Properties

Label 1764.2.k.m.1549.4
Level $1764$
Weight $2$
Character 1764.1549
Analytic conductor $14.086$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $8$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(361,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.k (of order \(3\), degree \(2\), 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: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: 8.0.12745506816.4
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 8x^{6} + 55x^{4} + 72x^{2} + 81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1549.4
Root \(0.581861 + 1.00781i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1549
Dual form 1764.2.k.m.361.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.87083 - 3.24037i) q^{5} +O(q^{10})\) \(q+(1.87083 - 3.24037i) q^{5} +(2.64575 + 4.58258i) q^{11} +4.24264 q^{13} +(1.87083 + 3.24037i) q^{17} +(1.41421 - 2.44949i) q^{19} +(-2.64575 + 4.58258i) q^{23} +(-4.50000 - 7.79423i) q^{25} +5.29150 q^{29} +(4.24264 + 7.34847i) q^{31} +(-2.00000 + 3.46410i) q^{37} -3.74166 q^{41} +8.00000 q^{43} +(-3.74166 + 6.48074i) q^{47} +(-5.29150 - 9.16515i) q^{53} +19.7990 q^{55} +(-3.74166 - 6.48074i) q^{59} +(4.94975 - 8.57321i) q^{61} +(7.93725 - 13.7477i) q^{65} +(-6.00000 - 10.3923i) q^{67} -15.8745 q^{71} +(0.707107 + 1.22474i) q^{73} +(2.00000 - 3.46410i) q^{79} +14.9666 q^{83} +14.0000 q^{85} +(-1.87083 + 3.24037i) q^{89} +(-5.29150 - 9.16515i) q^{95} +9.89949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 36 q^{25} - 16 q^{37} + 64 q^{43} - 48 q^{67} + 16 q^{79} + 112 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{3}\right)\)

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
\(5\) 1.87083 3.24037i 0.836660 1.44914i −0.0560116 0.998430i \(-0.517838\pi\)
0.892672 0.450708i \(-0.148828\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.64575 + 4.58258i 0.797724 + 1.38170i 0.921095 + 0.389338i \(0.127296\pi\)
−0.123371 + 0.992361i \(0.539370\pi\)
\(12\) 0 0
\(13\) 4.24264 1.17670 0.588348 0.808608i \(-0.299778\pi\)
0.588348 + 0.808608i \(0.299778\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.87083 + 3.24037i 0.453743 + 0.785905i 0.998615 0.0526138i \(-0.0167552\pi\)
−0.544872 + 0.838519i \(0.683422\pi\)
\(18\) 0 0
\(19\) 1.41421 2.44949i 0.324443 0.561951i −0.656957 0.753928i \(-0.728157\pi\)
0.981399 + 0.191977i \(0.0614899\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.64575 + 4.58258i −0.551677 + 0.955533i 0.446476 + 0.894795i \(0.352679\pi\)
−0.998154 + 0.0607377i \(0.980655\pi\)
\(24\) 0 0
\(25\) −4.50000 7.79423i −0.900000 1.55885i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.29150 0.982607 0.491304 0.870988i \(-0.336521\pi\)
0.491304 + 0.870988i \(0.336521\pi\)
\(30\) 0 0
\(31\) 4.24264 + 7.34847i 0.762001 + 1.31982i 0.941818 + 0.336124i \(0.109116\pi\)
−0.179817 + 0.983700i \(0.557551\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −2.00000 + 3.46410i −0.328798 + 0.569495i −0.982274 0.187453i \(-0.939977\pi\)
0.653476 + 0.756948i \(0.273310\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −3.74166 −0.584349 −0.292174 0.956365i \(-0.594379\pi\)
−0.292174 + 0.956365i \(0.594379\pi\)
\(42\) 0 0
\(43\) 8.00000 1.21999 0.609994 0.792406i \(-0.291172\pi\)
0.609994 + 0.792406i \(0.291172\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −3.74166 + 6.48074i −0.545777 + 0.945313i 0.452781 + 0.891622i \(0.350432\pi\)
−0.998558 + 0.0536913i \(0.982901\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −5.29150 9.16515i −0.726844 1.25893i −0.958211 0.286064i \(-0.907653\pi\)
0.231367 0.972867i \(-0.425680\pi\)
\(54\) 0 0
\(55\) 19.7990 2.66970
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −3.74166 6.48074i −0.487122 0.843721i 0.512768 0.858527i \(-0.328620\pi\)
−0.999890 + 0.0148066i \(0.995287\pi\)
\(60\) 0 0
\(61\) 4.94975 8.57321i 0.633750 1.09769i −0.353028 0.935613i \(-0.614848\pi\)
0.986778 0.162075i \(-0.0518186\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 7.93725 13.7477i 0.984495 1.70520i
\(66\) 0 0
\(67\) −6.00000 10.3923i −0.733017 1.26962i −0.955588 0.294706i \(-0.904778\pi\)
0.222571 0.974916i \(-0.428555\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −15.8745 −1.88396 −0.941979 0.335673i \(-0.891036\pi\)
