Properties

Label 2475.2.c.l.199.4
Level $2475$
Weight $2$
Character 2475.199
Analytic conductor $19.763$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(19.7629745003\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{3} \)
Twist minimal: no (minimal twist has level 55)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.4
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.l.199.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.41421i q^{2} -3.82843 q^{4} +2.00000i q^{7} -4.41421i q^{8} +O(q^{10})\) \(q+2.41421i q^{2} -3.82843 q^{4} +2.00000i q^{7} -4.41421i q^{8} -1.00000 q^{11} -1.17157i q^{13} -4.82843 q^{14} +3.00000 q^{16} +6.82843i q^{17} -2.41421i q^{22} +2.82843i q^{23} +2.82843 q^{26} -7.65685i q^{28} -3.65685 q^{29} -1.58579i q^{32} -16.4853 q^{34} +7.65685i q^{37} -6.00000 q^{41} -6.00000i q^{43} +3.82843 q^{44} -6.82843 q^{46} +2.82843i q^{47} +3.00000 q^{49} +4.48528i q^{52} -11.6569i q^{53} +8.82843 q^{56} -8.82843i q^{58} +1.65685 q^{59} -9.31371 q^{61} +9.82843 q^{64} -12.4853i q^{67} -26.1421i q^{68} -11.3137 q^{71} -1.17157i q^{73} -18.4853 q^{74} -2.00000i q^{77} -4.00000 q^{79} -14.4853i q^{82} +6.00000i q^{83} +14.4853 q^{86} +4.41421i q^{88} -13.3137 q^{89} +2.34315 q^{91} -10.8284i q^{92} -6.82843 q^{94} -3.65685i q^{97} +7.24264i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{4} + O(q^{10}) \) \( 4 q - 4 q^{4} - 4 q^{11} - 8 q^{14} + 12 q^{16} + 8 q^{29} - 32 q^{34} - 24 q^{41} + 4 q^{44} - 16 q^{46} + 12 q^{49} + 24 q^{56} - 16 q^{59} + 8 q^{61} + 28 q^{64} - 40 q^{74} - 16 q^{79} + 24 q^{86} - 8 q^{89} + 32 q^{91} - 16 q^{94} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2475\mathbb{Z}\right)^\times\).

\(n\) \(551\) \(2026\) \(2377\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.41421i 1.70711i 0.521005 + 0.853553i \(0.325557\pi\)
−0.521005 + 0.853553i \(0.674443\pi\)
\(3\) 0 0
\(4\) −3.82843 −1.91421
\(5\) 0 0
\(6\) 0 0
\(7\) 2.00000i 0.755929i 0.925820 + 0.377964i \(0.123376\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) − 4.41421i − 1.56066i
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) − 1.17157i − 0.324936i −0.986714 0.162468i \(-0.948055\pi\)
0.986714 0.162468i \(-0.0519454\pi\)
\(14\) −4.82843 −1.29045
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 6.82843i 1.65614i 0.560627 + 0.828068i \(0.310560\pi\)
−0.560627 + 0.828068i \(0.689440\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 2.41421i − 0.514712i
\(23\) 2.82843i 0.589768i 0.955533 + 0.294884i \(0.0952810\pi\)
−0.955533 + 0.294884i \(0.904719\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 2.82843 0.554700
\(27\) 0 0
\(28\) − 7.65685i − 1.44701i
\(29\) −3.65685 −0.679061 −0.339530 0.940595i \(-0.610268\pi\)
−0.339530 + 0.940595i \(0.610268\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) − 1.58579i − 0.280330i
\(33\) 0 0
\(34\) −16.4853 −2.82720
\(35\) 0 0
\(36\) 0 0
\(37\) 7.65685i 1.25878i 0.777090 + 0.629390i \(0.216695\pi\)
−0.777090 + 0.629390i \(0.783305\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) − 6.00000i − 0.914991i −0.889212 0.457496i \(-0.848747\pi\)
0.889212 0.457496i \(-0.151253\pi\)
\(44\) 3.82843 0.577157
\(45\) 0 0
\(46\) −6.82843 −1.00680
\(47\) 2.82843i 0.412568i 0.978492 + 0.206284i \(0.0661372\pi\)
−0.978492 + 0.206284i \(0.933863\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) 4.48528i 0.621997i
\(53\) − 11.6569i − 1.60119i −0.599204 0.800596i \(-0.704516\pi\)
0.599204 0.800596i \(-0.295484\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 8.82843 1.17975
\(57\) 0 0
\(58\) − 8.82843i − 1.15923i
\(59\) 1.65685 0.215704 0.107852 0.994167i \(-0.465603\pi\)
0.107852 + 0.994167i \(0.465603\pi\)
\(60\) 0 0
\(61\) −9.31371 −1.19250 −0.596249 0.802799i \(-0.703343\pi\)
−0.596249 + 0.802799i \(0.703343\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 9.82843 1.22855
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.4853i − 1.52532i −0.646800 0.762660i \(-0.723893\pi\)
0.646800 0.762660i \(-0.276107\pi\)
\(68\) − 26.1421i − 3.17020i
\(69\) 0 0
\(70\) 0 0
\(71\) −11.3137 −1.34269 −0.671345 0.741145i \(-0.734283\pi\)
−0.671345 + 0.741145i \(0.734283\pi\)
\(72\) 0 0
\(73\) − 1.17157i − 0.137122i −0.997647 0.0685611i \(-0.978159\pi\)
0.997647 0.0685611i \(-0.0218408\pi\)
\(74\) −18.4853 −2.14887
\(75\) 0 0
\(76\) 0 0
\(77\) − 2.00000i − 0.227921i
\(78\) 0 0
\(79\) −4.00000 −0.450035 −0.225018 0.974355i \(-0.572244\pi\)
