Properties

Label 126.3.b.a.71.4
Level $126$
Weight $3$
Character 126.71
Analytic conductor $3.433$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 126 = 2 \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 126.b (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: no
Analytic conductor: \(3.43325133094\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-2}, \sqrt{7})\)
Defining polynomial: \(x^{4} + 8 x^{2} + 9\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 71.4
Root \(1.16372i\) of defining polynomial
Character \(\chi\) \(=\) 126.71
Dual form 126.3.b.a.71.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} +6.06910i q^{5} -2.64575 q^{7} -2.82843i q^{8} +O(q^{10})\) \(q+1.41421i q^{2} -2.00000 q^{4} +6.06910i q^{5} -2.64575 q^{7} -2.82843i q^{8} -8.58301 q^{10} +12.1382i q^{11} -18.5830 q^{13} -3.74166i q^{14} +4.00000 q^{16} +10.9015i q^{17} +20.0000 q^{19} -12.1382i q^{20} -17.1660 q^{22} +12.1382i q^{23} -11.8340 q^{25} -26.2803i q^{26} +5.29150 q^{28} -41.8367i q^{29} +25.1660 q^{31} +5.65685i q^{32} -15.4170 q^{34} -16.0573i q^{35} +38.0000 q^{37} +28.2843i q^{38} +17.1660 q^{40} +60.6337i q^{41} +83.4980 q^{43} -24.2764i q^{44} -17.1660 q^{46} +16.9706i q^{47} +7.00000 q^{49} -16.7358i q^{50} +37.1660 q^{52} -94.0424i q^{53} -73.6680 q^{55} +7.48331i q^{56} +59.1660 q^{58} +58.2175i q^{59} +15.6680 q^{61} +35.5901i q^{62} -8.00000 q^{64} -112.782i q^{65} -132.664 q^{67} -21.8029i q^{68} +22.7085 q^{70} +12.1382i q^{71} -76.9150 q^{73} +53.7401i q^{74} -40.0000 q^{76} -32.1147i q^{77} +33.6680 q^{79} +24.2764i q^{80} -85.7490 q^{82} +60.5764i q^{83} -66.1621 q^{85} +118.084i q^{86} +34.3320 q^{88} +4.77506i q^{89} +49.1660 q^{91} -24.2764i q^{92} -24.0000 q^{94} +121.382i q^{95} -188.413 q^{97} +9.89949i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + O(q^{10}) \) \( 4 q - 8 q^{4} + 8 q^{10} - 32 q^{13} + 16 q^{16} + 80 q^{19} + 16 q^{22} - 132 q^{25} + 16 q^{31} - 104 q^{34} + 152 q^{37} - 16 q^{40} + 80 q^{43} + 16 q^{46} + 28 q^{49} + 64 q^{52} - 464 q^{55} + 152 q^{58} + 232 q^{61} - 32 q^{64} - 192 q^{67} + 112 q^{70} - 96 q^{73} - 160 q^{76} + 304 q^{79} - 216 q^{82} + 328 q^{85} - 32 q^{88} + 112 q^{91} - 96 q^{94} - 288 q^{97} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/126\mathbb{Z}\right)^\times\).

\(n\) \(29\) \(73\)
\(\chi(n)\) \(-1\) \(1\)

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 0.707107i
\(3\) 0 0
\(4\) −2.00000 −0.500000
\(5\) 6.06910i 1.21382i 0.794770 + 0.606910i \(0.207591\pi\)
−0.794770 + 0.606910i \(0.792409\pi\)
\(6\) 0 0
\(7\) −2.64575 −0.377964
\(8\) − 2.82843i − 0.353553i
\(9\) 0 0
\(10\) −8.58301 −0.858301
\(11\) 12.1382i 1.10347i 0.834019 + 0.551736i \(0.186035\pi\)
−0.834019 + 0.551736i \(0.813965\pi\)
\(12\) 0 0
\(13\) −18.5830 −1.42946 −0.714731 0.699399i \(-0.753451\pi\)
−0.714731 + 0.699399i \(0.753451\pi\)
\(14\) − 3.74166i − 0.267261i
\(15\) 0 0
\(16\) 4.00000 0.250000
\(17\) 10.9015i 0.641262i 0.947204 + 0.320631i \(0.103895\pi\)
−0.947204 + 0.320631i \(0.896105\pi\)
\(18\) 0 0
\(19\) 20.0000 1.05263 0.526316 0.850289i \(-0.323573\pi\)
0.526316 + 0.850289i \(0.323573\pi\)
\(20\) − 12.1382i − 0.606910i
\(21\) 0 0
\(22\) −17.1660 −0.780273
\(23\) 12.1382i 0.527748i 0.964557 + 0.263874i \(0.0850003\pi\)
−0.964557 + 0.263874i \(0.915000\pi\)
\(24\) 0 0
\(25\) −11.8340 −0.473360
\(26\) − 26.2803i − 1.01078i
\(27\) 0 0
\(28\) 5.29150 0.188982
\(29\) − 41.8367i − 1.44264i −0.692600 0.721322i \(-0.743535\pi\)
0.692600 0.721322i \(-0.256465\pi\)
\(30\) 0 0
\(31\) 25.1660 0.811807 0.405903 0.913916i \(-0.366957\pi\)
0.405903 + 0.913916i \(0.366957\pi\)
\(32\) 5.65685i 0.176777i
\(33\) 0 0
\(34\) −15.4170 −0.453441
\(35\) − 16.0573i − 0.458781i
\(36\) 0 0
\(37\) 38.0000 1.02703 0.513514 0.858082i \(-0.328344\pi\)
0.513514 + 0.858082i \(0.328344\pi\)
\(38\) 28.2843i 0.744323i
\(39\) 0 0
\(40\) 17.1660 0.429150
\(41\) 60.6337i 1.47887i 0.673227 + 0.739435i \(0.264908\pi\)
−0.673227 + 0.739435i \(0.735092\pi\)
\(42\) 0 0
\(43\) 83.4980 1.94181 0.970907 0.239455i \(-0.0769689\pi\)
0.970907 + 0.239455i \(0.0769689\pi\)
\(44\) − 24.2764i − 0.551736i
\(45\) 0 0
\(46\) −17.1660 −0.373174
\(47\) 16.9706i 0.361076i 0.983568 + 0.180538i \(0.0577838\pi\)
−0.983568 + 0.180538i \(0.942216\pi\)
\(48\) 0 0
\(49\) 7.00000 0.142857
\(50\) − 16.7358i − 0.334716i
\(51\) 0 0
\(52\) 37.1660 0.714731
\(53\) − 94.0424i − 1.77439i −0.461399 0.887193i \(-0.652652\pi\)
0.461399 0.887193i \(-0.347348\pi\)
\(54\) 0 0
\(55\) −73.6680 −1.33942
\(56\) 7.48331i 0.133631i
\(57\) 0 0
\(58\) 59.1660 1.02010
\(59\) 58.2175i 0.986738i 0.869820 + 0.493369i \(0.164235\pi\)
−0.869820 + 0.493369i \(0.835765\pi\)
\(60\) 0 0
\(61\) 15.6680 0.256852 0.128426 0.991719i \(-0.459008\pi\)
0.128426 + 0.991719i \(0.459008\pi\)
\(62\) 35.5901i 0.574034i
\(63\) 0 0
\(64\) −8.00000 −0.125000
