Properties

Label 825.4.c.j.199.1
Level $825$
Weight $4$
Character 825.199
Analytic conductor $48.677$
Analytic rank $0$
Dimension $4$
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: \(4\)
Coefficient field: \(\Q(i, \sqrt{17})\)
Defining polynomial: \(x^{4} + 9 x^{2} + 16\)
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 199.1
Root \(-2.56155i\) of defining polynomial
Character \(\chi\) \(=\) 825.199
Dual form 825.4.c.j.199.4

$q$-expansion

\(f(q)\) \(=\) \(q-2.56155i q^{2} -3.00000i q^{3} +1.43845 q^{4} -7.68466 q^{6} +6.24621i q^{7} -24.1771i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q-2.56155i q^{2} -3.00000i q^{3} +1.43845 q^{4} -7.68466 q^{6} +6.24621i q^{7} -24.1771i q^{8} -9.00000 q^{9} -11.0000 q^{11} -4.31534i q^{12} +49.1231i q^{13} +16.0000 q^{14} -50.4233 q^{16} +82.7083i q^{17} +23.0540i q^{18} +130.354 q^{19} +18.7386 q^{21} +28.1771i q^{22} +185.693i q^{23} -72.5312 q^{24} +125.831 q^{26} +27.0000i q^{27} +8.98485i q^{28} +8.90720 q^{29} +5.26137 q^{31} -64.2547i q^{32} +33.0000i q^{33} +211.862 q^{34} -12.9460 q^{36} -416.894i q^{37} -333.909i q^{38} +147.369 q^{39} -298.479 q^{41} -48.0000i q^{42} +513.633i q^{43} -15.8229 q^{44} +475.663 q^{46} +557.295i q^{47} +151.270i q^{48} +303.985 q^{49} +248.125 q^{51} +70.6610i q^{52} +168.064i q^{53} +69.1619 q^{54} +151.015 q^{56} -391.062i q^{57} -22.8163i q^{58} -618.773 q^{59} +786.405 q^{61} -13.4773i q^{62} -56.2159i q^{63} -567.978 q^{64} +84.5312 q^{66} -339.015i q^{67} +118.972i q^{68} +557.080 q^{69} +1120.71 q^{71} +217.594i q^{72} +123.430i q^{73} -1067.90 q^{74} +187.508 q^{76} -68.7083i q^{77} -377.494i q^{78} +309.835 q^{79} +81.0000 q^{81} +764.570i q^{82} +1021.22i q^{83} +26.9545 q^{84} +1315.70 q^{86} -26.7216i q^{87} +265.948i q^{88} +141.879 q^{89} -306.833 q^{91} +267.110i q^{92} -15.7841i q^{93} +1427.54 q^{94} -192.764 q^{96} +798.345i q^{97} -778.673i q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 14q^{4} - 6q^{6} - 36q^{9} + O(q^{10}) \) \( 4q + 14q^{4} - 6q^{6} - 36q^{9} - 44q^{11} + 64q^{14} - 78q^{16} + 340q^{19} - 24q^{21} - 18q^{24} + 124q^{26} + 316q^{29} + 120q^{31} + 732q^{34} - 126q^{36} + 540q^{39} + 76q^{41} - 154q^{44} + 1144q^{46} + 1084q^{49} - 96q^{51} + 54q^{54} + 736q^{56} - 496q^{59} + 144q^{61} - 1538q^{64} + 66q^{66} + 744q^{69} + 4120q^{71} - 2276q^{74} + 816q^{76} - 1284q^{79} + 324q^{81} - 288q^{84} + 2624q^{86} - 488q^{89} + 224q^{91} + 3896q^{94} + 738q^{96} + 396q^{99} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) − 2.56155i − 0.905646i −0.891601 0.452823i \(-0.850417\pi\)
0.891601 0.452823i \(-0.149583\pi\)
\(3\) − 3.00000i − 0.577350i
\(4\) 1.43845 0.179806
\(5\) 0 0
\(6\) −7.68466 −0.522875
\(7\) 6.24621i 0.337264i 0.985679 + 0.168632i \(0.0539349\pi\)
−0.985679 + 0.168632i \(0.946065\pi\)
\(8\) − 24.1771i − 1.06849i
\(9\) −9.00000 −0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) − 4.31534i − 0.103811i
\(13\) 49.1231i 1.04802i 0.851711 + 0.524011i \(0.175565\pi\)
−0.851711 + 0.524011i \(0.824435\pi\)
\(14\) 16.0000 0.305441
\(15\) 0 0
\(16\) −50.4233 −0.787864
\(17\) 82.7083i 1.17998i 0.807409 + 0.589992i \(0.200869\pi\)
−0.807409 + 0.589992i \(0.799131\pi\)
\(18\) 23.0540i 0.301882i
\(19\) 130.354 1.57396 0.786981 0.616977i \(-0.211643\pi\)
0.786981 + 0.616977i \(0.211643\pi\)
\(20\) 0 0
\(21\) 18.7386 0.194719
\(22\) 28.1771i 0.273062i
\(23\) 185.693i 1.68347i 0.539895 + 0.841733i \(0.318464\pi\)
−0.539895 + 0.841733i \(0.681536\pi\)
\(24\) −72.5312 −0.616891
\(25\) 0 0
\(26\) 125.831 0.949137
\(27\) 27.0000i 0.192450i
\(28\) 8.98485i 0.0606420i
\(29\) 8.90720 0.0570354 0.0285177 0.999593i \(-0.490921\pi\)
0.0285177 + 0.999593i \(0.490921\pi\)
\(30\) 0 0
\(31\) 5.26137 0.0304829 0.0152414 0.999884i \(-0.495148\pi\)
0.0152414 + 0.999884i \(0.495148\pi\)
\(32\) − 64.2547i − 0.354961i
\(33\) 33.0000i 0.174078i
\(34\) 211.862 1.06865
\(35\) 0 0
\(36\) −12.9460 −0.0599353
\(37\) − 416.894i − 1.85235i −0.377094 0.926175i \(-0.623077\pi\)
0.377094 0.926175i \(-0.376923\pi\)
\(38\) − 333.909i − 1.42545i
\(39\) 147.369 0.605076
\(40\) 0 0
\(41\) −298.479 −1.13694 −0.568471 0.822703i \(-0.692465\pi\)
−0.568471 + 0.822703i \(0.692465\pi\)
\(42\) − 48.0000i − 0.176347i
\(43\) 513.633i 1.82159i 0.412863 + 0.910793i \(0.364529\pi\)
−0.412863 + 0.910793i \(0.635471\pi\)
\(44\) −15.8229 −0.0542135
\(45\) 0 0
\(46\) 475.663 1.52462
\(47\) 557.295i 1.72957i 0.502140 + 0.864786i \(0.332546\pi\)
−0.502140 + 0.864786i \(0.667454\pi\)
\(48\) 151.270i 0.454873i
\(49\) 303.985 0.886253
\(50\) 0 0
\(51\) 248.125 0.681264
\(52\) 70.6610i 0.188441i
\(53\) 168.064i 0.435574i 0.975996 + 0.217787i \(0.0698838\pi\)
−0.975996 + 0.217787i \(0.930116\pi\)
\(54\) 69.1619 0.174292
\(55\) 0 0
\(56\) 151.015 0.360362
\(57\) − 391.062i − 0.908728i
\(58\) − 22.8163i − 0.0516539i
\(59\) −618.773 −1.36538 −0.682689 0.730709i \(-0.739190\pi\)
−0.682689 + 0.730709i \(0.739190\pi\)
\(60\) 0 0
\(61\) 786.405 1.65064 0.825319 0.564667i \(-0.190996\pi\)
