Properties

Label 147.6.a.l.1.2
Level $147$
Weight $6$
Character 147.1
Self dual yes
Analytic conductor $23.576$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [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: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 97x^{2} + 7x + 294 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 7 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.79080\) 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)
 
\(f(q)\) \(=\) \(q-0.790805 q^{2} -9.00000 q^{3} -31.3746 q^{4} +104.192 q^{5} +7.11724 q^{6} +50.1170 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-0.790805 q^{2} -9.00000 q^{3} -31.3746 q^{4} +104.192 q^{5} +7.11724 q^{6} +50.1170 q^{8} +81.0000 q^{9} -82.3953 q^{10} -497.660 q^{11} +282.372 q^{12} +206.551 q^{13} -937.725 q^{15} +964.355 q^{16} -63.1586 q^{17} -64.0552 q^{18} -1323.95 q^{19} -3268.98 q^{20} +393.552 q^{22} -194.437 q^{23} -451.053 q^{24} +7730.91 q^{25} -163.342 q^{26} -729.000 q^{27} +4323.14 q^{29} +741.558 q^{30} +7525.31 q^{31} -2366.36 q^{32} +4478.94 q^{33} +49.9461 q^{34} -2541.34 q^{36} +10355.6 q^{37} +1046.99 q^{38} -1858.96 q^{39} +5221.77 q^{40} +4180.92 q^{41} +5960.87 q^{43} +15613.9 q^{44} +8439.53 q^{45} +153.762 q^{46} +4389.74 q^{47} -8679.20 q^{48} -6113.64 q^{50} +568.427 q^{51} -6480.47 q^{52} +17784.8 q^{53} +576.497 q^{54} -51852.0 q^{55} +11915.6 q^{57} -3418.76 q^{58} -3500.47 q^{59} +29420.8 q^{60} +10632.5 q^{61} -5951.06 q^{62} -28988.0 q^{64} +21520.9 q^{65} -3541.97 q^{66} -13274.7 q^{67} +1981.58 q^{68} +1749.94 q^{69} +38811.1 q^{71} +4059.47 q^{72} -31375.6 q^{73} -8189.30 q^{74} -69578.2 q^{75} +41538.6 q^{76} +1470.08 q^{78} +39491.5 q^{79} +100478. q^{80} +6561.00 q^{81} -3306.29 q^{82} +102372. q^{83} -6580.60 q^{85} -4713.88 q^{86} -38908.2 q^{87} -24941.2 q^{88} +112821. q^{89} -6674.02 q^{90} +6100.40 q^{92} -67727.8 q^{93} -3471.43 q^{94} -137945. q^{95} +21297.2 q^{96} -30334.3 q^{97} -40310.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 3 q^{2} - 36 q^{3} + 69 q^{4} - 27 q^{6} + 123 q^{8} + 324 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 3 q^{2} - 36 q^{3} + 69 q^{4} - 27 q^{6} + 123 q^{8} + 324 q^{9} - 283 q^{10} + 402 q^{11} - 621 q^{12} - 462 q^{13} + 3273 q^{16} - 276 q^{17} + 243 q^{18} - 510 q^{19} - 4719 q^{20} + 1375 q^{22} + 6900 q^{23} - 1107 q^{24} + 2814 q^{25} + 15138 q^{26} - 2916 q^{27} + 540 q^{29} + 2547 q^{30} + 6410 q^{31} + 15519 q^{32} - 3618 q^{33} - 21144 q^{34} + 5589 q^{36} + 15250 q^{37} + 41250 q^{38} + 4158 q^{39} + 8547 q^{40} - 4308 q^{41} + 29198 q^{43} + 70743 q^{44} + 61800 q^{46} + 15060 q^{47} - 29457 q^{48} - 7302 q^{50} + 2484 q^{51} + 47476 q^{52} + 13692 q^{53} - 2187 q^{54} - 73124 q^{55} + 4590 q^{57} + 52309 q^{58} - 34830 q^{59} + 42471 q^{60} + 5364 q^{61} - 16029 q^{62} - 73487 q^{64} + 66864 q^{65} - 12375 q^{66} - 5994 q^{67} + 58272 q^{68} - 62100 q^{69} + 89268 q^{71} + 9963 q^{72} - 59638 q^{73} - 185442 q^{74} - 25326 q^{75} - 21308 q^{76} - 136242 q^{78} - 44062 q^{79} + 33381 q^{80} + 26244 q^{81} - 57596 q^{82} + 208446 q^{83} + 36324 q^{85} - 136968 q^{86} - 4860 q^{87} + 87597 q^{88} + 77520 q^{89} - 22923 q^{90} + 158256 q^{92} - 57690 q^{93} + 73722 q^{94} - 221376 q^{95} - 139671 q^{96} + 188630 q^{97} + 32562 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.790805 −0.139796 −0.0698979 0.997554i \(-0.522267\pi\)
−0.0698979 + 0.997554i \(0.522267\pi\)
\(3\) −9.00000 −0.577350
\(4\) −31.3746 −0.980457
\(5\) 104.192 1.86384 0.931919 0.362667i \(-0.118134\pi\)
0.931919 + 0.362667i \(0.118134\pi\)
\(6\) 7.11724 0.0807112
\(7\) 0 0
\(8\) 50.1170 0.276860
\(9\) 81.0000 0.333333
\(10\) −82.3953 −0.260557
\(11\) −497.660 −1.24008 −0.620041 0.784569i \(-0.712884\pi\)
−0.620041 + 0.784569i \(0.712884\pi\)
\(12\) 282.372 0.566067
\(13\) 206.551 0.338977 0.169488 0.985532i \(-0.445789\pi\)
0.169488 + 0.985532i \(0.445789\pi\)
\(14\) 0 0
\(15\) −937.725 −1.07609
\(16\) 964.355 0.941753
\(17\) −63.1586 −0.0530042 −0.0265021 0.999649i \(-0.508437\pi\)
−0.0265021 + 0.999649i \(0.508437\pi\)
\(18\) −64.0552 −0.0465986
\(19\) −1323.95 −0.841374 −0.420687 0.907206i \(-0.638211\pi\)
−0.420687 + 0.907206i \(0.638211\pi\)
\(20\) −3268.98 −1.82741
\(21\) 0 0
\(22\) 393.552 0.173358
\(23\) −194.437 −0.0766408 −0.0383204 0.999266i \(-0.512201\pi\)
−0.0383204 + 0.999266i \(0.512201\pi\)
\(24\) −451.053 −0.159845
\(25\) 7730.91 2.47389
\(26\) −163.342 −0.0473875
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) 4323.14 0.954562 0.477281 0.878751i \(-0.341622\pi\)
0.477281 + 0.878751i \(0.341622\pi\)
\(30\) 741.558 0.150433
\(31\) 7525.31 1.40644 0.703219 0.710974i \(-0.251745\pi\)
0.703219 + 0.710974i \(0.251745\pi\)
\(32\) −2366.36 −0.408513
\(33\) 4478.94 0.715962
\(34\) 49.9461 0.00740977
\(35\) 0 0
\(36\) −2541.34 −0.326819
\(37\) 10355.6 1.24358 0.621789 0.783185i \(-0.286406\pi\)
0.621789 + 0.783185i \(0.286406\pi\)
\(38\) 1046.99 0.117621
\(39\) −1858.96 −0.195708
\(40\) 5221.77 0.516022
\(41\) 4180.92 0.388429 0.194215 0.980959i \(-0.437784\pi\)
0.194215 + 0.980959i \(0.437784\pi\)
\(42\) 0 0
\(43\) 5960.87 0.491630 0.245815 0.969317i \(-0.420944\pi\)
