Properties

Label 64.12.e.a.17.18
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 17.18
Character \(\chi\) \(=\) 64.17
Dual form 64.12.e.a.49.18

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(435.014 + 435.014i) q^{3} +(-1517.34 + 1517.34i) q^{5} -59722.7i q^{7} +201326. i q^{9} +O(q^{10})\) \(q+(435.014 + 435.014i) q^{3} +(-1517.34 + 1517.34i) q^{5} -59722.7i q^{7} +201326. i q^{9} +(-56196.6 + 56196.6i) q^{11} +(-315796. - 315796. i) q^{13} -1.32013e6 q^{15} +8.11170e6 q^{17} +(1.33811e7 + 1.33811e7i) q^{19} +(2.59802e7 - 2.59802e7i) q^{21} +2.84613e7i q^{23} +4.42235e7i q^{25} +(-1.05184e7 + 1.05184e7i) q^{27} +(7.59695e7 + 7.59695e7i) q^{29} -5.67278e7 q^{31} -4.88926e7 q^{33} +(9.06196e7 + 9.06196e7i) q^{35} +(3.75680e8 - 3.75680e8i) q^{37} -2.74751e8i q^{39} +1.51696e7i q^{41} +(-1.08339e9 + 1.08339e9i) q^{43} +(-3.05481e8 - 3.05481e8i) q^{45} -2.79798e9 q^{47} -1.58947e9 q^{49} +(3.52870e9 + 3.52870e9i) q^{51} +(2.03657e9 - 2.03657e9i) q^{53} -1.70539e8i q^{55} +1.16419e10i q^{57} +(4.23123e9 - 4.23123e9i) q^{59} +(6.54407e9 + 6.54407e9i) q^{61} +1.20238e10 q^{63} +9.58341e8 q^{65} +(8.01903e9 + 8.01903e9i) q^{67} +(-1.23810e10 + 1.23810e10i) q^{69} +1.40718e10i q^{71} +6.29343e9i q^{73} +(-1.92378e10 + 1.92378e10i) q^{75} +(3.35621e9 + 3.35621e9i) q^{77} +3.46459e10 q^{79} +2.65131e10 q^{81} +(-4.29998e10 - 4.29998e10i) q^{83} +(-1.23082e10 + 1.23082e10i) q^{85} +6.60955e10i q^{87} +3.39308e9i q^{89} +(-1.88602e10 + 1.88602e10i) q^{91} +(-2.46773e10 - 2.46773e10i) q^{93} -4.06073e10 q^{95} +1.55551e10 q^{97} +(-1.13139e10 - 1.13139e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 42 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) 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
\(3\) 435.014 + 435.014i 1.03356 + 1.03356i 0.999417 + 0.0341434i \(0.0108703\pi\)
0.0341434 + 0.999417i \(0.489130\pi\)
\(4\) 0 0
\(5\) −1517.34 + 1517.34i −0.217144 + 0.217144i −0.807294 0.590150i \(-0.799069\pi\)
0.590150 + 0.807294i \(0.299069\pi\)
\(6\) 0 0
\(7\) 59722.7i 1.34307i −0.740971 0.671537i \(-0.765634\pi\)
0.740971 0.671537i \(-0.234366\pi\)
\(8\) 0 0
\(9\) 201326.i 1.13649i
\(10\) 0 0
\(11\) −56196.6 + 56196.6i −0.105208 + 0.105208i −0.757752 0.652543i \(-0.773702\pi\)
0.652543 + 0.757752i \(0.273702\pi\)
\(12\) 0 0
\(13\) −315796. 315796.i −0.235895 0.235895i 0.579253 0.815148i \(-0.303344\pi\)
−0.815148 + 0.579253i \(0.803344\pi\)
\(14\) 0 0
\(15\) −1.32013e6 −0.448863
\(16\) 0 0
\(17\) 8.11170e6 1.38562 0.692808 0.721122i \(-0.256373\pi\)
0.692808 + 0.721122i \(0.256373\pi\)
\(18\) 0 0
\(19\) 1.33811e7 + 1.33811e7i 1.23978 + 1.23978i 0.960089 + 0.279695i \(0.0902334\pi\)
0.279695 + 0.960089i \(0.409767\pi\)
\(20\) 0 0
\(21\) 2.59802e7 2.59802e7i 1.38815 1.38815i
\(22\) 0 0
\(23\) 2.84613e7i 0.922043i 0.887389 + 0.461022i \(0.152517\pi\)
−0.887389 + 0.461022i \(0.847483\pi\)
\(24\) 0 0
\(25\) 4.42235e7i 0.905697i
\(26\) 0 0
\(27\) −1.05184e7 + 1.05184e7i −0.141075 + 0.141075i
\(28\) 0 0
\(29\) 7.59695e7 + 7.59695e7i 0.687781 + 0.687781i 0.961741 0.273960i \(-0.0883335\pi\)
−0.273960 + 0.961741i \(0.588334\pi\)
\(30\) 0 0
\(31\) −5.67278e7 −0.355882 −0.177941 0.984041i \(-0.556944\pi\)
−0.177941 + 0.984041i \(0.556944\pi\)
\(32\) 0 0
\(33\) −4.88926e7 −0.217479
\(34\) 0 0
\(35\) 9.06196e7 + 9.06196e7i 0.291641 + 0.291641i
\(36\) 0 0
\(37\) 3.75680e8 3.75680e8i 0.890654 0.890654i −0.103931 0.994585i \(-0.533142\pi\)
0.994585 + 0.103931i \(0.0331421\pi\)
\(38\) 0 0
\(39\) 2.74751e8i 0.487623i
\(40\) 0 0
\(41\) 1.51696e7i 0.0204486i 0.999948 + 0.0102243i \(0.00325455\pi\)
−0.999948 + 0.0102243i \(0.996745\pi\)
\(42\) 0 0
\(43\) −1.08339e9 + 1.08339e9i −1.12385 + 1.12385i −0.132689 + 0.991158i \(0.542361\pi\)
−0.991158 + 0.132689i \(0.957639\pi\)
\(44\) 0 0
\(45\) −3.05481e8 3.05481e8i −0.246783 0.246783i
\(46\) 0 0
\(47\) −2.79798e9 −1.77954 −0.889768 0.456414i \(-0.849134\pi\)
−0.889768 + 0.456414i \(0.849134\pi\)
\(48\) 0 0
\(49\) −1.58947e9 −0.803848
\(50\) 0 0
\(51\) 3.52870e9 + 3.52870e9i 1.43212 + 1.43212i
\(52\) 0 0
\(53\) 2.03657e9 2.03657e9i 0.668932 0.668932i −0.288537 0.957469i \(-0.593169\pi\)
0.957469 + 0.288537i \(0.0931688\pi\)
\(54\) 0 0
\(55\) 1.70539e8i 0.0456908i
\(56\) 0 0
\(57\) 1.16419e10i 2.56278i
\(58\) 0 0
\(59\) 4.23123e9 4.23123e9i 0.770513 0.770513i −0.207683 0.978196i \(-0.566592\pi\)
0.978196 + 0.207683i \(0.0665922\pi\)
\(60\) 0 0
\(61\) 6.54407e9 + 6.54407e9i 0.992051 + 0.992051i 0.999969 0.00791726i \(-0.00252017\pi\)
−0.00791726 + 0.999969i \(0.502520\pi\)
\(62\) 0 0
\(63\) 1.20238e10 1.52640
\(64\) 0 0
\(65\) 9.58341e8 0.102446
\(66\) 0 0
\(67\) 8.01903e9 + 8.01903e9i 0.725622 + 0.725622i 0.969744 0.244122i \(-0.0784998\pi\)
−0.244122 + 0.969744i \(0.578500\pi\)
\(68\) 0 0
\(69\) −1.23810e10 + 1.23810e10i −0.952987 + 0.952987i
\(70\) 0 0
\(71\) 1.40718e10i 0.925608i 0.886461 + 0.462804i \(0.153157\pi\)
−0.886461 + 0.462804i \(0.846843\pi\)
\(72\) 0 0
\(73\) 6.29343e9i 0.355314i 0.984093 + 0.177657i \(0.0568517\pi\)
−0.984093 + 0.177657i \(0.943148\pi\)
\(74\) 0 0
\(75\) −1.92378e10 + 1.92378e10i −0.936092 + 0.936092i
\(76\) 0 0
\(77\) 3.35621e9 + 3.35621e9i 0.141303 + 0.141303i
\(78\) 0 0
