Properties

Label 128.12.e.b.33.6
Level $128$
Weight $12$
Character 128.33
Analytic conductor $98.348$
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] = [128,12,Mod(33,128)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(128, 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("128.33");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 128 = 2^{7} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 128.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: \(98.3479271116\)
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 33.6
Character \(\chi\) \(=\) 128.33
Dual form 128.12.e.b.97.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-255.917 - 255.917i) q^{3} +(4214.49 - 4214.49i) q^{5} -80345.3i q^{7} -46160.2i q^{9} +O(q^{10})\) \(q+(-255.917 - 255.917i) q^{3} +(4214.49 - 4214.49i) q^{5} -80345.3i q^{7} -46160.2i q^{9} +(-611105. + 611105. i) q^{11} +(-370364. - 370364. i) q^{13} -2.15712e6 q^{15} -725142. q^{17} +(1.13100e7 + 1.13100e7i) q^{19} +(-2.05617e7 + 2.05617e7i) q^{21} +4.89011e7i q^{23} +1.33043e7i q^{25} +(-5.71481e7 + 5.71481e7i) q^{27} +(-4.98291e7 - 4.98291e7i) q^{29} +4.20747e7 q^{31} +3.12784e8 q^{33} +(-3.38614e8 - 3.38614e8i) q^{35} +(1.05666e8 - 1.05666e8i) q^{37} +1.89565e8i q^{39} -4.49063e8i q^{41} +(-7.89791e8 + 7.89791e8i) q^{43} +(-1.94542e8 - 1.94542e8i) q^{45} +1.32368e9 q^{47} -4.47803e9 q^{49} +(1.85576e8 + 1.85576e8i) q^{51} +(-1.74345e9 + 1.74345e9i) q^{53} +5.15099e9i q^{55} -5.78882e9i q^{57} +(6.71604e8 - 6.71604e8i) q^{59} +(8.30682e8 + 8.30682e8i) q^{61} -3.70875e9 q^{63} -3.12179e9 q^{65} +(-3.88275e9 - 3.88275e9i) q^{67} +(1.25146e10 - 1.25146e10i) q^{69} +5.14233e9i q^{71} +4.42543e9i q^{73} +(3.40480e9 - 3.40480e9i) q^{75} +(4.90994e10 + 4.90994e10i) q^{77} +2.71827e10 q^{79} +2.10732e10 q^{81} +(-1.81378e10 - 1.81378e10i) q^{83} +(-3.05610e9 + 3.05610e9i) q^{85} +2.55042e10i q^{87} -5.56593e10i q^{89} +(-2.97570e10 + 2.97570e10i) q^{91} +(-1.07676e10 - 1.07676e10i) q^{93} +9.53314e10 q^{95} +2.61297e10 q^{97} +(2.82087e10 + 2.82087e10i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 42 q + 2 q^{3} + 2 q^{5}+O(q^{10}) \) 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 \(\left(\mathbb{Z}/128\mathbb{Z}\right)^\times\).

\(n\) \(5\) \(127\)
\(\chi(n)\) \(e\left(\frac{3}{4}\right)\) \(1\)

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
\(3\) −255.917 255.917i −0.608040 0.608040i 0.334394 0.942433i \(-0.391468\pi\)
−0.942433 + 0.334394i \(0.891468\pi\)
\(4\) 0 0
\(5\) 4214.49 4214.49i 0.603128 0.603128i −0.338013 0.941141i \(-0.609755\pi\)
0.941141 + 0.338013i \(0.109755\pi\)
\(6\) 0 0
\(7\) 80345.3i 1.80685i −0.428751 0.903423i \(-0.641046\pi\)
0.428751 0.903423i \(-0.358954\pi\)
\(8\) 0 0
\(9\) 46160.2i 0.260576i
\(10\) 0 0
\(11\) −611105. + 611105.i −1.14408 + 1.14408i −0.156382 + 0.987697i \(0.549983\pi\)
−0.987697 + 0.156382i \(0.950017\pi\)
\(12\) 0 0
\(13\) −370364. 370364.i −0.276656 0.276656i 0.555116 0.831773i \(-0.312674\pi\)
−0.831773 + 0.555116i \(0.812674\pi\)
\(14\) 0 0
\(15\) −2.15712e6 −0.733452
\(16\) 0 0
\(17\) −725142. −0.123867 −0.0619333 0.998080i \(-0.519727\pi\)
−0.0619333 + 0.998080i \(0.519727\pi\)
\(18\) 0 0
\(19\) 1.13100e7 + 1.13100e7i 1.04789 + 1.04789i 0.998794 + 0.0490973i \(0.0156345\pi\)
0.0490973 + 0.998794i \(0.484366\pi\)
\(20\) 0 0
\(21\) −2.05617e7 + 2.05617e7i −1.09863 + 1.09863i
\(22\) 0 0
\(23\) 4.89011e7i 1.58422i 0.610378 + 0.792110i \(0.291018\pi\)
−0.610378 + 0.792110i \(0.708982\pi\)
\(24\) 0 0
\(25\) 1.33043e7i 0.272472i
\(26\) 0 0
\(27\) −5.71481e7 + 5.71481e7i −0.766480 + 0.766480i
\(28\) 0 0
\(29\) −4.98291e7 4.98291e7i −0.451122 0.451122i 0.444605 0.895727i \(-0.353344\pi\)
−0.895727 + 0.444605i \(0.853344\pi\)
\(30\) 0 0
\(31\) 4.20747e7 0.263956 0.131978 0.991253i \(-0.457867\pi\)
0.131978 + 0.991253i \(0.457867\pi\)
\(32\) 0 0
\(33\) 3.12784e8 1.39129
\(34\) 0 0
\(35\) −3.38614e8 3.38614e8i −1.08976 1.08976i
\(36\) 0 0
\(37\) 1.05666e8 1.05666e8i 0.250510 0.250510i −0.570670 0.821180i \(-0.693316\pi\)
0.821180 + 0.570670i \(0.193316\pi\)
\(38\) 0 0
\(39\) 1.89565e8i 0.336436i
\(40\) 0 0
\(41\) 4.49063e8i 0.605335i −0.953096 0.302668i \(-0.902123\pi\)
0.953096 0.302668i \(-0.0978772\pi\)
\(42\) 0 0
\(43\) −7.89791e8 + 7.89791e8i −0.819286 + 0.819286i −0.986005 0.166718i \(-0.946683\pi\)
0.166718 + 0.986005i \(0.446683\pi\)
\(44\) 0 0
\(45\) −1.94542e8 1.94542e8i −0.157161 0.157161i
\(46\) 0 0
\(47\) 1.32368e9 0.841870 0.420935 0.907091i \(-0.361702\pi\)
0.420935 + 0.907091i \(0.361702\pi\)
\(48\) 0 0
\(49\) −4.47803e9 −2.26469
\(50\) 0 0
\(51\) 1.85576e8 + 1.85576e8i 0.0753158 + 0.0753158i
\(52\) 0 0
\(53\) −1.74345e9 + 1.74345e9i −0.572655 + 0.572655i −0.932870 0.360214i \(-0.882704\pi\)
0.360214 + 0.932870i \(0.382704\pi\)
\(54\) 0 0
\(55\) 5.15099e9i 1.38005i
\(56\) 0 0
\(57\) 5.78882e9i 1.27432i
\(58\) 0 0
\(59\) 6.71604e8 6.71604e8i 0.122300 0.122300i −0.643308 0.765608i \(-0.722438\pi\)
0.765608 + 0.643308i \(0.222438\pi\)
\(60\) 0 0
\(61\) 8.30682e8 + 8.30682e8i 0.125928 + 0.125928i 0.767262 0.641334i \(-0.221619\pi\)
−0.641334 + 0.767262i \(0.721619\pi\)
\(62\) 0 0
\(63\) −3.70875e9 −0.470820
\(64\) 0 0
\(65\) −3.12179e9 −0.333718
\(66\) 0 0