0.245815 + 0.969317i \(0.420944\pi\)
\(44\) 15613.9 1.21585
\(45\) 8439.53 0.621279
\(46\) 153.762 0.0107141
\(47\) 4389.74 0.289864 0.144932 0.989442i \(-0.453704\pi\)
0.144932 + 0.989442i \(0.453704\pi\)
\(48\) −8679.20 −0.543721
\(49\) 0 0
\(50\) −6113.64 −0.345840
\(51\) 568.427 0.0306020
\(52\) −6480.47 −0.332352
\(53\) 17784.8 0.869679 0.434839 0.900508i \(-0.356805\pi\)
0.434839 + 0.900508i \(0.356805\pi\)
\(54\) 576.497 0.0269037
\(55\) −51852.0 −2.31131
\(56\) 0 0
\(57\) 11915.6 0.485768
\(58\) −3418.76 −0.133444
\(59\) −3500.47 −0.130917 −0.0654585 0.997855i \(-0.520851\pi\)
−0.0654585 + 0.997855i \(0.520851\pi\)
\(60\) 29420.8 1.05506
\(61\) 10632.5 0.365855 0.182928 0.983126i \(-0.441443\pi\)
0.182928 + 0.983126i \(0.441443\pi\)
\(62\) −5951.06 −0.196614
\(63\) 0 0
\(64\) −28988.0 −0.884645
\(65\) 21520.9 0.631797
\(66\) −3541.97 −0.100089
\(67\) −13274.7 −0.361276 −0.180638 0.983550i \(-0.557816\pi\)
−0.180638 + 0.983550i \(0.557816\pi\)
\(68\) 1981.58 0.0519683
\(69\) 1749.94 0.0442486
\(70\) 0 0
\(71\) 38811.1 0.913713 0.456857 0.889540i \(-0.348975\pi\)
0.456857 + 0.889540i \(0.348975\pi\)
\(72\) 4059.47 0.0922866
\(73\) −31375.6 −0.689105 −0.344552 0.938767i \(-0.611969\pi\)
−0.344552 + 0.938767i \(0.611969\pi\)
\(74\) −8189.30 −0.173847
\(75\) −69578.2 −1.42830
\(76\) 41538.6 0.824931
\(77\) 0 0
\(78\) 1470.08 0.0273592
\(79\) 39491.5 0.711927 0.355964 0.934500i \(-0.384153\pi\)
0.355964 + 0.934500i \(0.384153\pi\)
\(80\) 100478. 1.75528
\(81\) 6561.00 0.111111
\(82\) −3306.29 −0.0543008
\(83\) 102372. 1.63112 0.815559 0.578675i \(-0.196430\pi\)
0.815559 + 0.578675i \(0.196430\pi\)
\(84\) 0 0
\(85\) −6580.60 −0.0987912
\(86\) −4713.88 −0.0687279
\(87\) −38908.2 −0.551117
\(88\) −24941.2 −0.343329
\(89\) 112821. 1.50978 0.754892 0.655849i \(-0.227689\pi\)
0.754892 + 0.655849i \(0.227689\pi\)
\(90\) −6674.02 −0.0868523
\(91\) 0 0
\(92\) 6100.40 0.0751430
\(93\) −67727.8 −0.812007
\(94\) −3471.43 −0.0405218
\(95\) −137945. −1.56818
\(96\) 21297.2 0.235855
\(97\) −30334.3 −0.327345 −0.163672 0.986515i \(-0.552334\pi\)
−0.163672 + 0.986515i \(0.552334\pi\)
\(98\) 0 0
\(99\) −40310.4 −0.413361
\(100\) −242554. −2.42554
\(101\) −106796. −1.04172 −0.520862 0.853641i \(-0.674390\pi\)
−0.520862 + 0.853641i \(0.674390\pi\)
\(102\) −449.515 −0.00427803
\(103\) −157536. −1.46314 −0.731570 0.681767i \(-0.761212\pi\)
−0.731570 + 0.681767i \(0.761212\pi\)
\(104\) 10351.7 0.0938490
\(105\) 0 0
\(106\) −14064.3 −0.121578
\(107\) −89324.3 −0.754241 −0.377121 0.926164i \(-0.623086\pi\)
−0.377121 + 0.926164i \(0.623086\pi\)
\(108\) 22872.1 0.188689
\(109\) 167605. 1.35120 0.675600 0.737268i \(-0.263885\pi\)
0.675600 + 0.737268i \(0.263885\pi\)
\(110\) 41004.8 0.323112
\(111\) −93200.8 −0.717980
\(112\) 0 0
\(113\) −115794. −0.853079 −0.426539 0.904469i \(-0.640267\pi\)
−0.426539 + 0.904469i \(0.640267\pi\)
\(114\) −9422.91 −0.0679083
\(115\) −20258.8 −0.142846
\(116\) −135637. −0.935907
\(117\) 16730.7 0.112992
\(118\) 2768.19 0.0183017
\(119\) 0 0
\(120\) −46995.9 −0.297925
\(121\) 86614.1 0.537805
\(122\) −8408.20 −0.0511451
\(123\) −37628.2 −0.224260
\(124\) −236104. −1.37895
\(125\) 479898. 2.74709
\(126\) 0 0
\(127\) 201513. 1.10865 0.554325 0.832300i \(-0.312977\pi\)
0.554325 + 0.832300i \(0.312977\pi\)
\(128\) 98647.4 0.532183
\(129\) −53647.8 −0.283843
\(130\) −17018.9 −0.0883227
\(131\) −38469.6 −0.195857 −0.0979285 0.995193i \(-0.531222\pi\)
−0.0979285 + 0.995193i \(0.531222\pi\)
\(132\) −140525. −0.701970
\(133\) 0 0
\(134\) 10497.7 0.0505049
\(135\) −75955.7 −0.358696
\(136\) −3165.32 −0.0146747
\(137\) −241722. −1.10031 −0.550155 0.835063i \(-0.685431\pi\)
−0.550155 + 0.835063i \(0.685431\pi\)
\(138\) −1383.86 −0.00618577
\(139\) 53112.2 0.233162 0.116581 0.993181i \(-0.462807\pi\)
0.116581 + 0.993181i \(0.462807\pi\)
\(140\) 0 0
\(141\) −39507.6 −0.167353
\(142\) −30692.0 −0.127733
\(143\) −102792. −0.420359
\(144\) 78112.8 0.313918
\(145\) 450435. 1.77915
\(146\) 24812.0 0.0963340
\(147\) 0 0
\(148\) −324905. −1.21927
\(149\) 129062. 0.476248 0.238124 0.971235i \(-0.423468\pi\)
0.238124 + 0.971235i \(0.423468\pi\)
\(150\) 55022.8 0.199671
\(151\) 153206. 0.546808 0.273404 0.961899i \(-0.411850\pi\)
0.273404 + 0.961899i \(0.411850\pi\)
\(152\) −66352.6 −0.232943
\(153\) −5115.85 −0.0176681
\(154\) 0 0
\(155\) 784075. 2.62137
\(156\) 58324.2 0.191884
\(157\) 151188. 0.489517 0.244758 0.969584i \(-0.421291\pi\)
0.244758 + 0.969584i \(0.421291\pi\)
\(158\) −31230.0 −0.0995245
\(159\) −160063. −0.502109
\(160\) −246555. −0.761402
\(161\) 0 0
\(162\) −5188.47 −0.0155329
\(163\) −32916.7 −0.0970392 −0.0485196 0.998822i \(-0.515450\pi\)
−0.0485196 + 0.998822i \(0.515450\pi\)
\(164\) −131175. −0.380838
\(165\) 466668. 1.33444
\(166\) −80956.1 −0.228023
\(167\) −217586. −0.603725 −0.301862 0.953352i \(-0.597608\pi\)
−0.301862 + 0.953352i \(0.597608\pi\)
\(168\) 0 0
\(169\) −328630. −0.885095
\(170\) 5203.97 0.0138106
\(171\) −107240. −0.280458
\(172\) −187020. −0.482022
\(173\) 421242. 1.07008 0.535041 0.844826i \(-0.320296\pi\)
0.535041 + 0.844826i \(0.320296\pi\)
\(174\) 30768.8 0.0770438
\(175\) 0 0
\(176\) −479921. −1.16785
\(177\) 31504.2 0.0755849
\(178\) −89219.4 −0.211062