−0.225018 + 0.974355i \(0.572244\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) − 14.4853i − 1.59963i
\(83\) 6.00000i 0.658586i 0.944228 + 0.329293i \(0.106810\pi\)
−0.944228 + 0.329293i \(0.893190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 14.4853 1.56199
\(87\) 0 0
\(88\) 4.41421i 0.470557i
\(89\) −13.3137 −1.41125 −0.705625 0.708585i \(-0.749334\pi\)
−0.705625 + 0.708585i \(0.749334\pi\)
\(90\) 0 0
\(91\) 2.34315 0.245628
\(92\) − 10.8284i − 1.12894i
\(93\) 0 0
\(94\) −6.82843 −0.704298
\(95\) 0 0
\(96\) 0 0
\(97\) − 3.65685i − 0.371297i −0.982616 0.185649i \(-0.940561\pi\)
0.982616 0.185649i \(-0.0594386\pi\)
\(98\) 7.24264i 0.731617i
\(99\) 0 0
\(100\) 0 0
\(101\) −9.31371 −0.926749 −0.463374 0.886163i \(-0.653361\pi\)
−0.463374 + 0.886163i \(0.653361\pi\)
\(102\) 0 0
\(103\) 6.82843i 0.672825i 0.941715 + 0.336412i \(0.109214\pi\)
−0.941715 + 0.336412i \(0.890786\pi\)
\(104\) −5.17157 −0.507114
\(105\) 0 0
\(106\) 28.1421 2.73341
\(107\) 7.65685i 0.740216i 0.928989 + 0.370108i \(0.120679\pi\)
−0.928989 + 0.370108i \(0.879321\pi\)
\(108\) 0 0
\(109\) 7.65685 0.733394 0.366697 0.930341i \(-0.380489\pi\)
0.366697 + 0.930341i \(0.380489\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 6.00000i 0.566947i
\(113\) − 19.6569i − 1.84916i −0.380986 0.924581i \(-0.624416\pi\)
0.380986 0.924581i \(-0.375584\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 14.0000 1.29987
\(117\) 0 0
\(118\) 4.00000i 0.368230i
\(119\) −13.6569 −1.25192
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 22.4853i − 2.03572i
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 4.34315i − 0.385392i −0.981259 0.192696i \(-0.938277\pi\)
0.981259 0.192696i \(-0.0617231\pi\)
\(128\) 20.5563i 1.81694i
\(129\) 0 0
\(130\) 0 0
\(131\) 11.3137 0.988483 0.494242 0.869325i \(-0.335446\pi\)
0.494242 + 0.869325i \(0.335446\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 30.1421 2.60388
\(135\) 0 0
\(136\) 30.1421 2.58467
\(137\) − 10.9706i − 0.937278i −0.883390 0.468639i \(-0.844744\pi\)
0.883390 0.468639i \(-0.155256\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 27.3137i − 2.29212i
\(143\) 1.17157i 0.0979718i
\(144\) 0 0
\(145\) 0 0
\(146\) 2.82843 0.234082
\(147\) 0 0
\(148\) − 29.3137i − 2.40957i
\(149\) 0.343146 0.0281116 0.0140558 0.999901i \(-0.495526\pi\)
0.0140558 + 0.999901i \(0.495526\pi\)
\(150\) 0 0
\(151\) −12.0000 −0.976546 −0.488273 0.872691i \(-0.662373\pi\)
−0.488273 + 0.872691i \(0.662373\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 4.82843 0.389086
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000i 1.11732i 0.829396 + 0.558661i \(0.188685\pi\)
−0.829396 + 0.558661i \(0.811315\pi\)
\(158\) − 9.65685i − 0.768258i
\(159\) 0 0
\(160\) 0 0
\(161\) −5.65685 −0.445823
\(162\) 0 0
\(163\) 16.4853i 1.29123i 0.763664 + 0.645613i \(0.223398\pi\)
−0.763664 + 0.645613i \(0.776602\pi\)
\(164\) 22.9706 1.79370
\(165\) 0 0
\(166\) −14.4853 −1.12428
\(167\) − 22.9706i − 1.77752i −0.458377 0.888758i \(-0.651569\pi\)
0.458377 0.888758i \(-0.348431\pi\)
\(168\) 0 0
\(169\) 11.6274 0.894417
\(170\) 0 0
\(171\) 0 0
\(172\) 22.9706i 1.75149i
\(173\) 22.1421i 1.68344i 0.539918 + 0.841718i \(0.318455\pi\)
−0.539918 + 0.841718i \(0.681545\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) − 32.1421i − 2.40915i
\(179\) 9.65685 0.721787 0.360894 0.932607i \(-0.382472\pi\)
0.360894 + 0.932607i \(0.382472\pi\)
\(180\) 0 0
\(181\) 21.3137 1.58424 0.792118 0.610368i \(-0.208979\pi\)
0.792118 + 0.610368i \(0.208979\pi\)
\(182\) 5.65685i 0.419314i
\(183\) 0 0
\(184\) 12.4853 0.920427
\(185\) 0 0
\(186\) 0 0
\(187\) − 6.82843i − 0.499344i
\(188\) − 10.8284i − 0.789744i
\(189\) 0 0
\(190\) 0 0
\(191\) −3.31371 −0.239772 −0.119886 0.992788i \(-0.538253\pi\)
−0.119886 + 0.992788i \(0.538253\pi\)
\(192\) 0 0
\(193\) − 1.17157i − 0.0843317i −0.999111 0.0421658i \(-0.986574\pi\)
0.999111 0.0421658i \(-0.0134258\pi\)
\(194\) 8.82843 0.633844
\(195\) 0 0
\(196\) −11.4853 −0.820377
\(197\) − 10.8284i − 0.771493i −0.922605 0.385747i \(-0.873944\pi\)
0.922605 0.385747i \(-0.126056\pi\)
\(198\) 0 0
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 22.4853i − 1.58206i
\(203\) − 7.31371i − 0.513322i
\(204\) 0 0
\(205\) 0 0
\(206\) −16.4853 −1.14858
\(207\) 0 0
\(208\) − 3.51472i − 0.243702i
\(209\) 0 0
\(210\) 0 0
\(211\) −16.0000 −1.10149 −0.550743 0.834675i \(-0.685655\pi\)
−0.550743 + 0.834675i \(0.685655\pi\)
\(212\) 44.6274i 3.06502i
\(213\) 0 0
