Properties

Label 3528.2.s.bl.3313.2
Level $3528$
Weight $2$
Character 3528.3313
Analytic conductor $28.171$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3528.s (of order \(3\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(28.1712218331\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{3})\)
Coefficient field: \(\Q(\sqrt{2}, \sqrt{-3})\)
Defining polynomial: \(x^{4} + 2 x^{2} + 4\)
Coefficient ring: \(\Z[a_1, \ldots, a_{25}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 1176)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 3313.2
Root \(0.707107 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 3528.3313
Dual form 3528.2.s.bl.361.2

$q$-expansion

\(f(q)\) \(=\) \(q+(1.70711 - 2.95680i) q^{5} +O(q^{10})\) \(q+(1.70711 - 2.95680i) q^{5} +(0.414214 + 0.717439i) q^{11} -4.24264 q^{13} +(3.70711 + 6.42090i) q^{17} +(-3.41421 + 5.91359i) q^{19} +(-2.41421 + 4.18154i) q^{23} +(-3.32843 - 5.76500i) q^{25} -2.82843 q^{29} +(1.41421 + 2.44949i) q^{31} +(0.828427 - 1.43488i) q^{37} -10.2426 q^{41} -11.3137 q^{43} +(-2.24264 + 3.88437i) q^{47} +(-1.00000 - 1.73205i) q^{53} +2.82843 q^{55} +(4.24264 + 7.34847i) q^{59} +(-5.53553 + 9.58783i) q^{61} +(-7.24264 + 12.5446i) q^{65} +(-5.65685 - 9.79796i) q^{67} +10.4853 q^{71} +(-3.87868 - 6.71807i) q^{73} +(-6.82843 + 11.8272i) q^{79} +4.00000 q^{83} +25.3137 q^{85} +(2.87868 - 4.98602i) q^{89} +(11.6569 + 20.1903i) q^{95} -0.242641 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{5} + O(q^{10}) \) \( 4 q + 4 q^{5} - 4 q^{11} + 12 q^{17} - 8 q^{19} - 4 q^{23} - 2 q^{25} - 8 q^{37} - 24 q^{41} + 8 q^{47} - 4 q^{53} - 8 q^{61} - 12 q^{65} + 8 q^{71} - 24 q^{73} - 16 q^{79} + 16 q^{83} + 56 q^{85} + 20 q^{89} + 24 q^{95} + 16 q^{97} + O(q^{100}) \)

Character values

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

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

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 1.70711 2.95680i 0.763441 1.32232i −0.177625 0.984098i \(-0.556842\pi\)
0.941067 0.338221i \(-0.109825\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0.414214 + 0.717439i 0.124890 + 0.216316i 0.921690 0.387927i \(-0.126809\pi\)
−0.796800 + 0.604243i \(0.793476\pi\)
\(12\) 0 0
\(13\) −4.24264 −1.17670 −0.588348 0.808608i \(-0.700222\pi\)
−0.588348 + 0.808608i \(0.700222\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 3.70711 + 6.42090i 0.899105 + 1.55730i 0.828640 + 0.559782i \(0.189115\pi\)
0.0704656 + 0.997514i \(0.477551\pi\)
\(18\) 0 0
\(19\) −3.41421 + 5.91359i −0.783274 + 1.35667i 0.146750 + 0.989174i \(0.453119\pi\)
−0.930025 + 0.367497i \(0.880215\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −2.41421 + 4.18154i −0.503398 + 0.871911i 0.496594 + 0.867983i \(0.334584\pi\)
−0.999992 + 0.00392850i \(0.998750\pi\)
\(24\) 0 0
\(25\) −3.32843 5.76500i −0.665685 1.15300i
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −2.82843 −0.525226 −0.262613 0.964901i \(-0.584584\pi\)
−0.262613 + 0.964901i \(0.584584\pi\)
\(30\) 0 0
\(31\) 1.41421 + 2.44949i 0.254000 + 0.439941i 0.964623 0.263631i \(-0.0849203\pi\)
−0.710623 + 0.703573i \(0.751587\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0.828427 1.43488i 0.136193 0.235892i −0.789860 0.613287i \(-0.789847\pi\)
0.926052 + 0.377395i \(0.123180\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −10.2426 −1.59963 −0.799816 0.600245i \(-0.795070\pi\)
−0.799816 + 0.600245i \(0.795070\pi\)
\(42\) 0 0
\(43\) −11.3137 −1.72532 −0.862662 0.505781i \(-0.831205\pi\)
