Properties

Label 882.6.a.z.1.1
Level $882$
Weight $6$
Character 882.1
Self dual yes
Analytic conductor $141.459$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [882,6,Mod(1,882)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(882, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 6, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("882.1");
 
S:= CuspForms(chi, 6);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 6 \)
Character orbit: \([\chi]\) \(=\) 882.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: \(141.458529075\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 294)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 882.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-4.00000 q^{2} +16.0000 q^{4} -61.0711 q^{5} -64.0000 q^{8} +O(q^{10})\) \(q-4.00000 q^{2} +16.0000 q^{4} -61.0711 q^{5} -64.0000 q^{8} +244.284 q^{10} +36.5442 q^{11} -34.5656 q^{13} +256.000 q^{16} -2061.04 q^{17} -452.238 q^{19} -977.137 q^{20} -146.177 q^{22} -1684.25 q^{23} +604.675 q^{25} +138.262 q^{26} +4765.22 q^{29} +5260.87 q^{31} -1024.00 q^{32} +8244.16 q^{34} -12822.0 q^{37} +1808.95 q^{38} +3908.55 q^{40} -7126.74 q^{41} +11141.7 q^{43} +584.706 q^{44} +6736.99 q^{46} -23443.2 q^{47} -2418.70 q^{50} -553.049 q^{52} +7030.30 q^{53} -2231.79 q^{55} -19060.9 q^{58} -44222.8 q^{59} -19380.5 q^{61} -21043.5 q^{62} +4096.00 q^{64} +2110.96 q^{65} +20944.1 q^{67} -32976.6 q^{68} -79843.4 q^{71} +37065.8 q^{73} +51287.8 q^{74} -7235.81 q^{76} +42072.0 q^{79} -15634.2 q^{80} +28506.9 q^{82} -6311.34 q^{83} +125870. q^{85} -44566.8 q^{86} -2338.83 q^{88} -51495.8 q^{89} -26948.0 q^{92} +93772.8 q^{94} +27618.7 q^{95} +127357. q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{2} + 32 q^{4} - 108 q^{5} - 128 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{2} + 32 q^{4} - 108 q^{5} - 128 q^{8} + 432 q^{10} + 124 q^{11} + 720 q^{13} + 512 q^{16} - 612 q^{17} + 2088 q^{19} - 1728 q^{20} - 496 q^{22} - 772 q^{23} - 318 q^{25} - 2880 q^{26} + 4592 q^{29} + 9792 q^{31} - 2048 q^{32} + 2448 q^{34} - 5992 q^{37} - 8352 q^{38} + 6912 q^{40} - 20196 q^{41} - 1136 q^{43} + 1984 q^{44} + 3088 q^{46} - 36936 q^{47} + 1272 q^{50} + 11520 q^{52} + 16708 q^{53} - 6336 q^{55} - 18368 q^{58} - 74592 q^{59} - 18648 q^{61} - 39168 q^{62} + 8192 q^{64} - 33300 q^{65} + 67344 q^{67} - 9792 q^{68} - 76548 q^{71} + 47304 q^{73} + 23968 q^{74} + 33408 q^{76} + 140656 q^{79} - 27648 q^{80} + 80784 q^{82} - 94104 q^{83} + 57868 q^{85} + 4544 q^{86} - 7936 q^{88} + 17604 q^{89} - 12352 q^{92} + 147744 q^{94} - 91592 q^{95} + 85176 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −4.00000 −0.707107
\(3\) 0 0
\(4\) 16.0000 0.500000
\(5\) −61.0711 −1.09247 −0.546236 0.837631i \(-0.683940\pi\)
−0.546236 + 0.837631i \(0.683940\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −64.0000 −0.353553
\(9\) 0 0
\(10\) 244.284 0.772495
\(11\) 36.5442 0.0910618 0.0455309 0.998963i \(-0.485502\pi\)
0.0455309 + 0.998963i \(0.485502\pi\)
\(12\) 0 0
\(13\) −34.5656 −0.0567264 −0.0283632 0.999598i \(-0.509030\pi\)
−0.0283632 + 0.999598i \(0.509030\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 0.250000
\(17\) −2061.04 −1.72967 −0.864836 0.502054i \(-0.832578\pi\)
−0.864836 + 0.502054i \(0.832578\pi\)
\(18\) 0 0
\(19\) −452.238 −0.287398 −0.143699 0.989621i \(-0.545900\pi\)
−0.143699 + 0.989621i \(0.545900\pi\)
\(20\) −977.137 −0.546236
\(21\) 0 0
\(22\) −146.177 −0.0643904
\(23\) −1684.25 −0.663875 −0.331938 0.943301i \(-0.607702\pi\)
−0.331938 + 0.943301i \(0.607702\pi\)
\(24\) 0 0
\(25\) 604.675 0.193496
\(26\) 138.262 0.0401116
\(27\) 0 0
\(28\) 0 0
\(29\) 4765.22 1.05217 0.526087 0.850431i \(-0.323659\pi\)
0.526087 + 0.850431i \(0.323659\pi\)
\(30\) 0 0
\(31\) 5260.87 0.983225 0.491613 0.870814i \(-0.336408\pi\)
0.491613 + 0.870814i \(0.336408\pi\)
\(32\) −1024.00 −0.176777
\(33\) 0 0
\(34\) 8244.16 1.22306
\(35\) 0 0
\(36\) 0 0
\(37\) −12822.0 −1.53975 −0.769875 0.638195i \(-0.779681\pi\)
−0.769875 + 0.638195i \(0.779681\pi\)
\(38\) 1808.95 0.203221
\(39\) 0 0
\(40\) 3908.55 0.386247
\(41\) −7126.74 −0.662111 −0.331056 0.943611i \(-0.607405\pi\)
−0.331056 + 0.943611i \(0.607405\pi\)
\(42\) 0 0
\(43\) 11141.7 0.918925 0.459462 0.888197i \(-0.348042\pi\)
0.459462 + 0.888197i \(0.348042\pi\)
\(44\) 584.706 0.0455309
\(45\) 0 0
\(46\) 6736.99 0.469431
\(47\) −23443.2 −1.54800 −0.774002 0.633183i \(-0.781748\pi\)
−0.774002 + 0.633183i \(0.781748\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −2418.70 −0.136822
\(51\) 0 0
\(52\) −553.049 −0.0283632
\(53\) 7030.30 0.343783 0.171891 0.985116i \(-0.445012\pi\)
