Properties

Label 294.6.a.b.1.1
Level $294$
Weight $6$
Character 294.1
Self dual yes
Analytic conductor $47.153$
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] = [294,6,Mod(1,294)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(294, 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("294.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 294 = 2 \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 294.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: \(47.1528430250\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 42)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 294.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -26.0000 q^{5} +36.0000 q^{6} -64.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} -9.00000 q^{3} +16.0000 q^{4} -26.0000 q^{5} +36.0000 q^{6} -64.0000 q^{8} +81.0000 q^{9} +104.000 q^{10} +664.000 q^{11} -144.000 q^{12} -318.000 q^{13} +234.000 q^{15} +256.000 q^{16} -1582.00 q^{17} -324.000 q^{18} -236.000 q^{19} -416.000 q^{20} -2656.00 q^{22} +2212.00 q^{23} +576.000 q^{24} -2449.00 q^{25} +1272.00 q^{26} -729.000 q^{27} -4954.00 q^{29} -936.000 q^{30} +7128.00 q^{31} -1024.00 q^{32} -5976.00 q^{33} +6328.00 q^{34} +1296.00 q^{36} +4358.00 q^{37} +944.000 q^{38} +2862.00 q^{39} +1664.00 q^{40} -10542.0 q^{41} -8452.00 q^{43} +10624.0 q^{44} -2106.00 q^{45} -8848.00 q^{46} -5352.00 q^{47} -2304.00 q^{48} +9796.00 q^{50} +14238.0 q^{51} -5088.00 q^{52} -33354.0 q^{53} +2916.00 q^{54} -17264.0 q^{55} +2124.00 q^{57} +19816.0 q^{58} +15436.0 q^{59} +3744.00 q^{60} +36762.0 q^{61} -28512.0 q^{62} +4096.00 q^{64} +8268.00 q^{65} +23904.0 q^{66} +40972.0 q^{67} -25312.0 q^{68} -19908.0 q^{69} -9092.00 q^{71} -5184.00 q^{72} +73454.0 q^{73} -17432.0 q^{74} +22041.0 q^{75} -3776.00 q^{76} -11448.0 q^{78} +89400.0 q^{79} -6656.00 q^{80} +6561.00 q^{81} +42168.0 q^{82} +6428.00 q^{83} +41132.0 q^{85} +33808.0 q^{86} +44586.0 q^{87} -42496.0 q^{88} +122658. q^{89} +8424.00 q^{90} +35392.0 q^{92} -64152.0 q^{93} +21408.0 q^{94} +6136.00 q^{95} +9216.00 q^{96} -21370.0 q^{97} +53784.0 q^{99} +O(q^{100})\)

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
\(5\) −26.0000 −0.465102 −0.232551 0.972584i \(-0.574707\pi\)
−0.232551 + 0.972584i \(0.574707\pi\)
\(6\) 36.0000 0.408248
\(7\) 0 0
\(8\) −64.0000 −0.353553
\(9\) 81.0000 0.333333
\(10\) 104.000 0.328877
\(11\) 664.000 1.65457 0.827287 0.561779i \(-0.189883\pi\)
0.827287 + 0.561779i \(0.189883\pi\)
\(12\) −144.000 −0.288675
\(13\) −318.000 −0.521878 −0.260939 0.965355i \(-0.584032\pi\)
−0.260939 + 0.965355i \(0.584032\pi\)
\(14\) 0 0
\(15\) 234.000 0.268527
\(16\) 256.000 0.250000
\(17\) −1582.00 −1.32765 −0.663826 0.747887i \(-0.731068\pi\)
−0.663826 + 0.747887i \(0.731068\pi\)
\(18\) −324.000 −0.235702
\(19\) −236.000 −0.149978 −0.0749891 0.997184i \(-0.523892\pi\)
−0.0749891 + 0.997184i \(0.523892\pi\)
\(20\) −416.000 −0.232551
\(21\) 0 0
\(22\) −2656.00 −1.16996
\(23\) 2212.00 0.871898 0.435949 0.899971i \(-0.356413\pi\)
0.435949 + 0.899971i \(0.356413\pi\)
\(24\) 576.000 0.204124
\(25\) −2449.00 −0.783680
\(26\) 1272.00 0.369023
\(27\) −729.000 −0.192450
\(28\) 0 0
\(29\) −4954.00 −1.09386 −0.546929 0.837179i \(-0.684203\pi\)
−0.546929 + 0.837179i \(0.684203\pi\)
\(30\) −936.000 −0.189877
\(31\) 7128.00 1.33218 0.666091 0.745871i \(-0.267966\pi\)
0.666091 + 0.745871i \(0.267966\pi\)
\(32\) −1024.00 −0.176777
\(33\) −5976.00 −0.955269
\(34\) 6328.00 0.938792
\(35\) 0 0
\(36\) 1296.00 0.166667
\(37\) 4358.00 0.523339 0.261669 0.965158i \(-0.415727\pi\)
0.261669 + 0.965158i \(0.415727\pi\)
\(38\) 944.000 0.106051
\(39\) 2862.00 0.301306
\(40\) 1664.00 0.164438
\(41\) −10542.0 −0.979407 −0.489704 0.871889i \(-0.662895\pi\)
−0.489704 + 0.871889i \(0.662895\pi\)
\(42\) 0 0
\(43\) −8452.00 −0.697089 −0.348545 0.937292i \(-0.613324\pi\)
−0.348545 + 0.937292i \(0.613324\pi\)
\(44\) 10624.0 0.827287
\(45\) −2106.00 −0.155034
\(46\) −8848.00 −0.616525
\(47\) −5352.00 −0.353404 −0.176702 0.984264i \(-0.556543\pi\)
−0.176702 + 0.984264i \(0.556543\pi\)
\(48\) −2304.00 −0.144338
\(49\) 0 0
\(50\) 9796.00 0.554145
\(51\) 14238.0 0.766520
\(52\) −5088.00 −0.260939
\(53\) −33354.0 −1.63102 −0.815508 0.578746i \(-0.803542\pi\)
−0.815508 + 0.578746i \(0.803542\pi\)
\(54\) 2916.00 0.136083
\(55\) −17264.0 −0.769546
\(56\) 0 0
\(57\) 2124.00 0.0865899
\(58\) 19816.0 0.773475
\(59\) 15436.0 0.577304 0.288652 0.957434i \(-0.406793\pi\)
0.288652 + 0.957434i \(0.406793\pi\)
\(60\) 3744.00 0.134263
\(61\) 36762.0 1.26495 0.632477 0.774579i \(-0.282038\pi\)
0.632477 + 0.774579i \(0.282038\pi\)
\(62\) −28512.0 −0.941995
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 8268.00 0.242726
\(66\) 23904.0 0.675477
\(67\) 40972.0 1.11506 0.557532 0.830155i \(-0.311748\pi\)
0.557532 + 0.830155i \(0.311748\pi\)
\(68\) −25312.0 −0.663826
\(69\) −19908.0 −0.503390
\(70\) 0 0
\(71\) −9092.00 −0.214049 −0.107025 0.994256i \(-0.534132\pi\)
−0.107025 + 0.994256i \(0.534132\pi\)
\(72\) −5184.00 −0.117851
\(73\) 73454.0 1.61327 0.806637 0.591047i \(-0.201285\pi\)