\(179\) 3494.06 0.00815075 0.00407538 0.999992i \(-0.498703\pi\)
0.00407538 + 0.999992i \(0.498703\pi\)
\(180\) −264787. −0.609138
\(181\) −594611. −1.34908 −0.674538 0.738240i \(-0.735657\pi\)
−0.674538 + 0.738240i \(0.735657\pi\)
\(182\) 0 0
\(183\) −95692.2 −0.211227
\(184\) −9744.61 −0.0212188
\(185\) 1.07897e6 2.31783
\(186\) 53559.5 0.113515
\(187\) 31431.5 0.0657296
\(188\) −137726. −0.284199
\(189\) 0 0
\(190\) 109088. 0.219226
\(191\) 828644. 1.64356 0.821778 0.569807i \(-0.192982\pi\)
0.821778 + 0.569807i \(0.192982\pi\)
\(192\) 260892. 0.510750
\(193\) 219739. 0.424633 0.212316 0.977201i \(-0.431899\pi\)
0.212316 + 0.977201i \(0.431899\pi\)
\(194\) 23988.5 0.0457614
\(195\) −193688. −0.364768
\(196\) 0 0
\(197\) 475612. 0.873146 0.436573 0.899669i \(-0.356192\pi\)
0.436573 + 0.899669i \(0.356192\pi\)
\(198\) 31877.7 0.0577862
\(199\) −627555. −1.12336 −0.561681 0.827354i \(-0.689845\pi\)
−0.561681 + 0.827354i \(0.689845\pi\)
\(200\) 387450. 0.684921
\(201\) 119473. 0.208583
\(202\) 84455.1 0.145629
\(203\) 0 0
\(204\) −17834.2 −0.0300039
\(205\) 435617. 0.723969
\(206\) 124580. 0.204541
\(207\) −15749.4 −0.0255469
\(208\) 199189. 0.319232
\(209\) 658879. 1.04337
\(210\) 0 0
\(211\) 570989. 0.882920 0.441460 0.897281i \(-0.354461\pi\)
0.441460 + 0.897281i \(0.354461\pi\)
\(212\) −557991. −0.852683
\(213\) −349300. −0.527533
\(214\) 70638.1 0.105440
\(215\) 621073. 0.916319
\(216\) −36535.3 −0.0532817
\(217\) 0 0
\(218\) −132543. −0.188892
\(219\) 282381. 0.397855
\(220\) 1.62684e6 2.26614
\(221\) −13045.5 −0.0179672
\(222\) 73703.7 0.100371
\(223\) −4233.11 −0.00570029 −0.00285015 0.999996i \(-0.500907\pi\)
−0.00285015 + 0.999996i \(0.500907\pi\)
\(224\) 0 0
\(225\) 626204. 0.824630
\(226\) 91570.3 0.119257
\(227\) −1.12986e6 −1.45533 −0.727664 0.685934i \(-0.759394\pi\)
−0.727664 + 0.685934i \(0.759394\pi\)
\(228\) −373847. −0.476274
\(229\) −804643. −1.01395 −0.506973 0.861962i \(-0.669236\pi\)
−0.506973 + 0.861962i \(0.669236\pi\)
\(230\) 16020.7 0.0199693
\(231\) 0 0
\(232\) 216663. 0.264280
\(233\) 1.16927e6 1.41100 0.705498 0.708712i \(-0.250723\pi\)
0.705498 + 0.708712i \(0.250723\pi\)
\(234\) −13230.7 −0.0157958
\(235\) 457374. 0.540259
\(236\) 109826. 0.128358
\(237\) −355423. −0.411031
\(238\) 0 0
\(239\) −1.70554e6 −1.93138 −0.965689 0.259700i \(-0.916376\pi\)
−0.965689 + 0.259700i \(0.916376\pi\)
\(240\) −904300. −1.01341
\(241\) −951196. −1.05494 −0.527470 0.849574i \(-0.676859\pi\)
−0.527470 + 0.849574i \(0.676859\pi\)
\(242\) −68494.9 −0.0751830
\(243\) −59049.0 −0.0641500
\(244\) −333590. −0.358705
\(245\) 0 0
\(246\) 29756.6 0.0313506
\(247\) −273465. −0.285206
\(248\) 377146. 0.389386
\(249\) −921346. −0.941726
\(250\) −379505. −0.384032
\(251\) −1.14498e6 −1.14713 −0.573566 0.819159i \(-0.694440\pi\)
−0.573566 + 0.819159i \(0.694440\pi\)
\(252\) 0 0
\(253\) 96763.6 0.0950409
\(254\) −159358. −0.154985
\(255\) 59225.4 0.0570371
\(256\) 849606. 0.810248
\(257\) −1.18374e6 −1.11795 −0.558976 0.829184i \(-0.688806\pi\)
−0.558976 + 0.829184i \(0.688806\pi\)
\(258\) 42425.0 0.0396800
\(259\) 0 0
\(260\) −675211. −0.619450
\(261\) 350174. 0.318187
\(262\) 30421.9 0.0273800
\(263\) 448316. 0.399664 0.199832 0.979830i \(-0.435960\pi\)
0.199832 + 0.979830i \(0.435960\pi\)
\(264\) 224471. 0.198221
\(265\) 1.85303e6 1.62094
\(266\) 0 0
\(267\) −1.01539e6 −0.871674
\(268\) 416490. 0.354216
\(269\) −747474. −0.629819 −0.314909 0.949122i \(-0.601974\pi\)
−0.314909 + 0.949122i \(0.601974\pi\)
\(270\) 60066.2 0.0501442
\(271\) 232637. 0.192423 0.0962114 0.995361i \(-0.469328\pi\)
0.0962114 + 0.995361i \(0.469328\pi\)
\(272\) −60907.3 −0.0499169
\(273\) 0 0
\(274\) 191155. 0.153819
\(275\) −3.84736e6 −3.06783
\(276\) −54903.6 −0.0433838
\(277\) −2.42924e6 −1.90227 −0.951134 0.308778i \(-0.900080\pi\)
−0.951134 + 0.308778i \(0.900080\pi\)
\(278\) −42001.4 −0.0325951
\(279\) 609550. 0.468812
\(280\) 0 0
\(281\) 2.51704e6 1.90163 0.950813 0.309766i \(-0.100251\pi\)
0.950813 + 0.309766i \(0.100251\pi\)
\(282\) 31242.8 0.0233953
\(283\) −260994. −0.193716 −0.0968579 0.995298i \(-0.530879\pi\)
−0.0968579 + 0.995298i \(0.530879\pi\)
\(284\) −1.21768e6 −0.895857
\(285\) 1.24151e6 0.905392
\(286\) 81288.6 0.0587645
\(287\) 0 0
\(288\) −191675. −0.136171
\(289\) −1.41587e6 −0.997191
\(290\) −356206. −0.248718
\(291\) 273009. 0.188993
\(292\) 984399. 0.675638
\(293\) −65011.7 −0.0442408 −0.0221204 0.999755i \(-0.507042\pi\)
−0.0221204 + 0.999755i \(0.507042\pi\)
\(294\) 0 0
\(295\) −364720. −0.244008
\(296\) 518994. 0.344297
\(297\) 362794. 0.238654
\(298\) −102063. −0.0665775
\(299\) −40161.3 −0.0259794
\(300\) 2.18299e6 1.40039
\(301\) 0 0
\(302\) −121156. −0.0764414
\(303\) 961167. 0.601440
\(304\) −1.27676e6 −0.792367
\(305\) 1.10781e6 0.681895
\(306\) 4045.64 0.00246992
\(307\) 2.35599e6 1.42668 0.713342 0.700816i \(-0.247181\pi\)
0.713342 + 0.700816i \(0.247181\pi\)
\(308\) 0 0
\(309\) 1.41782e6 0.844744
\(310\) −620051. −0.366457
\(311\) 2.11805e6 1.24176 0.620878 0.783907i \(-0.286776\pi\)
0.620878 + 0.783907i \(0.286776\pi\)
\(312\) −93165.6 −0.0541837
\(313\) 187514. 0.108186 0.0540931 0.998536i \(-0.482773\pi\)
0.0540931 + 0.998536i \(0.482773\pi\)
