Properties

Label 21.6.a.d.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)
 
\(f(q)\) \(=\) \(q+10.0000 q^{2} +9.00000 q^{3} +68.0000 q^{4} -106.000 q^{5} +90.0000 q^{6} -49.0000 q^{7} +360.000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+10.0000 q^{2} +9.00000 q^{3} +68.0000 q^{4} -106.000 q^{5} +90.0000 q^{6} -49.0000 q^{7} +360.000 q^{8} +81.0000 q^{9} -1060.00 q^{10} +92.0000 q^{11} +612.000 q^{12} +670.000 q^{13} -490.000 q^{14} -954.000 q^{15} +1424.00 q^{16} -222.000 q^{17} +810.000 q^{18} -908.000 q^{19} -7208.00 q^{20} -441.000 q^{21} +920.000 q^{22} -1176.00 q^{23} +3240.00 q^{24} +8111.00 q^{25} +6700.00 q^{26} +729.000 q^{27} -3332.00 q^{28} +1118.00 q^{29} -9540.00 q^{30} +3696.00 q^{31} +2720.00 q^{32} +828.000 q^{33} -2220.00 q^{34} +5194.00 q^{35} +5508.00 q^{36} +4182.00 q^{37} -9080.00 q^{38} +6030.00 q^{39} -38160.0 q^{40} -6662.00 q^{41} -4410.00 q^{42} -3700.00 q^{43} +6256.00 q^{44} -8586.00 q^{45} -11760.0 q^{46} -7056.00 q^{47} +12816.0 q^{48} +2401.00 q^{49} +81110.0 q^{50} -1998.00 q^{51} +45560.0 q^{52} -37578.0 q^{53} +7290.00 q^{54} -9752.00 q^{55} -17640.0 q^{56} -8172.00 q^{57} +11180.0 q^{58} +32700.0 q^{59} -64872.0 q^{60} -10802.0 q^{61} +36960.0 q^{62} -3969.00 q^{63} -18368.0 q^{64} -71020.0 q^{65} +8280.00 q^{66} +64996.0 q^{67} -15096.0 q^{68} -10584.0 q^{69} +51940.0 q^{70} -61320.0 q^{71} +29160.0 q^{72} +38922.0 q^{73} +41820.0 q^{74} +72999.0 q^{75} -61744.0 q^{76} -4508.00 q^{77} +60300.0 q^{78} -88096.0 q^{79} -150944. q^{80} +6561.00 q^{81} -66620.0 q^{82} +71892.0 q^{83} -29988.0 q^{84} +23532.0 q^{85} -37000.0 q^{86} +10062.0 q^{87} +33120.0 q^{88} +111818. q^{89} -85860.0 q^{90} -32830.0 q^{91} -79968.0 q^{92} +33264.0 q^{93} -70560.0 q^{94} +96248.0 q^{95} +24480.0 q^{96} -150846. q^{97} +24010.0 q^{98} +7452.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 10.0000 1.76777 0.883883 0.467707i \(-0.154920\pi\)
0.883883 + 0.467707i \(0.154920\pi\)
\(3\) 9.00000 0.577350
\(4\) 68.0000 2.12500
\(5\) −106.000 −1.89619 −0.948093 0.317994i \(-0.896991\pi\)
−0.948093 + 0.317994i \(0.896991\pi\)
\(6\) 90.0000 1.02062
\(7\) −49.0000 −0.377964
\(8\) 360.000 1.98874
\(9\) 81.0000 0.333333
\(10\) −1060.00 −3.35201
\(11\) 92.0000 0.229248 0.114624 0.993409i \(-0.463434\pi\)
0.114624 + 0.993409i \(0.463434\pi\)
\(12\) 612.000 1.22687
\(13\) 670.000 1.09955 0.549777 0.835312i \(-0.314713\pi\)
0.549777 + 0.835312i \(0.314713\pi\)
\(14\) −490.000 −0.668153
\(15\) −954.000 −1.09476
\(16\) 1424.00 1.39062
\(17\) −222.000 −0.186308 −0.0931538 0.995652i \(-0.529695\pi\)
−0.0931538 + 0.995652i \(0.529695\pi\)
\(18\) 810.000 0.589256
\(19\) −908.000 −0.577035 −0.288517 0.957475i \(-0.593162\pi\)
−0.288517 + 0.957475i \(0.593162\pi\)
\(20\) −7208.00 −4.02939
\(21\) −441.000 −0.218218
\(22\) 920.000 0.405258
\(23\) −1176.00 −0.463541 −0.231770 0.972771i \(-0.574452\pi\)
−0.231770 + 0.972771i \(0.574452\pi\)
\(24\) 3240.00 1.14820
\(25\) 8111.00 2.59552
\(26\) 6700.00 1.94375
\(27\) 729.000 0.192450
\(28\) −3332.00 −0.803175
\(29\) 1118.00 0.246858 0.123429 0.992353i \(-0.460611\pi\)
0.123429 + 0.992353i \(0.460611\pi\)
\(30\) −9540.00 −1.93529
\(31\) 3696.00 0.690761 0.345380 0.938463i \(-0.387750\pi\)
0.345380 + 0.938463i \(0.387750\pi\)
\(32\) 2720.00 0.469563
\(33\) 828.000 0.132357
\(34\) −2220.00 −0.329348
\(35\) 5194.00 0.716691
\(36\) 5508.00 0.708333
\(37\) 4182.00 0.502203 0.251102 0.967961i \(-0.419207\pi\)
0.251102 + 0.967961i \(0.419207\pi\)
\(38\) −9080.00 −1.02006
\(39\) 6030.00 0.634828
\(40\) −38160.0 −3.77102
\(41\) −6662.00 −0.618935 −0.309467 0.950910i \(-0.600151\pi\)
−0.309467 + 0.950910i \(0.600151\pi\)
\(42\) −4410.00 −0.385758
\(43\) −3700.00 −0.305162 −0.152581 0.988291i \(-0.548759\pi\)
−0.152581 + 0.988291i \(0.548759\pi\)
\(44\) 6256.00 0.487153
\(45\) −8586.00 −0.632062
\(46\) −11760.0 −0.819432
\(47\) −7056.00 −0.465923 −0.232961 0.972486i \(-0.574842\pi\)
−0.232961 + 0.972486i \(0.574842\pi\)
\(48\) 12816.0 0.802878
\(49\) 2401.00 0.142857
\(50\) 81110.0 4.58827
\(51\) −1998.00 −0.107565
\(52\) 45560.0 2.33655
\(53\) −37578.0 −1.83757 −0.918785 0.394758i \(-0.870828\pi\)
−0.918785 + 0.394758i \(0.870828\pi\)
\(54\) 7290.00 0.340207
\(55\) −9752.00 −0.434697
\(56\) −17640.0 −0.751672
\(57\) −8172.00 −0.333151
\(58\) 11180.0 0.436387
\(59\) 32700.0 1.22298 0.611488 0.791254i \(-0.290571\pi\)
0.611488 + 0.791254i \(0.290571\pi\)
\(60\) −64872.0 −2.32637
\(61\) −10802.0 −0.371689 −0.185844 0.982579i \(-0.559502\pi\)
−0.185844 + 0.982579i \(0.559502\pi\)
\(62\) 36960.0 1.22110
\(63\) −3969.00 −0.125988
\(64\) −18368.0 −0.560547
\(65\) −71020.0 −2.08496
\(66\) 8280.00 0.233976
\(67\) 64996.0 1.76889 0.884443 0.466649i \(-0.154539\pi\)
0.884443 + 0.466649i \(0.154539\pi\)
\(68\) −15096.0 −0.395904
\(69\) −10584.0 −0.267625
\(70\) 51940.0 1.26694
\(71\) −61320.0 −1.44363 −0.721816 0.692085i \(-0.756692\pi\)
−0.721816 + 0.692085i \(0.756692\pi\)
\(72\) 29160.0 0.662913
\(73\) 38922.0 0.854846 0.427423 0.904052i \(-0.359421\pi\)
0.427423 + 0.904052i \(0.359421\pi\)
\(74\) 41820.0 0.887779
\(75\) 72999.0 1.49852
\(76\) −61744.0 −1.22620
\(77\) −4508.00 −0.0866477
\(78\) 60300.0 1.12223
\(79\) −88096.0 −1.58814 −0.794069 0.607827i \(-0.792041\pi\)
−0.794069 + 0.607827i \(0.792041\pi\)
