Properties

Label 1617.4.a.k.1.1
Level $1617$
Weight $4$
Character 1617.1
Self dual yes
Analytic conductor $95.406$
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] = [1617,4,Mod(1,1617)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1617, 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("1617.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1617 = 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1617.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: \(95.4060884793\)
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: no (minimal twist has level 33)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-4.42443\) of defining polynomial
Character \(\chi\) \(=\) 1617.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.42443 q^{2} +3.00000 q^{3} +11.5756 q^{4} -2.84886 q^{5} -13.2733 q^{6} -15.8199 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.42443 q^{2} +3.00000 q^{3} +11.5756 q^{4} -2.84886 q^{5} -13.2733 q^{6} -15.8199 q^{8} +9.00000 q^{9} +12.6046 q^{10} -11.0000 q^{11} +34.7267 q^{12} -5.15114 q^{13} -8.54657 q^{15} -22.6107 q^{16} -121.942 q^{17} -39.8199 q^{18} -34.8489 q^{19} -32.9772 q^{20} +48.6687 q^{22} +116.244 q^{23} -47.4596 q^{24} -116.884 q^{25} +22.7909 q^{26} +27.0000 q^{27} -69.4534 q^{29} +37.8137 q^{30} -140.605 q^{31} +226.598 q^{32} -33.0000 q^{33} +539.524 q^{34} +104.180 q^{36} -420.070 q^{37} +154.186 q^{38} -15.4534 q^{39} +45.0685 q^{40} +322.058 q^{41} +321.035 q^{43} -127.331 q^{44} -25.6397 q^{45} -514.315 q^{46} +231.408 q^{47} -67.8322 q^{48} +517.145 q^{50} -365.826 q^{51} -59.6274 q^{52} +4.91916 q^{53} -119.460 q^{54} +31.3374 q^{55} -104.547 q^{57} +307.292 q^{58} -406.443 q^{59} -98.9315 q^{60} +556.431 q^{61} +622.095 q^{62} -821.683 q^{64} +14.6749 q^{65} +146.006 q^{66} +84.7452 q^{67} -1411.55 q^{68} +348.733 q^{69} +49.0808 q^{71} -142.379 q^{72} -785.884 q^{73} +1858.57 q^{74} -350.652 q^{75} -403.395 q^{76} +68.3726 q^{78} -383.118 q^{79} +64.4147 q^{80} +81.0000 q^{81} -1424.92 q^{82} +930.211 q^{83} +347.395 q^{85} -1420.40 q^{86} -208.360 q^{87} +174.018 q^{88} +732.559 q^{89} +113.441 q^{90} +1345.59 q^{92} -421.814 q^{93} -1023.85 q^{94} +99.2794 q^{95} +679.795 q^{96} +1171.49 q^{97} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + q^{2} + 6 q^{3} + 33 q^{4} + 14 q^{5} + 3 q^{6} + 57 q^{8} + 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + q^{2} + 6 q^{3} + 33 q^{4} + 14 q^{5} + 3 q^{6} + 57 q^{8} + 18 q^{9} + 104 q^{10} - 22 q^{11} + 99 q^{12} - 30 q^{13} + 42 q^{15} + 201 q^{16} - 106 q^{17} + 9 q^{18} - 50 q^{19} + 328 q^{20} - 11 q^{22} + 134 q^{23} + 171 q^{24} + 42 q^{25} - 112 q^{26} + 54 q^{27} - 198 q^{29} + 312 q^{30} - 360 q^{31} + 857 q^{32} - 66 q^{33} + 626 q^{34} + 297 q^{36} - 328 q^{37} + 72 q^{38} - 90 q^{39} + 1272 q^{40} + 782 q^{41} + 386 q^{43} - 363 q^{44} + 126 q^{45} - 418 q^{46} - 266 q^{47} + 603 q^{48} + 1379 q^{50} - 318 q^{51} - 592 q^{52} - 522 q^{53} + 27 q^{54} - 154 q^{55} - 150 q^{57} - 390 q^{58} + 172 q^{59} + 984 q^{60} + 778 q^{61} - 568 q^{62} + 809 q^{64} - 404 q^{65} - 33 q^{66} - 776 q^{67} - 1070 q^{68} + 402 q^{69} + 630 q^{71} + 513 q^{72} - 1296 q^{73} + 2358 q^{74} + 126 q^{75} - 728 q^{76} - 336 q^{78} + 652 q^{79} + 3832 q^{80} + 162 q^{81} + 1070 q^{82} + 324 q^{83} + 616 q^{85} - 1068 q^{86} - 594 q^{87} - 627 q^{88} + 756 q^{89} + 936 q^{90} + 1726 q^{92} - 1080 q^{93} - 3722 q^{94} - 156 q^{95} + 2571 q^{96} + 452 q^{97} - 198 q^{99}+O(q^{100}) \) 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 \(-0.540665\pi\)
