Properties

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

$q$-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 \(-0.813830\pi\)
0.833783 0.552092i \(-0.186170\pi\)
\(3\) 0 0
\(4\) −0.438447 −0.219224
\(5\) 0 0
\(6\) 0 0
\(7\) 1.00000i 0.377964i 0.981981 + 0.188982i \(0.0605189\pi\)
−0.981981 + 0.188982i \(0.939481\pi\)
\(8\) − 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 \(0.218001\pi\)
−0.774500 + 0.632574i \(0.781999\pi\)
\(14\) 1.56155 0.417343
\(15\) 0 0
\(16\) −4.68466 −1.17116
\(17\) 5.56155i 1.34887i 0.738332 + 0.674437i \(0.235614\pi\)
−0.738332 + 0.674437i \(0.764386\pi\)
\(18\) 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 \(-0.918179\pi\)
0.967145 0.254226i \(-0.0818206\pi\)
\(24\) 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 \(0.298476\pi\)
−0.591652 + 0.806193i \(0.701524\pi\)
\(38\) 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 \(0.199278\pi\)
−0.810347 + 0.585950i \(0.800722\pi\)
\(44\) 0.438447 0.0660984
\(45\) 0 0
\(46\) −3.80776 −0.561424
\(47\) 9.56155i 1.39470i 0.716733 + 0.697348i \(0.245637\pi\)
−0.716733 + 0.697348i \(0.754363\pi\)
\(48\) 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 \(-0.837283\pi\)
0.872162 0.489217i \(-0.162717\pi\)
\(54\) 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 \(0.112883\pi\)
−0.937774 + 0.347246i \(0.887117\pi\)
\(68\) − 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 \(0.00458650\pi\)
−0.999896 + 0.0144084i \(0.995413\pi\)
\(74\) 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 \(-0.899385\pi\)
0.950458 0.310854i \(-0.100615\pi\)
\(98\) − 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 \(0.0137557\pi\)
−0.999066 + 0.0432015i \(0.986244\pi\)
\(104\) 11.1231 1.09071
\(105\) 0 0
\(106\) −11.1231 −1.08037
\(107\) − 0.876894i − 0.0847726i −0.999101 0.0423863i \(-0.986504\pi\)
0.999101 0.0423863i \(-0.0134960\pi\)
\(108\) 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 \(0.0131326\pi\)
−0.999149 + 0.0412456i \(0.986867\pi\)
\(114\) 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 \(0.247422\pi\)
−0.712811 + 0.701356i \(0.752578\pi\)
\(128\) 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 \(0.366125\pi\)
−0.408291 + 0.912852i \(0.633875\pi\)
\(138\) 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 \(-0.873698\pi\)
0.922306 0.386460i \(-0.126302\pi\)
\(158\) − 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 \(0.243528\pi\)
−0.721337 + 0.692585i \(0.756472\pi\)
\(164\) −2.05398 −0.160389
\(165\) 0 0
\(166\) 0 0
\(167\) 9.36932i 0.725020i 0.931980 + 0.362510i \(0.118080\pi\)
−0.931980 + 0.362510i \(0.881920\pi\)
\(168\) 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 \(-0.767326\pi\)
0.744529 0.667590i \(-0.232674\pi\)
\(174\) 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 \(-0.839307\pi\)
0.875256 0.483660i \(-0.160693\pi\)
\(194\) −9.56155 −0.686479
\(195\) 0 0
\(196\) −2.63068 −0.187906
\(197\) − 10.9309i − 0.778792i −0.921070 0.389396i \(-0.872684\pi\)
0.921070 0.389396i \(-0.127316\pi\)
\(198\) 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 \(-0.987290\pi\)
0.999203 0.0399190i \(-0.0127100\pi\)
\(224\) −2.43845 −0.162926
\(225\) 0 0
\(226\) 1.36932 0.0910856
\(227\) − 12.4924i − 0.829151i −0.910015 0.414576i \(-0.863930\pi\)
0.910015 0.414576i \(-0.136070\pi\)
\(228\) 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 \(-0.956151\pi\)
0.990527 0.137321i \(-0.0438493\pi\)
\(234\) 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 \(0.0174200\pi\)
−0.998503 + 0.0546992i \(0.982580\pi\)
\(258\) 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 \(-0.771046\pi\)
0.752279 0.658844i \(-0.228954\pi\)
\(264\) 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 \(0.232100\pi\)
−0.745732 + 0.666246i \(0.767900\pi\)
\(278\) 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 \(-0.748845\pi\)
