Properties

Label 1764.4.k.c.361.1
Level $1764$
Weight $4$
Character 1764.361
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 361.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 1764.361
Dual form 1764.4.k.c.1549.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-7.00000 - 12.1244i) q^{5} +(2.00000 - 3.46410i) q^{11} -54.0000 q^{13} +(7.00000 - 12.1244i) q^{17} +(46.0000 + 79.6743i) q^{19} +(-76.0000 - 131.636i) q^{23} +(-35.5000 + 61.4878i) q^{25} +106.000 q^{29} +(-72.0000 + 124.708i) q^{31} +(-79.0000 - 136.832i) q^{37} -390.000 q^{41} -508.000 q^{43} +(264.000 + 457.261i) q^{47} +(303.000 - 524.811i) q^{53} -56.0000 q^{55} +(182.000 - 315.233i) q^{59} +(339.000 + 587.165i) q^{61} +(378.000 + 654.715i) q^{65} +(-422.000 + 730.925i) q^{67} +8.00000 q^{71} +(-211.000 + 365.463i) q^{73} +(-192.000 - 332.554i) q^{79} -548.000 q^{83} -196.000 q^{85} +(-597.000 - 1034.03i) q^{89} +(644.000 - 1115.44i) q^{95} +1502.00 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 14 q^{5} + 4 q^{11} - 108 q^{13} + 14 q^{17} + 92 q^{19} - 152 q^{23} - 71 q^{25} + 212 q^{29} - 144 q^{31} - 158 q^{37} - 780 q^{41} - 1016 q^{43} + 528 q^{47} + 606 q^{53} - 112 q^{55} + 364 q^{59}+ \cdots + 3004 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{2}{3}\right)\)

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
\(5\) −7.00000 12.1244i −0.626099 1.08444i −0.988327 0.152346i \(-0.951317\pi\)
0.362228 0.932089i \(-0.382016\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 2.00000 3.46410i 0.0548202 0.0949514i −0.837313 0.546724i \(-0.815875\pi\)
0.892133 + 0.451772i \(0.149208\pi\)
\(12\) 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 −0.811762 0.583988i \(-0.801491\pi\)
0.911630 + 0.411012i \(0.134825\pi\)
\(18\) 0 0
\(19\) 46.0000 + 79.6743i 0.555428 + 0.962029i 0.997870 + 0.0652319i \(0.0207787\pi\)
−0.442443 + 0.896797i \(0.645888\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −76.0000 131.636i −0.689004 1.19339i −0.972160 0.234316i \(-0.924715\pi\)
0.283156 0.959074i \(-0.408619\pi\)
\(24\) 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 −0.995651 0.0931587i \(-0.970304\pi\)
0.578503 + 0.815680i \(0.303637\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −79.0000 136.832i −0.351014 0.607974i 0.635413 0.772172i \(-0.280830\pi\)
−0.986427 + 0.164198i \(0.947496\pi\)
\(38\) 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 0.906179 + 0.422894i \(0.138986\pi\)
−0.0868522 + 0.996221i \(0.527681\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 303.000 524.811i 0.785288 1.36016i −0.143539 0.989645i \(-0.545848\pi\)
0.928827 0.370514i \(-0.120818\pi\)
\(54\) 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 −0.592319 0.805703i \(-0.701788\pi\)
0.993919 + 0.110112i \(0.0351210\pi\)
\(60\) 0 0
\(61\) 339.000 + 587.165i 0.711549 + 1.23244i 0.964275 + 0.264902i \(0.0853395\pi\)
−0.252726 + 0.967538i \(0.581327\pi\)
\(62\) 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 0.168357 + 0.985726i \(0.446154\pi\)
−0.937842 + 0.347061i \(0.887180\pi\)
\(68\) 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 −0.984113 0.177546i \(-0.943184\pi\)
0.645816 + 0.763494i \(0.276518\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −192.000 332.554i −0.273439 0.473610i 0.696301 0.717750i \(-0.254828\pi\)
−0.969740 + 0.244139i \(0.921495\pi\)
\(80\) 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 −0.964470 0.264192i \(-0.914895\pi\)
0.253438 0.967352i \(-0.418439\pi\)
\(90\) 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 −0.947245 0.320511i \(-0.896145\pi\)
