Properties

Label 2475.2.a.m.1.2
Level $2475$
Weight $2$
Character 2475.1
Self dual yes
Analytic conductor $19.763$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2475 = 3^{2} \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2475.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(19.7629745003\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.41421\) of defining polynomial
Character \(\chi\) \(=\) 2475.1

$q$-expansion

\(f(q)\) \(=\) \(q+0.414214 q^{2} -1.82843 q^{4} +4.82843 q^{7} -1.58579 q^{8} +O(q^{10})\) \(q+0.414214 q^{2} -1.82843 q^{4} +4.82843 q^{7} -1.58579 q^{8} +1.00000 q^{11} -5.65685 q^{13} +2.00000 q^{14} +3.00000 q^{16} -6.82843 q^{17} -1.17157 q^{19} +0.414214 q^{22} -4.00000 q^{23} -2.34315 q^{26} -8.82843 q^{28} -0.828427 q^{29} +4.41421 q^{32} -2.82843 q^{34} -0.343146 q^{37} -0.485281 q^{38} +0.828427 q^{41} +3.17157 q^{43} -1.82843 q^{44} -1.65685 q^{46} -4.00000 q^{47} +16.3137 q^{49} +10.3431 q^{52} -13.3137 q^{53} -7.65685 q^{56} -0.343146 q^{58} +4.00000 q^{59} -0.343146 q^{61} -4.17157 q^{64} -5.65685 q^{67} +12.4853 q^{68} -13.6569 q^{71} +11.3137 q^{73} -0.142136 q^{74} +2.14214 q^{76} +4.82843 q^{77} -8.48528 q^{79} +0.343146 q^{82} -10.0000 q^{83} +1.31371 q^{86} -1.58579 q^{88} +7.65685 q^{89} -27.3137 q^{91} +7.31371 q^{92} -1.65685 q^{94} -0.343146 q^{97} +6.75736 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} + 4q^{7} - 6q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} + 4q^{7} - 6q^{8} + 2q^{11} + 4q^{14} + 6q^{16} - 8q^{17} - 8q^{19} - 2q^{22} - 8q^{23} - 16q^{26} - 12q^{28} + 4q^{29} + 6q^{32} - 12q^{37} + 16q^{38} - 4q^{41} + 12q^{43} + 2q^{44} + 8q^{46} - 8q^{47} + 10q^{49} + 32q^{52} - 4q^{53} - 4q^{56} - 12q^{58} + 8q^{59} - 12q^{61} - 14q^{64} + 8q^{68} - 16q^{71} + 28q^{74} - 24q^{76} + 4q^{77} + 12q^{82} - 20q^{83} - 20q^{86} - 6q^{88} + 4q^{89} - 32q^{91} - 8q^{92} + 8q^{94} - 12q^{97} + 22q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.414214 0.292893 0.146447 0.989219i \(-0.453216\pi\)
0.146447 + 0.989219i \(0.453216\pi\)
\(3\) 0 0
\(4\) −1.82843 −0.914214
\(5\) 0 0
\(6\) 0 0
\(7\) 4.82843 1.82497 0.912487 0.409106i \(-0.134159\pi\)
0.912487 + 0.409106i \(0.134159\pi\)
\(8\) −1.58579 −0.560660
\(9\) 0 0
\(10\) 0 0
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −5.65685 −1.56893 −0.784465 0.620174i \(-0.787062\pi\)
−0.784465 + 0.620174i \(0.787062\pi\)
\(14\) 2.00000 0.534522
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) −6.82843 −1.65614 −0.828068 0.560627i \(-0.810560\pi\)
−0.828068 + 0.560627i \(0.810560\pi\)
\(18\) 0 0
\(19\) −1.17157 −0.268777 −0.134389 0.990929i \(-0.542907\pi\)
−0.134389 + 0.990929i \(0.542907\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.414214 0.0883106
\(23\) −4.00000 −0.834058 −0.417029 0.908893i \(-0.636929\pi\)
−0.417029 + 0.908893i \(0.636929\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −2.34315 −0.459529
\(27\) 0 0
\(28\) −8.82843 −1.66842
\(29\) −0.828427 −0.153835 −0.0769175 0.997037i \(-0.524508\pi\)
−0.0769175 + 0.997037i \(0.524508\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 4.41421 0.780330
\(33\) 0 0
\(34\) −2.82843 −0.485071
\(35\) 0 0
\(36\) 0 0
\(37\) −0.343146 −0.0564128 −0.0282064 0.999602i \(-0.508980\pi\)
−0.0282064 + 0.999602i \(0.508980\pi\)
\(38\) −0.485281 −0.0787230
\(39\) 0 0
\(40\) 0 0
\(41\) 0.828427 0.129379 0.0646893 0.997905i \(-0.479394\pi\)
0.0646893 + 0.997905i \(0.479394\pi\)
\(42\) 0 0
\(43\) 3.17157 0.483660 0.241830 0.970319i \(-0.422252\pi\)
0.241830 + 0.970319i \(0.422252\pi\)
\(44\) −1.82843 −0.275646
\(45\) 0 0
\(46\) −1.65685 −0.244290
\(47\) −4.00000 −0.583460 −0.291730 0.956501i \(-0.594231\pi\)
−0.291730 + 0.956501i \(0.594231\pi\)
\(48\) 0 0
\(49\) 16.3137 2.33053
\(50\) 0 0
\(51\) 0 0
\(52\) 10.3431 1.43434
\(53\) −13.3137 −1.82878 −0.914389 0.404836i \(-0.867329\pi\)
−0.914389 + 0.404836i \(0.867329\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −7.65685 −1.02319
\(57\) 0 0
\(58\) −0.343146 −0.0450572
\(59\) 4.00000 0.520756 0.260378 0.965507i \(-0.416153\pi\)
0.260378 + 0.965507i \(0.416153\pi\)
\(60\) 0 0
\(61\) −0.343146 −0.0439353 −0.0219677 0.999759i \(-0.506993\pi\)
−0.0219677 + 0.999759i \(0.506993\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) 0 0
\(66\) 0 0
\(67\) −5.65685 −0.691095 −0.345547 0.938401i \(-0.612307\pi\)
−0.345547 + 0.938401i \(0.612307\pi\)
