Properties

Label 1225.2.b.j.99.3
Level $1225$
Weight $2$
Character 1225.99
Analytic conductor $9.782$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 1225 = 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1225.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(9.78167424761\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
Defining polynomial: \(x^{4} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 245)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 99.3
Root \(0.707107 + 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1225.99
Dual form 1225.2.b.j.99.2

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.41421i q^{3} +3.41421 q^{6} +2.82843i q^{8} -2.82843 q^{9} +O(q^{10})\) \(q+1.41421i q^{2} -2.41421i q^{3} +3.41421 q^{6} +2.82843i q^{8} -2.82843 q^{9} -5.82843 q^{11} -1.58579i q^{13} -4.00000 q^{16} -5.24264i q^{17} -4.00000i q^{18} -6.00000 q^{19} -8.24264i q^{22} -4.58579i q^{23} +6.82843 q^{24} +2.24264 q^{26} -0.414214i q^{27} -2.65685 q^{29} +1.75736 q^{31} +14.0711i q^{33} +7.41421 q^{34} -6.24264i q^{37} -8.48528i q^{38} -3.82843 q^{39} +2.24264 q^{41} -2.00000i q^{43} +6.48528 q^{46} +1.24264i q^{47} +9.65685i q^{48} -12.6569 q^{51} +4.24264i q^{53} +0.585786 q^{54} +14.4853i q^{57} -3.75736i q^{58} -6.24264 q^{59} +2.82843 q^{61} +2.48528i q^{62} -8.00000 q^{64} -19.8995 q^{66} +0.242641i q^{67} -11.0711 q^{69} -8.82843 q^{71} -8.00000i q^{72} -8.48528i q^{73} +8.82843 q^{74} -5.41421i q^{78} +15.4853 q^{79} -9.48528 q^{81} +3.17157i q^{82} +2.82843 q^{86} +6.41421i q^{87} -16.4853i q^{88} +8.00000 q^{89} -4.24264i q^{93} -1.75736 q^{94} +4.75736i q^{97} +16.4853 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 8q^{6} + O(q^{10}) \) \( 4q + 8q^{6} - 12q^{11} - 16q^{16} - 24q^{19} + 16q^{24} - 8q^{26} + 12q^{29} + 24q^{31} + 24q^{34} - 4q^{39} - 8q^{41} - 8q^{46} - 28q^{51} + 8q^{54} - 8q^{59} - 32q^{64} - 40q^{66} - 16q^{69} - 24q^{71} + 24q^{74} + 28q^{79} - 4q^{81} + 32q^{89} - 24q^{94} + 32q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1225\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(1177\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) − 2.41421i − 1.39385i −0.717146 0.696923i \(-0.754552\pi\)
0.717146 0.696923i \(-0.245448\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 3.41421 1.39385
\(7\) 0 0
\(8\) 2.82843i 1.00000i
\(9\) −2.82843 −0.942809
\(10\) 0 0
\(11\) −5.82843 −1.75734 −0.878668 0.477432i \(-0.841568\pi\)
−0.878668 + 0.477432i \(0.841568\pi\)
\(12\) 0 0
\(13\) − 1.58579i − 0.439818i −0.975520 0.219909i \(-0.929424\pi\)
0.975520 0.219909i \(-0.0705760\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) − 5.24264i − 1.27153i −0.771884 0.635764i \(-0.780685\pi\)
0.771884 0.635764i \(-0.219315\pi\)
\(18\) − 4.00000i − 0.942809i
\(19\) −6.00000 −1.37649 −0.688247 0.725476i \(-0.741620\pi\)
−0.688247 + 0.725476i \(0.741620\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) − 8.24264i − 1.75734i
\(23\) − 4.58579i − 0.956203i −0.878305 0.478101i \(-0.841325\pi\)
0.878305 0.478101i \(-0.158675\pi\)
\(24\) 6.82843 1.39385
\(25\) 0 0
\(26\) 2.24264 0.439818
\(27\) − 0.414214i − 0.0797154i
\(28\) 0 0
\(29\) −2.65685 −0.493365 −0.246683 0.969096i \(-0.579341\pi\)
−0.246683 + 0.969096i \(0.579341\pi\)
\(30\) 0 0
\(31\) 1.75736 0.315631 0.157816 0.987469i \(-0.449555\pi\)
0.157816 + 0.987469i \(0.449555\pi\)
\(32\) 0 0
\(33\) 14.0711i 2.44946i
\(34\) 7.41421 1.27153
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.24264i − 1.02628i −0.858304 0.513142i \(-0.828481\pi\)
0.858304 0.513142i \(-0.171519\pi\)
\(38\) − 8.48528i − 1.37649i
\(39\) −3.82843 −0.613039
\(40\) 0 0
\(41\) 2.24264 0.350242 0.175121 0.984547i \(-0.443968\pi\)
0.175121 + 0.984547i \(0.443968\pi\)
\(42\) 0 0
\(43\) − 2.00000i − 0.304997i −0.988304 0.152499i \(-0.951268\pi\)
0.988304 0.152499i \(-0.0487319\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 6.48528 0.956203
\(47\) 1.24264i 0.181258i 0.995885 + 0.0906289i \(0.0288877\pi\)
−0.995885 + 0.0906289i \(0.971112\pi\)
\(48\) 9.65685i 1.39385i
\(49\) 0 0
\(50\) 0 0
\(51\) −12.6569 −1.77231
\(52\) 0 0
\(53\) 4.24264i 0.582772i 0.956606 + 0.291386i \(0.0941163\pi\)
−0.956606 + 0.291386i \(0.905884\pi\)
\(54\) 0.585786 0.0797154
\(55\) 0 0
\(56\) 0 0
\(57\) 14.4853i 1.91862i
\(58\) − 3.75736i − 0.493365i
\(59\) −6.24264 −0.812723 −0.406361 0.913712i \(-0.633203\pi\)
−0.406361 + 0.913712i \(0.633203\pi\)
\(60\) 0 0
\(61\) 2.82843 0.362143 0.181071 0.983470i \(-0.442043\pi\)
0.181071 + 0.983470i \(0.442043\pi\)
\(62\) 2.48528i 0.315631i
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) −19.8995 −2.44946
