Properties

Label 825.4.c.k.199.2
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.c (of order \(2\), degree \(1\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.36142572544.1
Defining polynomial: \(x^{6} + 53 x^{4} + 632 x^{2} + 484\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.2
Root \(-5.97123i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.k.199.5

$q$-expansion

\(f(q)\) \(=\) \(q-4.97123i q^{2} +3.00000i q^{3} -16.7131 q^{4} +14.9137 q^{6} -5.48376i q^{7} +43.3148i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-4.97123i q^{2} +3.00000i q^{3} -16.7131 q^{4} +14.9137 q^{6} -5.48376i q^{7} +43.3148i q^{8} -9.00000 q^{9} +11.0000 q^{11} -50.1393i q^{12} +24.5738i q^{13} -27.2610 q^{14} +81.6231 q^{16} +59.3687i q^{17} +44.7411i q^{18} -5.89234 q^{19} +16.4513 q^{21} -54.6835i q^{22} -68.4570i q^{23} -129.945 q^{24} +122.162 q^{26} -27.0000i q^{27} +91.6506i q^{28} +265.022 q^{29} -196.554 q^{31} -59.2482i q^{32} +33.0000i q^{33} +295.135 q^{34} +150.418 q^{36} -166.575i q^{37} +29.2922i q^{38} -73.7214 q^{39} +424.101 q^{41} -81.7830i q^{42} -177.351i q^{43} -183.844 q^{44} -340.316 q^{46} -141.148i q^{47} +244.869i q^{48} +312.928 q^{49} -178.106 q^{51} -410.704i q^{52} -339.828i q^{53} -134.223 q^{54} +237.528 q^{56} -17.6770i q^{57} -1317.48i q^{58} -416.784 q^{59} +662.146 q^{61} +977.115i q^{62} +49.3538i q^{63} +358.448 q^{64} +164.051 q^{66} -313.713i q^{67} -992.235i q^{68} +205.371 q^{69} -153.693 q^{71} -389.834i q^{72} +153.866i q^{73} -828.084 q^{74} +98.4793 q^{76} -60.3213i q^{77} +366.486i q^{78} -403.000 q^{79} +81.0000 q^{81} -2108.30i q^{82} -652.313i q^{83} -274.952 q^{84} -881.650 q^{86} +795.065i q^{87} +476.463i q^{88} -1226.77 q^{89} +134.757 q^{91} +1144.13i q^{92} -589.662i q^{93} -701.677 q^{94} +177.745 q^{96} -959.746i q^{97} -1555.64i q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q - 60q^{4} - 12q^{6} - 54q^{9} + O(q^{10}) \) \( 6q - 60q^{4} - 12q^{6} - 54q^{9} + 66q^{11} + 136q^{14} + 356q^{16} + 116q^{19} + 60q^{21} - 108q^{24} - 240q^{26} + 440q^{29} + 496q^{31} + 160q^{34} + 540q^{36} - 684q^{39} + 312q^{41} - 660q^{44} - 2512q^{46} - 558q^{49} - 624q^{51} + 108q^{54} - 3288q^{56} - 1096q^{59} + 828q^{61} + 116q^{64} - 132q^{66} - 720q^{69} - 1824q^{71} + 3224q^{74} - 5504q^{76} + 1084q^{79} + 486q^{81} - 4056q^{84} - 3096q^{86} - 1580q^{89} - 1544q^{91} + 848q^{94} - 1548q^{96} - 594q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/825\mathbb{Z}\right)^\times\).

\(n\) \(376\) \(551\) \(727\)
\(\chi(n)\) \(1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 4.97123i − 1.75759i −0.477195 0.878797i \(-0.658347\pi\)
0.477195 0.878797i \(-0.341653\pi\)
\(3\) 3.00000i 0.577350i
\(4\) −16.7131 −2.08914
\(5\) 0 0
\(6\) 14.9137 1.01475
\(7\) − 5.48376i − 0.296095i −0.988980 0.148048i \(-0.952701\pi\)
0.988980 0.148048i \(-0.0472989\pi\)
\(8\) 43.3148i 1.91426i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) − 50.1393i − 1.20616i
\(13\) 24.5738i 0.524272i 0.965031 + 0.262136i \(0.0844270\pi\)
−0.965031 + 0.262136i \(0.915573\pi\)
\(14\) −27.2610 −0.520415
\(15\) 0 0
\(16\) 81.6231 1.27536
\(17\) 59.3687i 0.847001i 0.905896 + 0.423501i \(0.139199\pi\)
−0.905896 + 0.423501i \(0.860801\pi\)
\(18\) 44.7411i 0.585865i
\(19\) −5.89234 −0.0711471 −0.0355736 0.999367i \(-0.511326\pi\)
−0.0355736 + 0.999367i \(0.511326\pi\)
\(20\) 0 0
\(21\) 16.4513 0.170951
\(22\) − 54.6835i − 0.529935i
\(23\) − 68.4570i − 0.620621i −0.950635 0.310310i \(-0.899567\pi\)
0.950635 0.310310i \(-0.100433\pi\)
\(24\) −129.945 −1.10520
\(25\) 0 0
\(26\) 122.162 0.921458
\(27\) − 27.0000i − 0.192450i
\(28\) 91.6506i 0.618584i
\(29\) 265.022 1.69701 0.848505 0.529187i \(-0.177503\pi\)
0.848505 + 0.529187i \(0.177503\pi\)
\(30\) 0 0
\(31\) −196.554 −1.13878 −0.569389 0.822068i \(-0.692820\pi\)
−0.569389 + 0.822068i \(0.692820\pi\)
\(32\) − 59.2482i − 0.327303i
\(33\) 33.0000i 0.174078i
\(34\) 295.135 1.48868
\(35\) 0 0
\(36\) 150.418 0.696379
\(37\) − 166.575i − 0.740131i −0.929006 0.370065i \(-0.879335\pi\)
0.929006 0.370065i \(-0.120665\pi\)
\(38\) 29.2922i 0.125048i
\(39\) −73.7214 −0.302689
\(40\) 0 0
\(41\) 424.101 1.61545 0.807725 0.589559i \(-0.200699\pi\)
0.807725 + 0.589559i \(0.200699\pi\)
\(42\) − 81.7830i − 0.300462i
\(43\) − 177.351i − 0.628970i −0.949262 0.314485i \(-0.898168\pi\)
0.949262 0.314485i \(-0.101832\pi\)
\(44\) −183.844 −0.629899
\(45\) 0 0
\(46\) −340.316 −1.09080
\(47\) − 141.148i − 0.438053i −0.975719 0.219026i \(-0.929712\pi\)
0.975719 0.219026i \(-0.0702881\pi\)
\(48\) 244.869i 0.736330i
\(49\) 312.928 0.912328
\(50\) 0 0
\(51\) −178.106 −0.489016
\(52\) − 410.704i − 1.09528i
\(53\) − 339.828i − 0.880736i −0.897817 0.440368i \(-0.854848\pi\)
0.897817 0.440368i \(-0.145152\pi\)
\(54\) −134.223 −0.338249
\(55\) 0 0
\(56\) 237.528 0.566804
\(57\) − 17.6770i − 0.0410768i
\(58\) − 1317.48i − 2.98266i
\(59\) −416.784 −0.919672 −0.459836 0.888004i \(-0.652092\pi\)