−0.941979 + 0.335673i \(0.891036\pi\)
\(72\) 0 0
\(73\) 0.707107 + 1.22474i 0.0827606 + 0.143346i 0.904435 0.426612i \(-0.140293\pi\)
−0.821674 + 0.569958i \(0.806960\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 2.00000 3.46410i 0.225018 0.389742i −0.731307 0.682048i \(-0.761089\pi\)
0.956325 + 0.292306i \(0.0944227\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 14.9666 1.64280 0.821401 0.570352i \(-0.193193\pi\)
0.821401 + 0.570352i \(0.193193\pi\)
\(84\) 0 0
\(85\) 14.0000 1.51851
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.87083 + 3.24037i −0.198307 + 0.343479i −0.947980 0.318331i \(-0.896878\pi\)
0.749672 + 0.661809i \(0.230211\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −5.29150 9.16515i −0.542897 0.940325i
\(96\) 0 0
\(97\) 9.89949 1.00514 0.502571 0.864536i \(-0.332388\pi\)
0.502571 + 0.864536i \(0.332388\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.61249 9.72111i −0.558463 0.967287i −0.997625 0.0688787i \(-0.978058\pi\)
0.439162 0.898408i \(-0.355275\pi\)
\(102\) 0 0
\(103\) −1.41421 + 2.44949i −0.139347 + 0.241355i −0.927249 0.374444i \(-0.877834\pi\)
0.787903 + 0.615800i \(0.211167\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −2.64575 + 4.58258i −0.255774 + 0.443014i −0.965106 0.261861i \(-0.915664\pi\)
0.709331 + 0.704875i \(0.248997\pi\)
\(108\) 0 0
\(109\) −8.00000 13.8564i −0.766261 1.32720i −0.939577 0.342337i \(-0.888782\pi\)
0.173316 0.984866i \(-0.444552\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 9.89949 + 17.1464i 0.923133 + 1.59891i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −8.50000 + 14.7224i −0.772727 + 1.33840i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −14.9666 −1.33866
\(126\) 0 0
\(127\) −8.00000 −0.709885 −0.354943 0.934888i \(-0.615500\pi\)
−0.354943 + 0.934888i \(0.615500\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 7.93725 + 13.7477i 0.678125 + 1.17455i 0.975545 + 0.219801i \(0.0705407\pi\)
−0.297419 + 0.954747i \(0.596126\pi\)
\(138\) 0 0
\(139\) 5.65685 0.479808 0.239904 0.970797i \(-0.422884\pi\)
0.239904 + 0.970797i \(0.422884\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 11.2250 + 19.4422i 0.938679 + 1.62584i
\(144\) 0 0
\(145\) 9.89949 17.1464i 0.822108 1.42393i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −5.29150 + 9.16515i −0.433497 + 0.750838i −0.997172 0.0751583i \(-0.976054\pi\)
0.563675 + 0.825997i \(0.309387\pi\)
\(150\) 0 0
\(151\) 2.00000 + 3.46410i 0.162758 + 0.281905i 0.935857 0.352381i \(-0.114628\pi\)
−0.773099 + 0.634285i \(0.781294\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 31.7490 2.55014
\(156\) 0 0
\(157\) −4.94975 8.57321i −0.395033 0.684217i 0.598073 0.801442i \(-0.295933\pi\)
−0.993105 + 0.117225i \(0.962600\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 10.0000 17.3205i 0.783260 1.35665i −0.146772 0.989170i \(-0.546888\pi\)
0.930033 0.367477i \(-0.119778\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −7.48331 −0.579076 −0.289538 0.957166i \(-0.593502\pi\)
−0.289538 + 0.957166i \(0.593502\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 5.61249 9.72111i 0.426709 0.739082i −0.569869 0.821736i \(-0.693006\pi\)
0.996578 + 0.0826532i \(0.0263394\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7.93725 + 13.7477i 0.593258 + 1.02755i 0.993790 + 0.111271i \(0.0354920\pi\)
−0.400532 + 0.916283i \(0.631175\pi\)
\(180\) 0 0
\(181\) −4.24264 −0.315353 −0.157676 0.987491i \(-0.550400\pi\)
−0.157676 + 0.987491i \(0.550400\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 7.48331 + 12.9615i 0.550184 + 0.952947i
\(186\) 0 0
\(187\) −9.89949 + 17.1464i −0.723923 + 1.25387i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2.64575 + 4.58258i −0.191440 + 0.331584i −0.945728 0.324960i \(-0.894649\pi\)
0.754288 + 0.656544i \(0.227982\pi\)
\(192\) 0 0
\(193\) −1.00000 1.73205i −0.0719816 0.124676i 0.827788 0.561041i \(-0.189599\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −10.5830 −0.754008 −0.377004 0.926212i \(-0.623046\pi\)
−0.377004 + 0.926212i \(0.623046\pi\)
\(198\) 0 0
\(199\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −7.00000 + 12.1244i −0.488901 + 0.846802i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 14.9666 1.03526