\(314\) −119560. −0.0684324
\(315\) 0 0
\(316\) −1.23903e6 −0.698014
\(317\) −1.00541e6 −0.561947 −0.280974 0.959716i \(-0.590657\pi\)
−0.280974 + 0.959716i \(0.590657\pi\)
\(318\) 126579. 0.0701928
\(319\) −2.15145e6 −1.18374
\(320\) −3.02031e6 −1.64883
\(321\) 803919. 0.435461
\(322\) 0 0
\(323\) 83619.1 0.0445964
\(324\) −205849. −0.108940
\(325\) 1.59683e6 0.838591
\(326\) 26030.7 0.0135657
\(327\) −1.50844e6 −0.780116
\(328\) 209535. 0.107540
\(329\) 0 0
\(330\) −369043. −0.186549
\(331\) 1.67984e6 0.842748 0.421374 0.906887i \(-0.361548\pi\)
0.421374 + 0.906887i \(0.361548\pi\)
\(332\) −3.21188e6 −1.59924
\(333\) 838807. 0.414526
\(334\) 172068. 0.0843982
\(335\) −1.38312e6 −0.673360
\(336\) 0 0
\(337\) −995036. −0.477270 −0.238635 0.971109i \(-0.576700\pi\)
−0.238635 + 0.971109i \(0.576700\pi\)
\(338\) 259882. 0.123733
\(339\) 1.04214e6 0.492525
\(340\) 206464. 0.0968605
\(341\) −3.74505e6 −1.74410
\(342\) 84806.2 0.0392069
\(343\) 0 0
\(344\) 298741. 0.136113
\(345\) 182329. 0.0824722
\(346\) −333121. −0.149593
\(347\) 4.05665e6 1.80861 0.904304 0.426890i \(-0.140391\pi\)
0.904304 + 0.426890i \(0.140391\pi\)
\(348\) 1.22073e6 0.540346
\(349\) −2.86202e6 −1.25779 −0.628897 0.777488i \(-0.716493\pi\)
−0.628897 + 0.777488i \(0.716493\pi\)
\(350\) 0 0
\(351\) −150576. −0.0652361
\(352\) 1.17764e6 0.506590
\(353\) −832685. −0.355667 −0.177834 0.984061i \(-0.556909\pi\)
−0.177834 + 0.984061i \(0.556909\pi\)
\(354\) −24913.7 −0.0105665
\(355\) 4.04379e6 1.70301
\(356\) −3.53972e6 −1.48028
\(357\) 0 0
\(358\) −2763.12 −0.00113944
\(359\) −2.69751e6 −1.10465 −0.552327 0.833627i \(-0.686260\pi\)
−0.552327 + 0.833627i \(0.686260\pi\)
\(360\) 422964. 0.172007
\(361\) −723243. −0.292090
\(362\) 470221. 0.188595
\(363\) −779527. −0.310502
\(364\) 0 0
\(365\) −3.26908e6 −1.28438
\(366\) 75673.8 0.0295286
\(367\) −1.45482e6 −0.563826 −0.281913 0.959440i \(-0.590969\pi\)
−0.281913 + 0.959440i \(0.590969\pi\)
\(368\) −187507. −0.0721767
\(369\) 338654. 0.129476
\(370\) −853257. −0.324023
\(371\) 0 0
\(372\) 2.12494e6 0.796138
\(373\) −2.04689e6 −0.761766 −0.380883 0.924623i \(-0.624380\pi\)
−0.380883 + 0.924623i \(0.624380\pi\)
\(374\) −24856.2 −0.00918872
\(375\) −4.31908e6 −1.58604
\(376\) 220000. 0.0802516
\(377\) 892950. 0.323574
\(378\) 0 0
\(379\) −416898. −0.149084 −0.0745421 0.997218i \(-0.523750\pi\)
−0.0745421 + 0.997218i \(0.523750\pi\)
\(380\) 4.32798e6 1.53754
\(381\) −1.81362e6 −0.640079
\(382\) −655296. −0.229762
\(383\) 3.87626e6 1.35025 0.675127 0.737701i \(-0.264089\pi\)
0.675127 + 0.737701i \(0.264089\pi\)
\(384\) −887827. −0.307256
\(385\) 0 0
\(386\) −173771. −0.0593619
\(387\) 482830. 0.163877
\(388\) 951729. 0.320947
\(389\) 2.83802e6 0.950915 0.475458 0.879739i \(-0.342283\pi\)
0.475458 + 0.879739i \(0.342283\pi\)
\(390\) 153170. 0.0509931
\(391\) 12280.4 0.00406228
\(392\) 0 0
\(393\) 346226. 0.113078
\(394\) −376116. −0.122062
\(395\) 4.11468e6 1.32692
\(396\) 1.26472e6 0.405283
\(397\) −4.34266e6 −1.38286 −0.691432 0.722442i \(-0.743020\pi\)
−0.691432 + 0.722442i \(0.743020\pi\)
\(398\) 496274. 0.157041
\(399\) 0 0
\(400\) 7.45534e6 2.32979
\(401\) 3.40304e6 1.05683 0.528417 0.848985i \(-0.322786\pi\)
0.528417 + 0.848985i \(0.322786\pi\)
\(402\) −94479.6 −0.0291590
\(403\) 1.55436e6 0.476749
\(404\) 3.35070e6 1.02137
\(405\) 683602. 0.207093
\(406\) 0 0
\(407\) −5.15359e6 −1.54214
\(408\) 28487.9 0.00847246
\(409\) 5.29431e6 1.56495 0.782477 0.622680i \(-0.213956\pi\)
0.782477 + 0.622680i \(0.213956\pi\)
\(410\) −344488. −0.101208
\(411\) 2.17550e6 0.635264
\(412\) 4.94262e6 1.43455
\(413\) 0 0
\(414\) 12454.7 0.00357136
\(415\) 1.06663e7 3.04014
\(416\) −488775. −0.138476
\(417\) −478010. −0.134616
\(418\) −521045. −0.145859
\(419\) −2.87267e6 −0.799376 −0.399688 0.916651i \(-0.630881\pi\)
−0.399688 + 0.916651i \(0.630881\pi\)
\(420\) 0 0
\(421\) 2.08688e6 0.573843 0.286921 0.957954i \(-0.407368\pi\)
0.286921 + 0.957954i \(0.407368\pi\)
\(422\) −451541. −0.123429
\(423\) 355569. 0.0966213
\(424\) 891319. 0.240779
\(425\) −488273. −0.131127
\(426\) 276228. 0.0737469
\(427\) 0 0
\(428\) 2.80252e6 0.739501
\(429\) 925130. 0.242694
\(430\) −491148. −0.128098
\(431\) −2.42863e6 −0.629751 −0.314876 0.949133i \(-0.601963\pi\)
−0.314876 + 0.949133i \(0.601963\pi\)
\(432\) −703015. −0.181240
\(433\) −956219. −0.245097 −0.122548 0.992463i \(-0.539107\pi\)
−0.122548 + 0.992463i \(0.539107\pi\)
\(434\) 0 0
\(435\) −4.05392e6 −1.02719
\(436\) −5.25853e6 −1.32479
\(437\) 257426. 0.0644836
\(438\) −223308. −0.0556185
\(439\) −2.64205e6 −0.654304 −0.327152 0.944972i \(-0.606089\pi\)
−0.327152 + 0.944972i \(0.606089\pi\)
\(440\) −2.59867e6 −0.639910
\(441\) 0 0
\(442\) 10316.4 0.00251174
\(443\) −3.43966e6 −0.832733 −0.416366 0.909197i \(-0.636697\pi\)
−0.416366 + 0.909197i \(0.636697\pi\)
\(444\) 2.92414e6 0.703949
\(445\) 1.17550e7 2.81399
\(446\) 3347.56 0.000796878 0
\(447\) −1.16156e6 −0.274962
\(448\) 0 0
\(449\) 4.39903e6 1.02977 0.514886 0.857259i \(-0.327834\pi\)
0.514886 + 0.857259i \(0.327834\pi\)
\(450\) −495205. −0.115280
\(451\) −2.08067e6 −0.481684
\(452\) 3.63299e6 0.836407
\(453\) −1.37886e6 −0.315699
\(454\) 893501. 0.203449
\(455\) 0 0
