Properties

Label 75.2.e.b.68.1
Level $75$
Weight $2$
Character 75.68
Analytic conductor $0.599$
Analytic rank $0$
Dimension $4$
CM discriminant -3
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 75.e (of order \(4\), degree \(2\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.598878015160\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(i)\)
Coefficient field: \(\Q(i, \sqrt{6})\)
Defining polynomial: \( x^{4} + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{4}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{4}]$

Embedding invariants

Embedding label 68.1
Root \(-1.22474 - 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.68
Dual form 75.2.e.b.32.1

$q$-expansion

\(f(q)\) \(=\) \(q+(-1.22474 - 1.22474i) q^{3} -2.00000i q^{4} +(1.22474 - 1.22474i) q^{7} +3.00000i q^{9} +O(q^{10})\) \(q+(-1.22474 - 1.22474i) q^{3} -2.00000i q^{4} +(1.22474 - 1.22474i) q^{7} +3.00000i q^{9} +(-2.44949 + 2.44949i) q^{12} +(3.67423 + 3.67423i) q^{13} -4.00000 q^{16} +1.00000i q^{19} -3.00000 q^{21} +(3.67423 - 3.67423i) q^{27} +(-2.44949 - 2.44949i) q^{28} +7.00000 q^{31} +6.00000 q^{36} +(-4.89898 + 4.89898i) q^{37} -9.00000i q^{39} +(-8.57321 - 8.57321i) q^{43} +(4.89898 + 4.89898i) q^{48} +4.00000i q^{49} +(7.34847 - 7.34847i) q^{52} +(1.22474 - 1.22474i) q^{57} -13.0000 q^{61} +(3.67423 + 3.67423i) q^{63} +8.00000i q^{64} +(-11.0227 + 11.0227i) q^{67} +(9.79796 + 9.79796i) q^{73} +2.00000 q^{76} -4.00000i q^{79} -9.00000 q^{81} +6.00000i q^{84} +9.00000 q^{91} +(-8.57321 - 8.57321i) q^{93} +(13.4722 - 13.4722i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 16 q^{16} - 12 q^{21} + 28 q^{31} + 24 q^{36} - 52 q^{61} + 8 q^{76} - 36 q^{81} + 36 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) −1.22474 1.22474i −0.707107 0.707107i
\(4\) 2.00000i 1.00000i
\(5\) 0 0
\(6\) 0 0
\(7\) 1.22474 1.22474i 0.462910 0.462910i −0.436698 0.899608i \(-0.643852\pi\)
0.899608 + 0.436698i \(0.143852\pi\)
\(8\) 0 0
\(9\) 3.00000i 1.00000i
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −2.44949 + 2.44949i −0.707107 + 0.707107i
\(13\) 3.67423 + 3.67423i 1.01905 + 1.01905i 0.999815 + 0.0192343i \(0.00612285\pi\)
0.0192343 + 0.999815i \(0.493877\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(18\) 0 0
\(19\) 1.00000i 0.229416i 0.993399 + 0.114708i \(0.0365932\pi\)
−0.993399 + 0.114708i \(0.963407\pi\)
\(20\) 0 0
\(21\) −3.00000 −0.654654
\(22\) 0 0
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 3.67423 3.67423i 0.707107 0.707107i
\(28\) −2.44949 2.44949i −0.462910 0.462910i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 6.00000 1.00000
\(37\) −4.89898 + 4.89898i −0.805387 + 0.805387i −0.983932 0.178545i \(-0.942861\pi\)
0.178545 + 0.983932i \(0.442861\pi\)
\(38\) 0 0
\(39\) 9.00000i 1.44115i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −8.57321 8.57321i −1.30740 1.30740i −0.923283 0.384120i \(-0.874505\pi\)
−0.384120 0.923283i \(-0.625495\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 4.89898 + 4.89898i 0.707107 + 0.707107i
\(49\) 4.00000i 0.571429i
\(50\) 0 0
\(51\) 0 0
\(52\) 7.34847 7.34847i 1.01905 1.01905i
\(53\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.22474 1.22474i 0.162221 0.162221i
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 3.67423 + 3.67423i 0.462910 + 0.462910i
\(64\) 8.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) −11.0227 + 11.0227i −1.34664 + 1.34664i −0.457352 + 0.889286i \(0.651202\pi\)
−0.889286 + 0.457352i \(0.848798\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 9.79796 + 9.79796i 1.14676 + 1.14676i 0.987185 + 0.159579i \(0.0510137\pi\)
0.159579 + 0.987185i \(0.448986\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) 4.00000i 0.450035i −0.974355 0.225018i \(-0.927756\pi\)
0.974355 0.225018i \(-0.0722440\pi\)
\(80\) 0 0
\(81\) −9.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 6.00000i 0.654654i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 9.00000 0.943456
\(92\) 0 0
\(93\) −8.57321 8.57321i −0.889001 0.889001i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 13.4722 13.4722i 1.36789 1.36789i 0.504457 0.863437i \(-0.331693\pi\)
