Properties

Label 1764.4.k.bb.361.2
Level $1764$
Weight $4$
Character 1764.361
Analytic conductor $104.079$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $2$

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Newspace parameters

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

Self dual: no
Analytic conductor: \(104.079369250\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\mathbb{Q}[x]/(x^{8} - \cdots)\)
Defining polynomial: \(x^{8} - 2 x^{7} + 27 x^{6} + 10 x^{5} + 446 x^{4} + 62 x^{3} + 3061 x^{2} + 2142 x + 14161\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 2^{8}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 588)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 361.2
Root \(-1.65506 + 2.86665i\) of defining polynomial
Character \(\chi\) \(=\) 1764.361
Dual form 1764.4.k.bb.1549.2

$q$-expansion

\(f(q)\) \(=\) \(q+(-4.08470 - 7.07491i) q^{5} +O(q^{10})\) \(q+(-4.08470 - 7.07491i) q^{5} +(18.9094 - 32.7521i) q^{11} -39.9319 q^{13} +(-4.96798 + 8.60480i) q^{17} +(-45.2229 - 78.3284i) q^{19} +(59.2973 + 102.706i) q^{23} +(29.1304 - 50.4554i) q^{25} +78.4061 q^{29} +(46.0055 - 79.6838i) q^{31} +(-166.218 - 287.897i) q^{37} +71.7451 q^{41} -115.947 q^{43} +(-153.964 - 266.673i) q^{47} +(-201.575 + 349.138i) q^{53} -308.958 q^{55} +(-296.855 + 514.168i) q^{59} +(166.585 + 288.534i) q^{61} +(163.110 + 282.515i) q^{65} +(371.756 - 643.900i) q^{67} +728.272 q^{71} +(-400.874 + 694.335i) q^{73} +(-533.953 - 924.833i) q^{79} +906.756 q^{83} +81.1709 q^{85} +(-556.634 - 964.119i) q^{89} +(-369.444 + 639.896i) q^{95} -1480.94 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 48q^{17} + 192q^{19} + 192q^{23} - 324q^{25} - 192q^{29} + 48q^{31} - 256q^{37} + 2016q^{41} - 224q^{43} - 864q^{47} - 648q^{53} - 4704q^{55} - 336q^{59} + 960q^{61} - 360q^{65} - 720q^{67} + 2688q^{71} + 672q^{73} + 1984q^{79} + 6240q^{83} + 1360q^{85} - 2160q^{89} - 3744q^{95} - 4032q^{97} + O(q^{100}) \)

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
\(5\) −4.08470 7.07491i −0.365347 0.632799i 0.623485 0.781835i \(-0.285716\pi\)
−0.988832 + 0.149036i \(0.952383\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 18.9094 32.7521i 0.518310 0.897739i −0.481464 0.876466i \(-0.659895\pi\)
0.999774 0.0212729i \(-0.00677189\pi\)
\(12\) 0 0
\(13\) −39.9319 −0.851933 −0.425966 0.904739i \(-0.640066\pi\)
−0.425966 + 0.904739i \(0.640066\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −4.96798 + 8.60480i −0.0708772 + 0.122763i −0.899286 0.437361i \(-0.855913\pi\)
0.828409 + 0.560124i \(0.189247\pi\)
\(18\) 0 0
\(19\) −45.2229 78.3284i −0.546045 0.945777i −0.998540 0.0540105i \(-0.982800\pi\)
0.452496 0.891767i \(-0.350534\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 59.2973 + 102.706i 0.537580 + 0.931116i 0.999034 + 0.0439513i \(0.0139947\pi\)
−0.461454 + 0.887164i \(0.652672\pi\)
\(24\) 0 0
\(25\) 29.1304 50.4554i 0.233043 0.403643i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 78.4061 0.502057 0.251028 0.967980i \(-0.419231\pi\)
0.251028 + 0.967980i \(0.419231\pi\)
\(30\) 0 0
\(31\) 46.0055 79.6838i 0.266543 0.461666i −0.701424 0.712744i \(-0.747452\pi\)
0.967967 + 0.251079i \(0.0807853\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −166.218 287.897i −0.738541 1.27919i −0.953152 0.302491i \(-0.902182\pi\)
0.214611 0.976700i \(-0.431152\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 71.7451 0.273285 0.136643 0.990620i \(-0.456369\pi\)
0.136643 + 0.990620i \(0.456369\pi\)
\(42\) 0 0