\(68\) 12.4853 1.51406
\(69\) 0 0
\(70\) 0 0
\(71\) −13.6569 −1.62077 −0.810385 0.585897i \(-0.800742\pi\)
−0.810385 + 0.585897i \(0.800742\pi\)
\(72\) 0 0
\(73\) 11.3137 1.32417 0.662085 0.749429i \(-0.269672\pi\)
0.662085 + 0.749429i \(0.269672\pi\)
\(74\) −0.142136 −0.0165229
\(75\) 0 0
\(76\) 2.14214 0.245720
\(77\) 4.82843 0.550250
\(78\) 0 0
\(79\) −8.48528 −0.954669 −0.477334 0.878722i \(-0.658397\pi\)
−0.477334 + 0.878722i \(0.658397\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0.343146 0.0378941
\(83\) −10.0000 −1.09764 −0.548821 0.835940i \(-0.684923\pi\)
−0.548821 + 0.835940i \(0.684923\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.31371 0.141661
\(87\) 0 0
\(88\) −1.58579 −0.169045
\(89\) 7.65685 0.811625 0.405812 0.913956i \(-0.366989\pi\)
0.405812 + 0.913956i \(0.366989\pi\)
\(90\) 0 0
\(91\) −27.3137 −2.86325
\(92\) 7.31371 0.762507
\(93\) 0 0
\(94\) −1.65685 −0.170891
\(95\) 0 0
\(96\) 0 0
\(97\) −0.343146 −0.0348412 −0.0174206 0.999848i \(-0.505545\pi\)
−0.0174206 + 0.999848i \(0.505545\pi\)
\(98\) 6.75736 0.682596
\(99\) 0 0
\(100\) 0 0
\(101\) −4.82843 −0.480446 −0.240223 0.970718i \(-0.577221\pi\)
−0.240223 + 0.970718i \(0.577221\pi\)
\(102\) 0 0
\(103\) −19.3137 −1.90304 −0.951518 0.307593i \(-0.900477\pi\)
−0.951518 + 0.307593i \(0.900477\pi\)
\(104\) 8.97056 0.879636
\(105\) 0 0
\(106\) −5.51472 −0.535637
\(107\) −5.31371 −0.513696 −0.256848 0.966452i \(-0.582684\pi\)
−0.256848 + 0.966452i \(0.582684\pi\)
\(108\) 0 0
\(109\) −5.31371 −0.508961 −0.254480 0.967078i \(-0.581904\pi\)
−0.254480 + 0.967078i \(0.581904\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 14.4853 1.36873
\(113\) 14.9706 1.40831 0.704156 0.710045i \(-0.251326\pi\)
0.704156 + 0.710045i \(0.251326\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.51472 0.140638
\(117\) 0 0
\(118\) 1.65685 0.152526
\(119\) −32.9706 −3.02241
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −0.142136 −0.0128684
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2.48528 −0.220533 −0.110267 0.993902i \(-0.535170\pi\)
−0.110267 + 0.993902i \(0.535170\pi\)
\(128\) −10.5563 −0.933058
\(129\) 0 0
\(130\) 0 0
\(131\) 19.3137 1.68745 0.843723 0.536778i \(-0.180359\pi\)
0.843723 + 0.536778i \(0.180359\pi\)
\(132\) 0 0
\(133\) −5.65685 −0.490511
\(134\) −2.34315 −0.202417
\(135\) 0 0
\(136\) 10.8284 0.928530
\(137\) 9.31371 0.795724 0.397862 0.917445i \(-0.369752\pi\)
0.397862 + 0.917445i \(0.369752\pi\)
\(138\) 0 0
\(139\) −16.4853 −1.39826 −0.699132 0.714993i \(-0.746430\pi\)
−0.699132 + 0.714993i \(0.746430\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −5.65685 −0.474713
\(143\) −5.65685 −0.473050
\(144\) 0 0
\(145\) 0 0
\(146\) 4.68629 0.387840
\(147\) 0 0
\(148\) 0.627417 0.0515734
\(149\) −18.4853 −1.51437 −0.757187 0.653199i \(-0.773427\pi\)
−0.757187 + 0.653199i \(0.773427\pi\)
\(150\) 0 0
\(151\) −0.485281 −0.0394916 −0.0197458 0.999805i \(-0.506286\pi\)
−0.0197458 + 0.999805i \(0.506286\pi\)
\(152\) 1.85786 0.150693
\(153\) 0 0
\(154\) 2.00000 0.161165
\(155\) 0 0
\(156\) 0 0
\(157\) −18.0000 −1.43656 −0.718278 0.695756i \(-0.755069\pi\)
−0.718278 + 0.695756i \(0.755069\pi\)
\(158\) −3.51472 −0.279616
\(159\) 0 0
\(160\) 0 0
\(161\) −19.3137 −1.52213
\(162\) 0 0
\(163\) −15.3137 −1.19946 −0.599731 0.800202i \(-0.704726\pi\)
−0.599731 + 0.800202i \(0.704726\pi\)
\(164\) −1.51472 −0.118280
\(165\) 0 0
\(166\) −4.14214 −0.321492
\(167\) 9.31371 0.720716 0.360358 0.932814i \(-0.382654\pi\)
0.360358 + 0.932814i \(0.382654\pi\)
\(168\) 0 0
\(169\) 19.0000 1.46154
\(170\) 0 0
\(171\) 0 0
\(172\) −5.79899 −0.442169
\(173\) −2.82843 −0.215041 −0.107521 0.994203i \(-0.534291\pi\)
−0.107521 + 0.994203i \(0.534291\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 3.00000 0.226134
\(177\) 0 0
\(178\) 3.17157 0.237719
\(179\) 6.34315 0.474109 0.237054 0.971496i \(-0.423818\pi\)
0.237054 + 0.971496i \(0.423818\pi\)
\(180\) 0 0
\(181\) −14.0000 −1.04061 −0.520306 0.853980i \(-0.674182\pi\)
−0.520306 + 0.853980i \(0.674182\pi\)
\(182\) −11.3137 −0.838628
\(183\) 0 0
\(184\) 6.34315 0.467623
\(185\) 0 0
\(186\) 0 0
\(187\) −6.82843 −0.499344
\(188\) 7.31371 0.533407
\(189\) 0 0
\(190\) 0 0
\(191\) 5.65685 0.409316 0.204658 0.978834i \(-0.434392\pi\)
0.204658 + 0.978834i \(0.434392\pi\)
\(192\) 0 0
\(193\) −2.34315 −0.168663 −0.0843317 0.996438i \(-0.526876\pi\)
