Properties

Label 5472.2.g.b.2737.4
Level $5472$
Weight $2$
Character 5472.2737
Analytic conductor $43.694$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [5472,2,Mod(2737,5472)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5472, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("5472.2737");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5472 = 2^{5} \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5472.g (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: \(43.6941399860\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + 48 x^{7} - 40 x^{6} + 32 x^{5} + 64 x^{4} - 128 x^{3} + 192 x^{2} - 256 x + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{13} \)
Twist minimal: no (minimal twist has level 152)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2737.4
Root \(0.340606 - 1.37258i\) of defining polynomial
Character \(\chi\) \(=\) 5472.2737
Dual form 5472.2.g.b.2737.13

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.13486i q^{5} -3.29464 q^{7} +O(q^{10})\) \(q-2.13486i q^{5} -3.29464 q^{7} -3.71210i q^{11} +2.32843i q^{13} +6.48822 q^{17} +1.00000i q^{19} +7.32651 q^{23} +0.442384 q^{25} +2.59857i q^{29} +1.34204 q^{31} +7.03360i q^{35} -3.72986i q^{37} -6.52385 q^{41} +1.97202i q^{43} +5.45991 q^{47} +3.85468 q^{49} -4.98640i q^{53} -7.92480 q^{55} +9.67136i q^{59} +8.15570i q^{61} +4.97088 q^{65} +0.524986i q^{67} +7.17489 q^{71} -6.33130 q^{73} +12.2300i q^{77} +8.75644 q^{79} -7.74008i q^{83} -13.8514i q^{85} +1.04368 q^{89} -7.67136i q^{91} +2.13486 q^{95} -0.117594 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 8 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 8 q^{7} + 8 q^{17} - 24 q^{25} - 16 q^{31} - 16 q^{41} + 24 q^{47} + 24 q^{49} - 16 q^{55} - 16 q^{65} + 48 q^{71} + 48 q^{79} + 16 q^{89} + 16 q^{95} + 32 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/5472\mathbb{Z}\right)^\times\).

\(n\) \(1217\) \(2053\) \(3745\) \(4447\)
\(\chi(n)\) \(1\) \(-1\) \(1\) \(1\)

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
\(5\) − 2.13486i − 0.954737i −0.878703 0.477369i \(-0.841591\pi\)
0.878703 0.477369i \(-0.158409\pi\)
\(6\) 0 0
\(7\) −3.29464 −1.24526 −0.622629 0.782517i \(-0.713936\pi\)
−0.622629 + 0.782517i \(0.713936\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.71210i − 1.11924i −0.828749 0.559620i \(-0.810947\pi\)
0.828749 0.559620i \(-0.189053\pi\)
\(12\) 0 0
\(13\) 2.32843i 0.645792i 0.946435 + 0.322896i \(0.104656\pi\)
−0.946435 + 0.322896i \(0.895344\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 6.48822 1.57362 0.786812 0.617192i \(-0.211730\pi\)
0.786812 + 0.617192i \(0.211730\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 7.32651 1.52768 0.763842 0.645404i \(-0.223311\pi\)
0.763842 + 0.645404i \(0.223311\pi\)
\(24\) 0 0
\(25\) 0.442384 0.0884769
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.59857i 0.482542i 0.970458 + 0.241271i \(0.0775643\pi\)
−0.970458 + 0.241271i \(0.922436\pi\)
\(30\) 0 0
\(31\) 1.34204 0.241037 0.120518 0.992711i \(-0.461544\pi\)
0.120518 + 0.992711i \(0.461544\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 7.03360i 1.18889i
\(36\) 0 0
\(37\) − 3.72986i − 0.613186i −0.951841 0.306593i \(-0.900811\pi\)
0.951841 0.306593i \(-0.0991890\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.52385 −1.01885 −0.509427 0.860514i \(-0.670143\pi\)
−0.509427 + 0.860514i \(0.670143\pi\)
\(42\) 0 0
\(43\) 1.97202i 0.300729i 0.988631 + 0.150365i \(0.0480448\pi\)
−0.988631 + 0.150365i \(0.951955\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.45991 0.796410 0.398205 0.917297i \(-0.369633\pi\)
0.398205 + 0.917297i \(0.369633\pi\)
\(48\) 0 0
\(49\) 3.85468 0.550669
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 4.98640i − 0.684935i −0.939530 0.342467i \(-0.888737\pi\)
0.939530 0.342467i \(-0.111263\pi\)
\(54\) 0 0
\(55\) −7.92480 −1.06858
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 9.67136i 1.25910i 0.776958 + 0.629552i \(0.216762\pi\)
−0.776958 + 0.629552i \(0.783238\pi\)
\(60\) 0 0
\(61\) 8.15570i 1.04423i 0.852875 + 0.522115i \(0.174857\pi\)
−0.852875 + 0.522115i \(0.825143\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 4.97088 0.616561
