Properties

Label 150.6.a.k.1.1
Level $150$
Weight $6$
Character 150.1
Self dual yes
Analytic conductor $24.058$
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] = [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: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 150.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -36.0000 q^{6} +1.00000 q^{7} +64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -36.0000 q^{6} +1.00000 q^{7} +64.0000 q^{8} +81.0000 q^{9} -210.000 q^{11} -144.000 q^{12} +667.000 q^{13} +4.00000 q^{14} +256.000 q^{16} -114.000 q^{17} +324.000 q^{18} +581.000 q^{19} -9.00000 q^{21} -840.000 q^{22} +4350.00 q^{23} -576.000 q^{24} +2668.00 q^{26} -729.000 q^{27} +16.0000 q^{28} -126.000 q^{29} +7583.00 q^{31} +1024.00 q^{32} +1890.00 q^{33} -456.000 q^{34} +1296.00 q^{36} +3742.00 q^{37} +2324.00 q^{38} -6003.00 q^{39} -2856.00 q^{41} -36.0000 q^{42} +18241.0 q^{43} -3360.00 q^{44} +17400.0 q^{46} +23370.0 q^{47} -2304.00 q^{48} -16806.0 q^{49} +1026.00 q^{51} +10672.0 q^{52} +21684.0 q^{53} -2916.00 q^{54} +64.0000 q^{56} -5229.00 q^{57} -504.000 q^{58} -32310.0 q^{59} -7165.00 q^{61} +30332.0 q^{62} +81.0000 q^{63} +4096.00 q^{64} +7560.00 q^{66} -59579.0 q^{67} -1824.00 q^{68} -39150.0 q^{69} -43080.0 q^{71} +5184.00 q^{72} +28942.0 q^{73} +14968.0 q^{74} +9296.00 q^{76} -210.000 q^{77} -24012.0 q^{78} +27608.0 q^{79} +6561.00 q^{81} -11424.0 q^{82} +1782.00 q^{83} -144.000 q^{84} +72964.0 q^{86} +1134.00 q^{87} -13440.0 q^{88} +50208.0 q^{89} +667.000 q^{91} +69600.0 q^{92} -68247.0 q^{93} +93480.0 q^{94} -9216.00 q^{96} -142793. q^{97} -67224.0 q^{98} -17010.0 q^{99} +O(q^{100})\)

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
\(7\) 1.00000 0.00771356 0.00385678 0.999993i \(-0.498772\pi\)
0.00385678 + 0.999993i \(0.498772\pi\)
\(8\) 64.0000 0.353553
\(9\) 81.0000 0.333333
\(10\) 0 0
\(11\) −210.000 −0.523284 −0.261642 0.965165i \(-0.584264\pi\)
−0.261642 + 0.965165i \(0.584264\pi\)
\(12\) −144.000 −0.288675
\(13\) 667.000 1.09463 0.547315 0.836927i \(-0.315650\pi\)
0.547315 + 0.836927i \(0.315650\pi\)
\(14\) 4.00000 0.00545431
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −114.000 −0.0956715 −0.0478357 0.998855i \(-0.515232\pi\)
−0.0478357 + 0.998855i \(0.515232\pi\)
\(18\) 324.000 0.235702
\(19\) 581.000 0.369226 0.184613 0.982811i \(-0.440897\pi\)
0.184613 + 0.982811i \(0.440897\pi\)
\(20\) 0 0
\(21\) −9.00000 −0.00445343
\(22\) −840.000 −0.370018
\(23\) 4350.00 1.71463 0.857314 0.514795i \(-0.172132\pi\)
0.857314 + 0.514795i \(0.172132\pi\)
\(24\) −576.000 −0.204124
\(25\) 0 0
\(26\) 2668.00 0.774020
\(27\) −729.000 −0.192450
\(28\) 16.0000 0.00385678
\(29\) −126.000 −0.0278212 −0.0139106 0.999903i \(-0.504428\pi\)
−0.0139106 + 0.999903i \(0.504428\pi\)
\(30\) 0 0
\(31\) 7583.00 1.41722 0.708609 0.705601i \(-0.249323\pi\)
0.708609 + 0.705601i \(0.249323\pi\)
\(32\) 1024.00 0.176777
\(33\) 1890.00 0.302118
\(34\) −456.000 −0.0676500
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) 3742.00 0.449365 0.224683 0.974432i \(-0.427865\pi\)
0.224683 + 0.974432i \(0.427865\pi\)
\(38\) 2324.00 0.261082
\(39\) −6003.00 −0.631985
\(40\) 0 0
\(41\) −2856.00 −0.265337 −0.132669 0.991160i \(-0.542355\pi\)
−0.132669 + 0.991160i \(0.542355\pi\)
\(42\) −36.0000 −0.00314905
\(43\) 18241.0 1.50445 0.752225 0.658907i \(-0.228981\pi\)
0.752225 + 0.658907i \(0.228981\pi\)
\(44\) −3360.00 −0.261642
\(45\) 0 0
\(46\) 17400.0 1.21242
\(47\) 23370.0 1.54317 0.771586 0.636126i \(-0.219464\pi\)
0.771586 + 0.636126i \(0.219464\pi\)
\(48\) −2304.00 −0.144338
\(49\) −16806.0 −0.999941
\(50\) 0 0
\(51\) 1026.00 0.0552360
\(52\) 10672.0 0.547315
\(53\) 21684.0 1.06035 0.530176 0.847888i \(-0.322126\pi\)
0.530176 + 0.847888i \(0.322126\pi\)
\(54\) −2916.00 −0.136083
\(55\) 0 0
\(56\) 64.0000 0.00272716
\(57\) −5229.00 −0.213173
\(58\) −504.000 −0.0196725
\(59\) −32310.0 −1.20839 −0.604195 0.796837i \(-0.706505\pi\)
−0.604195 + 0.796837i \(0.706505\pi\)
\(60\) 0 0
\(61\) −7165.00 −0.246542 −0.123271 0.992373i \(-0.539339\pi\)
−0.123271 + 0.992373i \(0.539339\pi\)
\(62\) 30332.0 1.00212
\(63\) 81.0000 0.00257119
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 7560.00 0.213630
\(67\) −59579.0 −1.62146 −0.810730 0.585420i \(-0.800929\pi\)
−0.810730 + 0.585420i \(0.800929\pi\)
\(68\) −1824.00 −0.0478357
\(69\) −39150.0 −0.989940
\(70\) 0 0
\(71\) −43080.0 −1.01421 −0.507107 0.861883i \(-0.669285\pi\)
−0.507107 + 0.861883i \(0.669285\pi\)
\(72\) 5184.00 0.117851
\(73\) 28942.0 0.635655 0.317827 0.948149i \(-0.397047\pi\)
0.317827 + 0.948149i \(0.397047\pi\)
\(74\) 14968.0 0.317749
\(75\) 0 0
\(76\) 9296.00 0.184613
\(77\) −210.000 −0.00403638
\(78\) −24012.0 −0.446881
\(79\) 27608.0 0.497700 0.248850 0.968542i \(-0.419947\pi\)
0.248850 + 0.968542i \(0.419947\pi\)
\(80\) 0 0
\(81\) 6561.00 0.111111
