Properties

Label 64.12.e.a.17.15
Level $64$
Weight $12$
Character 64.17
Analytic conductor $49.174$
Analytic rank $0$
Dimension $42$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [64,12,Mod(17,64)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(64, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.17");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 64.e (of order \(4\), degree \(2\), not 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: \(49.1739635558\)
Analytic rank: \(0\)
Dimension: \(42\)
Relative dimension: \(21\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.15
Character \(\chi\) \(=\) 64.17
Dual form 64.12.e.a.49.15

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(199.508 + 199.508i) q^{3} +(8937.54 - 8937.54i) q^{5} +55653.3i q^{7} -97540.0i q^{9} +O(q^{10})\) \(q+(199.508 + 199.508i) q^{3} +(8937.54 - 8937.54i) q^{5} +55653.3i q^{7} -97540.0i q^{9} +(146129. - 146129. i) q^{11} +(1.33136e6 + 1.33136e6i) q^{13} +3.56623e6 q^{15} -2.92311e6 q^{17} +(1.11550e7 + 1.11550e7i) q^{19} +(-1.11033e7 + 1.11033e7i) q^{21} -2.12560e7i q^{23} -1.10931e8i q^{25} +(5.48023e7 - 5.48023e7i) q^{27} +(-1.31742e7 - 1.31742e7i) q^{29} -1.84262e8 q^{31} +5.83078e7 q^{33} +(4.97404e8 + 4.97404e8i) q^{35} +(6.58484e7 - 6.58484e7i) q^{37} +5.31233e8i q^{39} +7.48317e8i q^{41} +(3.10180e8 - 3.10180e8i) q^{43} +(-8.71768e8 - 8.71768e8i) q^{45} +1.85656e9 q^{47} -1.11996e9 q^{49} +(-5.83185e8 - 5.83185e8i) q^{51} +(3.45179e9 - 3.45179e9i) q^{53} -2.61207e9i q^{55} +4.45104e9i q^{57} +(-8.57741e8 + 8.57741e8i) q^{59} +(6.80487e9 + 6.80487e9i) q^{61} +5.42842e9 q^{63} +2.37981e10 q^{65} +(5.85461e9 + 5.85461e9i) q^{67} +(4.24075e9 - 4.24075e9i) q^{69} -1.15906e8i q^{71} -1.23316e9i q^{73} +(2.21317e10 - 2.21317e10i) q^{75} +(8.13256e9 + 8.13256e9i) q^{77} +1.66808e9 q^{79} +4.58810e9 q^{81} +(1.21868e10 + 1.21868e10i) q^{83} +(-2.61254e10 + 2.61254e10i) q^{85} -5.25674e9i q^{87} -3.35444e10i q^{89} +(-7.40944e10 + 7.40944e10i) q^{91} +(-3.67618e10 - 3.67618e10i) q^{93} +1.99397e11 q^{95} +2.49231e10 q^{97} +(-1.42534e10 - 1.42534e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 42 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 42 q + 2 q^{3} - 2 q^{5} + 540846 q^{11} - 2 q^{13} + 6075004 q^{15} - 4 q^{17} + 11291290 q^{19} + 354292 q^{21} + 66463304 q^{27} + 77673206 q^{29} - 343549808 q^{31} - 4 q^{33} + 434731684 q^{35} - 522762058 q^{37} - 3824193658 q^{43} + 97301954 q^{45} + 4586900144 q^{47} - 8474257474 q^{49} - 7074245796 q^{51} - 2100608058 q^{53} - 955824746 q^{59} + 2150827022 q^{61} - 27758037828 q^{63} - 1884965292 q^{65} + 3186519018 q^{67} - 16193060732 q^{69} - 28890034486 q^{75} - 22711870540 q^{77} - 48011833792 q^{79} - 90656394430 q^{81} - 55713221118 q^{83} - 84575506252 q^{85} + 147369662716 q^{91} - 69689773328 q^{93} - 375702304500 q^{95} - 4 q^{97} + 286271331106 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/64\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(63\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) 199.508 + 199.508i 0.474017 + 0.474017i 0.903212 0.429195i \(-0.141203\pi\)
−0.429195 + 0.903212i \(0.641203\pi\)
\(4\) 0 0
\(5\) 8937.54 8937.54i 1.27904 1.27904i 0.337830 0.941207i \(-0.390307\pi\)
0.941207 0.337830i \(-0.109693\pi\)
\(6\) 0 0
\(7\) 55653.3i 1.25156i 0.780000 + 0.625780i \(0.215219\pi\)
−0.780000 + 0.625780i \(0.784781\pi\)
\(8\) 0 0
\(9\) 97540.0i 0.550616i
\(10\) 0 0
\(11\) 146129. 146129.i 0.273575 0.273575i −0.556963 0.830538i \(-0.688033\pi\)
0.830538 + 0.556963i \(0.188033\pi\)
\(12\) 0 0
\(13\) 1.33136e6 + 1.33136e6i 0.994502 + 0.994502i 0.999985 0.00548268i \(-0.00174520\pi\)
−0.00548268 + 0.999985i \(0.501745\pi\)
\(14\) 0 0
\(15\) 3.56623e6 1.21257
\(16\) 0 0
\(17\) −2.92311e6 −0.499317 −0.249659 0.968334i \(-0.580318\pi\)
−0.249659 + 0.968334i \(0.580318\pi\)
\(18\) 0 0
\(19\) 1.11550e7 + 1.11550e7i 1.03354 + 1.03354i 0.999418 + 0.0341197i \(0.0108627\pi\)
0.0341197 + 0.999418i \(0.489137\pi\)
\(20\) 0 0
\(21\) −1.11033e7 + 1.11033e7i −0.593261 + 0.593261i
\(22\) 0 0
\(23\) 2.12560e7i 0.688619i −0.938856 0.344309i \(-0.888113\pi\)
0.938856 0.344309i \(-0.111887\pi\)
\(24\) 0 0
\(25\) 1.10931e8i 2.27187i
\(26\) 0 0
\(27\) 5.48023e7 5.48023e7i 0.735018 0.735018i
\(28\) 0 0
\(29\) −1.31742e7 1.31742e7i −0.119271 0.119271i 0.644952 0.764223i \(-0.276877\pi\)
−0.764223 + 0.644952i \(0.776877\pi\)
\(30\) 0 0
\(31\) −1.84262e8 −1.15597 −0.577986 0.816047i \(-0.696161\pi\)
−0.577986 + 0.816047i \(0.696161\pi\)
\(32\) 0 0
\(33\) 5.83078e7 0.259358
\(34\) 0 0
\(35\) 4.97404e8 + 4.97404e8i 1.60079 + 1.60079i
\(36\) 0 0
\(37\) 6.58484e7 6.58484e7i 0.156112 0.156112i −0.624729 0.780841i \(-0.714791\pi\)
0.780841 + 0.624729i \(0.214791\pi\)
\(38\) 0 0
\(39\) 5.31233e8i 0.942822i
\(40\) 0 0
\(41\) 7.48317e8i 1.00873i 0.863491 + 0.504364i \(0.168273\pi\)
−0.863491 + 0.504364i \(0.831727\pi\)
\(42\) 0 0
\(43\) 3.10180e8 3.10180e8i 0.321764 0.321764i −0.527680 0.849443i \(-0.676938\pi\)
0.849443 + 0.527680i \(0.176938\pi\)
\(44\) 0 0
\(45\) −8.71768e8 8.71768e8i −0.704258 0.704258i
\(46\) 0 0
\(47\) 1.85656e9 1.18078 0.590392 0.807117i \(-0.298973\pi\)
0.590392 + 0.807117i \(0.298973\pi\)
\(48\) 0 0
\(49\) −1.11996e9 −0.566403
\(50\) 0 0
\(51\) −5.83185e8 5.83185e8i −0.236685 0.236685i
\(52\) 0 0
