Properties

Label 825.4.a.o.1.1
Level $825$
Weight $4$
Character 825.1
Self dual yes
Analytic conductor $48.677$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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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: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3368.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 15x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.03932\) of defining polynomial
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)
 
\(f(q)\) \(=\) \(q-4.03932 q^{2} +3.00000 q^{3} +8.31608 q^{4} -12.1180 q^{6} +6.49923 q^{7} -1.27677 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.03932 q^{2} +3.00000 q^{3} +8.31608 q^{4} -12.1180 q^{6} +6.49923 q^{7} -1.27677 q^{8} +9.00000 q^{9} +11.0000 q^{11} +24.9483 q^{12} -30.5385 q^{13} -26.2524 q^{14} -61.3714 q^{16} +43.1314 q^{17} -36.3539 q^{18} -6.46044 q^{19} +19.4977 q^{21} -44.4325 q^{22} -108.104 q^{23} -3.83030 q^{24} +123.355 q^{26} +27.0000 q^{27} +54.0481 q^{28} -274.857 q^{29} -68.5519 q^{31} +258.113 q^{32} +33.0000 q^{33} -174.221 q^{34} +74.8448 q^{36} +402.200 q^{37} +26.0958 q^{38} -91.6156 q^{39} -268.450 q^{41} -78.7573 q^{42} +30.5334 q^{43} +91.4769 q^{44} +436.666 q^{46} -31.5850 q^{47} -184.114 q^{48} -300.760 q^{49} +129.394 q^{51} -253.961 q^{52} -252.497 q^{53} -109.062 q^{54} -8.29800 q^{56} -19.3813 q^{57} +1110.24 q^{58} -558.331 q^{59} +335.270 q^{61} +276.903 q^{62} +58.4930 q^{63} -551.628 q^{64} -133.297 q^{66} -28.7521 q^{67} +358.684 q^{68} -324.312 q^{69} +89.0081 q^{71} -11.4909 q^{72} +717.622 q^{73} -1624.61 q^{74} -53.7256 q^{76} +71.4915 q^{77} +370.065 q^{78} +200.702 q^{79} +81.0000 q^{81} +1084.36 q^{82} -243.681 q^{83} +162.144 q^{84} -123.334 q^{86} -824.571 q^{87} -14.0444 q^{88} -312.832 q^{89} -198.477 q^{91} -899.002 q^{92} -205.656 q^{93} +127.582 q^{94} +774.338 q^{96} -27.9996 q^{97} +1214.87 q^{98} +99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 9 q^{3} + 7 q^{4} - 3 q^{6} - 16 q^{7} + 3 q^{8} + 27 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - q^{2} + 9 q^{3} + 7 q^{4} - 3 q^{6} - 16 q^{7} + 3 q^{8} + 27 q^{9} + 33 q^{11} + 21 q^{12} - 45 q^{13} - 116 q^{14} - 85 q^{16} + 58 q^{17} - 9 q^{18} - 169 q^{19} - 48 q^{21} - 11 q^{22} - 155 q^{23} + 9 q^{24} + 167 q^{26} + 81 q^{27} - 100 q^{28} - 277 q^{29} - 173 q^{31} - 97 q^{32} + 99 q^{33} - 146 q^{34} + 63 q^{36} - 60 q^{37} - 169 q^{38} - 135 q^{39} + 44 q^{41} - 348 q^{42} + 109 q^{43} + 77 q^{44} + 425 q^{46} - 270 q^{47} - 255 q^{48} - 427 q^{49} + 174 q^{51} - 45 q^{52} + 148 q^{53} - 27 q^{54} + 168 q^{56} - 507 q^{57} + 783 q^{58} - 684 q^{59} - 1038 q^{61} - 953 q^{62} - 144 q^{63} - 1129 q^{64} - 33 q^{66} - 314 q^{67} + 366 q^{68} - 465 q^{69} - 1459 q^{71} + 27 q^{72} + 1170 q^{73} - 1764 q^{74} + 211 q^{76} - 176 q^{77} + 501 q^{78} - 506 q^{79} + 243 q^{81} + 1040 q^{82} + 347 q^{83} - 300 q^{84} - 527 q^{86} - 831 q^{87} + 33 q^{88} - 607 q^{89} - 398 q^{91} - 687 q^{92} - 519 q^{93} - 1822 q^{94} - 291 q^{96} - 1263 q^{97} + 2273 q^{98} + 297 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.03932 −1.42811 −0.714057 0.700087i \(-0.753144\pi\)
−0.714057 + 0.700087i \(0.753144\pi\)
\(3\) 3.00000 0.577350
\(4\) 8.31608 1.03951
\(5\) 0 0
\(6\) −12.1180 −0.824522
\(7\) 6.49923 0.350925 0.175463 0.984486i \(-0.443858\pi\)
0.175463 + 0.984486i \(0.443858\pi\)
\(8\) −1.27677 −0.0564257
\(9\) 9.00000 0.333333
\(10\) 0 0
\(11\) 11.0000 0.301511
\(12\) 24.9483 0.600162
\(13\) −30.5385 −0.651528 −0.325764 0.945451i \(-0.605622\pi\)
−0.325764 + 0.945451i \(0.605622\pi\)
\(14\) −26.2524 −0.501161
\(15\) 0 0
\(16\) −61.3714 −0.958928
\(17\) 43.1314 0.615347 0.307674 0.951492i \(-0.400450\pi\)
0.307674 + 0.951492i \(0.400450\pi\)
\(18\) −36.3539 −0.476038
\(19\) −6.46044 −0.0780066 −0.0390033 0.999239i \(-0.512418\pi\)
−0.0390033 + 0.999239i \(0.512418\pi\)
\(20\) 0 0
\(21\) 19.4977 0.202607
\(22\) −44.4325 −0.430593
\(23\) −108.104 −0.980054 −0.490027 0.871707i \(-0.663013\pi\)
−0.490027 + 0.871707i \(0.663013\pi\)
\(24\) −3.83030 −0.0325774
\(25\) 0 0
\(26\) 123.355 0.930457
\(27\) 27.0000 0.192450
\(28\) 54.0481 0.364791
\(29\) −274.857 −1.75999 −0.879995 0.474984i \(-0.842454\pi\)
−0.879995 + 0.474984i \(0.842454\pi\)
\(30\) 0 0
\(31\) −68.5519 −0.397170 −0.198585 0.980084i \(-0.563635\pi\)
−0.198585 + 0.980084i \(0.563635\pi\)
\(32\) 258.113 1.42588
\(33\) 33.0000 0.174078
\(34\) −174.221 −0.878786
\(35\) 0 0
\(36\) 74.8448 0.346504
\(37\) 402.200 1.78706 0.893530 0.449004i \(-0.148221\pi\)
0.893530 + 0.449004i \(0.148221\pi\)
\(38\) 26.0958 0.111402
\(39\) −91.6156 −0.376160
\(40\) 0 0
\(41\) −268.450 −1.02256 −0.511280 0.859414i \(-0.670828\pi\)
−0.511280 + 0.859414i \(0.670828\pi\)
\(42\) −78.7573 −0.289346
\(43\) 30.5334 0.108286 0.0541431 0.998533i \(-0.482757\pi\)
0.0541431 + 0.998533i \(0.482757\pi\)
