Properties

Label 2475.2.c.n.199.1
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: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(0.866025 - 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 2475.199
Dual form 2475.2.c.n.199.4

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.73205i q^{8} +O(q^{10})\) \(q-1.73205i q^{2} -1.00000 q^{4} -2.00000i q^{7} -1.73205i q^{8} +1.00000 q^{11} +5.46410i q^{13} -3.46410 q^{14} -5.00000 q^{16} -5.46410 q^{19} -1.73205i q^{22} -6.92820i q^{23} +9.46410 q^{26} +2.00000i q^{28} -3.46410 q^{29} -10.9282 q^{31} +5.19615i q^{32} +4.92820i q^{37} +9.46410i q^{38} -3.46410 q^{41} -4.92820i q^{43} -1.00000 q^{44} -12.0000 q^{46} -6.92820i q^{47} +3.00000 q^{49} -5.46410i q^{52} -0.928203i q^{53} -3.46410 q^{56} +6.00000i q^{58} -6.92820 q^{59} +2.00000 q^{61} +18.9282i q^{62} -1.00000 q^{64} -8.00000i q^{67} -13.8564 q^{71} -8.39230i q^{73} +8.53590 q^{74} +5.46410 q^{76} -2.00000i q^{77} +6.53590 q^{79} +6.00000i q^{82} -8.53590i q^{83} -8.53590 q^{86} -1.73205i q^{88} +0.928203 q^{89} +10.9282 q^{91} +6.92820i q^{92} -12.0000 q^{94} +10.0000i q^{97} -5.19615i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 4q^{4} + O(q^{10}) \) \( 4q - 4q^{4} + 4q^{11} - 20q^{16} - 8q^{19} + 24q^{26} - 16q^{31} - 4q^{44} - 48q^{46} + 12q^{49} + 8q^{61} - 4q^{64} + 48q^{74} + 8q^{76} + 40q^{79} - 48q^{86} - 24q^{89} + 16q^{91} - 48q^{94} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 1.73205i − 1.22474i −0.790569 0.612372i \(-0.790215\pi\)
0.790569 0.612372i \(-0.209785\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 0 0
\(6\) 0 0
\(7\) − 2.00000i − 0.755929i −0.925820 0.377964i \(-0.876624\pi\)
0.925820 0.377964i \(-0.123376\pi\)
\(8\) − 1.73205i − 0.612372i
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) 5.46410i 1.51547i 0.652563 + 0.757735i \(0.273694\pi\)
−0.652563 + 0.757735i \(0.726306\pi\)
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −5.46410 −1.25355 −0.626775 0.779200i \(-0.715626\pi\)
−0.626775 + 0.779200i \(0.715626\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 1.73205i − 0.369274i
\(23\) − 6.92820i − 1.44463i −0.691564 0.722315i \(-0.743078\pi\)
0.691564 0.722315i \(-0.256922\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 9.46410 1.85606
\(27\) 0 0
\(28\) 2.00000i 0.377964i
\(29\) −3.46410 −0.643268 −0.321634 0.946864i \(-0.604232\pi\)
−0.321634 + 0.946864i \(0.604232\pi\)
\(30\) 0 0
\(31\) −10.9282 −1.96276 −0.981382 0.192068i \(-0.938481\pi\)
−0.981382 + 0.192068i \(0.938481\pi\)
\(32\) 5.19615i 0.918559i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 4.92820i 0.810192i 0.914274 + 0.405096i \(0.132762\pi\)
−0.914274 + 0.405096i \(0.867238\pi\)
\(38\) 9.46410i 1.53528i
\(39\) 0 0
\(40\) 0 0
\(41\) −3.46410 −0.541002 −0.270501 0.962720i \(-0.587189\pi\)
−0.270501 + 0.962720i \(0.587189\pi\)
\(42\) 0 0
\(43\) − 4.92820i − 0.751544i −0.926712 0.375772i \(-0.877378\pi\)
0.926712 0.375772i \(-0.122622\pi\)
\(44\) −1.00000 −0.150756
\(45\) 0 0
\(46\) −12.0000 −1.76930
\(47\) − 6.92820i − 1.01058i −0.862949 0.505291i \(-0.831385\pi\)
0.862949 0.505291i \(-0.168615\pi\)
\(48\) 0 0
\(49\) 3.00000 0.428571
\(50\) 0 0
\(51\) 0 0
\(52\) − 5.46410i − 0.757735i
\(53\) − 0.928203i − 0.127499i −0.997966 0.0637493i \(-0.979694\pi\)
0.997966 0.0637493i \(-0.0203058\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −3.46410 −0.462910
\(57\) 0 0
\(58\) 6.00000i 0.787839i
\(59\) −6.92820 −0.901975 −0.450988 0.892530i \(-0.648928\pi\)
−0.450988 + 0.892530i \(0.648928\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 18.9282i 2.40388i
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 0 0
\(66\) 0 0
\(67\) − 8.00000i − 0.977356i −0.872464 0.488678i \(-0.837479\pi\)
0.872464 0.488678i \(-0.162521\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −13.8564 −1.64445 −0.822226 0.569160i \(-0.807268\pi\)
−0.822226 + 0.569160i \(0.807268\pi\)
\(72\) 0 0
\(73\) − 8.39230i − 0.982245i −0.871091 0.491122i \(-0.836587\pi\)
0.871091 0.491122i \(-0.163413\pi\)
\(74\) 8.53590 0.992278
\(75\) 0 0
\(76\) 5.46410 0.626775
\(77\) − 2.00000i − 0.227921i
\(78\) 0 0
\(79\) 6.53590 0.735346 0.367673 0.929955i \(-0.380155\pi\)
0.367673 + 0.929955i \(0.380155\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 6.00000i 0.662589i
\(83\) − 8.53590i − 0.936937i −0.883480 0.468468i \(-0.844806\pi\)
