Properties

Label 147.6.a.n.1.6
Level $147$
Weight $6$
Character 147.1
Self dual yes
Analytic conductor $23.576$
Analytic rank $1$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [147,6,Mod(1,147)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(147, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("147.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 147 = 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 147.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: \(23.5764215125\)
Analytic rank: \(1\)
Dimension: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - 2x^{5} - 59x^{4} + 122x^{3} + 941x^{2} - 1856x - 2338 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2^{4}\cdot 7^{3} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(5.59680\) of defining polynomial
Character \(\chi\) \(=\) 147.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+10.0089 q^{2} -9.00000 q^{3} +68.1779 q^{4} -70.3512 q^{5} -90.0800 q^{6} +362.101 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+10.0089 q^{2} -9.00000 q^{3} +68.1779 q^{4} -70.3512 q^{5} -90.0800 q^{6} +362.101 q^{8} +81.0000 q^{9} -704.138 q^{10} -731.396 q^{11} -613.601 q^{12} -899.645 q^{13} +633.161 q^{15} +1442.53 q^{16} +1392.39 q^{17} +810.720 q^{18} +190.198 q^{19} -4796.40 q^{20} -7320.46 q^{22} +42.9770 q^{23} -3258.91 q^{24} +1824.30 q^{25} -9004.45 q^{26} -729.000 q^{27} -7746.59 q^{29} +6337.24 q^{30} -1179.22 q^{31} +2850.94 q^{32} +6582.56 q^{33} +13936.3 q^{34} +5522.41 q^{36} +9288.68 q^{37} +1903.67 q^{38} +8096.80 q^{39} -25474.2 q^{40} -13453.4 q^{41} +6033.68 q^{43} -49865.0 q^{44} -5698.45 q^{45} +430.152 q^{46} +3244.94 q^{47} -12982.8 q^{48} +18259.2 q^{50} -12531.5 q^{51} -61335.9 q^{52} +25675.8 q^{53} -7296.48 q^{54} +51454.6 q^{55} -1711.78 q^{57} -77534.8 q^{58} +26450.5 q^{59} +43167.6 q^{60} +6432.77 q^{61} -11802.7 q^{62} -17626.3 q^{64} +63291.1 q^{65} +65884.2 q^{66} -23815.7 q^{67} +94930.2 q^{68} -386.793 q^{69} -44680.7 q^{71} +29330.2 q^{72} +39601.3 q^{73} +92969.4 q^{74} -16418.7 q^{75} +12967.3 q^{76} +81040.0 q^{78} +23123.0 q^{79} -101484. q^{80} +6561.00 q^{81} -134654. q^{82} +17267.2 q^{83} -97956.3 q^{85} +60390.4 q^{86} +69719.3 q^{87} -264839. q^{88} -49929.7 q^{89} -57035.2 q^{90} +2930.08 q^{92} +10613.0 q^{93} +32478.3 q^{94} -13380.7 q^{95} -25658.5 q^{96} +16865.0 q^{97} -59243.1 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} - 54 q^{3} + 150 q^{4} - 100 q^{5} - 18 q^{6} - 114 q^{8} + 486 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 2 q^{2} - 54 q^{3} + 150 q^{4} - 100 q^{5} - 18 q^{6} - 114 q^{8} + 486 q^{9} - 864 q^{10} + 604 q^{11} - 1350 q^{12} - 1352 q^{13} + 900 q^{15} + 4578 q^{16} - 3028 q^{17} + 162 q^{18} - 1728 q^{19} - 452 q^{20} - 4116 q^{22} - 4484 q^{23} + 1026 q^{24} + 4806 q^{25} - 14172 q^{26} - 4374 q^{27} - 5320 q^{29} + 7776 q^{30} - 3976 q^{31} - 37326 q^{32} - 5436 q^{33} + 16336 q^{34} + 12150 q^{36} + 22680 q^{37} - 52744 q^{38} + 12168 q^{39} - 100600 q^{40} - 28756 q^{41} - 6768 q^{43} - 64940 q^{44} - 8100 q^{45} + 540 q^{46} - 51552 q^{47} - 41202 q^{48} - 40622 q^{50} + 27252 q^{51} - 119296 q^{52} + 80884 q^{53} - 1458 q^{54} - 11656 q^{55} + 15552 q^{57} - 70464 q^{58} - 8872 q^{59} + 4068 q^{60} - 50896 q^{61} - 11824 q^{62} + 199590 q^{64} + 3492 q^{65} + 37044 q^{66} + 6480 q^{67} - 37348 q^{68} + 40356 q^{69} - 110852 q^{71} - 9234 q^{72} - 64232 q^{73} - 27464 q^{74} - 43254 q^{75} + 194864 q^{76} + 127548 q^{78} + 111696 q^{79} + 308940 q^{80} + 39366 q^{81} + 189640 q^{82} - 101128 q^{83} - 23292 q^{85} + 3824 q^{86} + 47880 q^{87} - 97788 q^{88} + 35012 q^{89} - 69984 q^{90} - 449260 q^{92} + 35784 q^{93} + 121016 q^{94} - 119080 q^{95} + 335934 q^{96} - 70952 q^{97} + 48924 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.0089 1.76934 0.884669 0.466219i \(-0.154384\pi\)
0.884669 + 0.466219i \(0.154384\pi\)
\(3\) −9.00000 −0.577350
\(4\) 68.1779 2.13056
\(5\) −70.3512 −1.25848 −0.629241 0.777211i \(-0.716634\pi\)
−0.629241 + 0.777211i \(0.716634\pi\)
\(6\) −90.0800 −1.02153
\(7\) 0 0
\(8\) 362.101 2.00034
\(9\) 81.0000 0.333333
\(10\) −704.138 −2.22668
\(11\) −731.396 −1.82251 −0.911257 0.411838i \(-0.864887\pi\)
−0.911257 + 0.411838i \(0.864887\pi\)
\(12\) −613.601 −1.23008
\(13\) −899.645 −1.47643 −0.738215 0.674566i \(-0.764331\pi\)
−0.738215 + 0.674566i \(0.764331\pi\)
\(14\) 0 0
\(15\) 633.161 0.726584
\(16\) 1442.53 1.40872
\(17\) 1392.39 1.16853 0.584263 0.811564i \(-0.301384\pi\)
0.584263 + 0.811564i \(0.301384\pi\)
\(18\) 810.720 0.589780
\(19\) 190.198 0.120871 0.0604354 0.998172i \(-0.480751\pi\)
0.0604354 + 0.998172i \(0.480751\pi\)
\(20\) −4796.40 −2.68127
\(21\) 0 0
\(22\) −7320.46 −3.22464
\(23\) 42.9770 0.0169401 0.00847006 0.999964i \(-0.497304\pi\)
0.00847006 + 0.999964i \(0.497304\pi\)
\(24\) −3258.91 −1.15490
\(25\) 1824.30 0.583775
\(26\) −9004.45 −2.61230
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −7746.59 −1.71047 −0.855236 0.518239i \(-0.826588\pi\)
−0.855236 + 0.518239i \(0.826588\pi\)
\(30\) 6337.24 1.28557
\(31\) −1179.22 −0.220390 −0.110195 0.993910i \(-0.535148\pi\)
−0.110195 + 0.993910i \(0.535148\pi\)
\(32\) 2850.94 0.492168
\(33\) 6582.56 1.05223
