Properties

Label 338.6.a.a.1.1
Level $338$
Weight $6$
Character 338.1
Self dual yes
Analytic conductor $54.210$
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] = [338,6,Mod(1,338)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(338, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("338.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 338 = 2 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 338.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: \(54.2097310968\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 26)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 338.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -13.0000 q^{3} +16.0000 q^{4} +51.0000 q^{5} +52.0000 q^{6} -105.000 q^{7} -64.0000 q^{8} -74.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -13.0000 q^{3} +16.0000 q^{4} +51.0000 q^{5} +52.0000 q^{6} -105.000 q^{7} -64.0000 q^{8} -74.0000 q^{9} -204.000 q^{10} -120.000 q^{11} -208.000 q^{12} +420.000 q^{14} -663.000 q^{15} +256.000 q^{16} +1101.00 q^{17} +296.000 q^{18} -1170.00 q^{19} +816.000 q^{20} +1365.00 q^{21} +480.000 q^{22} -1050.00 q^{23} +832.000 q^{24} -524.000 q^{25} +4121.00 q^{27} -1680.00 q^{28} -4104.00 q^{29} +2652.00 q^{30} +9624.00 q^{31} -1024.00 q^{32} +1560.00 q^{33} -4404.00 q^{34} -5355.00 q^{35} -1184.00 q^{36} -8709.00 q^{37} +4680.00 q^{38} -3264.00 q^{40} -9480.00 q^{41} -5460.00 q^{42} -9995.00 q^{43} -1920.00 q^{44} -3774.00 q^{45} +4200.00 q^{46} +2943.00 q^{47} -3328.00 q^{48} -5782.00 q^{49} +2096.00 q^{50} -14313.0 q^{51} -750.000 q^{53} -16484.0 q^{54} -6120.00 q^{55} +6720.00 q^{56} +15210.0 q^{57} +16416.0 q^{58} +40938.0 q^{59} -10608.0 q^{60} -57920.0 q^{61} -38496.0 q^{62} +7770.00 q^{63} +4096.00 q^{64} -6240.00 q^{66} +22812.0 q^{67} +17616.0 q^{68} +13650.0 q^{69} +21420.0 q^{70} +63741.0 q^{71} +4736.00 q^{72} -58866.0 q^{73} +34836.0 q^{74} +6812.00 q^{75} -18720.0 q^{76} +12600.0 q^{77} +63202.0 q^{79} +13056.0 q^{80} -35591.0 q^{81} +37920.0 q^{82} +55458.0 q^{83} +21840.0 q^{84} +56151.0 q^{85} +39980.0 q^{86} +53352.0 q^{87} +7680.00 q^{88} +104778. q^{89} +15096.0 q^{90} -16800.0 q^{92} -125112. q^{93} -11772.0 q^{94} -59670.0 q^{95} +13312.0 q^{96} +160452. q^{97} +23128.0 q^{98} +8880.00 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) −13.0000 −0.833950 −0.416975 0.908918i \(-0.636910\pi\)
−0.416975 + 0.908918i \(0.636910\pi\)
\(4\) 16.0000 0.500000
\(5\) 51.0000 0.912316 0.456158 0.889899i \(-0.349225\pi\)
0.456158 + 0.889899i \(0.349225\pi\)
\(6\) 52.0000 0.589692
\(7\) −105.000 −0.809924 −0.404962 0.914334i \(-0.632715\pi\)
−0.404962 + 0.914334i \(0.632715\pi\)
\(8\) −64.0000 −0.353553
\(9\) −74.0000 −0.304527
\(10\) −204.000 −0.645105
\(11\) −120.000 −0.299020 −0.149510 0.988760i \(-0.547770\pi\)
−0.149510 + 0.988760i \(0.547770\pi\)
\(12\) −208.000 −0.416975
\(13\) 0 0
\(14\) 420.000 0.572703
\(15\) −663.000 −0.760826
\(16\) 256.000 0.250000
\(17\) 1101.00 0.923985 0.461993 0.886884i \(-0.347135\pi\)
0.461993 + 0.886884i \(0.347135\pi\)
\(18\) 296.000 0.215333
\(19\) −1170.00 −0.743536 −0.371768 0.928326i \(-0.621248\pi\)
−0.371768 + 0.928326i \(0.621248\pi\)
\(20\) 816.000 0.456158
\(21\) 1365.00 0.675436
\(22\) 480.000 0.211439
\(23\) −1050.00 −0.413875 −0.206938 0.978354i \(-0.566350\pi\)
−0.206938 + 0.978354i \(0.566350\pi\)
\(24\) 832.000 0.294846
\(25\) −524.000 −0.167680
\(26\) 0 0
\(27\) 4121.00 1.08791
\(28\) −1680.00 −0.404962
\(29\) −4104.00 −0.906176 −0.453088 0.891466i \(-0.649678\pi\)
−0.453088 + 0.891466i \(0.649678\pi\)
\(30\) 2652.00 0.537985
\(31\) 9624.00 1.79867 0.899335 0.437261i \(-0.144051\pi\)
0.899335 + 0.437261i \(0.144051\pi\)
\(32\) −1024.00 −0.176777
\(33\) 1560.00 0.249367
\(34\) −4404.00 −0.653356
\(35\) −5355.00 −0.738906
\(36\) −1184.00 −0.152263
\(37\) −8709.00 −1.04584 −0.522918 0.852383i \(-0.675157\pi\)
−0.522918 + 0.852383i \(0.675157\pi\)
\(38\) 4680.00 0.525759
\(39\) 0 0
\(40\) −3264.00 −0.322552
\(41\) −9480.00 −0.880742 −0.440371 0.897816i \(-0.645153\pi\)
−0.440371 + 0.897816i \(0.645153\pi\)
\(42\) −5460.00 −0.477606
\(43\) −9995.00 −0.824350 −0.412175 0.911105i \(-0.635231\pi\)
−0.412175 + 0.911105i \(0.635231\pi\)
\(44\) −1920.00 −0.149510
\(45\) −3774.00 −0.277825
\(46\) 4200.00 0.292654
\(47\) 2943.00 0.194333 0.0971663 0.995268i \(-0.469022\pi\)
0.0971663 + 0.995268i \(0.469022\pi\)
\(48\) −3328.00 −0.208488
\(49\) −5782.00 −0.344023
\(50\) 2096.00 0.118568
\(51\) −14313.0 −0.770558
\(52\) 0 0
\(53\) −750.000 −0.0366751 −0.0183376 0.999832i \(-0.505837\pi\)
−0.0183376 + 0.999832i \(0.505837\pi\)
\(54\) −16484.0 −0.769269
\(55\) −6120.00 −0.272800
\(56\) 6720.00 0.286351
\(57\) 15210.0 0.620072
\(58\) 16416.0 0.640763
\(59\) 40938.0 1.53108 0.765538 0.643391i \(-0.222473\pi\)
0.765538 + 0.643391i \(0.222473\pi\)
\(60\) −10608.0 −0.380413
\(61\) −57920.0 −1.99298 −0.996492 0.0836839i \(-0.973331\pi\)
−0.996492 + 0.0836839i \(0.973331\pi\)
\(62\) −38496.0 −1.27185
\(63\) 7770.00 0.246643
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) −6240.00 −0.176329
\(67\) 22812.0 0.620835 0.310418 0.950600i \(-0.399531\pi\)
0.310418 + 0.950600i \(0.399531\pi\)
