Properties

Label 400.6.a.z.1.3
Level $400$
Weight $6$
Character 400.1
Self dual yes
Analytic conductor $64.154$
Analytic rank $0$
Dimension $4$
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] = [400,6,Mod(1,400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(400, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("400.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 400 = 2^{4} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 400.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: \(64.1535279252\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: 4.4.1595208.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 20x^{2} + 33x - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{8}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 40)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-4.73066\) of defining polynomial
Character \(\chi\) \(=\) 400.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+5.49000 q^{3} +188.968 q^{7} -212.860 q^{9} +O(q^{10})\) \(q+5.49000 q^{3} +188.968 q^{7} -212.860 q^{9} +501.871 q^{11} +1061.48 q^{13} -29.5861 q^{17} -1578.33 q^{19} +1037.43 q^{21} +1295.86 q^{23} -2502.67 q^{27} -3586.63 q^{29} +3526.32 q^{31} +2755.27 q^{33} +8413.79 q^{37} +5827.54 q^{39} +7015.12 q^{41} -22694.2 q^{43} +3501.99 q^{47} +18901.8 q^{49} -162.428 q^{51} +27309.1 q^{53} -8665.04 q^{57} -7925.39 q^{59} -7020.54 q^{61} -40223.6 q^{63} -17631.2 q^{67} +7114.27 q^{69} -13432.9 q^{71} +39946.8 q^{73} +94837.3 q^{77} +93321.2 q^{79} +37985.3 q^{81} +58448.5 q^{83} -19690.6 q^{87} -13989.4 q^{89} +200586. q^{91} +19359.5 q^{93} -110640. q^{97} -106828. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 4 q^{3} - 148 q^{7} + 500 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 4 q^{3} - 148 q^{7} + 500 q^{9} + 368 q^{11} + 440 q^{13} - 672 q^{17} + 688 q^{19} + 992 q^{21} - 4492 q^{23} - 8152 q^{27} - 2936 q^{29} - 2112 q^{31} + 26864 q^{33} + 8792 q^{37} - 1504 q^{39} + 11800 q^{41} - 48276 q^{43} - 14724 q^{47} + 22500 q^{49} + 62400 q^{51} + 84296 q^{53} + 71024 q^{57} + 45840 q^{59} + 61928 q^{61} - 186292 q^{63} - 72700 q^{67} + 38368 q^{69} + 62816 q^{71} + 133072 q^{73} + 11440 q^{77} + 21632 q^{79} + 204836 q^{81} - 74660 q^{83} + 12472 q^{87} + 20952 q^{89} + 243808 q^{91} + 105600 q^{93} + 59456 q^{97} + 133424 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 5.49000 0.352184 0.176092 0.984374i \(-0.443654\pi\)
0.176092 + 0.984374i \(0.443654\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 188.968 1.45761 0.728807 0.684720i \(-0.240075\pi\)
0.728807 + 0.684720i \(0.240075\pi\)
\(8\) 0 0
\(9\) −212.860 −0.875967
\(10\) 0 0
\(11\) 501.871 1.25058 0.625288 0.780394i \(-0.284982\pi\)
0.625288 + 0.780394i \(0.284982\pi\)
\(12\) 0 0
\(13\) 1061.48 1.74203 0.871013 0.491259i \(-0.163463\pi\)
0.871013 + 0.491259i \(0.163463\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −29.5861 −0.0248294 −0.0124147 0.999923i \(-0.503952\pi\)
−0.0124147 + 0.999923i \(0.503952\pi\)
\(18\) 0 0
\(19\) −1578.33 −1.00303 −0.501515 0.865149i \(-0.667224\pi\)
−0.501515 + 0.865149i \(0.667224\pi\)
\(20\) 0 0
\(21\) 1037.43 0.513347
\(22\) 0 0
\(23\) 1295.86 0.510786 0.255393 0.966837i \(-0.417795\pi\)
0.255393 + 0.966837i \(0.417795\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −2502.67 −0.660685
\(28\) 0 0
\(29\) −3586.63 −0.791938 −0.395969 0.918264i \(-0.629591\pi\)
−0.395969 + 0.918264i \(0.629591\pi\)
\(30\) 0 0
\(31\) 3526.32 0.659048 0.329524 0.944147i \(-0.393112\pi\)
0.329524 + 0.944147i \(0.393112\pi\)
\(32\) 0 0
\(33\) 2755.27 0.440432
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 8413.79 1.01039 0.505193 0.863006i \(-0.331421\pi\)
0.505193 + 0.863006i \(0.331421\pi\)
\(38\) 0 0
\(39\) 5827.54 0.613513
\(40\) 0 0
\(41\) 7015.12 0.651742 0.325871 0.945414i \(-0.394343\pi\)
0.325871 + 0.945414i \(0.394343\pi\)
\(42\) 0 0
\(43\) −22694.2 −1.87173 −0.935865 0.352358i \(-0.885380\pi\)
−0.935865 + 0.352358i \(0.885380\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 3501.99 0.231244 0.115622 0.993293i \(-0.463114\pi\)
0.115622 + 0.993293i \(0.463114\pi\)
\(48\) 0 0
\(49\) 18901.8 1.12464
\(50\) 0 0
\(51\) −162.428 −0.00874449
\(52\) 0 0
\(53\) 27309.1 1.33542 0.667710 0.744422i \(-0.267275\pi\)
0.667710 + 0.744422i \(0.267275\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −8665.04 −0.353251