\(44\) 91.4769 0.313424
\(45\) 0 0
\(46\) 436.666 1.39963
\(47\) −31.5850 −0.0980243 −0.0490121 0.998798i \(-0.515607\pi\)
−0.0490121 + 0.998798i \(0.515607\pi\)
\(48\) −184.114 −0.553638
\(49\) −300.760 −0.876851
\(50\) 0 0
\(51\) 129.394 0.355271
\(52\) −253.961 −0.677271
\(53\) −252.497 −0.654398 −0.327199 0.944956i \(-0.606105\pi\)
−0.327199 + 0.944956i \(0.606105\pi\)
\(54\) −109.062 −0.274841
\(55\) 0 0
\(56\) −8.29800 −0.0198012
\(57\) −19.3813 −0.0450372
\(58\) 1110.24 2.51347
\(59\) −558.331 −1.23201 −0.616004 0.787743i \(-0.711250\pi\)
−0.616004 + 0.787743i \(0.711250\pi\)
\(60\) 0 0
\(61\) 335.270 0.703719 0.351860 0.936053i \(-0.385549\pi\)
0.351860 + 0.936053i \(0.385549\pi\)
\(62\) 276.903 0.567205
\(63\) 58.4930 0.116975
\(64\) −551.628 −1.07740
\(65\) 0 0
\(66\) −133.297 −0.248603
\(67\) −28.7521 −0.0524273 −0.0262136 0.999656i \(-0.508345\pi\)
−0.0262136 + 0.999656i \(0.508345\pi\)
\(68\) 358.684 0.639660
\(69\) −324.312 −0.565834
\(70\) 0 0
\(71\) 89.0081 0.148779 0.0743895 0.997229i \(-0.476299\pi\)
0.0743895 + 0.997229i \(0.476299\pi\)
\(72\) −11.4909 −0.0188086
\(73\) 717.622 1.15057 0.575283 0.817954i \(-0.304892\pi\)
0.575283 + 0.817954i \(0.304892\pi\)
\(74\) −1624.61 −2.55213
\(75\) 0 0
\(76\) −53.7256 −0.0810887
\(77\) 71.4915 0.105808
\(78\) 370.065 0.537200
\(79\) 200.702 0.285832 0.142916 0.989735i \(-0.454352\pi\)
0.142916 + 0.989735i \(0.454352\pi\)
\(80\) 0 0
\(81\) 81.0000 0.111111
\(82\) 1084.36 1.46033
\(83\) −243.681 −0.322258 −0.161129 0.986933i \(-0.551514\pi\)
−0.161129 + 0.986933i \(0.551514\pi\)
\(84\) 162.144 0.210612
\(85\) 0 0
\(86\) −123.334 −0.154645
\(87\) −824.571 −1.01613
\(88\) −14.0444 −0.0170130
\(89\) −312.832 −0.372585 −0.186293 0.982494i \(-0.559647\pi\)
−0.186293 + 0.982494i \(0.559647\pi\)
\(90\) 0 0
\(91\) −198.477 −0.228638
\(92\) −899.002 −1.01878
\(93\) −205.656 −0.229306
\(94\) 127.582 0.139990
\(95\) 0 0
\(96\) 774.338 0.823235
\(97\) −27.9996 −0.0293085 −0.0146543 0.999893i \(-0.504665\pi\)
−0.0146543 + 0.999893i \(0.504665\pi\)
\(98\) 1214.87 1.25224
\(99\) 99.0000 0.100504
\(100\) 0 0
\(101\) −228.278 −0.224896 −0.112448 0.993658i \(-0.535869\pi\)
−0.112448 + 0.993658i \(0.535869\pi\)
\(102\) −522.664 −0.507367
\(103\) 800.328 0.765618 0.382809 0.923827i \(-0.374957\pi\)
0.382809 + 0.923827i \(0.374957\pi\)
\(104\) 38.9906 0.0367629
\(105\) 0 0
\(106\) 1019.91 0.934555
\(107\) 1046.89 0.945857 0.472929 0.881101i \(-0.343197\pi\)
0.472929 + 0.881101i \(0.343197\pi\)
\(108\) 224.534 0.200054
\(109\) 317.023 0.278581 0.139291 0.990252i \(-0.455518\pi\)
0.139291 + 0.990252i \(0.455518\pi\)
\(110\) 0 0
\(111\) 1206.60 1.03176
\(112\) −398.867 −0.336512
\(113\) −1333.82 −1.11040 −0.555198 0.831718i \(-0.687358\pi\)
−0.555198 + 0.831718i \(0.687358\pi\)
\(114\) 78.2873 0.0643182
\(115\) 0 0
\(116\) −2285.74 −1.82953
\(117\) −274.847 −0.217176
\(118\) 2255.28 1.75945
\(119\) 280.321 0.215941
\(120\) 0 0
\(121\) 121.000 0.0909091
\(122\) −1354.26 −1.00499
\(123\) −805.351 −0.590375
\(124\) −570.083 −0.412863
\(125\) 0 0
\(126\) −236.272 −0.167054
\(127\) −564.888 −0.394691 −0.197345 0.980334i \(-0.563232\pi\)
−0.197345 + 0.980334i \(0.563232\pi\)
\(128\) 163.299 0.112763
\(129\) 91.6003 0.0625190
\(130\) 0 0
\(131\) −214.082 −0.142782 −0.0713908 0.997448i \(-0.522744\pi\)
−0.0713908 + 0.997448i \(0.522744\pi\)
\(132\) 274.431 0.180956
\(133\) −41.9879 −0.0273745
\(134\) 116.139 0.0748721
\(135\) 0 0
\(136\) −55.0688 −0.0347214
\(137\) 65.2288 0.0406779 0.0203389 0.999793i \(-0.493525\pi\)
0.0203389 + 0.999793i \(0.493525\pi\)
\(138\) 1310.00 0.808076
\(139\) −407.084 −0.248406 −0.124203 0.992257i \(-0.539637\pi\)
−0.124203 + 0.992257i \(0.539637\pi\)
\(140\) 0 0
\(141\) −94.7549 −0.0565943
\(142\) −359.532 −0.212474
\(143\) −335.924 −0.196443
\(144\) −552.343 −0.319643
\(145\) 0 0
\(146\) −2898.70 −1.64314
\(147\) −902.280 −0.506250
\(148\) 3344.73 1.85767
\(149\) 792.092 0.435508 0.217754 0.976004i \(-0.430127\pi\)
0.217754 + 0.976004i \(0.430127\pi\)
\(150\) 0 0
\(151\) −2861.80 −1.54232 −0.771159 0.636642i \(-0.780323\pi\)
−0.771159 + 0.636642i \(0.780323\pi\)
\(152\) 8.24848 0.00440158
\(153\) 388.183 0.205116
\(154\) −288.777 −0.151106
\(155\) 0 0
\(156\) −761.883 −0.391022
\(157\) −2991.55 −1.52071 −0.760355 0.649507i \(-0.774975\pi\)
−0.760355 + 0.649507i \(0.774975\pi\)
\(158\) −810.698 −0.408201
\(159\) −757.490 −0.377817
\(160\) 0 0
\(161\) −702.592 −0.343926
\(162\) −327.185 −0.158679
\(163\) −2907.60 −1.39718 −0.698592 0.715520i \(-0.746190\pi\)
−0.698592 + 0.715520i \(0.746190\pi\)
\(164\) −2232.46 −1.06296
\(165\) 0 0
\(166\) 984.304 0.460222
\(167\) −1100.39 −0.509885 −0.254942 0.966956i \(-0.582056\pi\)
−0.254942 + 0.966956i \(0.582056\pi\)
\(168\) −24.8940 −0.0114322
\(169\) −1264.40 −0.575511
\(170\) 0 0
\(171\) −58.1439 −0.0260022
\(172\) 253.919 0.112565
\(173\) −3754.75 −1.65011 −0.825054 0.565054i \(-0.808855\pi\)
