Properties

Label 4140.2.s.a.737.8
Level $4140$
Weight $2$
Character 4140.737
Analytic conductor $33.058$
Analytic rank $0$
Dimension $44$
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] = [4140,2,Mod(737,4140)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4140, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 2, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4140.737");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4140 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4140.s (of order \(4\), degree \(2\), 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: \(33.0580664368\)
Analytic rank: \(0\)
Dimension: \(44\)
Relative dimension: \(22\) over \(\Q(i)\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 737.8
Character \(\chi\) \(=\) 4140.737
Dual form 4140.2.s.a.2393.8

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.13155 + 1.92862i) q^{5} +(2.43117 + 2.43117i) q^{7} +O(q^{10})\) \(q+(-1.13155 + 1.92862i) q^{5} +(2.43117 + 2.43117i) q^{7} -5.50191i q^{11} +(1.03447 - 1.03447i) q^{13} +(0.543774 - 0.543774i) q^{17} -5.88294i q^{19} +(0.707107 + 0.707107i) q^{23} +(-2.43918 - 4.36468i) q^{25} -7.99217 q^{29} +3.34888 q^{31} +(-7.43979 + 1.93781i) q^{35} +(-3.41524 - 3.41524i) q^{37} -4.57386i q^{41} +(-1.70225 + 1.70225i) q^{43} +(-7.40852 + 7.40852i) q^{47} +4.82113i q^{49} +(-4.73776 - 4.73776i) q^{53} +(10.6111 + 6.22570i) q^{55} -11.2790 q^{59} -12.2845 q^{61} +(0.824546 + 3.16566i) q^{65} +(3.06469 + 3.06469i) q^{67} -10.1224i q^{71} +(-2.22335 + 2.22335i) q^{73} +(13.3761 - 13.3761i) q^{77} -6.61129i q^{79} +(-3.00107 - 3.00107i) q^{83} +(0.433427 + 1.66404i) q^{85} -2.16535 q^{89} +5.02993 q^{91} +(11.3460 + 6.65685i) q^{95} +(11.3198 + 11.3198i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 44 q - 12 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 44 q - 12 q^{7} - 4 q^{13} + 24 q^{25} - 48 q^{37} + 8 q^{43} + 40 q^{55} - 96 q^{61} - 44 q^{67} + 76 q^{73} + 72 q^{85} - 48 q^{91} - 20 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4140\mathbb{Z}\right)^\times\).

\(n\) \(461\) \(1657\) \(2071\) \(3961\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{4}\right)\) \(1\) \(1\)

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
\(5\) −1.13155 + 1.92862i −0.506046 + 0.862507i
\(6\) 0 0
\(7\) 2.43117 + 2.43117i 0.918894 + 0.918894i 0.996949 0.0780549i \(-0.0248709\pi\)
−0.0780549 + 0.996949i \(0.524871\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.50191i 1.65889i −0.558589 0.829444i \(-0.688657\pi\)
0.558589 0.829444i \(-0.311343\pi\)
\(12\) 0 0
\(13\) 1.03447 1.03447i 0.286910 0.286910i −0.548947 0.835857i \(-0.684971\pi\)
0.835857 + 0.548947i \(0.184971\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0.543774 0.543774i 0.131885 0.131885i −0.638083 0.769968i \(-0.720272\pi\)
0.769968 + 0.638083i \(0.220272\pi\)
\(18\) 0 0
\(19\) 5.88294i 1.34964i −0.737983 0.674819i \(-0.764222\pi\)
0.737983 0.674819i \(-0.235778\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.707107 + 0.707107i 0.147442 + 0.147442i
\(24\) 0 0
\(25\) −2.43918 4.36468i −0.487836 0.872936i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −7.99217 −1.48411 −0.742054 0.670340i \(-0.766148\pi\)
−0.742054 + 0.670340i \(0.766148\pi\)
\(30\) 0 0
\(31\) 3.34888 0.601476 0.300738 0.953707i \(-0.402767\pi\)
0.300738 + 0.953707i \(0.402767\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −7.43979 + 1.93781i −1.25755 + 0.327550i
\(36\) 0 0
\(37\) −3.41524 3.41524i −0.561462 0.561462i 0.368260 0.929723i \(-0.379954\pi\)
−0.929723 + 0.368260i \(0.879954\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 4.57386i 0.714316i −0.934044 0.357158i \(-0.883746\pi\)
0.934044 0.357158i \(-0.116254\pi\)
\(42\) 0 0
\(43\) −1.70225 + 1.70225i −0.259591 + 0.259591i −0.824888 0.565297i \(-0.808762\pi\)
0.565297 + 0.824888i \(0.308762\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −7.40852 + 7.40852i −1.08064 + 1.08064i −0.0841949 + 0.996449i \(0.526832\pi\)
−0.996449 + 0.0841949i \(0.973168\pi\)
\(48\) 0 0
\(49\) 4.82113i 0.688733i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −4.73776 4.73776i −0.650782 0.650782i 0.302400 0.953181i \(-0.402212\pi\)
−0.953181 + 0.302400i \(0.902212\pi\)
\(54\) 0 0
\(55\) 10.6111 + 6.22570i 1.43080 + 0.839473i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −11.2790 −1.46840 −0.734200 0.678934i \(-0.762442\pi\)
−0.734200 + 0.678934i \(0.762442\pi\)
\(60\) 0 0
\(61\) −12.2845 −1.57288 −0.786438 0.617670i \(-0.788077\pi\)
−0.786438 + 0.617670i \(0.788077\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0.824546 + 3.16566i 0.102272 + 0.392652i