\(58\) 0 0
\(59\) −7925.39 −0.296409 −0.148204 0.988957i \(-0.547349\pi\)
−0.148204 + 0.988957i \(0.547349\pi\)
\(60\) 0 0
\(61\) −7020.54 −0.241572 −0.120786 0.992679i \(-0.538541\pi\)
−0.120786 + 0.992679i \(0.538541\pi\)
\(62\) 0 0
\(63\) −40223.6 −1.27682
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −17631.2 −0.479837 −0.239919 0.970793i \(-0.577121\pi\)
−0.239919 + 0.970793i \(0.577121\pi\)
\(68\) 0 0
\(69\) 7114.27 0.179890
\(70\) 0 0
\(71\) −13432.9 −0.316246 −0.158123 0.987419i \(-0.550544\pi\)
−0.158123 + 0.987419i \(0.550544\pi\)
\(72\) 0 0
\(73\) 39946.8 0.877353 0.438677 0.898645i \(-0.355447\pi\)
0.438677 + 0.898645i \(0.355447\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 94837.3 1.82286
\(78\) 0 0
\(79\) 93321.2 1.68234 0.841168 0.540774i \(-0.181869\pi\)
0.841168 + 0.540774i \(0.181869\pi\)
\(80\) 0 0
\(81\) 37985.3 0.643284
\(82\) 0 0
\(83\) 58448.5 0.931275 0.465638 0.884975i \(-0.345825\pi\)
0.465638 + 0.884975i \(0.345825\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −19690.6 −0.278908
\(88\) 0 0
\(89\) −13989.4 −0.187208 −0.0936038 0.995610i \(-0.529839\pi\)
−0.0936038 + 0.995610i \(0.529839\pi\)
\(90\) 0 0
\(91\) 200586. 2.53920
\(92\) 0 0
\(93\) 19359.5 0.232106
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −110640. −1.19394 −0.596969 0.802264i \(-0.703629\pi\)
−0.596969 + 0.802264i \(0.703629\pi\)
\(98\) 0 0
\(99\) −106828. −1.09546
\(100\) 0 0
\(101\) 198468. 1.93592 0.967960 0.251103i \(-0.0807934\pi\)
0.967960 + 0.251103i \(0.0807934\pi\)
\(102\) 0 0
\(103\) −134780. −1.25179 −0.625897 0.779906i \(-0.715267\pi\)
−0.625897 + 0.779906i \(0.715267\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −39785.5 −0.335943 −0.167971 0.985792i \(-0.553722\pi\)
−0.167971 + 0.985792i \(0.553722\pi\)
\(108\) 0 0
\(109\) 92692.6 0.747272 0.373636 0.927575i \(-0.378111\pi\)
0.373636 + 0.927575i \(0.378111\pi\)
\(110\) 0 0
\(111\) 46191.7 0.355841
\(112\) 0 0
\(113\) −66923.1 −0.493037 −0.246519 0.969138i \(-0.579287\pi\)
−0.246519 + 0.969138i \(0.579287\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −225947. −1.52596
\(118\) 0 0
\(119\) −5590.81 −0.0361916
\(120\) 0 0
\(121\) 90823.1 0.563940
\(122\) 0 0
\(123\) 38513.0 0.229533
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 58998.9 0.324589 0.162295 0.986742i \(-0.448110\pi\)
0.162295 + 0.986742i \(0.448110\pi\)
\(128\) 0 0
\(129\) −124591. −0.659193
\(130\) 0 0
\(131\) 209380. 1.06600 0.533001 0.846115i \(-0.321064\pi\)
0.533001 + 0.846115i \(0.321064\pi\)
\(132\) 0 0
\(133\) −298254. −1.46203
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 326432. 1.48591 0.742953 0.669344i \(-0.233425\pi\)
0.742953 + 0.669344i \(0.233425\pi\)
\(138\) 0 0
\(139\) 233648. 1.02571 0.512856 0.858475i \(-0.328588\pi\)
0.512856 + 0.858475i \(0.328588\pi\)
\(140\) 0 0
\(141\) 19225.9 0.0814403
\(142\) 0 0
\(143\) 532727. 2.17854
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 103771. 0.396078
\(148\) 0 0
\(149\) 145336. 0.536299 0.268149 0.963377i \(-0.413588\pi\)
0.268149 + 0.963377i \(0.413588\pi\)
\(150\) 0 0
\(151\) −287343. −1.02555 −0.512777 0.858522i \(-0.671383\pi\)
−0.512777 + 0.858522i \(0.671383\pi\)
\(152\) 0 0
\(153\) 6297.69 0.0217497
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −42626.2 −0.138015 −0.0690077 0.997616i \(-0.521983\pi\)
−0.0690077 + 0.997616i \(0.521983\pi\)
\(158\) 0 0
\(159\) 149927. 0.470313
\(160\) 0 0
\(161\) 244876. 0.744528
\(162\) 0 0
\(163\) 221599. 0.653279 0.326639 0.945149i \(-0.394084\pi\)
0.326639 + 0.945149i \(0.394084\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 334835. 0.929051 0.464525 0.885560i \(-0.346225\pi\)
0.464525 + 0.885560i \(0.346225\pi\)
\(168\) 0 0
\(169\) 755454. 2.03466
\(170\) 0 0
\(171\) 335964. 0.878622
\(172\) 0 0
\(173\) −268475. −0.682006 −0.341003 0.940062i \(-0.610767\pi\)
−0.341003 + 0.940062i \(0.610767\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −43510.4 −0.104390
\(178\) 0 0
\(179\) 268052. 0.625297 0.312649 0.949869i \(-0.398784\pi\)
0.312649 + 0.949869i \(0.398784\pi\)
\(180\) 0 0
\(181\) −90565.5 −0.205479 −0.102739 0.994708i \(-0.532761\pi\)
−0.102739 + 0.994708i \(0.532761\pi\)
\(182\) 0 0
\(183\) −38542.8 −0.0850776
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −14848.4 −0.0310510
