Properties

Label 20.11.f.a.13.4
Level $20$
Weight $11$
Character 20.13
Analytic conductor $12.707$
Analytic rank $0$
Dimension $10$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,11,Mod(13,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3]))
 
N = Newforms(chi, 11, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.13");
 
S:= CuspForms(chi, 11);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 11 \)
Character orbit: \([\chi]\) \(=\) 20.f (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(12.7071450535\)
Analytic rank: \(0\)
Dimension: \(10\)
Relative dimension: \(5\) over \(\Q(i)\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} + \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} + 75402 x^{8} + 1918432665 x^{6} + 20025190470928 x^{4} + \cdots + 13\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 2^{22}\cdot 3^{2}\cdot 5^{6} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 13.4
Root \(75.9602i\) of defining polynomial
Character \(\chi\) \(=\) 20.13
Dual form 20.11.f.a.17.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(186.378 + 186.378i) q^{3} +(-473.688 + 3088.89i) q^{5} +(-7250.16 + 7250.16i) q^{7} +10424.8i q^{9} +O(q^{10})\) \(q+(186.378 + 186.378i) q^{3} +(-473.688 + 3088.89i) q^{5} +(-7250.16 + 7250.16i) q^{7} +10424.8i q^{9} -32274.9 q^{11} +(-123469. - 123469. i) q^{13} +(-663988. + 487417. i) q^{15} +(-1.34533e6 + 1.34533e6i) q^{17} +3.80845e6i q^{19} -2.70255e6 q^{21} +(-1.48258e6 - 1.48258e6i) q^{23} +(-9.31686e6 - 2.92634e6i) q^{25} +(9.06250e6 - 9.06250e6i) q^{27} +1.97669e7i q^{29} +3.84620e7 q^{31} +(-6.01534e6 - 6.01534e6i) q^{33} +(-1.89606e7 - 2.58293e7i) q^{35} +(6.70494e7 - 6.70494e7i) q^{37} -4.60238e7i q^{39} +1.28370e8 q^{41} +(-3.08862e7 - 3.08862e7i) q^{43} +(-3.22011e7 - 4.93811e6i) q^{45} +(-1.99277e8 + 1.99277e8i) q^{47} +1.77345e8i q^{49} -5.01480e8 q^{51} +(3.65599e8 + 3.65599e8i) q^{53} +(1.52882e7 - 9.96936e7i) q^{55} +(-7.09813e8 + 7.09813e8i) q^{57} -1.07362e9i q^{59} +2.62425e8 q^{61} +(-7.55816e7 - 7.55816e7i) q^{63} +(4.39867e8 - 3.22895e8i) q^{65} +(-5.30058e8 + 5.30058e8i) q^{67} -5.52641e8i q^{69} +2.34904e9 q^{71} +(1.92115e9 + 1.92115e9i) q^{73} +(-1.19106e9 - 2.28187e9i) q^{75} +(2.33998e8 - 2.33998e8i) q^{77} -3.63376e9i q^{79} +3.99368e9 q^{81} +(-1.23817e9 - 1.23817e9i) q^{83} +(-3.51831e9 - 4.79284e9i) q^{85} +(-3.68413e9 + 3.68413e9i) q^{87} +9.01544e9i q^{89} +1.79034e9 q^{91} +(7.16849e9 + 7.16849e9i) q^{93} +(-1.17639e10 - 1.80402e9i) q^{95} +(-9.05456e9 + 9.05456e9i) q^{97} -3.36460e8i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q + 62 q^{3} + 894 q^{5} + 22286 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 10 q + 62 q^{3} + 894 q^{5} + 22286 q^{7} - 201700 q^{11} + 239298 q^{13} + 213662 q^{15} + 1045442 q^{17} + 4578860 q^{21} - 4097986 q^{23} - 4233934 q^{25} - 4817488 q^{27} + 23221660 q^{31} + 31816220 q^{33} - 55388242 q^{35} - 87811974 q^{37} + 29776460 q^{41} + 156325470 q^{43} - 144135236 q^{45} - 450750018 q^{47} + 1632585820 q^{51} + 701393866 q^{53} - 1301185140 q^{55} - 2564330416 q^{57} + 2991488220 q^{61} + 3352397678 q^{63} - 2867494182 q^{65} - 6990333394 q^{67} + 9915200380 q^{71} + 8401915018 q^{73} - 10170758642 q^{75} - 19825815140 q^{77} + 26071184290 q^{81} + 16998617454 q^{83} - 24280829854 q^{85} - 36065578576 q^{87} + 52347612540 q^{91} + 26277966572 q^{93} - 23431125296 q^{95} - 48945511254 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{4}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 186.378 + 186.378i 0.766989 + 0.766989i 0.977575 0.210586i \(-0.0675372\pi\)
−0.210586 + 0.977575i \(0.567537\pi\)
\(4\) 0 0
\(5\) −473.688 + 3088.89i −0.151580 + 0.988445i
\(6\) 0 0
\(7\) −7250.16 + 7250.16i −0.431378 + 0.431378i −0.889097 0.457719i \(-0.848667\pi\)
0.457719 + 0.889097i \(0.348667\pi\)
\(8\) 0 0
\(9\) 10424.8i 0.176545i
\(10\) 0 0
\(11\) −32274.9 −0.200402 −0.100201 0.994967i \(-0.531949\pi\)
−0.100201 + 0.994967i \(0.531949\pi\)
\(12\) 0 0
\(13\) −123469. 123469.i −0.332537 0.332537i 0.521012 0.853549i \(-0.325555\pi\)
−0.853549 + 0.521012i \(0.825555\pi\)
\(14\) 0 0
\(15\) −663988. + 487417.i −0.874387 + 0.641866i
\(16\) 0 0
\(17\) −1.34533e6 + 1.34533e6i −0.947510 + 0.947510i −0.998689 0.0511797i \(-0.983702\pi\)
0.0511797 + 0.998689i \(0.483702\pi\)
\(18\) 0 0
\(19\) 3.80845e6i 1.53808i 0.639198 + 0.769042i \(0.279266\pi\)
−0.639198 + 0.769042i \(0.720734\pi\)
\(20\) 0 0
\(21\) −2.70255e6 −0.661724
\(22\) 0 0
\(23\) −1.48258e6 1.48258e6i −0.230345 0.230345i 0.582492 0.812837i \(-0.302078\pi\)
−0.812837 + 0.582492i \(0.802078\pi\)
\(24\) 0 0
\(25\) −9.31686e6 2.92634e6i −0.954047 0.299657i
\(26\) 0 0
\(27\) 9.06250e6 9.06250e6i 0.631581 0.631581i
\(28\) 0 0
\(29\) 1.97669e7i 0.963717i 0.876249 + 0.481859i \(0.160038\pi\)
−0.876249 + 0.481859i \(0.839962\pi\)
\(30\) 0 0
\(31\) 3.84620e7 1.34346 0.671729 0.740797i \(-0.265552\pi\)
0.671729 + 0.740797i \(0.265552\pi\)
\(32\) 0 0
\(33\) −6.01534e6 6.01534e6i −0.153706 0.153706i
\(34\) 0 0
\(35\) −1.89606e7 2.58293e7i −0.361005 0.491781i
\(36\) 0 0
\(37\) 6.70494e7 6.70494e7i 0.966910 0.966910i −0.0325595 0.999470i \(-0.510366\pi\)
0.999470 + 0.0325595i \(0.0103658\pi\)
\(38\) 0 0
\(39\) 4.60238e7i 0.510104i
\(40\) 0 0
\(41\) 1.28370e8 1.10801 0.554005 0.832513i \(-0.313099\pi\)
0.554005 + 0.832513i \(0.313099\pi\)
\(42\) 0 0
\(43\) −3.08862e7 3.08862e7i −0.210098 0.210098i 0.594211 0.804309i \(-0.297464\pi\)
−0.804309 + 0.594211i \(0.797464\pi\)
\(44\) 0 0
\(45\) −3.22011e7 4.93811e6i −0.174505 0.0267608i
