Properties

Label 165.4.a.h.1.1
Level $165$
Weight $4$
Character 165.1
Self dual yes
Analytic conductor $9.735$
Analytic rank $0$
Dimension $4$
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] = [165,4,Mod(1,165)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(165, 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("165.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 165.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: \(9.73531515095\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1540841.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 27x^{2} - 18x + 92 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.20196\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.20196 q^{2} +3.00000 q^{3} +9.65650 q^{4} +5.00000 q^{5} -12.6059 q^{6} +15.3793 q^{7} -6.96057 q^{8} +9.00000 q^{9} +O(q^{10})\) \(q-4.20196 q^{2} +3.00000 q^{3} +9.65650 q^{4} +5.00000 q^{5} -12.6059 q^{6} +15.3793 q^{7} -6.96057 q^{8} +9.00000 q^{9} -21.0098 q^{10} -11.0000 q^{11} +28.9695 q^{12} +24.4926 q^{13} -64.6233 q^{14} +15.0000 q^{15} -48.0040 q^{16} -54.9072 q^{17} -37.8177 q^{18} +119.454 q^{19} +48.2825 q^{20} +46.1379 q^{21} +46.2216 q^{22} -191.351 q^{23} -20.8817 q^{24} +25.0000 q^{25} -102.917 q^{26} +27.0000 q^{27} +148.510 q^{28} +225.140 q^{29} -63.0295 q^{30} +303.901 q^{31} +257.395 q^{32} -33.0000 q^{33} +230.718 q^{34} +76.8965 q^{35} +86.9085 q^{36} +109.382 q^{37} -501.939 q^{38} +73.4778 q^{39} -34.8029 q^{40} +348.133 q^{41} -193.870 q^{42} +92.7623 q^{43} -106.222 q^{44} +45.0000 q^{45} +804.051 q^{46} +306.728 q^{47} -144.012 q^{48} -106.477 q^{49} -105.049 q^{50} -164.722 q^{51} +236.513 q^{52} -216.854 q^{53} -113.453 q^{54} -55.0000 q^{55} -107.049 q^{56} +358.361 q^{57} -946.029 q^{58} -692.952 q^{59} +144.848 q^{60} -152.661 q^{61} -1276.98 q^{62} +138.414 q^{63} -697.535 q^{64} +122.463 q^{65} +138.665 q^{66} -62.9772 q^{67} -530.211 q^{68} -574.054 q^{69} -323.116 q^{70} +554.295 q^{71} -62.6452 q^{72} +122.911 q^{73} -459.619 q^{74} +75.0000 q^{75} +1153.50 q^{76} -169.172 q^{77} -308.751 q^{78} +476.528 q^{79} -240.020 q^{80} +81.0000 q^{81} -1462.84 q^{82} -913.360 q^{83} +445.531 q^{84} -274.536 q^{85} -389.784 q^{86} +675.419 q^{87} +76.5663 q^{88} +1603.65 q^{89} -189.088 q^{90} +376.679 q^{91} -1847.78 q^{92} +911.704 q^{93} -1288.86 q^{94} +597.268 q^{95} +772.186 q^{96} -498.916 q^{97} +447.412 q^{98} -99.0000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 4 q^{2} + 12 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} + 34 q^{7} + 48 q^{8} + 36 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 4 q^{2} + 12 q^{3} + 26 q^{4} + 20 q^{5} + 12 q^{6} + 34 q^{7} + 48 q^{8} + 36 q^{9} + 20 q^{10} - 44 q^{11} + 78 q^{12} + 2 q^{13} - 52 q^{14} + 60 q^{15} + 66 q^{16} + 74 q^{17} + 36 q^{18} + 136 q^{19} + 130 q^{20} + 102 q^{21} - 44 q^{22} - 64 q^{23} + 144 q^{24} + 100 q^{25} - 320 q^{26} + 108 q^{27} - 20 q^{28} + 52 q^{29} + 60 q^{30} + 492 q^{31} + 208 q^{32} - 132 q^{33} + 244 q^{34} + 170 q^{35} + 234 q^{36} - 4 q^{37} - 404 q^{38} + 6 q^{39} + 240 q^{40} + 268 q^{41} - 156 q^{42} + 546 q^{43} - 286 q^{44} + 180 q^{45} + 368 q^{46} - 276 q^{47} + 198 q^{48} - 496 q^{49} + 100 q^{50} + 222 q^{51} - 1084 q^{52} - 184 q^{53} + 108 q^{54} - 220 q^{55} - 852 q^{56} + 408 q^{57} - 444 q^{58} - 1032 q^{59} + 390 q^{60} + 116 q^{61} - 1240 q^{62} + 306 q^{63} - 918 q^{64} + 10 q^{65} - 132 q^{66} - 552 q^{67} - 720 q^{68} - 192 q^{69} - 260 q^{70} - 920 q^{71} + 432 q^{72} + 926 q^{73} - 2856 q^{74} + 300 q^{75} + 1572 q^{76} - 374 q^{77} - 960 q^{78} + 1152 q^{79} + 330 q^{80} + 324 q^{81} - 1924 q^{82} - 134 q^{83} - 60 q^{84} + 370 q^{85} + 236 q^{86} + 156 q^{87} - 528 q^{88} - 1064 q^{89} + 180 q^{90} + 2780 q^{91} - 4896 q^{92} + 1476 q^{93} - 1432 q^{94} + 680 q^{95} + 624 q^{96} - 1648 q^{97} - 188 q^{98} - 396 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.20196 −1.48562 −0.742809 0.669503i \(-0.766507\pi\)
−0.742809 + 0.669503i \(0.766507\pi\)
\(3\) 3.00000 0.577350
\(4\) 9.65650 1.20706
\(5\) 5.00000 0.447214
\(6\) −12.6059 −0.857722
\(7\) 15.3793 0.830404 0.415202 0.909729i \(-0.363711\pi\)
0.415202 + 0.909729i \(0.363711\pi\)
\(8\) −6.96057 −0.307617
\(9\) 9.00000 0.333333
\(10\) −21.0098 −0.664389
\(11\) −11.0000 −0.301511
\(12\) 28.9695 0.696898
\(13\) 24.4926 0.522540 0.261270 0.965266i \(-0.415859\pi\)
0.261270 + 0.965266i \(0.415859\pi\)
\(14\) −64.6233 −1.23366
\(15\) 15.0000 0.258199
\(16\) −48.0040 −0.750062
\(17\) −54.9072 −0.783350 −0.391675 0.920104i \(-0.628104\pi\)
−0.391675 + 0.920104i \(0.628104\pi\)
\(18\) −37.8177 −0.495206
\(19\) 119.454 1.44234 0.721172 0.692757i \(-0.243604\pi\)
0.721172 + 0.692757i \(0.243604\pi\)
\(20\) 48.2825 0.539815
\(21\) 46.1379 0.479434
\(22\) 46.2216 0.447931
\(23\) −191.351 −1.73476 −0.867380 0.497646i \(-0.834198\pi\)
−0.867380 + 0.497646i \(0.834198\pi\)
\(24\) −20.8817 −0.177603
\(25\) 25.0000 0.200000
\(26\) −102.917 −0.776296
\(27\) 27.0000 0.192450
\(28\) 148.510 1.00235
\(29\) 225.140 1.44163 0.720817 0.693125i \(-0.243767\pi\)
0.720817 + 0.693125i \(0.243767\pi\)
\(30\) −63.0295 −0.383585
\(31\) 303.901 1.76072 0.880359 0.474307i \(-0.157301\pi\)
0.880359 + 0.474307i \(0.157301\pi\)
\(32\) 257.395 1.42192
\(33\) −33.0000 −0.174078
\(34\) 230.718 1.16376
\(35\) 76.8965 0.371368
