Properties

Label 3528.2.s.bc.361.2
Level $3528$
Weight $2$
Character 3528.361
Analytic conductor $28.171$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3528,2,Mod(361,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 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("3528.361");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.s (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: \(28.1712218331\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 2x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1176)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.2
Root \(0.707107 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 3528.361
Dual form 3528.2.s.bc.3313.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.292893 - 0.507306i) q^{5} +O(q^{10})\) \(q+(-0.292893 - 0.507306i) q^{5} +(-2.41421 + 4.18154i) q^{11} -4.24264 q^{13} +(-2.29289 + 3.97141i) q^{17} +(0.585786 + 1.01461i) q^{19} +(0.414214 + 0.717439i) q^{23} +(2.32843 - 4.03295i) q^{25} +2.82843 q^{29} +(1.41421 - 2.44949i) q^{31} +(-4.82843 - 8.36308i) q^{37} +1.75736 q^{41} +11.3137 q^{43} +(-6.24264 - 10.8126i) q^{47} +(-1.00000 + 1.73205i) q^{53} +2.82843 q^{55} +(4.24264 - 7.34847i) q^{59} +(-1.53553 - 2.65962i) q^{61} +(1.24264 + 2.15232i) q^{65} +(5.65685 - 9.79796i) q^{67} -6.48528 q^{71} +(8.12132 - 14.0665i) q^{73} +(-1.17157 - 2.02922i) q^{79} -4.00000 q^{83} +2.68629 q^{85} +(-7.12132 - 12.3345i) q^{89} +(0.343146 - 0.594346i) q^{95} -8.24264 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{5} - 4 q^{11} - 12 q^{17} + 8 q^{19} - 4 q^{23} - 2 q^{25} - 8 q^{37} + 24 q^{41} - 8 q^{47} - 4 q^{53} + 8 q^{61} - 12 q^{65} + 8 q^{71} + 24 q^{73} - 16 q^{79} - 16 q^{83} + 56 q^{85} - 20 q^{89} + 24 q^{95} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) −0.292893 0.507306i −0.130986 0.226874i 0.793071 0.609129i \(-0.208481\pi\)
−0.924057 + 0.382255i \(0.875148\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) −2.41421 + 4.18154i −0.727913 + 1.26078i 0.229851 + 0.973226i \(0.426176\pi\)
−0.957764 + 0.287556i \(0.907157\pi\)
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −2.29289 + 3.97141i −0.556108 + 0.963208i 0.441708 + 0.897159i \(0.354373\pi\)
−0.997816 + 0.0660490i \(0.978961\pi\)
\(18\) 0 0
\(19\) 0.585786 + 1.01461i 0.134389 + 0.232768i 0.925364 0.379080i \(-0.123760\pi\)
−0.790975 + 0.611848i \(0.790426\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0.414214 + 0.717439i 0.0863695 + 0.149596i 0.905974 0.423333i \(-0.139140\pi\)
−0.819604 + 0.572930i \(0.805807\pi\)
\(24\) 0 0
\(25\) 2.32843 4.03295i 0.465685 0.806591i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 2.82843 0.525226 0.262613 0.964901i \(-0.415416\pi\)
0.262613 + 0.964901i \(0.415416\pi\)
\(30\) 0 0
\(31\) 1.41421 2.44949i 0.254000 0.439941i −0.710623 0.703573i \(-0.751587\pi\)
0.964623 + 0.263631i \(0.0849203\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −4.82843 8.36308i −0.793789 1.37488i −0.923606 0.383344i \(-0.874772\pi\)
0.129817 0.991538i \(-0.458561\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 1.75736 0.274453 0.137227 0.990540i \(-0.456181\pi\)
0.137227 + 0.990540i \(0.456181\pi\)
\(42\) 0 0
\(43\) 11.3137 1.72532 0.862662 0.505781i \(-0.168795\pi\)
0.862662 + 0.505781i \(0.168795\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −6.24264 10.8126i −0.910583 1.57718i −0.813243 0.581924i \(-0.802300\pi\)
−0.0973398 0.995251i \(-0.531033\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.00000 + 1.73205i −0.137361 + 0.237915i −0.926497 0.376303i \(-0.877195\pi\)
0.789136 + 0.614218i \(0.210529\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.24264 7.34847i 0.552345 0.956689i −0.445760 0.895152i \(-0.647067\pi\)
0.998105 0.0615367i \(-0.0196001\pi\)
\(60\) 0 0
\(61\) −1.53553 2.65962i −0.196605 0.340530i 0.750821 0.660506i \(-0.229658\pi\)
−0.947425 + 0.319976i \(0.896325\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.24264 + 2.15232i 0.154131 + 0.266962i
\(66\) 0 0
\(67\) 5.65685 9.79796i 0.691095 1.19701i −0.280385 0.959888i \(-0.590462\pi\)
0.971480 0.237124i \(-0.0762046\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −6.48528 −0.769661 −0.384831 0.922987i \(-0.625740\pi\)
−0.384831 + 0.922987i \(0.625740\pi\)
\(72\) 0 0
\(73\) 8.12132 14.0665i 0.950529 1.64636i 0.206245 0.978500i \(-0.433876\pi\)
