Properties

Label 525.4.d.b.274.2
Level $525$
Weight $4$
Character 525.274
Analytic conductor $30.976$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 525 = 3 \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 525.d (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(30.9760027530\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
Defining polynomial: \(x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 274.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 525.274
Dual form 525.4.d.b.274.1

$q$-expansion

\(f(q)\) \(=\) \(q+4.00000i q^{2} +3.00000i q^{3} -8.00000 q^{4} -12.0000 q^{6} -7.00000i q^{7} -9.00000 q^{9} +O(q^{10})\) \(q+4.00000i q^{2} +3.00000i q^{3} -8.00000 q^{4} -12.0000 q^{6} -7.00000i q^{7} -9.00000 q^{9} +62.0000 q^{11} -24.0000i q^{12} +62.0000i q^{13} +28.0000 q^{14} -64.0000 q^{16} +84.0000i q^{17} -36.0000i q^{18} -100.000 q^{19} +21.0000 q^{21} +248.000i q^{22} +42.0000i q^{23} -248.000 q^{26} -27.0000i q^{27} +56.0000i q^{28} +10.0000 q^{29} -48.0000 q^{31} -256.000i q^{32} +186.000i q^{33} -336.000 q^{34} +72.0000 q^{36} -246.000i q^{37} -400.000i q^{38} -186.000 q^{39} -248.000 q^{41} +84.0000i q^{42} -68.0000i q^{43} -496.000 q^{44} -168.000 q^{46} +324.000i q^{47} -192.000i q^{48} -49.0000 q^{49} -252.000 q^{51} -496.000i q^{52} -258.000i q^{53} +108.000 q^{54} -300.000i q^{57} +40.0000i q^{58} -120.000 q^{59} +622.000 q^{61} -192.000i q^{62} +63.0000i q^{63} +512.000 q^{64} -744.000 q^{66} +904.000i q^{67} -672.000i q^{68} -126.000 q^{69} -678.000 q^{71} +642.000i q^{73} +984.000 q^{74} +800.000 q^{76} -434.000i q^{77} -744.000i q^{78} -740.000 q^{79} +81.0000 q^{81} -992.000i q^{82} -468.000i q^{83} -168.000 q^{84} +272.000 q^{86} +30.0000i q^{87} -200.000 q^{89} +434.000 q^{91} -336.000i q^{92} -144.000i q^{93} -1296.00 q^{94} +768.000 q^{96} -1266.00i q^{97} -196.000i q^{98} -558.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 16q^{4} - 24q^{6} - 18q^{9} + O(q^{10}) \) \( 2q - 16q^{4} - 24q^{6} - 18q^{9} + 124q^{11} + 56q^{14} - 128q^{16} - 200q^{19} + 42q^{21} - 496q^{26} + 20q^{29} - 96q^{31} - 672q^{34} + 144q^{36} - 372q^{39} - 496q^{41} - 992q^{44} - 336q^{46} - 98q^{49} - 504q^{51} + 216q^{54} - 240q^{59} + 1244q^{61} + 1024q^{64} - 1488q^{66} - 252q^{69} - 1356q^{71} + 1968q^{74} + 1600q^{76} - 1480q^{79} + 162q^{81} - 336q^{84} + 544q^{86} - 400q^{89} + 868q^{91} - 2592q^{94} + 1536q^{96} - 1116q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/525\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(176\) \(451\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

Coefficient data

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

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −8.00000 −1.00000
\(5\) 0 0
\(6\) −12.0000 −0.816497
\(7\) − 7.00000i − 0.377964i
\(8\) 0 0
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 62.0000 1.69943 0.849714 0.527244i \(-0.176775\pi\)
0.849714 + 0.527244i \(0.176775\pi\)
\(12\) − 24.0000i − 0.577350i
\(13\) 62.0000i 1.32275i 0.750057 + 0.661373i \(0.230026\pi\)
−0.750057 + 0.661373i \(0.769974\pi\)
\(14\) 28.0000 0.534522
\(15\) 0 0
\(16\) −64.0000 −1.00000
\(17\) 84.0000i 1.19841i 0.800595 + 0.599206i \(0.204517\pi\)
−0.800595 + 0.599206i \(0.795483\pi\)
\(18\) − 36.0000i − 0.471405i
\(19\) −100.000 −1.20745 −0.603726 0.797192i \(-0.706318\pi\)
−0.603726 + 0.797192i \(0.706318\pi\)
\(20\) 0 0
\(21\) 21.0000 0.218218
\(22\) 248.000i 2.40335i
\(23\) 42.0000i 0.380765i 0.981710 + 0.190383i \(0.0609729\pi\)
−0.981710 + 0.190383i \(0.939027\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −248.000 −1.87065
\(27\) − 27.0000i − 0.192450i
\(28\) 56.0000i 0.377964i
\(29\) 10.0000 0.0640329 0.0320164 0.999487i \(-0.489807\pi\)
0.0320164 + 0.999487i \(0.489807\pi\)
\(30\) 0 0
\(31\) −48.0000 −0.278099 −0.139049 0.990285i \(-0.544405\pi\)
−0.139049 + 0.990285i \(0.544405\pi\)
\(32\) − 256.000i − 1.41421i
\(33\) 186.000i 0.981165i
\(34\) −336.000 −1.69481
\(35\) 0 0
\(36\) 72.0000 0.333333
\(37\) − 246.000i − 1.09303i −0.837449 0.546516i \(-0.815954\pi\)
0.837449 0.546516i \(-0.184046\pi\)
\(38\) − 400.000i − 1.70759i
\(39\) −186.000 −0.763688
\(40\) 0 0
\(41\) −248.000 −0.944661 −0.472330 0.881422i \(-0.656587\pi\)
−0.472330 + 0.881422i \(0.656587\pi\)
\(42\) 84.0000i 0.308607i
\(43\) − 68.0000i − 0.241161i −0.992704 0.120580i \(-0.961524\pi\)
0.992704 0.120580i \(-0.0384755\pi\)
\(44\) −496.000 −1.69943
\(45\) 0 0
\(46\) −168.000 −0.538484
\(47\) 324.000i 1.00554i 0.864421 + 0.502769i \(0.167685\pi\)
−0.864421 + 0.502769i \(0.832315\pi\)
\(48\) − 192.000i − 0.577350i
\(49\) −49.0000 −0.142857
\(50\) 0 0
\(51\) −252.000 −0.691903
\(52\) − 496.000i − 1.32275i
\(53\) − 258.000i − 0.668661i −0.942456 0.334330i \(-0.891490\pi\)
0.942456 0.334330i \(-0.108510\pi\)
\(54\) 108.000 0.272166
\(55\) 0 0
\(56\) 0 0
\(57\) − 300.000i − 0.697122i
\(58\) 40.0000i 0.0905562i
\(59\) −120.000 −0.264791 −0.132396 0.991197i \(-0.542267\pi\)
−0.132396 + 0.991197i \(0.542267\pi\)
