Properties

Label 21.6.a.a.1.1
Level $21$
Weight $6$
Character 21.1
Self dual yes
Analytic conductor $3.368$
Analytic rank $0$
Dimension $1$
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] = [21,6,Mod(1,21)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(21, 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("21.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 21.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: \(3.36806021607\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 21.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-6.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} +78.0000 q^{5} +54.0000 q^{6} +49.0000 q^{7} +168.000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-6.00000 q^{2} -9.00000 q^{3} +4.00000 q^{4} +78.0000 q^{5} +54.0000 q^{6} +49.0000 q^{7} +168.000 q^{8} +81.0000 q^{9} -468.000 q^{10} +444.000 q^{11} -36.0000 q^{12} -442.000 q^{13} -294.000 q^{14} -702.000 q^{15} -1136.00 q^{16} -126.000 q^{17} -486.000 q^{18} +2684.00 q^{19} +312.000 q^{20} -441.000 q^{21} -2664.00 q^{22} +4200.00 q^{23} -1512.00 q^{24} +2959.00 q^{25} +2652.00 q^{26} -729.000 q^{27} +196.000 q^{28} -5442.00 q^{29} +4212.00 q^{30} +80.0000 q^{31} +1440.00 q^{32} -3996.00 q^{33} +756.000 q^{34} +3822.00 q^{35} +324.000 q^{36} -5434.00 q^{37} -16104.0 q^{38} +3978.00 q^{39} +13104.0 q^{40} +7962.00 q^{41} +2646.00 q^{42} -11524.0 q^{43} +1776.00 q^{44} +6318.00 q^{45} -25200.0 q^{46} -13920.0 q^{47} +10224.0 q^{48} +2401.00 q^{49} -17754.0 q^{50} +1134.00 q^{51} -1768.00 q^{52} -9594.00 q^{53} +4374.00 q^{54} +34632.0 q^{55} +8232.00 q^{56} -24156.0 q^{57} +32652.0 q^{58} +27492.0 q^{59} -2808.00 q^{60} +49478.0 q^{61} -480.000 q^{62} +3969.00 q^{63} +27712.0 q^{64} -34476.0 q^{65} +23976.0 q^{66} -59356.0 q^{67} -504.000 q^{68} -37800.0 q^{69} -22932.0 q^{70} +32040.0 q^{71} +13608.0 q^{72} -61846.0 q^{73} +32604.0 q^{74} -26631.0 q^{75} +10736.0 q^{76} +21756.0 q^{77} -23868.0 q^{78} -65776.0 q^{79} -88608.0 q^{80} +6561.00 q^{81} -47772.0 q^{82} +40188.0 q^{83} -1764.00 q^{84} -9828.00 q^{85} +69144.0 q^{86} +48978.0 q^{87} +74592.0 q^{88} -7974.00 q^{89} -37908.0 q^{90} -21658.0 q^{91} +16800.0 q^{92} -720.000 q^{93} +83520.0 q^{94} +209352. q^{95} -12960.0 q^{96} -143662. q^{97} -14406.0 q^{98} +35964.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −6.00000 −1.06066 −0.530330 0.847791i \(-0.677932\pi\)
−0.530330 + 0.847791i \(0.677932\pi\)
\(3\) −9.00000 −0.577350
\(4\) 4.00000 0.125000
\(5\) 78.0000 1.39531 0.697653 0.716436i \(-0.254228\pi\)
0.697653 + 0.716436i \(0.254228\pi\)
\(6\) 54.0000 0.612372
\(7\) 49.0000 0.377964
\(8\) 168.000 0.928078
\(9\) 81.0000 0.333333
\(10\) −468.000 −1.47995
\(11\) 444.000 1.10637 0.553186 0.833058i \(-0.313412\pi\)
0.553186 + 0.833058i \(0.313412\pi\)
\(12\) −36.0000 −0.0721688
\(13\) −442.000 −0.725377 −0.362689 0.931910i \(-0.618141\pi\)
−0.362689 + 0.931910i \(0.618141\pi\)
\(14\) −294.000 −0.400892
\(15\) −702.000 −0.805581
\(16\) −1136.00 −1.10938
\(17\) −126.000 −0.105742 −0.0528711 0.998601i \(-0.516837\pi\)
−0.0528711 + 0.998601i \(0.516837\pi\)
\(18\) −486.000 −0.353553
\(19\) 2684.00 1.70568 0.852842 0.522169i \(-0.174877\pi\)
0.852842 + 0.522169i \(0.174877\pi\)
\(20\) 312.000 0.174413
\(21\) −441.000 −0.218218
\(22\) −2664.00 −1.17348
\(23\) 4200.00 1.65550 0.827751 0.561096i \(-0.189620\pi\)
0.827751 + 0.561096i \(0.189620\pi\)
\(24\) −1512.00 −0.535826
\(25\) 2959.00 0.946880
\(26\) 2652.00 0.769379
\(27\) −729.000 −0.192450
\(28\) 196.000 0.0472456
\(29\) −5442.00 −1.20161 −0.600805 0.799396i \(-0.705153\pi\)
−0.600805 + 0.799396i \(0.705153\pi\)
\(30\) 4212.00 0.854447
\(31\) 80.0000 0.0149515 0.00747577 0.999972i \(-0.497620\pi\)
0.00747577 + 0.999972i \(0.497620\pi\)
\(32\) 1440.00 0.248592
\(33\) −3996.00 −0.638764
\(34\) 756.000 0.112157
\(35\) 3822.00 0.527376
\(36\) 324.000 0.0416667
\(37\) −5434.00 −0.652552 −0.326276 0.945274i \(-0.605794\pi\)
−0.326276 + 0.945274i \(0.605794\pi\)
\(38\) −16104.0 −1.80915
\(39\) 3978.00 0.418797
\(40\) 13104.0 1.29495
\(41\) 7962.00 0.739712 0.369856 0.929089i \(-0.379407\pi\)
0.369856 + 0.929089i \(0.379407\pi\)
\(42\) 2646.00 0.231455
\(43\) −11524.0 −0.950456 −0.475228 0.879863i \(-0.657634\pi\)
−0.475228 + 0.879863i \(0.657634\pi\)
\(44\) 1776.00 0.138297
\(45\) 6318.00 0.465102
\(46\) −25200.0 −1.75592
\(47\) −13920.0 −0.919167 −0.459584 0.888134i \(-0.652001\pi\)
−0.459584 + 0.888134i \(0.652001\pi\)
\(48\) 10224.0 0.640498
\(49\) 2401.00 0.142857
\(50\) −17754.0 −1.00432
\(51\) 1134.00 0.0610503
\(52\) −1768.00 −0.0906721
\(53\) −9594.00 −0.469148 −0.234574 0.972098i \(-0.575370\pi\)
−0.234574 + 0.972098i \(0.575370\pi\)
\(54\) 4374.00 0.204124
\(55\) 34632.0 1.54373
\(56\) 8232.00 0.350780
\(57\) −24156.0 −0.984777
\(58\) 32652.0 1.27450
\(59\) 27492.0 1.02820 0.514098 0.857731i \(-0.328127\pi\)
0.514098 + 0.857731i \(0.328127\pi\)
\(60\) −2808.00 −0.100698
\(61\) 49478.0 1.70250 0.851251 0.524759i \(-0.175845\pi\)
0.851251 + 0.524759i \(0.175845\pi\)
\(62\) −480.000 −0.0158585
\(63\) 3969.00 0.125988
\(64\) 27712.0 0.845703
\(65\) −34476.0 −1.01212
\(66\) 23976.0 0.677512
\(67\) −59356.0 −1.61539 −0.807695 0.589600i \(-0.799285\pi\)
−0.807695 + 0.589600i \(0.799285\pi\)
\(68\) −504.000 −0.0132178
\(69\) −37800.0 −0.955805
\(70\) −22932.0 −0.559367