\(68\) 17616.0 0.461993
\(69\) 13650.0 0.345152
\(70\) 21420.0 0.522486
\(71\) 63741.0 1.50063 0.750314 0.661082i \(-0.229902\pi\)
0.750314 + 0.661082i \(0.229902\pi\)
\(72\) 4736.00 0.107666
\(73\) −58866.0 −1.29288 −0.646439 0.762966i \(-0.723742\pi\)
−0.646439 + 0.762966i \(0.723742\pi\)
\(74\) 34836.0 0.739518
\(75\) 6812.00 0.139837
\(76\) −18720.0 −0.371768
\(77\) 12600.0 0.242183
\(78\) 0 0
\(79\) 63202.0 1.13937 0.569683 0.821865i \(-0.307066\pi\)
0.569683 + 0.821865i \(0.307066\pi\)
\(80\) 13056.0 0.228079
\(81\) −35591.0 −0.602737
\(82\) 37920.0 0.622779
\(83\) 55458.0 0.883627 0.441813 0.897107i \(-0.354335\pi\)
0.441813 + 0.897107i \(0.354335\pi\)
\(84\) 21840.0 0.337718
\(85\) 56151.0 0.842966
\(86\) 39980.0 0.582903
\(87\) 53352.0 0.755705
\(88\) 7680.00 0.105719
\(89\) 104778. 1.40215 0.701076 0.713087i \(-0.252703\pi\)
0.701076 + 0.713087i \(0.252703\pi\)
\(90\) 15096.0 0.196452
\(91\) 0 0
\(92\) −16800.0 −0.206938
\(93\) −125112. −1.50000
\(94\) −11772.0 −0.137414
\(95\) −59670.0 −0.678339
\(96\) 13312.0 0.147423
\(97\) 160452. 1.73147 0.865737 0.500500i \(-0.166850\pi\)
0.865737 + 0.500500i \(0.166850\pi\)
\(98\) 23128.0 0.243261
\(99\) 8880.00 0.0910594
\(100\) −8384.00 −0.0838400
\(101\) 113124. 1.10345 0.551723 0.834027i \(-0.313970\pi\)
0.551723 + 0.834027i \(0.313970\pi\)
\(102\) 57252.0 0.544867
\(103\) −25046.0 −0.232619 −0.116310 0.993213i \(-0.537106\pi\)
−0.116310 + 0.993213i \(0.537106\pi\)
\(104\) 0 0
\(105\) 69615.0 0.616211
\(106\) 3000.00 0.0259332
\(107\) 24924.0 0.210455 0.105227 0.994448i \(-0.466443\pi\)
0.105227 + 0.994448i \(0.466443\pi\)
\(108\) 65936.0 0.543955
\(109\) 144831. 1.16760 0.583802 0.811896i \(-0.301565\pi\)
0.583802 + 0.811896i \(0.301565\pi\)
\(110\) 24480.0 0.192899
\(111\) 113217. 0.872176
\(112\) −26880.0 −0.202481
\(113\) 100266. 0.738682 0.369341 0.929294i \(-0.379583\pi\)
0.369341 + 0.929294i \(0.379583\pi\)
\(114\) −60840.0 −0.438457
\(115\) −53550.0 −0.377585
\(116\) −65664.0 −0.453088
\(117\) 0 0
\(118\) −163752. −1.08263
\(119\) −115605. −0.748358
\(120\) 42432.0 0.268993
\(121\) −146651. −0.910587
\(122\) 231680. 1.40925
\(123\) 123240. 0.734495
\(124\) 153984. 0.899335
\(125\) −186099. −1.06529
\(126\) −31080.0 −0.174403
\(127\) 202754. 1.11548 0.557738 0.830017i \(-0.311669\pi\)
0.557738 + 0.830017i \(0.311669\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 129935. 0.687467
\(130\) 0 0
\(131\) −303855. −1.54699 −0.773496 0.633801i \(-0.781494\pi\)
−0.773496 + 0.633801i \(0.781494\pi\)
\(132\) 24960.0 0.124684
\(133\) 122850. 0.602207
\(134\) −91248.0 −0.438997
\(135\) 210171. 0.992518
\(136\) −70464.0 −0.326678
\(137\) 63738.0 0.290133 0.145066 0.989422i \(-0.453660\pi\)
0.145066 + 0.989422i \(0.453660\pi\)
\(138\) −54600.0 −0.244059
\(139\) 13841.0 0.0607618 0.0303809 0.999538i \(-0.490328\pi\)
0.0303809 + 0.999538i \(0.490328\pi\)
\(140\) −85680.0 −0.369453
\(141\) −38259.0 −0.162064
\(142\) −254964. −1.06110
\(143\) 0 0
\(144\) −18944.0 −0.0761317
\(145\) −209304. −0.826718
\(146\) 235464. 0.914202
\(147\) 75166.0 0.286898
\(148\) −139344. −0.522918
\(149\) −276426. −1.02003 −0.510015 0.860165i \(-0.670360\pi\)
−0.510015 + 0.860165i \(0.670360\pi\)
\(150\) −27248.0 −0.0988796
\(151\) 321333. 1.14687 0.573433 0.819252i \(-0.305611\pi\)
0.573433 + 0.819252i \(0.305611\pi\)
\(152\) 74880.0 0.262880
\(153\) −81474.0 −0.281378
\(154\) −50400.0 −0.171249
\(155\) 490824. 1.64095
\(156\) 0 0
\(157\) 339506. 1.09925 0.549627 0.835410i \(-0.314770\pi\)
0.549627 + 0.835410i \(0.314770\pi\)
\(158\) −252808. −0.805653
\(159\) 9750.00 0.0305852
\(160\) −52224.0 −0.161276
\(161\) 110250. 0.335208
\(162\) 142364. 0.426199
\(163\) 395718. 1.16659 0.583293 0.812262i \(-0.301764\pi\)
0.583293 + 0.812262i \(0.301764\pi\)
\(164\) −151680. −0.440371
\(165\) 79560.0 0.227502
\(166\) −221832. −0.624819
\(167\) −426708. −1.18397 −0.591984 0.805950i \(-0.701655\pi\)
−0.591984 + 0.805950i \(0.701655\pi\)
\(168\) −87360.0 −0.238803
\(169\) 0 0
\(170\) −224604. −0.596067
\(171\) 86580.0 0.226427
\(172\) −159920. −0.412175
\(173\) −16026.0 −0.0407108 −0.0203554 0.999793i \(-0.506480\pi\)
−0.0203554 + 0.999793i \(0.506480\pi\)
\(174\) −213408. −0.534364
\(175\) 55020.0 0.135808
\(176\) −30720.0 −0.0747549
\(177\) −532194. −1.27684
\(178\) −419112. −0.991471
\(179\) 690045. 1.60970 0.804850 0.593479i \(-0.202246\pi\)
0.804850 + 0.593479i \(0.202246\pi\)
\(180\) −60384.0 −0.138912
\(181\) 96478.0 0.218893 0.109446 0.993993i \(-0.465092\pi\)
0.109446 + 0.993993i \(0.465092\pi\)
\(182\) 0 0
\(183\) 752960. 1.66205
\(184\) 67200.0 0.146327
\(185\) −444159. −0.954134
\(186\) 500448. 1.06066
\(187\) −132120. −0.276290
\(188\) 47088.0 0.0971663
\(189\) −432705. −0.881125
\(190\) 238680. 0.479658
\(191\) 708180. 1.40462 0.702312 0.711869i \(-0.252151\pi\)
0.702312 + 0.711869i \(0.252151\pi\)
\(192\) −53248.0 −0.104244
\(193\) 347862. 0.672224 0.336112 0.941822i \(-0.390888\pi\)
0.336112 + 0.941822i \(0.390888\pi\)
\(194\) −641808. −1.22434
\(195\) 0 0
\(196\) −92512.0 −0.172012
\(197\) 899589. 1.65150 0.825750 0.564036i \(-0.190752\pi\)
0.825750 + 0.564036i \(0.190752\pi\)
\(198\) −35520.0 −0.0643887