\(80\) −150944. −2.63688
\(81\) 6561.00 0.111111
\(82\) −66620.0 −1.09413
\(83\) 71892.0 1.14547 0.572737 0.819739i \(-0.305882\pi\)
0.572737 + 0.819739i \(0.305882\pi\)
\(84\) −29988.0 −0.463713
\(85\) 23532.0 0.353274
\(86\) −37000.0 −0.539455
\(87\) 10062.0 0.142523
\(88\) 33120.0 0.455915
\(89\) 111818. 1.49636 0.748181 0.663495i \(-0.230927\pi\)
0.748181 + 0.663495i \(0.230927\pi\)
\(90\) −85860.0 −1.11734
\(91\) −32830.0 −0.415592
\(92\) −79968.0 −0.985024
\(93\) 33264.0 0.398811
\(94\) −70560.0 −0.823643
\(95\) 96248.0 1.09416
\(96\) 24480.0 0.271102
\(97\) −150846. −1.62781 −0.813906 0.580996i \(-0.802663\pi\)
−0.813906 + 0.580996i \(0.802663\pi\)
\(98\) 24010.0 0.252538
\(99\) 7452.00 0.0764161
\(100\) 551548. 5.51548
\(101\) −137354. −1.33979 −0.669897 0.742454i \(-0.733662\pi\)
−0.669897 + 0.742454i \(0.733662\pi\)
\(102\) −19980.0 −0.190149
\(103\) 28760.0 0.267113 0.133557 0.991041i \(-0.457360\pi\)
0.133557 + 0.991041i \(0.457360\pi\)
\(104\) 241200. 2.18672
\(105\) 46746.0 0.413782
\(106\) −375780. −3.24840
\(107\) 22556.0 0.190460 0.0952298 0.995455i \(-0.469641\pi\)
0.0952298 + 0.995455i \(0.469641\pi\)
\(108\) 49572.0 0.408956
\(109\) 19998.0 0.161221 0.0806103 0.996746i \(-0.474313\pi\)
0.0806103 + 0.996746i \(0.474313\pi\)
\(110\) −97520.0 −0.768444
\(111\) 37638.0 0.289947
\(112\) −69776.0 −0.525607
\(113\) 17906.0 0.131918 0.0659588 0.997822i \(-0.478989\pi\)
0.0659588 + 0.997822i \(0.478989\pi\)
\(114\) −81720.0 −0.588933
\(115\) 124656. 0.878959
\(116\) 76024.0 0.524573
\(117\) 54270.0 0.366518
\(118\) 327000. 2.16194
\(119\) 10878.0 0.0704177
\(120\) −343440. −2.17720
\(121\) −152587. −0.947445
\(122\) −108020. −0.657059
\(123\) −59958.0 −0.357342
\(124\) 251328. 1.46787
\(125\) −528516. −3.02540
\(126\) −39690.0 −0.222718
\(127\) 66864.0 0.367860 0.183930 0.982939i \(-0.441118\pi\)
0.183930 + 0.982939i \(0.441118\pi\)
\(128\) −270720. −1.46048
\(129\) −33300.0 −0.176185
\(130\) −710200. −3.68572
\(131\) 153764. 0.782846 0.391423 0.920211i \(-0.371983\pi\)
0.391423 + 0.920211i \(0.371983\pi\)
\(132\) 56304.0 0.281258
\(133\) 44492.0 0.218099
\(134\) 649960. 3.12698
\(135\) −77274.0 −0.364921
\(136\) −79920.0 −0.370517
\(137\) 255978. 1.16520 0.582601 0.812758i \(-0.302035\pi\)
0.582601 + 0.812758i \(0.302035\pi\)
\(138\) −105840. −0.473099
\(139\) 282924. 1.24203 0.621016 0.783798i \(-0.286720\pi\)
0.621016 + 0.783798i \(0.286720\pi\)
\(140\) 353192. 1.52297
\(141\) −63504.0 −0.269001
\(142\) −613200. −2.55200
\(143\) 61640.0 0.252071
\(144\) 115344. 0.463542
\(145\) −118508. −0.468088
\(146\) 389220. 1.51117
\(147\) 21609.0 0.0824786
\(148\) 284376. 1.06718
\(149\) 408054. 1.50575 0.752873 0.658165i \(-0.228667\pi\)
0.752873 + 0.658165i \(0.228667\pi\)
\(150\) 729990. 2.64904
\(151\) 362504. 1.29381 0.646905 0.762571i \(-0.276063\pi\)
0.646905 + 0.762571i \(0.276063\pi\)
\(152\) −326880. −1.14757
\(153\) −17982.0 −0.0621025
\(154\) −45080.0 −0.153173
\(155\) −391776. −1.30981
\(156\) 410040. 1.34901
\(157\) −152786. −0.494691 −0.247346 0.968927i \(-0.579558\pi\)
−0.247346 + 0.968927i \(0.579558\pi\)
\(158\) −880960. −2.80746
\(159\) −338202. −1.06092
\(160\) −288320. −0.890379
\(161\) 57624.0 0.175202
\(162\) 65610.0 0.196419
\(163\) −150428. −0.443465 −0.221733 0.975107i \(-0.571171\pi\)
−0.221733 + 0.975107i \(0.571171\pi\)
\(164\) −453016. −1.31524
\(165\) −87768.0 −0.250973
\(166\) 718920. 2.02493
\(167\) −7288.00 −0.0202217 −0.0101108 0.999949i \(-0.503218\pi\)
−0.0101108 + 0.999949i \(0.503218\pi\)
\(168\) −158760. −0.433978
\(169\) 77607.0 0.209018
\(170\) 235320. 0.624506
\(171\) −73548.0 −0.192345
\(172\) −251600. −0.648469
\(173\) −289154. −0.734537 −0.367269 0.930115i \(-0.619707\pi\)
−0.367269 + 0.930115i \(0.619707\pi\)
\(174\) 100620. 0.251948
\(175\) −397439. −0.981014
\(176\) 131008. 0.318798
\(177\) 294300. 0.706085
\(178\) 1.11818e6 2.64522
\(179\) 199492. 0.465364 0.232682 0.972553i \(-0.425250\pi\)
0.232682 + 0.972553i \(0.425250\pi\)
\(180\) −583848. −1.34313
\(181\) 240550. 0.545769 0.272885 0.962047i \(-0.412022\pi\)
0.272885 + 0.962047i \(0.412022\pi\)
\(182\) −328300. −0.734670
\(183\) −97218.0 −0.214595
\(184\) −423360. −0.921861
\(185\) −443292. −0.952271
\(186\) 332640. 0.705005
\(187\) −20424.0 −0.0427107
\(188\) −479808. −0.990086
\(189\) −35721.0 −0.0727393
\(190\) 962480. 1.93423
\(191\) 290384. 0.575956 0.287978 0.957637i \(-0.407017\pi\)
0.287978 + 0.957637i \(0.407017\pi\)
\(192\) −165312. −0.323632
\(193\) −171454. −0.331325 −0.165663 0.986182i \(-0.552976\pi\)
−0.165663 + 0.986182i \(0.552976\pi\)
\(194\) −1.50846e6 −2.87759
\(195\) −639180. −1.20375
\(196\) 163268. 0.303571
\(197\) 401990. 0.737989 0.368994 0.929432i \(-0.379702\pi\)
0.368994 + 0.929432i \(0.379702\pi\)
\(198\) 74520.0 0.135086
\(199\) −259176. −0.463940 −0.231970 0.972723i \(-0.574517\pi\)
−0.231970 + 0.972723i \(0.574517\pi\)
\(200\) 2.91996e6 5.16181
\(201\) 584964. 1.02127
\(202\) −1.37354e6 −2.36844
\(203\) −54782.0 −0.0933035
\(204\) −135864. −0.228575
\(205\) 706172. 1.17362
\(206\) 287600. 0.472194
\(207\) −95256.0 −0.154514
\(208\) 954080. 1.52907
\(209\) −83536.0 −0.132284
\(210\) 467460. 0.731469
\(211\) −1.19179e6 −1.84286 −0.921431 0.388542i \(-0.872979\pi\)
