Properties

Label 4032.2.b.p.3583.8
Level $4032$
Weight $2$
Character 4032.3583
Analytic conductor $32.196$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4032,2,Mod(3583,4032)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4032, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4032.3583");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.b (of order \(2\), degree \(1\), not minimal)

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: no
Analytic conductor: \(32.1956820950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{16})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{9} \)
Twist minimal: no (minimal twist has level 224)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 3583.8
Root \(0.382683 + 0.923880i\) of defining polynomial
Character \(\chi\) \(=\) 4032.3583
Dual form 4032.2.b.p.3583.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.61313i q^{5} +(2.61313 + 0.414214i) q^{7} +O(q^{10})\) \(q+2.61313i q^{5} +(2.61313 + 0.414214i) q^{7} -2.00000i q^{11} -4.77791i q^{13} +3.06147i q^{17} -4.14386 q^{19} +7.65685i q^{23} -1.82843 q^{25} +3.65685 q^{29} -3.06147 q^{31} +(-1.08239 + 6.82843i) q^{35} +7.65685 q^{37} +9.55582i q^{41} -3.65685i q^{43} +7.39104 q^{47} +(6.65685 + 2.16478i) q^{49} -2.00000 q^{53} +5.22625 q^{55} -8.47343 q^{59} +2.61313i q^{61} +12.4853 q^{65} +15.6569i q^{67} -8.82843i q^{71} +12.6173i q^{73} +(0.828427 - 5.22625i) q^{77} -12.8284i q^{79} +11.5349 q^{83} -8.00000 q^{85} -2.16478i q^{89} +(1.97908 - 12.4853i) q^{91} -10.8284i q^{95} +13.5140i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 8 q^{25} - 16 q^{29} + 16 q^{37} + 8 q^{49} - 16 q^{53} + 32 q^{65} - 16 q^{77} - 64 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4032\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1793\) \(3781\)
\(\chi(n)\) \(-1\) \(-1\) \(1\) \(1\)

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 0
\(4\) 0 0
\(5\) 2.61313i 1.16863i 0.811529 + 0.584313i \(0.198636\pi\)
−0.811529 + 0.584313i \(0.801364\pi\)
\(6\) 0 0
\(7\) 2.61313 + 0.414214i 0.987669 + 0.156558i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000i 0.603023i −0.953463 0.301511i \(-0.902509\pi\)
0.953463 0.301511i \(-0.0974911\pi\)
\(12\) 0 0
\(13\) 4.77791i 1.32515i −0.748994 0.662577i \(-0.769463\pi\)
0.748994 0.662577i \(-0.230537\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.06147i 0.742515i 0.928530 + 0.371257i \(0.121073\pi\)
−0.928530 + 0.371257i \(0.878927\pi\)
\(18\) 0 0
\(19\) −4.14386 −0.950667 −0.475333 0.879806i \(-0.657673\pi\)
−0.475333 + 0.879806i \(0.657673\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.65685i 1.59656i 0.602284 + 0.798282i \(0.294258\pi\)
−0.602284 + 0.798282i \(0.705742\pi\)
\(24\) 0 0
\(25\) −1.82843 −0.365685
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 3.65685 0.679061 0.339530 0.940595i \(-0.389732\pi\)
0.339530 + 0.940595i \(0.389732\pi\)
\(30\) 0 0
\(31\) −3.06147 −0.549856 −0.274928 0.961465i \(-0.588654\pi\)
−0.274928 + 0.961465i \(0.588654\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.08239 + 6.82843i −0.182958 + 1.15421i
\(36\) 0 0
\(37\) 7.65685 1.25878 0.629390 0.777090i \(-0.283305\pi\)
0.629390 + 0.777090i \(0.283305\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 9.55582i 1.49237i 0.665740 + 0.746184i \(0.268116\pi\)
−0.665740 + 0.746184i \(0.731884\pi\)
\(42\) 0 0
\(43\) 3.65685i 0.557665i −0.960340 0.278833i \(-0.910053\pi\)
0.960340 0.278833i \(-0.0899474\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 7.39104 1.07809 0.539047 0.842276i \(-0.318785\pi\)
0.539047 + 0.842276i \(0.318785\pi\)
\(48\) 0 0
\(49\) 6.65685 + 2.16478i 0.950979 + 0.309255i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −2.00000 −0.274721 −0.137361 0.990521i \(-0.543862\pi\)
−0.137361 + 0.990521i \(0.543862\pi\)
\(54\) 0 0
\(55\) 5.22625 0.704708
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −8.47343 −1.10315 −0.551573 0.834126i \(-0.685972\pi\)
−0.551573 + 0.834126i \(0.685972\pi\)
\(60\) 0 0
\(61\) 2.61313i 0.334576i 0.985908 + 0.167288i \(0.0535011\pi\)
−0.985908 + 0.167288i \(0.946499\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 12.4853 1.54861
\(66\) 0 0
\(67\) 15.6569i 1.91279i 0.292078 + 0.956395i \(0.405653\pi\)
−0.292078 + 0.956395i \(0.594347\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.82843i 1.04774i −0.851798 0.523871i \(-0.824487\pi\)
