Properties

Label 72.22.a.d.1.2
Level $72$
Weight $22$
Character 72.1
Self dual yes
Analytic conductor $201.224$
Analytic rank $1$
Dimension $3$
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] = [72,22,Mod(1,72)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(72, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("72.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 72 = 2^{3} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 72.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: \(201.223687887\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 2295485x - 828958533 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{21}\cdot 3^{4}\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 24)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1283.97\) of defining polynomial
Character \(\chi\) \(=\) 72.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+8.90852e6 q^{5} +5.87822e8 q^{7} +O(q^{10})\) \(q+8.90852e6 q^{5} +5.87822e8 q^{7} +1.42332e11 q^{11} +1.49035e11 q^{13} -9.95199e12 q^{17} -5.02926e12 q^{19} -3.25343e14 q^{23} -3.97475e14 q^{25} -1.07038e15 q^{29} +3.98818e15 q^{31} +5.23663e15 q^{35} -3.35659e16 q^{37} -4.54703e16 q^{41} -8.38924e16 q^{43} +5.34085e16 q^{47} -2.13011e17 q^{49} -1.14205e18 q^{53} +1.26797e18 q^{55} +5.97497e18 q^{59} +4.12289e17 q^{61} +1.32769e18 q^{65} -2.42251e19 q^{67} -7.25973e16 q^{71} -2.97771e19 q^{73} +8.36661e19 q^{77} -5.78794e19 q^{79} +2.48286e20 q^{83} -8.86575e19 q^{85} +5.56792e19 q^{89} +8.76063e19 q^{91} -4.48032e19 q^{95} -8.39559e20 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2080026 q^{5} - 1205282064 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2080026 q^{5} - 1205282064 q^{7} - 13839247500 q^{11} + 718855551690 q^{13} - 2135189843046 q^{17} - 40122324686988 q^{19} - 278424417682632 q^{23} + 13\!\cdots\!01 q^{25}+ \cdots - 15\!\cdots\!22 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 8.90852e6 0.407963 0.203981 0.978975i \(-0.434612\pi\)
0.203981 + 0.978975i \(0.434612\pi\)
\(6\) 0 0
\(7\) 5.87822e8 0.786532 0.393266 0.919425i \(-0.371345\pi\)
0.393266 + 0.919425i \(0.371345\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 1.42332e11 1.65455 0.827276 0.561796i \(-0.189889\pi\)
0.827276 + 0.561796i \(0.189889\pi\)
\(12\) 0 0
\(13\) 1.49035e11 0.299836 0.149918 0.988698i \(-0.452099\pi\)
0.149918 + 0.988698i \(0.452099\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −9.95199e12 −1.19728 −0.598641 0.801018i \(-0.704292\pi\)
−0.598641 + 0.801018i \(0.704292\pi\)
\(18\) 0 0
\(19\) −5.02926e12 −0.188188 −0.0940938 0.995563i \(-0.529995\pi\)
−0.0940938 + 0.995563i \(0.529995\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −3.25343e14 −1.63757 −0.818783 0.574103i \(-0.805351\pi\)
−0.818783 + 0.574103i \(0.805351\pi\)
\(24\) 0 0
\(25\) −3.97475e14 −0.833566
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.07038e15 −0.472454 −0.236227 0.971698i \(-0.575911\pi\)
−0.236227 + 0.971698i \(0.575911\pi\)
\(30\) 0 0
\(31\) 3.98818e15 0.873932 0.436966 0.899478i \(-0.356053\pi\)
0.436966 + 0.899478i \(0.356053\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 5.23663e15 0.320876
\(36\) 0 0
\(37\) −3.35659e16 −1.14757 −0.573787 0.819004i \(-0.694526\pi\)
−0.573787 + 0.819004i \(0.694526\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −4.54703e16 −0.529051 −0.264526 0.964379i \(-0.585215\pi\)
−0.264526 + 0.964379i \(0.585215\pi\)
\(42\) 0 0
\(43\) −8.38924e16 −0.591976 −0.295988 0.955192i \(-0.595649\pi\)
−0.295988 + 0.955192i \(0.595649\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 5.34085e16 0.148110 0.0740548 0.997254i \(-0.476406\pi\)
0.0740548 + 0.997254i \(0.476406\pi\)
\(48\) 0 0
\(49\) −2.13011e17 −0.381367
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −1.14205e18 −0.896988 −0.448494 0.893786i \(-0.648039\pi\)
−0.448494 + 0.893786i \(0.648039\pi\)
\(54\) 0 0
\(55\) 1.26797e18 0.674996
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 5.97497e18 1.52191 0.760956 0.648803i \(-0.224730\pi\)
0.760956 + 0.648803i \(0.224730\pi\)
\(60\) 0 0
\(61\) 4.12289e17 0.0740011 0.0370006 0.999315i \(-0.488220\pi\)