−0.921431 + 0.388542i \(0.872979\pi\)
\(212\) −2.55530e6 −3.90484
\(213\) −551880. −0.833481
\(214\) 225560. 0.336688
\(215\) 392200. 0.578644
\(216\) 262440. 0.382733
\(217\) −181104. −0.261083
\(218\) 199980. 0.285000
\(219\) 350298. 0.493546
\(220\) −663136. −0.923732
\(221\) −148740. −0.204855
\(222\) 376380. 0.512559
\(223\) −218384. −0.294075 −0.147038 0.989131i \(-0.546974\pi\)
−0.147038 + 0.989131i \(0.546974\pi\)
\(224\) −133280. −0.177478
\(225\) 656991. 0.865173
\(226\) 179060. 0.233199
\(227\) 582852. 0.750747 0.375374 0.926874i \(-0.377514\pi\)
0.375374 + 0.926874i \(0.377514\pi\)
\(228\) −555696. −0.707946
\(229\) 961046. 1.21103 0.605516 0.795833i \(-0.292967\pi\)
0.605516 + 0.795833i \(0.292967\pi\)
\(230\) 1.24656e6 1.55379
\(231\) −40572.0 −0.0500261
\(232\) 402480. 0.490935
\(233\) 605994. 0.731271 0.365636 0.930758i \(-0.380852\pi\)
0.365636 + 0.930758i \(0.380852\pi\)
\(234\) 542700. 0.647918
\(235\) 747936. 0.883476
\(236\) 2.22360e6 2.59882
\(237\) −792864. −0.916912
\(238\) 108780. 0.124482
\(239\) 1.17014e6 1.32509 0.662544 0.749023i \(-0.269477\pi\)
0.662544 + 0.749023i \(0.269477\pi\)
\(240\) −1.35850e6 −1.52241
\(241\) −1.23691e6 −1.37181 −0.685907 0.727689i \(-0.740595\pi\)
−0.685907 + 0.727689i \(0.740595\pi\)
\(242\) −1.52587e6 −1.67486
\(243\) 59049.0 0.0641500
\(244\) −734536. −0.789839
\(245\) −254506. −0.270884
\(246\) −599580. −0.631698
\(247\) −608360. −0.634480
\(248\) 1.33056e6 1.37374
\(249\) 647028. 0.661340
\(250\) −5.28516e6 −5.34821
\(251\) 959708. 0.961512 0.480756 0.876854i \(-0.340362\pi\)
0.480756 + 0.876854i \(0.340362\pi\)
\(252\) −269892. −0.267725
\(253\) −108192. −0.106266
\(254\) 668640. 0.650291
\(255\) 211788. 0.203963
\(256\) −2.11942e6 −2.02124
\(257\) −1.21259e6 −1.14520 −0.572600 0.819835i \(-0.694065\pi\)
−0.572600 + 0.819835i \(0.694065\pi\)
\(258\) −333000. −0.311455
\(259\) −204918. −0.189815
\(260\) −4.82936e6 −4.43054
\(261\) 90558.0 0.0822859
\(262\) 1.53764e6 1.38389
\(263\) −1.25274e6 −1.11679 −0.558397 0.829574i \(-0.688583\pi\)
−0.558397 + 0.829574i \(0.688583\pi\)
\(264\) 298080. 0.263223
\(265\) 3.98327e6 3.48437
\(266\) 444920. 0.385547
\(267\) 1.00636e6 0.863925
\(268\) 4.41973e6 3.75888
\(269\) −136866. −0.115323 −0.0576614 0.998336i \(-0.518364\pi\)
−0.0576614 + 0.998336i \(0.518364\pi\)
\(270\) −772740. −0.645095
\(271\) −960896. −0.794791 −0.397396 0.917647i \(-0.630086\pi\)
−0.397396 + 0.917647i \(0.630086\pi\)
\(272\) −316128. −0.259084
\(273\) −295470. −0.239942
\(274\) 2.55978e6 2.05981
\(275\) 746212. 0.595019
\(276\) −719712. −0.568704
\(277\) 905830. 0.709328 0.354664 0.934994i \(-0.384595\pi\)
0.354664 + 0.934994i \(0.384595\pi\)
\(278\) 2.82924e6 2.19562
\(279\) 299376. 0.230254
\(280\) 1.86984e6 1.42531
\(281\) −33062.0 −0.0249783 −0.0124892 0.999922i \(-0.503976\pi\)
−0.0124892 + 0.999922i \(0.503976\pi\)
\(282\) −635040. −0.475530
\(283\) −863588. −0.640974 −0.320487 0.947253i \(-0.603847\pi\)
−0.320487 + 0.947253i \(0.603847\pi\)
\(284\) −4.16976e6 −3.06772
\(285\) 866232. 0.631716
\(286\) 616400. 0.445602
\(287\) 326438. 0.233935
\(288\) 220320. 0.156521
\(289\) −1.37057e6 −0.965289
\(290\) −1.18508e6 −0.827471
\(291\) −1.35761e6 −0.939818
\(292\) 2.64670e6 1.81655
\(293\) −1.33755e6 −0.910206 −0.455103 0.890439i \(-0.650398\pi\)
−0.455103 + 0.890439i \(0.650398\pi\)
\(294\) 216090. 0.145803
\(295\) −3.46620e6 −2.31899
\(296\) 1.50552e6 0.998751
\(297\) 67068.0 0.0441189
\(298\) 4.08054e6 2.66181
\(299\) −787920. −0.509688
\(300\) 4.96393e6 3.18436
\(301\) 181300. 0.115340
\(302\) 3.62504e6 2.28715
\(303\) −1.23619e6 −0.773530
\(304\) −1.29299e6 −0.802439
\(305\) 1.14501e6 0.704791
\(306\) −179820. −0.109783
\(307\) −1.32820e6 −0.804301 −0.402151 0.915573i \(-0.631737\pi\)
−0.402151 + 0.915573i \(0.631737\pi\)
\(308\) −306544. −0.184126
\(309\) 258840. 0.154218
\(310\) −3.91776e6 −2.31544
\(311\) −665832. −0.390359 −0.195179 0.980768i \(-0.562529\pi\)
−0.195179 + 0.980768i \(0.562529\pi\)
\(312\) 2.17080e6 1.26251
\(313\) −3.09021e6 −1.78290 −0.891451 0.453116i \(-0.850312\pi\)
−0.891451 + 0.453116i \(0.850312\pi\)
\(314\) −1.52786e6 −0.874499
\(315\) 420714. 0.238897
\(316\) −5.99053e6 −3.37479
\(317\) −974178. −0.544490 −0.272245 0.962228i \(-0.587766\pi\)
−0.272245 + 0.962228i \(0.587766\pi\)
\(318\) −3.38202e6 −1.87546
\(319\) 102856. 0.0565917
\(320\) 1.94701e6 1.06290
\(321\) 203004. 0.109962
\(322\) 576240. 0.309716
\(323\) 201576. 0.107506
\(324\) 446148. 0.236111
\(325\) 5.43437e6 2.85391
\(326\) −1.50428e6 −0.783943
\(327\) 179982. 0.0930807
\(328\) −2.39832e6 −1.23090
\(329\) 345744. 0.176102
\(330\) −877680. −0.443661
\(331\) 781772. 0.392202 0.196101 0.980584i \(-0.437172\pi\)
0.196101 + 0.980584i \(0.437172\pi\)
\(332\) 4.88866e6 2.43413
\(333\) 338742. 0.167401
\(334\) −72880.0 −0.0357472
\(335\) −6.88958e6 −3.35413
\(336\) −627984. −0.303459
\(337\) 348754. 0.167280 0.0836401 0.996496i \(-0.473345\pi\)
0.0836401 + 0.996496i \(0.473345\pi\)
\(338\) 776070. 0.369495
\(339\) 161154. 0.0761626
\(340\) 1.60018e6 0.750707
\(341\) 340032. 0.158356
\(342\) −735480. −0.340021
\(343\) −117649. −0.0539949
\(344\) −1.33200e6 −0.606887
\(345\) 1.12190e6 0.507467
\(346\) −2.89154e6 −1.29849
\(347\) 2.50625e6 1.11738 0.558690 0.829376i \(-0.311304\pi\)