−0.459836 + 0.888004i \(0.652092\pi\)
\(60\) 0 0
\(61\) 662.146 1.38982 0.694911 0.719096i \(-0.255444\pi\)
0.694911 + 0.719096i \(0.255444\pi\)
\(62\) 977.115i 2.00151i
\(63\) 49.3538i 0.0986984i
\(64\) 358.448 0.700094
\(65\) 0 0
\(66\) 164.051 0.305958
\(67\) − 313.713i − 0.572032i −0.958225 0.286016i \(-0.907669\pi\)
0.958225 0.286016i \(-0.0923311\pi\)
\(68\) − 992.235i − 1.76950i
\(69\) 205.371 0.358316
\(70\) 0 0
\(71\) −153.693 −0.256902 −0.128451 0.991716i \(-0.541000\pi\)
−0.128451 + 0.991716i \(0.541000\pi\)
\(72\) − 389.834i − 0.638088i
\(73\) 153.866i 0.246694i 0.992364 + 0.123347i \(0.0393628\pi\)
−0.992364 + 0.123347i \(0.960637\pi\)
\(74\) −828.084 −1.30085
\(75\) 0 0
\(76\) 98.4793 0.148636
\(77\) − 60.3213i − 0.0892760i
\(78\) 366.486i 0.532004i
\(79\) −403.000 −0.573938 −0.286969 0.957940i \(-0.592648\pi\)
−0.286969 + 0.957940i \(0.592648\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) − 2108.30i − 2.83931i
\(83\) − 652.313i − 0.862659i −0.902195 0.431329i \(-0.858045\pi\)
0.902195 0.431329i \(-0.141955\pi\)
\(84\) −274.952 −0.357139
\(85\) 0 0
\(86\) −881.650 −1.10547
\(87\) 795.065i 0.979769i
\(88\) 476.463i 0.577172i
\(89\) −1226.77 −1.46109 −0.730545 0.682864i \(-0.760734\pi\)
−0.730545 + 0.682864i \(0.760734\pi\)
\(90\) 0 0
\(91\) 134.757 0.155234
\(92\) 1144.13i 1.29656i
\(93\) − 589.662i − 0.657474i
\(94\) −701.677 −0.769919
\(95\) 0 0
\(96\) 177.745 0.188969
\(97\) − 959.746i − 1.00461i −0.864690 0.502306i \(-0.832485\pi\)
0.864690 0.502306i \(-0.167515\pi\)
\(98\) − 1555.64i − 1.60350i
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) −1258.52 −1.23988 −0.619939 0.784650i \(-0.712843\pi\)
−0.619939 + 0.784650i \(0.712843\pi\)
\(102\) 885.406i 0.859492i
\(103\) − 493.175i − 0.471786i −0.971779 0.235893i \(-0.924198\pi\)
0.971779 0.235893i \(-0.0758015\pi\)
\(104\) −1064.41 −1.00360
\(105\) 0 0
\(106\) −1689.36 −1.54798
\(107\) 1099.05i 0.992983i 0.868042 + 0.496492i \(0.165379\pi\)
−0.868042 + 0.496492i \(0.834621\pi\)
\(108\) 451.254i 0.402055i
\(109\) −1275.04 −1.12043 −0.560216 0.828347i \(-0.689282\pi\)
−0.560216 + 0.828347i \(0.689282\pi\)
\(110\) 0 0
\(111\) 499.726 0.427315
\(112\) − 447.601i − 0.377628i
\(113\) − 946.433i − 0.787902i −0.919131 0.393951i \(-0.871108\pi\)
0.919131 0.393951i \(-0.128892\pi\)
\(114\) −87.8765 −0.0721964
\(115\) 0 0
\(116\) −4429.34 −3.54529
\(117\) − 221.164i − 0.174757i
\(118\) 2071.93i 1.61641i
\(119\) 325.563 0.250793
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) − 3291.68i − 2.44274i
\(123\) 1272.30i 0.932681i
\(124\) 3285.03 2.37907
\(125\) 0 0
\(126\) 245.349 0.173472
\(127\) − 2260.77i − 1.57962i −0.613354 0.789808i \(-0.710180\pi\)
0.613354 0.789808i \(-0.289820\pi\)
\(128\) − 2255.91i − 1.55778i
\(129\) 532.052 0.363136
\(130\) 0 0
\(131\) −57.1277 −0.0381013 −0.0190507 0.999819i \(-0.506064\pi\)
−0.0190507 + 0.999819i \(0.506064\pi\)
\(132\) − 551.533i − 0.363672i
\(133\) 32.3122i 0.0210663i
\(134\) −1559.54 −1.00540
\(135\) 0 0
\(136\) −2571.54 −1.62138
\(137\) − 1919.27i − 1.19689i −0.801162 0.598447i \(-0.795785\pi\)
0.801162 0.598447i \(-0.204215\pi\)
\(138\) − 1020.95i − 0.629774i
\(139\) 1690.46 1.03153 0.515766 0.856729i \(-0.327507\pi\)
0.515766 + 0.856729i \(0.327507\pi\)
\(140\) 0 0
\(141\) 423.443 0.252910
\(142\) 764.044i 0.451529i
\(143\) 270.312i 0.158074i
\(144\) −734.608 −0.425120
\(145\) 0 0
\(146\) 764.904 0.433588
\(147\) 938.785i 0.526733i
\(148\) 2783.99i 1.54624i
\(149\) 2077.88 1.14246 0.571231 0.820789i \(-0.306466\pi\)
0.571231 + 0.820789i \(0.306466\pi\)
\(150\) 0 0
\(151\) 2711.16 1.46113 0.730566 0.682842i \(-0.239256\pi\)
0.730566 + 0.682842i \(0.239256\pi\)
\(152\) − 255.226i − 0.136194i
\(153\) − 534.318i − 0.282334i
\(154\) −299.871 −0.156911
\(155\) 0 0
\(156\) 1232.11 0.632359
\(157\) − 2341.23i − 1.19013i −0.803677 0.595066i \(-0.797126\pi\)
0.803677 0.595066i \(-0.202874\pi\)
\(158\) 2003.41i 1.00875i
\(159\) 1019.48 0.508493
\(160\) 0 0
\(161\) −375.402 −0.183763
\(162\) − 402.669i − 0.195288i
\(163\) 3523.01i 1.69291i 0.532464 + 0.846453i \(0.321266\pi\)
−0.532464 + 0.846453i \(0.678734\pi\)
\(164\) −7088.05 −3.37490
\(165\) 0 0
\(166\) −3242.80 −1.51620
\(167\) − 4011.90i − 1.85898i −0.368842 0.929492i \(-0.620246\pi\)
0.368842 0.929492i \(-0.379754\pi\)
\(168\) 712.584i 0.327244i
\(169\) 1593.13 0.725138
\(170\) 0 0
\(171\) 53.0310 0.0237157
\(172\) 2964.08i 1.31400i
\(173\) 3325.30i 1.46137i 0.682713 + 0.730687i \(0.260800\pi\)
−0.682713 + 0.730687i \(0.739200\pi\)
\(174\) 3952.45 1.72204
\(175\) 0 0
\(176\) 897.854 0.384536
\(177\) − 1250.35i − 0.530973i
\(178\) 6098.54i 2.56800i
\(179\) 26.6350 0.0111217 0.00556087 0.999985i \(-0.498230\pi\)
0.00556087 + 0.999985i \(0.498230\pi\)
\(180\) 0 0
\(181\) 1934.25 0.794320 0.397160 0.917749i \(-0.369996\pi\)
0.397160 + 0.917749i \(0.369996\pi\)
\(182\) − 669.906i − 0.272839i
\(183\) 1986.44i 0.802414i
\(184\) 2965.21 1.18803
\(185\) 0 0
\(186\) −2931.34 −1.15557
