Properties

Label 147.6.a.o.1.4
Level $147$
Weight $6$
Character 147.1
Self dual yes
Analytic conductor $23.576$
Analytic rank $0$
Dimension $6$
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] = [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: \(0\)
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 1.4
Root \(-5.10089\) 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)
 
\(f(q)\) \(=\) \(q+3.38033 q^{2} +9.00000 q^{3} -20.5734 q^{4} -54.5253 q^{5} +30.4230 q^{6} -177.715 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+3.38033 q^{2} +9.00000 q^{3} -20.5734 q^{4} -54.5253 q^{5} +30.4230 q^{6} -177.715 q^{8} +81.0000 q^{9} -184.313 q^{10} +481.911 q^{11} -185.160 q^{12} +512.622 q^{13} -490.728 q^{15} +57.6111 q^{16} +590.744 q^{17} +273.807 q^{18} +2451.02 q^{19} +1121.77 q^{20} +1629.02 q^{22} +1774.90 q^{23} -1599.44 q^{24} -151.993 q^{25} +1732.83 q^{26} +729.000 q^{27} -4246.44 q^{29} -1658.82 q^{30} +9767.37 q^{31} +5881.64 q^{32} +4337.20 q^{33} +1996.91 q^{34} -1666.44 q^{36} -9969.65 q^{37} +8285.24 q^{38} +4613.60 q^{39} +9689.98 q^{40} -3377.54 q^{41} -18223.4 q^{43} -9914.53 q^{44} -4416.55 q^{45} +5999.76 q^{46} +1320.64 q^{47} +518.500 q^{48} -513.788 q^{50} +5316.69 q^{51} -10546.4 q^{52} +34837.0 q^{53} +2464.26 q^{54} -26276.3 q^{55} +22059.1 q^{57} -14354.4 q^{58} +11592.5 q^{59} +10095.9 q^{60} +31406.1 q^{61} +33016.9 q^{62} +18038.3 q^{64} -27950.9 q^{65} +14661.2 q^{66} +27555.9 q^{67} -12153.6 q^{68} +15974.1 q^{69} -22868.5 q^{71} -14394.9 q^{72} +15936.4 q^{73} -33700.7 q^{74} -1367.94 q^{75} -50425.6 q^{76} +15595.5 q^{78} +87165.3 q^{79} -3141.26 q^{80} +6561.00 q^{81} -11417.2 q^{82} +90307.5 q^{83} -32210.5 q^{85} -61601.2 q^{86} -38218.0 q^{87} -85643.0 q^{88} -126570. q^{89} -14929.4 q^{90} -36515.7 q^{92} +87906.4 q^{93} +4464.20 q^{94} -133642. q^{95} +52934.7 q^{96} -16573.1 q^{97} +39034.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 2 q^{2} + 54 q^{3} + 150 q^{4} + 100 q^{5} + 18 q^{6} - 114 q^{8} + 486 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.38033 0.597564 0.298782 0.954321i \(-0.403420\pi\)
0.298782 + 0.954321i \(0.403420\pi\)
\(3\) 9.00000 0.577350
\(4\) −20.5734 −0.642918
\(5\) −54.5253 −0.975378 −0.487689 0.873017i \(-0.662160\pi\)
−0.487689 + 0.873017i \(0.662160\pi\)
\(6\) 30.4230 0.345004
\(7\) 0 0
\(8\) −177.715 −0.981748
\(9\) 81.0000 0.333333
\(10\) −184.313 −0.582850
\(11\) 481.911 1.20084 0.600420 0.799685i \(-0.295000\pi\)
0.600420 + 0.799685i \(0.295000\pi\)
\(12\) −185.160 −0.371189
\(13\) 512.622 0.841277 0.420638 0.907228i \(-0.361806\pi\)
0.420638 + 0.907228i \(0.361806\pi\)
\(14\) 0 0
\(15\) −490.728 −0.563135
\(16\) 57.6111 0.0562608
\(17\) 590.744 0.495766 0.247883 0.968790i \(-0.420265\pi\)
0.247883 + 0.968790i \(0.420265\pi\)
\(18\) 273.807 0.199188
\(19\) 2451.02 1.55762 0.778811 0.627259i \(-0.215823\pi\)
0.778811 + 0.627259i \(0.215823\pi\)
\(20\) 1121.77 0.627088
\(21\) 0 0
\(22\) 1629.02 0.717579
\(23\) 1774.90 0.699609 0.349804 0.936823i \(-0.386248\pi\)
0.349804 + 0.936823i \(0.386248\pi\)
\(24\) −1599.44 −0.566812
\(25\) −151.993 −0.0486379
\(26\) 1732.83 0.502716
\(27\) 729.000 0.192450
\(28\) 0 0
\(29\) −4246.44 −0.937627 −0.468814 0.883297i \(-0.655318\pi\)
−0.468814 + 0.883297i \(0.655318\pi\)
\(30\) −1658.82 −0.336509
\(31\) 9767.37 1.82547 0.912733 0.408558i \(-0.133968\pi\)
0.912733 + 0.408558i \(0.133968\pi\)
\(32\) 5881.64 1.01537
\(33\) 4337.20 0.693306
\(34\) 1996.91 0.296252
\(35\) 0 0
\(36\) −1666.44 −0.214306
\(37\) −9969.65 −1.19722 −0.598612 0.801039i \(-0.704281\pi\)
−0.598612 + 0.801039i \(0.704281\pi\)
\(38\) 8285.24 0.930778
\(39\) 4613.60 0.485711
\(40\) 9689.98 0.957575
\(41\) −3377.54 −0.313791 −0.156896 0.987615i \(-0.550149\pi\)
−0.156896 + 0.987615i \(0.550149\pi\)
\(42\) 0 0
\(43\) −18223.4 −1.50300 −0.751500 0.659733i \(-0.770669\pi\)
−0.751500 + 0.659733i \(0.770669\pi\)
\(44\) −9914.53 −0.772042
\(45\) −4416.55 −0.325126
\(46\) 5999.76 0.418061
\(47\) 1320.64 0.0872046 0.0436023 0.999049i \(-0.486117\pi\)
0.0436023 + 0.999049i \(0.486117\pi\)
\(48\) 518.500 0.0324822
\(49\) 0 0
\(50\) −513.788 −0.0290642
\(51\) 5316.69 0.286231
\(52\) −10546.4 −0.540872
\(53\) 34837.0 1.70353 0.851767 0.523921i \(-0.175531\pi\)
0.851767 + 0.523921i \(0.175531\pi\)
\(54\) 2464.26 0.115001
\(55\) −26276.3 −1.17127
\(56\) 0 0
\(57\) 22059.1 0.899293
\(58\) −14354.4 −0.560292
\(59\) 11592.5 0.433558 0.216779 0.976221i \(-0.430445\pi\)
0.216779 + 0.976221i \(0.430445\pi\)
\(60\) 10095.9 0.362049
\(61\) 31406.1 1.08066 0.540331 0.841452i \(-0.318299\pi\)
0.540331 + 0.841452i \(0.318299\pi\)
\(62\) 33016.9 1.09083
\(63\) 0 0
\(64\) 18038.3 0.550486
\(65\) −27950.9 −0.820563
\(66\) 14661.2 0.414294
\(67\) 27555.9 0.749942 0.374971 0.927036i \(-0.377653\pi\)
0.374971 + 0.927036i \(0.377653\pi\)
\(68\) −12153.6 −0.318737
\(69\) 15974.1 0.403919
\(70\) 0 0
\(71\) −22868.5 −0.538384 −0.269192 0.963086i \(-0.586757\pi\)
−0.269192 + 0.963086i \(0.586757\pi\)
