Properties

Label 384.8.a.m.1.2
Level $384$
Weight $8$
Character 384.1
Self dual yes
Analytic conductor $119.956$
Analytic rank $1$
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] = [384,8,Mod(1,384)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(384, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("384.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 384 = 2^{7} \cdot 3 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 384.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: \(119.955849786\)
Analytic rank: \(1\)
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: \( x^{4} - 620x^{2} - 700x + 83625 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{15}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-17.7724\) of defining polynomial
Character \(\chi\) \(=\) 384.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} -282.264 q^{5} +1362.23 q^{7} +729.000 q^{9} +O(q^{10})\) \(q-27.0000 q^{3} -282.264 q^{5} +1362.23 q^{7} +729.000 q^{9} +3814.16 q^{11} -14857.6 q^{13} +7621.12 q^{15} -4637.82 q^{17} -3218.11 q^{19} -36780.1 q^{21} +39347.2 q^{23} +1547.80 q^{25} -19683.0 q^{27} -154692. q^{29} +71211.7 q^{31} -102982. q^{33} -384507. q^{35} +261898. q^{37} +401156. q^{39} +325681. q^{41} +531827. q^{43} -205770. q^{45} +895746. q^{47} +1.03211e6 q^{49} +125221. q^{51} -1.02021e6 q^{53} -1.07660e6 q^{55} +86889.1 q^{57} +1.89747e6 q^{59} +3.00735e6 q^{61} +993062. q^{63} +4.19377e6 q^{65} -2.51954e6 q^{67} -1.06237e6 q^{69} -4.15228e6 q^{71} -4.35349e6 q^{73} -41790.5 q^{75} +5.19575e6 q^{77} -3.62436e6 q^{79} +531441. q^{81} -5.94931e6 q^{83} +1.30909e6 q^{85} +4.17669e6 q^{87} +1.21553e6 q^{89} -2.02394e7 q^{91} -1.92272e6 q^{93} +908357. q^{95} -1.04112e7 q^{97} +2.78053e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 108 q^{3} - 336 q^{5} - 680 q^{7} + 2916 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 108 q^{3} - 336 q^{5} - 680 q^{7} + 2916 q^{9} - 3856 q^{11} - 10680 q^{13} + 9072 q^{15} + 26232 q^{17} + 15456 q^{19} + 18360 q^{21} + 11312 q^{23} + 159052 q^{25} - 78732 q^{27} + 1856 q^{29} - 71752 q^{31} + 104112 q^{33} + 179040 q^{35} + 180088 q^{37} + 288360 q^{39} + 11224 q^{41} - 66688 q^{43} - 244944 q^{45} + 1334448 q^{47} + 2401140 q^{49} - 708264 q^{51} + 864576 q^{53} + 3304896 q^{55} - 417312 q^{57} - 1878448 q^{59} + 1901176 q^{61} - 495720 q^{63} + 4366944 q^{65} - 5505488 q^{67} - 305424 q^{69} + 967696 q^{71} + 3244760 q^{73} - 4294404 q^{75} + 8979488 q^{77} - 6471816 q^{79} + 2125764 q^{81} - 17019600 q^{83} - 12122592 q^{85} - 50112 q^{87} + 13559816 q^{89} - 6692304 q^{91} + 1937304 q^{93} - 22523904 q^{95} + 2180520 q^{97} - 2811024 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −27.0000 −0.577350
\(4\) 0 0
\(5\) −282.264 −1.00986 −0.504929 0.863161i \(-0.668481\pi\)
−0.504929 + 0.863161i \(0.668481\pi\)
\(6\) 0 0
\(7\) 1362.23 1.50109 0.750543 0.660821i \(-0.229792\pi\)
0.750543 + 0.660821i \(0.229792\pi\)
\(8\) 0 0
\(9\) 729.000 0.333333
\(10\) 0 0
\(11\) 3814.16 0.864022 0.432011 0.901868i \(-0.357804\pi\)
0.432011 + 0.901868i \(0.357804\pi\)
\(12\) 0 0
\(13\) −14857.6 −1.87563 −0.937817 0.347130i \(-0.887156\pi\)
−0.937817 + 0.347130i \(0.887156\pi\)
\(14\) 0 0
\(15\) 7621.12 0.583041
\(16\) 0 0
\(17\) −4637.82 −0.228951 −0.114476 0.993426i \(-0.536519\pi\)
−0.114476 + 0.993426i \(0.536519\pi\)
\(18\) 0 0
\(19\) −3218.11 −0.107638 −0.0538188 0.998551i \(-0.517139\pi\)
−0.0538188 + 0.998551i \(0.517139\pi\)
\(20\) 0 0
\(21\) −36780.1 −0.866653
\(22\) 0 0
\(23\) 39347.2 0.674320 0.337160 0.941447i \(-0.390534\pi\)
0.337160 + 0.941447i \(0.390534\pi\)
\(24\) 0 0
\(25\) 1547.80 0.0198118
\(26\) 0 0
\(27\) −19683.0 −0.192450
\(28\) 0 0
\(29\) −154692. −1.17781 −0.588906 0.808202i \(-0.700441\pi\)
−0.588906 + 0.808202i \(0.700441\pi\)
\(30\) 0 0
\(31\) 71211.7 0.429324 0.214662 0.976688i \(-0.431135\pi\)
0.214662 + 0.976688i \(0.431135\pi\)
\(32\) 0 0
\(33\) −102982. −0.498843
\(34\) 0 0
\(35\) −384507. −1.51588
\(36\) 0 0
\(37\) 261898. 0.850013 0.425006 0.905190i \(-0.360272\pi\)
0.425006 + 0.905190i \(0.360272\pi\)
\(38\) 0 0
\(39\) 401156. 1.08290
\(40\) 0 0
\(41\) 325681. 0.737987 0.368994 0.929432i \(-0.379702\pi\)
0.368994 + 0.929432i \(0.379702\pi\)
\(42\) 0 0
\(43\) 531827. 1.02007 0.510036 0.860153i \(-0.329632\pi\)
0.510036 + 0.860153i \(0.329632\pi\)
\(44\) 0 0
\(45\) −205770. −0.336619
\(46\) 0 0
