Properties

Label 825.4.a.j.1.1
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $0$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [825,4,Mod(1,825)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(825, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("825.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 825 = 3 \cdot 5^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 825.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(48.6765757547\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 825.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.00000 q^{2} +3.00000 q^{3} +17.0000 q^{4} +15.0000 q^{6} +3.00000 q^{7} +45.0000 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q+5.00000 q^{2} +3.00000 q^{3} +17.0000 q^{4} +15.0000 q^{6} +3.00000 q^{7} +45.0000 q^{8} +9.00000 q^{9} -11.0000 q^{11} +51.0000 q^{12} +32.0000 q^{13} +15.0000 q^{14} +89.0000 q^{16} +33.0000 q^{17} +45.0000 q^{18} +47.0000 q^{19} +9.00000 q^{21} -55.0000 q^{22} +113.000 q^{23} +135.000 q^{24} +160.000 q^{26} +27.0000 q^{27} +51.0000 q^{28} -54.0000 q^{29} +178.000 q^{31} +85.0000 q^{32} -33.0000 q^{33} +165.000 q^{34} +153.000 q^{36} +19.0000 q^{37} +235.000 q^{38} +96.0000 q^{39} +139.000 q^{41} +45.0000 q^{42} -308.000 q^{43} -187.000 q^{44} +565.000 q^{46} +195.000 q^{47} +267.000 q^{48} -334.000 q^{49} +99.0000 q^{51} +544.000 q^{52} +152.000 q^{53} +135.000 q^{54} +135.000 q^{56} +141.000 q^{57} -270.000 q^{58} -625.000 q^{59} +320.000 q^{61} +890.000 q^{62} +27.0000 q^{63} -287.000 q^{64} -165.000 q^{66} +200.000 q^{67} +561.000 q^{68} +339.000 q^{69} -947.000 q^{71} +405.000 q^{72} -448.000 q^{73} +95.0000 q^{74} +799.000 q^{76} -33.0000 q^{77} +480.000 q^{78} -721.000 q^{79} +81.0000 q^{81} +695.000 q^{82} +142.000 q^{83} +153.000 q^{84} -1540.00 q^{86} -162.000 q^{87} -495.000 q^{88} +404.000 q^{89} +96.0000 q^{91} +1921.00 q^{92} +534.000 q^{93} +975.000 q^{94} +255.000 q^{96} +79.0000 q^{97} -1670.00 q^{98} -99.0000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 5.00000 1.76777 0.883883 0.467707i \(-0.154920\pi\)
0.883883 + 0.467707i \(0.154920\pi\)
\(3\) 3.00000 0.577350
\(4\) 17.0000 2.12500
\(5\) 0 0
\(6\) 15.0000 1.02062
\(7\) 3.00000 0.161985 0.0809924 0.996715i \(-0.474191\pi\)
0.0809924 + 0.996715i \(0.474191\pi\)
\(8\) 45.0000 1.98874
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) −11.0000 −0.301511
\(12\) 51.0000 1.22687
\(13\) 32.0000 0.682708 0.341354 0.939935i \(-0.389115\pi\)
0.341354 + 0.939935i \(0.389115\pi\)
\(14\) 15.0000 0.286351
\(15\) 0 0
\(16\) 89.0000 1.39062
\(17\) 33.0000 0.470804 0.235402 0.971898i \(-0.424359\pi\)
0.235402 + 0.971898i \(0.424359\pi\)
\(18\) 45.0000 0.589256
\(19\) 47.0000 0.567502 0.283751 0.958898i \(-0.408421\pi\)
0.283751 + 0.958898i \(0.408421\pi\)
\(20\) 0 0
\(21\) 9.00000 0.0935220
\(22\) −55.0000 −0.533002
\(23\) 113.000 1.02444 0.512220 0.858854i \(-0.328823\pi\)
0.512220 + 0.858854i \(0.328823\pi\)
\(24\) 135.000 1.14820
\(25\) 0 0
\(26\) 160.000 1.20687
\(27\) 27.0000 0.192450
\(28\) 51.0000 0.344218
\(29\) −54.0000 −0.345778 −0.172889 0.984941i \(-0.555310\pi\)
−0.172889 + 0.984941i \(0.555310\pi\)
\(30\) 0 0
\(31\) 178.000 1.03128 0.515641 0.856805i \(-0.327554\pi\)
0.515641 + 0.856805i \(0.327554\pi\)
\(32\) 85.0000 0.469563
\(33\) −33.0000 −0.174078
\(34\) 165.000 0.832273
\(35\) 0 0
\(36\) 153.000 0.708333
\(37\) 19.0000 0.0844211 0.0422106 0.999109i \(-0.486560\pi\)
0.0422106 + 0.999109i \(0.486560\pi\)
\(38\) 235.000 1.00321
\(39\) 96.0000 0.394162
\(40\) 0 0
\(41\) 139.000 0.529467 0.264734 0.964322i \(-0.414716\pi\)
0.264734 + 0.964322i \(0.414716\pi\)
\(42\) 45.0000 0.165325
\(43\) −308.000 −1.09232 −0.546158 0.837682i \(-0.683910\pi\)
−0.546158 + 0.837682i \(0.683910\pi\)
\(44\) −187.000 −0.640712
\(45\) 0 0
\(46\) 565.000 1.81097
\(47\) 195.000 0.605185 0.302592 0.953120i \(-0.402148\pi\)
0.302592 + 0.953120i \(0.402148\pi\)
\(48\) 267.000 0.802878
\(49\) −334.000 −0.973761
\(50\) 0 0
\(51\) 99.0000 0.271819
\(52\) 544.000 1.45075
\(53\) 152.000 0.393940 0.196970 0.980410i \(-0.436890\pi\)
0.196970 + 0.980410i \(0.436890\pi\)
\(54\) 135.000 0.340207
\(55\) 0 0
\(56\) 135.000 0.322145
\(57\) 141.000 0.327647
\(58\) −270.000 −0.611254
\(59\) −625.000 −1.37912 −0.689560 0.724229i \(-0.742196\pi\)
−0.689560 + 0.724229i \(0.742196\pi\)
\(60\) 0 0
\(61\) 320.000 0.671669 0.335834 0.941921i \(-0.390982\pi\)
0.335834 + 0.941921i \(0.390982\pi\)
\(62\) 890.000 1.82307
\(63\) 27.0000 0.0539949
\(64\) −287.000 −0.560547
\(65\) 0 0
\(66\) −165.000 −0.307729
\(67\) 200.000 0.364685 0.182342 0.983235i \(-0.441632\pi\)
0.182342 + 0.983235i \(0.441632\pi\)
\(68\) 561.000 1.00046
\(69\) 339.000 0.591461
\(70\) 0 0