0.863437 0.504457i \(-0.168307\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −2.44949 2.44949i −0.241355 0.241355i 0.576055 0.817411i \(-0.304591\pi\)
−0.817411 + 0.576055i \(0.804591\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) −7.34847 7.34847i −0.707107 0.707107i
\(109\) 19.0000i 1.81987i −0.414751 0.909935i \(-0.636131\pi\)
0.414751 0.909935i \(-0.363869\pi\)
\(110\) 0 0
\(111\) 12.0000 1.13899
\(112\) −4.89898 + 4.89898i −0.462910 + 0.462910i
\(113\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −11.0227 + 11.0227i −1.01905 + 1.01905i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 11.0000 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 14.0000i 1.25724i
\(125\) 0 0
\(126\) 0 0
\(127\) 7.34847 7.34847i 0.652071 0.652071i −0.301420 0.953491i \(-0.597461\pi\)
0.953491 + 0.301420i \(0.0974607\pi\)
\(128\) 0 0
\(129\) 21.0000i 1.84895i
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 1.22474 + 1.22474i 0.106199 + 0.106199i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(138\) 0 0
\(139\) 16.0000i 1.35710i 0.734553 + 0.678551i \(0.237392\pi\)
−0.734553 + 0.678551i \(0.762608\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 12.0000i 1.00000i
\(145\) 0 0
\(146\) 0 0
\(147\) 4.89898 4.89898i 0.404061 0.404061i
\(148\) 9.79796 + 9.79796i 0.805387 + 0.805387i
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −23.0000 −1.87171 −0.935857 0.352381i \(-0.885372\pi\)
−0.935857 + 0.352381i \(0.885372\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −18.0000 −1.44115
\(157\) 1.22474 1.22474i 0.0977453 0.0977453i −0.656543 0.754288i \(-0.727982\pi\)
0.754288 + 0.656543i \(0.227982\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 3.67423 + 3.67423i 0.287788 + 0.287788i 0.836205 0.548417i \(-0.184769\pi\)
−0.548417 + 0.836205i \(0.684769\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 14.0000i 1.07692i
\(170\) 0 0
\(171\) −3.00000 −0.229416
\(172\) −17.1464 + 17.1464i −1.30740 + 1.30740i
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 7.00000 0.520306 0.260153 0.965567i \(-0.416227\pi\)
0.260153 + 0.965567i \(0.416227\pi\)
\(182\) 0 0
\(183\) 15.9217 + 15.9217i 1.17696 + 1.17696i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 9.00000i 0.654654i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 9.79796 9.79796i 0.707107 0.707107i
\(193\) −8.57321 8.57321i −0.617113 0.617113i 0.327677 0.944790i \(-0.393734\pi\)
−0.944790 + 0.327677i \(0.893734\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 8.00000 0.571429
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0 0
\(199\) 11.0000i 0.779769i 0.920864 + 0.389885i \(0.127485\pi\)
−0.920864 + 0.389885i \(0.872515\pi\)
\(200\) 0 0
\(201\) 27.0000 1.90443
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −14.6969 14.6969i −1.01905 1.01905i
\(209\) 0 0
\(210\) 0 0
\(211\) −13.0000 −0.894957 −0.447478 0.894295i \(-0.647678\pi\)
−0.447478 + 0.894295i \(0.647678\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 8.57321 8.57321i 0.581988 0.581988i
\(218\) 0 0
\(219\) 24.0000i 1.62177i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −20.8207 20.8207i −1.39425 1.39425i −0.815506 0.578749i \(-0.803541\pi\)
−0.578749 0.815506i \(-0.696459\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(228\) −2.44949 2.44949i −0.162221 0.162221i
\(229\) 29.0000i 1.91637i −0.286143 0.958187i \(-0.592373\pi\)
0.286143 0.958187i \(-0.407627\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4.89898 + 4.89898i −0.318223 + 0.318223i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 17.0000 1.09507 0.547533 0.836784i \(-0.315567\pi\)
0.547533 + 0.836784i \(0.315567\pi\)
\(242\) 0 0
\(243\) 11.0227 + 11.0227i 0.707107 + 0.707107i
\(244\) 26.0000i 1.66448i
\(245\) 0 0
\(246\) 0 0
\(247\) −3.67423 + 3.67423i −0.233786 + 0.233786i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 7.34847 7.34847i 0.462910 0.462910i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(258\) 0 0
