Properties

Label 102.6.a.d.1.1
Level $102$
Weight $6$
Character 102.1
Self dual yes
Analytic conductor $16.359$
Analytic rank $1$
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] = [102,6,Mod(1,102)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(102, 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("102.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 102 = 2 \cdot 3 \cdot 17 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 102.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: \(16.3591496209\)
Analytic rank: \(1\)
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 \(\chi\) \(=\) 102.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +25.0000 q^{5} -36.0000 q^{6} -188.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} +25.0000 q^{5} -36.0000 q^{6} -188.000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +100.000 q^{10} +135.000 q^{11} -144.000 q^{12} -1113.00 q^{13} -752.000 q^{14} -225.000 q^{15} +256.000 q^{16} +289.000 q^{17} +324.000 q^{18} -929.000 q^{19} +400.000 q^{20} +1692.00 q^{21} +540.000 q^{22} -1701.00 q^{23} -576.000 q^{24} -2500.00 q^{25} -4452.00 q^{26} -729.000 q^{27} -3008.00 q^{28} -4670.00 q^{29} -900.000 q^{30} -5490.00 q^{31} +1024.00 q^{32} -1215.00 q^{33} +1156.00 q^{34} -4700.00 q^{35} +1296.00 q^{36} +10808.0 q^{37} -3716.00 q^{38} +10017.0 q^{39} +1600.00 q^{40} +8911.00 q^{41} +6768.00 q^{42} -11839.0 q^{43} +2160.00 q^{44} +2025.00 q^{45} -6804.00 q^{46} -8578.00 q^{47} -2304.00 q^{48} +18537.0 q^{49} -10000.0 q^{50} -2601.00 q^{51} -17808.0 q^{52} +35606.0 q^{53} -2916.00 q^{54} +3375.00 q^{55} -12032.0 q^{56} +8361.00 q^{57} -18680.0 q^{58} +6138.00 q^{59} -3600.00 q^{60} +19688.0 q^{61} -21960.0 q^{62} -15228.0 q^{63} +4096.00 q^{64} -27825.0 q^{65} -4860.00 q^{66} -14724.0 q^{67} +4624.00 q^{68} +15309.0 q^{69} -18800.0 q^{70} +48396.0 q^{71} +5184.00 q^{72} +10322.0 q^{73} +43232.0 q^{74} +22500.0 q^{75} -14864.0 q^{76} -25380.0 q^{77} +40068.0 q^{78} +77710.0 q^{79} +6400.00 q^{80} +6561.00 q^{81} +35644.0 q^{82} -108258. q^{83} +27072.0 q^{84} +7225.00 q^{85} -47356.0 q^{86} +42030.0 q^{87} +8640.00 q^{88} -116488. q^{89} +8100.00 q^{90} +209244. q^{91} -27216.0 q^{92} +49410.0 q^{93} -34312.0 q^{94} -23225.0 q^{95} -9216.00 q^{96} +10524.0 q^{97} +74148.0 q^{98} +10935.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 4.00000 0.707107
\(3\) −9.00000 −0.577350
\(4\) 16.0000 0.500000
\(5\) 25.0000 0.447214 0.223607 0.974679i \(-0.428217\pi\)
0.223607 + 0.974679i \(0.428217\pi\)
\(6\) −36.0000 −0.408248
\(7\) −188.000 −1.45015 −0.725075 0.688670i \(-0.758195\pi\)
−0.725075 + 0.688670i \(0.758195\pi\)
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) 100.000 0.316228
\(11\) 135.000 0.336397 0.168198 0.985753i \(-0.446205\pi\)
0.168198 + 0.985753i \(0.446205\pi\)
\(12\) −144.000 −0.288675
\(13\) −1113.00 −1.82657 −0.913286 0.407319i \(-0.866464\pi\)
−0.913286 + 0.407319i \(0.866464\pi\)
\(14\) −752.000 −1.02541
\(15\) −225.000 −0.258199
\(16\) 256.000 0.250000
\(17\) 289.000 0.242536
\(18\) 324.000 0.235702
\(19\) −929.000 −0.590380 −0.295190 0.955439i \(-0.595383\pi\)
−0.295190 + 0.955439i \(0.595383\pi\)
\(20\) 400.000 0.223607
\(21\) 1692.00 0.837244
\(22\) 540.000 0.237869
\(23\) −1701.00 −0.670478 −0.335239 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(24\) −576.000 −0.204124
\(25\) −2500.00 −0.800000
\(26\) −4452.00 −1.29158
\(27\) −729.000 −0.192450
\(28\) −3008.00 −0.725075
\(29\) −4670.00 −1.03115 −0.515575 0.856844i \(-0.672422\pi\)
−0.515575 + 0.856844i \(0.672422\pi\)
\(30\) −900.000 −0.182574
\(31\) −5490.00 −1.02605 −0.513025 0.858374i \(-0.671475\pi\)
−0.513025 + 0.858374i \(0.671475\pi\)
\(32\) 1024.00 0.176777
\(33\) −1215.00 −0.194219
\(34\) 1156.00 0.171499
\(35\) −4700.00 −0.648527
\(36\) 1296.00 0.166667
\(37\) 10808.0 1.29790 0.648950 0.760831i \(-0.275209\pi\)
0.648950 + 0.760831i \(0.275209\pi\)
\(38\) −3716.00 −0.417462
\(39\) 10017.0 1.05457
\(40\) 1600.00 0.158114
\(41\) 8911.00 0.827879 0.413939 0.910304i \(-0.364153\pi\)
0.413939 + 0.910304i \(0.364153\pi\)
\(42\) 6768.00 0.592021
\(43\) −11839.0 −0.976436 −0.488218 0.872722i \(-0.662353\pi\)
−0.488218 + 0.872722i \(0.662353\pi\)
\(44\) 2160.00 0.168198
\(45\) 2025.00 0.149071
\(46\) −6804.00 −0.474100
\(47\) −8578.00 −0.566424 −0.283212 0.959057i \(-0.591400\pi\)
−0.283212 + 0.959057i \(0.591400\pi\)
\(48\) −2304.00 −0.144338
\(49\) 18537.0 1.10293
\(50\) −10000.0 −0.565685
\(51\) −2601.00 −0.140028
\(52\) −17808.0 −0.913286
\(53\) 35606.0 1.74114 0.870570 0.492045i \(-0.163751\pi\)
0.870570 + 0.492045i \(0.163751\pi\)
\(54\) −2916.00 −0.136083
\(55\) 3375.00 0.150441
\(56\) −12032.0 −0.512705
\(57\) 8361.00 0.340856
\(58\) −18680.0 −0.729133
\(59\) 6138.00 0.229560 0.114780 0.993391i \(-0.463384\pi\)
0.114780 + 0.993391i \(0.463384\pi\)
\(60\) −3600.00 −0.129099
\(61\) 19688.0 0.677450 0.338725 0.940885i \(-0.390004\pi\)
0.338725 + 0.940885i \(0.390004\pi\)
\(62\) −21960.0 −0.725526
\(63\) −15228.0 −0.483383
\(64\) 4096.00 0.125000
\(65\) −27825.0 −0.816868
\(66\) −4860.00 −0.137333
\(67\) −14724.0 −0.400718 −0.200359 0.979723i \(-0.564211\pi\)
−0.200359 + 0.979723i \(0.564211\pi\)
\(68\) 4624.00 0.121268
\(69\) 15309.0 0.387101
\(70\) −18800.0 −0.458578
