Properties

Label 1728.2.r.f.289.1
Level $1728$
Weight $2$
Character 1728.289
Analytic conductor $13.798$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1728,2,Mod(289,1728)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1728, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 3, 4]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1728.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1728 = 2^{6} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1728.r (of order \(6\), 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: \(13.7981494693\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(6\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 16x^{8} - 24x^{7} + 96x^{5} + 304x^{4} + 384x^{3} + 288x^{2} + 144x + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{6}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 576)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 289.1
Root \(-0.180407 - 0.673288i\) of defining polynomial
Character \(\chi\) \(=\) 1728.289
Dual form 1728.2.r.f.1441.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(1.50000 - 0.866025i) q^{5} +(-2.17731 + 3.77121i) q^{7} +O(q^{10})\) \(q+(1.50000 - 0.866025i) q^{5} +(-2.17731 + 3.77121i) q^{7} +(-5.05328 - 2.91751i) q^{11} +(3.48133 - 2.00994i) q^{13} -1.70739 q^{17} +2.00000i q^{19} +(-3.32123 - 5.75253i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(-4.06108 - 2.34467i) q^{29} +(-2.43071 - 4.21012i) q^{31} +7.54241i q^{35} -5.75194i q^{37} +(-0.646305 - 1.11943i) q^{41} +(3.06782 + 1.77121i) q^{43} +(5.30669 - 9.19145i) q^{47} +(-5.98133 - 10.3600i) q^{49} +0.506816i q^{53} -10.1066 q^{55} +(1.62152 - 0.936184i) q^{59} +(-1.06108 - 0.612617i) q^{61} +(3.48133 - 6.02983i) q^{65} +(1.36811 - 0.789879i) q^{67} -9.88549 q^{71} -7.96265 q^{73} +(22.0051 - 12.7046i) q^{77} +(-1.03339 + 1.78988i) q^{79} +(1.33577 + 0.771205i) q^{83} +(-2.56108 + 1.47864i) q^{85} -3.96265 q^{89} +17.5051i q^{91} +(1.73205 + 3.00000i) q^{95} +(-3.06108 + 5.30195i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 18 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 18 q^{5} - 6 q^{13} - 12 q^{25} - 18 q^{29} - 18 q^{41} - 24 q^{49} + 18 q^{61} - 6 q^{65} + 90 q^{77} + 48 q^{89} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times\).

\(n\) \(325\) \(703\) \(1217\)
\(\chi(n)\) \(-1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) 1.50000 0.866025i 0.670820 0.387298i −0.125567 0.992085i \(-0.540075\pi\)
0.796387 + 0.604787i \(0.206742\pi\)
\(6\) 0 0
\(7\) −2.17731 + 3.77121i −0.822944 + 1.42538i 0.0805357 + 0.996752i \(0.474337\pi\)
−0.903480 + 0.428630i \(0.858996\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −5.05328 2.91751i −1.52362 0.879663i −0.999609 0.0279511i \(-0.991102\pi\)
−0.524011 0.851712i \(-0.675565\pi\)
\(12\) 0 0
\(13\) 3.48133 2.00994i 0.965546 0.557458i 0.0676707 0.997708i \(-0.478443\pi\)
0.897876 + 0.440249i \(0.145110\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −1.70739 −0.414103 −0.207051 0.978330i \(-0.566387\pi\)
−0.207051 + 0.978330i \(0.566387\pi\)
\(18\) 0 0
\(19\) 2.00000i 0.458831i 0.973329 + 0.229416i \(0.0736815\pi\)
−0.973329 + 0.229416i \(0.926318\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.32123 5.75253i −0.692523 1.19949i −0.971008 0.239045i \(-0.923166\pi\)
0.278485 0.960441i \(-0.410168\pi\)
\(24\) 0 0
\(25\) −1.00000 + 1.73205i −0.200000 + 0.346410i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −4.06108 2.34467i −0.754124 0.435394i 0.0730578 0.997328i \(-0.476724\pi\)
−0.827182 + 0.561934i \(0.810058\pi\)
\(30\) 0 0
\(31\) −2.43071 4.21012i −0.436569 0.756160i 0.560853 0.827915i \(-0.310473\pi\)
−0.997422 + 0.0717553i \(0.977140\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 7.54241i 1.27490i
\(36\) 0 0
\(37\) 5.75194i 0.945613i −0.881166 0.472807i \(-0.843241\pi\)
0.881166 0.472807i \(-0.156759\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −0.646305 1.11943i −0.100936 0.174826i 0.811135 0.584860i \(-0.198850\pi\)
−0.912071 + 0.410033i \(0.865517\pi\)
\(42\) 0 0
\(43\) 3.06782 + 1.77121i 0.467838 + 0.270106i 0.715334 0.698783i \(-0.246274\pi\)
−0.247496 + 0.968889i \(0.579608\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.30669 9.19145i 0.774060 1.34071i −0.161262 0.986912i \(-0.551556\pi\)
0.935322 0.353799i \(-0.115110\pi\)
\(48\) 0 0
\(49\) −5.98133 10.3600i −0.854475 1.47999i
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0.506816i 0.0696166i 0.999394 + 0.0348083i \(0.0110821\pi\)
−0.999394 + 0.0348083i \(0.988918\pi\)
\(54\) 0 0
\(55\) −10.1066 −1.36277
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1.62152 0.936184i 0.211104 0.121881i −0.390721 0.920509i \(-0.627774\pi\)
0.601824 + 0.798629i \(0.294441\pi\)
\(60\) 0 0
\(61\) −1.06108 0.612617i −0.135858 0.0784376i 0.430531 0.902576i \(-0.358326\pi\)