\(46\) 0 0
\(47\) −1.99277e8 + 1.99277e8i −0.868897 + 0.868897i −0.992350 0.123453i \(-0.960603\pi\)
0.123453 + 0.992350i \(0.460603\pi\)
\(48\) 0 0
\(49\) 1.77345e8i 0.627827i
\(50\) 0 0
\(51\) −5.01480e8 −1.45346
\(52\) 0 0
\(53\) 3.65599e8 + 3.65599e8i 0.874231 + 0.874231i 0.992930 0.118699i \(-0.0378724\pi\)
−0.118699 + 0.992930i \(0.537872\pi\)
\(54\) 0 0
\(55\) 1.52882e7 9.96936e7i 0.0303769 0.198086i
\(56\) 0 0
\(57\) −7.09813e8 + 7.09813e8i −1.17969 + 1.17969i
\(58\) 0 0
\(59\) 1.07362e9i 1.50172i −0.660460 0.750861i \(-0.729639\pi\)
0.660460 0.750861i \(-0.270361\pi\)
\(60\) 0 0
\(61\) 2.62425e8 0.310710 0.155355 0.987859i \(-0.450348\pi\)
0.155355 + 0.987859i \(0.450348\pi\)
\(62\) 0 0
\(63\) −7.55816e7 7.55816e7i −0.0761577 0.0761577i
\(64\) 0 0
\(65\) 4.39867e8 3.22895e8i 0.379100 0.278288i
\(66\) 0 0
\(67\) −5.30058e8 + 5.30058e8i −0.392599 + 0.392599i −0.875613 0.483014i \(-0.839542\pi\)
0.483014 + 0.875613i \(0.339542\pi\)
\(68\) 0 0
\(69\) 5.52641e8i 0.353344i
\(70\) 0 0
\(71\) 2.34904e9 1.30196 0.650980 0.759095i \(-0.274358\pi\)
0.650980 + 0.759095i \(0.274358\pi\)
\(72\) 0 0
\(73\) 1.92115e9 + 1.92115e9i 0.926717 + 0.926717i 0.997492 0.0707753i \(-0.0225473\pi\)
−0.0707753 + 0.997492i \(0.522547\pi\)
\(74\) 0 0
\(75\) −1.19106e9 2.28187e9i −0.501910 0.961578i
\(76\) 0 0
\(77\) 2.33998e8 2.33998e8i 0.0864488 0.0864488i
\(78\) 0 0
\(79\) 3.63376e9i 1.18092i −0.807067 0.590460i \(-0.798946\pi\)
0.807067 0.590460i \(-0.201054\pi\)
\(80\) 0 0
\(81\) 3.99368e9 1.14538
\(82\) 0 0
\(83\) −1.23817e9 1.23817e9i −0.314334 0.314334i 0.532252 0.846586i \(-0.321346\pi\)
−0.846586 + 0.532252i \(0.821346\pi\)
\(84\) 0 0
\(85\) −3.51831e9 4.79284e9i −0.792938 1.08019i
\(86\) 0 0
\(87\) −3.68413e9 + 3.68413e9i −0.739161 + 0.739161i
\(88\) 0 0
\(89\) 9.01544e9i 1.61450i 0.590213 + 0.807248i \(0.299044\pi\)
−0.590213 + 0.807248i \(0.700956\pi\)
\(90\) 0 0
\(91\) 1.79034e9 0.286898
\(92\) 0 0
\(93\) 7.16849e9 + 7.16849e9i 1.03042 + 1.03042i
\(94\) 0 0
\(95\) −1.17639e10 1.80402e9i −1.52031 0.233143i
\(96\) 0 0
\(97\) −9.05456e9 + 9.05456e9i −1.05441 + 1.05441i −0.0559755 + 0.998432i \(0.517827\pi\)
−0.998432 + 0.0559755i \(0.982173\pi\)
\(98\) 0 0
\(99\) 3.36460e8i 0.0353799i
\(100\) 0 0
\(101\) 3.39687e9 0.323200 0.161600 0.986856i \(-0.448334\pi\)
0.161600 + 0.986856i \(0.448334\pi\)
\(102\) 0 0
\(103\) −5.76080e9 5.76080e9i −0.496931 0.496931i 0.413550 0.910481i \(-0.364289\pi\)
−0.910481 + 0.413550i \(0.864289\pi\)
\(104\) 0 0
\(105\) 1.28016e9 8.34787e9i 0.100304 0.654078i
\(106\) 0 0
\(107\) −1.05439e10 + 1.05439e10i −0.751766 + 0.751766i −0.974809 0.223043i \(-0.928401\pi\)
0.223043 + 0.974809i \(0.428401\pi\)
\(108\) 0 0
\(109\) 2.24234e10i 1.45737i −0.684850 0.728684i \(-0.740132\pi\)
0.684850 0.728684i \(-0.259868\pi\)
\(110\) 0 0
\(111\) 2.49931e10 1.48322
\(112\) 0 0
\(113\) 1.43697e10 + 1.43697e10i 0.779930 + 0.779930i 0.979819 0.199889i \(-0.0640580\pi\)
−0.199889 + 0.979819i \(0.564058\pi\)
\(114\) 0 0
\(115\) 5.28180e9 3.87724e9i 0.262599 0.192767i
\(116\) 0 0
\(117\) 1.28714e9 1.28714e9i 0.0587078 0.0587078i
\(118\) 0 0
\(119\) 1.95077e10i 0.817469i
\(120\) 0 0
\(121\) −2.48958e10 −0.959839
\(122\) 0 0
\(123\) 2.39254e10 + 2.39254e10i 0.849832 + 0.849832i
\(124\) 0 0
\(125\) 1.34524e10 2.73926e10i 0.440809 0.897601i
\(126\) 0 0
\(127\) −6.20236e9 + 6.20236e9i −0.187732 + 0.187732i −0.794715 0.606983i \(-0.792380\pi\)
0.606983 + 0.794715i \(0.292380\pi\)
\(128\) 0 0
\(129\) 1.15131e10i 0.322286i
\(130\) 0 0
\(131\) 3.92527e10 1.01745 0.508724 0.860929i \(-0.330117\pi\)
0.508724 + 0.860929i \(0.330117\pi\)
\(132\) 0 0
\(133\) −2.76119e10 2.76119e10i −0.663495 0.663495i
\(134\) 0 0
\(135\) 2.37003e10 + 3.22859e10i 0.528548 + 0.720018i
\(136\) 0 0
\(137\) 1.80246e9 1.80246e9i 0.0373477 0.0373477i −0.688186 0.725534i \(-0.741593\pi\)
0.725534 + 0.688186i \(0.241593\pi\)
\(138\) 0 0
\(139\) 1.12218e10i 0.216266i −0.994136 0.108133i \(-0.965513\pi\)
0.994136 0.108133i \(-0.0344873\pi\)
\(140\) 0 0
\(141\) −7.42819e10 −1.33287
\(142\) 0 0
\(143\) 3.98494e9 + 3.98494e9i 0.0666409 + 0.0666409i
\(144\) 0 0
\(145\) −6.10579e10 9.36337e9i −0.952581 0.146080i
\(146\) 0 0
\(147\) −3.30534e10 + 3.30534e10i −0.481536 + 0.481536i
\(148\) 0 0
\(149\) 1.11142e11i 1.51338i 0.653776 + 0.756688i \(0.273184\pi\)
−0.653776 + 0.756688i \(0.726816\pi\)
\(150\) 0 0
\(151\) −8.52396e9 −0.108582 −0.0542908 0.998525i \(-0.517290\pi\)
−0.0542908 + 0.998525i \(0.517290\pi\)
\(152\) 0 0
\(153\) −1.40248e10 1.40248e10i −0.167278 0.167278i
\(154\) 0 0
\(155\) −1.82190e10 + 1.18805e11i −0.203641 + 1.32793i
\(156\) 0 0
\(157\) 7.61882e10 7.61882e10i 0.798711 0.798711i −0.184181 0.982892i \(-0.558963\pi\)
0.982892 + 0.184181i \(0.0589634\pi\)
\(158\) 0 0
\(159\) 1.36280e11i 1.34105i
\(160\) 0 0
\(161\) 2.14979e10 0.198731
\(162\) 0 0
\(163\) −1.36707e11 1.36707e11i −1.18810 1.18810i −0.977592 0.210511i \(-0.932487\pi\)
−0.210511 0.977592i \(-0.567513\pi\)
\(164\) 0 0
\(165\) 2.14301e10 1.57313e10i 0.175229 0.128631i
\(166\) 0 0
\(167\) 1.44257e11 1.44257e11i 1.11059 1.11059i 0.117520 0.993071i \(-0.462506\pi\)
0.993071 0.117520i \(-0.0374943\pi\)
\(168\) 0 0
\(169\) 1.07369e11i 0.778838i
\(170\) 0 0
\(171\) −3.97024e10 −0.271541
\(172\) 0 0
\(173\) 2.24683e10 + 2.24683e10i 0.144990 + 0.144990i 0.775876 0.630885i \(-0.217308\pi\)
−0.630885 + 0.775876i \(0.717308\pi\)
\(174\) 0 0
\(175\) 8.87653e10 4.63323e10i 0.540820 0.282289i
\(176\) 0 0
\(177\) 2.00099e11 2.00099e11i 1.15180 1.15180i