−0.0843317 + 0.996438i \(0.526876\pi\)
\(194\) −0.142136 −0.0102047
\(195\) 0 0
\(196\) −29.8284 −2.13060
\(197\) 8.48528 0.604551 0.302276 0.953221i \(-0.402254\pi\)
0.302276 + 0.953221i \(0.402254\pi\)
\(198\) 0 0
\(199\) −10.3431 −0.733206 −0.366603 0.930377i \(-0.619479\pi\)
−0.366603 + 0.930377i \(0.619479\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −2.00000 −0.140720
\(203\) −4.00000 −0.280745
\(204\) 0 0
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 0 0
\(208\) −16.9706 −1.17670
\(209\) −1.17157 −0.0810394
\(210\) 0 0
\(211\) 6.82843 0.470088 0.235044 0.971985i \(-0.424477\pi\)
0.235044 + 0.971985i \(0.424477\pi\)
\(212\) 24.3431 1.67189
\(213\) 0 0
\(214\) −2.20101 −0.150458
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −2.20101 −0.149071
\(219\) 0 0
\(220\) 0 0
\(221\) 38.6274 2.59836
\(222\) 0 0
\(223\) 17.6569 1.18239 0.591195 0.806529i \(-0.298656\pi\)
0.591195 + 0.806529i \(0.298656\pi\)
\(224\) 21.3137 1.42408
\(225\) 0 0
\(226\) 6.20101 0.412485
\(227\) −14.0000 −0.929213 −0.464606 0.885517i \(-0.653804\pi\)
−0.464606 + 0.885517i \(0.653804\pi\)
\(228\) 0 0
\(229\) −2.00000 −0.132164 −0.0660819 0.997814i \(-0.521050\pi\)
−0.0660819 + 0.997814i \(0.521050\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.31371 0.0862492
\(233\) −13.1716 −0.862898 −0.431449 0.902137i \(-0.641998\pi\)
−0.431449 + 0.902137i \(0.641998\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −7.31371 −0.476082
\(237\) 0 0
\(238\) −13.6569 −0.885242
\(239\) −6.34315 −0.410304 −0.205152 0.978730i \(-0.565769\pi\)
−0.205152 + 0.978730i \(0.565769\pi\)
\(240\) 0 0
\(241\) −23.6569 −1.52387 −0.761936 0.647652i \(-0.775751\pi\)
−0.761936 + 0.647652i \(0.775751\pi\)
\(242\) 0.414214 0.0266267
\(243\) 0 0
\(244\) 0.627417 0.0401663
\(245\) 0 0
\(246\) 0 0
\(247\) 6.62742 0.421692
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 12.9706 0.818695 0.409347 0.912379i \(-0.365756\pi\)
0.409347 + 0.912379i \(0.365756\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) −1.02944 −0.0645926
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) −27.6569 −1.72519 −0.862594 0.505898i \(-0.831161\pi\)
−0.862594 + 0.505898i \(0.831161\pi\)
\(258\) 0 0
\(259\) −1.65685 −0.102952
\(260\) 0 0
\(261\) 0 0
\(262\) 8.00000 0.494242
\(263\) 18.0000 1.10993 0.554964 0.831875i \(-0.312732\pi\)
0.554964 + 0.831875i \(0.312732\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.34315 −0.143667
\(267\) 0 0
\(268\) 10.3431 0.631808
\(269\) 24.6274 1.50156 0.750780 0.660552i \(-0.229678\pi\)
0.750780 + 0.660552i \(0.229678\pi\)
\(270\) 0 0
\(271\) 27.7990 1.68867 0.844334 0.535817i \(-0.179996\pi\)
0.844334 + 0.535817i \(0.179996\pi\)
\(272\) −20.4853 −1.24210
\(273\) 0 0
\(274\) 3.85786 0.233062
\(275\) 0 0
\(276\) 0 0
\(277\) 13.6569 0.820561 0.410280 0.911959i \(-0.365431\pi\)
0.410280 + 0.911959i \(0.365431\pi\)
\(278\) −6.82843 −0.409542
\(279\) 0 0
\(280\) 0 0
\(281\) 16.8284 1.00390 0.501950 0.864897i \(-0.332616\pi\)
0.501950 + 0.864897i \(0.332616\pi\)
\(282\) 0 0
\(283\) 3.17157 0.188530 0.0942652 0.995547i \(-0.469950\pi\)
0.0942652 + 0.995547i \(0.469950\pi\)
\(284\) 24.9706 1.48173
\(285\) 0 0
\(286\) −2.34315 −0.138553
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) 29.6274 1.74279
\(290\) 0 0
\(291\) 0 0
\(292\) −20.6863 −1.21057
\(293\) −1.17157 −0.0684440 −0.0342220 0.999414i \(-0.510895\pi\)
−0.0342220 + 0.999414i \(0.510895\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.544156 0.0316284
\(297\) 0 0
\(298\) −7.65685 −0.443550
\(299\) 22.6274 1.30858
\(300\) 0 0
\(301\) 15.3137 0.882667
\(302\) −0.201010 −0.0115668
\(303\) 0 0
\(304\) −3.51472 −0.201583
\(305\) 0 0
\(306\) 0 0
\(307\) 8.82843 0.503865 0.251932 0.967745i \(-0.418934\pi\)
0.251932 + 0.967745i \(0.418934\pi\)
\(308\) −8.82843 −0.503046
\(309\) 0 0
\(310\) 0 0
\(311\) −19.3137 −1.09518 −0.547590 0.836747i \(-0.684455\pi\)
−0.547590 + 0.836747i \(0.684455\pi\)
\(312\) 0 0
\(313\) −4.34315 −0.245489 −0.122745 0.992438i \(-0.539170\pi\)
−0.122745 + 0.992438i \(0.539170\pi\)
\(314\) −7.45584 −0.420758
\(315\) 0 0
\(316\) 15.5147 0.872771
\(317\) 30.2843 1.70093 0.850467 0.526028i \(-0.176319\pi\)
0.850467 + 0.526028i \(0.176319\pi\)
\(318\) 0 0
\(319\) −0.828427 −0.0463830
\(320\) 0 0
\(321\) 0 0
\(322\) −8.00000 −0.445823
\(323\) 8.00000 0.445132
\(324\) 0 0
\(325\) 0 0
\(326\) −6.34315 −0.351314
\(327\) 0 0