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 14.9666 25.9230i 1.02072 1.76793i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 7.93725 + 13.7477i 0.533917 + 0.924772i
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −11.2250 19.4422i −0.745028 1.29043i −0.950182 0.311696i \(-0.899103\pi\)
0.205154 0.978730i \(-0.434230\pi\)
\(228\) 0 0
\(229\) −6.36396 + 11.0227i −0.420542 + 0.728401i −0.995993 0.0894361i \(-0.971494\pi\)
0.575450 + 0.817837i \(0.304827\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.93725 + 13.7477i −0.519987 + 0.900644i 0.479743 + 0.877409i \(0.340730\pi\)
−0.999730 + 0.0232346i \(0.992604\pi\)
\(234\) 0 0
\(235\) 14.0000 + 24.2487i 0.913259 + 1.58181i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −15.8745 −1.02684 −0.513418 0.858138i \(-0.671621\pi\)
−0.513418 + 0.858138i \(0.671621\pi\)
\(240\) 0 0
\(241\) −14.8492 25.7196i −0.956524 1.65675i −0.730842 0.682547i \(-0.760872\pi\)
−0.225682 0.974201i \(-0.572461\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.00000 10.3923i 0.381771 0.661247i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −7.48331 −0.472343 −0.236171 0.971711i \(-0.575893\pi\)
−0.236171 + 0.971711i \(0.575893\pi\)
\(252\) 0 0
\(253\) −28.0000 −1.76034
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −5.61249 + 9.72111i −0.350097 + 0.606386i −0.986266 0.165164i \(-0.947185\pi\)
0.636169 + 0.771550i \(0.280518\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −7.93725 13.7477i −0.489432 0.847721i 0.510494 0.859881i \(-0.329463\pi\)
−0.999926 + 0.0121601i \(0.996129\pi\)
\(264\) 0 0
\(265\) −39.5980 −2.43248
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −1.87083 3.24037i −0.114066 0.197569i 0.803340 0.595521i \(-0.203054\pi\)
−0.917406 + 0.397952i \(0.869721\pi\)
\(270\) 0 0
\(271\) 9.89949 17.1464i 0.601351 1.04157i −0.391265 0.920278i \(-0.627962\pi\)
0.992617 0.121293i \(-0.0387042\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 23.8118 41.2432i 1.43590 2.48706i
\(276\) 0 0
\(277\) 1.00000 + 1.73205i 0.0600842 + 0.104069i 0.894503 0.447062i \(-0.147530\pi\)
−0.834419 + 0.551131i \(0.814196\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −5.29150 −0.315665 −0.157832 0.987466i \(-0.550451\pi\)
−0.157832 + 0.987466i \(0.550451\pi\)
\(282\) 0 0
\(283\) 15.5563 + 26.9444i 0.924729 + 1.60168i 0.791996 + 0.610526i \(0.209042\pi\)
0.132733 + 0.991152i \(0.457625\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 1.50000 2.59808i 0.0882353 0.152828i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 26.1916 1.53013 0.765065 0.643953i \(-0.222707\pi\)
0.765065 + 0.643953i \(0.222707\pi\)
\(294\) 0 0
\(295\) −28.0000 −1.63022
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −11.2250 + 19.4422i −0.649157 + 1.12437i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −18.5203 32.0780i −1.06047 1.83678i
\(306\) 0 0
\(307\) −14.1421 −0.807134 −0.403567 0.914950i \(-0.632230\pi\)
−0.403567 + 0.914950i \(0.632230\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 3.74166 + 6.48074i 0.212170 + 0.367489i 0.952393 0.304872i \(-0.0986136\pi\)
−0.740223 + 0.672361i \(0.765280\pi\)
\(312\) 0 0
\(313\) −6.36396 + 11.0227i −0.359712 + 0.623040i −0.987913 0.155012i \(-0.950459\pi\)
0.628200 + 0.778052i \(0.283792\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −5.29150 + 9.16515i −0.297200 + 0.514766i −0.975494 0.220024i \(-0.929386\pi\)
0.678294 + 0.734791i \(0.262720\pi\)
\(318\) 0 0
\(319\) 14.0000 + 24.2487i 0.783850 + 1.35767i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 10.5830 0.588854
\(324\) 0 0
\(325\) −19.0919 33.0681i −1.05903 1.83429i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −2.00000 + 3.46410i −0.109930 + 0.190404i −0.915742 0.401768i \(-0.868396\pi\)
0.805812 + 0.592172i \(0.201729\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −44.8999 −2.45314
\(336\) 0 0
\(337\) 8.00000 0.435788 0.217894 0.975972i \(-0.430081\pi\)
0.217894 + 0.975972i \(0.430081\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −22.4499 + 38.8844i −1.21573 + 2.10571i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −7.93725 13.7477i −0.426094 0.738017i 0.570428 0.821348i \(-0.306777\pi\)