\(67\) 0.242641i 0.0296433i 0.999890 + 0.0148216i \(0.00471805\pi\)
−0.999890 + 0.0148216i \(0.995282\pi\)
\(68\) 0 0
\(69\) −11.0711 −1.33280
\(70\) 0 0
\(71\) −8.82843 −1.04774 −0.523871 0.851798i \(-0.675513\pi\)
−0.523871 + 0.851798i \(0.675513\pi\)
\(72\) − 8.00000i − 0.942809i
\(73\) − 8.48528i − 0.993127i −0.868000 0.496564i \(-0.834595\pi\)
0.868000 0.496564i \(-0.165405\pi\)
\(74\) 8.82843 1.02628
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) − 5.41421i − 0.613039i
\(79\) 15.4853 1.74223 0.871115 0.491079i \(-0.163397\pi\)
0.871115 + 0.491079i \(0.163397\pi\)
\(80\) 0 0
\(81\) −9.48528 −1.05392
\(82\) 3.17157i 0.350242i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 2.82843 0.304997
\(87\) 6.41421i 0.687676i
\(88\) − 16.4853i − 1.75734i
\(89\) 8.00000 0.847998 0.423999 0.905663i \(-0.360626\pi\)
0.423999 + 0.905663i \(0.360626\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 4.24264i − 0.439941i
\(94\) −1.75736 −0.181258
\(95\) 0 0
\(96\) 0 0
\(97\) 4.75736i 0.483037i 0.970396 + 0.241518i \(0.0776454\pi\)
−0.970396 + 0.241518i \(0.922355\pi\)
\(98\) 0 0
\(99\) 16.4853 1.65683
\(100\) 0 0
\(101\) −14.4853 −1.44134 −0.720670 0.693279i \(-0.756166\pi\)
−0.720670 + 0.693279i \(0.756166\pi\)
\(102\) − 17.8995i − 1.77231i
\(103\) − 10.7574i − 1.05995i −0.848012 0.529977i \(-0.822201\pi\)
0.848012 0.529977i \(-0.177799\pi\)
\(104\) 4.48528 0.439818
\(105\) 0 0
\(106\) −6.00000 −0.582772
\(107\) 14.4853i 1.40035i 0.713974 + 0.700173i \(0.246894\pi\)
−0.713974 + 0.700173i \(0.753106\pi\)
\(108\) 0 0
\(109\) −5.00000 −0.478913 −0.239457 0.970907i \(-0.576969\pi\)
−0.239457 + 0.970907i \(0.576969\pi\)
\(110\) 0 0
\(111\) −15.0711 −1.43048
\(112\) 0 0
\(113\) − 1.07107i − 0.100758i −0.998730 0.0503788i \(-0.983957\pi\)
0.998730 0.0503788i \(-0.0160429\pi\)
\(114\) −20.4853 −1.91862
\(115\) 0 0
\(116\) 0 0
\(117\) 4.48528i 0.414664i
\(118\) − 8.82843i − 0.812723i
\(119\) 0 0
\(120\) 0 0
\(121\) 22.9706 2.08823
\(122\) 4.00000i 0.362143i
\(123\) − 5.41421i − 0.488183i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 0.242641i − 0.0215309i −0.999942 0.0107654i \(-0.996573\pi\)
0.999942 0.0107654i \(-0.00342681\pi\)
\(128\) − 11.3137i − 1.00000i
\(129\) −4.82843 −0.425119
\(130\) 0 0
\(131\) 3.75736 0.328282 0.164141 0.986437i \(-0.447515\pi\)
0.164141 + 0.986437i \(0.447515\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −0.343146 −0.0296433
\(135\) 0 0
\(136\) 14.8284 1.27153
\(137\) 12.0000i 1.02523i 0.858619 + 0.512615i \(0.171323\pi\)
−0.858619 + 0.512615i \(0.828677\pi\)
\(138\) − 15.6569i − 1.33280i
\(139\) −16.2426 −1.37768 −0.688841 0.724912i \(-0.741880\pi\)
−0.688841 + 0.724912i \(0.741880\pi\)
\(140\) 0 0
\(141\) 3.00000 0.252646
\(142\) − 12.4853i − 1.04774i
\(143\) 9.24264i 0.772908i
\(144\) 11.3137 0.942809
\(145\) 0 0
\(146\) 12.0000 0.993127
\(147\) 0 0
\(148\) 0 0
\(149\) 14.8284 1.21479 0.607396 0.794399i \(-0.292214\pi\)
0.607396 + 0.794399i \(0.292214\pi\)
\(150\) 0 0
\(151\) 9.48528 0.771901 0.385951 0.922519i \(-0.373874\pi\)
0.385951 + 0.922519i \(0.373874\pi\)
\(152\) − 16.9706i − 1.37649i
\(153\) 14.8284i 1.19881i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 20.8284i − 1.66229i −0.556056 0.831145i \(-0.687686\pi\)
0.556056 0.831145i \(-0.312314\pi\)
\(158\) 21.8995i 1.74223i
\(159\) 10.2426 0.812294
\(160\) 0 0
\(161\) 0 0
\(162\) − 13.4142i − 1.05392i
\(163\) 1.75736i 0.137647i 0.997629 + 0.0688235i \(0.0219245\pi\)
−0.997629 + 0.0688235i \(0.978075\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 9.24264i 0.715217i 0.933872 + 0.357609i \(0.116408\pi\)
−0.933872 + 0.357609i \(0.883592\pi\)
\(168\) 0 0
\(169\) 10.4853 0.806560
\(170\) 0 0
\(171\) 16.9706 1.29777
\(172\) 0 0
\(173\) − 1.24264i − 0.0944762i −0.998884 0.0472381i \(-0.984958\pi\)
0.998884 0.0472381i \(-0.0150420\pi\)
\(174\) −9.07107 −0.687676
\(175\) 0 0
\(176\) 23.3137 1.75734
\(177\) 15.0711i 1.13281i
\(178\) 11.3137i 0.847998i
\(179\) −2.48528 −0.185759 −0.0928793 0.995677i \(-0.529607\pi\)
−0.0928793 + 0.995677i \(0.529607\pi\)
\(180\) 0 0
\(181\) 6.72792 0.500083 0.250041 0.968235i \(-0.419556\pi\)
0.250041 + 0.968235i \(0.419556\pi\)
\(182\) 0 0
\(183\) − 6.82843i − 0.504772i
\(184\) 12.9706 0.956203
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) 30.5563i 2.23450i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −19.9706 −1.44502 −0.722510 0.691361i \(-0.757011\pi\)
−0.722510 + 0.691361i \(0.757011\pi\)
\(192\) 19.3137i 1.39385i
\(193\) 16.0000i 1.15171i 0.817554 + 0.575853i \(0.195330\pi\)
−0.817554 + 0.575853i \(0.804670\pi\)
\(194\) −6.72792 −0.483037