0.558690 + 0.829376i \(0.311304\pi\)
\(348\) 684216. 0.302862
\(349\) 3.05861e6 1.34419 0.672094 0.740466i \(-0.265395\pi\)
0.672094 + 0.740466i \(0.265395\pi\)
\(350\) −3.97439e6 −1.73420
\(351\) 488430. 0.211609
\(352\) 250240. 0.107647
\(353\) −3.49291e6 −1.49194 −0.745969 0.665981i \(-0.768013\pi\)
−0.745969 + 0.665981i \(0.768013\pi\)
\(354\) 2.94300e6 1.24819
\(355\) 6.49992e6 2.73739
\(356\) 7.60362e6 3.17977
\(357\) 97902.0 0.0406557
\(358\) 1.99492e6 0.822655
\(359\) 2.12034e6 0.868301 0.434150 0.900840i \(-0.357049\pi\)
0.434150 + 0.900840i \(0.357049\pi\)
\(360\) −3.09096e6 −1.25701
\(361\) −1.65163e6 −0.667031
\(362\) 2.40550e6 0.964793
\(363\) −1.37328e6 −0.547008
\(364\) −2.23244e6 −0.883133
\(365\) −4.12573e6 −1.62095
\(366\) −972180. −0.379353
\(367\) 746592. 0.289346 0.144673 0.989479i \(-0.453787\pi\)
0.144673 + 0.989479i \(0.453787\pi\)
\(368\) −1.67462e6 −0.644611
\(369\) −539622. −0.206312
\(370\) −4.43292e6 −1.68339
\(371\) 1.84132e6 0.694536
\(372\) 2.26195e6 0.847473
\(373\) −939034. −0.349469 −0.174735 0.984616i \(-0.555907\pi\)
−0.174735 + 0.984616i \(0.555907\pi\)
\(374\) −204240. −0.0755026
\(375\) −4.75664e6 −1.74672
\(376\) −2.54016e6 −0.926598
\(377\) 749060. 0.271433
\(378\) −357210. −0.128586
\(379\) 5.16534e6 1.84714 0.923572 0.383424i \(-0.125255\pi\)
0.923572 + 0.383424i \(0.125255\pi\)
\(380\) 6.54486e6 2.32510
\(381\) 601776. 0.212384
\(382\) 2.90384e6 1.01816
\(383\) 400512. 0.139514 0.0697571 0.997564i \(-0.477778\pi\)
0.0697571 + 0.997564i \(0.477778\pi\)
\(384\) −2.43648e6 −0.843208
\(385\) 477848. 0.164300
\(386\) −1.71454e6 −0.585706
\(387\) −299700. −0.101721
\(388\) −1.02575e7 −3.45910
\(389\) 306822. 0.102805 0.0514023 0.998678i \(-0.483631\pi\)
0.0514023 + 0.998678i \(0.483631\pi\)
\(390\) −6.39180e6 −2.12795
\(391\) 261072. 0.0863611
\(392\) 864360. 0.284105
\(393\) 1.38388e6 0.451976
\(394\) 4.01990e6 1.30459
\(395\) 9.33818e6 3.01141
\(396\) 506736. 0.162384
\(397\) −3.83421e6 −1.22095 −0.610477 0.792034i \(-0.709022\pi\)
−0.610477 + 0.792034i \(0.709022\pi\)
\(398\) −2.59176e6 −0.820138
\(399\) 400428. 0.125919
\(400\) 1.15501e7 3.60940
\(401\) −3.29355e6 −1.02283 −0.511415 0.859334i \(-0.670878\pi\)
−0.511415 + 0.859334i \(0.670878\pi\)
\(402\) 5.84964e6 1.80536
\(403\) 2.47632e6 0.759529
\(404\) −9.34007e6 −2.84706
\(405\) −695466. −0.210687
\(406\) −547820. −0.164939
\(407\) 384744. 0.115129
\(408\) −719280. −0.213918
\(409\) −1.35473e6 −0.400445 −0.200223 0.979750i \(-0.564167\pi\)
−0.200223 + 0.979750i \(0.564167\pi\)
\(410\) 7.06172e6 2.07468
\(411\) 2.30380e6 0.672730
\(412\) 1.95568e6 0.567616
\(413\) −1.60230e6 −0.462241
\(414\) −952560. −0.273144
\(415\) −7.62055e6 −2.17203
\(416\) 1.82240e6 0.516310
\(417\) 2.54632e6 0.717088
\(418\) −835360. −0.233848
\(419\) 5.08199e6 1.41416 0.707080 0.707134i \(-0.250012\pi\)
0.707080 + 0.707134i \(0.250012\pi\)
\(420\) 3.17873e6 0.879286
\(421\) 628022. 0.172691 0.0863455 0.996265i \(-0.472481\pi\)
0.0863455 + 0.996265i \(0.472481\pi\)
\(422\) −1.19179e7 −3.25775
\(423\) −571536. −0.155308
\(424\) −1.35281e7 −3.65445
\(425\) −1.80064e6 −0.483565
\(426\) −5.51880e6 −1.47340
\(427\) 529298. 0.140485
\(428\) 1.53381e6 0.404726
\(429\) 554760. 0.145533
\(430\) 3.92200e6 1.02291
\(431\) −3.00086e6 −0.778132 −0.389066 0.921210i \(-0.627202\pi\)
−0.389066 + 0.921210i \(0.627202\pi\)
\(432\) 1.03810e6 0.267626
\(433\) 1.21496e6 0.311417 0.155709 0.987803i \(-0.450234\pi\)
0.155709 + 0.987803i \(0.450234\pi\)
\(434\) −1.81104e6 −0.461534
\(435\) −1.06657e6 −0.270251
\(436\) 1.35986e6 0.342594
\(437\) 1.06781e6 0.267479
\(438\) 3.50298e6 0.872474
\(439\) 4.00654e6 0.992219 0.496110 0.868260i \(-0.334761\pi\)
0.496110 + 0.868260i \(0.334761\pi\)
\(440\) −3.51072e6 −0.864499
\(441\) 194481. 0.0476190
\(442\) −1.48740e6 −0.362136
\(443\) −5.44751e6 −1.31883 −0.659415 0.751779i \(-0.729196\pi\)
−0.659415 + 0.751779i \(0.729196\pi\)
\(444\) 2.55938e6 0.616138
\(445\) −1.18527e7 −2.83738
\(446\) −2.18384e6 −0.519857
\(447\) 3.67249e6 0.869343
\(448\) 900032. 0.211867
\(449\) −1.81577e6 −0.425056 −0.212528 0.977155i \(-0.568170\pi\)
−0.212528 + 0.977155i \(0.568170\pi\)
\(450\) 6.56991e6 1.52942
\(451\) −612904. −0.141890
\(452\) 1.21761e6 0.280325
\(453\) 3.26254e6 0.746981
\(454\) 5.82852e6 1.32715
\(455\) 3.47998e6 0.788040
\(456\) −2.94192e6 −0.662550
\(457\) −5.30082e6 −1.18728 −0.593639 0.804731i \(-0.702309\pi\)
−0.593639 + 0.804731i \(0.702309\pi\)
\(458\) 9.61046e6 2.14082
\(459\) −161838. −0.0358549
\(460\) 8.47661e6 1.86779
\(461\) 3.20381e6 0.702124 0.351062 0.936352i \(-0.385821\pi\)
0.351062 + 0.936352i \(0.385821\pi\)
\(462\) −405720. −0.0884344
\(463\) −1.11853e6 −0.242490 −0.121245 0.992623i \(-0.538689\pi\)
−0.121245 + 0.992623i \(0.538689\pi\)
\(464\) 1.59203e6 0.343287
\(465\) −3.52598e6 −0.756220
\(466\) 6.05994e6 1.29272
\(467\) −3.85134e6 −0.817184 −0.408592 0.912717i \(-0.633980\pi\)
−0.408592 + 0.912717i \(0.633980\pi\)
\(468\) 3.69036e6 0.778850
\(469\) −3.18480e6 −0.668576
\(470\) 7.47936e6 1.56178
\(471\) −1.37507e6 −0.285610
\(472\) 1.17720e7 2.43218
\(473\) −340400. −0.0699579
\(474\) −7.92864e6 −1.62089
\(475\) −7.36479e6 −1.49770
\(476\) 739704. 0.149638
\(477\) −3.04382e6 −0.612523
\(478\) 1.17014e7 2.34245