−0.862662 + 0.505781i \(0.831205\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −2.24264 + 3.88437i −0.327123 + 0.566593i −0.981940 0.189194i \(-0.939412\pi\)
0.654817 + 0.755788i \(0.272746\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.00000 1.73205i −0.137361 0.237915i 0.789136 0.614218i \(-0.210529\pi\)
−0.926497 + 0.376303i \(0.877195\pi\)
\(54\) 0 0
\(55\) 2.82843 0.381385
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 4.24264 + 7.34847i 0.552345 + 0.956689i 0.998105 + 0.0615367i \(0.0196001\pi\)
−0.445760 + 0.895152i \(0.647067\pi\)
\(60\) 0 0
\(61\) −5.53553 + 9.58783i −0.708752 + 1.22760i 0.256568 + 0.966526i \(0.417408\pi\)
−0.965320 + 0.261069i \(0.915925\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) −7.24264 + 12.5446i −0.898339 + 1.55597i
\(66\) 0 0
\(67\) −5.65685 9.79796i −0.691095 1.19701i −0.971480 0.237124i \(-0.923795\pi\)
0.280385 0.959888i \(-0.409538\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 10.4853 1.24437 0.622187 0.782869i \(-0.286244\pi\)
0.622187 + 0.782869i \(0.286244\pi\)
\(72\) 0 0
\(73\) −3.87868 6.71807i −0.453965 0.786291i 0.544663 0.838655i \(-0.316658\pi\)
−0.998628 + 0.0523644i \(0.983324\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −6.82843 + 11.8272i −0.768258 + 1.33066i 0.170249 + 0.985401i \(0.445543\pi\)
−0.938507 + 0.345261i \(0.887790\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 4.00000 0.439057 0.219529 0.975606i \(-0.429548\pi\)
0.219529 + 0.975606i \(0.429548\pi\)
\(84\) 0 0
\(85\) 25.3137 2.74566
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 2.87868 4.98602i 0.305139 0.528517i −0.672153 0.740412i \(-0.734630\pi\)
0.977292 + 0.211895i \(0.0679636\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 11.6569 + 20.1903i 1.19597 + 2.07148i
\(96\) 0 0
\(97\) −0.242641 −0.0246364 −0.0123182 0.999924i \(-0.503921\pi\)
−0.0123182 + 0.999924i \(0.503921\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −5.36396 9.29065i −0.533734 0.924455i −0.999223 0.0394011i \(-0.987455\pi\)
0.465489 0.885053i \(-0.345878\pi\)
\(102\) 0 0
\(103\) −7.07107 + 12.2474i −0.696733 + 1.20678i 0.272860 + 0.962054i \(0.412030\pi\)
−0.969593 + 0.244723i \(0.921303\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.41421 11.1097i 0.620085 1.07402i −0.369384 0.929277i \(-0.620431\pi\)
0.989469 0.144743i \(-0.0462354\pi\)
\(108\) 0 0
\(109\) 1.65685 + 2.86976i 0.158698 + 0.274873i 0.934399 0.356227i \(-0.115937\pi\)
−0.775702 + 0.631100i \(0.782604\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 10.0000 0.940721 0.470360 0.882474i \(-0.344124\pi\)
0.470360 + 0.882474i \(0.344124\pi\)
\(114\) 0 0
\(115\) 8.24264 + 14.2767i 0.768630 + 1.33131i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.15685 8.93193i 0.468805 0.811994i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −5.65685 −0.505964
\(126\) 0 0
\(127\) 20.0000 1.77471 0.887357 0.461084i \(-0.152539\pi\)
0.887357 + 0.461084i \(0.152539\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −2.00000 + 3.46410i −0.174741 + 0.302660i −0.940072 0.340977i \(-0.889242\pi\)
0.765331 + 0.643637i \(0.222575\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.58579 7.94282i −0.391790 0.678600i 0.600896 0.799328i \(-0.294811\pi\)
−0.992686 + 0.120727i \(0.961477\pi\)
\(138\) 0 0
\(139\) −20.9706 −1.77870 −0.889350 0.457227i \(-0.848843\pi\)
−0.889350 + 0.457227i \(0.848843\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −1.75736 3.04384i −0.146958 0.254538i
\(144\) 0 0
