Properties

Label 150.6.a.i.1.1
Level $150$
Weight $6$
Character 150.1
Self dual yes
Analytic conductor $24.058$
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] = [150,6,Mod(1,150)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(150, 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("150.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 150 = 2 \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 150.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: \(24.0575729719\)
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 \(\chi\) \(=\) 150.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -36.0000 q^{6} -79.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -36.0000 q^{6} -79.0000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +150.000 q^{11} -144.000 q^{12} +137.000 q^{13} -316.000 q^{14} +256.000 q^{16} -2034.00 q^{17} +324.000 q^{18} -1969.00 q^{19} +711.000 q^{21} +600.000 q^{22} -1350.00 q^{23} -576.000 q^{24} +548.000 q^{26} -729.000 q^{27} -1264.00 q^{28} -2946.00 q^{29} +713.000 q^{31} +1024.00 q^{32} -1350.00 q^{33} -8136.00 q^{34} +1296.00 q^{36} -3238.00 q^{37} -7876.00 q^{38} -1233.00 q^{39} +6564.00 q^{41} +2844.00 q^{42} -19579.0 q^{43} +2400.00 q^{44} -5400.00 q^{46} -21150.0 q^{47} -2304.00 q^{48} -10566.0 q^{49} +18306.0 q^{51} +2192.00 q^{52} -25896.0 q^{53} -2916.00 q^{54} -5056.00 q^{56} +17721.0 q^{57} -11784.0 q^{58} +25350.0 q^{59} +50615.0 q^{61} +2852.00 q^{62} -6399.00 q^{63} +4096.00 q^{64} -5400.00 q^{66} -22519.0 q^{67} -32544.0 q^{68} +12150.0 q^{69} +33900.0 q^{71} +5184.00 q^{72} +82442.0 q^{73} -12952.0 q^{74} -31504.0 q^{76} -11850.0 q^{77} -4932.00 q^{78} -81472.0 q^{79} +6561.00 q^{81} +26256.0 q^{82} +25782.0 q^{83} +11376.0 q^{84} -78316.0 q^{86} +26514.0 q^{87} +9600.00 q^{88} +103728. q^{89} -10823.0 q^{91} -21600.0 q^{92} -6417.00 q^{93} -84600.0 q^{94} -9216.00 q^{96} -57343.0 q^{97} -42264.0 q^{98} +12150.0 q^{99} +O(q^{100})\)

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
\(5\) 0 0
\(6\) −36.0000 −0.408248
\(7\) −79.0000 −0.609371 −0.304686 0.952453i \(-0.598551\pi\)
−0.304686 + 0.952453i \(0.598551\pi\)
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) 150.000 0.373774 0.186887 0.982381i \(-0.440160\pi\)
0.186887 + 0.982381i \(0.440160\pi\)
\(12\) −144.000 −0.288675
\(13\) 137.000 0.224834 0.112417 0.993661i \(-0.464141\pi\)
0.112417 + 0.993661i \(0.464141\pi\)
\(14\) −316.000 −0.430891
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −2034.00 −1.70698 −0.853490 0.521109i \(-0.825519\pi\)
−0.853490 + 0.521109i \(0.825519\pi\)
\(18\) 324.000 0.235702
\(19\) −1969.00 −1.25130 −0.625650 0.780104i \(-0.715166\pi\)
−0.625650 + 0.780104i \(0.715166\pi\)
\(20\) 0 0
\(21\) 711.000 0.351821
\(22\) 600.000 0.264298
\(23\) −1350.00 −0.532126 −0.266063 0.963956i \(-0.585723\pi\)
−0.266063 + 0.963956i \(0.585723\pi\)
\(24\) −576.000 −0.204124
\(25\) 0 0
\(26\) 548.000 0.158982
\(27\) −729.000 −0.192450
\(28\) −1264.00 −0.304686
\(29\) −2946.00 −0.650486 −0.325243 0.945631i \(-0.605446\pi\)
−0.325243 + 0.945631i \(0.605446\pi\)
\(30\) 0 0
\(31\) 713.000 0.133256 0.0666278 0.997778i \(-0.478776\pi\)
0.0666278 + 0.997778i \(0.478776\pi\)
\(32\) 1024.00 0.176777
\(33\) −1350.00 −0.215799
\(34\) −8136.00 −1.20702
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) −3238.00 −0.388841 −0.194421 0.980918i \(-0.562283\pi\)
−0.194421 + 0.980918i \(0.562283\pi\)
\(38\) −7876.00 −0.884803
\(39\) −1233.00 −0.129808
\(40\) 0 0
\(41\) 6564.00 0.609830 0.304915 0.952380i \(-0.401372\pi\)
0.304915 + 0.952380i \(0.401372\pi\)
\(42\) 2844.00 0.248775
\(43\) −19579.0 −1.61480 −0.807401 0.590003i \(-0.799127\pi\)
−0.807401 + 0.590003i \(0.799127\pi\)
\(44\) 2400.00 0.186887
\(45\) 0 0
\(46\) −5400.00 −0.376270
\(47\) −21150.0 −1.39658 −0.698290 0.715815i \(-0.746055\pi\)
−0.698290 + 0.715815i \(0.746055\pi\)
\(48\) −2304.00 −0.144338
\(49\) −10566.0 −0.628667
\(50\) 0 0
\(51\) 18306.0 0.985526
\(52\) 2192.00 0.112417
\(53\) −25896.0 −1.26632 −0.633159 0.774021i \(-0.718242\pi\)
−0.633159 + 0.774021i \(0.718242\pi\)
\(54\) −2916.00 −0.136083
\(55\) 0 0
\(56\) −5056.00 −0.215445
\(57\) 17721.0 0.722439
\(58\) −11784.0 −0.459963
\(59\) 25350.0 0.948086 0.474043 0.880502i \(-0.342794\pi\)
0.474043 + 0.880502i \(0.342794\pi\)
\(60\) 0 0
\(61\) 50615.0 1.74163 0.870813 0.491615i \(-0.163593\pi\)
0.870813 + 0.491615i \(0.163593\pi\)
\(62\) 2852.00 0.0942259
\(63\) −6399.00 −0.203124
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) −5400.00 −0.152593
\(67\) −22519.0 −0.612861 −0.306431 0.951893i \(-0.599135\pi\)
−0.306431 + 0.951893i \(0.599135\pi\)
\(68\) −32544.0 −0.853490
\(69\) 12150.0 0.307223
\(70\) 0 0
\(71\) 33900.0 0.798094 0.399047 0.916931i \(-0.369341\pi\)
0.399047 + 0.916931i \(0.369341\pi\)
\(72\) 5184.00 0.117851
\(73\) 82442.0 1.81068 0.905339 0.424689i \(-0.139617\pi\)
