Properties

Label 4851.2.a.be.1.1
Level $4851$
Weight $2$
Character 4851.1
Self dual yes
Analytic conductor $38.735$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4851 = 3^{2} \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4851.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(38.7354300205\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 231)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 4851.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{5} +1.58579 q^{8} +O(q^{10})\) \(q-0.414214 q^{2} -1.82843 q^{4} +2.00000 q^{5} +1.58579 q^{8} -0.828427 q^{10} -1.00000 q^{11} +4.82843 q^{13} +3.00000 q^{16} +1.58579 q^{17} -1.24264 q^{19} -3.65685 q^{20} +0.414214 q^{22} +7.00000 q^{23} -1.00000 q^{25} -2.00000 q^{26} +5.24264 q^{29} -5.65685 q^{31} -4.41421 q^{32} -0.656854 q^{34} +7.48528 q^{37} +0.514719 q^{38} +3.17157 q^{40} +6.82843 q^{41} -11.2426 q^{43} +1.82843 q^{44} -2.89949 q^{46} -4.17157 q^{47} +0.414214 q^{50} -8.82843 q^{52} +12.8284 q^{53} -2.00000 q^{55} -2.17157 q^{58} -2.65685 q^{59} -4.00000 q^{61} +2.34315 q^{62} -4.17157 q^{64} +9.65685 q^{65} -8.82843 q^{67} -2.89949 q^{68} +9.82843 q^{71} +11.6569 q^{73} -3.10051 q^{74} +2.27208 q^{76} -9.31371 q^{79} +6.00000 q^{80} -2.82843 q^{82} -2.82843 q^{83} +3.17157 q^{85} +4.65685 q^{86} -1.58579 q^{88} -14.1421 q^{89} -12.7990 q^{92} +1.72792 q^{94} -2.48528 q^{95} +5.48528 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{4} + 4 q^{5} + 6 q^{8} + 4 q^{10} - 2 q^{11} + 4 q^{13} + 6 q^{16} + 6 q^{17} + 6 q^{19} + 4 q^{20} - 2 q^{22} + 14 q^{23} - 2 q^{25} - 4 q^{26} + 2 q^{29} - 6 q^{32} + 10 q^{34} - 2 q^{37} + 18 q^{38} + 12 q^{40} + 8 q^{41} - 14 q^{43} - 2 q^{44} + 14 q^{46} - 14 q^{47} - 2 q^{50} - 12 q^{52} + 20 q^{53} - 4 q^{55} - 10 q^{58} + 6 q^{59} - 8 q^{61} + 16 q^{62} - 14 q^{64} + 8 q^{65} - 12 q^{67} + 14 q^{68} + 14 q^{71} + 12 q^{73} - 26 q^{74} + 30 q^{76} + 4 q^{79} + 12 q^{80} + 12 q^{85} - 2 q^{86} - 6 q^{88} + 14 q^{92} - 22 q^{94} + 12 q^{95} - 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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 \(-0.546784\pi\)
−0.146447 + 0.989219i \(0.546784\pi\)
\(3\) 0 0
\(4\) −1.82843 −0.914214
\(5\) 2.00000 0.894427 0.447214 0.894427i \(-0.352416\pi\)
0.447214 + 0.894427i \(0.352416\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 1.58579 0.560660
\(9\) 0 0
\(10\) −0.828427 −0.261972
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) 4.82843 1.33916 0.669582 0.742738i \(-0.266473\pi\)
0.669582 + 0.742738i \(0.266473\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 3.00000 0.750000
\(17\) 1.58579 0.384610 0.192305 0.981335i \(-0.438404\pi\)
0.192305 + 0.981335i \(0.438404\pi\)
\(18\) 0 0
\(19\) −1.24264 −0.285081 −0.142541 0.989789i \(-0.545527\pi\)
−0.142541 + 0.989789i \(0.545527\pi\)
\(20\) −3.65685 −0.817697
\(21\) 0 0
\(22\) 0.414214 0.0883106
\(23\) 7.00000 1.45960 0.729800 0.683660i \(-0.239613\pi\)
0.729800 + 0.683660i \(0.239613\pi\)
\(24\) 0 0
\(25\) −1.00000 −0.200000
\(26\) −2.00000 −0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) 5.24264 0.973534 0.486767 0.873532i \(-0.338176\pi\)
0.486767 + 0.873532i \(0.338176\pi\)
\(30\) 0 0
\(31\) −5.65685 −1.01600 −0.508001 0.861357i \(-0.669615\pi\)
−0.508001 + 0.861357i \(0.669615\pi\)
\(32\) −4.41421 −0.780330
\(33\) 0 0
\(34\) −0.656854 −0.112650
\(35\) 0 0
\(36\) 0 0
\(37\) 7.48528 1.23057 0.615286 0.788304i \(-0.289040\pi\)
0.615286 + 0.788304i \(0.289040\pi\)
\(38\) 0.514719 0.0834984
\(39\) 0 0
\(40\) 3.17157 0.501470
\(41\) 6.82843 1.06642 0.533211 0.845983i \(-0.320985\pi\)
0.533211 + 0.845983i \(0.320985\pi\)
\(42\) 0 0
\(43\) −11.2426 −1.71449 −0.857243 0.514912i \(-0.827825\pi\)
−0.857243 + 0.514912i \(0.827825\pi\)
\(44\) 1.82843 0.275646
\(45\) 0 0
\(46\) −2.89949 −0.427507
\(47\) −4.17157 −0.608486 −0.304243 0.952594i \(-0.598404\pi\)
−0.304243 + 0.952594i \(0.598404\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0.414214 0.0585786
\(51\) 0 0
\(52\) −8.82843 −1.22428
\(53\) 12.8284 1.76212 0.881060 0.473005i \(-0.156831\pi\)
0.881060 + 0.473005i \(0.156831\pi\)
\(54\) 0 0
\(55\) −2.00000 −0.269680
\(56\) 0 0
\(57\) 0 0
\(58\) −2.17157 −0.285141
\(59\) −2.65685 −0.345893 −0.172946 0.984931i \(-0.555329\pi\)
−0.172946 + 0.984931i \(0.555329\pi\)
\(60\) 0 0
\(61\) −4.00000 −0.512148 −0.256074 0.966657i \(-0.582429\pi\)
−0.256074 + 0.966657i \(0.582429\pi\)
