Properties

Label 165.6.a.b.1.1
Level $165$
Weight $6$
Character 165.1
Self dual yes
Analytic conductor $26.463$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [165,6,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, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("165.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 165 = 3 \cdot 5 \cdot 11 \)
Weight: \( k \) \(=\) \( 6 \)
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: \(26.4633302691\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.3368.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 15x + 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.03932\) of defining polynomial
Character \(\chi\) \(=\) 165.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-8.07863 q^{2} +9.00000 q^{3} +33.2643 q^{4} -25.0000 q^{5} -72.7077 q^{6} -39.3760 q^{7} -10.2141 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-8.07863 q^{2} +9.00000 q^{3} +33.2643 q^{4} -25.0000 q^{5} -72.7077 q^{6} -39.3760 q^{7} -10.2141 q^{8} +81.0000 q^{9} +201.966 q^{10} +121.000 q^{11} +299.379 q^{12} -220.765 q^{13} +318.105 q^{14} -225.000 q^{15} -981.943 q^{16} -200.343 q^{17} -654.369 q^{18} +350.345 q^{19} -831.608 q^{20} -354.384 q^{21} -977.515 q^{22} +1385.09 q^{23} -91.9273 q^{24} +625.000 q^{25} +1783.48 q^{26} +729.000 q^{27} -1309.82 q^{28} +5506.64 q^{29} +1817.69 q^{30} -2450.86 q^{31} +8259.61 q^{32} +1089.00 q^{33} +1618.50 q^{34} +984.401 q^{35} +2694.41 q^{36} -4060.46 q^{37} -2830.31 q^{38} -1986.88 q^{39} +255.354 q^{40} +527.283 q^{41} +2862.94 q^{42} -12078.9 q^{43} +4024.99 q^{44} -2025.00 q^{45} -11189.6 q^{46} +563.023 q^{47} -8837.48 q^{48} -15256.5 q^{49} -5049.15 q^{50} -1803.09 q^{51} -7343.59 q^{52} -37203.0 q^{53} -5889.32 q^{54} -3025.00 q^{55} +402.192 q^{56} +3153.11 q^{57} -44486.2 q^{58} +2157.64 q^{59} -7484.48 q^{60} -39938.0 q^{61} +19799.6 q^{62} -3189.46 q^{63} -35304.2 q^{64} +5519.11 q^{65} -8797.63 q^{66} -38473.2 q^{67} -6664.29 q^{68} +12465.8 q^{69} -7952.62 q^{70} -13725.3 q^{71} -827.345 q^{72} -39736.3 q^{73} +32803.0 q^{74} +5625.00 q^{75} +11654.0 q^{76} -4764.50 q^{77} +16051.3 q^{78} +35672.9 q^{79} +24548.6 q^{80} +6561.00 q^{81} -4259.73 q^{82} -79999.7 q^{83} -11788.4 q^{84} +5008.58 q^{85} +97580.6 q^{86} +49559.8 q^{87} -1235.91 q^{88} -37783.0 q^{89} +16359.2 q^{90} +8692.83 q^{91} +46074.1 q^{92} -22057.7 q^{93} -4548.46 q^{94} -8758.63 q^{95} +74336.5 q^{96} -7616.35 q^{97} +123252. q^{98} +9801.00 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{2} + 27 q^{3} + 28 q^{4} - 75 q^{5} - 18 q^{6} - 232 q^{7} + 24 q^{8} + 243 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{2} + 27 q^{3} + 28 q^{4} - 75 q^{5} - 18 q^{6} - 232 q^{7} + 24 q^{8} + 243 q^{9} + 50 q^{10} + 363 q^{11} + 252 q^{12} + 450 q^{13} - 1504 q^{14} - 675 q^{15} - 1360 q^{16} - 334 q^{17} - 162 q^{18} - 4036 q^{19} - 700 q^{20} - 2088 q^{21} - 242 q^{22} - 7060 q^{23} + 216 q^{24} + 1875 q^{25} + 2932 q^{26} + 2187 q^{27} - 8320 q^{28} + 4042 q^{29} + 450 q^{30} - 608 q^{31} - 3104 q^{32} + 3267 q^{33} - 3644 q^{34} + 5800 q^{35} + 2268 q^{36} + 2250 q^{37} - 12632 q^{38} + 4050 q^{39} - 600 q^{40} + 10654 q^{41} - 13536 q^{42} - 35528 q^{43} + 3388 q^{44} - 6075 q^{45} - 41800 q^{46} - 2100 q^{47} - 12240 q^{48} + 7667 q^{49} - 1250 q^{50} - 3006 q^{51} - 14520 q^{52} - 12826 q^{53} - 1458 q^{54} - 9075 q^{55} + 17088 q^{56} - 36324 q^{57} - 17196 q^{58} - 81876 q^{59} - 6300 q^{60} - 62298 q^{61} + 109184 q^{62} - 18792 q^{63} - 72256 q^{64} - 11250 q^{65} - 2178 q^{66} - 46148 q^{67} - 35832 q^{68} - 63540 q^{69} + 37600 q^{70} - 64724 q^{71} + 1944 q^{72} + 810 q^{73} - 44796 q^{74} + 16875 q^{75} + 44656 q^{76} - 28072 q^{77} + 26388 q^{78} + 43876 q^{79} + 34000 q^{80} + 19683 q^{81} + 56060 q^{82} - 101024 q^{83} - 74880 q^{84} + 8350 q^{85} + 24128 q^{86} + 36378 q^{87} + 2904 q^{88} + 60022 q^{89} + 4050 q^{90} - 28568 q^{91} + 38256 q^{92} - 5472 q^{93} + 74552 q^{94} + 100900 q^{95} - 27936 q^{96} - 319746 q^{97} + 431134 q^{98} + 29403 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −8.07863 −1.42811 −0.714057 0.700087i \(-0.753144\pi\)
−0.714057 + 0.700087i \(0.753144\pi\)
\(3\) 9.00000 0.577350
\(4\) 33.2643 1.03951
\(5\) −25.0000 −0.447214
\(6\) −72.7077 −0.824522
\(7\) −39.3760 −0.303730 −0.151865 0.988401i \(-0.548528\pi\)
−0.151865 + 0.988401i \(0.548528\pi\)
\(8\) −10.2141 −0.0564257
\(9\) 81.0000 0.333333
\(10\) 201.966 0.638672
\(11\) 121.000 0.301511
\(12\) 299.379 0.600162
\(13\) −220.765 −0.362302 −0.181151 0.983455i \(-0.557982\pi\)
−0.181151 + 0.983455i \(0.557982\pi\)
\(14\) 318.105 0.433761
\(15\) −225.000 −0.258199
\(16\) −981.943 −0.958928
\(17\) −200.343 −0.168133 −0.0840664 0.996460i \(-0.526791\pi\)
−0.0840664 + 0.996460i \(0.526791\pi\)
\(18\) −654.369 −0.476038
\(19\) 350.345 0.222645 0.111322 0.993784i \(-0.464491\pi\)
0.111322 + 0.993784i \(0.464491\pi\)
\(20\) −831.608 −0.464883
\(21\) −354.384 −0.175358
\(22\) −977.515 −0.430593
\(23\) 1385.09 0.545957 0.272978 0.962020i \(-0.411991\pi\)
0.272978 + 0.962020i \(0.411991\pi\)
\(24\) −91.9273 −0.0325774
\(25\) 625.000 0.200000
\(26\) 1783.48 0.517409
\(27\) 729.000 0.192450
\(28\) −1309.82 −0.315730
\(29\) 5506.64 1.21588 0.607942 0.793982i \(-0.291995\pi\)
0.607942 + 0.793982i \(0.291995\pi\)
\(30\) 1817.69 0.368738
\(31\) −2450.86 −0.458051 −0.229025 0.973420i \(-0.573554\pi\)
−0.229025 + 0.973420i \(0.573554\pi\)
\(32\) 8259.61 1.42588
\(33\) 1089.00 0.174078