\(67\) −3.88275e9 3.88275e9i −0.351340 0.351340i 0.509268 0.860608i \(-0.329916\pi\)
−0.860608 + 0.509268i \(0.829916\pi\)
\(68\) 0 0
\(69\) 1.25146e10 1.25146e10i 0.963269 0.963269i
\(70\) 0 0
\(71\) 5.14233e9i 0.338251i 0.985595 + 0.169125i \(0.0540943\pi\)
−0.985595 + 0.169125i \(0.945906\pi\)
\(72\) 0 0
\(73\) 4.42543e9i 0.249850i 0.992166 + 0.124925i \(0.0398690\pi\)
−0.992166 + 0.124925i \(0.960131\pi\)
\(74\) 0 0
\(75\) 3.40480e9 3.40480e9i 0.165674 0.165674i
\(76\) 0 0
\(77\) 4.90994e10 + 4.90994e10i 2.06717 + 2.06717i
\(78\) 0 0
\(79\) 2.71827e10 0.993902 0.496951 0.867779i \(-0.334453\pi\)
0.496951 + 0.867779i \(0.334453\pi\)
\(80\) 0 0
\(81\) 2.10732e10 0.671525
\(82\) 0 0
\(83\) −1.81378e10 1.81378e10i −0.505423 0.505423i 0.407695 0.913118i \(-0.366333\pi\)
−0.913118 + 0.407695i \(0.866333\pi\)
\(84\) 0 0
\(85\) −3.05610e9 + 3.05610e9i −0.0747075 + 0.0747075i
\(86\) 0 0
\(87\) 2.55042e10i 0.548600i
\(88\) 0 0
\(89\) 5.56593e10i 1.05656i −0.849071 0.528278i \(-0.822838\pi\)
0.849071 0.528278i \(-0.177162\pi\)
\(90\) 0 0
\(91\) −2.97570e10 + 2.97570e10i −0.499875 + 0.499875i
\(92\) 0 0
\(93\) −1.07676e10 1.07676e10i −0.160496 0.160496i
\(94\) 0 0
\(95\) 9.53314e10 1.26403
\(96\) 0 0
\(97\) 2.61297e10 0.308951 0.154476 0.987997i \(-0.450631\pi\)
0.154476 + 0.987997i \(0.450631\pi\)
\(98\) 0 0
\(99\) 2.82087e10 + 2.82087e10i 0.298119 + 0.298119i
\(100\) 0 0
\(101\) 4.83525e10 4.83525e10i 0.457774 0.457774i −0.440150 0.897924i \(-0.645075\pi\)
0.897924 + 0.440150i \(0.145075\pi\)
\(102\) 0 0
\(103\) 8.13751e10i 0.691651i 0.938299 + 0.345825i \(0.112401\pi\)
−0.938299 + 0.345825i \(0.887599\pi\)
\(104\) 0 0
\(105\) 1.73314e11i 1.32523i
\(106\) 0 0
\(107\) 1.97606e10 1.97606e10i 0.136204 0.136204i −0.635718 0.771922i \(-0.719296\pi\)
0.771922 + 0.635718i \(0.219296\pi\)
\(108\) 0 0
\(109\) −1.02515e11 1.02515e11i −0.638175 0.638175i 0.311930 0.950105i \(-0.399025\pi\)
−0.950105 + 0.311930i \(0.899025\pi\)
\(110\) 0 0
\(111\) −5.40834e10 −0.304640
\(112\) 0 0
\(113\) 2.99733e11 1.53039 0.765196 0.643798i \(-0.222642\pi\)
0.765196 + 0.643798i \(0.222642\pi\)
\(114\) 0 0
\(115\) 2.06093e11 + 2.06093e11i 0.955488 + 0.955488i
\(116\) 0 0
\(117\) −1.70961e10 + 1.70961e10i −0.0720899 + 0.0720899i
\(118\) 0 0
\(119\) 5.82617e10i 0.223808i
\(120\) 0 0
\(121\) 4.61586e11i 1.61783i
\(122\) 0 0
\(123\) −1.14923e11 + 1.14923e11i −0.368068 + 0.368068i
\(124\) 0 0
\(125\) 2.61856e11 + 2.61856e11i 0.767464 + 0.767464i
\(126\) 0 0
\(127\) −6.80100e11 −1.82664 −0.913319 0.407245i \(-0.866490\pi\)
−0.913319 + 0.407245i \(0.866490\pi\)
\(128\) 0 0
\(129\) 4.04242e11 0.996317
\(130\) 0 0
\(131\) 4.89403e11 + 4.89403e11i 1.10834 + 1.10834i 0.993368 + 0.114976i \(0.0366790\pi\)
0.114976 + 0.993368i \(0.463321\pi\)
\(132\) 0 0
\(133\) 9.08702e11 9.08702e11i 1.89338 1.89338i
\(134\) 0 0
\(135\) 4.81700e11i 0.924572i
\(136\) 0 0
\(137\) 6.45344e11i 1.14243i 0.820802 + 0.571213i \(0.193527\pi\)
−0.820802 + 0.571213i \(0.806473\pi\)
\(138\) 0 0
\(139\) −7.22257e11 + 7.22257e11i −1.18062 + 1.18062i −0.201036 + 0.979584i \(0.564431\pi\)
−0.979584 + 0.201036i \(0.935569\pi\)
\(140\) 0 0
\(141\) −3.38752e11 3.38752e11i −0.511890 0.511890i
\(142\) 0 0
\(143\) 4.52662e11 0.633033
\(144\) 0 0
\(145\) −4.20008e11 −0.544169
\(146\) 0 0
\(147\) 1.14600e12 + 1.14600e12i 1.37702 + 1.37702i
\(148\) 0 0
\(149\) −7.59849e11 + 7.59849e11i −0.847623 + 0.847623i −0.989836 0.142213i \(-0.954578\pi\)
0.142213 + 0.989836i \(0.454578\pi\)
\(150\) 0 0
\(151\) 1.60628e12i 1.66513i 0.553925 + 0.832566i \(0.313129\pi\)
−0.553925 + 0.832566i \(0.686871\pi\)
\(152\) 0 0
\(153\) 3.34727e10i 0.0322766i
\(154\) 0 0
\(155\) 1.77323e11 1.77323e11i 0.159200 0.159200i
\(156\) 0 0
\(157\) 4.80944e11 + 4.80944e11i 0.402389 + 0.402389i 0.879074 0.476685i \(-0.158162\pi\)
−0.476685 + 0.879074i \(0.658162\pi\)
\(158\) 0 0
\(159\) 8.92358e11 0.696394
\(160\) 0 0
\(161\) 3.92897e12 2.86244
\(162\) 0 0
\(163\) −1.15153e12 1.15153e12i −0.783872 0.783872i 0.196610 0.980482i \(-0.437007\pi\)
−0.980482 + 0.196610i \(0.937007\pi\)
\(164\) 0 0
\(165\) 1.31822e12 1.31822e12i 0.839127 0.839127i
\(166\) 0 0
\(167\) 1.12515e12i 0.670301i −0.942165 0.335150i \(-0.891213\pi\)
0.942165 0.335150i \(-0.108787\pi\)
\(168\) 0 0
\(169\) 1.51782e12i 0.846923i
\(170\) 0 0
\(171\) 5.22070e11 5.22070e11i 0.273055 0.273055i
\(172\) 0 0
\(173\) 3.94285e11 + 3.94285e11i 0.193445 + 0.193445i 0.797183 0.603738i \(-0.206323\pi\)
−0.603738 + 0.797183i \(0.706323\pi\)
\(174\) 0 0
\(175\) 1.06894e12 0.492315
\(176\) 0 0
\(177\) −3.43750e11 −0.148727
\(178\) 0 0
\(179\) −4.94035e11 4.94035e11i −0.200940 0.200940i 0.599463 0.800403i \(-0.295381\pi\)
−0.800403 + 0.599463i \(0.795381\pi\)
\(180\) 0 0
\(181\) 1.41767e12 1.41767e12i 0.542430 0.542430i −0.381811 0.924240i \(-0.624699\pi\)
0.924240 + 0.381811i \(0.124699\pi\)
\(182\) 0 0
\(183\) 4.25171e11i 0.153138i
\(184\) 0 0
\(185\) 8.90656e11i 0.302180i
\(186\) 0 0
\(187\) 4.43138e11 4.43138e11i 0.141713 0.141713i
\(188\) 0 0
\(189\) 4.59158e12 + 4.59158e12i 1.38491 + 1.38491i
\(190\) 0 0
\(191\) 1.48907e12 0.423868 0.211934 0.977284i \(-0.432024\pi\)
0.211934 + 0.977284i \(0.432024\pi\)
\(192\) 0 0
\(193\) 4.59604e11 0.123543 0.0617716 0.998090i \(-0.480325\pi\)
0.0617716 + 0.998090i \(0.480325\pi\)