\(47\) 895746. 1.25847 0.629234 0.777216i \(-0.283369\pi\)
0.629234 + 0.777216i \(0.283369\pi\)
\(48\) 0 0
\(49\) 1.03211e6 1.25326
\(50\) 0 0
\(51\) 125221. 0.132185
\(52\) 0 0
\(53\) −1.02021e6 −0.941287 −0.470643 0.882323i \(-0.655978\pi\)
−0.470643 + 0.882323i \(0.655978\pi\)
\(54\) 0 0
\(55\) −1.07660e6 −0.872539
\(56\) 0 0
\(57\) 86889.1 0.0621446
\(58\) 0 0
\(59\) 1.89747e6 1.20280 0.601400 0.798948i \(-0.294610\pi\)
0.601400 + 0.798948i \(0.294610\pi\)
\(60\) 0 0
\(61\) 3.00735e6 1.69640 0.848202 0.529672i \(-0.177685\pi\)
0.848202 + 0.529672i \(0.177685\pi\)
\(62\) 0 0
\(63\) 993062. 0.500362
\(64\) 0 0
\(65\) 4.19377e6 1.89412
\(66\) 0 0
\(67\) −2.51954e6 −1.02343 −0.511717 0.859154i \(-0.670991\pi\)
−0.511717 + 0.859154i \(0.670991\pi\)
\(68\) 0 0
\(69\) −1.06237e6 −0.389319
\(70\) 0 0
\(71\) −4.15228e6 −1.37684 −0.688418 0.725314i \(-0.741695\pi\)
−0.688418 + 0.725314i \(0.741695\pi\)
\(72\) 0 0
\(73\) −4.35349e6 −1.30981 −0.654904 0.755712i \(-0.727291\pi\)
−0.654904 + 0.755712i \(0.727291\pi\)
\(74\) 0 0
\(75\) −41790.5 −0.0114384
\(76\) 0 0
\(77\) 5.19575e6 1.29697
\(78\) 0 0
\(79\) −3.62436e6 −0.827059 −0.413530 0.910491i \(-0.635704\pi\)
−0.413530 + 0.910491i \(0.635704\pi\)
\(80\) 0 0
\(81\) 531441. 0.111111
\(82\) 0 0
\(83\) −5.94931e6 −1.14207 −0.571036 0.820925i \(-0.693458\pi\)
−0.571036 + 0.820925i \(0.693458\pi\)
\(84\) 0 0
\(85\) 1.30909e6 0.231208
\(86\) 0 0
\(87\) 4.17669e6 0.680010
\(88\) 0 0
\(89\) 1.21553e6 0.182769 0.0913843 0.995816i \(-0.470871\pi\)
0.0913843 + 0.995816i \(0.470871\pi\)
\(90\) 0 0
\(91\) −2.02394e7 −2.81549
\(92\) 0 0
\(93\) −1.92272e6 −0.247870
\(94\) 0 0
\(95\) 908357. 0.108699
\(96\) 0 0
\(97\) −1.04112e7 −1.15824 −0.579119 0.815243i \(-0.696603\pi\)
−0.579119 + 0.815243i \(0.696603\pi\)
\(98\) 0 0
\(99\) 2.78053e6 0.288007
\(100\) 0 0
\(101\) −7.52279e6 −0.726531 −0.363266 0.931686i \(-0.618338\pi\)
−0.363266 + 0.931686i \(0.618338\pi\)
\(102\) 0 0
\(103\) 1.26662e7 1.14214 0.571068 0.820903i \(-0.306529\pi\)
0.571068 + 0.820903i \(0.306529\pi\)
\(104\) 0 0
\(105\) 1.03817e7 0.875196
\(106\) 0 0
\(107\) 2.03306e7 1.60438 0.802188 0.597072i \(-0.203669\pi\)
0.802188 + 0.597072i \(0.203669\pi\)
\(108\) 0 0
\(109\) −2.31105e7 −1.70930 −0.854648 0.519208i \(-0.826227\pi\)
−0.854648 + 0.519208i \(0.826227\pi\)
\(110\) 0 0
\(111\) −7.07124e6 −0.490755
\(112\) 0 0
\(113\) 1.55087e7 1.01111 0.505557 0.862793i \(-0.331287\pi\)
0.505557 + 0.862793i \(0.331287\pi\)
\(114\) 0 0
\(115\) −1.11063e7 −0.680967
\(116\) 0 0
\(117\) −1.08312e7 −0.625211
\(118\) 0 0
\(119\) −6.31776e6 −0.343676
\(120\) 0 0
\(121\) −4.93933e6 −0.253466
\(122\) 0 0
\(123\) −8.79339e6 −0.426077
\(124\) 0 0
\(125\) 2.16150e7 0.989850
\(126\) 0 0
\(127\) −4.52189e7 −1.95888 −0.979439 0.201742i \(-0.935340\pi\)
−0.979439 + 0.201742i \(0.935340\pi\)
\(128\) 0 0
\(129\) −1.43593e7 −0.588939
\(130\) 0 0
\(131\) 1.44228e7 0.560532 0.280266 0.959922i \(-0.409577\pi\)
0.280266 + 0.959922i \(0.409577\pi\)
\(132\) 0 0
\(133\) −4.38380e6 −0.161573
\(134\) 0 0
\(135\) 5.55580e6 0.194347
\(136\) 0 0
\(137\) 2.63066e7 0.874063 0.437032 0.899446i \(-0.356030\pi\)
0.437032 + 0.899446i \(0.356030\pi\)
\(138\) 0 0
\(139\) 1.79784e7 0.567806 0.283903 0.958853i \(-0.408371\pi\)
0.283903 + 0.958853i \(0.408371\pi\)
\(140\) 0 0
\(141\) −2.41851e7 −0.726577
\(142\) 0 0
\(143\) −5.66695e7 −1.62059
\(144\) 0 0
\(145\) 4.36640e7 1.18942
\(146\) 0 0
\(147\) −2.78671e7 −0.723571
\(148\) 0 0
\(149\) −5.16114e7 −1.27819 −0.639093 0.769129i \(-0.720690\pi\)
−0.639093 + 0.769129i \(0.720690\pi\)
\(150\) 0 0
\(151\) −6.67253e6 −0.157714 −0.0788572 0.996886i \(-0.525127\pi\)
−0.0788572 + 0.996886i \(0.525127\pi\)
\(152\) 0 0
\(153\) −3.38097e6 −0.0763171
\(154\) 0 0
\(155\) −2.01005e7 −0.433556
\(156\) 0 0
\(157\) −5.35184e7 −1.10371 −0.551854 0.833941i \(-0.686079\pi\)
−0.551854 + 0.833941i \(0.686079\pi\)
\(158\) 0 0
\(159\) 2.75455e7 0.543452
\(160\) 0 0
\(161\) 5.35997e7 1.01221
\(162\) 0 0
\(163\) 3.49634e7 0.632350 0.316175 0.948701i \(-0.397601\pi\)
0.316175 + 0.948701i \(0.397601\pi\)
\(164\) 0 0
\(165\) 2.90682e7 0.503761
\(166\) 0 0
\(167\) −3.78924e7 −0.629570 −0.314785 0.949163i \(-0.601932\pi\)
−0.314785 + 0.949163i \(0.601932\pi\)
\(168\) 0 0
\(169\) 1.58001e8 2.51800
\(170\) 0 0
\(171\) −2.34600e6 −0.0358792
\(172\) 0 0
\(173\) −2.40284e7 −0.352828 −0.176414 0.984316i \(-0.556450\pi\)
−0.176414 + 0.984316i \(0.556450\pi\)
\(174\) 0 0