\(214\) −18.4853 −1.26363
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 18.4853i 1.25198i
\(219\) 0 0
\(220\) 0 0
\(221\) 8.00000 0.538138
\(222\) 0 0
\(223\) − 10.8284i − 0.725125i −0.931959 0.362563i \(-0.881902\pi\)
0.931959 0.362563i \(-0.118098\pi\)
\(224\) 3.17157 0.211910
\(225\) 0 0
\(226\) 47.4558 3.15672
\(227\) 25.3137i 1.68013i 0.542486 + 0.840065i \(0.317483\pi\)
−0.542486 + 0.840065i \(0.682517\pi\)
\(228\) 0 0
\(229\) −1.31371 −0.0868123 −0.0434062 0.999058i \(-0.513821\pi\)
−0.0434062 + 0.999058i \(0.513821\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 16.1421i 1.05978i
\(233\) 6.14214i 0.402385i 0.979552 + 0.201192i \(0.0644816\pi\)
−0.979552 + 0.201192i \(0.935518\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −6.34315 −0.412904
\(237\) 0 0
\(238\) − 32.9706i − 2.13716i
\(239\) −23.3137 −1.50804 −0.754019 0.656852i \(-0.771887\pi\)
−0.754019 + 0.656852i \(0.771887\pi\)
\(240\) 0 0
\(241\) 6.00000 0.386494 0.193247 0.981150i \(-0.438098\pi\)
0.193247 + 0.981150i \(0.438098\pi\)
\(242\) 2.41421i 0.155192i
\(243\) 0 0
\(244\) 35.6569 2.28270
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −12.0000 −0.757433 −0.378717 0.925513i \(-0.623635\pi\)
−0.378717 + 0.925513i \(0.623635\pi\)
\(252\) 0 0
\(253\) − 2.82843i − 0.177822i
\(254\) 10.4853 0.657905
\(255\) 0 0
\(256\) −29.9706 −1.87316
\(257\) − 9.31371i − 0.580973i −0.956879 0.290487i \(-0.906183\pi\)
0.956879 0.290487i \(-0.0938172\pi\)
\(258\) 0 0
\(259\) −15.3137 −0.951548
\(260\) 0 0
\(261\) 0 0
\(262\) 27.3137i 1.68745i
\(263\) 10.9706i 0.676474i 0.941061 + 0.338237i \(0.109831\pi\)
−0.941061 + 0.338237i \(0.890169\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 47.7990i 2.91979i
\(269\) 17.3137 1.05564 0.527818 0.849358i \(-0.323010\pi\)
0.527818 + 0.849358i \(0.323010\pi\)
\(270\) 0 0
\(271\) 7.31371 0.444276 0.222138 0.975015i \(-0.428696\pi\)
0.222138 + 0.975015i \(0.428696\pi\)
\(272\) 20.4853i 1.24210i
\(273\) 0 0
\(274\) 26.4853 1.60003
\(275\) 0 0
\(276\) 0 0
\(277\) − 6.82843i − 0.410280i −0.978733 0.205140i \(-0.934235\pi\)
0.978733 0.205140i \(-0.0657650\pi\)
\(278\) 9.65685i 0.579180i
\(279\) 0 0
\(280\) 0 0
\(281\) −17.3137 −1.03285 −0.516425 0.856333i \(-0.672737\pi\)
−0.516425 + 0.856333i \(0.672737\pi\)
\(282\) 0 0
\(283\) 32.6274i 1.93950i 0.244103 + 0.969749i \(0.421507\pi\)
−0.244103 + 0.969749i \(0.578493\pi\)
\(284\) 43.3137 2.57020
\(285\) 0 0
\(286\) −2.82843 −0.167248
\(287\) − 12.0000i − 0.708338i
\(288\) 0 0
\(289\) −29.6274 −1.74279
\(290\) 0 0
\(291\) 0 0
\(292\) 4.48528i 0.262481i
\(293\) 9.17157i 0.535809i 0.963445 + 0.267905i \(0.0863312\pi\)
−0.963445 + 0.267905i \(0.913669\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 33.7990 1.96453
\(297\) 0 0
\(298\) 0.828427i 0.0479895i
\(299\) 3.31371 0.191637
\(300\) 0 0
\(301\) 12.0000 0.691669
\(302\) − 28.9706i − 1.66707i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 16.3431i 0.932753i 0.884586 + 0.466376i \(0.154441\pi\)
−0.884586 + 0.466376i \(0.845559\pi\)
\(308\) 7.65685i 0.436290i
\(309\) 0 0
\(310\) 0 0
\(311\) −4.68629 −0.265735 −0.132868 0.991134i \(-0.542419\pi\)
−0.132868 + 0.991134i \(0.542419\pi\)
\(312\) 0 0
\(313\) − 1.31371i − 0.0742552i −0.999311 0.0371276i \(-0.988179\pi\)
0.999311 0.0371276i \(-0.0118208\pi\)
\(314\) −33.7990 −1.90739
\(315\) 0 0
\(316\) 15.3137 0.861463
\(317\) − 1.31371i − 0.0737852i −0.999319 0.0368926i \(-0.988254\pi\)
0.999319 0.0368926i \(-0.0117459\pi\)
\(318\) 0 0
\(319\) 3.65685 0.204745
\(320\) 0 0
\(321\) 0 0
\(322\) − 13.6569i − 0.761067i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) −39.7990 −2.20426
\(327\) 0 0
\(328\) 26.4853i 1.46241i
\(329\) −5.65685 −0.311872
\(330\) 0 0
\(331\) −7.31371 −0.401998 −0.200999 0.979591i \(-0.564419\pi\)
−0.200999 + 0.979591i \(0.564419\pi\)
\(332\) − 22.9706i − 1.26067i
\(333\) 0 0
\(334\) 55.4558 3.03441
\(335\) 0 0
\(336\) 0 0
\(337\) 20.4853i 1.11590i 0.829873 + 0.557952i \(0.188413\pi\)
−0.829873 + 0.557952i \(0.811587\pi\)
\(338\) 28.0711i 1.52686i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 20.0000i 1.07990i
\(344\) −26.4853 −1.42799
\(345\) 0 0
\(346\) −53.4558 −2.87380
\(347\) 10.9706i 0.588931i 0.955662 + 0.294465i \(0.0951416\pi\)
−0.955662 + 0.294465i \(0.904858\pi\)
\(348\) 0 0
\(349\) −26.9706 −1.44370 −0.721851 0.692049i \(-0.756708\pi\)
−0.721851 + 0.692049i \(0.756708\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.58579i 0.0845227i