\(71\) −947.000 −1.58293 −0.791466 0.611213i \(-0.790682\pi\)
−0.791466 + 0.611213i \(0.790682\pi\)
\(72\) 405.000 0.662913
\(73\) −448.000 −0.718280 −0.359140 0.933284i \(-0.616930\pi\)
−0.359140 + 0.933284i \(0.616930\pi\)
\(74\) 95.0000 0.149237
\(75\) 0 0
\(76\) 799.000 1.20594
\(77\) −33.0000 −0.0488402
\(78\) 480.000 0.696786
\(79\) −721.000 −1.02682 −0.513410 0.858143i \(-0.671618\pi\)
−0.513410 + 0.858143i \(0.671618\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 695.000 0.935975
\(83\) 142.000 0.187789 0.0938947 0.995582i \(-0.470068\pi\)
0.0938947 + 0.995582i \(0.470068\pi\)
\(84\) 153.000 0.198734
\(85\) 0 0
\(86\) −1540.00 −1.93096
\(87\) −162.000 −0.199635
\(88\) −495.000 −0.599627
\(89\) 404.000 0.481168 0.240584 0.970628i \(-0.422661\pi\)
0.240584 + 0.970628i \(0.422661\pi\)
\(90\) 0 0
\(91\) 96.0000 0.110588
\(92\) 1921.00 2.17694
\(93\) 534.000 0.595411
\(94\) 975.000 1.06983
\(95\) 0 0
\(96\) 255.000 0.271102
\(97\) 79.0000 0.0826931 0.0413466 0.999145i \(-0.486835\pi\)
0.0413466 + 0.999145i \(0.486835\pi\)
\(98\) −1670.00 −1.72138
\(99\) −99.0000 −0.100504
\(100\) 0 0
\(101\) −545.000 −0.536926 −0.268463 0.963290i \(-0.586516\pi\)
−0.268463 + 0.963290i \(0.586516\pi\)
\(102\) 495.000 0.480513
\(103\) −1306.00 −1.24936 −0.624680 0.780881i \(-0.714770\pi\)
−0.624680 + 0.780881i \(0.714770\pi\)
\(104\) 1440.00 1.35773
\(105\) 0 0
\(106\) 760.000 0.696394
\(107\) 1938.00 1.75097 0.875484 0.483247i \(-0.160543\pi\)
0.875484 + 0.483247i \(0.160543\pi\)
\(108\) 459.000 0.408956
\(109\) −576.000 −0.506154 −0.253077 0.967446i \(-0.581443\pi\)
−0.253077 + 0.967446i \(0.581443\pi\)
\(110\) 0 0
\(111\) 57.0000 0.0487405
\(112\) 267.000 0.225260
\(113\) −1104.00 −0.919076 −0.459538 0.888158i \(-0.651985\pi\)
−0.459538 + 0.888158i \(0.651985\pi\)
\(114\) 705.000 0.579204
\(115\) 0 0
\(116\) −918.000 −0.734777
\(117\) 288.000 0.227569
\(118\) −3125.00 −2.43796
\(119\) 99.0000 0.0762632
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) 1600.00 1.18735
\(123\) 417.000 0.305688
\(124\) 3026.00 2.19147
\(125\) 0 0
\(126\) 135.000 0.0954504
\(127\) −1739.00 −1.21505 −0.607525 0.794301i \(-0.707837\pi\)
−0.607525 + 0.794301i \(0.707837\pi\)
\(128\) −2115.00 −1.46048
\(129\) −924.000 −0.630649
\(130\) 0 0
\(131\) 1818.00 1.21251 0.606257 0.795269i \(-0.292670\pi\)
0.606257 + 0.795269i \(0.292670\pi\)
\(132\) −561.000 −0.369915
\(133\) 141.000 0.0919267
\(134\) 1000.00 0.644678
\(135\) 0 0
\(136\) 1485.00 0.936307
\(137\) 870.000 0.542548 0.271274 0.962502i \(-0.412555\pi\)
0.271274 + 0.962502i \(0.412555\pi\)
\(138\) 1695.00 1.04557
\(139\) −636.000 −0.388092 −0.194046 0.980992i \(-0.562161\pi\)
−0.194046 + 0.980992i \(0.562161\pi\)
\(140\) 0 0
\(141\) 585.000 0.349403
\(142\) −4735.00 −2.79826
\(143\) −352.000 −0.205844
\(144\) 801.000 0.463542
\(145\) 0 0
\(146\) −2240.00 −1.26975
\(147\) −1002.00 −0.562201
\(148\) 323.000 0.179395
\(149\) −239.000 −0.131407 −0.0657035 0.997839i \(-0.520929\pi\)
−0.0657035 + 0.997839i \(0.520929\pi\)
\(150\) 0 0
\(151\) 1208.00 0.651031 0.325515 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 2115.00 1.12861
\(153\) 297.000 0.156935
\(154\) −165.000 −0.0863382
\(155\) 0 0
\(156\) 1632.00 0.837593
\(157\) −1874.00 −0.952621 −0.476310 0.879277i \(-0.658026\pi\)
−0.476310 + 0.879277i \(0.658026\pi\)
\(158\) −3605.00 −1.81518
\(159\) 456.000 0.227441
\(160\) 0 0
\(161\) 339.000 0.165944
\(162\) 405.000 0.196419
\(163\) −1904.00 −0.914925 −0.457463 0.889229i \(-0.651242\pi\)
−0.457463 + 0.889229i \(0.651242\pi\)
\(164\) 2363.00 1.12512
\(165\) 0 0
\(166\) 710.000 0.331968
\(167\) −1180.00 −0.546773 −0.273387 0.961904i \(-0.588144\pi\)
−0.273387 + 0.961904i \(0.588144\pi\)
\(168\) 405.000 0.185991
\(169\) −1173.00 −0.533910
\(170\) 0 0
\(171\) 423.000 0.189167
\(172\) −5236.00 −2.32117
\(173\) −3177.00 −1.39620 −0.698101 0.716000i \(-0.745971\pi\)
−0.698101 + 0.716000i \(0.745971\pi\)
\(174\) −810.000 −0.352908
\(175\) 0 0
\(176\) −979.000 −0.419289
\(177\) −1875.00 −0.796235
\(178\) 2020.00 0.850592
\(179\) 1787.00 0.746182 0.373091 0.927795i \(-0.378298\pi\)
0.373091 + 0.927795i \(0.378298\pi\)
\(180\) 0 0
\(181\) −835.000 −0.342901 −0.171450 0.985193i \(-0.554845\pi\)
−0.171450 + 0.985193i \(0.554845\pi\)
\(182\) 480.000 0.195494
\(183\) 960.000 0.387788
\(184\) 5085.00 2.03734
\(185\) 0 0
\(186\) 2670.00 1.05255
\(187\) −363.000 −0.141953
\(188\) 3315.00 1.28602
\(189\) 81.0000 0.0311740
\(190\) 0 0
\(191\) 3613.00 1.36873 0.684365 0.729139i \(-0.260079\pi\)
0.684365 + 0.729139i \(0.260079\pi\)
\(192\) −861.000 −0.323632
\(193\) 4204.00 1.56793 0.783965 0.620805i \(-0.213194\pi\)
0.783965 + 0.620805i \(0.213194\pi\)
\(194\) 395.000 0.146182
\(195\) 0 0
