Properties

Label 33.4.a.c.1.1
Level $33$
Weight $4$
Character 33.1
Self dual yes
Analytic conductor $1.947$
Analytic rank $0$
Dimension $2$
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] = [33,4,Mod(1,33)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(33, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("33.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 33 = 3 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 33.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: \(1.94706303019\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{97}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 24 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.42443\) of defining polynomial
Character \(\chi\) \(=\) 33.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.42443 q^{2} -3.00000 q^{3} +11.5756 q^{4} +2.84886 q^{5} +13.2733 q^{6} +31.6977 q^{7} -15.8199 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.42443 q^{2} -3.00000 q^{3} +11.5756 q^{4} +2.84886 q^{5} +13.2733 q^{6} +31.6977 q^{7} -15.8199 q^{8} +9.00000 q^{9} -12.6046 q^{10} -11.0000 q^{11} -34.7267 q^{12} +5.15114 q^{13} -140.244 q^{14} -8.54657 q^{15} -22.6107 q^{16} +121.942 q^{17} -39.8199 q^{18} +34.8489 q^{19} +32.9772 q^{20} -95.0931 q^{21} +48.6687 q^{22} +116.244 q^{23} +47.4596 q^{24} -116.884 q^{25} -22.7909 q^{26} -27.0000 q^{27} +366.919 q^{28} -69.4534 q^{29} +37.8137 q^{30} +140.605 q^{31} +226.598 q^{32} +33.0000 q^{33} -539.524 q^{34} +90.3023 q^{35} +104.180 q^{36} -420.070 q^{37} -154.186 q^{38} -15.4534 q^{39} -45.0685 q^{40} -322.058 q^{41} +420.733 q^{42} +321.035 q^{43} -127.331 q^{44} +25.6397 q^{45} -514.315 q^{46} -231.408 q^{47} +67.8322 q^{48} +661.745 q^{49} +517.145 q^{50} -365.826 q^{51} +59.6274 q^{52} +4.91916 q^{53} +119.460 q^{54} -31.3374 q^{55} -501.453 q^{56} -104.547 q^{57} +307.292 q^{58} +406.443 q^{59} -98.9315 q^{60} -556.431 q^{61} -622.095 q^{62} +285.279 q^{63} -821.683 q^{64} +14.6749 q^{65} -146.006 q^{66} +84.7452 q^{67} +1411.55 q^{68} -348.733 q^{69} -399.536 q^{70} +49.0808 q^{71} -142.379 q^{72} +785.884 q^{73} +1858.57 q^{74} +350.652 q^{75} +403.395 q^{76} -348.675 q^{77} +68.3726 q^{78} -383.118 q^{79} -64.4147 q^{80} +81.0000 q^{81} +1424.92 q^{82} -930.211 q^{83} -1100.76 q^{84} +347.395 q^{85} -1420.40 q^{86} +208.360 q^{87} +174.018 q^{88} -732.559 q^{89} -113.441 q^{90} +163.279 q^{91} +1345.59 q^{92} -421.814 q^{93} +1023.85 q^{94} +99.2794 q^{95} -679.795 q^{96} -1171.49 q^{97} -2927.84 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} - 3 q^{6} + 24 q^{7} + 57 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} - 6 q^{3} + 33 q^{4} - 14 q^{5} - 3 q^{6} + 24 q^{7} + 57 q^{8} + 18 q^{9} - 104 q^{10} - 22 q^{11} - 99 q^{12} + 30 q^{13} - 182 q^{14} + 42 q^{15} + 201 q^{16} + 106 q^{17} + 9 q^{18} + 50 q^{19} - 328 q^{20} - 72 q^{21} - 11 q^{22} + 134 q^{23} - 171 q^{24} + 42 q^{25} + 112 q^{26} - 54 q^{27} + 202 q^{28} - 198 q^{29} + 312 q^{30} + 360 q^{31} + 857 q^{32} + 66 q^{33} - 626 q^{34} + 220 q^{35} + 297 q^{36} - 328 q^{37} - 72 q^{38} - 90 q^{39} - 1272 q^{40} - 782 q^{41} + 546 q^{42} + 386 q^{43} - 363 q^{44} - 126 q^{45} - 418 q^{46} + 266 q^{47} - 603 q^{48} + 378 q^{49} + 1379 q^{50} - 318 q^{51} + 592 q^{52} - 522 q^{53} - 27 q^{54} + 154 q^{55} - 1062 q^{56} - 150 q^{57} - 390 q^{58} - 172 q^{59} + 984 q^{60} - 778 q^{61} + 568 q^{62} + 216 q^{63} + 809 q^{64} - 404 q^{65} + 33 q^{66} - 776 q^{67} + 1070 q^{68} - 402 q^{69} + 304 q^{70} + 630 q^{71} + 513 q^{72} + 1296 q^{73} + 2358 q^{74} - 126 q^{75} + 728 q^{76} - 264 q^{77} - 336 q^{78} + 652 q^{79} - 3832 q^{80} + 162 q^{81} - 1070 q^{82} - 324 q^{83} - 606 q^{84} + 616 q^{85} - 1068 q^{86} + 594 q^{87} - 627 q^{88} - 756 q^{89} - 936 q^{90} - 28 q^{91} + 1726 q^{92} - 1080 q^{93} + 3722 q^{94} - 156 q^{95} - 2571 q^{96} - 452 q^{97} - 4467 q^{98} - 198 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.42443 −1.56427 −0.782136 0.623108i \(-0.785870\pi\)
−0.782136 + 0.623108i \(0.785870\pi\)
\(3\) −3.00000 −0.577350
\(4\) 11.5756 1.44695
\(5\) 2.84886 0.254810 0.127405 0.991851i \(-0.459335\pi\)
0.127405 + 0.991851i \(0.459335\pi\)
\(6\) 13.2733 0.903133
\(7\) 31.6977 1.71152 0.855758 0.517377i \(-0.173091\pi\)
0.855758 + 0.517377i \(0.173091\pi\)
\(8\) −15.8199 −0.699146
\(9\) 9.00000 0.333333
\(10\) −12.6046 −0.398591
\(11\) −11.0000 −0.301511
\(12\) −34.7267 −0.835395
\(13\) 5.15114 0.109898 0.0549488 0.998489i \(-0.482500\pi\)
0.0549488 + 0.998489i \(0.482500\pi\)
\(14\) −140.244 −2.67728
\(15\) −8.54657 −0.147114
\(16\) −22.6107 −0.353293
\(17\) 121.942 1.73972 0.869861 0.493297i \(-0.164208\pi\)
0.869861 + 0.493297i \(0.164208\pi\)
\(18\) −39.8199 −0.521424
\(19\) 34.8489 0.420783 0.210391 0.977617i \(-0.432526\pi\)
0.210391 + 0.977617i \(0.432526\pi\)
\(20\) 32.9772 0.368696
\(21\) −95.0931 −0.988144
\(22\) 48.6687 0.471646
\(23\) 116.244 1.05385 0.526926 0.849911i \(-0.323344\pi\)
0.526926 + 0.849911i \(0.323344\pi\)
\(24\) 47.4596 0.403652
\(25\) −116.884 −0.935072
\(26\) −22.7909 −0.171910
\(27\) −27.0000 −0.192450
\(28\) 366.919 2.47647
\(29\) −69.4534 −0.444730 −0.222365 0.974963i \(-0.571378\pi\)
−0.222365 + 0.974963i \(0.571378\pi\)
\(30\) 37.8137 0.230127
\(31\) 140.605 0.814623 0.407312 0.913289i \(-0.366466\pi\)
0.407312 + 0.913289i \(0.366466\pi\)
\(32\) 226.598 1.25179
\(33\) 33.0000 0.174078
\(34\) −539.524 −2.72140