\(65\) − 112.782i − 1.73511i
\(66\) 0 0
\(67\) −132.664 −1.98006 −0.990030 0.140856i \(-0.955015\pi\)
−0.990030 + 0.140856i \(0.955015\pi\)
\(68\) − 21.8029i − 0.320631i
\(69\) 0 0
\(70\) 22.7085 0.324407
\(71\) 12.1382i 0.170961i 0.996340 + 0.0854803i \(0.0272425\pi\)
−0.996340 + 0.0854803i \(0.972758\pi\)
\(72\) 0 0
\(73\) −76.9150 −1.05363 −0.526815 0.849980i \(-0.676614\pi\)
−0.526815 + 0.849980i \(0.676614\pi\)
\(74\) 53.7401i 0.726218i
\(75\) 0 0
\(76\) −40.0000 −0.526316
\(77\) − 32.1147i − 0.417074i
\(78\) 0 0
\(79\) 33.6680 0.426177 0.213088 0.977033i \(-0.431648\pi\)
0.213088 + 0.977033i \(0.431648\pi\)
\(80\) 24.2764i 0.303455i
\(81\) 0 0
\(82\) −85.7490 −1.04572
\(83\) 60.5764i 0.729836i 0.931040 + 0.364918i \(0.118903\pi\)
−0.931040 + 0.364918i \(0.881097\pi\)
\(84\) 0 0
\(85\) −66.1621 −0.778377
\(86\) 118.084i 1.37307i
\(87\) 0 0
\(88\) 34.3320 0.390137
\(89\) 4.77506i 0.0536523i 0.999640 + 0.0268262i \(0.00854006\pi\)
−0.999640 + 0.0268262i \(0.991460\pi\)
\(90\) 0 0
\(91\) 49.1660 0.540286
\(92\) − 24.2764i − 0.263874i
\(93\) 0 0
\(94\) −24.0000 −0.255319
\(95\) 121.382i 1.27771i
\(96\) 0 0
\(97\) −188.413 −1.94240 −0.971201 0.238260i \(-0.923423\pi\)
−0.971201 + 0.238260i \(0.923423\pi\)
\(98\) 9.89949i 0.101015i
\(99\) 0 0
\(100\) 23.6680 0.236680
\(101\) − 106.713i − 1.05656i −0.849069 0.528282i \(-0.822836\pi\)
0.849069 0.528282i \(-0.177164\pi\)
\(102\) 0 0
\(103\) 131.498 1.27668 0.638340 0.769755i \(-0.279621\pi\)
0.638340 + 0.769755i \(0.279621\pi\)
\(104\) 52.5607i 0.505391i
\(105\) 0 0
\(106\) 132.996 1.25468
\(107\) − 82.3793i − 0.769900i −0.922937 0.384950i \(-0.874219\pi\)
0.922937 0.384950i \(-0.125781\pi\)
\(108\) 0 0
\(109\) 33.8301 0.310367 0.155184 0.987886i \(-0.450403\pi\)
0.155184 + 0.987886i \(0.450403\pi\)
\(110\) − 104.182i − 0.947111i
\(111\) 0 0
\(112\) −10.5830 −0.0944911
\(113\) 28.5190i 0.252381i 0.992006 + 0.126190i \(0.0402750\pi\)
−0.992006 + 0.126190i \(0.959725\pi\)
\(114\) 0 0
\(115\) −73.6680 −0.640591
\(116\) 83.6734i 0.721322i
\(117\) 0 0
\(118\) −82.3320 −0.697729
\(119\) − 28.8426i − 0.242374i
\(120\) 0 0
\(121\) −26.3360 −0.217653
\(122\) 22.1579i 0.181622i
\(123\) 0 0
\(124\) −50.3320 −0.405903
\(125\) 79.9059i 0.639247i
\(126\) 0 0
\(127\) 129.668 1.02101 0.510504 0.859875i \(-0.329459\pi\)
0.510504 + 0.859875i \(0.329459\pi\)
\(128\) − 11.3137i − 0.0883883i
\(129\) 0 0
\(130\) 159.498 1.22691
\(131\) 148.017i 1.12990i 0.825124 + 0.564952i \(0.191105\pi\)
−0.825124 + 0.564952i \(0.808895\pi\)
\(132\) 0 0
\(133\) −52.9150 −0.397857
\(134\) − 187.615i − 1.40011i
\(135\) 0 0
\(136\) 30.8340 0.226721
\(137\) 76.9573i 0.561732i 0.959747 + 0.280866i \(0.0906216\pi\)
−0.959747 + 0.280866i \(0.909378\pi\)
\(138\) 0 0
\(139\) 217.328 1.56351 0.781756 0.623585i \(-0.214324\pi\)
0.781756 + 0.623585i \(0.214324\pi\)
\(140\) 32.1147i 0.229390i
\(141\) 0 0
\(142\) −17.1660 −0.120887
\(143\) − 225.564i − 1.57737i
\(144\) 0 0
\(145\) 253.911 1.75111
\(146\) − 108.774i − 0.745029i
\(147\) 0 0
\(148\) −76.0000 −0.513514
\(149\) − 161.925i − 1.08674i −0.839492 0.543371i \(-0.817148\pi\)
0.839492 0.543371i \(-0.182852\pi\)
\(150\) 0 0
\(151\) −93.1660 −0.616993 −0.308497 0.951225i \(-0.599826\pi\)
−0.308497 + 0.951225i \(0.599826\pi\)
\(152\) − 56.5685i − 0.372161i
\(153\) 0 0
\(154\) 45.4170 0.294916
\(155\) 152.735i 0.985388i
\(156\) 0 0
\(157\) −184.996 −1.17832 −0.589159 0.808017i \(-0.700541\pi\)
−0.589159 + 0.808017i \(0.700541\pi\)
\(158\) 47.6137i 0.301353i
\(159\) 0 0
\(160\) −34.3320 −0.214575
\(161\) − 32.1147i − 0.199470i
\(162\) 0 0
\(163\) 86.9961 0.533718 0.266859 0.963736i \(-0.414014\pi\)
0.266859 + 0.963736i \(0.414014\pi\)
\(164\) − 121.267i − 0.739435i
\(165\) 0 0
\(166\) −85.6680 −0.516072
\(167\) − 60.5764i − 0.362733i −0.983416 0.181366i \(-0.941948\pi\)
0.983416 0.181366i \(-0.0580520\pi\)
\(168\) 0 0
\(169\) 176.328 1.04336
\(170\) − 93.5673i − 0.550396i
\(171\) 0 0
\(172\) −166.996 −0.970907
\(173\) − 162.572i − 0.939721i −0.882741 0.469860i \(-0.844304\pi\)
0.882741 0.469860i \(-0.155696\pi\)
\(174\) 0 0
\(175\) 31.3098 0.178913
\(176\) 48.5528i 0.275868i
\(177\) 0 0
\(178\) −6.75295 −0.0379379
\(179\) − 223.091i − 1.24632i −0.782095 0.623159i \(-0.785849\pi\)
0.782095 0.623159i \(-0.214151\pi\)
\(180\) 0 0
\(181\) 188.915 1.04373 0.521865 0.853028i \(-0.325237\pi\)
0.521865 + 0.853028i \(0.325237\pi\)
\(182\) 69.5312i 0.382040i
\(183\) 0 0
\(184\) 34.3320 0.186587
\(185\) 230.626i 1.24663i
\(186\) 0 0
\(187\) −132.324 −0.707616
\(188\) − 33.9411i − 0.180538i
\(189\) 0 0
\(190\) −171.660 −0.903474
\(191\) 228.038i 1.19391i 0.802273 + 0.596957i \(0.203624\pi\)
−0.802273 + 0.596957i \(0.796376\pi\)
\(192\) 0 0
\(193\) 134.000 0.694301 0.347150 0.937810i \(-0.387149\pi\)
0.347150 + 0.937810i \(0.387149\pi\)
\(194\) − 266.456i − 1.37349i
\(195\) 0 0
\(196\) −14.0000 −0.0714286