\(175\) 2.10845e6 0.0297392
\(176\) 0 0
\(177\) −5.12318e7 −0.694437
\(178\) 0 0
\(179\) −1.29176e8 −1.68344 −0.841720 0.539915i \(-0.818456\pi\)
−0.841720 + 0.539915i \(0.818456\pi\)
\(180\) 0 0
\(181\) −7.22849e7 −0.906092 −0.453046 0.891487i \(-0.649663\pi\)
−0.453046 + 0.891487i \(0.649663\pi\)
\(182\) 0 0
\(183\) −8.11984e7 −0.979420
\(184\) 0 0
\(185\) −7.39242e7 −0.858392
\(186\) 0 0
\(187\) −1.76894e7 −0.197819
\(188\) 0 0
\(189\) −2.68127e7 −0.288884
\(190\) 0 0
\(191\) −5.31417e7 −0.551846 −0.275923 0.961180i \(-0.588984\pi\)
−0.275923 + 0.961180i \(0.588984\pi\)
\(192\) 0 0
\(193\) −1.37647e8 −1.37821 −0.689105 0.724662i \(-0.741996\pi\)
−0.689105 + 0.724662i \(0.741996\pi\)
\(194\) 0 0
\(195\) −1.13232e8 −1.09357
\(196\) 0 0
\(197\) −6.55467e7 −0.610828 −0.305414 0.952220i \(-0.598795\pi\)
−0.305414 + 0.952220i \(0.598795\pi\)
\(198\) 0 0
\(199\) −3.65315e7 −0.328611 −0.164305 0.986410i \(-0.552538\pi\)
−0.164305 + 0.986410i \(0.552538\pi\)
\(200\) 0 0
\(201\) 6.80277e7 0.590880
\(202\) 0 0
\(203\) −2.10726e8 −1.76800
\(204\) 0 0
\(205\) −9.19279e7 −0.745262
\(206\) 0 0
\(207\) 2.86841e7 0.224773
\(208\) 0 0
\(209\) −1.22744e7 −0.0930012
\(210\) 0 0
\(211\) −1.37911e8 −1.01067 −0.505335 0.862923i \(-0.668631\pi\)
−0.505335 + 0.862923i \(0.668631\pi\)
\(212\) 0 0
\(213\) 1.12112e8 0.794917
\(214\) 0 0
\(215\) −1.50116e8 −1.03013
\(216\) 0 0
\(217\) 9.70064e7 0.644453
\(218\) 0 0
\(219\) 1.17544e8 0.756218
\(220\) 0 0
\(221\) 6.89071e7 0.429429
\(222\) 0 0
\(223\) −6.89464e7 −0.416337 −0.208168 0.978093i \(-0.566750\pi\)
−0.208168 + 0.978093i \(0.566750\pi\)
\(224\) 0 0
\(225\) 1.12834e6 0.00660394
\(226\) 0 0
\(227\) −1.41524e8 −0.803044 −0.401522 0.915849i \(-0.631519\pi\)
−0.401522 + 0.915849i \(0.631519\pi\)
\(228\) 0 0
\(229\) −1.38090e8 −0.759868 −0.379934 0.925014i \(-0.624053\pi\)
−0.379934 + 0.925014i \(0.624053\pi\)
\(230\) 0 0
\(231\) −1.40285e8 −0.748807
\(232\) 0 0
\(233\) 3.71716e8 1.92515 0.962577 0.271007i \(-0.0873566\pi\)
0.962577 + 0.271007i \(0.0873566\pi\)
\(234\) 0 0
\(235\) −2.52837e8 −1.27087
\(236\) 0 0
\(237\) 9.78577e7 0.477503
\(238\) 0 0
\(239\) 4.40837e7 0.208875 0.104437 0.994531i \(-0.466696\pi\)
0.104437 + 0.994531i \(0.466696\pi\)
\(240\) 0 0
\(241\) 1.62773e8 0.749069 0.374535 0.927213i \(-0.377802\pi\)
0.374535 + 0.927213i \(0.377802\pi\)
\(242\) 0 0
\(243\) −1.43489e7 −0.0641500
\(244\) 0 0
\(245\) −2.91328e8 −1.26561
\(246\) 0 0
\(247\) 4.78136e7 0.201889
\(248\) 0 0
\(249\) 1.60631e8 0.659376
\(250\) 0 0
\(251\) 1.77876e8 0.710001 0.355001 0.934866i \(-0.384481\pi\)
0.355001 + 0.934866i \(0.384481\pi\)
\(252\) 0 0
\(253\) 1.50077e8 0.582628
\(254\) 0 0
\(255\) −3.53454e7 −0.133488
\(256\) 0 0
\(257\) 3.50129e8 1.28665 0.643327 0.765592i \(-0.277554\pi\)
0.643327 + 0.765592i \(0.277554\pi\)
\(258\) 0 0
\(259\) 3.56764e8 1.27594
\(260\) 0 0
\(261\) −1.12771e8 −0.392604
\(262\) 0 0
\(263\) 3.08649e8 1.04621 0.523106 0.852268i \(-0.324773\pi\)
0.523106 + 0.852268i \(0.324773\pi\)
\(264\) 0 0
\(265\) 2.87967e8 0.950565
\(266\) 0 0
\(267\) −3.28194e7 −0.105521
\(268\) 0 0
\(269\) 2.10933e8 0.660712 0.330356 0.943856i \(-0.392831\pi\)
0.330356 + 0.943856i \(0.392831\pi\)
\(270\) 0 0
\(271\) −5.48682e8 −1.67467 −0.837334 0.546692i \(-0.815887\pi\)
−0.837334 + 0.546692i \(0.815887\pi\)
\(272\) 0 0
\(273\) 5.46465e8 1.62552
\(274\) 0 0
\(275\) 5.90355e6 0.0171178
\(276\) 0 0
\(277\) −5.01494e8 −1.41771 −0.708854 0.705355i \(-0.750788\pi\)
−0.708854 + 0.705355i \(0.750788\pi\)
\(278\) 0 0
\(279\) 5.19133e7 0.143108
\(280\) 0 0
\(281\) −5.11695e8 −1.37575 −0.687874 0.725830i \(-0.741456\pi\)
−0.687874 + 0.725830i \(0.741456\pi\)
\(282\) 0 0
\(283\) −2.43080e8 −0.637523 −0.318761 0.947835i \(-0.603267\pi\)
−0.318761 + 0.947835i \(0.603267\pi\)
\(284\) 0 0
\(285\) −2.45256e7 −0.0627571
\(286\) 0 0
\(287\) 4.43651e8 1.10778
\(288\) 0 0
\(289\) −3.88829e8 −0.947581
\(290\) 0 0
\(291\) 2.81101e8 0.668709
\(292\) 0 0
\(293\) −4.62895e8 −1.07509 −0.537547 0.843234i \(-0.680649\pi\)
−0.537547 + 0.843234i \(0.680649\pi\)
\(294\) 0 0
\(295\) −5.35588e8 −1.21466
\(296\) 0 0
\(297\) −7.50742e7 −0.166281
\(298\) 0 0
\(299\) −5.84606e8 −1.26478
\(300\) 0 0
\(301\) 7.24468e8 1.53122
\(302\) 0 0
\(303\) 2.03115e8 0.419463
\(304\) 0 0
\(305\) −8.48866e8 −1.71313
\(306\) 0 0
\(307\) 3.82064e8 0.753619 0.376810 0.926291i \(-0.377021\pi\)
0.376810 + 0.926291i \(0.377021\pi\)
\(308\) 0 0
\(309\) −3.41989e8 −0.659412