\(187\) 653.055i 0.255380i
\(188\) 2359.01i 0.915153i
\(189\) −148.061 −0.0569835
\(190\) 0 0
\(191\) −3341.59 −1.26591 −0.632956 0.774188i \(-0.718158\pi\)
−0.632956 + 0.774188i \(0.718158\pi\)
\(192\) 1075.34i 0.404200i
\(193\) − 1293.97i − 0.482600i −0.970451 0.241300i \(-0.922426\pi\)
0.970451 0.241300i \(-0.0775737\pi\)
\(194\) −4771.12 −1.76570
\(195\) 0 0
\(196\) −5230.01 −1.90598
\(197\) 2301.17i 0.832240i 0.909310 + 0.416120i \(0.136610\pi\)
−0.909310 + 0.416120i \(0.863390\pi\)
\(198\) 492.152i 0.176645i
\(199\) 3039.64 1.08279 0.541393 0.840770i \(-0.317897\pi\)
0.541393 + 0.840770i \(0.317897\pi\)
\(200\) 0 0
\(201\) 941.139 0.330263
\(202\) 6256.40i 2.17920i
\(203\) − 1453.31i − 0.502476i
\(204\) 2976.70 1.02162
\(205\) 0 0
\(206\) −2451.69 −0.829209
\(207\) 616.113i 0.206874i
\(208\) 2005.79i 0.668636i
\(209\) −64.8157 −0.0214517
\(210\) 0 0
\(211\) 2807.86 0.916119 0.458060 0.888921i \(-0.348545\pi\)
0.458060 + 0.888921i \(0.348545\pi\)
\(212\) 5679.58i 1.83998i
\(213\) − 461.080i − 0.148322i
\(214\) 5463.63 1.74526
\(215\) 0 0
\(216\) 1169.50 0.368400
\(217\) 1077.85i 0.337187i
\(218\) 6338.53i 1.96926i
\(219\) −461.598 −0.142429
\(220\) 0 0
\(221\) −1458.91 −0.444059
\(222\) − 2484.25i − 0.751046i
\(223\) 1443.09i 0.433348i 0.976244 + 0.216674i \(0.0695209\pi\)
−0.976244 + 0.216674i \(0.930479\pi\)
\(224\) −324.903 −0.0969129
\(225\) 0 0
\(226\) −4704.94 −1.38481
\(227\) − 3658.71i − 1.06977i −0.844926 0.534884i \(-0.820355\pi\)
0.844926 0.534884i \(-0.179645\pi\)
\(228\) 295.438i 0.0858151i
\(229\) 1695.50 0.489265 0.244632 0.969616i \(-0.421333\pi\)
0.244632 + 0.969616i \(0.421333\pi\)
\(230\) 0 0
\(231\) 180.964 0.0515435
\(232\) 11479.4i 3.24853i
\(233\) − 5782.24i − 1.62578i −0.582417 0.812890i \(-0.697893\pi\)
0.582417 0.812890i \(-0.302107\pi\)
\(234\) −1099.46 −0.307153
\(235\) 0 0
\(236\) 6965.76 1.92132
\(237\) − 1209.00i − 0.331363i
\(238\) − 1618.45i − 0.440792i
\(239\) 5836.16 1.57954 0.789770 0.613404i \(-0.210200\pi\)
0.789770 + 0.613404i \(0.210200\pi\)
\(240\) 0 0
\(241\) −982.308 −0.262556 −0.131278 0.991346i \(-0.541908\pi\)
−0.131278 + 0.991346i \(0.541908\pi\)
\(242\) − 601.519i − 0.159781i
\(243\) 243.000i 0.0641500i
\(244\) −11066.5 −2.90353
\(245\) 0 0
\(246\) 6324.91 1.63927
\(247\) − 144.797i − 0.0373005i
\(248\) − 8513.70i − 2.17992i
\(249\) 1956.94 0.498056
\(250\) 0 0
\(251\) 4320.23 1.08642 0.543209 0.839598i \(-0.317209\pi\)
0.543209 + 0.839598i \(0.317209\pi\)
\(252\) − 824.856i − 0.206195i
\(253\) − 753.027i − 0.187124i
\(254\) −11238.8 −2.77633
\(255\) 0 0
\(256\) −8347.07 −2.03786
\(257\) 6224.29i 1.51074i 0.655298 + 0.755371i \(0.272543\pi\)
−0.655298 + 0.755371i \(0.727457\pi\)
\(258\) − 2644.95i − 0.638246i
\(259\) −913.459 −0.219149
\(260\) 0 0
\(261\) −2385.20 −0.565670
\(262\) 283.995i 0.0669666i
\(263\) − 2588.95i − 0.607002i −0.952831 0.303501i \(-0.901844\pi\)
0.952831 0.303501i \(-0.0981555\pi\)
\(264\) −1429.39 −0.333230
\(265\) 0 0
\(266\) 160.631 0.0370260
\(267\) − 3680.30i − 0.843561i
\(268\) 5243.12i 1.19505i
\(269\) 5871.53 1.33083 0.665415 0.746473i \(-0.268254\pi\)
0.665415 + 0.746473i \(0.268254\pi\)
\(270\) 0 0
\(271\) −3306.93 −0.741261 −0.370630 0.928780i \(-0.620858\pi\)
−0.370630 + 0.928780i \(0.620858\pi\)
\(272\) 4845.85i 1.08023i
\(273\) 404.270i 0.0896247i
\(274\) −9541.14 −2.10366
\(275\) 0 0
\(276\) −3432.39 −0.748571
\(277\) 2861.37i 0.620660i 0.950629 + 0.310330i \(0.100440\pi\)
−0.950629 + 0.310330i \(0.899560\pi\)
\(278\) − 8403.66i − 1.81302i
\(279\) 1768.99 0.379593
\(280\) 0 0
\(281\) −2988.05 −0.634349 −0.317174 0.948367i \(-0.602734\pi\)
−0.317174 + 0.948367i \(0.602734\pi\)
\(282\) − 2105.03i − 0.444513i
\(283\) 1454.60i 0.305536i 0.988262 + 0.152768i \(0.0488188\pi\)
−0.988262 + 0.152768i \(0.951181\pi\)
\(284\) 2568.69 0.536704
\(285\) 0 0
\(286\) 1343.78 0.277830
\(287\) − 2325.67i − 0.478327i
\(288\) 533.234i 0.109101i
\(289\) 1388.36 0.282589
\(290\) 0 0
\(291\) 2879.24 0.580014
\(292\) − 2571.58i − 0.515378i
\(293\) − 8586.52i − 1.71205i −0.516937 0.856023i \(-0.672928\pi\)
0.516937 0.856023i \(-0.327072\pi\)
\(294\) 4666.92 0.925782
\(295\) 0 0
\(296\) 7215.19 1.41681
\(297\) − 297.000i − 0.0580259i
\(298\) − 10329.6i − 2.00798i
\(299\) 1682.25 0.325374
\(300\) 0 0
\(301\) −972.547 −0.186235
\(302\) − 13477.8i − 2.56808i
\(303\) − 3775.57i − 0.715844i
\(304\) −480.951 −0.0907382
\(305\) 0 0
\(306\) −2656.22 −0.496228
\(307\) − 529.029i − 0.0983495i −0.998790 0.0491748i \(-0.984341\pi\)
0.998790 0.0491748i \(-0.0156591\pi\)
\(308\) 1008.16i 0.186510i
\(309\) 1479.53 0.272386
\(310\) 0 0
\(311\) −5008.00 −0.913111 −0.456555 0.889695i \(-0.650917\pi\)
−0.456555 + 0.889695i \(0.650917\pi\)
\(312\) − 3193.23i − 0.579426i
\(313\) 7858.53i 1.41914i 0.704635 + 0.709570i \(0.251111\pi\)
−0.704635 + 0.709570i \(0.748889\pi\)
\(314\) −11638.8 −2.09177
\(315\) 0 0
\(316\) 6735.39 1.19904
\(317\) − 7747.04i − 1.37261i −0.727315 0.686304i \(-0.759232\pi\)
