Properties

Label 8624.2.a.co.1.2
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
Analytic rank $1$
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] = [8624,2,Mod(1,8624)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8624, 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("8624.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8624 = 2^{4} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8624.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: \(68.8629867032\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\Q(\zeta_{18})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - 3x - 1 \) 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 77)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-0.347296\) of defining polynomial
Character \(\chi\) \(=\) 8624.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.652704 q^{3} -3.53209 q^{5} -2.57398 q^{9} +O(q^{10})\) \(q+0.652704 q^{3} -3.53209 q^{5} -2.57398 q^{9} -1.00000 q^{11} -4.41147 q^{13} -2.30541 q^{15} +5.24897 q^{17} +1.81521 q^{19} +6.33275 q^{23} +7.47565 q^{25} -3.63816 q^{27} +1.92127 q^{29} +1.46791 q^{31} -0.652704 q^{33} +4.45336 q^{37} -2.87939 q^{39} -0.283119 q^{41} +3.41147 q^{43} +9.09152 q^{45} -4.55438 q^{47} +3.42602 q^{51} -7.23442 q^{53} +3.53209 q^{55} +1.18479 q^{57} +9.53983 q^{59} +1.14796 q^{61} +15.5817 q^{65} +0.694593 q^{67} +4.13341 q^{69} -9.46110 q^{71} -2.34730 q^{73} +4.87939 q^{75} +12.4192 q^{79} +5.34730 q^{81} -11.3327 q^{83} -18.5398 q^{85} +1.25402 q^{87} +3.46791 q^{89} +0.958111 q^{93} -6.41147 q^{95} -15.3473 q^{97} +2.57398 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 6 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + 3 q^{3} - 6 q^{5} - 3 q^{11} - 3 q^{13} - 9 q^{15} + 3 q^{17} + 9 q^{19} + 3 q^{25} + 6 q^{27} - 3 q^{29} + 9 q^{31} - 3 q^{33} - 3 q^{39} - 9 q^{41} - 3 q^{45} - 3 q^{47} + 18 q^{51} + 9 q^{53} + 6 q^{55} - 12 q^{61} + 15 q^{65} - 21 q^{69} + 9 q^{71} - 6 q^{73} + 9 q^{75} + 3 q^{79} + 15 q^{81} - 15 q^{83} - 27 q^{85} - 24 q^{87} + 15 q^{89} + 6 q^{93} - 9 q^{95} - 45 q^{97}+O(q^{100}) \) 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
\(3\) 0.652704 0.376839 0.188419 0.982089i \(-0.439664\pi\)
0.188419 + 0.982089i \(0.439664\pi\)
\(4\) 0 0
\(5\) −3.53209 −1.57960 −0.789799 0.613366i \(-0.789815\pi\)
−0.789799 + 0.613366i \(0.789815\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) −2.57398 −0.857993
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −4.41147 −1.22352 −0.611761 0.791042i \(-0.709539\pi\)
−0.611761 + 0.791042i \(0.709539\pi\)
\(14\) 0 0
\(15\) −2.30541 −0.595254
\(16\) 0 0
\(17\) 5.24897 1.27306 0.636531 0.771251i \(-0.280369\pi\)
0.636531 + 0.771251i \(0.280369\pi\)
\(18\) 0 0
\(19\) 1.81521 0.416437 0.208219 0.978082i \(-0.433233\pi\)
0.208219 + 0.978082i \(0.433233\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 6.33275 1.32047 0.660235 0.751059i \(-0.270457\pi\)
0.660235 + 0.751059i \(0.270457\pi\)
\(24\) 0 0
\(25\) 7.47565 1.49513
\(26\) 0 0
\(27\) −3.63816 −0.700163
\(28\) 0 0
\(29\) 1.92127 0.356772 0.178386 0.983961i \(-0.442912\pi\)
0.178386 + 0.983961i \(0.442912\pi\)
\(30\) 0 0
\(31\) 1.46791 0.263645 0.131822 0.991273i \(-0.457917\pi\)
0.131822 + 0.991273i \(0.457917\pi\)
\(32\) 0 0
\(33\) −0.652704 −0.113621
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.45336 0.732128 0.366064 0.930590i \(-0.380705\pi\)
0.366064 + 0.930590i \(0.380705\pi\)
\(38\) 0 0
\(39\) −2.87939 −0.461071
\(40\) 0 0
\(41\) −0.283119 −0.0442157 −0.0221078 0.999756i \(-0.507038\pi\)
−0.0221078 + 0.999756i \(0.507038\pi\)
\(42\) 0 0
\(43\) 3.41147 0.520245 0.260122 0.965576i \(-0.416237\pi\)
0.260122 + 0.965576i \(0.416237\pi\)
\(44\) 0 0
\(45\) 9.09152 1.35528
\(46\) 0 0
\(47\) −4.55438 −0.664324 −0.332162 0.943222i \(-0.607778\pi\)
−0.332162 + 0.943222i \(0.607778\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 3.42602 0.479739
\(52\) 0 0
\(53\) −7.23442 −0.993724 −0.496862 0.867829i \(-0.665515\pi\)
−0.496862 + 0.867829i \(0.665515\pi\)
\(54\) 0 0
\(55\) 3.53209 0.476267
\(56\) 0 0
\(57\) 1.18479 0.156930
\(58\) 0 0
\(59\) 9.53983 1.24198 0.620990 0.783818i \(-0.286731\pi\)
0.620990 + 0.783818i \(0.286731\pi\)
\(60\) 0 0
\(61\) 1.14796 0.146981 0.0734903 0.997296i \(-0.476586\pi\)
0.0734903 + 0.997296i \(0.476586\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 15.5817 1.93267
\(66\) 0 0