0.171891 + 0.985116i \(0.445012\pi\)
\(54\) 0 0
\(55\) −2231.79 −0.0994825
\(56\) 0 0
\(57\) 0 0
\(58\) −19060.9 −0.744000
\(59\) −44222.8 −1.65393 −0.826964 0.562255i \(-0.809934\pi\)
−0.826964 + 0.562255i \(0.809934\pi\)
\(60\) 0 0
\(61\) −19380.5 −0.666868 −0.333434 0.942773i \(-0.608207\pi\)
−0.333434 + 0.942773i \(0.608207\pi\)
\(62\) −21043.5 −0.695245
\(63\) 0 0
\(64\) 4096.00 0.125000
\(65\) 2110.96 0.0619721
\(66\) 0 0
\(67\) 20944.1 0.569999 0.285000 0.958528i \(-0.408007\pi\)
0.285000 + 0.958528i \(0.408007\pi\)
\(68\) −32976.6 −0.864836
\(69\) 0 0
\(70\) 0 0
\(71\) −79843.4 −1.87972 −0.939860 0.341560i \(-0.889045\pi\)
−0.939860 + 0.341560i \(0.889045\pi\)
\(72\) 0 0
\(73\) 37065.8 0.814079 0.407039 0.913411i \(-0.366561\pi\)
0.407039 + 0.913411i \(0.366561\pi\)
\(74\) 51287.8 1.08877
\(75\) 0 0
\(76\) −7235.81 −0.143699
\(77\) 0 0
\(78\) 0 0
\(79\) 42072.0 0.758448 0.379224 0.925305i \(-0.376191\pi\)
0.379224 + 0.925305i \(0.376191\pi\)
\(80\) −15634.2 −0.273118
\(81\) 0 0
\(82\) 28506.9 0.468184
\(83\) −6311.34 −0.100560 −0.0502801 0.998735i \(-0.516011\pi\)
−0.0502801 + 0.998735i \(0.516011\pi\)
\(84\) 0 0
\(85\) 125870. 1.88962
\(86\) −44566.8 −0.649778
\(87\) 0 0
\(88\) −2338.83 −0.0321952
\(89\) −51495.8 −0.689123 −0.344562 0.938764i \(-0.611972\pi\)
−0.344562 + 0.938764i \(0.611972\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −26948.0 −0.331938
\(93\) 0 0
\(94\) 93772.8 1.09460
\(95\) 27618.7 0.313974
\(96\) 0 0
\(97\) 127357. 1.37434 0.687171 0.726496i \(-0.258852\pi\)
0.687171 + 0.726496i \(0.258852\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 9674.81 0.0967481
\(101\) 65757.9 0.641423 0.320712 0.947177i \(-0.396078\pi\)
0.320712 + 0.947177i \(0.396078\pi\)
\(102\) 0 0
\(103\) −170010. −1.57900 −0.789499 0.613751i \(-0.789660\pi\)
−0.789499 + 0.613751i \(0.789660\pi\)
\(104\) 2212.20 0.0200558
\(105\) 0 0
\(106\) −28121.2 −0.243091
\(107\) 107933. 0.911369 0.455685 0.890141i \(-0.349395\pi\)
0.455685 + 0.890141i \(0.349395\pi\)
\(108\) 0 0
\(109\) −222698. −1.79536 −0.897679 0.440651i \(-0.854748\pi\)
−0.897679 + 0.440651i \(0.854748\pi\)
\(110\) 8927.16 0.0703448
\(111\) 0 0
\(112\) 0 0
\(113\) −181055. −1.33387 −0.666936 0.745115i \(-0.732395\pi\)
−0.666936 + 0.745115i \(0.732395\pi\)
\(114\) 0 0
\(115\) 102859. 0.725265
\(116\) 76243.5 0.526087
\(117\) 0 0
\(118\) 176891. 1.16950
\(119\) 0 0
\(120\) 0 0
\(121\) −159716. −0.991708
\(122\) 77521.9 0.471547
\(123\) 0 0
\(124\) 84173.9 0.491613
\(125\) 153919. 0.881083
\(126\) 0 0
\(127\) 91474.7 0.503259 0.251630 0.967824i \(-0.419034\pi\)
0.251630 + 0.967824i \(0.419034\pi\)
\(128\) −16384.0 −0.0883883
\(129\) 0 0
\(130\) −8443.83 −0.0438209
\(131\) −19343.9 −0.0984840 −0.0492420 0.998787i \(-0.515681\pi\)
−0.0492420 + 0.998787i \(0.515681\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −83776.3 −0.403050
\(135\) 0 0
\(136\) 131906. 0.611532
\(137\) 3077.04 0.0140065 0.00700327 0.999975i \(-0.497771\pi\)
0.00700327 + 0.999975i \(0.497771\pi\)
\(138\) 0 0
\(139\) −370001. −1.62430 −0.812149 0.583450i \(-0.801702\pi\)
−0.812149 + 0.583450i \(0.801702\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 319374. 1.32916
\(143\) −1263.17 −0.00516561
\(144\) 0 0
\(145\) −291017. −1.14947
\(146\) −148263. −0.575641
\(147\) 0 0
\(148\) −205151. −0.769875
\(149\) 454120. 1.67573 0.837866 0.545875i \(-0.183803\pi\)
0.837866 + 0.545875i \(0.183803\pi\)
\(150\) 0 0
\(151\) 177592. 0.633841 0.316921 0.948452i \(-0.397351\pi\)
0.316921 + 0.948452i \(0.397351\pi\)
\(152\) 28943.2 0.101610
\(153\) 0 0
\(154\) 0 0
\(155\) −321287. −1.07415
\(156\) 0 0
\(157\) 248981. 0.806153 0.403076 0.915166i \(-0.367941\pi\)
0.403076 + 0.915166i \(0.367941\pi\)
\(158\) −168288. −0.536303
\(159\) 0 0
\(160\) 62536.8 0.193124
\(161\) 0 0
\(162\) 0 0
\(163\) 202248. 0.596231 0.298116 0.954530i \(-0.403642\pi\)
0.298116 + 0.954530i \(0.403642\pi\)
\(164\) −114028. −0.331056
\(165\) 0 0
\(166\) 25245.3 0.0711068
\(167\) 475790. 1.32015 0.660076 0.751199i \(-0.270524\pi\)
0.660076 + 0.751199i \(0.270524\pi\)
\(168\) 0 0
\(169\) −370098. −0.996782
\(170\) −503479. −1.33616
\(171\) 0 0
\(172\) 178267. 0.459462
\(173\) −284703. −0.723231 −0.361616 0.932327i \(-0.617775\pi\)
−0.361616 + 0.932327i \(0.617775\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 9355.30 0.0227654
\(177\) 0 0
\(178\) 205983. 0.487284
\(179\) −347958. −0.811699 −0.405849 0.913940i \(-0.633024\pi\)
−0.405849 + 0.913940i \(0.633024\pi\)
\(180\) 0 0
\(181\) −379706. −0.861492 −0.430746 0.902473i \(-0.641750\pi\)