\(259\) 12.0000i 0.745644i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 22.0454 + 22.0454i 1.34664 + 1.34664i
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −28.0000 −1.70088 −0.850439 0.526073i \(-0.823664\pi\)
−0.850439 + 0.526073i \(0.823664\pi\)
\(272\) 0 0
\(273\) −11.0227 11.0227i −0.667124 0.667124i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −23.2702 + 23.2702i −1.39817 + 1.39817i −0.592869 + 0.805299i \(0.702005\pi\)
−0.805299 + 0.592869i \(0.797995\pi\)
\(278\) 0 0
\(279\) 21.0000i 1.25724i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 15.9217 + 15.9217i 0.946446 + 0.946446i 0.998637 0.0521913i \(-0.0166205\pi\)
−0.0521913 + 0.998637i \(0.516621\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.0000i 1.00000i
\(290\) 0 0
\(291\) −33.0000 −1.93449
\(292\) 19.5959 19.5959i 1.14676 1.14676i
\(293\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −21.0000 −1.21042
\(302\) 0 0
\(303\) 0 0
\(304\) 4.00000i 0.229416i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.22474 1.22474i 0.0698999 0.0698999i −0.671293 0.741192i \(-0.734261\pi\)
0.741192 + 0.671293i \(0.234261\pi\)
\(308\) 0 0
\(309\) 6.00000i 0.341328i
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 3.67423 + 3.67423i 0.207680 + 0.207680i 0.803281 0.595601i \(-0.203086\pi\)
−0.595601 + 0.803281i \(0.703086\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 18.0000i 1.00000i
\(325\) 0 0
\(326\) 0 0
\(327\) −23.2702 + 23.2702i −1.28684 + 1.28684i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 32.0000 1.75888 0.879440 0.476011i \(-0.157918\pi\)
0.879440 + 0.476011i \(0.157918\pi\)
\(332\) 0 0
\(333\) −14.6969 14.6969i −0.805387 0.805387i
\(334\) 0 0
\(335\) 0 0
\(336\) 12.0000 0.654654
\(337\) 25.7196 25.7196i 1.40104 1.40104i 0.604223 0.796815i \(-0.293484\pi\)
0.796815 0.604223i \(-0.206516\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 13.4722 + 13.4722i 0.727430 + 0.727430i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(348\) 0 0
\(349\) 14.0000i 0.749403i −0.927146 0.374701i \(-0.877745\pi\)
0.927146 0.374701i \(-0.122255\pi\)
\(350\) 0 0
\(351\) 27.0000 1.44115
\(352\) 0 0
\(353\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 18.0000 0.947368
\(362\) 0 0
\(363\) −13.4722 13.4722i −0.707107 0.707107i
\(364\) 18.0000i 0.943456i
\(365\) 0 0
\(366\) 0 0
\(367\) −11.0227 + 11.0227i −0.575380 + 0.575380i −0.933627 0.358247i \(-0.883375\pi\)
0.358247 + 0.933627i \(0.383375\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −17.1464 + 17.1464i −0.889001 + 0.889001i
\(373\) −20.8207 20.8207i −1.07805 1.07805i −0.996684 0.0813690i \(-0.974071\pi\)
−0.0813690 0.996684i \(-0.525929\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 29.0000i 1.48963i −0.667271 0.744815i \(-0.732538\pi\)
0.667271 0.744815i \(-0.267462\pi\)
\(380\) 0 0
\(381\) −18.0000 −0.922168
\(382\) 0 0
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 25.7196 25.7196i 1.30740 1.30740i
\(388\) −26.9444 26.9444i −1.36789 1.36789i
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 13.4722 13.4722i 0.676150 0.676150i −0.282977 0.959127i \(-0.591322\pi\)
0.959127 + 0.282977i \(0.0913219\pi\)
\(398\) 0 0
\(399\) 3.00000i 0.150188i
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 25.7196 + 25.7196i 1.28119 + 1.28119i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 31.0000i 1.53285i 0.642333 + 0.766426i \(0.277967\pi\)
−0.642333 + 0.766426i \(0.722033\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −4.89898 + 4.89898i −0.241355 + 0.241355i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 19.5959 19.5959i 0.959616 0.959616i
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 22.0000 1.07221 0.536107 0.844150i \(-0.319894\pi\)
0.536107 + 0.844150i \(0.319894\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −15.9217 + 15.9217i −0.770504 + 0.770504i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −14.6969 + 14.6969i −0.707107 + 0.707107i
\(433\) 15.9217 + 15.9217i 0.765147 + 0.765147i 0.977248 0.212101i \(-0.0680304\pi\)