\(66\) 0 0
\(67\) 3.06469 + 3.06469i 0.374412 + 0.374412i 0.869081 0.494669i \(-0.164711\pi\)
−0.494669 + 0.869081i \(0.664711\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.1224i 1.20130i −0.799510 0.600652i \(-0.794908\pi\)
0.799510 0.600652i \(-0.205092\pi\)
\(72\) 0 0
\(73\) −2.22335 + 2.22335i −0.260223 + 0.260223i −0.825145 0.564922i \(-0.808906\pi\)
0.564922 + 0.825145i \(0.308906\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 13.3761 13.3761i 1.52434 1.52434i
\(78\) 0 0
\(79\) 6.61129i 0.743828i −0.928267 0.371914i \(-0.878702\pi\)
0.928267 0.371914i \(-0.121298\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −3.00107 3.00107i −0.329410 0.329410i 0.522952 0.852362i \(-0.324831\pi\)
−0.852362 + 0.522952i \(0.824831\pi\)
\(84\) 0 0
\(85\) 0.433427 + 1.66404i 0.0470117 + 0.180491i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −2.16535 −0.229527 −0.114763 0.993393i \(-0.536611\pi\)
−0.114763 + 0.993393i \(0.536611\pi\)
\(90\) 0 0
\(91\) 5.02993 0.527280
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.3460 + 6.65685i 1.16407 + 0.682979i
\(96\) 0 0
\(97\) 11.3198 + 11.3198i 1.14936 + 1.14936i 0.986680 + 0.162676i \(0.0520126\pi\)
0.162676 + 0.986680i \(0.447987\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 13.0590i 1.29942i −0.760182 0.649710i \(-0.774890\pi\)
0.760182 0.649710i \(-0.225110\pi\)
\(102\) 0 0
\(103\) −5.36524 + 5.36524i −0.528653 + 0.528653i −0.920171 0.391518i \(-0.871950\pi\)
0.391518 + 0.920171i \(0.371950\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −7.53778 + 7.53778i −0.728704 + 0.728704i −0.970362 0.241657i \(-0.922309\pi\)
0.241657 + 0.970362i \(0.422309\pi\)
\(108\) 0 0
\(109\) 12.6981i 1.21625i −0.793840 0.608127i \(-0.791921\pi\)
0.793840 0.608127i \(-0.208079\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −7.86315 7.86315i −0.739703 0.739703i 0.232818 0.972520i \(-0.425206\pi\)
−0.972520 + 0.232818i \(0.925206\pi\)
\(114\) 0 0
\(115\) −2.16387 + 0.563614i −0.201782 + 0.0525573i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 2.64401 0.242376
\(120\) 0 0
\(121\) −19.2710 −1.75191
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 11.1779 + 0.234605i 0.999780 + 0.0209837i
\(126\) 0 0
\(127\) 9.27967 + 9.27967i 0.823437 + 0.823437i 0.986599 0.163162i \(-0.0521694\pi\)
−0.163162 + 0.986599i \(0.552169\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 19.1597i 1.67399i −0.547213 0.836993i \(-0.684311\pi\)
0.547213 0.836993i \(-0.315689\pi\)
\(132\) 0 0
\(133\) 14.3024 14.3024i 1.24017 1.24017i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.44031 + 2.44031i −0.208490 + 0.208490i −0.803625 0.595136i \(-0.797098\pi\)
0.595136 + 0.803625i \(0.297098\pi\)
\(138\) 0 0
\(139\) 7.32867i 0.621610i −0.950474 0.310805i \(-0.899401\pi\)
0.950474 0.310805i \(-0.100599\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −5.69156 5.69156i −0.475952 0.475952i
\(144\) 0 0
\(145\) 9.04356 15.4139i 0.751027 1.28005i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 15.9993 1.31071 0.655355 0.755321i \(-0.272519\pi\)
0.655355 + 0.755321i \(0.272519\pi\)
\(150\) 0 0
\(151\) 19.8746 1.61737 0.808686 0.588241i \(-0.200179\pi\)
0.808686 + 0.588241i \(0.200179\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −3.78943 + 6.45872i −0.304374 + 0.518777i
\(156\) 0 0
\(157\) −13.1539 13.1539i −1.04980 1.04980i −0.998693 0.0511019i \(-0.983727\pi\)
−0.0511019 0.998693i \(-0.516273\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3.43819i 0.270967i
\(162\) 0 0
\(163\) 1.13445 1.13445i 0.0888571 0.0888571i −0.661281 0.750138i \(-0.729987\pi\)
0.750138 + 0.661281i \(0.229987\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 15.1205 15.1205i 1.17006 1.17006i 0.187864 0.982195i \(-0.439844\pi\)
0.982195 0.187864i \(-0.0601563\pi\)
\(168\) 0 0
\(169\) 10.8597i 0.835365i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 13.8189 + 13.8189i 1.05063 + 1.05063i 0.998648 + 0.0519851i \(0.0165548\pi\)
0.0519851 + 0.998648i \(0.483445\pi\)
\(174\) 0 0
\(175\) 4.68121 16.5413i 0.353866 1.25040i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −12.0507 −0.900715 −0.450358 0.892848i \(-0.648704\pi\)
−0.450358 + 0.892848i \(0.648704\pi\)
\(180\) 0 0
\(181\) 16.3662 1.21649 0.608247 0.793748i \(-0.291873\pi\)
0.608247 + 0.793748i \(0.291873\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 10.4512 2.72219i 0.768391 0.200140i
\(186\) 0 0
\(187\) −2.99180 2.99180i −0.218782 0.218782i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 21.6033i 1.56316i 0.623806 + 0.781579i \(0.285585\pi\)
−0.623806 + 0.781579i \(0.714415\pi\)