0.883480 0.468468i \(-0.155194\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −8.53590 −0.920450
\(87\) 0 0
\(88\) − 1.73205i − 0.184637i
\(89\) 0.928203 0.0983893 0.0491947 0.998789i \(-0.484335\pi\)
0.0491947 + 0.998789i \(0.484335\pi\)
\(90\) 0 0
\(91\) 10.9282 1.14559
\(92\) 6.92820i 0.722315i
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) 10.0000i 1.01535i 0.861550 + 0.507673i \(0.169494\pi\)
−0.861550 + 0.507673i \(0.830506\pi\)
\(98\) − 5.19615i − 0.524891i
\(99\) 0 0
\(100\) 0 0
\(101\) 10.3923 1.03407 0.517036 0.855963i \(-0.327035\pi\)
0.517036 + 0.855963i \(0.327035\pi\)
\(102\) 0 0
\(103\) 8.00000i 0.788263i 0.919054 + 0.394132i \(0.128955\pi\)
−0.919054 + 0.394132i \(0.871045\pi\)
\(104\) 9.46410 0.928032
\(105\) 0 0
\(106\) −1.60770 −0.156153
\(107\) 8.53590i 0.825196i 0.910913 + 0.412598i \(0.135379\pi\)
−0.910913 + 0.412598i \(0.864621\pi\)
\(108\) 0 0
\(109\) 10.0000 0.957826 0.478913 0.877862i \(-0.341031\pi\)
0.478913 + 0.877862i \(0.341031\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 10.0000i 0.944911i
\(113\) 12.9282i 1.21618i 0.793867 + 0.608092i \(0.208065\pi\)
−0.793867 + 0.608092i \(0.791935\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 3.46410 0.321634
\(117\) 0 0
\(118\) 12.0000i 1.10469i
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) − 3.46410i − 0.313625i
\(123\) 0 0
\(124\) 10.9282 0.981382
\(125\) 0 0
\(126\) 0 0
\(127\) − 8.92820i − 0.792250i −0.918197 0.396125i \(-0.870355\pi\)
0.918197 0.396125i \(-0.129645\pi\)
\(128\) 12.1244i 1.07165i
\(129\) 0 0
\(130\) 0 0
\(131\) −18.9282 −1.65376 −0.826882 0.562375i \(-0.809888\pi\)
−0.826882 + 0.562375i \(0.809888\pi\)
\(132\) 0 0
\(133\) 10.9282i 0.947595i
\(134\) −13.8564 −1.19701
\(135\) 0 0
\(136\) 0 0
\(137\) − 18.0000i − 1.53784i −0.639343 0.768922i \(-0.720793\pi\)
0.639343 0.768922i \(-0.279207\pi\)
\(138\) 0 0
\(139\) −12.3923 −1.05110 −0.525551 0.850762i \(-0.676141\pi\)
−0.525551 + 0.850762i \(0.676141\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 24.0000i 2.01404i
\(143\) 5.46410i 0.456931i
\(144\) 0 0
\(145\) 0 0
\(146\) −14.5359 −1.20300
\(147\) 0 0
\(148\) − 4.92820i − 0.405096i
\(149\) −15.4641 −1.26687 −0.633434 0.773796i \(-0.718355\pi\)
−0.633434 + 0.773796i \(0.718355\pi\)
\(150\) 0 0
\(151\) −20.3923 −1.65950 −0.829751 0.558134i \(-0.811518\pi\)
−0.829751 + 0.558134i \(0.811518\pi\)
\(152\) 9.46410i 0.767640i
\(153\) 0 0
\(154\) −3.46410 −0.279145
\(155\) 0 0
\(156\) 0 0
\(157\) 3.07180i 0.245156i 0.992459 + 0.122578i \(0.0391162\pi\)
−0.992459 + 0.122578i \(0.960884\pi\)
\(158\) − 11.3205i − 0.900611i
\(159\) 0 0
\(160\) 0 0
\(161\) −13.8564 −1.09204
\(162\) 0 0
\(163\) 9.85641i 0.772013i 0.922496 + 0.386007i \(0.126146\pi\)
−0.922496 + 0.386007i \(0.873854\pi\)
\(164\) 3.46410 0.270501
\(165\) 0 0
\(166\) −14.7846 −1.14751
\(167\) − 10.3923i − 0.804181i −0.915600 0.402090i \(-0.868284\pi\)
0.915600 0.402090i \(-0.131716\pi\)
\(168\) 0 0
\(169\) −16.8564 −1.29665
\(170\) 0 0
\(171\) 0 0
\(172\) 4.92820i 0.375772i
\(173\) 12.0000i 0.912343i 0.889892 + 0.456172i \(0.150780\pi\)
−0.889892 + 0.456172i \(0.849220\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.00000 −0.376889
\(177\) 0 0
\(178\) − 1.60770i − 0.120502i
\(179\) 6.92820 0.517838 0.258919 0.965899i \(-0.416634\pi\)
0.258919 + 0.965899i \(0.416634\pi\)
\(180\) 0 0
\(181\) 15.8564 1.17860 0.589299 0.807915i \(-0.299404\pi\)
0.589299 + 0.807915i \(0.299404\pi\)
\(182\) − 18.9282i − 1.40305i
\(183\) 0 0
\(184\) −12.0000 −0.884652
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 6.92820i 0.505291i
\(189\) 0 0
\(190\) 0 0
\(191\) 18.9282 1.36960 0.684798 0.728733i \(-0.259890\pi\)
0.684798 + 0.728733i \(0.259890\pi\)
\(192\) 0 0
\(193\) 24.3923i 1.75580i 0.478847 + 0.877898i \(0.341055\pi\)
−0.478847 + 0.877898i \(0.658945\pi\)
\(194\) 17.3205 1.24354
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) − 12.0000i − 0.854965i −0.904024 0.427482i \(-0.859401\pi\)
0.904024 0.427482i \(-0.140599\pi\)
\(198\) 0 0
\(199\) 24.7846 1.75693 0.878467 0.477803i \(-0.158567\pi\)
0.878467 + 0.477803i \(0.158567\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) − 18.0000i − 1.26648i
\(203\) 6.92820i 0.486265i
\(204\) 0 0
\(205\) 0 0
\(206\) 13.8564 0.965422
\(207\) 0 0
\(208\) − 27.3205i − 1.89434i
\(209\) −5.46410 −0.377960
\(210\) 0 0