\(195\) 0 0
\(196\) 0 0
\(197\) 10.5858i 0.754206i 0.926171 + 0.377103i \(0.123080\pi\)
−0.926171 + 0.377103i \(0.876920\pi\)
\(198\) 23.3137i 1.65683i
\(199\) 4.58579 0.325078 0.162539 0.986702i \(-0.448032\pi\)
0.162539 + 0.986702i \(0.448032\pi\)
\(200\) 0 0
\(201\) 0.585786 0.0413182
\(202\) − 20.4853i − 1.44134i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 15.2132 1.05995
\(207\) 12.9706i 0.901516i
\(208\) 6.34315i 0.439818i
\(209\) 34.9706 2.41896
\(210\) 0 0
\(211\) 9.00000 0.619586 0.309793 0.950804i \(-0.399740\pi\)
0.309793 + 0.950804i \(0.399740\pi\)
\(212\) 0 0
\(213\) 21.3137i 1.46039i
\(214\) −20.4853 −1.40035
\(215\) 0 0
\(216\) 1.17157 0.0797154
\(217\) 0 0
\(218\) − 7.07107i − 0.478913i
\(219\) −20.4853 −1.38427
\(220\) 0 0
\(221\) −8.31371 −0.559241
\(222\) − 21.3137i − 1.43048i
\(223\) − 18.2132i − 1.21965i −0.792538 0.609823i \(-0.791240\pi\)
0.792538 0.609823i \(-0.208760\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 1.51472 0.100758
\(227\) − 9.72792i − 0.645665i −0.946456 0.322832i \(-0.895365\pi\)
0.946456 0.322832i \(-0.104635\pi\)
\(228\) 0 0
\(229\) 30.0416 1.98521 0.992603 0.121402i \(-0.0387390\pi\)
0.992603 + 0.121402i \(0.0387390\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) − 7.51472i − 0.493365i
\(233\) 14.8284i 0.971443i 0.874114 + 0.485721i \(0.161443\pi\)
−0.874114 + 0.485721i \(0.838557\pi\)
\(234\) −6.34315 −0.414664
\(235\) 0 0
\(236\) 0 0
\(237\) − 37.3848i − 2.42840i
\(238\) 0 0
\(239\) 0.514719 0.0332944 0.0166472 0.999861i \(-0.494701\pi\)
0.0166472 + 0.999861i \(0.494701\pi\)
\(240\) 0 0
\(241\) 24.7279 1.59287 0.796433 0.604727i \(-0.206718\pi\)
0.796433 + 0.604727i \(0.206718\pi\)
\(242\) 32.4853i 2.08823i
\(243\) 21.6569i 1.38929i
\(244\) 0 0
\(245\) 0 0
\(246\) 7.65685 0.488183
\(247\) 9.51472i 0.605407i
\(248\) 4.97056i 0.315631i
\(249\) 0 0
\(250\) 0 0
\(251\) −25.2132 −1.59144 −0.795722 0.605663i \(-0.792908\pi\)
−0.795722 + 0.605663i \(0.792908\pi\)
\(252\) 0 0
\(253\) 26.7279i 1.68037i
\(254\) 0.343146 0.0215309
\(255\) 0 0
\(256\) 0 0
\(257\) − 26.4853i − 1.65211i −0.563592 0.826053i \(-0.690581\pi\)
0.563592 0.826053i \(-0.309419\pi\)
\(258\) − 6.82843i − 0.425119i
\(259\) 0 0
\(260\) 0 0
\(261\) 7.51472 0.465149
\(262\) 5.31371i 0.328282i
\(263\) − 28.6274i − 1.76524i −0.470085 0.882621i \(-0.655777\pi\)
0.470085 0.882621i \(-0.344223\pi\)
\(264\) −39.7990 −2.44946
\(265\) 0 0
\(266\) 0 0
\(267\) − 19.3137i − 1.18198i
\(268\) 0 0
\(269\) 7.75736 0.472975 0.236487 0.971635i \(-0.424004\pi\)
0.236487 + 0.971635i \(0.424004\pi\)
\(270\) 0 0
\(271\) −23.3137 −1.41621 −0.708103 0.706109i \(-0.750449\pi\)
−0.708103 + 0.706109i \(0.750449\pi\)
\(272\) 20.9706i 1.27153i
\(273\) 0 0
\(274\) −16.9706 −1.02523
\(275\) 0 0
\(276\) 0 0
\(277\) − 27.2132i − 1.63508i −0.575870 0.817541i \(-0.695336\pi\)
0.575870 0.817541i \(-0.304664\pi\)
\(278\) − 22.9706i − 1.37768i
\(279\) −4.97056 −0.297580
\(280\) 0 0
\(281\) −20.3137 −1.21181 −0.605907 0.795535i \(-0.707190\pi\)
−0.605907 + 0.795535i \(0.707190\pi\)
\(282\) 4.24264i 0.252646i
\(283\) 6.55635i 0.389735i 0.980830 + 0.194867i \(0.0624276\pi\)
−0.980830 + 0.194867i \(0.937572\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −13.0711 −0.772908
\(287\) 0 0
\(288\) 0 0
\(289\) −10.4853 −0.616781
\(290\) 0 0
\(291\) 11.4853 0.673279
\(292\) 0 0
\(293\) − 0.272078i − 0.0158950i −0.999968 0.00794748i \(-0.997470\pi\)
0.999968 0.00794748i \(-0.00252979\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 17.6569 1.02628
\(297\) 2.41421i 0.140087i
\(298\) 20.9706i 1.21479i
\(299\) −7.27208 −0.420555
\(300\) 0 0
\(301\) 0 0
\(302\) 13.4142i 0.771901i
\(303\) 34.9706i 2.00901i
\(304\) 24.0000 1.37649
\(305\) 0 0
\(306\) −20.9706 −1.19881
\(307\) 11.1005i 0.633539i 0.948503 + 0.316770i \(0.102598\pi\)
−0.948503 + 0.316770i \(0.897402\pi\)
\(308\) 0 0
\(309\) −25.9706 −1.47741
\(310\) 0 0
\(311\) −10.0000 −0.567048 −0.283524 0.958965i \(-0.591504\pi\)
−0.283524 + 0.958965i \(0.591504\pi\)
\(312\) − 10.8284i − 0.613039i
\(313\) − 24.2132i − 1.36861i −0.729195 0.684306i \(-0.760105\pi\)
0.729195 0.684306i \(-0.239895\pi\)
\(314\) 29.4558 1.66229
\(315\) 0 0
\(316\) 0 0
\(317\) − 11.6569i − 0.654714i −0.944901 0.327357i \(-0.893842\pi\)
0.944901 0.327357i \(-0.106158\pi\)
\(318\) 14.4853i 0.812294i
\(319\) 15.4853 0.867009
\(320\) 0 0
\(321\) 34.9706 1.95187
\(322\) 0 0
\(323\) 31.4558i 1.75025i
\(324\) 0 0
\(325\) 0 0
\(326\) −2.48528 −0.137647
\(327\) 12.0711i 0.667532i
\(328\) 6.34315i 0.350242i
\(329\) 0 0
\(330\) 0 0
\(331\) 23.4558 1.28925 0.644625 0.764499i \(-0.277014\pi\)