−0.127405 + 0.991851i \(0.540665\pi\)
\(6\) −13.2733 −0.903133
\(7\) 0 0
\(8\) −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 \(-0.517500\pi\)
−0.0549488 + 0.998489i \(0.517500\pi\)
\(14\) 0 0
\(15\) −8.54657 −0.147114
\(16\) −22.6107 −0.353293
\(17\) −121.942 −1.73972 −0.869861 0.493297i \(-0.835792\pi\)
−0.869861 + 0.493297i \(0.835792\pi\)
\(18\) −39.8199 −0.521424
\(19\) −34.8489 −0.420783 −0.210391 0.977617i \(-0.567474\pi\)
−0.210391 + 0.977617i \(0.567474\pi\)
\(20\) −32.9772 −0.368696
\(21\) 0 0
\(22\) 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
\(28\) 0 0
\(29\) −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 \(-0.633534\pi\)
−0.407312 + 0.913289i \(0.633534\pi\)
\(32\) 226.598 1.25179
\(33\) −33.0000 −0.174078
\(34\) 539.524 2.72140
\(35\) 0 0
\(36\) 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 \(-0.289810\pi\)
0.613378 + 0.789789i \(0.289810\pi\)
\(42\) 0 0
\(43\) 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 \(-0.383088\pi\)
0.359088 + 0.933304i \(0.383088\pi\)
\(48\) −67.8322 −0.203974
\(49\) 0 0
\(50\) 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
\(56\) 0 0
\(57\) −104.547 −0.242939
\(58\) 307.292 0.695679
\(59\) −406.443 −0.896854 −0.448427 0.893820i \(-0.648016\pi\)
−0.448427 + 0.893820i \(0.648016\pi\)
\(60\) −98.9315 −0.212867
\(61\) 556.431 1.16793 0.583964 0.811779i \(-0.301501\pi\)
0.583964 + 0.811779i \(0.301501\pi\)
\(62\) 622.095 1.27429
\(63\) 0 0
\(64\) −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
\(70\) 0 0
\(71\) 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 \(-0.716947\pi\)
−0.630005 + 0.776591i \(0.716947\pi\)
\(74\) 1858.57 2.91966
\(75\) −350.652 −0.539864
\(76\) −403.395 −0.608850
\(77\) 0 0
\(78\) 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 \(-0.289122\pi\)
0.615084 + 0.788462i \(0.289122\pi\)
\(84\) 0 0
\(85\) 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 \(-0.356309\pi\)
0.436242 + 0.899829i \(0.356309\pi\)
\(90\) 113.441 0.132864
\(91\) 0 0
\(92\) 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 \(-0.289911\pi\)
0.613128 + 0.789984i \(0.289911\pi\)
\(98\) 0 0
\(99\) −99.0000 −0.100504
\(100\) −1353.00 −1.35300
\(101\) 1221.27 1.20318 0.601589 0.798806i \(-0.294535\pi\)
0.601589 + 0.798806i \(0.294535\pi\)
\(102\) 1618.57 1.57120
\(103\) −516.745 −0.494334 −0.247167 0.968973i \(-0.579500\pi\)
−0.247167 + 0.968973i \(0.579500\pi\)
\(104\) 81.4903 0.0768345
\(105\) 0 0
\(106\) −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
\(112\) 0 0
\(113\) −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
\(119\) 0 0
\(120\) 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
\(126\) 0 0
\(127\) −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 \(-0.541067\pi\)
−0.128659 + 0.991689i \(0.541067\pi\)
\(132\) −381.994 −0.251881
\(133\) 0 0
\(134\) −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 \(-0.391886\pi\)
0.333156 + 0.942872i \(0.391886\pi\)
\(140\) 0 0
\(141\) 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
\(147\) 0 0
\(148\) −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