\(67\) 0.694593 0.0848580 0.0424290 0.999099i \(-0.486490\pi\)
0.0424290 + 0.999099i \(0.486490\pi\)
\(68\) 0 0
\(69\) 4.13341 0.497604
\(70\) 0 0
\(71\) −9.46110 −1.12283 −0.561413 0.827536i \(-0.689742\pi\)
−0.561413 + 0.827536i \(0.689742\pi\)
\(72\) 0 0
\(73\) −2.34730 −0.274730 −0.137365 0.990520i \(-0.543863\pi\)
−0.137365 + 0.990520i \(0.543863\pi\)
\(74\) 0 0
\(75\) 4.87939 0.563423
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 12.4192 1.39727 0.698635 0.715478i \(-0.253791\pi\)
0.698635 + 0.715478i \(0.253791\pi\)
\(80\) 0 0
\(81\) 5.34730 0.594144
\(82\) 0 0
\(83\) −11.3327 −1.24393 −0.621965 0.783045i \(-0.713666\pi\)
−0.621965 + 0.783045i \(0.713666\pi\)
\(84\) 0 0
\(85\) −18.5398 −2.01093
\(86\) 0 0
\(87\) 1.25402 0.134445
\(88\) 0 0
\(89\) 3.46791 0.367598 0.183799 0.982964i \(-0.441160\pi\)
0.183799 + 0.982964i \(0.441160\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0.958111 0.0993515
\(94\) 0 0
\(95\) −6.41147 −0.657803
\(96\) 0 0
\(97\) −15.3473 −1.55828 −0.779141 0.626849i \(-0.784344\pi\)
−0.779141 + 0.626849i \(0.784344\pi\)
\(98\) 0 0
\(99\) 2.57398 0.258695
\(100\) 0 0
\(101\) −6.70233 −0.666907 −0.333454 0.942767i \(-0.608214\pi\)
−0.333454 + 0.942767i \(0.608214\pi\)
\(102\) 0 0
\(103\) 2.24897 0.221598 0.110799 0.993843i \(-0.464659\pi\)
0.110799 + 0.993843i \(0.464659\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.5321 1.11485 0.557425 0.830228i \(-0.311790\pi\)
0.557425 + 0.830228i \(0.311790\pi\)
\(108\) 0 0
\(109\) −11.9855 −1.14800 −0.573999 0.818856i \(-0.694609\pi\)
−0.573999 + 0.818856i \(0.694609\pi\)
\(110\) 0 0
\(111\) 2.90673 0.275894
\(112\) 0 0
\(113\) 5.00774 0.471089 0.235544 0.971864i \(-0.424313\pi\)
0.235544 + 0.971864i \(0.424313\pi\)
\(114\) 0 0
\(115\) −22.3678 −2.08581
\(116\) 0 0
\(117\) 11.3550 1.04977
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −0.184793 −0.0166622
\(124\) 0 0
\(125\) −8.74422 −0.782107
\(126\) 0 0
\(127\) 14.2344 1.26310 0.631550 0.775335i \(-0.282419\pi\)
0.631550 + 0.775335i \(0.282419\pi\)
\(128\) 0 0
\(129\) 2.22668 0.196048
\(130\) 0 0
\(131\) 5.11381 0.446795 0.223398 0.974727i \(-0.428285\pi\)
0.223398 + 0.974727i \(0.428285\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 12.8503 1.10598
\(136\) 0 0
\(137\) 19.7074 1.68372 0.841858 0.539699i \(-0.181462\pi\)
0.841858 + 0.539699i \(0.181462\pi\)
\(138\) 0 0
\(139\) −5.28581 −0.448336 −0.224168 0.974550i \(-0.571966\pi\)
−0.224168 + 0.974550i \(0.571966\pi\)
\(140\) 0 0
\(141\) −2.97266 −0.250343
\(142\) 0 0
\(143\) 4.41147 0.368906
\(144\) 0 0
\(145\) −6.78611 −0.563556
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −17.0155 −1.39396 −0.696981 0.717089i \(-0.745474\pi\)
−0.696981 + 0.717089i \(0.745474\pi\)
\(150\) 0 0
\(151\) −1.30272 −0.106014 −0.0530069 0.998594i \(-0.516881\pi\)
−0.0530069 + 0.998594i \(0.516881\pi\)
\(152\) 0 0
\(153\) −13.5107 −1.09228
\(154\) 0 0
\(155\) −5.18479 −0.416453
\(156\) 0 0
\(157\) −13.6946 −1.09295 −0.546474 0.837476i \(-0.684030\pi\)
−0.546474 + 0.837476i \(0.684030\pi\)
\(158\) 0 0
\(159\) −4.72193 −0.374474
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −16.4047 −1.28491 −0.642456 0.766322i \(-0.722085\pi\)
−0.642456 + 0.766322i \(0.722085\pi\)
\(164\) 0 0
\(165\) 2.30541 0.179476
\(166\) 0 0
\(167\) −8.24628 −0.638116 −0.319058 0.947735i \(-0.603367\pi\)
−0.319058 + 0.947735i \(0.603367\pi\)
\(168\) 0 0
\(169\) 6.46110 0.497008
\(170\) 0 0
\(171\) −4.67230 −0.357300
\(172\) 0 0
\(173\) −19.2344 −1.46237 −0.731183 0.682181i \(-0.761031\pi\)
−0.731183 + 0.682181i \(0.761031\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 6.22668 0.468026
\(178\) 0 0
\(179\) 1.32770 0.0992367 0.0496183 0.998768i \(-0.484200\pi\)
0.0496183 + 0.998768i \(0.484200\pi\)
\(180\) 0 0
\(181\) −17.6527 −1.31212 −0.656058 0.754711i \(-0.727777\pi\)
−0.656058 + 0.754711i \(0.727777\pi\)
\(182\) 0 0
\(183\) 0.749275 0.0553880
\(184\) 0 0
\(185\) −15.7297 −1.15647
\(186\) 0 0
\(187\) −5.24897 −0.383843
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −15.5526 −1.12535 −0.562674 0.826679i \(-0.690227\pi\)
−0.562674 + 0.826679i \(0.690227\pi\)
\(192\) 0 0
\(193\) −5.21894 −0.375668 −0.187834 0.982201i \(-0.560147\pi\)
−0.187834 + 0.982201i \(0.560147\pi\)
\(194\) 0 0
\(195\) 10.1702 0.728306
\(196\) 0 0
\(197\) −4.61856 −0.329058 −0.164529 0.986372i \(-0.552610\pi\)
−0.164529 + 0.986372i \(0.552610\pi\)