0.905339 + 0.424689i \(0.139617\pi\)
\(74\) −12952.0 −0.274952
\(75\) 0 0
\(76\) −31504.0 −0.625650
\(77\) −11850.0 −0.227767
\(78\) −4932.00 −0.0917881
\(79\) −81472.0 −1.46873 −0.734363 0.678757i \(-0.762519\pi\)
−0.734363 + 0.678757i \(0.762519\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) 26256.0 0.431215
\(83\) 25782.0 0.410791 0.205396 0.978679i \(-0.434152\pi\)
0.205396 + 0.978679i \(0.434152\pi\)
\(84\) 11376.0 0.175910
\(85\) 0 0
\(86\) −78316.0 −1.14184
\(87\) 26514.0 0.375558
\(88\) 9600.00 0.132149
\(89\) 103728. 1.38810 0.694050 0.719926i \(-0.255824\pi\)
0.694050 + 0.719926i \(0.255824\pi\)
\(90\) 0 0
\(91\) −10823.0 −0.137007
\(92\) −21600.0 −0.266063
\(93\) −6417.00 −0.0769351
\(94\) −84600.0 −0.987531
\(95\) 0 0
\(96\) −9216.00 −0.102062
\(97\) −57343.0 −0.618801 −0.309401 0.950932i \(-0.600128\pi\)
−0.309401 + 0.950932i \(0.600128\pi\)
\(98\) −42264.0 −0.444534
\(99\) 12150.0 0.124591
\(100\) 0 0
\(101\) −91032.0 −0.887954 −0.443977 0.896038i \(-0.646433\pi\)
−0.443977 + 0.896038i \(0.646433\pi\)
\(102\) 73224.0 0.696872
\(103\) 191636. 1.77985 0.889926 0.456104i \(-0.150756\pi\)
0.889926 + 0.456104i \(0.150756\pi\)
\(104\) 8768.00 0.0794909
\(105\) 0 0
\(106\) −103584. −0.895423
\(107\) 9288.00 0.0784265 0.0392132 0.999231i \(-0.487515\pi\)
0.0392132 + 0.999231i \(0.487515\pi\)
\(108\) −11664.0 −0.0962250
\(109\) −20635.0 −0.166356 −0.0831780 0.996535i \(-0.526507\pi\)
−0.0831780 + 0.996535i \(0.526507\pi\)
\(110\) 0 0
\(111\) 29142.0 0.224498
\(112\) −20224.0 −0.152343
\(113\) 176892. 1.30320 0.651602 0.758561i \(-0.274097\pi\)
0.651602 + 0.758561i \(0.274097\pi\)
\(114\) 70884.0 0.510841
\(115\) 0 0
\(116\) −47136.0 −0.325243
\(117\) 11097.0 0.0749447
\(118\) 101400. 0.670398
\(119\) 160686. 1.04019
\(120\) 0 0
\(121\) −138551. −0.860293
\(122\) 202460. 1.23151
\(123\) −59076.0 −0.352086
\(124\) 11408.0 0.0666278
\(125\) 0 0
\(126\) −25596.0 −0.143630
\(127\) −256480. −1.41106 −0.705528 0.708682i \(-0.749290\pi\)
−0.705528 + 0.708682i \(0.749290\pi\)
\(128\) 16384.0 0.0883883
\(129\) 176211. 0.932307
\(130\) 0 0
\(131\) 177600. 0.904200 0.452100 0.891967i \(-0.350675\pi\)
0.452100 + 0.891967i \(0.350675\pi\)
\(132\) −21600.0 −0.107899
\(133\) 155551. 0.762507
\(134\) −90076.0 −0.433358
\(135\) 0 0
\(136\) −130176. −0.603509
\(137\) 260886. 1.18754 0.593772 0.804634i \(-0.297638\pi\)
0.593772 + 0.804634i \(0.297638\pi\)
\(138\) 48600.0 0.217239
\(139\) −217684. −0.955629 −0.477815 0.878461i \(-0.658571\pi\)
−0.477815 + 0.878461i \(0.658571\pi\)
\(140\) 0 0
\(141\) 190350. 0.806316
\(142\) 135600. 0.564337
\(143\) 20550.0 0.0840372
\(144\) 20736.0 0.0833333
\(145\) 0 0
\(146\) 329768. 1.28034
\(147\) 95094.0 0.362961
\(148\) −51808.0 −0.194421
\(149\) 421794. 1.55645 0.778224 0.627987i \(-0.216121\pi\)
0.778224 + 0.627987i \(0.216121\pi\)
\(150\) 0 0
\(151\) −101917. −0.363751 −0.181876 0.983322i \(-0.558217\pi\)
−0.181876 + 0.983322i \(0.558217\pi\)
\(152\) −126016. −0.442402
\(153\) −164754. −0.568994
\(154\) −47400.0 −0.161056
\(155\) 0 0
\(156\) −19728.0 −0.0649040
\(157\) −101773. −0.329521 −0.164761 0.986334i \(-0.552685\pi\)
−0.164761 + 0.986334i \(0.552685\pi\)
\(158\) −325888. −1.03855
\(159\) 233064. 0.731109
\(160\) 0 0
\(161\) 106650. 0.324262
\(162\) 26244.0 0.0785674
\(163\) −202249. −0.596235 −0.298117 0.954529i \(-0.596359\pi\)
−0.298117 + 0.954529i \(0.596359\pi\)
\(164\) 105024. 0.304915
\(165\) 0 0
\(166\) 103128. 0.290473
\(167\) 231600. 0.642610 0.321305 0.946976i \(-0.395879\pi\)
0.321305 + 0.946976i \(0.395879\pi\)
\(168\) 45504.0 0.124387
\(169\) −352524. −0.949450
\(170\) 0 0
\(171\) −159489. −0.417100
\(172\) −313264. −0.807401
\(173\) −174222. −0.442576 −0.221288 0.975209i \(-0.571026\pi\)
−0.221288 + 0.975209i \(0.571026\pi\)
\(174\) 106056. 0.265560
\(175\) 0 0
\(176\) 38400.0 0.0934436
\(177\) −228150. −0.547378
\(178\) 414912. 0.981535
\(179\) −642066. −1.49778 −0.748888 0.662696i \(-0.769412\pi\)
−0.748888 + 0.662696i \(0.769412\pi\)
\(180\) 0 0
\(181\) −56071.0 −0.127216 −0.0636080 0.997975i \(-0.520261\pi\)
−0.0636080 + 0.997975i \(0.520261\pi\)
\(182\) −43292.0 −0.0968789
\(183\) −455535. −1.00553
\(184\) −86400.0 −0.188135
\(185\) 0 0
\(186\) −25668.0 −0.0544013
\(187\) −305100. −0.638026
\(188\) −338400. −0.698290
\(189\) 57591.0 0.117274
\(190\) 0 0
\(191\) −209694. −0.415913 −0.207957 0.978138i \(-0.566681\pi\)
−0.207957 + 0.978138i \(0.566681\pi\)
\(192\) −36864.0 −0.0721688
\(193\) −715597. −1.38285 −0.691425 0.722448i \(-0.743017\pi\)
−0.691425 + 0.722448i \(0.743017\pi\)
\(194\) −229372. −0.437558
\(195\) 0 0
\(196\) −169056. −0.314333
\(197\) 508854. 0.934174 0.467087 0.884211i \(-0.345304\pi\)
0.467087 + 0.884211i \(0.345304\pi\)
\(198\) 48600.0 0.0880995
\(199\) −986017. −1.76503 −0.882514 0.470286i \(-0.844151\pi\)
−0.882514 + 0.470286i \(0.844151\pi\)
\(200\) 0 0
\(201\) 202671. 0.353836
