Properties

Label 18.9.b.a.17.2
Level $18$
Weight $9$
Character 18.17
Analytic conductor $7.333$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [18,9,Mod(17,18)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(18, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("18.17");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 18 = 2 \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 18.b (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: \(7.33281498110\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-2}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 17.2
Root \(1.41421i\) of defining polynomial
Character \(\chi\) \(=\) 18.17
Dual form 18.9.b.a.17.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+11.3137i q^{2} -128.000 q^{4} +233.345i q^{5} -3532.00 q^{7} -1448.15i q^{8} +O(q^{10})\) \(q+11.3137i q^{2} -128.000 q^{4} +233.345i q^{5} -3532.00 q^{7} -1448.15i q^{8} -2640.00 q^{10} -20178.0i q^{11} -41824.0 q^{13} -39960.0i q^{14} +16384.0 q^{16} +94784.8i q^{17} -36304.0 q^{19} -29868.2i q^{20} +228288. q^{22} +413624. i q^{23} +336175. q^{25} -473185. i q^{26} +452096. q^{28} +269191. i q^{29} -471196. q^{31} +185364. i q^{32} -1.07237e6 q^{34} -824175. i q^{35} -3.00740e6 q^{37} -410733. i q^{38} +337920. q^{40} -1.71534e6i q^{41} +3.62372e6 q^{43} +2.58278e6i q^{44} -4.67962e6 q^{46} -6.01462e6i q^{47} +6.71022e6 q^{49} +3.80339e6i q^{50} +5.35347e6 q^{52} +1.02767e7i q^{53} +4.70844e6 q^{55} +5.11488e6i q^{56} -3.04555e6 q^{58} +2.68810e6i q^{59} -5.44063e6 q^{61} -5.33097e6i q^{62} -2.09715e6 q^{64} -9.75943e6i q^{65} -6.12158e6 q^{67} -1.21325e7i q^{68} +9.32448e6 q^{70} -2.11941e7i q^{71} -4.90312e7 q^{73} -3.40249e7i q^{74} +4.64691e6 q^{76} +7.12687e7i q^{77} +8.35776e6 q^{79} +3.82313e6i q^{80} +1.94068e7 q^{82} -5.13918e7i q^{83} -2.21176e7 q^{85} +4.09977e7i q^{86} -2.92209e7 q^{88} +1.07337e8i q^{89} +1.47722e8 q^{91} -5.29438e7i q^{92} +6.80477e7 q^{94} -8.47137e6i q^{95} +2.04313e7 q^{97} +7.59175e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 256 q^{4} - 7064 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 256 q^{4} - 7064 q^{7} - 5280 q^{10} - 83648 q^{13} + 32768 q^{16} - 72608 q^{19} + 456576 q^{22} + 672350 q^{25} + 904192 q^{28} - 942392 q^{31} - 2144736 q^{34} - 6014804 q^{37} + 675840 q^{40} + 7247440 q^{43} - 9359232 q^{46} + 13420446 q^{49} + 10706944 q^{52} + 9416880 q^{55} - 6091104 q^{58} - 10881260 q^{61} - 4194304 q^{64} - 12243152 q^{67} + 18648960 q^{70} - 98062304 q^{73} + 9293824 q^{76} + 16715512 q^{79} + 38813664 q^{82} - 44235180 q^{85} - 58441728 q^{88} + 295444736 q^{91} + 136095360 q^{94} + 40862656 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/18\mathbb{Z}\right)^\times\).

\(n\) \(11\)
\(\chi(n)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 11.3137i 0.707107i
\(3\) 0 0
\(4\) −128.000 −0.500000
\(5\) 233.345i 0.373352i 0.982421 + 0.186676i \(0.0597715\pi\)
−0.982421 + 0.186676i \(0.940228\pi\)
\(6\) 0 0
\(7\) −3532.00 −1.47105 −0.735527 0.677496i \(-0.763065\pi\)
−0.735527 + 0.677496i \(0.763065\pi\)
\(8\) − 1448.15i − 0.353553i
\(9\) 0 0
\(10\) −2640.00 −0.264000
\(11\) − 20178.0i − 1.37818i −0.724674 0.689092i \(-0.758009\pi\)
0.724674 0.689092i \(-0.241991\pi\)
\(12\) 0 0
\(13\) −41824.0 −1.46437 −0.732187 0.681103i \(-0.761500\pi\)
−0.732187 + 0.681103i \(0.761500\pi\)
\(14\) − 39960.0i − 1.04019i
\(15\) 0 0
\(16\) 16384.0 0.250000
\(17\) 94784.8i 1.13486i 0.823421 + 0.567431i \(0.192063\pi\)
−0.823421 + 0.567431i \(0.807937\pi\)
\(18\) 0 0
\(19\) −36304.0 −0.278574 −0.139287 0.990252i \(-0.544481\pi\)
−0.139287 + 0.990252i \(0.544481\pi\)
\(20\) − 29868.2i − 0.186676i
\(21\) 0 0
\(22\) 228288. 0.974524
\(23\) 413624.i 1.47807i 0.673669 + 0.739033i \(0.264717\pi\)
−0.673669 + 0.739033i \(0.735283\pi\)
\(24\) 0 0
\(25\) 336175. 0.860608
\(26\) − 473185.i − 1.03547i
\(27\) 0 0
\(28\) 452096. 0.735527
\(29\) 269191.i 0.380600i 0.981726 + 0.190300i \(0.0609461\pi\)
−0.981726 + 0.190300i \(0.939054\pi\)
\(30\) 0 0
\(31\) −471196. −0.510217 −0.255108 0.966912i \(-0.582111\pi\)
−0.255108 + 0.966912i \(0.582111\pi\)
\(32\) 185364.i 0.176777i
\(33\) 0 0
\(34\) −1.07237e6 −0.802469
\(35\) − 824175.i − 0.549221i
\(36\) 0 0
\(37\) −3.00740e6 −1.60467 −0.802333 0.596877i \(-0.796408\pi\)
−0.802333 + 0.596877i \(0.796408\pi\)
\(38\) − 410733.i − 0.196981i
\(39\) 0 0
\(40\) 337920. 0.132000
\(41\) − 1.71534e6i − 0.607036i −0.952826 0.303518i \(-0.901839\pi\)
0.952826 0.303518i \(-0.0981612\pi\)
\(42\) 0 0
\(43\) 3.62372e6 1.05994 0.529969 0.848017i \(-0.322203\pi\)
0.529969 + 0.848017i \(0.322203\pi\)
\(44\) 2.58278e6i 0.689092i
\(45\) 0 0
\(46\) −4.67962e6 −1.04515
\(47\) − 6.01462e6i − 1.23259i −0.787517 0.616293i \(-0.788634\pi\)
0.787517 0.616293i \(-0.211366\pi\)
\(48\) 0 0
\(49\) 6.71022e6 1.16400
\(50\) 3.80339e6i 0.608542i
\(51\) 0 0
\(52\) 5.35347e6 0.732187
\(53\) 1.02767e7i 1.30241i 0.758901 + 0.651206i \(0.225737\pi\)
−0.758901 + 0.651206i \(0.774263\pi\)
\(54\) 0 0
\(55\) 4.70844e6 0.514548