0.806637 + 0.591047i \(0.201285\pi\)
\(74\) −17432.0 −0.370056
\(75\) 22041.0 0.452458
\(76\) −3776.00 −0.0749891
\(77\) 0 0
\(78\) −11448.0 −0.213056
\(79\) 89400.0 1.61165 0.805823 0.592156i \(-0.201723\pi\)
0.805823 + 0.592156i \(0.201723\pi\)
\(80\) −6656.00 −0.116276
\(81\) 6561.00 0.111111
\(82\) 42168.0 0.692546
\(83\) 6428.00 0.102419 0.0512095 0.998688i \(-0.483692\pi\)
0.0512095 + 0.998688i \(0.483692\pi\)
\(84\) 0 0
\(85\) 41132.0 0.617494
\(86\) 33808.0 0.492916
\(87\) 44586.0 0.631539
\(88\) −42496.0 −0.584980
\(89\) 122658. 1.64142 0.820712 0.571342i \(-0.193577\pi\)
0.820712 + 0.571342i \(0.193577\pi\)
\(90\) 8424.00 0.109626
\(91\) 0 0
\(92\) 35392.0 0.435949
\(93\) −64152.0 −0.769135
\(94\) 21408.0 0.249894
\(95\) 6136.00 0.0697552
\(96\) 9216.00 0.102062
\(97\) −21370.0 −0.230608 −0.115304 0.993330i \(-0.536784\pi\)
−0.115304 + 0.993330i \(0.536784\pi\)
\(98\) 0 0
\(99\) 53784.0 0.551525
\(100\) −39184.0 −0.391840
\(101\) 36814.0 0.359095 0.179548 0.983749i \(-0.442537\pi\)
0.179548 + 0.983749i \(0.442537\pi\)
\(102\) −56952.0 −0.542012
\(103\) −104528. −0.970822 −0.485411 0.874286i \(-0.661330\pi\)
−0.485411 + 0.874286i \(0.661330\pi\)
\(104\) 20352.0 0.184512
\(105\) 0 0
\(106\) 133416. 1.15330
\(107\) 214440. 1.81070 0.905350 0.424667i \(-0.139609\pi\)
0.905350 + 0.424667i \(0.139609\pi\)
\(108\) −11664.0 −0.0962250
\(109\) 28798.0 0.232165 0.116082 0.993240i \(-0.462966\pi\)
0.116082 + 0.993240i \(0.462966\pi\)
\(110\) 69056.0 0.544151
\(111\) −39222.0 −0.302150
\(112\) 0 0
\(113\) −56014.0 −0.412668 −0.206334 0.978482i \(-0.566153\pi\)
−0.206334 + 0.978482i \(0.566153\pi\)
\(114\) −8496.00 −0.0612283
\(115\) −57512.0 −0.405521
\(116\) −79264.0 −0.546929
\(117\) −25758.0 −0.173959
\(118\) −61744.0 −0.408216
\(119\) 0 0
\(120\) −14976.0 −0.0949386
\(121\) 279845. 1.73762
\(122\) −147048. −0.894457
\(123\) 94878.0 0.565461
\(124\) 114048. 0.666091
\(125\) 144924. 0.829593
\(126\) 0 0
\(127\) 185400. 1.02000 0.510000 0.860174i \(-0.329645\pi\)
0.510000 + 0.860174i \(0.329645\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 76068.0 0.402465
\(130\) −33072.0 −0.171634
\(131\) −64532.0 −0.328547 −0.164273 0.986415i \(-0.552528\pi\)
−0.164273 + 0.986415i \(0.552528\pi\)
\(132\) −95616.0 −0.477635
\(133\) 0 0
\(134\) −163888. −0.788470
\(135\) 18954.0 0.0895089
\(136\) 101248. 0.469396
\(137\) 152930. 0.696131 0.348066 0.937470i \(-0.386839\pi\)
0.348066 + 0.937470i \(0.386839\pi\)
\(138\) 79632.0 0.355951
\(139\) 343460. 1.50778 0.753892 0.656998i \(-0.228174\pi\)
0.753892 + 0.656998i \(0.228174\pi\)
\(140\) 0 0
\(141\) 48168.0 0.204038
\(142\) 36368.0 0.151356
\(143\) −211152. −0.863486
\(144\) 20736.0 0.0833333
\(145\) 128804. 0.508756
\(146\) −293816. −1.14076
\(147\) 0 0
\(148\) 69728.0 0.261669
\(149\) −174858. −0.645238 −0.322619 0.946529i \(-0.604563\pi\)
−0.322619 + 0.946529i \(0.604563\pi\)
\(150\) −88164.0 −0.319936
\(151\) −452552. −1.61520 −0.807600 0.589731i \(-0.799234\pi\)
−0.807600 + 0.589731i \(0.799234\pi\)
\(152\) 15104.0 0.0530253
\(153\) −128142. −0.442551
\(154\) 0 0
\(155\) −185328. −0.619601
\(156\) 45792.0 0.150653
\(157\) 499066. 1.61588 0.807940 0.589265i \(-0.200583\pi\)
0.807940 + 0.589265i \(0.200583\pi\)
\(158\) −357600. −1.13961
\(159\) 300186. 0.941668
\(160\) 26624.0 0.0822192
\(161\) 0 0
\(162\) −26244.0 −0.0785674
\(163\) −475588. −1.40204 −0.701022 0.713139i \(-0.747273\pi\)
−0.701022 + 0.713139i \(0.747273\pi\)
\(164\) −168672. −0.489704
\(165\) 155376. 0.444298
\(166\) −25712.0 −0.0724212
\(167\) −120224. −0.333580 −0.166790 0.985992i \(-0.553340\pi\)
−0.166790 + 0.985992i \(0.553340\pi\)
\(168\) 0 0
\(169\) −270169. −0.727644
\(170\) −164528. −0.436634
\(171\) −19116.0 −0.0499927
\(172\) −135232. −0.348545
\(173\) −508874. −1.29269 −0.646346 0.763045i \(-0.723704\pi\)
−0.646346 + 0.763045i \(0.723704\pi\)
\(174\) −178344. −0.446566
\(175\) 0 0
\(176\) 169984. 0.413644
\(177\) −138924. −0.333307
\(178\) −490632. −1.16066
\(179\) 487560. 1.13735 0.568677 0.822561i \(-0.307456\pi\)
0.568677 + 0.822561i \(0.307456\pi\)
\(180\) −33696.0 −0.0775170
\(181\) 544410. 1.23518 0.617589 0.786501i \(-0.288109\pi\)
0.617589 + 0.786501i \(0.288109\pi\)
\(182\) 0 0
\(183\) −330858. −0.730321
\(184\) −141568. −0.308262
\(185\) −113308. −0.243406
\(186\) 256608. 0.543861
\(187\) −1.05045e6 −2.19670
\(188\) −85632.0 −0.176702
\(189\) 0 0
\(190\) −24544.0 −0.0493243
\(191\) 376404. 0.746570 0.373285 0.927717i \(-0.378231\pi\)
0.373285 + 0.927717i \(0.378231\pi\)
\(192\) −36864.0 −0.0721688
\(193\) 844946. 1.63281 0.816405 0.577480i \(-0.195964\pi\)
0.816405 + 0.577480i \(0.195964\pi\)
\(194\) 85480.0 0.163065
\(195\) −74412.0 −0.140138
\(196\) 0 0
\(197\) −492794. −0.904690 −0.452345 0.891843i \(-0.649412\pi\)
−0.452345 + 0.891843i \(0.649412\pi\)
\(198\) −215136. −0.389987
\(199\) 914776. 1.63750 0.818751 0.574148i \(-0.194667\pi\)
0.818751 + 0.574148i \(0.194667\pi\)
\(200\) 156736. 0.277073
\(201\) −368748. −0.643783
\(202\) −147256. −0.253919
