Properties

Label 76.8.e.a.45.7
Level $76$
Weight $8$
Character 76.45
Analytic conductor $23.741$
Analytic rank $0$
Dimension $22$
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] = [76,8,Mod(45,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 2]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.45");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 76.e (of order \(3\), 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: \(23.7412619368\)
Analytic rank: \(0\)
Dimension: \(22\)
Relative dimension: \(11\) over \(\Q(\zeta_{3})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 45.7
Character \(\chi\) \(=\) 76.45
Dual form 76.8.e.a.49.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(7.71048 + 13.3549i) q^{3} +(60.2613 + 104.376i) q^{5} -1390.32 q^{7} +(974.597 - 1688.05i) q^{9} +O(q^{10})\) \(q+(7.71048 + 13.3549i) q^{3} +(60.2613 + 104.376i) q^{5} -1390.32 q^{7} +(974.597 - 1688.05i) q^{9} +2042.99 q^{11} +(4315.46 - 7474.60i) q^{13} +(-929.286 + 1609.57i) q^{15} +(-946.772 - 1639.86i) q^{17} +(1273.75 + 29870.5i) q^{19} +(-10720.0 - 18567.6i) q^{21} +(12191.9 - 21117.1i) q^{23} +(31799.7 - 55078.6i) q^{25} +63784.1 q^{27} +(62492.6 - 108240. i) q^{29} +212857. q^{31} +(15752.4 + 27284.0i) q^{33} +(-83782.4 - 145115. i) q^{35} +117001. q^{37} +133097. q^{39} +(-88867.0 - 153922. i) q^{41} +(-68222.9 - 118166. i) q^{43} +234922. q^{45} +(651614. - 1.12863e6i) q^{47} +1.10945e6 q^{49} +(14600.1 - 25288.2i) q^{51} +(233530. - 404486. i) q^{53} +(123113. + 213238. i) q^{55} +(-389098. + 247327. i) q^{57} +(-1.20409e6 - 2.08555e6i) q^{59} +(-509201. + 881961. i) q^{61} +(-1.35500e6 + 2.34693e6i) q^{63} +1.04022e6 q^{65} +(-992579. + 1.71920e6i) q^{67} +376023. q^{69} +(2.20730e6 + 3.82316e6i) q^{71} +(-1.07716e6 - 1.86569e6i) q^{73} +980763. q^{75} -2.84041e6 q^{77} +(461512. + 799362. i) q^{79} +(-1.63964e6 - 2.83994e6i) q^{81} -7.74443e6 q^{83} +(114107. - 197640. i) q^{85} +1.92739e6 q^{87} +(-2.33221e6 + 4.03950e6i) q^{89} +(-5.99987e6 + 1.03921e7i) q^{91} +(1.64123e6 + 2.84270e6i) q^{93} +(-3.04100e6 + 1.93299e6i) q^{95} +(-8.09272e6 - 1.40170e7i) q^{97} +(1.99109e6 - 3.44867e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 22 q + 13 q^{3} + q^{5} + 560 q^{7} - 6002 q^{9} + 472 q^{11} - 567 q^{13} + 2995 q^{15} + 5589 q^{17} + 80912 q^{19} + 44412 q^{21} - 15425 q^{23} - 32806 q^{25} + 50290 q^{27} - 18919 q^{29} + 150296 q^{31} + 314618 q^{33} + 92808 q^{35} + 350100 q^{37} + 948810 q^{39} + 698891 q^{41} + 402545 q^{43} + 1477508 q^{45} - 653621 q^{47} - 1938490 q^{49} - 1386401 q^{51} - 106763 q^{53} + 414508 q^{55} + 1267563 q^{57} + 3136737 q^{59} + 2004581 q^{61} + 1465000 q^{63} - 7397638 q^{65} + 4344391 q^{67} + 1732238 q^{69} - 133823 q^{71} - 8349685 q^{73} - 12136824 q^{75} + 9147480 q^{77} - 94679 q^{79} - 838595 q^{81} - 2884080 q^{83} - 1421409 q^{85} - 31740598 q^{87} - 7039347 q^{89} + 1520096 q^{91} - 1993628 q^{93} + 1707587 q^{95} + 13308115 q^{97} + 6011488 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(e\left(\frac{1}{3}\right)\) \(1\)