\(79\) 3.46459e10 1.26678 0.633392 0.773831i \(-0.281662\pi\)
0.633392 + 0.773831i \(0.281662\pi\)
\(80\) 0 0
\(81\) 2.65131e10 0.844875
\(82\) 0 0
\(83\) −4.29998e10 4.29998e10i −1.19822 1.19822i −0.974699 0.223521i \(-0.928245\pi\)
−0.223521 0.974699i \(-0.571755\pi\)
\(84\) 0 0
\(85\) −1.23082e10 + 1.23082e10i −0.300878 + 0.300878i
\(86\) 0 0
\(87\) 6.60955e10i 1.42173i
\(88\) 0 0
\(89\) 3.39308e9i 0.0644093i 0.999481 + 0.0322047i \(0.0102528\pi\)
−0.999481 + 0.0322047i \(0.989747\pi\)
\(90\) 0 0
\(91\) −1.88602e10 + 1.88602e10i −0.316824 + 0.316824i
\(92\) 0 0
\(93\) −2.46773e10 2.46773e10i −0.367826 0.367826i
\(94\) 0 0
\(95\) −4.06073e10 −0.538423
\(96\) 0 0
\(97\) 1.55551e10 0.183920 0.0919601 0.995763i \(-0.470687\pi\)
0.0919601 + 0.995763i \(0.470687\pi\)
\(98\) 0 0
\(99\) −1.13139e10 1.13139e10i −0.119569 0.119569i
\(100\) 0 0
\(101\) 8.04221e10 8.04221e10i 0.761391 0.761391i −0.215183 0.976574i \(-0.569035\pi\)
0.976574 + 0.215183i \(0.0690347\pi\)
\(102\) 0 0
\(103\) 2.48411e10i 0.211138i 0.994412 + 0.105569i \(0.0336663\pi\)
−0.994412 + 0.105569i \(0.966334\pi\)
\(104\) 0 0
\(105\) 7.88415e10i 0.602856i
\(106\) 0 0
\(107\) −5.76499e10 + 5.76499e10i −0.397363 + 0.397363i −0.877302 0.479939i \(-0.840659\pi\)
0.479939 + 0.877302i \(0.340659\pi\)
\(108\) 0 0
\(109\) −5.69123e10 5.69123e10i −0.354291 0.354291i 0.507412 0.861704i \(-0.330602\pi\)
−0.861704 + 0.507412i \(0.830602\pi\)
\(110\) 0 0
\(111\) 3.26852e11 1.84109
\(112\) 0 0
\(113\) 1.29979e11 0.663653 0.331826 0.943340i \(-0.392335\pi\)
0.331826 + 0.943340i \(0.392335\pi\)
\(114\) 0 0
\(115\) −4.31854e10 4.31854e10i −0.200216 0.200216i
\(116\) 0 0
\(117\) 6.35782e10 6.35782e10i 0.268093 0.268093i
\(118\) 0 0
\(119\) 4.84453e11i 1.86099i
\(120\) 0 0
\(121\) 2.78996e11i 0.977862i
\(122\) 0 0
\(123\) −6.59899e9 + 6.59899e9i −0.0211348 + 0.0211348i
\(124\) 0 0
\(125\) −1.41191e11 1.41191e11i −0.413811 0.413811i
\(126\) 0 0
\(127\) 9.81625e9 0.0263648 0.0131824 0.999913i \(-0.495804\pi\)
0.0131824 + 0.999913i \(0.495804\pi\)
\(128\) 0 0
\(129\) −9.42576e11 −2.32313
\(130\) 0 0
\(131\) −3.01725e11 3.01725e11i −0.683312 0.683312i 0.277433 0.960745i \(-0.410516\pi\)
−0.960745 + 0.277433i \(0.910516\pi\)
\(132\) 0 0
\(133\) 7.99153e11 7.99153e11i 1.66512 1.66512i
\(134\) 0 0
\(135\) 3.19200e10i 0.0612671i
\(136\) 0 0
\(137\) 2.21227e11i 0.391629i 0.980641 + 0.195814i \(0.0627350\pi\)
−0.980641 + 0.195814i \(0.937265\pi\)
\(138\) 0 0
\(139\) −1.76441e11 + 1.76441e11i −0.288415 + 0.288415i −0.836453 0.548038i \(-0.815375\pi\)
0.548038 + 0.836453i \(0.315375\pi\)
\(140\) 0 0
\(141\) −1.21716e12 1.21716e12i −1.83926 1.83926i
\(142\) 0 0
\(143\) 3.54934e10 0.0496363
\(144\) 0 0
\(145\) −2.30543e11 −0.298695
\(146\) 0 0
\(147\) −6.91441e11 6.91441e11i −0.830825 0.830825i
\(148\) 0 0
\(149\) 1.96547e10 1.96547e10i 0.0219251 0.0219251i −0.696059 0.717984i \(-0.745065\pi\)
0.717984 + 0.696059i \(0.245065\pi\)
\(150\) 0 0
\(151\) 1.18330e12i 1.22666i −0.789828 0.613329i \(-0.789830\pi\)
0.789828 0.613329i \(-0.210170\pi\)
\(152\) 0 0
\(153\) 1.63310e12i 1.57474i
\(154\) 0 0
\(155\) 8.60753e10 8.60753e10i 0.0772777 0.0772777i
\(156\) 0 0
\(157\) 4.40152e11 + 4.40152e11i 0.368260 + 0.368260i 0.866842 0.498582i \(-0.166146\pi\)
−0.498582 + 0.866842i \(0.666146\pi\)
\(158\) 0 0
\(159\) 1.77187e12 1.38276
\(160\) 0 0
\(161\) 1.69978e12 1.23837
\(162\) 0 0
\(163\) −1.90450e11 1.90450e11i −0.129643 0.129643i 0.639308 0.768951i \(-0.279221\pi\)
−0.768951 + 0.639308i \(0.779221\pi\)
\(164\) 0 0
\(165\) 7.41867e10 7.41867e10i 0.0472242 0.0472242i
\(166\) 0 0
\(167\) 2.80229e12i 1.66945i 0.550669 + 0.834723i \(0.314372\pi\)
−0.550669 + 0.834723i \(0.685628\pi\)
\(168\) 0 0
\(169\) 1.59271e12i 0.888707i
\(170\) 0 0
\(171\) −2.69396e12 + 2.69396e12i −1.40901 + 1.40901i
\(172\) 0 0
\(173\) 4.15917e11 + 4.15917e11i 0.204058 + 0.204058i 0.801736 0.597678i \(-0.203910\pi\)
−0.597678 + 0.801736i \(0.703910\pi\)
\(174\) 0 0
\(175\) 2.64114e12 1.21642
\(176\) 0 0
\(177\) 3.68128e12 1.59274
\(178\) 0 0
\(179\) −1.25777e12 1.25777e12i −0.511575 0.511575i 0.403434 0.915009i \(-0.367817\pi\)
−0.915009 + 0.403434i \(0.867817\pi\)
\(180\) 0 0
\(181\) −7.02645e11 + 7.02645e11i −0.268846 + 0.268846i −0.828635 0.559789i \(-0.810882\pi\)
0.559789 + 0.828635i \(0.310882\pi\)
\(182\) 0 0
\(183\) 5.69352e12i 2.05069i
\(184\) 0 0
\(185\) 1.14007e12i 0.386800i
\(186\) 0 0
\(187\) −4.55850e11 + 4.55850e11i −0.145779 + 0.145779i
\(188\) 0 0
\(189\) 6.28187e11 + 6.28187e11i 0.189474 + 0.189474i
\(190\) 0 0
\(191\) 1.55093e12 0.441478 0.220739 0.975333i \(-0.429153\pi\)
0.220739 + 0.975333i \(0.429153\pi\)
\(192\) 0 0
\(193\) −4.58114e12 −1.23143 −0.615713 0.787971i \(-0.711132\pi\)
−0.615713 + 0.787971i \(0.711132\pi\)
\(194\) 0 0
\(195\) 4.16891e11 + 4.16891e11i 0.105885 + 0.105885i
\(196\) 0 0
\(197\) 8.42541e11 8.42541e11i 0.202315 0.202315i −0.598676 0.800991i \(-0.704306\pi\)
0.800991 + 0.598676i \(0.204306\pi\)
\(198\) 0 0
\(199\) 6.45938e12i 1.46723i −0.679564 0.733616i \(-0.737831\pi\)
0.679564 0.733616i \(-0.262169\pi\)
\(200\) 0 0
\(201\) 6.97677e12i 1.49995i
\(202\) 0 0
\(203\) 4.53710e12 4.53710e12i 0.923741 0.923741i
\(204\) 0 0