\(36\) 86.9085 0.402354
\(37\) 109.382 0.486008 0.243004 0.970025i \(-0.421867\pi\)
0.243004 + 0.970025i \(0.421867\pi\)
\(38\) −501.939 −2.14277
\(39\) 73.4778 0.301689
\(40\) −34.8029 −0.137570
\(41\) 348.133 1.32608 0.663040 0.748584i \(-0.269266\pi\)
0.663040 + 0.748584i \(0.269266\pi\)
\(42\) −193.870 −0.712256
\(43\) 92.7623 0.328979 0.164490 0.986379i \(-0.447402\pi\)
0.164490 + 0.986379i \(0.447402\pi\)
\(44\) −106.222 −0.363943
\(45\) 45.0000 0.149071
\(46\) 804.051 2.57719
\(47\) 306.728 0.951932 0.475966 0.879464i \(-0.342099\pi\)
0.475966 + 0.879464i \(0.342099\pi\)
\(48\) −144.012 −0.433048
\(49\) −106.477 −0.310429
\(50\) −105.049 −0.297124
\(51\) −164.722 −0.452267
\(52\) 236.513 0.630739
\(53\) −216.854 −0.562021 −0.281011 0.959705i \(-0.590670\pi\)
−0.281011 + 0.959705i \(0.590670\pi\)
\(54\) −113.453 −0.285907
\(55\) −55.0000 −0.134840
\(56\) −107.049 −0.255446
\(57\) 358.361 0.832737
\(58\) −946.029 −2.14172
\(59\) −692.952 −1.52906 −0.764532 0.644586i \(-0.777030\pi\)
−0.764532 + 0.644586i \(0.777030\pi\)
\(60\) 144.848 0.311662
\(61\) −152.661 −0.320431 −0.160215 0.987082i \(-0.551219\pi\)
−0.160215 + 0.987082i \(0.551219\pi\)
\(62\) −1276.98 −2.61576
\(63\) 138.414 0.276801
\(64\) −697.535 −1.36237
\(65\) 122.463 0.233687
\(66\) 138.665 0.258613
\(67\) −62.9772 −0.114834 −0.0574170 0.998350i \(-0.518286\pi\)
−0.0574170 + 0.998350i \(0.518286\pi\)
\(68\) −530.211 −0.945553
\(69\) −574.054 −1.00156
\(70\) −323.116 −0.551711
\(71\) 554.295 0.926518 0.463259 0.886223i \(-0.346680\pi\)
0.463259 + 0.886223i \(0.346680\pi\)
\(72\) −62.6452 −0.102539
\(73\) 122.911 0.197063 0.0985314 0.995134i \(-0.468586\pi\)
0.0985314 + 0.995134i \(0.468586\pi\)
\(74\) −459.619 −0.722022
\(75\) 75.0000 0.115470
\(76\) 1153.50 1.74100
\(77\) −169.172 −0.250376
\(78\) −308.751 −0.448195
\(79\) 476.528 0.678653 0.339326 0.940669i \(-0.389801\pi\)
0.339326 + 0.940669i \(0.389801\pi\)
\(80\) −240.020 −0.335438
\(81\) 81.0000 0.111111
\(82\) −1462.84 −1.97005
\(83\) −913.360 −1.20788 −0.603941 0.797029i \(-0.706404\pi\)
−0.603941 + 0.797029i \(0.706404\pi\)
\(84\) 445.531 0.578707
\(85\) −274.536 −0.350325
\(86\) −389.784 −0.488738
\(87\) 675.419 0.832328
\(88\) 76.5663 0.0927500
\(89\) 1603.65 1.90996 0.954980 0.296669i \(-0.0958759\pi\)
0.954980 + 0.296669i \(0.0958759\pi\)
\(90\) −189.088 −0.221463
\(91\) 376.679 0.433920
\(92\) −1847.78 −2.09397
\(93\) 911.704 1.01655
\(94\) −1288.86 −1.41421
\(95\) 597.268 0.645035
\(96\) 772.186 0.820947
\(97\) −498.916 −0.522239 −0.261120 0.965306i \(-0.584092\pi\)
−0.261120 + 0.965306i \(0.584092\pi\)
\(98\) 447.412 0.461178
\(99\) −99.0000 −0.100504
\(100\) 241.413 0.241413
\(101\) −570.956 −0.562497 −0.281248 0.959635i \(-0.590749\pi\)
−0.281248 + 0.959635i \(0.590749\pi\)
\(102\) 692.154 0.671897
\(103\) −1050.64 −1.00507 −0.502537 0.864556i \(-0.667600\pi\)
−0.502537 + 0.864556i \(0.667600\pi\)
\(104\) −170.483 −0.160742
\(105\) 230.690 0.214409
\(106\) 911.211 0.834949
\(107\) 1870.49 1.68997 0.844985 0.534791i \(-0.179609\pi\)
0.844985 + 0.534791i \(0.179609\pi\)
\(108\) 260.726 0.232299
\(109\) −657.285 −0.577583 −0.288791 0.957392i \(-0.593253\pi\)
−0.288791 + 0.957392i \(0.593253\pi\)
\(110\) 231.108 0.200321
\(111\) 328.146 0.280597
\(112\) −738.267 −0.622855
\(113\) −1144.38 −0.952691 −0.476345 0.879258i \(-0.658039\pi\)
−0.476345 + 0.879258i \(0.658039\pi\)
\(114\) −1505.82 −1.23713
\(115\) −956.756 −0.775808
\(116\) 2174.06 1.74014
\(117\) 220.433 0.174180
\(118\) 2911.76 2.27160
\(119\) −844.434 −0.650497
\(120\) −104.409 −0.0794263
\(121\) 121.000 0.0909091
\(122\) 641.477 0.476038
\(123\) 1044.40 0.765612
\(124\) 2934.62 2.12530
\(125\) 125.000 0.0894427
\(126\) −581.610 −0.411221
\(127\) −85.1754 −0.0595126 −0.0297563 0.999557i \(-0.509473\pi\)
−0.0297563 + 0.999557i \(0.509473\pi\)
\(128\) 871.854 0.602045
\(129\) 278.287 0.189936
\(130\) −514.585 −0.347170
\(131\) −541.509 −0.361159 −0.180580 0.983560i \(-0.557797\pi\)
−0.180580 + 0.983560i \(0.557797\pi\)
\(132\) −318.665 −0.210123
\(133\) 1837.11 1.19773
\(134\) 264.628 0.170600
\(135\) 135.000 0.0860663
\(136\) 382.185 0.240972
\(137\) −35.7727 −0.0223085 −0.0111543 0.999938i \(-0.503551\pi\)
−0.0111543 + 0.999938i \(0.503551\pi\)
\(138\) 2412.15 1.48794
\(139\) −1479.11 −0.902563 −0.451281 0.892382i \(-0.649033\pi\)
−0.451281 + 0.892382i \(0.649033\pi\)
\(140\) 742.552 0.448265
\(141\) 920.183 0.549598
\(142\) −2329.13 −1.37645
\(143\) −269.419 −0.157552
\(144\) −432.036 −0.250021
\(145\) 1125.70 0.644719
\(146\) −516.466 −0.292760
\(147\) −319.431 −0.179226
\(148\) 1056.25 0.586642
\(149\) 1779.29 0.978287 0.489143 0.872203i \(-0.337309\pi\)
0.489143 + 0.872203i \(0.337309\pi\)
\(150\) −315.147 −0.171544
\(151\) −182.059 −0.0981173 −0.0490587 0.998796i \(-0.515622\pi\)
−0.0490587 + 0.998796i \(0.515622\pi\)
\(152\) −831.465 −0.443689
\(153\) −494.165 −0.261117
\(154\) 710.856 0.371964
\(155\) 1519.51 0.787417
\(156\) 709.539 0.364157
\(157\) −1579.89 −0.803114 −0.401557 0.915834i \(-0.631531\pi\)
−0.401557 + 0.915834i \(0.631531\pi\)
\(158\) −2002.35 −1.00822
\(159\) −650.561 −0.324483
\(160\) 1286.98 0.635903
\(161\) −2942.85 −1.44055
\(162\) −340.359 −0.165069
\(163\) −1958.49 −0.941109 −0.470554 0.882371i \(-0.655946\pi\)
−0.470554 + 0.882371i \(0.655946\pi\)
\(164\) 3361.75 1.60066