0.0370006 + 0.999315i \(0.488220\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 1.32769e18 0.122322
\(66\) 0 0
\(67\) −2.42251e19 −1.62360 −0.811802 0.583933i \(-0.801513\pi\)
−0.811802 + 0.583933i \(0.801513\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −7.25973e16 −0.00264672 −0.00132336 0.999999i \(-0.500421\pi\)
−0.00132336 + 0.999999i \(0.500421\pi\)
\(72\) 0 0
\(73\) −2.97771e19 −0.810947 −0.405473 0.914107i \(-0.632893\pi\)
−0.405473 + 0.914107i \(0.632893\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.36661e19 1.30136
\(78\) 0 0
\(79\) −5.78794e19 −0.687764 −0.343882 0.939013i \(-0.611742\pi\)
−0.343882 + 0.939013i \(0.611742\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 2.48286e20 1.75643 0.878217 0.478262i \(-0.158733\pi\)
0.878217 + 0.478262i \(0.158733\pi\)
\(84\) 0 0
\(85\) −8.86575e19 −0.488446
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 5.56792e19 0.189277 0.0946385 0.995512i \(-0.469830\pi\)
0.0946385 + 0.995512i \(0.469830\pi\)
\(90\) 0 0
\(91\) 8.76063e19 0.235831
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −4.48032e19 −0.0767735
\(96\) 0 0
\(97\) −8.39559e20 −1.15597 −0.577987 0.816046i \(-0.696161\pi\)
−0.577987 + 0.816046i \(0.696161\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 7.25262e20 0.653312 0.326656 0.945143i \(-0.394078\pi\)
0.326656 + 0.945143i \(0.394078\pi\)
\(102\) 0 0
\(103\) −1.79494e21 −1.31601 −0.658004 0.753015i \(-0.728599\pi\)
−0.658004 + 0.753015i \(0.728599\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.59231e20 0.0782522 0.0391261 0.999234i \(-0.487543\pi\)
0.0391261 + 0.999234i \(0.487543\pi\)
\(108\) 0 0
\(109\) 2.97325e21 1.20297 0.601484 0.798885i \(-0.294577\pi\)
0.601484 + 0.798885i \(0.294577\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.38076e21 −0.382644 −0.191322 0.981527i \(-0.561277\pi\)
−0.191322 + 0.981527i \(0.561277\pi\)
\(114\) 0 0
\(115\) −2.89832e21 −0.668066
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −5.85000e21 −0.941700
\(120\) 0 0
\(121\) 1.28582e22 1.73754
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −7.78883e21 −0.748027
\(126\) 0 0
\(127\) 4.56871e21 0.371411 0.185706 0.982605i \(-0.440543\pi\)
0.185706 + 0.982605i \(0.440543\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 8.64013e21 0.507191 0.253595 0.967310i \(-0.418387\pi\)
0.253595 + 0.967310i \(0.418387\pi\)
\(132\) 0 0
\(133\) −2.95631e21 −0.148016
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −2.68277e22 −0.984052 −0.492026 0.870581i \(-0.663743\pi\)
−0.492026 + 0.870581i \(0.663743\pi\)
\(138\) 0 0
\(139\) 3.90524e22 1.23025 0.615123 0.788431i \(-0.289106\pi\)
0.615123 + 0.788431i \(0.289106\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 2.12126e22 0.496095
\(144\) 0 0
\(145\) −9.53551e21 −0.192744
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −3.86793e22 −0.587521 −0.293760 0.955879i \(-0.594907\pi\)
−0.293760 + 0.955879i \(0.594907\pi\)
\(150\) 0 0
\(151\) −1.07575e23 −1.42054 −0.710269 0.703930i \(-0.751427\pi\)
−0.710269 + 0.703930i \(0.751427\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 3.55288e22 0.356532
\(156\) 0 0
\(157\) 6.91463e22 0.606489 0.303244 0.952913i \(-0.401930\pi\)
0.303244 + 0.952913i \(0.401930\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −1.91244e23 −1.28800
\(162\) 0 0
\(163\) −2.70783e23 −1.60196 −0.800980 0.598691i \(-0.795688\pi\)
−0.800980 + 0.598691i \(0.795688\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.30715e23 1.05816 0.529081 0.848571i \(-0.322537\pi\)
0.529081 + 0.848571i \(0.322537\pi\)
\(168\) 0 0
\(169\) −2.24853e23 −0.910098
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 3.68393e23 1.16635 0.583173 0.812348i \(-0.301811\pi\)
0.583173 + 0.812348i \(0.301811\pi\)
\(174\) 0 0
\(175\) −2.33645e23 −0.655627
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 6.78508e23 1.50175 0.750876 0.660443i \(-0.229631\pi\)
0.750876 + 0.660443i \(0.229631\pi\)
\(180\) 0 0
\(181\) 9.41228e23 1.85383 0.926915 0.375272i \(-0.122451\pi\)
0.926915 + 0.375272i \(0.122451\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −2.99023e23 −0.468168
\(186\) 0 0
\(187\) −1.41649e24 −1.98096