−0.996522 + 0.0833311i \(0.973444\pi\)
\(348\) 0 0
\(349\) −29.6985 −1.58972 −0.794862 0.606791i \(-0.792457\pi\)
−0.794862 + 0.606791i \(0.792457\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −9.35414 16.2019i −0.497871 0.862338i 0.502126 0.864794i \(-0.332551\pi\)
−0.999997 + 0.00245682i \(0.999218\pi\)
\(354\) 0 0
\(355\) −29.6985 + 51.4393i −1.57623 + 2.73011i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 2.64575 4.58258i 0.139637 0.241859i −0.787722 0.616031i \(-0.788740\pi\)
0.927359 + 0.374172i \(0.122073\pi\)
\(360\) 0 0
\(361\) 5.50000 + 9.52628i 0.289474 + 0.501383i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5.29150 0.276970
\(366\) 0 0
\(367\) −5.65685 9.79796i −0.295285 0.511449i 0.679766 0.733429i \(-0.262081\pi\)
−0.975051 + 0.221980i \(0.928748\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −5.00000 + 8.66025i −0.258890 + 0.448411i −0.965945 0.258748i \(-0.916690\pi\)
0.707055 + 0.707159i \(0.250023\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 22.4499 1.15623
\(378\) 0 0
\(379\) −20.0000 −1.02733 −0.513665 0.857991i \(-0.671713\pi\)
−0.513665 + 0.857991i \(0.671713\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −7.48331 + 12.9615i −0.382380 + 0.662301i −0.991402 0.130852i \(-0.958229\pi\)
0.609022 + 0.793153i \(0.291562\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −2.64575 4.58258i −0.134145 0.232346i 0.791126 0.611654i \(-0.209495\pi\)
−0.925270 + 0.379308i \(0.876162\pi\)
\(390\) 0 0
\(391\) −19.7990 −1.00128
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −7.48331 12.9615i −0.376526 0.652163i
\(396\) 0 0
\(397\) 16.2635 28.1691i 0.816239 1.41377i −0.0921950 0.995741i \(-0.529388\pi\)
0.908434 0.418027i \(-0.137278\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −7.93725 + 13.7477i −0.396368 + 0.686529i −0.993275 0.115782i \(-0.963063\pi\)
0.596907 + 0.802310i \(0.296396\pi\)
\(402\) 0 0
\(403\) 18.0000 + 31.1769i 0.896644 + 1.55303i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −21.1660 −1.04916
\(408\) 0 0
\(409\) −10.6066 18.3712i −0.524463 0.908396i −0.999594 0.0284813i \(-0.990933\pi\)
0.475132 0.879915i \(-0.342400\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 28.0000 48.4974i 1.37447 2.38064i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 7.48331 0.365584 0.182792 0.983152i \(-0.441487\pi\)
0.182792 + 0.983152i \(0.441487\pi\)
\(420\) 0 0
\(421\) 6.00000 0.292422 0.146211 0.989253i \(-0.453292\pi\)
0.146211 + 0.989253i \(0.453292\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 16.8375 29.1633i 0.816737 1.41463i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 7.93725 + 13.7477i 0.382324 + 0.662205i 0.991394 0.130912i \(-0.0417905\pi\)
−0.609070 + 0.793117i \(0.708457\pi\)
\(432\) 0 0
\(433\) 4.24264 0.203888 0.101944 0.994790i \(-0.467494\pi\)
0.101944 + 0.994790i \(0.467494\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 7.48331 + 12.9615i 0.357975 + 0.620032i
\(438\) 0 0
\(439\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −2.64575 + 4.58258i −0.125703 + 0.217725i −0.922008 0.387172i \(-0.873452\pi\)
0.796304 + 0.604896i \(0.206785\pi\)
\(444\) 0 0
\(445\) 7.00000 + 12.1244i 0.331832 + 0.574750i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) −9.89949 17.1464i −0.466149 0.807394i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −9.00000 + 15.5885i −0.421002 + 0.729197i −0.996038 0.0889312i \(-0.971655\pi\)
0.575036 + 0.818128i \(0.304988\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 3.74166 0.174266 0.0871332 0.996197i \(-0.472229\pi\)
0.0871332 + 0.996197i \(0.472229\pi\)
\(462\) 0 0
\(463\) −8.00000 −0.371792 −0.185896 0.982569i \(-0.559519\pi\)
−0.185896 + 0.982569i \(0.559519\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 18.7083 32.4037i 0.865716 1.49946i −0.000617928 1.00000i \(-0.500197\pi\)
0.866334 0.499465i \(-0.166470\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 21.1660 + 36.6606i 0.973214 + 1.68566i
\(474\) 0 0
\(475\) −25.4558 −1.16799
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 18.7083 + 32.4037i 0.854803 + 1.48056i 0.876827 + 0.480806i \(0.159656\pi\)
−0.0220238 + 0.999757i \(0.507011\pi\)
\(480\) 0 0