\(192\) 0 0
\(193\) 3.48521 3.48521i 0.250871 0.250871i −0.570457 0.821328i \(-0.693234\pi\)
0.821328 + 0.570457i \(0.193234\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −8.46794 + 8.46794i −0.603315 + 0.603315i −0.941191 0.337875i \(-0.890292\pi\)
0.337875 + 0.941191i \(0.390292\pi\)
\(198\) 0 0
\(199\) 13.8671i 0.983012i −0.870874 0.491506i \(-0.836447\pi\)
0.870874 0.491506i \(-0.163553\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −19.4303 19.4303i −1.36374 1.36374i
\(204\) 0 0
\(205\) 8.82124 + 5.17556i 0.616102 + 0.361477i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −32.3674 −2.23890
\(210\) 0 0
\(211\) 12.4602 0.857796 0.428898 0.903353i \(-0.358902\pi\)
0.428898 + 0.903353i \(0.358902\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −1.35682 5.20919i −0.0925341 0.355264i
\(216\) 0 0
\(217\) 8.14167 + 8.14167i 0.552693 + 0.552693i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 1.12504i 0.0756781i
\(222\) 0 0
\(223\) 12.8400 12.8400i 0.859832 0.859832i −0.131486 0.991318i \(-0.541975\pi\)
0.991318 + 0.131486i \(0.0419748\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −0.525112 + 0.525112i −0.0348529 + 0.0348529i −0.724318 0.689466i \(-0.757845\pi\)
0.689466 + 0.724318i \(0.257845\pi\)
\(228\) 0 0
\(229\) 2.40850i 0.159158i 0.996829 + 0.0795791i \(0.0253576\pi\)
−0.996829 + 0.0795791i \(0.974642\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −6.22336 6.22336i −0.407706 0.407706i 0.473232 0.880938i \(-0.343087\pi\)
−0.880938 + 0.473232i \(0.843087\pi\)
\(234\) 0 0
\(235\) −5.90512 22.6714i −0.385208 1.47892i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.02140 0.0660692 0.0330346 0.999454i \(-0.489483\pi\)
0.0330346 + 0.999454i \(0.489483\pi\)
\(240\) 0 0
\(241\) −12.6183 −0.812815 −0.406407 0.913692i \(-0.633219\pi\)
−0.406407 + 0.913692i \(0.633219\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −9.29815 5.45536i −0.594037 0.348530i
\(246\) 0 0
\(247\) −6.08572 6.08572i −0.387225 0.387225i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 11.2549i 0.710404i 0.934790 + 0.355202i \(0.115588\pi\)
−0.934790 + 0.355202i \(0.884412\pi\)
\(252\) 0 0
\(253\) 3.89044 3.89044i 0.244590 0.244590i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 17.1697 17.1697i 1.07102 1.07102i 0.0737395 0.997278i \(-0.476507\pi\)
0.997278 0.0737395i \(-0.0234933\pi\)
\(258\) 0 0
\(259\) 16.6060i 1.03185i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 7.45359 + 7.45359i 0.459608 + 0.459608i 0.898527 0.438919i \(-0.144638\pi\)
−0.438919 + 0.898527i \(0.644638\pi\)
\(264\) 0 0
\(265\) 14.4984 3.77633i 0.890629 0.231978i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.25579 −0.0765668 −0.0382834 0.999267i \(-0.512189\pi\)
−0.0382834 + 0.999267i \(0.512189\pi\)
\(270\) 0 0
\(271\) −16.6861 −1.01361 −0.506803 0.862062i \(-0.669173\pi\)
−0.506803 + 0.862062i \(0.669173\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −24.0141 + 13.4201i −1.44810 + 0.809265i
\(276\) 0 0
\(277\) 9.10754 + 9.10754i 0.547219 + 0.547219i 0.925635 0.378416i \(-0.123531\pi\)
−0.378416 + 0.925635i \(0.623531\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 6.72230i 0.401019i −0.979692 0.200509i \(-0.935740\pi\)
0.979692 0.200509i \(-0.0642597\pi\)
\(282\) 0 0
\(283\) −17.1094 + 17.1094i −1.01705 + 1.01705i −0.0171964 + 0.999852i \(0.505474\pi\)
−0.999852 + 0.0171964i \(0.994526\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 11.1198 11.1198i 0.656381 0.656381i
\(288\) 0 0
\(289\) 16.4086i 0.965213i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −5.20891 5.20891i −0.304308 0.304308i 0.538389 0.842697i \(-0.319033\pi\)
−0.842697 + 0.538389i \(0.819033\pi\)
\(294\) 0 0
\(295\) 12.7628 21.7529i 0.743077 1.26650i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 1.46296 0.0846052
\(300\) 0 0
\(301\) −8.27692 −0.477074
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 13.9006 23.6923i 0.795947 1.35662i
\(306\) 0 0
\(307\) −20.2971 20.2971i −1.15842 1.15842i −0.984816 0.173601i \(-0.944460\pi\)
−0.173601 0.984816i \(-0.555540\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 15.1661i 0.859990i −0.902831 0.429995i \(-0.858515\pi\)
0.902831 0.429995i \(-0.141485\pi\)
\(312\) 0 0
\(313\) 12.6121 12.6121i 0.712879 0.712879i −0.254258 0.967137i \(-0.581831\pi\)
0.967137 + 0.254258i \(0.0818311\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −7.87599 + 7.87599i −0.442360 + 0.442360i −0.892804 0.450445i \(-0.851265\pi\)
0.450445 + 0.892804i \(0.351265\pi\)
\(318\) 0 0
\(319\) 43.9722i 2.46197i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −3.19899 3.19899i −0.177997 0.177997i