\(178\) 0 0
\(179\) 5.80959e10i 0.316141i 0.987428 + 0.158070i \(0.0505273\pi\)
−0.987428 + 0.158070i \(0.949473\pi\)
\(180\) 0 0
\(181\) −1.27285e11 −0.655214 −0.327607 0.944814i \(-0.606242\pi\)
−0.327607 + 0.944814i \(0.606242\pi\)
\(182\) 0 0
\(183\) 4.89103e10 + 4.89103e10i 0.238311 + 0.238311i
\(184\) 0 0
\(185\) 1.75348e11 + 2.38869e11i 0.809173 + 1.10230i
\(186\) 0 0
\(187\) 4.34203e10 4.34203e10i 0.189882 0.189882i
\(188\) 0 0
\(189\) 1.31409e11i 0.544900i
\(190\) 0 0
\(191\) −2.09552e10 −0.0824375 −0.0412187 0.999150i \(-0.513124\pi\)
−0.0412187 + 0.999150i \(0.513124\pi\)
\(192\) 0 0
\(193\) −1.68304e11 1.68304e11i −0.628502 0.628502i 0.319189 0.947691i \(-0.396589\pi\)
−0.947691 + 0.319189i \(0.896589\pi\)
\(194\) 0 0
\(195\) 1.42162e11 + 2.18009e10i 0.504210 + 0.0773217i
\(196\) 0 0
\(197\) −1.52718e11 + 1.52718e11i −0.514705 + 0.514705i −0.915965 0.401259i \(-0.868573\pi\)
0.401259 + 0.915965i \(0.368573\pi\)
\(198\) 0 0
\(199\) 3.50951e11i 1.12456i −0.826948 0.562278i \(-0.809925\pi\)
0.826948 0.562278i \(-0.190075\pi\)
\(200\) 0 0
\(201\) −1.97583e11 −0.602239
\(202\) 0 0
\(203\) −1.43314e11 1.43314e11i −0.415726 0.415726i
\(204\) 0 0
\(205\) −6.08073e10 + 3.96520e11i −0.167952 + 1.09521i
\(206\) 0 0
\(207\) 1.54556e10 1.54556e10i 0.0406662 0.0406662i
\(208\) 0 0
\(209\) 1.22917e11i 0.308235i
\(210\) 0 0
\(211\) −6.84043e11 −1.63558 −0.817788 0.575519i \(-0.804800\pi\)
−0.817788 + 0.575519i \(0.804800\pi\)
\(212\) 0 0
\(213\) 4.37809e11 + 4.37809e11i 0.998590 + 0.998590i
\(214\) 0 0
\(215\) 1.10035e11 8.07737e10i 0.239517 0.175824i
\(216\) 0 0
\(217\) −2.78856e11 + 2.78856e11i −0.579537 + 0.579537i
\(218\) 0 0
\(219\) 7.16122e11i 1.42156i
\(220\) 0 0
\(221\) 3.32212e11 0.630164
\(222\) 0 0
\(223\) −3.61330e11 3.61330e11i −0.655209 0.655209i 0.299033 0.954243i \(-0.403336\pi\)
−0.954243 + 0.299033i \(0.903336\pi\)
\(224\) 0 0
\(225\) 3.05066e10 9.71266e10i 0.0529031 0.168432i
\(226\) 0 0
\(227\) −2.54034e11 + 2.54034e11i −0.421466 + 0.421466i −0.885708 0.464242i \(-0.846327\pi\)
0.464242 + 0.885708i \(0.346327\pi\)
\(228\) 0 0
\(229\) 1.11963e11i 0.177785i −0.996041 0.0888926i \(-0.971667\pi\)
0.996041 0.0888926i \(-0.0283328\pi\)
\(230\) 0 0
\(231\) 8.72244e10 0.132611
\(232\) 0 0
\(233\) 2.00272e11 + 2.00272e11i 0.291636 + 0.291636i 0.837726 0.546090i \(-0.183884\pi\)
−0.546090 + 0.837726i \(0.683884\pi\)
\(234\) 0 0
\(235\) −5.21150e11 7.09941e11i −0.727149 0.990565i
\(236\) 0 0
\(237\) 6.77254e11 6.77254e11i 0.905754 0.905754i
\(238\) 0 0
\(239\) 4.47488e11i 0.573842i −0.957954 0.286921i \(-0.907368\pi\)
0.957954 0.286921i \(-0.0926316\pi\)
\(240\) 0 0
\(241\) 1.59973e12 1.96771 0.983854 0.178975i \(-0.0572780\pi\)
0.983854 + 0.178975i \(0.0572780\pi\)
\(242\) 0 0
\(243\) 2.09205e11 + 2.09205e11i 0.246911 + 0.246911i
\(244\) 0 0
\(245\) −5.47801e11 8.40065e10i −0.620572 0.0951661i
\(246\) 0 0
\(247\) 4.70224e11 4.70224e11i 0.511470 0.511470i
\(248\) 0 0
\(249\) 4.61538e11i 0.482181i
\(250\) 0 0
\(251\) 6.49953e11 0.652399 0.326199 0.945301i \(-0.394232\pi\)
0.326199 + 0.945301i \(0.394232\pi\)
\(252\) 0 0
\(253\) 4.78500e10 + 4.78500e10i 0.0461614 + 0.0461614i
\(254\) 0 0
\(255\) 2.37545e11 1.54902e12i 0.220316 1.43666i
\(256\) 0 0
\(257\) −6.14568e11 + 6.14568e11i −0.548156 + 0.548156i −0.925907 0.377751i \(-0.876697\pi\)
0.377751 + 0.925907i \(0.376697\pi\)
\(258\) 0 0
\(259\) 9.72238e11i 0.834207i
\(260\) 0 0
\(261\) −2.06067e11 −0.170140
\(262\) 0 0
\(263\) 1.57274e11 + 1.57274e11i 0.124991 + 0.124991i 0.766835 0.641844i \(-0.221830\pi\)
−0.641844 + 0.766835i \(0.721830\pi\)
\(264\) 0 0
\(265\) −1.30248e12 + 9.56117e11i −0.996645 + 0.731613i
\(266\) 0 0
\(267\) −1.68028e12 + 1.68028e12i −1.23830 + 1.23830i
\(268\) 0 0
\(269\) 1.16275e12i 0.825517i −0.910840 0.412759i \(-0.864565\pi\)
0.910840 0.412759i \(-0.135435\pi\)
\(270\) 0 0
\(271\) 2.12479e12 1.45368 0.726841 0.686806i \(-0.240988\pi\)
0.726841 + 0.686806i \(0.240988\pi\)
\(272\) 0 0
\(273\) 3.33680e11 + 3.33680e11i 0.220048 + 0.220048i
\(274\) 0 0
\(275\) 3.00701e11 + 9.44473e10i 0.191193 + 0.0600518i
\(276\) 0 0
\(277\) −1.10855e12 + 1.10855e12i −0.679764 + 0.679764i −0.959947 0.280183i \(-0.909605\pi\)
0.280183 + 0.959947i \(0.409605\pi\)
\(278\) 0 0
\(279\) 4.00960e11i 0.237181i
\(280\) 0 0
\(281\) −1.29651e12 −0.740020 −0.370010 0.929028i \(-0.620646\pi\)
−0.370010 + 0.929028i \(0.620646\pi\)
\(282\) 0 0
\(283\) 1.32307e12 + 1.32307e12i 0.728870 + 0.728870i 0.970395 0.241525i \(-0.0776475\pi\)
−0.241525 + 0.970395i \(0.577648\pi\)
\(284\) 0 0
\(285\) −1.85630e12 2.52876e12i −0.987245 1.34488i
\(286\) 0 0
\(287\) −9.30702e11 + 9.30702e11i −0.477971 + 0.477971i
\(288\) 0 0
\(289\) 1.60382e12i 0.795550i
\(290\) 0 0
\(291\) −3.37515e12 −1.61744
\(292\) 0 0
\(293\) −5.23005e11 5.23005e11i −0.242196 0.242196i 0.575562 0.817758i \(-0.304783\pi\)
−0.817758 + 0.575562i \(0.804783\pi\)
\(294\) 0 0
\(295\) 3.31629e12 + 5.08560e11i 1.48437 + 0.227631i
\(296\) 0 0
\(297\) −2.92491e11 + 2.92491e11i −0.126570 + 0.126570i
\(298\) 0 0
\(299\) 3.66103e11i 0.153196i
\(300\) 0 0
\(301\) 4.47861e11 0.181263
\(302\) 0 0
\(303\) 6.33103e11 + 6.33103e11i 0.247891 + 0.247891i
\(304\) 0 0
\(305\) −1.24307e11 + 8.10601e11i −0.0470975 + 0.307120i
\(306\) 0 0
\(307\) −2.47818e12 + 2.47818e12i −0.908742 + 0.908742i −0.996171 0.0874289i \(-0.972135\pi\)
0.0874289 + 0.996171i \(0.472135\pi\)
\(308\) 0 0
\(309\) 2.14738e12i 0.762282i
\(310\) 0 0
\(311\) 5.13390e12 1.76460 0.882298 0.470691i \(-0.155995\pi\)
0.882298 + 0.470691i \(0.155995\pi\)