\(202\) −364128. −0.627879
\(203\) 232734. 0.396387
\(204\) 292896. 0.492763
\(205\) 0 0
\(206\) 766544. 1.25855
\(207\) −109350. −0.177375
\(208\) 35072.0 0.0562085
\(209\) −295350. −0.467704
\(210\) 0 0
\(211\) 119495. 0.184775 0.0923876 0.995723i \(-0.470550\pi\)
0.0923876 + 0.995723i \(0.470550\pi\)
\(212\) −414336. −0.633159
\(213\) −305100. −0.460780
\(214\) 37152.0 0.0554559
\(215\) 0 0
\(216\) −46656.0 −0.0680414
\(217\) −56327.0 −0.0812021
\(218\) −82540.0 −0.117631
\(219\) −741978. −1.04540
\(220\) 0 0
\(221\) −278658. −0.383788
\(222\) 116568. 0.158744
\(223\) 48545.0 0.0653706 0.0326853 0.999466i \(-0.489594\pi\)
0.0326853 + 0.999466i \(0.489594\pi\)
\(224\) −80896.0 −0.107723
\(225\) 0 0
\(226\) 707568. 0.921504
\(227\) −287652. −0.370512 −0.185256 0.982690i \(-0.559311\pi\)
−0.185256 + 0.982690i \(0.559311\pi\)
\(228\) 283536. 0.361219
\(229\) 72065.0 0.0908104 0.0454052 0.998969i \(-0.485542\pi\)
0.0454052 + 0.998969i \(0.485542\pi\)
\(230\) 0 0
\(231\) 106650. 0.131502
\(232\) −188544. −0.229981
\(233\) −569148. −0.686808 −0.343404 0.939188i \(-0.611580\pi\)
−0.343404 + 0.939188i \(0.611580\pi\)
\(234\) 44388.0 0.0529939
\(235\) 0 0
\(236\) 405600. 0.474043
\(237\) 733248. 0.847969
\(238\) 642744. 0.735522
\(239\) 696504. 0.788731 0.394365 0.918954i \(-0.370964\pi\)
0.394365 + 0.918954i \(0.370964\pi\)
\(240\) 0 0
\(241\) 576137. 0.638974 0.319487 0.947591i \(-0.396489\pi\)
0.319487 + 0.947591i \(0.396489\pi\)
\(242\) −554204. −0.608319
\(243\) −59049.0 −0.0641500
\(244\) 809840. 0.870813
\(245\) 0 0
\(246\) −236304. −0.248962
\(247\) −269753. −0.281335
\(248\) 45632.0 0.0471130
\(249\) −232038. −0.237171
\(250\) 0 0
\(251\) −1.22492e6 −1.22723 −0.613613 0.789607i \(-0.710285\pi\)
−0.613613 + 0.789607i \(0.710285\pi\)
\(252\) −102384. −0.101562
\(253\) −202500. −0.198895
\(254\) −1.02592e6 −0.997767
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 972132. 0.918105 0.459053 0.888409i \(-0.348189\pi\)
0.459053 + 0.888409i \(0.348189\pi\)
\(258\) 704844. 0.659240
\(259\) 255802. 0.236949
\(260\) 0 0
\(261\) −238626. −0.216829
\(262\) 710400. 0.639366
\(263\) 1.76190e6 1.57070 0.785348 0.619055i \(-0.212484\pi\)
0.785348 + 0.619055i \(0.212484\pi\)
\(264\) −86400.0 −0.0762964
\(265\) 0 0
\(266\) 622204. 0.539174
\(267\) −933552. −0.801420
\(268\) −360304. −0.306431
\(269\) 1.74383e6 1.46935 0.734674 0.678421i \(-0.237335\pi\)
0.734674 + 0.678421i \(0.237335\pi\)
\(270\) 0 0
\(271\) −1.70426e6 −1.40965 −0.704826 0.709381i \(-0.748975\pi\)
−0.704826 + 0.709381i \(0.748975\pi\)
\(272\) −520704. −0.426745
\(273\) 97407.0 0.0791013
\(274\) 1.04354e6 0.839720
\(275\) 0 0
\(276\) 194400. 0.153611
\(277\) 1.46972e6 1.15089 0.575446 0.817840i \(-0.304829\pi\)
0.575446 + 0.817840i \(0.304829\pi\)
\(278\) −870736. −0.675732
\(279\) 57753.0 0.0444185
\(280\) 0 0
\(281\) −1.22530e6 −0.925715 −0.462858 0.886433i \(-0.653176\pi\)
−0.462858 + 0.886433i \(0.653176\pi\)
\(282\) 761400. 0.570151
\(283\) −637333. −0.473043 −0.236521 0.971626i \(-0.576007\pi\)
−0.236521 + 0.971626i \(0.576007\pi\)
\(284\) 542400. 0.399047
\(285\) 0 0
\(286\) 82200.0 0.0594233
\(287\) −518556. −0.371613
\(288\) 82944.0 0.0589256
\(289\) 2.71730e6 1.91378
\(290\) 0 0
\(291\) 516087. 0.357265
\(292\) 1.31907e6 0.905339
\(293\) 35094.0 0.0238816 0.0119408 0.999929i \(-0.496199\pi\)
0.0119408 + 0.999929i \(0.496199\pi\)
\(294\) 380376. 0.256652
\(295\) 0 0
\(296\) −207232. −0.137476
\(297\) −109350. −0.0719329
\(298\) 1.68718e6 1.10058
\(299\) −184950. −0.119640
\(300\) 0 0
\(301\) 1.54674e6 0.984014
\(302\) −407668. −0.257211
\(303\) 819288. 0.512661
\(304\) −504064. −0.312825
\(305\) 0 0
\(306\) −659016. −0.402339
\(307\) 2.88040e6 1.74424 0.872120 0.489293i \(-0.162745\pi\)
0.872120 + 0.489293i \(0.162745\pi\)
\(308\) −189600. −0.113884
\(309\) −1.72472e6 −1.02760
\(310\) 0 0
\(311\) −2.38305e6 −1.39712 −0.698558 0.715554i \(-0.746174\pi\)
−0.698558 + 0.715554i \(0.746174\pi\)
\(312\) −78912.0 −0.0458941
\(313\) −1.93081e6 −1.11399 −0.556993 0.830517i \(-0.688045\pi\)
−0.556993 + 0.830517i \(0.688045\pi\)
\(314\) −407092. −0.233007
\(315\) 0 0
\(316\) −1.30355e6 −0.734363
\(317\) −2.18995e6 −1.22401 −0.612007 0.790852i \(-0.709638\pi\)
−0.612007 + 0.790852i \(0.709638\pi\)
\(318\) 932256. 0.516972
\(319\) −441900. −0.243135
\(320\) 0 0
\(321\) −83592.0 −0.0452796
\(322\) 426600. 0.229288
\(323\) 4.00495e6 2.13595
\(324\) 104976. 0.0555556
\(325\) 0 0
\(326\) −808996. −0.421602
\(327\) 185715. 0.0960456
\(328\) 420096. 0.215608
\(329\) 1.67085e6 0.851036
\(330\) 0 0
\(331\) 1.41429e6 0.709527 0.354764 0.934956i \(-0.384561\pi\)
0.354764 + 0.934956i \(0.384561\pi\)
\(332\) 412512. 0.205396
\(333\) −262278. −0.129614
\(334\) 926400. 0.454394
\(335\) 0 0
\(336\) 182016. 0.0879552
\(337\) 1.38208e6 0.662916 0.331458 0.943470i \(-0.392459\pi\)
0.331458 + 0.943470i \(0.392459\pi\)