\(198\) 0 0
\(199\) −6.04963 −0.428847 −0.214423 0.976741i \(-0.568787\pi\)
−0.214423 + 0.976741i \(0.568787\pi\)
\(200\) 0 0
\(201\) 0.453363 0.0319778
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 1.00000 0.0698430
\(206\) 0 0
\(207\) −16.3004 −1.13295
\(208\) 0 0
\(209\) −1.81521 −0.125561
\(210\) 0 0
\(211\) −8.69459 −0.598560 −0.299280 0.954165i \(-0.596747\pi\)
−0.299280 + 0.954165i \(0.596747\pi\)
\(212\) 0 0
\(213\) −6.17530 −0.423124
\(214\) 0 0
\(215\) −12.0496 −0.821778
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −1.53209 −0.103529
\(220\) 0 0
\(221\) −23.1557 −1.55762
\(222\) 0 0
\(223\) −0.221629 −0.0148414 −0.00742069 0.999972i \(-0.502362\pi\)
−0.00742069 + 0.999972i \(0.502362\pi\)
\(224\) 0 0
\(225\) −19.2422 −1.28281
\(226\) 0 0
\(227\) −9.69965 −0.643788 −0.321894 0.946776i \(-0.604319\pi\)
−0.321894 + 0.946776i \(0.604319\pi\)
\(228\) 0 0
\(229\) −23.6946 −1.56578 −0.782891 0.622158i \(-0.786256\pi\)
−0.782891 + 0.622158i \(0.786256\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.8871 1.36836 0.684181 0.729313i \(-0.260160\pi\)
0.684181 + 0.729313i \(0.260160\pi\)
\(234\) 0 0
\(235\) 16.0865 1.04937
\(236\) 0 0
\(237\) 8.10607 0.526546
\(238\) 0 0
\(239\) −15.0300 −0.972212 −0.486106 0.873900i \(-0.661583\pi\)
−0.486106 + 0.873900i \(0.661583\pi\)
\(240\) 0 0
\(241\) −29.5398 −1.90283 −0.951414 0.307915i \(-0.900369\pi\)
−0.951414 + 0.307915i \(0.900369\pi\)
\(242\) 0 0
\(243\) 14.4047 0.924060
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −8.00774 −0.509520
\(248\) 0 0
\(249\) −7.39693 −0.468761
\(250\) 0 0
\(251\) 3.77063 0.238000 0.119000 0.992894i \(-0.462031\pi\)
0.119000 + 0.992894i \(0.462031\pi\)
\(252\) 0 0
\(253\) −6.33275 −0.398136
\(254\) 0 0
\(255\) −12.1010 −0.757795
\(256\) 0 0
\(257\) 8.66044 0.540224 0.270112 0.962829i \(-0.412939\pi\)
0.270112 + 0.962829i \(0.412939\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −4.94532 −0.306107
\(262\) 0 0
\(263\) 5.83481 0.359790 0.179895 0.983686i \(-0.442424\pi\)
0.179895 + 0.983686i \(0.442424\pi\)
\(264\) 0 0
\(265\) 25.5526 1.56969
\(266\) 0 0
\(267\) 2.26352 0.138525
\(268\) 0 0
\(269\) 21.9813 1.34023 0.670113 0.742259i \(-0.266246\pi\)
0.670113 + 0.742259i \(0.266246\pi\)
\(270\) 0 0
\(271\) 9.85204 0.598469 0.299235 0.954180i \(-0.403269\pi\)
0.299235 + 0.954180i \(0.403269\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −7.47565 −0.450799
\(276\) 0 0
\(277\) 23.6391 1.42034 0.710168 0.704033i \(-0.248619\pi\)
0.710168 + 0.704033i \(0.248619\pi\)
\(278\) 0 0
\(279\) −3.77837 −0.226205
\(280\) 0 0
\(281\) 16.2713 0.970662 0.485331 0.874331i \(-0.338699\pi\)
0.485331 + 0.874331i \(0.338699\pi\)
\(282\) 0 0
\(283\) 16.9982 1.01044 0.505220 0.862990i \(-0.331411\pi\)
0.505220 + 0.862990i \(0.331411\pi\)
\(284\) 0 0
\(285\) −4.18479 −0.247886
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 10.5517 0.620688
\(290\) 0 0
\(291\) −10.0172 −0.587221
\(292\) 0 0
\(293\) 13.7784 0.804941 0.402471 0.915433i \(-0.368152\pi\)
0.402471 + 0.915433i \(0.368152\pi\)
\(294\) 0 0
\(295\) −33.6955 −1.96183
\(296\) 0 0
\(297\) 3.63816 0.211107
\(298\) 0 0
\(299\) −27.9368 −1.61562
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) −4.37464 −0.251316
\(304\) 0 0
\(305\) −4.05468 −0.232170
\(306\) 0 0
\(307\) −11.6159 −0.662953 −0.331476 0.943464i \(-0.607547\pi\)
−0.331476 + 0.943464i \(0.607547\pi\)
\(308\) 0 0
\(309\) 1.46791 0.0835065
\(310\) 0 0
\(311\) 3.44831 0.195536 0.0977679 0.995209i \(-0.468830\pi\)
0.0977679 + 0.995209i \(0.468830\pi\)
\(312\) 0 0
\(313\) 19.4219 1.09779 0.548895 0.835891i \(-0.315049\pi\)
0.548895 + 0.835891i \(0.315049\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −32.8161 −1.84314 −0.921569 0.388214i \(-0.873092\pi\)
−0.921569 + 0.388214i \(0.873092\pi\)
\(318\) 0 0
\(319\) −1.92127 −0.107571
\(320\) 0 0
\(321\) 7.52704 0.420118
\(322\) 0 0
\(323\) 9.52797 0.530150
\(324\) 0 0
\(325\) −32.9786 −1.82933
\(326\) 0 0
\(327\) −7.82295 −0.432610
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6.36009 0.349582 0.174791 0.984606i \(-0.444075\pi\)
0.174791 + 0.984606i \(0.444075\pi\)
\(332\) 0 0
\(333\) −11.4629 −0.628161
\(334\) 0 0
\(335\) −2.45336 −0.134042
\(336\) 0 0
\(337\) −15.5449 −0.846784 −0.423392 0.905947i \(-0.639161\pi\)
−0.423392 + 0.905947i \(0.639161\pi\)
\(338\) 0 0
\(339\) 3.26857 0.177524
\(340\) 0 0
\(341\) −1.46791 −0.0794918