\(211\) −8.39230 −0.577750 −0.288875 0.957367i \(-0.593281\pi\)
−0.288875 + 0.957367i \(0.593281\pi\)
\(212\) 0.928203i 0.0637493i
\(213\) 0 0
\(214\) 14.7846 1.01066
\(215\) 0 0
\(216\) 0 0
\(217\) 21.8564i 1.48371i
\(218\) − 17.3205i − 1.17309i
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 9.85641i 0.660034i 0.943975 + 0.330017i \(0.107054\pi\)
−0.943975 + 0.330017i \(0.892946\pi\)
\(224\) 10.3923 0.694365
\(225\) 0 0
\(226\) 22.3923 1.48951
\(227\) 15.4641i 1.02639i 0.858272 + 0.513194i \(0.171538\pi\)
−0.858272 + 0.513194i \(0.828462\pi\)
\(228\) 0 0
\(229\) 23.8564 1.57648 0.788238 0.615371i \(-0.210994\pi\)
0.788238 + 0.615371i \(0.210994\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 6.00000i 0.393919i
\(233\) − 12.0000i − 0.786146i −0.919507 0.393073i \(-0.871412\pi\)
0.919507 0.393073i \(-0.128588\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 6.92820 0.450988
\(237\) 0 0
\(238\) 0 0
\(239\) −12.0000 −0.776215 −0.388108 0.921614i \(-0.626871\pi\)
−0.388108 + 0.921614i \(0.626871\pi\)
\(240\) 0 0
\(241\) 0.143594 0.00924967 0.00462484 0.999989i \(-0.498528\pi\)
0.00462484 + 0.999989i \(0.498528\pi\)
\(242\) − 1.73205i − 0.111340i
\(243\) 0 0
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 0 0
\(247\) − 29.8564i − 1.89972i
\(248\) 18.9282i 1.20194i
\(249\) 0 0
\(250\) 0 0
\(251\) 1.85641 0.117175 0.0585877 0.998282i \(-0.481340\pi\)
0.0585877 + 0.998282i \(0.481340\pi\)
\(252\) 0 0
\(253\) − 6.92820i − 0.435572i
\(254\) −15.4641 −0.970304
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) − 19.8564i − 1.23861i −0.785151 0.619304i \(-0.787415\pi\)
0.785151 0.619304i \(-0.212585\pi\)
\(258\) 0 0
\(259\) 9.85641 0.612447
\(260\) 0 0
\(261\) 0 0
\(262\) 32.7846i 2.02544i
\(263\) − 20.5359i − 1.26630i −0.774030 0.633149i \(-0.781762\pi\)
0.774030 0.633149i \(-0.218238\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 18.9282 1.16056
\(267\) 0 0
\(268\) 8.00000i 0.488678i
\(269\) 19.8564 1.21067 0.605333 0.795972i \(-0.293040\pi\)
0.605333 + 0.795972i \(0.293040\pi\)
\(270\) 0 0
\(271\) −11.6077 −0.705117 −0.352559 0.935790i \(-0.614688\pi\)
−0.352559 + 0.935790i \(0.614688\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −31.1769 −1.88347
\(275\) 0 0
\(276\) 0 0
\(277\) − 29.4641i − 1.77033i −0.465281 0.885163i \(-0.654047\pi\)
0.465281 0.885163i \(-0.345953\pi\)
\(278\) 21.4641i 1.28733i
\(279\) 0 0
\(280\) 0 0
\(281\) −3.46410 −0.206651 −0.103325 0.994648i \(-0.532948\pi\)
−0.103325 + 0.994648i \(0.532948\pi\)
\(282\) 0 0
\(283\) − 4.92820i − 0.292951i −0.989214 0.146476i \(-0.953207\pi\)
0.989214 0.146476i \(-0.0467930\pi\)
\(284\) 13.8564 0.822226
\(285\) 0 0
\(286\) 9.46410 0.559624
\(287\) 6.92820i 0.408959i
\(288\) 0 0
\(289\) 17.0000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 8.39230i 0.491122i
\(293\) 13.8564i 0.809500i 0.914427 + 0.404750i \(0.132641\pi\)
−0.914427 + 0.404750i \(0.867359\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 8.53590 0.496139
\(297\) 0 0
\(298\) 26.7846i 1.55159i
\(299\) 37.8564 2.18929
\(300\) 0 0
\(301\) −9.85641 −0.568114
\(302\) 35.3205i 2.03247i
\(303\) 0 0
\(304\) 27.3205 1.56694
\(305\) 0 0
\(306\) 0 0
\(307\) − 14.0000i − 0.799022i −0.916728 0.399511i \(-0.869180\pi\)
0.916728 0.399511i \(-0.130820\pi\)
\(308\) 2.00000i 0.113961i
\(309\) 0 0
\(310\) 0 0
\(311\) −5.07180 −0.287595 −0.143798 0.989607i \(-0.545931\pi\)
−0.143798 + 0.989607i \(0.545931\pi\)
\(312\) 0 0
\(313\) 20.9282i 1.18293i 0.806330 + 0.591466i \(0.201451\pi\)
−0.806330 + 0.591466i \(0.798549\pi\)
\(314\) 5.32051 0.300254
\(315\) 0 0
\(316\) −6.53590 −0.367673
\(317\) − 24.9282i − 1.40011i −0.714090 0.700054i \(-0.753159\pi\)
0.714090 0.700054i \(-0.246841\pi\)
\(318\) 0 0
\(319\) −3.46410 −0.193952
\(320\) 0 0
\(321\) 0 0
\(322\) 24.0000i 1.33747i
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 17.0718 0.945519
\(327\) 0 0
\(328\) 6.00000i 0.331295i
\(329\) −13.8564 −0.763928
\(330\) 0 0
\(331\) 9.85641 0.541757 0.270879 0.962614i \(-0.412686\pi\)
0.270879 + 0.962614i \(0.412686\pi\)
\(332\) 8.53590i 0.468468i
\(333\) 0 0
\(334\) −18.0000 −0.984916
\(335\) 0 0
\(336\) 0 0
\(337\) − 33.1769i − 1.80726i −0.428312 0.903631i \(-0.640892\pi\)
0.428312 0.903631i \(-0.359108\pi\)
\(338\) 29.1962i 1.58806i
\(339\) 0 0
\(340\) 0 0
\(341\) −10.9282 −0.591795
\(342\) 0 0
\(343\) − 20.0000i − 1.07990i
\(344\) −8.53590 −0.460225