\(481\) −8.48528 + 14.6969i −0.386896 + 0.670123i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 18.5203 32.0780i 0.840962 1.45659i
\(486\) 0 0
\(487\) 6.00000 + 10.3923i 0.271886 + 0.470920i 0.969345 0.245705i \(-0.0790193\pi\)
−0.697459 + 0.716625i \(0.745686\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 15.8745 0.716407 0.358203 0.933644i \(-0.383389\pi\)
0.358203 + 0.933644i \(0.383389\pi\)
\(492\) 0 0
\(493\) 9.89949 + 17.1464i 0.445851 + 0.772236i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −16.0000 + 27.7128i −0.716258 + 1.24060i 0.246214 + 0.969216i \(0.420813\pi\)
−0.962472 + 0.271380i \(0.912520\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 14.9666 0.667329 0.333665 0.942692i \(-0.391715\pi\)
0.333665 + 0.942692i \(0.391715\pi\)
\(504\) 0 0
\(505\) −42.0000 −1.86898
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 20.5791 35.6441i 0.912153 1.57990i 0.101137 0.994873i \(-0.467752\pi\)
0.811017 0.585023i \(-0.198915\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 5.29150 + 9.16515i 0.233171 + 0.403865i
\(516\) 0 0
\(517\) −39.5980 −1.74152
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −9.35414 16.2019i −0.409812 0.709816i 0.585056 0.810993i \(-0.301073\pi\)
−0.994868 + 0.101177i \(0.967739\pi\)
\(522\) 0 0
\(523\) −8.48528 + 14.6969i −0.371035 + 0.642652i −0.989725 0.142983i \(-0.954331\pi\)
0.618690 + 0.785635i \(0.287664\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −15.8745 + 27.4955i −0.691504 + 1.19772i
\(528\) 0 0
\(529\) −2.50000 4.33013i −0.108696 0.188266i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −15.8745 −0.687601
\(534\) 0 0
\(535\) 9.89949 + 17.1464i 0.427992 + 0.741305i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −5.00000 + 8.66025i −0.214967 + 0.372333i −0.953262 0.302144i \(-0.902298\pi\)
0.738296 + 0.674477i \(0.235631\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −59.8665 −2.56440
\(546\) 0 0
\(547\) 16.0000 0.684111 0.342055 0.939680i \(-0.388877\pi\)
0.342055 + 0.939680i \(0.388877\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 7.48331 12.9615i 0.318800 0.552178i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 15.8745 + 27.4955i 0.672624 + 1.16502i 0.977157 + 0.212518i \(0.0681664\pi\)
−0.304533 + 0.952502i \(0.598500\pi\)
\(558\) 0 0
\(559\) 33.9411 1.43556
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3.74166 6.48074i −0.157692 0.273131i 0.776344 0.630309i \(-0.217072\pi\)
−0.934036 + 0.357179i \(0.883739\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −13.2288 + 22.9129i −0.554578 + 0.960558i 0.443358 + 0.896345i \(0.353787\pi\)
−0.997936 + 0.0642132i \(0.979546\pi\)
\(570\) 0 0
\(571\) −20.0000 34.6410i −0.836974 1.44968i −0.892413 0.451219i \(-0.850989\pi\)
0.0554391 0.998462i \(-0.482344\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 47.6235 1.98604
\(576\) 0 0
\(577\) −10.6066 18.3712i −0.441559 0.764802i 0.556247 0.831017i \(-0.312241\pi\)
−0.997805 + 0.0662152i \(0.978908\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 28.0000 48.4974i 1.15964 2.00856i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 22.4499 0.926608 0.463304 0.886199i \(-0.346664\pi\)
0.463304 + 0.886199i \(0.346664\pi\)
\(588\) 0 0
\(589\) 24.0000 0.988903
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −13.0958 + 22.6826i −0.537780 + 0.931462i 0.461243 + 0.887274i \(0.347404\pi\)
−0.999023 + 0.0441886i \(0.985930\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −2.64575 4.58258i −0.108102 0.187239i 0.806899 0.590689i \(-0.201144\pi\)
−0.915002 + 0.403450i \(0.867811\pi\)
\(600\) 0 0
\(601\) −4.24264 −0.173061 −0.0865305 0.996249i \(-0.527578\pi\)
−0.0865305 + 0.996249i \(0.527578\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 31.8041 + 55.0863i 1.29302 + 2.23958i
\(606\) 0 0
\(607\) 8.48528 14.6969i 0.344407 0.596530i −0.640839 0.767675i \(-0.721413\pi\)
0.985246 + 0.171145i \(0.0547467\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −15.8745 + 27.4955i −0.642214 + 1.11235i
\(612\) 0 0
\(613\) 12.0000 + 20.7846i 0.484675 + 0.839482i 0.999845 0.0176058i \(-0.00560439\pi\)
−0.515170 + 0.857088i \(0.672271\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 26.4575 1.06514 0.532570 0.846386i \(-0.321226\pi\)