\(71\) 32040.0 0.754304 0.377152 0.926151i \(-0.376903\pi\)
0.377152 + 0.926151i \(0.376903\pi\)
\(72\) 13608.0 0.309359
\(73\) −61846.0 −1.35833 −0.679164 0.733987i \(-0.737657\pi\)
−0.679164 + 0.733987i \(0.737657\pi\)
\(74\) 32604.0 0.692136
\(75\) −26631.0 −0.546681
\(76\) 10736.0 0.213210
\(77\) 21756.0 0.418169
\(78\) −23868.0 −0.444201
\(79\) −65776.0 −1.18577 −0.592884 0.805288i \(-0.702011\pi\)
−0.592884 + 0.805288i \(0.702011\pi\)
\(80\) −88608.0 −1.54792
\(81\) 6561.00 0.111111
\(82\) −47772.0 −0.784583
\(83\) 40188.0 0.640326 0.320163 0.947362i \(-0.396262\pi\)
0.320163 + 0.947362i \(0.396262\pi\)
\(84\) −1764.00 −0.0272772
\(85\) −9828.00 −0.147543
\(86\) 69144.0 1.00811
\(87\) 48978.0 0.693750
\(88\) 74592.0 1.02680
\(89\) −7974.00 −0.106709 −0.0533545 0.998576i \(-0.516991\pi\)
−0.0533545 + 0.998576i \(0.516991\pi\)
\(90\) −37908.0 −0.493315
\(91\) −21658.0 −0.274167
\(92\) 16800.0 0.206938
\(93\) −720.000 −0.00863227
\(94\) 83520.0 0.974924
\(95\) 209352. 2.37995
\(96\) −12960.0 −0.143525
\(97\) −143662. −1.55029 −0.775144 0.631784i \(-0.782323\pi\)
−0.775144 + 0.631784i \(0.782323\pi\)
\(98\) −14406.0 −0.151523
\(99\) 35964.0 0.368791
\(100\) 11836.0 0.118360
\(101\) −2706.00 −0.0263952 −0.0131976 0.999913i \(-0.504201\pi\)
−0.0131976 + 0.999913i \(0.504201\pi\)
\(102\) −6804.00 −0.0647536
\(103\) 131768. 1.22382 0.611909 0.790928i \(-0.290402\pi\)
0.611909 + 0.790928i \(0.290402\pi\)
\(104\) −74256.0 −0.673206
\(105\) −34398.0 −0.304481
\(106\) 57564.0 0.497607
\(107\) −128916. −1.08855 −0.544274 0.838908i \(-0.683195\pi\)
−0.544274 + 0.838908i \(0.683195\pi\)
\(108\) −2916.00 −0.0240563
\(109\) −100978. −0.814068 −0.407034 0.913413i \(-0.633437\pi\)
−0.407034 + 0.913413i \(0.633437\pi\)
\(110\) −207792. −1.63737
\(111\) 48906.0 0.376751
\(112\) −55664.0 −0.419304
\(113\) 220146. 1.62186 0.810932 0.585140i \(-0.198960\pi\)
0.810932 + 0.585140i \(0.198960\pi\)
\(114\) 144936. 1.04451
\(115\) 327600. 2.30993
\(116\) −21768.0 −0.150201
\(117\) −35802.0 −0.241792
\(118\) −164952. −1.09057
\(119\) −6174.00 −0.0399668
\(120\) −117936. −0.747641
\(121\) 36085.0 0.224059
\(122\) −296868. −1.80578
\(123\) −71658.0 −0.427073
\(124\) 320.000 0.00186894
\(125\) −12948.0 −0.0741187
\(126\) −23814.0 −0.133631
\(127\) −74320.0 −0.408880 −0.204440 0.978879i \(-0.565537\pi\)
−0.204440 + 0.978879i \(0.565537\pi\)
\(128\) −212352. −1.14560
\(129\) 103716. 0.548746
\(130\) 206856. 1.07352
\(131\) −155316. −0.790748 −0.395374 0.918520i \(-0.629385\pi\)
−0.395374 + 0.918520i \(0.629385\pi\)
\(132\) −15984.0 −0.0798455
\(133\) 131516. 0.644688
\(134\) 356136. 1.71338
\(135\) −56862.0 −0.268527
\(136\) −21168.0 −0.0981369
\(137\) −264246. −1.20284 −0.601419 0.798934i \(-0.705398\pi\)
−0.601419 + 0.798934i \(0.705398\pi\)
\(138\) 226800. 1.01378
\(139\) 224612. 0.986043 0.493022 0.870017i \(-0.335892\pi\)
0.493022 + 0.870017i \(0.335892\pi\)
\(140\) 15288.0 0.0659220
\(141\) 125280. 0.530682
\(142\) −192240. −0.800061
\(143\) −196248. −0.802537
\(144\) −92016.0 −0.369792
\(145\) −424476. −1.67661
\(146\) 371076. 1.44072
\(147\) −21609.0 −0.0824786
\(148\) −21736.0 −0.0815690
\(149\) −82074.0 −0.302859 −0.151429 0.988468i \(-0.548388\pi\)
−0.151429 + 0.988468i \(0.548388\pi\)
\(150\) 159786. 0.579843
\(151\) −287032. −1.02444 −0.512222 0.858853i \(-0.671177\pi\)
−0.512222 + 0.858853i \(0.671177\pi\)
\(152\) 450912. 1.58301
\(153\) −10206.0 −0.0352474
\(154\) −130536. −0.443536
\(155\) 6240.00 0.0208620
\(156\) 15912.0 0.0523496
\(157\) 129878. 0.420520 0.210260 0.977646i \(-0.432569\pi\)
0.210260 + 0.977646i \(0.432569\pi\)
\(158\) 394656. 1.25770
\(159\) 86346.0 0.270863
\(160\) 112320. 0.346862
\(161\) 205800. 0.625721
\(162\) −39366.0 −0.117851
\(163\) 555284. 1.63699 0.818495 0.574513i \(-0.194809\pi\)
0.818495 + 0.574513i \(0.194809\pi\)
\(164\) 31848.0 0.0924640
\(165\) −311688. −0.891272
\(166\) −241128. −0.679168
\(167\) 43512.0 0.120731 0.0603654 0.998176i \(-0.480773\pi\)
0.0603654 + 0.998176i \(0.480773\pi\)
\(168\) −74088.0 −0.202523
\(169\) −175929. −0.473828
\(170\) 58968.0 0.156493
\(171\) 217404. 0.568561
\(172\) −46096.0 −0.118807
\(173\) −18330.0 −0.0465637 −0.0232818 0.999729i \(-0.507412\pi\)
−0.0232818 + 0.999729i \(0.507412\pi\)
\(174\) −293868. −0.735833
\(175\) 144991. 0.357887
\(176\) −504384. −1.22738
\(177\) −247428. −0.593630
\(178\) 47844.0 0.113182
\(179\) −153324. −0.357666 −0.178833 0.983879i \(-0.557232\pi\)
−0.178833 + 0.983879i \(0.557232\pi\)
\(180\) 25272.0 0.0581378
\(181\) −382066. −0.866846 −0.433423 0.901191i \(-0.642694\pi\)
−0.433423 + 0.901191i \(0.642694\pi\)
\(182\) 129948. 0.290798
\(183\) −445302. −0.982940
\(184\) 705600. 1.53643
\(185\) −423852. −0.910510
\(186\) 4320.00 0.00915591
\(187\) −55944.0 −0.116990
\(188\) −55680.0 −0.114896
\(189\) −35721.0 −0.0727393
\(190\) −1.25611e6 −2.52432
\(191\) −273408. −0.542285 −0.271143 0.962539i \(-0.587402\pi\)
−0.271143 + 0.962539i \(0.587402\pi\)
\(192\) −249408. −0.488267
\(193\) 153602. 0.296827 0.148414 0.988925i \(-0.452583\pi\)
0.148414 + 0.988925i \(0.452583\pi\)
\(194\) 861972. 1.64433
\(195\) 310284. 0.584350
\(196\) 9604.00 0.0178571
\(197\) 154422. 0.283494 0.141747 0.989903i \(-0.454728\pi\)
0.141747 + 0.989903i \(0.454728\pi\)
\(198\) −215784. −0.391162
\(199\) −366856. −0.656694 −0.328347 0.944557i \(-0.606492\pi\)