\(43\) −115.947 −0.411202 −0.205601 0.978636i \(-0.565915\pi\)
−0.205601 + 0.978636i \(0.565915\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −153.964 266.673i −0.477828 0.827623i 0.521849 0.853038i \(-0.325242\pi\)
−0.999677 + 0.0254154i \(0.991909\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −201.575 + 349.138i −0.522424 + 0.904865i 0.477236 + 0.878775i \(0.341639\pi\)
−0.999660 + 0.0260895i \(0.991695\pi\)
\(54\) 0 0
\(55\) −308.958 −0.757451
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −296.855 + 514.168i −0.655038 + 1.13456i 0.326847 + 0.945077i \(0.394014\pi\)
−0.981884 + 0.189481i \(0.939319\pi\)
\(60\) 0 0
\(61\) 166.585 + 288.534i 0.349657 + 0.605623i 0.986188 0.165628i \(-0.0529649\pi\)
−0.636532 + 0.771250i \(0.719632\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 163.110 + 282.515i 0.311251 + 0.539102i
\(66\) 0 0
\(67\) 371.756 643.900i 0.677869 1.17410i −0.297753 0.954643i \(-0.596237\pi\)
0.975622 0.219460i \(-0.0704295\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 728.272 1.21732 0.608662 0.793429i \(-0.291706\pi\)
0.608662 + 0.793429i \(0.291706\pi\)
\(72\) 0 0
\(73\) −400.874 + 694.335i −0.642723 + 1.11323i 0.342099 + 0.939664i \(0.388862\pi\)
−0.984822 + 0.173566i \(0.944471\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −533.953 924.833i −0.760435 1.31711i −0.942626 0.333849i \(-0.891652\pi\)
0.182191 0.983263i \(-0.441681\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 906.756 1.19915 0.599575 0.800319i \(-0.295336\pi\)
0.599575 + 0.800319i \(0.295336\pi\)
\(84\) 0 0
\(85\) 81.1709 0.103579
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −556.634 964.119i −0.662957 1.14827i −0.979835 0.199808i \(-0.935968\pi\)
0.316878 0.948466i \(-0.397365\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −369.444 + 639.896i −0.398991 + 0.691073i
\(96\) 0 0
\(97\) −1480.94 −1.55017 −0.775084 0.631858i \(-0.782293\pi\)
−0.775084 + 0.631858i \(0.782293\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 278.158 481.784i 0.274037 0.474647i −0.695854 0.718183i \(-0.744974\pi\)
0.969892 + 0.243536i \(0.0783075\pi\)
\(102\) 0 0
\(103\) 276.217 + 478.423i 0.264238 + 0.457674i 0.967364 0.253392i \(-0.0815462\pi\)
−0.703126 + 0.711066i \(0.748213\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 266.902 + 462.288i 0.241144 + 0.417674i 0.961040 0.276408i \(-0.0891440\pi\)
−0.719896 + 0.694082i \(0.755811\pi\)
\(108\) 0 0
\(109\) −547.314 + 947.976i −0.480947 + 0.833025i −0.999761 0.0218626i \(-0.993040\pi\)
0.518814 + 0.854887i \(0.326374\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1425.18 1.18646 0.593228 0.805035i \(-0.297853\pi\)
0.593228 + 0.805035i \(0.297853\pi\)
\(114\) 0 0
\(115\) 484.423 839.046i 0.392806 0.680360i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −49.6330 85.9668i −0.0372900 0.0645882i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −1497.13 −1.07126
\(126\) 0 0
\(127\) −786.485 −0.549522 −0.274761 0.961513i \(-0.588599\pi\)
−0.274761 + 0.961513i \(0.588599\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 13.8651 + 24.0151i 0.00924735 + 0.0160169i 0.870612 0.491970i \(-0.163723\pi\)
−0.861365 + 0.507987i \(0.830390\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −1181.49 + 2046.40i −0.736798 + 1.27617i 0.217132 + 0.976142i \(0.430330\pi\)
−0.953930 + 0.300029i \(0.903004\pi\)
\(138\) 0 0
\(139\) −2513.28 −1.53362 −0.766811 0.641873i \(-0.778158\pi\)
−0.766811 + 0.641873i \(0.778158\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −755.090 + 1307.85i −0.441565 + 0.764813i