\(328\) −1.31371 −0.0725374
\(329\) −19.3137 −1.06480
\(330\) 0 0
\(331\) 17.6569 0.970508 0.485254 0.874373i \(-0.338727\pi\)
0.485254 + 0.874373i \(0.338727\pi\)
\(332\) 18.2843 1.00348
\(333\) 0 0
\(334\) 3.85786 0.211093
\(335\) 0 0
\(336\) 0 0
\(337\) 19.3137 1.05208 0.526042 0.850458i \(-0.323675\pi\)
0.526042 + 0.850458i \(0.323675\pi\)
\(338\) 7.87006 0.428075
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 44.9706 2.42818
\(344\) −5.02944 −0.271169
\(345\) 0 0
\(346\) −1.17157 −0.0629841
\(347\) −6.68629 −0.358939 −0.179469 0.983764i \(-0.557438\pi\)
−0.179469 + 0.983764i \(0.557438\pi\)
\(348\) 0 0
\(349\) −22.9706 −1.22959 −0.614793 0.788688i \(-0.710760\pi\)
−0.614793 + 0.788688i \(0.710760\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 4.41421 0.235278
\(353\) 26.0000 1.38384 0.691920 0.721974i \(-0.256765\pi\)
0.691920 + 0.721974i \(0.256765\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −14.0000 −0.741999
\(357\) 0 0
\(358\) 2.62742 0.138863
\(359\) −12.0000 −0.633336 −0.316668 0.948536i \(-0.602564\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(360\) 0 0
\(361\) −17.6274 −0.927759
\(362\) −5.79899 −0.304788
\(363\) 0 0
\(364\) 49.9411 2.61763
\(365\) 0 0
\(366\) 0 0
\(367\) 1.65685 0.0864871 0.0432435 0.999065i \(-0.486231\pi\)
0.0432435 + 0.999065i \(0.486231\pi\)
\(368\) −12.0000 −0.625543
\(369\) 0 0
\(370\) 0 0
\(371\) −64.2843 −3.33747
\(372\) 0 0
\(373\) 34.6274 1.79294 0.896470 0.443105i \(-0.146123\pi\)
0.896470 + 0.443105i \(0.146123\pi\)
\(374\) −2.82843 −0.146254
\(375\) 0 0
\(376\) 6.34315 0.327123
\(377\) 4.68629 0.241356
\(378\) 0 0
\(379\) −0.686292 −0.0352524 −0.0176262 0.999845i \(-0.505611\pi\)
−0.0176262 + 0.999845i \(0.505611\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 2.34315 0.119886
\(383\) −8.00000 −0.408781 −0.204390 0.978889i \(-0.565521\pi\)
−0.204390 + 0.978889i \(0.565521\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.970563 −0.0494003
\(387\) 0 0
\(388\) 0.627417 0.0318523
\(389\) 12.3431 0.625822 0.312911 0.949782i \(-0.398696\pi\)
0.312911 + 0.949782i \(0.398696\pi\)
\(390\) 0 0
\(391\) 27.3137 1.38131
\(392\) −25.8701 −1.30664
\(393\) 0 0
\(394\) 3.51472 0.177069
\(395\) 0 0
\(396\) 0 0
\(397\) −18.9706 −0.952105 −0.476053 0.879417i \(-0.657933\pi\)
−0.476053 + 0.879417i \(0.657933\pi\)
\(398\) −4.28427 −0.214751
\(399\) 0 0
\(400\) 0 0
\(401\) 29.3137 1.46386 0.731928 0.681382i \(-0.238621\pi\)
0.731928 + 0.681382i \(0.238621\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 8.82843 0.439231
\(405\) 0 0
\(406\) −1.65685 −0.0822283
\(407\) −0.343146 −0.0170091
\(408\) 0 0
\(409\) −8.34315 −0.412542 −0.206271 0.978495i \(-0.566133\pi\)
−0.206271 + 0.978495i \(0.566133\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 35.3137 1.73978
\(413\) 19.3137 0.950365
\(414\) 0 0
\(415\) 0 0
\(416\) −24.9706 −1.22428
\(417\) 0 0
\(418\) −0.485281 −0.0237359
\(419\) 3.02944 0.147998 0.0739988 0.997258i \(-0.476424\pi\)
0.0739988 + 0.997258i \(0.476424\pi\)
\(420\) 0 0
\(421\) −6.00000 −0.292422 −0.146211 0.989253i \(-0.546708\pi\)
−0.146211 + 0.989253i \(0.546708\pi\)
\(422\) 2.82843 0.137686
\(423\) 0 0
\(424\) 21.1127 1.02532
\(425\) 0 0
\(426\) 0 0
\(427\) −1.65685 −0.0801808
\(428\) 9.71573 0.469627
\(429\) 0 0
\(430\) 0 0
\(431\) −10.3431 −0.498212 −0.249106 0.968476i \(-0.580137\pi\)
−0.249106 + 0.968476i \(0.580137\pi\)
\(432\) 0 0
\(433\) 4.34315 0.208718 0.104359 0.994540i \(-0.466721\pi\)
0.104359 + 0.994540i \(0.466721\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 9.71573 0.465299
\(437\) 4.68629 0.224176
\(438\) 0 0
\(439\) −3.51472 −0.167748 −0.0838742 0.996476i \(-0.526729\pi\)
−0.0838742 + 0.996476i \(0.526729\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 16.0000 0.761042
\(443\) 12.0000 0.570137 0.285069 0.958507i \(-0.407984\pi\)
0.285069 + 0.958507i \(0.407984\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 7.31371 0.346314
\(447\) 0 0
\(448\) −20.1421 −0.951626
\(449\) 2.97056 0.140190 0.0700948 0.997540i \(-0.477670\pi\)
0.0700948 + 0.997540i \(0.477670\pi\)
\(450\) 0 0
\(451\) 0.828427 0.0390091
\(452\) −27.3726 −1.28750
\(453\) 0 0
\(454\) −5.79899 −0.272160
\(455\) 0 0
\(456\) 0 0
\(457\) 0.686292 0.0321034 0.0160517 0.999871i \(-0.494890\pi\)
0.0160517 + 0.999871i \(0.494890\pi\)
\(458\) −0.828427 −0.0387099
\(459\) 0 0
\(460\) 0 0
