Properties

Label 4032.2.p.k.1567.10
Level $4032$
Weight $2$
Character 4032.1567
Analytic conductor $32.196$
Analytic rank $0$
Dimension $12$
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(1567,4032)]
 
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, 1, 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("4032.1567");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4032 = 2^{6} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4032.p (of order \(2\), degree \(1\), 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: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 15x^{10} + 90x^{8} - 247x^{6} + 270x^{4} + 21x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{14} \)
Twist minimal: no (minimal twist has level 1344)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 1567.10
Root \(-0.385124 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 4032.1567
Dual form 4032.2.p.k.1567.9

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.76873 q^{5} +(-0.480901 + 2.60168i) q^{7} +O(q^{10})\) \(q+2.76873 q^{5} +(-0.480901 + 2.60168i) q^{7} -5.75739 q^{11} +2.00000 q^{13} -6.71919i q^{17} -5.20336i q^{19} -4.43462i q^{23} +2.66589 q^{25} +1.54050i q^{29} -8.05068 q^{31} +(-1.33149 + 7.20336i) q^{35} +4.42590i q^{37} +0.209011i q^{41} -10.5530 q^{43} +4.58658 q^{47} +(-6.53747 - 2.50230i) q^{49} -8.05068i q^{53} -15.9407 q^{55} -5.53747i q^{59} -10.8692 q^{61} +5.53747 q^{65} +4.04280 q^{67} +1.10284i q^{71} +9.59118i q^{73} +(2.76873 - 14.9789i) q^{77} -14.7408i q^{79} +8.86925i q^{83} -18.6037i q^{85} -13.6474i q^{89} +(-0.961802 + 5.20336i) q^{91} -14.4067i q^{95} +4.58658i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 24 q^{13} + 36 q^{25} - 12 q^{49} - 72 q^{61}+O(q^{100}) \) 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
\(5\) 2.76873 1.23822 0.619108 0.785306i \(-0.287494\pi\)
0.619108 + 0.785306i \(0.287494\pi\)
\(6\) 0 0
\(7\) −0.480901 + 2.60168i −0.181763 + 0.983342i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.75739 −1.73592 −0.867959 0.496635i \(-0.834569\pi\)
−0.867959 + 0.496635i \(0.834569\pi\)
\(12\) 0 0
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.71919i 1.62964i −0.579712 0.814822i \(-0.696835\pi\)
0.579712 0.814822i \(-0.303165\pi\)
\(18\) 0 0
\(19\) 5.20336i 1.19373i −0.802341 0.596866i \(-0.796412\pi\)
0.802341 0.596866i \(-0.203588\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 4.43462i 0.924683i −0.886702 0.462342i \(-0.847009\pi\)
0.886702 0.462342i \(-0.152991\pi\)
\(24\) 0 0
\(25\) 2.66589 0.533178
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 1.54050i 0.286063i 0.989718 + 0.143032i \(0.0456851\pi\)
−0.989718 + 0.143032i \(0.954315\pi\)
\(30\) 0 0
\(31\) −8.05068 −1.44594 −0.722972 0.690877i \(-0.757225\pi\)
−0.722972 + 0.690877i \(0.757225\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −1.33149 + 7.20336i −0.225062 + 1.21759i
\(36\) 0 0
\(37\) 4.42590i 0.727614i 0.931474 + 0.363807i \(0.118523\pi\)
−0.931474 + 0.363807i \(0.881477\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.209011i 0.0326420i 0.999867 + 0.0163210i \(0.00519537\pi\)
−0.999867 + 0.0163210i \(0.994805\pi\)
\(42\) 0 0
\(43\) −10.5530 −1.60931 −0.804657 0.593740i \(-0.797651\pi\)
−0.804657 + 0.593740i \(0.797651\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 4.58658 0.669021 0.334511 0.942392i \(-0.391429\pi\)
0.334511 + 0.942392i \(0.391429\pi\)
\(48\) 0 0
\(49\) −6.53747 2.50230i −0.933924 0.357471i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 8.05068i 1.10585i −0.833232 0.552923i \(-0.813512\pi\)
0.833232 0.552923i \(-0.186488\pi\)
\(54\) 0 0
\(55\) −15.9407 −2.14944
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.53747i 0.720917i −0.932775 0.360459i \(-0.882620\pi\)
0.932775 0.360459i \(-0.117380\pi\)
\(60\) 0 0
\(61\) −10.8692 −1.39166 −0.695832 0.718204i \(-0.744964\pi\)
−0.695832 + 0.718204i \(0.744964\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 5.53747 0.686838
\(66\) 0 0
\(67\) 4.04280 0.493906 0.246953 0.969027i \(-0.420571\pi\)
0.246953 + 0.969027i \(0.420571\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 1.10284i 0.130884i 0.997856 + 0.0654418i \(0.0208457\pi\)
−0.997856 + 0.0654418i \(0.979154\pi\)
\(72\) 0 0
