Properties

Label 21.6.a.b.1.1
Level $21$
Weight $6$
Character 21.1
Self dual yes
Analytic conductor $3.368$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 21 = 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 21.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(3.36806021607\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 21.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -9.00000 q^{3} -31.0000 q^{4} -34.0000 q^{5} -9.00000 q^{6} -49.0000 q^{7} -63.0000 q^{8} +81.0000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -9.00000 q^{3} -31.0000 q^{4} -34.0000 q^{5} -9.00000 q^{6} -49.0000 q^{7} -63.0000 q^{8} +81.0000 q^{9} -34.0000 q^{10} -340.000 q^{11} +279.000 q^{12} +454.000 q^{13} -49.0000 q^{14} +306.000 q^{15} +929.000 q^{16} -798.000 q^{17} +81.0000 q^{18} +892.000 q^{19} +1054.00 q^{20} +441.000 q^{21} -340.000 q^{22} -3192.00 q^{23} +567.000 q^{24} -1969.00 q^{25} +454.000 q^{26} -729.000 q^{27} +1519.00 q^{28} -8242.00 q^{29} +306.000 q^{30} -2496.00 q^{31} +2945.00 q^{32} +3060.00 q^{33} -798.000 q^{34} +1666.00 q^{35} -2511.00 q^{36} +9798.00 q^{37} +892.000 q^{38} -4086.00 q^{39} +2142.00 q^{40} +19834.0 q^{41} +441.000 q^{42} -17236.0 q^{43} +10540.0 q^{44} -2754.00 q^{45} -3192.00 q^{46} +8928.00 q^{47} -8361.00 q^{48} +2401.00 q^{49} -1969.00 q^{50} +7182.00 q^{51} -14074.0 q^{52} +150.000 q^{53} -729.000 q^{54} +11560.0 q^{55} +3087.00 q^{56} -8028.00 q^{57} -8242.00 q^{58} -42396.0 q^{59} -9486.00 q^{60} +14758.0 q^{61} -2496.00 q^{62} -3969.00 q^{63} -26783.0 q^{64} -15436.0 q^{65} +3060.00 q^{66} -1676.00 q^{67} +24738.0 q^{68} +28728.0 q^{69} +1666.00 q^{70} +14568.0 q^{71} -5103.00 q^{72} +78378.0 q^{73} +9798.00 q^{74} +17721.0 q^{75} -27652.0 q^{76} +16660.0 q^{77} -4086.00 q^{78} -2272.00 q^{79} -31586.0 q^{80} +6561.00 q^{81} +19834.0 q^{82} -37764.0 q^{83} -13671.0 q^{84} +27132.0 q^{85} -17236.0 q^{86} +74178.0 q^{87} +21420.0 q^{88} -117286. q^{89} -2754.00 q^{90} -22246.0 q^{91} +98952.0 q^{92} +22464.0 q^{93} +8928.00 q^{94} -30328.0 q^{95} -26505.0 q^{96} +10002.0 q^{97} +2401.00 q^{98} -27540.0 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.176777 0.0883883 0.996086i \(-0.471828\pi\)
0.0883883 + 0.996086i \(0.471828\pi\)
\(3\) −9.00000 −0.577350
\(4\) −31.0000 −0.968750
\(5\) −34.0000 −0.608210 −0.304105 0.952638i \(-0.598357\pi\)
−0.304105 + 0.952638i \(0.598357\pi\)
\(6\) −9.00000 −0.102062
\(7\) −49.0000 −0.377964
\(8\) −63.0000 −0.348029
\(9\) 81.0000 0.333333
\(10\) −34.0000 −0.107517
\(11\) −340.000 −0.847222 −0.423611 0.905844i \(-0.639238\pi\)
−0.423611 + 0.905844i \(0.639238\pi\)
\(12\) 279.000 0.559308
\(13\) 454.000 0.745071 0.372535 0.928018i \(-0.378489\pi\)
0.372535 + 0.928018i \(0.378489\pi\)
\(14\) −49.0000 −0.0668153
\(15\) 306.000 0.351150
\(16\) 929.000 0.907227
\(17\) −798.000 −0.669700 −0.334850 0.942271i \(-0.608686\pi\)
−0.334850 + 0.942271i \(0.608686\pi\)
\(18\) 81.0000 0.0589256
\(19\) 892.000 0.566867 0.283433 0.958992i \(-0.408527\pi\)
0.283433 + 0.958992i \(0.408527\pi\)
\(20\) 1054.00 0.589204
\(21\) 441.000 0.218218
\(22\) −340.000 −0.149769
\(23\) −3192.00 −1.25818 −0.629091 0.777332i \(-0.716573\pi\)
−0.629091 + 0.777332i \(0.716573\pi\)
\(24\) 567.000 0.200935
\(25\) −1969.00 −0.630080
\(26\) 454.000 0.131711
\(27\) −729.000 −0.192450
\(28\) 1519.00 0.366153
\(29\) −8242.00 −1.81986 −0.909929 0.414764i \(-0.863864\pi\)
−0.909929 + 0.414764i \(0.863864\pi\)
\(30\) 306.000 0.0620752
\(31\) −2496.00 −0.466488 −0.233244 0.972418i \(-0.574934\pi\)
−0.233244 + 0.972418i \(0.574934\pi\)
\(32\) 2945.00 0.508406
\(33\) 3060.00 0.489144
\(34\) −798.000 −0.118387
\(35\) 1666.00 0.229882
\(36\) −2511.00 −0.322917
\(37\) 9798.00 1.17661 0.588306 0.808639i \(-0.299795\pi\)
0.588306 + 0.808639i \(0.299795\pi\)
\(38\) 892.000 0.100209
\(39\) −4086.00 −0.430167
\(40\) 2142.00 0.211675
\(41\) 19834.0 1.84268 0.921342 0.388754i \(-0.127094\pi\)
0.921342 + 0.388754i \(0.127094\pi\)
\(42\) 441.000 0.0385758
\(43\) −17236.0 −1.42156 −0.710780 0.703414i \(-0.751658\pi\)
−0.710780 + 0.703414i \(0.751658\pi\)
\(44\) 10540.0 0.820746
\(45\) −2754.00 −0.202737
\(46\) −3192.00 −0.222417
\(47\) 8928.00 0.589535 0.294767 0.955569i \(-0.404758\pi\)
0.294767 + 0.955569i \(0.404758\pi\)
\(48\) −8361.00 −0.523788
\(49\) 2401.00 0.142857
\(50\) −1969.00 −0.111383
\(51\) 7182.00 0.386652
\(52\) −14074.0 −0.721787
\(53\) 150.000 0.00733502 0.00366751 0.999993i \(-0.498833\pi\)
0.00366751 + 0.999993i \(0.498833\pi\)
\(54\) −729.000 −0.0340207
\(55\) 11560.0 0.515289
\(56\) 3087.00 0.131543
\(57\) −8028.00 −0.327281
\(58\) −8242.00 −0.321709
\(59\) −42396.0 −1.58560 −0.792802 0.609479i \(-0.791379\pi\)
−0.792802 + 0.609479i \(0.791379\pi\)
\(60\) −9486.00 −0.340177
\(61\) 14758.0 0.507812 0.253906 0.967229i \(-0.418285\pi\)
0.253906 + 0.967229i \(0.418285\pi\)
\(62\) −2496.00 −0.0824642
\(63\) −3969.00 −0.125988
\(64\) −26783.0 −0.817352
\(65\) −15436.0 −0.453160
\(66\) 3060.00 0.0864692
\(67\) −1676.00 −0.0456128 −0.0228064 0.999740i \(-0.507260\pi\)
−0.0228064 + 0.999740i \(0.507260\pi\)
\(68\) 24738.0 0.648772
\(69\) 28728.0 0.726411
\(70\) 1666.00 0.0406378
\(71\) 14568.0 0.342968 0.171484 0.985187i \(-0.445144\pi\)
0.171484 + 0.985187i \(0.445144\pi\)
\(72\) −5103.00 −0.116010
\(73\) 78378.0 1.72142 0.860710 0.509095i \(-0.170020\pi\)