\(71\) 48396.0 1.13937 0.569683 0.821864i \(-0.307066\pi\)
0.569683 + 0.821864i \(0.307066\pi\)
\(72\) 5184.00 0.117851
\(73\) 10322.0 0.226703 0.113351 0.993555i \(-0.463841\pi\)
0.113351 + 0.993555i \(0.463841\pi\)
\(74\) 43232.0 0.917753
\(75\) 22500.0 0.461880
\(76\) −14864.0 −0.295190
\(77\) −25380.0 −0.487826
\(78\) 40068.0 0.745695
\(79\) 77710.0 1.40091 0.700453 0.713698i \(-0.252981\pi\)
0.700453 + 0.713698i \(0.252981\pi\)
\(80\) 6400.00 0.111803
\(81\) 6561.00 0.111111
\(82\) 35644.0 0.585399
\(83\) −108258. −1.72490 −0.862452 0.506139i \(-0.831072\pi\)
−0.862452 + 0.506139i \(0.831072\pi\)
\(84\) 27072.0 0.418622
\(85\) 7225.00 0.108465
\(86\) −47356.0 −0.690445
\(87\) 42030.0 0.595335
\(88\) 8640.00 0.118934
\(89\) −116488. −1.55886 −0.779428 0.626491i \(-0.784490\pi\)
−0.779428 + 0.626491i \(0.784490\pi\)
\(90\) 8100.00 0.105409
\(91\) 209244. 2.64880
\(92\) −27216.0 −0.335239
\(93\) 49410.0 0.592390
\(94\) −34312.0 −0.400522
\(95\) −23225.0 −0.264026
\(96\) −9216.00 −0.102062
\(97\) 10524.0 0.113567 0.0567834 0.998387i \(-0.481916\pi\)
0.0567834 + 0.998387i \(0.481916\pi\)
\(98\) 74148.0 0.779892
\(99\) 10935.0 0.112132
\(100\) −40000.0 −0.400000
\(101\) 122896. 1.19877 0.599383 0.800462i \(-0.295413\pi\)
0.599383 + 0.800462i \(0.295413\pi\)
\(102\) −10404.0 −0.0990148
\(103\) 66661.0 0.619126 0.309563 0.950879i \(-0.399817\pi\)
0.309563 + 0.950879i \(0.399817\pi\)
\(104\) −71232.0 −0.645791
\(105\) 42300.0 0.374427
\(106\) 142424. 1.23117
\(107\) −162285. −1.37031 −0.685155 0.728397i \(-0.740266\pi\)
−0.685155 + 0.728397i \(0.740266\pi\)
\(108\) −11664.0 −0.0962250
\(109\) 125500. 1.01176 0.505880 0.862604i \(-0.331168\pi\)
0.505880 + 0.862604i \(0.331168\pi\)
\(110\) 13500.0 0.106378
\(111\) −97272.0 −0.749343
\(112\) −48128.0 −0.362537
\(113\) 77029.0 0.567490 0.283745 0.958900i \(-0.408423\pi\)
0.283745 + 0.958900i \(0.408423\pi\)
\(114\) 33444.0 0.241022
\(115\) −42525.0 −0.299847
\(116\) −74720.0 −0.515575
\(117\) −90153.0 −0.608857
\(118\) 24552.0 0.162324
\(119\) −54332.0 −0.351713
\(120\) −14400.0 −0.0912871
\(121\) −142826. −0.886837
\(122\) 78752.0 0.479029
\(123\) −80199.0 −0.477976
\(124\) −87840.0 −0.513025
\(125\) −140625. −0.804984
\(126\) −60912.0 −0.341803
\(127\) −256117. −1.40906 −0.704529 0.709675i \(-0.748842\pi\)
−0.704529 + 0.709675i \(0.748842\pi\)
\(128\) 16384.0 0.0883883
\(129\) 106551. 0.563746
\(130\) −111300. −0.577613
\(131\) −288335. −1.46798 −0.733988 0.679162i \(-0.762343\pi\)
−0.733988 + 0.679162i \(0.762343\pi\)
\(132\) −19440.0 −0.0971094
\(133\) 174652. 0.856139
\(134\) −58896.0 −0.283350
\(135\) −18225.0 −0.0860663
\(136\) 18496.0 0.0857493
\(137\) −144366. −0.657148 −0.328574 0.944478i \(-0.606568\pi\)
−0.328574 + 0.944478i \(0.606568\pi\)
\(138\) 61236.0 0.273722
\(139\) −165682. −0.727341 −0.363671 0.931528i \(-0.618477\pi\)
−0.363671 + 0.931528i \(0.618477\pi\)
\(140\) −75200.0 −0.324263
\(141\) 77202.0 0.327025
\(142\) 193584. 0.805654
\(143\) −150255. −0.614453
\(144\) 20736.0 0.0833333
\(145\) −116750. −0.461144
\(146\) 41288.0 0.160303
\(147\) −166833. −0.636779
\(148\) 172928. 0.648950
\(149\) −276306. −1.01959 −0.509794 0.860297i \(-0.670278\pi\)
−0.509794 + 0.860297i \(0.670278\pi\)
\(150\) 90000.0 0.326599
\(151\) 554808. 1.98016 0.990080 0.140504i \(-0.0448722\pi\)
0.990080 + 0.140504i \(0.0448722\pi\)
\(152\) −59456.0 −0.208731
\(153\) 23409.0 0.0808452
\(154\) −101520. −0.344945
\(155\) −137250. −0.458863
\(156\) 160272. 0.527286
\(157\) −403605. −1.30679 −0.653397 0.757015i \(-0.726657\pi\)
−0.653397 + 0.757015i \(0.726657\pi\)
\(158\) 310840. 0.990591
\(159\) −320454. −1.00525
\(160\) 25600.0 0.0790569
\(161\) 319788. 0.972294
\(162\) 26244.0 0.0785674
\(163\) −446490. −1.31626 −0.658132 0.752903i \(-0.728653\pi\)
−0.658132 + 0.752903i \(0.728653\pi\)
\(164\) 142576. 0.413939
\(165\) −30375.0 −0.0868573
\(166\) −433032. −1.21969
\(167\) 194423. 0.539457 0.269728 0.962936i \(-0.413066\pi\)
0.269728 + 0.962936i \(0.413066\pi\)
\(168\) 108288. 0.296011
\(169\) 867476. 2.33637
\(170\) 28900.0 0.0766965
\(171\) −75249.0 −0.196793
\(172\) −189424. −0.488218
\(173\) 80365.0 0.204151 0.102076 0.994777i \(-0.467452\pi\)
0.102076 + 0.994777i \(0.467452\pi\)
\(174\) 168120. 0.420965
\(175\) 470000. 1.16012
\(176\) 34560.0 0.0840992
\(177\) −55242.0 −0.132537
\(178\) −465952. −1.10228
\(179\) 39258.0 0.0915789 0.0457895 0.998951i \(-0.485420\pi\)
0.0457895 + 0.998951i \(0.485420\pi\)
\(180\) 32400.0 0.0745356
\(181\) −225662. −0.511991 −0.255995 0.966678i \(-0.582403\pi\)
−0.255995 + 0.966678i \(0.582403\pi\)
\(182\) 836976. 1.87299
\(183\) −177192. −0.391126
\(184\) −108864. −0.237050
\(185\) 270200. 0.580438
\(186\) 197640. 0.418883
\(187\) 39015.0 0.0815882
\(188\) −137248. −0.283212
\(189\) 137052. 0.279081
\(190\) −92900.0 −0.186695
\(191\) −163502. −0.324295 −0.162147 0.986767i \(-0.551842\pi\)
−0.162147 + 0.986767i \(0.551842\pi\)
\(192\) −36864.0 −0.0721688
\(193\) −449706. −0.869031 −0.434516 0.900664i \(-0.643080\pi\)
−0.434516 + 0.900664i \(0.643080\pi\)
\(194\) 42096.0 0.0803039
\(195\) 250425. 0.471619
\(196\) 296592. 0.551467
\(197\) 452233. 0.830227 0.415113 0.909770i \(-0.363742\pi\)
0.415113 + 0.909770i \(0.363742\pi\)
\(198\) 43740.0 0.0792895