\(56\) 5.11488e6i 0.520096i
\(57\) 0 0
\(58\) −3.04555e6 −0.269125
\(59\) 2.68810e6i 0.221839i 0.993829 + 0.110919i \(0.0353796\pi\)
−0.993829 + 0.110919i \(0.964620\pi\)
\(60\) 0 0
\(61\) −5.44063e6 −0.392943 −0.196472 0.980510i \(-0.562948\pi\)
−0.196472 + 0.980510i \(0.562948\pi\)
\(62\) − 5.33097e6i − 0.360778i
\(63\) 0 0
\(64\) −2.09715e6 −0.125000
\(65\) − 9.75943e6i − 0.546728i
\(66\) 0 0
\(67\) −6.12158e6 −0.303783 −0.151892 0.988397i \(-0.548536\pi\)
−0.151892 + 0.988397i \(0.548536\pi\)
\(68\) − 1.21325e7i − 0.567431i
\(69\) 0 0
\(70\) 9.32448e6 0.388358
\(71\) − 2.11941e7i − 0.834029i −0.908900 0.417014i \(-0.863076\pi\)
0.908900 0.417014i \(-0.136924\pi\)
\(72\) 0 0
\(73\) −4.90312e7 −1.72656 −0.863278 0.504729i \(-0.831593\pi\)
−0.863278 + 0.504729i \(0.831593\pi\)
\(74\) − 3.40249e7i − 1.13467i
\(75\) 0 0
\(76\) 4.64691e6 0.139287
\(77\) 7.12687e7i 2.02738i
\(78\) 0 0
\(79\) 8.35776e6 0.214576 0.107288 0.994228i \(-0.465783\pi\)
0.107288 + 0.994228i \(0.465783\pi\)
\(80\) 3.82313e6i 0.0933381i
\(81\) 0 0
\(82\) 1.94068e7 0.429239
\(83\) − 5.13918e7i − 1.08288i −0.840738 0.541441i \(-0.817879\pi\)
0.840738 0.541441i \(-0.182121\pi\)
\(84\) 0 0
\(85\) −2.21176e7 −0.423704
\(86\) 4.09977e7i 0.749490i
\(87\) 0 0
\(88\) −2.92209e7 −0.487262
\(89\) 1.07337e8i 1.71076i 0.517997 + 0.855382i \(0.326678\pi\)
−0.517997 + 0.855382i \(0.673322\pi\)
\(90\) 0 0
\(91\) 1.47722e8 2.15417
\(92\) − 5.29438e7i − 0.739033i
\(93\) 0 0
\(94\) 6.80477e7 0.871569
\(95\) − 8.47137e6i − 0.104006i
\(96\) 0 0
\(97\) 2.04313e7 0.230786 0.115393 0.993320i \(-0.463187\pi\)
0.115393 + 0.993320i \(0.463187\pi\)
\(98\) 7.59175e7i 0.823072i
\(99\) 0 0
\(100\) −4.30304e7 −0.430304
\(101\) − 1.69583e8i − 1.62966i −0.579700 0.814830i \(-0.696830\pi\)
0.579700 0.814830i \(-0.303170\pi\)
\(102\) 0 0
\(103\) −2.98252e7 −0.264993 −0.132497 0.991183i \(-0.542299\pi\)
−0.132497 + 0.991183i \(0.542299\pi\)
\(104\) 6.05676e7i 0.517735i
\(105\) 0 0
\(106\) −1.16267e8 −0.920945
\(107\) − 1.22823e8i − 0.937014i −0.883460 0.468507i \(-0.844792\pi\)
0.883460 0.468507i \(-0.155208\pi\)
\(108\) 0 0
\(109\) −4.88844e7 −0.346310 −0.173155 0.984895i \(-0.555396\pi\)
−0.173155 + 0.984895i \(0.555396\pi\)
\(110\) 5.32699e7i 0.363841i
\(111\) 0 0
\(112\) −5.78683e7 −0.367763
\(113\) 1.89245e8i 1.16068i 0.814376 + 0.580338i \(0.197080\pi\)
−0.814376 + 0.580338i \(0.802920\pi\)
\(114\) 0 0
\(115\) −9.65171e7 −0.551840
\(116\) − 3.44565e7i − 0.190300i
\(117\) 0 0
\(118\) −3.04124e7 −0.156864
\(119\) − 3.34780e8i − 1.66944i
\(120\) 0 0
\(121\) −1.92793e8 −0.899392
\(122\) − 6.15537e7i − 0.277853i
\(123\) 0 0
\(124\) 6.03131e7 0.255108
\(125\) 1.69595e8i 0.694662i
\(126\) 0 0
\(127\) 3.39908e8 1.30661 0.653306 0.757094i \(-0.273381\pi\)
0.653306 + 0.757094i \(0.273381\pi\)
\(128\) − 2.37266e7i − 0.0883883i
\(129\) 0 0
\(130\) 1.10415e8 0.386595
\(131\) 7.69237e7i 0.261201i 0.991435 + 0.130600i \(0.0416905\pi\)
−0.991435 + 0.130600i \(0.958309\pi\)
\(132\) 0 0
\(133\) 1.28226e8 0.409797
\(134\) − 6.92577e7i − 0.214807i
\(135\) 0 0
\(136\) 1.37263e8 0.401234
\(137\) 1.27859e8i 0.362952i 0.983395 + 0.181476i \(0.0580875\pi\)
−0.983395 + 0.181476i \(0.941913\pi\)
\(138\) 0 0
\(139\) 9.08929e6 0.0243484 0.0121742 0.999926i \(-0.496125\pi\)
0.0121742 + 0.999926i \(0.496125\pi\)
\(140\) 1.05494e8i 0.274611i
\(141\) 0 0
\(142\) 2.39784e8 0.589748
\(143\) 8.43925e8i 2.01818i
\(144\) 0 0
\(145\) −6.28145e7 −0.142098
\(146\) − 5.54724e8i − 1.22086i
\(147\) 0 0
\(148\) 3.84947e8 0.802333
\(149\) 1.85204e8i 0.375755i 0.982193 + 0.187877i \(0.0601607\pi\)
−0.982193 + 0.187877i \(0.939839\pi\)
\(150\) 0 0
\(151\) −3.35601e8 −0.645529 −0.322764 0.946479i \(-0.604612\pi\)
−0.322764 + 0.946479i \(0.604612\pi\)
\(152\) 5.25738e7i 0.0984907i
\(153\) 0 0
\(154\) −8.06313e8 −1.43358
\(155\) − 1.09951e8i − 0.190491i
\(156\) 0 0
\(157\) −2.01631e8 −0.331863 −0.165931 0.986137i \(-0.553063\pi\)
−0.165931 + 0.986137i \(0.553063\pi\)
\(158\) 9.45572e7i 0.151728i
\(159\) 0 0
\(160\) −4.32538e7 −0.0660000
\(161\) − 1.46092e9i − 2.17431i
\(162\) 0 0
\(163\) −5.27661e8 −0.747488 −0.373744 0.927532i \(-0.621926\pi\)
−0.373744 + 0.927532i \(0.621926\pi\)
\(164\) 2.19563e8i 0.303518i
\(165\) 0 0
\(166\) 5.81432e8 0.765714
\(167\) − 1.81615e8i − 0.233500i −0.993161 0.116750i \(-0.962752\pi\)
0.993161 0.116750i \(-0.0372476\pi\)
\(168\) 0 0
\(169\) 9.33516e8 1.14439
\(170\) − 2.50232e8i − 0.299604i
\(171\) 0 0
\(172\) −4.63836e8 −0.529969
\(173\) − 7.06460e7i − 0.0788684i −0.999222 0.0394342i \(-0.987444\pi\)
0.999222 0.0394342i \(-0.0125556\pi\)
\(174\) 0 0
\(175\) −1.18737e9 −1.26600
\(176\) − 3.30596e8i − 0.344546i
\(177\) 0 0
\(178\) −1.21438e9 −1.20969
\(179\) − 7.90371e7i − 0.0769873i −0.999259 0.0384936i \(-0.987744\pi\)
0.999259 0.0384936i \(-0.0122559\pi\)
\(180\) 0 0
\(181\) 5.48168e8 0.510739 0.255370 0.966844i \(-0.417803\pi\)
0.255370 + 0.966844i \(0.417803\pi\)
\(182\) 1.67129e9i 1.52323i
\(183\) 0 0
\(184\) 5.98991e8 0.522575
\(185\) − 7.01763e8i − 0.599106i