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −1.00355e24 −1.12380 −0.561899 0.827206i \(-0.689929\pi\)
−0.561899 + 0.827206i \(0.689929\pi\)
\(192\) 0 0
\(193\) −1.40841e24 −1.41377 −0.706885 0.707328i \(-0.749900\pi\)
−0.706885 + 0.707328i \(0.749900\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.96986e24 −1.59419 −0.797096 0.603853i \(-0.793631\pi\)
−0.797096 + 0.603853i \(0.793631\pi\)
\(198\) 0 0
\(199\) 1.33492e24 0.971620 0.485810 0.874064i \(-0.338525\pi\)
0.485810 + 0.874064i \(0.338525\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −6.29193e23 −0.371600
\(204\) 0 0
\(205\) −4.05073e23 −0.215833
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) −7.15826e23 −0.311366
\(210\) 0 0
\(211\) −1.21610e24 −0.478633 −0.239316 0.970942i \(-0.576923\pi\)
−0.239316 + 0.970942i \(0.576923\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −7.47357e23 −0.241504
\(216\) 0 0
\(217\) 2.34434e24 0.687375
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) −1.48320e24 −0.358989
\(222\) 0 0
\(223\) −5.72126e24 −1.25977 −0.629884 0.776689i \(-0.716898\pi\)
−0.629884 + 0.776689i \(0.716898\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −5.60839e24 −1.02463 −0.512314 0.858798i \(-0.671212\pi\)
−0.512314 + 0.858798i \(0.671212\pi\)
\(228\) 0 0
\(229\) −2.51872e24 −0.419670 −0.209835 0.977737i \(-0.567293\pi\)
−0.209835 + 0.977737i \(0.567293\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.98939e24 −0.276365 −0.138182 0.990407i \(-0.544126\pi\)
−0.138182 + 0.990407i \(0.544126\pi\)
\(234\) 0 0
\(235\) 4.75791e23 0.0604232
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −7.32699e24 −0.779378 −0.389689 0.920947i \(-0.627417\pi\)
−0.389689 + 0.920947i \(0.627417\pi\)
\(240\) 0 0
\(241\) −1.88926e25 −1.84125 −0.920625 0.390448i \(-0.872320\pi\)
−0.920625 + 0.390448i \(0.872320\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −1.89761e24 −0.155584
\(246\) 0 0
\(247\) −7.49537e23 −0.0564255
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.65459e25 1.05224 0.526122 0.850409i \(-0.323645\pi\)
0.526122 + 0.850409i \(0.323645\pi\)
\(252\) 0 0
\(253\) −4.63068e25 −2.70944
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 2.93804e25 1.45801 0.729004 0.684509i \(-0.239983\pi\)
0.729004 + 0.684509i \(0.239983\pi\)
\(258\) 0 0
\(259\) −1.97308e25 −0.902604
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −3.29044e25 −1.28150 −0.640749 0.767750i \(-0.721376\pi\)
−0.640749 + 0.767750i \(0.721376\pi\)
\(264\) 0 0
\(265\) −1.01739e25 −0.365938
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 4.30191e24 0.132209 0.0661047 0.997813i \(-0.478943\pi\)
0.0661047 + 0.997813i \(0.478943\pi\)
\(270\) 0 0
\(271\) −2.72610e25 −0.775110 −0.387555 0.921846i \(-0.626680\pi\)
−0.387555 + 0.921846i \(0.626680\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −5.65736e25 −1.37918
\(276\) 0 0
\(277\) −4.61030e25 −1.04158 −0.520788 0.853686i \(-0.674362\pi\)
−0.520788 + 0.853686i \(0.674362\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 7.74125e24 0.150451 0.0752254 0.997167i \(-0.476032\pi\)
0.0752254 + 0.997167i \(0.476032\pi\)
\(282\) 0 0
\(283\) −1.41856e25 −0.255911 −0.127955 0.991780i \(-0.540841\pi\)
−0.127955 + 0.991780i \(0.540841\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −2.67285e25 −0.416116
\(288\) 0 0
\(289\) 2.99502e25 0.433483
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −8.61121e25 −1.07883 −0.539416 0.842039i \(-0.681355\pi\)
−0.539416 + 0.842039i \(0.681355\pi\)
\(294\) 0 0
\(295\) 5.32282e25 0.620884
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −4.84876e25 −0.491002
\(300\) 0 0
\(301\) −4.93138e25 −0.465608
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 3.67288e24 0.0301897
\(306\) 0 0
\(307\) −9.91709e25 −0.761082 −0.380541 0.924764i \(-0.624262\pi\)
−0.380541 + 0.924764i \(0.624262\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.15943e26 0.776709 0.388355 0.921510i \(-0.373044\pi\)
0.388355 + 0.921510i \(0.373044\pi\)
\(312\) 0 0
\(313\) 2.06648e26 1.29424 0.647122 0.762386i \(-0.275972\pi\)
0.647122 + 0.762386i \(0.275972\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −2.90072e26 −1.58995 −0.794976 0.606641i \(-0.792517\pi\)