0.825319 + 0.564667i \(0.190996\pi\)
\(62\) − 13.4773i − 0.0276067i
\(63\) − 56.2159i − 0.112421i
\(64\) −567.978 −1.10933
\(65\) 0 0
\(66\) 84.5312 0.157653
\(67\) − 339.015i − 0.618169i −0.951035 0.309084i \(-0.899977\pi\)
0.951035 0.309084i \(-0.100023\pi\)
\(68\) 118.972i 0.212168i
\(69\) 557.080 0.971949
\(70\) 0 0
\(71\) 1120.71 1.87329 0.936645 0.350280i \(-0.113913\pi\)
0.936645 + 0.350280i \(0.113913\pi\)
\(72\) 217.594i 0.356162i
\(73\) 123.430i 0.197896i 0.995093 + 0.0989478i \(0.0315477\pi\)
−0.995093 + 0.0989478i \(0.968452\pi\)
\(74\) −1067.90 −1.67757
\(75\) 0 0
\(76\) 187.508 0.283008
\(77\) − 68.7083i − 0.101689i
\(78\) − 377.494i − 0.547985i
\(79\) 309.835 0.441255 0.220628 0.975358i \(-0.429189\pi\)
0.220628 + 0.975358i \(0.429189\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 764.570i 1.02967i
\(83\) 1021.22i 1.35053i 0.737577 + 0.675263i \(0.235970\pi\)
−0.737577 + 0.675263i \(0.764030\pi\)
\(84\) 26.9545 0.0350117
\(85\) 0 0
\(86\) 1315.70 1.64971
\(87\) − 26.7216i − 0.0329294i
\(88\) 265.948i 0.322161i
\(89\) 141.879 0.168979 0.0844894 0.996424i \(-0.473074\pi\)
0.0844894 + 0.996424i \(0.473074\pi\)
\(90\) 0 0
\(91\) −306.833 −0.353460
\(92\) 267.110i 0.302697i
\(93\) − 15.7841i − 0.0175993i
\(94\) 1427.54 1.56638
\(95\) 0 0
\(96\) −192.764 −0.204937
\(97\) 798.345i 0.835666i 0.908524 + 0.417833i \(0.137210\pi\)
−0.908524 + 0.417833i \(0.862790\pi\)
\(98\) − 778.673i − 0.802631i
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) 241.400 0.237823 0.118912 0.992905i \(-0.462059\pi\)
0.118912 + 0.992905i \(0.462059\pi\)
\(102\) − 635.585i − 0.616983i
\(103\) − 1168.38i − 1.11771i −0.829267 0.558853i \(-0.811242\pi\)
0.829267 0.558853i \(-0.188758\pi\)
\(104\) 1187.65 1.11980
\(105\) 0 0
\(106\) 430.506 0.394476
\(107\) − 2106.82i − 1.90350i −0.306882 0.951748i \(-0.599286\pi\)
0.306882 0.951748i \(-0.400714\pi\)
\(108\) 38.8381i 0.0346037i
\(109\) −493.792 −0.433914 −0.216957 0.976181i \(-0.569613\pi\)
−0.216957 + 0.976181i \(0.569613\pi\)
\(110\) 0 0
\(111\) −1250.68 −1.06945
\(112\) − 314.955i − 0.265718i
\(113\) − 170.000i − 0.141524i −0.997493 0.0707622i \(-0.977457\pi\)
0.997493 0.0707622i \(-0.0225431\pi\)
\(114\) −1001.73 −0.822986
\(115\) 0 0
\(116\) 12.8125 0.0102553
\(117\) − 442.108i − 0.349341i
\(118\) 1585.02i 1.23655i
\(119\) −516.614 −0.397966
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) − 2014.42i − 1.49489i
\(123\) 895.437i 0.656414i
\(124\) 7.56820 0.00548100
\(125\) 0 0
\(126\) −144.000 −0.101814
\(127\) − 948.182i − 0.662500i −0.943543 0.331250i \(-0.892530\pi\)
0.943543 0.331250i \(-0.107470\pi\)
\(128\) 940.868i 0.649702i
\(129\) 1540.90 1.05169
\(130\) 0 0
\(131\) −1484.84 −0.990312 −0.495156 0.868804i \(-0.664889\pi\)
−0.495156 + 0.868804i \(0.664889\pi\)
\(132\) 47.4688i 0.0313002i
\(133\) 814.220i 0.530841i
\(134\) −868.405 −0.559842
\(135\) 0 0
\(136\) 1999.65 1.26080
\(137\) 684.928i 0.427134i 0.976928 + 0.213567i \(0.0685082\pi\)
−0.976928 + 0.213567i \(0.931492\pi\)
\(138\) − 1426.99i − 0.880242i
\(139\) −830.483 −0.506767 −0.253384 0.967366i \(-0.581543\pi\)
−0.253384 + 0.967366i \(0.581543\pi\)
\(140\) 0 0
\(141\) 1671.89 0.998569
\(142\) − 2870.75i − 1.69654i
\(143\) − 540.354i − 0.315991i
\(144\) 453.810 0.262621
\(145\) 0 0
\(146\) 316.172 0.179223
\(147\) − 911.955i − 0.511679i
\(148\) − 599.680i − 0.333063i
\(149\) 1213.64 0.667285 0.333642 0.942700i \(-0.391722\pi\)
0.333642 + 0.942700i \(0.391722\pi\)
\(150\) 0 0
\(151\) 30.8466 0.0166242 0.00831212 0.999965i \(-0.497354\pi\)
0.00831212 + 0.999965i \(0.497354\pi\)
\(152\) − 3151.58i − 1.68176i
\(153\) − 744.375i − 0.393328i
\(154\) −176.000 −0.0920941
\(155\) 0 0
\(156\) 211.983 0.108796
\(157\) 345.239i 0.175497i 0.996143 + 0.0877485i \(0.0279672\pi\)
−0.996143 + 0.0877485i \(0.972033\pi\)
\(158\) − 793.659i − 0.399621i
\(159\) 504.193 0.251479
\(160\) 0 0
\(161\) −1159.88 −0.567772
\(162\) − 207.486i − 0.100627i
\(163\) − 1921.49i − 0.923331i −0.887054 0.461665i \(-0.847252\pi\)
0.887054 0.461665i \(-0.152748\pi\)
\(164\) −429.346 −0.204429
\(165\) 0 0
\(166\) 2615.91 1.22310
\(167\) 172.297i 0.0798369i 0.999203 + 0.0399185i \(0.0127098\pi\)
−0.999203 + 0.0399185i \(0.987290\pi\)
\(168\) − 453.045i − 0.208055i
\(169\) −216.080 −0.0983521
\(170\) 0 0
\(171\) −1173.19 −0.524654
\(172\) 738.833i 0.327532i
\(173\) 1025.29i 0.450587i 0.974291 + 0.225293i \(0.0723340\pi\)
−0.974291 + 0.225293i \(0.927666\pi\)
\(174\) −68.4488 −0.0298224
\(175\) 0 0
\(176\) 554.656 0.237550
\(177\) 1856.32i 0.788302i
\(178\) − 363.430i − 0.153035i
\(179\) −1658.29 −0.692437 −0.346219 0.938154i \(-0.612534\pi\)
−0.346219 + 0.938154i \(0.612534\pi\)
\(180\) 0 0
\(181\) −2021.81 −0.830275 −0.415137 0.909759i \(-0.636266\pi\)
−0.415137 + 0.909759i \(0.636266\pi\)
\(182\) 785.970i 0.320110i
\(183\) − 2359.22i − 0.952996i
\(184\) 4489.52 1.79876
\(185\) 0 0
\(186\) −40.4318 −0.0159387
\(187\) − 909.792i − 0.355778i
\(188\) 801.640i 0.310987i