\(34\) 1618.50 0.240113
\(35\) 984.401 0.135832
\(36\) 2694.41 0.346504
\(37\) −4060.46 −0.487609 −0.243804 0.969824i \(-0.578395\pi\)
−0.243804 + 0.969824i \(0.578395\pi\)
\(38\) −2830.31 −0.317962
\(39\) −1986.88 −0.209175
\(40\) 255.354 0.0252343
\(41\) 527.283 0.0489874 0.0244937 0.999700i \(-0.492203\pi\)
0.0244937 + 0.999700i \(0.492203\pi\)
\(42\) 2862.94 0.250432
\(43\) −12078.9 −0.996218 −0.498109 0.867114i \(-0.665972\pi\)
−0.498109 + 0.867114i \(0.665972\pi\)
\(44\) 4024.99 0.313424
\(45\) −2025.00 −0.149071
\(46\) −11189.6 −0.779689
\(47\) 563.023 0.0371776 0.0185888 0.999827i \(-0.494083\pi\)
0.0185888 + 0.999827i \(0.494083\pi\)
\(48\) −8837.48 −0.553638
\(49\) −15256.5 −0.907748
\(50\) −5049.15 −0.285623
\(51\) −1803.09 −0.0970715
\(52\) −7343.59 −0.376617
\(53\) −37203.0 −1.81923 −0.909617 0.415449i \(-0.863625\pi\)
−0.909617 + 0.415449i \(0.863625\pi\)
\(54\) −5889.32 −0.274841
\(55\) −3025.00 −0.134840
\(56\) 402.192 0.0171381
\(57\) 3153.11 0.128544
\(58\) −44486.2 −1.73642
\(59\) 2157.64 0.0806955 0.0403477 0.999186i \(-0.487153\pi\)
0.0403477 + 0.999186i \(0.487153\pi\)
\(60\) −7484.48 −0.268400
\(61\) −39938.0 −1.37424 −0.687119 0.726545i \(-0.741125\pi\)
−0.687119 + 0.726545i \(0.741125\pi\)
\(62\) 19799.6 0.654149
\(63\) −3189.46 −0.101243
\(64\) −35304.2 −1.07740
\(65\) 5519.11 0.162026
\(66\) −8797.63 −0.248603
\(67\) −38473.2 −1.04706 −0.523529 0.852008i \(-0.675385\pi\)
−0.523529 + 0.852008i \(0.675385\pi\)
\(68\) −6664.29 −0.174776
\(69\) 12465.8 0.315208
\(70\) −7952.62 −0.193984
\(71\) −13725.3 −0.323129 −0.161564 0.986862i \(-0.551654\pi\)
−0.161564 + 0.986862i \(0.551654\pi\)
\(72\) −827.345 −0.0188086
\(73\) −39736.3 −0.872731 −0.436366 0.899769i \(-0.643735\pi\)
−0.436366 + 0.899769i \(0.643735\pi\)
\(74\) 32803.0 0.696361
\(75\) 5625.00 0.115470
\(76\) 11654.0 0.231441
\(77\) −4764.50 −0.0915779
\(78\) 16051.3 0.298726
\(79\) 35672.9 0.643088 0.321544 0.946895i \(-0.395798\pi\)
0.321544 + 0.946895i \(0.395798\pi\)
\(80\) 24548.6 0.428846
\(81\) 6561.00 0.111111
\(82\) −4259.73 −0.0699596
\(83\) −79999.7 −1.27466 −0.637328 0.770593i \(-0.719960\pi\)
−0.637328 + 0.770593i \(0.719960\pi\)
\(84\) −11788.4 −0.182287
\(85\) 5008.58 0.0751913
\(86\) 97580.6 1.42271
\(87\) 49559.8 0.701991
\(88\) −1235.91 −0.0170130
\(89\) −37783.0 −0.505616 −0.252808 0.967516i \(-0.581354\pi\)
−0.252808 + 0.967516i \(0.581354\pi\)
\(90\) 16359.2 0.212891
\(91\) 8692.83 0.110042
\(92\) 46074.1 0.567528
\(93\) −22057.7 −0.264456
\(94\) −4548.46 −0.0530939
\(95\) −8758.63 −0.0995697
\(96\) 74336.5 0.823235
\(97\) −7616.35 −0.0821897 −0.0410948 0.999155i \(-0.513085\pi\)
−0.0410948 + 0.999155i \(0.513085\pi\)
\(98\) 123252. 1.29637
\(99\) 9801.00 0.100504
\(100\) 20790.2 0.207902
\(101\) 125963. 1.22868 0.614341 0.789041i \(-0.289422\pi\)
0.614341 + 0.789041i \(0.289422\pi\)
\(102\) 14566.5 0.138629
\(103\) −165220. −1.53451 −0.767257 0.641340i \(-0.778379\pi\)
−0.767257 + 0.641340i \(0.778379\pi\)
\(104\) 2254.92 0.0204431
\(105\) 8859.61 0.0784226
\(106\) 300550. 2.59807
\(107\) −13152.3 −0.111056 −0.0555279 0.998457i \(-0.517684\pi\)
−0.0555279 + 0.998457i \(0.517684\pi\)
\(108\) 24249.7 0.200054
\(109\) 37970.9 0.306115 0.153057 0.988217i \(-0.451088\pi\)
0.153057 + 0.988217i \(0.451088\pi\)
\(110\) 24437.9 0.192567
\(111\) −36544.2 −0.281521
\(112\) 38665.0 0.291255
\(113\) −85189.7 −0.627612 −0.313806 0.949487i \(-0.601604\pi\)
−0.313806 + 0.949487i \(0.601604\pi\)
\(114\) −25472.8 −0.183575
\(115\) −34627.3 −0.244159
\(116\) 183175. 1.26392
\(117\) −17881.9 −0.120767
\(118\) −17430.8 −0.115242
\(119\) 7888.73 0.0510669
\(120\) 2298.18 0.0145691
\(121\) 14641.0 0.0909091
\(122\) 322645. 1.96257
\(123\) 4745.55 0.0282829
\(124\) −81526.1 −0.476148
\(125\) −15625.0 −0.0894427
\(126\) 25766.5 0.144587
\(127\) −141237. −0.777033 −0.388517 0.921442i \(-0.627012\pi\)
−0.388517 + 0.921442i \(0.627012\pi\)
\(128\) 20902.2 0.112763
\(129\) −108710. −0.575167
\(130\) −44586.9 −0.231392
\(131\) 210469. 1.07155 0.535773 0.844362i \(-0.320020\pi\)
0.535773 + 0.844362i \(0.320020\pi\)
\(132\) 36224.9 0.180956
\(133\) −13795.2 −0.0676237
\(134\) 310811. 1.49532
\(135\) −18225.0 −0.0860663
\(136\) 2046.33 0.00948701
\(137\) 19565.1 0.0890594 0.0445297 0.999008i \(-0.485821\pi\)
0.0445297 + 0.999008i \(0.485821\pi\)
\(138\) −100707. −0.450154
\(139\) 132498. 0.581663 0.290831 0.956774i \(-0.406068\pi\)
0.290831 + 0.956774i \(0.406068\pi\)
\(140\) 32745.5 0.141199
\(141\) 5067.21 0.0214645
\(142\) 110882. 0.461465
\(143\) −26712.5 −0.109238
\(144\) −79537.3 −0.319643
\(145\) −137666. −0.543760
\(146\) 321015. 1.24636
\(147\) −137309. −0.524089
\(148\) −135069. −0.506874
\(149\) −153316. −0.565748 −0.282874 0.959157i \(-0.591288\pi\)
−0.282874 + 0.959157i \(0.591288\pi\)
\(150\) −45442.3 −0.164904
\(151\) 248151. 0.885672 0.442836 0.896603i \(-0.353972\pi\)
0.442836 + 0.896603i \(0.353972\pi\)
\(152\) −3578.47 −0.0125629
\(153\) −16227.8 −0.0560443
\(154\) 38490.7 0.130784
\(155\) 61271.4 0.204846
\(156\) −66092.3 −0.217440
\(157\) −180692. −0.585047 −0.292523 0.956258i \(-0.594495\pi\)
−0.292523 + 0.956258i \(0.594495\pi\)
\(158\) −288188. −0.918403
\(159\) −334827. −1.05033
\(160\) −206490. −0.637675
\(161\) −54539.4 −0.165823
\(162\) −53003.9 −0.158679
\(163\) −176045. −0.518985 −0.259493 0.965745i \(-0.583555\pi\)
−0.259493 + 0.965745i \(0.583555\pi\)
\(164\) 17539.7 0.0509229
\(165\) −27225.0 −0.0778499
\(166\) 646288. 1.82035