\(203\) 0 0
\(204\) 227808. 0.383260
\(205\) 274092. 0.455524
\(206\) 418112. 0.686475
\(207\) 179172. 0.290633
\(208\) −81408.0 −0.130469
\(209\) −156704. −0.248150
\(210\) 0 0
\(211\) 311780. 0.482106 0.241053 0.970512i \(-0.422507\pi\)
0.241053 + 0.970512i \(0.422507\pi\)
\(212\) −533664. −0.815508
\(213\) 81828.0 0.123581
\(214\) −857760. −1.28036
\(215\) 219752. 0.324218
\(216\) 46656.0 0.0680414
\(217\) 0 0
\(218\) −115192. −0.164165
\(219\) −661086. −0.931425
\(220\) −276224. −0.384773
\(221\) 503076. 0.692872
\(222\) 156888. 0.213652
\(223\) 1.28776e6 1.73409 0.867047 0.498226i \(-0.166015\pi\)
0.867047 + 0.498226i \(0.166015\pi\)
\(224\) 0 0
\(225\) −198369. −0.261227
\(226\) 224056. 0.291800
\(227\) −1.28905e6 −1.66037 −0.830187 0.557485i \(-0.811766\pi\)
−0.830187 + 0.557485i \(0.811766\pi\)
\(228\) 33984.0 0.0432950
\(229\) −678214. −0.854630 −0.427315 0.904103i \(-0.640540\pi\)
−0.427315 + 0.904103i \(0.640540\pi\)
\(230\) 230048. 0.286747
\(231\) 0 0
\(232\) 317056. 0.386737
\(233\) −1.11731e6 −1.34829 −0.674146 0.738598i \(-0.735488\pi\)
−0.674146 + 0.738598i \(0.735488\pi\)
\(234\) 103032. 0.123008
\(235\) 139152. 0.164369
\(236\) 246976. 0.288652
\(237\) −804600. −0.930485
\(238\) 0 0
\(239\) −1.26196e6 −1.42906 −0.714528 0.699606i \(-0.753359\pi\)
−0.714528 + 0.699606i \(0.753359\pi\)
\(240\) 59904.0 0.0671317
\(241\) −948218. −1.05164 −0.525818 0.850597i \(-0.676241\pi\)
−0.525818 + 0.850597i \(0.676241\pi\)
\(242\) −1.11938e6 −1.22868
\(243\) −59049.0 −0.0641500
\(244\) 588192. 0.632477
\(245\) 0 0
\(246\) −379512. −0.399841
\(247\) 75048.0 0.0782703
\(248\) −456192. −0.470997
\(249\) −57852.0 −0.0591317
\(250\) −579696. −0.586611
\(251\) 486396. 0.487310 0.243655 0.969862i \(-0.421653\pi\)
0.243655 + 0.969862i \(0.421653\pi\)
\(252\) 0 0
\(253\) 1.46877e6 1.44262
\(254\) −741600. −0.721249
\(255\) −370188. −0.356510
\(256\) 65536.0 0.0625000
\(257\) 1.03910e6 0.981349 0.490675 0.871343i \(-0.336750\pi\)
0.490675 + 0.871343i \(0.336750\pi\)
\(258\) −304272. −0.284585
\(259\) 0 0
\(260\) 132288. 0.121363
\(261\) −401274. −0.364619
\(262\) 258128. 0.232317
\(263\) 1.35104e6 1.20443 0.602213 0.798335i \(-0.294286\pi\)
0.602213 + 0.798335i \(0.294286\pi\)
\(264\) 382464. 0.337739
\(265\) 867204. 0.758589
\(266\) 0 0
\(267\) −1.10392e6 −0.947677
\(268\) 655552. 0.557532
\(269\) 1.11811e6 0.942115 0.471057 0.882103i \(-0.343872\pi\)
0.471057 + 0.882103i \(0.343872\pi\)
\(270\) −75816.0 −0.0632924
\(271\) 190104. 0.157242 0.0786209 0.996905i \(-0.474948\pi\)
0.0786209 + 0.996905i \(0.474948\pi\)
\(272\) −404992. −0.331913
\(273\) 0 0
\(274\) −611720. −0.492239
\(275\) −1.62614e6 −1.29666
\(276\) −318528. −0.251695
\(277\) −200506. −0.157010 −0.0785051 0.996914i \(-0.525015\pi\)
−0.0785051 + 0.996914i \(0.525015\pi\)
\(278\) −1.37384e6 −1.06616
\(279\) 577368. 0.444061
\(280\) 0 0
\(281\) 1.09237e6 0.825285 0.412643 0.910893i \(-0.364606\pi\)
0.412643 + 0.910893i \(0.364606\pi\)
\(282\) −192672. −0.144277
\(283\) −1.81258e6 −1.34534 −0.672669 0.739944i \(-0.734852\pi\)
−0.672669 + 0.739944i \(0.734852\pi\)
\(284\) −145472. −0.107025
\(285\) −55224.0 −0.0402732
\(286\) 844608. 0.610577
\(287\) 0 0
\(288\) −82944.0 −0.0589256
\(289\) 1.08287e6 0.762659
\(290\) −515216. −0.359745
\(291\) 192330. 0.133142
\(292\) 1.17526e6 0.806637
\(293\) −2.10031e6 −1.42927 −0.714634 0.699499i \(-0.753407\pi\)
−0.714634 + 0.699499i \(0.753407\pi\)
\(294\) 0 0
\(295\) −401336. −0.268505
\(296\) −278912. −0.185028
\(297\) −484056. −0.318423
\(298\) 699432. 0.456252
\(299\) −703416. −0.455024
\(300\) 352656. 0.226229
\(301\) 0 0
\(302\) 1.81021e6 1.14212
\(303\) −331326. −0.207324
\(304\) −60416.0 −0.0374945
\(305\) −955812. −0.588333
\(306\) 512568. 0.312931
\(307\) 1.64104e6 0.993743 0.496872 0.867824i \(-0.334482\pi\)
0.496872 + 0.867824i \(0.334482\pi\)
\(308\) 0 0
\(309\) 940752. 0.560504
\(310\) 741312. 0.438124
\(311\) 945232. 0.554163 0.277081 0.960846i \(-0.410633\pi\)
0.277081 + 0.960846i \(0.410633\pi\)
\(312\) −183168. −0.106528
\(313\) −415354. −0.239639 −0.119820 0.992796i \(-0.538232\pi\)
−0.119820 + 0.992796i \(0.538232\pi\)
\(314\) −1.99626e6 −1.14260
\(315\) 0 0
\(316\) 1.43040e6 0.805823
\(317\) 1.18481e6 0.662220 0.331110 0.943592i \(-0.392577\pi\)
0.331110 + 0.943592i \(0.392577\pi\)
\(318\) −1.20074e6 −0.665860
\(319\) −3.28946e6 −1.80987
\(320\) −106496. −0.0581378
\(321\) −1.92996e6 −1.04541
\(322\) 0 0
\(323\) 373352. 0.199119
\(324\) 104976. 0.0555556
\(325\) 778782. 0.408985
\(326\) 1.90235e6 0.991395
\(327\) −259182. −0.134040
\(328\) 674688. 0.346273
\(329\) 0 0
\(330\) −621504. −0.314166
\(331\) 1.37155e6 0.688083 0.344042 0.938954i \(-0.388204\pi\)
0.344042 + 0.938954i \(0.388204\pi\)
\(332\) 102848. 0.0512095
\(333\) 352998. 0.174446
\(334\) 480896. 0.235877
\(335\) −1.06527e6 −0.518619
\(336\) 0 0
\(337\) 963522. 0.462154 0.231077 0.972935i \(-0.425775\pi\)
0.231077 + 0.972935i \(0.425775\pi\)
\(338\) 1.08068e6 0.514522
\(339\) 504126. 0.238254
\(340\) 658112. 0.308747
\(341\) 4.73299e6 2.20419