\(60\) 0 0
\(61\) 622.000 1.30556 0.652778 0.757549i \(-0.273603\pi\)
0.652778 + 0.757549i \(0.273603\pi\)
\(62\) − 192.000i − 0.393291i
\(63\) 63.0000i 0.125988i
\(64\) 512.000 1.00000
\(65\) 0 0
\(66\) −744.000 −1.38758
\(67\) 904.000i 1.64838i 0.566316 + 0.824188i \(0.308368\pi\)
−0.566316 + 0.824188i \(0.691632\pi\)
\(68\) − 672.000i − 1.19841i
\(69\) −126.000 −0.219835
\(70\) 0 0
\(71\) −678.000 −1.13329 −0.566646 0.823961i \(-0.691759\pi\)
−0.566646 + 0.823961i \(0.691759\pi\)
\(72\) 0 0
\(73\) 642.000i 1.02932i 0.857394 + 0.514660i \(0.172082\pi\)
−0.857394 + 0.514660i \(0.827918\pi\)
\(74\) 984.000 1.54578
\(75\) 0 0
\(76\) 800.000 1.20745
\(77\) − 434.000i − 0.642323i
\(78\) − 744.000i − 1.08002i
\(79\) −740.000 −1.05388 −0.526940 0.849903i \(-0.676661\pi\)
−0.526940 + 0.849903i \(0.676661\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 992.000i − 1.33595i
\(83\) − 468.000i − 0.618912i −0.950914 0.309456i \(-0.899853\pi\)
0.950914 0.309456i \(-0.100147\pi\)
\(84\) −168.000 −0.218218
\(85\) 0 0
\(86\) 272.000 0.341052
\(87\) 30.0000i 0.0369694i
\(88\) 0 0
\(89\) −200.000 −0.238202 −0.119101 0.992882i \(-0.538001\pi\)
−0.119101 + 0.992882i \(0.538001\pi\)
\(90\) 0 0
\(91\) 434.000 0.499951
\(92\) − 336.000i − 0.380765i
\(93\) − 144.000i − 0.160560i
\(94\) −1296.00 −1.42204
\(95\) 0 0
\(96\) 768.000 0.816497
\(97\) − 1266.00i − 1.32518i −0.748981 0.662592i \(-0.769456\pi\)
0.748981 0.662592i \(-0.230544\pi\)
\(98\) − 196.000i − 0.202031i
\(99\) −558.000 −0.566476
\(100\) 0 0
\(101\) 232.000 0.228563 0.114281 0.993448i \(-0.463543\pi\)
0.114281 + 0.993448i \(0.463543\pi\)
\(102\) − 1008.00i − 0.978499i
\(103\) 1792.00i 1.71428i 0.515082 + 0.857141i \(0.327761\pi\)
−0.515082 + 0.857141i \(0.672239\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1032.00 0.945629
\(107\) − 1906.00i − 1.72206i −0.508558 0.861028i \(-0.669821\pi\)
0.508558 0.861028i \(-0.330179\pi\)
\(108\) 216.000i 0.192450i
\(109\) 90.0000 0.0790866 0.0395433 0.999218i \(-0.487410\pi\)
0.0395433 + 0.999218i \(0.487410\pi\)
\(110\) 0 0
\(111\) 738.000 0.631062
\(112\) 448.000i 0.377964i
\(113\) − 458.000i − 0.381283i −0.981660 0.190642i \(-0.938943\pi\)
0.981660 0.190642i \(-0.0610569\pi\)
\(114\) 1200.00 0.985880
\(115\) 0 0
\(116\) −80.0000 −0.0640329
\(117\) − 558.000i − 0.440916i
\(118\) − 480.000i − 0.374471i
\(119\) 588.000 0.452957
\(120\) 0 0
\(121\) 2513.00 1.88805
\(122\) 2488.00i 1.84634i
\(123\) − 744.000i − 0.545400i
\(124\) 384.000 0.278099
\(125\) 0 0
\(126\) −252.000 −0.178174
\(127\) 804.000i 0.561760i 0.959743 + 0.280880i \(0.0906262\pi\)
−0.959743 + 0.280880i \(0.909374\pi\)
\(128\) 0 0
\(129\) 204.000 0.139234
\(130\) 0 0
\(131\) 812.000 0.541563 0.270782 0.962641i \(-0.412718\pi\)
0.270782 + 0.962641i \(0.412718\pi\)
\(132\) − 1488.00i − 0.981165i
\(133\) 700.000i 0.456374i
\(134\) −3616.00 −2.33116
\(135\) 0 0
\(136\) 0 0
\(137\) 414.000i 0.258178i 0.991633 + 0.129089i \(0.0412053\pi\)
−0.991633 + 0.129089i \(0.958795\pi\)
\(138\) − 504.000i − 0.310894i
\(139\) 1620.00 0.988537 0.494268 0.869309i \(-0.335436\pi\)
0.494268 + 0.869309i \(0.335436\pi\)
\(140\) 0 0
\(141\) −972.000 −0.580547
\(142\) − 2712.00i − 1.60272i
\(143\) 3844.00i 2.24791i
\(144\) 576.000 0.333333
\(145\) 0 0
\(146\) −2568.00 −1.45568
\(147\) − 147.000i − 0.0824786i
\(148\) 1968.00i 1.09303i
\(149\) −2370.00 −1.30307 −0.651537 0.758617i \(-0.725875\pi\)
−0.651537 + 0.758617i \(0.725875\pi\)
\(150\) 0 0
\(151\) −568.000 −0.306114 −0.153057 0.988217i \(-0.548912\pi\)
−0.153057 + 0.988217i \(0.548912\pi\)
\(152\) 0 0
\(153\) − 756.000i − 0.399470i
\(154\) 1736.00 0.908382
\(155\) 0 0
\(156\) 1488.00 0.763688
\(157\) − 266.000i − 0.135217i −0.997712 0.0676086i \(-0.978463\pi\)
0.997712 0.0676086i \(-0.0215369\pi\)
\(158\) − 2960.00i − 1.49041i
\(159\) 774.000 0.386052
\(160\) 0 0
\(161\) 294.000 0.143916
\(162\) 324.000i 0.157135i
\(163\) 272.000i 0.130704i 0.997862 + 0.0653518i \(0.0208170\pi\)
−0.997862 + 0.0653518i \(0.979183\pi\)
\(164\) 1984.00 0.944661
\(165\) 0 0
\(166\) 1872.00 0.875273
\(167\) − 1876.00i − 0.869277i −0.900605 0.434638i \(-0.856876\pi\)
0.900605 0.434638i \(-0.143124\pi\)
\(168\) 0 0
\(169\) −1647.00 −0.749659
\(170\) 0 0
\(171\) 900.000 0.402484
\(172\) 544.000i 0.241161i
\(173\) 152.000i 0.0667997i 0.999442 + 0.0333998i \(0.0106335\pi\)
−0.999442 + 0.0333998i \(0.989367\pi\)
\(174\) −120.000 −0.0522826
\(175\) 0 0
\(176\) −3968.00 −1.69943
\(177\) − 360.000i − 0.152877i
\(178\) − 800.000i − 0.336868i
\(179\) −610.000 −0.254713 −0.127356 0.991857i \(-0.540649\pi\)
−0.127356 + 0.991857i \(0.540649\pi\)
\(180\) 0 0
\(181\) 1042.00 0.427907 0.213954 0.976844i \(-0.431366\pi\)
0.213954 + 0.976844i \(0.431366\pi\)
\(182\) 1736.00i 0.707038i
\(183\) 1866.00i 0.753763i
\(184\) 0 0
\(185\) 0 0
\(186\) 576.000 0.227067
\(187\) 5208.00i 2.03661i
\(188\) − 2592.00i − 1.00554i
\(189\) −189.000 −0.0727393
\(190\) 0 0
\(191\) −2038.00 −0.772065 −0.386033 0.922485i \(-0.626155\pi\)
