Properties

Label 342.10.a.l.1.4
Level $342$
Weight $10$
Character 342.1
Self dual yes
Analytic conductor $176.142$
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] = [342,10,Mod(1,342)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(342, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("342.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 342 = 2 \cdot 3^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 342.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: \(176.142255968\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 34433x^{2} - 2723303x - 48270488 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 38)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(219.264\) of defining polynomial
Character \(\chi\) \(=\) 342.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{2} +256.000 q^{4} +2418.56 q^{5} -1533.95 q^{7} +4096.00 q^{8} +O(q^{10})\) \(q+16.0000 q^{2} +256.000 q^{4} +2418.56 q^{5} -1533.95 q^{7} +4096.00 q^{8} +38696.9 q^{10} +5815.50 q^{11} -135546. q^{13} -24543.2 q^{14} +65536.0 q^{16} +448133. q^{17} +130321. q^{19} +619151. q^{20} +93048.1 q^{22} -2.07551e6 q^{23} +3.89629e6 q^{25} -2.16874e6 q^{26} -392691. q^{28} +643630. q^{29} -194155. q^{31} +1.04858e6 q^{32} +7.17012e6 q^{34} -3.70994e6 q^{35} +9.46515e6 q^{37} +2.08514e6 q^{38} +9.90641e6 q^{40} +2.34337e7 q^{41} +3.80295e7 q^{43} +1.48877e6 q^{44} -3.32081e7 q^{46} +2.28118e7 q^{47} -3.80006e7 q^{49} +6.23407e7 q^{50} -3.46998e7 q^{52} +6.65359e7 q^{53} +1.40651e7 q^{55} -6.28306e6 q^{56} +1.02981e7 q^{58} +9.14212e7 q^{59} -4.02765e7 q^{61} -3.10649e6 q^{62} +1.67772e7 q^{64} -3.27826e8 q^{65} +2.42739e8 q^{67} +1.14722e8 q^{68} -5.93591e7 q^{70} +1.83180e8 q^{71} -1.39336e8 q^{73} +1.51442e8 q^{74} +3.33622e7 q^{76} -8.92069e6 q^{77} +2.51527e8 q^{79} +1.58503e8 q^{80} +3.74939e8 q^{82} +1.30417e8 q^{83} +1.08383e9 q^{85} +6.08473e8 q^{86} +2.38203e7 q^{88} -7.10883e8 q^{89} +2.07921e8 q^{91} -5.31330e8 q^{92} +3.64988e8 q^{94} +3.15189e8 q^{95} -1.31644e9 q^{97} -6.08010e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q + 64 q^{2} + 1024 q^{4} + 1395 q^{5} + 12307 q^{7} + 16384 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 64 q^{2} + 1024 q^{4} + 1395 q^{5} + 12307 q^{7} + 16384 q^{8} + 22320 q^{10} + 104249 q^{11} + 120486 q^{13} + 196912 q^{14} + 262144 q^{16} + 412139 q^{17} + 521284 q^{19} + 357120 q^{20} + 1667984 q^{22} - 3010300 q^{23} + 9760585 q^{25} + 1927776 q^{26} + 3150592 q^{28} - 6153240 q^{29} + 12774024 q^{31} + 4194304 q^{32} + 6594224 q^{34} - 9823425 q^{35} + 20506048 q^{37} + 8340544 q^{38} + 5713920 q^{40} - 11620300 q^{41} + 7698327 q^{43} + 26687744 q^{44} - 48164800 q^{46} + 31581083 q^{47} + 18970383 q^{49} + 156169360 q^{50} + 30844416 q^{52} - 72549422 q^{53} + 21332505 q^{55} + 50409472 q^{56} - 98451840 q^{58} + 149234120 q^{59} + 129004373 q^{61} + 204384384 q^{62} + 67108864 q^{64} - 124691700 q^{65} + 132595266 q^{67} + 105507584 q^{68} - 157174800 q^{70} + 47138482 q^{71} - 39332795 q^{73} + 328096768 q^{74} + 133448704 q^{76} + 165933719 q^{77} - 307010840 q^{79} + 91422720 q^{80} - 185924800 q^{82} + 746568232 q^{83} - 105005985 q^{85} + 123173232 q^{86} + 427003904 q^{88} - 286943482 q^{89} + 3155781114 q^{91} - 770636800 q^{92} + 505297328 q^{94} + 181797795 q^{95} + 793519958 q^{97} + 303526128 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 16.0000 0.707107
\(3\) 0 0
\(4\) 256.000 0.500000
\(5\) 2418.56 1.73058 0.865289 0.501273i \(-0.167135\pi\)
0.865289 + 0.501273i \(0.167135\pi\)
\(6\) 0 0
\(7\) −1533.95 −0.241474 −0.120737 0.992685i \(-0.538526\pi\)
−0.120737 + 0.992685i \(0.538526\pi\)
\(8\) 4096.00 0.353553
\(9\) 0 0
\(10\) 38696.9 1.22370
\(11\) 5815.50 0.119762 0.0598812 0.998206i \(-0.480928\pi\)
0.0598812 + 0.998206i \(0.480928\pi\)
\(12\) 0 0
\(13\) −135546. −1.31626 −0.658130 0.752904i \(-0.728652\pi\)
−0.658130 + 0.752904i \(0.728652\pi\)
\(14\) −24543.2 −0.170748
\(15\) 0 0
\(16\) 65536.0 0.250000
\(17\) 448133. 1.30133 0.650663 0.759366i \(-0.274491\pi\)
0.650663 + 0.759366i \(0.274491\pi\)
\(18\) 0 0
\(19\) 130321. 0.229416
\(20\) 619151. 0.865289
\(21\) 0 0
\(22\) 93048.1 0.0846848
\(23\) −2.07551e6 −1.54650 −0.773249 0.634102i \(-0.781370\pi\)
−0.773249 + 0.634102i \(0.781370\pi\)
\(24\) 0 0
\(25\) 3.89629e6 1.99490
\(26\) −2.16874e6 −0.930737
\(27\) 0 0
\(28\) −392691. −0.120737
\(29\) 643630. 0.168984 0.0844920 0.996424i \(-0.473073\pi\)
0.0844920 + 0.996424i \(0.473073\pi\)
\(30\) 0 0
\(31\) −194155. −0.0377591 −0.0188796 0.999822i \(-0.506010\pi\)
−0.0188796 + 0.999822i \(0.506010\pi\)
\(32\) 1.04858e6 0.176777
\(33\) 0 0
\(34\) 7.17012e6 0.920177
\(35\) −3.70994e6 −0.417889
\(36\) 0 0
\(37\) 9.46515e6 0.830271 0.415135 0.909760i \(-0.363734\pi\)
0.415135 + 0.909760i \(0.363734\pi\)
\(38\) 2.08514e6 0.162221
\(39\) 0 0
\(40\) 9.90641e6 0.611852
