Properties

Label 63.6.a.c.1.1
Level $63$
Weight $6$
Character 63.1
Self dual yes
Analytic conductor $10.104$
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] = [63,6,Mod(1,63)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(63, 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("63.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 63 = 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 63.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: \(10.1041806482\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 21)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 63.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -31.0000 q^{4} +34.0000 q^{5} -49.0000 q^{7} +63.0000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} -31.0000 q^{4} +34.0000 q^{5} -49.0000 q^{7} +63.0000 q^{8} -34.0000 q^{10} +340.000 q^{11} +454.000 q^{13} +49.0000 q^{14} +929.000 q^{16} +798.000 q^{17} +892.000 q^{19} -1054.00 q^{20} -340.000 q^{22} +3192.00 q^{23} -1969.00 q^{25} -454.000 q^{26} +1519.00 q^{28} +8242.00 q^{29} -2496.00 q^{31} -2945.00 q^{32} -798.000 q^{34} -1666.00 q^{35} +9798.00 q^{37} -892.000 q^{38} +2142.00 q^{40} -19834.0 q^{41} -17236.0 q^{43} -10540.0 q^{44} -3192.00 q^{46} -8928.00 q^{47} +2401.00 q^{49} +1969.00 q^{50} -14074.0 q^{52} -150.000 q^{53} +11560.0 q^{55} -3087.00 q^{56} -8242.00 q^{58} +42396.0 q^{59} +14758.0 q^{61} +2496.00 q^{62} -26783.0 q^{64} +15436.0 q^{65} -1676.00 q^{67} -24738.0 q^{68} +1666.00 q^{70} -14568.0 q^{71} +78378.0 q^{73} -9798.00 q^{74} -27652.0 q^{76} -16660.0 q^{77} -2272.00 q^{79} +31586.0 q^{80} +19834.0 q^{82} +37764.0 q^{83} +27132.0 q^{85} +17236.0 q^{86} +21420.0 q^{88} +117286. q^{89} -22246.0 q^{91} -98952.0 q^{92} +8928.00 q^{94} +30328.0 q^{95} +10002.0 q^{97} -2401.00 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.176777 −0.0883883 0.996086i \(-0.528172\pi\)
−0.0883883 + 0.996086i \(0.528172\pi\)
\(3\) 0 0
\(4\) −31.0000 −0.968750
\(5\) 34.0000 0.608210 0.304105 0.952638i \(-0.401643\pi\)
0.304105 + 0.952638i \(0.401643\pi\)
\(6\) 0 0
\(7\) −49.0000 −0.377964
\(8\) 63.0000 0.348029
\(9\) 0 0
\(10\) −34.0000 −0.107517
\(11\) 340.000 0.847222 0.423611 0.905844i \(-0.360762\pi\)
0.423611 + 0.905844i \(0.360762\pi\)
\(12\) 0 0
\(13\) 454.000 0.745071 0.372535 0.928018i \(-0.378489\pi\)
0.372535 + 0.928018i \(0.378489\pi\)
\(14\) 49.0000 0.0668153
\(15\) 0 0
\(16\) 929.000 0.907227
\(17\) 798.000 0.669700 0.334850 0.942271i \(-0.391314\pi\)
0.334850 + 0.942271i \(0.391314\pi\)
\(18\) 0 0
\(19\) 892.000 0.566867 0.283433 0.958992i \(-0.408527\pi\)
0.283433 + 0.958992i \(0.408527\pi\)
\(20\) −1054.00 −0.589204
\(21\) 0 0
\(22\) −340.000 −0.149769
\(23\) 3192.00 1.25818 0.629091 0.777332i \(-0.283427\pi\)
0.629091 + 0.777332i \(0.283427\pi\)
\(24\) 0 0
\(25\) −1969.00 −0.630080
\(26\) −454.000 −0.131711
\(27\) 0 0
\(28\) 1519.00 0.366153
\(29\) 8242.00 1.81986 0.909929 0.414764i \(-0.136136\pi\)
0.909929 + 0.414764i \(0.136136\pi\)
\(30\) 0 0
\(31\) −2496.00 −0.466488 −0.233244 0.972418i \(-0.574934\pi\)
−0.233244 + 0.972418i \(0.574934\pi\)
\(32\) −2945.00 −0.508406
\(33\) 0 0
\(34\) −798.000 −0.118387
\(35\) −1666.00 −0.229882
\(36\) 0 0
\(37\) 9798.00 1.17661 0.588306 0.808639i \(-0.299795\pi\)
0.588306 + 0.808639i \(0.299795\pi\)
\(38\) −892.000 −0.100209
\(39\) 0 0
\(40\) 2142.00 0.211675
\(41\) −19834.0 −1.84268 −0.921342 0.388754i \(-0.872906\pi\)
−0.921342 + 0.388754i \(0.872906\pi\)
\(42\) 0 0
\(43\) −17236.0 −1.42156 −0.710780 0.703414i \(-0.751658\pi\)
−0.710780 + 0.703414i \(0.751658\pi\)
\(44\) −10540.0 −0.820746
\(45\) 0 0
\(46\) −3192.00 −0.222417
\(47\) −8928.00 −0.589535 −0.294767 0.955569i \(-0.595242\pi\)
−0.294767 + 0.955569i \(0.595242\pi\)
\(48\) 0 0
\(49\) 2401.00 0.142857
\(50\) 1969.00 0.111383
\(51\) 0 0
\(52\) −14074.0 −0.721787
\(53\) −150.000 −0.00733502 −0.00366751 0.999993i \(-0.501167\pi\)
−0.00366751 + 0.999993i \(0.501167\pi\)
\(54\) 0 0
\(55\) 11560.0 0.515289
\(56\) −3087.00 −0.131543
\(57\) 0 0
\(58\) −8242.00 −0.321709
\(59\) 42396.0 1.58560 0.792802 0.609479i \(-0.208621\pi\)
0.792802 + 0.609479i \(0.208621\pi\)
\(60\) 0 0
\(61\) 14758.0 0.507812 0.253906 0.967229i \(-0.418285\pi\)
0.253906 + 0.967229i \(0.418285\pi\)
\(62\) 2496.00 0.0824642
\(63\) 0 0
\(64\) −26783.0 −0.817352
\(65\) 15436.0 0.453160
\(66\) 0 0
\(67\) −1676.00 −0.0456128 −0.0228064 0.999740i \(-0.507260\pi\)
−0.0228064 + 0.999740i \(0.507260\pi\)
\(68\) −24738.0 −0.648772
\(69\) 0 0
\(70\) 1666.00 0.0406378
\(71\) −14568.0 −0.342968 −0.171484 0.985187i \(-0.554856\pi\)
−0.171484 + 0.985187i \(0.554856\pi\)
\(72\) 0 0