−0.212101 + 0.977248i \(0.568030\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −38.0000 −1.81987
\(437\) 0 0
\(438\) 0 0
\(439\) 41.0000i 1.95682i 0.206666 + 0.978412i \(0.433739\pi\)
−0.206666 + 0.978412i \(0.566261\pi\)
\(440\) 0 0
\(441\) −12.0000 −0.571429
\(442\) 0 0
\(443\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(444\) 24.0000i 1.13899i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 9.79796 + 9.79796i 0.462910 + 0.462910i
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 28.1691 + 28.1691i 1.32350 + 1.32350i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −29.3939 + 29.3939i −1.37499 + 1.37499i −0.522108 + 0.852879i \(0.674854\pi\)
−0.852879 + 0.522108i \(0.825146\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −26.9444 26.9444i −1.25221 1.25221i −0.954726 0.297486i \(-0.903852\pi\)
−0.297486 0.954726i \(-0.596148\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(468\) 22.0454 + 22.0454i 1.01905 + 1.01905i
\(469\) 27.0000i 1.24674i
\(470\) 0 0
\(471\) −3.00000 −0.138233
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −36.0000 −1.64146
\(482\) 0 0
\(483\) 0 0
\(484\) 22.0000i 1.00000i
\(485\) 0 0
\(486\) 0 0
\(487\) 25.7196 25.7196i 1.16547 1.16547i 0.182208 0.983260i \(-0.441675\pi\)
0.983260 0.182208i \(-0.0583245\pi\)
\(488\) 0 0
\(489\) 9.00000i 0.406994i
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −28.0000 −1.25724
\(497\) 0 0
\(498\) 0 0
\(499\) 11.0000i 0.492428i 0.969216 + 0.246214i \(0.0791865\pi\)
−0.969216 + 0.246214i \(0.920813\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 17.1464 17.1464i 0.761500 0.761500i
\(508\) −14.6969 14.6969i −0.652071 0.652071i
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 24.0000 1.06170
\(512\) 0 0
\(513\) 3.67423 + 3.67423i 0.162221 + 0.162221i
\(514\) 0 0
\(515\) 0 0
\(516\) 42.0000 1.84895
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −20.8207 20.8207i −0.910424 0.910424i 0.0858814 0.996305i \(-0.472629\pi\)
−0.996305 + 0.0858814i \(0.972629\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 23.0000i 1.00000i
\(530\) 0 0
\(531\) 0 0
\(532\) 2.44949 2.44949i 0.106199 0.106199i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 17.0000 0.730887 0.365444 0.930834i \(-0.380917\pi\)
0.365444 + 0.930834i \(0.380917\pi\)
\(542\) 0 0
\(543\) −8.57321 8.57321i −0.367912 0.367912i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −17.1464 + 17.1464i −0.733128 + 0.733128i −0.971238 0.238110i \(-0.923472\pi\)
0.238110 + 0.971238i \(0.423472\pi\)
\(548\) 0 0
\(549\) 39.0000i 1.66448i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −4.89898 4.89898i −0.208326 0.208326i
\(554\) 0 0
\(555\) 0 0
\(556\) 32.0000 1.35710
\(557\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 63.0000i 2.66462i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −11.0227 + 11.0227i −0.462910 + 0.462910i
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 47.0000 1.96689 0.983444 0.181210i \(-0.0580014\pi\)
0.983444 + 0.181210i \(0.0580014\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −24.0000 −1.00000
\(577\) −23.2702 + 23.2702i −0.968749 + 0.968749i −0.999526 0.0307771i \(-0.990202\pi\)
0.0307771 + 0.999526i \(0.490202\pi\)
\(578\) 0 0
\(579\) 21.0000i 0.872730i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(588\) −9.79796 9.79796i −0.404061 0.404061i
\(589\) 7.00000i 0.288430i
\(590\) 0 0
\(591\) 0 0
\(592\) 19.5959 19.5959i 0.805387 0.805387i
\(593\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 13.4722 13.4722i 0.551380 0.551380i
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −23.0000 −0.938190 −0.469095 0.883148i \(-0.655420\pi\)
−0.469095 + 0.883148i \(0.655420\pi\)
\(602\) 0 0
\(603\) −33.0681 33.0681i −1.34664 1.34664i
\(604\) 46.0000i 1.87171i
\(605\) 0 0
\(606\) 0 0
\(607\) 31.8434 31.8434i 1.29248 1.29248i 0.359235 0.933247i \(-0.383038\pi\)
0.933247 0.359235i \(-0.116962\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 34.2929 + 34.2929i 1.38508 + 1.38508i 0.835337 + 0.549739i \(0.185273\pi\)
0.549739 + 0.835337i \(0.314727\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) 0 0
