Properties

Label 1764.4.k.c.1549.1
Level $1764$
Weight $4$
Character 1764.1549
Analytic conductor $104.079$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,4,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, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.361");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 4 \)
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: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 1549.1
Root \(0.500000 - 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1549
Dual form 1764.4.k.c.361.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-7.00000 + 12.1244i) q^{5} +O(q^{10})\) \(q+(-7.00000 + 12.1244i) q^{5} +(2.00000 + 3.46410i) q^{11} -54.0000 q^{13} +(7.00000 + 12.1244i) q^{17} +(46.0000 - 79.6743i) q^{19} +(-76.0000 + 131.636i) q^{23} +(-35.5000 - 61.4878i) q^{25} +106.000 q^{29} +(-72.0000 - 124.708i) q^{31} +(-79.0000 + 136.832i) q^{37} -390.000 q^{41} -508.000 q^{43} +(264.000 - 457.261i) q^{47} +(303.000 + 524.811i) q^{53} -56.0000 q^{55} +(182.000 + 315.233i) q^{59} +(339.000 - 587.165i) q^{61} +(378.000 - 654.715i) q^{65} +(-422.000 - 730.925i) q^{67} +8.00000 q^{71} +(-211.000 - 365.463i) q^{73} +(-192.000 + 332.554i) q^{79} -548.000 q^{83} -196.000 q^{85} +(-597.000 + 1034.03i) q^{89} +(644.000 + 1115.44i) q^{95} +1502.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 14 q^{5} + 4 q^{11} - 108 q^{13} + 14 q^{17} + 92 q^{19} - 152 q^{23} - 71 q^{25} + 212 q^{29} - 144 q^{31} - 158 q^{37} - 780 q^{41} - 1016 q^{43} + 528 q^{47} + 606 q^{53} - 112 q^{55} + 364 q^{59} + 678 q^{61} + 756 q^{65} - 844 q^{67} + 16 q^{71} - 422 q^{73} - 384 q^{79} - 1096 q^{83} - 392 q^{85} - 1194 q^{89} + 1288 q^{95} + 3004 q^{97}+O(q^{100}) \) 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
\(5\) −7.00000 + 12.1244i −0.626099 + 1.08444i 0.362228 + 0.932089i \(0.382016\pi\)
−0.988327 + 0.152346i \(0.951317\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 + 3.46410i 0.0548202 + 0.0949514i 0.892133 0.451772i \(-0.149208\pi\)
−0.837313 + 0.546724i \(0.815875\pi\)
\(12\) 0 0
\(13\) −54.0000 −1.15207 −0.576035 0.817425i \(-0.695401\pi\)
−0.576035 + 0.817425i \(0.695401\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 7.00000 + 12.1244i 0.0998676 + 0.172976i 0.911630 0.411012i \(-0.134825\pi\)
−0.811762 + 0.583988i \(0.801491\pi\)
\(18\) 0 0
\(19\) 46.0000 79.6743i 0.555428 0.962029i −0.442443 0.896797i \(-0.645888\pi\)
0.997870 0.0652319i \(-0.0207787\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −76.0000 + 131.636i −0.689004 + 1.19339i 0.283156 + 0.959074i \(0.408619\pi\)
−0.972160 + 0.234316i \(0.924715\pi\)
\(24\) 0 0
\(25\) −35.5000 61.4878i −0.284000 0.491902i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 106.000 0.678748 0.339374 0.940651i \(-0.389785\pi\)
0.339374 + 0.940651i \(0.389785\pi\)
\(30\) 0 0
\(31\) −72.0000 124.708i −0.417148 0.722521i 0.578503 0.815680i \(-0.303637\pi\)
−0.995651 + 0.0931587i \(0.970304\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −79.0000 + 136.832i −0.351014 + 0.607974i −0.986427 0.164198i \(-0.947496\pi\)
0.635413 + 0.772172i \(0.280830\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −390.000 −1.48556 −0.742778 0.669538i \(-0.766492\pi\)
−0.742778 + 0.669538i \(0.766492\pi\)
\(42\) 0 0
\(43\) −508.000 −1.80161 −0.900806 0.434223i \(-0.857023\pi\)
−0.900806 + 0.434223i \(0.857023\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 264.000 457.261i 0.819327 1.41912i −0.0868522 0.996221i \(-0.527681\pi\)