0.860710 + 0.509095i \(0.170020\pi\)
\(74\) 9798.00 0.207998
\(75\) 17721.0 0.363777
\(76\) −27652.0 −0.549152
\(77\) 16660.0 0.320220
\(78\) −4086.00 −0.0760435
\(79\) −2272.00 −0.0409582 −0.0204791 0.999790i \(-0.506519\pi\)
−0.0204791 + 0.999790i \(0.506519\pi\)
\(80\) −31586.0 −0.551785
\(81\) 6561.00 0.111111
\(82\) 19834.0 0.325743
\(83\) −37764.0 −0.601704 −0.300852 0.953671i \(-0.597271\pi\)
−0.300852 + 0.953671i \(0.597271\pi\)
\(84\) −13671.0 −0.211399
\(85\) 27132.0 0.407319
\(86\) −17236.0 −0.251299
\(87\) 74178.0 1.05070
\(88\) 21420.0 0.294858
\(89\) −117286. −1.56954 −0.784768 0.619790i \(-0.787218\pi\)
−0.784768 + 0.619790i \(0.787218\pi\)
\(90\) −2754.00 −0.0358391
\(91\) −22246.0 −0.281610
\(92\) 98952.0 1.21886
\(93\) 22464.0 0.269327
\(94\) 8928.00 0.104216
\(95\) −30328.0 −0.344774
\(96\) −26505.0 −0.293528
\(97\) 10002.0 0.107934 0.0539669 0.998543i \(-0.482813\pi\)
0.0539669 + 0.998543i \(0.482813\pi\)
\(98\) 2401.00 0.0252538
\(99\) −27540.0 −0.282407
\(100\) 61039.0 0.610390
\(101\) −108770. −1.06098 −0.530488 0.847692i \(-0.677991\pi\)
−0.530488 + 0.847692i \(0.677991\pi\)
\(102\) 7182.00 0.0683510
\(103\) −199192. −1.85003 −0.925015 0.379930i \(-0.875948\pi\)
−0.925015 + 0.379930i \(0.875948\pi\)
\(104\) −28602.0 −0.259306
\(105\) −14994.0 −0.132722
\(106\) 150.000 0.00129666
\(107\) −79972.0 −0.675272 −0.337636 0.941277i \(-0.609627\pi\)
−0.337636 + 0.941277i \(0.609627\pi\)
\(108\) 22599.0 0.186436
\(109\) −46098.0 −0.371634 −0.185817 0.982584i \(-0.559493\pi\)
−0.185817 + 0.982584i \(0.559493\pi\)
\(110\) 11560.0 0.0910911
\(111\) −88182.0 −0.679317
\(112\) −45521.0 −0.342899
\(113\) 262706. 1.93541 0.967707 0.252078i \(-0.0811138\pi\)
0.967707 + 0.252078i \(0.0811138\pi\)
\(114\) −8028.00 −0.0578556
\(115\) 108528. 0.765239
\(116\) 255502. 1.76299
\(117\) 36774.0 0.248357
\(118\) −42396.0 −0.280298
\(119\) 39102.0 0.253123
\(120\) −19278.0 −0.122211
\(121\) −45451.0 −0.282215
\(122\) 14758.0 0.0897693
\(123\) −178506. −1.06387
\(124\) 77376.0 0.451910
\(125\) 173196. 0.991432
\(126\) −3969.00 −0.0222718
\(127\) 196608. 1.08166 0.540831 0.841131i \(-0.318110\pi\)
0.540831 + 0.841131i \(0.318110\pi\)
\(128\) −121023. −0.652894
\(129\) 155124. 0.820738
\(130\) −15436.0 −0.0801081
\(131\) −77140.0 −0.392737 −0.196368 0.980530i \(-0.562915\pi\)
−0.196368 + 0.980530i \(0.562915\pi\)
\(132\) −94860.0 −0.473858
\(133\) −43708.0 −0.214255
\(134\) −1676.00 −0.00806329
\(135\) 24786.0 0.117050
\(136\) 50274.0 0.233075
\(137\) 208170. 0.947582 0.473791 0.880637i \(-0.342885\pi\)
0.473791 + 0.880637i \(0.342885\pi\)
\(138\) 28728.0 0.128413
\(139\) −275580. −1.20979 −0.604896 0.796304i \(-0.706785\pi\)
−0.604896 + 0.796304i \(0.706785\pi\)
\(140\) −51646.0 −0.222698
\(141\) −80352.0 −0.340368
\(142\) 14568.0 0.0606288
\(143\) −154360. −0.631240
\(144\) 75249.0 0.302409
\(145\) 280228. 1.10686
\(146\) 78378.0 0.304307
\(147\) −21609.0 −0.0824786
\(148\) −303738. −1.13984
\(149\) −296106. −1.09265 −0.546326 0.837573i \(-0.683974\pi\)
−0.546326 + 0.837573i \(0.683974\pi\)
\(150\) 17721.0 0.0643073
\(151\) −426472. −1.52212 −0.761059 0.648683i \(-0.775320\pi\)
−0.761059 + 0.648683i \(0.775320\pi\)
\(152\) −56196.0 −0.197286
\(153\) −64638.0 −0.223233
\(154\) 16660.0 0.0566074
\(155\) 84864.0 0.283723
\(156\) 126666. 0.416724
\(157\) 178486. 0.577903 0.288952 0.957344i \(-0.406693\pi\)
0.288952 + 0.957344i \(0.406693\pi\)
\(158\) −2272.00 −0.00724045
\(159\) −1350.00 −0.00423488
\(160\) −100130. −0.309218
\(161\) 156408. 0.475548
\(162\) 6561.00 0.0196419
\(163\) 252772. 0.745178 0.372589 0.927996i \(-0.378470\pi\)
0.372589 + 0.927996i \(0.378470\pi\)
\(164\) −614854. −1.78510
\(165\) −104040. −0.297502
\(166\) −37764.0 −0.106367
\(167\) 508088. 1.40977 0.704884 0.709322i \(-0.250999\pi\)
0.704884 + 0.709322i \(0.250999\pi\)
\(168\) −27783.0 −0.0759462
\(169\) −165177. −0.444870
\(170\) 27132.0 0.0720045
\(171\) 72252.0 0.188956
\(172\) 534316. 1.37714
\(173\) −221834. −0.563525 −0.281762 0.959484i \(-0.590919\pi\)
−0.281762 + 0.959484i \(0.590919\pi\)
\(174\) 74178.0 0.185739
\(175\) 96481.0 0.238148
\(176\) −315860. −0.768622
\(177\) 381564. 0.915449
\(178\) −117286. −0.277457
\(179\) −113564. −0.264916 −0.132458 0.991189i \(-0.542287\pi\)
−0.132458 + 0.991189i \(0.542287\pi\)
\(180\) 85374.0 0.196401
\(181\) 663118. 1.50451 0.752254 0.658873i \(-0.228967\pi\)
0.752254 + 0.658873i \(0.228967\pi\)
\(182\) −22246.0 −0.0497821
\(183\) −132822. −0.293185
\(184\) 201096. 0.437884
\(185\) −333132. −0.715628
\(186\) 22464.0 0.0476107
\(187\) 271320. 0.567385
\(188\) −276768. −0.571112
\(189\) 35721.0 0.0727393
\(190\) −30328.0 −0.0609480
\(191\) 505664. 1.00295 0.501474 0.865173i \(-0.332791\pi\)
0.501474 + 0.865173i \(0.332791\pi\)
\(192\) 241047. 0.471899
\(193\) −432382. −0.835554 −0.417777 0.908550i \(-0.637191\pi\)
−0.417777 + 0.908550i \(0.637191\pi\)
\(194\) 10002.0 0.0190802
\(195\) 138924. 0.261632
\(196\) −74431.0 −0.138393
\(197\) −131962. −0.242261 −0.121130 0.992637i \(-0.538652\pi\)
−0.121130 + 0.992637i \(0.538652\pi\)
\(198\) −27540.0 −0.0499230
\(199\) 298536. 0.534397 0.267199 0.963642i \(-0.413902\pi\)
0.267199 + 0.963642i \(0.413902\pi\)
\(200\) 124047. 0.219286
\(201\) 15084.0 0.0263346
\(202\) −108770. −0.187556
\(203\) 403858. 0.687842