\(73\) 9.59118i 1.12256i 0.827625 + 0.561281i \(0.189691\pi\)
−0.827625 + 0.561281i \(0.810309\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 2.76873 14.9789i 0.315527 1.70700i
\(78\) 0 0
\(79\) 14.7408i 1.65847i −0.558898 0.829236i \(-0.688776\pi\)
0.558898 0.829236i \(-0.311224\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 8.86925i 0.973526i 0.873534 + 0.486763i \(0.161823\pi\)
−0.873534 + 0.486763i \(0.838177\pi\)
\(84\) 0 0
\(85\) 18.6037i 2.01785i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 13.6474i 1.44662i −0.690523 0.723311i \(-0.742620\pi\)
0.690523 0.723311i \(-0.257380\pi\)
\(90\) 0 0
\(91\) −0.961802 + 5.20336i −0.100824 + 0.545460i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 14.4067i 1.47810i
\(96\) 0 0
\(97\) 4.58658i 0.465696i 0.972513 + 0.232848i \(0.0748045\pi\)
−0.972513 + 0.232848i \(0.925195\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 2.76873 0.275499 0.137750 0.990467i \(-0.456013\pi\)
0.137750 + 0.990467i \(0.456013\pi\)
\(102\) 0 0
\(103\) 19.9835 1.96903 0.984515 0.175298i \(-0.0560888\pi\)
0.984515 + 0.175298i \(0.0560888\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −0.752791 −0.0727750 −0.0363875 0.999338i \(-0.511585\pi\)
−0.0363875 + 0.999338i \(0.511585\pi\)
\(108\) 0 0
\(109\) 8.85181i 0.847849i −0.905697 0.423925i \(-0.860652\pi\)
0.905697 0.423925i \(-0.139348\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −11.5375 −1.08535 −0.542677 0.839942i \(-0.682589\pi\)
−0.542677 + 0.839942i \(0.682589\pi\)
\(114\) 0 0
\(115\) 12.2783i 1.14496i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 17.4812 + 3.23127i 1.60250 + 0.296210i
\(120\) 0 0
\(121\) 22.1475 2.01341
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −6.46253 −0.578026
\(126\) 0 0
\(127\) 3.66589i 0.325295i 0.986684 + 0.162648i \(0.0520034\pi\)
−0.986684 + 0.162648i \(0.947997\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 3.13075i 0.273535i −0.990603 0.136768i \(-0.956329\pi\)
0.990603 0.136768i \(-0.0436713\pi\)
\(132\) 0 0
\(133\) 13.5375 + 2.50230i 1.17385 + 0.216977i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 19.4817 1.66443 0.832215 0.554453i \(-0.187073\pi\)
0.832215 + 0.554453i \(0.187073\pi\)
\(138\) 0 0
\(139\) 4.66822i 0.395953i −0.980207 0.197977i \(-0.936563\pi\)
0.980207 0.197977i \(-0.0634370\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −11.5148 −0.962914
\(144\) 0 0
\(145\) 4.26523i 0.354208i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 9.97428i 0.817125i −0.912730 0.408563i \(-0.866030\pi\)
0.912730 0.408563i \(-0.133970\pi\)
\(150\) 0 0
\(151\) 2.79664i 0.227587i 0.993504 + 0.113794i \(0.0363003\pi\)
−0.993504 + 0.113794i \(0.963700\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −22.2902 −1.79039
\(156\) 0 0
\(157\) −21.9442 −1.75134 −0.875668 0.482913i \(-0.839579\pi\)
−0.875668 + 0.482913i \(0.839579\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 11.5375 + 2.13261i 0.909280 + 0.168074i
\(162\) 0 0
\(163\) −7.89001 −0.617993 −0.308996 0.951063i \(-0.599993\pi\)
−0.308996 + 0.951063i \(0.599993\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.00460 0.387268 0.193634 0.981074i \(-0.437973\pi\)
0.193634 + 0.981074i \(0.437973\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.50723 −0.646793 −0.323396 0.946264i \(-0.604825\pi\)
−0.323396 + 0.946264i \(0.604825\pi\)
\(174\) 0 0
\(175\) −1.28203 + 6.93579i −0.0969123 + 0.524296i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −10.3440 −0.773144 −0.386572 0.922259i \(-0.626341\pi\)
−0.386572 + 0.922259i \(0.626341\pi\)
\(180\) 0 0
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 12.2542i 0.900943i
\(186\) 0 0
\(187\) 38.6850i 2.82893i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 3.50956i 0.253943i −0.991906 0.126971i \(-0.959474\pi\)
0.991906 0.126971i \(-0.0405257\pi\)
\(192\) 0 0
\(193\) 13.4090 0.965204 0.482602 0.875840i \(-0.339692\pi\)
0.482602 + 0.875840i \(0.339692\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 5.70905i 0.406753i −0.979101 0.203377i \(-0.934808\pi\)
0.979101 0.203377i \(-0.0651916\pi\)
\(198\) 0 0
\(199\) 4.80901 0.340902 0.170451 0.985366i \(-0.445478\pi\)
0.170451 + 0.985366i \(0.445478\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −4.00788 0.740827i −0.281298 0.0519959i