0.744284 0.667864i \(-0.232791\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.17157 2.02922i −0.131812 0.228306i 0.792563 0.609790i \(-0.208746\pi\)
−0.924375 + 0.381485i \(0.875413\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −4.00000 −0.439057 −0.219529 0.975606i \(-0.570452\pi\)
−0.219529 + 0.975606i \(0.570452\pi\)
\(84\) 0 0
\(85\) 2.68629 0.291369
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −7.12132 12.3345i −0.754858 1.30745i −0.945445 0.325783i \(-0.894372\pi\)
0.190586 0.981670i \(-0.438961\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0.343146 0.594346i 0.0352060 0.0609786i
\(96\) 0 0
\(97\) −8.24264 −0.836913 −0.418457 0.908237i \(-0.637429\pi\)
−0.418457 + 0.908237i \(0.637429\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −7.36396 + 12.7548i −0.732742 + 1.26915i 0.222966 + 0.974826i \(0.428426\pi\)
−0.955707 + 0.294319i \(0.904907\pi\)
\(102\) 0 0
\(103\) −7.07107 12.2474i −0.696733 1.20678i −0.969593 0.244723i \(-0.921303\pi\)
0.272860 0.962054i \(-0.412030\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 3.58579 + 6.21076i 0.346651 + 0.600417i 0.985652 0.168788i \(-0.0539855\pi\)
−0.639001 + 0.769206i \(0.720652\pi\)
\(108\) 0 0
\(109\) −9.65685 + 16.7262i −0.924959 + 1.60208i −0.133332 + 0.991071i \(0.542568\pi\)
−0.791627 + 0.611004i \(0.790766\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 0.242641 0.420266i 0.0226264 0.0391900i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −6.15685 10.6640i −0.559714 0.969453i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.00000 + 3.46410i 0.174741 + 0.302660i 0.940072 0.340977i \(-0.110758\pi\)
−0.765331 + 0.643637i \(0.777425\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −7.41421 + 12.8418i −0.633439 + 1.09715i 0.353405 + 0.935471i \(0.385024\pi\)
−0.986844 + 0.161678i \(0.948309\pi\)
\(138\) 0 0
\(139\) −12.9706 −1.10015 −0.550074 0.835116i \(-0.685401\pi\)
−0.550074 + 0.835116i \(0.685401\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 10.2426 17.7408i 0.856533 1.48356i
\(144\) 0 0
\(145\) −0.828427 1.43488i −0.0687971 0.119160i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −10.6569 18.4582i −0.873044 1.51216i −0.858832 0.512257i \(-0.828810\pi\)
−0.0142111 0.999899i \(-0.504524\pi\)
\(150\) 0 0
\(151\) −0.828427 + 1.43488i −0.0674164 + 0.116769i −0.897763 0.440478i \(-0.854809\pi\)
0.830347 + 0.557247i \(0.188142\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −1.65685 −0.133082
\(156\) 0 0
\(157\) −4.12132 + 7.13834i −0.328917 + 0.569701i −0.982297 0.187329i \(-0.940017\pi\)
0.653380 + 0.757030i \(0.273350\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 2.82843 + 4.89898i 0.221540 + 0.383718i 0.955276 0.295717i \(-0.0955585\pi\)
−0.733736 + 0.679435i \(0.762225\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −9.17157 −0.709718 −0.354859 0.934920i \(-0.615471\pi\)
−0.354859 + 0.934920i \(0.615471\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 9.70711 + 16.8132i 0.738018 + 1.27828i 0.953387 + 0.301751i \(0.0975712\pi\)
−0.215369 + 0.976533i \(0.569095\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 5.24264 9.08052i 0.391853 0.678710i −0.600841 0.799369i \(-0.705167\pi\)
0.992694 + 0.120659i \(0.0385007\pi\)
\(180\) 0 0
\(181\) −7.07107 −0.525588 −0.262794 0.964852i \(-0.584644\pi\)
−0.262794 + 0.964852i \(0.584644\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.82843 + 4.89898i −0.207950 + 0.360180i
\(186\) 0 0
\(187\) −11.0711 19.1757i −0.809597 1.40226i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −8.07107 13.9795i −0.584002 1.01152i −0.994999 0.0998844i \(-0.968153\pi\)
0.410997 0.911637i \(-0.365181\pi\)
\(192\) 0 0
\(193\) −1.00000 + 1.73205i −0.0719816 + 0.124676i −0.899770 0.436365i \(-0.856266\pi\)
0.827788 + 0.561041i \(0.189599\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 25.3137 1.80353 0.901764 0.432230i \(-0.142273\pi\)
0.901764 + 0.432230i \(0.142273\pi\)
\(198\) 0 0
\(199\) 2.82843 4.89898i 0.200502 0.347279i −0.748188 0.663486i \(-0.769076\pi\)
0.948690 + 0.316207i \(0.102409\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −0.514719 0.891519i −0.0359495 0.0622664i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) −9.65685 −0.664805 −0.332403 0.943138i \(-0.607859\pi\)
−0.332403 + 0.943138i \(0.607859\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −3.31371 5.73951i −0.225993 0.391431i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 9.72792 16.8493i 0.654371 1.13340i