\(338\) −1.41010e6 −0.671362
\(339\) −1.59203e6 −0.752405
\(340\) 0 0
\(341\) 106950. 0.0498075
\(342\) −637956. −0.294934
\(343\) 2.16247e6 0.992463
\(344\) −1.25306e6 −0.570919
\(345\) 0 0
\(346\) −696888. −0.312948
\(347\) −283758. −0.126510 −0.0632549 0.997997i \(-0.520148\pi\)
−0.0632549 + 0.997997i \(0.520148\pi\)
\(348\) 424224. 0.187779
\(349\) 2.13809e6 0.939642 0.469821 0.882762i \(-0.344318\pi\)
0.469821 + 0.882762i \(0.344318\pi\)
\(350\) 0 0
\(351\) −99873.0 −0.0432693
\(352\) 153600. 0.0660746
\(353\) 3.16015e6 1.34980 0.674901 0.737908i \(-0.264186\pi\)
0.674901 + 0.737908i \(0.264186\pi\)
\(354\) −912600. −0.387055
\(355\) 0 0
\(356\) 1.65965e6 0.694050
\(357\) −1.44617e6 −0.600551
\(358\) −2.56826e6 −1.05909
\(359\) −3.59564e6 −1.47245 −0.736225 0.676737i \(-0.763393\pi\)
−0.736225 + 0.676737i \(0.763393\pi\)
\(360\) 0 0
\(361\) 1.40086e6 0.565754
\(362\) −224284. −0.0899553
\(363\) 1.24696e6 0.496690
\(364\) −173168. −0.0685037
\(365\) 0 0
\(366\) −1.82214e6 −0.711015
\(367\) −4.62043e6 −1.79068 −0.895339 0.445385i \(-0.853067\pi\)
−0.895339 + 0.445385i \(0.853067\pi\)
\(368\) −345600. −0.133031
\(369\) 531684. 0.203277
\(370\) 0 0
\(371\) 2.04578e6 0.771658
\(372\) −102672. −0.0384676
\(373\) −3.51983e6 −1.30993 −0.654967 0.755657i \(-0.727318\pi\)
−0.654967 + 0.755657i \(0.727318\pi\)
\(374\) −1.22040e6 −0.451152
\(375\) 0 0
\(376\) −1.35360e6 −0.493765
\(377\) −403602. −0.146251
\(378\) 230364. 0.0829249
\(379\) 595061. 0.212796 0.106398 0.994324i \(-0.466068\pi\)
0.106398 + 0.994324i \(0.466068\pi\)
\(380\) 0 0
\(381\) 2.30832e6 0.814673
\(382\) −838776. −0.294095
\(383\) −1.30050e6 −0.453016 −0.226508 0.974009i \(-0.572731\pi\)
−0.226508 + 0.974009i \(0.572731\pi\)
\(384\) −147456. −0.0510310
\(385\) 0 0
\(386\) −2.86239e6 −0.977823
\(387\) −1.58590e6 −0.538267
\(388\) −917488. −0.309401
\(389\) 620724. 0.207981 0.103991 0.994578i \(-0.466839\pi\)
0.103991 + 0.994578i \(0.466839\pi\)
\(390\) 0 0
\(391\) 2.74590e6 0.908328
\(392\) −676224. −0.222267
\(393\) −1.59840e6 −0.522040
\(394\) 2.03542e6 0.660561
\(395\) 0 0
\(396\) 194400. 0.0622957
\(397\) 1.18622e6 0.377737 0.188869 0.982002i \(-0.439518\pi\)
0.188869 + 0.982002i \(0.439518\pi\)
\(398\) −3.94407e6 −1.24806
\(399\) −1.39996e6 −0.440233
\(400\) 0 0
\(401\) 3.13334e6 0.973077 0.486538 0.873659i \(-0.338259\pi\)
0.486538 + 0.873659i \(0.338259\pi\)
\(402\) 810684. 0.250200
\(403\) 97681.0 0.0299604
\(404\) −1.45651e6 −0.443977
\(405\) 0 0
\(406\) 930936. 0.280288
\(407\) −485700. −0.145339
\(408\) 1.17158e6 0.348436
\(409\) 567581. 0.167772 0.0838860 0.996475i \(-0.473267\pi\)
0.0838860 + 0.996475i \(0.473267\pi\)
\(410\) 0 0
\(411\) −2.34797e6 −0.685628
\(412\) 3.06618e6 0.889926
\(413\) −2.00265e6 −0.577737
\(414\) −437400. −0.125423
\(415\) 0 0
\(416\) 140288. 0.0397454
\(417\) 1.95916e6 0.551733
\(418\) −1.18140e6 −0.330717
\(419\) −6.09000e6 −1.69466 −0.847329 0.531068i \(-0.821791\pi\)
−0.847329 + 0.531068i \(0.821791\pi\)
\(420\) 0 0
\(421\) 3.07139e6 0.844558 0.422279 0.906466i \(-0.361230\pi\)
0.422279 + 0.906466i \(0.361230\pi\)
\(422\) 477980. 0.130656
\(423\) −1.71315e6 −0.465527
\(424\) −1.65734e6 −0.447711
\(425\) 0 0
\(426\) −1.22040e6 −0.325820
\(427\) −3.99858e6 −1.06130
\(428\) 148608. 0.0392132
\(429\) −184950. −0.0485189
\(430\) 0 0
\(431\) −669150. −0.173512 −0.0867562 0.996230i \(-0.527650\pi\)
−0.0867562 + 0.996230i \(0.527650\pi\)
\(432\) −186624. −0.0481125
\(433\) −3.25439e6 −0.834161 −0.417080 0.908870i \(-0.636947\pi\)
−0.417080 + 0.908870i \(0.636947\pi\)
\(434\) −225308. −0.0574186
\(435\) 0 0
\(436\) −330160. −0.0831780
\(437\) 2.65815e6 0.665849
\(438\) −2.96791e6 −0.739206
\(439\) 1.58759e6 0.393168 0.196584 0.980487i \(-0.437015\pi\)
0.196584 + 0.980487i \(0.437015\pi\)
\(440\) 0 0
\(441\) −855846. −0.209556
\(442\) −1.11463e6 −0.271379
\(443\) −213888. −0.0517818 −0.0258909 0.999665i \(-0.508242\pi\)
−0.0258909 + 0.999665i \(0.508242\pi\)
\(444\) 466272. 0.112249
\(445\) 0 0
\(446\) 194180. 0.0462240
\(447\) −3.79615e6 −0.898616
\(448\) −323584. −0.0761714
\(449\) 3.70724e6 0.867831 0.433916 0.900953i \(-0.357132\pi\)
0.433916 + 0.900953i \(0.357132\pi\)
\(450\) 0 0
\(451\) 984600. 0.227939
\(452\) 2.83027e6 0.651602
\(453\) 917253. 0.210012
\(454\) −1.15061e6 −0.261992
\(455\) 0 0
\(456\) 1.13414e6 0.255421
\(457\) 3.83193e6 0.858275 0.429138 0.903239i \(-0.358818\pi\)
0.429138 + 0.903239i \(0.358818\pi\)
\(458\) 288260. 0.0642127
\(459\) 1.48279e6 0.328509
\(460\) 0 0
\(461\) 5.58672e6 1.22435 0.612174 0.790723i \(-0.290295\pi\)
0.612174 + 0.790723i \(0.290295\pi\)
\(462\) 426600. 0.0929856
\(463\) −4.40142e6 −0.954203 −0.477101 0.878848i \(-0.658313\pi\)
−0.477101 + 0.878848i \(0.658313\pi\)
\(464\) −754176. −0.162621
\(465\) 0 0
\(466\) −2.27659e6 −0.485647
\(467\) 3.35225e6 0.711287 0.355643 0.934622i \(-0.384262\pi\)
0.355643 + 0.934622i \(0.384262\pi\)
