Properties

Label 1764.3.z.k.325.4
Level $1764$
Weight $3$
Character 1764.325
Analytic conductor $48.066$
Analytic rank $0$
Dimension $8$
CM no
Inner twists $4$

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Newspace parameters

Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 1764.z (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.0655186332\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.1
Defining polynomial: \(x^{8} - 4 x^{6} + 14 x^{4} - 8 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 7^{4} \)
Twist minimal: no (minimal twist has level 196)
Sato-Tate group: $\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label 325.4
Root \(-0.662827 - 0.382683i\) of defining polynomial
Character \(\chi\) \(=\) 1764.325
Dual form 1764.3.z.k.901.4

$q$-expansion

\(f(q)\) \(=\) \(q+(7.33820 - 4.23671i) q^{5} +O(q^{10})\) \(q+(7.33820 - 4.23671i) q^{5} +(1.94975 - 3.37706i) q^{11} +19.1886i q^{13} +(-11.5897 - 6.69133i) q^{17} +(5.80462 - 3.35130i) q^{19} +(13.0000 + 22.5167i) q^{23} +(23.3995 - 40.5291i) q^{25} +11.7990 q^{29} +(31.6825 + 18.2919i) q^{31} +(16.0000 + 27.7128i) q^{37} +20.9594i q^{41} +79.2965 q^{43} +(12.3858 - 7.15093i) q^{47} +(-6.89949 + 11.9503i) q^{53} -33.0421i q^{55} +(7.31869 + 4.22545i) q^{59} +(27.4114 - 15.8260i) q^{61} +(81.2965 + 140.810i) q^{65} +(15.6985 - 27.1906i) q^{67} -95.5980 q^{71} +(-15.8607 - 9.15721i) q^{73} +(-39.8995 - 69.1080i) q^{79} -141.381i q^{83} -113.397 q^{85} +(100.852 - 58.2269i) q^{89} +(28.3970 - 49.1850i) q^{95} -137.346i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8q + O(q^{10}) \) \( 8q - 24q^{11} + 104q^{23} + 108q^{25} - 64q^{29} + 128q^{37} + 80q^{43} + 24q^{53} + 96q^{65} - 112q^{67} - 448q^{71} - 240q^{79} - 432q^{85} - 248q^{95} + O(q^{100}) \)

Character values

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

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{6}\right)\)

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
\(5\) 7.33820 4.23671i 1.46764 0.847343i 0.468297 0.883571i \(-0.344868\pi\)
0.999344 + 0.0362281i \(0.0115343\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.94975 3.37706i 0.177250 0.307006i −0.763688 0.645586i \(-0.776613\pi\)
0.940938 + 0.338580i \(0.109947\pi\)
\(12\) 0 0
\(13\) 19.1886i 1.47604i 0.674777 + 0.738022i \(0.264240\pi\)
−0.674777 + 0.738022i \(0.735760\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −11.5897 6.69133i −0.681748 0.393607i 0.118765 0.992922i \(-0.462106\pi\)
−0.800513 + 0.599315i \(0.795440\pi\)
\(18\) 0 0
\(19\) 5.80462 3.35130i 0.305506 0.176384i −0.339408 0.940639i \(-0.610227\pi\)
0.644914 + 0.764255i \(0.276893\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 13.0000 + 22.5167i 0.565217 + 0.978985i 0.997029 + 0.0770216i \(0.0245410\pi\)
−0.431812 + 0.901964i \(0.642126\pi\)
\(24\) 0 0
\(25\) 23.3995 40.5291i 0.935980 1.62116i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 11.7990 0.406862 0.203431 0.979089i \(-0.434791\pi\)
0.203431 + 0.979089i \(0.434791\pi\)
\(30\) 0 0
\(31\) 31.6825 + 18.2919i 1.02202 + 0.590061i 0.914687 0.404163i \(-0.132437\pi\)
0.107328 + 0.994224i \(0.465770\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 16.0000 + 27.7128i 0.432432 + 0.748995i 0.997082 0.0763357i \(-0.0243221\pi\)
−0.564650 + 0.825331i \(0.690989\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 20.9594i 0.511205i 0.966782 + 0.255602i \(0.0822738\pi\)
−0.966782 + 0.255602i \(0.917726\pi\)
\(42\) 0 0
\(43\) 79.2965 1.84410 0.922052 0.387066i \(-0.126512\pi\)