\(165\) −165.000 −0.0778499
\(166\) 3837.90 1.79445
\(167\) −443.726 −0.205608 −0.102804 0.994702i \(-0.532781\pi\)
−0.102804 + 0.994702i \(0.532781\pi\)
\(168\) −321.146 −0.147482
\(169\) −1597.11 −0.726952
\(170\) 1153.59 0.520449
\(171\) 1075.08 0.480781
\(172\) 895.760 0.397099
\(173\) 809.958 0.355954 0.177977 0.984035i \(-0.443045\pi\)
0.177977 + 0.984035i \(0.443045\pi\)
\(174\) −2838.09 −1.23652
\(175\) 384.483 0.166081
\(176\) 528.043 0.226152
\(177\) −2078.86 −0.882805
\(178\) −6738.48 −2.83747
\(179\) 413.043 0.172471 0.0862355 0.996275i \(-0.472516\pi\)
0.0862355 + 0.996275i \(0.472516\pi\)
\(180\) 434.543 0.179938
\(181\) −3558.89 −1.46149 −0.730746 0.682649i \(-0.760828\pi\)
−0.730746 + 0.682649i \(0.760828\pi\)
\(182\) −1582.79 −0.644639
\(183\) −457.984 −0.185001
\(184\) 1331.91 0.533641
\(185\) 546.910 0.217349
\(186\) −3830.95 −1.51021
\(187\) 603.979 0.236189
\(188\) 2961.92 1.14904
\(189\) 415.241 0.159811
\(190\) −2509.70 −0.958277
\(191\) −2899.55 −1.09845 −0.549225 0.835675i \(-0.685077\pi\)
−0.549225 + 0.835675i \(0.685077\pi\)
\(192\) −2092.61 −0.786567
\(193\) −1391.37 −0.518929 −0.259464 0.965753i \(-0.583546\pi\)
−0.259464 + 0.965753i \(0.583546\pi\)
\(194\) 2096.43 0.775848
\(195\) 367.389 0.134919
\(196\) −1028.20 −0.374707
\(197\) 171.519 0.0620316 0.0310158 0.999519i \(-0.490126\pi\)
0.0310158 + 0.999519i \(0.490126\pi\)
\(198\) 415.994 0.149310
\(199\) 3602.86 1.28342 0.641709 0.766948i \(-0.278226\pi\)
0.641709 + 0.766948i \(0.278226\pi\)
\(200\) −174.014 −0.0615234
\(201\) −188.931 −0.0662995
\(202\) 2399.13 0.835656
\(203\) 3462.49 1.19714
\(204\) −1590.63 −0.545915
\(205\) 1740.67 0.593041
\(206\) 4414.75 1.49316
\(207\) −1722.16 −0.578253
\(208\) −1175.74 −0.391937
\(209\) −1313.99 −0.434883
\(210\) −969.349 −0.318531
\(211\) 3185.02 1.03918 0.519588 0.854417i \(-0.326086\pi\)
0.519588 + 0.854417i \(0.326086\pi\)
\(212\) −2094.05 −0.678395
\(213\) 1662.89 0.534925
\(214\) −7859.72 −2.51065
\(215\) 463.811 0.147124
\(216\) −187.936 −0.0592009
\(217\) 4673.79 1.46211
\(218\) 2761.89 0.858068
\(219\) 368.732 0.113774
\(220\) −531.108 −0.162760
\(221\) −1344.82 −0.409332
\(222\) −1378.86 −0.416860
\(223\) 5396.94 1.62065 0.810327 0.585978i \(-0.199289\pi\)
0.810327 + 0.585978i \(0.199289\pi\)
\(224\) 3958.56 1.18077
\(225\) 225.000 0.0666667
\(226\) 4808.64 1.41534
\(227\) −6799.75 −1.98817 −0.994086 0.108593i \(-0.965366\pi\)
−0.994086 + 0.108593i \(0.965366\pi\)
\(228\) 3460.51 1.00517
\(229\) −4596.19 −1.32631 −0.663154 0.748483i \(-0.730783\pi\)
−0.663154 + 0.748483i \(0.730783\pi\)
\(230\) 4020.26 1.15256
\(231\) −507.517 −0.144555
\(232\) −1567.10 −0.443471
\(233\) −5745.30 −1.61540 −0.807698 0.589597i \(-0.799287\pi\)
−0.807698 + 0.589597i \(0.799287\pi\)
\(234\) −926.253 −0.258765
\(235\) 1533.64 0.425717
\(236\) −6691.50 −1.84568
\(237\) 1429.58 0.391820
\(238\) 3548.28 0.966391
\(239\) −1.65110 −0.000446865 0 −0.000223432 1.00000i \(-0.500071\pi\)
−0.000223432 1.00000i \(0.500071\pi\)
\(240\) −720.059 −0.193665
\(241\) −6300.83 −1.68412 −0.842058 0.539387i \(-0.818656\pi\)
−0.842058 + 0.539387i \(0.818656\pi\)
\(242\) −508.438 −0.135056
\(243\) 243.000 0.0641500
\(244\) −1474.17 −0.386780
\(245\) −532.385 −0.138828
\(246\) −4388.53 −1.13741
\(247\) 2925.73 0.753682
\(248\) −2115.33 −0.541627
\(249\) −2740.08 −0.697371
\(250\) −525.246 −0.132878
\(251\) 6949.86 1.74769 0.873847 0.486201i \(-0.161618\pi\)
0.873847 + 0.486201i \(0.161618\pi\)
\(252\) 1336.59 0.334117
\(253\) 2104.86 0.523050
\(254\) 357.904 0.0884130
\(255\) −823.608 −0.202260
\(256\) 1916.78 0.467965
\(257\) 2795.78 0.678584 0.339292 0.940681i \(-0.389812\pi\)
0.339292 + 0.940681i \(0.389812\pi\)
\(258\) −1169.35 −0.282173
\(259\) 1682.22 0.403583
\(260\) 1182.56 0.282075
\(261\) 2026.26 0.480545
\(262\) 2275.40 0.536545
\(263\) −4185.49 −0.981325 −0.490662 0.871350i \(-0.663245\pi\)
−0.490662 + 0.871350i \(0.663245\pi\)
\(264\) 229.699 0.0535492
\(265\) −1084.27 −0.251344
\(266\) −7719.48 −1.77937
\(267\) 4810.95 1.10272
\(268\) −608.139 −0.138612
\(269\) −1924.32 −0.436164 −0.218082 0.975930i \(-0.569980\pi\)
−0.218082 + 0.975930i \(0.569980\pi\)
\(270\) −567.265 −0.127862
\(271\) 2472.36 0.554190 0.277095 0.960843i \(-0.410628\pi\)
0.277095 + 0.960843i \(0.410628\pi\)
\(272\) 2635.76 0.587561
\(273\) 1130.04 0.250524
\(274\) 150.316 0.0331420
\(275\) −275.000 −0.0603023
\(276\) −5543.35 −1.20895
\(277\) 1462.93 0.317324 0.158662 0.987333i \(-0.449282\pi\)
0.158662 + 0.987333i \(0.449282\pi\)
\(278\) 6215.16 1.34086
\(279\) 2735.11 0.586906
\(280\) −535.244 −0.114239
\(281\) −3956.44 −0.839934 −0.419967 0.907539i \(-0.637958\pi\)
−0.419967 + 0.907539i \(0.637958\pi\)
\(282\) −3866.57 −0.816493
\(283\) −4179.28 −0.877854 −0.438927 0.898523i \(-0.644641\pi\)
−0.438927 + 0.898523i \(0.644641\pi\)
\(284\) 5352.56 1.11837
\(285\) 1791.80 0.372411
\(286\) 1132.09 0.234062
\(287\) 5354.04 1.10118
\(288\) 2316.56 0.473974
\(289\) −1898.20 −0.386363
\(290\) −4730.15 −0.957806
\(291\) −1496.75 −0.301515
\(292\) 1186.89 0.237867
\(293\) 5677.09 1.13194 0.565971 0.824425i \(-0.308501\pi\)
0.565971 + 0.824425i \(0.308501\pi\)
\(294\) 1342.24 0.266261
\(295\) −3464.76 −0.683818
\(296\) −761.362 −0.149504
\(297\) −297.000 −0.0580259
\(298\) −7476.49 −1.45336
\(299\) −4686.69 −0.906482
\(300\) 724.238 0.139380
