Properties

Label 300.9.g.h.101.2
Level $300$
Weight $9$
Character 300.101
Analytic conductor $122.214$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,9,Mod(101,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.101");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 300.g (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(122.213583018\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 7378 x^{14} + 23156928 x^{12} - 101588726286 x^{10} + \cdots + 34\!\cdots\!81 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{17}]\)
Coefficient ring index: \( 2^{38}\cdot 3^{20}\cdot 5^{18} \)
Twist minimal: no (minimal twist has level 60)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 101.2
Root \(79.3447 - 16.2916i\) of defining polynomial
Character \(\chi\) \(=\) 300.101
Dual form 300.9.g.h.101.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-79.3447 + 16.2916i) q^{3} -3405.57 q^{7} +(6030.17 - 2585.30i) q^{9} +O(q^{10})\) \(q+(-79.3447 + 16.2916i) q^{3} -3405.57 q^{7} +(6030.17 - 2585.30i) q^{9} +24573.8i q^{11} -25174.6 q^{13} +127299. i q^{17} +110754. q^{19} +(270214. - 55482.1i) q^{21} -200166. i q^{23} +(-436343. + 303371. i) q^{27} +185716. i q^{29} +800734. q^{31} +(-400346. - 1.94980e6i) q^{33} +859139. q^{37} +(1.99747e6 - 410133. i) q^{39} +5.07451e6i q^{41} +4.48899e6 q^{43} +6.94005e6i q^{47} +5.83308e6 q^{49} +(-2.07390e6 - 1.01005e7i) q^{51} -3.64060e6i q^{53} +(-8.78777e6 + 1.80436e6i) q^{57} +6.92951e6i q^{59} -2.58815e6 q^{61} +(-2.05361e7 + 8.80442e6i) q^{63} +6.22264e6 q^{67} +(3.26103e6 + 1.58822e7i) q^{69} +3.07705e7i q^{71} +1.15853e7 q^{73} -8.36877e7i q^{77} +6.00742e7 q^{79} +(2.96791e7 - 3.11796e7i) q^{81} +4.00247e7i q^{83} +(-3.02560e6 - 1.47356e7i) q^{87} -4.48778e7i q^{89} +8.57336e7 q^{91} +(-6.35340e7 + 1.30452e7i) q^{93} -6.54198e7 q^{97} +(6.35307e7 + 1.48184e8i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 14756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 14756 q^{9} + 48024 q^{19} + 4724 q^{21} + 470104 q^{31} + 2849664 q^{39} + 19816920 q^{49} + 5026040 q^{51} + 18849944 q^{61} + 38669180 q^{69} + 90778632 q^{79} + 16242056 q^{81} + 22375296 q^{91} - 16968560 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) −79.3447 + 16.2916i −0.979564 + 0.201131i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −3405.57 −1.41839 −0.709197 0.705010i \(-0.750943\pi\)
−0.709197 + 0.705010i \(0.750943\pi\)
\(8\) 0 0
\(9\) 6030.17 2585.30i 0.919093 0.394041i
\(10\) 0 0
\(11\) 24573.8i 1.67842i 0.543805 + 0.839212i \(0.316983\pi\)
−0.543805 + 0.839212i \(0.683017\pi\)
\(12\) 0 0
\(13\) −25174.6 −0.881431 −0.440715 0.897647i \(-0.645275\pi\)
−0.440715 + 0.897647i \(0.645275\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 127299.i 1.52416i 0.647485 + 0.762078i \(0.275821\pi\)
−0.647485 + 0.762078i \(0.724179\pi\)
\(18\) 0 0
\(19\) 110754. 0.849857 0.424929 0.905227i \(-0.360299\pi\)
0.424929 + 0.905227i \(0.360299\pi\)
\(20\) 0 0
\(21\) 270214. 55482.1i 1.38941 0.285283i
\(22\) 0 0
\(23\) 200166.i 0.715286i −0.933858 0.357643i \(-0.883580\pi\)
0.933858 0.357643i \(-0.116420\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −436343. + 303371.i −0.821057 + 0.570846i
\(28\) 0 0
\(29\) 185716.i 0.262577i 0.991344 + 0.131288i \(0.0419114\pi\)
−0.991344 + 0.131288i \(0.958089\pi\)
\(30\) 0 0
\(31\) 800734. 0.867045 0.433522 0.901143i \(-0.357270\pi\)
0.433522 + 0.901143i \(0.357270\pi\)
\(32\) 0 0
\(33\) −400346. 1.94980e6i −0.337582 1.64412i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 859139. 0.458413 0.229206 0.973378i \(-0.426387\pi\)
0.229206 + 0.973378i \(0.426387\pi\)
\(38\) 0 0
\(39\) 1.99747e6 410133.i 0.863418 0.177283i
\(40\) 0 0
\(41\) 5.07451e6i 1.79580i 0.440196 + 0.897902i \(0.354909\pi\)
−0.440196 + 0.897902i \(0.645091\pi\)
\(42\) 0 0
\(43\) 4.48899e6 1.31303 0.656515 0.754313i \(-0.272030\pi\)
0.656515 + 0.754313i \(0.272030\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 6.94005e6i 1.42223i 0.703073 + 0.711117i \(0.251811\pi\)
−0.703073 + 0.711117i \(0.748189\pi\)
\(48\) 0 0
\(49\) 5.83308e6 1.01184
\(50\) 0 0
\(51\) −2.07390e6 1.01005e7i −0.306555 1.49301i
\(52\) 0 0
\(53\) 3.64060e6i 0.461392i −0.973026 0.230696i \(-0.925900\pi\)
0.973026 0.230696i \(-0.0741002\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −8.78777e6 + 1.80436e6i −0.832490 + 0.170932i
\(58\) 0 0
\(59\) 6.92951e6i 0.571866i 0.958250 + 0.285933i \(0.0923035\pi\)
−0.958250 + 0.285933i \(0.907696\pi\)
\(60\) 0 0
\(61\) −2.58815e6 −0.186926 −0.0934629 0.995623i \(-0.529794\pi\)
−0.0934629 + 0.995623i \(0.529794\pi\)
\(62\) 0 0
\(63\) −2.05361e7 + 8.80442e6i −1.30364 + 0.558906i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 6.22264e6 0.308799 0.154399 0.988009i \(-0.450656\pi\)