\(196\) −5678.00 −2.06924
\(197\) −4517.00 −1.63362 −0.816809 0.576908i \(-0.804259\pi\)
−0.816809 + 0.576908i \(0.804259\pi\)
\(198\) −495.000 −0.177667
\(199\) 4164.00 1.48331 0.741654 0.670783i \(-0.234042\pi\)
0.741654 + 0.670783i \(0.234042\pi\)
\(200\) 0 0
\(201\) 600.000 0.210551
\(202\) −2725.00 −0.949160
\(203\) −162.000 −0.0560107
\(204\) 1683.00 0.577616
\(205\) 0 0
\(206\) −6530.00 −2.20858
\(207\) 1017.00 0.341480
\(208\) 2848.00 0.949391
\(209\) −517.000 −0.171108
\(210\) 0 0
\(211\) 4660.00 1.52042 0.760208 0.649680i \(-0.225097\pi\)
0.760208 + 0.649680i \(0.225097\pi\)
\(212\) 2584.00 0.837122
\(213\) −2841.00 −0.913907
\(214\) 9690.00 3.09530
\(215\) 0 0
\(216\) 1215.00 0.382733
\(217\) 534.000 0.167052
\(218\) −2880.00 −0.894762
\(219\) −1344.00 −0.414699
\(220\) 0 0
\(221\) 1056.00 0.321422
\(222\) 285.000 0.0861619
\(223\) 3560.00 1.06904 0.534518 0.845157i \(-0.320493\pi\)
0.534518 + 0.845157i \(0.320493\pi\)
\(224\) 255.000 0.0760621
\(225\) 0 0
\(226\) −5520.00 −1.62471
\(227\) −4678.00 −1.36780 −0.683898 0.729577i \(-0.739717\pi\)
−0.683898 + 0.729577i \(0.739717\pi\)
\(228\) 2397.00 0.696251
\(229\) −4447.00 −1.28326 −0.641629 0.767015i \(-0.721741\pi\)
−0.641629 + 0.767015i \(0.721741\pi\)
\(230\) 0 0
\(231\) −99.0000 −0.0281979
\(232\) −2430.00 −0.687661
\(233\) 411.000 0.115560 0.0577801 0.998329i \(-0.481598\pi\)
0.0577801 + 0.998329i \(0.481598\pi\)
\(234\) 1440.00 0.402290
\(235\) 0 0
\(236\) −10625.0 −2.93063
\(237\) −2163.00 −0.592835
\(238\) 495.000 0.134815
\(239\) −6380.00 −1.72673 −0.863364 0.504582i \(-0.831647\pi\)
−0.863364 + 0.504582i \(0.831647\pi\)
\(240\) 0 0
\(241\) 7282.00 1.94637 0.973184 0.230027i \(-0.0738813\pi\)
0.973184 + 0.230027i \(0.0738813\pi\)
\(242\) 605.000 0.160706
\(243\) 243.000 0.0641500
\(244\) 5440.00 1.42730
\(245\) 0 0
\(246\) 2085.00 0.540385
\(247\) 1504.00 0.387438
\(248\) 8010.00 2.05095
\(249\) 426.000 0.108420
\(250\) 0 0
\(251\) −4728.00 −1.18896 −0.594480 0.804111i \(-0.702642\pi\)
−0.594480 + 0.804111i \(0.702642\pi\)
\(252\) 459.000 0.114739
\(253\) −1243.00 −0.308880
\(254\) −8695.00 −2.14792
\(255\) 0 0
\(256\) −8279.00 −2.02124
\(257\) 5418.00 1.31504 0.657521 0.753437i \(-0.271605\pi\)
0.657521 + 0.753437i \(0.271605\pi\)
\(258\) −4620.00 −1.11484
\(259\) 57.0000 0.0136749
\(260\) 0 0
\(261\) −486.000 −0.115259
\(262\) 9090.00 2.14344
\(263\) −3354.00 −0.786375 −0.393187 0.919458i \(-0.628628\pi\)
−0.393187 + 0.919458i \(0.628628\pi\)
\(264\) −1485.00 −0.346195
\(265\) 0 0
\(266\) 705.000 0.162505
\(267\) 1212.00 0.277802
\(268\) 3400.00 0.774955
\(269\) 1062.00 0.240711 0.120356 0.992731i \(-0.461597\pi\)
0.120356 + 0.992731i \(0.461597\pi\)
\(270\) 0 0
\(271\) −4821.00 −1.08065 −0.540323 0.841458i \(-0.681698\pi\)
−0.540323 + 0.841458i \(0.681698\pi\)
\(272\) 2937.00 0.654712
\(273\) 288.000 0.0638482
\(274\) 4350.00 0.959099
\(275\) 0 0
\(276\) 5763.00 1.25685
\(277\) 4.00000 0.000867642 0 0.000433821 1.00000i \(-0.499862\pi\)
0.000433821 1.00000i \(0.499862\pi\)
\(278\) −3180.00 −0.686057
\(279\) 1602.00 0.343761
\(280\) 0 0
\(281\) 4647.00 0.986537 0.493268 0.869877i \(-0.335802\pi\)
0.493268 + 0.869877i \(0.335802\pi\)
\(282\) 2925.00 0.617664
\(283\) −4283.00 −0.899639 −0.449820 0.893119i \(-0.648512\pi\)
−0.449820 + 0.893119i \(0.648512\pi\)
\(284\) −16099.0 −3.36373
\(285\) 0 0
\(286\) −1760.00 −0.363885
\(287\) 417.000 0.0857656
\(288\) 765.000 0.156521
\(289\) −3824.00 −0.778343
\(290\) 0 0
\(291\) 237.000 0.0477429
\(292\) −7616.00 −1.52634
\(293\) −6811.00 −1.35803 −0.679015 0.734124i \(-0.737593\pi\)
−0.679015 + 0.734124i \(0.737593\pi\)
\(294\) −5010.00 −0.993841
\(295\) 0 0
\(296\) 855.000 0.167891
\(297\) −297.000 −0.0580259
\(298\) −1195.00 −0.232297
\(299\) 3616.00 0.699394
\(300\) 0 0
\(301\) −924.000 −0.176938
\(302\) 6040.00 1.15087
\(303\) −1635.00 −0.309994
\(304\) 4183.00 0.789183
\(305\) 0 0
\(306\) 1485.00 0.277424
\(307\) −460.000 −0.0855166 −0.0427583 0.999085i \(-0.513615\pi\)
−0.0427583 + 0.999085i \(0.513615\pi\)
\(308\) −561.000 −0.103786
\(309\) −3918.00 −0.721318
\(310\) 0 0
\(311\) 8328.00 1.51845 0.759224 0.650829i \(-0.225579\pi\)
0.759224 + 0.650829i \(0.225579\pi\)
\(312\) 4320.00 0.783884
\(313\) −5929.00 −1.07069 −0.535346 0.844633i \(-0.679819\pi\)
−0.535346 + 0.844633i \(0.679819\pi\)
\(314\) −9370.00 −1.68401
\(315\) 0 0
\(316\) −12257.0 −2.18199
\(317\) 5040.00 0.892980 0.446490 0.894789i \(-0.352674\pi\)
0.446490 + 0.894789i \(0.352674\pi\)
\(318\) 2280.00 0.402063
\(319\) 594.000 0.104256
\(320\) 0 0
\(321\) 5814.00 1.01092
\(322\) 1695.00 0.293350
\(323\) 1551.00 0.267183
\(324\) 1377.00 0.236111
\(325\) 0 0
\(326\) −9520.00 −1.61737
\(327\) −1728.00 −0.292228
\(328\) 6255.00 1.05297