−0.386033 + 0.922485i \(0.626155\pi\)
\(192\) 1536.00i 0.577350i
\(193\) 2602.00i 0.970446i 0.874390 + 0.485223i \(0.161262\pi\)
−0.874390 + 0.485223i \(0.838738\pi\)
\(194\) 5064.00 1.87409
\(195\) 0 0
\(196\) 392.000 0.142857
\(197\) 2354.00i 0.851348i 0.904877 + 0.425674i \(0.139963\pi\)
−0.904877 + 0.425674i \(0.860037\pi\)
\(198\) − 2232.00i − 0.801118i
\(199\) −1680.00 −0.598452 −0.299226 0.954182i \(-0.596729\pi\)
−0.299226 + 0.954182i \(0.596729\pi\)
\(200\) 0 0
\(201\) −2712.00 −0.951690
\(202\) 928.000i 0.323237i
\(203\) − 70.0000i − 0.0242022i
\(204\) 2016.00 0.691903
\(205\) 0 0
\(206\) −7168.00 −2.42436
\(207\) − 378.000i − 0.126922i
\(208\) − 3968.00i − 1.32275i
\(209\) −6200.00 −2.05198
\(210\) 0 0
\(211\) −668.000 −0.217948 −0.108974 0.994045i \(-0.534757\pi\)
−0.108974 + 0.994045i \(0.534757\pi\)
\(212\) 2064.00i 0.668661i
\(213\) − 2034.00i − 0.654307i
\(214\) 7624.00 2.43535
\(215\) 0 0
\(216\) 0 0
\(217\) 336.000i 0.105111i
\(218\) 360.000i 0.111845i
\(219\) −1926.00 −0.594279
\(220\) 0 0
\(221\) −5208.00 −1.58519
\(222\) 2952.00i 0.892456i
\(223\) 1832.00i 0.550134i 0.961425 + 0.275067i \(0.0887000\pi\)
−0.961425 + 0.275067i \(0.911300\pi\)
\(224\) −1792.00 −0.534522
\(225\) 0 0
\(226\) 1832.00 0.539216
\(227\) 4944.00i 1.44557i 0.691072 + 0.722786i \(0.257139\pi\)
−0.691072 + 0.722786i \(0.742861\pi\)
\(228\) 2400.00i 0.697122i
\(229\) 5470.00 1.57846 0.789231 0.614096i \(-0.210479\pi\)
0.789231 + 0.614096i \(0.210479\pi\)
\(230\) 0 0
\(231\) 1302.00 0.370846
\(232\) 0 0
\(233\) 2802.00i 0.787833i 0.919146 + 0.393917i \(0.128880\pi\)
−0.919146 + 0.393917i \(0.871120\pi\)
\(234\) 2232.00 0.623549
\(235\) 0 0
\(236\) 960.000 0.264791
\(237\) − 2220.00i − 0.608458i
\(238\) 2352.00i 0.640578i
\(239\) 1170.00 0.316657 0.158328 0.987386i \(-0.449390\pi\)
0.158328 + 0.987386i \(0.449390\pi\)
\(240\) 0 0
\(241\) −2338.00 −0.624912 −0.312456 0.949932i \(-0.601152\pi\)
−0.312456 + 0.949932i \(0.601152\pi\)
\(242\) 10052.0i 2.67011i
\(243\) 243.000i 0.0641500i
\(244\) −4976.00 −1.30556
\(245\) 0 0
\(246\) 2976.00 0.771312
\(247\) − 6200.00i − 1.59715i
\(248\) 0 0
\(249\) 1404.00 0.357329
\(250\) 0 0
\(251\) 2792.00 0.702109 0.351055 0.936355i \(-0.385823\pi\)
0.351055 + 0.936355i \(0.385823\pi\)
\(252\) − 504.000i − 0.125988i
\(253\) 2604.00i 0.647083i
\(254\) −3216.00 −0.794448
\(255\) 0 0
\(256\) 4096.00 1.00000
\(257\) 7024.00i 1.70484i 0.522854 + 0.852422i \(0.324867\pi\)
−0.522854 + 0.852422i \(0.675133\pi\)
\(258\) 816.000i 0.196907i
\(259\) −1722.00 −0.413127
\(260\) 0 0
\(261\) −90.0000 −0.0213443
\(262\) 3248.00i 0.765886i
\(263\) − 2438.00i − 0.571610i −0.958288 0.285805i \(-0.907739\pi\)
0.958288 0.285805i \(-0.0922610\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2800.00 −0.645410
\(267\) − 600.000i − 0.137526i
\(268\) − 7232.00i − 1.64838i
\(269\) 6780.00 1.53674 0.768372 0.640004i \(-0.221067\pi\)
0.768372 + 0.640004i \(0.221067\pi\)
\(270\) 0 0
\(271\) −1928.00 −0.432168 −0.216084 0.976375i \(-0.569329\pi\)
−0.216084 + 0.976375i \(0.569329\pi\)
\(272\) − 5376.00i − 1.19841i
\(273\) 1302.00i 0.288647i
\(274\) −1656.00 −0.365119
\(275\) 0 0
\(276\) 1008.00 0.219835
\(277\) 5554.00i 1.20472i 0.798224 + 0.602360i \(0.205773\pi\)
−0.798224 + 0.602360i \(0.794227\pi\)
\(278\) 6480.00i 1.39800i
\(279\) 432.000 0.0926995
\(280\) 0 0
\(281\) 1942.00 0.412278 0.206139 0.978523i \(-0.433910\pi\)
0.206139 + 0.978523i \(0.433910\pi\)
\(282\) − 3888.00i − 0.821018i
\(283\) − 4828.00i − 1.01412i −0.861912 0.507058i \(-0.830733\pi\)
0.861912 0.507058i \(-0.169267\pi\)
\(284\) 5424.00 1.13329
\(285\) 0 0
\(286\) −15376.0 −3.17903
\(287\) 1736.00i 0.357048i
\(288\) 2304.00i 0.471405i
\(289\) −2143.00 −0.436190
\(290\) 0 0
\(291\) 3798.00 0.765095
\(292\) − 5136.00i − 1.02932i
\(293\) 6152.00i 1.22663i 0.789837 + 0.613317i \(0.210165\pi\)
−0.789837 + 0.613317i \(0.789835\pi\)
\(294\) 588.000 0.116642
\(295\) 0 0
\(296\) 0 0
\(297\) − 1674.00i − 0.327055i
\(298\) − 9480.00i − 1.84282i
\(299\) −2604.00 −0.503656
\(300\) 0 0
\(301\) −476.000 −0.0911501
\(302\) − 2272.00i − 0.432910i
\(303\) 696.000i 0.131961i
\(304\) 6400.00 1.20745
\(305\) 0 0
\(306\) 3024.00 0.564937
\(307\) 5884.00i 1.09387i 0.837176 + 0.546934i \(0.184205\pi\)
−0.837176 + 0.546934i \(0.815795\pi\)
\(308\) 3472.00i 0.642323i
\(309\) −5376.00 −0.989741
\(310\) 0 0
\(311\) 9132.00 1.66504 0.832521 0.553993i \(-0.186897\pi\)
0.832521 + 0.553993i \(0.186897\pi\)
\(312\) 0 0
\(313\) 9382.00i 1.69426i 0.531389 + 0.847128i \(0.321670\pi\)
−0.531389 + 0.847128i \(0.678330\pi\)
\(314\) 1064.00 0.191226
\(315\) 0 0
\(316\) 5920.00 1.05388
\(317\) 3114.00i 0.551734i 0.961196 + 0.275867i \(0.0889649\pi\)
−0.961196 + 0.275867i \(0.911035\pi\)
\(318\) 3096.00i 0.545959i
\(319\) 620.000 0.108819
\(320\) 0 0
\(321\) 5718.00 0.994229
\(322\) 1176.00i 0.203528i
\(323\) − 8400.00i − 1.44702i
\(324\) −648.000 −0.111111
\(325\) 0 0
\(326\) −1088.00 −0.184843
\(327\) 270.000i 0.0456607i
\(328\) 0 0