\(345\) 0 0
\(346\) 20.7846 1.11739
\(347\) 22.3923i 1.20208i 0.799218 + 0.601041i \(0.205247\pi\)
−0.799218 + 0.601041i \(0.794753\pi\)
\(348\) 0 0
\(349\) 8.14359 0.435917 0.217958 0.975958i \(-0.430060\pi\)
0.217958 + 0.975958i \(0.430060\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 5.19615i 0.276956i
\(353\) − 12.9282i − 0.688099i −0.938952 0.344049i \(-0.888201\pi\)
0.938952 0.344049i \(-0.111799\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −0.928203 −0.0491947
\(357\) 0 0
\(358\) − 12.0000i − 0.634220i
\(359\) −20.7846 −1.09697 −0.548485 0.836160i \(-0.684795\pi\)
−0.548485 + 0.836160i \(0.684795\pi\)
\(360\) 0 0
\(361\) 10.8564 0.571390
\(362\) − 27.4641i − 1.44348i
\(363\) 0 0
\(364\) −10.9282 −0.572793
\(365\) 0 0
\(366\) 0 0
\(367\) − 20.0000i − 1.04399i −0.852948 0.521996i \(-0.825188\pi\)
0.852948 0.521996i \(-0.174812\pi\)
\(368\) 34.6410i 1.80579i
\(369\) 0 0
\(370\) 0 0
\(371\) −1.85641 −0.0963798
\(372\) 0 0
\(373\) − 20.3923i − 1.05587i −0.849284 0.527937i \(-0.822966\pi\)
0.849284 0.527937i \(-0.177034\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −12.0000 −0.618853
\(377\) − 18.9282i − 0.974852i
\(378\) 0 0
\(379\) 17.8564 0.917222 0.458611 0.888637i \(-0.348347\pi\)
0.458611 + 0.888637i \(0.348347\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) − 32.7846i − 1.67741i
\(383\) 13.8564i 0.708029i 0.935240 + 0.354015i \(0.115184\pi\)
−0.935240 + 0.354015i \(0.884816\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 42.2487 2.15040
\(387\) 0 0
\(388\) − 10.0000i − 0.507673i
\(389\) 11.0718 0.561362 0.280681 0.959801i \(-0.409440\pi\)
0.280681 + 0.959801i \(0.409440\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 5.19615i − 0.262445i
\(393\) 0 0
\(394\) −20.7846 −1.04711
\(395\) 0 0
\(396\) 0 0
\(397\) − 2.00000i − 0.100377i −0.998740 0.0501886i \(-0.984018\pi\)
0.998740 0.0501886i \(-0.0159822\pi\)
\(398\) − 42.9282i − 2.15180i
\(399\) 0 0
\(400\) 0 0
\(401\) 7.85641 0.392330 0.196165 0.980571i \(-0.437151\pi\)
0.196165 + 0.980571i \(0.437151\pi\)
\(402\) 0 0
\(403\) − 59.7128i − 2.97451i
\(404\) −10.3923 −0.517036
\(405\) 0 0
\(406\) 12.0000 0.595550
\(407\) 4.92820i 0.244282i
\(408\) 0 0
\(409\) 6.78461 0.335477 0.167739 0.985831i \(-0.446354\pi\)
0.167739 + 0.985831i \(0.446354\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) − 8.00000i − 0.394132i
\(413\) 13.8564i 0.681829i
\(414\) 0 0
\(415\) 0 0
\(416\) −28.3923 −1.39205
\(417\) 0 0
\(418\) 9.46410i 0.462904i
\(419\) 30.9282 1.51094 0.755471 0.655182i \(-0.227408\pi\)
0.755471 + 0.655182i \(0.227408\pi\)
\(420\) 0 0
\(421\) 2.00000 0.0974740 0.0487370 0.998812i \(-0.484480\pi\)
0.0487370 + 0.998812i \(0.484480\pi\)
\(422\) 14.5359i 0.707596i
\(423\) 0 0
\(424\) −1.60770 −0.0780766
\(425\) 0 0
\(426\) 0 0
\(427\) − 4.00000i − 0.193574i
\(428\) − 8.53590i − 0.412598i
\(429\) 0 0
\(430\) 0 0
\(431\) −8.78461 −0.423140 −0.211570 0.977363i \(-0.567858\pi\)
−0.211570 + 0.977363i \(0.567858\pi\)
\(432\) 0 0
\(433\) 0.143594i 0.00690067i 0.999994 + 0.00345033i \(0.00109828\pi\)
−0.999994 + 0.00345033i \(0.998902\pi\)
\(434\) 37.8564 1.81717
\(435\) 0 0
\(436\) −10.0000 −0.478913
\(437\) 37.8564i 1.81092i
\(438\) 0 0
\(439\) −33.1769 −1.58345 −0.791724 0.610879i \(-0.790816\pi\)
−0.791724 + 0.610879i \(0.790816\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 12.0000i − 0.570137i −0.958507 0.285069i \(-0.907984\pi\)
0.958507 0.285069i \(-0.0920164\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 17.0718 0.808373
\(447\) 0 0
\(448\) 2.00000i 0.0944911i
\(449\) −26.7846 −1.26404 −0.632022 0.774950i \(-0.717775\pi\)
−0.632022 + 0.774950i \(0.717775\pi\)
\(450\) 0 0
\(451\) −3.46410 −0.163118
\(452\) − 12.9282i − 0.608092i
\(453\) 0 0
\(454\) 26.7846 1.25706
\(455\) 0 0
\(456\) 0 0
\(457\) − 12.3923i − 0.579688i −0.957074 0.289844i \(-0.906397\pi\)
0.957074 0.289844i \(-0.0936034\pi\)
\(458\) − 41.3205i − 1.93078i
\(459\) 0 0
\(460\) 0 0
\(461\) −36.2487 −1.68827 −0.844135 0.536130i \(-0.819886\pi\)
−0.844135 + 0.536130i \(0.819886\pi\)
\(462\) 0 0
\(463\) − 28.0000i − 1.30127i −0.759390 0.650635i \(-0.774503\pi\)
0.759390 0.650635i \(-0.225497\pi\)
\(464\) 17.3205 0.804084
\(465\) 0 0
\(466\) −20.7846 −0.962828
\(467\) 5.07180i 0.234695i 0.993091 + 0.117347i \(0.0374391\pi\)
−0.993091 + 0.117347i \(0.962561\pi\)
\(468\) 0 0
