Properties

Label 50.6.a.c.1.1
Level 50
Weight 6
Character 50.1
Self dual yes
Analytic conductor 8.019
Analytic rank 0
Dimension 1
CM no
Inner twists 1

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Newspace parameters

Level: \( N \) \(=\) \( 50 = 2 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 50.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(8.01919099065\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 10)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 50.1

$q$-expansion

\(f(q)\) \(=\) \(q-4.00000 q^{2} +14.0000 q^{3} +16.0000 q^{4} -56.0000 q^{6} +158.000 q^{7} -64.0000 q^{8} -47.0000 q^{9} +O(q^{10})\) \(q-4.00000 q^{2} +14.0000 q^{3} +16.0000 q^{4} -56.0000 q^{6} +158.000 q^{7} -64.0000 q^{8} -47.0000 q^{9} -148.000 q^{11} +224.000 q^{12} +684.000 q^{13} -632.000 q^{14} +256.000 q^{16} +2048.00 q^{17} +188.000 q^{18} +2220.00 q^{19} +2212.00 q^{21} +592.000 q^{22} -1246.00 q^{23} -896.000 q^{24} -2736.00 q^{26} -4060.00 q^{27} +2528.00 q^{28} -270.000 q^{29} -2048.00 q^{31} -1024.00 q^{32} -2072.00 q^{33} -8192.00 q^{34} -752.000 q^{36} -4372.00 q^{37} -8880.00 q^{38} +9576.00 q^{39} -2398.00 q^{41} -8848.00 q^{42} +2294.00 q^{43} -2368.00 q^{44} +4984.00 q^{46} -10682.0 q^{47} +3584.00 q^{48} +8157.00 q^{49} +28672.0 q^{51} +10944.0 q^{52} +2964.00 q^{53} +16240.0 q^{54} -10112.0 q^{56} +31080.0 q^{57} +1080.00 q^{58} -39740.0 q^{59} -42298.0 q^{61} +8192.00 q^{62} -7426.00 q^{63} +4096.00 q^{64} +8288.00 q^{66} +32098.0 q^{67} +32768.0 q^{68} -17444.0 q^{69} -4248.00 q^{71} +3008.00 q^{72} +30104.0 q^{73} +17488.0 q^{74} +35520.0 q^{76} -23384.0 q^{77} -38304.0 q^{78} +35280.0 q^{79} -45419.0 q^{81} +9592.00 q^{82} -27826.0 q^{83} +35392.0 q^{84} -9176.00 q^{86} -3780.00 q^{87} +9472.00 q^{88} -85210.0 q^{89} +108072. q^{91} -19936.0 q^{92} -28672.0 q^{93} +42728.0 q^{94} -14336.0 q^{96} -97232.0 q^{97} -32628.0 q^{98} +6956.00 q^{99} +O(q^{100})\)

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
\(3\) 14.0000 0.898100 0.449050 0.893507i \(-0.351762\pi\)
0.449050 + 0.893507i \(0.351762\pi\)
\(4\) 16.0000 0.500000
\(5\) 0 0
\(6\) −56.0000 −0.635053
\(7\) 158.000 1.21874 0.609371 0.792885i \(-0.291422\pi\)
0.609371 + 0.792885i \(0.291422\pi\)
\(8\) −64.0000 −0.353553
\(9\) −47.0000 −0.193416
\(10\) 0 0
\(11\) −148.000 −0.368791 −0.184395 0.982852i \(-0.559033\pi\)
−0.184395 + 0.982852i \(0.559033\pi\)
\(12\) 224.000 0.449050
\(13\) 684.000 1.12253 0.561265 0.827636i \(-0.310315\pi\)
0.561265 + 0.827636i \(0.310315\pi\)
\(14\) −632.000 −0.861781
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) 2048.00 1.71873 0.859365 0.511363i \(-0.170859\pi\)
0.859365 + 0.511363i \(0.170859\pi\)
\(18\) 188.000 0.136766
\(19\) 2220.00 1.41081 0.705406 0.708804i \(-0.250765\pi\)
0.705406 + 0.708804i \(0.250765\pi\)
\(20\) 0 0
\(21\) 2212.00 1.09455
\(22\) 592.000 0.260774
\(23\) −1246.00 −0.491132 −0.245566 0.969380i \(-0.578974\pi\)
−0.245566 + 0.969380i \(0.578974\pi\)
\(24\) −896.000 −0.317526
\(25\) 0 0
\(26\) −2736.00 −0.793748
\(27\) −4060.00 −1.07181
\(28\) 2528.00 0.609371
\(29\) −270.000 −0.0596168 −0.0298084 0.999556i \(-0.509490\pi\)
−0.0298084 + 0.999556i \(0.509490\pi\)
\(30\) 0 0
\(31\) −2048.00 −0.382759 −0.191380 0.981516i \(-0.561296\pi\)
−0.191380 + 0.981516i \(0.561296\pi\)
\(32\) −1024.00 −0.176777
\(33\) −2072.00 −0.331211
\(34\) −8192.00 −1.21533
\(35\) 0 0
\(36\) −752.000 −0.0967078
\(37\) −4372.00 −0.525020 −0.262510 0.964929i \(-0.584550\pi\)
−0.262510 + 0.964929i \(0.584550\pi\)
\(38\) −8880.00 −0.997594
\(39\) 9576.00 1.00814
\(40\) 0 0
\(41\) −2398.00 −0.222787 −0.111393 0.993776i \(-0.535531\pi\)
−0.111393 + 0.993776i \(0.535531\pi\)
\(42\) −8848.00 −0.773966
\(43\) 2294.00 0.189200 0.0946002 0.995515i \(-0.469843\pi\)
0.0946002 + 0.995515i \(0.469843\pi\)
\(44\) −2368.00 −0.184395
\(45\) 0 0
\(46\) 4984.00 0.347283
\(47\) −10682.0 −0.705355 −0.352678 0.935745i \(-0.614729\pi\)
−0.352678 + 0.935745i \(0.614729\pi\)
\(48\) 3584.00 0.224525
\(49\) 8157.00 0.485333
\(50\) 0 0
\(51\) 28672.0 1.54359
\(52\) 10944.0 0.561265
\(53\) 2964.00 0.144940 0.0724700 0.997371i \(-0.476912\pi\)
0.0724700 + 0.997371i \(0.476912\pi\)
\(54\) 16240.0 0.757882
\(55\) 0 0
\(56\) −10112.0 −0.430891
\(57\) 31080.0 1.26705
\(58\) 1080.00 0.0421555
\(59\) −39740.0 −1.48627 −0.743135 0.669141i \(-0.766662\pi\)
−0.743135 + 0.669141i \(0.766662\pi\)
\(60\) 0 0
\(61\) −42298.0 −1.45544 −0.727722 0.685873i \(-0.759421\pi\)
−0.727722 + 0.685873i \(0.759421\pi\)
\(62\) 8192.00 0.270652
\(63\) −7426.00 −0.235724
\(64\) 4096.00 0.125000
\(65\) 0 0
\(66\) 8288.00 0.234202
\(67\) 32098.0 0.873556 0.436778 0.899569i \(-0.356119\pi\)
0.436778 + 0.899569i \(0.356119\pi\)
\(68\) 32768.0 0.859365
\(69\) −17444.0 −0.441086
\(70\) 0 0
\(71\) −4248.00 −0.100009 −0.0500044 0.998749i \(-0.515924\pi\)
−0.0500044 + 0.998749i \(0.515924\pi\)
\(72\) 3008.00 0.0683828
\(73\) 30104.0 0.661176 0.330588 0.943775i \(-0.392753\pi\)