\(34\) 13936.3 2.06752
\(35\) 0 0
\(36\) 5522.41 0.710187
\(37\) 9288.68 1.11545 0.557724 0.830026i \(-0.311675\pi\)
0.557724 + 0.830026i \(0.311675\pi\)
\(38\) 1903.67 0.213861
\(39\) 8096.80 0.852417
\(40\) −25474.2 −2.51739
\(41\) −13453.4 −1.24989 −0.624945 0.780668i \(-0.714879\pi\)
−0.624945 + 0.780668i \(0.714879\pi\)
\(42\) 0 0
\(43\) 6033.68 0.497635 0.248818 0.968550i \(-0.419958\pi\)
0.248818 + 0.968550i \(0.419958\pi\)
\(44\) −49865.0 −3.88297
\(45\) −5698.45 −0.419494
\(46\) 430.152 0.0299728
\(47\) 3244.94 0.214270 0.107135 0.994244i \(-0.465832\pi\)
0.107135 + 0.994244i \(0.465832\pi\)
\(48\) −12982.8 −0.813327
\(49\) 0 0
\(50\) 18259.2 1.03289
\(51\) −12531.5 −0.674649
\(52\) −61335.9 −3.14562
\(53\) 25675.8 1.25555 0.627775 0.778395i \(-0.283966\pi\)
0.627775 + 0.778395i \(0.283966\pi\)
\(54\) −7296.48 −0.340509
\(55\) 51454.6 2.29360
\(56\) 0 0
\(57\) −1711.78 −0.0697848
\(58\) −77534.8 −3.02640
\(59\) 26450.5 0.989244 0.494622 0.869108i \(-0.335306\pi\)
0.494622 + 0.869108i \(0.335306\pi\)
\(60\) 43167.6 1.54803
\(61\) 6432.77 0.221347 0.110673 0.993857i \(-0.464699\pi\)
0.110673 + 0.993857i \(0.464699\pi\)
\(62\) −11802.7 −0.389944
\(63\) 0 0
\(64\) −17626.3 −0.537913
\(65\) 63291.1 1.85806
\(66\) 65884.2 1.86175
\(67\) −23815.7 −0.648150 −0.324075 0.946031i \(-0.605053\pi\)
−0.324075 + 0.946031i \(0.605053\pi\)
\(68\) 94930.2 2.48961
\(69\) −386.793 −0.00978039
\(70\) 0 0
\(71\) −44680.7 −1.05190 −0.525949 0.850516i \(-0.676290\pi\)
−0.525949 + 0.850516i \(0.676290\pi\)
\(72\) 29330.2 0.666781
\(73\) 39601.3 0.869765 0.434882 0.900487i \(-0.356790\pi\)
0.434882 + 0.900487i \(0.356790\pi\)
\(74\) 92969.4 1.97361
\(75\) −16418.7 −0.337042
\(76\) 12967.3 0.257523
\(77\) 0 0
\(78\) 81040.0 1.50821
\(79\) 23123.0 0.416846 0.208423 0.978039i \(-0.433167\pi\)
0.208423 + 0.978039i \(0.433167\pi\)
\(80\) −101484. −1.77285
\(81\) 6561.00 0.111111
\(82\) −134654. −2.21148
\(83\) 17267.2 0.275122 0.137561 0.990493i \(-0.456074\pi\)
0.137561 + 0.990493i \(0.456074\pi\)
\(84\) 0 0
\(85\) −97956.3 −1.47057
\(86\) 60390.4 0.880485
\(87\) 69719.3 0.987541
\(88\) −264839. −3.64565
\(89\) −49929.7 −0.668166 −0.334083 0.942544i \(-0.608427\pi\)
−0.334083 + 0.942544i \(0.608427\pi\)
\(90\) −57035.2 −0.742226
\(91\) 0 0
\(92\) 2930.08 0.0360920
\(93\) 10613.0 0.127242
\(94\) 32478.3 0.379117
\(95\) −13380.7 −0.152114
\(96\) −25658.5 −0.284153
\(97\) 16865.0 0.181994 0.0909970 0.995851i \(-0.470995\pi\)
0.0909970 + 0.995851i \(0.470995\pi\)
\(98\) 0 0
\(99\) −59243.1 −0.607505
\(100\) 124377. 1.24377
\(101\) −105449. −1.02858 −0.514290 0.857617i \(-0.671944\pi\)
−0.514290 + 0.857617i \(0.671944\pi\)
\(102\) −125426. −1.19368
\(103\) 20355.8 0.189058 0.0945290 0.995522i \(-0.469865\pi\)
0.0945290 + 0.995522i \(0.469865\pi\)
\(104\) −325762. −2.95337
\(105\) 0 0
\(106\) 256986. 2.22149
\(107\) −3775.43 −0.0318792 −0.0159396 0.999873i \(-0.505074\pi\)
−0.0159396 + 0.999873i \(0.505074\pi\)
\(108\) −49701.7 −0.410026
\(109\) 8536.98 0.0688237 0.0344118 0.999408i \(-0.489044\pi\)
0.0344118 + 0.999408i \(0.489044\pi\)
\(110\) 515004. 4.05815
\(111\) −83598.1 −0.644005
\(112\) 0 0
\(113\) −100869. −0.743125 −0.371562 0.928408i \(-0.621178\pi\)
−0.371562 + 0.928408i \(0.621178\pi\)
\(114\) −17133.0 −0.123473
\(115\) −3023.49 −0.0213188
\(116\) −528147. −3.64426
\(117\) −72871.2 −0.492143
\(118\) 264740. 1.75031
\(119\) 0 0
\(120\) 229268. 1.45342
\(121\) 373889. 2.32156
\(122\) 64384.9 0.391638
\(123\) 121081. 0.721625
\(124\) −80396.9 −0.469554
\(125\) 91506.2 0.523812
\(126\) 0 0
\(127\) −185451. −1.02028 −0.510142 0.860090i \(-0.670407\pi\)
−0.510142 + 0.860090i \(0.670407\pi\)
\(128\) −267650. −1.44392
\(129\) −54303.1 −0.287310
\(130\) 633474. 3.28754
\(131\) −124046. −0.631545 −0.315773 0.948835i \(-0.602264\pi\)
−0.315773 + 0.948835i \(0.602264\pi\)
\(132\) 448785. 2.24184
\(133\) 0 0
\(134\) −238368. −1.14680
\(135\) 51286.0 0.242195
\(136\) 504185. 2.33745
\(137\) 99520.5 0.453013 0.226507 0.974010i \(-0.427269\pi\)
0.226507 + 0.974010i \(0.427269\pi\)
\(138\) −3871.37 −0.0173048
\(139\) −254481. −1.11717 −0.558583 0.829449i \(-0.688655\pi\)
−0.558583 + 0.829449i \(0.688655\pi\)
\(140\) 0 0
\(141\) −29204.5 −0.123709
\(142\) −447204. −1.86116
\(143\) 657997. 2.69081
\(144\) 116845. 0.469575
\(145\) 544982. 2.15260
\(146\) 396365. 1.53891
\(147\) 0 0
\(148\) 633283. 2.37653
\(149\) −427260. −1.57662 −0.788309 0.615279i \(-0.789043\pi\)
−0.788309 + 0.615279i \(0.789043\pi\)
\(150\) −164333. −0.596342
\(151\) −150472. −0.537047 −0.268523 0.963273i \(-0.586536\pi\)
−0.268523 + 0.963273i \(0.586536\pi\)
\(152\) 68870.8 0.241783
\(153\) 112784. 0.389509
\(154\) 0 0
\(155\) 82959.8 0.277357
\(156\) 552023. 1.81613
\(157\) −165831. −0.536930 −0.268465 0.963290i \(-0.586516\pi\)
−0.268465 + 0.963290i \(0.586516\pi\)
\(158\) 231435. 0.737542
\(159\) −231082. −0.724892
\(160\) −200567. −0.619384
\(161\) 0 0
\(162\) 65668.3 0.196593
\(163\) 95718.2 0.282180 0.141090 0.989997i \(-0.454939\pi\)
0.141090 + 0.989997i \(0.454939\pi\)
\(164\) −917224. −2.66297
\(165\) −463091. −1.32421
\(166\) 172825. 0.486784
\(167\) −474209. −1.31577 −0.657883 0.753120i \(-0.728548\pi\)