\(197\) 188.560i 0.957157i 0.878045 + 0.478579i \(0.158848\pi\)
−0.878045 + 0.478579i \(0.841152\pi\)
\(198\) 0 0
\(199\) 102.494 0.515046 0.257523 0.966272i \(-0.417094\pi\)
0.257523 + 0.966272i \(0.417094\pi\)
\(200\) 33.4716i 0.167358i
\(201\) 0 0
\(202\) 150.915 0.747104
\(203\) 110.689i 0.545268i
\(204\) 0 0
\(205\) −367.992 −1.79508
\(206\) 185.966i 0.902749i
\(207\) 0 0
\(208\) −74.3320 −0.357365
\(209\) 242.764i 1.16155i
\(210\) 0 0
\(211\) −84.5020 −0.400483 −0.200242 0.979747i \(-0.564173\pi\)
−0.200242 + 0.979747i \(0.564173\pi\)
\(212\) 188.085i 0.887193i
\(213\) 0 0
\(214\) 116.502 0.544402
\(215\) 506.758i 2.35701i
\(216\) 0 0
\(217\) −66.5830 −0.306834
\(218\) 47.8429i 0.219463i
\(219\) 0 0
\(220\) 147.336 0.669709
\(221\) − 202.582i − 0.916660i
\(222\) 0 0
\(223\) −158.494 −0.710736 −0.355368 0.934727i \(-0.615644\pi\)
−0.355368 + 0.934727i \(0.615644\pi\)
\(224\) − 14.9666i − 0.0668153i
\(225\) 0 0
\(226\) −40.3320 −0.178460
\(227\) − 101.823i − 0.448561i −0.974525 0.224281i \(-0.927997\pi\)
0.974525 0.224281i \(-0.0720032\pi\)
\(228\) 0 0
\(229\) −268.915 −1.17430 −0.587151 0.809478i \(-0.699750\pi\)
−0.587151 + 0.809478i \(0.699750\pi\)
\(230\) − 104.182i − 0.452966i
\(231\) 0 0
\(232\) −118.332 −0.510052
\(233\) 26.2748i 0.112767i 0.998409 + 0.0563836i \(0.0179570\pi\)
−0.998409 + 0.0563836i \(0.982043\pi\)
\(234\) 0 0
\(235\) −102.996 −0.438281
\(236\) − 116.435i − 0.493369i
\(237\) 0 0
\(238\) 40.7895 0.171385
\(239\) − 92.2733i − 0.386081i −0.981191 0.193040i \(-0.938165\pi\)
0.981191 0.193040i \(-0.0618348\pi\)
\(240\) 0 0
\(241\) 343.247 1.42426 0.712131 0.702047i \(-0.247730\pi\)
0.712131 + 0.702047i \(0.247730\pi\)
\(242\) − 37.2447i − 0.153904i
\(243\) 0 0
\(244\) −31.3360 −0.128426
\(245\) 42.4837i 0.173403i
\(246\) 0 0
\(247\) −371.660 −1.50470
\(248\) − 71.1802i − 0.287017i
\(249\) 0 0
\(250\) −113.004 −0.452016
\(251\) − 356.382i − 1.41985i −0.704278 0.709924i \(-0.748729\pi\)
0.704278 0.709924i \(-0.251271\pi\)
\(252\) 0 0
\(253\) −147.336 −0.582356
\(254\) 183.378i 0.721961i
\(255\) 0 0
\(256\) 16.0000 0.0625000
\(257\) 254.730i 0.991169i 0.868560 + 0.495584i \(0.165046\pi\)
−0.868560 + 0.495584i \(0.834954\pi\)
\(258\) 0 0
\(259\) −100.539 −0.388180
\(260\) 225.564i 0.867555i
\(261\) 0 0
\(262\) −209.328 −0.798962
\(263\) − 261.979i − 0.996117i −0.867143 0.498059i \(-0.834046\pi\)
0.867143 0.498059i \(-0.165954\pi\)
\(264\) 0 0
\(265\) 570.753 2.15378
\(266\) − 74.8331i − 0.281328i
\(267\) 0 0
\(268\) 265.328 0.990030
\(269\) − 93.6246i − 0.348047i −0.984742 0.174023i \(-0.944323\pi\)
0.984742 0.174023i \(-0.0556768\pi\)
\(270\) 0 0
\(271\) 1.16601 0.00430262 0.00215131 0.999998i \(-0.499315\pi\)
0.00215131 + 0.999998i \(0.499315\pi\)
\(272\) 43.6058i 0.160316i
\(273\) 0 0
\(274\) −108.834 −0.397204
\(275\) − 143.643i − 0.522339i
\(276\) 0 0
\(277\) 32.0000 0.115523 0.0577617 0.998330i \(-0.481604\pi\)
0.0577617 + 0.998330i \(0.481604\pi\)
\(278\) 307.348i 1.10557i
\(279\) 0 0
\(280\) −45.4170 −0.162204
\(281\) − 166.757i − 0.593441i −0.954964 0.296721i \(-0.904107\pi\)
0.954964 0.296721i \(-0.0958930\pi\)
\(282\) 0 0
\(283\) 16.3399 0.0577381 0.0288691 0.999583i \(-0.490809\pi\)
0.0288691 + 0.999583i \(0.490809\pi\)
\(284\) − 24.2764i − 0.0854803i
\(285\) 0 0
\(286\) 318.996 1.11537
\(287\) − 160.422i − 0.558961i
\(288\) 0 0
\(289\) 170.158 0.588782
\(290\) 359.085i 1.23822i
\(291\) 0 0
\(292\) 153.830 0.526815
\(293\) − 368.921i − 1.25912i −0.776953 0.629558i \(-0.783236\pi\)
0.776953 0.629558i \(-0.216764\pi\)
\(294\) 0 0
\(295\) −353.328 −1.19772
\(296\) − 107.480i − 0.363109i
\(297\) 0 0
\(298\) 228.996 0.768443
\(299\) − 225.564i − 0.754396i
\(300\) 0 0
\(301\) −220.915 −0.733937
\(302\) − 131.757i − 0.436280i
\(303\) 0 0
\(304\) 80.0000 0.263158
\(305\) 95.0906i 0.311772i
\(306\) 0 0
\(307\) −192.664 −0.627570 −0.313785 0.949494i \(-0.601597\pi\)
−0.313785 + 0.949494i \(0.601597\pi\)
\(308\) 64.2293i 0.208537i
\(309\) 0 0
\(310\) −216.000 −0.696774
\(311\) − 131.276i − 0.422109i −0.977474 0.211055i \(-0.932310\pi\)
0.977474 0.211055i \(-0.0676898\pi\)
\(312\) 0 0
\(313\) 43.3281 0.138428 0.0692142 0.997602i \(-0.477951\pi\)
0.0692142 + 0.997602i \(0.477951\pi\)
\(314\) − 261.624i − 0.833197i
\(315\) 0 0
\(316\) −67.3360 −0.213088
\(317\) 251.724i 0.794083i 0.917801 + 0.397042i \(0.129963\pi\)
−0.917801 + 0.397042i \(0.870037\pi\)
\(318\) 0 0
\(319\) 507.822 1.59192
\(320\) − 48.5528i − 0.151728i
\(321\) 0 0
\(322\) 45.4170 0.141047
\(323\) 218.029i 0.675013i
\(324\) 0 0
\(325\) 219.911 0.676650
\(326\) 123.031i 0.377396i
\(327\) 0 0
\(328\) 171.498 0.522860
\(329\) − 44.8999i − 0.136474i
\(330\) 0 0
\(331\) 361.490 1.09212 0.546058 0.837748i \(-0.316128\pi\)
0.546058 + 0.837748i \(0.316128\pi\)
\(332\) − 121.153i − 0.364918i
\(333\) 0 0
\(334\) 85.6680 0.256491
\(335\) − 805.151i − 2.40344i