\(329\) 2268.00 0.380057
\(330\) 0 0
\(331\) 1532.00 0.254400 0.127200 0.991877i \(-0.459401\pi\)
0.127200 + 0.991877i \(0.459401\pi\)
\(332\) 3744.00i 0.618912i
\(333\) 2214.00i 0.364344i
\(334\) 7504.00 1.22934
\(335\) 0 0
\(336\) −1344.00 −0.218218
\(337\) − 4166.00i − 0.673402i −0.941612 0.336701i \(-0.890689\pi\)
0.941612 0.336701i \(-0.109311\pi\)
\(338\) − 6588.00i − 1.06018i
\(339\) 1374.00 0.220134
\(340\) 0 0
\(341\) −2976.00 −0.472608
\(342\) 3600.00i 0.569198i
\(343\) 343.000i 0.0539949i
\(344\) 0 0
\(345\) 0 0
\(346\) −608.000 −0.0944690
\(347\) − 11366.0i − 1.75838i −0.476469 0.879191i \(-0.658083\pi\)
0.476469 0.879191i \(-0.341917\pi\)
\(348\) − 240.000i − 0.0369694i
\(349\) −9310.00 −1.42795 −0.713973 0.700174i \(-0.753106\pi\)
−0.713973 + 0.700174i \(0.753106\pi\)
\(350\) 0 0
\(351\) 1674.00 0.254563
\(352\) − 15872.0i − 2.40335i
\(353\) 8572.00i 1.29247i 0.763139 + 0.646234i \(0.223657\pi\)
−0.763139 + 0.646234i \(0.776343\pi\)
\(354\) 1440.00 0.216201
\(355\) 0 0
\(356\) 1600.00 0.238202
\(357\) 1764.00i 0.261515i
\(358\) − 2440.00i − 0.360218i
\(359\) 4790.00 0.704196 0.352098 0.935963i \(-0.385468\pi\)
0.352098 + 0.935963i \(0.385468\pi\)
\(360\) 0 0
\(361\) 3141.00 0.457938
\(362\) 4168.00i 0.605153i
\(363\) 7539.00i 1.09007i
\(364\) −3472.00 −0.499951
\(365\) 0 0
\(366\) −7464.00 −1.06598
\(367\) 5424.00i 0.771473i 0.922609 + 0.385736i \(0.126053\pi\)
−0.922609 + 0.385736i \(0.873947\pi\)
\(368\) − 2688.00i − 0.380765i
\(369\) 2232.00 0.314887
\(370\) 0 0
\(371\) −1806.00 −0.252730
\(372\) 1152.00i 0.160560i
\(373\) − 1838.00i − 0.255142i −0.991829 0.127571i \(-0.959282\pi\)
0.991829 0.127571i \(-0.0407181\pi\)
\(374\) −20832.0 −2.88021
\(375\) 0 0
\(376\) 0 0
\(377\) 620.000i 0.0846993i
\(378\) − 756.000i − 0.102869i
\(379\) 4260.00 0.577365 0.288683 0.957425i \(-0.406783\pi\)
0.288683 + 0.957425i \(0.406783\pi\)
\(380\) 0 0
\(381\) −2412.00 −0.324332
\(382\) − 8152.00i − 1.09187i
\(383\) − 9048.00i − 1.20713i −0.797313 0.603566i \(-0.793746\pi\)
0.797313 0.603566i \(-0.206254\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −10408.0 −1.37242
\(387\) 612.000i 0.0803868i
\(388\) 10128.0i 1.32518i
\(389\) 11490.0 1.49760 0.748800 0.662796i \(-0.230631\pi\)
0.748800 + 0.662796i \(0.230631\pi\)
\(390\) 0 0
\(391\) −3528.00 −0.456314
\(392\) 0 0
\(393\) 2436.00i 0.312672i
\(394\) −9416.00 −1.20399
\(395\) 0 0
\(396\) 4464.00 0.566476
\(397\) − 1866.00i − 0.235899i −0.993020 0.117949i \(-0.962368\pi\)
0.993020 0.117949i \(-0.0376321\pi\)
\(398\) − 6720.00i − 0.846340i
\(399\) −2100.00 −0.263487
\(400\) 0 0
\(401\) 13662.0 1.70137 0.850683 0.525679i \(-0.176189\pi\)
0.850683 + 0.525679i \(0.176189\pi\)
\(402\) − 10848.0i − 1.34589i
\(403\) − 2976.00i − 0.367854i
\(404\) −1856.00 −0.228563
\(405\) 0 0
\(406\) 280.000 0.0342270
\(407\) − 15252.0i − 1.85753i
\(408\) 0 0
\(409\) 13210.0 1.59705 0.798524 0.601963i \(-0.205615\pi\)
0.798524 + 0.601963i \(0.205615\pi\)
\(410\) 0 0
\(411\) −1242.00 −0.149059
\(412\) − 14336.0i − 1.71428i
\(413\) 840.000i 0.100082i
\(414\) 1512.00 0.179495
\(415\) 0 0
\(416\) 15872.0 1.87065
\(417\) 4860.00i 0.570732i
\(418\) − 24800.0i − 2.90193i
\(419\) −6960.00 −0.811499 −0.405750 0.913984i \(-0.632990\pi\)
−0.405750 + 0.913984i \(0.632990\pi\)
\(420\) 0 0
\(421\) 8162.00 0.944873 0.472437 0.881365i \(-0.343375\pi\)
0.472437 + 0.881365i \(0.343375\pi\)
\(422\) − 2672.00i − 0.308225i
\(423\) − 2916.00i − 0.335179i
\(424\) 0 0
\(425\) 0 0
\(426\) 8136.00 0.925330
\(427\) − 4354.00i − 0.493454i
\(428\) 15248.0i 1.72206i
\(429\) −11532.0 −1.29783
\(430\) 0 0
\(431\) 16602.0 1.85543 0.927715 0.373290i \(-0.121770\pi\)
0.927715 + 0.373290i \(0.121770\pi\)
\(432\) 1728.00i 0.192450i
\(433\) − 7738.00i − 0.858810i −0.903112 0.429405i \(-0.858723\pi\)
0.903112 0.429405i \(-0.141277\pi\)
\(434\) −1344.00 −0.148650
\(435\) 0 0
\(436\) −720.000 −0.0790866
\(437\) − 4200.00i − 0.459756i
\(438\) − 7704.00i − 0.840437i
\(439\) 840.000 0.0913235 0.0456617 0.998957i \(-0.485460\pi\)
0.0456617 + 0.998957i \(0.485460\pi\)
\(440\) 0 0
\(441\) 441.000 0.0476190
\(442\) − 20832.0i − 2.24180i
\(443\) − 6618.00i − 0.709776i −0.934909 0.354888i \(-0.884519\pi\)
0.934909 0.354888i \(-0.115481\pi\)
\(444\) −5904.00 −0.631062
\(445\) 0 0
\(446\) −7328.00 −0.778006
\(447\) − 7110.00i − 0.752330i
\(448\) − 3584.00i − 0.377964i
\(449\) −3090.00 −0.324780 −0.162390 0.986727i \(-0.551920\pi\)
−0.162390 + 0.986727i \(0.551920\pi\)
\(450\) 0 0
\(451\) −15376.0 −1.60538
\(452\) 3664.00i 0.381283i
\(453\) − 1704.00i − 0.176735i
\(454\) −19776.0 −2.04435
\(455\) 0 0
\(456\) 0 0
\(457\) 5914.00i 0.605351i 0.953094 + 0.302675i \(0.0978798\pi\)
−0.953094 + 0.302675i \(0.902120\pi\)
\(458\) 21880.0i 2.23228i
\(459\) 2268.00 0.230634
\(460\) 0 0
\(461\) −15968.0 −1.61324 −0.806620 0.591070i \(-0.798706\pi\)
−0.806620 + 0.591070i \(0.798706\pi\)
\(462\) 5208.00i 0.524455i
\(463\) 1172.00i 0.117640i 0.998269 + 0.0588202i \(0.0187338\pi\)
−0.998269 + 0.0588202i \(0.981266\pi\)
