935386 0.353629i \(-0.115053\pi\)
0.935386 + 0.353629i \(0.115053\pi\)
\(402\) 0 0
\(403\) 2.66194 0.132601
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 41.8816 2.07599
\(408\) 0 0
\(409\) 9.42377 0.465976 0.232988 0.972480i \(-0.425150\pi\)
0.232988 + 0.972480i \(0.425150\pi\)
\(410\) 0 0
\(411\) 18.9869 0.936555
\(412\) 0 0
\(413\) 15.4159 0.758568
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −23.8068 −1.16582
\(418\) 0 0
\(419\) −27.4303 −1.34006 −0.670029 0.742335i \(-0.733718\pi\)
−0.670029 + 0.742335i \(0.733718\pi\)
\(420\) 0 0
\(421\) 18.4303 0.898239 0.449119 0.893472i \(-0.351738\pi\)
0.449119 + 0.893472i \(0.351738\pi\)
\(422\) 0 0
\(423\) −21.4172 −1.04134
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.56314 0.0756456
\(428\) 0 0
\(429\) 2.56314 0.123750
\(430\) 0 0
\(431\) 15.5117 0.747170 0.373585 0.927596i \(-0.378128\pi\)
0.373585 + 0.927596i \(0.378128\pi\)
\(432\) 0 0
\(433\) −31.8870 −1.53239 −0.766196 0.642607i \(-0.777853\pi\)
−0.766196 + 0.642607i \(0.777853\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 8.23163 0.393772
\(438\) 0 0
\(439\) −14.7278 −0.702920 −0.351460 0.936203i \(-0.614315\pi\)
−0.351460 + 0.936203i \(0.614315\pi\)
\(440\) 0 0
\(441\) 8.69596 0.414093
\(442\) 0 0
\(443\) 6.46087 0.306965 0.153483 0.988151i \(-0.450951\pi\)
0.153483 + 0.988151i \(0.450951\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −3.22378 −0.152480
\(448\) 0 0
\(449\) 21.2910 1.00478 0.502391 0.864641i \(-0.332454\pi\)
0.502391 + 0.864641i \(0.332454\pi\)
\(450\) 0 0
\(451\) 14.2975 0.673243
\(452\) 0 0
\(453\) 20.5751 0.966705
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 19.6465 0.919023 0.459512 0.888172i \(-0.348024\pi\)
0.459512 + 0.888172i \(0.348024\pi\)
\(458\) 0 0
\(459\) 6.55660 0.306036
\(460\) 0 0
\(461\) 1.16467 0.0542442 0.0271221 0.999632i \(-0.491366\pi\)
0.0271221 + 0.999632i \(0.491366\pi\)
\(462\) 0 0
\(463\) −5.15375 −0.239515 −0.119758 0.992803i \(-0.538212\pi\)
−0.119758 + 0.992803i \(0.538212\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.17122 −0.193021 −0.0965104 0.995332i \(-0.530768\pi\)
−0.0965104 + 0.995332i \(0.530768\pi\)
\(468\) 0 0
\(469\) 14.3040 0.660499
\(470\) 0 0
\(471\) 19.6146 0.903794
\(472\) 0 0
\(473\) 43.4018 1.99561
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 11.5631 0.529440
\(478\) 0 0
\(479\) 2.35552 0.107626 0.0538132 0.998551i \(-0.482862\pi\)
0.0538132 + 0.998551i \(0.482862\pi\)
\(480\) 0 0
\(481\) −2.60153 −0.118620
\(482\) 0 0
\(483\) 11.8277 0.538179
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 19.9056 0.902008 0.451004 0.892522i \(-0.351066\pi\)
0.451004 + 0.892522i \(0.351066\pi\)
\(488\) 0 0
\(489\) −9.93175 −0.449129
\(490\) 0 0
\(491\) −7.90557 −0.356773 −0.178387 0.983960i \(-0.557088\pi\)
−0.178387 + 0.983960i \(0.557088\pi\)
\(492\) 0 0
\(493\) −9.43032 −0.424720
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 2.54875 0.114327
\(498\) 0 0
\(499\) 35.7147 1.59881 0.799405 0.600792i \(-0.205148\pi\)
