Properties

Label 300.9.g.h.101.11
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 101.11
Root \(-51.0255 + 62.9079i\) of defining polynomial
Character \(\chi\) \(=\) 300.101
Dual form 300.9.g.h.101.12

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(51.0255 - 62.9079i) q^{3} +459.019 q^{7} +(-1353.80 - 6419.81i) q^{9} +O(q^{10})\) \(q+(51.0255 - 62.9079i) q^{3} +459.019 q^{7} +(-1353.80 - 6419.81i) q^{9} -9139.47i q^{11} -48094.6 q^{13} +53336.1i q^{17} -122304. q^{19} +(23421.7 - 28875.9i) q^{21} +263928. i q^{23} +(-472935. - 242410. i) q^{27} +1.04331e6i q^{29} +1.39629e6 q^{31} +(-574944. - 466346. i) q^{33} +2.84955e6 q^{37} +(-2.45405e6 + 3.02553e6i) q^{39} -2.57085e6i q^{41} +1.51750e6 q^{43} -2.59155e6i q^{47} -5.55410e6 q^{49} +(3.35526e6 + 2.72150e6i) q^{51} +2.74970e6i q^{53} +(-6.24065e6 + 7.69391e6i) q^{57} +1.78378e7i q^{59} -8.28811e6 q^{61} +(-621418. - 2.94681e6i) q^{63} +462829. q^{67} +(1.66031e7 + 1.34671e7i) q^{69} +3.15348e7i q^{71} -1.02650e7 q^{73} -4.19519e6i q^{77} +2.02668e7 q^{79} +(-3.93812e7 + 1.73822e7i) q^{81} +3.52318e7i q^{83} +(6.56326e7 + 5.32356e7i) q^{87} -4.31268e7i q^{89} -2.20763e7 q^{91} +(7.12462e7 - 8.78374e7i) q^{93} -5.78240e7 q^{97} +(-5.86737e7 + 1.23730e7i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 14756 q^{9}+O(q^{10}) \) 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
\(3\) 51.0255 62.9079i 0.629944 0.776640i
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 459.019 0.191178 0.0955890 0.995421i \(-0.469527\pi\)
0.0955890 + 0.995421i \(0.469527\pi\)
\(8\) 0 0
\(9\) −1353.80 6419.81i −0.206340 0.978480i
\(10\) 0 0
\(11\) 9139.47i 0.624238i −0.950043 0.312119i \(-0.898961\pi\)
0.950043 0.312119i \(-0.101039\pi\)
\(12\) 0 0
\(13\) −48094.6 −1.68392 −0.841962 0.539537i \(-0.818599\pi\)
−0.841962 + 0.539537i \(0.818599\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 53336.1i 0.638595i 0.947655 + 0.319297i \(0.103447\pi\)
−0.947655 + 0.319297i \(0.896553\pi\)
\(18\) 0 0
\(19\) −122304. −0.938486 −0.469243 0.883069i \(-0.655473\pi\)
−0.469243 + 0.883069i \(0.655473\pi\)
\(20\) 0 0
\(21\) 23421.7 28875.9i 0.120432 0.148477i
\(22\) 0 0
\(23\) 263928.i 0.943135i 0.881830 + 0.471568i \(0.156312\pi\)
−0.881830 + 0.471568i \(0.843688\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −472935. 242410.i −0.889910 0.456136i
\(28\) 0 0
\(29\) 1.04331e6i 1.47511i 0.675290 + 0.737553i \(0.264019\pi\)
−0.675290 + 0.737553i \(0.735981\pi\)
\(30\) 0 0
\(31\) 1.39629e6 1.51192 0.755958 0.654620i \(-0.227171\pi\)
0.755958 + 0.654620i \(0.227171\pi\)
\(32\) 0 0
\(33\) −574944. 466346.i −0.484808 0.393235i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 2.84955e6 1.52044 0.760221 0.649665i \(-0.225091\pi\)
0.760221 + 0.649665i \(0.225091\pi\)
\(38\) 0 0
\(39\) −2.45405e6 + 3.02553e6i −1.06078 + 1.30780i
\(40\) 0 0
\(41\) 2.57085e6i 0.909790i −0.890545 0.454895i \(-0.849677\pi\)
0.890545 0.454895i \(-0.150323\pi\)
\(42\) 0 0
\(43\) 1.51750e6 0.443868 0.221934 0.975062i \(-0.428763\pi\)
0.221934 + 0.975062i \(0.428763\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 2.59155e6i 0.531089i −0.964099 0.265545i \(-0.914448\pi\)
0.964099 0.265545i \(-0.0855518\pi\)
\(48\) 0 0
\(49\) −5.55410e6 −0.963451
\(50\) 0 0
\(51\) 3.35526e6 + 2.72150e6i 0.495958 + 0.402279i
\(52\) 0 0
\(53\) 2.74970e6i 0.348483i 0.984703 + 0.174241i \(0.0557473\pi\)
−0.984703 + 0.174241i \(0.944253\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −6.24065e6 + 7.69391e6i −0.591194 + 0.728866i
\(58\) 0 0
\(59\) 1.78378e7i 1.47208i 0.676936 + 0.736042i \(0.263307\pi\)
−0.676936 + 0.736042i \(0.736693\pi\)
\(60\) 0 0
\(61\) −8.28811e6 −0.598599 −0.299300 0.954159i \(-0.596753\pi\)
−0.299300 + 0.954159i \(0.596753\pi\)
\(62\) 0 0
\(63\) −621418. 2.94681e6i −0.0394477 0.187064i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 462829. 0.0229679 0.0114840 0.999934i \(-0.496344\pi\)
0.0114840 + 0.999934i \(0.496344\pi\)
\(68\) 0 0
\(69\) 1.66031e7 + 1.34671e7i 0.732477 + 0.594123i
\(70\) 0 0
\(71\) 3.15348e7i 1.24096i 0.784224 + 0.620478i \(0.213061\pi\)
−0.784224 + 0.620478i \(0.786939\pi\)
\(72\) 0 0
\(73\) −1.02650e7 −0.361468 −0.180734 0.983532i \(-0.557847\pi\)
−0.180734 + 0.983532i \(0.557847\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.19519e6i 0.119341i
\(78\) 0 0
\(79\) 2.02668e7 0.520328 0.260164 0.965564i \(-0.416223\pi\)
0.260164 + 0.965564i \(0.416223\pi\)
\(80\) 0 0
\(81\) −3.93812e7 + 1.73822e7i −0.914848 + 0.403799i
\(82\) 0 0
\(83\) 3.52318e7i 0.742373i 0.928558 + 0.371187i \(0.121049\pi\)
−0.928558 + 0.371187i \(0.878951\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 6.56326e7 + 5.32356e7i 1.14563 + 0.929234i