\(73\) 78378.0 1.72142 0.860710 0.509095i \(-0.170020\pi\)
0.860710 + 0.509095i \(0.170020\pi\)
\(74\) −9798.00 −0.207998
\(75\) 0 0
\(76\) −27652.0 −0.549152
\(77\) −16660.0 −0.320220
\(78\) 0 0
\(79\) −2272.00 −0.0409582 −0.0204791 0.999790i \(-0.506519\pi\)
−0.0204791 + 0.999790i \(0.506519\pi\)
\(80\) 31586.0 0.551785
\(81\) 0 0
\(82\) 19834.0 0.325743
\(83\) 37764.0 0.601704 0.300852 0.953671i \(-0.402729\pi\)
0.300852 + 0.953671i \(0.402729\pi\)
\(84\) 0 0
\(85\) 27132.0 0.407319
\(86\) 17236.0 0.251299
\(87\) 0 0
\(88\) 21420.0 0.294858
\(89\) 117286. 1.56954 0.784768 0.619790i \(-0.212782\pi\)
0.784768 + 0.619790i \(0.212782\pi\)
\(90\) 0 0
\(91\) −22246.0 −0.281610
\(92\) −98952.0 −1.21886
\(93\) 0 0
\(94\) 8928.00 0.104216
\(95\) 30328.0 0.344774
\(96\) 0 0
\(97\) 10002.0 0.107934 0.0539669 0.998543i \(-0.482813\pi\)
0.0539669 + 0.998543i \(0.482813\pi\)
\(98\) −2401.00 −0.0252538
\(99\) 0 0
\(100\) 61039.0 0.610390
\(101\) 108770. 1.06098 0.530488 0.847692i \(-0.322009\pi\)
0.530488 + 0.847692i \(0.322009\pi\)
\(102\) 0 0
\(103\) −199192. −1.85003 −0.925015 0.379930i \(-0.875948\pi\)
−0.925015 + 0.379930i \(0.875948\pi\)
\(104\) 28602.0 0.259306
\(105\) 0 0
\(106\) 150.000 0.00129666
\(107\) 79972.0 0.675272 0.337636 0.941277i \(-0.390373\pi\)
0.337636 + 0.941277i \(0.390373\pi\)
\(108\) 0 0
\(109\) −46098.0 −0.371634 −0.185817 0.982584i \(-0.559493\pi\)
−0.185817 + 0.982584i \(0.559493\pi\)
\(110\) −11560.0 −0.0910911
\(111\) 0 0
\(112\) −45521.0 −0.342899
\(113\) −262706. −1.93541 −0.967707 0.252078i \(-0.918886\pi\)
−0.967707 + 0.252078i \(0.918886\pi\)
\(114\) 0 0
\(115\) 108528. 0.765239
\(116\) −255502. −1.76299
\(117\) 0 0
\(118\) −42396.0 −0.280298
\(119\) −39102.0 −0.253123
\(120\) 0 0
\(121\) −45451.0 −0.282215
\(122\) −14758.0 −0.0897693
\(123\) 0 0
\(124\) 77376.0 0.451910
\(125\) −173196. −0.991432
\(126\) 0 0
\(127\) 196608. 1.08166 0.540831 0.841131i \(-0.318110\pi\)
0.540831 + 0.841131i \(0.318110\pi\)
\(128\) 121023. 0.652894
\(129\) 0 0
\(130\) −15436.0 −0.0801081
\(131\) 77140.0 0.392737 0.196368 0.980530i \(-0.437085\pi\)
0.196368 + 0.980530i \(0.437085\pi\)
\(132\) 0 0
\(133\) −43708.0 −0.214255
\(134\) 1676.00 0.00806329
\(135\) 0 0
\(136\) 50274.0 0.233075
\(137\) −208170. −0.947582 −0.473791 0.880637i \(-0.657115\pi\)
−0.473791 + 0.880637i \(0.657115\pi\)
\(138\) 0 0
\(139\) −275580. −1.20979 −0.604896 0.796304i \(-0.706785\pi\)
−0.604896 + 0.796304i \(0.706785\pi\)
\(140\) 51646.0 0.222698
\(141\) 0 0
\(142\) 14568.0 0.0606288
\(143\) 154360. 0.631240
\(144\) 0 0
\(145\) 280228. 1.10686
\(146\) −78378.0 −0.304307
\(147\) 0 0
\(148\) −303738. −1.13984
\(149\) 296106. 1.09265 0.546326 0.837573i \(-0.316026\pi\)
0.546326 + 0.837573i \(0.316026\pi\)
\(150\) 0 0
\(151\) −426472. −1.52212 −0.761059 0.648683i \(-0.775320\pi\)
−0.761059 + 0.648683i \(0.775320\pi\)
\(152\) 56196.0 0.197286
\(153\) 0 0
\(154\) 16660.0 0.0566074
\(155\) −84864.0 −0.283723
\(156\) 0 0
\(157\) 178486. 0.577903 0.288952 0.957344i \(-0.406693\pi\)
0.288952 + 0.957344i \(0.406693\pi\)
\(158\) 2272.00 0.00724045
\(159\) 0 0
\(160\) −100130. −0.309218
\(161\) −156408. −0.475548
\(162\) 0 0
\(163\) 252772. 0.745178 0.372589 0.927996i \(-0.378470\pi\)
0.372589 + 0.927996i \(0.378470\pi\)
\(164\) 614854. 1.78510
\(165\) 0 0
\(166\) −37764.0 −0.106367
\(167\) −508088. −1.40977 −0.704884 0.709322i \(-0.749001\pi\)
−0.704884 + 0.709322i \(0.749001\pi\)
\(168\) 0 0
\(169\) −165177. −0.444870
\(170\) −27132.0 −0.0720045
\(171\) 0 0
\(172\) 534316. 1.37714
\(173\) 221834. 0.563525 0.281762 0.959484i \(-0.409081\pi\)
0.281762 + 0.959484i \(0.409081\pi\)
\(174\) 0 0
\(175\) 96481.0 0.238148
\(176\) 315860. 0.768622
\(177\) 0 0
\(178\) −117286. −0.277457
\(179\) 113564. 0.264916 0.132458 0.991189i \(-0.457713\pi\)
0.132458 + 0.991189i \(0.457713\pi\)
\(180\) 0 0
\(181\) 663118. 1.50451 0.752254 0.658873i \(-0.228967\pi\)
0.752254 + 0.658873i \(0.228967\pi\)
\(182\) 22246.0 0.0497821
\(183\) 0 0
\(184\) 201096. 0.437884
\(185\) 333132. 0.715628
\(186\) 0 0
\(187\) 271320. 0.567385
\(188\) 276768. 0.571112
\(189\) 0 0
\(190\) −30328.0 −0.0609480
\(191\) −505664. −1.00295 −0.501474 0.865173i \(-0.667209\pi\)
−0.501474 + 0.865173i \(0.667209\pi\)
\(192\) 0 0
\(193\) −432382. −0.835554 −0.417777 0.908550i \(-0.637191\pi\)
−0.417777 + 0.908550i \(0.637191\pi\)
\(194\) −10002.0 −0.0190802
\(195\) 0 0
\(196\) −74431.0 −0.138393
\(197\) 131962. 0.242261 0.121130 0.992637i \(-0.461348\pi\)
0.121130 + 0.992637i \(0.461348\pi\)
\(198\) 0 0
\(199\) 298536. 0.534397 0.267199 0.963642i \(-0.413902\pi\)
0.267199 + 0.963642i \(0.413902\pi\)
\(200\) −124047. −0.219286
\(201\) 0 0