0.644625 + 0.764499i \(0.277014\pi\)
\(332\) 0 0
\(333\) 17.6569i 0.967590i
\(334\) −13.0711 −0.715217
\(335\) 0 0
\(336\) 0 0
\(337\) 13.7574i 0.749411i 0.927144 + 0.374706i \(0.122256\pi\)
−0.927144 + 0.374706i \(0.877744\pi\)
\(338\) 14.8284i 0.806560i
\(339\) −2.58579 −0.140441
\(340\) 0 0
\(341\) −10.2426 −0.554670
\(342\) 24.0000i 1.29777i
\(343\) 0 0
\(344\) 5.65685 0.304997
\(345\) 0 0
\(346\) 1.75736 0.0944762
\(347\) − 1.07107i − 0.0574979i −0.999587 0.0287490i \(-0.990848\pi\)
0.999587 0.0287490i \(-0.00915234\pi\)
\(348\) 0 0
\(349\) −22.9706 −1.22959 −0.614793 0.788688i \(-0.710760\pi\)
−0.614793 + 0.788688i \(0.710760\pi\)
\(350\) 0 0
\(351\) −0.656854 −0.0350603
\(352\) 0 0
\(353\) − 36.2132i − 1.92743i −0.266925 0.963717i \(-0.586008\pi\)
0.266925 0.963717i \(-0.413992\pi\)
\(354\) −21.3137 −1.13281
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) − 3.51472i − 0.185759i
\(359\) −11.3137 −0.597115 −0.298557 0.954392i \(-0.596505\pi\)
−0.298557 + 0.954392i \(0.596505\pi\)
\(360\) 0 0
\(361\) 17.0000 0.894737
\(362\) 9.51472i 0.500083i
\(363\) − 55.4558i − 2.91068i
\(364\) 0 0
\(365\) 0 0
\(366\) 9.65685 0.504772
\(367\) 29.8701i 1.55920i 0.626275 + 0.779602i \(0.284579\pi\)
−0.626275 + 0.779602i \(0.715421\pi\)
\(368\) 18.3431i 0.956203i
\(369\) −6.34315 −0.330211
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 0.485281i − 0.0251269i −0.999921 0.0125635i \(-0.996001\pi\)
0.999921 0.0125635i \(-0.00399918\pi\)
\(374\) −43.2132 −2.23450
\(375\) 0 0
\(376\) −3.51472 −0.181258
\(377\) 4.21320i 0.216991i
\(378\) 0 0
\(379\) −2.00000 −0.102733 −0.0513665 0.998680i \(-0.516358\pi\)
−0.0513665 + 0.998680i \(0.516358\pi\)
\(380\) 0 0
\(381\) −0.585786 −0.0300107
\(382\) − 28.2426i − 1.44502i
\(383\) 12.4853i 0.637968i 0.947760 + 0.318984i \(0.103342\pi\)
−0.947760 + 0.318984i \(0.896658\pi\)
\(384\) −27.3137 −1.39385
\(385\) 0 0
\(386\) −22.6274 −1.15171
\(387\) 5.65685i 0.287554i
\(388\) 0 0
\(389\) 35.1421 1.78178 0.890889 0.454222i \(-0.150083\pi\)
0.890889 + 0.454222i \(0.150083\pi\)
\(390\) 0 0
\(391\) −24.0416 −1.21584
\(392\) 0 0
\(393\) − 9.07107i − 0.457575i
\(394\) −14.9706 −0.754206
\(395\) 0 0
\(396\) 0 0
\(397\) 4.41421i 0.221543i 0.993846 + 0.110772i \(0.0353322\pi\)
−0.993846 + 0.110772i \(0.964668\pi\)
\(398\) 6.48528i 0.325078i
\(399\) 0 0
\(400\) 0 0
\(401\) 6.17157 0.308194 0.154097 0.988056i \(-0.450753\pi\)
0.154097 + 0.988056i \(0.450753\pi\)
\(402\) 0.828427i 0.0413182i
\(403\) − 2.78680i − 0.138820i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 36.3848i 1.80353i
\(408\) − 35.7990i − 1.77231i
\(409\) 14.4853 0.716251 0.358126 0.933673i \(-0.383416\pi\)
0.358126 + 0.933673i \(0.383416\pi\)
\(410\) 0 0
\(411\) 28.9706 1.42901
\(412\) 0 0
\(413\) 0 0
\(414\) −18.3431 −0.901516
\(415\) 0 0
\(416\) 0 0
\(417\) 39.2132i 1.92028i
\(418\) 49.4558i 2.41896i
\(419\) 6.72792 0.328681 0.164340 0.986404i \(-0.447450\pi\)
0.164340 + 0.986404i \(0.447450\pi\)
\(420\) 0 0
\(421\) −19.0000 −0.926003 −0.463002 0.886357i \(-0.653228\pi\)
−0.463002 + 0.886357i \(0.653228\pi\)
\(422\) 12.7279i 0.619586i
\(423\) − 3.51472i − 0.170891i
\(424\) −12.0000 −0.582772
\(425\) 0 0
\(426\) −30.1421 −1.46039
\(427\) 0 0
\(428\) 0 0
\(429\) 22.3137 1.07732
\(430\) 0 0
\(431\) 10.7990 0.520169 0.260085 0.965586i \(-0.416250\pi\)
0.260085 + 0.965586i \(0.416250\pi\)
\(432\) 1.65685i 0.0797154i
\(433\) 22.9706i 1.10389i 0.833879 + 0.551947i \(0.186115\pi\)
−0.833879 + 0.551947i \(0.813885\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 27.5147i 1.31621i
\(438\) − 28.9706i − 1.38427i
\(439\) −6.38478 −0.304729 −0.152364 0.988324i \(-0.548689\pi\)
−0.152364 + 0.988324i \(0.548689\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) − 11.7574i − 0.559241i
\(443\) − 20.8284i − 0.989588i −0.869010 0.494794i \(-0.835243\pi\)
0.869010 0.494794i \(-0.164757\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 25.7574 1.21965
\(447\) − 35.7990i − 1.69323i
\(448\) 0 0
\(449\) 29.8284 1.40769 0.703845 0.710353i \(-0.251465\pi\)
0.703845 + 0.710353i \(0.251465\pi\)
\(450\) 0 0
\(451\) −13.0711 −0.615493
\(452\) 0 0
\(453\) − 22.8995i − 1.07591i
\(454\) 13.7574 0.645665
\(455\) 0 0
\(456\) −40.9706 −1.91862
\(457\) − 20.2426i − 0.946911i −0.880818 0.473455i \(-0.843007\pi\)
0.880818 0.473455i \(-0.156993\pi\)
\(458\) 42.4853i 1.98521i
\(459\) −2.17157 −0.101360
\(460\) 0 0
\(461\) 36.9706 1.72189 0.860945 0.508697i \(-0.169873\pi\)
0.860945 + 0.508697i \(0.169873\pi\)
\(462\) 0 0
\(463\) − 29.4558i − 1.36893i −0.729046 0.684465i \(-0.760036\pi\)
0.729046 0.684465i \(-0.239964\pi\)
\(464\) 10.6274 0.493365