−0.657883 + 0.753120i \(0.728548\pi\)
\(168\) 0 0
\(169\) 438068. 1.17984
\(170\) −980434. −2.60193
\(171\) 15406.0 0.0402903
\(172\) 411363. 1.06024
\(173\) −258876. −0.657624 −0.328812 0.944395i \(-0.606648\pi\)
−0.328812 + 0.944395i \(0.606648\pi\)
\(174\) 697813. 1.74729
\(175\) 0 0
\(176\) −1.05506e6 −2.56742
\(177\) −238054. −0.571140
\(178\) −499741. −1.18221
\(179\) −542178. −1.26476 −0.632381 0.774657i \(-0.717922\pi\)
−0.632381 + 0.774657i \(0.717922\pi\)
\(180\) −388508. −0.893756
\(181\) −653930. −1.48366 −0.741831 0.670587i \(-0.766042\pi\)
−0.741831 + 0.670587i \(0.766042\pi\)
\(182\) 0 0
\(183\) −57894.9 −0.127795
\(184\) 15562.0 0.0338861
\(185\) −653470. −1.40377
\(186\) 106224. 0.225134
\(187\) −1.01839e6 −2.12965
\(188\) 221233. 0.456516
\(189\) 0 0
\(190\) −133925. −0.269141
\(191\) 769535. 1.52632 0.763159 0.646210i \(-0.223647\pi\)
0.763159 + 0.646210i \(0.223647\pi\)
\(192\) 158637. 0.310564
\(193\) 814230. 1.57345 0.786727 0.617301i \(-0.211774\pi\)
0.786727 + 0.617301i \(0.211774\pi\)
\(194\) 168800. 0.322009
\(195\) −569620. −1.07275
\(196\) 0 0
\(197\) −443688. −0.814539 −0.407269 0.913308i \(-0.633519\pi\)
−0.407269 + 0.913308i \(0.633519\pi\)
\(198\) −592957. −1.07488
\(199\) −618941. −1.10794 −0.553970 0.832536i \(-0.686888\pi\)
−0.553970 + 0.832536i \(0.686888\pi\)
\(200\) 660579. 1.16775
\(201\) 214341. 0.374210
\(202\) −1.05542e6 −1.81991
\(203\) 0 0
\(204\) −854372. −1.43738
\(205\) 946463. 1.57296
\(206\) 203739. 0.334508
\(207\) 3481.14 0.00564671
\(208\) −1.29777e6 −2.07988
\(209\) −139110. −0.220289
\(210\) 0 0
\(211\) 769805. 1.19035 0.595175 0.803596i \(-0.297083\pi\)
0.595175 + 0.803596i \(0.297083\pi\)
\(212\) 1.75052e6 2.67502
\(213\) 402126. 0.607314
\(214\) −37787.9 −0.0564050
\(215\) −424477. −0.626264
\(216\) −263971. −0.384966
\(217\) 0 0
\(218\) 85445.7 0.121772
\(219\) −356411. −0.502159
\(220\) 3.50807e6 4.88665
\(221\) −1.25266e6 −1.72525
\(222\) −836724. −1.13946
\(223\) 1.45664e6 1.96151 0.980755 0.195244i \(-0.0625498\pi\)
0.980755 + 0.195244i \(0.0625498\pi\)
\(224\) 0 0
\(225\) 147768. 0.194592
\(226\) −1.00959e6 −1.31484
\(227\) −949905. −1.22353 −0.611767 0.791038i \(-0.709541\pi\)
−0.611767 + 0.791038i \(0.709541\pi\)
\(228\) −116706. −0.148681
\(229\) 312283. 0.393513 0.196757 0.980452i \(-0.436959\pi\)
0.196757 + 0.980452i \(0.436959\pi\)
\(230\) −30261.7 −0.0377202
\(231\) 0 0
\(232\) −2.80505e6 −3.42153
\(233\) 396905. 0.478957 0.239479 0.970902i \(-0.423023\pi\)
0.239479 + 0.970902i \(0.423023\pi\)
\(234\) −729360. −0.870768
\(235\) −228286. −0.269655
\(236\) 1.80334e6 2.10764
\(237\) −208107. −0.240666
\(238\) 0 0
\(239\) −87033.3 −0.0985578 −0.0492789 0.998785i \(-0.515692\pi\)
−0.0492789 + 0.998785i \(0.515692\pi\)
\(240\) 913356. 1.02356
\(241\) −1.57173e6 −1.74315 −0.871576 0.490260i \(-0.836902\pi\)
−0.871576 + 0.490260i \(0.836902\pi\)
\(242\) 3.74222e6 4.10762
\(243\) −59049.0 −0.0641500
\(244\) 438573. 0.471593
\(245\) 0 0
\(246\) 1.21188e6 1.27680
\(247\) −171111. −0.178457
\(248\) −426997. −0.440855
\(249\) −155404. −0.158842
\(250\) 915875. 0.926801
\(251\) −385591. −0.386316 −0.193158 0.981168i \(-0.561873\pi\)
−0.193158 + 0.981168i \(0.561873\pi\)
\(252\) 0 0
\(253\) −31433.2 −0.0308736
\(254\) −1.85616e6 −1.80523
\(255\) 881607. 0.849033
\(256\) −2.11484e6 −2.01687
\(257\) −441809. −0.417255 −0.208627 0.977995i \(-0.566900\pi\)
−0.208627 + 0.977995i \(0.566900\pi\)
\(258\) −543514. −0.508348
\(259\) 0 0
\(260\) 4.31506e6 3.95870
\(261\) −627474. −0.570157
\(262\) −1.24156e6 −1.11742
\(263\) −600264. −0.535122 −0.267561 0.963541i \(-0.586218\pi\)
−0.267561 + 0.963541i \(0.586218\pi\)
\(264\) 2.38355e6 2.10482
\(265\) −1.80632e6 −1.58009
\(266\) 0 0
\(267\) 449368. 0.385766
\(268\) −1.62370e6 −1.38092
\(269\) 325609. 0.274357 0.137178 0.990546i \(-0.456197\pi\)
0.137178 + 0.990546i \(0.456197\pi\)
\(270\) 513316. 0.428525
\(271\) 1.75592e6 1.45239 0.726193 0.687491i \(-0.241288\pi\)
0.726193 + 0.687491i \(0.241288\pi\)
\(272\) 2.00857e6 1.64613
\(273\) 0 0
\(274\) 996090. 0.801534
\(275\) −1.33428e6 −1.06394
\(276\) −26370.7 −0.0208377
\(277\) −342363. −0.268094 −0.134047 0.990975i \(-0.542797\pi\)
−0.134047 + 0.990975i \(0.542797\pi\)
\(278\) −2.54707e6 −1.97664
\(279\) −95517.1 −0.0734633
\(280\) 0 0
\(281\) 930671. 0.703122 0.351561 0.936165i \(-0.385651\pi\)
0.351561 + 0.936165i \(0.385651\pi\)
\(282\) −292304. −0.218883
\(283\) −1.64760e6 −1.22288 −0.611442 0.791290i \(-0.709410\pi\)
−0.611442 + 0.791290i \(0.709410\pi\)
\(284\) −3.04623e6 −2.24113
\(285\) 120426. 0.0878229
\(286\) 6.58582e6 4.76096
\(287\) 0 0
\(288\) 230926. 0.164056
\(289\) 518891. 0.365453
\(290\) 5.45467e6 3.80867
\(291\) −151785. −0.105074
\(292\) 2.69993e6 1.85309
\(293\) −1.14378e6 −0.778347 −0.389174 0.921164i \(-0.627239\pi\)
−0.389174 + 0.921164i \(0.627239\pi\)
\(294\) 0 0
\(295\) −1.86082e6 −1.24495
\(296\) 3.36344e6 2.23128
\(297\) 533188. 0.350743
\(298\) −4.27640e6 −2.78957
\(299\) −38664.1 −0.0250109
\(300\) −1.11939e6 −0.718089
\(301\) 0 0
\(302\) −1.50605e6 −0.950218
\(303\) 949038. 0.593851
\(304\) 274367. 0.170274
\(305\) −452553. −0.278561
\(306\) 1.12884e6 0.689173
\(307\) 2.98831e6 1.80959 0.904795 0.425848i \(-0.140024\pi\)