\(479\) 1.43536e6 0.285839 0.142920 0.989734i \(-0.454351\pi\)
0.142920 + 0.989734i \(0.454351\pi\)
\(480\) −2.59488e6 −0.514060
\(481\) 2.80194e6 0.552200
\(482\) −1.23691e7 −2.42505
\(483\) 518616. 0.101153
\(484\) −1.03759e7 −2.01332
\(485\) 1.59897e7 3.08664
\(486\) 590490. 0.113402
\(487\) 4.61097e6 0.880987 0.440494 0.897756i \(-0.354803\pi\)
0.440494 + 0.897756i \(0.354803\pi\)
\(488\) −3.88872e6 −0.739192
\(489\) −1.35385e6 −0.256035
\(490\) −2.54506e6 −0.478859
\(491\) 7.40518e6 1.38622 0.693110 0.720832i \(-0.256240\pi\)
0.693110 + 0.720832i \(0.256240\pi\)
\(492\) −4.07714e6 −0.759352
\(493\) −248196. −0.0459915
\(494\) −6.08360e6 −1.12161
\(495\) −789912. −0.144899
\(496\) 5.26310e6 0.960589
\(497\) 3.00468e6 0.545641
\(498\) 6.47028e6 1.16909
\(499\) 3.93432e6 0.707325 0.353662 0.935373i \(-0.384936\pi\)
0.353662 + 0.935373i \(0.384936\pi\)
\(500\) −3.59391e7 −6.42898
\(501\) −65592.0 −0.0116750
\(502\) 9.59708e6 1.69973
\(503\) 3.40975e6 0.600901 0.300450 0.953797i \(-0.402863\pi\)
0.300450 + 0.953797i \(0.402863\pi\)
\(504\) −1.42884e6 −0.250557
\(505\) 1.45595e7 2.54050
\(506\) −1.08192e6 −0.187853
\(507\) 698463. 0.120677
\(508\) 4.54675e6 0.781703
\(509\) −7.72383e6 −1.32141 −0.660706 0.750645i \(-0.729743\pi\)
−0.660706 + 0.750645i \(0.729743\pi\)
\(510\) 2.11788e6 0.360559
\(511\) −1.90718e6 −0.323102
\(512\) −1.25312e7 −2.11260
\(513\) −661932. −0.111050
\(514\) −1.21259e7 −2.02445
\(515\) −3.04856e6 −0.506497
\(516\) −2.26440e6 −0.374394
\(517\) −649152. −0.106812
\(518\) −2.04918e6 −0.335549
\(519\) −2.60239e6 −0.424085
\(520\) −2.55672e7 −4.14643
\(521\) −4.77658e6 −0.770944 −0.385472 0.922719i \(-0.625961\pi\)
−0.385472 + 0.922719i \(0.625961\pi\)
\(522\) 905580. 0.145462
\(523\) −9.28754e6 −1.48473 −0.742363 0.669998i \(-0.766295\pi\)
−0.742363 + 0.669998i \(0.766295\pi\)
\(524\) 1.04560e7 1.66355
\(525\) −3.57695e6 −0.566389
\(526\) −1.25274e7 −1.97423
\(527\) −820512. −0.128694
\(528\) 1.17907e6 0.184058
\(529\) −5.05337e6 −0.785130
\(530\) 3.98327e7 6.15956
\(531\) 2.64870e6 0.407658
\(532\) 3.02546e6 0.463459
\(533\) −4.46354e6 −0.680552
\(534\) 1.00636e7 1.52722
\(535\) −2.39094e6 −0.361147
\(536\) 2.33986e7 3.51785
\(537\) 1.79543e6 0.268678
\(538\) −1.36866e6 −0.203864
\(539\) 220892. 0.0327498
\(540\) −5.25463e6 −0.775457
\(541\) 7.72917e6 1.13538 0.567688 0.823244i \(-0.307838\pi\)
0.567688 + 0.823244i \(0.307838\pi\)
\(542\) −9.60896e6 −1.40501
\(543\) 2.16495e6 0.315100
\(544\) −603840. −0.0874832
\(545\) −2.11979e6 −0.305704
\(546\) −2.95470e6 −0.424162
\(547\) −8.60361e6 −1.22945 −0.614727 0.788740i \(-0.710734\pi\)
−0.614727 + 0.788740i \(0.710734\pi\)
\(548\) 1.74065e7 2.47605
\(549\) −874962. −0.123896
\(550\) 7.46212e6 1.05185
\(551\) −1.01514e6 −0.142445
\(552\) −3.81024e6 −0.532236
\(553\) 4.31670e6 0.600260
\(554\) 9.05830e6 1.25393
\(555\) −3.98963e6 −0.549794
\(556\) 1.92388e7 2.63932
\(557\) −1.77723e6 −0.242721 −0.121360 0.992609i \(-0.538726\pi\)
−0.121360 + 0.992609i \(0.538726\pi\)
\(558\) 2.99376e6 0.407035
\(559\) −2.47900e6 −0.335542
\(560\) 7.39626e6 0.996648
\(561\) −183816. −0.0246590
\(562\) −330620. −0.0441559
\(563\) 2.68860e6 0.357482 0.178741 0.983896i \(-0.442798\pi\)
0.178741 + 0.983896i \(0.442798\pi\)
\(564\) −4.31827e6 −0.571626
\(565\) −1.89804e6 −0.250140
\(566\) −8.63588e6 −1.13309
\(567\) −321489. −0.0419961
\(568\) −2.20752e7 −2.87100
\(569\) 5.32630e6 0.689675 0.344838 0.938662i \(-0.387934\pi\)
0.344838 + 0.938662i \(0.387934\pi\)
\(570\) 8.66232e6 1.11673
\(571\) 1.33992e7 1.71984 0.859921 0.510427i \(-0.170513\pi\)
0.859921 + 0.510427i \(0.170513\pi\)
\(572\) 4.19152e6 0.535650
\(573\) 2.61346e6 0.332528
\(574\) 3.26438e6 0.413543
\(575\) −9.53854e6 −1.20313
\(576\) −1.48781e6 −0.186849
\(577\) −1.10502e6 −0.138176 −0.0690878 0.997611i \(-0.522009\pi\)
−0.0690878 + 0.997611i \(0.522009\pi\)
\(578\) −1.37057e7 −1.70641
\(579\) −1.54309e6 −0.191291
\(580\) −8.05854e6 −0.994687
\(581\) −3.52271e6 −0.432949
\(582\) −1.35761e7 −1.66138
\(583\) −3.45718e6 −0.421260
\(584\) 1.40119e7 1.70007
\(585\) −5.75262e6 −0.694986
\(586\) −1.33755e7 −1.60903
\(587\) 5.97288e6 0.715465 0.357732 0.933824i \(-0.383550\pi\)
0.357732 + 0.933824i \(0.383550\pi\)
\(588\) 1.46941e6 0.175267
\(589\) −3.35597e6 −0.398593
\(590\) −3.46620e7 −4.09943
\(591\) 3.61791e6 0.426078
\(592\) 5.95517e6 0.698377
\(593\) −1.11945e7 −1.30728 −0.653639 0.756807i \(-0.726758\pi\)
−0.653639 + 0.756807i \(0.726758\pi\)
\(594\) 670680. 0.0779919
\(595\) −1.15307e6 −0.133525
\(596\) 2.77477e7 3.19971
\(597\) −2.33258e6 −0.267856
\(598\) −7.87920e6 −0.901009
\(599\) 1.09055e7 1.24187 0.620937 0.783860i \(-0.286752\pi\)
0.620937 + 0.783860i \(0.286752\pi\)
\(600\) 2.62796e7 2.98017
\(601\) 7.39737e6 0.835394 0.417697 0.908586i \(-0.362837\pi\)
0.417697 + 0.908586i \(0.362837\pi\)
\(602\) 1.81300e6 0.203895
\(603\) 5.26468e6 0.589628
\(604\) 2.46503e7 2.74935
\(605\) 1.61742e7 1.79653
\(606\) −1.23619e7 −1.36742
\(607\) 7.13355e6 0.785840 0.392920 0.919573i \(-0.371465\pi\)
0.392920 + 0.919573i \(0.371465\pi\)
\(608\) −2.46976e6 −0.270954
\(609\) −493038. −0.0538688
\(610\) 1.14501e7 1.24591
\(611\) −4.72752e6 −0.512307
\(612\) −1.22278e6 −0.131968
\(613\) −1.71264e7 −1.84083 −0.920416 0.390939i \(-0.872150\pi\)