0.799405 + 0.600792i \(0.205148\pi\)
\(500\) 0 0
\(501\) −8.98691 −0.401506
\(502\) 0 0
\(503\) −30.7991 −1.37327 −0.686633 0.727004i \(-0.740912\pi\)
−0.686633 + 0.727004i \(0.740912\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 15.4238 0.684994
\(508\) 0 0
\(509\) 17.5631 0.778472 0.389236 0.921138i \(-0.372739\pi\)
0.389236 + 0.921138i \(0.372739\pi\)
\(510\) 0 0
\(511\) −3.00000 −0.132712
\(512\) 0 0
\(513\) 5.46980 0.241497
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 80.3887 3.53549
\(518\) 0 0
\(519\) 17.6763 0.775905
\(520\) 0 0
\(521\) −24.6081 −1.07810 −0.539050 0.842274i \(-0.681217\pi\)
−0.539050 + 0.842274i \(0.681217\pi\)
\(522\) 0 0
\(523\) −43.7147 −1.91151 −0.955756 0.294162i \(-0.904960\pi\)
−0.955756 + 0.294162i \(0.904960\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −8.75529 −0.381386
\(528\) 0 0
\(529\) 44.7597 1.94607
\(530\) 0 0
\(531\) 20.1030 0.872394
\(532\) 0 0
\(533\) −0.888109 −0.0384683
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 15.8857 0.685520
\(538\) 0 0
\(539\) −32.6399 −1.40590
\(540\) 0 0
\(541\) −8.38538 −0.360516 −0.180258 0.983619i \(-0.557693\pi\)
−0.180258 + 0.983619i \(0.557693\pi\)
\(542\) 0 0
\(543\) −20.0593 −0.860828
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 26.3040 1.12468 0.562340 0.826906i \(-0.309901\pi\)
0.562340 + 0.826906i \(0.309901\pi\)
\(548\) 0 0
\(549\) 2.03839 0.0869965
\(550\) 0 0
\(551\) −7.86718 −0.335153
\(552\) 0 0
\(553\) −9.27895 −0.394581
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −2.66740 −0.113021 −0.0565107 0.998402i \(-0.517998\pi\)
−0.0565107 + 0.998402i \(0.517998\pi\)
\(558\) 0 0
\(559\) −2.69596 −0.114027
\(560\) 0 0
\(561\) −8.43032 −0.355928
\(562\) 0 0
\(563\) 37.7751 1.59203 0.796016 0.605276i \(-0.206937\pi\)
0.796016 + 0.605276i \(0.206937\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.23817 0.0939943
\(568\) 0 0
\(569\) 38.8606 1.62912 0.814561 0.580078i \(-0.196978\pi\)
0.814561 + 0.580078i \(0.196978\pi\)
\(570\) 0 0
\(571\) −4.17122 −0.174560 −0.0872800 0.996184i \(-0.527817\pi\)
−0.0872800 + 0.996184i \(0.527817\pi\)
\(572\) 0 0
\(573\) −4.95398 −0.206955
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −39.3413 −1.63780 −0.818901 0.573935i \(-0.805416\pi\)
−0.818901 + 0.573935i \(0.805416\pi\)
\(578\) 0 0
\(579\) −8.49489 −0.353035
\(580\) 0 0
\(581\) −3.62116 −0.150231
\(582\) 0 0
\(583\) −43.4018 −1.79752
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −17.8672 −0.737457 −0.368729 0.929537i \(-0.620207\pi\)
−0.368729 + 0.929537i \(0.620207\pi\)
\(588\) 0 0
\(589\) −7.30404 −0.300958
\(590\) 0 0
\(591\) −24.5435 −1.00959
\(592\) 0 0
\(593\) 13.0803 0.537142 0.268571 0.963260i \(-0.413449\pi\)
0.268571 + 0.963260i \(0.413449\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 10.4776 0.428821
\(598\) 0 0
\(599\) −10.7278 −0.438326 −0.219163 0.975688i \(-0.570333\pi\)
−0.219163 + 0.975688i \(0.570333\pi\)
\(600\) 0 0
\(601\) 39.8026 1.62358 0.811791 0.583948i \(-0.198493\pi\)
0.811791 + 0.583948i \(0.198493\pi\)