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
\(3\) 7.71048 + 13.3549i 0.164876 + 0.285573i 0.936611 0.350370i \(-0.113944\pi\)
−0.771735 + 0.635944i \(0.780611\pi\)
\(4\) 0 0
\(5\) 60.2613 + 104.376i 0.215597 + 0.373425i 0.953457 0.301529i \(-0.0974969\pi\)
−0.737860 + 0.674954i \(0.764164\pi\)
\(6\) 0 0
\(7\) −1390.32 −1.53205 −0.766023 0.642814i \(-0.777767\pi\)
−0.766023 + 0.642814i \(0.777767\pi\)
\(8\) 0 0
\(9\) 974.597 1688.05i 0.445632 0.771857i
\(10\) 0 0
\(11\) 2042.99 0.462798 0.231399 0.972859i \(-0.425670\pi\)
0.231399 + 0.972859i \(0.425670\pi\)
\(12\) 0 0
\(13\) 4315.46 7474.60i 0.544786 0.943597i −0.453835 0.891086i \(-0.649944\pi\)
0.998620 0.0525106i \(-0.0167223\pi\)
\(14\) 0 0
\(15\) −929.286 + 1609.57i −0.0710935 + 0.123138i
\(16\) 0 0
\(17\) −946.772 1639.86i −0.0467384 0.0809533i 0.841710 0.539930i \(-0.181549\pi\)
−0.888448 + 0.458977i \(0.848216\pi\)
\(18\) 0 0
\(19\) 1273.75 + 29870.5i 0.0426038 + 0.999092i
\(20\) 0 0
\(21\) −10720.0 18567.6i −0.252597 0.437511i
\(22\) 0 0
\(23\) 12191.9 21117.1i 0.208942 0.361898i −0.742440 0.669913i \(-0.766331\pi\)
0.951381 + 0.308015i \(0.0996648\pi\)
\(24\) 0 0
\(25\) 31799.7 55078.6i 0.407036 0.705006i
\(26\) 0 0
\(27\) 63784.1 0.623647
\(28\) 0 0
\(29\) 62492.6 108240.i 0.475813 0.824132i −0.523803 0.851839i \(-0.675487\pi\)
0.999616 + 0.0277075i \(0.00882071\pi\)
\(30\) 0 0
\(31\) 212857. 1.28328 0.641642 0.767004i \(-0.278254\pi\)
0.641642 + 0.767004i \(0.278254\pi\)
\(32\) 0 0
\(33\) 15752.4 + 27284.0i 0.0763042 + 0.132163i
\(34\) 0 0
\(35\) −83782.4 145115.i −0.330305 0.572104i
\(36\) 0 0
\(37\) 117001. 0.379737 0.189869 0.981810i \(-0.439194\pi\)
0.189869 + 0.981810i \(0.439194\pi\)
\(38\) 0 0
\(39\) 133097. 0.359288
\(40\) 0 0
\(41\) −88867.0 153922.i −0.201371 0.348785i 0.747599 0.664150i \(-0.231206\pi\)
−0.948970 + 0.315365i \(0.897873\pi\)
\(42\) 0 0
\(43\) −68222.9 118166.i −0.130855 0.226648i 0.793151 0.609025i \(-0.208439\pi\)
−0.924006 + 0.382377i \(0.875106\pi\)
\(44\) 0 0
\(45\) 234922. 0.384308
\(46\) 0 0
\(47\) 651614. 1.12863e6i 0.915479 1.58566i 0.109279 0.994011i \(-0.465146\pi\)
0.806199 0.591644i \(-0.201521\pi\)
\(48\) 0 0
\(49\) 1.10945e6 1.34716
\(50\) 0 0
\(51\) 14600.1 25288.2i 0.0154121 0.0266945i
\(52\) 0 0
\(53\) 233530. 404486.i 0.215465 0.373197i −0.737951 0.674854i \(-0.764207\pi\)
0.953416 + 0.301658i \(0.0975399\pi\)
\(54\) 0 0