\(199\) −143116. −0.256186 −0.128093 0.991762i \(-0.540886\pi\)
−0.128093 + 0.991762i \(0.540886\pi\)
\(200\) 33536.0 0.0592838
\(201\) −296556. −0.517746
\(202\) −452496. −0.780255
\(203\) 430920. 0.733933
\(204\) −229008. −0.385279
\(205\) −483480. −0.803515
\(206\) 100184. 0.164487
\(207\) 77700.0 0.126036
\(208\) 0 0
\(209\) 140400. 0.222332
\(210\) −278460. −0.435727
\(211\) −339731. −0.525326 −0.262663 0.964888i \(-0.584601\pi\)
−0.262663 + 0.964888i \(0.584601\pi\)
\(212\) −12000.0 −0.0183376
\(213\) −828633. −1.25145
\(214\) −99696.0 −0.148814
\(215\) −509745. −0.752068
\(216\) −263744. −0.384634
\(217\) −1.01052e6 −1.45679
\(218\) −579324. −0.825620
\(219\) 765258. 1.07820
\(220\) −97920.0 −0.136400
\(221\) 0 0
\(222\) −452868. −0.616722
\(223\) 623757. 0.839950 0.419975 0.907536i \(-0.362039\pi\)
0.419975 + 0.907536i \(0.362039\pi\)
\(224\) 107520. 0.143176
\(225\) 38776.0 0.0510630
\(226\) −401064. −0.522327
\(227\) −177612. −0.228775 −0.114387 0.993436i \(-0.536490\pi\)
−0.114387 + 0.993436i \(0.536490\pi\)
\(228\) 243360. 0.310036
\(229\) 1.18705e6 1.49582 0.747911 0.663799i \(-0.231057\pi\)
0.747911 + 0.663799i \(0.231057\pi\)
\(230\) 214200. 0.266993
\(231\) −163800. −0.201969
\(232\) 262656. 0.320381
\(233\) 112317. 0.135536 0.0677682 0.997701i \(-0.478412\pi\)
0.0677682 + 0.997701i \(0.478412\pi\)
\(234\) 0 0
\(235\) 150093. 0.177293
\(236\) 655008. 0.765538
\(237\) −821626. −0.950174
\(238\) 462420. 0.529169
\(239\) −1.19805e6 −1.35669 −0.678346 0.734743i \(-0.737303\pi\)
−0.678346 + 0.734743i \(0.737303\pi\)
\(240\) −169728. −0.190207
\(241\) −1.16629e6 −1.29349 −0.646744 0.762707i \(-0.723870\pi\)
−0.646744 + 0.762707i \(0.723870\pi\)
\(242\) 586604. 0.643882
\(243\) −538720. −0.585258
\(244\) −926720. −0.996492
\(245\) −294882. −0.313858
\(246\) −492960. −0.519366
\(247\) 0 0
\(248\) −615936. −0.635926
\(249\) −720954. −0.736901
\(250\) 744396. 0.753276
\(251\) −648996. −0.650216 −0.325108 0.945677i \(-0.605401\pi\)
−0.325108 + 0.945677i \(0.605401\pi\)
\(252\) 124320. 0.123322
\(253\) 126000. 0.123757
\(254\) −811016. −0.788760
\(255\) −729963. −0.702992
\(256\) 65536.0 0.0625000
\(257\) 945885. 0.893317 0.446658 0.894705i \(-0.352614\pi\)
0.446658 + 0.894705i \(0.352614\pi\)
\(258\) −519740. −0.486113
\(259\) 914445. 0.847048
\(260\) 0 0
\(261\) 303696. 0.275955
\(262\) 1.21542e6 1.09389
\(263\) 1.01222e6 0.902375 0.451188 0.892429i \(-0.351000\pi\)
0.451188 + 0.892429i \(0.351000\pi\)
\(264\) −99840.0 −0.0881647
\(265\) −38250.0 −0.0334593
\(266\) −491400. −0.425825
\(267\) −1.36211e6 −1.16933
\(268\) 364992. 0.310418
\(269\) −1.01772e6 −0.857527 −0.428763 0.903417i \(-0.641051\pi\)
−0.428763 + 0.903417i \(0.641051\pi\)
\(270\) −840684. −0.701816
\(271\) −463461. −0.383345 −0.191673 0.981459i \(-0.561391\pi\)
−0.191673 + 0.981459i \(0.561391\pi\)
\(272\) 281856. 0.230996
\(273\) 0 0
\(274\) −254952. −0.205155
\(275\) 62880.0 0.0501396
\(276\) 218400. 0.172576
\(277\) −332528. −0.260393 −0.130196 0.991488i \(-0.541561\pi\)
−0.130196 + 0.991488i \(0.541561\pi\)
\(278\) −55364.0 −0.0429651
\(279\) −712176. −0.547743
\(280\) 342720. 0.261243
\(281\) 49122.0 0.0371116 0.0185558 0.999828i \(-0.494093\pi\)
0.0185558 + 0.999828i \(0.494093\pi\)
\(282\) 153036. 0.114596
\(283\) 1.55848e6 1.15674 0.578371 0.815774i \(-0.303689\pi\)
0.578371 + 0.815774i \(0.303689\pi\)
\(284\) 1.01986e6 0.750314
\(285\) 775710. 0.565701
\(286\) 0 0
\(287\) 995400. 0.713334
\(288\) 75776.0 0.0538332
\(289\) −207656. −0.146251
\(290\) 837216. 0.584578
\(291\) −2.08588e6 −1.44396
\(292\) −941856. −0.646439
\(293\) 218463. 0.148665 0.0743325 0.997234i \(-0.476317\pi\)
0.0743325 + 0.997234i \(0.476317\pi\)
\(294\) −300664. −0.202868
\(295\) 2.08784e6 1.39682
\(296\) 557376. 0.369759
\(297\) −494520. −0.325306
\(298\) 1.10570e6 0.721271
\(299\) 0 0
\(300\) 108992. 0.0699184
\(301\) 1.04948e6 0.667661
\(302\) −1.28533e6 −0.810957
\(303\) −1.47061e6 −0.920220
\(304\) −299520. −0.185884
\(305\) −2.95392e6 −1.81823
\(306\) 325896. 0.198964
\(307\) −321102. −0.194445 −0.0972226 0.995263i \(-0.530996\pi\)
−0.0972226 + 0.995263i \(0.530996\pi\)
\(308\) 201600. 0.121092
\(309\) 325598. 0.193993
\(310\) −1.96330e6 −1.16033
\(311\) −3.33725e6 −1.95654 −0.978269 0.207340i \(-0.933519\pi\)
−0.978269 + 0.207340i \(0.933519\pi\)
\(312\) 0 0
\(313\) 1.16568e6 0.672538 0.336269 0.941766i \(-0.390835\pi\)
0.336269 + 0.941766i \(0.390835\pi\)
\(314\) −1.35802e6 −0.777290
\(315\) 396270. 0.225017
\(316\) 1.01123e6 0.569683
\(317\) 73518.0 0.0410909 0.0205454 0.999789i \(-0.493460\pi\)
0.0205454 + 0.999789i \(0.493460\pi\)
\(318\) −39000.0 −0.0216270
\(319\) 492480. 0.270964
\(320\) 208896. 0.114039
\(321\) −324012. −0.175509
\(322\) −441000. −0.237028
\(323\) −1.28817e6 −0.687016
\(324\) −569456. −0.301368
\(325\) 0 0
\(326\) −1.58287e6 −0.824901
\(327\) −1.88280e6 −0.973723
\(328\) 606720. 0.311389
\(329\) −309015. −0.157395
\(330\) −318240. −0.160868
\(331\) 632682. 0.317406 0.158703 0.987326i \(-0.449269\pi\)
0.158703 + 0.987326i \(0.449269\pi\)
\(332\) 887328. 0.441813
\(333\) 644466. 0.318485
\(334\) 1.70683e6 0.837191
\(335\) 1.16341e6 0.566398