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −14.5996 −0.786014
\(346\) 0 0
\(347\) 19.7246 1.05887 0.529437 0.848350i \(-0.322403\pi\)
0.529437 + 0.848350i \(0.322403\pi\)
\(348\) 0 0
\(349\) 5.79385 0.310138 0.155069 0.987904i \(-0.450440\pi\)
0.155069 + 0.987904i \(0.450440\pi\)
\(350\) 0 0
\(351\) 16.0496 0.856666
\(352\) 0 0
\(353\) −33.1661 −1.76525 −0.882627 0.470073i \(-0.844227\pi\)
−0.882627 + 0.470073i \(0.844227\pi\)
\(354\) 0 0
\(355\) 33.4175 1.77361
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −4.20977 −0.222183 −0.111092 0.993810i \(-0.535435\pi\)
−0.111092 + 0.993810i \(0.535435\pi\)
\(360\) 0 0
\(361\) −15.7050 −0.826580
\(362\) 0 0
\(363\) 0.652704 0.0342581
\(364\) 0 0
\(365\) 8.29086 0.433963
\(366\) 0 0
\(367\) −19.4388 −1.01470 −0.507349 0.861741i \(-0.669374\pi\)
−0.507349 + 0.861741i \(0.669374\pi\)
\(368\) 0 0
\(369\) 0.728741 0.0379367
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 12.7142 0.658316 0.329158 0.944275i \(-0.393235\pi\)
0.329158 + 0.944275i \(0.393235\pi\)
\(374\) 0 0
\(375\) −5.70739 −0.294728
\(376\) 0 0
\(377\) −8.47565 −0.436518
\(378\) 0 0
\(379\) 22.8111 1.17173 0.585863 0.810410i \(-0.300755\pi\)
0.585863 + 0.810410i \(0.300755\pi\)
\(380\) 0 0
\(381\) 9.29086 0.475985
\(382\) 0 0
\(383\) 8.29860 0.424039 0.212019 0.977265i \(-0.431996\pi\)
0.212019 + 0.977265i \(0.431996\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −8.78106 −0.446366
\(388\) 0 0
\(389\) 11.3996 0.577983 0.288992 0.957332i \(-0.406680\pi\)
0.288992 + 0.957332i \(0.406680\pi\)
\(390\) 0 0
\(391\) 33.2404 1.68104
\(392\) 0 0
\(393\) 3.33780 0.168370
\(394\) 0 0
\(395\) −43.8658 −2.20713
\(396\) 0 0
\(397\) −6.58172 −0.330327 −0.165163 0.986266i \(-0.552815\pi\)
−0.165163 + 0.986266i \(0.552815\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 4.18984 0.209231 0.104615 0.994513i \(-0.466639\pi\)
0.104615 + 0.994513i \(0.466639\pi\)
\(402\) 0 0
\(403\) −6.47565 −0.322575
\(404\) 0 0
\(405\) −18.8871 −0.938509
\(406\) 0 0
\(407\) −4.45336 −0.220745
\(408\) 0 0
\(409\) −0.773318 −0.0382381 −0.0191191 0.999817i \(-0.506086\pi\)
−0.0191191 + 0.999817i \(0.506086\pi\)
\(410\) 0 0
\(411\) 12.8631 0.634489
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 40.0283 1.96491
\(416\) 0 0
\(417\) −3.45007 −0.168950
\(418\) 0 0
\(419\) 13.7547 0.671959 0.335979 0.941869i \(-0.390933\pi\)
0.335979 + 0.941869i \(0.390933\pi\)
\(420\) 0 0
\(421\) −5.80571 −0.282953 −0.141477 0.989942i \(-0.545185\pi\)
−0.141477 + 0.989942i \(0.545185\pi\)
\(422\) 0 0
\(423\) 11.7229 0.569985
\(424\) 0 0
\(425\) 39.2395 1.90339
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 2.87939 0.139018
\(430\) 0 0
\(431\) 18.0060 0.867318 0.433659 0.901077i \(-0.357222\pi\)
0.433659 + 0.901077i \(0.357222\pi\)
\(432\) 0 0
\(433\) −2.34049 −0.112477 −0.0562384 0.998417i \(-0.517911\pi\)
−0.0562384 + 0.998417i \(0.517911\pi\)
\(434\) 0 0
\(435\) −4.42932 −0.212370
\(436\) 0 0
\(437\) 11.4953 0.549892
\(438\) 0 0
\(439\) −40.0333 −1.91069 −0.955343 0.295498i \(-0.904514\pi\)
−0.955343 + 0.295498i \(0.904514\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −18.9760 −0.901575 −0.450787 0.892631i \(-0.648857\pi\)
−0.450787 + 0.892631i \(0.648857\pi\)
\(444\) 0 0
\(445\) −12.2490 −0.580657
\(446\) 0 0
\(447\) −11.1061 −0.525299
\(448\) 0 0
\(449\) −31.5371 −1.48833 −0.744165 0.667996i \(-0.767152\pi\)
−0.744165 + 0.667996i \(0.767152\pi\)
\(450\) 0 0
\(451\) 0.283119 0.0133315
\(452\) 0 0
\(453\) −0.850289 −0.0399501
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 11.6186 0.543493 0.271747 0.962369i \(-0.412399\pi\)
0.271747 + 0.962369i \(0.412399\pi\)
\(458\) 0 0
\(459\) −19.0966 −0.891352
\(460\) 0 0
\(461\) 12.9786 0.604476 0.302238 0.953233i \(-0.402266\pi\)
0.302238 + 0.953233i \(0.402266\pi\)
\(462\) 0 0
\(463\) −35.3705 −1.64381 −0.821904 0.569626i \(-0.807088\pi\)
−0.821904 + 0.569626i \(0.807088\pi\)
\(464\) 0 0
\(465\) −3.38413 −0.156935
\(466\) 0 0
\(467\) −39.8658 −1.84477 −0.922384 0.386274i \(-0.873762\pi\)
−0.922384 + 0.386274i \(0.873762\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −8.93851 −0.411865
\(472\) 0 0
\(473\) −3.41147 −0.156860
\(474\) 0 0
\(475\) 13.5699 0.622628
\(476\) 0 0
\(477\) 18.6212 0.852608
\(478\) 0 0
\(479\) −30.3800 −1.38810 −0.694049 0.719928i \(-0.744175\pi\)
−0.694049 + 0.719928i \(0.744175\pi\)
\(480\) 0 0
\(481\) −19.6459 −0.895776