\(222\) 0 0
\(223\) −2.34315 −0.156909 −0.0784543 0.996918i \(-0.524998\pi\)
−0.0784543 + 0.996918i \(0.524998\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 5.41421 9.37769i 0.359354 0.622419i −0.628499 0.777810i \(-0.716330\pi\)
0.987853 + 0.155391i \(0.0496637\pi\)
\(228\) 0 0
\(229\) −11.2929 19.5599i −0.746255 1.29255i −0.949606 0.313446i \(-0.898516\pi\)
0.203351 0.979106i \(-0.434817\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.58579 7.94282i −0.300425 0.520351i 0.675807 0.737078i \(-0.263795\pi\)
−0.976232 + 0.216727i \(0.930462\pi\)
\(234\) 0 0
\(235\) −3.65685 + 6.33386i −0.238547 + 0.413175i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 7.17157 0.463890 0.231945 0.972729i \(-0.425491\pi\)
0.231945 + 0.972729i \(0.425491\pi\)
\(240\) 0 0
\(241\) −2.94975 + 5.10911i −0.190010 + 0.329107i −0.945253 0.326338i \(-0.894185\pi\)
0.755243 + 0.655445i \(0.227519\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.48528 4.30463i −0.158135 0.273897i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 22.1421 1.39760 0.698800 0.715317i \(-0.253718\pi\)
0.698800 + 0.715317i \(0.253718\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0.292893 + 0.507306i 0.0182702 + 0.0316449i 0.875016 0.484094i \(-0.160851\pi\)
−0.856746 + 0.515739i \(0.827517\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 4.07107 7.05130i 0.251033 0.434802i −0.712778 0.701390i \(-0.752563\pi\)
0.963810 + 0.266589i \(0.0858965\pi\)
\(264\) 0 0
\(265\) 1.17157 0.0719691
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 9.94975 17.2335i 0.606647 1.05074i −0.385142 0.922857i \(-0.625848\pi\)
0.991789 0.127886i \(-0.0408191\pi\)
\(270\) 0 0
\(271\) −11.0711 19.1757i −0.672519 1.16484i −0.977187 0.212379i \(-0.931879\pi\)
0.304668 0.952459i \(-0.401455\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 11.2426 + 19.4728i 0.677957 + 1.17426i
\(276\) 0 0
\(277\) −8.31371 + 14.3998i −0.499522 + 0.865198i −1.00000 0.000551476i \(-0.999824\pi\)
0.500478 + 0.865750i \(0.333158\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6.82843 −0.407350 −0.203675 0.979039i \(-0.565289\pi\)
−0.203675 + 0.979039i \(0.565289\pi\)
\(282\) 0 0
\(283\) 1.07107 1.85514i 0.0636684 0.110277i −0.832434 0.554124i \(-0.813053\pi\)
0.896103 + 0.443847i \(0.146387\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −2.01472 3.48960i −0.118513 0.205270i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −10.2426 −0.598381 −0.299191 0.954193i \(-0.596717\pi\)
−0.299191 + 0.954193i \(0.596717\pi\)
\(294\) 0 0
\(295\) −4.97056 −0.289397
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.75736 3.04384i −0.101631 0.176030i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.899495 + 1.55797i −0.0515049 + 0.0892092i
\(306\) 0 0
\(307\) 28.4853 1.62574 0.812870 0.582445i \(-0.197904\pi\)
0.812870 + 0.582445i \(0.197904\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 7.89949 13.6823i 0.447939 0.775854i −0.550312 0.834959i \(-0.685491\pi\)
0.998252 + 0.0591052i \(0.0188247\pi\)
\(312\) 0 0
\(313\) 1.63604 + 2.83370i 0.0924744 + 0.160170i 0.908552 0.417772i \(-0.137189\pi\)
−0.816077 + 0.577943i \(0.803856\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.0000 19.0526i −0.617822 1.07010i −0.989882 0.141890i \(-0.954682\pi\)
0.372061 0.928208i \(-0.378651\pi\)
\(318\) 0 0
\(319\) −6.82843 + 11.8272i −0.382319 + 0.662195i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −5.37258 −0.298939
\(324\) 0 0
\(325\) −9.87868 + 17.1104i −0.547971 + 0.949113i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7.65685 + 13.2621i 0.420859 + 0.728949i 0.996024 0.0890887i \(-0.0283955\pi\)
−0.575165 + 0.818037i \(0.695062\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6.62742 −0.362094
\(336\) 0 0
\(337\) 21.6569 1.17972 0.589862 0.807504i \(-0.299182\pi\)
0.589862 + 0.807504i \(0.299182\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 6.82843 + 11.8272i 0.369780 + 0.640478i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 7.24264 12.5446i 0.388805 0.673431i −0.603484 0.797375i \(-0.706221\pi\)
0.992289 + 0.123945i \(0.0395545\pi\)
\(348\) 0 0
\(349\) −13.4142 −0.718046 −0.359023 0.933329i \(-0.616890\pi\)
−0.359023 + 0.933329i \(0.616890\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −15.8492 + 27.4517i −0.843570 + 1.46111i 0.0432872 + 0.999063i \(0.486217\pi\)