\(464\) −640.000 −0.0640329
\(465\) 0 0
\(466\) −11208.0 −1.11416
\(467\) 5304.00i 0.525567i 0.964855 + 0.262784i \(0.0846405\pi\)
−0.964855 + 0.262784i \(0.915359\pi\)
\(468\) 4464.00i 0.440916i
\(469\) 6328.00 0.623027
\(470\) 0 0
\(471\) 798.000 0.0780677
\(472\) 0 0
\(473\) − 4216.00i − 0.409835i
\(474\) 8880.00 0.860489
\(475\) 0 0
\(476\) −4704.00 −0.452957
\(477\) 2322.00i 0.222887i
\(478\) 4680.00i 0.447821i
\(479\) −5740.00 −0.547531 −0.273765 0.961796i \(-0.588269\pi\)
−0.273765 + 0.961796i \(0.588269\pi\)
\(480\) 0 0
\(481\) 15252.0 1.44580
\(482\) − 9352.00i − 0.883759i
\(483\) 882.000i 0.0830898i
\(484\) −20104.0 −1.88805
\(485\) 0 0
\(486\) −972.000 −0.0907218
\(487\) 8944.00i 0.832220i 0.909314 + 0.416110i \(0.136607\pi\)
−0.909314 + 0.416110i \(0.863393\pi\)
\(488\) 0 0
\(489\) −816.000 −0.0754617
\(490\) 0 0
\(491\) −5558.00 −0.510853 −0.255427 0.966828i \(-0.582216\pi\)
−0.255427 + 0.966828i \(0.582216\pi\)
\(492\) 5952.00i 0.545400i
\(493\) 840.000i 0.0767377i
\(494\) 24800.0 2.25871
\(495\) 0 0
\(496\) 3072.00 0.278099
\(497\) 4746.00i 0.428344i
\(498\) 5616.00i 0.505339i
\(499\) 19820.0 1.77809 0.889043 0.457823i \(-0.151371\pi\)
0.889043 + 0.457823i \(0.151371\pi\)
\(500\) 0 0
\(501\) 5628.00 0.501877
\(502\) 11168.0i 0.992933i
\(503\) − 1848.00i − 0.163814i −0.996640 0.0819068i \(-0.973899\pi\)
0.996640 0.0819068i \(-0.0261010\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −10416.0 −0.915114
\(507\) − 4941.00i − 0.432816i
\(508\) − 6432.00i − 0.561760i
\(509\) −340.000 −0.0296075 −0.0148038 0.999890i \(-0.504712\pi\)
−0.0148038 + 0.999890i \(0.504712\pi\)
\(510\) 0 0
\(511\) 4494.00 0.389047
\(512\) 16384.0i 1.41421i
\(513\) 2700.00i 0.232374i
\(514\) −28096.0 −2.41101
\(515\) 0 0
\(516\) −1632.00 −0.139234
\(517\) 20088.0i 1.70884i
\(518\) − 6888.00i − 0.584250i
\(519\) −456.000 −0.0385668
\(520\) 0 0
\(521\) 10212.0 0.858725 0.429363 0.903132i \(-0.358738\pi\)
0.429363 + 0.903132i \(0.358738\pi\)
\(522\) − 360.000i − 0.0301854i
\(523\) 9332.00i 0.780229i 0.920766 + 0.390115i \(0.127565\pi\)
−0.920766 + 0.390115i \(0.872435\pi\)
\(524\) −6496.00 −0.541563
\(525\) 0 0
\(526\) 9752.00 0.808379
\(527\) − 4032.00i − 0.333276i
\(528\) − 11904.0i − 0.981165i
\(529\) 10403.0 0.855018
\(530\) 0 0
\(531\) 1080.00 0.0882637
\(532\) − 5600.00i − 0.456374i
\(533\) − 15376.0i − 1.24955i
\(534\) 2400.00 0.194491
\(535\) 0 0
\(536\) 0 0
\(537\) − 1830.00i − 0.147058i
\(538\) 27120.0i 2.17328i
\(539\) −3038.00 −0.242775
\(540\) 0 0
\(541\) −8998.00 −0.715073 −0.357536 0.933899i \(-0.616383\pi\)
−0.357536 + 0.933899i \(0.616383\pi\)
\(542\) − 7712.00i − 0.611179i
\(543\) 3126.00i 0.247052i
\(544\) 21504.0 1.69481
\(545\) 0 0
\(546\) −5208.00 −0.408208
\(547\) − 3416.00i − 0.267016i −0.991048 0.133508i \(-0.957376\pi\)
0.991048 0.133508i \(-0.0426241\pi\)
\(548\) − 3312.00i − 0.258178i
\(549\) −5598.00 −0.435185
\(550\) 0 0
\(551\) −1000.00 −0.0773166
\(552\) 0 0
\(553\) 5180.00i 0.398329i
\(554\) −22216.0 −1.70373
\(555\) 0 0
\(556\) −12960.0 −0.988537
\(557\) − 526.000i − 0.0400132i −0.999800 0.0200066i \(-0.993631\pi\)
0.999800 0.0200066i \(-0.00636872\pi\)
\(558\) 1728.00i 0.131097i
\(559\) 4216.00 0.318994
\(560\) 0 0
\(561\) −15624.0 −1.17584
\(562\) 7768.00i 0.583049i
\(563\) 6712.00i 0.502446i 0.967929 + 0.251223i \(0.0808327\pi\)
−0.967929 + 0.251223i \(0.919167\pi\)
\(564\) 7776.00 0.580547
\(565\) 0 0
\(566\) 19312.0 1.43418
\(567\) − 567.000i − 0.0419961i
\(568\) 0 0
\(569\) −4190.00 −0.308706 −0.154353 0.988016i \(-0.549329\pi\)
−0.154353 + 0.988016i \(0.549329\pi\)
\(570\) 0 0
\(571\) 3032.00 0.222216 0.111108 0.993808i \(-0.464560\pi\)
0.111108 + 0.993808i \(0.464560\pi\)
\(572\) − 30752.0i − 2.24791i
\(573\) − 6114.00i − 0.445752i
\(574\) −6944.00 −0.504942
\(575\) 0 0
\(576\) −4608.00 −0.333333
\(577\) 5434.00i 0.392063i 0.980598 + 0.196032i \(0.0628055\pi\)
−0.980598 + 0.196032i \(0.937195\pi\)
\(578\) − 8572.00i − 0.616865i
\(579\) −7806.00 −0.560287
\(580\) 0 0
\(581\) −3276.00 −0.233927
\(582\) 15192.0i 1.08201i
\(583\) − 15996.0i − 1.13634i
\(584\) 0 0
\(585\) 0 0
\(586\) −24608.0 −1.73472
\(587\) 464.000i 0.0326258i 0.999867 + 0.0163129i \(0.00519278\pi\)
−0.999867 + 0.0163129i \(0.994807\pi\)
\(588\) 1176.00i 0.0824786i
\(589\) 4800.00 0.335790
\(590\) 0 0
\(591\) −7062.00 −0.491526
\(592\) 15744.0i 1.09303i
\(593\) − 11748.0i − 0.813546i −0.913529 0.406773i \(-0.866654\pi\)
0.913529 0.406773i \(-0.133346\pi\)
\(594\) 6696.00 0.462526
\(595\) 0 0
\(596\) 18960.0 1.30307
\(597\) − 5040.00i − 0.345517i
\(598\) − 10416.0i − 0.712277i
\(599\) −7650.00 −0.521821 −0.260910 0.965363i \(-0.584023\pi\)
−0.260910 + 0.965363i \(0.584023\pi\)
\(600\) 0 0
\(601\) −22878.0 −1.55277 −0.776384 0.630261i \(-0.782948\pi\)
−0.776384 + 0.630261i \(0.782948\pi\)
\(602\) − 1904.00i − 0.128906i
\(603\) − 8136.00i − 0.549459i
\(604\) 4544.00 0.306114
\(605\) 0 0
\(606\) −2784.00 −0.186621
\(607\) 704.000i 0.0470749i 0.999723 + 0.0235375i \(0.00749290\pi\)