−0.566388 + 0.824138i \(0.691660\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 3.48133 6.02983i 0.431805 0.747909i
\(66\) 0 0
\(67\) 1.36811 0.789879i 0.167141 0.0964990i −0.414096 0.910233i \(-0.635902\pi\)
0.581237 + 0.813734i \(0.302569\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −9.88549 −1.17319 −0.586596 0.809880i \(-0.699532\pi\)
−0.586596 + 0.809880i \(0.699532\pi\)
\(72\) 0 0
\(73\) −7.96265 −0.931958 −0.465979 0.884796i \(-0.654298\pi\)
−0.465979 + 0.884796i \(0.654298\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 22.0051 12.7046i 2.50771 1.44783i
\(78\) 0 0
\(79\) −1.03339 + 1.78988i −0.116265 + 0.201377i −0.918285 0.395921i \(-0.870426\pi\)
0.802020 + 0.597298i \(0.203759\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 1.33577 + 0.771205i 0.146619 + 0.0846508i 0.571515 0.820592i \(-0.306356\pi\)
−0.424896 + 0.905242i \(0.639689\pi\)
\(84\) 0 0
\(85\) −2.56108 + 1.47864i −0.277789 + 0.160381i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −3.96265 −0.420040 −0.210020 0.977697i \(-0.567353\pi\)
−0.210020 + 0.977697i \(0.567353\pi\)
\(90\) 0 0
\(91\) 17.5051i 1.83503i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 1.73205 + 3.00000i 0.177705 + 0.307794i
\(96\) 0 0
\(97\) −3.06108 + 5.30195i −0.310806 + 0.538332i −0.978537 0.206071i \(-0.933932\pi\)
0.667731 + 0.744403i \(0.267266\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 10.0051 + 5.77643i 0.995541 + 0.574776i 0.906926 0.421290i \(-0.138423\pi\)
0.0886151 + 0.996066i \(0.471756\pi\)
\(102\) 0 0
\(103\) −0.110533 0.191448i −0.0108911 0.0188639i 0.860528 0.509402i \(-0.170133\pi\)
−0.871420 + 0.490538i \(0.836800\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.329957i 0.0318982i 0.999873 + 0.0159491i \(0.00507697\pi\)
−0.999873 + 0.0159491i \(0.994923\pi\)
\(108\) 0 0
\(109\) 17.9253i 1.71693i −0.512873 0.858465i \(-0.671419\pi\)
0.512873 0.858465i \(-0.328581\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −9.04241 15.6619i −0.850638 1.47335i −0.880633 0.473799i \(-0.842882\pi\)
0.0299946 0.999550i \(-0.490451\pi\)
\(114\) 0 0
\(115\) −9.96368 5.75253i −0.929118 0.536426i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 3.71751 6.43892i 0.340784 0.590254i
\(120\) 0 0
\(121\) 11.5237 + 19.9597i 1.04761 + 1.81452i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 12.1244i 1.08444i
\(126\) 0 0
\(127\) −4.57568 −0.406026 −0.203013 0.979176i \(-0.565073\pi\)
−0.203013 + 0.979176i \(0.565073\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −11.7281 + 6.77121i −1.02469 + 0.591603i −0.915458 0.402414i \(-0.868171\pi\)
−0.109228 + 0.994017i \(0.534838\pi\)
\(132\) 0 0
\(133\) −7.54241 4.35461i −0.654010 0.377593i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −11.1887 + 19.3794i −0.955917 + 1.65570i −0.223660 + 0.974667i \(0.571800\pi\)
−0.732257 + 0.681029i \(0.761533\pi\)
\(138\) 0 0
\(139\) −9.99602 + 5.77121i −0.847851 + 0.489507i −0.859925 0.510420i \(-0.829490\pi\)
0.0120739 + 0.999927i \(0.496157\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −23.4561 −1.96150
\(144\) 0 0
\(145\) −8.12217 −0.674509
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1.93892 1.11943i 0.158842 0.0917076i −0.418472 0.908230i \(-0.637434\pi\)
0.577314 + 0.816522i \(0.304101\pi\)
\(150\) 0 0
\(151\) 8.51738 14.7525i 0.693134 1.20054i −0.277671 0.960676i \(-0.589563\pi\)
0.970806 0.239868i \(-0.0771041\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −7.29214 4.21012i −0.585719 0.338165i
\(156\) 0 0
\(157\) −0.0424108 + 0.0244859i −0.00338475 + 0.00195419i −0.501691 0.865047i \(-0.667289\pi\)
0.498307 + 0.867001i \(0.333955\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 28.9253 2.27963
\(162\) 0 0
\(163\) 19.0848i 1.49484i −0.664353 0.747419i \(-0.731293\pi\)
0.664353 0.747419i \(-0.268707\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.99602 + 17.3136i 0.773515 + 1.33977i 0.935625 + 0.352995i \(0.114837\pi\)
−0.162110 + 0.986773i \(0.551830\pi\)
\(168\) 0 0
\(169\) 1.57976 2.73622i 0.121520 0.210478i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −10.5000 6.06218i −0.798300 0.460899i 0.0445762 0.999006i \(-0.485806\pi\)
−0.842876 + 0.538107i \(0.819140\pi\)
\(174\) 0 0
\(175\) −4.35461 7.54241i −0.329178 0.570153i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.00000i 0.448461i −0.974536 0.224231i \(-0.928013\pi\)
0.974536 0.224231i \(-0.0719869\pi\)
\(180\) 0 0
\(181\) 5.75194i 0.427538i 0.976884 + 0.213769i \(0.0685740\pi\)
−0.976884 + 0.213769i \(0.931426\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −4.98133 8.62791i −0.366234 0.634337i
\(186\) 0 0
\(187\) 8.62791 + 4.98133i 0.630935 + 0.364271i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −0.110533 + 0.191448i −0.00799786 + 0.0138527i −0.869997 0.493058i \(-0.835879\pi\)
0.861999 + 0.506910i \(0.169212\pi\)
\(192\) 0 0
\(193\) −9.02374 15.6296i −0.649543 1.12504i −0.983232 0.182358i \(-0.941627\pi\)