0.532570 + 0.846386i \(0.321226\pi\)
\(618\) 0 0
\(619\) 5.65685 + 9.79796i 0.227368 + 0.393813i 0.957027 0.289998i \(-0.0936546\pi\)
−0.729659 + 0.683811i \(0.760321\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −5.50000 + 9.52628i −0.220000 + 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −14.9666 −0.596759
\(630\) 0 0
\(631\) −40.0000 −1.59237 −0.796187 0.605050i \(-0.793153\pi\)
−0.796187 + 0.605050i \(0.793153\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −14.9666 + 25.9230i −0.593933 + 1.02872i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −18.5203 32.0780i −0.731506 1.26701i −0.956239 0.292586i \(-0.905484\pi\)
0.224733 0.974420i \(-0.427849\pi\)
\(642\) 0 0
\(643\) 14.1421 0.557711 0.278856 0.960333i \(-0.410045\pi\)
0.278856 + 0.960333i \(0.410045\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −11.2250 19.4422i −0.441299 0.764353i 0.556487 0.830856i \(-0.312149\pi\)
−0.997786 + 0.0665037i \(0.978816\pi\)
\(648\) 0 0
\(649\) 19.7990 34.2929i 0.777178 1.34611i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 7.93725 13.7477i 0.310609 0.537990i −0.667886 0.744264i \(-0.732800\pi\)
0.978494 + 0.206274i \(0.0661338\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −47.6235 −1.85515 −0.927575 0.373638i \(-0.878110\pi\)
−0.927575 + 0.373638i \(0.878110\pi\)
\(660\) 0 0
\(661\) −10.6066 18.3712i −0.412549 0.714556i 0.582619 0.812746i \(-0.302028\pi\)
−0.995168 + 0.0981898i \(0.968695\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −14.0000 + 24.2487i −0.542082 + 0.938914i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 52.3832 2.02223
\(672\) 0 0
\(673\) 36.0000 1.38770 0.693849 0.720121i \(-0.255914\pi\)
0.693849 + 0.720121i \(0.255914\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −16.8375 + 29.1633i −0.647116 + 1.12084i 0.336692 + 0.941615i \(0.390692\pi\)
−0.983808 + 0.179223i \(0.942642\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 7.93725 + 13.7477i 0.303711 + 0.526042i 0.976973 0.213361i \(-0.0684411\pi\)
−0.673263 + 0.739403i \(0.735108\pi\)
\(684\) 0 0
\(685\) 59.3970 2.26944
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −22.4499 38.8844i −0.855275 1.48138i
\(690\) 0 0
\(691\) −8.48528 + 14.6969i −0.322795 + 0.559098i −0.981064 0.193685i \(-0.937956\pi\)
0.658268 + 0.752783i \(0.271289\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10.5830 18.3303i 0.401436 0.695308i
\(696\) 0 0
\(697\) −7.00000 12.1244i −0.265144 0.459243i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 15.8745 0.599572 0.299786 0.954006i \(-0.403085\pi\)
0.299786 + 0.954006i \(0.403085\pi\)
\(702\) 0 0
\(703\) 5.65685 + 9.79796i 0.213352 + 0.369537i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.00000 6.92820i 0.150223 0.260194i −0.781086 0.624423i \(-0.785334\pi\)
0.931309 + 0.364229i \(0.118667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −44.8999 −1.68151
\(714\) 0 0
\(715\) 84.0000 3.14142
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −14.9666 + 25.9230i −0.558161 + 0.966763i 0.439489 + 0.898248i \(0.355160\pi\)
−0.997650 + 0.0685154i \(0.978174\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −23.8118 41.2432i −0.884347 1.53173i
\(726\) 0 0
\(727\) 14.1421 0.524503 0.262251 0.965000i \(-0.415535\pi\)
0.262251 + 0.965000i \(0.415535\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 14.9666 + 25.9230i 0.553561 + 0.958795i
\(732\) 0 0
\(733\) −23.3345 + 40.4166i −0.861880 + 1.49282i 0.00823152 + 0.999966i \(0.497380\pi\)
−0.870112 + 0.492854i \(0.835954\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 31.7490 54.9909i 1.16949 2.02562i
\(738\) 0 0
\(739\) −6.00000 10.3923i −0.220714 0.382287i 0.734311 0.678813i \(-0.237505\pi\)
−0.955025 + 0.296526i \(0.904172\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −37.0405 −1.35888 −0.679442 0.733729i \(-0.737778\pi\)
−0.679442 + 0.733729i \(0.737778\pi\)
\(744\) 0 0
\(745\) 19.7990 + 34.2929i 0.725379 + 1.25639i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −16.0000 + 27.7128i −0.583848 + 1.01125i 0.411170 + 0.911559i \(0.365120\pi\)
−0.995018 + 0.0996961i \(0.968213\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 14.9666 0.544691
\(756\) 0 0
