Properties

Label 8624.2.a.bi.1.1
Level $8624$
Weight $2$
Character 8624.1
Self dual yes
Analytic conductor $68.863$
Analytic rank $0$
Dimension $2$
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: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{17}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 4 \) 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 616)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.56155\) 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)
 
\(f(q)\) \(=\) \(q-2.56155 q^{3} -0.561553 q^{5} +3.56155 q^{9} +O(q^{10})\) \(q-2.56155 q^{3} -0.561553 q^{5} +3.56155 q^{9} -1.00000 q^{11} -3.12311 q^{13} +1.43845 q^{15} +2.00000 q^{17} -5.12311 q^{19} +1.43845 q^{23} -4.68466 q^{25} -1.43845 q^{27} -2.00000 q^{29} -10.5616 q^{31} +2.56155 q^{33} +4.56155 q^{37} +8.00000 q^{39} +10.0000 q^{41} -4.00000 q^{43} -2.00000 q^{45} -6.24621 q^{47} -5.12311 q^{51} -4.24621 q^{53} +0.561553 q^{55} +13.1231 q^{57} -5.43845 q^{59} +4.24621 q^{61} +1.75379 q^{65} +2.56155 q^{67} -3.68466 q^{69} -3.68466 q^{71} +4.24621 q^{73} +12.0000 q^{75} -5.12311 q^{79} -7.00000 q^{81} -8.00000 q^{83} -1.12311 q^{85} +5.12311 q^{87} +12.5616 q^{89} +27.0540 q^{93} +2.87689 q^{95} -8.56155 q^{97} -3.56155 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - q^{3} + 3 q^{5} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - q^{3} + 3 q^{5} + 3 q^{9} - 2 q^{11} + 2 q^{13} + 7 q^{15} + 4 q^{17} - 2 q^{19} + 7 q^{23} + 3 q^{25} - 7 q^{27} - 4 q^{29} - 17 q^{31} + q^{33} + 5 q^{37} + 16 q^{39} + 20 q^{41} - 8 q^{43} - 4 q^{45} + 4 q^{47} - 2 q^{51} + 8 q^{53} - 3 q^{55} + 18 q^{57} - 15 q^{59} - 8 q^{61} + 20 q^{65} + q^{67} + 5 q^{69} + 5 q^{71} - 8 q^{73} + 24 q^{75} - 2 q^{79} - 14 q^{81} - 16 q^{83} + 6 q^{85} + 2 q^{87} + 21 q^{89} + 17 q^{93} + 14 q^{95} - 13 q^{97} - 3 q^{99}+O(q^{100}) \) 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
\(3\) −2.56155 −1.47891 −0.739457 0.673204i \(-0.764917\pi\)
−0.739457 + 0.673204i \(0.764917\pi\)
\(4\) 0 0
\(5\) −0.561553 −0.251134 −0.125567 0.992085i \(-0.540075\pi\)
−0.125567 + 0.992085i \(0.540075\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 3.56155 1.18718
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −3.12311 −0.866194 −0.433097 0.901347i \(-0.642579\pi\)
−0.433097 + 0.901347i \(0.642579\pi\)
\(14\) 0 0
\(15\) 1.43845 0.371405
\(16\) 0 0
\(17\) 2.00000 0.485071 0.242536 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) 0 0
\(19\) −5.12311 −1.17532 −0.587661 0.809108i \(-0.699951\pi\)
−0.587661 + 0.809108i \(0.699951\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 1.43845 0.299937 0.149968 0.988691i \(-0.452083\pi\)
0.149968 + 0.988691i \(0.452083\pi\)
\(24\) 0 0
\(25\) −4.68466 −0.936932
\(26\) 0 0
\(27\) −1.43845 −0.276829
\(28\) 0 0
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −10.5616 −1.89691 −0.948455 0.316911i \(-0.897355\pi\)
−0.948455 + 0.316911i \(0.897355\pi\)
\(32\) 0 0
\(33\) 2.56155 0.445909
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.56155 0.749915 0.374957 0.927042i \(-0.377657\pi\)
0.374957 + 0.927042i \(0.377657\pi\)
\(38\) 0 0
\(39\) 8.00000 1.28103
\(40\) 0 0
\(41\) 10.0000 1.56174 0.780869 0.624695i \(-0.214777\pi\)
0.780869 + 0.624695i \(0.214777\pi\)
\(42\) 0 0
\(43\) −4.00000 −0.609994 −0.304997 0.952353i \(-0.598656\pi\)
−0.304997 + 0.952353i \(0.598656\pi\)
\(44\) 0 0
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) −6.24621 −0.911104 −0.455552 0.890209i \(-0.650558\pi\)
−0.455552 + 0.890209i \(0.650558\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −5.12311 −0.717378
\(52\) 0 0
\(53\) −4.24621 −0.583262 −0.291631 0.956531i \(-0.594198\pi\)
−0.291631 + 0.956531i \(0.594198\pi\)
\(54\) 0 0
\(55\) 0.561553 0.0757198
\(56\) 0 0
\(57\) 13.1231 1.73820
\(58\) 0 0
\(59\) −5.43845 −0.708026 −0.354013 0.935241i \(-0.615183\pi\)
−0.354013 + 0.935241i \(0.615183\pi\)
\(60\) 0 0
\(61\) 4.24621 0.543672 0.271836 0.962344i \(-0.412369\pi\)
0.271836 + 0.962344i \(0.412369\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.75379 0.217531