\(312\) 0 0
\(313\) 3.01009e12 + 3.01009e12i 1.00198 + 1.00198i 0.999998 + 0.00198008i \(0.000630280\pi\)
0.00198008 + 0.999998i \(0.499370\pi\)
\(314\) 0 0
\(315\) 2.69266e11 1.97661e11i 0.0868216 0.0637337i
\(316\) 0 0
\(317\) 3.19749e12 3.19749e12i 0.998878 0.998878i −0.00112099 0.999999i \(-0.500357\pi\)
0.999999 + 0.00112099i \(0.000356822\pi\)
\(318\) 0 0
\(319\) 6.37976e11i 0.193130i
\(320\) 0 0
\(321\) −3.93031e12 −1.15319
\(322\) 0 0
\(323\) −5.12362e12 5.12362e12i −1.45735 1.45735i
\(324\) 0 0
\(325\) 7.89029e11 + 1.51165e12i 0.217609 + 0.416903i
\(326\) 0 0
\(327\) 4.17924e12 4.17924e12i 1.11779 1.11779i
\(328\) 0 0
\(329\) 2.88958e12i 0.749646i
\(330\) 0 0
\(331\) 4.37117e12 1.10016 0.550082 0.835110i \(-0.314596\pi\)
0.550082 + 0.835110i \(0.314596\pi\)
\(332\) 0 0
\(333\) 6.98978e11 + 6.98978e11i 0.170703 + 0.170703i
\(334\) 0 0
\(335\) −1.38621e12 1.88837e12i −0.328553 0.447573i
\(336\) 0 0
\(337\) 2.87588e12 2.87588e12i 0.661640 0.661640i −0.294127 0.955766i \(-0.595029\pi\)
0.955766 + 0.294127i \(0.0950287\pi\)
\(338\) 0 0
\(339\) 5.35641e12i 1.19640i
\(340\) 0 0
\(341\) −1.24136e12 −0.269231
\(342\) 0 0
\(343\) −3.33378e12 3.33378e12i −0.702208 0.702208i
\(344\) 0 0
\(345\) 1.70705e12 + 2.61779e11i 0.349261 + 0.0535599i
\(346\) 0 0
\(347\) −2.91536e12 + 2.91536e12i −0.579488 + 0.579488i −0.934762 0.355275i \(-0.884387\pi\)
0.355275 + 0.934762i \(0.384387\pi\)
\(348\) 0 0
\(349\) 8.62234e12i 1.66532i 0.553784 + 0.832660i \(0.313183\pi\)
−0.553784 + 0.832660i \(0.686817\pi\)
\(350\) 0 0
\(351\) −2.23787e12 −0.420048
\(352\) 0 0
\(353\) 1.32642e12 + 1.32642e12i 0.241996 + 0.241996i 0.817676 0.575679i \(-0.195262\pi\)
−0.575679 + 0.817676i \(0.695262\pi\)
\(354\) 0 0
\(355\) −1.11271e12 + 7.25591e12i −0.197351 + 1.28692i
\(356\) 0 0
\(357\) 3.63581e12 3.63581e12i 0.626990 0.626990i
\(358\) 0 0
\(359\) 4.42823e12i 0.742605i 0.928512 + 0.371303i \(0.121089\pi\)
−0.928512 + 0.371303i \(0.878911\pi\)
\(360\) 0 0
\(361\) −8.37322e12 −1.36570
\(362\) 0 0
\(363\) −4.64003e12 4.64003e12i −0.736186 0.736186i
\(364\) 0 0
\(365\) −6.84425e12 + 5.02420e12i −1.05648 + 0.775537i
\(366\) 0 0
\(367\) 2.04313e12 2.04313e12i 0.306878 0.306878i −0.536819 0.843697i \(-0.680374\pi\)
0.843697 + 0.536819i \(0.180374\pi\)
\(368\) 0 0
\(369\) 1.33823e12i 0.195614i
\(370\) 0 0
\(371\) −5.30131e12 −0.754247
\(372\) 0 0
\(373\) 3.01411e12 + 3.01411e12i 0.417461 + 0.417461i 0.884328 0.466867i \(-0.154617\pi\)
−0.466867 + 0.884328i \(0.654617\pi\)
\(374\) 0 0
\(375\) 7.61263e12 2.59815e12i 1.02655 0.350354i
\(376\) 0 0
\(377\) 2.44060e12 2.44060e12i 0.320472 0.320472i
\(378\) 0 0
\(379\) 7.63889e11i 0.0976864i 0.998806 + 0.0488432i \(0.0155535\pi\)
−0.998806 + 0.0488432i \(0.984447\pi\)
\(380\) 0 0
\(381\) −2.31197e12 −0.287977
\(382\) 0 0
\(383\) −9.82634e12 9.82634e12i −1.19233 1.19233i −0.976410 0.215924i \(-0.930724\pi\)
−0.215924 0.976410i \(-0.569276\pi\)
\(384\) 0 0
\(385\) 6.11953e11 + 8.33637e11i 0.0723459 + 0.0985538i
\(386\) 0 0
\(387\) 3.21983e11 3.21983e11i 0.0370919 0.0370919i
\(388\) 0 0
\(389\) 1.58595e12i 0.178050i 0.996029 + 0.0890250i \(0.0283751\pi\)
−0.996029 + 0.0890250i \(0.971625\pi\)
\(390\) 0 0
\(391\) 3.98911e12 0.436508
\(392\) 0 0
\(393\) 7.31585e12 + 7.31585e12i 0.780372 + 0.780372i
\(394\) 0 0
\(395\) 1.12243e13 + 1.72127e12i 1.16728 + 0.179004i
\(396\) 0 0
\(397\) 3.83039e12 3.83039e12i 0.388410 0.388410i −0.485710 0.874120i \(-0.661439\pi\)
0.874120 + 0.485710i \(0.161439\pi\)
\(398\) 0 0
\(399\) 1.02925e13i 1.01779i
\(400\) 0 0
\(401\) −6.10170e11 −0.0588477 −0.0294238 0.999567i \(-0.509367\pi\)
−0.0294238 + 0.999567i \(0.509367\pi\)
\(402\) 0 0
\(403\) −4.74885e12 4.74885e12i −0.446749 0.446749i
\(404\) 0 0
\(405\) −1.89176e12 + 1.23360e13i −0.173616 + 1.13214i
\(406\) 0 0
\(407\) −2.16401e12 + 2.16401e12i −0.193770 + 0.193770i
\(408\) 0 0
\(409\) 1.67942e13i 1.46738i −0.679486 0.733689i \(-0.737797\pi\)
0.679486 0.733689i \(-0.262203\pi\)
\(410\) 0 0
\(411\) 6.71880e11 0.0572905
\(412\) 0 0
\(413\) 7.78390e12 + 7.78390e12i 0.647809 + 0.647809i
\(414\) 0 0
\(415\) 4.41109e12 3.23807e12i 0.358348 0.263055i
\(416\) 0 0
\(417\) 2.09150e12 2.09150e12i 0.165874 0.165874i
\(418\) 0 0
\(419\) 1.70046e12i 0.131673i 0.997830 + 0.0658363i \(0.0209715\pi\)
−0.997830 + 0.0658363i \(0.979028\pi\)
\(420\) 0 0
\(421\) 1.12084e12 0.0847487 0.0423743 0.999102i \(-0.486508\pi\)
0.0423743 + 0.999102i \(0.486508\pi\)
\(422\) 0 0
\(423\) −2.07743e12 2.07743e12i −0.153400 0.153400i
\(424\) 0 0
\(425\) 1.64711e13 8.59735e12i 1.18790 0.620040i
\(426\) 0 0
\(427\) −1.90262e12 + 1.90262e12i −0.134033 + 0.134033i
\(428\) 0 0
\(429\) 1.48541e12i 0.102226i
\(430\) 0 0
\(431\) −1.32023e12 −0.0887695 −0.0443847 0.999015i \(-0.514133\pi\)
−0.0443847 + 0.999015i \(0.514133\pi\)
\(432\) 0 0
\(433\) 1.74616e12 + 1.74616e12i 0.114721 + 0.114721i 0.762137 0.647416i \(-0.224150\pi\)
−0.647416 + 0.762137i \(0.724150\pi\)
\(434\) 0 0
\(435\) −9.63475e12 1.31250e13i −0.618578 0.842662i
\(436\) 0 0
\(437\) 5.64632e12 5.64632e12i 0.354289 0.354289i
\(438\) 0 0
\(439\) 3.77138e12i 0.231301i 0.993290 + 0.115651i \(0.0368953\pi\)
−0.993290 + 0.115651i \(0.963105\pi\)
\(440\) 0 0
\(441\) −1.84879e12 −0.110840
\(442\) 0 0
\(443\) 5.11940e12 + 5.11940e12i 0.300055 + 0.300055i 0.841035 0.540980i \(-0.181947\pi\)
−0.540980 + 0.841035i \(0.681947\pi\)
\(444\) 0 0
\(445\) −2.78477e13 4.27051e12i −1.59584 0.244726i
\(446\) 0 0
\(447\) −2.07145e13 + 2.07145e13i −1.16074 + 1.16074i
\(448\) 0 0