\(204\) −222642. −0.374569
\(205\) −674356. −1.12074
\(206\) −199192. −0.327042
\(207\) −258552. −0.419394
\(208\) 421766. 0.675948
\(209\) −303280. −0.480262
\(210\) −14994.0 −0.0234622
\(211\) −1.17062e6 −1.81013 −0.905065 0.425273i \(-0.860178\pi\)
−0.905065 + 0.425273i \(0.860178\pi\)
\(212\) −4650.00 −0.00710581
\(213\) −131112. −0.198013
\(214\) −79972.0 −0.119372
\(215\) 586024. 0.864608
\(216\) 45927.0 0.0669782
\(217\) 122304. 0.176316
\(218\) −46098.0 −0.0656963
\(219\) −705402. −0.993863
\(220\) −358360. −0.499186
\(221\) −362292. −0.498974
\(222\) −88182.0 −0.120087
\(223\) 399376. 0.537799 0.268899 0.963168i \(-0.413340\pi\)
0.268899 + 0.963168i \(0.413340\pi\)
\(224\) −144305. −0.192159
\(225\) −159489. −0.210027
\(226\) 262706. 0.342136
\(227\) 707916. 0.911837 0.455918 0.890022i \(-0.349311\pi\)
0.455918 + 0.890022i \(0.349311\pi\)
\(228\) 248868. 0.317053
\(229\) −735778. −0.927167 −0.463584 0.886053i \(-0.653437\pi\)
−0.463584 + 0.886053i \(0.653437\pi\)
\(230\) 108528. 0.135276
\(231\) −149940. −0.184879
\(232\) 519246. 0.633364
\(233\) −208758. −0.251915 −0.125957 0.992036i \(-0.540200\pi\)
−0.125957 + 0.992036i \(0.540200\pi\)
\(234\) 36774.0 0.0439037
\(235\) −303552. −0.358561
\(236\) 1.31428e6 1.53605
\(237\) 20448.0 0.0236472
\(238\) 39102.0 0.0447462
\(239\) 713376. 0.807837 0.403919 0.914795i \(-0.367648\pi\)
0.403919 + 0.914795i \(0.367648\pi\)
\(240\) 284274. 0.318573
\(241\) −505246. −0.560351 −0.280176 0.959949i \(-0.590393\pi\)
−0.280176 + 0.959949i \(0.590393\pi\)
\(242\) −45451.0 −0.0498890
\(243\) −59049.0 −0.0641500
\(244\) −457498. −0.491943
\(245\) −81634.0 −0.0868872
\(246\) −178506. −0.188068
\(247\) 404968. 0.422356
\(248\) 157248. 0.162351
\(249\) 339876. 0.347394
\(250\) 173196. 0.175262
\(251\) 317108. 0.317704 0.158852 0.987302i \(-0.449221\pi\)
0.158852 + 0.987302i \(0.449221\pi\)
\(252\) 123039. 0.122051
\(253\) 1.08528e6 1.06596
\(254\) 196608. 0.191213
\(255\) −244188. −0.235166
\(256\) 736033. 0.701936
\(257\) −1.44285e6 −1.36266 −0.681329 0.731977i \(-0.738598\pi\)
−0.681329 + 0.731977i \(0.738598\pi\)
\(258\) 155124. 0.145087
\(259\) −480102. −0.444717
\(260\) 478516. 0.438999
\(261\) −667602. −0.606619
\(262\) −77140.0 −0.0694267
\(263\) 271496. 0.242033 0.121016 0.992651i \(-0.461385\pi\)
0.121016 + 0.992651i \(0.461385\pi\)
\(264\) −192780. −0.170236
\(265\) −5100.00 −0.00446124
\(266\) −43708.0 −0.0378754
\(267\) 1.05557e6 0.906172
\(268\) 51956.0 0.0441874
\(269\) 850614. 0.716724 0.358362 0.933583i \(-0.383335\pi\)
0.358362 + 0.933583i \(0.383335\pi\)
\(270\) 24786.0 0.0206917
\(271\) −540128. −0.446759 −0.223380 0.974732i \(-0.571709\pi\)
−0.223380 + 0.974732i \(0.571709\pi\)
\(272\) −741342. −0.607570
\(273\) 200214. 0.162588
\(274\) 208170. 0.167510
\(275\) 669460. 0.533818
\(276\) −890568. −0.703711
\(277\) 513574. 0.402164 0.201082 0.979574i \(-0.435554\pi\)
0.201082 + 0.979574i \(0.435554\pi\)
\(278\) −275580. −0.213863
\(279\) −202176. −0.155496
\(280\) −104958. −0.0800056
\(281\) −1.35642e6 −1.02478 −0.512388 0.858754i \(-0.671239\pi\)
−0.512388 + 0.858754i \(0.671239\pi\)
\(282\) −80352.0 −0.0601692
\(283\) 286756. 0.212837 0.106418 0.994321i \(-0.466062\pi\)
0.106418 + 0.994321i \(0.466062\pi\)
\(284\) −451608. −0.332251
\(285\) 272952. 0.199055
\(286\) −154360. −0.111589
\(287\) −971866. −0.696469
\(288\) 238545. 0.169469
\(289\) −783053. −0.551501
\(290\) 280228. 0.195667
\(291\) −90018.0 −0.0623156
\(292\) −2.42972e6 −1.66763
\(293\) −1.70727e6 −1.16180 −0.580901 0.813974i \(-0.697300\pi\)
−0.580901 + 0.813974i \(0.697300\pi\)
\(294\) −21609.0 −0.0145803
\(295\) 1.44146e6 0.964381
\(296\) −617274. −0.409495
\(297\) 247860. 0.163048
\(298\) −296106. −0.193155
\(299\) −1.44917e6 −0.937434
\(300\) −549351. −0.352409
\(301\) 844564. 0.537299
\(302\) −426472. −0.269075
\(303\) 978930. 0.612555
\(304\) 828668. 0.514276
\(305\) −501772. −0.308857
\(306\) −64638.0 −0.0394625
\(307\) −546788. −0.331111 −0.165555 0.986201i \(-0.552942\pi\)
−0.165555 + 0.986201i \(0.552942\pi\)
\(308\) −516460. −0.310213
\(309\) 1.79273e6 1.06812
\(310\) 84864.0 0.0501556
\(311\) 3.23426e6 1.89616 0.948079 0.318035i \(-0.103023\pi\)
0.948079 + 0.318035i \(0.103023\pi\)
\(312\) 257418. 0.149711
\(313\) 1.81313e6 1.04609 0.523044 0.852306i \(-0.324796\pi\)
0.523044 + 0.852306i \(0.324796\pi\)
\(314\) 178486. 0.102160
\(315\) 134946. 0.0766273
\(316\) 70432.0 0.0396782
\(317\) −1.27658e6 −0.713509 −0.356754 0.934198i \(-0.616117\pi\)
−0.356754 + 0.934198i \(0.616117\pi\)
\(318\) −1350.00 −0.000748628 0
\(319\) 2.80228e6 1.54182
\(320\) 910622. 0.497122
\(321\) 719748. 0.389868
\(322\) 156408. 0.0840658
\(323\) −711816. −0.379631
\(324\) −203391. −0.107639
\(325\) −893926. −0.469454
\(326\) 252772. 0.131730
\(327\) 414882. 0.214563
\(328\) −1.24954e6 −0.641307
\(329\) −437472. −0.222823
\(330\) −104040. −0.0525915
\(331\) −1.73621e6 −0.871029 −0.435515 0.900182i \(-0.643434\pi\)
−0.435515 + 0.900182i \(0.643434\pi\)
\(332\) 1.17068e6 0.582901
\(333\) 793638. 0.392204
\(334\) 508088. 0.249214
\(335\) 56984.0 0.0277422
\(336\) 409689. 0.197973
\(337\) 2.07215e6 0.993907 0.496953 0.867777i \(-0.334452\pi\)
0.496953 + 0.867777i \(0.334452\pi\)
\(338\) −165177. −0.0786426
\(339\) −2.36435e6 −1.11741
\(340\) −841092. −0.394590
\(341\) 848640. 0.395219