\(342\) 76464.0 0.0353502
\(343\) 0 0
\(344\) 540928. 0.246458
\(345\) 517608. 0.234128
\(346\) 2.03550e6 0.914071
\(347\) 2.57731e6 1.14906 0.574531 0.818483i \(-0.305185\pi\)
0.574531 + 0.818483i \(0.305185\pi\)
\(348\) 713376. 0.315770
\(349\) 3.06751e6 1.34810 0.674051 0.738684i \(-0.264553\pi\)
0.674051 + 0.738684i \(0.264553\pi\)
\(350\) 0 0
\(351\) 231822. 0.100435
\(352\) −679936. −0.292490
\(353\) 3.10144e6 1.32473 0.662364 0.749182i \(-0.269553\pi\)
0.662364 + 0.749182i \(0.269553\pi\)
\(354\) 555696. 0.235683
\(355\) 236392. 0.0995547
\(356\) 1.96253e6 0.820712
\(357\) 0 0
\(358\) −1.95024e6 −0.804230
\(359\) −327508. −0.134118 −0.0670588 0.997749i \(-0.521362\pi\)
−0.0670588 + 0.997749i \(0.521362\pi\)
\(360\) 134784. 0.0548128
\(361\) −2.42040e6 −0.977507
\(362\) −2.17764e6 −0.873403
\(363\) −2.51860e6 −1.00321
\(364\) 0 0
\(365\) −1.90980e6 −0.750337
\(366\) 1.32343e6 0.516415
\(367\) 2.86739e6 1.11128 0.555638 0.831424i \(-0.312474\pi\)
0.555638 + 0.831424i \(0.312474\pi\)
\(368\) 566272. 0.217974
\(369\) −853902. −0.326469
\(370\) 453232. 0.172114
\(371\) 0 0
\(372\) −1.02643e6 −0.384568
\(373\) 3.58029e6 1.33244 0.666218 0.745757i \(-0.267912\pi\)
0.666218 + 0.745757i \(0.267912\pi\)
\(374\) 4.20179e6 1.55330
\(375\) −1.30432e6 −0.478966
\(376\) 342528. 0.124947
\(377\) 1.57537e6 0.570860
\(378\) 0 0
\(379\) 1.64235e6 0.587310 0.293655 0.955912i \(-0.405128\pi\)
0.293655 + 0.955912i \(0.405128\pi\)
\(380\) 98176.0 0.0348776
\(381\) −1.66860e6 −0.588898
\(382\) −1.50562e6 −0.527905
\(383\) 2.05698e6 0.716527 0.358263 0.933621i \(-0.383369\pi\)
0.358263 + 0.933621i \(0.383369\pi\)
\(384\) 147456. 0.0510310
\(385\) 0 0
\(386\) −3.37978e6 −1.15457
\(387\) −684612. −0.232363
\(388\) −341920. −0.115304
\(389\) 616142. 0.206446 0.103223 0.994658i \(-0.467084\pi\)
0.103223 + 0.994658i \(0.467084\pi\)
\(390\) 297648. 0.0990927
\(391\) −3.49938e6 −1.15758
\(392\) 0 0
\(393\) 580788. 0.189686
\(394\) 1.97118e6 0.639713
\(395\) −2.32440e6 −0.749580
\(396\) 860544. 0.275762
\(397\) −2.19212e6 −0.698052 −0.349026 0.937113i \(-0.613487\pi\)
−0.349026 + 0.937113i \(0.613487\pi\)
\(398\) −3.65910e6 −1.15789
\(399\) 0 0
\(400\) −626944. −0.195920
\(401\) 3.28454e6 1.02003 0.510015 0.860165i \(-0.329640\pi\)
0.510015 + 0.860165i \(0.329640\pi\)
\(402\) 1.47499e6 0.455223
\(403\) −2.26670e6 −0.695236
\(404\) 589024. 0.179548
\(405\) −170586. −0.0516780
\(406\) 0 0
\(407\) 2.89371e6 0.865903
\(408\) −911232. −0.271006
\(409\) 3.61219e6 1.06773 0.533866 0.845569i \(-0.320739\pi\)
0.533866 + 0.845569i \(0.320739\pi\)
\(410\) −1.09637e6 −0.322104
\(411\) −1.37637e6 −0.401912
\(412\) −1.67245e6 −0.485411
\(413\) 0 0
\(414\) −716688. −0.205508
\(415\) −167128. −0.0476353
\(416\) 325632. 0.0922558
\(417\) −3.09114e6 −0.870520
\(418\) 626816. 0.175469
\(419\) −5.41489e6 −1.50680 −0.753398 0.657564i \(-0.771587\pi\)
−0.753398 + 0.657564i \(0.771587\pi\)
\(420\) 0 0
\(421\) 3.60629e6 0.991644 0.495822 0.868424i \(-0.334867\pi\)
0.495822 + 0.868424i \(0.334867\pi\)
\(422\) −1.24712e6 −0.340900
\(423\) −433512. −0.117801
\(424\) 2.13466e6 0.576651
\(425\) 3.87432e6 1.04045
\(426\) −327312. −0.0873852
\(427\) 0 0
\(428\) 3.43104e6 0.905350
\(429\) 1.90037e6 0.498534
\(430\) −879008. −0.229257
\(431\) −2.78214e6 −0.721416 −0.360708 0.932679i \(-0.617465\pi\)
−0.360708 + 0.932679i \(0.617465\pi\)
\(432\) −186624. −0.0481125
\(433\) −6.27619e6 −1.60871 −0.804353 0.594152i \(-0.797488\pi\)
−0.804353 + 0.594152i \(0.797488\pi\)
\(434\) 0 0
\(435\) −1.15924e6 −0.293730
\(436\) 460768. 0.116082
\(437\) −522032. −0.130766
\(438\) 2.64434e6 0.658617
\(439\) −641592. −0.158890 −0.0794452 0.996839i \(-0.525315\pi\)
−0.0794452 + 0.996839i \(0.525315\pi\)
\(440\) 1.10490e6 0.272076
\(441\) 0 0
\(442\) −2.01230e6 −0.489934
\(443\) 6.05546e6 1.46601 0.733006 0.680222i \(-0.238117\pi\)
0.733006 + 0.680222i \(0.238117\pi\)
\(444\) −627552. −0.151075
\(445\) −3.18911e6 −0.763430
\(446\) −5.15104e6 −1.22619
\(447\) 1.57372e6 0.372528
\(448\) 0 0
\(449\) −5.16681e6 −1.20950 −0.604752 0.796414i \(-0.706728\pi\)
−0.604752 + 0.796414i \(0.706728\pi\)
\(450\) 793476. 0.184715
\(451\) −6.99989e6 −1.62050
\(452\) −896224. −0.206334
\(453\) 4.07297e6 0.932536
\(454\) 5.15621e6 1.17406
\(455\) 0 0
\(456\) −135936. −0.0306142
\(457\) −227798. −0.0510222 −0.0255111 0.999675i \(-0.508121\pi\)
−0.0255111 + 0.999675i \(0.508121\pi\)
\(458\) 2.71286e6 0.604315
\(459\) 1.15328e6 0.255507
\(460\) −920192. −0.202761
\(461\) −585146. −0.128237 −0.0641183 0.997942i \(-0.520423\pi\)
−0.0641183 + 0.997942i \(0.520423\pi\)
\(462\) 0 0
\(463\) −3.41454e6 −0.740251 −0.370126 0.928982i \(-0.620685\pi\)
−0.370126 + 0.928982i \(0.620685\pi\)
\(464\) −1.26822e6 −0.273465
\(465\) 1.66795e6 0.357727
\(466\) 4.46924e6 0.953386
\(467\) −716300. −0.151986 −0.0759929 0.997108i \(-0.524213\pi\)
−0.0759929 + 0.997108i \(0.524213\pi\)
\(468\) −412128. −0.0869796
\(469\) 0 0
\(470\) −556608. −0.116226
\(471\) −4.49159e6 −0.932928
\(472\) −987904. −0.204108
\(473\) −5.61213e6 −1.15339
