Properties

Label 2475.2.c.i.199.3
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: \(\Q(i, \sqrt{17})\)
Defining polynomial: \( x^{4} + 9x^{2} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.3
Root \(1.56155i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.i.199.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.56155i q^{2} -0.438447 q^{4} -1.00000i q^{7} +2.43845i q^{8} +O(q^{10})\) \(q+1.56155i q^{2} -0.438447 q^{4} -1.00000i q^{7} +2.43845i q^{8} -1.00000 q^{11} -4.56155i q^{13} +1.56155 q^{14} -4.68466 q^{16} -5.56155i q^{17} -3.00000 q^{19} -1.56155i q^{22} +2.43845i q^{23} +7.12311 q^{26} +0.438447i q^{28} +3.12311 q^{29} -7.68466 q^{31} -2.43845i q^{32} +8.68466 q^{34} -9.80776i q^{37} -4.68466i q^{38} +4.68466 q^{41} -7.68466i q^{43} +0.438447 q^{44} -3.80776 q^{46} -9.56155i q^{47} +6.00000 q^{49} +2.00000i q^{52} +7.12311i q^{53} +2.43845 q^{56} +4.87689i q^{58} -6.43845 q^{59} -3.43845 q^{61} -12.0000i q^{62} -5.56155 q^{64} -5.68466i q^{67} +2.43845i q^{68} -11.8078 q^{71} -0.246211i q^{73} +15.3153 q^{74} +1.31534 q^{76} +1.00000i q^{77} +3.31534 q^{79} +7.31534i q^{82} +12.0000 q^{86} -2.43845i q^{88} +8.87689 q^{89} -4.56155 q^{91} -1.06913i q^{92} +14.9309 q^{94} +6.12311i q^{97} +9.36932i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 10 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 10 q^{4} - 4 q^{11} - 2 q^{14} + 6 q^{16} - 12 q^{19} + 12 q^{26} - 4 q^{29} - 6 q^{31} + 10 q^{34} - 6 q^{41} + 10 q^{44} + 26 q^{46} + 24 q^{49} + 18 q^{56} - 34 q^{59} - 22 q^{61} - 14 q^{64} - 6 q^{71} + 86 q^{74} + 30 q^{76} + 38 q^{79} + 48 q^{86} + 52 q^{89} - 10 q^{91} + 2 q^{94}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.56155i 1.10418i 0.833783 + 0.552092i \(0.186170\pi\)
−0.833783 + 0.552092i \(0.813830\pi\)
\(3\) 0 0
\(4\) −0.438447 −0.219224
\(5\) 0 0
\(6\) 0 0
\(7\) − 1.00000i − 0.377964i −0.981981 0.188982i \(-0.939481\pi\)
0.981981 0.188982i \(-0.0605189\pi\)
\(8\) 2.43845i 0.862121i
\(9\) 0 0
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) − 4.56155i − 1.26515i −0.774500 0.632574i \(-0.781999\pi\)
0.774500 0.632574i \(-0.218001\pi\)
\(14\) 1.56155 0.417343
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) − 5.56155i − 1.34887i −0.738332 0.674437i \(-0.764386\pi\)
0.738332 0.674437i \(-0.235614\pi\)
\(18\) 0 0
\(19\) −3.00000 −0.688247 −0.344124 0.938924i \(-0.611824\pi\)
−0.344124 + 0.938924i \(0.611824\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 1.56155i − 0.332924i
\(23\) 2.43845i 0.508451i 0.967145 + 0.254226i \(0.0818206\pi\)
−0.967145 + 0.254226i \(0.918179\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 7.12311 1.39696
\(27\) 0 0
\(28\) 0.438447i 0.0828587i
\(29\) 3.12311 0.579946 0.289973 0.957035i \(-0.406354\pi\)
0.289973 + 0.957035i \(0.406354\pi\)
\(30\) 0 0
\(31\) −7.68466 −1.38021 −0.690103 0.723711i \(-0.742435\pi\)
−0.690103 + 0.723711i \(0.742435\pi\)
\(32\) − 2.43845i − 0.431061i
\(33\) 0 0
\(34\) 8.68466 1.48941
\(35\) 0 0
\(36\) 0 0
\(37\) − 9.80776i − 1.61239i −0.591652 0.806193i \(-0.701524\pi\)
0.591652 0.806193i \(-0.298476\pi\)
\(38\) − 4.68466i − 0.759952i
\(39\) 0 0
\(40\) 0 0
\(41\) 4.68466 0.731621 0.365810 0.930689i \(-0.380792\pi\)
0.365810 + 0.930689i \(0.380792\pi\)
\(42\) 0 0
\(43\) − 7.68466i − 1.17190i −0.810347 0.585950i \(-0.800722\pi\)
0.810347 0.585950i \(-0.199278\pi\)
\(44\) 0.438447 0.0660984
\(45\) 0 0
\(46\) −3.80776 −0.561424
\(47\) − 9.56155i − 1.39470i −0.716733 0.697348i \(-0.754363\pi\)
0.716733 0.697348i \(-0.245637\pi\)
\(48\) 0 0
\(49\) 6.00000 0.857143
\(50\) 0 0
\(51\) 0 0
\(52\) 2.00000i 0.277350i
\(53\) 7.12311i 0.978434i 0.872162 + 0.489217i \(0.162717\pi\)
−0.872162 + 0.489217i \(0.837283\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 2.43845 0.325851
\(57\) 0 0
\(58\) 4.87689i 0.640368i
\(59\) −6.43845 −0.838214 −0.419107 0.907937i \(-0.637657\pi\)
−0.419107 + 0.907937i \(0.637657\pi\)
\(60\) 0 0
\(61\) −3.43845 −0.440248 −0.220124 0.975472i \(-0.570646\pi\)
−0.220124 + 0.975472i \(0.570646\pi\)
\(62\) − 12.0000i − 1.52400i
\(63\) 0 0
\(64\) −5.56155 −0.695194
\(65\) 0 0
\(66\) 0 0
\(67\) − 5.68466i − 0.694492i −0.937774 0.347246i \(-0.887117\pi\)
0.937774 0.347246i \(-0.112883\pi\)
\(68\) 2.43845i 0.295705i
\(69\) 0 0
\(70\) 0 0
\(71\) −11.8078 −1.40132 −0.700662 0.713493i \(-0.747112\pi\)
−0.700662 + 0.713493i \(0.747112\pi\)
\(72\) 0 0