\(456\) 597173. 0.134490
\(457\) −2.25110e6 −0.504202 −0.252101 0.967701i \(-0.581122\pi\)
−0.252101 + 0.967701i \(0.581122\pi\)
\(458\) 636316. 0.141745
\(459\) 46042.6 0.0102007
\(460\) 635611. 0.140054
\(461\) −1.85307e6 −0.406107 −0.203053 0.979168i \(-0.565087\pi\)
−0.203053 + 0.979168i \(0.565087\pi\)
\(462\) 0 0
\(463\) −3.01089e6 −0.652744 −0.326372 0.945241i \(-0.605826\pi\)
−0.326372 + 0.945241i \(0.605826\pi\)
\(464\) 4.16904e6 0.898962
\(465\) −7.05668e6 −1.51345
\(466\) −924667. −0.197252
\(467\) 1.06725e6 0.226452 0.113226 0.993569i \(-0.463882\pi\)
0.113226 + 0.993569i \(0.463882\pi\)
\(468\) −524918. −0.110784
\(469\) 0 0
\(470\) −361694. −0.0755260
\(471\) −1.36069e6 −0.282623
\(472\) −175433. −0.0362456
\(473\) −2.96648e6 −0.609662
\(474\) 281070. 0.0574605
\(475\) −1.02354e7 −2.08147
\(476\) 0 0
\(477\) 1.44057e6 0.289893
\(478\) 1.34875e6 0.269999
\(479\) 5.68788e6 1.13269 0.566346 0.824168i \(-0.308357\pi\)
0.566346 + 0.824168i \(0.308357\pi\)
\(480\) 2.21900e6 0.439596
\(481\) 2.13897e6 0.421544
\(482\) 752211. 0.147476
\(483\) 0 0
\(484\) −2.71749e6 −0.527295
\(485\) −3.16059e6 −0.610117
\(486\) 46696.2 0.00896791
\(487\) −6.84477e6 −1.30779 −0.653893 0.756587i \(-0.726865\pi\)
−0.653893 + 0.756587i \(0.726865\pi\)
\(488\) 532867. 0.101291
\(489\) 296250. 0.0560256
\(490\) 0 0
\(491\) 5.60132e6 1.04854 0.524272 0.851551i \(-0.324338\pi\)
0.524272 + 0.851551i \(0.324338\pi\)
\(492\) 1.18057e6 0.219877
\(493\) −273043. −0.0505958
\(494\) 216257. 0.0398706
\(495\) −4.20001e6 −0.770438
\(496\) 7.25708e6 1.32452
\(497\) 0 0
\(498\) 728605. 0.131649
\(499\) −2.33529e6 −0.419845 −0.209923 0.977718i \(-0.567321\pi\)
−0.209923 + 0.977718i \(0.567321\pi\)
\(500\) −1.50566e7 −2.69341
\(501\) 1.95827e6 0.348561
\(502\) 905455. 0.160364
\(503\) 1.08278e7 1.90819 0.954093 0.299510i \(-0.0968233\pi\)
0.954093 + 0.299510i \(0.0968233\pi\)
\(504\) 0 0
\(505\) −1.11273e7 −1.94161
\(506\) −76521.1 −0.0132863
\(507\) 2.95767e6 0.511010
\(508\) −6.32240e6 −1.08698
\(509\) −3.17847e6 −0.543781 −0.271890 0.962328i \(-0.587649\pi\)
−0.271890 + 0.962328i \(0.587649\pi\)
\(510\) −46835.7 −0.00797356
\(511\) 0 0
\(512\) −3.82859e6 −0.645452
\(513\) 965163. 0.161923
\(514\) 936107. 0.156285
\(515\) −1.64139e7 −2.72705
\(516\) 1.68318e6 0.278296
\(517\) −2.18460e6 −0.359455
\(518\) 0 0
\(519\) −3.79118e6 −0.617812
\(520\) 1.07856e6 0.174919
\(521\) 6.35837e6 1.02625 0.513123 0.858315i \(-0.328489\pi\)
0.513123 + 0.858315i \(0.328489\pi\)
\(522\) −276919. −0.0444813
\(523\) 7.22187e6 1.15450 0.577252 0.816566i \(-0.304125\pi\)
0.577252 + 0.816566i \(0.304125\pi\)
\(524\) 1.20697e6 0.192029
\(525\) 0 0
\(526\) −354531. −0.0558714
\(527\) −475288. −0.0745471
\(528\) 4.31929e6 0.674260
\(529\) −6.39854e6 −0.994126
\(530\) −1.46538e6 −0.226601
\(531\) −283538. −0.0436390
\(532\) 0 0
\(533\) 863574. 0.131668
\(534\) 802974. 0.121856
\(535\) −9.30685e6 −1.40578
\(536\) −665290. −0.100023
\(537\) −31446.5 −0.00470584
\(538\) 591106. 0.0880461
\(539\) 0 0
\(540\) 2.38308e6 0.351686
\(541\) −9.43165e6 −1.38546 −0.692731 0.721196i \(-0.743593\pi\)
−0.692731 + 0.721196i \(0.743593\pi\)
\(542\) −183971. −0.0268999
\(543\) 5.35150e6 0.778889
\(544\) 149456. 0.0216529
\(545\) 1.74630e7 2.51842
\(546\) 0 0
\(547\) 9.91568e6 1.41695 0.708474 0.705737i \(-0.249384\pi\)
0.708474 + 0.705737i \(0.249384\pi\)
\(548\) 7.58394e6 1.07881
\(549\) 861230. 0.121952
\(550\) 3.04251e6 0.428870
\(551\) −5.72364e6 −0.803144
\(552\) 87701.5 0.0122507
\(553\) 0 0
\(554\) 1.92106e6 0.265929
\(555\) −9.71075e6 −1.33820
\(556\) −1.66638e6 −0.228605
\(557\) 9.64613e6 1.31739 0.658696 0.752409i \(-0.271108\pi\)
0.658696 + 0.752409i \(0.271108\pi\)
\(558\) −482036. −0.0655381
\(559\) 1.23123e6 0.166651
\(560\) 0 0
\(561\) −282883. −0.0379490
\(562\) −1.99049e6 −0.265839
\(563\) −3.09504e6 −0.411524 −0.205762 0.978602i \(-0.565967\pi\)
−0.205762 + 0.978602i \(0.565967\pi\)
\(564\) 1.23954e6 0.164082
\(565\) −1.20648e7 −1.59000
\(566\) 206396. 0.0270807
\(567\) 0 0
\(568\) 1.94509e6 0.252970
\(569\) −1.23185e7 −1.59507 −0.797533 0.603275i \(-0.793862\pi\)
−0.797533 + 0.603275i \(0.793862\pi\)
\(570\) −981789. −0.126570
\(571\) −3.23510e6 −0.415238 −0.207619 0.978210i \(-0.566571\pi\)
−0.207619 + 0.978210i \(0.566571\pi\)
\(572\) 3.22507e6 0.412144
\(573\) −7.45780e6 −0.948908
\(574\) 0 0
\(575\) −1.50318e6 −0.189601
\(576\) −2.34803e6 −0.294882
\(577\) 9.05049e6 1.13170 0.565852 0.824507i \(-0.308547\pi\)
0.565852 + 0.824507i \(0.308547\pi\)
\(578\) 1.11968e6 0.139403
\(579\) −1.97765e6 −0.245162
\(580\) −1.41322e7 −1.74438
\(581\) 0 0
\(582\) −215897. −0.0264204
\(583\) −8.85077e6 −1.07847
\(584\) −1.57245e6 −0.190785
\(585\) 1.74320e6 0.210599
\(586\) 51411.6 0.00618468
\(587\) 2.13170e6 0.255347 0.127673 0.991816i \(-0.459249\pi\)
0.127673 + 0.991816i \(0.459249\pi\)
\(588\) 0 0
\(589\) −9.96318e6 −1.18334
\(590\) 288422. 0.0341113
\(591\) −4.28051e6 −0.504111
\(592\) 9.98652e6 1.17114
\(593\) 7.26933e6 0.848903 0.424451 0.905451i \(-0.360467\pi\)
0.424451 + 0.905451i \(0.360467\pi\)
\(594\) −286899. −0.0333629
\(595\) 0 0
\(596\) −4.04927e6 −0.466941
\(597\) 5.64800e6 0.648573
\(598\) 31759.8 0.00363182