\(353\) − 21.3137i − 1.13441i −0.823575 0.567207i \(-0.808024\pi\)
0.823575 0.567207i \(-0.191976\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 50.9706 2.70143
\(357\) 0 0
\(358\) 23.3137i 1.23217i
\(359\) 0.686292 0.0362211 0.0181105 0.999836i \(-0.494235\pi\)
0.0181105 + 0.999836i \(0.494235\pi\)
\(360\) 0 0
\(361\) −19.0000 −1.00000
\(362\) 51.4558i 2.70446i
\(363\) 0 0
\(364\) −8.97056 −0.470185
\(365\) 0 0
\(366\) 0 0
\(367\) 8.48528i 0.442928i 0.975169 + 0.221464i \(0.0710835\pi\)
−0.975169 + 0.221464i \(0.928916\pi\)
\(368\) 8.48528i 0.442326i
\(369\) 0 0
\(370\) 0 0
\(371\) 23.3137 1.21039
\(372\) 0 0
\(373\) 35.7990i 1.85360i 0.375554 + 0.926801i \(0.377453\pi\)
−0.375554 + 0.926801i \(0.622547\pi\)
\(374\) 16.4853 0.852434
\(375\) 0 0
\(376\) 12.4853 0.643879
\(377\) 4.28427i 0.220651i
\(378\) 0 0
\(379\) −33.6569 −1.72884 −0.864418 0.502773i \(-0.832313\pi\)
−0.864418 + 0.502773i \(0.832313\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 8.00000i − 0.409316i
\(383\) 5.85786i 0.299323i 0.988737 + 0.149661i \(0.0478184\pi\)
−0.988737 + 0.149661i \(0.952182\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.82843 0.143963
\(387\) 0 0
\(388\) 14.0000i 0.710742i
\(389\) 20.6274 1.04585 0.522926 0.852378i \(-0.324840\pi\)
0.522926 + 0.852378i \(0.324840\pi\)
\(390\) 0 0
\(391\) −19.3137 −0.976736
\(392\) − 13.2426i − 0.668854i
\(393\) 0 0
\(394\) 26.1421 1.31702
\(395\) 0 0
\(396\) 0 0
\(397\) 9.31371i 0.467442i 0.972304 + 0.233721i \(0.0750902\pi\)
−0.972304 + 0.233721i \(0.924910\pi\)
\(398\) − 24.9706i − 1.25166i
\(399\) 0 0
\(400\) 0 0
\(401\) 5.31371 0.265354 0.132677 0.991159i \(-0.457643\pi\)
0.132677 + 0.991159i \(0.457643\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 35.6569 1.77399
\(405\) 0 0
\(406\) 17.6569 0.876295
\(407\) − 7.65685i − 0.379536i
\(408\) 0 0
\(409\) −1.02944 −0.0509024 −0.0254512 0.999676i \(-0.508102\pi\)
−0.0254512 + 0.999676i \(0.508102\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 26.1421i − 1.28793i
\(413\) 3.31371i 0.163057i
\(414\) 0 0
\(415\) 0 0
\(416\) −1.85786 −0.0910893
\(417\) 0 0
\(418\) 0 0
\(419\) −25.6569 −1.25342 −0.626710 0.779253i \(-0.715599\pi\)
−0.626710 + 0.779253i \(0.715599\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) − 38.6274i − 1.88035i
\(423\) 0 0
\(424\) −51.4558 −2.49892
\(425\) 0 0
\(426\) 0 0
\(427\) − 18.6274i − 0.901444i
\(428\) − 29.3137i − 1.41693i
\(429\) 0 0
\(430\) 0 0
\(431\) 11.3137 0.544962 0.272481 0.962161i \(-0.412156\pi\)
0.272481 + 0.962161i \(0.412156\pi\)
\(432\) 0 0
\(433\) − 7.65685i − 0.367965i −0.982930 0.183982i \(-0.941101\pi\)
0.982930 0.183982i \(-0.0588990\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −29.3137 −1.40387
\(437\) 0 0
\(438\) 0 0
\(439\) 16.0000 0.763638 0.381819 0.924237i \(-0.375298\pi\)
0.381819 + 0.924237i \(0.375298\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 19.3137i 0.918659i
\(443\) 26.8284i 1.27466i 0.770592 + 0.637329i \(0.219961\pi\)
−0.770592 + 0.637329i \(0.780039\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 26.1421 1.23787
\(447\) 0 0
\(448\) 19.6569i 0.928699i
\(449\) 28.6274 1.35101 0.675506 0.737355i \(-0.263925\pi\)
0.675506 + 0.737355i \(0.263925\pi\)
\(450\) 0 0
\(451\) 6.00000 0.282529
\(452\) 75.2548i 3.53969i
\(453\) 0 0
\(454\) −61.1127 −2.86816
\(455\) 0 0
\(456\) 0 0
\(457\) − 0.485281i − 0.0227005i −0.999936 0.0113503i \(-0.996387\pi\)
0.999936 0.0113503i \(-0.00361298\pi\)
\(458\) − 3.17157i − 0.148198i
\(459\) 0 0
\(460\) 0 0
\(461\) −12.6274 −0.588117 −0.294059 0.955787i \(-0.595006\pi\)
−0.294059 + 0.955787i \(0.595006\pi\)
\(462\) 0 0
\(463\) − 6.14214i − 0.285449i −0.989762 0.142725i \(-0.954414\pi\)
0.989762 0.142725i \(-0.0455863\pi\)
\(464\) −10.9706 −0.509296
\(465\) 0 0
\(466\) −14.8284 −0.686914
\(467\) − 14.8284i − 0.686178i −0.939303 0.343089i \(-0.888527\pi\)
0.939303 0.343089i \(-0.111473\pi\)
\(468\) 0 0
\(469\) 24.9706 1.15303
\(470\) 0 0
\(471\) 0 0
\(472\) − 7.31371i − 0.336641i
\(473\) 6.00000i 0.275880i
\(474\) 0 0
\(475\) 0 0
\(476\) 52.2843 2.39645
\(477\) 0 0
\(478\) − 56.2843i − 2.57438i
\(479\) −36.0000 −1.64488 −0.822441 0.568850i \(-0.807388\pi\)
−0.822441 + 0.568850i \(0.807388\pi\)
\(480\) 0 0
\(481\) 8.97056 0.409022
\(482\) 14.4853i 0.659786i
\(483\) 0 0
\(484\) −3.82843 −0.174019
\(485\) 0 0
\(486\) 0 0
\(487\) 24.4853i 1.10953i 0.832006 + 0.554767i \(0.187193\pi\)
−0.832006 + 0.554767i \(0.812807\pi\)
\(488\) 41.1127i 1.86108i