\(189\) −168.648 −0.0649064
\(190\) 0 0
\(191\) 1440.22 0.545607 0.272803 0.962070i \(-0.412049\pi\)
0.272803 + 0.962070i \(0.412049\pi\)
\(192\) 1703.93i 0.640473i
\(193\) 2798.05i 1.04356i 0.853079 + 0.521782i \(0.174733\pi\)
−0.853079 + 0.521782i \(0.825267\pi\)
\(194\) 2045.00 0.756817
\(195\) 0 0
\(196\) 437.266 0.159354
\(197\) 458.943i 0.165982i 0.996550 + 0.0829908i \(0.0264472\pi\)
−0.996550 + 0.0829908i \(0.973553\pi\)
\(198\) − 253.594i − 0.0910208i
\(199\) 2371.04 0.844615 0.422308 0.906453i \(-0.361220\pi\)
0.422308 + 0.906453i \(0.361220\pi\)
\(200\) 0 0
\(201\) −1017.05 −0.356900
\(202\) − 618.358i − 0.215384i
\(203\) 55.6363i 0.0192360i
\(204\) 356.915 0.122495
\(205\) 0 0
\(206\) −2992.86 −1.01225
\(207\) − 1671.24i − 0.561155i
\(208\) − 2476.95i − 0.825699i
\(209\) −1433.90 −0.474568
\(210\) 0 0
\(211\) 4319.87 1.40944 0.704721 0.709484i \(-0.251072\pi\)
0.704721 + 0.709484i \(0.251072\pi\)
\(212\) 241.752i 0.0783187i
\(213\) − 3362.12i − 1.08154i
\(214\) −5396.73 −1.72389
\(215\) 0 0
\(216\) 652.781 0.205630
\(217\) 32.8636i 0.0102808i
\(218\) 1264.87i 0.392973i
\(219\) 370.290 0.114255
\(220\) 0 0
\(221\) −4062.89 −1.23665
\(222\) 3203.69i 0.968547i
\(223\) 3837.73i 1.15244i 0.817295 + 0.576219i \(0.195473\pi\)
−0.817295 + 0.576219i \(0.804527\pi\)
\(224\) 401.349 0.119715
\(225\) 0 0
\(226\) −435.464 −0.128171
\(227\) − 5003.71i − 1.46303i −0.681825 0.731515i \(-0.738813\pi\)
0.681825 0.731515i \(-0.261187\pi\)
\(228\) − 562.523i − 0.163395i
\(229\) 277.375 0.0800412 0.0400206 0.999199i \(-0.487258\pi\)
0.0400206 + 0.999199i \(0.487258\pi\)
\(230\) 0 0
\(231\) −206.125 −0.0587101
\(232\) − 215.350i − 0.0609415i
\(233\) − 2269.91i − 0.638225i −0.947717 0.319113i \(-0.896615\pi\)
0.947717 0.319113i \(-0.103385\pi\)
\(234\) −1132.48 −0.316379
\(235\) 0 0
\(236\) −890.072 −0.245503
\(237\) − 929.505i − 0.254759i
\(238\) 1323.33i 0.360416i
\(239\) 1617.11 0.437665 0.218832 0.975762i \(-0.429775\pi\)
0.218832 + 0.975762i \(0.429775\pi\)
\(240\) 0 0
\(241\) 5646.63 1.50926 0.754629 0.656151i \(-0.227817\pi\)
0.754629 + 0.656151i \(0.227817\pi\)
\(242\) − 309.948i − 0.0823314i
\(243\) − 243.000i − 0.0641500i
\(244\) 1131.20 0.296794
\(245\) 0 0
\(246\) 2293.71 0.594478
\(247\) 6403.40i 1.64955i
\(248\) − 127.204i − 0.0325705i
\(249\) 3063.66 0.779726
\(250\) 0 0
\(251\) −6217.61 −1.56355 −0.781777 0.623558i \(-0.785687\pi\)
−0.781777 + 0.623558i \(0.785687\pi\)
\(252\) − 80.8636i − 0.0202140i
\(253\) − 2042.62i − 0.507584i
\(254\) −2428.82 −0.599991
\(255\) 0 0
\(256\) −2133.74 −0.520933
\(257\) 7712.75i 1.87202i 0.351980 + 0.936008i \(0.385509\pi\)
−0.351980 + 0.936008i \(0.614491\pi\)
\(258\) − 3947.09i − 0.952462i
\(259\) 2604.01 0.624730
\(260\) 0 0
\(261\) −80.1648 −0.0190118
\(262\) 3803.49i 0.896871i
\(263\) − 206.347i − 0.0483798i −0.999707 0.0241899i \(-0.992299\pi\)
0.999707 0.0241899i \(-0.00770063\pi\)
\(264\) 797.844 0.186000
\(265\) 0 0
\(266\) 2085.67 0.480753
\(267\) − 425.636i − 0.0975600i
\(268\) − 487.655i − 0.111150i
\(269\) −1712.47 −0.388146 −0.194073 0.980987i \(-0.562170\pi\)
−0.194073 + 0.980987i \(0.562170\pi\)
\(270\) 0 0
\(271\) −477.081 −0.106940 −0.0534698 0.998569i \(-0.517028\pi\)
−0.0534698 + 0.998569i \(0.517028\pi\)
\(272\) − 4170.43i − 0.929666i
\(273\) 920.500i 0.204070i
\(274\) 1754.48 0.386832
\(275\) 0 0
\(276\) 801.329 0.174762
\(277\) 4283.48i 0.929130i 0.885539 + 0.464565i \(0.153789\pi\)
−0.885539 + 0.464565i \(0.846211\pi\)
\(278\) 2127.33i 0.458952i
\(279\) −47.3523 −0.0101610
\(280\) 0 0
\(281\) 3477.79 0.738319 0.369160 0.929366i \(-0.379646\pi\)
0.369160 + 0.929366i \(0.379646\pi\)
\(282\) − 4282.62i − 0.904350i
\(283\) 6568.27i 1.37966i 0.723973 + 0.689829i \(0.242314\pi\)
−0.723973 + 0.689829i \(0.757686\pi\)
\(284\) 1612.08 0.336829
\(285\) 0 0
\(286\) −1384.15 −0.286176
\(287\) − 1864.36i − 0.383449i
\(288\) 578.292i 0.118320i
\(289\) −1927.67 −0.392360
\(290\) 0 0
\(291\) 2395.03 0.482472
\(292\) 177.547i 0.0355828i
\(293\) − 8352.29i − 1.66534i −0.553766 0.832672i \(-0.686810\pi\)
0.553766 0.832672i \(-0.313190\pi\)
\(294\) −2336.02 −0.463399
\(295\) 0 0
\(296\) −10079.3 −1.97921
\(297\) − 297.000i − 0.0580259i
\(298\) − 3108.81i − 0.604324i
\(299\) −9121.83 −1.76431
\(300\) 0 0
\(301\) −3208.26 −0.614355
\(302\) − 79.0152i − 0.0150557i
\(303\) − 724.199i − 0.137307i
\(304\) −6572.89 −1.24007
\(305\) 0 0
\(306\) −1906.76 −0.356216
\(307\) 5383.89i 1.00090i 0.865767 + 0.500448i \(0.166831\pi\)
−0.865767 + 0.500448i \(0.833169\pi\)
\(308\) − 98.8333i − 0.0182843i
\(309\) −3505.14 −0.645308
\(310\) 0 0
\(311\) −1790.41 −0.326447 −0.163223 0.986589i \(-0.552189\pi\)
−0.163223 + 0.986589i \(0.552189\pi\)
\(312\) − 3562.96i − 0.646516i
\(313\) 809.076i 0.146108i 0.997328 + 0.0730538i \(0.0232745\pi\)
−0.997328 + 0.0730538i \(0.976726\pi\)
\(314\) 884.347 0.158938
\(315\) 0 0
\(316\) 445.682 0.0793403
\(317\) 10744.5i 1.90370i 0.306567 + 0.951849i \(0.400820\pi\)
−0.306567 + 0.951849i \(0.599180\pi\)