\(301\) 1426.62 0.273186
\(302\) 765.003 0.145765
\(303\) −1712.87 −0.324758
\(304\) −5734.24 −1.08185
\(305\) −763.306 −0.143301
\(306\) 2076.46 0.387920
\(307\) −1020.96 −0.189801 −0.0949007 0.995487i \(-0.530253\pi\)
−0.0949007 + 0.995487i \(0.530253\pi\)
\(308\) −1633.61 −0.302220
\(309\) −3151.92 −0.580280
\(310\) −6384.91 −1.16980
\(311\) −8338.99 −1.52045 −0.760227 0.649658i \(-0.774912\pi\)
−0.760227 + 0.649658i \(0.774912\pi\)
\(312\) −511.448 −0.0928045
\(313\) −5245.08 −0.947186 −0.473593 0.880744i \(-0.657043\pi\)
−0.473593 + 0.880744i \(0.657043\pi\)
\(314\) 6638.64 1.19312
\(315\) 692.069 0.123789
\(316\) 4601.59 0.819177
\(317\) 1076.71 0.190770 0.0953849 0.995440i \(-0.469592\pi\)
0.0953849 + 0.995440i \(0.469592\pi\)
\(318\) 2733.63 0.482058
\(319\) −2476.54 −0.434669
\(320\) −3487.68 −0.609272
\(321\) 5611.46 0.975704
\(322\) 12365.7 2.14011
\(323\) −6558.86 −1.12986
\(324\) 782.177 0.134118
\(325\) 612.315 0.104508
\(326\) 8229.50 1.39813
\(327\) −1971.86 −0.333468
\(328\) −2423.21 −0.407924
\(329\) 4717.26 0.790489
\(330\) 693.324 0.115655
\(331\) −1678.48 −0.278723 −0.139362 0.990242i \(-0.544505\pi\)
−0.139362 + 0.990242i \(0.544505\pi\)
\(332\) −8819.86 −1.45799
\(333\) 984.438 0.162003
\(334\) 1864.52 0.305455
\(335\) −314.886 −0.0513554
\(336\) −2214.80 −0.359605
\(337\) 6953.49 1.12398 0.561989 0.827145i \(-0.310036\pi\)
0.561989 + 0.827145i \(0.310036\pi\)
\(338\) 6711.01 1.07997
\(339\) −3433.14 −0.550036
\(340\) −2651.06 −0.422864
\(341\) −3342.91 −0.530877
\(342\) −4517.45 −0.714257
\(343\) −6912.64 −1.08819
\(344\) −645.679 −0.101200
\(345\) −2870.27 −0.447913
\(346\) −3403.41 −0.528811
\(347\) 5160.09 0.798294 0.399147 0.916887i \(-0.369306\pi\)
0.399147 + 0.916887i \(0.369306\pi\)
\(348\) 6522.19 1.00467
\(349\) −9631.12 −1.47720 −0.738599 0.674145i \(-0.764512\pi\)
−0.738599 + 0.674145i \(0.764512\pi\)
\(350\) −1615.58 −0.246733
\(351\) 661.300 0.100563
\(352\) −2831.35 −0.428726
\(353\) −7054.97 −1.06373 −0.531867 0.846828i \(-0.678509\pi\)
−0.531867 + 0.846828i \(0.678509\pi\)
\(354\) 8735.28 1.31151
\(355\) 2771.48 0.414351
\(356\) 15485.6 2.30544
\(357\) −2533.30 −0.375565
\(358\) −1735.59 −0.256226
\(359\) 13176.9 1.93719 0.968596 0.248640i \(-0.0799835\pi\)
0.968596 + 0.248640i \(0.0799835\pi\)
\(360\) −313.226 −0.0458568
\(361\) 7410.14 1.08035
\(362\) 14954.3 2.17122
\(363\) 363.000 0.0524864
\(364\) 3637.40 0.523769
\(365\) 614.553 0.0881292
\(366\) 1924.43 0.274841
\(367\) 652.403 0.0927934 0.0463967 0.998923i \(-0.485226\pi\)
0.0463967 + 0.998923i \(0.485226\pi\)
\(368\) 9185.62 1.30118
\(369\) 3133.20 0.442026
\(370\) −2298.10 −0.322898
\(371\) −3335.06 −0.466705
\(372\) 8803.87 1.22704
\(373\) −8214.36 −1.14028 −0.570139 0.821548i \(-0.693111\pi\)
−0.570139 + 0.821548i \(0.693111\pi\)
\(374\) −2537.90 −0.350887
\(375\) 375.000 0.0516398
\(376\) −2135.00 −0.292830
\(377\) 5514.26 0.753312
\(378\) −1744.83 −0.237419
\(379\) 12437.6 1.68569 0.842845 0.538157i \(-0.180879\pi\)
0.842845 + 0.538157i \(0.180879\pi\)
\(380\) 5767.52 0.778598
\(381\) −255.526 −0.0343596
\(382\) 12183.8 1.63188
\(383\) 3966.13 0.529137 0.264569 0.964367i \(-0.414770\pi\)
0.264569 + 0.964367i \(0.414770\pi\)
\(384\) 2615.56 0.347591
\(385\) −845.862 −0.111972
\(386\) 5846.50 0.770930
\(387\) 834.861 0.109660
\(388\) −4817.78 −0.630376
\(389\) −7656.45 −0.997937 −0.498969 0.866620i \(-0.666288\pi\)
−0.498969 + 0.866620i \(0.666288\pi\)
\(390\) −1543.76 −0.200439
\(391\) 10506.6 1.35892
\(392\) 741.141 0.0954930
\(393\) −1624.53 −0.208515
\(394\) −720.717 −0.0921554
\(395\) 2382.64 0.303503
\(396\) −955.994 −0.121314
\(397\) −5023.47 −0.635065 −0.317533 0.948247i \(-0.602854\pi\)
−0.317533 + 0.948247i \(0.602854\pi\)
\(398\) −15139.1 −1.90667
\(399\) 5511.34 0.691509
\(400\) −1200.10 −0.150012
\(401\) −10128.4 −1.26132 −0.630659 0.776060i \(-0.717216\pi\)
−0.630659 + 0.776060i \(0.717216\pi\)
\(402\) 793.883 0.0984958
\(403\) 7443.33 0.920046
\(404\) −5513.43 −0.678969
\(405\) 405.000 0.0496904
\(406\) −14549.3 −1.77849
\(407\) −1203.20 −0.146537
\(408\) 1146.56 0.139125
\(409\) 11872.6 1.43536 0.717680 0.696373i \(-0.245204\pi\)
0.717680 + 0.696373i \(0.245204\pi\)
\(410\) −7314.21 −0.881032
\(411\) −107.318 −0.0128798
\(412\) −10145.5 −1.21319
\(413\) −10657.1 −1.26974
\(414\) 7236.46 0.859064
\(415\) −4566.80 −0.540181
\(416\) 6304.28 0.743012
\(417\) −4437.32 −0.521095
\(418\) 5521.33 0.646070
\(419\) 12257.3 1.42914 0.714571 0.699563i \(-0.246622\pi\)
0.714571 + 0.699563i \(0.246622\pi\)
\(420\) 2227.65 0.258806
\(421\) 6571.84 0.760789 0.380394 0.924824i \(-0.375788\pi\)
0.380394 + 0.924824i \(0.375788\pi\)
\(422\) −13383.4 −1.54382
\(423\) 2760.55 0.317311
\(424\) 1509.43 0.172887
\(425\) −1372.68 −0.156670
\(426\) −6987.39 −0.794695
\(427\) −2347.82 −0.266087
\(428\) 18062.4 2.03990
\(429\) −808.256 −0.0909626
\(430\) −1948.92 −0.218570
\(431\) 6413.42 0.716760 0.358380 0.933576i \(-0.383329\pi\)
0.358380 + 0.933576i \(0.383329\pi\)
\(432\) −1296.11 −0.144349
\(433\) −10320.0 −1.14537 −0.572685 0.819775i \(-0.694098\pi\)
−0.572685 + 0.819775i \(0.694098\pi\)
\(434\) −19639.1 −2.17214
\(435\) 3377.10 0.372228
\(436\) −6347.08 −0.697179
\(437\) −22857.6 −2.50212
\(438\) −1549.40 −0.169025
\(439\) 5316.31 0.577981 0.288991 0.957332i \(-0.406680\pi\)
0.288991 + 0.957332i \(0.406680\pi\)