\(468\) 177552. 0.0374724
\(469\) 1.77900e6 0.373460
\(470\) 0 0
\(471\) 915957. 0.190249
\(472\) 1.62240e6 0.335199
\(473\) −2.93685e6 −0.603572
\(474\) 2.93299e6 0.599605
\(475\) 0 0
\(476\) 2.57098e6 0.520093
\(477\) −2.09758e6 −0.422106
\(478\) 2.78602e6 0.557717
\(479\) 909966. 0.181212 0.0906059 0.995887i \(-0.471120\pi\)
0.0906059 + 0.995887i \(0.471120\pi\)
\(480\) 0 0
\(481\) −443606. −0.0874248
\(482\) 2.30455e6 0.451823
\(483\) −959850. −0.187213
\(484\) −2.21682e6 −0.430146
\(485\) 0 0
\(486\) −236196. −0.0453609
\(487\) −4.03579e6 −0.771093 −0.385546 0.922689i \(-0.625987\pi\)
−0.385546 + 0.922689i \(0.625987\pi\)
\(488\) 3.23936e6 0.615757
\(489\) 1.82024e6 0.344236
\(490\) 0 0
\(491\) 379944. 0.0711239 0.0355620 0.999367i \(-0.488678\pi\)
0.0355620 + 0.999367i \(0.488678\pi\)
\(492\) −945216. −0.176043
\(493\) 5.99216e6 1.11037
\(494\) −1.07901e6 −0.198934
\(495\) 0 0
\(496\) 182528. 0.0333139
\(497\) −2.67810e6 −0.486335
\(498\) −928152. −0.167705
\(499\) −7.42433e6 −1.33477 −0.667384 0.744714i \(-0.732586\pi\)
−0.667384 + 0.744714i \(0.732586\pi\)
\(500\) 0 0
\(501\) −2.08440e6 −0.371011
\(502\) −4.89970e6 −0.867780
\(503\) 2.29988e6 0.405309 0.202654 0.979250i \(-0.435043\pi\)
0.202654 + 0.979250i \(0.435043\pi\)
\(504\) −409536. −0.0718151
\(505\) 0 0
\(506\) −810000. −0.140640
\(507\) 3.17272e6 0.548165
\(508\) −4.10368e6 −0.705528
\(509\) 6.73721e6 1.15262 0.576310 0.817231i \(-0.304492\pi\)
0.576310 + 0.817231i \(0.304492\pi\)
\(510\) 0 0
\(511\) −6.51292e6 −1.10338
\(512\) 262144. 0.0441942
\(513\) 1.43540e6 0.240813
\(514\) 3.88853e6 0.649198
\(515\) 0 0
\(516\) 2.81938e6 0.466153
\(517\) −3.17250e6 −0.522006
\(518\) 1.02321e6 0.167548
\(519\) 1.56800e6 0.255521
\(520\) 0 0
\(521\) −9.26806e6 −1.49587 −0.747936 0.663770i \(-0.768955\pi\)
−0.747936 + 0.663770i \(0.768955\pi\)
\(522\) −954504. −0.153321
\(523\) 1.10620e7 1.76839 0.884197 0.467113i \(-0.154706\pi\)
0.884197 + 0.467113i \(0.154706\pi\)
\(524\) 2.84160e6 0.452100
\(525\) 0 0
\(526\) 7.04760e6 1.11065
\(527\) −1.45024e6 −0.227465
\(528\) −345600. −0.0539497
\(529\) −4.61384e6 −0.716842
\(530\) 0 0
\(531\) 2.05335e6 0.316029
\(532\) 2.48882e6 0.381253
\(533\) 899268. 0.137111
\(534\) −3.73421e6 −0.566690
\(535\) 0 0
\(536\) −1.44122e6 −0.216679
\(537\) 5.77859e6 0.864742
\(538\) 6.97534e6 1.03899
\(539\) −1.58490e6 −0.234979
\(540\) 0 0
\(541\) −1.30754e7 −1.92072 −0.960358 0.278771i \(-0.910073\pi\)
−0.960358 + 0.278771i \(0.910073\pi\)
\(542\) −6.81702e6 −0.996774
\(543\) 504639. 0.0734482
\(544\) −2.08282e6 −0.301754
\(545\) 0 0
\(546\) 389628. 0.0559331
\(547\) −1.84605e6 −0.263800 −0.131900 0.991263i \(-0.542108\pi\)
−0.131900 + 0.991263i \(0.542108\pi\)
\(548\) 4.17418e6 0.593772
\(549\) 4.09982e6 0.580542
\(550\) 0 0
\(551\) 5.80067e6 0.813953
\(552\) 777600. 0.108620
\(553\) 6.43629e6 0.894999
\(554\) 5.87887e6 0.813803
\(555\) 0 0
\(556\) −3.48294e6 −0.477815
\(557\) 6.48782e6 0.886055 0.443027 0.896508i \(-0.353904\pi\)
0.443027 + 0.896508i \(0.353904\pi\)
\(558\) 231012. 0.0314086
\(559\) −2.68232e6 −0.363063
\(560\) 0 0
\(561\) 2.74590e6 0.368364
\(562\) −4.90121e6 −0.654579
\(563\) 3.01538e6 0.400932 0.200466 0.979701i \(-0.435754\pi\)
0.200466 + 0.979701i \(0.435754\pi\)
\(564\) 3.04560e6 0.403158
\(565\) 0 0
\(566\) −2.54933e6 −0.334492
\(567\) −518319. −0.0677079
\(568\) 2.16960e6 0.282169
\(569\) −3.49810e6 −0.452952 −0.226476 0.974017i \(-0.572720\pi\)
−0.226476 + 0.974017i \(0.572720\pi\)
\(570\) 0 0
\(571\) −1.23749e7 −1.58837 −0.794185 0.607676i \(-0.792102\pi\)
−0.794185 + 0.607676i \(0.792102\pi\)
\(572\) 328800. 0.0420186
\(573\) 1.88725e6 0.240128
\(574\) −2.07422e6 −0.262770
\(575\) 0 0
\(576\) 331776. 0.0416667
\(577\) 6.80687e6 0.851153 0.425577 0.904922i \(-0.360071\pi\)
0.425577 + 0.904922i \(0.360071\pi\)
\(578\) 1.08692e7 1.35325
\(579\) 6.44037e6 0.798389
\(580\) 0 0
\(581\) −2.03678e6 −0.250325
\(582\) 2.06435e6 0.252624
\(583\) −3.88440e6 −0.473317
\(584\) 5.27629e6 0.640172
\(585\) 0 0
\(586\) 140376. 0.0168869
\(587\) 7.77113e6 0.930870 0.465435 0.885082i \(-0.345898\pi\)
0.465435 + 0.885082i \(0.345898\pi\)
\(588\) 1.52150e6 0.181480
\(589\) −1.40390e6 −0.166743
\(590\) 0 0
\(591\) −4.57969e6 −0.539345
\(592\) −828928. −0.0972104
\(593\) −1.51222e7 −1.76595 −0.882976 0.469418i \(-0.844464\pi\)
−0.882976 + 0.469418i \(0.844464\pi\)
\(594\) −437400. −0.0508643
\(595\) 0 0
\(596\) 6.74870e6 0.778224
\(597\) 8.87415e6 1.01904
\(598\) −739800. −0.0845982
\(599\) −1.67012e6 −0.190187 −0.0950937 0.995468i \(-0.530315\pi\)
−0.0950937 + 0.995468i \(0.530315\pi\)
\(600\) 0 0
\(601\) 4.12220e6 0.465525 0.232763 0.972534i \(-0.425223\pi\)
0.232763 + 0.972534i \(0.425223\pi\)
\(602\) 6.18696e6 0.695803
\(603\) −1.82404e6 −0.204287
\(604\) −1.63067e6 −0.181876
\(605\) 0 0
\(606\) 3.27715e6 0.362506
\(607\) 6.81870e6 0.751155 0.375578 0.926791i \(-0.377444\pi\)