\(310\) 0 0
\(311\) −7.31257e7 −0.137851 −0.0689253 0.997622i \(-0.521957\pi\)
−0.0689253 + 0.997622i \(0.521957\pi\)
\(312\) 0 0
\(313\) −3.79042e8 −0.698687 −0.349343 0.936995i \(-0.613595\pi\)
−0.349343 + 0.936995i \(0.613595\pi\)
\(314\) 0 0
\(315\) −2.80305e8 −0.505294
\(316\) 0 0
\(317\) −6.03233e8 −1.06360 −0.531799 0.846871i \(-0.678484\pi\)
−0.531799 + 0.846871i \(0.678484\pi\)
\(318\) 0 0
\(319\) −5.90022e8 −1.01766
\(320\) 0 0
\(321\) −5.48925e8 −0.926287
\(322\) 0 0
\(323\) 1.49250e7 0.0246437
\(324\) 0 0
\(325\) −2.29966e7 −0.0371597
\(326\) 0 0
\(327\) 6.23984e8 0.986862
\(328\) 0 0
\(329\) 1.22021e9 1.88907
\(330\) 0 0
\(331\) −3.21985e8 −0.488021 −0.244010 0.969773i \(-0.578463\pi\)
−0.244010 + 0.969773i \(0.578463\pi\)
\(332\) 0 0
\(333\) 1.90923e8 0.283338
\(334\) 0 0
\(335\) 7.11176e8 1.03352
\(336\) 0 0
\(337\) −6.22663e8 −0.886233 −0.443117 0.896464i \(-0.646127\pi\)
−0.443117 + 0.896464i \(0.646127\pi\)
\(338\) 0 0
\(339\) −4.18734e8 −0.583767
\(340\) 0 0
\(341\) 2.71613e8 0.370946
\(342\) 0 0
\(343\) 2.84121e8 0.380167
\(344\) 0 0
\(345\) 2.99870e8 0.393157
\(346\) 0 0
\(347\) 1.23835e9 1.59108 0.795538 0.605903i \(-0.207188\pi\)
0.795538 + 0.605903i \(0.207188\pi\)
\(348\) 0 0
\(349\) 7.64841e7 0.0963124 0.0481562 0.998840i \(-0.484665\pi\)
0.0481562 + 0.998840i \(0.484665\pi\)
\(350\) 0 0
\(351\) 2.92443e8 0.360966
\(352\) 0 0
\(353\) −9.94452e8 −1.20330 −0.601648 0.798762i \(-0.705489\pi\)
−0.601648 + 0.798762i \(0.705489\pi\)
\(354\) 0 0
\(355\) 1.17204e9 1.39041
\(356\) 0 0
\(357\) 1.70579e8 0.198421
\(358\) 0 0
\(359\) −1.17743e9 −1.34308 −0.671542 0.740966i \(-0.734368\pi\)
−0.671542 + 0.740966i \(0.734368\pi\)
\(360\) 0 0
\(361\) −8.83515e8 −0.988414
\(362\) 0 0
\(363\) 1.33362e8 0.146338
\(364\) 0 0
\(365\) 1.22883e9 1.32272
\(366\) 0 0
\(367\) 1.89983e8 0.200624 0.100312 0.994956i \(-0.468016\pi\)
0.100312 + 0.994956i \(0.468016\pi\)
\(368\) 0 0
\(369\) 2.37421e8 0.245996
\(370\) 0 0
\(371\) −1.38975e9 −1.41295
\(372\) 0 0
\(373\) −5.55522e8 −0.554269 −0.277134 0.960831i \(-0.589385\pi\)
−0.277134 + 0.960831i \(0.589385\pi\)
\(374\) 0 0
\(375\) −5.83604e8 −0.571490
\(376\) 0 0
\(377\) 2.29836e9 2.20914
\(378\) 0 0
\(379\) 1.30691e9 1.23313 0.616565 0.787304i \(-0.288524\pi\)
0.616565 + 0.787304i \(0.288524\pi\)
\(380\) 0 0
\(381\) 1.22091e9 1.13096
\(382\) 0 0
\(383\) −1.42228e8 −0.129357 −0.0646784 0.997906i \(-0.520602\pi\)
−0.0646784 + 0.997906i \(0.520602\pi\)
\(384\) 0 0
\(385\) −1.46657e9 −1.30976
\(386\) 0 0
\(387\) 3.87702e8 0.340024
\(388\) 0 0
\(389\) 1.79969e9 1.55015 0.775077 0.631866i \(-0.217711\pi\)
0.775077 + 0.631866i \(0.217711\pi\)
\(390\) 0 0
\(391\) −1.82485e8 −0.154386
\(392\) 0 0
\(393\) −3.89416e8 −0.323623
\(394\) 0 0
\(395\) 1.02303e9 0.835212
\(396\) 0 0
\(397\) −2.73058e8 −0.219022 −0.109511 0.993986i \(-0.534929\pi\)
−0.109511 + 0.993986i \(0.534929\pi\)
\(398\) 0 0
\(399\) 1.18362e8 0.0932844
\(400\) 0 0
\(401\) −2.13136e9 −1.65064 −0.825320 0.564666i \(-0.809005\pi\)
−0.825320 + 0.564666i \(0.809005\pi\)
\(402\) 0 0
\(403\) −1.05804e9 −0.805255
\(404\) 0 0
\(405\) −1.50007e8 −0.112206
\(406\) 0 0
\(407\) 9.98920e8 0.734430
\(408\) 0 0
\(409\) −8.20609e7 −0.0593068 −0.0296534 0.999560i \(-0.509440\pi\)
−0.0296534 + 0.999560i \(0.509440\pi\)
\(410\) 0 0
\(411\) −7.10278e8 −0.504641
\(412\) 0 0
\(413\) 2.58479e9 1.80551
\(414\) 0 0
\(415\) 1.67928e9 1.15333
\(416\) 0 0
\(417\) −4.85418e8 −0.327823
\(418\) 0 0
\(419\) 1.97020e9 1.30847 0.654233 0.756293i \(-0.272992\pi\)
0.654233 + 0.756293i \(0.272992\pi\)
\(420\) 0 0
\(421\) −2.07781e9 −1.35712 −0.678559 0.734546i \(-0.737395\pi\)
−0.678559 + 0.734546i \(0.737395\pi\)
\(422\) 0 0
\(423\) 6.52999e8 0.419490
\(424\) 0 0
\(425\) −7.17841e6 −0.00453594
\(426\) 0 0
\(427\) 4.09669e9 2.54645
\(428\) 0 0
\(429\) 1.53008e9 0.935648
\(430\) 0 0
\(431\) −1.88880e9 −1.13636 −0.568181 0.822904i \(-0.692352\pi\)
−0.568181 + 0.822904i \(0.692352\pi\)
\(432\) 0 0
\(433\) −1.10906e9 −0.656517 −0.328258 0.944588i \(-0.606462\pi\)
−0.328258 + 0.944588i \(0.606462\pi\)
\(434\) 0 0
\(435\) −1.17893e9 −0.686713
\(436\) 0 0
\(437\) −1.26624e8 −0.0725822
\(438\) 0 0
\(439\) 1.94201e9 1.09553 0.547766 0.836632i \(-0.315479\pi\)
0.547766 + 0.836632i \(0.315479\pi\)
\(440\) 0 0
\(441\) 7.52411e8 0.417754
\(442\) 0 0
\(443\) −1.10899e9 −0.606057 −0.303028 0.952982i \(-0.597998\pi\)
−0.303028 + 0.952982i \(0.597998\pi\)