0.727315 0.686304i \(-0.240768\pi\)
\(318\) − 5068.09i − 0.893724i
\(319\) 2915.24 0.511668
\(320\) 0 0
\(321\) −3297.15 −0.573299
\(322\) 1866.21i 0.322980i
\(323\) − 349.820i − 0.0602617i
\(324\) −1353.76 −0.232126
\(325\) 0 0
\(326\) 17513.7 2.97544
\(327\) − 3825.13i − 0.646882i
\(328\) 18369.9i 3.09240i
\(329\) −774.019 −0.129705
\(330\) 0 0
\(331\) 1355.08 0.225022 0.112511 0.993650i \(-0.464111\pi\)
0.112511 + 0.993650i \(0.464111\pi\)
\(332\) 10902.2i 1.80221i
\(333\) 1499.18i 0.246710i
\(334\) −19944.1 −3.26734
\(335\) 0 0
\(336\) 1342.80 0.218024
\(337\) − 2596.97i − 0.419780i −0.977725 0.209890i \(-0.932689\pi\)
0.977725 0.209890i \(-0.0673106\pi\)
\(338\) − 7919.81i − 1.27450i
\(339\) 2839.30 0.454896
\(340\) 0 0
\(341\) −2162.09 −0.343355
\(342\) − 263.629i − 0.0416826i
\(343\) − 3596.95i − 0.566231i
\(344\) 7681.91 1.20401
\(345\) 0 0
\(346\) 16530.8 2.56850
\(347\) − 266.820i − 0.0412785i −0.999787 0.0206392i \(-0.993430\pi\)
0.999787 0.0206392i \(-0.00657014\pi\)
\(348\) − 13288.0i − 2.04687i
\(349\) 10549.4 1.61804 0.809021 0.587780i \(-0.199998\pi\)
0.809021 + 0.587780i \(0.199998\pi\)
\(350\) 0 0
\(351\) 663.492 0.100896
\(352\) − 651.730i − 0.0986856i
\(353\) 7152.63i 1.07846i 0.842159 + 0.539229i \(0.181284\pi\)
−0.842159 + 0.539229i \(0.818716\pi\)
\(354\) −6215.79 −0.933235
\(355\) 0 0
\(356\) 20503.1 3.05242
\(357\) 976.690i 0.144795i
\(358\) − 132.409i − 0.0195475i
\(359\) −359.182 −0.0528047 −0.0264024 0.999651i \(-0.508405\pi\)
−0.0264024 + 0.999651i \(0.508405\pi\)
\(360\) 0 0
\(361\) −6824.28 −0.994938
\(362\) − 9615.62i − 1.39609i
\(363\) 363.000i 0.0524864i
\(364\) −2252.20 −0.324306
\(365\) 0 0
\(366\) 9875.04 1.41032
\(367\) − 9042.47i − 1.28614i −0.765808 0.643070i \(-0.777661\pi\)
0.765808 0.643070i \(-0.222339\pi\)
\(368\) − 5587.67i − 0.791515i
\(369\) −3816.91 −0.538483
\(370\) 0 0
\(371\) −1863.54 −0.260781
\(372\) 9855.08i 1.37355i
\(373\) − 11929.0i − 1.65592i −0.560787 0.827960i \(-0.689501\pi\)
0.560787 0.827960i \(-0.310499\pi\)
\(374\) 3246.49 0.448855
\(375\) 0 0
\(376\) 6113.78 0.838549
\(377\) 6512.59i 0.889696i
\(378\) 736.047i 0.100154i
\(379\) −6556.08 −0.888557 −0.444279 0.895889i \(-0.646540\pi\)
−0.444279 + 0.895889i \(0.646540\pi\)
\(380\) 0 0
\(381\) 6782.32 0.911992
\(382\) 16611.8i 2.22496i
\(383\) 10703.1i 1.42795i 0.700172 + 0.713975i \(0.253107\pi\)
−0.700172 + 0.713975i \(0.746893\pi\)
\(384\) 6767.74 0.899388
\(385\) 0 0
\(386\) −6432.60 −0.848214
\(387\) 1596.15i 0.209657i
\(388\) 16040.3i 2.09878i
\(389\) 6450.30 0.840728 0.420364 0.907355i \(-0.361902\pi\)
0.420364 + 0.907355i \(0.361902\pi\)
\(390\) 0 0
\(391\) 4064.20 0.525667
\(392\) 13554.4i 1.74644i
\(393\) − 171.383i − 0.0219978i
\(394\) 11439.6 1.46274
\(395\) 0 0
\(396\) 1654.60 0.209966
\(397\) 12526.0i 1.58353i 0.610828 + 0.791764i \(0.290837\pi\)
−0.610828 + 0.791764i \(0.709163\pi\)
\(398\) − 15110.7i − 1.90310i
\(399\) −96.9365 −0.0121626
\(400\) 0 0
\(401\) 9119.52 1.13568 0.567839 0.823139i \(-0.307779\pi\)
0.567839 + 0.823139i \(0.307779\pi\)
\(402\) − 4678.62i − 0.580468i
\(403\) − 4830.08i − 0.597030i
\(404\) 21033.8 2.59028
\(405\) 0 0
\(406\) −7224.76 −0.883150
\(407\) − 1832.33i − 0.223158i
\(408\) − 7714.63i − 0.936106i
\(409\) 596.921 0.0721658 0.0360829 0.999349i \(-0.488512\pi\)
0.0360829 + 0.999349i \(0.488512\pi\)
\(410\) 0 0
\(411\) 5757.82 0.691027
\(412\) 8242.49i 0.985627i
\(413\) 2285.54i 0.272310i
\(414\) 3062.84 0.363600
\(415\) 0 0
\(416\) 1455.95 0.171596
\(417\) 5071.38i 0.595555i
\(418\) 322.214i 0.0377033i
\(419\) −11560.5 −1.34789 −0.673944 0.738782i \(-0.735401\pi\)
−0.673944 + 0.738782i \(0.735401\pi\)
\(420\) 0 0
\(421\) 15182.6 1.75761 0.878805 0.477182i \(-0.158342\pi\)
0.878805 + 0.477182i \(0.158342\pi\)
\(422\) − 13958.5i − 1.61017i
\(423\) 1270.33i 0.146018i
\(424\) 14719.6 1.68596
\(425\) 0 0
\(426\) −2292.13 −0.260691
\(427\) − 3631.05i − 0.411519i
\(428\) − 18368.5i − 2.07448i
\(429\) −810.935 −0.0912641
\(430\) 0 0
\(431\) −14902.7 −1.66551 −0.832757 0.553639i \(-0.813239\pi\)
−0.832757 + 0.553639i \(0.813239\pi\)
\(432\) − 2203.82i − 0.245443i
\(433\) − 2269.85i − 0.251922i −0.992035 0.125961i \(-0.959799\pi\)
0.992035 0.125961i \(-0.0402014\pi\)
\(434\) 5358.26 0.592637
\(435\) 0 0
\(436\) 21309.9 2.34074
\(437\) 403.372i 0.0441554i
\(438\) 2294.71i 0.250332i
\(439\) 13418.7 1.45886 0.729431 0.684054i \(-0.239785\pi\)
0.729431 + 0.684054i \(0.239785\pi\)
\(440\) 0 0
\(441\) −2816.36 −0.304109
\(442\) 7252.59i 0.780476i
\(443\) 11507.5i 1.23417i 0.786896 + 0.617086i \(0.211687\pi\)
−0.786896 + 0.617086i \(0.788313\pi\)
\(444\) −8351.98 −0.892719
\(445\) 0 0
\(446\) 7173.94 0.761650
\(447\) 6233.65i 0.659601i
\(448\) − 1965.64i − 0.207294i
\(449\) 16203.8 1.70313 0.851564 0.524250i \(-0.175654\pi\)
0.851564 + 0.524250i \(0.175654\pi\)
\(450\) 0 0
\(451\) 4665.11 0.487077
\(452\) 15817.8i 1.64604i
\(453\) 8133.48i 0.843585i
\(454\) −18188.3 −1.88022
\(455\) 0 0
\(456\) 765.677 0.0786318