\(329\) 585.000 0.0980307
\(330\) 0 0
\(331\) 10396.0 1.72633 0.863166 0.504920i \(-0.168478\pi\)
0.863166 + 0.504920i \(0.168478\pi\)
\(332\) 2414.00 0.399053
\(333\) 171.000 0.0281404
\(334\) −5900.00 −0.966568
\(335\) 0 0
\(336\) 801.000 0.130054
\(337\) −7236.00 −1.16964 −0.584822 0.811162i \(-0.698836\pi\)
−0.584822 + 0.811162i \(0.698836\pi\)
\(338\) −5865.00 −0.943828
\(339\) −3312.00 −0.530629
\(340\) 0 0
\(341\) −1958.00 −0.310943
\(342\) 2115.00 0.334404
\(343\) −2031.00 −0.319719
\(344\) −13860.0 −2.17233
\(345\) 0 0
\(346\) −15885.0 −2.46816
\(347\) 1468.00 0.227108 0.113554 0.993532i \(-0.463777\pi\)
0.113554 + 0.993532i \(0.463777\pi\)
\(348\) −2754.00 −0.424224
\(349\) 5690.00 0.872718 0.436359 0.899773i \(-0.356268\pi\)
0.436359 + 0.899773i \(0.356268\pi\)
\(350\) 0 0
\(351\) 864.000 0.131387
\(352\) −935.000 −0.141579
\(353\) 5376.00 0.810582 0.405291 0.914188i \(-0.367170\pi\)
0.405291 + 0.914188i \(0.367170\pi\)
\(354\) −9375.00 −1.40756
\(355\) 0 0
\(356\) 6868.00 1.02248
\(357\) 297.000 0.0440306
\(358\) 8935.00 1.31908
\(359\) 3734.00 0.548950 0.274475 0.961594i \(-0.411496\pi\)
0.274475 + 0.961594i \(0.411496\pi\)
\(360\) 0 0
\(361\) −4650.00 −0.677941
\(362\) −4175.00 −0.606169
\(363\) 363.000 0.0524864
\(364\) 1632.00 0.235000
\(365\) 0 0
\(366\) 4800.00 0.685519
\(367\) −10274.0 −1.46130 −0.730652 0.682750i \(-0.760784\pi\)
−0.730652 + 0.682750i \(0.760784\pi\)
\(368\) 10057.0 1.42461
\(369\) 1251.00 0.176489
\(370\) 0 0
\(371\) 456.000 0.0638122
\(372\) 9078.00 1.26525
\(373\) 13662.0 1.89649 0.948246 0.317537i \(-0.102856\pi\)
0.948246 + 0.317537i \(0.102856\pi\)
\(374\) −1815.00 −0.250940
\(375\) 0 0
\(376\) 8775.00 1.20355
\(377\) −1728.00 −0.236065
\(378\) 405.000 0.0551083
\(379\) −7906.00 −1.07151 −0.535757 0.844372i \(-0.679974\pi\)
−0.535757 + 0.844372i \(0.679974\pi\)
\(380\) 0 0
\(381\) −5217.00 −0.701509
\(382\) 18065.0 2.41960
\(383\) −3168.00 −0.422656 −0.211328 0.977415i \(-0.567779\pi\)
−0.211328 + 0.977415i \(0.567779\pi\)
\(384\) −6345.00 −0.843208
\(385\) 0 0
\(386\) 21020.0 2.77174
\(387\) −2772.00 −0.364105
\(388\) 1343.00 0.175723
\(389\) 10770.0 1.40375 0.701877 0.712298i \(-0.252345\pi\)
0.701877 + 0.712298i \(0.252345\pi\)
\(390\) 0 0
\(391\) 3729.00 0.482311
\(392\) −15030.0 −1.93656
\(393\) 5454.00 0.700046
\(394\) −22585.0 −2.88786
\(395\) 0 0
\(396\) −1683.00 −0.213571
\(397\) 5670.00 0.716799 0.358399 0.933568i \(-0.383323\pi\)
0.358399 + 0.933568i \(0.383323\pi\)
\(398\) 20820.0 2.62214
\(399\) 423.000 0.0530739
\(400\) 0 0
\(401\) 832.000 0.103611 0.0518056 0.998657i \(-0.483502\pi\)
0.0518056 + 0.998657i \(0.483502\pi\)
\(402\) 3000.00 0.372205
\(403\) 5696.00 0.704064
\(404\) −9265.00 −1.14097
\(405\) 0 0
\(406\) −810.000 −0.0990139
\(407\) −209.000 −0.0254539
\(408\) 4455.00 0.540577
\(409\) −5712.00 −0.690563 −0.345281 0.938499i \(-0.612217\pi\)
−0.345281 + 0.938499i \(0.612217\pi\)
\(410\) 0 0
\(411\) 2610.00 0.313240
\(412\) −22202.0 −2.65489
\(413\) −1875.00 −0.223396
\(414\) 5085.00 0.603657
\(415\) 0 0
\(416\) 2720.00 0.320574
\(417\) −1908.00 −0.224065
\(418\) −2585.00 −0.302480
\(419\) −4559.00 −0.531555 −0.265778 0.964034i \(-0.585629\pi\)
−0.265778 + 0.964034i \(0.585629\pi\)
\(420\) 0 0
\(421\) 6855.00 0.793568 0.396784 0.917912i \(-0.370126\pi\)
0.396784 + 0.917912i \(0.370126\pi\)
\(422\) 23300.0 2.68774
\(423\) 1755.00 0.201728
\(424\) 6840.00 0.783443
\(425\) 0 0
\(426\) −14205.0 −1.61557
\(427\) 960.000 0.108800
\(428\) 32946.0 3.72081
\(429\) −1056.00 −0.118844
\(430\) 0 0
\(431\) 10770.0 1.20365 0.601824 0.798628i \(-0.294441\pi\)
0.601824 + 0.798628i \(0.294441\pi\)
\(432\) 2403.00 0.267626
\(433\) 8498.00 0.943159 0.471579 0.881824i \(-0.343684\pi\)
0.471579 + 0.881824i \(0.343684\pi\)
\(434\) 2670.00 0.295309
\(435\) 0 0
\(436\) −9792.00 −1.07558
\(437\) 5311.00 0.581372
\(438\) −6720.00 −0.733091
\(439\) 9835.00 1.06925 0.534623 0.845091i \(-0.320454\pi\)
0.534623 + 0.845091i \(0.320454\pi\)
\(440\) 0 0
\(441\) −3006.00 −0.324587
\(442\) 5280.00 0.568199
\(443\) −10745.0 −1.15239 −0.576197 0.817311i \(-0.695464\pi\)
−0.576197 + 0.817311i \(0.695464\pi\)
\(444\) 969.000 0.103574
\(445\) 0 0
\(446\) 17800.0 1.88981
\(447\) −717.000 −0.0758679
\(448\) −861.000 −0.0908001
\(449\) 8356.00 0.878272 0.439136 0.898421i \(-0.355285\pi\)
0.439136 + 0.898421i \(0.355285\pi\)
\(450\) 0 0
\(451\) −1529.00 −0.159640
\(452\) −18768.0 −1.95304
\(453\) 3624.00 0.375873
\(454\) −23390.0 −2.41795
\(455\) 0 0
\(456\) 6345.00 0.651605
\(457\) −7058.00 −0.722449 −0.361225 0.932479i \(-0.617641\pi\)
−0.361225 + 0.932479i \(0.617641\pi\)
\(458\) −22235.0 −2.26850
\(459\) 891.000 0.0906064
\(460\) 0 0
\(461\) 646.000 0.0652651 0.0326326 0.999467i \(-0.489611\pi\)