\(474\) 3.21840e6 0.657952
\(475\) 577964. 0.117535
\(476\) 0 0
\(477\) −2.70167e6 −0.543672
\(478\) 5.04782e6 1.01050
\(479\) −5.24092e6 −1.04368 −0.521842 0.853042i \(-0.674755\pi\)
−0.521842 + 0.853042i \(0.674755\pi\)
\(480\) −239616. −0.0474693
\(481\) −1.38584e6 −0.273119
\(482\) 3.79287e6 0.743619
\(483\) 0 0
\(484\) 4.47752e6 0.868809
\(485\) 555620. 0.107256
\(486\) 236196. 0.0453609
\(487\) 1.11702e6 0.213421 0.106710 0.994290i \(-0.465968\pi\)
0.106710 + 0.994290i \(0.465968\pi\)
\(488\) −2.35277e6 −0.447229
\(489\) 4.28029e6 0.809471
\(490\) 0 0
\(491\) 1.34458e6 0.251699 0.125850 0.992049i \(-0.459834\pi\)
0.125850 + 0.992049i \(0.459834\pi\)
\(492\) 1.51805e6 0.282731
\(493\) 7.83723e6 1.45226
\(494\) −300192. −0.0553454
\(495\) −1.39838e6 −0.256515
\(496\) 1.82477e6 0.333045
\(497\) 0 0
\(498\) 231408. 0.0418124
\(499\) −6.54648e6 −1.17695 −0.588473 0.808517i \(-0.700271\pi\)
−0.588473 + 0.808517i \(0.700271\pi\)
\(500\) 2.31878e6 0.414797
\(501\) 1.08202e6 0.192592
\(502\) −1.94558e6 −0.344580
\(503\) 8.22050e6 1.44870 0.724350 0.689432i \(-0.242140\pi\)
0.724350 + 0.689432i \(0.242140\pi\)
\(504\) 0 0
\(505\) −957164. −0.167016
\(506\) −5.87507e6 −1.02009
\(507\) 2.43152e6 0.420105
\(508\) 2.96640e6 0.510000
\(509\) 5.11045e6 0.874308 0.437154 0.899387i \(-0.355987\pi\)
0.437154 + 0.899387i \(0.355987\pi\)
\(510\) 1.48075e6 0.252091
\(511\) 0 0
\(512\) −262144. −0.0441942
\(513\) 172044. 0.0288633
\(514\) −4.15639e6 −0.693919
\(515\) 2.71773e6 0.451531
\(516\) 1.21709e6 0.201232
\(517\) −3.55373e6 −0.584733
\(518\) 0 0
\(519\) 4.57987e6 0.746336
\(520\) −529152. −0.0858168
\(521\) −9.69999e6 −1.56559 −0.782793 0.622282i \(-0.786206\pi\)
−0.782793 + 0.622282i \(0.786206\pi\)
\(522\) 1.60510e6 0.257825
\(523\) 3.17295e6 0.507234 0.253617 0.967305i \(-0.418380\pi\)
0.253617 + 0.967305i \(0.418380\pi\)
\(524\) −1.03251e6 −0.164273
\(525\) 0 0
\(526\) −5.40418e6 −0.851658
\(527\) −1.12765e7 −1.76867
\(528\) −1.52986e6 −0.238817
\(529\) −1.54340e6 −0.239794
\(530\) −3.46882e6 −0.536403
\(531\) 1.25032e6 0.192435
\(532\) 0 0
\(533\) 3.35236e6 0.511131
\(534\) 4.41569e6 0.670109
\(535\) −5.57544e6 −0.842160
\(536\) −2.62221e6 −0.394235
\(537\) −4.38804e6 −0.656651
\(538\) −4.47244e6 −0.666176
\(539\) 0 0
\(540\) 303264. 0.0447545
\(541\) −6.62575e6 −0.973289 −0.486644 0.873600i \(-0.661779\pi\)
−0.486644 + 0.873600i \(0.661779\pi\)
\(542\) −760416. −0.111187
\(543\) −4.89969e6 −0.713131
\(544\) 1.61997e6 0.234698
\(545\) −748748. −0.107980
\(546\) 0 0
\(547\) 3.84707e6 0.549745 0.274873 0.961481i \(-0.411364\pi\)
0.274873 + 0.961481i \(0.411364\pi\)
\(548\) 2.44688e6 0.348066
\(549\) 2.97772e6 0.421651
\(550\) 6.50454e6 0.916875
\(551\) 1.16914e6 0.164055
\(552\) 1.27411e6 0.177975
\(553\) 0 0
\(554\) 802024. 0.111023
\(555\) 1.01977e6 0.140531
\(556\) 5.49536e6 0.753892
\(557\) 5.00176e6 0.683101 0.341550 0.939863i \(-0.389048\pi\)
0.341550 + 0.939863i \(0.389048\pi\)
\(558\) −2.30947e6 −0.313998
\(559\) 2.68774e6 0.363795
\(560\) 0 0
\(561\) 9.45403e6 1.26826
\(562\) −4.36948e6 −0.583565
\(563\) −2.27772e6 −0.302852 −0.151426 0.988469i \(-0.548386\pi\)
−0.151426 + 0.988469i \(0.548386\pi\)
\(564\) 770688. 0.102019
\(565\) 1.45636e6 0.191933
\(566\) 7.25032e6 0.951297
\(567\) 0 0
\(568\) 581888. 0.0756778
\(569\) 8.86979e6 1.14850 0.574252 0.818678i \(-0.305293\pi\)
0.574252 + 0.818678i \(0.305293\pi\)
\(570\) 220896. 0.0284774
\(571\) 1.40102e7 1.79826 0.899132 0.437678i \(-0.144199\pi\)
0.899132 + 0.437678i \(0.144199\pi\)
\(572\) −3.37843e6 −0.431743
\(573\) −3.38764e6 −0.431033
\(574\) 0 0
\(575\) −5.41719e6 −0.683289
\(576\) 331776. 0.0416667
\(577\) −8.75327e6 −1.09454 −0.547269 0.836957i \(-0.684332\pi\)
−0.547269 + 0.836957i \(0.684332\pi\)
\(578\) −4.33147e6 −0.539281
\(579\) −7.60451e6 −0.942703
\(580\) 2.06086e6 0.254378
\(581\) 0 0
\(582\) −769320. −0.0941455
\(583\) −2.21471e7 −2.69864
\(584\) −4.70106e6 −0.570379
\(585\) 669708. 0.0809088
\(586\) 8.40122e6 1.01064
\(587\) 1.06117e7 1.27113 0.635564 0.772048i \(-0.280768\pi\)
0.635564 + 0.772048i \(0.280768\pi\)
\(588\) 0 0
\(589\) −1.68221e6 −0.199798
\(590\) 1.60534e6 0.189862
\(591\) 4.43515e6 0.522323
\(592\) 1.11565e6 0.130835
\(593\) −1.88552e6 −0.220188 −0.110094 0.993921i \(-0.535115\pi\)
−0.110094 + 0.993921i \(0.535115\pi\)
\(594\) 1.93622e6 0.225159
\(595\) 0 0
\(596\) −2.79773e6 −0.322619
\(597\) −8.23298e6 −0.945413
\(598\) 2.81366e6 0.321751
\(599\) 1.27256e7 1.44915 0.724573 0.689198i \(-0.242037\pi\)
0.724573 + 0.689198i \(0.242037\pi\)
\(600\) −1.41062e6 −0.159968
\(601\) −7.18846e6 −0.811801 −0.405900 0.913917i \(-0.633042\pi\)
−0.405900 + 0.913917i \(0.633042\pi\)
\(602\) 0 0
\(603\) 3.31873e6 0.371688
\(604\) −7.24083e6 −0.807600
\(605\) −7.27597e6 −0.808170
\(606\) 1.32530e6 0.146600
\(607\) −1.08494e7 −1.19519 −0.597593 0.801800i \(-0.703876\pi\)
−0.597593 + 0.801800i \(0.703876\pi\)
\(608\) 241664. 0.0265126
\(609\) 0 0
\(610\) 3.82325e6 0.416014
\(611\) 1.70194e6 0.184434
\(612\) −2.05027e6 −0.221275