0.851798 0.523871i \(-0.175513\pi\)
\(72\) 0 0
\(73\) 12.6173i 1.47674i 0.674395 + 0.738371i \(0.264405\pi\)
−0.674395 + 0.738371i \(0.735595\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.828427 5.22625i 0.0944080 0.595587i
\(78\) 0 0
\(79\) 12.8284i 1.44331i −0.692252 0.721655i \(-0.743382\pi\)
0.692252 0.721655i \(-0.256618\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 11.5349 1.26612 0.633060 0.774103i \(-0.281799\pi\)
0.633060 + 0.774103i \(0.281799\pi\)
\(84\) 0 0
\(85\) −8.00000 −0.867722
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.16478i 0.229467i −0.993396 0.114733i \(-0.963399\pi\)
0.993396 0.114733i \(-0.0366014\pi\)
\(90\) 0 0
\(91\) 1.97908 12.4853i 0.207463 1.30881i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 10.8284i 1.11097i
\(96\) 0 0
\(97\) 13.5140i 1.37214i 0.727537 + 0.686068i \(0.240665\pi\)
−0.727537 + 0.686068i \(0.759335\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 5.67459i 0.564643i 0.959320 + 0.282322i \(0.0911045\pi\)
−0.959320 + 0.282322i \(0.908895\pi\)
\(102\) 0 0
\(103\) 16.9469 1.66982 0.834912 0.550384i \(-0.185519\pi\)
0.834912 + 0.550384i \(0.185519\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.343146i 0.0331732i 0.999862 + 0.0165866i \(0.00527991\pi\)
−0.999862 + 0.0165866i \(0.994720\pi\)
\(108\) 0 0
\(109\) 5.31371 0.508961 0.254480 0.967078i \(-0.418096\pi\)
0.254480 + 0.967078i \(0.418096\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −8.82843 −0.830509 −0.415254 0.909705i \(-0.636307\pi\)
−0.415254 + 0.909705i \(0.636307\pi\)
\(114\) 0 0
\(115\) −20.0083 −1.86579
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −1.26810 + 8.00000i −0.116247 + 0.733359i
\(120\) 0 0
\(121\) 7.00000 0.636364
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 8.28772i 0.741276i
\(126\) 0 0
\(127\) 3.65685i 0.324493i 0.986750 + 0.162247i \(0.0518740\pi\)
−0.986750 + 0.162247i \(0.948126\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −13.6997 −1.19695 −0.598473 0.801143i \(-0.704226\pi\)
−0.598473 + 0.801143i \(0.704226\pi\)
\(132\) 0 0
\(133\) −10.8284 1.71644i −0.938944 0.148834i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.00000 −0.170872 −0.0854358 0.996344i \(-0.527228\pi\)
−0.0854358 + 0.996344i \(0.527228\pi\)
\(138\) 0 0
\(139\) −1.97908 −0.167863 −0.0839315 0.996472i \(-0.526748\pi\)
−0.0839315 + 0.996472i \(0.526748\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −9.55582 −0.799098
\(144\) 0 0
\(145\) 9.55582i 0.793568i
\(146\) 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
\(151\) 8.34315i 0.678956i 0.940614 + 0.339478i \(0.110250\pi\)
−0.940614 + 0.339478i \(0.889750\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 8.00000i 0.642575i
\(156\) 0 0
\(157\) 5.67459i 0.452882i 0.974025 + 0.226441i \(0.0727090\pi\)
−0.974025 + 0.226441i \(0.927291\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −3.17157 + 20.0083i −0.249955 + 1.57688i
\(162\) 0 0
\(163\) 17.3137i 1.35611i −0.735009 0.678057i \(-0.762822\pi\)
0.735009 0.678057i \(-0.237178\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.49435 0.502548 0.251274 0.967916i \(-0.419150\pi\)
0.251274 + 0.967916i \(0.419150\pi\)
\(168\) 0 0
\(169\) −9.82843 −0.756033
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 10.9008i 0.828776i 0.910100 + 0.414388i \(0.136004\pi\)
−0.910100 + 0.414388i \(0.863996\pi\)
\(174\) 0 0
\(175\) −4.77791 0.757359i −0.361176 0.0572510i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 2.00000i 0.149487i −0.997203 0.0747435i \(-0.976186\pi\)
0.997203 0.0747435i \(-0.0238138\pi\)
\(180\) 0 0
\(181\) 1.71644i 0.127582i 0.997963 + 0.0637911i \(0.0203191\pi\)
−0.997963 + 0.0637911i \(0.979681\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 20.0083i 1.47104i
\(186\) 0 0
\(187\) 6.12293 0.447753
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.82843i 0.349373i 0.984624 + 0.174686i \(0.0558911\pi\)
−0.984624 + 0.174686i \(0.944109\pi\)
\(192\) 0 0
\(193\) −10.4853 −0.754747 −0.377374 0.926061i \(-0.623173\pi\)
−0.377374 + 0.926061i \(0.623173\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.65685 0.260540 0.130270 0.991479i \(-0.458416\pi\)
0.130270 + 0.991479i \(0.458416\pi\)
\(198\) 0 0
\(199\) −12.6173 −0.894416 −0.447208 0.894430i \(-0.647582\pi\)
−0.447208 + 0.894430i \(0.647582\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 9.55582 + 1.51472i 0.670687 + 0.106312i
