Properties

Label 720.3.c.b.449.3
Level $720$
Weight $3$
Character 720.449
Analytic conductor $19.619$
Analytic rank $0$
Dimension $4$
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] = [720,3,Mod(449,720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("720.449");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 720 = 2^{4} \cdot 3^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 720.c (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: \(19.6185790339\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{23})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 24x^{2} + 121 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 180)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 449.3
Root \(4.09827i\) of defining polynomial
Character \(\chi\) \(=\) 720.449
Dual form 720.3.c.b.449.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(4.79583 - 1.41421i) q^{5} +6.78233i q^{7} +O(q^{10})\) \(q+(4.79583 - 1.41421i) q^{5} +6.78233i q^{7} -9.89949i q^{11} -20.3470i q^{13} +19.1833 q^{17} -12.0000 q^{19} -9.59166 q^{23} +(21.0000 - 13.5647i) q^{25} -8.48528i q^{29} +38.0000 q^{31} +(9.59166 + 32.5269i) q^{35} -6.78233i q^{37} +69.2965i q^{41} -67.8233i q^{43} +76.7333 q^{47} +3.00000 q^{49} +(-14.0000 - 47.4763i) q^{55} +83.4386i q^{59} -70.0000 q^{61} +(-28.7750 - 97.5807i) q^{65} -108.517i q^{67} -118.794i q^{71} -13.5647i q^{73} +67.1416 q^{77} -30.0000 q^{79} +134.283 q^{83} +(92.0000 - 27.1293i) q^{85} +32.5269i q^{89} +138.000 q^{91} +(-57.5500 + 16.9706i) q^{95} -94.9526i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 48 q^{19} + 84 q^{25} + 152 q^{31} + 12 q^{49} - 56 q^{55} - 280 q^{61} - 120 q^{79} + 368 q^{85} + 552 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/720\mathbb{Z}\right)^\times\).

\(n\) \(181\) \(271\) \(577\) \(641\)
\(\chi(n)\) \(1\) \(1\) \(-1\) \(-1\)

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
\(5\) 4.79583 1.41421i 0.959166 0.282843i
\(6\) 0 0
\(7\) 6.78233i 0.968904i 0.874818 + 0.484452i \(0.160981\pi\)
−0.874818 + 0.484452i \(0.839019\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 9.89949i 0.899954i −0.893040 0.449977i \(-0.851432\pi\)
0.893040 0.449977i \(-0.148568\pi\)
\(12\) 0 0
\(13\) 20.3470i 1.56515i −0.622554 0.782577i \(-0.713905\pi\)
0.622554 0.782577i \(-0.286095\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 19.1833 1.12843 0.564215 0.825628i \(-0.309179\pi\)
0.564215 + 0.825628i \(0.309179\pi\)
\(18\) 0 0
\(19\) −12.0000 −0.631579 −0.315789 0.948829i \(-0.602269\pi\)
−0.315789 + 0.948829i \(0.602269\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −9.59166 −0.417029 −0.208514 0.978019i \(-0.566863\pi\)
−0.208514 + 0.978019i \(0.566863\pi\)
\(24\) 0 0
\(25\) 21.0000 13.5647i 0.840000 0.542586i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 8.48528i 0.292596i −0.989241 0.146298i \(-0.953264\pi\)
0.989241 0.146298i \(-0.0467358\pi\)
\(30\) 0 0
\(31\) 38.0000 1.22581 0.612903 0.790158i \(-0.290002\pi\)
0.612903 + 0.790158i \(0.290002\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 9.59166 + 32.5269i 0.274048 + 0.929340i
\(36\) 0 0
\(37\) 6.78233i 0.183306i −0.995791 0.0916531i \(-0.970785\pi\)
0.995791 0.0916531i \(-0.0292151\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 69.2965i 1.69016i 0.534642 + 0.845079i \(0.320447\pi\)
−0.534642 + 0.845079i \(0.679553\pi\)
\(42\) 0 0
\(43\) 67.8233i 1.57729i −0.614851 0.788643i \(-0.710784\pi\)
0.614851 0.788643i \(-0.289216\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 76.7333 1.63262 0.816312 0.577612i \(-0.196015\pi\)
0.816312 + 0.577612i \(0.196015\pi\)
\(48\) 0 0
\(49\) 3.00000 0.0612245
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) −14.0000 47.4763i −0.254545 0.863206i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 83.4386i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) −70.0000 −1.14754 −0.573770 0.819016i \(-0.694520\pi\)
−0.573770 + 0.819016i \(0.694520\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −28.7750 97.5807i −0.442692 1.50124i
\(66\) 0 0
\(67\) 108.517i 1.61966i −0.586664 0.809830i \(-0.699559\pi\)
0.586664 0.809830i \(-0.300441\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 118.794i 1.67315i −0.547849 0.836577i \(-0.684553\pi\)
0.547849 0.836577i \(-0.315447\pi\)
\(72\) 0 0
\(73\) 13.5647i 0.185817i −0.995675 0.0929086i \(-0.970384\pi\)