\(66\) 0 0
\(67\) 2.56155 0.312943 0.156472 0.987682i \(-0.449988\pi\)
0.156472 + 0.987682i \(0.449988\pi\)
\(68\) 0 0
\(69\) −3.68466 −0.443581
\(70\) 0 0
\(71\) −3.68466 −0.437289 −0.218644 0.975805i \(-0.570163\pi\)
−0.218644 + 0.975805i \(0.570163\pi\)
\(72\) 0 0
\(73\) 4.24621 0.496981 0.248491 0.968634i \(-0.420065\pi\)
0.248491 + 0.968634i \(0.420065\pi\)
\(74\) 0 0
\(75\) 12.0000 1.38564
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −5.12311 −0.576394 −0.288197 0.957571i \(-0.593056\pi\)
−0.288197 + 0.957571i \(0.593056\pi\)
\(80\) 0 0
\(81\) −7.00000 −0.777778
\(82\) 0 0
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) −1.12311 −0.121818
\(86\) 0 0
\(87\) 5.12311 0.549255
\(88\) 0 0
\(89\) 12.5616 1.33152 0.665761 0.746165i \(-0.268107\pi\)
0.665761 + 0.746165i \(0.268107\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 27.0540 2.80537
\(94\) 0 0
\(95\) 2.87689 0.295163
\(96\) 0 0
\(97\) −8.56155 −0.869294 −0.434647 0.900601i \(-0.643127\pi\)
−0.434647 + 0.900601i \(0.643127\pi\)
\(98\) 0 0
\(99\) −3.56155 −0.357950
\(100\) 0 0
\(101\) −13.3693 −1.33030 −0.665148 0.746711i \(-0.731632\pi\)
−0.665148 + 0.746711i \(0.731632\pi\)
\(102\) 0 0
\(103\) 16.4924 1.62505 0.812523 0.582929i \(-0.198093\pi\)
0.812523 + 0.582929i \(0.198093\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −17.1231 −1.65535 −0.827677 0.561205i \(-0.810338\pi\)
−0.827677 + 0.561205i \(0.810338\pi\)
\(108\) 0 0
\(109\) −9.36932 −0.897418 −0.448709 0.893678i \(-0.648116\pi\)
−0.448709 + 0.893678i \(0.648116\pi\)
\(110\) 0 0
\(111\) −11.6847 −1.10906
\(112\) 0 0
\(113\) 5.68466 0.534768 0.267384 0.963590i \(-0.413841\pi\)
0.267384 + 0.963590i \(0.413841\pi\)
\(114\) 0 0
\(115\) −0.807764 −0.0753244
\(116\) 0 0
\(117\) −11.1231 −1.02833
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −25.6155 −2.30967
\(124\) 0 0
\(125\) 5.43845 0.486430
\(126\) 0 0
\(127\) 5.12311 0.454602 0.227301 0.973825i \(-0.427010\pi\)
0.227301 + 0.973825i \(0.427010\pi\)
\(128\) 0 0
\(129\) 10.2462 0.902129
\(130\) 0 0
\(131\) 13.1231 1.14657 0.573286 0.819356i \(-0.305669\pi\)
0.573286 + 0.819356i \(0.305669\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0.807764 0.0695213
\(136\) 0 0
\(137\) −2.31534 −0.197813 −0.0989065 0.995097i \(-0.531534\pi\)
−0.0989065 + 0.995097i \(0.531534\pi\)
\(138\) 0 0
\(139\) −5.12311 −0.434536 −0.217268 0.976112i \(-0.569715\pi\)
−0.217268 + 0.976112i \(0.569715\pi\)
\(140\) 0 0
\(141\) 16.0000 1.34744
\(142\) 0 0
\(143\) 3.12311 0.261167
\(144\) 0 0
\(145\) 1.12311 0.0932688
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.0000 −0.819232 −0.409616 0.912258i \(-0.634337\pi\)
−0.409616 + 0.912258i \(0.634337\pi\)
\(150\) 0 0
\(151\) −5.12311 −0.416912 −0.208456 0.978032i \(-0.566844\pi\)
−0.208456 + 0.978032i \(0.566844\pi\)
\(152\) 0 0
\(153\) 7.12311 0.575869
\(154\) 0 0
\(155\) 5.93087 0.476379
\(156\) 0 0
\(157\) −16.5616 −1.32176 −0.660878 0.750493i \(-0.729816\pi\)
−0.660878 + 0.750493i \(0.729816\pi\)
\(158\) 0 0
\(159\) 10.8769 0.862594
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −16.4924 −1.29179 −0.645893 0.763428i \(-0.723515\pi\)
−0.645893 + 0.763428i \(0.723515\pi\)
\(164\) 0 0
\(165\) −1.43845 −0.111983
\(166\) 0 0
\(167\) −7.36932 −0.570255 −0.285127 0.958490i \(-0.592036\pi\)
−0.285127 + 0.958490i \(0.592036\pi\)
\(168\) 0 0
\(169\) −3.24621 −0.249709
\(170\) 0 0
\(171\) −18.2462 −1.39532
\(172\) 0 0
\(173\) 23.1231 1.75802 0.879009 0.476806i \(-0.158206\pi\)
0.879009 + 0.476806i \(0.158206\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 13.9309 1.04711
\(178\) 0 0
\(179\) 20.8078 1.55525 0.777623 0.628731i \(-0.216425\pi\)
0.777623 + 0.628731i \(0.216425\pi\)
\(180\) 0 0
\(181\) −7.93087 −0.589497 −0.294748 0.955575i \(-0.595236\pi\)
−0.294748 + 0.955575i \(0.595236\pi\)
\(182\) 0 0
\(183\) −10.8769 −0.804043
\(184\) 0 0
\(185\) −2.56155 −0.188329
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 27.0540 1.95756 0.978778 0.204921i \(-0.0656938\pi\)
0.978778 + 0.204921i \(0.0656938\pi\)
\(192\) 0 0
\(193\) 2.00000 0.143963 0.0719816 0.997406i \(-0.477068\pi\)