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 0 0
\(472\) 12.0000i 0.552345i
\(473\) − 4.92820i − 0.226599i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 20.7846i 0.950666i
\(479\) 12.0000 0.548294 0.274147 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) −26.9282 −1.22782
\(482\) − 0.248711i − 0.0113285i
\(483\) 0 0
\(484\) −1.00000 −0.0454545
\(485\) 0 0
\(486\) 0 0
\(487\) 31.7128i 1.43704i 0.695504 + 0.718522i \(0.255181\pi\)
−0.695504 + 0.718522i \(0.744819\pi\)
\(488\) − 3.46410i − 0.156813i
\(489\) 0 0
\(490\) 0 0
\(491\) 30.9282 1.39577 0.697885 0.716210i \(-0.254125\pi\)
0.697885 + 0.716210i \(0.254125\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) −51.7128 −2.32667
\(495\) 0 0
\(496\) 54.6410 2.45345
\(497\) 27.7128i 1.24309i
\(498\) 0 0
\(499\) −28.7846 −1.28858 −0.644288 0.764783i \(-0.722846\pi\)
−0.644288 + 0.764783i \(0.722846\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) − 3.21539i − 0.143510i
\(503\) − 31.1769i − 1.39011i −0.718957 0.695055i \(-0.755380\pi\)
0.718957 0.695055i \(-0.244620\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −12.0000 −0.533465
\(507\) 0 0
\(508\) 8.92820i 0.396125i
\(509\) 19.8564 0.880120 0.440060 0.897968i \(-0.354957\pi\)
0.440060 + 0.897968i \(0.354957\pi\)
\(510\) 0 0
\(511\) −16.7846 −0.742507
\(512\) − 8.66025i − 0.382733i
\(513\) 0 0
\(514\) −34.3923 −1.51698
\(515\) 0 0
\(516\) 0 0
\(517\) − 6.92820i − 0.304702i
\(518\) − 17.0718i − 0.750092i
\(519\) 0 0
\(520\) 0 0
\(521\) −6.00000 −0.262865 −0.131432 0.991325i \(-0.541958\pi\)
−0.131432 + 0.991325i \(0.541958\pi\)
\(522\) 0 0
\(523\) − 22.0000i − 0.961993i −0.876723 0.480996i \(-0.840275\pi\)
0.876723 0.480996i \(-0.159725\pi\)
\(524\) 18.9282 0.826882
\(525\) 0 0
\(526\) −35.5692 −1.55089
\(527\) 0 0
\(528\) 0 0
\(529\) −25.0000 −1.08696
\(530\) 0 0
\(531\) 0 0
\(532\) − 10.9282i − 0.473798i
\(533\) − 18.9282i − 0.819871i
\(534\) 0 0
\(535\) 0 0
\(536\) −13.8564 −0.598506
\(537\) 0 0
\(538\) − 34.3923i − 1.48276i
\(539\) 3.00000 0.129219
\(540\) 0 0
\(541\) 27.8564 1.19764 0.598820 0.800883i \(-0.295636\pi\)
0.598820 + 0.800883i \(0.295636\pi\)
\(542\) 20.1051i 0.863589i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 2.00000i − 0.0855138i −0.999086 0.0427569i \(-0.986386\pi\)
0.999086 0.0427569i \(-0.0136141\pi\)
\(548\) 18.0000i 0.768922i
\(549\) 0 0
\(550\) 0 0
\(551\) 18.9282 0.806369
\(552\) 0 0
\(553\) − 13.0718i − 0.555869i
\(554\) −51.0333 −2.16820
\(555\) 0 0
\(556\) 12.3923 0.525551
\(557\) 3.21539i 0.136240i 0.997677 + 0.0681202i \(0.0217001\pi\)
−0.997677 + 0.0681202i \(0.978300\pi\)
\(558\) 0 0
\(559\) 26.9282 1.13894
\(560\) 0 0
\(561\) 0 0
\(562\) 6.00000i 0.253095i
\(563\) − 10.3923i − 0.437983i −0.975727 0.218992i \(-0.929723\pi\)
0.975727 0.218992i \(-0.0702768\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −8.53590 −0.358791
\(567\) 0 0
\(568\) 24.0000i 1.00702i
\(569\) 5.32051 0.223047 0.111524 0.993762i \(-0.464427\pi\)
0.111524 + 0.993762i \(0.464427\pi\)
\(570\) 0 0
\(571\) 3.60770 0.150977 0.0754887 0.997147i \(-0.475948\pi\)
0.0754887 + 0.997147i \(0.475948\pi\)
\(572\) − 5.46410i − 0.228466i
\(573\) 0 0
\(574\) 12.0000 0.500870
\(575\) 0 0
\(576\) 0 0
\(577\) 18.7846i 0.782014i 0.920388 + 0.391007i \(0.127873\pi\)
−0.920388 + 0.391007i \(0.872127\pi\)
\(578\) − 29.4449i − 1.22474i
\(579\) 0 0
\(580\) 0 0
\(581\) −17.0718 −0.708257
\(582\) 0 0
\(583\) − 0.928203i − 0.0384422i
\(584\) −14.5359 −0.601500
\(585\) 0 0
\(586\) 24.0000 0.991431
\(587\) − 18.9282i − 0.781251i −0.920550 0.390625i \(-0.872259\pi\)
0.920550 0.390625i \(-0.127741\pi\)
\(588\) 0 0
\(589\) 59.7128 2.46042
\(590\) 0 0
\(591\) 0 0
\(592\) − 24.6410i − 1.01274i
\(593\) − 8.78461i − 0.360741i −0.983599 0.180370i \(-0.942270\pi\)
0.983599 0.180370i \(-0.0577296\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 15.4641 0.633434
\(597\) 0 0
\(598\) − 65.5692i − 2.68132i
\(599\) −37.8564 −1.54677 −0.773385 0.633936i \(-0.781438\pi\)
−0.773385 + 0.633936i \(0.781438\pi\)
\(600\) 0 0
\(601\) −32.6410 −1.33145 −0.665727 0.746195i \(-0.731879\pi\)
−0.665727 + 0.746195i \(0.731879\pi\)
\(602\) 17.0718i 0.695794i
\(603\) 0 0
\(604\) 20.3923 0.829751
\(605\) 0 0
\(606\) 0 0
\(607\) 18.7846i 0.762444i 0.924484 + 0.381222i \(0.124497\pi\)
−0.924484 + 0.381222i \(0.875503\pi\)
\(608\) − 28.3923i − 1.15146i
\(609\) 0 0
\(610\) 0 0
\(611\) 37.8564 1.53151