\(205\) −2.30175e10 2.30175e10i −0.00444029 0.00444029i
\(206\) 0 0
\(207\) −5.73001e12 −1.04790
\(208\) 0 0
\(209\) −1.50394e12 −0.260871
\(210\) 0 0
\(211\) 5.66233e12 + 5.66233e12i 0.932055 + 0.932055i 0.997834 0.0657787i \(-0.0209532\pi\)
−0.0657787 + 0.997834i \(0.520953\pi\)
\(212\) 0 0
\(213\) −6.12140e12 + 6.12140e12i −0.956672 + 0.956672i
\(214\) 0 0
\(215\) 3.28773e12i 0.488073i
\(216\) 0 0
\(217\) 3.38793e12i 0.477976i
\(218\) 0 0
\(219\) −2.73773e12 + 2.73773e12i −0.367238 + 0.367238i
\(220\) 0 0
\(221\) −2.56165e12 2.56165e12i −0.326860 0.326860i
\(222\) 0 0
\(223\) −2.46919e12 −0.299832 −0.149916 0.988699i \(-0.547900\pi\)
−0.149916 + 0.988699i \(0.547900\pi\)
\(224\) 0 0
\(225\) −8.90336e12 −1.02932
\(226\) 0 0
\(227\) 2.54818e12 + 2.54818e12i 0.280600 + 0.280600i 0.833348 0.552748i \(-0.186421\pi\)
−0.552748 + 0.833348i \(0.686421\pi\)
\(228\) 0 0
\(229\) 8.81027e12 8.81027e12i 0.924473 0.924473i −0.0728690 0.997342i \(-0.523215\pi\)
0.997342 + 0.0728690i \(0.0232155\pi\)
\(230\) 0 0
\(231\) 2.92000e12i 0.292090i
\(232\) 0 0
\(233\) 1.10524e13i 1.05438i −0.849747 0.527191i \(-0.823245\pi\)
0.849747 0.527191i \(-0.176755\pi\)
\(234\) 0 0
\(235\) 4.24549e12 4.24549e12i 0.386415 0.386415i
\(236\) 0 0
\(237\) 1.50714e13 + 1.50714e13i 1.30930 + 1.30930i
\(238\) 0 0
\(239\) −8.37084e12 −0.694353 −0.347177 0.937800i \(-0.612860\pi\)
−0.347177 + 0.937800i \(0.612860\pi\)
\(240\) 0 0
\(241\) 1.48928e13 1.18000 0.590002 0.807402i \(-0.299127\pi\)
0.590002 + 0.807402i \(0.299127\pi\)
\(242\) 0 0
\(243\) 1.33969e13 + 1.33969e13i 1.01430 + 1.01430i
\(244\) 0 0
\(245\) 2.41177e12 2.41177e12i 0.174551 0.174551i
\(246\) 0 0
\(247\) 8.45139e12i 0.584918i
\(248\) 0 0
\(249\) 3.74110e13i 2.47687i
\(250\) 0 0
\(251\) 7.98403e12 7.98403e12i 0.505844 0.505844i −0.407404 0.913248i \(-0.633566\pi\)
0.913248 + 0.407404i \(0.133566\pi\)
\(252\) 0 0
\(253\) −1.59943e12 1.59943e12i −0.0970067 0.0970067i
\(254\) 0 0
\(255\) −1.07085e13 −0.621952
\(256\) 0 0
\(257\) 2.70752e12 0.150640 0.0753200 0.997159i \(-0.476002\pi\)
0.0753200 + 0.997159i \(0.476002\pi\)
\(258\) 0 0
\(259\) −2.24366e13 2.24366e13i −1.19621 1.19621i
\(260\) 0 0
\(261\) −1.52947e13 + 1.52947e13i −0.781659 + 0.781659i
\(262\) 0 0
\(263\) 9.34160e11i 0.0457788i 0.999738 + 0.0228894i \(0.00728656\pi\)
−0.999738 + 0.0228894i \(0.992713\pi\)
\(264\) 0 0
\(265\) 6.18034e12i 0.290509i
\(266\) 0 0
\(267\) −1.47603e12 + 1.47603e12i −0.0665709 + 0.0665709i
\(268\) 0 0
\(269\) −1.46911e13 1.46911e13i −0.635942 0.635942i 0.313610 0.949552i \(-0.398462\pi\)
−0.949552 + 0.313610i \(0.898462\pi\)
\(270\) 0 0
\(271\) −3.98105e13 −1.65450 −0.827249 0.561835i \(-0.810095\pi\)
−0.827249 + 0.561835i \(0.810095\pi\)
\(272\) 0 0
\(273\) −1.64089e13 −0.654914
\(274\) 0 0
\(275\) −2.48521e12 2.48521e12i −0.0952869 0.0952869i
\(276\) 0 0
\(277\) −1.57182e13 + 1.57182e13i −0.579114 + 0.579114i −0.934659 0.355545i \(-0.884295\pi\)
0.355545 + 0.934659i \(0.384295\pi\)
\(278\) 0 0
\(279\) 1.14208e13i 0.404458i
\(280\) 0 0
\(281\) 5.65668e13i 1.92609i −0.269339 0.963045i \(-0.586805\pi\)
0.269339 0.963045i \(-0.413195\pi\)
\(282\) 0 0
\(283\) 2.30718e13 2.30718e13i 0.755538 0.755538i −0.219969 0.975507i \(-0.570596\pi\)
0.975507 + 0.219969i \(0.0705957\pi\)
\(284\) 0 0
\(285\) −1.76647e13 1.76647e13i −0.556493 0.556493i
\(286\) 0 0
\(287\) 9.05970e11 0.0274640
\(288\) 0 0
\(289\) 3.15279e13 0.919933
\(290\) 0 0
\(291\) 6.76670e12 + 6.76670e12i 0.190093 + 0.190093i
\(292\) 0 0
\(293\) −1.47734e13 + 1.47734e13i −0.399675 + 0.399675i −0.878118 0.478443i \(-0.841201\pi\)
0.478443 + 0.878118i \(0.341201\pi\)
\(294\) 0 0
\(295\) 1.28404e13i 0.334625i
\(296\) 0 0
\(297\) 1.18220e12i 0.0296845i
\(298\) 0 0
\(299\) 8.98797e12 8.98797e12i 0.217505 0.217505i
\(300\) 0 0
\(301\) 6.47027e13 + 6.47027e13i 1.50941 + 1.50941i
\(302\) 0 0
\(303\) 6.99694e13 1.57389
\(304\) 0 0
\(305\) −1.98592e13 −0.430836
\(306\) 0 0
\(307\) −3.74033e13 3.74033e13i −0.782796 0.782796i 0.197506 0.980302i \(-0.436716\pi\)
−0.980302 + 0.197506i \(0.936716\pi\)
\(308\) 0 0
\(309\) −1.08062e13 + 1.08062e13i −0.218223 + 0.218223i
\(310\) 0 0
\(311\) 5.07867e13i 0.989847i 0.868937 + 0.494924i \(0.164804\pi\)
−0.868937 + 0.494924i \(0.835196\pi\)
\(312\) 0 0
\(313\) 1.06893e13i 0.201120i 0.994931 + 0.100560i \(0.0320634\pi\)
−0.994931 + 0.100560i \(0.967937\pi\)
\(314\) 0 0
\(315\) −1.82441e13 + 1.82441e13i −0.331448 + 0.331448i
\(316\) 0 0
\(317\) −1.50024e13 1.50024e13i −0.263229 0.263229i 0.563135 0.826365i \(-0.309595\pi\)
−0.826365 + 0.563135i \(0.809595\pi\)
\(318\) 0 0
\(319\) −8.53846e12 −0.144721
\(320\) 0 0
\(321\) −5.01569e13 −0.821398
\(322\) 0 0
\(323\) 1.08543e14 + 1.08543e14i 1.71787 + 1.71787i
\(324\) 0 0
\(325\) 1.39656e13 1.39656e13i 0.213649 0.213649i
\(326\) 0 0
\(327\) 4.95153e13i 0.732363i
\(328\) 0 0
\(329\) 1.67103e14i 2.39005i
\(330\) 0 0
\(331\) −1.15849e12 + 1.15849e12i −0.0160265 + 0.0160265i −0.715075 0.699048i \(-0.753607\pi\)
0.699048 + 0.715075i \(0.253607\pi\)
\(332\) 0 0
\(333\) 7.56344e13 + 7.56344e13i 1.01222 + 1.01222i
\(334\) 0 0
\(335\) −2.43352e13 −0.315129
\(336\) 0 0
\(337\) 1.54894e14 1.94120 0.970602 0.240691i \(-0.0773741\pi\)