\(41\) 2.34337e7 1.29513 0.647565 0.762010i \(-0.275787\pi\)
0.647565 + 0.762010i \(0.275787\pi\)
\(42\) 0 0
\(43\) 3.80295e7 1.69634 0.848170 0.529724i \(-0.177704\pi\)
0.848170 + 0.529724i \(0.177704\pi\)
\(44\) 1.48877e6 0.0598812
\(45\) 0 0
\(46\) −3.32081e7 −1.09354
\(47\) 2.28118e7 0.681897 0.340948 0.940082i \(-0.389252\pi\)
0.340948 + 0.940082i \(0.389252\pi\)
\(48\) 0 0
\(49\) −3.80006e7 −0.941690
\(50\) 6.23407e7 1.41061
\(51\) 0 0
\(52\) −3.46998e7 −0.658130
\(53\) 6.65359e7 1.15828 0.579141 0.815227i \(-0.303388\pi\)
0.579141 + 0.815227i \(0.303388\pi\)
\(54\) 0 0
\(55\) 1.40651e7 0.207258
\(56\) −6.28306e6 −0.0853739
\(57\) 0 0
\(58\) 1.02981e7 0.119490
\(59\) 9.14212e7 0.982230 0.491115 0.871095i \(-0.336590\pi\)
0.491115 + 0.871095i \(0.336590\pi\)
\(60\) 0 0
\(61\) −4.02765e7 −0.372450 −0.186225 0.982507i \(-0.559625\pi\)
−0.186225 + 0.982507i \(0.559625\pi\)
\(62\) −3.10649e6 −0.0266997
\(63\) 0 0
\(64\) 1.67772e7 0.125000
\(65\) −3.27826e8 −2.27789
\(66\) 0 0
\(67\) 2.42739e8 1.47164 0.735822 0.677175i \(-0.236796\pi\)
0.735822 + 0.677175i \(0.236796\pi\)
\(68\) 1.14722e8 0.650663
\(69\) 0 0
\(70\) −5.93591e7 −0.295492
\(71\) 1.83180e8 0.855491 0.427745 0.903899i \(-0.359308\pi\)
0.427745 + 0.903899i \(0.359308\pi\)
\(72\) 0 0
\(73\) −1.39336e8 −0.574264 −0.287132 0.957891i \(-0.592702\pi\)
−0.287132 + 0.957891i \(0.592702\pi\)
\(74\) 1.51442e8 0.587090
\(75\) 0 0
\(76\) 3.33622e7 0.114708
\(77\) −8.92069e6 −0.0289195
\(78\) 0 0
\(79\) 2.51527e8 0.726546 0.363273 0.931683i \(-0.381659\pi\)
0.363273 + 0.931683i \(0.381659\pi\)
\(80\) 1.58503e8 0.432645
\(81\) 0 0
\(82\) 3.74939e8 0.915795
\(83\) 1.30417e8 0.301636 0.150818 0.988562i \(-0.451809\pi\)
0.150818 + 0.988562i \(0.451809\pi\)
\(84\) 0 0
\(85\) 1.08383e9 2.25205
\(86\) 6.08473e8 1.19949
\(87\) 0 0
\(88\) 2.38203e7 0.0423424
\(89\) −7.10883e8 −1.20100 −0.600500 0.799625i \(-0.705032\pi\)
−0.600500 + 0.799625i \(0.705032\pi\)
\(90\) 0 0
\(91\) 2.07921e8 0.317842
\(92\) −5.31330e8 −0.773249
\(93\) 0 0
\(94\) 3.64988e8 0.482174
\(95\) 3.15189e8 0.397022
\(96\) 0 0
\(97\) −1.31644e9 −1.50983 −0.754916 0.655822i \(-0.772322\pi\)
−0.754916 + 0.655822i \(0.772322\pi\)
\(98\) −6.08010e8 −0.665876
\(99\) 0 0
\(100\) 9.97451e8 0.997451
\(101\) −1.38054e9 −1.32009 −0.660044 0.751227i \(-0.729462\pi\)
−0.660044 + 0.751227i \(0.729462\pi\)
\(102\) 0 0
\(103\) −9.35066e8 −0.818606 −0.409303 0.912399i \(-0.634228\pi\)
−0.409303 + 0.912399i \(0.634228\pi\)
\(104\) −5.55197e8 −0.465368
\(105\) 0 0
\(106\) 1.06457e9 0.819030
\(107\) 4.27772e8 0.315490 0.157745 0.987480i \(-0.449578\pi\)
0.157745 + 0.987480i \(0.449578\pi\)
\(108\) 0 0
\(109\) −1.13801e8 −0.0772197 −0.0386098 0.999254i \(-0.512293\pi\)
−0.0386098 + 0.999254i \(0.512293\pi\)
\(110\) 2.25042e8 0.146554
\(111\) 0 0
\(112\) −1.00529e8 −0.0603684
\(113\) −1.37096e9 −0.790990 −0.395495 0.918468i \(-0.629427\pi\)
−0.395495 + 0.918468i \(0.629427\pi\)
\(114\) 0 0
\(115\) −5.01974e9 −2.67634
\(116\) 1.64769e8 0.0844920
\(117\) 0 0
\(118\) 1.46274e9 0.694541
\(119\) −6.87413e8 −0.314236
\(120\) 0 0
\(121\) −2.32413e9 −0.985657
\(122\) −6.44424e8 −0.263362
\(123\) 0 0
\(124\) −4.97038e7 −0.0188796
\(125\) 4.69966e9 1.72176
\(126\) 0 0
\(127\) 1.64706e9 0.561813 0.280906 0.959735i \(-0.409365\pi\)
0.280906 + 0.959735i \(0.409365\pi\)
\(128\) 2.68435e8 0.0883883
\(129\) 0 0
\(130\) −5.24521e9 −1.61071
\(131\) 3.65669e9 1.08485 0.542423 0.840106i \(-0.317507\pi\)
0.542423 + 0.840106i \(0.317507\pi\)
\(132\) 0 0
\(133\) −1.99906e8 −0.0553979
\(134\) 3.88382e9 1.04061
\(135\) 0 0
\(136\) 1.83555e9 0.460088
\(137\) 3.24163e9 0.786179 0.393089 0.919500i \(-0.371406\pi\)
0.393089 + 0.919500i \(0.371406\pi\)
\(138\) 0 0
\(139\) 8.69038e9 1.97457 0.987285 0.158963i \(-0.0508151\pi\)
0.987285 + 0.158963i \(0.0508151\pi\)
\(140\) −9.49746e8 −0.208945
\(141\) 0 0
\(142\) 2.93088e9 0.604923
\(143\) −7.88269e8 −0.157638
\(144\) 0 0
\(145\) 1.55666e9 0.292440
\(146\) −2.22938e9 −0.406066
\(147\) 0 0
\(148\) 2.42308e9 0.415135
\(149\) 7.11665e9 1.18287 0.591435 0.806353i \(-0.298562\pi\)
0.591435 + 0.806353i \(0.298562\pi\)
\(150\) 0 0
\(151\) 1.87879e9 0.294091 0.147046 0.989130i \(-0.453024\pi\)
0.147046 + 0.989130i \(0.453024\pi\)
\(152\) 5.33795e8 0.0811107
\(153\) 0 0
\(154\) −1.42731e8 −0.0204491
\(155\) −4.69576e8 −0.0653451
\(156\) 0 0
\(157\) 4.95963e9 0.651480 0.325740 0.945459i \(-0.394387\pi\)
0.325740 + 0.945459i \(0.394387\pi\)
\(158\) 4.02444e9 0.513746
\(159\) 0 0
\(160\) 2.53604e9 0.305926
\(161\) 3.18373e9 0.373439
\(162\) 0 0
\(163\) 2.87539e9 0.319046 0.159523 0.987194i \(-0.449004\pi\)
0.159523 + 0.987194i \(0.449004\pi\)
\(164\) 5.99902e9 0.647565
\(165\) 0 0
\(166\) 2.08668e9 0.213289
\(167\) −1.47841e10 −1.47085 −0.735427 0.677604i \(-0.763018\pi\)
−0.735427 + 0.677604i \(0.763018\pi\)
\(168\) 0 0
\(169\) 7.76823e9 0.732541
\(170\) 1.73413e10 1.59244