\(457\) − 4912.79i − 0.502868i −0.967874 0.251434i \(-0.919098\pi\)
0.967874 0.251434i \(-0.0809022\pi\)
\(458\) − 8428.70i − 0.859929i
\(459\) 1602.95 0.163005
\(460\) 0 0
\(461\) −14270.8 −1.44178 −0.720888 0.693051i \(-0.756266\pi\)
−0.720888 + 0.693051i \(0.756266\pi\)
\(462\) − 899.613i − 0.0905926i
\(463\) 6656.46i 0.668146i 0.942547 + 0.334073i \(0.108423\pi\)
−0.942547 + 0.334073i \(0.891577\pi\)
\(464\) 21631.9 2.16430
\(465\) 0 0
\(466\) −28744.8 −2.85746
\(467\) − 19347.7i − 1.91714i −0.284853 0.958571i \(-0.591945\pi\)
0.284853 0.958571i \(-0.408055\pi\)
\(468\) 3696.34i 0.365093i
\(469\) −1720.33 −0.169376
\(470\) 0 0
\(471\) 7023.70 0.687123
\(472\) − 18052.9i − 1.76050i
\(473\) − 1950.86i − 0.189642i
\(474\) −6010.22 −0.582402
\(475\) 0 0
\(476\) −5441.18 −0.523941
\(477\) 3058.45i 0.293579i
\(478\) − 29012.9i − 2.77619i
\(479\) −9826.69 −0.937355 −0.468678 0.883369i \(-0.655269\pi\)
−0.468678 + 0.883369i \(0.655269\pi\)
\(480\) 0 0
\(481\) 4093.39 0.388030
\(482\) 4883.28i 0.461467i
\(483\) − 1126.21i − 0.106095i
\(484\) −2022.29 −0.189922
\(485\) 0 0
\(486\) 1208.01 0.112750
\(487\) − 10278.1i − 0.956353i −0.878264 0.478176i \(-0.841298\pi\)
0.878264 0.478176i \(-0.158702\pi\)
\(488\) 28680.7i 2.66048i
\(489\) −10569.0 −0.977400
\(490\) 0 0
\(491\) 11397.1 1.04755 0.523773 0.851858i \(-0.324524\pi\)
0.523773 + 0.851858i \(0.324524\pi\)
\(492\) − 21264.1i − 1.94850i
\(493\) 15734.0i 1.43737i
\(494\) −719.819 −0.0655591
\(495\) 0 0
\(496\) −16043.3 −1.45235
\(497\) 842.817i 0.0760674i
\(498\) − 9728.39i − 0.875381i
\(499\) 9179.17 0.823479 0.411740 0.911302i \(-0.364921\pi\)
0.411740 + 0.911302i \(0.364921\pi\)
\(500\) 0 0
\(501\) 12035.7 1.07329
\(502\) − 21476.9i − 1.90948i
\(503\) 6782.80i 0.601253i 0.953742 + 0.300626i \(0.0971957\pi\)
−0.953742 + 0.300626i \(0.902804\pi\)
\(504\) −2137.75 −0.188935
\(505\) 0 0
\(506\) −3743.47 −0.328889
\(507\) 4779.39i 0.418659i
\(508\) 37784.6i 3.30004i
\(509\) −6814.24 −0.593391 −0.296695 0.954972i \(-0.595885\pi\)
−0.296695 + 0.954972i \(0.595885\pi\)
\(510\) 0 0
\(511\) 843.765 0.0730449
\(512\) 23447.9i 2.02395i
\(513\) 159.093i 0.0136923i
\(514\) 30942.4 2.65527
\(515\) 0 0
\(516\) −8892.23 −0.758641
\(517\) − 1552.62i − 0.132078i
\(518\) 4541.01i 0.385175i
\(519\) −9975.89 −0.843724
\(520\) 0 0
\(521\) 206.712 0.0173824 0.00869118 0.999962i \(-0.497233\pi\)
0.00869118 + 0.999962i \(0.497233\pi\)
\(522\) 11857.4i 0.994219i
\(523\) − 11375.9i − 0.951118i −0.879684 0.475559i \(-0.842246\pi\)
0.879684 0.475559i \(-0.157754\pi\)
\(524\) 954.781 0.0795989
\(525\) 0 0
\(526\) −12870.3 −1.06686
\(527\) − 11669.1i − 0.964547i
\(528\) 2693.56i 0.222012i
\(529\) 7480.63 0.614830
\(530\) 0 0
\(531\) 3751.06 0.306557
\(532\) − 540.036i − 0.0440104i
\(533\) 10421.8i 0.846936i
\(534\) −18295.6 −1.48264
\(535\) 0 0
\(536\) 13588.4 1.09502
\(537\) 79.9049i 0.00642114i
\(538\) − 29188.7i − 2.33906i
\(539\) 3442.21 0.275077
\(540\) 0 0
\(541\) −10228.1 −0.812831 −0.406415 0.913688i \(-0.633221\pi\)
−0.406415 + 0.913688i \(0.633221\pi\)
\(542\) 16439.5i 1.30284i
\(543\) 5802.76i 0.458601i
\(544\) 3517.49 0.277226
\(545\) 0 0
\(546\) 2009.72 0.157524
\(547\) 15538.9i 1.21461i 0.794467 + 0.607307i \(0.207750\pi\)
−0.794467 + 0.607307i \(0.792250\pi\)
\(548\) 32077.0i 2.50048i
\(549\) −5959.31 −0.463274
\(550\) 0 0
\(551\) −1561.60 −0.120737
\(552\) 8895.62i 0.685911i
\(553\) 2209.96i 0.169940i
\(554\) 14224.5 1.09087
\(555\) 0 0
\(556\) −28252.8 −2.15501
\(557\) 5191.17i 0.394896i 0.980313 + 0.197448i \(0.0632653\pi\)
−0.980313 + 0.197448i \(0.936735\pi\)
\(558\) − 8794.03i − 0.667170i
\(559\) 4358.17 0.329752
\(560\) 0 0
\(561\) −1959.17 −0.147444
\(562\) 14854.3i 1.11493i
\(563\) 13722.9i 1.02727i 0.858009 + 0.513635i \(0.171701\pi\)
−0.858009 + 0.513635i \(0.828299\pi\)
\(564\) −7077.04 −0.528364
\(565\) 0 0
\(566\) 7231.13 0.537009
\(567\) − 444.184i − 0.0328995i
\(568\) − 6657.20i − 0.491778i
\(569\) −23209.0 −1.70997 −0.854986 0.518652i \(-0.826434\pi\)
−0.854986 + 0.518652i \(0.826434\pi\)
\(570\) 0 0
\(571\) −13130.5 −0.962337 −0.481168 0.876628i \(-0.659787\pi\)
−0.481168 + 0.876628i \(0.659787\pi\)
\(572\) − 4517.75i − 0.330239i
\(573\) − 10024.8i − 0.730874i
\(574\) −11561.4 −0.840705
\(575\) 0 0
\(576\) −3226.03 −0.233365
\(577\) 12261.0i 0.884633i 0.896859 + 0.442317i \(0.145843\pi\)
−0.896859 + 0.442317i \(0.854157\pi\)
\(578\) − 6901.86i − 0.496677i
\(579\) 3881.90 0.278629
\(580\) 0 0
\(581\) −3577.13 −0.255429
\(582\) − 14313.3i − 1.01943i
\(583\) − 3738.11i − 0.265552i
\(584\) −6664.69 −0.472238
\(585\) 0 0
\(586\) −42685.5 −3.00908
\(587\) 3553.42i 0.249856i 0.992166 + 0.124928i \(0.0398700\pi\)
−0.992166 + 0.124928i \(0.960130\pi\)
\(588\) − 15690.0i − 1.10042i
\(589\) 1158.16 0.0810208
\(590\) 0 0
\(591\) −6903.50 −0.480494
\(592\) − 13596.4i − 0.943933i
\(593\) 1942.83i 0.134540i 0.997735 + 0.0672701i \(0.0214289\pi\)
−0.997735 + 0.0672701i \(0.978571\pi\)
\(594\) −1476.45 −0.101986