\(489\) 0 0
\(490\) 0 0
\(491\) 0.686292 0.0309719 0.0154860 0.999880i \(-0.495070\pi\)
0.0154860 + 0.999880i \(0.495070\pi\)
\(492\) 0 0
\(493\) − 24.9706i − 1.12462i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 22.6274i − 1.01498i
\(498\) 0 0
\(499\) −9.65685 −0.432300 −0.216150 0.976360i \(-0.569350\pi\)
−0.216150 + 0.976360i \(0.569350\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 28.9706i − 1.29302i
\(503\) − 16.6274i − 0.741380i −0.928757 0.370690i \(-0.879121\pi\)
0.928757 0.370690i \(-0.120879\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 6.82843 0.303561
\(507\) 0 0
\(508\) 16.6274i 0.737722i
\(509\) −13.3137 −0.590120 −0.295060 0.955479i \(-0.595340\pi\)
−0.295060 + 0.955479i \(0.595340\pi\)
\(510\) 0 0
\(511\) 2.34315 0.103655
\(512\) − 31.2426i − 1.38074i
\(513\) 0 0
\(514\) 22.4853 0.991783
\(515\) 0 0
\(516\) 0 0
\(517\) − 2.82843i − 0.124394i
\(518\) − 36.9706i − 1.62439i
\(519\) 0 0
\(520\) 0 0
\(521\) −25.3137 −1.10901 −0.554507 0.832179i \(-0.687093\pi\)
−0.554507 + 0.832179i \(0.687093\pi\)
\(522\) 0 0
\(523\) − 41.5980i − 1.81895i −0.415756 0.909476i \(-0.636483\pi\)
0.415756 0.909476i \(-0.363517\pi\)
\(524\) −43.3137 −1.89217
\(525\) 0 0
\(526\) −26.4853 −1.15481
\(527\) 0 0
\(528\) 0 0
\(529\) 15.0000 0.652174
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 7.02944i 0.304479i
\(534\) 0 0
\(535\) 0 0
\(536\) −55.1127 −2.38051
\(537\) 0 0
\(538\) 41.7990i 1.80208i
\(539\) −3.00000 −0.129219
\(540\) 0 0
\(541\) 6.00000 0.257960 0.128980 0.991647i \(-0.458830\pi\)
0.128980 + 0.991647i \(0.458830\pi\)
\(542\) 17.6569i 0.758427i
\(543\) 0 0
\(544\) 10.8284 0.464265
\(545\) 0 0
\(546\) 0 0
\(547\) 34.0000i 1.45374i 0.686778 + 0.726868i \(0.259025\pi\)
−0.686778 + 0.726868i \(0.740975\pi\)
\(548\) 42.0000i 1.79415i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) − 8.00000i − 0.340195i
\(554\) 16.4853 0.700392
\(555\) 0 0
\(556\) −15.3137 −0.649446
\(557\) 9.85786i 0.417691i 0.977949 + 0.208846i \(0.0669706\pi\)
−0.977949 + 0.208846i \(0.933029\pi\)
\(558\) 0 0
\(559\) −7.02944 −0.297314
\(560\) 0 0
\(561\) 0 0
\(562\) − 41.7990i − 1.76318i
\(563\) − 0.343146i − 0.0144619i −0.999974 0.00723093i \(-0.997698\pi\)
0.999974 0.00723093i \(-0.00230170\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −78.7696 −3.31093
\(567\) 0 0
\(568\) 49.9411i 2.09548i
\(569\) 31.6569 1.32712 0.663562 0.748121i \(-0.269044\pi\)
0.663562 + 0.748121i \(0.269044\pi\)
\(570\) 0 0
\(571\) −21.9411 −0.918208 −0.459104 0.888383i \(-0.651829\pi\)
−0.459104 + 0.888383i \(0.651829\pi\)
\(572\) − 4.48528i − 0.187539i
\(573\) 0 0
\(574\) 28.9706 1.20921
\(575\) 0 0
\(576\) 0 0
\(577\) 26.9706i 1.12280i 0.827545 + 0.561400i \(0.189737\pi\)
−0.827545 + 0.561400i \(0.810263\pi\)
\(578\) − 71.5269i − 2.97513i
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 0 0
\(583\) 11.6569i 0.482778i
\(584\) −5.17157 −0.214001
\(585\) 0 0
\(586\) −22.1421 −0.914683
\(587\) − 2.14214i − 0.0884154i −0.999022 0.0442077i \(-0.985924\pi\)
0.999022 0.0442077i \(-0.0140763\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 22.9706i 0.944084i
\(593\) − 3.51472i − 0.144332i −0.997393 0.0721661i \(-0.977009\pi\)
0.997393 0.0721661i \(-0.0229912\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −1.31371 −0.0538116
\(597\) 0 0
\(598\) 8.00000i 0.327144i
\(599\) −5.65685 −0.231133 −0.115566 0.993300i \(-0.536868\pi\)
−0.115566 + 0.993300i \(0.536868\pi\)
\(600\) 0 0
\(601\) 23.9411 0.976579 0.488289 0.872682i \(-0.337621\pi\)
0.488289 + 0.872682i \(0.337621\pi\)
\(602\) 28.9706i 1.18075i
\(603\) 0 0
\(604\) 45.9411 1.86932
\(605\) 0 0
\(606\) 0 0
\(607\) − 38.2843i − 1.55391i −0.629556 0.776955i \(-0.716763\pi\)
0.629556 0.776955i \(-0.283237\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 3.31371 0.134058
\(612\) 0 0
\(613\) − 25.4558i − 1.02815i −0.857745 0.514076i \(-0.828135\pi\)
0.857745 0.514076i \(-0.171865\pi\)
\(614\) −39.4558 −1.59231
\(615\) 0 0
\(616\) −8.82843 −0.355707
\(617\) 0.343146i 0.0138145i 0.999976 + 0.00690726i \(0.00219867\pi\)
−0.999976 + 0.00690726i \(0.997801\pi\)
\(618\) 0 0
\(619\) 14.3431 0.576500 0.288250 0.957555i \(-0.406927\pi\)
0.288250 + 0.957555i \(0.406927\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 11.3137i − 0.453638i
\(623\) − 26.6274i − 1.06680i
\(624\) 0 0
\(625\) 0 0
\(626\) 3.17157 0.126762
\(627\) 0 0
\(628\) − 53.5980i − 2.13879i
\(629\) −52.2843 −2.08471