\(72\) −14394.9 −0.327249
\(73\) 15936.4 0.350012 0.175006 0.984567i \(-0.444005\pi\)
0.175006 + 0.984567i \(0.444005\pi\)
\(74\) −33700.7 −0.715418
\(75\) −1367.94 −0.0280811
\(76\) −50425.6 −1.00142
\(77\) 0 0
\(78\) 15595.5 0.290243
\(79\) 87165.3 1.57136 0.785680 0.618633i \(-0.212313\pi\)
0.785680 + 0.618633i \(0.212313\pi\)
\(80\) −3141.26 −0.0548755
\(81\) 6561.00 0.111111
\(82\) −11417.2 −0.187510
\(83\) 90307.5 1.43889 0.719447 0.694548i \(-0.244395\pi\)
0.719447 + 0.694548i \(0.244395\pi\)
\(84\) 0 0
\(85\) −32210.5 −0.483559
\(86\) −61601.2 −0.898138
\(87\) −38218.0 −0.541339
\(88\) −85643.0 −1.17892
\(89\) −126570. −1.69377 −0.846884 0.531777i \(-0.821524\pi\)
−0.846884 + 0.531777i \(0.821524\pi\)
\(90\) −14929.4 −0.194283
\(91\) 0 0
\(92\) −36515.7 −0.449791
\(93\) 87906.4 1.05393
\(94\) 4464.20 0.0521103
\(95\) −133642. −1.51927
\(96\) 52934.7 0.586223
\(97\) −16573.1 −0.178844 −0.0894219 0.995994i \(-0.528502\pi\)
−0.0894219 + 0.995994i \(0.528502\pi\)
\(98\) 0 0
\(99\) 39034.8 0.400280
\(100\) 3127.02 0.0312702
\(101\) −76610.1 −0.747279 −0.373639 0.927574i \(-0.621890\pi\)
−0.373639 + 0.927574i \(0.621890\pi\)
\(102\) 17972.2 0.171041
\(103\) 80435.8 0.747062 0.373531 0.927618i \(-0.378147\pi\)
0.373531 + 0.927618i \(0.378147\pi\)
\(104\) −91100.8 −0.825922
\(105\) 0 0
\(106\) 117760. 1.01797
\(107\) 148042. 1.25005 0.625023 0.780606i \(-0.285090\pi\)
0.625023 + 0.780606i \(0.285090\pi\)
\(108\) −14998.0 −0.123730
\(109\) −124982. −1.00759 −0.503793 0.863825i \(-0.668062\pi\)
−0.503793 + 0.863825i \(0.668062\pi\)
\(110\) −88822.7 −0.699910
\(111\) −89726.9 −0.691218
\(112\) 0 0
\(113\) −194474. −1.43273 −0.716366 0.697724i \(-0.754196\pi\)
−0.716366 + 0.697724i \(0.754196\pi\)
\(114\) 74567.2 0.537385
\(115\) −96777.1 −0.682383
\(116\) 87363.6 0.602817
\(117\) 41522.4 0.280426
\(118\) 39186.5 0.259078
\(119\) 0 0
\(120\) 87209.8 0.552856
\(121\) 71187.4 0.442018
\(122\) 106163. 0.645765
\(123\) −30397.9 −0.181168
\(124\) −200948. −1.17362
\(125\) 178679. 1.02282
\(126\) 0 0
\(127\) 239766. 1.31910 0.659551 0.751660i \(-0.270746\pi\)
0.659551 + 0.751660i \(0.270746\pi\)
\(128\) −127237. −0.686417
\(129\) −164011. −0.867758
\(130\) −94483.2 −0.490339
\(131\) −349576. −1.77977 −0.889883 0.456188i \(-0.849214\pi\)
−0.889883 + 0.456188i \(0.849214\pi\)
\(132\) −89230.8 −0.445738
\(133\) 0 0
\(134\) 93148.1 0.448138
\(135\) −39748.9 −0.187712
\(136\) −104984. −0.486717
\(137\) −280966. −1.27894 −0.639472 0.768814i \(-0.720847\pi\)
−0.639472 + 0.768814i \(0.720847\pi\)
\(138\) 53997.8 0.241367
\(139\) 50042.9 0.219688 0.109844 0.993949i \(-0.464965\pi\)
0.109844 + 0.993949i \(0.464965\pi\)
\(140\) 0 0
\(141\) 11885.8 0.0503476
\(142\) −77303.2 −0.321719
\(143\) 247038. 1.01024
\(144\) 4666.50 0.0187536
\(145\) 231538. 0.914541
\(146\) 53870.3 0.209155
\(147\) 0 0
\(148\) 205109. 0.769717
\(149\) −328813. −1.21334 −0.606671 0.794953i \(-0.707495\pi\)
−0.606671 + 0.794953i \(0.707495\pi\)
\(150\) −4624.09 −0.0167802
\(151\) 511304. 1.82489 0.912445 0.409200i \(-0.134192\pi\)
0.912445 + 0.409200i \(0.134192\pi\)
\(152\) −435583. −1.52919
\(153\) 47850.2 0.165255
\(154\) 0 0
\(155\) −532569. −1.78052
\(156\) −94917.3 −0.312272
\(157\) −75442.1 −0.244267 −0.122134 0.992514i \(-0.538974\pi\)
−0.122134 + 0.992514i \(0.538974\pi\)
\(158\) 294647. 0.938988
\(159\) 313533. 0.983536
\(160\) −320698. −0.990367
\(161\) 0 0
\(162\) 22178.3 0.0663960
\(163\) 340753. 1.00455 0.502274 0.864709i \(-0.332497\pi\)
0.502274 + 0.864709i \(0.332497\pi\)
\(164\) 69487.4 0.201742
\(165\) −236487. −0.676235
\(166\) 305269. 0.859831
\(167\) −132392. −0.367342 −0.183671 0.982988i \(-0.558798\pi\)
−0.183671 + 0.982988i \(0.558798\pi\)
\(168\) 0 0
\(169\) −108512. −0.292253
\(170\) −108882. −0.288957
\(171\) 198532. 0.519207
\(172\) 374917. 0.966306
\(173\) 449322. 1.14141 0.570706 0.821154i \(-0.306670\pi\)
0.570706 + 0.821154i \(0.306670\pi\)
\(174\) −129189. −0.323485
\(175\) 0 0
\(176\) 27763.4 0.0675602
\(177\) 104332. 0.250315
\(178\) −427847. −1.01213
\(179\) −93666.8 −0.218501 −0.109250 0.994014i \(-0.534845\pi\)
−0.109250 + 0.994014i \(0.534845\pi\)
\(180\) 90863.3 0.209029
\(181\) 399795. 0.907071 0.453535 0.891238i \(-0.350163\pi\)
0.453535 + 0.891238i \(0.350163\pi\)
\(182\) 0 0
\(183\) 282655. 0.623921
\(184\) −315428. −0.686839
\(185\) 543598. 1.16775
\(186\) 297153. 0.629792
\(187\) 284686. 0.595336
\(188\) −27170.0 −0.0560654
\(189\) 0 0
\(190\) −451755. −0.907861
\(191\) 27822.7 0.0551844 0.0275922 0.999619i \(-0.491216\pi\)
0.0275922 + 0.999619i \(0.491216\pi\)
\(192\) 162345. 0.317823
\(193\) −66464.3 −0.128438 −0.0642192 0.997936i \(-0.520456\pi\)
−0.0642192 + 0.997936i \(0.520456\pi\)
\(194\) −56022.5 −0.106871
\(195\) −251558. −0.473752
\(196\) 0 0
\(197\) −492506. −0.904162 −0.452081 0.891977i \(-0.649318\pi\)
−0.452081 + 0.891977i \(0.649318\pi\)
\(198\) 131951. 0.239193
\(199\) 349206. 0.625099 0.312549 0.949902i \(-0.398817\pi\)
0.312549 + 0.949902i \(0.398817\pi\)
\(200\) 27011.6 0.0477501