0.922052 + 0.387066i \(0.126512\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 12.3858 7.15093i 0.263527 0.152148i −0.362415 0.932017i \(-0.618048\pi\)
0.625943 + 0.779869i \(0.284714\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −6.89949 + 11.9503i −0.130179 + 0.225477i −0.923746 0.383007i \(-0.874889\pi\)
0.793566 + 0.608484i \(0.208222\pi\)
\(54\) 0 0
\(55\) 33.0421i 0.600765i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 7.31869 + 4.22545i 0.124046 + 0.0716178i 0.560739 0.827993i \(-0.310517\pi\)
−0.436693 + 0.899610i \(0.643850\pi\)
\(60\) 0 0
\(61\) 27.4114 15.8260i 0.449368 0.259443i −0.258195 0.966093i \(-0.583128\pi\)
0.707563 + 0.706650i \(0.249794\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 81.2965 + 140.810i 1.25071 + 2.16630i
\(66\) 0 0
\(67\) 15.6985 27.1906i 0.234306 0.405829i −0.724765 0.688996i \(-0.758052\pi\)
0.959071 + 0.283167i \(0.0913850\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −95.5980 −1.34645 −0.673225 0.739438i \(-0.735092\pi\)
−0.673225 + 0.739438i \(0.735092\pi\)
\(72\) 0 0
\(73\) −15.8607 9.15721i −0.217270 0.125441i 0.387415 0.921905i \(-0.373368\pi\)
−0.604686 + 0.796464i \(0.706701\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −39.8995 69.1080i −0.505057 0.874784i −0.999983 0.00584908i \(-0.998138\pi\)
0.494926 0.868935i \(-0.335195\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 141.381i 1.70338i −0.524044 0.851691i \(-0.675577\pi\)
0.524044 0.851691i \(-0.324423\pi\)
\(84\) 0 0
\(85\) −113.397 −1.33408
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 100.852 58.2269i 1.13317 0.654235i 0.188439 0.982085i \(-0.439657\pi\)
0.944730 + 0.327850i \(0.106324\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 28.3970 49.1850i 0.298915 0.517737i
\(96\) 0 0
\(97\) 137.346i 1.41593i −0.706245 0.707967i \(-0.749612\pi\)
0.706245 0.707967i \(-0.250388\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 78.3906 + 45.2588i 0.776145 + 0.448107i 0.835062 0.550156i \(-0.185432\pi\)
−0.0589176 + 0.998263i \(0.518765\pi\)
\(102\) 0 0
\(103\) −12.3468 + 7.12840i −0.119871 + 0.0692078i −0.558737 0.829345i \(-0.688714\pi\)
0.438865 + 0.898553i \(0.355380\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −24.7990 42.9531i −0.231766 0.401431i 0.726562 0.687101i \(-0.241117\pi\)
−0.958328 + 0.285670i \(0.907784\pi\)
\(108\) 0 0
\(109\) −73.5980 + 127.475i −0.675211 + 1.16950i 0.301196 + 0.953562i \(0.402614\pi\)
−0.976407 + 0.215937i \(0.930719\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −74.3015 −0.657536 −0.328768 0.944411i \(-0.606633\pi\)
−0.328768 + 0.944411i \(0.606633\pi\)
\(114\) 0 0
\(115\) 190.793 + 110.155i 1.65907 + 0.957866i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 52.8970 + 91.6202i 0.437165 + 0.757192i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 184.712i 1.47770i
\(126\) 0 0
\(127\) −76.9949 −0.606259 −0.303130 0.952949i \(-0.598032\pi\)
−0.303130 + 0.952949i \(0.598032\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 134.398 77.5946i 1.02594 0.592325i 0.110119 0.993918i \(-0.464877\pi\)
0.915818 + 0.401593i \(0.131543\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −65.0503 + 112.670i −0.474819 + 0.822411i −0.999584 0.0288360i \(-0.990820\pi\)
0.524765 + 0.851247i \(0.324153\pi\)
\(138\) 0 0
\(139\) 200.740i 1.44417i −0.691804 0.722086i \(-0.743184\pi\)
0.691804 0.722086i \(-0.256816\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 64.8010 + 37.4129i 0.453154 + 0.261628i
\(144\) 0 0