\(324\) 0 0
\(325\) −7.03838 1.99187i −0.390419 0.110489i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −36.0227 −1.98600
\(330\) 0 0
\(331\) −4.60135 −0.252913 −0.126457 0.991972i \(-0.540360\pi\)
−0.126457 + 0.991972i \(0.540360\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −9.37850 + 2.44278i −0.512402 + 0.133463i
\(336\) 0 0
\(337\) −21.8178 21.8178i −1.18849 1.18849i −0.977484 0.211009i \(-0.932325\pi\)
−0.211009 0.977484i \(-0.567675\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 18.4252i 0.997782i
\(342\) 0 0
\(343\) 5.29719 5.29719i 0.286021 0.286021i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −18.9533 + 18.9533i −1.01747 + 1.01747i −0.0176207 + 0.999845i \(0.505609\pi\)
−0.999845 + 0.0176207i \(0.994391\pi\)
\(348\) 0 0
\(349\) 34.3433i 1.83836i 0.393842 + 0.919178i \(0.371146\pi\)
−0.393842 + 0.919178i \(0.628854\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 8.97379 + 8.97379i 0.477626 + 0.477626i 0.904372 0.426745i \(-0.140340\pi\)
−0.426745 + 0.904372i \(0.640340\pi\)
\(354\) 0 0
\(355\) 19.5222 + 11.4540i 1.03613 + 0.607915i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −8.12425 −0.428782 −0.214391 0.976748i \(-0.568777\pi\)
−0.214391 + 0.976748i \(0.568777\pi\)
\(360\) 0 0
\(361\) −15.6089 −0.821524
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.77217 6.80383i −0.0927594 0.356129i
\(366\) 0 0
\(367\) −4.56152 4.56152i −0.238109 0.238109i 0.577958 0.816067i \(-0.303850\pi\)
−0.816067 + 0.577958i \(0.803850\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 23.0366i 1.19600i
\(372\) 0 0
\(373\) 19.7177 19.7177i 1.02095 1.02095i 0.0211700 0.999776i \(-0.493261\pi\)
0.999776 0.0211700i \(-0.00673911\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −8.26765 + 8.26765i −0.425806 + 0.425806i
\(378\) 0 0
\(379\) 31.9209i 1.63967i 0.572603 + 0.819833i \(0.305934\pi\)
−0.572603 + 0.819833i \(0.694066\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −9.56531 9.56531i −0.488765 0.488765i 0.419152 0.907916i \(-0.362328\pi\)
−0.907916 + 0.419152i \(0.862328\pi\)
\(384\) 0 0
\(385\) 10.6617 + 40.9331i 0.543369 + 2.08614i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −3.48156 −0.176522 −0.0882610 0.996097i \(-0.528131\pi\)
−0.0882610 + 0.996097i \(0.528131\pi\)
\(390\) 0 0
\(391\) 0.769013 0.0388906
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 12.7507 + 7.48102i 0.641557 + 0.376411i
\(396\) 0 0
\(397\) −11.2108 11.2108i −0.562652 0.562652i 0.367408 0.930060i \(-0.380245\pi\)
−0.930060 + 0.367408i \(0.880245\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 5.63851i 0.281574i −0.990040 0.140787i \(-0.955037\pi\)
0.990040 0.140787i \(-0.0449632\pi\)
\(402\) 0 0
\(403\) 3.46431 3.46431i 0.172570 0.172570i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −18.7904 + 18.7904i −0.931404 + 0.931404i
\(408\) 0 0
\(409\) 5.91715i 0.292584i 0.989241 + 0.146292i \(0.0467340\pi\)
−0.989241 + 0.146292i \(0.953266\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −27.4211 27.4211i −1.34930 1.34930i
\(414\) 0 0
\(415\) 9.18381 2.39207i 0.450815 0.117422i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −14.6493 −0.715667 −0.357833 0.933785i \(-0.616484\pi\)
−0.357833 + 0.933785i \(0.616484\pi\)
\(420\) 0 0
\(421\) 24.3727 1.18785 0.593927 0.804519i \(-0.297577\pi\)
0.593927 + 0.804519i \(0.297577\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −3.69976 1.04704i −0.179465 0.0507887i
\(426\) 0 0
\(427\) −29.8658 29.8658i −1.44531 1.44531i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.77024i 0.0852693i 0.999091 + 0.0426346i \(0.0135751\pi\)
−0.999091 + 0.0426346i \(0.986425\pi\)
\(432\) 0 0
\(433\) −0.671303 + 0.671303i −0.0322608 + 0.0322608i −0.723053 0.690792i \(-0.757262\pi\)
0.690792 + 0.723053i \(0.257262\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 4.15986 4.15986i 0.198993 0.198993i
\(438\) 0 0
\(439\) 17.2380i 0.822723i −0.911472 0.411361i \(-0.865053\pi\)
0.911472 0.411361i \(-0.134947\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 12.6013 + 12.6013i 0.598704 + 0.598704i 0.939968 0.341264i \(-0.110855\pi\)
−0.341264 + 0.939968i \(0.610855\pi\)
\(444\) 0 0
\(445\) 2.45021 4.17615i 0.116151 0.197968i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 18.9451 0.894075 0.447038 0.894515i \(-0.352479\pi\)
0.447038 + 0.894515i \(0.352479\pi\)
\(450\) 0 0
\(451\) −25.1649 −1.18497
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −5.69163 + 9.70085i −0.266828 + 0.454783i
\(456\) 0 0
\(457\) 21.6660 + 21.6660i 1.01349 + 1.01349i 0.999908 + 0.0135859i \(0.00432466\pi\)