\(201\) 248003. 0.432979
\(202\) −258967. −0.446547
\(203\) 0 0
\(204\) −109382. −0.184023
\(205\) 184161. 0.306065
\(206\) 271900. 0.446417
\(207\) 143767. 0.233203
\(208\) 29532.7 0.0473309
\(209\) 1.18117e6 1.87046
\(210\) 0 0
\(211\) −218250. −0.337480 −0.168740 0.985661i \(-0.553970\pi\)
−0.168740 + 0.985661i \(0.553970\pi\)
\(212\) −716714. −1.09523
\(213\) −205817. −0.310836
\(214\) 500431. 0.746982
\(215\) 993638. 1.46599
\(216\) −129554. −0.188937
\(217\) 0 0
\(218\) −422481. −0.602096
\(219\) 143428. 0.202080
\(220\) 540593. 0.753032
\(221\) 302828. 0.417076
\(222\) −303306. −0.413047
\(223\) 1.24807e6 1.68065 0.840324 0.542084i \(-0.182365\pi\)
0.840324 + 0.542084i \(0.182365\pi\)
\(224\) 0 0
\(225\) −12311.5 −0.0162126
\(226\) −657386. −0.856149
\(227\) −520973. −0.671044 −0.335522 0.942032i \(-0.608913\pi\)
−0.335522 + 0.942032i \(0.608913\pi\)
\(228\) −453831. −0.578172
\(229\) −525651. −0.662383 −0.331191 0.943564i \(-0.607451\pi\)
−0.331191 + 0.943564i \(0.607451\pi\)
\(230\) −327139. −0.407767
\(231\) 0 0
\(232\) 754658. 0.920514
\(233\) 210205. 0.253661 0.126831 0.991924i \(-0.459520\pi\)
0.126831 + 0.991924i \(0.459520\pi\)
\(234\) 140359. 0.167572
\(235\) −72008.2 −0.0850575
\(236\) −238497. −0.278742
\(237\) 784488. 0.907225
\(238\) 0 0
\(239\) 805791. 0.912489 0.456244 0.889855i \(-0.349194\pi\)
0.456244 + 0.889855i \(0.349194\pi\)
\(240\) −28271.3 −0.0316824
\(241\) 1.13795e6 1.26206 0.631030 0.775758i \(-0.282632\pi\)
0.631030 + 0.775758i \(0.282632\pi\)
\(242\) 240637. 0.264134
\(243\) 59049.0 0.0641500
\(244\) −646130. −0.694777
\(245\) 0 0
\(246\) −102755. −0.108259
\(247\) 1.25644e6 1.31039
\(248\) −1.73581e6 −1.79215
\(249\) 812768. 0.830746
\(250\) 603994. 0.611199
\(251\) −1.16566e6 −1.16785 −0.583927 0.811806i \(-0.698484\pi\)
−0.583927 + 0.811806i \(0.698484\pi\)
\(252\) 0 0
\(253\) 855346. 0.840118
\(254\) 810489. 0.788248
\(255\) −289894. −0.279183
\(256\) −1.00733e6 −0.960664
\(257\) −1.02881e6 −0.971635 −0.485817 0.874060i \(-0.661478\pi\)
−0.485817 + 0.874060i \(0.661478\pi\)
\(258\) −554411. −0.518540
\(259\) 0 0
\(260\) 575043. 0.527554
\(261\) −343962. −0.312542
\(262\) −1.18168e6 −1.06352
\(263\) −657345. −0.586009 −0.293004 0.956111i \(-0.594655\pi\)
−0.293004 + 0.956111i \(0.594655\pi\)
\(264\) −770787. −0.680651
\(265\) −1.89950e6 −1.66159
\(266\) 0 0
\(267\) −1.13913e6 −0.977898
\(268\) −566918. −0.482151
\(269\) 5475.86 0.00461393 0.00230697 0.999997i \(-0.499266\pi\)
0.00230697 + 0.999997i \(0.499266\pi\)
\(270\) −134365. −0.112170
\(271\) −769147. −0.636189 −0.318094 0.948059i \(-0.603043\pi\)
−0.318094 + 0.948059i \(0.603043\pi\)
\(272\) 34033.4 0.0278922
\(273\) 0 0
\(274\) −949757. −0.764251
\(275\) −73247.3 −0.0584063
\(276\) −328642. −0.259687
\(277\) −1.17384e6 −0.919196 −0.459598 0.888127i \(-0.652007\pi\)
−0.459598 + 0.888127i \(0.652007\pi\)
\(278\) 169162. 0.131277
\(279\) 791157. 0.608488
\(280\) 0 0
\(281\) 837649. 0.632843 0.316422 0.948619i \(-0.397519\pi\)
0.316422 + 0.948619i \(0.397519\pi\)
\(282\) 40177.8 0.0300859
\(283\) −597221. −0.443271 −0.221635 0.975130i \(-0.571139\pi\)
−0.221635 + 0.975130i \(0.571139\pi\)
\(284\) 470483. 0.346137
\(285\) −1.20278e6 −0.877151
\(286\) 835071. 0.603682
\(287\) 0 0
\(288\) 476412. 0.338456
\(289\) −1.07088e6 −0.754216
\(290\) 782677. 0.546496
\(291\) −149158. −0.103256
\(292\) −327866. −0.225029
\(293\) −851321. −0.579328 −0.289664 0.957128i \(-0.593543\pi\)
−0.289664 + 0.957128i \(0.593543\pi\)
\(294\) 0 0
\(295\) −632084. −0.422883
\(296\) 1.77176e6 1.17537
\(297\) 351313. 0.231102
\(298\) −1.11150e6 −0.725049
\(299\) 909855. 0.588565
\(300\) 28143.1 0.0180538
\(301\) 0 0
\(302\) 1.72838e6 1.09049
\(303\) −689491. −0.431441
\(304\) 141206. 0.0876331
\(305\) −1.71243e6 −1.05405
\(306\) 161750. 0.0987506
\(307\) −600906. −0.363882 −0.181941 0.983309i \(-0.558238\pi\)
−0.181941 + 0.983309i \(0.558238\pi\)
\(308\) 0 0
\(309\) 723923. 0.431316
\(310\) −1.80026e6 −1.06397
\(311\) −465866. −0.273124 −0.136562 0.990632i \(-0.543605\pi\)
−0.136562 + 0.990632i \(0.543605\pi\)
\(312\) −819907. −0.476846
\(313\) −2.97366e6 −1.71565 −0.857827 0.513938i \(-0.828186\pi\)
−0.857827 + 0.513938i \(0.828186\pi\)
\(314\) −255019. −0.145965
\(315\) 0 0
\(316\) −1.79328e6 −1.01026
\(317\) 2.55521e6 1.42816 0.714082 0.700062i \(-0.246844\pi\)
0.714082 + 0.700062i \(0.246844\pi\)
\(318\) 1.05984e6 0.587725
\(319\) −2.04641e6 −1.12594
\(320\) −983544. −0.536932
\(321\) 1.33238e6 0.721714
\(322\) 0 0
\(323\) 1.44792e6 0.772216
\(324\) −134982. −0.0714353
\(325\) −77915.2 −0.0409179
\(326\) 1.15186e6 0.600281
\(327\) −1.12484e6 −0.581730
\(328\) 600241. 0.308064
\(329\) 0 0
\(330\) −799405. −0.404093
\(331\) 2.56138e6 1.28500 0.642502 0.766284i \(-0.277896\pi\)
0.642502 + 0.766284i \(0.277896\pi\)
\(332\) −1.85793e6 −0.925090
\(333\) −807542. −0.399075
\(334\) −447528. −0.219510
\(335\) −1.50249e6 −0.731477
\(336\) 0 0
\(337\) −1.21525e6 −0.582894 −0.291447 0.956587i \(-0.594137\pi\)