\(154\) 0 0
\(155\) 400.562 0.207574
\(156\) −178.882 −0.0918080
\(157\) 56.5343 0.0287384 0.0143692 0.999897i \(-0.495426\pi\)
0.0143692 + 0.999897i \(0.495426\pi\)
\(158\) 1695.08 0.853501
\(159\) 14.7575 0.00736066
\(160\) −645.547 −0.318968
\(161\) 0 0
\(162\) −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 \(-0.659107\pi\)
−0.479294 + 0.877654i \(0.659107\pi\)
\(168\) 0 0
\(169\) −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 \(-0.457628\pi\)
0.132723 + 0.991153i \(0.457628\pi\)
\(174\) 921.875 0.401650
\(175\) 0 0
\(176\) 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 \(-0.461385\pi\)
0.121015 + 0.992651i \(0.461385\pi\)
\(182\) 0 0
\(183\) 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
\(189\) 0 0
\(190\) −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
\(196\) 0 0
\(197\) −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 \(-0.501270\pi\)
−0.00398982 + 0.999992i \(0.501270\pi\)
\(200\) 1849.09 0.653752
\(201\) 254.236 0.0892159
\(202\) −5403.43 −1.88210
\(203\) 0 0
\(204\) −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
\(210\) 0 0
\(211\) −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
\(217\) 0 0
\(218\) −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 \(-0.696882\pi\)
−0.579832 + 0.814736i \(0.696882\pi\)
\(224\) 0 0
\(225\) −1051.96 −0.311691
\(226\) 2859.94 0.841771
\(227\) 872.721 0.255174 0.127587 0.991827i \(-0.459277\pi\)
0.127587 + 0.991827i \(0.459277\pi\)
\(228\) −1210.19 −0.351520
\(229\) −1841.72 −0.531459 −0.265730 0.964048i \(-0.585613\pi\)
−0.265730 + 0.964048i \(0.585613\pi\)
\(230\) 1465.21 0.420057
\(231\) 0 0
\(232\) 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
\(238\) 0 0
\(239\) 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 \(-0.678965\pi\)
−0.533078 + 0.846066i \(0.678965\pi\)
\(242\) −535.356 −0.142207
\(243\) 243.000 0.0641500
\(244\) 6441.00 1.68993
\(245\) 0 0
\(246\) −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 \(-0.258355\pi\)
0.688306 + 0.725421i \(0.258355\pi\)
\(252\) 0 0
\(253\) −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 \(-0.214801\pi\)
0.780822 + 0.624754i \(0.214801\pi\)
\(258\) −4261.19 −1.02826
\(259\) 0 0
\(260\) 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
\(266\) 0 0
\(267\) 2197.68 0.503729
\(268\) 980.974 0.223591
\(269\) −478.178 −0.108383 −0.0541914 0.998531i \(-0.517258\pi\)
−0.0541914 + 0.998531i \(0.517258\pi\)
\(270\) 340.323 0.0767090
\(271\) 122.323 0.0274192 0.0137096 0.999906i \(-0.495636\pi\)
0.0137096 + 0.999906i \(0.495636\pi\)
\(272\) 2757.20 0.614631
\(273\) 0 0
\(274\) −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
\(280\) 0 0
\(281\) 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 \(-0.534673\pi\)
−0.108715 + 0.994073i \(0.534673\pi\)
\(284\) 568.139 0.118707
\(285\) 297.838 0.0619032
\(286\) −250.699 −0.0518328
\(287\) 0 0
\(288\) 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 \(-0.290124\pi\)
0.612600 + 0.790393i \(0.290124\pi\)
\(294\) 0 0
\(295\) 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
\(301\) 0 0
\(302\) 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 \(-0.434865\pi\)
0.203203 + 0.979137i \(0.434865\pi\)
\(308\) 0 0
\(309\) −1550.24 −0.285404
\(310\) −1772.26 −0.324702
\(311\) 7484.83 1.36471 0.682357 0.731019i \(-0.260955\pi\)