\(199\) −37228.0 −0.0666403 −0.0333202 0.999445i \(-0.510608\pi\)
−0.0333202 + 0.999445i \(0.510608\pi\)
\(200\) −160000. −0.282843
\(201\) 132516. 0.231355
\(202\) 491584. 0.847655
\(203\) 877960. 1.49532
\(204\) −41616.0 −0.0700140
\(205\) 222775. 0.370239
\(206\) 266644. 0.437788
\(207\) −137781. −0.223493
\(208\) −284928. −0.456643
\(209\) −125415. −0.198602
\(210\) 169200. 0.264760
\(211\) −124402. −0.192363 −0.0961814 0.995364i \(-0.530663\pi\)
−0.0961814 + 0.995364i \(0.530663\pi\)
\(212\) 569696. 0.870570
\(213\) −435564. −0.657814
\(214\) −649140. −0.968956
\(215\) −295975. −0.436676
\(216\) −46656.0 −0.0680414
\(217\) 1.03212e6 1.48792
\(218\) 502000. 0.715422
\(219\) −92898.0 −0.130887
\(220\) 54000.0 0.0752206
\(221\) −321657. −0.443009
\(222\) −389088. −0.529865
\(223\) −1.20463e6 −1.62216 −0.811078 0.584938i \(-0.801119\pi\)
−0.811078 + 0.584938i \(0.801119\pi\)
\(224\) −192512. −0.256353
\(225\) −202500. −0.266667
\(226\) 308116. 0.401276
\(227\) 819497. 1.05556 0.527780 0.849381i \(-0.323025\pi\)
0.527780 + 0.849381i \(0.323025\pi\)
\(228\) 133776. 0.170428
\(229\) 366454. 0.461775 0.230888 0.972980i \(-0.425837\pi\)
0.230888 + 0.972980i \(0.425837\pi\)
\(230\) −170100. −0.212024
\(231\) 228420. 0.281646
\(232\) −298880. −0.364567
\(233\) 762015. 0.919547 0.459773 0.888036i \(-0.347931\pi\)
0.459773 + 0.888036i \(0.347931\pi\)
\(234\) −360612. −0.430527
\(235\) −214450. −0.253312
\(236\) 98208.0 0.114780
\(237\) −699390. −0.808814
\(238\) −217328. −0.248699
\(239\) −278752. −0.315663 −0.157831 0.987466i \(-0.550450\pi\)
−0.157831 + 0.987466i \(0.550450\pi\)
\(240\) −57600.0 −0.0645497
\(241\) 379920. 0.421356 0.210678 0.977555i \(-0.432433\pi\)
0.210678 + 0.977555i \(0.432433\pi\)
\(242\) −571304. −0.627089
\(243\) −59049.0 −0.0641500
\(244\) 315008. 0.338725
\(245\) 463425. 0.493247
\(246\) −320796. −0.337980
\(247\) 1.03398e6 1.07837
\(248\) −351360. −0.362763
\(249\) 974322. 0.995873
\(250\) −562500. −0.569210
\(251\) 976572. 0.978408 0.489204 0.872169i \(-0.337287\pi\)
0.489204 + 0.872169i \(0.337287\pi\)
\(252\) −243648. −0.241692
\(253\) −229635. −0.225547
\(254\) −1.02447e6 −0.996355
\(255\) −65025.0 −0.0626224
\(256\) 65536.0 0.0625000
\(257\) 907308. 0.856884 0.428442 0.903569i \(-0.359063\pi\)
0.428442 + 0.903569i \(0.359063\pi\)
\(258\) 426204. 0.398628
\(259\) −2.03190e6 −1.88215
\(260\) −445200. −0.408434
\(261\) −378270. −0.343717
\(262\) −1.15334e6 −1.03802
\(263\) −1.59365e6 −1.42070 −0.710351 0.703848i \(-0.751464\pi\)
−0.710351 + 0.703848i \(0.751464\pi\)
\(264\) −77760.0 −0.0686667
\(265\) 890150. 0.778661
\(266\) 698608. 0.605382
\(267\) 1.04839e6 0.900006
\(268\) −235584. −0.200359
\(269\) 1.24870e6 1.05215 0.526074 0.850439i \(-0.323663\pi\)
0.526074 + 0.850439i \(0.323663\pi\)
\(270\) −72900.0 −0.0608581
\(271\) 62459.0 0.0516621 0.0258310 0.999666i \(-0.491777\pi\)
0.0258310 + 0.999666i \(0.491777\pi\)
\(272\) 73984.0 0.0606339
\(273\) −1.88320e6 −1.52929
\(274\) −577464. −0.464674
\(275\) −337500. −0.269118
\(276\) 244944. 0.193550
\(277\) 1.14635e6 0.897669 0.448835 0.893615i \(-0.351839\pi\)
0.448835 + 0.893615i \(0.351839\pi\)
\(278\) −662728. −0.514308
\(279\) −444690. −0.342016
\(280\) −300800. −0.229289
\(281\) −2.14530e6 −1.62077 −0.810387 0.585896i \(-0.800743\pi\)
−0.810387 + 0.585896i \(0.800743\pi\)
\(282\) 308808. 0.231241
\(283\) −1.72019e6 −1.27676 −0.638380 0.769721i \(-0.720395\pi\)
−0.638380 + 0.769721i \(0.720395\pi\)
\(284\) 774336. 0.569683
\(285\) 209025. 0.152435
\(286\) −601020. −0.434484
\(287\) −1.67527e6 −1.20055
\(288\) 82944.0 0.0589256
\(289\) 83521.0 0.0588235
\(290\) −467000. −0.326078
\(291\) −94716.0 −0.0655678
\(292\) 165152. 0.113351
\(293\) −1.43433e6 −0.976069 −0.488035 0.872824i \(-0.662286\pi\)
−0.488035 + 0.872824i \(0.662286\pi\)
\(294\) −667332. −0.450271
\(295\) 153450. 0.102663
\(296\) 691712. 0.458877
\(297\) −98415.0 −0.0647396
\(298\) −1.10522e6 −0.720957
\(299\) 1.89321e6 1.22468
\(300\) 360000. 0.230940
\(301\) 2.22573e6 1.41598
\(302\) 2.21923e6 1.40018
\(303\) −1.10606e6 −0.692108
\(304\) −237824. −0.147595
\(305\) 492200. 0.302965
\(306\) 93636.0 0.0571662
\(307\) 1.94190e6 1.17593 0.587964 0.808887i \(-0.299930\pi\)
0.587964 + 0.808887i \(0.299930\pi\)
\(308\) −406080. −0.243913
\(309\) −599949. −0.357452
\(310\) −549000. −0.324465
\(311\) 739368. 0.433471 0.216735 0.976230i \(-0.430459\pi\)
0.216735 + 0.976230i \(0.430459\pi\)
\(312\) 641088. 0.372847
\(313\) −2.10696e6 −1.21561 −0.607806 0.794086i \(-0.707950\pi\)
−0.607806 + 0.794086i \(0.707950\pi\)
\(314\) −1.61442e6 −0.924043
\(315\) −380700. −0.216176
\(316\) 1.24336e6 0.700453
\(317\) 2.71118e6 1.51534 0.757671 0.652637i \(-0.226337\pi\)
0.757671 + 0.652637i \(0.226337\pi\)
\(318\) −1.28182e6 −0.710817
\(319\) −630450. −0.346876
\(320\) 102400. 0.0559017
\(321\) 1.46056e6 0.791149
\(322\) 1.27915e6 0.687515
\(323\) −268481. −0.143188
\(324\) 104976. 0.0555556
\(325\) 2.78250e6 1.46126
\(326\) −1.78596e6 −0.930739
\(327\) −1.12950e6 −0.584140
\(328\) 570304. 0.292699
\(329\) 1.61266e6 0.821399
\(330\) −121500. −0.0614174
\(331\) −23965.0 −0.0120229 −0.00601143 0.999982i \(-0.501914\pi\)
−0.00601143 + 0.999982i \(0.501914\pi\)
\(332\) −1.73213e6 −0.862452
\(333\) 875448. 0.432633