−0.920416 + 0.390939i \(0.872150\pi\)
\(614\) −1.32820e7 −1.42182
\(615\) 6.35555e6 0.677587
\(616\) −1.62288e6 −0.172320
\(617\) 2.29924e6 0.243149 0.121574 0.992582i \(-0.461206\pi\)
0.121574 + 0.992582i \(0.461206\pi\)
\(618\) 2.58840e6 0.272622
\(619\) −1.85176e6 −0.194249 −0.0971245 0.995272i \(-0.530965\pi\)
−0.0971245 + 0.995272i \(0.530965\pi\)
\(620\) −2.66408e7 −2.78335
\(621\) −857304. −0.0892084
\(622\) −6.65832e6 −0.690063
\(623\) −5.47908e6 −0.565572
\(624\) 8.58672e6 0.882807
\(625\) 3.06758e7 3.14120
\(626\) −3.09021e7 −3.15176
\(627\) −751824. −0.0763743
\(628\) −1.03894e7 −1.05122
\(629\) −928404. −0.0935643
\(630\) 4.20714e6 0.422314
\(631\) 9.25978e6 0.925822 0.462911 0.886405i \(-0.346805\pi\)
0.462911 + 0.886405i \(0.346805\pi\)
\(632\) −3.17146e7 −3.15839
\(633\) −1.07261e7 −1.06398
\(634\) −9.74178e6 −0.962532
\(635\) −7.08758e6 −0.697532
\(636\) −2.29977e7 −2.25446
\(637\) 1.60867e6 0.157079
\(638\) 1.02856e6 0.100041
\(639\) −4.96692e6 −0.481210
\(640\) 2.86963e7 2.76934
\(641\) 1.79419e7 1.72474 0.862369 0.506280i \(-0.168980\pi\)
0.862369 + 0.506280i \(0.168980\pi\)
\(642\) 2.03004e6 0.194387
\(643\) 6.70020e6 0.639087 0.319544 0.947572i \(-0.396470\pi\)
0.319544 + 0.947572i \(0.396470\pi\)
\(644\) 3.91843e6 0.372304
\(645\) 3.52980e6 0.334080
\(646\) 2.01576e6 0.190045
\(647\) −1.12549e7 −1.05701 −0.528507 0.848929i \(-0.677248\pi\)
−0.528507 + 0.848929i \(0.677248\pi\)
\(648\) 2.36196e6 0.220971
\(649\) 3.00840e6 0.280365
\(650\) 5.43437e7 5.04505
\(651\) −1.62994e6 −0.150736
\(652\) −1.02291e7 −0.942364
\(653\) 1.31704e7 1.20869 0.604347 0.796721i \(-0.293434\pi\)
0.604347 + 0.796721i \(0.293434\pi\)
\(654\) 1.79982e6 0.164545
\(655\) −1.62990e7 −1.48442
\(656\) −9.48669e6 −0.860706
\(657\) 3.15268e6 0.284949
\(658\) 3.45744e6 0.311308
\(659\) −1.43453e7 −1.28676 −0.643380 0.765547i \(-0.722468\pi\)
−0.643380 + 0.765547i \(0.722468\pi\)
\(660\) −5.96822e6 −0.533317
\(661\) −14138.0 −0.00125859 −0.000629295 1.00000i \(-0.500200\pi\)
−0.000629295 1.00000i \(0.500200\pi\)
\(662\) 7.81772e6 0.693322
\(663\) −1.33866e6 −0.118273
\(664\) 2.58811e7 2.27805
\(665\) −4.71615e6 −0.413555
\(666\) 3.38742e6 0.295926
\(667\) −1.31477e6 −0.114429
\(668\) −495584. −0.0429711
\(669\) −1.96546e6 −0.169784
\(670\) −6.88958e7 −5.92933
\(671\) −993784. −0.0852090
\(672\) −1.19952e6 −0.102467
\(673\) 1.37787e7 1.17266 0.586329 0.810073i \(-0.300573\pi\)
0.586329 + 0.810073i \(0.300573\pi\)
\(674\) 3.48754e6 0.295712
\(675\) 5.91292e6 0.499508
\(676\) 5.27728e6 0.444164
\(677\) 1.26155e7 1.05787 0.528936 0.848662i \(-0.322591\pi\)
0.528936 + 0.848662i \(0.322591\pi\)
\(678\) 1.61154e6 0.134638
\(679\) 7.39145e6 0.615255
\(680\) 8.47152e6 0.702569
\(681\) 5.24567e6 0.433444
\(682\) 3.40032e6 0.279936
\(683\) −1.08656e6 −0.0891258 −0.0445629 0.999007i \(-0.514190\pi\)
−0.0445629 + 0.999007i \(0.514190\pi\)
\(684\) −5.00126e6 −0.408733
\(685\) −2.71337e7 −2.20944
\(686\) −1.17649e6 −0.0954504
\(687\) 8.64941e6 0.699189
\(688\) −5.26880e6 −0.424366
\(689\) −2.51773e7 −2.02051
\(690\) 1.12190e7 0.897084
\(691\) 1.91229e7 1.52356 0.761780 0.647836i \(-0.224326\pi\)
0.761780 + 0.647836i \(0.224326\pi\)
\(692\) −1.96625e7 −1.56089
\(693\) −365148. −0.0288826
\(694\) 2.50625e7 1.97527
\(695\) −2.99899e7 −2.35512
\(696\) 3.62232e6 0.283442
\(697\) 1.47896e6 0.115312
\(698\) 3.05861e7 2.37621
\(699\) 5.45395e6 0.422200
\(700\) −2.70259e7 −2.08466
\(701\) 1.15000e7 0.883897 0.441948 0.897040i \(-0.354287\pi\)
0.441948 + 0.897040i \(0.354287\pi\)
\(702\) 4.88430e6 0.374076
\(703\) −3.79726e6 −0.289789
\(704\) −1.68986e6 −0.128504
\(705\) 6.73142e6 0.510075
\(706\) −3.49291e7 −2.63740
\(707\) 6.73035e6 0.506394
\(708\) 2.00124e7 1.50043
\(709\) 6.97551e6 0.521147 0.260574 0.965454i \(-0.416088\pi\)
0.260574 + 0.965454i \(0.416088\pi\)
\(710\) 6.49992e7 4.83907
\(711\) −7.13578e6 −0.529380
\(712\) 4.02545e7 2.97587
\(713\) −4.34650e6 −0.320196
\(714\) 979020. 0.0718697
\(715\) −6.53384e6 −0.477973
\(716\) 1.35655e7 0.988899
\(717\) 1.05313e7 0.765040
\(718\) 2.12034e7 1.53495
\(719\) −1.44264e6 −0.104072 −0.0520362 0.998645i \(-0.516571\pi\)
−0.0520362 + 0.998645i \(0.516571\pi\)
\(720\) −1.22265e7 −0.878961
\(721\) −1.40924e6 −0.100959
\(722\) −1.65164e7 −1.17916
\(723\) −1.11322e7 −0.792018
\(724\) 1.63574e7 1.15976
\(725\) 9.06810e6 0.640724
\(726\) −1.37328e7 −0.966982
\(727\) −1.90334e7 −1.33561 −0.667807 0.744334i \(-0.732767\pi\)
−0.667807 + 0.744334i \(0.732767\pi\)
\(728\) −1.18188e7 −0.826504
\(729\) 531441. 0.0370370
\(730\) −4.12573e7 −2.86546
\(731\) 821400. 0.0568540
\(732\) −6.61082e6 −0.456014
\(733\) 5.45585e6 0.375062 0.187531 0.982259i \(-0.439952\pi\)
0.187531 + 0.982259i \(0.439952\pi\)
\(734\) 7.46592e6 0.511497
\(735\) −2.29055e6 −0.156395
\(736\) −3.19872e6 −0.217662
\(737\) 5.97963e6 0.405514
\(738\) −5.39622e6 −0.364711
\(739\) 7.85197e6 0.528893 0.264446 0.964400i \(-0.414811\pi\)
0.264446 + 0.964400i \(0.414811\pi\)
\(740\) −3.01439e7 −2.02358
\(741\) −5.47524e6 −0.366317
\(742\) 1.84132e7 1.22778
\(743\) −1.48695e7 −0.988154 −0.494077 0.869418i \(-0.664494\pi\)
−0.494077 + 0.869418i \(0.664494\pi\)
\(744\) 1.19750e7 0.793130
\(745\) −4.32537e7 −2.85518
\(746\) −9.39034e6 −0.617781
\(747\) 5.82325e6 0.381825