\(171\) 0 0
\(172\) 9.73556e9 0.848170
\(173\) −1.45582e10 −1.23566 −0.617832 0.786310i \(-0.711989\pi\)
−0.617832 + 0.786310i \(0.711989\pi\)
\(174\) 0 0
\(175\) −5.97672e9 −0.481716
\(176\) 3.81125e8 0.0299406
\(177\) 0 0
\(178\) −1.13741e10 −0.849235
\(179\) 1.82403e10 1.32799 0.663994 0.747738i \(-0.268860\pi\)
0.663994 + 0.747738i \(0.268860\pi\)
\(180\) 0 0
\(181\) 2.81167e10 1.94720 0.973602 0.228252i \(-0.0733011\pi\)
0.973602 + 0.228252i \(0.0733011\pi\)
\(182\) 3.32673e9 0.224748
\(183\) 0 0
\(184\) −8.50129e9 −0.546770
\(185\) 2.28920e10 1.43685
\(186\) 0 0
\(187\) 2.60612e9 0.155850
\(188\) 5.83981e9 0.340948
\(189\) 0 0
\(190\) 5.04302e9 0.280737
\(191\) −2.82317e8 −0.0153492 −0.00767462 0.999971i \(-0.502443\pi\)
−0.00767462 + 0.999971i \(0.502443\pi\)
\(192\) 0 0
\(193\) 1.43117e10 0.742477 0.371239 0.928538i \(-0.378933\pi\)
0.371239 + 0.928538i \(0.378933\pi\)
\(194\) −2.10630e10 −1.06761
\(195\) 0 0
\(196\) −9.72816e9 −0.470845
\(197\) −2.02116e10 −0.956100 −0.478050 0.878333i \(-0.658656\pi\)
−0.478050 + 0.878333i \(0.658656\pi\)
\(198\) 0 0
\(199\) −3.95924e10 −1.78967 −0.894835 0.446398i \(-0.852707\pi\)
−0.894835 + 0.446398i \(0.852707\pi\)
\(200\) 1.59592e10 0.705304
\(201\) 0 0
\(202\) −2.20886e10 −0.933443
\(203\) −9.87296e8 −0.0408052
\(204\) 0 0
\(205\) 5.66757e10 2.24132
\(206\) −1.49611e10 −0.578842
\(207\) 0 0
\(208\) −8.88315e9 −0.329065
\(209\) 7.57882e8 0.0274754
\(210\) 0 0
\(211\) −1.31017e10 −0.455047 −0.227523 0.973773i \(-0.573063\pi\)
−0.227523 + 0.973773i \(0.573063\pi\)
\(212\) 1.70332e10 0.579141
\(213\) 0 0
\(214\) 6.84436e9 0.223085
\(215\) 9.19766e10 2.93565
\(216\) 0 0
\(217\) 2.97825e8 0.00911784
\(218\) −1.82082e9 −0.0546026
\(219\) 0 0
\(220\) 3.60067e9 0.103629
\(221\) −6.07426e10 −1.71288
\(222\) 0 0
\(223\) 6.50346e10 1.76105 0.880526 0.473997i \(-0.157189\pi\)
0.880526 + 0.473997i \(0.157189\pi\)
\(224\) −1.60846e9 −0.0426869
\(225\) 0 0
\(226\) −2.19353e10 −0.559315
\(227\) −1.02990e10 −0.257442 −0.128721 0.991681i \(-0.541087\pi\)
−0.128721 + 0.991681i \(0.541087\pi\)
\(228\) 0 0
\(229\) 1.71789e10 0.412797 0.206398 0.978468i \(-0.433826\pi\)
0.206398 + 0.978468i \(0.433826\pi\)
\(230\) −8.03158e10 −1.89246
\(231\) 0 0
\(232\) 2.63631e9 0.0597448
\(233\) 6.52465e10 1.45029 0.725146 0.688595i \(-0.241772\pi\)
0.725146 + 0.688595i \(0.241772\pi\)
\(234\) 0 0
\(235\) 5.51716e10 1.18008
\(236\) 2.34038e10 0.491115
\(237\) 0 0
\(238\) −1.09986e10 −0.222199
\(239\) 1.43024e10 0.283543 0.141771 0.989899i \(-0.454720\pi\)
0.141771 + 0.989899i \(0.454720\pi\)
\(240\) 0 0
\(241\) −4.79560e10 −0.915727 −0.457864 0.889022i \(-0.651385\pi\)
−0.457864 + 0.889022i \(0.651385\pi\)
\(242\) −3.71860e10 −0.696965
\(243\) 0 0
\(244\) −1.03108e10 −0.186225
\(245\) −9.19066e10 −1.62967
\(246\) 0 0
\(247\) −1.76645e10 −0.301971
\(248\) −7.95261e8 −0.0133499
\(249\) 0 0
\(250\) 7.51946e10 1.21747
\(251\) 1.25023e10 0.198819 0.0994096 0.995047i \(-0.468305\pi\)
0.0994096 + 0.995047i \(0.468305\pi\)
\(252\) 0 0
\(253\) −1.20701e10 −0.185212
\(254\) 2.63529e10 0.397261
\(255\) 0 0
\(256\) 4.29497e9 0.0625000
\(257\) −1.83653e10 −0.262602 −0.131301 0.991343i \(-0.541915\pi\)
−0.131301 + 0.991343i \(0.541915\pi\)
\(258\) 0 0
\(259\) −1.45191e10 −0.200489
\(260\) −8.39234e10 −1.13895
\(261\) 0 0
\(262\) 5.85071e10 0.767101
\(263\) −8.39157e10 −1.08154 −0.540770 0.841171i \(-0.681867\pi\)
−0.540770 + 0.841171i \(0.681867\pi\)
\(264\) 0 0
\(265\) 1.60921e11 2.00450
\(266\) −3.19849e9 −0.0391722
\(267\) 0 0
\(268\) 6.21412e10 0.735822
\(269\) −9.33233e9 −0.108669 −0.0543344 0.998523i \(-0.517304\pi\)
−0.0543344 + 0.998523i \(0.517304\pi\)
\(270\) 0 0
\(271\) −1.57854e11 −1.77785 −0.888924 0.458055i \(-0.848546\pi\)
−0.888924 + 0.458055i \(0.848546\pi\)
\(272\) 2.93688e10 0.325332
\(273\) 0 0
\(274\) 5.18661e10 0.555912
\(275\) 2.26589e10 0.238914
\(276\) 0 0
\(277\) −1.12981e11 −1.15305 −0.576525 0.817080i \(-0.695592\pi\)
−0.576525 + 0.817080i \(0.695592\pi\)
\(278\) 1.39046e11 1.39623
\(279\) 0 0
\(280\) −1.51959e10 −0.147746
\(281\) −7.83548e10 −0.749699 −0.374849 0.927086i \(-0.622306\pi\)
−0.374849 + 0.927086i \(0.622306\pi\)
\(282\) 0 0
\(283\) 8.46004e10 0.784032 0.392016 0.919958i \(-0.371778\pi\)
0.392016 + 0.919958i \(0.371778\pi\)
\(284\) 4.68941e10 0.427745
\(285\) 0 0
\(286\) −1.26123e10 −0.111467
\(287\) −3.59461e10 −0.312740
\(288\) 0 0
\(289\) 8.22349e10 0.693451
\(290\) 2.49065e10 0.206786
\(291\) 0 0
\(292\) −3.56701e10 −0.287132
\(293\) 1.06621e11 0.845157 0.422579 0.906326i \(-0.361125\pi\)
0.422579 + 0.906326i \(0.361125\pi\)
\(294\) 0 0
\(295\) 2.21107e11 1.69983
\(296\) 3.87693e10 0.293545
\(297\) 0 0
\(298\) 1.13866e11 0.836416
\(299\) 2.81327e11 2.03559
\(300\) 0 0
\(301\) −5.83354e10 −0.409622
\(302\) 3.00606e10 0.207954
\(303\) 0 0
\(304\) 8.54072e9 0.0573539
\(305\) −9.74110e10 −0.644553
\(306\) 0 0