\(461\) 28.1421 1.31071 0.655355 0.755321i \(-0.272519\pi\)
0.655355 + 0.755321i \(0.272519\pi\)
\(462\) 0 0
\(463\) 28.9706 1.34638 0.673188 0.739471i \(-0.264924\pi\)
0.673188 + 0.739471i \(0.264924\pi\)
\(464\) −2.48528 −0.115376
\(465\) 0 0
\(466\) −5.45584 −0.252737
\(467\) 22.6274 1.04707 0.523536 0.852004i \(-0.324613\pi\)
0.523536 + 0.852004i \(0.324613\pi\)
\(468\) 0 0
\(469\) −27.3137 −1.26123
\(470\) 0 0
\(471\) 0 0
\(472\) −6.34315 −0.291967
\(473\) 3.17157 0.145829
\(474\) 0 0
\(475\) 0 0
\(476\) 60.2843 2.76313
\(477\) 0 0
\(478\) −2.62742 −0.120175
\(479\) −3.02944 −0.138419 −0.0692093 0.997602i \(-0.522048\pi\)
−0.0692093 + 0.997602i \(0.522048\pi\)
\(480\) 0 0
\(481\) 1.94113 0.0885077
\(482\) −9.79899 −0.446332
\(483\) 0 0
\(484\) −1.82843 −0.0831103
\(485\) 0 0
\(486\) 0 0
\(487\) −20.9706 −0.950267 −0.475133 0.879914i \(-0.657600\pi\)
−0.475133 + 0.879914i \(0.657600\pi\)
\(488\) 0.544156 0.0246328
\(489\) 0 0
\(490\) 0 0
\(491\) −25.6569 −1.15788 −0.578939 0.815371i \(-0.696533\pi\)
−0.578939 + 0.815371i \(0.696533\pi\)
\(492\) 0 0
\(493\) 5.65685 0.254772
\(494\) 2.74517 0.123511
\(495\) 0 0
\(496\) 0 0
\(497\) −65.9411 −2.95786
\(498\) 0 0
\(499\) −33.6569 −1.50669 −0.753344 0.657627i \(-0.771560\pi\)
−0.753344 + 0.657627i \(0.771560\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 5.37258 0.239790
\(503\) −5.31371 −0.236927 −0.118463 0.992958i \(-0.537797\pi\)
−0.118463 + 0.992958i \(0.537797\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −1.65685 −0.0736562
\(507\) 0 0
\(508\) 4.54416 0.201614
\(509\) −41.3137 −1.83120 −0.915599 0.402093i \(-0.868283\pi\)
−0.915599 + 0.402093i \(0.868283\pi\)
\(510\) 0 0
\(511\) 54.6274 2.41657
\(512\) 22.7574 1.00574
\(513\) 0 0
\(514\) −11.4558 −0.505296
\(515\) 0 0
\(516\) 0 0
\(517\) −4.00000 −0.175920
\(518\) −0.686292 −0.0301539
\(519\) 0 0
\(520\) 0 0
\(521\) −12.6274 −0.553217 −0.276609 0.960983i \(-0.589211\pi\)
−0.276609 + 0.960983i \(0.589211\pi\)
\(522\) 0 0
\(523\) 26.4853 1.15812 0.579060 0.815285i \(-0.303420\pi\)
0.579060 + 0.815285i \(0.303420\pi\)
\(524\) −35.3137 −1.54269
\(525\) 0 0
\(526\) 7.45584 0.325090
\(527\) 0 0
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 0 0
\(531\) 0 0
\(532\) 10.3431 0.448432
\(533\) −4.68629 −0.202986
\(534\) 0 0
\(535\) 0 0
\(536\) 8.97056 0.387469
\(537\) 0 0
\(538\) 10.2010 0.439797
\(539\) 16.3137 0.702681
\(540\) 0 0
\(541\) −5.31371 −0.228454 −0.114227 0.993455i \(-0.536439\pi\)
−0.114227 + 0.993455i \(0.536439\pi\)
\(542\) 11.5147 0.494600
\(543\) 0 0
\(544\) −30.1421 −1.29233
\(545\) 0 0
\(546\) 0 0
\(547\) −20.1421 −0.861216 −0.430608 0.902539i \(-0.641701\pi\)
−0.430608 + 0.902539i \(0.641701\pi\)
\(548\) −17.0294 −0.727462
\(549\) 0 0
\(550\) 0 0
\(551\) 0.970563 0.0413474
\(552\) 0 0
\(553\) −40.9706 −1.74225
\(554\) 5.65685 0.240337
\(555\) 0 0
\(556\) 30.1421 1.27831
\(557\) 10.8284 0.458815 0.229408 0.973330i \(-0.426321\pi\)
0.229408 + 0.973330i \(0.426321\pi\)
\(558\) 0 0
\(559\) −17.9411 −0.758829
\(560\) 0 0
\(561\) 0 0
\(562\) 6.97056 0.294035
\(563\) 20.3431 0.857361 0.428681 0.903456i \(-0.358979\pi\)
0.428681 + 0.903456i \(0.358979\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.31371 0.0552193
\(567\) 0 0
\(568\) 21.6569 0.908701
\(569\) −15.4558 −0.647943 −0.323971 0.946067i \(-0.605018\pi\)
−0.323971 + 0.946067i \(0.605018\pi\)
\(570\) 0 0
\(571\) 0.485281 0.0203084 0.0101542 0.999948i \(-0.496768\pi\)
0.0101542 + 0.999948i \(0.496768\pi\)
\(572\) 10.3431 0.432469
\(573\) 0 0
\(574\) 1.65685 0.0691558
\(575\) 0 0
\(576\) 0 0
\(577\) −14.0000 −0.582828 −0.291414 0.956597i \(-0.594126\pi\)
−0.291414 + 0.956597i \(0.594126\pi\)
\(578\) 12.2721 0.510451
\(579\) 0 0
\(580\) 0 0
\(581\) −48.2843 −2.00317
\(582\) 0 0
\(583\) −13.3137 −0.551397
\(584\) −17.9411 −0.742409
\(585\) 0 0
\(586\) −0.485281 −0.0200468
\(587\) −30.6274 −1.26413 −0.632064 0.774916i \(-0.717792\pi\)
−0.632064 + 0.774916i \(0.717792\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −1.02944 −0.0423096
\(593\) −17.1716 −0.705152 −0.352576 0.935783i \(-0.614694\pi\)
−0.352576 + 0.935783i \(0.614694\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 33.7990 1.38446
\(597\) 0 0
\(598\) 9.37258 0.383273
\(599\) −4.68629 −0.191477 −0.0957383 0.995407i \(-0.530521\pi\)
−0.0957383 + 0.995407i \(0.530521\pi\)
\(600\) 0 0
\(601\) 17.3137 0.706241 0.353120 0.935578i \(-0.385121\pi\)