−0.328347 + 0.944557i \(0.606492\pi\)
\(200\) 497112. 0.878778
\(201\) 534204. 0.932646
\(202\) 16236.0 0.0279963
\(203\) −266658. −0.454166
\(204\) 4536.00 0.00763128
\(205\) 621036. 1.03212
\(206\) −790608. −1.29806
\(207\) 340200. 0.551834
\(208\) 502112. 0.804715
\(209\) 1.19170e6 1.88712
\(210\) 206388. 0.322951
\(211\) 520244. 0.804453 0.402227 0.915540i \(-0.368236\pi\)
0.402227 + 0.915540i \(0.368236\pi\)
\(212\) −38376.0 −0.0586435
\(213\) −288360. −0.435498
\(214\) 773496. 1.15458
\(215\) −898872. −1.32618
\(216\) −122472. −0.178609
\(217\) 3920.00 0.00565115
\(218\) 605868. 0.863449
\(219\) 556614. 0.784231
\(220\) 138528. 0.192966
\(221\) 55692.0 0.0767030
\(222\) −293436. −0.399605
\(223\) 304736. 0.410357 0.205178 0.978725i \(-0.434223\pi\)
0.205178 + 0.978725i \(0.434223\pi\)
\(224\) 70560.0 0.0939590
\(225\) 239679. 0.315627
\(226\) −1.32088e6 −1.72025
\(227\) 288588. 0.371718 0.185859 0.982576i \(-0.440493\pi\)
0.185859 + 0.982576i \(0.440493\pi\)
\(228\) −96624.0 −0.123097
\(229\) 772190. 0.973051 0.486525 0.873666i \(-0.338264\pi\)
0.486525 + 0.873666i \(0.338264\pi\)
\(230\) −1.96560e6 −2.45005
\(231\) −195804. −0.241430
\(232\) −914256. −1.11519
\(233\) 252234. 0.304378 0.152189 0.988351i \(-0.451368\pi\)
0.152189 + 0.988351i \(0.451368\pi\)
\(234\) 214812. 0.256460
\(235\) −1.08576e6 −1.28252
\(236\) 109968. 0.128525
\(237\) 591984. 0.684603
\(238\) 37044.0 0.0423912
\(239\) −1.45114e6 −1.64329 −0.821643 0.570002i \(-0.806942\pi\)
−0.821643 + 0.570002i \(0.806942\pi\)
\(240\) 797472. 0.893691
\(241\) −146398. −0.162365 −0.0811825 0.996699i \(-0.525870\pi\)
−0.0811825 + 0.996699i \(0.525870\pi\)
\(242\) −216510. −0.237651
\(243\) −59049.0 −0.0641500
\(244\) 197912. 0.212813
\(245\) 187278. 0.199329
\(246\) 429948. 0.452979
\(247\) −1.18633e6 −1.23726
\(248\) 13440.0 0.0138762
\(249\) −361692. −0.369692
\(250\) 77688.0 0.0786147
\(251\) 607860. 0.609003 0.304501 0.952512i \(-0.401510\pi\)
0.304501 + 0.952512i \(0.401510\pi\)
\(252\) 15876.0 0.0157485
\(253\) 1.86480e6 1.83160
\(254\) 445920. 0.433683
\(255\) 88452.0 0.0851838
\(256\) 387328. 0.369385
\(257\) 95586.0 0.0902737 0.0451369 0.998981i \(-0.485628\pi\)
0.0451369 + 0.998981i \(0.485628\pi\)
\(258\) −622296. −0.582033
\(259\) −266266. −0.246642
\(260\) −137904. −0.126515
\(261\) −440802. −0.400537
\(262\) 931896. 0.838715
\(263\) −2.20034e6 −1.96156 −0.980779 0.195121i \(-0.937490\pi\)
−0.980779 + 0.195121i \(0.937490\pi\)
\(264\) −671328. −0.592823
\(265\) −748332. −0.654605
\(266\) −789096. −0.683795
\(267\) 71766.0 0.0616085
\(268\) −237424. −0.201924
\(269\) 1.77025e6 1.49160 0.745801 0.666169i \(-0.232067\pi\)
0.745801 + 0.666169i \(0.232067\pi\)
\(270\) 341172. 0.284816
\(271\) −223504. −0.184868 −0.0924341 0.995719i \(-0.529465\pi\)
−0.0924341 + 0.995719i \(0.529465\pi\)
\(272\) 143136. 0.117308
\(273\) 194922. 0.158290
\(274\) 1.58548e6 1.27580
\(275\) 1.31380e6 1.04760
\(276\) −151200. −0.119476
\(277\) −342778. −0.268419 −0.134210 0.990953i \(-0.542850\pi\)
−0.134210 + 0.990953i \(0.542850\pi\)
\(278\) −1.34767e6 −1.04586
\(279\) 6480.00 0.00498384
\(280\) 642096. 0.489446
\(281\) 480378. 0.362925 0.181463 0.983398i \(-0.441917\pi\)
0.181463 + 0.983398i \(0.441917\pi\)
\(282\) −751680. −0.562873
\(283\) −29980.0 −0.0222518 −0.0111259 0.999938i \(-0.503542\pi\)
−0.0111259 + 0.999938i \(0.503542\pi\)
\(284\) 128160. 0.0942880
\(285\) −1.88417e6 −1.37407
\(286\) 1.17749e6 0.851219
\(287\) 390138. 0.279585
\(288\) 116640. 0.0828641
\(289\) −1.40398e6 −0.988819
\(290\) 2.54686e6 1.77832
\(291\) 1.29296e6 0.895060
\(292\) −247384. −0.169791
\(293\) −198066. −0.134785 −0.0673924 0.997727i \(-0.521468\pi\)
−0.0673924 + 0.997727i \(0.521468\pi\)
\(294\) 129654. 0.0874818
\(295\) 2.14438e6 1.43465
\(296\) −912912. −0.605619
\(297\) −323676. −0.212921
\(298\) 492444. 0.321230
\(299\) −1.85640e6 −1.20086
\(300\) −106524. −0.0683352
\(301\) −564676. −0.359239
\(302\) 1.72219e6 1.08659
\(303\) 24354.0 0.0152393
\(304\) −3.04902e6 −1.89224
\(305\) 3.85928e6 2.37551
\(306\) 61236.0 0.0373855
\(307\) −1.04564e6 −0.633191 −0.316595 0.948561i \(-0.602540\pi\)
−0.316595 + 0.948561i \(0.602540\pi\)
\(308\) 87024.0 0.0522712
\(309\) −1.18591e6 −0.706572
\(310\) −37440.0 −0.0221275
\(311\) 1.83718e6 1.07708 0.538542 0.842598i \(-0.318975\pi\)
0.538542 + 0.842598i \(0.318975\pi\)
\(312\) 668304. 0.388676
\(313\) −365494. −0.210872 −0.105436 0.994426i \(-0.533624\pi\)
−0.105436 + 0.994426i \(0.533624\pi\)
\(314\) −779268. −0.446029
\(315\) 309582. 0.175792
\(316\) −263104. −0.148221
\(317\) −28338.0 −0.0158388 −0.00791938 0.999969i \(-0.502521\pi\)
−0.00791938 + 0.999969i \(0.502521\pi\)
\(318\) −518076. −0.287293
\(319\) −2.41625e6 −1.32943
\(320\) 2.16154e6 1.18001
\(321\) 1.16024e6 0.628473
\(322\) −1.23480e6 −0.663677
\(323\) −338184. −0.180363
\(324\) 26244.0 0.0138889
\(325\) −1.30788e6 −0.686845
\(326\) −3.33170e6 −1.73629
\(327\) 908802. 0.470002
\(328\) 1.33762e6 0.686510
\(329\) −682080. −0.347413
\(330\) 1.87013e6 0.945337
\(331\) 1.93392e6 0.970214 0.485107 0.874455i \(-0.338781\pi\)
0.485107 + 0.874455i \(0.338781\pi\)
\(332\) 160752. 0.0800408
\(333\) −440154. −0.217517
\(334\) −261072. −0.128054
\(335\) −4.62977e6 −2.25397
\(336\) 500976. 0.242085
\(337\) −1.88817e6 −0.905664 −0.452832 0.891596i \(-0.649586\pi\)