\(307\) 1.65292e11 1.06201 0.531006 0.847368i \(-0.321814\pi\)
0.531006 + 0.847368i \(0.321814\pi\)
\(308\) −2.28370e9 −0.0144597
\(309\) 0 0
\(310\) −7.51322e9 −0.0462060
\(311\) 1.08969e11 0.660515 0.330257 0.943891i \(-0.392864\pi\)
0.330257 + 0.943891i \(0.392864\pi\)
\(312\) 0 0
\(313\) 9.48622e10 0.558655 0.279328 0.960196i \(-0.409888\pi\)
0.279328 + 0.960196i \(0.409888\pi\)
\(314\) 7.93541e10 0.460666
\(315\) 0 0
\(316\) 6.43910e10 0.363273
\(317\) −2.75000e11 −1.52956 −0.764780 0.644291i \(-0.777152\pi\)
−0.764780 + 0.644291i \(0.777152\pi\)
\(318\) 0 0
\(319\) 3.74303e9 0.0202379
\(320\) 4.05767e10 0.216322
\(321\) 0 0
\(322\) 5.09396e10 0.264061
\(323\) 5.84011e10 0.298545
\(324\) 0 0
\(325\) −5.28127e11 −2.62581
\(326\) 4.60063e10 0.225599
\(327\) 0 0
\(328\) 9.59844e10 0.457898
\(329\) −3.49921e10 −0.164660
\(330\) 0 0
\(331\) −1.28057e11 −0.586378 −0.293189 0.956055i \(-0.594716\pi\)
−0.293189 + 0.956055i \(0.594716\pi\)
\(332\) 3.33868e10 0.150818
\(333\) 0 0
\(334\) −2.36545e11 −1.04005
\(335\) 5.87078e11 2.54680
\(336\) 0 0
\(337\) −3.03698e10 −0.128265 −0.0641324 0.997941i \(-0.520428\pi\)
−0.0641324 + 0.997941i \(0.520428\pi\)
\(338\) 1.24292e11 0.517985
\(339\) 0 0
\(340\) 2.77462e11 1.12602
\(341\) −1.12911e9 −0.00452212
\(342\) 0 0
\(343\) 1.20191e11 0.468867
\(344\) 1.55769e11 0.599747
\(345\) 0 0
\(346\) −2.32931e11 −0.873747
\(347\) −4.03811e11 −1.49519 −0.747594 0.664156i \(-0.768791\pi\)
−0.747594 + 0.664156i \(0.768791\pi\)
\(348\) 0 0
\(349\) 6.29420e10 0.227105 0.113552 0.993532i \(-0.463777\pi\)
0.113552 + 0.993532i \(0.463777\pi\)
\(350\) −9.56275e10 −0.340625
\(351\) 0 0
\(352\) 6.09800e9 0.0211712
\(353\) −9.36033e10 −0.320852 −0.160426 0.987048i \(-0.551287\pi\)
−0.160426 + 0.987048i \(0.551287\pi\)
\(354\) 0 0
\(355\) 4.43031e11 1.48049
\(356\) −1.81986e11 −0.600500
\(357\) 0 0
\(358\) 2.91846e11 0.939030
\(359\) −3.18605e11 −1.01234 −0.506171 0.862433i \(-0.668940\pi\)
−0.506171 + 0.862433i \(0.668940\pi\)
\(360\) 0 0
\(361\) 1.69836e10 0.0526316
\(362\) 4.49868e11 1.37688
\(363\) 0 0
\(364\) 5.32277e10 0.158921
\(365\) −3.36993e11 −0.993810
\(366\) 0 0
\(367\) 9.81213e10 0.282336 0.141168 0.989986i \(-0.454914\pi\)
0.141168 + 0.989986i \(0.454914\pi\)
\(368\) −1.36021e11 −0.386625
\(369\) 0 0
\(370\) 3.66272e11 1.01601
\(371\) −1.02063e11 −0.279695
\(372\) 0 0
\(373\) 2.24989e11 0.601827 0.300913 0.953652i \(-0.402709\pi\)
0.300913 + 0.953652i \(0.402709\pi\)
\(374\) 4.16979e10 0.110203
\(375\) 0 0
\(376\) 9.34370e10 0.241087
\(377\) −8.72415e10 −0.222427
\(378\) 0 0
\(379\) 6.24000e11 1.55349 0.776744 0.629817i \(-0.216870\pi\)
0.776744 + 0.629817i \(0.216870\pi\)
\(380\) 8.06883e10 0.198511
\(381\) 0 0
\(382\) −4.51707e9 −0.0108536
\(383\) −5.36899e11 −1.27497 −0.637483 0.770465i \(-0.720024\pi\)
−0.637483 + 0.770465i \(0.720024\pi\)
\(384\) 0 0
\(385\) −2.15752e10 −0.0500474
\(386\) 2.28987e11 0.525011
\(387\) 0 0
\(388\) −3.37009e11 −0.754916
\(389\) 1.30155e11 0.288195 0.144097 0.989564i \(-0.453972\pi\)
0.144097 + 0.989564i \(0.453972\pi\)
\(390\) 0 0
\(391\) −9.30103e11 −2.01250
\(392\) −1.55650e11 −0.332938
\(393\) 0 0
\(394\) −3.23386e11 −0.676064
\(395\) 6.08333e11 1.25735
\(396\) 0 0
\(397\) −7.36706e11 −1.48846 −0.744230 0.667924i \(-0.767183\pi\)
−0.744230 + 0.667924i \(0.767183\pi\)
\(398\) −6.33478e11 −1.26549
\(399\) 0 0
\(400\) 2.55347e11 0.498726
\(401\) 5.90469e11 1.14037 0.570187 0.821515i \(-0.306871\pi\)
0.570187 + 0.821515i \(0.306871\pi\)
\(402\) 0 0
\(403\) 2.63170e10 0.0497008
\(404\) −3.53418e11 −0.660044
\(405\) 0 0
\(406\) −1.57967e10 −0.0288536
\(407\) 5.50446e10 0.0994352
\(408\) 0 0
\(409\) −4.27579e11 −0.755547 −0.377773 0.925898i \(-0.623310\pi\)
−0.377773 + 0.925898i \(0.623310\pi\)
\(410\) 9.06811e11 1.58486
\(411\) 0 0
\(412\) −2.39377e11 −0.409303
\(413\) −1.40236e11 −0.237183
\(414\) 0 0
\(415\) 3.15421e11 0.522005
\(416\) −1.42130e11 −0.232684
\(417\) 0 0
\(418\) 1.21261e10 0.0194280
\(419\) −9.75902e11 −1.54683 −0.773416 0.633899i \(-0.781454\pi\)
−0.773416 + 0.633899i \(0.781454\pi\)
\(420\) 0 0
\(421\) −6.82920e10 −0.105950 −0.0529750 0.998596i \(-0.516870\pi\)
−0.0529750 + 0.998596i \(0.516870\pi\)
\(422\) −2.09627e11 −0.321766
\(423\) 0 0
\(424\) 2.72531e11 0.409515
\(425\) 1.74606e12 2.59602
\(426\) 0 0
\(427\) 6.17821e10 0.0899368
\(428\) 1.09510e11 0.157745
\(429\) 0 0
\(430\) 1.47163e12 2.07582
\(431\) 2.97113e11 0.414738 0.207369 0.978263i \(-0.433510\pi\)
0.207369 + 0.978263i \(0.433510\pi\)
\(432\) 0 0
\(433\) −4.51168e11 −0.616798 −0.308399 0.951257i \(-0.599793\pi\)
−0.308399 + 0.951257i \(0.599793\pi\)
\(434\) 4.76520e9 0.00644729
\(435\) 0 0
\(436\) −2.91331e10 −0.0386098
\(437\) −2.70482e11 −0.354791
\(438\) 0 0
\(439\) 5.55482e11 0.713805 0.356903 0.934142i \(-0.383833\pi\)
0.356903 + 0.934142i \(0.383833\pi\)
\(440\) 5.76108e10 0.0732768
\(441\) 0 0
\(442\) −9.71881e11 −1.21119