\(599\) −2.20565e6 −0.251171 −0.125585 0.992083i \(-0.540081\pi\)
−0.125585 + 0.992083i \(0.540081\pi\)
\(600\) −3.48705e6 −0.395439
\(601\) 1.00121e7 1.13067 0.565336 0.824860i \(-0.308746\pi\)
0.565336 + 0.824860i \(0.308746\pi\)
\(602\) 0 0
\(603\) −1.07525e6 −0.120425
\(604\) −4.80679e6 −0.536121
\(605\) 9.02447e6 1.00238
\(606\) −760096. −0.0840789
\(607\) −6.32139e6 −0.696371 −0.348185 0.937426i \(-0.613202\pi\)
−0.348185 + 0.937426i \(0.613202\pi\)
\(608\) 3.13295e6 0.343712
\(609\) 0 0
\(610\) −876065. −0.0953261
\(611\) 906706. 0.0982570
\(612\) 160508. 0.0173228
\(613\) 1.43078e7 1.53788 0.768941 0.639319i \(-0.220784\pi\)
0.768941 + 0.639319i \(0.220784\pi\)
\(614\) −1.86313e6 −0.199445
\(615\) −3.92055e6 −0.417984
\(616\) 0 0
\(617\) 1.73991e7 1.83999 0.919993 0.391936i \(-0.128194\pi\)
0.919993 + 0.391936i \(0.128194\pi\)
\(618\) −1.12122e6 −0.118092
\(619\) −8.26736e6 −0.867242 −0.433621 0.901095i \(-0.642764\pi\)
−0.433621 + 0.901095i \(0.642764\pi\)
\(620\) −2.46001e7 −2.57014
\(621\) 141745. 0.0147495
\(622\) −1.67497e6 −0.173592
\(623\) 0 0
\(624\) −1.79270e6 −0.184309
\(625\) 2.58422e7 2.64625
\(626\) −148287. −0.0151240
\(627\) −5.92991e6 −0.602392
\(628\) −4.74346e6 −0.479950
\(629\) −654048. −0.0659148
\(630\) 0 0
\(631\) 2.83238e6 0.283190 0.141595 0.989925i \(-0.454777\pi\)
0.141595 + 0.989925i \(0.454777\pi\)
\(632\) 1.97919e6 0.197104
\(633\) −5.13890e6 −0.509754
\(634\) 795084. 0.0785579
\(635\) 2.09960e7 2.06634
\(636\) 5.02192e6 0.492297
\(637\) 0 0
\(638\) 1.70138e6 0.165481
\(639\) 3.14370e6 0.304571
\(640\) 1.02782e7 0.991902
\(641\) −3.57176e6 −0.343350 −0.171675 0.985154i \(-0.554918\pi\)
−0.171675 + 0.985154i \(0.554918\pi\)
\(642\) −635743. −0.0608757
\(643\) 5.96911e6 0.569354 0.284677 0.958624i \(-0.408114\pi\)
0.284677 + 0.958624i \(0.408114\pi\)
\(644\) 0 0
\(645\) −5.58966e6 −0.529037
\(646\) −66126.4 −0.00623439
\(647\) 1.68276e6 0.158038 0.0790189 0.996873i \(-0.474821\pi\)
0.0790189 + 0.996873i \(0.474821\pi\)
\(648\) 328817. 0.0307622
\(649\) 1.74204e6 0.162348
\(650\) −1.26278e6 −0.117232
\(651\) 0 0
\(652\) 1.03275e6 0.0951427
\(653\) 1.70305e7 1.56295 0.781475 0.623937i \(-0.214468\pi\)
0.781475 + 0.623937i \(0.214468\pi\)
\(654\) 1.19288e6 0.109057
\(655\) −4.00821e6 −0.365046
\(656\) 4.03189e6 0.365804
\(657\) −2.54143e6 −0.229702
\(658\) 0 0
\(659\) −1.99303e7 −1.78772 −0.893862 0.448342i \(-0.852015\pi\)
−0.893862 + 0.448342i \(0.852015\pi\)
\(660\) −1.46415e7 −1.30836
\(661\) −9.55735e6 −0.850812 −0.425406 0.905003i \(-0.639869\pi\)
−0.425406 + 0.905003i \(0.639869\pi\)
\(662\) −1.32843e6 −0.117813
\(663\) 117409. 0.0103734
\(664\) 5.13056e6 0.451591
\(665\) 0 0
\(666\) −663333. −0.0579490
\(667\) −840579. −0.0731584
\(668\) 6.82666e6 0.591926
\(669\) 38098.0 0.00329107
\(670\) 1.09378e6 0.0941330
\(671\) −5.29135e6 −0.453691
\(672\) 0 0
\(673\) 1.80397e7 1.53530 0.767648 0.640872i \(-0.221427\pi\)
0.767648 + 0.640872i \(0.221427\pi\)
\(674\) 786879. 0.0667204
\(675\) −5.63583e6 −0.476101
\(676\) 1.03106e7 0.867798
\(677\) 1.44741e7 1.21372 0.606861 0.794808i \(-0.292429\pi\)
0.606861 + 0.794808i \(0.292429\pi\)
\(678\) −824133. −0.0688530
\(679\) 0 0
\(680\) −329800. −0.0273513
\(681\) 1.01688e7 0.840234
\(682\) 2.96160e6 0.243818
\(683\) 2.68785e6 0.220472 0.110236 0.993905i \(-0.464839\pi\)
0.110236 + 0.993905i \(0.464839\pi\)
\(684\) 3.36463e6 0.274977
\(685\) −2.51854e7 −2.05080
\(686\) 0 0
\(687\) 7.24179e6 0.585402
\(688\) 5.74839e6 0.462994
\(689\) 3.67347e6 0.294801
\(690\) −144187. −0.0115293
\(691\) −4.67627e6 −0.372567 −0.186284 0.982496i \(-0.559644\pi\)
−0.186284 + 0.982496i \(0.559644\pi\)
\(692\) −1.32163e7 −1.04917
\(693\) 0 0
\(694\) −3.20802e6 −0.252836
\(695\) 5.53385e6 0.434576
\(696\) −1.94996e6 −0.152582
\(697\) −264061. −0.0205884
\(698\) 2.26330e6 0.175834
\(699\) −1.05235e7 −0.814639
\(700\) 0 0
\(701\) 6.34801e6 0.487913 0.243957 0.969786i \(-0.421555\pi\)
0.243957 + 0.969786i \(0.421555\pi\)
\(702\) 119076. 0.00911973
\(703\) −1.37104e7 −1.04631
\(704\) 1.44262e7 1.09703
\(705\) −4.11637e6 −0.311919
\(706\) 658492. 0.0497208
\(707\) 0 0
\(708\) −988433. −0.0741078
\(709\) −6.41052e6 −0.478936 −0.239468 0.970904i \(-0.576973\pi\)
−0.239468 + 0.970904i \(0.576973\pi\)
\(710\) −3.19785e6 −0.238074
\(711\) 3.19881e6 0.237309
\(712\) 5.65425e6 0.417998
\(713\) −1.46320e6 −0.107790
\(714\) 0 0
\(715\) −1.07101e7 −0.783481
\(716\) −109625. −0.00799146
\(717\) 1.53499e7 1.11508
\(718\) 2.13320e6 0.154426
\(719\) −7.34505e6 −0.529874 −0.264937 0.964266i \(-0.585351\pi\)
−0.264937 + 0.964266i \(0.585351\pi\)
\(720\) 8.13870e6 0.585092
\(721\) 0 0
\(722\) 571944. 0.0408329
\(723\) 8.56077e6 0.609070
\(724\) 1.86557e7 1.32271
\(725\) 3.34218e7 2.36148
\(726\) 616454. 0.0434069
\(727\) −1.57839e7 −1.10759 −0.553793 0.832655i \(-0.686820\pi\)
−0.553793 + 0.832655i \(0.686820\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 2.58520e6 0.179551
\(731\) −376480. −0.0260584
\(732\) 3.00231e6 0.207099
\(733\) −1.35367e7 −0.930579 −0.465289 0.885159i \(-0.654050\pi\)
−0.465289 + 0.885159i \(0.654050\pi\)
\(734\) 1.15048e6 0.0788205
\(735\) 0 0
\(736\) 460109. 0.0313088
\(737\) 6.60631e6 0.448012