\(202\) −108770. −0.187556
\(203\) −403858. −0.687842
\(204\) 0 0
\(205\) −674356. −1.12074
\(206\) 199192. 0.327042
\(207\) 0 0
\(208\) 421766. 0.675948
\(209\) 303280. 0.480262
\(210\) 0 0
\(211\) −1.17062e6 −1.81013 −0.905065 0.425273i \(-0.860178\pi\)
−0.905065 + 0.425273i \(0.860178\pi\)
\(212\) 4650.00 0.00710581
\(213\) 0 0
\(214\) −79972.0 −0.119372
\(215\) −586024. −0.864608
\(216\) 0 0
\(217\) 122304. 0.176316
\(218\) 46098.0 0.0656963
\(219\) 0 0
\(220\) −358360. −0.499186
\(221\) 362292. 0.498974
\(222\) 0 0
\(223\) 399376. 0.537799 0.268899 0.963168i \(-0.413340\pi\)
0.268899 + 0.963168i \(0.413340\pi\)
\(224\) 144305. 0.192159
\(225\) 0 0
\(226\) 262706. 0.342136
\(227\) −707916. −0.911837 −0.455918 0.890022i \(-0.650689\pi\)
−0.455918 + 0.890022i \(0.650689\pi\)
\(228\) 0 0
\(229\) −735778. −0.927167 −0.463584 0.886053i \(-0.653437\pi\)
−0.463584 + 0.886053i \(0.653437\pi\)
\(230\) −108528. −0.135276
\(231\) 0 0
\(232\) 519246. 0.633364
\(233\) 208758. 0.251915 0.125957 0.992036i \(-0.459800\pi\)
0.125957 + 0.992036i \(0.459800\pi\)
\(234\) 0 0
\(235\) −303552. −0.358561
\(236\) −1.31428e6 −1.53605
\(237\) 0 0
\(238\) 39102.0 0.0447462
\(239\) −713376. −0.807837 −0.403919 0.914795i \(-0.632352\pi\)
−0.403919 + 0.914795i \(0.632352\pi\)
\(240\) 0 0
\(241\) −505246. −0.560351 −0.280176 0.959949i \(-0.590393\pi\)
−0.280176 + 0.959949i \(0.590393\pi\)
\(242\) 45451.0 0.0498890
\(243\) 0 0
\(244\) −457498. −0.491943
\(245\) 81634.0 0.0868872
\(246\) 0 0
\(247\) 404968. 0.422356
\(248\) −157248. −0.162351
\(249\) 0 0
\(250\) 173196. 0.175262
\(251\) −317108. −0.317704 −0.158852 0.987302i \(-0.550779\pi\)
−0.158852 + 0.987302i \(0.550779\pi\)
\(252\) 0 0
\(253\) 1.08528e6 1.06596
\(254\) −196608. −0.191213
\(255\) 0 0
\(256\) 736033. 0.701936
\(257\) 1.44285e6 1.36266 0.681329 0.731977i \(-0.261402\pi\)
0.681329 + 0.731977i \(0.261402\pi\)
\(258\) 0 0
\(259\) −480102. −0.444717
\(260\) −478516. −0.438999
\(261\) 0 0
\(262\) −77140.0 −0.0694267
\(263\) −271496. −0.242033 −0.121016 0.992651i \(-0.538615\pi\)
−0.121016 + 0.992651i \(0.538615\pi\)
\(264\) 0 0
\(265\) −5100.00 −0.00446124
\(266\) 43708.0 0.0378754
\(267\) 0 0
\(268\) 51956.0 0.0441874
\(269\) −850614. −0.716724 −0.358362 0.933583i \(-0.616665\pi\)
−0.358362 + 0.933583i \(0.616665\pi\)
\(270\) 0 0
\(271\) −540128. −0.446759 −0.223380 0.974732i \(-0.571709\pi\)
−0.223380 + 0.974732i \(0.571709\pi\)
\(272\) 741342. 0.607570
\(273\) 0 0
\(274\) 208170. 0.167510
\(275\) −669460. −0.533818
\(276\) 0 0
\(277\) 513574. 0.402164 0.201082 0.979574i \(-0.435554\pi\)
0.201082 + 0.979574i \(0.435554\pi\)
\(278\) 275580. 0.213863
\(279\) 0 0
\(280\) −104958. −0.0800056
\(281\) 1.35642e6 1.02478 0.512388 0.858754i \(-0.328761\pi\)
0.512388 + 0.858754i \(0.328761\pi\)
\(282\) 0 0
\(283\) 286756. 0.212837 0.106418 0.994321i \(-0.466062\pi\)
0.106418 + 0.994321i \(0.466062\pi\)
\(284\) 451608. 0.332251
\(285\) 0 0
\(286\) −154360. −0.111589
\(287\) 971866. 0.696469
\(288\) 0 0
\(289\) −783053. −0.551501
\(290\) −280228. −0.195667
\(291\) 0 0
\(292\) −2.42972e6 −1.66763
\(293\) 1.70727e6 1.16180 0.580901 0.813974i \(-0.302700\pi\)
0.580901 + 0.813974i \(0.302700\pi\)
\(294\) 0 0
\(295\) 1.44146e6 0.964381
\(296\) 617274. 0.409495
\(297\) 0 0
\(298\) −296106. −0.193155
\(299\) 1.44917e6 0.937434
\(300\) 0 0
\(301\) 844564. 0.537299
\(302\) 426472. 0.269075
\(303\) 0 0
\(304\) 828668. 0.514276
\(305\) 501772. 0.308857
\(306\) 0 0
\(307\) −546788. −0.331111 −0.165555 0.986201i \(-0.552942\pi\)
−0.165555 + 0.986201i \(0.552942\pi\)
\(308\) 516460. 0.310213
\(309\) 0 0
\(310\) 84864.0 0.0501556
\(311\) −3.23426e6 −1.89616 −0.948079 0.318035i \(-0.896977\pi\)
−0.948079 + 0.318035i \(0.896977\pi\)
\(312\) 0 0
\(313\) 1.81313e6 1.04609 0.523044 0.852306i \(-0.324796\pi\)
0.523044 + 0.852306i \(0.324796\pi\)
\(314\) −178486. −0.102160
\(315\) 0 0
\(316\) 70432.0 0.0396782
\(317\) 1.27658e6 0.713509 0.356754 0.934198i \(-0.383883\pi\)
0.356754 + 0.934198i \(0.383883\pi\)
\(318\) 0 0
\(319\) 2.80228e6 1.54182
\(320\) −910622. −0.497122
\(321\) 0 0
\(322\) 156408. 0.0840658
\(323\) 711816. 0.379631
\(324\) 0 0
\(325\) −893926. −0.469454
\(326\) −252772. −0.131730
\(327\) 0 0
\(328\) −1.24954e6 −0.641307
\(329\) 437472. 0.222823
\(330\) 0 0
\(331\) −1.73621e6 −0.871029 −0.435515 0.900182i \(-0.643434\pi\)
−0.435515 + 0.900182i \(0.643434\pi\)
\(332\) −1.17068e6 −0.582901
\(333\) 0 0
\(334\) 508088. 0.249214
\(335\) −56984.0 −0.0277422
\(336\) 0 0
\(337\) 2.07215e6 0.993907 0.496953 0.867777i \(-0.334452\pi\)
0.496953 + 0.867777i \(0.334452\pi\)
\(338\) 165177. 0.0786426
\(339\) 0 0