\(613\) −4.90511e6 −0.527227 −0.263614 0.964628i \(-0.584914\pi\)
−0.263614 + 0.964628i \(0.584914\pi\)
\(614\) −6.56418e6 −0.702683
\(615\) −2.46683e6 −0.262997
\(616\) 0 0
\(617\) 2.58445e6 0.273310 0.136655 0.990619i \(-0.456365\pi\)
0.136655 + 0.990619i \(0.456365\pi\)
\(618\) −3.76301e6 −0.396336
\(619\) 4.99336e6 0.523801 0.261901 0.965095i \(-0.415651\pi\)
0.261901 + 0.965095i \(0.415651\pi\)
\(620\) −2.96525e6 −0.309800
\(621\) −1.61255e6 −0.167797
\(622\) −3.78093e6 −0.391852
\(623\) 0 0
\(624\) 732672. 0.0753266
\(625\) 3.88510e6 0.397834
\(626\) 1.66142e6 0.169450
\(627\) 1.41034e6 0.143269
\(628\) 7.98506e6 0.807940
\(629\) −6.89436e6 −0.694812
\(630\) 0 0
\(631\) −1.18219e7 −1.18199 −0.590997 0.806674i \(-0.701265\pi\)
−0.590997 + 0.806674i \(0.701265\pi\)
\(632\) −5.72160e6 −0.569803
\(633\) −2.80602e6 −0.278344
\(634\) −4.73926e6 −0.468260
\(635\) −4.82040e6 −0.474404
\(636\) 4.80298e6 0.470834
\(637\) 0 0
\(638\) 1.31578e7 1.27977
\(639\) −736452. −0.0713497
\(640\) 425984. 0.0411096
\(641\) −5.47007e6 −0.525833 −0.262916 0.964819i \(-0.584684\pi\)
−0.262916 + 0.964819i \(0.584684\pi\)
\(642\) 7.71984e6 0.739215
\(643\) −9.64934e6 −0.920386 −0.460193 0.887819i \(-0.652220\pi\)
−0.460193 + 0.887819i \(0.652220\pi\)
\(644\) 0 0
\(645\) −1.97777e6 −0.187187
\(646\) −1.49341e6 −0.140798
\(647\) −292368. −0.0274580 −0.0137290 0.999906i \(-0.504370\pi\)
−0.0137290 + 0.999906i \(0.504370\pi\)
\(648\) −419904. −0.0392837
\(649\) 1.02495e7 0.955193
\(650\) −3.11513e6 −0.289196
\(651\) 0 0
\(652\) −7.60941e6 −0.701022
\(653\) 6.94081e6 0.636982 0.318491 0.947926i \(-0.396824\pi\)
0.318491 + 0.947926i \(0.396824\pi\)
\(654\) 1.03673e6 0.0947808
\(655\) 1.67783e6 0.152808
\(656\) −2.69875e6 −0.244852
\(657\) 5.94977e6 0.537758
\(658\) 0 0
\(659\) −1.32912e7 −1.19221 −0.596104 0.802908i \(-0.703285\pi\)
−0.596104 + 0.802908i \(0.703285\pi\)
\(660\) 2.48602e6 0.222149
\(661\) −2.05219e6 −0.182690 −0.0913448 0.995819i \(-0.529117\pi\)
−0.0913448 + 0.995819i \(0.529117\pi\)
\(662\) −5.48619e6 −0.486548
\(663\) −4.52768e6 −0.400030
\(664\) −411392. −0.0362106
\(665\) 0 0
\(666\) −1.41199e6 −0.123352
\(667\) −1.09582e7 −0.953732
\(668\) −1.92358e6 −0.166790
\(669\) −1.15898e7 −1.00118
\(670\) 4.26109e6 0.366719
\(671\) 2.44100e7 2.09296
\(672\) 0 0
\(673\) −1.57039e7 −1.33650 −0.668252 0.743935i \(-0.732957\pi\)
−0.668252 + 0.743935i \(0.732957\pi\)
\(674\) −3.85409e6 −0.326792
\(675\) 1.78532e6 0.150819
\(676\) −4.32270e6 −0.363822
\(677\) 969534. 0.0813002 0.0406501 0.999173i \(-0.487057\pi\)
0.0406501 + 0.999173i \(0.487057\pi\)
\(678\) −2.01650e6 −0.168471
\(679\) 0 0
\(680\) −2.63245e6 −0.218317
\(681\) 1.16015e7 0.958617
\(682\) −1.89320e7 −1.55860
\(683\) −1.49908e7 −1.22962 −0.614812 0.788673i \(-0.710768\pi\)
−0.614812 + 0.788673i \(0.710768\pi\)
\(684\) −305856. −0.0249964
\(685\) −3.97618e6 −0.323772
\(686\) 0 0
\(687\) 6.10393e6 0.493421
\(688\) −2.16371e6 −0.174272
\(689\) 1.06066e7 0.851191
\(690\) −2.07043e6 −0.165553
\(691\) 7.16038e6 0.570481 0.285240 0.958456i \(-0.407927\pi\)
0.285240 + 0.958456i \(0.407927\pi\)
\(692\) −8.14198e6 −0.646346
\(693\) 0 0
\(694\) −1.03092e7 −0.812509
\(695\) −8.92996e6 −0.701274
\(696\) −2.85350e6 −0.223283
\(697\) 1.66774e7 1.30031
\(698\) −1.22701e7 −0.953253
\(699\) 1.00558e7 0.778437
\(700\) 0 0
\(701\) −91834.0 −0.00705844 −0.00352922 0.999994i \(-0.501123\pi\)
−0.00352922 + 0.999994i \(0.501123\pi\)
\(702\) −927288. −0.0710186
\(703\) −1.02849e6 −0.0784894
\(704\) 2.71974e6 0.206822
\(705\) −1.25237e6 −0.0948985
\(706\) −1.24058e7 −0.936725
\(707\) 0 0
\(708\) −2.22278e6 −0.166653
\(709\) 2.20981e7 1.65097 0.825487 0.564422i \(-0.190901\pi\)
0.825487 + 0.564422i \(0.190901\pi\)
\(710\) −945568. −0.0703958
\(711\) 7.24140e6 0.537216
\(712\) −7.85011e6 −0.580331
\(713\) 1.57671e7 1.16153
\(714\) 0 0
\(715\) 5.48995e6 0.401609
\(716\) 7.80096e6 0.568677
\(717\) 1.13576e7 0.825066
\(718\) 1.31003e6 0.0948355
\(719\) −1.58388e7 −1.14262 −0.571308 0.820736i \(-0.693564\pi\)
−0.571308 + 0.820736i \(0.693564\pi\)
\(720\) −539136. −0.0387585
\(721\) 0 0
\(722\) 9.68161e6 0.691202
\(723\) 8.53396e6 0.607163
\(724\) 8.71056e6 0.617589
\(725\) 1.21323e7 0.857235
\(726\) 1.00744e7 0.709379
\(727\) −6.31418e6 −0.443078 −0.221539 0.975151i \(-0.571108\pi\)
−0.221539 + 0.975151i \(0.571108\pi\)
\(728\) 0 0
\(729\) 531441. 0.0370370
\(730\) 7.63922e6 0.530569
\(731\) 1.33711e7 0.925492
\(732\) −5.29373e6 −0.365161
\(733\) −6.93003e6 −0.476404 −0.238202 0.971216i \(-0.576558\pi\)
−0.238202 + 0.971216i \(0.576558\pi\)
\(734\) −1.14696e7 −0.785791
\(735\) 0 0
\(736\) −2.26509e6 −0.154131
\(737\) 2.72054e7 1.84496
\(738\) 3.41561e6 0.230849
\(739\) 1.42331e7 0.958714 0.479357 0.877620i \(-0.340870\pi\)
0.479357 + 0.877620i \(0.340870\pi\)
\(740\) −1.81293e6 −0.121703
\(741\) −675432. −0.0451894
\(742\) 0 0
\(743\) −5.94460e6 −0.395048 −0.197524 0.980298i \(-0.563290\pi\)
−0.197524 + 0.980298i \(0.563290\pi\)
\(744\) 4.10573e6 0.271930
\(745\) 4.54631e6 0.300102
\(746\) −1.43212e7 −0.942175