\(145\) −4.82843 + 8.36308i −0.400979 + 0.694516i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0.656854 1.13770i 0.0538116 0.0932044i −0.837865 0.545878i \(-0.816196\pi\)
0.891676 + 0.452673i \(0.149530\pi\)
\(150\) 0 0
\(151\) 4.82843 + 8.36308i 0.392932 + 0.680578i 0.992835 0.119495i \(-0.0381275\pi\)
−0.599903 + 0.800073i \(0.704794\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 9.65685 0.775657
\(156\) 0 0
\(157\) −0.121320 0.210133i −0.00968242 0.0167704i 0.861144 0.508362i \(-0.169749\pi\)
−0.870826 + 0.491591i \(0.836415\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −2.82843 + 4.89898i −0.221540 + 0.383718i −0.955276 0.295717i \(-0.904442\pi\)
0.733736 + 0.679435i \(0.237775\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 14.8284 1.14746 0.573729 0.819045i \(-0.305496\pi\)
0.573729 + 0.819045i \(0.305496\pi\)
\(168\) 0 0
\(169\) 5.00000 0.384615
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −8.29289 + 14.3637i −0.630497 + 1.09205i 0.356953 + 0.934122i \(0.383816\pi\)
−0.987450 + 0.157931i \(0.949518\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −3.24264 5.61642i −0.242366 0.419791i 0.719022 0.694988i \(-0.244590\pi\)
−0.961388 + 0.275197i \(0.911257\pi\)
\(180\) 0 0
\(181\) −7.07107 −0.525588 −0.262794 0.964852i \(-0.584644\pi\)
−0.262794 + 0.964852i \(0.584644\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.82843 4.89898i −0.207950 0.360180i
\(186\) 0 0
\(187\) −3.07107 + 5.31925i −0.224579 + 0.388982i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 6.07107 10.5154i 0.439287 0.760867i −0.558348 0.829607i \(-0.688564\pi\)
0.997635 + 0.0687396i \(0.0218978\pi\)
\(192\) 0 0
\(193\) −1.00000 1.73205i −0.0719816 0.124676i 0.827788 0.561041i \(-0.189599\pi\)
−0.899770 + 0.436365i \(0.856266\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.68629 0.191390 0.0956952 0.995411i \(-0.469493\pi\)
0.0956952 + 0.995411i \(0.469493\pi\)
\(198\) 0 0
\(199\) 2.82843 + 4.89898i 0.200502 + 0.347279i 0.948690 0.316207i \(-0.102409\pi\)
−0.748188 + 0.663486i \(0.769076\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) −17.4853 + 30.2854i −1.22123 + 2.11522i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −5.65685 −0.391293
\(210\) 0 0
\(211\) 1.65685 0.114063 0.0570313 0.998372i \(-0.481837\pi\)
0.0570313 + 0.998372i \(0.481837\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −19.3137 + 33.4523i −1.31718 + 2.28143i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −15.7279 27.2416i −1.05797 1.83247i
\(222\) 0 0
\(223\) 13.6569 0.914531 0.457265 0.889330i \(-0.348829\pi\)
0.457265 + 0.889330i \(0.348829\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −2.58579 4.47871i −0.171625 0.297263i 0.767363 0.641213i \(-0.221568\pi\)
−0.938988 + 0.343950i \(0.888235\pi\)
\(228\) 0 0
\(229\) 12.7071 22.0094i 0.839709 1.45442i −0.0504286 0.998728i \(-0.516059\pi\)
0.890138 0.455691i \(-0.150608\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −7.41421 + 12.8418i −0.485721 + 0.841294i −0.999865 0.0164099i \(-0.994776\pi\)
0.514144 + 0.857704i \(0.328110\pi\)
\(234\) 0 0
\(235\) 7.65685 + 13.2621i 0.499478 + 0.865121i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 12.8284 0.829802 0.414901 0.909867i \(-0.363816\pi\)
0.414901 + 0.909867i \(0.363816\pi\)
\(240\) 0 0
\(241\) −6.94975 12.0373i −0.447673 0.775392i 0.550561 0.834795i \(-0.314414\pi\)
−0.998234 + 0.0594029i \(0.981080\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 14.4853 25.0892i 0.921676 1.59639i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 6.14214 0.387688 0.193844 0.981032i \(-0.437904\pi\)