0.0135859 + 0.999908i \(0.495675\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 8.90901i 0.414934i 0.978242 + 0.207467i \(0.0665220\pi\)
−0.978242 + 0.207467i \(0.933478\pi\)
\(462\) 0 0
\(463\) −25.0201 + 25.0201i −1.16278 + 1.16278i −0.178918 + 0.983864i \(0.557260\pi\)
−0.983864 + 0.178918i \(0.942740\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −6.09287 + 6.09287i −0.281944 + 0.281944i −0.833884 0.551940i \(-0.813888\pi\)
0.551940 + 0.833884i \(0.313888\pi\)
\(468\) 0 0
\(469\) 14.9016i 0.688090i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 9.36564 + 9.36564i 0.430633 + 0.430633i
\(474\) 0 0
\(475\) −25.6771 + 14.3495i −1.17815 + 0.658402i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 26.1992 1.19707 0.598537 0.801095i \(-0.295749\pi\)
0.598537 + 0.801095i \(0.295749\pi\)
\(480\) 0 0
\(481\) −7.06593 −0.322179
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −34.6407 + 9.02272i −1.57295 + 0.409701i
\(486\) 0 0
\(487\) −20.2014 20.2014i −0.915414 0.915414i 0.0812776 0.996692i \(-0.474100\pi\)
−0.996692 + 0.0812776i \(0.974100\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0.0638160i 0.00287998i 0.999999 + 0.00143999i \(0.000458363\pi\)
−0.999999 + 0.00143999i \(0.999542\pi\)
\(492\) 0 0
\(493\) −4.34594 + 4.34594i −0.195731 + 0.195731i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 24.6092 24.6092i 1.10387 1.10387i
\(498\) 0 0
\(499\) 30.0911i 1.34706i 0.739160 + 0.673530i \(0.235223\pi\)
−0.739160 + 0.673530i \(0.764777\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 29.7215 + 29.7215i 1.32522 + 1.32522i 0.909489 + 0.415729i \(0.136474\pi\)
0.415729 + 0.909489i \(0.363526\pi\)
\(504\) 0 0
\(505\) 25.1859 + 14.7769i 1.12076 + 0.657565i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 43.3528 1.92158 0.960790 0.277277i \(-0.0894320\pi\)
0.960790 + 0.277277i \(0.0894320\pi\)
\(510\) 0 0
\(511\) −10.8106 −0.478235
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.27648 16.4186i −0.188444 0.723489i
\(516\) 0 0
\(517\) 40.7610 + 40.7610i 1.79267 + 1.79267i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 3.24151i 0.142013i −0.997476 0.0710067i \(-0.977379\pi\)
0.997476 0.0710067i \(-0.0226212\pi\)
\(522\) 0 0
\(523\) 24.3439 24.3439i 1.06448 1.06448i 0.0667125 0.997772i \(-0.478749\pi\)
0.997772 0.0667125i \(-0.0212510\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.82103 1.82103i 0.0793254 0.0793254i
\(528\) 0 0
\(529\) 1.00000i 0.0434783i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.73151 4.73151i −0.204945 0.204945i
\(534\) 0 0
\(535\) −6.00814 23.0669i −0.259755 0.997270i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 26.5254 1.14253
\(540\) 0 0
\(541\) −2.11905 −0.0911050 −0.0455525 0.998962i \(-0.514505\pi\)
−0.0455525 + 0.998962i \(0.514505\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 24.4898 + 14.3685i 1.04903 + 0.615480i
\(546\) 0 0
\(547\) −3.30193 3.30193i −0.141180 0.141180i 0.632984 0.774165i \(-0.281830\pi\)
−0.774165 + 0.632984i \(0.781830\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 47.0174i 2.00301i
\(552\) 0 0
\(553\) 16.0731 16.0731i 0.683499 0.683499i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 10.5875 10.5875i 0.448605 0.448605i −0.446285 0.894891i \(-0.647253\pi\)
0.894891 + 0.446285i \(0.147253\pi\)
\(558\) 0 0
\(559\) 3.52186i 0.148959i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 22.5540 + 22.5540i 0.950536 + 0.950536i 0.998833 0.0482965i \(-0.0153793\pi\)
−0.0482965 + 0.998833i \(0.515379\pi\)
\(564\) 0 0
\(565\) 24.0626 6.26749i 1.01232 0.263675i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.18571 0.343163 0.171581 0.985170i \(-0.445112\pi\)
0.171581 + 0.985170i \(0.445112\pi\)
\(570\) 0 0
\(571\) 2.32436 0.0972716 0.0486358 0.998817i \(-0.484513\pi\)
0.0486358 + 0.998817i \(0.484513\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.36153 4.81105i 0.0567799 0.200635i
\(576\) 0 0
\(577\) −8.19241 8.19241i −0.341054 0.341054i 0.515709 0.856764i \(-0.327528\pi\)
−0.856764 + 0.515709i \(0.827528\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 14.5922i 0.605387i
\(582\) 0 0
\(583\) −26.0667 + 26.0667i −1.07957 + 1.07957i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −10.2250 + 10.2250i −0.422029 + 0.422029i −0.885902 0.463873i \(-0.846459\pi\)
0.463873 + 0.885902i \(0.346459\pi\)
\(588\) 0 0
\(589\) 19.7012i 0.811775i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.5865 + 23.5865i 0.968583 + 0.968583i 0.999521 0.0309386i \(-0.00984962\pi\)
−0.0309386 + 0.999521i \(0.509850\pi\)
\(594\) 0 0