0.995675 0.0929086i \(-0.0296164\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 67.1416 0.871969
\(78\) 0 0
\(79\) −30.0000 −0.379747 −0.189873 0.981809i \(-0.560808\pi\)
−0.189873 + 0.981809i \(0.560808\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 134.283 1.61787 0.808935 0.587897i \(-0.200044\pi\)
0.808935 + 0.587897i \(0.200044\pi\)
\(84\) 0 0
\(85\) 92.0000 27.1293i 1.08235 0.319168i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 32.5269i 0.365471i 0.983162 + 0.182735i \(0.0584952\pi\)
−0.983162 + 0.182735i \(0.941505\pi\)
\(90\) 0 0
\(91\) 138.000 1.51648
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −57.5500 + 16.9706i −0.605789 + 0.178638i
\(96\) 0 0
\(97\) 94.9526i 0.978893i −0.872033 0.489446i \(-0.837199\pi\)
0.872033 0.489446i \(-0.162801\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 79.1960i 0.784118i 0.919940 + 0.392059i \(0.128237\pi\)
−0.919940 + 0.392059i \(0.871763\pi\)
\(102\) 0 0
\(103\) 88.1703i 0.856022i 0.903773 + 0.428011i \(0.140786\pi\)
−0.903773 + 0.428011i \(0.859214\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −57.5500 −0.537850 −0.268925 0.963161i \(-0.586668\pi\)
−0.268925 + 0.963161i \(0.586668\pi\)
\(108\) 0 0
\(109\) 74.0000 0.678899 0.339450 0.940624i \(-0.389759\pi\)
0.339450 + 0.940624i \(0.389759\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 172.650 1.52788 0.763938 0.645290i \(-0.223263\pi\)
0.763938 + 0.645290i \(0.223263\pi\)
\(114\) 0 0
\(115\) −46.0000 + 13.5647i −0.400000 + 0.117954i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 130.108i 1.09334i
\(120\) 0 0
\(121\) 23.0000 0.190083
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 81.5291 94.7523i 0.652233 0.758018i
\(126\) 0 0
\(127\) 169.558i 1.33510i 0.744563 + 0.667552i \(0.232658\pi\)
−0.744563 + 0.667552i \(0.767342\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 165.463i 1.26308i −0.775345 0.631538i \(-0.782424\pi\)
0.775345 0.631538i \(-0.217576\pi\)
\(132\) 0 0
\(133\) 81.3880i 0.611940i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −115.100 −0.840146 −0.420073 0.907490i \(-0.637995\pi\)
−0.420073 + 0.907490i \(0.637995\pi\)
\(138\) 0 0
\(139\) −62.0000 −0.446043 −0.223022 0.974814i \(-0.571592\pi\)
−0.223022 + 0.974814i \(0.571592\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −201.425 −1.40857
\(144\) 0 0
\(145\) −12.0000 40.6940i −0.0827586 0.280648i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 158.392i 1.06303i 0.847048 + 0.531517i \(0.178378\pi\)
−0.847048 + 0.531517i \(0.821622\pi\)
\(150\) 0 0
\(151\) −70.0000 −0.463576 −0.231788 0.972766i \(-0.574458\pi\)
−0.231788 + 0.972766i \(0.574458\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 182.242 53.7401i 1.17575 0.346710i
\(156\) 0 0
\(157\) 47.4763i 0.302397i 0.988503 + 0.151198i \(0.0483132\pi\)
−0.988503 + 0.151198i \(0.951687\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 65.0538i 0.404061i
\(162\) 0 0
\(163\) 94.9526i 0.582531i −0.956642 0.291266i \(-0.905924\pi\)
0.956642 0.291266i \(-0.0940764\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −201.425 −1.20614 −0.603069 0.797689i \(-0.706055\pi\)
−0.603069 + 0.797689i \(0.706055\pi\)
\(168\) 0 0
\(169\) −245.000 −1.44970
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −268.567 −1.55241 −0.776204 0.630482i \(-0.782857\pi\)
−0.776204 + 0.630482i \(0.782857\pi\)
\(174\) 0 0
\(175\) 92.0000 + 142.429i 0.525714 + 0.813880i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 134.350i 0.750560i 0.926911 + 0.375280i \(0.122453\pi\)
−0.926911 + 0.375280i \(0.877547\pi\)
\(180\) 0 0
\(181\) −22.0000 −0.121547 −0.0607735 0.998152i \(-0.519357\pi\)
−0.0607735 + 0.998152i \(0.519357\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −9.59166 32.5269i −0.0518468 0.175821i
\(186\) 0 0
\(187\) 189.905i 1.01554i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 195.161i 1.02179i 0.859644 + 0.510894i \(0.170686\pi\)
−0.859644 + 0.510894i \(0.829314\pi\)
\(192\) 0 0
\(193\) 271.293i 1.40566i 0.711356 + 0.702832i \(0.248081\pi\)
−0.711356 + 0.702832i \(0.751919\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −163.058 −0.827707 −0.413853 0.910344i \(-0.635817\pi\)
−0.413853 + 0.910344i \(0.635817\pi\)
\(198\) 0 0
\(199\) −294.000 −1.47739 −0.738693 0.674042i \(-0.764557\pi\)
−0.738693 + 0.674042i \(0.764557\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 57.5500 0.283497
\(204\) 0 0