−0.452832 + 0.891596i \(0.649586\pi\)
\(338\) 1.05557e6 0.502570
\(339\) −1.98131e6 −0.936384
\(340\) −39312.0 −0.0184428
\(341\) 35520.0 0.0165420
\(342\) −1.30442e6 −0.603050
\(343\) 117649. 0.0539949
\(344\) −1.93603e6 −0.882097
\(345\) −2.94840e6 −1.33364
\(346\) 109980. 0.0493882
\(347\) 2.91937e6 1.30156 0.650782 0.759264i \(-0.274441\pi\)
0.650782 + 0.759264i \(0.274441\pi\)
\(348\) 195912. 0.0867187
\(349\) −780682. −0.343092 −0.171546 0.985176i \(-0.554876\pi\)
−0.171546 + 0.985176i \(0.554876\pi\)
\(350\) −869946. −0.379596
\(351\) 322218. 0.139599
\(352\) 639360. 0.275036
\(353\) 1.33437e6 0.569954 0.284977 0.958534i \(-0.408014\pi\)
0.284977 + 0.958534i \(0.408014\pi\)
\(354\) 1.48457e6 0.629639
\(355\) 2.49912e6 1.05249
\(356\) −31896.0 −0.0133386
\(357\) 55566.0 0.0230748
\(358\) 919944. 0.379362
\(359\) 1.01743e6 0.416648 0.208324 0.978060i \(-0.433199\pi\)
0.208324 + 0.978060i \(0.433199\pi\)
\(360\) 1.06142e6 0.431651
\(361\) 4.72776e6 1.90936
\(362\) 2.29240e6 0.919429
\(363\) −324765. −0.129361
\(364\) −86632.0 −0.0342709
\(365\) −4.82399e6 −1.89528
\(366\) 2.67181e6 1.04257
\(367\) 837680. 0.324648 0.162324 0.986737i \(-0.448101\pi\)
0.162324 + 0.986737i \(0.448101\pi\)
\(368\) −4.77120e6 −1.83657
\(369\) 644922. 0.246571
\(370\) 2.54311e6 0.965742
\(371\) −470106. −0.177321
\(372\) −2880.00 −0.00107903
\(373\) −1.51993e6 −0.565655 −0.282827 0.959171i \(-0.591272\pi\)
−0.282827 + 0.959171i \(0.591272\pi\)
\(374\) 335664. 0.124087
\(375\) 116532. 0.0427924
\(376\) −2.33856e6 −0.853059
\(377\) 2.40536e6 0.871620
\(378\) 214326. 0.0771517
\(379\) 2.64465e6 0.945737 0.472869 0.881133i \(-0.343219\pi\)
0.472869 + 0.881133i \(0.343219\pi\)
\(380\) 837408. 0.297494
\(381\) 668880. 0.236067
\(382\) 1.64045e6 0.575180
\(383\) 2.01336e6 0.701333 0.350667 0.936500i \(-0.385955\pi\)
0.350667 + 0.936500i \(0.385955\pi\)
\(384\) 1.91117e6 0.661410
\(385\) 1.69697e6 0.583474
\(386\) −921612. −0.314833
\(387\) −933444. −0.316819
\(388\) −574648. −0.193786
\(389\) −726234. −0.243334 −0.121667 0.992571i \(-0.538824\pi\)
−0.121667 + 0.992571i \(0.538824\pi\)
\(390\) −1.86170e6 −0.619796
\(391\) −529200. −0.175056
\(392\) 403368. 0.132583
\(393\) 1.39784e6 0.456538
\(394\) −926532. −0.300691
\(395\) −5.13053e6 −1.65451
\(396\) 143856. 0.0460988
\(397\) 4.57578e6 1.45710 0.728549 0.684993i \(-0.240195\pi\)
0.728549 + 0.684993i \(0.240195\pi\)
\(398\) 2.20114e6 0.696529
\(399\) −1.18364e6 −0.372211
\(400\) −3.36142e6 −1.05045
\(401\) −33870.0 −0.0105185 −0.00525926 0.999986i \(-0.501674\pi\)
−0.00525926 + 0.999986i \(0.501674\pi\)
\(402\) −3.20522e6 −0.989221
\(403\) −35360.0 −0.0108455
\(404\) −10824.0 −0.00329940
\(405\) 511758. 0.155034
\(406\) 1.59995e6 0.481716
\(407\) −2.41270e6 −0.721966
\(408\) 190512. 0.0566594
\(409\) −5.86178e6 −1.73269 −0.866346 0.499444i \(-0.833538\pi\)
−0.866346 + 0.499444i \(0.833538\pi\)
\(410\) −3.72622e6 −1.09473
\(411\) 2.37821e6 0.694459
\(412\) 527072. 0.152977
\(413\) 1.34711e6 0.388622
\(414\) −2.04120e6 −0.585308
\(415\) 3.13466e6 0.893451
\(416\) −636480. −0.180323
\(417\) −2.02151e6 −0.569292
\(418\) −7.15018e6 −2.00159
\(419\) 302748. 0.0842454 0.0421227 0.999112i \(-0.486588\pi\)
0.0421227 + 0.999112i \(0.486588\pi\)
\(420\) −137592. −0.0380601
\(421\) −5.36708e6 −1.47582 −0.737909 0.674900i \(-0.764187\pi\)
−0.737909 + 0.674900i \(0.764187\pi\)
\(422\) −3.12146e6 −0.853252
\(423\) −1.12752e6 −0.306389
\(424\) −1.61179e6 −0.435406
\(425\) −372834. −0.100125
\(426\) 1.73016e6 0.461915
\(427\) 2.42442e6 0.643485
\(428\) −515664. −0.136068
\(429\) 1.76623e6 0.463345
\(430\) 5.39323e6 1.40662
\(431\) 1.17706e6 0.305214 0.152607 0.988287i \(-0.451233\pi\)
0.152607 + 0.988287i \(0.451233\pi\)
\(432\) 828144. 0.213499
\(433\) −3.66249e6 −0.938766 −0.469383 0.882995i \(-0.655524\pi\)
−0.469383 + 0.882995i \(0.655524\pi\)
\(434\) −23520.0 −0.00599395
\(435\) 3.82028e6 0.967994
\(436\) −403912. −0.101758
\(437\) 1.12728e7 2.82376
\(438\) −3.33968e6 −0.831802
\(439\) −2.53674e6 −0.628225 −0.314113 0.949386i \(-0.601707\pi\)
−0.314113 + 0.949386i \(0.601707\pi\)
\(440\) 5.81818e6 1.43270
\(441\) 194481. 0.0476190
\(442\) −334152. −0.0813558
\(443\) 6.01504e6 1.45623 0.728113 0.685457i \(-0.240397\pi\)
0.728113 + 0.685457i \(0.240397\pi\)
\(444\) 195624. 0.0470939
\(445\) −621972. −0.148892
\(446\) −1.82842e6 −0.435249
\(447\) 738666. 0.174856
\(448\) 1.35789e6 0.319646
\(449\) 5.65965e6 1.32487 0.662436 0.749119i \(-0.269523\pi\)
0.662436 + 0.749119i \(0.269523\pi\)
\(450\) −1.43807e6 −0.334773
\(451\) 3.53513e6 0.818397
\(452\) 880584. 0.202733
\(453\) 2.58329e6 0.591463
\(454\) −1.73153e6 −0.394267
\(455\) −1.68932e6 −0.382547
\(456\) −4.05821e6 −0.913949
\(457\) −6.46159e6 −1.44727 −0.723634 0.690184i \(-0.757530\pi\)
−0.723634 + 0.690184i \(0.757530\pi\)
\(458\) −4.63314e6 −1.03208
\(459\) 91854.0 0.0203501
\(460\) 1.31040e6 0.288742
\(461\) −3.37353e6 −0.739320 −0.369660 0.929167i \(-0.620526\pi\)
−0.369660 + 0.929167i \(0.620526\pi\)
\(462\) 1.17482e6 0.256075
\(463\) −4.54974e6 −0.986358 −0.493179 0.869928i \(-0.664165\pi\)
−0.493179 + 0.869928i \(0.664165\pi\)
\(464\) 6.18211e6 1.33304
\(465\) −56160.0 −0.0120447
\(466\) −1.51340e6 −0.322842
\(467\) 2.01136e6 0.426773 0.213386 0.976968i \(-0.431551\pi\)
0.213386 + 0.976968i \(0.431551\pi\)
\(468\) −143208. −0.0302240