\(757\) −16.0000 −0.581530 −0.290765 0.956795i \(-0.593910\pi\)
−0.290765 + 0.956795i \(0.593910\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 20.5791 35.6441i 0.745992 1.29210i −0.203737 0.979026i \(-0.565309\pi\)
0.949730 0.313071i \(-0.101358\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −15.8745 27.4955i −0.573195 0.992803i
\(768\) 0 0
\(769\) 4.24264 0.152994 0.0764968 0.997070i \(-0.475627\pi\)
0.0764968 + 0.997070i \(0.475627\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −24.3208 42.1248i −0.874757 1.51512i −0.857021 0.515282i \(-0.827687\pi\)
−0.0177365 0.999843i \(-0.505646\pi\)
\(774\) 0 0
\(775\) 38.1838 66.1362i 1.37160 2.37568i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −5.29150 + 9.16515i −0.189588 + 0.328376i
\(780\) 0 0
\(781\) −42.0000 72.7461i −1.50288 2.60306i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −37.0405 −1.32203
\(786\) 0 0
\(787\) 19.7990 + 34.2929i 0.705758 + 1.22241i 0.966417 + 0.256978i \(0.0827266\pi\)
−0.260660 + 0.965431i \(0.583940\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 21.0000 36.3731i 0.745732 1.29165i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 11.2250 0.397609 0.198804 0.980039i \(-0.436294\pi\)
0.198804 + 0.980039i \(0.436294\pi\)
\(798\) 0 0
\(799\) −28.0000 −0.990569
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.74166 + 6.48074i −0.132040 + 0.228700i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −10.5830 18.3303i −0.372079 0.644459i 0.617806 0.786330i \(-0.288022\pi\)
−0.989885 + 0.141871i \(0.954688\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −37.4166 64.8074i −1.31065 2.27010i
\(816\) 0 0
\(817\) 11.3137 19.5959i 0.395817 0.685574i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 15.8745 27.4955i 0.554024 0.959598i −0.443955 0.896049i \(-0.646425\pi\)
0.997979 0.0635487i \(-0.0202418\pi\)
\(822\) 0 0
\(823\) −2.00000 3.46410i −0.0697156 0.120751i 0.829060 0.559159i \(-0.188876\pi\)
−0.898776 + 0.438408i \(0.855543\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −15.8745 −0.552011 −0.276005 0.961156i \(-0.589011\pi\)
−0.276005 + 0.961156i \(0.589011\pi\)
\(828\) 0 0
\(829\) 9.19239 + 15.9217i 0.319265 + 0.552983i 0.980335 0.197341i \(-0.0632307\pi\)
−0.661070 + 0.750324i \(0.729897\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −14.0000 + 24.2487i −0.484490 + 0.839161i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 7.48331 0.258353 0.129176 0.991622i \(-0.458767\pi\)
0.129176 + 0.991622i \(0.458767\pi\)
\(840\) 0 0
\(841\) −1.00000 −0.0344828
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 9.35414 16.2019i 0.321792 0.557361i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −10.5830 18.3303i −0.362781 0.628355i
\(852\) 0 0
\(853\) 55.1543 1.88845 0.944224 0.329304i \(-0.106814\pi\)
0.944224 + 0.329304i \(0.106814\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 20.5791 + 35.6441i 0.702969 + 1.21758i 0.967419 + 0.253179i \(0.0814762\pi\)
−0.264450 + 0.964399i \(0.585190\pi\)
\(858\) 0 0
\(859\) −9.89949 + 17.1464i −0.337766 + 0.585029i −0.984012 0.178101i \(-0.943005\pi\)
0.646246 + 0.763129i \(0.276338\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −2.64575 + 4.58258i −0.0900624 + 0.155993i −0.907537 0.419972i \(-0.862040\pi\)
0.817475 + 0.575964i \(0.195373\pi\)
\(864\) 0 0
\(865\) −21.0000 36.3731i −0.714021 1.23672i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 21.1660 0.718008
\(870\) 0 0
\(871\) −25.4558 44.0908i −0.862538 1.49396i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.00000 + 6.92820i −0.135070 + 0.233949i −0.925624 0.378444i \(-0.876459\pi\)
0.790554 + 0.612392i \(0.209793\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 56.1249 1.89089 0.945447 0.325775i \(-0.105625\pi\)
0.945447 + 0.325775i \(0.105625\pi\)
\(882\) 0 0
\(883\) 8.00000 0.269221 0.134611 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −18.7083 + 32.4037i −0.628163 + 1.08801i 0.359757 + 0.933046i \(0.382859\pi\)
−0.987920 + 0.154964i \(0.950474\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 10.5830 + 18.3303i 0.354147 + 0.613400i
\(894\) 0 0
\(895\) 59.3970 1.98542
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 22.4499 + 38.8844i 0.748748 + 1.29687i