\(205\) 98.0000 + 332.334i 0.478049 + 1.62114i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 118.794i 0.568392i
\(210\) 0 0
\(211\) 42.0000 0.199052 0.0995261 0.995035i \(-0.468267\pi\)
0.0995261 + 0.995035i \(0.468267\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −95.9166 325.269i −0.446124 1.51288i
\(216\) 0 0
\(217\) 257.729i 1.18769i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 390.323i 1.76617i
\(222\) 0 0
\(223\) 196.688i 0.882007i 0.897505 + 0.441004i \(0.145377\pi\)
−0.897505 + 0.441004i \(0.854623\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 268.567 1.18311 0.591556 0.806264i \(-0.298514\pi\)
0.591556 + 0.806264i \(0.298514\pi\)
\(228\) 0 0
\(229\) −422.000 −1.84279 −0.921397 0.388622i \(-0.872951\pi\)
−0.921397 + 0.388622i \(0.872951\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −211.017 −0.905651 −0.452825 0.891599i \(-0.649584\pi\)
−0.452825 + 0.891599i \(0.649584\pi\)
\(234\) 0 0
\(235\) 368.000 108.517i 1.56596 0.461776i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 70.7107i 0.295861i −0.988998 0.147930i \(-0.952739\pi\)
0.988998 0.147930i \(-0.0472611\pi\)
\(240\) 0 0
\(241\) 280.000 1.16183 0.580913 0.813966i \(-0.302696\pi\)
0.580913 + 0.813966i \(0.302696\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 14.3875 4.24264i 0.0587245 0.0173169i
\(246\) 0 0
\(247\) 244.164i 0.988518i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 80.6102i 0.321156i 0.987023 + 0.160578i \(0.0513358\pi\)
−0.987023 + 0.160578i \(0.948664\pi\)
\(252\) 0 0
\(253\) 94.9526i 0.375307i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 46.0000 0.177606
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −67.1416 −0.255291 −0.127646 0.991820i \(-0.540742\pi\)
−0.127646 + 0.991820i \(0.540742\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 263.044i 0.977858i 0.872324 + 0.488929i \(0.162612\pi\)
−0.872324 + 0.488929i \(0.837388\pi\)
\(270\) 0 0
\(271\) −322.000 −1.18819 −0.594096 0.804394i \(-0.702490\pi\)
−0.594096 + 0.804394i \(0.702490\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −134.283 207.889i −0.488303 0.755961i
\(276\) 0 0
\(277\) 33.9116i 0.122425i 0.998125 + 0.0612124i \(0.0194967\pi\)
−0.998125 + 0.0612124i \(0.980503\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 326.683i 1.16257i −0.813699 0.581287i \(-0.802549\pi\)
0.813699 0.581287i \(-0.197451\pi\)
\(282\) 0 0
\(283\) 108.517i 0.383453i 0.981448 + 0.191727i \(0.0614087\pi\)
−0.981448 + 0.191727i \(0.938591\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −469.991 −1.63760
\(288\) 0 0
\(289\) 79.0000 0.273356
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 201.425 0.687457 0.343729 0.939069i \(-0.388310\pi\)
0.343729 + 0.939069i \(0.388310\pi\)
\(294\) 0 0
\(295\) 118.000 + 400.157i 0.400000 + 1.35647i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 195.161i 0.652714i
\(300\) 0 0
\(301\) 460.000 1.52824
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −335.708 + 98.9949i −1.10068 + 0.324574i
\(306\) 0 0
\(307\) 162.776i 0.530215i 0.964219 + 0.265107i \(0.0854074\pi\)
−0.964219 + 0.265107i \(0.914593\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 178.191i 0.572961i 0.958086 + 0.286481i \(0.0924854\pi\)
−0.958086 + 0.286481i \(0.907515\pi\)
\(312\) 0 0
\(313\) 54.2586i 0.173350i −0.996237 0.0866751i \(-0.972376\pi\)
0.996237 0.0866751i \(-0.0276242\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 278.158 0.877471 0.438735 0.898616i \(-0.355427\pi\)
0.438735 + 0.898616i \(0.355427\pi\)
\(318\) 0 0
\(319\) −84.0000 −0.263323
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −230.200 −0.712693
\(324\) 0 0
\(325\) −276.000 427.287i −0.849231 1.31473i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 520.431i 1.58186i
\(330\) 0 0
\(331\) −40.0000 −0.120846 −0.0604230 0.998173i \(-0.519245\pi\)
−0.0604230 + 0.998173i \(0.519245\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −153.467 520.431i −0.458109 1.55352i
\(336\) 0 0
\(337\) 311.987i 0.925778i −0.886416 0.462889i \(-0.846813\pi\)
0.886416 0.462889i \(-0.153187\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 376.181i 1.10317i
\(342\) 0 0
\(343\) 352.681i 1.02822i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 402.850 1.16095 0.580475 0.814278i \(-0.302867\pi\)
0.580475 + 0.814278i \(0.302867\pi\)
\(348\) 0 0
\(349\) −406.000 −1.16332 −0.581662 0.813431i \(-0.697597\pi\)