\(66\) 0 0
\(67\) 0.524986i 0.0641372i 0.999486 + 0.0320686i \(0.0102095\pi\)
−0.999486 + 0.0320686i \(0.989790\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 7.17489 0.851503 0.425751 0.904840i \(-0.360010\pi\)
0.425751 + 0.904840i \(0.360010\pi\)
\(72\) 0 0
\(73\) −6.33130 −0.741022 −0.370511 0.928828i \(-0.620817\pi\)
−0.370511 + 0.928828i \(0.620817\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 12.2300i 1.39374i
\(78\) 0 0
\(79\) 8.75644 0.985176 0.492588 0.870263i \(-0.336051\pi\)
0.492588 + 0.870263i \(0.336051\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) − 7.74008i − 0.849585i −0.905291 0.424792i \(-0.860347\pi\)
0.905291 0.424792i \(-0.139653\pi\)
\(84\) 0 0
\(85\) − 13.8514i − 1.50240i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.04368 0.110630 0.0553148 0.998469i \(-0.482384\pi\)
0.0553148 + 0.998469i \(0.482384\pi\)
\(90\) 0 0
\(91\) − 7.67136i − 0.804178i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 2.13486 0.219032
\(96\) 0 0
\(97\) −0.117594 −0.0119398 −0.00596992 0.999982i \(-0.501900\pi\)
−0.00596992 + 0.999982i \(0.501900\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.6142i 1.05615i 0.849197 + 0.528077i \(0.177087\pi\)
−0.849197 + 0.528077i \(0.822913\pi\)
\(102\) 0 0
\(103\) −12.4190 −1.22368 −0.611840 0.790982i \(-0.709570\pi\)
−0.611840 + 0.790982i \(0.709570\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 16.5439i − 1.59936i −0.600430 0.799678i \(-0.705004\pi\)
0.600430 0.799678i \(-0.294996\pi\)
\(108\) 0 0
\(109\) − 17.4437i − 1.67080i −0.549641 0.835401i \(-0.685235\pi\)
0.549641 0.835401i \(-0.314765\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 14.3193 1.34705 0.673523 0.739166i \(-0.264780\pi\)
0.673523 + 0.739166i \(0.264780\pi\)
\(114\) 0 0
\(115\) − 15.6411i − 1.45854i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −21.3764 −1.95957
\(120\) 0 0
\(121\) −2.77968 −0.252698
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 11.6187i − 1.03921i
\(126\) 0 0
\(127\) −2.63985 −0.234248 −0.117124 0.993117i \(-0.537368\pi\)
−0.117124 + 0.993117i \(0.537368\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 15.3670i − 1.34263i −0.741174 0.671313i \(-0.765731\pi\)
0.741174 0.671313i \(-0.234269\pi\)
\(132\) 0 0
\(133\) − 3.29464i − 0.285682i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.91853 0.420218 0.210109 0.977678i \(-0.432618\pi\)
0.210109 + 0.977678i \(0.432618\pi\)
\(138\) 0 0
\(139\) − 9.80377i − 0.831545i −0.909469 0.415773i \(-0.863511\pi\)
0.909469 0.415773i \(-0.136489\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 8.64338 0.722796
\(144\) 0 0
\(145\) 5.54758 0.460701
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.724147i 0.0593244i 0.999560 + 0.0296622i \(0.00944316\pi\)
−0.999560 + 0.0296622i \(0.990557\pi\)
\(150\) 0 0
\(151\) 15.7252 1.27970 0.639850 0.768500i \(-0.278997\pi\)
0.639850 + 0.768500i \(0.278997\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 2.86505i − 0.230127i
\(156\) 0 0
\(157\) 0.141127i 0.0112632i 0.999984 + 0.00563160i \(0.00179260\pi\)
−0.999984 + 0.00563160i \(0.998207\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −24.1383 −1.90236
\(162\) 0 0
\(163\) − 8.41859i − 0.659395i −0.944087 0.329697i \(-0.893053\pi\)
0.944087 0.329697i \(-0.106947\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.16538 0.399709 0.199854 0.979826i \(-0.435953\pi\)
0.199854 + 0.979826i \(0.435953\pi\)
\(168\) 0 0
\(169\) 7.57839 0.582953
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.13988i 0.238721i 0.992851 + 0.119360i \(0.0380844\pi\)
−0.992851 + 0.119360i \(0.961916\pi\)
\(174\) 0 0
\(175\) −1.45750 −0.110177
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 18.1898i − 1.35957i −0.733410 0.679786i \(-0.762073\pi\)
0.733410 0.679786i \(-0.237927\pi\)
\(180\) 0 0
\(181\) − 14.0798i − 1.04654i −0.852166 0.523271i \(-0.824712\pi\)
0.852166 0.523271i \(-0.175288\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −7.96273 −0.585431
\(186\) 0 0
\(187\) − 24.0849i − 1.76126i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 4.87409 0.352677 0.176338 0.984330i \(-0.443575\pi\)
0.176338 + 0.984330i \(0.443575\pi\)