−0.291447 + 0.956587i \(0.594137\pi\)
\(338\) −366805. −0.174640
\(339\) −1.75026e6 −0.827188
\(340\) 662678. 0.310889
\(341\) 4.70701e6 2.19209
\(342\) 671105. 0.310259
\(343\) 0 0
\(344\) 3.23858e6 1.47557
\(345\) −870994. −0.393974
\(346\) 1.51886e6 0.682066
\(347\) 1.25318e6 0.558714 0.279357 0.960187i \(-0.409879\pi\)
0.279357 + 0.960187i \(0.409879\pi\)
\(348\) 786272. 0.348037
\(349\) 450403. 0.197942 0.0989709 0.995090i \(-0.468445\pi\)
0.0989709 + 0.995090i \(0.468445\pi\)
\(350\) 0 0
\(351\) 373702. 0.161904
\(352\) 2.83443e6 1.21929
\(353\) 545281. 0.232907 0.116454 0.993196i \(-0.462847\pi\)
0.116454 + 0.993196i \(0.462847\pi\)
\(354\) 352678. 0.149579
\(355\) 1.24691e6 0.525128
\(356\) 2.60396e6 1.08895
\(357\) 0 0
\(358\) −316625. −0.130568
\(359\) −1.01159e6 −0.414254 −0.207127 0.978314i \(-0.566411\pi\)
−0.207127 + 0.978314i \(0.566411\pi\)
\(360\) 784888. 0.319192
\(361\) 3.53138e6 1.42619
\(362\) 1.35144e6 0.542033
\(363\) 640686. 0.255199
\(364\) 0 0
\(365\) −868937. −0.341394
\(366\) 955468. 0.372832
\(367\) −1.09038e6 −0.422584 −0.211292 0.977423i \(-0.567767\pi\)
−0.211292 + 0.977423i \(0.567767\pi\)
\(368\) 102254. 0.0393605
\(369\) −273581. −0.104597
\(370\) 1.83754e6 0.697803
\(371\) 0 0
\(372\) −1.80853e6 −0.677592
\(373\) −3.13003e6 −1.16487 −0.582433 0.812879i \(-0.697899\pi\)
−0.582433 + 0.812879i \(0.697899\pi\)
\(374\) 962333. 0.355751
\(375\) 1.60811e6 0.590524
\(376\) −234698. −0.0856130
\(377\) −2.17682e6 −0.788804
\(378\) 0 0
\(379\) −1.27933e6 −0.457492 −0.228746 0.973486i \(-0.573463\pi\)
−0.228746 + 0.973486i \(0.573463\pi\)
\(380\) 2.74947e6 0.976766
\(381\) 2.15790e6 0.761584
\(382\) 94050.0 0.0329762
\(383\) 1.30666e6 0.455161 0.227580 0.973759i \(-0.426919\pi\)
0.227580 + 0.973759i \(0.426919\pi\)
\(384\) −1.14513e6 −0.396303
\(385\) 0 0
\(386\) −224671. −0.0767501
\(387\) −1.47610e6 −0.501000
\(388\) 340964. 0.114982
\(389\) −36488.2 −0.0122258 −0.00611292 0.999981i \(-0.501946\pi\)
−0.00611292 + 0.999981i \(0.501946\pi\)
\(390\) −850348. −0.283097
\(391\) 1.04851e6 0.346842
\(392\) 0 0
\(393\) −3.14618e6 −1.02755
\(394\) −1.66483e6 −0.540294
\(395\) −4.75271e6 −1.53267
\(396\) −803077. −0.257347
\(397\) 640204. 0.203865 0.101932 0.994791i \(-0.467497\pi\)
0.101932 + 0.994791i \(0.467497\pi\)
\(398\) 1.18043e6 0.373536
\(399\) 0 0
\(400\) −8756.50 −0.00273641
\(401\) −2.69149e6 −0.835858 −0.417929 0.908480i \(-0.637244\pi\)
−0.417929 + 0.908480i \(0.637244\pi\)
\(402\) 838333. 0.258733
\(403\) 5.00697e6 1.53572
\(404\) 1.57613e6 0.480439
\(405\) −357740. −0.108375
\(406\) 0 0
\(407\) −4.80449e6 −1.43768
\(408\) −944858. −0.281006
\(409\) 2.92572e6 0.864817 0.432409 0.901678i \(-0.357664\pi\)
0.432409 + 0.901678i \(0.357664\pi\)
\(410\) 622526. 0.182893
\(411\) −2.52869e6 −0.738399
\(412\) −1.65484e6 −0.480299
\(413\) 0 0
\(414\) 485981. 0.139354
\(415\) −4.92404e6 −1.40347
\(416\) 3.01506e6 0.854205
\(417\) 450386. 0.126837
\(418\) 3.99275e6 1.11772
\(419\) −5.52810e6 −1.53830 −0.769150 0.639068i \(-0.779320\pi\)
−0.769150 + 0.639068i \(0.779320\pi\)
\(420\) 0 0
\(421\) 5.77477e6 1.58792 0.793961 0.607968i \(-0.208015\pi\)
0.793961 + 0.607968i \(0.208015\pi\)
\(422\) −737756. −0.201666
\(423\) 106972. 0.0290682
\(424\) −6.19106e6 −1.67244
\(425\) −89789.1 −0.0241130
\(426\) −695729. −0.185744
\(427\) 0 0
\(428\) −3.04573e6 −0.803677
\(429\) 2.22334e6 0.583262
\(430\) 3.35882e6 0.876024
\(431\) 2.20964e6 0.572965 0.286483 0.958085i \(-0.407514\pi\)
0.286483 + 0.958085i \(0.407514\pi\)
\(432\) 41998.5 0.0108274
\(433\) 4.97215e6 1.27446 0.637228 0.770676i \(-0.280081\pi\)
0.637228 + 0.770676i \(0.280081\pi\)
\(434\) 0 0
\(435\) 2.08385e6 0.528010
\(436\) 2.57130e6 0.647794
\(437\) 4.35032e6 1.08973
\(438\) 484833. 0.120755
\(439\) −2.86405e6 −0.709282 −0.354641 0.935002i \(-0.615397\pi\)
−0.354641 + 0.935002i \(0.615397\pi\)
\(440\) 4.66971e6 1.14990
\(441\) 0 0
\(442\) 1.02366e6 0.249230
\(443\) −5.66234e6 −1.37084 −0.685419 0.728149i \(-0.740381\pi\)
−0.685419 + 0.728149i \(0.740381\pi\)
\(444\) 1.84598e6 0.444396
\(445\) 6.90124e6 1.65206
\(446\) 4.21889e6 1.00429
\(447\) −2.95932e6 −0.700523
\(448\) 0 0
\(449\) 7.54480e6 1.76617 0.883084 0.469215i \(-0.155463\pi\)
0.883084 + 0.469215i \(0.155463\pi\)
\(450\) −41616.8 −0.00968808
\(451\) −1.62767e6 −0.376813
\(452\) 4.00098e6 0.921129
\(453\) 4.60173e6 1.05360
\(454\) −1.76106e6 −0.400991
\(455\) 0 0
\(456\) −3.92025e6 −0.882879
\(457\) −589014. −0.131928 −0.0659638 0.997822i \(-0.521012\pi\)
−0.0659638 + 0.997822i \(0.521012\pi\)
\(458\) −1.77687e6 −0.395816
\(459\) 430652. 0.0954102
\(460\) 1.99103e6 0.438716
\(461\) −5.94951e6 −1.30385 −0.651927 0.758282i \(-0.726039\pi\)
−0.651927 + 0.758282i \(0.726039\pi\)
\(462\) 0 0
\(463\) −2.16083e6 −0.468456 −0.234228 0.972182i \(-0.575256\pi\)
−0.234228 + 0.972182i \(0.575256\pi\)
\(464\) −244642. −0.0527517
\(465\) −4.79312e6 −1.02798
\(466\) 710563. 0.151579
\(467\) 505766. 0.107314 0.0536571 0.998559i \(-0.482912\pi\)
0.0536571 + 0.998559i \(0.482912\pi\)