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 17.6569i 0.702352i
\(633\) 0 0
\(634\) 3.17157 0.125959
\(635\) 0 0
\(636\) 0 0
\(637\) − 3.51472i − 0.139258i
\(638\) 8.82843i 0.349521i
\(639\) 0 0
\(640\) 0 0
\(641\) −30.0000 −1.18493 −0.592464 0.805597i \(-0.701845\pi\)
−0.592464 + 0.805597i \(0.701845\pi\)
\(642\) 0 0
\(643\) − 1.45584i − 0.0574129i −0.999588 0.0287064i \(-0.990861\pi\)
0.999588 0.0287064i \(-0.00913880\pi\)
\(644\) 21.6569 0.853400
\(645\) 0 0
\(646\) 0 0
\(647\) 27.1127i 1.06591i 0.846144 + 0.532955i \(0.178919\pi\)
−0.846144 + 0.532955i \(0.821081\pi\)
\(648\) 0 0
\(649\) −1.65685 −0.0650372
\(650\) 0 0
\(651\) 0 0
\(652\) − 63.1127i − 2.47168i
\(653\) − 11.6569i − 0.456168i −0.973641 0.228084i \(-0.926754\pi\)
0.973641 0.228084i \(-0.0732461\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −18.0000 −0.702782
\(657\) 0 0
\(658\) − 13.6569i − 0.532400i
\(659\) 45.9411 1.78961 0.894806 0.446455i \(-0.147314\pi\)
0.894806 + 0.446455i \(0.147314\pi\)
\(660\) 0 0
\(661\) 44.6274 1.73581 0.867903 0.496734i \(-0.165468\pi\)
0.867903 + 0.496734i \(0.165468\pi\)
\(662\) − 17.6569i − 0.686253i
\(663\) 0 0
\(664\) 26.4853 1.02783
\(665\) 0 0
\(666\) 0 0
\(667\) − 10.3431i − 0.400488i
\(668\) 87.9411i 3.40254i
\(669\) 0 0
\(670\) 0 0
\(671\) 9.31371 0.359552
\(672\) 0 0
\(673\) − 12.4853i − 0.481272i −0.970615 0.240636i \(-0.922644\pi\)
0.970615 0.240636i \(-0.0773560\pi\)
\(674\) −49.4558 −1.90497
\(675\) 0 0
\(676\) −44.5147 −1.71210
\(677\) 22.8284i 0.877368i 0.898641 + 0.438684i \(0.144555\pi\)
−0.898641 + 0.438684i \(0.855445\pi\)
\(678\) 0 0
\(679\) 7.31371 0.280674
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 7.79899i 0.298420i 0.988806 + 0.149210i \(0.0476731\pi\)
−0.988806 + 0.149210i \(0.952327\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −48.2843 −1.84350
\(687\) 0 0
\(688\) − 18.0000i − 0.686244i
\(689\) −13.6569 −0.520285
\(690\) 0 0
\(691\) −39.3137 −1.49556 −0.747782 0.663944i \(-0.768881\pi\)
−0.747782 + 0.663944i \(0.768881\pi\)
\(692\) − 84.7696i − 3.22245i
\(693\) 0 0
\(694\) −26.4853 −1.00537
\(695\) 0 0
\(696\) 0 0
\(697\) − 40.9706i − 1.55187i
\(698\) − 65.1127i − 2.46455i
\(699\) 0 0
\(700\) 0 0
\(701\) 12.6274 0.476931 0.238465 0.971151i \(-0.423356\pi\)
0.238465 + 0.971151i \(0.423356\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −9.82843 −0.370423
\(705\) 0 0
\(706\) 51.4558 1.93657
\(707\) − 18.6274i − 0.700556i
\(708\) 0 0
\(709\) −24.6274 −0.924902 −0.462451 0.886645i \(-0.653030\pi\)
−0.462451 + 0.886645i \(0.653030\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 58.7696i 2.20248i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −36.9706 −1.38165
\(717\) 0 0
\(718\) 1.65685i 0.0618333i
\(719\) 18.3431 0.684084 0.342042 0.939685i \(-0.388882\pi\)
0.342042 + 0.939685i \(0.388882\pi\)
\(720\) 0 0
\(721\) −13.6569 −0.508608
\(722\) − 45.8701i − 1.70711i
\(723\) 0 0
\(724\) −81.5980 −3.03257
\(725\) 0 0
\(726\) 0 0
\(727\) 19.5147i 0.723761i 0.932225 + 0.361880i \(0.117865\pi\)
−0.932225 + 0.361880i \(0.882135\pi\)
\(728\) − 10.3431i − 0.383342i
\(729\) 0 0
\(730\) 0 0
\(731\) 40.9706 1.51535
\(732\) 0 0
\(733\) − 17.4558i − 0.644746i −0.946613 0.322373i \(-0.895519\pi\)
0.946613 0.322373i \(-0.104481\pi\)
\(734\) −20.4853 −0.756126
\(735\) 0 0
\(736\) 4.48528 0.165330
\(737\) 12.4853i 0.459901i
\(738\) 0 0
\(739\) −29.9411 −1.10140 −0.550701 0.834703i \(-0.685640\pi\)
−0.550701 + 0.834703i \(0.685640\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 56.2843i 2.06626i
\(743\) 49.5980i 1.81957i 0.415076 + 0.909787i \(0.363755\pi\)
−0.415076 + 0.909787i \(0.636245\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −86.4264 −3.16430
\(747\) 0 0
\(748\) 26.1421i 0.955851i
\(749\) −15.3137 −0.559551
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 8.48528i 0.309426i
\(753\) 0 0
\(754\) −10.3431 −0.376675
\(755\) 0 0
\(756\) 0 0
\(757\) − 13.3137i − 0.483895i −0.970289 0.241947i \(-0.922214\pi\)
0.970289 0.241947i \(-0.0777862\pi\)
\(758\) − 81.2548i − 2.95131i
\(759\) 0 0
\(760\) 0 0
\(761\) 30.0000 1.08750 0.543750 0.839248i \(-0.317004\pi\)
0.543750 + 0.839248i \(0.317004\pi\)
\(762\) 0 0
\(763\) 15.3137i 0.554393i
\(764\) 12.6863 0.458974
\(765\) 0 0
\(766\) −14.1421 −0.510976
\(767\) − 1.94113i − 0.0700900i
\(768\) 0 0
\(769\) 18.9706 0.684096 0.342048 0.939682i \(-0.388879\pi\)
0.342048 + 0.939682i \(0.388879\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.48528i 0.161429i