\(188\) 0 0
\(189\) −472924. −0.963023
\(190\) 0 0
\(191\) 706725. 1.40174 0.700870 0.713290i \(-0.252796\pi\)
0.700870 + 0.713290i \(0.252796\pi\)
\(192\) 0 0
\(193\) 236999. 0.457988 0.228994 0.973428i \(-0.426456\pi\)
0.228994 + 0.973428i \(0.426456\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −607901. −1.11601 −0.558004 0.829838i \(-0.688433\pi\)
−0.558004 + 0.829838i \(0.688433\pi\)
\(198\) 0 0
\(199\) −621060. −1.11173 −0.555867 0.831272i \(-0.687613\pi\)
−0.555867 + 0.831272i \(0.687613\pi\)
\(200\) 0 0
\(201\) −96795.1 −0.168991
\(202\) 0 0
\(203\) −677756. −1.15434
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −275837. −0.447431
\(208\) 0 0
\(209\) −792118. −1.25437
\(210\) 0 0
\(211\) −1.06177e6 −1.64181 −0.820906 0.571063i \(-0.806531\pi\)
−0.820906 + 0.571063i \(0.806531\pi\)
\(212\) 0 0
\(213\) −73746.8 −0.111377
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 666360. 0.960638
\(218\) 0 0
\(219\) 219308. 0.308989
\(220\) 0 0
\(221\) −31405.2 −0.0432534
\(222\) 0 0
\(223\) −240721. −0.324154 −0.162077 0.986778i \(-0.551819\pi\)
−0.162077 + 0.986778i \(0.551819\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 202494. 0.260824 0.130412 0.991460i \(-0.458370\pi\)
0.130412 + 0.991460i \(0.458370\pi\)
\(228\) 0 0
\(229\) −284279. −0.358225 −0.179112 0.983829i \(-0.557323\pi\)
−0.179112 + 0.983829i \(0.557323\pi\)
\(230\) 0 0
\(231\) 520657. 0.641980
\(232\) 0 0
\(233\) 345405. 0.416811 0.208406 0.978042i \(-0.433173\pi\)
0.208406 + 0.978042i \(0.433173\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 512333. 0.592491
\(238\) 0 0
\(239\) −1.59920e6 −1.81095 −0.905476 0.424398i \(-0.860486\pi\)
−0.905476 + 0.424398i \(0.860486\pi\)
\(240\) 0 0
\(241\) −42246.4 −0.0468541 −0.0234270 0.999726i \(-0.507458\pi\)
−0.0234270 + 0.999726i \(0.507458\pi\)
\(242\) 0 0
\(243\) 816688. 0.887239
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −1.67537e6 −1.74731
\(248\) 0 0
\(249\) 320882. 0.327980
\(250\) 0 0
\(251\) −802841. −0.804351 −0.402175 0.915563i \(-0.631746\pi\)
−0.402175 + 0.915563i \(0.631746\pi\)
\(252\) 0 0
\(253\) 650354. 0.638776
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −941153. −0.888848 −0.444424 0.895817i \(-0.646592\pi\)
−0.444424 + 0.895817i \(0.646592\pi\)
\(258\) 0 0
\(259\) 1.58993e6 1.47275
\(260\) 0 0
\(261\) 763449. 0.693711
\(262\) 0 0
\(263\) −1.53130e6 −1.36512 −0.682560 0.730830i \(-0.739133\pi\)
−0.682560 + 0.730830i \(0.739133\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −76801.7 −0.0659315
\(268\) 0 0
\(269\) 1.25712e6 1.05924 0.529621 0.848234i \(-0.322334\pi\)
0.529621 + 0.848234i \(0.322334\pi\)
\(270\) 0 0
\(271\) −843976. −0.698083 −0.349041 0.937107i \(-0.613493\pi\)
−0.349041 + 0.937107i \(0.613493\pi\)
\(272\) 0 0
\(273\) 1.10122e6 0.894265
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.52070e6 1.19082 0.595408 0.803423i \(-0.296990\pi\)
0.595408 + 0.803423i \(0.296990\pi\)
\(278\) 0 0
\(279\) −750612. −0.577305
\(280\) 0 0
\(281\) −1.97975e6 −1.49570 −0.747848 0.663870i \(-0.768913\pi\)
−0.747848 + 0.663870i \(0.768913\pi\)
\(282\) 0 0
\(283\) −2.02901e6 −1.50597 −0.752986 0.658036i \(-0.771387\pi\)
−0.752986 + 0.658036i \(0.771387\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.32563e6 0.949987
\(288\) 0 0
\(289\) −1.41898e6 −0.999384
\(290\) 0 0
\(291\) −607412. −0.420485
\(292\) 0 0
\(293\) 2.23140e6 1.51848 0.759238 0.650813i \(-0.225572\pi\)
0.759238 + 0.650813i \(0.225572\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −1.25602e6 −0.826236
\(298\) 0 0
\(299\) 1.37553e6 0.889802
\(300\) 0 0
\(301\) −4.28847e6 −2.72826
\(302\) 0 0
\(303\) 1.08959e6 0.681799
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 334318. 0.202448 0.101224 0.994864i \(-0.467724\pi\)
0.101224 + 0.994864i \(0.467724\pi\)
\(308\) 0 0
\(309\) −739943. −0.440861
\(310\) 0 0
\(311\) −2.39090e6 −1.40172 −0.700858 0.713300i \(-0.747200\pi\)
−0.700858 + 0.713300i \(0.747200\pi\)
\(312\) 0 0
\(313\) −1.88563e6 −1.08791 −0.543957 0.839113i \(-0.683075\pi\)
−0.543957 + 0.839113i \(0.683075\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 179636. 0.100403 0.0502014 0.998739i \(-0.484014\pi\)
0.0502014 + 0.998739i \(0.484014\pi\)
\(318\) 0 0
\(319\) −1.80002e6 −0.990378
\(320\) 0 0
\(321\) −218422. −0.118313