\(440\) 382.832 0.0414790
\(441\) −958.293 −0.103476
\(442\) 5650.88 0.608111
\(443\) −6180.96 −0.662904 −0.331452 0.943472i \(-0.607538\pi\)
−0.331452 + 0.943472i \(0.607538\pi\)
\(444\) 3168.74 0.338698
\(445\) 8018.25 0.854160
\(446\) −22677.8 −2.40767
\(447\) 5337.86 0.564814
\(448\) −10727.6 −1.13132
\(449\) −8468.27 −0.890072 −0.445036 0.895513i \(-0.646809\pi\)
−0.445036 + 0.895513i \(0.646809\pi\)
\(450\) −945.442 −0.0990413
\(451\) −3829.46 −0.399828
\(452\) −11050.7 −1.14996
\(453\) −546.176 −0.0566481
\(454\) 28572.3 2.95367
\(455\) 1883.40 0.194055
\(456\) −2494.40 −0.256164
\(457\) 11707.1 1.19833 0.599163 0.800627i \(-0.295500\pi\)
0.599163 + 0.800627i \(0.295500\pi\)
\(458\) 19313.0 1.97039
\(459\) −1482.49 −0.150756
\(460\) −9238.92 −0.936450
\(461\) −16533.1 −1.67034 −0.835169 0.549994i \(-0.814630\pi\)
−0.835169 + 0.549994i \(0.814630\pi\)
\(462\) 2132.57 0.214753
\(463\) −15065.7 −1.51223 −0.756117 0.654436i \(-0.772906\pi\)
−0.756117 + 0.654436i \(0.772906\pi\)
\(464\) −10807.6 −1.08132
\(465\) 4558.52 0.454616
\(466\) 24141.6 2.39986
\(467\) 17843.1 1.76806 0.884028 0.467433i \(-0.154821\pi\)
0.884028 + 0.467433i \(0.154821\pi\)
\(468\) 2128.62 0.210246
\(469\) −968.545 −0.0953587
\(470\) −6444.29 −0.632453
\(471\) −4739.67 −0.463678
\(472\) 4823.35 0.470366
\(473\) −1020.39 −0.0991910
\(474\) −6007.06 −0.582096
\(475\) 2986.34 0.288469
\(476\) −8154.28 −0.785191
\(477\) −1951.68 −0.187340
\(478\) 6.93786 0.000663871 0
\(479\) −6103.98 −0.582250 −0.291125 0.956685i \(-0.594030\pi\)
−0.291125 + 0.956685i \(0.594030\pi\)
\(480\) 3860.93 0.367139
\(481\) 2679.05 0.253959
\(482\) 26475.8 2.50195
\(483\) −8828.55 −0.831703
\(484\) 1168.44 0.109733
\(485\) −2494.58 −0.233552
\(486\) −1021.08 −0.0953025
\(487\) −487.182 −0.0453313 −0.0226656 0.999743i \(-0.507215\pi\)
−0.0226656 + 0.999743i \(0.507215\pi\)
\(488\) 1062.61 0.0985699
\(489\) −5875.47 −0.543350
\(490\) 2237.06 0.206245
\(491\) −12160.8 −1.11774 −0.558870 0.829256i \(-0.688765\pi\)
−0.558870 + 0.829256i \(0.688765\pi\)
\(492\) 10085.2 0.924142
\(493\) −12361.8 −1.12930
\(494\) −12293.8 −1.11968
\(495\) −495.000 −0.0449467
\(496\) −14588.5 −1.32065
\(497\) 8524.68 0.769384
\(498\) 11513.7 1.03603
\(499\) 12827.4 1.15076 0.575382 0.817885i \(-0.304853\pi\)
0.575382 + 0.817885i \(0.304853\pi\)
\(500\) 1207.06 0.107963
\(501\) −1331.18 −0.118708
\(502\) −29203.1 −2.59641
\(503\) 10573.7 0.937289 0.468644 0.883387i \(-0.344743\pi\)
0.468644 + 0.883387i \(0.344743\pi\)
\(504\) −963.439 −0.0851488
\(505\) −2854.78 −0.251556
\(506\) −8844.56 −0.777053
\(507\) −4791.34 −0.419706
\(508\) −822.497 −0.0718354
\(509\) −6591.24 −0.573971 −0.286986 0.957935i \(-0.592653\pi\)
−0.286986 + 0.957935i \(0.592653\pi\)
\(510\) 3460.77 0.300481
\(511\) 1890.28 0.163642
\(512\) −15029.1 −1.29726
\(513\) 3225.25 0.277579
\(514\) −11747.8 −1.00812
\(515\) −5253.20 −0.449483
\(516\) 2687.28 0.229265
\(517\) −3374.00 −0.287018
\(518\) −7068.63 −0.599571
\(519\) 2429.87 0.205510
\(520\) −852.413 −0.0718861
\(521\) −12331.0 −1.03691 −0.518454 0.855105i \(-0.673492\pi\)
−0.518454 + 0.855105i \(0.673492\pi\)
\(522\) −8514.26 −0.713907
\(523\) 17902.2 1.49677 0.748385 0.663265i \(-0.230830\pi\)
0.748385 + 0.663265i \(0.230830\pi\)
\(524\) −5229.08 −0.435942
\(525\) 1153.45 0.0958868
\(526\) 17587.3 1.45787
\(527\) −16686.4 −1.37926
\(528\) 1584.13 0.130569
\(529\) 24448.3 2.00939
\(530\) 4556.05 0.373401
\(531\) −6236.57 −0.509688
\(532\) 17740.1 1.44573
\(533\) 8526.68 0.692930
\(534\) −20215.4 −1.63822
\(535\) 9352.43 0.755777
\(536\) 438.357 0.0353249
\(537\) 1239.13 0.0995762
\(538\) 8085.93 0.647973
\(539\) 1171.25 0.0935977
\(540\) 1303.63 0.103887
\(541\) −7506.67 −0.596556 −0.298278 0.954479i \(-0.596412\pi\)
−0.298278 + 0.954479i \(0.596412\pi\)
\(542\) −10388.8 −0.823315
\(543\) −10676.7 −0.843793
\(544\) −14132.9 −1.11386
\(545\) −3286.43 −0.258303
\(546\) −4748.38 −0.372183
\(547\) −4579.71 −0.357978 −0.178989 0.983851i \(-0.557283\pi\)
−0.178989 + 0.983851i \(0.557283\pi\)
\(548\) −345.439 −0.0269278
\(549\) −1373.95 −0.106810
\(550\) 1155.54 0.0895862
\(551\) 26893.7 2.07933
\(552\) 3995.74 0.308098
\(553\) 7328.67 0.563556
\(554\) −6147.17 −0.471423
\(555\) 1640.73 0.125487
\(556\) −14283.0 −1.08945
\(557\) −1839.13 −0.139904 −0.0699519 0.997550i \(-0.522285\pi\)
−0.0699519 + 0.997550i \(0.522285\pi\)
\(558\) −11492.8 −0.871919
\(559\) 2271.99 0.171905
\(560\) −3691.34 −0.278549
\(561\) 1811.94 0.136364
\(562\) 16624.8 1.24782
\(563\) −1870.00 −0.139984 −0.0699921 0.997548i \(-0.522297\pi\)
−0.0699921 + 0.997548i \(0.522297\pi\)
\(564\) 8885.75 0.663400
\(565\) −5721.89 −0.426056
\(566\) 17561.2 1.30416
\(567\) 1245.72 0.0922672
\(568\) −3858.21 −0.285012
\(569\) −6673.53 −0.491685 −0.245843 0.969310i \(-0.579065\pi\)
−0.245843 + 0.969310i \(0.579065\pi\)
\(570\) −7529.09 −0.553261
\(571\) 15176.4 1.11228 0.556140 0.831088i \(-0.312282\pi\)
0.556140 + 0.831088i \(0.312282\pi\)
\(572\) −2601.64 −0.190175
\(573\) −8698.64 −0.634190
\(574\) −22497.5 −1.63594
\(575\) −4783.78 −0.346952
\(576\) −6277.82 −0.454124
\(577\) −13861.6 −1.00011 −0.500056 0.865993i \(-0.666687\pi\)
−0.500056 + 0.865993i \(0.666687\pi\)
\(578\) 7976.18 0.573988
\(579\) −4174.12 −0.299604
\(580\) 10870.3 0.778216
\(581\) −14046.8 −1.00303
\(582\) 6289.28 0.447936