\(194\) 0 0
\(195\) 7.98918e11 + 7.98918e11i 0.202914 + 0.202914i
\(196\) 0 0
\(197\) −2.16767e12 + 2.16767e12i −0.520509 + 0.520509i −0.917725 0.397216i \(-0.869976\pi\)
0.397216 + 0.917725i \(0.369976\pi\)
\(198\) 0 0
\(199\) 6.81559e12i 1.54814i 0.633097 + 0.774072i \(0.281783\pi\)
−0.633097 + 0.774072i \(0.718217\pi\)
\(200\) 0 0
\(201\) 1.98732e12i 0.427257i
\(202\) 0 0
\(203\) −4.00353e12 + 4.00353e12i −0.815108 + 0.815108i
\(204\) 0 0
\(205\) −1.89257e12 1.89257e12i −0.365095 0.365095i
\(206\) 0 0
\(207\) 2.25728e12 0.412809
\(208\) 0 0
\(209\) −1.38231e13 −2.39774
\(210\) 0 0
\(211\) −9.06969e11 9.06969e11i −0.149293 0.149293i 0.628509 0.777802i \(-0.283665\pi\)
−0.777802 + 0.628509i \(0.783665\pi\)
\(212\) 0 0
\(213\) 1.31601e12 1.31601e12i 0.205670 0.205670i
\(214\) 0 0
\(215\) 6.65713e12i 0.988270i
\(216\) 0 0
\(217\) 3.38051e12i 0.476928i
\(218\) 0 0
\(219\) 1.13254e12 1.13254e12i 0.151919 0.151919i
\(220\) 0 0
\(221\) 2.68567e11 + 2.68567e11i 0.0342685 + 0.0342685i
\(222\) 0 0
\(223\) −1.21567e13 −1.47618 −0.738090 0.674702i \(-0.764272\pi\)
−0.738090 + 0.674702i \(0.764272\pi\)
\(224\) 0 0
\(225\) 6.14129e11 0.0709996
\(226\) 0 0
\(227\) −5.41697e12 5.41697e12i −0.596506 0.596506i 0.342875 0.939381i \(-0.388599\pi\)
−0.939381 + 0.342875i \(0.888599\pi\)
\(228\) 0 0
\(229\) 6.76215e11 6.76215e11i 0.0709561 0.0709561i −0.670738 0.741694i \(-0.734023\pi\)
0.741694 + 0.670738i \(0.234023\pi\)
\(230\) 0 0
\(231\) 2.51307e13i 2.51385i
\(232\) 0 0
\(233\) 2.89696e12i 0.276366i −0.990407 0.138183i \(-0.955874\pi\)
0.990407 0.138183i \(-0.0441262\pi\)
\(234\) 0 0
\(235\) 5.57864e12 5.57864e12i 0.507756 0.507756i
\(236\) 0 0
\(237\) −6.95651e12 6.95651e12i −0.604332 0.604332i
\(238\) 0 0
\(239\) −2.40938e12 −0.199856 −0.0999278 0.994995i \(-0.531861\pi\)
−0.0999278 + 0.994995i \(0.531861\pi\)
\(240\) 0 0
\(241\) 6.75923e12 0.535555 0.267777 0.963481i \(-0.413711\pi\)
0.267777 + 0.963481i \(0.413711\pi\)
\(242\) 0 0
\(243\) 4.73063e12 + 4.73063e12i 0.358166 + 0.358166i
\(244\) 0 0
\(245\) −1.88726e13 + 1.88726e13i −1.36590 + 1.36590i
\(246\) 0 0
\(247\) 8.37761e12i 0.579811i
\(248\) 0 0
\(249\) 9.28353e12i 0.614634i
\(250\) 0 0
\(251\) 1.02005e13 1.02005e13i 0.646273 0.646273i −0.305817 0.952090i \(-0.598930\pi\)
0.952090 + 0.305817i \(0.0989297\pi\)
\(252\) 0 0
\(253\) −2.98837e13 2.98837e13i −1.81247 1.81247i
\(254\) 0 0
\(255\) 1.56422e12 0.0908502
\(256\) 0 0
\(257\) −1.59010e13 −0.884695 −0.442347 0.896844i \(-0.645854\pi\)
−0.442347 + 0.896844i \(0.645854\pi\)
\(258\) 0 0
\(259\) −8.48976e12 8.48976e12i −0.452633 0.452633i
\(260\) 0 0
\(261\) −2.30012e12 + 2.30012e12i −0.117551 + 0.117551i
\(262\) 0 0
\(263\) 2.14522e13i 1.05127i −0.850710 0.525636i \(-0.823827\pi\)
0.850710 0.525636i \(-0.176173\pi\)
\(264\) 0 0
\(265\) 1.46955e13i 0.690769i
\(266\) 0 0
\(267\) −1.42441e13 + 1.42441e13i −0.642428 + 0.642428i
\(268\) 0 0
\(269\) 1.79297e13 + 1.79297e13i 0.776130 + 0.776130i 0.979170 0.203040i \(-0.0650823\pi\)
−0.203040 + 0.979170i \(0.565082\pi\)
\(270\) 0 0
\(271\) 1.69476e13 0.704330 0.352165 0.935938i \(-0.385446\pi\)
0.352165 + 0.935938i \(0.385446\pi\)
\(272\) 0 0
\(273\) 1.52306e13 0.607888
\(274\) 0 0
\(275\) −8.13033e12 8.13033e12i −0.311730 0.311730i
\(276\) 0 0
\(277\) −6.82782e12 + 6.82782e12i −0.251561 + 0.251561i −0.821610 0.570050i \(-0.806924\pi\)
0.570050 + 0.821610i \(0.306924\pi\)
\(278\) 0 0
\(279\) 1.94218e12i 0.0687806i
\(280\) 0 0
\(281\) 4.34594e13i 1.47979i 0.672724 + 0.739893i \(0.265124\pi\)
−0.672724 + 0.739893i \(0.734876\pi\)
\(282\) 0 0
\(283\) 2.07572e13 2.07572e13i 0.679739 0.679739i −0.280202 0.959941i \(-0.590401\pi\)
0.959941 + 0.280202i \(0.0904013\pi\)
\(284\) 0 0
\(285\) −2.43969e13 2.43969e13i −0.768578 0.768578i
\(286\) 0 0
\(287\) −3.60801e13 −1.09375
\(288\) 0 0
\(289\) −3.37461e13 −0.984657
\(290\) 0 0
\(291\) −6.68703e12 6.68703e12i −0.187855 0.187855i
\(292\) 0 0
\(293\) −4.66490e12 + 4.66490e12i −0.126203 + 0.126203i −0.767387 0.641184i \(-0.778444\pi\)
0.641184 + 0.767387i \(0.278444\pi\)
\(294\) 0 0
\(295\) 5.66093e12i 0.147525i
\(296\) 0 0
\(297\) 6.98469e13i 1.75383i
\(298\) 0 0
\(299\) 1.81112e13 1.81112e13i 0.438284 0.438284i
\(300\) 0 0
\(301\) 6.34560e13 + 6.34560e13i 1.48032 + 1.48032i
\(302\) 0 0
\(303\) −2.47484e13 −0.556690
\(304\) 0 0
\(305\) 7.00180e12 0.151901
\(306\) 0 0
\(307\) 2.07573e13 + 2.07573e13i 0.434421 + 0.434421i 0.890129 0.455708i \(-0.150614\pi\)
−0.455708 + 0.890129i \(0.650614\pi\)
\(308\) 0 0
\(309\) 2.08253e13 2.08253e13i 0.420551 0.420551i
\(310\) 0 0
\(311\) 6.82615e13i 1.33043i −0.746650 0.665217i \(-0.768339\pi\)
0.746650 0.665217i \(-0.231661\pi\)
\(312\) 0 0
\(313\) 6.58243e13i 1.23849i 0.785198 + 0.619245i \(0.212561\pi\)
−0.785198 + 0.619245i \(0.787439\pi\)
\(314\) 0 0
\(315\) −1.56305e13 + 1.56305e13i −0.283965 + 0.283965i
\(316\) 0 0
\(317\) −1.67518e13 1.67518e13i −0.293924 0.293924i 0.544704 0.838628i \(-0.316642\pi\)
−0.838628 + 0.544704i \(0.816642\pi\)
\(318\) 0 0
\(319\) 6.09016e13 1.03224
\(320\) 0 0
\(321\) −1.01141e13 −0.165635
\(322\) 0 0
\(323\) −8.20133e12 8.20133e12i −0.129799 0.129799i
\(324\) 0 0
\(325\) 4.92744e12 4.92744e12i 0.0753811 0.0753811i
\(326\) 0 0
\(327\) 5.24704e13i 0.776071i
\(328\) 0 0