\(167\) 293818. 0.815244 0.407622 0.913151i \(-0.366358\pi\)
0.407622 + 0.913151i \(0.366358\pi\)
\(168\) 3619.73 0.00989471
\(169\) −322556. −0.868737
\(170\) −40462.5 −0.107382
\(171\) 28378.0 0.0742148
\(172\) −401795. −1.03558
\(173\) −20784.9 −0.0527999 −0.0263999 0.999651i \(-0.508404\pi\)
−0.0263999 + 0.999651i \(0.508404\pi\)
\(174\) −400375. −1.00252
\(175\) −24610.0 −0.0607459
\(176\) −118815. −0.289128
\(177\) 19418.8 0.0465896
\(178\) 305235. 0.722078
\(179\) −229326. −0.534958 −0.267479 0.963564i \(-0.586191\pi\)
−0.267479 + 0.963564i \(0.586191\pi\)
\(180\) −67360.3 −0.154961
\(181\) −90807.3 −0.206027 −0.103014 0.994680i \(-0.532849\pi\)
−0.103014 + 0.994680i \(0.532849\pi\)
\(182\) −70226.2 −0.157152
\(183\) −359442. −0.793416
\(184\) −14147.5 −0.0308060
\(185\) 101512. 0.218065
\(186\) 178196. 0.377673
\(187\) −24241.5 −0.0506939
\(188\) 18728.6 0.0386465
\(189\) −28705.1 −0.0584528
\(190\) 70757.8 0.142197
\(191\) −496794. −0.985356 −0.492678 0.870212i \(-0.663982\pi\)
−0.492678 + 0.870212i \(0.663982\pi\)
\(192\) −317738. −0.622036
\(193\) 271362. 0.524391 0.262196 0.965015i \(-0.415553\pi\)
0.262196 + 0.965015i \(0.415553\pi\)
\(194\) 61529.7 0.117376
\(195\) 49672.0 0.0935460
\(196\) −507498. −0.943614
\(197\) −190972. −0.350594 −0.175297 0.984516i \(-0.556089\pi\)
−0.175297 + 0.984516i \(0.556089\pi\)
\(198\) −79178.7 −0.143531
\(199\) 602637. 1.07876 0.539378 0.842064i \(-0.318660\pi\)
0.539378 + 0.842064i \(0.318660\pi\)
\(200\) −6383.84 −0.0112851
\(201\) −346258. −0.604519
\(202\) −1.01761e6 −1.75470
\(203\) −216830. −0.369300
\(204\) −59978.6 −0.100907
\(205\) −13182.1 −0.0219078
\(206\) 1.33476e6 2.19146
\(207\) 112192. 0.181986
\(208\) 216778. 0.347422
\(209\) 42391.8 0.0671299
\(210\) −71573.6 −0.111996
\(211\) 918176. 1.41978 0.709888 0.704315i \(-0.248746\pi\)
0.709888 + 0.704315i \(0.248746\pi\)
\(212\) −1.23753e6 −1.89111
\(213\) −123528. −0.186559
\(214\) 106252. 0.158600
\(215\) 301971. 0.445522
\(216\) −7446.11 −0.0108591
\(217\) 96505.0 0.139123
\(218\) −306753. −0.437167
\(219\) −357627. −0.503872
\(220\) −100625. −0.140168
\(221\) 44228.7 0.0609149
\(222\) 295227. 0.402044
\(223\) 959605. 1.29220 0.646101 0.763252i \(-0.276399\pi\)
0.646101 + 0.763252i \(0.276399\pi\)
\(224\) −325231. −0.433083
\(225\) 50625.0 0.0666667
\(226\) 688217. 0.896301
\(227\) −289954. −0.373477 −0.186739 0.982410i \(-0.559792\pi\)
−0.186739 + 0.982410i \(0.559792\pi\)
\(228\) 104886. 0.133623
\(229\) 111161. 0.140076 0.0700379 0.997544i \(-0.477688\pi\)
0.0700379 + 0.997544i \(0.477688\pi\)
\(230\) 279741. 0.348688
\(231\) −42880.5 −0.0528725
\(232\) −56245.6 −0.0686071
\(233\) −453216. −0.546909 −0.273455 0.961885i \(-0.588166\pi\)
−0.273455 + 0.961885i \(0.588166\pi\)
\(234\) 144462. 0.172470
\(235\) −14075.6 −0.0166263
\(236\) 71772.5 0.0838838
\(237\) 321056. 0.371287
\(238\) −63730.1 −0.0729294
\(239\) 1.26387e6 1.43123 0.715614 0.698496i \(-0.246147\pi\)
0.715614 + 0.698496i \(0.246147\pi\)
\(240\) 220937. 0.247594
\(241\) 134350. 0.149003 0.0745013 0.997221i \(-0.476264\pi\)
0.0745013 + 0.997221i \(0.476264\pi\)
\(242\) −118279. −0.129829
\(243\) 59049.0 0.0641500
\(244\) −1.32851e6 −1.42853
\(245\) 381413. 0.405957
\(246\) −38337.6 −0.0403912
\(247\) −77343.8 −0.0806646
\(248\) 25033.4 0.0258458
\(249\) −719997. −0.735923
\(250\) 126229. 0.127734
\(251\) −1.43370e6 −1.43639 −0.718197 0.695840i \(-0.755032\pi\)
−0.718197 + 0.695840i \(0.755032\pi\)
\(252\) −106095. −0.105243
\(253\) 167596. 0.164612
\(254\) 1.14100e6 1.10969
\(255\) 45077.2 0.0434117
\(256\) 960873. 0.916360
\(257\) −953935. −0.900920 −0.450460 0.892797i \(-0.648740\pi\)
−0.450460 + 0.892797i \(0.648740\pi\)
\(258\) 878226. 0.821404
\(259\) 159885. 0.148101
\(260\) 183590. 0.168428
\(261\) 446038. 0.405294
\(262\) −1.70030e6 −1.53029
\(263\) 322280. 0.287306 0.143653 0.989628i \(-0.454115\pi\)
0.143653 + 0.989628i \(0.454115\pi\)
\(264\) −11123.2 −0.00982245
\(265\) 930075. 0.813586
\(266\) 111446. 0.0965744
\(267\) −340047. −0.291918
\(268\) −1.27978e6 −1.08843
\(269\) 808737. 0.681438 0.340719 0.940165i \(-0.389329\pi\)
0.340719 + 0.940165i \(0.389329\pi\)
\(270\) 147233. 0.122913
\(271\) −909303. −0.752117 −0.376058 0.926596i \(-0.622721\pi\)
−0.376058 + 0.926596i \(0.622721\pi\)
\(272\) 196726. 0.161227
\(273\) 78235.5 0.0635327
\(274\) −158059. −0.127187
\(275\) 75625.0 0.0603023
\(276\) 414667. 0.327662
\(277\) 236933. 0.185535 0.0927675 0.995688i \(-0.470429\pi\)
0.0927675 + 0.995688i \(0.470429\pi\)
\(278\) −1.07040e6 −0.830681
\(279\) −198519. −0.152684
\(280\) −10054.8 −0.00766441
\(281\) −450779. −0.340563 −0.170282 0.985395i \(-0.554468\pi\)
−0.170282 + 0.985395i \(0.554468\pi\)
\(282\) −40936.1 −0.0306538
\(283\) 683413. 0.507245 0.253622 0.967303i \(-0.418378\pi\)
0.253622 + 0.967303i \(0.418378\pi\)
\(284\) −456563. −0.335896
\(285\) −78827.6 −0.0574866
\(286\) 215801. 0.156005
\(287\) −20762.3 −0.0148789
\(288\) 669028. 0.475295
\(289\) −1.37972e6 −0.971731
\(290\) 1.11215e6 0.776551
\(291\) −68547.1 −0.0474522
\(292\) −1.32180e6 −0.907213
\(293\) −1.81068e6 −1.23218 −0.616088 0.787678i \(-0.711283\pi\)
−0.616088 + 0.787678i \(0.711283\pi\)
\(294\) 1.10927e6 0.748459
\(295\) −53941.0 −0.0360881
\(296\) 41474.1 0.0275136
\(297\) 88209.0 0.0580259
\(298\) 1.23859e6 0.807953
\(299\) −305779. −0.197801
\(300\) 187112. 0.120032
\(301\) 475617. 0.302581
\(302\) −2.00472e6 −1.26484
\(303\) 1.13367e6 0.709380
\(304\) −344019. −0.213500
\(305\) 998450. 0.614578