\(336\) 349440. 0.168859
\(337\) 326843. 0.156771 0.0783853 0.996923i \(-0.475024\pi\)
0.0783853 + 0.996923i \(0.475024\pi\)
\(338\) 0 0
\(339\) −1.30346e6 −0.616024
\(340\) 898416. 0.421483
\(341\) −1.15488e6 −0.537837
\(342\) −346320. −0.160108
\(343\) 2.37184e6 1.08856
\(344\) 639680. 0.291452
\(345\) 696150. 0.314887
\(346\) 64104.0 0.0287869
\(347\) 2.96275e6 1.32090 0.660452 0.750868i \(-0.270365\pi\)
0.660452 + 0.750868i \(0.270365\pi\)
\(348\) 853632. 0.377853
\(349\) −866325. −0.380730 −0.190365 0.981713i \(-0.560967\pi\)
−0.190365 + 0.981713i \(0.560967\pi\)
\(350\) −220080. −0.0960308
\(351\) 0 0
\(352\) 122880. 0.0528597
\(353\) 1.66291e6 0.710282 0.355141 0.934813i \(-0.384433\pi\)
0.355141 + 0.934813i \(0.384433\pi\)
\(354\) 2.12878e6 0.902863
\(355\) 3.25079e6 1.36905
\(356\) 1.67645e6 0.701076
\(357\) 1.50286e6 0.624093
\(358\) −2.76018e6 −1.13823
\(359\) −625536. −0.256163 −0.128081 0.991764i \(-0.540882\pi\)
−0.128081 + 0.991764i \(0.540882\pi\)
\(360\) 241536. 0.0982258
\(361\) −1.10720e6 −0.447155
\(362\) −385912. −0.154781
\(363\) 1.90646e6 0.759385
\(364\) 0 0
\(365\) −3.00217e6 −1.17951
\(366\) −3.01184e6 −1.17525
\(367\) 1.08327e6 0.419829 0.209914 0.977720i \(-0.432681\pi\)
0.209914 + 0.977720i \(0.432681\pi\)
\(368\) −268800. −0.103469
\(369\) 701520. 0.268209
\(370\) 1.77664e6 0.674674
\(371\) 78750.0 0.0297041
\(372\) −2.00179e6 −0.750001
\(373\) −1.78896e6 −0.665775 −0.332888 0.942967i \(-0.608023\pi\)
−0.332888 + 0.942967i \(0.608023\pi\)
\(374\) 528480. 0.195366
\(375\) 2.41929e6 0.888401
\(376\) −188352. −0.0687069
\(377\) 0 0
\(378\) 1.73082e6 0.623049
\(379\) 868614. 0.310620 0.155310 0.987866i \(-0.450362\pi\)
0.155310 + 0.987866i \(0.450362\pi\)
\(380\) −954720. −0.339170
\(381\) −2.63580e6 −0.930251
\(382\) −2.83272e6 −0.993220
\(383\) −1.07972e6 −0.376108 −0.188054 0.982159i \(-0.560218\pi\)
−0.188054 + 0.982159i \(0.560218\pi\)
\(384\) 212992. 0.0737115
\(385\) 642600. 0.220947
\(386\) −1.39145e6 −0.475334
\(387\) 739630. 0.251037
\(388\) 2.56723e6 0.865737
\(389\) −1.28822e6 −0.431634 −0.215817 0.976434i \(-0.569241\pi\)
−0.215817 + 0.976434i \(0.569241\pi\)
\(390\) 0 0
\(391\) −1.15605e6 −0.382415
\(392\) 370048. 0.121631
\(393\) 3.95012e6 1.29011
\(394\) −3.59836e6 −1.16779
\(395\) 3.22330e6 1.03946
\(396\) 142080. 0.0455297
\(397\) 5.46909e6 1.74156 0.870781 0.491672i \(-0.163614\pi\)
0.870781 + 0.491672i \(0.163614\pi\)
\(398\) 572464. 0.181151
\(399\) −1.59705e6 −0.502211
\(400\) −134144. −0.0419200
\(401\) 1.58612e6 0.492577 0.246289 0.969196i \(-0.420789\pi\)
0.246289 + 0.969196i \(0.420789\pi\)
\(402\) 1.18622e6 0.366102
\(403\) 0 0
\(404\) 1.80998e6 0.551723
\(405\) −1.81514e6 −0.549886
\(406\) −1.72368e6 −0.518969
\(407\) 1.04508e6 0.312726
\(408\) 916032. 0.272433
\(409\) 6.44192e6 1.90418 0.952088 0.305825i \(-0.0989324\pi\)
0.952088 + 0.305825i \(0.0989324\pi\)
\(410\) 1.93392e6 0.568171
\(411\) −828594. −0.241956
\(412\) −400736. −0.116310
\(413\) −4.29849e6 −1.24005
\(414\) −310800. −0.0891210
\(415\) 2.82836e6 0.806147
\(416\) 0 0
\(417\) −179933. −0.0506723
\(418\) −561600. −0.157212
\(419\) −4.30545e6 −1.19807 −0.599037 0.800721i \(-0.704450\pi\)
−0.599037 + 0.800721i \(0.704450\pi\)
\(420\) 1.11384e6 0.308106
\(421\) 1.51346e6 0.416164 0.208082 0.978111i \(-0.433278\pi\)
0.208082 + 0.978111i \(0.433278\pi\)
\(422\) 1.35892e6 0.371462
\(423\) −217782. −0.0591795
\(424\) 48000.0 0.0129666
\(425\) −576924. −0.154934
\(426\) 3.31453e6 0.884908
\(427\) 6.08160e6 1.61417
\(428\) 398784. 0.105227
\(429\) 0 0
\(430\) 2.03898e6 0.531792
\(431\) −1.43116e6 −0.371105 −0.185552 0.982634i \(-0.559407\pi\)
−0.185552 + 0.982634i \(0.559407\pi\)
\(432\) 1.05498e6 0.271978
\(433\) −429613. −0.110118 −0.0550589 0.998483i \(-0.517535\pi\)
−0.0550589 + 0.998483i \(0.517535\pi\)
\(434\) 4.04208e6 1.03010
\(435\) 2.72095e6 0.689442
\(436\) 2.31730e6 0.583802
\(437\) 1.22850e6 0.307731
\(438\) −3.06103e6 −0.762399
\(439\) −552038. −0.136712 −0.0683562 0.997661i \(-0.521775\pi\)
−0.0683562 + 0.997661i \(0.521775\pi\)
\(440\) 391680. 0.0964494
\(441\) 427868. 0.104764
\(442\) 0 0
\(443\) 2.15255e6 0.521128 0.260564 0.965457i \(-0.416092\pi\)
0.260564 + 0.965457i \(0.416092\pi\)
\(444\) 1.81147e6 0.436088
\(445\) 5.34368e6 1.27921
\(446\) −2.49503e6 −0.593934
\(447\) 3.59354e6 0.850655
\(448\) −430080. −0.101240
\(449\) −1.40429e6 −0.328731 −0.164365 0.986400i \(-0.552558\pi\)
−0.164365 + 0.986400i \(0.552558\pi\)
\(450\) −155104. −0.0361070
\(451\) 1.13760e6 0.263359
\(452\) 1.60426e6 0.369341
\(453\) −4.17733e6 −0.956430
\(454\) 710448. 0.161768
\(455\) 0 0
\(456\) −973440. −0.219229
\(457\) −1.32818e6 −0.297485 −0.148743 0.988876i \(-0.547523\pi\)
−0.148743 + 0.988876i \(0.547523\pi\)
\(458\) −4.74820e6 −1.05771
\(459\) 4.53722e6 1.00521
\(460\) −856800. −0.188793
\(461\) 5.89070e6 1.29096 0.645482 0.763775i \(-0.276656\pi\)
0.645482 + 0.763775i \(0.276656\pi\)
\(462\) 655200. 0.142813
\(463\) −2.37139e6 −0.514104 −0.257052 0.966398i \(-0.582751\pi\)
−0.257052 + 0.966398i \(0.582751\pi\)
\(464\) −1.05062e6 −0.226544
\(465\) −6.38071e6 −1.36847
\(466\) −449268. −0.0958387