0.154399 + 0.988009i \(0.450656\pi\)
\(68\) 0 0
\(69\) 3.26103e6 + 1.58822e7i 0.143866 + 0.700669i
\(70\) 0 0
\(71\) 3.07705e7i 1.21088i 0.795890 + 0.605441i \(0.207003\pi\)
−0.795890 + 0.605441i \(0.792997\pi\)
\(72\) 0 0
\(73\) 1.15853e7 0.407959 0.203979 0.978975i \(-0.434612\pi\)
0.203979 + 0.978975i \(0.434612\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 8.36877e7i 2.38067i
\(78\) 0 0
\(79\) 6.00742e7 1.54234 0.771170 0.636630i \(-0.219672\pi\)
0.771170 + 0.636630i \(0.219672\pi\)
\(80\) 0 0
\(81\) 2.96791e7 3.11796e7i 0.689463 0.724320i
\(82\) 0 0
\(83\) 4.00247e7i 0.843365i 0.906744 + 0.421682i \(0.138560\pi\)
−0.906744 + 0.421682i \(0.861440\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −3.02560e6 1.47356e7i −0.0528123 0.257211i
\(88\) 0 0
\(89\) 4.48778e7i 0.715272i −0.933861 0.357636i \(-0.883583\pi\)
0.933861 0.357636i \(-0.116417\pi\)
\(90\) 0 0
\(91\) 8.57336e7 1.25022
\(92\) 0 0
\(93\) −6.35340e7 + 1.30452e7i −0.849326 + 0.174389i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.54198e7 −0.738962 −0.369481 0.929238i \(-0.620465\pi\)
−0.369481 + 0.929238i \(0.620465\pi\)
\(98\) 0 0
\(99\) 6.35307e7 + 1.48184e8i 0.661367 + 1.54263i
\(100\) 0 0
\(101\) 303020.i 0.00291197i −0.999999 0.00145598i \(-0.999537\pi\)
0.999999 0.00145598i \(-0.000463454\pi\)
\(102\) 0 0
\(103\) 9.21296e7 0.818559 0.409280 0.912409i \(-0.365780\pi\)
0.409280 + 0.912409i \(0.365780\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.98998e8i 1.51815i 0.651005 + 0.759074i \(0.274348\pi\)
−0.651005 + 0.759074i \(0.725652\pi\)
\(108\) 0 0
\(109\) −2.30400e8 −1.63222 −0.816108 0.577900i \(-0.803872\pi\)
−0.816108 + 0.577900i \(0.803872\pi\)
\(110\) 0 0
\(111\) −6.81681e7 + 1.39967e7i −0.449045 + 0.0922008i
\(112\) 0 0
\(113\) 2.90409e7i 0.178113i 0.996027 + 0.0890566i \(0.0283852\pi\)
−0.996027 + 0.0890566i \(0.971615\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.51807e8 + 6.50838e7i −0.810117 + 0.347320i
\(118\) 0 0
\(119\) 4.33525e8i 2.16186i
\(120\) 0 0
\(121\) −3.89512e8 −1.81710
\(122\) 0 0
\(123\) −8.26719e7 4.02636e8i −0.361191 1.75911i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −3.54560e8 −1.36293 −0.681467 0.731849i \(-0.738658\pi\)
−0.681467 + 0.731849i \(0.738658\pi\)
\(128\) 0 0
\(129\) −3.56177e8 + 7.31327e7i −1.28620 + 0.264091i
\(130\) 0 0
\(131\) 1.81000e8i 0.614603i 0.951612 + 0.307301i \(0.0994260\pi\)
−0.951612 + 0.307301i \(0.900574\pi\)
\(132\) 0 0
\(133\) −3.77181e8 −1.20543
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 4.06421e8i 1.15370i −0.816849 0.576851i \(-0.804281\pi\)
0.816849 0.576851i \(-0.195719\pi\)
\(138\) 0 0
\(139\) −1.47570e7 −0.0395312 −0.0197656 0.999805i \(-0.506292\pi\)
−0.0197656 + 0.999805i \(0.506292\pi\)
\(140\) 0 0
\(141\) −1.13064e8 5.50656e8i −0.286055 1.39317i
\(142\) 0 0
\(143\) 6.18634e8i 1.47941i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −4.62824e8 + 9.50301e7i −0.991167 + 0.203513i
\(148\) 0 0
\(149\) 7.35283e7i 0.149180i −0.997214 0.0745898i \(-0.976235\pi\)
0.997214 0.0745898i \(-0.0237647\pi\)
\(150\) 0 0
\(151\) 3.16954e8 0.609661 0.304830 0.952407i \(-0.401400\pi\)
0.304830 + 0.952407i \(0.401400\pi\)
\(152\) 0 0
\(153\) 3.29107e8 + 7.67635e8i 0.600580 + 1.40084i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −7.14521e8 −1.17602 −0.588012 0.808852i \(-0.700089\pi\)
−0.588012 + 0.808852i \(0.700089\pi\)
\(158\) 0 0
\(159\) 5.93112e7 + 2.88862e8i 0.0928000 + 0.451963i
\(160\) 0 0
\(161\) 6.81680e8i 1.01456i
\(162\) 0 0
\(163\) −7.11302e8 −1.00764 −0.503818 0.863810i \(-0.668072\pi\)
−0.503818 + 0.863810i \(0.668072\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 2.16724e8i 0.278638i −0.990248 0.139319i \(-0.955509\pi\)
0.990248 0.139319i \(-0.0444913\pi\)
\(168\) 0 0
\(169\) −1.81973e8 −0.223079
\(170\) 0 0
\(171\) 6.67867e8 2.86333e8i 0.781098 0.334879i
\(172\) 0 0
\(173\) 1.48836e9i 1.66159i 0.556581 + 0.830793i \(0.312113\pi\)
−0.556581 + 0.830793i \(0.687887\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) −1.12893e8 5.49820e8i −0.115020 0.560180i
\(178\) 0 0
\(179\) 5.05726e8i 0.492610i 0.969192 + 0.246305i \(0.0792165\pi\)
−0.969192 + 0.246305i \(0.920784\pi\)
\(180\) 0 0
\(181\) 4.18187e8 0.389634 0.194817 0.980840i \(-0.437589\pi\)
0.194817 + 0.980840i \(0.437589\pi\)
\(182\) 0 0
\(183\) 2.05356e8 4.21650e7i 0.183106 0.0375965i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −3.12822e9 −2.55818
\(188\) 0 0
\(189\) 1.48600e9 1.03315e9i 1.16458 0.809685i
\(190\) 0 0
\(191\) 6.58890e7i 0.0495085i −0.999694 0.0247542i \(-0.992120\pi\)
0.999694 0.0247542i \(-0.00788032\pi\)
\(192\) 0 0
\(193\) −2.16804e9 −1.56257 −0.781283 0.624176i \(-0.785435\pi\)