\(306\) 131099. 0.0800376
\(307\) −3.00860e6 −1.82187 −0.910937 0.412546i \(-0.864639\pi\)
−0.910937 + 0.412546i \(0.864639\pi\)
\(308\) −158488. −0.0951962
\(309\) −1.48698e6 −0.885952
\(310\) −494989. −0.292544
\(311\) −2.04083e6 −1.19648 −0.598241 0.801316i \(-0.704134\pi\)
−0.598241 + 0.801316i \(0.704134\pi\)
\(312\) 20294.3 0.0118029
\(313\) −1.73940e6 −1.00355 −0.501774 0.864999i \(-0.667319\pi\)
−0.501774 + 0.864999i \(0.667319\pi\)
\(314\) 1.45975e6 0.835513
\(315\) 79736.5 0.0452773
\(316\) 1.18663e6 0.668497
\(317\) 106704. 0.0596392 0.0298196 0.999555i \(-0.490507\pi\)
0.0298196 + 0.999555i \(0.490507\pi\)
\(318\) 2.70495e6 1.50000
\(319\) 666304. 0.366603
\(320\) 882605. 0.481827
\(321\) −118370. −0.0641181
\(322\) 440604. 0.236815
\(323\) −70189.3 −0.0374338
\(324\) 218247. 0.115501
\(325\) −137978. −0.0724604
\(326\) 1.42220e6 0.741170
\(327\) 341738. 0.176736
\(328\) −5385.74 −0.00276415
\(329\) −22169.6 −0.0112919
\(330\) 219941. 0.111179
\(331\) 1.54759e6 0.776402 0.388201 0.921575i \(-0.373097\pi\)
0.388201 + 0.921575i \(0.373097\pi\)
\(332\) −2.66114e6 −1.32502
\(333\) −328898. −0.162536
\(334\) −2.37365e6 −1.16426
\(335\) 961829. 0.468259
\(336\) 347985. 0.168156
\(337\) 1.22550e6 0.587814 0.293907 0.955834i \(-0.405044\pi\)
0.293907 + 0.955834i \(0.405044\pi\)
\(338\) 2.60581e6 1.24066
\(339\) −766708. −0.362352
\(340\) 166607. 0.0781621
\(341\) −296553. −0.138107
\(342\) −229255. −0.105987
\(343\) 1.26253e6 0.579440
\(344\) 123375. 0.0562123
\(345\) −311645. −0.140965
\(346\) 167914. 0.0754042
\(347\) 3.82684e6 1.70615 0.853074 0.521789i \(-0.174735\pi\)
0.853074 + 0.521789i \(0.174735\pi\)
\(348\) 1.64857e6 0.729727
\(349\) 456968. 0.200827 0.100413 0.994946i \(-0.467983\pi\)
0.100413 + 0.994946i \(0.467983\pi\)
\(350\) 198815. 0.0867521
\(351\) −160937. −0.0697251
\(352\) 999413. 0.429920
\(353\) 4.32830e6 1.84876 0.924380 0.381474i \(-0.124583\pi\)
0.924380 + 0.381474i \(0.124583\pi\)
\(354\) −156877. −0.0665352
\(355\) 343132. 0.144508
\(356\) −1.25683e6 −0.525594
\(357\) 70998.5 0.0294835
\(358\) 1.85264e6 0.763982
\(359\) −845380. −0.346191 −0.173095 0.984905i \(-0.555377\pi\)
−0.173095 + 0.984905i \(0.555377\pi\)
\(360\) 20683.6 0.00841145
\(361\) −2.35336e6 −0.950429
\(362\) 733599. 0.294230
\(363\) 131769. 0.0524864
\(364\) 289161. 0.114390
\(365\) 993408. 0.390297
\(366\) 2.90380e6 1.13309
\(367\) 2.96016e6 1.14723 0.573615 0.819125i \(-0.305540\pi\)
0.573615 + 0.819125i \(0.305540\pi\)
\(368\) −1.36008e6 −0.523534
\(369\) 42709.9 0.0163291
\(370\) −820075. −0.311422
\(371\) 1.46491e6 0.552555
\(372\) −733735. −0.274904
\(373\) 1.02299e6 0.380713 0.190357 0.981715i \(-0.439036\pi\)
0.190357 + 0.981715i \(0.439036\pi\)
\(374\) 195839. 0.0723968
\(375\) −140625. −0.0516398
\(376\) −5750.80 −0.00209777
\(377\) −1.21567e6 −0.440517
\(378\) 231898. 0.0834772
\(379\) −4.19402e6 −1.49980 −0.749899 0.661552i \(-0.769898\pi\)
−0.749899 + 0.661552i \(0.769898\pi\)
\(380\) −291350. −0.103504
\(381\) −1.27113e6 −0.448620
\(382\) 4.01342e6 1.40720
\(383\) −1.64287e6 −0.572276 −0.286138 0.958188i \(-0.592372\pi\)
−0.286138 + 0.958188i \(0.592372\pi\)
\(384\) 188120. 0.0651039
\(385\) 119113. 0.0409549
\(386\) −2.19223e6 −0.748891
\(387\) −978387. −0.332073
\(388\) −253353. −0.0854371
\(389\) 769820. 0.257938 0.128969 0.991649i \(-0.458833\pi\)
0.128969 + 0.991649i \(0.458833\pi\)
\(390\) −401282. −0.133594
\(391\) −277493. −0.0917933
\(392\) 155832. 0.0512203
\(393\) 1.89422e6 0.618657
\(394\) 1.54279e6 0.500688
\(395\) −891822. −0.287598
\(396\) 326024. 0.104475
\(397\) 4.86631e6 1.54961 0.774806 0.632199i \(-0.217847\pi\)
0.774806 + 0.632199i \(0.217847\pi\)
\(398\) −4.86848e6 −1.54059
\(399\) −124157. −0.0390426
\(400\) −613714. −0.191786
\(401\) 717064. 0.222688 0.111344 0.993782i \(-0.464484\pi\)
0.111344 + 0.993782i \(0.464484\pi\)
\(402\) 2.79730e6 0.863323
\(403\) 541062. 0.165953
\(404\) 4.19007e6 1.27723
\(405\) −164025. −0.0496904
\(406\) 1.75169e6 0.527402
\(407\) −491316. −0.147019
\(408\) 18417.0 0.00547733
\(409\) −3.80091e6 −1.12352 −0.561758 0.827302i \(-0.689875\pi\)
−0.561758 + 0.827302i \(0.689875\pi\)
\(410\) 106493. 0.0312869
\(411\) 176086. 0.0514185
\(412\) −5.49595e6 −1.59514
\(413\) −84959.4 −0.0245096
\(414\) −906361. −0.259896
\(415\) 1.99999e6 0.570044
\(416\) −1.82343e6 −0.516601
\(417\) 1.19248e6 0.335823
\(418\) −342468. −0.0958691
\(419\) −1.55888e6 −0.433787 −0.216894 0.976195i \(-0.569592\pi\)
−0.216894 + 0.976195i \(0.569592\pi\)
\(420\) 294709. 0.0815212
\(421\) 4.13561e6 1.13719 0.568596 0.822617i \(-0.307487\pi\)
0.568596 + 0.822617i \(0.307487\pi\)
\(422\) −7.41761e6 −2.02760
\(423\) 45604.9 0.0123925
\(424\) 379997. 0.102651
\(425\) −125215. −0.0336266
\(426\) 997935. 0.266427
\(427\) 1.57260e6 0.417397
\(428\) −437502. −0.115444
\(429\) −240413. −0.0630687
\(430\) −2.43952e6 −0.636257
\(431\) −5.25900e6 −1.36367 −0.681836 0.731505i \(-0.738818\pi\)
−0.681836 + 0.731505i \(0.738818\pi\)
\(432\) −715836. −0.184546
\(433\) −694549. −0.178026 −0.0890130 0.996030i \(-0.528371\pi\)
−0.0890130 + 0.996030i \(0.528371\pi\)
\(434\) −779629. −0.198684
\(435\) −1.23899e6 −0.313940
\(436\) 1.26308e6 0.318210
\(437\) 485260. 0.121554
\(438\) 2.88914e6 0.719586
\(439\) −7.80969e6 −1.93407 −0.967036 0.254640i \(-0.918043\pi\)
−0.967036 + 0.254640i \(0.918043\pi\)
\(440\) 30897.8 0.00760844
\(441\) −1.23578e6 −0.302583
\(442\) −357307. −0.0869934
\(443\) −5.43876e6 −1.31671 −0.658356 0.752707i \(-0.728748\pi\)