\(53\) 3.45179e9 3.45179e9i 1.13378 1.13378i 0.144232 0.989544i \(-0.453929\pi\)
0.989544 0.144232i \(-0.0460710\pi\)
\(54\) 0 0
\(55\) 2.61207e9i 0.699825i
\(56\) 0 0
\(57\) 4.45104e9i 0.979828i
\(58\) 0 0
\(59\) −8.57741e8 + 8.57741e8i −0.156196 + 0.156196i −0.780879 0.624683i \(-0.785228\pi\)
0.624683 + 0.780879i \(0.285228\pi\)
\(60\) 0 0
\(61\) 6.80487e9 + 6.80487e9i 1.03159 + 1.03159i 0.999485 + 0.0321015i \(0.0102200\pi\)
0.0321015 + 0.999485i \(0.489780\pi\)
\(62\) 0 0
\(63\) 5.42842e9 0.689129
\(64\) 0 0
\(65\) 2.37981e10 2.54401
\(66\) 0 0
\(67\) 5.85461e9 + 5.85461e9i 0.529769 + 0.529769i 0.920503 0.390735i \(-0.127779\pi\)
−0.390735 + 0.920503i \(0.627779\pi\)
\(68\) 0 0
\(69\) 4.24075e9 4.24075e9i 0.326417 0.326417i
\(70\) 0 0
\(71\) 1.15906e8i 0.00762402i −0.999993 0.00381201i \(-0.998787\pi\)
0.999993 0.00381201i \(-0.00121340\pi\)
\(72\) 0 0
\(73\) 1.23316e9i 0.0696213i −0.999394 0.0348106i \(-0.988917\pi\)
0.999394 0.0348106i \(-0.0110828\pi\)
\(74\) 0 0
\(75\) 2.21317e10 2.21317e10i 1.07691 1.07691i
\(76\) 0 0
\(77\) 8.13256e9 + 8.13256e9i 0.342396 + 0.342396i
\(78\) 0 0
\(79\) 1.66808e9 0.0609914 0.0304957 0.999535i \(-0.490291\pi\)
0.0304957 + 0.999535i \(0.490291\pi\)
\(80\) 0 0
\(81\) 4.58810e9 0.146206
\(82\) 0 0
\(83\) 1.21868e10 + 1.21868e10i 0.339593 + 0.339593i 0.856214 0.516621i \(-0.172810\pi\)
−0.516621 + 0.856214i \(0.672810\pi\)
\(84\) 0 0
\(85\) −2.61254e10 + 2.61254e10i −0.638645 + 0.638645i
\(86\) 0 0
\(87\) 5.25674e9i 0.113073i
\(88\) 0 0
\(89\) 3.35444e10i 0.636759i −0.947963 0.318379i \(-0.896861\pi\)
0.947963 0.318379i \(-0.103139\pi\)
\(90\) 0 0
\(91\) −7.40944e10 + 7.40944e10i −1.24468 + 1.24468i
\(92\) 0 0
\(93\) −3.67618e10 3.67618e10i −0.547950 0.547950i
\(94\) 0 0
\(95\) 1.99397e11 2.64387
\(96\) 0 0
\(97\) 2.49231e10 0.294685 0.147342 0.989086i \(-0.452928\pi\)
0.147342 + 0.989086i \(0.452928\pi\)
\(98\) 0 0
\(99\) −1.42534e10 1.42534e10i −0.150635 0.150635i
\(100\) 0 0
\(101\) 7.39401e10 7.39401e10i 0.700023 0.700023i −0.264392 0.964415i \(-0.585171\pi\)
0.964415 + 0.264392i \(0.0851713\pi\)
\(102\) 0 0
\(103\) 1.55937e10i 0.132539i 0.997802 + 0.0662694i \(0.0211097\pi\)
−0.997802 + 0.0662694i \(0.978890\pi\)
\(104\) 0 0
\(105\) 1.98472e11i 1.51760i
\(106\) 0 0
\(107\) −6.52098e10 + 6.52098e10i −0.449471 + 0.449471i −0.895179 0.445707i \(-0.852952\pi\)
0.445707 + 0.895179i \(0.352952\pi\)
\(108\) 0 0
\(109\) −1.02657e11 1.02657e11i −0.639061 0.639061i 0.311263 0.950324i \(-0.399248\pi\)
−0.950324 + 0.311263i \(0.899248\pi\)
\(110\) 0 0
\(111\) 2.62746e10 0.147999
\(112\) 0 0
\(113\) −2.05501e10 −0.104926 −0.0524630 0.998623i \(-0.516707\pi\)
−0.0524630 + 0.998623i \(0.516707\pi\)
\(114\) 0 0
\(115\) −1.89977e11 1.89977e11i −0.880769 0.880769i
\(116\) 0 0
\(117\) 1.29860e11 1.29860e11i 0.547589 0.547589i
\(118\) 0 0
\(119\) 1.62681e11i 0.624925i
\(120\) 0 0
\(121\) 2.42604e11i 0.850313i
\(122\) 0 0
\(123\) −1.49295e11 + 1.49295e11i −0.478154 + 0.478154i
\(124\) 0 0
\(125\) −5.55050e11 5.55050e11i −1.62677 1.62677i
\(126\) 0 0
\(127\) −6.09498e11 −1.63701 −0.818506 0.574498i \(-0.805197\pi\)
−0.818506 + 0.574498i \(0.805197\pi\)
\(128\) 0 0
\(129\) 1.23767e11 0.305043
\(130\) 0 0
\(131\) 9.65459e10 + 9.65459e10i 0.218646 + 0.218646i 0.807928 0.589282i \(-0.200589\pi\)
−0.589282 + 0.807928i \(0.700589\pi\)
\(132\) 0 0
\(133\) −6.20815e11 + 6.20815e11i −1.29353 + 1.29353i
\(134\) 0 0
\(135\) 9.79596e11i 1.88023i
\(136\) 0 0
\(137\) 4.17036e11i 0.738262i 0.929377 + 0.369131i \(0.120345\pi\)
−0.929377 + 0.369131i \(0.879655\pi\)
\(138\) 0 0
\(139\) −4.11704e11 + 4.11704e11i −0.672982 + 0.672982i −0.958403 0.285420i \(-0.907867\pi\)
0.285420 + 0.958403i \(0.407867\pi\)
\(140\) 0 0
\(141\) 3.70399e11 + 3.70399e11i 0.559711 + 0.559711i
\(142\) 0 0
\(143\) 3.89099e11 0.544142
\(144\) 0 0
\(145\) −2.35491e11 −0.305105
\(146\) 0 0
\(147\) −2.23442e11 2.23442e11i −0.268484 0.268484i
\(148\) 0 0
\(149\) 5.89614e11 5.89614e11i 0.657724 0.657724i −0.297117 0.954841i \(-0.596025\pi\)
0.954841 + 0.297117i \(0.0960253\pi\)
\(150\) 0 0
\(151\) 1.16918e12i 1.21201i 0.795459 + 0.606007i \(0.207230\pi\)
−0.795459 + 0.606007i \(0.792770\pi\)
\(152\) 0 0
\(153\) 2.85120e11i 0.274932i
\(154\) 0 0
\(155\) −1.64685e12 + 1.64685e12i −1.47853 + 1.47853i
\(156\) 0 0
\(157\) −1.50102e12 1.50102e12i −1.25585 1.25585i −0.953057 0.302791i \(-0.902081\pi\)
−0.302791 0.953057i \(-0.597919\pi\)
\(158\) 0 0
\(159\) 1.37732e12 1.07486
\(160\) 0 0
\(161\) 1.18297e12 0.861848
\(162\) 0 0
\(163\) −1.05489e12 1.05489e12i −0.718085 0.718085i 0.250128 0.968213i \(-0.419527\pi\)
−0.968213 + 0.250128i \(0.919527\pi\)
\(164\) 0 0
\(165\) 5.21129e11 5.21129e11i 0.331729 0.331729i
\(166\) 0 0
\(167\) 2.19763e12i 1.30923i −0.755965 0.654613i \(-0.772832\pi\)
0.755965 0.654613i \(-0.227168\pi\)
\(168\) 0 0
\(169\) 1.75286e12i 0.978070i
\(170\) 0 0
\(171\) 1.08806e12 1.08806e12i 0.569082 0.569082i
\(172\) 0 0
\(173\) −2.08631e12 2.08631e12i −1.02359 1.02359i −0.999715 0.0238746i \(-0.992400\pi\)
−0.0238746 0.999715i \(-0.507600\pi\)
\(174\) 0 0
\(175\) 6.17369e12 2.84339
\(176\) 0 0
\(177\) −3.42253e11 −0.148079
\(178\) 0 0
\(179\) 7.81494e11 + 7.81494e11i 0.317859 + 0.317859i 0.847944 0.530086i \(-0.177840\pi\)
−0.530086 + 0.847944i \(0.677840\pi\)
\(180\) 0 0
\(181\) 1.08737e12 1.08737e12i 0.416052 0.416052i −0.467789 0.883840i \(-0.654949\pi\)