\(144\) 0 0
\(145\) −320.266 554.716i −0.183425 0.317701i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 1132.92 + 1962.27i 0.622900 + 1.07889i 0.988943 + 0.148296i \(0.0473788\pi\)
−0.366043 + 0.930598i \(0.619288\pi\)
\(150\) 0 0
\(151\) −141.573 + 245.212i −0.0762984 + 0.132153i −0.901650 0.432466i \(-0.857644\pi\)
0.825352 + 0.564619i \(0.190977\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −751.675 −0.389522
\(156\) 0 0
\(157\) 96.2904 166.780i 0.0489478 0.0847801i −0.840513 0.541791i \(-0.817747\pi\)
0.889461 + 0.457011i \(0.151080\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 421.417 + 729.915i 0.202502 + 0.350744i 0.949334 0.314269i \(-0.101759\pi\)
−0.746832 + 0.665013i \(0.768426\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −3859.21 −1.78823 −0.894116 0.447836i \(-0.852195\pi\)
−0.894116 + 0.447836i \(0.852195\pi\)
\(168\) 0 0
\(169\) −602.441 −0.274211
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −755.329 1308.27i −0.331946 0.574947i 0.650948 0.759123i \(-0.274372\pi\)
−0.982893 + 0.184176i \(0.941038\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −1232.31 + 2134.43i −0.514566 + 0.891255i 0.485291 + 0.874353i \(0.338714\pi\)
−0.999857 + 0.0169022i \(0.994620\pi\)
\(180\) 0 0
\(181\) 3297.36 1.35409 0.677047 0.735940i \(-0.263259\pi\)
0.677047 + 0.735940i \(0.263259\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1357.90 + 2351.95i −0.539647 + 0.934696i
\(186\) 0 0
\(187\) 187.883 + 325.424i 0.0734727 + 0.127258i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −2364.85 4096.05i −0.895889 1.55173i −0.832700 0.553724i \(-0.813206\pi\)
−0.0631890 0.998002i \(-0.520127\pi\)
\(192\) 0 0
\(193\) −2276.59 + 3943.17i −0.849080 + 1.47065i 0.0329498 + 0.999457i \(0.489510\pi\)
−0.882030 + 0.471193i \(0.843823\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3109.06 1.12442 0.562212 0.826993i \(-0.309950\pi\)
0.562212 + 0.826993i \(0.309950\pi\)
\(198\) 0 0
\(199\) −221.756 + 384.092i −0.0789943 + 0.136822i −0.902816 0.430027i \(-0.858504\pi\)
0.823822 + 0.566849i \(0.191838\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −293.057 507.590i −0.0998439 0.172935i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −3420.56 −1.13208
\(210\) 0 0
\(211\) 4653.28 1.51822 0.759112 0.650960i \(-0.225633\pi\)
0.759112 + 0.650960i \(0.225633\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 473.607 + 820.311i 0.150231 + 0.260208i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 198.381 343.606i 0.0603826 0.104586i
\(222\) 0 0
\(223\) −2778.90 −0.834481 −0.417240 0.908796i \(-0.637003\pi\)
−0.417240 + 0.908796i \(0.637003\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −1608.61 + 2786.20i −0.470340 + 0.814653i −0.999425 0.0339159i \(-0.989202\pi\)
0.529084 + 0.848569i \(0.322536\pi\)
\(228\) 0 0
\(229\) −1864.01 3228.56i −0.537891 0.931655i −0.999017 0.0443203i \(-0.985888\pi\)
0.461126 0.887335i \(-0.347446\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1801.76 3120.73i −0.506596 0.877451i −0.999971 0.00763377i \(-0.997570\pi\)
0.493374 0.869817i \(-0.335763\pi\)
\(234\) 0 0
\(235\) −1257.79 + 2178.56i −0.349146 + 0.604739i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2348.76 −0.635684 −0.317842 0.948144i \(-0.602958\pi\)
−0.317842 + 0.948144i \(0.602958\pi\)
\(240\) 0 0
\(241\) −2546.70 + 4411.01i −0.680693 + 1.17900i 0.294076 + 0.955782i \(0.404988\pi\)