\(329\) 1.06351e14i 1.52113i
\(330\) 0 0
\(331\) 2.92920e13 2.92920e13i 0.405224 0.405224i −0.474845 0.880069i \(-0.657496\pi\)
0.880069 + 0.474845i \(0.157496\pi\)
\(332\) 0 0
\(333\) −4.87756e12 4.87756e12i −0.0652769 0.0652769i
\(334\) 0 0
\(335\) −3.27276e13 −0.423806
\(336\) 0 0
\(337\) −8.08695e13 −1.01349 −0.506746 0.862095i \(-0.669152\pi\)
−0.506746 + 0.862095i \(0.669152\pi\)
\(338\) 0 0
\(339\) −7.67066e13 7.67066e13i −0.930539 0.930539i
\(340\) 0 0
\(341\) −2.57121e13 + 2.57121e13i −0.301987 + 0.301987i
\(342\) 0 0
\(343\) 2.00920e14i 2.28510i
\(344\) 0 0
\(345\) 1.05485e14i 1.16195i
\(346\) 0 0
\(347\) −7.33385e13 + 7.33385e13i −0.782564 + 0.782564i −0.980263 0.197698i \(-0.936653\pi\)
0.197698 + 0.980263i \(0.436653\pi\)
\(348\) 0 0
\(349\) −2.93654e13 2.93654e13i −0.303596 0.303596i 0.538823 0.842419i \(-0.318869\pi\)
−0.842419 + 0.538823i \(0.818869\pi\)
\(350\) 0 0
\(351\) 4.23312e13 0.424103
\(352\) 0 0
\(353\) 5.76163e13 0.559480 0.279740 0.960076i \(-0.409752\pi\)
0.279740 + 0.960076i \(0.409752\pi\)
\(354\) 0 0
\(355\) 2.16723e13 + 2.16723e13i 0.204009 + 0.204009i
\(356\) 0 0
\(357\) 1.49102e13 1.49102e13i 0.136084 0.136084i
\(358\) 0 0
\(359\) 1.50109e14i 1.32858i 0.747475 + 0.664290i \(0.231266\pi\)
−0.747475 + 0.664290i \(0.768734\pi\)
\(360\) 0 0
\(361\) 1.39340e14i 1.19615i
\(362\) 0 0
\(363\) −1.18128e14 + 1.18128e14i −0.983706 + 0.983706i
\(364\) 0 0
\(365\) 1.86509e13 + 1.86509e13i 0.150692 + 0.150692i
\(366\) 0 0
\(367\) 2.30917e14 1.81047 0.905237 0.424908i \(-0.139693\pi\)
0.905237 + 0.424908i \(0.139693\pi\)
\(368\) 0 0
\(369\) −2.07288e13 −0.157736
\(370\) 0 0
\(371\) 1.40078e14 + 1.40078e14i 1.03470 + 1.03470i
\(372\) 0 0
\(373\) 7.43352e13 7.43352e13i 0.533084 0.533084i −0.388405 0.921489i \(-0.626974\pi\)
0.921489 + 0.388405i \(0.126974\pi\)
\(374\) 0 0
\(375\) 1.34027e14i 0.933297i
\(376\) 0 0
\(377\) 3.69098e13i 0.249611i
\(378\) 0 0
\(379\) −5.39364e13 + 5.39364e13i −0.354296 + 0.354296i −0.861705 0.507409i \(-0.830603\pi\)
0.507409 + 0.861705i \(0.330603\pi\)
\(380\) 0 0
\(381\) 1.74049e14 + 1.74049e14i 1.11067 + 1.11067i
\(382\) 0 0
\(383\) 1.62840e13 0.100965 0.0504823 0.998725i \(-0.483924\pi\)
0.0504823 + 0.998725i \(0.483924\pi\)
\(384\) 0 0
\(385\) 4.13857e14 2.49354
\(386\) 0 0
\(387\) 3.64569e13 + 3.64569e13i 0.213486 + 0.213486i
\(388\) 0 0
\(389\) 1.00553e14 1.00553e14i 0.572361 0.572361i −0.360426 0.932788i \(-0.617369\pi\)
0.932788 + 0.360426i \(0.117369\pi\)
\(390\) 0 0
\(391\) 3.54603e13i 0.196232i
\(392\) 0 0
\(393\) 2.50493e14i 1.34783i
\(394\) 0 0
\(395\) 1.14561e14 1.14561e14i 0.599451 0.599451i
\(396\) 0 0
\(397\) 5.36644e13 + 5.36644e13i 0.273110 + 0.273110i 0.830351 0.557241i \(-0.188140\pi\)
−0.557241 + 0.830351i \(0.688140\pi\)
\(398\) 0 0
\(399\) −4.65104e14 −2.30250
\(400\) 0 0
\(401\) 1.33118e14 0.641127 0.320563 0.947227i \(-0.396128\pi\)
0.320563 + 0.947227i \(0.396128\pi\)
\(402\) 0 0
\(403\) −1.55830e13 1.55830e13i −0.0730252 0.0730252i
\(404\) 0 0
\(405\) 8.88126e13 8.88126e13i 0.405016 0.405016i
\(406\) 0 0
\(407\) 1.29146e14i 0.573207i
\(408\) 0 0
\(409\) 3.22329e14i 1.39259i 0.717758 + 0.696293i \(0.245168\pi\)
−0.717758 + 0.696293i \(0.754832\pi\)
\(410\) 0 0
\(411\) 1.65154e14 1.65154e14i 0.694640 0.694640i
\(412\) 0 0
\(413\) −5.39602e13 5.39602e13i −0.220978 0.220978i
\(414\) 0 0
\(415\) −1.52883e14 −0.609669
\(416\) 0 0
\(417\) 3.69675e14 1.43573
\(418\) 0 0
\(419\) 2.82378e14 + 2.82378e14i 1.06820 + 1.06820i 0.997497 + 0.0707064i \(0.0225254\pi\)
0.0707064 + 0.997497i \(0.477475\pi\)
\(420\) 0 0
\(421\) 1.97312e14 1.97312e14i 0.727112 0.727112i −0.242931 0.970044i \(-0.578109\pi\)
0.970044 + 0.242931i \(0.0781089\pi\)
\(422\) 0 0
\(423\) 6.11014e13i 0.219371i
\(424\) 0 0
\(425\) 9.64752e12i 0.0337502i
\(426\) 0 0
\(427\) 6.67414e13 6.67414e13i 0.227532 0.227532i
\(428\) 0 0
\(429\) −1.15844e14 1.15844e14i −0.384909 0.384909i
\(430\) 0 0
\(431\) 3.78445e14 1.22568 0.612840 0.790207i \(-0.290027\pi\)
0.612840 + 0.790207i \(0.290027\pi\)
\(432\) 0 0
\(433\) −5.87357e14 −1.85447 −0.927233 0.374485i \(-0.877820\pi\)
−0.927233 + 0.374485i \(0.877820\pi\)
\(434\) 0 0
\(435\) 1.07487e14 + 1.07487e14i 0.330876 + 0.330876i
\(436\) 0 0
\(437\) −5.53070e14 + 5.53070e14i −1.66009 + 1.66009i
\(438\) 0 0
\(439\) 1.95351e14i 0.571821i −0.958256 0.285910i \(-0.907704\pi\)
0.958256 0.285910i \(-0.0922960\pi\)
\(440\) 0 0
\(441\) 2.06707e14i 0.590123i
\(442\) 0 0
\(443\) −3.24637e14 + 3.24637e14i −0.904018 + 0.904018i −0.995781 0.0917627i \(-0.970750\pi\)
0.0917627 + 0.995781i \(0.470750\pi\)
\(444\) 0 0
\(445\) −2.34575e14 2.34575e14i −0.637239 0.637239i
\(446\) 0 0
\(447\) 3.88916e14 1.03078
\(448\) 0 0
\(449\) −4.80220e13 −0.124190 −0.0620949 0.998070i \(-0.519778\pi\)
−0.0620949 + 0.998070i \(0.519778\pi\)
\(450\) 0 0
\(451\) 2.74425e14 + 2.74425e14i 0.692551 + 0.692551i
\(452\) 0 0
\(453\) 4.11075e14 4.11075e14i 1.01247 1.01247i
\(454\) 0 0
\(455\) 2.50821e14i 0.602978i
\(456\) 0 0
\(457\) 5.54781e13i 0.130192i 0.997879 + 0.0650958i \(0.0207353\pi\)
−0.997879 + 0.0650958i \(0.979265\pi\)
\(458\) 0 0
\(459\) 4.14405e13 4.14405e13i 0.0949413 0.0949413i
\(460\) 0 0
\(461\) 2.39250e13 + 2.39250e13i 0.0535177 + 0.0535177i 0.733359 0.679841i \(-0.237951\pi\)