0.704537 0.709667i \(-0.251155\pi\)
\(284\) 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 \(-0.640989\pi\)
0.428587 0.903500i \(-0.359011\pi\)
\(294\) 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 \(0.0541333\pi\)
−0.985574 + 0.169246i \(0.945867\pi\)
\(308\) 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 \(0.136907\pi\)
−0.908922 + 0.416967i \(0.863093\pi\)
\(314\) −15.1231 −0.853446
\(315\) 0 0
\(316\) −1.45360 −0.0817715
\(317\) 15.1231i 0.849398i 0.905335 + 0.424699i \(0.139620\pi\)
−0.905335 + 0.424699i \(0.860380\pi\)
\(318\) 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 \(0.0146107\pi\)
−0.998947 + 0.0458846i \(0.985389\pi\)
\(338\) 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 \(-0.977505\pi\)
0.997504 0.0706112i \(-0.0224950\pi\)
\(348\) 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 \(0.292367\pi\)
−0.607015 + 0.794690i \(0.707633\pi\)
\(354\) 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 \(-0.794489\pi\)
0.798719 0.601704i \(-0.205511\pi\)
\(368\) 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 \(-0.943604\pi\)
0.984346 0.176246i \(-0.0563955\pi\)
\(374\) −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 \(-0.953037\pi\)
0.989136 0.147002i \(-0.0469625\pi\)
\(384\) 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 \(0.304424\pi\)
−0.576484 + 0.817109i \(0.695576\pi\)
\(398\) 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 \(0.0925518\pi\)
−0.958026 + 0.286681i \(0.907448\pi\)
\(434\) −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 \(0.234690\pi\)
−0.740287 + 0.672291i \(0.765310\pi\)
\(444\) 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 \(-0.914325\pi\)
0.963996 0.265917i \(-0.0856747\pi\)
\(458\) 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 \(0.139370\pi\)
−0.905668 + 0.423987i \(0.860630\pi\)
\(464\) −14.6307 −0.679212
\(465\) 0 0
\(466\) −6.54640 −0.303256
\(467\) − 14.2462i − 0.659236i −0.944114 0.329618i \(-0.893080\pi\)
0.944114 0.329618i \(-0.106920\pi\)
\(468\) 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 \(0.192148\pi\)
−0.823269 + 0.567652i \(0.807852\pi\)
\(488\) 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 \(-0.880152\pi\)
0.929952 0.367680i \(-0.119848\pi\)
\(504\) 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 \(0.0506424\pi\)
−0.987371 + 0.158427i \(0.949358\pi\)
\(524\) −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 \(-0.940556\pi\)
0.982613 0.185665i \(-0.0594438\pi\)
\(548\) − 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 \(-0.951780\pi\)
0.988548 0.150908i \(-0.0482197\pi\)
\(558\) 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 \(0.132023\pi\)
−0.915213 + 0.402971i \(0.867977\pi\)
\(564\) 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 \(-0.975921\pi\)
0.997140 0.0755737i \(-0.0240788\pi\)
\(578\) 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 \(-0.552131\pi\)
0.163044 0.986619i \(-0.447869\pi\)
\(588\) 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 \(-0.692008\pi\)
0.567290 0.823518i \(-0.307992\pi\)
\(594\) 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 \(-0.933324\pi\)
0.978141 0.207940i \(-0.0666760\pi\)
\(608\) − 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 \(0.381186\pi\)
−0.364659 + 0.931141i \(0.618814\pi\)
\(614\) 9.26137 0.373758
\(615\) 0 0
\(616\) −2.43845 −0.0982478
\(617\) 38.2462i 1.53973i 0.638204 + 0.769867i \(0.279678\pi\)
−0.638204 + 0.769867i \(0.720322\pi\)
\(618\) 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 \(0.0591459\pi\)
−0.982787 + 0.184745i \(0.940854\pi\)
\(644\) −1.06913 −0.0421296
\(645\) 0 0
\(646\) −26.0540 −1.02508
\(647\) − 20.1922i − 0.793839i −0.917853 0.396919i \(-0.870079\pi\)
0.917853 0.396919i \(-0.129921\pi\)
\(648\) 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 \(-0.708886\pi\)
0.610138 0.792295i \(-0.291114\pi\)