\(595\) −2.99184 + 5.09930i −0.122653 + 0.209051i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −34.5916 −1.41337 −0.706687 0.707527i \(-0.749811\pi\)
−0.706687 + 0.707527i \(0.749811\pi\)
\(600\) 0 0
\(601\) −8.38323 −0.341959 −0.170980 0.985275i \(-0.554693\pi\)
−0.170980 + 0.985275i \(0.554693\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 21.8062 37.1666i 0.886547 1.51104i
\(606\) 0 0
\(607\) −26.5852 26.5852i −1.07906 1.07906i −0.996594 0.0824646i \(-0.973721\pi\)
−0.0824646 0.996594i \(-0.526279\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 15.3278i 0.620096i
\(612\) 0 0
\(613\) 32.8848 32.8848i 1.32820 1.32820i 0.421265 0.906938i \(-0.361586\pi\)
0.906938 0.421265i \(-0.138414\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 17.8211 17.8211i 0.717449 0.717449i −0.250633 0.968082i \(-0.580639\pi\)
0.968082 + 0.250633i \(0.0806387\pi\)
\(618\) 0 0
\(619\) 0.590511i 0.0237346i 0.999930 + 0.0118673i \(0.00377757\pi\)
−0.999930 + 0.0118673i \(0.996222\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −5.26433 5.26433i −0.210911 0.210911i
\(624\) 0 0
\(625\) −13.1008 + 21.2925i −0.524033 + 0.851698i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −3.71424 −0.148097
\(630\) 0 0
\(631\) −28.3481 −1.12852 −0.564260 0.825597i \(-0.690838\pi\)
−0.564260 + 0.825597i \(0.690838\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −28.3974 + 7.39655i −1.12692 + 0.293523i
\(636\) 0 0
\(637\) 4.98731 + 4.98731i 0.197604 + 0.197604i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12.2235i 0.482798i −0.970426 0.241399i \(-0.922394\pi\)
0.970426 0.241399i \(-0.0776062\pi\)
\(642\) 0 0
\(643\) 0.523304 0.523304i 0.0206371 0.0206371i −0.696713 0.717350i \(-0.745355\pi\)
0.717350 + 0.696713i \(0.245355\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 24.6991 24.6991i 0.971023 0.971023i −0.0285693 0.999592i \(-0.509095\pi\)
0.999592 + 0.0285693i \(0.00909513\pi\)
\(648\) 0 0
\(649\) 62.0560i 2.43591i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.4307 + 34.4307i 1.34738 + 1.34738i 0.888498 + 0.458881i \(0.151749\pi\)
0.458881 + 0.888498i \(0.348251\pi\)
\(654\) 0 0
\(655\) 36.9518 + 21.6802i 1.44382 + 0.847114i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −41.4303 −1.61390 −0.806948 0.590623i \(-0.798882\pi\)
−0.806948 + 0.590623i \(0.798882\pi\)
\(660\) 0 0
\(661\) 15.9008 0.618470 0.309235 0.950986i \(-0.399927\pi\)
0.309235 + 0.950986i \(0.399927\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 11.4000 + 43.7678i 0.442074 + 1.69724i
\(666\) 0 0
\(667\) −5.65132 5.65132i −0.218820 0.218820i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 67.5885i 2.60922i
\(672\) 0 0
\(673\) 11.1513 11.1513i 0.429851 0.429851i −0.458726 0.888578i \(-0.651694\pi\)
0.888578 + 0.458726i \(0.151694\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.0871 17.0871i 0.656710 0.656710i −0.297890 0.954600i \(-0.596283\pi\)
0.954600 + 0.297890i \(0.0962829\pi\)
\(678\) 0 0
\(679\) 55.0408i 2.11227i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −15.3471 15.3471i −0.587240 0.587240i 0.349643 0.936883i \(-0.386303\pi\)
−0.936883 + 0.349643i \(0.886303\pi\)
\(684\) 0 0
\(685\) −1.94510 7.46777i −0.0743184 0.285329i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −9.80214 −0.373432
\(690\) 0 0
\(691\) −22.9705 −0.873838 −0.436919 0.899501i \(-0.643930\pi\)
−0.436919 + 0.899501i \(0.643930\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 14.1342 + 8.29278i 0.536143 + 0.314563i
\(696\) 0 0
\(697\) −2.48714 2.48714i −0.0942073 0.0942073i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.15435i 0.345755i −0.984943 0.172877i \(-0.944694\pi\)
0.984943 0.172877i \(-0.0553065\pi\)
\(702\) 0 0
\(703\) −20.0917 + 20.0917i −0.757771 + 0.757771i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 31.7486 31.7486i 1.19403 1.19403i
\(708\) 0 0
\(709\) 20.5975i 0.773556i −0.922173 0.386778i \(-0.873588\pi\)
0.922173 0.386778i \(-0.126412\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.36801 + 2.36801i 0.0886828 + 0.0886828i
\(714\) 0 0
\(715\) 17.4172 4.53658i 0.651365 0.169658i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 35.5055 1.32413 0.662065 0.749446i \(-0.269680\pi\)
0.662065 + 0.749446i \(0.269680\pi\)
\(720\) 0 0
\(721\) −26.0876 −0.971552
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 19.4943 + 34.8832i 0.724001 + 1.29553i
\(726\) 0 0
\(727\) −23.9915 23.9915i −0.889796 0.889796i 0.104707 0.994503i \(-0.466610\pi\)
−0.994503 + 0.104707i \(0.966610\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1.85128i 0.0684722i
\(732\) 0 0
\(733\) 11.5305 11.5305i 0.425890 0.425890i −0.461336 0.887226i \(-0.652630\pi\)