0.0719816 + 0.997406i \(0.477068\pi\)
\(194\) 0 0
\(195\) −4.49242 −0.321709
\(196\) 0 0
\(197\) −20.2462 −1.44248 −0.721241 0.692684i \(-0.756428\pi\)
−0.721241 + 0.692684i \(0.756428\pi\)
\(198\) 0 0
\(199\) −14.2462 −1.00989 −0.504944 0.863152i \(-0.668487\pi\)
−0.504944 + 0.863152i \(0.668487\pi\)
\(200\) 0 0
\(201\) −6.56155 −0.462816
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −5.61553 −0.392205
\(206\) 0 0
\(207\) 5.12311 0.356080
\(208\) 0 0
\(209\) 5.12311 0.354373
\(210\) 0 0
\(211\) −25.1231 −1.72955 −0.864773 0.502163i \(-0.832538\pi\)
−0.864773 + 0.502163i \(0.832538\pi\)
\(212\) 0 0
\(213\) 9.43845 0.646712
\(214\) 0 0
\(215\) 2.24621 0.153190
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) −10.8769 −0.734992
\(220\) 0 0
\(221\) −6.24621 −0.420166
\(222\) 0 0
\(223\) −26.5616 −1.77869 −0.889347 0.457234i \(-0.848840\pi\)
−0.889347 + 0.457234i \(0.848840\pi\)
\(224\) 0 0
\(225\) −16.6847 −1.11231
\(226\) 0 0
\(227\) −10.2462 −0.680065 −0.340032 0.940414i \(-0.610438\pi\)
−0.340032 + 0.940414i \(0.610438\pi\)
\(228\) 0 0
\(229\) 2.31534 0.153002 0.0765010 0.997070i \(-0.475625\pi\)
0.0765010 + 0.997070i \(0.475625\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2.49242 −0.163284 −0.0816420 0.996662i \(-0.526016\pi\)
−0.0816420 + 0.996662i \(0.526016\pi\)
\(234\) 0 0
\(235\) 3.50758 0.228809
\(236\) 0 0
\(237\) 13.1231 0.852437
\(238\) 0 0
\(239\) 8.00000 0.517477 0.258738 0.965947i \(-0.416693\pi\)
0.258738 + 0.965947i \(0.416693\pi\)
\(240\) 0 0
\(241\) −5.36932 −0.345868 −0.172934 0.984933i \(-0.555325\pi\)
−0.172934 + 0.984933i \(0.555325\pi\)
\(242\) 0 0
\(243\) 22.2462 1.42710
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 16.0000 1.01806
\(248\) 0 0
\(249\) 20.4924 1.29865
\(250\) 0 0
\(251\) 15.0540 0.950198 0.475099 0.879932i \(-0.342412\pi\)
0.475099 + 0.879932i \(0.342412\pi\)
\(252\) 0 0
\(253\) −1.43845 −0.0904344
\(254\) 0 0
\(255\) 2.87689 0.180158
\(256\) 0 0
\(257\) 14.0000 0.873296 0.436648 0.899632i \(-0.356166\pi\)
0.436648 + 0.899632i \(0.356166\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −7.12311 −0.440909
\(262\) 0 0
\(263\) −9.61553 −0.592919 −0.296459 0.955045i \(-0.595806\pi\)
−0.296459 + 0.955045i \(0.595806\pi\)
\(264\) 0 0
\(265\) 2.38447 0.146477
\(266\) 0 0
\(267\) −32.1771 −1.96921
\(268\) 0 0
\(269\) −7.75379 −0.472757 −0.236378 0.971661i \(-0.575960\pi\)
−0.236378 + 0.971661i \(0.575960\pi\)
\(270\) 0 0
\(271\) 18.2462 1.10838 0.554189 0.832391i \(-0.313028\pi\)
0.554189 + 0.832391i \(0.313028\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 4.68466 0.282496
\(276\) 0 0
\(277\) 8.87689 0.533361 0.266680 0.963785i \(-0.414073\pi\)
0.266680 + 0.963785i \(0.414073\pi\)
\(278\) 0 0
\(279\) −37.6155 −2.25198
\(280\) 0 0
\(281\) 29.8617 1.78140 0.890701 0.454590i \(-0.150214\pi\)
0.890701 + 0.454590i \(0.150214\pi\)
\(282\) 0 0
\(283\) 28.4924 1.69370 0.846849 0.531833i \(-0.178497\pi\)
0.846849 + 0.531833i \(0.178497\pi\)
\(284\) 0 0
\(285\) −7.36932 −0.436521
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 21.9309 1.28561
\(292\) 0 0
\(293\) 12.2462 0.715431 0.357716 0.933831i \(-0.383556\pi\)
0.357716 + 0.933831i \(0.383556\pi\)
\(294\) 0 0
\(295\) 3.05398 0.177809
\(296\) 0 0
\(297\) 1.43845 0.0834672
\(298\) 0 0
\(299\) −4.49242 −0.259804
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 34.2462 1.96739
\(304\) 0 0
\(305\) −2.38447 −0.136534
\(306\) 0 0
\(307\) 20.4924 1.16956 0.584782 0.811190i \(-0.301180\pi\)
0.584782 + 0.811190i \(0.301180\pi\)
\(308\) 0 0
\(309\) −42.2462 −2.40330
\(310\) 0 0
\(311\) 18.7386 1.06257 0.531285 0.847193i \(-0.321709\pi\)
0.531285 + 0.847193i \(0.321709\pi\)
\(312\) 0 0
\(313\) −15.9309 −0.900466 −0.450233 0.892911i \(-0.648659\pi\)
−0.450233 + 0.892911i \(0.648659\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −13.0540 −0.733184 −0.366592 0.930382i \(-0.619476\pi\)
−0.366592 + 0.930382i \(0.619476\pi\)
\(318\) 0 0
\(319\) 2.00000 0.111979
\(320\) 0 0
\(321\) 43.8617 2.44812
\(322\) 0 0
\(323\) −10.2462 −0.570114
\(324\) 0 0