0.904795 + 0.425848i \(0.140024\pi\)
\(308\) 0 0
\(309\) −183202. −0.109153
\(310\) 830335. 0.490738
\(311\) 2.83751e6 1.66355 0.831775 0.555113i \(-0.187325\pi\)
0.831775 + 0.555113i \(0.187325\pi\)
\(312\) 2.93186e6 1.70513
\(313\) −2.14749e6 −1.23899 −0.619497 0.784999i \(-0.712664\pi\)
−0.619497 + 0.784999i \(0.712664\pi\)
\(314\) −1.65979e6 −0.950010
\(315\) 0 0
\(316\) 1.57648e6 0.888116
\(317\) −2.63357e6 −1.47196 −0.735982 0.677001i \(-0.763279\pi\)
−0.735982 + 0.677001i \(0.763279\pi\)
\(318\) −2.31288e6 −1.28258
\(319\) 5.66583e6 3.11736
\(320\) 1.24003e6 0.676953
\(321\) 33978.9 0.0184054
\(322\) 0 0
\(323\) 264829. 0.141241
\(324\) 447315. 0.236729
\(325\) −1.64122e6 −0.861902
\(326\) 958033. 0.499271
\(327\) −76832.8 −0.0397354
\(328\) −4.87148e6 −2.50021
\(329\) 0 0
\(330\) −4.63503e6 −2.34298
\(331\) 3.28434e6 1.64770 0.823849 0.566809i \(-0.191822\pi\)
0.823849 + 0.566809i \(0.191822\pi\)
\(332\) 1.17724e6 0.586164
\(333\) 752383. 0.371816
\(334\) −4.74631e6 −2.32804
\(335\) 1.67546e6 0.815685
\(336\) 0 0
\(337\) −238202. −0.114254 −0.0571270 0.998367i \(-0.518194\pi\)
−0.0571270 + 0.998367i \(0.518194\pi\)
\(338\) 4.38457e6 2.08754
\(339\) 907821. 0.429043
\(340\) −6.67846e6 −3.13313
\(341\) 862479. 0.401664
\(342\) 154197. 0.0712872
\(343\) 0 0
\(344\) 2.18480e6 0.995441
\(345\) 27211.4 0.0123084
\(346\) −2.59107e6 −1.16356
\(347\) 1.06436e6 0.474532 0.237266 0.971445i \(-0.423749\pi\)
0.237266 + 0.971445i \(0.423749\pi\)
\(348\) 4.75332e6 2.10402
\(349\) −1.85940e6 −0.817163 −0.408582 0.912722i \(-0.633977\pi\)
−0.408582 + 0.912722i \(0.633977\pi\)
\(350\) 0 0
\(351\) 655841. 0.284139
\(352\) −2.08517e6 −0.896983
\(353\) 1.66487e6 0.711123 0.355561 0.934653i \(-0.384290\pi\)
0.355561 + 0.934653i \(0.384290\pi\)
\(354\) −2.38266e6 −1.01054
\(355\) 3.14334e6 1.32379
\(356\) −3.40411e6 −1.42357
\(357\) 0 0
\(358\) −5.42660e6 −2.23779
\(359\) 2.67457e6 1.09526 0.547632 0.836720i \(-0.315530\pi\)
0.547632 + 0.836720i \(0.315530\pi\)
\(360\) −2.06341e6 −0.839131
\(361\) −2.43992e6 −0.985390
\(362\) −6.54511e6 −2.62510
\(363\) −3.36500e6 −1.34035
\(364\) 0 0
\(365\) −2.78600e6 −1.09458
\(366\) −579464. −0.226112
\(367\) 2.14384e6 0.830860 0.415430 0.909625i \(-0.363631\pi\)
0.415430 + 0.909625i \(0.363631\pi\)
\(368\) 61995.8 0.0238640
\(369\) −1.08972e6 −0.416630
\(370\) −6.54051e6 −2.48375
\(371\) 0 0
\(372\) 723573. 0.271097
\(373\) −759867. −0.282791 −0.141395 0.989953i \(-0.545159\pi\)
−0.141395 + 0.989953i \(0.545159\pi\)
\(374\) −1.01929e7 −3.76808
\(375\) −823555. −0.302423
\(376\) 1.17500e6 0.428614
\(377\) 6.96918e6 2.52539
\(378\) 0 0
\(379\) 3.03679e6 1.08597 0.542984 0.839743i \(-0.317294\pi\)
0.542984 + 0.839743i \(0.317294\pi\)
\(380\) −912265. −0.324087
\(381\) 1.66906e6 0.589061
\(382\) 7.70220e6 2.70057
\(383\) −2.50929e6 −0.874086 −0.437043 0.899441i \(-0.643974\pi\)
−0.437043 + 0.899441i \(0.643974\pi\)
\(384\) 2.40885e6 0.833646
\(385\) 0 0
\(386\) 8.14954e6 2.78397
\(387\) 488728. 0.165878
\(388\) 1.14982e6 0.387749
\(389\) −985137. −0.330082 −0.165041 0.986287i \(-0.552776\pi\)
−0.165041 + 0.986287i \(0.552776\pi\)
\(390\) −5.70127e6 −1.89806
\(391\) 59840.7 0.0197950
\(392\) 0 0
\(393\) 1.11641e6 0.364623
\(394\) −4.44082e6 −1.44119
\(395\) −1.62673e6 −0.524593
\(396\) −4.03907e6 −1.29432
\(397\) 2.90510e6 0.925092 0.462546 0.886595i \(-0.346936\pi\)
0.462546 + 0.886595i \(0.346936\pi\)
\(398\) −6.19491e6 −1.96032
\(399\) 0 0
\(400\) 2.63161e6 0.822377
\(401\) −4.23124e6 −1.31404 −0.657018 0.753875i \(-0.728182\pi\)
−0.657018 + 0.753875i \(0.728182\pi\)
\(402\) 2.14532e6 0.662104
\(403\) 1.06088e6 0.325390
\(404\) −7.18927e6 −2.19145
\(405\) −461574. −0.139831
\(406\) 0 0
\(407\) −6.79370e6 −2.03292
\(408\) −4.53767e6 −1.34953
\(409\) 4.13890e6 1.22342 0.611711 0.791081i \(-0.290482\pi\)
0.611711 + 0.791081i \(0.290482\pi\)
\(410\) 9.47304e6 2.78311
\(411\) −895684. −0.261547
\(412\) 1.38782e6 0.402799
\(413\) 0 0
\(414\) 34842.3 0.00999094
\(415\) −1.21477e6 −0.346236
\(416\) −2.56483e6 −0.726651
\(417\) 2.29033e6 0.644996
\(418\) −1.39234e6 −0.389766
\(419\) 5.99852e6 1.66920 0.834601 0.550855i \(-0.185698\pi\)
0.834601 + 0.550855i \(0.185698\pi\)
\(420\) 0 0
\(421\) −1.00759e6 −0.277062 −0.138531 0.990358i \(-0.544238\pi\)
−0.138531 + 0.990358i \(0.544238\pi\)
\(422\) 7.70489e6 2.10613
\(423\) 262840. 0.0714235
\(424\) 9.29722e6 2.51153
\(425\) 2.54013e6 0.682156
\(426\) 4.02484e6 1.07454
\(427\) 0 0
\(428\) −257401. −0.0679205
\(429\) −5.92197e6 −1.55354
\(430\) −4.24854e6 −1.10807
\(431\) 303149. 0.0786073 0.0393037 0.999227i \(-0.487486\pi\)
0.0393037 + 0.999227i \(0.487486\pi\)
\(432\) −1.05161e6 −0.271109
\(433\) 3.97130e6 1.01792 0.508960 0.860790i \(-0.330030\pi\)
0.508960 + 0.860790i \(0.330030\pi\)
\(434\) 0 0
\(435\) −4.90484e6 −1.24280
\(436\) 582033. 0.146633
\(437\) 8174.14 0.00204757
\(438\) −3.56728e6 −0.888489
\(439\) 3.74799e6 0.928192 0.464096 0.885785i \(-0.346379\pi\)
0.464096 + 0.885785i \(0.346379\pi\)
\(440\) 1.86318e7 4.58798
\(441\) 0 0
\(442\) −1.25377e7 −3.05255
\(443\) −1.31077e6 −0.317334 −0.158667 0.987332i \(-0.550720\pi\)
−0.158667 + 0.987332i \(0.550720\pi\)
\(444\) −5.69954e6 −1.37209
\(445\) 3.51262e6 0.840874