\(88\) 0 0
\(89\) 4.31268e7i 0.687365i −0.939086 0.343682i \(-0.888326\pi\)
0.939086 0.343682i \(-0.111674\pi\)
\(90\) 0 0
\(91\) −2.20763e7 −0.321929
\(92\) 0 0
\(93\) 7.12462e7 8.78374e7i 0.952423 1.17421i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −5.78240e7 −0.653162 −0.326581 0.945169i \(-0.605897\pi\)
−0.326581 + 0.945169i \(0.605897\pi\)
\(98\) 0 0
\(99\) −5.86737e7 + 1.23730e7i −0.610805 + 0.128805i
\(100\) 0 0
\(101\) 4.41147e7i 0.423934i 0.977277 + 0.211967i \(0.0679869\pi\)
−0.977277 + 0.211967i \(0.932013\pi\)
\(102\) 0 0
\(103\) 3.82195e7 0.339575 0.169788 0.985481i \(-0.445692\pi\)
0.169788 + 0.985481i \(0.445692\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 6.27380e7i 0.478625i −0.970943 0.239313i \(-0.923078\pi\)
0.970943 0.239313i \(-0.0769221\pi\)
\(108\) 0 0
\(109\) 3.48648e7 0.246991 0.123496 0.992345i \(-0.460590\pi\)
0.123496 + 0.992345i \(0.460590\pi\)
\(110\) 0 0
\(111\) 1.45400e8 1.79259e8i 0.957793 1.18084i
\(112\) 0 0
\(113\) 2.69695e8i 1.65409i 0.562135 + 0.827046i \(0.309980\pi\)
−0.562135 + 0.827046i \(0.690020\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 6.51102e7 + 3.08758e8i 0.347461 + 1.64769i
\(118\) 0 0
\(119\) 2.44823e7i 0.122085i
\(120\) 0 0
\(121\) 1.30829e8 0.610327
\(122\) 0 0
\(123\) −1.61727e8 1.31179e8i −0.706579 0.573117i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 4.94693e8 1.90161 0.950803 0.309796i \(-0.100261\pi\)
0.950803 + 0.309796i \(0.100261\pi\)
\(128\) 0 0
\(129\) 7.74310e7 9.54625e7i 0.279612 0.344726i
\(130\) 0 0
\(131\) 2.56571e8i 0.871208i −0.900138 0.435604i \(-0.856535\pi\)
0.900138 0.435604i \(-0.143465\pi\)
\(132\) 0 0
\(133\) −5.61400e7 −0.179418
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8.06580e7i 0.228963i 0.993425 + 0.114481i \(0.0365206\pi\)
−0.993425 + 0.114481i \(0.963479\pi\)
\(138\) 0 0
\(139\) −5.03868e8 −1.34976 −0.674882 0.737926i \(-0.735805\pi\)
−0.674882 + 0.737926i \(0.735805\pi\)
\(140\) 0 0
\(141\) −1.63029e8 1.32235e8i −0.412465 0.334557i
\(142\) 0 0
\(143\) 4.39559e8i 1.05117i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) −2.83401e8 + 3.49397e8i −0.606921 + 0.748255i
\(148\) 0 0
\(149\) 9.26379e8i 1.87951i −0.341854 0.939753i \(-0.611055\pi\)
0.341854 0.939753i \(-0.388945\pi\)
\(150\) 0 0
\(151\) 1.66966e8 0.321158 0.160579 0.987023i \(-0.448664\pi\)
0.160579 + 0.987023i \(0.448664\pi\)
\(152\) 0 0
\(153\) 3.42407e8 7.22062e7i 0.624852 0.131768i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 7.60014e8 1.25090 0.625451 0.780264i \(-0.284915\pi\)
0.625451 + 0.780264i \(0.284915\pi\)
\(158\) 0 0
\(159\) 1.72978e8 + 1.40305e8i 0.270646 + 0.219525i
\(160\) 0 0
\(161\) 1.21148e8i 0.180307i
\(162\) 0 0
\(163\) 1.02534e9 1.45250 0.726252 0.687429i \(-0.241261\pi\)
0.726252 + 0.687429i \(0.241261\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.26101e9i 1.62126i 0.585560 + 0.810629i \(0.300875\pi\)
−0.585560 + 0.810629i \(0.699125\pi\)
\(168\) 0 0
\(169\) 1.49736e9 1.83560
\(170\) 0 0
\(171\) 1.65575e8 + 7.85171e8i 0.193647 + 0.918290i
\(172\) 0 0
\(173\) 5.01044e7i 0.0559360i −0.999609 0.0279680i \(-0.991096\pi\)
0.999609 0.0279680i \(-0.00890365\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.12214e9 + 9.10181e8i 1.14328 + 0.927331i
\(178\) 0 0
\(179\) 1.00928e9i 0.983108i 0.870847 + 0.491554i \(0.163571\pi\)
−0.870847 + 0.491554i \(0.836429\pi\)
\(180\) 0 0
\(181\) −1.77364e9 −1.65254 −0.826268 0.563277i \(-0.809540\pi\)
−0.826268 + 0.563277i \(0.809540\pi\)
\(182\) 0 0
\(183\) −4.22905e8 + 5.21387e8i −0.377084 + 0.464896i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 4.87463e8 0.398635
\(188\) 0 0
\(189\) −2.17086e8 1.11271e8i −0.170131 0.0872033i
\(190\) 0 0
\(191\) 1.24397e9i 0.934712i 0.884069 + 0.467356i \(0.154793\pi\)
−0.884069 + 0.467356i \(0.845207\pi\)
\(192\) 0 0
\(193\) 2.76609e7 0.0199359 0.00996797 0.999950i \(-0.496827\pi\)
0.00996797 + 0.999950i \(0.496827\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 2.12130e9i 1.40844i 0.709984 + 0.704218i \(0.248702\pi\)
−0.709984 + 0.704218i \(0.751298\pi\)
\(198\) 0 0
\(199\) −7.80182e8 −0.497489 −0.248745 0.968569i \(-0.580018\pi\)
−0.248745 + 0.968569i \(0.580018\pi\)
\(200\) 0 0
\(201\) 2.36161e7 2.91156e7i 0.0144685 0.0178378i
\(202\) 0 0
\(203\) 4.78901e8i 0.282008i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 1.69437e9 3.57305e8i 0.922840 0.194607i
\(208\) 0 0
\(209\) 1.11780e9i 0.585839i
\(210\) 0 0
\(211\) 1.47843e9 0.745884 0.372942 0.927855i \(-0.378349\pi\)
0.372942 + 0.927855i \(0.378349\pi\)
\(212\) 0 0