\(204\) 0 0
\(205\) −24.9706 −1.74402
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 8.28772i 0.573274i
\(210\) 0 0
\(211\) 10.9706i 0.755245i 0.925960 + 0.377622i \(0.123258\pi\)
−0.925960 + 0.377622i \(0.876742\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 9.55582 0.651702
\(216\) 0 0
\(217\) −8.00000 1.26810i −0.543075 0.0860843i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 14.6274 0.983947
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −18.5545 −1.23151 −0.615753 0.787939i \(-0.711148\pi\)
−0.615753 + 0.787939i \(0.711148\pi\)
\(228\) 0 0
\(229\) 2.98454i 0.197224i 0.995126 + 0.0986121i \(0.0314403\pi\)
−0.995126 + 0.0986121i \(0.968560\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −18.0000 −1.17922 −0.589610 0.807688i \(-0.700718\pi\)
−0.589610 + 0.807688i \(0.700718\pi\)
\(234\) 0 0
\(235\) 19.3137i 1.25989i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.34315i 0.539673i −0.962906 0.269837i \(-0.913030\pi\)
0.962906 0.269837i \(-0.0869697\pi\)
\(240\) 0 0
\(241\) 23.9665i 1.54382i −0.635734 0.771908i \(-0.719302\pi\)
0.635734 0.771908i \(-0.280698\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −5.65685 + 17.3952i −0.361403 + 1.11134i
\(246\) 0 0
\(247\) 19.7990i 1.25978i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1.08239 −0.0683200 −0.0341600 0.999416i \(-0.510876\pi\)
−0.0341600 + 0.999416i \(0.510876\pi\)
\(252\) 0 0
\(253\) 15.3137 0.962765
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 6.12293i 0.381938i 0.981596 + 0.190969i \(0.0611630\pi\)
−0.981596 + 0.190969i \(0.938837\pi\)
\(258\) 0 0
\(259\) 20.0083 + 3.17157i 1.24326 + 0.197072i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.14214i 0.255415i −0.991812 0.127708i \(-0.959238\pi\)
0.991812 0.127708i \(-0.0407619\pi\)
\(264\) 0 0
\(265\) 5.22625i 0.321046i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 13.4370i 0.819271i 0.912249 + 0.409636i \(0.134344\pi\)
−0.912249 + 0.409636i \(0.865656\pi\)
\(270\) 0 0
\(271\) 1.79337 0.108939 0.0544696 0.998515i \(-0.482653\pi\)
0.0544696 + 0.998515i \(0.482653\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 3.65685i 0.220517i
\(276\) 0 0
\(277\) 26.9706 1.62050 0.810252 0.586082i \(-0.199330\pi\)
0.810252 + 0.586082i \(0.199330\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 28.6274 1.70777 0.853884 0.520463i \(-0.174241\pi\)
0.853884 + 0.520463i \(0.174241\pi\)
\(282\) 0 0
\(283\) 1.08239 0.0643415 0.0321708 0.999482i \(-0.489758\pi\)
0.0321708 + 0.999482i \(0.489758\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −3.95815 + 24.9706i −0.233642 + 1.47397i
\(288\) 0 0
\(289\) 7.62742 0.448672
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 5.67459i 0.331513i 0.986167 + 0.165757i \(0.0530066\pi\)
−0.986167 + 0.165757i \(0.946993\pi\)
\(294\) 0 0
\(295\) 22.1421i 1.28916i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 36.5838 2.11569
\(300\) 0 0
\(301\) 1.51472 9.55582i 0.0873069 0.550788i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −6.82843 −0.390995
\(306\) 0 0
\(307\) −11.5349 −0.658331 −0.329166 0.944272i \(-0.606767\pi\)
−0.329166 + 0.944272i \(0.606767\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 20.0083 1.13457 0.567284 0.823522i \(-0.307994\pi\)
0.567284 + 0.823522i \(0.307994\pi\)
\(312\) 0 0
\(313\) 15.6788i 0.886216i 0.896468 + 0.443108i \(0.146124\pi\)
−0.896468 + 0.443108i \(0.853876\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 27.6569 1.55336 0.776682 0.629893i \(-0.216901\pi\)
0.776682 + 0.629893i \(0.216901\pi\)
\(318\) 0 0
\(319\) 7.31371i 0.409489i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 12.6863i 0.705884i
\(324\) 0 0
\(325\) 8.73606i 0.484589i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 19.3137 + 3.06147i 1.06480 + 0.168784i
\(330\) 0 0
\(331\) 17.3137i 0.951647i −0.879541 0.475824i \(-0.842150\pi\)
0.879541 0.475824i \(-0.157850\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −40.9133 −2.23533
\(336\) 0 0
\(337\) 8.82843 0.480915 0.240458 0.970660i \(-0.422703\pi\)
0.240458 + 0.970660i \(0.422703\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.12293i 0.331576i
\(342\) 0 0
\(343\) 16.4985 + 8.41421i 0.890836 + 0.454325i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 19.6569i 1.05524i 0.849482 + 0.527618i \(0.176915\pi\)
−0.849482 + 0.527618i \(0.823085\pi\)
\(348\) 0 0