\(186\) 0 0
\(187\) 1.91257e9 1.56405
\(188\) 7.69872e8i 0.616293i
\(189\) 0 0
\(190\) 9.58426e7 0.0735435
\(191\) 2.24597e9i 1.68761i 0.536653 + 0.843803i \(0.319688\pi\)
−0.536653 + 0.843803i \(0.680312\pi\)
\(192\) 0 0
\(193\) 6.55575e8 0.472491 0.236245 0.971693i \(-0.424083\pi\)
0.236245 + 0.971693i \(0.424083\pi\)
\(194\) 2.31154e8i 0.163190i
\(195\) 0 0
\(196\) −8.58909e8 −0.582000
\(197\) − 4.48231e8i − 0.297603i −0.988867 0.148801i \(-0.952458\pi\)
0.988867 0.148801i \(-0.0475415\pi\)
\(198\) 0 0
\(199\) 7.34930e8 0.468634 0.234317 0.972160i \(-0.424715\pi\)
0.234317 + 0.972160i \(0.424715\pi\)
\(200\) − 4.86833e8i − 0.304271i
\(201\) 0 0
\(202\) 1.91861e9 1.15234
\(203\) − 9.50784e8i − 0.559883i
\(204\) 0 0
\(205\) 4.00266e8 0.226638
\(206\) − 3.37434e8i − 0.187379i
\(207\) 0 0
\(208\) −6.85244e8 −0.366094
\(209\) 7.32542e8i 0.383926i
\(210\) 0 0
\(211\) −3.26800e9 −1.64874 −0.824370 0.566051i \(-0.808471\pi\)
−0.824370 + 0.566051i \(0.808471\pi\)
\(212\) − 1.31541e9i − 0.651206i
\(213\) 0 0
\(214\) 1.38959e9 0.662569
\(215\) 8.45578e8i 0.395731i
\(216\) 0 0
\(217\) 1.66426e9 0.750556
\(218\) − 5.53064e8i − 0.244878i
\(219\) 0 0
\(220\) −6.02680e8 −0.257274
\(221\) − 3.96428e9i − 1.66186i
\(222\) 0 0
\(223\) −3.80841e9 −1.54001 −0.770005 0.638037i \(-0.779746\pi\)
−0.770005 + 0.638037i \(0.779746\pi\)
\(224\) − 6.54705e8i − 0.260048i
\(225\) 0 0
\(226\) −2.14107e9 −0.820722
\(227\) 4.34651e9i 1.63696i 0.574537 + 0.818478i \(0.305182\pi\)
−0.574537 + 0.818478i \(0.694818\pi\)
\(228\) 0 0
\(229\) −2.86397e9 −1.04142 −0.520711 0.853733i \(-0.674333\pi\)
−0.520711 + 0.853733i \(0.674333\pi\)
\(230\) − 1.09197e9i − 0.390209i
\(231\) 0 0
\(232\) 3.89831e8 0.134563
\(233\) 5.57102e8i 0.189022i 0.995524 + 0.0945108i \(0.0301287\pi\)
−0.995524 + 0.0945108i \(0.969871\pi\)
\(234\) 0 0
\(235\) 1.40348e9 0.460189
\(236\) − 3.44077e8i − 0.110919i
\(237\) 0 0
\(238\) 3.78760e9 1.18047
\(239\) 1.30831e9i 0.400977i 0.979696 + 0.200488i \(0.0642529\pi\)
−0.979696 + 0.200488i \(0.935747\pi\)
\(240\) 0 0
\(241\) −4.84182e9 −1.43529 −0.717647 0.696407i \(-0.754781\pi\)
−0.717647 + 0.696407i \(0.754781\pi\)
\(242\) − 2.18120e9i − 0.635967i
\(243\) 0 0
\(244\) 6.96401e8 0.196472
\(245\) 1.56580e9i 0.434582i
\(246\) 0 0
\(247\) 1.51838e9 0.407936
\(248\) 6.82365e8i 0.180389i
\(249\) 0 0
\(250\) −1.91875e9 −0.491201
\(251\) − 7.77711e9i − 1.95940i −0.200466 0.979701i \(-0.564246\pi\)
0.200466 0.979701i \(-0.435754\pi\)
\(252\) 0 0
\(253\) 8.34610e9 2.03705
\(254\) 3.84562e9i 0.923915i
\(255\) 0 0
\(256\) 2.68435e8 0.0625000
\(257\) 5.68198e9i 1.30247i 0.758877 + 0.651234i \(0.225749\pi\)
−0.758877 + 0.651234i \(0.774251\pi\)
\(258\) 0 0
\(259\) 1.06221e10 2.36055
\(260\) 1.24921e9i 0.273364i
\(261\) 0 0
\(262\) −8.70292e8 −0.184697
\(263\) − 1.04532e9i − 0.218486i −0.994015 0.109243i \(-0.965157\pi\)
0.994015 0.109243i \(-0.0348427\pi\)
\(264\) 0 0
\(265\) −2.39801e9 −0.486259
\(266\) 1.45071e9i 0.289770i
\(267\) 0 0
\(268\) 7.83562e8 0.151892
\(269\) 4.92684e9i 0.940935i 0.882417 + 0.470467i \(0.155915\pi\)
−0.882417 + 0.470467i \(0.844085\pi\)
\(270\) 0 0
\(271\) −6.59224e9 −1.22224 −0.611119 0.791539i \(-0.709280\pi\)
−0.611119 + 0.791539i \(0.709280\pi\)
\(272\) 1.55295e9i 0.283716i
\(273\) 0 0
\(274\) −1.44656e9 −0.256646
\(275\) − 6.78334e9i − 1.18608i
\(276\) 0 0
\(277\) −1.21590e8 −0.0206528 −0.0103264 0.999947i \(-0.503287\pi\)
−0.0103264 + 0.999947i \(0.503287\pi\)
\(278\) 1.02834e8i 0.0172169i
\(279\) 0 0
\(280\) −1.19353e9 −0.194179
\(281\) − 5.60638e9i − 0.899203i −0.893229 0.449601i \(-0.851566\pi\)
0.893229 0.449601i \(-0.148434\pi\)
\(282\) 0 0
\(283\) 1.62560e9 0.253435 0.126718 0.991939i \(-0.459556\pi\)
0.126718 + 0.991939i \(0.459556\pi\)
\(284\) 2.71284e9i 0.417014i
\(285\) 0 0
\(286\) −9.54792e9 −1.42707
\(287\) 6.05857e9i 0.892982i
\(288\) 0 0
\(289\) −2.00841e9 −0.287912
\(290\) − 7.10665e8i − 0.100478i
\(291\) 0 0
\(292\) 6.27599e9 0.863278
\(293\) − 4.21249e9i − 0.571569i −0.958294 0.285784i \(-0.907746\pi\)
0.958294 0.285784i \(-0.0922542\pi\)
\(294\) 0 0
\(295\) −6.27256e8 −0.0828241
\(296\) 4.35518e9i 0.567335i
\(297\) 0 0
\(298\) −2.09534e9 −0.265699
\(299\) − 1.72994e10i − 2.16444i
\(300\) 0 0
\(301\) −1.27990e10 −1.55923
\(302\) − 3.79689e9i − 0.456458i
\(303\) 0 0
\(304\) −5.94805e8 −0.0696434
\(305\) − 1.26955e9i − 0.146706i
\(306\) 0 0
\(307\) 5.88475e9 0.662483 0.331241 0.943546i \(-0.392533\pi\)
0.331241 + 0.943546i \(0.392533\pi\)
\(308\) − 9.12239e9i − 1.01369i
\(309\) 0 0
\(310\) 1.24396e9 0.134697
\(311\) 9.80006e9i 1.04758i 0.851847 + 0.523790i \(0.175482\pi\)
−0.851847 + 0.523790i \(0.824518\pi\)
\(312\) 0 0
\(313\) 3.76162e9 0.391920 0.195960 0.980612i \(-0.437218\pi\)
0.195960 + 0.980612i \(0.437218\pi\)
\(314\) − 2.28119e9i − 0.234662i
\(315\) 0 0
\(316\) −1.06979e9 −0.107288
\(317\) 1.69349e10i 1.67704i 0.544868 + 0.838522i \(0.316580\pi\)
−0.544868 + 0.838522i \(0.683420\pi\)
\(318\) 0 0
\(319\) 5.43174e9 0.524537
\(320\) − 4.89360e8i − 0.0466690i