\(213\) 1.98379e9 + 1.60908e9i 0.963776 + 0.781734i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 6.40921e8 0.289045
\(218\) 0 0
\(219\) −5.23779e8 + 6.45752e8i −0.227705 + 0.280730i
\(220\) 0 0
\(221\) 2.56517e9i 1.07535i
\(222\) 0 0
\(223\) −9.39456e8 −0.379889 −0.189945 0.981795i \(-0.560831\pi\)
−0.189945 + 0.981795i \(0.560831\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.93076e9i 1.85699i 0.371339 + 0.928497i \(0.378899\pi\)
−0.371339 + 0.928497i \(0.621101\pi\)
\(228\) 0 0
\(229\) −3.90618e9 −1.42040 −0.710199 0.704001i \(-0.751395\pi\)
−0.710199 + 0.704001i \(0.751395\pi\)
\(230\) 0 0
\(231\) −2.63910e8 2.14062e8i −0.0926847 0.0751780i
\(232\) 0 0
\(233\) 4.02506e7i 0.0136568i −0.999977 0.00682839i \(-0.997826\pi\)
0.999977 0.00682839i \(-0.00217356\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.03413e9 1.27494e9i 0.327778 0.404108i
\(238\) 0 0
\(239\) 5.06846e9i 1.55340i 0.629869 + 0.776702i \(0.283109\pi\)
−0.629869 + 0.776702i \(0.716891\pi\)
\(240\) 0 0
\(241\) 2.04398e9 0.605910 0.302955 0.953005i \(-0.402027\pi\)
0.302955 + 0.953005i \(0.402027\pi\)
\(242\) 0 0
\(243\) −9.15966e8 + 3.36432e9i −0.262697 + 0.964878i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 5.88218e9 1.58034
\(248\) 0 0
\(249\) 2.21636e9 + 1.79772e9i 0.576557 + 0.467654i
\(250\) 0 0
\(251\) 2.48750e9i 0.626711i 0.949636 + 0.313356i \(0.101453\pi\)
−0.949636 + 0.313356i \(0.898547\pi\)
\(252\) 0 0
\(253\) 2.41216e9 0.588741
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 4.66441e9i 1.06921i 0.845101 + 0.534606i \(0.179540\pi\)
−0.845101 + 0.534606i \(0.820460\pi\)
\(258\) 0 0
\(259\) 1.30800e9 0.290675
\(260\) 0 0
\(261\) 6.69788e9 1.41243e9i 1.44336 0.304373i
\(262\) 0 0
\(263\) 5.71598e9i 1.19473i −0.801971 0.597363i \(-0.796215\pi\)
0.801971 0.597363i \(-0.203785\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −2.71301e9 2.20057e9i −0.533835 0.433001i
\(268\) 0 0
\(269\) 5.53953e9i 1.05795i −0.848639 0.528973i \(-0.822577\pi\)
0.848639 0.528973i \(-0.177423\pi\)
\(270\) 0 0
\(271\) −8.41658e9 −1.56048 −0.780241 0.625479i \(-0.784904\pi\)
−0.780241 + 0.625479i \(0.784904\pi\)
\(272\) 0 0
\(273\) −1.12645e9 + 1.38877e9i −0.202798 + 0.250023i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.24754e9 0.211901 0.105951 0.994371i \(-0.466211\pi\)
0.105951 + 0.994371i \(0.466211\pi\)
\(278\) 0 0
\(279\) −1.89029e9 8.96389e9i −0.311969 1.47938i
\(280\) 0 0
\(281\) 6.20826e9i 0.995737i −0.867253 0.497868i \(-0.834116\pi\)
0.867253 0.497868i \(-0.165884\pi\)
\(282\) 0 0
\(283\) 4.82589e9 0.752370 0.376185 0.926545i \(-0.377236\pi\)
0.376185 + 0.926545i \(0.377236\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 1.18007e9i 0.173932i
\(288\) 0 0
\(289\) 4.13102e9 0.592197
\(290\) 0 0
\(291\) −2.95050e9 + 3.63758e9i −0.411456 + 0.507272i
\(292\) 0 0
\(293\) 4.75107e9i 0.644645i 0.946630 + 0.322323i \(0.104464\pi\)
−0.946630 + 0.322323i \(0.895536\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) −2.21550e9 + 4.32237e9i −0.284738 + 0.555516i
\(298\) 0 0
\(299\) 1.26935e10i 1.58817i
\(300\) 0 0
\(301\) 6.96559e8 0.0848579
\(302\) 0 0
\(303\) 2.77516e9 + 2.25097e9i 0.329244 + 0.267055i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −6.53537e9 −0.735727 −0.367863 0.929880i \(-0.619911\pi\)
−0.367863 + 0.929880i \(0.619911\pi\)
\(308\) 0 0
\(309\) 1.95017e9 2.40431e9i 0.213914 0.263728i
\(310\) 0 0
\(311\) 9.42520e9i 1.00751i 0.863847 + 0.503755i \(0.168048\pi\)
−0.863847 + 0.503755i \(0.831952\pi\)
\(312\) 0 0
\(313\) −1.16410e10 −1.21287 −0.606433 0.795134i \(-0.707400\pi\)
−0.606433 + 0.795134i \(0.707400\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 9.36294e9i 0.927203i −0.886044 0.463602i \(-0.846557\pi\)
0.886044 0.463602i \(-0.153443\pi\)
\(318\) 0 0
\(319\) 9.53534e9 0.920817
\(320\) 0 0
\(321\) −3.94671e9 3.20124e9i −0.371720 0.301507i
\(322\) 0 0
\(323\) 6.52324e9i 0.599312i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 1.77899e9 2.19327e9i 0.155591 0.191823i
\(328\) 0 0
\(329\) 1.18957e9i 0.101533i
\(330\) 0 0
\(331\) −1.04898e10 −0.873886 −0.436943 0.899489i \(-0.643939\pi\)
−0.436943 + 0.899489i \(0.643939\pi\)
\(332\) 0 0
\(333\) −3.85771e9 1.82936e10i −0.313728 1.48772i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −7.07467e9 −0.548512 −0.274256 0.961657i \(-0.588432\pi\)
−0.274256 + 0.961657i \(0.588432\pi\)
\(338\) 0 0
\(339\) 1.69659e10 + 1.37613e10i 1.28463 + 1.04199i
\(340\) 0 0
\(341\) 1.27613e10i 0.943796i
\(342\) 0 0
\(343\) −5.19559e9 −0.375369
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 8.94083e9i 0.616680i −0.951276 0.308340i \(-0.900227\pi\)
0.951276 0.308340i \(-0.0997734\pi\)