\(62\) 2.34315 0.297580
\(63\) 0 0
\(64\) −4.17157 −0.521447
\(65\) 9.65685 1.19779
\(66\) 0 0
\(67\) −8.82843 −1.07856 −0.539282 0.842125i \(-0.681304\pi\)
−0.539282 + 0.842125i \(0.681304\pi\)
\(68\) −2.89949 −0.351615
\(69\) 0 0
\(70\) 0 0
\(71\) 9.82843 1.16642 0.583210 0.812322i \(-0.301797\pi\)
0.583210 + 0.812322i \(0.301797\pi\)
\(72\) 0 0
\(73\) 11.6569 1.36433 0.682166 0.731198i \(-0.261038\pi\)
0.682166 + 0.731198i \(0.261038\pi\)
\(74\) −3.10051 −0.360426
\(75\) 0 0
\(76\) 2.27208 0.260625
\(77\) 0 0
\(78\) 0 0
\(79\) −9.31371 −1.04787 −0.523937 0.851757i \(-0.675537\pi\)
−0.523937 + 0.851757i \(0.675537\pi\)
\(80\) 6.00000 0.670820
\(81\) 0 0
\(82\) −2.82843 −0.312348
\(83\) −2.82843 −0.310460 −0.155230 0.987878i \(-0.549612\pi\)
−0.155230 + 0.987878i \(0.549612\pi\)
\(84\) 0 0
\(85\) 3.17157 0.344005
\(86\) 4.65685 0.502162
\(87\) 0 0
\(88\) −1.58579 −0.169045
\(89\) −14.1421 −1.49906 −0.749532 0.661968i \(-0.769721\pi\)
−0.749532 + 0.661968i \(0.769721\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −12.7990 −1.33439
\(93\) 0 0
\(94\) 1.72792 0.178222
\(95\) −2.48528 −0.254984
\(96\) 0 0
\(97\) 5.48528 0.556946 0.278473 0.960444i \(-0.410172\pi\)
0.278473 + 0.960444i \(0.410172\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.82843 0.182843
\(101\) 14.8995 1.48256 0.741278 0.671199i \(-0.234220\pi\)
0.741278 + 0.671199i \(0.234220\pi\)
\(102\) 0 0
\(103\) −4.48528 −0.441948 −0.220974 0.975280i \(-0.570924\pi\)
−0.220974 + 0.975280i \(0.570924\pi\)
\(104\) 7.65685 0.750816
\(105\) 0 0
\(106\) −5.31371 −0.516113
\(107\) 1.65685 0.160174 0.0800871 0.996788i \(-0.474480\pi\)
0.0800871 + 0.996788i \(0.474480\pi\)
\(108\) 0 0
\(109\) −1.17157 −0.112216 −0.0561082 0.998425i \(-0.517869\pi\)
−0.0561082 + 0.998425i \(0.517869\pi\)
\(110\) 0.828427 0.0789874
\(111\) 0 0
\(112\) 0 0
\(113\) −3.65685 −0.344008 −0.172004 0.985096i \(-0.555024\pi\)
−0.172004 + 0.985096i \(0.555024\pi\)
\(114\) 0 0
\(115\) 14.0000 1.30551
\(116\) −9.58579 −0.890018
\(117\) 0 0
\(118\) 1.10051 0.101310
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 1.65685 0.150005
\(123\) 0 0
\(124\) 10.3431 0.928842
\(125\) −12.0000 −1.07331
\(126\) 0 0
\(127\) −1.92893 −0.171165 −0.0855825 0.996331i \(-0.527275\pi\)
−0.0855825 + 0.996331i \(0.527275\pi\)
\(128\) 10.5563 0.933058
\(129\) 0 0
\(130\) −4.00000 −0.350823
\(131\) 10.8284 0.946084 0.473042 0.881040i \(-0.343156\pi\)
0.473042 + 0.881040i \(0.343156\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 3.65685 0.315904
\(135\) 0 0
\(136\) 2.51472 0.215635
\(137\) 16.4853 1.40843 0.704216 0.709985i \(-0.251299\pi\)
0.704216 + 0.709985i \(0.251299\pi\)
\(138\) 0 0
\(139\) 5.58579 0.473780 0.236890 0.971536i \(-0.423872\pi\)
0.236890 + 0.971536i \(0.423872\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4.07107 −0.341636
\(143\) −4.82843 −0.403773
\(144\) 0 0
\(145\) 10.4853 0.870755
\(146\) −4.82843 −0.399603
\(147\) 0 0
\(148\) −13.6863 −1.12501
\(149\) −20.2132 −1.65593 −0.827965 0.560780i \(-0.810501\pi\)
−0.827965 + 0.560780i \(0.810501\pi\)
\(150\) 0 0
\(151\) 0.899495 0.0731999 0.0365999 0.999330i \(-0.488347\pi\)
0.0365999 + 0.999330i \(0.488347\pi\)
\(152\) −1.97056 −0.159834
\(153\) 0 0
\(154\) 0 0
\(155\) −11.3137 −0.908739
\(156\) 0 0
\(157\) 7.00000 0.558661 0.279330 0.960195i \(-0.409888\pi\)
0.279330 + 0.960195i \(0.409888\pi\)
\(158\) 3.85786 0.306915
\(159\) 0 0
\(160\) −8.82843 −0.697948
\(161\) 0 0
\(162\) 0 0
\(163\) 12.9706 1.01593 0.507966 0.861377i \(-0.330397\pi\)
0.507966 + 0.861377i \(0.330397\pi\)
\(164\) −12.4853 −0.974937
\(165\) 0 0
\(166\) 1.17157 0.0909317
\(167\) −10.8284 −0.837929 −0.418964 0.908003i \(-0.637607\pi\)
−0.418964 + 0.908003i \(0.637607\pi\)
\(168\) 0 0
\(169\) 10.3137 0.793362
\(170\) −1.31371 −0.100757
\(171\) 0 0
\(172\) 20.5563 1.56741
\(173\) 9.17157 0.697302 0.348651 0.937253i \(-0.386640\pi\)
0.348651 + 0.937253i \(0.386640\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −3.00000 −0.226134
\(177\) 0 0
\(178\) 5.85786 0.439065
\(179\) −12.1716 −0.909746 −0.454873 0.890556i \(-0.650315\pi\)
−0.454873 + 0.890556i \(0.650315\pi\)
\(180\) 0 0
\(181\) −0.343146 −0.0255058 −0.0127529 0.999919i \(-0.504059\pi\)
−0.0127529 + 0.999919i \(0.504059\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 11.1005 0.818340
\(185\) 14.9706 1.10066
\(186\) 0 0
\(187\) −1.58579 −0.115964
\(188\) 7.62742 0.556287