−0.825054 + 0.565054i \(0.808855\pi\)
\(174\) 3330.71 1.45115
\(175\) 0 0
\(176\) −675.086 −0.289128
\(177\) −1674.99 −0.711301
\(178\) 1263.63 0.532094
\(179\) −3218.79 −1.34404 −0.672021 0.740532i \(-0.734574\pi\)
−0.672021 + 0.740532i \(0.734574\pi\)
\(180\) 0 0
\(181\) −456.885 −0.187624 −0.0938121 0.995590i \(-0.529905\pi\)
−0.0938121 + 0.995590i \(0.529905\pi\)
\(182\) 801.711 0.326521
\(183\) 1005.81 0.406292
\(184\) 138.024 0.0553002
\(185\) 0 0
\(186\) 830.709 0.327476
\(187\) 474.445 0.185534
\(188\) −262.663 −0.101897
\(189\) 175.479 0.0675356
\(190\) 0 0
\(191\) −4907.11 −1.85899 −0.929493 0.368841i \(-0.879755\pi\)
−0.929493 + 0.368841i \(0.879755\pi\)
\(192\) −1654.88 −0.622036
\(193\) −1462.17 −0.545333 −0.272667 0.962109i \(-0.587906\pi\)
−0.272667 + 0.962109i \(0.587906\pi\)
\(194\) 113.099 0.0418559
\(195\) 0 0
\(196\) −2501.15 −0.911496
\(197\) −2210.10 −0.799307 −0.399653 0.916666i \(-0.630869\pi\)
−0.399653 + 0.916666i \(0.630869\pi\)
\(198\) −399.892 −0.143531
\(199\) −395.272 −0.140804 −0.0704022 0.997519i \(-0.522428\pi\)
−0.0704022 + 0.997519i \(0.522428\pi\)
\(200\) 0 0
\(201\) −86.2563 −0.0302689
\(202\) 922.086 0.321177
\(203\) −1786.36 −0.617625
\(204\) 1076.05 0.369308
\(205\) 0 0
\(206\) −3232.78 −1.09339
\(207\) −972.936 −0.326685
\(208\) 1874.19 0.624769
\(209\) −71.0648 −0.0235199
\(210\) 0 0
\(211\) 1559.98 0.508974 0.254487 0.967076i \(-0.418093\pi\)
0.254487 + 0.967076i \(0.418093\pi\)
\(212\) −2099.78 −0.680253
\(213\) 267.024 0.0858976
\(214\) −4228.72 −1.35079
\(215\) 0 0
\(216\) −34.4727 −0.0108591
\(217\) −445.534 −0.139377
\(218\) −1280.56 −0.397846
\(219\) 2152.87 0.664280
\(220\) 0 0
\(221\) −1317.17 −0.400916
\(222\) −4873.84 −1.47347
\(223\) −756.215 −0.227085 −0.113542 0.993533i \(-0.536220\pi\)
−0.113542 + 0.993533i \(0.536220\pi\)
\(224\) 1677.53 0.500379
\(225\) 0 0
\(226\) 5387.70 1.58577
\(227\) 5758.69 1.68378 0.841890 0.539649i \(-0.181443\pi\)
0.841890 + 0.539649i \(0.181443\pi\)
\(228\) −161.177 −0.0468166
\(229\) −1912.63 −0.551921 −0.275961 0.961169i \(-0.588996\pi\)
−0.275961 + 0.961169i \(0.588996\pi\)
\(230\) 0 0
\(231\) 214.474 0.0610882
\(232\) 350.929 0.0993086
\(233\) −1034.86 −0.290970 −0.145485 0.989360i \(-0.546474\pi\)
−0.145485 + 0.989360i \(0.546474\pi\)
\(234\) 1110.19 0.310152
\(235\) 0 0
\(236\) −4643.13 −1.28069
\(237\) 602.105 0.165025
\(238\) −1132.30 −0.308388
\(239\) 2647.23 0.716465 0.358232 0.933632i \(-0.383380\pi\)
0.358232 + 0.933632i \(0.383380\pi\)
\(240\) 0 0
\(241\) 6734.00 1.79990 0.899948 0.435996i \(-0.143604\pi\)
0.899948 + 0.435996i \(0.143604\pi\)
\(242\) −488.757 −0.129829
\(243\) 243.000 0.0641500
\(244\) 2788.13 0.731523
\(245\) 0 0
\(246\) 3253.07 0.843123
\(247\) 197.292 0.0508235
\(248\) 87.5248 0.0224106
\(249\) −731.043 −0.186056
\(250\) 0 0
\(251\) −5377.57 −1.35231 −0.676154 0.736761i \(-0.736354\pi\)
−0.676154 + 0.736761i \(0.736354\pi\)
\(252\) 486.433 0.121597
\(253\) −1189.14 −0.295497
\(254\) 2281.76 0.563663
\(255\) 0 0
\(256\) 3753.41 0.916360
\(257\) −5882.33 −1.42774 −0.713871 0.700278i \(-0.753060\pi\)
−0.713871 + 0.700278i \(0.753060\pi\)
\(258\) −370.003 −0.0892843
\(259\) 2613.99 0.627124
\(260\) 0 0
\(261\) −2473.71 −0.586663
\(262\) 864.744 0.203909
\(263\) 4001.67 0.938227 0.469113 0.883138i \(-0.344574\pi\)
0.469113 + 0.883138i \(0.344574\pi\)
\(264\) −42.1333 −0.00982245
\(265\) 0 0
\(266\) 169.602 0.0390939
\(267\) −938.495 −0.215112
\(268\) −239.105 −0.0544987
\(269\) −1758.29 −0.398530 −0.199265 0.979946i \(-0.563856\pi\)
−0.199265 + 0.979946i \(0.563856\pi\)
\(270\) 0 0
\(271\) 6018.44 1.34906 0.674528 0.738249i \(-0.264347\pi\)
0.674528 + 0.738249i \(0.264347\pi\)
\(272\) −2647.03 −0.590074
\(273\) −595.431 −0.132004
\(274\) −263.480 −0.0580927
\(275\) 0 0
\(276\) −2697.01 −0.588191
\(277\) −2910.30 −0.631275 −0.315637 0.948880i \(-0.602218\pi\)
−0.315637 + 0.948880i \(0.602218\pi\)
\(278\) 1644.34 0.354752
\(279\) −616.967 −0.132390
\(280\) 0 0
\(281\) 6893.19 1.46339 0.731697 0.681631i \(-0.238729\pi\)
0.731697 + 0.681631i \(0.238729\pi\)
\(282\) 382.745 0.0808232
\(283\) −3120.16 −0.655385 −0.327693 0.944784i \(-0.606271\pi\)
−0.327693 + 0.944784i \(0.606271\pi\)
\(284\) 740.199 0.154657
\(285\) 0 0
\(286\) 1356.90 0.280543
\(287\) −1744.72 −0.358842
\(288\) 2323.01 0.475295
\(289\) −3052.68 −0.621348
\(290\) 0 0
\(291\) −83.9988 −0.0169213
\(292\) 5967.81 1.19603
\(293\) −8869.62 −1.76849 −0.884247 0.467020i \(-0.845328\pi\)
−0.884247 + 0.467020i \(0.845328\pi\)
\(294\) 3644.60 0.722983
\(295\) 0 0
\(296\) −513.515 −0.100836
\(297\) 297.000 0.0580259
\(298\) −3199.51 −0.621955
\(299\) 3301.34 0.638533
\(300\) 0 0
\(301\) 198.444 0.0380003
\(302\) 11559.7 2.20261
\(303\) −684.833 −0.129844
\(304\) 396.486 0.0748028
\(305\) 0 0
\(306\) −1567.99 −0.292929
\(307\) 5713.42 1.06216 0.531078 0.847323i \(-0.321787\pi\)