\(583\) 2385.39 0.169456
\(584\) −855.528 −0.0606198
\(585\) 1102.17 0.0778957
\(586\) −23854.9 −1.68163
\(587\) 26717.1 1.87859 0.939294 0.343113i \(-0.111482\pi\)
0.939294 + 0.343113i \(0.111482\pi\)
\(588\) −3084.59 −0.216337
\(589\) 36302.1 2.53956
\(590\) 14558.8 1.01589
\(591\) 514.557 0.0358140
\(592\) −5250.77 −0.364536
\(593\) 13459.5 0.932066 0.466033 0.884767i \(-0.345683\pi\)
0.466033 + 0.884767i \(0.345683\pi\)
\(594\) 1247.98 0.0862043
\(595\) −4222.17 −0.290911
\(596\) 17181.7 1.18085
\(597\) 10808.6 0.740982
\(598\) 19693.3 1.34669
\(599\) −8052.34 −0.549265 −0.274632 0.961549i \(-0.588556\pi\)
−0.274632 + 0.961549i \(0.588556\pi\)
\(600\) −522.043 −0.0355205
\(601\) −11113.5 −0.754292 −0.377146 0.926154i \(-0.623094\pi\)
−0.377146 + 0.926154i \(0.623094\pi\)
\(602\) −5994.60 −0.405850
\(603\) −566.794 −0.0382780
\(604\) −1758.05 −0.118434
\(605\) 605.000 0.0406558
\(606\) 7197.40 0.482466
\(607\) −27871.8 −1.86372 −0.931862 0.362813i \(-0.881816\pi\)
−0.931862 + 0.362813i \(0.881816\pi\)
\(608\) 30746.8 2.05090
\(609\) 10387.5 0.691169
\(610\) 3207.39 0.212891
\(611\) 7512.55 0.497423
\(612\) −4771.90 −0.315184
\(613\) 1266.99 0.0834798 0.0417399 0.999129i \(-0.486710\pi\)
0.0417399 + 0.999129i \(0.486710\pi\)
\(614\) 4290.02 0.281973
\(615\) 5222.00 0.342392
\(616\) 1177.54 0.0770200
\(617\) 23103.8 1.50750 0.753748 0.657164i \(-0.228244\pi\)
0.753748 + 0.657164i \(0.228244\pi\)
\(618\) 13244.3 0.862075
\(619\) −23151.2 −1.50327 −0.751636 0.659578i \(-0.770735\pi\)
−0.751636 + 0.659578i \(0.770735\pi\)
\(620\) 14673.1 0.950462
\(621\) −5166.48 −0.333855
\(622\) 35040.2 2.25881
\(623\) 24663.0 1.58604
\(624\) −3527.22 −0.226285
\(625\) 625.000 0.0400000
\(626\) 22039.6 1.40716
\(627\) −3941.97 −0.251080
\(628\) −15256.2 −0.969410
\(629\) −6005.86 −0.380714
\(630\) −2908.05 −0.183904
\(631\) 17741.1 1.11927 0.559636 0.828738i \(-0.310941\pi\)
0.559636 + 0.828738i \(0.310941\pi\)
\(632\) −3316.91 −0.208765
\(633\) 9555.07 0.599968
\(634\) −4524.30 −0.283411
\(635\) −425.877 −0.0266148
\(636\) −6282.14 −0.391672
\(637\) −2607.90 −0.162211
\(638\) 10406.3 0.645753
\(639\) 4988.66 0.308839
\(640\) 4359.27 0.269243
\(641\) 11428.1 0.704187 0.352094 0.935965i \(-0.385470\pi\)
0.352094 + 0.935965i \(0.385470\pi\)
\(642\) −23579.1 −1.44952
\(643\) 27018.9 1.65711 0.828554 0.559909i \(-0.189164\pi\)
0.828554 + 0.559909i \(0.189164\pi\)
\(644\) −28417.6 −1.73884
\(645\) 1391.43 0.0849421
\(646\) 27560.1 1.67854
\(647\) 8637.82 0.524865 0.262433 0.964950i \(-0.415475\pi\)
0.262433 + 0.964950i \(0.415475\pi\)
\(648\) −563.807 −0.0341796
\(649\) 7622.48 0.461030
\(650\) −2572.93 −0.155259
\(651\) 14021.4 0.844149
\(652\) −18912.2 −1.13598
\(653\) −2186.14 −0.131011 −0.0655055 0.997852i \(-0.520866\pi\)
−0.0655055 + 0.997852i \(0.520866\pi\)
\(654\) 8285.67 0.495406
\(655\) −2707.54 −0.161515
\(656\) −16711.8 −0.994641
\(657\) 1106.19 0.0656876
\(658\) −19821.7 −1.17436
\(659\) 32361.9 1.91296 0.956479 0.291800i \(-0.0942542\pi\)
0.956479 + 0.291800i \(0.0942542\pi\)
\(660\) −1593.32 −0.0939697
\(661\) 6029.07 0.354771 0.177385 0.984141i \(-0.443236\pi\)
0.177385 + 0.984141i \(0.443236\pi\)
\(662\) 7052.90 0.414076
\(663\) −4034.46 −0.236328
\(664\) 6357.51 0.371565
\(665\) 9185.56 0.535640
\(666\) −4136.57 −0.240674
\(667\) −43080.8 −2.50089
\(668\) −4284.84 −0.248182
\(669\) 16190.8 0.935685
\(670\) 1323.14 0.0762945
\(671\) 1679.27 0.0966135
\(672\) 11875.7 0.681718
\(673\) 7367.28 0.421973 0.210986 0.977489i \(-0.432332\pi\)
0.210986 + 0.977489i \(0.432332\pi\)
\(674\) −29218.3 −1.66980
\(675\) 675.000 0.0384900
\(676\) −15422.5 −0.877477
\(677\) −7017.15 −0.398362 −0.199181 0.979963i \(-0.563828\pi\)
−0.199181 + 0.979963i \(0.563828\pi\)
\(678\) 14425.9 0.817144
\(679\) −7672.97 −0.433670
\(680\) 1910.93 0.107766
\(681\) −20399.2 −1.14787
\(682\) 14046.8 0.788680
\(683\) 16608.1 0.930442 0.465221 0.885195i \(-0.345975\pi\)
0.465221 + 0.885195i \(0.345975\pi\)
\(684\) 10381.5 0.580333
\(685\) −178.864 −0.00997668
\(686\) 29046.7 1.61663
\(687\) −13788.6 −0.765745
\(688\) −4452.96 −0.246755
\(689\) −5311.31 −0.293679
\(690\) 12060.8 0.665428
\(691\) 3349.66 0.184409 0.0922047 0.995740i \(-0.470609\pi\)
0.0922047 + 0.995740i \(0.470609\pi\)
\(692\) 7821.36 0.429658
\(693\) −1522.55 −0.0834588
\(694\) −21682.5 −1.18596
\(695\) −7395.54 −0.403638
\(696\) −4701.31 −0.256038
\(697\) −19115.0 −1.03878
\(698\) 40469.6 2.19455
\(699\) −17235.9 −0.932649
\(700\) 3712.76 0.200470
\(701\) 14755.7 0.795030 0.397515 0.917596i \(-0.369873\pi\)
0.397515 + 0.917596i \(0.369873\pi\)
\(702\) −2778.76 −0.149398
\(703\) 13066.1 0.700990
\(704\) 7672.89 0.410771
\(705\) 4600.91 0.245788
\(706\) 29644.7 1.58030
\(707\) −8780.90 −0.467100
\(708\) −20074.5 −1.06560
\(709\) 27937.6 1.47985 0.739927 0.672687i \(-0.234860\pi\)
0.739927 + 0.672687i \(0.234860\pi\)
\(710\) −11645.6 −0.615568
\(711\) 4288.75 0.226218
\(712\) −11162.3 −0.587536
\(713\) −58151.9 −3.05442
\(714\) 10644.8 0.557946
\(715\) −1347.09 −0.0704593
\(716\) 3988.56 0.208183
\(717\) −4.95329 −0.000257997 0
\(718\) −55369.0 −2.87793
\(719\) −23650.0 −1.22670 −0.613349 0.789812i \(-0.710178\pi\)
−0.613349 + 0.789812i \(0.710178\pi\)
\(720\) −2160.18 −0.111813
\(721\) −16158.1 −0.834618
\(722\) −31137.2 −1.60499
\(723\) −18902.5 −0.972325
\(724\) −34366.4 −1.76411