\(189\) 0 0
\(190\) 1.02944 0.0746832
\(191\) 10.3431 0.748404 0.374202 0.927347i \(-0.377917\pi\)
0.374202 + 0.927347i \(0.377917\pi\)
\(192\) 0 0
\(193\) 16.1421 1.16194 0.580968 0.813926i \(-0.302674\pi\)
0.580968 + 0.813926i \(0.302674\pi\)
\(194\) −2.27208 −0.163126
\(195\) 0 0
\(196\) 0 0
\(197\) 13.5858 0.967947 0.483974 0.875083i \(-0.339193\pi\)
0.483974 + 0.875083i \(0.339193\pi\)
\(198\) 0 0
\(199\) 19.7990 1.40351 0.701757 0.712417i \(-0.252399\pi\)
0.701757 + 0.712417i \(0.252399\pi\)
\(200\) −1.58579 −0.112132
\(201\) 0 0
\(202\) −6.17157 −0.434230
\(203\) 0 0
\(204\) 0 0
\(205\) 13.6569 0.953836
\(206\) 1.85786 0.129444
\(207\) 0 0
\(208\) 14.4853 1.00437
\(209\) 1.24264 0.0859553
\(210\) 0 0
\(211\) −18.9706 −1.30599 −0.652994 0.757363i \(-0.726487\pi\)
−0.652994 + 0.757363i \(0.726487\pi\)
\(212\) −23.4558 −1.61095
\(213\) 0 0
\(214\) −0.686292 −0.0469139
\(215\) −22.4853 −1.53348
\(216\) 0 0
\(217\) 0 0
\(218\) 0.485281 0.0328674
\(219\) 0 0
\(220\) 3.65685 0.246545
\(221\) 7.65685 0.515056
\(222\) 0 0
\(223\) 10.9706 0.734643 0.367322 0.930094i \(-0.380275\pi\)
0.367322 + 0.930094i \(0.380275\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.51472 0.100758
\(227\) 18.9706 1.25912 0.629560 0.776952i \(-0.283235\pi\)
0.629560 + 0.776952i \(0.283235\pi\)
\(228\) 0 0
\(229\) −4.34315 −0.287003 −0.143502 0.989650i \(-0.545836\pi\)
−0.143502 + 0.989650i \(0.545836\pi\)
\(230\) −5.79899 −0.382374
\(231\) 0 0
\(232\) 8.31371 0.545822
\(233\) −7.58579 −0.496961 −0.248481 0.968637i \(-0.579931\pi\)
−0.248481 + 0.968637i \(0.579931\pi\)
\(234\) 0 0
\(235\) −8.34315 −0.544247
\(236\) 4.85786 0.316220
\(237\) 0 0
\(238\) 0 0
\(239\) −10.4853 −0.678236 −0.339118 0.940744i \(-0.610129\pi\)
−0.339118 + 0.940744i \(0.610129\pi\)
\(240\) 0 0
\(241\) −15.3137 −0.986443 −0.493221 0.869904i \(-0.664181\pi\)
−0.493221 + 0.869904i \(0.664181\pi\)
\(242\) −0.414214 −0.0266267
\(243\) 0 0
\(244\) 7.31371 0.468212
\(245\) 0 0
\(246\) 0 0
\(247\) −6.00000 −0.381771
\(248\) −8.97056 −0.569631
\(249\) 0 0
\(250\) 4.97056 0.314366
\(251\) 19.4853 1.22990 0.614950 0.788566i \(-0.289176\pi\)
0.614950 + 0.788566i \(0.289176\pi\)
\(252\) 0 0
\(253\) −7.00000 −0.440086
\(254\) 0.798990 0.0501331
\(255\) 0 0
\(256\) 3.97056 0.248160
\(257\) 29.4558 1.83741 0.918703 0.394950i \(-0.129238\pi\)
0.918703 + 0.394950i \(0.129238\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −17.6569 −1.09503
\(261\) 0 0
\(262\) −4.48528 −0.277102
\(263\) −18.0000 −1.10993 −0.554964 0.831875i \(-0.687268\pi\)
−0.554964 + 0.831875i \(0.687268\pi\)
\(264\) 0 0
\(265\) 25.6569 1.57609
\(266\) 0 0
\(267\) 0 0
\(268\) 16.1421 0.986038
\(269\) −11.7990 −0.719397 −0.359699 0.933069i \(-0.617120\pi\)
−0.359699 + 0.933069i \(0.617120\pi\)
\(270\) 0 0
\(271\) 10.9706 0.666414 0.333207 0.942854i \(-0.391869\pi\)
0.333207 + 0.942854i \(0.391869\pi\)
\(272\) 4.75736 0.288457
\(273\) 0 0
\(274\) −6.82843 −0.412520
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −14.3431 −0.861796 −0.430898 0.902401i \(-0.641803\pi\)
−0.430898 + 0.902401i \(0.641803\pi\)
\(278\) −2.31371 −0.138767
\(279\) 0 0
\(280\) 0 0
\(281\) −11.5858 −0.691150 −0.345575 0.938391i \(-0.612316\pi\)
−0.345575 + 0.938391i \(0.612316\pi\)
\(282\) 0 0
\(283\) 31.6569 1.88180 0.940902 0.338678i \(-0.109980\pi\)
0.940902 + 0.338678i \(0.109980\pi\)
\(284\) −17.9706 −1.06636
\(285\) 0 0
\(286\) 2.00000 0.118262
\(287\) 0 0
\(288\) 0 0
\(289\) −14.4853 −0.852075
\(290\) −4.34315 −0.255038
\(291\) 0 0
\(292\) −21.3137 −1.24729
\(293\) 19.7279 1.15252 0.576259 0.817267i \(-0.304512\pi\)
0.576259 + 0.817267i \(0.304512\pi\)
\(294\) 0 0
\(295\) −5.31371 −0.309376
\(296\) 11.8701 0.689933
\(297\) 0 0
\(298\) 8.37258 0.485011
\(299\) 33.7990 1.95465
\(300\) 0 0
\(301\) 0 0
\(302\) −0.372583 −0.0214397
\(303\) 0 0
\(304\) −3.72792 −0.213811
\(305\) −8.00000 −0.458079
\(306\) 0 0
\(307\) 5.31371 0.303269 0.151635 0.988437i \(-0.451546\pi\)
0.151635 + 0.988437i \(0.451546\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 4.68629 0.266163
\(311\) 15.6274 0.886150 0.443075 0.896485i \(-0.353888\pi\)
0.443075 + 0.896485i \(0.353888\pi\)
\(312\) 0 0
\(313\) −34.7990 −1.96696 −0.983478 0.181030i \(-0.942057\pi\)
−0.983478 + 0.181030i \(0.942057\pi\)
\(314\) −2.89949 −0.163628
\(315\) 0 0
\(316\) 17.0294 0.957981
\(317\) 2.68629 0.150877 0.0754386 0.997150i \(-0.475964\pi\)