\(446\) 1.45794e7 3.47057
\(447\) 3.84534e6 0.910261
\(448\) 0 0
\(449\) 6.31311e6 1.47784 0.738920 0.673793i \(-0.235336\pi\)
0.738920 + 0.673793i \(0.235336\pi\)
\(450\) 1.47899e6 0.344298
\(451\) 9.83976e6 2.27794
\(452\) −6.87704e6 −1.58327
\(453\) 1.35424e6 0.310064
\(454\) −9.50750e6 −2.16484
\(455\) 0 0
\(456\) −619837. −0.139594
\(457\) −583476. −0.130687 −0.0653435 0.997863i \(-0.520814\pi\)
−0.0653435 + 0.997863i \(0.520814\pi\)
\(458\) 3.12561e6 0.696259
\(459\) −1.01505e6 −0.224883
\(460\) −206135. −0.0454210
\(461\) −2.93381e6 −0.642953 −0.321476 0.946918i \(-0.604179\pi\)
−0.321476 + 0.946918i \(0.604179\pi\)
\(462\) 0 0
\(463\) 3.76716e6 0.816699 0.408349 0.912826i \(-0.366105\pi\)
0.408349 + 0.912826i \(0.366105\pi\)
\(464\) −1.11747e7 −2.40958
\(465\) −746638. −0.160132
\(466\) 3.97258e6 0.847438
\(467\) −2.90596e6 −0.616592 −0.308296 0.951290i \(-0.599759\pi\)
−0.308296 + 0.951290i \(0.599759\pi\)
\(468\) −4.96821e6 −1.04854
\(469\) 0 0
\(470\) −2.28489e6 −0.477111
\(471\) 1.49248e6 0.309996
\(472\) 9.57774e6 1.97883
\(473\) −4.41301e6 −0.906947
\(474\) −2.08292e6 −0.425820
\(475\) 346977. 0.0705613
\(476\) 0 0
\(477\) 2.07974e6 0.418517
\(478\) −871107. −0.174382
\(479\) −7.06622e6 −1.40718 −0.703588 0.710608i \(-0.748420\pi\)
−0.703588 + 0.710608i \(0.748420\pi\)
\(480\) 1.80510e6 0.357601
\(481\) −8.35651e6 −1.64688
\(482\) −1.57313e7 −3.08423
\(483\) 0 0
\(484\) 2.54910e7 4.94622
\(485\) −1.18647e6 −0.229036
\(486\) −591015. −0.113503
\(487\) 2.72981e6 0.521566 0.260783 0.965397i \(-0.416019\pi\)
0.260783 + 0.965397i \(0.416019\pi\)
\(488\) 2.32931e6 0.442770
\(489\) −861464. −0.162916
\(490\) 0 0
\(491\) −1.02859e7 −1.92548 −0.962740 0.270428i \(-0.912835\pi\)
−0.962740 + 0.270428i \(0.912835\pi\)
\(492\) 8.25502e6 1.53746
\(493\) −1.07863e7 −1.99873
\(494\) −1.71263e6 −0.315751
\(495\) 4.16782e6 0.764533
\(496\) −1.70107e6 −0.310469
\(497\) 0 0
\(498\) −1.55543e6 −0.281045
\(499\) −3.63591e6 −0.653674 −0.326837 0.945081i \(-0.605983\pi\)
−0.326837 + 0.945081i \(0.605983\pi\)
\(500\) 6.23870e6 1.11601
\(501\) 4.26788e6 0.759658
\(502\) −3.85934e6 −0.683524
\(503\) −1.65499e6 −0.291658 −0.145829 0.989310i \(-0.546585\pi\)
−0.145829 + 0.989310i \(0.546585\pi\)
\(504\) 0 0
\(505\) 7.41845e6 1.29445
\(506\) −314612. −0.0546259
\(507\) −3.94261e6 −0.681183
\(508\) −1.26437e7 −2.17377
\(509\) 731110. 0.125080 0.0625401 0.998042i \(-0.480080\pi\)
0.0625401 + 0.998042i \(0.480080\pi\)
\(510\) 8.82391e6 1.50223
\(511\) 0 0
\(512\) −1.26024e7 −2.12460
\(513\) −138654. −0.0232616
\(514\) −4.42202e6 −0.738265
\(515\) −1.43206e6 −0.237926
\(516\) −3.70227e6 −0.612130
\(517\) −2.37334e6 −0.390511
\(518\) 0 0
\(519\) 2.32989e6 0.379679
\(520\) 2.29178e7 3.71675
\(521\) −4.73095e6 −0.763579 −0.381790 0.924249i \(-0.624692\pi\)
−0.381790 + 0.924249i \(0.624692\pi\)
\(522\) −6.28032e6 −1.00880
\(523\) 9.72563e6 1.55476 0.777380 0.629031i \(-0.216548\pi\)
0.777380 + 0.629031i \(0.216548\pi\)
\(524\) −8.45719e6 −1.34554
\(525\) 0 0
\(526\) −6.00798e6 −0.946812
\(527\) −1.64194e6 −0.257531
\(528\) 9.49557e6 1.48230
\(529\) −6.43450e6 −0.999713
\(530\) −1.80793e7 −2.79571
\(531\) 2.14249e6 0.329748
\(532\) 0 0
\(533\) 1.21033e7 1.84538
\(534\) 4.49767e6 0.682550
\(535\) 265606. 0.0401193
\(536\) −8.62367e6 −1.29652
\(537\) 4.87960e6 0.730211
\(538\) 3.25898e6 0.485430
\(539\) 0 0
\(540\) 3.49658e6 0.516010
\(541\) 8.19002e6 1.20307 0.601536 0.798845i \(-0.294556\pi\)
0.601536 + 0.798845i \(0.294556\pi\)
\(542\) 1.75748e7 2.56976
\(543\) 5.88537e6 0.856592
\(544\) 3.96962e6 0.575111
\(545\) −600587. −0.0866133
\(546\) 0 0
\(547\) −4.17136e6 −0.596087 −0.298043 0.954552i \(-0.596334\pi\)
−0.298043 + 0.954552i \(0.596334\pi\)
\(548\) 6.78510e6 0.965172
\(549\) 521054. 0.0737823
\(550\) −1.33547e7 −1.88247
\(551\) −1.47339e6 −0.206746
\(552\) −140058. −0.0195641
\(553\) 0 0
\(554\) −3.42668e6 −0.474350
\(555\) 5.88123e6 0.810468
\(556\) −1.73500e7 −2.38019
\(557\) −1.26658e6 −0.172980 −0.0864899 0.996253i \(-0.527565\pi\)
−0.0864899 + 0.996253i \(0.527565\pi\)
\(558\) −956020. −0.129981
\(559\) −5.42817e6 −0.734723
\(560\) 0 0
\(561\) 9.16549e6 1.22956
\(562\) 9.31499e6 1.24406
\(563\) 6.37431e6 0.847544 0.423772 0.905769i \(-0.360706\pi\)
0.423772 + 0.905769i \(0.360706\pi\)
\(564\) −1.99110e6 −0.263570
\(565\) 7.09626e6 0.935208
\(566\) −1.64906e7 −2.16369
\(567\) 0 0
\(568\) −1.61789e7 −2.10416
\(569\) 7.64040e6 0.989317 0.494659 0.869087i \(-0.335293\pi\)
0.494659 + 0.869087i \(0.335293\pi\)
\(570\) 1.20533e6 0.155388
\(571\) 790914. 0.101517 0.0507585 0.998711i \(-0.483836\pi\)
0.0507585 + 0.998711i \(0.483836\pi\)
\(572\) 4.48608e7 5.73294
\(573\) −6.92582e6 −0.881221
\(574\) 0 0
\(575\) 78402.8 0.00988922
\(576\) −1.42773e6 −0.179304
\(577\) 8.55670e6 1.06996 0.534979 0.844865i \(-0.320320\pi\)
0.534979 + 0.844865i \(0.320320\pi\)
\(578\) 5.19352e6 0.646610
\(579\) −7.32807e6 −0.908434
\(580\) 3.71558e7 4.58623
\(581\) 0 0
\(582\) −1.51920e6 −0.185912
\(583\) −1.87792e7 −2.28826
\(584\) 1.43396e7 1.73983
\(585\) 5.12658e6 0.619353
\(586\) −1.14480e7 −1.37716
\(587\) 1.17153e7 1.40333 0.701663 0.712509i \(-0.252441\pi\)
0.701663 + 0.712509i \(0.252441\pi\)