0.193844 + 0.981032i \(0.437904\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −1.70711 + 2.95680i −0.106486 + 0.184440i −0.914345 0.404937i \(-0.867293\pi\)
0.807858 + 0.589377i \(0.200627\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −10.0711 17.4436i −0.621009 1.07562i −0.989298 0.145907i \(-0.953390\pi\)
0.368290 0.929711i \(-0.379943\pi\)
\(264\) 0 0
\(265\) −6.82843 −0.419467
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.0502525 0.0870399i −0.00306395 0.00530692i 0.864489 0.502651i \(-0.167642\pi\)
−0.867553 + 0.497344i \(0.834309\pi\)
\(270\) 0 0
\(271\) −3.07107 + 5.31925i −0.186554 + 0.323121i −0.944099 0.329662i \(-0.893065\pi\)
0.757545 + 0.652783i \(0.226399\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 2.75736 4.77589i 0.166275 0.287997i
\(276\) 0 0
\(277\) 14.3137 + 24.7921i 0.860027 + 1.48961i 0.871901 + 0.489682i \(0.162887\pi\)
−0.0118739 + 0.999930i \(0.503780\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −1.17157 −0.0698902 −0.0349451 0.999389i \(-0.511126\pi\)
−0.0349451 + 0.999389i \(0.511126\pi\)
\(282\) 0 0
\(283\) 13.0711 + 22.6398i 0.776994 + 1.34579i 0.933667 + 0.358143i \(0.116590\pi\)
−0.156672 + 0.987651i \(0.550077\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −18.9853 + 32.8835i −1.11678 + 1.93432i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.75736 0.102666 0.0513330 0.998682i \(-0.483653\pi\)
0.0513330 + 0.998682i \(0.483653\pi\)
\(294\) 0 0
\(295\) 28.9706 1.68673
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 10.2426 17.7408i 0.592347 1.02598i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 18.8995 + 32.7349i 1.08218 + 1.87439i
\(306\) 0 0
\(307\) −11.5147 −0.657180 −0.328590 0.944473i \(-0.606573\pi\)
−0.328590 + 0.944473i \(0.606573\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 11.8995 + 20.6105i 0.674758 + 1.16872i 0.976539 + 0.215339i \(0.0690855\pi\)
−0.301781 + 0.953377i \(0.597581\pi\)
\(312\) 0 0
\(313\) −14.3640 + 24.8791i −0.811899 + 1.40625i 0.0996342 + 0.995024i \(0.468233\pi\)
−0.911533 + 0.411226i \(0.865101\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −11.0000 + 19.0526i −0.617822 + 1.07010i 0.372061 + 0.928208i \(0.378651\pi\)
−0.989882 + 0.141890i \(0.954682\pi\)
\(318\) 0 0
\(319\) −1.17157 2.02922i −0.0655955 0.113615i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −50.6274 −2.81698
\(324\) 0 0
\(325\) 14.1213 + 24.4588i 0.783310 + 1.35673i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −3.65685 + 6.33386i −0.200999 + 0.348140i −0.948851 0.315726i \(-0.897752\pi\)
0.747852 + 0.663866i \(0.231085\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −38.6274 −2.11044
\(336\) 0 0
\(337\) 10.3431 0.563427 0.281714 0.959499i \(-0.409097\pi\)
0.281714 + 0.959499i \(0.409097\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −1.17157 + 2.02922i −0.0634442 + 0.109889i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.24264 2.15232i −0.0667084 0.115542i 0.830742 0.556657i \(-0.187916\pi\)
−0.897451 + 0.441115i \(0.854583\pi\)
\(348\) 0 0
\(349\) 10.5858 0.566644 0.283322 0.959025i \(-0.408563\pi\)
0.283322 + 0.959025i \(0.408563\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −13.8492 23.9876i −0.737121 1.27673i −0.953787 0.300485i \(-0.902852\pi\)
0.216666 0.976246i \(-0.430482\pi\)
\(354\) 0 0