\(321\) 0 0
\(322\) 1.65284e10 1.53747
\(323\) − 3.44107e9i − 0.316143i
\(324\) 0 0
\(325\) −1.40602e10 −1.26025
\(326\) − 5.96980e9i − 0.528554i
\(327\) 0 0
\(328\) −2.48407e9 −0.214620
\(329\) 2.12436e10i 1.81320i
\(330\) 0 0
\(331\) 3.18998e9 0.265752 0.132876 0.991133i \(-0.457579\pi\)
0.132876 + 0.991133i \(0.457579\pi\)
\(332\) 6.57815e9i 0.541441i
\(333\) 0 0
\(334\) 2.05474e9 0.165109
\(335\) − 1.42844e9i − 0.113418i
\(336\) 0 0
\(337\) 1.66378e10 1.28996 0.644978 0.764201i \(-0.276866\pi\)
0.644978 + 0.764201i \(0.276866\pi\)
\(338\) 1.05615e10i 0.809208i
\(339\) 0 0
\(340\) 2.83105e9 0.211852
\(341\) 9.50779e9i 0.703173i
\(342\) 0 0
\(343\) −3.33923e9 −0.241251
\(344\) − 5.24771e9i − 0.374745i
\(345\) 0 0
\(346\) 7.99268e8 0.0557684
\(347\) 1.07257e9i 0.0739792i 0.999316 + 0.0369896i \(0.0117768\pi\)
−0.999316 + 0.0369896i \(0.988223\pi\)
\(348\) 0 0
\(349\) −2.32302e9 −0.156586 −0.0782928 0.996930i \(-0.524947\pi\)
−0.0782928 + 0.996930i \(0.524947\pi\)
\(350\) − 1.34336e10i − 0.895198i
\(351\) 0 0
\(352\) 3.74027e9 0.243631
\(353\) − 1.98506e10i − 1.27843i −0.769030 0.639213i \(-0.779260\pi\)
0.769030 0.639213i \(-0.220740\pi\)
\(354\) 0 0
\(355\) 4.94554e9 0.311387
\(356\) − 1.37392e10i − 0.855382i
\(357\) 0 0
\(358\) 8.94203e8 0.0544382
\(359\) 1.03679e10i 0.624187i 0.950051 + 0.312094i \(0.101030\pi\)
−0.950051 + 0.312094i \(0.898970\pi\)
\(360\) 0 0
\(361\) −1.56656e10 −0.922397
\(362\) 6.20181e9i 0.361147i
\(363\) 0 0
\(364\) −1.89085e10 −1.07709
\(365\) − 1.14412e10i − 0.644614i
\(366\) 0 0
\(367\) −2.02880e10 −1.11834 −0.559171 0.829052i \(-0.688880\pi\)
−0.559171 + 0.829052i \(0.688880\pi\)
\(368\) 6.77681e9i 0.369517i
\(369\) 0 0
\(370\) 7.93954e9 0.423632
\(371\) − 3.62972e10i − 1.91592i
\(372\) 0 0
\(373\) 2.22111e10 1.14745 0.573726 0.819047i \(-0.305497\pi\)
0.573726 + 0.819047i \(0.305497\pi\)
\(374\) 2.16382e10i 1.10595i
\(375\) 0 0
\(376\) −8.71010e9 −0.435785
\(377\) − 1.12587e10i − 0.557341i
\(378\) 0 0
\(379\) 1.23790e10 0.599967 0.299983 0.953944i \(-0.403019\pi\)
0.299983 + 0.953944i \(0.403019\pi\)
\(380\) 1.08433e9i 0.0520031i
\(381\) 0 0
\(382\) −2.54103e10 −1.19332
\(383\) 2.90368e10i 1.34944i 0.738074 + 0.674720i \(0.235735\pi\)
−0.738074 + 0.674720i \(0.764265\pi\)
\(384\) 0 0
\(385\) −1.66302e10 −0.756928
\(386\) 7.41699e9i 0.334101i
\(387\) 0 0
\(388\) −2.61521e9 −0.115393
\(389\) − 2.93738e10i − 1.28281i −0.767203 0.641404i \(-0.778352\pi\)
0.767203 0.641404i \(-0.221648\pi\)
\(390\) 0 0
\(391\) −3.92052e10 −1.67740
\(392\) − 9.71744e9i − 0.411536i
\(393\) 0 0
\(394\) 5.07116e9 0.210437
\(395\) 1.95024e9i 0.0801125i
\(396\) 0 0
\(397\) 2.58158e10 1.03926 0.519630 0.854391i \(-0.326070\pi\)
0.519630 + 0.854391i \(0.326070\pi\)
\(398\) 8.31479e9i 0.331374i
\(399\) 0 0
\(400\) 5.50789e9 0.215152
\(401\) − 6.54944e9i − 0.253295i −0.991948 0.126647i \(-0.959578\pi\)
0.991948 0.126647i \(-0.0404217\pi\)
\(402\) 0 0
\(403\) 1.97073e10 0.747149
\(404\) 2.17066e10i 0.814830i
\(405\) 0 0
\(406\) 1.07569e10 0.395897
\(407\) 6.06834e10i 2.21153i
\(408\) 0 0
\(409\) 4.41813e10 1.57887 0.789433 0.613836i \(-0.210374\pi\)
0.789433 + 0.613836i \(0.210374\pi\)
\(410\) 4.52849e9i 0.160257i
\(411\) 0 0
\(412\) 3.81763e9 0.132497
\(413\) − 9.49438e9i − 0.326337i
\(414\) 0 0
\(415\) 1.19920e10 0.404297
\(416\) − 7.75266e9i − 0.258867i
\(417\) 0 0
\(418\) −8.28777e9 −0.271477
\(419\) − 4.52526e10i − 1.46821i −0.679038 0.734103i \(-0.737603\pi\)
0.679038 0.734103i \(-0.262397\pi\)
\(420\) 0 0
\(421\) 2.19193e10 0.697747 0.348873 0.937170i \(-0.386564\pi\)
0.348873 + 0.937170i \(0.386564\pi\)
\(422\) − 3.69732e10i − 1.16584i
\(423\) 0 0
\(424\) 1.48822e10 0.460472
\(425\) 3.18643e10i 0.976672i
\(426\) 0 0
\(427\) 1.92163e10 0.578041
\(428\) 1.57214e10i 0.468507i
\(429\) 0 0
\(430\) −9.56662e9 −0.279824
\(431\) 9.62022e9i 0.278789i 0.990237 + 0.139395i \(0.0445157\pi\)
−0.990237 + 0.139395i \(0.955484\pi\)
\(432\) 0 0
\(433\) −4.64805e10 −1.32227 −0.661134 0.750268i \(-0.729924\pi\)
−0.661134 + 0.750268i \(0.729924\pi\)
\(434\) 1.88290e10i 0.530724i
\(435\) 0 0
\(436\) 6.25721e9 0.173155
\(437\) − 1.50162e10i − 0.411750i
\(438\) 0 0
\(439\) −3.72274e10 −1.00232 −0.501158 0.865356i \(-0.667092\pi\)
−0.501158 + 0.865356i \(0.667092\pi\)
\(440\) − 6.81855e9i − 0.181920i
\(441\) 0 0
\(442\) 4.48507e10 1.17511
\(443\) 5.33802e10i 1.38601i 0.720935 + 0.693003i \(0.243713\pi\)
−0.720935 + 0.693003i \(0.756287\pi\)
\(444\) 0 0
\(445\) −2.50466e10 −0.638718
\(446\) − 4.30872e10i − 1.08895i
\(447\) 0 0
\(448\) 7.40714e9 0.183882
\(449\) − 4.16287e9i − 0.102425i −0.998688 0.0512126i \(-0.983691\pi\)
0.998688 0.0512126i \(-0.0163086\pi\)
\(450\) 0 0
\(451\) −3.46121e10 −0.836607
\(452\) − 2.42234e10i − 0.580338i
\(453\) 0 0
\(454\) −4.91751e10 −1.15750
\(455\) 3.44703e10i 0.804266i
\(456\) 0 0
\(457\) −2.91975e10 −0.669393 −0.334697 0.942326i \(-0.608634\pi\)
−0.334697 + 0.942326i \(0.608634\pi\)
\(458\) − 3.24021e10i − 0.736396i
\(459\) 0 0
\(460\) 1.23542e10 0.275920