\(595\) 0 0
\(596\) −34727.9 −2.38676
\(597\) 9118.92i 0.625147i
\(598\) − 8362.84i − 0.571876i
\(599\) −18585.4 −1.26774 −0.633871 0.773439i \(-0.718535\pi\)
−0.633871 + 0.773439i \(0.718535\pi\)
\(600\) 0 0
\(601\) 8010.10 0.543658 0.271829 0.962346i \(-0.412371\pi\)
0.271829 + 0.962346i \(0.412371\pi\)
\(602\) 4834.75i 0.327325i
\(603\) 2823.42i 0.190677i
\(604\) −45311.9 −3.05251
\(605\) 0 0
\(606\) −18769.2 −1.25816
\(607\) 1537.69i 0.102822i 0.998678 + 0.0514111i \(0.0163719\pi\)
−0.998678 + 0.0514111i \(0.983628\pi\)
\(608\) 349.110i 0.0232867i
\(609\) 4359.94 0.290105
\(610\) 0 0
\(611\) 3468.53 0.229659
\(612\) 8930.11i 0.589834i
\(613\) 2298.75i 0.151461i 0.997128 + 0.0757304i \(0.0241289\pi\)
−0.997128 + 0.0757304i \(0.975871\pi\)
\(614\) −2629.93 −0.172859
\(615\) 0 0
\(616\) 2612.81 0.170898
\(617\) − 6089.51i − 0.397333i −0.980067 0.198666i \(-0.936339\pi\)
0.980067 0.198666i \(-0.0636610\pi\)
\(618\) − 7355.06i − 0.478744i
\(619\) −12255.4 −0.795775 −0.397887 0.917434i \(-0.630257\pi\)
−0.397887 + 0.917434i \(0.630257\pi\)
\(620\) 0 0
\(621\) −1848.34 −0.119439
\(622\) 24895.9i 1.60488i
\(623\) 6727.29i 0.432622i
\(624\) −6017.36 −0.386037
\(625\) 0 0
\(626\) 39066.6 2.49427
\(627\) − 194.447i − 0.0123851i
\(628\) 39129.3i 2.48635i
\(629\) 9889.36 0.626891
\(630\) 0 0
\(631\) −25746.5 −1.62433 −0.812165 0.583428i \(-0.801711\pi\)
−0.812165 + 0.583428i \(0.801711\pi\)
\(632\) − 17455.9i − 1.09867i
\(633\) 8423.58i 0.528922i
\(634\) −38512.3 −2.41249
\(635\) 0 0
\(636\) −17038.8 −1.06231
\(637\) 7689.84i 0.478308i
\(638\) − 14492.3i − 0.899305i
\(639\) 1383.24 0.0856340
\(640\) 0 0
\(641\) −598.022 −0.0368494 −0.0184247 0.999830i \(-0.505865\pi\)
−0.0184247 + 0.999830i \(0.505865\pi\)
\(642\) 16390.9i 1.00763i
\(643\) 14610.9i 0.896109i 0.894006 + 0.448054i \(0.147883\pi\)
−0.894006 + 0.448054i \(0.852117\pi\)
\(644\) 6274.13 0.383906
\(645\) 0 0
\(646\) −1739.04 −0.105916
\(647\) 17252.3i 1.04831i 0.851622 + 0.524156i \(0.175619\pi\)
−0.851622 + 0.524156i \(0.824381\pi\)
\(648\) 3508.50i 0.212696i
\(649\) −4584.63 −0.277292
\(650\) 0 0
\(651\) −3233.56 −0.194675
\(652\) − 58880.5i − 3.53671i
\(653\) − 17768.6i − 1.06484i −0.846481 0.532418i \(-0.821283\pi\)
0.846481 0.532418i \(-0.178717\pi\)
\(654\) −19015.6 −1.13696
\(655\) 0 0
\(656\) 34616.4 2.06028
\(657\) − 1384.80i − 0.0822314i
\(658\) 3847.82i 0.227969i
\(659\) 17594.9 1.04006 0.520031 0.854147i \(-0.325920\pi\)
0.520031 + 0.854147i \(0.325920\pi\)
\(660\) 0 0
\(661\) 23355.9 1.37434 0.687170 0.726497i \(-0.258853\pi\)
0.687170 + 0.726497i \(0.258853\pi\)
\(662\) − 6736.44i − 0.395497i
\(663\) − 4376.74i − 0.256378i
\(664\) 28254.8 1.65136
\(665\) 0 0
\(666\) 7452.76 0.433617
\(667\) − 18142.6i − 1.05320i
\(668\) 67051.4i 3.88368i
\(669\) −4329.28 −0.250194
\(670\) 0 0
\(671\) 7283.61 0.419047
\(672\) − 974.708i − 0.0559527i
\(673\) − 2340.47i − 0.134054i −0.997751 0.0670271i \(-0.978649\pi\)
0.997751 0.0670271i \(-0.0213514\pi\)
\(674\) −12910.1 −0.737803
\(675\) 0 0
\(676\) −26626.1 −1.51491
\(677\) 16562.2i 0.940234i 0.882604 + 0.470117i \(0.155788\pi\)
−0.882604 + 0.470117i \(0.844212\pi\)
\(678\) − 14114.8i − 0.799522i
\(679\) −5263.01 −0.297461
\(680\) 0 0
\(681\) 10976.1 0.617630
\(682\) 10748.3i 0.603478i
\(683\) 22303.4i 1.24951i 0.780821 + 0.624755i \(0.214801\pi\)
−0.780821 + 0.624755i \(0.785199\pi\)
\(684\) −886.313 −0.0495454
\(685\) 0 0
\(686\) −17881.3 −0.995204
\(687\) 5086.49i 0.282477i
\(688\) − 14475.9i − 0.802163i
\(689\) 8350.87 0.461745
\(690\) 0 0
\(691\) 6244.91 0.343802 0.171901 0.985114i \(-0.445009\pi\)
0.171901 + 0.985114i \(0.445009\pi\)
\(692\) − 55576.0i − 3.05301i
\(693\) 542.892i 0.0297587i
\(694\) −1326.42 −0.0725508
\(695\) 0 0
\(696\) −34438.1 −1.87554
\(697\) 25178.3i 1.36829i
\(698\) − 52443.5i − 2.84386i
\(699\) 17346.7 0.938645
\(700\) 0 0
\(701\) −15277.7 −0.823154 −0.411577 0.911375i \(-0.635022\pi\)
−0.411577 + 0.911375i \(0.635022\pi\)
\(702\) − 3298.37i − 0.177335i
\(703\) 981.519i 0.0526582i
\(704\) 3942.93 0.211086
\(705\) 0 0
\(706\) 35557.3 1.89549
\(707\) 6901.43i 0.367122i
\(708\) 20897.3i 1.10928i
\(709\) 11732.1 0.621449 0.310725 0.950500i \(-0.399428\pi\)
0.310725 + 0.950500i \(0.399428\pi\)
\(710\) 0 0
\(711\) 3627.00 0.191313
\(712\) − 53137.2i − 2.79691i
\(713\) 13455.5i 0.706750i
\(714\) 4855.35 0.254491
\(715\) 0 0
\(716\) −445.153 −0.0232349
\(717\) 17508.5i 0.911947i
\(718\) 1785.57i 0.0928093i
\(719\) 19006.2 0.985828 0.492914 0.870078i \(-0.335932\pi\)
0.492914 + 0.870078i \(0.335932\pi\)
\(720\) 0 0
\(721\) −2704.45 −0.139694
\(722\) 33925.1i 1.74870i
\(723\) − 2946.92i − 0.151587i
\(724\) −32327.4 −1.65945
\(725\) 0 0
\(726\) 1804.56 0.0922498
\(727\) − 2650.77i − 0.135229i −0.997712 0.0676146i \(-0.978461\pi\)
0.997712 0.0676146i \(-0.0215388\pi\)
\(728\) 5836.96i 0.297160i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) 10529.1 0.532738
\(732\) − 33199.5i − 1.67635i