0.970602 + 0.240691i \(0.0773741\pi\)
\(338\) 0 0
\(339\) 5.65425e13 + 5.65425e13i 0.685925 + 0.685925i
\(340\) 0 0
\(341\) 3.18791e12 3.18791e12i 0.0374418 0.0374418i
\(342\) 0 0
\(343\) 2.31639e13i 0.263447i
\(344\) 0 0
\(345\) 3.75725e13i 0.413871i
\(346\) 0 0
\(347\) −4.18262e13 + 4.18262e13i −0.446309 + 0.446309i −0.894126 0.447816i \(-0.852202\pi\)
0.447816 + 0.894126i \(0.352202\pi\)
\(348\) 0 0
\(349\) −1.30512e14 1.30512e14i −1.34931 1.34931i −0.886412 0.462897i \(-0.846810\pi\)
−0.462897 0.886412i \(-0.653190\pi\)
\(350\) 0 0
\(351\) 6.64335e12 0.0665576
\(352\) 0 0
\(353\) −1.77057e14 −1.71930 −0.859652 0.510880i \(-0.829320\pi\)
−0.859652 + 0.510880i \(0.829320\pi\)
\(354\) 0 0
\(355\) −2.13516e13 2.13516e13i −0.200990 0.200990i
\(356\) 0 0
\(357\) 2.10743e14 2.10743e14i 1.92344 1.92344i
\(358\) 0 0
\(359\) 1.06853e14i 0.945727i 0.881136 + 0.472863i \(0.156780\pi\)
−0.881136 + 0.472863i \(0.843220\pi\)
\(360\) 0 0
\(361\) 2.41616e14i 2.07413i
\(362\) 0 0
\(363\) −1.21367e14 + 1.21367e14i −1.01068 + 1.01068i
\(364\) 0 0
\(365\) −9.54928e12 9.54928e12i −0.0771542 0.0771542i
\(366\) 0 0
\(367\) −4.42951e13 −0.347290 −0.173645 0.984808i \(-0.555554\pi\)
−0.173645 + 0.984808i \(0.555554\pi\)
\(368\) 0 0
\(369\) −3.05404e12 −0.0232397
\(370\) 0 0
\(371\) −1.21629e14 1.21629e14i −0.898425 0.898425i
\(372\) 0 0
\(373\) −9.20225e13 + 9.20225e13i −0.659926 + 0.659926i −0.955362 0.295436i \(-0.904535\pi\)
0.295436 + 0.955362i \(0.404535\pi\)
\(374\) 0 0
\(375\) 1.22840e14i 0.855397i
\(376\) 0 0
\(377\) 4.79818e13i 0.324488i
\(378\) 0 0
\(379\) 4.11973e13 4.11973e13i 0.270616 0.270616i −0.558732 0.829348i \(-0.688712\pi\)
0.829348 + 0.558732i \(0.188712\pi\)
\(380\) 0 0
\(381\) 4.27020e12 + 4.27020e12i 0.0272496 + 0.0272496i
\(382\) 0 0
\(383\) 2.44621e13 0.151670 0.0758351 0.997120i \(-0.475838\pi\)
0.0758351 + 0.997120i \(0.475838\pi\)
\(384\) 0 0
\(385\) −1.01850e13 −0.0613661
\(386\) 0 0
\(387\) −2.18114e14 2.18114e14i −1.27724 1.27724i
\(388\) 0 0
\(389\) 4.33424e13 4.33424e13i 0.246712 0.246712i −0.572908 0.819620i \(-0.694185\pi\)
0.819620 + 0.572908i \(0.194185\pi\)
\(390\) 0 0
\(391\) 2.30870e14i 1.27760i
\(392\) 0 0
\(393\) 2.62509e14i 1.41249i
\(394\) 0 0
\(395\) −5.25696e13 + 5.25696e13i −0.275075 + 0.275075i
\(396\) 0 0
\(397\) −2.24044e13 2.24044e13i −0.114021 0.114021i 0.647794 0.761815i \(-0.275692\pi\)
−0.761815 + 0.647794i \(0.775692\pi\)
\(398\) 0 0
\(399\) 6.95285e14 3.44201
\(400\) 0 0
\(401\) 3.28823e14 1.58368 0.791842 0.610727i \(-0.209123\pi\)
0.791842 + 0.610727i \(0.209123\pi\)
\(402\) 0 0
\(403\) 1.79144e13 + 1.79144e13i 0.0839508 + 0.0839508i
\(404\) 0 0
\(405\) −4.02294e13 + 4.02294e13i −0.183460 + 0.183460i
\(406\) 0 0
\(407\) 4.22239e13i 0.187409i
\(408\) 0 0
\(409\) 1.11279e14i 0.480767i 0.970678 + 0.240384i \(0.0772732\pi\)
−0.970678 + 0.240384i \(0.922727\pi\)
\(410\) 0 0
\(411\) −9.62366e13 + 9.62366e13i −0.404772 + 0.404772i
\(412\) 0 0
\(413\) −2.52700e14 2.52700e14i −1.03486 1.03486i
\(414\) 0 0
\(415\) 1.30491e14 0.520373
\(416\) 0 0
\(417\) −1.53508e14 −0.596188
\(418\) 0 0
\(419\) −1.31929e14 1.31929e14i −0.499072 0.499072i 0.412077 0.911149i \(-0.364803\pi\)
−0.911149 + 0.412077i \(0.864803\pi\)
\(420\) 0 0
\(421\) 1.11228e13 1.11228e13i 0.0409887 0.0409887i −0.686315 0.727304i \(-0.740773\pi\)
0.727304 + 0.686315i \(0.240773\pi\)
\(422\) 0 0
\(423\) 5.63308e14i 2.02243i
\(424\) 0 0
\(425\) 3.58728e14i 1.25495i
\(426\) 0 0
\(427\) 3.90830e14 3.90830e14i 1.33240 1.33240i
\(428\) 0 0
\(429\) 1.54401e13 + 1.54401e13i 0.0513021 + 0.0513021i
\(430\) 0 0
\(431\) −4.45018e14 −1.44129 −0.720647 0.693302i \(-0.756155\pi\)
−0.720647 + 0.693302i \(0.756155\pi\)
\(432\) 0 0
\(433\) −1.12463e13 −0.0355080 −0.0177540 0.999842i \(-0.505652\pi\)
−0.0177540 + 0.999842i \(0.505652\pi\)
\(434\) 0 0
\(435\) −1.00289e14 1.00289e14i −0.308720 0.308720i
\(436\) 0 0
\(437\) −3.80842e14 + 3.80842e14i −1.14313 + 1.14313i
\(438\) 0 0
\(439\) 5.67001e13i 0.165970i 0.996551 + 0.0829849i \(0.0264453\pi\)
−0.996551 + 0.0829849i \(0.973555\pi\)
\(440\) 0 0
\(441\) 3.20002e14i 0.913568i
\(442\) 0 0
\(443\) 2.60481e13 2.60481e13i 0.0725362 0.0725362i −0.669908 0.742444i \(-0.733666\pi\)
0.742444 + 0.669908i \(0.233666\pi\)
\(444\) 0 0
\(445\) −5.14845e12 5.14845e12i −0.0139861 0.0139861i
\(446\) 0 0
\(447\) 1.71001e13 0.0453219
\(448\) 0 0
\(449\) −3.91719e14 −1.01302 −0.506512 0.862233i \(-0.669065\pi\)
−0.506512 + 0.862233i \(0.669065\pi\)
\(450\) 0 0
\(451\) −8.52481e11 8.52481e11i −0.00215136 0.00215136i
\(452\) 0 0
\(453\) 5.14753e14 5.14753e14i 1.26782 1.26782i
\(454\) 0 0
\(455\) 5.72347e13i 0.137593i
\(456\) 0 0
\(457\) 1.98305e14i 0.465365i −0.972553 0.232683i \(-0.925250\pi\)
0.972553 0.232683i \(-0.0747504\pi\)
\(458\) 0 0
\(459\) −8.53222e13 + 8.53222e13i −0.195475 + 0.195475i
\(460\) 0 0
\(461\) −2.94593e14 2.94593e14i −0.658972 0.658972i 0.296165 0.955137i \(-0.404292\pi\)
−0.955137 + 0.296165i \(0.904292\pi\)
\(462\) 0 0
\(463\) −5.44141e14 −1.18855 −0.594273 0.804263i \(-0.702560\pi\)
−0.594273 + 0.804263i \(0.702560\pi\)
\(464\) 0 0
\(465\) 7.48878e13 0.159742
\(466\) 0 0
\(467\) 6.92417e13 + 6.92417e13i 0.144253 + 0.144253i 0.775545 0.631292i \(-0.217475\pi\)