0.751193 + 0.660083i \(0.229479\pi\)
\(102\) 0 0
\(103\) 580.000 + 1004.59i 0.554846 + 0.961021i 0.997916 + 0.0645337i \(0.0205560\pi\)
−0.443070 + 0.896487i \(0.646111\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 162.000 + 280.592i 0.146366 + 0.253513i 0.929882 0.367859i \(-0.119909\pi\)
−0.783516 + 0.621372i \(0.786576\pi\)
\(108\) 0 0
\(109\) 469.000 812.332i 0.412129 0.713828i −0.582993 0.812477i \(-0.698119\pi\)
0.995122 + 0.0986487i \(0.0314520\pi\)
\(110\) 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 0.999225 0.0393543i \(-0.0125301\pi\)
−0.533694 + 0.845677i \(0.679197\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1405.00 2433.53i 0.876184 1.51760i 0.0206885 0.999786i \(-0.493414\pi\)
0.855496 0.517810i \(-0.173253\pi\)
\(138\) 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 0.990731 0.135838i \(-0.0433727\pi\)
−0.613005 + 0.790079i \(0.710039\pi\)
\(150\) 0 0
\(151\) −1052.00 + 1822.12i −0.566957 + 0.981999i 0.429907 + 0.902873i \(0.358546\pi\)
−0.996865 + 0.0791258i \(0.974787\pi\)
\(152\) 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 −0.0945650 0.995519i \(-0.530146\pi\)
0.909427 0.415864i \(-0.136521\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −166.000 287.520i −0.0797676 0.138162i 0.823382 0.567488i \(-0.192084\pi\)
−0.903150 + 0.429326i \(0.858751\pi\)
\(164\) 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 0.956899 + 0.290422i \(0.0937958\pi\)
−0.226936 + 0.973910i \(0.572871\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −450.000 + 779.423i −0.187903 + 0.325457i −0.944551 0.328365i \(-0.893502\pi\)
0.756648 + 0.653822i \(0.226836\pi\)
\(180\) 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 0.930825 + 0.365466i \(0.119090\pi\)
−0.148910 + 0.988851i \(0.547576\pi\)
\(192\) 0 0
\(193\) 671.000 1162.21i 0.250257 0.433458i −0.713339 0.700819i \(-0.752818\pi\)
0.963597 + 0.267361i \(0.0861515\pi\)
\(194\) 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 −0.799092 0.601208i \(-0.794686\pi\)
0.920208 + 0.391430i \(0.128020\pi\)
\(200\) 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 −0.576582 0.817039i \(-0.695614\pi\)
0.995868 + 0.0908148i \(0.0289471\pi\)
\(228\) 0 0
\(229\) 2399.00 + 4155.19i 0.692272 + 1.19905i 0.971092 + 0.238707i \(0.0767237\pi\)
−0.278819 + 0.960344i \(0.589943\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −2563.00 4439.25i −0.720634 1.24817i −0.960746 0.277429i \(-0.910518\pi\)
0.240112 0.970745i \(-0.422816\pi\)
\(234\) 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 −0.915278 0.402822i \(-0.868029\pi\)
0.806493 + 0.591243i \(0.201363\pi\)
\(242\) 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 0.993695 0.112113i \(-0.0357619\pi\)
−0.593940 + 0.804509i \(0.702429\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3540.00 + 6131.46i −0.829984 + 1.43757i 0.0680662 + 0.997681i \(0.478317\pi\)
−0.898050 + 0.439893i \(0.855016\pi\)
\(264\) 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 −0.0404764 + 0.999180i \(0.512888\pi\)
−0.845078 + 0.534644i \(0.820446\pi\)
\(270\) 0 0
\(271\) 1584.00 + 2743.57i 0.355060 + 0.614981i 0.987128 0.159931i \(-0.0511272\pi\)
−0.632069 + 0.774912i \(0.717794\pi\)
\(272\) 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 −0.0269403 0.999637i \(-0.508576\pi\)
0.879181 0.476488i \(-0.158090\pi\)
\(278\) 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 −0.979897 0.199503i \(-0.936067\pi\)
0.662723 + 0.748864i \(0.269401\pi\)
\(284\) 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 0.109780 + 0.993956i \(0.464985\pi\)