\(192\) 0 0
\(193\) −9.99845 −0.719704 −0.359852 0.933009i \(-0.617173\pi\)
−0.359852 + 0.933009i \(0.617173\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 19.2209i − 1.36943i −0.728810 0.684716i \(-0.759926\pi\)
0.728810 0.684716i \(-0.240074\pi\)
\(198\) 0 0
\(199\) 11.7323 0.831683 0.415842 0.909437i \(-0.363487\pi\)
0.415842 + 0.909437i \(0.363487\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 8.56137i − 0.600890i
\(204\) 0 0
\(205\) 13.9275i 0.972737i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 3.71210 0.256771
\(210\) 0 0
\(211\) − 0.399383i − 0.0274947i −0.999906 0.0137473i \(-0.995624\pi\)
0.999906 0.0137473i \(-0.00437605\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 4.20997 0.287118
\(216\) 0 0
\(217\) −4.42153 −0.300153
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 15.1074i 1.01623i
\(222\) 0 0
\(223\) 19.2548 1.28939 0.644697 0.764438i \(-0.276983\pi\)
0.644697 + 0.764438i \(0.276983\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 20.0414i 1.33019i 0.746758 + 0.665096i \(0.231609\pi\)
−0.746758 + 0.665096i \(0.768391\pi\)
\(228\) 0 0
\(229\) 18.2013i 1.20277i 0.798958 + 0.601387i \(0.205385\pi\)
−0.798958 + 0.601387i \(0.794615\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −14.2376 −0.932739 −0.466369 0.884590i \(-0.654438\pi\)
−0.466369 + 0.884590i \(0.654438\pi\)
\(234\) 0 0
\(235\) − 11.6561i − 0.760362i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 8.31368 0.537767 0.268884 0.963173i \(-0.413345\pi\)
0.268884 + 0.963173i \(0.413345\pi\)
\(240\) 0 0
\(241\) −26.1776 −1.68625 −0.843125 0.537718i \(-0.819287\pi\)
−0.843125 + 0.537718i \(0.819287\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 8.22920i − 0.525744i
\(246\) 0 0
\(247\) −2.32843 −0.148155
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 29.4740i − 1.86038i −0.367075 0.930191i \(-0.619641\pi\)
0.367075 0.930191i \(-0.380359\pi\)
\(252\) 0 0
\(253\) − 27.1967i − 1.70984i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.28195 −0.0799657 −0.0399828 0.999200i \(-0.512730\pi\)
−0.0399828 + 0.999200i \(0.512730\pi\)
\(258\) 0 0
\(259\) 12.2886i 0.763575i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −29.2736 −1.80509 −0.902544 0.430597i \(-0.858303\pi\)
−0.902544 + 0.430597i \(0.858303\pi\)
\(264\) 0 0
\(265\) −10.6453 −0.653933
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 14.1850i 0.864875i 0.901664 + 0.432437i \(0.142346\pi\)
−0.901664 + 0.432437i \(0.857654\pi\)
\(270\) 0 0
\(271\) −26.3185 −1.59874 −0.799369 0.600840i \(-0.794833\pi\)
−0.799369 + 0.600840i \(0.794833\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 1.64217i − 0.0990269i
\(276\) 0 0
\(277\) 3.58863i 0.215620i 0.994172 + 0.107810i \(0.0343838\pi\)
−0.994172 + 0.107810i \(0.965616\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4.42597 0.264031 0.132016 0.991248i \(-0.457855\pi\)
0.132016 + 0.991248i \(0.457855\pi\)
\(282\) 0 0
\(283\) − 0.493151i − 0.0293148i −0.999893 0.0146574i \(-0.995334\pi\)
0.999893 0.0146574i \(-0.00466576\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 21.4938 1.26874
\(288\) 0 0
\(289\) 25.0970 1.47630
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) − 12.7228i − 0.743275i −0.928378 0.371637i \(-0.878796\pi\)
0.928378 0.371637i \(-0.121204\pi\)
\(294\) 0 0
\(295\) 20.6470 1.20211
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 17.0593i 0.986565i
\(300\) 0 0
\(301\) − 6.49709i − 0.374486i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 17.4113 0.996966
\(306\) 0 0
\(307\) 10.2166i 0.583092i 0.956557 + 0.291546i \(0.0941697\pi\)
−0.956557 + 0.291546i \(0.905830\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0.184934 0.0104867 0.00524333 0.999986i \(-0.498331\pi\)
0.00524333 + 0.999986i \(0.498331\pi\)
\(312\) 0 0
\(313\) −19.0733 −1.07809 −0.539044 0.842278i \(-0.681214\pi\)
−0.539044 + 0.842278i \(0.681214\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 0.911635i − 0.0512025i −0.999672 0.0256013i \(-0.991850\pi\)
0.999672 0.0256013i \(-0.00815002\pi\)
\(318\) 0 0
\(319\) 9.64615 0.540081
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 6.48822i 0.361014i
\(324\) 0 0