\(449\) 8.94777e11i 0.0490324i −0.999699 0.0245162i \(-0.992195\pi\)
0.999699 0.0245162i \(-0.00780453\pi\)
\(450\) 0 0
\(451\) −4.14312e12 −0.222047
\(452\) 0 0
\(453\) −1.58868e12 1.58868e12i −0.0832810 0.0832810i
\(454\) 0 0
\(455\) −8.48061e11 + 5.53015e12i −0.0434881 + 0.283583i
\(456\) 0 0
\(457\) −1.40263e13 + 1.40263e13i −0.703660 + 0.703660i −0.965194 0.261534i \(-0.915772\pi\)
0.261534 + 0.965194i \(0.415772\pi\)
\(458\) 0 0
\(459\) 2.43841e13i 1.19686i
\(460\) 0 0
\(461\) 1.77271e13 0.851397 0.425699 0.904865i \(-0.360028\pi\)
0.425699 + 0.904865i \(0.360028\pi\)
\(462\) 0 0
\(463\) 7.62363e12 + 7.62363e12i 0.358308 + 0.358308i 0.863189 0.504881i \(-0.168464\pi\)
−0.504881 + 0.863189i \(0.668464\pi\)
\(464\) 0 0
\(465\) −2.55383e13 + 1.87471e13i −1.17470 + 0.862320i
\(466\) 0 0
\(467\) −1.07668e13 + 1.07668e13i −0.484733 + 0.484733i −0.906639 0.421906i \(-0.861361\pi\)
0.421906 + 0.906639i \(0.361361\pi\)
\(468\) 0 0
\(469\) 7.68602e12i 0.338717i
\(470\) 0 0
\(471\) 2.83997e13 1.22521
\(472\) 0 0
\(473\) 9.96849e11 + 9.96849e11i 0.0421040 + 0.0421040i
\(474\) 0 0
\(475\) 1.11448e13 3.54828e13i 0.460898 1.46740i
\(476\) 0 0
\(477\) −3.81131e12 + 3.81131e12i −0.154341 + 0.154341i
\(478\) 0 0
\(479\) 2.92794e13i 1.16114i 0.814211 + 0.580569i \(0.197170\pi\)
−0.814211 + 0.580569i \(0.802830\pi\)
\(480\) 0 0
\(481\) −1.65570e13 −0.643067
\(482\) 0 0
\(483\) 4.00674e12 + 4.00674e12i 0.152425 + 0.152425i
\(484\) 0 0
\(485\) −2.36795e13 3.22576e13i −0.882397 1.20205i
\(486\) 0 0
\(487\) 1.67002e13 1.67002e13i 0.609644 0.609644i −0.333209 0.942853i \(-0.608131\pi\)
0.942853 + 0.333209i \(0.108131\pi\)
\(488\) 0 0
\(489\) 5.09586e13i 1.82252i
\(490\) 0 0
\(491\) 3.55034e13 1.24412 0.622059 0.782970i \(-0.286296\pi\)
0.622059 + 0.782970i \(0.286296\pi\)
\(492\) 0 0
\(493\) −2.65930e13 2.65930e13i −0.913131 0.913131i
\(494\) 0 0
\(495\) 1.03929e12 + 1.59377e11i 0.0349711 + 0.00536290i
\(496\) 0 0
\(497\) −1.70309e13 + 1.70309e13i −0.561637 + 0.561637i
\(498\) 0 0
\(499\) 1.76933e13i 0.571883i −0.958247 0.285941i \(-0.907694\pi\)
0.958247 0.285941i \(-0.0923062\pi\)
\(500\) 0 0
\(501\) 5.37727e13 1.70362
\(502\) 0 0
\(503\) 2.72856e13 + 2.72856e13i 0.847411 + 0.847411i 0.989809 0.142399i \(-0.0454815\pi\)
−0.142399 + 0.989809i \(0.545481\pi\)
\(504\) 0 0
\(505\) −1.60906e12 + 1.04926e13i −0.0489908 + 0.319466i
\(506\) 0 0
\(507\) 2.00114e13 2.00114e13i 0.597361 0.597361i
\(508\) 0 0
\(509\) 6.27994e12i 0.183809i −0.995768 0.0919045i \(-0.970705\pi\)
0.995768 0.0919045i \(-0.0292954\pi\)
\(510\) 0 0
\(511\) −2.78573e13 −0.799530
\(512\) 0 0
\(513\) 3.45141e13 + 3.45141e13i 0.971425 + 0.971425i
\(514\) 0 0
\(515\) 2.05233e13 1.50657e13i 0.566514 0.415864i
\(516\) 0 0
\(517\) 6.43165e12 6.43165e12i 0.174128 0.174128i
\(518\) 0 0
\(519\) 8.37521e12i 0.222412i
\(520\) 0 0
\(521\) −2.02225e13 −0.526800 −0.263400 0.964687i \(-0.584844\pi\)
−0.263400 + 0.964687i \(0.584844\pi\)
\(522\) 0 0
\(523\) 9.99902e12 + 9.99902e12i 0.255534 + 0.255534i 0.823235 0.567701i \(-0.192167\pi\)
−0.567701 + 0.823235i \(0.692167\pi\)
\(524\) 0 0
\(525\) 2.51793e13 + 7.90858e12i 0.631316 + 0.198290i
\(526\) 0 0
\(527\) −5.17441e13 + 5.17441e13i −1.27294 + 1.27294i
\(528\) 0 0
\(529\) 3.70304e13i 0.893883i
\(530\) 0 0
\(531\) 1.11923e13 0.265122
\(532\) 0 0
\(533\) −1.58496e13 1.58496e13i −0.368454 0.368454i
\(534\) 0 0
\(535\) −2.75744e13 3.75635e13i −0.629126 0.857032i
\(536\) 0 0
\(537\) −1.08278e13 + 1.08278e13i −0.242476 + 0.242476i
\(538\) 0 0
\(539\) 5.72380e12i 0.125817i
\(540\) 0 0
\(541\) −3.80088e13 −0.820159 −0.410079 0.912050i \(-0.634499\pi\)
−0.410079 + 0.912050i \(0.634499\pi\)
\(542\) 0 0
\(543\) −2.37231e13 2.37231e13i −0.502542 0.502542i
\(544\) 0 0
\(545\) 6.92635e13 + 1.06217e13i 1.44053 + 0.220908i
\(546\) 0 0
\(547\) 1.19171e13 1.19171e13i 0.243352 0.243352i −0.574883 0.818236i \(-0.694952\pi\)
0.818236 + 0.574883i \(0.194952\pi\)
\(548\) 0 0
\(549\) 2.73573e12i 0.0548544i
\(550\) 0 0
\(551\) −7.52814e13 −1.48228
\(552\) 0 0
\(553\) 2.63454e13 + 2.63454e13i 0.509423 + 0.509423i
\(554\) 0 0
\(555\) −1.18389e13 + 7.72010e13i −0.224827 + 1.46608i
\(556\) 0 0
\(557\) 2.13146e13 2.13146e13i 0.397559 0.397559i −0.479812 0.877371i \(-0.659295\pi\)
0.877371 + 0.479812i \(0.159295\pi\)
\(558\) 0 0
\(559\) 7.62696e12i 0.139731i
\(560\) 0 0
\(561\) 1.61852e13 0.291276
\(562\) 0 0
\(563\) −3.72944e13 3.72944e13i −0.659328 0.659328i 0.295893 0.955221i \(-0.404383\pi\)
−0.955221 + 0.295893i \(0.904383\pi\)
\(564\) 0 0
\(565\) −5.11932e13 + 3.75797e13i −0.889140 + 0.652696i
\(566\) 0 0
\(567\) −2.89549e13 + 2.89549e13i −0.494090 + 0.494090i
\(568\) 0 0
\(569\) 7.65415e13i 1.28332i 0.766988 + 0.641661i \(0.221754\pi\)
−0.766988 + 0.641661i \(0.778246\pi\)
\(570\) 0 0
\(571\) −6.05290e12 −0.0997202 −0.0498601 0.998756i \(-0.515878\pi\)
−0.0498601 + 0.998756i \(0.515878\pi\)
\(572\) 0 0
\(573\) −3.90559e12 3.90559e12i −0.0632287 0.0632287i
\(574\) 0 0
\(575\) 9.47444e12 + 1.81515e13i 0.150735 + 0.288784i
\(576\) 0 0
\(577\) −3.45547e13 + 3.45547e13i −0.540292 + 0.540292i −0.923615 0.383323i \(-0.874780\pi\)
0.383323 + 0.923615i \(0.374780\pi\)
\(578\) 0 0
\(579\) 6.27363e13i 0.964109i
\(580\) 0 0
\(581\) 1.79539e13 0.271193
\(582\) 0 0
\(583\) −1.17997e13 1.17997e13i −0.175197 0.175197i
\(584\) 0 0
\(585\) 3.36613e12 + 4.58553e12i 0.0491305 + 0.0669284i
\(586\) 0 0
\(587\) 8.97080e13 8.97080e13i 1.28718 1.28718i 0.350694 0.936490i \(-0.385946\pi\)
0.936490 0.350694i \(-0.114054\pi\)
\(588\) 0 0
\(589\) 1.46481e14i 2.06635i
\(590\) 0 0