\(348\) 0 0
\(349\) −2.42213e10 −1.63266 −0.816329 0.577588i \(-0.803994\pi\)
−0.816329 + 0.577588i \(0.803994\pi\)
\(350\) 0 0
\(351\) 2.27456e10 + 1.16586e10i 1.49854 + 0.768099i
\(352\) 0 0
\(353\) 2.34151e10i 1.50798i 0.656884 + 0.753991i \(0.271874\pi\)
−0.656884 + 0.753991i \(0.728126\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 1.54013e9 + 1.24922e9i 0.0948164 + 0.0769070i
\(358\) 0 0
\(359\) 7.48179e9i 0.450431i 0.974309 + 0.225215i \(0.0723086\pi\)
−0.974309 + 0.225215i \(0.927691\pi\)
\(360\) 0 0
\(361\) −2.02518e9 −0.119244
\(362\) 0 0
\(363\) 6.67561e9 8.23017e9i 0.384472 0.474004i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −3.08337e10 −1.69966 −0.849830 0.527057i \(-0.823295\pi\)
−0.849830 + 0.527057i \(0.823295\pi\)
\(368\) 0 0
\(369\) −1.65044e10 + 3.48040e9i −0.890211 + 0.187726i
\(370\) 0 0
\(371\) 1.26216e9i 0.0666223i
\(372\) 0 0
\(373\) 2.68065e10 1.38486 0.692428 0.721487i \(-0.256541\pi\)
0.692428 + 0.721487i \(0.256541\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 5.01777e10i 2.48397i
\(378\) 0 0
\(379\) −2.56615e10 −1.24373 −0.621863 0.783126i \(-0.713624\pi\)
−0.621863 + 0.783126i \(0.713624\pi\)
\(380\) 0 0
\(381\) 2.52419e10 3.11201e10i 1.19791 1.47686i
\(382\) 0 0
\(383\) 1.75285e10i 0.814612i −0.913292 0.407306i \(-0.866468\pi\)
0.913292 0.407306i \(-0.133532\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.05438e9 9.74204e9i −0.0915877 0.434316i
\(388\) 0 0
\(389\) 3.24519e10i 1.41723i 0.705594 + 0.708617i \(0.250680\pi\)
−0.705594 + 0.708617i \(0.749320\pi\)
\(390\) 0 0
\(391\) −1.40769e10 −0.602281
\(392\) 0 0
\(393\) −1.61403e10 1.30916e10i −0.676615 0.548813i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 4.58427e9 0.184548 0.0922738 0.995734i \(-0.470586\pi\)
0.0922738 + 0.995734i \(0.470586\pi\)
\(398\) 0 0
\(399\) −2.86457e9 + 3.53165e9i −0.113023 + 0.139343i
\(400\) 0 0
\(401\) 3.56669e10i 1.37939i 0.724100 + 0.689695i \(0.242256\pi\)
−0.724100 + 0.689695i \(0.757744\pi\)
\(402\) 0 0
\(403\) −6.71538e10 −2.54595
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.60434e10i 0.949117i
\(408\) 0 0
\(409\) 1.87379e10 0.669620 0.334810 0.942286i \(-0.391328\pi\)
0.334810 + 0.942286i \(0.391328\pi\)
\(410\) 0 0
\(411\) 5.07402e9 + 4.11561e9i 0.177822 + 0.144234i
\(412\) 0 0
\(413\) 8.18787e9i 0.281430i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2.57101e10 + 3.16973e10i −0.850276 + 1.04828i
\(418\) 0 0
\(419\) 2.82911e10i 0.917898i 0.888463 + 0.458949i \(0.151774\pi\)
−0.888463 + 0.458949i \(0.848226\pi\)
\(420\) 0 0
\(421\) −2.06273e10 −0.656618 −0.328309 0.944570i \(-0.606479\pi\)
−0.328309 + 0.944570i \(0.606479\pi\)
\(422\) 0 0
\(423\) −1.66372e10 + 3.50843e9i −0.519661 + 0.109585i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −3.80440e9 −0.114439
\(428\) 0 0
\(429\) 2.76517e10 + 2.24287e10i 0.816381 + 0.662178i
\(430\) 0 0
\(431\) 1.72749e10i 0.500618i −0.968166 0.250309i \(-0.919468\pi\)
0.968166 0.250309i \(-0.0805322\pi\)
\(432\) 0 0
\(433\) 3.35377e10 0.954075 0.477037 0.878883i \(-0.341711\pi\)
0.477037 + 0.878883i \(0.341711\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.22796e10i 0.885120i
\(438\) 0 0
\(439\) −3.58022e6 −9.63945e−5 −4.81972e−5 1.00000i \(-0.500015\pi\)
−4.81972e−5 1.00000i \(0.500015\pi\)
\(440\) 0 0
\(441\) 7.51912e9 + 3.56563e10i 0.198798 + 0.942718i
\(442\) 0 0
\(443\) 1.35038e10i 0.350623i −0.984513 0.175312i \(-0.943907\pi\)
0.984513 0.175312i \(-0.0560933\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −5.82765e10 4.72690e10i −1.45970 1.18398i
\(448\) 0 0
\(449\) 6.60570e10i 1.62530i −0.582752 0.812650i \(-0.698024\pi\)
0.582752 0.812650i \(-0.301976\pi\)
\(450\) 0 0
\(451\) −2.34962e10 −0.567925
\(452\) 0 0
\(453\) 8.51950e9 1.05034e10i 0.202312 0.249424i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2.96351e10 −0.679425 −0.339712 0.940529i \(-0.610330\pi\)
−0.339712 + 0.940529i \(0.610330\pi\)
\(458\) 0 0
\(459\) 1.29292e10 2.52245e10i 0.291286 0.568292i
\(460\) 0 0
\(461\) 4.06549e10i 0.900139i 0.892994 + 0.450069i \(0.148601\pi\)
−0.892994 + 0.450069i \(0.851399\pi\)
\(462\) 0 0
\(463\) 4.18983e10 0.911743 0.455872 0.890046i \(-0.349328\pi\)
0.455872 + 0.890046i \(0.349328\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 4.13906e10i 0.870230i 0.900375 + 0.435115i \(0.143292\pi\)
−0.900375 + 0.435115i \(0.856708\pi\)
\(468\) 0 0
\(469\) 2.12447e8 0.00439096
\(470\) 0 0
\(471\) 3.87801e10 4.78109e10i 0.787999 0.971501i
\(472\) 0 0
\(473\) 1.38691e10i 0.277079i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 1.76525e10 3.72253e9i 0.340984 0.0719059i
\(478\) 0 0
