Properties

Label 64.24.e.a.17.36
Level $64$
Weight $24$
Character 64.17
Analytic conductor $214.531$
Analytic rank $0$
Dimension $90$
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] = [64,24,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, 24, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("64.17");
 
S:= CuspForms(chi, 24);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 64 = 2^{6} \)
Weight: \( k \) \(=\) \( 24 \)
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: \(214.530583901\)
Analytic rank: \(0\)
Dimension: \(90\)
Relative dimension: \(45\) over \(\Q(i)\)
Twist minimal: no (minimal twist has level 16)
Sato-Tate group: $\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label 17.36
Character \(\chi\) \(=\) 64.17
Dual form 64.24.e.a.49.36

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(270677. + 270677. i) q^{3} +(-1.51855e8 + 1.51855e8i) q^{5} +9.83375e9i q^{7} +5.23892e10i q^{9} +O(q^{10})\) \(q+(270677. + 270677. i) q^{3} +(-1.51855e8 + 1.51855e8i) q^{5} +9.83375e9i q^{7} +5.23892e10i q^{9} +(-1.93402e11 + 1.93402e11i) q^{11} +(-3.48061e12 - 3.48061e12i) q^{13} -8.22076e13 q^{15} -4.03593e13 q^{17} +(3.81860e14 + 3.81860e14i) q^{19} +(-2.66177e15 + 2.66177e15i) q^{21} +5.51116e14i q^{23} -3.41992e16i q^{25} +(1.13019e16 - 1.13019e16i) q^{27} +(-2.35476e16 - 2.35476e16i) q^{29} +8.66833e16 q^{31} -1.04699e17 q^{33} +(-1.49331e18 - 1.49331e18i) q^{35} +(-7.22561e17 + 7.22561e17i) q^{37} -1.88425e18i q^{39} -3.41831e17i q^{41} +(4.79520e18 - 4.79520e18i) q^{43} +(-7.95558e18 - 7.95558e18i) q^{45} -1.40438e19 q^{47} -6.93340e19 q^{49} +(-1.09244e19 - 1.09244e19i) q^{51} +(-1.58841e19 + 1.58841e19i) q^{53} -5.87383e19i q^{55} +2.06722e20i q^{57} +(8.07394e19 - 8.07394e19i) q^{59} +(-2.60908e20 - 2.60908e20i) q^{61} -5.15183e20 q^{63} +1.05710e21 q^{65} +(-1.00472e21 - 1.00472e21i) q^{67} +(-1.49175e20 + 1.49175e20i) q^{69} +1.00814e21i q^{71} -1.06993e21i q^{73} +(9.25694e21 - 9.25694e21i) q^{75} +(-1.90187e21 - 1.90187e21i) q^{77} -2.18097e21 q^{79} +1.10504e22 q^{81} +(1.33029e22 + 1.33029e22i) q^{83} +(6.12878e21 - 6.12878e21i) q^{85} -1.27476e22i q^{87} -1.63580e22i q^{89} +(3.42275e22 - 3.42275e22i) q^{91} +(2.34632e22 + 2.34632e22i) q^{93} -1.15975e23 q^{95} +5.68132e22 q^{97} +(-1.01322e22 - 1.01322e22i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 90 q + 2 q^{3} - 2 q^{5}+O(q^{10}) \) Copy content Toggle raw display \( 90 q + 2 q^{3} - 2 q^{5} - 975574266674 q^{11} - 2 q^{13} + 69198046875004 q^{15} - 4 q^{17} - 33497748322214 q^{19} + 188286357652 q^{21} + 68\!\cdots\!12 q^{27}+ \cdots - 15\!\cdots\!18 q^{99}+O(q^{100}) \) 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
\(3\) 270677. + 270677.i 0.882180 + 0.882180i 0.993756 0.111576i \(-0.0355897\pi\)
−0.111576 + 0.993756i \(0.535590\pi\)
\(4\) 0 0
\(5\) −1.51855e8 + 1.51855e8i −1.39083 + 1.39083i −0.567372 + 0.823462i \(0.692040\pi\)
−0.823462 + 0.567372i \(0.807960\pi\)
\(6\) 0 0
\(7\) 9.83375e9i 1.87971i 0.341567 + 0.939857i \(0.389042\pi\)
−0.341567 + 0.939857i \(0.610958\pi\)
\(8\) 0 0
\(9\) 5.23892e10i 0.556484i
\(10\) 0 0
\(11\) −1.93402e11 + 1.93402e11i −0.204383 + 0.204383i −0.801875 0.597492i \(-0.796164\pi\)
0.597492 + 0.801875i \(0.296164\pi\)
\(12\) 0 0
\(13\) −3.48061e12 3.48061e12i −0.538651 0.538651i 0.384482 0.923133i \(-0.374380\pi\)
−0.923133 + 0.384482i \(0.874380\pi\)
\(14\) 0 0
\(15\) −8.22076e13 −2.45393
\(16\) 0 0
\(17\) −4.03593e13 −0.285615 −0.142808 0.989750i \(-0.545613\pi\)
−0.142808 + 0.989750i \(0.545613\pi\)
\(18\) 0 0
\(19\) 3.81860e14 + 3.81860e14i 0.752035 + 0.752035i 0.974859 0.222823i \(-0.0715273\pi\)
−0.222823 + 0.974859i \(0.571527\pi\)
\(20\) 0 0
\(21\) −2.66177e15 + 2.66177e15i −1.65825 + 1.65825i
\(22\) 0 0
\(23\) 5.51116e14i 0.120607i 0.998180 + 0.0603035i \(0.0192069\pi\)
−0.998180 + 0.0603035i \(0.980793\pi\)
\(24\) 0 0
\(25\) 3.41992e16i 2.86884i
\(26\) 0 0
\(27\) 1.13019e16 1.13019e16i 0.391261 0.391261i
\(28\) 0 0
\(29\) −2.35476e16 2.35476e16i −0.358401 0.358401i 0.504822 0.863223i \(-0.331558\pi\)
−0.863223 + 0.504822i \(0.831558\pi\)
\(30\) 0 0
\(31\) 8.66833e16 0.612740 0.306370 0.951913i \(-0.400886\pi\)
0.306370 + 0.951913i \(0.400886\pi\)
\(32\) 0 0
\(33\) −1.04699e17 −0.360606
\(34\) 0 0
\(35\) −1.49331e18 1.49331e18i −2.61437 2.61437i
\(36\) 0 0
\(37\) −7.22561e17 + 7.22561e17i −0.667659 + 0.667659i −0.957174 0.289515i \(-0.906506\pi\)
0.289515 + 0.957174i \(0.406506\pi\)
\(38\) 0 0
\(39\) 1.88425e18i 0.950374i
\(40\) 0 0
\(41\) 3.41831e17i 0.0970056i −0.998823 0.0485028i \(-0.984555\pi\)
0.998823 0.0485028i \(-0.0154450\pi\)
\(42\) 0 0
\(43\) 4.79520e18 4.79520e18i 0.786900 0.786900i −0.194085 0.980985i \(-0.562174\pi\)
0.980985 + 0.194085i \(0.0621738\pi\)
\(44\) 0 0
\(45\) −7.95558e18 7.95558e18i −0.773977 0.773977i
\(46\) 0 0
\(47\) −1.40438e19 −0.828627 −0.414314 0.910134i \(-0.635978\pi\)
−0.414314 + 0.910134i \(0.635978\pi\)
\(48\) 0 0
\(49\) −6.93340e19 −2.53333
\(50\) 0 0
\(51\) −1.09244e19 1.09244e19i −0.251964 0.251964i
\(52\) 0 0
\(53\) −1.58841e19 + 1.58841e19i −0.235391 + 0.235391i −0.814939 0.579547i \(-0.803229\pi\)
0.579547 + 0.814939i \(0.303229\pi\)
\(54\) 0 0
\(55\) 5.87383e19i 0.568526i
\(56\) 0 0
\(57\) 2.06722e20i 1.32686i
\(58\) 0 0