\(336\) 0 0
\(337\) −298.834 −0.886748 −0.443374 0.896337i \(-0.646219\pi\)
−0.443374 + 0.896337i \(0.646219\pi\)
\(338\) 249.366i 0.737768i
\(339\) 0 0
\(340\) 132.324 0.389189
\(341\) 305.470i 0.895807i
\(342\) 0 0
\(343\) −18.5203 −0.0539949
\(344\) − 236.168i − 0.686535i
\(345\) 0 0
\(346\) 229.911 0.664483
\(347\) − 206.120i − 0.594006i −0.954876 0.297003i \(-0.904013\pi\)
0.954876 0.297003i \(-0.0959872\pi\)
\(348\) 0 0
\(349\) 434.324 1.24448 0.622241 0.782826i \(-0.286222\pi\)
0.622241 + 0.782826i \(0.286222\pi\)
\(350\) 44.2787i 0.126511i
\(351\) 0 0
\(352\) −68.6640 −0.195068
\(353\) 185.439i 0.525324i 0.964888 + 0.262662i \(0.0846005\pi\)
−0.964888 + 0.262662i \(0.915400\pi\)
\(354\) 0 0
\(355\) −73.6680 −0.207515
\(356\) − 9.55012i − 0.0268262i
\(357\) 0 0
\(358\) 315.498 0.881279
\(359\) 516.767i 1.43946i 0.694254 + 0.719731i \(0.255735\pi\)
−0.694254 + 0.719731i \(0.744265\pi\)
\(360\) 0 0
\(361\) 39.0000 0.108033
\(362\) 267.166i 0.738028i
\(363\) 0 0
\(364\) −98.3320 −0.270143
\(365\) − 466.805i − 1.27892i
\(366\) 0 0
\(367\) −117.490 −0.320137 −0.160068 0.987106i \(-0.551171\pi\)
−0.160068 + 0.987106i \(0.551171\pi\)
\(368\) 48.5528i 0.131937i
\(369\) 0 0
\(370\) −326.154 −0.881498
\(371\) 248.813i 0.670655i
\(372\) 0 0
\(373\) −402.664 −1.07953 −0.539764 0.841816i \(-0.681487\pi\)
−0.539764 + 0.841816i \(0.681487\pi\)
\(374\) − 187.135i − 0.500360i
\(375\) 0 0
\(376\) 48.0000 0.127660
\(377\) 777.451i 2.06221i
\(378\) 0 0
\(379\) 398.834 1.05233 0.526166 0.850382i \(-0.323629\pi\)
0.526166 + 0.850382i \(0.323629\pi\)
\(380\) − 242.764i − 0.638853i
\(381\) 0 0
\(382\) −322.494 −0.844225
\(383\) − 744.804i − 1.94466i −0.233614 0.972329i \(-0.575055\pi\)
0.233614 0.972329i \(-0.424945\pi\)
\(384\) 0 0
\(385\) 194.907 0.506252
\(386\) 189.505i 0.490945i
\(387\) 0 0
\(388\) 376.826 0.971201
\(389\) 535.162i 1.37574i 0.725834 + 0.687869i \(0.241454\pi\)
−0.725834 + 0.687869i \(0.758546\pi\)
\(390\) 0 0
\(391\) −132.324 −0.338425
\(392\) − 19.7990i − 0.0505076i
\(393\) 0 0
\(394\) −266.664 −0.676812
\(395\) 204.334i 0.517302i
\(396\) 0 0
\(397\) −94.3241 −0.237592 −0.118796 0.992919i \(-0.537904\pi\)
−0.118796 + 0.992919i \(0.537904\pi\)
\(398\) 144.949i 0.364192i
\(399\) 0 0
\(400\) −47.3360 −0.118340
\(401\) 103.593i 0.258335i 0.991623 + 0.129168i \(0.0412306\pi\)
−0.991623 + 0.129168i \(0.958769\pi\)
\(402\) 0 0
\(403\) −467.660 −1.16045
\(404\) 213.426i 0.528282i
\(405\) 0 0
\(406\) −156.539 −0.385563
\(407\) 461.252i 1.13330i
\(408\) 0 0
\(409\) −9.75689 −0.0238555 −0.0119277 0.999929i \(-0.503797\pi\)
−0.0119277 + 0.999929i \(0.503797\pi\)
\(410\) − 520.419i − 1.26932i
\(411\) 0 0
\(412\) −262.996 −0.638340
\(413\) − 154.029i − 0.372952i
\(414\) 0 0
\(415\) −367.644 −0.885890
\(416\) − 105.121i − 0.252696i
\(417\) 0 0
\(418\) −343.320 −0.821340
\(419\) − 339.411i − 0.810051i −0.914305 0.405025i \(-0.867263\pi\)
0.914305 0.405025i \(-0.132737\pi\)
\(420\) 0 0
\(421\) −599.320 −1.42356 −0.711782 0.702401i \(-0.752111\pi\)
−0.711782 + 0.702401i \(0.752111\pi\)
\(422\) − 119.504i − 0.283184i
\(423\) 0 0
\(424\) −265.992 −0.627340
\(425\) − 129.008i − 0.303548i
\(426\) 0 0
\(427\) −41.4536 −0.0970810
\(428\) 164.759i 0.384950i
\(429\) 0 0
\(430\) −716.664 −1.66666
\(431\) − 710.978i − 1.64960i −0.565424 0.824800i \(-0.691288\pi\)
0.565424 0.824800i \(-0.308712\pi\)
\(432\) 0 0
\(433\) 377.984 0.872943 0.436471 0.899718i \(-0.356228\pi\)
0.436471 + 0.899718i \(0.356228\pi\)
\(434\) − 94.1626i − 0.216964i
\(435\) 0 0
\(436\) −67.6601 −0.155184
\(437\) 242.764i 0.555524i
\(438\) 0 0
\(439\) 528.146 1.20307 0.601533 0.798848i \(-0.294557\pi\)
0.601533 + 0.798848i \(0.294557\pi\)
\(440\) 208.365i 0.473556i
\(441\) 0 0
\(442\) 286.494 0.648177
\(443\) − 36.6438i − 0.0827174i −0.999144 0.0413587i \(-0.986831\pi\)
0.999144 0.0413587i \(-0.0131686\pi\)
\(444\) 0 0
\(445\) −28.9803 −0.0651243
\(446\) − 224.144i − 0.502566i
\(447\) 0 0
\(448\) 21.1660 0.0472456
\(449\) 397.612i 0.885550i 0.896633 + 0.442775i \(0.146006\pi\)
−0.896633 + 0.442775i \(0.853994\pi\)
\(450\) 0 0
\(451\) −735.984 −1.63189
\(452\) − 57.0381i − 0.126190i
\(453\) 0 0
\(454\) 144.000 0.317181
\(455\) 298.393i 0.655810i
\(456\) 0 0
\(457\) 344.324 0.753445 0.376722 0.926326i \(-0.377051\pi\)
0.376722 + 0.926326i \(0.377051\pi\)
\(458\) − 380.303i − 0.830357i
\(459\) 0 0
\(460\) 147.336 0.320296
\(461\) 370.936i 0.804634i 0.915500 + 0.402317i \(0.131795\pi\)
−0.915500 + 0.402317i \(0.868205\pi\)
\(462\) 0 0
\(463\) 78.3320 0.169184 0.0845918 0.996416i \(-0.473041\pi\)
0.0845918 + 0.996416i \(0.473041\pi\)
\(464\) − 167.347i − 0.360661i
\(465\) 0 0
\(466\) −37.1581 −0.0797385
\(467\) 399.758i 0.856014i 0.903775 + 0.428007i \(0.140784\pi\)
−0.903775 + 0.428007i \(0.859216\pi\)
\(468\) 0 0
\(469\) 350.996 0.748392
\(470\) − 145.658i − 0.309912i
\(471\) 0 0