\(443\) 1.61691e12 1.99466 0.997329 0.0730386i \(-0.0232696\pi\)
0.997329 + 0.0730386i \(0.0232696\pi\)
\(444\) 0 0
\(445\) −1.71931e12 −2.07842
\(446\) 1.04055e12 1.24525
\(447\) 0 0
\(448\) −2.57354e10 −0.0301842
\(449\) −4.52893e11 −0.525880 −0.262940 0.964812i \(-0.584692\pi\)
−0.262940 + 0.964812i \(0.584692\pi\)
\(450\) 0 0
\(451\) 1.36279e11 0.155108
\(452\) −3.50965e11 −0.395495
\(453\) 0 0
\(454\) −1.64784e11 −0.182039
\(455\) 5.02868e11 0.550051
\(456\) 0 0
\(457\) 1.10310e12 1.18302 0.591511 0.806297i \(-0.298532\pi\)
0.591511 + 0.806297i \(0.298532\pi\)
\(458\) 2.74863e11 0.291891
\(459\) 0 0
\(460\) −1.28505e12 −1.33817
\(461\) −2.22092e11 −0.229023 −0.114511 0.993422i \(-0.536530\pi\)
−0.114511 + 0.993422i \(0.536530\pi\)
\(462\) 0 0
\(463\) 9.97229e11 1.00851 0.504255 0.863555i \(-0.331767\pi\)
0.504255 + 0.863555i \(0.331767\pi\)
\(464\) 4.21809e10 0.0422460
\(465\) 0 0
\(466\) 1.04394e12 1.02551
\(467\) −8.73582e11 −0.849919 −0.424959 0.905212i \(-0.639712\pi\)
−0.424959 + 0.905212i \(0.639712\pi\)
\(468\) 0 0
\(469\) −3.72349e11 −0.355363
\(470\) 8.82745e11 0.834440
\(471\) 0 0
\(472\) 3.74461e11 0.347271
\(473\) 2.21161e11 0.203158
\(474\) 0 0
\(475\) 5.07769e11 0.457662
\(476\) −1.75978e11 −0.157118
\(477\) 0 0
\(478\) 2.28838e11 0.200495
\(479\) −1.60658e12 −1.39442 −0.697209 0.716868i \(-0.745575\pi\)
−0.697209 + 0.716868i \(0.745575\pi\)
\(480\) 0 0
\(481\) −1.28296e12 −1.09285
\(482\) −7.67296e11 −0.647517
\(483\) 0 0
\(484\) −5.94977e11 −0.492828
\(485\) −3.18389e12 −2.61288
\(486\) 0 0
\(487\) −9.25381e11 −0.745487 −0.372744 0.927934i \(-0.621583\pi\)
−0.372744 + 0.927934i \(0.621583\pi\)
\(488\) −1.64973e11 −0.131681
\(489\) 0 0
\(490\) −1.47051e12 −1.15235
\(491\) 9.28628e11 0.721066 0.360533 0.932746i \(-0.382595\pi\)
0.360533 + 0.932746i \(0.382595\pi\)
\(492\) 0 0
\(493\) 2.88432e11 0.219903
\(494\) −2.82632e11 −0.213526
\(495\) 0 0
\(496\) −1.27242e10 −0.00943978
\(497\) −2.80989e11 −0.206579
\(498\) 0 0
\(499\) 1.19609e12 0.863601 0.431800 0.901969i \(-0.357878\pi\)
0.431800 + 0.901969i \(0.357878\pi\)
\(500\) 1.20311e12 0.860878
\(501\) 0 0
\(502\) 2.00037e11 0.140586
\(503\) −1.02039e12 −0.710741 −0.355370 0.934726i \(-0.615645\pi\)
−0.355370 + 0.934726i \(0.615645\pi\)
\(504\) 0 0
\(505\) −3.33892e12 −2.28452
\(506\) −1.93122e11 −0.130965
\(507\) 0 0
\(508\) 4.21646e11 0.280906
\(509\) 5.49549e11 0.362891 0.181446 0.983401i \(-0.441922\pi\)
0.181446 + 0.983401i \(0.441922\pi\)
\(510\) 0 0
\(511\) 2.13735e11 0.138670
\(512\) 6.87195e10 0.0441942
\(513\) 0 0
\(514\) −2.93844e11 −0.185688
\(515\) −2.26151e12 −1.41666
\(516\) 0 0
\(517\) 1.32662e11 0.0816656
\(518\) −2.32305e11 −0.141767
\(519\) 0 0
\(520\) −1.34277e12 −0.805356
\(521\) −1.28517e12 −0.764172 −0.382086 0.924127i \(-0.624794\pi\)
−0.382086 + 0.924127i \(0.624794\pi\)
\(522\) 0 0
\(523\) 1.65568e12 0.967654 0.483827 0.875164i \(-0.339246\pi\)
0.483827 + 0.875164i \(0.339246\pi\)
\(524\) 9.36114e11 0.542423
\(525\) 0 0
\(526\) −1.34265e12 −0.764764
\(527\) −8.70074e10 −0.0491370
\(528\) 0 0
\(529\) 2.50659e12 1.39166
\(530\) 2.57473e12 1.41740
\(531\) 0 0
\(532\) −5.11759e10 −0.0276989
\(533\) −3.17634e12 −1.70473
\(534\) 0 0
\(535\) 1.03459e12 0.545980
\(536\) 9.94259e11 0.520305
\(537\) 0 0
\(538\) −1.49317e11 −0.0768404
\(539\) −2.20993e11 −0.112779
\(540\) 0 0
\(541\) −1.91314e12 −0.960192 −0.480096 0.877216i \(-0.659398\pi\)
−0.480096 + 0.877216i \(0.659398\pi\)
\(542\) −2.52567e12 −1.25713
\(543\) 0 0
\(544\) 4.69901e11 0.230044
\(545\) −2.75235e11 −0.133635
\(546\) 0 0
\(547\) −8.87680e11 −0.423949 −0.211974 0.977275i \(-0.567989\pi\)
−0.211974 + 0.977275i \(0.567989\pi\)
\(548\) 8.29858e11 0.393089
\(549\) 0 0
\(550\) 3.62543e11 0.168938
\(551\) 8.38785e10 0.0387676
\(552\) 0 0
\(553\) −3.85830e11 −0.175442
\(554\) −1.80770e12 −0.815329
\(555\) 0 0
\(556\) 2.22474e12 0.987285
\(557\) 1.28351e12 0.565001 0.282501 0.959267i \(-0.408836\pi\)
0.282501 + 0.959267i \(0.408836\pi\)
\(558\) 0 0
\(559\) −5.15475e12 −2.23283
\(560\) −2.43135e11 −0.104472
\(561\) 0 0
\(562\) −1.25368e12 −0.530117
\(563\) −4.00016e12 −1.67799 −0.838995 0.544139i \(-0.816856\pi\)
−0.838995 + 0.544139i \(0.816856\pi\)
\(564\) 0 0
\(565\) −3.31574e12 −1.36887
\(566\) 1.35361e12 0.554394
\(567\) 0 0
\(568\) 7.50305e11 0.302462
\(569\) 1.23909e12 0.495563 0.247782 0.968816i \(-0.420298\pi\)
0.247782 + 0.968816i \(0.420298\pi\)
\(570\) 0 0
\(571\) −5.04024e12 −1.98421 −0.992107 0.125395i \(-0.959980\pi\)
−0.992107 + 0.125395i \(0.959980\pi\)
\(572\) −2.01797e11 −0.0788192
\(573\) 0 0
\(574\) −5.75137e11 −0.221140
\(575\) −8.08679e12 −3.08511
\(576\) 0 0
\(577\) −4.83898e12 −1.81745 −0.908725 0.417395i \(-0.862943\pi\)
−0.908725 + 0.417395i \(0.862943\pi\)
\(578\) 1.31576e12 0.490344
\(579\) 0 0
\(580\) 3.98504e11 0.146220
\(581\) −2.00053e11 −0.0728372
\(582\) 0 0
\(583\) 3.86940e11 0.138719