\(479\) 7.07355e10i 1.34368i −0.740697 0.671839i \(-0.765505\pi\)
0.740697 0.671839i \(-0.234495\pi\)
\(480\) 0 0
\(481\) −1.37048e11 −2.56031
\(482\) 0 0
\(483\) 7.62115e9 + 6.18163e9i 0.140034 + 0.113583i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.97837e10 0.351715 0.175857 0.984416i \(-0.443730\pi\)
0.175857 + 0.984416i \(0.443730\pi\)
\(488\) 0 0
\(489\) 5.23185e10 6.45019e10i 0.914997 1.12807i
\(490\) 0 0
\(491\) 8.33438e10i 1.43399i −0.697076 0.716997i \(-0.745516\pi\)
0.697076 0.716997i \(-0.254484\pi\)
\(492\) 0 0
\(493\) −5.56463e10 −0.941995
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.44751e10i 0.237244i
\(498\) 0 0
\(499\) −4.62385e10 −0.745764 −0.372882 0.927879i \(-0.621630\pi\)
−0.372882 + 0.927879i \(0.621630\pi\)
\(500\) 0 0
\(501\) 7.93274e10 + 6.43436e10i 1.25913 + 1.02130i
\(502\) 0 0
\(503\) 8.85743e10i 1.38368i −0.722051 0.691840i \(-0.756800\pi\)
0.722051 0.691840i \(-0.243200\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 7.64033e10 9.41954e10i 1.15633 1.42560i
\(508\) 0 0
\(509\) 4.40674e10i 0.656518i 0.944588 + 0.328259i \(0.106462\pi\)
−0.944588 + 0.328259i \(0.893538\pi\)
\(510\) 0 0
\(511\) −4.71185e9 −0.0691047
\(512\) 0 0
\(513\) 5.78420e10 + 2.96478e10i 0.835168 + 0.428078i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −2.36854e10 −0.331526
\(518\) 0 0
\(519\) −3.15196e9 2.55660e9i −0.0434422 0.0352366i
\(520\) 0 0
\(521\) 1.51676e10i 0.205857i 0.994689 + 0.102929i \(0.0328213\pi\)
−0.994689 + 0.102929i \(0.967179\pi\)
\(522\) 0 0
\(523\) −2.51178e10 −0.335719 −0.167859 0.985811i \(-0.553685\pi\)
−0.167859 + 0.985811i \(0.553685\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 7.44724e10i 0.965502i
\(528\) 0 0
\(529\) 8.65301e9 0.110496
\(530\) 0 0
\(531\) 1.14515e11 2.41487e10i 1.44040 0.303750i
\(532\) 0 0
\(533\) 1.23644e11i 1.53202i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 6.34919e10 + 5.14992e10i 0.763521 + 0.619303i
\(538\) 0 0
\(539\) 5.07616e10i 0.601423i
\(540\) 0 0
\(541\) 2.12945e10 0.248587 0.124294 0.992245i \(-0.460334\pi\)
0.124294 + 0.992245i \(0.460334\pi\)
\(542\) 0 0
\(543\) −9.05008e10 + 1.11576e11i −1.04101 + 1.28343i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.20728e10 −0.358251 −0.179125 0.983826i \(-0.557327\pi\)
−0.179125 + 0.983826i \(0.557327\pi\)
\(548\) 0 0
\(549\) 1.12204e10 + 5.32081e10i 0.123515 + 0.585718i
\(550\) 0 0
\(551\) 1.27602e11i 1.38437i
\(552\) 0 0
\(553\) 9.30285e9 0.0994754
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.22209e11i 1.26965i 0.772657 + 0.634824i \(0.218927\pi\)
−0.772657 + 0.634824i \(0.781073\pi\)
\(558\) 0 0
\(559\) −7.29833e10 −0.747440
\(560\) 0 0
\(561\) 2.48731e10 3.06653e10i 0.251118 0.309596i
\(562\) 0 0
\(563\) 1.65935e11i 1.65160i −0.563965 0.825798i \(-0.690725\pi\)
0.563965 0.825798i \(-0.309275\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −1.80767e10 + 7.97877e9i −0.174899 + 0.0771976i
\(568\) 0 0
\(569\) 4.71831e10i 0.450130i −0.974344 0.225065i \(-0.927741\pi\)
0.974344 0.225065i \(-0.0722594\pi\)
\(570\) 0 0
\(571\) −2.58965e10 −0.243611 −0.121805 0.992554i \(-0.538868\pi\)
−0.121805 + 0.992554i \(0.538868\pi\)
\(572\) 0 0
\(573\) 7.82557e10 + 6.34744e10i 0.725935 + 0.588817i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 8.31317e10 0.750004 0.375002 0.927024i \(-0.377642\pi\)
0.375002 + 0.927024i \(0.377642\pi\)
\(578\) 0 0
\(579\) 1.41141e9 1.74009e9i 0.0125585 0.0154831i
\(580\) 0 0
\(581\) 1.61720e10i 0.141925i
\(582\) 0 0
\(583\) 2.51308e10 0.217536
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 1.44597e10i 0.121789i −0.998144 0.0608944i \(-0.980605\pi\)
0.998144 0.0608944i \(-0.0193953\pi\)
\(588\) 0 0
\(589\) −1.70772e11 −1.41891
\(590\) 0 0
\(591\) 1.33446e11 + 1.08240e11i 1.09385 + 0.887237i
\(592\) 0 0
\(593\) 1.00123e11i 0.809687i 0.914386 + 0.404843i \(0.132674\pi\)
−0.914386 + 0.404843i \(0.867326\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −3.98092e10 + 4.90796e10i −0.313391 + 0.386370i
\(598\) 0 0
\(599\) 5.36658e10i 0.416860i 0.978037 + 0.208430i \(0.0668354\pi\)
−0.978037 + 0.208430i \(0.933165\pi\)
\(600\) 0 0
\(601\) −2.45026e11 −1.87808 −0.939042 0.343802i \(-0.888285\pi\)
−0.939042 + 0.343802i \(0.888285\pi\)
\(602\) 0 0
\(603\) −6.26576e8 2.97127e9i −0.00473920 0.0224736i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 2.54344e11 1.87355 0.936777 0.349926i \(-0.113793\pi\)
0.936777 + 0.349926i \(0.113793\pi\)
\(608\) 0 0
\(609\) 3.01266e10 + 2.44361e10i 0.219019 + 0.177649i
\(610\) 0 0
\(611\) 1.24639e11i 0.894314i
\(612\) 0 0
\(613\) 1.26140e11 0.893330 0.446665 0.894701i \(-0.352612\pi\)