\(59\) 8.07394e19 8.07394e19i 0.348568 0.348568i −0.511008 0.859576i \(-0.670728\pi\)
0.859576 + 0.511008i \(0.170728\pi\)
\(60\) 0 0
\(61\) −2.60908e20 2.60908e20i −0.767704 0.767704i 0.209998 0.977702i \(-0.432654\pi\)
−0.977702 + 0.209998i \(0.932654\pi\)
\(62\) 0 0
\(63\) −5.15183e20 −1.04603
\(64\) 0 0
\(65\) 1.05710e21 1.49835
\(66\) 0 0
\(67\) −1.00472e21 1.00472e21i −1.00504 1.00504i −0.999987 0.00505403i \(-0.998391\pi\)
−0.00505403 0.999987i \(-0.501609\pi\)
\(68\) 0 0
\(69\) −1.49175e20 + 1.49175e20i −0.106397 + 0.106397i
\(70\) 0 0
\(71\) 1.00814e21i 0.517668i 0.965922 + 0.258834i \(0.0833383\pi\)
−0.965922 + 0.258834i \(0.916662\pi\)
\(72\) 0 0
\(73\) 1.06993e21i 0.399156i −0.979882 0.199578i \(-0.936043\pi\)
0.979882 0.199578i \(-0.0639571\pi\)
\(74\) 0 0
\(75\) 9.25694e21 9.25694e21i 2.53083 2.53083i
\(76\) 0 0
\(77\) −1.90187e21 1.90187e21i −0.384182 0.384182i
\(78\) 0 0
\(79\) −2.18097e21 −0.328049 −0.164025 0.986456i \(-0.552448\pi\)
−0.164025 + 0.986456i \(0.552448\pi\)
\(80\) 0 0
\(81\) 1.10504e22 1.24681
\(82\) 0 0
\(83\) 1.33029e22 + 1.33029e22i 1.13383 + 1.13383i 0.989535 + 0.144292i \(0.0460903\pi\)
0.144292 + 0.989535i \(0.453910\pi\)
\(84\) 0 0
\(85\) 6.12878e21 6.12878e21i 0.397243 0.397243i
\(86\) 0 0
\(87\) 1.27476e22i 0.632349i
\(88\) 0 0
\(89\) 1.63580e22i 0.624806i −0.949950 0.312403i \(-0.898866\pi\)
0.949950 0.312403i \(-0.101134\pi\)
\(90\) 0 0
\(91\) 3.42275e22 3.42275e22i 1.01251 1.01251i
\(92\) 0 0
\(93\) 2.34632e22 + 2.34632e22i 0.540547 + 0.540547i
\(94\) 0 0
\(95\) −1.15975e23 −2.09191
\(96\) 0 0
\(97\) 5.68132e22 0.806445 0.403222 0.915102i \(-0.367890\pi\)
0.403222 + 0.915102i \(0.367890\pi\)
\(98\) 0 0
\(99\) −1.01322e22 1.01322e22i −0.113736 0.113736i
\(100\) 0 0
\(101\) −5.48094e22 + 5.48094e22i −0.488831 + 0.488831i −0.907937 0.419106i \(-0.862344\pi\)
0.419106 + 0.907937i \(0.362344\pi\)
\(102\) 0 0
\(103\) 2.88609e22i 0.205439i −0.994710 0.102719i \(-0.967246\pi\)
0.994710 0.102719i \(-0.0327543\pi\)
\(104\) 0 0
\(105\) 8.08409e23i 4.61269i
\(106\) 0 0
\(107\) −1.73458e23 + 1.73458e23i −0.796676 + 0.796676i −0.982570 0.185894i \(-0.940482\pi\)
0.185894 + 0.982570i \(0.440482\pi\)
\(108\) 0 0
\(109\) 1.73035e23 + 1.73035e23i 0.642288 + 0.642288i 0.951118 0.308829i \(-0.0999371\pi\)
−0.308829 + 0.951118i \(0.599937\pi\)
\(110\) 0 0
\(111\) −3.91162e23 −1.17799
\(112\) 0 0
\(113\) −4.37190e23 −1.07218 −0.536091 0.844160i \(-0.680100\pi\)
−0.536091 + 0.844160i \(0.680100\pi\)
\(114\) 0 0
\(115\) −8.36899e22 8.36899e22i −0.167744 0.167744i
\(116\) 0 0
\(117\) 1.82347e23 1.82347e23i 0.299751 0.299751i
\(118\) 0 0
\(119\) 3.96884e23i 0.536875i
\(120\) 0 0
\(121\) 8.20622e23i 0.916455i
\(122\) 0 0
\(123\) 9.25258e22 9.25258e22i 0.0855764 0.0855764i
\(124\) 0 0
\(125\) 3.38307e24 + 3.38307e24i 2.59924 + 2.59924i
\(126\) 0 0
\(127\) −2.94304e23 −0.188388 −0.0941939 0.995554i \(-0.530027\pi\)
−0.0941939 + 0.995554i \(0.530027\pi\)
\(128\) 0 0
\(129\) 2.59590e24 1.38837
\(130\) 0 0
\(131\) 1.75054e24 + 1.75054e24i 0.784424 + 0.784424i 0.980574 0.196150i \(-0.0628439\pi\)
−0.196150 + 0.980574i \(0.562844\pi\)
\(132\) 0 0
\(133\) −3.75512e24 + 3.75512e24i −1.41361 + 1.41361i
\(134\) 0 0
\(135\) 3.43249e24i 1.08836i
\(136\) 0 0
\(137\) 3.08993e23i 0.0827299i 0.999144 + 0.0413649i \(0.0131706\pi\)
−0.999144 + 0.0413649i \(0.986829\pi\)
\(138\) 0 0
\(139\) −4.09547e24 + 4.09547e24i −0.928182 + 0.928182i −0.997588 0.0694065i \(-0.977889\pi\)
0.0694065 + 0.997588i \(0.477889\pi\)
\(140\) 0 0
\(141\) −3.80134e24 3.80134e24i −0.730999 0.730999i
\(142\) 0 0
\(143\) 1.34631e24 0.220182
\(144\) 0 0
\(145\) 7.15166e24 0.996953
\(146\) 0 0
\(147\) −1.87671e25 1.87671e25i −2.23485 2.23485i
\(148\) 0 0
\(149\) 1.10897e25 1.10897e25i 1.13052 1.13052i 0.140432 0.990090i \(-0.455151\pi\)
0.990090 0.140432i \(-0.0448492\pi\)
\(150\) 0 0
\(151\) 4.05535e23i 0.0354645i −0.999843 0.0177322i \(-0.994355\pi\)
0.999843 0.0177322i \(-0.00564464\pi\)
\(152\) 0 0
\(153\) 2.11439e24i 0.158940i
\(154\) 0 0
\(155\) −1.31633e25 + 1.31633e25i −0.852219 + 0.852219i
\(156\) 0 0
\(157\) −1.50527e25 1.50527e25i −0.840944 0.840944i 0.148037 0.988982i \(-0.452704\pi\)
−0.988982 + 0.148037i \(0.952704\pi\)
\(158\) 0 0
\(159\) −8.59894e24 −0.415315
\(160\) 0 0
\(161\) −5.41954e24 −0.226707
\(162\) 0 0
\(163\) 6.44894e23 + 6.44894e23i 0.0234062 + 0.0234062i 0.718713 0.695307i \(-0.244732\pi\)
−0.695307 + 0.718713i \(0.744732\pi\)
\(164\) 0 0
\(165\) 1.58991e25 1.58991e25i 0.501542 0.501542i
\(166\) 0 0
\(167\) 5.06935e24i 0.139224i 0.997574 + 0.0696118i \(0.0221761\pi\)
−0.997574 + 0.0696118i \(0.977824\pi\)
\(168\) 0 0
\(169\) 1.75246e25i 0.419711i
\(170\) 0 0
\(171\) −2.00054e25 + 2.00054e25i −0.418496 + 0.418496i
\(172\) 0 0
\(173\) 6.69878e25 + 6.69878e25i 1.22593 + 1.22593i 0.965490 + 0.260438i \(0.0838671\pi\)
0.260438 + 0.965490i \(0.416133\pi\)
\(174\) 0 0
\(175\) 3.36306e26 5.39259
\(176\) 0 0
\(177\) 4.37086e25 0.614999
\(178\) 0 0
\(179\) −1.01063e26 1.01063e26i −1.24963 1.24963i −0.955884 0.293746i \(-0.905098\pi\)
−0.293746 0.955884i \(-0.594902\pi\)
\(180\) 0 0
\(181\) −5.35488e25 + 5.35488e25i −0.582701 + 0.582701i −0.935645 0.352943i \(-0.885181\pi\)
0.352943 + 0.935645i \(0.385181\pi\)
\(182\) 0 0
\(183\) 1.41244e26i 1.35451i