\(340\) −841092. −0.394590
\(341\) −848640. −0.395219
\(342\) 0 0
\(343\) −117649. −0.0539949
\(344\) −1.08587e6 −0.494744
\(345\) 0 0
\(346\) −221834. −0.0996180
\(347\) 1.65146e6 0.736282 0.368141 0.929770i \(-0.379994\pi\)
0.368141 + 0.929770i \(0.379994\pi\)
\(348\) 0 0
\(349\) 1.26645e6 0.556578 0.278289 0.960497i \(-0.410233\pi\)
0.278289 + 0.960497i \(0.410233\pi\)
\(350\) −96481.0 −0.0420990
\(351\) 0 0
\(352\) −1.00130e6 −0.430732
\(353\) −573218. −0.244840 −0.122420 0.992478i \(-0.539066\pi\)
−0.122420 + 0.992478i \(0.539066\pi\)
\(354\) 0 0
\(355\) −495312. −0.208597
\(356\) −3.63587e6 −1.52049
\(357\) 0 0
\(358\) −113564. −0.0468310
\(359\) −4.46322e6 −1.82773 −0.913866 0.406016i \(-0.866918\pi\)
−0.913866 + 0.406016i \(0.866918\pi\)
\(360\) 0 0
\(361\) −1.68044e6 −0.678662
\(362\) −663118. −0.265962
\(363\) 0 0
\(364\) 689626. 0.272810
\(365\) 2.66485e6 1.04699
\(366\) 0 0
\(367\) −4.50797e6 −1.74709 −0.873546 0.486742i \(-0.838185\pi\)
−0.873546 + 0.486742i \(0.838185\pi\)
\(368\) 2.96537e6 1.14146
\(369\) 0 0
\(370\) −333132. −0.126506
\(371\) 7350.00 0.00277238
\(372\) 0 0
\(373\) 1.66535e6 0.619774 0.309887 0.950773i \(-0.399709\pi\)
0.309887 + 0.950773i \(0.399709\pi\)
\(374\) −271320. −0.100300
\(375\) 0 0
\(376\) −562464. −0.205175
\(377\) 3.74187e6 1.35592
\(378\) 0 0
\(379\) −2.53232e6 −0.905568 −0.452784 0.891620i \(-0.649569\pi\)
−0.452784 + 0.891620i \(0.649569\pi\)
\(380\) −940168. −0.334000
\(381\) 0 0
\(382\) 505664. 0.177298
\(383\) −796368. −0.277407 −0.138703 0.990334i \(-0.544293\pi\)
−0.138703 + 0.990334i \(0.544293\pi\)
\(384\) 0 0
\(385\) −566440. −0.194761
\(386\) 432382. 0.147706
\(387\) 0 0
\(388\) −310062. −0.104561
\(389\) −1.94799e6 −0.652699 −0.326349 0.945249i \(-0.605819\pi\)
−0.326349 + 0.945249i \(0.605819\pi\)
\(390\) 0 0
\(391\) 2.54722e6 0.842605
\(392\) 151263. 0.0497184
\(393\) 0 0
\(394\) −131962. −0.0428261
\(395\) −77248.0 −0.0249112
\(396\) 0 0
\(397\) 1.08116e6 0.344281 0.172140 0.985072i \(-0.444932\pi\)
0.172140 + 0.985072i \(0.444932\pi\)
\(398\) −298536. −0.0944689
\(399\) 0 0
\(400\) −1.82920e6 −0.571625
\(401\) −2.76770e6 −0.859524 −0.429762 0.902942i \(-0.641402\pi\)
−0.429762 + 0.902942i \(0.641402\pi\)
\(402\) 0 0
\(403\) −1.13318e6 −0.347566
\(404\) −3.37187e6 −1.02782
\(405\) 0 0
\(406\) 403858. 0.121594
\(407\) 3.33132e6 0.996851
\(408\) 0 0
\(409\) 2.36350e6 0.698630 0.349315 0.937005i \(-0.386414\pi\)
0.349315 + 0.937005i \(0.386414\pi\)
\(410\) 674356. 0.198121
\(411\) 0 0
\(412\) 6.17495e6 1.79222
\(413\) −2.07740e6 −0.599302
\(414\) 0 0
\(415\) 1.28398e6 0.365963
\(416\) −1.33703e6 −0.378798
\(417\) 0 0
\(418\) −303280. −0.0848991
\(419\) 2.98669e6 0.831104 0.415552 0.909569i \(-0.363588\pi\)
0.415552 + 0.909569i \(0.363588\pi\)
\(420\) 0 0
\(421\) −3.46331e6 −0.952326 −0.476163 0.879357i \(-0.657973\pi\)
−0.476163 + 0.879357i \(0.657973\pi\)
\(422\) 1.17062e6 0.319989
\(423\) 0 0
\(424\) −9450.00 −0.00255280
\(425\) −1.57126e6 −0.421965
\(426\) 0 0
\(427\) −723142. −0.191935
\(428\) −2.47913e6 −0.654169
\(429\) 0 0
\(430\) 586024. 0.152843
\(431\) −2.33693e6 −0.605971 −0.302986 0.952995i \(-0.597983\pi\)
−0.302986 + 0.952995i \(0.597983\pi\)
\(432\) 0 0
\(433\) −3.50838e6 −0.899264 −0.449632 0.893214i \(-0.648445\pi\)
−0.449632 + 0.893214i \(0.648445\pi\)
\(434\) −122304. −0.0311685
\(435\) 0 0
\(436\) 1.42904e6 0.360021
\(437\) 2.84726e6 0.713221
\(438\) 0 0
\(439\) 3.54833e6 0.878744 0.439372 0.898305i \(-0.355201\pi\)
0.439372 + 0.898305i \(0.355201\pi\)
\(440\) 728280. 0.179336
\(441\) 0 0
\(442\) −362292. −0.0882070
\(443\) −1.76833e6 −0.428109 −0.214055 0.976822i \(-0.568667\pi\)
−0.214055 + 0.976822i \(0.568667\pi\)
\(444\) 0 0
\(445\) 3.98772e6 0.954608
\(446\) −399376. −0.0950703
\(447\) 0 0
\(448\) 1.31237e6 0.308930
\(449\) 5.52579e6 1.29354 0.646768 0.762687i \(-0.276120\pi\)
0.646768 + 0.762687i \(0.276120\pi\)
\(450\) 0 0
\(451\) −6.74356e6 −1.56116
\(452\) 8.14389e6 1.87493
\(453\) 0 0
\(454\) 707916. 0.161191
\(455\) −756364. −0.171278
\(456\) 0 0
\(457\) −2.96226e6 −0.663488 −0.331744 0.943369i \(-0.607637\pi\)
−0.331744 + 0.943369i \(0.607637\pi\)
\(458\) 735778. 0.163902
\(459\) 0 0
\(460\) −3.36437e6 −0.741325
\(461\) −2.11884e6 −0.464350 −0.232175 0.972674i \(-0.574584\pi\)
−0.232175 + 0.972674i \(0.574584\pi\)
\(462\) 0 0
\(463\) 3.19226e6 0.692062 0.346031 0.938223i \(-0.387529\pi\)
0.346031 + 0.938223i \(0.387529\pi\)
\(464\) 7.65682e6 1.65102
\(465\) 0 0
\(466\) −208758. −0.0445326
\(467\) 7.42621e6 1.57571 0.787853 0.615863i \(-0.211193\pi\)
0.787853 + 0.615863i \(0.211193\pi\)
\(468\) 0 0
\(469\) 82124.0 0.0172400
\(470\) 303552. 0.0633853