\(325\) 14.6307 0.811564
\(326\) 0 0
\(327\) 24.0000 1.32720
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −12.8078 −0.703978 −0.351989 0.936004i \(-0.614495\pi\)
−0.351989 + 0.936004i \(0.614495\pi\)
\(332\) 0 0
\(333\) 16.2462 0.890287
\(334\) 0 0
\(335\) −1.43845 −0.0785908
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) 0 0
\(339\) −14.5616 −0.790875
\(340\) 0 0
\(341\) 10.5616 0.571940
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 2.06913 0.111398
\(346\) 0 0
\(347\) −6.87689 −0.369171 −0.184586 0.982816i \(-0.559094\pi\)
−0.184586 + 0.982816i \(0.559094\pi\)
\(348\) 0 0
\(349\) 12.2462 0.655525 0.327762 0.944760i \(-0.393705\pi\)
0.327762 + 0.944760i \(0.393705\pi\)
\(350\) 0 0
\(351\) 4.49242 0.239788
\(352\) 0 0
\(353\) 20.5616 1.09438 0.547191 0.837008i \(-0.315697\pi\)
0.547191 + 0.837008i \(0.315697\pi\)
\(354\) 0 0
\(355\) 2.06913 0.109818
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.24621 −0.118550 −0.0592752 0.998242i \(-0.518879\pi\)
−0.0592752 + 0.998242i \(0.518879\pi\)
\(360\) 0 0
\(361\) 7.24621 0.381380
\(362\) 0 0
\(363\) −2.56155 −0.134447
\(364\) 0 0
\(365\) −2.38447 −0.124809
\(366\) 0 0
\(367\) −0.315342 −0.0164607 −0.00823035 0.999966i \(-0.502620\pi\)
−0.00823035 + 0.999966i \(0.502620\pi\)
\(368\) 0 0
\(369\) 35.6155 1.85407
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −11.6155 −0.601429 −0.300715 0.953714i \(-0.597225\pi\)
−0.300715 + 0.953714i \(0.597225\pi\)
\(374\) 0 0
\(375\) −13.9309 −0.719387
\(376\) 0 0
\(377\) 6.24621 0.321696
\(378\) 0 0
\(379\) 25.9309 1.33198 0.665990 0.745961i \(-0.268009\pi\)
0.665990 + 0.745961i \(0.268009\pi\)
\(380\) 0 0
\(381\) −13.1231 −0.672317
\(382\) 0 0
\(383\) 9.93087 0.507444 0.253722 0.967277i \(-0.418345\pi\)
0.253722 + 0.967277i \(0.418345\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −14.2462 −0.724176
\(388\) 0 0
\(389\) −29.0540 −1.47310 −0.736548 0.676386i \(-0.763545\pi\)
−0.736548 + 0.676386i \(0.763545\pi\)
\(390\) 0 0
\(391\) 2.87689 0.145491
\(392\) 0 0
\(393\) −33.6155 −1.69568
\(394\) 0 0
\(395\) 2.87689 0.144752
\(396\) 0 0
\(397\) 10.4924 0.526600 0.263300 0.964714i \(-0.415189\pi\)
0.263300 + 0.964714i \(0.415189\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 2.00000 0.0998752 0.0499376 0.998752i \(-0.484098\pi\)
0.0499376 + 0.998752i \(0.484098\pi\)
\(402\) 0 0
\(403\) 32.9848 1.64309
\(404\) 0 0
\(405\) 3.93087 0.195326
\(406\) 0 0
\(407\) −4.56155 −0.226108
\(408\) 0 0
\(409\) −34.4924 −1.70554 −0.852770 0.522286i \(-0.825079\pi\)
−0.852770 + 0.522286i \(0.825079\pi\)
\(410\) 0 0
\(411\) 5.93087 0.292548
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 4.49242 0.220524
\(416\) 0 0
\(417\) 13.1231 0.642641
\(418\) 0 0
\(419\) 18.7386 0.915442 0.457721 0.889096i \(-0.348666\pi\)
0.457721 + 0.889096i \(0.348666\pi\)
\(420\) 0 0
\(421\) 30.9848 1.51011 0.755054 0.655662i \(-0.227610\pi\)
0.755054 + 0.655662i \(0.227610\pi\)
\(422\) 0 0
\(423\) −22.2462 −1.08165
\(424\) 0 0
\(425\) −9.36932 −0.454479
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) −8.00000 −0.386244
\(430\) 0 0
\(431\) 14.7386 0.709935 0.354968 0.934879i \(-0.384492\pi\)
0.354968 + 0.934879i \(0.384492\pi\)
\(432\) 0 0
\(433\) 1.68466 0.0809595 0.0404798 0.999180i \(-0.487111\pi\)
0.0404798 + 0.999180i \(0.487111\pi\)
\(434\) 0 0
\(435\) −2.87689 −0.137937
\(436\) 0 0
\(437\) −7.36932 −0.352522
\(438\) 0 0
\(439\) 17.6155 0.840743 0.420372 0.907352i \(-0.361900\pi\)
0.420372 + 0.907352i \(0.361900\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −12.1771 −0.578551 −0.289275 0.957246i \(-0.593414\pi\)
−0.289275 + 0.957246i \(0.593414\pi\)
\(444\) 0 0
\(445\) −7.05398 −0.334390
\(446\) 0 0
\(447\) 25.6155 1.21157
\(448\) 0 0
\(449\) −17.6847 −0.834591 −0.417295 0.908771i \(-0.637022\pi\)
−0.417295 + 0.908771i \(0.637022\pi\)
\(450\) 0 0
\(451\) −10.0000 −0.470882
\(452\) 0 0
\(453\) 13.1231 0.616577
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 37.8617 1.77110 0.885549 0.464546i \(-0.153783\pi\)
0.885549 + 0.464546i \(0.153783\pi\)
\(458\) 0 0