0.531078 + 0.847323i \(0.321787\pi\)
\(308\) 594.529 0.109988
\(309\) 2400.98 0.442030
\(310\) 0 0
\(311\) 1984.01 0.361746 0.180873 0.983506i \(-0.442108\pi\)
0.180873 + 0.983506i \(0.442108\pi\)
\(312\) 116.972 0.0212251
\(313\) 2675.50 0.483156 0.241578 0.970381i \(-0.422335\pi\)
0.241578 + 0.970381i \(0.422335\pi\)
\(314\) 12083.8 2.17175
\(315\) 0 0
\(316\) 1669.05 0.297125
\(317\) 7083.76 1.25509 0.627545 0.778580i \(-0.284060\pi\)
0.627545 + 0.778580i \(0.284060\pi\)
\(318\) 3059.74 0.539565
\(319\) −3023.43 −0.530657
\(320\) 0 0
\(321\) 3140.67 0.546091
\(322\) 2837.99 0.491165
\(323\) −278.648 −0.0480012
\(324\) 673.603 0.115501
\(325\) 0 0
\(326\) 11744.7 1.99534
\(327\) 951.070 0.160839
\(328\) 342.749 0.0576986
\(329\) −205.278 −0.0343992
\(330\) 0 0
\(331\) 7812.39 1.29730 0.648652 0.761085i \(-0.275333\pi\)
0.648652 + 0.761085i \(0.275333\pi\)
\(332\) −2026.47 −0.334991
\(333\) 3619.80 0.595686
\(334\) 4444.82 0.728173
\(335\) 0 0
\(336\) −1196.60 −0.194285
\(337\) 11866.4 1.91811 0.959056 0.283218i \(-0.0914019\pi\)
0.959056 + 0.283218i \(0.0914019\pi\)
\(338\) 5107.30 0.821895
\(339\) −4001.45 −0.641088
\(340\) 0 0
\(341\) −754.071 −0.119751
\(342\) 234.862 0.0371341
\(343\) −4183.94 −0.658635
\(344\) −38.9841 −0.00611012
\(345\) 0 0
\(346\) 15166.6 2.35654
\(347\) −3485.50 −0.539225 −0.269613 0.962969i \(-0.586896\pi\)
−0.269613 + 0.962969i \(0.586896\pi\)
\(348\) −6857.21 −1.05628
\(349\) −6366.76 −0.976518 −0.488259 0.872699i \(-0.662368\pi\)
−0.488259 + 0.872699i \(0.662368\pi\)
\(350\) 0 0
\(351\) −824.541 −0.125387
\(352\) 2839.24 0.429920
\(353\) −8223.15 −1.23987 −0.619935 0.784653i \(-0.712841\pi\)
−0.619935 + 0.784653i \(0.712841\pi\)
\(354\) 6765.83 1.01582
\(355\) 0 0
\(356\) −2601.54 −0.387306
\(357\) 840.962 0.124674
\(358\) 13001.7 1.91945
\(359\) −9377.54 −1.37863 −0.689314 0.724462i \(-0.742088\pi\)
−0.689314 + 0.724462i \(0.742088\pi\)
\(360\) 0 0
\(361\) −6817.26 −0.993915
\(362\) 1845.50 0.267949
\(363\) 363.000 0.0524864
\(364\) −1650.55 −0.237671
\(365\) 0 0
\(366\) −4062.78 −0.580232
\(367\) 310.895 0.0442196 0.0221098 0.999756i \(-0.492962\pi\)
0.0221098 + 0.999756i \(0.492962\pi\)
\(368\) 6634.49 0.939801
\(369\) −2416.05 −0.340853
\(370\) 0 0
\(371\) −1641.03 −0.229645
\(372\) −1710.25 −0.238366
\(373\) 4738.51 0.657777 0.328889 0.944369i \(-0.393326\pi\)
0.328889 + 0.944369i \(0.393326\pi\)
\(374\) −1916.44 −0.264964
\(375\) 0 0
\(376\) 40.3267 0.00553109
\(377\) 8393.74 1.14668
\(378\) −708.816 −0.0964486
\(379\) −4735.47 −0.641806 −0.320903 0.947112i \(-0.603986\pi\)
−0.320903 + 0.947112i \(0.603986\pi\)
\(380\) 0 0
\(381\) −1694.66 −0.227875
\(382\) 19821.4 2.65484
\(383\) −3632.14 −0.484579 −0.242289 0.970204i \(-0.577898\pi\)
−0.242289 + 0.970204i \(0.577898\pi\)
\(384\) 489.896 0.0651039
\(385\) 0 0
\(386\) 5906.17 0.778798
\(387\) 274.801 0.0360954
\(388\) −232.847 −0.0304665
\(389\) 6557.50 0.854701 0.427350 0.904086i \(-0.359447\pi\)
0.427350 + 0.904086i \(0.359447\pi\)
\(390\) 0 0
\(391\) −4662.68 −0.603073
\(392\) 384.001 0.0494769
\(393\) −642.245 −0.0824350
\(394\) 8927.31 1.14150
\(395\) 0 0
\(396\) 823.292 0.104475
\(397\) −4084.97 −0.516420 −0.258210 0.966089i \(-0.583133\pi\)
−0.258210 + 0.966089i \(0.583133\pi\)
\(398\) 1596.63 0.201085
\(399\) −125.964 −0.0158047
\(400\) 0 0
\(401\) −6676.04 −0.831386 −0.415693 0.909505i \(-0.636461\pi\)
−0.415693 + 0.909505i \(0.636461\pi\)
\(402\) 348.417 0.0432275
\(403\) 2093.47 0.258768
\(404\) −1898.38 −0.233782
\(405\) 0 0
\(406\) 7215.67 0.882039
\(407\) 4424.20 0.538819
\(408\) −165.206 −0.0200464
\(409\) 1718.56 0.207769 0.103884 0.994589i \(-0.466873\pi\)
0.103884 + 0.994589i \(0.466873\pi\)
\(410\) 0 0
\(411\) 195.686 0.0234854
\(412\) 6655.60 0.795868
\(413\) −3628.72 −0.432343
\(414\) 3930.00 0.466543
\(415\) 0 0
\(416\) −7882.39 −0.929004
\(417\) −1221.25 −0.143417
\(418\) 287.053 0.0335891
\(419\) 5732.23 0.668348 0.334174 0.942511i \(-0.391543\pi\)
0.334174 + 0.942511i \(0.391543\pi\)
\(420\) 0 0
\(421\) −9647.38 −1.11683 −0.558414 0.829563i \(-0.688590\pi\)
−0.558414 + 0.829563i \(0.688590\pi\)
\(422\) −6301.26 −0.726873
\(423\) −284.265 −0.0326748
\(424\) 322.379 0.0369248
\(425\) 0 0
\(426\) −1078.60 −0.122672
\(427\) 2178.99 0.246953
\(428\) 8706.03 0.983229
\(429\) −1007.77 −0.113417
\(430\) 0 0
\(431\) −3054.57 −0.341376 −0.170688 0.985325i \(-0.554599\pi\)
−0.170688 + 0.985325i \(0.554599\pi\)
\(432\) −1657.03 −0.184546
\(433\) −11022.8 −1.22337 −0.611686 0.791100i \(-0.709509\pi\)
−0.611686 + 0.791100i \(0.709509\pi\)
\(434\) 1799.65 0.199046
\(435\) 0 0
\(436\) 2636.39 0.289588
\(437\) 698.399 0.0764507
\(438\) −8696.11 −0.948667
\(439\) −7589.30 −0.825097 −0.412548 0.910936i \(-0.635361\pi\)
−0.412548 + 0.910936i \(0.635361\pi\)
\(440\) 0 0
\(441\) −2706.84 −0.292284
\(442\) 5320.47 0.572554
\(443\) −4043.30 −0.433641 −0.216821 0.976211i \(-0.569569\pi\)