0.353120 + 0.935578i \(0.385121\pi\)
\(602\) 6.34315 0.258527
\(603\) 0 0
\(604\) 0.887302 0.0361038
\(605\) 0 0
\(606\) 0 0
\(607\) −18.4853 −0.750294 −0.375147 0.926965i \(-0.622408\pi\)
−0.375147 + 0.926965i \(0.622408\pi\)
\(608\) −5.17157 −0.209735
\(609\) 0 0
\(610\) 0 0
\(611\) 22.6274 0.915407
\(612\) 0 0
\(613\) −21.9411 −0.886194 −0.443097 0.896474i \(-0.646120\pi\)
−0.443097 + 0.896474i \(0.646120\pi\)
\(614\) 3.65685 0.147579
\(615\) 0 0
\(616\) −7.65685 −0.308503
\(617\) 11.6569 0.469287 0.234644 0.972081i \(-0.424608\pi\)
0.234644 + 0.972081i \(0.424608\pi\)
\(618\) 0 0
\(619\) −25.6569 −1.03124 −0.515618 0.856819i \(-0.672438\pi\)
−0.515618 + 0.856819i \(0.672438\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −8.00000 −0.320771
\(623\) 36.9706 1.48119
\(624\) 0 0
\(625\) 0 0
\(626\) −1.79899 −0.0719021
\(627\) 0 0
\(628\) 32.9117 1.31332
\(629\) 2.34315 0.0934273
\(630\) 0 0
\(631\) 34.3431 1.36718 0.683590 0.729867i \(-0.260418\pi\)
0.683590 + 0.729867i \(0.260418\pi\)
\(632\) 13.4558 0.535245
\(633\) 0 0
\(634\) 12.5442 0.498192
\(635\) 0 0
\(636\) 0 0
\(637\) −92.2843 −3.65644
\(638\) −0.343146 −0.0135853
\(639\) 0 0
\(640\) 0 0
\(641\) 26.9706 1.06527 0.532637 0.846344i \(-0.321201\pi\)
0.532637 + 0.846344i \(0.321201\pi\)
\(642\) 0 0
\(643\) 29.9411 1.18076 0.590381 0.807124i \(-0.298977\pi\)
0.590381 + 0.807124i \(0.298977\pi\)
\(644\) 35.3137 1.39156
\(645\) 0 0
\(646\) 3.31371 0.130376
\(647\) 27.3137 1.07381 0.536906 0.843642i \(-0.319593\pi\)
0.536906 + 0.843642i \(0.319593\pi\)
\(648\) 0 0
\(649\) 4.00000 0.157014
\(650\) 0 0
\(651\) 0 0
\(652\) 28.0000 1.09656
\(653\) −26.9706 −1.05544 −0.527720 0.849418i \(-0.676953\pi\)
−0.527720 + 0.849418i \(0.676953\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.48528 0.0970339
\(657\) 0 0
\(658\) −8.00000 −0.311872
\(659\) −7.31371 −0.284902 −0.142451 0.989802i \(-0.545498\pi\)
−0.142451 + 0.989802i \(0.545498\pi\)
\(660\) 0 0
\(661\) −13.3137 −0.517843 −0.258922 0.965898i \(-0.583367\pi\)
−0.258922 + 0.965898i \(0.583367\pi\)
\(662\) 7.31371 0.284255
\(663\) 0 0
\(664\) 15.8579 0.615404
\(665\) 0 0
\(666\) 0 0
\(667\) 3.31371 0.128307
\(668\) −17.0294 −0.658889
\(669\) 0 0
\(670\) 0 0
\(671\) −0.343146 −0.0132470
\(672\) 0 0
\(673\) −29.6569 −1.14319 −0.571594 0.820537i \(-0.693675\pi\)
−0.571594 + 0.820537i \(0.693675\pi\)
\(674\) 8.00000 0.308148
\(675\) 0 0
\(676\) −34.7401 −1.33616
\(677\) −21.4558 −0.824615 −0.412308 0.911045i \(-0.635277\pi\)
−0.412308 + 0.911045i \(0.635277\pi\)
\(678\) 0 0
\(679\) −1.65685 −0.0635842
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 24.0000 0.918334 0.459167 0.888350i \(-0.348148\pi\)
0.459167 + 0.888350i \(0.348148\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 18.6274 0.711198
\(687\) 0 0
\(688\) 9.51472 0.362745
\(689\) 75.3137 2.86922
\(690\) 0 0
\(691\) 20.0000 0.760836 0.380418 0.924815i \(-0.375780\pi\)
0.380418 + 0.924815i \(0.375780\pi\)
\(692\) 5.17157 0.196594
\(693\) 0 0
\(694\) −2.76955 −0.105131
\(695\) 0 0
\(696\) 0 0
\(697\) −5.65685 −0.214269
\(698\) −9.51472 −0.360137
\(699\) 0 0
\(700\) 0 0
\(701\) −7.85786 −0.296787 −0.148394 0.988928i \(-0.547410\pi\)
−0.148394 + 0.988928i \(0.547410\pi\)
\(702\) 0 0
\(703\) 0.402020 0.0151625
\(704\) −4.17157 −0.157222
\(705\) 0 0
\(706\) 10.7696 0.405317
\(707\) −23.3137 −0.876802
\(708\) 0 0
\(709\) 29.3137 1.10090 0.550450 0.834868i \(-0.314456\pi\)
0.550450 + 0.834868i \(0.314456\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −12.1421 −0.455046
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −11.5980 −0.433437
\(717\) 0 0
\(718\) −4.97056 −0.185500
\(719\) −31.5980 −1.17841 −0.589203 0.807985i \(-0.700558\pi\)
−0.589203 + 0.807985i \(0.700558\pi\)
\(720\) 0 0
\(721\) −93.2548 −3.47299
\(722\) −7.30152 −0.271734
\(723\) 0 0
\(724\) 25.5980 0.951341
\(725\) 0 0
\(726\) 0 0
\(727\) 33.9411 1.25881 0.629403 0.777079i \(-0.283299\pi\)
0.629403 + 0.777079i \(0.283299\pi\)
\(728\) 43.3137 1.60531
\(729\) 0 0
\(730\) 0 0
\(731\) −21.6569 −0.801008
\(732\) 0 0
\(733\) 17.6569 0.652171 0.326085 0.945340i \(-0.394270\pi\)
0.326085 + 0.945340i \(0.394270\pi\)
\(734\) 0.686292 0.0253315
\(735\) 0 0
\(736\) −17.6569 −0.650840
\(737\) −5.65685 −0.208373
\(738\) 0 0
\(739\) 47.1127 1.73307 0.866534 0.499118i \(-0.166342\pi\)