\(471\) 0 0
\(472\) 2.67095e6 0.551837
\(473\) −5.86024e6 −1.20438
\(474\) 0 0
\(475\) −1.75635e6 −0.357171
\(476\) 1.21216e6 0.245213
\(477\) 0 0
\(478\) 713376. 0.142807
\(479\) 3.39685e6 0.676453 0.338226 0.941065i \(-0.390173\pi\)
0.338226 + 0.941065i \(0.390173\pi\)
\(480\) 0 0
\(481\) 4.44829e6 0.876659
\(482\) 505246. 0.0990570
\(483\) 0 0
\(484\) 1.40898e6 0.273396
\(485\) 340068. 0.0656465
\(486\) 0 0
\(487\) −3.71382e6 −0.709574 −0.354787 0.934947i \(-0.615447\pi\)
−0.354787 + 0.934947i \(0.615447\pi\)
\(488\) 929754. 0.176733
\(489\) 0 0
\(490\) −81634.0 −0.0153596
\(491\) −5.57494e6 −1.04361 −0.521803 0.853066i \(-0.674740\pi\)
−0.521803 + 0.853066i \(0.674740\pi\)
\(492\) 0 0
\(493\) 6.57712e6 1.21876
\(494\) −404968. −0.0746626
\(495\) 0 0
\(496\) −2.31878e6 −0.423210
\(497\) 713832. 0.129630
\(498\) 0 0
\(499\) 3.92698e6 0.706004 0.353002 0.935623i \(-0.385161\pi\)
0.353002 + 0.935623i \(0.385161\pi\)
\(500\) 5.36908e6 0.960450
\(501\) 0 0
\(502\) 317108. 0.0561627
\(503\) −6.42079e6 −1.13154 −0.565768 0.824564i \(-0.691420\pi\)
−0.565768 + 0.824564i \(0.691420\pi\)
\(504\) 0 0
\(505\) 3.69818e6 0.645297
\(506\) −1.08528e6 −0.188437
\(507\) 0 0
\(508\) −6.09485e6 −1.04786
\(509\) −146278. −0.0250256 −0.0125128 0.999922i \(-0.503983\pi\)
−0.0125128 + 0.999922i \(0.503983\pi\)
\(510\) 0 0
\(511\) −3.84052e6 −0.650636
\(512\) −4.60877e6 −0.776980
\(513\) 0 0
\(514\) −1.44285e6 −0.240886
\(515\) −6.77253e6 −1.12521
\(516\) 0 0
\(517\) −3.03552e6 −0.499467
\(518\) 480102. 0.0786157
\(519\) 0 0
\(520\) 972468. 0.157713
\(521\) −7.70937e6 −1.24430 −0.622149 0.782899i \(-0.713740\pi\)
−0.622149 + 0.782899i \(0.713740\pi\)
\(522\) 0 0
\(523\) −569420. −0.0910287 −0.0455144 0.998964i \(-0.514493\pi\)
−0.0455144 + 0.998964i \(0.514493\pi\)
\(524\) −2.39134e6 −0.380464
\(525\) 0 0
\(526\) 271496. 0.0427857
\(527\) −1.99181e6 −0.312407
\(528\) 0 0
\(529\) 3.75252e6 0.583021
\(530\) 5100.00 0.000788643 0
\(531\) 0 0
\(532\) 1.35495e6 0.207560
\(533\) −9.00464e6 −1.37293
\(534\) 0 0
\(535\) 2.71905e6 0.410707
\(536\) −105588. −0.0158746
\(537\) 0 0
\(538\) 850614. 0.126700
\(539\) 816340. 0.121032
\(540\) 0 0
\(541\) −9.44802e6 −1.38787 −0.693933 0.720040i \(-0.744124\pi\)
−0.693933 + 0.720040i \(0.744124\pi\)
\(542\) 540128. 0.0789766
\(543\) 0 0
\(544\) −2.35011e6 −0.340479
\(545\) −1.56733e6 −0.226032
\(546\) 0 0
\(547\) −1.35321e6 −0.193374 −0.0966869 0.995315i \(-0.530825\pi\)
−0.0966869 + 0.995315i \(0.530825\pi\)
\(548\) 6.45327e6 0.917970
\(549\) 0 0
\(550\) 669460. 0.0943665
\(551\) 7.35186e6 1.03162
\(552\) 0 0
\(553\) 111328. 0.0154807
\(554\) −513574. −0.0710933
\(555\) 0 0
\(556\) 8.54298e6 1.17199
\(557\) −8.19390e6 −1.11906 −0.559529 0.828811i \(-0.689018\pi\)
−0.559529 + 0.828811i \(0.689018\pi\)
\(558\) 0 0
\(559\) −7.82514e6 −1.05916
\(560\) −1.54771e6 −0.208555
\(561\) 0 0
\(562\) −1.35642e6 −0.181157
\(563\) 1.05796e7 1.40669 0.703347 0.710847i \(-0.251688\pi\)
0.703347 + 0.710847i \(0.251688\pi\)
\(564\) 0 0
\(565\) −8.93200e6 −1.17714
\(566\) −286756. −0.0376246
\(567\) 0 0
\(568\) −917784. −0.119363
\(569\) 1.20205e7 1.55648 0.778238 0.627969i \(-0.216114\pi\)
0.778238 + 0.627969i \(0.216114\pi\)
\(570\) 0 0
\(571\) −2.48948e6 −0.319534 −0.159767 0.987155i \(-0.551074\pi\)
−0.159767 + 0.987155i \(0.551074\pi\)
\(572\) −4.78516e6 −0.611514
\(573\) 0 0
\(574\) −971866. −0.123119
\(575\) −6.28505e6 −0.792755
\(576\) 0 0
\(577\) 8.21322e6 1.02701 0.513504 0.858087i \(-0.328347\pi\)
0.513504 + 0.858087i \(0.328347\pi\)
\(578\) 783053. 0.0974926
\(579\) 0 0
\(580\) −8.68707e6 −1.07227
\(581\) −1.85044e6 −0.227423
\(582\) 0 0
\(583\) −51000.0 −0.00621439
\(584\) 4.93781e6 0.599105
\(585\) 0 0
\(586\) −1.70727e6 −0.205380
\(587\) 1.21827e6 0.145931 0.0729655 0.997334i \(-0.476754\pi\)
0.0729655 + 0.997334i \(0.476754\pi\)
\(588\) 0 0
\(589\) −2.22643e6 −0.264436
\(590\) −1.44146e6 −0.170480
\(591\) 0 0
\(592\) 9.10234e6 1.06745
\(593\) 8.42379e6 0.983718 0.491859 0.870675i \(-0.336317\pi\)
0.491859 + 0.870675i \(0.336317\pi\)
\(594\) 0 0
\(595\) −1.32947e6 −0.153952
\(596\) −9.17929e6 −1.05851
\(597\) 0 0
\(598\) −1.44917e6 −0.165717
\(599\) −8.21254e6 −0.935212 −0.467606 0.883937i \(-0.654883\pi\)
−0.467606 + 0.883937i \(0.654883\pi\)
\(600\) 0 0
\(601\) 3.25478e6 0.367566 0.183783 0.982967i \(-0.441166\pi\)
0.183783 + 0.982967i \(0.441166\pi\)
\(602\) −844564. −0.0949820
\(603\) 0 0
\(604\) 1.32206e7 1.47455
\(605\) −1.54533e6 −0.171646
\(606\) 0 0
\(607\) 7.82101e6 0.861571 0.430785 0.902454i \(-0.358237\pi\)
0.430785 + 0.902454i \(0.358237\pi\)
\(608\) −2.62694e6 −0.288198
\(609\) 0 0
\(610\) −501772. −0.0545986