\(602\) 0 0
\(603\) 18.6530 0.759609
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 43.0318 1.74661 0.873304 0.487175i \(-0.161973\pi\)
0.873304 + 0.487175i \(0.161973\pi\)
\(608\) 0 0
\(609\) −11.3040 −0.458063
\(610\) 0 0
\(611\) −4.99346 −0.202014
\(612\) 0 0
\(613\) 15.4722 0.624915 0.312458 0.949932i \(-0.398848\pi\)
0.312458 + 0.949932i \(0.398848\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −35.5237 −1.43013 −0.715064 0.699059i \(-0.753603\pi\)
−0.715064 + 0.699059i \(0.753603\pi\)
\(618\) 0 0
\(619\) 15.0449 0.604707 0.302354 0.953196i \(-0.402228\pi\)
0.302354 + 0.953196i \(0.402228\pi\)
\(620\) 0 0
\(621\) 45.0253 1.80680
\(622\) 0 0
\(623\) 6.81986 0.273232
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −7.03293 −0.280868
\(628\) 0 0
\(629\) 8.55660 0.341174
\(630\) 0 0
\(631\) −35.6399 −1.41880 −0.709402 0.704805i \(-0.751035\pi\)
−0.709402 + 0.704805i \(0.751035\pi\)
\(632\) 0 0
\(633\) −22.4567 −0.892574
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.02748 0.0803315
\(638\) 0 0
\(639\) 3.32367 0.131482
\(640\) 0 0
\(641\) 6.98691 0.275966 0.137983 0.990435i \(-0.455938\pi\)
0.137983 + 0.990435i \(0.455938\pi\)
\(642\) 0 0
\(643\) 3.74744 0.147785 0.0738924 0.997266i \(-0.476458\pi\)
0.0738924 + 0.997266i \(0.476458\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 34.1658 1.34319 0.671597 0.740916i \(-0.265609\pi\)
0.671597 + 0.740916i \(0.265609\pi\)
\(648\) 0 0
\(649\) −75.4556 −2.96189
\(650\) 0 0
\(651\) −10.4949 −0.411327
\(652\) 0 0
\(653\) 13.5446 0.530041 0.265020 0.964243i \(-0.414621\pi\)
0.265020 + 0.964243i \(0.414621\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −3.91211 −0.152626
\(658\) 0 0
\(659\) 22.6334 0.881671 0.440836 0.897588i \(-0.354682\pi\)
0.440836 + 0.897588i \(0.354682\pi\)
\(660\) 0 0
\(661\) −3.96161 −0.154089 −0.0770443 0.997028i \(-0.524548\pi\)
−0.0770443 + 0.997028i \(0.524548\pi\)
\(662\) 0 0
\(663\) 0.523661 0.0203373
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −64.7597 −2.50750
\(668\) 0 0
\(669\) 5.11973 0.197940
\(670\) 0 0
\(671\) −7.65102 −0.295365
\(672\) 0 0
\(673\) −28.9605 −1.11635 −0.558173 0.829725i \(-0.688497\pi\)
−0.558173 + 0.829725i \(0.688497\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −24.7518 −0.951290 −0.475645 0.879637i \(-0.657785\pi\)
−0.475645 + 0.879637i \(0.657785\pi\)
\(678\) 0 0
\(679\) −1.31058 −0.0502955
\(680\) 0 0
\(681\) −25.8475 −0.990480
\(682\) 0 0
\(683\) 40.0593 1.53283 0.766414 0.642347i \(-0.222039\pi\)
0.766414 + 0.642347i \(0.222039\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −32.8648 −1.25387
\(688\) 0 0
\(689\) 2.69596 0.102708
\(690\) 0 0
\(691\) −35.2162 −1.33969 −0.669843 0.742503i \(-0.733639\pi\)
−0.669843 + 0.742503i \(0.733639\pi\)
\(692\) 0 0
\(693\) 10.9935 0.417607
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.92104 0.110642
\(698\) 0 0
\(699\) −9.43032 −0.356687
\(700\) 0 0
\(701\) −38.1283 −1.44008 −0.720042 0.693930i \(-0.755878\pi\)
−0.720042 + 0.693930i \(0.755878\pi\)