0.446665 + 0.894701i \(0.352612\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 2.04776e11i 1.41299i 0.707718 + 0.706495i \(0.249725\pi\)
−0.707718 + 0.706495i \(0.750275\pi\)
\(618\) 0 0
\(619\) 6.98668e10 0.475892 0.237946 0.971278i \(-0.423526\pi\)
0.237946 + 0.971278i \(0.423526\pi\)
\(620\) 0 0
\(621\) 6.39787e10 1.24821e11i 0.430198 0.839306i
\(622\) 0 0
\(623\) 1.97960e10i 0.131409i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 7.03183e10 + 5.70362e10i 0.454986 + 0.369046i
\(628\) 0 0
\(629\) 1.51984e11i 0.970946i
\(630\) 0 0
\(631\) −2.51868e11 −1.58875 −0.794373 0.607430i \(-0.792201\pi\)
−0.794373 + 0.607430i \(0.792201\pi\)
\(632\) 0 0
\(633\) 7.54377e10 9.30050e10i 0.469866 0.579284i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 2.67122e11 1.62238
\(638\) 0 0
\(639\) 2.02447e11 4.26917e10i 1.21425 0.256059i
\(640\) 0 0
\(641\) 2.41559e11i 1.43084i 0.698695 + 0.715420i \(0.253765\pi\)
−0.698695 + 0.715420i \(0.746235\pi\)
\(642\) 0 0
\(643\) 6.40818e9 0.0374879 0.0187439 0.999824i \(-0.494033\pi\)
0.0187439 + 0.999824i \(0.494033\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 1.87104e11i 1.06774i 0.845566 + 0.533871i \(0.179263\pi\)
−0.845566 + 0.533871i \(0.820737\pi\)
\(648\) 0 0
\(649\) 1.63028e11 0.918931
\(650\) 0 0
\(651\) 3.27033e10 4.03190e10i 0.182082 0.224484i
\(652\) 0 0
\(653\) 2.67029e11i 1.46861i 0.678821 + 0.734304i \(0.262491\pi\)
−0.678821 + 0.734304i \(0.737509\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.38968e10 + 6.58997e10i 0.0745852 + 0.353689i
\(658\) 0 0
\(659\) 7.73297e10i 0.410020i 0.978760 + 0.205010i \(0.0657226\pi\)
−0.978760 + 0.205010i \(0.934277\pi\)
\(660\) 0 0
\(661\) −2.86028e9 −0.0149831 −0.00749157 0.999972i \(-0.502385\pi\)
−0.00749157 + 0.999972i \(0.502385\pi\)
\(662\) 0 0
\(663\) −1.61370e11 1.30889e11i −0.835156 0.677408i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2.75360e11 −1.39122
\(668\) 0 0
\(669\) −4.79362e10 + 5.90992e10i −0.239309 + 0.295037i
\(670\) 0 0
\(671\) 7.57489e10i 0.373668i
\(672\) 0 0
\(673\) 3.48736e11 1.69995 0.849976 0.526821i \(-0.176616\pi\)
0.849976 + 0.526821i \(0.176616\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 2.13457e11i 1.01615i −0.861314 0.508073i \(-0.830358\pi\)
0.861314 0.508073i \(-0.169642\pi\)
\(678\) 0 0
\(679\) −2.65423e10 −0.124870
\(680\) 0 0
\(681\) 3.10184e11 + 2.51595e11i 1.44222 + 1.16980i
\(682\) 0 0
\(683\) 5.28052e10i 0.242657i 0.992612 + 0.121329i \(0.0387155\pi\)
−0.992612 + 0.121329i \(0.961284\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.99315e11 + 2.45729e11i −0.894772 + 1.10314i
\(688\) 0 0
\(689\) 1.32245e11i 0.586818i
\(690\) 0 0
\(691\) −3.18883e11 −1.39868 −0.699341 0.714788i \(-0.746523\pi\)
−0.699341 + 0.714788i \(0.746523\pi\)
\(692\) 0 0
\(693\) −2.69323e10 + 5.67943e9i −0.116772 + 0.0246247i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 1.37119e11 0.580987
\(698\) 0 0
\(699\) −2.53208e9 2.05381e9i −0.0106064 0.00860302i
\(700\) 0 0
\(701\) 3.03198e11i 1.25561i −0.778371 0.627804i \(-0.783954\pi\)
0.778371 0.627804i \(-0.216046\pi\)
\(702\) 0 0
\(703\) −3.48513e11 −1.42691
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 2.02495e10i 0.0810468i
\(708\) 0 0
\(709\) −4.44301e11 −1.75830 −0.879149 0.476547i \(-0.841888\pi\)
−0.879149 + 0.476547i \(0.841888\pi\)
\(710\) 0 0
\(711\) −2.74372e10 1.30109e11i −0.107365 0.509131i
\(712\) 0 0
\(713\) 3.68519e11i 1.42594i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 3.18846e11 + 2.58621e11i 1.20644 + 0.978558i
\(718\) 0 0
\(719\) 3.95363e11i 1.47938i −0.672946 0.739692i \(-0.734971\pi\)
0.672946 0.739692i \(-0.265029\pi\)
\(720\) 0 0
\(721\) 1.75435e10 0.0649194
\(722\) 0 0
\(723\) 1.04295e11 1.28582e11i 0.381690 0.470574i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.95379e10 0.0699423 0.0349712 0.999388i \(-0.488866\pi\)
0.0349712 + 0.999388i \(0.488866\pi\)
\(728\) 0 0
\(729\) 1.64905e11 + 2.29288e11i 0.583879 + 0.811841i
\(730\) 0 0
\(731\) 8.09373e10i 0.283452i
\(732\) 0 0
\(733\) −2.26211e11 −0.783607 −0.391803 0.920049i \(-0.628149\pi\)
−0.391803 + 0.920049i \(0.628149\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.23001e9i 0.0143374i
\(738\) 0 0
\(739\) −1.33510e11 −0.447649 −0.223824 0.974630i \(-0.571854\pi\)
−0.223824 + 0.974630i \(0.571854\pi\)
\(740\) 0 0
\(741\) 3.00141e11 3.70035e11i 0.995526 1.22736i
\(742\) 0 0
\(743\) 2.51159e11i 0.824125i −0.911156 0.412062i \(-0.864809\pi\)
0.911156 0.412062i \(-0.135191\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 2.26181e11 4.76967e10i 0.726398 0.153181i
\(748\) 0 0
\(749\) 2.87979e10i 0.0915027i
\(750\) 0 0
\(751\) 1.05457e11 0.331525 0.165762 0.986166i \(-0.446992\pi\)