\(469\) −2.90844e6 −0.610560
\(470\) 6.51456e6 1.36032
\(471\) −1.16890e6 −0.242787
\(472\) 4.61866e6 0.954247
\(473\) −5.11666e6 −1.05156
\(474\) −3.55190e6 −0.726132
\(475\) 7.94196e6 1.61508
\(476\) −24696.0 −0.00499585
\(477\) −777114. −0.156383
\(478\) 8.70682e6 1.74297
\(479\) −7.60402e6 −1.51427 −0.757137 0.653257i \(-0.773402\pi\)
−0.757137 + 0.653257i \(0.773402\pi\)
\(480\) −1.01088e6 −0.200261
\(481\) 2.40183e6 0.473347
\(482\) 878388. 0.172214
\(483\) −1.85220e6 −0.361260
\(484\) 144340. 0.0280074
\(485\) −1.12056e7 −2.16313
\(486\) 354294. 0.0680414
\(487\) 673112. 0.128607 0.0643035 0.997930i \(-0.479517\pi\)
0.0643035 + 0.997930i \(0.479517\pi\)
\(488\) 8.31230e6 1.58005
\(489\) −4.99756e6 −0.945117
\(490\) −1.12367e6 −0.211421
\(491\) −2.47170e6 −0.462692 −0.231346 0.972872i \(-0.574313\pi\)
−0.231346 + 0.972872i \(0.574313\pi\)
\(492\) −286632. −0.0533841
\(493\) 685692. 0.127061
\(494\) 7.11797e6 1.31232
\(495\) 2.80519e6 0.514576
\(496\) −90880.0 −0.0165869
\(497\) 1.56996e6 0.285100
\(498\) 2.17015e6 0.392118
\(499\) 6.08152e6 1.09335 0.546677 0.837343i \(-0.315892\pi\)
0.546677 + 0.837343i \(0.315892\pi\)
\(500\) −51792.0 −0.00926483
\(501\) −391608. −0.0697039
\(502\) −3.64716e6 −0.645945
\(503\) −846216. −0.149129 −0.0745644 0.997216i \(-0.523757\pi\)
−0.0745644 + 0.997216i \(0.523757\pi\)
\(504\) 666792. 0.116927
\(505\) −211068. −0.0368293
\(506\) −1.11888e7 −1.94271
\(507\) 1.58336e6 0.273565
\(508\) −297280. −0.0511101
\(509\) −7.66785e6 −1.31183 −0.655917 0.754833i \(-0.727718\pi\)
−0.655917 + 0.754833i \(0.727718\pi\)
\(510\) −530712. −0.0903511
\(511\) −3.03045e6 −0.513400
\(512\) 4.47130e6 0.753804
\(513\) −1.95664e6 −0.328259
\(514\) −573516. −0.0957498
\(515\) 1.02779e7 1.70760
\(516\) 414864. 0.0685933
\(517\) −6.18048e6 −1.01694
\(518\) 1.59760e6 0.261603
\(519\) 164970. 0.0268835
\(520\) −5.79197e6 −0.939329
\(521\) −9.68938e6 −1.56387 −0.781937 0.623357i \(-0.785768\pi\)
−0.781937 + 0.623357i \(0.785768\pi\)
\(522\) 2.64481e6 0.424833
\(523\) −7.51678e6 −1.20165 −0.600824 0.799381i \(-0.705161\pi\)
−0.600824 + 0.799381i \(0.705161\pi\)
\(524\) −621264. −0.0988435
\(525\) −1.30492e6 −0.206626
\(526\) 1.32021e7 2.08055
\(527\) −10080.0 −0.00158101
\(528\) 4.53946e6 0.708629
\(529\) 1.12037e7 1.74069
\(530\) 4.48999e6 0.694314
\(531\) 2.22685e6 0.342732
\(532\) 526064. 0.0805860
\(533\) −3.51920e6 −0.536570
\(534\) −430596. −0.0653457
\(535\) −1.00554e7 −1.51886
\(536\) −9.97181e6 −1.49921
\(537\) 1.37992e6 0.206499
\(538\) −1.06215e7 −1.58208
\(539\) 1.06604e6 0.158053
\(540\) −227448. −0.0335659
\(541\) 7.34325e6 1.07869 0.539343 0.842086i \(-0.318673\pi\)
0.539343 + 0.842086i \(0.318673\pi\)
\(542\) 1.34102e6 0.196082
\(543\) 3.43859e6 0.500474
\(544\) −181440. −0.0262867
\(545\) −7.87628e6 −1.13587
\(546\) −1.16953e6 −0.167892
\(547\) 2.18296e6 0.311945 0.155973 0.987761i \(-0.450149\pi\)
0.155973 + 0.987761i \(0.450149\pi\)
\(548\) −1.05698e6 −0.150355
\(549\) 4.00772e6 0.567501
\(550\) −7.88278e6 −1.11115
\(551\) −1.46063e7 −2.04957
\(552\) −6.35040e6 −0.887061
\(553\) −3.22302e6 −0.448178
\(554\) 2.05667e6 0.284702
\(555\) 3.81467e6 0.525683
\(556\) 898448. 0.123255
\(557\) 1.25466e7 1.71351 0.856755 0.515724i \(-0.172477\pi\)
0.856755 + 0.515724i \(0.172477\pi\)
\(558\) −38880.0 −0.00528617
\(559\) 5.09361e6 0.689439
\(560\) −4.34179e6 −0.585058
\(561\) 503496. 0.0675443
\(562\) −2.88227e6 −0.384940
\(563\) 5.15972e6 0.686050 0.343025 0.939326i \(-0.388549\pi\)
0.343025 + 0.939326i \(0.388549\pi\)
\(564\) 501120. 0.0663352
\(565\) 1.71714e7 2.26300
\(566\) 179880. 0.0236016
\(567\) 321489. 0.0419961
\(568\) 5.38272e6 0.700053
\(569\) 1.17452e7 1.52083 0.760414 0.649439i \(-0.224996\pi\)
0.760414 + 0.649439i \(0.224996\pi\)
\(570\) 1.13050e7 1.45742
\(571\) −7.54728e6 −0.968725 −0.484362 0.874867i \(-0.660948\pi\)
−0.484362 + 0.874867i \(0.660948\pi\)
\(572\) −784992. −0.100317
\(573\) 2.46067e6 0.313089
\(574\) −2.34083e6 −0.296544
\(575\) 1.24278e7 1.56756
\(576\) 2.24467e6 0.281901
\(577\) 9.28483e6 1.16101 0.580503 0.814258i \(-0.302856\pi\)
0.580503 + 0.814258i \(0.302856\pi\)
\(578\) 8.42389e6 1.04880
\(579\) −1.38242e6 −0.171373
\(580\) −1.69790e6 −0.209577
\(581\) 1.96921e6 0.242020
\(582\) −7.75775e6 −0.949354
\(583\) −4.25974e6 −0.519053
\(584\) −1.03901e7 −1.26063
\(585\) −2.79256e6 −0.337374
\(586\) 1.18840e6 0.142961
\(587\) 1.47623e6 0.176831 0.0884155 0.996084i \(-0.471820\pi\)
0.0884155 + 0.996084i \(0.471820\pi\)
\(588\) −86436.0 −0.0103098
\(589\) 214720. 0.0255026
\(590\) −1.28663e7 −1.52168
\(591\) −1.38980e6 −0.163675
\(592\) 6.17302e6 0.723925
\(593\) −1.24007e7 −1.44813 −0.724067 0.689729i \(-0.757730\pi\)
−0.724067 + 0.689729i \(0.757730\pi\)
\(594\) 1.94206e6 0.225837
\(595\) −481572. −0.0557659
\(596\) −328296. −0.0378573
\(597\) 3.30170e6 0.379142
\(598\) 1.11384e7 1.27371
\(599\) −3.69127e6 −0.420348 −0.210174 0.977664i \(-0.567403\pi\)
−0.210174 + 0.977664i \(0.567403\pi\)
\(600\) −4.47401e6 −0.507363
\(601\) 9.12223e6 1.03018 0.515092 0.857135i \(-0.327758\pi\)
0.515092 + 0.857135i \(0.327758\pi\)
\(602\) 3.38806e6 0.381030
\(603\) −4.80784e6 −0.538464
\(604\) −1.14813e6 −0.128055
\(605\) 2.81463e6 0.312632
\(606\) −146124. −0.0161637
\(607\) −5.67914e6 −0.625620 −0.312810 0.949816i \(-0.601270\pi\)
−0.312810 + 0.949816i \(0.601270\pi\)