−0.430746 + 0.902473i \(0.641750\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 107792. 0.234715
\(185\) 783051. 1.68213
\(186\) 0 0
\(187\) −75318.9 −0.157507
\(188\) −375091. −0.774002
\(189\) 0 0
\(190\) −110475. −0.222013
\(191\) −194492. −0.385762 −0.192881 0.981222i \(-0.561783\pi\)
−0.192881 + 0.981222i \(0.561783\pi\)
\(192\) 0 0
\(193\) 877401. 1.69553 0.847763 0.530375i \(-0.177949\pi\)
0.847763 + 0.530375i \(0.177949\pi\)
\(194\) −509430. −0.971806
\(195\) 0 0
\(196\) 0 0
\(197\) 752368. 1.38123 0.690613 0.723225i \(-0.257341\pi\)
0.690613 + 0.723225i \(0.257341\pi\)
\(198\) 0 0
\(199\) 155453. 0.278270 0.139135 0.990273i \(-0.455568\pi\)
0.139135 + 0.990273i \(0.455568\pi\)
\(200\) −38699.2 −0.0684112
\(201\) 0 0
\(202\) −263032. −0.453555
\(203\) 0 0
\(204\) 0 0
\(205\) 435237. 0.723339
\(206\) 680041. 1.11652
\(207\) 0 0
\(208\) −8848.79 −0.0141816
\(209\) −16526.7 −0.0261709
\(210\) 0 0
\(211\) 3898.23 0.00602783 0.00301391 0.999995i \(-0.499041\pi\)
0.00301391 + 0.999995i \(0.499041\pi\)
\(212\) 112485. 0.171891
\(213\) 0 0
\(214\) −431731. −0.644435
\(215\) −680435. −1.00390
\(216\) 0 0
\(217\) 0 0
\(218\) 890794. 1.26951
\(219\) 0 0
\(220\) −35708.6 −0.0497413
\(221\) 71241.0 0.0981182
\(222\) 0 0
\(223\) −782574. −1.05381 −0.526906 0.849924i \(-0.676648\pi\)
−0.526906 + 0.849924i \(0.676648\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 724220. 0.943190
\(227\) 796543. 1.02599 0.512997 0.858391i \(-0.328535\pi\)
0.512997 + 0.858391i \(0.328535\pi\)
\(228\) 0 0
\(229\) −387045. −0.487722 −0.243861 0.969810i \(-0.578414\pi\)
−0.243861 + 0.969810i \(0.578414\pi\)
\(230\) −411435. −0.512840
\(231\) 0 0
\(232\) −304974. −0.372000
\(233\) 383759. 0.463093 0.231547 0.972824i \(-0.425621\pi\)
0.231547 + 0.972824i \(0.425621\pi\)
\(234\) 0 0
\(235\) 1.43170e6 1.69115
\(236\) −707565. −0.826964
\(237\) 0 0
\(238\) 0 0
\(239\) 465409. 0.527036 0.263518 0.964654i \(-0.415117\pi\)
0.263518 + 0.964654i \(0.415117\pi\)
\(240\) 0 0
\(241\) 348744. 0.386781 0.193390 0.981122i \(-0.438052\pi\)
0.193390 + 0.981122i \(0.438052\pi\)
\(242\) 638862. 0.701243
\(243\) 0 0
\(244\) −310088. −0.333434
\(245\) 0 0
\(246\) 0 0
\(247\) 15631.9 0.0163030
\(248\) −336695. −0.347623
\(249\) 0 0
\(250\) −615676. −0.623020
\(251\) −186543. −0.186894 −0.0934468 0.995624i \(-0.529789\pi\)
−0.0934468 + 0.995624i \(0.529789\pi\)
\(252\) 0 0
\(253\) −61549.4 −0.0604537
\(254\) −365899. −0.355858
\(255\) 0 0
\(256\) 65536.0 0.0625000
\(257\) 1.55355e6 1.46721 0.733605 0.679577i \(-0.237836\pi\)
0.733605 + 0.679577i \(0.237836\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 33775.3 0.0309860
\(261\) 0 0
\(262\) 77375.6 0.0696387
\(263\) 1.25960e6 1.12290 0.561451 0.827510i \(-0.310243\pi\)
0.561451 + 0.827510i \(0.310243\pi\)
\(264\) 0 0
\(265\) −429348. −0.375573
\(266\) 0 0
\(267\) 0 0
\(268\) 335105. 0.285000
\(269\) −170461. −0.143630 −0.0718149 0.997418i \(-0.522879\pi\)
−0.0718149 + 0.997418i \(0.522879\pi\)
\(270\) 0 0
\(271\) −2.22743e6 −1.84239 −0.921194 0.389104i \(-0.872785\pi\)
−0.921194 + 0.389104i \(0.872785\pi\)
\(272\) −527626. −0.432418
\(273\) 0 0
\(274\) −12308.1 −0.00990412
\(275\) 22097.3 0.0176201
\(276\) 0 0
\(277\) 1.98339e6 1.55314 0.776568 0.630034i \(-0.216959\pi\)
0.776568 + 0.630034i \(0.216959\pi\)
\(278\) 1.48000e6 1.14855
\(279\) 0 0
\(280\) 0 0
\(281\) 1.16687e6 0.881573 0.440786 0.897612i \(-0.354700\pi\)
0.440786 + 0.897612i \(0.354700\pi\)
\(282\) 0 0
\(283\) 1.90090e6 1.41089 0.705445 0.708765i \(-0.250747\pi\)
0.705445 + 0.708765i \(0.250747\pi\)
\(284\) −1.27749e6 −0.939860
\(285\) 0 0
\(286\) 5052.68 0.00365264
\(287\) 0 0
\(288\) 0 0
\(289\) 2.82802e6 1.99177
\(290\) 1.16407e6 0.812799
\(291\) 0 0
\(292\) 593053. 0.407039
\(293\) 1.65009e6 1.12289 0.561445 0.827514i \(-0.310245\pi\)
0.561445 + 0.827514i \(0.310245\pi\)
\(294\) 0 0
\(295\) 2.70073e6 1.80687
\(296\) 820605. 0.544384
\(297\) 0 0
\(298\) −1.81648e6 −1.18492
\(299\) 58217.0 0.0376593
\(300\) 0 0
\(301\) 0 0
\(302\) −710367. −0.448194
\(303\) 0 0
\(304\) −115773. −0.0718494
\(305\) 1.18359e6 0.728535
\(306\) 0 0
\(307\) −597936. −0.362084 −0.181042 0.983475i \(-0.557947\pi\)
−0.181042 + 0.983475i \(0.557947\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 1.28515e6 0.759536
\(311\) 230480. 0.135124 0.0675620 0.997715i \(-0.478478\pi\)
0.0675620 + 0.997715i \(0.478478\pi\)
\(312\) 0 0
\(313\) −442577. −0.255346 −0.127673 0.991816i \(-0.540751\pi\)
−0.127673 + 0.991816i \(0.540751\pi\)
\(314\) −995925. −0.570036
\(315\) 0 0
\(316\) 673152. 0.379224