0.906179 0.422894i \(-0.138986\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 303.000 + 524.811i 0.785288 + 1.36016i 0.928827 + 0.370514i \(0.120818\pi\)
−0.143539 + 0.989645i \(0.545848\pi\)
\(54\) 0 0
\(55\) −56.0000 −0.137292
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 182.000 + 315.233i 0.401600 + 0.695591i 0.993919 0.110112i \(-0.0351210\pi\)
−0.592319 + 0.805703i \(0.701788\pi\)
\(60\) 0 0
\(61\) 339.000 587.165i 0.711549 1.23244i −0.252726 0.967538i \(-0.581327\pi\)
0.964275 0.264902i \(-0.0853395\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 378.000 654.715i 0.721310 1.24935i
\(66\) 0 0
\(67\) −422.000 730.925i −0.769485 1.33279i −0.937842 0.347061i \(-0.887180\pi\)
0.168357 0.985726i \(-0.446154\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 8.00000 0.0133722 0.00668609 0.999978i \(-0.497872\pi\)
0.00668609 + 0.999978i \(0.497872\pi\)
\(72\) 0 0
\(73\) −211.000 365.463i −0.338297 0.585948i 0.645816 0.763494i \(-0.276518\pi\)
−0.984113 + 0.177546i \(0.943184\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −192.000 + 332.554i −0.273439 + 0.473610i −0.969740 0.244139i \(-0.921495\pi\)
0.696301 + 0.717750i \(0.254828\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −548.000 −0.724709 −0.362354 0.932040i \(-0.618027\pi\)
−0.362354 + 0.932040i \(0.618027\pi\)
\(84\) 0 0
\(85\) −196.000 −0.250108
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −597.000 + 1034.03i −0.711032 + 1.23154i 0.253438 + 0.967352i \(0.418439\pi\)
−0.964470 + 0.264192i \(0.914895\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 644.000 + 1115.44i 0.695505 + 1.20465i
\(96\) 0 0
\(97\) 1502.00 1.57222 0.786108 0.618089i \(-0.212093\pi\)
0.786108 + 0.618089i \(0.212093\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −199.000 344.678i −0.196052 0.339572i 0.751193 0.660083i \(-0.229479\pi\)
−0.947245 + 0.320511i \(0.896145\pi\)
\(102\) 0 0
\(103\) 580.000 1004.59i 0.554846 0.961021i −0.443070 0.896487i \(-0.646111\pi\)
0.997916 0.0645337i \(-0.0205560\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 162.000 280.592i 0.146366 0.253513i −0.783516 0.621372i \(-0.786576\pi\)
0.929882 + 0.367859i \(0.119909\pi\)
\(108\) 0 0
\(109\) 469.000 + 812.332i 0.412129 + 0.713828i 0.995122 0.0986487i \(-0.0314520\pi\)
−0.582993 + 0.812477i \(0.698119\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 622.000 0.517813 0.258906 0.965902i \(-0.416638\pi\)
0.258906 + 0.965902i \(0.416638\pi\)
\(114\) 0 0
\(115\) −1064.00 1842.90i −0.862770 1.49436i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 657.500 1138.82i 0.493989 0.855615i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −756.000 −0.540950
\(126\) 0 0
\(127\) 1200.00 0.838447 0.419224 0.907883i \(-0.362302\pi\)
0.419224 + 0.907883i \(0.362302\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 698.000 1208.97i 0.465531 0.806323i −0.533694 0.845677i \(-0.679197\pi\)
0.999225 + 0.0393543i \(0.0125301\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1405.00 + 2433.53i 0.876184 + 1.51760i 0.855496 + 0.517810i \(0.173253\pi\)
0.0206885 + 0.999786i \(0.493414\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.00244083 −0.00122042 0.999999i \(-0.500388\pi\)
−0.00122042 + 0.999999i \(0.500388\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −108.000 187.061i −0.0631567 0.109391i
\(144\) 0 0