−0.216821 + 0.976211i \(0.569569\pi\)
\(444\) 10034.2 1.07252
\(445\) 0 0
\(446\) 3054.59 0.324303
\(447\) 2376.28 0.251441
\(448\) −3585.16 −0.378086
\(449\) 14619.3 1.53658 0.768291 0.640101i \(-0.221107\pi\)
0.768291 + 0.640101i \(0.221107\pi\)
\(450\) 0 0
\(451\) −2952.96 −0.308313
\(452\) −11092.1 −1.15427
\(453\) −8585.40 −0.890458
\(454\) −23261.2 −2.40463
\(455\) 0 0
\(456\) 24.7454 0.00254125
\(457\) 10910.8 1.11682 0.558411 0.829565i \(-0.311411\pi\)
0.558411 + 0.829565i \(0.311411\pi\)
\(458\) 7725.71 0.788207
\(459\) 1164.55 0.118424
\(460\) 0 0
\(461\) 16371.0 1.65395 0.826977 0.562236i \(-0.190059\pi\)
0.826977 + 0.562236i \(0.190059\pi\)
\(462\) −866.331 −0.0872410
\(463\) 16023.5 1.60837 0.804186 0.594377i \(-0.202601\pi\)
0.804186 + 0.594377i \(0.202601\pi\)
\(464\) 16868.4 1.68770
\(465\) 0 0
\(466\) 4180.13 0.415538
\(467\) 15097.3 1.49597 0.747985 0.663716i \(-0.231022\pi\)
0.747985 + 0.663716i \(0.231022\pi\)
\(468\) −2285.65 −0.225757
\(469\) −186.866 −0.0183981
\(470\) 0 0
\(471\) −8974.65 −0.877983
\(472\) 712.859 0.0695170
\(473\) 335.868 0.0326495
\(474\) −2432.09 −0.235675
\(475\) 0 0
\(476\) 2331.17 0.224473
\(477\) −2272.47 −0.218133
\(478\) −10693.0 −1.02319
\(479\) −10459.0 −0.997667 −0.498834 0.866698i \(-0.666238\pi\)
−0.498834 + 0.866698i \(0.666238\pi\)
\(480\) 0 0
\(481\) −12282.6 −1.16432
\(482\) −27200.8 −2.57046
\(483\) −2107.78 −0.198566
\(484\) 1006.25 0.0945010
\(485\) 0 0
\(486\) −981.554 −0.0916136
\(487\) 7939.86 0.738788 0.369394 0.929273i \(-0.379565\pi\)
0.369394 + 0.929273i \(0.379565\pi\)
\(488\) −428.061 −0.0397078
\(489\) −8722.81 −0.806665
\(490\) 0 0
\(491\) 11492.7 1.05633 0.528167 0.849141i \(-0.322880\pi\)
0.528167 + 0.849141i \(0.322880\pi\)
\(492\) −6697.37 −0.613701
\(493\) −11855.0 −1.08300
\(494\) −796.927 −0.0725818
\(495\) 0 0
\(496\) 4207.13 0.380858
\(497\) 578.484 0.0522103
\(498\) 2952.91 0.265709
\(499\) 4813.54 0.431831 0.215915 0.976412i \(-0.430726\pi\)
0.215915 + 0.976412i \(0.430726\pi\)
\(500\) 0 0
\(501\) −3301.17 −0.294382
\(502\) 21721.7 1.93125
\(503\) −17756.3 −1.57399 −0.786993 0.616962i \(-0.788363\pi\)
−0.786993 + 0.616962i \(0.788363\pi\)
\(504\) −74.6820 −0.00660040
\(505\) 0 0
\(506\) 4803.33 0.422004
\(507\) −3793.19 −0.332271
\(508\) −4697.66 −0.410285
\(509\) 3183.05 0.277183 0.138592 0.990350i \(-0.455742\pi\)
0.138592 + 0.990350i \(0.455742\pi\)
\(510\) 0 0
\(511\) 4663.99 0.403763
\(512\) −16467.6 −1.42143
\(513\) −174.432 −0.0150124
\(514\) 23760.6 2.03898
\(515\) 0 0
\(516\) 761.756 0.0649892
\(517\) −347.435 −0.0295554
\(518\) −10558.7 −0.895605
\(519\) −11264.3 −0.952690
\(520\) 0 0
\(521\) 7246.60 0.609365 0.304683 0.952454i \(-0.401450\pi\)
0.304683 + 0.952454i \(0.401450\pi\)
\(522\) 9992.12 0.837822
\(523\) 17985.0 1.50369 0.751845 0.659340i \(-0.229164\pi\)
0.751845 + 0.659340i \(0.229164\pi\)
\(524\) −1780.32 −0.148423
\(525\) 0 0
\(526\) −16164.0 −1.33990
\(527\) −2956.74 −0.244398
\(528\) −2025.26 −0.166928
\(529\) −480.530 −0.0394945
\(530\) 0 0
\(531\) −5024.98 −0.410670
\(532\) −349.175 −0.0284561
\(533\) 8198.09 0.666226
\(534\) 3790.88 0.307205
\(535\) 0 0
\(536\) 36.7097 0.00295825
\(537\) −9656.36 −0.775983
\(538\) 7102.28 0.569147
\(539\) −3308.36 −0.264381
\(540\) 0 0
\(541\) −14549.3 −1.15624 −0.578119 0.815952i \(-0.696213\pi\)
−0.578119 + 0.815952i \(0.696213\pi\)
\(542\) −24310.4 −1.92661
\(543\) −1370.65 −0.108325
\(544\) 11132.8 0.877414
\(545\) 0 0
\(546\) 2405.13 0.188517
\(547\) −5487.29 −0.428920 −0.214460 0.976733i \(-0.568799\pi\)
−0.214460 + 0.976733i \(0.568799\pi\)
\(548\) 542.448 0.0422851
\(549\) 3017.43 0.234573
\(550\) 0 0
\(551\) 1775.70 0.137291
\(552\) 414.071 0.0319276
\(553\) 1304.41 0.100306
\(554\) 11755.6 0.901532
\(555\) 0 0
\(556\) −3385.35 −0.258221
\(557\) 7366.60 0.560382 0.280191 0.959944i \(-0.409602\pi\)
0.280191 + 0.959944i \(0.409602\pi\)
\(558\) 2492.13 0.189068
\(559\) −932.446 −0.0705515
\(560\) 0 0
\(561\) 1423.34 0.107118
\(562\) −27843.8 −2.08989
\(563\) 4516.14 0.338068 0.169034 0.985610i \(-0.445935\pi\)
0.169034 + 0.985610i \(0.445935\pi\)
\(564\) −787.990 −0.0588304
\(565\) 0 0
\(566\) 12603.3 0.935965
\(567\) 526.437 0.0389917
\(568\) −113.643 −0.00839496
\(569\) 2174.35 0.160200 0.0800999 0.996787i \(-0.474476\pi\)
0.0800999 + 0.996787i \(0.474476\pi\)
\(570\) 0 0
\(571\) −25339.7 −1.85715 −0.928575 0.371145i \(-0.878965\pi\)
−0.928575 + 0.371145i \(0.878965\pi\)
\(572\) −2793.57 −0.204205
\(573\) −14721.3 −1.07329
\(574\) 7047.48 0.512467
\(575\) 0 0
\(576\) −4964.65 −0.359133
\(577\) 21909.1 1.58074 0.790370 0.612630i \(-0.209888\pi\)
0.790370 + 0.612630i \(0.209888\pi\)
\(578\) 12330.8 0.887356
\(579\) −4386.51 −0.314848
\(580\) 0 0
\(581\) −1583.74 −0.113089
\(582\) 339.298 0.0241655
\(583\) −2777.46 −0.197308
\(584\) −916.237 −0.0649215
\(585\) 0 0
\(586\) 35827.2 2.52561
\(587\) 922.235 0.0648462 0.0324231 0.999474i \(-0.489678\pi\)