\(342\) 72252.0 0.0334029
\(343\) −117649. −0.0539949
\(344\) 1.08587e6 0.494744
\(345\) −976752. −0.441811
\(346\) −221834. −0.0996180
\(347\) −1.65146e6 −0.736282 −0.368141 0.929770i \(-0.620006\pi\)
−0.368141 + 0.929770i \(0.620006\pi\)
\(348\) −2.29952e6 −1.01786
\(349\) 1.26645e6 0.556578 0.278289 0.960497i \(-0.410233\pi\)
0.278289 + 0.960497i \(0.410233\pi\)
\(350\) 96481.0 0.0420990
\(351\) −330966. −0.143389
\(352\) −1.00130e6 −0.430732
\(353\) 573218. 0.244840 0.122420 0.992478i \(-0.460934\pi\)
0.122420 + 0.992478i \(0.460934\pi\)
\(354\) 381564. 0.161830
\(355\) −495312. −0.208597
\(356\) 3.63587e6 1.52049
\(357\) −351918. −0.146141
\(358\) −113564. −0.0468310
\(359\) 4.46322e6 1.82773 0.913866 0.406016i \(-0.133082\pi\)
0.913866 + 0.406016i \(0.133082\pi\)
\(360\) 173502. 0.0705583
\(361\) −1.68044e6 −0.678662
\(362\) 663118. 0.265962
\(363\) 409059. 0.162937
\(364\) 689626. 0.272810
\(365\) −2.66485e6 −1.04699
\(366\) −132822. −0.0518283
\(367\) −4.50797e6 −1.74709 −0.873546 0.486742i \(-0.838185\pi\)
−0.873546 + 0.486742i \(0.838185\pi\)
\(368\) −2.96537e6 −1.14146
\(369\) 1.60655e6 0.614228
\(370\) −333132. −0.126506
\(371\) −7350.00 −0.00277238
\(372\) −696384. −0.260910
\(373\) 1.66535e6 0.619774 0.309887 0.950773i \(-0.399709\pi\)
0.309887 + 0.950773i \(0.399709\pi\)
\(374\) 271320. 0.100300
\(375\) −1.55876e6 −0.572403
\(376\) −562464. −0.205175
\(377\) −3.74187e6 −1.35592
\(378\) 35721.0 0.0128586
\(379\) −2.53232e6 −0.905568 −0.452784 0.891620i \(-0.649569\pi\)
−0.452784 + 0.891620i \(0.649569\pi\)
\(380\) 940168. 0.334000
\(381\) −1.76947e6 −0.624498
\(382\) 505664. 0.177298
\(383\) 796368. 0.277407 0.138703 0.990334i \(-0.455707\pi\)
0.138703 + 0.990334i \(0.455707\pi\)
\(384\) 1.08921e6 0.376949
\(385\) −566440. −0.194761
\(386\) −432382. −0.147706
\(387\) −1.39612e6 −0.473853
\(388\) −310062. −0.104561
\(389\) 1.94799e6 0.652699 0.326349 0.945249i \(-0.394181\pi\)
0.326349 + 0.945249i \(0.394181\pi\)
\(390\) 138924. 0.0462504
\(391\) 2.54722e6 0.842605
\(392\) −151263. −0.0497184
\(393\) 694260. 0.226747
\(394\) −131962. −0.0428261
\(395\) 77248.0 0.0249112
\(396\) 853740. 0.273582
\(397\) 1.08116e6 0.344281 0.172140 0.985072i \(-0.444932\pi\)
0.172140 + 0.985072i \(0.444932\pi\)
\(398\) 298536. 0.0944689
\(399\) 393372. 0.123700
\(400\) −1.82920e6 −0.571625
\(401\) 2.76770e6 0.859524 0.429762 0.902942i \(-0.358598\pi\)
0.429762 + 0.902942i \(0.358598\pi\)
\(402\) 15084.0 0.00465534
\(403\) −1.13318e6 −0.347566
\(404\) 3.37187e6 1.02782
\(405\) −223074. −0.0675789
\(406\) 403858. 0.121594
\(407\) −3.33132e6 −0.996851
\(408\) −452466. −0.134566
\(409\) 2.36350e6 0.698630 0.349315 0.937005i \(-0.386414\pi\)
0.349315 + 0.937005i \(0.386414\pi\)
\(410\) −674356. −0.198121
\(411\) −1.87353e6 −0.547087
\(412\) 6.17495e6 1.79222
\(413\) 2.07740e6 0.599302
\(414\) −258552. −0.0741391
\(415\) 1.28398e6 0.365963
\(416\) 1.33703e6 0.378798
\(417\) 2.48022e6 0.698474
\(418\) −303280. −0.0848991
\(419\) −2.98669e6 −0.831104 −0.415552 0.909569i \(-0.636412\pi\)
−0.415552 + 0.909569i \(0.636412\pi\)
\(420\) 464814. 0.128575
\(421\) −3.46331e6 −0.952326 −0.476163 0.879357i \(-0.657973\pi\)
−0.476163 + 0.879357i \(0.657973\pi\)
\(422\) −1.17062e6 −0.319989
\(423\) 723168. 0.196512
\(424\) −9450.00 −0.00255280
\(425\) 1.57126e6 0.421965
\(426\) −131112. −0.0350041
\(427\) −723142. −0.191935
\(428\) 2.47913e6 0.654169
\(429\) 1.38924e6 0.364447
\(430\) 586024. 0.152843
\(431\) 2.33693e6 0.605971 0.302986 0.952995i \(-0.402017\pi\)
0.302986 + 0.952995i \(0.402017\pi\)
\(432\) −677241. −0.174596
\(433\) −3.50838e6 −0.899264 −0.449632 0.893214i \(-0.648445\pi\)
−0.449632 + 0.893214i \(0.648445\pi\)
\(434\) 122304. 0.0311685
\(435\) −2.52205e6 −0.639044
\(436\) 1.42904e6 0.360021
\(437\) −2.84726e6 −0.713221
\(438\) −705402. −0.175692
\(439\) 3.54833e6 0.878744 0.439372 0.898305i \(-0.355201\pi\)
0.439372 + 0.898305i \(0.355201\pi\)
\(440\) −728280. −0.179336
\(441\) 194481. 0.0476190
\(442\) −362292. −0.0882070
\(443\) 1.76833e6 0.428109 0.214055 0.976822i \(-0.431333\pi\)
0.214055 + 0.976822i \(0.431333\pi\)
\(444\) 2.73364e6 0.658088
\(445\) 3.98772e6 0.954608
\(446\) 399376. 0.0950703
\(447\) 2.66495e6 0.630842
\(448\) 1.31237e6 0.308930
\(449\) −5.52579e6 −1.29354 −0.646768 0.762687i \(-0.723880\pi\)
−0.646768 + 0.762687i \(0.723880\pi\)
\(450\) −159489. −0.0371278
\(451\) −6.74356e6 −1.56116
\(452\) −8.14389e6 −1.87493
\(453\) 3.83825e6 0.878795
\(454\) 707916. 0.161191
\(455\) 756364. 0.171278
\(456\) 505764. 0.113903
\(457\) −2.96226e6 −0.663488 −0.331744 0.943369i \(-0.607637\pi\)
−0.331744 + 0.943369i \(0.607637\pi\)
\(458\) −735778. −0.163902
\(459\) 581742. 0.128884
\(460\) −3.36437e6 −0.741325
\(461\) 2.11884e6 0.464350 0.232175 0.972674i \(-0.425416\pi\)
0.232175 + 0.972674i \(0.425416\pi\)
\(462\) −149940. −0.0326823
\(463\) 3.19226e6 0.692062 0.346031 0.938223i \(-0.387529\pi\)
0.346031 + 0.938223i \(0.387529\pi\)
\(464\) −7.65682e6 −1.65102
\(465\) −763776. −0.163807
\(466\) −208758. −0.0445326
\(467\) −7.42621e6 −1.57571 −0.787853 0.615863i \(-0.788807\pi\)
−0.787853 + 0.615863i \(0.788807\pi\)
\(468\) −1.13999e6 −0.240596
\(469\) 82124.0 0.0172400
\(470\) −303552. −0.0633853
\(471\) −1.60637e6 −0.333653
\(472\) 2.67095e6 0.551837
\(473\) 5.86024e6 1.20438