\(459\) −2.87689 −0.134282
\(460\) 0 0
\(461\) −14.0000 −0.652045 −0.326023 0.945362i \(-0.605709\pi\)
−0.326023 + 0.945362i \(0.605709\pi\)
\(462\) 0 0
\(463\) 39.5464 1.83788 0.918938 0.394401i \(-0.129048\pi\)
0.918938 + 0.394401i \(0.129048\pi\)
\(464\) 0 0
\(465\) −15.1922 −0.704523
\(466\) 0 0
\(467\) −30.4233 −1.40782 −0.703911 0.710288i \(-0.748565\pi\)
−0.703911 + 0.710288i \(0.748565\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 42.4233 1.95476
\(472\) 0 0
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 24.0000 1.10120
\(476\) 0 0
\(477\) −15.1231 −0.692439
\(478\) 0 0
\(479\) 20.4924 0.936323 0.468161 0.883643i \(-0.344917\pi\)
0.468161 + 0.883643i \(0.344917\pi\)
\(480\) 0 0
\(481\) −14.2462 −0.649571
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 4.80776 0.218309
\(486\) 0 0
\(487\) 17.4384 0.790211 0.395106 0.918636i \(-0.370708\pi\)
0.395106 + 0.918636i \(0.370708\pi\)
\(488\) 0 0
\(489\) 42.2462 1.91044
\(490\) 0 0
\(491\) 17.1231 0.772755 0.386377 0.922341i \(-0.373726\pi\)
0.386377 + 0.922341i \(0.373726\pi\)
\(492\) 0 0
\(493\) −4.00000 −0.180151
\(494\) 0 0
\(495\) 2.00000 0.0898933
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 32.4924 1.45456 0.727280 0.686341i \(-0.240784\pi\)
0.727280 + 0.686341i \(0.240784\pi\)
\(500\) 0 0
\(501\) 18.8769 0.843357
\(502\) 0 0
\(503\) 14.7386 0.657163 0.328582 0.944476i \(-0.393429\pi\)
0.328582 + 0.944476i \(0.393429\pi\)
\(504\) 0 0
\(505\) 7.50758 0.334083
\(506\) 0 0
\(507\) 8.31534 0.369297
\(508\) 0 0
\(509\) 14.8078 0.656343 0.328171 0.944618i \(-0.393568\pi\)
0.328171 + 0.944618i \(0.393568\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 7.36932 0.325363
\(514\) 0 0
\(515\) −9.26137 −0.408105
\(516\) 0 0
\(517\) 6.24621 0.274708
\(518\) 0 0
\(519\) −59.2311 −2.59995
\(520\) 0 0
\(521\) −22.3153 −0.977653 −0.488826 0.872381i \(-0.662575\pi\)
−0.488826 + 0.872381i \(0.662575\pi\)
\(522\) 0 0
\(523\) 2.24621 0.0982200 0.0491100 0.998793i \(-0.484362\pi\)
0.0491100 + 0.998793i \(0.484362\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −21.1231 −0.920137
\(528\) 0 0
\(529\) −20.9309 −0.910038
\(530\) 0 0
\(531\) −19.3693 −0.840557
\(532\) 0 0
\(533\) −31.2311 −1.35277
\(534\) 0 0
\(535\) 9.61553 0.415716
\(536\) 0 0
\(537\) −53.3002 −2.30007
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −15.1231 −0.650193 −0.325097 0.945681i \(-0.605397\pi\)
−0.325097 + 0.945681i \(0.605397\pi\)
\(542\) 0 0
\(543\) 20.3153 0.871815
\(544\) 0 0
\(545\) 5.26137 0.225372
\(546\) 0 0
\(547\) −34.7386 −1.48532 −0.742658 0.669670i \(-0.766435\pi\)
−0.742658 + 0.669670i \(0.766435\pi\)
\(548\) 0 0
\(549\) 15.1231 0.645438
\(550\) 0 0
\(551\) 10.2462 0.436503
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 6.56155 0.278522
\(556\) 0 0
\(557\) 40.2462 1.70529 0.852643 0.522493i \(-0.174998\pi\)
0.852643 + 0.522493i \(0.174998\pi\)
\(558\) 0 0
\(559\) 12.4924 0.528373
\(560\) 0 0
\(561\) 5.12311 0.216298
\(562\) 0 0
\(563\) −30.7386 −1.29548 −0.647739 0.761862i \(-0.724285\pi\)
−0.647739 + 0.761862i \(0.724285\pi\)
\(564\) 0 0
\(565\) −3.19224 −0.134298
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 34.0000 1.42535 0.712677 0.701492i \(-0.247483\pi\)
0.712677 + 0.701492i \(0.247483\pi\)
\(570\) 0 0
\(571\) 40.4924 1.69456 0.847278 0.531150i \(-0.178240\pi\)
0.847278 + 0.531150i \(0.178240\pi\)
\(572\) 0 0
\(573\) −69.3002 −2.89506
\(574\) 0 0
\(575\) −6.73863 −0.281020
\(576\) 0 0
\(577\) −6.31534 −0.262911 −0.131456 0.991322i \(-0.541965\pi\)
−0.131456 + 0.991322i \(0.541965\pi\)
\(578\) 0 0
\(579\) −5.12311 −0.212909
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 4.24621 0.175860
\(584\) 0 0
\(585\) 6.24621 0.258249
\(586\) 0 0
\(587\) 4.00000 0.165098 0.0825488 0.996587i \(-0.473694\pi\)
0.0825488 + 0.996587i \(0.473694\pi\)
\(588\) 0 0
\(589\) 54.1080 2.22948
\(590\) 0 0
\(591\) 51.8617 2.13331
\(592\) 0 0
\(593\) −8.24621 −0.338631 −0.169316 0.985562i \(-0.554156\pi\)
−0.169316 + 0.985562i \(0.554156\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 36.4924 1.49354