\(467\) −7.17827e6 −1.52310 −0.761548 0.648108i \(-0.775560\pi\)
−0.761548 + 0.648108i \(0.775560\pi\)
\(468\) 0 0
\(469\) −2.39526e6 −0.502829
\(470\) −600372. −0.125365
\(471\) −4.41358e6 −0.916724
\(472\) −2.62003e6 −0.541317
\(473\) 1.19940e6 0.246497
\(474\) 3.28650e6 0.671875
\(475\) 613080. 0.124676
\(476\) −1.84968e6 −0.374179
\(477\) 55500.0 0.0111686
\(478\) 4.79221e6 0.959326
\(479\) −7.25193e6 −1.44416 −0.722079 0.691810i \(-0.756814\pi\)
−0.722079 + 0.691810i \(0.756814\pi\)
\(480\) 678912. 0.134496
\(481\) 0 0
\(482\) 4.66514e6 0.914634
\(483\) −1.43325e6 −0.279547
\(484\) −2.34642e6 −0.455294
\(485\) 8.18305e6 1.57965
\(486\) 2.15488e6 0.413840
\(487\) −2.53364e6 −0.484087 −0.242043 0.970265i \(-0.577818\pi\)
−0.242043 + 0.970265i \(0.577818\pi\)
\(488\) 3.70688e6 0.704626
\(489\) −5.14433e6 −0.972875
\(490\) 1.17953e6 0.221931
\(491\) −8.46186e6 −1.58403 −0.792013 0.610504i \(-0.790967\pi\)
−0.792013 + 0.610504i \(0.790967\pi\)
\(492\) 1.97184e6 0.367248
\(493\) −4.51850e6 −0.837293
\(494\) 0 0
\(495\) 452880. 0.0830750
\(496\) 2.46374e6 0.449667
\(497\) −6.69280e6 −1.21539
\(498\) 2.88382e6 0.521068
\(499\) −1.95383e6 −0.351265 −0.175633 0.984456i \(-0.556197\pi\)
−0.175633 + 0.984456i \(0.556197\pi\)
\(500\) −2.97758e6 −0.532646
\(501\) 5.54720e6 0.987370
\(502\) 2.59598e6 0.459772
\(503\) 119778. 0.0211085 0.0105542 0.999944i \(-0.496640\pi\)
0.0105542 + 0.999944i \(0.496640\pi\)
\(504\) −497280. −0.0872016
\(505\) 5.76932e6 1.00669
\(506\) −504000. −0.0875093
\(507\) 0 0
\(508\) 3.24406e6 0.557738
\(509\) −1.03653e7 −1.77332 −0.886661 0.462420i \(-0.846981\pi\)
−0.886661 + 0.462420i \(0.846981\pi\)
\(510\) 2.91985e6 0.497090
\(511\) 6.18093e6 1.04713
\(512\) −262144. −0.0441942
\(513\) −4.82157e6 −0.808900
\(514\) −3.78354e6 −0.631670
\(515\) −1.27735e6 −0.212222
\(516\) 2.07896e6 0.343734
\(517\) −353160. −0.0581092
\(518\) −3.65778e6 −0.598954
\(519\) 208338. 0.0339508
\(520\) 0 0
\(521\) −1.04899e7 −1.69307 −0.846537 0.532330i \(-0.821316\pi\)
−0.846537 + 0.532330i \(0.821316\pi\)
\(522\) −1.21478e6 −0.195129
\(523\) 4.42662e6 0.707649 0.353824 0.935312i \(-0.384881\pi\)
0.353824 + 0.935312i \(0.384881\pi\)
\(524\) −4.86168e6 −0.773496
\(525\) −715260. −0.113257
\(526\) −4.04890e6 −0.638076
\(527\) 1.05960e7 1.66194
\(528\) 399360. 0.0623419
\(529\) −5.33384e6 −0.828707
\(530\) 153000. 0.0236593
\(531\) −3.02941e6 −0.466253
\(532\) 1.96560e6 0.301104
\(533\) 0 0
\(534\) 5.44846e6 0.826838
\(535\) 1.27112e6 0.192001
\(536\) −1.45997e6 −0.219498
\(537\) −8.97058e6 −1.34241
\(538\) 4.07088e6 0.606363
\(539\) 693840. 0.102870
\(540\) 3.36274e6 0.496259
\(541\) −2.26377e6 −0.332536 −0.166268 0.986081i \(-0.553172\pi\)
−0.166268 + 0.986081i \(0.553172\pi\)
\(542\) 1.85384e6 0.271066
\(543\) −1.25421e6 −0.182546
\(544\) −1.12742e6 −0.163339
\(545\) 7.38638e6 1.06522
\(546\) 0 0
\(547\) 7.21090e6 1.03044 0.515218 0.857059i \(-0.327711\pi\)
0.515218 + 0.857059i \(0.327711\pi\)
\(548\) 1.01981e6 0.145066
\(549\) 4.28608e6 0.606917
\(550\) −251520. −0.0354540
\(551\) 4.80168e6 0.673774
\(552\) −873600. −0.122030
\(553\) −6.63621e6 −0.922799
\(554\) 1.33011e6 0.184125
\(555\) 5.77407e6 0.795700
\(556\) 221456. 0.0303809
\(557\) −273507. −0.0373534 −0.0186767 0.999826i \(-0.505945\pi\)
−0.0186767 + 0.999826i \(0.505945\pi\)
\(558\) 2.84870e6 0.387313
\(559\) 0 0
\(560\) −1.37088e6 −0.184727
\(561\) 1.71756e6 0.230412
\(562\) −196488. −0.0262419
\(563\) 959349. 0.127557 0.0637787 0.997964i \(-0.479685\pi\)
0.0637787 + 0.997964i \(0.479685\pi\)
\(564\) −612144. −0.0810319
\(565\) 5.11357e6 0.673911
\(566\) −6.23394e6 −0.817940
\(567\) 3.73706e6 0.488171
\(568\) −4.07942e6 −0.530552
\(569\) 1.19403e7 1.54609 0.773044 0.634352i \(-0.218733\pi\)
0.773044 + 0.634352i \(0.218733\pi\)
\(570\) −3.10284e6 −0.400011
\(571\) −7.20205e6 −0.924413 −0.462206 0.886772i \(-0.652942\pi\)
−0.462206 + 0.886772i \(0.652942\pi\)
\(572\) 0 0
\(573\) −9.20634e6 −1.17139
\(574\) −3.98160e6 −0.504403
\(575\) 550200. 0.0693986
\(576\) −303104. −0.0380658
\(577\) −1.66990e6 −0.208810 −0.104405 0.994535i \(-0.533294\pi\)
−0.104405 + 0.994535i \(0.533294\pi\)
\(578\) 830624. 0.103415
\(579\) −4.52221e6 −0.560601
\(580\) −3.34886e6 −0.413359
\(581\) −5.82309e6 −0.715671
\(582\) 8.34350e6 1.02104
\(583\) 90000.0 0.0109666
\(584\) 3.76742e6 0.457101
\(585\) 0 0
\(586\) −873852. −0.105122
\(587\) 8.29913e6 0.994117 0.497059 0.867717i \(-0.334413\pi\)
0.497059 + 0.867717i \(0.334413\pi\)
\(588\) 1.20266e6 0.143449
\(589\) −1.12601e7 −1.33738
\(590\) −8.35135e6 −0.987704
\(591\) −1.16947e7 −1.37727
\(592\) −2.22950e6 −0.261459
\(593\) 4.48969e6 0.524300 0.262150 0.965027i \(-0.415568\pi\)
0.262150 + 0.965027i \(0.415568\pi\)
\(594\) 1.97808e6 0.230026
\(595\) −5.89586e6 −0.682738
\(596\) −4.42282e6 −0.510015
\(597\) 1.86051e6 0.213646
\(598\) 0 0
\(599\) 1.38261e6 0.157446 0.0787232 0.996897i \(-0.474916\pi\)
0.0787232 + 0.996897i \(0.474916\pi\)
\(600\) −435968. −0.0494398
\(601\) 1.04021e7 1.17472 0.587359 0.809327i \(-0.300168\pi\)
0.587359 + 0.809327i \(0.300168\pi\)
\(602\) −4.19790e6 −0.472107
\(603\) −1.68809e6 −0.189061
\(604\) 5.14133e6 0.573433
\(605\) −7.47920e6 −0.830743