\(474\) 20448.0 0.00418028
\(475\) −1.75635e6 −0.357171
\(476\) −1.21216e6 −0.245213
\(477\) 12150.0 0.00244501
\(478\) 713376. 0.142807
\(479\) −3.39685e6 −0.676453 −0.338226 0.941065i \(-0.609827\pi\)
−0.338226 + 0.941065i \(0.609827\pi\)
\(480\) 901170. 0.178527
\(481\) 4.44829e6 0.876659
\(482\) −505246. −0.0990570
\(483\) −1.40767e6 −0.274558
\(484\) 1.40898e6 0.273396
\(485\) −340068. −0.0656465
\(486\) −59049.0 −0.0113402
\(487\) −3.71382e6 −0.709574 −0.354787 0.934947i \(-0.615447\pi\)
−0.354787 + 0.934947i \(0.615447\pi\)
\(488\) −929754. −0.176733
\(489\) −2.27495e6 −0.430229
\(490\) −81634.0 −0.0153596
\(491\) 5.57494e6 1.04361 0.521803 0.853066i \(-0.325260\pi\)
0.521803 + 0.853066i \(0.325260\pi\)
\(492\) 5.53369e6 1.03063
\(493\) 6.57712e6 1.21876
\(494\) 404968. 0.0746626
\(495\) 936360. 0.171763
\(496\) −2.31878e6 −0.423210
\(497\) −713832. −0.129630
\(498\) 339876. 0.0614111
\(499\) 3.92698e6 0.706004 0.353002 0.935623i \(-0.385161\pi\)
0.353002 + 0.935623i \(0.385161\pi\)
\(500\) −5.36908e6 −0.960450
\(501\) −4.57279e6 −0.813930
\(502\) 317108. 0.0561627
\(503\) 6.42079e6 1.13154 0.565768 0.824564i \(-0.308580\pi\)
0.565768 + 0.824564i \(0.308580\pi\)
\(504\) 250047. 0.0438475
\(505\) 3.69818e6 0.645297
\(506\) 1.08528e6 0.188437
\(507\) 1.48659e6 0.256846
\(508\) −6.09485e6 −1.04786
\(509\) 146278. 0.0250256 0.0125128 0.999922i \(-0.496017\pi\)
0.0125128 + 0.999922i \(0.496017\pi\)
\(510\) −244188. −0.0415718
\(511\) −3.84052e6 −0.650636
\(512\) 4.60877e6 0.776980
\(513\) −650268. −0.109094
\(514\) −1.44285e6 −0.240886
\(515\) 6.77253e6 1.12521
\(516\) −4.80884e6 −0.795090
\(517\) −3.03552e6 −0.499467
\(518\) −480102. −0.0786157
\(519\) 1.99651e6 0.325351
\(520\) 972468. 0.157713
\(521\) 7.70937e6 1.24430 0.622149 0.782899i \(-0.286260\pi\)
0.622149 + 0.782899i \(0.286260\pi\)
\(522\) −667602. −0.107236
\(523\) −569420. −0.0910287 −0.0455144 0.998964i \(-0.514493\pi\)
−0.0455144 + 0.998964i \(0.514493\pi\)
\(524\) 2.39134e6 0.380464
\(525\) −868329. −0.137495
\(526\) 271496. 0.0427857
\(527\) 1.99181e6 0.312407
\(528\) 2.84274e6 0.443764
\(529\) 3.75252e6 0.583021
\(530\) −5100.00 −0.000788643 0
\(531\) −3.43408e6 −0.528535
\(532\) 1.35495e6 0.207560
\(533\) 9.00464e6 1.37293
\(534\) 1.05557e6 0.160190
\(535\) 2.71905e6 0.410707
\(536\) 105588. 0.0158746
\(537\) 1.02208e6 0.152949
\(538\) 850614. 0.126700
\(539\) −816340. −0.121032
\(540\) −768366. −0.113392
\(541\) −9.44802e6 −1.38787 −0.693933 0.720040i \(-0.744124\pi\)
−0.693933 + 0.720040i \(0.744124\pi\)
\(542\) −540128. −0.0789766
\(543\) −5.96806e6 −0.868628
\(544\) −2.35011e6 −0.340479
\(545\) 1.56733e6 0.226032
\(546\) 200214. 0.0287417
\(547\) −1.35321e6 −0.193374 −0.0966869 0.995315i \(-0.530825\pi\)
−0.0966869 + 0.995315i \(0.530825\pi\)
\(548\) −6.45327e6 −0.917970
\(549\) 1.19540e6 0.169271
\(550\) 669460. 0.0943665
\(551\) −7.35186e6 −1.03162
\(552\) −1.80986e6 −0.252812
\(553\) 111328. 0.0154807
\(554\) 513574. 0.0710933
\(555\) 2.99819e6 0.413168
\(556\) 8.54298e6 1.17199
\(557\) 8.19390e6 1.11906 0.559529 0.828811i \(-0.310982\pi\)
0.559529 + 0.828811i \(0.310982\pi\)
\(558\) −202176. −0.0274881
\(559\) −7.82514e6 −1.05916
\(560\) 1.54771e6 0.208555
\(561\) −2.44188e6 −0.327580
\(562\) −1.35642e6 −0.181157
\(563\) −1.05796e7 −1.40669 −0.703347 0.710847i \(-0.748312\pi\)
−0.703347 + 0.710847i \(0.748312\pi\)
\(564\) 2.49091e6 0.329732
\(565\) −8.93200e6 −1.17714
\(566\) 286756. 0.0376246
\(567\) −321489. −0.0419961
\(568\) −917784. −0.119363
\(569\) −1.20205e7 −1.55648 −0.778238 0.627969i \(-0.783886\pi\)
−0.778238 + 0.627969i \(0.783886\pi\)
\(570\) 272952. 0.0351884
\(571\) −2.48948e6 −0.319534 −0.159767 0.987155i \(-0.551074\pi\)
−0.159767 + 0.987155i \(0.551074\pi\)
\(572\) 4.78516e6 0.611514
\(573\) −4.55098e6 −0.579053
\(574\) −971866. −0.123119
\(575\) 6.28505e6 0.792755
\(576\) −2.16942e6 −0.272451
\(577\) 8.21322e6 1.02701 0.513504 0.858087i \(-0.328347\pi\)
0.513504 + 0.858087i \(0.328347\pi\)
\(578\) −783053. −0.0974926
\(579\) 3.89144e6 0.482407
\(580\) −8.68707e6 −1.07227
\(581\) 1.85044e6 0.227423
\(582\) −90018.0 −0.0110159
\(583\) −51000.0 −0.00621439
\(584\) −4.93781e6 −0.599105
\(585\) −1.25032e6 −0.151053
\(586\) −1.70727e6 −0.205380
\(587\) −1.21827e6 −0.145931 −0.0729655 0.997334i \(-0.523246\pi\)
−0.0729655 + 0.997334i \(0.523246\pi\)
\(588\) 669879. 0.0799012
\(589\) −2.22643e6 −0.264436
\(590\) 1.44146e6 0.170480
\(591\) 1.18766e6 0.139869
\(592\) 9.10234e6 1.06745
\(593\) −8.42379e6 −0.983718 −0.491859 0.870675i \(-0.663683\pi\)
−0.491859 + 0.870675i \(0.663683\pi\)
\(594\) 247860. 0.0288231
\(595\) −1.32947e6 −0.153952
\(596\) 9.17929e6 1.05851
\(597\) −2.68682e6 −0.308534
\(598\) −1.44917e6 −0.165717
\(599\) 8.21254e6 0.935212 0.467606 0.883937i \(-0.345117\pi\)
0.467606 + 0.883937i \(0.345117\pi\)
\(600\) −1.11642e6 −0.126605
\(601\) 3.25478e6 0.367566 0.183783 0.982967i \(-0.441166\pi\)
0.183783 + 0.982967i \(0.441166\pi\)
\(602\) 844564. 0.0949820
\(603\) −135756. −0.0152043
\(604\) 1.32206e7 1.47455
\(605\) 1.54533e6 0.171646
\(606\) 978930. 0.108285
\(607\) 7.82101e6 0.861571 0.430785 0.902454i \(-0.358237\pi\)
0.430785 + 0.902454i \(0.358237\pi\)
\(608\) 2.62694e6 0.288198
\(609\) −3.63472e6 −0.397126
\(610\) −501772. −0.0545986
\(611\) 4.05331e6 0.439245