−0.794976 + 0.606641i \(0.792517\pi\)
\(318\) 0 0
\(319\) −1.52350e26 −0.781699
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 5.00511e25 0.225314
\(324\) 0 0
\(325\) −5.92379e25 −0.249934
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 3.13947e25 0.116493
\(330\) 0 0
\(331\) 2.56275e26 0.892303 0.446151 0.894958i \(-0.352794\pi\)
0.446151 + 0.894958i \(0.352794\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) −2.15810e26 −0.662370
\(336\) 0 0
\(337\) 1.42732e26 0.411535 0.205768 0.978601i \(-0.434031\pi\)
0.205768 + 0.978601i \(0.434031\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 5.67648e26 1.44597
\(342\) 0 0
\(343\) −4.53538e26 −1.08649
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 4.74021e26 1.00540 0.502699 0.864462i \(-0.332341\pi\)
0.502699 + 0.864462i \(0.332341\pi\)
\(348\) 0 0
\(349\) 8.88859e26 1.77487 0.887434 0.460935i \(-0.152486\pi\)
0.887434 + 0.460935i \(0.152486\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.86433e26 0.861764 0.430882 0.902408i \(-0.358202\pi\)
0.430882 + 0.902408i \(0.358202\pi\)
\(354\) 0 0
\(355\) −6.46734e23 −0.00107976
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 4.36541e26 0.647938 0.323969 0.946068i \(-0.394983\pi\)
0.323969 + 0.946068i \(0.394983\pi\)
\(360\) 0 0
\(361\) −6.88916e26 −0.964585
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −2.65270e26 −0.330836
\(366\) 0 0
\(367\) −1.13289e27 −1.33412 −0.667060 0.745004i \(-0.732447\pi\)
−0.667060 + 0.745004i \(0.732447\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −6.71319e26 −0.705510
\(372\) 0 0
\(373\) 6.51576e26 0.647176 0.323588 0.946198i \(-0.395111\pi\)
0.323588 + 0.946198i \(0.395111\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.59525e26 −0.141659
\(378\) 0 0
\(379\) 1.71451e27 1.44022 0.720109 0.693861i \(-0.244092\pi\)
0.720109 + 0.693861i \(0.244092\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.63419e27 1.22946 0.614731 0.788737i \(-0.289265\pi\)
0.614731 + 0.788737i \(0.289265\pi\)
\(384\) 0 0
\(385\) 7.45341e26 0.530906
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 2.64030e27 1.68726 0.843632 0.536922i \(-0.180413\pi\)
0.843632 + 0.536922i \(0.180413\pi\)
\(390\) 0 0
\(391\) 3.23781e27 1.96063
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −5.15620e26 −0.280582
\(396\) 0 0
\(397\) 2.94709e27 1.52087 0.760437 0.649411i \(-0.224985\pi\)
0.760437 + 0.649411i \(0.224985\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −1.81255e27 −0.841927 −0.420963 0.907078i \(-0.638308\pi\)
−0.420963 + 0.907078i \(0.638308\pi\)
\(402\) 0 0
\(403\) 5.94381e26 0.262037
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −4.77752e27 −1.89872
\(408\) 0 0
\(409\) 2.59753e27 0.980542 0.490271 0.871570i \(-0.336898\pi\)
0.490271 + 0.871570i \(0.336898\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 3.51222e27 1.19703
\(414\) 0 0
\(415\) 2.21186e27 0.716560
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 3.40142e26 0.0996353 0.0498176 0.998758i \(-0.484136\pi\)
0.0498176 + 0.998758i \(0.484136\pi\)
\(420\) 0 0
\(421\) −4.22891e27 −1.17833 −0.589164 0.808014i \(-0.700543\pi\)
−0.589164 + 0.808014i \(0.700543\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 3.95567e27 0.998014
\(426\) 0 0
\(427\) 2.42353e26 0.0582043
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −7.72212e27 −1.68161 −0.840805 0.541339i \(-0.817918\pi\)
−0.840805 + 0.541339i \(0.817918\pi\)
\(432\) 0 0
\(433\) 3.14232e27 0.651819 0.325910 0.945401i \(-0.394330\pi\)
0.325910 + 0.945401i \(0.394330\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.63623e27 0.308170
\(438\) 0 0
\(439\) −3.34873e27 −0.601177 −0.300588 0.953754i \(-0.597183\pi\)
−0.300588 + 0.953754i \(0.597183\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −5.82708e27 −0.951068 −0.475534 0.879697i \(-0.657745\pi\)
−0.475534 + 0.879697i \(0.657745\pi\)
\(444\) 0 0
\(445\) 4.96019e26 0.0772179
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.14216e28 1.61860 0.809300 0.587396i \(-0.199847\pi\)
0.809300 + 0.587396i \(0.199847\pi\)
\(450\) 0 0
\(451\) −6.47190e27 −0.875343