0.866534 + 0.499118i \(0.166342\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −26.6274 −0.977523
\(743\) 47.6569 1.74836 0.874180 0.485602i \(-0.161399\pi\)
0.874180 + 0.485602i \(0.161399\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 14.3431 0.525140
\(747\) 0 0
\(748\) 12.4853 0.456507
\(749\) −25.6569 −0.937481
\(750\) 0 0
\(751\) −36.2843 −1.32403 −0.662016 0.749490i \(-0.730299\pi\)
−0.662016 + 0.749490i \(0.730299\pi\)
\(752\) −12.0000 −0.437595
\(753\) 0 0
\(754\) 1.94113 0.0706916
\(755\) 0 0
\(756\) 0 0
\(757\) −8.62742 −0.313569 −0.156784 0.987633i \(-0.550113\pi\)
−0.156784 + 0.987633i \(0.550113\pi\)
\(758\) −0.284271 −0.0103252
\(759\) 0 0
\(760\) 0 0
\(761\) 23.1716 0.839969 0.419984 0.907531i \(-0.362036\pi\)
0.419984 + 0.907531i \(0.362036\pi\)
\(762\) 0 0
\(763\) −25.6569 −0.928840
\(764\) −10.3431 −0.374202
\(765\) 0 0
\(766\) −3.31371 −0.119729
\(767\) −22.6274 −0.817029
\(768\) 0 0
\(769\) 33.3137 1.20132 0.600662 0.799503i \(-0.294904\pi\)
0.600662 + 0.799503i \(0.294904\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 4.28427 0.154194
\(773\) −7.65685 −0.275398 −0.137699 0.990474i \(-0.543971\pi\)
−0.137699 + 0.990474i \(0.543971\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0.544156 0.0195341
\(777\) 0 0
\(778\) 5.11270 0.183299
\(779\) −0.970563 −0.0347740
\(780\) 0 0
\(781\) −13.6569 −0.488681
\(782\) 11.3137 0.404577
\(783\) 0 0
\(784\) 48.9411 1.74790
\(785\) 0 0
\(786\) 0 0
\(787\) −8.14214 −0.290236 −0.145118 0.989414i \(-0.546356\pi\)
−0.145118 + 0.989414i \(0.546356\pi\)
\(788\) −15.5147 −0.552689
\(789\) 0 0
\(790\) 0 0
\(791\) 72.2843 2.57013
\(792\) 0 0
\(793\) 1.94113 0.0689314
\(794\) −7.85786 −0.278865
\(795\) 0 0
\(796\) 18.9117 0.670307
\(797\) 1.02944 0.0364645 0.0182323 0.999834i \(-0.494196\pi\)
0.0182323 + 0.999834i \(0.494196\pi\)
\(798\) 0 0
\(799\) 27.3137 0.966290
\(800\) 0 0
\(801\) 0 0
\(802\) 12.1421 0.428754
\(803\) 11.3137 0.399252
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 7.65685 0.269367
\(809\) 56.4264 1.98385 0.991923 0.126838i \(-0.0404829\pi\)
0.991923 + 0.126838i \(0.0404829\pi\)
\(810\) 0 0
\(811\) −16.4853 −0.578877 −0.289438 0.957197i \(-0.593468\pi\)
−0.289438 + 0.957197i \(0.593468\pi\)
\(812\) 7.31371 0.256661
\(813\) 0 0
\(814\) −0.142136 −0.00498185
\(815\) 0 0
\(816\) 0 0
\(817\) −3.71573 −0.129997
\(818\) −3.45584 −0.120831
\(819\) 0 0
\(820\) 0 0
\(821\) 7.17157 0.250290 0.125145 0.992138i \(-0.460060\pi\)
0.125145 + 0.992138i \(0.460060\pi\)
\(822\) 0 0
\(823\) −16.0000 −0.557725 −0.278862 0.960331i \(-0.589957\pi\)
−0.278862 + 0.960331i \(0.589957\pi\)
\(824\) 30.6274 1.06696
\(825\) 0 0
\(826\) 8.00000 0.278356
\(827\) 18.6863 0.649786 0.324893 0.945751i \(-0.394672\pi\)
0.324893 + 0.945751i \(0.394672\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 23.5980 0.818113
\(833\) −111.397 −3.85968
\(834\) 0 0
\(835\) 0 0
\(836\) 2.14214 0.0740873
\(837\) 0 0
\(838\) 1.25483 0.0433475
\(839\) −22.6274 −0.781185 −0.390593 0.920564i \(-0.627730\pi\)
−0.390593 + 0.920564i \(0.627730\pi\)
\(840\) 0 0
\(841\) −28.3137 −0.976335
\(842\) −2.48528 −0.0856485
\(843\) 0 0
\(844\) −12.4853 −0.429761
\(845\) 0 0
\(846\) 0 0
\(847\) 4.82843 0.165907
\(848\) −39.9411 −1.37158
\(849\) 0 0
\(850\) 0 0
\(851\) 1.37258 0.0470515
\(852\) 0 0
\(853\) 31.3137 1.07216 0.536080 0.844167i \(-0.319904\pi\)
0.536080 + 0.844167i \(0.319904\pi\)
\(854\) −0.686292 −0.0234844
\(855\) 0 0
\(856\) 8.42641 0.288009
\(857\) −11.5147 −0.393335 −0.196668 0.980470i \(-0.563012\pi\)
−0.196668 + 0.980470i \(0.563012\pi\)
\(858\) 0 0
\(859\) −19.0294 −0.649276 −0.324638 0.945838i \(-0.605242\pi\)
−0.324638 + 0.945838i \(0.605242\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −4.28427 −0.145923
\(863\) −43.3137 −1.47442 −0.737208 0.675666i \(-0.763856\pi\)
−0.737208 + 0.675666i \(0.763856\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 1.79899 0.0611322
\(867\) 0 0
\(868\) 0 0
\(869\) −8.48528 −0.287843
\(870\) 0 0
\(871\) 32.0000 1.08428
\(872\) 8.42641 0.285354
\(873\) 0 0
\(874\) 1.94113 0.0656595
\(875\) 0 0
\(876\) 0 0
\(877\) 42.6274 1.43943 0.719713 0.694272i \(-0.244273\pi\)
0.719713 + 0.694272i \(0.244273\pi\)
\(878\) −1.45584 −0.0491324
\(879\) 0 0
\(880\) 0 0
\(881\) 13.0294 0.438973 0.219486 0.975616i \(-0.429562\pi\)
0.219486 + 0.975616i \(0.429562\pi\)
\(882\) 0 0
\(883\) 50.6274 1.70375 0.851874 0.523747i \(-0.175466\pi\)