−0.781283 + 0.624176i \(0.785435\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.87186e9i 1.24282i −0.783484 0.621412i \(-0.786559\pi\)
0.783484 0.621412i \(-0.213441\pi\)
\(198\) 0 0
\(199\) 9.42274e8 0.600848 0.300424 0.953806i \(-0.402872\pi\)
0.300424 + 0.953806i \(0.402872\pi\)
\(200\) 0 0
\(201\) −4.93734e8 + 1.01377e8i −0.302488 + 0.0621089i
\(202\) 0 0
\(203\) 6.32467e8i 0.372438i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −5.17491e8 1.20704e9i −0.281852 0.657415i
\(208\) 0 0
\(209\) 2.72165e9i 1.42642i
\(210\) 0 0
\(211\) −9.71708e8 −0.490237 −0.245118 0.969493i \(-0.578827\pi\)
−0.245118 + 0.969493i \(0.578827\pi\)
\(212\) 0 0
\(213\) −5.01301e8 2.44148e9i −0.243545 1.18614i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.72695e9 −1.22981
\(218\) 0 0
\(219\) −9.19233e8 + 1.88743e8i −0.399622 + 0.0820530i
\(220\) 0 0
\(221\) 3.20470e9i 1.34344i
\(222\) 0 0
\(223\) −7.54076e8 −0.304927 −0.152463 0.988309i \(-0.548721\pi\)
−0.152463 + 0.988309i \(0.548721\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 3.53094e9i 1.32980i 0.746931 + 0.664902i \(0.231527\pi\)
−0.746931 + 0.664902i \(0.768473\pi\)
\(228\) 0 0
\(229\) 2.50568e9 0.911136 0.455568 0.890201i \(-0.349436\pi\)
0.455568 + 0.890201i \(0.349436\pi\)
\(230\) 0 0
\(231\) 1.36341e9 + 6.64018e9i 0.478825 + 2.33202i
\(232\) 0 0
\(233\) 3.87132e9i 1.31351i −0.754102 0.656757i \(-0.771928\pi\)
0.754102 0.656757i \(-0.228072\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −4.76657e9 + 9.78705e8i −1.51082 + 0.310212i
\(238\) 0 0
\(239\) 1.52584e9i 0.467645i −0.972279 0.233823i \(-0.924876\pi\)
0.972279 0.233823i \(-0.0751235\pi\)
\(240\) 0 0
\(241\) −2.57111e9 −0.762171 −0.381086 0.924540i \(-0.624450\pi\)
−0.381086 + 0.924540i \(0.624450\pi\)
\(242\) 0 0
\(243\) −1.84692e9 + 2.95746e9i −0.529691 + 0.848191i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.78819e9 −0.749091
\(248\) 0 0
\(249\) −6.52065e8 3.17575e9i −0.169627 0.826130i
\(250\) 0 0
\(251\) 4.56222e9i 1.14943i −0.818355 0.574714i \(-0.805113\pi\)
0.818355 0.574714i \(-0.194887\pi\)
\(252\) 0 0
\(253\) 4.91885e9 1.20055
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 3.22319e9i 0.738844i 0.929262 + 0.369422i \(0.120444\pi\)
−0.929262 + 0.369422i \(0.879556\pi\)
\(258\) 0 0
\(259\) −2.92585e9 −0.650210
\(260\) 0 0
\(261\) 4.80131e8 + 1.11990e9i 0.103466 + 0.241333i
\(262\) 0 0
\(263\) 3.25948e7i 0.00681280i −0.999994 0.00340640i \(-0.998916\pi\)
0.999994 0.00340640i \(-0.00108429\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 7.31130e8 + 3.56081e9i 0.143863 + 0.700655i
\(268\) 0 0
\(269\) 2.55752e9i 0.488438i 0.969720 + 0.244219i \(0.0785316\pi\)
−0.969720 + 0.244219i \(0.921468\pi\)
\(270\) 0 0
\(271\) 7.24486e9 1.34324 0.671618 0.740897i \(-0.265599\pi\)
0.671618 + 0.740897i \(0.265599\pi\)
\(272\) 0 0
\(273\) −6.80251e9 + 1.39674e9i −1.22467 + 0.251457i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.29340e9 0.729259 0.364630 0.931153i \(-0.381196\pi\)
0.364630 + 0.931153i \(0.381196\pi\)
\(278\) 0 0
\(279\) 4.82856e9 2.07014e9i 0.796895 0.341651i
\(280\) 0 0
\(281\) 1.81158e9i 0.290557i 0.989391 + 0.145279i \(0.0464079\pi\)
−0.989391 + 0.145279i \(0.953592\pi\)
\(282\) 0 0
\(283\) 2.90902e9 0.453524 0.226762 0.973950i \(-0.427186\pi\)
0.226762 + 0.973950i \(0.427186\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.72816e10i 2.54716i
\(288\) 0 0
\(289\) −9.22930e9 −1.32305
\(290\) 0 0
\(291\) 5.19072e9 1.06579e9i 0.723861 0.148628i
\(292\) 0 0
\(293\) 1.24005e9i 0.168255i 0.996455 + 0.0841276i \(0.0268103\pi\)
−0.996455 + 0.0841276i \(0.973190\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −7.45498e9 1.07226e10i −0.958122 1.37808i
\(298\) 0 0
\(299\) 5.03910e9i 0.630476i
\(300\) 0 0
\(301\) −1.52875e10 −1.86240
\(302\) 0 0
\(303\) 4.93668e6 + 2.40431e7i 0.000585686 + 0.00285246i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 3.66791e9 0.412919 0.206460 0.978455i \(-0.433806\pi\)
0.206460 + 0.978455i \(0.433806\pi\)
\(308\) 0 0
\(309\) −7.31000e9 + 1.50094e9i −0.801832 + 0.164637i
\(310\) 0 0
\(311\) 1.25169e8i 0.0133800i −0.999978 0.00668998i \(-0.997870\pi\)
0.999978 0.00668998i \(-0.00212950\pi\)
\(312\) 0 0
\(313\) −5.49432e9 −0.572449 −0.286224 0.958163i \(-0.592400\pi\)
−0.286224 + 0.958163i \(0.592400\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.19225e10i 1.18068i 0.807155 + 0.590340i \(0.201006\pi\)
−0.807155 + 0.590340i \(0.798994\pi\)
\(318\) 0 0
\(319\) −4.56374e9 −0.440715
\(320\) 0 0
\(321\) −3.24200e9 1.57894e10i −0.305346 1.48712i
\(322\) 0 0
\(323\) 1.40989e10i 1.29532i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.82811e10 3.75359e9i 1.59886 0.328289i
\(328\) 0 0
\(329\) 2.36348e10i 2.01729i
\(330\) 0 0