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 7.80443e26 0.0962103
\(456\) 0 0
\(457\) 1.70411e27 0.200622 0.100311 0.994956i \(-0.468016\pi\)
0.100311 + 0.994956i \(0.468016\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.57190e28 1.68875 0.844374 0.535753i \(-0.179972\pi\)
0.844374 + 0.535753i \(0.179972\pi\)
\(462\) 0 0
\(463\) −2.45032e27 −0.251549 −0.125774 0.992059i \(-0.540142\pi\)
−0.125774 + 0.992059i \(0.540142\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −4.79077e26 −0.0449344 −0.0224672 0.999748i \(-0.507152\pi\)
−0.0224672 + 0.999748i \(0.507152\pi\)
\(468\) 0 0
\(469\) −1.42400e28 −1.27702
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −1.19406e28 −0.979455
\(474\) 0 0
\(475\) 1.99901e27 0.156867
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 5.29156e27 0.380243 0.190121 0.981761i \(-0.439112\pi\)
0.190121 + 0.981761i \(0.439112\pi\)
\(480\) 0 0
\(481\) −5.00252e27 −0.344085
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −7.47922e27 −0.471594
\(486\) 0 0
\(487\) −2.99405e28 −1.80803 −0.904013 0.427506i \(-0.859392\pi\)
−0.904013 + 0.427506i \(0.859392\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.72464e27 −0.0955749 −0.0477875 0.998858i \(-0.515217\pi\)
−0.0477875 + 0.998858i \(0.515217\pi\)
\(492\) 0 0
\(493\) 1.06524e28 0.565660
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) −4.26743e25 −0.00208173
\(498\) 0 0
\(499\) −1.82237e28 −0.852276 −0.426138 0.904658i \(-0.640126\pi\)
−0.426138 + 0.904658i \(0.640126\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −3.98628e28 −1.71437 −0.857183 0.515011i \(-0.827788\pi\)
−0.857183 + 0.515011i \(0.827788\pi\)
\(504\) 0 0
\(505\) 6.46101e27 0.266527
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 1.48470e28 0.563771 0.281886 0.959448i \(-0.409040\pi\)
0.281886 + 0.959448i \(0.409040\pi\)
\(510\) 0 0
\(511\) −1.75036e28 −0.637836
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −1.59902e28 −0.536882
\(516\) 0 0
\(517\) 7.60176e27 0.245055
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −4.40264e28 −1.30893 −0.654465 0.756092i \(-0.727106\pi\)
−0.654465 + 0.756092i \(0.727106\pi\)
\(522\) 0 0
\(523\) −3.71320e28 −1.06043 −0.530214 0.847864i \(-0.677888\pi\)
−0.530214 + 0.847864i \(0.677888\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) −3.96904e28 −1.04634
\(528\) 0 0
\(529\) 6.63763e28 1.68162
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −6.77669e27 −0.158629
\(534\) 0 0
\(535\) 1.41851e27 0.0319240
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) −3.03184e28 −0.630992
\(540\) 0 0
\(541\) −4.12076e28 −0.824909 −0.412454 0.910978i \(-0.635328\pi\)
−0.412454 + 0.910978i \(0.635328\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 2.64873e28 0.490766
\(546\) 0 0
\(547\) −2.13981e28 −0.381512 −0.190756 0.981637i \(-0.561094\pi\)
−0.190756 + 0.981637i \(0.561094\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 5.38322e27 0.0889099
\(552\) 0 0
\(553\) −3.40228e28 −0.540948
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 9.11122e28 1.34307 0.671533 0.740975i \(-0.265636\pi\)
0.671533 + 0.740975i \(0.265636\pi\)
\(558\) 0 0
\(559\) −1.25029e28 −0.177496
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −7.72184e28 −1.01714 −0.508572 0.861019i \(-0.669826\pi\)
−0.508572 + 0.861019i \(0.669826\pi\)
\(564\) 0 0
\(565\) −1.23005e28 −0.156104
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 8.18518e28 0.964605 0.482303 0.876005i \(-0.339801\pi\)
0.482303 + 0.876005i \(0.339801\pi\)
\(570\) 0 0
\(571\) 3.83570e27 0.0435678 0.0217839 0.999763i \(-0.493065\pi\)
0.0217839 + 0.999763i \(0.493065\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.29316e29 1.36502
\(576\) 0 0
\(577\) 1.15551e29 1.17606 0.588028 0.808841i \(-0.299905\pi\)
0.588028 + 0.808841i \(0.299905\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 1.45948e29 1.38149
\(582\) 0 0
\(583\) −1.62550e29 −1.48411
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −1.04413e29 −0.887271 −0.443636 0.896207i \(-0.646312\pi\)
−0.443636 + 0.896207i \(0.646312\pi\)
\(588\) 0 0
\(589\) −2.00576e28 −0.164463