\(612\) 2.00378e6 0.216257
\(613\) −9.51670e6 −1.02290 −0.511452 0.859312i \(-0.670892\pi\)
−0.511452 + 0.859312i \(0.670892\pi\)
\(614\) −546788. −0.0585326
\(615\) 6.06920e6 0.647059
\(616\) −1.04958e6 −0.111446
\(617\) −7.04895e6 −0.745438 −0.372719 0.927944i \(-0.621574\pi\)
−0.372719 + 0.927944i \(0.621574\pi\)
\(618\) 1.79273e6 0.188818
\(619\) −6.32174e6 −0.663147 −0.331574 0.943429i \(-0.607580\pi\)
−0.331574 + 0.943429i \(0.607580\pi\)
\(620\) −2.63078e6 −0.274856
\(621\) 2.32697e6 0.242137
\(622\) 3.23426e6 0.335197
\(623\) 5.74701e6 0.593229
\(624\) −3.79589e6 −0.390259
\(625\) 264461. 0.0270808
\(626\) 1.81313e6 0.184924
\(627\) 2.72952e6 0.277279
\(628\) −5.53307e6 −0.559844
\(629\) −7.81880e6 −0.787977
\(630\) 134946. 0.0135459
\(631\) 8.61236e6 0.861090 0.430545 0.902569i \(-0.358321\pi\)
0.430545 + 0.902569i \(0.358321\pi\)
\(632\) 143136. 0.0142546
\(633\) 1.05356e7 1.04508
\(634\) −1.27658e6 −0.126132
\(635\) −6.68467e6 −0.657879
\(636\) 41850.0 0.00410254
\(637\) 1.09005e6 0.106439
\(638\) 2.80228e6 0.272559
\(639\) 1.18001e6 0.114323
\(640\) 4.11478e6 0.397097
\(641\) −5.22829e6 −0.502590 −0.251295 0.967910i \(-0.580857\pi\)
−0.251295 + 0.967910i \(0.580857\pi\)
\(642\) 719748. 0.0689196
\(643\) 1.61373e7 1.53923 0.769615 0.638508i \(-0.220448\pi\)
0.769615 + 0.638508i \(0.220448\pi\)
\(644\) −4.84865e6 −0.460687
\(645\) −5.27422e6 −0.499182
\(646\) −711816. −0.0671099
\(647\) −1.58749e7 −1.49090 −0.745451 0.666560i \(-0.767766\pi\)
−0.745451 + 0.666560i \(0.767766\pi\)
\(648\) −413343. −0.0386699
\(649\) 1.44146e7 1.34336
\(650\) −893926. −0.0829886
\(651\) −1.10074e6 −0.101796
\(652\) −7.83593e6 −0.721891
\(653\) −5.94112e6 −0.545237 −0.272619 0.962122i \(-0.587890\pi\)
−0.272619 + 0.962122i \(0.587890\pi\)
\(654\) 414882. 0.0379298
\(655\) 2.62276e6 0.238867
\(656\) 1.84258e7 1.67173
\(657\) 6.34862e6 0.573807
\(658\) −437472. −0.0393900
\(659\) −7.64430e6 −0.685684 −0.342842 0.939393i \(-0.611390\pi\)
−0.342842 + 0.939393i \(0.611390\pi\)
\(660\) 3.22524e6 0.288205
\(661\) −7.58688e6 −0.675398 −0.337699 0.941254i \(-0.609649\pi\)
−0.337699 + 0.941254i \(0.609649\pi\)
\(662\) −1.73621e6 −0.153978
\(663\) 3.26063e6 0.288083
\(664\) 2.37913e6 0.209410
\(665\) 1.48607e6 0.130312
\(666\) 793638. 0.0693325
\(667\) 2.63085e7 2.28971
\(668\) −1.57507e7 −1.36571
\(669\) −3.59438e6 −0.310498
\(670\) 56984.0 0.00490417
\(671\) −5.01772e6 −0.430229
\(672\) 1.29874e6 0.110943
\(673\) −2.06681e7 −1.75899 −0.879494 0.475910i \(-0.842119\pi\)
−0.879494 + 0.475910i \(0.842119\pi\)
\(674\) 2.07215e6 0.175700
\(675\) 1.43540e6 0.121259
\(676\) 5.12049e6 0.430968
\(677\) 7.89541e6 0.662068 0.331034 0.943619i \(-0.392602\pi\)
0.331034 + 0.943619i \(0.392602\pi\)
\(678\) −2.36435e6 −0.197532
\(679\) −490098. −0.0407951
\(680\) −1.70932e6 −0.141759
\(681\) −6.37124e6 −0.526449
\(682\) 848640. 0.0698655
\(683\) −1.96015e7 −1.60782 −0.803911 0.594750i \(-0.797251\pi\)
−0.803911 + 0.594750i \(0.797251\pi\)
\(684\) −2.23981e6 −0.183051
\(685\) −7.07778e6 −0.576329
\(686\) −117649. −0.00954504
\(687\) 6.62200e6 0.535300
\(688\) −1.60122e7 −1.28968
\(689\) 68100.0 0.00546511
\(690\) −976752. −0.0781019
\(691\) −1.72710e7 −1.37601 −0.688005 0.725706i \(-0.741513\pi\)
−0.688005 + 0.725706i \(0.741513\pi\)
\(692\) 6.87685e6 0.545914
\(693\) 1.34946e6 0.106740
\(694\) −1.65146e6 −0.130158
\(695\) 9.36972e6 0.735808
\(696\) −4.67321e6 −0.365673
\(697\) −1.58275e7 −1.23405
\(698\) 1.26645e6 0.0983900
\(699\) 1.87882e6 0.145443
\(700\) −2.99091e6 −0.230706
\(701\) −5.36344e6 −0.412238 −0.206119 0.978527i \(-0.566083\pi\)
−0.206119 + 0.978527i \(0.566083\pi\)
\(702\) −330966. −0.0253478
\(703\) 8.73982e6 0.666982
\(704\) 9.10622e6 0.692479
\(705\) 2.73197e6 0.207015
\(706\) 573218. 0.0432821
\(707\) 5.32973e6 0.401011
\(708\) −1.18285e7 −0.886841
\(709\) −1.73733e7 −1.29798 −0.648988 0.760798i \(-0.724808\pi\)
−0.648988 + 0.760798i \(0.724808\pi\)
\(710\) −495312. −0.0368751
\(711\) −184032. −0.0136527
\(712\) 7.38902e6 0.546244
\(713\) 7.96723e6 0.586926
\(714\) −351918. −0.0258343
\(715\) 5.24824e6 0.383927
\(716\) 3.52048e6 0.256637
\(717\) −6.42038e6 −0.466405
\(718\) 4.46322e6 0.323100
\(719\) 424608. 0.0306313 0.0153157 0.999883i \(-0.495125\pi\)
0.0153157 + 0.999883i \(0.495125\pi\)
\(720\) −2.55847e6 −0.183928
\(721\) 9.76041e6 0.699246
\(722\) −1.68044e6 −0.119972
\(723\) 4.54721e6 0.323519
\(724\) −2.05567e7 −1.45749
\(725\) 1.62285e7 1.14666
\(726\) 409059. 0.0288034
\(727\) 2.18290e7 1.53179 0.765893 0.642968i \(-0.222297\pi\)
0.765893 + 0.642968i \(0.222297\pi\)
\(728\) 1.40150e6 0.0980086
\(729\) 531441. 0.0370370
\(730\) −2.66485e6 −0.185083
\(731\) 1.37543e7 0.952020
\(732\) 4.11748e6 0.284023
\(733\) 2.17675e7 1.49640 0.748202 0.663470i \(-0.230917\pi\)
0.748202 + 0.663470i \(0.230917\pi\)
\(734\) −4.50797e6 −0.308845
\(735\) 734706. 0.0501644
\(736\) −9.40044e6 −0.639667
\(737\) 569840. 0.0386442
\(738\) 1.60655e6 0.108581
\(739\) 6.21786e6 0.418822 0.209411 0.977828i \(-0.432845\pi\)
0.209411 + 0.977828i \(0.432845\pi\)
\(740\) 1.03271e7 0.693264
\(741\) −3.64471e6 −0.243847
\(742\) −7350.00 −0.000490092 0
\(743\) 3.77647e6 0.250966 0.125483 0.992096i \(-0.459952\pi\)
0.125483 + 0.992096i \(0.459952\pi\)
\(744\) −1.41523e6 −0.0937336
\(745\) 1.00676e7 0.664562
\(746\) 1.66535e6 0.109562