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 54.2080 2.46146
\(486\) 0 0
\(487\) 10.4311 0.472677 0.236339 0.971671i \(-0.424053\pi\)
0.236339 + 0.971671i \(0.424053\pi\)
\(488\) 0 0
\(489\) −10.7074 −0.484205
\(490\) 0 0
\(491\) 10.2763 0.463763 0.231882 0.972744i \(-0.425512\pi\)
0.231882 + 0.972744i \(0.425512\pi\)
\(492\) 0 0
\(493\) 10.0847 0.454193
\(494\) 0 0
\(495\) −9.09152 −0.408633
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 36.5185 1.63479 0.817396 0.576077i \(-0.195417\pi\)
0.817396 + 0.576077i \(0.195417\pi\)
\(500\) 0 0
\(501\) −5.38238 −0.240467
\(502\) 0 0
\(503\) −31.3628 −1.39840 −0.699199 0.714928i \(-0.746460\pi\)
−0.699199 + 0.714928i \(0.746460\pi\)
\(504\) 0 0
\(505\) 23.6732 1.05345
\(506\) 0 0
\(507\) 4.21719 0.187292
\(508\) 0 0
\(509\) 11.5645 0.512587 0.256293 0.966599i \(-0.417499\pi\)
0.256293 + 0.966599i \(0.417499\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) −6.60401 −0.291574
\(514\) 0 0
\(515\) −7.94356 −0.350035
\(516\) 0 0
\(517\) 4.55438 0.200301
\(518\) 0 0
\(519\) −12.5544 −0.551076
\(520\) 0 0
\(521\) 30.5604 1.33887 0.669437 0.742869i \(-0.266535\pi\)
0.669437 + 0.742869i \(0.266535\pi\)
\(522\) 0 0
\(523\) −4.45512 −0.194809 −0.0974044 0.995245i \(-0.531054\pi\)
−0.0974044 + 0.995245i \(0.531054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.70502 0.335636
\(528\) 0 0
\(529\) 17.1037 0.743639
\(530\) 0 0
\(531\) −24.5553 −1.06561
\(532\) 0 0
\(533\) 1.24897 0.0540989
\(534\) 0 0
\(535\) −40.7324 −1.76101
\(536\) 0 0
\(537\) 0.866592 0.0373962
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −16.6705 −0.716723 −0.358361 0.933583i \(-0.616664\pi\)
−0.358361 + 0.933583i \(0.616664\pi\)
\(542\) 0 0
\(543\) −11.5220 −0.494456
\(544\) 0 0
\(545\) 42.3337 1.81338
\(546\) 0 0
\(547\) −36.2080 −1.54814 −0.774071 0.633098i \(-0.781783\pi\)
−0.774071 + 0.633098i \(0.781783\pi\)
\(548\) 0 0
\(549\) −2.95481 −0.126108
\(550\) 0 0
\(551\) 3.48751 0.148573
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −10.2668 −0.435802
\(556\) 0 0
\(557\) −0.669940 −0.0283863 −0.0141931 0.999899i \(-0.504518\pi\)
−0.0141931 + 0.999899i \(0.504518\pi\)
\(558\) 0 0
\(559\) −15.0496 −0.636532
\(560\) 0 0
\(561\) −3.42602 −0.144647
\(562\) 0 0
\(563\) −13.0341 −0.549324 −0.274662 0.961541i \(-0.588566\pi\)
−0.274662 + 0.961541i \(0.588566\pi\)
\(564\) 0 0
\(565\) −17.6878 −0.744131
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 27.4415 1.15041 0.575204 0.818010i \(-0.304923\pi\)
0.575204 + 0.818010i \(0.304923\pi\)
\(570\) 0 0
\(571\) −16.2395 −0.679601 −0.339800 0.940498i \(-0.610360\pi\)
−0.339800 + 0.940498i \(0.610360\pi\)
\(572\) 0 0
\(573\) −10.1513 −0.424075
\(574\) 0 0
\(575\) 47.3414 1.97427
\(576\) 0 0
\(577\) −23.7178 −0.987386 −0.493693 0.869636i \(-0.664353\pi\)
−0.493693 + 0.869636i \(0.664353\pi\)
\(578\) 0 0
\(579\) −3.40642 −0.141566
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 7.23442 0.299619
\(584\) 0 0
\(585\) −40.1070 −1.65822
\(586\) 0 0
\(587\) −44.5904 −1.84044 −0.920221 0.391399i \(-0.871991\pi\)
−0.920221 + 0.391399i \(0.871991\pi\)
\(588\) 0 0
\(589\) 2.66456 0.109791
\(590\) 0 0
\(591\) −3.01455 −0.124002
\(592\) 0 0
\(593\) 42.8435 1.75937 0.879685 0.475556i \(-0.157753\pi\)
0.879685 + 0.475556i \(0.157753\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.94862 −0.161606
\(598\) 0 0
\(599\) 0.633103 0.0258679 0.0129339 0.999916i \(-0.495883\pi\)
0.0129339 + 0.999916i \(0.495883\pi\)
\(600\) 0 0
\(601\) 3.45842 0.141072 0.0705359 0.997509i \(-0.477529\pi\)
0.0705359 + 0.997509i \(0.477529\pi\)
\(602\) 0 0
\(603\) −1.78787 −0.0728075
\(604\) 0 0
\(605\) −3.53209 −0.143600
\(606\) 0 0
\(607\) 48.2749 1.95942 0.979708 0.200428i \(-0.0642332\pi\)
0.979708 + 0.200428i \(0.0642332\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 20.0915 0.812816
\(612\) 0 0
\(613\) −21.3155 −0.860925 −0.430463 0.902608i \(-0.641650\pi\)
−0.430463 + 0.902608i \(0.641650\pi\)
\(614\) 0 0
\(615\) 0.652704 0.0263196
\(616\) 0 0
\(617\) −12.4243 −0.500182 −0.250091 0.968222i \(-0.580461\pi\)
−0.250091 + 0.968222i \(0.580461\pi\)
\(618\) 0 0
\(619\) 36.3405 1.46065 0.730324 0.683101i \(-0.239369\pi\)
0.730324 + 0.683101i \(0.239369\pi\)
\(620\) 0 0
\(621\) −23.0395 −0.924544
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −6.49289 −0.259716
\(626\) 0 0
\(627\) −1.18479 −0.0473161
\(628\) 0 0
\(629\) 23.3756 0.932045