\(702\) 0 0
\(703\) 7.13828 0.269225
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −13.0264 −0.489908
\(708\) 0 0
\(709\) −6.08789 −0.228635 −0.114318 0.993444i \(-0.536468\pi\)
−0.114318 + 0.993444i \(0.536468\pi\)
\(710\) 0 0
\(711\) −12.1001 −0.453789
\(712\) 0 0
\(713\) −60.1241 −2.25167
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 11.1527 0.416504
\(718\) 0 0
\(719\) 12.1647 0.453666 0.226833 0.973934i \(-0.427163\pi\)
0.226833 + 0.973934i \(0.427163\pi\)
\(720\) 0 0
\(721\) −11.6763 −0.434849
\(722\) 0 0
\(723\) 26.2580 0.976546
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −25.6859 −0.952639 −0.476320 0.879272i \(-0.658029\pi\)
−0.476320 + 0.879272i \(0.658029\pi\)
\(728\) 0 0
\(729\) 22.5884 0.836609
\(730\) 0 0
\(731\) 8.86718 0.327964
\(732\) 0 0
\(733\) 43.4502 1.60487 0.802434 0.596741i \(-0.203538\pi\)
0.802434 + 0.596741i \(0.203538\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −70.0133 −2.57897
\(738\) 0 0
\(739\) 43.4172 1.59713 0.798564 0.601910i \(-0.205593\pi\)
0.798564 + 0.601910i \(0.205593\pi\)
\(740\) 0 0
\(741\) 0.436861 0.0160485
\(742\) 0 0
\(743\) 26.6709 0.978459 0.489230 0.872155i \(-0.337278\pi\)
0.489230 + 0.872155i \(0.337278\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −4.72214 −0.172774
\(748\) 0 0
\(749\) 19.6146 0.716703
\(750\) 0 0
\(751\) 1.52674 0.0557114 0.0278557 0.999612i \(-0.491132\pi\)
0.0278557 + 0.999612i \(0.491132\pi\)
\(752\) 0 0
\(753\) −8.90666 −0.324577
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 9.38191 0.340991 0.170496 0.985358i \(-0.445463\pi\)
0.170496 + 0.985358i \(0.445463\pi\)
\(758\) 0 0
\(759\) −57.8925 −2.10136
\(760\) 0 0
\(761\) −6.70251 −0.242966 −0.121483 0.992594i \(-0.538765\pi\)
−0.121483 + 0.992594i \(0.538765\pi\)
\(762\) 0 0
\(763\) −11.1448 −0.403470
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 4.68703 0.169239
\(768\) 0 0
\(769\) −36.0637 −1.30049 −0.650245 0.759724i \(-0.725334\pi\)
−0.650245 + 0.759724i \(0.725334\pi\)
\(770\) 0 0
\(771\) −6.05802 −0.218174
\(772\) 0 0
\(773\) 11.6763 0.419968 0.209984 0.977705i \(-0.432659\pi\)
0.209984 + 0.977705i \(0.432659\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 10.2567 0.367958
\(778\) 0 0
\(779\) 2.43686 0.0873096
\(780\) 0 0
\(781\) −12.4753 −0.446400
\(782\) 0 0
\(783\) −43.0318 −1.53783
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 43.1880 1.53949 0.769743 0.638354i \(-0.220384\pi\)
0.769743 + 0.638354i \(0.220384\pi\)
\(788\) 0 0
\(789\) −21.2975 −0.758211
\(790\) 0 0
\(791\) 3.55660 0.126458
\(792\) 0 0
\(793\) 0.475254 0.0168768
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −29.1658 −1.03310 −0.516552 0.856256i \(-0.672785\pi\)
−0.516552 + 0.856256i \(0.672785\pi\)
\(798\) 0 0
\(799\) 16.4238 0.581031
\(800\) 0 0
\(801\) 8.89335 0.314231
\(802\) 0 0
\(803\) 14.6840 0.518186
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −11.0013 −0.387264
\(808\) 0 0
\(809\) 2.43032 0.0854454 0.0427227 0.999087i \(-0.486397\pi\)
0.0427227 + 0.999087i \(0.486397\pi\)