−0.658356 + 0.752707i \(0.728748\pi\)
\(444\) −1.21562e6 −0.292644
\(445\) 944575. 0.226119
\(446\) −7.75229e6 −1.84541
\(447\) −1.37985e6 −0.326635
\(448\) 1.39014e6 0.327238
\(449\) 5.46277e6 1.27878 0.639392 0.768881i \(-0.279186\pi\)
0.639392 + 0.768881i \(0.279186\pi\)
\(450\) −408981. −0.0952076
\(451\) 63801.3 0.0147703
\(452\) −2.83378e6 −0.652409
\(453\) 2.23336e6 0.511343
\(454\) 2.34243e6 0.533368
\(455\) −217321. −0.0492122
\(456\) −32206.3 −0.00725318
\(457\) 3.07144e6 0.687941 0.343970 0.938980i \(-0.388228\pi\)
0.343970 + 0.938980i \(0.388228\pi\)
\(458\) −898028. −0.200044
\(459\) −146050. −0.0323572
\(460\) −1.15185e6 −0.253806
\(461\) 3.85641e6 0.845143 0.422572 0.906329i \(-0.361127\pi\)
0.422572 + 0.906329i \(0.361127\pi\)
\(462\) 346416. 0.0755080
\(463\) 3.20594e6 0.695029 0.347514 0.937675i \(-0.387026\pi\)
0.347514 + 0.937675i \(0.387026\pi\)
\(464\) −5.40721e6 −1.16594
\(465\) 551442. 0.118268
\(466\) 3.66136e6 0.781049
\(467\) −2.43952e6 −0.517622 −0.258811 0.965928i \(-0.583331\pi\)
−0.258811 + 0.965928i \(0.583331\pi\)
\(468\) −594830. −0.125539
\(469\) 1.51492e6 0.318023
\(470\) 113711. 0.0237443
\(471\) −1.62623e6 −0.337777
\(472\) −22038.5 −0.00455330
\(473\) −1.46154e6 −0.300371
\(474\) −2.59369e6 −0.530240
\(475\) 218966. 0.0445289
\(476\) 262413. 0.0530846
\(477\) −3.01344e6 −0.606411
\(478\) −1.02104e7 −2.04396
\(479\) 4.69186e6 0.934342 0.467171 0.884167i \(-0.345273\pi\)
0.467171 + 0.884167i \(0.345273\pi\)
\(480\) −1.85841e6 −0.368162
\(481\) 896406. 0.176662
\(482\) −1.08536e6 −0.212793
\(483\) −490854. −0.0957381
\(484\) 487023. 0.0945010
\(485\) 190409. 0.0367564
\(486\) −477035. −0.0916136
\(487\) 5.73305e6 1.09538 0.547688 0.836683i \(-0.315508\pi\)
0.547688 + 0.836683i \(0.315508\pi\)
\(488\) 407932. 0.0775423
\(489\) −1.58441e6 −0.299636
\(490\) −3.08130e6 −0.579754
\(491\) −693872. −0.129890 −0.0649449 0.997889i \(-0.520687\pi\)
−0.0649449 + 0.997889i \(0.520687\pi\)
\(492\) 157858. 0.0294004
\(493\) −1.10322e6 −0.204430
\(494\) 624832. 0.115198
\(495\) −245025. −0.0449467
\(496\) 2.40660e6 0.439238
\(497\) 540448. 0.0981438
\(498\) 5.81659e6 1.05098
\(499\) 1.23153e6 0.221407 0.110704 0.993853i \(-0.464690\pi\)
0.110704 + 0.993853i \(0.464690\pi\)
\(500\) −519755. −0.0929767
\(501\) 2.64436e6 0.470681
\(502\) 1.15823e7 2.05133
\(503\) −7.61551e6 −1.34208 −0.671041 0.741420i \(-0.734153\pi\)
−0.671041 + 0.741420i \(0.734153\pi\)
\(504\) 32577.6 0.00571272
\(505\) −3.14907e6 −0.549483
\(506\) −1.35395e6 −0.235085
\(507\) −2.90300e6 −0.501566
\(508\) −4.69816e6 −0.807734
\(509\) −1.55496e6 −0.266026 −0.133013 0.991114i \(-0.542465\pi\)
−0.133013 + 0.991114i \(0.542465\pi\)
\(510\) −364163. −0.0619969
\(511\) 1.56466e6 0.265074
\(512\) −8.43141e6 −1.42143
\(513\) 255402. 0.0428480
\(514\) 7.70649e6 1.28662
\(515\) 4.13051e6 0.686255
\(516\) −3.61616e6 −0.597892
\(517\) 68125.8 0.0112095
\(518\) −1.29165e6 −0.211505
\(519\) −187064. −0.0304840
\(520\) −56373.0 −0.00914245
\(521\) −3.26766e6 −0.527403 −0.263701 0.964604i \(-0.584943\pi\)
−0.263701 + 0.964604i \(0.584943\pi\)
\(522\) −3.60338e6 −0.578807
\(523\) −8.35303e6 −1.33533 −0.667667 0.744460i \(-0.732707\pi\)
−0.667667 + 0.744460i \(0.732707\pi\)
\(524\) 7.00112e6 1.11388
\(525\) −221490. −0.0350717
\(526\) −2.60358e6 −0.410305
\(527\) 491012. 0.0770133
\(528\) −1.06934e6 −0.166928
\(529\) −4.51787e6 −0.701931
\(530\) −7.51374e6 −1.16189
\(531\) 174769. 0.0268985
\(532\) −458888. −0.0702956
\(533\) −116405. −0.0177482
\(534\) 2.74711e6 0.416892
\(535\) 328807. 0.0496657
\(536\) 392970. 0.0590810
\(537\) −2.06393e6 −0.308858
\(538\) −6.53349e6 −0.973172
\(539\) −1.84604e6 −0.273696
\(540\) −606243. −0.0894668
\(541\) 4.72814e6 0.694540 0.347270 0.937765i \(-0.387109\pi\)
0.347270 + 0.937765i \(0.387109\pi\)
\(542\) 7.34593e6 1.07411
\(543\) −817266. −0.118950
\(544\) −1.65476e6 −0.239738
\(545\) −949272. −0.136899
\(546\) −632036. −0.0907320
\(547\) −3.84639e6 −0.549648 −0.274824 0.961495i \(-0.588620\pi\)
−0.274824 + 0.961495i \(0.588620\pi\)
\(548\) 650819. 0.0925782
\(549\) −3.23498e6 −0.458079
\(550\) −610947. −0.0861185
\(551\) 1.92923e6 0.270710
\(552\) −127328. −0.0177859
\(553\) −1.40466e6 −0.195325
\(554\) −1.91409e6 −0.264965
\(555\) 913604. 0.125900
\(556\) 4.40745e6 0.604644
\(557\) 1.34095e7 1.83137 0.915684 0.401900i \(-0.131650\pi\)
0.915684 + 0.401900i \(0.131650\pi\)
\(558\) 1.60376e6 0.218050
\(559\) 2.66658e6 0.360932
\(560\) −966625. −0.130253
\(561\) −218174. −0.0292682
\(562\) 3.64168e6 0.486363
\(563\) 1.51163e6 0.200990 0.100495 0.994938i \(-0.467957\pi\)
0.100495 + 0.994938i \(0.467957\pi\)
\(564\) 168557. 0.0223126
\(565\) 2.12974e6 0.280677
\(566\) −5.52105e6 −0.724403
\(567\) −258346. −0.0337477
\(568\) 140192. 0.0182328
\(569\) 1.39221e7 1.80271 0.901353 0.433085i \(-0.142575\pi\)
0.901353 + 0.433085i \(0.142575\pi\)
\(570\) 636820. 0.0820974
\(571\) −3.44073e6 −0.441632 −0.220816 0.975316i \(-0.570872\pi\)
−0.220816 + 0.975316i \(0.570872\pi\)
\(572\) −888574. −0.113554
\(573\) −4.47115e6 −0.568896
\(574\) 167731. 0.0212488
\(575\) 865681. 0.109191
\(576\) −2.85964e6 −0.359133
\(577\) 3.96778e6 0.496144 0.248072 0.968742i \(-0.420203\pi\)
0.248072 + 0.968742i \(0.420203\pi\)
\(578\) 1.11463e7 1.38774
\(579\) 2.44226e6 0.302758
\(580\) −4.57937e6 −0.565244
\(581\) 3.15007e6 0.387151
\(582\) 553767. 0.0677672
\(583\) −4.50156e6 −0.548519
\(584\) 405872. 0.0492445
\(585\) 447048. 0.0540088
\(586\) 1.46278e7 1.75969