\(331\) −1.33201e10 −1.10967 −0.554837 0.831959i \(-0.687220\pi\)
−0.554837 + 0.831959i \(0.687220\pi\)
\(332\) 0 0
\(333\) 5.18075e9 2.22113e9i 0.421324 0.180633i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 9.00994e9 0.698558 0.349279 0.937019i \(-0.386427\pi\)
0.349279 + 0.937019i \(0.386427\pi\)
\(338\) 0 0
\(339\) −4.73122e8 2.30424e9i −0.0358240 0.174473i
\(340\) 0 0
\(341\) 1.96771e10i 1.45527i
\(342\) 0 0
\(343\) −2.32530e8 −0.0167998
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 9.53927e9i 0.657956i 0.944338 + 0.328978i \(0.106704\pi\)
−0.944338 + 0.328978i \(0.893296\pi\)
\(348\) 0 0
\(349\) −2.23914e9 −0.150931 −0.0754657 0.997148i \(-0.524044\pi\)
−0.0754657 + 0.997148i \(0.524044\pi\)
\(350\) 0 0
\(351\) 1.09847e10 7.63723e9i 0.723705 0.503162i
\(352\) 0 0
\(353\) 1.38508e9i 0.0892021i −0.999005 0.0446011i \(-0.985798\pi\)
0.999005 0.0446011i \(-0.0142017\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 7.06282e9 + 3.43980e10i 0.434816 + 2.11768i
\(358\) 0 0
\(359\) 2.18160e10i 1.31340i 0.754150 + 0.656702i \(0.228049\pi\)
−0.754150 + 0.656702i \(0.771951\pi\)
\(360\) 0 0
\(361\) −4.71705e9 −0.277742
\(362\) 0 0
\(363\) 3.09058e10 6.34578e9i 1.77997 0.365475i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 6.43071e8 0.0354482 0.0177241 0.999843i \(-0.494358\pi\)
0.0177241 + 0.999843i \(0.494358\pi\)
\(368\) 0 0
\(369\) 1.31192e10 + 3.06002e10i 0.707620 + 1.65051i
\(370\) 0 0
\(371\) 1.23983e10i 0.654435i
\(372\) 0 0
\(373\) 1.52291e10 0.786752 0.393376 0.919378i \(-0.371307\pi\)
0.393376 + 0.919378i \(0.371307\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 4.67531e9i 0.231443i
\(378\) 0 0
\(379\) 3.15729e10 1.53023 0.765116 0.643892i \(-0.222682\pi\)
0.765116 + 0.643892i \(0.222682\pi\)
\(380\) 0 0
\(381\) 2.81325e10 5.77634e9i 1.33508 0.274128i
\(382\) 0 0
\(383\) 1.83628e10i 0.853384i −0.904397 0.426692i \(-0.859679\pi\)
0.904397 0.426692i \(-0.140321\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.70694e10 1.16054e10i 1.20680 0.517388i
\(388\) 0 0
\(389\) 3.15515e10i 1.37791i −0.724802 0.688957i \(-0.758069\pi\)
0.724802 0.688957i \(-0.241931\pi\)
\(390\) 0 0
\(391\) 2.54810e10 1.09021
\(392\) 0 0
\(393\) −2.94878e9 1.43614e10i −0.123615 0.602043i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.82543e10 1.94256 0.971280 0.237941i \(-0.0764724\pi\)
0.971280 + 0.237941i \(0.0764724\pi\)
\(398\) 0 0
\(399\) 2.99273e10 6.14488e9i 1.18080 0.242450i
\(400\) 0 0
\(401\) 4.44335e9i 0.171843i −0.996302 0.0859217i \(-0.972617\pi\)
0.996302 0.0859217i \(-0.0273835\pi\)
\(402\) 0 0
\(403\) −2.01581e10 −0.764240
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.11123e10i 0.769410i
\(408\) 0 0
\(409\) −9.06587e9 −0.323979 −0.161989 0.986793i \(-0.551791\pi\)
−0.161989 + 0.986793i \(0.551791\pi\)
\(410\) 0 0
\(411\) 6.62124e9 + 3.22474e10i 0.232045 + 1.13013i
\(412\) 0 0
\(413\) 2.35989e10i 0.811132i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.17089e9 2.40416e8i 0.0387234 0.00795094i
\(418\) 0 0
\(419\) 1.48532e10i 0.481906i −0.970537 0.240953i \(-0.922540\pi\)
0.970537 0.240953i \(-0.0774600\pi\)
\(420\) 0 0
\(421\) −1.82155e10 −0.579846 −0.289923 0.957050i \(-0.593630\pi\)
−0.289923 + 0.957050i \(0.593630\pi\)
\(422\) 0 0
\(423\) 1.79421e10 + 4.18497e10i 0.560419 + 1.30717i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 8.81410e9 0.265135
\(428\) 0 0
\(429\) 1.00785e10 + 4.90854e10i 0.297556 + 1.44918i
\(430\) 0 0
\(431\) 2.12650e10i 0.616249i −0.951346 0.308125i \(-0.900299\pi\)
0.951346 0.308125i \(-0.0997014\pi\)
\(432\) 0 0
\(433\) 4.54370e9 0.129258 0.0646291 0.997909i \(-0.479414\pi\)
0.0646291 + 0.997909i \(0.479414\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 2.21693e10i 0.607892i
\(438\) 0 0
\(439\) −2.38568e10 −0.642324 −0.321162 0.947024i \(-0.604073\pi\)
−0.321162 + 0.947024i \(0.604073\pi\)
\(440\) 0 0
\(441\) 3.51745e10 1.50803e10i 0.929979 0.398708i
\(442\) 0 0
\(443\) 3.94183e10i 1.02349i 0.859138 + 0.511744i \(0.171000\pi\)
−0.859138 + 0.511744i \(0.829000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 1.19789e9 + 5.83408e9i 0.0300046 + 0.146131i
\(448\) 0 0
\(449\) 2.16040e10i 0.531555i 0.964034 + 0.265778i \(0.0856287\pi\)
−0.964034 + 0.265778i \(0.914371\pi\)
\(450\) 0 0
\(451\) −1.24700e11 −3.01412
\(452\) 0 0
\(453\) −2.51486e10 + 5.16368e9i −0.597202 + 0.122622i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 5.37441e10 1.23216 0.616078 0.787685i \(-0.288721\pi\)
0.616078 + 0.787685i \(0.288721\pi\)
\(458\) 0 0
\(459\) −3.86189e10 5.55461e10i −0.870059 1.25142i
\(460\) 0 0
\(461\) 1.60037e10i 0.354337i 0.984181 + 0.177168i \(0.0566937\pi\)
−0.984181 + 0.177168i \(0.943306\pi\)
\(462\) 0 0
\(463\) 6.87672e10 1.49643 0.748217 0.663454i \(-0.230910\pi\)