\(355\) 17.8995 31.0028i 0.950007 1.64546i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 14.4142 24.9662i 0.760753 1.31766i −0.181710 0.983352i \(-0.558163\pi\)
0.942463 0.334311i \(-0.108504\pi\)
\(360\) 0 0
\(361\) −13.8137 23.9260i −0.727037 1.25927i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −26.4853 −1.38630
\(366\) 0 0
\(367\) −2.34315 4.05845i −0.122311 0.211849i 0.798368 0.602171i \(-0.205697\pi\)
−0.920679 + 0.390321i \(0.872364\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.65685 + 4.60181i −0.137567 + 0.238273i −0.926575 0.376110i \(-0.877261\pi\)
0.789008 + 0.614383i \(0.210595\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 12.0000 0.618031
\(378\) 0 0
\(379\) −23.3137 −1.19754 −0.598772 0.800919i \(-0.704345\pi\)
−0.598772 + 0.800919i \(0.704345\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −4.48528 + 7.76874i −0.229187 + 0.396964i −0.957567 0.288209i \(-0.906940\pi\)
0.728380 + 0.685173i \(0.240273\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −17.8995 31.0028i −0.907540 1.57191i −0.817470 0.575971i \(-0.804624\pi\)
−0.0900701 0.995935i \(-0.528709\pi\)
\(390\) 0 0
\(391\) −35.7990 −1.81043
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 23.3137 + 40.3805i 1.17304 + 2.03176i
\(396\) 0 0
\(397\) 8.12132 14.0665i 0.407597 0.705979i −0.587023 0.809571i \(-0.699700\pi\)
0.994620 + 0.103591i \(0.0330334\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0.585786 1.01461i 0.0292528 0.0506673i −0.851028 0.525120i \(-0.824021\pi\)
0.880281 + 0.474452i \(0.157354\pi\)
\(402\) 0 0
\(403\) −6.00000 10.3923i −0.298881 0.517678i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.37258 0.0680364
\(408\) 0 0
\(409\) 4.12132 + 7.13834i 0.203786 + 0.352968i 0.949745 0.313024i \(-0.101342\pi\)
−0.745959 + 0.665992i \(0.768009\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 6.82843 11.8272i 0.335194 0.580574i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −7.51472 −0.367118 −0.183559 0.983009i \(-0.558762\pi\)
−0.183559 + 0.983009i \(0.558762\pi\)
\(420\) 0 0
\(421\) −25.3137 −1.23371 −0.616857 0.787075i \(-0.711594\pi\)
−0.616857 + 0.787075i \(0.711594\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 24.6777 42.7430i 1.19704 2.07334i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −4.07107 7.05130i −0.196096 0.339649i 0.751163 0.660117i \(-0.229493\pi\)
−0.947259 + 0.320468i \(0.896160\pi\)
\(432\) 0 0
\(433\) 0.928932 0.0446416 0.0223208 0.999751i \(-0.492894\pi\)
0.0223208 + 0.999751i \(0.492894\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −16.4853 28.5533i −0.788598 1.36589i
\(438\) 0 0
\(439\) −6.34315 + 10.9867i −0.302742 + 0.524364i −0.976756 0.214354i \(-0.931235\pi\)
0.674014 + 0.738718i \(0.264569\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.58579 + 9.67487i −0.265389 + 0.459667i −0.967665 0.252237i \(-0.918834\pi\)
0.702277 + 0.711904i \(0.252167\pi\)
\(444\) 0 0
\(445\) −9.82843 17.0233i −0.465912 0.806983i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 16.6274 0.784696 0.392348 0.919817i \(-0.371663\pi\)
0.392348 + 0.919817i \(0.371663\pi\)
\(450\) 0 0
\(451\) −4.24264 7.34847i −0.199778 0.346026i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 12.3137 21.3280i 0.576011 0.997680i −0.419920 0.907561i \(-0.637942\pi\)
0.995931 0.0901192i \(-0.0287248\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −16.3848 −0.763115 −0.381558 0.924345i \(-0.624612\pi\)
−0.381558 + 0.924345i \(0.624612\pi\)
\(462\) 0 0
\(463\) 1.65685 0.0770005 0.0385003 0.999259i \(-0.487742\pi\)