\(630\) 0 0
\(631\) −15.9195 −0.633746 −0.316873 0.948468i \(-0.602633\pi\)
−0.316873 + 0.948468i \(0.602633\pi\)
\(632\) 0 0
\(633\) −5.67499 −0.225561
\(634\) 0 0
\(635\) −50.2772 −1.99519
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 24.3527 0.963377
\(640\) 0 0
\(641\) −12.4739 −0.492689 −0.246345 0.969182i \(-0.579230\pi\)
−0.246345 + 0.969182i \(0.579230\pi\)
\(642\) 0 0
\(643\) −36.8331 −1.45255 −0.726277 0.687402i \(-0.758751\pi\)
−0.726277 + 0.687402i \(0.758751\pi\)
\(644\) 0 0
\(645\) −7.86484 −0.309678
\(646\) 0 0
\(647\) 20.4587 0.804316 0.402158 0.915570i \(-0.368260\pi\)
0.402158 + 0.915570i \(0.368260\pi\)
\(648\) 0 0
\(649\) −9.53983 −0.374471
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −15.5594 −0.608888 −0.304444 0.952530i \(-0.598471\pi\)
−0.304444 + 0.952530i \(0.598471\pi\)
\(654\) 0 0
\(655\) −18.0624 −0.705757
\(656\) 0 0
\(657\) 6.04189 0.235717
\(658\) 0 0
\(659\) 23.6673 0.921945 0.460973 0.887414i \(-0.347501\pi\)
0.460973 + 0.887414i \(0.347501\pi\)
\(660\) 0 0
\(661\) −20.1976 −0.785595 −0.392798 0.919625i \(-0.628493\pi\)
−0.392798 + 0.919625i \(0.628493\pi\)
\(662\) 0 0
\(663\) −15.1138 −0.586972
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.1669 0.471106
\(668\) 0 0
\(669\) −0.144658 −0.00559281
\(670\) 0 0
\(671\) −1.14796 −0.0443163
\(672\) 0 0
\(673\) 7.92633 0.305537 0.152769 0.988262i \(-0.451181\pi\)
0.152769 + 0.988262i \(0.451181\pi\)
\(674\) 0 0
\(675\) −27.1976 −1.04684
\(676\) 0 0
\(677\) −36.0564 −1.38576 −0.692881 0.721052i \(-0.743659\pi\)
−0.692881 + 0.721052i \(0.743659\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −6.33099 −0.242604
\(682\) 0 0
\(683\) 12.3568 0.472819 0.236410 0.971653i \(-0.424029\pi\)
0.236410 + 0.971653i \(0.424029\pi\)
\(684\) 0 0
\(685\) −69.6082 −2.65959
\(686\) 0 0
\(687\) −15.4655 −0.590047
\(688\) 0 0
\(689\) 31.9145 1.21584
\(690\) 0 0
\(691\) 8.00000 0.304334 0.152167 0.988355i \(-0.451375\pi\)
0.152167 + 0.988355i \(0.451375\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 18.6699 0.708191
\(696\) 0 0
\(697\) −1.48608 −0.0562893
\(698\) 0 0
\(699\) 13.6331 0.515651
\(700\) 0 0
\(701\) 12.7611 0.481981 0.240991 0.970527i \(-0.422528\pi\)
0.240991 + 0.970527i \(0.422528\pi\)
\(702\) 0 0
\(703\) 8.08378 0.304885
\(704\) 0 0
\(705\) 10.4997 0.395441
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.70502 0.176701 0.0883504 0.996089i \(-0.471840\pi\)
0.0883504 + 0.996089i \(0.471840\pi\)
\(710\) 0 0
\(711\) −31.9668 −1.19885
\(712\) 0 0
\(713\) 9.29591 0.348135
\(714\) 0 0
\(715\) −15.5817 −0.582723
\(716\) 0 0
\(717\) −9.81016 −0.366367
\(718\) 0 0
\(719\) 52.6219 1.96246 0.981232 0.192831i \(-0.0617669\pi\)
0.981232 + 0.192831i \(0.0617669\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) −19.2808 −0.717059
\(724\) 0 0
\(725\) 14.3628 0.533420
\(726\) 0 0
\(727\) −22.3901 −0.830404 −0.415202 0.909729i \(-0.636289\pi\)
−0.415202 + 0.909729i \(0.636289\pi\)
\(728\) 0 0
\(729\) −6.63991 −0.245923
\(730\) 0 0
\(731\) 17.9067 0.662304
\(732\) 0 0
\(733\) 30.2158 1.11604 0.558022 0.829826i \(-0.311560\pi\)
0.558022 + 0.829826i \(0.311560\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −0.694593 −0.0255857
\(738\) 0 0
\(739\) −12.7716 −0.469810 −0.234905 0.972018i \(-0.575478\pi\)
−0.234905 + 0.972018i \(0.575478\pi\)
\(740\) 0 0
\(741\) −5.22668 −0.192007
\(742\) 0 0
\(743\) 37.4989 1.37570 0.687850 0.725853i \(-0.258555\pi\)
0.687850 + 0.725853i \(0.258555\pi\)
\(744\) 0 0
\(745\) 60.1002 2.20190
\(746\) 0 0
\(747\) 29.1702 1.06728
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −36.7847 −1.34229 −0.671146 0.741325i \(-0.734198\pi\)
−0.671146 + 0.741325i \(0.734198\pi\)
\(752\) 0 0
\(753\) 2.46110 0.0896876
\(754\) 0 0
\(755\) 4.60132 0.167459
\(756\) 0 0
\(757\) −2.27126 −0.0825503 −0.0412752 0.999148i \(-0.513142\pi\)
−0.0412752 + 0.999148i \(0.513142\pi\)
\(758\) 0 0
\(759\) −4.13341 −0.150033
\(760\) 0 0
\(761\) −10.1908 −0.369415 −0.184708 0.982793i \(-0.559134\pi\)
−0.184708 + 0.982793i \(0.559134\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 47.7211 1.72536
\(766\) 0 0
\(767\) −42.0847 −1.51959
\(768\) 0 0
\(769\) 27.7802 1.00178 0.500891 0.865511i \(-0.333006\pi\)
0.500891 + 0.865511i \(0.333006\pi\)
\(770\) 0 0
\(771\) 5.65270 0.203577
\(772\) 0 0
\(773\) 10.7820 0.387801 0.193901 0.981021i \(-0.437886\pi\)
0.193901 + 0.981021i \(0.437886\pi\)