\(145\) −742.000 + 1285.18i −0.424964 + 0.736059i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 687.000 1189.92i 0.377726 0.654241i −0.613005 0.790079i \(-0.710039\pi\)
0.990731 + 0.135838i \(0.0433727\pi\)
\(150\) 0 0
\(151\) −1052.00 1822.12i −0.566957 0.981999i −0.996865 0.0791258i \(-0.974787\pi\)
0.429907 0.902873i \(-0.358546\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2016.00 1.04470
\(156\) 0 0
\(157\) 1603.00 + 2776.48i 0.814862 + 1.41138i 0.909427 + 0.415864i \(0.136521\pi\)
−0.0945650 + 0.995519i \(0.530146\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −166.000 + 287.520i −0.0797676 + 0.138162i −0.903150 0.429326i \(-0.858751\pi\)
0.823382 + 0.567488i \(0.192084\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1496.00 0.693197 0.346599 0.938014i \(-0.387337\pi\)
0.346599 + 0.938014i \(0.387337\pi\)
\(168\) 0 0
\(169\) 719.000 0.327264
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1661.00 2876.94i 0.729962 1.26433i −0.226936 0.973910i \(-0.572871\pi\)
0.956899 0.290422i \(-0.0937958\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −450.000 779.423i −0.187903 0.325457i 0.756648 0.653822i \(-0.226836\pi\)
−0.944551 + 0.328365i \(0.893502\pi\)
\(180\) 0 0
\(181\) −1902.00 −0.781075 −0.390537 0.920587i \(-0.627711\pi\)
−0.390537 + 0.920587i \(0.627711\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1106.00 1915.65i −0.439539 0.761304i
\(186\) 0 0
\(187\) −28.0000 + 48.4974i −0.0109495 + 0.0189651i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 2064.00 3574.95i 0.781915 1.35432i −0.148910 0.988851i \(-0.547576\pi\)
0.930825 0.365466i \(-0.119090\pi\)
\(192\) 0 0
\(193\) 671.000 + 1162.21i 0.250257 + 0.433458i 0.963597 0.267361i \(-0.0861515\pi\)
−0.713339 + 0.700819i \(0.752818\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3506.00 1.26798 0.633990 0.773341i \(-0.281416\pi\)
0.633990 + 0.773341i \(0.281416\pi\)
\(198\) 0 0
\(199\) 340.000 + 588.897i 0.121115 + 0.209778i 0.920208 0.391430i \(-0.128020\pi\)
−0.799092 + 0.601208i \(0.794686\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 2730.00 4728.50i 0.930105 1.61099i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 368.000 0.121795
\(210\) 0 0
\(211\) 5372.00 1.75272 0.876360 0.481657i \(-0.159965\pi\)
0.876360 + 0.481657i \(0.159965\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 3556.00 6159.17i 1.12799 1.95373i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −378.000 654.715i −0.115054 0.199280i
\(222\) 0 0
\(223\) 1072.00 0.321912 0.160956 0.986962i \(-0.448542\pi\)
0.160956 + 0.986962i \(0.448542\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 1434.00 + 2483.76i 0.419286 + 0.726225i 0.995868 0.0908148i \(-0.0289471\pi\)
−0.576582 + 0.817039i \(0.695614\pi\)
\(228\) 0 0
\(229\) 2399.00 4155.19i 0.692272 1.19905i −0.278819 0.960344i \(-0.589943\pi\)
0.971092 0.238707i \(-0.0767237\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2563.00 + 4439.25i −0.720634 + 1.24817i 0.240112 + 0.970745i \(0.422816\pi\)
−0.960746 + 0.277429i \(0.910518\pi\)
\(234\) 0 0
\(235\) 3696.00 + 6401.66i 1.02596 + 1.77701i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 528.000 0.142902 0.0714508 0.997444i \(-0.477237\pi\)
0.0714508 + 0.997444i \(0.477237\pi\)
\(240\) 0 0
\(241\) −407.000 704.945i −0.108785 0.188421i 0.806493 0.591243i \(-0.201363\pi\)
−0.915278 + 0.402822i \(0.868029\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2484.00 + 4302.41i −0.639891 + 1.10832i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −1932.00 −0.485844 −0.242922 0.970046i \(-0.578106\pi\)