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −1.30270e29 −0.994879 −0.497439 0.867499i \(-0.665726\pi\)
−0.497439 + 0.867499i \(0.665726\pi\)
\(594\) 0 0
\(595\) −5.21148e28 −0.384179
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.35818e29 −0.933205 −0.466603 0.884467i \(-0.654522\pi\)
−0.466603 + 0.884467i \(0.654522\pi\)
\(600\) 0 0
\(601\) −5.67794e28 −0.376712 −0.188356 0.982101i \(-0.560316\pi\)
−0.188356 + 0.982101i \(0.560316\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.14548e29 0.708853
\(606\) 0 0
\(607\) 2.93166e29 1.75239 0.876197 0.481953i \(-0.160073\pi\)
0.876197 + 0.481953i \(0.160073\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 7.95976e27 0.0444086
\(612\) 0 0
\(613\) −1.03374e29 −0.557282 −0.278641 0.960395i \(-0.589884\pi\)
−0.278641 + 0.960395i \(0.589884\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.61665e29 1.31750 0.658750 0.752362i \(-0.271085\pi\)
0.658750 + 0.752362i \(0.271085\pi\)
\(618\) 0 0
\(619\) 2.52208e29 1.22746 0.613729 0.789517i \(-0.289669\pi\)
0.613729 + 0.789517i \(0.289669\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 3.27294e28 0.148872
\(624\) 0 0
\(625\) 1.20144e29 0.528399
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 3.34048e29 1.37397
\(630\) 0 0
\(631\) 4.18556e29 1.66512 0.832559 0.553936i \(-0.186875\pi\)
0.832559 + 0.553936i \(0.186875\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 4.07005e28 0.151522
\(636\) 0 0
\(637\) −3.17462e28 −0.114348
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −4.16106e29 −1.40344 −0.701721 0.712451i \(-0.747585\pi\)
−0.701721 + 0.712451i \(0.747585\pi\)
\(642\) 0 0
\(643\) 1.16723e29 0.381015 0.190508 0.981686i \(-0.438987\pi\)
0.190508 + 0.981686i \(0.438987\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.12199e29 −0.649007 −0.324503 0.945885i \(-0.605197\pi\)
−0.324503 + 0.945885i \(0.605197\pi\)
\(648\) 0 0
\(649\) 8.50432e29 2.51808
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −2.19143e29 −0.608329 −0.304165 0.952619i \(-0.598377\pi\)
−0.304165 + 0.952619i \(0.598377\pi\)
\(654\) 0 0
\(655\) 7.69707e28 0.206915
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 7.81698e28 0.197125 0.0985627 0.995131i \(-0.468576\pi\)
0.0985627 + 0.995131i \(0.468576\pi\)
\(660\) 0 0
\(661\) −4.98165e29 −1.21691 −0.608454 0.793589i \(-0.708210\pi\)
−0.608454 + 0.793589i \(0.708210\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −2.63363e28 −0.0603849
\(666\) 0 0
\(667\) 3.48241e29 0.773674
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 5.86820e28 0.122439
\(672\) 0 0
\(673\) 7.45598e29 1.50781 0.753904 0.656984i \(-0.228168\pi\)
0.753904 + 0.656984i \(0.228168\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.07199e29 −0.203708 −0.101854 0.994799i \(-0.532477\pi\)
−0.101854 + 0.994799i \(0.532477\pi\)
\(678\) 0 0
\(679\) −4.93511e29 −0.909210
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 6.20112e29 1.07412 0.537060 0.843544i \(-0.319535\pi\)
0.537060 + 0.843544i \(0.319535\pi\)
\(684\) 0 0
\(685\) −2.38995e29 −0.401457
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −1.70205e29 −0.268950
\(690\) 0 0
\(691\) −1.11755e30 −1.71296 −0.856482 0.516177i \(-0.827355\pi\)
−0.856482 + 0.516177i \(0.827355\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 3.47899e29 0.501895
\(696\) 0 0
\(697\) 4.52520e29 0.633423
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 9.02930e29 1.19019 0.595093 0.803657i \(-0.297115\pi\)
0.595093 + 0.803657i \(0.297115\pi\)
\(702\) 0 0
\(703\) 1.68812e29 0.215959
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 4.26325e29 0.513851
\(708\) 0 0
\(709\) 1.25054e29 0.146322 0.0731612 0.997320i \(-0.476691\pi\)
0.0731612 + 0.997320i \(0.476691\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −1.29753e30 −1.43112
\(714\) 0 0
\(715\) 1.88973e29 0.202388
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 7.25984e29 0.733287 0.366643 0.930362i \(-0.380507\pi\)
0.366643 + 0.930362i \(0.380507\pi\)
\(720\) 0 0
\(721\) −1.05510e30 −1.03508
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.25450e29 0.393822
\(726\) 0 0
\(727\) −3.73992e29 −0.336318 −0.168159 0.985760i \(-0.553782\pi\)