\(465\) 0 0
\(466\) −20.9706 −0.971443
\(467\) 19.7279i 0.912899i 0.889749 + 0.456450i \(0.150879\pi\)
−0.889749 + 0.456450i \(0.849121\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −50.2843 −2.31698
\(472\) − 17.6569i − 0.812723i
\(473\) 11.6569i 0.535983i
\(474\) 52.8701 2.42840
\(475\) 0 0
\(476\) 0 0
\(477\) − 12.0000i − 0.549442i
\(478\) 0.727922i 0.0332944i
\(479\) −20.2426 −0.924910 −0.462455 0.886643i \(-0.653031\pi\)
−0.462455 + 0.886643i \(0.653031\pi\)
\(480\) 0 0
\(481\) −9.89949 −0.451378
\(482\) 34.9706i 1.59287i
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) −30.6274 −1.38929
\(487\) − 31.6985i − 1.43640i −0.695839 0.718198i \(-0.744967\pi\)
0.695839 0.718198i \(-0.255033\pi\)
\(488\) 8.00000i 0.362143i
\(489\) 4.24264 0.191859
\(490\) 0 0
\(491\) 19.2843 0.870287 0.435143 0.900361i \(-0.356698\pi\)
0.435143 + 0.900361i \(0.356698\pi\)
\(492\) 0 0
\(493\) 13.9289i 0.627328i
\(494\) −13.4558 −0.605407
\(495\) 0 0
\(496\) −7.02944 −0.315631
\(497\) 0 0
\(498\) 0 0
\(499\) −3.00000 −0.134298 −0.0671492 0.997743i \(-0.521390\pi\)
−0.0671492 + 0.997743i \(0.521390\pi\)
\(500\) 0 0
\(501\) 22.3137 0.996903
\(502\) − 35.6569i − 1.59144i
\(503\) 32.7574i 1.46058i 0.683138 + 0.730289i \(0.260615\pi\)
−0.683138 + 0.730289i \(0.739385\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −37.7990 −1.68037
\(507\) − 25.3137i − 1.12422i
\(508\) 0 0
\(509\) −17.2132 −0.762962 −0.381481 0.924377i \(-0.624586\pi\)
−0.381481 + 0.924377i \(0.624586\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) − 22.6274i − 1.00000i
\(513\) 2.48528i 0.109728i
\(514\) 37.4558 1.65211
\(515\) 0 0
\(516\) 0 0
\(517\) − 7.24264i − 0.318531i
\(518\) 0 0
\(519\) −3.00000 −0.131685
\(520\) 0 0
\(521\) −18.9706 −0.831115 −0.415558 0.909567i \(-0.636414\pi\)
−0.415558 + 0.909567i \(0.636414\pi\)
\(522\) 10.6274i 0.465149i
\(523\) − 15.5147i − 0.678411i −0.940712 0.339206i \(-0.889842\pi\)
0.940712 0.339206i \(-0.110158\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 40.4853 1.76524
\(527\) − 9.21320i − 0.401333i
\(528\) − 56.2843i − 2.44946i
\(529\) 1.97056 0.0856766
\(530\) 0 0
\(531\) 17.6569 0.766242
\(532\) 0 0
\(533\) − 3.55635i − 0.154043i
\(534\) 27.3137 1.18198
\(535\) 0 0
\(536\) −0.686292 −0.0296433
\(537\) 6.00000i 0.258919i
\(538\) 10.9706i 0.472975i
\(539\) 0 0
\(540\) 0 0
\(541\) 11.9706 0.514655 0.257327 0.966324i \(-0.417158\pi\)
0.257327 + 0.966324i \(0.417158\pi\)
\(542\) − 32.9706i − 1.41621i
\(543\) − 16.2426i − 0.697038i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) − 7.51472i − 0.321306i −0.987011 0.160653i \(-0.948640\pi\)
0.987011 0.160653i \(-0.0513600\pi\)
\(548\) 0 0
\(549\) −8.00000 −0.341432
\(550\) 0 0
\(551\) 15.9411 0.679115
\(552\) − 31.3137i − 1.33280i
\(553\) 0 0
\(554\) 38.4853 1.63508
\(555\) 0 0
\(556\) 0 0
\(557\) 31.7990i 1.34737i 0.739020 + 0.673683i \(0.235289\pi\)
−0.739020 + 0.673683i \(0.764711\pi\)
\(558\) − 7.02944i − 0.297580i
\(559\) −3.17157 −0.134143
\(560\) 0 0
\(561\) 73.7696 3.11455
\(562\) − 28.7279i − 1.21181i
\(563\) − 35.9411i − 1.51474i −0.652987 0.757369i \(-0.726484\pi\)
0.652987 0.757369i \(-0.273516\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −9.27208 −0.389735
\(567\) 0 0
\(568\) − 24.9706i − 1.04774i
\(569\) 2.14214 0.0898030 0.0449015 0.998991i \(-0.485703\pi\)
0.0449015 + 0.998991i \(0.485703\pi\)
\(570\) 0 0
\(571\) −34.4853 −1.44316 −0.721582 0.692329i \(-0.756585\pi\)
−0.721582 + 0.692329i \(0.756585\pi\)
\(572\) 0 0
\(573\) 48.2132i 2.01414i
\(574\) 0 0
\(575\) 0 0
\(576\) 22.6274 0.942809
\(577\) − 9.72792i − 0.404979i −0.979284 0.202489i \(-0.935097\pi\)
0.979284 0.202489i \(-0.0649032\pi\)
\(578\) − 14.8284i − 0.616781i
\(579\) 38.6274 1.60530
\(580\) 0 0
\(581\) 0 0
\(582\) 16.2426i 0.673279i
\(583\) − 24.7279i − 1.02413i
\(584\) 24.0000 0.993127
\(585\) 0 0
\(586\) 0.384776 0.0158950
\(587\) 13.4558i 0.555382i 0.960670 + 0.277691i \(0.0895692\pi\)
−0.960670 + 0.277691i \(0.910431\pi\)
\(588\) 0 0
\(589\) −10.5442 −0.434464
\(590\) 0 0
\(591\) 25.5563 1.05125
\(592\) 24.9706i 1.02628i
\(593\) − 10.7574i − 0.441752i −0.975302 0.220876i \(-0.929108\pi\)
0.975302 0.220876i \(-0.0708916\pi\)
\(594\) −3.41421 −0.140087
\(595\) 0 0
\(596\) 0 0
\(597\) − 11.0711i − 0.453109i
\(598\) − 10.2843i − 0.420555i
\(599\) 12.1716 0.497317 0.248658 0.968591i \(-0.420010\pi\)
0.248658 + 0.968591i \(0.420010\pi\)
\(600\) 0 0
\(601\) 22.9706 0.936989 0.468494 0.883466i \(-0.344797\pi\)
0.468494 + 0.883466i \(0.344797\pi\)
\(602\) 0 0
\(603\) − 0.686292i − 0.0279480i
\(604\) 0 0
\(605\) 0 0
\(606\) −49.4558 −2.00901