−0.631292 + 0.775545i \(0.717475\pi\)
\(468\) 0 0
\(469\) 4.78918e14 4.78918e14i 0.974564 0.974564i
\(470\) 0 0
\(471\) 3.82944e14i 0.761238i
\(472\) 0 0
\(473\) 1.21765e14i 0.236476i
\(474\) 0 0
\(475\) −5.91757e14 + 5.91757e14i −1.12287 + 1.12287i
\(476\) 0 0
\(477\) 4.10015e14 + 4.10015e14i 0.760237 + 0.760237i
\(478\) 0 0
\(479\) −5.45377e14 −0.988216 −0.494108 0.869401i \(-0.664505\pi\)
−0.494108 + 0.869401i \(0.664505\pi\)
\(480\) 0 0
\(481\) −2.37277e14 −0.420201
\(482\) 0 0
\(483\) 7.39429e14 + 7.39429e14i 1.27993 + 1.27993i
\(484\) 0 0
\(485\) −2.36025e13 + 2.36025e13i −0.0399372 + 0.0399372i
\(486\) 0 0
\(487\) 2.73024e14i 0.451640i −0.974169 0.225820i \(-0.927494\pi\)
0.974169 0.225820i \(-0.0725061\pi\)
\(488\) 0 0
\(489\) 1.65696e14i 0.267987i
\(490\) 0 0
\(491\) −2.44712e13 + 2.44712e13i −0.0386996 + 0.0386996i −0.726192 0.687492i \(-0.758712\pi\)
0.687492 + 0.726192i \(0.258712\pi\)
\(492\) 0 0
\(493\) 6.16242e14 + 6.16242e14i 0.953001 + 0.953001i
\(494\) 0 0
\(495\) 3.43340e13 0.0519273
\(496\) 0 0
\(497\) 8.40402e14 1.24316
\(498\) 0 0
\(499\) 5.44937e14 + 5.44937e14i 0.788484 + 0.788484i 0.981246 0.192762i \(-0.0617444\pi\)
−0.192762 + 0.981246i \(0.561744\pi\)
\(500\) 0 0
\(501\) −1.21903e15 + 1.21903e15i −1.72547 + 1.72547i
\(502\) 0 0
\(503\) 2.12390e14i 0.294110i 0.989128 + 0.147055i \(0.0469794\pi\)
−0.989128 + 0.147055i \(0.953021\pi\)
\(504\) 0 0
\(505\) 2.44055e14i 0.330663i
\(506\) 0 0
\(507\) 6.92848e14 6.92848e14i 0.918532 0.918532i
\(508\) 0 0
\(509\) 2.91460e14 + 2.91460e14i 0.378121 + 0.378121i 0.870424 0.492303i \(-0.163845\pi\)
−0.492303 + 0.870424i \(0.663845\pi\)
\(510\) 0 0
\(511\) 3.75861e14 0.477212
\(512\) 0 0
\(513\) −2.81495e14 −0.349804
\(514\) 0 0
\(515\) −3.76924e13 3.76924e13i −0.0458473 0.0458473i
\(516\) 0 0
\(517\) 1.57237e14 1.57237e14i 0.187222 0.187222i
\(518\) 0 0
\(519\) 3.61859e14i 0.421812i
\(520\) 0 0
\(521\) 1.21124e15i 1.38236i 0.722681 + 0.691182i \(0.242910\pi\)
−0.722681 + 0.691182i \(0.757090\pi\)
\(522\) 0 0
\(523\) 7.39703e14 7.39703e14i 0.826605 0.826605i −0.160440 0.987046i \(-0.551291\pi\)
0.987046 + 0.160440i \(0.0512914\pi\)
\(524\) 0 0
\(525\) 1.14893e15 + 1.14893e15i 1.25724 + 1.25724i
\(526\) 0 0
\(527\) −4.60159e14 −0.493116
\(528\) 0 0
\(529\) 1.42765e14 0.149836
\(530\) 0 0
\(531\) 8.51858e14 + 8.51858e14i 0.875684 + 0.875684i
\(532\) 0 0
\(533\) 4.79051e12 4.79051e12i 0.00482372 0.00482372i
\(534\) 0 0
\(535\) 1.74949e14i 0.172570i
\(536\) 0 0
\(537\) 1.09429e15i 1.05749i
\(538\) 0 0
\(539\) 8.93228e13 8.93228e13i 0.0845716 0.0845716i
\(540\) 0 0
\(541\) −9.36873e14 9.36873e14i −0.869152 0.869152i 0.123227 0.992379i \(-0.460676\pi\)
−0.992379 + 0.123227i \(0.960676\pi\)
\(542\) 0 0
\(543\) −6.11320e14 −0.555737
\(544\) 0 0
\(545\) 1.72711e14 0.153865
\(546\) 0 0
\(547\) 7.01752e13 + 7.01752e13i 0.0612708 + 0.0612708i 0.737078 0.675807i \(-0.236205\pi\)
−0.675807 + 0.737078i \(0.736205\pi\)
\(548\) 0 0
\(549\) −1.31750e15 + 1.31750e15i −1.12746 + 1.12746i
\(550\) 0 0
\(551\) 2.03311e15i 1.70540i
\(552\) 0 0
\(553\) 2.06914e15i 1.70138i
\(554\) 0 0
\(555\) −4.95946e14 + 4.95946e14i −0.399781 + 0.399781i
\(556\) 0 0
\(557\) −4.21492e14 4.21492e14i −0.333109 0.333109i 0.520657 0.853766i \(-0.325687\pi\)
−0.853766 + 0.520657i \(0.825687\pi\)
\(558\) 0 0
\(559\) 6.84259e14 0.530219
\(560\) 0 0
\(561\) −3.96602e14 −0.301342
\(562\) 0 0
\(563\) −1.18164e15 1.18164e15i −0.880416 0.880416i 0.113160 0.993577i \(-0.463903\pi\)
−0.993577 + 0.113160i \(0.963903\pi\)
\(564\) 0 0
\(565\) −1.97222e14 + 1.97222e14i −0.144108 + 0.144108i
\(566\) 0 0
\(567\) 1.58343e15i 1.13473i
\(568\) 0 0
\(569\) 9.73112e14i 0.683983i −0.939703 0.341992i \(-0.888899\pi\)
0.939703 0.341992i \(-0.111101\pi\)
\(570\) 0 0
\(571\) 1.00987e15 1.00987e15i 0.696253 0.696253i −0.267347 0.963600i \(-0.586147\pi\)
0.963600 + 0.267347i \(0.0861470\pi\)
\(572\) 0 0
\(573\) 6.74676e14 + 6.74676e14i 0.456294 + 0.456294i
\(574\) 0 0
\(575\) −1.25866e15 −0.835092
\(576\) 0 0
\(577\) 1.63029e15 1.06120 0.530602 0.847621i \(-0.321966\pi\)
0.530602 + 0.847621i \(0.321966\pi\)
\(578\) 0 0
\(579\) −1.99286e15 1.99286e15i −1.27275 1.27275i
\(580\) 0 0
\(581\) −2.56806e15 + 2.56806e15i −1.60930 + 1.60930i
\(582\) 0 0
\(583\) 2.28897e14i 0.140755i
\(584\) 0 0
\(585\) 1.92939e14i 0.116430i
\(586\) 0 0
\(587\) −7.41304e14 + 7.41304e14i −0.439022 + 0.439022i −0.891683 0.452661i \(-0.850475\pi\)
0.452661 + 0.891683i \(0.350475\pi\)
\(588\) 0 0
\(589\) −7.59078e14 7.59078e14i −0.441217 0.441217i
\(590\) 0 0
\(591\) 7.33034e14 0.418209
\(592\) 0 0
\(593\) −3.32085e15 −1.85972 −0.929861 0.367910i \(-0.880073\pi\)
−0.929861 + 0.367910i \(0.880073\pi\)
\(594\) 0 0
\(595\) 7.35079e14 + 7.35079e14i 0.404102 + 0.404102i
\(596\) 0 0
\(597\) 2.80992e15 2.80992e15i 1.51647 1.51647i
\(598\) 0 0
\(599\) 2.87597e15i 1.52383i −0.647675 0.761916i \(-0.724259\pi\)
0.647675 0.761916i \(-0.275741\pi\)
\(600\) 0 0
\(601\) 3.01761e14i 0.156984i −0.996915 0.0784918i \(-0.974990\pi\)
0.996915 0.0784918i \(-0.0250104\pi\)
\(602\) 0 0
\(603\) −1.61444e15 + 1.61444e15i −0.824665 + 0.824665i
\(604\) 0 0
\(605\) −4.23331e14 4.23331e14i −0.212337 0.212337i