\(317\) 1.26248e6 0.705627 0.352813 0.935694i \(-0.385225\pi\)
0.352813 + 0.935694i \(0.385225\pi\)
\(318\) 0 0
\(319\) 174141. 0.0958129
\(320\) −250147. −0.136559
\(321\) 0 0
\(322\) 0 0
\(323\) 932080. 0.497104
\(324\) 0 0
\(325\) −20901.0 −0.0109763
\(326\) −808991. −0.421599
\(327\) 0 0
\(328\) 456111. 0.234092
\(329\) 0 0
\(330\) 0 0
\(331\) −468936. −0.235257 −0.117629 0.993058i \(-0.537529\pi\)
−0.117629 + 0.993058i \(0.537529\pi\)
\(332\) −100981. −0.0502801
\(333\) 0 0
\(334\) −1.90316e6 −0.933489
\(335\) −1.27908e6 −0.622708
\(336\) 0 0
\(337\) −2.46798e6 −1.18377 −0.591884 0.806023i \(-0.701616\pi\)
−0.591884 + 0.806023i \(0.701616\pi\)
\(338\) 1.48039e6 0.704831
\(339\) 0 0
\(340\) 2.01392e6 0.944810
\(341\) 192254. 0.0895343
\(342\) 0 0
\(343\) 0 0
\(344\) −713068. −0.324889
\(345\) 0 0
\(346\) 1.13881e6 0.511402
\(347\) 4.28250e6 1.90930 0.954648 0.297736i \(-0.0962315\pi\)
0.954648 + 0.297736i \(0.0962315\pi\)
\(348\) 0 0
\(349\) 1.84999e6 0.813028 0.406514 0.913645i \(-0.366744\pi\)
0.406514 + 0.913645i \(0.366744\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −37421.2 −0.0160976
\(353\) −2.27380e6 −0.971213 −0.485607 0.874177i \(-0.661401\pi\)
−0.485607 + 0.874177i \(0.661401\pi\)
\(354\) 0 0
\(355\) 4.87612e6 2.05354
\(356\) −823933. −0.344562
\(357\) 0 0
\(358\) 1.39183e6 0.573958
\(359\) 4.39992e6 1.80181 0.900905 0.434017i \(-0.142904\pi\)
0.900905 + 0.434017i \(0.142904\pi\)
\(360\) 0 0
\(361\) −2.27158e6 −0.917403
\(362\) 1.51882e6 0.609167
\(363\) 0 0
\(364\) 0 0
\(365\) −2.26365e6 −0.889359
\(366\) 0 0
\(367\) 1.89743e6 0.735362 0.367681 0.929952i \(-0.380152\pi\)
0.367681 + 0.929952i \(0.380152\pi\)
\(368\) −431168. −0.165969
\(369\) 0 0
\(370\) −3.13220e6 −1.18945
\(371\) 0 0
\(372\) 0 0
\(373\) 4.56260e6 1.69801 0.849005 0.528385i \(-0.177202\pi\)
0.849005 + 0.528385i \(0.177202\pi\)
\(374\) 301276. 0.111374
\(375\) 0 0
\(376\) 1.50037e6 0.547302
\(377\) −164713. −0.0596861
\(378\) 0 0
\(379\) 4.99400e6 1.78587 0.892936 0.450184i \(-0.148642\pi\)
0.892936 + 0.450184i \(0.148642\pi\)
\(380\) 441898. 0.156987
\(381\) 0 0
\(382\) 777970. 0.272775
\(383\) −3.67907e6 −1.28157 −0.640783 0.767722i \(-0.721390\pi\)
−0.640783 + 0.767722i \(0.721390\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −3.50960e6 −1.19892
\(387\) 0 0
\(388\) 2.03772e6 0.687171
\(389\) −257234. −0.0861894 −0.0430947 0.999071i \(-0.513722\pi\)
−0.0430947 + 0.999071i \(0.513722\pi\)
\(390\) 0 0
\(391\) 3.47130e6 1.14829
\(392\) 0 0
\(393\) 0 0
\(394\) −3.00947e6 −0.976674
\(395\) −2.56938e6 −0.828583
\(396\) 0 0
\(397\) −433280. −0.137972 −0.0689862 0.997618i \(-0.521976\pi\)
−0.0689862 + 0.997618i \(0.521976\pi\)
\(398\) −621811. −0.196766
\(399\) 0 0
\(400\) 154797. 0.0483740
\(401\) 1.13824e6 0.353488 0.176744 0.984257i \(-0.443444\pi\)
0.176744 + 0.984257i \(0.443444\pi\)
\(402\) 0 0
\(403\) −181845. −0.0557749
\(404\) 1.05213e6 0.320712
\(405\) 0 0
\(406\) 0 0
\(407\) −468568. −0.140212
\(408\) 0 0
\(409\) 5.02438e6 1.48516 0.742581 0.669756i \(-0.233601\pi\)
0.742581 + 0.669756i \(0.233601\pi\)
\(410\) −1.74095e6 −0.511478
\(411\) 0 0
\(412\) −2.72016e6 −0.789499
\(413\) 0 0
\(414\) 0 0
\(415\) 385440. 0.109859
\(416\) 35395.2 0.0100279
\(417\) 0 0
\(418\) 66106.6 0.0185056
\(419\) −2.57295e6 −0.715974 −0.357987 0.933727i \(-0.616537\pi\)
−0.357987 + 0.933727i \(0.616537\pi\)
\(420\) 0 0
\(421\) 336425. 0.0925089 0.0462545 0.998930i \(-0.485271\pi\)
0.0462545 + 0.998930i \(0.485271\pi\)
\(422\) −15592.9 −0.00426232
\(423\) 0 0
\(424\) −449939. −0.121546
\(425\) −1.24626e6 −0.334685
\(426\) 0 0
\(427\) 0 0
\(428\) 1.72693e6 0.455685
\(429\) 0 0
\(430\) 2.72174e6 0.709864
\(431\) −236410. −0.0613018 −0.0306509 0.999530i \(-0.509758\pi\)
−0.0306509 + 0.999530i \(0.509758\pi\)
\(432\) 0 0
\(433\) 2.33004e6 0.597232 0.298616 0.954373i \(-0.403475\pi\)
0.298616 + 0.954373i \(0.403475\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −3.56317e6 −0.897679
\(437\) 761681. 0.190796
\(438\) 0 0
\(439\) −3.19962e6 −0.792387 −0.396193 0.918167i \(-0.629669\pi\)
−0.396193 + 0.918167i \(0.629669\pi\)
\(440\) 142835. 0.0351724
\(441\) 0 0
\(442\) −284964. −0.0693800
\(443\) −1.13240e6 −0.274152 −0.137076 0.990561i \(-0.543770\pi\)
−0.137076 + 0.990561i \(0.543770\pi\)
\(444\) 0 0
\(445\) 3.14490e6 0.752848
\(446\) 3.13029e6 0.745157
\(447\) 0 0
\(448\) 0 0
\(449\) 8.23687e6 1.92817 0.964087 0.265585i \(-0.0855651\pi\)
0.964087 + 0.265585i \(0.0855651\pi\)
\(450\) 0 0
\(451\) −260441. −0.0602931
\(452\) −2.89688e6 −0.666936
\(453\) 0 0
\(454\) −3.18617e6 −0.725487
\(455\) 0 0
\(456\) 0 0