\(611\) −4.05331e6 −0.439245
\(612\) 0 0
\(613\) −9.51670e6 −1.02290 −0.511452 0.859312i \(-0.670892\pi\)
−0.511452 + 0.859312i \(0.670892\pi\)
\(614\) 546788. 0.0585326
\(615\) 0 0
\(616\) −1.04958e6 −0.111446
\(617\) 7.04895e6 0.745438 0.372719 0.927944i \(-0.378426\pi\)
0.372719 + 0.927944i \(0.378426\pi\)
\(618\) 0 0
\(619\) −6.32174e6 −0.663147 −0.331574 0.943429i \(-0.607580\pi\)
−0.331574 + 0.943429i \(0.607580\pi\)
\(620\) 2.63078e6 0.274856
\(621\) 0 0
\(622\) 3.23426e6 0.335197
\(623\) −5.74701e6 −0.593229
\(624\) 0 0
\(625\) 264461. 0.0270808
\(626\) −1.81313e6 −0.184924
\(627\) 0 0
\(628\) −5.53307e6 −0.559844
\(629\) 7.81880e6 0.787977
\(630\) 0 0
\(631\) 8.61236e6 0.861090 0.430545 0.902569i \(-0.358321\pi\)
0.430545 + 0.902569i \(0.358321\pi\)
\(632\) −143136. −0.0142546
\(633\) 0 0
\(634\) −1.27658e6 −0.126132
\(635\) 6.68467e6 0.657879
\(636\) 0 0
\(637\) 1.09005e6 0.106439
\(638\) −2.80228e6 −0.272559
\(639\) 0 0
\(640\) 4.11478e6 0.397097
\(641\) 5.22829e6 0.502590 0.251295 0.967910i \(-0.419143\pi\)
0.251295 + 0.967910i \(0.419143\pi\)
\(642\) 0 0
\(643\) 1.61373e7 1.53923 0.769615 0.638508i \(-0.220448\pi\)
0.769615 + 0.638508i \(0.220448\pi\)
\(644\) 4.84865e6 0.460687
\(645\) 0 0
\(646\) −711816. −0.0671099
\(647\) 1.58749e7 1.49090 0.745451 0.666560i \(-0.232234\pi\)
0.745451 + 0.666560i \(0.232234\pi\)
\(648\) 0 0
\(649\) 1.44146e7 1.34336
\(650\) 893926. 0.0829886
\(651\) 0 0
\(652\) −7.83593e6 −0.721891
\(653\) 5.94112e6 0.545237 0.272619 0.962122i \(-0.412110\pi\)
0.272619 + 0.962122i \(0.412110\pi\)
\(654\) 0 0
\(655\) 2.62276e6 0.238867
\(656\) −1.84258e7 −1.67173
\(657\) 0 0
\(658\) −437472. −0.0393900
\(659\) 7.64430e6 0.685684 0.342842 0.939393i \(-0.388610\pi\)
0.342842 + 0.939393i \(0.388610\pi\)
\(660\) 0 0
\(661\) −7.58688e6 −0.675398 −0.337699 0.941254i \(-0.609649\pi\)
−0.337699 + 0.941254i \(0.609649\pi\)
\(662\) 1.73621e6 0.153978
\(663\) 0 0
\(664\) 2.37913e6 0.209410
\(665\) −1.48607e6 −0.130312
\(666\) 0 0
\(667\) 2.63085e7 2.28971
\(668\) 1.57507e7 1.36571
\(669\) 0 0
\(670\) 56984.0 0.00490417
\(671\) 5.01772e6 0.430229
\(672\) 0 0
\(673\) −2.06681e7 −1.75899 −0.879494 0.475910i \(-0.842119\pi\)
−0.879494 + 0.475910i \(0.842119\pi\)
\(674\) −2.07215e6 −0.175700
\(675\) 0 0
\(676\) 5.12049e6 0.430968
\(677\) −7.89541e6 −0.662068 −0.331034 0.943619i \(-0.607398\pi\)
−0.331034 + 0.943619i \(0.607398\pi\)
\(678\) 0 0
\(679\) −490098. −0.0407951
\(680\) 1.70932e6 0.141759
\(681\) 0 0
\(682\) 848640. 0.0698655
\(683\) 1.96015e7 1.60782 0.803911 0.594750i \(-0.202749\pi\)
0.803911 + 0.594750i \(0.202749\pi\)
\(684\) 0 0
\(685\) −7.07778e6 −0.576329
\(686\) 117649. 0.00954504
\(687\) 0 0
\(688\) −1.60122e7 −1.28968
\(689\) −68100.0 −0.00546511
\(690\) 0 0
\(691\) −1.72710e7 −1.37601 −0.688005 0.725706i \(-0.741513\pi\)
−0.688005 + 0.725706i \(0.741513\pi\)
\(692\) −6.87685e6 −0.545914
\(693\) 0 0
\(694\) −1.65146e6 −0.130158
\(695\) −9.36972e6 −0.735808
\(696\) 0 0
\(697\) −1.58275e7 −1.23405
\(698\) −1.26645e6 −0.0983900
\(699\) 0 0
\(700\) −2.99091e6 −0.230706
\(701\) 5.36344e6 0.412238 0.206119 0.978527i \(-0.433917\pi\)
0.206119 + 0.978527i \(0.433917\pi\)
\(702\) 0 0
\(703\) 8.73982e6 0.666982
\(704\) −9.10622e6 −0.692479
\(705\) 0 0
\(706\) 573218. 0.0432821
\(707\) −5.32973e6 −0.401011
\(708\) 0 0
\(709\) −1.73733e7 −1.29798 −0.648988 0.760798i \(-0.724808\pi\)
−0.648988 + 0.760798i \(0.724808\pi\)
\(710\) 495312. 0.0368751
\(711\) 0 0
\(712\) 7.38902e6 0.546244
\(713\) −7.96723e6 −0.586926
\(714\) 0 0
\(715\) 5.24824e6 0.383927
\(716\) −3.52048e6 −0.256637
\(717\) 0 0
\(718\) 4.46322e6 0.323100
\(719\) −424608. −0.0306313 −0.0153157 0.999883i \(-0.504875\pi\)
−0.0153157 + 0.999883i \(0.504875\pi\)
\(720\) 0 0
\(721\) 9.76041e6 0.699246
\(722\) 1.68044e6 0.119972
\(723\) 0 0
\(724\) −2.05567e7 −1.45749
\(725\) −1.62285e7 −1.14666
\(726\) 0 0
\(727\) 2.18290e7 1.53179 0.765893 0.642968i \(-0.222297\pi\)
0.765893 + 0.642968i \(0.222297\pi\)
\(728\) −1.40150e6 −0.0980086
\(729\) 0 0
\(730\) −2.66485e6 −0.185083
\(731\) −1.37543e7 −0.952020
\(732\) 0 0
\(733\) 2.17675e7 1.49640 0.748202 0.663470i \(-0.230917\pi\)
0.748202 + 0.663470i \(0.230917\pi\)
\(734\) 4.50797e6 0.308845
\(735\) 0 0
\(736\) −9.40044e6 −0.639667
\(737\) −569840. −0.0386442
\(738\) 0 0
\(739\) 6.21786e6 0.418822 0.209411 0.977828i \(-0.432845\pi\)
0.209411 + 0.977828i \(0.432845\pi\)
\(740\) −1.03271e7 −0.693264
\(741\) 0 0
\(742\) −7350.00 −0.000490092 0
\(743\) −3.77647e6 −0.250966 −0.125483 0.992096i \(-0.540048\pi\)
−0.125483 + 0.992096i \(0.540048\pi\)
\(744\) 0 0
\(745\) 1.00676e7 0.664562