0.883840 + 0.467789i \(0.154949\pi\)
\(182\) 0 0
\(183\) 2.71525e12i 0.977978i
\(184\) 0 0
\(185\) 1.17705e12i 0.399346i
\(186\) 0 0
\(187\) −4.27151e11 + 4.27151e11i −0.136601 + 0.136601i
\(188\) 0 0
\(189\) 3.04993e12 + 3.04993e12i 0.919919 + 0.919919i
\(190\) 0 0
\(191\) −2.35610e12 −0.670672 −0.335336 0.942099i \(-0.608850\pi\)
−0.335336 + 0.942099i \(0.608850\pi\)
\(192\) 0 0
\(193\) 5.05945e12 1.36000 0.679999 0.733213i \(-0.261980\pi\)
0.679999 + 0.733213i \(0.261980\pi\)
\(194\) 0 0
\(195\) 4.74792e12 + 4.74792e12i 1.20590 + 1.20590i
\(196\) 0 0
\(197\) −2.28419e12 + 2.28419e12i −0.548490 + 0.548490i −0.926004 0.377514i \(-0.876779\pi\)
0.377514 + 0.926004i \(0.376779\pi\)
\(198\) 0 0
\(199\) 1.32865e12i 0.301799i −0.988549 0.150899i \(-0.951783\pi\)
0.988549 0.150899i \(-0.0482170\pi\)
\(200\) 0 0
\(201\) 2.33608e12i 0.502239i
\(202\) 0 0
\(203\) 7.33190e11 7.33190e11i 0.149275 0.149275i
\(204\) 0 0
\(205\) 6.68812e12 + 6.68812e12i 1.29020 + 1.29020i
\(206\) 0 0
\(207\) −2.07331e12 −0.379165
\(208\) 0 0
\(209\) 3.26015e12 0.565500
\(210\) 0 0
\(211\) −7.82026e12 7.82026e12i −1.28726 1.28726i −0.936442 0.350823i \(-0.885902\pi\)
−0.350823 0.936442i \(-0.614098\pi\)
\(212\) 0 0
\(213\) 2.31242e10 2.31242e10i 0.00361392 0.00361392i
\(214\) 0 0
\(215\) 5.54449e12i 0.823096i
\(216\) 0 0
\(217\) 1.02548e13i 1.44677i
\(218\) 0 0
\(219\) 2.46025e11 2.46025e11i 0.0330017 0.0330017i
\(220\) 0 0
\(221\) −3.89170e12 3.89170e12i −0.496572 0.496572i
\(222\) 0 0
\(223\) −1.45466e13 −1.76638 −0.883190 0.469015i \(-0.844609\pi\)
−0.883190 + 0.469015i \(0.844609\pi\)
\(224\) 0 0
\(225\) −1.08202e13 −1.25093
\(226\) 0 0
\(227\) 1.14517e13 + 1.14517e13i 1.26104 + 1.26104i 0.950591 + 0.310446i \(0.100478\pi\)
0.310446 + 0.950591i \(0.399522\pi\)
\(228\) 0 0
\(229\) 8.19588e12 8.19588e12i 0.860004 0.860004i −0.131335 0.991338i \(-0.541926\pi\)
0.991338 + 0.131335i \(0.0419262\pi\)
\(230\) 0 0
\(231\) 3.24502e12i 0.324603i
\(232\) 0 0
\(233\) 1.19894e13i 1.14378i 0.820332 + 0.571888i \(0.193789\pi\)
−0.820332 + 0.571888i \(0.806211\pi\)
\(234\) 0 0
\(235\) 1.65931e13 1.65931e13i 1.51027 1.51027i
\(236\) 0 0
\(237\) 3.32796e11 + 3.32796e11i 0.0289109 + 0.0289109i
\(238\) 0 0
\(239\) 1.05045e13 0.871339 0.435669 0.900107i \(-0.356512\pi\)
0.435669 + 0.900107i \(0.356512\pi\)
\(240\) 0 0
\(241\) −4.17569e12 −0.330853 −0.165426 0.986222i \(-0.552900\pi\)
−0.165426 + 0.986222i \(0.552900\pi\)
\(242\) 0 0
\(243\) −8.79270e12 8.79270e12i −0.665714 0.665714i
\(244\) 0 0
\(245\) −1.00097e13 + 1.00097e13i −0.724450 + 0.724450i
\(246\) 0 0
\(247\) 2.97027e13i 2.05571i
\(248\) 0 0
\(249\) 4.86271e12i 0.321945i
\(250\) 0 0
\(251\) −1.23422e13 + 1.23422e13i −0.781962 + 0.781962i −0.980162 0.198199i \(-0.936491\pi\)
0.198199 + 0.980162i \(0.436491\pi\)
\(252\) 0 0
\(253\) −3.10612e12 3.10612e12i −0.188389 0.188389i
\(254\) 0 0
\(255\) −1.04245e13 −0.605457
\(256\) 0 0
\(257\) −1.61488e13 −0.898477 −0.449238 0.893412i \(-0.648305\pi\)
−0.449238 + 0.893412i \(0.648305\pi\)
\(258\) 0 0
\(259\) 3.66468e12 + 3.66468e12i 0.195383 + 0.195383i
\(260\) 0 0
\(261\) −1.28501e12 + 1.28501e12i −0.0656728 + 0.0656728i
\(262\) 0 0
\(263\) 2.33354e13i 1.14356i −0.820407 0.571780i \(-0.806253\pi\)
0.820407 0.571780i \(-0.193747\pi\)
\(264\) 0 0
\(265\) 6.17011e13i 2.90028i
\(266\) 0 0
\(267\) 6.69238e12 6.69238e12i 0.301834 0.301834i
\(268\) 0 0
\(269\) 2.23132e12 + 2.23132e12i 0.0965881 + 0.0965881i 0.753750 0.657162i \(-0.228243\pi\)
−0.657162 + 0.753750i \(0.728243\pi\)
\(270\) 0 0
\(271\) −3.01445e13 −1.25279 −0.626393 0.779508i \(-0.715469\pi\)
−0.626393 + 0.779508i \(0.715469\pi\)
\(272\) 0 0
\(273\) −2.95649e13 −1.18000
\(274\) 0 0
\(275\) −1.62103e13 1.62103e13i −0.621528 0.621528i
\(276\) 0 0
\(277\) −1.38774e13 + 1.38774e13i −0.511292 + 0.511292i −0.914922 0.403630i \(-0.867748\pi\)
0.403630 + 0.914922i \(0.367748\pi\)
\(278\) 0 0
\(279\) 1.79729e13i 0.636497i
\(280\) 0 0
\(281\) 1.23343e13i 0.419981i −0.977703 0.209990i \(-0.932657\pi\)
0.977703 0.209990i \(-0.0673433\pi\)
\(282\) 0 0
\(283\) −5.76116e12 + 5.76116e12i −0.188662 + 0.188662i −0.795118 0.606455i \(-0.792591\pi\)
0.606455 + 0.795118i \(0.292591\pi\)
\(284\) 0 0
\(285\) 3.97814e13 + 3.97814e13i 1.25324 + 1.25324i
\(286\) 0 0
\(287\) −4.16463e13 −1.26248
\(288\) 0 0
\(289\) −2.57273e13 −0.750683
\(290\) 0 0
\(291\) 4.97237e12 + 4.97237e12i 0.139686 + 0.139686i
\(292\) 0 0
\(293\) −3.65192e13 + 3.65192e13i −0.987982 + 0.987982i −0.999929 0.0119465i \(-0.996197\pi\)
0.0119465 + 0.999929i \(0.496197\pi\)
\(294\) 0 0
\(295\) 1.53322e13i 0.399561i
\(296\) 0 0
\(297\) 1.60164e13i 0.402165i
\(298\) 0 0
\(299\) 2.82993e13 2.82993e13i 0.684833 0.684833i
\(300\) 0 0
\(301\) 1.72625e13 + 1.72625e13i 0.402707 + 0.402707i
\(302\) 0 0
\(303\) 2.95033e13 0.663646
\(304\) 0 0
\(305\) 1.21638e14 2.63887
\(306\) 0 0
\(307\) 3.86138e13 + 3.86138e13i 0.808131 + 0.808131i 0.984351 0.176220i \(-0.0563871\pi\)
−0.176220 + 0.984351i \(0.556387\pi\)
\(308\) 0 0
\(309\) −3.11106e12 + 3.11106e12i −0.0628256 + 0.0628256i
\(310\) 0 0
\(311\) 2.21250e13i 0.431221i −0.976479 0.215611i \(-0.930826\pi\)
0.976479 0.215611i \(-0.0691742\pi\)
\(312\) 0 0
\(313\) 4.91648e13i 0.925039i 0.886609 + 0.462520i \(0.153055\pi\)
−0.886609 + 0.462520i \(0.846945\pi\)
\(314\) 0 0