\(774\) 0 0
\(775\) 10.9736 0.394183
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −0.513919 −0.0184131
\(780\) 0 0
\(781\) 9.46110 0.338545
\(782\) 0 0
\(783\) −6.98990 −0.249798
\(784\) 0 0
\(785\) 48.3705 1.72642
\(786\) 0 0
\(787\) −21.3473 −0.760949 −0.380474 0.924791i \(-0.624239\pi\)
−0.380474 + 0.924791i \(0.624239\pi\)
\(788\) 0 0
\(789\) 3.80840 0.135583
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −5.06418 −0.179834
\(794\) 0 0
\(795\) 16.6783 0.591518
\(796\) 0 0
\(797\) −39.3096 −1.39242 −0.696209 0.717839i \(-0.745132\pi\)
−0.696209 + 0.717839i \(0.745132\pi\)
\(798\) 0 0
\(799\) −23.9058 −0.845726
\(800\) 0 0
\(801\) −8.92633 −0.315396
\(802\) 0 0
\(803\) 2.34730 0.0828343
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 14.3473 0.505049
\(808\) 0 0
\(809\) 3.24661 0.114145 0.0570723 0.998370i \(-0.481823\pi\)
0.0570723 + 0.998370i \(0.481823\pi\)
\(810\) 0 0
\(811\) 33.4056 1.17303 0.586515 0.809939i \(-0.300500\pi\)
0.586515 + 0.809939i \(0.300500\pi\)
\(812\) 0 0
\(813\) 6.43047 0.225526
\(814\) 0 0
\(815\) 57.9427 2.02965
\(816\) 0 0
\(817\) 6.19253 0.216649
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −24.1976 −0.844502 −0.422251 0.906479i \(-0.638760\pi\)
−0.422251 + 0.906479i \(0.638760\pi\)
\(822\) 0 0
\(823\) 46.7151 1.62839 0.814193 0.580594i \(-0.197179\pi\)
0.814193 + 0.580594i \(0.197179\pi\)
\(824\) 0 0
\(825\) −4.87939 −0.169878
\(826\) 0 0
\(827\) −1.89992 −0.0660667 −0.0330333 0.999454i \(-0.510517\pi\)
−0.0330333 + 0.999454i \(0.510517\pi\)
\(828\) 0 0
\(829\) −50.9701 −1.77026 −0.885132 0.465340i \(-0.845932\pi\)
−0.885132 + 0.465340i \(0.845932\pi\)
\(830\) 0 0
\(831\) 15.4293 0.535237
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 29.1266 1.00797
\(836\) 0 0
\(837\) −5.34049 −0.184594
\(838\) 0 0
\(839\) 42.9368 1.48234 0.741171 0.671317i \(-0.234271\pi\)
0.741171 + 0.671317i \(0.234271\pi\)
\(840\) 0 0
\(841\) −25.3087 −0.872714
\(842\) 0 0
\(843\) 10.6203 0.365783
\(844\) 0 0
\(845\) −22.8212 −0.785073
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 11.0948 0.380773
\(850\) 0 0
\(851\) 28.2020 0.966753
\(852\) 0 0
\(853\) −52.3560 −1.79263 −0.896317 0.443414i \(-0.853767\pi\)
−0.896317 + 0.443414i \(0.853767\pi\)
\(854\) 0 0
\(855\) 16.5030 0.564390
\(856\) 0 0
\(857\) 0.978036 0.0334091 0.0167045 0.999860i \(-0.494683\pi\)
0.0167045 + 0.999860i \(0.494683\pi\)
\(858\) 0 0
\(859\) −34.0147 −1.16057 −0.580283 0.814415i \(-0.697058\pi\)
−0.580283 + 0.814415i \(0.697058\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 22.6955 0.772565 0.386282 0.922381i \(-0.373759\pi\)
0.386282 + 0.922381i \(0.373759\pi\)
\(864\) 0 0
\(865\) 67.9377 2.30995
\(866\) 0 0
\(867\) 6.88713 0.233899
\(868\) 0 0
\(869\) −12.4192 −0.421293
\(870\) 0 0
\(871\) −3.06418 −0.103826
\(872\) 0 0
\(873\) 39.5036 1.33699
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.22844 −0.0414813 −0.0207407 0.999785i \(-0.506602\pi\)
−0.0207407 + 0.999785i \(0.506602\pi\)
\(878\) 0 0
\(879\) 8.99319 0.303333
\(880\) 0 0
\(881\) −9.13516 −0.307771 −0.153886 0.988089i \(-0.549179\pi\)
−0.153886 + 0.988089i \(0.549179\pi\)
\(882\) 0 0
\(883\) −44.8256 −1.50850 −0.754251 0.656586i \(-0.772000\pi\)
−0.754251 + 0.656586i \(0.772000\pi\)
\(884\) 0 0
\(885\) −21.9932 −0.739293
\(886\) 0 0
\(887\) −23.0178 −0.772864 −0.386432 0.922318i \(-0.626293\pi\)
−0.386432 + 0.922318i \(0.626293\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −5.34730 −0.179141
\(892\) 0 0
\(893\) −8.26714 −0.276649
\(894\) 0 0
\(895\) −4.68954 −0.156754
\(896\) 0 0
\(897\) −18.2344 −0.608830
\(898\) 0 0
\(899\) 2.82026 0.0940609
\(900\) 0 0
\(901\) −37.9733 −1.26507
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 62.3509 2.07261
\(906\) 0 0
\(907\) 28.3209 0.940380 0.470190 0.882565i \(-0.344185\pi\)
0.470190 + 0.882565i \(0.344185\pi\)
\(908\) 0 0
\(909\) 17.2517 0.572201
\(910\) 0 0
\(911\) −2.97596 −0.0985978 −0.0492989 0.998784i \(-0.515699\pi\)
−0.0492989 + 0.998784i \(0.515699\pi\)
\(912\) 0 0
\(913\) 11.3327 0.375059
\(914\) 0 0
\(915\) −2.64651 −0.0874908
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −21.7939 −0.718913 −0.359456 0.933162i \(-0.617038\pi\)
−0.359456 + 0.933162i \(0.617038\pi\)
\(920\) 0 0
\(921\) −7.58172 −0.249826
\(922\) 0 0
\(923\) 41.7374 1.37380
\(924\) 0 0
\(925\) 33.2918 1.09463
\(926\) 0 0
\(927\) −5.78880 −0.190129
\(928\) 0 0