\(654\) 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 \(-0.761244\pi\)
0.731638 0.681693i \(-0.238756\pi\)
\(674\) 2.63068 0.101330
\(675\) 0 0
\(676\) 3.42329 0.131665
\(677\) 20.8769i 0.802364i 0.915998 + 0.401182i \(0.131401\pi\)
−0.915998 + 0.401182i \(0.868599\pi\)
\(678\) 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 \(0.00117070\pi\)
−0.999993 + 0.00367785i \(0.998829\pi\)
\(684\) 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 \(0.0417574\pi\)
−0.991408 + 0.130809i \(0.958243\pi\)
\(728\) 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 \(-0.988240\pi\)
0.999318 0.0369358i \(-0.0117597\pi\)
\(734\) −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 \(0.0578998\pi\)
−0.983502 + 0.180896i \(0.942100\pi\)
\(744\) 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 \(-0.670727\pi\)
0.511006 0.859577i \(-0.329273\pi\)
\(758\) 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 \(-0.910035\pi\)
0.960325 0.278884i \(-0.0899646\pi\)
\(774\) 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 \(0.407184\pi\)
−0.287476 + 0.957788i \(0.592816\pi\)
\(788\) 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 \(0.360456\pi\)
−0.424481 + 0.905437i \(0.639544\pi\)
\(798\) 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 \(0.223480\pi\)
−0.763498 + 0.645810i \(0.776520\pi\)
\(824\) 2.13826 0.0744898
\(825\) 0 0
\(826\) −10.0540 −0.349823
\(827\) 26.2462i 0.912670i 0.889808 + 0.456335i \(0.150838\pi\)
−0.889808 + 0.456335i \(0.849162\pi\)
\(828\) 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 \(-0.850583\pi\)
0.891836 0.452358i \(-0.149417\pi\)
\(854\) −5.36932 −0.183734
\(855\) 0 0
\(856\) −2.13826 −0.0730842
\(857\) 9.56155i 0.326616i 0.986575 + 0.163308i \(0.0522165\pi\)
−0.986575 + 0.163308i \(0.947784\pi\)
\(858\) 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 \(-0.922041\pi\)
0.970158 0.242473i \(-0.0779587\pi\)
\(864\) 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 \(-0.872738\pi\)
0.921137 0.389239i \(-0.127262\pi\)
\(878\) − 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 \(0.0871509\pi\)
−0.962752 + 0.270385i \(0.912849\pi\)
\(884\) 11.1231 0.374111
\(885\) 0 0
\(886\) 44.1922 1.48467
\(887\) − 29.8617i − 1.00266i −0.865256 0.501330i \(-0.832844\pi\)
0.865256 0.501330i \(-0.167156\pi\)
\(888\) 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 \(-0.845459\pi\)
0.884440 0.466655i \(-0.154541\pi\)
\(908\) 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 \(0.118762\pi\)
−0.931201 + 0.364505i \(0.881238\pi\)
\(938\) 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 \(0.0686815\pi\)
−0.976812 + 0.214099i \(0.931319\pi\)
\(948\) 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 \(-0.917585\pi\)
0.966668 0.256032i \(-0.0824152\pi\)
\(954\) 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 \(0.250165\pi\)
−0.706741 + 0.707472i \(0.749835\pi\)
\(968\) − 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 \(0.0803624\pi\)
−0.968299 + 0.249793i \(0.919638\pi\)
\(978\) 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 \(0.0352549\pi\)
−0.993873 + 0.110530i \(0.964745\pi\)
\(984\) 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 \(-0.849610\pi\)
0.890450 0.455081i \(-0.150390\pi\)
\(998\) − 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.199.2 4
3.2 odd 2 2475.2.c.j.199.3 4
5.2 odd 4 2475.2.a.p.1.2 2
5.3 odd 4 2475.2.a.v.1.1 yes 2
5.4 even 2 inner 2475.2.c.i.199.3 4
15.2 even 4 2475.2.a.u.1.1 yes 2
15.8 even 4 2475.2.a.q.1.2 yes 2
15.14 odd 2 2475.2.c.j.199.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2475.2.a.p.1.2 2 5.2 odd 4
2475.2.a.q.1.2 yes 2 15.8 even 4
2475.2.a.u.1.1 yes 2 15.2 even 4
2475.2.a.v.1.1 yes 2 5.3 odd 4
2475.2.c.i.199.2 4 1.1 even 1 trivial
2475.2.c.i.199.3 4 5.4 even 2 inner
2475.2.c.j.199.2 4 15.14 odd 2
2475.2.c.j.199.3 4 3.2 odd 2