\(607\) − 24.8995i − 1.01064i −0.862932 0.505320i \(-0.831375\pi\)
0.862932 0.505320i \(-0.168625\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.97056 0.0797204
\(612\) 0 0
\(613\) − 43.9411i − 1.77477i −0.461034 0.887383i \(-0.652521\pi\)
0.461034 0.887383i \(-0.347479\pi\)
\(614\) −15.6985 −0.633539
\(615\) 0 0
\(616\) 0 0
\(617\) 7.41421i 0.298485i 0.988801 + 0.149242i \(0.0476835\pi\)
−0.988801 + 0.149242i \(0.952316\pi\)
\(618\) − 36.7279i − 1.47741i
\(619\) −31.0711 −1.24885 −0.624426 0.781084i \(-0.714667\pi\)
−0.624426 + 0.781084i \(0.714667\pi\)
\(620\) 0 0
\(621\) −1.89949 −0.0762241
\(622\) − 14.1421i − 0.567048i
\(623\) 0 0
\(624\) 15.3137 0.613039
\(625\) 0 0
\(626\) 34.2426 1.36861
\(627\) − 84.4264i − 3.37167i
\(628\) 0 0
\(629\) −32.7279 −1.30495
\(630\) 0 0
\(631\) 8.45584 0.336622 0.168311 0.985734i \(-0.446169\pi\)
0.168311 + 0.985734i \(0.446169\pi\)
\(632\) 43.7990i 1.74223i
\(633\) − 21.7279i − 0.863607i
\(634\) 16.4853 0.654714
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 21.8995i 0.867009i
\(639\) 24.9706 0.987820
\(640\) 0 0
\(641\) −0.686292 −0.0271069 −0.0135534 0.999908i \(-0.504314\pi\)
−0.0135534 + 0.999908i \(0.504314\pi\)
\(642\) 49.4558i 1.95187i
\(643\) − 27.7279i − 1.09348i −0.837302 0.546741i \(-0.815868\pi\)
0.837302 0.546741i \(-0.184132\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −44.4853 −1.75025
\(647\) − 28.4853i − 1.11987i −0.828536 0.559936i \(-0.810826\pi\)
0.828536 0.559936i \(-0.189174\pi\)
\(648\) − 26.8284i − 1.05392i
\(649\) 36.3848 1.42823
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) − 1.02944i − 0.0402850i −0.999797 0.0201425i \(-0.993588\pi\)
0.999797 0.0201425i \(-0.00641199\pi\)
\(654\) −17.0711 −0.667532
\(655\) 0 0
\(656\) −8.97056 −0.350242
\(657\) 24.0000i 0.936329i
\(658\) 0 0
\(659\) 13.9706 0.544216 0.272108 0.962267i \(-0.412279\pi\)
0.272108 + 0.962267i \(0.412279\pi\)
\(660\) 0 0
\(661\) −37.4558 −1.45686 −0.728432 0.685118i \(-0.759750\pi\)
−0.728432 + 0.685118i \(0.759750\pi\)
\(662\) 33.1716i 1.28925i
\(663\) 20.0711i 0.779496i
\(664\) 0 0
\(665\) 0 0
\(666\) −24.9706 −0.967590
\(667\) 12.1838i 0.471757i
\(668\) 0 0
\(669\) −43.9706 −1.70000
\(670\) 0 0
\(671\) −16.4853 −0.636407
\(672\) 0 0
\(673\) − 20.4853i − 0.789650i −0.918756 0.394825i \(-0.870805\pi\)
0.918756 0.394825i \(-0.129195\pi\)
\(674\) −19.4558 −0.749411
\(675\) 0 0
\(676\) 0 0
\(677\) 1.78680i 0.0686722i 0.999410 + 0.0343361i \(0.0109317\pi\)
−0.999410 + 0.0343361i \(0.989068\pi\)
\(678\) − 3.65685i − 0.140441i
\(679\) 0 0
\(680\) 0 0
\(681\) −23.4853 −0.899958
\(682\) − 14.4853i − 0.554670i
\(683\) 7.79899i 0.298420i 0.988806 + 0.149210i \(0.0476731\pi\)
−0.988806 + 0.149210i \(0.952327\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 72.5269i − 2.76707i
\(688\) 8.00000i 0.304997i
\(689\) 6.72792 0.256313
\(690\) 0 0
\(691\) 9.17157 0.348903 0.174452 0.984666i \(-0.444185\pi\)
0.174452 + 0.984666i \(0.444185\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.51472 0.0574979
\(695\) 0 0
\(696\) −18.1421 −0.687676
\(697\) − 11.7574i − 0.445342i
\(698\) − 32.4853i − 1.22959i
\(699\) 35.7990 1.35404
\(700\) 0 0
\(701\) −4.45584 −0.168295 −0.0841475 0.996453i \(-0.526817\pi\)
−0.0841475 + 0.996453i \(0.526817\pi\)
\(702\) − 0.928932i − 0.0350603i
\(703\) 37.4558i 1.41267i
\(704\) 46.6274 1.75734
\(705\) 0 0
\(706\) 51.2132 1.92743
\(707\) 0 0
\(708\) 0 0
\(709\) −17.0000 −0.638448 −0.319224 0.947679i \(-0.603422\pi\)
−0.319224 + 0.947679i \(0.603422\pi\)
\(710\) 0 0
\(711\) −43.7990 −1.64259
\(712\) 22.6274i 0.847998i
\(713\) − 8.05887i − 0.301807i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 1.24264i − 0.0464073i
\(718\) − 16.0000i − 0.597115i
\(719\) −17.2132 −0.641944 −0.320972 0.947089i \(-0.604010\pi\)
−0.320972 + 0.947089i \(0.604010\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 24.0416i 0.894737i
\(723\) − 59.6985i − 2.22021i
\(724\) 0 0
\(725\) 0 0
\(726\) 78.4264 2.91068
\(727\) − 29.3137i − 1.08719i −0.839349 0.543593i \(-0.817064\pi\)
0.839349 0.543593i \(-0.182936\pi\)
\(728\) 0 0
\(729\) 23.8284 0.882534
\(730\) 0 0
\(731\) −10.4853 −0.387812
\(732\) 0 0
\(733\) 44.6985i 1.65098i 0.564420 + 0.825488i \(0.309100\pi\)
−0.564420 + 0.825488i \(0.690900\pi\)
\(734\) −42.2426 −1.55920
\(735\) 0 0
\(736\) 0 0
\(737\) − 1.41421i − 0.0520932i
\(738\) − 8.97056i − 0.330211i
\(739\) 3.97056 0.146060 0.0730298 0.997330i \(-0.476733\pi\)
0.0730298 + 0.997330i \(0.476733\pi\)
\(740\) 0 0
\(741\) 22.9706 0.843845
\(742\) 0 0
\(743\) − 36.7279i − 1.34742i −0.738997 0.673708i \(-0.764700\pi\)
0.738997 0.673708i \(-0.235300\pi\)
\(744\) 12.0000 0.439941