0.330588 + 0.943775i \(0.392753\pi\)
\(74\) 17488.0 0.371245
\(75\) 0 0
\(76\) 35520.0 0.705406
\(77\) −23384.0 −0.449461
\(78\) −38304.0 −0.712866
\(79\) 35280.0 0.636005 0.318003 0.948090i \(-0.396988\pi\)
0.318003 + 0.948090i \(0.396988\pi\)
\(80\) 0 0
\(81\) −45419.0 −0.769175
\(82\) 9592.00 0.157534
\(83\) −27826.0 −0.443359 −0.221680 0.975120i \(-0.571154\pi\)
−0.221680 + 0.975120i \(0.571154\pi\)
\(84\) 35392.0 0.547277
\(85\) 0 0
\(86\) −9176.00 −0.133785
\(87\) −3780.00 −0.0535419
\(88\) 9472.00 0.130387
\(89\) −85210.0 −1.14029 −0.570145 0.821544i \(-0.693113\pi\)
−0.570145 + 0.821544i \(0.693113\pi\)
\(90\) 0 0
\(91\) 108072. 1.36807
\(92\) −19936.0 −0.245566
\(93\) −28672.0 −0.343756
\(94\) 42728.0 0.498762
\(95\) 0 0
\(96\) −14336.0 −0.158763
\(97\) −97232.0 −1.04925 −0.524626 0.851333i \(-0.675795\pi\)
−0.524626 + 0.851333i \(0.675795\pi\)
\(98\) −32628.0 −0.343183
\(99\) 6956.00 0.0713299
\(100\) 0 0
\(101\) −4298.00 −0.0419240 −0.0209620 0.999780i \(-0.506673\pi\)
−0.0209620 + 0.999780i \(0.506673\pi\)
\(102\) −114688. −1.09148
\(103\) 124114. 1.15273 0.576365 0.817192i \(-0.304471\pi\)
0.576365 + 0.817192i \(0.304471\pi\)
\(104\) −43776.0 −0.396874
\(105\) 0 0
\(106\) −11856.0 −0.102488
\(107\) −42342.0 −0.357530 −0.178765 0.983892i \(-0.557210\pi\)
−0.178765 + 0.983892i \(0.557210\pi\)
\(108\) −64960.0 −0.535904
\(109\) −35990.0 −0.290145 −0.145073 0.989421i \(-0.546342\pi\)
−0.145073 + 0.989421i \(0.546342\pi\)
\(110\) 0 0
\(111\) −61208.0 −0.471521
\(112\) 40448.0 0.304686
\(113\) −228816. −1.68574 −0.842869 0.538118i \(-0.819135\pi\)
−0.842869 + 0.538118i \(0.819135\pi\)
\(114\) −124320. −0.895940
\(115\) 0 0
\(116\) −4320.00 −0.0298084
\(117\) −32148.0 −0.217115
\(118\) 158960. 1.05095
\(119\) 323584. 2.09469
\(120\) 0 0
\(121\) −139147. −0.863993
\(122\) 169192. 1.02915
\(123\) −33572.0 −0.200085
\(124\) −32768.0 −0.191380
\(125\) 0 0
\(126\) 29704.0 0.166682
\(127\) 175238. 0.964093 0.482047 0.876146i \(-0.339894\pi\)
0.482047 + 0.876146i \(0.339894\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 32116.0 0.169921
\(130\) 0 0
\(131\) 299652. 1.52559 0.762797 0.646638i \(-0.223826\pi\)
0.762797 + 0.646638i \(0.223826\pi\)
\(132\) −33152.0 −0.165606
\(133\) 350760. 1.71942
\(134\) −128392. −0.617698
\(135\) 0 0
\(136\) −131072. −0.607663
\(137\) 107928. 0.491284 0.245642 0.969361i \(-0.421001\pi\)
0.245642 + 0.969361i \(0.421001\pi\)
\(138\) 69776.0 0.311895
\(139\) −196460. −0.862456 −0.431228 0.902243i \(-0.641920\pi\)
−0.431228 + 0.902243i \(0.641920\pi\)
\(140\) 0 0
\(141\) −149548. −0.633480
\(142\) 16992.0 0.0707170
\(143\) −101232. −0.413978
\(144\) −12032.0 −0.0483539
\(145\) 0 0
\(146\) −120416. −0.467522
\(147\) 114198. 0.435878
\(148\) −69952.0 −0.262510
\(149\) 138850. 0.512366 0.256183 0.966628i \(-0.417535\pi\)
0.256183 + 0.966628i \(0.417535\pi\)
\(150\) 0 0
\(151\) 416152. 1.48528 0.742642 0.669688i \(-0.233572\pi\)
0.742642 + 0.669688i \(0.233572\pi\)
\(152\) −142080. −0.498797
\(153\) −96256.0 −0.332429
\(154\) 93536.0 0.317817
\(155\) 0 0
\(156\) 153216. 0.504072
\(157\) 433108. 1.40232 0.701160 0.713004i \(-0.252666\pi\)
0.701160 + 0.713004i \(0.252666\pi\)
\(158\) −141120. −0.449724
\(159\) 41496.0 0.130171
\(160\) 0 0
\(161\) −196868. −0.598564
\(162\) 181676. 0.543889
\(163\) 149134. 0.439651 0.219825 0.975539i \(-0.429451\pi\)
0.219825 + 0.975539i \(0.429451\pi\)
\(164\) −38368.0 −0.111393
\(165\) 0 0
\(166\) 111304. 0.313502
\(167\) −559602. −1.55270 −0.776351 0.630301i \(-0.782932\pi\)
−0.776351 + 0.630301i \(0.782932\pi\)
\(168\) −141568. −0.386983
\(169\) 96563.0 0.260072
\(170\) 0 0
\(171\) −104340. −0.272873
\(172\) 36704.0 0.0946002
\(173\) 343804. 0.873365 0.436682 0.899616i \(-0.356153\pi\)
0.436682 + 0.899616i \(0.356153\pi\)
\(174\) 15120.0 0.0378598
\(175\) 0 0
\(176\) −37888.0 −0.0921977
\(177\) −556360. −1.33482
\(178\) 340840. 0.806307
\(179\) 23980.0 0.0559392 0.0279696 0.999609i \(-0.491096\pi\)
0.0279696 + 0.999609i \(0.491096\pi\)
\(180\) 0 0
\(181\) −651898. −1.47905 −0.739526 0.673128i \(-0.764950\pi\)
−0.739526 + 0.673128i \(0.764950\pi\)
\(182\) −432288. −0.967375
\(183\) −592172. −1.30713
\(184\) 79744.0 0.173641
\(185\) 0 0
\(186\) 114688. 0.243072
\(187\) −303104. −0.633852
\(188\) −170912. −0.352678
\(189\) −641480. −1.30626
\(190\) 0 0
\(191\) 202752. 0.402144 0.201072 0.979576i \(-0.435557\pi\)
0.201072 + 0.979576i \(0.435557\pi\)
\(192\) 57344.0 0.112263
\(193\) −452656. −0.874732 −0.437366 0.899284i \(-0.644089\pi\)
−0.437366 + 0.899284i \(0.644089\pi\)
\(194\) 388928. 0.741933
\(195\) 0 0
\(196\) 130512. 0.242667
\(197\) 337468. 0.619537 0.309768 0.950812i \(-0.399748\pi\)
0.309768 + 0.950812i \(0.399748\pi\)
\(198\) −27824.0 −0.0504379
\(199\) −561000. −1.00422 −0.502112 0.864803i \(-0.667443\pi\)
−0.502112 + 0.864803i \(0.667443\pi\)
\(200\) 0 0
\(201\) 449372. 0.784541
\(202\) 17192.0 0.0296448
\(203\) −42660.0 −0.0726576
\(204\) 458752. 0.771796