\(35\) 90.3023 0.436111
\(36\) 104.180 0.482315
\(37\) −420.070 −1.86646 −0.933232 0.359276i \(-0.883024\pi\)
−0.933232 + 0.359276i \(0.883024\pi\)
\(38\) −154.186 −0.658219
\(39\) −15.4534 −0.0634495
\(40\) −45.0685 −0.178149
\(41\) −322.058 −1.22676 −0.613378 0.789789i \(-0.710190\pi\)
−0.613378 + 0.789789i \(0.710190\pi\)
\(42\) 420.733 1.54573
\(43\) 321.035 1.13854 0.569272 0.822149i \(-0.307225\pi\)
0.569272 + 0.822149i \(0.307225\pi\)
\(44\) −127.331 −0.436271
\(45\) 25.6397 0.0849365
\(46\) −514.315 −1.64851
\(47\) −231.408 −0.718176 −0.359088 0.933304i \(-0.616912\pi\)
−0.359088 + 0.933304i \(0.616912\pi\)
\(48\) 67.8322 0.203974
\(49\) 661.745 1.92929
\(50\) 517.145 1.46271
\(51\) −365.826 −1.00443
\(52\) 59.6274 0.159016
\(53\) 4.91916 0.0127490 0.00637452 0.999980i \(-0.497971\pi\)
0.00637452 + 0.999980i \(0.497971\pi\)
\(54\) 119.460 0.301044
\(55\) −31.3374 −0.0768280
\(56\) −501.453 −1.19660
\(57\) −104.547 −0.242939
\(58\) 307.292 0.695679
\(59\) 406.443 0.896854 0.448427 0.893820i \(-0.351984\pi\)
0.448427 + 0.893820i \(0.351984\pi\)
\(60\) −98.9315 −0.212867
\(61\) −556.431 −1.16793 −0.583964 0.811779i \(-0.698499\pi\)
−0.583964 + 0.811779i \(0.698499\pi\)
\(62\) −622.095 −1.27429
\(63\) 285.279 0.570505
\(64\) −821.683 −1.60485
\(65\) 14.6749 0.0280030
\(66\) −146.006 −0.272305
\(67\) 84.7452 0.154526 0.0772632 0.997011i \(-0.475382\pi\)
0.0772632 + 0.997011i \(0.475382\pi\)
\(68\) 1411.55 2.51728
\(69\) −348.733 −0.608442
\(70\) −399.536 −0.682196
\(71\) 49.0808 0.0820398 0.0410199 0.999158i \(-0.486939\pi\)
0.0410199 + 0.999158i \(0.486939\pi\)
\(72\) −142.379 −0.233049
\(73\) 785.884 1.26001 0.630005 0.776591i \(-0.283053\pi\)
0.630005 + 0.776591i \(0.283053\pi\)
\(74\) 1858.57 2.91966
\(75\) 350.652 0.539864
\(76\) 403.395 0.608850
\(77\) −348.675 −0.516041
\(78\) 68.3726 0.0992522
\(79\) −383.118 −0.545622 −0.272811 0.962068i \(-0.587953\pi\)
−0.272811 + 0.962068i \(0.587953\pi\)
\(80\) −64.4147 −0.0900223
\(81\) 81.0000 0.111111
\(82\) 1424.92 1.91898
\(83\) −930.211 −1.23017 −0.615084 0.788462i \(-0.710878\pi\)
−0.615084 + 0.788462i \(0.710878\pi\)
\(84\) −1100.76 −1.42979
\(85\) 347.395 0.443298
\(86\) −1420.40 −1.78099
\(87\) 208.360 0.256765
\(88\) 174.018 0.210800
\(89\) −732.559 −0.872484 −0.436242 0.899829i \(-0.643691\pi\)
−0.436242 + 0.899829i \(0.643691\pi\)
\(90\) −113.441 −0.132864
\(91\) 163.279 0.188092
\(92\) 1345.59 1.52487
\(93\) −421.814 −0.470323
\(94\) 1023.85 1.12342
\(95\) 99.2794 0.107220
\(96\) −679.795 −0.722722
\(97\) −1171.49 −1.22626 −0.613128 0.789984i \(-0.710089\pi\)
−0.613128 + 0.789984i \(0.710089\pi\)
\(98\) −2927.84 −3.01793
\(99\) −99.0000 −0.100504
\(100\) −1353.00 −1.35300
\(101\) −1221.27 −1.20318 −0.601589 0.798806i \(-0.705465\pi\)
−0.601589 + 0.798806i \(0.705465\pi\)
\(102\) 1618.57 1.57120
\(103\) 516.745 0.494334 0.247167 0.968973i \(-0.420500\pi\)
0.247167 + 0.968973i \(0.420500\pi\)
\(104\) −81.4903 −0.0768345
\(105\) −270.907 −0.251789
\(106\) −21.7645 −0.0199430
\(107\) −152.025 −0.137353 −0.0686765 0.997639i \(-0.521878\pi\)
−0.0686765 + 0.997639i \(0.521878\pi\)
\(108\) −312.540 −0.278465
\(109\) 2170.32 1.90714 0.953572 0.301164i \(-0.0973752\pi\)
0.953572 + 0.301164i \(0.0973752\pi\)
\(110\) 138.650 0.120180
\(111\) 1260.21 1.07760
\(112\) −716.708 −0.604666
\(113\) −646.397 −0.538123 −0.269062 0.963123i \(-0.586714\pi\)
−0.269062 + 0.963123i \(0.586714\pi\)
\(114\) 462.559 0.380023
\(115\) 331.163 0.268532
\(116\) −803.963 −0.643501
\(117\) 46.3603 0.0366326
\(118\) −1798.28 −1.40292
\(119\) 3865.28 2.97756
\(120\) 135.206 0.102854
\(121\) 121.000 0.0909091
\(122\) 2461.89 1.82696
\(123\) 966.174 0.708268
\(124\) 1627.58 1.17872
\(125\) −689.093 −0.493075
\(126\) −1262.20 −0.892425
\(127\) −993.304 −0.694027 −0.347014 0.937860i \(-0.612804\pi\)
−0.347014 + 0.937860i \(0.612804\pi\)
\(128\) 1822.69 1.25863
\(129\) −963.105 −0.657339
\(130\) −64.9279 −0.0438043
\(131\) 385.814 0.257318 0.128659 0.991689i \(-0.458933\pi\)
0.128659 + 0.991689i \(0.458933\pi\)
\(132\) 381.994 0.251881
\(133\) 1104.63 0.720177
\(134\) −374.949 −0.241721
\(135\) −76.9192 −0.0490381
\(136\) −1929.11 −1.21632
\(137\) 884.840 0.551803 0.275901 0.961186i \(-0.411024\pi\)
0.275901 + 0.961186i \(0.411024\pi\)
\(138\) 1542.94 0.951769
\(139\) −1091.94 −0.666312 −0.333156 0.942872i \(-0.608114\pi\)
−0.333156 + 0.942872i \(0.608114\pi\)
\(140\) 1045.30 0.631029
\(141\) 694.223 0.414639
\(142\) −217.155 −0.128333
\(143\) −56.6626 −0.0331354
\(144\) −203.497 −0.117764
\(145\) −197.863 −0.113322
\(146\) −3477.09 −1.97100
\(147\) −1985.24 −1.11387
\(148\) −4862.55 −2.70067
\(149\) 297.014 0.163304 0.0816522 0.996661i \(-0.473980\pi\)
0.0816522 + 0.996661i \(0.473980\pi\)
\(150\) −1551.43 −0.844494
\(151\) −1887.86 −1.01743 −0.508716 0.860935i \(-0.669880\pi\)
−0.508716 + 0.860935i \(0.669880\pi\)
\(152\) −551.304 −0.294189
\(153\) 1097.48 0.579907
\(154\) 1542.69 0.807229
\(155\) 400.562 0.207574
\(156\) −178.882 −0.0918080
\(157\) −56.5343 −0.0287384 −0.0143692 0.999897i \(-0.504574\pi\)
−0.0143692 + 0.999897i \(0.504574\pi\)
\(158\) 1695.08 0.853501
\(159\) −14.7575 −0.00736066
\(160\) 645.547 0.318968
\(161\) 3684.68 1.80369
\(162\) −358.379 −0.173808
\(163\) −49.2338 −0.0236582 −0.0118291 0.999930i \(-0.503765\pi\)
−0.0118291 + 0.999930i \(0.503765\pi\)
\(164\) −3728.01 −1.77505
\(165\) 94.0123 0.0443567