\(619\) 49.0000i 1.96948i −0.174042 0.984738i \(-0.555683\pi\)
0.174042 0.984738i \(-0.444317\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 36.0000i 1.44115i
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) −2.44949 2.44949i −0.0977453 0.0977453i
\(629\) 0 0
\(630\) 0 0
\(631\) −43.0000 −1.71180 −0.855901 0.517139i \(-0.826997\pi\)
−0.855901 + 0.517139i \(0.826997\pi\)
\(632\) 0 0
\(633\) 15.9217 + 15.9217i 0.632830 + 0.632830i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −14.6969 + 14.6969i −0.582314 + 0.582314i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 22.0454 + 22.0454i 0.869386 + 0.869386i 0.992404 0.123018i \(-0.0392574\pi\)
−0.123018 + 0.992404i \(0.539257\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −21.0000 −0.823055
\(652\) 7.34847 7.34847i 0.287788 0.287788i
\(653\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −29.3939 + 29.3939i −1.14676 + 1.14676i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) −38.0000 −1.47803 −0.739014 0.673690i \(-0.764708\pi\)
−0.739014 + 0.673690i \(0.764708\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 51.0000i 1.97177i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 9.79796 + 9.79796i 0.377684 + 0.377684i 0.870266 0.492582i \(-0.163947\pi\)
−0.492582 + 0.870266i \(0.663947\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 28.0000 1.07692
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) 0 0
\(679\) 33.0000i 1.26642i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(684\) 6.00000i 0.229416i
\(685\) 0 0
\(686\) 0 0
\(687\) −35.5176 + 35.5176i −1.35508 + 1.35508i
\(688\) 34.2929 + 34.2929i 1.30740 + 1.30740i
\(689\) 0 0
\(690\) 0 0
\(691\) −8.00000 −0.304334 −0.152167 0.988355i \(-0.548625\pi\)
−0.152167 + 0.988355i \(0.548625\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −4.89898 4.89898i −0.184769 0.184769i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 31.0000i 1.16423i 0.813107 + 0.582115i \(0.197775\pi\)
−0.813107 + 0.582115i \(0.802225\pi\)
\(710\) 0 0
\(711\) 12.0000 0.450035
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −6.00000 −0.223452
\(722\) 0 0
\(723\) −20.8207 20.8207i −0.774329 0.774329i
\(724\) 14.0000i 0.520306i
\(725\) 0 0
\(726\) 0 0
\(727\) 37.9671 37.9671i 1.40812 1.40812i 0.638498 0.769624i \(-0.279556\pi\)
0.769624 0.638498i \(-0.220444\pi\)
\(728\) 0 0
\(729\) 27.0000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 31.8434 31.8434i 1.17696 1.17696i
\(733\) −14.6969 14.6969i −0.542844 0.542844i 0.381518 0.924362i \(-0.375402\pi\)
−0.924362 + 0.381518i \(0.875402\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 16.0000i 0.588570i 0.955718 + 0.294285i \(0.0950814\pi\)
−0.955718 + 0.294285i \(0.904919\pi\)
\(740\) 0 0
\(741\) 9.00000 0.330623
\(742\) 0 0
\(743\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 52.0000 1.89751 0.948753 0.316017i \(-0.102346\pi\)
0.948753 + 0.316017i \(0.102346\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −18.0000 −0.654654
\(757\) 1.22474 1.22474i 0.0445141 0.0445141i −0.684499 0.729013i \(-0.739979\pi\)
0.729013 + 0.684499i \(0.239979\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −23.2702 23.2702i −0.842436 0.842436i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −19.5959 19.5959i −0.707107 0.707107i
\(769\) 49.0000i 1.76699i −0.468445 0.883493i \(-0.655186\pi\)
0.468445 0.883493i \(-0.344814\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −17.1464 + 17.1464i −0.617113 + 0.617113i
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 14.6969 14.6969i 0.527250 0.527250i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 16.0000i 0.571429i
\(785\) 0 0
\(786\) 0 0
\(787\) −35.5176 + 35.5176i −1.26607 + 1.26607i −0.317962 + 0.948103i \(0.602999\pi\)
−0.948103 + 0.317962i \(0.897001\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −47.7650 47.7650i −1.69619 1.69619i
\(794\) 0 0
\(795\) 0 0
\(796\) 22.0000 0.779769
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 54.0000i 1.90443i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0