\(184\) 0 0
\(185\) 2.19450e26i 1.85720i
\(186\) 0 0
\(187\) 7.80558e24 7.80558e24i 0.0583750 0.0583750i
\(188\) 0 0
\(189\) 1.11140e26 + 1.11140e26i 0.735459 + 0.735459i
\(190\) 0 0
\(191\) −1.09204e26 −0.640260 −0.320130 0.947374i \(-0.603727\pi\)
−0.320130 + 0.947374i \(0.603727\pi\)
\(192\) 0 0
\(193\) −2.07215e26 −1.07774 −0.538869 0.842390i \(-0.681148\pi\)
−0.538869 + 0.842390i \(0.681148\pi\)
\(194\) 0 0
\(195\) 2.86133e26 + 2.86133e26i 1.32181 + 1.32181i
\(196\) 0 0
\(197\) −2.24730e26 + 2.24730e26i −0.923207 + 0.923207i −0.997255 0.0740475i \(-0.976408\pi\)
0.0740475 + 0.997255i \(0.476408\pi\)
\(198\) 0 0
\(199\) 2.12830e26i 0.778434i −0.921146 0.389217i \(-0.872746\pi\)
0.921146 0.389217i \(-0.127254\pi\)
\(200\) 0 0
\(201\) 5.43909e26i 1.77326i
\(202\) 0 0
\(203\) 2.31561e26 2.31561e26i 0.673692 0.673692i
\(204\) 0 0
\(205\) 5.19088e25 + 5.19088e25i 0.134919 + 0.134919i
\(206\) 0 0
\(207\) −2.88725e25 −0.0671159
\(208\) 0 0
\(209\) −1.47705e26 −0.307407
\(210\) 0 0
\(211\) −2.97898e25 2.97898e25i −0.0555673 0.0555673i 0.678777 0.734344i \(-0.262510\pi\)
−0.734344 + 0.678777i \(0.762510\pi\)
\(212\) 0 0
\(213\) −2.72882e26 + 2.72882e26i −0.456677 + 0.456677i
\(214\) 0 0
\(215\) 1.45635e27i 2.18889i
\(216\) 0 0
\(217\) 8.52423e26i 1.15178i
\(218\) 0 0
\(219\) 2.89606e26 2.89606e26i 0.352127 0.352127i
\(220\) 0 0
\(221\) 1.40475e26 + 1.40475e26i 0.153847 + 0.153847i
\(222\) 0 0
\(223\) −1.35645e26 −0.133937 −0.0669683 0.997755i \(-0.521333\pi\)
−0.0669683 + 0.997755i \(0.521333\pi\)
\(224\) 0 0
\(225\) 1.79167e27 1.59646
\(226\) 0 0
\(227\) 3.47947e26 + 3.47947e26i 0.280037 + 0.280037i 0.833124 0.553087i \(-0.186550\pi\)
−0.553087 + 0.833124i \(0.686550\pi\)
\(228\) 0 0
\(229\) −5.33106e26 + 5.33106e26i −0.387887 + 0.387887i −0.873933 0.486046i \(-0.838439\pi\)
0.486046 + 0.873933i \(0.338439\pi\)
\(230\) 0 0
\(231\) 1.02958e27i 0.677836i
\(232\) 0 0
\(233\) 2.28776e27i 1.36401i −0.731349 0.682004i \(-0.761109\pi\)
0.731349 0.682004i \(-0.238891\pi\)
\(234\) 0 0
\(235\) 2.13263e27 2.13263e27i 1.15248 1.15248i
\(236\) 0 0
\(237\) −5.90340e26 5.90340e26i −0.289399 0.289399i
\(238\) 0 0
\(239\) 2.59509e27 1.15499 0.577493 0.816396i \(-0.304031\pi\)
0.577493 + 0.816396i \(0.304031\pi\)
\(240\) 0 0
\(241\) 2.19632e27 0.888177 0.444089 0.895983i \(-0.353528\pi\)
0.444089 + 0.895983i \(0.353528\pi\)
\(242\) 0 0
\(243\) 1.92710e27 + 1.92710e27i 0.708650 + 0.708650i
\(244\) 0 0
\(245\) 1.05287e28 1.05287e28i 3.52344 3.52344i
\(246\) 0 0
\(247\) 2.65822e27i 0.810169i
\(248\) 0 0
\(249\) 7.20156e27i 2.00048i
\(250\) 0 0
\(251\) −2.68559e27 + 2.68559e27i −0.680444 + 0.680444i −0.960100 0.279656i \(-0.909779\pi\)
0.279656 + 0.960100i \(0.409779\pi\)
\(252\) 0 0
\(253\) −1.06587e26 1.06587e26i −0.0246501 0.0246501i
\(254\) 0 0
\(255\) 3.31785e27 0.700881
\(256\) 0 0
\(257\) 8.37554e26 0.161727 0.0808634 0.996725i \(-0.474232\pi\)
0.0808634 + 0.996725i \(0.474232\pi\)
\(258\) 0 0
\(259\) −7.10549e27 7.10549e27i −1.25501 1.25501i
\(260\) 0 0
\(261\) 1.23364e27 1.23364e27i 0.199445 0.199445i
\(262\) 0 0
\(263\) 1.07425e28i 1.59080i −0.606087 0.795399i \(-0.707262\pi\)
0.606087 0.795399i \(-0.292738\pi\)
\(264\) 0 0
\(265\) 4.82418e27i 0.654781i
\(266\) 0 0
\(267\) 4.42774e27 4.42774e27i 0.551191 0.551191i
\(268\) 0 0
\(269\) 7.28942e27 + 7.28942e27i 0.832802 + 0.832802i 0.987899 0.155097i \(-0.0495689\pi\)
−0.155097 + 0.987899i \(0.549569\pi\)
\(270\) 0 0
\(271\) −4.44202e27 −0.466051 −0.233026 0.972471i \(-0.574863\pi\)
−0.233026 + 0.972471i \(0.574863\pi\)
\(272\) 0 0
\(273\) 1.85292e28 1.78643
\(274\) 0 0
\(275\) 6.61419e27 + 6.61419e27i 0.586342 + 0.586342i
\(276\) 0 0
\(277\) 5.43923e27 5.43923e27i 0.443629 0.443629i −0.449601 0.893230i \(-0.648434\pi\)
0.893230 + 0.449601i \(0.148434\pi\)
\(278\) 0 0
\(279\) 4.54127e27i 0.340980i
\(280\) 0 0
\(281\) 7.05886e27i 0.488216i 0.969748 + 0.244108i \(0.0784952\pi\)
−0.969748 + 0.244108i \(0.921505\pi\)
\(282\) 0 0
\(283\) 2.17754e28 2.17754e28i 1.38811 1.38811i 0.558809 0.829297i \(-0.311259\pi\)
0.829297 0.558809i \(-0.188741\pi\)
\(284\) 0 0
\(285\) −3.13918e28 3.13918e28i −1.84544 1.84544i
\(286\) 0 0
\(287\) 3.36148e27 0.182343
\(288\) 0 0
\(289\) −1.83387e28 −0.918424
\(290\) 0 0
\(291\) 1.53780e28 + 1.53780e28i 0.711430 + 0.711430i
\(292\) 0 0
\(293\) 9.89148e27 9.89148e27i 0.422945 0.422945i −0.463271 0.886216i \(-0.653325\pi\)
0.886216 + 0.463271i \(0.153325\pi\)
\(294\) 0 0
\(295\) 2.45214e28i 0.969599i
\(296\) 0 0
\(297\) 4.37160e27i 0.159934i
\(298\) 0 0
\(299\) 1.91822e27 1.91822e27i 0.0649651 0.0649651i
\(300\) 0 0
\(301\) 4.71548e28 + 4.71548e28i 1.47915 + 1.47915i
\(302\) 0 0
\(303\) −2.96713e28 −0.862475
\(304\) 0 0
\(305\) 7.92405e28 2.13550
\(306\) 0 0
\(307\) 4.79524e28 + 4.79524e28i 1.19872 + 1.19872i 0.974549 + 0.224176i \(0.0719690\pi\)
0.224176 + 0.974549i \(0.428031\pi\)
\(308\) 0 0
\(309\) 7.81199e27 7.81199e27i 0.181234 0.181234i
\(310\) 0 0
\(311\) 3.61799e27i 0.0779332i −0.999241 0.0389666i \(-0.987593\pi\)
0.999241 0.0389666i \(-0.0124066\pi\)
\(312\) 0 0
\(313\) 4.60366e28i 0.921178i −0.887614 0.460589i \(-0.847638\pi\)
0.887614 0.460589i \(-0.152362\pi\)
\(314\) 0 0
\(315\) 7.82332e28 7.82332e28i 1.45486 1.45486i
\(316\) 0 0