\(204\) 0 0
\(205\) 0.578696i 0.0404179i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 29.9578i 2.07222i
\(210\) 0 0
\(211\) −14.4002 −0.991350 −0.495675 0.868508i \(-0.665079\pi\)
−0.495675 + 0.868508i \(0.665079\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −29.2184 −1.99268
\(216\) 0 0
\(217\) 3.87158 20.9453i 0.262820 1.42186i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 13.4384i 0.903964i
\(222\) 0 0
\(223\) 12.0586 0.807501 0.403750 0.914869i \(-0.367706\pi\)
0.403750 + 0.914869i \(0.367706\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 17.7385i 1.17735i 0.808372 + 0.588673i \(0.200349\pi\)
−0.808372 + 0.588673i \(0.799651\pi\)
\(228\) 0 0
\(229\) 11.4817 0.758729 0.379365 0.925247i \(-0.376143\pi\)
0.379365 + 0.925247i \(0.376143\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 20.6077 1.35006 0.675029 0.737791i \(-0.264131\pi\)
0.675029 + 0.737791i \(0.264131\pi\)
\(234\) 0 0
\(235\) 12.6990 0.828392
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 18.6403i 1.20574i −0.797839 0.602871i \(-0.794023\pi\)
0.797839 0.602871i \(-0.205977\pi\)
\(240\) 0 0
\(241\) 14.5958i 0.940197i 0.882614 + 0.470098i \(0.155782\pi\)
−0.882614 + 0.470098i \(0.844218\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −18.1005 6.92820i −1.15640 0.442627i
\(246\) 0 0
\(247\) 10.4067i 0.662164i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.4817i 0.850954i −0.904969 0.425477i \(-0.860106\pi\)
0.904969 0.425477i \(-0.139894\pi\)
\(252\) 0 0
\(253\) 25.5319i 1.60517i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 23.6566i 1.47566i −0.674988 0.737829i \(-0.735851\pi\)
0.674988 0.737829i \(-0.264149\pi\)
\(258\) 0 0
\(259\) −11.5148 2.12842i −0.715494 0.132254i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 10.8972i 0.671947i 0.941871 + 0.335974i \(0.109065\pi\)
−0.941871 + 0.335974i \(0.890935\pi\)
\(264\) 0 0
\(265\) 22.2902i 1.36928i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −14.0447 −0.856320 −0.428160 0.903703i \(-0.640838\pi\)
−0.428160 + 0.903703i \(0.640838\pi\)
\(270\) 0 0
\(271\) 26.9117 1.63477 0.817384 0.576093i \(-0.195423\pi\)
0.817384 + 0.576093i \(0.195423\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −15.3486 −0.925553
\(276\) 0 0
\(277\) 2.50230i 0.150349i −0.997170 0.0751743i \(-0.976049\pi\)
0.997170 0.0751743i \(-0.0239513\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 16.3509 0.975413 0.487707 0.873008i \(-0.337834\pi\)
0.487707 + 0.873008i \(0.337834\pi\)
\(282\) 0 0
\(283\) 14.9419i 0.888201i 0.895977 + 0.444101i \(0.146477\pi\)
−0.895977 + 0.444101i \(0.853523\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.543780 0.100514i −0.0320983 0.00593313i
\(288\) 0 0
\(289\) −28.1475 −1.65574
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8.50723 −0.496998 −0.248499 0.968632i \(-0.579937\pi\)
−0.248499 + 0.968632i \(0.579937\pi\)
\(294\) 0 0
\(295\) 15.3318i 0.892651i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 8.86925i 0.512922i
\(300\) 0 0
\(301\) 5.07494 27.4555i 0.292515 1.58251i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −30.0941 −1.72318
\(306\) 0 0
\(307\) 12.5351i 0.715418i −0.933833 0.357709i \(-0.883558\pi\)
0.933833 0.357709i \(-0.116442\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −16.5194 −0.936728 −0.468364 0.883536i \(-0.655156\pi\)
−0.468364 + 0.883536i \(0.655156\pi\)
\(312\) 0 0
\(313\) 4.68325i 0.264713i 0.991202 + 0.132357i \(0.0422544\pi\)
−0.991202 + 0.132357i \(0.957746\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.97428i 0.560212i −0.959969 0.280106i \(-0.909630\pi\)
0.959969 0.280106i \(-0.0903695\pi\)
\(318\) 0 0
\(319\) 8.86925i 0.496583i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −34.9624 −1.94536
\(324\) 0 0
\(325\) 5.33178 0.295754
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −2.20569 + 11.9328i −0.121604 + 0.657877i
\(330\) 0 0
\(331\) −3.62478 −0.199236 −0.0996178 0.995026i \(-0.531762\pi\)
−0.0996178 + 0.995026i \(0.531762\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 11.1934 0.611563
\(336\) 0 0
\(337\) −2.00000 −0.108947 −0.0544735 0.998515i \(-0.517348\pi\)
−0.0544735 + 0.998515i \(0.517348\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 46.3509 2.51004