\(315\) 4.85168e13 4.85168e13i 0.881422 0.881422i
\(316\) 0 0
\(317\) −2.50121e13 2.50121e13i −0.438858 0.438858i 0.452770 0.891627i \(-0.350436\pi\)
−0.891627 + 0.452770i \(0.850436\pi\)
\(318\) 0 0
\(319\) −3.85027e12 −0.0652594
\(320\) 0 0
\(321\) −2.60198e13 −0.426114
\(322\) 0 0
\(323\) −3.26074e13 3.26074e13i −0.516063 0.516063i
\(324\) 0 0
\(325\) 1.47689e14 1.47689e14i 2.25938 2.25938i
\(326\) 0 0
\(327\) 4.09618e13i 0.605852i
\(328\) 0 0
\(329\) 1.03324e14i 1.47782i
\(330\) 0 0
\(331\) −6.15855e13 + 6.15855e13i −0.851970 + 0.851970i −0.990376 0.138405i \(-0.955802\pi\)
0.138405 + 0.990376i \(0.455802\pi\)
\(332\) 0 0
\(333\) −6.42285e12 6.42285e12i −0.0859577 0.0859577i
\(334\) 0 0
\(335\) 1.04652e14 1.35519
\(336\) 0 0
\(337\) −1.50805e11 −0.00188996 −0.000944980 1.00000i \(-0.500301\pi\)
−0.000944980 1.00000i \(0.500301\pi\)
\(338\) 0 0
\(339\) −4.09992e12 4.09992e12i −0.0497367 0.0497367i
\(340\) 0 0
\(341\) −2.69261e13 + 2.69261e13i −0.316245 + 0.316245i
\(342\) 0 0
\(343\) 4.77151e13i 0.542673i
\(344\) 0 0
\(345\) 7.58038e13i 0.834999i
\(346\) 0 0
\(347\) −1.74401e13 + 1.74401e13i −0.186096 + 0.186096i −0.794006 0.607910i \(-0.792008\pi\)
0.607910 + 0.794006i \(0.292008\pi\)
\(348\) 0 0
\(349\) −3.25744e13 3.25744e13i −0.336773 0.336773i 0.518378 0.855151i \(-0.326536\pi\)
−0.855151 + 0.518378i \(0.826536\pi\)
\(350\) 0 0
\(351\) 1.45923e14 1.46195
\(352\) 0 0
\(353\) −1.61148e12 −0.0156481 −0.00782407 0.999969i \(-0.502491\pi\)
−0.00782407 + 0.999969i \(0.502491\pi\)
\(354\) 0 0
\(355\) −1.03591e12 1.03591e12i −0.00975141 0.00975141i
\(356\) 0 0
\(357\) 3.24562e13 3.24562e13i 0.296225 0.296225i
\(358\) 0 0
\(359\) 5.33566e13i 0.472246i 0.971723 + 0.236123i \(0.0758769\pi\)
−0.971723 + 0.236123i \(0.924123\pi\)
\(360\) 0 0
\(361\) 1.32379e14i 1.13640i
\(362\) 0 0
\(363\) −4.84015e13 + 4.84015e13i −0.403063 + 0.403063i
\(364\) 0 0
\(365\) −1.10214e13 1.10214e13i −0.0890482 0.0890482i
\(366\) 0 0
\(367\) −9.63988e13 −0.755802 −0.377901 0.925846i \(-0.623354\pi\)
−0.377901 + 0.925846i \(0.623354\pi\)
\(368\) 0 0
\(369\) 7.29908e13 0.555422
\(370\) 0 0
\(371\) 1.92104e14 + 1.92104e14i 1.41899 + 1.41899i
\(372\) 0 0
\(373\) −6.67716e13 + 6.67716e13i −0.478843 + 0.478843i −0.904761 0.425919i \(-0.859951\pi\)
0.425919 + 0.904761i \(0.359951\pi\)
\(374\) 0 0
\(375\) 2.21474e14i 1.54224i
\(376\) 0 0
\(377\) 3.50792e13i 0.237231i
\(378\) 0 0
\(379\) −3.97407e13 + 3.97407e13i −0.261048 + 0.261048i −0.825480 0.564432i \(-0.809095\pi\)
0.564432 + 0.825480i \(0.309095\pi\)
\(380\) 0 0
\(381\) −1.21600e14 1.21600e14i −0.775971 0.775971i
\(382\) 0 0
\(383\) 1.57401e14 0.975918 0.487959 0.872866i \(-0.337741\pi\)
0.487959 + 0.872866i \(0.337741\pi\)
\(384\) 0 0
\(385\) 1.45370e14 0.875873
\(386\) 0 0
\(387\) −3.02549e13 3.02549e13i −0.177168 0.177168i
\(388\) 0 0
\(389\) −1.16568e14 + 1.16568e14i −0.663525 + 0.663525i −0.956209 0.292684i \(-0.905452\pi\)
0.292684 + 0.956209i \(0.405452\pi\)
\(390\) 0 0
\(391\) 6.21337e13i 0.343839i
\(392\) 0 0
\(393\) 3.85234e13i 0.207284i
\(394\) 0 0
\(395\) 1.49086e13 1.49086e13i 0.0780102 0.0780102i
\(396\) 0 0
\(397\) −6.93873e13 6.93873e13i −0.353128 0.353128i 0.508144 0.861272i \(-0.330332\pi\)
−0.861272 + 0.508144i \(0.830332\pi\)
\(398\) 0 0
\(399\) −2.47715e14 −1.22631
\(400\) 0 0
\(401\) −1.22810e14 −0.591482 −0.295741 0.955268i \(-0.595566\pi\)
−0.295741 + 0.955268i \(0.595566\pi\)
\(402\) 0 0
\(403\) −2.45319e14 2.45319e14i −1.14962 1.14962i
\(404\) 0 0
\(405\) 4.10063e13 4.10063e13i 0.187003 0.187003i
\(406\) 0 0
\(407\) 1.92447e13i 0.0854166i
\(408\) 0 0
\(409\) 7.71261e13i 0.333214i −0.986023 0.166607i \(-0.946719\pi\)
0.986023 0.166607i \(-0.0532811\pi\)
\(410\) 0 0
\(411\) −8.32021e13 + 8.32021e13i −0.349948 + 0.349948i
\(412\) 0 0
\(413\) −4.77361e13 4.77361e13i −0.195489 0.195489i
\(414\) 0 0
\(415\) 2.17839e14 0.868704
\(416\) 0 0
\(417\) −1.64277e14 −0.638010
\(418\) 0 0
\(419\) −3.70961e14 3.70961e14i −1.40330 1.40330i −0.789314 0.613989i \(-0.789564\pi\)
−0.613989 0.789314i \(-0.710436\pi\)
\(420\) 0 0
\(421\) 1.97013e13 1.97013e13i 0.0726011 0.0726011i −0.669874 0.742475i \(-0.733652\pi\)
0.742475 + 0.669874i \(0.233652\pi\)
\(422\) 0 0
\(423\) 1.81089e14i 0.650159i
\(424\) 0 0
\(425\) 3.24265e14i 1.13438i
\(426\) 0 0
\(427\) −3.78713e14 + 3.78713e14i −1.29109 + 1.29109i
\(428\) 0 0
\(429\) 7.76285e13 + 7.76285e13i 0.257932 + 0.257932i
\(430\) 0 0
\(431\) −8.37088e13 −0.271110 −0.135555 0.990770i \(-0.543282\pi\)
−0.135555 + 0.990770i \(0.543282\pi\)
\(432\) 0 0
\(433\) 3.95547e14 1.24886 0.624431 0.781080i \(-0.285331\pi\)
0.624431 + 0.781080i \(0.285331\pi\)
\(434\) 0 0
\(435\) −4.69823e13 4.69823e13i −0.144625 0.144625i
\(436\) 0 0
\(437\) 2.37112e14 2.37112e14i 0.711713 0.711713i
\(438\) 0 0
\(439\) 4.95762e14i 1.45117i −0.688132 0.725586i \(-0.741569\pi\)
0.688132 0.725586i \(-0.258431\pi\)
\(440\) 0 0
\(441\) 1.09241e14i 0.311871i
\(442\) 0 0
\(443\) −9.47829e13 + 9.47829e13i −0.263943 + 0.263943i −0.826654 0.562711i \(-0.809758\pi\)
0.562711 + 0.826654i \(0.309758\pi\)
\(444\) 0 0
\(445\) −2.99805e14 2.99805e14i −0.814438 0.814438i
\(446\) 0 0
\(447\) 2.35266e14 0.623544
\(448\) 0 0
\(449\) −3.01226e14 −0.779001 −0.389500 0.921026i \(-0.627352\pi\)
−0.389500 + 0.921026i \(0.627352\pi\)
\(450\) 0 0
\(451\) 1.09351e14 + 1.09351e14i 0.275963 + 0.275963i