\(322\) 0 0
\(323\) 46696.7 0.0249046
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 508882. 0.263177
\(328\) 0 0
\(329\) 661763. 0.337064
\(330\) 0 0
\(331\) −168620. −0.0845937 −0.0422968 0.999105i \(-0.513468\pi\)
−0.0422968 + 0.999105i \(0.513468\pi\)
\(332\) 0 0
\(333\) −1.79096e6 −0.885064
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.74240e6 0.835744 0.417872 0.908506i \(-0.362776\pi\)
0.417872 + 0.908506i \(0.362776\pi\)
\(338\) 0 0
\(339\) −367407. −0.173640
\(340\) 0 0
\(341\) 1.76976e6 0.824190
\(342\) 0 0
\(343\) 395842. 0.181672
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −2.89447e6 −1.29046 −0.645232 0.763987i \(-0.723239\pi\)
−0.645232 + 0.763987i \(0.723239\pi\)
\(348\) 0 0
\(349\) 63803.8 0.0280403 0.0140202 0.999902i \(-0.495537\pi\)
0.0140202 + 0.999902i \(0.495537\pi\)
\(350\) 0 0
\(351\) −2.65654e6 −1.15093
\(352\) 0 0
\(353\) −2.79206e6 −1.19258 −0.596290 0.802769i \(-0.703359\pi\)
−0.596290 + 0.802769i \(0.703359\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −30693.6 −0.0127461
\(358\) 0 0
\(359\) −1.48391e6 −0.607674 −0.303837 0.952724i \(-0.598268\pi\)
−0.303837 + 0.952724i \(0.598268\pi\)
\(360\) 0 0
\(361\) 15032.0 0.00607083
\(362\) 0 0
\(363\) 498619. 0.198610
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.08188e6 −0.419289 −0.209644 0.977778i \(-0.567231\pi\)
−0.209644 + 0.977778i \(0.567231\pi\)
\(368\) 0 0
\(369\) −1.49324e6 −0.570904
\(370\) 0 0
\(371\) 5.16053e6 1.94652
\(372\) 0 0
\(373\) −127750. −0.0475434 −0.0237717 0.999717i \(-0.507567\pi\)
−0.0237717 + 0.999717i \(0.507567\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −3.80714e6 −1.37958
\(378\) 0 0
\(379\) 1.61295e6 0.576798 0.288399 0.957510i \(-0.406877\pi\)
0.288399 + 0.957510i \(0.406877\pi\)
\(380\) 0 0
\(381\) 323904. 0.114315
\(382\) 0 0
\(383\) 3.96656e6 1.38171 0.690855 0.722993i \(-0.257234\pi\)
0.690855 + 0.722993i \(0.257234\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.83068e6 1.63957
\(388\) 0 0
\(389\) −3.54852e6 −1.18898 −0.594488 0.804104i \(-0.702645\pi\)
−0.594488 + 0.804104i \(0.702645\pi\)
\(390\) 0 0
\(391\) −38339.5 −0.0126825
\(392\) 0 0
\(393\) 1.14950e6 0.375428
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −580854. −0.184965 −0.0924827 0.995714i \(-0.529480\pi\)
−0.0924827 + 0.995714i \(0.529480\pi\)
\(398\) 0 0
\(399\) −1.63741e6 −0.514903
\(400\) 0 0
\(401\) 2.10413e6 0.653449 0.326725 0.945120i \(-0.394055\pi\)
0.326725 + 0.945120i \(0.394055\pi\)
\(402\) 0 0
\(403\) 3.74313e6 1.14808
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 4.22263e6 1.26356
\(408\) 0 0
\(409\) 3.23687e6 0.956790 0.478395 0.878145i \(-0.341219\pi\)
0.478395 + 0.878145i \(0.341219\pi\)
\(410\) 0 0
\(411\) 1.79211e6 0.523312
\(412\) 0 0
\(413\) −1.49764e6 −0.432049
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.28273e6 0.361239
\(418\) 0 0
\(419\) 2.03059e6 0.565049 0.282525 0.959260i \(-0.408828\pi\)
0.282525 + 0.959260i \(0.408828\pi\)
\(420\) 0 0
\(421\) 2.15883e6 0.593625 0.296813 0.954936i \(-0.404076\pi\)
0.296813 + 0.954936i \(0.404076\pi\)
\(422\) 0 0
\(423\) −745434. −0.202562
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.32665e6 −0.352118
\(428\) 0 0
\(429\) 2.92467e6 0.767245
\(430\) 0 0
\(431\) −5.49812e6 −1.42568 −0.712838 0.701328i \(-0.752591\pi\)
−0.712838 + 0.701328i \(0.752591\pi\)
\(432\) 0 0
\(433\) 3.87201e6 0.992470 0.496235 0.868188i \(-0.334716\pi\)
0.496235 + 0.868188i \(0.334716\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −2.04530e6 −0.512334
\(438\) 0 0
\(439\) −3.15465e6 −0.781250 −0.390625 0.920550i \(-0.627741\pi\)
−0.390625 + 0.920550i \(0.627741\pi\)
\(440\) 0 0
\(441\) −4.02343e6 −0.985144
\(442\) 0 0
\(443\) 4.07021e6 0.985389 0.492694 0.870202i \(-0.336012\pi\)
0.492694 + 0.870202i \(0.336012\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 797893. 0.188876
\(448\) 0 0
\(449\) 957718. 0.224193 0.112096 0.993697i \(-0.464243\pi\)
0.112096 + 0.993697i \(0.464243\pi\)
\(450\) 0 0
\(451\) 3.52068e6 0.815052
\(452\) 0 0
\(453\) −1.57751e6 −0.361183
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 7.38167e6 1.65335 0.826673 0.562682i \(-0.190231\pi\)
0.826673 + 0.562682i \(0.190231\pi\)
\(458\) 0 0
\(459\) 74044.2 0.0164044
\(460\) 0 0
\(461\) −2.96862e6 −0.650582 −0.325291 0.945614i \(-0.605462\pi\)