\(166\) 4115.65 1.92432
\(167\) 2068.75 0.958589 0.479294 0.877654i \(-0.340893\pi\)
0.479294 + 0.877654i \(0.340893\pi\)
\(168\) 1504.36 0.690857
\(169\) −2170.47 −0.987923
\(170\) −1537.03 −0.693438
\(171\) 313.640 0.140261
\(172\) 3716.17 1.64741
\(173\) −604.012 −0.265446 −0.132723 0.991153i \(-0.542372\pi\)
−0.132723 + 0.991153i \(0.542372\pi\)
\(174\) −921.875 −0.401650
\(175\) −3704.96 −1.60039
\(176\) 248.718 0.106522
\(177\) −1219.33 −0.517799
\(178\) 3241.15 1.36480
\(179\) −2132.02 −0.890251 −0.445126 0.895468i \(-0.646841\pi\)
−0.445126 + 0.895468i \(0.646841\pi\)
\(180\) 296.794 0.122899
\(181\) −589.371 −0.242031 −0.121015 0.992651i \(-0.538615\pi\)
−0.121015 + 0.992651i \(0.538615\pi\)
\(182\) −722.418 −0.294226
\(183\) 1669.29 0.674304
\(184\) −1838.97 −0.736796
\(185\) −1196.72 −0.475593
\(186\) 1866.28 0.735713
\(187\) −1341.36 −0.524546
\(188\) −2678.68 −1.03916
\(189\) −855.838 −0.329381
\(190\) −439.255 −0.167720
\(191\) −2160.90 −0.818624 −0.409312 0.912395i \(-0.634231\pi\)
−0.409312 + 0.912395i \(0.634231\pi\)
\(192\) 2465.05 0.926560
\(193\) −1490.91 −0.556052 −0.278026 0.960574i \(-0.589680\pi\)
−0.278026 + 0.960574i \(0.589680\pi\)
\(194\) 5183.18 1.91820
\(195\) −44.0246 −0.0161675
\(196\) 7660.08 2.79157
\(197\) −230.529 −0.0833732 −0.0416866 0.999131i \(-0.513273\pi\)
−0.0416866 + 0.999131i \(0.513273\pi\)
\(198\) 438.018 0.157215
\(199\) 22.4007 0.00797963 0.00398982 0.999992i \(-0.498730\pi\)
0.00398982 + 0.999992i \(0.498730\pi\)
\(200\) 1849.09 0.653752
\(201\) −254.236 −0.0892159
\(202\) 5403.43 1.88210
\(203\) −2201.51 −0.761163
\(204\) −4234.65 −1.45336
\(205\) −917.497 −0.312589
\(206\) −2286.30 −0.773273
\(207\) 1046.20 0.351284
\(208\) −116.471 −0.0388260
\(209\) −383.337 −0.126871
\(210\) 1198.61 0.393866
\(211\) −1051.64 −0.343117 −0.171558 0.985174i \(-0.554880\pi\)
−0.171558 + 0.985174i \(0.554880\pi\)
\(212\) 56.9421 0.0184472
\(213\) −147.243 −0.0473657
\(214\) 672.622 0.214857
\(215\) 914.583 0.290112
\(216\) 427.136 0.134551
\(217\) 4456.84 1.39424
\(218\) −9602.42 −2.98329
\(219\) −2357.65 −0.727467
\(220\) −362.749 −0.111166
\(221\) 628.141 0.191191
\(222\) −5575.71 −1.68566
\(223\) 3861.80 1.15966 0.579832 0.814736i \(-0.303118\pi\)
0.579832 + 0.814736i \(0.303118\pi\)
\(224\) 7182.65 2.14246
\(225\) −1051.96 −0.311691
\(226\) 2859.94 0.841771
\(227\) −872.721 −0.255174 −0.127587 0.991827i \(-0.540723\pi\)
−0.127587 + 0.991827i \(0.540723\pi\)
\(228\) −1210.19 −0.351520
\(229\) 1841.72 0.531459 0.265730 0.964048i \(-0.414387\pi\)
0.265730 + 0.964048i \(0.414387\pi\)
\(230\) −1465.21 −0.420057
\(231\) 1046.02 0.297937
\(232\) 1098.74 0.310931
\(233\) 3932.14 1.10559 0.552796 0.833317i \(-0.313561\pi\)
0.552796 + 0.833317i \(0.313561\pi\)
\(234\) −205.118 −0.0573033
\(235\) −659.248 −0.182998
\(236\) 4704.81 1.29770
\(237\) 1149.35 0.315015
\(238\) −17101.7 −4.65772
\(239\) 4772.10 1.29155 0.645777 0.763526i \(-0.276534\pi\)
0.645777 + 0.763526i \(0.276534\pi\)
\(240\) 193.244 0.0519744
\(241\) 3988.84 1.06616 0.533078 0.846066i \(-0.321035\pi\)
0.533078 + 0.846066i \(0.321035\pi\)
\(242\) −535.356 −0.142207
\(243\) −243.000 −0.0641500
\(244\) −6441.00 −1.68993
\(245\) 1885.22 0.491601
\(246\) −4274.77 −1.10792
\(247\) 179.511 0.0462431
\(248\) −2224.34 −0.569540
\(249\) 2790.63 0.710238
\(250\) 3048.84 0.771303
\(251\) −5474.22 −1.37661 −0.688306 0.725421i \(-0.741645\pi\)
−0.688306 + 0.725421i \(0.741645\pi\)
\(252\) 3302.27 0.825491
\(253\) −1278.69 −0.317749
\(254\) 4394.80 1.08565
\(255\) −1042.19 −0.255938
\(256\) −1490.90 −0.363989
\(257\) −6434.01 −1.56164 −0.780822 0.624754i \(-0.785199\pi\)
−0.780822 + 0.624754i \(0.785199\pi\)
\(258\) 4261.19 1.02826
\(259\) −13315.3 −3.19448
\(260\) 169.870 0.0405188
\(261\) −625.081 −0.148243
\(262\) −1707.01 −0.402516
\(263\) 7589.00 1.77931 0.889654 0.456636i \(-0.150946\pi\)
0.889654 + 0.456636i \(0.150946\pi\)
\(264\) −522.055 −0.121706
\(265\) 14.0140 0.00324858
\(266\) −4887.35 −1.12655
\(267\) 2197.68 0.503729
\(268\) 980.974 0.223591
\(269\) 478.178 0.108383 0.0541914 0.998531i \(-0.482742\pi\)
0.0541914 + 0.998531i \(0.482742\pi\)
\(270\) 340.323 0.0767090
\(271\) −122.323 −0.0274192 −0.0137096 0.999906i \(-0.504364\pi\)
−0.0137096 + 0.999906i \(0.504364\pi\)
\(272\) −2757.20 −0.614631
\(273\) −489.838 −0.108595
\(274\) −3914.91 −0.863170
\(275\) 1285.72 0.281935
\(276\) −4036.78 −0.880383
\(277\) 8199.41 1.77854 0.889269 0.457385i \(-0.151214\pi\)
0.889269 + 0.457385i \(0.151214\pi\)
\(278\) 4831.22 1.04229
\(279\) 1265.44 0.271541
\(280\) −1428.57 −0.304905
\(281\) 6943.79 1.47413 0.737067 0.675820i \(-0.236210\pi\)
0.737067 + 0.675820i \(0.236210\pi\)
\(282\) −3071.54 −0.648609
\(283\) 1035.14 0.217429 0.108715 0.994073i \(-0.465327\pi\)
0.108715 + 0.994073i \(0.465327\pi\)
\(284\) 568.139 0.118707
\(285\) −297.838 −0.0619032
\(286\) 250.699 0.0518328
\(287\) −10208.5 −2.09961
\(288\) 2039.39 0.417264
\(289\) 9956.85 2.02663
\(290\) 875.430 0.177266
\(291\) 3514.47 0.707979
\(292\) 9097.06 1.82317
\(293\) −6144.81 −1.22520 −0.612600 0.790393i \(-0.709876\pi\)
−0.612600 + 0.790393i \(0.709876\pi\)
\(294\) 8783.53 1.74240
\(295\) 1157.90 0.228527
\(296\) 6645.45 1.30493
\(297\) 297.000 0.0580259
\(298\) −1314.12 −0.255452
\(299\) 598.791 0.115816
\(300\) 4059.00 0.781154
\(301\) 10176.1 1.94864
\(302\) 8352.72 1.59154
\(303\) 3663.81 0.694655