\(606\) 0 0
\(607\) 1.10741e15 0.545470 0.272735 0.962089i \(-0.412072\pi\)
0.272735 + 0.962089i \(0.412072\pi\)
\(608\) 0 0
\(609\) 3.94740e15 1.90948
\(610\) 0 0
\(611\) 8.83592e14 + 8.83592e14i 0.419783 + 0.419783i
\(612\) 0 0
\(613\) 2.34187e15 2.34187e15i 1.09277 1.09277i 0.0975435 0.995231i \(-0.468902\pi\)
0.995231 0.0975435i \(-0.0310985\pi\)
\(614\) 0 0
\(615\) 2.00258e13i 0.00917861i
\(616\) 0 0
\(617\) 6.98011e14i 0.314264i −0.987578 0.157132i \(-0.949775\pi\)
0.987578 0.157132i \(-0.0502248\pi\)
\(618\) 0 0
\(619\) 2.28130e15 2.28130e15i 1.00898 1.00898i 0.00902486 0.999959i \(-0.497127\pi\)
0.999959 0.00902486i \(-0.00287274\pi\)
\(620\) 0 0
\(621\) −2.99367e14 2.99367e14i −0.130077 0.130077i
\(622\) 0 0
\(623\) 2.02644e14 0.0865065
\(624\) 0 0
\(625\) −1.73088e15 −0.725984
\(626\) 0 0
\(627\) −6.54235e14 6.54235e14i −0.269626 0.269626i
\(628\) 0 0
\(629\) 3.04741e15 3.04741e15i 1.23410 1.23410i
\(630\) 0 0
\(631\) 3.06944e15i 1.22151i 0.791819 + 0.610755i \(0.209134\pi\)
−0.791819 + 0.610755i \(0.790866\pi\)
\(632\) 0 0
\(633\) 4.92638e15i 1.92667i
\(634\) 0 0
\(635\) −1.48946e13 + 1.48946e13i −0.00572497 + 0.00572497i
\(636\) 0 0
\(637\) 5.01949e14 + 5.01949e14i 0.189624 + 0.189624i
\(638\) 0 0
\(639\) −2.83302e15 −1.05195
\(640\) 0 0
\(641\) −2.26554e15 −0.826899 −0.413449 0.910527i \(-0.635676\pi\)
−0.413449 + 0.910527i \(0.635676\pi\)
\(642\) 0 0
\(643\) 4.93160e14 + 4.93160e14i 0.176941 + 0.176941i 0.790021 0.613080i \(-0.210070\pi\)
−0.613080 + 0.790021i \(0.710070\pi\)
\(644\) 0 0
\(645\) 1.43021e15 1.43021e15i 0.504453 0.504453i
\(646\) 0 0
\(647\) 2.61317e15i 0.906139i −0.891475 0.453069i \(-0.850329\pi\)
0.891475 0.453069i \(-0.149671\pi\)
\(648\) 0 0
\(649\) 4.75561e14i 0.162129i
\(650\) 0 0
\(651\) −1.47380e15 + 1.47380e15i −0.494017 + 0.494017i
\(652\) 0 0
\(653\) −3.24689e15 3.24689e15i −1.07015 1.07015i −0.997346 0.0728038i \(-0.976805\pi\)
−0.0728038 0.997346i \(-0.523195\pi\)
\(654\) 0 0
\(655\) 9.15639e14 0.296754
\(656\) 0 0
\(657\) −1.26703e15 −0.403812
\(658\) 0 0
\(659\) 8.89857e13 + 8.89857e13i 0.0278901 + 0.0278901i 0.720914 0.693024i \(-0.243722\pi\)
−0.693024 + 0.720914i \(0.743722\pi\)
\(660\) 0 0
\(661\) 2.37602e14 2.37602e14i 0.0732388 0.0732388i −0.669539 0.742777i \(-0.733508\pi\)
0.742777 + 0.669539i \(0.233508\pi\)
\(662\) 0 0
\(663\) 2.22870e15i 0.675659i
\(664\) 0 0
\(665\) 2.42517e15i 0.723142i
\(666\) 0 0
\(667\) −2.16219e15 + 2.16219e15i −0.634164 + 0.634164i
\(668\) 0 0
\(669\) −1.07413e15 1.07413e15i −0.309894 0.309894i
\(670\) 0 0
\(671\) −7.35510e14 −0.208744
\(672\) 0 0
\(673\) 4.98817e14 0.139270 0.0696352 0.997573i \(-0.477816\pi\)
0.0696352 + 0.997573i \(0.477816\pi\)
\(674\) 0 0
\(675\) −4.65160e14 4.65160e14i −0.127771 0.127771i
\(676\) 0 0
\(677\) −3.03419e15 + 3.03419e15i −0.819984 + 0.819984i −0.986105 0.166121i \(-0.946876\pi\)
0.166121 + 0.986105i \(0.446876\pi\)
\(678\) 0 0
\(679\) 9.28995e14i 0.247019i
\(680\) 0 0
\(681\) 2.21698e15i 0.580034i
\(682\) 0 0
\(683\) −3.12787e15 + 3.12787e15i −0.805257 + 0.805257i −0.983912 0.178655i \(-0.942825\pi\)
0.178655 + 0.983912i \(0.442825\pi\)
\(684\) 0 0
\(685\) −3.35676e14 3.35676e14i −0.0850398 0.0850398i
\(686\) 0 0
\(687\) 7.66517e15 1.91100
\(688\) 0 0
\(689\) −1.28628e15 −0.315595
\(690\) 0 0
\(691\) −2.77334e15 2.77334e15i −0.669689 0.669689i 0.287955 0.957644i \(-0.407025\pi\)
−0.957644 + 0.287955i \(0.907025\pi\)
\(692\) 0 0
\(693\) −6.75694e14 + 6.75694e14i −0.160590 + 0.160590i
\(694\) 0 0
\(695\) 5.35441e14i 0.125255i
\(696\) 0 0
\(697\) 1.23051e14i 0.0283339i
\(698\) 0 0
\(699\) 4.80793e15 4.80793e15i 1.08977 1.08977i
\(700\) 0 0
\(701\) 3.58804e15 + 3.58804e15i 0.800587 + 0.800587i 0.983187 0.182600i \(-0.0584514\pi\)
−0.182600 + 0.983187i \(0.558451\pi\)
\(702\) 0 0
\(703\) 1.00540e16 2.20844
\(704\) 0 0
\(705\) 3.69369e15 0.798767
\(706\) 0 0
\(707\) −4.80302e15 4.80302e15i −1.02260 1.02260i
\(708\) 0 0
\(709\) −2.93575e15 + 2.93575e15i −0.615410 + 0.615410i −0.944351 0.328940i \(-0.893308\pi\)
0.328940 + 0.944351i \(0.393308\pi\)
\(710\) 0 0
\(711\) 6.97513e15i 1.43969i
\(712\) 0 0
\(713\) 1.61454e15i 0.328139i
\(714\) 0 0
\(715\) −5.38555e13 + 5.38555e13i −0.0107782 + 0.0107782i
\(716\) 0 0
\(717\) −3.64143e15 3.64143e15i −0.717656 0.717656i
\(718\) 0 0
\(719\) −4.19879e15 −0.814920 −0.407460 0.913223i \(-0.633585\pi\)
−0.407460 + 0.913223i \(0.633585\pi\)
\(720\) 0 0
\(721\) 1.48357e15 0.283573
\(722\) 0 0
\(723\) 6.47858e15 + 6.47858e15i 1.21960 + 1.21960i
\(724\) 0 0
\(725\) −3.35964e15 + 3.35964e15i −0.622922 + 0.622922i
\(726\) 0 0
\(727\) 4.12785e15i 0.753849i 0.926244 + 0.376925i \(0.123018\pi\)
−0.926244 + 0.376925i \(0.876982\pi\)
\(728\) 0 0
\(729\) 6.95891e15i 1.25181i
\(730\) 0 0
\(731\) −8.78811e15 + 8.78811e15i −1.55722 + 1.55722i
\(732\) 0 0
\(733\) 4.75984e15 + 4.75984e15i 0.830845 + 0.830845i 0.987632 0.156787i \(-0.0501136\pi\)
−0.156787 + 0.987632i \(0.550114\pi\)
\(734\) 0 0
\(735\) 2.09830e15 0.360817
\(736\) 0 0
\(737\) −9.01285e14 −0.152683
\(738\) 0 0
\(739\) −3.24543e15 3.24543e15i −0.541661 0.541661i 0.382355 0.924016i \(-0.375113\pi\)
−0.924016 + 0.382355i \(0.875113\pi\)