−0.242922 + 0.970046i \(0.578106\pi\)
\(252\) 0 0
\(253\) −608.000 −0.151086
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 1647.00 2852.69i 0.399755 0.692396i −0.593940 0.804509i \(-0.702429\pi\)
0.993695 + 0.112113i \(0.0357619\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3540.00 6131.46i −0.829984 1.43757i −0.898050 0.439893i \(-0.855016\pi\)
0.0680662 0.997681i \(-0.478317\pi\)
\(264\) 0 0
\(265\) −8484.00 −1.96667
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3907.00 6767.12i −0.885554 1.53382i −0.845078 0.534644i \(-0.820446\pi\)
−0.0404764 0.999180i \(-0.512888\pi\)
\(270\) 0 0
\(271\) 1584.00 2743.57i 0.355060 0.614981i −0.632069 0.774912i \(-0.717794\pi\)
0.987128 + 0.159931i \(0.0511272\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 142.000 245.951i 0.0311379 0.0539324i
\(276\) 0 0
\(277\) 3929.00 + 6805.23i 0.852241 + 1.47612i 0.879181 + 0.476488i \(0.158090\pi\)
−0.0269403 + 0.999637i \(0.508576\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −6730.00 −1.42875 −0.714374 0.699764i \(-0.753288\pi\)
−0.714374 + 0.699764i \(0.753288\pi\)
\(282\) 0 0
\(283\) −1510.00 2615.40i −0.317174 0.549361i 0.662723 0.748864i \(-0.269401\pi\)
−0.979897 + 0.199503i \(0.936067\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2358.50 4085.04i 0.480053 0.831476i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6834.00 −1.36262 −0.681308 0.731997i \(-0.738589\pi\)
−0.681308 + 0.731997i \(0.738589\pi\)
\(294\) 0 0
\(295\) −5096.00 −1.00576
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4104.00 7108.34i 0.793781 1.37487i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 4746.00 + 8220.31i 0.891001 + 1.54326i
\(306\) 0 0
\(307\) −2332.00 −0.433532 −0.216766 0.976224i \(-0.569551\pi\)
−0.216766 + 0.976224i \(0.569551\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −4420.00 7655.66i −0.805901 1.39586i −0.915681 0.401906i \(-0.868348\pi\)
0.109780 0.993956i \(-0.464985\pi\)
\(312\) 0 0
\(313\) −523.000 + 905.863i −0.0944464 + 0.163586i −0.909377 0.415972i \(-0.863441\pi\)
0.814931 + 0.579558i \(0.196775\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3771.00 6531.56i 0.668140 1.15725i −0.310283 0.950644i \(-0.600424\pi\)
0.978424 0.206609i \(-0.0662427\pi\)
\(318\) 0 0
\(319\) 212.000 + 367.195i 0.0372092 + 0.0644482i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 1288.00 0.221877
\(324\) 0 0
\(325\) 1917.00 + 3320.34i 0.327188 + 0.566706i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1378.00 + 2386.77i −0.228827 + 0.396340i −0.957461 0.288564i \(-0.906822\pi\)
0.728634 + 0.684904i \(0.240156\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 11816.0 1.92710
\(336\) 0 0
\(337\) 3954.00 0.639134 0.319567 0.947564i \(-0.396463\pi\)
0.319567 + 0.947564i \(0.396463\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 288.000 498.831i 0.0457363 0.0792176i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3450.00 + 5975.58i 0.533734 + 0.924454i 0.999223 + 0.0394010i \(0.0125450\pi\)
−0.465489 + 0.885053i \(0.654122\pi\)
\(348\) 0 0
\(349\) 2426.00 0.372094 0.186047 0.982541i \(-0.440432\pi\)
0.186047 + 0.982541i \(0.440432\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 735.000 + 1273.06i 0.110822 + 0.191949i 0.916102 0.400946i \(-0.131318\pi\)
−0.805280 + 0.592895i \(0.797985\pi\)
\(354\) 0 0