0.375578 + 0.926791i \(0.377444\pi\)
\(608\) −2.01626e6 −0.221201
\(609\) −2.09461e6 −0.228854
\(610\) 0 0
\(611\) −2.89755e6 −0.313999
\(612\) −2.63606e6 −0.284497
\(613\) 1.59466e7 1.71402 0.857010 0.515300i \(-0.172319\pi\)
0.857010 + 0.515300i \(0.172319\pi\)
\(614\) 1.15216e7 1.23336
\(615\) 0 0
\(616\) −758400. −0.0805279
\(617\) −8.21952e6 −0.869228 −0.434614 0.900617i \(-0.643115\pi\)
−0.434614 + 0.900617i \(0.643115\pi\)
\(618\) −6.89890e6 −0.726622
\(619\) −1.23323e7 −1.29366 −0.646828 0.762636i \(-0.723905\pi\)
−0.646828 + 0.762636i \(0.723905\pi\)
\(620\) 0 0
\(621\) 984150. 0.102408
\(622\) −9.53220e6 −0.987910
\(623\) −8.19451e6 −0.845869
\(624\) −315648. −0.0324520
\(625\) 0 0
\(626\) −7.72325e6 −0.787706
\(627\) 2.65815e6 0.270029
\(628\) −1.62837e6 −0.164761
\(629\) 6.58609e6 0.663745
\(630\) 0 0
\(631\) −3.01912e6 −0.301861 −0.150931 0.988544i \(-0.548227\pi\)
−0.150931 + 0.988544i \(0.548227\pi\)
\(632\) −5.21421e6 −0.519273
\(633\) −1.07546e6 −0.106680
\(634\) −8.75981e6 −0.865509
\(635\) 0 0
\(636\) 3.72902e6 0.365555
\(637\) −1.44754e6 −0.141346
\(638\) −1.76760e6 −0.171922
\(639\) 2.74590e6 0.266031
\(640\) 0 0
\(641\) −1.18052e7 −1.13482 −0.567411 0.823435i \(-0.692055\pi\)
−0.567411 + 0.823435i \(0.692055\pi\)
\(642\) −334368. −0.0320175
\(643\) −2.85360e6 −0.272186 −0.136093 0.990696i \(-0.543455\pi\)
−0.136093 + 0.990696i \(0.543455\pi\)
\(644\) 1.70640e6 0.162131
\(645\) 0 0
\(646\) 1.60198e7 1.51034
\(647\) −3.40192e6 −0.319494 −0.159747 0.987158i \(-0.551068\pi\)
−0.159747 + 0.987158i \(0.551068\pi\)
\(648\) 419904. 0.0392837
\(649\) 3.80250e6 0.354370
\(650\) 0 0
\(651\) 506943. 0.0468821
\(652\) −3.23598e6 −0.298117
\(653\) −9.39166e6 −0.861905 −0.430953 0.902375i \(-0.641822\pi\)
−0.430953 + 0.902375i \(0.641822\pi\)
\(654\) 742860. 0.0679145
\(655\) 0 0
\(656\) 1.68038e6 0.152458
\(657\) 6.67780e6 0.603560
\(658\) 6.68340e6 0.601773
\(659\) 9.36659e6 0.840171 0.420086 0.907484i \(-0.362000\pi\)
0.420086 + 0.907484i \(0.362000\pi\)
\(660\) 0 0
\(661\) 1.57748e7 1.40430 0.702150 0.712029i \(-0.252223\pi\)
0.702150 + 0.712029i \(0.252223\pi\)
\(662\) 5.65717e6 0.501712
\(663\) 2.50792e6 0.221580
\(664\) 1.65005e6 0.145237
\(665\) 0 0
\(666\) −1.04911e6 −0.0916508
\(667\) 3.97710e6 0.346140
\(668\) 3.70560e6 0.321305
\(669\) −436905. −0.0377417
\(670\) 0 0
\(671\) 7.59225e6 0.650975
\(672\) 728064. 0.0621937
\(673\) −6.64515e6 −0.565545 −0.282773 0.959187i \(-0.591254\pi\)
−0.282773 + 0.959187i \(0.591254\pi\)
\(674\) 5.52832e6 0.468753
\(675\) 0 0
\(676\) −5.64038e6 −0.474725
\(677\) −1.57780e7 −1.32306 −0.661532 0.749917i \(-0.730094\pi\)
−0.661532 + 0.749917i \(0.730094\pi\)
\(678\) −6.36811e6 −0.532030
\(679\) 4.53010e6 0.377080
\(680\) 0 0
\(681\) 2.58887e6 0.213915
\(682\) 427800. 0.0352192
\(683\) −1.96654e7 −1.61306 −0.806532 0.591190i \(-0.798658\pi\)
−0.806532 + 0.591190i \(0.798658\pi\)
\(684\) −2.55182e6 −0.208550
\(685\) 0 0
\(686\) 8.64987e6 0.701777
\(687\) −648585. −0.0524294
\(688\) −5.01222e6 −0.403701
\(689\) −3.54775e6 −0.284712
\(690\) 0 0
\(691\) 990464. 0.0789121 0.0394560 0.999221i \(-0.487437\pi\)
0.0394560 + 0.999221i \(0.487437\pi\)
\(692\) −2.78755e6 −0.221288
\(693\) −959850. −0.0759225
\(694\) −1.13503e6 −0.0894560
\(695\) 0 0
\(696\) 1.69690e6 0.132780
\(697\) −1.33512e7 −1.04097
\(698\) 8.55236e6 0.664427
\(699\) 5.12233e6 0.396529
\(700\) 0 0
\(701\) 6.52919e6 0.501839 0.250920 0.968008i \(-0.419267\pi\)
0.250920 + 0.968008i \(0.419267\pi\)
\(702\) −399492. −0.0305960
\(703\) 6.37562e6 0.486558
\(704\) 614400. 0.0467218
\(705\) 0 0
\(706\) 1.26406e7 0.954455
\(707\) 7.19153e6 0.541094
\(708\) −3.65040e6 −0.273689
\(709\) −2.02106e7 −1.50996 −0.754979 0.655749i \(-0.772353\pi\)
−0.754979 + 0.655749i \(0.772353\pi\)
\(710\) 0 0
\(711\) −6.59923e6 −0.489575
\(712\) 6.63859e6 0.490768
\(713\) −962550. −0.0709087
\(714\) −5.78470e6 −0.424654
\(715\) 0 0
\(716\) −1.02731e7 −0.748888
\(717\) −6.26854e6 −0.455374
\(718\) −1.43826e7 −1.04118
\(719\) −2.27720e7 −1.64277 −0.821387 0.570371i \(-0.806800\pi\)
−0.821387 + 0.570371i \(0.806800\pi\)
\(720\) 0 0
\(721\) −1.51392e7 −1.08459
\(722\) 5.60345e6 0.400048
\(723\) −5.18523e6 −0.368912
\(724\) −897136. −0.0636080
\(725\) 0 0
\(726\) 4.98784e6 0.351213
\(727\) −7.09186e6 −0.497650 −0.248825 0.968548i \(-0.580044\pi\)
−0.248825 + 0.968548i \(0.580044\pi\)
\(728\) −692672. −0.0484394
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) 3.98237e7 2.75644
\(732\) −7.28856e6 −0.502764
\(733\) −1.38982e7 −0.955432 −0.477716 0.878514i \(-0.658535\pi\)
−0.477716 + 0.878514i \(0.658535\pi\)
\(734\) −1.84817e7 −1.26620
\(735\) 0 0
\(736\) −1.38240e6 −0.0940674
\(737\) −3.37785e6 −0.229072
\(738\) 2.12674e6 0.143738
\(739\) 1.54857e7 1.04309 0.521543 0.853225i \(-0.325357\pi\)
0.521543 + 0.853225i \(0.325357\pi\)
\(740\) 0 0