\(468\) −854255. −0.180291
\(469\) 0 0
\(470\) −243412. −0.0508272
\(471\) −678979. −0.141028
\(472\) −2.06016e6 −0.425644
\(473\) −8.78208e6 −1.80486
\(474\) 2.65183e6 0.542125
\(475\) −372538. −0.0757594
\(476\) 0 0
\(477\) 2.82179e6 0.567845
\(478\) 2.72384e6 0.545270
\(479\) −547067. −0.108944 −0.0544718 0.998515i \(-0.517348\pi\)
−0.0544718 + 0.998515i \(0.517348\pi\)
\(480\) −2.88628e6 −0.571789
\(481\) −5.11066e6 −1.00720
\(482\) 3.84665e6 0.754162
\(483\) 0 0
\(484\) −1.46456e6 −0.284181
\(485\) 903652. 0.174440
\(486\) 199605. 0.0383337
\(487\) 2.37889e6 0.454519 0.227259 0.973834i \(-0.427024\pi\)
0.227259 + 0.973834i \(0.427024\pi\)
\(488\) −5.58135e6 −1.06094
\(489\) 3.06677e6 0.579976
\(490\) 0 0
\(491\) −7.19719e6 −1.34728 −0.673642 0.739058i \(-0.735271\pi\)
−0.673642 + 0.739058i \(0.735271\pi\)
\(492\) 625386. 0.116476
\(493\) −2.50856e6 −0.464844
\(494\) 4.24720e6 0.783042
\(495\) −2.12838e6 −0.390424
\(496\) 562709. 0.102702
\(497\) 0 0
\(498\) 2.74742e6 0.496423
\(499\) 2.22236e6 0.399543 0.199771 0.979843i \(-0.435980\pi\)
0.199771 + 0.979843i \(0.435980\pi\)
\(500\) −3.67603e6 −0.657588
\(501\) −1.19153e6 −0.212085
\(502\) −3.94032e6 −0.697867
\(503\) −305857. −0.0539013 −0.0269506 0.999637i \(-0.508580\pi\)
−0.0269506 + 0.999637i \(0.508580\pi\)
\(504\) 0 0
\(505\) 4.17719e6 0.728879
\(506\) 2.89135e6 0.502024
\(507\) −976604. −0.168732
\(508\) −4.93280e6 −0.848074
\(509\) −9.06420e6 −1.55073 −0.775363 0.631516i \(-0.782433\pi\)
−0.775363 + 0.631516i \(0.782433\pi\)
\(510\) −979938. −0.166830
\(511\) 0 0
\(512\) 666475. 0.112359
\(513\) 1.78679e6 0.299764
\(514\) −3.47772e6 −0.580614
\(515\) −4.38579e6 −0.728668
\(516\) 3.37426e6 0.557897
\(517\) 636431. 0.104719
\(518\) 0 0
\(519\) 4.04390e6 0.658995
\(520\) 4.96730e6 0.805586
\(521\) 8.44649e6 1.36327 0.681635 0.731692i \(-0.261269\pi\)
0.681635 + 0.731692i \(0.261269\pi\)
\(522\) −1.16270e6 −0.186764
\(523\) 3.91585e6 0.625997 0.312998 0.949754i \(-0.398667\pi\)
0.312998 + 0.949754i \(0.398667\pi\)
\(524\) 7.19195e6 1.14424
\(525\) 0 0
\(526\) −2.22204e6 −0.350178
\(527\) 5.77001e6 0.905003
\(528\) 249871. 0.0390059
\(529\) −3.28606e6 −0.510548
\(530\) −6.42092e6 −0.992905
\(531\) 938992. 0.144519
\(532\) 0 0
\(533\) −1.73140e6 −0.263985
\(534\) −3.85062e6 −0.584356
\(535\) −8.07204e6 −1.21927
\(536\) −4.89711e6 −0.736254
\(537\) −843001. −0.126152
\(538\) 18510.2 0.00275712
\(539\) 0 0
\(540\) 817769. 0.120683
\(541\) −2.94802e6 −0.433049 −0.216524 0.976277i \(-0.569472\pi\)
−0.216524 + 0.976277i \(0.569472\pi\)
\(542\) −2.59997e6 −0.380163
\(543\) 3.59816e6 0.523698
\(544\) 3.47454e6 0.503385
\(545\) 6.81469e6 0.982776
\(546\) 0 0
\(547\) −5.50670e6 −0.786906 −0.393453 0.919345i \(-0.628720\pi\)
−0.393453 + 0.919345i \(0.628720\pi\)
\(548\) 5.78041e6 0.822256
\(549\) 2.54390e6 0.360221
\(550\) −247600. −0.0349015
\(551\) −1.04081e7 −1.46047
\(552\) −2.83885e6 −0.396547
\(553\) 0 0
\(554\) −3.96796e6 −0.549278
\(555\) 4.89238e6 0.674199
\(556\) −1.02955e6 −0.141241
\(557\) −4.97235e6 −0.679085 −0.339542 0.940591i \(-0.610272\pi\)
−0.339542 + 0.940591i \(0.610272\pi\)
\(558\) 2.67437e6 0.363611
\(559\) −9.34174e6 −1.26444
\(560\) 0 0
\(561\) 2.56217e6 0.343717
\(562\) 2.83153e6 0.378164
\(563\) 1.31397e7 1.74709 0.873544 0.486745i \(-0.161816\pi\)
0.873544 + 0.486745i \(0.161816\pi\)
\(564\) −244530. −0.0323694
\(565\) 1.06037e7 1.39746
\(566\) −2.01880e6 −0.264882
\(567\) 0 0
\(568\) 4.06409e6 0.528558
\(569\) −1.38589e7 −1.79452 −0.897261 0.441500i \(-0.854447\pi\)
−0.897261 + 0.441500i \(0.854447\pi\)
\(570\) −4.06580e6 −0.524154
\(571\) −1.06977e7 −1.37309 −0.686547 0.727085i \(-0.740874\pi\)
−0.686547 + 0.727085i \(0.740874\pi\)
\(572\) −5.08241e6 −0.649501
\(573\) 250404. 0.0318607
\(574\) 0 0
\(575\) −269774. −0.0340275
\(576\) 1.46110e6 0.183495
\(577\) −5.61898e6 −0.702616 −0.351308 0.936260i \(-0.614263\pi\)
−0.351308 + 0.936260i \(0.614263\pi\)
\(578\) −3.61993e6 −0.450692
\(579\) −598178. −0.0741540
\(580\) −4.76353e6 −0.587975
\(581\) 0 0
\(582\) −504203. −0.0617018
\(583\) 1.67883e7 2.04567
\(584\) −2.83214e6 −0.343624
\(585\) −2.26402e6 −0.273521
\(586\) −2.87775e6 −0.346185
\(587\) −6.86832e6 −0.822726 −0.411363 0.911471i \(-0.634947\pi\)
−0.411363 + 0.911471i \(0.634947\pi\)
\(588\) 0 0
\(589\) 2.39400e7 2.84338
\(590\) −2.13665e6 −0.252699
\(591\) −4.43256e6 −0.522018
\(592\) −574362. −0.0673568
\(593\) 4.07737e6 0.476149 0.238075 0.971247i \(-0.423484\pi\)
0.238075 + 0.971247i \(0.423484\pi\)
\(594\) 1.18755e6 0.138098
\(595\) 0 0
\(596\) 6.76479e6 0.780079
\(597\) 3.14285e6 0.360901
\(598\) 3.07561e6 0.351705
\(599\) 582993. 0.0663890 0.0331945 0.999449i \(-0.489432\pi\)
0.0331945 + 0.999449i \(0.489432\pi\)
\(600\) 243104. 0.0275686
\(601\) −6.87250e6 −0.776119 −0.388060 0.921634i \(-0.626855\pi\)
−0.388060 + 0.921634i \(0.626855\pi\)
\(602\) 0 0
\(603\) 2.23203e6 0.249981
\(604\) −1.05192e7 −1.17325
\(605\) −3.88151e6 −0.431134
\(606\) −2.33071e6 −0.257814