−0.974770 + 0.223213i \(0.928345\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 1805.84 + 3127.80i 0.465193 + 0.805738i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −5939.02 −1.49350 −0.746749 0.665106i \(-0.768386\pi\)
−0.746749 + 0.665106i \(0.768386\pi\)
\(252\) 0 0
\(253\) 4485.11 1.11453
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 757.944 + 1312.80i 0.183966 + 0.318638i 0.943228 0.332147i \(-0.107773\pi\)
−0.759262 + 0.650785i \(0.774440\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3216.96 + 5571.94i −0.754244 + 1.30639i 0.191505 + 0.981492i \(0.438663\pi\)
−0.945749 + 0.324898i \(0.894670\pi\)
\(264\) 0 0
\(265\) 3293.50 0.763464
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 3474.90 6018.69i 0.787614 1.36419i −0.139811 0.990178i \(-0.544650\pi\)
0.927425 0.374009i \(-0.122017\pi\)
\(270\) 0 0
\(271\) −480.997 833.111i −0.107817 0.186745i 0.807068 0.590458i \(-0.201053\pi\)
−0.914886 + 0.403713i \(0.867719\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −1101.68 1908.16i −0.241577 0.418424i
\(276\) 0 0
\(277\) −380.005 + 658.188i −0.0824271 + 0.142768i −0.904292 0.426915i \(-0.859600\pi\)
0.821865 + 0.569682i \(0.192934\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 4412.07 0.936662 0.468331 0.883553i \(-0.344855\pi\)
0.468331 + 0.883553i \(0.344855\pi\)
\(282\) 0 0
\(283\) −1301.07 + 2253.53i −0.273289 + 0.473351i −0.969702 0.244291i \(-0.921445\pi\)
0.696413 + 0.717641i \(0.254778\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2407.14 + 4169.29i 0.489953 + 0.848623i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −9332.18 −1.86072 −0.930361 0.366644i \(-0.880507\pi\)
−0.930361 + 0.366644i \(0.880507\pi\)
\(294\) 0 0
\(295\) 4850.26 0.957264
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −2367.85 4101.24i −0.457982 0.793248i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 1360.90 2357.15i 0.255492 0.442525i
\(306\) 0 0
\(307\) −4895.51 −0.910103 −0.455051 0.890465i \(-0.650379\pi\)
−0.455051 + 0.890465i \(0.650379\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −3752.64 + 6499.77i −0.684221 + 1.18511i 0.289460 + 0.957190i \(0.406524\pi\)
−0.973681 + 0.227916i \(0.926809\pi\)
\(312\) 0 0
\(313\) −3174.64 5498.63i −0.573294 0.992974i −0.996225 0.0868124i \(-0.972332\pi\)
0.422931 0.906162i \(-0.361001\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3999.80 6927.86i −0.708679 1.22747i −0.965347 0.260968i \(-0.915958\pi\)
0.256669 0.966499i \(-0.417375\pi\)
\(318\) 0 0
\(319\) 1482.61 2567.96i 0.260221 0.450716i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 898.666 0.154808
\(324\) 0 0
\(325\) −1163.23 + 2014.78i −0.198537 + 0.343876i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1060.78 + 1837.32i 0.176150 + 0.305101i 0.940559 0.339631i \(-0.110302\pi\)
−0.764409 + 0.644732i \(0.776969\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −6074.05 −0.990629
\(336\) 0 0
\(337\) −9114.19 −1.47324 −0.736620 0.676307i \(-0.763579\pi\)
−0.736620 + 0.676307i \(0.763579\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1739.87 3013.55i −0.276303 0.478572i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −836.534 + 1448.92i −0.129416 + 0.224156i −0.923451 0.383717i \(-0.874644\pi\)
0.794034 + 0.607873i \(0.207977\pi\)
\(348\) 0 0
\(349\) 3467.56 0.531845 0.265923 0.963994i \(-0.414323\pi\)
0.265923 + 0.963994i \(0.414323\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1992.12 + 3450.45i −0.300368 + 0.520252i −0.976219 0.216785i \(-0.930443\pi\)
0.675851 + 0.737038i \(0.263776\pi\)
\(354\) 0 0