\(349\) 8.73606i 0.467631i 0.972281 + 0.233815i \(0.0751211\pi\)
−0.972281 + 0.233815i \(0.924879\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 29.5641i 1.57354i −0.617246 0.786770i \(-0.711752\pi\)
0.617246 0.786770i \(-0.288248\pi\)
\(354\) 0 0
\(355\) 23.0698 1.22442
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.9706i 0.790116i −0.918656 0.395058i \(-0.870724\pi\)
0.918656 0.395058i \(-0.129276\pi\)
\(360\) 0 0
\(361\) −1.82843 −0.0962330
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −32.9706 −1.72576
\(366\) 0 0
\(367\) −10.4525 −0.545616 −0.272808 0.962068i \(-0.587952\pi\)
−0.272808 + 0.962068i \(0.587952\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −5.22625 0.828427i −0.271333 0.0430098i
\(372\) 0 0
\(373\) −6.97056 −0.360922 −0.180461 0.983582i \(-0.557759\pi\)
−0.180461 + 0.983582i \(0.557759\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 17.4721i 0.899860i
\(378\) 0 0
\(379\) 5.31371i 0.272947i 0.990644 + 0.136473i \(0.0435768\pi\)
−0.990644 + 0.136473i \(0.956423\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −11.7206 −0.598895 −0.299447 0.954113i \(-0.596802\pi\)
−0.299447 + 0.954113i \(0.596802\pi\)
\(384\) 0 0
\(385\) 13.6569 + 2.16478i 0.696018 + 0.110328i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 19.6569 0.996642 0.498321 0.866993i \(-0.333950\pi\)
0.498321 + 0.866993i \(0.333950\pi\)
\(390\) 0 0
\(391\) −23.4412 −1.18547
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 33.5223 1.68669
\(396\) 0 0
\(397\) 33.9706i 1.70494i 0.522778 + 0.852469i \(0.324896\pi\)
−0.522778 + 0.852469i \(0.675104\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 15.1716 0.757632 0.378816 0.925472i \(-0.376331\pi\)
0.378816 + 0.925472i \(0.376331\pi\)
\(402\) 0 0
\(403\) 14.6274i 0.728644i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 15.3137i 0.759072i
\(408\) 0 0
\(409\) 0.896683i 0.0443381i 0.999754 + 0.0221691i \(0.00705721\pi\)
−0.999754 + 0.0221691i \(0.992943\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −22.1421 3.50981i −1.08954 0.172706i
\(414\) 0 0
\(415\) 30.1421i 1.47962i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −34.6047 −1.69055 −0.845275 0.534332i \(-0.820563\pi\)
−0.845275 + 0.534332i \(0.820563\pi\)
\(420\) 0 0
\(421\) 21.3137 1.03877 0.519383 0.854541i \(-0.326162\pi\)
0.519383 + 0.854541i \(0.326162\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 5.59767i 0.271527i
\(426\) 0 0
\(427\) −1.08239 + 6.82843i −0.0523806 + 0.330451i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 12.3431i 0.594548i 0.954792 + 0.297274i \(0.0960775\pi\)
−0.954792 + 0.297274i \(0.903922\pi\)
\(432\) 0 0
\(433\) 5.59767i 0.269007i −0.990913 0.134503i \(-0.957056\pi\)
0.990913 0.134503i \(-0.0429439\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 31.7289i 1.51780i
\(438\) 0 0
\(439\) −22.5445 −1.07599 −0.537996 0.842948i \(-0.680818\pi\)
−0.537996 + 0.842948i \(0.680818\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 10.0000i 0.475114i −0.971374 0.237557i \(-0.923653\pi\)
0.971374 0.237557i \(-0.0763467\pi\)
\(444\) 0 0
\(445\) 5.65685 0.268161
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −21.3137 −1.00586 −0.502928 0.864328i \(-0.667744\pi\)
−0.502928 + 0.864328i \(0.667744\pi\)
\(450\) 0 0
\(451\) 19.1116 0.899932
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 32.6256 + 5.17157i 1.52951 + 0.242447i
\(456\) 0 0
\(457\) 37.1127 1.73606 0.868029 0.496513i \(-0.165386\pi\)
0.868029 + 0.496513i \(0.165386\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10.3756i 0.483239i 0.970371 + 0.241619i \(0.0776786\pi\)
−0.970371 + 0.241619i \(0.922321\pi\)
\(462\) 0 0
\(463\) 16.1421i 0.750189i −0.926987 0.375094i \(-0.877610\pi\)
0.926987 0.375094i \(-0.122390\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8.10201 −0.374916 −0.187458 0.982273i \(-0.560025\pi\)
−0.187458 + 0.982273i \(0.560025\pi\)
\(468\) 0 0
\(469\) −6.48528 + 40.9133i −0.299462 + 1.88920i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −7.31371 −0.336285
\(474\) 0 0
\(475\) 7.57675 0.347645
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −30.0894 −1.37482 −0.687410 0.726269i \(-0.741252\pi\)
−0.687410 + 0.726269i \(0.741252\pi\)
\(480\) 0 0
\(481\) 36.5838i 1.66808i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −35.3137 −1.60351
\(486\) 0 0
\(487\) 30.2843i 1.37231i −0.727455 0.686156i \(-0.759297\pi\)