\(205\) 0 0
\(206\) −496456. −0.815103
\(207\) 58562.0 0.0949927
\(208\) 175104. 0.280632
\(209\) −328560. −0.520294
\(210\) 0 0
\(211\) −805548. −1.24562 −0.622810 0.782373i \(-0.714009\pi\)
−0.622810 + 0.782373i \(0.714009\pi\)
\(212\) 47424.0 0.0724700
\(213\) −59472.0 −0.0898180
\(214\) 169368. 0.252812
\(215\) 0 0
\(216\) 259840. 0.378941
\(217\) −323584. −0.466485
\(218\) 143960. 0.205164
\(219\) 421456. 0.593802
\(220\) 0 0
\(221\) 1.40083e6 1.92932
\(222\) 244832. 0.333415
\(223\) 1.21855e6 1.64090 0.820451 0.571717i \(-0.193722\pi\)
0.820451 + 0.571717i \(0.193722\pi\)
\(224\) −161792. −0.215445
\(225\) 0 0
\(226\) 915264. 1.19200
\(227\) 564338. 0.726900 0.363450 0.931614i \(-0.381599\pi\)
0.363450 + 0.931614i \(0.381599\pi\)
\(228\) 497280. 0.633525
\(229\) 560330. 0.706082 0.353041 0.935608i \(-0.385148\pi\)
0.353041 + 0.935608i \(0.385148\pi\)
\(230\) 0 0
\(231\) −327376. −0.403661
\(232\) 17280.0 0.0210777
\(233\) −293576. −0.354267 −0.177134 0.984187i \(-0.556682\pi\)
−0.177134 + 0.984187i \(0.556682\pi\)
\(234\) 128592. 0.153523
\(235\) 0 0
\(236\) −635840. −0.743135
\(237\) 493920. 0.571197
\(238\) −1.29434e6 −1.48117
\(239\) 584240. 0.661602 0.330801 0.943701i \(-0.392681\pi\)
0.330801 + 0.943701i \(0.392681\pi\)
\(240\) 0 0
\(241\) −563798. −0.625289 −0.312645 0.949870i \(-0.601215\pi\)
−0.312645 + 0.949870i \(0.601215\pi\)
\(242\) 556588. 0.610936
\(243\) 350714. 0.381011
\(244\) −676768. −0.727722
\(245\) 0 0
\(246\) 134288. 0.141481
\(247\) 1.51848e6 1.58368
\(248\) 131072. 0.135326
\(249\) −389564. −0.398181
\(250\) 0 0
\(251\) −1.01975e6 −1.02167 −0.510833 0.859680i \(-0.670663\pi\)
−0.510833 + 0.859680i \(0.670663\pi\)
\(252\) −118816. −0.117862
\(253\) 184408. 0.181125
\(254\) −700952. −0.681717
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 657408. 0.620872 0.310436 0.950594i \(-0.399525\pi\)
0.310436 + 0.950594i \(0.399525\pi\)
\(258\) −128464. −0.120152
\(259\) −690776. −0.639864
\(260\) 0 0
\(261\) 12690.0 0.0115308
\(262\) −1.19861e6 −1.07876
\(263\) −562366. −0.501337 −0.250668 0.968073i \(-0.580650\pi\)
−0.250668 + 0.968073i \(0.580650\pi\)
\(264\) 132608. 0.117101
\(265\) 0 0
\(266\) −1.40304e6 −1.21581
\(267\) −1.19294e6 −1.02410
\(268\) 513568. 0.436778
\(269\) 366570. 0.308870 0.154435 0.988003i \(-0.450644\pi\)
0.154435 + 0.988003i \(0.450644\pi\)
\(270\) 0 0
\(271\) 1.16075e6 0.960099 0.480050 0.877241i \(-0.340619\pi\)
0.480050 + 0.877241i \(0.340619\pi\)
\(272\) 524288. 0.429682
\(273\) 1.51301e6 1.22867
\(274\) −431712. −0.347390
\(275\) 0 0
\(276\) −279104. −0.220543
\(277\) −2.51501e6 −1.96943 −0.984715 0.174172i \(-0.944275\pi\)
−0.984715 + 0.174172i \(0.944275\pi\)
\(278\) 785840. 0.609849
\(279\) 96256.0 0.0740316
\(280\) 0 0
\(281\) 2.08600e6 1.57597 0.787987 0.615692i \(-0.211124\pi\)
0.787987 + 0.615692i \(0.211124\pi\)
\(282\) 598192. 0.447938
\(283\) −2.23803e6 −1.66111 −0.830556 0.556935i \(-0.811977\pi\)
−0.830556 + 0.556935i \(0.811977\pi\)
\(284\) −67968.0 −0.0500044
\(285\) 0 0
\(286\) 404928. 0.292727
\(287\) −378884. −0.271520
\(288\) 48128.0 0.0341914
\(289\) 2.77445e6 1.95403
\(290\) 0 0
\(291\) −1.36125e6 −0.942334
\(292\) 481664. 0.330588
\(293\) −975756. −0.664006 −0.332003 0.943278i \(-0.607724\pi\)
−0.332003 + 0.943278i \(0.607724\pi\)
\(294\) −456792. −0.308212
\(295\) 0 0
\(296\) 279808. 0.185623
\(297\) 600880. 0.395273
\(298\) −555400. −0.362297
\(299\) −852264. −0.551310
\(300\) 0 0
\(301\) 362452. 0.230587
\(302\) −1.66461e6 −1.05025
\(303\) −60172.0 −0.0376520
\(304\) 568320. 0.352703
\(305\) 0 0
\(306\) 385024. 0.235063
\(307\) 87858.0 0.0532029 0.0266015 0.999646i \(-0.491531\pi\)
0.0266015 + 0.999646i \(0.491531\pi\)
\(308\) −374144. −0.224730
\(309\) 1.73760e6 1.03527
\(310\) 0 0
\(311\) 599352. 0.351383 0.175692 0.984445i \(-0.443784\pi\)
0.175692 + 0.984445i \(0.443784\pi\)
\(312\) −612864. −0.356433
\(313\) −2.09342e6 −1.20780 −0.603900 0.797060i \(-0.706387\pi\)
−0.603900 + 0.797060i \(0.706387\pi\)
\(314\) −1.73243e6 −0.991590
\(315\) 0 0
\(316\) 564480. 0.318003
\(317\) −2.41625e6 −1.35050 −0.675249 0.737590i \(-0.735964\pi\)
−0.675249 + 0.737590i \(0.735964\pi\)
\(318\) −165984. −0.0920446
\(319\) 39960.0 0.0219861
\(320\) 0 0
\(321\) −592788. −0.321097
\(322\) 787472. 0.423249
\(323\) 4.54656e6 2.42480
\(324\) −726704. −0.384587
\(325\) 0 0
\(326\) −596536. −0.310880
\(327\) −503860. −0.260580
\(328\) 153472. 0.0787670
\(329\) −1.68776e6 −0.859647
\(330\) 0 0
\(331\) −1.64095e6 −0.823237 −0.411618 0.911356i \(-0.635036\pi\)
−0.411618 + 0.911356i \(0.635036\pi\)
\(332\) −445216. −0.221680
\(333\) 205484. 0.101547
\(334\) 2.23841e6 1.09793
\(335\) 0 0
\(336\) 566272. 0.273638
\(337\) 2.18773e6 1.04935 0.524673 0.851304i \(-0.324188\pi\)
0.524673 + 0.851304i \(0.324188\pi\)
\(338\) −386252. −0.183899
\(339\) −3.20342e6 −1.51396
\(340\) 0 0
\(341\) 303104. 0.141158
\(342\) 417360. 0.192950
\(343\) −1.36670e6 −0.627246
\(344\) −146816. −0.0668925