\(745\) 0 0
\(746\) 0.686292 0.0251269
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 12.5147 0.456669 0.228334 0.973583i \(-0.426672\pi\)
0.228334 + 0.973583i \(0.426672\pi\)
\(752\) − 4.97056i − 0.181258i
\(753\) 60.8701i 2.21823i
\(754\) −5.95837 −0.216991
\(755\) 0 0
\(756\) 0 0
\(757\) − 16.4853i − 0.599168i −0.954070 0.299584i \(-0.903152\pi\)
0.954070 0.299584i \(-0.0968478\pi\)
\(758\) − 2.82843i − 0.102733i
\(759\) 64.5269 2.34218
\(760\) 0 0
\(761\) 12.7279 0.461387 0.230693 0.973026i \(-0.425901\pi\)
0.230693 + 0.973026i \(0.425901\pi\)
\(762\) − 0.828427i − 0.0300107i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −17.6569 −0.637968
\(767\) 9.89949i 0.357450i
\(768\) 0 0
\(769\) 3.17157 0.114370 0.0571849 0.998364i \(-0.481788\pi\)
0.0571849 + 0.998364i \(0.481788\pi\)
\(770\) 0 0
\(771\) −63.9411 −2.30278
\(772\) 0 0
\(773\) − 36.2132i − 1.30250i −0.758864 0.651249i \(-0.774245\pi\)
0.758864 0.651249i \(-0.225755\pi\)
\(774\) −8.00000 −0.287554
\(775\) 0 0
\(776\) −13.4558 −0.483037
\(777\) 0 0
\(778\) 49.6985i 1.78178i
\(779\) −13.4558 −0.482106
\(780\) 0 0
\(781\) 51.4558 1.84123
\(782\) − 34.0000i − 1.21584i
\(783\) 1.10051i 0.0393288i
\(784\) 0 0
\(785\) 0 0
\(786\) 12.8284 0.457575
\(787\) 45.7279i 1.63002i 0.579444 + 0.815012i \(0.303270\pi\)
−0.579444 + 0.815012i \(0.696730\pi\)
\(788\) 0 0
\(789\) −69.1127 −2.46048
\(790\) 0 0
\(791\) 0 0
\(792\) 46.6274i 1.65683i
\(793\) − 4.48528i − 0.159277i
\(794\) −6.24264 −0.221543
\(795\) 0 0
\(796\) 0 0
\(797\) 55.1838i 1.95471i 0.211608 + 0.977355i \(0.432130\pi\)
−0.211608 + 0.977355i \(0.567870\pi\)
\(798\) 0 0
\(799\) 6.51472 0.230474
\(800\) 0 0
\(801\) −22.6274 −0.799500
\(802\) 8.72792i 0.308194i
\(803\) 49.4558i 1.74526i
\(804\) 0 0
\(805\) 0 0
\(806\) 3.94113 0.138820
\(807\) − 18.7279i − 0.659254i
\(808\) − 40.9706i − 1.44134i
\(809\) −30.5980 −1.07577 −0.537884 0.843019i \(-0.680776\pi\)
−0.537884 + 0.843019i \(0.680776\pi\)
\(810\) 0 0
\(811\) −23.3553 −0.820117 −0.410058 0.912059i \(-0.634492\pi\)
−0.410058 + 0.912059i \(0.634492\pi\)
\(812\) 0 0
\(813\) 56.2843i 1.97398i
\(814\) −51.4558 −1.80353
\(815\) 0 0
\(816\) 50.6274 1.77231
\(817\) 12.0000i 0.419827i
\(818\) 20.4853i 0.716251i
\(819\) 0 0
\(820\) 0 0
\(821\) −6.51472 −0.227365 −0.113683 0.993517i \(-0.536265\pi\)
−0.113683 + 0.993517i \(0.536265\pi\)
\(822\) 40.9706i 1.42901i
\(823\) 8.72792i 0.304236i 0.988362 + 0.152118i \(0.0486094\pi\)
−0.988362 + 0.152118i \(0.951391\pi\)
\(824\) 30.4264 1.05995
\(825\) 0 0
\(826\) 0 0
\(827\) − 6.04163i − 0.210088i −0.994468 0.105044i \(-0.966502\pi\)
0.994468 0.105044i \(-0.0334984\pi\)
\(828\) 0 0
\(829\) −54.0416 −1.87694 −0.938472 0.345356i \(-0.887758\pi\)
−0.938472 + 0.345356i \(0.887758\pi\)
\(830\) 0 0
\(831\) −65.6985 −2.27906
\(832\) 12.6863i 0.439818i
\(833\) 0 0
\(834\) −55.4558 −1.92028
\(835\) 0 0
\(836\) 0 0
\(837\) − 0.727922i − 0.0251607i
\(838\) 9.51472i 0.328681i
\(839\) 48.7279 1.68227 0.841137 0.540822i \(-0.181887\pi\)
0.841137 + 0.540822i \(0.181887\pi\)
\(840\) 0 0
\(841\) −21.9411 −0.756591
\(842\) − 26.8701i − 0.926003i
\(843\) 49.0416i 1.68908i
\(844\) 0 0
\(845\) 0 0
\(846\) 4.97056 0.170891
\(847\) 0 0
\(848\) − 16.9706i − 0.582772i
\(849\) 15.8284 0.543230
\(850\) 0 0
\(851\) −28.6274 −0.981335
\(852\) 0 0
\(853\) 22.9706i 0.786497i 0.919432 + 0.393249i \(0.128649\pi\)
−0.919432 + 0.393249i \(0.871351\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −40.9706 −1.40035
\(857\) − 5.51472i − 0.188379i −0.995554 0.0941896i \(-0.969974\pi\)
0.995554 0.0941896i \(-0.0300260\pi\)
\(858\) 31.5563i 1.07732i
\(859\) −48.7696 −1.66400 −0.831998 0.554779i \(-0.812803\pi\)
−0.831998 + 0.554779i \(0.812803\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 15.2721i 0.520169i
\(863\) 18.3848i 0.625825i 0.949782 + 0.312913i \(0.101305\pi\)
−0.949782 + 0.312913i \(0.898695\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −32.4853 −1.10389
\(867\) 25.3137i 0.859699i
\(868\) 0 0
\(869\) −90.2548 −3.06169
\(870\) 0 0
\(871\) 0.384776 0.0130376
\(872\) − 14.1421i − 0.478913i
\(873\) − 13.4558i − 0.455411i
\(874\) −38.9117 −1.31621
\(875\) 0 0
\(876\) 0 0
\(877\) 46.9706i 1.58608i 0.609167 + 0.793042i \(0.291504\pi\)
−0.609167 + 0.793042i \(0.708496\pi\)
\(878\) − 9.02944i − 0.304729i
\(879\) −0.656854 −0.0221551
\(880\) 0 0
\(881\) −19.0294 −0.641118 −0.320559 0.947229i \(-0.603871\pi\)
−0.320559 + 0.947229i \(0.603871\pi\)
\(882\) 0 0
\(883\) − 48.4853i − 1.63166i −0.578292 0.815830i \(-0.696281\pi\)
0.578292 0.815830i \(-0.303719\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 29.4558 0.989588