−0.886857 + 0.462044i \(0.847116\pi\)
\(354\) 0 0
\(355\) 1.89949 + 3.29002i 0.100815 + 0.174616i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 11.5858 + 20.0672i 0.611474 + 1.05910i 0.990992 + 0.133920i \(0.0427566\pi\)
−0.379518 + 0.925184i \(0.623910\pi\)
\(360\) 0 0
\(361\) 8.81371 15.2658i 0.463879 0.803463i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −9.51472 −0.498023
\(366\) 0 0
\(367\) 13.6569 23.6544i 0.712882 1.23475i −0.250889 0.968016i \(-0.580723\pi\)
0.963771 0.266732i \(-0.0859438\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 8.65685 + 14.9941i 0.448235 + 0.776366i 0.998271 0.0587751i \(-0.0187195\pi\)
−0.550036 + 0.835141i \(0.685386\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) −0.686292 −0.0352524 −0.0176262 0.999845i \(-0.505611\pi\)
−0.0176262 + 0.999845i \(0.505611\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −12.4853 21.6251i −0.637968 1.10499i −0.985878 0.167465i \(-0.946442\pi\)
0.347910 0.937528i \(-0.386892\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.89949 3.29002i 0.0963082 0.166811i −0.813846 0.581081i \(-0.802630\pi\)
0.910154 + 0.414270i \(0.135963\pi\)
\(390\) 0 0
\(391\) −3.79899 −0.192123
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.686292 + 1.18869i −0.0345311 + 0.0598096i
\(396\) 0 0
\(397\) −3.87868 6.71807i −0.194665 0.337170i 0.752125 0.659020i \(-0.229029\pi\)
−0.946791 + 0.321850i \(0.895695\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 3.41421 + 5.91359i 0.170498 + 0.295311i 0.938594 0.345024i \(-0.112129\pi\)
−0.768096 + 0.640334i \(0.778796\pi\)
\(402\) 0 0
\(403\) −6.00000 + 10.3923i −0.298881 + 0.517678i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 46.6274 2.31124
\(408\) 0 0
\(409\) 0.121320 0.210133i 0.00599890 0.0103904i −0.863010 0.505186i \(-0.831424\pi\)
0.869009 + 0.494796i \(0.164757\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 1.17157 + 2.02922i 0.0575103 + 0.0996107i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 24.4853 1.19618 0.598092 0.801427i \(-0.295926\pi\)
0.598092 + 0.801427i \(0.295926\pi\)
\(420\) 0 0
\(421\) −2.68629 −0.130922 −0.0654609 0.997855i \(-0.520852\pi\)
−0.0654609 + 0.997855i \(0.520852\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 10.6777 + 18.4943i 0.517943 + 0.897104i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 10.0711 17.4436i 0.485106 0.840229i −0.514747 0.857342i \(-0.672114\pi\)
0.999854 + 0.0171133i \(0.00544758\pi\)
\(432\) 0 0
\(433\) −15.0711 −0.724269 −0.362135 0.932126i \(-0.617952\pi\)
−0.362135 + 0.932126i \(0.617952\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −0.485281 + 0.840532i −0.0232142 + 0.0402081i
\(438\) 0 0
\(439\) 17.6569 + 30.5826i 0.842716 + 1.45963i 0.887590 + 0.460634i \(0.152378\pi\)
−0.0448746 + 0.998993i \(0.514289\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −8.41421 14.5738i −0.399771 0.692424i 0.593926 0.804520i \(-0.297577\pi\)
−0.993697 + 0.112095i \(0.964244\pi\)
\(444\) 0 0
\(445\) −4.17157 + 7.22538i −0.197752 + 0.342516i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −28.6274 −1.35101 −0.675506 0.737355i \(-0.736075\pi\)
−0.675506 + 0.737355i \(0.736075\pi\)
\(450\) 0 0
\(451\) −4.24264 + 7.34847i −0.199778 + 0.346026i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −10.3137 17.8639i −0.482455 0.835636i 0.517342 0.855779i \(-0.326921\pi\)
−0.999797 + 0.0201422i \(0.993588\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −20.3848 −0.949414 −0.474707 0.880144i \(-0.657446\pi\)
−0.474707 + 0.880144i \(0.657446\pi\)
\(462\) 0 0
\(463\) −9.65685 −0.448792 −0.224396 0.974498i \(-0.572041\pi\)
−0.224396 + 0.974498i \(0.572041\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 6.58579 + 11.4069i 0.304754 + 0.527849i 0.977206 0.212291i \(-0.0680925\pi\)
−0.672453 + 0.740140i \(0.734759\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −27.3137 + 47.3087i −1.25589 + 2.17526i
\(474\) 0 0
\(475\) 5.45584 0.250331
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −8.58579 + 14.8710i −0.392295 + 0.679474i −0.992752 0.120183i \(-0.961652\pi\)
0.600457 + 0.799657i \(0.294985\pi\)
\(480\) 0 0
\(481\) 20.4853 + 35.4815i 0.934048 + 1.61782i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 2.41421 + 4.18154i 0.109624 + 0.189874i
\(486\) 0 0