\(748\) −1.38883e6 −0.0907603
\(749\) −1.10524e6 −0.0719869
\(750\) −4.75664e7 −3.08779
\(751\) −2.51309e7 −1.62595 −0.812977 0.582295i \(-0.802155\pi\)
−0.812977 + 0.582295i \(0.802155\pi\)
\(752\) −1.00477e7 −0.647924
\(753\) 8.63737e6 0.555129
\(754\) 7.49060e6 0.479831
\(755\) −3.84254e7 −2.45330
\(756\) −2.42903e6 −0.154571
\(757\) −1.97874e7 −1.25501 −0.627507 0.778611i \(-0.715925\pi\)
−0.627507 + 0.778611i \(0.715925\pi\)
\(758\) 5.16534e7 3.26532
\(759\) −973728. −0.0613526
\(760\) 3.46493e7 2.17601
\(761\) −1.49642e7 −0.936678 −0.468339 0.883549i \(-0.655147\pi\)
−0.468339 + 0.883549i \(0.655147\pi\)
\(762\) 6.01776e6 0.375446
\(763\) −979902. −0.0609356
\(764\) 1.97461e7 1.22391
\(765\) 1.90609e6 0.117758
\(766\) 4.00512e6 0.246629
\(767\) 2.19090e7 1.34473
\(768\) −1.90748e7 −1.16696
\(769\) 5.06419e6 0.308812 0.154406 0.988007i \(-0.450654\pi\)
0.154406 + 0.988007i \(0.450654\pi\)
\(770\) 4.77848e6 0.290444
\(771\) −1.09133e7 −0.661181
\(772\) −1.16589e7 −0.704066
\(773\) 2.63025e7 1.58324 0.791622 0.611011i \(-0.209237\pi\)
0.791622 + 0.611011i \(0.209237\pi\)
\(774\) −2.99700e6 −0.179818
\(775\) 2.99783e7 1.79288
\(776\) −5.43046e7 −3.23729
\(777\) −1.84426e6 −0.109590
\(778\) 3.06822e6 0.181735
\(779\) 6.04910e6 0.357147
\(780\) −4.34642e7 −2.55797
\(781\) −5.64144e6 −0.330950
\(782\) 2.61072e6 0.152666
\(783\) 815022. 0.0475078
\(784\) 3.41902e6 0.198661
\(785\) 1.61953e7 0.938027
\(786\) 1.38388e7 0.798989
\(787\) 1.00525e7 0.578545 0.289273 0.957247i \(-0.406587\pi\)
0.289273 + 0.957247i \(0.406587\pi\)
\(788\) 2.73353e7 1.56823
\(789\) −1.12747e7 −0.644781
\(790\) 9.33818e7 5.32346
\(791\) −877394. −0.0498601
\(792\) 2.68272e6 0.151972
\(793\) −7.23734e6 −0.408692
\(794\) −3.83421e7 −2.15836
\(795\) 3.58494e7 2.01170
\(796\) −1.76240e7 −0.985873
\(797\) 2.78516e7 1.55312 0.776560 0.630044i \(-0.216963\pi\)
0.776560 + 0.630044i \(0.216963\pi\)
\(798\) 4.00428e6 0.222596
\(799\) 1.56643e6 0.0868050
\(800\) 2.20619e7 1.21876
\(801\) 9.05726e6 0.498787
\(802\) −3.29355e7 −1.80812
\(803\) 3.58082e6 0.195972
\(804\) 3.97776e7 2.17019
\(805\) −6.10814e6 −0.332215
\(806\) 2.47632e7 1.34267
\(807\) −1.23179e6 −0.0665816
\(808\) −4.94474e7 −2.66450
\(809\) −1.96936e7 −1.05792 −0.528961 0.848646i \(-0.677418\pi\)
−0.528961 + 0.848646i \(0.677418\pi\)
\(810\) −6.95466e6 −0.372446
\(811\) 5.05617e6 0.269942 0.134971 0.990850i \(-0.456906\pi\)
0.134971 + 0.990850i \(0.456906\pi\)
\(812\) −3.72518e6 −0.198270
\(813\) −8.64806e6 −0.458873
\(814\) 3.84744e6 0.203522
\(815\) 1.59454e7 0.840893
\(816\) −2.84515e6 −0.149582
\(817\) 3.35960e6 0.176089
\(818\) −1.35473e7 −0.707894
\(819\) −2.65923e6 −0.138531
\(820\) 4.80197e7 2.49393
\(821\) −2.82324e6 −0.146181 −0.0730904 0.997325i \(-0.523286\pi\)
−0.0730904 + 0.997325i \(0.523286\pi\)
\(822\) 2.30380e7 1.18923
\(823\) 2.54741e7 1.31099 0.655494 0.755201i \(-0.272461\pi\)
0.655494 + 0.755201i \(0.272461\pi\)
\(824\) 1.03536e7 0.531219
\(825\) 6.71591e6 0.343534
\(826\) −1.60230e7 −0.817135
\(827\) −1.75616e7 −0.892893 −0.446446 0.894810i \(-0.647311\pi\)
−0.446446 + 0.894810i \(0.647311\pi\)
\(828\) −6.47741e6 −0.328341
\(829\) 9.14728e6 0.462280 0.231140 0.972920i \(-0.425754\pi\)
0.231140 + 0.972920i \(0.425754\pi\)
\(830\) −7.62055e7 −3.83965
\(831\) 8.15247e6 0.409531
\(832\) −1.23066e7 −0.616351
\(833\) −533022. −0.0266154
\(834\) 2.54632e7 1.26764
\(835\) 772528. 0.0383441
\(836\) −5.68045e6 −0.281104
\(837\) 2.69438e6 0.132937
\(838\) 5.08199e7 2.49991
\(839\) −1.09891e7 −0.538961 −0.269481 0.963006i \(-0.586852\pi\)
−0.269481 + 0.963006i \(0.586852\pi\)
\(840\) 1.68286e7 0.822903
\(841\) −1.92612e7 −0.939061
\(842\) 6.28022e6 0.305277
\(843\) −297558. −0.0144212
\(844\) −8.10416e7 −3.91608
\(845\) −8.22634e6 −0.396337
\(846\) −5.71536e6 −0.274548
\(847\) 7.47676e6 0.358101
\(848\) −5.35111e7 −2.55537
\(849\) −7.77229e6 −0.370067
\(850\) −1.80064e7 −0.854831
\(851\) −4.91803e6 −0.232792
\(852\) −3.75278e7 −1.77115
\(853\) −1.34854e7 −0.634586 −0.317293 0.948328i \(-0.602774\pi\)
−0.317293 + 0.948328i \(0.602774\pi\)
\(854\) 5.29298e6 0.248345
\(855\) 7.79609e6 0.364722
\(856\) 8.12016e6 0.378774
\(857\) 1.07032e7 0.497808 0.248904 0.968528i \(-0.419930\pi\)
0.248904 + 0.968528i \(0.419930\pi\)
\(858\) 5.54760e6 0.257269
\(859\) −1.33747e7 −0.618446 −0.309223 0.950990i \(-0.600069\pi\)
−0.309223 + 0.950990i \(0.600069\pi\)
\(860\) 2.66696e7 1.22962
\(861\) 2.93794e6 0.135063
\(862\) −3.00086e7 −1.37556
\(863\) 1.99768e7 0.913058 0.456529 0.889708i \(-0.349092\pi\)
0.456529 + 0.889708i \(0.349092\pi\)
\(864\) 1.98288e6 0.0903675
\(865\) 3.06503e7 1.39282
\(866\) 1.21496e7 0.550513
\(867\) −1.23352e7 −0.557310
\(868\) −1.23151e7 −0.554802
\(869\) −8.10483e6 −0.364078
\(870\) −1.06657e7 −0.477740
\(871\) 4.35473e7 1.94498
\(872\) 7.19928e6 0.320625
\(873\) −1.22185e7 −0.542604
\(874\) 1.06781e7 0.472840
\(875\) 2.58973e7 1.14349
\(876\) 2.38203e7 1.04878
\(877\) 8.81107e6 0.386838 0.193419 0.981116i \(-0.438042\pi\)
0.193419 + 0.981116i \(0.438042\pi\)
\(878\) 4.00654e7 1.75401
\(879\) −1.20379e7 −0.525508
\(880\) −1.38868e7 −0.604501
\(881\) 4.01078e7 1.74096 0.870481 0.492202i \(-0.163808\pi\)
0.870481 + 0.492202i \(0.163808\pi\)
\(882\) 1.94481e6 0.0841794