\(584\) −5.70722e11 −0.203033
\(585\) 0 0
\(586\) 1.70593e12 0.597617
\(587\) 1.95960e12 0.681233 0.340617 0.940202i \(-0.389364\pi\)
0.340617 + 0.940202i \(0.389364\pi\)
\(588\) 0 0
\(589\) −2.53025e10 −0.00866254
\(590\) 3.53772e12 1.20196
\(591\) 0 0
\(592\) 6.20308e11 0.207568
\(593\) 4.89414e12 1.62529 0.812643 0.582762i \(-0.198028\pi\)
0.812643 + 0.582762i \(0.198028\pi\)
\(594\) 0 0
\(595\) −1.66255e12 −0.543810
\(596\) 1.82186e12 0.591435
\(597\) 0 0
\(598\) 4.50123e12 1.43938
\(599\) −4.50082e12 −1.42847 −0.714235 0.699906i \(-0.753225\pi\)
−0.714235 + 0.699906i \(0.753225\pi\)
\(600\) 0 0
\(601\) −5.43075e12 −1.69795 −0.848974 0.528435i \(-0.822779\pi\)
−0.848974 + 0.528435i \(0.822779\pi\)
\(602\) −9.33366e11 −0.289646
\(603\) 0 0
\(604\) 4.80970e11 0.147046
\(605\) −5.62104e12 −1.70576
\(606\) 0 0
\(607\) 5.89921e12 1.76378 0.881891 0.471454i \(-0.156271\pi\)
0.881891 + 0.471454i \(0.156271\pi\)
\(608\) 1.36651e11 0.0405554
\(609\) 0 0
\(610\) −1.55858e12 −0.455768
\(611\) −3.09205e12 −0.897554
\(612\) 0 0
\(613\) −2.44465e11 −0.0699268 −0.0349634 0.999389i \(-0.511131\pi\)
−0.0349634 + 0.999389i \(0.511131\pi\)
\(614\) 2.64467e12 0.750955
\(615\) 0 0
\(616\) −3.65391e10 −0.0102246
\(617\) 6.79288e11 0.188699 0.0943497 0.995539i \(-0.469923\pi\)
0.0943497 + 0.995539i \(0.469923\pi\)
\(618\) 0 0
\(619\) 1.31874e12 0.361037 0.180519 0.983572i \(-0.442222\pi\)
0.180519 + 0.983572i \(0.442222\pi\)
\(620\) −1.20211e11 −0.0326726
\(621\) 0 0
\(622\) 1.74351e12 0.467054
\(623\) 1.09046e12 0.290010
\(624\) 0 0
\(625\) 3.75646e12 0.984732
\(626\) 1.51780e12 0.395029
\(627\) 0 0
\(628\) 1.26967e12 0.325740
\(629\) 4.24164e12 1.08045
\(630\) 0 0
\(631\) −6.44908e12 −1.61944 −0.809721 0.586815i \(-0.800382\pi\)
−0.809721 + 0.586815i \(0.800382\pi\)
\(632\) 1.03026e12 0.256873
\(633\) 0 0
\(634\) −4.40001e12 −1.08156
\(635\) 3.98350e12 0.972261
\(636\) 0 0
\(637\) 5.15083e12 1.23951
\(638\) 5.98885e10 0.0143104
\(639\) 0 0
\(640\) 6.49226e11 0.152963
\(641\) 5.38181e12 1.25912 0.629561 0.776951i \(-0.283235\pi\)
0.629561 + 0.776951i \(0.283235\pi\)
\(642\) 0 0
\(643\) −7.16838e12 −1.65376 −0.826878 0.562381i \(-0.809885\pi\)
−0.826878 + 0.562381i \(0.809885\pi\)
\(644\) 8.15034e11 0.186719
\(645\) 0 0
\(646\) 9.34417e11 0.211103
\(647\) 1.24135e12 0.278501 0.139250 0.990257i \(-0.455531\pi\)
0.139250 + 0.990257i \(0.455531\pi\)
\(648\) 0 0
\(649\) 5.31661e11 0.117634
\(650\) −8.45003e12 −1.85673
\(651\) 0 0
\(652\) 7.36100e11 0.159523
\(653\) −6.46167e12 −1.39071 −0.695353 0.718669i \(-0.744752\pi\)
−0.695353 + 0.718669i \(0.744752\pi\)
\(654\) 0 0
\(655\) 8.84392e12 1.87741
\(656\) 1.53575e12 0.323782
\(657\) 0 0
\(658\) −5.59874e11 −0.116432
\(659\) −3.48331e12 −0.719462 −0.359731 0.933056i \(-0.617131\pi\)
−0.359731 + 0.933056i \(0.617131\pi\)
\(660\) 0 0
\(661\) −2.69783e12 −0.549678 −0.274839 0.961490i \(-0.588625\pi\)
−0.274839 + 0.961490i \(0.588625\pi\)
\(662\) −2.04891e12 −0.414632
\(663\) 0 0
\(664\) 5.34189e11 0.106645
\(665\) −4.83484e11 −0.0958704
\(666\) 0 0
\(667\) −1.33586e12 −0.261333
\(668\) −3.78472e12 −0.735427
\(669\) 0 0
\(670\) 9.39325e12 1.80086
\(671\) −2.34228e11 −0.0446054
\(672\) 0 0
\(673\) −7.15485e12 −1.34441 −0.672207 0.740364i \(-0.734653\pi\)
−0.672207 + 0.740364i \(0.734653\pi\)
\(674\) −4.85917e11 −0.0906970
\(675\) 0 0
\(676\) 1.98867e12 0.366271
\(677\) −3.99069e12 −0.730127 −0.365064 0.930983i \(-0.618953\pi\)
−0.365064 + 0.930983i \(0.618953\pi\)
\(678\) 0 0
\(679\) 2.01935e12 0.364585
\(680\) 4.43938e12 0.796219
\(681\) 0 0
\(682\) −1.80658e10 −0.00319762
\(683\) 7.70306e12 1.35447 0.677236 0.735766i \(-0.263177\pi\)
0.677236 + 0.735766i \(0.263177\pi\)
\(684\) 0 0
\(685\) 7.84008e12 1.36054
\(686\) 1.92306e12 0.331539
\(687\) 0 0
\(688\) 2.49230e12 0.424085
\(689\) −9.01868e12 −1.52460
\(690\) 0 0
\(691\) −6.94842e12 −1.15940 −0.579702 0.814828i \(-0.696831\pi\)
−0.579702 + 0.814828i \(0.696831\pi\)
\(692\) −3.72690e12 −0.617832
\(693\) 0 0
\(694\) −6.46098e12 −1.05726
\(695\) 2.10182e13 3.41715
\(696\) 0 0
\(697\) 1.05014e13 1.68539
\(698\) 1.00707e12 0.160587
\(699\) 0 0
\(700\) −1.53004e12 −0.240858
\(701\) −8.89057e12 −1.39059 −0.695294 0.718725i \(-0.744726\pi\)
−0.695294 + 0.718725i \(0.744726\pi\)
\(702\) 0 0
\(703\) 1.23351e12 0.190477
\(704\) 9.75680e10 0.0149703
\(705\) 0 0
\(706\) −1.49765e12 −0.226877
\(707\) 2.11768e12 0.318767
\(708\) 0 0
\(709\) 5.07939e12 0.754924 0.377462 0.926025i \(-0.376797\pi\)
0.377462 + 0.926025i \(0.376797\pi\)
\(710\) 7.08850e12 1.04687
\(711\) 0 0
\(712\) −2.91178e12 −0.424617
\(713\) 4.02972e11 0.0583944
\(714\) 0 0
\(715\) −1.90647e12 −0.272806
\(716\) 4.66953e12 0.663994
\(717\) 0 0
\(718\) −5.09768e12 −0.715834
\(719\) 2.62697e11 0.0366585 0.0183293 0.999832i \(-0.494165\pi\)
0.0183293 + 0.999832i \(0.494165\pi\)
\(720\) 0 0
\(721\) 1.43434e12 0.197672