0.0326326 + 0.999467i \(0.489611\pi\)
\(462\) −495.000 −0.0498474
\(463\) −8982.00 −0.901574 −0.450787 0.892631i \(-0.648857\pi\)
−0.450787 + 0.892631i \(0.648857\pi\)
\(464\) −4806.00 −0.480847
\(465\) 0 0
\(466\) 2055.00 0.204283
\(467\) −13476.0 −1.33532 −0.667661 0.744466i \(-0.732704\pi\)
−0.667661 + 0.744466i \(0.732704\pi\)
\(468\) 4896.00 0.483585
\(469\) 600.000 0.0590734
\(470\) 0 0
\(471\) −5622.00 −0.549996
\(472\) −28125.0 −2.74271
\(473\) 3388.00 0.329345
\(474\) −10815.0 −1.04799
\(475\) 0 0
\(476\) 1683.00 0.162059
\(477\) 1368.00 0.131313
\(478\) −31900.0 −3.05245
\(479\) 12996.0 1.23967 0.619835 0.784732i \(-0.287199\pi\)
0.619835 + 0.784732i \(0.287199\pi\)
\(480\) 0 0
\(481\) 608.000 0.0576350
\(482\) 36410.0 3.44073
\(483\) 1017.00 0.0958077
\(484\) 2057.00 0.193182
\(485\) 0 0
\(486\) 1215.00 0.113402
\(487\) −6026.00 −0.560707 −0.280353 0.959897i \(-0.590452\pi\)
−0.280353 + 0.959897i \(0.590452\pi\)
\(488\) 14400.0 1.33577
\(489\) −5712.00 −0.528232
\(490\) 0 0
\(491\) 11698.0 1.07520 0.537600 0.843200i \(-0.319331\pi\)
0.537600 + 0.843200i \(0.319331\pi\)
\(492\) 7089.00 0.649587
\(493\) −1782.00 −0.162794
\(494\) 7520.00 0.684900
\(495\) 0 0
\(496\) 15842.0 1.43413
\(497\) −2841.00 −0.256411
\(498\) 2130.00 0.191662
\(499\) −17052.0 −1.52976 −0.764882 0.644170i \(-0.777203\pi\)
−0.764882 + 0.644170i \(0.777203\pi\)
\(500\) 0 0
\(501\) −3540.00 −0.315680
\(502\) −23640.0 −2.10180
\(503\) −932.000 −0.0826160 −0.0413080 0.999146i \(-0.513152\pi\)
−0.0413080 + 0.999146i \(0.513152\pi\)
\(504\) 1215.00 0.107382
\(505\) 0 0
\(506\) −6215.00 −0.546029
\(507\) −3519.00 −0.308253
\(508\) −29563.0 −2.58198
\(509\) 4384.00 0.381763 0.190882 0.981613i \(-0.438865\pi\)
0.190882 + 0.981613i \(0.438865\pi\)
\(510\) 0 0
\(511\) −1344.00 −0.116350
\(512\) −24475.0 −2.11260
\(513\) 1269.00 0.109216
\(514\) 27090.0 2.32469
\(515\) 0 0
\(516\) −15708.0 −1.34013
\(517\) −2145.00 −0.182470
\(518\) 285.000 0.0241741
\(519\) −9531.00 −0.806097
\(520\) 0 0
\(521\) −2322.00 −0.195257 −0.0976283 0.995223i \(-0.531126\pi\)
−0.0976283 + 0.995223i \(0.531126\pi\)
\(522\) −2430.00 −0.203751
\(523\) −9749.00 −0.815094 −0.407547 0.913184i \(-0.633616\pi\)
−0.407547 + 0.913184i \(0.633616\pi\)
\(524\) 30906.0 2.57659
\(525\) 0 0
\(526\) −16770.0 −1.39013
\(527\) 5874.00 0.485532
\(528\) −2937.00 −0.242077
\(529\) 602.000 0.0494781
\(530\) 0 0
\(531\) −5625.00 −0.459707
\(532\) 2397.00 0.195344
\(533\) 4448.00 0.361471
\(534\) 6060.00 0.491090
\(535\) 0 0
\(536\) 9000.00 0.725263
\(537\) 5361.00 0.430809
\(538\) 5310.00 0.425521
\(539\) 3674.00 0.293600
\(540\) 0 0
\(541\) 4208.00 0.334410 0.167205 0.985922i \(-0.446526\pi\)
0.167205 + 0.985922i \(0.446526\pi\)
\(542\) −24105.0 −1.91033
\(543\) −2505.00 −0.197974
\(544\) 2805.00 0.221072
\(545\) 0 0
\(546\) 1440.00 0.112869
\(547\) 10179.0 0.795654 0.397827 0.917461i \(-0.369764\pi\)
0.397827 + 0.917461i \(0.369764\pi\)
\(548\) 14790.0 1.15292
\(549\) 2880.00 0.223890
\(550\) 0 0
\(551\) −2538.00 −0.196229
\(552\) 15255.0 1.17626
\(553\) −2163.00 −0.166329
\(554\) 20.0000 0.00153379
\(555\) 0 0
\(556\) −10812.0 −0.824696
\(557\) −2314.00 −0.176028 −0.0880138 0.996119i \(-0.528052\pi\)
−0.0880138 + 0.996119i \(0.528052\pi\)
\(558\) 8010.00 0.607689
\(559\) −9856.00 −0.745732
\(560\) 0 0
\(561\) −1089.00 −0.0819565
\(562\) 23235.0 1.74397
\(563\) 24330.0 1.82129 0.910646 0.413188i \(-0.135585\pi\)
0.910646 + 0.413188i \(0.135585\pi\)
\(564\) 9945.00 0.742482
\(565\) 0 0
\(566\) −21415.0 −1.59035
\(567\) 243.000 0.0179983
\(568\) −42615.0 −3.14804
\(569\) 3445.00 0.253817 0.126909 0.991914i \(-0.459495\pi\)
0.126909 + 0.991914i \(0.459495\pi\)
\(570\) 0 0
\(571\) −13056.0 −0.956877 −0.478438 0.878121i \(-0.658797\pi\)
−0.478438 + 0.878121i \(0.658797\pi\)
\(572\) −5984.00 −0.437419
\(573\) 10839.0 0.790237
\(574\) 2085.00 0.151614
\(575\) 0 0
\(576\) −2583.00 −0.186849
\(577\) −17347.0 −1.25159 −0.625793 0.779989i \(-0.715225\pi\)
−0.625793 + 0.779989i \(0.715225\pi\)
\(578\) −19120.0 −1.37593
\(579\) 12612.0 0.905245
\(580\) 0 0
\(581\) 426.000 0.0304190
\(582\) 1185.00 0.0843983
\(583\) −1672.00 −0.118777
\(584\) −20160.0 −1.42847
\(585\) 0 0
\(586\) −34055.0 −2.40068
\(587\) 8379.00 0.589162 0.294581 0.955626i \(-0.404820\pi\)
0.294581 + 0.955626i \(0.404820\pi\)
\(588\) −17034.0 −1.19468
\(589\) 8366.00 0.585255
\(590\) 0 0
\(591\) −13551.0 −0.943170
\(592\) 1691.00 0.117398
\(593\) 1958.00 0.135591 0.0677955 0.997699i \(-0.478403\pi\)
0.0677955 + 0.997699i \(0.478403\pi\)
\(594\) −1485.00 −0.102576
\(595\) 0 0
\(596\) −4063.00 −0.279240
\(597\) 12492.0 0.856388
\(598\) 18080.0 1.23636
\(599\) 23583.0 1.60864 0.804320 0.594196i \(-0.202530\pi\)