\(608\) 3.86496e6 0.424020
\(609\) 2.39992e6 0.262213
\(610\) −2.31557e7 −2.51961
\(611\) 6.15264e6 0.666743
\(612\) −40824.0 −0.00440592
\(613\) −1.40106e7 −1.50593 −0.752966 0.658060i \(-0.771377\pi\)
−0.752966 + 0.658060i \(0.771377\pi\)
\(614\) 6.27382e6 0.671600
\(615\) −5.58932e6 −0.595897
\(616\) 3.65501e6 0.388094
\(617\) −253686. −0.0268277 −0.0134139 0.999910i \(-0.504270\pi\)
−0.0134139 + 0.999910i \(0.504270\pi\)
\(618\) 7.11547e6 0.749433
\(619\) 4.30034e6 0.451103 0.225552 0.974231i \(-0.427582\pi\)
0.225552 + 0.974231i \(0.427582\pi\)
\(620\) 24960.0 0.00260775
\(621\) −3.06180e6 −0.318602
\(622\) −1.10231e7 −1.14242
\(623\) −390726. −0.0403322
\(624\) −4.51901e6 −0.464603
\(625\) −1.02568e7 −1.05030
\(626\) 2.19296e6 0.223664
\(627\) −1.07253e7 −1.08953
\(628\) 519512. 0.0525650
\(629\) 684684. 0.0690023
\(630\) −1.85749e6 −0.186456
\(631\) 1.04150e7 1.04132 0.520662 0.853763i \(-0.325685\pi\)
0.520662 + 0.853763i \(0.325685\pi\)
\(632\) −1.10504e7 −1.10048
\(633\) −4.68220e6 −0.464451
\(634\) 170028. 0.0167995
\(635\) −5.79696e6 −0.570514
\(636\) 345384. 0.0338579
\(637\) −1.06124e6 −0.103625
\(638\) 1.44975e7 1.41007
\(639\) 2.59524e6 0.251435
\(640\) −1.65635e7 −1.59846
\(641\) 4.52714e6 0.435190 0.217595 0.976039i \(-0.430179\pi\)
0.217595 + 0.976039i \(0.430179\pi\)
\(642\) −6.96146e6 −0.666596
\(643\) 1.49687e7 1.42776 0.713882 0.700266i \(-0.246935\pi\)
0.713882 + 0.700266i \(0.246935\pi\)
\(644\) 823200. 0.0782151
\(645\) 8.08985e6 0.765669
\(646\) 2.02910e6 0.191304
\(647\) −1.73020e7 −1.62493 −0.812465 0.583010i \(-0.801875\pi\)
−0.812465 + 0.583010i \(0.801875\pi\)
\(648\) 1.10225e6 0.103120
\(649\) 1.22064e7 1.13757
\(650\) 7.84727e6 0.728509
\(651\) −35280.0 −0.00326269
\(652\) 2.22114e6 0.204624
\(653\) 4.07470e6 0.373949 0.186975 0.982365i \(-0.440132\pi\)
0.186975 + 0.982365i \(0.440132\pi\)
\(654\) −5.45281e6 −0.498513
\(655\) −1.21146e7 −1.10334
\(656\) −9.04483e6 −0.820618
\(657\) −5.00953e6 −0.452776
\(658\) 4.09248e6 0.368487
\(659\) −3.79475e6 −0.340384 −0.170192 0.985411i \(-0.554439\pi\)
−0.170192 + 0.985411i \(0.554439\pi\)
\(660\) −1.24675e6 −0.111409
\(661\) 1.64261e7 1.46228 0.731142 0.682225i \(-0.238988\pi\)
0.731142 + 0.682225i \(0.238988\pi\)
\(662\) −1.16035e7 −1.02907
\(663\) −501228. −0.0442845
\(664\) 6.75158e6 0.594272
\(665\) 1.02582e7 0.899537
\(666\) 2.64092e6 0.230712
\(667\) −2.28564e7 −1.98927
\(668\) 174048. 0.0150913
\(669\) −2.74262e6 −0.236920
\(670\) 2.77786e7 2.39069
\(671\) 2.19682e7 1.88360
\(672\) −635040. −0.0542473
\(673\) 5.50675e6 0.468660 0.234330 0.972157i \(-0.424710\pi\)
0.234330 + 0.972157i \(0.424710\pi\)
\(674\) 1.13290e7 0.960602
\(675\) −2.15711e6 −0.182227
\(676\) −703716. −0.0592285
\(677\) 1.83957e7 1.54257 0.771286 0.636488i \(-0.219614\pi\)
0.771286 + 0.636488i \(0.219614\pi\)
\(678\) 1.18879e7 0.993185
\(679\) −7.03944e6 −0.585954
\(680\) −1.65110e6 −0.136931
\(681\) −2.59729e6 −0.214612
\(682\) −213120. −0.0175454
\(683\) 1.75835e6 0.144229 0.0721146 0.997396i \(-0.477025\pi\)
0.0721146 + 0.997396i \(0.477025\pi\)
\(684\) 869616. 0.0710702
\(685\) −2.06112e7 −1.67833
\(686\) −705894. −0.0572703
\(687\) −6.94971e6 −0.561791
\(688\) 1.30913e7 1.05441
\(689\) 4.24055e6 0.340309
\(690\) 1.76904e7 1.41454
\(691\) −5.36314e6 −0.427291 −0.213646 0.976911i \(-0.568534\pi\)
−0.213646 + 0.976911i \(0.568534\pi\)
\(692\) −73320.0 −0.00582046
\(693\) 1.76224e6 0.139390
\(694\) −1.75162e7 −1.38052
\(695\) 1.75197e7 1.37583
\(696\) 8.22830e6 0.643854
\(697\) −1.00321e6 −0.0782187
\(698\) 4.68409e6 0.363904
\(699\) −2.27011e6 −0.175733
\(700\) 579964. 0.0447359
\(701\) −2.12606e7 −1.63411 −0.817054 0.576561i \(-0.804394\pi\)
−0.817054 + 0.576561i \(0.804394\pi\)
\(702\) −1.93331e6 −0.148067
\(703\) −1.45849e7 −1.11305
\(704\) 1.23041e7 0.935662
\(705\) 9.77184e6 0.740463
\(706\) −8.00622e6 −0.604527
\(707\) −132594. −0.00997643
\(708\) −989712. −0.0742037
\(709\) 2.07729e6 0.155196 0.0775980 0.996985i \(-0.475275\pi\)
0.0775980 + 0.996985i \(0.475275\pi\)
\(710\) −1.49947e7 −1.11633
\(711\) −5.32786e6 −0.395256
\(712\) −1.33963e6 −0.0990343
\(713\) 336000. 0.0247523
\(714\) −333396. −0.0244746
\(715\) −1.53073e7 −1.11979
\(716\) −613296. −0.0447082
\(717\) 1.30602e7 0.948752
\(718\) −6.10459e6 −0.441922
\(719\) 4.23619e6 0.305600 0.152800 0.988257i \(-0.451171\pi\)
0.152800 + 0.988257i \(0.451171\pi\)
\(720\) −7.17725e6 −0.515973
\(721\) 6.45663e6 0.462560
\(722\) −2.83665e7 −2.02518
\(723\) 1.31758e6 0.0937415
\(724\) −1.52826e6 −0.108356
\(725\) −1.61029e7 −1.13778
\(726\) 1.94859e6 0.137208
\(727\) 2.14524e7 1.50536 0.752678 0.658389i \(-0.228762\pi\)
0.752678 + 0.658389i \(0.228762\pi\)
\(728\) −3.63854e6 −0.254448
\(729\) 531441. 0.0370370
\(730\) 2.89439e7 2.01025
\(731\) 1.45202e6 0.100503
\(732\) −1.78121e6 −0.122867
\(733\) −1.48892e7 −1.02355 −0.511777 0.859118i \(-0.671013\pi\)
−0.511777 + 0.859118i \(0.671013\pi\)
\(734\) −5.02608e6 −0.344341
\(735\) −1.68550e6 −0.115083
\(736\) 6.04800e6 0.411545
\(737\) −2.63541e7 −1.78722
\(738\) −3.86953e6 −0.261528
\(739\) 6.99324e6 0.471050 0.235525 0.971868i \(-0.424319\pi\)
0.235525 + 0.971868i \(0.424319\pi\)
\(740\) −1.69541e6 −0.113814
\(741\) 1.06770e7 0.714335
\(742\) 2.82064e6 0.188078
\(743\) 1.90428e6 0.126549 0.0632745 0.997996i \(-0.479846\pi\)
0.0632745 + 0.997996i \(0.479846\pi\)