\(746\) −1.66535e6 −0.109562
\(747\) 0 0
\(748\) −8.41092e6 −0.549654
\(749\) −3.91863e6 −0.255229
\(750\) 0 0
\(751\) −2.88795e6 −0.186849 −0.0934244 0.995626i \(-0.529781\pi\)
−0.0934244 + 0.995626i \(0.529781\pi\)
\(752\) −8.29411e6 −0.534842
\(753\) 0 0
\(754\) −3.74187e6 −0.239696
\(755\) −1.45000e7 −0.925768
\(756\) 0 0
\(757\) 1.25519e6 0.0796104 0.0398052 0.999207i \(-0.487326\pi\)
0.0398052 + 0.999207i \(0.487326\pi\)
\(758\) 2.53232e6 0.160083
\(759\) 0 0
\(760\) 1.91066e6 0.119991
\(761\) 1.42623e7 0.892746 0.446373 0.894847i \(-0.352716\pi\)
0.446373 + 0.894847i \(0.352716\pi\)
\(762\) 0 0
\(763\) 2.25880e6 0.140465
\(764\) 1.56756e7 0.971606
\(765\) 0 0
\(766\) 796368. 0.0490390
\(767\) 1.92478e7 1.18139
\(768\) 0 0
\(769\) −2.02261e7 −1.23338 −0.616689 0.787207i \(-0.711526\pi\)
−0.616689 + 0.787207i \(0.711526\pi\)
\(770\) 566440. 0.0344292
\(771\) 0 0
\(772\) 1.34038e7 0.809443
\(773\) −2.62288e7 −1.57881 −0.789406 0.613872i \(-0.789611\pi\)
−0.789406 + 0.613872i \(0.789611\pi\)
\(774\) 0 0
\(775\) 4.91462e6 0.293925
\(776\) 630126. 0.0375641
\(777\) 0 0
\(778\) 1.94799e6 0.115382
\(779\) −1.76919e7 −1.04456
\(780\) 0 0
\(781\) −4.95312e6 −0.290570
\(782\) −2.54722e6 −0.148953
\(783\) 0 0
\(784\) 2.23053e6 0.129604
\(785\) 6.06852e6 0.351487
\(786\) 0 0
\(787\) −9.92829e6 −0.571397 −0.285698 0.958320i \(-0.592226\pi\)
−0.285698 + 0.958320i \(0.592226\pi\)
\(788\) −4.09082e6 −0.234690
\(789\) 0 0
\(790\) 77248.0 0.00440372
\(791\) 1.28726e7 0.731518
\(792\) 0 0
\(793\) 6.70013e6 0.378356
\(794\) −1.08116e6 −0.0608608
\(795\) 0 0
\(796\) −9.25462e6 −0.517697
\(797\) −1.09033e7 −0.608014 −0.304007 0.952670i \(-0.598325\pi\)
−0.304007 + 0.952670i \(0.598325\pi\)
\(798\) 0 0
\(799\) −7.12454e6 −0.394812
\(800\) 5.79871e6 0.320336
\(801\) 0 0
\(802\) 2.76770e6 0.151944
\(803\) 2.66485e7 1.45843
\(804\) 0 0
\(805\) −5.31787e6 −0.289233
\(806\) 1.13318e6 0.0614416
\(807\) 0 0
\(808\) 6.85251e6 0.369251
\(809\) −6.06398e6 −0.325751 −0.162876 0.986647i \(-0.552077\pi\)
−0.162876 + 0.986647i \(0.552077\pi\)
\(810\) 0 0
\(811\) −8.59438e6 −0.458841 −0.229421 0.973327i \(-0.573683\pi\)
−0.229421 + 0.973327i \(0.573683\pi\)
\(812\) 1.25196e7 0.666347
\(813\) 0 0
\(814\) −3.33132e6 −0.176220
\(815\) 8.59425e6 0.453225
\(816\) 0 0
\(817\) −1.53745e7 −0.805835
\(818\) −2.36350e6 −0.123501
\(819\) 0 0
\(820\) 2.09050e7 1.08572
\(821\) 2.01396e6 0.104278 0.0521391 0.998640i \(-0.483396\pi\)
0.0521391 + 0.998640i \(0.483396\pi\)
\(822\) 0 0
\(823\) −2.64679e7 −1.36213 −0.681067 0.732221i \(-0.738484\pi\)
−0.681067 + 0.732221i \(0.738484\pi\)
\(824\) −1.25491e7 −0.643864
\(825\) 0 0
\(826\) 2.07740e6 0.105943
\(827\) 3.90229e6 0.198407 0.0992033 0.995067i \(-0.468371\pi\)
0.0992033 + 0.995067i \(0.468371\pi\)
\(828\) 0 0
\(829\) −1.95595e7 −0.988487 −0.494244 0.869323i \(-0.664555\pi\)
−0.494244 + 0.869323i \(0.664555\pi\)
\(830\) −1.28398e6 −0.0646937
\(831\) 0 0
\(832\) −1.21595e7 −0.608985
\(833\) 1.91600e6 0.0956715
\(834\) 0 0
\(835\) −1.72750e7 −0.857436
\(836\) −9.40168e6 −0.465254
\(837\) 0 0
\(838\) −2.98669e6 −0.146920
\(839\) 2.45448e7 1.20380 0.601901 0.798570i \(-0.294410\pi\)
0.601901 + 0.798570i \(0.294410\pi\)
\(840\) 0 0
\(841\) 4.74194e7 2.31188
\(842\) 3.46331e6 0.168349
\(843\) 0 0
\(844\) 3.62892e7 1.75356
\(845\) −5.61602e6 −0.270574
\(846\) 0 0
\(847\) 2.22710e6 0.106667
\(848\) −139350. −0.00665453
\(849\) 0 0
\(850\) 1.57126e6 0.0745936
\(851\) 3.12752e7 1.48039
\(852\) 0 0
\(853\) 3.38305e7 1.59197 0.795987 0.605314i \(-0.206952\pi\)
0.795987 + 0.605314i \(0.206952\pi\)
\(854\) 723142. 0.0339296
\(855\) 0 0
\(856\) 5.03824e6 0.235014
\(857\) −3.18009e7 −1.47907 −0.739534 0.673120i \(-0.764954\pi\)
−0.739534 + 0.673120i \(0.764954\pi\)
\(858\) 0 0
\(859\) 638420. 0.0295205 0.0147602 0.999891i \(-0.495301\pi\)
0.0147602 + 0.999891i \(0.495301\pi\)
\(860\) 1.81667e7 0.837589
\(861\) 0 0
\(862\) 2.33693e6 0.107122
\(863\) 4.22256e6 0.192996 0.0964981 0.995333i \(-0.469236\pi\)
0.0964981 + 0.995333i \(0.469236\pi\)
\(864\) 0 0
\(865\) 7.54236e6 0.342742
\(866\) 3.50838e6 0.158969
\(867\) 0 0
\(868\) −3.79142e6 −0.170806
\(869\) −772480. −0.0347007
\(870\) 0 0
\(871\) −760904. −0.0339848
\(872\) −2.90417e6 −0.129340
\(873\) 0 0
\(874\) −2.84726e6 −0.126081
\(875\) 8.48660e6 0.374726
\(876\) 0 0
\(877\) −2.45043e7 −1.07583 −0.537915 0.842999i \(-0.680788\pi\)
−0.537915 + 0.842999i \(0.680788\pi\)
\(878\) −3.54833e6 −0.155341
\(879\) 0 0
\(880\) 1.07392e7 0.467484
\(881\) 2.77630e7 1.20511 0.602555 0.798078i \(-0.294150\pi\)
0.602555 + 0.798078i \(0.294150\pi\)
\(882\) 0 0
\(883\) 3.30170e7 1.42507 0.712534 0.701638i \(-0.247548\pi\)