0.0324231 + 0.999474i \(0.489678\pi\)
\(588\) −7503.44 −0.526253
\(589\) 442.875 0.0309819
\(590\) 0 0
\(591\) −6630.31 −0.461480
\(592\) −24683.6 −1.71366
\(593\) 9075.86 0.628501 0.314250 0.949340i \(-0.398247\pi\)
0.314250 + 0.949340i \(0.398247\pi\)
\(594\) −1199.68 −0.0828676
\(595\) 0 0
\(596\) 6587.10 0.452715
\(597\) −1185.82 −0.0812935
\(598\) −13335.2 −0.911898
\(599\) 19987.0 1.36335 0.681675 0.731655i \(-0.261252\pi\)
0.681675 + 0.731655i \(0.261252\pi\)
\(600\) 0 0
\(601\) 11116.7 0.754509 0.377255 0.926110i \(-0.376868\pi\)
0.377255 + 0.926110i \(0.376868\pi\)
\(602\) −801.577 −0.0542688
\(603\) −258.769 −0.0174758
\(604\) −23799.0 −1.60326
\(605\) 0 0
\(606\) 2766.26 0.185432
\(607\) −19355.0 −1.29423 −0.647114 0.762393i \(-0.724024\pi\)
−0.647114 + 0.762393i \(0.724024\pi\)
\(608\) −1667.52 −0.111228
\(609\) −5359.08 −0.356586
\(610\) 0 0
\(611\) 964.559 0.0638656
\(612\) 3228.16 0.213220
\(613\) 12325.5 0.812111 0.406056 0.913848i \(-0.366904\pi\)
0.406056 + 0.913848i \(0.366904\pi\)
\(614\) −23078.3 −1.51688
\(615\) 0 0
\(616\) −91.2780 −0.00597029
\(617\) −28207.0 −1.84047 −0.920236 0.391363i \(-0.872004\pi\)
−0.920236 + 0.391363i \(0.872004\pi\)
\(618\) −9698.34 −0.631269
\(619\) 17236.9 1.11924 0.559621 0.828748i \(-0.310947\pi\)
0.559621 + 0.828748i \(0.310947\pi\)
\(620\) 0 0
\(621\) −2918.81 −0.188611
\(622\) −8014.06 −0.516615
\(623\) −2033.16 −0.130750
\(624\) 5622.58 0.360711
\(625\) 0 0
\(626\) −10807.2 −0.690003
\(627\) −213.194 −0.0135792
\(628\) −24878.0 −1.58080
\(629\) 17347.4 1.09966
\(630\) 0 0
\(631\) −23582.7 −1.48782 −0.743908 0.668282i \(-0.767030\pi\)
−0.743908 + 0.668282i \(0.767030\pi\)
\(632\) −256.250 −0.0161283
\(633\) 4679.94 0.293856
\(634\) −28613.6 −1.79241
\(635\) 0 0
\(636\) −6299.35 −0.392744
\(637\) 9184.77 0.571294
\(638\) 12212.6 0.757839
\(639\) 801.073 0.0495930
\(640\) 0 0
\(641\) −13003.1 −0.801233 −0.400617 0.916246i \(-0.631204\pi\)
−0.400617 + 0.916246i \(0.631204\pi\)
\(642\) −12686.2 −0.779880
\(643\) 21269.1 1.30446 0.652232 0.758019i \(-0.273833\pi\)
0.652232 + 0.758019i \(0.273833\pi\)
\(644\) −5842.82 −0.357514
\(645\) 0 0
\(646\) 1125.55 0.0685511
\(647\) −29247.2 −1.77716 −0.888582 0.458717i \(-0.848309\pi\)
−0.888582 + 0.458717i \(0.848309\pi\)
\(648\) −103.418 −0.00626952
\(649\) −6141.64 −0.371465
\(650\) 0 0
\(651\) −1336.60 −0.0804694
\(652\) −24179.9 −1.45239
\(653\) 1678.30 0.100577 0.0502885 0.998735i \(-0.483986\pi\)
0.0502885 + 0.998735i \(0.483986\pi\)
\(654\) −3841.67 −0.229696
\(655\) 0 0
\(656\) 16475.2 0.980561
\(657\) 6458.60 0.383522
\(658\) 829.182 0.0491260
\(659\) −7620.41 −0.450454 −0.225227 0.974306i \(-0.572312\pi\)
−0.225227 + 0.974306i \(0.572312\pi\)
\(660\) 0 0
\(661\) −22762.2 −1.33941 −0.669703 0.742629i \(-0.733579\pi\)
−0.669703 + 0.742629i \(0.733579\pi\)
\(662\) −31556.7 −1.85270
\(663\) −3951.51 −0.231469
\(664\) 311.124 0.0181837
\(665\) 0 0
\(666\) −14621.5 −0.850708
\(667\) 29713.2 1.72488
\(668\) −9150.93 −0.530030
\(669\) −2268.65 −0.131107
\(670\) 0 0
\(671\) 3687.96 0.212179
\(672\) 5032.60 0.288894
\(673\) −698.528 −0.0400093 −0.0200047 0.999800i \(-0.506368\pi\)
−0.0200047 + 0.999800i \(0.506368\pi\)
\(674\) −47932.1 −2.73928
\(675\) 0 0
\(676\) −10514.8 −0.598250
\(677\) 10860.1 0.616524 0.308262 0.951302i \(-0.400253\pi\)
0.308262 + 0.951302i \(0.400253\pi\)
\(678\) 16163.1 0.915547
\(679\) −181.976 −0.0102851
\(680\) 0 0
\(681\) 17276.1 0.972131
\(682\) 3045.93 0.171019
\(683\) 6534.95 0.366110 0.183055 0.983103i \(-0.441401\pi\)
0.183055 + 0.983103i \(0.441401\pi\)
\(684\) −483.530 −0.0270296
\(685\) 0 0
\(686\) 16900.3 0.940606
\(687\) −5737.88 −0.318652
\(688\) −1873.88 −0.103839
\(689\) 7710.88 0.426359
\(690\) 0 0
\(691\) −10915.3 −0.600924 −0.300462 0.953794i \(-0.597141\pi\)
−0.300462 + 0.953794i \(0.597141\pi\)
\(692\) −31224.9 −1.71530
\(693\) 643.423 0.0352693
\(694\) 14079.0 0.770076
\(695\) 0 0
\(696\) 1052.79 0.0573359
\(697\) −11578.6 −0.629229
\(698\) 25717.3 1.39458
\(699\) −3104.58 −0.167991
\(700\) 0 0
\(701\) 18405.4 0.991674 0.495837 0.868416i \(-0.334861\pi\)
0.495837 + 0.868416i \(0.334861\pi\)
\(702\) 3330.58 0.179067
\(703\) −2598.39 −0.139403
\(704\) −6067.91 −0.324848
\(705\) 0 0
\(706\) 33215.9 1.77068
\(707\) −1483.63 −0.0789217
\(708\) −13929.4 −0.739405
\(709\) −21436.2 −1.13547 −0.567737 0.823210i \(-0.692181\pi\)
−0.567737 + 0.823210i \(0.692181\pi\)
\(710\) 0 0
\(711\) 1806.32 0.0952773
\(712\) 399.413 0.0210234
\(713\) 7410.73 0.389248
\(714\) −3396.91 −0.178048
\(715\) 0 0
\(716\) −26767.7 −1.39715
\(717\) 7941.69 0.413651
\(718\) 37878.9 1.96884
\(719\) 28641.7 1.48561 0.742807 0.669506i \(-0.233494\pi\)
0.742807 + 0.669506i \(0.233494\pi\)
\(720\) 0 0
\(721\) 5201.51 0.268675
\(722\) 27537.1 1.41942
\(723\) 20202.0 1.03917
\(724\) −3799.49 −0.195037
\(725\) 0 0
\(726\) −1466.27 −0.0749566