\(722\) 2.71737e11 0.0372161
\(723\) 0 0
\(724\) 7.19789e12 0.973602
\(725\) 2.50777e12 0.337106
\(726\) 0 0
\(727\) −5.69375e12 −0.755951 −0.377976 0.925816i \(-0.623380\pi\)
−0.377976 + 0.925816i \(0.623380\pi\)
\(728\) 8.51644e11 0.112374
\(729\) 0 0
\(730\) −5.39189e12 −0.702730
\(731\) 1.70423e13 2.20749
\(732\) 0 0
\(733\) −1.28415e13 −1.64304 −0.821521 0.570178i \(-0.806874\pi\)
−0.821521 + 0.570178i \(0.806874\pi\)
\(734\) 1.56994e12 0.199642
\(735\) 0 0
\(736\) −2.17633e12 −0.273385
\(737\) 1.41165e12 0.176248
\(738\) 0 0
\(739\) 2.66001e12 0.328083 0.164042 0.986453i \(-0.447547\pi\)
0.164042 + 0.986453i \(0.447547\pi\)
\(740\) 5.86035e12 0.718425
\(741\) 0 0
\(742\) −1.63300e12 −0.197774
\(743\) −1.02254e13 −1.23092 −0.615462 0.788166i \(-0.711031\pi\)
−0.615462 + 0.788166i \(0.711031\pi\)
\(744\) 0 0
\(745\) 1.72120e13 2.04705
\(746\) 3.59982e12 0.425556
\(747\) 0 0
\(748\) 6.67166e11 0.0779250
\(749\) −6.56181e11 −0.0761826
\(750\) 0 0
\(751\) 7.08317e12 0.812546 0.406273 0.913752i \(-0.366828\pi\)
0.406273 + 0.913752i \(0.366828\pi\)
\(752\) 1.49499e12 0.170474
\(753\) 0 0
\(754\) −1.39586e12 −0.157280
\(755\) 4.54396e12 0.508948
\(756\) 0 0
\(757\) 1.55185e13 1.71758 0.858792 0.512325i \(-0.171216\pi\)
0.858792 + 0.512325i \(0.171216\pi\)
\(758\) 9.98399e12 1.09848
\(759\) 0 0
\(760\) 1.29101e12 0.140368
\(761\) −3.36663e12 −0.363885 −0.181943 0.983309i \(-0.558239\pi\)
−0.181943 + 0.983309i \(0.558239\pi\)
\(762\) 0 0
\(763\) 1.74565e11 0.0186465
\(764\) −7.22732e10 −0.00767462
\(765\) 0 0
\(766\) −8.59039e12 −0.901537
\(767\) −1.23918e13 −1.29287
\(768\) 0 0
\(769\) −9.31790e12 −0.960836 −0.480418 0.877040i \(-0.659515\pi\)
−0.480418 + 0.877040i \(0.659515\pi\)
\(770\) −3.45203e11 −0.0353888
\(771\) 0 0
\(772\) 3.66379e12 0.371239
\(773\) 7.33840e12 0.739254 0.369627 0.929180i \(-0.379486\pi\)
0.369627 + 0.929180i \(0.379486\pi\)
\(774\) 0 0
\(775\) −7.56487e11 −0.0753258
\(776\) −5.39214e12 −0.533806
\(777\) 0 0
\(778\) 2.08247e12 0.203784
\(779\) 3.05390e12 0.297123
\(780\) 0 0
\(781\) 1.06528e12 0.102456
\(782\) −1.48817e13 −1.42305
\(783\) 0 0
\(784\) −2.49041e12 −0.235423
\(785\) 1.19951e13 1.12744
\(786\) 0 0
\(787\) 1.81540e13 1.68689 0.843443 0.537219i \(-0.180525\pi\)
0.843443 + 0.537219i \(0.180525\pi\)
\(788\) −5.17417e12 −0.478050
\(789\) 0 0
\(790\) 9.73333e12 0.889077
\(791\) 2.10298e12 0.191003
\(792\) 0 0
\(793\) 5.45932e12 0.490241
\(794\) −1.17873e13 −1.05250
\(795\) 0 0
\(796\) −1.01356e13 −0.894835
\(797\) −5.44638e12 −0.478129 −0.239065 0.971004i \(-0.576841\pi\)
−0.239065 + 0.971004i \(0.576841\pi\)
\(798\) 0 0
\(799\) 1.02227e13 0.887371
\(800\) 4.08556e12 0.352652
\(801\) 0 0
\(802\) 9.44751e12 0.806367
\(803\) −8.10312e11 −0.0687753
\(804\) 0 0
\(805\) 7.70002e12 0.646265
\(806\) 4.21072e11 0.0351438
\(807\) 0 0
\(808\) −5.65469e12 −0.466722
\(809\) −2.44611e11 −0.0200774 −0.0100387 0.999950i \(-0.503195\pi\)
−0.0100387 + 0.999950i \(0.503195\pi\)
\(810\) 0 0
\(811\) 1.86901e13 1.51711 0.758555 0.651609i \(-0.225906\pi\)
0.758555 + 0.651609i \(0.225906\pi\)
\(812\) −2.52748e11 −0.0204026
\(813\) 0 0
\(814\) 8.80714e11 0.0703113
\(815\) 6.95430e12 0.552134
\(816\) 0 0
\(817\) 4.95605e12 0.389167
\(818\) −6.84126e12 −0.534252
\(819\) 0 0
\(820\) 1.45090e13 1.12066
\(821\) −2.98096e12 −0.228987 −0.114494 0.993424i \(-0.536525\pi\)
−0.114494 + 0.993424i \(0.536525\pi\)
\(822\) 0 0
\(823\) −1.01707e13 −0.772774 −0.386387 0.922337i \(-0.626277\pi\)
−0.386387 + 0.922337i \(0.626277\pi\)
\(824\) −3.83003e12 −0.289421
\(825\) 0 0
\(826\) −2.24377e12 −0.167713
\(827\) −1.99682e13 −1.48445 −0.742223 0.670153i \(-0.766229\pi\)
−0.742223 + 0.670153i \(0.766229\pi\)
\(828\) 0 0
\(829\) 2.35288e13 1.73023 0.865115 0.501574i \(-0.167245\pi\)
0.865115 + 0.501574i \(0.167245\pi\)
\(830\) 5.04674e12 0.369113
\(831\) 0 0
\(832\) −2.27409e12 −0.164533
\(833\) −1.70293e13 −1.22545
\(834\) 0 0
\(835\) −3.57561e13 −2.54543
\(836\) 1.94018e11 0.0137377
\(837\) 0 0
\(838\) −1.56144e13 −1.09378
\(839\) −9.16607e12 −0.638637 −0.319319 0.947647i \(-0.603454\pi\)
−0.319319 + 0.947647i \(0.603454\pi\)
\(840\) 0 0
\(841\) −1.40929e13 −0.971444
\(842\) −1.09267e12 −0.0749179
\(843\) 0 0
\(844\) −3.35403e12 −0.227523
\(845\) 1.87879e13 1.26772
\(846\) 0 0
\(847\) 3.56509e12 0.238010
\(848\) 4.36050e12 0.289571
\(849\) 0 0
\(850\) 2.79369e13 1.83566
\(851\) −1.96450e13 −1.28401
\(852\) 0 0
\(853\) 9.94411e12 0.643125 0.321562 0.946888i \(-0.395792\pi\)
0.321562 + 0.946888i \(0.395792\pi\)
\(854\) 9.88514e11 0.0635949
\(855\) 0 0
\(856\) 1.75216e12 0.111543
\(857\) 1.90800e13 1.20827 0.604137 0.796880i \(-0.293518\pi\)
0.604137 + 0.796880i \(0.293518\pi\)
\(858\) 0 0
\(859\) 3.10124e12 0.194342 0.0971710 0.995268i \(-0.469021\pi\)
0.0971710 + 0.995268i \(0.469021\pi\)
\(860\) 2.35460e13 1.46782
\(861\) 0 0
\(862\) 4.75381e12 0.293264