\(612\) 0 0
\(613\) − 20.3923i − 0.823637i −0.911266 0.411819i \(-0.864894\pi\)
0.911266 0.411819i \(-0.135106\pi\)
\(614\) −24.2487 −0.978598
\(615\) 0 0
\(616\) −3.46410 −0.139573
\(617\) − 36.9282i − 1.48667i −0.668917 0.743337i \(-0.733242\pi\)
0.668917 0.743337i \(-0.266758\pi\)
\(618\) 0 0
\(619\) −20.0000 −0.803868 −0.401934 0.915669i \(-0.631662\pi\)
−0.401934 + 0.915669i \(0.631662\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 8.78461i 0.352231i
\(623\) − 1.85641i − 0.0743754i
\(624\) 0 0
\(625\) 0 0
\(626\) 36.2487 1.44879
\(627\) 0 0
\(628\) − 3.07180i − 0.122578i
\(629\) 0 0
\(630\) 0 0
\(631\) −21.0718 −0.838855 −0.419427 0.907789i \(-0.637769\pi\)
−0.419427 + 0.907789i \(0.637769\pi\)
\(632\) − 11.3205i − 0.450306i
\(633\) 0 0
\(634\) −43.1769 −1.71477
\(635\) 0 0
\(636\) 0 0
\(637\) 16.3923i 0.649487i
\(638\) 6.00000i 0.237542i
\(639\) 0 0
\(640\) 0 0
\(641\) 12.9282 0.510633 0.255317 0.966857i \(-0.417820\pi\)
0.255317 + 0.966857i \(0.417820\pi\)
\(642\) 0 0
\(643\) 37.5692i 1.48159i 0.671734 + 0.740793i \(0.265550\pi\)
−0.671734 + 0.740793i \(0.734450\pi\)
\(644\) 13.8564 0.546019
\(645\) 0 0
\(646\) 0 0
\(647\) 27.7128i 1.08950i 0.838597 + 0.544752i \(0.183376\pi\)
−0.838597 + 0.544752i \(0.816624\pi\)
\(648\) 0 0
\(649\) −6.92820 −0.271956
\(650\) 0 0
\(651\) 0 0
\(652\) − 9.85641i − 0.386007i
\(653\) − 19.8564i − 0.777041i −0.921440 0.388521i \(-0.872986\pi\)
0.921440 0.388521i \(-0.127014\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 17.3205 0.676252
\(657\) 0 0
\(658\) 24.0000i 0.935617i
\(659\) −15.7128 −0.612084 −0.306042 0.952018i \(-0.599005\pi\)
−0.306042 + 0.952018i \(0.599005\pi\)
\(660\) 0 0
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) − 17.0718i − 0.663514i
\(663\) 0 0
\(664\) −14.7846 −0.573754
\(665\) 0 0
\(666\) 0 0
\(667\) 24.0000i 0.929284i
\(668\) 10.3923i 0.402090i
\(669\) 0 0
\(670\) 0 0
\(671\) 2.00000 0.0772091
\(672\) 0 0
\(673\) − 3.32051i − 0.127996i −0.997950 0.0639981i \(-0.979615\pi\)
0.997950 0.0639981i \(-0.0203852\pi\)
\(674\) −57.4641 −2.21343
\(675\) 0 0
\(676\) 16.8564 0.648323
\(677\) 8.78461i 0.337620i 0.985649 + 0.168810i \(0.0539924\pi\)
−0.985649 + 0.168810i \(0.946008\pi\)
\(678\) 0 0
\(679\) 20.0000 0.767530
\(680\) 0 0
\(681\) 0 0
\(682\) 18.9282i 0.724798i
\(683\) 32.7846i 1.25447i 0.778831 + 0.627234i \(0.215813\pi\)
−0.778831 + 0.627234i \(0.784187\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −34.6410 −1.32260
\(687\) 0 0
\(688\) 24.6410i 0.939430i
\(689\) 5.07180 0.193220
\(690\) 0 0
\(691\) 47.7128 1.81508 0.907540 0.419965i \(-0.137958\pi\)
0.907540 + 0.419965i \(0.137958\pi\)
\(692\) − 12.0000i − 0.456172i
\(693\) 0 0
\(694\) 38.7846 1.47224
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) − 14.1051i − 0.533887i
\(699\) 0 0
\(700\) 0 0
\(701\) −39.4641 −1.49054 −0.745269 0.666764i \(-0.767679\pi\)
−0.745269 + 0.666764i \(0.767679\pi\)
\(702\) 0 0
\(703\) − 26.9282i − 1.01562i
\(704\) −1.00000 −0.0376889
\(705\) 0 0
\(706\) −22.3923 −0.842746
\(707\) − 20.7846i − 0.781686i
\(708\) 0 0
\(709\) 11.8564 0.445277 0.222638 0.974901i \(-0.428533\pi\)
0.222638 + 0.974901i \(0.428533\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) − 1.60770i − 0.0602509i
\(713\) 75.7128i 2.83547i
\(714\) 0 0
\(715\) 0 0
\(716\) −6.92820 −0.258919
\(717\) 0 0
\(718\) 36.0000i 1.34351i
\(719\) −5.07180 −0.189146 −0.0945731 0.995518i \(-0.530149\pi\)
−0.0945731 + 0.995518i \(0.530149\pi\)
\(720\) 0 0
\(721\) 16.0000 0.595871
\(722\) − 18.8038i − 0.699807i
\(723\) 0 0
\(724\) −15.8564 −0.589299
\(725\) 0 0
\(726\) 0 0
\(727\) − 8.00000i − 0.296704i −0.988935 0.148352i \(-0.952603\pi\)
0.988935 0.148352i \(-0.0473968\pi\)
\(728\) − 18.9282i − 0.701526i
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 53.9615i 1.99311i 0.0829082 + 0.996557i \(0.473579\pi\)
−0.0829082 + 0.996557i \(0.526421\pi\)
\(734\) −34.6410 −1.27862
\(735\) 0 0
\(736\) 36.0000 1.32698
\(737\) − 8.00000i − 0.294684i
\(738\) 0 0
\(739\) −17.4641 −0.642427 −0.321214 0.947007i \(-0.604091\pi\)
−0.321214 + 0.947007i \(0.604091\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 3.21539i 0.118041i
\(743\) − 25.6077i − 0.939455i −0.882811 0.469728i \(-0.844352\pi\)
0.882811 0.469728i \(-0.155648\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −35.3205 −1.29318
\(747\) 0 0
\(748\) 0 0