0.165762 + 0.986166i \(0.446992\pi\)
\(752\) 0 0
\(753\) 1.56483e11 + 1.26926e11i 0.486729 + 0.394793i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −7.46755e10 −0.227402 −0.113701 0.993515i \(-0.536271\pi\)
−0.113701 + 0.993515i \(0.536271\pi\)
\(758\) 0 0
\(759\) 1.23082e11 1.51744e11i 0.370874 0.457240i
\(760\) 0 0
\(761\) 4.79443e11i 1.42955i −0.699356 0.714773i \(-0.746530\pi\)
0.699356 0.714773i \(-0.253470\pi\)
\(762\) 0 0
\(763\) 1.60036e10 0.0472193
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 8.57899e11i 2.47888i
\(768\) 0 0
\(769\) 1.54805e11 0.442669 0.221334 0.975198i \(-0.428959\pi\)
0.221334 + 0.975198i \(0.428959\pi\)
\(770\) 0 0
\(771\) 2.93428e11 + 2.38004e11i 0.830393 + 0.673544i
\(772\) 0 0
\(773\) 5.73981e10i 0.160761i −0.996764 0.0803803i \(-0.974387\pi\)
0.996764 0.0803803i \(-0.0256135\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 6.67412e10 8.22833e10i 0.183109 0.225750i
\(778\) 0 0
\(779\) 3.14426e11i 0.853825i
\(780\) 0 0
\(781\) 2.88211e11 0.774652
\(782\) 0 0
\(783\) 2.52909e11 4.93419e11i 0.672849 1.31271i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) −3.98550e11 −1.03893 −0.519463 0.854493i \(-0.673868\pi\)
−0.519463 + 0.854493i \(0.673868\pi\)
\(788\) 0 0
\(789\) −3.59580e11 2.91661e11i −0.927872 0.752611i
\(790\) 0 0
\(791\) 1.23795e11i 0.316226i
\(792\) 0 0
\(793\) 3.98613e11 1.00800
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 3.04742e11i 0.755264i 0.925956 + 0.377632i \(0.123262\pi\)
−0.925956 + 0.377632i \(0.876738\pi\)
\(798\) 0 0
\(799\) 1.38223e11 0.339151
\(800\) 0 0
\(801\) −2.76866e11 + 5.83849e10i −0.672573 + 0.141831i
\(802\) 0 0
\(803\) 9.38171e10i 0.225642i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −3.48480e11 2.82657e11i −0.821643 0.666447i
\(808\) 0 0
\(809\) 1.47553e11i 0.344471i −0.985056 0.172236i \(-0.944901\pi\)
0.985056 0.172236i \(-0.0550990\pi\)
\(810\) 0 0
\(811\) −4.18158e11 −0.966622 −0.483311 0.875449i \(-0.660566\pi\)
−0.483311 + 0.875449i \(0.660566\pi\)
\(812\) 0 0
\(813\) −4.29460e11 + 5.29469e11i −0.983017 + 1.21193i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.85597e11 −0.416564
\(818\) 0 0
\(819\) 2.98868e10 + 1.41726e11i 0.0664269 + 0.315002i
\(820\) 0 0
\(821\) 1.74589e11i 0.384277i −0.981368 0.192138i \(-0.938458\pi\)
0.981368 0.192138i \(-0.0615422\pi\)
\(822\) 0 0
\(823\) −5.90822e11 −1.28783 −0.643913 0.765099i \(-0.722690\pi\)
−0.643913 + 0.765099i \(0.722690\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 4.79472e10i 0.102504i 0.998686 + 0.0512521i \(0.0163212\pi\)
−0.998686 + 0.0512521i \(0.983679\pi\)
\(828\) 0 0
\(829\) 1.65827e11 0.351104 0.175552 0.984470i \(-0.443829\pi\)
0.175552 + 0.984470i \(0.443829\pi\)
\(830\) 0 0
\(831\) 6.36561e10 7.84798e10i 0.133486 0.164571i
\(832\) 0 0
\(833\) 2.96234e11i 0.615255i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −6.60352e11 3.38473e11i −1.34547 0.689640i
\(838\) 0 0
\(839\) 5.82814e11i 1.17620i −0.808787 0.588101i \(-0.799876\pi\)
0.808787 0.588101i \(-0.200124\pi\)
\(840\) 0 0
\(841\) −5.88258e11 −1.17594
\(842\) 0 0
\(843\) −3.90548e11 3.16779e11i −0.773329 0.627259i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 6.00529e10 0.116681
\(848\) 0 0
\(849\) 2.46243e11 3.03586e11i 0.473952 0.584321i
\(850\) 0 0
\(851\) 7.52076e11i 1.43398i
\(852\) 0 0
\(853\) 4.89760e11 0.925097 0.462549 0.886594i \(-0.346935\pi\)
0.462549 + 0.886594i \(0.346935\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 1.11714e11i 0.207103i −0.994624 0.103551i \(-0.966979\pi\)
0.994624 0.103551i \(-0.0330206\pi\)
\(858\) 0 0
\(859\) −9.17638e10 −0.168538 −0.0842692 0.996443i \(-0.526856\pi\)
−0.0842692 + 0.996443i \(0.526856\pi\)
\(860\) 0 0
\(861\) −7.42355e10 6.02135e10i −0.135082 0.109567i
\(862\) 0 0
\(863\) 5.85110e11i 1.05486i −0.849599 0.527429i \(-0.823156\pi\)
0.849599 0.527429i \(-0.176844\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 2.10787e11 2.59874e11i 0.373051 0.459924i
\(868\) 0 0
\(869\) 1.85228e11i 0.324809i
\(870\) 0 0
\(871\) −2.22596e10 −0.0386762
\(872\) 0 0
\(873\) 7.82819e10 + 3.71219e11i 0.134773 + 0.639106i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −3.71675e11 −0.628297 −0.314148 0.949374i \(-0.601719\pi\)
−0.314148 + 0.949374i \(0.601719\pi\)
\(878\) 0 0
\(879\) 2.98880e11 + 2.42426e11i 0.500658 + 0.406091i
\(880\) 0 0
\(881\) 8.23900e10i 0.136764i 0.997659 + 0.0683818i \(0.0217836\pi\)
−0.997659 + 0.0683818i \(0.978216\pi\)
\(882\) 0 0
\(883\) −9.90170e11 −1.62880 −0.814399 0.580306i \(-0.802933\pi\)
−0.814399 + 0.580306i \(0.802933\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 2.88048e11i 0.465339i −0.972556 0.232670i \(-0.925254\pi\)