\(145\) 86.5834 49.9889i 0.597127 0.344751i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 84.6985 + 146.702i 0.568446 + 0.984578i 0.996720 + 0.0809286i \(0.0257886\pi\)
−0.428274 + 0.903649i \(0.640878\pi\)
\(150\) 0 0
\(151\) 1.10051 1.90613i 0.00728811 0.0126234i −0.862358 0.506298i \(-0.831013\pi\)
0.869646 + 0.493675i \(0.164347\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 309.990 1.99993
\(156\) 0 0
\(157\) −139.504 80.5426i −0.888560 0.513010i −0.0150888 0.999886i \(-0.504803\pi\)
−0.873471 + 0.486876i \(0.838136\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 111.146 + 192.510i 0.681876 + 1.18104i 0.974408 + 0.224787i \(0.0721688\pi\)
−0.292532 + 0.956256i \(0.594498\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 207.084i 1.24003i −0.784592 0.620013i \(-0.787127\pi\)
0.784592 0.620013i \(-0.212873\pi\)
\(168\) 0 0
\(169\) −199.201 −1.17870
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −240.608 + 138.915i −1.39080 + 0.802976i −0.993403 0.114673i \(-0.963418\pi\)
−0.397392 + 0.917649i \(0.630085\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 170.196 294.788i 0.950815 1.64686i 0.207149 0.978309i \(-0.433581\pi\)
0.743666 0.668551i \(-0.233085\pi\)
\(180\) 0 0
\(181\) 184.174i 1.01753i 0.860904 + 0.508767i \(0.169899\pi\)
−0.860904 + 0.508767i \(0.830101\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 234.823 + 135.575i 1.26931 + 0.732837i
\(186\) 0 0
\(187\) −45.1941 + 26.0928i −0.241679 + 0.139534i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 114.794 + 198.829i 0.601015 + 1.04099i 0.992668 + 0.120876i \(0.0385703\pi\)
−0.391652 + 0.920113i \(0.628096\pi\)
\(192\) 0 0
\(193\) 20.7487 35.9379i 0.107506 0.186207i −0.807253 0.590205i \(-0.799047\pi\)
0.914759 + 0.403999i \(0.132380\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −253.598 −1.28730 −0.643650 0.765320i \(-0.722581\pi\)
−0.643650 + 0.765320i \(0.722581\pi\)
\(198\) 0 0
\(199\) −97.2990 56.1756i −0.488940 0.282289i 0.235195 0.971948i \(-0.424427\pi\)
−0.724135 + 0.689659i \(0.757761\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 88.7990 + 153.804i 0.433166 + 0.750265i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 26.1367i 0.125056i
\(210\) 0 0
\(211\) 353.789 1.67672 0.838362 0.545113i \(-0.183513\pi\)
0.838362 + 0.545113i \(0.183513\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 581.894 335.956i 2.70648 1.56259i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 128.397 222.390i 0.580982 1.00629i
\(222\) 0 0
\(223\) 126.653i 0.567951i 0.958832 + 0.283976i \(0.0916534\pi\)
−0.958832 + 0.283976i \(0.908347\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 195.589 + 112.923i 0.861626 + 0.497460i 0.864557 0.502535i \(-0.167599\pi\)
−0.00293018 + 0.999996i \(0.500933\pi\)
\(228\) 0 0
\(229\) 223.680 129.141i 0.976767 0.563937i 0.0754744 0.997148i \(-0.475953\pi\)
0.901292 + 0.433211i \(0.142620\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 28.8492 + 49.9684i 0.123816 + 0.214456i 0.921270 0.388924i \(-0.127153\pi\)
−0.797453 + 0.603381i \(0.793820\pi\)
\(234\) 0 0
\(235\) 60.5929 104.950i 0.257842 0.446596i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −82.6030 −0.345619 −0.172810 0.984955i \(-0.555285\pi\)
−0.172810 + 0.984955i \(0.555285\pi\)
\(240\) 0 0
\(241\) −113.160 65.3328i −0.469543 0.271091i 0.246506 0.969141i \(-0.420718\pi\)
−0.716048 + 0.698051i \(0.754051\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 64.3066 + 111.382i 0.260350 + 0.450940i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 149.854i 0.597029i 0.954405 + 0.298514i \(0.0964911\pi\)