\(461\) − 1.65767e10i − 0.367025i −0.983017 0.183512i \(-0.941253\pi\)
0.983017 0.183512i \(-0.0587467\pi\)
\(462\) 0 0
\(463\) −3.66847e10 −0.798291 −0.399146 0.916888i \(-0.630693\pi\)
−0.399146 + 0.916888i \(0.630693\pi\)
\(464\) 4.41043e9i 0.0951501i
\(465\) 0 0
\(466\) −6.30289e9 −0.133658
\(467\) − 1.80368e10i − 0.379220i −0.981860 0.189610i \(-0.939278\pi\)
0.981860 0.189610i \(-0.0607223\pi\)
\(468\) 0 0
\(469\) 2.16214e10 0.446882
\(470\) 1.58786e10i 0.325402i
\(471\) 0 0
\(472\) 3.89279e9 0.0784319
\(473\) − 7.31194e10i − 1.46079i
\(474\) 0 0
\(475\) −1.22045e10 −0.239743
\(476\) 4.28518e10i 0.834722i
\(477\) 0 0
\(478\) −1.48018e10 −0.283533
\(479\) 2.85119e9i 0.0541606i 0.999633 + 0.0270803i \(0.00862099\pi\)
−0.999633 + 0.0270803i \(0.991379\pi\)
\(480\) 0 0
\(481\) 1.25782e11 2.34983
\(482\) − 5.47790e10i − 1.01491i
\(483\) 0 0
\(484\) 2.46775e10 0.449696
\(485\) 4.76755e9i 0.0861645i
\(486\) 0 0
\(487\) −1.01307e11 −1.80104 −0.900522 0.434810i \(-0.856816\pi\)
−0.900522 + 0.434810i \(0.856816\pi\)
\(488\) 7.87887e9i 0.138926i
\(489\) 0 0
\(490\) −1.77150e10 −0.307296
\(491\) − 1.17469e10i − 0.202114i −0.994881 0.101057i \(-0.967778\pi\)
0.994881 0.101057i \(-0.0322225\pi\)
\(492\) 0 0
\(493\) −2.55153e10 −0.431929
\(494\) 1.71785e10i 0.288454i
\(495\) 0 0
\(496\) −7.72008e9 −0.127554
\(497\) 7.48575e10i 1.22690i
\(498\) 0 0
\(499\) 1.14667e11 1.84942 0.924708 0.380676i \(-0.124309\pi\)
0.924708 + 0.380676i \(0.124309\pi\)
\(500\) − 2.17082e10i − 0.347331i
\(501\) 0 0
\(502\) 8.79880e10 1.38551
\(503\) − 1.63604e10i − 0.255577i −0.991801 0.127789i \(-0.959212\pi\)
0.991801 0.127789i \(-0.0407879\pi\)
\(504\) 0 0
\(505\) 3.95714e10 0.608437
\(506\) 9.44253e10i 1.44041i
\(507\) 0 0
\(508\) −4.35083e10 −0.653306
\(509\) 3.92710e10i 0.585061i 0.956256 + 0.292531i \(0.0944973\pi\)
−0.956256 + 0.292531i \(0.905503\pi\)
\(510\) 0 0
\(511\) 1.73178e11 2.53986
\(512\) 3.03700e9i 0.0441942i
\(513\) 0 0
\(514\) −6.42843e10 −0.920984
\(515\) − 6.95958e9i − 0.0989359i
\(516\) 0 0
\(517\) −1.21363e11 −1.69873
\(518\) 1.20176e11i 1.66916i
\(519\) 0 0
\(520\) −1.41332e10 −0.193297
\(521\) − 3.71996e10i − 0.504879i −0.967613 0.252439i \(-0.918767\pi\)
0.967613 0.252439i \(-0.0812328\pi\)
\(522\) 0 0
\(523\) 3.03301e10 0.405385 0.202692 0.979242i \(-0.435031\pi\)
0.202692 + 0.979242i \(0.435031\pi\)
\(524\) − 9.84623e9i − 0.130600i
\(525\) 0 0
\(526\) 1.18264e10 0.154493
\(527\) − 4.46622e10i − 0.579026i
\(528\) 0 0
\(529\) −9.27734e10 −1.18468
\(530\) − 2.71304e10i − 0.343837i
\(531\) 0 0
\(532\) −1.64129e10 −0.204898
\(533\) 7.17423e10i 0.888928i
\(534\) 0 0
\(535\) 2.86603e10 0.349836
\(536\) 8.86499e9i 0.107404i
\(537\) 0 0
\(538\) −5.57409e10 −0.665341
\(539\) − 1.35399e11i − 1.60421i
\(540\) 0 0
\(541\) 7.54262e10 0.880508 0.440254 0.897873i \(-0.354888\pi\)
0.440254 + 0.897873i \(0.354888\pi\)
\(542\) − 7.45826e10i − 0.864252i
\(543\) 0 0
\(544\) −1.75697e10 −0.200617
\(545\) − 1.14069e10i − 0.129295i
\(546\) 0 0
\(547\) −2.04221e10 −0.228113 −0.114057 0.993474i \(-0.536385\pi\)
−0.114057 + 0.993474i \(0.536385\pi\)
\(548\) − 1.63660e10i − 0.181476i
\(549\) 0 0
\(550\) 7.67447e10 0.838683
\(551\) − 9.77272e9i − 0.106025i
\(552\) 0 0
\(553\) −2.95196e10 −0.315653
\(554\) − 1.37563e9i − 0.0146037i
\(555\) 0 0
\(556\) −1.16343e9 −0.0121742
\(557\) 2.53750e10i 0.263625i 0.991275 + 0.131812i \(0.0420796\pi\)
−0.991275 + 0.131812i \(0.957920\pi\)
\(558\) 0 0
\(559\) −1.51558e11 −1.55215
\(560\) − 1.35033e10i − 0.137305i
\(561\) 0 0
\(562\) 6.34290e10 0.635832
\(563\) − 1.03444e11i − 1.02960i −0.857309 0.514802i \(-0.827866\pi\)
0.857309 0.514802i \(-0.172134\pi\)
\(564\) 0 0
\(565\) −4.41595e10 −0.433341
\(566\) 1.83915e10i 0.179206i
\(567\) 0 0
\(568\) −3.06923e10 −0.294874
\(569\) − 9.50812e10i − 0.907080i −0.891236 0.453540i \(-0.850161\pi\)
0.891236 0.453540i \(-0.149839\pi\)
\(570\) 0 0
\(571\) −1.03046e11 −0.969368 −0.484684 0.874689i \(-0.661065\pi\)
−0.484684 + 0.874689i \(0.661065\pi\)
\(572\) − 1.08022e11i − 1.00909i
\(573\) 0 0
\(574\) −6.85449e10 −0.631434
\(575\) 1.39050e11i 1.27204i
\(576\) 0 0
\(577\) −8.28869e10 −0.747795 −0.373898 0.927470i \(-0.621979\pi\)
−0.373898 + 0.927470i \(0.621979\pi\)
\(578\) − 2.27225e10i − 0.203585i
\(579\) 0 0
\(580\) 8.04026e9 0.0710490
\(581\) 1.81516e11i 1.59298i
\(582\) 0 0
\(583\) 2.07363e11 1.79497
\(584\) 7.10047e10i 0.610430i
\(585\) 0 0
\(586\) 4.76589e10 0.404160
\(587\) − 1.12503e10i − 0.0947573i −0.998877 0.0473787i \(-0.984913\pi\)
0.998877 0.0473787i \(-0.0150867\pi\)
\(588\) 0 0
\(589\) 1.71063e10 0.142133
\(590\) − 7.09659e9i − 0.0585655i
\(591\) 0 0
\(592\) −4.92733e10 −0.401166
\(593\) 1.94961e11i 1.57663i 0.615272 + 0.788315i \(0.289046\pi\)
−0.615272 + 0.788315i \(0.710954\pi\)
\(594\) 0 0
\(595\) 7.81193e10 0.623291
\(596\) − 2.37061e10i − 0.187877i
\(597\) 0 0
\(598\) 1.95720e11 1.53049
\(599\) − 3.03920e10i − 0.236076i −0.993009 0.118038i \(-0.962340\pi\)
0.993009 0.118038i \(-0.0376605\pi\)
\(600\) 0 0