\(810\) 0 0
\(811\) 42.2779 1.48458 0.742288 0.670081i \(-0.233741\pi\)
0.742288 + 0.670081i \(0.233741\pi\)
\(812\) 0 0
\(813\) −24.8462 −0.871396
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 7.39738 0.258802
\(818\) 0 0
\(819\) −0.682874 −0.0238616
\(820\) 0 0
\(821\) 52.5136 1.83274 0.916369 0.400334i \(-0.131106\pi\)
0.916369 + 0.400334i \(0.131106\pi\)
\(822\) 0 0
\(823\) −36.3150 −1.26586 −0.632930 0.774209i \(-0.718148\pi\)
−0.632930 + 0.774209i \(0.718148\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 46.3150 1.61053 0.805264 0.592916i \(-0.202023\pi\)
0.805264 + 0.592916i \(0.202023\pi\)
\(828\) 0 0
\(829\) 20.0899 0.697750 0.348875 0.937169i \(-0.386564\pi\)
0.348875 + 0.937169i \(0.386564\pi\)
\(830\) 0 0
\(831\) −29.4753 −1.02249
\(832\) 0 0
\(833\) −6.66849 −0.231049
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −39.9516 −1.38093
\(838\) 0 0
\(839\) 10.5930 0.365711 0.182855 0.983140i \(-0.441466\pi\)
0.182855 + 0.983140i \(0.441466\pi\)
\(840\) 0 0
\(841\) 32.8925 1.13422
\(842\) 0 0
\(843\) −4.47634 −0.154173
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −28.0779 −0.964767
\(848\) 0 0
\(849\) −15.6212 −0.536117
\(850\) 0 0
\(851\) 58.7597 2.01426
\(852\) 0 0
\(853\) −40.0648 −1.37179 −0.685896 0.727700i \(-0.740590\pi\)
−0.685896 + 0.727700i \(0.740590\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 38.3019 1.30837 0.654183 0.756336i \(-0.273012\pi\)
0.654183 + 0.756336i \(0.273012\pi\)
\(858\) 0 0
\(859\) 39.9056 1.36156 0.680780 0.732488i \(-0.261641\pi\)
0.680780 + 0.732488i \(0.261641\pi\)
\(860\) 0 0
\(861\) 3.50143 0.119328
\(862\) 0 0
\(863\) 23.3315 0.794214 0.397107 0.917772i \(-0.370014\pi\)
0.397107 + 0.917772i \(0.370014\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 18.6554 0.633571
\(868\) 0 0
\(869\) 45.4172 1.54067
\(870\) 0 0
\(871\) 4.34898 0.147359
\(872\) 0 0
\(873\) −1.70905 −0.0578426
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 10.4105 0.351537 0.175768 0.984432i \(-0.443759\pi\)
0.175768 + 0.984432i \(0.443759\pi\)
\(878\) 0 0
\(879\) −23.8541 −0.804578
\(880\) 0 0
\(881\) −23.7662 −0.800704 −0.400352 0.916361i \(-0.631112\pi\)
−0.400352 + 0.916361i \(0.631112\pi\)
\(882\) 0 0
\(883\) 28.3908 0.955428 0.477714 0.878515i \(-0.341466\pi\)
0.477714 + 0.878515i \(0.341466\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 32.5644 1.09341 0.546703 0.837326i \(-0.315883\pi\)
0.546703 + 0.837326i \(0.315883\pi\)
\(888\) 0 0
\(889\) −13.6146 −0.456620
\(890\) 0 0
\(891\) −10.9551 −0.367008
\(892\) 0 0
\(893\) 13.7014 0.458501
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 3.59607 0.120069
\(898\) 0 0
\(899\) 57.4622 1.91647
\(900\) 0 0
\(901\) −8.86718 −0.295409
\(902\) 0 0
\(903\) 10.6290 0.353711
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −57.2109 −1.89966 −0.949829 0.312771i \(-0.898743\pi\)
−0.949829 + 0.312771i \(0.898743\pi\)
\(908\) 0 0
\(909\) −16.9869 −0.563420
\(910\) 0 0
\(911\) 42.0617 1.39357 0.696783 0.717282i \(-0.254614\pi\)