\(82\) −11424.0 −0.187622
\(83\) 1782.00 0.0283931 0.0141965 0.999899i \(-0.495481\pi\)
0.0141965 + 0.999899i \(0.495481\pi\)
\(84\) −144.000 −0.00222671
\(85\) 0 0
\(86\) 72964.0 1.06381
\(87\) 1134.00 0.0160626
\(88\) −13440.0 −0.185009
\(89\) 50208.0 0.671890 0.335945 0.941882i \(-0.390944\pi\)
0.335945 + 0.941882i \(0.390944\pi\)
\(90\) 0 0
\(91\) 667.000 0.00844350
\(92\) 69600.0 0.857314
\(93\) −68247.0 −0.818231
\(94\) 93480.0 1.09119
\(95\) 0 0
\(96\) −9216.00 −0.102062
\(97\) −142793. −1.54091 −0.770456 0.637494i \(-0.779971\pi\)
−0.770456 + 0.637494i \(0.779971\pi\)
\(98\) −67224.0 −0.707065
\(99\) −17010.0 −0.174428
\(100\) 0 0
\(101\) −164052. −1.60021 −0.800107 0.599857i \(-0.795224\pi\)
−0.800107 + 0.599857i \(0.795224\pi\)
\(102\) 4104.00 0.0390577
\(103\) 107236. 0.995973 0.497986 0.867185i \(-0.334073\pi\)
0.497986 + 0.867185i \(0.334073\pi\)
\(104\) 42688.0 0.387010
\(105\) 0 0
\(106\) 86736.0 0.749782
\(107\) −21012.0 −0.177422 −0.0887111 0.996057i \(-0.528275\pi\)
−0.0887111 + 0.996057i \(0.528275\pi\)
\(108\) −11664.0 −0.0962250
\(109\) 237545. 1.91505 0.957524 0.288354i \(-0.0931079\pi\)
0.957524 + 0.288354i \(0.0931079\pi\)
\(110\) 0 0
\(111\) −33678.0 −0.259441
\(112\) 256.000 0.00192839
\(113\) −233388. −1.71942 −0.859711 0.510781i \(-0.829356\pi\)
−0.859711 + 0.510781i \(0.829356\pi\)
\(114\) −20916.0 −0.150736
\(115\) 0 0
\(116\) −2016.00 −0.0139106
\(117\) 54027.0 0.364877
\(118\) −129240. −0.854460
\(119\) −114.000 −0.000737968 0
\(120\) 0 0
\(121\) −116951. −0.726174
\(122\) −28660.0 −0.174332
\(123\) 25704.0 0.153193
\(124\) 121328. 0.708609
\(125\) 0 0
\(126\) 324.000 0.00181810
\(127\) −138800. −0.763625 −0.381813 0.924240i \(-0.624700\pi\)
−0.381813 + 0.924240i \(0.624700\pi\)
\(128\) 16384.0 0.0883883
\(129\) −164169. −0.868594
\(130\) 0 0
\(131\) −161340. −0.821417 −0.410709 0.911767i \(-0.634719\pi\)
−0.410709 + 0.911767i \(0.634719\pi\)
\(132\) 30240.0 0.151059
\(133\) 581.000 0.00284805
\(134\) −238316. −1.14655
\(135\) 0 0
\(136\) −7296.00 −0.0338250
\(137\) −69054.0 −0.314331 −0.157166 0.987572i \(-0.550236\pi\)
−0.157166 + 0.987572i \(0.550236\pi\)
\(138\) −156600. −0.699994
\(139\) 224396. 0.985095 0.492547 0.870286i \(-0.336066\pi\)
0.492547 + 0.870286i \(0.336066\pi\)
\(140\) 0 0
\(141\) −210330. −0.890950
\(142\) −172320. −0.717158
\(143\) −140070. −0.572803
\(144\) 20736.0 0.0833333
\(145\) 0 0
\(146\) 115768. 0.449476
\(147\) 151254. 0.577316
\(148\) 59872.0 0.224683
\(149\) −407646. −1.50424 −0.752121 0.659025i \(-0.770969\pi\)
−0.752121 + 0.659025i \(0.770969\pi\)
\(150\) 0 0
\(151\) 216053. 0.771113 0.385556 0.922684i \(-0.374010\pi\)
0.385556 + 0.922684i \(0.374010\pi\)
\(152\) 37184.0 0.130541
\(153\) −9234.00 −0.0318905
\(154\) −840.000 −0.00285415
\(155\) 0 0
\(156\) −96048.0 −0.315993
\(157\) 188617. 0.610705 0.305353 0.952239i \(-0.401226\pi\)
0.305353 + 0.952239i \(0.401226\pi\)
\(158\) 110432. 0.351927
\(159\) −195156. −0.612194
\(160\) 0 0
\(161\) 4350.00 0.0132259
\(162\) 26244.0 0.0785674
\(163\) 422251. 1.24481 0.622403 0.782697i \(-0.286157\pi\)
0.622403 + 0.782697i \(0.286157\pi\)
\(164\) −45696.0 −0.132669
\(165\) 0 0
\(166\) 7128.00 0.0200769
\(167\) −41700.0 −0.115703 −0.0578515 0.998325i \(-0.518425\pi\)
−0.0578515 + 0.998325i \(0.518425\pi\)
\(168\) −576.000 −0.00157452
\(169\) 73596.0 0.198215
\(170\) 0 0
\(171\) 47061.0 0.123075
\(172\) 291856. 0.752225
\(173\) −266142. −0.676080 −0.338040 0.941132i \(-0.609764\pi\)
−0.338040 + 0.941132i \(0.609764\pi\)
\(174\) 4536.00 0.0113579
\(175\) 0 0
\(176\) −53760.0 −0.130821
\(177\) 290790. 0.697664
\(178\) 200832. 0.475098
\(179\) −51846.0 −0.120944 −0.0604718 0.998170i \(-0.519261\pi\)
−0.0604718 + 0.998170i \(0.519261\pi\)
\(180\) 0 0
\(181\) 499349. 1.13294 0.566471 0.824082i \(-0.308308\pi\)
0.566471 + 0.824082i \(0.308308\pi\)
\(182\) 2668.00 0.00597045
\(183\) 64485.0 0.142341
\(184\) 278400. 0.606212
\(185\) 0 0
\(186\) −272988. −0.578577
\(187\) 23940.0 0.0500634
\(188\) 373920. 0.771586
\(189\) −729.000 −0.00148448
\(190\) 0 0
\(191\) 705906. 1.40011 0.700057 0.714087i \(-0.253158\pi\)
0.700057 + 0.714087i \(0.253158\pi\)
\(192\) −36864.0 −0.0721688
\(193\) 519973. 1.00482 0.502409 0.864630i \(-0.332447\pi\)
0.502409 + 0.864630i \(0.332447\pi\)
\(194\) −571172. −1.08959
\(195\) 0 0
\(196\) −268896. −0.499970
\(197\) 200394. 0.367891 0.183945 0.982936i \(-0.441113\pi\)
0.183945 + 0.982936i \(0.441113\pi\)
\(198\) −68040.0 −0.123339
\(199\) 553673. 0.991107 0.495554 0.868577i \(-0.334965\pi\)
0.495554 + 0.868577i \(0.334965\pi\)
\(200\) 0 0
\(201\) 536211. 0.936150
\(202\) −656208. −1.13152
\(203\) −126.000 −0.000214600 0
\(204\) 16416.0 0.0276180
\(205\) 0 0
\(206\) 428944. 0.704259
\(207\) 352350. 0.571542
\(208\) 170752. 0.273658
\(209\) −122010. −0.193210
\(210\) 0 0
\(211\) 777365. 1.20204 0.601020 0.799234i \(-0.294761\pi\)