\(73\) − 0.246211i − 0.0288168i −0.999896 0.0144084i \(-0.995413\pi\)
0.999896 0.0144084i \(-0.00458650\pi\)
\(74\) 15.3153 1.78037
\(75\) 0 0
\(76\) 1.31534 0.150880
\(77\) 1.00000i 0.113961i
\(78\) 0 0
\(79\) 3.31534 0.373005 0.186503 0.982454i \(-0.440285\pi\)
0.186503 + 0.982454i \(0.440285\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 7.31534i 0.807844i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 12.0000 1.29399
\(87\) 0 0
\(88\) − 2.43845i − 0.259939i
\(89\) 8.87689 0.940949 0.470474 0.882414i \(-0.344083\pi\)
0.470474 + 0.882414i \(0.344083\pi\)
\(90\) 0 0
\(91\) −4.56155 −0.478181
\(92\) − 1.06913i − 0.111465i
\(93\) 0 0
\(94\) 14.9309 1.54000
\(95\) 0 0
\(96\) 0 0
\(97\) 6.12311i 0.621707i 0.950458 + 0.310854i \(0.100615\pi\)
−0.950458 + 0.310854i \(0.899385\pi\)
\(98\) 9.36932i 0.946444i
\(99\) 0 0
\(100\) 0 0
\(101\) 5.56155 0.553395 0.276698 0.960957i \(-0.410760\pi\)
0.276698 + 0.960957i \(0.410760\pi\)
\(102\) 0 0
\(103\) − 0.876894i − 0.0864030i −0.999066 0.0432015i \(-0.986244\pi\)
0.999066 0.0432015i \(-0.0137557\pi\)
\(104\) 11.1231 1.09071
\(105\) 0 0
\(106\) −11.1231 −1.08037
\(107\) 0.876894i 0.0847726i 0.999101 + 0.0423863i \(0.0134960\pi\)
−0.999101 + 0.0423863i \(0.986504\pi\)
\(108\) 0 0
\(109\) −12.8078 −1.22676 −0.613381 0.789787i \(-0.710191\pi\)
−0.613381 + 0.789787i \(0.710191\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 4.68466i 0.442659i
\(113\) − 0.876894i − 0.0824913i −0.999149 0.0412456i \(-0.986867\pi\)
0.999149 0.0412456i \(-0.0131326\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.36932 −0.127138
\(117\) 0 0
\(118\) − 10.0540i − 0.925543i
\(119\) −5.56155 −0.509827
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 5.36932i − 0.486115i
\(123\) 0 0
\(124\) 3.36932 0.302574
\(125\) 0 0
\(126\) 0 0
\(127\) − 15.8078i − 1.40271i −0.712811 0.701356i \(-0.752578\pi\)
0.712811 0.701356i \(-0.247422\pi\)
\(128\) − 13.5616i − 1.19868i
\(129\) 0 0
\(130\) 0 0
\(131\) 16.4924 1.44095 0.720475 0.693481i \(-0.243924\pi\)
0.720475 + 0.693481i \(0.243924\pi\)
\(132\) 0 0
\(133\) 3.00000i 0.260133i
\(134\) 8.87689 0.766847
\(135\) 0 0
\(136\) 13.5616 1.16289
\(137\) − 21.3693i − 1.82570i −0.408291 0.912852i \(-0.633875\pi\)
0.408291 0.912852i \(-0.366125\pi\)
\(138\) 0 0
\(139\) −8.00000 −0.678551 −0.339276 0.940687i \(-0.610182\pi\)
−0.339276 + 0.940687i \(0.610182\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) − 18.4384i − 1.54732i
\(143\) 4.56155i 0.381456i
\(144\) 0 0
\(145\) 0 0
\(146\) 0.384472 0.0318191
\(147\) 0 0
\(148\) 4.30019i 0.353473i
\(149\) 11.8078 0.967330 0.483665 0.875253i \(-0.339305\pi\)
0.483665 + 0.875253i \(0.339305\pi\)
\(150\) 0 0
\(151\) −4.31534 −0.351178 −0.175589 0.984464i \(-0.556183\pi\)
−0.175589 + 0.984464i \(0.556183\pi\)
\(152\) − 7.31534i − 0.593353i
\(153\) 0 0
\(154\) −1.56155 −0.125834
\(155\) 0 0
\(156\) 0 0
\(157\) 9.68466i 0.772920i 0.922306 + 0.386460i \(0.126302\pi\)
−0.922306 + 0.386460i \(0.873698\pi\)
\(158\) 5.17708i 0.411866i
\(159\) 0 0
\(160\) 0 0
\(161\) 2.43845 0.192177
\(162\) 0 0
\(163\) − 17.6847i − 1.38517i −0.721337 0.692585i \(-0.756472\pi\)
0.721337 0.692585i \(-0.243528\pi\)
\(164\) −2.05398 −0.160389
\(165\) 0 0
\(166\) 0 0
\(167\) − 9.36932i − 0.725020i −0.931980 0.362510i \(-0.881920\pi\)
0.931980 0.362510i \(-0.118080\pi\)
\(168\) 0 0
\(169\) −7.80776 −0.600597
\(170\) 0 0
\(171\) 0 0
\(172\) 3.36932i 0.256908i
\(173\) 17.5616i 1.33518i 0.744529 + 0.667590i \(0.232674\pi\)
−0.744529 + 0.667590i \(0.767326\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.68466 0.353119
\(177\) 0 0
\(178\) 13.8617i 1.03898i
\(179\) −10.4384 −0.780206 −0.390103 0.920771i \(-0.627561\pi\)
−0.390103 + 0.920771i \(0.627561\pi\)
\(180\) 0 0
\(181\) 20.1231 1.49574 0.747869 0.663846i \(-0.231077\pi\)
0.747869 + 0.663846i \(0.231077\pi\)
\(182\) − 7.12311i − 0.528000i
\(183\) 0 0
\(184\) −5.94602 −0.438347
\(185\) 0 0
\(186\) 0 0
\(187\) 5.56155i 0.406701i
\(188\) 4.19224i 0.305750i
\(189\) 0 0
\(190\) 0 0
\(191\) 26.9309 1.94865 0.974325 0.225148i \(-0.0722864\pi\)
0.974325 + 0.225148i \(0.0722864\pi\)
\(192\) 0 0
\(193\) 13.4384i 0.967321i 0.875256 + 0.483660i \(0.160693\pi\)
−0.875256 + 0.483660i \(0.839307\pi\)
\(194\) −9.56155 −0.686479
\(195\) 0 0
\(196\) −2.63068 −0.187906
\(197\) 10.9309i 0.778792i 0.921070 + 0.389396i \(0.127316\pi\)
−0.921070 + 0.389396i \(0.872684\pi\)
\(198\) 0 0