\(355\) −56.0000 + 96.9948i −0.00837231 + 0.0145013i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 3436.00 5951.33i 0.505140 0.874928i −0.494843 0.868983i \(-0.664774\pi\)
0.999982 0.00594499i \(-0.00189236\pi\)
\(360\) 0 0
\(361\) −802.500 1389.97i −0.117000 0.202649i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 5908.00 0.847230
\(366\) 0 0
\(367\) −3536.00 6124.53i −0.502937 0.871112i −0.999994 0.00339411i \(-0.998920\pi\)
0.497058 0.867717i \(-0.334414\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 409.000 708.409i 0.0567754 0.0983378i −0.836241 0.548362i \(-0.815251\pi\)
0.893016 + 0.450025i \(0.148585\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −5724.00 −0.781966
\(378\) 0 0
\(379\) −5132.00 −0.695549 −0.347775 0.937578i \(-0.613063\pi\)
−0.347775 + 0.937578i \(0.613063\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 4288.00 7427.03i 0.572080 0.990871i −0.424272 0.905535i \(-0.639470\pi\)
0.996352 0.0853367i \(-0.0271966\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1865.00 3230.27i −0.243083 0.421032i 0.718508 0.695519i \(-0.244825\pi\)
−0.961591 + 0.274487i \(0.911492\pi\)
\(390\) 0 0
\(391\) −2128.00 −0.275237
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −2688.00 4655.75i −0.342400 0.593054i
\(396\) 0 0
\(397\) 3339.00 5783.32i 0.422115 0.731124i −0.574031 0.818833i \(-0.694621\pi\)
0.996146 + 0.0877090i \(0.0279546\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1527.00 + 2644.84i −0.190161 + 0.329369i −0.945304 0.326192i \(-0.894234\pi\)
0.755142 + 0.655561i \(0.227568\pi\)
\(402\) 0 0
\(403\) 3888.00 + 6734.21i 0.480583 + 0.832395i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −632.000 −0.0769707
\(408\) 0 0
\(409\) 133.000 + 230.363i 0.0160793 + 0.0278501i 0.873953 0.486010i \(-0.161548\pi\)
−0.857874 + 0.513860i \(0.828215\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 3836.00 6644.15i 0.453739 0.785900i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 8844.00 1.03116 0.515582 0.856840i \(-0.327576\pi\)
0.515582 + 0.856840i \(0.327576\pi\)
\(420\) 0 0
\(421\) −4482.00 −0.518858 −0.259429 0.965762i \(-0.583534\pi\)
−0.259429 + 0.965762i \(0.583534\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 497.000 860.829i 0.0567248 0.0982502i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 4968.00 + 8604.83i 0.555221 + 0.961671i 0.997886 + 0.0649838i \(0.0206996\pi\)
−0.442666 + 0.896687i \(0.645967\pi\)
\(432\) 0 0
\(433\) 11758.0 1.30497 0.652487 0.757800i \(-0.273726\pi\)
0.652487 + 0.757800i \(0.273726\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 6992.00 + 12110.5i 0.765384 + 1.32568i
\(438\) 0 0
\(439\) −2052.00 + 3554.17i −0.223090 + 0.386404i −0.955745 0.294197i \(-0.904948\pi\)
0.732655 + 0.680601i \(0.238281\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4874.00 8442.02i 0.522733 0.905400i −0.476917 0.878948i \(-0.658246\pi\)
0.999650 0.0264519i \(-0.00842087\pi\)
\(444\) 0 0
\(445\) −8358.00 14476.5i −0.890353 1.54214i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 478.000 0.0502410 0.0251205 0.999684i \(-0.492003\pi\)
0.0251205 + 0.999684i \(0.492003\pi\)
\(450\) 0 0
\(451\) −780.000 1351.00i −0.0814385 0.141056i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5587.00 9676.97i 0.571879 0.990524i −0.424494 0.905431i \(-0.639548\pi\)
0.996373 0.0850931i \(-0.0271188\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −11674.0 −1.17942 −0.589710 0.807615i \(-0.700758\pi\)