−0.954405 + 0.298514i \(0.903509\pi\)
\(252\) 0 0
\(253\) 101.387 0.400739
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −4.52272 + 2.61119i −0.0175981 + 0.0101603i −0.508773 0.860901i \(-0.669901\pi\)
0.491175 + 0.871061i \(0.336568\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −66.7035 + 115.534i −0.253626 + 0.439292i −0.964521 0.264005i \(-0.914956\pi\)
0.710896 + 0.703297i \(0.248290\pi\)
\(264\) 0 0
\(265\) 116.925i 0.441225i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −3.57252 2.06260i −0.0132808 0.00766765i 0.493345 0.869834i \(-0.335774\pi\)
−0.506626 + 0.862166i \(0.669107\pi\)
\(270\) 0 0
\(271\) 273.455 157.879i 1.00906 0.582580i 0.0981429 0.995172i \(-0.468710\pi\)
0.910916 + 0.412592i \(0.135376\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −91.2462 158.043i −0.331804 0.574702i
\(276\) 0 0
\(277\) 130.095 225.332i 0.469659 0.813473i −0.529740 0.848160i \(-0.677710\pi\)
0.999398 + 0.0346877i \(0.0110436\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 372.894 1.32703 0.663513 0.748165i \(-0.269065\pi\)
0.663513 + 0.748165i \(0.269065\pi\)
\(282\) 0 0
\(283\) −69.2476 39.9801i −0.244691 0.141273i 0.372640 0.927976i \(-0.378453\pi\)
−0.617331 + 0.786704i \(0.711786\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −54.9523 95.1801i −0.190146 0.329343i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 112.038i 0.382382i 0.981553 + 0.191191i \(0.0612351\pi\)
−0.981553 + 0.191191i \(0.938765\pi\)
\(294\) 0 0
\(295\) 71.6081 0.242739
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −432.062 + 249.451i −1.44502 + 0.834285i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 134.101 232.269i 0.439674 0.761537i
\(306\) 0 0
\(307\) 146.177i 0.476148i −0.971247 0.238074i \(-0.923484\pi\)
0.971247 0.238074i \(-0.0765160\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −366.290 211.477i −1.17778 0.679992i −0.222280 0.974983i \(-0.571350\pi\)
−0.955500 + 0.294991i \(0.904683\pi\)
\(312\) 0 0
\(313\) −75.9634 + 43.8575i −0.242695 + 0.140120i −0.616415 0.787422i \(-0.711415\pi\)
0.373720 + 0.927542i \(0.378082\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 215.693 + 373.592i 0.680421 + 1.17852i 0.974853 + 0.222851i \(0.0715364\pi\)
−0.294432 + 0.955673i \(0.595130\pi\)
\(318\) 0 0
\(319\) 23.0051 39.8459i 0.0721161 0.124909i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −89.6985 −0.277704
\(324\) 0 0
\(325\) 777.696 + 449.003i 2.39291 + 1.38155i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 170.744 + 295.737i 0.515842 + 0.893464i 0.999831 + 0.0183904i \(0.00585416\pi\)
−0.483989 + 0.875074i \(0.660813\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 266.040i 0.794149i
\(336\) 0 0
\(337\) −391.377 −1.16136 −0.580678 0.814134i \(-0.697212\pi\)
−0.580678 + 0.814134i \(0.697212\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 123.546 71.3291i 0.362304 0.209176i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −304.241 + 526.961i −0.876776 + 1.51862i −0.0219166 + 0.999760i \(0.506977\pi\)
−0.854859 + 0.518860i \(0.826357\pi\)
\(348\) 0 0
\(349\) 93.1627i 0.266942i 0.991053 + 0.133471i \(0.0426123\pi\)
−0.991053 + 0.133471i \(0.957388\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 80.0022 + 46.1893i 0.226635 + 0.130848i 0.609019 0.793156i \(-0.291563\pi\)
−0.382384 + 0.924004i \(0.624897\pi\)
\(354\) 0 0