−0.325291 + 0.945614i \(0.605462\pi\)
\(462\) 0 0
\(463\) −6.84593e6 −1.48416 −0.742079 0.670313i \(-0.766160\pi\)
−0.742079 + 0.670313i \(0.766160\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.20816e6 −0.892895 −0.446447 0.894810i \(-0.647311\pi\)
−0.446447 + 0.894810i \(0.647311\pi\)
\(468\) 0 0
\(469\) −3.33172e6 −0.699417
\(470\) 0 0
\(471\) −234018. −0.0486068
\(472\) 0 0
\(473\) −1.13895e7 −2.34074
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −5.81301e6 −1.16978
\(478\) 0 0
\(479\) 5.73077e6 1.14123 0.570617 0.821217i \(-0.306704\pi\)
0.570617 + 0.821217i \(0.306704\pi\)
\(480\) 0 0
\(481\) 8.93110e6 1.76012
\(482\) 0 0
\(483\) 1.34437e6 0.262211
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −6.63249e6 −1.26723 −0.633613 0.773650i \(-0.718429\pi\)
−0.633613 + 0.773650i \(0.718429\pi\)
\(488\) 0 0
\(489\) 1.21658e6 0.230074
\(490\) 0 0
\(491\) 3.26463e6 0.611125 0.305563 0.952172i \(-0.401156\pi\)
0.305563 + 0.952172i \(0.401156\pi\)
\(492\) 0 0
\(493\) 106114. 0.0196633
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −2.53839e6 −0.460965
\(498\) 0 0
\(499\) 8.51527e6 1.53090 0.765450 0.643495i \(-0.222516\pi\)
0.765450 + 0.643495i \(0.222516\pi\)
\(500\) 0 0
\(501\) 1.83824e6 0.327196
\(502\) 0 0
\(503\) −3.24174e6 −0.571292 −0.285646 0.958335i \(-0.592208\pi\)
−0.285646 + 0.958335i \(0.592208\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 4.14744e6 0.716573
\(508\) 0 0
\(509\) 1.04569e7 1.78900 0.894499 0.447070i \(-0.147533\pi\)
0.894499 + 0.447070i \(0.147533\pi\)
\(510\) 0 0
\(511\) 7.54865e6 1.27884
\(512\) 0 0
\(513\) 3.95004e6 0.662687
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1.75755e6 0.289188
\(518\) 0 0
\(519\) −1.47393e6 −0.240191
\(520\) 0 0
\(521\) −328142. −0.0529625 −0.0264812 0.999649i \(-0.508430\pi\)
−0.0264812 + 0.999649i \(0.508430\pi\)
\(522\) 0 0
\(523\) −9.42102e6 −1.50606 −0.753032 0.657984i \(-0.771410\pi\)
−0.753032 + 0.657984i \(0.771410\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −104330. −0.0163637
\(528\) 0 0
\(529\) −4.75709e6 −0.739098
\(530\) 0 0
\(531\) 1.68700e6 0.259644
\(532\) 0 0
\(533\) 7.44643e6 1.13535
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.47160e6 0.220219
\(538\) 0 0
\(539\) 9.48624e6 1.40644
\(540\) 0 0
\(541\) 6.08041e6 0.893182 0.446591 0.894738i \(-0.352638\pi\)
0.446591 + 0.894738i \(0.352638\pi\)
\(542\) 0 0
\(543\) −497205. −0.0723662
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.32182e6 −0.617587 −0.308794 0.951129i \(-0.599925\pi\)
−0.308794 + 0.951129i \(0.599925\pi\)
\(548\) 0 0
\(549\) 1.49439e6 0.211609
\(550\) 0 0
\(551\) 5.66089e6 0.794338
\(552\) 0 0
\(553\) 1.76347e7 2.45220
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 7.41019e6 1.01203 0.506013 0.862526i \(-0.331119\pi\)
0.506013 + 0.862526i \(0.331119\pi\)
\(558\) 0 0
\(559\) −2.40895e7 −3.26061
\(560\) 0 0
\(561\) −81517.7 −0.0109356
\(562\) 0 0
\(563\) −1.08039e7 −1.43652 −0.718260 0.695775i \(-0.755061\pi\)
−0.718260 + 0.695775i \(0.755061\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 7.17799e6 0.937660
\(568\) 0 0
\(569\) 1.77404e6 0.229712 0.114856 0.993382i \(-0.463359\pi\)
0.114856 + 0.993382i \(0.463359\pi\)
\(570\) 0 0
\(571\) 70981.0 0.00911070 0.00455535 0.999990i \(-0.498550\pi\)
0.00455535 + 0.999990i \(0.498550\pi\)
\(572\) 0 0
\(573\) 3.87992e6 0.493669
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −4.04410e6 −0.505688 −0.252844 0.967507i \(-0.581366\pi\)
−0.252844 + 0.967507i \(0.581366\pi\)
\(578\) 0 0
\(579\) 1.30113e6 0.161296
\(580\) 0 0
\(581\) 1.10449e7 1.35744
\(582\) 0 0
\(583\) 1.37056e7 1.67004
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.86862e6 0.343620 0.171810 0.985130i \(-0.445038\pi\)
0.171810 + 0.985130i \(0.445038\pi\)
\(588\) 0 0
\(589\) −5.56570e6 −0.661046
\(590\) 0 0
\(591\) −3.33737e6 −0.393040
\(592\) 0 0
\(593\) −3.23131e6 −0.377347 −0.188674 0.982040i \(-0.560419\pi\)
−0.188674 + 0.982040i \(0.560419\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.40962e6 −0.391534
\(598\) 0 0
\(599\) −3.80078e6 −0.432819 −0.216409 0.976303i \(-0.569435\pi\)
−0.216409 + 0.976303i \(0.569435\pi\)
\(600\) 0 0
\(601\) 5.45289e6 0.615801 0.307900 0.951419i \(-0.400374\pi\)
0.307900 + 0.951419i \(0.400374\pi\)
\(602\) 0 0
\(603\) 3.75297e6 0.420321