\(747\) 520668. 0.0341397
\(748\) −1.68072e7 −1.09835
\(749\) 0 0
\(750\) 5.21726e6 0.338680
\(751\) −682752. −0.0441736 −0.0220868 0.999756i \(-0.507031\pi\)
−0.0220868 + 0.999756i \(0.507031\pi\)
\(752\) −1.37011e6 −0.0883510
\(753\) −4.37756e6 −0.281349
\(754\) −6.30149e6 −0.403659
\(755\) 1.17664e7 0.751233
\(756\) 0 0
\(757\) 1.46333e7 0.928116 0.464058 0.885805i \(-0.346393\pi\)
0.464058 + 0.885805i \(0.346393\pi\)
\(758\) −6.56939e6 −0.415291
\(759\) −1.32189e7 −0.832897
\(760\) −392704. −0.0246622
\(761\) 1.16367e7 0.728399 0.364200 0.931321i \(-0.381343\pi\)
0.364200 + 0.931321i \(0.381343\pi\)
\(762\) 6.67440e6 0.416414
\(763\) 0 0
\(764\) 6.02246e6 0.373285
\(765\) 3.33169e6 0.205831
\(766\) −8.22790e6 −0.506661
\(767\) −4.90865e6 −0.301282
\(768\) −589824. −0.0360844
\(769\) −1.91472e7 −1.16759 −0.583793 0.811902i \(-0.698432\pi\)
−0.583793 + 0.811902i \(0.698432\pi\)
\(770\) 0 0
\(771\) −9.35188e6 −0.566582
\(772\) 1.35191e7 0.816405
\(773\) 5.39261e6 0.324601 0.162301 0.986741i \(-0.448109\pi\)
0.162301 + 0.986741i \(0.448109\pi\)
\(774\) 2.73845e6 0.164305
\(775\) −1.74565e7 −1.04400
\(776\) 1.36768e6 0.0815324
\(777\) 0 0
\(778\) −2.46457e6 −0.145979
\(779\) 2.48791e6 0.146890
\(780\) −1.19059e6 −0.0700691
\(781\) −6.03709e6 −0.354160
\(782\) 1.39975e7 0.818530
\(783\) 3.61147e6 0.210513
\(784\) 0 0
\(785\) −1.29757e7 −0.751549
\(786\) −2.32315e6 −0.134129
\(787\) −3.04348e6 −0.175159 −0.0875796 0.996158i \(-0.527913\pi\)
−0.0875796 + 0.996158i \(0.527913\pi\)
\(788\) −7.88470e6 −0.452345
\(789\) −1.21594e7 −0.695376
\(790\) 9.29760e6 0.530033
\(791\) 0 0
\(792\) −3.44218e6 −0.194993
\(793\) −1.16903e7 −0.660151
\(794\) 8.76847e6 0.493597
\(795\) −7.80484e6 −0.437972
\(796\) 1.46364e7 0.818751
\(797\) −2.29652e7 −1.28063 −0.640316 0.768111i \(-0.721197\pi\)
−0.640316 + 0.768111i \(0.721197\pi\)
\(798\) 0 0
\(799\) 8.46686e6 0.469197
\(800\) 2.50778e6 0.138536
\(801\) 9.93530e6 0.547141
\(802\) −1.31382e7 −0.721271
\(803\) 4.87735e7 2.66928
\(804\) −5.89997e6 −0.321892
\(805\) 0 0
\(806\) 9.06682e6 0.491606
\(807\) −1.00630e7 −0.543930
\(808\) −2.35610e6 −0.126959
\(809\) 1.90787e7 1.02489 0.512445 0.858720i \(-0.328740\pi\)
0.512445 + 0.858720i \(0.328740\pi\)
\(810\) 682344. 0.0365419
\(811\) −1.09414e7 −0.584147 −0.292074 0.956396i \(-0.594345\pi\)
−0.292074 + 0.956396i \(0.594345\pi\)
\(812\) 0 0
\(813\) −1.71094e6 −0.0907836
\(814\) −1.15748e7 −0.612286
\(815\) 1.23653e7 0.652094
\(816\) 3.64493e6 0.191630
\(817\) 1.99467e6 0.104548
\(818\) −1.44488e7 −0.755001
\(819\) 0 0
\(820\) 4.38547e6 0.227762
\(821\) 2.12594e7 1.10076 0.550380 0.834914i \(-0.314483\pi\)
0.550380 + 0.834914i \(0.314483\pi\)
\(822\) 5.50548e6 0.284194
\(823\) −1.42256e7 −0.732103 −0.366052 0.930595i \(-0.619291\pi\)
−0.366052 + 0.930595i \(0.619291\pi\)
\(824\) 6.68979e6 0.343237
\(825\) 1.46352e7 0.748625
\(826\) 0 0
\(827\) 2.76103e6 0.140381 0.0701904 0.997534i \(-0.477639\pi\)
0.0701904 + 0.997534i \(0.477639\pi\)
\(828\) 2.86675e6 0.145316
\(829\) 3.82147e7 1.93127 0.965637 0.259895i \(-0.0836880\pi\)
0.965637 + 0.259895i \(0.0836880\pi\)
\(830\) 668512. 0.0336832
\(831\) 1.80455e6 0.0906499
\(832\) −1.30253e6 −0.0652347
\(833\) 0 0
\(834\) 1.23646e7 0.615550
\(835\) 3.12582e6 0.155149
\(836\) −2.50726e6 −0.124075
\(837\) −5.19631e6 −0.256378
\(838\) 2.16596e7 1.06547
\(839\) −1.06044e7 −0.520094 −0.260047 0.965596i \(-0.583738\pi\)
−0.260047 + 0.965596i \(0.583738\pi\)
\(840\) 0 0
\(841\) 4.03097e6 0.196526
\(842\) −1.44252e7 −0.701198
\(843\) −9.83133e6 −0.476479
\(844\) 4.98848e6 0.241053
\(845\) 7.02439e6 0.338429
\(846\) 1.73405e6 0.0832981
\(847\) 0 0
\(848\) −8.53862e6 −0.407754
\(849\) 1.63132e7 0.776731
\(850\) −1.54973e7 −0.735712
\(851\) 9.63990e6 0.456298
\(852\) 1.30925e6 0.0617907
\(853\) 4.07009e7 1.91527 0.957637 0.287977i \(-0.0929826\pi\)
0.957637 + 0.287977i \(0.0929826\pi\)
\(854\) 0 0
\(855\) 497016. 0.0232517
\(856\) −1.37242e7 −0.640179
\(857\) 3.10120e7 1.44237 0.721187 0.692741i \(-0.243597\pi\)
0.721187 + 0.692741i \(0.243597\pi\)
\(858\) −7.60147e6 −0.352517
\(859\) −1.09104e7 −0.504495 −0.252247 0.967663i \(-0.581170\pi\)
−0.252247 + 0.967663i \(0.581170\pi\)
\(860\) 3.51603e6 0.162109
\(861\) 0 0
\(862\) 1.11286e7 0.510118
\(863\) 1.04089e7 0.475751 0.237875 0.971296i \(-0.423549\pi\)
0.237875 + 0.971296i \(0.423549\pi\)
\(864\) 746496. 0.0340207
\(865\) 1.32307e7 0.601234
\(866\) 2.51048e7 1.13753
\(867\) −9.74580e6 −0.440321
\(868\) 0 0
\(869\) 5.93616e7 2.66659
\(870\) 4.63694e6 0.207699
\(871\) −1.30291e7 −0.581928
\(872\) −1.84307e6 −0.0820826
\(873\) −1.73097e6 −0.0768695
\(874\) 2.08813e6 0.0924652
\(875\) 0 0
\(876\) −1.05774e7 −0.465712
\(877\) 1.64064e7 0.720299 0.360150 0.932895i \(-0.382726\pi\)
0.360150 + 0.932895i \(0.382726\pi\)
\(878\) 2.56637e6 0.112352
\(879\) 1.89028e7 0.825188
\(880\) −4.41958e6 −0.192387
\(881\) −1.48577e7 −0.644927 −0.322464 0.946582i \(-0.604511\pi\)
−0.322464 + 0.946582i \(0.604511\pi\)
\(882\) 0 0
\(883\) −2.72018e7 −1.17407 −0.587037 0.809560i \(-0.699706\pi\)
