Properties

Label 338.6.a.c.1.1
Level $338$
Weight $6$
Character 338.1
Self dual yes
Analytic conductor $54.210$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

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: \(1\)
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 \(-0.650775\pi\)
−0.456158 + 0.889899i \(0.650775\pi\)
\(6\) −52.0000 −0.589692
\(7\) 105.000 0.809924 0.404962 0.914334i \(-0.367285\pi\)
0.404962 + 0.914334i \(0.367285\pi\)
\(8\) 64.0000 0.353553
\(9\) −74.0000 −0.304527
\(10\) −204.000 −0.645105
\(11\) 120.000 0.299020 0.149510 0.988760i \(-0.452230\pi\)
0.149510 + 0.988760i \(0.452230\pi\)
\(12\) −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 \(-0.378752\pi\)
0.371768 + 0.928326i \(0.378752\pi\)
\(20\) −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 \(-0.855949\pi\)
−0.899335 + 0.437261i \(0.855949\pi\)
\(32\) 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 \(-0.324843\pi\)
0.522918 + 0.852383i \(0.324843\pi\)
\(38\) 4680.00 0.525759
\(39\) 0 0
\(40\) −3264.00 −0.322552
\(41\) 9480.00 0.880742 0.440371 0.897816i \(-0.354847\pi\)
0.440371 + 0.897816i \(0.354847\pi\)
\(42\) −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 \(-0.530978\pi\)
−0.0971663 + 0.995268i \(0.530978\pi\)
\(48\) −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 \(-0.777527\pi\)
−0.765538 + 0.643391i \(0.777527\pi\)
\(60\) 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 \(-0.600469\pi\)
−0.310418 + 0.950600i \(0.600469\pi\)
\(68\) 17616.0 0.461993
\(69\) 13650.0 0.345152
\(70\) −21420.0 −0.522486
\(71\) −63741.0 −1.50063 −0.750314 0.661082i \(-0.770098\pi\)
−0.750314 + 0.661082i \(0.770098\pi\)
\(72\) −4736.00 −0.107666
\(73\) 58866.0 1.29288 0.646439 0.762966i \(-0.276258\pi\)
0.646439 + 0.762966i \(0.276258\pi\)
\(74\) 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 \(-0.645665\pi\)
−0.441813 + 0.897107i \(0.645665\pi\)
\(84\) −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 \(-0.747297\pi\)
−0.701076 + 0.713087i \(0.747297\pi\)
\(90\) 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 \(-0.833150\pi\)
−0.865737 + 0.500500i \(0.833150\pi\)
\(98\) −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 \(-0.698435\pi\)
−0.583802 + 0.811896i \(0.698435\pi\)
\(110\) −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 \(-0.546340\pi\)
−0.145066 + 0.989422i \(0.546340\pi\)
\(138\) 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 \(-0.329640\pi\)
0.510015 + 0.860165i \(0.329640\pi\)
\(150\) 27248.0 0.0988796
\(151\) −321333. −1.14687 −0.573433 0.819252i \(-0.694389\pi\)
−0.573433 + 0.819252i \(0.694389\pi\)
\(152\) 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 \(-0.698236\pi\)
−0.583293 + 0.812262i \(0.698236\pi\)
\(164\) 151680. 0.440371
\(165\) 79560.0 0.227502
\(166\) −221832. −0.624819
\(167\) 426708. 1.18397 0.591984 0.805950i \(-0.298345\pi\)
0.591984 + 0.805950i \(0.298345\pi\)
\(168\) −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 \(-0.609112\pi\)
−0.336112 + 0.941822i \(0.609112\pi\)
\(194\) −641808. −1.22434
\(195\) 0 0
\(196\) −92512.0 −0.172012
\(197\) −899589. −1.65150 −0.825750 0.564036i \(-0.809248\pi\)
−0.825750 + 0.564036i \(0.809248\pi\)
\(198\) −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 \(-0.637961\pi\)
−0.419975 + 0.907536i \(0.637961\pi\)
\(224\) 107520. 0.143176
\(225\) 38776.0 0.0510630
\(226\) 401064. 0.522327
\(227\) 177612. 0.228775 0.114387 0.993436i \(-0.463510\pi\)
0.114387 + 0.993436i \(0.463510\pi\)
\(228\) −243360. −0.310036
\(229\) −1.18705e6 −1.49582 −0.747911 0.663799i \(-0.768943\pi\)
−0.747911 + 0.663799i \(0.768943\pi\)
\(230\) 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 \(-0.262697\pi\)
0.678346 + 0.734743i \(0.262697\pi\)
\(240\) 169728. 0.190207
\(241\) 1.16629e6 1.29349 0.646744 0.762707i \(-0.276130\pi\)
0.646744 + 0.762707i \(0.276130\pi\)
\(242\) −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 \(-0.438609\pi\)