\(304\) −787.958 −0.148659
\(305\) −1585.19 −0.297599
\(306\) −4855.71 −0.907133
\(307\) −2186.09 −0.406406 −0.203203 0.979137i \(-0.565135\pi\)
−0.203203 + 0.979137i \(0.565135\pi\)
\(308\) −4036.11 −0.746684
\(309\) −1550.24 −0.285404
\(310\) −1772.26 −0.324702
\(311\) −7484.83 −1.36471 −0.682357 0.731019i \(-0.739045\pi\)
−0.682357 + 0.731019i \(0.739045\pi\)
\(312\) 244.471 0.0443604
\(313\) −6833.33 −1.23400 −0.617001 0.786962i \(-0.711653\pi\)
−0.617001 + 0.786962i \(0.711653\pi\)
\(314\) 250.132 0.0449546
\(315\) 812.721 0.145370
\(316\) −4434.81 −0.789485
\(317\) 924.265 0.163760 0.0818800 0.996642i \(-0.473908\pi\)
0.0818800 + 0.996642i \(0.473908\pi\)
\(318\) 65.2934 0.0115141
\(319\) 763.988 0.134091
\(320\) −2340.86 −0.408931
\(321\) 456.074 0.0793008
\(322\) −16302.6 −2.82145
\(323\) 4249.54 0.732046
\(324\) 937.621 0.160772
\(325\) −602.086 −0.102762
\(326\) 217.831 0.0370078
\(327\) −6510.95 −1.10109
\(328\) 5094.91 0.857681
\(329\) −7335.10 −1.22917
\(330\) −415.951 −0.0693859
\(331\) −9820.46 −1.63076 −0.815380 0.578927i \(-0.803472\pi\)
−0.815380 + 0.578927i \(0.803472\pi\)
\(332\) −10767.7 −1.77999
\(333\) −3780.63 −0.622154
\(334\) −9153.02 −1.49949
\(335\) 241.427 0.0393748
\(336\) 2150.12 0.349104
\(337\) 600.808 0.0971161 0.0485580 0.998820i \(-0.484537\pi\)
0.0485580 + 0.998820i \(0.484537\pi\)
\(338\) 9603.07 1.54538
\(339\) 1939.19 0.310686
\(340\) 4021.30 0.641428
\(341\) −1546.65 −0.245618
\(342\) −1387.68 −0.219406
\(343\) 10103.5 1.59049
\(344\) −5078.73 −0.796008
\(345\) −993.490 −0.155037
\(346\) 2672.41 0.415230
\(347\) −3143.41 −0.486303 −0.243152 0.969988i \(-0.578181\pi\)
−0.243152 + 0.969988i \(0.578181\pi\)
\(348\) 2411.89 0.371525
\(349\) 720.663 0.110533 0.0552667 0.998472i \(-0.482399\pi\)
0.0552667 + 0.998472i \(0.482399\pi\)
\(350\) 16392.3 2.50345
\(351\) −139.081 −0.0211498
\(352\) −2492.58 −0.377429
\(353\) 1207.12 0.182007 0.0910034 0.995851i \(-0.470993\pi\)
0.0910034 + 0.995851i \(0.470993\pi\)
\(354\) 5394.83 0.809978
\(355\) 139.824 0.0209045
\(356\) −8479.79 −1.26244
\(357\) −11595.8 −1.71910
\(358\) 9432.99 1.39260
\(359\) 8748.31 1.28612 0.643062 0.765814i \(-0.277664\pi\)
0.643062 + 0.765814i \(0.277664\pi\)
\(360\) −405.617 −0.0593830
\(361\) −5644.56 −0.822942
\(362\) 2607.63 0.378602
\(363\) −363.000 −0.0524864
\(364\) 1890.05 0.272158
\(365\) 2238.87 0.321063
\(366\) −7385.66 −1.05479
\(367\) −6730.45 −0.957293 −0.478647 0.878008i \(-0.658872\pi\)
−0.478647 + 0.878008i \(0.658872\pi\)
\(368\) −2628.37 −0.372318
\(369\) −2898.52 −0.408919
\(370\) 5294.81 0.743956
\(371\) 155.926 0.0218202
\(372\) −4882.73 −0.680532
\(373\) −227.394 −0.0315657 −0.0157828 0.999875i \(-0.505024\pi\)
−0.0157828 + 0.999875i \(0.505024\pi\)
\(374\) 5934.76 0.820533
\(375\) 2067.28 0.284677
\(376\) 3660.84 0.502110
\(377\) −357.764 −0.0488748
\(378\) 3786.60 0.515242
\(379\) 11356.2 1.53913 0.769565 0.638568i \(-0.220473\pi\)
0.769565 + 0.638568i \(0.220473\pi\)
\(380\) 1149.22 0.155141
\(381\) 2979.91 0.400697
\(382\) 9560.74 1.28055
\(383\) 10753.6 1.43468 0.717338 0.696725i \(-0.245360\pi\)
0.717338 + 0.696725i \(0.245360\pi\)
\(384\) −5468.07 −0.726670
\(385\) −993.325 −0.131492
\(386\) 6596.43 0.869817
\(387\) 2889.32 0.379515
\(388\) −13560.7 −1.77433
\(389\) −11727.1 −1.52850 −0.764252 0.644918i \(-0.776891\pi\)
−0.764252 + 0.644918i \(0.776891\pi\)
\(390\) 194.784 0.0252904
\(391\) 14175.1 1.83341
\(392\) −10468.7 −1.34885
\(393\) −1157.44 −0.148563
\(394\) 1019.96 0.130418
\(395\) −1091.45 −0.139030
\(396\) −1145.98 −0.145424
\(397\) −359.905 −0.0454990 −0.0227495 0.999741i \(-0.507242\pi\)
−0.0227495 + 0.999741i \(0.507242\pi\)
\(398\) −99.1105 −0.0124823
\(399\) −3313.89 −0.415794
\(400\) 2642.83 0.330354
\(401\) −4066.71 −0.506438 −0.253219 0.967409i \(-0.581489\pi\)
−0.253219 + 0.967409i \(0.581489\pi\)
\(402\) 1124.85 0.139558
\(403\) 724.274 0.0895252
\(404\) −14136.9 −1.74093
\(405\) 230.757 0.0283122
\(406\) 9740.45 1.19067
\(407\) 4620.77 0.562760
\(408\) 5787.32 0.702242
\(409\) −13488.8 −1.63076 −0.815379 0.578927i \(-0.803472\pi\)
−0.815379 + 0.578927i \(0.803472\pi\)
\(410\) 4059.40 0.488975
\(411\) −2654.52 −0.318584
\(412\) 5981.62 0.715275
\(413\) 12883.3 1.53498
\(414\) −4628.83 −0.549504
\(415\) −2650.04 −0.313459
\(416\) 1167.24 0.137569
\(417\) 3275.83 0.384695
\(418\) 1696.05 0.198460
\(419\) −7040.12 −0.820841 −0.410420 0.911896i \(-0.634618\pi\)
−0.410420 + 0.911896i \(0.634618\pi\)
\(420\) −3135.90 −0.364325
\(421\) 9171.74 1.06177 0.530883 0.847445i \(-0.321860\pi\)
0.530883 + 0.847445i \(0.321860\pi\)
\(422\) 4652.89 0.536728
\(423\) −2082.67 −0.239392
\(424\) −77.8204 −0.00891343
\(425\) −14253.1 −1.62677
\(426\) 651.464 0.0740928
\(427\) −17637.6 −1.99893
\(428\) −1759.77 −0.198742
\(429\) 169.988 0.0191307
\(430\) −4046.51 −0.453814
\(431\) 992.995 0.110976 0.0554882 0.998459i \(-0.482328\pi\)
0.0554882 + 0.998459i \(0.482328\pi\)
\(432\) 610.490 0.0679912
\(433\) 3790.21 0.420660 0.210330 0.977630i \(-0.432546\pi\)
0.210330 + 0.977630i \(0.432546\pi\)
\(434\) −19719.0 −2.18097
\(435\) 593.589 0.0654262
\(436\) 25122.7 2.75954
\(437\) 4050.98 0.443443
\(438\) 10431.3 1.13796
\(439\) −5136.97 −0.558483 −0.279242 0.960221i \(-0.590083\pi\)
−0.279242 + 0.960221i \(0.590083\pi\)
\(440\) 495.754 0.0537139
\(441\) 5955.71 0.643095
\(442\) −2779.16 −0.299075
\(443\) 10676.8 1.14508 0.572541 0.819876i \(-0.305958\pi\)