\(900\) 0 0
\(901\) 19.7990 34.2929i 0.659600 1.14246i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) −7.93725 + 13.7477i −0.263843 + 0.456990i
\(906\) 0 0
\(907\) −8.00000 13.8564i −0.265636 0.460094i 0.702094 0.712084i \(-0.252248\pi\)
−0.967730 + 0.251990i \(0.918915\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 5.29150 0.175315 0.0876577 0.996151i \(-0.472062\pi\)
0.0876577 + 0.996151i \(0.472062\pi\)
\(912\) 0 0
\(913\) 39.5980 + 68.5857i 1.31050 + 2.26986i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −2.00000 + 3.46410i −0.0659739 + 0.114270i −0.897126 0.441776i \(-0.854349\pi\)
0.831152 + 0.556046i \(0.187682\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −67.3498 −2.21685
\(924\) 0 0
\(925\) 36.0000 1.18367
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −9.35414 + 16.2019i −0.306899 + 0.531566i −0.977682 0.210088i \(-0.932625\pi\)
0.670783 + 0.741654i \(0.265958\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 37.0405 + 64.1561i 1.21135 + 2.09813i
\(936\) 0 0
\(937\) 35.3553 1.15501 0.577504 0.816388i \(-0.304027\pi\)
0.577504 + 0.816388i \(0.304027\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 13.0958 + 22.6826i 0.426911 + 0.739431i 0.996597 0.0824312i \(-0.0262685\pi\)
−0.569686 + 0.821863i \(0.692935\pi\)
\(942\) 0 0
\(943\) 9.89949 17.1464i 0.322372 0.558365i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −23.8118 + 41.2432i −0.773778 + 1.34022i 0.161700 + 0.986840i \(0.448302\pi\)
−0.935479 + 0.353383i \(0.885031\pi\)
\(948\) 0 0
\(949\) 3.00000 + 5.19615i 0.0973841 + 0.168674i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 42.3320 1.37127 0.685634 0.727946i \(-0.259525\pi\)
0.685634 + 0.727946i \(0.259525\pi\)
\(954\) 0 0
\(955\) 9.89949 + 17.1464i 0.320340 + 0.554845i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −20.5000 + 35.5070i −0.661290 + 1.14539i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −7.48331 −0.240896
\(966\) 0 0
\(967\) 48.0000 1.54358 0.771788 0.635880i \(-0.219363\pi\)
0.771788 + 0.635880i \(0.219363\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.48331 12.9615i 0.240151 0.415954i −0.720606 0.693345i \(-0.756136\pi\)
0.960757 + 0.277391i \(0.0894697\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 7.93725 + 13.7477i 0.253935 + 0.439829i 0.964606 0.263696i \(-0.0849417\pi\)
−0.710671 + 0.703525i \(0.751608\pi\)
\(978\) 0 0
\(979\) −19.7990 −0.632778
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −7.48331 12.9615i −0.238681 0.413407i 0.721655 0.692253i \(-0.243382\pi\)
−0.960336 + 0.278846i \(0.910048\pi\)
\(984\) 0 0
\(985\) −19.7990 + 34.2929i −0.630848 + 1.09266i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −21.1660 + 36.6606i −0.673040 + 1.16574i
\(990\) 0 0
\(991\) −22.0000 38.1051i −0.698853 1.21045i −0.968864 0.247592i \(-0.920361\pi\)
0.270011 0.962857i \(-0.412973\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.707107 + 1.22474i 0.0223943 + 0.0387881i 0.877005 0.480481i \(-0.159538\pi\)
−0.854611 + 0.519269i \(0.826204\pi\)
\(998\) 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 1764.2.k.m.1549.4 8
3.2 odd 2 inner 1764.2.k.m.1549.1 8
7.2 even 3 1764.2.a.m.1.1 4
7.3 odd 6 inner 1764.2.k.m.361.2 8
7.4 even 3 inner 1764.2.k.m.361.4 8
7.5 odd 6 1764.2.a.m.1.3 yes 4
7.6 odd 2 inner 1764.2.k.m.1549.2 8
21.2 odd 6 1764.2.a.m.1.4 yes 4
21.5 even 6 1764.2.a.m.1.2 yes 4
21.11 odd 6 inner 1764.2.k.m.361.1 8
21.17 even 6 inner 1764.2.k.m.361.3 8
21.20 even 2 inner 1764.2.k.m.1549.3 8
28.19 even 6 7056.2.a.cy.1.4 4
28.23 odd 6 7056.2.a.cy.1.2 4
84.23 even 6 7056.2.a.cy.1.3 4
84.47 odd 6 7056.2.a.cy.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.2.a.m.1.1 4 7.2 even 3
1764.2.a.m.1.2 yes 4 21.5 even 6
1764.2.a.m.1.3 yes 4 7.5 odd 6
1764.2.a.m.1.4 yes 4 21.2 odd 6
1764.2.k.m.361.1 8 21.11 odd 6 inner
1764.2.k.m.361.2 8 7.3 odd 6 inner
1764.2.k.m.361.3 8 21.17 even 6 inner
1764.2.k.m.361.4 8 7.4 even 3 inner
1764.2.k.m.1549.1 8 3.2 odd 2 inner
1764.2.k.m.1549.2 8 7.6 odd 2 inner
1764.2.k.m.1549.3 8 21.20 even 2 inner
1764.2.k.m.1549.4 8 1.1 even 1 trivial
7056.2.a.cy.1.1 4 84.47 odd 6
7056.2.a.cy.1.2 4 28.23 odd 6
7056.2.a.cy.1.3 4 84.23 even 6
7056.2.a.cy.1.4 4 28.19 even 6