\(355\) −2974.78 5152.46i −0.444746 0.770322i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 1085.47 + 1880.08i 0.159579 + 0.276398i 0.934717 0.355394i \(-0.115653\pi\)
−0.775138 + 0.631792i \(0.782320\pi\)
\(360\) 0 0
\(361\) −660.724 + 1144.41i −0.0963295 + 0.166848i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 6549.81 0.939268
\(366\) 0 0
\(367\) 1796.06 3110.86i 0.255459 0.442468i −0.709561 0.704644i \(-0.751107\pi\)
0.965020 + 0.262176i \(0.0844400\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −3719.90 6443.05i −0.516378 0.894393i −0.999819 0.0190163i \(-0.993947\pi\)
0.483441 0.875377i \(-0.339387\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3130.91 −0.427719
\(378\) 0 0
\(379\) 11243.9 1.52390 0.761951 0.647635i \(-0.224242\pi\)
0.761951 + 0.647635i \(0.224242\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −1770.62 3066.81i −0.236226 0.409156i 0.723402 0.690427i \(-0.242577\pi\)
−0.959628 + 0.281271i \(0.909244\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −5279.10 + 9143.66i −0.688074 + 1.19178i 0.284386 + 0.958710i \(0.408210\pi\)
−0.972460 + 0.233070i \(0.925123\pi\)
\(390\) 0 0
\(391\) −1178.35 −0.152409
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −4362.08 + 7555.34i −0.555645 + 0.962406i
\(396\) 0 0
\(397\) −270.287 468.151i −0.0341696 0.0591834i 0.848435 0.529300i \(-0.177545\pi\)
−0.882604 + 0.470116i \(0.844212\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1083.52 + 1876.70i 0.134933 + 0.233711i 0.925572 0.378572i \(-0.123585\pi\)
−0.790639 + 0.612283i \(0.790251\pi\)
\(402\) 0 0
\(403\) −1837.09 + 3181.93i −0.227077 + 0.393308i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −12572.3 −1.53117
\(408\) 0 0
\(409\) −478.845 + 829.384i −0.0578908 + 0.100270i −0.893518 0.449027i \(-0.851771\pi\)
0.835628 + 0.549296i \(0.185104\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −3703.83 6415.22i −0.438105 0.758821i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 6464.42 0.753718 0.376859 0.926271i \(-0.377004\pi\)
0.376859 + 0.926271i \(0.377004\pi\)
\(420\) 0 0
\(421\) −6201.23 −0.717885 −0.358943 0.933360i \(-0.616863\pi\)
−0.358943 + 0.933360i \(0.616863\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 289.439 + 501.322i 0.0330349 + 0.0572181i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 707.796 1225.94i 0.0791029 0.137010i −0.823760 0.566938i \(-0.808128\pi\)
0.902863 + 0.429928i \(0.141461\pi\)
\(432\) 0 0
\(433\) −8905.73 −0.988411 −0.494206 0.869345i \(-0.664541\pi\)
−0.494206 + 0.869345i \(0.664541\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 5363.19 9289.32i 0.587085 1.01686i
\(438\) 0 0
\(439\) 5438.25 + 9419.33i 0.591238 + 1.02405i 0.994066 + 0.108779i \(0.0346940\pi\)
−0.402828 + 0.915276i \(0.631973\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 5701.48 + 9875.25i 0.611479 + 1.05911i 0.990991 + 0.133927i \(0.0427586\pi\)
−0.379512 + 0.925187i \(0.623908\pi\)
\(444\) 0 0
\(445\) −4547.37 + 7876.28i −0.484418 + 0.839037i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 12689.0 1.33370 0.666852 0.745190i \(-0.267641\pi\)
0.666852 + 0.745190i \(0.267641\pi\)
\(450\) 0 0
\(451\) 1356.66 2349.80i 0.141646 0.245339i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1135.02 1965.92i −0.116180 0.201229i 0.802071 0.597229i \(-0.203732\pi\)
−0.918251 + 0.395999i \(0.870398\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 10731.2 1.08417 0.542085 0.840324i \(-0.317635\pi\)
0.542085 + 0.840324i \(0.317635\pi\)
\(462\) 0 0