\(738\) −267809. −0.0181003
\(739\) −7.33702e6 −0.494207 −0.247103 0.968989i \(-0.579479\pi\)
−0.247103 + 0.968989i \(0.579479\pi\)
\(740\) −3.38524e7 −2.27253
\(741\) 2.46118e6 0.164664
\(742\) 0 0
\(743\) 1.20844e7 0.803068 0.401534 0.915844i \(-0.368477\pi\)
0.401534 + 0.915844i \(0.368477\pi\)
\(744\) −3.39431e6 −0.224812
\(745\) 1.34472e7 0.887649
\(746\) 1.61869e6 0.106492
\(747\) 8.29212e6 0.543706
\(748\) −986151. −0.0644450
\(749\) 0 0
\(750\) 3.41555e6 0.221721
\(751\) 8.78176e6 0.568175 0.284087 0.958798i \(-0.408309\pi\)
0.284087 + 0.958798i \(0.408309\pi\)
\(752\) 4.23327e6 0.272980
\(753\) 1.03048e7 0.662297
\(754\) −706149. −0.0452343
\(755\) 1.59628e7 1.01916
\(756\) 0 0
\(757\) 465874. 0.0295480 0.0147740 0.999891i \(-0.495297\pi\)
0.0147740 + 0.999891i \(0.495297\pi\)
\(758\) 329685. 0.0208414
\(759\) −870872. −0.0548719
\(760\) −6.91339e6 −0.434167
\(761\) −1.49978e7 −0.938786 −0.469393 0.882989i \(-0.655527\pi\)
−0.469393 + 0.882989i \(0.655527\pi\)
\(762\) 1.43422e6 0.0894804
\(763\) 0 0
\(764\) −2.59984e7 −1.61144
\(765\) −533029. −0.0329304
\(766\) −3.06536e6 −0.188760
\(767\) −723026. −0.0443778
\(768\) −7.64646e6 −0.467797
\(769\) 1.85606e7 1.13181 0.565907 0.824469i \(-0.308526\pi\)
0.565907 + 0.824469i \(0.308526\pi\)
\(770\) 0 0
\(771\) 1.06537e7 0.645450
\(772\) −6.89422e6 −0.416334
\(773\) −1.22760e7 −0.738941 −0.369470 0.929243i \(-0.620461\pi\)
−0.369470 + 0.929243i \(0.620461\pi\)
\(774\) −381825. −0.0229093
\(775\) 5.81775e7 3.47937
\(776\) −1.52027e6 −0.0906286
\(777\) 0 0
\(778\) −2.24432e6 −0.132934
\(779\) −5.53534e6 −0.326814
\(780\) 6.07690e6 0.357640
\(781\) −1.93147e7 −1.13308
\(782\) −9711.39 −0.000567890 0
\(783\) −3.15157e6 −0.183706
\(784\) 0 0
\(785\) 1.57525e7 0.912380
\(786\) −273797. −0.0158079
\(787\) −5.26991e6 −0.303296 −0.151648 0.988435i \(-0.548458\pi\)
−0.151648 + 0.988435i \(0.548458\pi\)
\(788\) −1.49221e7 −0.856082
\(789\) −4.03484e6 −0.230746
\(790\) −3.25391e6 −0.185497
\(791\) 0 0
\(792\) −2.02024e6 −0.114443
\(793\) 2.19615e6 0.124016
\(794\) 3.43420e6 0.193319
\(795\) −1.66772e7 −0.935850
\(796\) 1.96893e7 1.10141
\(797\) −1.11889e7 −0.623940 −0.311970 0.950092i \(-0.600989\pi\)
−0.311970 + 0.950092i \(0.600989\pi\)
\(798\) 0 0
\(799\) −277250. −0.0153640
\(800\) −1.82941e7 −1.01062
\(801\) 9.13850e6 0.503261
\(802\) −2.69114e6 −0.147741
\(803\) 1.56144e7 0.854547
\(804\) −3.74841e6 −0.204507
\(805\) 0 0
\(806\) −1.22920e6 −0.0666476
\(807\) 6.72727e6 0.363626
\(808\) −5.35231e6 −0.288412
\(809\) 7.05051e6 0.378747 0.189374 0.981905i \(-0.439354\pi\)
0.189374 + 0.981905i \(0.439354\pi\)
\(810\) −540596. −0.0289508
\(811\) 2.54873e7 1.36073 0.680364 0.732875i \(-0.261822\pi\)
0.680364 + 0.732875i \(0.261822\pi\)
\(812\) 0 0
\(813\) −2.09374e6 −0.111095
\(814\) 4.07548e6 0.215585
\(815\) −3.42964e6 −0.180865
\(816\) 548166. 0.0288195
\(817\) −7.89192e6 −0.413645
\(818\) −4.18677e6 −0.218774
\(819\) 0 0
\(820\) −1.36673e7 −0.709820
\(821\) −2.34879e7 −1.21615 −0.608073 0.793881i \(-0.708057\pi\)
−0.608073 + 0.793881i \(0.708057\pi\)
\(822\) −1.72040e6 −0.0888073
\(823\) −1.20903e7 −0.622213 −0.311107 0.950375i \(-0.600700\pi\)
−0.311107 + 0.950375i \(0.600700\pi\)
\(824\) −7.89521e6 −0.405084
\(825\) 3.46263e7 1.77121
\(826\) 0 0
\(827\) 3.33335e6 0.169480 0.0847398 0.996403i \(-0.472994\pi\)
0.0847398 + 0.996403i \(0.472994\pi\)
\(828\) 494132. 0.0250477
\(829\) 7.70534e6 0.389408 0.194704 0.980862i \(-0.437625\pi\)
0.194704 + 0.980862i \(0.437625\pi\)
\(830\) −8.43496e6 −0.424999
\(831\) 2.18632e7 1.09828
\(832\) −5.98752e6 −0.299874
\(833\) 0 0
\(834\) 378013. 0.0188188
\(835\) −2.26706e7 −1.12524
\(836\) −2.06721e7 −1.02298
\(837\) −5.48595e6 −0.270669
\(838\) 2.27172e6 0.111749
\(839\) −2.40843e7 −1.18121 −0.590607 0.806959i \(-0.701112\pi\)
−0.590607 + 0.806959i \(0.701112\pi\)
\(840\) 0 0
\(841\) −1.82163e6 −0.0888116
\(842\) −1.65032e6 −0.0802208
\(843\) −2.26534e7 −1.09790
\(844\) −1.79146e7 −0.865665
\(845\) −3.42405e7 −1.64967
\(846\) −281186. −0.0135073
\(847\) 0 0
\(848\) 1.71509e7 0.819023
\(849\) 2.34895e6 0.111842
\(850\) 386129. 0.0183310
\(851\) −2.01352e6 −0.0953088
\(852\) 1.09592e7 0.517223
\(853\) 2.82938e7 1.33143 0.665716 0.746206i \(-0.268126\pi\)
0.665716 + 0.746206i \(0.268126\pi\)
\(854\) 0 0
\(855\) −1.11736e7 −0.522728
\(856\) −4.47666e6 −0.208819
\(857\) −2.16176e7 −1.00544 −0.502719 0.864450i \(-0.667667\pi\)
−0.502719 + 0.864450i \(0.667667\pi\)
\(858\) −731598. −0.0339277
\(859\) −3.25382e7 −1.50456 −0.752282 0.658841i \(-0.771047\pi\)
−0.752282 + 0.658841i \(0.771047\pi\)
\(860\) −1.94859e7 −0.898411
\(861\) 0 0
\(862\) 1.92058e6 0.0880366
\(863\) 8.06384e6 0.368566 0.184283 0.982873i \(-0.441004\pi\)
0.184283 + 0.982873i \(0.441004\pi\)
\(864\) 1.72508e6 0.0786184
\(865\) 4.38900e7 1.99446
\(866\) 756183. 0.0342635
\(867\) 1.27428e7 0.575728
\(868\) 0 0
\(869\) −1.96533e7 −0.882849
\(870\) 3.20586e6 0.143597
\(871\) −2.74192e6 −0.122464
\(872\) 8.39983e6 0.374093
\(873\) −2.45708e6 −0.109115
\(874\) −203574. −0.00901454
\(875\) 0 0
\(876\) −8.85959e6 −0.390080
\(877\) 1.70312e7 0.747732 0.373866 0.927483i \(-0.378032\pi\)
0.373866 + 0.927483i \(0.378032\pi\)