\(588\) 0 0
\(589\) −224286. −0.0266387
\(590\) −1.86248e7 −2.20273
\(591\) 3.99319e6 0.470274
\(592\) 1.33992e7 1.57136
\(593\) −1.55617e7 −1.81728 −0.908639 0.417583i \(-0.862877\pi\)
−0.908639 + 0.417583i \(0.862877\pi\)
\(594\) 5.33662e6 0.620583
\(595\) 0 0
\(596\) −2.91297e7 −3.35908
\(597\) 5.57047e6 0.639670
\(598\) −386984. −0.0442528
\(599\) −9.18930e6 −1.04644 −0.523221 0.852197i \(-0.675270\pi\)
−0.523221 + 0.852197i \(0.675270\pi\)
\(600\) −5.94521e6 −0.674200
\(601\) −1.61226e7 −1.82075 −0.910373 0.413789i \(-0.864205\pi\)
−0.910373 + 0.413789i \(0.864205\pi\)
\(602\) 0 0
\(603\) −1.92907e6 −0.216050
\(604\) −1.02588e7 −1.14421
\(605\) −2.63036e7 −2.92164
\(606\) 9.49882e6 1.05072
\(607\) −1.04056e7 −1.14629 −0.573144 0.819455i \(-0.694276\pi\)
−0.573144 + 0.819455i \(0.694276\pi\)
\(608\) 542243. 0.0594887
\(609\) 0 0
\(610\) −4.52956e6 −0.492868
\(611\) −2.91929e6 −0.316355
\(612\) 7.68935e6 0.829871
\(613\) −1.13921e7 −1.22449 −0.612243 0.790670i \(-0.709732\pi\)
−0.612243 + 0.790670i \(0.709732\pi\)
\(614\) 2.99097e7 3.20178
\(615\) −8.51816e6 −0.908151
\(616\) 0 0
\(617\) 6.20517e6 0.656206 0.328103 0.944642i \(-0.393591\pi\)
0.328103 + 0.944642i \(0.393591\pi\)
\(618\) −1.83365e6 −0.193128
\(619\) 9.85608e6 1.03390 0.516949 0.856016i \(-0.327068\pi\)
0.516949 + 0.856016i \(0.327068\pi\)
\(620\) 5.65602e6 0.590925
\(621\) −31330.2 −0.00326013
\(622\) 2.84003e7 2.94338
\(623\) 0 0
\(624\) 1.16799e7 1.20082
\(625\) −1.21385e7 −1.24298
\(626\) −2.14940e7 −2.19220
\(627\) 1.25199e6 0.127184
\(628\) −1.13060e7 −1.14396
\(629\) 1.29335e7 1.30343
\(630\) 0 0
\(631\) 7.26873e6 0.726750 0.363375 0.931643i \(-0.381624\pi\)
0.363375 + 0.931643i \(0.381624\pi\)
\(632\) 8.37284e6 0.833835
\(633\) −6.92824e6 −0.687248
\(634\) −2.63591e7 −2.60440
\(635\) 1.30467e7 1.28401
\(636\) −1.57547e7 −1.54443
\(637\) 0 0
\(638\) 5.67087e7 5.51566
\(639\) −3.61913e6 −0.350633
\(640\) 1.88295e7 1.81714
\(641\) 8.30119e6 0.797986 0.398993 0.916954i \(-0.369360\pi\)
0.398993 + 0.916954i \(0.369360\pi\)
\(642\) 340091. 0.0325655
\(643\) 4.14322e6 0.395194 0.197597 0.980283i \(-0.436686\pi\)
0.197597 + 0.980283i \(0.436686\pi\)
\(644\) 0 0
\(645\) 3.82029e6 0.361574
\(646\) 2.65065e6 0.249903
\(647\) 1.47906e7 1.38907 0.694535 0.719459i \(-0.255610\pi\)
0.694535 + 0.719459i \(0.255610\pi\)
\(648\) 2.37574e6 0.222260
\(649\) −1.93458e7 −1.80291
\(650\) −1.64268e7 −1.52500
\(651\) 0 0
\(652\) 6.52587e6 0.601200
\(653\) −6.08931e6 −0.558837 −0.279419 0.960169i \(-0.590142\pi\)
−0.279419 + 0.960169i \(0.590142\pi\)
\(654\) −769011. −0.0703053
\(655\) 8.72679e6 0.794788
\(656\) −1.94070e7 −1.76075
\(657\) 3.20770e6 0.289922
\(658\) 0 0
\(659\) 4.14447e6 0.371754 0.185877 0.982573i \(-0.440487\pi\)
0.185877 + 0.982573i \(0.440487\pi\)
\(660\) −3.15726e7 −2.82131
\(661\) −1.80378e7 −1.60576 −0.802881 0.596140i \(-0.796700\pi\)
−0.802881 + 0.596140i \(0.796700\pi\)
\(662\) 3.28726e7 2.91534
\(663\) 1.12739e7 0.996071
\(664\) 6.25245e6 0.550339
\(665\) 0 0
\(666\) 7.53052e6 0.657869
\(667\) −332926. −0.0289756
\(668\) −3.23306e7 −2.80332
\(669\) −1.31098e7 −1.13248
\(670\) 1.67695e7 1.44322
\(671\) −4.70490e6 −0.403408
\(672\) 0 0
\(673\) 5.62885e6 0.479051 0.239526 0.970890i \(-0.423008\pi\)
0.239526 + 0.970890i \(0.423008\pi\)
\(674\) −2.38414e6 −0.202154
\(675\) −1.32991e6 −0.112347
\(676\) 2.98666e7 2.51373
\(677\) 1.50877e7 1.26518 0.632588 0.774488i \(-0.281993\pi\)
0.632588 + 0.774488i \(0.281993\pi\)
\(678\) 9.08628e6 0.759123
\(679\) 0 0
\(680\) −3.54700e7 −2.94164
\(681\) 8.54915e6 0.706407
\(682\) 8.63246e6 0.710679
\(683\) 7.20414e6 0.590922 0.295461 0.955355i \(-0.404527\pi\)
0.295461 + 0.955355i \(0.404527\pi\)
\(684\) 1.05035e6 0.0858408
\(685\) −7.00139e6 −0.570109
\(686\) 0 0
\(687\) −2.81055e6 −0.227195
\(688\) 8.70378e6 0.701031
\(689\) −2.30991e7 −1.85373
\(690\) 272356. 0.0217778
\(691\) −9.08625e6 −0.723918 −0.361959 0.932194i \(-0.617892\pi\)
−0.361959 + 0.932194i \(0.617892\pi\)
\(692\) −1.76497e7 −1.40111
\(693\) 0 0
\(694\) 1.06531e7 0.839607
\(695\) 1.79030e7 1.40593
\(696\) 2.52454e7 1.97542
\(697\) −1.87324e7 −1.46053
\(698\) −1.86105e7 −1.44584
\(699\) −3.57215e6 −0.276526
\(700\) 0 0
\(701\) −1.41913e7 −1.09075 −0.545376 0.838192i \(-0.683613\pi\)
−0.545376 + 0.838192i \(0.683613\pi\)
\(702\) 6.56424e6 0.502738
\(703\) 1.76669e6 0.134825
\(704\) 1.28918e7 0.980354
\(705\) 2.05457e6 0.155686
\(706\) 1.66635e7 1.25822
\(707\) 0 0
\(708\) −1.62300e7 −1.21685
\(709\) −2.21070e6 −0.165164 −0.0825819 0.996584i \(-0.526317\pi\)
−0.0825819 + 0.996584i \(0.526317\pi\)
\(710\) 3.14613e7 2.34224
\(711\) 1.87296e6 0.138949
\(712\) −1.80796e7 −1.33656
\(713\) −50679.5 −0.00373343
\(714\) 0 0
\(715\) −4.62909e7 −3.38634
\(716\) −3.69645e7 −2.69465
\(717\) 783300. 0.0569024
\(718\) 2.67695e7 1.93789
\(719\) 2.99082e6 0.215758 0.107879 0.994164i \(-0.465594\pi\)
0.107879 + 0.994164i \(0.465594\pi\)
\(720\) −8.22021e6 −0.590951
\(721\) 0 0
\(722\) −2.44209e7 −1.74349
\(723\) 1.41456e7 1.00641
\(724\) −4.45836e7 −3.16103
\(725\) −1.41321e7 −0.998530
\(726\) −3.36799e7 −2.37154
\(727\) −8.38066e6 −0.588088 −0.294044 0.955792i \(-0.595001\pi\)