\(606\) 5.88245e6 0.650694
\(607\) −4.78668e6 −0.527306 −0.263653 0.964618i \(-0.584927\pi\)
−0.263653 + 0.964618i \(0.584927\pi\)
\(608\) 1.19808e6 0.131440
\(609\) −5.60196e6 −0.612064
\(610\) 1.18157e7 1.28568
\(611\) 0 0
\(612\) −1.30358e6 −0.140689
\(613\) 1.04783e7 1.12627 0.563134 0.826366i \(-0.309596\pi\)
0.563134 + 0.826366i \(0.309596\pi\)
\(614\) 1.28441e6 0.137493
\(615\) 6.28524e6 0.670091
\(616\) −806400. −0.0856246
\(617\) −1.79106e7 −1.89407 −0.947036 0.321128i \(-0.895938\pi\)
−0.947036 + 0.321128i \(0.895938\pi\)
\(618\) −1.30239e6 −0.137174
\(619\) 4.43222e6 0.464938 0.232469 0.972604i \(-0.425320\pi\)
0.232469 + 0.972604i \(0.425320\pi\)
\(620\) 7.85318e6 0.820477
\(621\) −4.32705e6 −0.450260
\(622\) 1.33490e7 1.38348
\(623\) −1.10017e7 −1.13564
\(624\) 0 0
\(625\) −7.85355e6 −0.804203
\(626\) −4.66270e6 −0.475556
\(627\) −1.82520e6 −0.185414
\(628\) 5.43210e6 0.549627
\(629\) −9.58861e6 −0.966338
\(630\) −1.58508e6 −0.159111
\(631\) −1.43291e7 −1.43267 −0.716335 0.697756i \(-0.754182\pi\)
−0.716335 + 0.697756i \(0.754182\pi\)
\(632\) −4.04493e6 −0.402827
\(633\) 4.41650e6 0.438096
\(634\) −294072. −0.0290556
\(635\) 1.03405e7 1.01767
\(636\) 156000. 0.0152926
\(637\) 0 0
\(638\) −1.96992e6 −0.191601
\(639\) −4.71683e6 −0.456981
\(640\) −835584. −0.0806381
\(641\) −6.65869e6 −0.640094 −0.320047 0.947402i \(-0.603699\pi\)
−0.320047 + 0.947402i \(0.603699\pi\)
\(642\) 1.29605e6 0.124103
\(643\) 1.55224e7 1.48058 0.740291 0.672286i \(-0.234688\pi\)
0.740291 + 0.672286i \(0.234688\pi\)
\(644\) 1.76400e6 0.167604
\(645\) 6.62668e6 0.627187
\(646\) 5.15268e6 0.485794
\(647\) 2.44454e6 0.229581 0.114791 0.993390i \(-0.463380\pi\)
0.114791 + 0.993390i \(0.463380\pi\)
\(648\) 2.27782e6 0.213100
\(649\) −4.91256e6 −0.457821
\(650\) 0 0
\(651\) 1.31368e7 1.21489
\(652\) 6.33149e6 0.583293
\(653\) 1.16500e7 1.06916 0.534580 0.845118i \(-0.320470\pi\)
0.534580 + 0.845118i \(0.320470\pi\)
\(654\) 7.53121e6 0.688526
\(655\) −1.54966e7 −1.41135
\(656\) −2.42688e6 −0.220185
\(657\) 4.35608e6 0.393716
\(658\) 1.23606e6 0.111295
\(659\) 1.33185e7 1.19465 0.597326 0.801999i \(-0.296230\pi\)
0.597326 + 0.801999i \(0.296230\pi\)
\(660\) 1.27296e6 0.113751
\(661\) 1.35722e7 1.20822 0.604112 0.796900i \(-0.293528\pi\)
0.604112 + 0.796900i \(0.293528\pi\)
\(662\) −2.53073e6 −0.224440
\(663\) 0 0
\(664\) −3.54931e6 −0.312409
\(665\) 6.26535e6 0.549403
\(666\) −2.57786e6 −0.225203
\(667\) 4.30920e6 0.375044
\(668\) −6.82733e6 −0.591984
\(669\) −8.10884e6 −0.700476
\(670\) −4.65365e6 −0.400504
\(671\) 6.95040e6 0.595941
\(672\) −1.39776e6 −0.119401
\(673\) 1.58674e7 1.35042 0.675209 0.737626i \(-0.264053\pi\)
0.675209 + 0.737626i \(0.264053\pi\)
\(674\) −1.30737e6 −0.110854
\(675\) −2.15940e6 −0.182421
\(676\) 0 0
\(677\) −2.24264e7 −1.88056 −0.940281 0.340398i \(-0.889438\pi\)
−0.940281 + 0.340398i \(0.889438\pi\)
\(678\) 5.21383e6 0.435595
\(679\) −1.68475e7 −1.40236
\(680\) −3.59366e6 −0.298034
\(681\) 2.30896e6 0.190787
\(682\) 4.61952e6 0.380308
\(683\) −8.11034e6 −0.665254 −0.332627 0.943059i \(-0.607935\pi\)
−0.332627 + 0.943059i \(0.607935\pi\)
\(684\) 1.38528e6 0.113213
\(685\) 3.25064e6 0.264693
\(686\) −9.48738e6 −0.769726
\(687\) −1.54316e7 −1.24744
\(688\) −2.55872e6 −0.206088
\(689\) 0 0
\(690\) −2.78460e6 −0.222659
\(691\) 2.00020e7 1.59359 0.796797 0.604246i \(-0.206526\pi\)
0.796797 + 0.604246i \(0.206526\pi\)
\(692\) −256416. −0.0203554
\(693\) −932400. −0.0737512
\(694\) −1.18510e7 −0.934020
\(695\) 705891. 0.0554339
\(696\) −3.41453e6 −0.267182
\(697\) −1.04375e7 −0.813793
\(698\) 3.46530e6 0.269217
\(699\) −1.46012e6 −0.113031
\(700\) 880320. 0.0679040
\(701\) 2.22272e6 0.170840 0.0854200 0.996345i \(-0.472777\pi\)
0.0854200 + 0.996345i \(0.472777\pi\)
\(702\) 0 0
\(703\) 1.01895e7 0.777617
\(704\) −491520. −0.0373774
\(705\) −1.95121e6 −0.147853
\(706\) −6.65162e6 −0.502245
\(707\) −1.18780e7 −0.893708
\(708\) −8.51510e6 −0.638420
\(709\) −2.03634e7 −1.52137 −0.760684 0.649122i \(-0.775136\pi\)
−0.760684 + 0.649122i \(0.775136\pi\)
\(710\) −1.30032e7 −0.968062
\(711\) −4.67695e6 −0.346967
\(712\) −6.70579e6 −0.495736
\(713\) −1.01052e7 −0.744425
\(714\) −6.01146e6 −0.441301
\(715\) 0 0
\(716\) 1.10407e7 0.804850
\(717\) 1.55747e7 1.13141
\(718\) 2.50214e6 0.181135
\(719\) 1.98255e7 1.43022 0.715108 0.699014i \(-0.246377\pi\)
0.715108 + 0.699014i \(0.246377\pi\)
\(720\) −966144. −0.0694561
\(721\) 2.62983e6 0.188404
\(722\) 4.42880e6 0.316186
\(723\) 1.51617e7 1.07870
\(724\) 1.54365e6 0.109446
\(725\) 2.15050e6 0.151948
\(726\) −7.62585e6 −0.536966
\(727\) 9.24667e6 0.648857 0.324429 0.945910i \(-0.394828\pi\)
0.324429 + 0.945910i \(0.394828\pi\)
\(728\) 0 0
\(729\) 1.56520e7 1.09081
\(730\) 1.20087e7 0.834041
\(731\) −1.10045e7 −0.761687
\(732\) 1.20474e7 0.831025
\(733\) −1.48114e7 −1.01821 −0.509105 0.860704i \(-0.670024\pi\)
−0.509105 + 0.860704i \(0.670024\pi\)
\(734\) −4.33309e6 −0.296864
\(735\) 3.83347e6 0.261742
\(736\) 1.07520e6 0.0731635
\(737\) −2.73744e6 −0.185642
\(738\) −2.80608e6 −0.189653
\(739\) 5.67210e6 0.382061 0.191031 0.981584i \(-0.438817\pi\)
0.191031 + 0.981584i \(0.438817\pi\)