\(463\) −3307.74 −0.332017 −0.166008 0.986124i \(-0.553088\pi\)
−0.166008 + 0.986124i \(0.553088\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4623.35 + 8007.87i 0.458122 + 0.793490i 0.998862 0.0476996i \(-0.0151890\pi\)
−0.540740 + 0.841190i \(0.681856\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −2192.48 + 3797.49i −0.213130 + 0.369152i
\(474\) 0 0
\(475\) −5269.45 −0.509008
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −7611.66 + 13183.8i −0.726066 + 1.25758i 0.232468 + 0.972604i \(0.425320\pi\)
−0.958534 + 0.284978i \(0.908014\pi\)
\(480\) 0 0
\(481\) 6637.39 + 11496.3i 0.629187 + 1.08978i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6049.19 + 10477.5i 0.566349 + 0.980946i
\(486\) 0 0
\(487\) 3879.36 6719.25i 0.360966 0.625212i −0.627154 0.778895i \(-0.715780\pi\)
0.988120 + 0.153683i \(0.0491136\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 8342.30 0.766767 0.383384 0.923589i \(-0.374759\pi\)
0.383384 + 0.923589i \(0.374759\pi\)
\(492\) 0 0
\(493\) −389.520 + 674.668i −0.0355844 + 0.0616340i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 1293.47 + 2240.36i 0.116040 + 0.200987i 0.918195 0.396129i \(-0.129647\pi\)
−0.802155 + 0.597116i \(0.796313\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 409.682 0.0363157 0.0181578 0.999835i \(-0.494220\pi\)
0.0181578 + 0.999835i \(0.494220\pi\)
\(504\) 0 0
\(505\) −4544.77 −0.400475
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −2195.25 3802.29i −0.191165 0.331107i 0.754472 0.656333i \(-0.227893\pi\)
−0.945637 + 0.325225i \(0.894560\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 2256.53 3908.43i 0.193077 0.334419i
\(516\) 0 0
\(517\) −11645.5 −0.990652
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 9714.76 16826.5i 0.816912 1.41493i −0.0910341 0.995848i \(-0.529017\pi\)
0.907947 0.419086i \(-0.137649\pi\)
\(522\) 0 0
\(523\) 8974.35 + 15544.0i 0.750326 + 1.29960i 0.947664 + 0.319268i \(0.103437\pi\)
−0.197338 + 0.980336i \(0.563230\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 457.109 + 791.735i 0.0377836 + 0.0654431i
\(528\) 0 0
\(529\) −948.833 + 1643.43i −0.0779841 + 0.135072i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −2864.92 −0.232821
\(534\) 0 0
\(535\) 2180.43 3776.62i 0.176202 0.305192i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −11547.0 20000.0i −0.917641 1.58940i −0.802989 0.595994i \(-0.796758\pi\)
−0.114652 0.993406i \(-0.536575\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 8942.47 0.702850
\(546\) 0 0
\(547\) −2266.68 −0.177178 −0.0885889 0.996068i \(-0.528236\pi\)
−0.0885889 + 0.996068i \(0.528236\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −3545.75 6141.42i −0.274145 0.474834i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −5519.27 + 9559.65i −0.419854 + 0.727209i −0.995924 0.0901913i \(-0.971252\pi\)
0.576070 + 0.817400i \(0.304585\pi\)
\(558\) 0 0
\(559\) 4629.97 0.350316
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 3529.68 6113.58i 0.264224 0.457650i −0.703136 0.711055i \(-0.748217\pi\)
0.967360 + 0.253406i \(0.0815508\pi\)
\(564\) 0 0
\(565\) −5821.43 10083.0i −0.433468 0.750788i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −676.403 1171.56i −0.0498353 0.0863172i 0.840032 0.542537i \(-0.182536\pi\)
−0.889867 + 0.456220i \(0.849203\pi\)
\(570\) 0 0
\(571\) 10419.6 18047.3i 0.763655 1.32269i −0.177300 0.984157i \(-0.556736\pi\)
0.940955 0.338532i \(-0.109930\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 6909.42 0.501117
\(576\) 0 0