\(883\) 1.49664e7 0.645976 0.322988 0.946403i \(-0.395313\pi\)
0.322988 + 0.946403i \(0.395313\pi\)
\(884\) −1.01143e7 −0.435317
\(885\) −3.11958e7 −1.33887
\(886\) −5.44751e7 −2.33138
\(887\) −575144. −0.0245453 −0.0122726 0.999925i \(-0.503907\pi\)
−0.0122726 + 0.999925i \(0.503907\pi\)
\(888\) 1.35497e7 0.576629
\(889\) −3.27634e6 −0.139038
\(890\) −1.18527e8 −5.01583
\(891\) 603612. 0.0254720
\(892\) −1.48501e7 −0.624910
\(893\) 6.40685e6 0.268854
\(894\) 3.67249e7 1.53680
\(895\) −2.11462e7 −0.882417
\(896\) 1.32653e7 0.552009
\(897\) −7.09128e6 −0.294268
\(898\) −1.81577e7 −0.751400
\(899\) 4.13213e6 0.170520
\(900\) 4.46754e7 1.83849
\(901\) 8.34232e6 0.342353
\(902\) −6.12904e6 −0.250828
\(903\) 1.63170e6 0.0665918
\(904\) 6.44616e6 0.262349
\(905\) −2.54983e7 −1.03488
\(906\) 3.26254e7 1.32049
\(907\) −3.33367e7 −1.34557 −0.672783 0.739840i \(-0.734901\pi\)
−0.672783 + 0.739840i \(0.734901\pi\)
\(908\) 3.96339e7 1.59534
\(909\) −1.11257e7 −0.446598
\(910\) 3.47998e7 1.39307
\(911\) 2.17451e7 0.868090 0.434045 0.900891i \(-0.357086\pi\)
0.434045 + 0.900891i \(0.357086\pi\)
\(912\) −1.16369e7 −0.463288
\(913\) 6.61406e6 0.262598
\(914\) −5.30082e7 −2.09883
\(915\) 1.03051e7 0.406911
\(916\) 6.53511e7 2.57344
\(917\) −7.53444e6 −0.295888
\(918\) −1.61838e6 −0.0633831
\(919\) −4.03986e6 −0.157789 −0.0788947 0.996883i \(-0.525139\pi\)
−0.0788947 + 0.996883i \(0.525139\pi\)
\(920\) 4.48762e7 1.74802
\(921\) −1.19538e7 −0.464364
\(922\) 3.20381e7 1.24119
\(923\) −4.10844e7 −1.58735
\(924\) −2.75890e6 −0.106305
\(925\) 3.39202e7 1.30348
\(926\) −1.11853e7 −0.428666
\(927\) 2.32956e6 0.0890378
\(928\) 3.04096e6 0.115915
\(929\) 7.69679e6 0.292597 0.146299 0.989240i \(-0.453264\pi\)
0.146299 + 0.989240i \(0.453264\pi\)
\(930\) −3.52598e7 −1.33682
\(931\) −2.18011e6 −0.0824335
\(932\) 4.12076e7 1.55395
\(933\) −5.99249e6 −0.225374
\(934\) −3.85134e7 −1.44459
\(935\) 2.16494e6 0.0809874
\(936\) 1.95372e7 0.728908
\(937\) −453558. −0.0168766 −0.00843828 0.999964i \(-0.502686\pi\)
−0.00843828 + 0.999964i \(0.502686\pi\)
\(938\) −3.18480e7 −1.18189
\(939\) −2.78119e7 −1.02936
\(940\) 5.08596e7 1.87739
\(941\) −2.43852e7 −0.897745 −0.448873 0.893596i \(-0.648174\pi\)
−0.448873 + 0.893596i \(0.648174\pi\)
\(942\) −1.37507e7 −0.504892
\(943\) 7.83451e6 0.286901
\(944\) 4.65648e7 1.70070
\(945\) 3.78643e6 0.137927
\(946\) −3.40400e6 −0.123669
\(947\) −2.18745e7 −0.792615 −0.396308 0.918118i \(-0.629709\pi\)
−0.396308 + 0.918118i \(0.629709\pi\)
\(948\) −5.39148e7 −1.94844
\(949\) 2.60777e7 0.939949
\(950\) −7.36479e7 −2.64759
\(951\) −8.76760e6 −0.314362
\(952\) 3.91608e6 0.140042
\(953\) −3.93319e7 −1.40286 −0.701428 0.712741i \(-0.747454\pi\)
−0.701428 + 0.712741i \(0.747454\pi\)
\(954\) −3.04382e7 −1.08280
\(955\) −3.07807e7 −1.09212
\(956\) 7.95698e7 2.81581
\(957\) 925704. 0.0326732
\(958\) 1.43536e7 0.505297
\(959\) −1.25429e7 −0.440405
\(960\) 1.75231e7 0.613666
\(961\) −1.49687e7 −0.522849
\(962\) 2.80194e7 0.976160
\(963\) 1.82704e6 0.0634865
\(964\) −8.41099e7 −2.91511
\(965\) 1.81741e7 0.628254
\(966\) 5.18616e6 0.178815
\(967\) 3.85234e7 1.32483 0.662413 0.749139i \(-0.269532\pi\)
0.662413 + 0.749139i \(0.269532\pi\)
\(968\) −5.49313e7 −1.88422
\(969\) 1.81418e6 0.0620686
\(970\) 1.59897e8 5.45645
\(971\) 2.43543e7 0.828947 0.414473 0.910061i \(-0.363966\pi\)
0.414473 + 0.910061i \(0.363966\pi\)
\(972\) 4.01533e6 0.136319
\(973\) −1.38633e7 −0.469444
\(974\) 4.61097e7 1.55738
\(975\) 4.89093e7 1.64771
\(976\) −1.53820e7 −0.516880
\(977\) 1.62534e7 0.544763 0.272382 0.962189i \(-0.412189\pi\)
0.272382 + 0.962189i \(0.412189\pi\)
\(978\) −1.35385e7 −0.452610
\(979\) 1.02873e7 0.343038
\(980\) −1.73064e7 −0.575628
\(981\) 1.61984e6 0.0537402
\(982\) 7.40518e7 2.45051
\(983\) −252456. −0.00833301 −0.00416650 0.999991i \(-0.501326\pi\)
−0.00416650 + 0.999991i \(0.501326\pi\)
\(984\) −2.15849e7 −0.710660
\(985\) −4.26109e7 −1.39936
\(986\) −2.48196e6 −0.0813022
\(987\) 3.11170e6 0.101673
\(988\) −4.13685e7 −1.34827
\(989\) 4.35120e6 0.141455
\(990\) −7.89912e6 −0.256148
\(991\) −2.49728e7 −0.807761 −0.403880 0.914812i \(-0.632339\pi\)
−0.403880 + 0.914812i \(0.632339\pi\)
\(992\) 1.00531e7 0.324356
\(993\) 7.03595e6 0.226438
\(994\) 3.00468e7 0.964567
\(995\) 2.74727e7 0.879717
\(996\) 4.39979e7 1.40535
\(997\) −3.60983e7 −1.15013 −0.575067 0.818106i \(-0.695024\pi\)
−0.575067 + 0.818106i \(0.695024\pi\)
\(998\) 3.93432e7 1.25039
\(999\) 3.04868e6 0.0966491
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.d.1.1 1
3.2 odd 2 63.6.a.a.1.1 1
4.3 odd 2 336.6.a.a.1.1 1
5.2 odd 4 525.6.d.a.274.2 2
5.3 odd 4 525.6.d.a.274.1 2
5.4 even 2 525.6.a.a.1.1 1
7.2 even 3 147.6.e.a.67.1 2
7.3 odd 6 147.6.e.b.79.1 2
7.4 even 3 147.6.e.a.79.1 2
7.5 odd 6 147.6.e.b.67.1 2
7.6 odd 2 147.6.a.g.1.1 1
12.11 even 2 1008.6.a.bc.1.1 1
21.20 even 2 441.6.a.b.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.d.1.1 1 1.1 even 1 trivial
63.6.a.a.1.1 1 3.2 odd 2
147.6.a.g.1.1 1 7.6 odd 2
147.6.e.a.67.1 2 7.2 even 3
147.6.e.a.79.1 2 7.4 even 3
147.6.e.b.67.1 2 7.5 odd 6
147.6.e.b.79.1 2 7.3 odd 6
336.6.a.a.1.1 1 4.3 odd 2
441.6.a.b.1.1 1 21.20 even 2
525.6.a.a.1.1 1 5.4 even 2
525.6.d.a.274.1 2 5.3 odd 4
525.6.d.a.274.2 2 5.2 odd 4
1008.6.a.bc.1.1 1 12.11 even 2