0.601020 + 0.799234i \(0.294761\pi\)
\(212\) 346944. 0.530176
\(213\) 387720. 0.585557
\(214\) −84048.0 −0.125456
\(215\) 0 0
\(216\) −46656.0 −0.0680414
\(217\) 7583.00 0.0109318
\(218\) 950180. 1.35414
\(219\) −260478. −0.366996
\(220\) 0 0
\(221\) −76038.0 −0.104725
\(222\) −134712. −0.183453
\(223\) 776185. 1.04521 0.522604 0.852575i \(-0.324960\pi\)
0.522604 + 0.852575i \(0.324960\pi\)
\(224\) 1024.00 0.00136358
\(225\) 0 0
\(226\) −933552. −1.21581
\(227\) −919992. −1.18500 −0.592501 0.805569i \(-0.701860\pi\)
−0.592501 + 0.805569i \(0.701860\pi\)
\(228\) −83664.0 −0.106586
\(229\) −1.13220e6 −1.42670 −0.713350 0.700808i \(-0.752823\pi\)
−0.713350 + 0.700808i \(0.752823\pi\)
\(230\) 0 0
\(231\) 1890.00 0.00233041
\(232\) −8064.00 −0.00983627
\(233\) −823128. −0.993293 −0.496647 0.867953i \(-0.665436\pi\)
−0.496647 + 0.867953i \(0.665436\pi\)
\(234\) 216108. 0.258007
\(235\) 0 0
\(236\) −516960. −0.604195
\(237\) −248472. −0.287347
\(238\) −456.000 −0.000521822 0
\(239\) 1.32836e6 1.50426 0.752129 0.659016i \(-0.229027\pi\)
0.752129 + 0.659016i \(0.229027\pi\)
\(240\) 0 0
\(241\) 94937.0 0.105291 0.0526457 0.998613i \(-0.483235\pi\)
0.0526457 + 0.998613i \(0.483235\pi\)
\(242\) −467804. −0.513482
\(243\) −59049.0 −0.0641500
\(244\) −114640. −0.123271
\(245\) 0 0
\(246\) 102816. 0.108324
\(247\) 387527. 0.404166
\(248\) 485312. 0.501062
\(249\) −16038.0 −0.0163928
\(250\) 0 0
\(251\) −128124. −0.128365 −0.0641824 0.997938i \(-0.520444\pi\)
−0.0641824 + 0.997938i \(0.520444\pi\)
\(252\) 1296.00 0.00128559
\(253\) −913500. −0.897237
\(254\) −555200. −0.539964
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 1.72285e6 1.62710 0.813552 0.581493i \(-0.197531\pi\)
0.813552 + 0.581493i \(0.197531\pi\)
\(258\) −656676. −0.614189
\(259\) 3742.00 0.00346621
\(260\) 0 0
\(261\) −10206.0 −0.00927373
\(262\) −645360. −0.580830
\(263\) −692160. −0.617045 −0.308523 0.951217i \(-0.599835\pi\)
−0.308523 + 0.951217i \(0.599835\pi\)
\(264\) 120960. 0.106815
\(265\) 0 0
\(266\) 2324.00 0.00201387
\(267\) −451872. −0.387916
\(268\) −953264. −0.810730
\(269\) 690894. 0.582144 0.291072 0.956701i \(-0.405988\pi\)
0.291072 + 0.956701i \(0.405988\pi\)
\(270\) 0 0
\(271\) −1.04690e6 −0.865925 −0.432963 0.901412i \(-0.642532\pi\)
−0.432963 + 0.901412i \(0.642532\pi\)
\(272\) −29184.0 −0.0239179
\(273\) −6003.00 −0.00487486
\(274\) −276216. −0.222266
\(275\) 0 0
\(276\) −626400. −0.494970
\(277\) 1.07137e6 0.838955 0.419478 0.907766i \(-0.362213\pi\)
0.419478 + 0.907766i \(0.362213\pi\)
\(278\) 897584. 0.696567
\(279\) 614223. 0.472406
\(280\) 0 0
\(281\) −1.40416e6 −1.06084 −0.530422 0.847734i \(-0.677966\pi\)
−0.530422 + 0.847734i \(0.677966\pi\)
\(282\) −841320. −0.629997
\(283\) −1.68203e6 −1.24844 −0.624221 0.781248i \(-0.714584\pi\)
−0.624221 + 0.781248i \(0.714584\pi\)
\(284\) −689280. −0.507107
\(285\) 0 0
\(286\) −560280. −0.405033
\(287\) −2856.00 −0.00204670
\(288\) 82944.0 0.0589256
\(289\) −1.40686e6 −0.990847
\(290\) 0 0
\(291\) 1.28514e6 0.889645
\(292\) 463072. 0.317827
\(293\) −1.76293e6 −1.19968 −0.599839 0.800120i \(-0.704769\pi\)
−0.599839 + 0.800120i \(0.704769\pi\)
\(294\) 605016. 0.408224
\(295\) 0 0
\(296\) 239488. 0.158875
\(297\) 153090. 0.100706
\(298\) −1.63058e6 −1.06366
\(299\) 2.90145e6 1.87688
\(300\) 0 0
\(301\) 18241.0 0.0116047
\(302\) 864212. 0.545259
\(303\) 1.47647e6 0.923884
\(304\) 148736. 0.0923065
\(305\) 0 0
\(306\) −36936.0 −0.0225500
\(307\) −471905. −0.285765 −0.142882 0.989740i \(-0.545637\pi\)
−0.142882 + 0.989740i \(0.545637\pi\)
\(308\) −3360.00 −0.00201819
\(309\) −965124. −0.575025
\(310\) 0 0
\(311\) −1.64439e6 −0.964060 −0.482030 0.876155i \(-0.660100\pi\)
−0.482030 + 0.876155i \(0.660100\pi\)
\(312\) −384192. −0.223440
\(313\) −2.37908e6 −1.37262 −0.686308 0.727311i \(-0.740770\pi\)
−0.686308 + 0.727311i \(0.740770\pi\)
\(314\) 754468. 0.431834
\(315\) 0 0
\(316\) 441728. 0.248850
\(317\) −1.32679e6 −0.741574 −0.370787 0.928718i \(-0.620912\pi\)
−0.370787 + 0.928718i \(0.620912\pi\)
\(318\) −780624. −0.432887
\(319\) 26460.0 0.0145584
\(320\) 0 0
\(321\) 189108. 0.102435
\(322\) 17400.0 0.00935211
\(323\) −66234.0 −0.0353244
\(324\) 104976. 0.0555556
\(325\) 0 0
\(326\) 1.68900e6 0.880211
\(327\) −2.13790e6 −1.10565
\(328\) −182784. −0.0938110
\(329\) 23370.0 0.0119033
\(330\) 0 0
\(331\) 661172. 0.331699 0.165850 0.986151i \(-0.446963\pi\)
0.165850 + 0.986151i \(0.446963\pi\)
\(332\) 28512.0 0.0141965
\(333\) 303102. 0.149788
\(334\) −166800. −0.0818144
\(335\) 0 0
\(336\) −2304.00 −0.00111336
\(337\) 1.53675e6 0.737104 0.368552 0.929607i \(-0.379854\pi\)
0.368552 + 0.929607i \(0.379854\pi\)
\(338\) 294384. 0.140159
\(339\) 2.10049e6 0.992709
\(340\) 0 0
\(341\) −1.59243e6 −0.741608
\(342\) 188244. 0.0870274
\(343\) −33613.0 −0.0154267
\(344\) 1.16742e6 0.531903
\(345\) 0 0