0.727455 0.686156i \(-0.240703\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 30.2843i 1.36671i −0.730086 0.683355i \(-0.760520\pi\)
0.730086 0.683355i \(-0.239480\pi\)
\(492\) 0 0
\(493\) 11.1953i 0.504213i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.65685 23.0698i 0.164032 1.03482i
\(498\) 0 0
\(499\) 17.3137i 0.775068i −0.921855 0.387534i \(-0.873327\pi\)
0.921855 0.387534i \(-0.126673\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 17.4721 0.779043 0.389522 0.921017i \(-0.372640\pi\)
0.389522 + 0.921017i \(0.372640\pi\)
\(504\) 0 0
\(505\) −14.8284 −0.659856
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 26.0543i 1.15484i −0.816448 0.577419i \(-0.804060\pi\)
0.816448 0.577419i \(-0.195940\pi\)
\(510\) 0 0
\(511\) −5.22625 + 32.9706i −0.231196 + 1.45853i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 44.2843i 1.95140i
\(516\) 0 0
\(517\) 14.7821i 0.650115i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0.896683i 0.0392844i 0.999807 + 0.0196422i \(0.00625271\pi\)
−0.999807 + 0.0196422i \(0.993747\pi\)
\(522\) 0 0
\(523\) 18.9259 0.827573 0.413787 0.910374i \(-0.364206\pi\)
0.413787 + 0.910374i \(0.364206\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 9.37258i 0.408276i
\(528\) 0 0
\(529\) −35.6274 −1.54902
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 45.6569 1.97762
\(534\) 0 0
\(535\) −0.896683 −0.0387670
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 4.32957 13.3137i 0.186488 0.573462i
\(540\) 0 0
\(541\) −25.3137 −1.08832 −0.544161 0.838981i \(-0.683152\pi\)
−0.544161 + 0.838981i \(0.683152\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 13.8854i 0.594785i
\(546\) 0 0
\(547\) 33.3137i 1.42439i −0.701981 0.712196i \(-0.747701\pi\)
0.701981 0.712196i \(-0.252299\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −15.1535 −0.645560
\(552\) 0 0
\(553\) 5.31371 33.5223i 0.225962 1.42551i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5.31371 −0.225149 −0.112575 0.993643i \(-0.535910\pi\)
−0.112575 + 0.993643i \(0.535910\pi\)
\(558\) 0 0
\(559\) −17.4721 −0.738992
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −4.51528 −0.190296 −0.0951481 0.995463i \(-0.530332\pi\)
−0.0951481 + 0.995463i \(0.530332\pi\)
\(564\) 0 0
\(565\) 23.0698i 0.970553i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −7.85786 −0.329419 −0.164709 0.986342i \(-0.552669\pi\)
−0.164709 + 0.986342i \(0.552669\pi\)
\(570\) 0 0
\(571\) 8.34315i 0.349150i −0.984644 0.174575i \(-0.944145\pi\)
0.984644 0.174575i \(-0.0558551\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 14.0000i 0.583840i
\(576\) 0 0
\(577\) 35.6871i 1.48567i 0.669473 + 0.742836i \(0.266520\pi\)
−0.669473 + 0.742836i \(0.733480\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 30.1421 + 4.77791i 1.25051 + 0.198221i
\(582\) 0 0
\(583\) 4.00000i 0.165663i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 29.0070 1.19725 0.598624 0.801030i \(-0.295714\pi\)
0.598624 + 0.801030i \(0.295714\pi\)
\(588\) 0 0
\(589\) 12.6863 0.522730
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.79337i 0.0736447i −0.999322 0.0368224i \(-0.988276\pi\)
0.999322 0.0368224i \(-0.0117236\pi\)
\(594\) 0 0
\(595\) −20.9050 3.31371i −0.857022 0.135849i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 31.1716i 1.27364i 0.771014 + 0.636818i \(0.219750\pi\)
−0.771014 + 0.636818i \(0.780250\pi\)
\(600\) 0 0
\(601\) 42.1814i 1.72062i −0.509774 0.860308i \(-0.670271\pi\)
0.509774 0.860308i \(-0.329729\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 18.2919i 0.743671i
\(606\) 0 0
\(607\) 17.3183 0.702927 0.351464 0.936202i \(-0.385684\pi\)
0.351464 + 0.936202i \(0.385684\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 35.3137i 1.42864i
\(612\) 0 0
\(613\) 5.31371 0.214619 0.107309 0.994226i \(-0.465776\pi\)
0.107309 + 0.994226i \(0.465776\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −13.1127 −0.527897 −0.263949 0.964537i \(-0.585025\pi\)
−0.263949 + 0.964537i \(0.585025\pi\)
\(618\) 0 0
\(619\) 2.87576 0.115586 0.0577932 0.998329i \(-0.481594\pi\)
0.0577932 + 0.998329i \(0.481594\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0.896683 5.65685i 0.0359248 0.226637i
\(624\) 0 0
\(625\) −30.7990 −1.23196
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 23.4412i 0.934662i
\(630\) 0 0
\(631\) 26.4853i 1.05436i −0.849753 0.527181i \(-0.823249\pi\)