0.0754386 + 0.997150i \(0.475964\pi\)
\(318\) 0 0
\(319\) −5.24264 −0.293532
\(320\) −8.34315 −0.466396
\(321\) 0 0
\(322\) 0 0
\(323\) −1.97056 −0.109645
\(324\) 0 0
\(325\) −4.82843 −0.267833
\(326\) −5.37258 −0.297560
\(327\) 0 0
\(328\) 10.8284 0.597900
\(329\) 0 0
\(330\) 0 0
\(331\) 5.51472 0.303116 0.151558 0.988448i \(-0.451571\pi\)
0.151558 + 0.988448i \(0.451571\pi\)
\(332\) 5.17157 0.283827
\(333\) 0 0
\(334\) 4.48528 0.245424
\(335\) −17.6569 −0.964697
\(336\) 0 0
\(337\) 32.1421 1.75089 0.875447 0.483314i \(-0.160567\pi\)
0.875447 + 0.483314i \(0.160567\pi\)
\(338\) −4.27208 −0.232370
\(339\) 0 0
\(340\) −5.79899 −0.314494
\(341\) 5.65685 0.306336
\(342\) 0 0
\(343\) 0 0
\(344\) −17.8284 −0.961244
\(345\) 0 0
\(346\) −3.79899 −0.204235
\(347\) 18.9706 1.01839 0.509197 0.860650i \(-0.329943\pi\)
0.509197 + 0.860650i \(0.329943\pi\)
\(348\) 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
\(353\) 9.31371 0.495719 0.247859 0.968796i \(-0.420273\pi\)
0.247859 + 0.968796i \(0.420273\pi\)
\(354\) 0 0
\(355\) 19.6569 1.04328
\(356\) 25.8579 1.37046
\(357\) 0 0
\(358\) 5.04163 0.266458
\(359\) 8.00000 0.422224 0.211112 0.977462i \(-0.432292\pi\)
0.211112 + 0.977462i \(0.432292\pi\)
\(360\) 0 0
\(361\) −17.4558 −0.918729
\(362\) 0.142136 0.00747048
\(363\) 0 0
\(364\) 0 0
\(365\) 23.3137 1.22030
\(366\) 0 0
\(367\) 24.1421 1.26021 0.630105 0.776510i \(-0.283012\pi\)
0.630105 + 0.776510i \(0.283012\pi\)
\(368\) 21.0000 1.09470
\(369\) 0 0
\(370\) −6.20101 −0.322375
\(371\) 0 0
\(372\) 0 0
\(373\) 9.65685 0.500013 0.250006 0.968244i \(-0.419567\pi\)
0.250006 + 0.968244i \(0.419567\pi\)
\(374\) 0.656854 0.0339651
\(375\) 0 0
\(376\) −6.61522 −0.341154
\(377\) 25.3137 1.30372
\(378\) 0 0
\(379\) 13.1716 0.676578 0.338289 0.941042i \(-0.390152\pi\)
0.338289 + 0.941042i \(0.390152\pi\)
\(380\) 4.54416 0.233110
\(381\) 0 0
\(382\) −4.28427 −0.219202
\(383\) 20.3137 1.03798 0.518991 0.854780i \(-0.326308\pi\)
0.518991 + 0.854780i \(0.326308\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.68629 −0.340323
\(387\) 0 0
\(388\) −10.0294 −0.509168
\(389\) −27.7990 −1.40946 −0.704732 0.709473i \(-0.748933\pi\)
−0.704732 + 0.709473i \(0.748933\pi\)
\(390\) 0 0
\(391\) 11.1005 0.561377
\(392\) 0 0
\(393\) 0 0
\(394\) −5.62742 −0.283505
\(395\) −18.6274 −0.937247
\(396\) 0 0
\(397\) 0.514719 0.0258330 0.0129165 0.999917i \(-0.495888\pi\)
0.0129165 + 0.999917i \(0.495888\pi\)
\(398\) −8.20101 −0.411079
\(399\) 0 0
\(400\) −3.00000 −0.150000
\(401\) −24.4853 −1.22274 −0.611368 0.791346i \(-0.709381\pi\)
−0.611368 + 0.791346i \(0.709381\pi\)
\(402\) 0 0
\(403\) −27.3137 −1.36059
\(404\) −27.2426 −1.35537
\(405\) 0 0
\(406\) 0 0
\(407\) −7.48528 −0.371032
\(408\) 0 0
\(409\) 31.7990 1.57236 0.786179 0.617998i \(-0.212056\pi\)
0.786179 + 0.617998i \(0.212056\pi\)
\(410\) −5.65685 −0.279372
\(411\) 0 0
\(412\) 8.20101 0.404035
\(413\) 0 0
\(414\) 0 0
\(415\) −5.65685 −0.277684
\(416\) −21.3137 −1.04499
\(417\) 0 0
\(418\) −0.514719 −0.0251757
\(419\) 24.7990 1.21151 0.605755 0.795651i \(-0.292871\pi\)
0.605755 + 0.795651i \(0.292871\pi\)
\(420\) 0 0
\(421\) −7.00000 −0.341159 −0.170580 0.985344i \(-0.554564\pi\)
−0.170580 + 0.985344i \(0.554564\pi\)
\(422\) 7.85786 0.382515
\(423\) 0 0
\(424\) 20.3431 0.987950
\(425\) −1.58579 −0.0769219
\(426\) 0 0
\(427\) 0 0
\(428\) −3.02944 −0.146433
\(429\) 0 0
\(430\) 9.31371 0.449147
\(431\) 26.9706 1.29913 0.649563 0.760308i \(-0.274952\pi\)
0.649563 + 0.760308i \(0.274952\pi\)
\(432\) 0 0
\(433\) 29.1421 1.40048 0.700241 0.713907i \(-0.253076\pi\)
0.700241 + 0.713907i \(0.253076\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.14214 0.102590
\(437\) −8.69848 −0.416105
\(438\) 0 0
\(439\) 11.1005 0.529798 0.264899 0.964276i \(-0.414661\pi\)
0.264899 + 0.964276i \(0.414661\pi\)
\(440\) −3.17157 −0.151199
\(441\) 0 0
\(442\) −3.17157 −0.150856
\(443\) 11.6274 0.552435 0.276218 0.961095i \(-0.410919\pi\)
0.276218 + 0.961095i \(0.410919\pi\)
\(444\) 0 0
\(445\) −28.2843 −1.34080
\(446\) −4.54416 −0.215172
\(447\) 0 0
\(448\) 0 0
\(449\) 4.00000 0.188772 0.0943858 0.995536i \(-0.469911\pi\)
0.0943858 + 0.995536i \(0.469911\pi\)
\(450\) 0 0
\(451\) −6.82843 −0.321538
\(452\) 6.68629 0.314497
\(453\) 0 0
\(454\) −7.85786 −0.368788
\(455\) 0 0
\(456\) 0 0
\(457\) 13.1716 0.616140 0.308070 0.951364i \(-0.400317\pi\)
0.308070 + 0.951364i \(0.400317\pi\)
\(458\) 1.79899 0.0840613
\(459\) 0 0