\(346\) −1.06457e6 −0.478061
\(347\) −3.86576e6 −1.72350 −0.861749 0.507334i \(-0.830631\pi\)
−0.861749 + 0.507334i \(0.830631\pi\)
\(348\) 18144.0 0.00803128
\(349\) −4.23391e6 −1.86071 −0.930354 0.366663i \(-0.880500\pi\)
−0.930354 + 0.366663i \(0.880500\pi\)
\(350\) 0 0
\(351\) −486243. −0.210662
\(352\) −215040. −0.0925044
\(353\) 2.41459e6 1.03135 0.515675 0.856784i \(-0.327541\pi\)
0.515675 + 0.856784i \(0.327541\pi\)
\(354\) 1.16316e6 0.493323
\(355\) 0 0
\(356\) 803328. 0.335945
\(357\) 1026.00 0.000426066 0
\(358\) −207384. −0.0855200
\(359\) 2.94802e6 1.20724 0.603620 0.797272i \(-0.293724\pi\)
0.603620 + 0.797272i \(0.293724\pi\)
\(360\) 0 0
\(361\) −2.13854e6 −0.863672
\(362\) 1.99740e6 0.801111
\(363\) 1.05256e6 0.419257
\(364\) 10672.0 0.00422175
\(365\) 0 0
\(366\) 257940. 0.100651
\(367\) −679793. −0.263458 −0.131729 0.991286i \(-0.542053\pi\)
−0.131729 + 0.991286i \(0.542053\pi\)
\(368\) 1.11360e6 0.428657
\(369\) −231336. −0.0884458
\(370\) 0 0
\(371\) 21684.0 0.00817908
\(372\) −1.09195e6 −0.409116
\(373\) 3.60888e6 1.34308 0.671538 0.740970i \(-0.265634\pi\)
0.671538 + 0.740970i \(0.265634\pi\)
\(374\) 95760.0 0.0354002
\(375\) 0 0
\(376\) 1.49568e6 0.545593
\(377\) −84042.0 −0.0304539
\(378\) −2916.00 −0.00104968
\(379\) −386809. −0.138324 −0.0691622 0.997605i \(-0.522033\pi\)
−0.0691622 + 0.997605i \(0.522033\pi\)
\(380\) 0 0
\(381\) 1.24920e6 0.440879
\(382\) 2.82362e6 0.990030
\(383\) 645060. 0.224700 0.112350 0.993669i \(-0.464162\pi\)
0.112350 + 0.993669i \(0.464162\pi\)
\(384\) −147456. −0.0510310
\(385\) 0 0
\(386\) 2.07989e6 0.710514
\(387\) 1.47752e6 0.501483
\(388\) −2.28469e6 −0.770456
\(389\) −812976. −0.272398 −0.136199 0.990682i \(-0.543489\pi\)
−0.136199 + 0.990682i \(0.543489\pi\)
\(390\) 0 0
\(391\) −495900. −0.164041
\(392\) −1.07558e6 −0.353532
\(393\) 1.45206e6 0.474245
\(394\) 801576. 0.260138
\(395\) 0 0
\(396\) −272160. −0.0872140
\(397\) 988813. 0.314875 0.157437 0.987529i \(-0.449677\pi\)
0.157437 + 0.987529i \(0.449677\pi\)
\(398\) 2.21469e6 0.700819
\(399\) −5229.00 −0.00164432
\(400\) 0 0
\(401\) 2.14580e6 0.666391 0.333195 0.942858i \(-0.391873\pi\)
0.333195 + 0.942858i \(0.391873\pi\)
\(402\) 2.14484e6 0.661958
\(403\) 5.05786e6 1.55133
\(404\) −2.62483e6 −0.800107
\(405\) 0 0
\(406\) −504.000 −0.000151745 0
\(407\) −785820. −0.235146
\(408\) 65664.0 0.0195289
\(409\) −5.77658e6 −1.70751 −0.853753 0.520678i \(-0.825679\pi\)
−0.853753 + 0.520678i \(0.825679\pi\)
\(410\) 0 0
\(411\) 621486. 0.181479
\(412\) 1.71578e6 0.497986
\(413\) −32310.0 −0.00932099
\(414\) 1.40940e6 0.404141
\(415\) 0 0
\(416\) 683008. 0.193505
\(417\) −2.01956e6 −0.568745
\(418\) −488040. −0.136620
\(419\) −5.07318e6 −1.41171 −0.705855 0.708357i \(-0.749437\pi\)
−0.705855 + 0.708357i \(0.749437\pi\)
\(420\) 0 0
\(421\) −3.01297e6 −0.828494 −0.414247 0.910164i \(-0.635955\pi\)
−0.414247 + 0.910164i \(0.635955\pi\)
\(422\) 3.10946e6 0.849970
\(423\) 1.89297e6 0.514390
\(424\) 1.38778e6 0.374891
\(425\) 0 0
\(426\) 1.55088e6 0.414051
\(427\) −7165.00 −0.00190172
\(428\) −336192. −0.0887111
\(429\) 1.26063e6 0.330708
\(430\) 0 0
\(431\) 5.59449e6 1.45067 0.725333 0.688398i \(-0.241686\pi\)
0.725333 + 0.688398i \(0.241686\pi\)
\(432\) −186624. −0.0481125
\(433\) −5.63454e6 −1.44424 −0.722119 0.691769i \(-0.756832\pi\)
−0.722119 + 0.691769i \(0.756832\pi\)
\(434\) 30332.0 0.00772995
\(435\) 0 0
\(436\) 3.80072e6 0.957524
\(437\) 2.52735e6 0.633085
\(438\) −1.04191e6 −0.259505
\(439\) 860663. 0.213143 0.106572 0.994305i \(-0.466013\pi\)
0.106572 + 0.994305i \(0.466013\pi\)
\(440\) 0 0
\(441\) −1.36129e6 −0.333314
\(442\) −304152. −0.0740517
\(443\) 2.75335e6 0.666580 0.333290 0.942824i \(-0.391841\pi\)
0.333290 + 0.942824i \(0.391841\pi\)
\(444\) −538848. −0.129721
\(445\) 0 0
\(446\) 3.10474e6 0.739074
\(447\) 3.66881e6 0.868474
\(448\) 4096.00 0.000964195 0
\(449\) −2.83168e6 −0.662869 −0.331434 0.943478i \(-0.607533\pi\)
−0.331434 + 0.943478i \(0.607533\pi\)
\(450\) 0 0
\(451\) 599760. 0.138847
\(452\) −3.73421e6 −0.859711
\(453\) −1.94448e6 −0.445202
\(454\) −3.67997e6 −0.837924
\(455\) 0 0
\(456\) −334656. −0.0753679
\(457\) 4.66235e6 1.04427 0.522136 0.852862i \(-0.325135\pi\)
0.522136 + 0.852862i \(0.325135\pi\)
\(458\) −4.52878e6 −1.00883
\(459\) 83106.0 0.0184120
\(460\) 0 0
\(461\) −5.27154e6 −1.15527 −0.577637 0.816294i \(-0.696025\pi\)
−0.577637 + 0.816294i \(0.696025\pi\)
\(462\) 7560.00 0.00164785
\(463\) 7.16954e6 1.55431 0.777157 0.629307i \(-0.216661\pi\)
0.777157 + 0.629307i \(0.216661\pi\)
\(464\) −32256.0 −0.00695530
\(465\) 0 0
\(466\) −3.29251e6 −0.702365
\(467\) −4.30441e6 −0.913316 −0.456658 0.889642i \(-0.650954\pi\)
−0.456658 + 0.889642i \(0.650954\pi\)
\(468\) 864432. 0.182438
\(469\) −59579.0 −0.0125072
\(470\) 0 0
\(471\) −1.69755e6 −0.352591
\(472\) −2.06784e6 −0.427230
\(473\) −3.83061e6 −0.787254