\(345\) 0 0
\(346\) −1.37522e6 −0.617562
\(347\) 2.74502e6 1.22383 0.611916 0.790923i \(-0.290399\pi\)
0.611916 + 0.790923i \(0.290399\pi\)
\(348\) −60480.0 −0.0267709
\(349\) −2.65115e6 −1.16512 −0.582560 0.812788i \(-0.697949\pi\)
−0.582560 + 0.812788i \(0.697949\pi\)
\(350\) 0 0
\(351\) −2.77704e6 −1.20313
\(352\) 151552. 0.0651936
\(353\) 3.05766e6 1.30603 0.653015 0.757345i \(-0.273504\pi\)
0.653015 + 0.757345i \(0.273504\pi\)
\(354\) 2.22544e6 0.943860
\(355\) 0 0
\(356\) −1.36336e6 −0.570145
\(357\) 4.53018e6 1.88124
\(358\) −95920.0 −0.0395550
\(359\) 3.79356e6 1.55350 0.776749 0.629810i \(-0.216867\pi\)
0.776749 + 0.629810i \(0.216867\pi\)
\(360\) 0 0
\(361\) 2.45230e6 0.990389
\(362\) 2.60759e6 1.04585
\(363\) −1.94806e6 −0.775953
\(364\) 1.72915e6 0.684037
\(365\) 0 0
\(366\) 2.36869e6 0.924283
\(367\) −3.11060e6 −1.20553 −0.602767 0.797917i \(-0.705935\pi\)
−0.602767 + 0.797917i \(0.705935\pi\)
\(368\) −318976. −0.122783
\(369\) 112706. 0.0430905
\(370\) 0 0
\(371\) 468312. 0.176645
\(372\) −458752. −0.171878
\(373\) −1.41520e6 −0.526677 −0.263339 0.964703i \(-0.584824\pi\)
−0.263339 + 0.964703i \(0.584824\pi\)
\(374\) 1.21242e6 0.448201
\(375\) 0 0
\(376\) 683648. 0.249381
\(377\) −184680. −0.0669216
\(378\) 2.56592e6 0.923663
\(379\) −3.90262e6 −1.39559 −0.697796 0.716297i \(-0.745836\pi\)
−0.697796 + 0.716297i \(0.745836\pi\)
\(380\) 0 0
\(381\) 2.45333e6 0.865852
\(382\) −811008. −0.284359
\(383\) 695674. 0.242331 0.121165 0.992632i \(-0.461337\pi\)
0.121165 + 0.992632i \(0.461337\pi\)
\(384\) −229376. −0.0793816
\(385\) 0 0
\(386\) 1.81062e6 0.618529
\(387\) −107818. −0.0365943
\(388\) −1.55571e6 −0.524626
\(389\) 498290. 0.166958 0.0834792 0.996510i \(-0.473397\pi\)
0.0834792 + 0.996510i \(0.473397\pi\)
\(390\) 0 0
\(391\) −2.55181e6 −0.844124
\(392\) −522048. −0.171591
\(393\) 4.19513e6 1.37014
\(394\) −1.34987e6 −0.438079
\(395\) 0 0
\(396\) 111296. 0.0356649
\(397\) 1.09567e6 0.348901 0.174451 0.984666i \(-0.444185\pi\)
0.174451 + 0.984666i \(0.444185\pi\)
\(398\) 2.24400e6 0.710093
\(399\) 4.91064e6 1.54421
\(400\) 0 0
\(401\) −2.49160e6 −0.773779 −0.386890 0.922126i \(-0.626451\pi\)
−0.386890 + 0.922126i \(0.626451\pi\)
\(402\) −1.79749e6 −0.554755
\(403\) −1.40083e6 −0.429659
\(404\) −68768.0 −0.0209620
\(405\) 0 0
\(406\) 170640. 0.0513766
\(407\) 647056. 0.193623
\(408\) −1.83501e6 −0.545742
\(409\) −3.63349e6 −1.07403 −0.537014 0.843573i \(-0.680448\pi\)
−0.537014 + 0.843573i \(0.680448\pi\)
\(410\) 0 0
\(411\) 1.51099e6 0.441222
\(412\) 1.98582e6 0.576365
\(413\) −6.27892e6 −1.81138
\(414\) −234248. −0.0671700
\(415\) 0 0
\(416\) −700416. −0.198437
\(417\) −2.75044e6 −0.774572
\(418\) 1.31424e6 0.367904
\(419\) −3.64378e6 −1.01395 −0.506976 0.861960i \(-0.669237\pi\)
−0.506976 + 0.861960i \(0.669237\pi\)
\(420\) 0 0
\(421\) −1.82530e6 −0.501913 −0.250957 0.967998i \(-0.580745\pi\)
−0.250957 + 0.967998i \(0.580745\pi\)
\(422\) 3.22219e6 0.880786
\(423\) 502054. 0.136427
\(424\) −189696. −0.0512441
\(425\) 0 0
\(426\) 237888. 0.0635109
\(427\) −6.68308e6 −1.77381
\(428\) −677472. −0.178765
\(429\) −1.41725e6 −0.371794
\(430\) 0 0
\(431\) 2.85435e6 0.740141 0.370070 0.929004i \(-0.379334\pi\)
0.370070 + 0.929004i \(0.379334\pi\)
\(432\) −1.03936e6 −0.267952
\(433\) −587776. −0.150658 −0.0753290 0.997159i \(-0.524001\pi\)
−0.0753290 + 0.997159i \(0.524001\pi\)
\(434\) 1.29434e6 0.329855
\(435\) 0 0
\(436\) −575840. −0.145073
\(437\) −2.76612e6 −0.692895
\(438\) −1.68582e6 −0.419882
\(439\) 6.11604e6 1.51464 0.757319 0.653045i \(-0.226509\pi\)
0.757319 + 0.653045i \(0.226509\pi\)
\(440\) 0 0
\(441\) −383379. −0.0938711
\(442\) −5.60333e6 −1.36424
\(443\) −2.35771e6 −0.570795 −0.285398 0.958409i \(-0.592126\pi\)
−0.285398 + 0.958409i \(0.592126\pi\)
\(444\) −979328. −0.235760
\(445\) 0 0
\(446\) −4.87422e6 −1.16029
\(447\) 1.94390e6 0.460156
\(448\) 647168. 0.152343
\(449\) 5.49735e6 1.28688 0.643439 0.765497i \(-0.277507\pi\)
0.643439 + 0.765497i \(0.277507\pi\)
\(450\) 0 0
\(451\) 354904. 0.0821617
\(452\) −3.66106e6 −0.842869
\(453\) 5.82613e6 1.33393
\(454\) −2.25735e6 −0.513996
\(455\) 0 0
\(456\) −1.98912e6 −0.447970
\(457\) −1.16039e6 −0.259905 −0.129952 0.991520i \(-0.541482\pi\)
−0.129952 + 0.991520i \(0.541482\pi\)
\(458\) −2.24132e6 −0.499275
\(459\) −8.31488e6 −1.84215
\(460\) 0 0
\(461\) −2.30330e6 −0.504775 −0.252387 0.967626i \(-0.581216\pi\)
−0.252387 + 0.967626i \(0.581216\pi\)
\(462\) 1.30950e6 0.285431
\(463\) 2.71343e6 0.588257 0.294128 0.955766i \(-0.404971\pi\)
0.294128 + 0.955766i \(0.404971\pi\)
\(464\) −69120.0 −0.0149042
\(465\) 0 0
\(466\) 1.17430e6 0.250505
\(467\) 4.05050e6 0.859441 0.429721 0.902962i \(-0.358612\pi\)
0.429721 + 0.902962i \(0.358612\pi\)
\(468\) −514368. −0.108557
\(469\) 5.07148e6 1.06464
\(470\) 0 0
\(471\) 6.06351e6 1.25942
\(472\) 2.54336e6 0.525476
\(473\) −339512. −0.0697754
\(474\) −1.97568e6 −0.403897
\(475\) 0 0
\(476\) 5.17734e6 1.04734