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −1.50483e7 −1.65774 −0.828870 0.559441i \(-0.811016\pi\)
−0.828870 + 0.559441i \(0.811016\pi\)
\(608\) 0 0
\(609\) −3.72088e6 −0.406539
\(610\) 0 0
\(611\) 3.71731e6 0.402833
\(612\) 0 0
\(613\) −2.30782e6 −0.248056 −0.124028 0.992279i \(-0.539581\pi\)
−0.124028 + 0.992279i \(0.539581\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 381761. 0.0403718 0.0201859 0.999796i \(-0.493574\pi\)
0.0201859 + 0.999796i \(0.493574\pi\)
\(618\) 0 0
\(619\) 1.07208e7 1.12460 0.562302 0.826932i \(-0.309916\pi\)
0.562302 + 0.826932i \(0.309916\pi\)
\(620\) 0 0
\(621\) −3.24311e6 −0.337468
\(622\) 0 0
\(623\) −2.64354e6 −0.272876
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.34873e6 −0.441767
\(628\) 0 0
\(629\) −248931. −0.0250872
\(630\) 0 0
\(631\) 4.56189e6 0.456112 0.228056 0.973648i \(-0.426763\pi\)
0.228056 + 0.973648i \(0.426763\pi\)
\(632\) 0 0
\(633\) −5.82910e6 −0.578219
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.00639e7 1.95915
\(638\) 0 0
\(639\) 2.85933e6 0.277021
\(640\) 0 0
\(641\) −2.70983e6 −0.260494 −0.130247 0.991482i \(-0.541577\pi\)
−0.130247 + 0.991482i \(0.541577\pi\)
\(642\) 0 0
\(643\) −6.10481e6 −0.582297 −0.291149 0.956678i \(-0.594037\pi\)
−0.291149 + 0.956678i \(0.594037\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.22081e7 1.14654 0.573269 0.819367i \(-0.305675\pi\)
0.573269 + 0.819367i \(0.305675\pi\)
\(648\) 0 0
\(649\) −3.97752e6 −0.370681
\(650\) 0 0
\(651\) 3.65832e6 0.338321
\(652\) 0 0
\(653\) 6.26514e6 0.574973 0.287487 0.957785i \(-0.407180\pi\)
0.287487 + 0.957785i \(0.407180\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −8.50307e6 −0.768532
\(658\) 0 0
\(659\) 1.85107e7 1.66039 0.830194 0.557474i \(-0.188229\pi\)
0.830194 + 0.557474i \(0.188229\pi\)
\(660\) 0 0
\(661\) −8.46041e6 −0.753161 −0.376581 0.926384i \(-0.622900\pi\)
−0.376581 + 0.926384i \(0.622900\pi\)
\(662\) 0 0
\(663\) −172414. −0.0152331
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −4.64777e6 −0.404511
\(668\) 0 0
\(669\) −1.32156e6 −0.114162
\(670\) 0 0
\(671\) −3.52340e6 −0.302104
\(672\) 0 0
\(673\) −7.25182e6 −0.617177 −0.308588 0.951196i \(-0.599857\pi\)
−0.308588 + 0.951196i \(0.599857\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −4.74893e6 −0.398221 −0.199110 0.979977i \(-0.563805\pi\)
−0.199110 + 0.979977i \(0.563805\pi\)
\(678\) 0 0
\(679\) −2.09073e7 −1.74030
\(680\) 0 0
\(681\) 1.11169e6 0.0918578
\(682\) 0 0
\(683\) 1.37712e7 1.12959 0.564793 0.825233i \(-0.308956\pi\)
0.564793 + 0.825233i \(0.308956\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.56069e6 −0.126161
\(688\) 0 0
\(689\) 2.89882e7 2.32634
\(690\) 0 0
\(691\) −1.64305e7 −1.30905 −0.654526 0.756040i \(-0.727132\pi\)
−0.654526 + 0.756040i \(0.727132\pi\)
\(692\) 0 0
\(693\) −2.01871e7 −1.59676
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −207550. −0.0161823
\(698\) 0 0
\(699\) 1.89628e6 0.146794
\(700\) 0 0
\(701\) −1.05170e7 −0.808342 −0.404171 0.914683i \(-0.632440\pi\)
−0.404171 + 0.914683i \(0.632440\pi\)
\(702\) 0 0
\(703\) −1.32797e7 −1.01345
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 3.75041e7 2.82182
\(708\) 0 0
\(709\) −1.32691e6 −0.0991349 −0.0495674 0.998771i \(-0.515784\pi\)
−0.0495674 + 0.998771i \(0.515784\pi\)
\(710\) 0 0
\(711\) −1.98643e7 −1.47367
\(712\) 0 0
\(713\) 4.56962e6 0.336633
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −8.77958e6 −0.637787
\(718\) 0 0
\(719\) −4.65753e6 −0.335995 −0.167998 0.985787i \(-0.553730\pi\)
−0.167998 + 0.985787i \(0.553730\pi\)
\(720\) 0 0
\(721\) −2.54691e7 −1.82463
\(722\) 0 0
\(723\) −231933. −0.0165012
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.47884e6 −0.103773 −0.0518867 0.998653i \(-0.516523\pi\)
−0.0518867 + 0.998653i \(0.516523\pi\)
\(728\) 0 0
\(729\) −4.74681e6 −0.330814
\(730\) 0 0
\(731\) 671432. 0.0464739
\(732\) 0 0
\(733\) −1.41118e7 −0.970115 −0.485057 0.874482i \(-0.661201\pi\)
−0.485057 + 0.874482i \(0.661201\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −8.84856e6 −0.600073
\(738\) 0 0
\(739\) 5.94045e6 0.400137 0.200068 0.979782i \(-0.435884\pi\)
0.200068 + 0.979782i \(0.435884\pi\)
\(740\) 0 0
\(741\) −9.19780e6 −0.615373
\(742\) 0 0
\(743\) 1.46589e7 0.974156 0.487078 0.873359i \(-0.338063\pi\)