0.682357 + 0.731019i \(0.260955\pi\)
\(312\) 244.471 0.0443604
\(313\) 6833.33 1.23400 0.617001 0.786962i \(-0.288347\pi\)
0.617001 + 0.786962i \(0.288347\pi\)
\(314\) −250.132 −0.0449546
\(315\) 0 0
\(316\) −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
\(322\) 0 0
\(323\) 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
\(329\) 0 0
\(330\) −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
\(336\) 0 0
\(337\) 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
\(343\) 0 0
\(344\) −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 \(-0.517601\pi\)
−0.0552667 + 0.998472i \(0.517601\pi\)
\(350\) 0 0
\(351\) −139.081 −0.0211498
\(352\) −2492.58 −0.377429
\(353\) −1207.12 −0.182007 −0.0910034 0.995851i \(-0.529007\pi\)
−0.0910034 + 0.995851i \(0.529007\pi\)
\(354\) 5394.83 0.809978
\(355\) −139.824 −0.0209045
\(356\) 8479.79 1.26244
\(357\) 0 0
\(358\) 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
\(364\) 0 0
\(365\) 2238.87 0.321063
\(366\) −7385.66 −1.05479
\(367\) 6730.45 0.957293 0.478647 0.878008i \(-0.341128\pi\)
0.478647 + 0.878008i \(0.341128\pi\)
\(368\) −2628.37 −0.372318
\(369\) 2898.52 0.408919
\(370\) −5294.81 −0.743956
\(371\) 0 0
\(372\) −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
\(378\) 0 0
\(379\) 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 \(-0.754640\pi\)
−0.717338 + 0.696725i \(0.754640\pi\)
\(384\) 5468.07 0.726670
\(385\) 0 0
\(386\) 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
\(392\) 0 0
\(393\) −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 \(-0.492758\pi\)
0.0227495 + 0.999741i \(0.492758\pi\)
\(398\) 99.1105 0.0124823
\(399\) 0 0
\(400\) 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
\(406\) 0 0
\(407\) 4620.77 0.562760
\(408\) 5787.32 0.702242
\(409\) 13488.8 1.63076 0.815379 0.578927i \(-0.196528\pi\)
0.815379 + 0.578927i \(0.196528\pi\)
\(410\) 4059.40 0.488975
\(411\) 2654.52 0.318584
\(412\) −5981.62 −0.715275
\(413\) 0 0
\(414\) −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 \(-0.365382\pi\)
0.410420 + 0.911896i \(0.365382\pi\)
\(420\) 0 0
\(421\) 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
\(427\) 0 0
\(428\) −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 \(-0.567454\pi\)
−0.210330 + 0.977630i \(0.567454\pi\)
\(434\) 0 0
\(435\) 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 \(-0.409917\pi\)
0.279242 + 0.960221i \(0.409917\pi\)
\(440\) −495.754 −0.0537139
\(441\) 0 0
\(442\) −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
\(448\) 0 0
\(449\) 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
\(455\) 0 0
\(456\) 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 \(-0.499504\pi\)
0.00155673 + 0.999999i \(0.499504\pi\)
\(462\) 0 0
\(463\) 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 \(-0.728432\pi\)
−0.657609 + 0.753360i \(0.728432\pi\)
\(468\) −536.647 −0.0530053
\(469\) 0 0
\(470\) 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
\(476\) 0 0
\(477\) 44.2724 0.00424968
\(478\) −21113.8 −2.02034
\(479\) −2496.68 −0.238155 −0.119077 0.992885i \(-0.537994\pi\)
−0.119077 + 0.992885i \(0.537994\pi\)
\(480\) −1936.64 −0.184156
\(481\) 2163.84 0.205120
\(482\) 17648.3 1.66776
\(483\) 0 0
\(484\) 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
\(490\) 0 0