0.887226 + 0.461336i \(0.152630\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 16.8617 16.8617i 0.621108 0.621108i
\(738\) 0 0
\(739\) 29.7815i 1.09553i −0.836633 0.547764i \(-0.815479\pi\)
0.836633 0.547764i \(-0.184521\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.19752 + 4.19752i 0.153992 + 0.153992i 0.779898 0.625906i \(-0.215271\pi\)
−0.625906 + 0.779898i \(0.715271\pi\)
\(744\) 0 0
\(745\) −18.1040 + 30.8565i −0.663279 + 1.13050i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −36.6512 −1.33920
\(750\) 0 0
\(751\) 31.6039 1.15324 0.576621 0.817012i \(-0.304371\pi\)
0.576621 + 0.817012i \(0.304371\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −22.4891 + 38.3306i −0.818464 + 1.39499i
\(756\) 0 0
\(757\) −25.9099 25.9099i −0.941712 0.941712i 0.0566800 0.998392i \(-0.481949\pi\)
−0.998392 + 0.0566800i \(0.981949\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 13.5166i 0.489977i 0.969526 + 0.244989i \(0.0787842\pi\)
−0.969526 + 0.244989i \(0.921216\pi\)
\(762\) 0 0
\(763\) 30.8711 30.8711i 1.11761 1.11761i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −11.6678 + 11.6678i −0.421299 + 0.421299i
\(768\) 0 0
\(769\) 16.5764i 0.597760i −0.954291 0.298880i \(-0.903387\pi\)
0.954291 0.298880i \(-0.0966130\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 22.2301 + 22.2301i 0.799562 + 0.799562i 0.983026 0.183464i \(-0.0587311\pi\)
−0.183464 + 0.983026i \(0.558731\pi\)
\(774\) 0 0
\(775\) −8.16851 14.6168i −0.293421 0.525050i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −26.9077 −0.964068
\(780\) 0 0
\(781\) −55.6924 −1.99283
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 40.2532 10.4846i 1.43670 0.374211i
\(786\) 0 0
\(787\) 5.85030 + 5.85030i 0.208541 + 0.208541i 0.803647 0.595106i \(-0.202890\pi\)
−0.595106 + 0.803647i \(0.702890\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 38.2332i 1.35942i
\(792\) 0 0
\(793\) −12.7080 + 12.7080i −0.451274 + 0.451274i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 2.35974 2.35974i 0.0835861 0.0835861i −0.664078 0.747664i \(-0.731176\pi\)
0.747664 + 0.664078i \(0.231176\pi\)
\(798\) 0 0
\(799\) 8.05713i 0.285041i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 12.2327 + 12.2327i 0.431681 + 0.431681i
\(804\) 0 0
\(805\) −6.63097 3.89049i −0.233711 0.137122i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 5.25285 0.184680 0.0923402 0.995728i \(-0.470565\pi\)
0.0923402 + 0.995728i \(0.470565\pi\)
\(810\) 0 0
\(811\) −1.60570 −0.0563836 −0.0281918 0.999603i \(-0.508975\pi\)
−0.0281918 + 0.999603i \(0.508975\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0.904238 + 3.47162i 0.0316741 + 0.121606i
\(816\) 0 0
\(817\) 10.0142 + 10.0142i 0.350354 + 0.350354i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 8.95002i 0.312358i 0.987729 + 0.156179i \(0.0499176\pi\)
−0.987729 + 0.156179i \(0.950082\pi\)
\(822\) 0 0
\(823\) 1.39852 1.39852i 0.0487494 0.0487494i −0.682312 0.731061i \(-0.739025\pi\)
0.731061 + 0.682312i \(0.239025\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −0.842785 + 0.842785i −0.0293065 + 0.0293065i −0.721608 0.692302i \(-0.756597\pi\)
0.692302 + 0.721608i \(0.256597\pi\)
\(828\) 0 0
\(829\) 10.1812i 0.353607i −0.984246 0.176804i \(-0.943424\pi\)
0.984246 0.176804i \(-0.0565757\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.62161 + 2.62161i 0.0908333 + 0.0908333i
\(834\) 0 0
\(835\) 12.0521 + 46.2714i 0.417080 + 1.60129i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 8.69579 0.300212 0.150106 0.988670i \(-0.452039\pi\)
0.150106 + 0.988670i \(0.452039\pi\)
\(840\) 0 0
\(841\) 34.8748 1.20258
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −20.9444 12.2884i −0.720508 0.422733i
\(846\) 0 0
\(847\) −46.8510 46.8510i −1.60982 1.60982i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.82988i 0.165566i
\(852\) 0 0
\(853\) −18.3988 + 18.3988i −0.629962 + 0.629962i −0.948058 0.318096i \(-0.896957\pi\)
0.318096 + 0.948058i \(0.396957\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −8.82302 + 8.82302i −0.301388 + 0.301388i −0.841557 0.540168i \(-0.818361\pi\)
0.540168 + 0.841557i \(0.318361\pi\)
\(858\) 0 0
\(859\) 5.70365i 0.194606i 0.995255 + 0.0973031i \(0.0310216\pi\)
−0.995255 + 0.0973031i \(0.968978\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −29.8104 29.8104i −1.01476 1.01476i −0.999889 0.0148680i \(-0.995267\pi\)
−0.0148680 0.999889i \(-0.504733\pi\)
\(864\) 0 0
\(865\) −42.2883 + 11.0147i −1.43785 + 0.374510i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −36.3747 −1.23393
\(870\) 0 0
\(871\) 6.34066 0.214845