\(199\) −6.80776 −0.482590 −0.241295 0.970452i \(-0.577572\pi\)
−0.241295 + 0.970452i \(0.577572\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 8.68466i 0.611050i
\(203\) − 3.12311i − 0.219199i
\(204\) 0 0
\(205\) 0 0
\(206\) 1.36932 0.0954048
\(207\) 0 0
\(208\) 21.3693i 1.48170i
\(209\) 3.00000 0.207514
\(210\) 0 0
\(211\) 15.6847 1.07978 0.539888 0.841737i \(-0.318467\pi\)
0.539888 + 0.841737i \(0.318467\pi\)
\(212\) − 3.12311i − 0.214496i
\(213\) 0 0
\(214\) −1.36932 −0.0936046
\(215\) 0 0
\(216\) 0 0
\(217\) 7.68466i 0.521669i
\(218\) − 20.0000i − 1.35457i
\(219\) 0 0
\(220\) 0 0
\(221\) −25.3693 −1.70652
\(222\) 0 0
\(223\) 1.19224i 0.0798380i 0.999203 + 0.0399190i \(0.0127100\pi\)
−0.999203 + 0.0399190i \(0.987290\pi\)
\(224\) −2.43845 −0.162926
\(225\) 0 0
\(226\) 1.36932 0.0910856
\(227\) 12.4924i 0.829151i 0.910015 + 0.414576i \(0.136070\pi\)
−0.910015 + 0.414576i \(0.863930\pi\)
\(228\) 0 0
\(229\) −0.123106 −0.00813505 −0.00406752 0.999992i \(-0.501295\pi\)
−0.00406752 + 0.999992i \(0.501295\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 7.61553i 0.499984i
\(233\) 4.19224i 0.274643i 0.990527 + 0.137321i \(0.0438493\pi\)
−0.990527 + 0.137321i \(0.956151\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 2.82292 0.183756
\(237\) 0 0
\(238\) − 8.68466i − 0.562943i
\(239\) −16.4924 −1.06681 −0.533403 0.845861i \(-0.679087\pi\)
−0.533403 + 0.845861i \(0.679087\pi\)
\(240\) 0 0
\(241\) 23.0540 1.48504 0.742519 0.669826i \(-0.233631\pi\)
0.742519 + 0.669826i \(0.233631\pi\)
\(242\) 1.56155i 0.100380i
\(243\) 0 0
\(244\) 1.50758 0.0965128
\(245\) 0 0
\(246\) 0 0
\(247\) 13.6847i 0.870734i
\(248\) − 18.7386i − 1.18990i
\(249\) 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
\(253\) − 2.43845i − 0.153304i
\(254\) 24.6847 1.54885
\(255\) 0 0
\(256\) 10.0540 0.628373
\(257\) − 1.75379i − 0.109398i −0.998503 0.0546992i \(-0.982580\pi\)
0.998503 0.0546992i \(-0.0174200\pi\)
\(258\) 0 0
\(259\) −9.80776 −0.609425
\(260\) 0 0
\(261\) 0 0
\(262\) 25.7538i 1.59107i
\(263\) 21.3693i 1.31769i 0.752279 + 0.658844i \(0.228954\pi\)
−0.752279 + 0.658844i \(0.771046\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.68466 −0.287235
\(267\) 0 0
\(268\) 2.49242i 0.152249i
\(269\) 4.49242 0.273908 0.136954 0.990577i \(-0.456269\pi\)
0.136954 + 0.990577i \(0.456269\pi\)
\(270\) 0 0
\(271\) 17.1771 1.04343 0.521717 0.853119i \(-0.325292\pi\)
0.521717 + 0.853119i \(0.325292\pi\)
\(272\) 26.0540i 1.57975i
\(273\) 0 0
\(274\) 33.3693 2.01591
\(275\) 0 0
\(276\) 0 0
\(277\) − 22.1771i − 1.33249i −0.745732 0.666246i \(-0.767900\pi\)
0.745732 0.666246i \(-0.232100\pi\)
\(278\) − 12.4924i − 0.749246i
\(279\) 0 0
\(280\) 0 0
\(281\) −23.8078 −1.42025 −0.710126 0.704075i \(-0.751362\pi\)
−0.710126 + 0.704075i \(0.751362\pi\)
\(282\) 0 0
\(283\) 23.8769i 1.41933i 0.704537 + 0.709667i \(0.251155\pi\)
−0.704537 + 0.709667i \(0.748845\pi\)
\(284\) 5.17708 0.307203
\(285\) 0 0
\(286\) −7.12311 −0.421198
\(287\) − 4.68466i − 0.276527i
\(288\) 0 0
\(289\) −13.9309 −0.819463
\(290\) 0 0
\(291\) 0 0
\(292\) 0.107951i 0.00631733i
\(293\) 30.9309i 1.80700i 0.428587 + 0.903500i \(0.359011\pi\)
−0.428587 + 0.903500i \(0.640989\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 23.9157 1.39007
\(297\) 0 0
\(298\) 18.4384i 1.06811i
\(299\) 11.1231 0.643266
\(300\) 0 0
\(301\) −7.68466 −0.442936
\(302\) − 6.73863i − 0.387765i
\(303\) 0 0
\(304\) 14.0540 0.806051
\(305\) 0 0
\(306\) 0 0
\(307\) − 5.93087i − 0.338493i −0.985574 0.169246i \(-0.945867\pi\)
0.985574 0.169246i \(-0.0541333\pi\)
\(308\) − 0.438447i − 0.0249828i
\(309\) 0 0
\(310\) 0 0
\(311\) −30.2462 −1.71511 −0.857553 0.514396i \(-0.828016\pi\)
−0.857553 + 0.514396i \(0.828016\pi\)
\(312\) 0 0
\(313\) − 14.7538i − 0.833933i −0.908922 0.416967i \(-0.863093\pi\)
0.908922 0.416967i \(-0.136907\pi\)
\(314\) −15.1231 −0.853446
\(315\) 0 0
\(316\) −1.45360 −0.0817715
\(317\) − 15.1231i − 0.849398i −0.905335 0.424699i \(-0.860380\pi\)
0.905335 0.424699i \(-0.139620\pi\)
\(318\) 0 0
\(319\) −3.12311 −0.174860
\(320\) 0 0
\(321\) 0 0
\(322\) 3.80776i 0.212198i
\(323\) 16.6847i 0.928359i
\(324\) 0 0
\(325\) 0 0
\(326\) 27.6155 1.52948
\(327\) 0 0
\(328\) 11.4233i 0.630746i
\(329\) −9.56155 −0.527145
\(330\) 0 0
\(331\) 6.24621 0.343323 0.171661 0.985156i \(-0.445086\pi\)
0.171661 + 0.985156i \(0.445086\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 14.6307 0.800555
\(335\) 0 0
\(336\) 0 0