\(491\) −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
\(497\) 0 0
\(498\) −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 \(-0.262124\pi\)
0.679667 + 0.733520i \(0.262124\pi\)
\(504\) 0 0
\(505\) −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 \(-0.397169\pi\)
0.317464 + 0.948270i \(0.397169\pi\)
\(510\) −4611.08 −0.400357
\(511\) 0 0
\(512\) −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
\(518\) 0 0
\(519\) 1812.04 0.153255
\(520\) −232.154 −0.0195782
\(521\) −16794.3 −1.41223 −0.706114 0.708098i \(-0.749553\pi\)
−0.706114 + 0.708098i \(0.749553\pi\)
\(522\) 2765.63 0.231893
\(523\) 21009.4 1.75655 0.878275 0.478157i \(-0.158695\pi\)
0.878275 + 0.478157i \(0.158695\pi\)
\(524\) −4466.01 −0.372326
\(525\) 0 0
\(526\) −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
\(532\) 0 0
\(533\) −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
\(539\) 0 0
\(540\) −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
\(546\) 0 0
\(547\) 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
\(553\) 0 0
\(554\) −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
\(560\) 0 0
\(561\) 4024.09 0.302847
\(562\) −30722.3 −2.30595
\(563\) −2090.88 −0.156518 −0.0782592 0.996933i \(-0.524936\pi\)
−0.0782592 + 0.996933i \(0.524936\pi\)
\(564\) 8036.03 0.599961
\(565\) 1841.49 0.137119
\(566\) 4579.89 0.340119
\(567\) 0 0
\(568\) −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
\(574\) 0 0
\(575\) −13587.1 −0.985428
\(576\) −7395.15 −0.534950
\(577\) 15729.1 1.13486 0.567429 0.823423i \(-0.307938\pi\)
0.567429 + 0.823423i \(0.307938\pi\)
\(578\) −44053.4 −3.17021
\(579\) −4472.73 −0.321037
\(580\) 2290.38 0.163970
\(581\) 0 0
\(582\) −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 \(-0.685057\pi\)
−0.549173 + 0.835709i \(0.685057\pi\)
\(588\) 0 0
\(589\) 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 \(-0.494560\pi\)
0.0170888 + 0.999854i \(0.494560\pi\)
\(594\) 1314.06 0.0907683
\(595\) 0 0
\(596\) 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 \(-0.638905\pi\)
−0.422664 + 0.906286i \(0.638905\pi\)
\(602\) 0 0
\(603\) 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 \(-0.454690\pi\)
0.141867 + 0.989886i \(0.454690\pi\)
\(608\) −7896.70 −0.526732
\(609\) 0 0
\(610\) 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
\(616\) 0 0
\(617\) 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 \(-0.422052\pi\)
0.242440 + 0.970167i \(0.422052\pi\)
\(620\) 4636.74 0.300348
\(621\) 3138.60 0.202814
\(622\) −33116.1 −2.13478
\(623\) 0 0
\(624\) 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
\(630\) 0 0
\(631\) −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
\(637\) 0 0
\(638\) −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 \(-0.353783\pi\)
0.443369 + 0.896339i \(0.353783\pi\)
\(644\) 0 0
\(645\) −2743.75 −0.167496
\(646\) −18801.8 −1.14512
\(647\) −15792.8 −0.959625 −0.479813 0.877371i \(-0.659295\pi\)
−0.479813 + 0.877371i \(0.659295\pi\)
\(648\) −1281.41 −0.0776828
\(649\) 4470.87 0.270412
\(650\) −2663.89 −0.160748
\(651\) 0 0
\(652\) −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
\(658\) 0 0
\(659\) 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 \(-0.530332\pi\)
−0.0951464 + 0.995463i \(0.530332\pi\)
\(662\) 43449.9 2.55095
\(663\) 1884.42 0.110384
\(664\) −14715.8 −0.860066
\(665\) 0 0