−0.915681 + 0.401906i \(0.868348\pi\)
\(312\) 0 0
\(313\) −523.000 905.863i −0.0944464 0.163586i 0.814931 0.579558i \(-0.196775\pi\)
−0.909377 + 0.415972i \(0.863441\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 3771.00 + 6531.56i 0.668140 + 1.15725i 0.978424 + 0.206609i \(0.0662427\pi\)
−0.310283 + 0.950644i \(0.600424\pi\)
\(318\) 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 0.728634 0.684904i \(-0.240156\pi\)
−0.957461 + 0.288564i \(0.906822\pi\)
\(332\) 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 −0.465489 0.885053i \(-0.654122\pi\)
0.999223 0.0394010i \(-0.0125450\pi\)
\(348\) 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 −0.805280 0.592895i \(-0.797985\pi\)
0.916102 + 0.400946i \(0.131318\pi\)
\(354\) 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 0.999982 + 0.00594499i \(0.00189236\pi\)
−0.494843 + 0.868983i \(0.664774\pi\)
\(360\) 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 0.497058 + 0.867717i \(0.334414\pi\)
−0.999994 + 0.00339411i \(0.998920\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 409.000 + 708.409i 0.0567754 + 0.0983378i 0.893016 0.450025i \(-0.148585\pi\)
−0.836241 + 0.548362i \(0.815251\pi\)
\(374\) 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 0.996352 + 0.0853367i \(0.0271966\pi\)
−0.424272 + 0.905535i \(0.639470\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1865.00 + 3230.27i −0.243083 + 0.421032i −0.961591 0.274487i \(-0.911492\pi\)
0.718508 + 0.695519i \(0.244825\pi\)
\(390\) 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 0.996146 0.0877090i \(-0.0279546\pi\)
−0.574031 + 0.818833i \(0.694621\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1527.00 2644.84i −0.190161 0.329369i 0.755142 0.655561i \(-0.227568\pi\)
−0.945304 + 0.326192i \(0.894234\pi\)
\(402\) 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 −0.857874 0.513860i \(-0.828215\pi\)
0.873953 + 0.486010i \(0.161548\pi\)
\(410\) 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 −0.442666 0.896687i \(-0.645967\pi\)
0.997886 0.0649838i \(-0.0206996\pi\)
\(432\) 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 0.732655 0.680601i \(-0.238281\pi\)
−0.955745 + 0.294197i \(0.904948\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4874.00 + 8442.02i 0.522733 + 0.905400i 0.999650 + 0.0264519i \(0.00842087\pi\)
−0.476917 + 0.878948i \(0.658246\pi\)
\(444\) 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 0.996373 + 0.0850931i \(0.0271188\pi\)
−0.424494 + 0.905431i \(0.639548\pi\)
\(458\) 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 −0.903713 0.428139i \(-0.859170\pi\)
0.0810777 0.996708i \(-0.474164\pi\)
\(468\) 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 −0.585754 0.810489i \(-0.699202\pi\)
0.994781 + 0.102034i \(0.0325350\pi\)
\(480\) 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 −0.998478 0.0551605i \(-0.982433\pi\)
0.547009 + 0.837127i \(0.315766\pi\)
\(488\) 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 −0.918164 0.396201i \(-0.870328\pi\)
0.115961 0.993254i \(-0.463005\pi\)
\(500\) 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 −0.988511 0.151149i \(-0.951703\pi\)
0.363357 0.931650i \(-0.381631\pi\)
\(510\) 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 −0.952861 0.303407i \(-0.901876\pi\)
0.739189 + 0.673498i \(0.235209\pi\)
\(522\) 0 0
\(523\) −878.000 1520.74i −0.0734078 0.127146i 0.826985 0.562224i \(-0.190054\pi\)
−0.900393 + 0.435078i \(0.856721\pi\)
\(524\) 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 −0.984325 0.176367i \(-0.943566\pi\)