\(337\) − 1.68466i − 0.0917692i −0.998947 0.0458846i \(-0.985389\pi\)
0.998947 0.0458846i \(-0.0146107\pi\)
\(338\) − 12.1922i − 0.663170i
\(339\) 0 0
\(340\) 0 0
\(341\) 7.68466 0.416148
\(342\) 0 0
\(343\) − 13.0000i − 0.701934i
\(344\) 18.7386 1.01032
\(345\) 0 0
\(346\) −27.4233 −1.47429
\(347\) 2.63068i 0.141222i 0.997504 + 0.0706112i \(0.0224950\pi\)
−0.997504 + 0.0706112i \(0.977505\pi\)
\(348\) 0 0
\(349\) −31.3693 −1.67916 −0.839581 0.543235i \(-0.817199\pi\)
−0.839581 + 0.543235i \(0.817199\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.43845i 0.129970i
\(353\) − 29.8617i − 1.58938i −0.607015 0.794690i \(-0.707633\pi\)
0.607015 0.794690i \(-0.292367\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −3.89205 −0.206278
\(357\) 0 0
\(358\) − 16.3002i − 0.861492i
\(359\) −9.75379 −0.514785 −0.257393 0.966307i \(-0.582863\pi\)
−0.257393 + 0.966307i \(0.582863\pi\)
\(360\) 0 0
\(361\) −10.0000 −0.526316
\(362\) 31.4233i 1.65157i
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) 0 0
\(366\) 0 0
\(367\) 23.0540i 1.20341i 0.798719 + 0.601704i \(0.205511\pi\)
−0.798719 + 0.601704i \(0.794489\pi\)
\(368\) − 11.4233i − 0.595480i
\(369\) 0 0
\(370\) 0 0
\(371\) 7.12311 0.369813
\(372\) 0 0
\(373\) 6.80776i 0.352493i 0.984346 + 0.176246i \(0.0563955\pi\)
−0.984346 + 0.176246i \(0.943604\pi\)
\(374\) −8.68466 −0.449073
\(375\) 0 0
\(376\) 23.3153 1.20240
\(377\) − 14.2462i − 0.733717i
\(378\) 0 0
\(379\) −29.3002 −1.50505 −0.752525 0.658564i \(-0.771164\pi\)
−0.752525 + 0.658564i \(0.771164\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 42.0540i 2.15167i
\(383\) 5.75379i 0.294005i 0.989136 + 0.147002i \(0.0469625\pi\)
−0.989136 + 0.147002i \(0.953037\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −20.9848 −1.06810
\(387\) 0 0
\(388\) − 2.68466i − 0.136293i
\(389\) −33.8617 −1.71686 −0.858429 0.512932i \(-0.828559\pi\)
−0.858429 + 0.512932i \(0.828559\pi\)
\(390\) 0 0
\(391\) 13.5616 0.685837
\(392\) 14.6307i 0.738961i
\(393\) 0 0
\(394\) −17.0691 −0.859930
\(395\) 0 0
\(396\) 0 0
\(397\) − 32.5616i − 1.63422i −0.576484 0.817109i \(-0.695576\pi\)
0.576484 0.817109i \(-0.304424\pi\)
\(398\) − 10.6307i − 0.532868i
\(399\) 0 0
\(400\) 0 0
\(401\) −35.6155 −1.77855 −0.889277 0.457368i \(-0.848792\pi\)
−0.889277 + 0.457368i \(0.848792\pi\)
\(402\) 0 0
\(403\) 35.0540i 1.74616i
\(404\) −2.43845 −0.121317
\(405\) 0 0
\(406\) 4.87689 0.242036
\(407\) 9.80776i 0.486153i
\(408\) 0 0
\(409\) 29.0540 1.43663 0.718313 0.695720i \(-0.244914\pi\)
0.718313 + 0.695720i \(0.244914\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.384472i 0.0189416i
\(413\) 6.43845i 0.316815i
\(414\) 0 0
\(415\) 0 0
\(416\) −11.1231 −0.545355
\(417\) 0 0
\(418\) 4.68466i 0.229134i
\(419\) −29.5616 −1.44418 −0.722088 0.691801i \(-0.756818\pi\)
−0.722088 + 0.691801i \(0.756818\pi\)
\(420\) 0 0
\(421\) 20.0540 0.977371 0.488685 0.872460i \(-0.337477\pi\)
0.488685 + 0.872460i \(0.337477\pi\)
\(422\) 24.4924i 1.19227i
\(423\) 0 0
\(424\) −17.3693 −0.843529
\(425\) 0 0
\(426\) 0 0
\(427\) 3.43845i 0.166398i
\(428\) − 0.384472i − 0.0185841i
\(429\) 0 0
\(430\) 0 0
\(431\) −38.2462 −1.84226 −0.921128 0.389261i \(-0.872730\pi\)
−0.921128 + 0.389261i \(0.872730\pi\)
\(432\) 0 0
\(433\) − 11.9309i − 0.573361i −0.958026 0.286681i \(-0.907448\pi\)
0.958026 0.286681i \(-0.0925518\pi\)
\(434\) −12.0000 −0.576018
\(435\) 0 0
\(436\) 5.61553 0.268935
\(437\) − 7.31534i − 0.349940i
\(438\) 0 0
\(439\) 37.7386 1.80117 0.900583 0.434683i \(-0.143140\pi\)
0.900583 + 0.434683i \(0.143140\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 39.6155i − 1.88432i
\(443\) − 28.3002i − 1.34458i −0.740287 0.672291i \(-0.765310\pi\)
0.740287 0.672291i \(-0.234690\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1.86174 −0.0881559
\(447\) 0 0
\(448\) 5.56155i 0.262759i
\(449\) −4.49242 −0.212011 −0.106005 0.994366i \(-0.533806\pi\)
−0.106005 + 0.994366i \(0.533806\pi\)
\(450\) 0 0
\(451\) −4.68466 −0.220592
\(452\) 0.384472i 0.0180840i
\(453\) 0 0
\(454\) −19.5076 −0.915536
\(455\) 0 0
\(456\) 0 0
\(457\) 11.3693i 0.531834i 0.963996 + 0.265917i \(0.0856747\pi\)
−0.963996 + 0.265917i \(0.914325\pi\)
\(458\) − 0.192236i − 0.00898260i
\(459\) 0 0
\(460\) 0 0
\(461\) 41.8617 1.94970 0.974848 0.222872i \(-0.0715431\pi\)
0.974848 + 0.222872i \(0.0715431\pi\)
\(462\) 0 0
\(463\) − 18.2462i − 0.847973i −0.905668 0.423987i \(-0.860630\pi\)