\(773\) − 26.2843i − 0.945380i −0.881229 0.472690i \(-0.843283\pi\)
0.881229 0.472690i \(-0.156717\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −16.1421 −0.579469
\(777\) 0 0
\(778\) 49.7990i 1.78538i
\(779\) 0 0
\(780\) 0 0
\(781\) 11.3137 0.404836
\(782\) − 46.6274i − 1.66739i
\(783\) 0 0
\(784\) 9.00000 0.321429
\(785\) 0 0
\(786\) 0 0
\(787\) 14.9706i 0.533643i 0.963746 + 0.266821i \(0.0859734\pi\)
−0.963746 + 0.266821i \(0.914027\pi\)
\(788\) 41.4558i 1.47680i
\(789\) 0 0
\(790\) 0 0
\(791\) 39.3137 1.39783
\(792\) 0 0
\(793\) 10.9117i 0.387485i
\(794\) −22.4853 −0.797973
\(795\) 0 0
\(796\) 39.5980 1.40351
\(797\) 32.6274i 1.15572i 0.816135 + 0.577861i \(0.196113\pi\)
−0.816135 + 0.577861i \(0.803887\pi\)
\(798\) 0 0
\(799\) −19.3137 −0.683270
\(800\) 0 0
\(801\) 0 0
\(802\) 12.8284i 0.452988i
\(803\) 1.17157i 0.0413439i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 41.1127i 1.44634i
\(809\) −10.9706 −0.385704 −0.192852 0.981228i \(-0.561774\pi\)
−0.192852 + 0.981228i \(0.561774\pi\)
\(810\) 0 0
\(811\) 53.9411 1.89413 0.947065 0.321043i \(-0.104033\pi\)
0.947065 + 0.321043i \(0.104033\pi\)
\(812\) 28.0000i 0.982607i
\(813\) 0 0
\(814\) 18.4853 0.647909
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) − 2.48528i − 0.0868958i
\(819\) 0 0
\(820\) 0 0
\(821\) 41.3137 1.44186 0.720929 0.693009i \(-0.243715\pi\)
0.720929 + 0.693009i \(0.243715\pi\)
\(822\) 0 0
\(823\) 19.5147i 0.680240i 0.940382 + 0.340120i \(0.110468\pi\)
−0.940382 + 0.340120i \(0.889532\pi\)
\(824\) 30.1421 1.05005
\(825\) 0 0
\(826\) −8.00000 −0.278356
\(827\) 22.2843i 0.774900i 0.921891 + 0.387450i \(0.126644\pi\)
−0.921891 + 0.387450i \(0.873356\pi\)
\(828\) 0 0
\(829\) −18.0000 −0.625166 −0.312583 0.949890i \(-0.601194\pi\)
−0.312583 + 0.949890i \(0.601194\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 11.5147i − 0.399201i
\(833\) 20.4853i 0.709773i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) − 61.9411i − 2.13972i
\(839\) 26.3431 0.909466 0.454733 0.890628i \(-0.349735\pi\)
0.454733 + 0.890628i \(0.349735\pi\)
\(840\) 0 0
\(841\) −15.6274 −0.538876
\(842\) − 14.4853i − 0.499196i
\(843\) 0 0
\(844\) 61.2548 2.10848
\(845\) 0 0
\(846\) 0 0
\(847\) 2.00000i 0.0687208i
\(848\) − 34.9706i − 1.20089i
\(849\) 0 0
\(850\) 0 0
\(851\) −21.6569 −0.742387
\(852\) 0 0
\(853\) − 15.5147i − 0.531214i −0.964081 0.265607i \(-0.914428\pi\)
0.964081 0.265607i \(-0.0855723\pi\)
\(854\) 44.9706 1.53886
\(855\) 0 0
\(856\) 33.7990 1.15523
\(857\) 24.7696i 0.846112i 0.906104 + 0.423056i \(0.139043\pi\)
−0.906104 + 0.423056i \(0.860957\pi\)
\(858\) 0 0
\(859\) −24.2843 −0.828569 −0.414284 0.910148i \(-0.635968\pi\)
−0.414284 + 0.910148i \(0.635968\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 27.3137i 0.930309i
\(863\) 9.17157i 0.312204i 0.987741 + 0.156102i \(0.0498929\pi\)
−0.987741 + 0.156102i \(0.950107\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 18.4853 0.628155
\(867\) 0 0
\(868\) 0 0
\(869\) 4.00000 0.135691
\(870\) 0 0
\(871\) −14.6274 −0.495631
\(872\) − 33.7990i − 1.14458i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 49.4558i 1.67001i 0.550246 + 0.835003i \(0.314534\pi\)
−0.550246 + 0.835003i \(0.685466\pi\)
\(878\) 38.6274i 1.30361i
\(879\) 0 0
\(880\) 0 0
\(881\) 7.37258 0.248389 0.124194 0.992258i \(-0.460365\pi\)
0.124194 + 0.992258i \(0.460365\pi\)
\(882\) 0 0
\(883\) 37.1716i 1.25092i 0.780255 + 0.625462i \(0.215089\pi\)
−0.780255 + 0.625462i \(0.784911\pi\)
\(884\) −30.6274 −1.03011
\(885\) 0 0
\(886\) −64.7696 −2.17598
\(887\) 38.2843i 1.28546i 0.766093 + 0.642730i \(0.222198\pi\)
−0.766093 + 0.642730i \(0.777802\pi\)
\(888\) 0 0
\(889\) 8.68629 0.291329
\(890\) 0 0
\(891\) 0 0
\(892\) 41.4558i 1.38804i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) −41.1127 −1.37348
\(897\) 0 0
\(898\) 69.1127i 2.30632i
\(899\) 0 0
\(900\) 0 0
\(901\) 79.5980 2.65179
\(902\) 14.4853i 0.482307i
\(903\) 0 0
\(904\) −86.7696 −2.88591
\(905\) 0 0
\(906\) 0 0
\(907\) 27.5147i 0.913611i 0.889567 + 0.456806i \(0.151007\pi\)
−0.889567 + 0.456806i \(0.848993\pi\)
\(908\) − 96.9117i − 3.21613i
\(909\) 0 0
\(910\) 0 0
\(911\) 9.94113 0.329364 0.164682 0.986347i \(-0.447340\pi\)
0.164682 + 0.986347i \(0.447340\pi\)
\(912\) 0 0
\(913\) − 6.00000i − 0.198571i
\(914\) 1.17157 0.0387522
\(915\) 0 0
\(916\) 5.02944 0.166177
\(917\) 22.6274i 0.747223i
\(918\) 0 0
\(919\) 32.0000 1.05558 0.527791 0.849374i \(-0.323020\pi\)