0.333689 0.942683i \(-0.391706\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 9.82080i 0.699703i −0.936805 0.349852i \(-0.886232\pi\)
0.936805 0.349852i \(-0.113768\pi\)
\(198\) 0 0
\(199\) 0.604760 0.0428703 0.0214352 0.999770i \(-0.493176\pi\)
0.0214352 + 0.999770i \(0.493176\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 17.6844 10.2101i 1.24120 0.716610i
\(204\) 0 0
\(205\) −1.93892 1.11943i −0.135420 0.0781846i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 5.83502 10.1066i 0.403617 0.699085i
\(210\) 0 0
\(211\) 0.363941 0.210121i 0.0250547 0.0144654i −0.487420 0.873167i \(-0.662062\pi\)
0.512475 + 0.858702i \(0.328729\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 6.13564 0.418447
\(216\) 0 0
\(217\) 21.1696 1.43709
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −5.94398 + 3.43176i −0.399835 + 0.230845i
\(222\) 0 0
\(223\) −11.3933 + 19.7339i −0.762955 + 1.32148i 0.178366 + 0.983964i \(0.442919\pi\)
−0.941321 + 0.337513i \(0.890414\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.2494 + 5.91751i 0.680279 + 0.392759i 0.799960 0.600053i \(-0.204854\pi\)
−0.119681 + 0.992812i \(0.538187\pi\)
\(228\) 0 0
\(229\) 13.5848 7.84320i 0.897710 0.518293i 0.0212537 0.999774i \(-0.493234\pi\)
0.876457 + 0.481481i \(0.159901\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0.829557 0.0543461 0.0271731 0.999631i \(-0.491349\pi\)
0.0271731 + 0.999631i \(0.491349\pi\)
\(234\) 0 0
\(235\) 18.3829i 1.19917i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 3.60698 + 6.24747i 0.233316 + 0.404115i 0.958782 0.284143i \(-0.0917090\pi\)
−0.725466 + 0.688258i \(0.758376\pi\)
\(240\) 0 0
\(241\) 10.4627 18.1218i 0.673959 1.16733i −0.302813 0.953050i \(-0.597926\pi\)
0.976772 0.214281i \(-0.0687408\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −17.9440 10.3600i −1.14640 0.661874i
\(246\) 0 0
\(247\) 4.01989 + 6.96265i 0.255779 + 0.443023i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 20.2553i 1.27850i 0.768999 + 0.639250i \(0.220755\pi\)
−0.768999 + 0.639250i \(0.779245\pi\)
\(252\) 0 0
\(253\) 38.7588i 2.43675i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −3.37237 5.84111i −0.210363 0.364359i 0.741465 0.670991i \(-0.234131\pi\)
−0.951828 + 0.306632i \(0.900798\pi\)
\(258\) 0 0
\(259\) 21.6917 + 12.5237i 1.34786 + 0.778187i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 13.2067 22.8747i 0.814361 1.41051i −0.0954250 0.995437i \(-0.530421\pi\)
0.909786 0.415078i \(-0.136246\pi\)
\(264\) 0 0
\(265\) 0.438916 + 0.760225i 0.0269624 + 0.0467002i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 23.1704i 1.41272i 0.707851 + 0.706362i \(0.249665\pi\)
−0.707851 + 0.706362i \(0.750335\pi\)
\(270\) 0 0
\(271\) 17.7626 1.07900 0.539502 0.841984i \(-0.318613\pi\)
0.539502 + 0.841984i \(0.318613\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 10.1066 5.83502i 0.609448 0.351865i
\(276\) 0 0
\(277\) 9.04241 + 5.22064i 0.543306 + 0.313678i 0.746418 0.665478i \(-0.231772\pi\)
−0.203112 + 0.979156i \(0.565106\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −13.4440 + 23.2857i −0.802001 + 1.38911i 0.116297 + 0.993215i \(0.462898\pi\)
−0.918297 + 0.395891i \(0.870436\pi\)
\(282\) 0 0
\(283\) 16.1317 9.31362i 0.958927 0.553637i 0.0630847 0.998008i \(-0.479906\pi\)
0.895843 + 0.444371i \(0.146573\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 5.62882 0.332259
\(288\) 0 0
\(289\) −14.0848 −0.828519
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −1.00506 + 0.580274i −0.0587165 + 0.0339000i −0.529071 0.848578i \(-0.677459\pi\)
0.470354 + 0.882478i \(0.344126\pi\)
\(294\) 0 0
\(295\) 1.62152 2.80855i 0.0944084 0.163520i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −23.1245 13.3510i −1.33733 0.772106i
\(300\) 0 0
\(301\) −13.3592 + 7.71291i −0.770009 + 0.444565i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −2.12217 −0.121515
\(306\) 0 0
\(307\) 19.0848i 1.08923i 0.838687 + 0.544614i \(0.183324\pi\)
−0.838687 + 0.544614i \(0.816676\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 5.56009 + 9.63036i 0.315284 + 0.546088i 0.979498 0.201455i \(-0.0645669\pi\)
−0.664214 + 0.747543i \(0.731234\pi\)
\(312\) 0 0
\(313\) −13.4627 + 23.3180i −0.760954 + 1.31801i 0.181405 + 0.983408i \(0.441935\pi\)
−0.942359 + 0.334603i \(0.891398\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −19.4440 11.2260i −1.09208 0.630514i −0.157953 0.987447i \(-0.550489\pi\)
−0.934130 + 0.356932i \(0.883823\pi\)
\(318\) 0 0
\(319\) 13.6812 + 23.6965i 0.765999 + 1.32675i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 3.41478i 0.190003i
\(324\) 0 0
\(325\) 8.03978i 0.445967i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 23.1086 + 40.0252i 1.27402 + 2.20666i
\(330\) 0 0
\(331\) −19.5634 11.2949i −1.07530 0.620826i −0.145677 0.989332i \(-0.546536\pi\)