\(477\) −139308. −0.0280337
\(478\) −2.33696e6 −0.467823
\(479\) 5.60528e6 1.11624 0.558121 0.829759i \(-0.311522\pi\)
0.558121 + 0.829759i \(0.311522\pi\)
\(480\) 0 0
\(481\) −2.99045e6 −0.589350
\(482\) 2.25519e6 0.442146
\(483\) −2.75615e6 −0.537570
\(484\) −2.22635e6 −0.431997
\(485\) 0 0
\(486\) −1.40286e6 −0.269415
\(487\) 7.13168e6 1.36260 0.681301 0.732003i \(-0.261414\pi\)
0.681301 + 0.732003i \(0.261414\pi\)
\(488\) 2.70707e6 0.514577
\(489\) 2.08788e6 0.394850
\(490\) 0 0
\(491\) 5.88145e6 1.10098 0.550492 0.834841i \(-0.314440\pi\)
0.550492 + 0.834841i \(0.314440\pi\)
\(492\) −537152. −0.100042
\(493\) −552960. −0.102465
\(494\) −6.07392e6 −1.11983
\(495\) 0 0
\(496\) −524288. −0.0956898
\(497\) −671184. −0.121885
\(498\) 1.55826e6 0.281556
\(499\) 1.75710e6 0.315897 0.157948 0.987447i \(-0.449512\pi\)
0.157948 + 0.987447i \(0.449512\pi\)
\(500\) 0 0
\(501\) −7.83443e6 −1.39448
\(502\) 4.07899e6 0.722426
\(503\) 4.91411e6 0.866015 0.433007 0.901390i \(-0.357452\pi\)
0.433007 + 0.901390i \(0.357452\pi\)
\(504\) 475264. 0.0833410
\(505\) 0 0
\(506\) −737632. −0.128075
\(507\) 1.35188e6 0.233571
\(508\) 2.80381e6 0.482047
\(509\) −5.75499e6 −0.984578 −0.492289 0.870432i \(-0.663840\pi\)
−0.492289 + 0.870432i \(0.663840\pi\)
\(510\) 0 0
\(511\) 4.75643e6 0.805803
\(512\) −262144. −0.0441942
\(513\) −9.01320e6 −1.51212
\(514\) −2.62963e6 −0.439023
\(515\) 0 0
\(516\) 513856. 0.0849605
\(517\) 1.58094e6 0.260128
\(518\) 2.76310e6 0.452452
\(519\) 4.81326e6 0.784369
\(520\) 0 0
\(521\) −1.61980e6 −0.261437 −0.130718 0.991420i \(-0.541728\pi\)
−0.130718 + 0.991420i \(0.541728\pi\)
\(522\) −50760.0 −0.00815352
\(523\) 1.19117e7 1.90422 0.952112 0.305751i \(-0.0989074\pi\)
0.952112 + 0.305751i \(0.0989074\pi\)
\(524\) 4.79443e6 0.762797
\(525\) 0 0
\(526\) 2.24946e6 0.354499
\(527\) −4.19430e6 −0.657860
\(528\) −530432. −0.0828028
\(529\) −4.88383e6 −0.758789
\(530\) 0 0
\(531\) 1.86778e6 0.287468
\(532\) 5.61216e6 0.859708
\(533\) −1.64023e6 −0.250085
\(534\) 4.77176e6 0.724145
\(535\) 0 0
\(536\) −2.05427e6 −0.308849
\(537\) 335720. 0.0502391
\(538\) −1.46628e6 −0.218404
\(539\) −1.20724e6 −0.178986
\(540\) 0 0
\(541\) 4.07630e6 0.598788 0.299394 0.954130i \(-0.403215\pi\)
0.299394 + 0.954130i \(0.403215\pi\)
\(542\) −4.64301e6 −0.678893
\(543\) −9.12657e6 −1.32834
\(544\) −2.09715e6 −0.303831
\(545\) 0 0
\(546\) −6.05203e6 −0.868800
\(547\) 1.23680e7 1.76739 0.883694 0.468065i \(-0.155049\pi\)
0.883694 + 0.468065i \(0.155049\pi\)
\(548\) 1.72685e6 0.245642
\(549\) 1.98801e6 0.281505
\(550\) 0 0
\(551\) −599400. −0.0841081
\(552\) 1.11642e6 0.155947
\(553\) 5.57424e6 0.775127
\(554\) 1.00600e7 1.39260
\(555\) 0 0
\(556\) −3.14336e6 −0.431228
\(557\) 130308. 0.0177964 0.00889822 0.999960i \(-0.497168\pi\)
0.00889822 + 0.999960i \(0.497168\pi\)
\(558\) −385024. −0.0523483
\(559\) 1.56910e6 0.212383
\(560\) 0 0
\(561\) −4.24346e6 −0.569262
\(562\) −8.34401e6 −1.11438
\(563\) −5.91687e6 −0.786721 −0.393361 0.919384i \(-0.628688\pi\)
−0.393361 + 0.919384i \(0.628688\pi\)
\(564\) −2.39277e6 −0.316740
\(565\) 0 0
\(566\) 8.95210e6 1.17458
\(567\) −7.17620e6 −0.937426
\(568\) 271872. 0.0353585
\(569\) −9.03013e6 −1.16927 −0.584633 0.811298i \(-0.698761\pi\)
−0.584633 + 0.811298i \(0.698761\pi\)
\(570\) 0 0
\(571\) −1.07093e7 −1.37459 −0.687294 0.726379i \(-0.741202\pi\)
−0.687294 + 0.726379i \(0.741202\pi\)
\(572\) −1.61971e6 −0.206989
\(573\) 2.83853e6 0.361166
\(574\) 1.51554e6 0.191994
\(575\) 0 0
\(576\) −192512. −0.0241770
\(577\) −1.22051e6 −0.152617 −0.0763084 0.997084i \(-0.524313\pi\)
−0.0763084 + 0.997084i \(0.524313\pi\)
\(578\) −1.10978e7 −1.38171
\(579\) −6.33718e6 −0.785597
\(580\) 0 0
\(581\) −4.39651e6 −0.540341
\(582\) 5.44499e6 0.666331
\(583\) −438672. −0.0534526
\(584\) −1.92666e6 −0.233761
\(585\) 0 0
\(586\) 3.90302e6 0.469523
\(587\) −1.47104e7 −1.76210 −0.881049 0.473026i \(-0.843162\pi\)
−0.881049 + 0.473026i \(0.843162\pi\)
\(588\) 1.82717e6 0.217939
\(589\) −4.54656e6 −0.540001
\(590\) 0 0
\(591\) 4.72455e6 0.556406
\(592\) −1.11923e6 −0.131255
\(593\) 8.52014e6 0.994970 0.497485 0.867472i \(-0.334257\pi\)
0.497485 + 0.867472i \(0.334257\pi\)
\(594\) −2.40352e6 −0.279500
\(595\) 0 0
\(596\) 2.22160e6 0.256183
\(597\) −7.85400e6 −0.901893
\(598\) 3.40906e6 0.389835
\(599\) 2.90100e6 0.330355 0.165177 0.986264i \(-0.447180\pi\)
0.165177 + 0.986264i \(0.447180\pi\)
\(600\) 0 0
\(601\) 5.72760e6 0.646825 0.323412 0.946258i \(-0.395170\pi\)
0.323412 + 0.946258i \(0.395170\pi\)
\(602\) −1.44981e6 −0.163049
\(603\) −1.50861e6 −0.168959
\(604\) 6.65843e6 0.742642
\(605\) 0 0
\(606\) 240688. 0.0266240
\(607\) −8.79924e6 −0.969334 −0.484667 0.874699i \(-0.661059\pi\)
−0.484667 + 0.874699i \(0.661059\pi\)
\(608\) −2.27328e6 −0.249399
\(609\) −597240. −0.0652538
\(610\) 0 0
\(611\) −7.30649e6 −0.791782
\(612\) −1.54010e6 −0.166215
\(613\) 1.03408e6 0.111149 0.0555744 0.998455i \(-0.482301\pi\)
0.0555744 + 0.998455i \(0.482301\pi\)