\(317\) 4.10975e28 + 4.10975e28i 0.710614 + 0.710614i 0.966664 0.256050i \(-0.0824211\pi\)
−0.256050 + 0.966664i \(0.582421\pi\)
\(318\) 0 0
\(319\) 9.10831e27 0.146502
\(320\) 0 0
\(321\) −9.39024e28 −1.40562
\(322\) 0 0
\(323\) −1.54116e28 1.54116e28i −0.214793 0.214793i
\(324\) 0 0
\(325\) −1.19034e29 + 1.19034e29i −1.54530 + 1.54530i
\(326\) 0 0
\(327\) 9.36735e28i 1.13323i
\(328\) 0 0
\(329\) 1.38103e29i 1.55758i
\(330\) 0 0
\(331\) −9.48582e28 + 9.48582e28i −0.997821 + 0.997821i −0.999998 0.00217642i \(-0.999307\pi\)
0.00217642 + 0.999998i \(0.499307\pi\)
\(332\) 0 0
\(333\) −3.78544e28 3.78544e28i −0.371542 0.371542i
\(334\) 0 0
\(335\) 3.05144e29 2.79569
\(336\) 0 0
\(337\) −1.33751e29 −1.14434 −0.572168 0.820136i \(-0.693898\pi\)
−0.572168 + 0.820136i \(0.693898\pi\)
\(338\) 0 0
\(339\) −1.18338e29 1.18338e29i −0.945858 0.945858i
\(340\) 0 0
\(341\) −1.67647e28 + 1.67647e28i −0.125234 + 0.125234i
\(342\) 0 0
\(343\) 4.12676e29i 2.88222i
\(344\) 0 0
\(345\) 4.53059e28i 0.295962i
\(346\) 0 0
\(347\) −1.94151e27 + 1.94151e27i −0.0118673 + 0.0118673i −0.713016 0.701148i \(-0.752671\pi\)
0.701148 + 0.713016i \(0.252671\pi\)
\(348\) 0 0
\(349\) 2.03246e28 + 2.03246e28i 0.116286 + 0.116286i 0.762855 0.646569i \(-0.223797\pi\)
−0.646569 + 0.762855i \(0.723797\pi\)
\(350\) 0 0
\(351\) −7.86748e28 −0.421506
\(352\) 0 0
\(353\) −2.69206e28 −0.135106 −0.0675532 0.997716i \(-0.521519\pi\)
−0.0675532 + 0.997716i \(0.521519\pi\)
\(354\) 0 0
\(355\) −1.53092e29 1.53092e29i −0.719990 0.719990i
\(356\) 0 0
\(357\) 1.07427e29 1.07427e29i 0.473621 0.473621i
\(358\) 0 0
\(359\) 2.01350e29i 0.832465i −0.909258 0.416233i \(-0.863350\pi\)
0.909258 0.416233i \(-0.136650\pi\)
\(360\) 0 0
\(361\) 3.38052e28i 0.131115i
\(362\) 0 0
\(363\) −2.22124e29 + 2.22124e29i −0.808479 + 0.808479i
\(364\) 0 0
\(365\) 1.62475e29 + 1.62475e29i 0.555159 + 0.555159i
\(366\) 0 0
\(367\) 6.32569e28 0.202978 0.101489 0.994837i \(-0.467639\pi\)
0.101489 + 0.994837i \(0.467639\pi\)
\(368\) 0 0
\(369\) 1.79082e28 0.0539821
\(370\) 0 0
\(371\) −1.56201e29 1.56201e29i −0.442469 0.442469i
\(372\) 0 0
\(373\) 2.80897e29 2.80897e29i 0.747989 0.747989i −0.226112 0.974101i \(-0.572602\pi\)
0.974101 + 0.226112i \(0.0726016\pi\)
\(374\) 0 0
\(375\) 1.83144e30i 4.58599i
\(376\) 0 0
\(377\) 1.63920e29i 0.386106i
\(378\) 0 0
\(379\) −2.80353e29 + 2.80353e29i −0.621377 + 0.621377i −0.945883 0.324507i \(-0.894802\pi\)
0.324507 + 0.945883i \(0.394802\pi\)
\(380\) 0 0
\(381\) −7.96613e28 7.96613e28i −0.166192 0.166192i
\(382\) 0 0
\(383\) −3.35364e29 −0.658765 −0.329383 0.944197i \(-0.606841\pi\)
−0.329383 + 0.944197i \(0.606841\pi\)
\(384\) 0 0
\(385\) 5.77618e29 1.06867
\(386\) 0 0
\(387\) 2.51217e29 + 2.51217e29i 0.437897 + 0.437897i
\(388\) 0 0
\(389\) −3.08430e29 + 3.08430e29i −0.506682 + 0.506682i −0.913507 0.406824i \(-0.866636\pi\)
0.406824 + 0.913507i \(0.366636\pi\)
\(390\) 0 0
\(391\) 2.22427e28i 0.0344472i
\(392\) 0 0
\(393\) 9.47660e29i 1.38401i
\(394\) 0 0
\(395\) 3.31193e29 3.31193e29i 0.456262 0.456262i
\(396\) 0 0
\(397\) −2.25999e29 2.25999e29i −0.293776 0.293776i 0.544794 0.838570i \(-0.316608\pi\)
−0.838570 + 0.544794i \(0.816608\pi\)
\(398\) 0 0
\(399\) −2.03285e30 −2.49412
\(400\) 0 0
\(401\) −1.34782e30 −1.56125 −0.780624 0.625000i \(-0.785099\pi\)
−0.780624 + 0.625000i \(0.785099\pi\)
\(402\) 0 0
\(403\) −3.01711e29 3.01711e29i −0.330053 0.330053i
\(404\) 0 0
\(405\) −1.67806e30 + 1.67806e30i −1.73410 + 1.73410i
\(406\) 0 0
\(407\) 2.79489e29i 0.272916i
\(408\) 0 0
\(409\) 1.63159e30i 1.50589i −0.658084 0.752944i \(-0.728633\pi\)
0.658084 0.752944i \(-0.271367\pi\)
\(410\) 0 0
\(411\) −8.36374e28 + 8.36374e28i −0.0729827 + 0.0729827i
\(412\) 0 0
\(413\) 7.93971e29 + 7.93971e29i 0.655208 + 0.655208i
\(414\) 0 0
\(415\) −4.04022e30 −3.15393
\(416\) 0 0
\(417\) −2.21710e30 −1.63765
\(418\) 0 0
\(419\) 5.98958e29 + 5.98958e29i 0.418731 + 0.418731i 0.884766 0.466035i \(-0.154318\pi\)
−0.466035 + 0.884766i \(0.654318\pi\)
\(420\) 0 0
\(421\) 5.98556e29 5.98556e29i 0.396151 0.396151i −0.480722 0.876873i \(-0.659625\pi\)
0.876873 + 0.480722i \(0.159625\pi\)
\(422\) 0 0
\(423\) 7.35743e29i 0.461118i
\(424\) 0 0
\(425\) 1.38026e30i 0.819383i
\(426\) 0 0
\(427\) 2.56570e30 2.56570e30i 1.44307 1.44307i
\(428\) 0 0
\(429\) 3.64417e29 + 3.64417e29i 0.194240 + 0.194240i
\(430\) 0 0
\(431\) −5.71960e29 −0.288986 −0.144493 0.989506i \(-0.546155\pi\)
−0.144493 + 0.989506i \(0.546155\pi\)
\(432\) 0 0
\(433\) −2.64080e30 −1.26510 −0.632551 0.774519i \(-0.717992\pi\)
−0.632551 + 0.774519i \(0.717992\pi\)
\(434\) 0 0
\(435\) 1.93579e30 + 1.93579e30i 0.879493 + 0.879493i
\(436\) 0 0
\(437\) −2.10449e29 + 2.10449e29i −0.0907008 + 0.0907008i
\(438\) 0 0
\(439\) 2.25796e29i 0.0923366i −0.998934 0.0461683i \(-0.985299\pi\)
0.998934 0.0461683i \(-0.0147010\pi\)
\(440\) 0 0
\(441\) 3.63235e30i 1.40976i
\(442\) 0 0
\(443\) 5.63800e29 5.63800e29i 0.207722 0.207722i −0.595577 0.803299i \(-0.703076\pi\)
0.803299 + 0.595577i \(0.203076\pi\)
\(444\) 0 0
\(445\) 2.48405e30 + 2.48405e30i 0.869001 + 0.869001i
\(446\) 0 0
\(447\) 6.00348e30 1.99465
\(448\) 0 0
\(449\) 3.18945e30 1.00666 0.503330 0.864094i \(-0.332108\pi\)
0.503330 + 0.864094i \(0.332108\pi\)
\(450\) 0 0
\(451\) 6.61107e28 + 6.61107e28i 0.0198263 + 0.0198263i