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 26.6049 + 27.7456i 0.899410 + 0.937974i
\(876\) 0 0
\(877\) −0.684837 0.684837i −0.0231253 0.0231253i 0.695450 0.718575i \(-0.255205\pi\)
−0.718575 + 0.695450i \(0.755205\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 28.4255i 0.957680i −0.877902 0.478840i \(-0.841057\pi\)
0.877902 0.478840i \(-0.158943\pi\)
\(882\) 0 0
\(883\) −24.1329 + 24.1329i −0.812136 + 0.812136i −0.984954 0.172818i \(-0.944713\pi\)
0.172818 + 0.984954i \(0.444713\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −32.1374 + 32.1374i −1.07907 + 1.07907i −0.0824752 + 0.996593i \(0.526283\pi\)
−0.996593 + 0.0824752i \(0.973717\pi\)
\(888\) 0 0
\(889\) 45.1208i 1.51330i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 43.5839 + 43.5839i 1.45848 + 1.45848i
\(894\) 0 0
\(895\) 13.6361 23.2414i 0.455803 0.776873i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −26.7648 −0.892656
\(900\) 0 0
\(901\) −5.15255 −0.171656
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −18.5193 + 31.5643i −0.615601 + 1.04923i
\(906\) 0 0
\(907\) −25.4365 25.4365i −0.844604 0.844604i 0.144850 0.989454i \(-0.453730\pi\)
−0.989454 + 0.144850i \(0.953730\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 14.4306i 0.478108i 0.971006 + 0.239054i \(0.0768373\pi\)
−0.971006 + 0.239054i \(0.923163\pi\)
\(912\) 0 0
\(913\) −16.5116 + 16.5116i −0.546455 + 0.546455i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 46.5803 46.5803i 1.53822 1.53822i
\(918\) 0 0
\(919\) 1.67732i 0.0553298i −0.999617 0.0276649i \(-0.991193\pi\)
0.999617 0.0276649i \(-0.00880713\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −10.4713 10.4713i −0.344667 0.344667i
\(924\) 0 0
\(925\) −6.57605 + 23.2368i −0.216219 + 0.764022i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −15.4353 −0.506417 −0.253209 0.967412i \(-0.581486\pi\)
−0.253209 + 0.967412i \(0.581486\pi\)
\(930\) 0 0
\(931\) 28.3624 0.929540
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 9.15543 2.38468i 0.299414 0.0779872i
\(936\) 0 0
\(937\) −27.7315 27.7315i −0.905950 0.905950i 0.0899926 0.995942i \(-0.471316\pi\)
−0.995942 + 0.0899926i \(0.971316\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.0993439i 0.00323852i 0.999999 + 0.00161926i \(0.000515427\pi\)
−0.999999 + 0.00161926i \(0.999485\pi\)
\(942\) 0 0
\(943\) 3.23420 3.23420i 0.105320 0.105320i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 23.6359 23.6359i 0.768062 0.768062i −0.209703 0.977765i \(-0.567250\pi\)
0.977765 + 0.209703i \(0.0672496\pi\)
\(948\) 0 0
\(949\) 4.59997i 0.149321i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 18.5781 + 18.5781i 0.601803 + 0.601803i 0.940791 0.338988i \(-0.110085\pi\)
−0.338988 + 0.940791i \(0.610085\pi\)
\(954\) 0 0
\(955\) −41.6646 24.4452i −1.34823 0.791029i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −11.8656 −0.383160
\(960\) 0 0
\(961\) −19.7850 −0.638227
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 2.77796 + 10.6654i 0.0894257 + 0.343330i
\(966\) 0 0
\(967\) 31.6457 + 31.6457i 1.01766 + 1.01766i 0.999841 + 0.0178154i \(0.00567113\pi\)
0.0178154 + 0.999841i \(0.494329\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.33918i 0.235526i 0.993042 + 0.117763i \(0.0375722\pi\)
−0.993042 + 0.117763i \(0.962428\pi\)
\(972\) 0 0
\(973\) 17.8172 17.8172i 0.571194 0.571194i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −11.0849 + 11.0849i −0.354637 + 0.354637i −0.861831 0.507195i \(-0.830682\pi\)
0.507195 + 0.861831i \(0.330682\pi\)
\(978\) 0 0
\(979\) 11.9136i 0.380759i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 19.4968 + 19.4968i 0.621852 + 0.621852i 0.946005 0.324153i \(-0.105079\pi\)
−0.324153 + 0.946005i \(0.605079\pi\)
\(984\) 0 0
\(985\) −6.74955 25.9134i −0.215058 0.825669i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2.40735 −0.0765493
\(990\) 0 0
\(991\) −23.7812 −0.755435 −0.377717 0.925921i \(-0.623291\pi\)
−0.377717 + 0.925921i \(0.623291\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 26.7444 + 15.6913i 0.847854 + 0.497449i
\(996\) 0 0
\(997\) 6.63985 + 6.63985i 0.210286 + 0.210286i 0.804389 0.594103i \(-0.202493\pi\)
−0.594103 + 0.804389i \(0.702493\pi\)
\(998\) 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 4140.2.s.a.737.8 44
3.2 odd 2 inner 4140.2.s.a.737.15 yes 44
5.3 odd 4 inner 4140.2.s.a.2393.15 yes 44
15.8 even 4 inner 4140.2.s.a.2393.8 yes 44
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4140.2.s.a.737.8 44 1.1 even 1 trivial
4140.2.s.a.737.15 yes 44 3.2 odd 2 inner
4140.2.s.a.2393.8 yes 44 15.8 even 4 inner
4140.2.s.a.2393.15 yes 44 5.3 odd 4 inner