\(460\) −25.5980 −1.19351
\(461\) −15.2426 −0.709921 −0.354960 0.934881i \(-0.615506\pi\)
−0.354960 + 0.934881i \(0.615506\pi\)
\(462\) 0 0
\(463\) −26.6274 −1.23748 −0.618741 0.785595i \(-0.712357\pi\)
−0.618741 + 0.785595i \(0.712357\pi\)
\(464\) 15.7279 0.730150
\(465\) 0 0
\(466\) 3.14214 0.145557
\(467\) −14.3137 −0.662359 −0.331180 0.943568i \(-0.607447\pi\)
−0.331180 + 0.943568i \(0.607447\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 3.45584 0.159406
\(471\) 0 0
\(472\) −4.21320 −0.193928
\(473\) 11.2426 0.516937
\(474\) 0 0
\(475\) 1.24264 0.0570163
\(476\) 0 0
\(477\) 0 0
\(478\) 4.34315 0.198651
\(479\) −14.0000 −0.639676 −0.319838 0.947472i \(-0.603629\pi\)
−0.319838 + 0.947472i \(0.603629\pi\)
\(480\) 0 0
\(481\) 36.1421 1.64794
\(482\) 6.34315 0.288922
\(483\) 0 0
\(484\) −1.82843 −0.0831103
\(485\) 10.9706 0.498148
\(486\) 0 0
\(487\) 12.6274 0.572203 0.286101 0.958199i \(-0.407641\pi\)
0.286101 + 0.958199i \(0.407641\pi\)
\(488\) −6.34315 −0.287141
\(489\) 0 0
\(490\) 0 0
\(491\) 18.4853 0.834229 0.417115 0.908854i \(-0.363041\pi\)
0.417115 + 0.908854i \(0.363041\pi\)
\(492\) 0 0
\(493\) 8.31371 0.374431
\(494\) 2.48528 0.111818
\(495\) 0 0
\(496\) −16.9706 −0.762001
\(497\) 0 0
\(498\) 0 0
\(499\) 7.79899 0.349131 0.174565 0.984646i \(-0.444148\pi\)
0.174565 + 0.984646i \(0.444148\pi\)
\(500\) 21.9411 0.981237
\(501\) 0 0
\(502\) −8.07107 −0.360229
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 29.7990 1.32604
\(506\) 2.89949 0.128898
\(507\) 0 0
\(508\) 3.52691 0.156481
\(509\) 12.0000 0.531891 0.265945 0.963988i \(-0.414316\pi\)
0.265945 + 0.963988i \(0.414316\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −22.7574 −1.00574
\(513\) 0 0
\(514\) −12.2010 −0.538163
\(515\) −8.97056 −0.395290
\(516\) 0 0
\(517\) 4.17157 0.183466
\(518\) 0 0
\(519\) 0 0
\(520\) 15.3137 0.671551
\(521\) −36.8284 −1.61348 −0.806741 0.590905i \(-0.798771\pi\)
−0.806741 + 0.590905i \(0.798771\pi\)
\(522\) 0 0
\(523\) 34.2843 1.49915 0.749573 0.661921i \(-0.230259\pi\)
0.749573 + 0.661921i \(0.230259\pi\)
\(524\) −19.7990 −0.864923
\(525\) 0 0
\(526\) 7.45584 0.325090
\(527\) −8.97056 −0.390764
\(528\) 0 0
\(529\) 26.0000 1.13043
\(530\) −10.6274 −0.461625
\(531\) 0 0
\(532\) 0 0
\(533\) 32.9706 1.42811
\(534\) 0 0
\(535\) 3.31371 0.143264
\(536\) −14.0000 −0.604708
\(537\) 0 0
\(538\) 4.88730 0.210707
\(539\) 0 0
\(540\) 0 0
\(541\) 8.34315 0.358700 0.179350 0.983785i \(-0.442601\pi\)
0.179350 + 0.983785i \(0.442601\pi\)
\(542\) −4.54416 −0.195188
\(543\) 0 0
\(544\) −7.00000 −0.300123
\(545\) −2.34315 −0.100369
\(546\) 0 0
\(547\) 18.2132 0.778740 0.389370 0.921081i \(-0.372693\pi\)
0.389370 + 0.921081i \(0.372693\pi\)
\(548\) −30.1421 −1.28761
\(549\) 0 0
\(550\) −0.414214 −0.0176621
\(551\) −6.51472 −0.277536
\(552\) 0 0
\(553\) 0 0
\(554\) 5.94113 0.252414
\(555\) 0 0
\(556\) −10.2132 −0.433136
\(557\) −22.5563 −0.955743 −0.477872 0.878430i \(-0.658592\pi\)
−0.477872 + 0.878430i \(0.658592\pi\)
\(558\) 0 0
\(559\) −54.2843 −2.29598
\(560\) 0 0
\(561\) 0 0
\(562\) 4.79899 0.202433
\(563\) −19.1716 −0.807985 −0.403993 0.914762i \(-0.632378\pi\)
−0.403993 + 0.914762i \(0.632378\pi\)
\(564\) 0 0
\(565\) −7.31371 −0.307690
\(566\) −13.1127 −0.551168
\(567\) 0 0
\(568\) 15.5858 0.653965
\(569\) 7.24264 0.303627 0.151814 0.988409i \(-0.451489\pi\)
0.151814 + 0.988409i \(0.451489\pi\)
\(570\) 0 0
\(571\) −18.0711 −0.756251 −0.378125 0.925754i \(-0.623431\pi\)
−0.378125 + 0.925754i \(0.623431\pi\)
\(572\) 8.82843 0.369135
\(573\) 0 0
\(574\) 0 0
\(575\) −7.00000 −0.291920
\(576\) 0 0
\(577\) −3.65685 −0.152237 −0.0761184 0.997099i \(-0.524253\pi\)
−0.0761184 + 0.997099i \(0.524253\pi\)
\(578\) 6.00000 0.249567
\(579\) 0 0
\(580\) −19.1716 −0.796056
\(581\) 0 0
\(582\) 0 0
\(583\) −12.8284 −0.531299
\(584\) 18.4853 0.764926
\(585\) 0 0
\(586\) −8.17157 −0.337565
\(587\) −31.3137 −1.29246 −0.646228 0.763145i \(-0.723654\pi\)
−0.646228 + 0.763145i \(0.723654\pi\)
\(588\) 0 0
\(589\) 7.02944 0.289643
\(590\) 2.20101 0.0906142
\(591\) 0 0
\(592\) 22.4558 0.922930
\(593\) 47.7279 1.95995 0.979975 0.199118i \(-0.0638079\pi\)
0.979975 + 0.199118i \(0.0638079\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 36.9584 1.51387
\(597\) 0 0
\(598\) −14.0000 −0.572503
\(599\) 46.6274 1.90514 0.952572 0.304312i \(-0.0984267\pi\)
0.952572 + 0.304312i \(0.0984267\pi\)
\(600\) 0 0