−0.581662 + 0.813431i \(0.697597\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 537.133 1.52162 0.760812 0.648973i \(-0.224801\pi\)
0.760812 + 0.648973i \(0.224801\pi\)
\(354\) 0 0
\(355\) −168.000 569.716i −0.473239 1.60483i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 161.220i 0.449082i 0.974465 + 0.224541i \(0.0720882\pi\)
−0.974465 + 0.224541i \(0.927912\pi\)
\(360\) 0 0
\(361\) −217.000 −0.601108
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −19.1833 65.0538i −0.0525571 0.178230i
\(366\) 0 0
\(367\) 250.946i 0.683777i 0.939740 + 0.341889i \(0.111067\pi\)
−0.939740 + 0.341889i \(0.888933\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 101.735i 0.272748i −0.990657 0.136374i \(-0.956455\pi\)
0.990657 0.136374i \(-0.0435449\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −172.650 −0.457957
\(378\) 0 0
\(379\) −538.000 −1.41953 −0.709763 0.704441i \(-0.751198\pi\)
−0.709763 + 0.704441i \(0.751198\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 268.567 0.701218 0.350609 0.936522i \(-0.385975\pi\)
0.350609 + 0.936522i \(0.385975\pi\)
\(384\) 0 0
\(385\) 322.000 94.9526i 0.836364 0.246630i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 494.975i 1.27243i 0.771512 + 0.636214i \(0.219501\pi\)
−0.771512 + 0.636214i \(0.780499\pi\)
\(390\) 0 0
\(391\) −184.000 −0.470588
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −143.875 + 42.4264i −0.364240 + 0.107409i
\(396\) 0 0
\(397\) 278.076i 0.700442i 0.936667 + 0.350221i \(0.113894\pi\)
−0.936667 + 0.350221i \(0.886106\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 247.487i 0.617175i −0.951196 0.308588i \(-0.900144\pi\)
0.951196 0.308588i \(-0.0998563\pi\)
\(402\) 0 0
\(403\) 773.186i 1.91857i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −67.1416 −0.164967
\(408\) 0 0
\(409\) −242.000 −0.591687 −0.295844 0.955236i \(-0.595601\pi\)
−0.295844 + 0.955236i \(0.595601\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −565.908 −1.37024
\(414\) 0 0
\(415\) 644.000 189.905i 1.55181 0.457603i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 790.545i 1.88674i 0.331738 + 0.943372i \(0.392365\pi\)
−0.331738 + 0.943372i \(0.607635\pi\)
\(420\) 0 0
\(421\) 358.000 0.850356 0.425178 0.905110i \(-0.360211\pi\)
0.425178 + 0.905110i \(0.360211\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 402.850 260.215i 0.947882 0.612271i
\(426\) 0 0
\(427\) 474.763i 1.11186i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 14.1421i 0.0328124i −0.999865 0.0164062i \(-0.994778\pi\)
0.999865 0.0164062i \(-0.00522249\pi\)
\(432\) 0 0
\(433\) 257.729i 0.595216i −0.954688 0.297608i \(-0.903811\pi\)
0.954688 0.297608i \(-0.0961888\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 115.100 0.263387
\(438\) 0 0
\(439\) 690.000 1.57175 0.785877 0.618383i \(-0.212212\pi\)
0.785877 + 0.618383i \(0.212212\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 402.850 0.909368 0.454684 0.890653i \(-0.349752\pi\)
0.454684 + 0.890653i \(0.349752\pi\)
\(444\) 0 0
\(445\) 46.0000 + 155.994i 0.103371 + 0.350547i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 80.6102i 0.179533i 0.995963 + 0.0897663i \(0.0286120\pi\)
−0.995963 + 0.0897663i \(0.971388\pi\)
\(450\) 0 0
\(451\) 686.000 1.52106
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 661.825 195.161i 1.45456 0.428926i
\(456\) 0 0
\(457\) 339.116i 0.742049i 0.928623 + 0.371025i \(0.120993\pi\)
−0.928623 + 0.371025i \(0.879007\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 644.881i 1.39888i 0.714694 + 0.699438i \(0.246566\pi\)
−0.714694 + 0.699438i \(0.753434\pi\)
\(462\) 0 0
\(463\) 264.511i 0.571298i −0.958334 0.285649i \(-0.907791\pi\)
0.958334 0.285649i \(-0.0922091\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 460.400 0.985867 0.492933 0.870067i \(-0.335925\pi\)
0.492933 + 0.870067i \(0.335925\pi\)
\(468\) 0 0
\(469\) 736.000 1.56930
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −671.416 −1.41949
\(474\) 0 0
\(475\) −252.000 + 162.776i −0.530526 + 0.342686i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 79.1960i 0.165336i −0.996577 0.0826680i \(-0.973656\pi\)
0.996577 0.0826680i \(-0.0263441\pi\)
\(480\) 0 0
\(481\) −138.000 −0.286902
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −134.283 455.377i −0.276873 0.938921i
\(486\) 0 0
\(487\) 793.533i 1.62943i −0.579861 0.814715i \(-0.696893\pi\)