−0.589710 + 0.807615i \(0.700758\pi\)
\(462\) 0 0
\(463\) 10528.0 1.05676 0.528378 0.849009i \(-0.322801\pi\)
0.528378 + 0.849009i \(0.322801\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −8302.00 + 14379.5i −0.822635 + 1.42485i 0.0810777 + 0.996708i \(0.474164\pi\)
−0.903713 + 0.428139i \(0.859170\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1016.00 1759.76i −0.0987648 0.171066i
\(474\) 0 0
\(475\) −6532.00 −0.630966
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 4288.00 + 7427.03i 0.409027 + 0.708455i 0.994781 0.102034i \(-0.0325350\pi\)
−0.585754 + 0.810489i \(0.699202\pi\)
\(480\) 0 0
\(481\) 4266.00 7388.93i 0.404393 0.700429i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −10514.0 + 18210.8i −0.984363 + 1.70497i
\(486\) 0 0
\(487\) −4852.00 8403.91i −0.451468 0.781966i 0.547009 0.837127i \(-0.315766\pi\)
−0.998478 + 0.0551605i \(0.982433\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 4092.00 0.376109 0.188054 0.982159i \(-0.439782\pi\)
0.188054 + 0.982159i \(0.439782\pi\)
\(492\) 0 0
\(493\) 742.000 + 1285.18i 0.0677850 + 0.117407i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −8942.00 + 15488.0i −0.802202 + 1.38945i 0.115961 + 0.993254i \(0.463005\pi\)
−0.918164 + 0.396201i \(0.870328\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −7704.00 −0.682911 −0.341456 0.939898i \(-0.610920\pi\)
−0.341456 + 0.939898i \(0.610920\pi\)
\(504\) 0 0
\(505\) 5572.00 0.490992
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −7179.00 + 12434.4i −0.625154 + 1.08280i 0.363357 + 0.931650i \(0.381631\pi\)
−0.988511 + 0.151149i \(0.951703\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 8120.00 + 14064.3i 0.694777 + 1.20339i
\(516\) 0 0
\(517\) 2112.00 0.179663
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2541.00 4401.14i −0.213672 0.370091i 0.739189 0.673498i \(-0.235209\pi\)
−0.952861 + 0.303407i \(0.901876\pi\)
\(522\) 0 0
\(523\) −878.000 + 1520.74i −0.0734078 + 0.127146i −0.900393 0.435078i \(-0.856721\pi\)
0.826985 + 0.562224i \(0.190054\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1008.00 1745.91i 0.0833191 0.144313i
\(528\) 0 0
\(529\) −5468.50 9471.72i −0.449453 0.778476i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 21060.0 1.71146
\(534\) 0 0
\(535\) 2268.00 + 3928.29i 0.183279 + 0.317448i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −8115.00 + 14055.6i −0.644900 + 1.11700i 0.339424 + 0.940633i \(0.389768\pi\)
−0.984325 + 0.176367i \(0.943566\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −13132.0 −1.03213
\(546\) 0 0
\(547\) 17676.0 1.38167 0.690833 0.723014i \(-0.257244\pi\)
0.690833 + 0.723014i \(0.257244\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 4876.00 8445.48i 0.376996 0.652976i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −6125.00 10608.8i −0.465933 0.807019i 0.533310 0.845920i \(-0.320948\pi\)
−0.999243 + 0.0389004i \(0.987614\pi\)
\(558\) 0 0
\(559\) 27432.0 2.07558
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 5026.00 + 8705.29i 0.376236 + 0.651659i 0.990511 0.137432i \(-0.0438850\pi\)
−0.614276 + 0.789092i \(0.710552\pi\)
\(564\) 0 0
\(565\) −4354.00 + 7541.35i −0.324202 + 0.561534i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 12837.0 22234.3i 0.945791 1.63816i 0.191631 0.981467i \(-0.438622\pi\)
0.754160 0.656691i \(-0.228044\pi\)
\(570\) 0 0
\(571\) −1866.00 3232.01i −0.136759 0.236874i 0.789509 0.613739i \(-0.210335\pi\)