\(614\) −351432. −0.0376201
\(615\) 0 0
\(616\) 1.49658e6 0.158908
\(617\) 1.29854e7 1.37323 0.686616 0.727020i \(-0.259095\pi\)
0.686616 + 0.727020i \(0.259095\pi\)
\(618\) −6.95038e6 −0.732045
\(619\) 7.92002e6 0.830806 0.415403 0.909637i \(-0.363641\pi\)
0.415403 + 0.909637i \(0.363641\pi\)
\(620\) 0 0
\(621\) 5.05876e6 0.526399
\(622\) −2.39741e6 −0.248465
\(623\) −1.34632e7 −1.38972
\(624\) 2.45146e6 0.252036
\(625\) 0 0
\(626\) 8.37366e6 0.854043
\(627\) −4.59984e6 −0.467276
\(628\) 6.92973e6 0.701160
\(629\) −8.95386e6 −0.902368
\(630\) 0 0
\(631\) 1.68218e7 1.68189 0.840945 0.541120i \(-0.181999\pi\)
0.840945 + 0.541120i \(0.181999\pi\)
\(632\) −2.25792e6 −0.224862
\(633\) −1.12777e7 −1.11869
\(634\) 9.66501e6 0.954947
\(635\) 0 0
\(636\) 663936. 0.0650854
\(637\) 5.57939e6 0.544801
\(638\) −159840. −0.0155465
\(639\) 199656. 0.0193433
\(640\) 0 0
\(641\) −1.55154e7 −1.49148 −0.745741 0.666236i \(-0.767904\pi\)
−0.745741 + 0.666236i \(0.767904\pi\)
\(642\) 2.37115e6 0.227050
\(643\) 1.05801e7 1.00916 0.504582 0.863364i \(-0.331646\pi\)
0.504582 + 0.863364i \(0.331646\pi\)
\(644\) −3.14989e6 −0.299282
\(645\) 0 0
\(646\) −1.81862e7 −1.71460
\(647\) 1.37883e7 1.29494 0.647471 0.762090i \(-0.275827\pi\)
0.647471 + 0.762090i \(0.275827\pi\)
\(648\) 2.90682e6 0.271944
\(649\) 5.88152e6 0.548123
\(650\) 0 0
\(651\) −4.53018e6 −0.418950
\(652\) 2.38614e6 0.219825
\(653\) −1.58924e6 −0.145850 −0.0729248 0.997337i \(-0.523233\pi\)
−0.0729248 + 0.997337i \(0.523233\pi\)
\(654\) 2.01544e6 0.184258
\(655\) 0 0
\(656\) −613888. −0.0556967
\(657\) −1.41489e6 −0.127882
\(658\) 6.75102e6 0.607862
\(659\) −9.12434e6 −0.818442 −0.409221 0.912435i \(-0.634199\pi\)
−0.409221 + 0.912435i \(0.634199\pi\)
\(660\) 0 0
\(661\) 6.50310e6 0.578918 0.289459 0.957190i \(-0.406525\pi\)
0.289459 + 0.957190i \(0.406525\pi\)
\(662\) 6.56379e6 0.582116
\(663\) 1.96116e7 1.73273
\(664\) 1.78086e6 0.156751
\(665\) 0 0
\(666\) −821936. −0.0718046
\(667\) 336420. 0.0292797
\(668\) −8.95363e6 −0.776351
\(669\) 1.70598e7 1.47369
\(670\) 0 0
\(671\) 6.26010e6 0.536754
\(672\) −2.26509e6 −0.193492
\(673\) −2.17810e6 −0.185370 −0.0926850 0.995695i \(-0.529545\pi\)
−0.0926850 + 0.995695i \(0.529545\pi\)
\(674\) −8.75091e6 −0.741999
\(675\) 0 0
\(676\) 1.54501e6 0.130036
\(677\) 3.98419e6 0.334094 0.167047 0.985949i \(-0.446577\pi\)
0.167047 + 0.985949i \(0.446577\pi\)
\(678\) 1.28137e7 1.07053
\(679\) −1.53627e7 −1.27877
\(680\) 0 0
\(681\) 7.90073e6 0.652829
\(682\) −1.21242e6 −0.0998138
\(683\) −5.91563e6 −0.485231 −0.242616 0.970122i \(-0.578005\pi\)
−0.242616 + 0.970122i \(0.578005\pi\)
\(684\) −1.66944e6 −0.136436
\(685\) 0 0
\(686\) 5.46680e6 0.443530
\(687\) 7.84462e6 0.634133
\(688\) 587264. 0.0473001
\(689\) 2.02738e6 0.162700
\(690\) 0 0
\(691\) −1.55471e7 −1.23867 −0.619335 0.785127i \(-0.712598\pi\)
−0.619335 + 0.785127i \(0.712598\pi\)
\(692\) 5.50086e6 0.436682
\(693\) 1.09905e6 0.0869328
\(694\) −1.09801e7 −0.865379
\(695\) 0 0
\(696\) 241920. 0.0189299
\(697\) −4.91110e6 −0.382910
\(698\) 1.06046e7 0.823864
\(699\) −4.11006e6 −0.318167
\(700\) 0 0
\(701\) −2.27103e7 −1.74553 −0.872766 0.488139i \(-0.837676\pi\)
−0.872766 + 0.488139i \(0.837676\pi\)
\(702\) 1.11082e7 0.850745
\(703\) −9.70584e6 −0.740704
\(704\) −606208. −0.0460988
\(705\) 0 0
\(706\) −1.22307e7 −0.923502
\(707\) −679084. −0.0510946
\(708\) −8.90176e6 −0.667410
\(709\) 6.29841e6 0.470560 0.235280 0.971928i \(-0.424399\pi\)
0.235280 + 0.971928i \(0.424399\pi\)
\(710\) 0 0
\(711\) −1.65816e6 −0.123013
\(712\) 5.45344e6 0.403154
\(713\) 2.55181e6 0.187985
\(714\) −1.81207e7 −1.33024
\(715\) 0 0
\(716\) 383680. 0.0279696
\(717\) 8.17936e6 0.594185
\(718\) −1.51742e7 −1.09849
\(719\) 2.11911e7 1.52873 0.764367 0.644782i \(-0.223052\pi\)
0.764367 + 0.644782i \(0.223052\pi\)
\(720\) 0 0
\(721\) 1.96100e7 1.40488
\(722\) −9.80920e6 −0.700311
\(723\) −7.89317e6 −0.561572
\(724\) −1.04304e7 −0.739526
\(725\) 0 0
\(726\) 7.79223e6 0.548682
\(727\) 1.35610e7 0.951605 0.475803 0.879552i \(-0.342158\pi\)
0.475803 + 0.879552i \(0.342158\pi\)
\(728\) −6.91661e6 −0.483687
\(729\) 1.59468e7 1.11136
\(730\) 0 0
\(731\) 4.69811e6 0.325185
\(732\) −9.47475e6 −0.653567
\(733\) 2.69413e7 1.85208 0.926038 0.377429i \(-0.123192\pi\)
0.926038 + 0.377429i \(0.123192\pi\)
\(734\) 1.24424e7 0.852441
\(735\) 0 0
\(736\) 1.27590e6 0.0868207
\(737\) −4.75050e6 −0.322160
\(738\) −450824. −0.0304696
\(739\) 2.77414e6 0.186860 0.0934302 0.995626i \(-0.470217\pi\)
0.0934302 + 0.995626i \(0.470217\pi\)
\(740\) 0 0
\(741\) 2.12587e7 1.42230
\(742\) −1.87325e6 −0.124907
\(743\) 1.85538e7 1.23299 0.616497 0.787358i \(-0.288551\pi\)
0.616497 + 0.787358i \(0.288551\pi\)
\(744\) 1.83501e6 0.121536
\(745\) 0 0
\(746\) 5.66078e6 0.372417
\(747\) 1.30782e6 0.0857526
\(748\) −4.84966e6 −0.316926
\(749\) −6.69004e6 −0.435736
\(750\) 0 0
\(751\) −2.19285e6 −0.141876 −0.0709380 0.997481i \(-0.522599\pi\)