0.572541 + 0.819876i \(0.305958\pi\)
\(444\) 14587.7 1.55923
\(445\) −2086.96 −0.222317
\(446\) −17086.2 −1.81403
\(447\) −891.042 −0.0942838
\(448\) −26045.5 −2.74672
\(449\) 10529.9 1.10676 0.553379 0.832929i \(-0.313338\pi\)
0.553379 + 0.832929i \(0.313338\pi\)
\(450\) 4654.30 0.487569
\(451\) 3542.64 0.369881
\(452\) −7482.42 −0.778636
\(453\) 5663.59 0.587414
\(454\) 3861.29 0.399162
\(455\) 465.160 0.0479275
\(456\) 1653.91 0.169850
\(457\) −14072.5 −1.44045 −0.720225 0.693741i \(-0.755961\pi\)
−0.720225 + 0.693741i \(0.755961\pi\)
\(458\) −8148.55 −0.831347
\(459\) −3292.43 −0.334810
\(460\) 3833.41 0.388551
\(461\) −30.8173 −0.00311346 −0.00155673 0.999999i \(-0.500496\pi\)
−0.00155673 + 0.999999i \(0.500496\pi\)
\(462\) −4628.06 −0.466054
\(463\) 17591.3 1.76573 0.882867 0.469622i \(-0.155610\pi\)
0.882867 + 0.469622i \(0.155610\pi\)
\(464\) 1570.39 0.157120
\(465\) −1201.69 −0.119843
\(466\) −17397.5 −1.72945
\(467\) 13273.1 1.31522 0.657609 0.753360i \(-0.271568\pi\)
0.657609 + 0.753360i \(0.271568\pi\)
\(468\) 536.647 0.0530053
\(469\) 2686.23 0.264474
\(470\) 2916.79 0.286259
\(471\) 169.603 0.0165921
\(472\) −6429.87 −0.627031
\(473\) −3531.39 −0.343284
\(474\) −5085.23 −0.492769
\(475\) −4073.27 −0.393462
\(476\) 44742.9 4.30837
\(477\) 44.2724 0.00424968
\(478\) −21113.8 −2.02034
\(479\) 2496.68 0.238155 0.119077 0.992885i \(-0.462006\pi\)
0.119077 + 0.992885i \(0.462006\pi\)
\(480\) −1936.64 −0.184156
\(481\) −2163.84 −0.205120
\(482\) −17648.3 −1.66776
\(483\) −11054.0 −1.04136
\(484\) 1400.64 0.131541
\(485\) −3337.41 −0.312462
\(486\) 1075.14 0.100348
\(487\) −3464.42 −0.322357 −0.161178 0.986925i \(-0.551529\pi\)
−0.161178 + 0.986925i \(0.551529\pi\)
\(488\) 8802.65 0.816552
\(489\) 147.701 0.0136591
\(490\) −8341.01 −0.768997
\(491\) −16224.6 −1.49125 −0.745625 0.666366i \(-0.767849\pi\)
−0.745625 + 0.666366i \(0.767849\pi\)
\(492\) 11184.0 1.02483
\(493\) −8469.29 −0.773707
\(494\) −794.236 −0.0723367
\(495\) −282.037 −0.0256093
\(496\) −3179.17 −0.287800
\(497\) 1555.75 0.140412
\(498\) −12347.0 −1.11100
\(499\) 9993.81 0.896562 0.448281 0.893893i \(-0.352036\pi\)
0.448281 + 0.893893i \(0.352036\pi\)
\(500\) −7976.65 −0.713453
\(501\) −6206.24 −0.553441
\(502\) 24220.3 2.15340
\(503\) −15334.8 −1.35933 −0.679667 0.733520i \(-0.737876\pi\)
−0.679667 + 0.733520i \(0.737876\pi\)
\(504\) −4513.08 −0.398866
\(505\) −3479.23 −0.306581
\(506\) 5657.46 0.497045
\(507\) 6511.40 0.570377
\(508\) −11498.1 −1.00422
\(509\) −7291.23 −0.634927 −0.317464 0.948270i \(-0.602831\pi\)
−0.317464 + 0.948270i \(0.602831\pi\)
\(510\) 4611.08 0.400357
\(511\) 24910.7 2.15653
\(512\) −7985.14 −0.689251
\(513\) −940.919 −0.0809797
\(514\) 28466.8 2.44283
\(515\) 1472.13 0.125961
\(516\) −11148.5 −0.951134
\(517\) 2545.49 0.216538
\(518\) 58912.5 4.99704
\(519\) 1812.04 0.153255
\(520\) −232.154 −0.0195782
\(521\) 16794.3 1.41223 0.706114 0.708098i \(-0.250447\pi\)
0.706114 + 0.708098i \(0.250447\pi\)
\(522\) 2765.63 0.231893
\(523\) −21009.4 −1.75655 −0.878275 0.478157i \(-0.841305\pi\)
−0.878275 + 0.478157i \(0.841305\pi\)
\(524\) 4466.01 0.372326
\(525\) 11114.9 0.923986
\(526\) −33577.0 −2.78332
\(527\) 17145.6 1.41722
\(528\) −746.154 −0.0615003
\(529\) 1345.73 0.110605
\(530\) −62.0039 −0.00508166
\(531\) 3657.99 0.298951
\(532\) 12786.7 1.04206
\(533\) −1658.97 −0.134818
\(534\) −9723.46 −0.787969
\(535\) −433.097 −0.0349989
\(536\) −1340.66 −0.108036
\(537\) 6396.07 0.513987
\(538\) −2115.66 −0.169540
\(539\) −7279.20 −0.581702
\(540\) −890.383 −0.0709555
\(541\) −16802.8 −1.33532 −0.667662 0.744464i \(-0.732705\pi\)
−0.667662 + 0.744464i \(0.732705\pi\)
\(542\) 541.211 0.0428911
\(543\) 1768.11 0.139737
\(544\) 27631.9 2.17777
\(545\) 6182.93 0.485959
\(546\) 2167.25 0.169872
\(547\) 16784.5 1.31198 0.655990 0.754770i \(-0.272251\pi\)
0.655990 + 0.754770i \(0.272251\pi\)
\(548\) 10242.5 0.798429
\(549\) −5007.88 −0.389309
\(550\) −5688.59 −0.441023
\(551\) −2420.37 −0.187135
\(552\) 5516.91 0.425390
\(553\) −12144.0 −0.933840
\(554\) −36277.7 −2.78212
\(555\) 3590.16 0.274584
\(556\) −12639.9 −0.964117
\(557\) 18127.0 1.37893 0.689467 0.724317i \(-0.257845\pi\)
0.689467 + 0.724317i \(0.257845\pi\)
\(558\) −5598.85 −0.424764
\(559\) 1653.70 0.125123
\(560\) −2041.80 −0.154075
\(561\) 4024.09 0.302847
\(562\) −30722.3 −2.30595
\(563\) 2090.88 0.156518 0.0782592 0.996933i \(-0.475064\pi\)
0.0782592 + 0.996933i \(0.475064\pi\)
\(564\) 8036.03 0.599961
\(565\) −1841.49 −0.137119
\(566\) −4579.89 −0.340119
\(567\) 2567.51 0.190168
\(568\) −776.452 −0.0573578
\(569\) 6249.23 0.460424 0.230212 0.973140i \(-0.426058\pi\)
0.230212 + 0.973140i \(0.426058\pi\)
\(570\) 1317.76 0.0968335
\(571\) 6048.79 0.443317 0.221659 0.975124i \(-0.428853\pi\)
0.221659 + 0.975124i \(0.428853\pi\)
\(572\) −655.902 −0.0479451
\(573\) 6482.69 0.472633
\(574\) 45166.8 3.28437
\(575\) −13587.1 −0.985428
\(576\) −7395.15 −0.534950
\(577\) −15729.1 −1.13486 −0.567429 0.823423i \(-0.692062\pi\)
−0.567429 + 0.823423i \(0.692062\pi\)
\(578\) −44053.4 −3.17021
\(579\) 4472.73 0.321037
\(580\) −2290.38 −0.163970
\(581\) −29485.6 −2.10545
\(582\) −15549.5 −1.10747
\(583\) −54.1108 −0.00384398
\(584\) −12432.6 −0.880931
\(585\) 132.074 0.00933433
\(586\) 27187.3 1.91655
\(587\) 15620.5 1.09835 0.549173 0.835709i \(-0.314943\pi\)
0.549173 + 0.835709i \(0.314943\pi\)