\(472\) 164.664 0.348864
\(473\) 1013.52i 2.14274i
\(474\) 0 0
\(475\) −236.680 −0.498273
\(476\) 57.6851i 0.121187i
\(477\) 0 0
\(478\) 130.494 0.273000
\(479\) 703.328i 1.46833i 0.678973 + 0.734163i \(0.262425\pi\)
−0.678973 + 0.734163i \(0.737575\pi\)
\(480\) 0 0
\(481\) −706.154 −1.46810
\(482\) 485.425i 1.00711i
\(483\) 0 0
\(484\) 52.6719 0.108826
\(485\) − 1143.50i − 2.35773i
\(486\) 0 0
\(487\) −82.5098 −0.169425 −0.0847124 0.996405i \(-0.526997\pi\)
−0.0847124 + 0.996405i \(0.526997\pi\)
\(488\) − 44.3157i − 0.0908109i
\(489\) 0 0
\(490\) −60.0810 −0.122614
\(491\) 184.203i 0.375158i 0.982249 + 0.187579i \(0.0600641\pi\)
−0.982249 + 0.187579i \(0.939936\pi\)
\(492\) 0 0
\(493\) 456.081 0.925114
\(494\) − 525.607i − 1.06398i
\(495\) 0 0
\(496\) 100.664 0.202952
\(497\) − 32.1147i − 0.0646170i
\(498\) 0 0
\(499\) 752.810 1.50864 0.754319 0.656508i \(-0.227967\pi\)
0.754319 + 0.656508i \(0.227967\pi\)
\(500\) − 159.812i − 0.319623i
\(501\) 0 0
\(502\) 504.000 1.00398
\(503\) 662.540i 1.31718i 0.752504 + 0.658588i \(0.228846\pi\)
−0.752504 + 0.658588i \(0.771154\pi\)
\(504\) 0 0
\(505\) 647.652 1.28248
\(506\) − 208.365i − 0.411788i
\(507\) 0 0
\(508\) −259.336 −0.510504
\(509\) − 949.115i − 1.86467i −0.361601 0.932333i \(-0.617770\pi\)
0.361601 0.932333i \(-0.382230\pi\)
\(510\) 0 0
\(511\) 203.498 0.398235
\(512\) 22.6274i 0.0441942i
\(513\) 0 0
\(514\) −360.243 −0.700862
\(515\) 798.075i 1.54966i
\(516\) 0 0
\(517\) −205.992 −0.398437
\(518\) − 142.183i − 0.274485i
\(519\) 0 0
\(520\) −318.996 −0.613454
\(521\) − 714.344i − 1.37110i −0.728025 0.685551i \(-0.759561\pi\)
0.728025 0.685551i \(-0.240439\pi\)
\(522\) 0 0
\(523\) −232.000 −0.443595 −0.221797 0.975093i \(-0.571192\pi\)
−0.221797 + 0.975093i \(0.571192\pi\)
\(524\) − 296.035i − 0.564952i
\(525\) 0 0
\(526\) 370.494 0.704361
\(527\) 274.346i 0.520581i
\(528\) 0 0
\(529\) 381.664 0.721482
\(530\) 807.167i 1.52296i
\(531\) 0 0
\(532\) 105.830 0.198929
\(533\) − 1126.76i − 2.11399i
\(534\) 0 0
\(535\) 499.969 0.934521
\(536\) 375.231i 0.700057i
\(537\) 0 0
\(538\) 132.405 0.246106
\(539\) 84.9674i 0.157639i
\(540\) 0 0
\(541\) 165.668 0.306225 0.153113 0.988209i \(-0.451070\pi\)
0.153113 + 0.988209i \(0.451070\pi\)
\(542\) 1.64899i 0.00304241i
\(543\) 0 0
\(544\) −61.6680 −0.113360
\(545\) 205.318i 0.376730i
\(546\) 0 0
\(547\) 295.676 0.540541 0.270270 0.962784i \(-0.412887\pi\)
0.270270 + 0.962784i \(0.412887\pi\)
\(548\) − 153.915i − 0.280866i
\(549\) 0 0
\(550\) 203.142 0.369350
\(551\) − 836.734i − 1.51857i
\(552\) 0 0
\(553\) −89.0771 −0.161080
\(554\) 45.2548i 0.0816874i
\(555\) 0 0
\(556\) −434.656 −0.781756
\(557\) − 76.8426i − 0.137958i −0.997618 0.0689790i \(-0.978026\pi\)
0.997618 0.0689790i \(-0.0219742\pi\)
\(558\) 0 0
\(559\) −1551.64 −2.77575
\(560\) − 64.2293i − 0.114695i
\(561\) 0 0
\(562\) 235.830 0.419626
\(563\) − 1016.33i − 1.80521i −0.430470 0.902605i \(-0.641652\pi\)
0.430470 0.902605i \(-0.358348\pi\)
\(564\) 0 0
\(565\) −173.085 −0.306345
\(566\) 23.1081i 0.0408270i
\(567\) 0 0
\(568\) 34.3320 0.0604437
\(569\) 586.533i 1.03081i 0.856946 + 0.515406i \(0.172359\pi\)
−0.856946 + 0.515406i \(0.827641\pi\)
\(570\) 0 0
\(571\) 951.644 1.66663 0.833314 0.552800i \(-0.186441\pi\)
0.833314 + 0.552800i \(0.186441\pi\)
\(572\) 451.129i 0.788686i
\(573\) 0 0
\(574\) 226.871 0.395245
\(575\) − 143.643i − 0.249815i
\(576\) 0 0
\(577\) −148.672 −0.257664 −0.128832 0.991666i \(-0.541123\pi\)
−0.128832 + 0.991666i \(0.541123\pi\)
\(578\) 240.640i 0.416332i
\(579\) 0 0
\(580\) −507.822 −0.875555
\(581\) − 160.270i − 0.275852i
\(582\) 0 0
\(583\) 1141.51 1.95799
\(584\) 217.549i 0.372515i
\(585\) 0 0
\(586\) 521.733 0.890330
\(587\) − 332.564i − 0.566548i −0.959039 0.283274i \(-0.908579\pi\)
0.959039 0.283274i \(-0.0914206\pi\)
\(588\) 0 0
\(589\) 503.320 0.854533
\(590\) − 499.681i − 0.846918i
\(591\) 0 0
\(592\) 152.000 0.256757
\(593\) − 217.251i − 0.366359i −0.983079 0.183180i \(-0.941361\pi\)
0.983079 0.183180i \(-0.0586390\pi\)
\(594\) 0 0
\(595\) 175.048 0.294199
\(596\) 323.849i 0.543371i
\(597\) 0 0
\(598\) 318.996 0.533438
\(599\) 172.179i 0.287444i 0.989618 + 0.143722i \(0.0459072\pi\)
−0.989618 + 0.143722i \(0.954093\pi\)
\(600\) 0 0
\(601\) −418.000 −0.695507 −0.347754 0.937586i \(-0.613055\pi\)
−0.347754 + 0.937586i \(0.613055\pi\)
\(602\) − 312.421i − 0.518972i
\(603\) 0 0
\(604\) 186.332 0.308497
\(605\) − 159.836i − 0.264191i
\(606\) 0 0
\(607\) 627.158 1.03321 0.516605 0.856224i \(-0.327196\pi\)
0.516605 + 0.856224i \(0.327196\pi\)
\(608\) 113.137i 0.186081i
\(609\) 0 0
\(610\) −134.478 −0.220456
\(611\) − 315.364i − 0.516144i
\(612\) 0 0
\(613\) −279.328 −0.455674 −0.227837 0.973699i \(-0.573165\pi\)
−0.227837 + 0.973699i \(0.573165\pi\)
\(614\) − 272.468i − 0.443759i
\(615\) 0 0
\(616\) −90.8340 −0.147458
\(617\) − 358.380i − 0.580843i −0.956899 0.290422i \(-0.906204\pi\)