−0.999723 + 0.0235375i \(0.992507\pi\)
\(608\) 25600.0i 1.70759i
\(609\) 210.000 0.0139731
\(610\) 0 0
\(611\) −20088.0 −1.33007
\(612\) 6048.00i 0.399470i
\(613\) − 24958.0i − 1.64444i −0.569167 0.822222i \(-0.692734\pi\)
0.569167 0.822222i \(-0.307266\pi\)
\(614\) −23536.0 −1.54696
\(615\) 0 0
\(616\) 0 0
\(617\) − 8826.00i − 0.575886i −0.957648 0.287943i \(-0.907029\pi\)
0.957648 0.287943i \(-0.0929713\pi\)
\(618\) − 21504.0i − 1.39971i
\(619\) −21220.0 −1.37787 −0.688937 0.724821i \(-0.741922\pi\)
−0.688937 + 0.724821i \(0.741922\pi\)
\(620\) 0 0
\(621\) 1134.00 0.0732783
\(622\) 36528.0i 2.35473i
\(623\) 1400.00i 0.0900318i
\(624\) 11904.0 0.763688
\(625\) 0 0
\(626\) −37528.0 −2.39604
\(627\) − 18600.0i − 1.18471i
\(628\) 2128.00i 0.135217i
\(629\) 20664.0 1.30990
\(630\) 0 0
\(631\) −3268.00 −0.206176 −0.103088 0.994672i \(-0.532872\pi\)
−0.103088 + 0.994672i \(0.532872\pi\)
\(632\) 0 0
\(633\) − 2004.00i − 0.125832i
\(634\) −12456.0 −0.780270
\(635\) 0 0
\(636\) −6192.00 −0.386052
\(637\) − 3038.00i − 0.188964i
\(638\) 2480.00i 0.153894i
\(639\) 6102.00 0.377764
\(640\) 0 0
\(641\) 13062.0 0.804864 0.402432 0.915450i \(-0.368165\pi\)
0.402432 + 0.915450i \(0.368165\pi\)
\(642\) 22872.0i 1.40605i
\(643\) 28012.0i 1.71802i 0.511961 + 0.859009i \(0.328919\pi\)
−0.511961 + 0.859009i \(0.671081\pi\)
\(644\) −2352.00 −0.143916
\(645\) 0 0
\(646\) 33600.0 2.04640
\(647\) 3844.00i 0.233575i 0.993157 + 0.116788i \(0.0372597\pi\)
−0.993157 + 0.116788i \(0.962740\pi\)
\(648\) 0 0
\(649\) −7440.00 −0.449993
\(650\) 0 0
\(651\) −1008.00 −0.0606861
\(652\) − 2176.00i − 0.130704i
\(653\) 28482.0i 1.70687i 0.521198 + 0.853436i \(0.325485\pi\)
−0.521198 + 0.853436i \(0.674515\pi\)
\(654\) −1080.00 −0.0645739
\(655\) 0 0
\(656\) 15872.0 0.944661
\(657\) − 5778.00i − 0.343107i
\(658\) 9072.00i 0.537482i
\(659\) 9330.00 0.551510 0.275755 0.961228i \(-0.411072\pi\)
0.275755 + 0.961228i \(0.411072\pi\)
\(660\) 0 0
\(661\) 8782.00 0.516763 0.258381 0.966043i \(-0.416811\pi\)
0.258381 + 0.966043i \(0.416811\pi\)
\(662\) 6128.00i 0.359776i
\(663\) − 15624.0i − 0.915212i
\(664\) 0 0
\(665\) 0 0
\(666\) −8856.00 −0.515260
\(667\) 420.000i 0.0243815i
\(668\) 15008.0i 0.869277i
\(669\) −5496.00 −0.317620
\(670\) 0 0
\(671\) 38564.0 2.21870
\(672\) − 5376.00i − 0.308607i
\(673\) 10562.0i 0.604956i 0.953156 + 0.302478i \(0.0978138\pi\)
−0.953156 + 0.302478i \(0.902186\pi\)
\(674\) 16664.0 0.952334
\(675\) 0 0
\(676\) 13176.0 0.749659
\(677\) − 26016.0i − 1.47692i −0.674296 0.738461i \(-0.735553\pi\)
0.674296 0.738461i \(-0.264447\pi\)
\(678\) 5496.00i 0.311317i
\(679\) −8862.00 −0.500872
\(680\) 0 0
\(681\) −14832.0 −0.834601
\(682\) − 11904.0i − 0.668369i
\(683\) − 8898.00i − 0.498496i −0.968440 0.249248i \(-0.919817\pi\)
0.968440 0.249248i \(-0.0801834\pi\)
\(684\) −7200.00 −0.402484
\(685\) 0 0
\(686\) −1372.00 −0.0763604
\(687\) 16410.0i 0.911325i
\(688\) 4352.00i 0.241161i
\(689\) 15996.0 0.884469
\(690\) 0 0
\(691\) 30572.0 1.68309 0.841544 0.540189i \(-0.181647\pi\)
0.841544 + 0.540189i \(0.181647\pi\)
\(692\) − 1216.00i − 0.0667997i
\(693\) 3906.00i 0.214108i
\(694\) 45464.0 2.48673
\(695\) 0 0
\(696\) 0 0
\(697\) − 20832.0i − 1.13209i
\(698\) − 37240.0i − 2.01942i
\(699\) −8406.00 −0.454856
\(700\) 0 0
\(701\) −30618.0 −1.64968 −0.824840 0.565366i \(-0.808735\pi\)
−0.824840 + 0.565366i \(0.808735\pi\)
\(702\) 6696.00i 0.360006i
\(703\) 24600.0i 1.31978i
\(704\) 31744.0 1.69943
\(705\) 0 0
\(706\) −34288.0 −1.82783
\(707\) − 1624.00i − 0.0863887i
\(708\) 2880.00i 0.152877i
\(709\) 8130.00 0.430647 0.215323 0.976543i \(-0.430919\pi\)
0.215323 + 0.976543i \(0.430919\pi\)
\(710\) 0 0
\(711\) 6660.00 0.351293
\(712\) 0 0
\(713\) − 2016.00i − 0.105890i
\(714\) −7056.00 −0.369838
\(715\) 0 0
\(716\) 4880.00 0.254713
\(717\) 3510.00i 0.182822i
\(718\) 19160.0i 0.995884i
\(719\) 27840.0 1.44403 0.722014 0.691878i \(-0.243216\pi\)
0.722014 + 0.691878i \(0.243216\pi\)
\(720\) 0 0
\(721\) 12544.0 0.647938
\(722\) 12564.0i 0.647623i
\(723\) − 7014.00i − 0.360793i
\(724\) −8336.00 −0.427907
\(725\) 0 0
\(726\) −30156.0 −1.54159
\(727\) 14624.0i 0.746044i 0.927822 + 0.373022i \(0.121678\pi\)
−0.927822 + 0.373022i \(0.878322\pi\)
\(728\) 0 0
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 5712.00 0.289010
\(732\) − 14928.0i − 0.753763i
\(733\) 20862.0i 1.05124i 0.850721 + 0.525618i \(0.176166\pi\)
−0.850721 + 0.525618i \(0.823834\pi\)
\(734\) −21696.0 −1.09103
\(735\) 0 0
\(736\) 10752.0 0.538484
\(737\) 56048.0i 2.80130i
\(738\) 8928.00i 0.445317i
\(739\) 13920.0 0.692903 0.346452 0.938068i \(-0.387386\pi\)
0.346452 + 0.938068i \(0.387386\pi\)
\(740\) 0 0
\(741\) 18600.0 0.922116
\(742\) − 7224.00i − 0.357414i
\(743\) − 25578.0i − 1.26294i −0.775400 0.631471i \(-0.782452\pi\)
0.775400 0.631471i \(-0.217548\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 7352.00 0.360826
\(747\) 4212.00i 0.206304i
\(748\) − 41664.0i − 2.03661i
\(749\) −13342.0 −0.650876
\(750\) 0 0