0.191673 + 0.981459i \(0.438609\pi\)
\(272\) 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 \(-0.505907\pi\)
−0.0185558 + 0.999828i \(0.505907\pi\)
\(282\) 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 \(-0.523683\pi\)
−0.0743325 + 0.997234i \(0.523683\pi\)
\(294\) 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 \(-0.469004\pi\)
0.0972226 + 0.995263i \(0.469004\pi\)
\(308\) 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 \(-0.506540\pi\)
−0.0205454 + 0.999789i \(0.506540\pi\)
\(318\) 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 \(-0.550731\pi\)
−0.158703 + 0.987326i \(0.550731\pi\)
\(332\) −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 \(-0.439033\pi\)
0.190365 + 0.981713i \(0.439033\pi\)
\(350\) −220080. −0.0960308
\(351\) 0 0
\(352\) 122880. 0.0528597
\(353\) −1.66291e6 −0.710282 −0.355141 0.934813i \(-0.615567\pi\)
−0.355141 + 0.934813i \(0.615567\pi\)
\(354\) 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 \(-0.459118\pi\)
0.128081 + 0.991764i \(0.459118\pi\)
\(360\) 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 \(-0.549638\pi\)
−0.155310 + 0.987866i \(0.549638\pi\)
\(380\) −954720. −0.339170
\(381\) −2.63580e6 −0.930251
\(382\) 2.83272e6 0.993220
\(383\) 1.07972e6 0.376108 0.188054 0.982159i \(-0.439782\pi\)
0.188054 + 0.982159i \(0.439782\pi\)
\(384\) −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 \(-0.836386\pi\)
−0.870781 + 0.491672i \(0.836386\pi\)
\(398\) −572464. −0.181151
\(399\) −1.59705e6 −0.502211
\(400\) −134144. −0.0419200
\(401\) −1.58612e6 −0.492577 −0.246289 0.969196i \(-0.579211\pi\)
−0.246289 + 0.969196i \(0.579211\pi\)
\(402\) 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 \(-0.901068\pi\)
−0.952088 + 0.305825i \(0.901068\pi\)
\(410\) −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 \(-0.566722\pi\)
−0.208082 + 0.978111i \(0.566722\pi\)
\(422\) −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 \(-0.440593\pi\)
0.185552 + 0.982634i \(0.440593\pi\)
\(432\) 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 \(-0.447442\pi\)
0.164365 + 0.986400i \(0.447442\pi\)
\(450\) 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 \(-0.452477\pi\)
0.148743 + 0.988876i \(0.452477\pi\)
\(458\) −4.74820e6 −1.05771
\(459\) 4.53722e6 1.00521
\(460\) 856800. 0.188793
\(461\) −5.89070e6 −1.29096 −0.645482 0.763775i \(-0.723344\pi\)
−0.645482 + 0.763775i \(0.723344\pi\)
\(462\) −655200. −0.142813
\(463\) 2.37139e6 0.514104 0.257052 0.966398i \(-0.417249\pi\)
0.257052 + 0.966398i \(0.417249\pi\)
\(464\) −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 \(-0.243186\pi\)
0.722079 + 0.691810i \(0.243186\pi\)
\(480\) 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 \(-0.422182\pi\)
0.242043 + 0.970265i \(0.422182\pi\)
\(488\) −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 \(-0.443803\pi\)
0.175633 + 0.984456i \(0.443803\pi\)
\(500\) 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 \(-0.153019\pi\)
0.886661 + 0.462420i \(0.153019\pi\)
\(510\) 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 \(-0.446828\pi\)
0.166268 + 0.986081i \(0.446828\pi\)
\(542\) 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 \(-0.494055\pi\)
0.0186767 + 0.999826i \(0.494055\pi\)
\(558\) 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 \(-0.466706\pi\)
0.104405 + 0.994535i \(0.466706\pi\)
\(578\) −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 \(-0.665587\pi\)
−0.497059 + 0.867717i \(0.665587\pi\)
\(588\) 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 \(-0.584432\pi\)
−0.262150 + 0.965027i \(0.584432\pi\)
\(594\) 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 \(-0.690404\pi\)
−0.563134 + 0.826366i \(0.690404\pi\)
\(614\) 1.28441e6 0.137493
\(615\) 6.28524e6 0.670091
\(616\) 806400. 0.0856246
\(617\) 1.79106e7 1.89407 0.947036 0.321128i \(-0.104062\pi\)