\(887\) − 30.9706i − 1.03989i −0.854200 0.519945i \(-0.825952\pi\)
0.854200 0.519945i \(-0.174048\pi\)
\(888\) − 42.6274i − 1.43048i
\(889\) 0 0
\(890\) 0 0
\(891\) 55.2843 1.85209
\(892\) 0 0
\(893\) − 7.45584i − 0.249500i
\(894\) 50.6274 1.69323
\(895\) 0 0
\(896\) 0 0
\(897\) 17.5563i 0.586189i
\(898\) 42.1838i 1.40769i
\(899\) −4.66905 −0.155721
\(900\) 0 0
\(901\) 22.2426 0.741010
\(902\) − 18.4853i − 0.615493i
\(903\) 0 0
\(904\) 3.02944 0.100758
\(905\) 0 0
\(906\) 32.3848 1.07591
\(907\) 32.1838i 1.06864i 0.845281 + 0.534322i \(0.179433\pi\)
−0.845281 + 0.534322i \(0.820567\pi\)
\(908\) 0 0
\(909\) 40.9706 1.35891
\(910\) 0 0
\(911\) −5.65685 −0.187420 −0.0937100 0.995600i \(-0.529873\pi\)
−0.0937100 + 0.995600i \(0.529873\pi\)
\(912\) − 57.9411i − 1.91862i
\(913\) 0 0
\(914\) 28.6274 0.946911
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) − 3.07107i − 0.101360i
\(919\) 38.4558 1.26854 0.634271 0.773111i \(-0.281301\pi\)
0.634271 + 0.773111i \(0.281301\pi\)
\(920\) 0 0
\(921\) 26.7990 0.883057
\(922\) 52.2843i 1.72189i
\(923\) 14.0000i 0.460816i
\(924\) 0 0
\(925\) 0 0
\(926\) 41.6569 1.36893
\(927\) 30.4264i 0.999334i
\(928\) 0 0
\(929\) −54.7279 −1.79556 −0.897782 0.440439i \(-0.854823\pi\)
−0.897782 + 0.440439i \(0.854823\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 24.1421i 0.790378i
\(934\) −27.8995 −0.912899
\(935\) 0 0
\(936\) −12.6863 −0.414664
\(937\) − 17.4437i − 0.569859i −0.958548 0.284930i \(-0.908030\pi\)
0.958548 0.284930i \(-0.0919702\pi\)
\(938\) 0 0
\(939\) −58.4558 −1.90763
\(940\) 0 0
\(941\) −4.97056 −0.162036 −0.0810179 0.996713i \(-0.525817\pi\)
−0.0810179 + 0.996713i \(0.525817\pi\)
\(942\) − 71.1127i − 2.31698i
\(943\) − 10.2843i − 0.334902i
\(944\) 24.9706 0.812723
\(945\) 0 0
\(946\) −16.4853 −0.535983
\(947\) − 43.7574i − 1.42192i −0.703231 0.710962i \(-0.748260\pi\)
0.703231 0.710962i \(-0.251740\pi\)
\(948\) 0 0
\(949\) −13.4558 −0.436795
\(950\) 0 0
\(951\) −28.1421 −0.912571
\(952\) 0 0
\(953\) 29.0122i 0.939797i 0.882720 + 0.469899i \(0.155710\pi\)
−0.882720 + 0.469899i \(0.844290\pi\)
\(954\) 16.9706 0.549442
\(955\) 0 0
\(956\) 0 0
\(957\) − 37.3848i − 1.20848i
\(958\) − 28.6274i − 0.924910i
\(959\) 0 0
\(960\) 0 0
\(961\) −27.9117 −0.900377
\(962\) − 14.0000i − 0.451378i
\(963\) − 40.9706i − 1.32026i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 24.4264i 0.785500i 0.919645 + 0.392750i \(0.128476\pi\)
−0.919645 + 0.392750i \(0.871524\pi\)
\(968\) 64.9706i 2.08823i
\(969\) 75.9411 2.43958
\(970\) 0 0
\(971\) −42.7279 −1.37120 −0.685602 0.727976i \(-0.740461\pi\)
−0.685602 + 0.727976i \(0.740461\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 44.8284 1.43640
\(975\) 0 0
\(976\) −11.3137 −0.362143
\(977\) − 30.7696i − 0.984405i −0.870481 0.492203i \(-0.836192\pi\)
0.870481 0.492203i \(-0.163808\pi\)
\(978\) 6.00000i 0.191859i
\(979\) −46.6274 −1.49022
\(980\) 0 0
\(981\) 14.1421 0.451524
\(982\) 27.2721i 0.870287i
\(983\) 42.2132i 1.34639i 0.739464 + 0.673196i \(0.235079\pi\)
−0.739464 + 0.673196i \(0.764921\pi\)
\(984\) 15.3137 0.488183
\(985\) 0 0
\(986\) −19.6985 −0.627328
\(987\) 0 0
\(988\) 0 0
\(989\) −9.17157 −0.291639
\(990\) 0 0
\(991\) −47.9411 −1.52290 −0.761450 0.648224i \(-0.775512\pi\)
−0.761450 + 0.648224i \(0.775512\pi\)
\(992\) 0 0
\(993\) − 56.6274i − 1.79702i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.72792i 0.118064i 0.998256 + 0.0590322i \(0.0188015\pi\)
−0.998256 + 0.0590322i \(0.981199\pi\)
\(998\) − 4.24264i − 0.134298i
\(999\) −2.58579 −0.0818107
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 1225.2.b.j.99.3 4
5.2 odd 4 1225.2.a.p.1.1 2
5.3 odd 4 245.2.a.f.1.2 yes 2
5.4 even 2 inner 1225.2.b.j.99.2 4
7.6 odd 2 1225.2.b.i.99.4 4
15.8 even 4 2205.2.a.t.1.1 2
20.3 even 4 3920.2.a.br.1.1 2
35.3 even 12 245.2.e.g.226.1 4
35.13 even 4 245.2.a.e.1.2 2
35.18 odd 12 245.2.e.f.226.1 4
35.23 odd 12 245.2.e.f.116.1 4
35.27 even 4 1225.2.a.r.1.1 2
35.33 even 12 245.2.e.g.116.1 4
35.34 odd 2 1225.2.b.i.99.1 4
105.83 odd 4 2205.2.a.v.1.1 2
140.83 odd 4 3920.2.a.bw.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
245.2.a.e.1.2 2 35.13 even 4
245.2.a.f.1.2 yes 2 5.3 odd 4
245.2.e.f.116.1 4 35.23 odd 12
245.2.e.f.226.1 4 35.18 odd 12
245.2.e.g.116.1 4 35.33 even 12
245.2.e.g.226.1 4 35.3 even 12
1225.2.a.p.1.1 2 5.2 odd 4
1225.2.a.r.1.1 2 35.27 even 4
1225.2.b.i.99.1 4 35.34 odd 2
1225.2.b.i.99.4 4 7.6 odd 2
1225.2.b.j.99.2 4 5.4 even 2 inner
1225.2.b.j.99.3 4 1.1 even 1 trivial
2205.2.a.t.1.1 2 15.8 even 4
2205.2.a.v.1.1 2 105.83 odd 4
3920.2.a.br.1.1 2 20.3 even 4
3920.2.a.bw.1.2 2 140.83 odd 4