\(457\) −365132. −0.0817824 −0.0408912 0.999164i \(-0.513020\pi\)
−0.0408912 + 0.999164i \(0.513020\pi\)
\(458\) 1.54818e6 0.344872
\(459\) 0 0
\(460\) 1.64574e6 0.362633
\(461\) 332567. 0.0728832 0.0364416 0.999336i \(-0.488398\pi\)
0.0364416 + 0.999336i \(0.488398\pi\)
\(462\) 0 0
\(463\) 1.69992e6 0.368533 0.184266 0.982876i \(-0.441009\pi\)
0.184266 + 0.982876i \(0.441009\pi\)
\(464\) 1.21990e6 0.263044
\(465\) 0 0
\(466\) −1.53504e6 −0.327456
\(467\) −990465. −0.210159 −0.105079 0.994464i \(-0.533510\pi\)
−0.105079 + 0.994464i \(0.533510\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −5.72681e6 −1.19583
\(471\) 0 0
\(472\) 2.83026e6 0.584752
\(473\) 407164. 0.0836789
\(474\) 0 0
\(475\) −273457. −0.0556103
\(476\) 0 0
\(477\) 0 0
\(478\) −1.86164e6 −0.372671
\(479\) −6.08065e6 −1.21091 −0.605454 0.795880i \(-0.707009\pi\)
−0.605454 + 0.795880i \(0.707009\pi\)
\(480\) 0 0
\(481\) 443198. 0.0873445
\(482\) −1.39498e6 −0.273495
\(483\) 0 0
\(484\) −2.55545e6 −0.495854
\(485\) −7.77785e6 −1.50143
\(486\) 0 0
\(487\) −6.71098e6 −1.28222 −0.641112 0.767448i \(-0.721526\pi\)
−0.641112 + 0.767448i \(0.721526\pi\)
\(488\) 1.24035e6 0.235773
\(489\) 0 0
\(490\) 0 0
\(491\) 914042. 0.171105 0.0855525 0.996334i \(-0.472734\pi\)
0.0855525 + 0.996334i \(0.472734\pi\)
\(492\) 0 0
\(493\) −9.82130e6 −1.81992
\(494\) −62527.5 −0.0115280
\(495\) 0 0
\(496\) 1.34678e6 0.245806
\(497\) 0 0
\(498\) 0 0
\(499\) −7.75492e6 −1.39420 −0.697101 0.716972i \(-0.745527\pi\)
−0.697101 + 0.716972i \(0.745527\pi\)
\(500\) 2.46270e6 0.440542
\(501\) 0 0
\(502\) 746172. 0.132154
\(503\) −3.79381e6 −0.668584 −0.334292 0.942470i \(-0.608497\pi\)
−0.334292 + 0.942470i \(0.608497\pi\)
\(504\) 0 0
\(505\) −4.01591e6 −0.700737
\(506\) 246198. 0.0427472
\(507\) 0 0
\(508\) 1.46360e6 0.251630
\(509\) −8.53160e6 −1.45961 −0.729804 0.683657i \(-0.760389\pi\)
−0.729804 + 0.683657i \(0.760389\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −262144. −0.0441942
\(513\) 0 0
\(514\) −6.21419e6 −1.03747
\(515\) 1.03827e7 1.72501
\(516\) 0 0
\(517\) −856712. −0.140964
\(518\) 0 0
\(519\) 0 0
\(520\) −135101. −0.0219104
\(521\) 3.19066e6 0.514976 0.257488 0.966282i \(-0.417105\pi\)
0.257488 + 0.966282i \(0.417105\pi\)
\(522\) 0 0
\(523\) −9.60678e6 −1.53576 −0.767880 0.640593i \(-0.778689\pi\)
−0.767880 + 0.640593i \(0.778689\pi\)
\(524\) −309502. −0.0492420
\(525\) 0 0
\(526\) −5.03838e6 −0.794012
\(527\) −1.08429e7 −1.70066
\(528\) 0 0
\(529\) −3.59965e6 −0.559270
\(530\) 1.71739e6 0.265570
\(531\) 0 0
\(532\) 0 0
\(533\) 246340. 0.0375592
\(534\) 0 0
\(535\) −6.59158e6 −0.995646
\(536\) −1.34042e6 −0.201525
\(537\) 0 0
\(538\) 681844. 0.101562
\(539\) 0 0
\(540\) 0 0
\(541\) 1.02366e7 1.50371 0.751855 0.659328i \(-0.229159\pi\)
0.751855 + 0.659328i \(0.229159\pi\)
\(542\) 8.90972e6 1.30276
\(543\) 0 0
\(544\) 2.11050e6 0.305766
\(545\) 1.36004e7 1.96138
\(546\) 0 0
\(547\) −9.27757e6 −1.32576 −0.662882 0.748724i \(-0.730667\pi\)
−0.662882 + 0.748724i \(0.730667\pi\)
\(548\) 49232.6 0.00700327
\(549\) 0 0
\(550\) −88389.4 −0.0124593
\(551\) −2.15501e6 −0.302392
\(552\) 0 0
\(553\) 0 0
\(554\) −7.93357e6 −1.09823
\(555\) 0 0
\(556\) −5.92001e6 −0.812149
\(557\) −620098. −0.0846881 −0.0423441 0.999103i \(-0.513483\pi\)
−0.0423441 + 0.999103i \(0.513483\pi\)
\(558\) 0 0
\(559\) −385119. −0.0521273
\(560\) 0 0
\(561\) 0 0
\(562\) −4.66750e6 −0.623366
\(563\) −7.36664e6 −0.979487 −0.489744 0.871867i \(-0.662910\pi\)
−0.489744 + 0.871867i \(0.662910\pi\)
\(564\) 0 0
\(565\) 1.10572e7 1.45722
\(566\) −7.60360e6 −0.997650
\(567\) 0 0
\(568\) 5.10998e6 0.664581
\(569\) −3.68222e6 −0.476792 −0.238396 0.971168i \(-0.576622\pi\)
−0.238396 + 0.971168i \(0.576622\pi\)
\(570\) 0 0
\(571\) 1.03320e7 1.32615 0.663077 0.748551i \(-0.269250\pi\)
0.663077 + 0.748551i \(0.269250\pi\)
\(572\) −20210.7 −0.00258281
\(573\) 0 0
\(574\) 0 0
\(575\) −1.01842e6 −0.128457
\(576\) 0 0
\(577\) −1.25123e7 −1.56458 −0.782292 0.622911i \(-0.785950\pi\)
−0.782292 + 0.622911i \(0.785950\pi\)
\(578\) −1.13121e7 −1.40839
\(579\) 0 0
\(580\) −4.65627e6 −0.574736
\(581\) 0 0
\(582\) 0 0
\(583\) 256916. 0.0313055
\(584\) −2.37221e6 −0.287820
\(585\) 0 0
\(586\) −6.60034e6 −0.794004
\(587\) 1.60865e7 1.92694 0.963468 0.267823i \(-0.0863042\pi\)
0.963468 + 0.267823i \(0.0863042\pi\)
\(588\) 0 0
\(589\) −2.37916e6 −0.282577
\(590\) −1.08029e7 −1.27765
\(591\) 0 0
\(592\) −3.28242e6 −0.384937
\(593\) 9.46330e6 1.10511 0.552556 0.833476i \(-0.313653\pi\)
0.552556 + 0.833476i \(0.313653\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 7.26592e6 0.837866
\(597\) 0 0