\(474\) −993888. −0.203185
\(475\) 0 0
\(476\) −1824.00 −0.000368984 0
\(477\) 1.75640e6 0.353450
\(478\) 5.31346e6 1.06367
\(479\) 190506. 0.0379376 0.0189688 0.999820i \(-0.493962\pi\)
0.0189688 + 0.999820i \(0.493962\pi\)
\(480\) 0 0
\(481\) 2.49591e6 0.491889
\(482\) 379748. 0.0744523
\(483\) −39150.0 −0.00763597
\(484\) −1.87122e6 −0.363087
\(485\) 0 0
\(486\) −236196. −0.0453609
\(487\) 128887. 0.0246256 0.0123128 0.999924i \(-0.496081\pi\)
0.0123128 + 0.999924i \(0.496081\pi\)
\(488\) −458560. −0.0871659
\(489\) −3.80026e6 −0.718689
\(490\) 0 0
\(491\) −304296. −0.0569630 −0.0284815 0.999594i \(-0.509067\pi\)
−0.0284815 + 0.999594i \(0.509067\pi\)
\(492\) 411264. 0.0765963
\(493\) 14364.0 0.00266169
\(494\) 1.55011e6 0.285788
\(495\) 0 0
\(496\) 1.94125e6 0.354305
\(497\) −43080.0 −0.00782321
\(498\) −64152.0 −0.0115914
\(499\) 6.41974e6 1.15416 0.577080 0.816688i \(-0.304192\pi\)
0.577080 + 0.816688i \(0.304192\pi\)
\(500\) 0 0
\(501\) 375300. 0.0668012
\(502\) −512496. −0.0907677
\(503\) 1.55534e6 0.274098 0.137049 0.990564i \(-0.456238\pi\)
0.137049 + 0.990564i \(0.456238\pi\)
\(504\) 5184.00 0.000909052 0
\(505\) 0 0
\(506\) −3.65400e6 −0.634442
\(507\) −662364. −0.114440
\(508\) −2.22080e6 −0.381813
\(509\) −541506. −0.0926422 −0.0463211 0.998927i \(-0.514750\pi\)
−0.0463211 + 0.998927i \(0.514750\pi\)
\(510\) 0 0
\(511\) 28942.0 0.00490316
\(512\) 262144. 0.0441942
\(513\) −423549. −0.0710576
\(514\) 6.89141e6 1.15054
\(515\) 0 0
\(516\) −2.62670e6 −0.434297
\(517\) −4.90770e6 −0.807517
\(518\) 14968.0 0.00245098
\(519\) 2.39528e6 0.390335
\(520\) 0 0
\(521\) −888762. −0.143447 −0.0717235 0.997425i \(-0.522850\pi\)
−0.0717235 + 0.997425i \(0.522850\pi\)
\(522\) −40824.0 −0.00655752
\(523\) −7.69950e6 −1.23086 −0.615430 0.788192i \(-0.711018\pi\)
−0.615430 + 0.788192i \(0.711018\pi\)
\(524\) −2.58144e6 −0.410709
\(525\) 0 0
\(526\) −2.76864e6 −0.436317
\(527\) −864462. −0.135587
\(528\) 483840. 0.0755296
\(529\) 1.24862e7 1.93995
\(530\) 0 0
\(531\) −2.61711e6 −0.402796
\(532\) 9296.00 0.00142402
\(533\) −1.90495e6 −0.290446
\(534\) −1.80749e6 −0.274298
\(535\) 0 0
\(536\) −3.81306e6 −0.573273
\(537\) 466614. 0.0698268
\(538\) 2.76358e6 0.411638
\(539\) 3.52926e6 0.523253
\(540\) 0 0
\(541\) −3.01878e6 −0.443443 −0.221721 0.975110i \(-0.571168\pi\)
−0.221721 + 0.975110i \(0.571168\pi\)
\(542\) −4.18758e6 −0.612301
\(543\) −4.49414e6 −0.654104
\(544\) −116736. −0.0169125
\(545\) 0 0
\(546\) −24012.0 −0.00344704
\(547\) −7.23481e6 −1.03385 −0.516926 0.856030i \(-0.672924\pi\)
−0.516926 + 0.856030i \(0.672924\pi\)
\(548\) −1.10486e6 −0.157166
\(549\) −580365. −0.0821808
\(550\) 0 0
\(551\) −73206.0 −0.0102723
\(552\) −2.50560e6 −0.349997
\(553\) 27608.0 0.00383904
\(554\) 4.28547e6 0.593231
\(555\) 0 0
\(556\) 3.59034e6 0.492547
\(557\) 9.88712e6 1.35030 0.675152 0.737678i \(-0.264078\pi\)
0.675152 + 0.737678i \(0.264078\pi\)
\(558\) 2.45689e6 0.334042
\(559\) 1.21667e7 1.64682
\(560\) 0 0
\(561\) −215460. −0.0289041
\(562\) −5.61665e6 −0.750130
\(563\) −946362. −0.125831 −0.0629153 0.998019i \(-0.520040\pi\)
−0.0629153 + 0.998019i \(0.520040\pi\)
\(564\) −3.36528e6 −0.445475
\(565\) 0 0
\(566\) −6.72813e6 −0.882782
\(567\) 6561.00 0.000857062 0
\(568\) −2.75712e6 −0.358579
\(569\) −4.51552e6 −0.584692 −0.292346 0.956313i \(-0.594436\pi\)
−0.292346 + 0.956313i \(0.594436\pi\)
\(570\) 0 0
\(571\) 7.57824e6 0.972699 0.486349 0.873764i \(-0.338328\pi\)
0.486349 + 0.873764i \(0.338328\pi\)
\(572\) −2.24112e6 −0.286401
\(573\) −6.35315e6 −0.808356
\(574\) −11424.0 −0.00144723
\(575\) 0 0
\(576\) 331776. 0.0416667
\(577\) −1.11688e7 −1.39659 −0.698295 0.715811i \(-0.746057\pi\)
−0.698295 + 0.715811i \(0.746057\pi\)
\(578\) −5.62744e6 −0.700635
\(579\) −4.67976e6 −0.580132
\(580\) 0 0
\(581\) 1782.00 0.000219012 0
\(582\) 5.14055e6 0.629074
\(583\) −4.55364e6 −0.554865
\(584\) 1.85229e6 0.224738
\(585\) 0 0
\(586\) −7.05170e6 −0.848301
\(587\) −1.01053e6 −0.121047 −0.0605236 0.998167i \(-0.519277\pi\)
−0.0605236 + 0.998167i \(0.519277\pi\)
\(588\) 2.42006e6 0.288658
\(589\) 4.40572e6 0.523274
\(590\) 0 0
\(591\) −1.80355e6 −0.212402
\(592\) 957952. 0.112341
\(593\) −1.13674e7 −1.32746 −0.663732 0.747970i \(-0.731029\pi\)
−0.663732 + 0.747970i \(0.731029\pi\)
\(594\) 612360. 0.0712100
\(595\) 0 0
\(596\) −6.52234e6 −0.752121
\(597\) −4.98306e6 −0.572216
\(598\) 1.16058e7 1.32716
\(599\) 2.56732e6 0.292356 0.146178 0.989258i \(-0.453303\pi\)
0.146178 + 0.989258i \(0.453303\pi\)
\(600\) 0 0
\(601\) 1.59837e7 1.80506 0.902530 0.430626i \(-0.141707\pi\)
0.902530 + 0.430626i \(0.141707\pi\)
\(602\) 72964.0 0.00820573
\(603\) −4.82590e6 −0.540487
\(604\) 3.45685e6 0.385556
\(605\) 0 0
\(606\) 5.90587e6 0.653285
\(607\) 1.01626e7 1.11952 0.559760 0.828655i \(-0.310893\pi\)
0.559760 + 0.828655i \(0.310893\pi\)
\(608\) 594944. 0.0652705