0.956899 0.290422i \(-0.0937955\pi\)
\(618\) 0 0
\(619\) −983.644 −1.58909 −0.794543 0.607208i \(-0.792290\pi\)
−0.794543 + 0.607208i \(0.792290\pi\)
\(620\) − 305.470i − 0.492694i
\(621\) 0 0
\(622\) 185.652 0.298476
\(623\) − 12.6336i − 0.0202787i
\(624\) 0 0
\(625\) −780.806 −1.24929
\(626\) 61.2752i 0.0978836i
\(627\) 0 0
\(628\) 369.992 0.589159
\(629\) 414.256i 0.658594i
\(630\) 0 0
\(631\) −298.996 −0.473845 −0.236922 0.971529i \(-0.576139\pi\)
−0.236922 + 0.971529i \(0.576139\pi\)
\(632\) − 95.2274i − 0.150676i
\(633\) 0 0
\(634\) −355.992 −0.561502
\(635\) 786.968i 1.23932i
\(636\) 0 0
\(637\) −130.081 −0.204209
\(638\) 718.169i 1.12566i
\(639\) 0 0
\(640\) 68.6640 0.107288
\(641\) 311.957i 0.486672i 0.969942 + 0.243336i \(0.0782418\pi\)
−0.969942 + 0.243336i \(0.921758\pi\)
\(642\) 0 0
\(643\) −604.000 −0.939347 −0.469673 0.882840i \(-0.655628\pi\)
−0.469673 + 0.882840i \(0.655628\pi\)
\(644\) 64.2293i 0.0997350i
\(645\) 0 0
\(646\) −308.340 −0.477306
\(647\) − 179.600i − 0.277588i −0.990321 0.138794i \(-0.955677\pi\)
0.990321 0.138794i \(-0.0443226\pi\)
\(648\) 0 0
\(649\) −706.656 −1.08884
\(650\) 311.001i 0.478463i
\(651\) 0 0
\(652\) −173.992 −0.266859
\(653\) − 481.892i − 0.737966i −0.929436 0.368983i \(-0.879706\pi\)
0.929436 0.368983i \(-0.120294\pi\)
\(654\) 0 0
\(655\) −898.332 −1.37150
\(656\) 242.535i 0.369718i
\(657\) 0 0
\(658\) 63.4980 0.0965016
\(659\) − 877.408i − 1.33142i −0.746209 0.665711i \(-0.768128\pi\)
0.746209 0.665711i \(-0.231872\pi\)
\(660\) 0 0
\(661\) −521.644 −0.789175 −0.394587 0.918858i \(-0.629112\pi\)
−0.394587 + 0.918858i \(0.629112\pi\)
\(662\) 511.224i 0.772242i
\(663\) 0 0
\(664\) 171.336 0.258036
\(665\) − 321.147i − 0.482927i
\(666\) 0 0
\(667\) 507.822 0.761353
\(668\) 121.153i 0.181366i
\(669\) 0 0
\(670\) 1138.66 1.69949
\(671\) 190.181i 0.283429i
\(672\) 0 0
\(673\) −659.992 −0.980672 −0.490336 0.871534i \(-0.663126\pi\)
−0.490336 + 0.871534i \(0.663126\pi\)
\(674\) − 422.615i − 0.627025i
\(675\) 0 0
\(676\) −352.656 −0.521681
\(677\) − 1016.28i − 1.50115i −0.660787 0.750573i \(-0.729778\pi\)
0.660787 0.750573i \(-0.270222\pi\)
\(678\) 0 0
\(679\) 498.494 0.734159
\(680\) 187.135i 0.275198i
\(681\) 0 0
\(682\) −432.000 −0.633431
\(683\) 235.114i 0.344238i 0.985076 + 0.172119i \(0.0550613\pi\)
−0.985076 + 0.172119i \(0.944939\pi\)
\(684\) 0 0
\(685\) −467.061 −0.681841
\(686\) − 26.1916i − 0.0381802i
\(687\) 0 0
\(688\) 333.992 0.485454
\(689\) 1747.59i 2.53642i
\(690\) 0 0
\(691\) −50.9803 −0.0737776 −0.0368888 0.999319i \(-0.511745\pi\)
−0.0368888 + 0.999319i \(0.511745\pi\)
\(692\) 325.143i 0.469860i
\(693\) 0 0
\(694\) 291.498 0.420026
\(695\) 1318.99i 1.89782i
\(696\) 0 0
\(697\) −660.996 −0.948344
\(698\) 614.227i 0.879982i
\(699\) 0 0
\(700\) −62.6196 −0.0894566
\(701\) − 141.530i − 0.201898i −0.994892 0.100949i \(-0.967812\pi\)
0.994892 0.100949i \(-0.0321879\pi\)
\(702\) 0 0
\(703\) 760.000 1.08108
\(704\) − 97.1056i − 0.137934i
\(705\) 0 0
\(706\) −262.251 −0.371460
\(707\) 282.336i 0.399344i
\(708\) 0 0
\(709\) −55.4980 −0.0782765 −0.0391382 0.999234i \(-0.512461\pi\)
−0.0391382 + 0.999234i \(0.512461\pi\)
\(710\) − 104.182i − 0.146736i
\(711\) 0 0
\(712\) 13.5059 0.0189690
\(713\) 305.470i 0.428429i
\(714\) 0 0
\(715\) 1368.97 1.91465
\(716\) 446.182i 0.623159i
\(717\) 0 0
\(718\) −730.818 −1.01785
\(719\) 1009.03i 1.40338i 0.712484 + 0.701688i \(0.247570\pi\)
−0.712484 + 0.701688i \(0.752430\pi\)
\(720\) 0 0
\(721\) −347.911 −0.482540
\(722\) 55.1543i 0.0763910i
\(723\) 0 0
\(724\) −377.830 −0.521865
\(725\) 495.095i 0.682890i
\(726\) 0 0
\(727\) −365.182 −0.502313 −0.251157 0.967946i \(-0.580811\pi\)
−0.251157 + 0.967946i \(0.580811\pi\)
\(728\) − 139.062i − 0.191020i
\(729\) 0 0
\(730\) 660.162 0.904332
\(731\) 910.251i 1.24521i
\(732\) 0 0
\(733\) −353.077 −0.481688 −0.240844 0.970564i \(-0.577424\pi\)
−0.240844 + 0.970564i \(0.577424\pi\)
\(734\) − 166.156i − 0.226371i
\(735\) 0 0
\(736\) −68.6640 −0.0932935
\(737\) − 1610.30i − 2.18494i
\(738\) 0 0
\(739\) 329.684 0.446121 0.223061 0.974805i \(-0.428395\pi\)
0.223061 + 0.974805i \(0.428395\pi\)
\(740\) − 461.252i − 0.623313i
\(741\) 0 0
\(742\) −351.875 −0.474224
\(743\) 112.061i 0.150822i 0.997153 + 0.0754112i \(0.0240270\pi\)
−0.997153 + 0.0754112i \(0.975973\pi\)
\(744\) 0 0
\(745\) 982.737 1.31911
\(746\) − 569.453i − 0.763342i
\(747\) 0 0
\(748\) 264.648 0.353808
\(749\) 217.955i 0.290995i
\(750\) 0 0
\(751\) −144.826 −0.192844 −0.0964222 0.995341i \(-0.530740\pi\)
−0.0964222 + 0.995341i \(0.530740\pi\)
\(752\) 67.8823i 0.0902690i
\(753\) 0 0
\(754\) −1099.48 −1.45820
\(755\) − 565.434i − 0.748919i
\(756\) 0 0
\(757\) 78.1699 0.103263 0.0516314 0.998666i \(-0.483558\pi\)
0.0516314 + 0.998666i \(0.483558\pi\)
\(758\) 564.036i 0.744111i
\(759\) 0 0
\(760\) 343.320 0.451737