0.851874 + 0.523747i \(0.175466\pi\)
\(884\) −70.6274 −2.37546
\(885\) 0 0
\(886\) 4.97056 0.166989
\(887\) −4.34315 −0.145829 −0.0729143 0.997338i \(-0.523230\pi\)
−0.0729143 + 0.997338i \(0.523230\pi\)
\(888\) 0 0
\(889\) −12.0000 −0.402467
\(890\) 0 0
\(891\) 0 0
\(892\) −32.2843 −1.08096
\(893\) 4.68629 0.156821
\(894\) 0 0
\(895\) 0 0
\(896\) −50.9706 −1.70281
\(897\) 0 0
\(898\) 1.23045 0.0410606
\(899\) 0 0
\(900\) 0 0
\(901\) 90.9117 3.02871
\(902\) 0.343146 0.0114255
\(903\) 0 0
\(904\) −23.7401 −0.789584
\(905\) 0 0
\(906\) 0 0
\(907\) −7.02944 −0.233409 −0.116704 0.993167i \(-0.537233\pi\)
−0.116704 + 0.993167i \(0.537233\pi\)
\(908\) 25.5980 0.849499
\(909\) 0 0
\(910\) 0 0
\(911\) 15.0294 0.497947 0.248974 0.968510i \(-0.419907\pi\)
0.248974 + 0.968510i \(0.419907\pi\)
\(912\) 0 0
\(913\) −10.0000 −0.330952
\(914\) 0.284271 0.00940286
\(915\) 0 0
\(916\) 3.65685 0.120826
\(917\) 93.2548 3.07955
\(918\) 0 0
\(919\) 28.4853 0.939643 0.469821 0.882762i \(-0.344318\pi\)
0.469821 + 0.882762i \(0.344318\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 11.6569 0.383898
\(923\) 77.2548 2.54287
\(924\) 0 0
\(925\) 0 0
\(926\) 12.0000 0.394344
\(927\) 0 0
\(928\) −3.65685 −0.120042
\(929\) −33.5980 −1.10231 −0.551157 0.834402i \(-0.685813\pi\)
−0.551157 + 0.834402i \(0.685813\pi\)
\(930\) 0 0
\(931\) −19.1127 −0.626393
\(932\) 24.0833 0.788873
\(933\) 0 0
\(934\) 9.37258 0.306680
\(935\) 0 0
\(936\) 0 0
\(937\) −44.9706 −1.46912 −0.734562 0.678541i \(-0.762612\pi\)
−0.734562 + 0.678541i \(0.762612\pi\)
\(938\) −11.3137 −0.369406
\(939\) 0 0
\(940\) 0 0
\(941\) −38.7696 −1.26385 −0.631926 0.775029i \(-0.717735\pi\)
−0.631926 + 0.775029i \(0.717735\pi\)
\(942\) 0 0
\(943\) −3.31371 −0.107909
\(944\) 12.0000 0.390567
\(945\) 0 0
\(946\) 1.31371 0.0427123
\(947\) −38.6274 −1.25522 −0.627611 0.778527i \(-0.715967\pi\)
−0.627611 + 0.778527i \(0.715967\pi\)
\(948\) 0 0
\(949\) −64.0000 −2.07753
\(950\) 0 0
\(951\) 0 0
\(952\) 52.2843 1.69454
\(953\) 27.7990 0.900498 0.450249 0.892903i \(-0.351335\pi\)
0.450249 + 0.892903i \(0.351335\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 11.5980 0.375105
\(957\) 0 0
\(958\) −1.25483 −0.0405418
\(959\) 44.9706 1.45218
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0.804041 0.0259233
\(963\) 0 0
\(964\) 43.2548 1.39314
\(965\) 0 0
\(966\) 0 0
\(967\) 39.4558 1.26881 0.634407 0.772999i \(-0.281244\pi\)
0.634407 + 0.772999i \(0.281244\pi\)
\(968\) −1.58579 −0.0509691
\(969\) 0 0
\(970\) 0 0
\(971\) −10.6274 −0.341050 −0.170525 0.985353i \(-0.554546\pi\)
−0.170525 + 0.985353i \(0.554546\pi\)
\(972\) 0 0
\(973\) −79.5980 −2.55179
\(974\) −8.68629 −0.278327
\(975\) 0 0
\(976\) −1.02944 −0.0329515
\(977\) 25.3137 0.809857 0.404929 0.914348i \(-0.367296\pi\)
0.404929 + 0.914348i \(0.367296\pi\)
\(978\) 0 0
\(979\) 7.65685 0.244714
\(980\) 0 0
\(981\) 0 0
\(982\) −10.6274 −0.339135
\(983\) 14.6274 0.466542 0.233271 0.972412i \(-0.425057\pi\)
0.233271 + 0.972412i \(0.425057\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 2.34315 0.0746210
\(987\) 0 0
\(988\) −12.1177 −0.385517
\(989\) −12.6863 −0.403401
\(990\) 0 0
\(991\) 14.6274 0.464655 0.232328 0.972638i \(-0.425366\pi\)
0.232328 + 0.972638i \(0.425366\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −27.3137 −0.866338
\(995\) 0 0
\(996\) 0 0
\(997\) 16.6863 0.528460 0.264230 0.964460i \(-0.414882\pi\)
0.264230 + 0.964460i \(0.414882\pi\)
\(998\) −13.9411 −0.441299
\(999\) 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.a.m.1.2 2
3.2 odd 2 825.2.a.g.1.1 2
5.2 odd 4 2475.2.c.m.199.3 4
5.3 odd 4 2475.2.c.m.199.2 4
5.4 even 2 495.2.a.d.1.1 2
15.2 even 4 825.2.c.e.199.2 4
15.8 even 4 825.2.c.e.199.3 4
15.14 odd 2 165.2.a.a.1.2 2
20.19 odd 2 7920.2.a.cg.1.2 2
33.32 even 2 9075.2.a.v.1.2 2
55.54 odd 2 5445.2.a.m.1.2 2
60.59 even 2 2640.2.a.bb.1.2 2
105.104 even 2 8085.2.a.ba.1.2 2
165.164 even 2 1815.2.a.k.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.a.a.1.2 2 15.14 odd 2
495.2.a.d.1.1 2 5.4 even 2
825.2.a.g.1.1 2 3.2 odd 2
825.2.c.e.199.2 4 15.2 even 4
825.2.c.e.199.3 4 15.8 even 4
1815.2.a.k.1.1 2 165.164 even 2
2475.2.a.m.1.2 2 1.1 even 1 trivial
2475.2.c.m.199.2 4 5.3 odd 4
2475.2.c.m.199.3 4 5.2 odd 4
2640.2.a.bb.1.2 2 60.59 even 2
5445.2.a.m.1.2 2 55.54 odd 2
7920.2.a.cg.1.2 2 20.19 odd 2
8085.2.a.ba.1.2 2 105.104 even 2
9075.2.a.v.1.2 2 33.32 even 2