\(588\) −22980.2 −1.61172
\(589\) 4899.91 0.342780
\(590\) −5123.04 −0.357478
\(591\) 691.587 0.0481355
\(592\) 9498.09 0.659407
\(593\) −493.541 −0.0341776 −0.0170888 0.999854i \(-0.505440\pi\)
−0.0170888 + 0.999854i \(0.505440\pi\)
\(594\) −1314.06 −0.0907683
\(595\) 11011.6 0.758711
\(596\) 3438.11 0.236293
\(597\) −67.2022 −0.00460704
\(598\) −2649.31 −0.181168
\(599\) −12455.1 −0.849585 −0.424793 0.905291i \(-0.639653\pi\)
−0.424793 + 0.905291i \(0.639653\pi\)
\(600\) −5547.27 −0.377444
\(601\) 12454.8 0.845329 0.422664 0.906286i \(-0.361095\pi\)
0.422664 + 0.906286i \(0.361095\pi\)
\(602\) −45023.3 −3.04820
\(603\) 762.707 0.0515088
\(604\) −21853.1 −1.47217
\(605\) 344.712 0.0231645
\(606\) −16210.3 −1.08663
\(607\) −4243.19 −0.283733 −0.141867 0.989886i \(-0.545310\pi\)
−0.141867 + 0.989886i \(0.545310\pi\)
\(608\) 7896.70 0.526732
\(609\) 6604.54 0.439458
\(610\) 7013.57 0.465526
\(611\) −1192.01 −0.0789259
\(612\) 12703.9 0.839095
\(613\) 5733.14 0.377748 0.188874 0.982001i \(-0.439516\pi\)
0.188874 + 0.982001i \(0.439516\pi\)
\(614\) 9672.18 0.635729
\(615\) 2752.49 0.180473
\(616\) 5515.99 0.360788
\(617\) 15642.1 1.02063 0.510314 0.859988i \(-0.329529\pi\)
0.510314 + 0.859988i \(0.329529\pi\)
\(618\) 6858.91 0.446449
\(619\) −7467.40 −0.484879 −0.242440 0.970167i \(-0.577948\pi\)
−0.242440 + 0.970167i \(0.577948\pi\)
\(620\) 4636.74 0.300348
\(621\) −3138.60 −0.202814
\(622\) 33116.1 2.13478
\(623\) −23220.4 −1.49327
\(624\) 349.413 0.0224162
\(625\) 12647.4 0.809432
\(626\) 30233.6 1.93031
\(627\) 1150.01 0.0732489
\(628\) −654.416 −0.0415829
\(629\) −51224.2 −3.24713
\(630\) −3595.82 −0.227399
\(631\) −1486.38 −0.0937745 −0.0468872 0.998900i \(-0.514930\pi\)
−0.0468872 + 0.998900i \(0.514930\pi\)
\(632\) 6060.87 0.381469
\(633\) 3154.91 0.198099
\(634\) −4089.35 −0.256165
\(635\) −2829.78 −0.176845
\(636\) −170.826 −0.0106505
\(637\) 3408.74 0.212024
\(638\) −3380.21 −0.209755
\(639\) 441.728 0.0273466
\(640\) 5192.58 0.320711
\(641\) 12386.0 0.763211 0.381606 0.924325i \(-0.375371\pi\)
0.381606 + 0.924325i \(0.375371\pi\)
\(642\) −2017.87 −0.124048
\(643\) −14458.1 −0.886737 −0.443369 0.896339i \(-0.646217\pi\)
−0.443369 + 0.896339i \(0.646217\pi\)
\(644\) 42652.3 2.60984
\(645\) −2743.75 −0.167496
\(646\) −18801.8 −1.14512
\(647\) 15792.8 0.959625 0.479813 0.877371i \(-0.340705\pi\)
0.479813 + 0.877371i \(0.340705\pi\)
\(648\) −1281.41 −0.0776828
\(649\) −4470.87 −0.270412
\(650\) 2663.89 0.160748
\(651\) −13370.5 −0.804965
\(652\) −569.909 −0.0342321
\(653\) −3179.93 −0.190567 −0.0952837 0.995450i \(-0.530376\pi\)
−0.0952837 + 0.995450i \(0.530376\pi\)
\(654\) 28807.3 1.72240
\(655\) 1099.13 0.0655672
\(656\) 7281.96 0.433404
\(657\) 7072.96 0.420003
\(658\) 32453.6 1.92276
\(659\) 11593.5 0.685308 0.342654 0.939462i \(-0.388674\pi\)
0.342654 + 0.939462i \(0.388674\pi\)
\(660\) 1088.25 0.0641817
\(661\) 3233.88 0.190293 0.0951464 0.995463i \(-0.469668\pi\)
0.0951464 + 0.995463i \(0.469668\pi\)
\(662\) 43449.9 2.55095
\(663\) −1884.42 −0.110384
\(664\) 14715.8 0.860066
\(665\) 3146.93 0.183508
\(666\) 16727.1 0.973219
\(667\) −8073.56 −0.468680
\(668\) 23946.9 1.38703
\(669\) −11585.4 −0.669532
\(670\) −1068.18 −0.0615929
\(671\) 6120.74 0.352144
\(672\) −21548.0 −1.23695
\(673\) −5495.72 −0.314776 −0.157388 0.987537i \(-0.550307\pi\)
−0.157388 + 0.987537i \(0.550307\pi\)
\(674\) −2658.23 −0.151916
\(675\) 3155.87 0.179955
\(676\) −25124.4 −1.42947
\(677\) 33836.7 1.92090 0.960451 0.278448i \(-0.0898200\pi\)
0.960451 + 0.278448i \(0.0898200\pi\)
\(678\) −8579.82 −0.485997
\(679\) −37133.6 −2.09876
\(680\) −5495.75 −0.309930
\(681\) 2618.16 0.147325
\(682\) 6843.04 0.384214
\(683\) −21080.3 −1.18099 −0.590493 0.807043i \(-0.701067\pi\)
−0.590493 + 0.807043i \(0.701067\pi\)
\(684\) 3630.56 0.202950
\(685\) 2520.78 0.140605
\(686\) −44702.2 −2.48796
\(687\) −5525.16 −0.306838
\(688\) −7258.84 −0.402239
\(689\) 25.3393 0.00140109
\(690\) 4395.63 0.242520
\(691\) 11811.3 0.650253 0.325127 0.945671i \(-0.394593\pi\)
0.325127 + 0.945671i \(0.394593\pi\)
\(692\) −6991.79 −0.384087
\(693\) −3138.07 −0.172014
\(694\) 13907.8 0.760711
\(695\) −3110.79 −0.169783
\(696\) −3296.23 −0.179516
\(697\) −39272.4 −2.13422
\(698\) −3188.52 −0.172904
\(699\) −11796.4 −0.638313
\(700\) −42887.0 −2.31568
\(701\) 4244.99 0.228718 0.114359 0.993440i \(-0.463519\pi\)
0.114359 + 0.993440i \(0.463519\pi\)
\(702\) 615.353 0.0330841
\(703\) −14639.0 −0.785376
\(704\) 9038.51 0.483880
\(705\) 1977.74 0.105654
\(706\) −5340.81 −0.284708
\(707\) −38711.5 −2.05926
\(708\) −14114.4 −0.749227
\(709\) −898.822 −0.0476107 −0.0238053 0.999717i \(-0.507578\pi\)
−0.0238053 + 0.999717i \(0.507578\pi\)
\(710\) −618.643 −0.0327004
\(711\) −3448.06 −0.181874
\(712\) 11589.0 0.609993
\(713\) 16344.5 0.858493
\(714\) 51305.0 2.68913
\(715\) −161.424 −0.00844322
\(716\) −24679.4 −1.28815
\(717\) −14316.3 −0.745679
\(718\) −38706.3 −2.01185
\(719\) 10741.8 0.557165 0.278582 0.960412i \(-0.410135\pi\)
0.278582 + 0.960412i \(0.410135\pi\)
\(720\) −579.733 −0.0300074
\(721\) 16379.6 0.846061
\(722\) 24973.9 1.28730
\(723\) −11966.5 −0.615546
\(724\) −6822.30 −0.350206
\(725\) 8117.99 0.415855
\(726\) 1606.07 0.0821030
\(727\) 16794.2 0.856758 0.428379 0.903599i \(-0.359085\pi\)
0.428379 + 0.903599i \(0.359085\pi\)
\(728\) −2583.06 −0.131503
\(729\) 729.000 0.0370370
\(730\) −9905.73 −0.502229