0.972556 0.232670i \(-0.0747461\pi\)
\(888\) 0 0
\(889\) 2.27073e11 0.363545
\(890\) 0 0
\(891\) 1.58864e11 + 3.59923e11i 0.252067 + 0.571083i
\(892\) 0 0
\(893\) 3.16958e11i 0.498420i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) −7.98521e11 6.47692e11i −1.23344 1.00046i
\(898\) 0 0
\(899\) 1.45677e12i 2.23024i
\(900\) 0 0
\(901\) −1.46658e11 −0.222539
\(902\) 0 0
\(903\) 3.55423e10 4.38190e10i 0.0534557 0.0659040i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −3.59541e11 −0.531274 −0.265637 0.964073i \(-0.585582\pi\)
−0.265637 + 0.964073i \(0.585582\pi\)
\(908\) 0 0
\(909\) 2.83208e11 5.97223e10i 0.414811 0.0874744i
\(910\) 0 0
\(911\) 1.59446e11i 0.231494i 0.993279 + 0.115747i \(0.0369263\pi\)
−0.993279 + 0.115747i \(0.963074\pi\)
\(912\) 0 0
\(913\) 3.22000e11 0.463418
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 1.17771e11i 0.166556i
\(918\) 0 0
\(919\) −6.75378e11 −0.946858 −0.473429 0.880832i \(-0.656984\pi\)
−0.473429 + 0.880832i \(0.656984\pi\)
\(920\) 0 0
\(921\) −3.33470e11 + 4.11126e11i −0.463467 + 0.571395i
\(922\) 0 0
\(923\) 1.51665e12i 2.08968i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −5.17414e10 2.45362e11i −0.0700680 0.332268i
\(928\) 0 0
\(929\) 7.68419e11i 1.03166i 0.856692 + 0.515829i \(0.172516\pi\)
−0.856692 + 0.515829i \(0.827484\pi\)
\(930\) 0 0
\(931\) 6.79292e11 0.904185
\(932\) 0 0
\(933\) 5.92919e11 + 4.80926e11i 0.782472 + 0.634675i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 2.45957e11 0.319080 0.159540 0.987191i \(-0.448999\pi\)
0.159540 + 0.987191i \(0.448999\pi\)
\(938\) 0 0
\(939\) −5.93988e11 + 7.32311e11i −0.764039 + 0.941961i
\(940\) 0 0
\(941\) 2.05589e11i 0.262205i 0.991369 + 0.131102i \(0.0418517\pi\)
−0.991369 + 0.131102i \(0.958148\pi\)
\(942\) 0 0
\(943\) 6.78519e11 0.858055
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.25097e12i 1.55542i −0.628624 0.777709i \(-0.716382\pi\)
0.628624 0.777709i \(-0.283618\pi\)
\(948\) 0 0
\(949\) 4.93693e11 0.608684
\(950\) 0 0
\(951\) −5.89002e11 4.77749e11i −0.720103 0.584087i
\(952\) 0 0
\(953\) 9.19955e11i 1.11531i −0.830074 0.557654i \(-0.811702\pi\)
0.830074 0.557654i \(-0.188298\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 4.86545e11 5.99848e11i 0.580064 0.715143i
\(958\) 0 0
\(959\) 3.70235e10i 0.0437727i
\(960\) 0 0
\(961\) 1.09672e12 1.28589
\(962\) 0 0
\(963\) −4.02766e11 + 8.49345e10i −0.468326 + 0.0987595i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −7.65204e11 −0.875127 −0.437563 0.899188i \(-0.644158\pi\)
−0.437563 + 0.899188i \(0.644158\pi\)
\(968\) 0 0
\(969\) −4.10363e11 3.32852e11i −0.465450 0.377534i
\(970\) 0 0
\(971\) 1.23934e12i 1.39417i −0.716990 0.697084i \(-0.754481\pi\)
0.716990 0.697084i \(-0.245519\pi\)
\(972\) 0 0
\(973\) −2.31285e11 −0.258045
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1.59613e12i 1.75182i 0.482473 + 0.875911i \(0.339739\pi\)
−0.482473 + 0.875911i \(0.660261\pi\)
\(978\) 0 0
\(979\) −3.94156e11 −0.429079
\(980\) 0 0
\(981\) −4.71998e10 2.23825e11i −0.0509641 0.241676i
\(982\) 0 0
\(983\) 1.82288e12i 1.95229i 0.217130 + 0.976143i \(0.430331\pi\)
−0.217130 + 0.976143i \(0.569669\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −7.48332e10 6.06983e10i −0.0788543 0.0639599i
\(988\) 0 0
\(989\) 4.00510e11i 0.418628i
\(990\) 0 0
\(991\) 9.32448e11 0.966786 0.483393 0.875404i \(-0.339404\pi\)
0.483393 + 0.875404i \(0.339404\pi\)
\(992\) 0 0
\(993\) −5.35247e11 + 6.59890e11i −0.550500 + 0.678695i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −8.66862e10 −0.0877343 −0.0438671 0.999037i \(-0.513968\pi\)
−0.0438671 + 0.999037i \(0.513968\pi\)
\(998\) 0 0
\(999\) −1.34765e12 6.90759e11i −1.35306 0.693529i
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.101.11 16
3.2 odd 2 inner 300.9.g.h.101.12 16
5.2 odd 4 60.9.b.a.29.3 16
5.3 odd 4 60.9.b.a.29.14 yes 16
5.4 even 2 inner 300.9.g.h.101.6 16
15.2 even 4 60.9.b.a.29.13 yes 16
15.8 even 4 60.9.b.a.29.4 yes 16
15.14 odd 2 inner 300.9.g.h.101.5 16
20.3 even 4 240.9.c.d.209.3 16
20.7 even 4 240.9.c.d.209.14 16
60.23 odd 4 240.9.c.d.209.13 16
60.47 odd 4 240.9.c.d.209.4 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
60.9.b.a.29.3 16 5.2 odd 4
60.9.b.a.29.4 yes 16 15.8 even 4
60.9.b.a.29.13 yes 16 15.2 even 4
60.9.b.a.29.14 yes 16 5.3 odd 4
240.9.c.d.209.3 16 20.3 even 4
240.9.c.d.209.4 16 60.47 odd 4
240.9.c.d.209.13 16 60.23 odd 4
240.9.c.d.209.14 16 20.7 even 4
300.9.g.h.101.5 16 15.14 odd 2 inner
300.9.g.h.101.6 16 5.4 even 2 inner
300.9.g.h.101.11 16 1.1 even 1 trivial
300.9.g.h.101.12 16 3.2 odd 2 inner