\(325\) 1.03006i 0.0571376i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −17.9885 −0.991736
\(330\) 0 0
\(331\) − 2.45230i − 0.134791i −0.997726 0.0673953i \(-0.978531\pi\)
0.997726 0.0673953i \(-0.0214689\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.12077 0.0612342
\(336\) 0 0
\(337\) 35.1467 1.91456 0.957282 0.289156i \(-0.0933746\pi\)
0.957282 + 0.289156i \(0.0933746\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 4.98177i − 0.269778i
\(342\) 0 0
\(343\) 10.3627 0.559533
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 5.30643i 0.284864i 0.989805 + 0.142432i \(0.0454922\pi\)
−0.989805 + 0.142432i \(0.954508\pi\)
\(348\) 0 0
\(349\) − 26.8913i − 1.43946i −0.694255 0.719729i \(-0.744266\pi\)
0.694255 0.719729i \(-0.255734\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 3.21383 0.171055 0.0855275 0.996336i \(-0.472742\pi\)
0.0855275 + 0.996336i \(0.472742\pi\)
\(354\) 0 0
\(355\) − 15.3174i − 0.812961i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −15.6827 −0.827699 −0.413850 0.910345i \(-0.635816\pi\)
−0.413850 + 0.910345i \(0.635816\pi\)
\(360\) 0 0
\(361\) −1.00000 −0.0526316
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 13.5164i 0.707481i
\(366\) 0 0
\(367\) −14.3509 −0.749111 −0.374555 0.927205i \(-0.622205\pi\)
−0.374555 + 0.927205i \(0.622205\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 16.4284i 0.852921i
\(372\) 0 0
\(373\) − 9.08701i − 0.470508i −0.971934 0.235254i \(-0.924408\pi\)
0.971934 0.235254i \(-0.0755921\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −6.05060 −0.311622
\(378\) 0 0
\(379\) 18.2006i 0.934900i 0.884020 + 0.467450i \(0.154827\pi\)
−0.884020 + 0.467450i \(0.845173\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −22.4154 −1.14537 −0.572686 0.819775i \(-0.694099\pi\)
−0.572686 + 0.819775i \(0.694099\pi\)
\(384\) 0 0
\(385\) 26.1094 1.33066
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 34.9213i − 1.77058i −0.465038 0.885291i \(-0.653959\pi\)
0.465038 0.885291i \(-0.346041\pi\)
\(390\) 0 0
\(391\) 47.5360 2.40400
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 18.6937i − 0.940584i
\(396\) 0 0
\(397\) 31.3443i 1.57313i 0.617510 + 0.786563i \(0.288142\pi\)
−0.617510 + 0.786563i \(0.711858\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.64321 −0.0820579 −0.0410290 0.999158i \(-0.513064\pi\)
−0.0410290 + 0.999158i \(0.513064\pi\)
\(402\) 0 0
\(403\) 3.12484i 0.155659i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −13.8456 −0.686302
\(408\) 0 0
\(409\) −2.46645 −0.121958 −0.0609791 0.998139i \(-0.519422\pi\)
−0.0609791 + 0.998139i \(0.519422\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) − 31.8637i − 1.56791i
\(414\) 0 0
\(415\) −16.5240 −0.811130
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.25063i 0.207657i 0.994595 + 0.103828i \(0.0331093\pi\)
−0.994595 + 0.103828i \(0.966891\pi\)
\(420\) 0 0
\(421\) 31.4900i 1.53473i 0.641212 + 0.767364i \(0.278432\pi\)
−0.641212 + 0.767364i \(0.721568\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 2.87029 0.139229
\(426\) 0 0
\(427\) − 26.8701i − 1.30034i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −21.0564 −1.01425 −0.507125 0.861872i \(-0.669292\pi\)
−0.507125 + 0.861872i \(0.669292\pi\)
\(432\) 0 0
\(433\) −7.53125 −0.361929 −0.180964 0.983490i \(-0.557922\pi\)
−0.180964 + 0.983490i \(0.557922\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.32651i 0.350475i
\(438\) 0 0
\(439\) −0.823995 −0.0393271 −0.0196636 0.999807i \(-0.506260\pi\)
−0.0196636 + 0.999807i \(0.506260\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 8.58526i − 0.407898i −0.978982 0.203949i \(-0.934622\pi\)
0.978982 0.203949i \(-0.0653777\pi\)
\(444\) 0 0
\(445\) − 2.22810i − 0.105622i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −2.49451 −0.117723 −0.0588615 0.998266i \(-0.518747\pi\)
−0.0588615 + 0.998266i \(0.518747\pi\)
\(450\) 0 0
\(451\) 24.2172i 1.14034i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) −16.3773 −0.767778
\(456\) 0 0
\(457\) 14.5004 0.678301 0.339150 0.940732i \(-0.389860\pi\)
0.339150 + 0.940732i \(0.389860\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 23.1930i 1.08020i 0.841600 + 0.540102i \(0.181614\pi\)