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −43.1231 −1.75903 −0.879514 0.475873i \(-0.842132\pi\)
−0.879514 + 0.475873i \(0.842132\pi\)
\(602\) 0 0
\(603\) 9.12311 0.371522
\(604\) 0 0
\(605\) −0.561553 −0.0228304
\(606\) 0 0
\(607\) −48.3542 −1.96263 −0.981317 0.192396i \(-0.938374\pi\)
−0.981317 + 0.192396i \(0.938374\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 19.5076 0.789192
\(612\) 0 0
\(613\) 23.6155 0.953822 0.476911 0.878952i \(-0.341756\pi\)
0.476911 + 0.878952i \(0.341756\pi\)
\(614\) 0 0
\(615\) 14.3845 0.580038
\(616\) 0 0
\(617\) −26.4924 −1.06654 −0.533272 0.845944i \(-0.679038\pi\)
−0.533272 + 0.845944i \(0.679038\pi\)
\(618\) 0 0
\(619\) 27.1922 1.09295 0.546474 0.837476i \(-0.315970\pi\)
0.546474 + 0.837476i \(0.315970\pi\)
\(620\) 0 0
\(621\) −2.06913 −0.0830313
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 20.3693 0.814773
\(626\) 0 0
\(627\) −13.1231 −0.524086
\(628\) 0 0
\(629\) 9.12311 0.363762
\(630\) 0 0
\(631\) 37.9309 1.51000 0.755002 0.655722i \(-0.227636\pi\)
0.755002 + 0.655722i \(0.227636\pi\)
\(632\) 0 0
\(633\) 64.3542 2.55785
\(634\) 0 0
\(635\) −2.87689 −0.114166
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −13.1231 −0.519142
\(640\) 0 0
\(641\) 34.8078 1.37482 0.687412 0.726268i \(-0.258747\pi\)
0.687412 + 0.726268i \(0.258747\pi\)
\(642\) 0 0
\(643\) 43.5464 1.71730 0.858651 0.512560i \(-0.171303\pi\)
0.858651 + 0.512560i \(0.171303\pi\)
\(644\) 0 0
\(645\) −5.75379 −0.226555
\(646\) 0 0
\(647\) −36.1771 −1.42227 −0.711134 0.703057i \(-0.751818\pi\)
−0.711134 + 0.703057i \(0.751818\pi\)
\(648\) 0 0
\(649\) 5.43845 0.213478
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 43.3002 1.69447 0.847234 0.531220i \(-0.178266\pi\)
0.847234 + 0.531220i \(0.178266\pi\)
\(654\) 0 0
\(655\) −7.36932 −0.287943
\(656\) 0 0
\(657\) 15.1231 0.590009
\(658\) 0 0
\(659\) 21.6155 0.842021 0.421011 0.907056i \(-0.361675\pi\)
0.421011 + 0.907056i \(0.361675\pi\)
\(660\) 0 0
\(661\) 34.6695 1.34849 0.674244 0.738509i \(-0.264470\pi\)
0.674244 + 0.738509i \(0.264470\pi\)
\(662\) 0 0
\(663\) 16.0000 0.621389
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.87689 −0.111394
\(668\) 0 0
\(669\) 68.0388 2.63053
\(670\) 0 0
\(671\) −4.24621 −0.163923
\(672\) 0 0
\(673\) 33.3693 1.28629 0.643146 0.765743i \(-0.277629\pi\)
0.643146 + 0.765743i \(0.277629\pi\)
\(674\) 0 0
\(675\) 6.73863 0.259370
\(676\) 0 0
\(677\) 28.2462 1.08559 0.542795 0.839865i \(-0.317366\pi\)
0.542795 + 0.839865i \(0.317366\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 26.2462 1.00576
\(682\) 0 0
\(683\) 32.4924 1.24329 0.621644 0.783300i \(-0.286465\pi\)
0.621644 + 0.783300i \(0.286465\pi\)
\(684\) 0 0
\(685\) 1.30019 0.0496776
\(686\) 0 0
\(687\) −5.93087 −0.226277
\(688\) 0 0
\(689\) 13.2614 0.505218
\(690\) 0 0
\(691\) −28.1771 −1.07191 −0.535953 0.844248i \(-0.680048\pi\)
−0.535953 + 0.844248i \(0.680048\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 2.87689 0.109127
\(696\) 0 0
\(697\) 20.0000 0.757554
\(698\) 0 0
\(699\) 6.38447 0.241483
\(700\) 0 0
\(701\) 36.1080 1.36378 0.681889 0.731455i \(-0.261159\pi\)
0.681889 + 0.731455i \(0.261159\pi\)
\(702\) 0 0
\(703\) −23.3693 −0.881390
\(704\) 0 0
\(705\) −8.98485 −0.338389
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.30019 0.123941 0.0619706 0.998078i \(-0.480262\pi\)
0.0619706 + 0.998078i \(0.480262\pi\)
\(710\) 0 0
\(711\) −18.2462 −0.684286
\(712\) 0 0
\(713\) −15.1922 −0.568954
\(714\) 0 0
\(715\) −1.75379 −0.0655880
\(716\) 0 0
\(717\) −20.4924 −0.765304
\(718\) 0 0
\(719\) −41.3002 −1.54024 −0.770119 0.637901i \(-0.779803\pi\)
−0.770119 + 0.637901i \(0.779803\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 13.7538 0.511509
\(724\) 0 0
\(725\) 9.36932 0.347968
\(726\) 0 0
\(727\) 3.19224 0.118393 0.0591967 0.998246i \(-0.481146\pi\)
0.0591967 + 0.998246i \(0.481146\pi\)
\(728\) 0 0
\(729\) −35.9848 −1.33277
\(730\) 0 0
\(731\) −8.00000 −0.295891
\(732\) 0 0
\(733\) 23.1231 0.854071 0.427036 0.904235i \(-0.359558\pi\)
0.427036 + 0.904235i \(0.359558\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.56155 −0.0943560