0.905668 0.423987i \(-0.139370\pi\)
\(464\) −14.6307 −0.679212
\(465\) 0 0
\(466\) −6.54640 −0.303256
\(467\) 14.2462i 0.659236i 0.944114 + 0.329618i \(0.106920\pi\)
−0.944114 + 0.329618i \(0.893080\pi\)
\(468\) 0 0
\(469\) −5.68466 −0.262493
\(470\) 0 0
\(471\) 0 0
\(472\) − 15.6998i − 0.722642i
\(473\) 7.68466i 0.353341i
\(474\) 0 0
\(475\) 0 0
\(476\) 2.43845 0.111766
\(477\) 0 0
\(478\) − 25.7538i − 1.17795i
\(479\) 21.3693 0.976389 0.488195 0.872735i \(-0.337656\pi\)
0.488195 + 0.872735i \(0.337656\pi\)
\(480\) 0 0
\(481\) −44.7386 −2.03991
\(482\) 36.0000i 1.63976i
\(483\) 0 0
\(484\) −0.438447 −0.0199294
\(485\) 0 0
\(486\) 0 0
\(487\) − 25.0540i − 1.13530i −0.823269 0.567652i \(-0.807852\pi\)
0.823269 0.567652i \(-0.192148\pi\)
\(488\) − 8.38447i − 0.379547i
\(489\) 0 0
\(490\) 0 0
\(491\) −16.4924 −0.744293 −0.372146 0.928174i \(-0.621378\pi\)
−0.372146 + 0.928174i \(0.621378\pi\)
\(492\) 0 0
\(493\) − 17.3693i − 0.782275i
\(494\) −21.3693 −0.961451
\(495\) 0 0
\(496\) 36.0000 1.61645
\(497\) 11.8078i 0.529651i
\(498\) 0 0
\(499\) 16.8078 0.752419 0.376209 0.926535i \(-0.377227\pi\)
0.376209 + 0.926535i \(0.377227\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 18.7386i − 0.836346i
\(503\) 16.4924i 0.735361i 0.929952 + 0.367680i \(0.119848\pi\)
−0.929952 + 0.367680i \(0.880152\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 3.80776 0.169276
\(507\) 0 0
\(508\) 6.93087i 0.307508i
\(509\) 8.87689 0.393461 0.196731 0.980458i \(-0.436968\pi\)
0.196731 + 0.980458i \(0.436968\pi\)
\(510\) 0 0
\(511\) −0.246211 −0.0108917
\(512\) − 11.4233i − 0.504843i
\(513\) 0 0
\(514\) 2.73863 0.120796
\(515\) 0 0
\(516\) 0 0
\(517\) 9.56155i 0.420517i
\(518\) − 15.3153i − 0.672917i
\(519\) 0 0
\(520\) 0 0
\(521\) 15.6155 0.684129 0.342064 0.939677i \(-0.388874\pi\)
0.342064 + 0.939677i \(0.388874\pi\)
\(522\) 0 0
\(523\) − 7.24621i − 0.316855i −0.987371 0.158427i \(-0.949358\pi\)
0.987371 0.158427i \(-0.0506424\pi\)
\(524\) −7.23106 −0.315890
\(525\) 0 0
\(526\) −33.3693 −1.45497
\(527\) 42.7386i 1.86172i
\(528\) 0 0
\(529\) 17.0540 0.741477
\(530\) 0 0
\(531\) 0 0
\(532\) − 1.31534i − 0.0570273i
\(533\) − 21.3693i − 0.925608i
\(534\) 0 0
\(535\) 0 0
\(536\) 13.8617 0.598736
\(537\) 0 0
\(538\) 7.01515i 0.302445i
\(539\) −6.00000 −0.258438
\(540\) 0 0
\(541\) −19.6847 −0.846310 −0.423155 0.906057i \(-0.639077\pi\)
−0.423155 + 0.906057i \(0.639077\pi\)
\(542\) 26.8229i 1.15214i
\(543\) 0 0
\(544\) −13.5616 −0.581447
\(545\) 0 0
\(546\) 0 0
\(547\) 8.68466i 0.371329i 0.982613 + 0.185665i \(0.0594438\pi\)
−0.982613 + 0.185665i \(0.940556\pi\)
\(548\) 9.36932i 0.400237i
\(549\) 0 0
\(550\) 0 0
\(551\) −9.36932 −0.399146
\(552\) 0 0
\(553\) − 3.31534i − 0.140983i
\(554\) 34.6307 1.47132
\(555\) 0 0
\(556\) 3.50758 0.148754
\(557\) 7.12311i 0.301816i 0.988548 + 0.150908i \(0.0482197\pi\)
−0.988548 + 0.150908i \(0.951780\pi\)
\(558\) 0 0
\(559\) −35.0540 −1.48263
\(560\) 0 0
\(561\) 0 0
\(562\) − 37.1771i − 1.56822i
\(563\) − 19.1231i − 0.805943i −0.915213 0.402971i \(-0.867977\pi\)
0.915213 0.402971i \(-0.132023\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −37.2850 −1.56721
\(567\) 0 0
\(568\) − 28.7926i − 1.20811i
\(569\) −9.06913 −0.380198 −0.190099 0.981765i \(-0.560881\pi\)
−0.190099 + 0.981765i \(0.560881\pi\)
\(570\) 0 0
\(571\) 0.946025 0.0395899 0.0197950 0.999804i \(-0.493699\pi\)
0.0197950 + 0.999804i \(0.493699\pi\)
\(572\) − 2.00000i − 0.0836242i
\(573\) 0 0
\(574\) 7.31534 0.305336
\(575\) 0 0
\(576\) 0 0
\(577\) 3.63068i 0.151147i 0.997140 + 0.0755737i \(0.0240788\pi\)
−0.997140 + 0.0755737i \(0.975921\pi\)
\(578\) − 21.7538i − 0.904838i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 7.12311i − 0.295009i
\(584\) 0.600373 0.0248436
\(585\) 0 0
\(586\) −48.3002 −1.99526
\(587\) 47.8078i 1.97324i 0.163044 + 0.986619i \(0.447869\pi\)
−0.163044 + 0.986619i \(0.552131\pi\)
\(588\) 0 0
\(589\) 23.0540 0.949923
\(590\) 0 0
\(591\) 0 0
\(592\) 45.9460i 1.88837i
\(593\) 40.1080i 1.64704i 0.567290 + 0.823518i \(0.307992\pi\)
−0.567290 + 0.823518i \(0.692008\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −5.17708 −0.212061
\(597\) 0 0
\(598\) 17.3693i 0.710284i
\(599\) 21.1771 0.865272 0.432636 0.901569i \(-0.357584\pi\)
0.432636 + 0.901569i \(0.357584\pi\)
\(600\) 0 0
\(601\) −24.4233 −0.996247 −0.498123 0.867106i \(-0.665977\pi\)
−0.498123 + 0.867106i \(0.665977\pi\)
\(602\) − 12.0000i − 0.489083i