0.804320 + 0.594196i \(0.202530\pi\)
\(600\) 0 0
\(601\) −15328.0 −1.04034 −0.520168 0.854064i \(-0.674131\pi\)
−0.520168 + 0.854064i \(0.674131\pi\)
\(602\) −4620.00 −0.312786
\(603\) 1800.00 0.121562
\(604\) 20536.0 1.38344
\(605\) 0 0
\(606\) −8175.00 −0.547998
\(607\) 160.000 0.0106988 0.00534942 0.999986i \(-0.498297\pi\)
0.00534942 + 0.999986i \(0.498297\pi\)
\(608\) 3995.00 0.266478
\(609\) −486.000 −0.0323378
\(610\) 0 0
\(611\) 6240.00 0.413164
\(612\) 5049.00 0.333486
\(613\) −5948.00 −0.391904 −0.195952 0.980613i \(-0.562780\pi\)
−0.195952 + 0.980613i \(0.562780\pi\)
\(614\) −2300.00 −0.151173
\(615\) 0 0
\(616\) −1485.00 −0.0971304
\(617\) 334.000 0.0217931 0.0108965 0.999941i \(-0.496531\pi\)
0.0108965 + 0.999941i \(0.496531\pi\)
\(618\) −19590.0 −1.27512
\(619\) −7202.00 −0.467646 −0.233823 0.972279i \(-0.575124\pi\)
−0.233823 + 0.972279i \(0.575124\pi\)
\(620\) 0 0
\(621\) 3051.00 0.197154
\(622\) 41640.0 2.68426
\(623\) 1212.00 0.0779418
\(624\) 8544.00 0.548131
\(625\) 0 0
\(626\) −29645.0 −1.89274
\(627\) −1551.00 −0.0987894
\(628\) −31858.0 −2.02432
\(629\) 627.000 0.0397458
\(630\) 0 0
\(631\) 10306.0 0.650199 0.325099 0.945680i \(-0.394602\pi\)
0.325099 + 0.945680i \(0.394602\pi\)
\(632\) −32445.0 −2.04208
\(633\) 13980.0 0.877812
\(634\) 25200.0 1.57858
\(635\) 0 0
\(636\) 7752.00 0.483313
\(637\) −10688.0 −0.664794
\(638\) 2970.00 0.184300
\(639\) −8523.00 −0.527644
\(640\) 0 0
\(641\) −1228.00 −0.0756678 −0.0378339 0.999284i \(-0.512046\pi\)
−0.0378339 + 0.999284i \(0.512046\pi\)
\(642\) 29070.0 1.78707
\(643\) −18454.0 −1.13181 −0.565906 0.824470i \(-0.691473\pi\)
−0.565906 + 0.824470i \(0.691473\pi\)
\(644\) 5763.00 0.352630
\(645\) 0 0
\(646\) 7755.00 0.472316
\(647\) 17647.0 1.07230 0.536148 0.844124i \(-0.319879\pi\)
0.536148 + 0.844124i \(0.319879\pi\)
\(648\) 3645.00 0.220971
\(649\) 6875.00 0.415820
\(650\) 0 0
\(651\) 1602.00 0.0964475
\(652\) −32368.0 −1.94422
\(653\) 25918.0 1.55322 0.776608 0.629984i \(-0.216939\pi\)
0.776608 + 0.629984i \(0.216939\pi\)
\(654\) −8640.00 −0.516591
\(655\) 0 0
\(656\) 12371.0 0.736290
\(657\) −4032.00 −0.239427
\(658\) 2925.00 0.173295
\(659\) 12864.0 0.760410 0.380205 0.924902i \(-0.375853\pi\)
0.380205 + 0.924902i \(0.375853\pi\)
\(660\) 0 0
\(661\) −11419.0 −0.671933 −0.335966 0.941874i \(-0.609063\pi\)
−0.335966 + 0.941874i \(0.609063\pi\)
\(662\) 51980.0 3.05175
\(663\) 3168.00 0.185573
\(664\) 6390.00 0.373464
\(665\) 0 0
\(666\) 855.000 0.0497456
\(667\) −6102.00 −0.354228
\(668\) −20060.0 −1.16189
\(669\) 10680.0 0.617209
\(670\) 0 0
\(671\) −3520.00 −0.202516
\(672\) 765.000 0.0439145
\(673\) 15784.0 0.904054 0.452027 0.892004i \(-0.350701\pi\)
0.452027 + 0.892004i \(0.350701\pi\)
\(674\) −36180.0 −2.06766
\(675\) 0 0
\(676\) −19941.0 −1.13456
\(677\) −26050.0 −1.47885 −0.739426 0.673238i \(-0.764903\pi\)
−0.739426 + 0.673238i \(0.764903\pi\)
\(678\) −16560.0 −0.938028
\(679\) 237.000 0.0133950
\(680\) 0 0
\(681\) −14034.0 −0.789698
\(682\) −9790.00 −0.549675
\(683\) −15095.0 −0.845672 −0.422836 0.906206i \(-0.638965\pi\)
−0.422836 + 0.906206i \(0.638965\pi\)
\(684\) 7191.00 0.401981
\(685\) 0 0
\(686\) −10155.0 −0.565189
\(687\) −13341.0 −0.740889
\(688\) −27412.0 −1.51900
\(689\) 4864.00 0.268946
\(690\) 0 0
\(691\) 15896.0 0.875126 0.437563 0.899188i \(-0.355842\pi\)
0.437563 + 0.899188i \(0.355842\pi\)
\(692\) −54009.0 −2.96693
\(693\) −297.000 −0.0162801
\(694\) 7340.00 0.401473
\(695\) 0 0
\(696\) −7290.00 −0.397021
\(697\) 4587.00 0.249275
\(698\) 28450.0 1.54276
\(699\) 1233.00 0.0667187
\(700\) 0 0
\(701\) 10529.0 0.567296 0.283648 0.958928i \(-0.408455\pi\)
0.283648 + 0.958928i \(0.408455\pi\)
\(702\) 4320.00 0.232262
\(703\) 893.000 0.0479092
\(704\) 3157.00 0.169011
\(705\) 0 0
\(706\) 26880.0 1.43292
\(707\) −1635.00 −0.0869738
\(708\) −31875.0 −1.69200
\(709\) −16087.0 −0.852130 −0.426065 0.904693i \(-0.640100\pi\)
−0.426065 + 0.904693i \(0.640100\pi\)
\(710\) 0 0
\(711\) −6489.00 −0.342274
\(712\) 18180.0 0.956916
\(713\) 20114.0 1.05649
\(714\) 1485.00 0.0778358
\(715\) 0 0
\(716\) 30379.0 1.58564
\(717\) −19140.0 −0.996927
\(718\) 18670.0 0.970415
\(719\) 24336.0 1.26228 0.631140 0.775669i \(-0.282587\pi\)
0.631140 + 0.775669i \(0.282587\pi\)
\(720\) 0 0
\(721\) −3918.00 −0.202377
\(722\) −23250.0 −1.19844
\(723\) 21846.0 1.12374
\(724\) −14195.0 −0.728664
\(725\) 0 0
\(726\) 1815.00 0.0927837
\(727\) 13960.0 0.712170 0.356085 0.934454i \(-0.384111\pi\)
0.356085 + 0.934454i \(0.384111\pi\)
\(728\) 4320.00 0.219931
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) −10164.0 −0.514267
\(732\) 16320.0 0.824050
\(733\) 9252.00 0.466208 0.233104 0.972452i \(-0.425112\pi\)