−0.929625 + 0.368506i \(0.879869\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.36811 2.36964i 0.0747478 0.129467i
\(336\) 0 0
\(337\) 7.04241 + 12.1978i 0.383625 + 0.664457i 0.991577 0.129516i \(-0.0413423\pi\)
−0.607953 + 0.793973i \(0.708009\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 28.3665i 1.53613i
\(342\) 0 0
\(343\) 21.6104 1.16685
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 6.05745 3.49727i 0.325181 0.187743i −0.328519 0.944497i \(-0.606549\pi\)
0.653699 + 0.756754i \(0.273216\pi\)
\(348\) 0 0
\(349\) 8.16458 + 4.71382i 0.437040 + 0.252325i 0.702341 0.711840i \(-0.252138\pi\)
−0.265301 + 0.964166i \(0.585471\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −3.81128 + 6.60134i −0.202854 + 0.351354i −0.949447 0.313928i \(-0.898355\pi\)
0.746593 + 0.665281i \(0.231688\pi\)
\(354\) 0 0
\(355\) −14.8282 + 8.56108i −0.787001 + 0.454375i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −10.8344 −0.571820 −0.285910 0.958257i \(-0.592296\pi\)
−0.285910 + 0.958257i \(0.592296\pi\)
\(360\) 0 0
\(361\) 15.0000 0.789474
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −11.9440 + 6.89586i −0.625176 + 0.360946i
\(366\) 0 0
\(367\) 11.4747 19.8747i 0.598973 1.03745i −0.394000 0.919110i \(-0.628909\pi\)
0.992973 0.118341i \(-0.0377575\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.91131 1.10349i −0.0992302 0.0572906i
\(372\) 0 0
\(373\) 0.0984305 0.0568289i 0.00509654 0.00294249i −0.497450 0.867493i \(-0.665730\pi\)
0.502546 + 0.864550i \(0.332397\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −18.8506 −0.970856
\(378\) 0 0
\(379\) 16.2070i 0.832497i −0.909251 0.416249i \(-0.863345\pi\)
0.909251 0.416249i \(-0.136655\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 7.03874 + 12.1914i 0.359663 + 0.622954i 0.987904 0.155064i \(-0.0495585\pi\)
−0.628242 + 0.778018i \(0.716225\pi\)
\(384\) 0 0
\(385\) 22.0051 38.1139i 1.12148 1.94246i
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −25.0051 14.4367i −1.26781 0.731969i −0.293234 0.956041i \(-0.594732\pi\)
−0.974573 + 0.224072i \(0.928065\pi\)
\(390\) 0 0
\(391\) 5.67063 + 9.82181i 0.286776 + 0.496710i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 3.57976i 0.180117i
\(396\) 0 0
\(397\) 2.85934i 0.143506i −0.997422 0.0717531i \(-0.977141\pi\)
0.997422 0.0717531i \(-0.0228594\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 10.7739 + 18.6610i 0.538025 + 0.931886i 0.999010 + 0.0444786i \(0.0141626\pi\)
−0.460986 + 0.887408i \(0.652504\pi\)
\(402\) 0 0
\(403\) −16.9242 9.77121i −0.843056 0.486738i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −16.7813 + 29.0661i −0.831821 + 1.44076i
\(408\) 0 0
\(409\) 10.4627 + 18.1218i 0.517345 + 0.896068i 0.999797 + 0.0201454i \(0.00641290\pi\)
−0.482452 + 0.875922i \(0.660254\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 8.15344i 0.401204i
\(414\) 0 0
\(415\) 2.67153 0.131140
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.9678 6.33229i 0.535814 0.309353i −0.207567 0.978221i \(-0.566554\pi\)
0.743381 + 0.668868i \(0.233221\pi\)
\(420\) 0 0
\(421\) 23.1086 + 13.3417i 1.12624 + 0.650236i 0.942987 0.332829i \(-0.108003\pi\)
0.183255 + 0.983065i \(0.441337\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 1.70739 2.95729i 0.0828205 0.143449i
\(426\) 0 0
\(427\) 4.62061 2.66771i 0.223607 0.129100i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 23.6772 1.14049 0.570246 0.821474i \(-0.306848\pi\)
0.570246 + 0.821474i \(0.306848\pi\)
\(432\) 0 0
\(433\) 7.96265 0.382661 0.191330 0.981526i \(-0.438720\pi\)
0.191330 + 0.981526i \(0.438720\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 11.5051 6.64245i 0.550362 0.317752i
\(438\) 0 0
\(439\) −5.33903 + 9.24747i −0.254818 + 0.441358i −0.964846 0.262816i \(-0.915349\pi\)
0.710028 + 0.704173i \(0.248682\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 11.1566 + 6.44125i 0.530065 + 0.306033i 0.741043 0.671458i \(-0.234332\pi\)
−0.210978 + 0.977491i \(0.567665\pi\)
\(444\) 0 0
\(445\) −5.94398 + 3.43176i −0.281772 + 0.162681i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.547875 0.0258558 0.0129279 0.999916i \(-0.495885\pi\)
0.0129279 + 0.999916i \(0.495885\pi\)
\(450\) 0 0
\(451\) 7.54241i 0.355158i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 15.1598 + 26.2576i 0.710704 + 1.23098i
\(456\) 0 0
\(457\) −9.90157 + 17.1500i −0.463176 + 0.802244i −0.999117 0.0420111i \(-0.986624\pi\)
0.535941 + 0.844255i \(0.319957\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 12.1833 + 7.03400i 0.567431 + 0.327606i 0.756122 0.654430i \(-0.227091\pi\)
−0.188692 + 0.982036i \(0.560425\pi\)
\(462\) 0 0
\(463\) −6.22954 10.7899i −0.289511 0.501448i 0.684182 0.729311i \(-0.260159\pi\)
−0.973693 + 0.227863i \(0.926826\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 29.3401i 1.35770i −0.734278 0.678849i \(-0.762479\pi\)
0.734278 0.678849i \(-0.237521\pi\)
\(468\) 0 0