\(598\) −232868. −0.0266291
\(599\) −5.01498e6 −0.571087 −0.285544 0.958366i \(-0.592174\pi\)
−0.285544 + 0.958366i \(0.592174\pi\)
\(600\) 0 0
\(601\) 1.26473e7 1.42828 0.714139 0.700004i \(-0.246818\pi\)
0.714139 + 0.700004i \(0.246818\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2.84147e6 0.316921
\(605\) 9.75400e6 1.08341
\(606\) 0 0
\(607\) −2.87702e6 −0.316936 −0.158468 0.987364i \(-0.550655\pi\)
−0.158468 + 0.987364i \(0.550655\pi\)
\(608\) 463092. 0.0508052
\(609\) 0 0
\(610\) −4.73434e6 −0.515152
\(611\) 810328. 0.0878128
\(612\) 0 0
\(613\) −3.72582e6 −0.400471 −0.200236 0.979748i \(-0.564171\pi\)
−0.200236 + 0.979748i \(0.564171\pi\)
\(614\) 2.39174e6 0.256032
\(615\) 0 0
\(616\) 0 0
\(617\) 1.15861e7 1.22525 0.612625 0.790374i \(-0.290114\pi\)
0.612625 + 0.790374i \(0.290114\pi\)
\(618\) 0 0
\(619\) 1.85608e6 0.194702 0.0973508 0.995250i \(-0.468963\pi\)
0.0973508 + 0.995250i \(0.468963\pi\)
\(620\) −5.14059e6 −0.537073
\(621\) 0 0
\(622\) −921921. −0.0955471
\(623\) 0 0
\(624\) 0 0
\(625\) −1.12896e7 −1.15606
\(626\) 1.77031e6 0.180557
\(627\) 0 0
\(628\) 3.98370e6 0.403076
\(629\) 2.64266e7 2.66326
\(630\) 0 0
\(631\) 8.92135e6 0.891984 0.445992 0.895037i \(-0.352851\pi\)
0.445992 + 0.895037i \(0.352851\pi\)
\(632\) −2.69261e6 −0.268152
\(633\) 0 0
\(634\) −5.04990e6 −0.498953
\(635\) −5.58646e6 −0.549797
\(636\) 0 0
\(637\) 0 0
\(638\) −696563. −0.0677499
\(639\) 0 0
\(640\) 1.00059e6 0.0965618
\(641\) −1.31866e7 −1.26762 −0.633810 0.773489i \(-0.718510\pi\)
−0.633810 + 0.773489i \(0.718510\pi\)
\(642\) 0 0
\(643\) 9.67813e6 0.923132 0.461566 0.887106i \(-0.347288\pi\)
0.461566 + 0.887106i \(0.347288\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) −3.72832e6 −0.351505
\(647\) −1.72491e6 −0.161996 −0.0809982 0.996714i \(-0.525811\pi\)
−0.0809982 + 0.996714i \(0.525811\pi\)
\(648\) 0 0
\(649\) −1.61609e6 −0.150610
\(650\) 83603.8 0.00776145
\(651\) 0 0
\(652\) 3.23596e6 0.298116
\(653\) −597510. −0.0548356 −0.0274178 0.999624i \(-0.508728\pi\)
−0.0274178 + 0.999624i \(0.508728\pi\)
\(654\) 0 0
\(655\) 1.18135e6 0.107591
\(656\) −1.82444e6 −0.165528
\(657\) 0 0
\(658\) 0 0
\(659\) −1.47605e7 −1.32400 −0.662001 0.749503i \(-0.730293\pi\)
−0.662001 + 0.749503i \(0.730293\pi\)
\(660\) 0 0
\(661\) −1.69041e7 −1.50484 −0.752418 0.658686i \(-0.771112\pi\)
−0.752418 + 0.658686i \(0.771112\pi\)
\(662\) 1.87574e6 0.166352
\(663\) 0 0
\(664\) 403925. 0.0355534
\(665\) 0 0
\(666\) 0 0
\(667\) −8.02581e6 −0.698512
\(668\) 7.61264e6 0.660076
\(669\) 0 0
\(670\) 5.11631e6 0.440321
\(671\) −708243. −0.0607262
\(672\) 0 0
\(673\) −1.58960e7 −1.35285 −0.676425 0.736512i \(-0.736472\pi\)
−0.676425 + 0.736512i \(0.736472\pi\)
\(674\) 9.87191e6 0.837050
\(675\) 0 0
\(676\) −5.92157e6 −0.498391
\(677\) −1.95361e6 −0.163820 −0.0819099 0.996640i \(-0.526102\pi\)
−0.0819099 + 0.996640i \(0.526102\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −8.05567e6 −0.668081
\(681\) 0 0
\(682\) −769016. −0.0633103
\(683\) 9.23970e6 0.757890 0.378945 0.925419i \(-0.376287\pi\)
0.378945 + 0.925419i \(0.376287\pi\)
\(684\) 0 0
\(685\) −187918. −0.0153018
\(686\) 0 0
\(687\) 0 0
\(688\) 2.85227e6 0.229731
\(689\) −243006. −0.0195016
\(690\) 0 0
\(691\) 1.45698e7 1.16080 0.580401 0.814331i \(-0.302896\pi\)
0.580401 + 0.814331i \(0.302896\pi\)
\(692\) −4.55525e6 −0.361616
\(693\) 0 0
\(694\) −1.71300e7 −1.35008
\(695\) 2.25964e7 1.77450
\(696\) 0 0
\(697\) 1.46885e7 1.14524
\(698\) −7.39995e6 −0.574898
\(699\) 0 0
\(700\) 0 0
\(701\) 3.70190e6 0.284531 0.142266 0.989829i \(-0.454561\pi\)
0.142266 + 0.989829i \(0.454561\pi\)
\(702\) 0 0
\(703\) 5.79858e6 0.442520
\(704\) 149685. 0.0113827
\(705\) 0 0
\(706\) 9.09518e6 0.686751
\(707\) 0 0
\(708\) 0 0
\(709\) −2.52937e7 −1.88971 −0.944857 0.327484i \(-0.893799\pi\)
−0.944857 + 0.327484i \(0.893799\pi\)
\(710\) −1.95045e7 −1.45207
\(711\) 0 0
\(712\) 3.29573e6 0.243642
\(713\) −8.86061e6 −0.652739
\(714\) 0 0
\(715\) 77143.1 0.00564329
\(716\) −5.56734e6 −0.405849
\(717\) 0 0
\(718\) −1.75997e7 −1.27407
\(719\) 1.45185e7 1.04737 0.523683 0.851913i \(-0.324557\pi\)
0.523683 + 0.851913i \(0.324557\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 9.08632e6 0.648702
\(723\) 0 0
\(724\) −6.07530e6 −0.430746
\(725\) 2.88141e6 0.203592
\(726\) 0 0
\(727\) 1.42855e7 1.00244 0.501222 0.865319i \(-0.332884\pi\)
0.501222 + 0.865319i \(0.332884\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 9.05460e6 0.628872
\(731\) −2.29635e7 −1.58944
\(732\) 0 0
\(733\) 8.21556e6 0.564777 0.282389 0.959300i \(-0.408873\pi\)
0.282389 + 0.959300i \(0.408873\pi\)