\(355\) −701.518 + 405.021i −1.97611 + 1.14091i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −144.588 250.434i −0.402752 0.697586i 0.591305 0.806448i \(-0.298613\pi\)
−0.994057 + 0.108861i \(0.965280\pi\)
\(360\) 0 0
\(361\) −158.038 + 273.729i −0.437777 + 0.758253i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −155.186 −0.425167
\(366\) 0 0
\(367\) −458.968 264.986i −1.25060 0.722032i −0.279368 0.960184i \(-0.590125\pi\)
−0.971228 + 0.238153i \(0.923458\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −232.899 403.394i −0.624395 1.08148i −0.988657 0.150188i \(-0.952012\pi\)
0.364262 0.931297i \(-0.381321\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 226.406i 0.600546i
\(378\) 0 0
\(379\) 91.8793 0.242426 0.121213 0.992627i \(-0.461322\pi\)
0.121213 + 0.992627i \(0.461322\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 269.572 155.638i 0.703844 0.406364i −0.104934 0.994479i \(-0.533463\pi\)
0.808777 + 0.588115i \(0.200130\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −271.296 + 469.899i −0.697420 + 1.20797i 0.271938 + 0.962315i \(0.412336\pi\)
−0.969358 + 0.245652i \(0.920998\pi\)
\(390\) 0 0
\(391\) 347.949i 0.889895i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −585.581 338.086i −1.48248 0.855913i
\(396\) 0 0
\(397\) −141.834 + 81.8877i −0.357263 + 0.206266i −0.667880 0.744269i \(-0.732798\pi\)
0.310616 + 0.950535i \(0.399465\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −71.7035 124.194i −0.178812 0.309711i 0.762662 0.646797i \(-0.223892\pi\)
−0.941474 + 0.337086i \(0.890559\pi\)
\(402\) 0 0
\(403\) −350.995 + 607.941i −0.870955 + 1.50854i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 124.784 0.306594
\(408\) 0 0
\(409\) 251.150 + 145.001i 0.614058 + 0.354526i 0.774552 0.632510i \(-0.217975\pi\)
−0.160494 + 0.987037i \(0.551309\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −598.990 1037.48i −1.44335 2.49995i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 340.170i 0.811860i −0.913904 0.405930i \(-0.866948\pi\)
0.913904 0.405930i \(-0.133052\pi\)
\(420\) 0 0
\(421\) −6.18081 −0.0146813 −0.00734063 0.999973i \(-0.502337\pi\)
−0.00734063 + 0.999973i \(0.502337\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −542.387 + 313.147i −1.27621 + 0.736817i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −14.2965 + 24.7622i −0.0331705 + 0.0574529i −0.882134 0.470998i \(-0.843894\pi\)
0.848964 + 0.528451i \(0.177227\pi\)
\(432\) 0 0
\(433\) 243.736i 0.562900i −0.959576 0.281450i \(-0.909185\pi\)
0.959576 0.281450i \(-0.0908153\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 150.920 + 87.1337i 0.345355 + 0.199391i
\(438\) 0 0
\(439\) 428.605 247.455i 0.976321 0.563679i 0.0751637 0.997171i \(-0.476052\pi\)
0.901157 + 0.433492i \(0.142719\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 39.6985 + 68.7598i 0.0896128 + 0.155214i 0.907348 0.420381i \(-0.138104\pi\)
−0.817735 + 0.575595i \(0.804770\pi\)
\(444\) 0 0
\(445\) 493.382 854.562i 1.10872 1.92036i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −804.362 −1.79145 −0.895726 0.444607i \(-0.853343\pi\)
−0.895726 + 0.444607i \(0.853343\pi\)
\(450\) 0 0
\(451\) 70.7812 + 40.8655i 0.156943 + 0.0906109i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −268.739 465.469i −0.588050 1.01853i −0.994488 0.104853i \(-0.966563\pi\)
0.406438 0.913678i \(-0.366771\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 524.278i 1.13726i 0.822593 + 0.568631i \(0.192527\pi\)
−0.822593 + 0.568631i \(0.807473\pi\)
\(462\) 0 0
\(463\) −506.995 −1.09502 −0.547511 0.836799i \(-0.684424\pi\)