\(334\) 777692. 0.381453
\(335\) −368100. −0.179207
\(336\) 433152. 0.209311
\(337\) −2.95570e6 −1.41771 −0.708853 0.705357i \(-0.750787\pi\)
−0.708853 + 0.705357i \(0.750787\pi\)
\(338\) 3.46990e6 1.65206
\(339\) −693261. −0.327640
\(340\) 115600. 0.0542326
\(341\) −741150. −0.345160
\(342\) −300996. −0.139154
\(343\) −325240. −0.149269
\(344\) −757696. −0.345222
\(345\) 382725. 0.173117
\(346\) 321460. 0.144357
\(347\) −1.99041e6 −0.887400 −0.443700 0.896175i \(-0.646334\pi\)
−0.443700 + 0.896175i \(0.646334\pi\)
\(348\) 672480. 0.297667
\(349\) 475581. 0.209007 0.104504 0.994525i \(-0.466675\pi\)
0.104504 + 0.994525i \(0.466675\pi\)
\(350\) 1.88000e6 0.820328
\(351\) 811377. 0.351524
\(352\) 138240. 0.0594671
\(353\) −1.98088e6 −0.846099 −0.423049 0.906107i \(-0.639040\pi\)
−0.423049 + 0.906107i \(0.639040\pi\)
\(354\) −220968. −0.0937176
\(355\) 1.20990e6 0.509540
\(356\) −1.86381e6 −0.779428
\(357\) 488988. 0.203062
\(358\) 157032. 0.0647561
\(359\) −3.20778e6 −1.31362 −0.656808 0.754058i \(-0.728094\pi\)
−0.656808 + 0.754058i \(0.728094\pi\)
\(360\) 129600. 0.0527046
\(361\) −1.61306e6 −0.651451
\(362\) −902648. −0.362032
\(363\) 1.28543e6 0.512016
\(364\) 3.34790e6 1.32440
\(365\) 258050. 0.101385
\(366\) −708768. −0.276568
\(367\) 491980. 0.190670 0.0953350 0.995445i \(-0.469608\pi\)
0.0953350 + 0.995445i \(0.469608\pi\)
\(368\) −435456. −0.167620
\(369\) 721791. 0.275960
\(370\) 1.08080e6 0.410432
\(371\) −6.69393e6 −2.52491
\(372\) 790560. 0.296195
\(373\) 1.81206e6 0.674373 0.337186 0.941438i \(-0.390525\pi\)
0.337186 + 0.941438i \(0.390525\pi\)
\(374\) 156060. 0.0576916
\(375\) 1.26562e6 0.464758
\(376\) −548992. −0.200261
\(377\) 5.19771e6 1.88347
\(378\) 548208. 0.197340
\(379\) 4.61391e6 1.64995 0.824976 0.565168i \(-0.191189\pi\)
0.824976 + 0.565168i \(0.191189\pi\)
\(380\) −371600. −0.132013
\(381\) 2.30505e6 0.813520
\(382\) −654008. −0.229311
\(383\) −3.44867e6 −1.20131 −0.600655 0.799508i \(-0.705093\pi\)
−0.600655 + 0.799508i \(0.705093\pi\)
\(384\) −147456. −0.0510310
\(385\) −634500. −0.218162
\(386\) −1.79882e6 −0.614498
\(387\) −958959. −0.325479
\(388\) 168384. 0.0567834
\(389\) −515080. −0.172584 −0.0862920 0.996270i \(-0.527502\pi\)
−0.0862920 + 0.996270i \(0.527502\pi\)
\(390\) 1.00170e6 0.333485
\(391\) −491589. −0.162615
\(392\) 1.18637e6 0.389946
\(393\) 2.59502e6 0.847537
\(394\) 1.80893e6 0.587059
\(395\) 1.94275e6 0.626504
\(396\) 174960. 0.0560662
\(397\) −3.42470e6 −1.09055 −0.545275 0.838257i \(-0.683575\pi\)
−0.545275 + 0.838257i \(0.683575\pi\)
\(398\) −148912. −0.0471218
\(399\) −1.57187e6 −0.494292
\(400\) −640000. −0.200000
\(401\) 2.46651e6 0.765987 0.382994 0.923751i \(-0.374893\pi\)
0.382994 + 0.923751i \(0.374893\pi\)
\(402\) 530064. 0.163592
\(403\) 6.11037e6 1.87415
\(404\) 1.96634e6 0.599383
\(405\) 164025. 0.0496904
\(406\) 3.51184e6 1.05735
\(407\) 1.45908e6 0.436609
\(408\) −166464. −0.0495074
\(409\) −5.40107e6 −1.59651 −0.798254 0.602320i \(-0.794243\pi\)
−0.798254 + 0.602320i \(0.794243\pi\)
\(410\) 891100. 0.261798
\(411\) 1.29929e6 0.379405
\(412\) 1.06658e6 0.309563
\(413\) −1.15394e6 −0.332897
\(414\) −551124. −0.158033
\(415\) −2.70645e6 −0.771400
\(416\) −1.13971e6 −0.322895
\(417\) 1.49114e6 0.419931
\(418\) −501660. −0.140433
\(419\) 1.16768e6 0.324930 0.162465 0.986714i \(-0.448055\pi\)
0.162465 + 0.986714i \(0.448055\pi\)
\(420\) 676800. 0.187213
\(421\) −468257. −0.128759 −0.0643797 0.997925i \(-0.520507\pi\)
−0.0643797 + 0.997925i \(0.520507\pi\)
\(422\) −497608. −0.136021
\(423\) −694818. −0.188808
\(424\) 2.27878e6 0.615586
\(425\) −722500. −0.194029
\(426\) −1.74226e6 −0.465145
\(427\) −3.70134e6 −0.982403
\(428\) −2.59656e6 −0.685155
\(429\) 1.35230e6 0.354755
\(430\) −1.18390e6 −0.308776
\(431\) −3.86744e6 −1.00284 −0.501419 0.865205i \(-0.667188\pi\)
−0.501419 + 0.865205i \(0.667188\pi\)
\(432\) −186624. −0.0481125
\(433\) −4.33460e6 −1.11104 −0.555520 0.831503i \(-0.687481\pi\)
−0.555520 + 0.831503i \(0.687481\pi\)
\(434\) 4.12848e6 1.05212
\(435\) 1.05075e6 0.266242
\(436\) 2.00800e6 0.505880
\(437\) 1.58023e6 0.395837
\(438\) −371592. −0.0925510
\(439\) −5.76408e6 −1.42748 −0.713738 0.700412i \(-0.752999\pi\)
−0.713738 + 0.700412i \(0.752999\pi\)
\(440\) 216000. 0.0531890
\(441\) 1.50150e6 0.367644
\(442\) −1.28663e6 −0.313255
\(443\) 4.91399e6 1.18967 0.594833 0.803849i \(-0.297218\pi\)
0.594833 + 0.803849i \(0.297218\pi\)
\(444\) −1.55635e6 −0.374671
\(445\) −2.91220e6 −0.697142
\(446\) −4.81853e6 −1.14704
\(447\) 2.48675e6 0.588659
\(448\) −770048. −0.181269
\(449\) 920726. 0.215533 0.107767 0.994176i \(-0.465630\pi\)
0.107767 + 0.994176i \(0.465630\pi\)
\(450\) −810000. −0.188562
\(451\) 1.20298e6 0.278496
\(452\) 1.23246e6 0.283745
\(453\) −4.99327e6 −1.14325
\(454\) 3.27799e6 0.746393
\(455\) 5.23110e6 1.18458
\(456\) 535104. 0.120511
\(457\) −514955. −0.115340 −0.0576698 0.998336i \(-0.518367\pi\)
−0.0576698 + 0.998336i \(0.518367\pi\)
\(458\) 1.46582e6 0.326524
\(459\) −210681. −0.0466760
\(460\) −680400. −0.149924
\(461\) −6.11783e6 −1.34074 −0.670371 0.742026i \(-0.733865\pi\)
−0.670371 + 0.742026i \(0.733865\pi\)
\(462\) 913680. 0.199154
\(463\) 7.85456e6 1.70282 0.851411 0.524499i \(-0.175747\pi\)
0.851411 + 0.524499i \(0.175747\pi\)