\(487\) 14.4853 25.0892i 0.656391 1.13690i −0.325152 0.945662i \(-0.605416\pi\)
0.981543 0.191241i \(-0.0612510\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −24.8284 −1.12049 −0.560246 0.828327i \(-0.689293\pi\)
−0.560246 + 0.828327i \(0.689293\pi\)
\(492\) 0 0
\(493\) −6.48528 + 11.2328i −0.292082 + 0.505902i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −4.48528 7.76874i −0.200789 0.347776i 0.747994 0.663705i \(-0.231017\pi\)
−0.948783 + 0.315929i \(0.897684\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 3.31371 0.147751 0.0738755 0.997267i \(-0.476463\pi\)
0.0738755 + 0.997267i \(0.476463\pi\)
\(504\) 0 0
\(505\) 8.62742 0.383915
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.192388 0.333226i −0.00852746 0.0147700i 0.861730 0.507367i \(-0.169381\pi\)
−0.870258 + 0.492597i \(0.836048\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −4.14214 + 7.17439i −0.182524 + 0.316141i
\(516\) 0 0
\(517\) 60.2843 2.65130
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.87868 + 4.98602i −0.126117 + 0.218441i −0.922169 0.386787i \(-0.873585\pi\)
0.796052 + 0.605228i \(0.206918\pi\)
\(522\) 0 0
\(523\) 14.4853 + 25.0892i 0.633397 + 1.09708i 0.986852 + 0.161625i \(0.0516734\pi\)
−0.353455 + 0.935451i \(0.614993\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 6.48528 + 11.2328i 0.282503 + 0.489310i
\(528\) 0 0
\(529\) 11.1569 19.3242i 0.485081 0.840184i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −7.45584 −0.322948
\(534\) 0 0
\(535\) 2.10051 3.63818i 0.0908128 0.157292i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9.00000 + 15.5885i 0.386940 + 0.670200i 0.992036 0.125952i \(-0.0401986\pi\)
−0.605096 + 0.796152i \(0.706865\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 11.3137 0.484626
\(546\) 0 0
\(547\) −36.9706 −1.58075 −0.790374 0.612625i \(-0.790114\pi\)
−0.790374 + 0.612625i \(0.790114\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 1.65685 + 2.86976i 0.0705844 + 0.122256i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6.31371 + 10.9357i −0.267520 + 0.463359i −0.968221 0.250097i \(-0.919538\pi\)
0.700700 + 0.713456i \(0.252871\pi\)
\(558\) 0 0
\(559\) −48.0000 −2.03018
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −15.0711 + 26.1039i −0.635170 + 1.10015i 0.351309 + 0.936259i \(0.385737\pi\)
−0.986479 + 0.163887i \(0.947597\pi\)
\(564\) 0 0
\(565\) −2.92893 5.07306i −0.123221 0.213425i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −17.0711 29.5680i −0.715656 1.23955i −0.962706 0.270550i \(-0.912794\pi\)
0.247049 0.969003i \(-0.420539\pi\)
\(570\) 0 0
\(571\) 15.1716 26.2779i 0.634911 1.09970i −0.351624 0.936141i \(-0.614370\pi\)
0.986534 0.163556i \(-0.0522964\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 3.85786 0.160884
\(576\) 0 0
\(577\) −14.6066 + 25.2994i −0.608081 + 1.05323i 0.383476 + 0.923551i \(0.374727\pi\)
−0.991556 + 0.129676i \(0.958606\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −4.82843 8.36308i −0.199973 0.346363i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.79899 0.156801 0.0784005 0.996922i \(-0.475019\pi\)
0.0784005 + 0.996922i \(0.475019\pi\)
\(588\) 0 0
\(589\) 3.31371 0.136539
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −22.2929 38.6124i −0.915459 1.58562i −0.806227 0.591606i \(-0.798494\pi\)
−0.109232 0.994016i \(-0.534839\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −13.7279 + 23.7775i −0.560908 + 0.971521i 0.436510 + 0.899699i \(0.356214\pi\)
−0.997418 + 0.0718211i \(0.977119\pi\)
\(600\) 0 0
\(601\) 3.75736 0.153266 0.0766329 0.997059i \(-0.475583\pi\)
0.0766329 + 0.997059i \(0.475583\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −3.60660 + 6.24682i −0.146629 + 0.253969i
\(606\) 0 0
\(607\) −4.48528 7.76874i −0.182052 0.315323i 0.760527 0.649306i \(-0.224941\pi\)
−0.942579 + 0.333983i \(0.891607\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 26.4853 + 45.8739i 1.07148 + 1.85586i
\(612\) 0 0
\(613\) 10.8284 18.7554i 0.437356 0.757523i −0.560129 0.828406i \(-0.689248\pi\)
0.997485 + 0.0708828i \(0.0225816\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −12.4853 −0.502639 −0.251319 0.967904i \(-0.580864\pi\)
−0.251319 + 0.967904i \(0.580864\pi\)
\(618\) 0 0
\(619\) 11.1716 19.3497i 0.449023 0.777731i −0.549299 0.835626i \(-0.685105\pi\)
0.998323 + 0.0578943i \(0.0184386\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −9.98528 17.2950i −0.399411 0.691801i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 44.2843 1.76573