0.233104 + 0.972452i \(0.425112\pi\)
\(734\) −51370.0 −2.58324
\(735\) 0 0
\(736\) 9605.00 0.481039
\(737\) −2200.00 −0.109957
\(738\) 6255.00 0.311992
\(739\) 28453.0 1.41632 0.708160 0.706052i \(-0.249526\pi\)
0.708160 + 0.706052i \(0.249526\pi\)
\(740\) 0 0
\(741\) 4512.00 0.223688
\(742\) 2280.00 0.112805
\(743\) 512.000 0.0252806 0.0126403 0.999920i \(-0.495976\pi\)
0.0126403 + 0.999920i \(0.495976\pi\)
\(744\) 24030.0 1.18412
\(745\) 0 0
\(746\) 68310.0 3.35256
\(747\) 1278.00 0.0625965
\(748\) −6171.00 −0.301650
\(749\) 5814.00 0.283630
\(750\) 0 0
\(751\) 772.000 0.0375109 0.0187554 0.999824i \(-0.494030\pi\)
0.0187554 + 0.999824i \(0.494030\pi\)
\(752\) 17355.0 0.841585
\(753\) −14184.0 −0.686446
\(754\) −8640.00 −0.417308
\(755\) 0 0
\(756\) 1377.00 0.0662447
\(757\) 8058.00 0.386886 0.193443 0.981111i \(-0.438034\pi\)
0.193443 + 0.981111i \(0.438034\pi\)
\(758\) −39530.0 −1.89419
\(759\) −3729.00 −0.178332
\(760\) 0 0
\(761\) 18650.0 0.888386 0.444193 0.895931i \(-0.353490\pi\)
0.444193 + 0.895931i \(0.353490\pi\)
\(762\) −26085.0 −1.24010
\(763\) −1728.00 −0.0819893
\(764\) 61421.0 2.90855
\(765\) 0 0
\(766\) −15840.0 −0.747157
\(767\) −20000.0 −0.941536
\(768\) −24837.0 −1.16696
\(769\) −7144.00 −0.335005 −0.167503 0.985872i \(-0.553570\pi\)
−0.167503 + 0.985872i \(0.553570\pi\)
\(770\) 0 0
\(771\) 16254.0 0.759239
\(772\) 71468.0 3.33185
\(773\) −1904.00 −0.0885927 −0.0442963 0.999018i \(-0.514105\pi\)
−0.0442963 + 0.999018i \(0.514105\pi\)
\(774\) −13860.0 −0.643653
\(775\) 0 0
\(776\) 3555.00 0.164455
\(777\) 171.000 0.00789523
\(778\) 53850.0 2.48151
\(779\) 6533.00 0.300474
\(780\) 0 0
\(781\) 10417.0 0.477272
\(782\) 18645.0 0.852614
\(783\) −1458.00 −0.0665449
\(784\) −29726.0 −1.35414
\(785\) 0 0
\(786\) 27270.0 1.23752
\(787\) 7555.00 0.342194 0.171097 0.985254i \(-0.445269\pi\)
0.171097 + 0.985254i \(0.445269\pi\)
\(788\) −76789.0 −3.47144
\(789\) −10062.0 −0.454014
\(790\) 0 0
\(791\) −3312.00 −0.148876
\(792\) −4455.00 −0.199876
\(793\) 10240.0 0.458554
\(794\) 28350.0 1.26713
\(795\) 0 0
\(796\) 70788.0 3.15203
\(797\) 24950.0 1.10888 0.554438 0.832225i \(-0.312933\pi\)
0.554438 + 0.832225i \(0.312933\pi\)
\(798\) 2115.00 0.0938223
\(799\) 6435.00 0.284924
\(800\) 0 0
\(801\) 3636.00 0.160389
\(802\) 4160.00 0.183160
\(803\) 4928.00 0.216570
\(804\) 10200.0 0.447421
\(805\) 0 0
\(806\) 28480.0 1.24462
\(807\) 3186.00 0.138975
\(808\) −24525.0 −1.06781
\(809\) 19893.0 0.864525 0.432262 0.901748i \(-0.357715\pi\)
0.432262 + 0.901748i \(0.357715\pi\)
\(810\) 0 0
\(811\) 34503.0 1.49391 0.746957 0.664872i \(-0.231514\pi\)
0.746957 + 0.664872i \(0.231514\pi\)
\(812\) −2754.00 −0.119023
\(813\) −14463.0 −0.623911
\(814\) −1045.00 −0.0449966
\(815\) 0 0
\(816\) 8811.00 0.377998
\(817\) −14476.0 −0.619891
\(818\) −28560.0 −1.22075
\(819\) 864.000 0.0368628
\(820\) 0 0
\(821\) 16890.0 0.717984 0.358992 0.933341i \(-0.383120\pi\)
0.358992 + 0.933341i \(0.383120\pi\)
\(822\) 13050.0 0.553736
\(823\) 34692.0 1.46936 0.734682 0.678411i \(-0.237331\pi\)
0.734682 + 0.678411i \(0.237331\pi\)
\(824\) −58770.0 −2.48465
\(825\) 0 0
\(826\) −9375.00 −0.394913
\(827\) 41424.0 1.74178 0.870891 0.491476i \(-0.163543\pi\)
0.870891 + 0.491476i \(0.163543\pi\)
\(828\) 17289.0 0.725645
\(829\) −18494.0 −0.774817 −0.387408 0.921908i \(-0.626630\pi\)
−0.387408 + 0.921908i \(0.626630\pi\)
\(830\) 0 0
\(831\) 12.0000 0.000500933 0
\(832\) −9184.00 −0.382690
\(833\) −11022.0 −0.458451
\(834\) −9540.00 −0.396095
\(835\) 0 0
\(836\) −8789.00 −0.363605
\(837\) 4806.00 0.198470
\(838\) −22795.0 −0.939666
\(839\) 6680.00 0.274874 0.137437 0.990511i \(-0.456114\pi\)
0.137437 + 0.990511i \(0.456114\pi\)
\(840\) 0 0
\(841\) −21473.0 −0.880438
\(842\) 34275.0 1.40284
\(843\) 13941.0 0.569577
\(844\) 79220.0 3.23088
\(845\) 0 0
\(846\) 8775.00 0.356608
\(847\) 363.000 0.0147259
\(848\) 13528.0 0.547822
\(849\) −12849.0 −0.519407
\(850\) 0 0
\(851\) 2147.00 0.0864844
\(852\) −48297.0 −1.94205
\(853\) 43358.0 1.74039 0.870193 0.492711i \(-0.163994\pi\)
0.870193 + 0.492711i \(0.163994\pi\)
\(854\) 4800.00 0.192333
\(855\) 0 0
\(856\) 87210.0 3.48222
\(857\) 15585.0 0.621206 0.310603 0.950540i \(-0.399469\pi\)
0.310603 + 0.950540i \(0.399469\pi\)
\(858\) −5280.00 −0.210089
\(859\) −17036.0 −0.676672 −0.338336 0.941025i \(-0.609864\pi\)
−0.338336 + 0.941025i \(0.609864\pi\)
\(860\) 0 0
\(861\) 1251.00 0.0495168
\(862\) 53850.0 2.12777
\(863\) 28064.0 1.10696 0.553482 0.832861i \(-0.313299\pi\)
0.553482 + 0.832861i \(0.313299\pi\)
\(864\) 2295.00 0.0903675
\(865\) 0 0
\(866\) 42490.0 1.66729
\(867\) −11472.0 −0.449377
\(868\) 9078.00 0.354985
\(869\) 7931.00 0.309598
\(870\) 0 0
\(871\) 6400.00 0.248973
\(872\) −25920.0 −1.00661