0.527791 + 0.849374i \(0.323020\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 30.4853i − 1.00398i
\(923\) 13.2548i 0.436288i
\(924\) 0 0
\(925\) 0 0
\(926\) 14.8284 0.487292
\(927\) 0 0
\(928\) 5.79899i 0.190361i
\(929\) 5.31371 0.174337 0.0871686 0.996194i \(-0.472218\pi\)
0.0871686 + 0.996194i \(0.472218\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 23.5147i − 0.770250i
\(933\) 0 0
\(934\) 35.7990 1.17138
\(935\) 0 0
\(936\) 0 0
\(937\) 1.45584i 0.0475604i 0.999717 + 0.0237802i \(0.00757018\pi\)
−0.999717 + 0.0237802i \(0.992430\pi\)
\(938\) 60.2843i 1.96835i
\(939\) 0 0
\(940\) 0 0
\(941\) 6.68629 0.217967 0.108983 0.994044i \(-0.465240\pi\)
0.108983 + 0.994044i \(0.465240\pi\)
\(942\) 0 0
\(943\) − 16.9706i − 0.552638i
\(944\) 4.97056 0.161778
\(945\) 0 0
\(946\) −14.4853 −0.470957
\(947\) 41.1716i 1.33790i 0.743309 + 0.668948i \(0.233255\pi\)
−0.743309 + 0.668948i \(0.766745\pi\)
\(948\) 0 0
\(949\) −1.37258 −0.0445559
\(950\) 0 0
\(951\) 0 0
\(952\) 60.2843i 1.95382i
\(953\) − 53.1716i − 1.72240i −0.508269 0.861198i \(-0.669715\pi\)
0.508269 0.861198i \(-0.330285\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 89.2548 2.88671
\(957\) 0 0
\(958\) − 86.9117i − 2.80799i
\(959\) 21.9411 0.708516
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 21.6569i 0.698245i
\(963\) 0 0
\(964\) −22.9706 −0.739832
\(965\) 0 0
\(966\) 0 0
\(967\) 14.9706i 0.481421i 0.970597 + 0.240710i \(0.0773804\pi\)
−0.970597 + 0.240710i \(0.922620\pi\)
\(968\) − 4.41421i − 0.141878i
\(969\) 0 0
\(970\) 0 0
\(971\) 8.68629 0.278756 0.139378 0.990239i \(-0.455490\pi\)
0.139378 + 0.990239i \(0.455490\pi\)
\(972\) 0 0
\(973\) 8.00000i 0.256468i
\(974\) −59.1127 −1.89409
\(975\) 0 0
\(976\) −27.9411 −0.894374
\(977\) 32.3431i 1.03475i 0.855759 + 0.517374i \(0.173091\pi\)
−0.855759 + 0.517374i \(0.826909\pi\)
\(978\) 0 0
\(979\) 13.3137 0.425508
\(980\) 0 0
\(981\) 0 0
\(982\) 1.65685i 0.0528723i
\(983\) 21.8579i 0.697158i 0.937279 + 0.348579i \(0.113336\pi\)
−0.937279 + 0.348579i \(0.886664\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 60.2843 1.91984
\(987\) 0 0
\(988\) 0 0
\(989\) 16.9706 0.539633
\(990\) 0 0
\(991\) −57.9411 −1.84056 −0.920280 0.391260i \(-0.872039\pi\)
−0.920280 + 0.391260i \(0.872039\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 54.6274 1.73268
\(995\) 0 0
\(996\) 0 0
\(997\) 41.4558i 1.31292i 0.754361 + 0.656460i \(0.227947\pi\)
−0.754361 + 0.656460i \(0.772053\pi\)
\(998\) − 23.3137i − 0.737983i
\(999\) 0 0
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 \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 2475.2.c.l.199.4 4
3.2 odd 2 275.2.b.d.199.1 4
5.2 odd 4 495.2.a.b.1.1 2
5.3 odd 4 2475.2.a.x.1.2 2
5.4 even 2 inner 2475.2.c.l.199.1 4
12.11 even 2 4400.2.b.q.4049.3 4
15.2 even 4 55.2.a.b.1.2 2
15.8 even 4 275.2.a.c.1.1 2
15.14 odd 2 275.2.b.d.199.4 4
20.7 even 4 7920.2.a.ch.1.2 2
55.32 even 4 5445.2.a.y.1.2 2
60.23 odd 4 4400.2.a.bn.1.1 2
60.47 odd 4 880.2.a.m.1.2 2
60.59 even 2 4400.2.b.q.4049.2 4
105.62 odd 4 2695.2.a.f.1.2 2
120.77 even 4 3520.2.a.bn.1.2 2
120.107 odd 4 3520.2.a.bo.1.1 2
165.2 odd 20 605.2.g.l.81.2 8
165.17 odd 20 605.2.g.l.366.2 8
165.32 odd 4 605.2.a.d.1.1 2
165.47 even 20 605.2.g.f.251.2 8
165.62 odd 20 605.2.g.l.511.1 8
165.92 even 20 605.2.g.f.511.2 8
165.98 odd 4 3025.2.a.o.1.2 2
165.107 odd 20 605.2.g.l.251.1 8
165.137 even 20 605.2.g.f.366.1 8
165.152 even 20 605.2.g.f.81.1 8
195.77 even 4 9295.2.a.g.1.1 2
660.527 even 4 9680.2.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
55.2.a.b.1.2 2 15.2 even 4
275.2.a.c.1.1 2 15.8 even 4
275.2.b.d.199.1 4 3.2 odd 2
275.2.b.d.199.4 4 15.14 odd 2
495.2.a.b.1.1 2 5.2 odd 4
605.2.a.d.1.1 2 165.32 odd 4
605.2.g.f.81.1 8 165.152 even 20
605.2.g.f.251.2 8 165.47 even 20
605.2.g.f.366.1 8 165.137 even 20
605.2.g.f.511.2 8 165.92 even 20
605.2.g.l.81.2 8 165.2 odd 20
605.2.g.l.251.1 8 165.107 odd 20
605.2.g.l.366.2 8 165.17 odd 20
605.2.g.l.511.1 8 165.62 odd 20
880.2.a.m.1.2 2 60.47 odd 4
2475.2.a.x.1.2 2 5.3 odd 4
2475.2.c.l.199.1 4 5.4 even 2 inner
2475.2.c.l.199.4 4 1.1 even 1 trivial
2695.2.a.f.1.2 2 105.62 odd 4
3025.2.a.o.1.2 2 165.98 odd 4
3520.2.a.bn.1.2 2 120.77 even 4
3520.2.a.bo.1.1 2 120.107 odd 4
4400.2.a.bn.1.1 2 60.23 odd 4
4400.2.b.q.4049.2 4 60.59 even 2
4400.2.b.q.4049.3 4 12.11 even 2
5445.2.a.y.1.2 2 55.32 even 4
7920.2.a.ch.1.2 2 20.7 even 4
9295.2.a.g.1.1 2 195.77 even 4
9680.2.a.bn.1.2 2 660.527 even 4