\(452\) 0 0
\(453\) 1.09769e29 1.09769e29i 0.0312861 0.0312861i
\(454\) 0 0
\(455\) 1.03953e31i 2.81646i
\(456\) 0 0
\(457\) 4.87740e28i 0.0125647i 0.999980 + 0.00628235i \(0.00199975\pi\)
−0.999980 + 0.00628235i \(0.998000\pi\)
\(458\) 0 0
\(459\) −4.56135e29 + 4.56135e29i −0.111750 + 0.111750i
\(460\) 0 0
\(461\) −2.25019e30 2.25019e30i −0.524394 0.524394i 0.394501 0.918896i \(-0.370917\pi\)
−0.918896 + 0.394501i \(0.870917\pi\)
\(462\) 0 0
\(463\) −1.96033e29 −0.0434658 −0.0217329 0.999764i \(-0.506918\pi\)
−0.0217329 + 0.999764i \(0.506918\pi\)
\(464\) 0 0
\(465\) −7.12603e30 −1.50362
\(466\) 0 0
\(467\) −2.11336e30 2.11336e30i −0.424452 0.424452i 0.462281 0.886733i \(-0.347031\pi\)
−0.886733 + 0.462281i \(0.847031\pi\)
\(468\) 0 0
\(469\) 9.88015e30 9.88015e30i 1.88919 1.88919i
\(470\) 0 0
\(471\) 8.14883e30i 1.48373i
\(472\) 0 0
\(473\) 1.85480e30i 0.321658i
\(474\) 0 0
\(475\) 1.30593e31 1.30593e31i 2.15747 2.15747i
\(476\) 0 0
\(477\) −8.32157e29 8.32157e29i −0.130992 0.130992i
\(478\) 0 0
\(479\) −7.36090e30 −1.10426 −0.552131 0.833757i \(-0.686185\pi\)
−0.552131 + 0.833757i \(0.686185\pi\)
\(480\) 0 0
\(481\) 5.02991e30 0.719270
\(482\) 0 0
\(483\) −1.46695e30 1.46695e30i −0.199996 0.199996i
\(484\) 0 0
\(485\) −8.62739e30 + 8.62739e30i −1.12163 + 1.12163i
\(486\) 0 0
\(487\) 1.98979e30i 0.246731i 0.992361 + 0.123365i \(0.0393687\pi\)
−0.992361 + 0.123365i \(0.960631\pi\)
\(488\) 0 0
\(489\) 3.49116e29i 0.0412970i
\(490\) 0 0
\(491\) −7.44622e30 + 7.44622e30i −0.840425 + 0.840425i −0.988914 0.148489i \(-0.952559\pi\)
0.148489 + 0.988914i \(0.452559\pi\)
\(492\) 0 0
\(493\) 9.50366e29 + 9.50366e29i 0.102365 + 0.102365i
\(494\) 0 0
\(495\) 3.07725e30 0.316376
\(496\) 0 0
\(497\) −9.91384e30 −0.973069
\(498\) 0 0
\(499\) −7.80043e30 7.80043e30i −0.731076 0.731076i 0.239757 0.970833i \(-0.422932\pi\)
−0.970833 + 0.239757i \(0.922932\pi\)
\(500\) 0 0
\(501\) −1.37216e30 + 1.37216e30i −0.122820 + 0.122820i
\(502\) 0 0
\(503\) 1.33936e31i 1.14515i 0.819851 + 0.572577i \(0.194056\pi\)
−0.819851 + 0.572577i \(0.805944\pi\)
\(504\) 0 0
\(505\) 1.66462e31i 1.35977i
\(506\) 0 0
\(507\) 4.74350e30 4.74350e30i 0.370261 0.370261i
\(508\) 0 0
\(509\) 9.56104e30 + 9.56104e30i 0.713264 + 0.713264i 0.967217 0.253952i \(-0.0817306\pi\)
−0.253952 + 0.967217i \(0.581731\pi\)
\(510\) 0 0
\(511\) 1.05214e31 0.750299
\(512\) 0 0
\(513\) 8.63146e30 0.588484
\(514\) 0 0
\(515\) 4.38269e30 + 4.38269e30i 0.285731 + 0.285731i
\(516\) 0 0
\(517\) 2.71610e30 2.71610e30i 0.169357 0.169357i
\(518\) 0 0
\(519\) 3.62641e31i 2.16298i
\(520\) 0 0
\(521\) 2.57417e31i 1.46894i −0.678641 0.734470i \(-0.737431\pi\)
0.678641 0.734470i \(-0.262569\pi\)
\(522\) 0 0
\(523\) 9.61915e30 9.61915e30i 0.525251 0.525251i −0.393901 0.919153i \(-0.628875\pi\)
0.919153 + 0.393901i \(0.128875\pi\)
\(524\) 0 0
\(525\) 9.10305e31 + 9.10305e31i 4.75724 + 4.75724i
\(526\) 0 0
\(527\) −3.49848e30 −0.175008
\(528\) 0 0
\(529\) 2.05767e31 0.985454
\(530\) 0 0
\(531\) 4.22987e30 + 4.22987e30i 0.193972 + 0.193972i
\(532\) 0 0
\(533\) −1.18978e30 + 1.18978e30i −0.0522521 + 0.0522521i
\(534\) 0 0
\(535\) 5.26811e31i 2.21609i
\(536\) 0 0
\(537\) 5.47108e31i 2.20480i
\(538\) 0 0
\(539\) 1.34093e31 1.34093e31i 0.517769 0.517769i
\(540\) 0 0
\(541\) −1.95813e31 1.95813e31i −0.724557 0.724557i 0.244973 0.969530i \(-0.421221\pi\)
−0.969530 + 0.244973i \(0.921221\pi\)
\(542\) 0 0
\(543\) −2.89889e31 −1.02810
\(544\) 0 0
\(545\) −5.25527e31 −1.78663
\(546\) 0 0
\(547\) 1.30856e31 + 1.30856e31i 0.426520 + 0.426520i 0.887441 0.460921i \(-0.152481\pi\)
−0.460921 + 0.887441i \(0.652481\pi\)
\(548\) 0 0
\(549\) 1.36688e31 1.36688e31i 0.427215 0.427215i
\(550\) 0 0
\(551\) 1.79838e31i 0.539061i
\(552\) 0 0
\(553\) 2.14472e31i 0.616639i
\(554\) 0 0
\(555\) 5.94000e31 5.94000e31i 1.63839 1.63839i
\(556\) 0 0
\(557\) 1.67716e30 + 1.67716e30i 0.0443854 + 0.0443854i 0.728951 0.684566i \(-0.240008\pi\)
−0.684566 + 0.728951i \(0.740008\pi\)
\(558\) 0 0
\(559\) −3.33805e31 −0.847728
\(560\) 0 0
\(561\) 4.22559e30 0.102994
\(562\) 0 0
\(563\) 7.89860e30 + 7.89860e30i 0.184801 + 0.184801i 0.793444 0.608643i \(-0.208286\pi\)
−0.608643 + 0.793444i \(0.708286\pi\)
\(564\) 0 0
\(565\) 6.63897e31 6.63897e31i 1.49123 1.49123i
\(566\) 0 0
\(567\) 1.08667e32i 2.34365i
\(568\) 0 0
\(569\) 4.64202e31i 0.961426i −0.876878 0.480713i \(-0.840378\pi\)
0.876878 0.480713i \(-0.159622\pi\)
\(570\) 0 0
\(571\) 1.74215e31 1.74215e31i 0.346553 0.346553i −0.512271 0.858824i \(-0.671196\pi\)
0.858824 + 0.512271i \(0.171196\pi\)
\(572\) 0 0
\(573\) −2.95592e31 2.95592e31i −0.564825 0.564825i
\(574\) 0 0
\(575\) 1.88477e31 0.346002
\(576\) 0 0
\(577\) −1.76828e31 −0.311909 −0.155955 0.987764i \(-0.549845\pi\)
−0.155955 + 0.987764i \(0.549845\pi\)
\(578\) 0 0
\(579\) −5.60885e31 5.60885e31i −0.950759 0.950759i
\(580\) 0 0
\(581\) −1.30817e32 + 1.30817e32i −2.13127 + 2.13127i
\(582\) 0 0
\(583\) 6.14404e30i 0.0962201i
\(584\) 0 0
\(585\) 5.53806e31i 0.833806i
\(586\) 0 0
\(587\) 5.35811e31 5.35811e31i 0.775664 0.775664i −0.203427 0.979090i \(-0.565208\pi\)
0.979090 + 0.203427i \(0.0652078\pi\)
\(588\) 0 0
\(589\) 3.31009e31 + 3.31009e31i 0.460802 + 0.460802i
\(590\) 0 0
\(591\) −1.21659e32 −1.62887
\(592\) 0 0