0.339424 0.940633i \(-0.389768\pi\)
\(542\) 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 −0.999243 0.0389004i \(-0.987614\pi\)
0.533310 + 0.845920i \(0.320948\pi\)
\(558\) 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 −0.614276 0.789092i \(-0.710552\pi\)
0.990511 + 0.137432i \(0.0438850\pi\)
\(564\) 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 0.754160 + 0.656691i \(0.228044\pi\)
0.191631 + 0.981467i \(0.438622\pi\)
\(570\) 0 0
\(571\) −1866.00 + 3232.01i −0.136759 + 0.236874i −0.926268 0.376865i \(-0.877002\pi\)
0.789509 + 0.613739i \(0.210335\pi\)
\(572\) 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 −0.887092 0.461593i \(-0.847278\pi\)
0.843297 + 0.537448i \(0.180612\pi\)
\(578\) 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 0.739405 0.673261i \(-0.235107\pi\)
−0.952763 + 0.303713i \(0.901774\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1236.00 2140.81i 0.0843098 0.146029i −0.820787 0.571234i \(-0.806465\pi\)
0.905097 + 0.425205i \(0.139798\pi\)
\(600\) 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 0.610767 0.791810i \(-0.290861\pi\)
−0.991111 + 0.133035i \(0.957528\pi\)
\(608\) 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 0.306018 + 0.952026i \(0.401003\pi\)
−0.977488 + 0.210993i \(0.932330\pi\)
\(614\) 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 −0.932180 0.361996i \(-0.882095\pi\)
0.779588 + 0.626293i \(0.215429\pi\)
\(620\) 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 −0.993871 0.110546i \(-0.964740\pi\)
0.592671 + 0.805444i \(0.298073\pi\)
\(642\) 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 −0.576597 0.817029i \(-0.695620\pi\)
0.995866 + 0.0908335i \(0.0289531\pi\)
\(648\) 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 0.823043 0.567980i \(-0.192275\pi\)
−0.903406 + 0.428786i \(0.858941\pi\)
\(654\) 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 −0.982891 0.184186i \(-0.941035\pi\)
0.650956 + 0.759116i \(0.274368\pi\)
\(662\) 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 0.786757 0.617263i \(-0.211758\pi\)
−0.927944 + 0.372721i \(0.878425\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −4494.00 + 7783.84i −0.251769 + 0.436076i −0.964013 0.265855i \(-0.914346\pi\)
0.712244 + 0.701932i \(0.247679\pi\)
\(684\) 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 0.971385 0.237512i \(-0.0763319\pi\)
−0.691384 + 0.722488i \(0.742999\pi\)
\(692\) 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 0.808213 0.588890i \(-0.200435\pi\)
−0.914100 + 0.405488i \(0.867102\pi\)
\(710\) 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 −0.977628 0.210342i \(-0.932542\pi\)
0.306652 0.951822i \(-0.400791\pi\)
\(720\) 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 0.948820 0.315817i \(-0.102278\pi\)
−0.747915 + 0.663794i \(0.768945\pi\)
\(734\) 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 −0.135050 0.990839i \(-0.543119\pi\)
0.925616 0.378463i \(-0.123547\pi\)
\(740\) 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 0.888240 + 0.459379i \(0.151928\pi\)
−0.0462858 + 0.998928i \(0.514738\pi\)
\(752\) 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 −0.999035 0.0439244i \(-0.986014\pi\)
0.461478 0.887152i \(-0.347319\pi\)
\(762\) 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 −0.898709 0.438546i \(-0.855493\pi\)
0.829147 + 0.559031i \(0.188827\pi\)
\(774\) 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 0.273137 + 0.961975i \(0.411939\pi\)
−0.969663 + 0.244444i \(0.921394\pi\)
\(788\) 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
\(795\) 0 0
\(796\) 0