\(630\) 0 0
\(631\) −36.9706 −1.47177 −0.735887 0.677104i \(-0.763235\pi\)
−0.735887 + 0.677104i \(0.763235\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −5.85786 10.1461i −0.232462 0.402636i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −13.5563 + 23.4803i −0.535444 + 0.927416i 0.463698 + 0.885993i \(0.346522\pi\)
−0.999142 + 0.0414223i \(0.986811\pi\)
\(642\) 0 0
\(643\) 7.79899 0.307562 0.153781 0.988105i \(-0.450855\pi\)
0.153781 + 0.988105i \(0.450855\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 19.8995 34.4669i 0.782330 1.35504i −0.148251 0.988950i \(-0.547364\pi\)
0.930581 0.366085i \(-0.119302\pi\)
\(648\) 0 0
\(649\) 20.4853 + 35.4815i 0.804118 + 1.39277i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 8.92893 + 15.4654i 0.349416 + 0.605206i 0.986146 0.165881i \(-0.0530466\pi\)
−0.636730 + 0.771087i \(0.719713\pi\)
\(654\) 0 0
\(655\) 1.17157 2.02922i 0.0457771 0.0792883i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −38.4853 −1.49917 −0.749587 0.661906i \(-0.769748\pi\)
−0.749587 + 0.661906i \(0.769748\pi\)
\(660\) 0 0
\(661\) 9.53553 16.5160i 0.370889 0.642399i −0.618813 0.785538i \(-0.712386\pi\)
0.989703 + 0.143139i \(0.0457196\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.17157 + 2.02922i 0.0453635 + 0.0785719i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 14.8284 0.572445
\(672\) 0 0
\(673\) −0.686292 −0.0264546 −0.0132273 0.999913i \(-0.504211\pi\)
−0.0132273 + 0.999913i \(0.504211\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 15.8492 + 27.4517i 0.609136 + 1.05505i 0.991383 + 0.130994i \(0.0418169\pi\)
−0.382247 + 0.924060i \(0.624850\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 18.8995 32.7349i 0.723169 1.25257i −0.236554 0.971618i \(-0.576018\pi\)
0.959723 0.280947i \(-0.0906486\pi\)
\(684\) 0 0
\(685\) 8.68629 0.331886
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.24264 7.34847i 0.161632 0.279954i
\(690\) 0 0
\(691\) 15.6569 + 27.1185i 0.595615 + 1.03164i 0.993460 + 0.114182i \(0.0364248\pi\)
−0.397845 + 0.917453i \(0.630242\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.79899 + 6.58004i 0.144104 + 0.249595i
\(696\) 0 0
\(697\) −4.02944 + 6.97919i −0.152626 + 0.264356i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −22.1421 −0.836297 −0.418148 0.908379i \(-0.637321\pi\)
−0.418148 + 0.908379i \(0.637321\pi\)
\(702\) 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
\(709\) 19.3137 + 33.4523i 0.725342 + 1.25633i 0.958833 + 0.283970i \(0.0916515\pi\)
−0.233492 + 0.972359i \(0.575015\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 2.34315 0.0877515
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −19.3137 33.4523i −0.720280 1.24756i −0.960888 0.276939i \(-0.910680\pi\)
0.240608 0.970622i \(-0.422653\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 6.58579 11.4069i 0.244590 0.423642i
\(726\) 0 0
\(727\) 38.1421 1.41461 0.707307 0.706907i \(-0.249910\pi\)
0.707307 + 0.706907i \(0.249910\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −25.9411 + 44.9313i −0.959467 + 1.66185i
\(732\) 0 0
\(733\) 20.0208 + 34.6771i 0.739486 + 1.28083i 0.952727 + 0.303827i \(0.0982646\pi\)
−0.213241 + 0.977000i \(0.568402\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 27.3137 + 47.3087i 1.00611 + 1.74264i
\(738\) 0 0
\(739\) 0.485281 0.840532i 0.0178514 0.0309195i −0.856962 0.515380i \(-0.827651\pi\)
0.874813 + 0.484461i \(0.160984\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −0.828427 −0.0303920 −0.0151960 0.999885i \(-0.504837\pi\)
−0.0151960 + 0.999885i \(0.504837\pi\)
\(744\) 0 0
\(745\) −6.24264 + 10.8126i −0.228713 + 0.396142i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 18.1421 + 31.4231i 0.662016 + 1.14665i 0.980085 + 0.198578i \(0.0636322\pi\)
−0.318069 + 0.948067i \(0.603034\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0.970563 0.0353224
\(756\) 0 0
\(757\) −25.9411 −0.942846 −0.471423 0.881907i \(-0.656260\pi\)
−0.471423 + 0.881907i \(0.656260\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.05025 + 10.4793i 0.219321 + 0.379876i 0.954601 0.297888i \(-0.0962823\pi\)
−0.735279 + 0.677764i \(0.762949\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −18.0000 + 31.1769i −0.649942 + 1.12573i
\(768\) 0 0
\(769\) 18.8701 0.680472 0.340236 0.940340i \(-0.389493\pi\)