\(747\) −3.05888e6 −0.200568
\(748\) −8.41092e6 −0.549654
\(749\) 3.91863e6 0.255229
\(750\) −1.55876e6 −0.101188
\(751\) −2.88795e6 −0.186849 −0.0934244 0.995626i \(-0.529781\pi\)
−0.0934244 + 0.995626i \(0.529781\pi\)
\(752\) 8.29411e6 0.534842
\(753\) −2.85397e6 −0.183427
\(754\) −3.74187e6 −0.239696
\(755\) 1.45000e7 0.925768
\(756\) −1.10735e6 −0.0704662
\(757\) 1.25519e6 0.0796104 0.0398052 0.999207i \(-0.487326\pi\)
0.0398052 + 0.999207i \(0.487326\pi\)
\(758\) −2.53232e6 −0.160083
\(759\) −9.76752e6 −0.615432
\(760\) 1.91066e6 0.119991
\(761\) −1.42623e7 −0.892746 −0.446373 0.894847i \(-0.647284\pi\)
−0.446373 + 0.894847i \(0.647284\pi\)
\(762\) −1.76947e6 −0.110397
\(763\) 2.25880e6 0.140465
\(764\) −1.56756e7 −0.971606
\(765\) 2.19769e6 0.135773
\(766\) 796368. 0.0490390
\(767\) −1.92478e7 −1.18139
\(768\) −6.62430e6 −0.405263
\(769\) −2.02261e7 −1.23338 −0.616689 0.787207i \(-0.711526\pi\)
−0.616689 + 0.787207i \(0.711526\pi\)
\(770\) −566440. −0.0344292
\(771\) 1.29856e7 0.786732
\(772\) 1.34038e7 0.809443
\(773\) 2.62288e7 1.57881 0.789406 0.613872i \(-0.210389\pi\)
0.789406 + 0.613872i \(0.210389\pi\)
\(774\) −1.39612e6 −0.0837663
\(775\) 4.91462e6 0.293925
\(776\) −630126. −0.0375641
\(777\) 4.32092e6 0.256758
\(778\) 1.94799e6 0.115382
\(779\) 1.76919e7 1.04456
\(780\) −4.30664e6 −0.253456
\(781\) −4.95312e6 −0.290570
\(782\) 2.54722e6 0.148953
\(783\) 6.00842e6 0.350232
\(784\) 2.23053e6 0.129604
\(785\) −6.06852e6 −0.351487
\(786\) 694260. 0.0400835
\(787\) −9.92829e6 −0.571397 −0.285698 0.958320i \(-0.592226\pi\)
−0.285698 + 0.958320i \(0.592226\pi\)
\(788\) 4.09082e6 0.234690
\(789\) −2.44346e6 −0.139738
\(790\) 77248.0 0.00440372
\(791\) −1.28726e7 −0.731518
\(792\) 1.73502e6 0.0982860
\(793\) 6.70013e6 0.378356
\(794\) 1.08116e6 0.0608608
\(795\) 45900.0 0.00257570
\(796\) −9.25462e6 −0.517697
\(797\) 1.09033e7 0.608014 0.304007 0.952670i \(-0.401675\pi\)
0.304007 + 0.952670i \(0.401675\pi\)
\(798\) 393372. 0.0218674
\(799\) −7.12454e6 −0.394812
\(800\) −5.79871e6 −0.320336
\(801\) −9.50017e6 −0.523179
\(802\) 2.76770e6 0.151944
\(803\) −2.66485e7 −1.45843
\(804\) −467604. −0.0255116
\(805\) −5.31787e6 −0.289233
\(806\) −1.13318e6 −0.0614416
\(807\) −7.65553e6 −0.413801
\(808\) 6.85251e6 0.369251
\(809\) 6.06398e6 0.325751 0.162876 0.986647i \(-0.447923\pi\)
0.162876 + 0.986647i \(0.447923\pi\)
\(810\) −223074. −0.0119464
\(811\) −8.59438e6 −0.458841 −0.229421 0.973327i \(-0.573683\pi\)
−0.229421 + 0.973327i \(0.573683\pi\)
\(812\) −1.25196e7 −0.666347
\(813\) 4.86115e6 0.257937
\(814\) −3.33132e6 −0.176220
\(815\) −8.59425e6 −0.453225
\(816\) 6.67208e6 0.350781
\(817\) −1.53745e7 −0.805835
\(818\) 2.36350e6 0.123501
\(819\) −1.80193e6 −0.0938701
\(820\) 2.09050e7 1.08572
\(821\) −2.01396e6 −0.104278 −0.0521391 0.998640i \(-0.516604\pi\)
−0.0521391 + 0.998640i \(0.516604\pi\)
\(822\) −1.87353e6 −0.0967122
\(823\) −2.64679e7 −1.36213 −0.681067 0.732221i \(-0.738484\pi\)
−0.681067 + 0.732221i \(0.738484\pi\)
\(824\) 1.25491e7 0.643864
\(825\) −6.02514e6 −0.308200
\(826\) 2.07740e6 0.105943
\(827\) −3.90229e6 −0.198407 −0.0992033 0.995067i \(-0.531629\pi\)
−0.0992033 + 0.995067i \(0.531629\pi\)
\(828\) 8.01511e6 0.406288
\(829\) −1.95595e7 −0.988487 −0.494244 0.869323i \(-0.664555\pi\)
−0.494244 + 0.869323i \(0.664555\pi\)
\(830\) 1.28398e6 0.0646937
\(831\) −4.62217e6 −0.232190
\(832\) −1.21595e7 −0.608985
\(833\) −1.91600e6 −0.0956715
\(834\) 2.48022e6 0.123474
\(835\) −1.72750e7 −0.857436
\(836\) 9.40168e6 0.465254
\(837\) 1.81958e6 0.0897756
\(838\) −2.98669e6 −0.146920
\(839\) −2.45448e7 −1.20380 −0.601901 0.798570i \(-0.705590\pi\)
−0.601901 + 0.798570i \(0.705590\pi\)
\(840\) 944622. 0.0461913
\(841\) 4.74194e7 2.31188
\(842\) −3.46331e6 −0.168349
\(843\) 1.22078e7 0.591655
\(844\) 3.62892e7 1.75356
\(845\) 5.61602e6 0.270574
\(846\) 723168. 0.0347387
\(847\) 2.22710e6 0.106667
\(848\) 139350. 0.00665453
\(849\) −2.58080e6 −0.122881
\(850\) 1.57126e6 0.0745936
\(851\) −3.12752e7 −1.48039
\(852\) 4.06447e6 0.191825
\(853\) 3.38305e7 1.59197 0.795987 0.605314i \(-0.206952\pi\)
0.795987 + 0.605314i \(0.206952\pi\)
\(854\) −723142. −0.0339296
\(855\) −2.45657e6 −0.114925
\(856\) 5.03824e6 0.235014
\(857\) 3.18009e7 1.47907 0.739534 0.673120i \(-0.235046\pi\)
0.739534 + 0.673120i \(0.235046\pi\)
\(858\) 1.38924e6 0.0644257
\(859\) 638420. 0.0295205 0.0147602 0.999891i \(-0.495301\pi\)
0.0147602 + 0.999891i \(0.495301\pi\)
\(860\) −1.81667e7 −0.837589
\(861\) 8.74679e6 0.402106
\(862\) 2.33693e6 0.107122
\(863\) −4.22256e6 −0.192996 −0.0964981 0.995333i \(-0.530764\pi\)
−0.0964981 + 0.995333i \(0.530764\pi\)
\(864\) −2.14690e6 −0.0978427
\(865\) 7.54236e6 0.342742
\(866\) −3.50838e6 −0.158969
\(867\) 7.04748e6 0.318409
\(868\) −3.79142e6 −0.170806
\(869\) 772480. 0.0347007
\(870\) −2.52205e6 −0.112968
\(871\) −760904. −0.0339848
\(872\) 2.90417e6 0.129340
\(873\) 810162. 0.0359779
\(874\) −2.84726e6 −0.126081
\(875\) −8.48660e6 −0.374726
\(876\) 2.18675e7 0.962805
\(877\) −2.45043e7 −1.07583 −0.537915 0.842999i \(-0.680788\pi\)
−0.537915 + 0.842999i \(0.680788\pi\)
\(878\) 3.54833e6 0.155341
\(879\) 1.53654e7 0.670767
\(880\) 1.07392e7 0.467484
\(881\) −2.77630e7 −1.20511 −0.602555 0.798078i \(-0.705850\pi\)
−0.602555 + 0.798078i \(0.705850\pi\)