\(878\) 2.08934e6 0.0914690
\(879\) 585106. 0.0255424
\(880\) −5.00038e7 −2.17669
\(881\) −1.24740e7 −0.541459 −0.270729 0.962655i \(-0.587265\pi\)
−0.270729 + 0.962655i \(0.587265\pi\)
\(882\) 0 0
\(883\) 7.28955e6 0.314629 0.157315 0.987549i \(-0.449716\pi\)
0.157315 + 0.987549i \(0.449716\pi\)
\(884\) 409298. 0.0176160
\(885\) 3.28248e6 0.140878
\(886\) 2.72010e6 0.116413
\(887\) −1.87675e7 −0.800937 −0.400469 0.916310i \(-0.631153\pi\)
−0.400469 + 0.916310i \(0.631153\pi\)
\(888\) −4.67094e6 −0.198780
\(889\) 0 0
\(890\) −9.29592e6 −0.393385
\(891\) −3.26514e6 −0.137787
\(892\) 132812. 0.00558889
\(893\) −5.81181e6 −0.243884
\(894\) 918566. 0.0384385
\(895\) 364052. 0.0151917
\(896\) 0 0
\(897\) 361452. 0.0149992
\(898\) −3.47878e6 −0.143958
\(899\) 3.25330e7 1.34253
\(900\) −1.96469e7 −0.808515
\(901\) −1.12326e6 −0.0460966
\(902\) 1.64541e6 0.0673375
\(903\) 0 0
\(904\) −5.80323e6 −0.236183
\(905\) −6.19535e7 −2.51446
\(906\) 1.09041e6 0.0441335
\(907\) 1.26367e6 0.0510052 0.0255026 0.999675i \(-0.491881\pi\)
0.0255026 + 0.999675i \(0.491881\pi\)
\(908\) 3.54490e7 1.42689
\(909\) −8.65051e6 −0.347242
\(910\) 0 0
\(911\) −2.47718e7 −0.988921 −0.494461 0.869200i \(-0.664634\pi\)
−0.494461 + 0.869200i \(0.664634\pi\)
\(912\) 1.14909e7 0.457473
\(913\) −5.09463e7 −2.02272
\(914\) 1.78018e6 0.0704854
\(915\) −9.97033e6 −0.393692
\(916\) 2.52454e7 0.994130
\(917\) 0 0
\(918\) −36410.7 −0.00142601
\(919\) −9.36676e6 −0.365848 −0.182924 0.983127i \(-0.558556\pi\)
−0.182924 + 0.983127i \(0.558556\pi\)
\(920\) −1.01531e6 −0.0395483
\(921\) −2.12039e7 −0.823696
\(922\) 1.46542e6 0.0567721
\(923\) 8.01648e6 0.309727
\(924\) 0 0
\(925\) 8.00586e7 3.07648
\(926\) 2.38103e6 0.0912509
\(927\) −1.27604e7 −0.487713
\(928\) −1.02301e7 −0.389951
\(929\) 6.87350e6 0.261300 0.130650 0.991429i \(-0.458294\pi\)
0.130650 + 0.991429i \(0.458294\pi\)
\(930\) 5.58046e6 0.211574
\(931\) 0 0
\(932\) −3.66855e7 −1.38342
\(933\) −1.90625e7 −0.716928
\(934\) −843990. −0.0316570
\(935\) 3.27490e6 0.122509
\(936\) 838490. 0.0312830
\(937\) 4.26197e7 1.58585 0.792923 0.609322i \(-0.208558\pi\)
0.792923 + 0.609322i \(0.208558\pi\)
\(938\) 0 0
\(939\) −1.68762e6 −0.0624614
\(940\) −1.43499e7 −0.529701
\(941\) −4.50226e6 −0.165751 −0.0828755 0.996560i \(-0.526410\pi\)
−0.0828755 + 0.996560i \(0.526410\pi\)
\(942\) 1.07604e6 0.0395095
\(943\) −812926. −0.0297695
\(944\) −3.37569e6 −0.123291
\(945\) 0 0
\(946\) 2.34591e6 0.0852282
\(947\) 3.37725e7 1.22374 0.611870 0.790959i \(-0.290418\pi\)
0.611870 + 0.790959i \(0.290418\pi\)
\(948\) 1.11513e7 0.402999
\(949\) −6.48068e6 −0.233590
\(950\) 8.09419e6 0.290981
\(951\) 9.04870e6 0.324440
\(952\) 0 0
\(953\) 4.81813e7 1.71849 0.859244 0.511566i \(-0.170934\pi\)
0.859244 + 0.511566i \(0.170934\pi\)
\(954\) −1.13921e6 −0.0405258
\(955\) 8.63379e7 3.06332
\(956\) 5.35107e7 1.89363
\(957\) 1.93631e7 0.683430
\(958\) −4.49800e6 −0.158346
\(959\) 0 0
\(960\) 2.71828e7 0.951955
\(961\) 2.80012e7 0.978066
\(962\) −1.69151e6 −0.0589301
\(963\) −7.23527e6 −0.251414
\(964\) 2.98434e7 1.03432
\(965\) 2.28950e7 0.791446
\(966\) 0 0
\(967\) −3.54640e7 −1.21961 −0.609805 0.792551i \(-0.708752\pi\)
−0.609805 + 0.792551i \(0.708752\pi\)
\(968\) 4.34084e6 0.148897
\(969\) −752572. −0.0257477
\(970\) 2.49941e6 0.0852919
\(971\) 7.84296e6 0.266951 0.133476 0.991052i \(-0.457386\pi\)
0.133476 + 0.991052i \(0.457386\pi\)
\(972\) 1.85264e6 0.0628964
\(973\) 0 0
\(974\) 5.41288e6 0.182823
\(975\) −1.43715e7 −0.484161
\(976\) 1.02535e7 0.344545
\(977\) −2.47534e7 −0.829658 −0.414829 0.909899i \(-0.636159\pi\)
−0.414829 + 0.909899i \(0.636159\pi\)
\(978\) −234276. −0.00783215
\(979\) −5.61464e7 −1.87226
\(980\) 0 0
\(981\) 1.35760e7 0.450400
\(982\) −4.42955e6 −0.146582
\(983\) −4.53129e7 −1.49568 −0.747838 0.663881i \(-0.768908\pi\)
−0.747838 + 0.663881i \(0.768908\pi\)
\(984\) −1.88581e6 −0.0620885
\(985\) 4.95548e7 1.62740
\(986\) 215924. 0.00707308
\(987\) 0 0
\(988\) 8.57985e6 0.279632
\(989\) −1.15902e6 −0.0376789
\(990\) 3.32139e6 0.107704
\(991\) −3.74143e7 −1.21019 −0.605095 0.796154i \(-0.706865\pi\)
−0.605095 + 0.796154i \(0.706865\pi\)
\(992\) −1.78076e7 −0.574548
\(993\) −1.51186e7 −0.486561
\(994\) 0 0
\(995\) −6.53861e7 −2.09376
\(996\) 2.89069e7 0.923322
\(997\) −4.65278e7 −1.48243 −0.741216 0.671267i \(-0.765751\pi\)
−0.741216 + 0.671267i \(0.765751\pi\)
\(998\) 1.84676e6 0.0586926
\(999\) −7.54927e6 −0.239327
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.l.1.2 4
3.2 odd 2 441.6.a.v.1.3 4
7.2 even 3 147.6.e.o.67.3 8
7.3 odd 6 21.6.e.c.16.3 yes 8
7.4 even 3 147.6.e.o.79.3 8
7.5 odd 6 21.6.e.c.4.3 8
7.6 odd 2 147.6.a.m.1.2 4
21.5 even 6 63.6.e.e.46.2 8
21.17 even 6 63.6.e.e.37.2 8
21.20 even 2 441.6.a.w.1.3 4
28.3 even 6 336.6.q.j.289.4 8
28.19 even 6 336.6.q.j.193.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.e.c.4.3 8 7.5 odd 6
21.6.e.c.16.3 yes 8 7.3 odd 6
63.6.e.e.37.2 8 21.17 even 6
63.6.e.e.46.2 8 21.5 even 6
147.6.a.l.1.2 4 1.1 even 1 trivial
147.6.a.m.1.2 4 7.6 odd 2
147.6.e.o.67.3 8 7.2 even 3
147.6.e.o.79.3 8 7.4 even 3
336.6.q.j.193.4 8 28.19 even 6
336.6.q.j.289.4 8 28.3 even 6
441.6.a.v.1.3 4 3.2 odd 2
441.6.a.w.1.3 4 21.20 even 2