Properties

Label 300.11.k.c.157.1
Level $300$
Weight $11$
Character 300.157
Analytic conductor $190.607$
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] = [300,11,Mod(157,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.157");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 300.k (of order \(4\), degree \(2\), 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: \(190.607175802\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 63831600 x^{13} + 120528248672 x^{12} - 17600989215600 x^{11} + \cdots + 38\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{44}\cdot 3^{12}\cdot 5^{16} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 157.1
Root \(-381.708 - 381.708i\) of defining polynomial
Character \(\chi\) \(=\) 300.157
Dual form 300.11.k.c.193.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-99.2043 + 99.2043i) q^{3} +(-17583.1 - 17583.1i) q^{7} -19683.0i q^{9} +O(q^{10})\) \(q+(-99.2043 + 99.2043i) q^{3} +(-17583.1 - 17583.1i) q^{7} -19683.0i q^{9} -245351. q^{11} +(-25882.8 + 25882.8i) q^{13} +(1.01646e6 + 1.01646e6i) q^{17} -2.08723e6i q^{19} +3.48864e6 q^{21} +(4.16272e6 - 4.16272e6i) q^{23} +(1.95264e6 + 1.95264e6i) q^{27} -2.48858e7i q^{29} +4.39967e7 q^{31} +(2.43399e7 - 2.43399e7i) q^{33} +(-8.21011e7 - 8.21011e7i) q^{37} -5.13538e6i q^{39} +3.41660e7 q^{41} +(5.69545e7 - 5.69545e7i) q^{43} +(-1.56181e8 - 1.56181e8i) q^{47} +3.35854e8i q^{49} -2.01674e8 q^{51} +(9.10269e7 - 9.10269e7i) q^{53} +(2.07062e8 + 2.07062e8i) q^{57} -1.50313e8i q^{59} +1.58473e9 q^{61} +(-3.46088e8 + 3.46088e8i) q^{63} +(9.08379e7 + 9.08379e7i) q^{67} +8.25920e8i q^{69} -3.33819e9 q^{71} +(6.15460e8 - 6.15460e8i) q^{73} +(4.31403e9 + 4.31403e9i) q^{77} -1.25540e9i q^{79} -3.87420e8 q^{81} +(2.20145e9 - 2.20145e9i) q^{83} +(2.46878e9 + 2.46878e9i) q^{87} +5.39933e9i q^{89} +9.10200e8 q^{91} +(-4.36467e9 + 4.36467e9i) q^{93} +(-6.66166e9 - 6.66166e9i) q^{97} +4.82925e9i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 331104 q^{11} + 555984 q^{21} - 140804816 q^{31} + 29553600 q^{41} + 471062304 q^{51} + 3576862832 q^{61} + 1853192640 q^{71} - 6198727824 q^{81} + 7033272240 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(1\) \(1\) \(e\left(\frac{1}{4}\right)\)

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
\(3\) −99.2043 + 99.2043i −0.408248 + 0.408248i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) −17583.1 17583.1i −1.04618 1.04618i −0.998881 0.0472951i \(-0.984940\pi\)
−0.0472951 0.998881i \(-0.515060\pi\)
\(8\) 0 0
\(9\) 19683.0i 0.333333i
\(10\) 0 0
\(11\) −245351. −1.52344 −0.761719 0.647908i \(-0.775644\pi\)
−0.761719 + 0.647908i \(0.775644\pi\)
\(12\) 0 0
\(13\) −25882.8 + 25882.8i −0.0697100 + 0.0697100i −0.741102 0.671392i \(-0.765697\pi\)
0.671392 + 0.741102i \(0.265697\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 1.01646e6 + 1.01646e6i 0.715886 + 0.715886i 0.967760 0.251874i \(-0.0810470\pi\)
−0.251874 + 0.967760i \(0.581047\pi\)
\(18\) 0 0
\(19\) 2.08723e6i 0.842952i −0.906840 0.421476i \(-0.861512\pi\)
0.906840 0.421476i \(-0.138488\pi\)
\(20\) 0 0
\(21\) 3.48864e6 0.854199
\(22\) 0 0
\(23\) 4.16272e6 4.16272e6i 0.646752 0.646752i −0.305454 0.952207i \(-0.598808\pi\)
0.952207 + 0.305454i \(0.0988084\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 1.95264e6 + 1.95264e6i 0.136083 + 0.136083i
\(28\) 0 0
\(29\) 2.48858e7i 1.21328i −0.794975 0.606642i \(-0.792516\pi\)
0.794975 0.606642i \(-0.207484\pi\)
\(30\) 0 0
\(31\) 4.39967e7 1.53678 0.768391 0.639981i \(-0.221058\pi\)
0.768391 + 0.639981i \(0.221058\pi\)
\(32\) 0 0
\(33\) 2.43399e7 2.43399e7i 0.621941 0.621941i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −8.21011e7 8.21011e7i −1.18397 1.18397i −0.978707 0.205263i \(-0.934195\pi\)
−0.205263 0.978707i \(-0.565805\pi\)
\(38\) 0 0
\(39\) 5.13538e6i 0.0569180i
\(40\) 0 0
\(41\) 3.41660e7 0.294900 0.147450 0.989070i \(-0.452893\pi\)
0.147450 + 0.989070i \(0.452893\pi\)
\(42\) 0 0
\(43\) 5.69545e7 5.69545e7i 0.387424 0.387424i −0.486344 0.873767i \(-0.661670\pi\)
0.873767 + 0.486344i \(0.161670\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) −1.56181e8 1.56181e8i −0.680989 0.680989i 0.279234 0.960223i \(-0.409920\pi\)
−0.960223 + 0.279234i \(0.909920\pi\)
\(48\) 0 0
\(49\) 3.35854e8i 1.18897i
\(50\) 0 0
\(51\) −2.01674e8 −0.584518
\(52\) 0 0
\(53\) 9.10269e7 9.10269e7i 0.217666 0.217666i −0.589848 0.807514i \(-0.700812\pi\)
0.807514 + 0.589848i \(0.200812\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 2.07062e8 + 2.07062e8i 0.344134 + 0.344134i
\(58\) 0 0
\(59\) 1.50313e8i 0.210251i −0.994459 0.105125i \(-0.966476\pi\)
0.994459 0.105125i \(-0.0335244\pi\)
\(60\) 0 0
\(61\) 1.58473e9 1.87632 0.938160 0.346202i \(-0.112529\pi\)
0.938160 + 0.346202i \(0.112529\pi\)
\(62\) 0 0
\(63\) −3.46088e8 + 3.46088e8i −0.348725 + 0.348725i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 9.08379e7 + 9.08379e7i 0.0672811 + 0.0672811i 0.739947 0.672666i \(-0.234851\pi\)