\(741\) 2.42778e6 0.162429
\(742\) 8.18314e6 0.545645
\(743\) 159276. 0.0105847 0.00529235 0.999986i \(-0.498315\pi\)
0.00529235 + 0.999986i \(0.498315\pi\)
\(744\) −410688. −0.0272007
\(745\) 0 0
\(746\) −1.40793e7 −0.926263
\(747\) 2.08834e6 0.136930
\(748\) −4.88160e6 −0.319013
\(749\) −733752. −0.0477908
\(750\) 0 0
\(751\) 1.15554e7 0.747625 0.373812 0.927504i \(-0.378050\pi\)
0.373812 + 0.927504i \(0.378050\pi\)
\(752\) −5.41440e6 −0.349145
\(753\) 1.10243e7 0.708540
\(754\) −1.61441e6 −0.103415
\(755\) 0 0
\(756\) 921456. 0.0586368
\(757\) −6.34402e6 −0.402369 −0.201185 0.979553i \(-0.564479\pi\)
−0.201185 + 0.979553i \(0.564479\pi\)
\(758\) 2.38024e6 0.150469
\(759\) 1.82250e6 0.114832
\(760\) 0 0
\(761\) −1.17748e7 −0.737040 −0.368520 0.929620i \(-0.620135\pi\)
−0.368520 + 0.929620i \(0.620135\pi\)
\(762\) 9.23328e6 0.576061
\(763\) 1.63016e6 0.101373
\(764\) −3.35510e6 −0.207957
\(765\) 0 0
\(766\) −5.20200e6 −0.320331
\(767\) 3.47295e6 0.213162
\(768\) −589824. −0.0360844
\(769\) −2.47869e7 −1.51149 −0.755745 0.654865i \(-0.772725\pi\)
−0.755745 + 0.654865i \(0.772725\pi\)
\(770\) 0 0
\(771\) −8.74919e6 −0.530068
\(772\) −1.14496e7 −0.691425
\(773\) −1.38017e7 −0.830774 −0.415387 0.909645i \(-0.636354\pi\)
−0.415387 + 0.909645i \(0.636354\pi\)
\(774\) −6.34360e6 −0.380613
\(775\) 0 0
\(776\) −3.66995e6 −0.218779
\(777\) −2.30222e6 −0.136802
\(778\) 2.48290e6 0.147065
\(779\) −1.29245e7 −0.763081
\(780\) 0 0
\(781\) 5.08500e6 0.298307
\(782\) 1.09836e7 0.642285
\(783\) 2.14763e6 0.125186
\(784\) −2.70490e6 −0.157167
\(785\) 0 0
\(786\) −6.39360e6 −0.369138
\(787\) −1.53795e7 −0.885129 −0.442565 0.896737i \(-0.645931\pi\)
−0.442565 + 0.896737i \(0.645931\pi\)
\(788\) 8.14166e6 0.467087
\(789\) −1.58571e7 −0.906841
\(790\) 0 0
\(791\) −1.39745e7 −0.794135
\(792\) 777600. 0.0440497
\(793\) 6.93426e6 0.391577
\(794\) 4.74489e6 0.267101
\(795\) 0 0
\(796\) −1.57763e7 −0.882514
\(797\) −2.33978e7 −1.30475 −0.652377 0.757894i \(-0.726228\pi\)
−0.652377 + 0.757894i \(0.726228\pi\)
\(798\) −5.59984e6 −0.311292
\(799\) 4.30191e7 2.38393
\(800\) 0 0
\(801\) 8.40197e6 0.462700
\(802\) 1.25334e7 0.688069
\(803\) 1.23663e7 0.676785
\(804\) 3.24274e6 0.176918
\(805\) 0 0
\(806\) 390724. 0.0211852
\(807\) −1.56945e7 −0.848328
\(808\) −5.82605e6 −0.313939
\(809\) −2.80188e7 −1.50515 −0.752573 0.658509i \(-0.771187\pi\)
−0.752573 + 0.658509i \(0.771187\pi\)
\(810\) 0 0
\(811\) −1.65641e6 −0.0884332 −0.0442166 0.999022i \(-0.514079\pi\)
−0.0442166 + 0.999022i \(0.514079\pi\)
\(812\) 3.72374e6 0.198194
\(813\) 1.53383e7 0.813862
\(814\) −1.94280e6 −0.102770
\(815\) 0 0
\(816\) 4.68634e6 0.246381
\(817\) 3.85511e7 2.02060
\(818\) 2.27032e6 0.118633
\(819\) −876663. −0.0456691
\(820\) 0 0
\(821\) 2.97382e7 1.53978 0.769888 0.638179i \(-0.220312\pi\)
0.769888 + 0.638179i \(0.220312\pi\)
\(822\) −9.39190e6 −0.484812
\(823\) −1.60108e7 −0.823972 −0.411986 0.911190i \(-0.635165\pi\)
−0.411986 + 0.911190i \(0.635165\pi\)
\(824\) 1.22647e7 0.629273
\(825\) 0 0
\(826\) −8.01060e6 −0.408522
\(827\) 2.54748e7 1.29523 0.647614 0.761969i \(-0.275767\pi\)
0.647614 + 0.761969i \(0.275767\pi\)
\(828\) −1.74960e6 −0.0886876
\(829\) 1.89971e7 0.960064 0.480032 0.877251i \(-0.340625\pi\)
0.480032 + 0.877251i \(0.340625\pi\)
\(830\) 0 0
\(831\) −1.32275e7 −0.664467
\(832\) 561152. 0.0281043
\(833\) 2.14912e7 1.07312
\(834\) 7.83662e6 0.390134
\(835\) 0 0
\(836\) −4.72560e6 −0.233852
\(837\) −519777. −0.0256450
\(838\) −2.43600e7 −1.19830
\(839\) 3.98539e7 1.95463 0.977317 0.211782i \(-0.0679267\pi\)
0.977317 + 0.211782i \(0.0679267\pi\)
\(840\) 0 0
\(841\) −1.18322e7 −0.576868
\(842\) 1.22856e7 0.597193
\(843\) 1.10277e7 0.534462
\(844\) 1.91192e6 0.0923876
\(845\) 0 0
\(846\) −6.85260e6 −0.329177
\(847\) 1.09455e7 0.524238
\(848\) −6.62938e6 −0.316580
\(849\) 5.73600e6 0.273111
\(850\) 0 0
\(851\) 4.37130e6 0.206912
\(852\) −4.88160e6 −0.230390
\(853\) 2.19179e7 1.03140 0.515700 0.856769i \(-0.327532\pi\)
0.515700 + 0.856769i \(0.327532\pi\)
\(854\) −1.59943e7 −0.750450
\(855\) 0 0
\(856\) 594432. 0.0277280
\(857\) −2.72213e7 −1.26607 −0.633033 0.774125i \(-0.718190\pi\)
−0.633033 + 0.774125i \(0.718190\pi\)
\(858\) −739800. −0.0343081
\(859\) 3.76214e7 1.73961 0.869805 0.493396i \(-0.164245\pi\)
0.869805 + 0.493396i \(0.164245\pi\)
\(860\) 0 0
\(861\) 4.66700e6 0.214551
\(862\) −2.67660e6 −0.122692
\(863\) 2.51995e7 1.15177 0.575885 0.817531i \(-0.304658\pi\)
0.575885 + 0.817531i \(0.304658\pi\)
\(864\) −746496. −0.0340207
\(865\) 0 0
\(866\) −1.30176e7 −0.589841
\(867\) −2.44557e7 −1.10492
\(868\) −901232. −0.0406011
\(869\) −1.22208e7 −0.548972
\(870\) 0 0
\(871\) −3.08510e6 −0.137792
\(872\) −1.32064e6 −0.0588157
\(873\) −4.64478e6 −0.206267
\(874\) 1.06326e7 0.470826
\(875\) 0 0
\(876\) −1.18716e7 −0.522698
\(877\) 2.77307e6 0.121748 0.0608739 0.998145i \(-0.480611\pi\)