\(577\) 5007.32 8672.92i 0.361278 0.625751i −0.626894 0.779105i \(-0.715674\pi\)
0.988171 + 0.153353i \(0.0490073\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 7623.34 + 13204.0i 0.541555 + 0.938001i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 20864.8 1.46709 0.733546 0.679640i \(-0.237864\pi\)
0.733546 + 0.679640i \(0.237864\pi\)
\(588\) 0 0
\(589\) −8322.01 −0.582177
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −8773.84 15196.7i −0.607586 1.05237i −0.991637 0.129058i \(-0.958805\pi\)
0.384051 0.923312i \(-0.374529\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −65.3409 + 113.174i −0.00445703 + 0.00771980i −0.868245 0.496135i \(-0.834752\pi\)
0.863788 + 0.503855i \(0.168085\pi\)
\(600\) 0 0
\(601\) 5964.47 0.404818 0.202409 0.979301i \(-0.435123\pi\)
0.202409 + 0.979301i \(0.435123\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −405.472 + 702.298i −0.0272476 + 0.0471942i
\(606\) 0 0
\(607\) −4955.82 8583.73i −0.331384 0.573975i 0.651399 0.758735i \(-0.274182\pi\)
−0.982784 + 0.184761i \(0.940849\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 6148.07 + 10648.8i 0.407077 + 0.705079i
\(612\) 0 0
\(613\) 3241.26 5614.03i 0.213562 0.369900i −0.739265 0.673415i \(-0.764827\pi\)
0.952827 + 0.303515i \(0.0981602\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −7189.36 −0.469097 −0.234548 0.972104i \(-0.575361\pi\)
−0.234548 + 0.972104i \(0.575361\pi\)
\(618\) 0 0
\(619\) 930.696 1612.01i 0.0604327 0.104672i −0.834226 0.551422i \(-0.814085\pi\)
0.894659 + 0.446750i \(0.147419\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2474.04 + 4285.16i 0.158338 + 0.274250i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3303.06 0.209383
\(630\) 0 0
\(631\) −29032.0 −1.83161 −0.915803 0.401627i \(-0.868445\pi\)
−0.915803 + 0.401627i \(0.868445\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3212.56 + 5564.32i 0.200766 + 0.347737i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 12228.9 21181.0i 0.753527 1.30515i −0.192576 0.981282i \(-0.561684\pi\)
0.946103 0.323866i \(-0.104983\pi\)
\(642\) 0 0
\(643\) −10968.2 −0.672695 −0.336348 0.941738i \(-0.609192\pi\)
−0.336348 + 0.941738i \(0.609192\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2742.37 + 4749.92i −0.166636 + 0.288622i −0.937235 0.348698i \(-0.886624\pi\)
0.770599 + 0.637320i \(0.219957\pi\)
\(648\) 0 0
\(649\) 11226.7 + 19445.2i 0.679025 + 1.17611i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −5164.98 8946.01i −0.309527 0.536117i 0.668732 0.743504i \(-0.266837\pi\)
−0.978259 + 0.207387i \(0.933504\pi\)
\(654\) 0 0
\(655\) 113.270 196.189i 0.00675698 0.0117034i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −26822.2 −1.58550 −0.792751 0.609546i \(-0.791352\pi\)
−0.792751 + 0.609546i \(0.791352\pi\)
\(660\) 0 0
\(661\) 1249.38 2163.99i 0.0735177 0.127336i −0.826923 0.562315i \(-0.809911\pi\)
0.900441 + 0.434979i \(0.143244\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4649.27 + 8052.77i 0.269896 + 0.467473i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 12600.1 0.724922
\(672\) 0 0
\(673\) 10092.1 0.578044 0.289022 0.957322i \(-0.406670\pi\)
0.289022 + 0.957322i \(0.406670\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13183.2 + 22833.9i 0.748406 + 1.29628i 0.948586 + 0.316518i \(0.102514\pi\)
−0.200181 + 0.979759i \(0.564153\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 11285.4 19546.9i 0.632245 1.09508i −0.354847 0.934924i \(-0.615467\pi\)
0.987092 0.160155i \(-0.0511996\pi\)
\(684\) 0 0