\(749\) 17.0718 0.623790
\(750\) 0 0
\(751\) 26.9282 0.982624 0.491312 0.870984i \(-0.336517\pi\)
0.491312 + 0.870984i \(0.336517\pi\)
\(752\) 34.6410i 1.26323i
\(753\) 0 0
\(754\) −32.7846 −1.19395
\(755\) 0 0
\(756\) 0 0
\(757\) − 34.7846i − 1.26427i −0.774859 0.632134i \(-0.782179\pi\)
0.774859 0.632134i \(-0.217821\pi\)
\(758\) − 30.9282i − 1.12336i
\(759\) 0 0
\(760\) 0 0
\(761\) 32.5359 1.17943 0.589713 0.807613i \(-0.299241\pi\)
0.589713 + 0.807613i \(0.299241\pi\)
\(762\) 0 0
\(763\) − 20.0000i − 0.724049i
\(764\) −18.9282 −0.684798
\(765\) 0 0
\(766\) 24.0000 0.867155
\(767\) − 37.8564i − 1.36692i
\(768\) 0 0
\(769\) −50.4974 −1.82098 −0.910492 0.413527i \(-0.864297\pi\)
−0.910492 + 0.413527i \(0.864297\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) − 24.3923i − 0.877898i
\(773\) 4.14359i 0.149035i 0.997220 + 0.0745174i \(0.0237416\pi\)
−0.997220 + 0.0745174i \(0.976258\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 17.3205 0.621770
\(777\) 0 0
\(778\) − 19.1769i − 0.687526i
\(779\) 18.9282 0.678173
\(780\) 0 0
\(781\) −13.8564 −0.495821
\(782\) 0 0
\(783\) 0 0
\(784\) −15.0000 −0.535714
\(785\) 0 0
\(786\) 0 0
\(787\) − 22.7846i − 0.812184i −0.913832 0.406092i \(-0.866891\pi\)
0.913832 0.406092i \(-0.133109\pi\)
\(788\) 12.0000i 0.427482i
\(789\) 0 0
\(790\) 0 0
\(791\) 25.8564 0.919348
\(792\) 0 0
\(793\) 10.9282i 0.388072i
\(794\) −3.46410 −0.122936
\(795\) 0 0
\(796\) −24.7846 −0.878467
\(797\) − 52.6410i − 1.86464i −0.361634 0.932320i \(-0.617781\pi\)
0.361634 0.932320i \(-0.382219\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) − 13.6077i − 0.480504i
\(803\) − 8.39230i − 0.296158i
\(804\) 0 0
\(805\) 0 0
\(806\) −103.426 −3.64301
\(807\) 0 0
\(808\) − 18.0000i − 0.633238i
\(809\) −15.4641 −0.543689 −0.271844 0.962341i \(-0.587634\pi\)
−0.271844 + 0.962341i \(0.587634\pi\)
\(810\) 0 0
\(811\) 12.3923 0.435153 0.217576 0.976043i \(-0.430185\pi\)
0.217576 + 0.976043i \(0.430185\pi\)
\(812\) − 6.92820i − 0.243132i
\(813\) 0 0
\(814\) 8.53590 0.299183
\(815\) 0 0
\(816\) 0 0
\(817\) 26.9282i 0.942099i
\(818\) − 11.7513i − 0.410874i
\(819\) 0 0
\(820\) 0 0
\(821\) −20.5359 −0.716708 −0.358354 0.933586i \(-0.616662\pi\)
−0.358354 + 0.933586i \(0.616662\pi\)
\(822\) 0 0
\(823\) − 33.5692i − 1.17015i −0.810979 0.585075i \(-0.801065\pi\)
0.810979 0.585075i \(-0.198935\pi\)
\(824\) 13.8564 0.482711
\(825\) 0 0
\(826\) 24.0000 0.835067
\(827\) 22.3923i 0.778657i 0.921099 + 0.389328i \(0.127293\pi\)
−0.921099 + 0.389328i \(0.872707\pi\)
\(828\) 0 0
\(829\) −29.7128 −1.03197 −0.515984 0.856598i \(-0.672574\pi\)
−0.515984 + 0.856598i \(0.672574\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) − 5.46410i − 0.189434i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 5.46410 0.188980
\(837\) 0 0
\(838\) − 53.5692i − 1.85052i
\(839\) 56.7846 1.96042 0.980211 0.197954i \(-0.0634298\pi\)
0.980211 + 0.197954i \(0.0634298\pi\)
\(840\) 0 0
\(841\) −17.0000 −0.586207
\(842\) − 3.46410i − 0.119381i
\(843\) 0 0
\(844\) 8.39230 0.288875
\(845\) 0 0
\(846\) 0 0
\(847\) − 2.00000i − 0.0687208i
\(848\) 4.64102i 0.159373i
\(849\) 0 0
\(850\) 0 0
\(851\) 34.1436 1.17043
\(852\) 0 0
\(853\) 3.60770i 0.123525i 0.998091 + 0.0617626i \(0.0196722\pi\)
−0.998091 + 0.0617626i \(0.980328\pi\)
\(854\) −6.92820 −0.237078
\(855\) 0 0
\(856\) 14.7846 0.505328
\(857\) − 37.8564i − 1.29315i −0.762850 0.646575i \(-0.776201\pi\)
0.762850 0.646575i \(-0.223799\pi\)
\(858\) 0 0
\(859\) 7.71281 0.263158 0.131579 0.991306i \(-0.457995\pi\)
0.131579 + 0.991306i \(0.457995\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 15.2154i 0.518238i
\(863\) 37.8564i 1.28865i 0.764753 + 0.644324i \(0.222861\pi\)
−0.764753 + 0.644324i \(0.777139\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0.248711 0.00845155
\(867\) 0 0
\(868\) − 21.8564i − 0.741855i
\(869\) 6.53590 0.221715
\(870\) 0 0
\(871\) 43.7128 1.48115
\(872\) − 17.3205i − 0.586546i
\(873\) 0 0
\(874\) 65.5692 2.21791
\(875\) 0 0
\(876\) 0 0
\(877\) 34.2487i 1.15650i 0.815861 + 0.578248i \(0.196264\pi\)
−0.815861 + 0.578248i \(0.803736\pi\)
\(878\) 57.4641i 1.93932i
\(879\) 0 0
\(880\) 0 0
\(881\) 0.928203 0.0312720 0.0156360 0.999878i \(-0.495023\pi\)
0.0156360 + 0.999878i \(0.495023\pi\)
\(882\) 0 0
\(883\) − 4.00000i − 0.134611i −0.997732 0.0673054i \(-0.978560\pi\)