\(731\) 39147.7 1.98075
\(732\) 19323.0 0.975681
\(733\) 8659.40 0.436347 0.218173 0.975910i \(-0.429990\pi\)
0.218173 + 0.975910i \(0.429990\pi\)
\(734\) 29778.4 1.49747
\(735\) −5655.65 −0.283826
\(736\) 26340.8 1.31920
\(737\) −932.197 −0.0465915
\(738\) 12824.3 0.639660
\(739\) 16705.7 0.831567 0.415783 0.909464i \(-0.363507\pi\)
0.415783 + 0.909464i \(0.363507\pi\)
\(740\) −13852.7 −0.688157
\(741\) −538.534 −0.0266984
\(742\) −689.884 −0.0341327
\(743\) 1292.12 0.0637996 0.0318998 0.999491i \(-0.489844\pi\)
0.0318998 + 0.999491i \(0.489844\pi\)
\(744\) 6673.03 0.328824
\(745\) 846.151 0.0416115
\(746\) 1006.09 0.0493773
\(747\) −8371.90 −0.410056
\(748\) −15527.0 −0.758990
\(749\) −4818.83 −0.235082
\(750\) −9146.53 −0.445312
\(751\) −14980.4 −0.727886 −0.363943 0.931421i \(-0.618570\pi\)
−0.363943 + 0.931421i \(0.618570\pi\)
\(752\) 5232.30 0.253726
\(753\) 16422.7 0.794787
\(754\) 1582.90 0.0764535
\(755\) −5378.25 −0.259251
\(756\) −9906.82 −0.476597
\(757\) 3003.41 0.144202 0.0721010 0.997397i \(-0.477030\pi\)
0.0721010 + 0.997397i \(0.477030\pi\)
\(758\) −50244.8 −2.40762
\(759\) 3836.06 0.183452
\(760\) −1570.59 −0.0749621
\(761\) −20375.0 −0.970555 −0.485277 0.874360i \(-0.661281\pi\)
−0.485277 + 0.874360i \(0.661281\pi\)
\(762\) −13184.4 −0.626799
\(763\) 68794.1 3.26411
\(764\) −25013.6 −1.18450
\(765\) 3126.56 0.147766
\(766\) −47578.4 −2.24422
\(767\) 2093.65 0.0985621
\(768\) 4472.70 0.210149
\(769\) −12372.4 −0.580184 −0.290092 0.956999i \(-0.593686\pi\)
−0.290092 + 0.956999i \(0.593686\pi\)
\(770\) 4394.90 0.205690
\(771\) 19302.0 0.901615
\(772\) −17258.1 −0.804578
\(773\) 21023.6 0.978225 0.489113 0.872221i \(-0.337321\pi\)
0.489113 + 0.872221i \(0.337321\pi\)
\(774\) −12783.6 −0.593664
\(775\) −16434.4 −0.761732
\(776\) 18532.8 0.857331
\(777\) 39945.8 1.84433
\(778\) 51885.7 2.39099
\(779\) −11223.4 −0.516198
\(780\) −509.610 −0.0233935
\(781\) −539.889 −0.0247359
\(782\) −62716.6 −2.86795
\(783\) 1875.24 0.0855884
\(784\) −14962.5 −0.681602
\(785\) −161.058 −0.00732281
\(786\) 5121.02 0.232393
\(787\) 30286.2 1.37177 0.685886 0.727709i \(-0.259415\pi\)
0.685886 + 0.727709i \(0.259415\pi\)
\(788\) −2668.51 −0.120637
\(789\) −22767.0 −1.02728
\(790\) 4829.03 0.217480
\(791\) −20489.3 −0.921007
\(792\) 1566.17 0.0702668
\(793\) −2866.25 −0.128353
\(794\) 1592.37 0.0711729
\(795\) −42.0420 −0.00187557
\(796\) 259.301 0.0115461
\(797\) 32337.8 1.43722 0.718610 0.695413i \(-0.244779\pi\)
0.718610 + 0.695413i \(0.244779\pi\)
\(798\) 14662.1 0.650415
\(799\) −28218.3 −1.24943
\(800\) −26485.7 −1.17052
\(801\) −6593.03 −0.290828
\(802\) 17992.9 0.792207
\(803\) −8644.72 −0.379907
\(804\) −2942.92 −0.129091
\(805\) 10497.1 0.459596
\(806\) −3204.50 −0.140042
\(807\) −1434.53 −0.0625749
\(808\) 19320.3 0.841197
\(809\) −891.707 −0.0387525 −0.0193762 0.999812i \(-0.506168\pi\)
−0.0193762 + 0.999812i \(0.506168\pi\)
\(810\) −1020.97 −0.0442879
\(811\) −10114.9 −0.437957 −0.218978 0.975730i \(-0.570272\pi\)
−0.218978 + 0.975730i \(0.570272\pi\)
\(812\) −25483.8 −1.10136
\(813\) 366.970 0.0158305
\(814\) −20444.3 −0.880309
\(815\) −140.260 −0.00602833
\(816\) 8271.59 0.354857
\(817\) 11187.7 0.479080
\(818\) 59680.4 2.55095
\(819\) 1469.51 0.0626972
\(820\) −10620.6 −0.452300
\(821\) 10833.5 0.460525 0.230262 0.973129i \(-0.426042\pi\)
0.230262 + 0.973129i \(0.426042\pi\)
\(822\) 11744.7 0.498351
\(823\) 31958.5 1.35359 0.676794 0.736173i \(-0.263369\pi\)
0.676794 + 0.736173i \(0.263369\pi\)
\(824\) −8174.84 −0.345612
\(825\) −3857.17 −0.162775
\(826\) −57001.3 −2.40112
\(827\) 34847.3 1.46525 0.732624 0.680634i \(-0.238296\pi\)
0.732624 + 0.680634i \(0.238296\pi\)
\(828\) 12110.3 0.508289
\(829\) 6537.91 0.273910 0.136955 0.990577i \(-0.456268\pi\)
0.136955 + 0.990577i \(0.456268\pi\)
\(830\) 11724.9 0.490334
\(831\) −24598.2 −1.02684
\(832\) −4232.60 −0.176369
\(833\) 80694.5 3.35642
\(834\) −14493.7 −0.601768
\(835\) 5893.56 0.244258
\(836\) −4437.35 −0.183575
\(837\) −3796.32 −0.156774
\(838\) 31148.5 1.28402
\(839\) 2710.34 0.111527 0.0557635 0.998444i \(-0.482241\pi\)
0.0557635 + 0.998444i \(0.482241\pi\)
\(840\) 4285.71 0.176037
\(841\) −19565.2 −0.802215
\(842\) −40579.7 −1.66089
\(843\) −20831.4 −0.851092
\(844\) −12173.3 −0.496471
\(845\) −6183.35 −0.251732
\(846\) 9214.62 0.374474
\(847\) 3835.42 0.155592
\(848\) −111.226 −0.00450414
\(849\) −3105.41 −0.125533
\(850\) 63061.7 2.54470
\(851\) −48830.8 −1.96698
\(852\) −1704.42 −0.0685356
\(853\) 9759.32 0.391738 0.195869 0.980630i \(-0.437247\pi\)
0.195869 + 0.980630i \(0.437247\pi\)
\(854\) 78036.2 3.12687
\(855\) 893.515 0.0357398
\(856\) 2405.01 0.0960298
\(857\) −13649.8 −0.544072 −0.272036 0.962287i \(-0.587697\pi\)
−0.272036 + 0.962287i \(0.587697\pi\)
\(858\) −752.098 −0.0299257
\(859\) 7796.42 0.309674 0.154837 0.987940i \(-0.450515\pi\)
0.154837 + 0.987940i \(0.450515\pi\)
\(860\) 10586.8 0.419776
\(861\) 30625.5 1.21221
\(862\) −4393.43 −0.173597
\(863\) 7183.57 0.283350 0.141675 0.989913i \(-0.454751\pi\)
0.141675 + 0.989913i \(0.454751\pi\)
\(864\) −6118.16 −0.240907
\(865\) −1720.75 −0.0676383
\(866\) −16769.5 −0.658026
\(867\) −29870.6 −1.17008
\(868\) 51590.5 2.01739
\(869\) 4214.30 0.164511
\(870\) −2626.29 −0.102344
\(871\) 436.534 0.0169821
\(872\) −34334.1 −1.33337
\(873\) −10543.4 −0.408752
\(874\) −17923.3 −0.693666
\(875\) −21842.7 −0.843905
\(876\) −27291.2 −1.05261