−0.672666 + 0.739947i \(0.734851\pi\)
\(68\) 0 0
\(69\) 8.25920e8i 0.528071i
\(70\) 0 0
\(71\) −3.33819e9 −1.85020 −0.925100 0.379723i \(-0.876019\pi\)
−0.925100 + 0.379723i \(0.876019\pi\)
\(72\) 0 0
\(73\) 6.15460e8 6.15460e8i 0.296883 0.296883i −0.542909 0.839792i \(-0.682677\pi\)
0.839792 + 0.542909i \(0.182677\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.31403e9 + 4.31403e9i 1.59378 + 1.59378i
\(78\) 0 0
\(79\) 1.25540e9i 0.407988i −0.978972 0.203994i \(-0.934608\pi\)
0.978972 0.203994i \(-0.0653923\pi\)
\(80\) 0 0
\(81\) −3.87420e8 −0.111111
\(82\) 0 0
\(83\) 2.20145e9 2.20145e9i 0.558881 0.558881i −0.370108 0.928989i \(-0.620679\pi\)
0.928989 + 0.370108i \(0.120679\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 2.46878e9 + 2.46878e9i 0.495321 + 0.495321i
\(88\) 0 0
\(89\) 5.39933e9i 0.966918i 0.875367 + 0.483459i \(0.160620\pi\)
−0.875367 + 0.483459i \(0.839380\pi\)
\(90\) 0 0
\(91\) 9.10200e8 0.145858
\(92\) 0 0
\(93\) −4.36467e9 + 4.36467e9i −0.627388 + 0.627388i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −6.66166e9 6.66166e9i −0.775754 0.775754i 0.203352 0.979106i \(-0.434817\pi\)
−0.979106 + 0.203352i \(0.934817\pi\)
\(98\) 0 0
\(99\) 4.82925e9i 0.507813i
\(100\) 0 0
\(101\) −6.10333e9 −0.580711 −0.290355 0.956919i \(-0.593774\pi\)
−0.290355 + 0.956919i \(0.593774\pi\)
\(102\) 0 0
\(103\) 6.03828e9 6.03828e9i 0.520868 0.520868i −0.396966 0.917833i \(-0.629937\pi\)
0.917833 + 0.396966i \(0.129937\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −1.44897e10 1.44897e10i −1.03309 1.03309i −0.999433 0.0336592i \(-0.989284\pi\)
−0.0336592 0.999433i \(-0.510716\pi\)
\(108\) 0 0
\(109\) 2.56984e10i 1.67022i 0.550081 + 0.835111i \(0.314597\pi\)
−0.550081 + 0.835111i \(0.685403\pi\)
\(110\) 0 0
\(111\) 1.62896e10 0.966707
\(112\) 0 0
\(113\) 2.51562e10 2.51562e10i 1.36538 1.36538i 0.498474 0.866904i \(-0.333894\pi\)
0.866904 0.498474i \(-0.166106\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 5.09452e8 + 5.09452e8i 0.0232367 + 0.0232367i
\(118\) 0 0
\(119\) 3.57448e10i 1.49788i
\(120\) 0 0
\(121\) 3.42598e10 1.32086
\(122\) 0 0
\(123\) −3.38942e9 + 3.38942e9i −0.120392 + 0.120392i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 2.56680e10 + 2.56680e10i 0.776914 + 0.776914i 0.979305 0.202391i \(-0.0648711\pi\)
−0.202391 + 0.979305i \(0.564871\pi\)
\(128\) 0 0
\(129\) 1.13003e10i 0.316330i
\(130\) 0 0
\(131\) −3.75211e10 −0.972565 −0.486283 0.873802i \(-0.661647\pi\)
−0.486283 + 0.873802i \(0.661647\pi\)
\(132\) 0 0
\(133\) −3.67000e10 + 3.67000e10i −0.881876 + 0.881876i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −5.09695e9 5.09695e9i −0.105611 0.105611i 0.652327 0.757938i \(-0.273793\pi\)
−0.757938 + 0.652327i \(0.773793\pi\)
\(138\) 0 0
\(139\) 2.00719e9i 0.0386825i −0.999813 0.0193412i \(-0.993843\pi\)
0.999813 0.0193412i \(-0.00615689\pi\)
\(140\) 0 0
\(141\) 3.09878e10 0.556025
\(142\) 0 0
\(143\) 6.35038e9 6.35038e9i 0.106199 0.106199i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −3.33182e10 3.33182e10i −0.485394 0.485394i
\(148\) 0 0
\(149\) 3.97699e10i 0.541531i 0.962645 + 0.270765i \(0.0872768\pi\)
−0.962645 + 0.270765i \(0.912723\pi\)
\(150\) 0 0
\(151\) −9.92476e10 −1.26426 −0.632128 0.774864i \(-0.717819\pi\)
−0.632128 + 0.774864i \(0.717819\pi\)
\(152\) 0 0
\(153\) 2.00069e10 2.00069e10i 0.238629 0.238629i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 2.76032e10 + 2.76032e10i 0.289375 + 0.289375i 0.836833 0.547458i \(-0.184404\pi\)
−0.547458 + 0.836833i \(0.684404\pi\)
\(158\) 0 0
\(159\) 1.80605e10i 0.177723i
\(160\) 0 0
\(161\) −1.46387e11 −1.35323
\(162\) 0 0
\(163\) −1.34584e11 + 1.34584e11i −1.16965 + 1.16965i −0.187355 + 0.982292i \(0.559991\pi\)
−0.982292 + 0.187355i \(0.940009\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) −1.12320e11 1.12320e11i −0.864717 0.864717i 0.127164 0.991882i \(-0.459412\pi\)
−0.991882 + 0.127164i \(0.959412\pi\)
\(168\) 0 0
\(169\) 1.36519e11i 0.990281i
\(170\) 0 0
\(171\) −4.10830e10 −0.280984
\(172\) 0 0
\(173\) 2.60619e10 2.60619e10i 0.168180 0.168180i −0.617999 0.786179i \(-0.712056\pi\)
0.786179 + 0.617999i \(0.212056\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.49117e10 + 1.49117e10i 0.0858345 + 0.0858345i
\(178\) 0 0
\(179\) 4.37045e9i 0.0237827i −0.999929 0.0118913i \(-0.996215\pi\)
0.999929 0.0118913i \(-0.00378522\pi\)
\(180\) 0 0
\(181\) −1.96965e11 −1.01390 −0.506952 0.861974i \(-0.669228\pi\)
−0.506952 + 0.861974i \(0.669228\pi\)
\(182\) 0 0
\(183\) −1.57212e11 + 1.57212e11i −0.766005 + 0.766005i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.49388e11 2.49388e11i −1.09061 1.09061i
\(188\) 0 0
\(189\) 6.86668e10i 0.284733i
\(190\) 0 0
\(191\) 2.91592e11 1.14712 0.573559 0.819164i \(-0.305562\pi\)
0.573559 + 0.819164i \(0.305562\pi\)
\(192\) 0 0
\(193\) 2.04238e11 2.04238e11i 0.762693 0.762693i −0.214116 0.976808i \(-0.568687\pi\)