\(601\) −1.19015e11 −0.912227 −0.456114 0.889921i \(-0.650759\pi\)
−0.456114 + 0.889921i \(0.650759\pi\)
\(602\) − 1.44804e11i − 1.10254i
\(603\) 0 0
\(604\) 4.29569e10 0.322764
\(605\) − 4.49873e10i − 0.335790i
\(606\) 0 0
\(607\) −1.13080e11 −0.832973 −0.416486 0.909142i \(-0.636739\pi\)
−0.416486 + 0.909142i \(0.636739\pi\)
\(608\) − 6.72945e9i − 0.0492453i
\(609\) 0 0
\(610\) 1.43633e10 0.103737
\(611\) 2.51556e11i 1.80497i
\(612\) 0 0
\(613\) 1.93018e11 1.36696 0.683481 0.729968i \(-0.260465\pi\)
0.683481 + 0.729968i \(0.260465\pi\)
\(614\) 6.65783e10i 0.468446i
\(615\) 0 0
\(616\) 1.03208e11 0.716788
\(617\) 2.28160e11i 1.57434i 0.616734 + 0.787172i \(0.288455\pi\)
−0.616734 + 0.787172i \(0.711545\pi\)
\(618\) 0 0
\(619\) 1.21474e11 0.827413 0.413706 0.910410i \(-0.364234\pi\)
0.413706 + 0.910410i \(0.364234\pi\)
\(620\) 1.40738e10i 0.0952453i
\(621\) 0 0
\(622\) −1.10875e11 −0.740751
\(623\) − 3.79115e11i − 2.51663i
\(624\) 0 0
\(625\) 9.17441e10 0.601254
\(626\) 4.25579e10i 0.277129i
\(627\) 0 0
\(628\) 2.58087e10 0.165931
\(629\) − 2.85056e11i − 1.82107i
\(630\) 0 0
\(631\) 3.03109e11 1.91197 0.955986 0.293412i \(-0.0947908\pi\)
0.955986 + 0.293412i \(0.0947908\pi\)
\(632\) − 1.21033e10i − 0.0758641i
\(633\) 0 0
\(634\) −1.91596e11 −1.18585
\(635\) 7.93160e10i 0.487827i
\(636\) 0 0
\(637\) −2.80648e11 −1.70453
\(638\) 6.14531e10i 0.370904i
\(639\) 0 0
\(640\) 5.53648e9 0.0330000
\(641\) 2.84037e10i 0.168246i 0.996455 + 0.0841228i \(0.0268088\pi\)
−0.996455 + 0.0841228i \(0.973191\pi\)
\(642\) 0 0
\(643\) −3.38883e11 −1.98247 −0.991234 0.132118i \(-0.957822\pi\)
−0.991234 + 0.132118i \(0.957822\pi\)
\(644\) 1.86998e11i 1.08716i
\(645\) 0 0
\(646\) 3.89312e10 0.223547
\(647\) − 1.91011e11i − 1.09004i −0.838424 0.545019i \(-0.816523\pi\)
0.838424 0.545019i \(-0.183477\pi\)
\(648\) 0 0
\(649\) 5.42405e10 0.305735
\(650\) − 1.59073e11i − 0.891133i
\(651\) 0 0
\(652\) 6.75406e10 0.373744
\(653\) − 4.76514e10i − 0.262073i −0.991378 0.131037i \(-0.958169\pi\)
0.991378 0.131037i \(-0.0418306\pi\)
\(654\) 0 0
\(655\) −1.79498e10 −0.0975200
\(656\) − 2.81041e10i − 0.151759i
\(657\) 0 0
\(658\) −2.40344e11 −1.28213
\(659\) 8.16161e10i 0.432747i 0.976311 + 0.216374i \(0.0694229\pi\)
−0.976311 + 0.216374i \(0.930577\pi\)
\(660\) 0 0
\(661\) 5.88857e10 0.308463 0.154232 0.988035i \(-0.450710\pi\)
0.154232 + 0.988035i \(0.450710\pi\)
\(662\) 3.60905e10i 0.187915i
\(663\) 0 0
\(664\) −7.44233e10 −0.382857
\(665\) 2.99209e10i 0.152999i
\(666\) 0 0
\(667\) −1.11344e11 −0.562552
\(668\) 2.32468e10i 0.116750i
\(669\) 0 0
\(670\) 1.61610e10 0.0801988
\(671\) 1.09781e11i 0.541548i
\(672\) 0 0
\(673\) 1.65634e8 0.000807400 0 0.000403700 1.00000i \(-0.499871\pi\)
0.000403700 1.00000i \(0.499871\pi\)
\(674\) 1.88235e11i 0.912137i
\(675\) 0 0
\(676\) −1.19490e11 −0.572196
\(677\) − 2.58779e11i − 1.23190i −0.787786 0.615949i \(-0.788773\pi\)
0.787786 0.615949i \(-0.211227\pi\)
\(678\) 0 0
\(679\) −7.21635e10 −0.339499
\(680\) 3.20297e10i 0.149802i
\(681\) 0 0
\(682\) −1.07568e11 −0.497218
\(683\) − 1.24132e11i − 0.570428i −0.958464 0.285214i \(-0.907935\pi\)
0.958464 0.285214i \(-0.0920647\pi\)
\(684\) 0 0
\(685\) −2.98353e10 −0.135509
\(686\) − 3.77791e10i − 0.170591i
\(687\) 0 0
\(688\) 5.93710e10 0.264985
\(689\) − 4.29811e11i − 1.90722i
\(690\) 0 0
\(691\) 1.19734e11 0.525175 0.262588 0.964908i \(-0.415424\pi\)
0.262588 + 0.964908i \(0.415424\pi\)
\(692\) 9.04269e9i 0.0394342i
\(693\) 0 0
\(694\) −1.21348e10 −0.0523112
\(695\) 2.12094e9i 0.00909054i
\(696\) 0 0
\(697\) 1.62588e11 0.688902
\(698\) − 2.62820e10i − 0.110723i
\(699\) 0 0
\(700\) 1.51983e11 0.633000
\(701\) 1.43654e11i 0.594902i 0.954737 + 0.297451i \(0.0961365\pi\)
−0.954737 + 0.297451i \(0.903864\pi\)
\(702\) 0 0
\(703\) 1.09181e11 0.447018
\(704\) 4.23163e10i 0.172273i
\(705\) 0 0
\(706\) 2.24584e11 0.903984
\(707\) 5.98967e11i 2.39732i
\(708\) 0 0
\(709\) 4.14241e11 1.63934 0.819669 0.572838i \(-0.194158\pi\)
0.819669 + 0.572838i \(0.194158\pi\)
\(710\) 5.59524e10i 0.220184i
\(711\) 0 0
\(712\) 1.55441e11 0.604847
\(713\) − 1.94898e11i − 0.754134i
\(714\) 0 0
\(715\) −1.96926e11 −0.753492
\(716\) 1.01168e10i 0.0384936i
\(717\) 0 0
\(718\) −1.17300e11 −0.441367
\(719\) 3.07755e11i 1.15157i 0.817602 + 0.575784i \(0.195303\pi\)
−0.817602 + 0.575784i \(0.804697\pi\)
\(720\) 0 0
\(721\) 1.05343e11 0.389819
\(722\) − 1.77236e11i − 0.652233i
\(723\) 0 0
\(724\) −7.01655e10 −0.255370
\(725\) 9.04954e10i 0.327548i
\(726\) 0 0
\(727\) −1.59396e10 −0.0570610 −0.0285305 0.999593i \(-0.509083\pi\)
−0.0285305 + 0.999593i \(0.509083\pi\)
\(728\) − 2.13925e11i − 0.761615i
\(729\) 0 0
\(730\) 1.29442e11 0.455811
\(731\) 3.43474e11i 1.20288i
\(732\) 0 0
\(733\) 1.67831e10 0.0581374 0.0290687 0.999577i \(-0.490746\pi\)
0.0290687 + 0.999577i \(0.490746\pi\)
\(734\) − 2.29532e11i − 0.790787i
\(735\) 0 0
\(736\) −7.66708e10 −0.261288
\(737\) 1.23521e11i 0.418670i
\(738\) 0 0
\(739\) −3.67552e11 −1.23237 −0.616185 0.787601i \(-0.711323\pi\)
−0.616185 + 0.787601i \(0.711323\pi\)