0.712534 + 0.701638i \(0.247548\pi\)
\(884\) −1.12311e7 −0.483381
\(885\) 0 0
\(886\) 1.76833e6 0.0756797
\(887\) −4.34462e6 −0.185414 −0.0927070 0.995693i \(-0.529552\pi\)
−0.0927070 + 0.995693i \(0.529552\pi\)
\(888\) 0 0
\(889\) −9.63379e6 −0.408830
\(890\) −3.98772e6 −0.168752
\(891\) 0 0
\(892\) −1.23807e7 −0.520993
\(893\) −7.96378e6 −0.334188
\(894\) 0 0
\(895\) 3.86118e6 0.161125
\(896\) −5.93013e6 −0.246771
\(897\) 0 0
\(898\) −5.52579e6 −0.228667
\(899\) −2.05720e7 −0.848942
\(900\) 0 0
\(901\) −119700. −0.00491227
\(902\) 6.74356e6 0.275977
\(903\) 0 0
\(904\) −1.65505e7 −0.673580
\(905\) 2.25460e7 0.915057
\(906\) 0 0
\(907\) 1.96499e7 0.793128 0.396564 0.918007i \(-0.370203\pi\)
0.396564 + 0.918007i \(0.370203\pi\)
\(908\) 2.19454e7 0.883342
\(909\) 0 0
\(910\) 756364. 0.0302780
\(911\) 7.26518e6 0.290035 0.145018 0.989429i \(-0.453676\pi\)
0.145018 + 0.989429i \(0.453676\pi\)
\(912\) 0 0
\(913\) 1.28398e7 0.509777
\(914\) 2.96226e6 0.117289
\(915\) 0 0
\(916\) 2.28091e7 0.898193
\(917\) −3.77986e6 −0.148440
\(918\) 0 0
\(919\) 9.82532e6 0.383758 0.191879 0.981419i \(-0.438542\pi\)
0.191879 + 0.981419i \(0.438542\pi\)
\(920\) 6.83726e6 0.266326
\(921\) 0 0
\(922\) 2.11884e6 0.0820863
\(923\) −6.61387e6 −0.255536
\(924\) 0 0
\(925\) −1.92923e7 −0.741359
\(926\) −3.19226e6 −0.122340
\(927\) 0 0
\(928\) −2.42727e7 −0.925226
\(929\) −2.71152e7 −1.03080 −0.515399 0.856951i \(-0.672356\pi\)
−0.515399 + 0.856951i \(0.672356\pi\)
\(930\) 0 0
\(931\) 2.14169e6 0.0809809
\(932\) −6.47150e6 −0.244042
\(933\) 0 0
\(934\) −7.42621e6 −0.278548
\(935\) 9.22488e6 0.345089
\(936\) 0 0
\(937\) −4.53522e7 −1.68752 −0.843761 0.536720i \(-0.819663\pi\)
−0.843761 + 0.536720i \(0.819663\pi\)
\(938\) −82124.0 −0.00304764
\(939\) 0 0
\(940\) 9.41011e6 0.347356
\(941\) −4.65780e7 −1.71477 −0.857387 0.514672i \(-0.827914\pi\)
−0.857387 + 0.514672i \(0.827914\pi\)
\(942\) 0 0
\(943\) −6.33101e7 −2.31843
\(944\) 3.93859e7 1.43850
\(945\) 0 0
\(946\) 5.86024e6 0.212906
\(947\) −2.53799e7 −0.919632 −0.459816 0.888014i \(-0.652085\pi\)
−0.459816 + 0.888014i \(0.652085\pi\)
\(948\) 0 0
\(949\) 3.55836e7 1.28258
\(950\) 1.75635e6 0.0631396
\(951\) 0 0
\(952\) −2.46343e6 −0.0880942
\(953\) −1.52948e7 −0.545520 −0.272760 0.962082i \(-0.587937\pi\)
−0.272760 + 0.962082i \(0.587937\pi\)
\(954\) 0 0
\(955\) −1.71926e7 −0.610004
\(956\) 2.21147e7 0.782592
\(957\) 0 0
\(958\) −3.39685e6 −0.119581
\(959\) 1.02003e7 0.358152
\(960\) 0 0
\(961\) −2.23991e7 −0.782389
\(962\) −4.44829e6 −0.154973
\(963\) 0 0
\(964\) 1.56626e7 0.542840
\(965\) −1.47010e7 −0.508192
\(966\) 0 0
\(967\) −5.71465e6 −0.196527 −0.0982637 0.995160i \(-0.531329\pi\)
−0.0982637 + 0.995160i \(0.531329\pi\)
\(968\) −2.86341e6 −0.0982190
\(969\) 0 0
\(970\) −340068. −0.0116048
\(971\) −1.30250e7 −0.443332 −0.221666 0.975123i \(-0.571149\pi\)
−0.221666 + 0.975123i \(0.571149\pi\)
\(972\) 0 0
\(973\) 1.35034e7 0.457258
\(974\) 3.71382e6 0.125436
\(975\) 0 0
\(976\) 1.37102e7 0.460700
\(977\) 1.70360e7 0.570992 0.285496 0.958380i \(-0.407842\pi\)
0.285496 + 0.958380i \(0.407842\pi\)
\(978\) 0 0
\(979\) 3.98772e7 1.32974
\(980\) −2.53065e6 −0.0841720
\(981\) 0 0
\(982\) 5.57494e6 0.184485
\(983\) 1.36985e7 0.452156 0.226078 0.974109i \(-0.427410\pi\)
0.226078 + 0.974109i \(0.427410\pi\)
\(984\) 0 0
\(985\) 4.48671e6 0.147346
\(986\) −6.57712e6 −0.215448
\(987\) 0 0
\(988\) −1.25540e7 −0.409157
\(989\) −5.50173e7 −1.78858
\(990\) 0 0
\(991\) −3.49088e7 −1.12915 −0.564574 0.825383i \(-0.690959\pi\)
−0.564574 + 0.825383i \(0.690959\pi\)
\(992\) 7.35072e6 0.237165
\(993\) 0 0
\(994\) −713832. −0.0229155
\(995\) 1.01502e7 0.325026
\(996\) 0 0
\(997\) 875662. 0.0278996 0.0139498 0.999903i \(-0.495559\pi\)
0.0139498 + 0.999903i \(0.495559\pi\)
\(998\) −3.92698e6 −0.124805
\(999\) 0 0
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 63.6.a.c.1.1 1
3.2 odd 2 21.6.a.b.1.1 1
4.3 odd 2 1008.6.a.t.1.1 1
7.6 odd 2 441.6.a.d.1.1 1
12.11 even 2 336.6.a.l.1.1 1
15.2 even 4 525.6.d.d.274.2 2
15.8 even 4 525.6.d.d.274.1 2
15.14 odd 2 525.6.a.c.1.1 1
21.2 odd 6 147.6.e.f.67.1 2
21.5 even 6 147.6.e.e.67.1 2
21.11 odd 6 147.6.e.f.79.1 2
21.17 even 6 147.6.e.e.79.1 2
21.20 even 2 147.6.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.b.1.1 1 3.2 odd 2
63.6.a.c.1.1 1 1.1 even 1 trivial
147.6.a.e.1.1 1 21.20 even 2
147.6.e.e.67.1 2 21.5 even 6
147.6.e.e.79.1 2 21.17 even 6
147.6.e.f.67.1 2 21.2 odd 6
147.6.e.f.79.1 2 21.11 odd 6
336.6.a.l.1.1 1 12.11 even 2
441.6.a.d.1.1 1 7.6 odd 2
525.6.a.c.1.1 1 15.14 odd 2
525.6.d.d.274.1 2 15.8 even 4
525.6.d.d.274.2 2 15.2 even 4
1008.6.a.t.1.1 1 4.3 odd 2