\(318\) − 1291.52i − 0.227751i
\(319\) −97.9792 −0.0171968
\(320\) 0 0
\(321\) −6320.46 −1.09898
\(322\) 2971.09i 0.514200i
\(323\) 10781.4i 1.85725i
\(324\) 116.514 0.0199784
\(325\) 0 0
\(326\) −4922.00 −0.836210
\(327\) 1481.37i 0.250521i
\(328\) 7216.35i 1.21481i
\(329\) −3480.98 −0.583322
\(330\) 0 0
\(331\) 3399.12 0.564449 0.282224 0.959348i \(-0.408928\pi\)
0.282224 + 0.959348i \(0.408928\pi\)
\(332\) 1468.97i 0.242832i
\(333\) 3752.05i 0.617450i
\(334\) 441.349 0.0723039
\(335\) 0 0
\(336\) −944.864 −0.153412
\(337\) − 11840.0i − 1.91384i −0.290356 0.956919i \(-0.593774\pi\)
0.290356 0.956919i \(-0.406226\pi\)
\(338\) 553.499i 0.0890721i
\(339\) −510.000 −0.0817091
\(340\) 0 0
\(341\) −57.8750 −0.00919093
\(342\) 3005.18i 0.475151i
\(343\) 4041.20i 0.636165i
\(344\) 12418.1 1.94634
\(345\) 0 0
\(346\) 2626.34 0.408072
\(347\) − 2076.67i − 0.321272i −0.987014 0.160636i \(-0.948645\pi\)
0.987014 0.160636i \(-0.0513545\pi\)
\(348\) − 38.4376i − 0.00592090i
\(349\) −5837.37 −0.895322 −0.447661 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(350\) 0 0
\(351\) −1326.32 −0.201692
\(352\) 706.802i 0.107025i
\(353\) 2423.64i 0.365431i 0.983166 + 0.182715i \(0.0584887\pi\)
−0.983166 + 0.182715i \(0.941511\pi\)
\(354\) 4755.06 0.713922
\(355\) 0 0
\(356\) 204.085 0.0303834
\(357\) 1549.84i 0.229765i
\(358\) 4247.79i 0.627103i
\(359\) 3882.22 0.570740 0.285370 0.958417i \(-0.407884\pi\)
0.285370 + 0.958417i \(0.407884\pi\)
\(360\) 0 0
\(361\) 10133.2 1.47736
\(362\) 5178.96i 0.751935i
\(363\) − 363.000i − 0.0524864i
\(364\) −441.363 −0.0635542
\(365\) 0 0
\(366\) −6043.26 −0.863077
\(367\) 5666.65i 0.805986i 0.915203 + 0.402993i \(0.132030\pi\)
−0.915203 + 0.402993i \(0.867970\pi\)
\(368\) − 9363.26i − 1.32634i
\(369\) 2686.31 0.378981
\(370\) 0 0
\(371\) −1049.77 −0.146903
\(372\) − 22.7046i − 0.00316446i
\(373\) − 174.771i − 0.0242608i −0.999926 0.0121304i \(-0.996139\pi\)
0.999926 0.0121304i \(-0.00386132\pi\)
\(374\) −2330.48 −0.322209
\(375\) 0 0
\(376\) 13473.8 1.84802
\(377\) 437.550i 0.0597744i
\(378\) 432.000i 0.0587822i
\(379\) −252.686 −0.0342470 −0.0171235 0.999853i \(-0.505451\pi\)
−0.0171235 + 0.999853i \(0.505451\pi\)
\(380\) 0 0
\(381\) −2844.55 −0.382495
\(382\) − 3689.21i − 0.494126i
\(383\) 11014.5i 1.46950i 0.678340 + 0.734748i \(0.262700\pi\)
−0.678340 + 0.734748i \(0.737300\pi\)
\(384\) 2822.61 0.375105
\(385\) 0 0
\(386\) 7167.35 0.945099
\(387\) − 4622.69i − 0.607196i
\(388\) 1148.38i 0.150258i
\(389\) −8099.40 −1.05567 −0.527835 0.849347i \(-0.676996\pi\)
−0.527835 + 0.849347i \(0.676996\pi\)
\(390\) 0 0
\(391\) −15358.4 −1.98646
\(392\) − 7349.47i − 0.946949i
\(393\) 4454.51i 0.571757i
\(394\) 1175.61 0.150320
\(395\) 0 0
\(396\) 142.406 0.0180712
\(397\) − 424.353i − 0.0536465i −0.999640 0.0268232i \(-0.991461\pi\)
0.999640 0.0268232i \(-0.00853912\pi\)
\(398\) − 6073.54i − 0.764922i
\(399\) 2442.66 0.306481
\(400\) 0 0
\(401\) −5904.18 −0.735263 −0.367632 0.929972i \(-0.619831\pi\)
−0.367632 + 0.929972i \(0.619831\pi\)
\(402\) 2605.22i 0.323225i
\(403\) 258.455i 0.0319468i
\(404\) 347.241 0.0427620
\(405\) 0 0
\(406\) 142.515 0.0174210
\(407\) 4585.83i 0.558504i
\(408\) − 5998.94i − 0.727921i
\(409\) −1370.47 −0.165686 −0.0828430 0.996563i \(-0.526400\pi\)
−0.0828430 + 0.996563i \(0.526400\pi\)
\(410\) 0 0
\(411\) 2054.78 0.246606
\(412\) − 1680.65i − 0.200970i
\(413\) − 3864.98i − 0.460493i
\(414\) −4280.97 −0.508208
\(415\) 0 0
\(416\) 3156.39 0.372007
\(417\) 2491.45i 0.292582i
\(418\) 3673.00i 0.429790i
\(419\) −1268.73 −0.147927 −0.0739635 0.997261i \(-0.523565\pi\)
−0.0739635 + 0.997261i \(0.523565\pi\)
\(420\) 0 0
\(421\) −12241.9 −1.41719 −0.708594 0.705617i \(-0.750670\pi\)
−0.708594 + 0.705617i \(0.750670\pi\)
\(422\) − 11065.6i − 1.27646i
\(423\) − 5015.66i − 0.576524i
\(424\) 4063.31 0.465405
\(425\) 0 0
\(426\) −8612.26 −0.979496
\(427\) 4912.05i 0.556700i
\(428\) − 3030.55i − 0.342260i
\(429\) −1621.06 −0.182437
\(430\) 0 0
\(431\) 8050.11 0.899675 0.449838 0.893110i \(-0.351482\pi\)
0.449838 + 0.893110i \(0.351482\pi\)
\(432\) − 1361.43i − 0.151624i
\(433\) 16565.7i 1.83856i 0.393600 + 0.919282i \(0.371230\pi\)
−0.393600 + 0.919282i \(0.628770\pi\)
\(434\) 84.1819 0.00931073
\(435\) 0 0
\(436\) −710.293 −0.0780203
\(437\) 24205.9i 2.64971i
\(438\) − 948.517i − 0.103475i
\(439\) −4705.80 −0.511607 −0.255804 0.966729i \(-0.582340\pi\)
−0.255804 + 0.966729i \(0.582340\pi\)
\(440\) 0 0
\(441\) −2735.86 −0.295418
\(442\) 10407.3i 1.11997i
\(443\) − 15094.0i − 1.61882i −0.587246 0.809408i \(-0.699788\pi\)
0.587246 0.809408i \(-0.300212\pi\)
\(444\) −1799.04 −0.192294
\(445\) 0 0
\(446\) 9830.56 1.04370
\(447\) − 3640.93i − 0.385257i
\(448\) − 3547.71i − 0.374138i
\(449\) −973.478 −0.102319 −0.0511595 0.998690i \(-0.516292\pi\)
−0.0511595 + 0.998690i \(0.516292\pi\)
\(450\) 0 0
\(451\) 3283.27 0.342801
\(452\) − 244.536i − 0.0254469i
\(453\) − 92.5398i − 0.00959801i
\(454\) −12817.3 −1.32499
\(455\) 0 0