−0.547511 + 0.836799i \(0.684424\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −178.234 + 102.903i −0.381657 + 0.220350i −0.678539 0.734564i \(-0.737387\pi\)
0.296882 + 0.954914i \(0.404053\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 154.608 267.789i 0.326867 0.566150i
\(474\) 0 0
\(475\) 313.675i 0.660368i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −429.577 248.016i −0.896820 0.517779i −0.0206527 0.999787i \(-0.506574\pi\)
−0.876167 + 0.482008i \(0.839908\pi\)
\(480\) 0 0
\(481\) −531.769 + 307.017i −1.10555 + 0.638289i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −581.894 1007.87i −1.19978 2.07808i
\(486\) 0 0
\(487\) −236.688 + 409.956i −0.486013 + 0.841799i −0.999871 0.0160761i \(-0.994883\pi\)
0.513858 + 0.857875i \(0.328216\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 143.417 0.292092 0.146046 0.989278i \(-0.453345\pi\)
0.146046 + 0.989278i \(0.453345\pi\)
\(492\) 0 0
\(493\) −136.747 78.9509i −0.277377 0.160144i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 223.497 + 387.109i 0.447891 + 0.775770i 0.998249 0.0591594i \(-0.0188420\pi\)
−0.550358 + 0.834929i \(0.685509\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 660.346i 1.31282i 0.754406 + 0.656408i \(0.227925\pi\)
−0.754406 + 0.656408i \(0.772075\pi\)
\(504\) 0 0
\(505\) 766.995 1.51880
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −421.228 + 243.196i −0.827559 + 0.477792i −0.853016 0.521884i \(-0.825229\pi\)
0.0254569 + 0.999676i \(0.491896\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −60.4020 + 104.619i −0.117285 + 0.203144i
\(516\) 0 0
\(517\) 55.7701i 0.107872i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 89.0106 + 51.3903i 0.170846 + 0.0986378i 0.582985 0.812483i \(-0.301885\pi\)
−0.412139 + 0.911121i \(0.635218\pi\)
\(522\) 0 0
\(523\) 447.960 258.630i 0.856520 0.494512i −0.00632508 0.999980i \(-0.502013\pi\)
0.862846 + 0.505468i \(0.168680\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −244.794 423.996i −0.464505 0.804546i
\(528\) 0 0
\(529\) −73.5000 + 127.306i −0.138941 + 0.240654i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −402.181 −0.754561
\(534\) 0 0
\(535\) −363.960 210.132i −0.680299 0.392771i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 225.201 + 390.060i 0.416268 + 0.720997i 0.995561 0.0941223i \(-0.0300045\pi\)
−0.579293 + 0.815120i \(0.696671\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 1247.25i 2.28854i
\(546\) 0 0
\(547\) 195.095 0.356664 0.178332 0.983970i \(-0.442930\pi\)
0.178332 + 0.983970i \(0.442930\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 68.4886 39.5419i 0.124299 0.0717639i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −344.980 + 597.523i −0.619353 + 1.07275i 0.370251 + 0.928932i \(0.379272\pi\)
−0.989604 + 0.143820i \(0.954062\pi\)
\(558\) 0 0
\(559\) 1521.59i 2.72198i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −724.755 418.438i −1.28731 0.743229i −0.309136 0.951018i \(-0.600040\pi\)
−0.978174 + 0.207789i \(0.933373\pi\)
\(564\) 0 0
\(565\) −545.240 + 314.794i −0.965026 + 0.557158i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 3.90960 + 6.77162i 0.00687100 + 0.0119009i 0.869440 0.494038i \(-0.164480\pi\)
−0.862569 + 0.505939i \(0.831146\pi\)
\(570\) 0 0
\(571\) −0.854293 + 1.47968i −0.00149613 + 0.00259138i −0.866773 0.498704i \(-0.833810\pi\)
0.865276 + 0.501295i \(0.167143\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1216.77 2.11613
\(576\) 0 0
\(577\) 187.824 + 108.440i 0.325518 + 0.187938i 0.653849 0.756625i \(-0.273153\pi\)