−0.926268 + 0.376865i \(0.877002\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 10792.0 0.782709
\(576\) 0 0
\(577\) −607.000 1051.35i −0.0437950 0.0758552i 0.843297 0.537448i \(-0.180612\pi\)
−0.887092 + 0.461593i \(0.847278\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1212.00 + 2099.25i −0.0860993 + 0.149128i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 7108.00 0.499793 0.249897 0.968273i \(-0.419603\pi\)
0.249897 + 0.968273i \(0.419603\pi\)
\(588\) 0 0
\(589\) −13248.0 −0.926782
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −3081.00 + 5336.45i −0.213358 + 0.369548i −0.952763 0.303713i \(-0.901774\pi\)
0.739405 + 0.673261i \(0.235107\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1236.00 + 2140.81i 0.0843098 + 0.146029i 0.905097 0.425205i \(-0.139798\pi\)
−0.820787 + 0.571234i \(0.806465\pi\)
\(600\) 0 0
\(601\) 13750.0 0.933235 0.466617 0.884459i \(-0.345472\pi\)
0.466617 + 0.884459i \(0.345472\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 9205.00 + 15943.5i 0.618573 + 1.07140i
\(606\) 0 0
\(607\) −5688.00 + 9851.90i −0.380344 + 0.658775i −0.991111 0.133035i \(-0.957528\pi\)
0.610767 + 0.791810i \(0.290861\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −14256.0 + 24692.1i −0.943921 + 1.63492i
\(612\) 0 0
\(613\) −10191.0 17651.3i −0.671469 1.16302i −0.977488 0.210993i \(-0.932330\pi\)
0.306018 0.952026i \(-0.401003\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −21178.0 −1.38184 −0.690919 0.722932i \(-0.742794\pi\)
−0.690919 + 0.722932i \(0.742794\pi\)
\(618\) 0 0
\(619\) −2350.00 4070.32i −0.152592 0.264297i 0.779588 0.626293i \(-0.215429\pi\)
−0.932180 + 0.361996i \(0.882095\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 9729.50 16852.0i 0.622688 1.07853i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −2212.00 −0.140220
\(630\) 0 0
\(631\) −21736.0 −1.37131 −0.685655 0.727927i \(-0.740484\pi\)
−0.685655 + 0.727927i \(0.740484\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −8400.00 + 14549.2i −0.524951 + 0.909242i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −6511.00 11277.4i −0.401200 0.694898i 0.592671 0.805444i \(-0.298073\pi\)
−0.993871 + 0.110546i \(0.964740\pi\)
\(642\) 0 0
\(643\) −3308.00 −0.202885 −0.101442 0.994841i \(-0.532346\pi\)
−0.101442 + 0.994841i \(0.532346\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 6900.00 + 11951.2i 0.419269 + 0.726195i 0.995866 0.0908335i \(-0.0289531\pi\)
−0.576597 + 0.817029i \(0.695620\pi\)
\(648\) 0 0
\(649\) −728.000 + 1260.93i −0.0440316 + 0.0762649i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1341.00 + 2322.68i −0.0803635 + 0.139194i −0.903406 0.428786i \(-0.858941\pi\)
0.823043 + 0.567980i \(0.192275\pi\)
\(654\) 0 0
\(655\) 9772.00 + 16925.6i 0.582937 + 1.00968i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −23836.0 −1.40898 −0.704491 0.709713i \(-0.748825\pi\)
−0.704491 + 0.709713i \(0.748825\pi\)
\(660\) 0 0
\(661\) −5641.00 9770.50i −0.331936 0.574929i 0.650956 0.759116i \(-0.274368\pi\)
−0.982891 + 0.184186i \(0.941035\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −8056.00 + 13953.4i −0.467661 + 0.810012i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 2712.00 0.156029
\(672\) 0 0
\(673\) −13726.0 −0.786179 −0.393089 0.919500i \(-0.628594\pi\)
−0.393089 + 0.919500i \(0.628594\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −2487.00 + 4307.61i −0.141186 + 0.244542i −0.927944 0.372721i \(-0.878425\pi\)