\(733\) − 8854.22i − 0.446164i −0.974800 0.223082i \(-0.928388\pi\)
0.974800 0.223082i \(-0.0716117\pi\)
\(734\) −44952.2 −2.26051
\(735\) 0 0
\(736\) −4055.96 −0.203131
\(737\) − 3450.84i − 0.172474i
\(738\) 18974.7i 0.946435i
\(739\) −17174.8 −0.854921 −0.427460 0.904034i \(-0.640592\pi\)
−0.427460 + 0.904034i \(0.640592\pi\)
\(740\) 0 0
\(741\) 434.391 0.0215354
\(742\) 9264.06i 0.458348i
\(743\) − 9800.30i − 0.483901i −0.970289 0.241950i \(-0.922213\pi\)
0.970289 0.241950i \(-0.0777871\pi\)
\(744\) 25541.1 1.25858
\(745\) 0 0
\(746\) −59301.6 −2.91044
\(747\) 5870.82i 0.287553i
\(748\) − 10914.6i − 0.533525i
\(749\) 6026.92 0.294017
\(750\) 0 0
\(751\) 2781.25 0.135139 0.0675695 0.997715i \(-0.478476\pi\)
0.0675695 + 0.997715i \(0.478476\pi\)
\(752\) − 11520.9i − 0.558675i
\(753\) 12960.7i 0.627243i
\(754\) 32375.6 1.56372
\(755\) 0 0
\(756\) 2474.57 0.119046
\(757\) − 3010.87i − 0.144560i −0.997384 0.0722800i \(-0.976972\pi\)
0.997384 0.0722800i \(-0.0230275\pi\)
\(758\) 32591.8i 1.56172i
\(759\) 2259.08 0.108036
\(760\) 0 0
\(761\) −2709.54 −0.129068 −0.0645340 0.997916i \(-0.520556\pi\)
−0.0645340 + 0.997916i \(0.520556\pi\)
\(762\) − 33716.5i − 1.60291i
\(763\) 6992.03i 0.331754i
\(764\) 55848.4 2.64466
\(765\) 0 0
\(766\) 53207.7 2.50976
\(767\) − 10242.0i − 0.482159i
\(768\) − 25041.2i − 1.17656i
\(769\) 28121.1 1.31869 0.659345 0.751841i \(-0.270834\pi\)
0.659345 + 0.751841i \(0.270834\pi\)
\(770\) 0 0
\(771\) −18672.9 −0.872227
\(772\) 21626.2i 1.00822i
\(773\) 4274.61i 0.198897i 0.995043 + 0.0994483i \(0.0317078\pi\)
−0.995043 + 0.0994483i \(0.968292\pi\)
\(774\) 7934.85 0.368491
\(775\) 0 0
\(776\) 41571.2 1.92309
\(777\) − 2740.38i − 0.126526i
\(778\) − 32065.9i − 1.47766i
\(779\) −2498.95 −0.114935
\(780\) 0 0
\(781\) −1690.63 −0.0774589
\(782\) − 20204.1i − 0.923909i
\(783\) − 7155.59i − 0.326590i
\(784\) 25542.2 1.16355
\(785\) 0 0
\(786\) −851.985 −0.0386632
\(787\) − 19107.9i − 0.865469i −0.901522 0.432734i \(-0.857549\pi\)
0.901522 0.432734i \(-0.142451\pi\)
\(788\) − 38459.6i − 1.73866i
\(789\) 7766.85 0.350453
\(790\) 0 0
\(791\) −5190.01 −0.233294
\(792\) − 4288.17i − 0.192391i
\(793\) 16271.4i 0.728645i
\(794\) 62269.4 2.78320
\(795\) 0 0
\(796\) −50801.9 −2.26209
\(797\) − 27518.4i − 1.22303i −0.791234 0.611513i \(-0.790561\pi\)
0.791234 0.611513i \(-0.209439\pi\)
\(798\) 481.893i 0.0213770i
\(799\) 8379.74 0.371031
\(800\) 0 0
\(801\) 11040.9 0.487030
\(802\) − 45335.2i − 1.99606i
\(803\) 1692.53i 0.0743811i
\(804\) −15729.4 −0.689965
\(805\) 0 0
\(806\) −24011.4 −1.04934
\(807\) 17614.6i 0.768356i
\(808\) − 54512.7i − 2.37345i
\(809\) −21986.6 −0.955511 −0.477755 0.878493i \(-0.658550\pi\)
−0.477755 + 0.878493i \(0.658550\pi\)
\(810\) 0 0
\(811\) 11835.1 0.512437 0.256218 0.966619i \(-0.417523\pi\)
0.256218 + 0.966619i \(0.417523\pi\)
\(812\) 24289.4i 1.04974i
\(813\) − 9920.79i − 0.427967i
\(814\) −9108.93 −0.392221
\(815\) 0 0
\(816\) −14537.6 −0.623672
\(817\) 1045.01i 0.0447494i
\(818\) − 2967.43i − 0.126838i
\(819\) −1212.81 −0.0517448
\(820\) 0 0
\(821\) −7890.58 −0.335424 −0.167712 0.985836i \(-0.553638\pi\)
−0.167712 + 0.985836i \(0.553638\pi\)
\(822\) − 28623.4i − 1.21455i
\(823\) 1000.74i 0.0423861i 0.999775 + 0.0211930i \(0.00674646\pi\)
−0.999775 + 0.0211930i \(0.993254\pi\)
\(824\) 21361.8 0.903123
\(825\) 0 0
\(826\) 11362.0 0.478611
\(827\) − 22764.2i − 0.957181i −0.878038 0.478590i \(-0.841148\pi\)
0.878038 0.478590i \(-0.158852\pi\)
\(828\) − 10297.2i − 0.432188i
\(829\) −39835.8 −1.66894 −0.834471 0.551051i \(-0.814227\pi\)
−0.834471 + 0.551051i \(0.814227\pi\)
\(830\) 0 0
\(831\) −8584.10 −0.358338
\(832\) 8808.43i 0.367040i
\(833\) 18578.1i 0.772742i
\(834\) 25211.0 1.04674
\(835\) 0 0
\(836\) 1083.27 0.0448155
\(837\) 5306.96i 0.219158i
\(838\) 57469.7i 2.36904i
\(839\) −23330.9 −0.960037 −0.480019 0.877258i \(-0.659370\pi\)
−0.480019 + 0.877258i \(0.659370\pi\)
\(840\) 0 0
\(841\) 45847.5 1.87984
\(842\) − 75476.0i − 3.08916i
\(843\) − 8964.14i − 0.366241i
\(844\) −46928.1 −1.91390
\(845\) 0 0
\(846\) 6315.09 0.256640
\(847\) − 663.535i − 0.0269177i
\(848\) − 27737.8i − 1.12326i
\(849\) −4363.79 −0.176402
\(850\) 0 0
\(851\) −11403.3 −0.459341
\(852\) 7706.08i 0.309866i
\(853\) 22937.3i 0.920701i 0.887737 + 0.460351i \(0.152276\pi\)
−0.887737 + 0.460351i \(0.847724\pi\)
\(854\) −18050.8 −0.723284
\(855\) 0 0
\(856\) −47605.2 −1.90083
\(857\) 38472.3i 1.53348i 0.641961 + 0.766738i \(0.278121\pi\)
−0.641961 + 0.766738i \(0.721879\pi\)
\(858\) 4031.34i 0.160405i
\(859\) 22970.3 0.912384 0.456192 0.889881i \(-0.349213\pi\)
0.456192 + 0.889881i \(0.349213\pi\)
\(860\) 0 0
\(861\) 6977.00 0.276162
\(862\) 74084.5i 2.92730i
\(863\) − 33587.6i − 1.32484i −0.749133 0.662420i \(-0.769530\pi\)
0.749133 0.662420i \(-0.230470\pi\)
\(864\) −1599.70 −0.0629895
\(865\) 0 0
\(866\) −11284.0 −0.442777
\(867\) 4165.08i 0.163153i
\(868\) − 18014.3i − 0.704430i
\(869\) −4433.00 −0.173049
\(870\) 0 0
\(871\) 7709.12 0.299901
\(872\) − 55228.3i − 2.14480i