\(55\) 123113. + 213238.i 0.0997780 + 0.172821i
\(56\) 0 0
\(57\) −389098. + 247327.i −0.278290 + 0.176893i
\(58\) 0 0
\(59\) −1.20409e6 2.08555e6i −0.763269 1.32202i −0.941157 0.337969i \(-0.890260\pi\)
0.177888 0.984051i \(-0.443073\pi\)
\(60\) 0 0
\(61\) −509201. + 881961.i −0.287233 + 0.497502i −0.973148 0.230179i \(-0.926069\pi\)
0.685915 + 0.727682i \(0.259402\pi\)
\(62\) 0 0
\(63\) −1.35500e6 + 2.34693e6i −0.682728 + 1.18252i
\(64\) 0 0
\(65\) 1.04022e6 0.469817
\(66\) 0 0
\(67\) −992579. + 1.71920e6i −0.403184 + 0.698335i −0.994108 0.108392i \(-0.965430\pi\)
0.590924 + 0.806727i \(0.298763\pi\)
\(68\) 0 0
\(69\) 376023. 0.137798
\(70\) 0 0
\(71\) 2.20730e6 + 3.82316e6i 0.731910 + 1.26770i 0.956066 + 0.293152i \(0.0947041\pi\)
−0.224156 + 0.974553i \(0.571963\pi\)
\(72\) 0 0
\(73\) −1.07716e6 1.86569e6i −0.324078 0.561320i 0.657247 0.753675i \(-0.271721\pi\)
−0.981325 + 0.192355i \(0.938387\pi\)
\(74\) 0 0
\(75\) 980763. 0.268441
\(76\) 0 0
\(77\) −2.84041e6 −0.709028
\(78\) 0 0
\(79\) 461512. + 799362.i 0.105314 + 0.182410i 0.913867 0.406014i \(-0.133082\pi\)
−0.808552 + 0.588424i \(0.799748\pi\)
\(80\) 0 0
\(81\) −1.63964e6 2.83994e6i −0.342808 0.593760i
\(82\) 0 0
\(83\) −7.74443e6 −1.48668 −0.743338 0.668916i \(-0.766759\pi\)
−0.743338 + 0.668916i \(0.766759\pi\)
\(84\) 0 0
\(85\) 114107. 197640.i 0.0201533 0.0349066i
\(86\) 0 0
\(87\) 1.92739e6 0.313800
\(88\) 0 0
\(89\) −2.33221e6 + 4.03950e6i −0.350673 + 0.607383i −0.986367 0.164558i \(-0.947380\pi\)
0.635695 + 0.771940i \(0.280714\pi\)
\(90\) 0 0
\(91\) −5.99987e6 + 1.03921e7i −0.834636 + 1.44563i
\(92\) 0 0
\(93\) 1.64123e6 + 2.84270e6i 0.211583 + 0.366472i
\(94\) 0 0
\(95\) −3.04100e6 + 1.93299e6i −0.363901 + 0.231311i
\(96\) 0 0
\(97\) −8.09272e6 1.40170e7i −0.900313 1.55939i −0.827088 0.562072i \(-0.810004\pi\)
−0.0732246 0.997315i \(-0.523329\pi\)
\(98\) 0 0
\(99\) 1.99109e6 3.44867e6i 0.206238 0.357214i
\(100\) 0 0
\(101\) −8.31748e6 + 1.44063e7i −0.803280 + 1.39132i 0.114166 + 0.993462i \(0.463580\pi\)
−0.917446 + 0.397860i \(0.869753\pi\)
\(102\) 0 0
\(103\) 1.90128e7 1.71441 0.857206 0.514974i \(-0.172198\pi\)
0.857206 + 0.514974i \(0.172198\pi\)
\(104\) 0 0
\(105\) 1.29201e6 2.23782e6i 0.108919 0.188652i
\(106\) 0 0
\(107\) −3.14505e6 −0.248190 −0.124095 0.992270i \(-0.539603\pi\)
−0.124095 + 0.992270i \(0.539603\pi\)
\(108\) 0 0
\(109\) −760400. 1.31705e6i −0.0562405 0.0974114i 0.836534 0.547914i \(-0.184578\pi\)
−0.892775 + 0.450503i \(0.851245\pi\)
\(110\) 0 0
\(111\) 902133. + 1.56254e6i 0.0626095 + 0.108443i
\(112\) 0 0
\(113\) 1.92682e7 1.25622 0.628110 0.778125i \(-0.283829\pi\)
0.628110 + 0.778125i \(0.283829\pi\)
\(114\) 0 0
\(115\) 2.93880e6 0.180189
\(116\) 0 0
\(117\) −8.41168e6 1.45694e7i −0.485548 0.840993i