\(740\) 8.98257e10i 0.299553i
\(741\) 0 0
\(742\) 4.10656e11 1.35476
\(743\) − 3.34339e11i − 1.09706i −0.836130 0.548531i \(-0.815187\pi\)
0.836130 0.548531i \(-0.184813\pi\)
\(744\) 0 0
\(745\) −4.32164e10 −0.140289
\(746\) 2.51290e11i 0.811371i
\(747\) 0 0
\(748\) −2.44809e11 −0.782025
\(749\) 4.33812e11i 1.37840i
\(750\) 0 0
\(751\) −1.79372e11 −0.563892 −0.281946 0.959430i \(-0.590980\pi\)
−0.281946 + 0.959430i \(0.590980\pi\)
\(752\) − 9.85436e10i − 0.308146i
\(753\) 0 0
\(754\) 1.27377e11 0.394100
\(755\) − 7.83109e10i − 0.241010i
\(756\) 0 0
\(757\) −1.95683e11 −0.595894 −0.297947 0.954582i \(-0.596302\pi\)
−0.297947 + 0.954582i \(0.596302\pi\)
\(758\) 1.40052e11i 0.424241i
\(759\) 0 0
\(760\) −1.22678e10 −0.0367717
\(761\) − 3.29292e11i − 0.981843i −0.871204 0.490922i \(-0.836660\pi\)
0.871204 0.490922i \(-0.163340\pi\)
\(762\) 0 0
\(763\) 1.72660e11 0.509440
\(764\) − 2.87485e11i − 0.843803i
\(765\) 0 0
\(766\) −3.28514e11 −0.954198
\(767\) − 1.12427e11i − 0.324855i
\(768\) 0 0
\(769\) −8.67201e10 −0.247979 −0.123989 0.992284i \(-0.539569\pi\)
−0.123989 + 0.992284i \(0.539569\pi\)
\(770\) − 1.88149e11i − 0.535229i
\(771\) 0 0
\(772\) −8.39136e10 −0.236245
\(773\) − 7.04193e8i − 0.00197230i −1.00000 0.000986152i \(-0.999686\pi\)
1.00000 0.000986152i \(-0.000313902\pi\)
\(774\) 0 0
\(775\) −1.58404e11 −0.439097
\(776\) − 2.95877e10i − 0.0815952i
\(777\) 0 0
\(778\) 3.32326e11 0.907082
\(779\) 6.22736e10i 0.169104i
\(780\) 0 0
\(781\) −4.27654e11 −1.14945
\(782\) − 4.43557e11i − 1.18610i
\(783\) 0 0
\(784\) 1.09940e11 0.291000
\(785\) − 4.70496e10i − 0.123902i
\(786\) 0 0
\(787\) −4.04090e11 −1.05337 −0.526683 0.850062i \(-0.676565\pi\)
−0.526683 + 0.850062i \(0.676565\pi\)
\(788\) 5.73736e10i 0.148801i
\(789\) 0 0
\(790\) −2.20645e10 −0.0566481
\(791\) − 6.68414e11i − 1.70742i
\(792\) 0 0
\(793\) 2.27549e11 0.575416
\(794\) 2.92073e11i 0.734868i
\(795\) 0 0
\(796\) −9.40711e10 −0.234317
\(797\) − 3.97163e11i − 0.984318i −0.870505 0.492159i \(-0.836208\pi\)
0.870505 0.492159i \(-0.163792\pi\)
\(798\) 0 0
\(799\) 5.70095e11 1.39881
\(800\) 6.23147e10i 0.152135i
\(801\) 0 0
\(802\) 7.40984e10 0.179107
\(803\) 9.89351e11i 2.37951i
\(804\) 0 0
\(805\) 3.40898e11 0.811786
\(806\) 2.22963e11i 0.528314i
\(807\) 0 0
\(808\) −2.45583e11 −0.576172
\(809\) − 5.98958e11i − 1.39831i −0.714972 0.699153i \(-0.753561\pi\)
0.714972 0.699153i \(-0.246439\pi\)
\(810\) 0 0
\(811\) −1.62271e11 −0.375109 −0.187554 0.982254i \(-0.560056\pi\)
−0.187554 + 0.982254i \(0.560056\pi\)
\(812\) 1.21700e11i 0.279942i
\(813\) 0 0
\(814\) −6.86554e11 −1.56378
\(815\) − 1.23127e11i − 0.279077i
\(816\) 0 0
\(817\) −1.31556e11 −0.295271
\(818\) 4.99855e11i 1.11643i
\(819\) 0 0
\(820\) −5.12340e10 −0.113319
\(821\) − 2.35494e11i − 0.518332i −0.965833 0.259166i \(-0.916552\pi\)
0.965833 0.259166i \(-0.0834476\pi\)
\(822\) 0 0
\(823\) 3.64579e11 0.794681 0.397340 0.917671i \(-0.369933\pi\)
0.397340 + 0.917671i \(0.369933\pi\)
\(824\) 4.31916e10i 0.0936893i
\(825\) 0 0
\(826\) 1.07417e11 0.230755
\(827\) 7.48202e11i 1.59955i 0.600302 + 0.799774i \(0.295047\pi\)
−0.600302 + 0.799774i \(0.704953\pi\)
\(828\) 0 0
\(829\) 4.91506e11 1.04066 0.520332 0.853964i \(-0.325808\pi\)
0.520332 + 0.853964i \(0.325808\pi\)
\(830\) 1.35674e11i 0.285881i
\(831\) 0 0
\(832\) 8.77113e10 0.183047
\(833\) 6.36027e11i 1.32098i
\(834\) 0 0
\(835\) 4.23791e10 0.0871778
\(836\) − 9.37654e10i − 0.191963i
\(837\) 0 0
\(838\) 5.11974e11 1.03818
\(839\) − 6.85496e10i − 0.138343i −0.997605 0.0691715i \(-0.977964\pi\)
0.997605 0.0691715i \(-0.0220356\pi\)
\(840\) 0 0
\(841\) 4.27782e11 0.855143
\(842\) 2.47988e11i 0.493381i
\(843\) 0 0
\(844\) 4.18304e11 0.824370
\(845\) 2.17832e11i 0.427262i
\(846\) 0 0
\(847\) 6.80944e11 1.32305
\(848\) 1.68373e11i 0.325603i
\(849\) 0 0
\(850\) −3.60503e11 −0.690611
\(851\) − 1.24393e12i − 2.37180i
\(852\) 0 0
\(853\) −7.31087e11 −1.38093 −0.690467 0.723364i \(-0.742595\pi\)
−0.690467 + 0.723364i \(0.742595\pi\)
\(854\) 2.17408e11i 0.408736i
\(855\) 0 0
\(856\) −1.77867e11 −0.331284
\(857\) 8.31857e11i 1.54215i 0.636747 + 0.771073i \(0.280280\pi\)
−0.636747 + 0.771073i \(0.719720\pi\)
\(858\) 0 0
\(859\) −2.64366e11 −0.485549 −0.242775 0.970083i \(-0.578058\pi\)
−0.242775 + 0.970083i \(0.578058\pi\)
\(860\) − 1.08234e11i − 0.197865i
\(861\) 0 0
\(862\) −1.08840e11 −0.197134
\(863\) − 7.30956e11i − 1.31780i −0.752232 0.658898i \(-0.771023\pi\)
0.752232 0.658898i \(-0.228977\pi\)
\(864\) 0 0
\(865\) 1.64849e10 0.0294457
\(866\) − 5.25867e11i − 0.934984i
\(867\) 0 0
\(868\) −2.13026e11 −0.375278
\(869\) − 1.68643e11i − 0.295725i
\(870\) 0 0
\(871\) 2.56029e11 0.444853
\(872\) 7.07922e10i 0.122439i
\(873\) 0 0
\(874\) 1.69889e11 0.291151
\(875\) − 5.99011e11i − 1.02189i
\(876\) 0 0
\(877\) 6.63180e11 1.12107 0.560536 0.828130i \(-0.310595\pi\)
0.560536 + 0.828130i \(0.310595\pi\)
\(878\) − 4.21180e11i − 0.708744i
\(879\) 0 0
\(880\) 7.71431e10 0.128637
\(881\) 4.47616e11i 0.743023i 0.928428 + 0.371511i \(0.121160\pi\)