\(666\) 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
\(672\) 0 0
\(673\) −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 \(-0.910180\pi\)
−0.960451 + 0.278448i \(0.910180\pi\)
\(678\) 8579.82 0.485997
\(679\) 0 0
\(680\) −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
\(686\) 0 0
\(687\) −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 \(-0.605407\pi\)
−0.325127 + 0.945671i \(0.605407\pi\)
\(692\) 6991.79 0.384087
\(693\) 0 0
\(694\) 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
\(700\) 0 0
\(701\) 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
\(707\) 0 0
\(708\) −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
\(714\) 0 0
\(715\) −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 \(-0.589865\pi\)
−0.278582 + 0.960412i \(0.589865\pi\)
\(720\) 579.733 0.0300074
\(721\) 0 0
\(722\) 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 \(-0.640915\pi\)
−0.428379 + 0.903599i \(0.640915\pi\)
\(728\) 0 0
\(729\) 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 \(-0.570010\pi\)
−0.218173 + 0.975910i \(0.570010\pi\)
\(734\) −29778.4 −1.49747
\(735\) 0 0
\(736\) 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
\(742\) 0 0
\(743\) 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
\(749\) 0 0
\(750\) −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
\(756\) 0 0
\(757\) 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 \(-0.338719\pi\)
0.485277 + 0.874360i \(0.338719\pi\)
\(762\) 13184.4 0.626799
\(763\) 0 0
\(764\) −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 \(-0.406314\pi\)
0.290092 + 0.956999i \(0.406314\pi\)
\(770\) 0 0
\(771\) 19302.0 0.901615
\(772\) −17258.1 −0.804578
\(773\) −21023.6 −0.978225 −0.489113 0.872221i \(-0.662679\pi\)
−0.489113 + 0.872221i \(0.662679\pi\)
\(774\) −12783.6 −0.593664
\(775\) 16434.4 0.761732
\(776\) −18532.8 −0.857331
\(777\) 0 0
\(778\) 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
\(784\) 0 0
\(785\) −161.058 −0.00732281
\(786\) 5121.02 0.232393
\(787\) −30286.2 −1.37177 −0.685886 0.727709i \(-0.740585\pi\)
−0.685886 + 0.727709i \(0.740585\pi\)
\(788\) −2668.51 −0.120637
\(789\) 22767.0 1.02728
\(790\) −4829.03 −0.217480
\(791\) 0 0
\(792\) 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 \(-0.755221\pi\)
−0.718610 + 0.695413i \(0.755221\pi\)
\(798\) 0 0
\(799\) −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
\(805\) 0 0
\(806\) −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 \(-0.429728\pi\)
0.218978 + 0.975730i \(0.429728\pi\)
\(812\) 0 0
\(813\) 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
\(819\) 0 0
\(820\) −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
\(826\) 0 0
\(827\) 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 \(-0.543732\pi\)
−0.136955 + 0.990577i \(0.543732\pi\)
\(830\) 11724.9 0.490334
\(831\) 24598.2 1.02684
\(832\) 4232.60 0.176369
\(833\) 0 0
\(834\) −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 \(-0.517759\pi\)
−0.0557635 + 0.998444i \(0.517759\pi\)
\(840\) 0 0
\(841\) −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
\(847\) 0 0
\(848\) −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 \(-0.562753\pi\)
−0.195869 + 0.980630i \(0.562753\pi\)
\(854\) 0 0
\(855\) 893.515 0.0357398
\(856\) 2405.01 0.0960298
\(857\) 13649.8 0.544072 0.272036 0.962287i \(-0.412303\pi\)