\(464\) −1.19552e6 −0.257788
\(465\) 1.23525e6 0.264925
\(466\) 3.04806e6 0.650218
\(467\) −2.50736e6 −0.532015 −0.266008 0.963971i \(-0.585705\pi\)
−0.266008 + 0.963971i \(0.585705\pi\)
\(468\) −1.44245e6 −0.304429
\(469\) 2.76811e6 0.581101
\(470\) −857800. −0.179119
\(471\) 3.63244e6 0.754478
\(472\) 392832. 0.0811618
\(473\) −1.59826e6 −0.328470
\(474\) −2.79756e6 −0.571918
\(475\) 2.32250e6 0.472304
\(476\) −869312. −0.175856
\(477\) 2.88409e6 0.580380
\(478\) −1.11501e6 −0.223207
\(479\) −256969. −0.0511731 −0.0255866 0.999673i \(-0.508145\pi\)
−0.0255866 + 0.999673i \(0.508145\pi\)
\(480\) −230400. −0.0456435
\(481\) −1.20293e7 −2.37071
\(482\) 1.51968e6 0.297944
\(483\) −2.87809e6 −0.561354
\(484\) −2.28522e6 −0.443419
\(485\) 263100. 0.0507886
\(486\) −236196. −0.0453609
\(487\) 1.47589e6 0.281989 0.140994 0.990010i \(-0.454970\pi\)
0.140994 + 0.990010i \(0.454970\pi\)
\(488\) 1.26003e6 0.239515
\(489\) 4.01841e6 0.759945
\(490\) 1.85370e6 0.348778
\(491\) 1.88665e6 0.353174 0.176587 0.984285i \(-0.443494\pi\)
0.176587 + 0.984285i \(0.443494\pi\)
\(492\) −1.28318e6 −0.238988
\(493\) −1.34963e6 −0.250091
\(494\) 4.13591e6 0.762524
\(495\) 273375. 0.0501471
\(496\) −1.40544e6 −0.256512
\(497\) −9.09845e6 −1.65225
\(498\) 3.89729e6 0.704189
\(499\) −1.00548e7 −1.80768 −0.903841 0.427869i \(-0.859265\pi\)
−0.903841 + 0.427869i \(0.859265\pi\)
\(500\) −2.25000e6 −0.402492
\(501\) −1.74981e6 −0.311455
\(502\) 3.90629e6 0.691839
\(503\) −6.05457e6 −1.06700 −0.533499 0.845801i \(-0.679123\pi\)
−0.533499 + 0.845801i \(0.679123\pi\)
\(504\) −974592. −0.170902
\(505\) 3.07240e6 0.536104
\(506\) −918540. −0.159486
\(507\) −7.80728e6 −1.34890
\(508\) −4.09787e6 −0.704529
\(509\) 7.26288e6 1.24255 0.621276 0.783592i \(-0.286615\pi\)
0.621276 + 0.783592i \(0.286615\pi\)
\(510\) −260100. −0.0442807
\(511\) −1.94054e6 −0.328753
\(512\) 262144. 0.0441942
\(513\) 677241. 0.113619
\(514\) 3.62923e6 0.605908
\(515\) 1.66653e6 0.276881
\(516\) 1.70482e6 0.281873
\(517\) −1.15803e6 −0.190543
\(518\) −8.12762e6 −1.33088
\(519\) −723285. −0.117867
\(520\) −1.78080e6 −0.288806
\(521\) −1.90029e6 −0.306708 −0.153354 0.988171i \(-0.549008\pi\)
−0.153354 + 0.988171i \(0.549008\pi\)
\(522\) −1.51308e6 −0.243044
\(523\) 5.24950e6 0.839197 0.419598 0.907710i \(-0.362171\pi\)
0.419598 + 0.907710i \(0.362171\pi\)
\(524\) −4.61336e6 −0.733988
\(525\) −4.23000e6 −0.669795
\(526\) −6.37459e6 −1.00459
\(527\) −1.58661e6 −0.248853
\(528\) −311040. −0.0485547
\(529\) −3.54294e6 −0.550459
\(530\) 3.56060e6 0.550597
\(531\) 497178. 0.0765201
\(532\) 2.79443e6 0.428070
\(533\) −9.91794e6 −1.51218
\(534\) 4.19357e6 0.636401
\(535\) −4.05712e6 −0.612821
\(536\) −942336. −0.141675
\(537\) −353322. −0.0528731
\(538\) 4.99480e6 0.743981
\(539\) 2.50250e6 0.371023
\(540\) −291600. −0.0430331
\(541\) 1.19372e7 1.75351 0.876757 0.480933i \(-0.159702\pi\)
0.876757 + 0.480933i \(0.159702\pi\)
\(542\) 249836. 0.0365306
\(543\) 2.03096e6 0.295598
\(544\) 295936. 0.0428746
\(545\) 3.13750e6 0.452473
\(546\) −7.53278e6 −1.08137
\(547\) 3.92109e6 0.560323 0.280161 0.959953i \(-0.409612\pi\)
0.280161 + 0.959953i \(0.409612\pi\)
\(548\) −2.30986e6 −0.328574
\(549\) 1.59473e6 0.225817
\(550\) −1.35000e6 −0.190295
\(551\) 4.33843e6 0.608770
\(552\) 979776. 0.136861
\(553\) −1.46095e7 −2.03152
\(554\) 4.58538e6 0.634748
\(555\) −2.43180e6 −0.335116
\(556\) −2.65091e6 −0.363671
\(557\) 8.62416e6 1.17782 0.588910 0.808199i \(-0.299557\pi\)
0.588910 + 0.808199i \(0.299557\pi\)
\(558\) −1.77876e6 −0.241842
\(559\) 1.31768e7 1.78353
\(560\) −1.20320e6 −0.162132
\(561\) −351135. −0.0471050
\(562\) −8.58120e6 −1.14606
\(563\) −1.20126e6 −0.159723 −0.0798614 0.996806i \(-0.525448\pi\)
−0.0798614 + 0.996806i \(0.525448\pi\)
\(564\) 1.23523e6 0.163512
\(565\) 1.92572e6 0.253789
\(566\) −6.88074e6 −0.902806
\(567\) −1.23347e6 −0.161128
\(568\) 3.09734e6 0.402827
\(569\) −1.10716e6 −0.143360 −0.0716800 0.997428i \(-0.522836\pi\)
−0.0716800 + 0.997428i \(0.522836\pi\)
\(570\) 836100. 0.107788
\(571\) 1.30319e7 1.67270 0.836349 0.548198i \(-0.184686\pi\)
0.836349 + 0.548198i \(0.184686\pi\)
\(572\) −2.40408e6 −0.307227
\(573\) 1.47152e6 0.187232
\(574\) −6.70107e6 −0.848916
\(575\) 4.25250e6 0.536383
\(576\) 331776. 0.0416667
\(577\) 8.13114e6 1.01674 0.508372 0.861137i \(-0.330247\pi\)
0.508372 + 0.861137i \(0.330247\pi\)
\(578\) 334084. 0.0415945
\(579\) 4.04735e6 0.501735
\(580\) −1.86800e6 −0.230572
\(581\) 2.03525e7 2.50137
\(582\) −378864. −0.0463635
\(583\) 4.80681e6 0.585714
\(584\) 660608. 0.0801515
\(585\) −2.25382e6 −0.272289
\(586\) −5.73733e6 −0.690185
\(587\) 1.68467e6 0.201799 0.100900 0.994897i \(-0.467828\pi\)
0.100900 + 0.994897i \(0.467828\pi\)
\(588\) −2.66933e6 −0.318389
\(589\) 5.10021e6 0.605759
\(590\) 613800. 0.0725934
\(591\) −4.07010e6 −0.479332
\(592\) 2.76685e6 0.324475
\(593\) −113022. −0.0131986 −0.00659928 0.999978i \(-0.502101\pi\)
−0.00659928 + 0.999978i \(0.502101\pi\)
\(594\) −393660. −0.0457778
\(595\) −1.35830e6 −0.157291
\(596\) −4.42090e6 −0.509794
\(597\) 335052. 0.0384748
\(598\) 7.57285e6 0.865977
\(599\) 8.26813e6 0.941543 0.470772 0.882255i \(-0.343976\pi\)
0.470772 + 0.882255i \(0.343976\pi\)
\(600\) 1.44000e6 0.163299
\(601\) −7.67922e6 −0.867224 −0.433612 0.901100i \(-0.642761\pi\)