\(342\) 0 0
\(343\) 9.65406 15.8050i 0.521270 0.853392i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 15.7666 0.846395 0.423197 0.906037i \(-0.360908\pi\)
0.423197 + 0.906037i \(0.360908\pi\)
\(348\) 0 0
\(349\) 17.1308 0.916988 0.458494 0.888697i \(-0.348389\pi\)
0.458494 + 0.888697i \(0.348389\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 26.6678i 1.41938i 0.704513 + 0.709691i \(0.251165\pi\)
−0.704513 + 0.709691i \(0.748835\pi\)
\(354\) 0 0
\(355\) 3.05348i 0.162062i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 25.1028i 1.32488i −0.749116 0.662439i \(-0.769521\pi\)
0.749116 0.662439i \(-0.230479\pi\)
\(360\) 0 0
\(361\) −8.07494 −0.424997
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 26.5554i 1.38997i
\(366\) 0 0
\(367\) −22.0678 −1.15193 −0.575964 0.817475i \(-0.695373\pi\)
−0.575964 + 0.817475i \(0.695373\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 20.9453 + 3.87158i 1.08742 + 0.201002i
\(372\) 0 0
\(373\) 19.9486i 1.03290i −0.856318 0.516449i \(-0.827254\pi\)
0.856318 0.516449i \(-0.172746\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.08100i 0.158679i
\(378\) 0 0
\(379\) 21.6497 1.11207 0.556036 0.831158i \(-0.312322\pi\)
0.556036 + 0.831158i \(0.312322\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −29.2184 −1.49299 −0.746495 0.665391i \(-0.768265\pi\)
−0.746495 + 0.665391i \(0.768265\pi\)
\(384\) 0 0
\(385\) 7.66589 41.4725i 0.390690 2.11364i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 5.70905i 0.289460i 0.989471 + 0.144730i \(0.0462314\pi\)
−0.989471 + 0.144730i \(0.953769\pi\)
\(390\) 0 0
\(391\) −29.7971 −1.50690
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 40.8134i 2.05355i
\(396\) 0 0
\(397\) 13.2760 0.666302 0.333151 0.942874i \(-0.391888\pi\)
0.333151 + 0.942874i \(0.391888\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −26.1452 −1.30563 −0.652815 0.757518i \(-0.726412\pi\)
−0.652815 + 0.757518i \(0.726412\pi\)
\(402\) 0 0
\(403\) −16.1014 −0.802066
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 25.4817i 1.26308i
\(408\) 0 0
\(409\) 14.5958i 0.721715i 0.932621 + 0.360857i \(0.117516\pi\)
−0.932621 + 0.360857i \(0.882484\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 14.4067 + 2.66297i 0.708908 + 0.131036i
\(414\) 0 0
\(415\) 24.5566i 1.20544i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 28.6124i 1.39781i 0.715216 + 0.698904i \(0.246328\pi\)
−0.715216 + 0.698904i \(0.753672\pi\)
\(420\) 0 0
\(421\) 12.5115i 0.609773i 0.952389 + 0.304887i \(0.0986186\pi\)
−0.952389 + 0.304887i \(0.901381\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 17.9126i 0.868890i
\(426\) 0 0
\(427\) 5.22703 28.2783i 0.252954 1.36848i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 35.4537i 1.70775i 0.520481 + 0.853873i \(0.325753\pi\)
−0.520481 + 0.853873i \(0.674247\pi\)
\(432\) 0 0
\(433\) 36.8860i 1.77263i −0.463086 0.886313i \(-0.653258\pi\)
0.463086 0.886313i \(-0.346742\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −23.0749 −1.10382
\(438\) 0 0
\(439\) 10.9710 0.523617 0.261809 0.965120i \(-0.415681\pi\)
0.261809 + 0.965120i \(0.415681\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −21.5374 −1.02327 −0.511636 0.859202i \(-0.670961\pi\)
−0.511636 + 0.859202i \(0.670961\pi\)
\(444\) 0 0
\(445\) 37.7860i 1.79123i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −17.2760 −0.815303 −0.407652 0.913138i \(-0.633652\pi\)
−0.407652 + 0.913138i \(0.633652\pi\)
\(450\) 0 0
\(451\) 1.20336i 0.0566639i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −2.66297 + 14.4067i −0.124842 + 0.675397i
\(456\) 0 0
\(457\) 2.00000 0.0935561 0.0467780 0.998905i \(-0.485105\pi\)
0.0467780 + 0.998905i \(0.485105\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −37.8437 −1.76256 −0.881278 0.472599i \(-0.843316\pi\)
−0.881278 + 0.472599i \(0.843316\pi\)
\(462\) 0 0
\(463\) 2.99767i 0.139313i −0.997571 0.0696567i \(-0.977810\pi\)
0.997571 0.0696567i \(-0.0221904\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 8.66822i 0.401117i −0.979682 0.200559i \(-0.935724\pi\)
0.979682 0.200559i \(-0.0642757\pi\)
\(468\) 0 0
\(469\) −1.94419 + 10.5181i −0.0897741 + 0.485679i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 60.7576 2.79364