\(751\) 33472.0 1.62638 0.813189 0.581999i \(-0.197729\pi\)
0.813189 + 0.581999i \(0.197729\pi\)
\(752\) − 20736.0i − 1.00554i
\(753\) 8376.00i 0.405363i
\(754\) −2480.00 −0.119783
\(755\) 0 0
\(756\) 1512.00 0.0727393
\(757\) 25934.0i 1.24516i 0.782556 + 0.622581i \(0.213916\pi\)
−0.782556 + 0.622581i \(0.786084\pi\)
\(758\) 17040.0i 0.816518i
\(759\) −7812.00 −0.373594
\(760\) 0 0
\(761\) 26952.0 1.28385 0.641925 0.766768i \(-0.278136\pi\)
0.641925 + 0.766768i \(0.278136\pi\)
\(762\) − 9648.00i − 0.458675i
\(763\) − 630.000i − 0.0298919i
\(764\) 16304.0 0.772065
\(765\) 0 0
\(766\) 36192.0 1.70714
\(767\) − 7440.00i − 0.350251i
\(768\) 12288.0i 0.577350i
\(769\) −23450.0 −1.09965 −0.549824 0.835281i \(-0.685305\pi\)
−0.549824 + 0.835281i \(0.685305\pi\)
\(770\) 0 0
\(771\) −21072.0 −0.984293
\(772\) − 20816.0i − 0.970446i
\(773\) − 39568.0i − 1.84109i −0.390637 0.920545i \(-0.627745\pi\)
0.390637 0.920545i \(-0.372255\pi\)
\(774\) −2448.00 −0.113684
\(775\) 0 0
\(776\) 0 0
\(777\) − 5166.00i − 0.238519i
\(778\) 45960.0i 2.11793i
\(779\) 24800.0 1.14063
\(780\) 0 0
\(781\) −42036.0 −1.92595
\(782\) − 14112.0i − 0.645325i
\(783\) − 270.000i − 0.0123231i
\(784\) 3136.00 0.142857
\(785\) 0 0
\(786\) −9744.00 −0.442184
\(787\) − 12356.0i − 0.559649i −0.960051 0.279825i \(-0.909724\pi\)
0.960051 0.279825i \(-0.0902763\pi\)
\(788\) − 18832.0i − 0.851348i
\(789\) 7314.00 0.330019
\(790\) 0 0
\(791\) −3206.00 −0.144112
\(792\) 0 0
\(793\) 38564.0i 1.72692i
\(794\) 7464.00 0.333611
\(795\) 0 0
\(796\) 13440.0 0.598452
\(797\) − 21736.0i − 0.966033i −0.875611 0.483017i \(-0.839541\pi\)
0.875611 0.483017i \(-0.160459\pi\)
\(798\) − 8400.00i − 0.372628i
\(799\) −27216.0 −1.20505
\(800\) 0 0
\(801\) 1800.00 0.0794006
\(802\) 54648.0i 2.40609i
\(803\) 39804.0i 1.74926i
\(804\) 21696.0 0.951690
\(805\) 0 0
\(806\) 11904.0 0.520224
\(807\) 20340.0i 0.887239i
\(808\) 0 0
\(809\) 38310.0 1.66490 0.832452 0.554097i \(-0.186936\pi\)
0.832452 + 0.554097i \(0.186936\pi\)
\(810\) 0 0
\(811\) 2132.00 0.0923115 0.0461558 0.998934i \(-0.485303\pi\)
0.0461558 + 0.998934i \(0.485303\pi\)
\(812\) 560.000i 0.0242022i
\(813\) − 5784.00i − 0.249513i
\(814\) 61008.0 2.62694
\(815\) 0 0
\(816\) 16128.0 0.691903
\(817\) 6800.00i 0.291190i
\(818\) 52840.0i 2.25857i
\(819\) −3906.00 −0.166650
\(820\) 0 0
\(821\) 5002.00 0.212632 0.106316 0.994332i \(-0.466094\pi\)
0.106316 + 0.994332i \(0.466094\pi\)
\(822\) − 4968.00i − 0.210802i
\(823\) 3612.00i 0.152985i 0.997070 + 0.0764923i \(0.0243721\pi\)
−0.997070 + 0.0764923i \(0.975628\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −3360.00 −0.141537
\(827\) − 27666.0i − 1.16329i −0.813443 0.581645i \(-0.802409\pi\)
0.813443 0.581645i \(-0.197591\pi\)
\(828\) 3024.00i 0.126922i
\(829\) −12890.0 −0.540034 −0.270017 0.962856i \(-0.587029\pi\)
−0.270017 + 0.962856i \(0.587029\pi\)
\(830\) 0 0
\(831\) −16662.0 −0.695546
\(832\) 31744.0i 1.32275i
\(833\) − 4116.00i − 0.171202i
\(834\) −19440.0 −0.807137
\(835\) 0 0
\(836\) 49600.0 2.05198
\(837\) 1296.00i 0.0535201i
\(838\) − 27840.0i − 1.14763i
\(839\) 9340.00 0.384330 0.192165 0.981363i \(-0.438449\pi\)
0.192165 + 0.981363i \(0.438449\pi\)
\(840\) 0 0
\(841\) −24289.0 −0.995900
\(842\) 32648.0i 1.33625i
\(843\) 5826.00i 0.238029i
\(844\) 5344.00 0.217948
\(845\) 0 0
\(846\) 11664.0 0.474015
\(847\) − 17591.0i − 0.713617i
\(848\) 16512.0i 0.668661i
\(849\) 14484.0 0.585500
\(850\) 0 0
\(851\) 10332.0 0.416188
\(852\) 16272.0i 0.654307i
\(853\) 33082.0i 1.32791i 0.747773 + 0.663954i \(0.231123\pi\)
−0.747773 + 0.663954i \(0.768877\pi\)
\(854\) 17416.0 0.697849
\(855\) 0 0
\(856\) 0 0
\(857\) 7544.00i 0.300698i 0.988633 + 0.150349i \(0.0480397\pi\)
−0.988633 + 0.150349i \(0.951960\pi\)
\(858\) − 46128.0i − 1.83541i
\(859\) −8180.00 −0.324910 −0.162455 0.986716i \(-0.551941\pi\)
−0.162455 + 0.986716i \(0.551941\pi\)
\(860\) 0 0
\(861\) −5208.00 −0.206142
\(862\) 66408.0i 2.62397i
\(863\) − 10518.0i − 0.414875i −0.978248 0.207437i \(-0.933488\pi\)
0.978248 0.207437i \(-0.0665123\pi\)
\(864\) −6912.00 −0.272166
\(865\) 0 0
\(866\) 30952.0 1.21454
\(867\) − 6429.00i − 0.251834i
\(868\) − 2688.00i − 0.105111i
\(869\) −45880.0 −1.79099
\(870\) 0 0
\(871\) −56048.0 −2.18038
\(872\) 0 0
\(873\) 11394.0i 0.441728i
\(874\) 16800.0 0.650193
\(875\) 0 0
\(876\) 15408.0 0.594279
\(877\) 14134.0i 0.544209i 0.962268 + 0.272104i \(0.0877196\pi\)
−0.962268 + 0.272104i \(0.912280\pi\)
\(878\) 3360.00i 0.129151i
\(879\) −18456.0 −0.708197
\(880\) 0 0
\(881\) 6492.00 0.248265 0.124132 0.992266i \(-0.460385\pi\)
0.124132 + 0.992266i \(0.460385\pi\)
\(882\) 1764.00i 0.0673435i
\(883\) − 38228.0i − 1.45694i −0.685080 0.728468i \(-0.740233\pi\)
0.685080 0.728468i \(-0.259767\pi\)
\(884\) 41664.0 1.58519
\(885\) 0 0
\(886\) 26472.0 1.00377
\(887\) − 43076.0i − 1.63061i −0.579032 0.815305i \(-0.696569\pi\)
0.579032 0.815305i \(-0.303431\pi\)
\(888\) 0 0
\(889\) 5628.00 0.212325
\(890\) 0 0
\(891\) 5022.00 0.188825
\(892\) − 14656.0i − 0.550134i