\(593\) −9.64929e31 −1.24270 −0.621350 0.783533i \(-0.713416\pi\)
−0.621350 + 0.783533i \(0.713416\pi\)
\(594\) 0 0
\(595\) 6.02690e31 + 6.02690e31i 0.746704 + 0.746704i
\(596\) 0 0
\(597\) 5.76082e31 5.76082e31i 0.686719 0.686719i
\(598\) 0 0
\(599\) 1.21951e32i 1.39886i 0.714699 + 0.699432i \(0.246564\pi\)
−0.714699 + 0.699432i \(0.753436\pi\)
\(600\) 0 0
\(601\) 1.50501e32i 1.66143i 0.556699 + 0.830714i \(0.312068\pi\)
−0.556699 + 0.830714i \(0.687932\pi\)
\(602\) 0 0
\(603\) 5.26364e31 5.26364e31i 0.559290 0.559290i
\(604\) 0 0
\(605\) −1.24616e32 1.24616e32i −1.27464 1.27464i
\(606\) 0 0
\(607\) −8.41878e31 −0.829047 −0.414523 0.910039i \(-0.636052\pi\)
−0.414523 + 0.910039i \(0.636052\pi\)
\(608\) 0 0
\(609\) 1.25357e32 1.18864
\(610\) 0 0
\(611\) 4.88810e31 + 4.88810e31i 0.446341 + 0.446341i
\(612\) 0 0
\(613\) 7.36701e31 7.36701e31i 0.647882 0.647882i −0.304599 0.952481i \(-0.598522\pi\)
0.952481 + 0.304599i \(0.0985224\pi\)
\(614\) 0 0
\(615\) 2.81011e31i 0.238045i
\(616\) 0 0
\(617\) 1.78250e32i 1.45462i 0.686310 + 0.727309i \(0.259229\pi\)
−0.686310 + 0.727309i \(0.740771\pi\)
\(618\) 0 0
\(619\) −1.05467e32 + 1.05467e32i −0.829225 + 0.829225i −0.987410 0.158185i \(-0.949436\pi\)
0.158185 + 0.987410i \(0.449436\pi\)
\(620\) 0 0
\(621\) 6.22863e30 + 6.22863e30i 0.0471888 + 0.0471888i
\(622\) 0 0
\(623\) 1.60860e32 1.17446
\(624\) 0 0
\(625\) −6.19790e32 −4.36138
\(626\) 0 0
\(627\) −3.99804e31 3.99804e31i −0.271188 0.271188i
\(628\) 0 0
\(629\) 2.91621e31 2.91621e31i 0.190694 0.190694i
\(630\) 0 0
\(631\) 1.35896e32i 0.856780i −0.903594 0.428390i \(-0.859081\pi\)
0.903594 0.428390i \(-0.140919\pi\)
\(632\) 0 0
\(633\) 1.61268e31i 0.0980408i
\(634\) 0 0
\(635\) 4.46916e31 4.46916e31i 0.262016 0.262016i
\(636\) 0 0
\(637\) 2.41325e32 + 2.41325e32i 1.36458 + 1.36458i
\(638\) 0 0
\(639\) −5.28159e31 −0.288074
\(640\) 0 0
\(641\) 2.63596e32 1.38698 0.693492 0.720464i \(-0.256071\pi\)
0.693492 + 0.720464i \(0.256071\pi\)
\(642\) 0 0
\(643\) −4.62276e31 4.62276e31i −0.234680 0.234680i 0.579963 0.814643i \(-0.303067\pi\)
−0.814643 + 0.579963i \(0.803067\pi\)
\(644\) 0 0
\(645\) −3.94202e32 + 3.94202e32i −1.93100 + 1.93100i
\(646\) 0 0
\(647\) 1.71570e32i 0.811040i 0.914086 + 0.405520i \(0.132910\pi\)
−0.914086 + 0.405520i \(0.867090\pi\)
\(648\) 0 0
\(649\) 3.12303e31i 0.142483i
\(650\) 0 0
\(651\) −2.30731e32 + 2.30731e32i −1.01607 + 1.01607i
\(652\) 0 0
\(653\) 1.18613e32 + 1.18613e32i 0.504233 + 0.504233i 0.912750 0.408517i \(-0.133954\pi\)
−0.408517 + 0.912750i \(0.633954\pi\)
\(654\) 0 0
\(655\) −5.31656e32 −2.18201
\(656\) 0 0
\(657\) 5.60528e31 0.222124
\(658\) 0 0
\(659\) 3.26973e31 + 3.26973e31i 0.125121 + 0.125121i 0.766894 0.641773i \(-0.221801\pi\)
−0.641773 + 0.766894i \(0.721801\pi\)
\(660\) 0 0
\(661\) −1.06496e32 + 1.06496e32i −0.393566 + 0.393566i −0.875956 0.482391i \(-0.839769\pi\)
0.482391 + 0.875956i \(0.339769\pi\)
\(662\) 0 0
\(663\) 7.60469e31i 0.271441i
\(664\) 0 0
\(665\) 1.14047e33i 3.93220i
\(666\) 0 0
\(667\) 1.29775e31 1.29775e31i 0.0432257 0.0432257i
\(668\) 0 0
\(669\) −3.67161e31 3.67161e31i −0.118156 0.118156i
\(670\) 0 0
\(671\) 1.00920e32 0.313812
\(672\) 0 0
\(673\) −2.76608e32 −0.831172 −0.415586 0.909554i \(-0.636423\pi\)
−0.415586 + 0.909554i \(0.636423\pi\)
\(674\) 0 0
\(675\) −3.86514e32 3.86514e32i −1.12246 1.12246i
\(676\) 0 0
\(677\) 3.32911e32 3.32911e32i 0.934455 0.934455i −0.0635250 0.997980i \(-0.520234\pi\)
0.997980 + 0.0635250i \(0.0202343\pi\)
\(678\) 0 0
\(679\) 5.58687e32i 1.51589i
\(680\) 0 0
\(681\) 1.88363e32i 0.494087i
\(682\) 0 0
\(683\) −3.04233e32 + 3.04233e32i −0.771558 + 0.771558i −0.978379 0.206821i \(-0.933688\pi\)
0.206821 + 0.978379i \(0.433688\pi\)
\(684\) 0 0
\(685\) −4.69223e31 4.69223e31i −0.115063 0.115063i
\(686\) 0 0
\(687\) −2.88599e32 −0.684372
\(688\) 0 0
\(689\) 1.10573e32 0.253588
\(690\) 0 0
\(691\) 3.88350e32 + 3.88350e32i 0.861441 + 0.861441i 0.991506 0.130065i \(-0.0415184\pi\)
−0.130065 + 0.991506i \(0.541518\pi\)
\(692\) 0 0
\(693\) 9.96373e31 9.96373e31i 0.213791 0.213791i
\(694\) 0 0
\(695\) 1.24384e33i 2.58189i
\(696\) 0 0
\(697\) 1.37961e31i 0.0277063i
\(698\) 0 0
\(699\) 6.19244e32 6.19244e32i 1.20330 1.20330i
\(700\) 0 0
\(701\) −2.88920e32 2.88920e32i −0.543275 0.543275i 0.381213 0.924487i \(-0.375507\pi\)
−0.924487 + 0.381213i \(0.875507\pi\)
\(702\) 0 0
\(703\) −5.51835e32 −1.00421
\(704\) 0 0
\(705\) 1.15451e33 2.03339
\(706\) 0 0
\(707\) −5.38982e32 5.38982e32i −0.918864 0.918864i
\(708\) 0 0
\(709\) −2.50043e32 + 2.50043e32i −0.412651 + 0.412651i −0.882661 0.470010i \(-0.844250\pi\)
0.470010 + 0.882661i \(0.344250\pi\)
\(710\) 0 0
\(711\) 1.14260e32i 0.182554i
\(712\) 0 0
\(713\) 4.77726e31i 0.0739008i
\(714\) 0 0
\(715\) −2.04445e32 + 2.04445e32i −0.306237 + 0.306237i
\(716\) 0 0
\(717\) 7.02432e32 + 7.02432e32i 1.01891 + 1.01891i
\(718\) 0 0
\(719\) 6.60858e32 0.928380 0.464190 0.885736i \(-0.346345\pi\)
0.464190 + 0.885736i \(0.346345\pi\)
\(720\) 0 0
\(721\) 2.83811e32 0.386166
\(722\) 0 0
\(723\) 5.94494e32 + 5.94494e32i 0.783532 + 0.783532i
\(724\) 0 0
\(725\) −8.05309e32 + 8.05309e32i −1.02819 + 1.02819i
\(726\) 0 0
\(727\) 1.64076e31i 0.0202955i 0.999949 + 0.0101478i \(0.00323019\pi\)
−0.999949 + 0.0101478i \(0.996770\pi\)
\(728\) 0 0