\(474\) 0 0
\(475\) 13.8716i 0.636472i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −4.26523 −0.194883 −0.0974417 0.995241i \(-0.531066\pi\)
−0.0974417 + 0.995241i \(0.531066\pi\)
\(480\) 0 0
\(481\) 8.85181i 0.403608i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 12.6990i 0.576633i
\(486\) 0 0
\(487\) 22.9419i 1.03959i 0.854290 + 0.519797i \(0.173993\pi\)
−0.854290 + 0.519797i \(0.826007\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −9.25641 −0.417736 −0.208868 0.977944i \(-0.566978\pi\)
−0.208868 + 0.977944i \(0.566978\pi\)
\(492\) 0 0
\(493\) 10.3509 0.466181
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.86925 0.530359i −0.128703 0.0237899i
\(498\) 0 0
\(499\) −5.54838 −0.248380 −0.124190 0.992258i \(-0.539633\pi\)
−0.124190 + 0.992258i \(0.539633\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 9.26983 0.413321 0.206661 0.978413i \(-0.433740\pi\)
0.206661 + 0.978413i \(0.433740\pi\)
\(504\) 0 0
\(505\) 7.66589 0.341128
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 26.0447 1.15441 0.577205 0.816599i \(-0.304143\pi\)
0.577205 + 0.816599i \(0.304143\pi\)
\(510\) 0 0
\(511\) −24.9532 4.61241i −1.10386 0.204041i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 55.3290 2.43808
\(516\) 0 0
\(517\) −26.4067 −1.16137
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 33.5960i 1.47187i −0.677054 0.735933i \(-0.736744\pi\)
0.677054 0.735933i \(-0.263256\pi\)
\(522\) 0 0
\(523\) 26.8181i 1.17267i 0.810067 + 0.586337i \(0.199430\pi\)
−0.810067 + 0.586337i \(0.800570\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 54.0941i 2.35637i
\(528\) 0 0
\(529\) 3.33411 0.144961
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0.418022i 0.0181065i
\(534\) 0 0
\(535\) −2.08428 −0.0901112
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 37.6388 + 14.4067i 1.62122 + 0.620541i
\(540\) 0 0
\(541\) 28.2217i 1.21334i 0.794952 + 0.606672i \(0.207496\pi\)
−0.794952 + 0.606672i \(0.792504\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 24.5083i 1.04982i
\(546\) 0 0
\(547\) 1.28315 0.0548635 0.0274318 0.999624i \(-0.491267\pi\)
0.0274318 + 0.999624i \(0.491267\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 8.01576 0.341483
\(552\) 0 0
\(553\) 38.3509 + 7.08888i 1.63085 + 0.301450i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 40.7681i 1.72740i −0.504007 0.863700i \(-0.668141\pi\)
0.504007 0.863700i \(-0.331859\pi\)
\(558\) 0 0
\(559\) −21.1060 −0.892687
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 19.9442i 0.840547i −0.907397 0.420274i \(-0.861934\pi\)
0.907397 0.420274i \(-0.138066\pi\)
\(564\) 0 0
\(565\) −31.9442 −1.34390
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −26.8692 −1.12642 −0.563209 0.826315i \(-0.690433\pi\)
−0.563209 + 0.826315i \(0.690433\pi\)
\(570\) 0 0
\(571\) −29.7622 −1.24551 −0.622754 0.782418i \(-0.713986\pi\)
−0.622754 + 0.782418i \(0.713986\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.8222i 0.493021i
\(576\) 0 0
\(577\) 30.7240i 1.27906i 0.768768 + 0.639528i \(0.220870\pi\)
−0.768768 + 0.639528i \(0.779130\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −23.0749 4.26523i −0.957310 0.176952i
\(582\) 0 0
\(583\) 46.3509i 1.91966i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.33178i 0.137517i 0.997633 + 0.0687586i \(0.0219038\pi\)
−0.997633 + 0.0687586i \(0.978096\pi\)
\(588\) 0 0
\(589\) 41.8906i 1.72607i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.29001i 0.135104i −0.997716 0.0675522i \(-0.978481\pi\)
0.997716 0.0675522i \(-0.0215189\pi\)
\(594\) 0 0
\(595\) 48.4007 + 8.94652i 1.98424 + 0.366771i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 3.71059i 0.151611i −0.997123 0.0758053i \(-0.975847\pi\)
0.997123 0.0758053i \(-0.0241527\pi\)
\(600\) 0 0
\(601\) 13.3417i 0.544220i 0.962266 + 0.272110i \(0.0877214\pi\)
−0.962266 + 0.272110i \(0.912279\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 61.3207 2.49304
\(606\) 0 0
\(607\) 26.7510 1.08579 0.542895 0.839801i \(-0.317328\pi\)
0.542895 + 0.839801i \(0.317328\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.17315 0.371106
\(612\) 0 0
\(613\) 30.2791i 1.22296i 0.791259 + 0.611481i \(0.209426\pi\)