−0.679841 + 0.733359i \(0.737951\pi\)
\(462\) 0 0
\(463\) −7.76509e14 −1.69610 −0.848049 0.529918i \(-0.822223\pi\)
−0.848049 + 0.529918i \(0.822223\pi\)
\(464\) 0 0
\(465\) −9.07601e13 −0.193599
\(466\) 0 0
\(467\) −4.52807e14 4.52807e14i −0.943344 0.943344i 0.0551351 0.998479i \(-0.482441\pi\)
−0.998479 + 0.0551351i \(0.982441\pi\)
\(468\) 0 0
\(469\) −3.11960e14 + 3.11960e14i −0.634817 + 0.634817i
\(470\) 0 0
\(471\) 2.46163e14i 0.489337i
\(472\) 0 0
\(473\) 9.65290e14i 1.87466i
\(474\) 0 0
\(475\) −1.50471e14 + 1.50471e14i −0.285521 + 0.285521i
\(476\) 0 0
\(477\) 8.04782e13 + 8.04782e13i 0.149220 + 0.149220i
\(478\) 0 0
\(479\) 1.36724e14 0.247742 0.123871 0.992298i \(-0.460469\pi\)
0.123871 + 0.992298i \(0.460469\pi\)
\(480\) 0 0
\(481\) −7.82697e13 −0.138610
\(482\) 0 0
\(483\) −1.00549e15 1.00549e15i −1.74048 1.74048i
\(484\) 0 0
\(485\) 1.10123e14 1.10123e14i 0.186337 0.186337i
\(486\) 0 0
\(487\) 1.79388e14i 0.296746i −0.988931 0.148373i \(-0.952596\pi\)
0.988931 0.148373i \(-0.0474037\pi\)
\(488\) 0 0
\(489\) 5.89394e14i 0.953250i
\(490\) 0 0
\(491\) 3.42827e14 3.42827e14i 0.542159 0.542159i −0.382002 0.924161i \(-0.624765\pi\)
0.924161 + 0.382002i \(0.124765\pi\)
\(492\) 0 0
\(493\) 3.61332e13 + 3.61332e13i 0.0558790 + 0.0558790i
\(494\) 0 0
\(495\) 2.37771e14 0.359608
\(496\) 0 0
\(497\) 4.13162e14 0.611167
\(498\) 0 0
\(499\) −1.85485e14 1.85485e14i −0.268383 0.268383i 0.560065 0.828448i \(-0.310776\pi\)
−0.828448 + 0.560065i \(0.810776\pi\)
\(500\) 0 0
\(501\) −2.87945e14 + 2.87945e14i −0.407569 + 0.407569i
\(502\) 0 0
\(503\) 3.66232e14i 0.507145i 0.967316 + 0.253572i \(0.0816056\pi\)
−0.967316 + 0.253572i \(0.918394\pi\)
\(504\) 0 0
\(505\) 4.07562e14i 0.552193i
\(506\) 0 0
\(507\) −3.88436e14 + 3.88436e14i −0.514963 + 0.514963i
\(508\) 0 0
\(509\) −7.50462e13 7.50462e13i −0.0973600 0.0973600i 0.656749 0.754109i \(-0.271931\pi\)
−0.754109 + 0.656749i \(0.771931\pi\)
\(510\) 0 0
\(511\) 3.55562e14 0.451441
\(512\) 0 0
\(513\) −1.29268e15 −1.60638
\(514\) 0 0
\(515\) 3.42954e14 + 3.42954e14i 0.417154 + 0.417154i
\(516\) 0 0
\(517\) −8.08908e14 + 8.08908e14i −0.963166 + 0.963166i
\(518\) 0 0
\(519\) 2.01809e14i 0.235244i
\(520\) 0 0
\(521\) 8.52126e14i 0.972516i 0.873815 + 0.486258i \(0.161638\pi\)
−0.873815 + 0.486258i \(0.838362\pi\)
\(522\) 0 0
\(523\) −3.84070e14 + 3.84070e14i −0.429192 + 0.429192i −0.888353 0.459161i \(-0.848150\pi\)
0.459161 + 0.888353i \(0.348150\pi\)
\(524\) 0 0
\(525\) −2.73559e14 2.73559e14i −0.299347 0.299347i
\(526\) 0 0
\(527\) −3.05102e13 −0.0326954
\(528\) 0 0
\(529\) −1.43851e15 −1.50975
\(530\) 0 0
\(531\) −3.10014e13 3.10014e13i −0.0318685 0.0318685i
\(532\) 0 0
\(533\) −1.66317e14 + 1.66317e14i −0.167470 + 0.167470i
\(534\) 0 0
\(535\) 1.66562e14i 0.164297i
\(536\) 0 0
\(537\) 2.52864e14i 0.244359i
\(538\) 0 0
\(539\) 2.73655e15 2.73655e15i 2.59098 2.59098i
\(540\) 0 0
\(541\) −1.73890e13 1.73890e13i −0.0161320 0.0161320i 0.698995 0.715127i \(-0.253631\pi\)
−0.715127 + 0.698995i \(0.753631\pi\)
\(542\) 0 0
\(543\) −7.25612e14 −0.659637
\(544\) 0 0
\(545\) −8.64092e14 −0.769803
\(546\) 0 0
\(547\) −1.23494e15 1.23494e15i −1.07824 1.07824i −0.996667 0.0815740i \(-0.974005\pi\)
−0.0815740 0.996667i \(-0.525995\pi\)
\(548\) 0 0
\(549\) 3.83445e13 3.83445e13i 0.0328137 0.0328137i
\(550\) 0 0
\(551\) 1.12713e15i 0.945454i
\(552\) 0 0
\(553\) 2.18400e15i 1.79583i
\(554\) 0 0
\(555\) −2.27934e14 + 2.27934e14i −0.183737 + 0.183737i
\(556\) 0 0
\(557\) 1.19349e15 + 1.19349e15i 0.943223 + 0.943223i 0.998473 0.0552494i \(-0.0175954\pi\)
−0.0552494 + 0.998473i \(0.517595\pi\)
\(558\) 0 0
\(559\) 5.85021e14 0.453321
\(560\) 0 0
\(561\) −2.26813e14 −0.172334
\(562\) 0 0
\(563\) 5.15004e14 + 5.15004e14i 0.383720 + 0.383720i 0.872440 0.488720i \(-0.162536\pi\)
−0.488720 + 0.872440i \(0.662536\pi\)
\(564\) 0 0
\(565\) 1.26322e15 1.26322e15i 0.923023 0.923023i
\(566\) 0 0
\(567\) 1.69313e15i 1.21334i
\(568\) 0 0
\(569\) 1.36483e15i 0.959315i 0.877456 + 0.479657i \(0.159239\pi\)
−0.877456 + 0.479657i \(0.840761\pi\)
\(570\) 0 0
\(571\) −1.77410e15 + 1.77410e15i −1.22315 + 1.22315i −0.256639 + 0.966507i \(0.582615\pi\)
−0.966507 + 0.256639i \(0.917385\pi\)
\(572\) 0 0
\(573\) −3.81077e14 3.81077e14i −0.257729 0.257729i
\(574\) 0 0
\(575\) −6.50596e14 −0.431656
\(576\) 0 0
\(577\) 3.65991e14 0.238234 0.119117 0.992880i \(-0.461994\pi\)
0.119117 + 0.992880i \(0.461994\pi\)
\(578\) 0 0
\(579\) −1.17620e14 1.17620e14i −0.0751191 0.0751191i
\(580\) 0 0
\(581\) −1.45728e15 + 1.45728e15i −0.913220 + 0.913220i
\(582\) 0 0
\(583\) 2.13087e15i 1.31032i
\(584\) 0 0
\(585\) 1.44102e14i 0.0869589i
\(586\) 0 0
\(587\) 1.82609e15 1.82609e15i 1.08146 1.08146i 0.0850900 0.996373i \(-0.472882\pi\)
0.996373 0.0850900i \(-0.0271178\pi\)
\(588\) 0 0
\(589\) 4.75864e14 + 4.75864e14i 0.276598 + 0.276598i
\(590\) 0 0
\(591\) 1.10949e15 0.632981
\(592\) 0 0
\(593\) 3.34557e15 1.87357 0.936784 0.349909i \(-0.113787\pi\)
0.936784 + 0.349909i \(0.113787\pi\)
\(594\) 0 0
\(595\) 2.45543e14 + 2.45543e14i 0.134985 + 0.134985i
\(596\) 0 0
\(597\) 1.74422e15 1.74422e15i 0.941333 0.941333i
\(598\) 0 0
\(599\) 3.39597e15i 1.79935i 0.436556 + 0.899677i \(0.356198\pi\)
−0.436556 + 0.899677i \(0.643802\pi\)
\(600\) 0 0