\(729\) 2.92431e30i 0.00350475i
\(730\) 0 0
\(731\) −1.93531e32 + 1.93531e32i −0.224751 + 0.224751i
\(732\) 0 0
\(733\) −5.35292e32 5.35292e32i −0.602414 0.602414i 0.338538 0.940953i \(-0.390068\pi\)
−0.940953 + 0.338538i \(0.890068\pi\)
\(734\) 0 0
\(735\) 5.69978e33 6.21661
\(736\) 0 0
\(737\) 3.88629e32 0.410827
\(738\) 0 0
\(739\) −5.31681e32 5.31681e32i −0.544804 0.544804i 0.380130 0.924933i \(-0.375879\pi\)
−0.924933 + 0.380130i \(0.875879\pi\)
\(740\) 0 0
\(741\) 7.19519e32 7.19519e32i 0.714715 0.714715i
\(742\) 0 0
\(743\) 5.11766e32i 0.492834i 0.969164 + 0.246417i \(0.0792532\pi\)
−0.969164 + 0.246417i \(0.920747\pi\)
\(744\) 0 0
\(745\) 3.36807e33i 3.14474i
\(746\) 0 0
\(747\) −6.96926e32 + 6.96926e32i −0.630957 + 0.630957i
\(748\) 0 0
\(749\) −1.70575e33 1.70575e33i −1.49752 1.49752i
\(750\) 0 0
\(751\) −1.09027e33 −0.928267 −0.464133 0.885765i \(-0.653634\pi\)
−0.464133 + 0.885765i \(0.653634\pi\)
\(752\) 0 0
\(753\) −1.45386e33 −1.20055
\(754\) 0 0
\(755\) 6.15827e31 + 6.15827e31i 0.0493252 + 0.0493252i
\(756\) 0 0
\(757\) 1.46727e33 1.46727e33i 1.14001 1.14001i 0.151557 0.988449i \(-0.451571\pi\)
0.988449 0.151557i \(-0.0484287\pi\)
\(758\) 0 0
\(759\) 5.77013e31i 0.0434916i
\(760\) 0 0
\(761\) 9.58389e32i 0.700839i 0.936593 + 0.350419i \(0.113961\pi\)
−0.936593 + 0.350419i \(0.886039\pi\)
\(762\) 0 0
\(763\) −1.70159e33 + 1.70159e33i −1.20732 + 1.20732i
\(764\) 0 0
\(765\) 3.21082e32 + 3.21082e32i 0.221060 + 0.221060i
\(766\) 0 0
\(767\) −5.62045e32 −0.375512
\(768\) 0 0
\(769\) −8.03545e32 −0.521023 −0.260512 0.965471i \(-0.583891\pi\)
−0.260512 + 0.965471i \(0.583891\pi\)
\(770\) 0 0
\(771\) 2.26707e32 + 2.26707e32i 0.142672 + 0.142672i
\(772\) 0 0
\(773\) −9.91141e32 + 9.91141e32i −0.605440 + 0.605440i −0.941751 0.336311i \(-0.890821\pi\)
0.336311 + 0.941751i \(0.390821\pi\)
\(774\) 0 0
\(775\) 2.96450e33i 1.75785i
\(776\) 0 0
\(777\) 3.84659e33i 2.21429i
\(778\) 0 0
\(779\) 1.30532e32 1.30532e32i 0.0729516 0.0729516i
\(780\) 0 0
\(781\) −1.94977e32 1.94977e32i −0.105803 0.105803i
\(782\) 0 0
\(783\) −5.32263e32 −0.280457
\(784\) 0 0
\(785\) 4.57165e33 2.33923
\(786\) 0 0
\(787\) 2.09360e33 + 2.09360e33i 1.04036 + 1.04036i 0.999150 + 0.0412119i \(0.0131219\pi\)
0.0412119 + 0.999150i \(0.486878\pi\)
\(788\) 0 0
\(789\) 2.90775e33 2.90775e33i 1.40337 1.40337i
\(790\) 0 0
\(791\) 4.29922e33i 2.01540i
\(792\) 0 0
\(793\) 1.81624e33i 0.827049i
\(794\) 0 0
\(795\) 1.30580e33 1.30580e33i 0.577635 0.577635i
\(796\) 0 0
\(797\) −3.26822e33 3.26822e33i −1.40456 1.40456i −0.784756 0.619805i \(-0.787212\pi\)
−0.619805 0.784756i \(-0.712788\pi\)
\(798\) 0 0
\(799\) 5.66798e32 0.236669
\(800\) 0 0
\(801\) 8.56982e32 0.347695
\(802\) 0 0
\(803\) 2.06927e32 + 2.06927e32i 0.0815807 + 0.0815807i
\(804\) 0 0
\(805\) 8.22986e32 8.22986e32i 0.315312 0.315312i
\(806\) 0 0
\(807\) 3.94616e33i 1.46936i
\(808\) 0 0
\(809\) 1.60492e33i 0.580824i 0.956902 + 0.290412i \(0.0937924\pi\)
−0.956902 + 0.290412i \(0.906208\pi\)
\(810\) 0 0
\(811\) 2.22096e33 2.22096e33i 0.781270 0.781270i −0.198775 0.980045i \(-0.563696\pi\)
0.980045 + 0.198775i \(0.0636964\pi\)
\(812\) 0 0
\(813\) −1.20235e33 1.20235e33i −0.411141 0.411141i
\(814\) 0 0
\(815\) −1.95861e32 −0.0651083
\(816\) 0 0
\(817\) 3.66219e33 1.18355
\(818\) 0 0
\(819\) 1.79315e33 + 1.79315e33i 0.563446 + 0.563446i
\(820\) 0 0
\(821\) −3.89933e33 + 3.89933e33i −1.19136 + 1.19136i −0.214676 + 0.976685i \(0.568870\pi\)
−0.976685 + 0.214676i \(0.931130\pi\)
\(822\) 0 0
\(823\) 1.16137e33i 0.345042i −0.985006 0.172521i \(-0.944809\pi\)
0.985006 0.172521i \(-0.0551913\pi\)
\(824\) 0 0
\(825\) 3.58062e33i 1.03452i
\(826\) 0 0
\(827\) 1.77941e33 1.77941e33i 0.499992 0.499992i −0.411444 0.911435i \(-0.634975\pi\)
0.911435 + 0.411444i \(0.134975\pi\)
\(828\) 0 0
\(829\) 3.65065e32 + 3.65065e32i 0.0997683 + 0.0997683i 0.755229 0.655461i \(-0.227526\pi\)
−0.655461 + 0.755229i \(0.727526\pi\)
\(830\) 0 0
\(831\) 2.94455e33 0.782721
\(832\) 0 0
\(833\) 2.79827e33 0.723557
\(834\) 0 0
\(835\) −7.69808e32 7.69808e32i −0.193637 0.193637i
\(836\) 0 0
\(837\) 9.79682e32 9.79682e32i 0.239741 0.239741i
\(838\) 0 0
\(839\) 4.96177e32i 0.118134i −0.998254 0.0590669i \(-0.981187\pi\)
0.998254 0.0590669i \(-0.0188125\pi\)
\(840\) 0 0
\(841\) 3.20774e33i 0.743097i
\(842\) 0 0
\(843\) −1.91067e33 + 1.91067e33i −0.430695 + 0.430695i
\(844\) 0 0
\(845\) 2.66120e33 + 2.66120e33i 0.583748 + 0.583748i
\(846\) 0 0
\(847\) −8.06979e33 −1.72267
\(848\) 0 0
\(849\) 1.17882e34 2.44912
\(850\) 0 0
\(851\) −3.98215e32 3.98215e32i −0.0805244 0.0805244i
\(852\) 0 0
\(853\) 1.43738e33 1.43738e33i 0.282917 0.282917i −0.551354 0.834271i \(-0.685889\pi\)
0.834271 + 0.551354i \(0.185889\pi\)
\(854\) 0 0
\(855\) 6.07584e33i 1.16412i
\(856\) 0 0
\(857\) 4.00126e33i 0.746307i 0.927770 + 0.373153i \(0.121723\pi\)
−0.927770 + 0.373153i \(0.878277\pi\)
\(858\) 0 0
\(859\) 5.56085e33 5.56085e33i 1.00976 1.00976i 0.00981133 0.999952i \(-0.496877\pi\)
0.999952 0.00981133i \(-0.00312309\pi\)
\(860\) 0 0
\(861\) 9.09876e32 + 9.09876e32i 0.160859 + 0.160859i
\(862\) 0 0
\(863\) −6.88999e33 −1.18603 −0.593014 0.805192i \(-0.702062\pi\)
−0.593014 + 0.805192i \(0.702062\pi\)
\(864\) 0 0
\(865\) −2.03449e34 −3.41013
\(866\) 0 0