0.748217 + 0.663454i \(0.230910\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.62821e10i 0.552577i 0.961075 + 0.276289i \(0.0891046\pi\)
−0.961075 + 0.276289i \(0.910895\pi\)
\(468\) 0 0
\(469\) −2.11916e10 −0.437999
\(470\) 0 0
\(471\) 5.66935e10 1.16407e10i 1.15199 0.236535i
\(472\) 0 0
\(473\) 1.10311e11i 2.20382i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −9.41206e9 2.19534e10i −0.181807 0.424062i
\(478\) 0 0
\(479\) 5.74718e10i 1.09172i 0.837875 + 0.545862i \(0.183798\pi\)
−0.837875 + 0.545862i \(0.816202\pi\)
\(480\) 0 0
\(481\) −2.16284e10 −0.404059
\(482\) 0 0
\(483\) −1.11057e10 5.40877e10i −0.204059 0.993826i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 7.95448e10 1.41415 0.707076 0.707138i \(-0.250014\pi\)
0.707076 + 0.707138i \(0.250014\pi\)
\(488\) 0 0
\(489\) 5.64381e10 1.15882e10i 0.987044 0.202666i
\(490\) 0 0
\(491\) 8.80767e10i 1.51543i −0.652587 0.757714i \(-0.726316\pi\)
0.652587 0.757714i \(-0.273684\pi\)
\(492\) 0 0
\(493\) −2.36414e10 −0.400208
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.04791e11i 1.71751i
\(498\) 0 0
\(499\) −1.06941e11 −1.72481 −0.862404 0.506221i \(-0.831042\pi\)
−0.862404 + 0.506221i \(0.831042\pi\)
\(500\) 0 0
\(501\) 3.53077e9 + 1.71959e10i 0.0560427 + 0.272944i
\(502\) 0 0
\(503\) 6.57043e10i 1.02641i −0.858265 0.513206i \(-0.828458\pi\)
0.858265 0.513206i \(-0.171542\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 1.44386e10 2.96462e9i 0.218521 0.0448681i
\(508\) 0 0
\(509\) 8.25888e10i 1.23041i −0.788367 0.615205i \(-0.789073\pi\)
0.788367 0.615205i \(-0.210927\pi\)
\(510\) 0 0
\(511\) −3.94545e10 −0.578646
\(512\) 0 0
\(513\) −4.83269e10 + 3.35996e10i −0.697781 + 0.485138i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.70543e11 −2.38711
\(518\) 0 0
\(519\) −2.42477e10 1.18093e11i −0.334196 1.62763i
\(520\) 0 0
\(521\) 8.47463e10i 1.15019i 0.818086 + 0.575095i \(0.195035\pi\)
−0.818086 + 0.575095i \(0.804965\pi\)
\(522\) 0 0
\(523\) 1.24977e11 1.67041 0.835207 0.549935i \(-0.185348\pi\)
0.835207 + 0.549935i \(0.185348\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 1.01933e11i 1.32151i
\(528\) 0 0
\(529\) 3.82444e10 0.488365
\(530\) 0 0
\(531\) 1.79149e10 + 4.17861e10i 0.225339 + 0.525598i
\(532\) 0 0
\(533\) 1.27749e11i 1.58288i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −8.23908e9 4.01267e10i −0.0990790 0.482543i
\(538\) 0 0
\(539\) 1.43341e11i 1.69830i
\(540\) 0 0
\(541\) 3.07875e10 0.359406 0.179703 0.983721i \(-0.442486\pi\)
0.179703 + 0.983721i \(0.442486\pi\)
\(542\) 0 0
\(543\) −3.31810e10 + 6.81294e9i −0.381671 + 0.0783673i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −4.05128e10 −0.452526 −0.226263 0.974066i \(-0.572651\pi\)
−0.226263 + 0.974066i \(0.572651\pi\)
\(548\) 0 0
\(549\) −1.56070e10 + 6.69114e9i −0.171802 + 0.0736564i
\(550\) 0 0
\(551\) 2.05688e10i 0.223153i
\(552\) 0 0
\(553\) −2.04587e11 −2.18765
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.74913e11i 1.81719i 0.417679 + 0.908595i \(0.362844\pi\)
−0.417679 + 0.908595i \(0.637156\pi\)
\(558\) 0 0
\(559\) −1.13008e11 −1.15735
\(560\) 0 0
\(561\) 2.48208e11 5.09637e10i 2.50590 0.514528i
\(562\) 0 0
\(563\) 1.26724e11i 1.26132i −0.776060 0.630659i \(-0.782785\pi\)
0.776060 0.630659i \(-0.217215\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.01074e11 + 1.06184e11i −0.977932 + 1.02737i
\(568\) 0 0
\(569\) 4.40561e10i 0.420298i 0.977669 + 0.210149i \(0.0673949\pi\)
−0.977669 + 0.210149i \(0.932605\pi\)
\(570\) 0 0
\(571\) −1.97761e11 −1.86036 −0.930180 0.367105i \(-0.880349\pi\)
−0.930180 + 0.367105i \(0.880349\pi\)
\(572\) 0 0
\(573\) 1.07344e9 + 5.22794e9i 0.00995767 + 0.0484967i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −8.70837e10 −0.785658 −0.392829 0.919612i \(-0.628504\pi\)
−0.392829 + 0.919612i \(0.628504\pi\)
\(578\) 0 0
\(579\) 1.72023e11 3.53209e10i 1.53063 0.314280i
\(580\) 0 0
\(581\) 1.36307e11i 1.19622i
\(582\) 0 0
\(583\) 8.94634e10 0.774410
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.21607e11i 1.02425i −0.858910 0.512126i \(-0.828858\pi\)
0.858910 0.512126i \(-0.171142\pi\)
\(588\) 0 0
\(589\) 8.86847e10 0.736865
\(590\) 0 0
\(591\) 3.04956e10 + 1.48523e11i 0.249970 + 1.21743i
\(592\) 0 0
\(593\) 1.34327e11i 1.08629i 0.839640 + 0.543144i \(0.182766\pi\)
−0.839640 + 0.543144i \(0.817234\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −7.47644e10 + 1.53511e10i −0.588569 + 0.120849i
\(598\) 0 0
\(599\) 1.21066e11i 0.940406i −0.882558 0.470203i \(-0.844181\pi\)
0.882558 0.470203i \(-0.155819\pi\)
\(600\) 0 0
\(601\) −1.45545e11 −1.11558 −0.557789 0.829983i \(-0.688350\pi\)
−0.557789 + 0.829983i \(0.688350\pi\)
\(602\) 0 0