0.849753 0.527181i \(-0.176751\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −9.55582 −0.379211
\(636\) 0 0
\(637\) 10.3431 31.8059i 0.409810 1.26019i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −28.1421 −1.11155 −0.555774 0.831334i \(-0.687578\pi\)
−0.555774 + 0.831334i \(0.687578\pi\)
\(642\) 0 0
\(643\) −17.1326 −0.675642 −0.337821 0.941210i \(-0.609690\pi\)
−0.337821 + 0.941210i \(0.609690\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −4.70099 −0.184815 −0.0924074 0.995721i \(-0.529456\pi\)
−0.0924074 + 0.995721i \(0.529456\pi\)
\(648\) 0 0
\(649\) 16.9469i 0.665222i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 2.68629 0.105123 0.0525614 0.998618i \(-0.483261\pi\)
0.0525614 + 0.998618i \(0.483261\pi\)
\(654\) 0 0
\(655\) 35.7990i 1.39878i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 17.0294i 0.663373i −0.943390 0.331686i \(-0.892382\pi\)
0.943390 0.331686i \(-0.107618\pi\)
\(660\) 0 0
\(661\) 30.9092i 1.20223i −0.799164 0.601114i \(-0.794724\pi\)
0.799164 0.601114i \(-0.205276\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 4.48528 28.2960i 0.173932 1.09727i
\(666\) 0 0
\(667\) 28.0000i 1.08416i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.22625 0.201757
\(672\) 0 0
\(673\) 6.68629 0.257738 0.128869 0.991662i \(-0.458865\pi\)
0.128869 + 0.991662i \(0.458865\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 24.4148i 0.938338i −0.883109 0.469169i \(-0.844554\pi\)
0.883109 0.469169i \(-0.155446\pi\)
\(678\) 0 0
\(679\) −5.59767 + 35.3137i −0.214819 + 1.35522i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 43.6569i 1.67048i 0.549883 + 0.835242i \(0.314672\pi\)
−0.549883 + 0.835242i \(0.685328\pi\)
\(684\) 0 0
\(685\) 5.22625i 0.199685i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 9.55582i 0.364048i
\(690\) 0 0
\(691\) −32.4399 −1.23407 −0.617036 0.786935i \(-0.711667\pi\)
−0.617036 + 0.786935i \(0.711667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 5.17157i 0.196169i
\(696\) 0 0
\(697\) −29.2548 −1.10811
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.31371 0.351774 0.175887 0.984410i \(-0.443721\pi\)
0.175887 + 0.984410i \(0.443721\pi\)
\(702\) 0 0
\(703\) −31.7289 −1.19668
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −2.35049 + 14.8284i −0.0883994 + 0.557680i
\(708\) 0 0
\(709\) 18.0000 0.676004 0.338002 0.941145i \(-0.390249\pi\)
0.338002 + 0.941145i \(0.390249\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 23.4412i 0.877880i
\(714\) 0 0
\(715\) 24.9706i 0.933846i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −34.4190 −1.28361 −0.641806 0.766867i \(-0.721814\pi\)
−0.641806 + 0.766867i \(0.721814\pi\)
\(720\) 0 0
\(721\) 44.2843 + 7.01962i 1.64923 + 0.261424i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6.68629 −0.248323
\(726\) 0 0
\(727\) 42.1814 1.56442 0.782211 0.623013i \(-0.214092\pi\)
0.782211 + 0.623013i \(0.214092\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 11.1953 0.414075
\(732\) 0 0
\(733\) 24.4148i 0.901782i −0.892579 0.450891i \(-0.851106\pi\)
0.892579 0.450891i \(-0.148894\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 31.3137 1.15346
\(738\) 0 0
\(739\) 16.6274i 0.611649i 0.952088 + 0.305825i \(0.0989322\pi\)
−0.952088 + 0.305825i \(0.901068\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 24.3431i 0.893063i −0.894768 0.446532i \(-0.852659\pi\)
0.894768 0.446532i \(-0.147341\pi\)
\(744\) 0 0
\(745\) 26.1313i 0.957375i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −0.142136 + 0.896683i −0.00519352 + 0.0327641i
\(750\) 0 0
\(751\) 14.9706i 0.546284i 0.961974 + 0.273142i \(0.0880628\pi\)
−0.961974 + 0.273142i \(0.911937\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −21.8017 −0.793445
\(756\) 0 0
\(757\) −49.3137 −1.79234 −0.896169 0.443714i \(-0.853661\pi\)
−0.896169 + 0.443714i \(0.853661\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 7.76245i 0.281389i 0.990053 + 0.140694i \(0.0449335\pi\)
−0.990053 + 0.140694i \(0.955067\pi\)
\(762\) 0 0
\(763\) 13.8854 + 2.20101i 0.502685 + 0.0796819i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 40.4853i 1.46184i
\(768\) 0 0
\(769\) 13.5140i 0.487326i −0.969860 0.243663i \(-0.921651\pi\)
0.969860 0.243663i \(-0.0783491\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 26.9510i 0.969361i −0.874691 0.484680i \(-0.838936\pi\)
0.874691 0.484680i \(-0.161064\pi\)