−0.0709380 + 0.997481i \(0.522599\pi\)
\(752\) −2.73459e6 −0.176339
\(753\) −1.42765e7 −0.917558
\(754\) 738720. 0.0473207
\(755\) 0 0
\(756\) −1.02637e7 −0.653128
\(757\) −9.48749e6 −0.601744 −0.300872 0.953665i \(-0.597278\pi\)
−0.300872 + 0.953665i \(0.597278\pi\)
\(758\) 1.56105e7 0.986832
\(759\) 2.58171e6 0.162668
\(760\) 0 0
\(761\) 9.69580e6 0.606907 0.303453 0.952846i \(-0.401860\pi\)
0.303453 + 0.952846i \(0.401860\pi\)
\(762\) −9.81333e6 −0.612250
\(763\) −5.68642e6 −0.353612
\(764\) 3.24403e6 0.201072
\(765\) 0 0
\(766\) −2.78270e6 −0.171354
\(767\) −2.71822e7 −1.66838
\(768\) 917504. 0.0561313
\(769\) 9.32787e6 0.568809 0.284405 0.958704i \(-0.408204\pi\)
0.284405 + 0.958704i \(0.408204\pi\)
\(770\) 0 0
\(771\) 9.20371e6 0.557606
\(772\) −7.24250e6 −0.437366
\(773\) −9.68080e6 −0.582723 −0.291362 0.956613i \(-0.594108\pi\)
−0.291362 + 0.956613i \(0.594108\pi\)
\(774\) 431272. 0.0258761
\(775\) 0 0
\(776\) 6.22285e6 0.370967
\(777\) −9.67086e6 −0.574662
\(778\) −1.99316e6 −0.118057
\(779\) −5.32356e6 −0.314310
\(780\) 0 0
\(781\) 628704. 0.0368824
\(782\) 1.02072e7 0.596886
\(783\) 1.09620e6 0.0638977
\(784\) 2.08819e6 0.121333
\(785\) 0 0
\(786\) −1.67805e7 −0.968833
\(787\) −5.52302e6 −0.317863 −0.158931 0.987290i \(-0.550805\pi\)
−0.158931 + 0.987290i \(0.550805\pi\)
\(788\) 5.39949e6 0.309768
\(789\) −7.87312e6 −0.450251
\(790\) 0 0
\(791\) −3.61529e7 −2.05448
\(792\) −445184. −0.0252189
\(793\) −2.89318e7 −1.63378
\(794\) −4.38267e6 −0.246711
\(795\) 0 0
\(796\) −8.97600e6 −0.502112
\(797\) −1.71119e7 −0.954230 −0.477115 0.878841i \(-0.658318\pi\)
−0.477115 + 0.878841i \(0.658318\pi\)
\(798\) −1.96426e7 −1.09192
\(799\) −2.18767e7 −1.21232
\(800\) 0 0
\(801\) 4.00487e6 0.220550
\(802\) 9.96639e6 0.547145
\(803\) −4.45539e6 −0.243836
\(804\) 7.18995e6 0.392271
\(805\) 0 0
\(806\) 5.60333e6 0.303814
\(807\) 5.13198e6 0.277397
\(808\) 275072. 0.0148224
\(809\) −1.45309e7 −0.780586 −0.390293 0.920691i \(-0.627626\pi\)
−0.390293 + 0.920691i \(0.627626\pi\)
\(810\) 0 0
\(811\) −2.13545e7 −1.14009 −0.570044 0.821614i \(-0.693074\pi\)
−0.570044 + 0.821614i \(0.693074\pi\)
\(812\) −682560. −0.0363288
\(813\) 1.62505e7 0.862266
\(814\) −2.58822e6 −0.136912
\(815\) 0 0
\(816\) 7.34003e6 0.385898
\(817\) 5.09268e6 0.266926
\(818\) 1.45340e7 0.759453
\(819\) −5.07938e6 −0.264607
\(820\) 0 0
\(821\) 3.67967e7 1.90525 0.952623 0.304154i \(-0.0983737\pi\)
0.952623 + 0.304154i \(0.0983737\pi\)
\(822\) −6.04397e6 −0.311991
\(823\) −3.30668e7 −1.70174 −0.850870 0.525376i \(-0.823925\pi\)
−0.850870 + 0.525376i \(0.823925\pi\)
\(824\) −7.94330e6 −0.407552
\(825\) 0 0
\(826\) 2.51157e7 1.28084
\(827\) 1.77309e7 0.901505 0.450752 0.892649i \(-0.351156\pi\)
0.450752 + 0.892649i \(0.351156\pi\)
\(828\) 936992. 0.0474963
\(829\) 1.29375e7 0.653830 0.326915 0.945054i \(-0.393991\pi\)
0.326915 + 0.945054i \(0.393991\pi\)
\(830\) 0 0
\(831\) −3.52102e7 −1.76875
\(832\) 2.80166e6 0.140316
\(833\) 1.67055e7 0.834157
\(834\) 1.10018e7 0.547705
\(835\) 0 0
\(836\) −5.25696e6 −0.260147
\(837\) 8.31488e6 0.410244
\(838\) 1.45751e7 0.716972
\(839\) 3.31812e7 1.62738 0.813688 0.581302i \(-0.197457\pi\)
0.813688 + 0.581302i \(0.197457\pi\)
\(840\) 0 0
\(841\) −2.04382e7 −0.996446
\(842\) 7.30119e6 0.354906
\(843\) 2.92040e7 1.41538
\(844\) −1.28888e7 −0.622810
\(845\) 0 0
\(846\) −2.00822e6 −0.0964683
\(847\) −2.19852e7 −1.05299
\(848\) 758784. 0.0362350
\(849\) −3.13324e7 −1.49185
\(850\) 0 0
\(851\) 5.44751e6 0.257854
\(852\) −951552. −0.0449090
\(853\) −5.17224e6 −0.243392 −0.121696 0.992567i \(-0.538833\pi\)
−0.121696 + 0.992567i \(0.538833\pi\)
\(854\) 2.67323e7 1.25427
\(855\) 0 0
\(856\) 2.70989e6 0.126406
\(857\) −1.05320e7 −0.489845 −0.244922 0.969543i \(-0.578762\pi\)
−0.244922 + 0.969543i \(0.578762\pi\)
\(858\) 5.66899e6 0.262898
\(859\) −1.14741e7 −0.530563 −0.265282 0.964171i \(-0.585465\pi\)
−0.265282 + 0.964171i \(0.585465\pi\)
\(860\) 0 0
\(861\) −5.30438e6 −0.243852
\(862\) −1.14174e7 −0.523359
\(863\) 1.92722e7 0.880856 0.440428 0.897788i \(-0.354827\pi\)
0.440428 + 0.897788i \(0.354827\pi\)
\(864\) 4.15744e6 0.189471
\(865\) 0 0
\(866\) 2.35110e6 0.106531
\(867\) 3.88423e7 1.75492
\(868\) −5.17734e6 −0.233243
\(869\) −5.22144e6 −0.234553
\(870\) 0 0
\(871\) 2.19550e7 0.980593
\(872\) 2.30336e6 0.102582
\(873\) 4.56990e6 0.202942
\(874\) 1.10645e7 0.489951
\(875\) 0 0
\(876\) 6.74330e6 0.296901
\(877\) 2.30524e7 1.01208 0.506042 0.862509i \(-0.331108\pi\)
0.506042 + 0.862509i \(0.331108\pi\)
\(878\) −2.44642e7 −1.07101
\(879\) −1.36606e7 −0.596344
\(880\) 0 0
\(881\) 2.26690e7 0.983994 0.491997 0.870597i \(-0.336267\pi\)
0.491997 + 0.870597i \(0.336267\pi\)
\(882\) 1.53352e6 0.0663769
\(883\) 3.67337e6 0.158549 0.0792745 0.996853i \(-0.474740\pi\)
0.0792745 + 0.996853i \(0.474740\pi\)
\(884\) 2.24133e7 0.964662
\(885\) 0 0
\(886\) 9.43082e6 0.403613
\(887\) 3.39649e7 1.44951 0.724755 0.689007i \(-0.241953\pi\)
0.724755 + 0.689007i \(0.241953\pi\)