\(734\) −7.58973e6 −0.519979
\(735\) 0 0
\(736\) 1.72467e6 0.117358
\(737\) 765384. 0.0519052
\(738\) 0 0
\(739\) −3.43896e6 −0.231641 −0.115821 0.993270i \(-0.536950\pi\)
−0.115821 + 0.993270i \(0.536950\pi\)
\(740\) 1.25288e7 0.841067
\(741\) 0 0
\(742\) 0 0
\(743\) −1.53588e6 −0.102067 −0.0510334 0.998697i \(-0.516252\pi\)
−0.0510334 + 0.998697i \(0.516252\pi\)
\(744\) 0 0
\(745\) −2.77336e7 −1.83069
\(746\) −1.82504e7 −1.20067
\(747\) 0 0
\(748\) −1.20510e6 −0.0787535
\(749\) 0 0
\(750\) 0 0
\(751\) −2.01146e7 −1.30140 −0.650700 0.759335i \(-0.725524\pi\)
−0.650700 + 0.759335i \(0.725524\pi\)
\(752\) −6.00146e6 −0.387001
\(753\) 0 0
\(754\) 658850. 0.0422044
\(755\) −1.08457e7 −0.692454
\(756\) 0 0
\(757\) 202045. 0.0128147 0.00640733 0.999979i \(-0.497960\pi\)
0.00640733 + 0.999979i \(0.497960\pi\)
\(758\) −1.99760e7 −1.26280
\(759\) 0 0
\(760\) −1.76759e6 −0.111007
\(761\) −1.08202e6 −0.0677290 −0.0338645 0.999426i \(-0.510781\pi\)
−0.0338645 + 0.999426i \(0.510781\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −3.11188e6 −0.192881
\(765\) 0 0
\(766\) 1.47163e7 0.906205
\(767\) 1.52859e6 0.0938214
\(768\) 0 0
\(769\) −1.19305e7 −0.727517 −0.363759 0.931493i \(-0.618507\pi\)
−0.363759 + 0.931493i \(0.618507\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.40384e7 0.847763
\(773\) −6.58657e6 −0.396470 −0.198235 0.980155i \(-0.563521\pi\)
−0.198235 + 0.980155i \(0.563521\pi\)
\(774\) 0 0
\(775\) 3.18112e6 0.190250
\(776\) −8.15087e6 −0.485903
\(777\) 0 0
\(778\) 1.02894e6 0.0609451
\(779\) 3.22298e6 0.190289
\(780\) 0 0
\(781\) −2.91781e6 −0.171171
\(782\) −1.38852e7 −0.811961
\(783\) 0 0
\(784\) 0 0
\(785\) −1.52055e7 −0.880700
\(786\) 0 0
\(787\) 1.92720e7 1.10915 0.554575 0.832134i \(-0.312881\pi\)
0.554575 + 0.832134i \(0.312881\pi\)
\(788\) 1.20379e7 0.690613
\(789\) 0 0
\(790\) 1.02775e7 0.585897
\(791\) 0 0
\(792\) 0 0
\(793\) 669897. 0.0378290
\(794\) 1.73312e6 0.0975613
\(795\) 0 0
\(796\) 2.48724e6 0.139135
\(797\) 1.54906e7 0.863816 0.431908 0.901918i \(-0.357841\pi\)
0.431908 + 0.901918i \(0.357841\pi\)
\(798\) 0 0
\(799\) 4.83174e7 2.67754
\(800\) −619188. −0.0342056
\(801\) 0 0
\(802\) −4.55298e6 −0.249954
\(803\) 1.35454e6 0.0741315
\(804\) 0 0
\(805\) 0 0
\(806\) 727380. 0.0394388
\(807\) 0 0
\(808\) −4.20851e6 −0.226777
\(809\) 1.77071e6 0.0951210 0.0475605 0.998868i \(-0.484855\pi\)
0.0475605 + 0.998868i \(0.484855\pi\)
\(810\) 0 0
\(811\) −6.80547e6 −0.363334 −0.181667 0.983360i \(-0.558149\pi\)
−0.181667 + 0.983360i \(0.558149\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 1.87427e6 0.0991451
\(815\) −1.23515e7 −0.651366
\(816\) 0 0
\(817\) −5.03869e6 −0.264097
\(818\) −2.00975e7 −1.05017
\(819\) 0 0
\(820\) 6.96380e6 0.361669
\(821\) −2.27127e7 −1.17601 −0.588006 0.808857i \(-0.700087\pi\)
−0.588006 + 0.808857i \(0.700087\pi\)
\(822\) 0 0
\(823\) −3.03576e7 −1.56231 −0.781156 0.624336i \(-0.785370\pi\)
−0.781156 + 0.624336i \(0.785370\pi\)
\(824\) 1.08807e7 0.558260
\(825\) 0 0
\(826\) 0 0
\(827\) 5.80776e6 0.295287 0.147644 0.989041i \(-0.452831\pi\)
0.147644 + 0.989041i \(0.452831\pi\)
\(828\) 0 0
\(829\) 9.04916e6 0.457322 0.228661 0.973506i \(-0.426565\pi\)
0.228661 + 0.973506i \(0.426565\pi\)
\(830\) −1.54176e6 −0.0776822
\(831\) 0 0
\(832\) −141581. −0.00709080
\(833\) 0 0
\(834\) 0 0
\(835\) −2.90570e7 −1.44223
\(836\) −264426. −0.0130855
\(837\) 0 0
\(838\) 1.02918e7 0.506270
\(839\) −8.17101e6 −0.400747 −0.200374 0.979720i \(-0.564216\pi\)
−0.200374 + 0.979720i \(0.564216\pi\)
\(840\) 0 0
\(841\) 2.19614e6 0.107071
\(842\) −1.34570e6 −0.0654137
\(843\) 0 0
\(844\) 62371.6 0.00301391
\(845\) 2.26023e7 1.08896
\(846\) 0 0
\(847\) 0 0
\(848\) 1.79976e6 0.0859457
\(849\) 0 0
\(850\) 4.98504e6 0.236658
\(851\) 2.15954e7 1.02220
\(852\) 0 0
\(853\) −2.88437e7 −1.35731 −0.678654 0.734458i \(-0.737436\pi\)
−0.678654 + 0.734458i \(0.737436\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −6.90770e6 −0.322218
\(857\) −2.69462e7 −1.25327 −0.626637 0.779311i \(-0.715569\pi\)
−0.626637 + 0.779311i \(0.715569\pi\)
\(858\) 0 0
\(859\) 7.49124e6 0.346394 0.173197 0.984887i \(-0.444590\pi\)
0.173197 + 0.984887i \(0.444590\pi\)
\(860\) −1.08870e7 −0.501950
\(861\) 0 0
\(862\) 945641. 0.0433469
\(863\) 1.41552e6 0.0646977 0.0323489 0.999477i \(-0.489701\pi\)
0.0323489 + 0.999477i \(0.489701\pi\)
\(864\) 0 0
\(865\) 1.73871e7 0.790110
\(866\) −9.32015e6 −0.422307
\(867\) 0 0
\(868\) 0 0
\(869\) 1.53749e6 0.0690656
\(870\) 0 0
\(871\) −723944. −0.0323340
\(872\) 1.42527e7 0.634755
\(873\) 0 0
\(874\) −3.04672e6 −0.134913
\(875\) 0 0