\(873\) 8637.71i 0.334871i
\(874\) 2005.25 0.0776073
\(875\) 0 0
\(876\) 7714.74 0.297554
\(877\) 704.130i 0.0271115i 0.999908 + 0.0135558i \(0.00431506\pi\)
−0.999908 + 0.0135558i \(0.995685\pi\)
\(878\) − 66707.5i − 2.56409i
\(879\) 25759.5 0.988450
\(880\) 0 0
\(881\) 9746.33 0.372715 0.186358 0.982482i \(-0.440332\pi\)
0.186358 + 0.982482i \(0.440332\pi\)
\(882\) 14000.7i 0.534501i
\(883\) − 8774.24i − 0.334402i −0.985923 0.167201i \(-0.946527\pi\)
0.985923 0.167201i \(-0.0534728\pi\)
\(884\) 24383.0 0.927701
\(885\) 0 0
\(886\) 57206.4 2.16917
\(887\) 13505.1i 0.511224i 0.966779 + 0.255612i \(0.0822770\pi\)
−0.966779 + 0.255612i \(0.917723\pi\)
\(888\) 21645.6i 0.817993i
\(889\) −12397.5 −0.467717
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) − 24118.6i − 0.905324i
\(893\) 831.689i 0.0311662i
\(894\) 30988.9 1.15931
\(895\) 0 0
\(896\) −12370.9 −0.461252
\(897\) 5046.75i 0.187855i
\(898\) − 80552.8i − 2.99341i
\(899\) −52091.1 −1.93252
\(900\) 0 0
\(901\) 20175.1 0.745984
\(902\) − 23191.3i − 0.856083i
\(903\) − 2917.64i − 0.107523i
\(904\) 40994.6 1.50825
\(905\) 0 0
\(906\) 40433.4 1.48268
\(907\) 2523.23i 0.0923732i 0.998933 + 0.0461866i \(0.0147069\pi\)
−0.998933 + 0.0461866i \(0.985293\pi\)
\(908\) 61148.4i 2.23489i
\(909\) 11326.7 0.413293
\(910\) 0 0
\(911\) −32285.3 −1.17416 −0.587080 0.809529i \(-0.699723\pi\)
−0.587080 + 0.809529i \(0.699723\pi\)
\(912\) − 1442.85i − 0.0523877i
\(913\) − 7175.45i − 0.260101i
\(914\) −24422.6 −0.883838
\(915\) 0 0
\(916\) −28337.0 −1.02214
\(917\) 313.274i 0.0112816i
\(918\) − 7968.65i − 0.286497i
\(919\) 40956.5 1.47011 0.735055 0.678008i \(-0.237156\pi\)
0.735055 + 0.678008i \(0.237156\pi\)
\(920\) 0 0
\(921\) 1587.09 0.0567821
\(922\) 70943.6i 2.53406i
\(923\) − 3776.83i − 0.134687i
\(924\) −3024.47 −0.107682
\(925\) 0 0
\(926\) 33090.8 1.17433
\(927\) 4438.58i 0.157262i
\(928\) − 15702.1i − 0.555437i
\(929\) 6728.41 0.237623 0.118812 0.992917i \(-0.462092\pi\)
0.118812 + 0.992917i \(0.462092\pi\)
\(930\) 0 0
\(931\) −1843.88 −0.0649095
\(932\) 96639.1i 3.39648i
\(933\) − 15024.0i − 0.527185i
\(934\) −96181.9 −3.36956
\(935\) 0 0
\(936\) 9579.69 0.334532
\(937\) 36377.8i 1.26831i 0.773205 + 0.634157i \(0.218653\pi\)
−0.773205 + 0.634157i \(0.781347\pi\)
\(938\) 8552.13i 0.297694i
\(939\) −23575.6 −0.819340
\(940\) 0 0
\(941\) 25907.0 0.897496 0.448748 0.893658i \(-0.351870\pi\)
0.448748 + 0.893658i \(0.351870\pi\)
\(942\) − 34916.4i − 1.20768i
\(943\) − 29032.7i − 1.00258i
\(944\) −34019.2 −1.17291
\(945\) 0 0
\(946\) −9698.15 −0.333313
\(947\) − 37096.3i − 1.27293i −0.771304 0.636466i \(-0.780395\pi\)
0.771304 0.636466i \(-0.219605\pi\)
\(948\) 20206.2i 0.692263i
\(949\) −3781.07 −0.129335
\(950\) 0 0
\(951\) 23241.1 0.792476
\(952\) 14101.7i 0.480084i
\(953\) − 15323.7i − 0.520865i −0.965492 0.260432i \(-0.916135\pi\)
0.965492 0.260432i \(-0.0838651\pi\)
\(954\) 15204.3 0.515992
\(955\) 0 0
\(956\) −97540.4 −3.29988
\(957\) 8745.72i 0.295412i
\(958\) 48850.7i 1.64749i
\(959\) −10524.8 −0.354395
\(960\) 0 0
\(961\) 8842.46 0.296817
\(962\) − 20349.2i − 0.682000i
\(963\) − 9891.45i − 0.330994i
\(964\) 16417.4 0.548516
\(965\) 0 0
\(966\) −5598.62 −0.186473
\(967\) 53388.0i 1.77543i 0.460391 + 0.887716i \(0.347709\pi\)
−0.460391 + 0.887716i \(0.652291\pi\)
\(968\) 5241.10i 0.174024i
\(969\) 1049.46 0.0347921
\(970\) 0 0
\(971\) −36325.7 −1.20056 −0.600281 0.799789i \(-0.704945\pi\)
−0.600281 + 0.799789i \(0.704945\pi\)
\(972\) − 4061.28i − 0.134018i
\(973\) − 9270.07i − 0.305432i
\(974\) −51094.6 −1.68088
\(975\) 0 0
\(976\) 54046.4 1.77252
\(977\) − 48608.7i − 1.59174i −0.605468 0.795870i \(-0.707014\pi\)
0.605468 0.795870i \(-0.292986\pi\)
\(978\) 52541.1i 1.71787i
\(979\) −13494.4 −0.440535
\(980\) 0 0
\(981\) 11475.4 0.373477
\(982\) − 56657.7i − 1.84116i
\(983\) 31762.6i 1.03059i 0.857013 + 0.515295i \(0.172318\pi\)
−0.857013 + 0.515295i \(0.827682\pi\)
\(984\) −55109.6 −1.78540
\(985\) 0 0
\(986\) 78217.2 2.52631
\(987\) − 2322.06i − 0.0748854i
\(988\) 2420.01i 0.0779258i
\(989\) −12140.9 −0.390352
\(990\) 0 0
\(991\) −5073.82 −0.162639 −0.0813195 0.996688i \(-0.525913\pi\)
−0.0813195 + 0.996688i \(0.525913\pi\)
\(992\) 11645.5i 0.372726i
\(993\) 4065.25i 0.129916i
\(994\) 4189.83 0.133696
\(995\) 0 0
\(996\) −32706.5 −1.04051
\(997\) − 24607.0i − 0.781657i −0.920464 0.390828i \(-0.872189\pi\)
0.920464 0.390828i \(-0.127811\pi\)
\(998\) − 45631.8i − 1.44734i
\(999\) −4497.54 −0.142438
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 825.4.c.k.199.2 6
5.2 odd 4 165.4.a.e.1.3 3
5.3 odd 4 825.4.a.r.1.1 3
5.4 even 2 inner 825.4.c.k.199.5 6
15.2 even 4 495.4.a.k.1.1 3
15.8 even 4 2475.4.a.t.1.3 3
55.32 even 4 1815.4.a.r.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.e.1.3 3 5.2 odd 4
495.4.a.k.1.1 3 15.2 even 4
825.4.a.r.1.1 3 5.3 odd 4
825.4.c.k.199.2 6 1.1 even 1 trivial
825.4.c.k.199.5 6 5.4 even 2 inner
1815.4.a.r.1.1 3 55.32 even 4
2475.4.a.t.1.3 3 15.8 even 4