\(444\) 0 0
\(445\) −3.43101e8 −0.184570
\(446\) 0 0
\(447\) 1.39351e9 0.737961
\(448\) 0 0
\(449\) 2.19688e9 1.14537 0.572683 0.819777i \(-0.305902\pi\)
0.572683 + 0.819777i \(0.305902\pi\)
\(450\) 0 0
\(451\) 1.24220e9 0.637637
\(452\) 0 0
\(453\) 1.80158e8 0.0910564
\(454\) 0 0
\(455\) 5.71286e9 2.84324
\(456\) 0 0
\(457\) 2.84116e8 0.139248 0.0696240 0.997573i \(-0.477820\pi\)
0.0696240 + 0.997573i \(0.477820\pi\)
\(458\) 0 0
\(459\) 9.12863e7 0.0440617
\(460\) 0 0
\(461\) 1.29061e9 0.613541 0.306770 0.951784i \(-0.400752\pi\)
0.306770 + 0.951784i \(0.400752\pi\)
\(462\) 0 0
\(463\) 4.12483e9 1.93140 0.965701 0.259658i \(-0.0836098\pi\)
0.965701 + 0.259658i \(0.0836098\pi\)
\(464\) 0 0
\(465\) 5.42713e8 0.250314
\(466\) 0 0
\(467\) 2.27629e9 1.03424 0.517118 0.855914i \(-0.327005\pi\)
0.517118 + 0.855914i \(0.327005\pi\)
\(468\) 0 0
\(469\) −3.43219e9 −1.53626
\(470\) 0 0
\(471\) 1.44500e9 0.637226
\(472\) 0 0
\(473\) 2.02848e9 0.881365
\(474\) 0 0
\(475\) −4.98099e6 −0.00213249
\(476\) 0 0
\(477\) −7.43730e8 −0.313762
\(478\) 0 0
\(479\) 6.83485e8 0.284155 0.142077 0.989856i \(-0.454622\pi\)
0.142077 + 0.989856i \(0.454622\pi\)
\(480\) 0 0
\(481\) −3.89118e9 −1.59431
\(482\) 0 0
\(483\) −1.44719e9 −0.584402
\(484\) 0 0
\(485\) 2.93869e9 1.16966
\(486\) 0 0
\(487\) −3.70483e9 −1.45351 −0.726753 0.686899i \(-0.758972\pi\)
−0.726753 + 0.686899i \(0.758972\pi\)
\(488\) 0 0
\(489\) −9.44013e8 −0.365087
\(490\) 0 0
\(491\) −2.89597e9 −1.10410 −0.552050 0.833811i \(-0.686154\pi\)
−0.552050 + 0.833811i \(0.686154\pi\)
\(492\) 0 0
\(493\) 7.17436e8 0.269661
\(494\) 0 0
\(495\) −7.84841e8 −0.290846
\(496\) 0 0
\(497\) −5.65634e9 −2.06675
\(498\) 0 0
\(499\) 1.72733e9 0.622334 0.311167 0.950355i \(-0.399280\pi\)
0.311167 + 0.950355i \(0.399280\pi\)
\(500\) 0 0
\(501\) 1.02309e9 0.363483
\(502\) 0 0
\(503\) 2.71038e9 0.949604 0.474802 0.880093i \(-0.342520\pi\)
0.474802 + 0.880093i \(0.342520\pi\)
\(504\) 0 0
\(505\) 2.12341e9 0.733693
\(506\) 0 0
\(507\) −4.26603e9 −1.45377
\(508\) 0 0
\(509\) 2.23564e9 0.751432 0.375716 0.926735i \(-0.377397\pi\)
0.375716 + 0.926735i \(0.377397\pi\)
\(510\) 0 0
\(511\) −5.93043e9 −1.96613
\(512\) 0 0
\(513\) 6.33421e7 0.0207149
\(514\) 0 0
\(515\) −3.57522e9 −1.15339
\(516\) 0 0
\(517\) 3.41652e9 1.08734
\(518\) 0 0
\(519\) 6.48767e8 0.203706
\(520\) 0 0
\(521\) −3.88054e9 −1.20215 −0.601076 0.799192i \(-0.705261\pi\)
−0.601076 + 0.799192i \(0.705261\pi\)
\(522\) 0 0
\(523\) 2.91868e9 0.892135 0.446067 0.894999i \(-0.352824\pi\)
0.446067 + 0.894999i \(0.352824\pi\)
\(524\) 0 0
\(525\) −5.69281e7 −0.0171700
\(526\) 0 0
\(527\) −3.30267e8 −0.0982943
\(528\) 0 0
\(529\) −1.85662e9 −0.545292
\(530\) 0 0
\(531\) 1.38326e9 0.400933
\(532\) 0 0
\(533\) −4.83885e9 −1.38419
\(534\) 0 0
\(535\) −5.73858e9 −1.62019
\(536\) 0 0
\(537\) 3.48776e9 0.971934
\(538\) 0 0
\(539\) 3.93665e9 1.08285
\(540\) 0 0
\(541\) 3.32189e9 0.901976 0.450988 0.892530i \(-0.351072\pi\)
0.450988 + 0.892530i \(0.351072\pi\)
\(542\) 0 0
\(543\) 1.95169e9 0.523133
\(544\) 0 0
\(545\) 6.52326e9 1.72614
\(546\) 0 0
\(547\) 2.18673e9 0.571267 0.285634 0.958339i \(-0.407796\pi\)
0.285634 + 0.958339i \(0.407796\pi\)
\(548\) 0 0
\(549\) 2.19236e9 0.565468
\(550\) 0 0
\(551\) 4.97817e8 0.126777
\(552\) 0 0
\(553\) −4.93720e9 −1.24149
\(554\) 0 0
\(555\) 1.99595e9 0.495593
\(556\) 0 0
\(557\) 4.20230e9 1.03037 0.515186 0.857078i \(-0.327723\pi\)
0.515186 + 0.857078i \(0.327723\pi\)
\(558\) 0 0
\(559\) −7.90170e9 −1.91328
\(560\) 0 0
\(561\) 4.77614e8 0.114211
\(562\) 0 0
\(563\) 9.23236e8 0.218039 0.109019 0.994040i \(-0.465229\pi\)
0.109019 + 0.994040i \(0.465229\pi\)
\(564\) 0 0
\(565\) −4.37754e9 −1.02108
\(566\) 0 0
\(567\) 7.23942e8 0.166787
\(568\) 0 0
\(569\) −4.79957e8 −0.109222 −0.0546109 0.998508i \(-0.517392\pi\)
−0.0546109 + 0.998508i \(0.517392\pi\)
\(570\) 0 0
\(571\) 3.38377e9 0.760632 0.380316 0.924857i \(-0.375815\pi\)
0.380316 + 0.924857i \(0.375815\pi\)
\(572\) 0 0
\(573\) 1.43482e9 0.318609
\(574\) 0 0
\(575\) 6.09015e7 0.0133595
\(576\) 0 0
\(577\) −3.30613e9 −0.716481 −0.358241 0.933629i \(-0.616623\pi\)
−0.358241 + 0.933629i \(0.616623\pi\)
\(578\) 0 0
\(579\) 3.71646e9 0.795710
\(580\) 0 0
\(581\) −8.10431e9 −1.71435
\(582\) 0 0
\(583\) −3.89123e9 −0.813293
\(584\) 0 0
\(585\) 3.05726e9 0.631374
\(586\) 0 0
\(587\) −7.73199e9 −1.57782 −0.788910 0.614508i \(-0.789354\pi\)
−0.788910 + 0.614508i \(0.789354\pi\)