0.340236 + 0.940340i \(0.389493\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 18.6777 32.3507i 0.671789 1.16357i −0.305607 0.952158i \(-0.598859\pi\)
0.977396 0.211415i \(-0.0678072\pi\)
\(774\) 0 0
\(775\) −6.58579 11.4069i −0.236568 0.409749i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.02944 + 1.78304i 0.0368834 + 0.0638840i
\(780\) 0 0
\(781\) 15.6569 27.1185i 0.560246 0.970375i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 4.82843 0.172334
\(786\) 0 0
\(787\) −3.65685 + 6.33386i −0.130353 + 0.225778i −0.923813 0.382845i \(-0.874944\pi\)
0.793460 + 0.608623i \(0.208278\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 6.51472 + 11.2838i 0.231344 + 0.400700i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −36.5858 −1.29594 −0.647968 0.761668i \(-0.724381\pi\)
−0.647968 + 0.761668i \(0.724381\pi\)
\(798\) 0 0
\(799\) 57.2548 2.02553
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 39.2132 + 67.9193i 1.38380 + 2.39682i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 20.3137 35.1844i 0.714192 1.23702i −0.249078 0.968483i \(-0.580128\pi\)
0.963270 0.268533i \(-0.0865390\pi\)
\(810\) 0 0
\(811\) −14.3431 −0.503656 −0.251828 0.967772i \(-0.581032\pi\)
−0.251828 + 0.967772i \(0.581032\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 1.65685 2.86976i 0.0580371 0.100523i
\(816\) 0 0
\(817\) 6.62742 + 11.4790i 0.231864 + 0.401600i
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −16.6569 28.8505i −0.581328 1.00689i −0.995322 0.0966104i \(-0.969200\pi\)
0.413994 0.910280i \(-0.364133\pi\)
\(822\) 0 0
\(823\) −4.48528 + 7.76874i −0.156347 + 0.270801i −0.933549 0.358451i \(-0.883305\pi\)
0.777202 + 0.629252i \(0.216639\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −11.8579 −0.412338 −0.206169 0.978516i \(-0.566100\pi\)
−0.206169 + 0.978516i \(0.566100\pi\)
\(828\) 0 0
\(829\) −1.19239 + 2.06528i −0.0414134 + 0.0717300i −0.885989 0.463706i \(-0.846519\pi\)
0.844576 + 0.535436i \(0.179853\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 2.68629 + 4.65279i 0.0929630 + 0.161017i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −15.7990 −0.545442 −0.272721 0.962093i \(-0.587924\pi\)
−0.272721 + 0.962093i \(0.587924\pi\)
\(840\) 0 0
\(841\) −21.0000 −0.724138
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1.46447 2.53653i −0.0503792 0.0872593i
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 4.00000 6.92820i 0.137118 0.237496i
\(852\) 0 0
\(853\) −19.0711 −0.652981 −0.326490 0.945200i \(-0.605866\pi\)
−0.326490 + 0.945200i \(0.605866\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −15.6066 + 27.0314i −0.533111 + 0.923376i 0.466141 + 0.884711i \(0.345644\pi\)
−0.999252 + 0.0386654i \(0.987689\pi\)
\(858\) 0 0
\(859\) 26.7279 + 46.2941i 0.911945 + 1.57953i 0.811314 + 0.584611i \(0.198753\pi\)
0.100631 + 0.994924i \(0.467914\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −21.0416 36.4452i −0.716265 1.24061i −0.962469 0.271390i \(-0.912517\pi\)
0.246204 0.969218i \(-0.420817\pi\)
\(864\) 0 0
\(865\) 5.68629 9.84895i 0.193340 0.334874i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 11.3137 0.383791
\(870\) 0 0
\(871\) −24.0000 + 41.5692i −0.813209 + 1.40852i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −14.1421 24.4949i −0.477546 0.827134i 0.522123 0.852870i \(-0.325140\pi\)
−0.999669 + 0.0257364i \(0.991807\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −18.0416 −0.607838 −0.303919 0.952698i \(-0.598295\pi\)
−0.303919 + 0.952698i \(0.598295\pi\)
\(882\) 0 0
\(883\) −21.6569 −0.728811 −0.364406 0.931240i \(-0.618728\pi\)
−0.364406 + 0.931240i \(0.618728\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0.100505 + 0.174080i 0.00337463 + 0.00584503i 0.867708 0.497075i \(-0.165592\pi\)
−0.864333 + 0.502920i \(0.832259\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 7.31371 12.6677i 0.244744 0.423909i
\(894\) 0 0
\(895\) −6.14214 −0.205309
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 4.00000 6.92820i 0.133407 0.231069i
\(900\) 0 0
\(901\) −4.58579 7.94282i −0.152775 0.264614i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 2.07107 + 3.58719i 0.0688446 + 0.119242i
\(906\) 0 0
\(907\) −21.1716 + 36.6702i −0.702991 + 1.21762i 0.264421 + 0.964407i \(0.414819\pi\)
−0.967412 + 0.253208i \(0.918514\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 10.4853 0.347393 0.173696 0.984799i \(-0.444429\pi\)