0.579861 0.814715i \(-0.303107\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 564.271i 1.14923i −0.818424 0.574614i \(-0.805152\pi\)
0.818424 0.574614i \(-0.194848\pi\)
\(492\) 0 0
\(493\) 162.776i 0.330174i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 805.700 1.62113
\(498\) 0 0
\(499\) 72.0000 0.144289 0.0721443 0.997394i \(-0.477016\pi\)
0.0721443 + 0.997394i \(0.477016\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −230.200 −0.457654 −0.228827 0.973467i \(-0.573489\pi\)
−0.228827 + 0.973467i \(0.573489\pi\)
\(504\) 0 0
\(505\) 112.000 + 379.810i 0.221782 + 0.752100i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 492.146i 0.966889i −0.875375 0.483444i \(-0.839386\pi\)
0.875375 0.483444i \(-0.160614\pi\)
\(510\) 0 0
\(511\) 92.0000 0.180039
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 124.692 + 422.850i 0.242120 + 0.821068i
\(516\) 0 0
\(517\) 759.621i 1.46929i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 69.2965i 0.133007i −0.997786 0.0665033i \(-0.978816\pi\)
0.997786 0.0665033i \(-0.0211843\pi\)
\(522\) 0 0
\(523\) 339.116i 0.648406i −0.945987 0.324203i \(-0.894904\pi\)
0.945987 0.324203i \(-0.105096\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 728.966 1.38324
\(528\) 0 0
\(529\) −437.000 −0.826087
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 1409.97 2.64536
\(534\) 0 0
\(535\) −276.000 + 81.3880i −0.515888 + 0.152127i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 29.6985i 0.0550992i
\(540\) 0 0
\(541\) −590.000 −1.09057 −0.545287 0.838250i \(-0.683579\pi\)
−0.545287 + 0.838250i \(0.683579\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 354.892 104.652i 0.651177 0.192022i
\(546\) 0 0
\(547\) 230.599i 0.421571i 0.977532 + 0.210785i \(0.0676021\pi\)
−0.977532 + 0.210785i \(0.932398\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 101.823i 0.184797i
\(552\) 0 0
\(553\) 203.470i 0.367938i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −690.600 −1.23986 −0.619928 0.784659i \(-0.712838\pi\)
−0.619928 + 0.784659i \(0.712838\pi\)
\(558\) 0 0
\(559\) −1380.00 −2.46869
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −19.1833 −0.0340734 −0.0170367 0.999855i \(-0.505423\pi\)
−0.0170367 + 0.999855i \(0.505423\pi\)
\(564\) 0 0
\(565\) 828.000 244.164i 1.46549 0.432148i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 108.894i 0.191379i 0.995411 + 0.0956893i \(0.0305055\pi\)
−0.995411 + 0.0956893i \(0.969494\pi\)
\(570\) 0 0
\(571\) 704.000 1.23292 0.616462 0.787384i \(-0.288565\pi\)
0.616462 + 0.787384i \(0.288565\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −201.425 + 130.108i −0.350304 + 0.226274i
\(576\) 0 0
\(577\) 1125.87i 1.95124i −0.219462 0.975621i \(-0.570430\pi\)
0.219462 0.975621i \(-0.429570\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 910.754i 1.56756i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −978.350 −1.66669 −0.833347 0.552750i \(-0.813578\pi\)
−0.833347 + 0.552750i \(0.813578\pi\)
\(588\) 0 0
\(589\) −456.000 −0.774194
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 184.000 + 623.974i 0.309244 + 1.04870i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 296.985i 0.495801i 0.968785 + 0.247901i \(0.0797406\pi\)
−0.968785 + 0.247901i \(0.920259\pi\)
\(600\) 0 0
\(601\) 20.0000 0.0332779 0.0166389 0.999862i \(-0.494703\pi\)
0.0166389 + 0.999862i \(0.494703\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 110.304 32.5269i 0.182321 0.0537635i
\(606\) 0 0
\(607\) 6.78233i 0.0111735i −0.999984 0.00558676i \(-0.998222\pi\)
0.999984 0.00558676i \(-0.00177833\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1561.29i 2.55531i
\(612\) 0 0
\(613\) 47.4763i 0.0774491i 0.999250 + 0.0387246i \(0.0123295\pi\)
−0.999250 + 0.0387246i \(0.987671\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −38.3667 −0.0621826 −0.0310913 0.999517i \(-0.509898\pi\)
−0.0310913 + 0.999517i \(0.509898\pi\)
\(618\) 0 0
\(619\) −178.000 −0.287561 −0.143780 0.989610i \(-0.545926\pi\)
−0.143780 + 0.989610i \(0.545926\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) −220.608 −0.354106
\(624\) 0 0
\(625\) 257.000 569.716i 0.411200 0.911545i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 130.108i 0.206848i
\(630\) 0 0
\(631\) 154.000 0.244057 0.122029 0.992527i \(-0.461060\pi\)
0.122029 + 0.992527i \(0.461060\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 239.792 + 813.173i 0.377625 + 1.28059i