−0.841600 + 0.540102i \(0.818386\pi\)
\(462\) 0 0
\(463\) −26.0637 −1.21128 −0.605642 0.795737i \(-0.707084\pi\)
−0.605642 + 0.795737i \(0.707084\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 12.1953i 0.564332i 0.959366 + 0.282166i \(0.0910529\pi\)
−0.959366 + 0.282166i \(0.908947\pi\)
\(468\) 0 0
\(469\) − 1.72964i − 0.0798674i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 7.32031 0.336588
\(474\) 0 0
\(475\) 0.442384i 0.0202980i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 18.1883 0.831045 0.415523 0.909583i \(-0.363599\pi\)
0.415523 + 0.909583i \(0.363599\pi\)
\(480\) 0 0
\(481\) 8.68475 0.395990
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0.251046i 0.0113994i
\(486\) 0 0
\(487\) 2.13693 0.0968334 0.0484167 0.998827i \(-0.484582\pi\)
0.0484167 + 0.998827i \(0.484582\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 2.32773i − 0.105049i −0.998620 0.0525244i \(-0.983273\pi\)
0.998620 0.0525244i \(-0.0167267\pi\)
\(492\) 0 0
\(493\) 16.8601i 0.759341i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −23.6387 −1.06034
\(498\) 0 0
\(499\) − 16.9382i − 0.758260i −0.925343 0.379130i \(-0.876223\pi\)
0.925343 0.379130i \(-0.123777\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 8.22413 0.366696 0.183348 0.983048i \(-0.441307\pi\)
0.183348 + 0.983048i \(0.441307\pi\)
\(504\) 0 0
\(505\) 22.6598 1.00835
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 19.8145i − 0.878261i −0.898423 0.439131i \(-0.855287\pi\)
0.898423 0.439131i \(-0.144713\pi\)
\(510\) 0 0
\(511\) 20.8594 0.922764
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 26.5128i 1.16829i
\(516\) 0 0
\(517\) − 20.2677i − 0.891374i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 36.0978 1.58147 0.790737 0.612156i \(-0.209698\pi\)
0.790737 + 0.612156i \(0.209698\pi\)
\(522\) 0 0
\(523\) − 4.19475i − 0.183424i −0.995786 0.0917119i \(-0.970766\pi\)
0.995786 0.0917119i \(-0.0292339\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 8.70743 0.379301
\(528\) 0 0
\(529\) 30.6778 1.33382
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) − 15.1903i − 0.657967i
\(534\) 0 0
\(535\) −35.3188 −1.52696
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 14.3090i − 0.616331i
\(540\) 0 0
\(541\) − 6.12211i − 0.263210i −0.991302 0.131605i \(-0.957987\pi\)
0.991302 0.131605i \(-0.0420131\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −37.2398 −1.59518
\(546\) 0 0
\(547\) 4.88879i 0.209029i 0.994523 + 0.104515i \(0.0333289\pi\)
−0.994523 + 0.104515i \(0.966671\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.59857 −0.110703
\(552\) 0 0
\(553\) −28.8493 −1.22680
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 28.5987i − 1.21176i −0.795554 0.605882i \(-0.792820\pi\)
0.795554 0.605882i \(-0.207180\pi\)
\(558\) 0 0
\(559\) −4.59171 −0.194209
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 9.83345i 0.414431i 0.978295 + 0.207215i \(0.0664401\pi\)
−0.978295 + 0.207215i \(0.933560\pi\)
\(564\) 0 0
\(565\) − 30.5697i − 1.28608i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 10.3543 0.434075 0.217037 0.976163i \(-0.430361\pi\)
0.217037 + 0.976163i \(0.430361\pi\)
\(570\) 0 0
\(571\) 36.7988i 1.53998i 0.638055 + 0.769991i \(0.279739\pi\)
−0.638055 + 0.769991i \(0.720261\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.24113 0.135165
\(576\) 0 0
\(577\) −8.22871 −0.342566 −0.171283 0.985222i \(-0.554791\pi\)
−0.171283 + 0.985222i \(0.554791\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 25.5008i 1.05795i
\(582\) 0 0
\(583\) −18.5100 −0.766606
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 7.18047i − 0.296370i −0.988960 0.148185i \(-0.952657\pi\)
0.988960 0.148185i \(-0.0473431\pi\)
\(588\) 0 0
\(589\) 1.34204i 0.0552976i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 24.7687 1.01713 0.508564 0.861024i \(-0.330177\pi\)
0.508564 + 0.861024i \(0.330177\pi\)
\(594\) 0 0
\(595\) 45.6355i 1.87087i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 41.9307 1.71324 0.856621 0.515946i \(-0.172560\pi\)
0.856621 + 0.515946i \(0.172560\pi\)
\(600\) 0 0
\(601\) 2.10027 0.0856718 0.0428359 0.999082i \(-0.486361\pi\)