\(725\) 5628.49 0.288327
\(726\) −1525.31 −0.0779748
\(727\) −9880.21 −0.504039 −0.252020 0.967722i \(-0.581095\pi\)
−0.252020 + 0.967722i \(0.581095\pi\)
\(728\) −2621.90 −0.133481
\(729\) 729.000 0.0370370
\(730\) −2582.33 −0.130926
\(731\) −5093.32 −0.257706
\(732\) −4422.52 −0.223308
\(733\) −7315.37 −0.368621 −0.184311 0.982868i \(-0.559005\pi\)
−0.184311 + 0.982868i \(0.559005\pi\)
\(734\) −2741.38 −0.137856
\(735\) −1597.15 −0.0801523
\(736\) −49252.9 −2.46670
\(737\) 692.749 0.0346238
\(738\) −13165.6 −0.656683
\(739\) 18857.1 0.938661 0.469331 0.883022i \(-0.344495\pi\)
0.469331 + 0.883022i \(0.344495\pi\)
\(740\) 5281.24 0.262354
\(741\) 8777.18 0.435139
\(742\) 14013.8 0.693346
\(743\) 18305.4 0.903851 0.451925 0.892056i \(-0.350737\pi\)
0.451925 + 0.892056i \(0.350737\pi\)
\(744\) −6345.98 −0.312708
\(745\) 8896.43 0.437503
\(746\) 34516.5 1.69402
\(747\) −8220.24 −0.402627
\(748\) 5832.33 0.285095
\(749\) 28766.8 1.40336
\(750\) −1575.74 −0.0767170
\(751\) −8697.79 −0.422619 −0.211310 0.977419i \(-0.567773\pi\)
−0.211310 + 0.977419i \(0.567773\pi\)
\(752\) −14724.1 −0.714008
\(753\) 20849.6 1.00903
\(754\) −23170.7 −1.11913
\(755\) −910.293 −0.0438794
\(756\) 4009.78 0.192902
\(757\) −1545.04 −0.0741814 −0.0370907 0.999312i \(-0.511809\pi\)
−0.0370907 + 0.999312i \(0.511809\pi\)
\(758\) −52262.3 −2.50429
\(759\) 6314.59 0.301983
\(760\) −4157.33 −0.198424
\(761\) 10359.5 0.493472 0.246736 0.969083i \(-0.420642\pi\)
0.246736 + 0.969083i \(0.420642\pi\)
\(762\) 1073.71 0.0510452
\(763\) −10108.6 −0.479627
\(764\) −27999.5 −1.32590
\(765\) −2470.82 −0.116775
\(766\) −16665.5 −0.786096
\(767\) −16972.2 −0.798997
\(768\) 5750.35 0.270179
\(769\) −35625.3 −1.67059 −0.835293 0.549806i \(-0.814702\pi\)
−0.835293 + 0.549806i \(0.814702\pi\)
\(770\) 3554.28 0.166347
\(771\) 8387.35 0.391781
\(772\) −13435.8 −0.626380
\(773\) −290.303 −0.0135077 −0.00675387 0.999977i \(-0.502150\pi\)
−0.00675387 + 0.999977i \(0.502150\pi\)
\(774\) −3508.05 −0.162913
\(775\) 7597.53 0.352144
\(776\) 3472.74 0.160650
\(777\) 5046.66 0.233009
\(778\) 32172.1 1.48255
\(779\) 41585.7 1.91266
\(780\) 3547.69 0.162856
\(781\) −6097.25 −0.279356
\(782\) −44148.2 −2.01884
\(783\) 6078.77 0.277443
\(784\) 5111.32 0.232841
\(785\) −7899.45 −0.359164
\(786\) 6826.20 0.309774
\(787\) −6676.06 −0.302383 −0.151192 0.988504i \(-0.548311\pi\)
−0.151192 + 0.988504i \(0.548311\pi\)
\(788\) 1656.28 0.0748761
\(789\) −12556.5 −0.566568
\(790\) −10011.8 −0.450889
\(791\) −17599.7 −0.791119
\(792\) 689.097 0.0309167
\(793\) −3739.07 −0.167438
\(794\) 21108.4 0.943465
\(795\) −3252.80 −0.145113
\(796\) 34791.1 1.54917
\(797\) 22698.2 1.00880 0.504399 0.863471i \(-0.331714\pi\)
0.504399 + 0.863471i \(0.331714\pi\)
\(798\) −23158.4 −1.02732
\(799\) −16841.5 −0.745696
\(800\) 6434.89 0.284385
\(801\) 14432.8 0.636654
\(802\) 42559.2 1.87384
\(803\) −1352.02 −0.0594167
\(804\) −1824.42 −0.0800277
\(805\) −14714.2 −0.644235
\(806\) −31276.6 −1.36684
\(807\) −5772.97 −0.251819
\(808\) 3974.18 0.173034
\(809\) 40727.4 1.76996 0.884982 0.465625i \(-0.154171\pi\)
0.884982 + 0.465625i \(0.154171\pi\)
\(810\) −1701.80 −0.0738210
\(811\) −10202.3 −0.441740 −0.220870 0.975303i \(-0.570890\pi\)
−0.220870 + 0.975303i \(0.570890\pi\)
\(812\) 33435.6 1.44502
\(813\) 7417.09 0.319962
\(814\) 5055.81 0.217698
\(815\) −9792.45 −0.420877
\(816\) 7907.28 0.339228
\(817\) 11080.8 0.474501
\(818\) −49888.3 −2.13240
\(819\) 3390.11 0.144640
\(820\) 16808.7 0.715837
\(821\) 9771.65 0.415387 0.207694 0.978194i \(-0.433404\pi\)
0.207694 + 0.978194i \(0.433404\pi\)
\(822\) 450.947 0.0191345
\(823\) −43074.7 −1.82441 −0.912204 0.409736i \(-0.865621\pi\)
−0.912204 + 0.409736i \(0.865621\pi\)
\(824\) 7313.06 0.309178
\(825\) −825.000 −0.0348155
\(826\) 44780.9 1.88635
\(827\) 22853.5 0.960937 0.480468 0.877012i \(-0.340467\pi\)
0.480468 + 0.877012i \(0.340467\pi\)
\(828\) −16630.1 −0.697988
\(829\) −4593.74 −0.192457 −0.0962287 0.995359i \(-0.530678\pi\)
−0.0962287 + 0.995359i \(0.530678\pi\)
\(830\) 19189.5 0.802504
\(831\) 4388.78 0.183207
\(832\) −17084.4 −0.711895
\(833\) 5846.35 0.243174
\(834\) 18645.5 0.774148
\(835\) −2218.63 −0.0919507
\(836\) −12688.5 −0.524931
\(837\) 8205.33 0.338850
\(838\) −51504.9 −2.12316
\(839\) 21907.5 0.901469 0.450734 0.892658i \(-0.351162\pi\)
0.450734 + 0.892658i \(0.351162\pi\)
\(840\) −1605.73 −0.0659560
\(841\) 26298.9 1.07831
\(842\) −27614.7 −1.13024
\(843\) −11869.3 −0.484936
\(844\) 30756.2 1.25435
\(845\) −7985.56 −0.325103
\(846\) −11599.7 −0.471403
\(847\) 1860.90 0.0754913
\(848\) 10409.8 0.421551
\(849\) −12537.8 −0.506829
\(850\) 5767.95 0.232752
\(851\) −20930.4 −0.843107
\(852\) 16057.7 0.645689
\(853\) −8159.43 −0.327519 −0.163759 0.986500i \(-0.552362\pi\)
−0.163759 + 0.986500i \(0.552362\pi\)
\(854\) 9865.47 0.395304
\(855\) 5375.41 0.215012
\(856\) −13019.7 −0.519863
\(857\) 42705.7 1.70222 0.851108 0.524990i \(-0.175931\pi\)
0.851108 + 0.524990i \(0.175931\pi\)
\(858\) 3396.26 0.135136
\(859\) −27898.2 −1.10812 −0.554060 0.832477i \(-0.686922\pi\)
−0.554060 + 0.832477i \(0.686922\pi\)
\(860\) 4478.80 0.177588
\(861\) 16062.1 0.635768
\(862\) −26949.0 −1.06483
\(863\) −19559.0 −0.771492 −0.385746 0.922605i \(-0.626056\pi\)
−0.385746 + 0.922605i \(0.626056\pi\)
\(864\) 6949.68 0.273649
\(865\) 4049.79 0.159187