\(888\) 3.91731e6 0.166708
\(889\) 2.76876e7 1.17498
\(890\) 0 0
\(891\) 6.72201e6 0.283665
\(892\) 1.94969e7 0.820451
\(893\) −2.37140e7 −0.995123
\(894\) −7.77560e6 −0.325379
\(895\) 0 0
\(896\) −2.58867e6 −0.107723
\(897\) −1.19317e7 −0.495132
\(898\) −2.19894e7 −0.909960
\(899\) 552960. 0.0228189
\(900\) 0 0
\(901\) 6.07027e6 0.249113
\(902\) −1.41962e6 −0.0580971
\(903\) 5.07433e6 0.207090
\(904\) 1.46442e7 0.595999
\(905\) 0 0
\(906\) −2.33045e7 −0.943234
\(907\) −2.13327e7 −0.861050 −0.430525 0.902579i \(-0.641672\pi\)
−0.430525 + 0.902579i \(0.641672\pi\)
\(908\) 9.02941e6 0.363450
\(909\) 202006. 0.00810876
\(910\) 0 0
\(911\) −1.03512e7 −0.413235 −0.206617 0.978422i \(-0.566246\pi\)
−0.206617 + 0.978422i \(0.566246\pi\)
\(912\) 7.95648e6 0.316763
\(913\) 4.11825e6 0.163507
\(914\) 4.64157e6 0.183780
\(915\) 0 0
\(916\) 8.96528e6 0.353041
\(917\) 4.73450e7 1.85931
\(918\) 3.32595e7 1.30259
\(919\) −2.59019e7 −1.01168 −0.505839 0.862628i \(-0.668817\pi\)
−0.505839 + 0.862628i \(0.668817\pi\)
\(920\) 0 0
\(921\) 1.23001e6 0.0477816
\(922\) 9.21319e6 0.356930
\(923\) −2.90563e6 −0.112263
\(924\) −5.23802e6 −0.201831
\(925\) 0 0
\(926\) −1.08537e7 −0.415960
\(927\) −5.83336e6 −0.222956
\(928\) 276480. 0.0105389
\(929\) −3.13230e7 −1.19076 −0.595379 0.803445i \(-0.702998\pi\)
−0.595379 + 0.803445i \(0.702998\pi\)
\(930\) 0 0
\(931\) 1.81085e7 0.684714
\(932\) −4.69722e6 −0.177134
\(933\) 8.39093e6 0.315577
\(934\) −1.62020e7 −0.607717
\(935\) 0 0
\(936\) 2.05747e6 0.0767617
\(937\) −2.08461e7 −0.775667 −0.387833 0.921729i \(-0.626776\pi\)
−0.387833 + 0.921729i \(0.626776\pi\)
\(938\) −2.02859e7 −0.752814
\(939\) −2.93078e7 −1.08472
\(940\) 0 0
\(941\) −3.82929e7 −1.40976 −0.704878 0.709328i \(-0.748998\pi\)
−0.704878 + 0.709328i \(0.748998\pi\)
\(942\) −2.42540e7 −0.890547
\(943\) 2.98791e6 0.109418
\(944\) −1.01734e7 −0.371568
\(945\) 0 0
\(946\) 1.35805e6 0.0493387
\(947\) −4.25088e7 −1.54029 −0.770147 0.637866i \(-0.779817\pi\)
−0.770147 + 0.637866i \(0.779817\pi\)
\(948\) 7.90272e6 0.285598
\(949\) 2.05911e7 0.742189
\(950\) 0 0
\(951\) −3.38275e7 −1.21288
\(952\) −2.07094e7 −0.740585
\(953\) 3.91855e7 1.39763 0.698816 0.715302i \(-0.253711\pi\)
0.698816 + 0.715302i \(0.253711\pi\)
\(954\) 557232. 0.0198228
\(955\) 0 0
\(956\) 9.34784e6 0.330801
\(957\) 559440. 0.0197458
\(958\) −2.24211e7 −0.789303
\(959\) 1.70526e7 0.598749
\(960\) 0 0
\(961\) −2.44348e7 −0.853495
\(962\) 1.19618e7 0.416734
\(963\) 1.99007e6 0.0691518
\(964\) −9.02077e6 −0.312645
\(965\) 0 0
\(966\) 1.10246e7 0.380120
\(967\) −1.84836e7 −0.635653 −0.317827 0.948149i \(-0.602953\pi\)
−0.317827 + 0.948149i \(0.602953\pi\)
\(968\) 8.90541e6 0.305468
\(969\) 6.36518e7 2.17772
\(970\) 0 0
\(971\) 3.95031e7 1.34457 0.672284 0.740294i \(-0.265314\pi\)
0.672284 + 0.740294i \(0.265314\pi\)
\(972\) 5.61142e6 0.190505
\(973\) −3.10407e7 −1.05111
\(974\) −2.85267e7 −0.963506
\(975\) 0 0
\(976\) −1.08283e7 −0.363861
\(977\) 3.29043e7 1.10285 0.551425 0.834225i \(-0.314084\pi\)
0.551425 + 0.834225i \(0.314084\pi\)
\(978\) −8.35150e6 −0.279201
\(979\) 1.26111e7 0.420529
\(980\) 0 0
\(981\) 1.69153e6 0.0561186
\(982\) −2.35258e7 −0.778513
\(983\) −2.65797e7 −0.877338 −0.438669 0.898649i \(-0.644550\pi\)
−0.438669 + 0.898649i \(0.644550\pi\)
\(984\) 2.14861e6 0.0707407
\(985\) 0 0
\(986\) 2.21184e6 0.0724538
\(987\) −2.36286e7 −0.772049
\(988\) 2.42957e7 0.791839
\(989\) −2.85832e6 −0.0929225
\(990\) 0 0
\(991\) 1.92964e7 0.624153 0.312077 0.950057i \(-0.398975\pi\)
0.312077 + 0.950057i \(0.398975\pi\)
\(992\) 2.09715e6 0.0676629
\(993\) −2.29733e7 −0.739349
\(994\) 2.68474e6 0.0861858
\(995\) 0 0
\(996\) −6.23302e6 −0.199090
\(997\) −5.12017e7 −1.63135 −0.815674 0.578511i \(-0.803634\pi\)
−0.815674 + 0.578511i \(0.803634\pi\)
\(998\) −7.02840e6 −0.223373
\(999\) 1.77503e7 0.562720
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 50.6.a.c.1.1 1
3.2 odd 2 450.6.a.w.1.1 1
4.3 odd 2 400.6.a.c.1.1 1
5.2 odd 4 10.6.b.a.9.1 2
5.3 odd 4 10.6.b.a.9.2 yes 2
5.4 even 2 50.6.a.e.1.1 1
15.2 even 4 90.6.c.a.19.2 2
15.8 even 4 90.6.c.a.19.1 2
15.14 odd 2 450.6.a.c.1.1 1
20.3 even 4 80.6.c.c.49.1 2
20.7 even 4 80.6.c.c.49.2 2
20.19 odd 2 400.6.a.k.1.1 1
40.3 even 4 320.6.c.a.129.2 2
40.13 odd 4 320.6.c.b.129.1 2
40.27 even 4 320.6.c.a.129.1 2
40.37 odd 4 320.6.c.b.129.2 2
60.23 odd 4 720.6.f.a.289.1 2
60.47 odd 4 720.6.f.a.289.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
10.6.b.a.9.1 2 5.2 odd 4
10.6.b.a.9.2 yes 2 5.3 odd 4
50.6.a.c.1.1 1 1.1 even 1 trivial
50.6.a.e.1.1 1 5.4 even 2
80.6.c.c.49.1 2 20.3 even 4
80.6.c.c.49.2 2 20.7 even 4
90.6.c.a.19.1 2 15.8 even 4
90.6.c.a.19.2 2 15.2 even 4
320.6.c.a.129.1 2 40.27 even 4
320.6.c.a.129.2 2 40.3 even 4
320.6.c.b.129.1 2 40.13 odd 4
320.6.c.b.129.2 2 40.37 odd 4
400.6.a.c.1.1 1 4.3 odd 2
400.6.a.k.1.1 1 20.19 odd 2
450.6.a.c.1.1 1 15.14 odd 2
450.6.a.w.1.1 1 3.2 odd 2
720.6.f.a.289.1 2 60.23 odd 4
720.6.f.a.289.2 2 60.47 odd 4