0.976808 + 0.214116i \(0.0686869\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.68621e10 + 2.68621e10i 0.0905335 + 0.0905335i 0.750923 0.660390i \(-0.229609\pi\)
−0.660390 + 0.750923i \(0.729609\pi\)
\(198\) 0 0
\(199\) 1.32496e11i 0.424557i −0.977209 0.212278i \(-0.931912\pi\)
0.977209 0.212278i \(-0.0680884\pi\)
\(200\) 0 0
\(201\) −1.80230e10 −0.0549348
\(202\) 0 0
\(203\) −4.37570e11 + 4.37570e11i −1.26931 + 1.26931i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −8.19348e10 8.19348e10i −0.215584 0.215584i
\(208\) 0 0
\(209\) 5.12105e11i 1.28418i
\(210\) 0 0
\(211\) 3.62184e11 0.865998 0.432999 0.901394i \(-0.357455\pi\)
0.432999 + 0.901394i \(0.357455\pi\)
\(212\) 0 0
\(213\) 3.31162e11 3.31162e11i 0.755341 0.755341i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −7.73598e11 7.73598e11i −1.60774 1.60774i
\(218\) 0 0
\(219\) 1.22113e11i 0.242404i
\(220\) 0 0
\(221\) −5.26175e10 −0.0998088
\(222\) 0 0
\(223\) 4.84416e11 4.84416e11i 0.878404 0.878404i −0.114966 0.993369i \(-0.536676\pi\)
0.993369 + 0.114966i \(0.0366758\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) −4.13763e11 4.13763e11i −0.686471 0.686471i 0.274979 0.961450i \(-0.411329\pi\)
−0.961450 + 0.274979i \(0.911329\pi\)
\(228\) 0 0
\(229\) 2.11849e11i 0.336395i 0.985753 + 0.168197i \(0.0537946\pi\)
−0.985753 + 0.168197i \(0.946205\pi\)
\(230\) 0 0
\(231\) −8.55941e11 −1.30132
\(232\) 0 0
\(233\) −5.33034e11 + 5.33034e11i −0.776203 + 0.776203i −0.979183 0.202980i \(-0.934937\pi\)
0.202980 + 0.979183i \(0.434937\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.24541e11 + 1.24541e11i 0.166560 + 0.166560i
\(238\) 0 0
\(239\) 1.42513e12i 1.82753i 0.406244 + 0.913765i \(0.366838\pi\)
−0.406244 + 0.913765i \(0.633162\pi\)
\(240\) 0 0
\(241\) −1.05869e12 −1.30221 −0.651107 0.758986i \(-0.725695\pi\)
−0.651107 + 0.758986i \(0.725695\pi\)
\(242\) 0 0
\(243\) 3.84338e10 3.84338e10i 0.0453609 0.0453609i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.40235e10 + 5.40235e10i 0.0587622 + 0.0587622i
\(248\) 0 0
\(249\) 4.36788e11i 0.456324i
\(250\) 0 0
\(251\) −1.21882e12 −1.22341 −0.611704 0.791087i \(-0.709516\pi\)
−0.611704 + 0.791087i \(0.709516\pi\)
\(252\) 0 0
\(253\) −1.02133e12 + 1.02133e12i −0.985287 + 0.985287i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −2.16474e11 2.16474e11i −0.193081 0.193081i 0.603945 0.797026i \(-0.293595\pi\)
−0.797026 + 0.603945i \(0.793595\pi\)
\(258\) 0 0
\(259\) 2.88718e12i 2.47728i
\(260\) 0 0
\(261\) −4.89828e11 −0.404428
\(262\) 0 0
\(263\) 4.22079e11 4.22079e11i 0.335440 0.335440i −0.519208 0.854648i \(-0.673773\pi\)
0.854648 + 0.519208i \(0.173773\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −5.35637e11 5.35637e11i −0.394743 0.394743i
\(268\) 0 0
\(269\) 1.97700e12i 1.40361i 0.712370 + 0.701804i \(0.247622\pi\)
−0.712370 + 0.701804i \(0.752378\pi\)
\(270\) 0 0
\(271\) −1.14114e12 −0.780718 −0.390359 0.920663i \(-0.627649\pi\)
−0.390359 + 0.920663i \(0.627649\pi\)
\(272\) 0 0
\(273\) −9.02958e10 + 9.02958e10i −0.0595462 + 0.0595462i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.80834e10 + 1.80834e10i 0.0110887 + 0.0110887i 0.712629 0.701541i \(-0.247504\pi\)
−0.701541 + 0.712629i \(0.747504\pi\)
\(278\) 0 0
\(279\) 8.65988e11i 0.512260i
\(280\) 0 0
\(281\) 4.83399e11 0.275914 0.137957 0.990438i \(-0.455946\pi\)
0.137957 + 0.990438i \(0.455946\pi\)
\(282\) 0 0
\(283\) 1.30550e12 1.30550e12i 0.719192 0.719192i −0.249248 0.968440i \(-0.580183\pi\)
0.968440 + 0.249248i \(0.0801834\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −6.00744e11 6.00744e11i −0.308517 0.308517i
\(288\) 0 0
\(289\) 5.03680e10i 0.0249842i
\(290\) 0 0
\(291\) 1.32173e12 0.633400
\(292\) 0 0
\(293\) −1.79363e11 + 1.79363e11i −0.0830606 + 0.0830606i −0.747416 0.664356i \(-0.768706\pi\)
0.664356 + 0.747416i \(0.268706\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −4.79082e11 4.79082e11i −0.207314 0.207314i
\(298\) 0 0
\(299\) 2.15486e11i 0.0901702i
\(300\) 0 0
\(301\) −2.00287e12 −0.810626
\(302\) 0 0
\(303\) 6.05477e11 6.05477e11i 0.237074 0.237074i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −2.02370e12 2.02370e12i −0.742085 0.742085i 0.230894 0.972979i \(-0.425835\pi\)
−0.972979 + 0.230894i \(0.925835\pi\)
\(308\) 0 0
\(309\) 1.19805e12i 0.425287i
\(310\) 0 0
\(311\) −4.23387e11 −0.145524 −0.0727622 0.997349i \(-0.523181\pi\)
−0.0727622 + 0.997349i \(0.523181\pi\)
\(312\) 0 0
\(313\) −3.30038e12 + 3.30038e12i −1.09861 + 1.09861i −0.104032 + 0.994574i \(0.533174\pi\)
−0.994574 + 0.104032i \(0.966826\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −3.57425e12 3.57425e12i −1.11658 1.11658i −0.992240 0.124336i \(-0.960320\pi\)
−0.124336 0.992240i \(-0.539680\pi\)
\(318\) 0 0
\(319\) 6.10577e12i 1.84836i
\(320\) 0 0
\(321\) 2.87487e12 0.843517
\(322\) 0 0