−0.791259 + 0.611481i \(0.790574\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.48165 0.301200 0.150600 0.988595i \(-0.451879\pi\)
0.150600 + 0.988595i \(0.451879\pi\)
\(618\) 0 0
\(619\) 16.1452i 0.648931i −0.945898 0.324465i \(-0.894816\pi\)
0.945898 0.324465i \(-0.105184\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 35.5061 + 6.56305i 1.42252 + 0.262943i
\(624\) 0 0
\(625\) −31.2225 −1.24890
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 29.7385 1.18575
\(630\) 0 0
\(631\) 5.00233i 0.199140i −0.995031 0.0995698i \(-0.968253\pi\)
0.995031 0.0995698i \(-0.0317467\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 10.1499i 0.402785i
\(636\) 0 0
\(637\) −13.0749 5.00460i −0.518048 0.198289i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −28.5519 −1.12773 −0.563867 0.825866i \(-0.690687\pi\)
−0.563867 + 0.825866i \(0.690687\pi\)
\(642\) 0 0
\(643\) 8.27830i 0.326464i 0.986588 + 0.163232i \(0.0521919\pi\)
−0.986588 + 0.163232i \(0.947808\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.9486 0.784259 0.392130 0.919910i \(-0.371739\pi\)
0.392130 + 0.919910i \(0.371739\pi\)
\(648\) 0 0
\(649\) 31.8814i 1.25145i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 32.6825i 1.27896i 0.768806 + 0.639482i \(0.220851\pi\)
−0.768806 + 0.639482i \(0.779149\pi\)
\(654\) 0 0
\(655\) 8.66822i 0.338695i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −3.93046 −0.153109 −0.0765545 0.997065i \(-0.524392\pi\)
−0.0765545 + 0.997065i \(0.524392\pi\)
\(660\) 0 0
\(661\) −11.6827 −0.454404 −0.227202 0.973848i \(-0.572958\pi\)
−0.227202 + 0.973848i \(0.572958\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 37.4817 + 6.92820i 1.45348 + 0.268664i
\(666\) 0 0
\(667\) 6.83153 0.264518
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 62.5785 2.41582
\(672\) 0 0
\(673\) −17.6659 −0.680970 −0.340485 0.940250i \(-0.610591\pi\)
−0.340485 + 0.940250i \(0.610591\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −30.6571 −1.17825 −0.589124 0.808043i \(-0.700527\pi\)
−0.589124 + 0.808043i \(0.700527\pi\)
\(678\) 0 0
\(679\) −11.9328 2.20569i −0.457939 0.0846466i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 23.8790 0.913706 0.456853 0.889542i \(-0.348977\pi\)
0.456853 + 0.889542i \(0.348977\pi\)
\(684\) 0 0
\(685\) 53.9395 2.06092
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 16.1014i 0.613413i
\(690\) 0 0
\(691\) 44.5566i 1.69501i −0.530785 0.847506i \(-0.678103\pi\)
0.530785 0.847506i \(-0.321897\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 12.9251i 0.490276i
\(696\) 0 0
\(697\) 1.40439 0.0531949
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 46.0940i 1.74095i 0.492215 + 0.870474i \(0.336188\pi\)
−0.492215 + 0.870474i \(0.663812\pi\)
\(702\) 0 0
\(703\) 23.0296 0.868576
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −1.33149 + 7.20336i −0.0500757 + 0.270910i
\(708\) 0 0
\(709\) 23.8656i 0.896292i −0.893960 0.448146i \(-0.852085\pi\)
0.893960 0.448146i \(-0.147915\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 35.7017i 1.33704i
\(714\) 0 0
\(715\) −31.8814 −1.19230
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 3.42919 0.127887 0.0639435 0.997954i \(-0.479632\pi\)
0.0639435 + 0.997954i \(0.479632\pi\)
\(720\) 0 0
\(721\) −9.61007 + 51.9906i −0.357898 + 1.93623i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.10680i 0.152523i
\(726\) 0 0
\(727\) 13.0553 0.484193 0.242097 0.970252i \(-0.422165\pi\)
0.242097 + 0.970252i \(0.422165\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 70.9075i 2.62261i
\(732\) 0 0
\(733\) −18.1452 −0.670209 −0.335104 0.942181i \(-0.608772\pi\)
−0.335104 + 0.942181i \(0.608772\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −23.2760 −0.857381
\(738\) 0 0
\(739\) 53.1131 1.95380 0.976898 0.213705i \(-0.0685532\pi\)
0.976898 + 0.213705i \(0.0685532\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 21.2481i 0.779516i 0.920917 + 0.389758i \(0.127441\pi\)
−0.920917 + 0.389758i \(0.872559\pi\)
\(744\) 0 0
\(745\) 27.6161i 1.01178i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0.362018 1.95852i 0.0132278 0.0715628i
\(750\) 0 0
\(751\) 9.61474i 0.350847i −0.984493 0.175423i \(-0.943871\pi\)
0.984493 0.175423i \(-0.0561294\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 7.74316i 0.281802i