0.0428359 + 0.999082i \(0.486361\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 5.93422i 0.241260i
\(606\) 0 0
\(607\) 11.2877 0.458152 0.229076 0.973409i \(-0.426430\pi\)
0.229076 + 0.973409i \(0.426430\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.7130i 0.514315i
\(612\) 0 0
\(613\) − 31.4983i − 1.27220i −0.771605 0.636102i \(-0.780546\pi\)
0.771605 0.636102i \(-0.219454\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 11.4563 0.461214 0.230607 0.973047i \(-0.425929\pi\)
0.230607 + 0.973047i \(0.425929\pi\)
\(618\) 0 0
\(619\) − 37.6816i − 1.51455i −0.653095 0.757276i \(-0.726530\pi\)
0.653095 0.757276i \(-0.273470\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −3.43855 −0.137763
\(624\) 0 0
\(625\) −22.5924 −0.903695
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) − 24.2002i − 0.964925i
\(630\) 0 0
\(631\) −6.99670 −0.278534 −0.139267 0.990255i \(-0.544475\pi\)
−0.139267 + 0.990255i \(0.544475\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 5.63569i 0.223646i
\(636\) 0 0
\(637\) 8.97538i 0.355617i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 6.20749 0.245181 0.122591 0.992457i \(-0.460880\pi\)
0.122591 + 0.992457i \(0.460880\pi\)
\(642\) 0 0
\(643\) − 29.4008i − 1.15945i −0.814811 0.579726i \(-0.803159\pi\)
0.814811 0.579726i \(-0.196841\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 5.62285 0.221057 0.110529 0.993873i \(-0.464746\pi\)
0.110529 + 0.993873i \(0.464746\pi\)
\(648\) 0 0
\(649\) 35.9011 1.40924
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 34.9447i 1.36749i 0.729721 + 0.683745i \(0.239650\pi\)
−0.729721 + 0.683745i \(0.760350\pi\)
\(654\) 0 0
\(655\) −32.8064 −1.28185
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 37.8768i 1.47547i 0.675090 + 0.737736i \(0.264105\pi\)
−0.675090 + 0.737736i \(0.735895\pi\)
\(660\) 0 0
\(661\) − 6.77264i − 0.263425i −0.991288 0.131713i \(-0.957952\pi\)
0.991288 0.131713i \(-0.0420476\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −7.03360 −0.272751
\(666\) 0 0
\(667\) 19.0385i 0.737172i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 30.2748 1.16874
\(672\) 0 0
\(673\) 24.6355 0.949630 0.474815 0.880086i \(-0.342515\pi\)
0.474815 + 0.880086i \(0.342515\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.2956i 1.08749i 0.839251 + 0.543744i \(0.182994\pi\)
−0.839251 + 0.543744i \(0.817006\pi\)
\(678\) 0 0
\(679\) 0.387430 0.0148682
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 34.4333i 1.31755i 0.752339 + 0.658777i \(0.228926\pi\)
−0.752339 + 0.658777i \(0.771074\pi\)
\(684\) 0 0
\(685\) − 10.5004i − 0.401198i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 11.6105 0.442325
\(690\) 0 0
\(691\) − 2.19928i − 0.0836646i −0.999125 0.0418323i \(-0.986680\pi\)
0.999125 0.0418323i \(-0.0133195\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −20.9297 −0.793907
\(696\) 0 0
\(697\) −42.3282 −1.60329
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 36.0608i − 1.36200i −0.732285 0.680999i \(-0.761546\pi\)
0.732285 0.680999i \(-0.238454\pi\)
\(702\) 0 0
\(703\) 3.72986 0.140675
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 34.9701i − 1.31518i
\(708\) 0 0
\(709\) − 10.7299i − 0.402970i −0.979492 0.201485i \(-0.935423\pi\)
0.979492 0.201485i \(-0.0645768\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 9.83244 0.368228
\(714\) 0 0
\(715\) − 18.4524i − 0.690080i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 29.1037 1.08539 0.542693 0.839931i \(-0.317405\pi\)
0.542693 + 0.839931i \(0.317405\pi\)
\(720\) 0 0
\(721\) 40.9162 1.52380
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.14957i 0.0426938i
\(726\) 0 0
\(727\) 40.5850 1.50521 0.752607 0.658470i \(-0.228796\pi\)
0.752607 + 0.658470i \(0.228796\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 12.7949i 0.473235i
\(732\) 0 0
\(733\) 18.6206i 0.687766i 0.939012 + 0.343883i \(0.111742\pi\)
−0.939012 + 0.343883i \(0.888258\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.94880 0.0717849
\(738\) 0 0
\(739\) − 7.89939i − 0.290584i −0.989389 0.145292i \(-0.953588\pi\)
0.989389 0.145292i \(-0.0464121\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −14.4064 −0.528521 −0.264261 0.964451i \(-0.585128\pi\)