\(591\) −5.69266e13 −0.789547
\(592\) 0 0
\(593\) −5.66464e13 5.66464e13i −0.772501 0.772501i 0.206042 0.978543i \(-0.433942\pi\)
−0.978543 + 0.206042i \(0.933942\pi\)
\(594\) 0 0
\(595\) 6.02572e13 + 9.24057e12i 0.808023 + 0.123912i
\(596\) 0 0
\(597\) 6.54097e13 6.54097e13i 0.862523 0.862523i
\(598\) 0 0
\(599\) 2.21070e13i 0.286679i 0.989674 + 0.143339i \(0.0457840\pi\)
−0.989674 + 0.143339i \(0.954216\pi\)
\(600\) 0 0
\(601\) −1.42854e12 −0.0182188 −0.00910942 0.999959i \(-0.502900\pi\)
−0.00910942 + 0.999959i \(0.502900\pi\)
\(602\) 0 0
\(603\) −5.52576e12 5.52576e12i −0.0693115 0.0693115i
\(604\) 0 0
\(605\) 1.17928e13 7.69003e13i 0.145493 0.948748i
\(606\) 0 0
\(607\) 1.10881e14 1.10881e14i 1.34560 1.34560i 0.455220 0.890379i \(-0.349561\pi\)
0.890379 0.455220i \(-0.150439\pi\)
\(608\) 0 0
\(609\) 5.34211e13i 0.637715i
\(610\) 0 0
\(611\) 4.92090e13 0.577881
\(612\) 0 0
\(613\) 6.08895e13 + 6.08895e13i 0.703460 + 0.703460i 0.965152 0.261691i \(-0.0842802\pi\)
−0.261691 + 0.965152i \(0.584280\pi\)
\(614\) 0 0
\(615\) −8.52360e13 + 6.25697e13i −0.968829 + 0.711194i
\(616\) 0 0
\(617\) 8.12126e13 8.12126e13i 0.908234 0.908234i −0.0878960 0.996130i \(-0.528014\pi\)
0.996130 + 0.0878960i \(0.0280143\pi\)
\(618\) 0 0
\(619\) 9.32089e13i 1.02566i −0.858490 0.512831i \(-0.828597\pi\)
0.858490 0.512831i \(-0.171403\pi\)
\(620\) 0 0
\(621\) −2.68717e13 −0.290963
\(622\) 0 0
\(623\) −6.53634e13 6.53634e13i −0.696457 0.696457i
\(624\) 0 0
\(625\) 7.82405e13 + 5.45287e13i 0.820411 + 0.571774i
\(626\) 0 0
\(627\) 2.29091e13 2.29091e13i 0.236413 0.236413i
\(628\) 0 0
\(629\) 1.80407e14i 1.83231i
\(630\) 0 0
\(631\) 5.46630e13 0.546445 0.273222 0.961951i \(-0.411910\pi\)
0.273222 + 0.961951i \(0.411910\pi\)
\(632\) 0 0
\(633\) −1.27491e14 1.27491e14i −1.25447 1.25447i
\(634\) 0 0
\(635\) −1.62204e13 2.20964e13i −0.157106 0.214019i
\(636\) 0 0
\(637\) 2.18966e13 2.18966e13i 0.208776 0.208776i
\(638\) 0 0
\(639\) 2.44883e13i 0.229855i
\(640\) 0 0
\(641\) −4.03783e12 −0.0373128 −0.0186564 0.999826i \(-0.505939\pi\)
−0.0186564 + 0.999826i \(0.505939\pi\)
\(642\) 0 0
\(643\) 8.71348e13 + 8.71348e13i 0.792751 + 0.792751i 0.981941 0.189190i \(-0.0605861\pi\)
−0.189190 + 0.981941i \(0.560586\pi\)
\(644\) 0 0
\(645\) 3.55626e13 + 5.45360e12i 0.318562 + 0.0488522i
\(646\) 0 0
\(647\) −1.50155e14 + 1.50155e14i −1.32440 + 1.32440i −0.414225 + 0.910174i \(0.635947\pi\)
−0.910174 + 0.414225i \(0.864053\pi\)
\(648\) 0 0
\(649\) 3.46509e13i 0.300947i
\(650\) 0 0
\(651\) −1.03945e14 −0.888998
\(652\) 0 0
\(653\) −1.04798e13 1.04798e13i −0.0882644 0.0882644i 0.661596 0.749860i \(-0.269879\pi\)
−0.749860 + 0.661596i \(0.769879\pi\)
\(654\) 0 0
\(655\) −1.85935e13 + 1.21247e14i −0.154225 + 1.00569i
\(656\) 0 0
\(657\) −2.00276e13 + 2.00276e13i −0.163607 + 0.163607i
\(658\) 0 0
\(659\) 1.12979e13i 0.0909013i −0.998967 0.0454507i \(-0.985528\pi\)
0.998967 0.0454507i \(-0.0144724\pi\)
\(660\) 0 0
\(661\) −9.56609e13 −0.758101 −0.379050 0.925376i \(-0.623749\pi\)
−0.379050 + 0.925376i \(0.623749\pi\)
\(662\) 0 0
\(663\) 6.19171e13 + 6.19171e13i 0.483329 + 0.483329i
\(664\) 0 0
\(665\) 9.83695e13 7.22107e13i 0.756401 0.555256i
\(666\) 0 0
\(667\) 2.93060e13 2.93060e13i 0.221987 0.221987i
\(668\) 0 0
\(669\) 1.34688e14i 1.00508i
\(670\) 0 0
\(671\) −8.46972e12 −0.0622668
\(672\) 0 0
\(673\) 7.19067e13 + 7.19067e13i 0.520827 + 0.520827i 0.917821 0.396994i \(-0.129947\pi\)
−0.396994 + 0.917821i \(0.629947\pi\)
\(674\) 0 0
\(675\) −1.10954e14 + 5.79141e13i −0.791816 + 0.413300i
\(676\) 0 0
\(677\) −1.71244e14 + 1.71244e14i −1.20413 + 1.20413i −0.231227 + 0.972900i \(0.574274\pi\)
−0.972900 + 0.231227i \(0.925726\pi\)
\(678\) 0 0
\(679\) 1.31294e14i 0.909696i
\(680\) 0 0
\(681\) −9.46929e13 −0.646520
\(682\) 0 0
\(683\) −1.53293e14 1.53293e14i −1.03138 1.03138i −0.999491 0.0318872i \(-0.989848\pi\)
−0.0318872 0.999491i \(-0.510152\pi\)
\(684\) 0 0
\(685\) 4.71381e12 + 6.42142e12i 0.0312550 + 0.0425773i
\(686\) 0 0
\(687\) 2.08674e13 2.08674e13i 0.136359 0.136359i
\(688\) 0 0
\(689\) 9.02801e13i 0.581428i
\(690\) 0 0
\(691\) 5.85081e13 0.371386 0.185693 0.982608i \(-0.440547\pi\)
0.185693 + 0.982608i \(0.440547\pi\)
\(692\) 0 0
\(693\) 2.43939e12 + 2.43939e12i 0.0152621 + 0.0152621i
\(694\) 0 0
\(695\) 3.46629e13 + 5.31563e12i 0.213767 + 0.0327817i
\(696\) 0 0
\(697\) −1.72700e14 + 1.72700e14i −1.04985 + 1.04985i
\(698\) 0 0
\(699\) 7.46529e13i 0.447364i
\(700\) 0 0
\(701\) 1.06770e14 0.630754 0.315377 0.948966i \(-0.397869\pi\)
0.315377 + 0.948966i \(0.397869\pi\)
\(702\) 0 0
\(703\) 2.55354e14 + 2.55354e14i 1.48719 + 1.48719i
\(704\) 0 0
\(705\) 3.51865e13 2.29449e14i 0.202037 1.31747i
\(706\) 0 0
\(707\) −2.46279e13 + 2.46279e13i −0.139421 + 0.139421i
\(708\) 0 0
\(709\) 1.39815e14i 0.780409i 0.920728 + 0.390204i \(0.127596\pi\)
−0.920728 + 0.390204i \(0.872404\pi\)
\(710\) 0 0
\(711\) 3.78813e13 0.208486
\(712\) 0 0
\(713\) −5.70229e13 5.70229e13i −0.309458 0.309458i
\(714\) 0 0
\(715\) −1.41966e13 + 1.04214e13i −0.0759723 + 0.0557694i
\(716\) 0 0
\(717\) 8.34021e13 8.34021e13i 0.440130 0.440130i
\(718\) 0 0
\(719\) 6.98686e13i 0.363612i −0.983334 0.181806i \(-0.941806\pi\)
0.983334 0.181806i \(-0.0581942\pi\)
\(720\) 0 0
\(721\) 8.35335e13 0.428730
\(722\) 0 0
\(723\) 2.98154e14 + 2.98154e14i 1.50921 + 1.50921i
\(724\) 0 0
\(725\) 5.78448e13 1.84166e14i 0.288785 0.919431i
\(726\) 0 0
\(727\) 5.53649e13 5.53649e13i 0.272623 0.272623i −0.557532 0.830155i \(-0.688252\pi\)
0.830155 + 0.557532i \(0.188252\pi\)
\(728\) 0 0