0.947036 + 0.321128i \(0.104062\pi\)
\(618\) 1.30239e6 0.137174
\(619\) −4.43222e6 −0.464938 −0.232469 0.972604i \(-0.574680\pi\)
−0.232469 + 0.972604i \(0.574680\pi\)
\(620\) 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 \(-0.245818\pi\)
0.716335 + 0.697756i \(0.245818\pi\)
\(632\) 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 \(-0.765312\pi\)
−0.740291 + 0.672286i \(0.765312\pi\)
\(644\) −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 \(-0.706472\pi\)
−0.604112 + 0.796900i \(0.706472\pi\)
\(662\) −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 \(-0.392065\pi\)
0.332627 + 0.943059i \(0.392065\pi\)
\(684\) −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 \(-0.793474\pi\)
−0.796797 + 0.604246i \(0.793474\pi\)
\(692\) −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 \(-0.224864\pi\)
0.760684 + 0.649122i \(0.224864\pi\)
\(710\) 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 \(-0.329976\pi\)
0.509105 + 0.860704i \(0.329976\pi\)
\(734\) 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 \(-0.561183\pi\)
−0.191031 + 0.981584i \(0.561183\pi\)
\(740\) −7.10654e6 −0.477067
\(741\) 0 0
\(742\) −315000. −0.0210039
\(743\) 2.75704e7 1.83219 0.916095 0.400960i \(-0.131323\pi\)
0.916095 + 0.400960i \(0.131323\pi\)
\(744\) 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 \(-0.486354\pi\)
0.0428586 + 0.999081i \(0.486354\pi\)
\(762\) −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 \(-0.392805\pi\)
0.330433 + 0.943829i \(0.392805\pi\)
\(770\) −2.57040e6 −0.156233
\(771\) −1.22965e7 −0.744982
\(772\) −5.56579e6 −0.336112
\(773\) 2.05445e7 1.23665 0.618325 0.785922i \(-0.287812\pi\)
0.618325 + 0.785922i \(0.287812\pi\)
\(774\) 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 \(-0.626639\pi\)
−0.387435 + 0.921897i \(0.626639\pi\)
\(788\) −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 \(-0.263608\pi\)
0.676241 + 0.736681i \(0.263608\pi\)
\(812\) −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 \(-0.394300\pi\)
0.325996 + 0.945371i \(0.394300\pi\)
\(822\) 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 \(-0.225604\pi\)
0.759171 + 0.650891i \(0.225604\pi\)
\(828\) 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 \(-0.616759\pi\)
−0.358639 + 0.933476i \(0.616759\pi\)
\(840\) 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 \(-0.618048\pi\)
−0.362416 + 0.932016i \(0.618048\pi\)
\(854\) −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 \(-0.856094\pi\)
−0.899535 + 0.436849i \(0.856094\pi\)
\(864\) 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 \(-0.722970\pi\)
−0.644585 + 0.764532i \(0.722970\pi\)
\(878\) −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 \(-0.691217\pi\)
−0.565243 + 0.824925i \(0.691217\pi\)
\(930\) −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 \(-0.551804\pi\)
−0.162029 + 0.986786i \(0.551804\pi\)
\(942\) −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 \(-0.447815\pi\)
0.163210 + 0.986591i \(0.447815\pi\)
\(948\) −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 \(-0.491613\pi\)
0.0263446 + 0.999653i \(0.491613\pi\)
\(968\) −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 \(-0.607521\pi\)
−0.331399 + 0.943491i \(0.607521\pi\)
\(978\) 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 \(-0.416567\pi\)
0.259121 + 0.965845i \(0.416567\pi\)
\(984\) −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.c.1.1 1
13.5 odd 4 26.6.b.a.25.1 2
13.8 odd 4 26.6.b.a.25.2 yes 2
13.12 even 2 338.6.a.a.1.1 1
39.5 even 4 234.6.b.b.181.2 2
39.8 even 4 234.6.b.b.181.1 2
52.31 even 4 208.6.f.b.129.2 2
52.47 even 4 208.6.f.b.129.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
26.6.b.a.25.1 2 13.5 odd 4
26.6.b.a.25.2 yes 2 13.8 odd 4
208.6.f.b.129.1 2 52.47 even 4
208.6.f.b.129.2 2 52.31 even 4
234.6.b.b.181.1 2 39.8 even 4
234.6.b.b.181.2 2 39.5 even 4
338.6.a.a.1.1 1 13.12 even 2
338.6.a.c.1.1 1 1.1 even 1 trivial