\(601\) 3.43538e15i 1.78717i 0.448898 + 0.893583i \(0.351817\pi\)
−0.448898 + 0.893583i \(0.648183\pi\)
\(602\) 0 0
\(603\) −1.79228e14 + 1.79228e14i −0.0915507 + 0.0915507i
\(604\) 0 0
\(605\) −1.94535e15 1.94535e15i −0.975760 0.975760i
\(606\) 0 0
\(607\) 1.96296e14 0.0966884 0.0483442 0.998831i \(-0.484606\pi\)
0.0483442 + 0.998831i \(0.484606\pi\)
\(608\) 0 0
\(609\) 2.04914e15 0.991236
\(610\) 0 0
\(611\) −4.90244e14 4.90244e14i −0.232909 0.232909i
\(612\) 0 0
\(613\) −1.99692e15 + 1.99692e15i −0.931813 + 0.931813i −0.997819 0.0660064i \(-0.978974\pi\)
0.0660064 + 0.997819i \(0.478974\pi\)
\(614\) 0 0
\(615\) 9.68681e14i 0.443984i
\(616\) 0 0
\(617\) 2.13465e15i 0.961079i −0.876973 0.480539i \(-0.840441\pi\)
0.876973 0.480539i \(-0.159559\pi\)
\(618\) 0 0
\(619\) −1.66182e15 + 1.66182e15i −0.734998 + 0.734998i −0.971605 0.236608i \(-0.923964\pi\)
0.236608 + 0.971605i \(0.423964\pi\)
\(620\) 0 0
\(621\) −2.79460e15 2.79460e15i −1.21427 1.21427i
\(622\) 0 0
\(623\) −4.47196e15 −1.90903
\(624\) 0 0
\(625\) 1.55756e15 0.653287
\(626\) 0 0
\(627\) 3.53757e15 + 3.53757e15i 1.45792 + 1.45792i
\(628\) 0 0
\(629\) −7.66229e13 + 7.66229e13i −0.0310299 + 0.0310299i
\(630\) 0 0
\(631\) 1.28852e15i 0.512778i 0.966574 + 0.256389i \(0.0825328\pi\)
−0.966574 + 0.256389i \(0.917467\pi\)
\(632\) 0 0
\(633\) 4.64217e14i 0.181552i
\(634\) 0 0
\(635\) −2.86627e15 + 2.86627e15i −1.10170 + 1.10170i
\(636\) 0 0
\(637\) 1.65850e15 + 1.65850e15i 0.626541 + 0.626541i
\(638\) 0 0
\(639\) 2.37371e14 0.0881399
\(640\) 0 0
\(641\) −2.37556e15 −0.867055 −0.433528 0.901140i \(-0.642731\pi\)
−0.433528 + 0.901140i \(0.642731\pi\)
\(642\) 0 0
\(643\) 3.12329e15 + 3.12329e15i 1.12060 + 1.12060i 0.991651 + 0.128951i \(0.0411611\pi\)
0.128951 + 0.991651i \(0.458839\pi\)
\(644\) 0 0
\(645\) 1.70367e15 1.70367e15i 0.600907 0.600907i
\(646\) 0 0
\(647\) 2.99929e15i 1.04003i 0.854158 + 0.520014i \(0.174073\pi\)
−0.854158 + 0.520014i \(0.825927\pi\)
\(648\) 0 0
\(649\) 8.20841e14i 0.279842i
\(650\) 0 0
\(651\) −8.65128e14 + 8.65128e14i −0.289991 + 0.289991i
\(652\) 0 0
\(653\) −1.19197e15 1.19197e15i −0.392866 0.392866i 0.482842 0.875708i \(-0.339605\pi\)
−0.875708 + 0.482842i \(0.839605\pi\)
\(654\) 0 0
\(655\) 4.12517e15 1.33695
\(656\) 0 0
\(657\) 2.04279e14 0.0651049
\(658\) 0 0
\(659\) −2.22810e15 2.22810e15i −0.698337 0.698337i 0.265715 0.964052i \(-0.414392\pi\)
−0.964052 + 0.265715i \(0.914392\pi\)
\(660\) 0 0
\(661\) 1.31704e15 1.31704e15i 0.405967 0.405967i −0.474362 0.880330i \(-0.657321\pi\)
0.880330 + 0.474362i \(0.157321\pi\)
\(662\) 0 0
\(663\) 1.37461e14i 0.0416732i
\(664\) 0 0
\(665\) 7.65942e15i 2.28390i
\(666\) 0 0
\(667\) 2.43670e15 2.43670e15i 0.714677 0.714677i
\(668\) 0 0
\(669\) 3.11111e15 + 3.11111e15i 0.897576 + 0.897576i
\(670\) 0 0
\(671\) −1.01527e15 −0.288142
\(672\) 0 0
\(673\) 1.44297e14 0.0402879 0.0201439 0.999797i \(-0.493588\pi\)
0.0201439 + 0.999797i \(0.493588\pi\)
\(674\) 0 0
\(675\) −7.60315e14 7.60315e14i −0.208845 0.208845i
\(676\) 0 0
\(677\) 1.26299e14 1.26299e14i 0.0341320 0.0341320i −0.689835 0.723967i \(-0.742317\pi\)
0.723967 + 0.689835i \(0.242317\pi\)
\(678\) 0 0
\(679\) 2.09940e15i 0.558227i
\(680\) 0 0
\(681\) 2.77259e15i 0.725398i
\(682\) 0 0
\(683\) 2.21846e14 2.21846e14i 0.0571133 0.0571133i −0.677973 0.735087i \(-0.737141\pi\)
0.735087 + 0.677973i \(0.237141\pi\)
\(684\) 0 0
\(685\) 2.71979e15 + 2.71979e15i 0.689029 + 0.689029i
\(686\) 0 0
\(687\) −3.46109e14 −0.0862882
\(688\) 0 0
\(689\) 1.29143e15 0.316857
\(690\) 0 0
\(691\) 4.10228e15 + 4.10228e15i 0.990596 + 0.990596i 0.999956 0.00936034i \(-0.00297953\pi\)
−0.00936034 + 0.999956i \(0.502980\pi\)
\(692\) 0 0
\(693\) 2.26644e15 2.26644e15i 0.538655 0.538655i
\(694\) 0 0
\(695\) 6.08788e15i 1.42413i
\(696\) 0 0
\(697\) 3.25635e14i 0.0749808i
\(698\) 0 0
\(699\) −7.41381e14 + 7.41381e14i −0.168042 + 0.168042i
\(700\) 0 0
\(701\) 3.04410e15 + 3.04410e15i 0.679218 + 0.679218i 0.959823 0.280605i \(-0.0905351\pi\)
−0.280605 + 0.959823i \(0.590535\pi\)
\(702\) 0 0
\(703\) 2.39016e15 0.525015
\(704\) 0 0
\(705\) −2.85533e15 −0.617471
\(706\) 0 0
\(707\) −3.88489e15 3.88489e15i −0.827127 0.827127i
\(708\) 0 0
\(709\) 4.62095e15 4.62095e15i 0.968672 0.968672i −0.0308521 0.999524i \(-0.509822\pi\)
0.999524 + 0.0308521i \(0.00982209\pi\)
\(710\) 0 0
\(711\) 1.25476e15i 0.258987i
\(712\) 0 0
\(713\) 2.05750e15i 0.418165i
\(714\) 0 0
\(715\) 1.90774e15 1.90774e15i 0.381800 0.381800i
\(716\) 0 0
\(717\) 6.16600e14 + 6.16600e14i 0.121520 + 0.121520i
\(718\) 0 0
\(719\) −1.71684e12 −0.000333212 −0.000166606 1.00000i \(-0.500053\pi\)
−0.000166606 1.00000i \(0.500053\pi\)
\(720\) 0 0
\(721\) 6.53810e15 1.24971
\(722\) 0 0
\(723\) −1.72980e15 1.72980e15i −0.325638 0.325638i
\(724\) 0 0
\(725\) 6.62942e14 6.62942e14i 0.122918 0.122918i
\(726\) 0 0
\(727\) 5.86270e14i 0.107068i −0.998566 0.0535339i \(-0.982951\pi\)
0.998566 0.0535339i \(-0.0170485\pi\)
\(728\) 0 0
\(729\) 6.15434e15i 1.10708i
\(730\) 0 0
\(731\) 5.72711e14 5.72711e14i 0.101482 0.101482i
\(732\) 0 0
\(733\) 3.03937e15 + 3.03937e15i 0.530531 + 0.530531i 0.920731 0.390199i \(-0.127594\pi\)
−0.390199 + 0.920731i \(0.627594\pi\)
\(734\) 0 0
\(735\) 9.65964e15 1.66104
\(736\) 0 0
\(737\) 4.74553e15 0.803921