\(729\) 1.57840e14i 0.766621i
\(730\) 0 0
\(731\) 8.31043e13 0.398140
\(732\) 0 0
\(733\) −2.77070e14 2.77070e14i −1.30939 1.30939i −0.921852 0.387541i \(-0.873324\pi\)
−0.387541 0.921852i \(-0.626676\pi\)
\(734\) 0 0
\(735\) −8.64412e13 1.17755e14i −0.402981 0.548964i
\(736\) 0 0
\(737\) 1.71076e13 1.71076e13i 0.0786775 0.0786775i
\(738\) 0 0
\(739\) 1.82183e14i 0.826581i −0.910599 0.413291i \(-0.864379\pi\)
0.910599 0.413291i \(-0.135621\pi\)
\(740\) 0 0
\(741\) 1.75279e14 0.784584
\(742\) 0 0
\(743\) −4.92009e13 4.92009e13i −0.217285 0.217285i 0.590069 0.807353i \(-0.299101\pi\)
−0.807353 + 0.590069i \(0.799101\pi\)
\(744\) 0 0
\(745\) −3.43305e14 5.26466e13i −1.49589 0.229398i
\(746\) 0 0
\(747\) 1.29077e13 1.29077e13i 0.0554941 0.0554941i
\(748\) 0 0
\(749\) 1.52890e14i 0.648590i
\(750\) 0 0
\(751\) 3.76620e14 1.57654 0.788269 0.615331i \(-0.210978\pi\)
0.788269 + 0.615331i \(0.210978\pi\)
\(752\) 0 0
\(753\) 1.21137e14 + 1.21137e14i 0.500383 + 0.500383i
\(754\) 0 0
\(755\) 4.03770e12 2.63296e13i 0.0164588 0.107327i
\(756\) 0 0
\(757\) −9.39645e13 + 9.39645e13i −0.377993 + 0.377993i −0.870378 0.492384i \(-0.836125\pi\)
0.492384 + 0.870378i \(0.336125\pi\)
\(758\) 0 0
\(759\) 1.78364e13i 0.0708106i
\(760\) 0 0
\(761\) 3.12361e14 1.22386 0.611932 0.790910i \(-0.290393\pi\)
0.611932 + 0.790910i \(0.290393\pi\)
\(762\) 0 0
\(763\) 1.62573e14 + 1.62573e14i 0.628676 + 0.628676i
\(764\) 0 0
\(765\) 4.99645e13 3.66777e13i 0.190701 0.139989i
\(766\) 0 0
\(767\) −1.32558e14 + 1.32558e14i −0.499378 + 0.499378i
\(768\) 0 0
\(769\) 3.65626e14i 1.35958i 0.733406 + 0.679790i \(0.237929\pi\)
−0.733406 + 0.679790i \(0.762071\pi\)
\(770\) 0 0
\(771\) −2.29084e14 −0.840860
\(772\) 0 0
\(773\) −2.98060e14 2.98060e14i −1.07996 1.07996i −0.996512 0.0834454i \(-0.973408\pi\)
−0.0834454 0.996512i \(-0.526592\pi\)
\(774\) 0 0
\(775\) −3.58346e14 1.12553e14i −1.28172 0.402577i
\(776\) 0 0
\(777\) −1.81204e14 + 1.81204e14i −0.639828 + 0.639828i
\(778\) 0 0
\(779\) 4.88890e14i 1.70421i
\(780\) 0 0
\(781\) −7.58148e13 −0.260915
\(782\) 0 0
\(783\) 1.79138e14 + 1.79138e14i 0.608666 + 0.608666i
\(784\) 0 0
\(785\) 1.99248e14 + 2.71427e14i 0.668413 + 0.910550i
\(786\) 0 0
\(787\) 1.35390e14 1.35390e14i 0.448450 0.448450i −0.446389 0.894839i \(-0.647290\pi\)
0.894839 + 0.446389i \(0.147290\pi\)
\(788\) 0 0
\(789\) 5.86251e13i 0.191734i
\(790\) 0 0
\(791\) −2.08365e14 −0.672889
\(792\) 0 0
\(793\) −3.24012e13 3.24012e13i −0.103323 0.103323i
\(794\) 0 0
\(795\) −4.20953e14 6.45541e13i −1.32556 0.203277i
\(796\) 0 0
\(797\) −3.63733e14 + 3.63733e14i −1.13108 + 1.13108i −0.141077 + 0.989999i \(0.545056\pi\)
−0.989999 + 0.141077i \(0.954944\pi\)
\(798\) 0 0
\(799\) 5.36187e14i 1.64658i
\(800\) 0 0
\(801\) −9.39843e13 −0.285031
\(802\) 0 0
\(803\) −6.20049e13 6.20049e13i −0.185716 0.185716i
\(804\) 0 0
\(805\) −1.01833e13 + 6.64045e13i −0.0301237 + 0.196435i
\(806\) 0 0
\(807\) 2.16712e14 2.16712e14i 0.633163 0.633163i
\(808\) 0 0
\(809\) 2.70403e14i 0.780314i 0.920748 + 0.390157i \(0.127579\pi\)
−0.920748 + 0.390157i \(0.872421\pi\)
\(810\) 0 0
\(811\) −5.25247e14 −1.49713 −0.748565 0.663061i \(-0.769257\pi\)
−0.748565 + 0.663061i \(0.769257\pi\)
\(812\) 0 0
\(813\) 3.96015e14 + 3.96015e14i 1.11496 + 1.11496i
\(814\) 0 0
\(815\) 4.87031e14 3.57517e14i 1.35447 0.994281i
\(816\) 0 0
\(817\) 1.17629e14 1.17629e14i 0.323149 0.323149i
\(818\) 0 0
\(819\) 1.86639e13i 0.0506505i
\(820\) 0 0
\(821\) 6.84037e14 1.83385 0.916925 0.399061i \(-0.130664\pi\)
0.916925 + 0.399061i \(0.130664\pi\)
\(822\) 0 0
\(823\) −2.05629e14 2.05629e14i −0.544610 0.544610i 0.380267 0.924877i \(-0.375832\pi\)
−0.924877 + 0.380267i \(0.875832\pi\)
\(824\) 0 0
\(825\) 3.84412e13 + 7.36471e13i 0.100584 + 0.192702i
\(826\) 0 0
\(827\) −4.20671e14 + 4.20671e14i −1.08746 + 1.08746i −0.0916761 + 0.995789i \(0.529222\pi\)
−0.995789 + 0.0916761i \(0.970778\pi\)
\(828\) 0 0
\(829\) 6.52280e14i 1.66595i −0.553311 0.832975i \(-0.686636\pi\)
0.553311 0.832975i \(-0.313364\pi\)
\(830\) 0 0
\(831\) −4.13221e14 −1.04274
\(832\) 0 0
\(833\) −2.38588e14 2.38588e14i −0.594872 0.594872i
\(834\) 0 0
\(835\) 3.77261e14 + 5.13926e14i 0.929414 + 1.26610i
\(836\) 0 0
\(837\) 3.48562e14 3.48562e14i 0.848502 0.848502i
\(838\) 0 0
\(839\) 3.02853e13i 0.0728488i −0.999336 0.0364244i \(-0.988403\pi\)
0.999336 0.0364244i \(-0.0115968\pi\)
\(840\) 0 0
\(841\) 2.99750e13 0.0712491
\(842\) 0 0
\(843\) −2.41641e14 2.41641e14i −0.567587 0.567587i
\(844\) 0 0
\(845\) 3.31653e14 + 5.08597e13i 0.769839 + 0.118056i
\(846\) 0 0
\(847\) 1.80498e14 1.80498e14i 0.414053 0.414053i
\(848\) 0 0
\(849\) 4.93183e14i 1.11807i
\(850\) 0 0
\(851\) −1.98812e14 −0.445445
\(852\) 0 0
\(853\) 2.91456e14 + 2.91456e14i 0.645399 + 0.645399i 0.951878 0.306479i \(-0.0991508\pi\)
−0.306479 + 0.951878i \(0.599151\pi\)
\(854\) 0 0
\(855\) 1.88066e13 1.22636e14i 0.0411603 0.268404i
\(856\) 0 0
\(857\) 3.05558e14 3.05558e14i 0.660982 0.660982i −0.294629 0.955612i \(-0.595196\pi\)
0.955612 + 0.294629i \(0.0951961\pi\)
\(858\) 0 0
\(859\) 5.89475e14i 1.26037i −0.776443 0.630187i \(-0.782978\pi\)
0.776443 0.630187i \(-0.217022\pi\)
\(860\) 0 0
\(861\) −3.46926e14 −0.733197
\(862\) 0 0
\(863\) 4.31410e14 + 4.31410e14i 0.901231 + 0.901231i 0.995543 0.0943117i \(-0.0300650\pi\)
−0.0943117 + 0.995543i \(0.530065\pi\)
\(864\) 0 0
\(865\) −8.00450e13 + 5.87591e13i −0.165293 + 0.121337i
\(866\) 0 0
\(867\) 2.98918e14 2.98918e14i 0.610178 0.610178i
\(868\) 0 0
\(869\) 1.17279e14i 0.236658i
\(870\) 0 0
\(871\) 1.30891e14 0.261108