−0.433612 + 0.901100i \(0.642761\pi\)
\(602\) 8.90293e6 1.00125
\(603\) −1.19264e6 −0.133573
\(604\) 8.87693e6 0.990080
\(605\) −3.57065e6 −0.396606
\(606\) −4.42426e6 −0.489394
\(607\) 2.95861e6 0.325924 0.162962 0.986632i \(-0.447895\pi\)
0.162962 + 0.986632i \(0.447895\pi\)
\(608\) −951296. −0.104365
\(609\) −7.90164e6 −0.863324
\(610\) 1.96880e6 0.214228
\(611\) 9.54731e6 1.03461
\(612\) 374544. 0.0404226
\(613\) −1.12745e7 −1.21185 −0.605924 0.795523i \(-0.707196\pi\)
−0.605924 + 0.795523i \(0.707196\pi\)
\(614\) 7.76760e6 0.831507
\(615\) −2.00498e6 −0.213757
\(616\) −1.62432e6 −0.172472
\(617\) 1.63830e7 1.73252 0.866262 0.499589i \(-0.166516\pi\)
0.866262 + 0.499589i \(0.166516\pi\)
\(618\) −2.39980e6 −0.252757
\(619\) 3.35390e6 0.351823 0.175911 0.984406i \(-0.443713\pi\)
0.175911 + 0.984406i \(0.443713\pi\)
\(620\) −2.19600e6 −0.229432
\(621\) 1.24003e6 0.129034
\(622\) 2.95747e6 0.306510
\(623\) 2.18997e7 2.26058
\(624\) 2.56435e6 0.263643
\(625\) 4.29688e6 0.440000
\(626\) −8.42782e6 −0.859567
\(627\) 1.12874e6 0.114663
\(628\) −6.45768e6 −0.653397
\(629\) 3.12351e6 0.314787
\(630\) −1.52280e6 −0.152859
\(631\) 1.51263e7 1.51237 0.756185 0.654358i \(-0.227061\pi\)
0.756185 + 0.654358i \(0.227061\pi\)
\(632\) 4.97344e6 0.495295
\(633\) 1.11962e6 0.111061
\(634\) 1.08447e7 1.07151
\(635\) −6.40292e6 −0.630150
\(636\) −5.12726e6 −0.502624
\(637\) −2.06317e7 −2.01459
\(638\) −2.52180e6 −0.245278
\(639\) 3.92008e6 0.379789
\(640\) 409600. 0.0395285
\(641\) −1.24739e7 −1.19910 −0.599551 0.800336i \(-0.704654\pi\)
−0.599551 + 0.800336i \(0.704654\pi\)
\(642\) 5.84226e6 0.559427
\(643\) −1.08684e7 −1.03667 −0.518334 0.855178i \(-0.673448\pi\)
−0.518334 + 0.855178i \(0.673448\pi\)
\(644\) 5.11661e6 0.486147
\(645\) 2.66378e6 0.252115
\(646\) −1.07392e6 −0.101249
\(647\) 3.37275e6 0.316755 0.158377 0.987379i \(-0.449374\pi\)
0.158377 + 0.987379i \(0.449374\pi\)
\(648\) 419904. 0.0392837
\(649\) 828630. 0.0772234
\(650\) 1.11300e7 1.03327
\(651\) −9.28908e6 −0.859054
\(652\) −7.14384e6 −0.658132
\(653\) 1.77220e7 1.62641 0.813203 0.581981i \(-0.197722\pi\)
0.813203 + 0.581981i \(0.197722\pi\)
\(654\) −4.51800e6 −0.413049
\(655\) −7.20838e6 −0.656499
\(656\) 2.28122e6 0.206970
\(657\) 836082. 0.0755676
\(658\) 6.45066e6 0.580817
\(659\) 1.27338e7 1.14221 0.571103 0.820878i \(-0.306516\pi\)
0.571103 + 0.820878i \(0.306516\pi\)
\(660\) −486000. −0.0434287
\(661\) −1.72968e7 −1.53980 −0.769898 0.638167i \(-0.779693\pi\)
−0.769898 + 0.638167i \(0.779693\pi\)
\(662\) −95860.0 −0.00850144
\(663\) 2.89491e6 0.255771
\(664\) −6.92851e6 −0.609845
\(665\) 4.36630e6 0.382877
\(666\) 3.50179e6 0.305918
\(667\) 7.94367e6 0.691364
\(668\) 3.11077e6 0.269728
\(669\) 1.08417e7 0.936552
\(670\) −1.47240e6 −0.126718
\(671\) 2.65788e6 0.227892
\(672\) 1.73261e6 0.148005
\(673\) −1.34307e7 −1.14303 −0.571517 0.820590i \(-0.693645\pi\)
−0.571517 + 0.820590i \(0.693645\pi\)
\(674\) −1.18228e7 −1.00247
\(675\) 1.82250e6 0.153960
\(676\) 1.38796e7 1.16818
\(677\) −1.75486e7 −1.47153 −0.735767 0.677235i \(-0.763178\pi\)
−0.735767 + 0.677235i \(0.763178\pi\)
\(678\) −2.77304e6 −0.231677
\(679\) −1.97851e6 −0.164689
\(680\) 462400. 0.0383482
\(681\) −7.37547e6 −0.609428
\(682\) −2.96460e6 −0.244065
\(683\) −1.61614e7 −1.32565 −0.662824 0.748775i \(-0.730642\pi\)
−0.662824 + 0.748775i \(0.730642\pi\)
\(684\) −1.20398e6 −0.0983967
\(685\) −3.60915e6 −0.293886
\(686\) −1.30096e6 −0.105549
\(687\) −3.29809e6 −0.266606
\(688\) −3.03078e6 −0.244109
\(689\) −3.96295e7 −3.18032
\(690\) 1.53090e6 0.122412
\(691\) 1.13905e7 0.907498 0.453749 0.891129i \(-0.350086\pi\)
0.453749 + 0.891129i \(0.350086\pi\)
\(692\) 1.28584e6 0.102076
\(693\) −2.05578e6 −0.162609
\(694\) −7.96165e6 −0.627486
\(695\) −4.14205e6 −0.325277
\(696\) 2.68992e6 0.210483
\(697\) 2.57528e6 0.200790
\(698\) 1.90232e6 0.147790
\(699\) −6.85814e6 −0.530900
\(700\) 7.52000e6 0.580060
\(701\) −2.22903e7 −1.71325 −0.856626 0.515938i \(-0.827443\pi\)
−0.856626 + 0.515938i \(0.827443\pi\)
\(702\) 3.24551e6 0.248565
\(703\) −1.00406e7 −0.766254
\(704\) 552960. 0.0420496
\(705\) 1.93005e6 0.146250
\(706\) −7.92351e6 −0.598282
\(707\) −2.31044e7 −1.73839
\(708\) −883872. −0.0662684
\(709\) 1.24843e7 0.932715 0.466357 0.884596i \(-0.345566\pi\)
0.466357 + 0.884596i \(0.345566\pi\)
\(710\) 4.83960e6 0.360299
\(711\) 6.29451e6 0.466969
\(712\) −7.45523e6 −0.551139
\(713\) 9.33849e6 0.687944
\(714\) 1.95595e6 0.143586
\(715\) −3.75638e6 −0.274792
\(716\) 628128. 0.0457895
\(717\) 2.50877e6 0.182248
\(718\) −1.28311e7 −0.928867
\(719\) −1.44584e7 −1.04303 −0.521515 0.853242i \(-0.674633\pi\)
−0.521515 + 0.853242i \(0.674633\pi\)
\(720\) 518400. 0.0372678
\(721\) −1.25323e7 −0.897825
\(722\) −6.45223e6 −0.460646
\(723\) −3.41928e6 −0.243270
\(724\) −3.61059e6 −0.255995
\(725\) 1.16750e7 0.824920
\(726\) 5.14174e6 0.362050
\(727\) 8.35149e6 0.586041 0.293020 0.956106i \(-0.405340\pi\)
0.293020 + 0.956106i \(0.405340\pi\)
\(728\) 1.33916e7 0.936493
\(729\) 531441. 0.0370370
\(730\) 1.03220e6 0.0716897
\(731\) −3.42147e6 −0.236821
\(732\) −2.83507e6 −0.195563
\(733\) 1.58985e7 1.09294 0.546470 0.837479i \(-0.315971\pi\)
0.546470 + 0.837479i \(0.315971\pi\)
\(734\) 1.96792e6 0.134824