\(609\) 1134.00 0.000123900 0
\(610\) 0 0
\(611\) 1.55878e7 1.68920
\(612\) −147744. −0.0159452
\(613\) −5.83042e6 −0.626684 −0.313342 0.949640i \(-0.601449\pi\)
−0.313342 + 0.949640i \(0.601449\pi\)
\(614\) −1.88762e6 −0.202066
\(615\) 0 0
\(616\) −13440.0 −0.00142708
\(617\) 1.51029e7 1.59716 0.798578 0.601891i \(-0.205586\pi\)
0.798578 + 0.601891i \(0.205586\pi\)
\(618\) −3.86050e6 −0.406604
\(619\) −5.38515e6 −0.564900 −0.282450 0.959282i \(-0.591147\pi\)
−0.282450 + 0.959282i \(0.591147\pi\)
\(620\) 0 0
\(621\) −3.17115e6 −0.329980
\(622\) −6.57756e6 −0.681693
\(623\) 50208.0 0.00518266
\(624\) −1.53677e6 −0.157996
\(625\) 0 0
\(626\) −9.51633e6 −0.970585
\(627\) 1.09809e6 0.111550
\(628\) 3.01787e6 0.305353
\(629\) −426588. −0.0429914
\(630\) 0 0
\(631\) −5.59345e6 −0.559251 −0.279625 0.960109i \(-0.590210\pi\)
−0.279625 + 0.960109i \(0.590210\pi\)
\(632\) 1.76691e6 0.175963
\(633\) −6.99628e6 −0.693998
\(634\) −5.30717e6 −0.524372
\(635\) 0 0
\(636\) −3.12250e6 −0.306097
\(637\) −1.12096e7 −1.09457
\(638\) 105840. 0.0102943
\(639\) −3.48948e6 −0.338072
\(640\) 0 0
\(641\) −1.00072e7 −0.961985 −0.480993 0.876725i \(-0.659724\pi\)
−0.480993 + 0.876725i \(0.659724\pi\)
\(642\) 756432. 0.0724323
\(643\) 9.95036e6 0.949098 0.474549 0.880229i \(-0.342611\pi\)
0.474549 + 0.880229i \(0.342611\pi\)
\(644\) 69600.0 0.00661294
\(645\) 0 0
\(646\) −264936. −0.0249781
\(647\) 9.16904e6 0.861119 0.430560 0.902562i \(-0.358316\pi\)
0.430560 + 0.902562i \(0.358316\pi\)
\(648\) 419904. 0.0392837
\(649\) 6.78510e6 0.632331
\(650\) 0 0
\(651\) −68247.0 −0.00631148
\(652\) 6.75602e6 0.622403
\(653\) 1.77127e7 1.62555 0.812777 0.582575i \(-0.197955\pi\)
0.812777 + 0.582575i \(0.197955\pi\)
\(654\) −8.55162e6 −0.781815
\(655\) 0 0
\(656\) −731136. −0.0663344
\(657\) 2.34430e6 0.211885
\(658\) 93480.0 0.00841693
\(659\) −2.07949e7 −1.86527 −0.932637 0.360816i \(-0.882498\pi\)
−0.932637 + 0.360816i \(0.882498\pi\)
\(660\) 0 0
\(661\) −1.51745e6 −0.135086 −0.0675429 0.997716i \(-0.521516\pi\)
−0.0675429 + 0.997716i \(0.521516\pi\)
\(662\) 2.64469e6 0.234547
\(663\) 684342. 0.0604630
\(664\) 114048. 0.0100385
\(665\) 0 0
\(666\) 1.21241e6 0.105916
\(667\) −548100. −0.0477029
\(668\) −667200. −0.0578515
\(669\) −6.98566e6 −0.603452
\(670\) 0 0
\(671\) 1.50465e6 0.129012
\(672\) −9216.00 −0.000787262 0
\(673\) 2.90483e6 0.247219 0.123610 0.992331i \(-0.460553\pi\)
0.123610 + 0.992331i \(0.460553\pi\)
\(674\) 6.14700e6 0.521211
\(675\) 0 0
\(676\) 1.17754e6 0.0991077
\(677\) 7.06049e6 0.592057 0.296028 0.955179i \(-0.404338\pi\)
0.296028 + 0.955179i \(0.404338\pi\)
\(678\) 8.40197e6 0.701951
\(679\) −142793. −0.0118859
\(680\) 0 0
\(681\) 8.27993e6 0.684162
\(682\) −6.36972e6 −0.524396
\(683\) −3.81415e6 −0.312857 −0.156429 0.987689i \(-0.549998\pi\)
−0.156429 + 0.987689i \(0.549998\pi\)
\(684\) 752976. 0.0615376
\(685\) 0 0
\(686\) −134452. −0.0109083
\(687\) 1.01898e7 0.823705
\(688\) 4.66970e6 0.376112
\(689\) 1.44632e7 1.16069
\(690\) 0 0
\(691\) −8.95166e6 −0.713195 −0.356597 0.934258i \(-0.616063\pi\)
−0.356597 + 0.934258i \(0.616063\pi\)
\(692\) −4.25827e6 −0.338040
\(693\) −17010.0 −0.00134546
\(694\) −1.54630e7 −1.21870
\(695\) 0 0
\(696\) 72576.0 0.00567897
\(697\) 325584. 0.0253852
\(698\) −1.69356e7 −1.31572
\(699\) 7.40815e6 0.573478
\(700\) 0 0
\(701\) 8.45297e6 0.649702 0.324851 0.945765i \(-0.394686\pi\)
0.324851 + 0.945765i \(0.394686\pi\)
\(702\) −1.94497e6 −0.148960
\(703\) 2.17410e6 0.165917
\(704\) −860160. −0.0654105
\(705\) 0 0
\(706\) 9.65834e6 0.729274
\(707\) −164052. −0.0123433
\(708\) 4.65264e6 0.348832
\(709\) −1.18033e7 −0.881834 −0.440917 0.897548i \(-0.645347\pi\)
−0.440917 + 0.897548i \(0.645347\pi\)
\(710\) 0 0
\(711\) 2.23625e6 0.165900
\(712\) 3.21331e6 0.237549
\(713\) 3.29861e7 2.43000
\(714\) 4104.00 0.000301274 0
\(715\) 0 0
\(716\) −829536. −0.0604718
\(717\) −1.19553e7 −0.868484
\(718\) 1.17921e7 0.853648
\(719\) −2.68089e7 −1.93400 −0.967000 0.254778i \(-0.917998\pi\)
−0.967000 + 0.254778i \(0.917998\pi\)
\(720\) 0 0
\(721\) 107236. 0.00768250
\(722\) −8.55415e6 −0.610709
\(723\) −854433. −0.0607900
\(724\) 7.98958e6 0.566471
\(725\) 0 0
\(726\) 4.21024e6 0.296459
\(727\) 2.00638e6 0.140792 0.0703958 0.997519i \(-0.477574\pi\)
0.0703958 + 0.997519i \(0.477574\pi\)
\(728\) 42688.0 0.00298523
\(729\) 531441. 0.0370370
\(730\) 0 0
\(731\) −2.07947e6 −0.143933
\(732\) 1.03176e6 0.0711707
\(733\) 9.16554e6 0.630083 0.315042 0.949078i \(-0.397981\pi\)
0.315042 + 0.949078i \(0.397981\pi\)
\(734\) −2.71917e6 −0.186293
\(735\) 0 0
\(736\) 4.45440e6 0.303106
\(737\) 1.25116e7 0.848484
\(738\) −925344. −0.0625406
\(739\) 1.55897e7 1.05009 0.525047 0.851073i \(-0.324048\pi\)
0.525047 + 0.851073i \(0.324048\pi\)
\(740\) 0 0
\(741\) −3.48774e6 −0.233345
\(742\) 86736.0 0.00578349
\(743\) 3.78304e6 0.251402 0.125701 0.992068i \(-0.459882\pi\)