−0.928428 + 0.371511i \(0.878840\pi\)
\(882\) 0 0
\(883\) −2.27693e11 −0.374548 −0.187274 0.982308i \(-0.559965\pi\)
−0.187274 + 0.982308i \(0.559965\pi\)
\(884\) 5.07428e11i 0.830932i
\(885\) 0 0
\(886\) −6.03928e11 −0.980054
\(887\) 1.92088e11i 0.310316i 0.987890 + 0.155158i \(0.0495887\pi\)
−0.987890 + 0.155158i \(0.950411\pi\)
\(888\) 0 0
\(889\) −1.20056e12 −1.92210
\(890\) − 2.83370e11i − 0.451642i
\(891\) 0 0
\(892\) 4.87476e11 0.770005
\(893\) 2.18355e11i 0.343366i
\(894\) 0 0
\(895\) 1.84429e10 0.0287434
\(896\) 8.38022e10i 0.130024i
\(897\) 0 0
\(898\) 4.70975e10 0.0724256
\(899\) − 1.26842e11i − 0.194189i
\(900\) 0 0
\(901\) −9.74072e11 −1.47806
\(902\) − 3.91591e11i − 0.591571i
\(903\) 0 0
\(904\) 2.74056e11 0.410361
\(905\) 1.27912e11i 0.190686i
\(906\) 0 0
\(907\) 6.50953e11 0.961879 0.480939 0.876754i \(-0.340296\pi\)
0.480939 + 0.876754i \(0.340296\pi\)
\(908\) − 5.56353e11i − 0.818478i
\(909\) 0 0
\(910\) −3.89987e11 −0.568702
\(911\) − 2.94797e11i − 0.428005i −0.976833 0.214003i \(-0.931350\pi\)
0.976833 0.214003i \(-0.0686501\pi\)
\(912\) 0 0
\(913\) −1.03698e12 −1.49241
\(914\) − 3.30332e11i − 0.473333i
\(915\) 0 0
\(916\) 3.66588e11 0.520711
\(917\) − 2.71694e11i − 0.384241i
\(918\) 0 0
\(919\) −8.25357e11 −1.15712 −0.578562 0.815639i \(-0.696386\pi\)
−0.578562 + 0.815639i \(0.696386\pi\)
\(920\) 1.39772e11i 0.195105i
\(921\) 0 0
\(922\) 1.87544e11 0.259526
\(923\) 8.86421e11i 1.22133i
\(924\) 0 0
\(925\) −1.01101e12 −1.38099
\(926\) − 4.15040e11i − 0.564477i
\(927\) 0 0
\(928\) −4.98983e10 −0.0672813
\(929\) 1.07779e12i 1.44701i 0.690320 + 0.723504i \(0.257470\pi\)
−0.690320 + 0.723504i \(0.742530\pi\)
\(930\) 0 0
\(931\) −2.43608e11 −0.324259
\(932\) − 7.13091e10i − 0.0945108i
\(933\) 0 0
\(934\) 2.04063e11 0.268149
\(935\) 4.46289e11i 0.583942i
\(936\) 0 0
\(937\) 1.44214e12 1.87090 0.935449 0.353463i \(-0.114996\pi\)
0.935449 + 0.353463i \(0.114996\pi\)
\(938\) 2.44618e11i 0.315993i
\(939\) 0 0
\(940\) −1.79646e11 −0.230094
\(941\) 4.67480e11i 0.596217i 0.954532 + 0.298109i \(0.0963558\pi\)
−0.954532 + 0.298109i \(0.903644\pi\)
\(942\) 0 0
\(943\) 7.09504e11 0.897239
\(944\) 4.40419e10i 0.0554597i
\(945\) 0 0
\(946\) 8.27252e11 1.03294
\(947\) − 1.07029e12i − 1.33077i −0.746502 0.665384i \(-0.768268\pi\)
0.746502 0.665384i \(-0.231732\pi\)
\(948\) 0 0
\(949\) 2.05068e12 2.52832
\(950\) − 1.38078e11i − 0.169524i
\(951\) 0 0
\(952\) −4.84813e11 −0.590237
\(953\) − 5.68036e11i − 0.688659i −0.938849 0.344330i \(-0.888106\pi\)
0.938849 0.344330i \(-0.111894\pi\)
\(954\) 0 0
\(955\) −5.24087e11 −0.630072
\(956\) − 1.67464e11i − 0.200488i
\(957\) 0 0
\(958\) −3.22575e10 −0.0382973
\(959\) − 4.51598e11i − 0.533922i
\(960\) 0 0
\(961\) −6.30865e11 −0.739679
\(962\) 1.42306e12i 1.66158i
\(963\) 0 0
\(964\) 6.19753e11 0.717647
\(965\) 1.52975e11i 0.176406i
\(966\) 0 0
\(967\) 5.58775e11 0.639044 0.319522 0.947579i \(-0.396478\pi\)
0.319522 + 0.947579i \(0.396478\pi\)
\(968\) 2.79194e11i 0.317983i
\(969\) 0 0
\(970\) −5.39387e10 −0.0609275
\(971\) 7.11343e11i 0.800207i 0.916470 + 0.400103i \(0.131026\pi\)
−0.916470 + 0.400103i \(0.868974\pi\)
\(972\) 0 0
\(973\) −3.21034e10 −0.0358178
\(974\) − 1.14616e12i − 1.27353i
\(975\) 0 0
\(976\) −8.91393e10 −0.0982358
\(977\) 2.73431e11i 0.300103i 0.988678 + 0.150051i \(0.0479439\pi\)
−0.988678 + 0.150051i \(0.952056\pi\)
\(978\) 0 0
\(979\) 2.16585e12 2.35775
\(980\) − 2.00422e11i − 0.217291i
\(981\) 0 0
\(982\) 1.32901e11 0.142916
\(983\) 2.95740e11i 0.316735i 0.987380 + 0.158368i \(0.0506231\pi\)
−0.987380 + 0.158368i \(0.949377\pi\)
\(984\) 0 0
\(985\) 1.04593e11 0.111111
\(986\) − 2.88672e11i − 0.305420i
\(987\) 0 0
\(988\) −1.94352e11 −0.203968
\(989\) 1.49886e12i 1.56666i
\(990\) 0 0
\(991\) −1.29048e12 −1.33800 −0.669000 0.743263i \(-0.733277\pi\)
−0.669000 + 0.743263i \(0.733277\pi\)
\(992\) − 8.73427e10i − 0.0901945i
\(993\) 0 0
\(994\) −8.46916e11 −0.867550
\(995\) 1.71492e11i 0.174966i
\(996\) 0 0
\(997\) −2.21939e11 −0.224623 −0.112311 0.993673i \(-0.535825\pi\)
−0.112311 + 0.993673i \(0.535825\pi\)
\(998\) 1.29730e12i 1.30774i
\(999\) 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 18.9.b.a.17.2 yes 2
3.2 odd 2 inner 18.9.b.a.17.1 2
4.3 odd 2 144.9.e.d.17.2 2
5.2 odd 4 450.9.b.a.449.1 4
5.3 odd 4 450.9.b.a.449.4 4
5.4 even 2 450.9.d.b.251.1 2
9.2 odd 6 162.9.d.d.53.1 4
9.4 even 3 162.9.d.d.107.1 4
9.5 odd 6 162.9.d.d.107.2 4
9.7 even 3 162.9.d.d.53.2 4
12.11 even 2 144.9.e.d.17.1 2
15.2 even 4 450.9.b.a.449.3 4
15.8 even 4 450.9.b.a.449.2 4
15.14 odd 2 450.9.d.b.251.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
18.9.b.a.17.1 2 3.2 odd 2 inner
18.9.b.a.17.2 yes 2 1.1 even 1 trivial
144.9.e.d.17.1 2 12.11 even 2
144.9.e.d.17.2 2 4.3 odd 2
162.9.d.d.53.1 4 9.2 odd 6
162.9.d.d.53.2 4 9.7 even 3
162.9.d.d.107.1 4 9.4 even 3
162.9.d.d.107.2 4 9.5 odd 6
450.9.b.a.449.1 4 5.2 odd 4
450.9.b.a.449.2 4 15.8 even 4
450.9.b.a.449.3 4 15.2 even 4
450.9.b.a.449.4 4 5.3 odd 4
450.9.d.b.251.1 2 5.4 even 2
450.9.d.b.251.2 2 15.14 odd 2