\(469\) 6.87923i 0.317653i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −10.3350 17.9008i −0.475205 0.823079i
\(474\) 0 0
\(475\) −3.46410 2.00000i −0.158944 0.0917663i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −9.96368 + 17.2576i −0.455252 + 0.788520i −0.998703 0.0509214i \(-0.983784\pi\)
0.543451 + 0.839441i \(0.317118\pi\)
\(480\) 0 0
\(481\) −11.5611 20.0244i −0.527140 0.913033i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 10.6039i 0.481499i
\(486\) 0 0
\(487\) −7.46828 −0.338420 −0.169210 0.985580i \(-0.554122\pi\)
−0.169210 + 0.985580i \(0.554122\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 30.2186 17.4467i 1.36375 0.787359i 0.373626 0.927580i \(-0.378114\pi\)
0.990120 + 0.140220i \(0.0447811\pi\)
\(492\) 0 0
\(493\) 6.93385 + 4.00326i 0.312285 + 0.180298i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 21.5237 37.2802i 0.965472 1.67225i
\(498\) 0 0
\(499\) 32.4480 18.7339i 1.45257 0.838643i 0.453945 0.891030i \(-0.350016\pi\)
0.998627 + 0.0523870i \(0.0166830\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −10.8991 −0.485968 −0.242984 0.970030i \(-0.578126\pi\)
−0.242984 + 0.970030i \(0.578126\pi\)
\(504\) 0 0
\(505\) 20.0101 0.890439
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −4.06108 + 2.34467i −0.180004 + 0.103926i −0.587295 0.809373i \(-0.699807\pi\)
0.407290 + 0.913299i \(0.366474\pi\)
\(510\) 0 0
\(511\) 17.3371 30.0288i 0.766950 1.32840i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −0.331598 0.191448i −0.0146119 0.00843621i
\(516\) 0 0
\(517\) −53.6323 + 30.9646i −2.35875 + 1.36182i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 26.4249 1.15770 0.578848 0.815435i \(-0.303502\pi\)
0.578848 + 0.815435i \(0.303502\pi\)
\(522\) 0 0
\(523\) 13.8880i 0.607278i −0.952787 0.303639i \(-0.901798\pi\)
0.952787 0.303639i \(-0.0982017\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.15018 + 7.18832i 0.180785 + 0.313128i
\(528\) 0 0
\(529\) −10.5611 + 18.2923i −0.459178 + 0.795319i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −4.50000 2.59808i −0.194917 0.112535i
\(534\) 0 0
\(535\) 0.285751 + 0.494936i 0.0123541 + 0.0213979i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 69.8023i 3.00660i
\(540\) 0 0
\(541\) 25.0161i 1.07553i −0.843096 0.537763i \(-0.819269\pi\)
0.843096 0.537763i \(-0.180731\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −15.5237 26.8879i −0.664964 1.15175i
\(546\) 0 0
\(547\) 12.4236 + 7.17277i 0.531195 + 0.306686i 0.741503 0.670950i \(-0.234113\pi\)
−0.210308 + 0.977635i \(0.567447\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4.68934 8.12217i 0.199772 0.346016i
\(552\) 0 0
\(553\) −4.50000 7.79423i −0.191359 0.331444i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 32.9912i 1.39788i −0.715179 0.698941i \(-0.753655\pi\)
0.715179 0.698941i \(-0.246345\pi\)
\(558\) 0 0
\(559\) 14.2401 0.602292
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −21.6459 + 12.4973i −0.912266 + 0.526697i −0.881160 0.472819i \(-0.843236\pi\)
−0.0311065 + 0.999516i \(0.509903\pi\)
\(564\) 0 0
\(565\) −27.1272 15.6619i −1.14125 0.658902i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.79261 + 4.83694i −0.117072 + 0.202775i −0.918606 0.395174i \(-0.870684\pi\)
0.801534 + 0.597949i \(0.204018\pi\)
\(570\) 0 0
\(571\) 4.83221 2.78988i 0.202222 0.116753i −0.395470 0.918479i \(-0.629418\pi\)
0.597691 + 0.801726i \(0.296085\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 13.2849 0.554019
\(576\) 0 0
\(577\) −9.04748 −0.376651 −0.188326 0.982107i \(-0.560306\pi\)
−0.188326 + 0.982107i \(0.560306\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) −5.81675 + 3.35830i −0.241319 + 0.139326i
\(582\) 0 0
\(583\) 1.47864 2.56108i 0.0612391 0.106069i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 6.24617 + 3.60623i 0.257807 + 0.148845i 0.623334 0.781956i \(-0.285778\pi\)
−0.365527 + 0.930801i \(0.619111\pi\)
\(588\) 0 0
\(589\) 8.42024 4.86143i 0.346950 0.200312i
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 25.0475 1.02858 0.514288 0.857617i \(-0.328056\pi\)
0.514288 + 0.857617i \(0.328056\pi\)
\(594\) 0 0
\(595\) 12.8778i 0.527940i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −20.1026 34.8187i −0.821369 1.42265i −0.904663 0.426128i \(-0.859877\pi\)
0.0832941 0.996525i \(-0.473456\pi\)
\(600\) 0 0
\(601\) 5.48133 9.49394i 0.223588 0.387266i −0.732307 0.680975i \(-0.761556\pi\)
0.955895 + 0.293709i \(0.0948897\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 34.5712 + 19.9597i 1.40552 + 0.811477i
\(606\) 0 0
\(607\) −14.2370 24.6592i −0.577861 1.00088i −0.995724 0.0923747i \(-0.970554\pi\)
0.417863 0.908510i \(-0.362779\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 42.6646i 1.72602i
\(612\) 0 0
\(613\) 14.3632i 0.580125i −0.957008 0.290063i \(-0.906324\pi\)