0.125701 + 0.992068i \(0.459882\pi\)
\(744\) −4.36781e6 −0.289289
\(745\) 0 0
\(746\) 1.44355e7 0.949698
\(747\) 144342. 0.00946436
\(748\) 383040. 0.0250317
\(749\) −21012.0 −0.00136856
\(750\) 0 0
\(751\) 1.24726e7 0.806972 0.403486 0.914986i \(-0.367798\pi\)
0.403486 + 0.914986i \(0.367798\pi\)
\(752\) 5.98272e6 0.385793
\(753\) 1.15312e6 0.0741115
\(754\) −336168. −0.0215342
\(755\) 0 0
\(756\) −11664.0 −0.000742238 0
\(757\) −1.87121e6 −0.118682 −0.0593408 0.998238i \(-0.518900\pi\)
−0.0593408 + 0.998238i \(0.518900\pi\)
\(758\) −1.54724e6 −0.0978101
\(759\) 8.22150e6 0.518020
\(760\) 0 0
\(761\) 2.53638e7 1.58764 0.793821 0.608152i \(-0.208089\pi\)
0.793821 + 0.608152i \(0.208089\pi\)
\(762\) 4.99680e6 0.311749
\(763\) 237545. 0.0147718
\(764\) 1.12945e7 0.700057
\(765\) 0 0
\(766\) 2.58024e6 0.158887
\(767\) −2.15508e7 −1.32274
\(768\) −589824. −0.0360844
\(769\) −1.43729e7 −0.876452 −0.438226 0.898865i \(-0.644393\pi\)
−0.438226 + 0.898865i \(0.644393\pi\)
\(770\) 0 0
\(771\) −1.55057e7 −0.939409
\(772\) 8.31957e6 0.502409
\(773\) 1.64185e7 0.988289 0.494145 0.869380i \(-0.335481\pi\)
0.494145 + 0.869380i \(0.335481\pi\)
\(774\) 5.91008e6 0.354602
\(775\) 0 0
\(776\) −9.13875e6 −0.544794
\(777\) −33678.0 −0.00200121
\(778\) −3.25190e6 −0.192614
\(779\) −1.65934e6 −0.0979695
\(780\) 0 0
\(781\) 9.04680e6 0.530722
\(782\) −1.98360e6 −0.115994
\(783\) 91854.0 0.00535419
\(784\) −4.30234e6 −0.249985
\(785\) 0 0
\(786\) 5.80824e6 0.335342
\(787\) 1.46320e7 0.842106 0.421053 0.907036i \(-0.361661\pi\)
0.421053 + 0.907036i \(0.361661\pi\)
\(788\) 3.20630e6 0.183945
\(789\) 6.22944e6 0.356251
\(790\) 0 0
\(791\) −233388. −0.0132629
\(792\) −1.08864e6 −0.0616696
\(793\) −4.77906e6 −0.269873
\(794\) 3.95525e6 0.222650
\(795\) 0 0
\(796\) 8.85877e6 0.495554
\(797\) −2.61796e7 −1.45988 −0.729941 0.683511i \(-0.760452\pi\)
−0.729941 + 0.683511i \(0.760452\pi\)
\(798\) −20916.0 −0.00116271
\(799\) −2.66418e6 −0.147637
\(800\) 0 0
\(801\) 4.06685e6 0.223963
\(802\) 8.58322e6 0.471210
\(803\) −6.07782e6 −0.332628
\(804\) 8.57938e6 0.468075
\(805\) 0 0
\(806\) 2.02314e7 1.09696
\(807\) −6.21805e6 −0.336101
\(808\) −1.04993e7 −0.565761
\(809\) 2.99051e7 1.60648 0.803238 0.595658i \(-0.203109\pi\)
0.803238 + 0.595658i \(0.203109\pi\)
\(810\) 0 0
\(811\) −3.12873e7 −1.67038 −0.835190 0.549961i \(-0.814642\pi\)
−0.835190 + 0.549961i \(0.814642\pi\)
\(812\) −2016.00 −0.000107300 0
\(813\) 9.42206e6 0.499942
\(814\) −3.14328e6 −0.166273
\(815\) 0 0
\(816\) 262656. 0.0138090
\(817\) 1.05980e7 0.555482
\(818\) −2.31063e7 −1.20739
\(819\) 54027.0 0.00281450
\(820\) 0 0
\(821\) −3.24291e7 −1.67910 −0.839550 0.543282i \(-0.817182\pi\)
−0.839550 + 0.543282i \(0.817182\pi\)
\(822\) 2.48594e6 0.128325
\(823\) −1.91700e7 −0.986560 −0.493280 0.869871i \(-0.664202\pi\)
−0.493280 + 0.869871i \(0.664202\pi\)
\(824\) 6.86310e6 0.352130
\(825\) 0 0
\(826\) −129240. −0.00659093
\(827\) 98052.0 0.00498532 0.00249266 0.999997i \(-0.499207\pi\)
0.00249266 + 0.999997i \(0.499207\pi\)
\(828\) 5.63760e6 0.285771
\(829\) 5.77426e6 0.291817 0.145908 0.989298i \(-0.453390\pi\)
0.145908 + 0.989298i \(0.453390\pi\)
\(830\) 0 0
\(831\) −9.64230e6 −0.484371
\(832\) 2.73203e6 0.136829
\(833\) 1.91588e6 0.0956658
\(834\) −8.07826e6 −0.402163
\(835\) 0 0
\(836\) −1.95216e6 −0.0966050
\(837\) −5.52801e6 −0.272744
\(838\) −2.02927e7 −0.998229
\(839\) −1.69354e7 −0.830599 −0.415299 0.909685i \(-0.636323\pi\)
−0.415299 + 0.909685i \(0.636323\pi\)
\(840\) 0 0
\(841\) −2.04953e7 −0.999226
\(842\) −1.20519e7 −0.585834
\(843\) 1.26375e7 0.612478
\(844\) 1.24378e7 0.601020
\(845\) 0 0
\(846\) 7.57188e6 0.363729
\(847\) −116951. −0.00560138
\(848\) 5.55110e6 0.265088
\(849\) 1.51383e7 0.720788
\(850\) 0 0
\(851\) 1.62777e7 0.770494
\(852\) 6.20352e6 0.292779
\(853\) −5.04391e6 −0.237353 −0.118677 0.992933i \(-0.537865\pi\)
−0.118677 + 0.992933i \(0.537865\pi\)
\(854\) −28660.0 −0.00134472
\(855\) 0 0
\(856\) −1.34477e6 −0.0627282
\(857\) −1.28829e7 −0.599188 −0.299594 0.954067i \(-0.596851\pi\)
−0.299594 + 0.954067i \(0.596851\pi\)
\(858\) 5.04252e6 0.233846
\(859\) −3.59522e7 −1.66243 −0.831213 0.555954i \(-0.812353\pi\)
−0.831213 + 0.555954i \(0.812353\pi\)
\(860\) 0 0
\(861\) 25704.0 0.00118166
\(862\) 2.23780e7 1.02578
\(863\) −2.20345e7 −1.00711 −0.503554 0.863964i \(-0.667975\pi\)
−0.503554 + 0.863964i \(0.667975\pi\)
\(864\) −746496. −0.0340207
\(865\) 0 0
\(866\) −2.25382e7 −1.02123
\(867\) 1.26617e7 0.572066
\(868\) 121328. 0.00546590
\(869\) −5.79768e6 −0.260438
\(870\) 0 0
\(871\) −3.97392e7 −1.77490
\(872\) 1.52029e7 0.677072
\(873\) −1.15662e7 −0.513637
\(874\) 1.01094e7 0.447658
\(875\) 0 0
\(876\) −4.16765e6 −0.183498
\(877\) −1.34531e7 −0.590641 −0.295320 0.955398i \(-0.595426\pi\)
−0.295320 + 0.955398i \(0.595426\pi\)
\(878\) 3.44265e6 0.150715