\(735\) −4.17082e6 −0.284776
\(736\) −1.74182e6 −0.118525
\(737\) −1.98774e6 −0.134800
\(738\) 2.88716e6 0.195133
\(739\) −109329. −0.00736418 −0.00368209 0.999993i \(-0.501172\pi\)
−0.00368209 + 0.999993i \(0.501172\pi\)
\(740\) 4.32320e6 0.290219
\(741\) −9.30579e6 −0.622598
\(742\) −2.67757e7 −1.78538
\(743\) 9.80891e6 0.651852 0.325926 0.945395i \(-0.394324\pi\)
0.325926 + 0.945395i \(0.394324\pi\)
\(744\) 3.16224e6 0.209441
\(745\) −6.90765e6 −0.455974
\(746\) 7.24823e6 0.476854
\(747\) −8.76890e6 −0.574968
\(748\) 624240. 0.0407941
\(749\) 3.05096e7 1.98715
\(750\) 5.06250e6 0.328634
\(751\) 1.74987e7 1.13216 0.566078 0.824352i \(-0.308460\pi\)
0.566078 + 0.824352i \(0.308460\pi\)
\(752\) −2.19597e6 −0.141606
\(753\) −8.78915e6 −0.564884
\(754\) 2.07908e7 1.33181
\(755\) 1.38702e7 0.885555
\(756\) 2.19283e6 0.139541
\(757\) 1.67029e7 1.05938 0.529691 0.848191i \(-0.322308\pi\)
0.529691 + 0.848191i \(0.322308\pi\)
\(758\) 1.84556e7 1.16669
\(759\) 2.06672e6 0.130220
\(760\) −1.48640e6 −0.0933473
\(761\) −2.24151e6 −0.140307 −0.0701536 0.997536i \(-0.522349\pi\)
−0.0701536 + 0.997536i \(0.522349\pi\)
\(762\) 9.22021e6 0.575246
\(763\) −2.35940e7 −1.46720
\(764\) −2.61603e6 −0.162147
\(765\) 585225. 0.0361551
\(766\) −1.37947e7 −0.849454
\(767\) −6.83159e6 −0.419308
\(768\) −589824. −0.0360844
\(769\) −1.31165e7 −0.799837 −0.399919 0.916551i \(-0.630962\pi\)
−0.399919 + 0.916551i \(0.630962\pi\)
\(770\) −2.53800e6 −0.154264
\(771\) −8.16577e6 −0.494722
\(772\) −7.19530e6 −0.434516
\(773\) −2.82980e7 −1.70336 −0.851680 0.524063i \(-0.824416\pi\)
−0.851680 + 0.524063i \(0.824416\pi\)
\(774\) −3.83584e6 −0.230148
\(775\) 1.37250e7 0.820839
\(776\) 673536. 0.0401519
\(777\) 1.82871e7 1.08666
\(778\) −2.06032e6 −0.122035
\(779\) −8.27832e6 −0.488763
\(780\) 4.00680e6 0.235809
\(781\) 6.53346e6 0.383280
\(782\) −1.96636e6 −0.114986
\(783\) 3.40443e6 0.198445
\(784\) 4.74547e6 0.275733
\(785\) −1.00901e7 −0.584416
\(786\) 1.03801e7 0.599299
\(787\) −2.31191e6 −0.133056 −0.0665279 0.997785i \(-0.521192\pi\)
−0.0665279 + 0.997785i \(0.521192\pi\)
\(788\) 7.23573e6 0.415113
\(789\) 1.43428e7 0.820243
\(790\) 7.77100e6 0.443006
\(791\) −1.44815e7 −0.822945
\(792\) 699840. 0.0396448
\(793\) −2.19127e7 −1.23741
\(794\) −1.36988e7 −0.771136
\(795\) −8.01135e6 −0.449560
\(796\) −595648. −0.0333202
\(797\) −4.14006e6 −0.230867 −0.115433 0.993315i \(-0.536826\pi\)
−0.115433 + 0.993315i \(0.536826\pi\)
\(798\) −6.28747e6 −0.349517
\(799\) −2.47904e6 −0.137378
\(800\) −2.56000e6 −0.141421
\(801\) −9.43553e6 −0.519619
\(802\) 9.86603e6 0.541635
\(803\) 1.39347e6 0.0762621
\(804\) 2.12026e6 0.115677
\(805\) 7.99470e6 0.434823
\(806\) 2.44415e7 1.32523
\(807\) −1.12383e7 −0.607458
\(808\) 7.86534e6 0.423828
\(809\) 3.97495e6 0.213531 0.106765 0.994284i \(-0.465951\pi\)
0.106765 + 0.994284i \(0.465951\pi\)
\(810\) 656100. 0.0351364
\(811\) 8.40205e6 0.448573 0.224286 0.974523i \(-0.427995\pi\)
0.224286 + 0.974523i \(0.427995\pi\)
\(812\) 1.40474e7 0.747661
\(813\) −562131. −0.0298271
\(814\) 5.83632e6 0.308729
\(815\) −1.11622e7 −0.588651
\(816\) −665856. −0.0350070
\(817\) 1.09984e7 0.576469
\(818\) −2.16043e7 −1.12890
\(819\) 1.69488e7 0.882934
\(820\) 3.56440e6 0.185119
\(821\) −1.91180e7 −0.989886 −0.494943 0.868925i \(-0.664811\pi\)
−0.494943 + 0.868925i \(0.664811\pi\)
\(822\) 5.19718e6 0.268280
\(823\) −2.00766e7 −1.03322 −0.516608 0.856222i \(-0.672806\pi\)
−0.516608 + 0.856222i \(0.672806\pi\)
\(824\) 4.26630e6 0.218894
\(825\) 3.03750e6 0.155375
\(826\) −4.61578e6 −0.235394
\(827\) −787327. −0.0400305 −0.0200153 0.999800i \(-0.506371\pi\)
−0.0200153 + 0.999800i \(0.506371\pi\)
\(828\) −2.20450e6 −0.111746
\(829\) 2.59446e7 1.31118 0.655588 0.755119i \(-0.272421\pi\)
0.655588 + 0.755119i \(0.272421\pi\)
\(830\) −1.08258e7 −0.545462
\(831\) −1.03171e7 −0.518270
\(832\) −4.55885e6 −0.228321
\(833\) 5.35719e6 0.267501
\(834\) 5.96455e6 0.296936
\(835\) 4.86058e6 0.241252
\(836\) −2.00664e6 −0.0993010
\(837\) 4.00221e6 0.197463
\(838\) 4.67074e6 0.229760
\(839\) −1.64947e7 −0.808984 −0.404492 0.914541i \(-0.632552\pi\)
−0.404492 + 0.914541i \(0.632552\pi\)
\(840\) 2.70720e6 0.132380
\(841\) 1.29775e6 0.0632705
\(842\) −1.87303e6 −0.0910467
\(843\) 1.93077e7 0.935754
\(844\) −1.99043e6 −0.0961814
\(845\) 2.16869e7 1.04485
\(846\) −2.77927e6 −0.133507
\(847\) 2.68513e7 1.28605
\(848\) 9.11514e6 0.435285
\(849\) 1.54817e7 0.737138
\(850\) −2.89000e6 −0.137199
\(851\) −1.83844e7 −0.870213
\(852\) −6.96902e6 −0.328907
\(853\) −4.05828e6 −0.190972 −0.0954859 0.995431i \(-0.530440\pi\)
−0.0954859 + 0.995431i \(0.530440\pi\)
\(854\) −1.48054e7 −0.694664
\(855\) −1.88122e6 −0.0880087
\(856\) −1.03862e7 −0.484478
\(857\) −1.93630e7 −0.900575 −0.450288 0.892884i \(-0.648679\pi\)
−0.450288 + 0.892884i \(0.648679\pi\)
\(858\) 5.40918e6 0.250849
\(859\) 1.83312e7 0.847632 0.423816 0.905748i \(-0.360690\pi\)
0.423816 + 0.905748i \(0.360690\pi\)
\(860\) −4.73560e6 −0.218338
\(861\) 1.50774e7 0.693137
\(862\) −1.54698e7 −0.709113
\(863\) −3.14626e7 −1.43803 −0.719015 0.694994i \(-0.755407\pi\)
−0.719015 + 0.694994i \(0.755407\pi\)
\(864\) −746496. −0.0340207
\(865\) 2.00912e6 0.0912991
\(866\) −1.73384e7 −0.785623
\(867\) −751689. −0.0339618
\(868\) 1.65139e7 0.743962