0.957008 0.290063i \(-0.0936761\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.316348 0.547931i −0.0127357 0.0220589i 0.859587 0.510989i \(-0.170721\pi\)
−0.872323 + 0.488930i \(0.837387\pi\)
\(618\) 0 0
\(619\) −28.9516 16.7152i −1.16366 0.671840i −0.211482 0.977382i \(-0.567829\pi\)
−0.952179 + 0.305542i \(0.901162\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 8.62791 14.9440i 0.345670 0.598718i
\(624\) 0 0
\(625\) 5.50000 + 9.52628i 0.220000 + 0.381051i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 9.82080i 0.391581i
\(630\) 0 0
\(631\) −7.59765 −0.302458 −0.151229 0.988499i \(-0.548323\pi\)
−0.151229 + 0.988499i \(0.548323\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −6.86352 + 3.96265i −0.272370 + 0.157253i
\(636\) 0 0
\(637\) −41.6459 24.0443i −1.65007 0.952669i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −11.8774 + 20.5723i −0.469130 + 0.812558i −0.999377 0.0352856i \(-0.988766\pi\)
0.530247 + 0.847843i \(0.322099\pi\)
\(642\) 0 0
\(643\) −12.4236 + 7.17277i −0.489939 + 0.282867i −0.724549 0.689223i \(-0.757952\pi\)
0.234610 + 0.972090i \(0.424619\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −28.5781 −1.12352 −0.561761 0.827299i \(-0.689876\pi\)
−0.561761 + 0.827299i \(0.689876\pi\)
\(648\) 0 0
\(649\) −10.9253 −0.428856
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −14.7443 + 8.51265i −0.576990 + 0.333126i −0.759936 0.649997i \(-0.774770\pi\)
0.182946 + 0.983123i \(0.441437\pi\)
\(654\) 0 0
\(655\) −11.7281 + 20.3136i −0.458254 + 0.793719i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 10.8209 + 6.24747i 0.421524 + 0.243367i 0.695729 0.718304i \(-0.255081\pi\)
−0.274205 + 0.961671i \(0.588415\pi\)
\(660\) 0 0
\(661\) 30.1272 17.3940i 1.17181 0.676547i 0.217707 0.976014i \(-0.430142\pi\)
0.954107 + 0.299467i \(0.0968090\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −15.0848 −0.584964
\(666\) 0 0
\(667\) 31.1487i 1.20608i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 3.57463 + 6.19145i 0.137997 + 0.239018i
\(672\) 0 0
\(673\) −13.0237 + 22.5578i −0.502028 + 0.869538i 0.497969 + 0.867195i \(0.334079\pi\)
−0.999997 + 0.00234354i \(0.999254\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 28.8829 + 16.6755i 1.11006 + 0.640893i 0.938845 0.344340i \(-0.111897\pi\)
0.171215 + 0.985234i \(0.445231\pi\)
\(678\) 0 0
\(679\) −13.3298 23.0880i −0.511552 0.886034i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 26.4249i 1.01112i −0.862791 0.505560i \(-0.831286\pi\)
0.862791 0.505560i \(-0.168714\pi\)
\(684\) 0 0
\(685\) 38.7588i 1.48090i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1.01867 + 1.76439i 0.0388084 + 0.0672181i
\(690\) 0 0
\(691\) −17.6521 10.1914i −0.671518 0.387701i 0.125134 0.992140i \(-0.460064\pi\)
−0.796651 + 0.604439i \(0.793397\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −9.99602 + 17.3136i −0.379171 + 0.656743i
\(696\) 0 0
\(697\) 1.10349 + 1.91131i 0.0417978 + 0.0723960i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 49.8696i 1.88355i −0.336247 0.941774i \(-0.609158\pi\)
0.336247 0.941774i \(-0.390842\pi\)
\(702\) 0 0
\(703\) 11.5039 0.433877
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −43.5682 + 25.1541i −1.63855 + 0.946017i
\(708\) 0 0
\(709\) 8.57469 + 4.95060i 0.322029 + 0.185924i 0.652297 0.757964i \(-0.273806\pi\)
−0.330267 + 0.943887i \(0.607139\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −16.1459 + 27.9655i −0.604669 + 1.04732i
\(714\) 0 0
\(715\) −35.1842 + 20.3136i −1.31582 + 0.759686i
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −25.5562 −0.953084 −0.476542 0.879152i \(-0.658110\pi\)
−0.476542 + 0.879152i \(0.658110\pi\)
\(720\) 0 0
\(721\) 0.962653 0.0358511
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 8.12217 4.68934i 0.301650 0.174158i
\(726\) 0 0
\(727\) −20.0702 + 34.7627i −0.744364 + 1.28928i 0.206128 + 0.978525i \(0.433914\pi\)
−0.950491 + 0.310751i \(0.899420\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −5.23796 3.02414i −0.193733 0.111852i
\(732\) 0 0
\(733\) 40.9965 23.6694i 1.51424 0.874247i 0.514380 0.857562i \(-0.328022\pi\)
0.999861 0.0166852i \(-0.00531131\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −9.21792 −0.339546
\(738\) 0 0
\(739\) 37.9627i 1.39648i 0.715864 + 0.698239i \(0.246033\pi\)
−0.715864 + 0.698239i \(0.753967\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −7.25980 12.5743i −0.266336 0.461308i 0.701577 0.712594i \(-0.252480\pi\)
−0.967913 + 0.251286i \(0.919146\pi\)
\(744\) 0 0
\(745\) 1.93892 3.35830i 0.0710364 0.123039i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.24434 0.718418i −0.0454671 0.0262504i
\(750\) 0 0
\(751\) 18.3216 + 31.7339i 0.668563 + 1.15798i 0.978306 + 0.207165i \(0.0664236\pi\)
−0.309743 + 0.950820i \(0.600243\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 29.5051i 1.07380i
\(756\) 0 0
\(757\) 2.02727i 0.0736822i −0.999321 0.0368411i \(-0.988270\pi\)