\(761\) − 1465.50i − 1.92576i −0.269928 0.962880i \(-0.587000\pi\)
0.269928 0.962880i \(-0.413000\pi\)
\(762\) 0 0
\(763\) −89.5059 −0.117308
\(764\) − 456.076i − 0.596957i
\(765\) 0 0
\(766\) 1053.31 1.37508
\(767\) − 1081.86i − 1.41050i
\(768\) 0 0
\(769\) 729.320 0.948401 0.474200 0.880417i \(-0.342737\pi\)
0.474200 + 0.880417i \(0.342737\pi\)
\(770\) 275.640i 0.357974i
\(771\) 0 0
\(772\) −268.000 −0.347150
\(773\) 434.559i 0.562172i 0.959683 + 0.281086i \(0.0906947\pi\)
−0.959683 + 0.281086i \(0.909305\pi\)
\(774\) 0 0
\(775\) −297.814 −0.384277
\(776\) 532.913i 0.686743i
\(777\) 0 0
\(778\) −756.834 −0.972794
\(779\) 1212.67i 1.55671i
\(780\) 0 0
\(781\) −147.336 −0.188650
\(782\) − 187.135i − 0.239303i
\(783\) 0 0
\(784\) 28.0000 0.0357143
\(785\) − 1122.76i − 1.43027i
\(786\) 0 0
\(787\) −15.3517 −0.0195066 −0.00975331 0.999952i \(-0.503105\pi\)
−0.00975331 + 0.999952i \(0.503105\pi\)
\(788\) − 377.120i − 0.478579i
\(789\) 0 0
\(790\) −288.972 −0.365788
\(791\) − 75.4543i − 0.0953910i
\(792\) 0 0
\(793\) −291.158 −0.367160
\(794\) − 133.394i − 0.168003i
\(795\) 0 0
\(796\) −204.988 −0.257523
\(797\) 1043.48i 1.30927i 0.755947 + 0.654633i \(0.227177\pi\)
−0.755947 + 0.654633i \(0.772823\pi\)
\(798\) 0 0
\(799\) −185.004 −0.231544
\(800\) − 66.9432i − 0.0836789i
\(801\) 0 0
\(802\) −146.502 −0.182671
\(803\) − 933.610i − 1.16265i
\(804\) 0 0
\(805\) 194.907 0.242121
\(806\) − 661.371i − 0.820560i
\(807\) 0 0
\(808\) −301.830 −0.373552
\(809\) − 1041.31i − 1.28716i −0.765378 0.643581i \(-0.777448\pi\)
0.765378 0.643581i \(-0.222552\pi\)
\(810\) 0 0
\(811\) 502.316 0.619379 0.309689 0.950838i \(-0.399775\pi\)
0.309689 + 0.950838i \(0.399775\pi\)
\(812\) − 221.379i − 0.272634i
\(813\) 0 0
\(814\) −652.308 −0.801362
\(815\) 527.988i 0.647838i
\(816\) 0 0
\(817\) 1669.96 2.04402
\(818\) − 13.7983i − 0.0168684i
\(819\) 0 0
\(820\) 735.984 0.897542
\(821\) 23.1137i 0.0281531i 0.999901 + 0.0140765i \(0.00448085\pi\)
−0.999901 + 0.0140765i \(0.995519\pi\)
\(822\) 0 0
\(823\) −600.664 −0.729847 −0.364923 0.931038i \(-0.618905\pi\)
−0.364923 + 0.931038i \(0.618905\pi\)
\(824\) − 371.933i − 0.451375i
\(825\) 0 0
\(826\) 217.830 0.263717
\(827\) 1309.21i 1.58308i 0.611118 + 0.791540i \(0.290720\pi\)
−0.611118 + 0.791540i \(0.709280\pi\)
\(828\) 0 0
\(829\) −621.919 −0.750204 −0.375102 0.926984i \(-0.622392\pi\)
−0.375102 + 0.926984i \(0.622392\pi\)
\(830\) − 519.928i − 0.626419i
\(831\) 0 0
\(832\) 148.664 0.178683
\(833\) 76.3102i 0.0916089i
\(834\) 0 0
\(835\) 367.644 0.440293
\(836\) − 485.528i − 0.580775i
\(837\) 0 0
\(838\) 480.000 0.572792
\(839\) − 1190.30i − 1.41871i −0.704851 0.709355i \(-0.748986\pi\)
0.704851 0.709355i \(-0.251014\pi\)
\(840\) 0 0
\(841\) −909.308 −1.08122
\(842\) − 847.567i − 1.00661i
\(843\) 0 0
\(844\) 169.004 0.200242
\(845\) 1070.15i 1.26645i
\(846\) 0 0
\(847\) 69.6784 0.0822649
\(848\) − 376.170i − 0.443596i
\(849\) 0 0
\(850\) 182.445 0.214641
\(851\) 461.252i 0.542011i
\(852\) 0 0
\(853\) 137.012 0.160623 0.0803117 0.996770i \(-0.474408\pi\)
0.0803117 + 0.996770i \(0.474408\pi\)
\(854\) − 58.6242i − 0.0686466i
\(855\) 0 0
\(856\) −233.004 −0.272201
\(857\) − 466.141i − 0.543922i −0.962308 0.271961i \(-0.912328\pi\)
0.962308 0.271961i \(-0.0876722\pi\)
\(858\) 0 0
\(859\) 23.9843 0.0279211 0.0139606 0.999903i \(-0.495556\pi\)
0.0139606 + 0.999903i \(0.495556\pi\)
\(860\) − 1013.52i − 1.17851i
\(861\) 0 0
\(862\) 1005.47 1.16644
\(863\) − 0.114603i 0 0.000132796i −1.00000 6.63982e-5i \(-0.999979\pi\)
1.00000 6.63982e-5i \(-2.11352e-5\pi\)
\(864\) 0 0
\(865\) 986.664 1.14065
\(866\) 534.550i 0.617264i
\(867\) 0 0
\(868\) 133.166 0.153417
\(869\) 408.669i 0.470275i
\(870\) 0 0
\(871\) 2465.30 2.83042
\(872\) − 95.6858i − 0.109731i
\(873\) 0 0
\(874\) −343.320 −0.392815
\(875\) − 211.411i − 0.241613i
\(876\) 0 0
\(877\) 997.304 1.13718 0.568589 0.822622i \(-0.307490\pi\)
0.568589 + 0.822622i \(0.307490\pi\)
\(878\) 746.912i 0.850697i
\(879\) 0 0
\(880\) −294.672 −0.334854
\(881\) − 935.649i − 1.06203i −0.847362 0.531015i \(-0.821811\pi\)
0.847362 0.531015i \(-0.178189\pi\)
\(882\) 0 0
\(883\) −1549.47 −1.75478 −0.877392 0.479774i \(-0.840719\pi\)
−0.877392 + 0.479774i \(0.840719\pi\)
\(884\) 405.164i 0.458330i
\(885\) 0 0
\(886\) 51.8222 0.0584900
\(887\) 894.493i 1.00845i 0.863573 + 0.504224i \(0.168221\pi\)
−0.863573 + 0.504224i \(0.831779\pi\)
\(888\) 0 0
\(889\) −343.069 −0.385905
\(890\) − 40.9844i − 0.0460498i
\(891\) 0 0
\(892\) 316.988 0.355368
\(893\) 339.411i 0.380080i
\(894\) 0 0
\(895\) 1353.96 1.51281
\(896\) 29.9333i 0.0334077i
\(897\) 0 0
\(898\) −562.308 −0.626179
\(899\) − 1052.86i − 1.17115i
\(900\) 0 0
\(901\) 1025.20 1.13785
\(902\) − 1040.84i − 1.15392i
\(903\) 0 0
\(904\) 80.6640 0.0892301
\(905\) 1146.54i 1.26690i
\(906\) 0 0
\(907\) −135.838 −0.149766 −0.0748831 0.997192i \(-0.523858\pi\)