\(863\) 2.40338e13 1.47494 0.737470 0.675380i \(-0.236020\pi\)
0.737470 + 0.675380i \(0.236020\pi\)
\(864\) 0 0
\(865\) −3.52099e13 −2.13842
\(866\) −7.21869e12 −0.436142
\(867\) 0 0
\(868\) 7.62431e10 0.00455892
\(869\) 1.46276e12 0.0870129
\(870\) 0 0
\(871\) −3.29023e13 −1.93707
\(872\) −4.66130e11 −0.0273013
\(873\) 0 0
\(874\) −4.32772e12 −0.250875
\(875\) −7.20905e12 −0.415759
\(876\) 0 0
\(877\) −5.20893e12 −0.297338 −0.148669 0.988887i \(-0.547499\pi\)
−0.148669 + 0.988887i \(0.547499\pi\)
\(878\) 8.88771e12 0.504736
\(879\) 0 0
\(880\) 9.21772e11 0.0518145
\(881\) 7.36387e12 0.411826 0.205913 0.978570i \(-0.433984\pi\)
0.205913 + 0.978570i \(0.433984\pi\)
\(882\) 0 0
\(883\) 2.98715e13 1.65362 0.826808 0.562485i \(-0.190155\pi\)
0.826808 + 0.562485i \(0.190155\pi\)
\(884\) −1.55501e13 −0.856442
\(885\) 0 0
\(886\) 2.58705e13 1.41044
\(887\) 1.43208e13 0.776802 0.388401 0.921490i \(-0.373027\pi\)
0.388401 + 0.921490i \(0.373027\pi\)
\(888\) 0 0
\(889\) −2.52650e12 −0.135663
\(890\) −2.75090e13 −1.46967
\(891\) 0 0
\(892\) 1.66489e13 0.880526
\(893\) 2.97285e12 0.156438
\(894\) 0 0
\(895\) 4.41153e13 2.29819
\(896\) −4.11766e11 −0.0213435
\(897\) 0 0
\(898\) −7.24628e12 −0.371853
\(899\) −1.24964e11 −0.00638069
\(900\) 0 0
\(901\) 2.98169e13 1.50730
\(902\) 2.18046e12 0.109678
\(903\) 0 0
\(904\) −5.61544e12 −0.279657
\(905\) 6.80020e13 3.36979
\(906\) 0 0
\(907\) −3.98348e13 −1.95448 −0.977238 0.212146i \(-0.931955\pi\)
−0.977238 + 0.212146i \(0.931955\pi\)
\(908\) −2.63655e12 −0.128721
\(909\) 0 0
\(910\) 8.04589e12 0.388945
\(911\) 8.77892e12 0.422288 0.211144 0.977455i \(-0.432281\pi\)
0.211144 + 0.977455i \(0.432281\pi\)
\(912\) 0 0
\(913\) 7.58442e11 0.0361247
\(914\) 1.76496e13 0.836522
\(915\) 0 0
\(916\) 4.39780e12 0.206398
\(917\) −5.60918e12 −0.261962
\(918\) 0 0
\(919\) 2.89837e12 0.134040 0.0670200 0.997752i \(-0.478651\pi\)
0.0670200 + 0.997752i \(0.478651\pi\)
\(920\) −2.05608e13 −0.946228
\(921\) 0 0
\(922\) −3.55347e12 −0.161943
\(923\) −2.48293e13 −1.12605
\(924\) 0 0
\(925\) 3.68790e13 1.65631
\(926\) 1.59557e13 0.713125
\(927\) 0 0
\(928\) 6.74895e11 0.0298724
\(929\) −1.09141e13 −0.480749 −0.240375 0.970680i \(-0.577270\pi\)
−0.240375 + 0.970680i \(0.577270\pi\)
\(930\) 0 0
\(931\) −4.95228e12 −0.216039
\(932\) 1.67031e13 0.725146
\(933\) 0 0
\(934\) −1.39773e13 −0.600983
\(935\) 6.30304e12 0.269711
\(936\) 0 0
\(937\) 3.31892e13 1.40659 0.703296 0.710897i \(-0.251711\pi\)
0.703296 + 0.710897i \(0.251711\pi\)
\(938\) −5.95759e12 −0.251280
\(939\) 0 0
\(940\) 1.41239e13 0.590038
\(941\) 4.06875e13 1.69164 0.845819 0.533471i \(-0.179112\pi\)
0.845819 + 0.533471i \(0.179112\pi\)
\(942\) 0 0
\(943\) −4.86368e13 −2.00292
\(944\) 5.99138e12 0.245557
\(945\) 0 0
\(946\) 3.53857e12 0.143654
\(947\) 1.90875e13 0.771211 0.385606 0.922664i \(-0.373993\pi\)
0.385606 + 0.922664i \(0.373993\pi\)
\(948\) 0 0
\(949\) 1.88865e13 0.755882
\(950\) 8.12430e12 0.323616
\(951\) 0 0
\(952\) −2.81564e12 −0.111099
\(953\) 2.79671e12 0.109832 0.0549161 0.998491i \(-0.482511\pi\)
0.0549161 + 0.998491i \(0.482511\pi\)
\(954\) 0 0
\(955\) −6.82800e11 −0.0265631
\(956\) 3.66141e12 0.141771
\(957\) 0 0
\(958\) −2.57053e13 −0.986002
\(959\) −4.97250e12 −0.189842
\(960\) 0 0
\(961\) −2.64019e13 −0.998574
\(962\) −2.05274e13 −0.772764
\(963\) 0 0
\(964\) −1.22767e13 −0.457864
\(965\) 3.46136e13 1.28492
\(966\) 0 0
\(967\) −1.90809e10 −0.000701745 0 −0.000350873 1.00000i \(-0.500112\pi\)
−0.000350873 1.00000i \(0.500112\pi\)
\(968\) −9.51963e12 −0.348482
\(969\) 0 0
\(970\) −5.09422e13 −1.84759
\(971\) 4.13526e12 0.149285 0.0746426 0.997210i \(-0.476218\pi\)
0.0746426 + 0.997210i \(0.476218\pi\)
\(972\) 0 0
\(973\) −1.33306e13 −0.476807
\(974\) −1.48061e13 −0.527139
\(975\) 0 0
\(976\) −2.63956e12 −0.0931124
\(977\) −3.74336e13 −1.31443 −0.657213 0.753705i \(-0.728265\pi\)
−0.657213 + 0.753705i \(0.728265\pi\)
\(978\) 0 0
\(979\) −4.13414e12 −0.143835
\(980\) −2.35281e13 −0.814835
\(981\) 0 0
\(982\) 1.48581e13 0.509871
\(983\) −1.37412e13 −0.469390 −0.234695 0.972069i \(-0.575409\pi\)
−0.234695 + 0.972069i \(0.575409\pi\)
\(984\) 0 0
\(985\) −4.88830e13 −1.65461
\(986\) 4.61491e12 0.155495
\(987\) 0 0
\(988\) −4.52211e12 −0.150985
\(989\) −7.89307e13 −2.62339
\(990\) 0 0
\(991\) −2.92261e13 −0.962584 −0.481292 0.876560i \(-0.659832\pi\)
−0.481292 + 0.876560i \(0.659832\pi\)
\(992\) −2.03587e11 −0.00667494
\(993\) 0 0
\(994\) −4.49582e12 −0.146073
\(995\) −9.57564e13 −3.09716
\(996\) 0 0
\(997\) 1.27081e13 0.407335 0.203668 0.979040i \(-0.434714\pi\)
0.203668 + 0.979040i \(0.434714\pi\)
\(998\) 1.91375e13 0.610658
\(999\) 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 342.10.a.l.1.4 4
3.2 odd 2 38.10.a.d.1.2 4
12.11 even 2 304.10.a.e.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
38.10.a.d.1.2 4 3.2 odd 2
304.10.a.e.1.3 4 12.11 even 2
342.10.a.l.1.4 4 1.1 even 1 trivial