\(636\) 0 0
\(637\) 61.0410i 0.0958257i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 807.516i 1.25978i 0.776686 + 0.629888i \(0.216899\pi\)
−0.776686 + 0.629888i \(0.783101\pi\)
\(642\) 0 0
\(643\) 1017.35i 1.58219i 0.611692 + 0.791096i \(0.290489\pi\)
−0.611692 + 0.791096i \(0.709511\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −805.700 −1.24529 −0.622643 0.782506i \(-0.713941\pi\)
−0.622643 + 0.782506i \(0.713941\pi\)
\(648\) 0 0
\(649\) 826.000 1.27273
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 67.1416 0.102820 0.0514101 0.998678i \(-0.483628\pi\)
0.0514101 + 0.998678i \(0.483628\pi\)
\(654\) 0 0
\(655\) −234.000 793.533i −0.357252 1.21150i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 524.673i 0.796166i 0.917349 + 0.398083i \(0.130324\pi\)
−0.917349 + 0.398083i \(0.869676\pi\)
\(660\) 0 0
\(661\) 74.0000 0.111952 0.0559758 0.998432i \(-0.482173\pi\)
0.0559758 + 0.998432i \(0.482173\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −115.100 390.323i −0.173083 0.586952i
\(666\) 0 0
\(667\) 81.3880i 0.122021i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 692.965i 1.03273i
\(672\) 0 0
\(673\) 691.798i 1.02793i −0.857811 0.513966i \(-0.828176\pi\)
0.857811 0.513966i \(-0.171824\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 278.158 0.410869 0.205434 0.978671i \(-0.434139\pi\)
0.205434 + 0.978671i \(0.434139\pi\)
\(678\) 0 0
\(679\) 644.000 0.948454
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 422.033 0.617911 0.308955 0.951077i \(-0.400021\pi\)
0.308955 + 0.951077i \(0.400021\pi\)
\(684\) 0 0
\(685\) −552.000 + 162.776i −0.805839 + 0.237629i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 68.0000 0.0984081 0.0492041 0.998789i \(-0.484332\pi\)
0.0492041 + 0.998789i \(0.484332\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −297.342 + 87.6812i −0.427830 + 0.126160i
\(696\) 0 0
\(697\) 1329.34i 1.90723i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 130.108i 0.185603i −0.995685 0.0928015i \(-0.970418\pi\)
0.995685 0.0928015i \(-0.0295822\pi\)
\(702\) 0 0
\(703\) 81.3880i 0.115772i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −537.133 −0.759736
\(708\) 0 0
\(709\) −210.000 −0.296192 −0.148096 0.988973i \(-0.547314\pi\)
−0.148096 + 0.988973i \(0.547314\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −364.483 −0.511197
\(714\) 0 0
\(715\) −966.000 + 284.858i −1.35105 + 0.398403i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 121.622i 0.169155i −0.996417 0.0845774i \(-0.973046\pi\)
0.996417 0.0845774i \(-0.0269540\pi\)
\(720\) 0 0
\(721\) −598.000 −0.829404
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −115.100 178.191i −0.158759 0.245781i
\(726\) 0 0
\(727\) 20.3470i 0.0279876i −0.999902 0.0139938i \(-0.995545\pi\)
0.999902 0.0139938i \(-0.00445451\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 1301.08i 1.77986i
\(732\) 0 0
\(733\) 1417.51i 1.93384i 0.255074 + 0.966922i \(0.417900\pi\)
−0.255074 + 0.966922i \(0.582100\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −1074.27 −1.45762
\(738\) 0 0
\(739\) −416.000 −0.562923 −0.281461 0.959573i \(-0.590819\pi\)
−0.281461 + 0.959573i \(0.590819\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 805.700 1.08439 0.542194 0.840254i \(-0.317594\pi\)
0.542194 + 0.840254i \(0.317594\pi\)
\(744\) 0 0
\(745\) 224.000 + 759.621i 0.300671 + 1.01963i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 390.323i 0.521125i
\(750\) 0 0
\(751\) −86.0000 −0.114514 −0.0572570 0.998359i \(-0.518235\pi\)
−0.0572570 + 0.998359i \(0.518235\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −335.708 + 98.9949i −0.444647 + 0.131119i
\(756\) 0 0
\(757\) 47.4763i 0.0627164i −0.999508 0.0313582i \(-0.990017\pi\)
0.999508 0.0313582i \(-0.00998326\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 369.110i 0.485033i 0.970147 + 0.242516i \(0.0779728\pi\)
−0.970147 + 0.242516i \(0.922027\pi\)
\(762\) 0 0
\(763\) 501.892i 0.657788i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1697.72 2.21346
\(768\) 0 0
\(769\) 384.000 0.499350 0.249675 0.968330i \(-0.419676\pi\)
0.249675 + 0.968330i \(0.419676\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −201.425 −0.260576 −0.130288 0.991476i \(-0.541590\pi\)
−0.130288 + 0.991476i \(0.541590\pi\)
\(774\) 0 0