\(774\) 0 0
\(775\) 5.59767 0.201074
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 39.5980i 1.41874i
\(780\) 0 0
\(781\) −17.6569 −0.631812
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −14.8284 −0.529249
\(786\) 0 0
\(787\) 4.51528 0.160952 0.0804761 0.996757i \(-0.474356\pi\)
0.0804761 + 0.996757i \(0.474356\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −23.0698 3.65685i −0.820267 0.130023i
\(792\) 0 0
\(793\) 12.4853 0.443365
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 41.3617i 1.46511i −0.680710 0.732553i \(-0.738329\pi\)
0.680710 0.732553i \(-0.261671\pi\)
\(798\) 0 0
\(799\) 22.6274i 0.800500i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 25.2346 0.890509
\(804\) 0 0
\(805\) −52.2843 8.28772i −1.84278 0.292104i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 17.5147 0.615785 0.307892 0.951421i \(-0.400376\pi\)
0.307892 + 0.951421i \(0.400376\pi\)
\(810\) 0 0
\(811\) 38.5628 1.35412 0.677062 0.735926i \(-0.263253\pi\)
0.677062 + 0.735926i \(0.263253\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 45.2429 1.58479
\(816\) 0 0
\(817\) 15.1535i 0.530154i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −49.5980 −1.73098 −0.865491 0.500925i \(-0.832993\pi\)
−0.865491 + 0.500925i \(0.832993\pi\)
\(822\) 0 0
\(823\) 25.1127i 0.875374i −0.899127 0.437687i \(-0.855798\pi\)
0.899127 0.437687i \(-0.144202\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 43.2548i 1.50412i 0.659096 + 0.752059i \(0.270939\pi\)
−0.659096 + 0.752059i \(0.729061\pi\)
\(828\) 0 0
\(829\) 6.19986i 0.215330i 0.994187 + 0.107665i \(0.0343374\pi\)
−0.994187 + 0.107665i \(0.965663\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −6.62742 + 20.3797i −0.229626 + 0.706116i
\(834\) 0 0
\(835\) 16.9706i 0.587291i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 42.1814 1.45626 0.728132 0.685437i \(-0.240389\pi\)
0.728132 + 0.685437i \(0.240389\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 25.6829i 0.883519i
\(846\) 0 0
\(847\) 18.2919 + 2.89949i 0.628516 + 0.0996278i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 58.6274i 2.00972i
\(852\) 0 0
\(853\) 31.8059i 1.08901i 0.838757 + 0.544506i \(0.183283\pi\)
−0.838757 + 0.544506i \(0.816717\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 7.01962i 0.239786i −0.992787 0.119893i \(-0.961745\pi\)
0.992787 0.119893i \(-0.0382551\pi\)
\(858\) 0 0
\(859\) −2.50434 −0.0854470 −0.0427235 0.999087i \(-0.513603\pi\)
−0.0427235 + 0.999087i \(0.513603\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 27.4558i 0.934608i 0.884097 + 0.467304i \(0.154775\pi\)
−0.884097 + 0.467304i \(0.845225\pi\)
\(864\) 0 0
\(865\) −28.4853 −0.968529
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −25.6569 −0.870349
\(870\) 0 0
\(871\) 74.8070 2.53474
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −3.43289 + 21.6569i −0.116053 + 0.732135i
\(876\) 0 0
\(877\) −0.343146 −0.0115872 −0.00579360 0.999983i \(-0.501844\pi\)
−0.00579360 + 0.999983i \(0.501844\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 27.7708i 0.935621i −0.883829 0.467811i \(-0.845043\pi\)
0.883829 0.467811i \(-0.154957\pi\)
\(882\) 0 0
\(883\) 24.3431i 0.819212i −0.912263 0.409606i \(-0.865666\pi\)
0.912263 0.409606i \(-0.134334\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 55.1701 1.85243 0.926216 0.376993i \(-0.123042\pi\)
0.926216 + 0.376993i \(0.123042\pi\)
\(888\) 0 0
\(889\) −1.51472 + 9.55582i −0.0508020 + 0.320492i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −30.6274 −1.02491
\(894\) 0 0
\(895\) 5.22625 0.174694
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −11.1953 −0.373386
\(900\) 0 0
\(901\) 6.12293i 0.203985i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −4.48528 −0.149096
\(906\) 0 0
\(907\) 26.0000i 0.863316i 0.902037 + 0.431658i \(0.142071\pi\)
−0.902037 + 0.431658i \(0.857929\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 24.3431i 0.806524i −0.915084 0.403262i \(-0.867876\pi\)
0.915084 0.403262i \(-0.132124\pi\)
\(912\) 0 0
\(913\) 23.0698i 0.763499i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −35.7990 5.67459i −1.18219 0.187392i
\(918\) 0 0
\(919\) 29.7990i 0.982978i −0.870884 0.491489i \(-0.836453\pi\)
0.870884 0.491489i \(-0.163547\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −42.1814 −1.38842
\(924\) 0 0
\(925\) −14.0000 −0.460317