\(877\) 17063.1 0.656991 0.328495 0.944506i \(-0.393458\pi\)
0.328495 + 0.944506i \(0.393458\pi\)
\(878\) 22728.2 0.873620
\(879\) 18434.4 0.707369
\(880\) 708.562 0.0271428
\(881\) −32174.9 −1.23042 −0.615210 0.788363i \(-0.710929\pi\)
−0.615210 + 0.788363i \(0.710929\pi\)
\(882\) −26350.6 −1.00598
\(883\) 2843.68 0.108378 0.0541889 0.998531i \(-0.482743\pi\)
0.0541889 + 0.998531i \(0.482743\pi\)
\(884\) 7271.09 0.276644
\(885\) −3473.69 −0.131940
\(886\) −47238.9 −1.79122
\(887\) −31417.8 −1.18930 −0.594649 0.803985i \(-0.702709\pi\)
−0.594649 + 0.803985i \(0.702709\pi\)
\(888\) −19936.4 −0.753401
\(889\) −31485.5 −1.18784
\(890\) 9233.59 0.347765
\(891\) −891.000 −0.0335013
\(892\) 44702.5 1.67797
\(893\) −8064.30 −0.302196
\(894\) 3942.35 0.147485
\(895\) −6073.83 −0.226845
\(896\) 57775.1 2.15416
\(897\) −1796.37 −0.0668664
\(898\) −46588.6 −1.73127
\(899\) −9765.47 −0.362288
\(900\) −12177.0 −0.451000
\(901\) 599.852 0.0221798
\(902\) −15674.1 −0.578594
\(903\) −30528.2 −1.12505
\(904\) 10225.9 0.376227
\(905\) −1679.03 −0.0616718
\(906\) −25058.1 −0.918875
\(907\) 12253.1 0.448573 0.224287 0.974523i \(-0.427995\pi\)
0.224287 + 0.974523i \(0.427995\pi\)
\(908\) −10102.2 −0.369223
\(909\) −10991.4 −0.401059
\(910\) −2058.07 −0.0749717
\(911\) −48422.4 −1.76104 −0.880518 0.474012i \(-0.842805\pi\)
−0.880518 + 0.474012i \(0.842805\pi\)
\(912\) 2363.87 0.0858286
\(913\) 10232.3 0.370909
\(914\) 62262.9 2.25326
\(915\) 4755.57 0.171819
\(916\) 21318.9 0.768993
\(917\) 12229.4 0.440404
\(918\) 14567.1 0.523733
\(919\) 5546.18 0.199077 0.0995385 0.995034i \(-0.468263\pi\)
0.0995385 + 0.995034i \(0.468263\pi\)
\(920\) −5238.96 −0.187743
\(921\) 6558.26 0.234638
\(922\) 136.349 0.00487030
\(923\) 252.822 0.00901598
\(924\) 12108.3 0.431098
\(925\) 49099.5 1.74528
\(926\) −77831.3 −2.76209
\(927\) 4650.71 0.164778
\(928\) −15738.0 −0.556709
\(929\) −35684.5 −1.26025 −0.630125 0.776494i \(-0.716996\pi\)
−0.630125 + 0.776494i \(0.716996\pi\)
\(930\) 5316.78 0.187467
\(931\) 23061.1 0.811811
\(932\) 45516.7 1.59973
\(933\) 22454.5 0.787918
\(934\) −58726.0 −2.05736
\(935\) −3821.35 −0.133659
\(936\) −733.413 −0.0256115
\(937\) −48903.6 −1.70503 −0.852514 0.522705i \(-0.824923\pi\)
−0.852514 + 0.522705i \(0.824923\pi\)
\(938\) −11885.0 −0.413710
\(939\) 20500.0 0.712451
\(940\) −7631.17 −0.264789
\(941\) −23741.9 −0.822490 −0.411245 0.911525i \(-0.634906\pi\)
−0.411245 + 0.911525i \(0.634906\pi\)
\(942\) −750.396 −0.0259546
\(943\) −37437.4 −1.29282
\(944\) −9189.97 −0.316852
\(945\) −2438.16 −0.0839295
\(946\) 15624.4 0.536989
\(947\) 37612.4 1.29064 0.645321 0.763911i \(-0.276724\pi\)
0.645321 + 0.763911i \(0.276724\pi\)
\(948\) 13304.4 0.455810
\(949\) 4048.20 0.138472
\(950\) 18021.9 0.615482
\(951\) −2772.80 −0.0945469
\(952\) −61148.2 −2.08175
\(953\) −48294.3 −1.64156 −0.820779 0.571246i \(-0.806460\pi\)
−0.820779 + 0.571246i \(0.806460\pi\)
\(954\) −195.880 −0.00664765
\(955\) −6156.09 −0.208593
\(956\) 55239.8 1.86881
\(957\) −2291.96 −0.0774176
\(958\) −11046.4 −0.372539
\(959\) 28047.4 0.944419
\(960\) 7022.57 0.236096
\(961\) −10021.4 −0.336389
\(962\) 9573.76 0.320863
\(963\) −1368.22 −0.0457843
\(964\) 46173.1 1.54267
\(965\) −4247.39 −0.141687
\(966\) 48907.8 1.62897
\(967\) 1840.92 0.0612204 0.0306102 0.999531i \(-0.490255\pi\)
0.0306102 + 0.999531i \(0.490255\pi\)
\(968\) −1914.20 −0.0635587
\(969\) −12748.6 −0.422647
\(970\) 14766.1 0.488775
\(971\) 31461.8 1.03981 0.519906 0.854223i \(-0.325967\pi\)
0.519906 + 0.854223i \(0.325967\pi\)
\(972\) −2812.86 −0.0928217
\(973\) −34612.1 −1.14040
\(974\) 15328.1 0.504254
\(975\) 1806.26 0.0593298
\(976\) 12581.3 0.412620
\(977\) −7040.11 −0.230535 −0.115268 0.993334i \(-0.536773\pi\)
−0.115268 + 0.993334i \(0.536773\pi\)
\(978\) −653.494 −0.0213665
\(979\) 8058.15 0.263064
\(980\) 21822.5 0.711320
\(981\) 19532.9 0.635715
\(982\) 71784.4 2.33272
\(983\) −24610.9 −0.798541 −0.399270 0.916833i \(-0.630737\pi\)
−0.399270 + 0.916833i \(0.630737\pi\)
\(984\) −15284.7 −0.495183
\(985\) −656.744 −0.0212443
\(986\) 37471.8 1.21029
\(987\) 22005.3 0.709662
\(988\) 2077.95 0.0669112
\(989\) 37318.5 1.19986
\(990\) 1247.85 0.0400599
\(991\) −40003.3 −1.28229 −0.641144 0.767421i \(-0.721540\pi\)
−0.641144 + 0.767421i \(0.721540\pi\)
\(992\) 31860.8 1.01974
\(993\) 29461.4 0.941519
\(994\) −6883.31 −0.219643
\(995\) 63.8165 0.00203329
\(996\) 32303.2 1.02768
\(997\) −7342.61 −0.233242 −0.116621 0.993176i \(-0.537206\pi\)
−0.116621 + 0.993176i \(0.537206\pi\)
\(998\) −44216.9 −1.40247
\(999\) 11341.9 0.359201
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 33.4.a.c.1.1 2
3.2 odd 2 99.4.a.f.1.2 2
4.3 odd 2 528.4.a.p.1.2 2
5.2 odd 4 825.4.c.h.199.2 4
5.3 odd 4 825.4.c.h.199.3 4
5.4 even 2 825.4.a.l.1.2 2
7.6 odd 2 1617.4.a.k.1.1 2
8.3 odd 2 2112.4.a.bg.1.1 2
8.5 even 2 2112.4.a.bn.1.1 2
11.10 odd 2 363.4.a.i.1.2 2
12.11 even 2 1584.4.a.bj.1.1 2
15.14 odd 2 2475.4.a.p.1.1 2
33.32 even 2 1089.4.a.u.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
33.4.a.c.1.1 2 1.1 even 1 trivial
99.4.a.f.1.2 2 3.2 odd 2
363.4.a.i.1.2 2 11.10 odd 2
528.4.a.p.1.2 2 4.3 odd 2
825.4.a.l.1.2 2 5.4 even 2
825.4.c.h.199.2 4 5.2 odd 4
825.4.c.h.199.3 4 5.3 odd 4
1089.4.a.u.1.1 2 33.32 even 2
1584.4.a.bj.1.1 2 12.11 even 2
1617.4.a.k.1.1 2 7.6 odd 2
2112.4.a.bg.1.1 2 8.3 odd 2
2112.4.a.bn.1.1 2 8.5 even 2
2475.4.a.p.1.1 2 15.14 odd 2