\(323\) 2.12158e12 2.12158e12i 0.603457 0.603457i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) −2.54940e12 2.54940e12i −0.681865 0.681865i
\(328\) 0 0
\(329\) 5.49230e12i 1.42487i
\(330\) 0 0
\(331\) −9.13628e11 −0.229948 −0.114974 0.993369i \(-0.536678\pi\)
−0.114974 + 0.993369i \(0.536678\pi\)
\(332\) 0 0
\(333\) −1.61600e12 + 1.61600e12i −0.394657 + 0.394657i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3.53730e12 3.53730e12i −0.813808 0.813808i 0.171395 0.985202i \(-0.445173\pi\)
−0.985202 + 0.171395i \(0.945173\pi\)
\(338\) 0 0
\(339\) 4.99121e12i 1.11483i
\(340\) 0 0
\(341\) −1.07947e13 −2.34119
\(342\) 0 0
\(343\) 9.38566e11 9.38566e11i 0.197694 0.197694i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −4.02321e12 4.02321e12i −0.799697 0.799697i 0.183351 0.983048i \(-0.441306\pi\)
−0.983048 + 0.183351i \(0.941306\pi\)
\(348\) 0 0
\(349\) 4.01641e12i 0.775730i −0.921716 0.387865i \(-0.873213\pi\)
0.921716 0.387865i \(-0.126787\pi\)
\(350\) 0 0
\(351\) −1.01080e11 −0.0189727
\(352\) 0 0
\(353\) 4.80344e11 4.80344e11i 0.0876353 0.0876353i −0.661930 0.749565i \(-0.730262\pi\)
0.749565 + 0.661930i \(0.230262\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 3.54604e12 + 3.54604e12i 0.611509 + 0.611509i
\(358\) 0 0
\(359\) 1.16724e13i 1.95743i −0.205217 0.978716i \(-0.565790\pi\)
0.205217 0.978716i \(-0.434210\pi\)
\(360\) 0 0
\(361\) 1.77453e12 0.289432
\(362\) 0 0
\(363\) −3.39872e12 + 3.39872e12i −0.539240 + 0.539240i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 3.48825e11 + 3.48825e11i 0.0523935 + 0.0523935i 0.732818 0.680425i \(-0.238205\pi\)
−0.680425 + 0.732818i \(0.738205\pi\)
\(368\) 0 0
\(369\) 6.72490e11i 0.0983000i
\(370\) 0 0
\(371\) −3.20107e12 −0.455434
\(372\) 0 0
\(373\) −3.19857e12 + 3.19857e12i −0.443008 + 0.443008i −0.893022 0.450013i \(-0.851419\pi\)
0.450013 + 0.893022i \(0.351419\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 6.44116e11 + 6.44116e11i 0.0845780 + 0.0845780i
\(378\) 0 0
\(379\) 8.46898e12i 1.08302i 0.840695 + 0.541508i \(0.182147\pi\)
−0.840695 + 0.541508i \(0.817853\pi\)
\(380\) 0 0
\(381\) −5.09275e12 −0.634348
\(382\) 0 0
\(383\) −7.15121e12 + 7.15121e12i −0.867732 + 0.867732i −0.992221 0.124489i \(-0.960271\pi\)
0.124489 + 0.992221i \(0.460271\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −1.12104e12 1.12104e12i −0.129141 0.129141i
\(388\) 0 0
\(389\) 6.37482e12i 0.715682i 0.933783 + 0.357841i \(0.116487\pi\)
−0.933783 + 0.357841i \(0.883513\pi\)
\(390\) 0 0
\(391\) 8.46243e12 0.926001
\(392\) 0 0
\(393\) 3.72225e12 3.72225e12i 0.397048 0.397048i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 2.32306e12 + 2.32306e12i 0.235563 + 0.235563i 0.815010 0.579447i \(-0.196731\pi\)
−0.579447 + 0.815010i \(0.696731\pi\)
\(398\) 0 0
\(399\) 7.28159e12i 0.720049i
\(400\) 0 0
\(401\) −1.57234e13 −1.51644 −0.758219 0.652000i \(-0.773930\pi\)
−0.758219 + 0.652000i \(0.773930\pi\)
\(402\) 0 0
\(403\) −1.13876e12 + 1.13876e12i −0.107129 + 0.107129i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.01436e13 + 2.01436e13i 1.80370 + 1.80370i
\(408\) 0 0
\(409\) 4.69349e12i 0.410090i −0.978753 0.205045i \(-0.934266\pi\)
0.978753 0.205045i \(-0.0657340\pi\)
\(410\) 0 0
\(411\) 1.01128e12 0.0862307
\(412\) 0 0
\(413\) −2.64297e12 + 2.64297e12i −0.219959 + 0.219959i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.99122e11 + 1.99122e11i 0.0157920 + 0.0157920i
\(418\) 0 0
\(419\) 5.55630e12i 0.430244i 0.976587 + 0.215122i \(0.0690150\pi\)
−0.976587 + 0.215122i \(0.930985\pi\)
\(420\) 0 0
\(421\) 5.57057e12 0.421201 0.210600 0.977572i \(-0.432458\pi\)
0.210600 + 0.977572i \(0.432458\pi\)
\(422\) 0 0
\(423\) −3.07412e12 + 3.07412e12i −0.226996 + 0.226996i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −2.78645e13 2.78645e13i −1.96296 1.96296i
\(428\) 0 0
\(429\) 1.25997e12i 0.0867110i
\(430\) 0 0
\(431\) 1.81802e13 1.22240 0.611199 0.791477i \(-0.290687\pi\)
0.611199 + 0.791477i \(0.290687\pi\)
\(432\) 0 0
\(433\) 7.48843e12 7.48843e12i 0.491984 0.491984i −0.416947 0.908931i \(-0.636900\pi\)
0.908931 + 0.416947i \(0.136900\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −8.68856e12 8.68856e12i −0.545181 0.545181i
\(438\) 0 0
\(439\) 1.55826e13i 0.955688i −0.878445 0.477844i \(-0.841418\pi\)
0.878445 0.477844i \(-0.158582\pi\)
\(440\) 0 0
\(441\) 6.61062e12 0.396323
\(442\) 0 0
\(443\) 1.09695e13 1.09695e13i 0.642935 0.642935i −0.308341 0.951276i \(-0.599774\pi\)
0.951276 + 0.308341i \(0.0997736\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −3.94535e12 3.94535e12i −0.221079 0.221079i
\(448\) 0 0
\(449\) 8.28951e12i 0.454252i 0.973865 + 0.227126i \(0.0729330\pi\)
−0.973865 + 0.227126i \(0.927067\pi\)
\(450\) 0 0
\(451\) −8.38267e12 −0.449262
\(452\) 0 0
\(453\) 9.84579e12 9.84579e12i 0.516131 0.516131i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.58586e13 + 2.58586e13i 1.29725 + 1.29725i 0.930202 + 0.367048i \(0.119632\pi\)
0.367048 + 0.930202i \(0.380368\pi\)
\(458\) 0 0