0.487078 + 0.873359i \(0.338063\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −1.24413e7 −0.815766
\(748\) 0 0
\(749\) −7.51817e6 −0.489674
\(750\) 0 0
\(751\) 6.08263e6 0.393543 0.196771 0.980449i \(-0.436954\pi\)
0.196771 + 0.980449i \(0.436954\pi\)
\(752\) 0 0
\(753\) −4.40760e6 −0.283279
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −8.11757e6 −0.514856 −0.257428 0.966297i \(-0.582875\pi\)
−0.257428 + 0.966297i \(0.582875\pi\)
\(758\) 0 0
\(759\) 3.57044e6 0.224966
\(760\) 0 0
\(761\) 1.63567e7 1.02384 0.511921 0.859033i \(-0.328934\pi\)
0.511921 + 0.859033i \(0.328934\pi\)
\(762\) 0 0
\(763\) 1.75159e7 1.08923
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −8.41267e6 −0.516352
\(768\) 0 0
\(769\) −2.09586e7 −1.27805 −0.639024 0.769187i \(-0.720661\pi\)
−0.639024 + 0.769187i \(0.720661\pi\)
\(770\) 0 0
\(771\) −5.16693e6 −0.313038
\(772\) 0 0
\(773\) 8.95651e6 0.539126 0.269563 0.962983i \(-0.413121\pi\)
0.269563 + 0.962983i \(0.413121\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 8.72873e6 0.518679
\(778\) 0 0
\(779\) −1.10722e7 −0.653717
\(780\) 0 0
\(781\) −6.74160e6 −0.395490
\(782\) 0 0
\(783\) 8.97614e6 0.523221
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −2.42009e7 −1.39282 −0.696411 0.717643i \(-0.745221\pi\)
−0.696411 + 0.717643i \(0.745221\pi\)
\(788\) 0 0
\(789\) −8.40683e6 −0.480773
\(790\) 0 0
\(791\) −1.26463e7 −0.718657
\(792\) 0 0
\(793\) −7.45219e6 −0.420824
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −9.91659e6 −0.552989 −0.276494 0.961016i \(-0.589173\pi\)
−0.276494 + 0.961016i \(0.589173\pi\)
\(798\) 0 0
\(799\) −103610. −0.00574164
\(800\) 0 0
\(801\) 2.97778e6 0.163988
\(802\) 0 0
\(803\) 2.00481e7 1.09720
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 6.90158e6 0.373048
\(808\) 0 0
\(809\) −9.15657e6 −0.491883 −0.245941 0.969285i \(-0.579097\pi\)
−0.245941 + 0.969285i \(0.579097\pi\)
\(810\) 0 0
\(811\) −1.49145e7 −0.796264 −0.398132 0.917328i \(-0.630341\pi\)
−0.398132 + 0.917328i \(0.630341\pi\)
\(812\) 0 0
\(813\) −4.63343e6 −0.245853
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 3.58189e7 1.87740
\(818\) 0 0
\(819\) −4.26967e7 −2.22426
\(820\) 0 0
\(821\) 9.39549e6 0.486476 0.243238 0.969967i \(-0.421790\pi\)
0.243238 + 0.969967i \(0.421790\pi\)
\(822\) 0 0
\(823\) 1.59508e7 0.820884 0.410442 0.911887i \(-0.365374\pi\)
0.410442 + 0.911887i \(0.365374\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.68518e7 0.856805 0.428403 0.903588i \(-0.359077\pi\)
0.428403 + 0.903588i \(0.359077\pi\)
\(828\) 0 0
\(829\) −3.92620e6 −0.198420 −0.0992101 0.995067i \(-0.531632\pi\)
−0.0992101 + 0.995067i \(0.531632\pi\)
\(830\) 0 0
\(831\) 8.34865e6 0.419386
\(832\) 0 0
\(833\) −559229. −0.0279240
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −8.82521e6 −0.435423
\(838\) 0 0
\(839\) 1.23626e7 0.606322 0.303161 0.952939i \(-0.401958\pi\)
0.303161 + 0.952939i \(0.401958\pi\)
\(840\) 0 0
\(841\) −7.64726e6 −0.372834
\(842\) 0 0
\(843\) −1.08688e7 −0.526760
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.71626e7 0.822006
\(848\) 0 0
\(849\) −1.11392e7 −0.530379
\(850\) 0 0
\(851\) 1.09031e7 0.516091
\(852\) 0 0
\(853\) 3.17612e6 0.149460 0.0747299 0.997204i \(-0.476191\pi\)
0.0747299 + 0.997204i \(0.476191\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.97233e7 −0.917334 −0.458667 0.888608i \(-0.651673\pi\)
−0.458667 + 0.888608i \(0.651673\pi\)
\(858\) 0 0
\(859\) −4.18456e6 −0.193494 −0.0967470 0.995309i \(-0.530844\pi\)
−0.0967470 + 0.995309i \(0.530844\pi\)
\(860\) 0 0
\(861\) 7.27771e6 0.334570
\(862\) 0 0
\(863\) −2.89684e7 −1.32403 −0.662015 0.749490i \(-0.730299\pi\)
−0.662015 + 0.749490i \(0.730299\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −7.79021e6 −0.351966
\(868\) 0 0
\(869\) 4.68352e7 2.10389
\(870\) 0 0
\(871\) −1.87152e7 −0.835889
\(872\) 0 0
\(873\) 2.35508e7 1.04585
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2.78801e7 −1.22404 −0.612019 0.790843i \(-0.709642\pi\)
−0.612019 + 0.790843i \(0.709642\pi\)
\(878\) 0 0
\(879\) 1.22504e7 0.534783
\(880\) 0 0
\(881\) −4.21471e7 −1.82948 −0.914741 0.404040i \(-0.867606\pi\)
−0.914741 + 0.404040i \(0.867606\pi\)
\(882\) 0 0
\(883\) −4.38615e7 −1.89314 −0.946568 0.322505i \(-0.895475\pi\)
−0.946568 + 0.322505i \(0.895475\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) −2.32386e7 −0.991747 −0.495874 0.868395i \(-0.665152\pi\)