\(587\) 3.64595e6 0.436733 0.218366 0.975867i \(-0.429927\pi\)
0.218366 + 0.975867i \(0.429927\pi\)
\(588\) −4.56748e6 −0.544796
\(589\) −858645. −0.101982
\(590\) 435770. 0.0515380
\(591\) −1.71875e6 −0.202416
\(592\) 3.98714e6 0.467582
\(593\) −486759. −0.0568431 −0.0284215 0.999596i \(-0.509048\pi\)
−0.0284215 + 0.999596i \(0.509048\pi\)
\(594\) −712608. −0.0828676
\(595\) −197218. −0.0228378
\(596\) −5.09997e6 −0.588101
\(597\) 5.42373e6 0.622820
\(598\) 2.47027e6 0.282483
\(599\) 1.37413e7 1.56481 0.782404 0.622771i \(-0.213993\pi\)
0.782404 + 0.622771i \(0.213993\pi\)
\(600\) −57454.5 −0.00651548
\(601\) 7.84470e6 0.885911 0.442955 0.896544i \(-0.353930\pi\)
0.442955 + 0.896544i \(0.353930\pi\)
\(602\) −3.84234e6 −0.432120
\(603\) −3.11633e6 −0.349019
\(604\) 8.25457e6 0.920666
\(605\) −366025. −0.0406558
\(606\) −9.15848e6 −1.01308
\(607\) −3.41681e6 −0.376399 −0.188200 0.982131i \(-0.560265\pi\)
−0.188200 + 0.982131i \(0.560265\pi\)
\(608\) 2.89371e6 0.317465
\(609\) −1.95147e6 −0.213215
\(610\) −8.06611e6 −0.877687
\(611\) −124296. −0.0134695
\(612\) −539807. −0.0582586
\(613\) −1.32402e7 −1.42313 −0.711564 0.702621i \(-0.752013\pi\)
−0.711564 + 0.702621i \(0.752013\pi\)
\(614\) 2.43054e7 2.60184
\(615\) −118639. −0.0126485
\(616\) 48665.3 0.00516735
\(617\) 1.72143e7 1.82044 0.910220 0.414126i \(-0.135913\pi\)
0.910220 + 0.414126i \(0.135913\pi\)
\(618\) 1.20128e7 1.26524
\(619\) 1.60142e7 1.67988 0.839940 0.542679i \(-0.182590\pi\)
0.839940 + 0.542679i \(0.182590\pi\)
\(620\) 2.03815e6 0.212940
\(621\) 1.00973e6 0.105069
\(622\) 1.64871e7 1.70871
\(623\) 1.48774e6 0.153571
\(624\) 1.95100e6 0.200584
\(625\) 390625. 0.0400000
\(626\) 1.40520e7 1.43318
\(627\) 381526. 0.0387574
\(628\) −6.01061e6 −0.608162
\(629\) 813486. 0.0819830
\(630\) −644162. −0.0646612
\(631\) −7.75049e6 −0.774918 −0.387459 0.921887i \(-0.626647\pi\)
−0.387459 + 0.921887i \(0.626647\pi\)
\(632\) −364368. −0.0362867
\(633\) 8.26358e6 0.819708
\(634\) −862021. −0.0851716
\(635\) 3.53093e6 0.347500
\(636\) −1.11378e7 −1.09183
\(637\) 3.36810e6 0.328879
\(638\) −5.38282e6 −0.523550
\(639\) −1.11175e6 −0.107710
\(640\) −522556. −0.0504293
\(641\) 8.77648e6 0.843675 0.421838 0.906671i \(-0.361385\pi\)
0.421838 + 0.906671i \(0.361385\pi\)
\(642\) 956272. 0.0915680
\(643\) −1.51011e7 −1.44040 −0.720198 0.693768i \(-0.755949\pi\)
−0.720198 + 0.693768i \(0.755949\pi\)
\(644\) −1.81422e6 −0.172375
\(645\) 2.71774e6 0.257222
\(646\) 567034. 0.0534598
\(647\) 1.57910e7 1.48303 0.741516 0.670936i \(-0.234107\pi\)
0.741516 + 0.670936i \(0.234107\pi\)
\(648\) −67015.0 −0.00626952
\(649\) 261075. 0.0243306
\(650\) 1.11467e6 0.103482
\(651\) 868545. 0.0803230
\(652\) −5.85603e6 −0.539491
\(653\) 1.33260e7 1.22297 0.611484 0.791256i \(-0.290573\pi\)
0.611484 + 0.791256i \(0.290573\pi\)
\(654\) −2.76078e6 −0.252399
\(655\) −5.26173e6 −0.479210
\(656\) −517762. −0.0469754
\(657\) −3.21864e6 −0.290910
\(658\) 179100. 0.0161262
\(659\) −882213. −0.0791334 −0.0395667 0.999217i \(-0.512598\pi\)
−0.0395667 + 0.999217i \(0.512598\pi\)
\(660\) −905622. −0.0809258
\(661\) −5.59700e6 −0.498255 −0.249127 0.968471i \(-0.580144\pi\)
−0.249127 + 0.968471i \(0.580144\pi\)
\(662\) −1.25024e7 −1.10879
\(663\) 398058. 0.0351692
\(664\) 817128. 0.0719234
\(665\) 344880. 0.0302422
\(666\) 2.65704e6 0.232120
\(667\) 7.62720e6 0.663820
\(668\) 9.77367e6 0.847454
\(669\) 8.63644e6 0.746053
\(670\) −7.77027e6 −0.668727
\(671\) −4.83250e6 −0.414348
\(672\) −2.92708e6 −0.250041
\(673\) 1.29320e6 0.110060 0.0550300 0.998485i \(-0.482475\pi\)
0.0550300 + 0.998485i \(0.482475\pi\)
\(674\) −9.90040e6 −0.839466
\(675\) 455625. 0.0384900
\(676\) −1.07296e7 −0.903062
\(677\) 1.48798e7 1.24775 0.623873 0.781526i \(-0.285558\pi\)
0.623873 + 0.781526i \(0.285558\pi\)
\(678\) 6.19395e6 0.517480
\(679\) 299902. 0.0249634
\(680\) −51158.4 −0.00424272
\(681\) −2.60958e6 −0.215627
\(682\) 2.39575e6 0.197233
\(683\) 8.62925e6 0.707817 0.353909 0.935280i \(-0.384852\pi\)
0.353909 + 0.935280i \(0.384852\pi\)
\(684\) 943974. 0.0771471
\(685\) −489127. −0.0398286
\(686\) −1.01996e7 −0.827506
\(687\) 1.00045e6 0.0808729
\(688\) 1.18607e7 0.955302
\(689\) 8.21310e6 0.659112
\(690\) 2.51767e6 0.201315
\(691\) −1.79734e7 −1.43198 −0.715988 0.698113i \(-0.754023\pi\)
−0.715988 + 0.698113i \(0.754023\pi\)
\(692\) −691396. −0.0548860
\(693\) −385925. −0.0305260
\(694\) −3.09157e7 −2.43658
\(695\) −3.31244e6 −0.260127
\(696\) −506211. −0.0396103
\(697\) −105638. −0.00823639
\(698\) −3.69167e6 −0.286804
\(699\) −4.07894e6 −0.315758
\(700\) −818636. −0.0631460
\(701\) −2.13485e7 −1.64086 −0.820432 0.571744i \(-0.806267\pi\)
−0.820432 + 0.571744i \(0.806267\pi\)
\(702\) 1.30015e6 0.0995754
\(703\) −1.42256e6 −0.108563
\(704\) −4.27181e6 −0.324848
\(705\) −126680. −0.00959922
\(706\) −3.49667e7 −2.64024
\(707\) −4.95992e6 −0.373187
\(708\) 645953. 0.0484303
\(709\) 6.90604e6 0.515957 0.257979 0.966151i \(-0.416944\pi\)
0.257979 + 0.966151i \(0.416944\pi\)
\(710\) −2.77204e6 −0.206373
\(711\) 2.88950e6 0.214363
\(712\) 385921. 0.0285298
\(713\) −3.39466e6 −0.250076
\(714\) −573571. −0.0421058
\(715\) 667813. 0.0488528
\(716\) −7.62837e6 −0.556095
\(717\) 1.13749e7 0.826320
\(718\) 6.82951e6 0.494400
\(719\) −1.71863e7 −1.23983 −0.619913 0.784670i \(-0.712832\pi\)
−0.619913 + 0.784670i \(0.712832\pi\)
\(720\) 1.98843e6 0.142949
\(721\) 6.50573e6 0.466077
\(722\) 1.90119e7 1.35732
\(723\) 1.20915e6 0.0860266
\(724\) −3.02065e6 −0.214167