\(727\) 1723.13 0.0879055 0.0439527 0.999034i \(-0.486005\pi\)
0.0439527 + 0.999034i \(0.486005\pi\)
\(728\) 253.409 0.0129010
\(729\) 729.000 0.0370370
\(730\) 0 0
\(731\) 1316.95 0.0666335
\(732\) 8364.39 0.422345
\(733\) 18813.4 0.948005 0.474003 0.880523i \(-0.342809\pi\)
0.474003 + 0.880523i \(0.342809\pi\)
\(734\) −1255.80 −0.0631506
\(735\) 0 0
\(736\) −27903.0 −1.39744
\(737\) −316.273 −0.0158074
\(738\) 9759.21 0.486777
\(739\) −10893.3 −0.542242 −0.271121 0.962545i \(-0.587394\pi\)
−0.271121 + 0.962545i \(0.587394\pi\)
\(740\) 0 0
\(741\) 591.877 0.0293430
\(742\) 6628.65 0.327959
\(743\) −11888.6 −0.587013 −0.293507 0.955957i \(-0.594822\pi\)
−0.293507 + 0.955957i \(0.594822\pi\)
\(744\) 262.574 0.0129388
\(745\) 0 0
\(746\) −19140.4 −0.939381
\(747\) −2193.13 −0.107419
\(748\) 3945.53 0.192865
\(749\) 6803.98 0.331925
\(750\) 0 0
\(751\) −32134.8 −1.56140 −0.780702 0.624903i \(-0.785138\pi\)
−0.780702 + 0.624903i \(0.785138\pi\)
\(752\) 1938.41 0.0939982
\(753\) −16132.7 −0.780755
\(754\) −33905.0 −1.63759
\(755\) 0 0
\(756\) 1459.30 0.0702040
\(757\) −17366.1 −0.833794 −0.416897 0.908954i \(-0.636882\pi\)
−0.416897 + 0.908954i \(0.636882\pi\)
\(758\) 19128.1 0.916573
\(759\) −3567.43 −0.170605
\(760\) 0 0
\(761\) −27393.9 −1.30490 −0.652450 0.757832i \(-0.726259\pi\)
−0.652450 + 0.757832i \(0.726259\pi\)
\(762\) 6845.28 0.325431
\(763\) 2060.41 0.0977611
\(764\) −40808.0 −1.93243
\(765\) 0 0
\(766\) 14671.4 0.692034
\(767\) 17050.6 0.802689
\(768\) 11260.2 0.529060
\(769\) −12091.3 −0.567000 −0.283500 0.958972i \(-0.591496\pi\)
−0.283500 + 0.958972i \(0.591496\pi\)
\(770\) 0 0
\(771\) −17647.0 −0.824307
\(772\) −12159.5 −0.566879
\(773\) −18895.8 −0.879216 −0.439608 0.898190i \(-0.644883\pi\)
−0.439608 + 0.898190i \(0.644883\pi\)
\(774\) −1110.01 −0.0515483
\(775\) 0 0
\(776\) 35.7490 0.00165375
\(777\) 7841.96 0.362070
\(778\) −26487.8 −1.22061
\(779\) 1734.31 0.0797664
\(780\) 0 0
\(781\) 979.089 0.0448586
\(782\) 18834.0 0.861258
\(783\) −7421.14 −0.338710
\(784\) 18458.1 0.840838
\(785\) 0 0
\(786\) 2594.23 0.117727
\(787\) −23705.9 −1.07373 −0.536863 0.843669i \(-0.680391\pi\)
−0.536863 + 0.843669i \(0.680391\pi\)
\(788\) −18379.4 −0.830888
\(789\) 12005.0 0.541685
\(790\) 0 0
\(791\) −8668.77 −0.389666
\(792\) −126.400 −0.00567100
\(793\) −10238.6 −0.458493
\(794\) 16500.5 0.737507
\(795\) 0 0
\(796\) −3287.11 −0.146368
\(797\) 37248.0 1.65545 0.827725 0.561135i \(-0.189635\pi\)
0.827725 + 0.561135i \(0.189635\pi\)
\(798\) 508.807 0.0225709
\(799\) −1362.30 −0.0603189
\(800\) 0 0
\(801\) −2815.49 −0.124195
\(802\) 26966.7 1.18731
\(803\) 7893.84 0.346909
\(804\) −717.315 −0.0314648
\(805\) 0 0
\(806\) −8456.21 −0.369550
\(807\) −5274.86 −0.230092
\(808\) 291.458 0.0126899
\(809\) −24861.0 −1.08043 −0.540214 0.841528i \(-0.681657\pi\)
−0.540214 + 0.841528i \(0.681657\pi\)
\(810\) 0 0
\(811\) 4131.24 0.178875 0.0894375 0.995992i \(-0.471493\pi\)
0.0894375 + 0.995992i \(0.471493\pi\)
\(812\) −14855.5 −0.642027
\(813\) 18055.3 0.778878
\(814\) −17870.7 −0.769495
\(815\) 0 0
\(816\) −7941.10 −0.340679
\(817\) −197.259 −0.00844704
\(818\) −6941.81 −0.296717
\(819\) −1786.29 −0.0762126
\(820\) 0 0
\(821\) 25297.3 1.07537 0.537687 0.843145i \(-0.319298\pi\)
0.537687 + 0.843145i \(0.319298\pi\)
\(822\) −790.439 −0.0335398
\(823\) 18155.5 0.768969 0.384484 0.923131i \(-0.374379\pi\)
0.384484 + 0.923131i \(0.374379\pi\)
\(824\) −1021.83 −0.0432005
\(825\) 0 0
\(826\) 14657.6 0.617435
\(827\) 29566.8 1.24321 0.621607 0.783329i \(-0.286480\pi\)
0.621607 + 0.783329i \(0.286480\pi\)
\(828\) −8091.02 −0.339592
\(829\) 27249.4 1.14163 0.570814 0.821079i \(-0.306628\pi\)
0.570814 + 0.821079i \(0.306628\pi\)
\(830\) 0 0
\(831\) −8730.90 −0.364467
\(832\) 16845.9 0.701956
\(833\) −12972.2 −0.539568
\(834\) 4933.02 0.204816
\(835\) 0 0
\(836\) −590.981 −0.0244492
\(837\) −1850.90 −0.0764355
\(838\) −23154.3 −0.954478
\(839\) −15506.1 −0.638058 −0.319029 0.947745i \(-0.603357\pi\)
−0.319029 + 0.947745i \(0.603357\pi\)
\(840\) 0 0
\(841\) 51157.5 2.09756
\(842\) 38968.8 1.59496
\(843\) 20679.6 0.844890
\(844\) 12972.9 0.529084
\(845\) 0 0
\(846\) 1148.24 0.0466633
\(847\) 786.406 0.0319023
\(848\) 15496.1 0.627520
\(849\) −9360.47 −0.378387
\(850\) 0 0
\(851\) −43479.4 −1.75141
\(852\) 2220.60 0.0892915
\(853\) −8698.16 −0.349143 −0.174572 0.984644i \(-0.555854\pi\)
−0.174572 + 0.984644i \(0.555854\pi\)
\(854\) −8801.64 −0.352677
\(855\) 0 0
\(856\) −1336.64 −0.0533707
\(857\) 28468.8 1.13474 0.567372 0.823462i \(-0.307960\pi\)
0.567372 + 0.823462i \(0.307960\pi\)
\(858\) 4070.71 0.161972
\(859\) −21154.3 −0.840251 −0.420125 0.907466i \(-0.638014\pi\)
−0.420125 + 0.907466i \(0.638014\pi\)
\(860\) 0 0
\(861\) −5234.16 −0.207177
\(862\) 12338.4 0.487525
\(863\) 45959.5 1.81284 0.906420 0.422378i \(-0.138805\pi\)
0.906420 + 0.422378i \(0.138805\pi\)
\(864\) 6969.04 0.274412
\(865\) 0 0
\(866\) 44524.5 1.74712