0.173696 + 0.984799i \(0.444429\pi\)
\(912\) 0 0
\(913\) 9.65685 16.7262i 0.319595 0.553555i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −4.34315 7.52255i −0.143267 0.248146i 0.785458 0.618915i \(-0.212427\pi\)
−0.928725 + 0.370769i \(0.879094\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 27.5147 0.905658
\(924\) 0 0
\(925\) −44.9706 −1.47862
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.94975 3.37706i −0.0639691 0.110798i 0.832267 0.554375i \(-0.187043\pi\)
−0.896236 + 0.443577i \(0.853709\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −6.48528 + 11.2328i −0.212091 + 0.367353i
\(936\) 0 0
\(937\) 19.3553 0.632311 0.316156 0.948707i \(-0.397608\pi\)
0.316156 + 0.948707i \(0.397608\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −5.36396 + 9.29065i −0.174860 + 0.302867i −0.940113 0.340863i \(-0.889281\pi\)
0.765253 + 0.643730i \(0.222614\pi\)
\(942\) 0 0
\(943\) 0.727922 + 1.26080i 0.0237044 + 0.0410572i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 7.38478 + 12.7908i 0.239973 + 0.415645i 0.960706 0.277567i \(-0.0895281\pi\)
−0.720733 + 0.693212i \(0.756195\pi\)
\(948\) 0 0
\(949\) −34.4558 + 59.6793i −1.11848 + 1.93727i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −2.00000 −0.0647864 −0.0323932 0.999475i \(-0.510313\pi\)
−0.0323932 + 0.999475i \(0.510313\pi\)
\(954\) 0 0
\(955\) −4.72792 + 8.18900i −0.152992 + 0.264990i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 11.5000 + 19.9186i 0.370968 + 0.642535i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1.17157 0.0377143
\(966\) 0 0
\(967\) −8.68629 −0.279332 −0.139666 0.990199i \(-0.544603\pi\)
−0.139666 + 0.990199i \(0.544603\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 10.0000 + 17.3205i 0.320915 + 0.555842i 0.980677 0.195633i \(-0.0626762\pi\)
−0.659762 + 0.751475i \(0.729343\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 15.8995 27.5387i 0.508670 0.881042i −0.491280 0.871002i \(-0.663471\pi\)
0.999950 0.0100402i \(-0.00319596\pi\)
\(978\) 0 0
\(979\) 68.7696 2.19788
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −23.3137 + 40.3805i −0.743592 + 1.28794i 0.207258 + 0.978286i \(0.433546\pi\)
−0.950850 + 0.309652i \(0.899787\pi\)
\(984\) 0 0
\(985\) −7.41421 12.8418i −0.236236 0.409174i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 4.68629 + 8.11689i 0.149015 + 0.258102i
\(990\) 0 0
\(991\) −13.3137 + 23.0600i −0.422924 + 0.732526i −0.996224 0.0868198i \(-0.972330\pi\)
0.573300 + 0.819345i \(0.305663\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −3.31371 −0.105052
\(996\) 0 0
\(997\) −8.80761 + 15.2552i −0.278940 + 0.483138i −0.971122 0.238585i \(-0.923316\pi\)
0.692182 + 0.721723i \(0.256650\pi\)
\(998\) 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 3528.2.s.bc.361.2 4
3.2 odd 2 1176.2.q.m.361.1 4
7.2 even 3 inner 3528.2.s.bc.3313.2 4
7.3 odd 6 3528.2.a.bc.1.2 2
7.4 even 3 3528.2.a.bm.1.1 2
7.5 odd 6 3528.2.s.bl.3313.1 4
7.6 odd 2 3528.2.s.bl.361.1 4
12.11 even 2 2352.2.q.bg.1537.1 4
21.2 odd 6 1176.2.q.m.961.1 4
21.5 even 6 1176.2.q.n.961.2 4
21.11 odd 6 1176.2.a.m.1.2 yes 2
21.17 even 6 1176.2.a.l.1.1 2
21.20 even 2 1176.2.q.n.361.2 4
28.3 even 6 7056.2.a.ce.1.2 2
28.11 odd 6 7056.2.a.cw.1.1 2
84.11 even 6 2352.2.a.z.1.2 2
84.23 even 6 2352.2.q.bg.961.1 4
84.47 odd 6 2352.2.q.ba.961.2 4
84.59 odd 6 2352.2.a.bg.1.1 2
84.83 odd 2 2352.2.q.ba.1537.2 4
168.11 even 6 9408.2.a.ed.1.1 2
168.53 odd 6 9408.2.a.dr.1.1 2
168.59 odd 6 9408.2.a.dh.1.2 2
168.101 even 6 9408.2.a.dv.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1176.2.a.l.1.1 2 21.17 even 6
1176.2.a.m.1.2 yes 2 21.11 odd 6
1176.2.q.m.361.1 4 3.2 odd 2
1176.2.q.m.961.1 4 21.2 odd 6
1176.2.q.n.361.2 4 21.20 even 2
1176.2.q.n.961.2 4 21.5 even 6
2352.2.a.z.1.2 2 84.11 even 6
2352.2.a.bg.1.1 2 84.59 odd 6
2352.2.q.ba.961.2 4 84.47 odd 6
2352.2.q.ba.1537.2 4 84.83 odd 2
2352.2.q.bg.961.1 4 84.23 even 6
2352.2.q.bg.1537.1 4 12.11 even 2
3528.2.a.bc.1.2 2 7.3 odd 6
3528.2.a.bm.1.1 2 7.4 even 3
3528.2.s.bc.361.2 4 1.1 even 1 trivial
3528.2.s.bc.3313.2 4 7.2 even 3 inner
3528.2.s.bl.361.1 4 7.6 odd 2
3528.2.s.bl.3313.1 4 7.5 odd 6
7056.2.a.ce.1.2 2 28.3 even 6
7056.2.a.cw.1.1 2 28.11 odd 6
9408.2.a.dh.1.2 2 168.59 odd 6
9408.2.a.dr.1.1 2 168.53 odd 6
9408.2.a.dv.1.2 2 168.101 even 6
9408.2.a.ed.1.1 2 168.11 even 6