\(603\) 3.75236e10 1.60874e10i 0.283815 0.121679i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −1.20190e11 −0.885346 −0.442673 0.896683i \(-0.645970\pi\)
−0.442673 + 0.896683i \(0.645970\pi\)
\(608\) 0 0
\(609\) 1.03039e10 + 5.01829e10i 0.0749086 + 0.364827i
\(610\) 0 0
\(611\) 1.74713e11i 1.25360i
\(612\) 0 0
\(613\) −1.50533e11 −1.06608 −0.533041 0.846089i \(-0.678951\pi\)
−0.533041 + 0.846089i \(0.678951\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.44657e11i 1.68817i −0.536206 0.844087i \(-0.680143\pi\)
0.536206 0.844087i \(-0.319857\pi\)
\(618\) 0 0
\(619\) 4.61556e10 0.314385 0.157192 0.987568i \(-0.449756\pi\)
0.157192 + 0.987568i \(0.449756\pi\)
\(620\) 0 0
\(621\) 6.07247e10 + 8.73413e10i 0.408319 + 0.587291i
\(622\) 0 0
\(623\) 1.52834e11i 1.01454i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −4.43400e10 2.15949e11i −0.286897 1.39727i
\(628\) 0 0
\(629\) 1.09368e11i 0.698692i
\(630\) 0 0
\(631\) −1.75958e11 −1.10992 −0.554960 0.831877i \(-0.687267\pi\)
−0.554960 + 0.831877i \(0.687267\pi\)
\(632\) 0 0
\(633\) 7.70999e10 1.58307e10i 0.480218 0.0986016i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −1.46845e11 −0.891871
\(638\) 0 0
\(639\) 7.95511e10 + 1.85551e11i 0.477137 + 1.11291i
\(640\) 0 0
\(641\) 2.37563e11i 1.40717i 0.710612 + 0.703584i \(0.248418\pi\)
−0.710612 + 0.703584i \(0.751582\pi\)
\(642\) 0 0
\(643\) −1.28962e11 −0.754427 −0.377214 0.926126i \(-0.623118\pi\)
−0.377214 + 0.926126i \(0.623118\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.61483e11i 0.921529i −0.887523 0.460764i \(-0.847575\pi\)
0.887523 0.460764i \(-0.152425\pi\)
\(648\) 0 0
\(649\) −1.70284e11 −0.959834
\(650\) 0 0
\(651\) 2.16369e11 4.44264e10i 1.20468 0.247353i
\(652\) 0 0
\(653\) 1.46894e11i 0.807887i −0.914784 0.403943i \(-0.867639\pi\)
0.914784 0.403943i \(-0.132361\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 6.98613e10 2.99515e10i 0.374952 0.160752i
\(658\) 0 0
\(659\) 1.59905e11i 0.847855i −0.905696 0.423927i \(-0.860651\pi\)
0.905696 0.423927i \(-0.139349\pi\)
\(660\) 0 0
\(661\) −2.69483e10 −0.141165 −0.0705824 0.997506i \(-0.522486\pi\)
−0.0705824 + 0.997506i \(0.522486\pi\)
\(662\) 0 0
\(663\) 5.22096e10 + 2.54276e11i 0.270207 + 1.31598i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.71740e10 0.187818
\(668\) 0 0
\(669\) 5.98320e10 1.22851e10i 0.298696 0.0613302i
\(670\) 0 0
\(671\) 6.36005e10i 0.313741i
\(672\) 0 0
\(673\) −2.57651e11 −1.25595 −0.627974 0.778234i \(-0.716116\pi\)
−0.627974 + 0.778234i \(0.716116\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1.99005e11i 0.947346i 0.880701 + 0.473673i \(0.157072\pi\)
−0.880701 + 0.473673i \(0.842928\pi\)
\(678\) 0 0
\(679\) 2.22792e11 1.04814
\(680\) 0 0
\(681\) −5.75247e10 2.80162e11i −0.267464 1.30263i
\(682\) 0 0
\(683\) 3.67991e11i 1.69104i 0.533943 + 0.845521i \(0.320710\pi\)
−0.533943 + 0.845521i \(0.679290\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.98812e11 + 4.08215e10i −0.892517 + 0.183257i
\(688\) 0 0
\(689\) 9.16505e10i 0.406685i
\(690\) 0 0
\(691\) 3.29138e11 1.44366 0.721831 0.692069i \(-0.243301\pi\)
0.721831 + 0.692069i \(0.243301\pi\)
\(692\) 0 0
\(693\) −2.16358e11 5.04651e11i −0.938080 2.18805i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −6.45981e11 −2.73709
\(698\) 0 0
\(699\) 6.30699e10 + 3.07168e11i 0.264188 + 1.28667i
\(700\) 0 0
\(701\) 1.29137e11i 0.534782i 0.963588 + 0.267391i \(0.0861616\pi\)
−0.963588 + 0.267391i \(0.913838\pi\)
\(702\) 0 0
\(703\) 9.51533e10 0.389585
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.03196e9i 0.00413032i
\(708\) 0 0
\(709\) −2.17473e11 −0.860639 −0.430320 0.902677i \(-0.641599\pi\)
−0.430320 + 0.902677i \(0.641599\pi\)
\(710\) 0 0
\(711\) 3.62258e11 1.55310e11i 1.41755 0.607745i
\(712\) 0 0
\(713\) 1.60280e11i 0.620186i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 2.48583e10 + 1.21067e11i 0.0940578 + 0.458089i
\(718\) 0 0
\(719\) 1.91891e11i 0.718022i −0.933333 0.359011i \(-0.883114\pi\)
0.933333 0.359011i \(-0.116886\pi\)
\(720\) 0 0
\(721\) −3.13753e11 −1.16104
\(722\) 0 0
\(723\) 2.04004e11 4.18875e10i 0.746596 0.153296i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.64178e11 1.30370 0.651848 0.758349i \(-0.273994\pi\)
0.651848 + 0.758349i \(0.273994\pi\)
\(728\) 0 0
\(729\) 9.83615e10 2.64748e11i 0.348269 0.937395i
\(730\) 0 0
\(731\) 5.71444e11i 2.00126i
\(732\) 0 0
\(733\) −2.73816e11 −0.948511 −0.474256 0.880387i \(-0.657283\pi\)
−0.474256 + 0.880387i \(0.657283\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 1.52914e11i 0.518295i
\(738\) 0 0
\(739\) 9.00973e10 0.302088 0.151044 0.988527i \(-0.451736\pi\)
0.151044 + 0.988527i \(0.451736\pi\)
\(740\) 0 0
\(741\) 2.21228e11 4.54240e10i 0.733783 0.150665i
\(742\) 0 0
\(743\) 2.11015e11i 0.692400i −0.938161 0.346200i \(-0.887472\pi\)