\(452\) 0 0
\(453\) −2.33261e14 + 2.33261e14i −0.574515 + 0.574515i
\(454\) 0 0
\(455\) 1.32444e15i 3.18398i
\(456\) 0 0
\(457\) 1.00211e14i 0.235166i −0.993063 0.117583i \(-0.962485\pi\)
0.993063 0.117583i \(-0.0375146\pi\)
\(458\) 0 0
\(459\) −1.60193e14 + 1.60193e14i −0.367007 + 0.367007i
\(460\) 0 0
\(461\) 3.90890e14 + 3.90890e14i 0.874377 + 0.874377i 0.992946 0.118568i \(-0.0378305\pi\)
−0.118568 + 0.992946i \(0.537831\pi\)
\(462\) 0 0
\(463\) 7.10495e13 0.155191 0.0775953 0.996985i \(-0.475276\pi\)
0.0775953 + 0.996985i \(0.475276\pi\)
\(464\) 0 0
\(465\) −6.57121e14 −1.40170
\(466\) 0 0
\(467\) 4.92262e14 + 4.92262e14i 1.02554 + 1.02554i 0.999665 + 0.0258775i \(0.00823799\pi\)
0.0258775 + 0.999665i \(0.491762\pi\)
\(468\) 0 0
\(469\) −3.25828e14 + 3.25828e14i −0.663037 + 0.663037i
\(470\) 0 0
\(471\) 5.98930e14i 1.19059i
\(472\) 0 0
\(473\) 9.06525e13i 0.176053i
\(474\) 0 0
\(475\) 1.23744e15 1.23744e15i 2.34807 2.34807i
\(476\) 0 0
\(477\) −3.36688e14 3.36688e14i −0.624275 0.624275i
\(478\) 0 0
\(479\) −1.05196e15 −1.90614 −0.953069 0.302752i \(-0.902095\pi\)
−0.953069 + 0.302752i \(0.902095\pi\)
\(480\) 0 0
\(481\) 1.75335e14 0.310507
\(482\) 0 0
\(483\) 2.36012e14 + 2.36012e14i 0.408530 + 0.408530i
\(484\) 0 0
\(485\) 2.22752e14 2.22752e14i 0.376913 0.376913i
\(486\) 0 0
\(487\) 1.75551e14i 0.290398i −0.989403 0.145199i \(-0.953618\pi\)
0.989403 0.145199i \(-0.0463822\pi\)
\(488\) 0 0
\(489\) 4.20919e14i 0.680769i
\(490\) 0 0
\(491\) 3.90969e14 3.90969e14i 0.618293 0.618293i −0.326800 0.945093i \(-0.605970\pi\)
0.945093 + 0.326800i \(0.105970\pi\)
\(492\) 0 0
\(493\) 3.85098e13 + 3.85098e13i 0.0595543 + 0.0595543i
\(494\) 0 0
\(495\) −2.54781e14 −0.385335
\(496\) 0 0
\(497\) 6.45054e12 0.00954192
\(498\) 0 0
\(499\) 4.65711e14 + 4.65711e14i 0.673851 + 0.673851i 0.958602 0.284751i \(-0.0919109\pi\)
−0.284751 + 0.958602i \(0.591911\pi\)
\(500\) 0 0
\(501\) 4.38445e14 4.38445e14i 0.620595 0.620595i
\(502\) 0 0
\(503\) 1.00596e15i 1.39302i 0.717546 + 0.696511i \(0.245265\pi\)
−0.717546 + 0.696511i \(0.754735\pi\)
\(504\) 0 0
\(505\) 1.32169e15i 1.79071i
\(506\) 0 0
\(507\) −3.49709e14 + 3.49709e14i −0.463621 + 0.463621i
\(508\) 0 0
\(509\) 9.28220e14 + 9.28220e14i 1.20421 + 1.20421i 0.972873 + 0.231339i \(0.0743105\pi\)
0.231339 + 0.972873i \(0.425690\pi\)
\(510\) 0 0
\(511\) 6.86292e13 0.0871352
\(512\) 0 0
\(513\) 1.22264e15 1.51934
\(514\) 0 0
\(515\) 1.39369e14 + 1.39369e14i 0.169522 + 0.169522i
\(516\) 0 0
\(517\) 2.71297e14 2.71297e14i 0.323033 0.323033i
\(518\) 0 0
\(519\) 8.32473e14i 0.970397i
\(520\) 0 0
\(521\) 3.89954e14i 0.445047i 0.974927 + 0.222523i \(0.0714294\pi\)
−0.974927 + 0.222523i \(0.928571\pi\)
\(522\) 0 0
\(523\) 8.43298e14 8.43298e14i 0.942370 0.942370i −0.0560573 0.998428i \(-0.517853\pi\)
0.998428 + 0.0560573i \(0.0178530\pi\)
\(524\) 0 0
\(525\) 1.23170e15 + 1.23170e15i 1.34781 + 1.34781i
\(526\) 0 0
\(527\) 5.38619e14 0.577196
\(528\) 0 0
\(529\) 5.00991e14 0.525804
\(530\) 0 0
\(531\) 8.36641e13 + 8.36641e13i 0.0860041 + 0.0860041i
\(532\) 0 0
\(533\) −9.96276e14 + 9.96276e14i −1.00318 + 1.00318i
\(534\) 0 0
\(535\) 1.16563e15i 1.14978i
\(536\) 0 0
\(537\) 3.11829e14i 0.301341i
\(538\) 0 0
\(539\) −1.63659e14 + 1.63659e14i −0.154954 + 0.154954i
\(540\) 0 0
\(541\) 7.75321e14 + 7.75321e14i 0.719277 + 0.719277i 0.968457 0.249180i \(-0.0801610\pi\)
−0.249180 + 0.968457i \(0.580161\pi\)
\(542\) 0 0
\(543\) 4.33880e14 0.394431
\(544\) 0 0
\(545\) −1.83500e15 −1.63477
\(546\) 0 0
\(547\) −1.27728e15 1.27728e15i −1.11521 1.11521i −0.992435 0.122774i \(-0.960821\pi\)
−0.122774 0.992435i \(-0.539179\pi\)
\(548\) 0 0
\(549\) 6.63747e14 6.63747e14i 0.568008 0.568008i
\(550\) 0 0
\(551\) 2.93918e14i 0.246543i
\(552\) 0 0
\(553\) 9.28343e13i 0.0763344i
\(554\) 0 0
\(555\) 2.34830e14 2.34830e14i 0.189297 0.189297i
\(556\) 0 0
\(557\) −6.89209e14 6.89209e14i −0.544687 0.544687i 0.380212 0.924899i \(-0.375851\pi\)
−0.924899 + 0.380212i \(0.875851\pi\)
\(558\) 0 0
\(559\) 8.25920e14 0.639990
\(560\) 0 0
\(561\) −1.70440e14 −0.129502
\(562\) 0 0
\(563\) −7.63745e14 7.63745e14i −0.569052 0.569052i 0.362811 0.931863i \(-0.381817\pi\)
−0.931863 + 0.362811i \(0.881817\pi\)
\(564\) 0 0
\(565\) −1.83668e14 + 1.83668e14i −0.134204 + 0.134204i
\(566\) 0 0
\(567\) 2.55343e14i 0.182985i
\(568\) 0 0
\(569\) 1.49261e14i 0.104913i −0.998623 0.0524563i \(-0.983295\pi\)
0.998623 0.0524563i \(-0.0167050\pi\)
\(570\) 0 0
\(571\) −1.12561e15 + 1.12561e15i −0.776052 + 0.776052i −0.979157 0.203105i \(-0.934897\pi\)
0.203105 + 0.979157i \(0.434897\pi\)
\(572\) 0 0
\(573\) −4.70061e14 4.70061e14i −0.317910 0.317910i
\(574\) 0 0
\(575\) −2.35796e15 −1.56445
\(576\) 0 0
\(577\) −1.10552e15 −0.719615 −0.359807 0.933027i \(-0.617158\pi\)
−0.359807 + 0.933027i \(0.617158\pi\)
\(578\) 0 0
\(579\) 1.00940e15 + 1.00940e15i 0.644662 + 0.644662i
\(580\) 0 0
\(581\) −6.78233e14 + 6.78233e14i −0.425021 + 0.425021i
\(582\) 0 0
\(583\) 1.00881e15i 0.620345i
\(584\) 0 0
\(585\) 2.32127e15i 1.40077i
\(586\) 0 0
\(587\) 5.74857e14 5.74857e14i 0.340448 0.340448i −0.516088 0.856536i \(-0.672612\pi\)
0.856536 + 0.516088i \(0.172612\pi\)
\(588\) 0 0
\(589\) −2.05545e15 2.05545e15i −1.19474 1.19474i
\(590\) 0 0
\(591\) −9.11431e14 −0.519987
\(592\) 0 0
\(593\) 1.70041e15 0.952252 0.476126 0.879377i \(-0.342041\pi\)