\(588\) 0 0
\(589\) −2.29167e8 −0.0462114
\(590\) 0 0
\(591\) 1.76976e9 0.352662
\(592\) 0 0
\(593\) 7.78755e9 1.53359 0.766795 0.641891i \(-0.221850\pi\)
0.766795 + 0.641891i \(0.221850\pi\)
\(594\) 0 0
\(595\) 1.78327e9 0.347063
\(596\) 0 0
\(597\) 9.86351e8 0.189724
\(598\) 0 0
\(599\) −2.28551e8 −0.0434500 −0.0217250 0.999764i \(-0.506916\pi\)
−0.0217250 + 0.999764i \(0.506916\pi\)
\(600\) 0 0
\(601\) −6.38900e9 −1.20053 −0.600264 0.799802i \(-0.704938\pi\)
−0.600264 + 0.799802i \(0.704938\pi\)
\(602\) 0 0
\(603\) −1.83675e9 −0.341145
\(604\) 0 0
\(605\) 1.39419e9 0.255964
\(606\) 0 0
\(607\) 5.84357e9 1.06052 0.530259 0.847836i \(-0.322095\pi\)
0.530259 + 0.847836i \(0.322095\pi\)
\(608\) 0 0
\(609\) 5.68960e9 1.02075
\(610\) 0 0
\(611\) −1.33087e10 −2.36043
\(612\) 0 0
\(613\) −6.34845e9 −1.11316 −0.556578 0.830796i \(-0.687886\pi\)
−0.556578 + 0.830796i \(0.687886\pi\)
\(614\) 0 0
\(615\) 2.48205e9 0.430277
\(616\) 0 0
\(617\) 3.22881e9 0.553406 0.276703 0.960955i \(-0.410758\pi\)
0.276703 + 0.960955i \(0.410758\pi\)
\(618\) 0 0
\(619\) −3.22164e9 −0.545959 −0.272980 0.962020i \(-0.588009\pi\)
−0.272980 + 0.962020i \(0.588009\pi\)
\(620\) 0 0
\(621\) −7.74471e8 −0.129773
\(622\) 0 0
\(623\) 1.65583e9 0.274351
\(624\) 0 0
\(625\) −6.22204e9 −1.01942
\(626\) 0 0
\(627\) 3.31409e8 0.0536943
\(628\) 0 0
\(629\) −1.21463e9 −0.194611
\(630\) 0 0
\(631\) −4.72340e9 −0.748431 −0.374216 0.927342i \(-0.622088\pi\)
−0.374216 + 0.927342i \(0.622088\pi\)
\(632\) 0 0
\(633\) 3.72359e9 0.583510
\(634\) 0 0
\(635\) 1.27637e10 1.97819
\(636\) 0 0
\(637\) −1.53348e10 −2.35066
\(638\) 0 0
\(639\) −3.02701e9 −0.458945
\(640\) 0 0
\(641\) −3.27899e9 −0.491742 −0.245871 0.969303i \(-0.579074\pi\)
−0.245871 + 0.969303i \(0.579074\pi\)
\(642\) 0 0
\(643\) −7.67620e9 −1.13870 −0.569348 0.822097i \(-0.692804\pi\)
−0.569348 + 0.822097i \(0.692804\pi\)
\(644\) 0 0
\(645\) 4.05312e9 0.594744
\(646\) 0 0
\(647\) −1.62477e9 −0.235845 −0.117923 0.993023i \(-0.537623\pi\)
−0.117923 + 0.993023i \(0.537623\pi\)
\(648\) 0 0
\(649\) 7.23727e9 1.03925
\(650\) 0 0
\(651\) −2.61917e9 −0.372075
\(652\) 0 0
\(653\) −1.03058e10 −1.44838 −0.724192 0.689598i \(-0.757787\pi\)
−0.724192 + 0.689598i \(0.757787\pi\)
\(654\) 0 0
\(655\) −4.07104e9 −0.566058
\(656\) 0 0
\(657\) −3.17369e9 −0.436602
\(658\) 0 0
\(659\) −2.92364e9 −0.397947 −0.198973 0.980005i \(-0.563761\pi\)
−0.198973 + 0.980005i \(0.563761\pi\)
\(660\) 0 0
\(661\) −5.43749e9 −0.732307 −0.366154 0.930554i \(-0.619325\pi\)
−0.366154 + 0.930554i \(0.619325\pi\)
\(662\) 0 0
\(663\) −1.86049e9 −0.247931
\(664\) 0 0
\(665\) 1.23739e9 0.163166
\(666\) 0 0
\(667\) −6.08671e9 −0.794222
\(668\) 0 0
\(669\) 1.86155e9 0.240372
\(670\) 0 0
\(671\) 1.14705e10 1.46573
\(672\) 0 0
\(673\) −3.90783e9 −0.494177 −0.247089 0.968993i \(-0.579474\pi\)
−0.247089 + 0.968993i \(0.579474\pi\)
\(674\) 0 0
\(675\) −3.04653e7 −0.00381278
\(676\) 0 0
\(677\) 7.18167e9 0.889540 0.444770 0.895645i \(-0.353285\pi\)
0.444770 + 0.895645i \(0.353285\pi\)
\(678\) 0 0
\(679\) −1.41823e10 −1.73862
\(680\) 0 0
\(681\) 3.82115e9 0.463638
\(682\) 0 0
\(683\) −1.25450e10 −1.50660 −0.753302 0.657674i \(-0.771540\pi\)
−0.753302 + 0.657674i \(0.771540\pi\)
\(684\) 0 0
\(685\) −7.42540e9 −0.882679
\(686\) 0 0
\(687\) 3.72843e9 0.438710
\(688\) 0 0
\(689\) 1.51578e10 1.76551
\(690\) 0 0
\(691\) 1.50101e10 1.73065 0.865327 0.501208i \(-0.167111\pi\)
0.865327 + 0.501208i \(0.167111\pi\)
\(692\) 0 0
\(693\) 3.78770e9 0.432324
\(694\) 0 0
\(695\) −5.07466e9 −0.573403
\(696\) 0 0
\(697\) −1.51045e9 −0.168963
\(698\) 0 0
\(699\) −1.00363e10 −1.11149
\(700\) 0 0
\(701\) −1.24579e10 −1.36594 −0.682968 0.730448i \(-0.739311\pi\)
−0.682968 + 0.730448i \(0.739311\pi\)
\(702\) 0 0
\(703\) −8.42816e8 −0.0914933
\(704\) 0 0
\(705\) 6.82659e9 0.733739
\(706\) 0 0
\(707\) −1.02477e10 −1.09059
\(708\) 0 0
\(709\) 6.91477e9 0.728645 0.364323 0.931273i \(-0.381301\pi\)
0.364323 + 0.931273i \(0.381301\pi\)
\(710\) 0 0
\(711\) −2.64216e9 −0.275686
\(712\) 0 0
\(713\) 2.80198e9 0.289502
\(714\) 0 0
\(715\) 1.59957e10 1.63656
\(716\) 0 0
\(717\) −1.19026e9 −0.120594
\(718\) 0 0
\(719\) −9.16082e9 −0.919143 −0.459572 0.888141i \(-0.651997\pi\)
−0.459572 + 0.888141i \(0.651997\pi\)
\(720\) 0 0
\(721\) 1.72543e10 1.71444
\(722\) 0 0
\(723\) −4.39486e9 −0.432475
\(724\) 0 0
\(725\) −2.39432e8 −0.0233346
\(726\) 0 0
\(727\) 1.28357e9 0.123894 0.0619469 0.998079i \(-0.480269\pi\)