\(879\) 1.58663e7 0.692635
\(880\) 0 0
\(881\) 2.36331e7 1.02584 0.512922 0.858435i \(-0.328563\pi\)
0.512922 + 0.858435i \(0.328563\pi\)
\(882\) −5.44514e6 −0.235688
\(883\) −3.49588e7 −1.50888 −0.754439 0.656370i \(-0.772091\pi\)
−0.754439 + 0.656370i \(0.772091\pi\)
\(884\) −1.21661e6 −0.0523625
\(885\) 0 0
\(886\) 1.10134e7 0.471343
\(887\) −3.11168e7 −1.32796 −0.663982 0.747749i \(-0.731135\pi\)
−0.663982 + 0.747749i \(0.731135\pi\)
\(888\) −2.15539e6 −0.0917263
\(889\) −138800. −0.00589027
\(890\) 0 0
\(891\) −1.37781e6 −0.0581427
\(892\) 1.24190e7 0.522604
\(893\) 1.35780e7 0.569779
\(894\) 1.46753e7 0.614104
\(895\) 0 0
\(896\) 16384.0 0.000681789 0
\(897\) −2.61130e7 −1.08362
\(898\) −1.13267e7 −0.468719
\(899\) −955458. −0.0394287
\(900\) 0 0
\(901\) −2.47198e6 −0.101445
\(902\) 2.39904e6 0.0981796
\(903\) −164169. −0.00669995
\(904\) −1.49368e7 −0.607907
\(905\) 0 0
\(906\) −7.77791e6 −0.314805
\(907\) 1.97320e7 0.796438 0.398219 0.917290i \(-0.369628\pi\)
0.398219 + 0.917290i \(0.369628\pi\)
\(908\) −1.47199e7 −0.592501
\(909\) −1.32882e7 −0.533405
\(910\) 0 0
\(911\) −4.40720e7 −1.75941 −0.879704 0.475521i \(-0.842260\pi\)
−0.879704 + 0.475521i \(0.842260\pi\)
\(912\) −1.33862e6 −0.0532932
\(913\) −374220. −0.0148576
\(914\) 1.86494e7 0.738412
\(915\) 0 0
\(916\) −1.81151e7 −0.713350
\(917\) −161340. −0.00633605
\(918\) 332424. 0.0130192
\(919\) −4.33621e7 −1.69364 −0.846821 0.531879i \(-0.821486\pi\)
−0.846821 + 0.531879i \(0.821486\pi\)
\(920\) 0 0
\(921\) 4.24714e6 0.164986
\(922\) −2.10862e7 −0.816902
\(923\) −2.87344e7 −1.11019
\(924\) 30240.0 0.00116520
\(925\) 0 0
\(926\) 2.86781e7 1.09907
\(927\) 8.68612e6 0.331991
\(928\) −129024. −0.00491814
\(929\) 1.65235e7 0.628147 0.314074 0.949399i \(-0.398306\pi\)
0.314074 + 0.949399i \(0.398306\pi\)
\(930\) 0 0
\(931\) −9.76429e6 −0.369204
\(932\) −1.31700e7 −0.496647
\(933\) 1.47995e7 0.556600
\(934\) −1.72176e7 −0.645812
\(935\) 0 0
\(936\) 3.45773e6 0.129003
\(937\) −2.03546e7 −0.757381 −0.378691 0.925523i \(-0.623626\pi\)
−0.378691 + 0.925523i \(0.623626\pi\)
\(938\) −238316. −0.00884395
\(939\) 2.14117e7 0.792480
\(940\) 0 0
\(941\) −5.12406e7 −1.88643 −0.943215 0.332184i \(-0.892214\pi\)
−0.943215 + 0.332184i \(0.892214\pi\)
\(942\) −6.79021e6 −0.249319
\(943\) −1.24236e7 −0.454955
\(944\) −8.27136e6 −0.302097
\(945\) 0 0
\(946\) −1.53224e7 −0.556673
\(947\) −2.79736e7 −1.01361 −0.506807 0.862059i \(-0.669174\pi\)
−0.506807 + 0.862059i \(0.669174\pi\)
\(948\) −3.97555e6 −0.143673
\(949\) 1.93043e7 0.695807
\(950\) 0 0
\(951\) 1.19411e7 0.428148
\(952\) −7296.00 −0.000260911 0
\(953\) −2.68337e7 −0.957080 −0.478540 0.878066i \(-0.658834\pi\)
−0.478540 + 0.878066i \(0.658834\pi\)
\(954\) 7.02562e6 0.249927
\(955\) 0 0
\(956\) 2.12538e7 0.752129
\(957\) −238140. −0.00840529
\(958\) 762024. 0.0268259
\(959\) −69054.0 −0.00242461
\(960\) 0 0
\(961\) 2.88727e7 1.00851
\(962\) 9.98366e6 0.347818
\(963\) −1.70197e6 −0.0591407
\(964\) 1.51899e6 0.0526457
\(965\) 0 0
\(966\) −156600. −0.00539944
\(967\) 5.24831e7 1.80490 0.902450 0.430794i \(-0.141766\pi\)
0.902450 + 0.430794i \(0.141766\pi\)
\(968\) −7.48486e6 −0.256741
\(969\) 596106. 0.0203945
\(970\) 0 0
\(971\) 5.01029e7 1.70536 0.852678 0.522437i \(-0.174977\pi\)
0.852678 + 0.522437i \(0.174977\pi\)
\(972\) −944784. −0.0320750
\(973\) 224396. 0.00759859
\(974\) 515548. 0.0174129
\(975\) 0 0
\(976\) −1.83424e6 −0.0616356
\(977\) 4.50181e7 1.50887 0.754433 0.656377i \(-0.227912\pi\)
0.754433 + 0.656377i \(0.227912\pi\)
\(978\) −1.52010e7 −0.508190
\(979\) −1.05437e7 −0.351589
\(980\) 0 0
\(981\) 1.92411e7 0.638349
\(982\) −1.21718e6 −0.0402789
\(983\) 3.26907e7 1.07905 0.539523 0.841971i \(-0.318605\pi\)
0.539523 + 0.841971i \(0.318605\pi\)
\(984\) 1.64506e6 0.0541618
\(985\) 0 0
\(986\) 57456.0 0.00188210
\(987\) −210330. −0.00687240
\(988\) 6.20043e6 0.202083
\(989\) 7.93484e7 2.57957
\(990\) 0 0
\(991\) 6.24606e6 0.202033 0.101016 0.994885i \(-0.467791\pi\)
0.101016 + 0.994885i \(0.467791\pi\)
\(992\) 7.76499e6 0.250531
\(993\) −5.95055e6 −0.191507
\(994\) −172320. −0.00553184
\(995\) 0 0
\(996\) −256608. −0.00819638
\(997\) −2.85010e7 −0.908077 −0.454039 0.890982i \(-0.650017\pi\)
−0.454039 + 0.890982i \(0.650017\pi\)
\(998\) 2.56790e7 0.816115
\(999\) −2.72792e6 −0.0864804
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.k.1.1 yes 1
3.2 odd 2 450.6.a.g.1.1 1
5.2 odd 4 150.6.c.a.49.2 2
5.3 odd 4 150.6.c.a.49.1 2
5.4 even 2 150.6.a.e.1.1 1
15.2 even 4 450.6.c.k.199.1 2
15.8 even 4 450.6.c.k.199.2 2
15.14 odd 2 450.6.a.r.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
150.6.a.e.1.1 1 5.4 even 2
150.6.a.k.1.1 yes 1 1.1 even 1 trivial
150.6.c.a.49.1 2 5.3 odd 4
150.6.c.a.49.2 2 5.2 odd 4
450.6.a.g.1.1 1 3.2 odd 2
450.6.a.r.1.1 1 15.14 odd 2
450.6.c.k.199.1 2 15.2 even 4
450.6.c.k.199.2 2 15.8 even 4