\(876\) 0 0
\(877\) 2.55232e7 1.12056 0.560281 0.828303i \(-0.310693\pi\)
0.560281 + 0.828303i \(0.310693\pi\)
\(878\) 1.27985e7 0.560302
\(879\) 0 0
\(880\) −571338. −0.0248706
\(881\) −2.07516e6 −0.0900764 −0.0450382 0.998985i \(-0.514341\pi\)
−0.0450382 + 0.998985i \(0.514341\pi\)
\(882\) 0 0
\(883\) 1.80641e7 0.779677 0.389839 0.920883i \(-0.372531\pi\)
0.389839 + 0.920883i \(0.372531\pi\)
\(884\) 1.13986e6 0.0490591
\(885\) 0 0
\(886\) 4.52961e6 0.193855
\(887\) −9.28389e6 −0.396206 −0.198103 0.980181i \(-0.563478\pi\)
−0.198103 + 0.980181i \(0.563478\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −1.25796e7 −0.532344
\(891\) 0 0
\(892\) −1.25212e7 −0.526906
\(893\) 1.06019e7 0.444893
\(894\) 0 0
\(895\) 2.12502e7 0.886758
\(896\) 0 0
\(897\) 0 0
\(898\) −3.29475e7 −1.36343
\(899\) 2.50692e7 1.03452
\(900\) 0 0
\(901\) −1.44897e7 −0.594631
\(902\) 1.04176e6 0.0426336
\(903\) 0 0
\(904\) 1.15875e7 0.471595
\(905\) 2.31891e7 0.941156
\(906\) 0 0
\(907\) −4.20818e7 −1.69854 −0.849270 0.527958i \(-0.822958\pi\)
−0.849270 + 0.527958i \(0.822958\pi\)
\(908\) 1.27447e7 0.512997
\(909\) 0 0
\(910\) 0 0
\(911\) −2.33510e7 −0.932199 −0.466100 0.884732i \(-0.654341\pi\)
−0.466100 + 0.884732i \(0.654341\pi\)
\(912\) 0 0
\(913\) −230642. −0.00915719
\(914\) 1.46053e6 0.0578289
\(915\) 0 0
\(916\) −6.19272e6 −0.243861
\(917\) 0 0
\(918\) 0 0
\(919\) 1.54594e7 0.603817 0.301908 0.953337i \(-0.402376\pi\)
0.301908 + 0.953337i \(0.402376\pi\)
\(920\) −6.58296e6 −0.256420
\(921\) 0 0
\(922\) −1.33027e6 −0.0515362
\(923\) 2.75983e6 0.106630
\(924\) 0 0
\(925\) −7.75312e6 −0.297935
\(926\) −6.79968e6 −0.260592
\(927\) 0 0
\(928\) −4.87958e6 −0.186000
\(929\) 1.49993e7 0.570204 0.285102 0.958497i \(-0.407972\pi\)
0.285102 + 0.958497i \(0.407972\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 6.14014e6 0.231547
\(933\) 0 0
\(934\) 3.96186e6 0.148605
\(935\) 4.59981e6 0.172072
\(936\) 0 0
\(937\) −1.52742e7 −0.568343 −0.284172 0.958773i \(-0.591719\pi\)
−0.284172 + 0.958773i \(0.591719\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 2.29072e7 0.845576
\(941\) −5.13600e6 −0.189082 −0.0945412 0.995521i \(-0.530138\pi\)
−0.0945412 + 0.995521i \(0.530138\pi\)
\(942\) 0 0
\(943\) 1.20032e7 0.439559
\(944\) −1.13210e7 −0.413482
\(945\) 0 0
\(946\) −1.62865e6 −0.0591699
\(947\) 4.50054e7 1.63076 0.815379 0.578928i \(-0.196529\pi\)
0.815379 + 0.578928i \(0.196529\pi\)
\(948\) 0 0
\(949\) −1.28120e6 −0.0461798
\(950\) 1.09383e6 0.0393224
\(951\) 0 0
\(952\) 0 0
\(953\) 4.49596e7 1.60358 0.801789 0.597607i \(-0.203882\pi\)
0.801789 + 0.597607i \(0.203882\pi\)
\(954\) 0 0
\(955\) 1.18779e7 0.421434
\(956\) 7.44655e6 0.263518
\(957\) 0 0
\(958\) 2.43226e7 0.856242
\(959\) 0 0
\(960\) 0 0
\(961\) −952428. −0.0332678
\(962\) −1.77279e6 −0.0617619
\(963\) 0 0
\(964\) 5.57991e6 0.193390
\(965\) −5.35838e7 −1.85232
\(966\) 0 0
\(967\) 1.04364e7 0.358907 0.179454 0.983766i \(-0.442567\pi\)
0.179454 + 0.983766i \(0.442567\pi\)
\(968\) 1.02218e7 0.350622
\(969\) 0 0
\(970\) 3.11114e7 1.06167
\(971\) −1.72207e7 −0.586142 −0.293071 0.956091i \(-0.594677\pi\)
−0.293071 + 0.956091i \(0.594677\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 2.68439e7 0.906669
\(975\) 0 0
\(976\) −4.96140e6 −0.166717
\(977\) −3.24376e7 −1.08721 −0.543603 0.839343i \(-0.682940\pi\)
−0.543603 + 0.839343i \(0.682940\pi\)
\(978\) 0 0
\(979\) −1.88187e6 −0.0627528
\(980\) 0 0
\(981\) 0 0
\(982\) −3.65617e6 −0.120989
\(983\) 3.19804e7 1.05560 0.527801 0.849368i \(-0.323017\pi\)
0.527801 + 0.849368i \(0.323017\pi\)
\(984\) 0 0
\(985\) −4.59479e7 −1.50895
\(986\) 3.92852e7 1.28688
\(987\) 0 0
\(988\) 250110. 0.00815152
\(989\) −1.87654e7 −0.610051
\(990\) 0 0
\(991\) 5.49614e7 1.77776 0.888880 0.458139i \(-0.151484\pi\)
0.888880 + 0.458139i \(0.151484\pi\)
\(992\) −5.38713e6 −0.173811
\(993\) 0 0
\(994\) 0 0
\(995\) −9.49367e6 −0.304002
\(996\) 0 0
\(997\) −7.51010e6 −0.239281 −0.119640 0.992817i \(-0.538174\pi\)
−0.119640 + 0.992817i \(0.538174\pi\)
\(998\) 3.10197e7 0.985850
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 882.6.a.z.1.1 2
3.2 odd 2 294.6.a.t.1.2 2
7.6 odd 2 882.6.a.bj.1.2 2
21.2 odd 6 294.6.e.v.67.1 4
21.5 even 6 294.6.e.u.67.2 4
21.11 odd 6 294.6.e.v.79.1 4
21.17 even 6 294.6.e.u.79.2 4
21.20 even 2 294.6.a.u.1.1 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
294.6.a.t.1.2 2 3.2 odd 2
294.6.a.u.1.1 yes 2 21.20 even 2
294.6.e.u.67.2 4 21.5 even 6
294.6.e.u.79.2 4 21.17 even 6
294.6.e.v.67.1 4 21.2 odd 6
294.6.e.v.79.1 4 21.11 odd 6
882.6.a.z.1.1 2 1.1 even 1 trivial
882.6.a.bj.1.2 2 7.6 odd 2