0.0608739 + 0.998145i \(0.480611\pi\)
\(878\) 6.35037e6 0.278012
\(879\) −315846. −0.0137881
\(880\) 0 0
\(881\) −1.37827e7 −0.598268 −0.299134 0.954211i \(-0.596698\pi\)
−0.299134 + 0.954211i \(0.596698\pi\)
\(882\) −3.42338e6 −0.148178
\(883\) −4.96322e6 −0.214221 −0.107110 0.994247i \(-0.534160\pi\)
−0.107110 + 0.994247i \(0.534160\pi\)
\(884\) −4.45853e6 −0.191894
\(885\) 0 0
\(886\) −855552. −0.0366153
\(887\) 4.12640e7 1.76101 0.880505 0.474037i \(-0.157204\pi\)
0.880505 + 0.474037i \(0.157204\pi\)
\(888\) 1.86509e6 0.0793719
\(889\) 2.02619e7 0.859857
\(890\) 0 0
\(891\) 984150. 0.0415305
\(892\) 776720. 0.0326853
\(893\) 4.16444e7 1.74754
\(894\) −1.51846e7 −0.635417
\(895\) 0 0
\(896\) −1.29434e6 −0.0538613
\(897\) 1.66455e6 0.0690742
\(898\) 1.48290e7 0.613649
\(899\) −2.10050e6 −0.0866808
\(900\) 0 0
\(901\) 5.26725e7 2.16158
\(902\) 3.93840e6 0.161177
\(903\) −1.39207e7 −0.568121
\(904\) 1.13211e7 0.460752
\(905\) 0 0
\(906\) 3.66901e6 0.148501
\(907\) 1.66308e7 0.671268 0.335634 0.941992i \(-0.391049\pi\)
0.335634 + 0.941992i \(0.391049\pi\)
\(908\) −4.60243e6 −0.185256
\(909\) −7.37359e6 −0.295985
\(910\) 0 0
\(911\) 3.64995e7 1.45711 0.728554 0.684989i \(-0.240193\pi\)
0.728554 + 0.684989i \(0.240193\pi\)
\(912\) 4.53658e6 0.180610
\(913\) 3.86730e6 0.153543
\(914\) 1.53277e7 0.606892
\(915\) 0 0
\(916\) 1.15304e6 0.0454052
\(917\) −1.40304e7 −0.550994
\(918\) 5.93114e6 0.232291
\(919\) 6.83190e6 0.266841 0.133421 0.991060i \(-0.457404\pi\)
0.133421 + 0.991060i \(0.457404\pi\)
\(920\) 0 0
\(921\) −2.59236e7 −1.00704
\(922\) 2.23469e7 0.865744
\(923\) 4.64430e6 0.179439
\(924\) 1.70640e6 0.0657508
\(925\) 0 0
\(926\) −1.76057e7 −0.674723
\(927\) 1.55225e7 0.593284
\(928\) −3.01670e6 −0.114991
\(929\) 5.05115e6 0.192022 0.0960111 0.995380i \(-0.469392\pi\)
0.0960111 + 0.995380i \(0.469392\pi\)
\(930\) 0 0
\(931\) 2.08045e7 0.786651
\(932\) −9.10637e6 −0.343404
\(933\) 2.14474e7 0.806625
\(934\) 1.34090e7 0.502956
\(935\) 0 0
\(936\) 710208. 0.0264970
\(937\) 8.30023e6 0.308845 0.154423 0.988005i \(-0.450648\pi\)
0.154423 + 0.988005i \(0.450648\pi\)
\(938\) 7.11600e6 0.264076
\(939\) 1.73773e7 0.643160
\(940\) 0 0
\(941\) −4.87332e7 −1.79412 −0.897059 0.441910i \(-0.854301\pi\)
−0.897059 + 0.441910i \(0.854301\pi\)
\(942\) 3.66383e6 0.134526
\(943\) −8.86140e6 −0.324506
\(944\) 6.48960e6 0.237022
\(945\) 0 0
\(946\) −1.17474e7 −0.426790
\(947\) −7.34777e6 −0.266245 −0.133122 0.991100i \(-0.542500\pi\)
−0.133122 + 0.991100i \(0.542500\pi\)
\(948\) 1.17320e7 0.423985
\(949\) 1.12946e7 0.407102
\(950\) 0 0
\(951\) 1.97096e7 0.706685
\(952\) 1.02839e7 0.367761
\(953\) −3.51437e7 −1.25347 −0.626737 0.779231i \(-0.715610\pi\)
−0.626737 + 0.779231i \(0.715610\pi\)
\(954\) −8.39030e6 −0.298474
\(955\) 0 0
\(956\) 1.11441e7 0.394365
\(957\) 3.97710e6 0.140374
\(958\) 3.63986e6 0.128136
\(959\) −2.06100e7 −0.723655
\(960\) 0 0
\(961\) −2.81208e7 −0.982243
\(962\) −1.77442e6 −0.0618187
\(963\) 752328. 0.0261422
\(964\) 9.21819e6 0.319487
\(965\) 0 0
\(966\) −3.83940e6 −0.132379
\(967\) 2.31186e6 0.0795050 0.0397525 0.999210i \(-0.487343\pi\)
0.0397525 + 0.999210i \(0.487343\pi\)
\(968\) −8.86726e6 −0.304159
\(969\) −3.60445e7 −1.23319
\(970\) 0 0
\(971\) −3.73588e7 −1.27158 −0.635791 0.771861i \(-0.719326\pi\)
−0.635791 + 0.771861i \(0.719326\pi\)
\(972\) −944784. −0.0320750
\(973\) 1.71970e7 0.582333
\(974\) −1.61432e7 −0.545245
\(975\) 0 0
\(976\) 1.29574e7 0.435406
\(977\) −1.02666e7 −0.344104 −0.172052 0.985088i \(-0.555040\pi\)
−0.172052 + 0.985088i \(0.555040\pi\)
\(978\) 7.28096e6 0.243412
\(979\) 1.55592e7 0.518837
\(980\) 0 0
\(981\) −1.67144e6 −0.0554520
\(982\) 1.51978e6 0.0502922
\(983\) 2.98404e7 0.984966 0.492483 0.870322i \(-0.336089\pi\)
0.492483 + 0.870322i \(0.336089\pi\)
\(984\) −3.78086e6 −0.124481
\(985\) 0 0
\(986\) 2.39687e7 0.785148
\(987\) −1.50376e7 −0.491346
\(988\) −4.31605e6 −0.140668
\(989\) 2.64316e7 0.859278
\(990\) 0 0
\(991\) −2.34222e7 −0.757605 −0.378803 0.925477i \(-0.623664\pi\)
−0.378803 + 0.925477i \(0.623664\pi\)
\(992\) 730112. 0.0235565
\(993\) −1.27286e7 −0.409646
\(994\) −1.07124e7 −0.343891
\(995\) 0 0
\(996\) −3.71261e6 −0.118585
\(997\) −2.60709e7 −0.830651 −0.415326 0.909673i \(-0.636332\pi\)
−0.415326 + 0.909673i \(0.636332\pi\)
\(998\) −2.96973e7 −0.943824
\(999\) 2.36050e6 0.0748326
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 150.6.a.i.1.1 yes 1
3.2 odd 2 450.6.a.e.1.1 1
5.2 odd 4 150.6.c.c.49.2 2
5.3 odd 4 150.6.c.c.49.1 2
5.4 even 2 150.6.a.g.1.1 1
15.2 even 4 450.6.c.g.199.1 2
15.8 even 4 450.6.c.g.199.2 2
15.14 odd 2 450.6.a.t.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.6.a.g.1.1 1 5.4 even 2
150.6.a.i.1.1 yes 1 1.1 even 1 trivial
150.6.c.c.49.1 2 5.3 odd 4
150.6.c.c.49.2 2 5.2 odd 4
450.6.a.e.1.1 1 3.2 odd 2
450.6.a.t.1.1 1 15.14 odd 2
450.6.c.g.199.1 2 15.2 even 4
450.6.c.g.199.2 2 15.8 even 4