0.938161 0.346200i \(-0.112528\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.03476e11 + 2.41356e11i 0.332320 + 0.775131i
\(748\) 0 0
\(749\) 6.77701e11i 2.15333i
\(750\) 0 0
\(751\) −5.45794e11 −1.71581 −0.857904 0.513809i \(-0.828234\pi\)
−0.857904 + 0.513809i \(0.828234\pi\)
\(752\) 0 0
\(753\) 7.43258e10 + 3.61988e11i 0.231185 + 1.12594i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.61963e11 0.493211 0.246605 0.969116i \(-0.420685\pi\)
0.246605 + 0.969116i \(0.420685\pi\)
\(758\) 0 0
\(759\) −3.90285e11 + 8.01359e10i −1.17602 + 0.241468i
\(760\) 0 0
\(761\) 6.37983e10i 0.190226i 0.995466 + 0.0951131i \(0.0303213\pi\)
−0.995466 + 0.0951131i \(0.969679\pi\)
\(762\) 0 0
\(763\) 7.84644e11 2.31513
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 1.74447e11i 0.504061i
\(768\) 0 0
\(769\) −5.24097e11 −1.49867 −0.749336 0.662190i \(-0.769627\pi\)
−0.749336 + 0.662190i \(0.769627\pi\)
\(770\) 0 0
\(771\) −5.25108e10 2.55743e11i −0.148604 0.723746i
\(772\) 0 0
\(773\) 6.47397e11i 1.81323i −0.421960 0.906614i \(-0.638658\pi\)
0.421960 0.906614i \(-0.361342\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 2.32151e11 4.76668e10i 0.636923 0.130777i
\(778\) 0 0
\(779\) 5.62024e11i 1.52618i
\(780\) 0 0
\(781\) −7.56149e11 −2.03237
\(782\) 0 0
\(783\) −5.63408e10 8.10358e10i −0.149891 0.215591i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −3.64667e10 −0.0950600 −0.0475300 0.998870i \(-0.515135\pi\)
−0.0475300 + 0.998870i \(0.515135\pi\)
\(788\) 0 0
\(789\) 5.31022e8 + 2.58623e9i 0.00137026 + 0.00667358i
\(790\) 0 0
\(791\) 9.89007e10i 0.252635i
\(792\) 0 0
\(793\) 6.51554e10 0.164762
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.73965e11i 0.431150i 0.976487 + 0.215575i \(0.0691626\pi\)
−0.976487 + 0.215575i \(0.930837\pi\)
\(798\) 0 0
\(799\) −8.83462e11 −2.16771
\(800\) 0 0
\(801\) −1.16023e11 2.70621e11i −0.281846 0.657401i
\(802\) 0 0
\(803\) 2.84695e11i 0.684727i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −4.16660e10 2.02926e11i −0.0982400 0.478457i
\(808\) 0 0
\(809\) 2.21348e11i 0.516751i 0.966045 + 0.258376i \(0.0831872\pi\)
−0.966045 + 0.258376i \(0.916813\pi\)
\(810\) 0 0
\(811\) 5.02335e11 1.16121 0.580604 0.814186i \(-0.302817\pi\)
0.580604 + 0.814186i \(0.302817\pi\)
\(812\) 0 0
\(813\) −5.74841e11 + 1.18030e11i −1.31579 + 0.270166i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 4.97175e11 1.11589
\(818\) 0 0
\(819\) 5.16988e11 2.21647e11i 1.14907 0.492637i
\(820\) 0 0
\(821\) 3.69491e11i 0.813264i −0.913592 0.406632i \(-0.866703\pi\)
0.913592 0.406632i \(-0.133297\pi\)
\(822\) 0 0
\(823\) 3.33488e11 0.726910 0.363455 0.931612i \(-0.381597\pi\)
0.363455 + 0.931612i \(0.381597\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 3.18933e11i 0.681832i 0.940094 + 0.340916i \(0.110737\pi\)
−0.940094 + 0.340916i \(0.889263\pi\)
\(828\) 0 0
\(829\) 5.18645e11 1.09813 0.549063 0.835781i \(-0.314985\pi\)
0.549063 + 0.835781i \(0.314985\pi\)
\(830\) 0 0
\(831\) −3.40658e11 + 6.99462e10i −0.714356 + 0.146676i
\(832\) 0 0
\(833\) 7.42546e11i 1.54221i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −3.49395e11 + 2.42920e11i −0.711893 + 0.494949i
\(838\) 0 0
\(839\) 9.44187e11i 1.90551i −0.303747 0.952753i \(-0.598238\pi\)
0.303747 0.952753i \(-0.401762\pi\)
\(840\) 0 0
\(841\) 4.65756e11 0.931053
\(842\) 0 0
\(843\) −2.95135e10 1.43739e11i −0.0584400 0.284620i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.32651e12 2.57737
\(848\) 0 0
\(849\) −2.30815e11 + 4.73925e10i −0.444256 + 0.0912177i
\(850\) 0 0
\(851\) 1.71971e11i 0.327896i
\(852\) 0 0
\(853\) 1.68425e11 0.318134 0.159067 0.987268i \(-0.449151\pi\)
0.159067 + 0.987268i \(0.449151\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 6.34379e11i 1.17605i −0.808843 0.588025i \(-0.799906\pi\)
0.808843 0.588025i \(-0.200094\pi\)
\(858\) 0 0
\(859\) 5.11818e11 0.940033 0.470016 0.882658i \(-0.344248\pi\)
0.470016 + 0.882658i \(0.344248\pi\)
\(860\) 0 0
\(861\) 2.81544e11 + 1.37120e12i 0.512312 + 2.49511i
\(862\) 0 0
\(863\) 2.33148e11i 0.420329i 0.977666 + 0.210164i \(0.0673999\pi\)
−0.977666 + 0.210164i \(0.932600\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 7.32296e11 1.50360e11i 1.29602 0.266107i
\(868\) 0 0
\(869\) 1.47625e12i 2.58870i
\(870\) 0 0
\(871\) −1.56652e11 −0.272185
\(872\) 0 0
\(873\) −3.94492e11 + 1.69130e11i −0.679175 + 0.291181i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.06198e11 −0.179522 −0.0897611 0.995963i \(-0.528610\pi\)
−0.0897611 + 0.995963i \(0.528610\pi\)
\(878\) 0 0
\(879\) −2.02024e10 9.83913e10i −0.0338413 0.164817i
\(880\) 0 0
\(881\) 1.10796e12i 1.83916i −0.392902 0.919580i \(-0.628529\pi\)
0.392902 0.919580i \(-0.371471\pi\)
\(882\) 0 0