\(459\) 3.96954e12i 0.194839i
\(460\) 0 0
\(461\) 2.93741e13 1.41078 0.705392 0.708817i \(-0.250771\pi\)
0.705392 + 0.708817i \(0.250771\pi\)
\(462\) 0 0
\(463\) 1.41383e13 1.41383e13i 0.664494 0.664494i −0.291942 0.956436i \(-0.594301\pi\)
0.956436 + 0.291942i \(0.0943013\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −5.42118e12 5.42118e12i −0.244067 0.244067i 0.574463 0.818530i \(-0.305211\pi\)
−0.818530 + 0.574463i \(0.805211\pi\)
\(468\) 0 0
\(469\) 3.19442e12i 0.140776i
\(470\) 0 0
\(471\) −5.47672e12 −0.236274
\(472\) 0 0
\(473\) −1.39739e13 + 1.39739e13i −0.590216 + 0.590216i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −1.79168e12 1.79168e12i −0.0725553 0.0725553i
\(478\) 0 0
\(479\) 2.01084e13i 0.797442i −0.917072 0.398721i \(-0.869454\pi\)
0.917072 0.398721i \(-0.130546\pi\)
\(480\) 0 0
\(481\) 4.25002e12 0.165069
\(482\) 0 0
\(483\) 1.45222e13 1.45222e13i 0.552455 0.552455i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 3.04347e13 + 3.04347e13i 1.11103 + 1.11103i 0.993012 + 0.118015i \(0.0376532\pi\)
0.118015 + 0.993012i \(0.462347\pi\)
\(488\) 0 0
\(489\) 2.67026e13i 0.955013i
\(490\) 0 0
\(491\) 2.96582e13 1.03929 0.519645 0.854382i \(-0.326064\pi\)
0.519645 + 0.854382i \(0.326064\pi\)
\(492\) 0 0
\(493\) 2.52953e13 2.52953e13i 0.868572 0.868572i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 5.86956e13 + 5.86956e13i 1.93564 + 1.93564i
\(498\) 0 0
\(499\) 4.43963e13i 1.43497i −0.696572 0.717487i \(-0.745292\pi\)
0.696572 0.717487i \(-0.254708\pi\)
\(500\) 0 0
\(501\) 2.22852e13 0.706039
\(502\) 0 0
\(503\) −2.95547e13 + 2.95547e13i −0.917881 + 0.917881i −0.996875 0.0789944i \(-0.974829\pi\)
0.0789944 + 0.996875i \(0.474829\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.35432e13 1.35432e13i −0.404281 0.404281i
\(508\) 0 0
\(509\) 1.33159e13i 0.389747i 0.980828 + 0.194873i \(0.0624296\pi\)
−0.980828 + 0.194873i \(0.937570\pi\)
\(510\) 0 0
\(511\) −2.16434e13 −0.621184
\(512\) 0 0
\(513\) 4.07561e12 4.07561e12i 0.114711 0.114711i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 3.83193e13 + 3.83193e13i 1.03744 + 1.03744i
\(518\) 0 0
\(519\) 5.17090e12i 0.137319i
\(520\) 0 0
\(521\) −1.57642e13 −0.410661 −0.205331 0.978693i \(-0.565827\pi\)
−0.205331 + 0.978693i \(0.565827\pi\)
\(522\) 0 0
\(523\) −1.20803e13 + 1.20803e13i −0.308723 + 0.308723i −0.844414 0.535691i \(-0.820051\pi\)
0.535691 + 0.844414i \(0.320051\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 4.47207e13 + 4.47207e13i 1.10016 + 1.10016i
\(528\) 0 0
\(529\) 6.77005e12i 0.163423i
\(530\) 0 0
\(531\) −2.95862e12 −0.0700836
\(532\) 0 0
\(533\) −8.84313e11 + 8.84313e11i −0.0205575 + 0.0205575i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 4.33568e11 + 4.33568e11i 0.00970924 + 0.00970924i
\(538\) 0 0
\(539\) 8.24022e13i 1.81132i
\(540\) 0 0
\(541\) 5.08818e13 1.09793 0.548966 0.835844i \(-0.315021\pi\)
0.548966 + 0.835844i \(0.315021\pi\)
\(542\) 0 0
\(543\) 1.95398e13 1.95398e13i 0.413924 0.413924i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.49260e13 1.49260e13i −0.304794 0.304794i 0.538092 0.842886i \(-0.319145\pi\)
−0.842886 + 0.538092i \(0.819145\pi\)
\(548\) 0 0
\(549\) 3.11923e13i 0.625440i
\(550\) 0 0
\(551\) −5.19425e13 −1.02274
\(552\) 0 0
\(553\) −2.20738e13 + 2.20738e13i −0.426827 + 0.426827i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.17195e12 + 1.17195e12i 0.0218592 + 0.0218592i 0.717952 0.696093i \(-0.245080\pi\)
−0.696093 + 0.717952i \(0.745080\pi\)
\(558\) 0 0
\(559\) 2.94829e12i 0.0540146i
\(560\) 0 0
\(561\) 4.94808e13 0.890477
\(562\) 0 0
\(563\) 5.14141e13 5.14141e13i 0.908951 0.908951i −0.0872368 0.996188i \(-0.527804\pi\)
0.996188 + 0.0872368i \(0.0278037\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 6.81205e12 + 6.81205e12i 0.116242 + 0.116242i
\(568\) 0 0
\(569\) 3.04638e13i 0.510767i 0.966840 + 0.255384i \(0.0822018\pi\)
−0.966840 + 0.255384i \(0.917798\pi\)
\(570\) 0 0
\(571\) 7.28538e13 1.20025 0.600125 0.799906i \(-0.295117\pi\)
0.600125 + 0.799906i \(0.295117\pi\)
\(572\) 0 0
\(573\) −2.89272e13 + 2.89272e13i −0.468309 + 0.468309i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −7.24079e13 7.24079e13i −1.13216 1.13216i −0.989818 0.142339i \(-0.954538\pi\)
−0.142339 0.989818i \(-0.545462\pi\)
\(578\) 0 0
\(579\) 4.05226e13i 0.622736i
\(580\) 0 0
\(581\) −7.74167e13 −1.16938
\(582\) 0 0
\(583\) −2.23336e13 + 2.23336e13i −0.331600 + 0.331600i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 8.23536e13 + 8.23536e13i 1.18166 + 1.18166i 0.979314 + 0.202345i \(0.0648563\pi\)
0.202345 + 0.979314i \(0.435144\pi\)
\(588\) 0 0
\(589\) 9.18314e13i 1.29543i
\(590\) 0 0
\(591\) −5.32968e12 −0.0739203
\(592\) 0 0
\(593\) −5.89458e13 + 5.89458e13i −0.803858 + 0.803858i −0.983696 0.179839i \(-0.942442\pi\)
0.179839 + 0.983696i \(0.442442\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.31441e13 + 1.31441e13i 0.173325 + 0.173325i