\(775\) 798.000 515.457i 1.02968 0.665106i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 831.558i 1.06747i
\(780\) 0 0
\(781\) −1176.00 −1.50576
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 67.1416 + 227.688i 0.0855308 + 0.290049i
\(786\) 0 0
\(787\) 474.763i 0.603257i 0.953426 + 0.301628i \(0.0975302\pi\)
−0.953426 + 0.301628i \(0.902470\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1170.97i 1.48037i
\(792\) 0 0
\(793\) 1424.29i 1.79608i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −498.766 −0.625805 −0.312902 0.949785i \(-0.601301\pi\)
−0.312902 + 0.949785i \(0.601301\pi\)
\(798\) 0 0
\(799\) 1472.00 1.84230
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −134.283 −0.167227
\(804\) 0 0
\(805\) −92.0000 311.987i −0.114286 0.387562i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1480.68i 1.83026i −0.403157 0.915131i \(-0.632087\pi\)
0.403157 0.915131i \(-0.367913\pi\)
\(810\) 0 0
\(811\) 602.000 0.742293 0.371147 0.928574i \(-0.378965\pi\)
0.371147 + 0.928574i \(0.378965\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −134.283 455.377i −0.164765 0.558744i
\(816\) 0 0
\(817\) 813.880i 0.996181i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1405.73i 1.71221i −0.516798 0.856107i \(-0.672876\pi\)
0.516798 0.856107i \(-0.327124\pi\)
\(822\) 0 0
\(823\) 1281.86i 1.55755i −0.627306 0.778773i \(-0.715842\pi\)
0.627306 0.778773i \(-0.284158\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −1170.18 −1.41497 −0.707487 0.706727i \(-0.750171\pi\)
−0.707487 + 0.706727i \(0.750171\pi\)
\(828\) 0 0
\(829\) 1074.00 1.29554 0.647768 0.761837i \(-0.275702\pi\)
0.647768 + 0.761837i \(0.275702\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 57.5500 0.0690876
\(834\) 0 0
\(835\) −966.000 + 284.858i −1.15689 + 0.341147i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 743.876i 0.886623i −0.896368 0.443311i \(-0.853804\pi\)
0.896368 0.443311i \(-0.146196\pi\)
\(840\) 0 0
\(841\) 769.000 0.914388
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1174.98 + 346.482i −1.39051 + 0.410038i
\(846\) 0 0
\(847\) 155.994i 0.184172i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 65.0538i 0.0764440i
\(852\) 0 0
\(853\) 169.558i 0.198779i −0.995049 0.0993894i \(-0.968311\pi\)
0.995049 0.0993894i \(-0.0316889\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 115.100 0.134306 0.0671528 0.997743i \(-0.478608\pi\)
0.0671528 + 0.997743i \(0.478608\pi\)
\(858\) 0 0
\(859\) −154.000 −0.179278 −0.0896391 0.995974i \(-0.528571\pi\)
−0.0896391 + 0.995974i \(0.528571\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −1064.67 −1.23369 −0.616845 0.787085i \(-0.711589\pi\)
−0.616845 + 0.787085i \(0.711589\pi\)
\(864\) 0 0
\(865\) −1288.00 + 379.810i −1.48902 + 0.439087i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 296.985i 0.341755i
\(870\) 0 0
\(871\) −2208.00 −2.53502
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 642.641 + 552.958i 0.734447 + 0.631951i
\(876\) 0 0
\(877\) 1390.38i 1.58538i 0.609625 + 0.792690i \(0.291320\pi\)
−0.609625 + 0.792690i \(0.708680\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 861.256i 0.977589i 0.872399 + 0.488795i \(0.162563\pi\)
−0.872399 + 0.488795i \(0.837437\pi\)
\(882\) 0 0
\(883\) 1017.35i 1.15215i 0.817396 + 0.576076i \(0.195417\pi\)
−0.817396 + 0.576076i \(0.804583\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 604.275 0.681257 0.340628 0.940198i \(-0.389360\pi\)
0.340628 + 0.940198i \(0.389360\pi\)
\(888\) 0 0
\(889\) −1150.00 −1.29359
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −920.800 −1.03113
\(894\) 0 0
\(895\) 190.000 + 644.321i 0.212291 + 0.719912i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 322.441i 0.358666i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −105.508 + 31.1127i −0.116584 + 0.0343787i
\(906\) 0 0
\(907\) 935.962i 1.03193i 0.856609 + 0.515966i \(0.172567\pi\)
−0.856609 + 0.515966i \(0.827433\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 763.675i 0.838282i 0.907921 + 0.419141i \(0.137669\pi\)
−0.907921 + 0.419141i \(0.862331\pi\)
\(912\) 0 0
\(913\) 1329.34i 1.45601i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1122.22 1.22380
\(918\) 0 0
\(919\) 54.0000 0.0587595 0.0293798 0.999568i \(-0.490647\pi\)
0.0293798 + 0.999568i \(0.490647\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −2417.10 −2.61874
\(924\) 0 0