\(607\) −2.40563e6 −0.265006 −0.132503 0.991183i \(-0.542301\pi\)
−0.132503 + 0.991183i \(0.542301\pi\)
\(608\) 1.44160e7 1.58156
\(609\) 0 0
\(610\) −5.78858e6 −0.629865
\(611\) 676989. 0.0733632
\(612\) −984440. −0.106246
\(613\) −1.16245e7 −1.24946 −0.624731 0.780840i \(-0.714791\pi\)
−0.624731 + 0.780840i \(0.714791\pi\)
\(614\) −2.03126e6 −0.217443
\(615\) 1.65745e6 0.176707
\(616\) 0 0
\(617\) −1.37370e7 −1.45271 −0.726353 0.687322i \(-0.758786\pi\)
−0.726353 + 0.687322i \(0.758786\pi\)
\(618\) 2.44710e6 0.257739
\(619\) 1.64008e7 1.72044 0.860220 0.509923i \(-0.170326\pi\)
0.860220 + 0.509923i \(0.170326\pi\)
\(620\) 1.09567e7 1.14473
\(621\) 1.29390e6 0.134640
\(622\) −1.57478e6 −0.163209
\(623\) 0 0
\(624\) 265794. 0.0273265
\(625\) −9.26754e6 −0.948996
\(626\) −1.00519e7 −1.02521
\(627\) 1.06305e7 1.07991
\(628\) 1.55210e6 0.157044
\(629\) −5.88951e6 −0.593543
\(630\) 0 0
\(631\) −7.28252e6 −0.728129 −0.364065 0.931374i \(-0.618611\pi\)
−0.364065 + 0.931374i \(0.618611\pi\)
\(632\) −1.54906e7 −1.54268
\(633\) −1.96425e6 −0.194844
\(634\) 8.63745e6 0.853419
\(635\) −1.30733e7 −1.28662
\(636\) −6.45042e6 −0.632332
\(637\) 0 0
\(638\) −6.91754e6 −0.672821
\(639\) −1.85235e6 −0.179461
\(640\) 6.93763e6 0.669516
\(641\) −107514. −0.0103352 −0.00516762 0.999987i \(-0.501645\pi\)
−0.00516762 + 0.999987i \(0.501645\pi\)
\(642\) 4.50388e6 0.431270
\(643\) 9.75913e6 0.930858 0.465429 0.885085i \(-0.345900\pi\)
0.465429 + 0.885085i \(0.345900\pi\)
\(644\) 0 0
\(645\) 8.94274e6 0.846392
\(646\) 4.89445e6 0.461448
\(647\) −6.75588e6 −0.634485 −0.317243 0.948344i \(-0.602757\pi\)
−0.317243 + 0.948344i \(0.602757\pi\)
\(648\) −1.16599e6 −0.109083
\(649\) 5.58655e6 0.520633
\(650\) −263379. −0.0244511
\(651\) 0 0
\(652\) −7.01043e6 −0.645841
\(653\) 2.20346e6 0.202219 0.101110 0.994875i \(-0.467761\pi\)
0.101110 + 0.994875i \(0.467761\pi\)
\(654\) −3.80233e6 −0.347620
\(655\) 1.90607e7 1.73595
\(656\) −194584. −0.0176542
\(657\) 1.29085e6 0.116671
\(658\) 0 0
\(659\) −1.39630e7 −1.25247 −0.626233 0.779636i \(-0.715404\pi\)
−0.626233 + 0.779636i \(0.715404\pi\)
\(660\) 4.86534e6 0.434763
\(661\) −51052.8 −0.00454482 −0.00227241 0.999997i \(-0.500723\pi\)
−0.00227241 + 0.999997i \(0.500723\pi\)
\(662\) 8.65832e6 0.767872
\(663\) 2.72545e6 0.240799
\(664\) −1.60490e7 −1.41263
\(665\) 0 0
\(666\) −2.72976e6 −0.238473
\(667\) −7.53703e6 −0.655972
\(668\) 2.72375e6 0.236170
\(669\) 1.12326e7 0.970323
\(670\) −5.07893e6 −0.437104
\(671\) 1.51350e7 1.29770
\(672\) 0 0
\(673\) 6.71688e6 0.571650 0.285825 0.958282i \(-0.407732\pi\)
0.285825 + 0.958282i \(0.407732\pi\)
\(674\) −4.10793e6 −0.348316
\(675\) −110803. −0.00936036
\(676\) 2.23245e6 0.187895
\(677\) 1.45655e7 1.22139 0.610693 0.791867i \(-0.290891\pi\)
0.610693 + 0.791867i \(0.290891\pi\)
\(678\) −5.91647e6 −0.494298
\(679\) 0 0
\(680\) 5.72429e6 0.474733
\(681\) −4.68876e6 −0.387427
\(682\) 1.59112e7 1.30991
\(683\) −1.62565e6 −0.133344 −0.0666722 0.997775i \(-0.521238\pi\)
−0.0666722 + 0.997775i \(0.521238\pi\)
\(684\) −4.08448e6 −0.333808
\(685\) 1.53197e7 1.24745
\(686\) 0 0
\(687\) −4.73086e6 −0.382427
\(688\) −1.04987e6 −0.0845600
\(689\) 1.78582e7 1.43314
\(690\) −2.94425e6 −0.235424
\(691\) −2.81556e6 −0.224321 −0.112160 0.993690i \(-0.535777\pi\)
−0.112160 + 0.993690i \(0.535777\pi\)
\(692\) −9.24407e6 −0.733834
\(693\) 0 0
\(694\) 4.23616e6 0.333867
\(695\) −2.72860e6 −0.214278
\(696\) 6.79192e6 0.531459
\(697\) −1.99526e6 −0.155567
\(698\) 1.52251e6 0.118283
\(699\) 1.89185e6 0.146451
\(700\) 0 0
\(701\) 1.40996e7 1.08371 0.541855 0.840472i \(-0.317722\pi\)
0.541855 + 0.840472i \(0.317722\pi\)
\(702\) 1.26323e6 0.0967478
\(703\) −2.44358e7 −1.86482
\(704\) 8.69287e6 0.661046
\(705\) −648074. −0.0491079
\(706\) 1.84323e6 0.139177
\(707\) 0 0
\(708\) −2.14647e6 −0.160932
\(709\) −8.44943e6 −0.631265 −0.315633 0.948881i \(-0.602217\pi\)
−0.315633 + 0.948881i \(0.602217\pi\)
\(710\) 4.21498e6 0.313797
\(711\) 7.06039e6 0.523787
\(712\) 2.24933e7 1.66285
\(713\) 1.73361e7 1.27711
\(714\) 0 0
\(715\) −1.34698e7 −0.985365
\(716\) 1.92704e6 0.140478
\(717\) 7.25212e6 0.526826
\(718\) −3.41950e6 −0.247543
\(719\) 1.82657e7 1.31769 0.658846 0.752278i \(-0.271045\pi\)
0.658846 + 0.752278i \(0.271045\pi\)
\(720\) −254442. −0.0182918
\(721\) 0 0
\(722\) 1.19372e7 0.852237
\(723\) 1.02415e7 0.728651
\(724\) −8.22513e6 −0.583172
\(725\) 645431. 0.0456042
\(726\) 2.16573e6 0.152498
\(727\) 1.37476e6 0.0964700 0.0482350 0.998836i \(-0.484640\pi\)
0.0482350 + 0.998836i \(0.484640\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) −2.93729e6 −0.204005
\(731\) −1.07654e7 −0.745136
\(732\) −5.81517e6 −0.401130
\(733\) 1.64625e7 1.13171 0.565857 0.824503i \(-0.308545\pi\)
0.565857 + 0.824503i \(0.308545\pi\)
\(734\) −3.68585e6 −0.252521
\(735\) 0 0
\(736\) 1.04393e7 0.710360
\(737\) 1.32795e7 0.900561
\(738\) −924793. −0.0625034
\(739\) 2.72997e6 0.183885 0.0919425 0.995764i \(-0.470692\pi\)
0.0919425 + 0.995764i \(0.470692\pi\)
\(740\) −1.11836e7 −0.750765