\(598\) 0 0
\(599\) 7.95878e13i 1.03208i −0.856565 0.516039i \(-0.827406\pi\)
0.856565 0.516039i \(-0.172594\pi\)
\(600\) 0 0
\(601\) 1.52648e14 1.94679 0.973396 0.229130i \(-0.0735882\pi\)
0.973396 + 0.229130i \(0.0735882\pi\)
\(602\) 0 0
\(603\) 1.78796e12 1.78796e12i 0.0224270 0.0224270i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1.73177e13 + 1.73177e13i 0.210158 + 0.210158i 0.804335 0.594176i \(-0.202522\pi\)
−0.594176 + 0.804335i \(0.702522\pi\)
\(608\) 0 0
\(609\) 8.68177e13i 1.03639i
\(610\) 0 0
\(611\) 8.08484e12 0.0949435
\(612\) 0 0
\(613\) 7.17442e13 7.17442e13i 0.828866 0.828866i −0.158494 0.987360i \(-0.550664\pi\)
0.987360 + 0.158494i \(0.0506637\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −3.30842e13 3.30842e13i −0.369994 0.369994i 0.497481 0.867475i \(-0.334258\pi\)
−0.867475 + 0.497481i \(0.834258\pi\)
\(618\) 0 0
\(619\) 1.50831e14i 1.65973i −0.557965 0.829865i \(-0.688418\pi\)
0.557965 0.829865i \(-0.311582\pi\)
\(620\) 0 0
\(621\) 1.62566e13 0.176024
\(622\) 0 0
\(623\) 9.49369e13 9.49369e13i 1.01157 1.01157i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) −5.08030e13 5.08030e13i −0.524266 0.524266i
\(628\) 0 0
\(629\) 1.66904e14i 1.69517i
\(630\) 0 0
\(631\) 6.51439e13 0.651219 0.325609 0.945504i \(-0.394431\pi\)
0.325609 + 0.945504i \(0.394431\pi\)
\(632\) 0 0
\(633\) −3.59302e13 + 3.59302e13i −0.353542 + 0.353542i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −8.69286e12 8.69286e12i −0.0828830 0.0828830i
\(638\) 0 0
\(639\) 6.57055e13i 0.616733i
\(640\) 0 0
\(641\) −8.33777e13 −0.770477 −0.385238 0.922817i \(-0.625881\pi\)
−0.385238 + 0.922817i \(0.625881\pi\)
\(642\) 0 0
\(643\) 9.00080e13 9.00080e13i 0.818891 0.818891i −0.167056 0.985947i \(-0.553426\pi\)
0.985947 + 0.167056i \(0.0534262\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.21631e14 + 1.21631e14i 1.07281 + 1.07281i 0.997132 + 0.0756761i \(0.0241115\pi\)
0.0756761 + 0.997132i \(0.475889\pi\)
\(648\) 0 0
\(649\) 3.68796e13i 0.320304i
\(650\) 0 0
\(651\) 1.53489e14 1.31272
\(652\) 0 0
\(653\) −1.39661e14 + 1.39661e14i −1.17628 + 1.17628i −0.195595 + 0.980685i \(0.562664\pi\)
−0.980685 + 0.195595i \(0.937336\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.21141e13 1.21141e13i −0.0989610 0.0989610i
\(658\) 0 0
\(659\) 1.95289e14i 1.57127i 0.618693 + 0.785633i \(0.287663\pi\)
−0.618693 + 0.785633i \(0.712337\pi\)
\(660\) 0 0
\(661\) 8.92478e12 0.0707278 0.0353639 0.999375i \(-0.488741\pi\)
0.0353639 + 0.999375i \(0.488741\pi\)
\(662\) 0 0
\(663\) 5.21988e12 5.21988e12i 0.0407468 0.0407468i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −1.03593e14 1.03593e14i −0.784694 0.784694i
\(668\) 0 0
\(669\) 9.61124e13i 0.717214i
\(670\) 0 0
\(671\) −3.88816e14 −2.85846
\(672\) 0 0
\(673\) −3.20625e13 + 3.20625e13i −0.232232 + 0.232232i −0.813624 0.581392i \(-0.802508\pi\)
0.581392 + 0.813624i \(0.302508\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.96987e13 + 4.96987e13i 0.349463 + 0.349463i 0.859910 0.510446i \(-0.170520\pi\)
−0.510446 + 0.859910i \(0.670520\pi\)
\(678\) 0 0
\(679\) 2.34265e14i 1.62315i
\(680\) 0 0
\(681\) 8.20942e13 0.560502
\(682\) 0 0
\(683\) −2.01777e14 + 2.01777e14i −1.35759 + 1.35759i −0.480704 + 0.876883i \(0.659619\pi\)
−0.876883 + 0.480704i \(0.840381\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −2.10163e13 2.10163e13i −0.137333 0.137333i
\(688\) 0 0
\(689\) 4.71207e12i 0.0303470i
\(690\) 0 0
\(691\) −6.96254e13 −0.441955 −0.220977 0.975279i \(-0.570925\pi\)
−0.220977 + 0.975279i \(0.570925\pi\)
\(692\) 0 0
\(693\) 8.49131e13 8.49131e13i 0.531261 0.531261i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 3.47282e13 + 3.47282e13i 0.211115 + 0.211115i
\(698\) 0 0
\(699\) 1.05759e14i 0.633767i
\(700\) 0 0
\(701\) −1.15316e14 −0.681242 −0.340621 0.940201i \(-0.610637\pi\)
−0.340621 + 0.940201i \(0.610637\pi\)
\(702\) 0 0
\(703\) −1.71364e14 + 1.71364e14i −0.998029 + 0.998029i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.07315e14 + 1.07315e14i 0.607526 + 0.607526i
\(708\) 0 0
\(709\) 5.88271e13i 0.328357i 0.986431 + 0.164178i \(0.0524973\pi\)
−0.986431 + 0.164178i \(0.947503\pi\)
\(710\) 0 0
\(711\) −2.47101e13 −0.135996
\(712\) 0 0
\(713\) 1.83146e14 1.83146e14i 0.993917 0.993917i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.41379e14 1.41379e14i −0.746086 0.746086i
\(718\) 0 0
\(719\) 1.84801e14i 0.961745i 0.876790 + 0.480873i \(0.159680\pi\)
−0.876790 + 0.480873i \(0.840320\pi\)
\(720\) 0 0
\(721\) −2.12343e14 −1.08984
\(722\) 0 0
\(723\) 1.05026e14 1.05026e14i 0.531627 0.531627i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 3.34341e13 + 3.34341e13i 0.164633 + 0.164633i 0.784616 0.619983i \(-0.212860\pi\)
−0.619983 + 0.784616i \(0.712860\pi\)
\(728\) 0 0
\(729\) 7.62560e12i 0.0370370i
\(730\) 0 0
\(731\) 1.15783e14 0.554702
\(732\) 0 0
\(733\) −1.34429e13 + 1.34429e13i −0.0635291 + 0.0635291i −0.738157 0.674628i \(-0.764304\pi\)