\(744\) −120960. −0.00801142
\(745\) −6.40177e6 −0.422581
\(746\) 9.11958e6 0.599968
\(747\) 3.25523e6 0.213442
\(748\) −223776. −0.0146238
\(749\) −6.31688e6 −0.411432
\(750\) −699192. −0.0453882
\(751\) 1.95361e7 1.26398 0.631988 0.774978i \(-0.282239\pi\)
0.631988 + 0.774978i \(0.282239\pi\)
\(752\) 1.58131e7 1.01970
\(753\) −5.47074e6 −0.351608
\(754\) −1.44322e7 −0.924493
\(755\) −2.23885e7 −1.42941
\(756\) −142884. −0.00909241
\(757\) 1.25183e6 0.0793973 0.0396986 0.999212i \(-0.487360\pi\)
0.0396986 + 0.999212i \(0.487360\pi\)
\(758\) −1.58679e7 −1.00311
\(759\) −1.67832e7 −1.05748
\(760\) 3.51711e7 2.20878
\(761\) 2.04472e7 1.27989 0.639944 0.768422i \(-0.278958\pi\)
0.639944 + 0.768422i \(0.278958\pi\)
\(762\) −4.01328e6 −0.250387
\(763\) −4.94792e6 −0.307689
\(764\) −1.09363e6 −0.0677857
\(765\) −796068. −0.0491809
\(766\) −1.20802e7 −0.743876
\(767\) −1.21515e7 −0.745831
\(768\) −3.48595e6 −0.213264
\(769\) 2.21064e6 0.134804 0.0674020 0.997726i \(-0.478529\pi\)
0.0674020 + 0.997726i \(0.478529\pi\)
\(770\) −1.01818e7 −0.618868
\(771\) −860274. −0.0521196
\(772\) 614408. 0.0371034
\(773\) 1.29151e7 0.777405 0.388703 0.921363i \(-0.372923\pi\)
0.388703 + 0.921363i \(0.372923\pi\)
\(774\) 5.60066e6 0.336037
\(775\) 236720. 0.0141573
\(776\) −2.41352e7 −1.43879
\(777\) 2.39639e6 0.142399
\(778\) 4.35740e6 0.258095
\(779\) 2.13700e7 1.26171
\(780\) 1.24114e6 0.0730437
\(781\) 1.42258e7 0.834541
\(782\) 3.17520e6 0.185675
\(783\) 3.96722e6 0.231250
\(784\) −2.72754e6 −0.158482
\(785\) 1.01305e7 0.586754
\(786\) −8.38706e6 −0.484232
\(787\) −1.35499e7 −0.779830 −0.389915 0.920851i \(-0.627496\pi\)
−0.389915 + 0.920851i \(0.627496\pi\)
\(788\) 617688. 0.0354367
\(789\) 1.98031e7 1.13251
\(790\) 3.07832e7 1.75487
\(791\) 1.07872e7 0.613007
\(792\) 6.04195e6 0.342266
\(793\) −2.18693e7 −1.23496
\(794\) −2.74547e7 −1.54549
\(795\) 6.73499e6 0.377937
\(796\) −1.46742e6 −0.0820867
\(797\) −2.45956e7 −1.37155 −0.685776 0.727813i \(-0.740537\pi\)
−0.685776 + 0.727813i \(0.740537\pi\)
\(798\) 7.10186e6 0.394789
\(799\) 1.75392e6 0.0971948
\(800\) 4.26096e6 0.235387
\(801\) −645894. −0.0355697
\(802\) 203220. 0.0111566
\(803\) −2.74596e7 −1.50282
\(804\) 2.13682e6 0.116581
\(805\) 1.60524e7 0.873072
\(806\) 212160. 0.0115034
\(807\) −1.59322e7 −0.861177
\(808\) −454608. −0.0244968
\(809\) 1.55237e7 0.833920 0.416960 0.908925i \(-0.363095\pi\)
0.416960 + 0.908925i \(0.363095\pi\)
\(810\) −3.07055e6 −0.164438
\(811\) −2.66262e7 −1.42153 −0.710766 0.703429i \(-0.751651\pi\)
−0.710766 + 0.703429i \(0.751651\pi\)
\(812\) −1.06663e6 −0.0567707
\(813\) 2.01154e6 0.106734
\(814\) 1.44762e7 0.765760
\(815\) 4.33122e7 2.28410
\(816\) −1.28822e6 −0.0677276
\(817\) −3.09304e7 −1.62118
\(818\) 3.51707e7 1.83780
\(819\) −1.75430e6 −0.0913889
\(820\) 2.48414e6 0.129016
\(821\) −1.23891e7 −0.641477 −0.320739 0.947168i \(-0.603931\pi\)
−0.320739 + 0.947168i \(0.603931\pi\)
\(822\) −1.42693e7 −0.736585
\(823\) −3.65630e6 −0.188166 −0.0940831 0.995564i \(-0.529992\pi\)
−0.0940831 + 0.995564i \(0.529992\pi\)
\(824\) 2.21370e7 1.13580
\(825\) −1.18242e7 −0.604833
\(826\) −8.08265e6 −0.412196
\(827\) 2.80463e7 1.42597 0.712987 0.701178i \(-0.247342\pi\)
0.712987 + 0.701178i \(0.247342\pi\)
\(828\) 1.36080e6 0.0689792
\(829\) 2.11153e7 1.06712 0.533558 0.845763i \(-0.320855\pi\)
0.533558 + 0.845763i \(0.320855\pi\)
\(830\) −1.88080e7 −0.947648
\(831\) 3.08500e6 0.154972
\(832\) −1.22487e7 −0.613454
\(833\) −302526. −0.0151060
\(834\) 1.21290e7 0.603826
\(835\) 3.39394e6 0.168456
\(836\) 4.76678e6 0.235890
\(837\) −58320.0 −0.00287742
\(838\) −1.81649e6 −0.0893557
\(839\) 1.33947e7 0.656944 0.328472 0.944514i \(-0.393466\pi\)
0.328472 + 0.944514i \(0.393466\pi\)
\(840\) −5.77886e6 −0.282582
\(841\) 9.10422e6 0.443867
\(842\) 3.22025e7 1.56534
\(843\) −4.32340e6 −0.209535
\(844\) 2.08098e6 0.100557
\(845\) −1.37225e7 −0.661135
\(846\) 6.76512e6 0.324975
\(847\) 1.76816e6 0.0846865
\(848\) 1.08988e7 0.520461
\(849\) 269820. 0.0128471
\(850\) 2.23700e6 0.106199
\(851\) −2.28228e7 −1.08030
\(852\) −1.15344e6 −0.0544372
\(853\) 3.01513e7 1.41884 0.709420 0.704786i \(-0.248957\pi\)
0.709420 + 0.704786i \(0.248957\pi\)
\(854\) −1.45465e7 −0.682519
\(855\) 1.69575e7 0.793317
\(856\) −2.16579e7 −1.01026
\(857\) 2.39894e7 1.11575 0.557875 0.829925i \(-0.311617\pi\)
0.557875 + 0.829925i \(0.311617\pi\)
\(858\) −1.05974e7 −0.491452
\(859\) −8.87576e6 −0.410414 −0.205207 0.978719i \(-0.565787\pi\)
−0.205207 + 0.978719i \(0.565787\pi\)
\(860\) −3.59549e6 −0.165772
\(861\) −3.51124e6 −0.161418
\(862\) −7.06234e6 −0.323728
\(863\) −8.71286e6 −0.398230 −0.199115 0.979976i \(-0.563807\pi\)
−0.199115 + 0.979976i \(0.563807\pi\)
\(864\) −1.04976e6 −0.0478416
\(865\) −1.42974e6 −0.0649706
\(866\) 2.19750e7 0.995711
\(867\) 1.26358e7 0.570895
\(868\) 15680.0 0.000706394 0
\(869\) −2.92045e7 −1.31190
\(870\) −2.29217e7 −1.02671
\(871\) 2.62354e7 1.17177
\(872\) −1.69643e7 −0.755518
\(873\) −1.16366e7 −0.516763
\(874\) −6.76368e7 −2.99505
\(875\) −634452. −0.0280142
\(876\) 2.22646e6 0.0980288
\(877\) −2.95788e7 −1.29862 −0.649310 0.760524i \(-0.724942\pi\)
−0.649310 + 0.760524i \(0.724942\pi\)
\(878\) 1.52205e7 0.666333
\(879\) 1.78259e6 0.0778180
\(880\) −3.93420e7 −1.71257
\(881\) 2.45670e7 1.06638 0.533190 0.845995i \(-0.320993\pi\)