−0.294044 + 0.955792i \(0.595001\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) −2.78847e7 −1.93669
\(731\) 8.40123e6 0.581499
\(732\) −3.94715e6 −0.272274
\(733\) −7.70245e6 −0.529504 −0.264752 0.964317i \(-0.585290\pi\)
−0.264752 + 0.964317i \(0.585290\pi\)
\(734\) 2.14575e7 1.47007
\(735\) 0 0
\(736\) 122525. 0.00833739
\(737\) 1.74187e7 1.18126
\(738\) −1.09069e7 −0.737160
\(739\) −7.59899e6 −0.511852 −0.255926 0.966696i \(-0.582380\pi\)
−0.255926 + 0.966696i \(0.582380\pi\)
\(740\) −4.45522e7 −2.99082
\(741\) 1.53999e6 0.103032
\(742\) 0 0
\(743\) −2.40124e7 −1.59574 −0.797871 0.602828i \(-0.794041\pi\)
−0.797871 + 0.602828i \(0.794041\pi\)
\(744\) 3.84298e6 0.254528
\(745\) 3.00583e7 1.98414
\(746\) −7.60543e6 −0.500353
\(747\) 1.39864e6 0.0917074
\(748\) −6.94316e7 −4.53736
\(749\) 0 0
\(750\) −8.24288e6 −0.535089
\(751\) 1.69479e7 1.09652 0.548259 0.836308i \(-0.315291\pi\)
0.548259 + 0.836308i \(0.315291\pi\)
\(752\) 4.68094e6 0.301848
\(753\) 3.47032e6 0.223040
\(754\) 6.97538e7 4.46827
\(755\) 1.05859e7 0.675863
\(756\) 0 0
\(757\) 4.56402e6 0.289473 0.144736 0.989470i \(-0.453767\pi\)
0.144736 + 0.989470i \(0.453767\pi\)
\(758\) 3.03949e7 1.92145
\(759\) 282899. 0.0178249
\(760\) −4.84514e6 −0.304279
\(761\) −4.74018e6 −0.296711 −0.148355 0.988934i \(-0.547398\pi\)
−0.148355 + 0.988934i \(0.547398\pi\)
\(762\) 1.67055e7 1.04225
\(763\) 0 0
\(764\) 5.24653e7 3.25191
\(765\) −7.93446e6 −0.490189
\(766\) −2.51152e7 −1.54655
\(767\) −2.37960e7 −1.46055
\(768\) 1.90335e7 1.16444
\(769\) −1.39469e7 −0.850476 −0.425238 0.905082i \(-0.639810\pi\)
−0.425238 + 0.905082i \(0.639810\pi\)
\(770\) 0 0
\(771\) 3.97628e6 0.240902
\(772\) 5.55125e7 3.35234
\(773\) 1.49561e7 0.900264 0.450132 0.892962i \(-0.351377\pi\)
0.450132 + 0.892962i \(0.351377\pi\)
\(774\) 4.89162e6 0.293495
\(775\) −2.15125e6 −0.128658
\(776\) 6.10683e6 0.364050
\(777\) 0 0
\(778\) −9.86012e6 −0.584028
\(779\) −2.55881e6 −0.151075
\(780\) −3.88355e7 −2.28556
\(781\) 3.26793e7 1.91710
\(782\) 598940. 0.0350240
\(783\) 5.64727e6 0.329180
\(784\) 0 0
\(785\) 1.16664e7 0.675716
\(786\) 1.11741e7 0.645141
\(787\) 2.18273e7 1.25621 0.628105 0.778128i \(-0.283831\pi\)
0.628105 + 0.778128i \(0.283831\pi\)
\(788\) −3.02497e7 −1.73542
\(789\) 5.40238e6 0.308953
\(790\) −1.62818e7 −0.928183
\(791\) 0 0
\(792\) −2.14520e7 −1.21522
\(793\) −5.78721e6 −0.326803
\(794\) 2.90768e7 1.63680
\(795\) 1.62569e7 0.912263
\(796\) −4.21981e7 −2.36053
\(797\) −8.79229e6 −0.490294 −0.245147 0.969486i \(-0.578836\pi\)
−0.245147 + 0.969486i \(0.578836\pi\)
\(798\) 0 0
\(799\) 4.51822e6 0.250381
\(800\) 5.20096e6 0.287315
\(801\) −4.04431e6 −0.222722
\(802\) −4.23501e7 −2.32497
\(803\) −2.89642e7 −1.58516
\(804\) 1.46133e7 0.797276
\(805\) 0 0
\(806\) 1.06183e7 0.575725
\(807\) −2.93048e6 −0.158400
\(808\) −3.81831e7 −2.05751
\(809\) −2.87451e6 −0.154416 −0.0772081 0.997015i \(-0.524601\pi\)
−0.0772081 + 0.997015i \(0.524601\pi\)
\(810\) −4.61985e6 −0.247409
\(811\) −660356. −0.0352554 −0.0176277 0.999845i \(-0.505611\pi\)
−0.0176277 + 0.999845i \(0.505611\pi\)
\(812\) 0 0
\(813\) −1.58033e7 −0.838535
\(814\) −6.79974e7 −3.59693
\(815\) −6.73389e6 −0.355118
\(816\) −1.80771e7 −0.950394
\(817\) 1.14759e6 0.0601496
\(818\) 4.14258e7 2.16465
\(819\) 0 0
\(820\) 6.45278e7 3.35129
\(821\) −3.05738e7 −1.58304 −0.791519 0.611144i \(-0.790710\pi\)
−0.791519 + 0.611144i \(0.790710\pi\)
\(822\) −8.96481e6 −0.462766
\(823\) 1.11281e7 0.572693 0.286346 0.958126i \(-0.407559\pi\)
0.286346 + 0.958126i \(0.407559\pi\)
\(824\) 7.37085e6 0.378181
\(825\) 1.20085e7 0.614264
\(826\) 0 0
\(827\) −2.78304e7 −1.41500 −0.707500 0.706714i \(-0.750177\pi\)
−0.707500 + 0.706714i \(0.750177\pi\)
\(828\) 237337. 0.0120307
\(829\) 7.06133e6 0.356862 0.178431 0.983952i \(-0.442898\pi\)
0.178431 + 0.983952i \(0.442898\pi\)
\(830\) −1.21585e7 −0.612609
\(831\) 3.08127e6 0.154784
\(832\) 1.58574e7 0.794190
\(833\) 0 0
\(834\) 2.29236e7 1.14122
\(835\) 3.33612e7 1.65587
\(836\) −9.48422e6 −0.469338
\(837\) 859654. 0.0424141
\(838\) 6.00385e7 2.95338
\(839\) −2.27299e7 −1.11479 −0.557395 0.830247i \(-0.688199\pi\)
−0.557395 + 0.830247i \(0.688199\pi\)
\(840\) 0 0
\(841\) 3.94986e7 1.92571
\(842\) −1.00848e7 −0.490217
\(843\) −8.37604e6 −0.405947
\(844\) 5.24837e7 2.53611
\(845\) −3.08186e7 −1.48481
\(846\) 2.63074e6 0.126372
\(847\) 0 0
\(848\) 3.70382e7 1.76872
\(849\) 1.48284e7 0.706032
\(850\) 2.54239e7 1.20696
\(851\) 399200. 0.0188958
\(852\) 2.74161e7 1.29392
\(853\) −7.94309e6 −0.373781 −0.186890 0.982381i \(-0.559841\pi\)
−0.186890 + 0.982381i \(0.559841\pi\)
\(854\) 0 0
\(855\) −1.08383e6 −0.0507046
\(856\) −1.36709e6 −0.0637693
\(857\) −9.91766e6 −0.461272 −0.230636 0.973040i \(-0.574081\pi\)
−0.230636 + 0.973040i \(0.574081\pi\)
\(858\) −5.92724e7 −2.74874
\(859\) −61754.4 −0.00285552 −0.00142776 0.999999i \(-0.500454\pi\)
−0.00142776 + 0.999999i \(0.500454\pi\)
\(860\) −2.89399e7 −1.33429
\(861\) 0 0
\(862\) 3.03419e6 0.139083
\(863\) 3.99007e7 1.82370 0.911851 0.410522i \(-0.134653\pi\)
0.911851 + 0.410522i \(0.134653\pi\)
\(864\) −2.07834e6 −0.0947177
\(865\) 1.82123e7 0.827607
\(866\) 3.97484e7 1.80104
\(867\) −4.67002e6 −0.210994
\(868\) 0 0
\(869\) −1.69120e7 −0.759708
\(870\) −4.90920e7 −2.19894