−0.328332 + 0.944563i \(0.606486\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 26.9045 + 46.6000i 0.0461484 + 0.0799315i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 50.9983i 0.0868795i −0.999056 0.0434398i \(-0.986168\pi\)
0.999056 0.0434398i \(-0.0138317\pi\)
\(588\) 0 0
\(589\) 245.206 0.416309
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −563.586 + 325.387i −0.950398 + 0.548713i −0.893205 0.449650i \(-0.851549\pi\)
−0.0571937 + 0.998363i \(0.518215\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −16.8944 + 29.2620i −0.0282044 + 0.0488515i −0.879783 0.475375i \(-0.842312\pi\)
0.851579 + 0.524227i \(0.175646\pi\)
\(600\) 0 0
\(601\) 415.133i 0.690737i 0.938467 + 0.345368i \(0.112246\pi\)
−0.938467 + 0.345368i \(0.887754\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 776.338 + 448.219i 1.28320 + 0.740857i
\(606\) 0 0
\(607\) 377.083 217.709i 0.621225 0.358664i −0.156121 0.987738i \(-0.549899\pi\)
0.777346 + 0.629074i \(0.216566\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 137.216 + 237.665i 0.224576 + 0.388978i
\(612\) 0 0
\(613\) −212.382 + 367.856i −0.346463 + 0.600092i −0.985618 0.168986i \(-0.945951\pi\)
0.639155 + 0.769078i \(0.279284\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 726.563 1.17757 0.588787 0.808289i \(-0.299606\pi\)
0.588787 + 0.808289i \(0.299606\pi\)
\(618\) 0 0
\(619\) −613.943 354.460i −0.991830 0.572633i −0.0860094 0.996294i \(-0.527412\pi\)
−0.905821 + 0.423661i \(0.860745\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −197.585 342.228i −0.316137 0.547565i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 428.245i 0.680835i
\(630\) 0 0
\(631\) 207.176 0.328329 0.164165 0.986433i \(-0.447507\pi\)
0.164165 + 0.986433i \(0.447507\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −565.005 + 326.206i −0.889771 + 0.513710i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 456.291 790.320i 0.711843 1.23295i −0.252321 0.967643i \(-0.581194\pi\)
0.964165 0.265305i \(-0.0854726\pi\)
\(642\) 0 0
\(643\) 656.917i 1.02164i 0.859686 + 0.510822i \(0.170659\pi\)
−0.859686 + 0.510822i \(0.829341\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −33.0014 19.0534i −0.0510068 0.0294488i 0.474280 0.880374i \(-0.342709\pi\)
−0.525287 + 0.850925i \(0.676042\pi\)
\(648\) 0 0
\(649\) 28.5392 16.4771i 0.0439741 0.0253885i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −188.291 326.130i −0.288348 0.499434i 0.685067 0.728480i \(-0.259773\pi\)
−0.973416 + 0.229046i \(0.926439\pi\)
\(654\) 0 0
\(655\) 657.492 1138.81i 1.00381 1.73864i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 954.291 1.44809 0.724045 0.689753i \(-0.242281\pi\)
0.724045 + 0.689753i \(0.242281\pi\)
\(660\) 0 0
\(661\) −1010.11 583.187i −1.52815 0.882281i −0.999439 0.0334815i \(-0.989341\pi\)
−0.528716 0.848799i \(-0.677326\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 153.387 + 265.674i 0.229965 + 0.398312i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 123.427i 0.183945i
\(672\) 0 0
\(673\) −774.101 −1.15022 −0.575112 0.818075i \(-0.695041\pi\)
−0.575112 + 0.818075i \(0.695041\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 486.263 280.744i 0.718261 0.414688i −0.0958511 0.995396i \(-0.530557\pi\)
0.814112 + 0.580707i \(0.197224\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 176.080 304.980i 0.257804 0.446530i −0.707849 0.706364i \(-0.750334\pi\)
0.965653 + 0.259834i \(0.0836677\pi\)
\(684\) 0 0