0.476126 + 0.879377i \(0.342041\pi\)
\(594\) 0 0
\(595\) −1.45397e15 1.45397e15i −0.799303 0.799303i
\(596\) 0 0
\(597\) 2.65076e14 2.65076e14i 0.143058 0.143058i
\(598\) 0 0
\(599\) 9.06284e14i 0.480194i 0.970749 + 0.240097i \(0.0771792\pi\)
−0.970749 + 0.240097i \(0.922821\pi\)
\(600\) 0 0
\(601\) 3.33138e15i 1.73306i 0.499124 + 0.866531i \(0.333655\pi\)
−0.499124 + 0.866531i \(0.666345\pi\)
\(602\) 0 0
\(603\) 5.71058e14 5.71058e14i 0.291699 0.291699i
\(604\) 0 0
\(605\) 2.16829e15 + 2.16829e15i 1.08758 + 1.08758i
\(606\) 0 0
\(607\) −1.62819e15 −0.801986 −0.400993 0.916081i \(-0.631335\pi\)
−0.400993 + 0.916081i \(0.631335\pi\)
\(608\) 0 0
\(609\) 2.92555e14 0.141518
\(610\) 0 0
\(611\) 2.47174e15 + 2.47174e15i 1.17429 + 1.17429i
\(612\) 0 0
\(613\) −1.39720e15 + 1.39720e15i −0.651968 + 0.651968i −0.953467 0.301499i \(-0.902513\pi\)
0.301499 + 0.953467i \(0.402513\pi\)
\(614\) 0 0
\(615\) 2.66867e15i 1.22315i
\(616\) 0 0
\(617\) 1.42742e15i 0.642661i −0.946967 0.321331i \(-0.895870\pi\)
0.946967 0.321331i \(-0.104130\pi\)
\(618\) 0 0
\(619\) 5.96980e14 5.96980e14i 0.264035 0.264035i −0.562656 0.826691i \(-0.690220\pi\)
0.826691 + 0.562656i \(0.190220\pi\)
\(620\) 0 0
\(621\) −1.16488e15 1.16488e15i −0.506147 0.506147i
\(622\) 0 0
\(623\) 1.86686e15 0.796942
\(624\) 0 0
\(625\) −4.50500e15 −1.88953
\(626\) 0 0
\(627\) 6.50426e14 + 6.50426e14i 0.268057 + 0.268057i
\(628\) 0 0
\(629\) −1.92482e14 + 1.92482e14i −0.0779493 + 0.0779493i
\(630\) 0 0
\(631\) 3.18386e15i 1.26704i −0.773724 0.633522i \(-0.781608\pi\)
0.773724 0.633522i \(-0.218392\pi\)
\(632\) 0 0
\(633\) 3.12041e15i 1.22037i
\(634\) 0 0
\(635\) −5.44742e15 + 5.44742e15i −2.09380 + 2.09380i
\(636\) 0 0
\(637\) −1.49107e15 1.49107e15i −0.563289 0.563289i
\(638\) 0 0
\(639\) −1.13054e13 −0.00419791
\(640\) 0 0
\(641\) 3.07527e15 1.12244 0.561222 0.827665i \(-0.310331\pi\)
0.561222 + 0.827665i \(0.310331\pi\)
\(642\) 0 0
\(643\) 2.71246e14 + 2.71246e14i 0.0973203 + 0.0973203i 0.754091 0.656770i \(-0.228078\pi\)
−0.656770 + 0.754091i \(0.728078\pi\)
\(644\) 0 0
\(645\) 1.10617e15 1.10617e15i 0.390161 0.390161i
\(646\) 0 0
\(647\) 4.50034e15i 1.56053i −0.625451 0.780264i \(-0.715085\pi\)
0.625451 0.780264i \(-0.284915\pi\)
\(648\) 0 0
\(649\) 2.50682e14i 0.0854627i
\(650\) 0 0
\(651\) 2.04592e15 2.04592e15i 0.685792 0.685792i
\(652\) 0 0
\(653\) −2.95719e15 2.95719e15i −0.974668 0.974668i 0.0250186 0.999687i \(-0.492035\pi\)
−0.999687 + 0.0250186i \(0.992035\pi\)
\(654\) 0 0
\(655\) 1.72577e15 0.559313
\(656\) 0 0
\(657\) −1.20282e14 −0.0383346
\(658\) 0 0
\(659\) −2.05647e15 2.05647e15i −0.644545 0.644545i 0.307124 0.951669i \(-0.400633\pi\)
−0.951669 + 0.307124i \(0.900633\pi\)
\(660\) 0 0
\(661\) 2.94861e15 2.94861e15i 0.908887 0.908887i −0.0872957 0.996182i \(-0.527823\pi\)
0.996182 + 0.0872957i \(0.0278225\pi\)
\(662\) 0 0
\(663\) 1.55285e15i 0.470767i
\(664\) 0 0
\(665\) 1.10971e16i 3.30896i
\(666\) 0 0
\(667\) −2.80032e14 + 2.80032e14i −0.0821326 + 0.0821326i
\(668\) 0 0
\(669\) −2.90216e15 2.90216e15i −0.837294 0.837294i
\(670\) 0 0
\(671\) 1.98878e15 0.564432
\(672\) 0 0
\(673\) −3.67663e15 −1.02652 −0.513259 0.858234i \(-0.671562\pi\)
−0.513259 + 0.858234i \(0.671562\pi\)
\(674\) 0 0
\(675\) −6.07929e15 6.07929e15i −1.66987 1.66987i
\(676\) 0 0
\(677\) −1.77059e15 + 1.77059e15i −0.478499 + 0.478499i −0.904651 0.426152i \(-0.859869\pi\)
0.426152 + 0.904651i \(0.359869\pi\)
\(678\) 0 0
\(679\) 1.38705e15i 0.368816i
\(680\) 0 0
\(681\) 4.56942e15i 1.19551i
\(682\) 0 0
\(683\) −2.83214e15 + 2.83214e15i −0.729124 + 0.729124i −0.970445 0.241321i \(-0.922419\pi\)
0.241321 + 0.970445i \(0.422419\pi\)
\(684\) 0 0
\(685\) 3.72728e15 + 3.72728e15i 0.944264 + 0.944264i
\(686\) 0 0
\(687\) 3.27029e15 0.815312
\(688\) 0 0
\(689\) 9.19112e15 2.25508
\(690\) 0 0
\(691\) 2.15938e15 + 2.15938e15i 0.521435 + 0.521435i 0.918005 0.396570i \(-0.129800\pi\)
−0.396570 + 0.918005i \(0.629800\pi\)
\(692\) 0 0
\(693\) 7.93250e14 7.93250e14i 0.188529 0.188529i
\(694\) 0 0
\(695\) 7.35925e15i 1.72154i
\(696\) 0 0
\(697\) 2.18741e15i 0.503675i
\(698\) 0 0
\(699\) −2.39199e15 + 2.39199e15i −0.542169 + 0.542169i
\(700\) 0 0
\(701\) −2.82397e14 2.82397e14i −0.0630102 0.0630102i 0.674900 0.737910i \(-0.264187\pi\)
−0.737910 + 0.674900i \(0.764187\pi\)
\(702\) 0 0
\(703\) 1.46908e15 0.322695
\(704\) 0 0
\(705\) 6.62091e15 1.43178
\(706\) 0 0
\(707\) 4.11501e15 + 4.11501e15i 0.876121 + 0.876121i
\(708\) 0 0
\(709\) 5.08406e15 5.08406e15i 1.06575 1.06575i 0.0680718 0.997680i \(-0.478315\pi\)
0.997680 0.0680718i \(-0.0216847\pi\)
\(710\) 0 0
\(711\) 1.62705e14i 0.0335828i
\(712\) 0 0
\(713\) 3.91668e15i 0.796024i
\(714\) 0 0
\(715\) 3.47759e15 3.47759e15i 0.695978 0.695978i
\(716\) 0 0
\(717\) 2.09573e15 + 2.09573e15i 0.413029 + 0.413029i
\(718\) 0 0
\(719\) 5.28231e15 1.02522 0.512608 0.858623i \(-0.328679\pi\)
0.512608 + 0.858623i \(0.328679\pi\)
\(720\) 0 0
\(721\) −8.67838e14 −0.165880
\(722\) 0 0
\(723\) −8.33084e14 8.33084e14i −0.156830 0.156830i
\(724\) 0 0
\(725\) −1.46144e15 + 1.46144e15i −0.270970 + 0.270970i
\(726\) 0 0
\(727\) 1.04603e16i 1.91032i −0.296087 0.955161i \(-0.595682\pi\)
0.296087 0.955161i \(-0.404318\pi\)
\(728\) 0 0
\(729\) 4.32120e15i 0.777325i
\(730\) 0 0
\(731\) −9.06691e14 + 9.06691e14i −0.160662 + 0.160662i
\(732\) 0 0