\(866\) 43364.1 1.70158
\(867\) −5694.61 −0.223067
\(868\) 45132.5 1.76486
\(869\) −5241.81 −0.204621
\(870\) −14190.4 −0.552990
\(871\) −1542.47 −0.0600054
\(872\) 4575.08 0.177674
\(873\) −4490.24 −0.174080
\(874\) 96046.7 3.71720
\(875\) 1922.41 0.0742736
\(876\) 3560.66 0.137333
\(877\) −21973.8 −0.846068 −0.423034 0.906114i \(-0.639035\pi\)
−0.423034 + 0.906114i \(0.639035\pi\)
\(878\) −22339.0 −0.858660
\(879\) 17031.3 0.653527
\(880\) 2640.22 0.101138
\(881\) 8425.63 0.322210 0.161105 0.986937i \(-0.448494\pi\)
0.161105 + 0.986937i \(0.448494\pi\)
\(882\) 4026.71 0.153726
\(883\) 8349.89 0.318229 0.159114 0.987260i \(-0.449136\pi\)
0.159114 + 0.987260i \(0.449136\pi\)
\(884\) −12986.3 −0.494089
\(885\) −10394.3 −0.394802
\(886\) 25972.2 0.984823
\(887\) −40472.6 −1.53206 −0.766030 0.642805i \(-0.777771\pi\)
−0.766030 + 0.642805i \(0.777771\pi\)
\(888\) −2284.08 −0.0863163
\(889\) −1309.94 −0.0494195
\(890\) −33692.4 −1.26896
\(891\) −891.000 −0.0335013
\(892\) 52115.6 1.95623
\(893\) 36639.7 1.37301
\(894\) −22429.5 −0.839099
\(895\) 2065.22 0.0771314
\(896\) 13408.5 0.499940
\(897\) −14060.1 −0.523358
\(898\) 35583.4 1.32231
\(899\) 68420.3 2.53831
\(900\) 2172.71 0.0804709
\(901\) 11906.8 0.440259
\(902\) 16091.3 0.593992
\(903\) 4279.86 0.157724
\(904\) 7965.53 0.293064
\(905\) −17794.4 −0.653599
\(906\) 2295.01 0.0841574
\(907\) −40036.9 −1.46572 −0.732858 0.680382i \(-0.761814\pi\)
−0.732858 + 0.680382i \(0.761814\pi\)
\(908\) −65661.8 −2.39985
\(909\) −5138.60 −0.187499
\(910\) −7913.96 −0.288291
\(911\) −40429.0 −1.47033 −0.735166 0.677887i \(-0.762896\pi\)
−0.735166 + 0.677887i \(0.762896\pi\)
\(912\) −17202.7 −0.624604
\(913\) 10047.0 0.364190
\(914\) −49192.8 −1.78026
\(915\) −2289.92 −0.0827349
\(916\) −44383.1 −1.60094
\(917\) −8328.03 −0.299908
\(918\) 6229.39 0.223966
\(919\) −42179.2 −1.51400 −0.756998 0.653417i \(-0.773335\pi\)
−0.756998 + 0.653417i \(0.773335\pi\)
\(920\) 6659.57 0.238652
\(921\) −3062.87 −0.109582
\(922\) 69471.7 2.48148
\(923\) 13576.1 0.484143
\(924\) −4900.84 −0.174487
\(925\) 2734.55 0.0972016
\(926\) 63305.7 2.24660
\(927\) −9455.76 −0.335025
\(928\) 57950.0 2.04989
\(929\) −36655.4 −1.29454 −0.647269 0.762262i \(-0.724089\pi\)
−0.647269 + 0.762262i \(0.724089\pi\)
\(930\) −19154.7 −0.675385
\(931\) −12719.1 −0.447744
\(932\) −55479.5 −1.94988
\(933\) −25017.0 −0.877834
\(934\) −74976.3 −2.62666
\(935\) 3019.89 0.105627
\(936\) −1534.34 −0.0535807
\(937\) 30247.9 1.05459 0.527297 0.849681i \(-0.323205\pi\)
0.527297 + 0.849681i \(0.323205\pi\)
\(938\) 4069.79 0.141667
\(939\) −15735.2 −0.546858
\(940\) 14809.6 0.513867
\(941\) −14708.3 −0.509541 −0.254770 0.967002i \(-0.582000\pi\)
−0.254770 + 0.967002i \(0.582000\pi\)
\(942\) 19915.9 0.688849
\(943\) −66615.7 −2.30043
\(944\) 33264.5 1.14689
\(945\) 2076.21 0.0714698
\(946\) 4287.62 0.147360
\(947\) 14714.8 0.504929 0.252464 0.967606i \(-0.418759\pi\)
0.252464 + 0.967606i \(0.418759\pi\)
\(948\) 13804.8 0.472952
\(949\) 3010.40 0.102973
\(950\) −12548.5 −0.428554
\(951\) 3230.13 0.110141
\(952\) 5877.75 0.200104
\(953\) −13658.7 −0.464270 −0.232135 0.972684i \(-0.574571\pi\)
−0.232135 + 0.972684i \(0.574571\pi\)
\(954\) 8200.90 0.278316
\(955\) −14497.7 −0.491242
\(956\) −15.9438 −0.000539394 0
\(957\) −7429.61 −0.250956
\(958\) 25648.7 0.865002
\(959\) −550.159 −0.0185251
\(960\) −10463.0 −0.351763
\(961\) 62565.0 2.10013
\(962\) −11257.3 −0.377286
\(963\) 16834.4 0.563323
\(964\) −60844.0 −2.03283
\(965\) −6956.87 −0.232072
\(966\) 37097.2 1.23559
\(967\) −17355.6 −0.577164 −0.288582 0.957455i \(-0.593184\pi\)
−0.288582 + 0.957455i \(0.593184\pi\)
\(968\) −842.229 −0.0279652
\(969\) −19676.6 −0.652324
\(970\) 10482.1 0.346970
\(971\) 9629.08 0.318241 0.159120 0.987259i \(-0.449134\pi\)
0.159120 + 0.987259i \(0.449134\pi\)
\(972\) 2346.53 0.0774331
\(973\) −22747.6 −0.749492
\(974\) 2047.12 0.0673450
\(975\) 1836.94 0.0603378
\(976\) 7328.34 0.240343
\(977\) −16039.2 −0.525221 −0.262610 0.964902i \(-0.584583\pi\)
−0.262610 + 0.964902i \(0.584583\pi\)
\(978\) 24688.5 0.807210
\(979\) −17640.1 −0.575875
\(980\) −5140.98 −0.167574
\(981\) −5915.57 −0.192528
\(982\) 51099.3 1.66053
\(983\) −56471.0 −1.83229 −0.916147 0.400843i \(-0.868717\pi\)
−0.916147 + 0.400843i \(0.868717\pi\)
\(984\) −7269.62 −0.235515
\(985\) 857.596 0.0277414
\(986\) 51943.8 1.67772
\(987\) 14151.8 0.456389
\(988\) 28252.3 0.909742
\(989\) −17750.2 −0.570701
\(990\) 2079.97 0.0667736
\(991\) 7278.03 0.233294 0.116647 0.993173i \(-0.462785\pi\)
0.116647 + 0.993173i \(0.462785\pi\)
\(992\) 78222.8 2.50361
\(993\) −5035.43 −0.160921
\(994\) −35820.4 −1.14301
\(995\) 18014.3 0.573962
\(996\) −26459.6 −0.841771
\(997\) −6208.44 −0.197215 −0.0986075 0.995126i \(-0.531439\pi\)
−0.0986075 + 0.995126i \(0.531439\pi\)
\(998\) −53900.1 −1.70960
\(999\) 2953.31 0.0935323
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 165.4.a.h.1.1 4
3.2 odd 2 495.4.a.m.1.4 4
5.2 odd 4 825.4.c.p.199.2 8
5.3 odd 4 825.4.c.p.199.7 8
5.4 even 2 825.4.a.t.1.4 4
11.10 odd 2 1815.4.a.t.1.4 4
15.14 odd 2 2475.4.a.be.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.4.a.h.1.1 4 1.1 even 1 trivial
495.4.a.m.1.4 4 3.2 odd 2
825.4.a.t.1.4 4 5.4 even 2
825.4.c.p.199.2 8 5.2 odd 4
825.4.c.p.199.7 8 5.3 odd 4
1815.4.a.t.1.4 4 11.10 odd 2
2475.4.a.be.1.1 4 15.14 odd 2