\(872\) 0 0
\(873\) −9.43921e13 9.43921e13i −0.186151 0.186151i
\(874\) 0 0
\(875\) 1.01068e14 + 2.96133e14i 0.197050 + 0.577360i
\(876\) 0 0
\(877\) −5.06135e13 + 5.06135e13i −0.0975593 + 0.0975593i −0.754202 0.656643i \(-0.771976\pi\)
0.656643 + 0.754202i \(0.271976\pi\)
\(878\) 0 0
\(879\) 1.94954e14i 0.371524i
\(880\) 0 0
\(881\) 4.33506e14 0.816800 0.408400 0.912803i \(-0.366087\pi\)
0.408400 + 0.912803i \(0.366087\pi\)
\(882\) 0 0
\(883\) −5.45003e14 5.45003e14i −1.01530 1.01530i −0.999881 0.0154219i \(-0.995091\pi\)
−0.0154219 0.999881i \(-0.504909\pi\)
\(884\) 0 0
\(885\) 5.23300e14 + 7.12869e14i 0.963905 + 1.31309i
\(886\) 0 0
\(887\) 4.99448e14 4.99448e14i 0.909645 0.909645i −0.0865986 0.996243i \(-0.527600\pi\)
0.996243 + 0.0865986i \(0.0275998\pi\)
\(888\) 0 0
\(889\) 8.99362e13i 0.161967i
\(890\) 0 0
\(891\) −1.28896e14 −0.229535
\(892\) 0 0
\(893\) −7.58937e14 7.58937e14i −1.33644 1.33644i
\(894\) 0 0
\(895\) −1.79452e14 2.75193e13i −0.312488 0.0479207i
\(896\) 0 0
\(897\) −6.82338e13 + 6.82338e13i −0.117500 + 0.117500i
\(898\) 0 0
\(899\) 7.60277e14i 1.29471i
\(900\) 0 0
\(901\) −9.83703e14 −1.65668
\(902\) 0 0
\(903\) 8.34715e13 + 8.34715e13i 0.139027 + 0.139027i
\(904\) 0 0
\(905\) 6.02932e13 3.93168e14i 0.0993174 0.647643i
\(906\) 0 0
\(907\) 9.77914e13 9.77914e13i 0.159318 0.159318i −0.622947 0.782264i \(-0.714065\pi\)
0.782264 + 0.622947i \(0.214065\pi\)
\(908\) 0 0
\(909\) 3.54117e13i 0.0570595i
\(910\) 0 0
\(911\) 6.85912e14 1.09314 0.546571 0.837413i \(-0.315933\pi\)
0.546571 + 0.837413i \(0.315933\pi\)
\(912\) 0 0
\(913\) 3.99619e13 + 3.99619e13i 0.0629930 + 0.0629930i
\(914\) 0 0
\(915\) −1.74247e14 + 1.27910e14i −0.271681 + 0.199434i
\(916\) 0 0
\(917\) −2.84588e14 + 2.84588e14i −0.438905 + 0.438905i
\(918\) 0 0
\(919\) 6.85058e14i 1.04508i 0.852615 + 0.522540i \(0.175015\pi\)
−0.852615 + 0.522540i \(0.824985\pi\)
\(920\) 0 0
\(921\) −9.23757e14 −1.39399
\(922\) 0 0
\(923\) −2.90032e14 2.90032e14i −0.432950 0.432950i
\(924\) 0 0
\(925\) −8.20899e14 + 4.28481e14i −1.21222 + 0.632736i
\(926\) 0 0
\(927\) 6.00553e13 6.00553e13i 0.0877309 0.0877309i
\(928\) 0 0
\(929\) 1.11675e15i 1.61391i 0.590614 + 0.806954i \(0.298886\pi\)
−0.590614 + 0.806954i \(0.701114\pi\)
\(930\) 0 0
\(931\) −6.75411e14 −0.965651
\(932\) 0 0
\(933\) 9.56848e14 + 9.56848e14i 1.35343 + 1.35343i
\(934\) 0 0
\(935\) 1.13553e14 + 1.54688e14i 0.158906 + 0.216471i
\(936\) 0 0
\(937\) 8.19683e14 8.19683e14i 1.13488 1.13488i 0.145520 0.989355i \(-0.453514\pi\)
0.989355 0.145520i \(-0.0464856\pi\)
\(938\) 0 0
\(939\) 1.12203e15i 1.53701i
\(940\) 0 0
\(941\) −3.97374e14 −0.538582 −0.269291 0.963059i \(-0.586789\pi\)
−0.269291 + 0.963059i \(0.586789\pi\)
\(942\) 0 0
\(943\) −1.90318e14 1.90318e14i −0.255224 0.255224i
\(944\) 0 0
\(945\) −4.05909e14 6.22470e13i −0.538604 0.0825960i
\(946\) 0 0
\(947\) −5.07249e14 + 5.07249e14i −0.665995 + 0.665995i −0.956786 0.290792i \(-0.906081\pi\)
0.290792 + 0.956786i \(0.406081\pi\)
\(948\) 0 0
\(949\) 4.74404e14i 0.616335i
\(950\) 0 0
\(951\) 1.19189e15 1.53226
\(952\) 0 0
\(953\) −6.83456e14 6.83456e14i −0.869453 0.869453i 0.122958 0.992412i \(-0.460762\pi\)
−0.992412 + 0.122958i \(0.960762\pi\)
\(954\) 0 0
\(955\) 9.92622e12 6.47283e13i 0.0124959 0.0814849i
\(956\) 0 0
\(957\) 1.18905e14 1.18905e14i 0.148129 0.148129i
\(958\) 0 0
\(959\) 2.61363e13i 0.0322219i
\(960\) 0 0
\(961\) 6.59700e14 0.804877
\(962\) 0 0
\(963\) −1.09918e14 1.09918e14i −0.132721 0.132721i
\(964\) 0 0
\(965\) 5.99594e14 4.40148e14i 0.716508 0.525971i
\(966\) 0 0
\(967\) −2.78260e14 + 2.78260e14i −0.329093 + 0.329093i −0.852241 0.523149i \(-0.824757\pi\)
0.523149 + 0.852241i \(0.324757\pi\)
\(968\) 0 0
\(969\) 1.90986e15i 2.23554i
\(970\) 0 0
\(971\) −1.23796e14 −0.143420 −0.0717101 0.997426i \(-0.522846\pi\)
−0.0717101 + 0.997426i \(0.522846\pi\)
\(972\) 0 0
\(973\) 8.13599e13 + 8.13599e13i 0.0932924 + 0.0932924i
\(974\) 0 0
\(975\) −1.34681e14 + 4.28797e14i −0.152857 + 0.486664i
\(976\) 0 0
\(977\) 4.70589e14 4.70589e14i 0.528651 0.528651i −0.391519 0.920170i \(-0.628050\pi\)
0.920170 + 0.391519i \(0.128050\pi\)
\(978\) 0 0
\(979\) 2.90972e14i 0.323548i
\(980\) 0 0
\(981\) 2.33760e14 0.257291
\(982\) 0 0
\(983\) −8.15213e14 8.15213e14i −0.888185 0.888185i 0.106164 0.994349i \(-0.466143\pi\)
−0.994349 + 0.106164i \(0.966143\pi\)
\(984\) 0 0
\(985\) −3.99388e14 5.44069e14i −0.430739 0.586777i
\(986\) 0 0
\(987\) 5.38556e14 5.38556e14i 0.574970 0.574970i
\(988\) 0 0
\(989\) 9.15824e13i 0.0967900i
\(990\) 0 0
\(991\) 4.87602e14 0.510150 0.255075 0.966921i \(-0.417900\pi\)
0.255075 + 0.966921i \(0.417900\pi\)
\(992\) 0 0
\(993\) 8.14692e14 + 8.14692e14i 0.843815 + 0.843815i
\(994\) 0 0
\(995\) 1.08405e15 + 1.66241e14i 1.11156 + 0.170461i
\(996\) 0 0
\(997\) −2.05598e14 + 2.05598e14i −0.208710 + 0.208710i −0.803719 0.595009i \(-0.797149\pi\)
0.595009 + 0.803719i \(0.297149\pi\)
\(998\) 0 0
\(999\) 1.21527e15i 1.22136i
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 20.11.f.a.13.4 10
3.2 odd 2 180.11.l.a.73.3 10
4.3 odd 2 80.11.p.e.33.2 10
5.2 odd 4 inner 20.11.f.a.17.4 yes 10
5.3 odd 4 100.11.f.b.57.2 10
5.4 even 2 100.11.f.b.93.2 10
15.2 even 4 180.11.l.a.37.3 10
20.7 even 4 80.11.p.e.17.2 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.11.f.a.13.4 10 1.1 even 1 trivial
20.11.f.a.17.4 yes 10 5.2 odd 4 inner
80.11.p.e.17.2 10 20.7 even 4
80.11.p.e.33.2 10 4.3 odd 2
100.11.f.b.57.2 10 5.3 odd 4
100.11.f.b.93.2 10 5.4 even 2
180.11.l.a.37.3 10 15.2 even 4
180.11.l.a.73.3 10 3.2 odd 2