\(929\) −37.0651 −1.21607 −0.608033 0.793911i \(-0.708041\pi\)
−0.608033 + 0.793911i \(0.708041\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 2.25072 0.0736854
\(934\) 0 0
\(935\) 18.5398 0.606317
\(936\) 0 0
\(937\) −39.8340 −1.30132 −0.650660 0.759369i \(-0.725508\pi\)
−0.650660 + 0.759369i \(0.725508\pi\)
\(938\) 0 0
\(939\) 12.6767 0.413690
\(940\) 0 0
\(941\) −13.6919 −0.446343 −0.223172 0.974779i \(-0.571641\pi\)
−0.223172 + 0.974779i \(0.571641\pi\)
\(942\) 0 0
\(943\) −1.79292 −0.0583855
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 47.1334 1.53163 0.765815 0.643061i \(-0.222336\pi\)
0.765815 + 0.643061i \(0.222336\pi\)
\(948\) 0 0
\(949\) 10.3550 0.336139
\(950\) 0 0
\(951\) −21.4192 −0.694566
\(952\) 0 0
\(953\) −26.8990 −0.871344 −0.435672 0.900106i \(-0.643489\pi\)
−0.435672 + 0.900106i \(0.643489\pi\)
\(954\) 0 0
\(955\) 54.9332 1.77760
\(956\) 0 0
\(957\) −1.25402 −0.0405368
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −28.8452 −0.930492
\(962\) 0 0
\(963\) −29.6833 −0.956532
\(964\) 0 0
\(965\) 18.4338 0.593404
\(966\) 0 0
\(967\) 39.4270 1.26789 0.633943 0.773380i \(-0.281435\pi\)
0.633943 + 0.773380i \(0.281435\pi\)
\(968\) 0 0
\(969\) 6.21894 0.199781
\(970\) 0 0
\(971\) 16.0814 0.516077 0.258039 0.966135i \(-0.416924\pi\)
0.258039 + 0.966135i \(0.416924\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −21.5253 −0.689361
\(976\) 0 0
\(977\) 27.4279 0.877496 0.438748 0.898610i \(-0.355422\pi\)
0.438748 + 0.898610i \(0.355422\pi\)
\(978\) 0 0
\(979\) −3.46791 −0.110835
\(980\) 0 0
\(981\) 30.8503 0.984974
\(982\) 0 0
\(983\) −7.46616 −0.238133 −0.119067 0.992886i \(-0.537990\pi\)
−0.119067 + 0.992886i \(0.537990\pi\)
\(984\) 0 0
\(985\) 16.3131 0.519780
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 21.6040 0.686967
\(990\) 0 0
\(991\) 24.2704 0.770976 0.385488 0.922713i \(-0.374033\pi\)
0.385488 + 0.922713i \(0.374033\pi\)
\(992\) 0 0
\(993\) 4.15125 0.131736
\(994\) 0 0
\(995\) 21.3678 0.677406
\(996\) 0 0
\(997\) 33.1162 1.04880 0.524400 0.851472i \(-0.324290\pi\)
0.524400 + 0.851472i \(0.324290\pi\)
\(998\) 0 0
\(999\) −16.2020 −0.512610
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 8624.2.a.co.1.2 3
4.3 odd 2 539.2.a.g.1.3 3
7.3 odd 6 1232.2.q.m.177.2 6
7.5 odd 6 1232.2.q.m.529.2 6
7.6 odd 2 8624.2.a.ch.1.2 3
12.11 even 2 4851.2.a.bk.1.1 3
28.3 even 6 77.2.e.a.23.1 6
28.11 odd 6 539.2.e.m.177.1 6
28.19 even 6 77.2.e.a.67.1 yes 6
28.23 odd 6 539.2.e.m.67.1 6
28.27 even 2 539.2.a.j.1.3 3
44.43 even 2 5929.2.a.u.1.1 3
84.47 odd 6 693.2.i.h.298.3 6
84.59 odd 6 693.2.i.h.100.3 6
84.83 odd 2 4851.2.a.bj.1.1 3
308.3 even 30 847.2.n.g.9.3 24
308.19 odd 30 847.2.n.f.130.3 24
308.31 even 30 847.2.n.g.807.3 24
308.47 even 30 847.2.n.g.130.1 24
308.59 even 30 847.2.n.g.632.1 24
308.75 even 30 847.2.n.g.81.1 24
308.87 odd 6 847.2.e.c.485.3 6
308.103 even 30 847.2.n.g.753.3 24
308.115 even 30 847.2.n.g.366.1 24
308.131 odd 6 847.2.e.c.606.3 6
308.159 even 30 847.2.n.g.487.3 24
308.171 odd 30 847.2.n.f.366.3 24
308.215 odd 30 847.2.n.f.487.1 24
308.227 odd 30 847.2.n.f.632.3 24
308.255 odd 30 847.2.n.f.807.1 24
308.271 odd 30 847.2.n.f.753.1 24
308.283 odd 30 847.2.n.f.9.1 24
308.299 odd 30 847.2.n.f.81.3 24
308.307 odd 2 5929.2.a.x.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
77.2.e.a.23.1 6 28.3 even 6
77.2.e.a.67.1 yes 6 28.19 even 6
539.2.a.g.1.3 3 4.3 odd 2
539.2.a.j.1.3 3 28.27 even 2
539.2.e.m.67.1 6 28.23 odd 6
539.2.e.m.177.1 6 28.11 odd 6
693.2.i.h.100.3 6 84.59 odd 6
693.2.i.h.298.3 6 84.47 odd 6
847.2.e.c.485.3 6 308.87 odd 6
847.2.e.c.606.3 6 308.131 odd 6
847.2.n.f.9.1 24 308.283 odd 30
847.2.n.f.81.3 24 308.299 odd 30
847.2.n.f.130.3 24 308.19 odd 30
847.2.n.f.366.3 24 308.171 odd 30
847.2.n.f.487.1 24 308.215 odd 30
847.2.n.f.632.3 24 308.227 odd 30
847.2.n.f.753.1 24 308.271 odd 30
847.2.n.f.807.1 24 308.255 odd 30
847.2.n.g.9.3 24 308.3 even 30
847.2.n.g.81.1 24 308.75 even 30
847.2.n.g.130.1 24 308.47 even 30
847.2.n.g.366.1 24 308.115 even 30
847.2.n.g.487.3 24 308.159 even 30
847.2.n.g.632.1 24 308.59 even 30
847.2.n.g.753.3 24 308.103 even 30
847.2.n.g.807.3 24 308.31 even 30
1232.2.q.m.177.2 6 7.3 odd 6
1232.2.q.m.529.2 6 7.5 odd 6
4851.2.a.bj.1.1 3 84.83 odd 2
4851.2.a.bk.1.1 3 12.11 even 2
5929.2.a.u.1.1 3 44.43 even 2
5929.2.a.x.1.1 3 308.307 odd 2
8624.2.a.ch.1.2 3 7.6 odd 2
8624.2.a.co.1.2 3 1.1 even 1 trivial