\(456\) −9454.75 −0.970963
\(457\) 62.6577i 0.00641358i 0.999995 + 0.00320679i \(0.00102075\pi\)
−0.999995 + 0.00320679i \(0.998979\pi\)
\(458\) − 710.510i − 0.0724890i
\(459\) −2233.12 −0.227088
\(460\) 0 0
\(461\) −11866.2 −1.19884 −0.599419 0.800436i \(-0.704602\pi\)
−0.599419 + 0.800436i \(0.704602\pi\)
\(462\) 528.000i 0.0531705i
\(463\) 13144.8i 1.31942i 0.751519 + 0.659711i \(0.229321\pi\)
−0.751519 + 0.659711i \(0.770679\pi\)
\(464\) −449.131 −0.0449361
\(465\) 0 0
\(466\) −5814.48 −0.578006
\(467\) − 10176.8i − 1.00840i −0.863586 0.504201i \(-0.831787\pi\)
0.863586 0.504201i \(-0.168213\pi\)
\(468\) − 635.949i − 0.0628136i
\(469\) 2117.56 0.208486
\(470\) 0 0
\(471\) 1035.72 0.101323
\(472\) 14960.1i 1.45889i
\(473\) − 5649.96i − 0.549229i
\(474\) −2380.98 −0.230721
\(475\) 0 0
\(476\) −743.121 −0.0715565
\(477\) − 1512.58i − 0.145191i
\(478\) − 4142.30i − 0.396369i
\(479\) −3431.25 −0.327302 −0.163651 0.986518i \(-0.552327\pi\)
−0.163651 + 0.986518i \(0.552327\pi\)
\(480\) 0 0
\(481\) 20479.1 1.94130
\(482\) − 14464.1i − 1.36685i
\(483\) 3479.64i 0.327803i
\(484\) 174.052 0.0163460
\(485\) 0 0
\(486\) −622.457 −0.0580972
\(487\) − 2833.20i − 0.263624i −0.991275 0.131812i \(-0.957921\pi\)
0.991275 0.131812i \(-0.0420795\pi\)
\(488\) − 19013.0i − 1.76368i
\(489\) −5764.48 −0.533085
\(490\) 0 0
\(491\) −2667.29 −0.245159 −0.122580 0.992459i \(-0.539117\pi\)
−0.122580 + 0.992459i \(0.539117\pi\)
\(492\) 1288.04i 0.118027i
\(493\) 736.700i 0.0673008i
\(494\) 16402.7 1.49391
\(495\) 0 0
\(496\) −265.295 −0.0240164
\(497\) 7000.18i 0.631793i
\(498\) − 7847.74i − 0.706156i
\(499\) 11137.0 0.999120 0.499560 0.866279i \(-0.333495\pi\)
0.499560 + 0.866279i \(0.333495\pi\)
\(500\) 0 0
\(501\) 516.892 0.0460939
\(502\) 15926.7i 1.41603i
\(503\) − 8780.30i − 0.778319i −0.921170 0.389159i \(-0.872766\pi\)
0.921170 0.389159i \(-0.127234\pi\)
\(504\) −1359.14 −0.120121
\(505\) 0 0
\(506\) −5232.29 −0.459691
\(507\) 648.239i 0.0567836i
\(508\) − 1363.91i − 0.119121i
\(509\) −13597.4 −1.18408 −0.592039 0.805910i \(-0.701677\pi\)
−0.592039 + 0.805910i \(0.701677\pi\)
\(510\) 0 0
\(511\) −770.969 −0.0667430
\(512\) 12992.6i 1.12148i
\(513\) 3519.56i 0.302909i
\(514\) 19756.6 1.69538
\(515\) 0 0
\(516\) 2216.50 0.189101
\(517\) − 6130.25i − 0.521486i
\(518\) − 6670.30i − 0.565784i
\(519\) 3075.88 0.260146
\(520\) 0 0
\(521\) 14001.3 1.17736 0.588682 0.808364i \(-0.299647\pi\)
0.588682 + 0.808364i \(0.299647\pi\)
\(522\) 205.346i 0.0172180i
\(523\) 14749.8i 1.23320i 0.787275 + 0.616602i \(0.211491\pi\)
−0.787275 + 0.616602i \(0.788509\pi\)
\(524\) −2135.86 −0.178064
\(525\) 0 0
\(526\) −528.568 −0.0438149
\(527\) 435.159i 0.0359693i
\(528\) − 1663.97i − 0.137150i
\(529\) −22315.0 −1.83406
\(530\) 0 0
\(531\) 5568.95 0.455126
\(532\) 1171.21i 0.0954483i
\(533\) − 14662.2i − 1.19154i
\(534\) −1090.29 −0.0883548
\(535\) 0 0
\(536\) −8196.40 −0.660505
\(537\) 4974.86i 0.399779i
\(538\) 4386.59i 0.351523i
\(539\) −3343.83 −0.267215
\(540\) 0 0
\(541\) 1484.06 0.117939 0.0589694 0.998260i \(-0.481219\pi\)
0.0589694 + 0.998260i \(0.481219\pi\)
\(542\) 1222.07i 0.0968493i
\(543\) 6065.42i 0.479359i
\(544\) 5314.40 0.418847
\(545\) 0 0
\(546\) 2357.91 0.184815
\(547\) − 16562.2i − 1.29460i −0.762234 0.647302i \(-0.775897\pi\)
0.762234 0.647302i \(-0.224103\pi\)
\(548\) 985.233i 0.0768012i
\(549\) −7077.65 −0.550212
\(550\) 0 0
\(551\) 1161.09 0.0897716
\(552\) − 13468.6i − 1.03851i
\(553\) 1935.30i 0.148819i
\(554\) 10972.3 0.841463
\(555\) 0 0
\(556\) −1194.61 −0.0911197
\(557\) − 8821.52i − 0.671059i −0.942030 0.335529i \(-0.891085\pi\)
0.942030 0.335529i \(-0.108915\pi\)
\(558\) 121.295i 0.00920223i
\(559\) −25231.2 −1.90906
\(560\) 0 0
\(561\) −2729.37 −0.205409
\(562\) − 8908.55i − 0.668656i
\(563\) 5985.53i 0.448064i 0.974582 + 0.224032i \(0.0719220\pi\)
−0.974582 + 0.224032i \(0.928078\pi\)
\(564\) 2404.92 0.179549
\(565\) 0 0
\(566\) 16825.0 1.24948
\(567\) 505.943i 0.0374737i
\(568\) − 27095.5i − 2.00158i
\(569\) 3453.08 0.254413 0.127206 0.991876i \(-0.459399\pi\)
0.127206 + 0.991876i \(0.459399\pi\)
\(570\) 0 0
\(571\) −21484.5 −1.57460 −0.787302 0.616568i \(-0.788523\pi\)
−0.787302 + 0.616568i \(0.788523\pi\)
\(572\) − 777.271i − 0.0568170i
\(573\) − 4320.67i − 0.315006i
\(574\) −4775.67 −0.347269
\(575\) 0 0
\(576\) 5111.80 0.369777
\(577\) − 13294.4i − 0.959189i −0.877490 0.479594i \(-0.840784\pi\)
0.877490 0.479594i \(-0.159216\pi\)
\(578\) 4937.82i 0.355340i
\(579\) 8394.14 0.602502
\(580\) 0 0
\(581\) −6378.77 −0.455483
\(582\) − 6135.01i − 0.436949i
\(583\) − 1848.71i − 0.131330i
\(584\) 2984.18 0.211449
\(585\) 0 0
\(586\) −21394.8 −1.50821
\(587\) 6695.73i 0.470805i 0.971898 + 0.235402i \(0.0756408\pi\)
−0.971898 + 0.235402i \(0.924359\pi\)
\(588\) − 1311.80i − 0.0920028i
\(589\) 685.841 0.0479789
\(590\) 0 0
\(591\) 1376.83 0.0958295
\(592\) 21021.2i 1.45940i
\(593\) 10239.6i 0.709088i 0.935039 + 0.354544i \(0.115364\pi\)
−0.935039 + 0.354544i \(0.884636\pi\)