−0.587037 + 0.809560i \(0.699706\pi\)
\(884\) 8.04922e6 0.346436
\(885\) 3.61202e6 0.155022
\(886\) −2.42218e7 −1.03663
\(887\) −2.71242e7 −1.15757 −0.578785 0.815480i \(-0.696473\pi\)
−0.578785 + 0.815480i \(0.696473\pi\)
\(888\) 2.51021e6 0.106826
\(889\) 0 0
\(890\) 1.27564e7 0.539827
\(891\) 4.35650e6 0.183842
\(892\) 2.06042e7 0.867047
\(893\) 1.26307e6 0.0530029
\(894\) −6.29489e6 −0.263417
\(895\) −1.26766e7 −0.528986
\(896\) 0 0
\(897\) 6.33074e6 0.262708
\(898\) 2.06673e7 0.855248
\(899\) −3.53121e7 −1.45722
\(900\) −3.17390e6 −0.130613
\(901\) 5.27660e7 2.16542
\(902\) 2.79996e7 1.14587
\(903\) 0 0
\(904\) 3.58490e6 0.145900
\(905\) −1.41547e7 −0.574484
\(906\) −1.62919e7 −0.659402
\(907\) −8.42269e6 −0.339964 −0.169982 0.985447i \(-0.554371\pi\)
−0.169982 + 0.985447i \(0.554371\pi\)
\(908\) −2.06248e7 −0.830187
\(909\) 2.98193e6 0.119698
\(910\) 0 0
\(911\) 3.08637e7 1.23212 0.616060 0.787700i \(-0.288728\pi\)
0.616060 + 0.787700i \(0.288728\pi\)
\(912\) 543744. 0.0216475
\(913\) 4.26819e6 0.169460
\(914\) 911192. 0.0360782
\(915\) 8.60231e6 0.339674
\(916\) −1.08514e7 −0.427315
\(917\) 0 0
\(918\) −4.61311e6 −0.180671
\(919\) 4.93895e6 0.192906 0.0964531 0.995338i \(-0.469250\pi\)
0.0964531 + 0.995338i \(0.469250\pi\)
\(920\) 3.68077e6 0.143373
\(921\) −1.47694e7 −0.573738
\(922\) 2.34058e6 0.0906770
\(923\) 2.89126e6 0.111707
\(924\) 0 0
\(925\) −1.06727e7 −0.410130
\(926\) 1.36581e7 0.523437
\(927\) −8.46677e6 −0.323607
\(928\) 5.07290e6 0.193369
\(929\) −5.62575e6 −0.213866 −0.106933 0.994266i \(-0.534103\pi\)
−0.106933 + 0.994266i \(0.534103\pi\)
\(930\) −6.67181e6 −0.252951
\(931\) 0 0
\(932\) −1.78770e7 −0.674146
\(933\) −8.50709e6 −0.319946
\(934\) 2.86520e6 0.107470
\(935\) 2.73116e7 1.02169
\(936\) 1.64851e6 0.0615039
\(937\) −2.60073e7 −0.967714 −0.483857 0.875147i \(-0.660764\pi\)
−0.483857 + 0.875147i \(0.660764\pi\)
\(938\) 0 0
\(939\) 3.73819e6 0.138356
\(940\) 2.22643e6 0.0821845
\(941\) −3.02160e6 −0.111241 −0.0556203 0.998452i \(-0.517714\pi\)
−0.0556203 + 0.998452i \(0.517714\pi\)
\(942\) 1.79664e7 0.659680
\(943\) −2.33189e7 −0.853943
\(944\) 3.95162e6 0.144326
\(945\) 0 0
\(946\) 2.24485e7 0.815567
\(947\) −3.48282e7 −1.26199 −0.630995 0.775787i \(-0.717353\pi\)
−0.630995 + 0.775787i \(0.717353\pi\)
\(948\) −1.28736e7 −0.465242
\(949\) −2.33584e7 −0.841932
\(950\) −2.31186e6 −0.0831097
\(951\) −1.06633e7 −0.382333
\(952\) 0 0
\(953\) −9.39009e6 −0.334917 −0.167459 0.985879i \(-0.553556\pi\)
−0.167459 + 0.985879i \(0.553556\pi\)
\(954\) 1.08067e7 0.384434
\(955\) −9.78650e6 −0.347232
\(956\) −2.01913e7 −0.714528
\(957\) 2.96051e7 1.04493
\(958\) 2.09637e7 0.737996
\(959\) 0 0
\(960\) 958464. 0.0335659
\(961\) 2.21792e7 0.774708
\(962\) 5.54338e6 0.193124
\(963\) 1.73696e7 0.603566
\(964\) −1.51715e7 −0.525818
\(965\) −2.19686e7 −0.759423
\(966\) 0 0
\(967\) 1.44768e7 0.497860 0.248930 0.968521i \(-0.419921\pi\)
0.248930 + 0.968521i \(0.419921\pi\)
\(968\) −1.79101e7 −0.614340
\(969\) −3.36017e6 −0.114961
\(970\) −2.22248e6 −0.0758418
\(971\) −9.24976e6 −0.314834 −0.157417 0.987532i \(-0.550317\pi\)
−0.157417 + 0.987532i \(0.550317\pi\)
\(972\) −944784. −0.0320750
\(973\) 0 0
\(974\) −4.46806e6 −0.150911
\(975\) −7.00904e6 −0.236128
\(976\) 9.41107e6 0.316238
\(977\) −4.97780e7 −1.66840 −0.834202 0.551459i \(-0.814071\pi\)
−0.834202 + 0.551459i \(0.814071\pi\)
\(978\) −1.71212e7 −0.572382
\(979\) 8.14449e7 2.71586
\(980\) 0 0
\(981\) 2.33264e6 0.0773882
\(982\) −5.37830e6 −0.177978
\(983\) 8.95601e6 0.295618 0.147809 0.989016i \(-0.452778\pi\)
0.147809 + 0.989016i \(0.452778\pi\)
\(984\) −6.07219e6 −0.199921
\(985\) 1.28126e7 0.420773
\(986\) −3.13489e7 −1.02690
\(987\) 0 0
\(988\) 1.20077e6 0.0391351
\(989\) −1.86958e7 −0.607790
\(990\) 5.59354e6 0.181384
\(991\) 2.62400e7 0.848751 0.424376 0.905486i \(-0.360494\pi\)
0.424376 + 0.905486i \(0.360494\pi\)
\(992\) −7.29907e6 −0.235499
\(993\) −1.23439e7 −0.397265
\(994\) 0 0
\(995\) −2.37842e7 −0.761606
\(996\) −925632. −0.0295658
\(997\) −2.80506e7 −0.893727 −0.446863 0.894602i \(-0.647459\pi\)
−0.446863 + 0.894602i \(0.647459\pi\)
\(998\) 2.61859e7 0.832226
\(999\) −3.17698e6 −0.100717
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 294.6.a.b.1.1 1
3.2 odd 2 882.6.a.s.1.1 1
7.2 even 3 294.6.e.p.67.1 2
7.3 odd 6 294.6.e.i.79.1 2
7.4 even 3 294.6.e.p.79.1 2
7.5 odd 6 294.6.e.i.67.1 2
7.6 odd 2 42.6.a.d.1.1 1
21.20 even 2 126.6.a.i.1.1 1
28.27 even 2 336.6.a.h.1.1 1
35.13 even 4 1050.6.g.i.799.2 2
35.27 even 4 1050.6.g.i.799.1 2
35.34 odd 2 1050.6.a.k.1.1 1
84.83 odd 2 1008.6.a.j.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
42.6.a.d.1.1 1 7.6 odd 2
126.6.a.i.1.1 1 21.20 even 2
294.6.a.b.1.1 1 1.1 even 1 trivial
294.6.e.i.67.1 2 7.5 odd 6
294.6.e.i.79.1 2 7.3 odd 6
294.6.e.p.67.1 2 7.2 even 3
294.6.e.p.79.1 2 7.4 even 3
336.6.a.h.1.1 1 28.27 even 2
882.6.a.s.1.1 1 3.2 odd 2
1008.6.a.j.1.1 1 84.83 odd 2
1050.6.a.k.1.1 1 35.34 odd 2
1050.6.g.i.799.1 2 35.27 even 4
1050.6.g.i.799.2 2 35.13 even 4