0.0619469 + 0.998079i \(0.480269\pi\)
\(728\) 0 0
\(729\) 3.87420e8 0.0370370
\(730\) 0 0
\(731\) −2.46652e9 −0.233547
\(732\) 0 0
\(733\) −1.81802e10 −1.70505 −0.852523 0.522690i \(-0.824928\pi\)
−0.852523 + 0.522690i \(0.824928\pi\)
\(734\) 0 0
\(735\) 7.86587e9 0.730703
\(736\) 0 0
\(737\) −9.60995e9 −0.884270
\(738\) 0 0
\(739\) 1.74466e10 1.59021 0.795107 0.606469i \(-0.207414\pi\)
0.795107 + 0.606469i \(0.207414\pi\)
\(740\) 0 0
\(741\) −1.29097e9 −0.116560
\(742\) 0 0
\(743\) 8.52480e9 0.762471 0.381236 0.924478i \(-0.375499\pi\)
0.381236 + 0.924478i \(0.375499\pi\)
\(744\) 0 0
\(745\) 1.45680e10 1.29079
\(746\) 0 0
\(747\) −4.33705e9 −0.380691
\(748\) 0 0
\(749\) 2.76948e10 2.40831
\(750\) 0 0
\(751\) −1.26184e10 −1.08708 −0.543542 0.839382i \(-0.682917\pi\)
−0.543542 + 0.839382i \(0.682917\pi\)
\(752\) 0 0
\(753\) −4.80265e9 −0.409920
\(754\) 0 0
\(755\) 1.88341e9 0.159269
\(756\) 0 0
\(757\) 1.38035e10 1.15652 0.578259 0.815854i \(-0.303732\pi\)
0.578259 + 0.815854i \(0.303732\pi\)
\(758\) 0 0
\(759\) −4.05207e9 −0.336380
\(760\) 0 0
\(761\) 1.26732e10 1.04241 0.521205 0.853432i \(-0.325483\pi\)
0.521205 + 0.853432i \(0.325483\pi\)
\(762\) 0 0
\(763\) −3.14817e10 −2.56580
\(764\) 0 0
\(765\) 9.54326e8 0.0770693
\(766\) 0 0
\(767\) −2.81920e10 −2.25601
\(768\) 0 0
\(769\) 1.76471e10 1.39936 0.699681 0.714455i \(-0.253325\pi\)
0.699681 + 0.714455i \(0.253325\pi\)
\(770\) 0 0
\(771\) −9.45347e9 −0.742850
\(772\) 0 0
\(773\) 2.20323e10 1.71566 0.857829 0.513935i \(-0.171813\pi\)
0.857829 + 0.513935i \(0.171813\pi\)
\(774\) 0 0
\(775\) 1.10221e8 0.00850569
\(776\) 0 0
\(777\) −9.63261e9 −0.736666
\(778\) 0 0
\(779\) −1.04808e9 −0.0794351
\(780\) 0 0
\(781\) −1.58375e10 −1.18962
\(782\) 0 0
\(783\) 3.04481e9 0.226670
\(784\) 0 0
\(785\) 1.51063e10 1.11459
\(786\) 0 0
\(787\) −3.57574e9 −0.261489 −0.130745 0.991416i \(-0.541737\pi\)
−0.130745 + 0.991416i \(0.541737\pi\)
\(788\) 0 0
\(789\) −8.33351e9 −0.604030
\(790\) 0 0
\(791\) 2.11263e10 1.51777
\(792\) 0 0
\(793\) −4.46821e10 −3.18183
\(794\) 0 0
\(795\) −7.77511e9 −0.548809
\(796\) 0 0
\(797\) −2.23299e9 −0.156237 −0.0781183 0.996944i \(-0.524891\pi\)
−0.0781183 + 0.996944i \(0.524891\pi\)
\(798\) 0 0
\(799\) −4.15431e9 −0.288128
\(800\) 0 0
\(801\) 8.86123e8 0.0609228
\(802\) 0 0
\(803\) −1.66049e10 −1.13170
\(804\) 0 0
\(805\) −1.51293e10 −1.02219
\(806\) 0 0
\(807\) −5.69520e9 −0.381462
\(808\) 0 0
\(809\) −1.89092e10 −1.25560 −0.627802 0.778373i \(-0.716045\pi\)
−0.627802 + 0.778373i \(0.716045\pi\)
\(810\) 0 0
\(811\) 9.97440e9 0.656620 0.328310 0.944570i \(-0.393521\pi\)
0.328310 + 0.944570i \(0.393521\pi\)
\(812\) 0 0
\(813\) 1.48144e10 0.966870
\(814\) 0 0
\(815\) −9.86891e9 −0.638583
\(816\) 0 0
\(817\) −1.71148e9 −0.109798
\(818\) 0 0
\(819\) −1.47546e10 −0.938496
\(820\) 0 0
\(821\) 2.26783e10 1.43024 0.715120 0.699002i \(-0.246372\pi\)
0.715120 + 0.699002i \(0.246372\pi\)
\(822\) 0 0
\(823\) −3.64946e9 −0.228207 −0.114103 0.993469i \(-0.536400\pi\)
−0.114103 + 0.993469i \(0.536400\pi\)
\(824\) 0 0
\(825\) −1.59396e8 −0.00988299
\(826\) 0 0
\(827\) −1.51192e9 −0.0929521 −0.0464761 0.998919i \(-0.514799\pi\)
−0.0464761 + 0.998919i \(0.514799\pi\)
\(828\) 0 0
\(829\) 1.26210e10 0.769398 0.384699 0.923042i \(-0.374305\pi\)
0.384699 + 0.923042i \(0.374305\pi\)
\(830\) 0 0
\(831\) 1.35403e10 0.818514
\(832\) 0 0
\(833\) −4.78676e9 −0.286936
\(834\) 0 0
\(835\) 1.06956e10 0.635776
\(836\) 0 0
\(837\) −1.40166e9 −0.0826235
\(838\) 0 0
\(839\) −1.67517e10 −0.979244 −0.489622 0.871935i \(-0.662865\pi\)
−0.489622 + 0.871935i \(0.662865\pi\)
\(840\) 0 0
\(841\) 6.67984e9 0.387240
\(842\) 0 0
\(843\) 1.38158e10 0.794289
\(844\) 0 0
\(845\) −4.45979e10 −2.54282
\(846\) 0 0
\(847\) −6.72848e9 −0.380474
\(848\) 0 0
\(849\) 6.56315e9 0.368074
\(850\) 0 0
\(851\) 1.03049e10 0.573181
\(852\) 0 0
\(853\) −3.42742e10 −1.89080 −0.945401 0.325909i \(-0.894330\pi\)
−0.945401 + 0.325909i \(0.894330\pi\)
\(854\) 0 0
\(855\) 6.62192e8 0.0362329
\(856\) 0 0
\(857\) 6.94990e9 0.377177 0.188589 0.982056i \(-0.439609\pi\)
0.188589 + 0.982056i \(0.439609\pi\)
\(858\) 0 0
\(859\) 2.55440e9 0.137503 0.0687516 0.997634i \(-0.478098\pi\)
0.0687516 + 0.997634i \(0.478098\pi\)
\(860\) 0 0
\(861\) −1.19786e10 −0.639579
\(862\) 0 0
\(863\) 6.10627e9 0.323399 0.161699 0.986840i \(-0.448303\pi\)
0.161699 + 0.986840i \(0.448303\pi\)
\(864\) 0 0