\(594\) −760.781 −0.0525509
\(595\) 0 0
\(596\) 1745.76 0.119982
\(597\) − 7113.11i − 0.487639i
\(598\) 23366.0i 1.59784i
\(599\) 23890.8 1.62963 0.814817 0.579719i \(-0.196838\pi\)
0.814817 + 0.579719i \(0.196838\pi\)
\(600\) 0 0
\(601\) −11343.8 −0.769920 −0.384960 0.922933i \(-0.625785\pi\)
−0.384960 + 0.922933i \(0.625785\pi\)
\(602\) 8218.12i 0.556388i
\(603\) 3051.14i 0.206056i
\(604\) 44.3712 0.00298914
\(605\) 0 0
\(606\) −1855.07 −0.124352
\(607\) − 26032.5i − 1.74074i −0.492399 0.870369i \(-0.663880\pi\)
0.492399 0.870369i \(-0.336120\pi\)
\(608\) − 8375.87i − 0.558695i
\(609\) 166.909 0.0111059
\(610\) 0 0
\(611\) −27376.1 −1.81263
\(612\) − 1070.74i − 0.0707226i
\(613\) 4568.13i 0.300987i 0.988611 + 0.150493i \(0.0480863\pi\)
−0.988611 + 0.150493i \(0.951914\pi\)
\(614\) 13791.1 0.906457
\(615\) 0 0
\(616\) −1661.17 −0.108653
\(617\) − 12755.9i − 0.832308i −0.909294 0.416154i \(-0.863378\pi\)
0.909294 0.416154i \(-0.136622\pi\)
\(618\) 8978.59i 0.584421i
\(619\) 1138.94 0.0739545 0.0369772 0.999316i \(-0.488227\pi\)
0.0369772 + 0.999316i \(0.488227\pi\)
\(620\) 0 0
\(621\) −5013.72 −0.323983
\(622\) 4586.24i 0.295645i
\(623\) 886.205i 0.0569904i
\(624\) −7430.85 −0.476718
\(625\) 0 0
\(626\) 2072.49 0.132322
\(627\) 4301.69i 0.273992i
\(628\) 496.607i 0.0315554i
\(629\) 34480.6 2.18574
\(630\) 0 0
\(631\) 7997.36 0.504548 0.252274 0.967656i \(-0.418822\pi\)
0.252274 + 0.967656i \(0.418822\pi\)
\(632\) − 7490.91i − 0.471475i
\(633\) − 12959.6i − 0.813742i
\(634\) 27522.7 1.72408
\(635\) 0 0
\(636\) 725.255 0.0452173
\(637\) 14932.7i 0.928814i
\(638\) 250.979i 0.0155742i
\(639\) −10086.4 −0.624430
\(640\) 0 0
\(641\) −573.115 −0.0353146 −0.0176573 0.999844i \(-0.505621\pi\)
−0.0176573 + 0.999844i \(0.505621\pi\)
\(642\) 16190.2i 0.995290i
\(643\) 16027.8i 0.983009i 0.870875 + 0.491504i \(0.163553\pi\)
−0.870875 + 0.491504i \(0.836447\pi\)
\(644\) −1668.42 −0.102089
\(645\) 0 0
\(646\) 27617.1 1.68201
\(647\) − 2622.74i − 0.159367i −0.996820 0.0796837i \(-0.974609\pi\)
0.996820 0.0796837i \(-0.0253910\pi\)
\(648\) − 1958.34i − 0.118721i
\(649\) 6806.50 0.411677
\(650\) 0 0
\(651\) 98.5908 0.00593560
\(652\) − 2763.97i − 0.166020i
\(653\) − 3102.00i − 0.185897i −0.995671 0.0929484i \(-0.970371\pi\)
0.995671 0.0929484i \(-0.0296292\pi\)
\(654\) 3794.62 0.226883
\(655\) 0 0
\(656\) 15050.3 0.895755
\(657\) − 1110.87i − 0.0659652i
\(658\) 8916.73i 0.528283i
\(659\) 20840.2 1.23190 0.615948 0.787787i \(-0.288773\pi\)
0.615948 + 0.787787i \(0.288773\pi\)
\(660\) 0 0
\(661\) 18242.9 1.07348 0.536738 0.843749i \(-0.319657\pi\)
0.536738 + 0.843749i \(0.319657\pi\)
\(662\) − 8707.03i − 0.511191i
\(663\) 12188.7i 0.713980i
\(664\) 24690.2 1.44302
\(665\) 0 0
\(666\) 9611.06 0.559191
\(667\) 1654.01i 0.0960171i
\(668\) 247.841i 0.0143551i
\(669\) 11513.2 0.665361
\(670\) 0 0
\(671\) −8650.46 −0.497686
\(672\) − 1204.05i − 0.0691177i
\(673\) 12746.6i 0.730084i 0.930991 + 0.365042i \(0.118945\pi\)
−0.930991 + 0.365042i \(0.881055\pi\)
\(674\) −30328.7 −1.73326
\(675\) 0 0
\(676\) −310.819 −0.0176843
\(677\) − 7683.11i − 0.436168i −0.975930 0.218084i \(-0.930019\pi\)
0.975930 0.218084i \(-0.0699807\pi\)
\(678\) 1306.39i 0.0739995i
\(679\) −4986.63 −0.281840
\(680\) 0 0
\(681\) −15011.1 −0.844681
\(682\) 148.250i 0.00832373i
\(683\) 21397.1i 1.19874i 0.800473 + 0.599368i \(0.204582\pi\)
−0.800473 + 0.599368i \(0.795418\pi\)
\(684\) −1687.57 −0.0943359
\(685\) 0 0
\(686\) 10351.8 0.576140
\(687\) − 832.124i − 0.0462118i
\(688\) − 25899.0i − 1.43516i
\(689\) −8255.84 −0.456491
\(690\) 0 0
\(691\) 26137.5 1.43895 0.719477 0.694516i \(-0.244381\pi\)
0.719477 + 0.694516i \(0.244381\pi\)
\(692\) 1474.83i 0.0810181i
\(693\) 618.375i 0.0338963i
\(694\) −5319.50 −0.290959
\(695\) 0 0
\(696\) −646.051 −0.0351846
\(697\) − 24686.7i − 1.34157i
\(698\) 14952.7i 0.810845i
\(699\) −6809.72 −0.368479
\(700\) 0 0
\(701\) 13382.4 0.721036 0.360518 0.932752i \(-0.382600\pi\)
0.360518 + 0.932752i \(0.382600\pi\)
\(702\) 3397.45i 0.182662i
\(703\) − 54343.9i − 2.91553i
\(704\) 6247.76 0.334476
\(705\) 0 0
\(706\) 6208.27 0.330951
\(707\) 1507.83i 0.0802092i
\(708\) 2670.22i 0.141741i
\(709\) −18164.6 −0.962179 −0.481090 0.876671i \(-0.659759\pi\)
−0.481090 + 0.876671i \(0.659759\pi\)
\(710\) 0 0
\(711\) −2788.52 −0.147085
\(712\) − 3430.21i − 0.180552i
\(713\) 977.000i 0.0513169i
\(714\) 3970.00 0.208086
\(715\) 0 0
\(716\) −2385.36 −0.124504
\(717\) − 4851.32i − 0.252686i
\(718\) − 9944.50i − 0.516888i
\(719\) −9665.62 −0.501344 −0.250672 0.968072i \(-0.580652\pi\)
−0.250672 + 0.968072i \(0.580652\pi\)
\(720\) 0 0
\(721\) 7297.94 0.376962
\(722\) − 25956.7i − 1.33796i
\(723\) − 16939.9i − 0.871371i
\(724\) −2908.26 −0.149288
\(725\) 0 0
\(726\) −929.844 −0.0475341
\(727\) − 29779.6i − 1.51921i −0.650385 0.759605i \(-0.725392\pi\)
0.650385 0.759605i \(-0.274608\pi\)
\(728\) 7418.33i 0.377667i
\(729\) −729.000 −0.0370370
\(730\) 0 0
\(731\) −42481.7 −2.14944
\(732\) − 3393.61i − 0.171354i