\(738\) 0 0
\(739\) −3.93959e15 3.93959e15i −0.657516 0.657516i 0.297276 0.954792i \(-0.403922\pi\)
−0.954792 + 0.297276i \(0.903922\pi\)
\(740\) 0 0
\(741\) −2.14397e15 + 2.14397e15i −0.352548 + 0.352548i
\(742\) 0 0
\(743\) 1.16532e15i 0.188803i −0.995534 0.0944013i \(-0.969906\pi\)
0.995534 0.0944013i \(-0.0300937\pi\)
\(744\) 0 0
\(745\) 6.40475e15i 1.02245i
\(746\) 0 0
\(747\) −8.37243e14 + 8.37243e14i −0.131701 + 0.131701i
\(748\) 0 0
\(749\) −1.58767e15 1.58767e15i −0.246099 0.246099i
\(750\) 0 0
\(751\) −7.91358e15 −1.20880 −0.604399 0.796682i \(-0.706587\pi\)
−0.604399 + 0.796682i \(0.706587\pi\)
\(752\) 0 0
\(753\) −5.22096e15 −0.785919
\(754\) 0 0
\(755\) 6.76966e15 + 6.76966e15i 1.00429 + 1.00429i
\(756\) 0 0
\(757\) −4.02992e15 + 4.02992e15i −0.589209 + 0.589209i −0.937417 0.348208i \(-0.886790\pi\)
0.348208 + 0.937417i \(0.386790\pi\)
\(758\) 0 0
\(759\) 1.52955e16i 2.20411i
\(760\) 0 0
\(761\) 7.01563e15i 0.996440i −0.867051 0.498220i \(-0.833987\pi\)
0.867051 0.498220i \(-0.166013\pi\)
\(762\) 0 0
\(763\) −8.23656e15 + 8.23656e15i −1.15308 + 1.15308i
\(764\) 0 0
\(765\) 1.41070e14 + 1.41070e14i 0.0194669 + 0.0194669i
\(766\) 0 0
\(767\) −4.97476e14 −0.0676702
\(768\) 0 0
\(769\) −3.74998e15 −0.502845 −0.251422 0.967877i \(-0.580898\pi\)
−0.251422 + 0.967877i \(0.580898\pi\)
\(770\) 0 0
\(771\) 4.06935e15 + 4.06935e15i 0.537930 + 0.537930i
\(772\) 0 0
\(773\) −9.30098e15 + 9.30098e15i −1.21211 + 1.21211i −0.241777 + 0.970332i \(0.577730\pi\)
−0.970332 + 0.241777i \(0.922270\pi\)
\(774\) 0 0
\(775\) 5.59775e14i 0.0719208i
\(776\) 0 0
\(777\) 4.34534e15i 0.550438i
\(778\) 0 0
\(779\) 5.07888e15 5.07888e15i 0.634326 0.634326i
\(780\) 0 0
\(781\) −3.14250e15 3.14250e15i −0.386985 0.386985i
\(782\) 0 0
\(783\) 5.69527e15 0.691552
\(784\) 0 0
\(785\) 4.05386e15 0.485385
\(786\) 0 0
\(787\) −1.11730e16 1.11730e16i −1.31919 1.31919i −0.914417 0.404774i \(-0.867350\pi\)
−0.404774 0.914417i \(-0.632650\pi\)
\(788\) 0 0
\(789\) −5.48998e15 + 5.48998e15i −0.639215 + 0.639215i
\(790\) 0 0
\(791\) 2.40821e16i 2.76518i
\(792\) 0 0
\(793\) 6.15310e14i 0.0696773i
\(794\) 0 0
\(795\) 3.76083e15 3.76083e15i 0.420015 0.420015i
\(796\) 0 0
\(797\) −4.36170e15 4.36170e15i −0.480435 0.480435i 0.424836 0.905271i \(-0.360332\pi\)
−0.905271 + 0.424836i \(0.860332\pi\)
\(798\) 0 0
\(799\) −9.59857e14 −0.104280
\(800\) 0 0
\(801\) −2.56924e15 −0.275313
\(802\) 0 0
\(803\) −2.70440e15 2.70440e15i −0.285848 0.285848i
\(804\) 0 0
\(805\) 1.65586e16 1.65586e16i 1.72642 1.72642i
\(806\) 0 0
\(807\) 9.17700e15i 0.943836i
\(808\) 0 0
\(809\) 8.84145e15i 0.897029i 0.893775 + 0.448515i \(0.148047\pi\)
−0.893775 + 0.448515i \(0.851953\pi\)
\(810\) 0 0
\(811\) 8.54171e14 8.54171e14i 0.0854929 0.0854929i −0.663067 0.748560i \(-0.730746\pi\)
0.748560 + 0.663067i \(0.230746\pi\)
\(812\) 0 0
\(813\) −4.33716e15 4.33716e15i −0.428260 0.428260i
\(814\) 0 0
\(815\) −9.70625e15 −0.945550
\(816\) 0 0
\(817\) −1.78650e16 −1.71705
\(818\) 0 0
\(819\) 1.37359e15 + 1.37359e15i 0.130255 + 0.130255i
\(820\) 0 0
\(821\) 2.55379e15 2.55379e15i 0.238945 0.238945i −0.577468 0.816413i \(-0.695959\pi\)
0.816413 + 0.577468i \(0.195959\pi\)
\(822\) 0 0
\(823\) 1.46679e15i 0.135415i −0.997705 0.0677076i \(-0.978431\pi\)
0.997705 0.0677076i \(-0.0215685\pi\)
\(824\) 0 0
\(825\) 4.16137e15i 0.379088i
\(826\) 0 0
\(827\) −3.01585e15 + 3.01585e15i −0.271100 + 0.271100i −0.829543 0.558443i \(-0.811399\pi\)
0.558443 + 0.829543i \(0.311399\pi\)
\(828\) 0 0
\(829\) −1.28217e16 1.28217e16i −1.13736 1.13736i −0.988922 0.148433i \(-0.952577\pi\)
−0.148433 0.988922i \(-0.547423\pi\)
\(830\) 0 0
\(831\) 3.49471e15 0.305918
\(832\) 0 0
\(833\) 3.24721e15 0.280519
\(834\) 0 0
\(835\) −4.74193e15 4.74193e15i −0.404277 0.404277i
\(836\) 0 0
\(837\) −2.40449e15 + 2.40449e15i −0.202317 + 0.202317i
\(838\) 0 0
\(839\) 4.46711e15i 0.370967i −0.982647 0.185484i \(-0.940615\pi\)
0.982647 0.185484i \(-0.0593852\pi\)
\(840\) 0 0
\(841\) 7.23463e15i 0.592978i
\(842\) 0 0
\(843\) 1.11220e16 1.11220e16i 0.899769 0.899769i
\(844\) 0 0
\(845\) −6.39684e15 6.39684e15i −0.510803 0.510803i
\(846\) 0 0
\(847\) −3.70863e16 −2.92317
\(848\) 0 0
\(849\) −1.06242e16 −0.826617
\(850\) 0 0
\(851\) 5.16718e15 + 5.16718e15i 0.396864 + 0.396864i
\(852\) 0 0
\(853\) −1.63160e16 + 1.63160e16i −1.23707 + 1.23707i −0.275876 + 0.961193i \(0.588968\pi\)
−0.961193 + 0.275876i \(0.911032\pi\)
\(854\) 0 0
\(855\) 4.40051e15i 0.329374i
\(856\) 0 0
\(857\) 1.20290e16i 0.888861i −0.895813 0.444430i \(-0.853406\pi\)
0.895813 0.444430i \(-0.146594\pi\)
\(858\) 0 0
\(859\) −7.64963e15 + 7.64963e15i −0.558057 + 0.558057i −0.928754 0.370697i \(-0.879119\pi\)
0.370697 + 0.928754i \(0.379119\pi\)
\(860\) 0 0
\(861\) 9.23350e15 + 9.23350e15i 0.665042 + 0.665042i
\(862\) 0 0
\(863\) 1.49922e16 1.06612 0.533061 0.846077i \(-0.321041\pi\)
0.533061 + 0.846077i \(0.321041\pi\)
\(864\) 0 0
\(865\) 3.32342e15 0.233344
\(866\) 0 0
\(867\) 8.63618e15 + 8.63618e15i 0.598711 + 0.598711i
\(868\) 0 0
\(869\) −1.66115e16 + 1.66115e16i −1.13710 + 1.13710i
\(870\) 0 0
\(871\) 2.87606e15i 0.194401i
\(872\) 0 0
\(873\) 1.20615e15i 0.0805051i
\(874\) 0 0
\(875\) 2.10389e16 2.10389e16i 1.38669 1.38669i
\(876\) 0 0