\(725\) 3.44165e6 0.243177
\(726\) −1.06451e6 −0.0749566
\(727\) −2.43516e7 −1.70880 −0.854401 0.519615i \(-0.826075\pi\)
−0.854401 + 0.519615i \(0.826075\pi\)
\(728\) −88789.8 −0.00620919
\(729\) 531441. 0.0370370
\(730\) −8.02538e6 −0.557389
\(731\) 2.41992e6 0.167497
\(732\) −1.19566e7 −0.824765
\(733\) −1.93102e7 −1.32747 −0.663737 0.747966i \(-0.731031\pi\)
−0.663737 + 0.747966i \(0.731031\pi\)
\(734\) −2.39141e7 −1.63838
\(735\) 3.43272e6 0.234380
\(736\) 1.14403e7 0.778472
\(737\) −4.65525e6 −0.315700
\(738\) −345038. −0.0233199
\(739\) −9.23336e6 −0.621940 −0.310970 0.950420i \(-0.600654\pi\)
−0.310970 + 0.950420i \(0.600654\pi\)
\(740\) 3.37672e6 0.226681
\(741\) −696094. −0.0465717
\(742\) −1.18345e7 −0.789112
\(743\) 8.02302e6 0.533170 0.266585 0.963811i \(-0.414105\pi\)
0.266585 + 0.963811i \(0.414105\pi\)
\(744\) 225300. 0.0149221
\(745\) 3.83291e6 0.253010
\(746\) −8.26434e6 −0.543702
\(747\) −6.47998e6 −0.424885
\(748\) −806379. −0.0526969
\(749\) 517885. 0.0337309
\(750\) 1.13606e6 0.0737475
\(751\) 1.19930e6 0.0775938 0.0387969 0.999247i \(-0.487647\pi\)
0.0387969 + 0.999247i \(0.487647\pi\)
\(752\) −552856. −0.0356507
\(753\) −1.29033e7 −0.829302
\(754\) 9.82096e6 0.629109
\(755\) −6.20377e6 −0.396085
\(756\) −954858. −0.0607623
\(757\) −1.40172e7 −0.889040 −0.444520 0.895769i \(-0.646626\pi\)
−0.444520 + 0.895769i \(0.646626\pi\)
\(758\) 3.38820e7 2.14188
\(759\) 1.50836e6 0.0950389
\(760\) 89461.9 0.00561829
\(761\) −1.98659e7 −1.24350 −0.621750 0.783215i \(-0.713578\pi\)
−0.621750 + 0.783215i \(0.713578\pi\)
\(762\) 1.02690e7 0.640681
\(763\) −1.49514e6 −0.0929761
\(764\) −1.65255e7 −1.02429
\(765\) 405695. 0.0250638
\(766\) 1.32721e7 0.817275
\(767\) −476331. −0.0292361
\(768\) 8.64785e6 0.529060
\(769\) −2.09738e7 −1.27897 −0.639486 0.768803i \(-0.720853\pi\)
−0.639486 + 0.768803i \(0.720853\pi\)
\(770\) −962267. −0.0584883
\(771\) −8.58542e6 −0.520146
\(772\) 9.02668e6 0.545111
\(773\) 7.69041e6 0.462914 0.231457 0.972845i \(-0.425651\pi\)
0.231457 + 0.972845i \(0.425651\pi\)
\(774\) 7.90403e6 0.474238
\(775\) −1.53178e6 −0.0916101
\(776\) 77794.4 0.00463761
\(777\) 1.43896e6 0.0855062
\(778\) −6.21909e6 −0.368365
\(779\) 184731. 0.0109068
\(780\) 1.65231e6 0.0972421
\(781\) −1.66076e6 −0.0974270
\(782\) 2.24177e6 0.131091
\(783\) 4.01434e6 0.233997
\(784\) 1.49810e7 0.870466
\(785\) 4.51731e6 0.261641
\(786\) −1.53027e7 −0.883513
\(787\) −1.39975e6 −0.0805589 −0.0402794 0.999188i \(-0.512825\pi\)
−0.0402794 + 0.999188i \(0.512825\pi\)
\(788\) −6.35256e6 −0.364446
\(789\) 2.90052e6 0.165876
\(790\) 7.20470e6 0.410722
\(791\) 3.35443e6 0.190624
\(792\) −100109. −0.00567100
\(793\) 8.81689e6 0.497889
\(794\) −3.93131e7 −2.21302
\(795\) 8.37068e6 0.469724
\(796\) 2.00463e7 1.12138
\(797\) 1.22917e7 0.685432 0.342716 0.939439i \(-0.388653\pi\)
0.342716 + 0.939439i \(0.388653\pi\)
\(798\) 1.00302e6 0.0557573
\(799\) −112798. −0.00625078
\(800\) 5.16225e6 0.285177
\(801\) −3.06042e6 −0.168539
\(802\) −5.79290e6 −0.318024
\(803\) −4.80810e6 −0.263138
\(804\) −1.15181e7 −0.628404
\(805\) 1.36348e6 0.0741584
\(806\) −4.37104e6 −0.236999
\(807\) 7.27863e6 0.393429
\(808\) −1.28660e6 −0.0693292
\(809\) −1.19607e7 −0.642515 −0.321258 0.946992i \(-0.604106\pi\)
−0.321258 + 0.946992i \(0.604106\pi\)
\(810\) 1.32510e6 0.0709636
\(811\) −1.15112e7 −0.614565 −0.307282 0.951618i \(-0.599420\pi\)
−0.307282 + 0.951618i \(0.599420\pi\)
\(812\) −7.21270e6 −0.383891
\(813\) −8.18373e6 −0.434235
\(814\) 3.96916e6 0.209961
\(815\) 4.40113e6 0.232097
\(816\) 1.77053e6 0.0930846
\(817\) −4.23177e6 −0.221803
\(818\) 3.07062e7 1.60451
\(819\) 704120. 0.0366806
\(820\) −438493. −0.0227734
\(821\) 2.77684e7 1.43778 0.718890 0.695124i \(-0.244651\pi\)
0.718890 + 0.695124i \(0.244651\pi\)
\(822\) −1.42253e6 −0.0734315
\(823\) −2.63447e7 −1.35580 −0.677898 0.735156i \(-0.737109\pi\)
−0.677898 + 0.735156i \(0.737109\pi\)
\(824\) 1.68758e6 0.0865860
\(825\) 680625. 0.0348155
\(826\) 686356. 0.0350025
\(827\) −3.32551e7 −1.69081 −0.845406 0.534125i \(-0.820641\pi\)
−0.845406 + 0.534125i \(0.820641\pi\)
\(828\) 3.73200e6 0.189176
\(829\) 2.84581e7 1.43820 0.719101 0.694905i \(-0.244554\pi\)
0.719101 + 0.694905i \(0.244554\pi\)
\(830\) −1.61572e7 −0.814087
\(831\) 2.13239e6 0.107119
\(832\) 7.79391e6 0.390344
\(833\) 3.05654e6 0.152622
\(834\) −9.63360e6 −0.479594
\(835\) −7.34545e6 −0.364588
\(836\) 1.41013e6 0.0697822
\(837\) −1.78667e6 −0.0881519
\(838\) 1.25936e7 0.619498
\(839\) −9.47931e6 −0.464913 −0.232457 0.972607i \(-0.574676\pi\)
−0.232457 + 0.972607i \(0.574676\pi\)
\(840\) −90493.3 −0.00442505
\(841\) 9.81196e6 0.478372
\(842\) −3.34101e7 −1.62404
\(843\) −4.05701e6 −0.196624
\(844\) 3.05425e7 1.47587
\(845\) 8.06390e6 0.388511
\(846\) −368425. −0.0176980
\(847\) −576505. −0.0276118
\(848\) 3.65312e7 1.74451
\(849\) 6.15072e6 0.292858
\(850\) 1.01156e6 0.0480226
\(851\) −5.62411e6 −0.266213
\(852\) −4.10907e6 −0.193930
\(853\) 2.13145e7 1.00300 0.501501 0.865157i \(-0.332781\pi\)
0.501501 + 0.865157i \(0.332781\pi\)
\(854\) −1.27045e7 −0.596090
\(855\) −709449. −0.0331899
\(856\) 134339. 0.00626640
\(857\) 2.21044e7 1.02808 0.514039 0.857767i \(-0.328149\pi\)
0.514039 + 0.857767i \(0.328149\pi\)
\(858\) 1.94221e6 0.0900693
\(859\) −7.07193e6 −0.327006 −0.163503 0.986543i \(-0.552279\pi\)
−0.163503 + 0.986543i \(0.552279\pi\)
\(860\) 1.00449e7 0.463125
\(861\) −186861. −0.00859035
\(862\) 4.24855e7 1.94748
\(863\) −5.45080e6 −0.249134 −0.124567 0.992211i \(-0.539754\pi\)