0.0385003 + 0.999259i \(0.487742\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −9.41421 + 16.3059i −0.435638 + 0.754547i −0.997347 0.0727876i \(-0.976810\pi\)
0.561710 + 0.827334i \(0.310144\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −4.68629 8.11689i −0.215476 0.373215i
\(474\) 0 0
\(475\) 45.4558 2.08566
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 11.4142 + 19.7700i 0.521529 + 0.903314i 0.999686 + 0.0250403i \(0.00797140\pi\)
−0.478158 + 0.878274i \(0.658695\pi\)
\(480\) 0 0
\(481\) −3.51472 + 6.08767i −0.160257 + 0.277574i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.414214 + 0.717439i −0.0188085 + 0.0325772i
\(486\) 0 0
\(487\) −2.48528 4.30463i −0.112619 0.195062i 0.804207 0.594350i \(-0.202591\pi\)
−0.916825 + 0.399288i \(0.869257\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −19.1716 −0.865201 −0.432600 0.901586i \(-0.642404\pi\)
−0.432600 + 0.901586i \(0.642404\pi\)
\(492\) 0 0
\(493\) −10.4853 18.1610i −0.472233 0.817932i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 12.4853 21.6251i 0.558918 0.968074i −0.438669 0.898649i \(-0.644550\pi\)
0.997587 0.0694257i \(-0.0221167\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 19.3137 0.861156 0.430578 0.902553i \(-0.358310\pi\)
0.430578 + 0.902553i \(0.358310\pi\)
\(504\) 0 0
\(505\) −36.6274 −1.62990
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −18.1924 + 31.5101i −0.806363 + 1.39666i 0.109003 + 0.994041i \(0.465234\pi\)
−0.915367 + 0.402621i \(0.868099\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 24.1421 + 41.8154i 1.06383 + 1.84261i
\(516\) 0 0
\(517\) −3.71573 −0.163418
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 7.12132 + 12.3345i 0.311991 + 0.540384i 0.978793 0.204851i \(-0.0656709\pi\)
−0.666803 + 0.745234i \(0.732338\pi\)
\(522\) 0 0
\(523\) 2.48528 4.30463i 0.108674 0.188228i −0.806559 0.591153i \(-0.798673\pi\)
0.915233 + 0.402924i \(0.132006\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −10.4853 + 18.1610i −0.456746 + 0.791107i
\(528\) 0 0
\(529\) −0.156854 0.271680i −0.00681975 0.0118122i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 43.4558 1.88228
\(534\) 0 0
\(535\) −21.8995 37.9310i −0.946798 1.63990i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 9.00000 15.5885i 0.386940 0.670200i −0.605096 0.796152i \(-0.706865\pi\)
0.992036 + 0.125952i \(0.0401986\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 11.3137 0.484626
\(546\) 0 0
\(547\) −3.02944 −0.129529 −0.0647647 0.997901i \(-0.520630\pi\)
−0.0647647 + 0.997901i \(0.520630\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 9.65685 16.7262i 0.411396 0.712558i
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 16.3137 + 28.2562i 0.691234 + 1.19725i 0.971434 + 0.237311i \(0.0762660\pi\)
−0.280200 + 0.959942i \(0.590401\pi\)
\(558\) 0 0
\(559\) 48.0000 2.03018
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0.928932 + 1.60896i 0.0391498 + 0.0678095i 0.884936 0.465712i \(-0.154202\pi\)
−0.845787 + 0.533521i \(0.820868\pi\)
\(564\) 0 0
\(565\) 17.0711 29.5680i 0.718185 1.24393i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −2.92893 + 5.07306i −0.122787 + 0.212674i −0.920866 0.389880i \(-0.872517\pi\)
0.798079 + 0.602553i \(0.205850\pi\)
\(570\) 0 0
\(571\) 20.8284 + 36.0759i 0.871643 + 1.50973i 0.860297 + 0.509794i \(0.170278\pi\)
0.0113458 + 0.999936i \(0.496388\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 32.1421 1.34042
\(576\) 0 0
\(577\) −6.60660 11.4430i −0.275036 0.476377i 0.695108 0.718905i \(-0.255357\pi\)