\(925\) −92.0000 142.429i −0.0994595 0.153977i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 148.492i 0.159841i 0.996801 + 0.0799206i \(0.0254667\pi\)
−0.996801 + 0.0799206i \(0.974533\pi\)
\(930\) 0 0
\(931\) −36.0000 −0.0386681
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −268.567 910.754i −0.287237 0.974068i
\(936\) 0 0
\(937\) 786.750i 0.839648i −0.907606 0.419824i \(-0.862092\pi\)
0.907606 0.419824i \(-0.137908\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 342.240i 0.363698i −0.983326 0.181849i \(-0.941792\pi\)
0.983326 0.181849i \(-0.0582082\pi\)
\(942\) 0 0
\(943\) 664.668i 0.704844i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 402.850 0.425396 0.212698 0.977118i \(-0.431775\pi\)
0.212698 + 0.977118i \(0.431775\pi\)
\(948\) 0 0
\(949\) −276.000 −0.290832
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1074.27 −1.12725 −0.563623 0.826032i \(-0.690593\pi\)
−0.563623 + 0.826032i \(0.690593\pi\)
\(954\) 0 0
\(955\) 276.000 + 935.962i 0.289005 + 0.980064i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 780.646i 0.814021i
\(960\) 0 0
\(961\) 483.000 0.502601
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 383.667 + 1301.08i 0.397582 + 1.34827i
\(966\) 0 0
\(967\) 400.157i 0.413813i 0.978361 + 0.206907i \(0.0663396\pi\)
−0.978361 + 0.206907i \(0.933660\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 683.065i 0.703466i 0.936100 + 0.351733i \(0.114407\pi\)
−0.936100 + 0.351733i \(0.885593\pi\)
\(972\) 0 0
\(973\) 420.504i 0.432173i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1611.40 1.64933 0.824667 0.565618i \(-0.191363\pi\)
0.824667 + 0.565618i \(0.191363\pi\)
\(978\) 0 0
\(979\) 322.000 0.328907
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 959.166 0.975754 0.487877 0.872912i \(-0.337771\pi\)
0.487877 + 0.872912i \(0.337771\pi\)
\(984\) 0 0
\(985\) −782.000 + 230.599i −0.793909 + 0.234111i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 650.538i 0.657774i
\(990\) 0 0
\(991\) −1054.00 −1.06357 −0.531786 0.846879i \(-0.678479\pi\)
−0.531786 + 0.846879i \(0.678479\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1409.97 + 415.779i −1.41706 + 0.417868i
\(996\) 0 0
\(997\) 1200.47i 1.20408i −0.798464 0.602042i \(-0.794354\pi\)
0.798464 0.602042i \(-0.205646\pi\)
\(998\) 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 720.3.c.b.449.3 4
3.2 odd 2 inner 720.3.c.b.449.2 4
4.3 odd 2 180.3.b.a.89.3 yes 4
5.2 odd 4 3600.3.l.r.1601.1 4
5.3 odd 4 3600.3.l.r.1601.3 4
5.4 even 2 inner 720.3.c.b.449.1 4
8.3 odd 2 2880.3.c.c.449.2 4
8.5 even 2 2880.3.c.f.449.2 4
12.11 even 2 180.3.b.a.89.2 yes 4
15.2 even 4 3600.3.l.r.1601.2 4
15.8 even 4 3600.3.l.r.1601.4 4
15.14 odd 2 inner 720.3.c.b.449.4 4
20.3 even 4 900.3.g.c.701.2 4
20.7 even 4 900.3.g.c.701.4 4
20.19 odd 2 180.3.b.a.89.1 4
24.5 odd 2 2880.3.c.f.449.3 4
24.11 even 2 2880.3.c.c.449.3 4
36.7 odd 6 1620.3.t.c.1349.2 8
36.11 even 6 1620.3.t.c.1349.3 8
36.23 even 6 1620.3.t.c.269.4 8
36.31 odd 6 1620.3.t.c.269.1 8
40.19 odd 2 2880.3.c.c.449.4 4
40.29 even 2 2880.3.c.f.449.4 4
60.23 odd 4 900.3.g.c.701.1 4
60.47 odd 4 900.3.g.c.701.3 4
60.59 even 2 180.3.b.a.89.4 yes 4
120.29 odd 2 2880.3.c.f.449.1 4
120.59 even 2 2880.3.c.c.449.1 4
180.59 even 6 1620.3.t.c.269.2 8
180.79 odd 6 1620.3.t.c.1349.4 8
180.119 even 6 1620.3.t.c.1349.1 8
180.139 odd 6 1620.3.t.c.269.3 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
180.3.b.a.89.1 4 20.19 odd 2
180.3.b.a.89.2 yes 4 12.11 even 2
180.3.b.a.89.3 yes 4 4.3 odd 2
180.3.b.a.89.4 yes 4 60.59 even 2
720.3.c.b.449.1 4 5.4 even 2 inner
720.3.c.b.449.2 4 3.2 odd 2 inner
720.3.c.b.449.3 4 1.1 even 1 trivial
720.3.c.b.449.4 4 15.14 odd 2 inner
900.3.g.c.701.1 4 60.23 odd 4
900.3.g.c.701.2 4 20.3 even 4
900.3.g.c.701.3 4 60.47 odd 4
900.3.g.c.701.4 4 20.7 even 4
1620.3.t.c.269.1 8 36.31 odd 6
1620.3.t.c.269.2 8 180.59 even 6
1620.3.t.c.269.3 8 180.139 odd 6
1620.3.t.c.269.4 8 36.23 even 6
1620.3.t.c.1349.1 8 180.119 even 6
1620.3.t.c.1349.2 8 36.7 odd 6
1620.3.t.c.1349.3 8 36.11 even 6
1620.3.t.c.1349.4 8 180.79 odd 6
2880.3.c.c.449.1 4 120.59 even 2
2880.3.c.c.449.2 4 8.3 odd 2
2880.3.c.c.449.3 4 24.11 even 2
2880.3.c.c.449.4 4 40.19 odd 2
2880.3.c.f.449.1 4 120.29 odd 2
2880.3.c.f.449.2 4 8.5 even 2
2880.3.c.f.449.3 4 24.5 odd 2
2880.3.c.f.449.4 4 40.29 even 2
3600.3.l.r.1601.1 4 5.2 odd 4
3600.3.l.r.1601.2 4 15.2 even 4
3600.3.l.r.1601.3 4 5.3 odd 4
3600.3.l.r.1601.4 4 15.8 even 4