\(740\) 0 0
\(741\) 3.67647e15 3.67647e15i 0.604548 0.604548i
\(742\) 0 0
\(743\) 9.92853e15i 1.60859i 0.594227 + 0.804297i \(0.297458\pi\)
−0.594227 + 0.804297i \(0.702542\pi\)
\(744\) 0 0
\(745\) 5.96457e13i 0.00952182i
\(746\) 0 0
\(747\) 8.65699e15 8.65699e15i 1.36177 1.36177i
\(748\) 0 0
\(749\) 3.44300e15 + 3.44300e15i 0.533688 + 0.533688i
\(750\) 0 0
\(751\) −2.11436e14 −0.0322969 −0.0161484 0.999870i \(-0.505140\pi\)
−0.0161484 + 0.999870i \(0.505140\pi\)
\(752\) 0 0
\(753\) 6.94632e15 1.04564
\(754\) 0 0
\(755\) 1.79548e15 + 1.79548e15i 0.266361 + 0.266361i
\(756\) 0 0
\(757\) 7.13066e15 7.13066e15i 1.04256 1.04256i 0.0435106 0.999053i \(-0.486146\pi\)
0.999053 0.0435106i \(-0.0138542\pi\)
\(758\) 0 0
\(759\) 1.39155e15i 0.200525i
\(760\) 0 0
\(761\) 7.63959e15i 1.08506i 0.840036 + 0.542531i \(0.182534\pi\)
−0.840036 + 0.542531i \(0.817466\pi\)
\(762\) 0 0
\(763\) −3.39896e15 + 3.39896e15i −0.475840 + 0.475840i
\(764\) 0 0
\(765\) −2.47797e15 2.47797e15i −0.341946 0.341946i
\(766\) 0 0
\(767\) −2.67241e15 −0.363521
\(768\) 0 0
\(769\) 5.21778e15 0.699667 0.349833 0.936812i \(-0.386238\pi\)
0.349833 + 0.936812i \(0.386238\pi\)
\(770\) 0 0
\(771\) 1.17781e15 + 1.17781e15i 0.155696 + 0.155696i
\(772\) 0 0
\(773\) −3.63573e15 + 3.63573e15i −0.473811 + 0.473811i −0.903145 0.429335i \(-0.858748\pi\)
0.429335 + 0.903145i \(0.358748\pi\)
\(774\) 0 0
\(775\) 2.50870e15i 0.322321i
\(776\) 0 0
\(777\) 1.95205e16i 2.47272i
\(778\) 0 0
\(779\) −2.02986e14 + 2.02986e14i −0.0253518 + 0.0253518i
\(780\) 0 0
\(781\) −7.90785e14 7.90785e14i −0.0973818 0.0973818i
\(782\) 0 0
\(783\) −1.59816e15 −0.194057
\(784\) 0 0
\(785\) −1.33572e15 −0.159931
\(786\) 0 0
\(787\) 6.27028e15 + 6.27028e15i 0.740331 + 0.740331i 0.972642 0.232311i \(-0.0746286\pi\)
−0.232311 + 0.972642i \(0.574629\pi\)
\(788\) 0 0
\(789\) −4.06372e14 + 4.06372e14i −0.0473152 + 0.0473152i
\(790\) 0 0
\(791\) 7.76268e15i 0.891335i
\(792\) 0 0
\(793\) 4.13319e15i 0.468040i
\(794\) 0 0
\(795\) −2.68853e15 + 2.68853e15i −0.300259 + 0.300259i
\(796\) 0 0
\(797\) 5.11428e15 + 5.11428e15i 0.563331 + 0.563331i 0.930252 0.366921i \(-0.119588\pi\)
−0.366921 + 0.930252i \(0.619588\pi\)
\(798\) 0 0
\(799\) −2.26964e16 −2.46575
\(800\) 0 0
\(801\) −6.83116e14 −0.0732008
\(802\) 0 0
\(803\) −3.53670e14 3.53670e14i −0.0373820 0.0373820i
\(804\) 0 0
\(805\) −2.57915e15 + 2.57915e15i −0.268905 + 0.268905i
\(806\) 0 0
\(807\) 1.27817e16i 1.31457i
\(808\) 0 0
\(809\) 5.91750e15i 0.600373i −0.953881 0.300187i \(-0.902951\pi\)
0.953881 0.300187i \(-0.0970489\pi\)
\(810\) 0 0
\(811\) 5.44663e15 5.44663e15i 0.545147 0.545147i −0.379886 0.925033i \(-0.624037\pi\)
0.925033 + 0.379886i \(0.124037\pi\)
\(812\) 0 0
\(813\) −1.73181e16 1.73181e16i −1.71002 1.71002i
\(814\) 0 0
\(815\) 5.77954e14 0.0563023
\(816\) 0 0
\(817\) −2.89937e16 −2.78665
\(818\) 0 0
\(819\) −3.79706e15 3.79706e15i −0.360069 0.360069i
\(820\) 0 0
\(821\) −7.94880e14 + 7.94880e14i −0.0743728 + 0.0743728i −0.743315 0.668942i \(-0.766747\pi\)
0.668942 + 0.743315i \(0.266747\pi\)
\(822\) 0 0
\(823\) 5.95135e15i 0.549435i −0.961525 0.274717i \(-0.911416\pi\)
0.961525 0.274717i \(-0.0885843\pi\)
\(824\) 0 0
\(825\) 2.16220e15i 0.196970i
\(826\) 0 0
\(827\) −2.73320e15 + 2.73320e15i −0.245692 + 0.245692i −0.819200 0.573508i \(-0.805582\pi\)
0.573508 + 0.819200i \(0.305582\pi\)
\(828\) 0 0
\(829\) 1.31586e16 + 1.31586e16i 1.16724 + 1.16724i 0.982855 + 0.184382i \(0.0590285\pi\)
0.184382 + 0.982855i \(0.440971\pi\)
\(830\) 0 0
\(831\) −1.36753e16 −1.19710
\(832\) 0 0
\(833\) −1.28933e16 −1.11382
\(834\) 0 0
\(835\) −4.25203e15 4.25203e15i −0.362510 0.362510i
\(836\) 0 0
\(837\) 5.96686e14 5.96686e14i 0.0502060 0.0502060i
\(838\) 0 0
\(839\) 1.78995e15i 0.148645i −0.997234 0.0743226i \(-0.976321\pi\)
0.997234 0.0743226i \(-0.0236795\pi\)
\(840\) 0 0
\(841\) 6.57772e14i 0.0539135i
\(842\) 0 0
\(843\) 2.46073e16 2.46073e16i 1.99073 1.99073i
\(844\) 0 0
\(845\) 2.41668e15 + 2.41668e15i 0.192977 + 0.192977i
\(846\) 0 0
\(847\) 1.66624e16 1.31334
\(848\) 0 0
\(849\) 2.00731e16 1.56179
\(850\) 0 0
\(851\) 1.06923e16 + 1.06923e16i 0.821221 + 0.821221i
\(852\) 0 0
\(853\) −3.28258e15 + 3.28258e15i −0.248883 + 0.248883i −0.820512 0.571629i \(-0.806312\pi\)
0.571629 + 0.820512i \(0.306312\pi\)
\(854\) 0 0
\(855\) 8.17532e15i 0.611915i
\(856\) 0 0
\(857\) 9.76700e15i 0.721717i −0.932621 0.360858i \(-0.882484\pi\)
0.932621 0.360858i \(-0.117516\pi\)
\(858\) 0 0
\(859\) 1.01771e16 1.01771e16i 0.742439 0.742439i −0.230608 0.973047i \(-0.574071\pi\)
0.973047 + 0.230608i \(0.0740714\pi\)
\(860\) 0 0
\(861\) 3.94109e14 + 3.94109e14i 0.0283857 + 0.0283857i
\(862\) 0 0
\(863\) −1.35368e16 −0.962627 −0.481314 0.876548i \(-0.659840\pi\)
−0.481314 + 0.876548i \(0.659840\pi\)
\(864\) 0 0
\(865\) −1.26217e15 −0.0886198
\(866\) 0 0
\(867\) 1.37150e16 + 1.37150e16i 0.950807 + 0.950807i
\(868\) 0 0
\(869\) −1.94698e15 + 1.94698e15i −0.133276 + 0.133276i
\(870\) 0 0
\(871\) 5.06476e15i 0.342341i
\(872\) 0 0
\(873\) 3.13166e15i 0.209024i
\(874\) 0 0
\(875\) −8.43230e15 + 8.43230e15i −0.555778 + 0.555778i
\(876\) 0 0
\(877\) 1.80704e16 + 1.80704e16i 1.17617 + 1.17617i 0.980713 + 0.195454i \(0.0626179\pi\)