\(756\) 0 0
\(757\) 16.8676i 0.613062i 0.951861 + 0.306531i \(0.0991684\pi\)
−0.951861 + 0.306531i \(0.900832\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20.1576i 0.730712i −0.930868 0.365356i \(-0.880947\pi\)
0.930868 0.365356i \(-0.119053\pi\)
\(762\) 0 0
\(763\) 23.0296 + 4.25684i 0.833726 + 0.154108i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11.0749i 0.399893i
\(768\) 0 0
\(769\) 35.7286i 1.28841i 0.764855 + 0.644203i \(0.222811\pi\)
−0.764855 + 0.644203i \(0.777189\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −15.6938 −0.564467 −0.282233 0.959346i \(-0.591075\pi\)
−0.282233 + 0.959346i \(0.591075\pi\)
\(774\) 0 0
\(775\) −21.4622 −0.770946
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.08756 0.0389659
\(780\) 0 0
\(781\) 6.34951i 0.227203i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −60.7576 −2.16853
\(786\) 0 0
\(787\) 4.00000i 0.142585i 0.997455 + 0.0712923i \(0.0227123\pi\)
−0.997455 + 0.0712923i \(0.977288\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 5.54838 30.0168i 0.197278 1.06727i
\(792\) 0 0
\(793\) −21.7385 −0.771957
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 35.4705 1.25643 0.628215 0.778039i \(-0.283786\pi\)
0.628215 + 0.778039i \(0.283786\pi\)
\(798\) 0 0
\(799\) 30.8181i 1.09027i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 55.2201i 1.94868i
\(804\) 0 0
\(805\) 31.9442 + 5.90464i 1.12588 + 0.208111i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −20.6077 −0.724530 −0.362265 0.932075i \(-0.617996\pi\)
−0.362265 + 0.932075i \(0.617996\pi\)
\(810\) 0 0
\(811\) 35.2201i 1.23675i −0.785884 0.618373i \(-0.787792\pi\)
0.785884 0.618373i \(-0.212208\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −21.8453 −0.765208
\(816\) 0 0
\(817\) 54.9109i 1.92109i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 34.1612i 1.19224i 0.802897 + 0.596118i \(0.203291\pi\)
−0.802897 + 0.596118i \(0.796709\pi\)
\(822\) 0 0
\(823\) 7.81110i 0.272278i −0.990690 0.136139i \(-0.956531\pi\)
0.990690 0.136139i \(-0.0434694\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 5.66072 0.196842 0.0984212 0.995145i \(-0.468621\pi\)
0.0984212 + 0.995145i \(0.468621\pi\)
\(828\) 0 0
\(829\) −50.5566 −1.75590 −0.877951 0.478750i \(-0.841090\pi\)
−0.877951 + 0.478750i \(0.841090\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −16.8134 + 43.9265i −0.582551 + 1.52196i
\(834\) 0 0
\(835\) 13.8564 0.479521
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −27.2679 −0.941394 −0.470697 0.882295i \(-0.655998\pi\)
−0.470697 + 0.882295i \(0.655998\pi\)
\(840\) 0 0
\(841\) 26.6269 0.918168
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −24.9186 −0.857226
\(846\) 0 0
\(847\) −10.6508 + 57.6208i −0.365965 + 1.97987i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 19.6272 0.672812
\(852\) 0 0
\(853\) −34.8692 −1.19390 −0.596950 0.802278i \(-0.703621\pi\)
−0.596950 + 0.802278i \(0.703621\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 21.6363i 0.739082i −0.929214 0.369541i \(-0.879515\pi\)
0.929214 0.369541i \(-0.120485\pi\)
\(858\) 0 0
\(859\) 22.5398i 0.769048i 0.923115 + 0.384524i \(0.125634\pi\)
−0.923115 + 0.384524i \(0.874366\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 25.6594i 0.873457i −0.899593 0.436729i \(-0.856137\pi\)
0.899593 0.436729i \(-0.143863\pi\)
\(864\) 0 0
\(865\) −23.5543 −0.800869
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 84.8687i 2.87897i
\(870\) 0 0
\(871\) 8.08560 0.273970
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 3.10784 16.8134i 0.105064 0.568398i
\(876\) 0 0
\(877\) 14.9440i 0.504622i −0.967646 0.252311i \(-0.918809\pi\)
0.967646 0.252311i \(-0.0811906\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 10.9844i 0.370075i 0.982732 + 0.185037i \(0.0592406\pi\)
−0.982732 + 0.185037i \(0.940759\pi\)
\(882\) 0 0
\(883\) −0.613612 −0.0206497 −0.0103249 0.999947i \(-0.503287\pi\)
−0.0103249 + 0.999947i \(0.503287\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 41.0545 1.37848 0.689238 0.724535i \(-0.257945\pi\)
0.689238 + 0.724535i \(0.257945\pi\)
\(888\) 0 0
\(889\) −9.53747 1.76293i −0.319876 0.0591268i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 23.8656i 0.798632i
\(894\) 0 0