\(738\) 0 0
\(739\) 20.6307 0.758912 0.379456 0.925210i \(-0.376111\pi\)
0.379456 + 0.925210i \(0.376111\pi\)
\(740\) 0 0
\(741\) −40.9848 −1.50562
\(742\) 0 0
\(743\) −16.0000 −0.586983 −0.293492 0.955962i \(-0.594817\pi\)
−0.293492 + 0.955962i \(0.594817\pi\)
\(744\) 0 0
\(745\) 5.61553 0.205737
\(746\) 0 0
\(747\) −28.4924 −1.04248
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 37.3002 1.36110 0.680552 0.732700i \(-0.261740\pi\)
0.680552 + 0.732700i \(0.261740\pi\)
\(752\) 0 0
\(753\) −38.5616 −1.40526
\(754\) 0 0
\(755\) 2.87689 0.104701
\(756\) 0 0
\(757\) −26.9848 −0.980781 −0.490390 0.871503i \(-0.663146\pi\)
−0.490390 + 0.871503i \(0.663146\pi\)
\(758\) 0 0
\(759\) 3.68466 0.133745
\(760\) 0 0
\(761\) 40.1080 1.45391 0.726956 0.686684i \(-0.240934\pi\)
0.726956 + 0.686684i \(0.240934\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −4.00000 −0.144620
\(766\) 0 0
\(767\) 16.9848 0.613287
\(768\) 0 0
\(769\) 18.6307 0.671840 0.335920 0.941891i \(-0.390953\pi\)
0.335920 + 0.941891i \(0.390953\pi\)
\(770\) 0 0
\(771\) −35.8617 −1.29153
\(772\) 0 0
\(773\) −13.5076 −0.485834 −0.242917 0.970047i \(-0.578104\pi\)
−0.242917 + 0.970047i \(0.578104\pi\)
\(774\) 0 0
\(775\) 49.4773 1.77728
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −51.2311 −1.83554
\(780\) 0 0
\(781\) 3.68466 0.131847
\(782\) 0 0
\(783\) 2.87689 0.102812
\(784\) 0 0
\(785\) 9.30019 0.331938
\(786\) 0 0
\(787\) −10.8769 −0.387719 −0.193860 0.981029i \(-0.562101\pi\)
−0.193860 + 0.981029i \(0.562101\pi\)
\(788\) 0 0
\(789\) 24.6307 0.876876
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −13.2614 −0.470925
\(794\) 0 0
\(795\) −6.10795 −0.216627
\(796\) 0 0
\(797\) −12.0691 −0.427511 −0.213755 0.976887i \(-0.568570\pi\)
−0.213755 + 0.976887i \(0.568570\pi\)
\(798\) 0 0
\(799\) −12.4924 −0.441950
\(800\) 0 0
\(801\) 44.7386 1.58076
\(802\) 0 0
\(803\) −4.24621 −0.149846
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 19.8617 0.699166
\(808\) 0 0
\(809\) −20.7386 −0.729132 −0.364566 0.931178i \(-0.618783\pi\)
−0.364566 + 0.931178i \(0.618783\pi\)
\(810\) 0 0
\(811\) −5.12311 −0.179897 −0.0899483 0.995946i \(-0.528670\pi\)
−0.0899483 + 0.995946i \(0.528670\pi\)
\(812\) 0 0
\(813\) −46.7386 −1.63920
\(814\) 0 0
\(815\) 9.26137 0.324412
\(816\) 0 0
\(817\) 20.4924 0.716939
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −36.2462 −1.26500 −0.632501 0.774560i \(-0.717971\pi\)
−0.632501 + 0.774560i \(0.717971\pi\)
\(822\) 0 0
\(823\) 52.6695 1.83594 0.917972 0.396646i \(-0.129826\pi\)
0.917972 + 0.396646i \(0.129826\pi\)
\(824\) 0 0
\(825\) −12.0000 −0.417786
\(826\) 0 0
\(827\) −47.2311 −1.64238 −0.821192 0.570651i \(-0.806691\pi\)
−0.821192 + 0.570651i \(0.806691\pi\)
\(828\) 0 0
\(829\) 6.17708 0.214539 0.107269 0.994230i \(-0.465789\pi\)
0.107269 + 0.994230i \(0.465789\pi\)
\(830\) 0 0
\(831\) −22.7386 −0.788794
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 4.13826 0.143210
\(836\) 0 0
\(837\) 15.1922 0.525120
\(838\) 0 0
\(839\) 7.68466 0.265304 0.132652 0.991163i \(-0.457651\pi\)
0.132652 + 0.991163i \(0.457651\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) 0 0
\(843\) −76.4924 −2.63454
\(844\) 0 0
\(845\) 1.82292 0.0627103
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −72.9848 −2.50483
\(850\) 0 0
\(851\) 6.56155 0.224927
\(852\) 0 0
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) 10.2462 0.350413
\(856\) 0 0
\(857\) −32.8769 −1.12305 −0.561527 0.827459i \(-0.689786\pi\)
−0.561527 + 0.827459i \(0.689786\pi\)
\(858\) 0 0
\(859\) 36.8078 1.25586 0.627932 0.778268i \(-0.283901\pi\)
0.627932 + 0.778268i \(0.283901\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −51.2311 −1.74393 −0.871963 0.489572i \(-0.837153\pi\)
−0.871963 + 0.489572i \(0.837153\pi\)
\(864\) 0 0
\(865\) −12.9848 −0.441498
\(866\) 0 0
\(867\) 33.3002 1.13093
\(868\) 0 0
\(869\) 5.12311 0.173789
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) −30.4924 −1.03201
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −44.2462 −1.49409 −0.747044 0.664774i \(-0.768528\pi\)