\(601\) 2.48528 0.101377 0.0506884 0.998715i \(-0.483858\pi\)
0.0506884 + 0.998715i \(0.483858\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.64466 −0.0669203
\(605\) 2.00000 0.0813116
\(606\) 0 0
\(607\) −41.3137 −1.67687 −0.838436 0.545000i \(-0.816530\pi\)
−0.838436 + 0.545000i \(0.816530\pi\)
\(608\) 5.48528 0.222458
\(609\) 0 0
\(610\) 3.31371 0.134168
\(611\) −20.1421 −0.814864
\(612\) 0 0
\(613\) 30.8284 1.24515 0.622574 0.782561i \(-0.286087\pi\)
0.622574 + 0.782561i \(0.286087\pi\)
\(614\) −2.20101 −0.0888255
\(615\) 0 0
\(616\) 0 0
\(617\) −18.8284 −0.758004 −0.379002 0.925396i \(-0.623733\pi\)
−0.379002 + 0.925396i \(0.623733\pi\)
\(618\) 0 0
\(619\) −10.6274 −0.427152 −0.213576 0.976926i \(-0.568511\pi\)
−0.213576 + 0.976926i \(0.568511\pi\)
\(620\) 20.6863 0.830781
\(621\) 0 0
\(622\) −6.47309 −0.259547
\(623\) 0 0
\(624\) 0 0
\(625\) −19.0000 −0.760000
\(626\) 14.4142 0.576108
\(627\) 0 0
\(628\) −12.7990 −0.510735
\(629\) 11.8701 0.473290
\(630\) 0 0
\(631\) −30.2843 −1.20560 −0.602799 0.797893i \(-0.705948\pi\)
−0.602799 + 0.797893i \(0.705948\pi\)
\(632\) −14.7696 −0.587501
\(633\) 0 0
\(634\) −1.11270 −0.0441909
\(635\) −3.85786 −0.153095
\(636\) 0 0
\(637\) 0 0
\(638\) 2.17157 0.0859734
\(639\) 0 0
\(640\) 21.1127 0.834553
\(641\) −4.48528 −0.177158 −0.0885790 0.996069i \(-0.528233\pi\)
−0.0885790 + 0.996069i \(0.528233\pi\)
\(642\) 0 0
\(643\) 2.00000 0.0788723 0.0394362 0.999222i \(-0.487444\pi\)
0.0394362 + 0.999222i \(0.487444\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0.816234 0.0321143
\(647\) −3.31371 −0.130275 −0.0651377 0.997876i \(-0.520749\pi\)
−0.0651377 + 0.997876i \(0.520749\pi\)
\(648\) 0 0
\(649\) 2.65685 0.104291
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) −23.7157 −0.928780
\(653\) 31.1127 1.21753 0.608767 0.793349i \(-0.291664\pi\)
0.608767 + 0.793349i \(0.291664\pi\)
\(654\) 0 0
\(655\) 21.6569 0.846203
\(656\) 20.4853 0.799816
\(657\) 0 0
\(658\) 0 0
\(659\) −17.5147 −0.682277 −0.341138 0.940013i \(-0.610812\pi\)
−0.341138 + 0.940013i \(0.610812\pi\)
\(660\) 0 0
\(661\) −21.9706 −0.854556 −0.427278 0.904120i \(-0.640527\pi\)
−0.427278 + 0.904120i \(0.640527\pi\)
\(662\) −2.28427 −0.0887807
\(663\) 0 0
\(664\) −4.48528 −0.174063
\(665\) 0 0
\(666\) 0 0
\(667\) 36.6985 1.42097
\(668\) 19.7990 0.766046
\(669\) 0 0
\(670\) 7.31371 0.282553
\(671\) 4.00000 0.154418
\(672\) 0 0
\(673\) 32.8284 1.26544 0.632721 0.774379i \(-0.281938\pi\)
0.632721 + 0.774379i \(0.281938\pi\)
\(674\) −13.3137 −0.512825
\(675\) 0 0
\(676\) −18.8579 −0.725302
\(677\) −50.0122 −1.92212 −0.961062 0.276332i \(-0.910881\pi\)
−0.961062 + 0.276332i \(0.910881\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 5.02944 0.192870
\(681\) 0 0
\(682\) −2.34315 −0.0897237
\(683\) −42.7990 −1.63766 −0.818829 0.574038i \(-0.805376\pi\)
−0.818829 + 0.574038i \(0.805376\pi\)
\(684\) 0 0
\(685\) 32.9706 1.25974
\(686\) 0 0
\(687\) 0 0
\(688\) −33.7279 −1.28586
\(689\) 61.9411 2.35977
\(690\) 0 0
\(691\) 39.4558 1.50097 0.750486 0.660887i \(-0.229820\pi\)
0.750486 + 0.660887i \(0.229820\pi\)
\(692\) −16.7696 −0.637483
\(693\) 0 0
\(694\) −7.85786 −0.298280
\(695\) 11.1716 0.423762
\(696\) 0 0
\(697\) 10.8284 0.410156
\(698\) 9.51472 0.360137
\(699\) 0 0
\(700\) 0 0
\(701\) −16.8995 −0.638285 −0.319143 0.947707i \(-0.603395\pi\)
−0.319143 + 0.947707i \(0.603395\pi\)
\(702\) 0 0
\(703\) −9.30152 −0.350813
\(704\) 4.17157 0.157222
\(705\) 0 0
\(706\) −3.85786 −0.145193
\(707\) 0 0
\(708\) 0 0
\(709\) −47.7696 −1.79402 −0.897012 0.442007i \(-0.854267\pi\)
−0.897012 + 0.442007i \(0.854267\pi\)
\(710\) −8.14214 −0.305569
\(711\) 0 0
\(712\) −22.4264 −0.840465
\(713\) −39.5980 −1.48296
\(714\) 0 0
\(715\) −9.65685 −0.361146
\(716\) 22.2548 0.831702
\(717\) 0 0
\(718\) −3.31371 −0.123667
\(719\) −50.7990 −1.89448 −0.947241 0.320521i \(-0.896142\pi\)
−0.947241 + 0.320521i \(0.896142\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 7.23045 0.269089
\(723\) 0 0
\(724\) 0.627417 0.0233178
\(725\) −5.24264 −0.194707
\(726\) 0 0
\(727\) −50.0833 −1.85749 −0.928743 0.370725i \(-0.879109\pi\)
−0.928743 + 0.370725i \(0.879109\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −9.65685 −0.357416
\(731\) −17.8284 −0.659408
\(732\) 0 0
\(733\) −1.65685 −0.0611973 −0.0305987 0.999532i \(-0.509741\pi\)
−0.0305987 + 0.999532i \(0.509741\pi\)
\(734\) −10.0000 −0.369107
\(735\) 0 0
\(736\) −30.8995 −1.13897
\(737\) 8.82843 0.325199