\(603\) 0 0
\(604\) 1.89205 0.0769864
\(605\) 0 0
\(606\) 0 0
\(607\) 10.2462i 0.415881i 0.978141 + 0.207940i \(0.0666760\pi\)
−0.978141 + 0.207940i \(0.933324\pi\)
\(608\) 7.31534i 0.296676i
\(609\) 0 0
\(610\) 0 0
\(611\) −43.6155 −1.76450
\(612\) 0 0
\(613\) − 46.1080i − 1.86228i −0.364659 0.931141i \(-0.618814\pi\)
0.364659 0.931141i \(-0.381186\pi\)
\(614\) 9.26137 0.373758
\(615\) 0 0
\(616\) −2.43845 −0.0982478
\(617\) − 38.2462i − 1.53973i −0.638204 0.769867i \(-0.720322\pi\)
0.638204 0.769867i \(-0.279678\pi\)
\(618\) 0 0
\(619\) 38.1771 1.53447 0.767233 0.641368i \(-0.221633\pi\)
0.767233 + 0.641368i \(0.221633\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 47.2311i − 1.89379i
\(623\) − 8.87689i − 0.355645i
\(624\) 0 0
\(625\) 0 0
\(626\) 23.0388 0.920816
\(627\) 0 0
\(628\) − 4.24621i − 0.169442i
\(629\) −54.5464 −2.17491
\(630\) 0 0
\(631\) −10.3153 −0.410647 −0.205324 0.978694i \(-0.565825\pi\)
−0.205324 + 0.978694i \(0.565825\pi\)
\(632\) 8.08429i 0.321576i
\(633\) 0 0
\(634\) 23.6155 0.937892
\(635\) 0 0
\(636\) 0 0
\(637\) − 27.3693i − 1.08441i
\(638\) − 4.87689i − 0.193078i
\(639\) 0 0
\(640\) 0 0
\(641\) −6.73863 −0.266160 −0.133080 0.991105i \(-0.542487\pi\)
−0.133080 + 0.991105i \(0.542487\pi\)
\(642\) 0 0
\(643\) − 9.36932i − 0.369490i −0.982787 0.184745i \(-0.940854\pi\)
0.982787 0.184745i \(-0.0591459\pi\)
\(644\) −1.06913 −0.0421296
\(645\) 0 0
\(646\) −26.0540 −1.02508
\(647\) 20.1922i 0.793839i 0.917853 + 0.396919i \(0.129921\pi\)
−0.917853 + 0.396919i \(0.870079\pi\)
\(648\) 0 0
\(649\) 6.43845 0.252731
\(650\) 0 0
\(651\) 0 0
\(652\) 7.75379i 0.303662i
\(653\) 40.4924i 1.58459i 0.610138 + 0.792295i \(0.291114\pi\)
−0.610138 + 0.792295i \(0.708886\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −21.9460 −0.856848
\(657\) 0 0
\(658\) − 14.9309i − 0.582066i
\(659\) −33.3693 −1.29988 −0.649942 0.759984i \(-0.725207\pi\)
−0.649942 + 0.759984i \(0.725207\pi\)
\(660\) 0 0
\(661\) −13.8078 −0.537060 −0.268530 0.963271i \(-0.586538\pi\)
−0.268530 + 0.963271i \(0.586538\pi\)
\(662\) 9.75379i 0.379092i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 7.61553i 0.294874i
\(668\) 4.10795i 0.158941i
\(669\) 0 0
\(670\) 0 0
\(671\) 3.43845 0.132740
\(672\) 0 0
\(673\) 35.3693i 1.36339i 0.731638 + 0.681693i \(0.238756\pi\)
−0.731638 + 0.681693i \(0.761244\pi\)
\(674\) 2.63068 0.101330
\(675\) 0 0
\(676\) 3.42329 0.131665
\(677\) − 20.8769i − 0.802364i −0.915998 0.401182i \(-0.868599\pi\)
0.915998 0.401182i \(-0.131401\pi\)
\(678\) 0 0
\(679\) 6.12311 0.234983
\(680\) 0 0
\(681\) 0 0
\(682\) 12.0000i 0.459504i
\(683\) − 0.192236i − 0.00735570i −0.999993 0.00367785i \(-0.998829\pi\)
0.999993 0.00367785i \(-0.00117070\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 20.3002 0.775065
\(687\) 0 0
\(688\) 36.0000i 1.37249i
\(689\) 32.4924 1.23786
\(690\) 0 0
\(691\) −20.4924 −0.779568 −0.389784 0.920906i \(-0.627450\pi\)
−0.389784 + 0.920906i \(0.627450\pi\)
\(692\) − 7.69981i − 0.292703i
\(693\) 0 0
\(694\) −4.10795 −0.155936
\(695\) 0 0
\(696\) 0 0
\(697\) − 26.0540i − 0.986865i
\(698\) − 48.9848i − 1.85410i
\(699\) 0 0
\(700\) 0 0
\(701\) 26.9309 1.01716 0.508582 0.861013i \(-0.330170\pi\)
0.508582 + 0.861013i \(0.330170\pi\)
\(702\) 0 0
\(703\) 29.4233i 1.10972i
\(704\) 5.56155 0.209609
\(705\) 0 0
\(706\) 46.6307 1.75497
\(707\) − 5.56155i − 0.209164i
\(708\) 0 0
\(709\) 2.12311 0.0797349 0.0398675 0.999205i \(-0.487306\pi\)
0.0398675 + 0.999205i \(0.487306\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 21.6458i 0.811212i
\(713\) − 18.7386i − 0.701767i
\(714\) 0 0
\(715\) 0 0
\(716\) 4.57671 0.171040
\(717\) 0 0
\(718\) − 15.2311i − 0.568418i
\(719\) −32.9848 −1.23013 −0.615064 0.788478i \(-0.710870\pi\)
−0.615064 + 0.788478i \(0.710870\pi\)
\(720\) 0 0
\(721\) −0.876894 −0.0326573
\(722\) − 15.6155i − 0.581150i
\(723\) 0 0
\(724\) −8.82292 −0.327901
\(725\) 0 0
\(726\) 0 0
\(727\) − 7.05398i − 0.261617i −0.991408 0.130809i \(-0.958243\pi\)
0.991408 0.130809i \(-0.0417574\pi\)
\(728\) − 11.1231i − 0.412250i
\(729\) 0 0
\(730\) 0 0
\(731\) −42.7386 −1.58075
\(732\) 0 0
\(733\) 2.00000i 0.0738717i 0.999318 + 0.0369358i \(0.0117597\pi\)
−0.999318 + 0.0369358i \(0.988240\pi\)
\(734\) −36.0000 −1.32878
\(735\) 0 0
\(736\) 5.94602 0.219173
\(737\) 5.68466i 0.209397i
\(738\) 0 0
\(739\) −0.684658 −0.0251856 −0.0125928 0.999921i \(-0.504009\pi\)