0.696783 + 0.717282i \(0.254614\pi\)
\(912\) 0 0
\(913\) 17.7243 0.586590
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.87372 −0.0618757
\(918\) 0 0
\(919\) −5.68287 −0.187461 −0.0937304 0.995598i \(-0.529879\pi\)
−0.0937304 + 0.995598i \(0.529879\pi\)
\(920\) 0 0
\(921\) 9.61462 0.316813
\(922\) 0 0
\(923\) 0.774918 0.0255067
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −15.2264 −0.500100
\(928\) 0 0
\(929\) −30.9289 −1.01474 −0.507372 0.861727i \(-0.669383\pi\)
−0.507372 + 0.861727i \(0.669383\pi\)
\(930\) 0 0
\(931\) −5.56314 −0.182325
\(932\) 0 0
\(933\) −19.3921 −0.634870
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 44.5357 1.45492 0.727458 0.686152i \(-0.240701\pi\)
0.727458 + 0.686152i \(0.240701\pi\)
\(938\) 0 0
\(939\) 8.55660 0.279234
\(940\) 0 0
\(941\) −2.08134 −0.0678498 −0.0339249 0.999424i \(-0.510801\pi\)
−0.0339249 + 0.999424i \(0.510801\pi\)
\(942\) 0 0
\(943\) 20.0593 0.653221
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 52.7189 1.71313 0.856567 0.516036i \(-0.172593\pi\)
0.856567 + 0.516036i \(0.172593\pi\)
\(948\) 0 0
\(949\) −0.912115 −0.0296085
\(950\) 0 0
\(951\) −4.74744 −0.153946
\(952\) 0 0
\(953\) −3.22071 −0.104329 −0.0521645 0.998639i \(-0.516612\pi\)
−0.0521645 + 0.998639i \(0.516612\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 55.3293 1.78854
\(958\) 0 0
\(959\) 18.9869 0.613119
\(960\) 0 0
\(961\) 22.3490 0.720935
\(962\) 0 0
\(963\) 25.5782 0.824247
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.65956 0.0855256 0.0427628 0.999085i \(-0.486384\pi\)
0.0427628 + 0.999085i \(0.486384\pi\)
\(968\) 0 0
\(969\) −1.43686 −0.0461586
\(970\) 0 0
\(971\) −35.1263 −1.12726 −0.563628 0.826029i \(-0.690595\pi\)
−0.563628 + 0.826029i \(0.690595\pi\)
\(972\) 0 0
\(973\) −23.8068 −0.763210
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −18.1668 −0.581209 −0.290604 0.956843i \(-0.593856\pi\)
−0.290604 + 0.956843i \(0.593856\pi\)
\(978\) 0 0
\(979\) −33.3808 −1.06686
\(980\) 0 0
\(981\) −14.5333 −0.464012
\(982\) 0 0
\(983\) 48.4556 1.54549 0.772747 0.634714i \(-0.218882\pi\)
0.772747 + 0.634714i \(0.218882\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 19.6870 0.626645
\(988\) 0 0
\(989\) 60.8925 1.93627
\(990\) 0 0
\(991\) −29.8157 −0.947127 −0.473563 0.880760i \(-0.657033\pi\)
−0.473563 + 0.880760i \(0.657033\pi\)
\(992\) 0 0
\(993\) −7.13044 −0.226278
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −29.6894 −0.940273 −0.470137 0.882594i \(-0.655795\pi\)
−0.470137 + 0.882594i \(0.655795\pi\)
\(998\) 0 0
\(999\) 39.0449 1.23533
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 7600.2.a.by.1.1 3
4.3 odd 2 1900.2.a.f.1.3 3
5.4 even 2 7600.2.a.bj.1.3 3
20.3 even 4 1900.2.c.g.1749.5 6
20.7 even 4 1900.2.c.g.1749.2 6
20.19 odd 2 1900.2.a.h.1.1 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1900.2.a.f.1.3 3 4.3 odd 2
1900.2.a.h.1.1 yes 3 20.19 odd 2
1900.2.c.g.1749.2 6 20.7 even 4
1900.2.c.g.1749.5 6 20.3 even 4
7600.2.a.bj.1.3 3 5.4 even 2
7600.2.a.by.1.1 3 1.1 even 1 trivial