\(741\) 1.13080e7 0.756555
\(742\) 0 0
\(743\) −2.34396e7 −1.55768 −0.778839 0.627224i \(-0.784191\pi\)
−0.778839 + 0.627224i \(0.784191\pi\)
\(744\) −1.56223e7 −1.03470
\(745\) 1.79286e7 1.18347
\(746\) −1.05805e7 −0.696082
\(747\) 7.31491e6 0.479631
\(748\) −5.85695e6 −0.382752
\(749\) 0 0
\(750\) 5.43595e6 0.352876
\(751\) 1.79407e7 1.16075 0.580376 0.814349i \(-0.302906\pi\)
0.580376 + 0.814349i \(0.302906\pi\)
\(752\) 76083.4 0.00490620
\(753\) −1.04910e7 −0.674260
\(754\) −7.35837e6 −0.471361
\(755\) −2.78790e7 −1.77996
\(756\) 0 0
\(757\) −2.30823e7 −1.46399 −0.731997 0.681308i \(-0.761412\pi\)
−0.731997 + 0.681308i \(0.761412\pi\)
\(758\) −4.32455e6 −0.273381
\(759\) 7.69811e6 0.485043
\(760\) 2.37503e7 1.49154
\(761\) 1.91587e7 1.19924 0.599618 0.800286i \(-0.295319\pi\)
0.599618 + 0.800286i \(0.295319\pi\)
\(762\) 7.29440e6 0.455095
\(763\) 0 0
\(764\) −572407. −0.0354790
\(765\) −2.60905e6 −0.161186
\(766\) 4.41693e6 0.271988
\(767\) 5.94257e6 0.364742
\(768\) −9.06596e6 −0.554639
\(769\) 1.73017e7 1.05505 0.527526 0.849539i \(-0.323120\pi\)
0.527526 + 0.849539i \(0.323120\pi\)
\(770\) 0 0
\(771\) −9.25931e6 −0.560974
\(772\) 1.36739e6 0.0825753
\(773\) 3.12287e7 1.87977 0.939885 0.341491i \(-0.110932\pi\)
0.939885 + 0.341491i \(0.110932\pi\)
\(774\) −4.98970e6 −0.299379
\(775\) −1.48458e6 −0.0887868
\(776\) 2.94529e6 0.175580
\(777\) 0 0
\(778\) −123342. −0.00730572
\(779\) −8.27840e6 −0.488768
\(780\) 5.17539e6 0.304584
\(781\) −1.10206e7 −0.646514
\(782\) 3.54432e6 0.207260
\(783\) −3.09566e6 −0.180446
\(784\) 0 0
\(785\) 4.11350e6 0.238253
\(786\) −1.06351e7 −0.614026
\(787\) −2.48691e7 −1.43128 −0.715638 0.698471i \(-0.753864\pi\)
−0.715638 + 0.698471i \(0.753864\pi\)
\(788\) 1.01325e7 0.581302
\(789\) −5.91611e6 −0.338332
\(790\) −1.60657e7 −0.915868
\(791\) 0 0
\(792\) −6.93708e6 −0.392974
\(793\) 1.60995e7 0.909136
\(794\) 2.16410e6 0.121822
\(795\) −1.70955e7 −0.959319
\(796\) −7.18433e6 −0.401887
\(797\) 584574. 0.0325982 0.0162991 0.999867i \(-0.494812\pi\)
0.0162991 + 0.999867i \(0.494812\pi\)
\(798\) 0 0
\(799\) 780159. 0.0432331
\(800\) −893970. −0.0493853
\(801\) −1.02521e7 −0.564590
\(802\) −9.09814e6 −0.499478
\(803\) 7.67993e6 0.420309
\(804\) −5.10226e6 −0.278370
\(805\) 0 0
\(806\) 1.69252e7 0.917691
\(807\) 49282.7 0.00266386
\(808\) 1.36148e7 0.733639
\(809\) −1.10552e7 −0.593878 −0.296939 0.954897i \(-0.595966\pi\)
−0.296939 + 0.954897i \(0.595966\pi\)
\(810\) −1.20928e6 −0.0647612
\(811\) 1.62780e7 0.869060 0.434530 0.900657i \(-0.356914\pi\)
0.434530 + 0.900657i \(0.356914\pi\)
\(812\) 0 0
\(813\) −6.92232e6 −0.367304
\(814\) −1.62408e7 −0.859103
\(815\) −1.85796e7 −0.979813
\(816\) 306300. 0.0161036
\(817\) −4.46659e7 −2.34111
\(818\) 9.88989e6 0.516783
\(819\) 0 0
\(820\) −3.78882e6 −0.196775
\(821\) 3.28574e7 1.70128 0.850638 0.525752i \(-0.176216\pi\)
0.850638 + 0.525752i \(0.176216\pi\)
\(822\) −8.54781e6 −0.441240
\(823\) 1.06046e7 0.545752 0.272876 0.962049i \(-0.412025\pi\)
0.272876 + 0.962049i \(0.412025\pi\)
\(824\) −1.42947e7 −0.733426
\(825\) −659226. −0.0337209
\(826\) 0 0
\(827\) 1.57582e7 0.801203 0.400602 0.916252i \(-0.368801\pi\)
0.400602 + 0.916252i \(0.368801\pi\)
\(828\) −2.95777e6 −0.149930
\(829\) −2.70135e7 −1.36520 −0.682598 0.730794i \(-0.739150\pi\)
−0.682598 + 0.730794i \(0.739150\pi\)
\(830\) −1.66449e7 −0.838660
\(831\) −1.05645e7 −0.530698
\(832\) 9.24684e6 0.463111
\(833\) 0 0
\(834\) 1.52245e6 0.0757930
\(835\) 7.21871e6 0.358297
\(836\) −2.43007e7 −1.20255
\(837\) 7.12042e6 0.351311
\(838\) −1.86868e7 −0.919232
\(839\) −3.50490e7 −1.71898 −0.859489 0.511154i \(-0.829218\pi\)
−0.859489 + 0.511154i \(0.829218\pi\)
\(840\) 0 0
\(841\) −2.47888e6 −0.120855
\(842\) 1.95206e7 0.948885
\(843\) 7.53884e6 0.365372
\(844\) 4.49013e6 0.216972
\(845\) 5.91663e6 0.285057
\(846\) 361600. 0.0173701
\(847\) 0 0
\(848\) 2.00699e6 0.0958422
\(849\) −5.37499e6 −0.255922
\(850\) −303517. −0.0144091
\(851\) −1.76952e7 −0.837589
\(852\) 4.23434e6 0.199842
\(853\) −538313. −0.0253316 −0.0126658 0.999920i \(-0.504032\pi\)
−0.0126658 + 0.999920i \(0.504032\pi\)
\(854\) 0 0
\(855\) −1.08250e7 −0.506423
\(856\) −2.63094e7 −1.22723
\(857\) −3.68275e6 −0.171285 −0.0856426 0.996326i \(-0.527294\pi\)
−0.0856426 + 0.996326i \(0.527294\pi\)
\(858\) 7.51564e6 0.348536
\(859\) −1.88883e7 −0.873394 −0.436697 0.899609i \(-0.643852\pi\)
−0.436697 + 0.899609i \(0.643852\pi\)
\(860\) −2.04425e7 −0.942513
\(861\) 0 0
\(862\) 7.46931e6 0.342383
\(863\) 2.78544e7 1.27311 0.636557 0.771230i \(-0.280358\pi\)
0.636557 + 0.771230i \(0.280358\pi\)
\(864\) 4.28771e6 0.195408
\(865\) −2.44994e7 −1.11331
\(866\) 1.68075e7 0.761568
\(867\) −9.63791e6 −0.435447
\(868\) 0 0
\(869\) 4.20059e7 1.88695
\(870\) 7.04409e6 0.315520
\(871\) 1.41258e7 0.630909
\(872\) 2.22113e7 0.989195
\(873\) −1.34242e6 −0.0596146
\(874\) 1.47055e7 0.651180
\(875\) 0 0
\(876\) −2.95079e6 −0.129921
\(877\) −1.75052e7 −0.768544 −0.384272 0.923220i \(-0.625548\pi\)
−0.384272 + 0.923220i \(0.625548\pi\)