0.272036 + 0.962287i \(0.412303\pi\)
\(858\) −752.098 −0.0299257
\(859\) −7796.42 −0.309674 −0.154837 0.987940i \(-0.549485\pi\)
−0.154837 + 0.987940i \(0.549485\pi\)
\(860\) −10586.8 −0.419776
\(861\) 0 0
\(862\) −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
\(868\) 0 0
\(869\) 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
\(875\) 0 0
\(876\) −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 \(-0.289071\pi\)
0.615210 + 0.788363i \(0.289071\pi\)
\(882\) 0 0
\(883\) 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 \(-0.297291\pi\)
0.594649 + 0.803985i \(0.297291\pi\)
\(888\) 19936.4 0.753401
\(889\) 0 0
\(890\) 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
\(896\) 0 0
\(897\) −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
\(903\) 0 0
\(904\) 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
\(910\) 0 0
\(911\) −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
\(917\) 0 0
\(918\) 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
\(924\) 0 0
\(925\) 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 \(-0.283004\pi\)
0.630125 + 0.776494i \(0.283004\pi\)
\(930\) −5316.78 −0.187467
\(931\) 0 0
\(932\) 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 \(-0.175077\pi\)
0.852514 + 0.522705i \(0.175077\pi\)
\(938\) 0 0
\(939\) 20500.0 0.712451
\(940\) −7631.17 −0.264789
\(941\) 23741.9 0.822490 0.411245 0.911525i \(-0.365094\pi\)
0.411245 + 0.911525i \(0.365094\pi\)
\(942\) −750.396 −0.0259546
\(943\) 37437.4 1.29282
\(944\) 9189.97 0.316852
\(945\) 0 0
\(946\) 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
\(952\) 0 0
\(953\) −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
\(959\) 0 0
\(960\) 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
\(966\) 0 0
\(967\) 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 \(-0.674033\pi\)
−0.519906 + 0.854223i \(0.674033\pi\)
\(972\) 2812.86 0.0928217
\(973\) 0 0
\(974\) 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
\(980\) 0 0
\(981\) 19532.9 0.635715
\(982\) 71784.4 2.33272
\(983\) 24610.9 0.798541 0.399270 0.916833i \(-0.369263\pi\)
0.399270 + 0.916833i \(0.369263\pi\)
\(984\) −15284.7 −0.495183
\(985\) 656.744 0.0212443
\(986\) −37471.8 −1.21029
\(987\) 0 0
\(988\) 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
\(994\) 0 0
\(995\) 63.8165 0.00203329
\(996\) 32303.2 1.02768
\(997\) 7342.61 0.233242 0.116621 0.993176i \(-0.462794\pi\)
0.116621 + 0.993176i \(0.462794\pi\)
\(998\) −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 1617.4.a.k.1.1 2
7.6 odd 2 33.4.a.c.1.1 2
21.20 even 2 99.4.a.f.1.2 2
28.27 even 2 528.4.a.p.1.2 2
35.13 even 4 825.4.c.h.199.3 4
35.27 even 4 825.4.c.h.199.2 4
35.34 odd 2 825.4.a.l.1.2 2
56.13 odd 2 2112.4.a.bn.1.1 2
56.27 even 2 2112.4.a.bg.1.1 2
77.76 even 2 363.4.a.i.1.2 2
84.83 odd 2 1584.4.a.bj.1.1 2
105.104 even 2 2475.4.a.p.1.1 2
231.230 odd 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 7.6 odd 2
99.4.a.f.1.2 2 21.20 even 2
363.4.a.i.1.2 2 77.76 even 2
528.4.a.p.1.2 2 28.27 even 2
825.4.a.l.1.2 2 35.34 odd 2
825.4.c.h.199.2 4 35.27 even 4
825.4.c.h.199.3 4 35.13 even 4
1089.4.a.u.1.1 2 231.230 odd 2
1584.4.a.bj.1.1 2 84.83 odd 2
1617.4.a.k.1.1 2 1.1 even 1 trivial
2112.4.a.bg.1.1 2 56.27 even 2
2112.4.a.bn.1.1 2 56.13 odd 2
2475.4.a.p.1.1 2 105.104 even 2