−0.0748831 + 0.997192i \(0.523858\pi\)
\(908\) 203.647i 0.224281i
\(909\) 0 0
\(910\) −421.992 −0.463728
\(911\) 1242.01i 1.36335i 0.731655 + 0.681675i \(0.238748\pi\)
−0.731655 + 0.681675i \(0.761252\pi\)
\(912\) 0 0
\(913\) −735.289 −0.805355
\(914\) 486.948i 0.532766i
\(915\) 0 0
\(916\) 537.830 0.587151
\(917\) − 391.617i − 0.427063i
\(918\) 0 0
\(919\) −388.162 −0.422374 −0.211187 0.977446i \(-0.567733\pi\)
−0.211187 + 0.977446i \(0.567733\pi\)
\(920\) 208.365i 0.226483i
\(921\) 0 0
\(922\) −524.583 −0.568962
\(923\) − 225.564i − 0.244382i
\(924\) 0 0
\(925\) −449.692 −0.486153
\(926\) 110.778i 0.119631i
\(927\) 0 0
\(928\) 236.664 0.255026
\(929\) 621.694i 0.669207i 0.942359 + 0.334604i \(0.108602\pi\)
−0.942359 + 0.334604i \(0.891398\pi\)
\(930\) 0 0
\(931\) 140.000 0.150376
\(932\) − 52.5495i − 0.0563836i
\(933\) 0 0
\(934\) −565.344 −0.605293
\(935\) − 803.089i − 0.858918i
\(936\) 0 0
\(937\) 1262.00 1.34685 0.673426 0.739255i \(-0.264822\pi\)
0.673426 + 0.739255i \(0.264822\pi\)
\(938\) 496.383i 0.529193i
\(939\) 0 0
\(940\) 205.992 0.219141
\(941\) − 672.410i − 0.714569i −0.933996 0.357285i \(-0.883703\pi\)
0.933996 0.357285i \(-0.116297\pi\)
\(942\) 0 0
\(943\) −735.984 −0.780471
\(944\) 232.870i 0.246684i
\(945\) 0 0
\(946\) −1433.33 −1.51515
\(947\) − 1159.75i − 1.22465i −0.790605 0.612327i \(-0.790234\pi\)
0.790605 0.612327i \(-0.209766\pi\)
\(948\) 0 0
\(949\) 1429.31 1.50612
\(950\) − 334.716i − 0.352332i
\(951\) 0 0
\(952\) −81.5791 −0.0856923
\(953\) 163.104i 0.171148i 0.996332 + 0.0855740i \(0.0272724\pi\)
−0.996332 + 0.0855740i \(0.972728\pi\)
\(954\) 0 0
\(955\) −1383.98 −1.44920
\(956\) 184.547i 0.193040i
\(957\) 0 0
\(958\) −994.656 −1.03826
\(959\) − 203.610i − 0.212315i
\(960\) 0 0
\(961\) −327.672 −0.340970
\(962\) − 998.653i − 1.03810i
\(963\) 0 0
\(964\) −686.494 −0.712131
\(965\) 813.260i 0.842756i
\(966\) 0 0
\(967\) 887.012 0.917282 0.458641 0.888622i \(-0.348336\pi\)
0.458641 + 0.888622i \(0.348336\pi\)
\(968\) 74.4893i 0.0769518i
\(969\) 0 0
\(970\) 1617.15 1.66717
\(971\) 1416.32i 1.45862i 0.684183 + 0.729310i \(0.260159\pi\)
−0.684183 + 0.729310i \(0.739841\pi\)
\(972\) 0 0
\(973\) −574.996 −0.590952
\(974\) − 116.687i − 0.119801i
\(975\) 0 0
\(976\) 62.6719 0.0642130
\(977\) − 339.051i − 0.347032i −0.984831 0.173516i \(-0.944487\pi\)
0.984831 0.173516i \(-0.0555129\pi\)
\(978\) 0 0
\(979\) −57.9606 −0.0592039
\(980\) − 84.9674i − 0.0867014i
\(981\) 0 0
\(982\) −260.502 −0.265277
\(983\) 487.887i 0.496324i 0.968718 + 0.248162i \(0.0798266\pi\)
−0.968718 + 0.248162i \(0.920173\pi\)
\(984\) 0 0
\(985\) −1144.39 −1.16182
\(986\) 644.996i 0.654154i
\(987\) 0 0
\(988\) 743.320 0.752348
\(989\) 1013.52i 1.02479i
\(990\) 0 0
\(991\) −937.474 −0.945988 −0.472994 0.881066i \(-0.656827\pi\)
−0.472994 + 0.881066i \(0.656827\pi\)
\(992\) 142.360i 0.143509i
\(993\) 0 0
\(994\) 45.4170 0.0456911
\(995\) 622.047i 0.625173i
\(996\) 0 0
\(997\) 461.012 0.462399 0.231200 0.972906i \(-0.425735\pi\)
0.231200 + 0.972906i \(0.425735\pi\)
\(998\) 1064.63i 1.06677i
\(999\) 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 126.3.b.a.71.4 yes 4
3.2 odd 2 inner 126.3.b.a.71.1 4
4.3 odd 2 1008.3.d.a.449.3 4
5.2 odd 4 3150.3.c.b.449.1 8
5.3 odd 4 3150.3.c.b.449.7 8
5.4 even 2 3150.3.e.e.701.2 4
7.2 even 3 882.3.s.e.557.4 8
7.3 odd 6 882.3.s.i.863.2 8
7.4 even 3 882.3.s.e.863.1 8
7.5 odd 6 882.3.s.i.557.3 8
7.6 odd 2 882.3.b.f.197.3 4
8.3 odd 2 4032.3.d.j.449.2 4
8.5 even 2 4032.3.d.i.449.2 4
9.2 odd 6 1134.3.q.c.701.2 8
9.4 even 3 1134.3.q.c.1079.2 8
9.5 odd 6 1134.3.q.c.1079.3 8
9.7 even 3 1134.3.q.c.701.3 8
12.11 even 2 1008.3.d.a.449.2 4
15.2 even 4 3150.3.c.b.449.6 8
15.8 even 4 3150.3.c.b.449.4 8
15.14 odd 2 3150.3.e.e.701.4 4
21.2 odd 6 882.3.s.e.557.1 8
21.5 even 6 882.3.s.i.557.2 8
21.11 odd 6 882.3.s.e.863.4 8
21.17 even 6 882.3.s.i.863.3 8
21.20 even 2 882.3.b.f.197.2 4
24.5 odd 2 4032.3.d.i.449.3 4
24.11 even 2 4032.3.d.j.449.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
126.3.b.a.71.1 4 3.2 odd 2 inner
126.3.b.a.71.4 yes 4 1.1 even 1 trivial
882.3.b.f.197.2 4 21.20 even 2
882.3.b.f.197.3 4 7.6 odd 2
882.3.s.e.557.1 8 21.2 odd 6
882.3.s.e.557.4 8 7.2 even 3
882.3.s.e.863.1 8 7.4 even 3
882.3.s.e.863.4 8 21.11 odd 6
882.3.s.i.557.2 8 21.5 even 6
882.3.s.i.557.3 8 7.5 odd 6
882.3.s.i.863.2 8 7.3 odd 6
882.3.s.i.863.3 8 21.17 even 6
1008.3.d.a.449.2 4 12.11 even 2
1008.3.d.a.449.3 4 4.3 odd 2
1134.3.q.c.701.2 8 9.2 odd 6
1134.3.q.c.701.3 8 9.7 even 3
1134.3.q.c.1079.2 8 9.4 even 3
1134.3.q.c.1079.3 8 9.5 odd 6
3150.3.c.b.449.1 8 5.2 odd 4
3150.3.c.b.449.4 8 15.8 even 4
3150.3.c.b.449.6 8 15.2 even 4
3150.3.c.b.449.7 8 5.3 odd 4
3150.3.e.e.701.2 4 5.4 even 2
3150.3.e.e.701.4 4 15.14 odd 2
4032.3.d.i.449.2 4 8.5 even 2
4032.3.d.i.449.3 4 24.5 odd 2
4032.3.d.j.449.2 4 8.3 odd 2
4032.3.d.j.449.3 4 24.11 even 2