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 57.8602i 1.89833i −0.314777 0.949166i \(-0.601930\pi\)
0.314777 0.949166i \(-0.398070\pi\)
\(930\) 0 0
\(931\) −27.5851 8.97056i −0.904064 0.293998i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 16.0000i 0.523256i
\(936\) 0 0
\(937\) 44.7176i 1.46086i 0.682987 + 0.730431i \(0.260681\pi\)
−0.682987 + 0.730431i \(0.739319\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 24.7862i 0.808008i −0.914757 0.404004i \(-0.867618\pi\)
0.914757 0.404004i \(-0.132382\pi\)
\(942\) 0 0
\(943\) −73.1675 −2.38266
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 28.3431i 0.921028i −0.887652 0.460514i \(-0.847665\pi\)
0.887652 0.460514i \(-0.152335\pi\)
\(948\) 0 0
\(949\) 60.2843 1.95691
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.627417 −0.0203240 −0.0101620 0.999948i \(-0.503235\pi\)
−0.0101620 + 0.999948i \(0.503235\pi\)
\(954\) 0 0
\(955\) −12.6173 −0.408286
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −5.22625 0.828427i −0.168764 0.0267513i
\(960\) 0 0
\(961\) −21.6274 −0.697659
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 27.3994i 0.882017i
\(966\) 0 0
\(967\) 17.0294i 0.547630i −0.961782 0.273815i \(-0.911714\pi\)
0.961782 0.273815i \(-0.0882856\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −4.14386 −0.132983 −0.0664914 0.997787i \(-0.521180\pi\)
−0.0664914 + 0.997787i \(0.521180\pi\)
\(972\) 0 0
\(973\) −5.17157 0.819760i −0.165793 0.0262803i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −16.6274 −0.531958 −0.265979 0.963979i \(-0.585695\pi\)
−0.265979 + 0.963979i \(0.585695\pi\)
\(978\) 0 0
\(979\) −4.32957 −0.138374
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −33.5223 −1.06920 −0.534598 0.845107i \(-0.679537\pi\)
−0.534598 + 0.845107i \(0.679537\pi\)
\(984\) 0 0
\(985\) 9.55582i 0.304474i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 28.0000 0.890348
\(990\) 0 0
\(991\) 14.4853i 0.460140i 0.973174 + 0.230070i \(0.0738955\pi\)
−0.973174 + 0.230070i \(0.926104\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 32.9706i 1.04524i
\(996\) 0 0
\(997\) 17.3952i 0.550911i −0.961314 0.275456i \(-0.911171\pi\)
0.961314 0.275456i \(-0.0888287\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 4032.2.b.p.3583.8 8
3.2 odd 2 448.2.f.d.447.3 8
4.3 odd 2 inner 4032.2.b.p.3583.7 8
7.6 odd 2 inner 4032.2.b.p.3583.1 8
8.3 odd 2 2016.2.b.b.1567.1 8
8.5 even 2 2016.2.b.b.1567.2 8
12.11 even 2 448.2.f.d.447.5 8
21.20 even 2 448.2.f.d.447.6 8
24.5 odd 2 224.2.f.a.223.6 yes 8
24.11 even 2 224.2.f.a.223.4 yes 8
28.27 even 2 inner 4032.2.b.p.3583.2 8
48.5 odd 4 1792.2.e.f.895.6 8
48.11 even 4 1792.2.e.g.895.4 8
48.29 odd 4 1792.2.e.g.895.3 8
48.35 even 4 1792.2.e.f.895.5 8
56.13 odd 2 2016.2.b.b.1567.7 8
56.27 even 2 2016.2.b.b.1567.8 8
84.83 odd 2 448.2.f.d.447.4 8
168.5 even 6 1568.2.p.a.31.5 16
168.11 even 6 1568.2.p.a.607.5 16
168.53 odd 6 1568.2.p.a.607.3 16
168.59 odd 6 1568.2.p.a.607.4 16
168.83 odd 2 224.2.f.a.223.5 yes 8
168.101 even 6 1568.2.p.a.607.6 16
168.107 even 6 1568.2.p.a.31.6 16
168.125 even 2 224.2.f.a.223.3 8
168.131 odd 6 1568.2.p.a.31.3 16
168.149 odd 6 1568.2.p.a.31.4 16
336.83 odd 4 1792.2.e.f.895.4 8
336.125 even 4 1792.2.e.g.895.6 8
336.251 odd 4 1792.2.e.g.895.5 8
336.293 even 4 1792.2.e.f.895.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
224.2.f.a.223.3 8 168.125 even 2
224.2.f.a.223.4 yes 8 24.11 even 2
224.2.f.a.223.5 yes 8 168.83 odd 2
224.2.f.a.223.6 yes 8 24.5 odd 2
448.2.f.d.447.3 8 3.2 odd 2
448.2.f.d.447.4 8 84.83 odd 2
448.2.f.d.447.5 8 12.11 even 2
448.2.f.d.447.6 8 21.20 even 2
1568.2.p.a.31.3 16 168.131 odd 6
1568.2.p.a.31.4 16 168.149 odd 6
1568.2.p.a.31.5 16 168.5 even 6
1568.2.p.a.31.6 16 168.107 even 6
1568.2.p.a.607.3 16 168.53 odd 6
1568.2.p.a.607.4 16 168.59 odd 6
1568.2.p.a.607.5 16 168.11 even 6
1568.2.p.a.607.6 16 168.101 even 6
1792.2.e.f.895.3 8 336.293 even 4
1792.2.e.f.895.4 8 336.83 odd 4
1792.2.e.f.895.5 8 48.35 even 4
1792.2.e.f.895.6 8 48.5 odd 4
1792.2.e.g.895.3 8 48.29 odd 4
1792.2.e.g.895.4 8 48.11 even 4
1792.2.e.g.895.5 8 336.251 odd 4
1792.2.e.g.895.6 8 336.125 even 4
2016.2.b.b.1567.1 8 8.3 odd 2
2016.2.b.b.1567.2 8 8.5 even 2
2016.2.b.b.1567.7 8 56.13 odd 2
2016.2.b.b.1567.8 8 56.27 even 2
4032.2.b.p.3583.1 8 7.6 odd 2 inner
4032.2.b.p.3583.2 8 28.27 even 2 inner
4032.2.b.p.3583.7 8 4.3 odd 2 inner
4032.2.b.p.3583.8 8 1.1 even 1 trivial