\(871\) 2.14256e7 0.956949
\(872\) 3.09125e6 0.137671
\(873\) 1.36606e6 0.0606647
\(874\) 81814.0 0.00362284
\(875\) 0 0
\(876\) −2.42994e7 −1.06988
\(877\) −1.89184e7 −0.830586 −0.415293 0.909688i \(-0.636321\pi\)
−0.415293 + 0.909688i \(0.636321\pi\)
\(878\) 3.75133e7 1.64229
\(879\) 1.02940e7 0.449379
\(880\) 7.42250e7 3.23105
\(881\) 9.12061e6 0.395899 0.197949 0.980212i \(-0.436572\pi\)
0.197949 + 0.980212i \(0.436572\pi\)
\(882\) 0 0
\(883\) −4.49665e7 −1.94083 −0.970414 0.241449i \(-0.922377\pi\)
−0.970414 + 0.241449i \(0.922377\pi\)
\(884\) −8.54035e7 −3.67574
\(885\) 1.67474e7 0.718769
\(886\) −1.31193e7 −0.561472
\(887\) 2.82985e7 1.20769 0.603845 0.797102i \(-0.293635\pi\)
0.603845 + 0.797102i \(0.293635\pi\)
\(888\) −3.02709e7 −1.28823
\(889\) 0 0
\(890\) 3.51574e7 1.48779
\(891\) −4.79869e6 −0.202502
\(892\) 9.93107e7 4.17911
\(893\) 617181. 0.0258990
\(894\) 3.84876e7 1.61056
\(895\) 3.81429e7 1.59168
\(896\) 0 0
\(897\) 347977. 0.0144401
\(898\) 6.31873e7 2.61480
\(899\) 9.13496e6 0.376971
\(900\) 1.00745e7 0.414589
\(901\) 3.57507e7 1.46714
\(902\) 9.84850e7 4.03045
\(903\) 0 0
\(904\) −3.65247e7 −1.48650
\(905\) 4.60048e7 1.86716
\(906\) 1.35545e7 0.548608
\(907\) −768537. −0.0310204 −0.0155102 0.999880i \(-0.504937\pi\)
−0.0155102 + 0.999880i \(0.504937\pi\)
\(908\) −6.47626e7 −2.60681
\(909\) −8.54135e6 −0.342860
\(910\) 0 0
\(911\) 3.45594e6 0.137965 0.0689827 0.997618i \(-0.478025\pi\)
0.0689827 + 0.997618i \(0.478025\pi\)
\(912\) −2.46930e6 −0.0983076
\(913\) −1.26291e7 −0.501414
\(914\) −5.83994e6 −0.231229
\(915\) 4.07298e6 0.160827
\(916\) 2.12908e7 0.838404
\(917\) 0 0
\(918\) −1.01595e7 −0.397894
\(919\) 5.65327e6 0.220806 0.110403 0.993887i \(-0.464786\pi\)
0.110403 + 0.993887i \(0.464786\pi\)
\(920\) −1.09481e6 −0.0426450
\(921\) −2.68948e7 −1.04477
\(922\) −2.93641e7 −1.13760
\(923\) 4.01967e7 1.55305
\(924\) 0 0
\(925\) 1.69453e7 0.651171
\(926\) 3.77051e7 1.44502
\(927\) 1.64882e6 0.0630194
\(928\) −2.20851e7 −0.841839
\(929\) 2.42253e7 0.920937 0.460469 0.887676i \(-0.347681\pi\)
0.460469 + 0.887676i \(0.347681\pi\)
\(930\) −7.47302e6 −0.283327
\(931\) 0 0
\(932\) 2.70602e7 1.02045
\(933\) −2.55376e7 −0.960451
\(934\) −2.90855e7 −1.09096
\(935\) 7.16448e7 2.68013
\(936\) −2.63867e7 −0.984455
\(937\) −3.02477e7 −1.12549 −0.562746 0.826630i \(-0.690255\pi\)
−0.562746 + 0.826630i \(0.690255\pi\)
\(938\) 0 0
\(939\) 1.93274e7 0.715334
\(940\) −1.55640e7 −0.574517
\(941\) −1.38949e7 −0.511541 −0.255770 0.966738i \(-0.582329\pi\)
−0.255770 + 0.966738i \(0.582329\pi\)
\(942\) 1.49381e7 0.548489
\(943\) −578187. −0.0211733
\(944\) 3.81557e7 1.39357
\(945\) 0 0
\(946\) −4.41693e7 −1.60470
\(947\) −1.78680e7 −0.647441 −0.323721 0.946153i \(-0.604934\pi\)
−0.323721 + 0.946153i \(0.604934\pi\)
\(948\) −1.41883e7 −0.512754
\(949\) −3.56271e7 −1.28415
\(950\) 3.47286e6 0.124847
\(951\) 2.37022e7 0.849839
\(952\) 0 0
\(953\) 1.37348e7 0.489880 0.244940 0.969538i \(-0.421232\pi\)
0.244940 + 0.969538i \(0.421232\pi\)
\(954\) 2.08159e7 0.740498
\(955\) −5.41378e7 −1.92084
\(956\) −5.93375e6 −0.209983
\(957\) −5.09924e7 −1.79981
\(958\) −7.07250e7 −2.48977
\(959\) 0 0
\(960\) −1.11603e7 −0.390839
\(961\) −2.72386e7 −0.951428
\(962\) −8.36394e7 −2.91389
\(963\) −305810. −0.0106264
\(964\) −1.07157e8 −3.71389
\(965\) −5.72821e7 −1.98016
\(966\) 0 0
\(967\) −7.63195e6 −0.262464 −0.131232 0.991352i \(-0.541893\pi\)
−0.131232 + 0.991352i \(0.541893\pi\)
\(968\) 1.35386e8 4.64391
\(969\) −2.38346e6 −0.0815454
\(970\) −1.18753e7 −0.405242
\(971\) 2.22480e7 0.757257 0.378629 0.925549i \(-0.376396\pi\)
0.378629 + 0.925549i \(0.376396\pi\)
\(972\) −4.02584e6 −0.136675
\(973\) 0 0
\(974\) 2.73223e7 0.922827
\(975\) 1.47710e7 0.497619
\(976\) 9.27949e6 0.311817
\(977\) −1.82385e7 −0.611297 −0.305648 0.952144i \(-0.598873\pi\)
−0.305648 + 0.952144i \(0.598873\pi\)
\(978\) −8.62230e6 −0.288254
\(979\) 3.65184e7 1.21774
\(980\) 0 0
\(981\) 691495. 0.0229412
\(982\) −1.02951e8 −3.40683
\(983\) 3.84459e7 1.26902 0.634508 0.772917i \(-0.281203\pi\)
0.634508 + 0.772917i \(0.281203\pi\)
\(984\) 4.38433e7 1.44350
\(985\) 3.12140e7 1.02508
\(986\) −1.07959e8 −3.53643
\(987\) 0 0
\(988\) −1.16660e7 −0.380214
\(989\) 259309. 0.00843000
\(990\) 4.17153e7 1.35272
\(991\) −2.49658e7 −0.807533 −0.403767 0.914862i \(-0.632299\pi\)
−0.403767 + 0.914862i \(0.632299\pi\)
\(992\) −3.36189e6 −0.108469
\(993\) −2.95590e7 −0.951299
\(994\) 0 0
\(995\) 4.35433e7 1.39432
\(996\) −1.05951e7 −0.338422
\(997\) −3.22763e7 −1.02836 −0.514181 0.857682i \(-0.671904\pi\)
−0.514181 + 0.857682i \(0.671904\pi\)
\(998\) −3.63914e7 −1.15657
\(999\) −6.77145e6 −0.214668
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 147.6.a.n.1.6 6
3.2 odd 2 441.6.a.bb.1.1 6
7.2 even 3 147.6.e.q.67.1 12
7.3 odd 6 147.6.e.p.79.1 12
7.4 even 3 147.6.e.q.79.1 12
7.5 odd 6 147.6.e.p.67.1 12
7.6 odd 2 147.6.a.o.1.6 yes 6
21.20 even 2 441.6.a.ba.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.n.1.6 6 1.1 even 1 trivial
147.6.a.o.1.6 yes 6 7.6 odd 2
147.6.e.p.67.1 12 7.5 odd 6
147.6.e.p.79.1 12 7.3 odd 6
147.6.e.q.67.1 12 7.2 even 3
147.6.e.q.79.1 12 7.4 even 3
441.6.a.ba.1.1 6 21.20 even 2
441.6.a.bb.1.1 6 3.2 odd 2