−0.495874 + 0.868395i \(0.665152\pi\)
\(888\) 0 0
\(889\) 1.11489e7 0.473126
\(890\) 0 0
\(891\) 1.90637e7 0.804476
\(892\) 0 0
\(893\) −5.52731e6 −0.231945
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 7.55168e6 0.313374
\(898\) 0 0
\(899\) −1.26476e7 −0.521925
\(900\) 0 0
\(901\) −807970. −0.0331576
\(902\) 0 0
\(903\) −2.35437e7 −0.960848
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −7.74916e6 −0.312778 −0.156389 0.987696i \(-0.549985\pi\)
−0.156389 + 0.987696i \(0.549985\pi\)
\(908\) 0 0
\(909\) −4.22459e7 −1.69580
\(910\) 0 0
\(911\) −1.54221e7 −0.615668 −0.307834 0.951440i \(-0.599604\pi\)
−0.307834 + 0.951440i \(0.599604\pi\)
\(912\) 0 0
\(913\) 2.93336e7 1.16463
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 3.95661e7 1.55382
\(918\) 0 0
\(919\) 2.40744e7 0.940302 0.470151 0.882586i \(-0.344199\pi\)
0.470151 + 0.882586i \(0.344199\pi\)
\(920\) 0 0
\(921\) 1.83540e6 0.0712989
\(922\) 0 0
\(923\) −1.42588e7 −0.550909
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 2.86893e7 1.09653
\(928\) 0 0
\(929\) −1.51619e7 −0.576387 −0.288193 0.957572i \(-0.593055\pi\)
−0.288193 + 0.957572i \(0.593055\pi\)
\(930\) 0 0
\(931\) −2.98333e7 −1.12804
\(932\) 0 0
\(933\) −1.31260e7 −0.493662
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −4.90390e6 −0.182470 −0.0912352 0.995829i \(-0.529082\pi\)
−0.0912352 + 0.995829i \(0.529082\pi\)
\(938\) 0 0
\(939\) −1.03521e7 −0.383146
\(940\) 0 0
\(941\) −1.62296e7 −0.597495 −0.298748 0.954332i \(-0.596569\pi\)
−0.298748 + 0.954332i \(0.596569\pi\)
\(942\) 0 0
\(943\) 9.09062e6 0.332900
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −4.16817e7 −1.51033 −0.755163 0.655537i \(-0.772442\pi\)
−0.755163 + 0.655537i \(0.772442\pi\)
\(948\) 0 0
\(949\) 4.24028e7 1.52837
\(950\) 0 0
\(951\) 986202. 0.0353602
\(952\) 0 0
\(953\) −1.61135e7 −0.574723 −0.287362 0.957822i \(-0.592778\pi\)
−0.287362 + 0.957822i \(0.592778\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −9.88212e6 −0.348795
\(958\) 0 0
\(959\) 6.16851e7 2.16588
\(960\) 0 0
\(961\) −1.61942e7 −0.565655
\(962\) 0 0
\(963\) 8.46873e6 0.294274
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.51707e7 0.521722 0.260861 0.965376i \(-0.415994\pi\)
0.260861 + 0.965376i \(0.415994\pi\)
\(968\) 0 0
\(969\) 256365. 0.00877099
\(970\) 0 0
\(971\) 2.82884e7 0.962854 0.481427 0.876486i \(-0.340119\pi\)
0.481427 + 0.876486i \(0.340119\pi\)
\(972\) 0 0
\(973\) 4.41519e7 1.49509
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 2.88218e7 0.966017 0.483009 0.875616i \(-0.339544\pi\)
0.483009 + 0.875616i \(0.339544\pi\)
\(978\) 0 0
\(979\) −7.02086e6 −0.234117
\(980\) 0 0
\(981\) −1.97305e7 −0.654586
\(982\) 0 0
\(983\) 2.26448e6 0.0747453 0.0373727 0.999301i \(-0.488101\pi\)
0.0373727 + 0.999301i \(0.488101\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 3.63308e6 0.118709
\(988\) 0 0
\(989\) −2.94085e7 −0.956053
\(990\) 0 0
\(991\) −995592. −0.0322031 −0.0161015 0.999870i \(-0.505125\pi\)
−0.0161015 + 0.999870i \(0.505125\pi\)
\(992\) 0 0
\(993\) −925721. −0.0297925
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 5.57133e7 1.77509 0.887547 0.460718i \(-0.152408\pi\)
0.887547 + 0.460718i \(0.152408\pi\)
\(998\) 0 0
\(999\) −2.10569e7 −0.667546
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 400.6.a.z.1.3 4
4.3 odd 2 200.6.a.k.1.2 4
5.2 odd 4 80.6.c.d.49.3 8
5.3 odd 4 80.6.c.d.49.6 8
5.4 even 2 400.6.a.ba.1.2 4
15.2 even 4 720.6.f.n.289.8 8
15.8 even 4 720.6.f.n.289.7 8
20.3 even 4 40.6.c.a.9.3 8
20.7 even 4 40.6.c.a.9.6 yes 8
20.19 odd 2 200.6.a.j.1.3 4
40.3 even 4 320.6.c.j.129.6 8
40.13 odd 4 320.6.c.i.129.3 8
40.27 even 4 320.6.c.j.129.3 8
40.37 odd 4 320.6.c.i.129.6 8
60.23 odd 4 360.6.f.b.289.7 8
60.47 odd 4 360.6.f.b.289.8 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
40.6.c.a.9.3 8 20.3 even 4
40.6.c.a.9.6 yes 8 20.7 even 4
80.6.c.d.49.3 8 5.2 odd 4
80.6.c.d.49.6 8 5.3 odd 4
200.6.a.j.1.3 4 20.19 odd 2
200.6.a.k.1.2 4 4.3 odd 2
320.6.c.i.129.3 8 40.13 odd 4
320.6.c.i.129.6 8 40.37 odd 4
320.6.c.j.129.3 8 40.27 even 4
320.6.c.j.129.6 8 40.3 even 4
360.6.f.b.289.7 8 60.23 odd 4
360.6.f.b.289.8 8 60.47 odd 4
400.6.a.z.1.3 4 1.1 even 1 trivial
400.6.a.ba.1.2 4 5.4 even 2
720.6.f.n.289.7 8 15.8 even 4
720.6.f.n.289.8 8 15.2 even 4