−0.970144 + 0.242528i \(0.922023\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0.828427 1.43488i 0.0343099 0.0594266i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 35.7990 1.47758 0.738791 0.673934i \(-0.235397\pi\)
0.738791 + 0.673934i \(0.235397\pi\)
\(588\) 0 0
\(589\) −19.3137 −0.795807
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 23.7071 41.0619i 0.973534 1.68621i 0.288844 0.957376i \(-0.406729\pi\)
0.684690 0.728835i \(-0.259938\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 11.7279 + 20.3134i 0.479190 + 0.829981i 0.999715 0.0238650i \(-0.00759718\pi\)
−0.520525 + 0.853846i \(0.674264\pi\)
\(600\) 0 0
\(601\) −12.2426 −0.499388 −0.249694 0.968325i \(-0.580330\pi\)
−0.249694 + 0.968325i \(0.580330\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −17.6066 30.4955i −0.715810 1.23982i
\(606\) 0 0
\(607\) −12.4853 + 21.6251i −0.506762 + 0.877737i 0.493207 + 0.869912i \(0.335824\pi\)
−0.999969 + 0.00782569i \(0.997509\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 9.51472 16.4800i 0.384924 0.666708i
\(612\) 0 0
\(613\) 5.17157 + 8.95743i 0.208878 + 0.361787i 0.951361 0.308077i \(-0.0996856\pi\)
−0.742483 + 0.669864i \(0.766352\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.48528 0.180571 0.0902853 0.995916i \(-0.471222\pi\)
0.0902853 + 0.995916i \(0.471222\pi\)
\(618\) 0 0
\(619\) −16.8284 29.1477i −0.676392 1.17154i −0.976060 0.217501i \(-0.930209\pi\)
0.299668 0.954043i \(-0.403124\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 6.98528 12.0989i 0.279411 0.483954i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 12.2843 0.489806
\(630\) 0 0
\(631\) −3.02944 −0.120600 −0.0603000 0.998180i \(-0.519206\pi\)
−0.0603000 + 0.998180i \(0.519206\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 34.1421 59.1359i 1.35489 2.34674i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 17.5563 + 30.4085i 0.693434 + 1.20106i 0.970706 + 0.240272i \(0.0772365\pi\)
−0.277272 + 0.960792i \(0.589430\pi\)
\(642\) 0 0
\(643\) 31.7990 1.25403 0.627015 0.779007i \(-0.284277\pi\)
0.627015 + 0.779007i \(0.284277\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −0.100505 0.174080i −0.00395126 0.00684379i 0.864043 0.503418i \(-0.167924\pi\)
−0.867994 + 0.496574i \(0.834591\pi\)
\(648\) 0 0
\(649\) −3.51472 + 6.08767i −0.137965 + 0.238962i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 23.0711 39.9603i 0.902841 1.56377i 0.0790508 0.996871i \(-0.474811\pi\)
0.823790 0.566895i \(-0.191856\pi\)
\(654\) 0 0
\(655\) 6.82843 + 11.8272i 0.266809 + 0.462126i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −21.5147 −0.838094 −0.419047 0.907964i \(-0.637636\pi\)
−0.419047 + 0.907964i \(0.637636\pi\)
\(660\) 0 0
\(661\) −2.46447 4.26858i −0.0958566 0.166029i 0.814109 0.580712i \(-0.197226\pi\)
−0.909966 + 0.414683i \(0.863892\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 6.82843 11.8272i 0.264398 0.457950i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) −9.17157 −0.354065
\(672\) 0 0
\(673\) −23.3137 −0.898677 −0.449339 0.893361i \(-0.648340\pi\)
−0.449339 + 0.893361i \(0.648340\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 13.8492 23.9876i 0.532270 0.921918i −0.467021 0.884246i \(-0.654673\pi\)
0.999290 0.0376716i \(-0.0119941\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −0.899495 1.55797i −0.0344182 0.0596141i 0.848303 0.529511i \(-0.177624\pi\)
−0.882721 + 0.469897i \(0.844291\pi\)
\(684\) 0 0
\(685\) −31.3137 −1.19644