−0.264261 + 0.964451i \(0.585128\pi\)
\(744\) 0 0
\(745\) 1.54595 0.0566392
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 54.5061i 1.99161i
\(750\) 0 0
\(751\) −14.0914 −0.514204 −0.257102 0.966384i \(-0.582768\pi\)
−0.257102 + 0.966384i \(0.582768\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 33.5711i − 1.22178i
\(756\) 0 0
\(757\) 50.2311i 1.82568i 0.408318 + 0.912840i \(0.366116\pi\)
−0.408318 + 0.912840i \(0.633884\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −23.7483 −0.860875 −0.430437 0.902620i \(-0.641641\pi\)
−0.430437 + 0.902620i \(0.641641\pi\)
\(762\) 0 0
\(763\) 57.4707i 2.08058i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −22.5191 −0.813119
\(768\) 0 0
\(769\) −47.4035 −1.70942 −0.854708 0.519109i \(-0.826264\pi\)
−0.854708 + 0.519109i \(0.826264\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) − 11.8508i − 0.426242i −0.977026 0.213121i \(-0.931637\pi\)
0.977026 0.213121i \(-0.0683628\pi\)
\(774\) 0 0
\(775\) 0.593696 0.0213262
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) − 6.52385i − 0.233741i
\(780\) 0 0
\(781\) − 26.6339i − 0.953036i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0.301287 0.0107534
\(786\) 0 0
\(787\) − 26.3903i − 0.940713i −0.882477 0.470356i \(-0.844125\pi\)
0.882477 0.470356i \(-0.155875\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −47.1770 −1.67742
\(792\) 0 0
\(793\) −18.9900 −0.674356
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 14.1893i − 0.502610i −0.967908 0.251305i \(-0.919140\pi\)
0.967908 0.251305i \(-0.0808597\pi\)
\(798\) 0 0
\(799\) 35.4251 1.25325
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 23.5024i 0.829382i
\(804\) 0 0
\(805\) 51.5317i 1.81625i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 29.9998 1.05474 0.527368 0.849637i \(-0.323179\pi\)
0.527368 + 0.849637i \(0.323179\pi\)
\(810\) 0 0
\(811\) 39.8833i 1.40049i 0.713901 + 0.700246i \(0.246927\pi\)
−0.713901 + 0.700246i \(0.753073\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −17.9725 −0.629549
\(816\) 0 0
\(817\) −1.97202 −0.0689921
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) − 56.4395i − 1.96975i −0.173256 0.984877i \(-0.555429\pi\)
0.173256 0.984877i \(-0.444571\pi\)
\(822\) 0 0
\(823\) 41.7360 1.45482 0.727412 0.686201i \(-0.240723\pi\)
0.727412 + 0.686201i \(0.240723\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 17.4248i 0.605921i 0.953003 + 0.302960i \(0.0979750\pi\)
−0.953003 + 0.302960i \(0.902025\pi\)
\(828\) 0 0
\(829\) − 52.2723i − 1.81549i −0.419519 0.907747i \(-0.637801\pi\)
0.419519 0.907747i \(-0.362199\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 25.0100 0.866547
\(834\) 0 0
\(835\) − 11.0273i − 0.381617i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 40.8429 1.41005 0.705027 0.709180i \(-0.250935\pi\)
0.705027 + 0.709180i \(0.250935\pi\)
\(840\) 0 0
\(841\) 22.2474 0.767153
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 16.1788i − 0.556567i
\(846\) 0 0
\(847\) 9.15805 0.314674
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 27.3269i − 0.936754i
\(852\) 0 0
\(853\) − 42.8153i − 1.46597i −0.680245 0.732984i \(-0.738127\pi\)
0.680245 0.732984i \(-0.261873\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −23.2947 −0.795732 −0.397866 0.917444i \(-0.630249\pi\)
−0.397866 + 0.917444i \(0.630249\pi\)
\(858\) 0 0
\(859\) − 17.5879i − 0.600091i −0.953925 0.300045i \(-0.902998\pi\)
0.953925 0.300045i \(-0.0970018\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −11.8886 −0.404692 −0.202346 0.979314i \(-0.564857\pi\)
−0.202346 + 0.979314i \(0.564857\pi\)
\(864\) 0 0
\(865\) 6.70319 0.227915
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 32.5048i − 1.10265i
\(870\) 0 0
\(871\) −1.22239 −0.0414193
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 38.2795i 1.29408i
\(876\) 0 0
\(877\) 28.6698i 0.968111i 0.875037 + 0.484056i \(0.160837\pi\)
−0.875037 + 0.484056i \(0.839163\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 44.2297 1.49014 0.745069 0.666988i \(-0.232417\pi\)
0.745069 + 0.666988i \(0.232417\pi\)
\(882\) 0 0