\(877\) −7.04625e15 7.04625e15i −0.458628 0.458628i 0.439577 0.898205i \(-0.355128\pi\)
−0.898205 + 0.439577i \(0.855128\pi\)
\(878\) 0 0
\(879\) 2.38765e15 0.153473
\(880\) 0 0
\(881\) −9.40033e15 −0.596727 −0.298363 0.954452i \(-0.596441\pi\)
−0.298363 + 0.954452i \(0.596441\pi\)
\(882\) 0 0
\(883\) 7.11010e15 + 7.11010e15i 0.445751 + 0.445751i 0.893939 0.448189i \(-0.147931\pi\)
−0.448189 + 0.893939i \(0.647931\pi\)
\(884\) 0 0
\(885\) −1.44873e15 + 1.44873e15i −0.0897013 + 0.0897013i
\(886\) 0 0
\(887\) 9.85237e15i 0.602505i 0.953544 + 0.301253i \(0.0974047\pi\)
−0.953544 + 0.301253i \(0.902595\pi\)
\(888\) 0 0
\(889\) 5.46428e16i 3.30045i
\(890\) 0 0
\(891\) −1.28779e16 + 1.28779e16i −0.768277 + 0.768277i
\(892\) 0 0
\(893\) 1.49708e16 + 1.49708e16i 0.882188 + 0.882188i
\(894\) 0 0
\(895\) −4.16421e15 −0.242385
\(896\) 0 0
\(897\) −9.26993e15 −0.532989
\(898\) 0 0
\(899\) −2.09655e15 2.09655e15i −0.119077 0.119077i
\(900\) 0 0
\(901\) 1.26425e15 1.26425e15i 0.0709328 0.0709328i
\(902\) 0 0
\(903\) 3.24789e16i 1.80019i
\(904\) 0 0
\(905\) 1.19495e16i 0.654309i
\(906\) 0 0
\(907\) −1.16098e16 + 1.16098e16i −0.628038 + 0.628038i −0.947574 0.319536i \(-0.896473\pi\)
0.319536 + 0.947574i \(0.396473\pi\)
\(908\) 0 0
\(909\) −2.23196e15 2.23196e15i −0.119285 0.119285i
\(910\) 0 0
\(911\) 7.25287e15 0.382965 0.191482 0.981496i \(-0.438671\pi\)
0.191482 + 0.981496i \(0.438671\pi\)
\(912\) 0 0
\(913\) 2.21682e16 1.15649
\(914\) 0 0
\(915\) −1.79188e15 1.79188e15i −0.0923618 0.0923618i
\(916\) 0 0
\(917\) 3.93212e16 3.93212e16i 2.00261 2.00261i
\(918\) 0 0
\(919\) 1.55963e16i 0.784849i 0.919784 + 0.392425i \(0.128364\pi\)
−0.919784 + 0.392425i \(0.871636\pi\)
\(920\) 0 0
\(921\) 1.06243e16i 0.528290i
\(922\) 0 0
\(923\) 1.90453e15 1.90453e15i 0.0935792 0.0935792i
\(924\) 0 0
\(925\) 1.40581e15 + 1.40581e15i 0.0682571 + 0.0682571i
\(926\) 0 0
\(927\) 3.75629e15 0.180227
\(928\) 0 0
\(929\) 1.27007e16 0.602201 0.301101 0.953592i \(-0.402646\pi\)
0.301101 + 0.953592i \(0.402646\pi\)
\(930\) 0 0
\(931\) −5.06464e16 5.06464e16i −2.37315 2.37315i
\(932\) 0 0
\(933\) −1.74693e16 + 1.74693e16i −0.808957 + 0.808957i
\(934\) 0 0
\(935\) 3.73520e15i 0.170942i
\(936\) 0 0
\(937\) 3.55502e16i 1.60796i −0.594657 0.803979i \(-0.702712\pi\)
0.594657 0.803979i \(-0.297288\pi\)
\(938\) 0 0
\(939\) 1.68455e16 1.68455e16i 0.753050 0.753050i
\(940\) 0 0
\(941\) 1.37174e16 + 1.37174e16i 0.606078 + 0.606078i 0.941919 0.335841i \(-0.109021\pi\)
−0.335841 + 0.941919i \(0.609021\pi\)
\(942\) 0 0
\(943\) 2.19597e16 0.958985
\(944\) 0 0
\(945\) 3.87023e16 1.67056
\(946\) 0 0
\(947\) −1.05720e16 1.05720e16i −0.451058 0.451058i 0.444648 0.895706i \(-0.353329\pi\)
−0.895706 + 0.444648i \(0.853329\pi\)
\(948\) 0 0
\(949\) 1.63902e15 1.63902e15i 0.0691226 0.0691226i
\(950\) 0 0
\(951\) 8.57413e15i 0.357435i
\(952\) 0 0
\(953\) 1.45464e15i 0.0599440i −0.999551 0.0299720i \(-0.990458\pi\)
0.999551 0.0299720i \(-0.00954181\pi\)
\(954\) 0 0
\(955\) 6.27566e15 6.27566e15i 0.255647 0.255647i
\(956\) 0 0
\(957\) −1.55857e16 1.55857e16i −0.627642 0.627642i
\(958\) 0 0
\(959\) 5.18503e16 2.06419
\(960\) 0 0
\(961\) −2.36382e16 −0.930327
\(962\) 0 0
\(963\) −9.12154e14 9.12154e14i −0.0354914 0.0354914i
\(964\) 0 0
\(965\) 1.93700e15 1.93700e15i 0.0745124 0.0745124i
\(966\) 0 0
\(967\) 1.07005e15i 0.0406967i 0.999793 + 0.0203484i \(0.00647753\pi\)
−0.999793 + 0.0203484i \(0.993522\pi\)
\(968\) 0 0
\(969\) 4.19772e15i 0.157846i
\(970\) 0 0
\(971\) −7.63799e15 + 7.63799e15i −0.283971 + 0.283971i −0.834690 0.550720i \(-0.814353\pi\)
0.550720 + 0.834690i \(0.314353\pi\)
\(972\) 0 0
\(973\) 5.80299e16 + 5.80299e16i 2.13320 + 2.13320i
\(974\) 0 0
\(975\) −2.52203e15 −0.0916694
\(976\) 0 0
\(977\) −3.17074e16 −1.13957 −0.569784 0.821794i \(-0.692973\pi\)
−0.569784 + 0.821794i \(0.692973\pi\)
\(978\) 0 0
\(979\) 3.40137e16 + 3.40137e16i 1.20878 + 1.20878i
\(980\) 0 0
\(981\) −4.73209e15 + 4.73209e15i −0.166293 + 0.166293i
\(982\) 0 0
\(983\) 4.52427e16i 1.57219i 0.618107 + 0.786094i \(0.287900\pi\)
−0.618107 + 0.786094i \(0.712100\pi\)
\(984\) 0 0
\(985\) 1.82712e16i 0.627868i
\(986\) 0 0
\(987\) −2.72171e16 + 2.72171e16i −0.924907 + 0.924907i
\(988\) 0 0
\(989\) −3.86217e16 3.86217e16i −1.29793 1.29793i
\(990\) 0 0
\(991\) −1.90319e16 −0.632523 −0.316261 0.948672i \(-0.602428\pi\)
−0.316261 + 0.948672i \(0.602428\pi\)
\(992\) 0 0
\(993\) −1.49926e16 −0.492785
\(994\) 0 0
\(995\) 2.87242e16 + 2.87242e16i 0.933730 + 0.933730i
\(996\) 0 0
\(997\) 1.23350e16 1.23350e16i 0.396568 0.396568i −0.480453 0.877021i \(-0.659528\pi\)
0.877021 + 0.480453i \(0.159528\pi\)
\(998\) 0 0
\(999\) 1.20772e16i 0.384022i
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 128.12.e.b.33.6 42
4.3 odd 2 128.12.e.a.33.16 42
8.3 odd 2 64.12.e.a.17.6 42
8.5 even 2 16.12.e.a.13.15 yes 42
16.3 odd 4 64.12.e.a.49.6 42
16.5 even 4 inner 128.12.e.b.97.6 42
16.11 odd 4 128.12.e.a.97.16 42
16.13 even 4 16.12.e.a.5.15 42
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.12.e.a.5.15 42 16.13 even 4
16.12.e.a.13.15 yes 42 8.5 even 2
64.12.e.a.17.6 42 8.3 odd 2
64.12.e.a.49.6 42 16.3 odd 4
128.12.e.a.33.16 42 4.3 odd 2
128.12.e.a.97.16 42 16.11 odd 4
128.12.e.b.33.6 42 1.1 even 1 trivial
128.12.e.b.97.6 42 16.5 even 4 inner