\(733\) 2.63519e15 + 2.63519e15i 0.459982 + 0.459982i 0.898649 0.438668i \(-0.144549\pi\)
−0.438668 + 0.898649i \(0.644549\pi\)
\(734\) 0 0
\(735\) −3.99404e15 −0.686803
\(736\) 0 0
\(737\) 1.71105e15 0.289863
\(738\) 0 0
\(739\) −1.96675e15 1.96675e15i −0.328250 0.328250i 0.523671 0.851921i \(-0.324562\pi\)
−0.851921 + 0.523671i \(0.824562\pi\)
\(740\) 0 0
\(741\) −5.92592e15 + 5.92592e15i −0.974442 + 0.974442i
\(742\) 0 0
\(743\) 3.54440e15i 0.574255i −0.957892 0.287127i \(-0.907300\pi\)
0.957892 0.287127i \(-0.0927003\pi\)
\(744\) 0 0
\(745\) 1.05394e16i 1.68251i
\(746\) 0 0
\(747\) 1.18870e15 1.18870e15i 0.186985 0.186985i
\(748\) 0 0
\(749\) −3.62914e15 3.62914e15i −0.562541 0.562541i
\(750\) 0 0
\(751\) 7.22921e15 1.10426 0.552130 0.833758i \(-0.313815\pi\)
0.552130 + 0.833758i \(0.313815\pi\)
\(752\) 0 0
\(753\) −4.92473e15 −0.741327
\(754\) 0 0
\(755\) 1.04496e16 + 1.04496e16i 1.55021 + 1.55021i
\(756\) 0 0
\(757\) 1.87506e15 1.87506e15i 0.274150 0.274150i −0.556618 0.830768i \(-0.687901\pi\)
0.830768 + 0.556618i \(0.187901\pi\)
\(758\) 0 0
\(759\) 1.23939e15i 0.178599i
\(760\) 0 0
\(761\) 1.29960e16i 1.84584i −0.384994 0.922919i \(-0.625796\pi\)
0.384994 0.922919i \(-0.374204\pi\)
\(762\) 0 0
\(763\) 5.71320e15 5.71320e15i 0.799824 0.799824i
\(764\) 0 0
\(765\) 2.54828e15 + 2.54828e15i 0.351648 + 0.351648i
\(766\) 0 0
\(767\) −2.28392e15 −0.310675
\(768\) 0 0
\(769\) 1.09185e16 1.46408 0.732042 0.681259i \(-0.238567\pi\)
0.732042 + 0.681259i \(0.238567\pi\)
\(770\) 0 0
\(771\) −3.22181e15 3.22181e15i −0.425893 0.425893i
\(772\) 0 0
\(773\) −7.36281e15 + 7.36281e15i −0.959526 + 0.959526i −0.999212 0.0396865i \(-0.987364\pi\)
0.0396865 + 0.999212i \(0.487364\pi\)
\(774\) 0 0
\(775\) 2.04405e16i 2.62622i
\(776\) 0 0
\(777\) 1.46227e15i 0.185230i
\(778\) 0 0
\(779\) −8.34750e15 + 8.34750e15i −1.04256 + 1.04256i
\(780\) 0 0
\(781\) −1.69372e13 1.69372e13i −0.00208574 0.00208574i
\(782\) 0 0
\(783\) −1.44396e15 −0.175333
\(784\) 0 0
\(785\) −2.68308e16 −3.21255
\(786\) 0 0
\(787\) −2.03453e15 2.03453e15i −0.240216 0.240216i 0.576723 0.816940i \(-0.304331\pi\)
−0.816940 + 0.576723i \(0.804331\pi\)
\(788\) 0 0
\(789\) 4.65561e15 4.65561e15i 0.542067 0.542067i
\(790\) 0 0
\(791\) 1.14368e15i 0.131321i
\(792\) 0 0
\(793\) 1.81194e16i 2.05183i
\(794\) 0 0
\(795\) 1.23099e16 1.23099e16i 1.37478 1.37478i
\(796\) 0 0
\(797\) 1.65013e15 + 1.65013e15i 0.181760 + 0.181760i 0.792122 0.610363i \(-0.208976\pi\)
−0.610363 + 0.792122i \(0.708976\pi\)
\(798\) 0 0
\(799\) −5.42693e15 −0.589585
\(800\) 0 0
\(801\) −3.27192e15 −0.350610
\(802\) 0 0
\(803\) −1.80200e14 1.80200e14i −0.0190466 0.0190466i
\(804\) 0 0
\(805\) 1.05728e16 1.05728e16i 1.10234 1.10234i
\(806\) 0 0
\(807\) 8.90332e14i 0.0915688i
\(808\) 0 0
\(809\) 8.91512e15i 0.904504i 0.891890 + 0.452252i \(0.149379\pi\)
−0.891890 + 0.452252i \(0.850621\pi\)
\(810\) 0 0
\(811\) −1.44005e15 + 1.44005e15i −0.144133 + 0.144133i −0.775491 0.631358i \(-0.782498\pi\)
0.631358 + 0.775491i \(0.282498\pi\)
\(812\) 0 0
\(813\) −6.01407e15 6.01407e15i −0.593841 0.593841i
\(814\) 0 0
\(815\) −1.88563e16 −1.83691
\(816\) 0 0
\(817\) 6.92014e15 0.665110
\(818\) 0 0
\(819\) 7.22716e15 + 7.22716e15i 0.685341 + 0.685341i
\(820\) 0 0
\(821\) −2.49954e15 + 2.49954e15i −0.233869 + 0.233869i −0.814305 0.580437i \(-0.802882\pi\)
0.580437 + 0.814305i \(0.302882\pi\)
\(822\) 0 0
\(823\) 7.96258e15i 0.735114i −0.930001 0.367557i \(-0.880194\pi\)
0.930001 0.367557i \(-0.119806\pi\)
\(824\) 0 0
\(825\) 6.46816e15i 0.589229i
\(826\) 0 0
\(827\) −4.10254e15 + 4.10254e15i −0.368784 + 0.368784i −0.867034 0.498249i \(-0.833976\pi\)
0.498249 + 0.867034i \(0.333976\pi\)
\(828\) 0 0
\(829\) −2.13377e15 2.13377e15i −0.189277 0.189277i 0.606107 0.795383i \(-0.292730\pi\)
−0.795383 + 0.606107i \(0.792730\pi\)
\(830\) 0 0
\(831\) −5.53730e15 −0.484722
\(832\) 0 0
\(833\) 3.27378e15 0.282815
\(834\) 0 0
\(835\) −1.96414e16 1.96414e16i −1.67455 1.67455i
\(836\) 0 0
\(837\) −1.00980e16 + 1.00980e16i −0.849660 + 0.849660i
\(838\) 0 0
\(839\) 1.44656e16i 1.20128i 0.799519 + 0.600641i \(0.205088\pi\)
−0.799519 + 0.600641i \(0.794912\pi\)
\(840\) 0 0
\(841\) 1.18534e16i 0.971549i
\(842\) 0 0
\(843\) 2.46079e15 2.46079e15i 0.199078 0.199078i
\(844\) 0 0
\(845\) 1.56662e16 + 1.56662e16i 1.25099 + 1.25099i
\(846\) 0 0
\(847\) −1.35017e16 −1.06422
\(848\) 0 0
\(849\) −2.29880e15 −0.178858
\(850\) 0 0
\(851\) −1.39968e15 1.39968e15i −0.107502 0.107502i
\(852\) 0 0
\(853\) −1.03276e14 + 1.03276e14i −0.00783032 + 0.00783032i −0.711011 0.703181i \(-0.751762\pi\)
0.703181 + 0.711011i \(0.251762\pi\)
\(854\) 0 0
\(855\) 1.94492e16i 1.45576i
\(856\) 0 0
\(857\) 4.13417e14i 0.0305488i 0.999883 + 0.0152744i \(0.00486218\pi\)
−0.999883 + 0.0152744i \(0.995138\pi\)
\(858\) 0 0
\(859\) −1.18177e16 + 1.18177e16i −0.862124 + 0.862124i −0.991585 0.129461i \(-0.958675\pi\)
0.129461 + 0.991585i \(0.458675\pi\)
\(860\) 0 0
\(861\) −8.30878e15 8.30878e15i −0.598439 0.598439i
\(862\) 0 0
\(863\) −2.93105e15 −0.208432 −0.104216 0.994555i \(-0.533233\pi\)
−0.104216 + 0.994555i \(0.533233\pi\)
\(864\) 0 0
\(865\) −3.72930e16 −2.61842
\(866\) 0 0
\(867\) −5.13281e15 5.13281e15i −0.355836 0.355836i
\(868\) 0 0
\(869\) 2.43755e14 2.43755e14i 0.0166857 0.0166857i
\(870\) 0 0
\(871\) 1.55891e16i 1.05371i
\(872\) 0 0
\(873\) 2.43100e15i 0.162258i
\(874\) 0 0