\(883\) 26.5869i 0.894719i 0.894354 + 0.447359i \(0.147636\pi\)
−0.894354 + 0.447359i \(0.852364\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 16.4124 0.551074 0.275537 0.961290i \(-0.411144\pi\)
0.275537 + 0.961290i \(0.411144\pi\)
\(888\) 0 0
\(889\) 8.69735 0.291700
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 5.45991i 0.182709i
\(894\) 0 0
\(895\) −38.8327 −1.29803
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 3.48737i 0.116310i
\(900\) 0 0
\(901\) − 32.3529i − 1.07783i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −30.0583 −0.999172
\(906\) 0 0
\(907\) 47.3380i 1.57183i 0.618333 + 0.785916i \(0.287808\pi\)
−0.618333 + 0.785916i \(0.712192\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.30896 0.175894 0.0879469 0.996125i \(-0.471969\pi\)
0.0879469 + 0.996125i \(0.471969\pi\)
\(912\) 0 0
\(913\) −28.7320 −0.950889
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 50.6290i 1.67192i
\(918\) 0 0
\(919\) −49.2513 −1.62465 −0.812326 0.583204i \(-0.801799\pi\)
−0.812326 + 0.583204i \(0.801799\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 16.7063i 0.549893i
\(924\) 0 0
\(925\) − 1.65003i − 0.0542528i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 29.2990 0.961270 0.480635 0.876921i \(-0.340406\pi\)
0.480635 + 0.876921i \(0.340406\pi\)
\(930\) 0 0
\(931\) 3.85468i 0.126332i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −51.4179 −1.68154
\(936\) 0 0
\(937\) −6.35345 −0.207558 −0.103779 0.994600i \(-0.533093\pi\)
−0.103779 + 0.994600i \(0.533093\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) − 2.07523i − 0.0676505i −0.999428 0.0338253i \(-0.989231\pi\)
0.999428 0.0338253i \(-0.0107690\pi\)
\(942\) 0 0
\(943\) −47.7970 −1.55649
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 13.3618i 0.434201i 0.976149 + 0.217101i \(0.0696600\pi\)
−0.976149 + 0.217101i \(0.930340\pi\)
\(948\) 0 0
\(949\) − 14.7420i − 0.478546i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 52.6796 1.70646 0.853231 0.521534i \(-0.174640\pi\)
0.853231 + 0.521534i \(0.174640\pi\)
\(954\) 0 0
\(955\) − 10.4055i − 0.336714i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −16.2048 −0.523281
\(960\) 0 0
\(961\) −29.1989 −0.941901
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 21.3453i 0.687128i
\(966\) 0 0
\(967\) 7.17519 0.230739 0.115369 0.993323i \(-0.463195\pi\)
0.115369 + 0.993323i \(0.463195\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 14.6991i 0.471718i 0.971787 + 0.235859i \(0.0757903\pi\)
−0.971787 + 0.235859i \(0.924210\pi\)
\(972\) 0 0
\(973\) 32.3000i 1.03549i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −19.5470 −0.625365 −0.312682 0.949858i \(-0.601228\pi\)
−0.312682 + 0.949858i \(0.601228\pi\)
\(978\) 0 0
\(979\) − 3.87424i − 0.123821i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 24.0339 0.766562 0.383281 0.923632i \(-0.374794\pi\)
0.383281 + 0.923632i \(0.374794\pi\)
\(984\) 0 0
\(985\) −41.0339 −1.30745
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 14.4480i 0.459419i
\(990\) 0 0
\(991\) −50.5419 −1.60552 −0.802758 0.596305i \(-0.796635\pi\)
−0.802758 + 0.596305i \(0.796635\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 25.0469i − 0.794039i
\(996\) 0 0
\(997\) − 21.9225i − 0.694293i −0.937811 0.347147i \(-0.887151\pi\)
0.937811 0.347147i \(-0.112849\pi\)
\(998\) 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 5472.2.g.b.2737.4 16
3.2 odd 2 608.2.c.b.305.2 16
4.3 odd 2 1368.2.g.b.685.13 16
8.3 odd 2 1368.2.g.b.685.14 16
8.5 even 2 inner 5472.2.g.b.2737.13 16
12.11 even 2 152.2.c.b.77.4 yes 16
24.5 odd 2 608.2.c.b.305.15 16
24.11 even 2 152.2.c.b.77.3 16
48.5 odd 4 4864.2.a.bn.1.1 8
48.11 even 4 4864.2.a.bo.1.8 8
48.29 odd 4 4864.2.a.bp.1.8 8
48.35 even 4 4864.2.a.bq.1.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
152.2.c.b.77.3 16 24.11 even 2
152.2.c.b.77.4 yes 16 12.11 even 2
608.2.c.b.305.2 16 3.2 odd 2
608.2.c.b.305.15 16 24.5 odd 2
1368.2.g.b.685.13 16 4.3 odd 2
1368.2.g.b.685.14 16 8.3 odd 2
4864.2.a.bn.1.1 8 48.5 odd 4
4864.2.a.bo.1.8 8 48.11 even 4
4864.2.a.bp.1.8 8 48.29 odd 4
4864.2.a.bq.1.1 8 48.35 even 4
5472.2.g.b.2737.4 16 1.1 even 1 trivial
5472.2.g.b.2737.13 16 8.5 even 2 inner