\(878\) −9.68143e6 −0.423841
\(879\) −7.66189e6 −0.334475
\(880\) −1.51381e6 −0.0658968
\(881\) −5.66292e6 −0.245811 −0.122905 0.992418i \(-0.539221\pi\)
−0.122905 + 0.992418i \(0.539221\pi\)
\(882\) 0 0
\(883\) 1.43933e7 0.621239 0.310620 0.950534i \(-0.399463\pi\)
0.310620 + 0.950534i \(0.399463\pi\)
\(884\) −6.23020e6 −0.268146
\(885\) −5.68876e6 −0.244151
\(886\) −1.91406e7 −0.819163
\(887\) 5.50193e6 0.234804 0.117402 0.993084i \(-0.462543\pi\)
0.117402 + 0.993084i \(0.462543\pi\)
\(888\) 1.59458e7 0.678602
\(889\) 0 0
\(890\) 2.33285e7 0.987214
\(891\) 3.16182e6 0.133427
\(892\) −2.56770e7 −1.08052
\(893\) 3.23691e6 0.135832
\(894\) −1.00035e7 −0.418607
\(895\) 5.10721e6 0.213121
\(896\) 0 0
\(897\) 8.18869e6 0.339808
\(898\) 2.55039e7 1.05540
\(899\) −4.14766e7 −1.71161
\(900\) 253288. 0.0104234
\(901\) 2.05797e7 0.844554
\(902\) −5.50208e6 −0.225170
\(903\) 0 0
\(904\) 3.45610e7 1.40658
\(905\) −2.17990e7 −0.884737
\(906\) 1.55554e7 0.629593
\(907\) 6.01555e6 0.242805 0.121402 0.992603i \(-0.461261\pi\)
0.121402 + 0.992603i \(0.461261\pi\)
\(908\) 1.07182e7 0.431426
\(909\) −6.20542e6 −0.249093
\(910\) 0 0
\(911\) −1.47513e7 −0.588892 −0.294446 0.955668i \(-0.595135\pi\)
−0.294446 + 0.955668i \(0.595135\pi\)
\(912\) 1.27085e6 0.0505950
\(913\) 4.35202e7 1.72788
\(914\) −1.99106e6 −0.0788351
\(915\) −1.54119e7 −0.608559
\(916\) 1.08144e7 0.425858
\(917\) 0 0
\(918\) 1.45575e6 0.0570137
\(919\) −3.54001e7 −1.38266 −0.691331 0.722538i \(-0.742976\pi\)
−0.691331 + 0.722538i \(0.742976\pi\)
\(920\) 1.71988e7 0.669928
\(921\) −5.40816e6 −0.210088
\(922\) −2.01113e7 −0.779135
\(923\) −1.17229e7 −0.452930
\(924\) 0 0
\(925\) 1.51532e6 0.0582305
\(926\) −7.30434e6 −0.279932
\(927\) 6.51530e6 0.249021
\(928\) −2.49760e7 −0.952036
\(929\) −3.83266e7 −1.45700 −0.728502 0.685044i \(-0.759783\pi\)
−0.728502 + 0.685044i \(0.759783\pi\)
\(930\) −1.62023e7 −0.614285
\(931\) 0 0
\(932\) −4.32463e6 −0.163083
\(933\) −4.19280e6 −0.157688
\(934\) 1.70966e6 0.0641271
\(935\) −1.55226e7 −0.580677
\(936\) −7.37917e6 −0.275307
\(937\) 1.48693e7 0.553274 0.276637 0.960974i \(-0.410780\pi\)
0.276637 + 0.960974i \(0.410780\pi\)
\(938\) 0 0
\(939\) −2.67629e7 −0.990534
\(940\) 1.48145e6 0.0546849
\(941\) −3.50894e6 −0.129182 −0.0645909 0.997912i \(-0.520574\pi\)
−0.0645909 + 0.997912i \(0.520574\pi\)
\(942\) −2.29517e6 −0.0842730
\(943\) −5.99481e6 −0.219531
\(944\) 667856. 0.0243923
\(945\) 0 0
\(946\) −2.96863e7 −1.07852
\(947\) 3.89774e7 1.41234 0.706168 0.708044i \(-0.250422\pi\)
0.706168 + 0.708044i \(0.250422\pi\)
\(948\) −1.61395e7 −0.583271
\(949\) 8.16935e6 0.294457
\(950\) −1.25930e6 −0.0452711
\(951\) 2.29969e7 0.824551
\(952\) 0 0
\(953\) 7.11438e6 0.253749 0.126875 0.991919i \(-0.459505\pi\)
0.126875 + 0.991919i \(0.459505\pi\)
\(954\) 9.53860e6 0.339323
\(955\) −1.51704e6 −0.0538256
\(956\) −1.65778e7 −0.586655
\(957\) −1.84177e7 −0.650062
\(958\) −1.84927e6 −0.0651008
\(959\) 0 0
\(960\) −8.85190e6 −0.309998
\(961\) 6.67724e7 2.33232
\(962\) −1.72757e7 −0.601865
\(963\) 1.19914e7 0.416682
\(964\) −2.34114e7 −0.811401
\(965\) 3.62398e6 0.125276
\(966\) 0 0
\(967\) −1.99357e7 −0.685592 −0.342796 0.939410i \(-0.611374\pi\)
−0.342796 + 0.939410i \(0.611374\pi\)
\(968\) −1.26511e7 −0.433950
\(969\) 1.30313e7 0.445839
\(970\) 3.05464e6 0.104239
\(971\) 4.95888e7 1.68786 0.843928 0.536456i \(-0.180237\pi\)
0.843928 + 0.536456i \(0.180237\pi\)
\(972\) −1.21484e6 −0.0412432
\(973\) 0 0
\(974\) 8.04143e6 0.271604
\(975\) −701236. −0.0236240
\(976\) 1.80934e6 0.0607989
\(977\) −355348. −0.0119102 −0.00595508 0.999982i \(-0.501896\pi\)
−0.00595508 + 0.999982i \(0.501896\pi\)
\(978\) 1.03667e7 0.346572
\(979\) −6.09953e7 −2.03395
\(980\) 0 0
\(981\) −1.01236e7 −0.335862
\(982\) −2.43289e7 −0.805088
\(983\) −991754. −0.0327356 −0.0163678 0.999866i \(-0.505210\pi\)
−0.0163678 + 0.999866i \(0.505210\pi\)
\(984\) 5.40217e6 0.177861
\(985\) 2.68541e7 0.881900
\(986\) −8.47976e6 −0.277774
\(987\) 0 0
\(988\) −2.58493e7 −0.842474
\(989\) −3.23448e7 −1.05151
\(990\) −7.19464e6 −0.233303
\(991\) 1.06674e7 0.345045 0.172523 0.985006i \(-0.444808\pi\)
0.172523 + 0.985006i \(0.444808\pi\)
\(992\) 5.74481e7 1.85352
\(993\) 2.30524e7 0.741898
\(994\) 0 0
\(995\) −1.90405e7 −0.609707
\(996\) −1.67214e7 −0.534101
\(997\) 5.58283e7 1.77876 0.889379 0.457171i \(-0.151137\pi\)
0.889379 + 0.457171i \(0.151137\pi\)
\(998\) 7.51232e6 0.238752
\(999\) −7.26788e6 −0.230406
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.o.1.4 yes 6
3.2 odd 2 441.6.a.ba.1.3 6
7.2 even 3 147.6.e.p.67.3 12
7.3 odd 6 147.6.e.q.79.3 12
7.4 even 3 147.6.e.p.79.3 12
7.5 odd 6 147.6.e.q.67.3 12
7.6 odd 2 147.6.a.n.1.4 6
21.20 even 2 441.6.a.bb.1.3 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.6.a.n.1.4 6 7.6 odd 2
147.6.a.o.1.4 yes 6 1.1 even 1 trivial
147.6.e.p.67.3 12 7.2 even 3
147.6.e.p.79.3 12 7.4 even 3
147.6.e.q.67.3 12 7.5 odd 6
147.6.e.q.79.3 12 7.3 odd 6
441.6.a.ba.1.3 6 3.2 odd 2
441.6.a.bb.1.3 6 21.20 even 2