0.533190 + 0.845995i \(0.320993\pi\)
\(882\) −1.16689e6 −0.0505076
\(883\) 1.45682e7 0.628788 0.314394 0.949293i \(-0.398199\pi\)
0.314394 + 0.949293i \(0.398199\pi\)
\(884\) 222768. 0.00958787
\(885\) −1.92994e7 −0.828295
\(886\) −3.60902e7 −1.54456
\(887\) 1.61714e7 0.690141 0.345070 0.938577i \(-0.387855\pi\)
0.345070 + 0.938577i \(0.387855\pi\)
\(888\) 8.21621e6 0.349654
\(889\) −3.64168e6 −0.154542
\(890\) 3.73183e6 0.157924
\(891\) 2.91308e6 0.122930
\(892\) 1.21894e6 0.0512946
\(893\) −3.73613e7 −1.56781
\(894\) −4.43200e6 −0.185462
\(895\) −1.19593e7 −0.499054
\(896\) −1.04052e7 −0.432995
\(897\) 1.67076e7 0.693319
\(898\) −3.39579e7 −1.40524
\(899\) −435360. −0.0179659
\(900\) 958716. 0.0394533
\(901\) 1.20884e6 0.0496087
\(902\) −2.12108e7 −0.868041
\(903\) 5.08208e6 0.207407
\(904\) 3.69845e7 1.50522
\(905\) −2.98011e7 −1.20952
\(906\) −1.54997e7 −0.627341
\(907\) 3.14446e7 1.26919 0.634596 0.772844i \(-0.281167\pi\)
0.634596 + 0.772844i \(0.281167\pi\)
\(908\) 1.15435e6 0.0464648
\(909\) −219186. −0.00879839
\(910\) 1.01359e7 0.405752
\(911\) 1.51427e7 0.604514 0.302257 0.953227i \(-0.402260\pi\)
0.302257 + 0.953227i \(0.402260\pi\)
\(912\) 2.74412e7 1.09249
\(913\) 1.78435e7 0.708439
\(914\) 3.87695e7 1.53506
\(915\) −3.47336e7 −1.37150
\(916\) 3.08876e6 0.121631
\(917\) −7.61048e6 −0.298875
\(918\) −551124. −0.0215845
\(919\) 4.14876e7 1.62043 0.810214 0.586134i \(-0.199351\pi\)
0.810214 + 0.586134i \(0.199351\pi\)
\(920\) 5.50368e7 2.14380
\(921\) 9.41072e6 0.365573
\(922\) 2.02412e7 0.784167
\(923\) −1.41617e7 −0.547155
\(924\) −783216. −0.0301788
\(925\) −1.60792e7 −0.617889
\(926\) 2.72985e7 1.04619
\(927\) 1.06732e7 0.407939
\(928\) −7.83648e6 −0.298711
\(929\) −1.78495e7 −0.678556 −0.339278 0.940686i \(-0.610183\pi\)
−0.339278 + 0.940686i \(0.610183\pi\)
\(930\) 336960. 0.0127753
\(931\) 6.44428e6 0.243669
\(932\) 1.00894e6 0.0380473
\(933\) −1.65346e7 −0.621855
\(934\) −1.20681e7 −0.452661
\(935\) −4.36363e6 −0.163237
\(936\) −6.01474e6 −0.224402
\(937\) 2.96399e7 1.10288 0.551439 0.834215i \(-0.314079\pi\)
0.551439 + 0.834215i \(0.314079\pi\)
\(938\) 1.74507e7 0.647597
\(939\) 3.28945e6 0.121747
\(940\) −4.34304e6 −0.160315
\(941\) −3.22282e7 −1.18648 −0.593242 0.805024i \(-0.702152\pi\)
−0.593242 + 0.805024i \(0.702152\pi\)
\(942\) 7.01341e6 0.257515
\(943\) 3.34404e7 1.22459
\(944\) −3.12309e7 −1.14066
\(945\) −2.78624e6 −0.101494
\(946\) 3.06999e7 1.11535
\(947\) 4.84885e7 1.75697 0.878484 0.477772i \(-0.158556\pi\)
0.878484 + 0.477772i \(0.158556\pi\)
\(948\) 2.36794e6 0.0855754
\(949\) 2.73359e7 0.985300
\(950\) −4.76517e7 −1.71305
\(951\) 255042. 0.00914451
\(952\) −1.03723e6 −0.0370923
\(953\) −2.03264e7 −0.724983 −0.362491 0.931987i \(-0.618074\pi\)
−0.362491 + 0.931987i \(0.618074\pi\)
\(954\) 4.66268e6 0.165869
\(955\) −2.13258e7 −0.756654
\(956\) −5.80454e6 −0.205411
\(957\) 2.17462e7 0.767546
\(958\) 4.56241e7 1.60613
\(959\) −1.29481e7 −0.454630
\(960\) −1.94538e7 −0.681282
\(961\) −2.86228e7 −0.999776
\(962\) −1.44110e7 −0.502060
\(963\) −1.04422e7 −0.362849
\(964\) −585592. −0.0202956
\(965\) 1.19810e7 0.414165
\(966\) 1.11132e7 0.383174
\(967\) −3.66292e6 −0.125968 −0.0629841 0.998015i \(-0.520062\pi\)
−0.0629841 + 0.998015i \(0.520062\pi\)
\(968\) 6.06228e6 0.207945
\(969\) 3.04366e6 0.104132
\(970\) 6.72338e7 2.29434
\(971\) 1.48741e6 0.0506271 0.0253136 0.999680i \(-0.491942\pi\)
0.0253136 + 0.999680i \(0.491942\pi\)
\(972\) −236196. −0.00801875
\(973\) 1.10060e7 0.372689
\(974\) −4.03867e6 −0.136408
\(975\) 1.17709e7 0.396550
\(976\) −5.62070e7 −1.88871
\(977\) 4.07930e7 1.36725 0.683627 0.729831i \(-0.260401\pi\)
0.683627 + 0.729831i \(0.260401\pi\)
\(978\) 2.99853e7 1.00245
\(979\) −3.54046e6 −0.118060
\(980\) 749112. 0.0249162
\(981\) −8.17922e6 −0.271356
\(982\) 1.48302e7 0.490759
\(983\) −9.26326e6 −0.305759 −0.152880 0.988245i \(-0.548855\pi\)
−0.152880 + 0.988245i \(0.548855\pi\)
\(984\) −1.20385e7 −0.396357
\(985\) 1.20449e7 0.395561
\(986\) −4.11415e6 −0.134768
\(987\) 6.13872e6 0.200579
\(988\) −4.74531e6 −0.154658
\(989\) −4.84008e7 −1.57348
\(990\) −1.68312e7 −0.545790
\(991\) −5.22051e7 −1.68861 −0.844303 0.535866i \(-0.819985\pi\)
−0.844303 + 0.535866i \(0.819985\pi\)
\(992\) 115200. 0.00371684
\(993\) −1.74052e7 −0.560153
\(994\) −9.41976e6 −0.302394
\(995\) −2.86148e7 −0.916289
\(996\) −1.44677e6 −0.0462116
\(997\) −1.86609e7 −0.594560 −0.297280 0.954790i \(-0.596079\pi\)
−0.297280 + 0.954790i \(0.596079\pi\)
\(998\) −3.64891e7 −1.15968
\(999\) 3.96139e6 0.125584
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 21.6.a.a.1.1 1
3.2 odd 2 63.6.a.d.1.1 1
4.3 odd 2 336.6.a.r.1.1 1
5.2 odd 4 525.6.d.b.274.1 2
5.3 odd 4 525.6.d.b.274.2 2
5.4 even 2 525.6.a.d.1.1 1
7.2 even 3 147.6.e.j.67.1 2
7.3 odd 6 147.6.e.i.79.1 2
7.4 even 3 147.6.e.j.79.1 2
7.5 odd 6 147.6.e.i.67.1 2
7.6 odd 2 147.6.a.b.1.1 1
12.11 even 2 1008.6.a.c.1.1 1
21.20 even 2 441.6.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.a.1.1 1 1.1 even 1 trivial
63.6.a.d.1.1 1 3.2 odd 2
147.6.a.b.1.1 1 7.6 odd 2
147.6.e.i.67.1 2 7.5 odd 6
147.6.e.i.79.1 2 7.3 odd 6
147.6.e.j.67.1 2 7.2 even 3
147.6.e.j.79.1 2 7.4 even 3
336.6.a.r.1.1 1 4.3 odd 2
441.6.a.j.1.1 1 21.20 even 2
525.6.a.d.1.1 1 5.4 even 2
525.6.d.b.274.1 2 5.2 odd 4
525.6.d.b.274.2 2 5.3 odd 4
1008.6.a.c.1.1 1 12.11 even 2