\(867\) −9158.05 −0.358735
\(868\) −3705.10 −0.144884
\(869\) 2207.72 0.0861815
\(870\) 0 0
\(871\) 878.047 0.0341579
\(872\) −404.765 −0.0157191
\(873\) −251.996 −0.00976951
\(874\) −2821.06 −0.109180
\(875\) 0 0
\(876\) 17903.4 0.690526
\(877\) 7315.10 0.281657 0.140829 0.990034i \(-0.455023\pi\)
0.140829 + 0.990034i \(0.455023\pi\)
\(878\) 30655.6 1.17833
\(879\) −26608.9 −1.02104
\(880\) 0 0
\(881\) 4652.19 0.177907 0.0889536 0.996036i \(-0.471648\pi\)
0.0889536 + 0.996036i \(0.471648\pi\)
\(882\) 10933.8 0.417415
\(883\) 38954.3 1.48462 0.742309 0.670058i \(-0.233731\pi\)
0.742309 + 0.670058i \(0.233731\pi\)
\(884\) −10953.7 −0.416756
\(885\) 0 0
\(886\) 16332.2 0.619289
\(887\) 16917.5 0.640398 0.320199 0.947350i \(-0.396250\pi\)
0.320199 + 0.947350i \(0.396250\pi\)
\(888\) −1540.55 −0.0582177
\(889\) −3671.33 −0.138507
\(890\) 0 0
\(891\) 891.000 0.0335013
\(892\) −6288.75 −0.236057
\(893\) 204.053 0.00764654
\(894\) −9598.53 −0.359086
\(895\) 0 0
\(896\) 1061.32 0.0395715
\(897\) 9904.01 0.368657
\(898\) −59051.8 −2.19441
\(899\) 18842.0 0.699016
\(900\) 0 0
\(901\) −10890.5 −0.402682
\(902\) 11927.9 0.440306
\(903\) 595.331 0.0219395
\(904\) 1702.97 0.0626549
\(905\) 0 0
\(906\) 34679.2 1.27168
\(907\) −38985.4 −1.42722 −0.713610 0.700543i \(-0.752941\pi\)
−0.713610 + 0.700543i \(0.752941\pi\)
\(908\) 47889.8 1.75031
\(909\) −2054.50 −0.0749653
\(910\) 0 0
\(911\) 27759.8 1.00957 0.504787 0.863244i \(-0.331571\pi\)
0.504787 + 0.863244i \(0.331571\pi\)
\(912\) 1189.46 0.0431874
\(913\) −2680.49 −0.0971646
\(914\) −44072.3 −1.59495
\(915\) 0 0
\(916\) −15905.6 −0.573728
\(917\) −1391.37 −0.0501057
\(918\) −4703.98 −0.169122
\(919\) 29705.5 1.06626 0.533130 0.846033i \(-0.321016\pi\)
0.533130 + 0.846033i \(0.321016\pi\)
\(920\) 0 0
\(921\) 17140.3 0.613236
\(922\) −66127.6 −2.36204
\(923\) −2718.18 −0.0969338
\(924\) 1783.59 0.0635019
\(925\) 0 0
\(926\) −64724.1 −2.29694
\(927\) 7202.95 0.255206
\(928\) −70944.1 −2.50954
\(929\) 2940.01 0.103831 0.0519153 0.998651i \(-0.483467\pi\)
0.0519153 + 0.998651i \(0.483467\pi\)
\(930\) 0 0
\(931\) 1943.04 0.0684002
\(932\) −8605.99 −0.302466
\(933\) 5952.04 0.208854
\(934\) −60982.6 −2.13642
\(935\) 0 0
\(936\) 350.916 0.0122543
\(937\) 20041.9 0.698762 0.349381 0.936981i \(-0.386392\pi\)
0.349381 + 0.936981i \(0.386392\pi\)
\(938\) 754.813 0.0262745
\(939\) 8026.49 0.278951
\(940\) 0 0
\(941\) −14446.1 −0.500457 −0.250229 0.968187i \(-0.580506\pi\)
−0.250229 + 0.968187i \(0.580506\pi\)
\(942\) 36251.4 1.25386
\(943\) 29020.6 1.00216
\(944\) 34265.6 1.18141
\(945\) 0 0
\(946\) −1356.68 −0.0466272
\(947\) −3901.33 −0.133871 −0.0669357 0.997757i \(-0.521322\pi\)
−0.0669357 + 0.997757i \(0.521322\pi\)
\(948\) 5007.16 0.171545
\(949\) −21915.1 −0.749626
\(950\) 0 0
\(951\) 21251.3 0.724627
\(952\) −357.904 −0.0121846
\(953\) −613.963 −0.0208691 −0.0104345 0.999946i \(-0.503321\pi\)
−0.0104345 + 0.999946i \(0.503321\pi\)
\(954\) 9179.22 0.311518
\(955\) 0 0
\(956\) 22014.6 0.744773
\(957\) −9070.29 −0.306375
\(958\) 42247.1 1.42478
\(959\) 423.937 0.0142749
\(960\) 0 0
\(961\) −25091.6 −0.842256
\(962\) 49613.3 1.66278
\(963\) 9422.02 0.315286
\(964\) 56000.5 1.87101
\(965\) 0 0
\(966\) 8513.98 0.283574
\(967\) 20617.4 0.685637 0.342819 0.939402i \(-0.388618\pi\)
0.342819 + 0.939402i \(0.388618\pi\)
\(968\) −154.489 −0.00512961
\(969\) −835.943 −0.0277135
\(970\) 0 0
\(971\) 22816.0 0.754069 0.377035 0.926199i \(-0.376944\pi\)
0.377035 + 0.926199i \(0.376944\pi\)
\(972\) 2020.81 0.0666846
\(973\) −2645.73 −0.0871719
\(974\) −32071.6 −1.05507
\(975\) 0 0
\(976\) −20576.0 −0.674816
\(977\) −22650.2 −0.741702 −0.370851 0.928692i \(-0.620934\pi\)
−0.370851 + 0.928692i \(0.620934\pi\)
\(978\) 35234.2 1.15201
\(979\) −3441.15 −0.112339
\(980\) 0 0
\(981\) 2853.21 0.0928604
\(982\) −46422.8 −1.50857
\(983\) 8963.01 0.290820 0.145410 0.989372i \(-0.453550\pi\)
0.145410 + 0.989372i \(0.453550\pi\)
\(984\) 1028.25 0.0333123
\(985\) 0 0
\(986\) 47886.0 1.54665
\(987\) −615.834 −0.0198604
\(988\) 1640.70 0.0528316
\(989\) −3300.78 −0.106126
\(990\) 0 0
\(991\) 422.204 0.0135336 0.00676678 0.999977i \(-0.497846\pi\)
0.00676678 + 0.999977i \(0.497846\pi\)
\(992\) −17694.1 −0.566319
\(993\) 23437.2 0.748999
\(994\) −2336.68 −0.0745623
\(995\) 0 0
\(996\) −6079.41 −0.193407
\(997\) 12577.3 0.399524 0.199762 0.979844i \(-0.435983\pi\)
0.199762 + 0.979844i \(0.435983\pi\)
\(998\) −19443.4 −0.616704
\(999\) 10859.4 0.343920
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.o.1.1 3
3.2 odd 2 2475.4.a.y.1.3 3
5.2 odd 4 825.4.c.n.199.1 6
5.3 odd 4 825.4.c.n.199.6 6
5.4 even 2 825.4.a.q.1.3 yes 3
15.14 odd 2 2475.4.a.v.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
825.4.a.o.1.1 3 1.1 even 1 trivial
825.4.a.q.1.3 yes 3 5.4 even 2
825.4.c.n.199.1 6 5.2 odd 4
825.4.c.n.199.6 6 5.3 odd 4
2475.4.a.v.1.1 3 15.14 odd 2
2475.4.a.y.1.3 3 3.2 odd 2