\(738\) 0 0
\(739\) −28.3431 −1.04262 −0.521310 0.853368i \(-0.674556\pi\)
−0.521310 + 0.853368i \(0.674556\pi\)
\(740\) −27.3726 −1.00624
\(741\) 0 0
\(742\) 0 0
\(743\) 5.79899 0.212744 0.106372 0.994326i \(-0.466077\pi\)
0.106372 + 0.994326i \(0.466077\pi\)
\(744\) 0 0
\(745\) −40.4264 −1.48111
\(746\) −4.00000 −0.146450
\(747\) 0 0
\(748\) 2.89949 0.106016
\(749\) 0 0
\(750\) 0 0
\(751\) −6.68629 −0.243986 −0.121993 0.992531i \(-0.538929\pi\)
−0.121993 + 0.992531i \(0.538929\pi\)
\(752\) −12.5147 −0.456365
\(753\) 0 0
\(754\) −10.4853 −0.381851
\(755\) 1.79899 0.0654719
\(756\) 0 0
\(757\) −30.3137 −1.10177 −0.550885 0.834581i \(-0.685710\pi\)
−0.550885 + 0.834581i \(0.685710\pi\)
\(758\) −5.45584 −0.198165
\(759\) 0 0
\(760\) −3.94113 −0.142960
\(761\) 39.7990 1.44271 0.721356 0.692564i \(-0.243519\pi\)
0.721356 + 0.692564i \(0.243519\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −18.9117 −0.684201
\(765\) 0 0
\(766\) −8.41421 −0.304018
\(767\) −12.8284 −0.463208
\(768\) 0 0
\(769\) −5.79899 −0.209117 −0.104558 0.994519i \(-0.533343\pi\)
−0.104558 + 0.994519i \(0.533343\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −29.5147 −1.06226
\(773\) 40.3431 1.45104 0.725521 0.688200i \(-0.241599\pi\)
0.725521 + 0.688200i \(0.241599\pi\)
\(774\) 0 0
\(775\) 5.65685 0.203200
\(776\) 8.69848 0.312257
\(777\) 0 0
\(778\) 11.5147 0.412823
\(779\) −8.48528 −0.304017
\(780\) 0 0
\(781\) −9.82843 −0.351689
\(782\) −4.59798 −0.164423
\(783\) 0 0
\(784\) 0 0
\(785\) 14.0000 0.499681
\(786\) 0 0
\(787\) −35.5269 −1.26640 −0.633199 0.773989i \(-0.718258\pi\)
−0.633199 + 0.773989i \(0.718258\pi\)
\(788\) −24.8406 −0.884910
\(789\) 0 0
\(790\) 7.71573 0.274513
\(791\) 0 0
\(792\) 0 0
\(793\) −19.3137 −0.685850
\(794\) −0.213203 −0.00756631
\(795\) 0 0
\(796\) −36.2010 −1.28311
\(797\) −16.8284 −0.596093 −0.298047 0.954551i \(-0.596335\pi\)
−0.298047 + 0.954551i \(0.596335\pi\)
\(798\) 0 0
\(799\) −6.61522 −0.234030
\(800\) 4.41421 0.156066
\(801\) 0 0
\(802\) 10.1421 0.358131
\(803\) −11.6569 −0.411361
\(804\) 0 0
\(805\) 0 0
\(806\) 11.3137 0.398508
\(807\) 0 0
\(808\) 23.6274 0.831210
\(809\) −5.17157 −0.181823 −0.0909114 0.995859i \(-0.528978\pi\)
−0.0909114 + 0.995859i \(0.528978\pi\)
\(810\) 0 0
\(811\) −18.9706 −0.666147 −0.333073 0.942901i \(-0.608086\pi\)
−0.333073 + 0.942901i \(0.608086\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 3.10051 0.108673
\(815\) 25.9411 0.908678
\(816\) 0 0
\(817\) 13.9706 0.488768
\(818\) −13.1716 −0.460533
\(819\) 0 0
\(820\) −24.9706 −0.872010
\(821\) −33.1716 −1.15770 −0.578848 0.815436i \(-0.696498\pi\)
−0.578848 + 0.815436i \(0.696498\pi\)
\(822\) 0 0
\(823\) −38.8284 −1.35347 −0.676737 0.736225i \(-0.736607\pi\)
−0.676737 + 0.736225i \(0.736607\pi\)
\(824\) −7.11270 −0.247783
\(825\) 0 0
\(826\) 0 0
\(827\) 14.6863 0.510692 0.255346 0.966850i \(-0.417811\pi\)
0.255346 + 0.966850i \(0.417811\pi\)
\(828\) 0 0
\(829\) −11.8284 −0.410818 −0.205409 0.978676i \(-0.565853\pi\)
−0.205409 + 0.978676i \(0.565853\pi\)
\(830\) 2.34315 0.0813318
\(831\) 0 0
\(832\) −20.1421 −0.698303
\(833\) 0 0
\(834\) 0 0
\(835\) −21.6569 −0.749466
\(836\) −2.27208 −0.0785815
\(837\) 0 0
\(838\) −10.2721 −0.354843
\(839\) 24.9706 0.862080 0.431040 0.902333i \(-0.358147\pi\)
0.431040 + 0.902333i \(0.358147\pi\)
\(840\) 0 0
\(841\) −1.51472 −0.0522317
\(842\) 2.89949 0.0999232
\(843\) 0 0
\(844\) 34.6863 1.19395
\(845\) 20.6274 0.709605
\(846\) 0 0
\(847\) 0 0
\(848\) 38.4853 1.32159
\(849\) 0 0
\(850\) 0.656854 0.0225299
\(851\) 52.3970 1.79614
\(852\) 0 0
\(853\) 4.48528 0.153573 0.0767866 0.997048i \(-0.475534\pi\)
0.0767866 + 0.997048i \(0.475534\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 2.62742 0.0898033
\(857\) −26.3553 −0.900281 −0.450141 0.892958i \(-0.648626\pi\)
−0.450141 + 0.892958i \(0.648626\pi\)
\(858\) 0 0
\(859\) −7.51472 −0.256399 −0.128199 0.991748i \(-0.540920\pi\)
−0.128199 + 0.991748i \(0.540920\pi\)
\(860\) 41.1127 1.40193
\(861\) 0 0
\(862\) −11.1716 −0.380505
\(863\) 42.6274 1.45105 0.725527 0.688194i \(-0.241596\pi\)
0.725527 + 0.688194i \(0.241596\pi\)
\(864\) 0 0
\(865\) 18.3431 0.623686
\(866\) −12.0711 −0.410192
\(867\) 0 0
\(868\) 0 0
\(869\) 9.31371 0.315946
\(870\) 0 0
\(871\) −42.6274 −1.44437
\(872\) −1.85786 −0.0629152
\(873\) 0 0
\(874\) 3.60303 0.121874
\(875\) 0 0
\(876\) 0 0
\(877\) −14.4853 −0.489133 −0.244567 0.969632i \(-0.578646\pi\)