0.999321 0.0368411i \(-0.0117295\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −8.60349 14.9017i −0.311876 0.540186i 0.666892 0.745154i \(-0.267624\pi\)
−0.978769 + 0.204968i \(0.934291\pi\)
\(762\) 0 0
\(763\) 67.5999 + 39.0288i 2.44728 + 1.41294i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.76336 6.51832i 0.135887 0.235363i
\(768\) 0 0
\(769\) −13.4253 23.2533i −0.484129 0.838536i 0.515705 0.856766i \(-0.327530\pi\)
−0.999834 + 0.0182304i \(0.994197\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 4.41305i 0.158726i 0.996846 + 0.0793632i \(0.0252887\pi\)
−0.996846 + 0.0793632i \(0.974711\pi\)
\(774\) 0 0
\(775\) 9.72286 0.349255
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.23887 1.29261i 0.0802157 0.0463126i
\(780\) 0 0
\(781\) 49.9541 + 28.8410i 1.78750 + 1.03201i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.0424108 + 0.0734576i −0.00151371 + 0.00262182i
\(786\) 0 0
\(787\) −29.7765 + 17.1914i −1.06142 + 0.612809i −0.925824 0.377956i \(-0.876627\pi\)
−0.135593 + 0.990765i \(0.543294\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 78.7524 2.80011
\(792\) 0 0
\(793\) −4.92531 −0.174903
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 19.8269 11.4471i 0.702304 0.405475i −0.105901 0.994377i \(-0.533773\pi\)
0.808205 + 0.588901i \(0.200439\pi\)
\(798\) 0 0
\(799\) −9.06058 + 15.6934i −0.320540 + 0.555192i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 40.2375 + 23.2311i 1.41995 + 0.819809i
\(804\) 0 0
\(805\) 43.3880 25.0501i 1.52922 0.882898i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −34.7922 −1.22323 −0.611614 0.791156i \(-0.709480\pi\)
−0.611614 + 0.791156i \(0.709480\pi\)
\(810\) 0 0
\(811\) 11.6810i 0.410174i −0.978744 0.205087i \(-0.934252\pi\)
0.978744 0.205087i \(-0.0657478\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −16.5279 28.6272i −0.578948 1.00277i
\(816\) 0 0
\(817\) −3.54241 + 6.13564i −0.123933 + 0.214659i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 13.9389 + 8.04764i 0.486472 + 0.280864i 0.723109 0.690733i \(-0.242712\pi\)
−0.236638 + 0.971598i \(0.576046\pi\)
\(822\) 0 0
\(823\) −6.40163 11.0880i −0.223147 0.386502i 0.732615 0.680643i \(-0.238300\pi\)
−0.955762 + 0.294142i \(0.904966\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 25.9253i 0.901511i 0.892647 + 0.450756i \(0.148845\pi\)
−0.892647 + 0.450756i \(0.851155\pi\)
\(828\) 0 0
\(829\) 49.0708i 1.70430i −0.523299 0.852149i \(-0.675299\pi\)
0.523299 0.852149i \(-0.324701\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 10.2125 + 17.6885i 0.353841 + 0.612870i
\(834\) 0 0
\(835\) 29.9881 + 17.3136i 1.03778 + 0.599162i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 15.1275 26.2016i 0.522259 0.904579i −0.477406 0.878683i \(-0.658423\pi\)
0.999665 0.0258959i \(-0.00824385\pi\)
\(840\) 0 0
\(841\) −3.50506 6.07095i −0.120864 0.209343i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 5.47244i 0.188258i
\(846\) 0 0
\(847\) −100.363 −3.44851
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) −33.0882 + 19.1035i −1.13425 + 0.654859i
\(852\) 0 0
\(853\) −25.9389 14.9758i −0.888132 0.512763i −0.0148007 0.999890i \(-0.504711\pi\)
−0.873331 + 0.487127i \(0.838045\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 11.9016 20.6141i 0.406550 0.704165i −0.587951 0.808897i \(-0.700065\pi\)
0.994501 + 0.104732i \(0.0333984\pi\)
\(858\) 0 0
\(859\) 28.5000 16.4545i 0.972406 0.561419i 0.0724371 0.997373i \(-0.476922\pi\)
0.899969 + 0.435954i \(0.143589\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −36.3906 −1.23875 −0.619375 0.785095i \(-0.712614\pi\)
−0.619375 + 0.785095i \(0.712614\pi\)
\(864\) 0 0
\(865\) −21.0000 −0.714021
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 10.4440 6.02983i 0.354288 0.204548i
\(870\) 0 0
\(871\) 3.17523 5.49965i 0.107588 0.186349i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −45.7234 26.3984i −1.54573 0.892430i
\(876\) 0 0
\(877\) −42.3529 + 24.4525i −1.43016 + 0.825701i −0.997132 0.0756818i \(-0.975887\pi\)
−0.433024 + 0.901383i \(0.642553\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −39.9627 −1.34638 −0.673188 0.739471i \(-0.735076\pi\)
−0.673188 + 0.739471i \(0.735076\pi\)
\(882\) 0 0
\(883\) 10.0000i 0.336527i −0.985742 0.168263i \(-0.946184\pi\)
0.985742 0.168263i \(-0.0538159\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −22.5949 39.1354i −0.758661 1.31404i −0.943534 0.331277i \(-0.892521\pi\)
0.184873 0.982762i \(-0.440813\pi\)
\(888\) 0 0
\(889\) 9.96265 17.2558i 0.334137 0.578742i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 18.3829 + 10.6134i 0.615160 + 0.355163i
\(894\) 0 0
\(895\) −5.19615 9.00000i −0.173688 0.300837i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 22.7969i 0.760318i
\(900\) 0 0
\(901\) 0.865333i 0.0288284i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 4.98133 + 8.62791i 0.165585 + 0.286801i