−0.747044 + 0.664774i \(0.768528\pi\)
\(878\) 0 0
\(879\) −31.3693 −1.05806
\(880\) 0 0
\(881\) 19.9309 0.671488 0.335744 0.941953i \(-0.391012\pi\)
0.335744 + 0.941953i \(0.391012\pi\)
\(882\) 0 0
\(883\) 52.0000 1.74994 0.874970 0.484178i \(-0.160881\pi\)
0.874970 + 0.484178i \(0.160881\pi\)
\(884\) 0 0
\(885\) −7.82292 −0.262965
\(886\) 0 0
\(887\) −31.3693 −1.05328 −0.526639 0.850089i \(-0.676548\pi\)
−0.526639 + 0.850089i \(0.676548\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 7.00000 0.234509
\(892\) 0 0
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −11.6847 −0.390575
\(896\) 0 0
\(897\) 11.5076 0.384227
\(898\) 0 0
\(899\) 21.1231 0.704495
\(900\) 0 0
\(901\) −8.49242 −0.282924
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.45360 0.148043
\(906\) 0 0
\(907\) −1.75379 −0.0582336 −0.0291168 0.999576i \(-0.509269\pi\)
−0.0291168 + 0.999576i \(0.509269\pi\)
\(908\) 0 0
\(909\) −47.6155 −1.57931
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 8.00000 0.264761
\(914\) 0 0
\(915\) 6.10795 0.201923
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 38.7386 1.27787 0.638935 0.769261i \(-0.279375\pi\)
0.638935 + 0.769261i \(0.279375\pi\)
\(920\) 0 0
\(921\) −52.4924 −1.72968
\(922\) 0 0
\(923\) 11.5076 0.378777
\(924\) 0 0
\(925\) −21.3693 −0.702619
\(926\) 0 0
\(927\) 58.7386 1.92923
\(928\) 0 0
\(929\) 14.0000 0.459325 0.229663 0.973270i \(-0.426238\pi\)
0.229663 + 0.973270i \(0.426238\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −48.0000 −1.57145
\(934\) 0 0
\(935\) 1.12311 0.0367295
\(936\) 0 0
\(937\) 12.8769 0.420670 0.210335 0.977629i \(-0.432545\pi\)
0.210335 + 0.977629i \(0.432545\pi\)
\(938\) 0 0
\(939\) 40.8078 1.33171
\(940\) 0 0
\(941\) −4.73863 −0.154475 −0.0772375 0.997013i \(-0.524610\pi\)
−0.0772375 + 0.997013i \(0.524610\pi\)
\(942\) 0 0
\(943\) 14.3845 0.468423
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −17.9309 −0.582675 −0.291337 0.956620i \(-0.594100\pi\)
−0.291337 + 0.956620i \(0.594100\pi\)
\(948\) 0 0
\(949\) −13.2614 −0.430482
\(950\) 0 0
\(951\) 33.4384 1.08432
\(952\) 0 0
\(953\) −54.3542 −1.76070 −0.880352 0.474321i \(-0.842694\pi\)
−0.880352 + 0.474321i \(0.842694\pi\)
\(954\) 0 0
\(955\) −15.1922 −0.491609
\(956\) 0 0
\(957\) −5.12311 −0.165606
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 80.5464 2.59827
\(962\) 0 0
\(963\) −60.9848 −1.96521
\(964\) 0 0
\(965\) −1.12311 −0.0361540
\(966\) 0 0
\(967\) 37.1231 1.19380 0.596899 0.802316i \(-0.296399\pi\)
0.596899 + 0.802316i \(0.296399\pi\)
\(968\) 0 0
\(969\) 26.2462 0.843150
\(970\) 0 0
\(971\) −47.0540 −1.51003 −0.755017 0.655705i \(-0.772371\pi\)
−0.755017 + 0.655705i \(0.772371\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) −37.4773 −1.20023
\(976\) 0 0
\(977\) 4.42329 0.141514 0.0707568 0.997494i \(-0.477459\pi\)
0.0707568 + 0.997494i \(0.477459\pi\)
\(978\) 0 0
\(979\) −12.5616 −0.401469
\(980\) 0 0
\(981\) −33.3693 −1.06540
\(982\) 0 0
\(983\) 47.6847 1.52090 0.760452 0.649394i \(-0.224977\pi\)
0.760452 + 0.649394i \(0.224977\pi\)
\(984\) 0 0
\(985\) 11.3693 0.362257
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −5.75379 −0.182960
\(990\) 0 0
\(991\) 3.50758 0.111422 0.0557109 0.998447i \(-0.482257\pi\)
0.0557109 + 0.998447i \(0.482257\pi\)
\(992\) 0 0
\(993\) 32.8078 1.04112
\(994\) 0 0
\(995\) 8.00000 0.253617
\(996\) 0 0
\(997\) −13.3693 −0.423411 −0.211705 0.977334i \(-0.567902\pi\)
−0.211705 + 0.977334i \(0.567902\pi\)
\(998\) 0 0
\(999\) −6.56155 −0.207598
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.bi.1.1 2
4.3 odd 2 4312.2.a.t.1.2 2
7.6 odd 2 1232.2.a.o.1.2 2
28.27 even 2 616.2.a.f.1.1 2
56.13 odd 2 4928.2.a.bo.1.1 2
56.27 even 2 4928.2.a.bs.1.2 2
84.83 odd 2 5544.2.a.bf.1.1 2
308.307 odd 2 6776.2.a.l.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
616.2.a.f.1.1 2 28.27 even 2
1232.2.a.o.1.2 2 7.6 odd 2
4312.2.a.t.1.2 2 4.3 odd 2
4928.2.a.bo.1.1 2 56.13 odd 2
4928.2.a.bs.1.2 2 56.27 even 2
5544.2.a.bf.1.1 2 84.83 odd 2
6776.2.a.l.1.1 2 308.307 odd 2
8624.2.a.bi.1.1 2 1.1 even 1 trivial