−0.124567 + 0.992211i \(0.539754\pi\)
\(864\) 6.02125e6 0.274412
\(865\) 519623. 0.0236128
\(866\) 5.61101e6 0.254241
\(867\) −1.24175e7 −0.561029
\(868\) 3.21017e6 0.144620
\(869\) 4.31642e6 0.193898
\(870\) 1.00094e7 0.448342
\(871\) 8.49351e6 0.379351
\(872\) −387840. −0.0172727
\(873\) −616924. −0.0273966
\(874\) −3.92023e6 −0.173593
\(875\) 615251. 0.0271664
\(876\) −1.18962e7 −0.523780
\(877\) −2.02823e7 −0.890467 −0.445234 0.895414i \(-0.646879\pi\)
−0.445234 + 0.895414i \(0.646879\pi\)
\(878\) 6.30917e7 2.76208
\(879\) −1.62961e7 −0.711397
\(880\) 2.97038e6 0.129302
\(881\) −6.58178e6 −0.285696 −0.142848 0.989745i \(-0.545626\pi\)
−0.142848 + 0.989745i \(0.545626\pi\)
\(882\) 9.98340e6 0.432123
\(883\) −1.63860e7 −0.707247 −0.353623 0.935388i \(-0.615051\pi\)
−0.353623 + 0.935388i \(0.615051\pi\)
\(884\) 1.47124e6 0.0633217
\(885\) −485469. −0.0208355
\(886\) 4.39378e7 1.88042
\(887\) −2.92929e6 −0.125013 −0.0625063 0.998045i \(-0.519909\pi\)
−0.0625063 + 0.998045i \(0.519909\pi\)
\(888\) 373267. 0.0158850
\(889\) 5.56136e6 0.236008
\(890\) −7.63087e6 −0.322923
\(891\) 793881. 0.0335013
\(892\) 3.19206e7 1.34326
\(893\) 197252. 0.00827739
\(894\) 1.11473e7 0.466472
\(895\) 5.73314e6 0.239241
\(896\) −823047. −0.0342495
\(897\) −2.75201e6 −0.114201
\(898\) −4.41317e7 −1.82625
\(899\) −1.34960e7 −0.556936
\(900\) 1.68401e6 0.0693007
\(901\) 7.45337e6 0.305873
\(902\) −515427. −0.0210936
\(903\) 4.28056e6 0.174695
\(904\) 870140. 0.0354134
\(905\) 2.27018e6 0.0921381
\(906\) −1.80425e7 −0.730256
\(907\) −7.88236e6 −0.318154 −0.159077 0.987266i \(-0.550852\pi\)
−0.159077 + 0.987266i \(0.550852\pi\)
\(908\) −9.64512e6 −0.388234
\(909\) 1.02030e7 0.409561
\(910\) 1.75566e6 0.0702807
\(911\) −2.16713e7 −0.865146 −0.432573 0.901599i \(-0.642394\pi\)
−0.432573 + 0.901599i \(0.642394\pi\)
\(912\) −3.09617e6 −0.123264
\(913\) −9.67996e6 −0.384323
\(914\) −2.48130e7 −0.982458
\(915\) 8.98605e6 0.354827
\(916\) 3.69769e6 0.145610
\(917\) −8.28745e6 −0.325460
\(918\) 1.17989e6 0.0462097
\(919\) 4.32037e7 1.68746 0.843728 0.536771i \(-0.180356\pi\)
0.843728 + 0.536771i \(0.180356\pi\)
\(920\) 353688. 0.0137769
\(921\) −2.70774e7 −1.05186
\(922\) −3.11545e7 −1.20696
\(923\) 3.03006e6 0.117070
\(924\) −1.42639e6 −0.0549616
\(925\) −2.53779e6 −0.0975217
\(926\) −2.58996e7 −0.992580
\(927\) −1.33829e7 −0.511504
\(928\) 4.54827e7 1.73371
\(929\) 3.70288e7 1.40767 0.703834 0.710364i \(-0.251470\pi\)
0.703834 + 0.710364i \(0.251470\pi\)
\(930\) −4.45490e6 −0.168900
\(931\) −5.34505e6 −0.202105
\(932\) −1.50759e7 −0.568518
\(933\) −1.83675e7 −0.690790
\(934\) 1.97080e7 0.739224
\(935\) 606038. 0.0226710
\(936\) 182648. 0.00681438
\(937\) 1.22598e7 0.456180 0.228090 0.973640i \(-0.426752\pi\)
0.228090 + 0.973640i \(0.426752\pi\)
\(938\) −1.22385e7 −0.454173
\(939\) −1.56546e7 −0.579399
\(940\) −468215. −0.0172833
\(941\) −1.25646e7 −0.462565 −0.231283 0.972887i \(-0.574292\pi\)
−0.231283 + 0.972887i \(0.574292\pi\)
\(942\) 1.31377e7 0.482384
\(943\) 730335. 0.0267450
\(944\) −2.11868e6 −0.0773812
\(945\) 717628. 0.0261409
\(946\) 1.18073e7 0.428964
\(947\) 3.16122e7 1.14546 0.572730 0.819744i \(-0.305884\pi\)
0.572730 + 0.819744i \(0.305884\pi\)
\(948\) 1.06797e7 0.385957
\(949\) 8.77237e6 0.316192
\(950\) −1.76894e6 −0.0635924
\(951\) 960334. 0.0344327
\(952\) −80576.6 −0.00288148
\(953\) −2.84509e7 −1.01476 −0.507380 0.861722i \(-0.669386\pi\)
−0.507380 + 0.861722i \(0.669386\pi\)
\(954\) 2.43445e7 0.866024
\(955\) 1.24199e7 0.440665
\(956\) 4.20419e7 1.48778
\(957\) 5.99673e6 0.211658
\(958\) −3.79038e7 −1.33435
\(959\) −770395. −0.0270500
\(960\) 7.94344e6 0.278183
\(961\) −2.26225e7 −0.790190
\(962\) −7.24174e6 −0.252293
\(963\) −1.06533e6 −0.0370186
\(964\) 4.46905e6 0.154890
\(965\) −6.78405e6 −0.234515
\(966\) 3.96543e6 0.136725
\(967\) −4.73439e7 −1.62816 −0.814081 0.580751i \(-0.802759\pi\)
−0.814081 + 0.580751i \(0.802759\pi\)
\(968\) −149545. −0.00512961
\(969\) −631704. −0.0216124
\(970\) −1.53824e6 −0.0524923
\(971\) −4.33248e7 −1.47465 −0.737324 0.675539i \(-0.763911\pi\)
−0.737324 + 0.675539i \(0.763911\pi\)
\(972\) 1.96423e6 0.0666846
\(973\) −5.21723e6 −0.176668
\(974\) −4.63152e7 −1.56432
\(975\) −1.24180e6 −0.0418350
\(976\) 3.92168e7 1.31780
\(977\) −4.64678e7 −1.55745 −0.778727 0.627363i \(-0.784134\pi\)
−0.778727 + 0.627363i \(0.784134\pi\)
\(978\) 1.27998e7 0.427915
\(979\) −4.57174e6 −0.152449
\(980\) 1.26875e7 0.421997
\(981\) 3.07564e6 0.102038
\(982\) 5.60553e6 0.185498
\(983\) −1.02443e7 −0.338141 −0.169071 0.985604i \(-0.554077\pi\)
−0.169071 + 0.985604i \(0.554077\pi\)
\(984\) −48471.7 −0.00159588
\(985\) 4.77430e6 0.156790
\(986\) 8.91250e6 0.291949
\(987\) −199527. −0.00651940
\(988\) −2.57279e6 −0.0838517
\(989\) −1.67303e7 −0.543892
\(990\) 1.97947e6 0.0641890
\(991\) −6.52638e6 −0.211100 −0.105550 0.994414i \(-0.533660\pi\)
−0.105550 + 0.994414i \(0.533660\pi\)
\(992\) −2.02431e7 −0.653127
\(993\) 1.39283e7 0.448256
\(994\) −4.36608e6 −0.140161
\(995\) −1.50659e7 −0.482434
\(996\) −2.39502e7 −0.765000
\(997\) 1.08535e7 0.345804 0.172902 0.984939i \(-0.444686\pi\)
0.172902 + 0.984939i \(0.444686\pi\)
\(998\) −9.94904e6 −0.316195
\(999\) −2.96008e6 −0.0938403
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 165.6.a.b.1.1 3
3.2 odd 2 495.6.a.d.1.3 3
5.4 even 2 825.6.a.i.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.6.a.b.1.1 3 1.1 even 1 trivial
495.6.a.d.1.3 3 3.2 odd 2
825.6.a.i.1.3 3 5.4 even 2