\(740\) −7.10654e6 −0.477067
\(741\) 0 0
\(742\) −315000. −0.0210039
\(743\) −2.75704e7 −1.83219 −0.916095 0.400960i \(-0.868677\pi\)
−0.916095 + 0.400960i \(0.868677\pi\)
\(744\) 8.00717e6 0.530330
\(745\) −1.40977e7 −0.930590
\(746\) 7.15582e6 0.470774
\(747\) −4.10389e6 −0.269088
\(748\) −2.11392e6 −0.138145
\(749\) −2.61702e6 −0.170452
\(750\) −9.67715e6 −0.628195
\(751\) 4.09636e6 0.265032 0.132516 0.991181i \(-0.457694\pi\)
0.132516 + 0.991181i \(0.457694\pi\)
\(752\) 753408. 0.0485831
\(753\) 8.43695e6 0.542248
\(754\) 0 0
\(755\) 1.63880e7 1.04630
\(756\) −6.92328e6 −0.440562
\(757\) 1.09396e7 0.693844 0.346922 0.937894i \(-0.387227\pi\)
0.346922 + 0.937894i \(0.387227\pi\)
\(758\) −3.47446e6 −0.219641
\(759\) −1.63800e6 −0.103207
\(760\) 3.81888e6 0.239829
\(761\) −1.36940e6 −0.0857172 −0.0428586 0.999081i \(-0.513646\pi\)
−0.0428586 + 0.999081i \(0.513646\pi\)
\(762\) 1.05432e7 0.657787
\(763\) −1.52073e7 −0.945670
\(764\) 1.13309e7 0.702312
\(765\) −4.15517e6 −0.256706
\(766\) 4.31886e6 0.265948
\(767\) 0 0
\(768\) −851968. −0.0521219
\(769\) −1.08375e7 −0.660867 −0.330433 0.943829i \(-0.607195\pi\)
−0.330433 + 0.943829i \(0.607195\pi\)
\(770\) −2.57040e6 −0.156233
\(771\) −1.22965e7 −0.744982
\(772\) 5.56579e6 0.336112
\(773\) −2.05445e7 −1.23665 −0.618325 0.785922i \(-0.712188\pi\)
−0.618325 + 0.785922i \(0.712188\pi\)
\(774\) −2.95852e6 −0.177510
\(775\) −5.04298e6 −0.301601
\(776\) −1.02689e7 −0.612168
\(777\) −1.18878e7 −0.706396
\(778\) 5.15287e6 0.305211
\(779\) 1.10916e7 0.654863
\(780\) 0 0
\(781\) −7.64892e6 −0.448717
\(782\) 4.62420e6 0.270408
\(783\) −1.69126e7 −0.985838
\(784\) −1.48019e6 −0.0860058
\(785\) 1.73148e7 1.00287
\(786\) −1.58005e7 −0.912249
\(787\) 1.34637e7 0.774869 0.387435 0.921897i \(-0.373361\pi\)
0.387435 + 0.921897i \(0.373361\pi\)
\(788\) 1.43934e7 0.825750
\(789\) −1.31589e7 −0.752536
\(790\) −1.28932e7 −0.735010
\(791\) −1.05279e7 −0.598276
\(792\) −568320. −0.0321944
\(793\) 0 0
\(794\) −2.18764e7 −1.23147
\(795\) 497250. 0.0279034
\(796\) −2.28986e6 −0.128093
\(797\) −2.02451e7 −1.12895 −0.564475 0.825450i \(-0.690921\pi\)
−0.564475 + 0.825450i \(0.690921\pi\)
\(798\) 6.38820e6 0.355117
\(799\) 3.24024e6 0.179560
\(800\) 536576. 0.0296419
\(801\) −7.75357e6 −0.426993
\(802\) −6.34447e6 −0.348305
\(803\) 7.06392e6 0.386596
\(804\) −4.74490e6 −0.258873
\(805\) 5.62275e6 0.305815
\(806\) 0 0
\(807\) 1.32304e7 0.715135
\(808\) −7.23994e6 −0.390127
\(809\) 2.48958e7 1.33738 0.668689 0.743542i \(-0.266856\pi\)
0.668689 + 0.743542i \(0.266856\pi\)
\(810\) 7.26056e6 0.388828
\(811\) −2.53328e7 −1.35248 −0.676241 0.736681i \(-0.736392\pi\)
−0.676241 + 0.736681i \(0.736392\pi\)
\(812\) 6.89472e6 0.366967
\(813\) 6.02499e6 0.319691
\(814\) −4.18032e6 −0.221130
\(815\) 2.01816e7 1.06429
\(816\) −3.66413e6 −0.192639
\(817\) 1.16942e7 0.612934
\(818\) −2.57677e7 −1.34646
\(819\) 0 0
\(820\) −7.73568e6 −0.401757
\(821\) −1.25922e7 −0.651993 −0.325996 0.945371i \(-0.605700\pi\)
−0.325996 + 0.945371i \(0.605700\pi\)
\(822\) 3.31438e6 0.171089
\(823\) 3.32776e7 1.71259 0.856294 0.516490i \(-0.172761\pi\)
0.856294 + 0.516490i \(0.172761\pi\)
\(824\) 1.60294e6 0.0822433
\(825\) −817440. −0.0418139
\(826\) 1.71940e7 0.876851
\(827\) −2.98630e7 −1.51834 −0.759171 0.650891i \(-0.774396\pi\)
−0.759171 + 0.650891i \(0.774396\pi\)
\(828\) 1.24320e6 0.0630181
\(829\) −3.04980e7 −1.54129 −0.770647 0.637262i \(-0.780067\pi\)
−0.770647 + 0.637262i \(0.780067\pi\)
\(830\) −1.13134e7 −0.570032
\(831\) 4.32286e6 0.217155
\(832\) 0 0
\(833\) −6.36598e6 −0.317872
\(834\) 719732. 0.0358307
\(835\) −2.17621e7 −1.08015
\(836\) 2.24640e6 0.111166
\(837\) 3.96605e7 1.95679
\(838\) 1.72218e7 0.847167
\(839\) 1.46249e7 0.717278 0.358639 0.933476i \(-0.383241\pi\)
0.358639 + 0.933476i \(0.383241\pi\)
\(840\) −4.45536e6 −0.217864
\(841\) −3.66833e6 −0.178846
\(842\) −6.05382e6 −0.294272
\(843\) −638586. −0.0309493
\(844\) −5.43570e6 −0.262663
\(845\) 0 0
\(846\) 871128. 0.0418462
\(847\) 1.53984e7 0.737506
\(848\) −192000. −0.00916878
\(849\) −2.02603e7 −0.964665
\(850\) 2.30770e6 0.109555
\(851\) 9.14445e6 0.432846
\(852\) −1.32581e7 −0.625725
\(853\) 1.54032e7 0.724832 0.362416 0.932016i \(-0.381952\pi\)
0.362416 + 0.932016i \(0.381952\pi\)
\(854\) −2.43264e7 −1.14139
\(855\) 4.41558e6 0.206572
\(856\) −1.59514e6 −0.0744069
\(857\) 2.59910e7 1.20884 0.604422 0.796664i \(-0.293404\pi\)
0.604422 + 0.796664i \(0.293404\pi\)
\(858\) 0 0
\(859\) 1.26690e7 0.585815 0.292908 0.956141i \(-0.405377\pi\)
0.292908 + 0.956141i \(0.405377\pi\)
\(860\) −8.15592e6 −0.376034
\(861\) −1.29402e7 −0.594885
\(862\) 5.72466e6 0.262411
\(863\) 3.93618e7 1.79907 0.899535 0.436849i \(-0.143906\pi\)
0.899535 + 0.436849i \(0.143906\pi\)
\(864\) −4.21990e6 −0.192317
\(865\) −817326. −0.0371411
\(866\) 1.71845e6 0.0778651
\(867\) 2.69953e6 0.121966
\(868\) −1.61683e7 −0.728393
\(869\) −7.58424e6 −0.340693
\(870\) −1.08838e7 −0.487509
\(871\) 0 0
\(872\) −9.26918e6 −0.412810
\(873\) −1.18734e7 −0.527280
\(874\) −4.91400e6 −0.217599
\(875\) 1.95404e7 0.862806
\(876\) 1.22441e7 0.539098
\(877\) 2.93636e7 1.28917 0.644585 0.764532i \(-0.277030\pi\)