\(895\) −28.6397 −0.957320
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 12.4021i 0.413632i
\(900\) 0 0
\(901\) −54.0941 −1.80213
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −5.53747 −0.184072
\(906\) 0 0
\(907\) 41.6681 1.38357 0.691784 0.722105i \(-0.256825\pi\)
0.691784 + 0.722105i \(0.256825\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 34.8972i 1.15619i −0.815968 0.578097i \(-0.803795\pi\)
0.815968 0.578097i \(-0.196205\pi\)
\(912\) 0 0
\(913\) 51.0637i 1.68996i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 8.14521 + 1.50558i 0.268979 + 0.0497187i
\(918\) 0 0
\(919\) 14.5398i 0.479624i −0.970819 0.239812i \(-0.922914\pi\)
0.970819 0.239812i \(-0.0770858\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 2.20569i 0.0726011i
\(924\) 0 0
\(925\) 11.7990i 0.387948i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 9.38217i 0.307819i −0.988085 0.153909i \(-0.950814\pi\)
0.988085 0.153909i \(-0.0491864\pi\)
\(930\) 0 0
\(931\) −13.0204 + 34.0168i −0.426725 + 1.11486i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 107.109i 3.50282i
\(936\) 0 0
\(937\) 0.321348i 0.0104980i −0.999986 0.00524899i \(-0.998329\pi\)
0.999986 0.00524899i \(-0.00167081\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 27.6938 0.902792 0.451396 0.892324i \(-0.350926\pi\)
0.451396 + 0.892324i \(0.350926\pi\)
\(942\) 0 0
\(943\) 0.926885 0.0301835
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 21.8587 0.710314 0.355157 0.934807i \(-0.384427\pi\)
0.355157 + 0.934807i \(0.384427\pi\)
\(948\) 0 0
\(949\) 19.1824i 0.622686i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −37.7432 −1.22262 −0.611310 0.791391i \(-0.709357\pi\)
−0.611310 + 0.791391i \(0.709357\pi\)
\(954\) 0 0
\(955\) 9.71704i 0.314436i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −9.36875 + 50.6850i −0.302533 + 1.63670i
\(960\) 0 0
\(961\) 33.8134 1.09076
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 37.1261 1.19513
\(966\) 0 0
\(967\) 34.4840i 1.10893i −0.832207 0.554465i \(-0.812923\pi\)
0.832207 0.554465i \(-0.187077\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 50.9633i 1.63549i 0.575580 + 0.817745i \(0.304776\pi\)
−0.575580 + 0.817745i \(0.695224\pi\)
\(972\) 0 0
\(973\) 12.1452 + 2.24495i 0.389358 + 0.0719698i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 46.6124 1.49126 0.745631 0.666359i \(-0.232148\pi\)
0.745631 + 0.666359i \(0.232148\pi\)
\(978\) 0 0
\(979\) 78.5734i 2.51122i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 7.31938 0.233452 0.116726 0.993164i \(-0.462760\pi\)
0.116726 + 0.993164i \(0.462760\pi\)
\(984\) 0 0
\(985\) 15.8069i 0.503648i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 46.7985i 1.48811i
\(990\) 0 0
\(991\) 37.0917i 1.17826i 0.808039 + 0.589129i \(0.200529\pi\)
−0.808039 + 0.589129i \(0.799471\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 13.3149 0.422110
\(996\) 0 0
\(997\) 53.0191 1.67913 0.839566 0.543257i \(-0.182809\pi\)
0.839566 + 0.543257i \(0.182809\pi\)
\(998\) 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.p.k.1567.10 12
3.2 odd 2 1344.2.p.d.223.7 yes 12
4.3 odd 2 inner 4032.2.p.k.1567.11 12
7.6 odd 2 4032.2.p.j.1567.4 12
8.3 odd 2 4032.2.p.j.1567.3 12
8.5 even 2 4032.2.p.j.1567.2 12
12.11 even 2 1344.2.p.d.223.2 yes 12
21.20 even 2 1344.2.p.c.223.6 yes 12
24.5 odd 2 1344.2.p.c.223.5 12
24.11 even 2 1344.2.p.c.223.12 yes 12
28.27 even 2 4032.2.p.j.1567.1 12
56.13 odd 2 inner 4032.2.p.k.1567.12 12
56.27 even 2 inner 4032.2.p.k.1567.9 12
84.83 odd 2 1344.2.p.c.223.11 yes 12
168.83 odd 2 1344.2.p.d.223.1 yes 12
168.125 even 2 1344.2.p.d.223.8 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1344.2.p.c.223.5 12 24.5 odd 2
1344.2.p.c.223.6 yes 12 21.20 even 2
1344.2.p.c.223.11 yes 12 84.83 odd 2
1344.2.p.c.223.12 yes 12 24.11 even 2
1344.2.p.d.223.1 yes 12 168.83 odd 2
1344.2.p.d.223.2 yes 12 12.11 even 2
1344.2.p.d.223.7 yes 12 3.2 odd 2
1344.2.p.d.223.8 yes 12 168.125 even 2
4032.2.p.j.1567.1 12 28.27 even 2
4032.2.p.j.1567.2 12 8.5 even 2
4032.2.p.j.1567.3 12 8.3 odd 2
4032.2.p.j.1567.4 12 7.6 odd 2
4032.2.p.k.1567.9 12 56.27 even 2 inner
4032.2.p.k.1567.10 12 1.1 even 1 trivial
4032.2.p.k.1567.11 12 4.3 odd 2 inner
4032.2.p.k.1567.12 12 56.13 odd 2 inner