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 4.24264 + 7.34847i 0.161632 + 0.279954i
\(690\) 0 0
\(691\) −4.34315 + 7.52255i −0.165221 + 0.286171i −0.936734 0.350043i \(-0.886167\pi\)
0.771513 + 0.636214i \(0.219500\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −35.7990 + 62.0057i −1.35793 + 2.35201i
\(696\) 0 0
\(697\) −37.9706 65.7669i −1.43824 2.49110i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 6.14214 0.231985 0.115993 0.993250i \(-0.462995\pi\)
0.115993 + 0.993250i \(0.462995\pi\)
\(702\) 0 0
\(703\) 5.65685 + 9.79796i 0.213352 + 0.369537i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −3.31371 + 5.73951i −0.124449 + 0.215552i −0.921517 0.388337i \(-0.873050\pi\)
0.797068 + 0.603889i \(0.206383\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −13.6569 −0.511453
\(714\) 0 0
\(715\) −12.0000 −0.448775
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) −3.31371 + 5.73951i −0.123580 + 0.214048i −0.921177 0.389143i \(-0.872771\pi\)
0.797597 + 0.603191i \(0.206104\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 9.41421 + 16.3059i 0.349635 + 0.605586i
\(726\) 0 0
\(727\) −9.85786 −0.365608 −0.182804 0.983149i \(-0.558517\pi\)
−0.182804 + 0.983149i \(0.558517\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) −41.9411 72.6442i −1.55125 2.68684i
\(732\) 0 0
\(733\) 4.02082 6.96426i 0.148512 0.257231i −0.782166 0.623071i \(-0.785885\pi\)
0.930678 + 0.365840i \(0.119218\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.68629 8.11689i 0.172622 0.298990i
\(738\) 0 0
\(739\) −16.4853 28.5533i −0.606421 1.05035i −0.991825 0.127604i \(-0.959271\pi\)
0.385404 0.922748i \(-0.374062\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 4.82843 0.177138 0.0885689 0.996070i \(-0.471771\pi\)
0.0885689 + 0.996070i \(0.471771\pi\)
\(744\) 0 0
\(745\) −2.24264 3.88437i −0.0821640 0.142312i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −10.1421 + 17.5667i −0.370092 + 0.641018i −0.989579 0.143989i \(-0.954007\pi\)
0.619488 + 0.785006i \(0.287340\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 32.9706 1.19992
\(756\) 0 0
\(757\) 41.9411 1.52438 0.762188 0.647356i \(-0.224125\pi\)
0.762188 + 0.647356i \(0.224125\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −15.9497 + 27.6258i −0.578178 + 1.00143i 0.417510 + 0.908672i \(0.362903\pi\)
−0.995688 + 0.0927614i \(0.970431\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −18.0000 31.1769i −0.649942 1.12573i
\(768\) 0 0
\(769\) 34.8701 1.25745 0.628723 0.777629i \(-0.283578\pi\)
0.628723 + 0.777629i \(0.283578\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 16.6777 + 28.8866i 0.599854 + 1.03898i 0.992842 + 0.119434i \(0.0381080\pi\)
−0.392988 + 0.919544i \(0.628559\pi\)
\(774\) 0 0
\(775\) 9.41421 16.3059i 0.338169 0.585725i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 34.9706 60.5708i 1.25295 2.17017i
\(780\) 0 0
\(781\) 4.34315 + 7.52255i 0.155410 + 0.269178i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.828427 −0.0295678
\(786\) 0 0
\(787\) −7.65685 13.2621i −0.272937 0.472741i 0.696675 0.717387i \(-0.254662\pi\)
−0.969613 + 0.244645i \(0.921329\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 23.4853 40.6777i 0.833987 1.44451i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 39.4142 1.39612 0.698062 0.716038i \(-0.254046\pi\)
0.698062 + 0.716038i \(0.254046\pi\)
\(798\) 0 0
\(799\) −33.2548 −1.17647
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 3.21320 5.56543i 0.113391 0.196400i
\(804\) 0 0
\(805\) 0 0