0.644585 + 0.764532i \(0.277030\pi\)
\(878\) 2.20815e6 0.0966702
\(879\) −2.84002e6 −0.123979
\(880\) −1.56672e6 −0.0682001
\(881\) 2.47421e7 1.07398 0.536990 0.843589i \(-0.319561\pi\)
0.536990 + 0.843589i \(0.319561\pi\)
\(882\) −1.71147e6 −0.0740796
\(883\) −1.56178e7 −0.674092 −0.337046 0.941488i \(-0.609428\pi\)
−0.337046 + 0.941488i \(0.609428\pi\)
\(884\) 0 0
\(885\) −2.71419e7 −1.16488
\(886\) −8.61020e6 −0.368493
\(887\) 1.41193e6 0.0602566 0.0301283 0.999546i \(-0.490408\pi\)
0.0301283 + 0.999546i \(0.490408\pi\)
\(888\) −7.24589e6 −0.308361
\(889\) −2.12892e7 −0.903450
\(890\) −2.13747e7 −0.904535
\(891\) 4.27092e6 0.180230
\(892\) 9.98011e6 0.419975
\(893\) −3.44331e6 −0.144493
\(894\) −1.43742e7 −0.601504
\(895\) 3.51923e7 1.46855
\(896\) 1.72032e6 0.0715878
\(897\) 0 0
\(898\) 5.61715e6 0.232448
\(899\) −3.94969e7 −1.62991
\(900\) 620416. 0.0255315
\(901\) −825750. −0.0338873
\(902\) −4.55040e6 −0.186223
\(903\) −1.36432e7 −0.556796
\(904\) −6.41702e6 −0.261164
\(905\) 4.92038e6 0.199700
\(906\) 1.67093e7 0.676298
\(907\) 1.48543e7 0.599563 0.299781 0.954008i \(-0.403086\pi\)
0.299781 + 0.954008i \(0.403086\pi\)
\(908\) −2.84179e6 −0.114387
\(909\) −8.37118e6 −0.336029
\(910\) 0 0
\(911\) 4.29118e7 1.71309 0.856547 0.516069i \(-0.172605\pi\)
0.856547 + 0.516069i \(0.172605\pi\)
\(912\) 3.89376e6 0.155018
\(913\) −6.65496e6 −0.264222
\(914\) 5.31271e6 0.210354
\(915\) 3.84010e7 1.51631
\(916\) 1.89928e7 0.747911
\(917\) 3.19048e7 1.25295
\(918\) −1.81489e7 −0.710793
\(919\) 4.34706e7 1.69788 0.848939 0.528491i \(-0.177242\pi\)
0.848939 + 0.528491i \(0.177242\pi\)
\(920\) 3.42720e6 0.133497
\(921\) 4.17433e6 0.162158
\(922\) −2.35628e7 −0.912850
\(923\) 0 0
\(924\) −2.62080e6 −0.100984
\(925\) 4.56352e6 0.175366
\(926\) 9.48557e6 0.363526
\(927\) 1.85340e6 0.0708387
\(928\) 4.20250e6 0.160191
\(929\) 2.97375e7 1.13049 0.565243 0.824925i \(-0.308783\pi\)
0.565243 + 0.824925i \(0.308783\pi\)
\(930\) 2.55228e7 0.967658
\(931\) 6.76494e6 0.255794
\(932\) 1.79707e6 0.0677682
\(933\) 4.33843e7 1.63166
\(934\) 2.87131e7 1.07699
\(935\) −6.73812e6 −0.252063
\(936\) 0 0
\(937\) 1.46550e7 0.545303 0.272651 0.962113i \(-0.412099\pi\)
0.272651 + 0.962113i \(0.412099\pi\)
\(938\) 9.58104e6 0.355554
\(939\) −1.51538e7 −0.560863
\(940\) 2.40149e6 0.0886463
\(941\) 8.80233e6 0.324059 0.162029 0.986786i \(-0.448196\pi\)
0.162029 + 0.986786i \(0.448196\pi\)
\(942\) 1.76543e7 0.648222
\(943\) 9.95400e6 0.364518
\(944\) 1.04801e7 0.382769
\(945\) −2.20680e7 −0.803864
\(946\) −4.79760e6 −0.174300
\(947\) −9.00847e6 −0.326420 −0.163210 0.986591i \(-0.552185\pi\)
−0.163210 + 0.986591i \(0.552185\pi\)
\(948\) −1.31460e7 −0.475087
\(949\) 0 0
\(950\) −2.45232e6 −0.0881593
\(951\) −955734. −0.0342678
\(952\) 7.39872e6 0.264584
\(953\) −1.31122e7 −0.467675 −0.233837 0.972276i \(-0.575128\pi\)
−0.233837 + 0.972276i \(0.575128\pi\)
\(954\) −222000. −0.00789736
\(955\) 3.61172e7 1.28146
\(956\) −1.91688e7 −0.678346
\(957\) −6.40224e6 −0.225971
\(958\) 2.90077e7 1.02117
\(959\) −6.69249e6 −0.234986
\(960\) −2.71565e6 −0.0951033
\(961\) 6.39922e7 2.23521
\(962\) 0 0
\(963\) −1.84438e6 −0.0640890
\(964\) −1.86606e7 −0.646744
\(965\) 1.77410e7 0.613280
\(966\) 5.73300e6 0.197669
\(967\) −1.53210e6 −0.0526892 −0.0263446 0.999653i \(-0.508387\pi\)
−0.0263446 + 0.999653i \(0.508387\pi\)
\(968\) 9.38566e6 0.321941
\(969\) 1.67462e7 0.572937
\(970\) −3.27322e7 −1.11698
\(971\) 2.37514e7 0.808426 0.404213 0.914665i \(-0.367545\pi\)
0.404213 + 0.914665i \(0.367545\pi\)
\(972\) −8.61952e6 −0.292629
\(973\) −1.45330e6 −0.0492124
\(974\) 1.01346e7 0.342301
\(975\) 0 0
\(976\) −1.48275e7 −0.498246
\(977\) 1.97751e7 0.662798 0.331399 0.943491i \(-0.392479\pi\)
0.331399 + 0.943491i \(0.392479\pi\)
\(978\) 2.05773e7 0.687926
\(979\) −1.25734e7 −0.419271
\(980\) −4.71811e6 −0.156929
\(981\) −1.07175e7 −0.355566
\(982\) 3.38475e7 1.12008
\(983\) −1.57006e7 −0.518241 −0.259121 0.965845i \(-0.583433\pi\)
−0.259121 + 0.965845i \(0.583433\pi\)
\(984\) −7.88736e6 −0.259683
\(985\) 4.58790e7 1.50669
\(986\) 1.80740e7 0.592055
\(987\) 4.01720e6 0.131259
\(988\) 0 0
\(989\) 1.04948e7 0.341178
\(990\) −1.81152e6 −0.0587429
\(991\) 1.65835e7 0.536406 0.268203 0.963362i \(-0.413570\pi\)
0.268203 + 0.963362i \(0.413570\pi\)
\(992\) −9.85498e6 −0.317963
\(993\) −8.22487e6 −0.264701
\(994\) 2.67712e7 0.859414
\(995\) −7.29892e6 −0.233723
\(996\) −1.15353e7 −0.368451
\(997\) −2.08432e7 −0.664091 −0.332045 0.943263i \(-0.607739\pi\)
−0.332045 + 0.943263i \(0.607739\pi\)
\(998\) 7.81531e6 0.248382
\(999\) −3.58898e7 −1.13778
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 338.6.a.a.1.1 1
13.5 odd 4 26.6.b.a.25.2 yes 2
13.8 odd 4 26.6.b.a.25.1 2
13.12 even 2 338.6.a.c.1.1 1
39.5 even 4 234.6.b.b.181.1 2
39.8 even 4 234.6.b.b.181.2 2
52.31 even 4 208.6.f.b.129.1 2
52.47 even 4 208.6.f.b.129.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.6.b.a.25.1 2 13.8 odd 4
26.6.b.a.25.2 yes 2 13.5 odd 4
208.6.f.b.129.1 2 52.31 even 4
208.6.f.b.129.2 2 52.47 even 4
234.6.b.b.181.1 2 39.5 even 4
234.6.b.b.181.2 2 39.8 even 4
338.6.a.a.1.1 1 1.1 even 1 trivial
338.6.a.c.1.1 1 13.12 even 2