−0.0125928 + 0.999921i \(0.504009\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 11.1231i 0.408342i
\(743\) − 9.86174i − 0.361792i −0.983502 0.180896i \(-0.942100\pi\)
0.983502 0.180896i \(-0.0578998\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −10.6307 −0.389217
\(747\) 0 0
\(748\) − 2.43845i − 0.0891585i
\(749\) 0.876894 0.0320410
\(750\) 0 0
\(751\) −22.7386 −0.829745 −0.414872 0.909880i \(-0.636174\pi\)
−0.414872 + 0.909880i \(0.636174\pi\)
\(752\) 44.7926i 1.63342i
\(753\) 0 0
\(754\) 22.2462 0.810159
\(755\) 0 0
\(756\) 0 0
\(757\) 47.3002i 1.71915i 0.511006 + 0.859577i \(0.329273\pi\)
−0.511006 + 0.859577i \(0.670727\pi\)
\(758\) − 45.7538i − 1.66185i
\(759\) 0 0
\(760\) 0 0
\(761\) 45.3693 1.64464 0.822318 0.569028i \(-0.192680\pi\)
0.822318 + 0.569028i \(0.192680\pi\)
\(762\) 0 0
\(763\) 12.8078i 0.463672i
\(764\) −11.8078 −0.427190
\(765\) 0 0
\(766\) −8.98485 −0.324636
\(767\) 29.3693i 1.06046i
\(768\) 0 0
\(769\) 31.0540 1.11983 0.559917 0.828548i \(-0.310833\pi\)
0.559917 + 0.828548i \(0.310833\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 5.89205i − 0.212059i
\(773\) 15.5076i 0.557769i 0.960325 + 0.278884i \(0.0899646\pi\)
−0.960325 + 0.278884i \(0.910035\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −14.9309 −0.535987
\(777\) 0 0
\(778\) − 52.8769i − 1.89573i
\(779\) −14.0540 −0.503536
\(780\) 0 0
\(781\) 11.8078 0.422515
\(782\) 21.1771i 0.757291i
\(783\) 0 0
\(784\) −28.1080 −1.00386
\(785\) 0 0
\(786\) 0 0
\(787\) − 53.7386i − 1.91558i −0.287476 0.957788i \(-0.592816\pi\)
0.287476 0.957788i \(-0.407184\pi\)
\(788\) − 4.79261i − 0.170730i
\(789\) 0 0
\(790\) 0 0
\(791\) −0.876894 −0.0311788
\(792\) 0 0
\(793\) 15.6847i 0.556979i
\(794\) 50.8466 1.80448
\(795\) 0 0
\(796\) 2.98485 0.105795
\(797\) − 51.1231i − 1.81087i −0.424481 0.905437i \(-0.639544\pi\)
0.424481 0.905437i \(-0.360456\pi\)
\(798\) 0 0
\(799\) −53.1771 −1.88127
\(800\) 0 0
\(801\) 0 0
\(802\) − 55.6155i − 1.96385i
\(803\) 0.246211i 0.00868861i
\(804\) 0 0
\(805\) 0 0
\(806\) −54.7386 −1.92809
\(807\) 0 0
\(808\) 13.5616i 0.477094i
\(809\) −7.31534 −0.257194 −0.128597 0.991697i \(-0.541047\pi\)
−0.128597 + 0.991697i \(0.541047\pi\)
\(810\) 0 0
\(811\) −1.00000 −0.0351147 −0.0175574 0.999846i \(-0.505589\pi\)
−0.0175574 + 0.999846i \(0.505589\pi\)
\(812\) 1.36932i 0.0480536i
\(813\) 0 0
\(814\) −15.3153 −0.536802
\(815\) 0 0
\(816\) 0 0
\(817\) 23.0540i 0.806557i
\(818\) 45.3693i 1.58630i
\(819\) 0 0
\(820\) 0 0
\(821\) −16.8769 −0.589008 −0.294504 0.955650i \(-0.595154\pi\)
−0.294504 + 0.955650i \(0.595154\pi\)
\(822\) 0 0
\(823\) − 37.0540i − 1.29162i −0.763498 0.645810i \(-0.776520\pi\)
0.763498 0.645810i \(-0.223480\pi\)
\(824\) 2.13826 0.0744898
\(825\) 0 0
\(826\) −10.0540 −0.349823
\(827\) − 26.2462i − 0.912670i −0.889808 0.456335i \(-0.849162\pi\)
0.889808 0.456335i \(-0.150838\pi\)
\(828\) 0 0
\(829\) 0.738634 0.0256538 0.0128269 0.999918i \(-0.495917\pi\)
0.0128269 + 0.999918i \(0.495917\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 25.3693i 0.879523i
\(833\) − 33.3693i − 1.15618i
\(834\) 0 0
\(835\) 0 0
\(836\) −1.31534 −0.0454920
\(837\) 0 0
\(838\) − 46.1619i − 1.59464i
\(839\) 12.4924 0.431286 0.215643 0.976472i \(-0.430815\pi\)
0.215643 + 0.976472i \(0.430815\pi\)
\(840\) 0 0
\(841\) −19.2462 −0.663662
\(842\) 31.3153i 1.07920i
\(843\) 0 0
\(844\) −6.87689 −0.236712
\(845\) 0 0
\(846\) 0 0
\(847\) − 1.00000i − 0.0343604i
\(848\) − 33.3693i − 1.14591i
\(849\) 0 0
\(850\) 0 0
\(851\) 23.9157 0.819820
\(852\) 0 0
\(853\) 26.4233i 0.904716i 0.891836 + 0.452358i \(0.149417\pi\)
−0.891836 + 0.452358i \(0.850583\pi\)
\(854\) −5.36932 −0.183734
\(855\) 0 0
\(856\) −2.13826 −0.0730842
\(857\) − 9.56155i − 0.326616i −0.986575 0.163308i \(-0.947784\pi\)
0.986575 0.163308i \(-0.0522165\pi\)
\(858\) 0 0
\(859\) 34.3542 1.17215 0.586074 0.810257i \(-0.300673\pi\)
0.586074 + 0.810257i \(0.300673\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) − 59.7235i − 2.03419i
\(863\) 14.2462i 0.484947i 0.970158 + 0.242473i \(0.0779587\pi\)
−0.970158 + 0.242473i \(0.922041\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 18.6307 0.633096
\(867\) 0 0
\(868\) − 3.36932i − 0.114362i
\(869\) −3.31534 −0.112465
\(870\) 0 0
\(871\) −25.9309 −0.878634
\(872\) − 31.2311i − 1.05762i
\(873\) 0 0
\(874\) 11.4233 0.386399
\(875\) 0 0
\(876\) 0 0
\(877\) 23.0540i 0.778477i 0.921137 + 0.389239i \(0.127262\pi\)