0.786757 + 0.617263i \(0.211758\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −4494.00 7783.84i −0.251769 0.436076i 0.712244 0.701932i \(-0.247679\pi\)
−0.964013 + 0.265855i \(0.914346\pi\)
\(684\) 0 0
\(685\) −39340.0 −2.19431
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −16362.0 28339.8i −0.904706 1.56700i
\(690\) 0 0
\(691\) 5086.00 8809.21i 0.280001 0.484976i −0.691384 0.722488i \(-0.742999\pi\)
0.971385 + 0.237512i \(0.0763319\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 28.0000 48.4974i 0.00152820 0.00264692i
\(696\) 0 0
\(697\) −2730.00 4728.50i −0.148359 0.256965i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −27446.0 −1.47877 −0.739387 0.673280i \(-0.764885\pi\)
−0.739387 + 0.673280i \(0.764885\pi\)
\(702\) 0 0
\(703\) 7268.00 + 12588.5i 0.389926 + 0.675371i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1999.00 + 3462.37i −0.105887 + 0.183402i −0.914100 0.405488i \(-0.867102\pi\)
0.808213 + 0.588890i \(0.200435\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 21888.0 1.14967
\(714\) 0 0
\(715\) 3024.00 0.158169
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −12936.0 + 22405.8i −0.670976 + 1.16216i 0.306652 + 0.951822i \(0.400791\pi\)
−0.977628 + 0.210342i \(0.932542\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −3763.00 6517.71i −0.192765 0.333878i
\(726\) 0 0
\(727\) −12088.0 −0.616670 −0.308335 0.951278i \(-0.599772\pi\)
−0.308335 + 0.951278i \(0.599772\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −3556.00 6159.17i −0.179923 0.311635i
\(732\) 0 0
\(733\) 3987.00 6905.69i 0.200905 0.347977i −0.747915 0.663794i \(-0.768945\pi\)
0.948820 + 0.315817i \(0.102278\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1688.00 2923.70i 0.0843667 0.146127i
\(738\) 0 0
\(739\) 15882.0 + 27508.4i 0.790567 + 1.36930i 0.925616 + 0.378463i \(0.123547\pi\)
−0.135050 + 0.990839i \(0.543119\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −888.000 −0.0438460 −0.0219230 0.999760i \(-0.506979\pi\)
−0.0219230 + 0.999760i \(0.506979\pi\)
\(744\) 0 0
\(745\) 9618.00 + 16658.9i 0.472988 + 0.819240i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 17328.0 30013.0i 0.841954 1.45831i −0.0462858 0.998928i \(-0.514738\pi\)
0.888240 0.459379i \(-0.151928\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 29456.0 1.41989
\(756\) 0 0
\(757\) −22866.0 −1.09786 −0.548929 0.835869i \(-0.684964\pi\)
−0.548929 + 0.835869i \(0.684964\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −11285.0 + 19546.2i −0.537557 + 0.931076i 0.461478 + 0.887152i \(0.347319\pi\)
−0.999035 + 0.0439244i \(0.986014\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −9828.00 17022.6i −0.462671 0.801369i
\(768\) 0 0
\(769\) 1790.00 0.0839389 0.0419695 0.999119i \(-0.486637\pi\)
0.0419695 + 0.999119i \(0.486637\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1495.00 2589.42i −0.0695620 0.120485i 0.829147 0.559031i \(-0.188827\pi\)
−0.898709 + 0.438546i \(0.855493\pi\)
\(774\) 0 0
\(775\) −5112.00 + 8854.24i −0.236940 + 0.410392i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −17940.0 + 31073.0i −0.825118 + 1.42915i
\(780\) 0 0
\(781\) 16.0000 + 27.7128i 0.000733067 + 0.00126971i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −44884.0 −2.04074
\(786\) 0 0
\(787\) −15378.0 26635.5i −0.696527 1.20642i −0.969663 0.244444i \(-0.921394\pi\)
0.273137 0.961975i \(-0.411939\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −18306.0 + 31706.9i −0.819754 + 1.41986i
\(794\) 0 0