\(867\) −4.96387e33 4.96387e33i −0.810215 0.810215i
\(868\) 0 0
\(869\) 4.21805e32 4.21805e32i 0.0670477 0.0670477i
\(870\) 0 0
\(871\) 6.99407e33i 1.08273i
\(872\) 0 0
\(873\) 2.97640e33i 0.448774i
\(874\) 0 0
\(875\) −3.32683e34 + 3.32683e34i −4.88583 + 4.88583i
\(876\) 0 0
\(877\) −5.71470e33 5.71470e33i −0.817519 0.817519i 0.168229 0.985748i \(-0.446195\pi\)
−0.985748 + 0.168229i \(0.946195\pi\)
\(878\) 0 0
\(879\) 5.35480e33 0.746227
\(880\) 0 0
\(881\) −6.34532e33 −0.861452 −0.430726 0.902483i \(-0.641742\pi\)
−0.430726 + 0.902483i \(0.641742\pi\)
\(882\) 0 0
\(883\) −8.69292e33 8.69292e33i −1.14979 1.14979i −0.986594 0.163194i \(-0.947820\pi\)
−0.163194 0.986594i \(-0.552180\pi\)
\(884\) 0 0
\(885\) −6.63739e33 + 6.63739e33i −0.855361 + 0.855361i
\(886\) 0 0
\(887\) 1.18530e34i 1.48836i −0.667982 0.744178i \(-0.732842\pi\)
0.667982 0.744178i \(-0.267158\pi\)
\(888\) 0 0
\(889\) 2.89411e33i 0.354115i
\(890\) 0 0
\(891\) −2.13717e33 + 2.13717e33i −0.254827 + 0.254827i
\(892\) 0 0
\(893\) −5.36277e33 5.36277e33i −0.623157 0.623157i
\(894\) 0 0
\(895\) 3.06938e34 3.47605
\(896\) 0 0
\(897\) 1.03844e33 0.114622
\(898\) 0 0
\(899\) −2.04119e33 2.04119e33i −0.219607 0.219607i
\(900\) 0 0
\(901\) 6.41073e32 6.41073e32i 0.0672314 0.0672314i
\(902\) 0 0
\(903\) 2.55275e34i 2.60975i
\(904\) 0 0
\(905\) 1.62633e34i 1.62088i
\(906\) 0 0
\(907\) −2.64414e33 + 2.64414e33i −0.256921 + 0.256921i −0.823801 0.566879i \(-0.808151\pi\)
0.566879 + 0.823801i \(0.308151\pi\)
\(908\) 0 0
\(909\) −2.87142e33 2.87142e33i −0.272027 0.272027i
\(910\) 0 0
\(911\) −7.75129e33 −0.715999 −0.358000 0.933722i \(-0.616541\pi\)
−0.358000 + 0.933722i \(0.616541\pi\)
\(912\) 0 0
\(913\) −5.14560e33 −0.463470
\(914\) 0 0
\(915\) 2.14486e34 + 2.14486e34i 1.88389 + 1.88389i
\(916\) 0 0
\(917\) −1.72143e34 + 1.72143e34i −1.47449 + 1.47449i
\(918\) 0 0
\(919\) 1.55693e34i 1.30059i −0.759680 0.650297i \(-0.774645\pi\)
0.759680 0.650297i \(-0.225355\pi\)
\(920\) 0 0
\(921\) 2.59593e34i 2.11498i
\(922\) 0 0
\(923\) 3.50896e33 3.50896e33i 0.278842 0.278842i
\(924\) 0 0
\(925\) 2.47110e34 + 2.47110e34i 1.91540 + 1.91540i
\(926\) 0 0
\(927\) 1.51200e33 0.114323
\(928\) 0 0
\(929\) 1.86088e34 1.37258 0.686290 0.727328i \(-0.259238\pi\)
0.686290 + 0.727328i \(0.259238\pi\)
\(930\) 0 0
\(931\) −2.64759e34 2.64759e34i −1.90515 1.90515i
\(932\) 0 0
\(933\) 9.79306e32 9.79306e32i 0.0687511 0.0687511i
\(934\) 0 0
\(935\) 2.37064e33i 0.162380i
\(936\) 0 0
\(937\) 3.57732e33i 0.239085i 0.992829 + 0.119543i \(0.0381428\pi\)
−0.992829 + 0.119543i \(0.961857\pi\)
\(938\) 0 0
\(939\) 1.24611e34 1.24611e34i 0.812645 0.812645i
\(940\) 0 0
\(941\) −7.01549e33 7.01549e33i −0.446455 0.446455i 0.447719 0.894174i \(-0.352236\pi\)
−0.894174 + 0.447719i \(0.852236\pi\)
\(942\) 0 0
\(943\) 1.88388e32 0.0116996
\(944\) 0 0
\(945\) −3.37543e34 −2.04580
\(946\) 0 0
\(947\) 1.89121e34 + 1.89121e34i 1.11870 + 1.11870i 0.991932 + 0.126773i \(0.0404619\pi\)
0.126773 + 0.991932i \(0.459538\pi\)
\(948\) 0 0
\(949\) −3.72401e33 + 3.72401e33i −0.215006 + 0.215006i
\(950\) 0 0
\(951\) 2.22483e34i 1.25378i
\(952\) 0 0
\(953\) 7.18956e33i 0.395489i −0.980254 0.197744i \(-0.936638\pi\)
0.980254 0.197744i \(-0.0633616\pi\)
\(954\) 0 0
\(955\) 1.65833e34 1.65833e34i 0.890496 0.890496i
\(956\) 0 0
\(957\) 2.46541e33 + 2.46541e33i 0.129242 + 0.129242i
\(958\) 0 0
\(959\) −3.03856e33 −0.155509
\(960\) 0 0
\(961\) −1.24993e34 −0.624550
\(962\) 0 0
\(963\) −9.08734e33 9.08734e33i −0.443337 0.443337i
\(964\) 0 0
\(965\) 3.14668e34 3.14668e34i 1.49895 1.49895i
\(966\) 0 0
\(967\) 3.64651e33i 0.169618i −0.996397 0.0848092i \(-0.972972\pi\)
0.996397 0.0848092i \(-0.0270281\pi\)
\(968\) 0 0
\(969\) 8.34316e33i 0.378972i
\(970\) 0 0
\(971\) 2.24084e34 2.24084e34i 0.994008 0.994008i −0.00597423 0.999982i \(-0.501902\pi\)
0.999982 + 0.00597423i \(0.00190167\pi\)
\(972\) 0 0
\(973\) −4.02738e34 4.02738e34i −1.74472 1.74472i
\(974\) 0 0
\(975\) −6.44397e34 −2.72647
\(976\) 0 0
\(977\) 3.46442e34 1.43167 0.715836 0.698268i \(-0.246046\pi\)
0.715836 + 0.698268i \(0.246046\pi\)
\(978\) 0 0
\(979\) 3.16367e33 + 3.16367e33i 0.127700 + 0.127700i
\(980\) 0 0
\(981\) −9.06518e33 + 9.06518e33i −0.357423 + 0.357423i
\(982\) 0 0
\(983\) 2.88177e34i 1.10993i −0.831875 0.554963i \(-0.812733\pi\)
0.831875 0.554963i \(-0.187267\pi\)
\(984\) 0 0
\(985\) 6.82529e34i 2.56806i
\(986\) 0 0
\(987\) 3.73814e34 3.73814e34i 1.37407 1.37407i
\(988\) 0 0
\(989\) 2.64271e33 + 2.64271e33i 0.0949057 + 0.0949057i
\(990\) 0 0
\(991\) −3.19719e34 −1.12182 −0.560908 0.827878i \(-0.689548\pi\)
−0.560908 + 0.827878i \(0.689548\pi\)
\(992\) 0 0
\(993\) −5.13519e34 −1.76052
\(994\) 0 0
\(995\) 3.23193e34 + 3.23193e34i 1.08267 + 1.08267i
\(996\) 0 0
\(997\) −1.97178e34 + 1.97178e34i −0.645453 + 0.645453i −0.951891 0.306438i \(-0.900863\pi\)
0.306438 + 0.951891i \(0.400863\pi\)
\(998\) 0 0
\(999\) 1.63326e34i 0.522458i
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 64.24.e.a.17.36 90
4.3 odd 2 16.24.e.a.13.28 yes 90
16.5 even 4 inner 64.24.e.a.49.36 90
16.11 odd 4 16.24.e.a.5.28 90
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
16.24.e.a.5.28 90 16.11 odd 4
16.24.e.a.13.28 yes 90 4.3 odd 2
64.24.e.a.17.36 90 1.1 even 1 trivial
64.24.e.a.49.36 90 16.5 even 4 inner