−0.921137 + 0.389239i \(0.872738\pi\)
\(878\) 58.9309i 1.98882i
\(879\) 0 0
\(880\) 0 0
\(881\) −26.2462 −0.884257 −0.442129 0.896952i \(-0.645777\pi\)
−0.442129 + 0.896952i \(0.645777\pi\)
\(882\) 0 0
\(883\) − 16.0691i − 0.540769i −0.962752 0.270385i \(-0.912849\pi\)
0.962752 0.270385i \(-0.0871509\pi\)
\(884\) 11.1231 0.374111
\(885\) 0 0
\(886\) 44.1922 1.48467
\(887\) 29.8617i 1.00266i 0.865256 + 0.501330i \(0.167156\pi\)
−0.865256 + 0.501330i \(0.832844\pi\)
\(888\) 0 0
\(889\) −15.8078 −0.530175
\(890\) 0 0
\(891\) 0 0
\(892\) − 0.522732i − 0.0175024i
\(893\) 28.6847i 0.959895i
\(894\) 0 0
\(895\) 0 0
\(896\) −13.5616 −0.453060
\(897\) 0 0
\(898\) − 7.01515i − 0.234099i
\(899\) −24.0000 −0.800445
\(900\) 0 0
\(901\) 39.6155 1.31978
\(902\) − 7.31534i − 0.243574i
\(903\) 0 0
\(904\) 2.13826 0.0711175
\(905\) 0 0
\(906\) 0 0
\(907\) 28.1080i 0.933309i 0.884440 + 0.466655i \(0.154541\pi\)
−0.884440 + 0.466655i \(0.845459\pi\)
\(908\) − 5.47727i − 0.181770i
\(909\) 0 0
\(910\) 0 0
\(911\) 54.1619 1.79446 0.897232 0.441559i \(-0.145574\pi\)
0.897232 + 0.441559i \(0.145574\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −17.7538 −0.587243
\(915\) 0 0
\(916\) 0.0539753 0.00178339
\(917\) − 16.4924i − 0.544628i
\(918\) 0 0
\(919\) 33.1080 1.09213 0.546065 0.837743i \(-0.316125\pi\)
0.546065 + 0.837743i \(0.316125\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 65.3693i 2.15282i
\(923\) 53.8617i 1.77288i
\(924\) 0 0
\(925\) 0 0
\(926\) 28.4924 0.936319
\(927\) 0 0
\(928\) − 7.61553i − 0.249992i
\(929\) 59.2311 1.94331 0.971654 0.236408i \(-0.0759701\pi\)
0.971654 + 0.236408i \(0.0759701\pi\)
\(930\) 0 0
\(931\) −18.0000 −0.589926
\(932\) − 1.83807i − 0.0602081i
\(933\) 0 0
\(934\) −22.2462 −0.727918
\(935\) 0 0
\(936\) 0 0
\(937\) − 22.3153i − 0.729010i −0.931201 0.364505i \(-0.881238\pi\)
0.931201 0.364505i \(-0.118762\pi\)
\(938\) − 8.87689i − 0.289841i
\(939\) 0 0
\(940\) 0 0
\(941\) 14.8229 0.483213 0.241607 0.970374i \(-0.422326\pi\)
0.241607 + 0.970374i \(0.422326\pi\)
\(942\) 0 0
\(943\) 11.4233i 0.371994i
\(944\) 30.1619 0.981687
\(945\) 0 0
\(946\) −12.0000 −0.390154
\(947\) − 13.1771i − 0.428198i −0.976812 0.214099i \(-0.931319\pi\)
0.976812 0.214099i \(-0.0686815\pi\)
\(948\) 0 0
\(949\) −1.12311 −0.0364576
\(950\) 0 0
\(951\) 0 0
\(952\) − 13.5616i − 0.439532i
\(953\) 15.8078i 0.512064i 0.966668 + 0.256032i \(0.0824152\pi\)
−0.966668 + 0.256032i \(0.917585\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 7.23106 0.233869
\(957\) 0 0
\(958\) 33.3693i 1.07811i
\(959\) −21.3693 −0.690051
\(960\) 0 0
\(961\) 28.0540 0.904967
\(962\) − 69.8617i − 2.25243i
\(963\) 0 0
\(964\) −10.1080 −0.325555
\(965\) 0 0
\(966\) 0 0
\(967\) − 44.0000i − 1.41494i −0.706741 0.707472i \(-0.749835\pi\)
0.706741 0.707472i \(-0.250165\pi\)
\(968\) 2.43845i 0.0783747i
\(969\) 0 0
\(970\) 0 0
\(971\) −37.1771 −1.19307 −0.596535 0.802587i \(-0.703456\pi\)
−0.596535 + 0.802587i \(0.703456\pi\)
\(972\) 0 0
\(973\) 8.00000i 0.256468i
\(974\) 39.1231 1.25359
\(975\) 0 0
\(976\) 16.1080 0.515603
\(977\) − 15.6155i − 0.499585i −0.968299 0.249793i \(-0.919638\pi\)
0.968299 0.249793i \(-0.0803624\pi\)
\(978\) 0 0
\(979\) −8.87689 −0.283707
\(980\) 0 0
\(981\) 0 0
\(982\) − 25.7538i − 0.821836i
\(983\) − 6.93087i − 0.221060i −0.993873 0.110530i \(-0.964745\pi\)
0.993873 0.110530i \(-0.0352549\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 27.1231 0.863776
\(987\) 0 0
\(988\) − 6.00000i − 0.190885i
\(989\) 18.7386 0.595854
\(990\) 0 0
\(991\) 49.6847 1.57829 0.789143 0.614210i \(-0.210525\pi\)
0.789143 + 0.614210i \(0.210525\pi\)
\(992\) 18.7386i 0.594952i
\(993\) 0 0
\(994\) −18.4384 −0.584832
\(995\) 0 0
\(996\) 0 0
\(997\) 28.7386i 0.910162i 0.890450 + 0.455081i \(0.150390\pi\)
−0.890450 + 0.455081i \(0.849610\pi\)
\(998\) 26.2462i 0.830809i
\(999\) 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.i.199.3 4
3.2 odd 2 2475.2.c.j.199.2 4
5.2 odd 4 2475.2.a.v.1.1 yes 2
5.3 odd 4 2475.2.a.p.1.2 2
5.4 even 2 inner 2475.2.c.i.199.2 4
15.2 even 4 2475.2.a.q.1.2 yes 2
15.8 even 4 2475.2.a.u.1.1 yes 2
15.14 odd 2 2475.2.c.j.199.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2475.2.a.p.1.2 2 5.3 odd 4
2475.2.a.q.1.2 yes 2 15.2 even 4
2475.2.a.u.1.1 yes 2 15.8 even 4
2475.2.a.v.1.1 yes 2 5.2 odd 4
2475.2.c.i.199.2 4 5.4 even 2 inner
2475.2.c.i.199.3 4 1.1 even 1 trivial
2475.2.c.j.199.2 4 3.2 odd 2
2475.2.c.j.199.3 4 15.14 odd 2