0.195454 + 0.980713i \(0.437382\pi\)
\(878\) 0 0
\(879\) −1.28532e16 −0.826177
\(880\) 0 0
\(881\) 1.07538e16 0.682642 0.341321 0.939947i \(-0.389126\pi\)
0.341321 + 0.939947i \(0.389126\pi\)
\(882\) 0 0
\(883\) −1.44950e15 1.44950e15i −0.0908731 0.0908731i 0.660209 0.751082i \(-0.270468\pi\)
−0.751082 + 0.660209i \(0.770468\pi\)
\(884\) 0 0
\(885\) −5.58576e15 + 5.58576e15i −0.345855 + 0.345855i
\(886\) 0 0
\(887\) 2.98383e14i 0.0182471i 0.999958 + 0.00912354i \(0.00290415\pi\)
−0.999958 + 0.00912354i \(0.997096\pi\)
\(888\) 0 0
\(889\) 5.86252e14i 0.0354099i
\(890\) 0 0
\(891\) −1.48995e15 + 1.48995e15i −0.0888880 + 0.0888880i
\(892\) 0 0
\(893\) −3.74400e16 3.74400e16i −2.20624 2.20624i
\(894\) 0 0
\(895\) 3.81693e15 0.222171
\(896\) 0 0
\(897\) 7.81978e15 0.449610
\(898\) 0 0
\(899\) −4.30958e15 4.30958e15i −0.244769 0.244769i
\(900\) 0 0
\(901\) 1.65201e16 1.65201e16i 0.926883 0.926883i
\(902\) 0 0
\(903\) 5.62931e16i 3.12013i
\(904\) 0 0
\(905\) 2.13230e15i 0.116757i
\(906\) 0 0
\(907\) −1.80050e16 + 1.80050e16i −0.973986 + 0.973986i −0.999670 0.0256843i \(-0.991824\pi\)
0.0256843 + 0.999670i \(0.491824\pi\)
\(908\) 0 0
\(909\) 1.61911e16 + 1.61911e16i 0.865316 + 0.865316i
\(910\) 0 0
\(911\) 2.73035e16 1.44168 0.720838 0.693104i \(-0.243757\pi\)
0.720838 + 0.693104i \(0.243757\pi\)
\(912\) 0 0
\(913\) 4.83289e15 0.252126
\(914\) 0 0
\(915\) −8.63901e15 8.63901e15i −0.445295 0.445295i
\(916\) 0 0
\(917\) −1.80198e16 + 1.80198e16i −0.917739 + 0.917739i
\(918\) 0 0
\(919\) 5.96410e15i 0.300130i −0.988676 0.150065i \(-0.952052\pi\)
0.988676 0.150065i \(-0.0479483\pi\)
\(920\) 0 0
\(921\) 3.25418e16i 1.61813i
\(922\) 0 0
\(923\) 4.44381e15 4.44381e15i 0.218346 0.218346i
\(924\) 0 0
\(925\) 1.66139e16 + 1.66139e16i 0.806662 + 0.806662i
\(926\) 0 0
\(927\) −5.00117e15 −0.239957
\(928\) 0 0
\(929\) 1.74073e16 0.825365 0.412683 0.910875i \(-0.364592\pi\)
0.412683 + 0.910875i \(0.364592\pi\)
\(930\) 0 0
\(931\) −2.12688e16 2.12688e16i −0.996598 0.996598i
\(932\) 0 0
\(933\) −2.20929e16 + 2.20929e16i −1.02307 + 1.02307i
\(934\) 0 0
\(935\) 1.38336e15i 0.0633099i
\(936\) 0 0
\(937\) 8.96273e15i 0.405390i 0.979242 + 0.202695i \(0.0649699\pi\)
−0.979242 + 0.202695i \(0.935030\pi\)
\(938\) 0 0
\(939\) −4.64999e15 + 4.64999e15i −0.207870 + 0.207870i
\(940\) 0 0
\(941\) 4.72270e15 + 4.72270e15i 0.208664 + 0.208664i 0.803699 0.595036i \(-0.202862\pi\)
−0.595036 + 0.803699i \(0.702862\pi\)
\(942\) 0 0
\(943\) −4.31747e14 −0.0188545
\(944\) 0 0
\(945\) −1.90635e15 −0.0822862
\(946\) 0 0
\(947\) 1.11972e16 + 1.11972e16i 0.477732 + 0.477732i 0.904406 0.426674i \(-0.140315\pi\)
−0.426674 + 0.904406i \(0.640315\pi\)
\(948\) 0 0
\(949\) 1.98744e15 1.98744e15i 0.0838167 0.0838167i
\(950\) 0 0
\(951\) 1.30525e16i 0.544127i
\(952\) 0 0
\(953\) 1.58414e16i 0.652804i −0.945231 0.326402i \(-0.894164\pi\)
0.945231 0.326402i \(-0.105836\pi\)
\(954\) 0 0
\(955\) −2.35329e15 + 2.35329e15i −0.0958642 + 0.0958642i
\(956\) 0 0
\(957\) −3.71435e15 3.71435e15i −0.149578 0.149578i
\(958\) 0 0
\(959\) 1.32122e16 0.525986
\(960\) 0 0
\(961\) −2.21904e16 −0.873348
\(962\) 0 0
\(963\) −1.16064e16 1.16064e16i −0.451601 0.451601i
\(964\) 0 0
\(965\) 6.95115e15 6.95115e15i 0.267397 0.267397i
\(966\) 0 0
\(967\) 3.29506e16i 1.25319i 0.779345 + 0.626596i \(0.215552\pi\)
−0.779345 + 0.626596i \(0.784448\pi\)
\(968\) 0 0
\(969\) 9.44356e16i 3.55103i
\(970\) 0 0
\(971\) 1.54480e16 1.54480e16i 0.574335 0.574335i −0.359002 0.933337i \(-0.616883\pi\)
0.933337 + 0.359002i \(0.116883\pi\)
\(972\) 0 0
\(973\) 1.05375e16 + 1.05375e16i 0.387362 + 0.387362i
\(974\) 0 0
\(975\) 1.21505e16 0.441639
\(976\) 0 0
\(977\) −4.75813e15 −0.171008 −0.0855040 0.996338i \(-0.527250\pi\)
−0.0855040 + 0.996338i \(0.527250\pi\)
\(978\) 0 0
\(979\) −1.90679e14 1.90679e14i −0.00677640 0.00677640i
\(980\) 0 0
\(981\) 1.14580e16 1.14580e16i 0.402650 0.402650i
\(982\) 0 0
\(983\) 1.45977e16i 0.507270i −0.967300 0.253635i \(-0.918374\pi\)
0.967300 0.253635i \(-0.0816262\pi\)
\(984\) 0 0
\(985\) 2.55684e15i 0.0878628i
\(986\) 0 0
\(987\) −7.26920e16 + 7.26920e16i −2.47026 + 2.47026i
\(988\) 0 0
\(989\) −3.08346e16 3.08346e16i −1.03623 1.03623i
\(990\) 0 0
\(991\) −3.14429e16 −1.04500 −0.522501 0.852639i \(-0.675001\pi\)
−0.522501 + 0.852639i \(0.675001\pi\)
\(992\) 0 0
\(993\) −1.00792e15 −0.0331286
\(994\) 0 0
\(995\) 9.80108e15 + 9.80108e15i 0.318601 + 0.318601i
\(996\) 0 0
\(997\) 1.86303e16 1.86303e16i 0.598958 0.598958i −0.341077 0.940035i \(-0.610792\pi\)
0.940035 + 0.341077i \(0.110792\pi\)
\(998\) 0 0
\(999\) 7.90311e15i 0.251297i
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.17.18 42
4.3 odd 2 16.12.e.a.13.3 yes 42
8.3 odd 2 128.12.e.b.33.18 42
8.5 even 2 128.12.e.a.33.4 42
16.3 odd 4 128.12.e.b.97.18 42
16.5 even 4 inner 64.12.e.a.49.18 42
16.11 odd 4 16.12.e.a.5.3 42
16.13 even 4 128.12.e.a.97.4 42
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.12.e.a.5.3 42 16.11 odd 4
16.12.e.a.13.3 yes 42 4.3 odd 2
64.12.e.a.17.18 42 1.1 even 1 trivial
64.12.e.a.49.18 42 16.5 even 4 inner
128.12.e.a.33.4 42 8.5 even 2
128.12.e.a.97.4 42 16.13 even 4
128.12.e.b.33.18 42 8.3 odd 2
128.12.e.b.97.18 42 16.3 odd 4