−0.168159 + 0.985760i \(0.553782\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 8.34897e29 0.708762
\(732\) 0 0
\(733\) −1.77085e30 −1.46080 −0.730401 0.683019i \(-0.760667\pi\)
−0.730401 + 0.683019i \(0.760667\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −3.44802e30 −2.68634
\(738\) 0 0
\(739\) −1.88567e30 −1.42791 −0.713953 0.700194i \(-0.753097\pi\)
−0.713953 + 0.700194i \(0.753097\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 8.39029e29 0.600336 0.300168 0.953886i \(-0.402957\pi\)
0.300168 + 0.953886i \(0.402957\pi\)
\(744\) 0 0
\(745\) −3.44576e29 −0.239687
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 9.35992e28 0.0615479
\(750\) 0 0
\(751\) 1.32480e30 0.847095 0.423547 0.905874i \(-0.360785\pi\)
0.423547 + 0.905874i \(0.360785\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −9.58333e29 −0.579527
\(756\) 0 0
\(757\) −8.91190e29 −0.524159 −0.262080 0.965046i \(-0.584408\pi\)
−0.262080 + 0.965046i \(0.584408\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −2.22977e30 −1.24086 −0.620429 0.784263i \(-0.713041\pi\)
−0.620429 + 0.784263i \(0.713041\pi\)
\(762\) 0 0
\(763\) 1.74774e30 0.946172
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.90482e29 0.456325
\(768\) 0 0
\(769\) −1.65220e30 −0.823827 −0.411913 0.911223i \(-0.635139\pi\)
−0.411913 + 0.911223i \(0.635139\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 7.95379e29 0.375569 0.187784 0.982210i \(-0.439869\pi\)
0.187784 + 0.982210i \(0.439869\pi\)
\(774\) 0 0
\(775\) −1.58521e30 −0.728480
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.28682e29 0.0995609
\(780\) 0 0
\(781\) −1.03329e28 −0.00437913
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 6.15992e29 0.247425
\(786\) 0 0
\(787\) 2.00720e30 0.784977 0.392489 0.919757i \(-0.371614\pi\)
0.392489 + 0.919757i \(0.371614\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −8.11641e29 −0.300962
\(792\) 0 0
\(793\) 6.14457e28 0.0221882
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −3.69070e30 −1.26414 −0.632071 0.774910i \(-0.717795\pi\)
−0.632071 + 0.774910i \(0.717795\pi\)
\(798\) 0 0
\(799\) −5.31521e29 −0.177329
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −4.23824e30 −1.34175
\(804\) 0 0
\(805\) −1.70370e30 −0.525455
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −4.72614e30 −1.38372 −0.691859 0.722033i \(-0.743208\pi\)
−0.691859 + 0.722033i \(0.743208\pi\)
\(810\) 0 0
\(811\) 2.65583e30 0.757673 0.378837 0.925464i \(-0.376324\pi\)
0.378837 + 0.925464i \(0.376324\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −2.41228e30 −0.653540
\(816\) 0 0
\(817\) 4.21917e29 0.111403
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 5.10610e30 1.28082 0.640408 0.768035i \(-0.278765\pi\)
0.640408 + 0.768035i \(0.278765\pi\)
\(822\) 0 0
\(823\) −4.35473e30 −1.06479 −0.532394 0.846497i \(-0.678707\pi\)
−0.532394 + 0.846497i \(0.678707\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.07345e30 0.946573 0.473287 0.880909i \(-0.343068\pi\)
0.473287 + 0.880909i \(0.343068\pi\)
\(828\) 0 0
\(829\) −2.77876e30 −0.629548 −0.314774 0.949167i \(-0.601929\pi\)
−0.314774 + 0.949167i \(0.601929\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 2.11988e30 0.456604
\(834\) 0 0
\(835\) 2.05533e30 0.431691
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −3.15950e30 −0.631128 −0.315564 0.948904i \(-0.602194\pi\)
−0.315564 + 0.948904i \(0.602194\pi\)
\(840\) 0 0
\(841\) −3.98713e30 −0.776787
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −2.00311e30 −0.371286
\(846\) 0 0
\(847\) 7.55836e30 1.36663
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1.09204e31 1.87923
\(852\) 0 0
\(853\) −1.12969e31 −1.89668 −0.948338 0.317260i \(-0.897237\pi\)
−0.948338 + 0.317260i \(0.897237\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.09526e31 1.75073 0.875366 0.483462i \(-0.160621\pi\)
0.875366 + 0.483462i \(0.160621\pi\)
\(858\) 0 0
\(859\) −8.09790e29 −0.126312 −0.0631559 0.998004i \(-0.520117\pi\)
−0.0631559 + 0.998004i \(0.520117\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 9.41039e30 1.39796 0.698979 0.715142i \(-0.253638\pi\)
0.698979 + 0.715142i \(0.253638\pi\)
\(864\) 0 0