\(733\) − 35029.5i − 1.76513i −0.470187 0.882567i \(-0.655814\pi\)
0.470187 0.882567i \(-0.344186\pi\)
\(734\) 14515.4 0.729937
\(735\) 0 0
\(736\) 11931.7 0.597564
\(737\) 3729.17i 0.186385i
\(738\) − 6881.13i − 0.343222i
\(739\) −23297.3 −1.15968 −0.579842 0.814729i \(-0.696886\pi\)
−0.579842 + 0.814729i \(0.696886\pi\)
\(740\) 0 0
\(741\) 19210.2 0.952368
\(742\) 2689.03i 0.133042i
\(743\) − 21570.4i − 1.06506i −0.846411 0.532530i \(-0.821241\pi\)
0.846411 0.532530i \(-0.178759\pi\)
\(744\) −381.613 −0.0188046
\(745\) 0 0
\(746\) −447.684 −0.0219717
\(747\) − 9190.99i − 0.450175i
\(748\) − 1308.69i − 0.0639710i
\(749\) 13159.6 0.641980
\(750\) 0 0
\(751\) 28554.8 1.38746 0.693729 0.720236i \(-0.255966\pi\)
0.693729 + 0.720236i \(0.255966\pi\)
\(752\) − 28100.7i − 1.36267i
\(753\) 18652.8i 0.902719i
\(754\) 1120.81 0.0541344
\(755\) 0 0
\(756\) −242.591 −0.0116706
\(757\) 7812.81i 0.375114i 0.982254 + 0.187557i \(0.0600570\pi\)
−0.982254 + 0.187557i \(0.939943\pi\)
\(758\) 647.268i 0.0310156i
\(759\) −6127.87 −0.293054
\(760\) 0 0
\(761\) 2875.13 0.136956 0.0684778 0.997653i \(-0.478186\pi\)
0.0684778 + 0.997653i \(0.478186\pi\)
\(762\) 7286.45i 0.346405i
\(763\) − 3084.33i − 0.146344i
\(764\) 2071.69 0.0981033
\(765\) 0 0
\(766\) 28214.3 1.33084
\(767\) − 30396.0i − 1.43095i
\(768\) 6401.22i 0.300761i
\(769\) 27657.7 1.29696 0.648479 0.761233i \(-0.275406\pi\)
0.648479 + 0.761233i \(0.275406\pi\)
\(770\) 0 0
\(771\) 23138.2 1.08081
\(772\) 4024.84i 0.187639i
\(773\) 3929.35i 0.182832i 0.995813 + 0.0914160i \(0.0291393\pi\)
−0.995813 + 0.0914160i \(0.970861\pi\)
\(774\) −11841.3 −0.549904
\(775\) 0 0
\(776\) 19301.6 0.892898
\(777\) − 7812.02i − 0.360688i
\(778\) 20747.0i 0.956064i
\(779\) −38908.0 −1.78950
\(780\) 0 0
\(781\) −12327.8 −0.564818
\(782\) 39341.3i 1.79903i
\(783\) 240.495i 0.0109765i
\(784\) −15327.9 −0.698247
\(785\) 0 0
\(786\) 11410.5 0.517809
\(787\) − 21125.7i − 0.956860i −0.878126 0.478430i \(-0.841206\pi\)
0.878126 0.478430i \(-0.158794\pi\)
\(788\) 660.166i 0.0298445i
\(789\) −619.040 −0.0279321
\(790\) 0 0
\(791\) 1061.86 0.0477310
\(792\) − 2393.53i − 0.107387i
\(793\) 38630.7i 1.72991i
\(794\) −1087.00 −0.0485847
\(795\) 0 0
\(796\) 3410.61 0.151867
\(797\) 11696.3i 0.519828i 0.965632 + 0.259914i \(0.0836942\pi\)
−0.965632 + 0.259914i \(0.916306\pi\)
\(798\) − 6257.00i − 0.277563i
\(799\) −46093.0 −2.04087
\(800\) 0 0
\(801\) −1276.91 −0.0563263
\(802\) 15123.9i 0.665888i
\(803\) − 1357.73i − 0.0596678i
\(804\) −1462.97 −0.0641727
\(805\) 0 0
\(806\) 662.045 0.0289324
\(807\) 5137.42i 0.224096i
\(808\) − 5836.34i − 0.254111i
\(809\) −14310.2 −0.621902 −0.310951 0.950426i \(-0.600648\pi\)
−0.310951 + 0.950426i \(0.600648\pi\)
\(810\) 0 0
\(811\) 21697.9 0.939477 0.469739 0.882806i \(-0.344348\pi\)
0.469739 + 0.882806i \(0.344348\pi\)
\(812\) 80.0299i 0.00345874i
\(813\) 1431.24i 0.0617416i
\(814\) 11746.9 0.505807
\(815\) 0 0
\(816\) −12511.3 −0.536743
\(817\) 66954.1i 2.86711i
\(818\) 3510.54i 0.150053i
\(819\) 2761.50 0.117820
\(820\) 0 0
\(821\) 3613.00 0.153587 0.0767934 0.997047i \(-0.475532\pi\)
0.0767934 + 0.997047i \(0.475532\pi\)
\(822\) − 5263.44i − 0.223338i
\(823\) 4763.98i 0.201776i 0.994898 + 0.100888i \(0.0321684\pi\)
−0.994898 + 0.100888i \(0.967832\pi\)
\(824\) −28248.0 −1.19425
\(825\) 0 0
\(826\) −9900.36 −0.417043
\(827\) − 33571.7i − 1.41161i −0.708405 0.705806i \(-0.750585\pi\)
0.708405 0.705806i \(-0.249415\pi\)
\(828\) − 2403.99i − 0.100899i
\(829\) −17980.5 −0.753303 −0.376652 0.926355i \(-0.622925\pi\)
−0.376652 + 0.926355i \(0.622925\pi\)
\(830\) 0 0
\(831\) 12850.4 0.536434
\(832\) − 27900.9i − 1.16261i
\(833\) 25142.1i 1.04576i
\(834\) 6381.98 0.264976
\(835\) 0 0
\(836\) −2062.58 −0.0853301
\(837\) 142.057i 0.00586643i
\(838\) 3249.91i 0.133969i
\(839\) 40139.5 1.65169 0.825847 0.563895i \(-0.190698\pi\)
0.825847 + 0.563895i \(0.190698\pi\)
\(840\) 0 0
\(841\) −24309.7 −0.996747
\(842\) 31358.4i 1.28347i
\(843\) − 10433.4i − 0.426269i
\(844\) 6213.91 0.253426
\(845\) 0 0
\(846\) −12847.9 −0.522127
\(847\) 755.792i 0.0306603i
\(848\) − 8474.36i − 0.343173i
\(849\) 19704.8 0.796546
\(850\) 0 0
\(851\) 77414.4 3.11837
\(852\) − 4836.24i − 0.194468i
\(853\) 15369.2i 0.616919i 0.951237 + 0.308459i \(0.0998134\pi\)
−0.951237 + 0.308459i \(0.900187\pi\)
\(854\) 12582.5 0.504173
\(855\) 0 0
\(856\) −50936.8 −2.03386
\(857\) 10324.9i 0.411541i 0.978600 + 0.205770i \(0.0659700\pi\)
−0.978600 + 0.205770i \(0.934030\pi\)
\(858\) 4152.44i 0.165224i
\(859\) 27112.5 1.07691 0.538455 0.842655i \(-0.319008\pi\)
0.538455 + 0.842655i \(0.319008\pi\)
\(860\) 0 0
\(861\) −5593.09 −0.221384
\(862\) − 20620.8i − 0.814787i
\(863\) − 30463.6i − 1.20161i −0.799394 0.600807i \(-0.794846\pi\)
0.799394 0.600807i \(-0.205154\pi\)
\(864\) 1734.88 0.0683122
\(865\) 0 0
\(866\) 42434.0 1.66509
\(867\) 5783.00i 0.226529i
\(868\) 47.2726i 0.00184854i
\(869\) −3408.19 −0.133044
\(870\) 0 0
\(871\) 16653.5 0.647855
\(872\) 11938.4i 0.463631i