\(893\) − 32400.0i − 1.21414i
\(894\) 28440.0 1.06396
\(895\) 0 0
\(896\) 0 0
\(897\) − 7812.00i − 0.290786i
\(898\) − 12360.0i − 0.459308i
\(899\) −480.000 −0.0178074
\(900\) 0 0
\(901\) 21672.0 0.801331
\(902\) − 61504.0i − 2.27035i
\(903\) − 1428.00i − 0.0526255i
\(904\) 0 0
\(905\) 0 0
\(906\) 6816.00 0.249941
\(907\) − 32236.0i − 1.18013i −0.807355 0.590065i \(-0.799102\pi\)
0.807355 0.590065i \(-0.200898\pi\)
\(908\) − 39552.0i − 1.44557i
\(909\) −2088.00 −0.0761877
\(910\) 0 0
\(911\) −46518.0 −1.69178 −0.845889 0.533359i \(-0.820930\pi\)
−0.845889 + 0.533359i \(0.820930\pi\)
\(912\) 19200.0i 0.697122i
\(913\) − 29016.0i − 1.05180i
\(914\) −23656.0 −0.856095
\(915\) 0 0
\(916\) −43760.0 −1.57846
\(917\) − 5684.00i − 0.204692i
\(918\) 9072.00i 0.326166i
\(919\) −17840.0 −0.640356 −0.320178 0.947357i \(-0.603743\pi\)
−0.320178 + 0.947357i \(0.603743\pi\)
\(920\) 0 0
\(921\) −17652.0 −0.631545
\(922\) − 63872.0i − 2.28147i
\(923\) − 42036.0i − 1.49906i
\(924\) −10416.0 −0.370846
\(925\) 0 0
\(926\) −4688.00 −0.166369
\(927\) − 16128.0i − 0.571427i
\(928\) − 2560.00i − 0.0905562i
\(929\) −7000.00 −0.247215 −0.123607 0.992331i \(-0.539446\pi\)
−0.123607 + 0.992331i \(0.539446\pi\)
\(930\) 0 0
\(931\) 4900.00 0.172493
\(932\) − 22416.0i − 0.787833i
\(933\) 27396.0i 0.961313i
\(934\) −21216.0 −0.743264
\(935\) 0 0
\(936\) 0 0
\(937\) 36114.0i 1.25912i 0.776953 + 0.629559i \(0.216764\pi\)
−0.776953 + 0.629559i \(0.783236\pi\)
\(938\) 25312.0i 0.881094i
\(939\) −28146.0 −0.978179
\(940\) 0 0
\(941\) −4748.00 −0.164485 −0.0822425 0.996612i \(-0.526208\pi\)
−0.0822425 + 0.996612i \(0.526208\pi\)
\(942\) 3192.00i 0.110404i
\(943\) − 10416.0i − 0.359694i
\(944\) 7680.00 0.264791
\(945\) 0 0
\(946\) 16864.0 0.579594
\(947\) 42694.0i 1.46501i 0.680759 + 0.732507i \(0.261650\pi\)
−0.680759 + 0.732507i \(0.738350\pi\)
\(948\) 17760.0i 0.608458i
\(949\) −39804.0 −1.36153
\(950\) 0 0
\(951\) −9342.00 −0.318544
\(952\) 0 0
\(953\) 16742.0i 0.569073i 0.958665 + 0.284537i \(0.0918397\pi\)
−0.958665 + 0.284537i \(0.908160\pi\)
\(954\) −9288.00 −0.315210
\(955\) 0 0
\(956\) −9360.00 −0.316657
\(957\) 1860.00i 0.0628268i
\(958\) − 22960.0i − 0.774326i
\(959\) 2898.00 0.0975822
\(960\) 0 0
\(961\) −27487.0 −0.922661
\(962\) 61008.0i 2.04467i
\(963\) 17154.0i 0.574019i
\(964\) 18704.0 0.624912
\(965\) 0 0
\(966\) −3528.00 −0.117507
\(967\) − 9956.00i − 0.331089i −0.986202 0.165545i \(-0.947062\pi\)
0.986202 0.165545i \(-0.0529382\pi\)
\(968\) 0 0
\(969\) 25200.0 0.835439
\(970\) 0 0
\(971\) −26388.0 −0.872123 −0.436061 0.899917i \(-0.643627\pi\)
−0.436061 + 0.899917i \(0.643627\pi\)
\(972\) − 1944.00i − 0.0641500i
\(973\) − 11340.0i − 0.373632i
\(974\) −35776.0 −1.17694
\(975\) 0 0
\(976\) −39808.0 −1.30556
\(977\) − 786.000i − 0.0257383i −0.999917 0.0128692i \(-0.995904\pi\)
0.999917 0.0128692i \(-0.00409650\pi\)
\(978\) − 3264.00i − 0.106719i
\(979\) −12400.0 −0.404807
\(980\) 0 0
\(981\) −810.000 −0.0263622
\(982\) − 22232.0i − 0.722456i
\(983\) − 51888.0i − 1.68359i −0.539796 0.841796i \(-0.681499\pi\)
0.539796 0.841796i \(-0.318501\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −3360.00 −0.108524
\(987\) 6804.00i 0.219426i
\(988\) 49600.0i 1.59715i
\(989\) 2856.00 0.0918256
\(990\) 0 0
\(991\) −51928.0 −1.66453 −0.832264 0.554379i \(-0.812956\pi\)
−0.832264 + 0.554379i \(0.812956\pi\)
\(992\) 12288.0i 0.393291i
\(993\) 4596.00i 0.146878i
\(994\) −18984.0 −0.605771
\(995\) 0 0
\(996\) −11232.0 −0.357329
\(997\) − 386.000i − 0.0122615i −0.999981 0.00613076i \(-0.998049\pi\)
0.999981 0.00613076i \(-0.00195149\pi\)
\(998\) 79280.0i 2.51459i
\(999\) −6642.00 −0.210354
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 525.4.d.b.274.2 2
5.2 odd 4 525.4.a.b.1.1 1
5.3 odd 4 21.4.a.b.1.1 1
5.4 even 2 inner 525.4.d.b.274.1 2
15.2 even 4 1575.4.a.k.1.1 1
15.8 even 4 63.4.a.a.1.1 1
20.3 even 4 336.4.a.h.1.1 1
35.3 even 12 147.4.e.b.79.1 2
35.13 even 4 147.4.a.g.1.1 1
35.18 odd 12 147.4.e.c.79.1 2
35.23 odd 12 147.4.e.c.67.1 2
35.33 even 12 147.4.e.b.67.1 2
40.3 even 4 1344.4.a.i.1.1 1
40.13 odd 4 1344.4.a.w.1.1 1
60.23 odd 4 1008.4.a.m.1.1 1
105.23 even 12 441.4.e.m.361.1 2
105.38 odd 12 441.4.e.n.226.1 2
105.53 even 12 441.4.e.m.226.1 2
105.68 odd 12 441.4.e.n.361.1 2
105.83 odd 4 441.4.a.b.1.1 1
140.83 odd 4 2352.4.a.l.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.4.a.b.1.1 1 5.3 odd 4
63.4.a.a.1.1 1 15.8 even 4
147.4.a.g.1.1 1 35.13 even 4
147.4.e.b.67.1 2 35.33 even 12
147.4.e.b.79.1 2 35.3 even 12
147.4.e.c.67.1 2 35.23 odd 12
147.4.e.c.79.1 2 35.18 odd 12
336.4.a.h.1.1 1 20.3 even 4
441.4.a.b.1.1 1 105.83 odd 4
441.4.e.m.226.1 2 105.53 even 12
441.4.e.m.361.1 2 105.23 even 12
441.4.e.n.226.1 2 105.38 odd 12
441.4.e.n.361.1 2 105.68 odd 12
525.4.a.b.1.1 1 5.2 odd 4
525.4.d.b.274.1 2 5.4 even 2 inner
525.4.d.b.274.2 2 1.1 even 1 trivial
1008.4.a.m.1.1 1 60.23 odd 4
1344.4.a.i.1.1 1 40.3 even 4
1344.4.a.w.1.1 1 40.13 odd 4
1575.4.a.k.1.1 1 15.2 even 4
2352.4.a.l.1.1 1 140.83 odd 4