\(882\) 194481. 0.00841794
\(883\) 3.30170e7 1.42507 0.712534 0.701638i \(-0.247548\pi\)
0.712534 + 0.701638i \(0.247548\pi\)
\(884\) 1.12311e7 0.483381
\(885\) −1.29732e7 −0.556786
\(886\) 1.76833e6 0.0756797
\(887\) 4.34462e6 0.185414 0.0927070 0.995693i \(-0.470448\pi\)
0.0927070 + 0.995693i \(0.470448\pi\)
\(888\) 5.55547e6 0.236422
\(889\) −9.63379e6 −0.408830
\(890\) 3.98772e6 0.168752
\(891\) −2.23074e6 −0.0941358
\(892\) −1.23807e7 −0.520993
\(893\) 7.96378e6 0.334188
\(894\) 2.66495e6 0.111518
\(895\) 3.86118e6 0.161125
\(896\) 5.93013e6 0.246771
\(897\) 1.30425e7 0.541228
\(898\) −5.52579e6 −0.228667
\(899\) 2.05720e7 0.848942
\(900\) 4.94416e6 0.203463
\(901\) −119700. −0.00491227
\(902\) −6.74356e6 −0.275977
\(903\) −7.60108e6 −0.310210
\(904\) −1.65505e7 −0.673580
\(905\) −2.25460e7 −0.915057
\(906\) 3.83825e6 0.155350
\(907\) 1.96499e7 0.793128 0.396564 0.918007i \(-0.370203\pi\)
0.396564 + 0.918007i \(0.370203\pi\)
\(908\) −2.19454e7 −0.883342
\(909\) −8.81037e6 −0.353659
\(910\) 756364. 0.0302780
\(911\) −7.26518e6 −0.290035 −0.145018 0.989429i \(-0.546324\pi\)
−0.145018 + 0.989429i \(0.546324\pi\)
\(912\) −7.45801e6 −0.296918
\(913\) 1.28398e7 0.509777
\(914\) −2.96226e6 −0.117289
\(915\) 4.51595e6 0.178318
\(916\) 2.28091e7 0.898193
\(917\) 3.77986e6 0.148440
\(918\) 581742. 0.0227837
\(919\) 9.82532e6 0.383758 0.191879 0.981419i \(-0.438542\pi\)
0.191879 + 0.981419i \(0.438542\pi\)
\(920\) −6.83726e6 −0.266326
\(921\) 4.92109e6 0.191167
\(922\) 2.11884e6 0.0820863
\(923\) 6.61387e6 0.255536
\(924\) 4.64814e6 0.179102
\(925\) −1.92923e7 −0.741359
\(926\) 3.19226e6 0.122340
\(927\) −1.61346e7 −0.616677
\(928\) −2.42727e7 −0.925226
\(929\) 2.71152e7 1.03080 0.515399 0.856951i \(-0.327644\pi\)
0.515399 + 0.856951i \(0.327644\pi\)
\(930\) −763776. −0.0289573
\(931\) 2.14169e6 0.0809809
\(932\) 6.47150e6 0.244042
\(933\) −2.91084e7 −1.09475
\(934\) −7.42621e6 −0.278548
\(935\) −9.22488e6 −0.345089
\(936\) −2.31676e6 −0.0864354
\(937\) −4.53522e7 −1.68752 −0.843761 0.536720i \(-0.819663\pi\)
−0.843761 + 0.536720i \(0.819663\pi\)
\(938\) 82124.0 0.00304764
\(939\) −1.63182e7 −0.603959
\(940\) 9.41011e6 0.347356
\(941\) 4.65780e7 1.71477 0.857387 0.514672i \(-0.172086\pi\)
0.857387 + 0.514672i \(0.172086\pi\)
\(942\) −1.60637e6 −0.0589820
\(943\) −6.33101e7 −2.31843
\(944\) −3.93859e7 −1.43850
\(945\) −1.21451e6 −0.0442408
\(946\) 5.86024e6 0.212906
\(947\) 2.53799e7 0.919632 0.459816 0.888014i \(-0.347915\pi\)
0.459816 + 0.888014i \(0.347915\pi\)
\(948\) −633888. −0.0229082
\(949\) 3.55836e7 1.28258
\(950\) −1.75635e6 −0.0631396
\(951\) 1.14892e7 0.411944
\(952\) −2.46343e6 −0.0880942
\(953\) 1.52948e7 0.545520 0.272760 0.962082i \(-0.412063\pi\)
0.272760 + 0.962082i \(0.412063\pi\)
\(954\) 12150.0 0.000432220 0
\(955\) −1.71926e7 −0.610004
\(956\) −2.21147e7 −0.782592
\(957\) −2.52205e7 −0.890173
\(958\) −3.39685e6 −0.119581
\(959\) −1.02003e7 −0.358152
\(960\) −8.19560e6 −0.287014
\(961\) −2.23991e7 −0.782389
\(962\) 4.44829e6 0.154973
\(963\) −6.47773e6 −0.225091
\(964\) 1.56626e7 0.542840
\(965\) 1.47010e7 0.508192
\(966\) −1.40767e6 −0.0485354
\(967\) −5.71465e6 −0.196527 −0.0982637 0.995160i \(-0.531329\pi\)
−0.0982637 + 0.995160i \(0.531329\pi\)
\(968\) 2.86341e6 0.0982190
\(969\) 6.40634e6 0.219180
\(970\) −340068. −0.0116048
\(971\) 1.30250e7 0.443332 0.221666 0.975123i \(-0.428851\pi\)
0.221666 + 0.975123i \(0.428851\pi\)
\(972\) 1.83052e6 0.0621453
\(973\) 1.35034e7 0.457258
\(974\) −3.71382e6 −0.125436
\(975\) 8.04533e6 0.271039
\(976\) 1.37102e7 0.460700
\(977\) −1.70360e7 −0.570992 −0.285496 0.958380i \(-0.592158\pi\)
−0.285496 + 0.958380i \(0.592158\pi\)
\(978\) −2.27495e6 −0.0760544
\(979\) 3.98772e7 1.32974
\(980\) 2.53065e6 0.0841720
\(981\) −3.73394e6 −0.123878
\(982\) 5.57494e6 0.184485
\(983\) −1.36985e7 −0.452156 −0.226078 0.974109i \(-0.572590\pi\)
−0.226078 + 0.974109i \(0.572590\pi\)
\(984\) 1.12459e7 0.370259
\(985\) 4.48671e6 0.147346
\(986\) 6.57712e6 0.215448
\(987\) 3.93725e6 0.128647
\(988\) −1.25540e7 −0.409157
\(989\) 5.50173e7 1.78858
\(990\) 936360. 0.0303637
\(991\) −3.49088e7 −1.12915 −0.564574 0.825383i \(-0.690959\pi\)
−0.564574 + 0.825383i \(0.690959\pi\)
\(992\) −7.35072e6 −0.237165
\(993\) 1.56259e7 0.502889
\(994\) −713832. −0.0229155
\(995\) −1.01502e7 −0.325026
\(996\) −1.05362e7 −0.336538
\(997\) 875662. 0.0278996 0.0139498 0.999903i \(-0.495559\pi\)
0.0139498 + 0.999903i \(0.495559\pi\)
\(998\) 3.92698e6 0.124805
\(999\) −7.14274e6 −0.226439
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 21.6.a.b.1.1 1
3.2 odd 2 63.6.a.c.1.1 1
4.3 odd 2 336.6.a.l.1.1 1
5.2 odd 4 525.6.d.d.274.2 2
5.3 odd 4 525.6.d.d.274.1 2
5.4 even 2 525.6.a.c.1.1 1
7.2 even 3 147.6.e.f.67.1 2
7.3 odd 6 147.6.e.e.79.1 2
7.4 even 3 147.6.e.f.79.1 2
7.5 odd 6 147.6.e.e.67.1 2
7.6 odd 2 147.6.a.e.1.1 1
12.11 even 2 1008.6.a.t.1.1 1
21.20 even 2 441.6.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
21.6.a.b.1.1 1 1.1 even 1 trivial
63.6.a.c.1.1 1 3.2 odd 2
147.6.a.e.1.1 1 7.6 odd 2
147.6.e.e.67.1 2 7.5 odd 6
147.6.e.e.79.1 2 7.3 odd 6
147.6.e.f.67.1 2 7.2 even 3
147.6.e.f.79.1 2 7.4 even 3
336.6.a.l.1.1 1 4.3 odd 2
441.6.a.d.1.1 1 21.20 even 2
525.6.a.c.1.1 1 5.4 even 2
525.6.d.d.274.1 2 5.3 odd 4
525.6.d.d.274.2 2 5.2 odd 4
1008.6.a.t.1.1 1 12.11 even 2