\(685\) 1102.40i 1.60934i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −229.309 132.391i −0.332814 0.192150i
\(690\) 0 0
\(691\) −907.163 + 523.751i −1.31283 + 0.757961i −0.982563 0.185929i \(-0.940471\pi\)
−0.330263 + 0.943889i \(0.607137\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −850.477 1473.07i −1.22371 2.11952i
\(696\) 0 0
\(697\) 140.246 242.914i 0.201214 0.348513i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) −809.990 −1.15548 −0.577739 0.816222i \(-0.696065\pi\)
−0.577739 + 0.816222i \(0.696065\pi\)
\(702\) 0 0
\(703\) 185.748 + 107.241i 0.264221 + 0.152548i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 229.980 + 398.337i 0.324372 + 0.561829i 0.981385 0.192050i \(-0.0615137\pi\)
−0.657013 + 0.753879i \(0.728180\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 951.178i 1.33405i
\(714\) 0 0
\(715\) 634.030 0.886756
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −74.5098 + 43.0183i −0.103630 + 0.0598307i −0.550919 0.834559i \(-0.685723\pi\)
0.447289 + 0.894389i \(0.352389\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 276.090 478.203i 0.380814 0.659590i
\(726\) 0 0
\(727\) 78.4124i 0.107858i −0.998545 0.0539288i \(-0.982826\pi\)
0.998545 0.0539288i \(-0.0171744\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −919.024 530.599i −1.25721 0.725853i
\(732\) 0 0
\(733\) −449.960 + 259.785i −0.613861 + 0.354413i −0.774475 0.632604i \(-0.781986\pi\)
0.160614 + 0.987017i \(0.448653\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −61.2162 106.030i −0.0830613 0.143866i
\(738\) 0 0
\(739\) −581.739 + 1007.60i −0.787197 + 1.36347i 0.140481 + 0.990083i \(0.455135\pi\)
−0.927678 + 0.373382i \(0.878198\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 66.8040 0.0899112 0.0449556 0.998989i \(-0.485685\pi\)
0.0449556 + 0.998989i \(0.485685\pi\)
\(744\) 0 0
\(745\) 1243.07 + 717.687i 1.66855 + 0.963338i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.69343 2.93311i −0.00225491 0.00390561i 0.864896 0.501952i \(-0.167384\pi\)
−0.867151 + 0.498046i \(0.834051\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 18.6501i 0.0247021i
\(756\) 0 0
\(757\) −700.402 −0.925234 −0.462617 0.886558i \(-0.653089\pi\)
−0.462617 + 0.886558i \(0.653089\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 529.962 305.974i 0.696403 0.402068i −0.109604 0.993975i \(-0.534958\pi\)
0.806006 + 0.591907i \(0.201625\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −81.0803 + 140.435i −0.105711 + 0.183097i
\(768\) 0 0
\(769\) 638.310i 0.830052i −0.909810 0.415026i \(-0.863773\pi\)
0.909810 0.415026i \(-0.136227\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 411.367 + 237.503i 0.532169 + 0.307248i 0.741899 0.670511i \(-0.233925\pi\)
−0.209730 + 0.977759i \(0.567259\pi\)
\(774\) 0 0
\(775\) 1482.71 856.042i 1.91317 1.10457i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 70.2412 + 121.661i 0.0901684 + 0.156176i
\(780\) 0 0
\(781\) −186.392 + 322.840i −0.238658 + 0.413368i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −1364.94 −1.73878
\(786\) 0 0
\(787\) −1277.76 737.717i −1.62359 0.937378i −0.985950 0.167041i \(-0.946579\pi\)
−0.637637 0.770337i \(-0.720088\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 303.678 + 525.986i 0.382949 + 0.663287i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 330.869i 0.415143i 0.978220 + 0.207572i \(0.0665560\pi\)
−0.978220 + 0.207572i \(0.933444\pi\)
\(798\) 0 0
\(799\) −191.397 −0.239546
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −61.8489 + 35.7085i −0.0770223 + 0.0444688i
\(804\) 0 0