−0.244567 + 0.969632i \(0.578646\pi\)
\(878\) −4.59798 −0.155174
\(879\) 0 0
\(880\) −6.00000 −0.202260
\(881\) −39.3137 −1.32451 −0.662256 0.749277i \(-0.730401\pi\)
−0.662256 + 0.749277i \(0.730401\pi\)
\(882\) 0 0
\(883\) 28.3431 0.953823 0.476911 0.878951i \(-0.341756\pi\)
0.476911 + 0.878951i \(0.341756\pi\)
\(884\) −14.0000 −0.470871
\(885\) 0 0
\(886\) −4.81623 −0.161805
\(887\) −9.31371 −0.312724 −0.156362 0.987700i \(-0.549977\pi\)
−0.156362 + 0.987700i \(0.549977\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 11.7157 0.392712
\(891\) 0 0
\(892\) −20.0589 −0.671621
\(893\) 5.18377 0.173468
\(894\) 0 0
\(895\) −24.3431 −0.813702
\(896\) 0 0
\(897\) 0 0
\(898\) −1.65685 −0.0552899
\(899\) −29.6569 −0.989111
\(900\) 0 0
\(901\) 20.3431 0.677728
\(902\) 2.82843 0.0941763
\(903\) 0 0
\(904\) −5.79899 −0.192872
\(905\) −0.686292 −0.0228131
\(906\) 0 0
\(907\) −34.4853 −1.14506 −0.572532 0.819882i \(-0.694039\pi\)
−0.572532 + 0.819882i \(0.694039\pi\)
\(908\) −34.6863 −1.15111
\(909\) 0 0
\(910\) 0 0
\(911\) 21.4853 0.711839 0.355920 0.934517i \(-0.384168\pi\)
0.355920 + 0.934517i \(0.384168\pi\)
\(912\) 0 0
\(913\) 2.82843 0.0936073
\(914\) −5.45584 −0.180463
\(915\) 0 0
\(916\) 7.94113 0.262382
\(917\) 0 0
\(918\) 0 0
\(919\) −14.3553 −0.473539 −0.236769 0.971566i \(-0.576089\pi\)
−0.236769 + 0.971566i \(0.576089\pi\)
\(920\) 22.2010 0.731946
\(921\) 0 0
\(922\) 6.31371 0.207931
\(923\) 47.4558 1.56203
\(924\) 0 0
\(925\) −7.48528 −0.246115
\(926\) 11.0294 0.362450
\(927\) 0 0
\(928\) −23.1421 −0.759678
\(929\) 17.1716 0.563381 0.281691 0.959505i \(-0.409105\pi\)
0.281691 + 0.959505i \(0.409105\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 13.8701 0.454329
\(933\) 0 0
\(934\) 5.92893 0.194001
\(935\) −3.17157 −0.103722
\(936\) 0 0
\(937\) 34.1421 1.11537 0.557687 0.830051i \(-0.311689\pi\)
0.557687 + 0.830051i \(0.311689\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 15.2548 0.497558
\(941\) −2.07107 −0.0675149 −0.0337574 0.999430i \(-0.510747\pi\)
−0.0337574 + 0.999430i \(0.510747\pi\)
\(942\) 0 0
\(943\) 47.7990 1.55655
\(944\) −7.97056 −0.259420
\(945\) 0 0
\(946\) −4.65685 −0.151407
\(947\) −20.5147 −0.666639 −0.333319 0.942814i \(-0.608169\pi\)
−0.333319 + 0.942814i \(0.608169\pi\)
\(948\) 0 0
\(949\) 56.2843 1.82706
\(950\) −0.514719 −0.0166997
\(951\) 0 0
\(952\) 0 0
\(953\) −14.1421 −0.458109 −0.229054 0.973414i \(-0.573563\pi\)
−0.229054 + 0.973414i \(0.573563\pi\)
\(954\) 0 0
\(955\) 20.6863 0.669393
\(956\) 19.1716 0.620053
\(957\) 0 0
\(958\) 5.79899 0.187357
\(959\) 0 0
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) −14.9706 −0.482670
\(963\) 0 0
\(964\) 28.0000 0.901819
\(965\) 32.2843 1.03927
\(966\) 0 0
\(967\) 27.0416 0.869600 0.434800 0.900527i \(-0.356819\pi\)
0.434800 + 0.900527i \(0.356819\pi\)
\(968\) 1.58579 0.0509691
\(969\) 0 0
\(970\) −4.54416 −0.145904
\(971\) 28.9706 0.929710 0.464855 0.885387i \(-0.346107\pi\)
0.464855 + 0.885387i \(0.346107\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −5.23045 −0.167594
\(975\) 0 0
\(976\) −12.0000 −0.384111
\(977\) 7.17157 0.229439 0.114719 0.993398i \(-0.463403\pi\)
0.114719 + 0.993398i \(0.463403\pi\)
\(978\) 0 0
\(979\) 14.1421 0.451985
\(980\) 0 0
\(981\) 0 0
\(982\) −7.65685 −0.244340
\(983\) 35.9706 1.14728 0.573641 0.819107i \(-0.305530\pi\)
0.573641 + 0.819107i \(0.305530\pi\)
\(984\) 0 0
\(985\) 27.1716 0.865758
\(986\) −3.44365 −0.109668
\(987\) 0 0
\(988\) 10.9706 0.349020
\(989\) −78.6985 −2.50247
\(990\) 0 0
\(991\) −47.3137 −1.50297 −0.751485 0.659750i \(-0.770662\pi\)
−0.751485 + 0.659750i \(0.770662\pi\)
\(992\) 24.9706 0.792816
\(993\) 0 0
\(994\) 0 0
\(995\) 39.5980 1.25534
\(996\) 0 0
\(997\) 38.1421 1.20797 0.603987 0.796994i \(-0.293578\pi\)
0.603987 + 0.796994i \(0.293578\pi\)
\(998\) −3.23045 −0.102258
\(999\) 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 4851.2.a.be.1.1 2
3.2 odd 2 1617.2.a.n.1.2 2
7.2 even 3 693.2.i.f.298.2 4
7.4 even 3 693.2.i.f.100.2 4
7.6 odd 2 4851.2.a.bd.1.1 2
21.2 odd 6 231.2.i.d.67.1 4
21.11 odd 6 231.2.i.d.100.1 yes 4
21.20 even 2 1617.2.a.m.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
231.2.i.d.67.1 4 21.2 odd 6
231.2.i.d.100.1 yes 4 21.11 odd 6
693.2.i.f.100.2 4 7.4 even 3
693.2.i.f.298.2 4 7.2 even 3
1617.2.a.m.1.2 2 21.20 even 2
1617.2.a.n.1.2 2 3.2 odd 2
4851.2.a.bd.1.1 2 7.6 odd 2
4851.2.a.be.1.1 2 1.1 even 1 trivial