\(869\) 1.04908e7 0.471261
\(870\) 4.20300e6 0.188261
\(871\) 1.63878e7 0.731940
\(872\) 8.03200e6 0.357711
\(873\) 852444. 0.0378556
\(874\) 6.32092e6 0.279899
\(875\) 2.64375e7 1.16735
\(876\) −1.48637e6 −0.0654434
\(877\) −2.78956e7 −1.22472 −0.612360 0.790579i \(-0.709780\pi\)
−0.612360 + 0.790579i \(0.709780\pi\)
\(878\) −2.30563e7 −1.00938
\(879\) 1.29090e7 0.563534
\(880\) 864000. 0.0376103
\(881\) 3.96320e7 1.72031 0.860153 0.510036i \(-0.170368\pi\)
0.860153 + 0.510036i \(0.170368\pi\)
\(882\) 6.00599e6 0.259964
\(883\) 3.45310e7 1.49042 0.745208 0.666832i \(-0.232350\pi\)
0.745208 + 0.666832i \(0.232350\pi\)
\(884\) −5.14651e6 −0.221504
\(885\) −1.38105e6 −0.0592722
\(886\) 1.96560e7 0.841221
\(887\) −4.02361e7 −1.71715 −0.858573 0.512692i \(-0.828648\pi\)
−0.858573 + 0.512692i \(0.828648\pi\)
\(888\) −6.22541e6 −0.264933
\(889\) 4.81500e7 2.04335
\(890\) −1.16488e7 −0.492954
\(891\) 885735. 0.0373774
\(892\) −1.92741e7 −0.811078
\(893\) 7.96896e6 0.334405
\(894\) 9.94702e6 0.416245
\(895\) 981450. 0.0409553
\(896\) −3.08019e6 −0.128176
\(897\) −1.70389e7 −0.707068
\(898\) 3.68290e6 0.152405
\(899\) 2.56383e7 1.05801
\(900\) −3.24000e6 −0.133333
\(901\) 1.02901e7 0.422288
\(902\) 4.81194e6 0.196926
\(903\) −2.00316e7 −0.817516
\(904\) 4.92986e6 0.200638
\(905\) −5.64155e6 −0.228969
\(906\) −1.99731e7 −0.808397
\(907\) −3.50673e6 −0.141542 −0.0707708 0.997493i \(-0.522546\pi\)
−0.0707708 + 0.997493i \(0.522546\pi\)
\(908\) 1.31120e7 0.527780
\(909\) 9.95458e6 0.399589
\(910\) 2.09244e7 0.837625
\(911\) 2.71634e7 1.08439 0.542197 0.840251i \(-0.317593\pi\)
0.542197 + 0.840251i \(0.317593\pi\)
\(912\) 2.14042e6 0.0852140
\(913\) −1.46148e7 −0.580252
\(914\) −2.05982e6 −0.0815575
\(915\) −4.42980e6 −0.174917
\(916\) 5.86326e6 0.230888
\(917\) 5.42070e7 2.12879
\(918\) −842724. −0.0330049
\(919\) 3.61774e7 1.41302 0.706510 0.707703i \(-0.250269\pi\)
0.706510 + 0.707703i \(0.250269\pi\)
\(920\) −2.72160e6 −0.106012
\(921\) −1.74771e7 −0.678923
\(922\) −2.44713e7 −0.948047
\(923\) −5.38647e7 −2.08114
\(924\) 3.65472e6 0.140823
\(925\) −2.70200e7 −1.03832
\(926\) 3.14182e7 1.20408
\(927\) 5.39954e6 0.206375
\(928\) −4.78208e6 −0.182283
\(929\) 5.76520e6 0.219167 0.109583 0.993978i \(-0.465048\pi\)
0.109583 + 0.993978i \(0.465048\pi\)
\(930\) 4.94100e6 0.187330
\(931\) −1.72209e7 −0.651150
\(932\) 1.21922e7 0.459773
\(933\) −6.65431e6 −0.250264
\(934\) −1.00294e7 −0.376192
\(935\) 975375. 0.0364874
\(936\) −5.76979e6 −0.215264
\(937\) −1.41673e7 −0.527155 −0.263578 0.964638i \(-0.584902\pi\)
−0.263578 + 0.964638i \(0.584902\pi\)
\(938\) 1.10724e7 0.410900
\(939\) 1.89626e7 0.701833
\(940\) −3.43120e6 −0.126656
\(941\) −3.59561e7 −1.32373 −0.661864 0.749624i \(-0.730234\pi\)
−0.661864 + 0.749624i \(0.730234\pi\)
\(942\) 1.45298e7 0.533497
\(943\) −1.51576e7 −0.555075
\(944\) 1.57133e6 0.0573901
\(945\) 3.42630e6 0.124809
\(946\) −6.39306e6 −0.232263
\(947\) −1.70356e7 −0.617280 −0.308640 0.951179i \(-0.599874\pi\)
−0.308640 + 0.951179i \(0.599874\pi\)
\(948\) −1.11902e7 −0.404407
\(949\) −1.14884e7 −0.414089
\(950\) 9.29000e6 0.333969
\(951\) −2.44006e7 −0.874883
\(952\) −3.47725e6 −0.124349
\(953\) −1.91214e7 −0.682003 −0.341002 0.940063i \(-0.610766\pi\)
−0.341002 + 0.940063i \(0.610766\pi\)
\(954\) 1.15363e7 0.410390
\(955\) −4.08755e6 −0.145029
\(956\) −4.46003e6 −0.157831
\(957\) 5.67405e6 0.200269
\(958\) −1.02788e6 −0.0361849
\(959\) 2.71408e7 0.952963
\(960\) −921600. −0.0322749
\(961\) 1.51095e6 0.0527766
\(962\) −4.81172e7 −1.67634
\(963\) −1.31451e7 −0.456770
\(964\) 6.07872e6 0.210678
\(965\) −1.12426e7 −0.388643
\(966\) −1.15124e7 −0.396937
\(967\) 5.27790e7 1.81508 0.907538 0.419969i \(-0.137959\pi\)
0.907538 + 0.419969i \(0.137959\pi\)
\(968\) −9.14086e6 −0.313544
\(969\) 2.41633e6 0.0826697
\(970\) 1.05240e6 0.0359130
\(971\) 4.11049e7 1.39909 0.699546 0.714588i \(-0.253386\pi\)
0.699546 + 0.714588i \(0.253386\pi\)
\(972\) −944784. −0.0320750
\(973\) 3.11482e7 1.05475
\(974\) 5.90356e6 0.199396
\(975\) −2.50425e7 −0.843657
\(976\) 5.04013e6 0.169362
\(977\) −6.94601e6 −0.232809 −0.116404 0.993202i \(-0.537137\pi\)
−0.116404 + 0.993202i \(0.537137\pi\)
\(978\) 1.60736e7 0.537362
\(979\) −1.57259e7 −0.524395
\(980\) 7.41480e6 0.246623
\(981\) 1.01655e7 0.337253
\(982\) 7.54662e6 0.249732
\(983\) −3.57573e7 −1.18027 −0.590134 0.807306i \(-0.700925\pi\)
−0.590134 + 0.807306i \(0.700925\pi\)
\(984\) −5.13274e6 −0.168990
\(985\) 1.13058e7 0.371289
\(986\) −5.39852e6 −0.176841
\(987\) −1.45140e7 −0.474235
\(988\) 1.65436e7 0.539186
\(989\) 2.01381e7 0.654679
\(990\) 1.09350e6 0.0354594
\(991\) 4.09547e7 1.32471 0.662353 0.749192i \(-0.269558\pi\)
0.662353 + 0.749192i \(0.269558\pi\)
\(992\) −5.62176e6 −0.181382
\(993\) 215685. 0.00694140
\(994\) −3.63938e7 −1.16832
\(995\) −930700. −0.0298025
\(996\) 1.55892e7 0.497937
\(997\) 2.10064e7 0.669288 0.334644 0.942345i \(-0.391384\pi\)
0.334644 + 0.942345i \(0.391384\pi\)
\(998\) −4.02192e7 −1.27822
\(999\) −7.87903e6 −0.249781
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 102.6.a.d.1.1 1
3.2 odd 2 306.6.a.a.1.1 1
4.3 odd 2 816.6.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
102.6.a.d.1.1 1 1.1 even 1 trivial
306.6.a.a.1.1 1 3.2 odd 2
816.6.a.d.1.1 1 4.3 odd 2