\(865\) 6.78235e9 0.356306
\(866\) 0 0
\(867\) 1.04984e10 0.547086
\(868\) 0 0
\(869\) −1.38239e10 −0.714598
\(870\) 0 0
\(871\) 3.74345e10 1.91959
\(872\) 0 0
\(873\) −7.58973e9 −0.386079
\(874\) 0 0
\(875\) 2.94444e10 1.48585
\(876\) 0 0
\(877\) −1.66173e9 −0.0831882 −0.0415941 0.999135i \(-0.513244\pi\)
−0.0415941 + 0.999135i \(0.513244\pi\)
\(878\) 0 0
\(879\) 1.24982e10 0.620705
\(880\) 0 0
\(881\) 1.86590e10 0.919332 0.459666 0.888092i \(-0.347969\pi\)
0.459666 + 0.888092i \(0.347969\pi\)
\(882\) 0 0
\(883\) −8.45215e8 −0.0413147 −0.0206573 0.999787i \(-0.506576\pi\)
−0.0206573 + 0.999787i \(0.506576\pi\)
\(884\) 0 0
\(885\) 1.44609e10 0.701282
\(886\) 0 0
\(887\) 3.73395e10 1.79654 0.898269 0.439447i \(-0.144825\pi\)
0.898269 + 0.439447i \(0.144825\pi\)
\(888\) 0 0
\(889\) −6.15984e10 −2.94044
\(890\) 0 0
\(891\) 2.02700e9 0.0960025
\(892\) 0 0
\(893\) −2.88261e9 −0.135458
\(894\) 0 0
\(895\) 3.64618e10 1.70003
\(896\) 0 0
\(897\) 1.57844e10 0.730220
\(898\) 0 0
\(899\) −1.10159e10 −0.505663
\(900\) 0 0
\(901\) 4.73153e9 0.215509
\(902\) 0 0
\(903\) −1.95606e10 −0.884048
\(904\) 0 0
\(905\) 2.04034e10 0.915024
\(906\) 0 0
\(907\) −1.22684e10 −0.545960 −0.272980 0.962020i \(-0.588009\pi\)
−0.272980 + 0.962020i \(0.588009\pi\)
\(908\) 0 0
\(909\) −5.48412e9 −0.242177
\(910\) 0 0
\(911\) −3.21931e10 −1.41074 −0.705372 0.708837i \(-0.749220\pi\)
−0.705372 + 0.708837i \(0.749220\pi\)
\(912\) 0 0
\(913\) −2.26917e10 −0.986776
\(914\) 0 0
\(915\) 2.29194e10 0.989074
\(916\) 0 0
\(917\) 1.96471e10 0.841407
\(918\) 0 0
\(919\) −3.10394e10 −1.31919 −0.659597 0.751620i \(-0.729273\pi\)
−0.659597 + 0.751620i \(0.729273\pi\)
\(920\) 0 0
\(921\) −1.03157e10 −0.435102
\(922\) 0 0
\(923\) 6.16931e10 2.58244
\(924\) 0 0
\(925\) 4.05364e8 0.0168403
\(926\) 0 0
\(927\) 9.23369e9 0.380712
\(928\) 0 0
\(929\) 7.41429e9 0.303399 0.151700 0.988427i \(-0.451525\pi\)
0.151700 + 0.988427i \(0.451525\pi\)
\(930\) 0 0
\(931\) −3.32146e9 −0.134898
\(932\) 0 0
\(933\) 1.97439e9 0.0795880
\(934\) 0 0
\(935\) 4.99308e9 0.199769
\(936\) 0 0
\(937\) 6.86618e9 0.272663 0.136332 0.990663i \(-0.456469\pi\)
0.136332 + 0.990663i \(0.456469\pi\)
\(938\) 0 0
\(939\) 1.02341e10 0.403387
\(940\) 0 0
\(941\) −7.55613e9 −0.295621 −0.147811 0.989016i \(-0.547223\pi\)
−0.147811 + 0.989016i \(0.547223\pi\)
\(942\) 0 0
\(943\) 1.28146e10 0.497640
\(944\) 0 0
\(945\) 7.56825e9 0.291732
\(946\) 0 0
\(947\) −2.18917e10 −0.837636 −0.418818 0.908070i \(-0.637556\pi\)
−0.418818 + 0.908070i \(0.637556\pi\)
\(948\) 0 0
\(949\) 6.46826e10 2.45672
\(950\) 0 0
\(951\) 1.62873e10 0.614069
\(952\) 0 0
\(953\) −1.91574e10 −0.716986 −0.358493 0.933532i \(-0.616709\pi\)
−0.358493 + 0.933532i \(0.616709\pi\)
\(954\) 0 0
\(955\) 1.50000e10 0.557286
\(956\) 0 0
\(957\) 1.59306e10 0.587543
\(958\) 0 0
\(959\) 3.58355e10 1.31204
\(960\) 0 0
\(961\) −2.24415e10 −0.815681
\(962\) 0 0
\(963\) 1.48210e10 0.534792
\(964\) 0 0
\(965\) 3.88527e10 1.39180
\(966\) 0 0
\(967\) −4.49432e10 −1.59835 −0.799174 0.601099i \(-0.794730\pi\)
−0.799174 + 0.601099i \(0.794730\pi\)
\(968\) 0 0
\(969\) −4.02976e8 −0.0142281
\(970\) 0 0
\(971\) 1.58327e10 0.554994 0.277497 0.960726i \(-0.410495\pi\)
0.277497 + 0.960726i \(0.410495\pi\)
\(972\) 0 0
\(973\) 2.44907e10 0.852327
\(974\) 0 0
\(975\) 6.20909e8 0.0214542
\(976\) 0 0
\(977\) −2.34667e10 −0.805048 −0.402524 0.915409i \(-0.631867\pi\)
−0.402524 + 0.915409i \(0.631867\pi\)
\(978\) 0 0
\(979\) 4.63624e9 0.157916
\(980\) 0 0
\(981\) −1.68476e10 −0.569765
\(982\) 0 0
\(983\) −1.54097e10 −0.517438 −0.258719 0.965953i \(-0.583300\pi\)
−0.258719 + 0.965953i \(0.583300\pi\)
\(984\) 0 0
\(985\) 1.85015e10 0.616849
\(986\) 0 0
\(987\) −3.29456e10 −1.09066
\(988\) 0 0
\(989\) 2.09259e10 0.687856
\(990\) 0 0
\(991\) −2.37648e10 −0.775669 −0.387835 0.921729i \(-0.626777\pi\)
−0.387835 + 0.921729i \(0.626777\pi\)
\(992\) 0 0
\(993\) 8.69361e9 0.281759
\(994\) 0 0
\(995\) 1.03115e10 0.331850
\(996\) 0 0
\(997\) 3.48653e10 1.11419 0.557096 0.830448i \(-0.311916\pi\)
0.557096 + 0.830448i \(0.311916\pi\)
\(998\) 0 0
\(999\) −5.15493e9 −0.163585
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 384.8.a.m.1.2 4
4.3 odd 2 384.8.a.q.1.2 yes 4
8.3 odd 2 384.8.a.p.1.3 yes 4
8.5 even 2 384.8.a.t.1.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
384.8.a.m.1.2 4 1.1 even 1 trivial
384.8.a.p.1.3 yes 4 8.3 odd 2
384.8.a.q.1.2 yes 4 4.3 odd 2
384.8.a.t.1.3 yes 4 8.5 even 2