\(865\) 3.28184e30 0.475825
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −8.23811e30 −1.13794
\(870\) 0 0
\(871\) −3.61040e30 −0.486816
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −4.57845e30 −0.588347
\(876\) 0 0
\(877\) 9.07878e30 1.13902 0.569510 0.821984i \(-0.307133\pi\)
0.569510 + 0.821984i \(0.307133\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −3.03572e30 −0.363090 −0.181545 0.983383i \(-0.558110\pi\)
−0.181545 + 0.983383i \(0.558110\pi\)
\(882\) 0 0
\(883\) 6.61886e28 0.00773030 0.00386515 0.999993i \(-0.498770\pi\)
0.00386515 + 0.999993i \(0.498770\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 3.50101e30 0.389937 0.194969 0.980809i \(-0.437539\pi\)
0.194969 + 0.980809i \(0.437539\pi\)
\(888\) 0 0
\(889\) 2.68559e30 0.292127
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −2.68605e29 −0.0278724
\(894\) 0 0
\(895\) 6.04451e30 0.612659
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −4.26888e30 −0.412892
\(900\) 0 0
\(901\) 1.13656e31 1.07395
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 8.38495e30 0.756293
\(906\) 0 0
\(907\) 5.24604e29 0.0462333 0.0231167 0.999733i \(-0.492641\pi\)
0.0231167 + 0.999733i \(0.492641\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.16565e31 0.980902 0.490451 0.871469i \(-0.336832\pi\)
0.490451 + 0.871469i \(0.336832\pi\)
\(912\) 0 0
\(913\) 3.53391e31 2.90611
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 5.07886e30 0.398922
\(918\) 0 0
\(919\) −7.15232e30 −0.549078 −0.274539 0.961576i \(-0.588525\pi\)
−0.274539 + 0.961576i \(0.588525\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −1.08196e28 −0.000793582 0
\(924\) 0 0
\(925\) 1.33416e31 0.956579
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −9.72648e30 −0.666486 −0.333243 0.942841i \(-0.608143\pi\)
−0.333243 + 0.942841i \(0.608143\pi\)
\(930\) 0 0
\(931\) 1.07129e30 0.0717686
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −1.26188e31 −0.808160
\(936\) 0 0
\(937\) −1.38653e31 −0.868288 −0.434144 0.900844i \(-0.642949\pi\)
−0.434144 + 0.900844i \(0.642949\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −2.03835e31 −1.22064 −0.610320 0.792155i \(-0.708959\pi\)
−0.610320 + 0.792155i \(0.708959\pi\)
\(942\) 0 0
\(943\) 1.47934e31 0.866356
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 3.05609e30 0.171195 0.0855975 0.996330i \(-0.472720\pi\)
0.0855975 + 0.996330i \(0.472720\pi\)
\(948\) 0 0
\(949\) −4.43784e30 −0.243151
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.54521e31 0.810053 0.405026 0.914305i \(-0.367262\pi\)
0.405026 + 0.914305i \(0.367262\pi\)
\(954\) 0 0
\(955\) −8.94014e30 −0.458468
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −1.57699e31 −0.773988
\(960\) 0 0
\(961\) −4.91989e30 −0.236243
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −1.25469e31 −0.576766
\(966\) 0 0
\(967\) 2.30504e31 1.03681 0.518406 0.855135i \(-0.326526\pi\)
0.518406 + 0.855135i \(0.326526\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −5.03493e29 −0.0216866 −0.0108433 0.999941i \(-0.503452\pi\)
−0.0108433 + 0.999941i \(0.503452\pi\)
\(972\) 0 0
\(973\) 2.29558e31 0.967628
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.24415e31 1.30981 0.654905 0.755711i \(-0.272709\pi\)
0.654905 + 0.755711i \(0.272709\pi\)
\(978\) 0 0
\(979\) 7.92495e30 0.313169
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) −1.25577e31 −0.475444 −0.237722 0.971333i \(-0.576401\pi\)
−0.237722 + 0.971333i \(0.576401\pi\)
\(984\) 0 0
\(985\) −1.75486e31 −0.650371
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 2.72938e31 0.969399
\(990\) 0 0
\(991\) −1.56643e31 −0.544676 −0.272338 0.962202i \(-0.587797\pi\)
−0.272338 + 0.962202i \(0.587797\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.18921e31 0.396385
\(996\) 0 0
\(997\) 4.16202e31 1.35833 0.679165 0.733986i \(-0.262342\pi\)
0.679165 + 0.733986i \(0.262342\pi\)
\(998\) 0 0
\(999\) 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 72.22.a.d.1.2 3
3.2 odd 2 24.22.a.c.1.2 3
12.11 even 2 48.22.a.l.1.2 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
24.22.a.c.1.2 3 3.2 odd 2
48.22.a.l.1.2 3 12.11 even 2
72.22.a.d.1.2 3 1.1 even 1 trivial