\(873\) 711.000 0.0275644
\(874\) 26555.0 1.02773
\(875\) 0 0
\(876\) −22848.0 −0.881236
\(877\) −22654.0 −0.872259 −0.436130 0.899884i \(-0.643651\pi\)
−0.436130 + 0.899884i \(0.643651\pi\)
\(878\) 49175.0 1.89018
\(879\) −20433.0 −0.784059
\(880\) 0 0
\(881\) −22380.0 −0.855847 −0.427924 0.903815i \(-0.640755\pi\)
−0.427924 + 0.903815i \(0.640755\pi\)
\(882\) −15030.0 −0.573794
\(883\) 35174.0 1.34054 0.670271 0.742116i \(-0.266178\pi\)
0.670271 + 0.742116i \(0.266178\pi\)
\(884\) 17952.0 0.683022
\(885\) 0 0
\(886\) −53725.0 −2.03716
\(887\) 30868.0 1.16848 0.584242 0.811579i \(-0.301392\pi\)
0.584242 + 0.811579i \(0.301392\pi\)
\(888\) 2565.00 0.0969322
\(889\) −5217.00 −0.196820
\(890\) 0 0
\(891\) −891.000 −0.0335013
\(892\) 60520.0 2.27170
\(893\) 9165.00 0.343443
\(894\) −3585.00 −0.134117
\(895\) 0 0
\(896\) −6345.00 −0.236575
\(897\) 10848.0 0.403795
\(898\) 41780.0 1.55258
\(899\) −9612.00 −0.356594
\(900\) 0 0
\(901\) 5016.00 0.185469
\(902\) −7645.00 −0.282207
\(903\) −2772.00 −0.102155
\(904\) −49680.0 −1.82780
\(905\) 0 0
\(906\) 18120.0 0.664455
\(907\) 10070.0 0.368654 0.184327 0.982865i \(-0.440990\pi\)
0.184327 + 0.982865i \(0.440990\pi\)
\(908\) −79526.0 −2.90657
\(909\) −4905.00 −0.178975
\(910\) 0 0
\(911\) 1885.00 0.0685542 0.0342771 0.999412i \(-0.489087\pi\)
0.0342771 + 0.999412i \(0.489087\pi\)
\(912\) 12549.0 0.455635
\(913\) −1562.00 −0.0566207
\(914\) −35290.0 −1.27712
\(915\) 0 0
\(916\) −75599.0 −2.72692
\(917\) 5454.00 0.196409
\(918\) 4455.00 0.160171
\(919\) 23703.0 0.850805 0.425403 0.905004i \(-0.360133\pi\)
0.425403 + 0.905004i \(0.360133\pi\)
\(920\) 0 0
\(921\) −1380.00 −0.0493730
\(922\) 3230.00 0.115374
\(923\) −30304.0 −1.08068
\(924\) −1683.00 −0.0599206
\(925\) 0 0
\(926\) −44910.0 −1.59377
\(927\) −11754.0 −0.416453
\(928\) −4590.00 −0.162364
\(929\) 53804.0 1.90016 0.950082 0.312001i \(-0.100999\pi\)
0.950082 + 0.312001i \(0.100999\pi\)
\(930\) 0 0
\(931\) −15698.0 −0.552611
\(932\) 6987.00 0.245565
\(933\) 24984.0 0.876677
\(934\) −67380.0 −2.36054
\(935\) 0 0
\(936\) 12960.0 0.452576
\(937\) 1326.00 0.0462311 0.0231155 0.999733i \(-0.492641\pi\)
0.0231155 + 0.999733i \(0.492641\pi\)
\(938\) 3000.00 0.104428
\(939\) −17787.0 −0.618165
\(940\) 0 0
\(941\) −27109.0 −0.939137 −0.469569 0.882896i \(-0.655591\pi\)
−0.469569 + 0.882896i \(0.655591\pi\)
\(942\) −28110.0 −0.972265
\(943\) 15707.0 0.542408
\(944\) −55625.0 −1.91784
\(945\) 0 0
\(946\) 16940.0 0.582206
\(947\) 31143.0 1.06865 0.534325 0.845279i \(-0.320566\pi\)
0.534325 + 0.845279i \(0.320566\pi\)
\(948\) −36771.0 −1.25977
\(949\) −14336.0 −0.490375
\(950\) 0 0
\(951\) 15120.0 0.515562
\(952\) 4455.00 0.151667
\(953\) −879.000 −0.0298779 −0.0149389 0.999888i \(-0.504755\pi\)
−0.0149389 + 0.999888i \(0.504755\pi\)
\(954\) 6840.00 0.232131
\(955\) 0 0
\(956\) −108460. −3.66930
\(957\) 1782.00 0.0601921
\(958\) 64980.0 2.19145
\(959\) 2610.00 0.0878846
\(960\) 0 0
\(961\) 1893.00 0.0635427
\(962\) 3040.00 0.101885
\(963\) 17442.0 0.583656
\(964\) 123794. 4.13603
\(965\) 0 0
\(966\) 5085.00 0.169366
\(967\) −14824.0 −0.492976 −0.246488 0.969146i \(-0.579277\pi\)
−0.246488 + 0.969146i \(0.579277\pi\)
\(968\) 5445.00 0.180794
\(969\) 4653.00 0.154258
\(970\) 0 0
\(971\) 34089.0 1.12664 0.563320 0.826239i \(-0.309524\pi\)
0.563320 + 0.826239i \(0.309524\pi\)
\(972\) 4131.00 0.136319
\(973\) −1908.00 −0.0628650
\(974\) −30130.0 −0.991199
\(975\) 0 0
\(976\) 28480.0 0.934040
\(977\) 33446.0 1.09522 0.547611 0.836733i \(-0.315537\pi\)
0.547611 + 0.836733i \(0.315537\pi\)
\(978\) −28560.0 −0.933792
\(979\) −4444.00 −0.145077
\(980\) 0 0
\(981\) −5184.00 −0.168718
\(982\) 58490.0 1.90070
\(983\) −52025.0 −1.68804 −0.844018 0.536315i \(-0.819816\pi\)
−0.844018 + 0.536315i \(0.819816\pi\)
\(984\) 18765.0 0.607933
\(985\) 0 0
\(986\) −8910.00 −0.287781
\(987\) 1755.00 0.0565980
\(988\) 25568.0 0.823306
\(989\) −34804.0 −1.11901
\(990\) 0 0
\(991\) −41260.0 −1.32257 −0.661285 0.750135i \(-0.729989\pi\)
−0.661285 + 0.750135i \(0.729989\pi\)
\(992\) 15130.0 0.484252
\(993\) 31188.0 0.996698
\(994\) −14205.0 −0.453275
\(995\) 0 0
\(996\) 7242.00 0.230393
\(997\) −190.000 −0.00603547 −0.00301773 0.999995i \(-0.500961\pi\)
−0.00301773 + 0.999995i \(0.500961\pi\)
\(998\) −85260.0 −2.70427
\(999\) 513.000 0.0162468
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.a.j.1.1 yes 1
3.2 odd 2 2475.4.a.a.1.1 1
5.2 odd 4 825.4.c.b.199.2 2
5.3 odd 4 825.4.c.b.199.1 2
5.4 even 2 825.4.a.a.1.1 1
15.14 odd 2 2475.4.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.a.1.1 1 5.4 even 2
825.4.a.j.1.1 yes 1 1.1 even 1 trivial
825.4.c.b.199.1 2 5.3 odd 4
825.4.c.b.199.2 2 5.2 odd 4
2475.4.a.a.1.1 1 3.2 odd 2
2475.4.a.k.1.1 1 15.14 odd 2