\(906\) 0 0
\(907\) −34.9726 20.1914i −1.16125 0.670446i −0.209644 0.977778i \(-0.567231\pi\)
−0.951602 + 0.307332i \(0.900564\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4.26071 7.37976i 0.141164 0.244503i −0.786771 0.617244i \(-0.788249\pi\)
0.927935 + 0.372742i \(0.121582\pi\)
\(912\) 0 0
\(913\) −4.50000 7.79423i −0.148928 0.257951i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 58.9720i 1.94743i
\(918\) 0 0
\(919\) 2.00834 0.0662490 0.0331245 0.999451i \(-0.489454\pi\)
0.0331245 + 0.999451i \(0.489454\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −34.4146 + 19.8693i −1.13277 + 0.654006i
\(924\) 0 0
\(925\) 9.96265 + 5.75194i 0.327570 + 0.189123i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 14.9864 25.9572i 0.491688 0.851628i −0.508267 0.861200i \(-0.669714\pi\)
0.999954 + 0.00957195i \(0.00304689\pi\)
\(930\) 0 0
\(931\) 20.7199 11.9627i 0.679068 0.392060i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 17.2558 0.564326
\(936\) 0 0
\(937\) −45.0475 −1.47164 −0.735818 0.677179i \(-0.763202\pi\)
−0.735818 + 0.677179i \(0.763202\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −46.3880 + 26.7821i −1.51220 + 0.873072i −0.512306 + 0.858803i \(0.671209\pi\)
−0.999898 + 0.0142686i \(0.995458\pi\)
\(942\) 0 0
\(943\) −4.29305 + 7.43578i −0.139801 + 0.242142i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 20.6417 + 11.9175i 0.670766 + 0.387267i 0.796367 0.604814i \(-0.206752\pi\)
−0.125601 + 0.992081i \(0.540086\pi\)
\(948\) 0 0
\(949\) −27.7206 + 16.0045i −0.899849 + 0.519528i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 32.4732 1.05191 0.525955 0.850513i \(-0.323708\pi\)
0.525955 + 0.850513i \(0.323708\pi\)
\(954\) 0 0
\(955\) 0.382896i 0.0123902i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −48.7225 84.3899i −1.57333 2.72509i
\(960\) 0 0
\(961\) 3.68325 6.37958i 0.118815 0.205793i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −27.0712 15.6296i −0.871453 0.503134i
\(966\) 0 0
\(967\) 4.83221 + 8.36964i 0.155393 + 0.269149i 0.933202 0.359352i \(-0.117002\pi\)
−0.777809 + 0.628501i \(0.783669\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 49.8133i 1.59858i −0.600943 0.799292i \(-0.705208\pi\)
0.600943 0.799292i \(-0.294792\pi\)
\(972\) 0 0
\(973\) 50.2627i 1.61135i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −3.59029 6.21856i −0.114863 0.198949i 0.802862 0.596165i \(-0.203310\pi\)
−0.917725 + 0.397216i \(0.869976\pi\)
\(978\) 0 0
\(979\) 20.0244 + 11.5611i 0.639982 + 0.369494i
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 2.38174 4.12530i 0.0759658 0.131577i −0.825540 0.564344i \(-0.809129\pi\)
0.901506 + 0.432767i \(0.142463\pi\)
\(984\) 0 0
\(985\) −8.50506 14.7312i −0.270994 0.469375i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 23.5303i 0.748220i
\(990\) 0 0
\(991\) −28.5781 −0.907815 −0.453907 0.891049i \(-0.649970\pi\)
−0.453907 + 0.891049i \(0.649970\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0.907140 0.523738i 0.0287583 0.0166036i
\(996\) 0 0
\(997\) −32.2494 18.6192i −1.02135 0.589676i −0.106854 0.994275i \(-0.534078\pi\)
−0.914494 + 0.404599i \(0.867411\pi\)
\(998\) 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 1728.2.r.f.289.1 12
3.2 odd 2 576.2.r.e.97.4 yes 12
4.3 odd 2 inner 1728.2.r.f.289.6 12
8.3 odd 2 1728.2.r.e.289.6 12
8.5 even 2 1728.2.r.e.289.1 12
9.2 odd 6 5184.2.d.q.2593.6 12
9.4 even 3 1728.2.r.e.1441.1 12
9.5 odd 6 576.2.r.f.481.3 yes 12
9.7 even 3 5184.2.d.r.2593.12 12
12.11 even 2 576.2.r.e.97.3 12
24.5 odd 2 576.2.r.f.97.3 yes 12
24.11 even 2 576.2.r.f.97.4 yes 12
36.7 odd 6 5184.2.d.r.2593.7 12
36.11 even 6 5184.2.d.q.2593.1 12
36.23 even 6 576.2.r.f.481.4 yes 12
36.31 odd 6 1728.2.r.e.1441.6 12
72.5 odd 6 576.2.r.e.481.4 yes 12
72.11 even 6 5184.2.d.q.2593.7 12
72.13 even 6 inner 1728.2.r.f.1441.1 12
72.29 odd 6 5184.2.d.q.2593.12 12
72.43 odd 6 5184.2.d.r.2593.1 12
72.59 even 6 576.2.r.e.481.3 yes 12
72.61 even 6 5184.2.d.r.2593.6 12
72.67 odd 6 inner 1728.2.r.f.1441.6 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
576.2.r.e.97.3 12 12.11 even 2
576.2.r.e.97.4 yes 12 3.2 odd 2
576.2.r.e.481.3 yes 12 72.59 even 6
576.2.r.e.481.4 yes 12 72.5 odd 6
576.2.r.f.97.3 yes 12 24.5 odd 2
576.2.r.f.97.4 yes 12 24.11 even 2
576.2.r.f.481.3 yes 12 9.5 odd 6
576.2.r.f.481.4 yes 12 36.23 even 6
1728.2.r.e.289.1 12 8.5 even 2
1728.2.r.e.289.6 12 8.3 odd 2
1728.2.r.e.1441.1 12 9.4 even 3
1728.2.r.e.1441.6 12 36.31 odd 6
1728.2.r.f.289.1 12 1.1 even 1 trivial
1728.2.r.f.289.6 12 4.3 odd 2 inner
1728.2.r.f.1441.1 12 72.13 even 6 inner
1728.2.r.f.1441.6 12 72.67 odd 6 inner
5184.2.d.q.2593.1 12 36.11 even 6
5184.2.d.q.2593.6 12 9.2 odd 6
5184.2.d.q.2593.7 12 72.11 even 6
5184.2.d.q.2593.12 12 72.29 odd 6
5184.2.d.r.2593.1 12 72.43 odd 6
5184.2.d.r.2593.6 12 72.61 even 6
5184.2.d.r.2593.7 12 36.7 odd 6
5184.2.d.r.2593.12 12 9.7 even 3