\(685\) 19304.1 1.07675
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 8049.28 13941.8i 0.445070 0.770884i
\(690\) 0 0
\(691\) −5965.75 10333.0i −0.328434 0.568864i 0.653768 0.756695i \(-0.273187\pi\)
−0.982201 + 0.187832i \(0.939854\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 10266.0 + 17781.2i 0.560304 + 0.970475i
\(696\) 0 0
\(697\) −356.428 + 617.352i −0.0193697 + 0.0335493i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −16592.6 −0.893997 −0.446999 0.894535i \(-0.647507\pi\)
−0.446999 + 0.894535i \(0.647507\pi\)
\(702\) 0 0
\(703\) −15033.7 + 26039.1i −0.806553 + 1.39699i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −9357.00 16206.8i −0.495641 0.858476i 0.504346 0.863502i \(-0.331734\pi\)
−0.999987 + 0.00502575i \(0.998400\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 10912.0 0.573152
\(714\) 0 0
\(715\) 12337.3 0.645298
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 12427.4 + 21524.9i 0.644594 + 1.11647i 0.984395 + 0.175972i \(0.0563070\pi\)
−0.339801 + 0.940497i \(0.610360\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2284.00 3956.01i 0.117001 0.202652i
\(726\) 0 0
\(727\) 22506.0 1.14814 0.574071 0.818805i \(-0.305363\pi\)
0.574071 + 0.818805i \(0.305363\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 576.020 997.696i 0.0291448 0.0504803i
\(732\) 0 0
\(733\) −11896.0 20604.5i −0.599440 1.03826i −0.992904 0.118920i \(-0.962057\pi\)
0.393464 0.919340i \(-0.371277\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −14059.4 24351.6i −0.702692 1.21710i
\(738\) 0 0
\(739\) −2401.62 + 4159.72i −0.119547 + 0.207061i −0.919588 0.392884i \(-0.871477\pi\)
0.800041 + 0.599945i \(0.204811\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −5076.10 −0.250638 −0.125319 0.992116i \(-0.539995\pi\)
−0.125319 + 0.992116i \(0.539995\pi\)
\(744\) 0 0
\(745\) 9255.24 16030.6i 0.455149 0.788341i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 4579.36 + 7931.68i 0.222508 + 0.385394i 0.955569 0.294768i \(-0.0952425\pi\)
−0.733061 + 0.680163i \(0.761909\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 2313.14 0.111502
\(756\) 0 0
\(757\) 33682.2 1.61717 0.808587 0.588377i \(-0.200233\pi\)
0.808587 + 0.588377i \(0.200233\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −6673.14 11558.2i −0.317873 0.550571i 0.662171 0.749352i \(-0.269635\pi\)
−0.980044 + 0.198781i \(0.936302\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 11854.0 20531.7i 0.558048 0.966568i
\(768\) 0 0
\(769\) 15530.6 0.728281 0.364140 0.931344i \(-0.381363\pi\)
0.364140 + 0.931344i \(0.381363\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 5507.62 9539.47i 0.256268 0.443869i −0.708971 0.705238i \(-0.750840\pi\)
0.965239 + 0.261368i \(0.0841737\pi\)
\(774\) 0 0
\(775\) −2680.32 4642.45i −0.124232 0.215176i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −3244.52 5619.67i −0.149226 0.258467i
\(780\) 0 0
\(781\) 13771.2 23852.4i 0.630951 1.09284i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1573.27 −0.0715317
\(786\) 0 0
\(787\) 10596.3 18353.3i 0.479946 0.831291i −0.519789 0.854294i \(-0.673990\pi\)
0.999735 + 0.0230036i \(0.00732292\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −6652.07 11521.7i −0.297884 0.515950i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 18189.9 0.808432 0.404216 0.914663i \(-0.367544\pi\)
0.404216 + 0.914663i \(0.367544\pi\)
\(798\) 0 0
\(799\) 3059.56 0.135468
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 15160.6 + 26259.0i 0.666260 + 1.15400i
\(804\) 0