Properties

Label 75.2.e.b.68.2
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 68.2
Root \(1.22474 + 1.22474i\) of defining polynomial
Character \(\chi\) \(=\) 75.68
Dual form 75.2.e.b.32.2

$q$-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 −0.899608 0.436698i \(-0.856148\pi\)
0.436698 + 0.899608i \(0.356148\pi\)
\(8\) 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 \(-0.993877\pi\)
−0.0192343 0.999815i \(-0.506123\pi\)
\(14\) 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 −0.178545 0.983932i \(-0.557139\pi\)
0.983932 + 0.178545i \(0.0571389\pi\)
\(38\) 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 \(0.125495\pi\)
0.384120 + 0.923283i \(0.374505\pi\)
\(44\) 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 \(-0.348798\pi\)
0.889286 0.457352i \(-0.151202\pi\)
\(68\) 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 \(-0.948986\pi\)
−0.159579 0.987185i \(-0.551014\pi\)
\(74\) 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 \(0.668307\pi\)
−0.863437 + 0.504457i \(0.831693\pi\)
\(98\) 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 0.817411 0.576055i \(-0.195409\pi\)
−0.576055 + 0.817411i \(0.695409\pi\)
\(104\) 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 −0.953491 0.301420i \(-0.902539\pi\)
0.301420 + 0.953491i \(0.402539\pi\)
\(128\) 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 −0.754288 0.656543i \(-0.772018\pi\)
0.656543 + 0.754288i \(0.272018\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −3.67423 3.67423i −0.287788 0.287788i 0.548417 0.836205i \(-0.315231\pi\)
−0.836205 + 0.548417i \(0.815231\pi\)
\(164\) 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 0.944790 0.327677i \(-0.106266\pi\)
−0.327677 + 0.944790i \(0.606266\pi\)
\(194\) 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 \(0.196459\pi\)
0.578749 + 0.815506i \(0.303541\pi\)
\(224\) 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 \(-0.297995\pi\)
0.805299 0.592869i \(-0.202005\pi\)
\(278\) 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 0.0521913 0.998637i \(-0.483379\pi\)
−0.998637 + 0.0521913i \(0.983379\pi\)
\(284\) 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 −0.741192 0.671293i \(-0.765739\pi\)
0.671293 + 0.741192i \(0.265739\pi\)
\(308\) 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 0.595601 0.803281i \(-0.296914\pi\)
−0.803281 + 0.595601i \(0.796914\pi\)
\(314\) 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 \(0.706516\pi\)
−0.796815 + 0.604223i \(0.793484\pi\)
\(338\) 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 −0.358247 0.933627i \(-0.616625\pi\)
0.933627 + 0.358247i \(0.116625\pi\)
\(368\) 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 \(0.0259292\pi\)
0.0813690 + 0.996684i \(0.474071\pi\)
\(374\) 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 −0.959127 0.282977i \(-0.908678\pi\)
0.282977 + 0.959127i \(0.408678\pi\)
\(398\) 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 0.212101 0.977248i \(-0.431970\pi\)
−0.977248 + 0.212101i \(0.931970\pi\)
\(434\) 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 \(-0.325146\pi\)
0.852879 0.522108i \(-0.174854\pi\)
\(458\) 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 \(0.0961480\pi\)
0.297486 + 0.954726i \(0.403852\pi\)
\(464\) 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 \(0.558325\pi\)
−0.983260 + 0.182208i \(0.941675\pi\)
\(488\) 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 0.996305 0.0858814i \(-0.0273706\pi\)
−0.0858814 + 0.996305i \(0.527371\pi\)
\(524\) 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 −0.238110 0.971238i \(-0.576528\pi\)
0.971238 + 0.238110i \(0.0765278\pi\)
\(548\) 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 −0.0307771 0.999526i \(-0.509798\pi\)
0.999526 + 0.0307771i \(0.00979822\pi\)
\(578\) 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 \(0.616962\pi\)
−0.933247 + 0.359235i \(0.883038\pi\)
\(608\) 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 \(-0.814727\pi\)
−0.549739 0.835337i \(-0.685273\pi\)
\(614\) 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 0.123018 0.992404i \(-0.460743\pi\)
−0.992404 + 0.123018i \(0.960743\pi\)
\(644\) 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 0.492582 0.870266i \(-0.336053\pi\)
−0.870266 + 0.492582i \(0.836053\pi\)
\(674\) 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 \(0.720444\pi\)
−0.769624 + 0.638498i \(0.779556\pi\)
\(728\) 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 0.924362 0.381518i \(-0.124598\pi\)
−0.381518 + 0.924362i \(0.624598\pi\)
\(734\) 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 −0.729013 0.684499i \(-0.760021\pi\)
0.684499 + 0.729013i \(0.260021\pi\)
\(758\) 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 \(-0.397001\pi\)
0.948103 0.317962i \(-0.102999\pi\)
\(788\) 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
\(808\)