\(873\) − 7185.10i − 0.278555i
\(874\) 62004.6 2.39970
\(875\) 0 0
\(876\) 532.642 0.0205437
\(877\) − 5086.12i − 0.195833i −0.995195 0.0979167i \(-0.968782\pi\)
0.995195 0.0979167i \(-0.0312179\pi\)
\(878\) 12054.2i 0.463335i
\(879\) −25056.9 −0.961487
\(880\) 0 0
\(881\) −10625.5 −0.406338 −0.203169 0.979144i \(-0.565124\pi\)
−0.203169 + 0.979144i \(0.565124\pi\)
\(882\) 7008.06i 0.267544i
\(883\) − 13112.2i − 0.499728i −0.968281 0.249864i \(-0.919614\pi\)
0.968281 0.249864i \(-0.0803859\pi\)
\(884\) −5844.25 −0.222357
\(885\) 0 0
\(886\) −38664.0 −1.46607
\(887\) − 14442.8i − 0.546719i −0.961912 0.273360i \(-0.911865\pi\)
0.961912 0.273360i \(-0.0881349\pi\)
\(888\) 30237.8i 1.14270i
\(889\) 5922.54 0.223437
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) 5520.38i 0.207215i
\(893\) 72645.8i 2.72228i
\(894\) −9326.43 −0.348906
\(895\) 0 0
\(896\) −5876.86 −0.219121
\(897\) 27365.5i 1.01863i
\(898\) 2493.61i 0.0926648i
\(899\) 46.8641 0.00173860
\(900\) 0 0
\(901\) −13900.3 −0.513970
\(902\) − 8410.27i − 0.310456i
\(903\) 9624.77i 0.354698i
\(904\) −4110.10 −0.151217
\(905\) 0 0
\(906\) −237.045 −0.00869239
\(907\) − 44981.9i − 1.64675i −0.567499 0.823374i \(-0.692089\pi\)
0.567499 0.823374i \(-0.307911\pi\)
\(908\) − 7197.57i − 0.263062i
\(909\) −2172.60 −0.0792745
\(910\) 0 0
\(911\) 6841.96 0.248830 0.124415 0.992230i \(-0.460295\pi\)
0.124415 + 0.992230i \(0.460295\pi\)
\(912\) 19718.7i 0.715954i
\(913\) − 11233.4i − 0.407199i
\(914\) 160.501 0.00580843
\(915\) 0 0
\(916\) 398.989 0.0143919
\(917\) − 9274.61i − 0.333996i
\(918\) 5720.27i 0.205661i
\(919\) 4753.54 0.170625 0.0853127 0.996354i \(-0.472811\pi\)
0.0853127 + 0.996354i \(0.472811\pi\)
\(920\) 0 0
\(921\) 16151.7 0.577868
\(922\) 30395.9i 1.08572i
\(923\) 55052.7i 1.96325i
\(924\) −296.500 −0.0105564
\(925\) 0 0
\(926\) 33671.2 1.19493
\(927\) 10515.4i 0.372569i
\(928\) − 572.330i − 0.0202453i
\(929\) 7507.93 0.265153 0.132576 0.991173i \(-0.457675\pi\)
0.132576 + 0.991173i \(0.457675\pi\)
\(930\) 0 0
\(931\) 39625.7 1.39493
\(932\) − 3265.14i − 0.114757i
\(933\) 5371.24i 0.188474i
\(934\) −26068.3 −0.913256
\(935\) 0 0
\(936\) −10688.9 −0.373266
\(937\) 8540.47i 0.297764i 0.988855 + 0.148882i \(0.0475675\pi\)
−0.988855 + 0.148882i \(0.952433\pi\)
\(938\) − 5424.24i − 0.188814i
\(939\) 2427.23 0.0843552
\(940\) 0 0
\(941\) 9101.13 0.315290 0.157645 0.987496i \(-0.449610\pi\)
0.157645 + 0.987496i \(0.449610\pi\)
\(942\) − 2653.04i − 0.0917630i
\(943\) − 55425.5i − 1.91400i
\(944\) 31200.6 1.07573
\(945\) 0 0
\(946\) −14472.7 −0.497407
\(947\) 47540.0i 1.63130i 0.578546 + 0.815650i \(0.303620\pi\)
−0.578546 + 0.815650i \(0.696380\pi\)
\(948\) − 1337.04i − 0.0458072i
\(949\) −6063.26 −0.207399
\(950\) 0 0
\(951\) 32233.6 1.09910
\(952\) 12490.2i 0.425221i
\(953\) − 47370.7i − 1.61016i −0.593164 0.805082i \(-0.702121\pi\)
0.593164 0.805082i \(-0.297879\pi\)
\(954\) −3874.55 −0.131492
\(955\) 0 0
\(956\) 2326.12 0.0786947
\(957\) 293.938i 0.00992859i
\(958\) 8789.33i 0.296420i
\(959\) −4278.20 −0.144057
\(960\) 0 0
\(961\) −29763.3 −0.999071
\(962\) − 52458.4i − 1.75813i
\(963\) 18961.4i 0.634498i
\(964\) 8122.38 0.271374
\(965\) 0 0
\(966\) 8913.27 0.296874
\(967\) − 36171.6i − 1.20290i −0.798912 0.601448i \(-0.794591\pi\)
0.798912 0.601448i \(-0.205409\pi\)
\(968\) − 2925.43i − 0.0971351i
\(969\) 32344.1 1.07228
\(970\) 0 0
\(971\) −31713.2 −1.04812 −0.524060 0.851681i \(-0.675583\pi\)
−0.524060 + 0.851681i \(0.675583\pi\)
\(972\) − 349.543i − 0.0115346i
\(973\) − 5187.37i − 0.170914i
\(974\) −7257.40 −0.238750
\(975\) 0 0
\(976\) −39653.1 −1.30048
\(977\) 22800.5i 0.746626i 0.927706 + 0.373313i \(0.121778\pi\)
−0.927706 + 0.373313i \(0.878222\pi\)
\(978\) 14766.0i 0.482786i
\(979\) −1560.67 −0.0509490
\(980\) 0 0
\(981\) 4444.12 0.144638
\(982\) 6832.41i 0.222027i
\(983\) 44597.4i 1.44704i 0.690305 + 0.723518i \(0.257476\pi\)
−0.690305 + 0.723518i \(0.742524\pi\)
\(984\) 21649.1 0.701369
\(985\) 0 0
\(986\) 1887.10 0.0609507
\(987\) 10443.0i 0.336781i
\(988\) 9210.95i 0.296599i
\(989\) −95378.1 −3.06658
\(990\) 0 0
\(991\) −34788.1 −1.11512 −0.557558 0.830138i \(-0.688261\pi\)
−0.557558 + 0.830138i \(0.688261\pi\)
\(992\) − 338.068i − 0.0108202i
\(993\) − 10197.4i − 0.325885i
\(994\) 17931.3 0.572180
\(995\) 0 0
\(996\) 4406.92 0.140199
\(997\) − 20360.5i − 0.646765i −0.946268 0.323383i \(-0.895180\pi\)
0.946268 0.323383i \(-0.104820\pi\)
\(998\) − 28528.0i − 0.904849i
\(999\) 11256.1 0.356485
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.j.199.1 4
5.2 odd 4 825.4.a.m.1.2 2
5.3 odd 4 165.4.a.c.1.1 2
5.4 even 2 inner 825.4.c.j.199.4 4
15.2 even 4 2475.4.a.n.1.1 2
15.8 even 4 495.4.a.d.1.2 2
55.43 even 4 1815.4.a.n.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.c.1.1 2 5.3 odd 4
495.4.a.d.1.2 2 15.8 even 4
825.4.a.m.1.2 2 5.2 odd 4
825.4.c.j.199.1 4 1.1 even 1 trivial
825.4.c.j.199.4 4 5.4 even 2 inner
1815.4.a.n.1.2 2 55.43 even 4
2475.4.a.n.1.1 2 15.2 even 4