0.674628 + 0.738157i \(0.264304\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) −2.22872e13 2.22872e13i −0.102499 0.102499i
\(738\) 0 0
\(739\) 9.19414e12i 0.0417147i −0.999782 0.0208573i \(-0.993360\pi\)
0.999782 0.0208573i \(-0.00663958\pi\)
\(740\) 0 0
\(741\) −1.07187e13 −0.0479791
\(742\) 0 0
\(743\) −3.45280e13 + 3.45280e13i −0.152485 + 0.152485i −0.779227 0.626742i \(-0.784388\pi\)
0.626742 + 0.779227i \(0.284388\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −4.33312e13 4.33312e13i −0.186294 0.186294i
\(748\) 0 0
\(749\) 5.09546e14i 2.16159i
\(750\) 0 0
\(751\) −2.78104e14 −1.16415 −0.582073 0.813136i \(-0.697758\pi\)
−0.582073 + 0.813136i \(0.697758\pi\)
\(752\) 0 0
\(753\) 1.20912e14 1.20912e14i 0.499454 0.499454i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.20330e14 2.20330e14i −0.886327 0.886327i 0.107841 0.994168i \(-0.465606\pi\)
−0.994168 + 0.107841i \(0.965606\pi\)
\(758\) 0 0
\(759\) 2.02640e14i 0.804483i
\(760\) 0 0
\(761\) 4.15151e14 1.62661 0.813303 0.581840i \(-0.197667\pi\)
0.813303 + 0.581840i \(0.197667\pi\)
\(762\) 0 0
\(763\) 4.51858e14 4.51858e14i 1.74735 1.74735i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 3.89054e12 + 3.89054e12i 0.0146566 + 0.0146566i
\(768\) 0 0
\(769\) 3.28902e14i 1.22302i 0.791235 + 0.611512i \(0.209438\pi\)
−0.791235 + 0.611512i \(0.790562\pi\)
\(770\) 0 0
\(771\) 4.29503e13 0.157650
\(772\) 0 0
\(773\) −3.53823e14 + 3.53823e14i −1.28200 + 1.28200i −0.342475 + 0.939527i \(0.611265\pi\)
−0.939527 + 0.342475i \(0.888735\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −2.86421e14 2.86421e14i −1.01135 1.01135i
\(778\) 0 0
\(779\) 7.13124e13i 0.248587i
\(780\) 0 0
\(781\) 8.19028e14 2.81867
\(782\) 0 0
\(783\) 4.85931e13 4.85931e13i 0.165107 0.165107i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −3.42715e14 3.42715e14i −1.13517 1.13517i −0.989305 0.145862i \(-0.953404\pi\)
−0.145862 0.989305i \(-0.546596\pi\)
\(788\) 0 0
\(789\) 8.37441e13i 0.273885i
\(790\) 0 0
\(791\) −8.84648e14 −2.85685
\(792\) 0 0
\(793\) −4.10174e13 + 4.10174e13i −0.130798 + 0.130798i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.34436e14 + 3.34436e14i 1.03997 + 1.03997i 0.999167 + 0.0408030i \(0.0129916\pi\)
0.0408030 + 0.999167i \(0.487008\pi\)
\(798\) 0 0
\(799\) 3.17503e14i 0.975021i
\(800\) 0 0
\(801\) 1.06275e14 0.322306
\(802\) 0 0
\(803\) −1.51004e14 + 1.51004e14i −0.452283 + 0.452283i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1.96127e14 1.96127e14i −0.573020 0.573020i
\(808\) 0 0
\(809\) 4.58044e14i 1.32180i 0.750475 + 0.660898i \(0.229825\pi\)
−0.750475 + 0.660898i \(0.770175\pi\)
\(810\) 0 0
\(811\) −2.39895e14 −0.683780 −0.341890 0.939740i \(-0.611067\pi\)
−0.341890 + 0.939740i \(0.611067\pi\)
\(812\) 0 0
\(813\) 1.13206e14 1.13206e14i 0.318727 0.318727i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.18877e14 1.18877e14i −0.326579 0.326579i
\(818\) 0 0
\(819\) 1.79155e13i 0.0486193i
\(820\) 0 0
\(821\) −4.01069e14 −1.07523 −0.537617 0.843189i \(-0.680675\pi\)
−0.537617 + 0.843189i \(0.680675\pi\)
\(822\) 0 0
\(823\) 1.07810e14 1.07810e14i 0.285534 0.285534i −0.549777 0.835311i \(-0.685287\pi\)
0.835311 + 0.549777i \(0.185287\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −2.85157e14 2.85157e14i −0.737150 0.737150i 0.234876 0.972025i \(-0.424532\pi\)
−0.972025 + 0.234876i \(0.924532\pi\)
\(828\) 0 0
\(829\) 4.81396e14i 1.22950i 0.788720 + 0.614752i \(0.210744\pi\)
−0.788720 + 0.614752i \(0.789256\pi\)
\(830\) 0 0
\(831\) −3.58790e12 −0.00905388
\(832\) 0 0
\(833\) −3.41381e14 + 3.41381e14i −0.851165 + 0.851165i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 8.59097e13 + 8.59097e13i 0.209129 + 0.209129i
\(838\) 0 0
\(839\) 6.74082e14i 1.62145i 0.585430 + 0.810723i \(0.300926\pi\)
−0.585430 + 0.810723i \(0.699074\pi\)
\(840\) 0 0
\(841\) −1.98598e14 −0.472058
\(842\) 0 0
\(843\) −4.79553e13 + 4.79553e13i −0.112641 + 0.112641i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −6.02393e14 6.02393e14i −1.38186 1.38186i
\(848\) 0 0
\(849\) 2.59023e14i 0.587218i
\(850\) 0 0
\(851\) −6.83528e14 −1.53147
\(852\) 0 0
\(853\) −2.47427e13 + 2.47427e13i −0.0547900 + 0.0547900i −0.733971 0.679181i \(-0.762335\pi\)
0.679181 + 0.733971i \(0.262335\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 4.73644e14 + 4.73644e14i 1.02458 + 1.02458i 0.999690 + 0.0248941i \(0.00792485\pi\)
0.0248941 + 0.999690i \(0.492075\pi\)
\(858\) 0 0
\(859\) 9.26066e13i 0.198005i 0.995087 + 0.0990025i \(0.0315652\pi\)
−0.995087 + 0.0990025i \(0.968435\pi\)
\(860\) 0 0
\(861\) 1.19193e14 0.251903
\(862\) 0 0
\(863\) −5.31608e14 + 5.31608e14i −1.11055 + 1.11055i −0.117473 + 0.993076i \(0.537479\pi\)
−0.993076 + 0.117473i \(0.962521\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −4.99672e12 4.99672e12i −0.0101998 0.0101998i
\(868\) 0 0
\(869\) 3.08014e14i 0.621544i
\(870\) 0 0