0.997732 0.0673054i \(-0.0214402\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −20.7846 −0.698273
\(887\) 12.2487i 0.411271i 0.978629 + 0.205636i \(0.0659262\pi\)
−0.978629 + 0.205636i \(0.934074\pi\)
\(888\) 0 0
\(889\) −17.8564 −0.598885
\(890\) 0 0
\(891\) 0 0
\(892\) − 9.85641i − 0.330017i
\(893\) 37.8564i 1.26682i
\(894\) 0 0
\(895\) 0 0
\(896\) 24.2487 0.810093
\(897\) 0 0
\(898\) 46.3923i 1.54813i
\(899\) 37.8564 1.26258
\(900\) 0 0
\(901\) 0 0
\(902\) 6.00000i 0.199778i
\(903\) 0 0
\(904\) 22.3923 0.744757
\(905\) 0 0
\(906\) 0 0
\(907\) − 18.1436i − 0.602448i −0.953553 0.301224i \(-0.902605\pi\)
0.953553 0.301224i \(-0.0973952\pi\)
\(908\) − 15.4641i − 0.513194i
\(909\) 0 0
\(910\) 0 0
\(911\) −18.9282 −0.627119 −0.313560 0.949568i \(-0.601522\pi\)
−0.313560 + 0.949568i \(0.601522\pi\)
\(912\) 0 0
\(913\) − 8.53590i − 0.282497i
\(914\) −21.4641 −0.709969
\(915\) 0 0
\(916\) −23.8564 −0.788238
\(917\) 37.8564i 1.25013i
\(918\) 0 0
\(919\) 32.3923 1.06852 0.534262 0.845319i \(-0.320590\pi\)
0.534262 + 0.845319i \(0.320590\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 62.7846i 2.06770i
\(923\) − 75.7128i − 2.49212i
\(924\) 0 0
\(925\) 0 0
\(926\) −48.4974 −1.59372
\(927\) 0 0
\(928\) − 18.0000i − 0.590879i
\(929\) 2.78461 0.0913601 0.0456800 0.998956i \(-0.485455\pi\)
0.0456800 + 0.998956i \(0.485455\pi\)
\(930\) 0 0
\(931\) −16.3923 −0.537236
\(932\) 12.0000i 0.393073i
\(933\) 0 0
\(934\) 8.78461 0.287441
\(935\) 0 0
\(936\) 0 0
\(937\) 20.3923i 0.666188i 0.942894 + 0.333094i \(0.108093\pi\)
−0.942894 + 0.333094i \(0.891907\pi\)
\(938\) 27.7128i 0.904855i
\(939\) 0 0
\(940\) 0 0
\(941\) −27.4641 −0.895304 −0.447652 0.894208i \(-0.647740\pi\)
−0.447652 + 0.894208i \(0.647740\pi\)
\(942\) 0 0
\(943\) 24.0000i 0.781548i
\(944\) 34.6410 1.12747
\(945\) 0 0
\(946\) −8.53590 −0.277526
\(947\) − 18.9282i − 0.615084i −0.951535 0.307542i \(-0.900494\pi\)
0.951535 0.307542i \(-0.0995065\pi\)
\(948\) 0 0
\(949\) 45.8564 1.48856
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 3.21539i 0.104157i 0.998643 + 0.0520784i \(0.0165846\pi\)
−0.998643 + 0.0520784i \(0.983415\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 12.0000 0.388108
\(957\) 0 0
\(958\) − 20.7846i − 0.671520i
\(959\) −36.0000 −1.16250
\(960\) 0 0
\(961\) 88.4256 2.85244
\(962\) 46.6410i 1.50377i
\(963\) 0 0
\(964\) −0.143594 −0.00462484
\(965\) 0 0
\(966\) 0 0
\(967\) − 22.7846i − 0.732704i −0.930476 0.366352i \(-0.880607\pi\)
0.930476 0.366352i \(-0.119393\pi\)
\(968\) − 1.73205i − 0.0556702i
\(969\) 0 0
\(970\) 0 0
\(971\) −25.8564 −0.829772 −0.414886 0.909873i \(-0.636178\pi\)
−0.414886 + 0.909873i \(0.636178\pi\)
\(972\) 0 0
\(973\) 24.7846i 0.794558i
\(974\) 54.9282 1.76001
\(975\) 0 0
\(976\) −10.0000 −0.320092
\(977\) 47.5692i 1.52187i 0.648826 + 0.760937i \(0.275260\pi\)
−0.648826 + 0.760937i \(0.724740\pi\)
\(978\) 0 0
\(979\) 0.928203 0.0296655
\(980\) 0 0
\(981\) 0 0
\(982\) − 53.5692i − 1.70946i
\(983\) − 24.0000i − 0.765481i −0.923856 0.382741i \(-0.874980\pi\)
0.923856 0.382741i \(-0.125020\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 29.8564i 0.949859i
\(989\) −34.1436 −1.08570
\(990\) 0 0
\(991\) −7.21539 −0.229204 −0.114602 0.993411i \(-0.536559\pi\)
−0.114602 + 0.993411i \(0.536559\pi\)
\(992\) − 56.7846i − 1.80291i
\(993\) 0 0
\(994\) 48.0000 1.52247
\(995\) 0 0
\(996\) 0 0
\(997\) − 27.6077i − 0.874344i −0.899378 0.437172i \(-0.855980\pi\)
0.899378 0.437172i \(-0.144020\pi\)
\(998\) 49.8564i 1.57818i
\(999\) 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.n.199.1 4
3.2 odd 2 825.2.c.c.199.4 4
5.2 odd 4 495.2.a.c.1.2 2
5.3 odd 4 2475.2.a.r.1.1 2
5.4 even 2 inner 2475.2.c.n.199.4 4
15.2 even 4 165.2.a.b.1.1 2
15.8 even 4 825.2.a.e.1.2 2
15.14 odd 2 825.2.c.c.199.1 4
20.7 even 4 7920.2.a.bz.1.2 2
55.32 even 4 5445.2.a.s.1.1 2
60.47 odd 4 2640.2.a.x.1.2 2
105.62 odd 4 8085.2.a.bd.1.1 2
165.32 odd 4 1815.2.a.i.1.2 2
165.98 odd 4 9075.2.a.bh.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.b.1.1 2 15.2 even 4
495.2.a.c.1.2 2 5.2 odd 4
825.2.a.e.1.2 2 15.8 even 4
825.2.c.c.199.1 4 15.14 odd 2
825.2.c.c.199.4 4 3.2 odd 2
1815.2.a.i.1.2 2 165.32 odd 4
2475.2.a.r.1.1 2 5.3 odd 4
2475.2.c.n.199.1 4 1.1 even 1 trivial
2475.2.c.n.199.4 4 5.4 even 2 inner
2640.2.a.x.1.2 2 60.47 odd 4
5445.2.a.s.1.1 2 55.32 even 4
7920.2.a.bz.1.2 2 20.7 even 4
8085.2.a.bd.1.1 2 105.62 odd 4
9075.2.a.bh.1.1 2 165.98 odd 4