\(875\) 3.08904e16 3.08904e16i 2.03600 2.03600i
\(876\) 0 0
\(877\) 1.13600e16 + 1.13600e16i 0.739402 + 0.739402i 0.972462 0.233060i \(-0.0748740\pi\)
−0.233060 + 0.972462i \(0.574874\pi\)
\(878\) 0 0
\(879\) −1.45717e16 −0.936640
\(880\) 0 0
\(881\) 1.42576e16 0.905062 0.452531 0.891749i \(-0.350521\pi\)
0.452531 + 0.891749i \(0.350521\pi\)
\(882\) 0 0
\(883\) 1.57729e16 + 1.57729e16i 0.988842 + 0.988842i 0.999938 0.0110969i \(-0.00353232\pi\)
−0.0110969 + 0.999938i \(0.503532\pi\)
\(884\) 0 0
\(885\) −3.05890e15 + 3.05890e15i −0.189399 + 0.189399i
\(886\) 0 0
\(887\) 1.79870e16i 1.09997i −0.835175 0.549984i \(-0.814634\pi\)
0.835175 0.549984i \(-0.185366\pi\)
\(888\) 0 0
\(889\) 3.39206e16i 2.04882i
\(890\) 0 0
\(891\) 6.70453e14 6.70453e14i 0.0399983 0.0399983i
\(892\) 0 0
\(893\) 2.07100e16 + 2.07100e16i 1.22038 + 1.22038i
\(894\) 0 0
\(895\) 1.39693e16 0.813106
\(896\) 0 0
\(897\) 1.12919e16 0.649245
\(898\) 0 0
\(899\) 2.42752e15 + 2.42752e15i 0.137874 + 0.137874i
\(900\) 0 0
\(901\) −1.00900e16 + 1.00900e16i −0.566113 + 0.566113i
\(902\) 0 0
\(903\) 6.88804e15i 0.381780i
\(904\) 0 0
\(905\) 1.94369e16i 1.06429i
\(906\) 0 0
\(907\) −1.21117e15 + 1.21117e15i −0.0655186 + 0.0655186i −0.739107 0.673588i \(-0.764752\pi\)
0.673588 + 0.739107i \(0.264752\pi\)
\(908\) 0 0
\(909\) −7.21212e15 7.21212e15i −0.385444 0.385444i
\(910\) 0 0
\(911\) 2.23895e16 1.18221 0.591104 0.806595i \(-0.298692\pi\)
0.591104 + 0.806595i \(0.298692\pi\)
\(912\) 0 0
\(913\) 3.56167e15 0.185808
\(914\) 0 0
\(915\) 2.42677e16 + 2.42677e16i 1.25087 + 1.25087i
\(916\) 0 0
\(917\) −5.37310e15 + 5.37310e15i −0.273649 + 0.273649i
\(918\) 0 0
\(919\) 2.32306e16i 1.16903i 0.811384 + 0.584514i \(0.198715\pi\)
−0.811384 + 0.584514i \(0.801285\pi\)
\(920\) 0 0
\(921\) 1.54075e16i 0.766135i
\(922\) 0 0
\(923\) 1.54312e14 1.54312e14i 0.00758211 0.00758211i
\(924\) 0 0
\(925\) −7.30465e15 7.30465e15i −0.354666 0.354666i
\(926\) 0 0
\(927\) 1.52100e15 0.0729780
\(928\) 0 0
\(929\) −1.44121e16 −0.683345 −0.341673 0.939819i \(-0.610993\pi\)
−0.341673 + 0.939819i \(0.610993\pi\)
\(930\) 0 0
\(931\) −1.24932e16 1.24932e16i −0.585399 0.585399i
\(932\) 0 0
\(933\) 4.41411e15 4.41411e15i 0.204406 0.204406i
\(934\) 0 0
\(935\) 7.63537e15i 0.349435i
\(936\) 0 0
\(937\) 2.81562e16i 1.27352i 0.771061 + 0.636762i \(0.219727\pi\)
−0.771061 + 0.636762i \(0.780273\pi\)
\(938\) 0 0
\(939\) −9.80877e15 + 9.80877e15i −0.438484 + 0.438484i
\(940\) 0 0
\(941\) −6.14074e15 6.14074e15i −0.271318 0.271318i 0.558313 0.829631i \(-0.311449\pi\)
−0.829631 + 0.558313i \(0.811449\pi\)
\(942\) 0 0
\(943\) 1.59062e16 0.694630
\(944\) 0 0
\(945\) 5.45178e16 2.35322
\(946\) 0 0
\(947\) −1.03061e16 1.03061e16i −0.439714 0.439714i 0.452202 0.891916i \(-0.350639\pi\)
−0.891916 + 0.452202i \(0.850639\pi\)
\(948\) 0 0
\(949\) 1.64177e15 1.64177e15i 0.0692385 0.0692385i
\(950\) 0 0
\(951\) 9.98022e15i 0.416052i
\(952\) 0 0
\(953\) 9.28274e15i 0.382530i 0.981538 + 0.191265i \(0.0612589\pi\)
−0.981538 + 0.191265i \(0.938741\pi\)
\(954\) 0 0
\(955\) −2.10578e16 + 2.10578e16i −0.857815 + 0.857815i
\(956\) 0 0
\(957\) −7.68161e14 7.68161e14i −0.0309340 0.0309340i
\(958\) 0 0
\(959\) −2.32094e16 −0.923979
\(960\) 0 0
\(961\) 8.54412e15 0.336271
\(962\) 0 0
\(963\) 6.36056e15 + 6.36056e15i 0.247486 + 0.247486i
\(964\) 0 0
\(965\) 4.52191e16 4.52191e16i 1.73949 1.73949i
\(966\) 0 0
\(967\) 1.09617e16i 0.416901i 0.978033 + 0.208451i \(0.0668421\pi\)
−0.978033 + 0.208451i \(0.933158\pi\)
\(968\) 0 0
\(969\) 1.30109e16i 0.489245i
\(970\) 0 0
\(971\) 2.45150e16 2.45150e16i 0.911436 0.911436i −0.0849489 0.996385i \(-0.527073\pi\)
0.996385 + 0.0849489i \(0.0270727\pi\)
\(972\) 0 0
\(973\) −2.29127e16 2.29127e16i −0.842278 0.842278i
\(974\) 0 0
\(975\) 5.89303e16 2.14197
\(976\) 0 0
\(977\) −3.81818e16 −1.37226 −0.686130 0.727479i \(-0.740692\pi\)
−0.686130 + 0.727479i \(0.740692\pi\)
\(978\) 0 0
\(979\) −4.90181e15 4.90181e15i −0.174201 0.174201i
\(980\) 0 0
\(981\) −1.00132e16 + 1.00132e16i −0.351877 + 0.351877i
\(982\) 0 0
\(983\) 2.94742e16i 1.02423i −0.858917 0.512115i \(-0.828862\pi\)
0.858917 0.512115i \(-0.171138\pi\)
\(984\) 0 0
\(985\) 4.08302e16i 1.40308i
\(986\) 0 0
\(987\) −2.06139e16 + 2.06139e16i −0.700512 + 0.700512i
\(988\) 0 0
\(989\) −6.59319e15 6.59319e15i −0.221573 0.221573i
\(990\) 0 0
\(991\) −2.13654e15 −0.0710077 −0.0355038 0.999370i \(-0.511304\pi\)
−0.0355038 + 0.999370i \(0.511304\pi\)
\(992\) 0 0
\(993\) −2.45736e16 −0.807696
\(994\) 0 0
\(995\) −1.18748e16 1.18748e16i −0.386012 0.386012i
\(996\) 0 0
\(997\) 2.64209e16 2.64209e16i 0.849423 0.849423i −0.140638 0.990061i \(-0.544915\pi\)
0.990061 + 0.140638i \(0.0449154\pi\)
\(998\) 0 0
\(999\) 7.21729e15i 0.229490i
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 64.12.e.a.17.15 42
4.3 odd 2 16.12.e.a.13.8 yes 42
8.3 odd 2 128.12.e.b.33.15 42
8.5 even 2 128.12.e.a.33.7 42
16.3 odd 4 128.12.e.b.97.15 42
16.5 even 4 inner 64.12.e.a.49.15 42
16.11 odd 4 16.12.e.a.5.8 42
16.13 even 4 128.12.e.a.97.7 42
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.12.e.a.5.8 42 16.11 odd 4
16.12.e.a.13.8 yes 42 4.3 odd 2
64.12.e.a.17.15 42 1.1 even 1 trivial
64.12.e.a.49.15 42 16.5 even 4 inner
128.12.e.a.33.7 42 8.5 even 2
128.12.e.a.97.7 42 16.13 even 4
128.12.e.b.33.15 42 8.3 odd 2
128.12.e.b.97.15 42 16.3 odd 4