\(871\) −4.70228e12 −0.00938033
\(872\) 0 0
\(873\) −1.31122e14 + 1.31122e14i −0.258585 + 0.258585i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −7.18639e14 7.18639e14i −1.38520 1.38520i −0.835094 0.550107i \(-0.814587\pi\)
−0.550107 0.835094i \(-0.685413\pi\)
\(878\) 0 0
\(879\) 3.55872e13i 0.0678187i
\(880\) 0 0
\(881\) −2.44985e14 −0.461593 −0.230797 0.973002i \(-0.574133\pi\)
−0.230797 + 0.973002i \(0.574133\pi\)
\(882\) 0 0
\(883\) 2.71238e14 2.71238e14i 0.505297 0.505297i −0.407782 0.913079i \(-0.633698\pi\)
0.913079 + 0.407782i \(0.133698\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.10781e14 + 2.10781e14i 0.383895 + 0.383895i 0.872503 0.488608i \(-0.162495\pi\)
−0.488608 + 0.872503i \(0.662495\pi\)
\(888\) 0 0
\(889\) 9.02644e14i 1.62558i
\(890\) 0 0
\(891\) 9.50541e13 0.169271
\(892\) 0 0
\(893\) −3.25987e14 + 3.25987e14i −0.574041 + 0.574041i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −2.13771e13 2.13771e13i −0.0368118 0.0368118i
\(898\) 0 0
\(899\) 1.09490e15i 1.86455i
\(900\) 0 0
\(901\) 1.85050e14 0.311648
\(902\) 0 0
\(903\) 1.98694e14 1.98694e14i 0.330937 0.330937i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 6.15726e14 + 6.15726e14i 1.00312 + 1.00312i 0.999995 + 0.00312037i \(0.000993247\pi\)
0.00312037 + 0.999995i \(0.499007\pi\)
\(908\) 0 0
\(909\) 1.20132e14i 0.193570i
\(910\) 0 0
\(911\) −1.02185e15 −1.62853 −0.814267 0.580490i \(-0.802861\pi\)
−0.814267 + 0.580490i \(0.802861\pi\)
\(912\) 0 0
\(913\) −5.40129e14 + 5.40129e14i −0.851420 + 0.851420i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 6.59736e14 + 6.59736e14i 1.01747 + 1.01747i
\(918\) 0 0
\(919\) 1.05732e15i 1.61298i 0.591249 + 0.806489i \(0.298635\pi\)
−0.591249 + 0.806489i \(0.701365\pi\)
\(920\) 0 0
\(921\) 4.01519e14 0.605910
\(922\) 0 0
\(923\) 8.64017e13 8.64017e13i 0.128977 0.128977i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −1.18852e14 1.18852e14i −0.173623 0.173623i
\(928\) 0 0
\(929\) 2.32216e14i 0.335594i −0.985822 0.167797i \(-0.946335\pi\)
0.985822 0.167797i \(-0.0536653\pi\)
\(930\) 0 0
\(931\) 7.01006e14 1.00224
\(932\) 0 0
\(933\) 4.20019e13 4.20019e13i 0.0594101 0.0594101i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −4.15284e14 4.15284e14i −0.574973 0.574973i 0.358541 0.933514i \(-0.383274\pi\)
−0.933514 + 0.358541i \(0.883274\pi\)
\(938\) 0 0
\(939\) 6.54823e14i 0.897008i
\(940\) 0 0
\(941\) −6.25302e14 −0.847504 −0.423752 0.905778i \(-0.639287\pi\)
−0.423752 + 0.905778i \(0.639287\pi\)
\(942\) 0 0
\(943\) 1.42223e14 1.42223e14i 0.190727 0.190727i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −2.21564e14 2.21564e14i −0.290903 0.290903i 0.546534 0.837437i \(-0.315947\pi\)
−0.837437 + 0.546534i \(0.815947\pi\)
\(948\) 0 0
\(949\) 3.18597e13i 0.0413914i
\(950\) 0 0
\(951\) 7.09162e14 0.911681
\(952\) 0 0
\(953\) 7.17069e14 7.17069e14i 0.912213 0.912213i −0.0842326 0.996446i \(-0.526844\pi\)
0.996446 + 0.0842326i \(0.0268439\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) −6.05719e14 6.05719e14i −0.754591 0.754591i
\(958\) 0 0
\(959\) 1.79240e14i 0.220975i
\(960\) 0 0
\(961\) 1.11608e15 1.36170
\(962\) 0 0
\(963\) −2.85200e14 + 2.85200e14i −0.344364 + 0.344364i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 2.11433e14 + 2.11433e14i 0.250058 + 0.250058i 0.820994 0.570936i \(-0.193420\pi\)
−0.570936 + 0.820994i \(0.693420\pi\)
\(968\) 0 0
\(969\) 4.20939e14i 0.492721i
\(970\) 0 0
\(971\) −1.45574e15 −1.68651 −0.843255 0.537514i \(-0.819363\pi\)
−0.843255 + 0.537514i \(0.819363\pi\)
\(972\) 0 0
\(973\) −3.52925e13 + 3.52925e13i −0.0404687 + 0.0404687i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 3.31622e14 + 3.31622e14i 0.372538 + 0.372538i 0.868401 0.495863i \(-0.165148\pi\)
−0.495863 + 0.868401i \(0.665148\pi\)
\(978\) 0 0
\(979\) 1.32473e15i 1.47304i
\(980\) 0 0
\(981\) 5.05822e14 0.556741
\(982\) 0 0
\(983\) 9.11608e14 9.11608e14i 0.993209 0.993209i −0.00676795 0.999977i \(-0.502154\pi\)
0.999977 + 0.00676795i \(0.00215432\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −5.44860e14 5.44860e14i −0.581700 0.581700i
\(988\) 0 0
\(989\) 4.74171e14i 0.501134i
\(990\) 0 0
\(991\) 7.69307e14 0.804881 0.402440 0.915446i \(-0.368162\pi\)
0.402440 + 0.915446i \(0.368162\pi\)
\(992\) 0 0
\(993\) 9.06358e13 9.06358e13i 0.0938758 0.0938758i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.07674e14 1.07674e14i −0.109304 0.109304i 0.650340 0.759644i \(-0.274627\pi\)
−0.759644 + 0.650340i \(0.774627\pi\)
\(998\) 0 0
\(999\) 3.20628e14i 0.322236i
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 300.11.k.c.157.1 16
5.2 odd 4 inner 300.11.k.c.193.8 yes 16
5.3 odd 4 inner 300.11.k.c.193.1 yes 16
5.4 even 2 inner 300.11.k.c.157.8 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.11.k.c.157.1 16 1.1 even 1 trivial
300.11.k.c.157.8 yes 16 5.4 even 2 inner
300.11.k.c.193.1 yes 16 5.3 odd 4 inner
300.11.k.c.193.8 yes 16 5.2 odd 4 inner