\(883\) 4.28093e11 0.704198 0.352099 0.935963i \(-0.385468\pi\)
0.352099 + 0.935963i \(0.385468\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 4.09034e11i 0.660793i 0.943842 + 0.330396i \(0.107182\pi\)
−0.943842 + 0.330396i \(0.892818\pi\)
\(888\) 0 0
\(889\) 1.20748e12 1.93318
\(890\) 0 0
\(891\) 7.66202e11 + 7.29329e11i 1.21572 + 1.15721i
\(892\) 0 0
\(893\) 7.68640e11i 1.20870i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −8.20949e10 3.99826e11i −0.126808 0.617591i
\(898\) 0 0
\(899\) 1.48709e11i 0.227666i
\(900\) 0 0
\(901\) 4.63445e11 0.703233
\(902\) 0 0
\(903\) 1.21299e12 2.49058e11i 1.82434 0.374585i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 6.86396e11 1.01425 0.507126 0.861872i \(-0.330708\pi\)
0.507126 + 0.861872i \(0.330708\pi\)
\(908\) 0 0
\(909\) −7.83399e8 1.82726e9i −0.00114743 0.00267637i
\(910\) 0 0
\(911\) 6.11887e11i 0.888378i 0.895933 + 0.444189i \(0.146508\pi\)
−0.895933 + 0.444189i \(0.853492\pi\)
\(912\) 0 0
\(913\) −9.83558e11 −1.41552
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.16409e11i 0.871749i
\(918\) 0 0
\(919\) −4.23039e11 −0.593087 −0.296544 0.955019i \(-0.595834\pi\)
−0.296544 + 0.955019i \(0.595834\pi\)
\(920\) 0 0
\(921\) −2.91029e11 + 5.97561e10i −0.404481 + 0.0830507i
\(922\) 0 0
\(923\) 7.74634e11i 1.06731i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 5.55557e11 2.38183e11i 0.752332 0.322546i
\(928\) 0 0
\(929\) 1.56450e11i 0.210045i −0.994470 0.105022i \(-0.966509\pi\)
0.994470 0.105022i \(-0.0334914\pi\)
\(930\) 0 0
\(931\) 6.46039e11 0.859923
\(932\) 0 0
\(933\) 2.03920e9 + 9.93149e9i 0.00269112 + 0.0131065i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.01239e12 1.31338 0.656691 0.754160i \(-0.271956\pi\)
0.656691 + 0.754160i \(0.271956\pi\)
\(938\) 0 0
\(939\) 4.35945e11 8.95112e10i 0.560751 0.115137i
\(940\) 0 0
\(941\) 1.25449e12i 1.59996i −0.600026 0.799980i \(-0.704843\pi\)
0.600026 0.799980i \(-0.295157\pi\)
\(942\) 0 0
\(943\) 1.01575e12 1.28451
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.29119e12i 1.60543i 0.596365 + 0.802713i \(0.296611\pi\)
−0.596365 + 0.802713i \(0.703389\pi\)
\(948\) 0 0
\(949\) −2.91655e11 −0.359587
\(950\) 0 0
\(951\) −1.94237e11 9.45991e11i −0.237471 1.15655i
\(952\) 0 0
\(953\) 1.35954e12i 1.64824i −0.566418 0.824118i \(-0.691671\pi\)
0.566418 0.824118i \(-0.308329\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 3.62108e11 7.43505e10i 0.431709 0.0886413i
\(958\) 0 0
\(959\) 1.38409e12i 1.63641i
\(960\) 0 0
\(961\) −2.11716e11 −0.248233
\(962\) 0 0
\(963\) 5.14470e11 + 1.19999e12i 0.598212 + 1.39532i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.18542e12 −1.35571 −0.677855 0.735195i \(-0.737090\pi\)
−0.677855 + 0.735195i \(0.737090\pi\)
\(968\) 0 0
\(969\) −2.29694e11 1.11867e12i −0.260528 1.26885i
\(970\) 0 0
\(971\) 2.30844e11i 0.259682i 0.991535 + 0.129841i \(0.0414467\pi\)
−0.991535 + 0.129841i \(0.958553\pi\)
\(972\) 0 0
\(973\) 5.02561e10 0.0560709
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.05133e12i 1.15388i 0.816785 + 0.576942i \(0.195754\pi\)
−0.816785 + 0.576942i \(0.804246\pi\)
\(978\) 0 0
\(979\) 1.10282e12 1.20053
\(980\) 0 0
\(981\) −1.38935e12 + 5.95655e11i −1.50016 + 0.643160i
\(982\) 0 0
\(983\) 1.65893e11i 0.177669i −0.996046 0.0888347i \(-0.971686\pi\)
0.996046 0.0888347i \(-0.0283143\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 3.85048e11 + 1.87530e12i 0.405739 + 1.97607i
\(988\) 0 0
\(989\) 8.98545e11i 0.939193i
\(990\) 0 0
\(991\) −4.77605e11 −0.495192 −0.247596 0.968863i \(-0.579641\pi\)
−0.247596 + 0.968863i \(0.579641\pi\)
\(992\) 0 0
\(993\) 1.05688e12 2.17006e11i 1.08700 0.223190i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 3.99520e11 0.404350 0.202175 0.979349i \(-0.435199\pi\)
0.202175 + 0.979349i \(0.435199\pi\)
\(998\) 0 0
\(999\) −3.74880e11 + 2.60638e11i −0.376383 + 0.261683i
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 300.9.g.h.101.2 16
3.2 odd 2 inner 300.9.g.h.101.1 16
5.2 odd 4 60.9.b.a.29.10 yes 16
5.3 odd 4 60.9.b.a.29.7 16
5.4 even 2 inner 300.9.g.h.101.15 16
15.2 even 4 60.9.b.a.29.8 yes 16
15.8 even 4 60.9.b.a.29.9 yes 16
15.14 odd 2 inner 300.9.g.h.101.16 16
20.3 even 4 240.9.c.d.209.10 16
20.7 even 4 240.9.c.d.209.7 16
60.23 odd 4 240.9.c.d.209.8 16
60.47 odd 4 240.9.c.d.209.9 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.9.b.a.29.7 16 5.3 odd 4
60.9.b.a.29.8 yes 16 15.2 even 4
60.9.b.a.29.9 yes 16 15.8 even 4
60.9.b.a.29.10 yes 16 5.2 odd 4
240.9.c.d.209.7 16 20.7 even 4
240.9.c.d.209.8 16 60.23 odd 4
240.9.c.d.209.9 16 60.47 odd 4
240.9.c.d.209.10 16 20.3 even 4
300.9.g.h.101.1 16 3.2 odd 2 inner
300.9.g.h.101.2 16 1.1 even 1 trivial
300.9.g.h.101.15 16 5.4 even 2 inner
300.9.g.h.101.16 16 15.14 odd 2 inner