\(118\) 0 0
\(119\) 1.31632e6 + 2.27993e6i 0.0716054 + 0.124024i
\(120\) 0 0
\(121\) −1.53134e7 −0.785818
\(122\) 0 0
\(123\) 1.37041e6 2.37363e6i 0.0664024 0.115012i
\(124\) 0 0
\(125\) 1.70810e7 0.782217
\(126\) 0 0
\(127\) −4.77865e6 + 8.27687e6i −0.207011 + 0.358553i −0.950771 0.309893i \(-0.899707\pi\)
0.743761 + 0.668446i \(0.233040\pi\)
\(128\) 0 0
\(129\) 1.05206e6 1.82223e6i 0.0431497 0.0747374i
\(130\) 0 0
\(131\) 1.30471e7 + 2.25982e7i 0.507065 + 0.878263i 0.999967 + 0.00817779i \(0.00260310\pi\)
−0.492901 + 0.870085i \(0.664064\pi\)
\(132\) 0 0
\(133\) −1.77093e6 4.15296e7i −0.0652709 1.53065i
\(134\) 0 0
\(135\) 3.84371e6 + 6.65750e6i 0.134457 + 0.232886i
\(136\) 0 0
\(137\) 2.55600e7 4.42712e7i 0.849256 1.47095i −0.0326178 0.999468i \(-0.510384\pi\)
0.881873 0.471486i \(-0.156282\pi\)
\(138\) 0 0
\(139\) 4.64040e6 8.03741e6i 0.146556 0.253843i −0.783396 0.621522i \(-0.786514\pi\)
0.929952 + 0.367680i \(0.119848\pi\)
\(140\) 0 0
\(141\) 2.00970e7 0.603761
\(142\) 0 0
\(143\) 8.81644e6 1.52705e7i 0.252126 0.436695i
\(144\) 0 0
\(145\) 1.50635e7 0.410335
\(146\) 0 0
\(147\) 8.55436e6 + 1.48166e7i 0.222114 + 0.384714i
\(148\) 0 0
\(149\) −2.51057e7 4.34844e7i −0.621757 1.07692i −0.989158 0.146853i \(-0.953086\pi\)
0.367401 0.930063i \(-0.380248\pi\)
\(150\) 0 0
\(151\) 2.40690e7 0.568903 0.284451 0.958690i \(-0.408189\pi\)
0.284451 + 0.958690i \(0.408189\pi\)
\(152\) 0 0
\(153\) −3.69088e6 −0.0833125
\(154\) 0 0
\(155\) 1.28271e7 + 2.22171e7i 0.276672 + 0.479211i
\(156\) 0 0
\(157\) −1.51003e7 2.61545e7i −0.311413 0.539384i 0.667255 0.744829i \(-0.267469\pi\)
−0.978669 + 0.205445i \(0.934136\pi\)
\(158\) 0 0
\(159\) 7.20251e6 0.142100
\(160\) 0 0
\(161\) −1.69507e7 + 2.93595e7i −0.320108 + 0.554444i
\(162\) 0 0
\(163\) −1.25527e7 −0.227029 −0.113514 0.993536i \(-0.536211\pi\)
−0.113514 + 0.993536i \(0.536211\pi\)
\(164\) 0 0
\(165\) −1.89852e6 + 3.28834e6i −0.0329020 + 0.0569879i
\(166\) 0 0
\(167\) −1.56134e7 + 2.70432e7i −0.259412 + 0.449315i −0.966085 0.258226i \(-0.916862\pi\)
0.706673 + 0.707541i \(0.250195\pi\)
\(168\) 0 0
\(169\) −5.87219e6 1.01709e7i −0.0935830 0.162090i
\(170\) 0 0
\(171\) 5.16644e7 + 2.69616e7i 0.790142 + 0.412343i
\(172\) 0 0
\(173\) −347200. 601369.i −0.00509822 0.00883038i 0.863465 0.504409i \(-0.168289\pi\)
−0.868563 + 0.495578i \(0.834956\pi\)
\(174\) 0 0
\(175\) −4.42117e7 + 7.65769e7i −0.623597 + 1.08010i
\(176\) 0 0
\(177\) 1.85682e7 3.21611e7i 0.251689 0.435938i
\(178\) 0 0
\(179\) −6.73363e7 −0.877533 −0.438767 0.898601i \(-0.644584\pi\)
−0.438767 + 0.898601i \(0.644584\pi\)
\(180\) 0 0
\(181\) −65005.9 + 112593.i −0.000814850 + 0.00141136i −0.866433 0.499294i \(-0.833593\pi\)
0.865618 + 0.500706i \(0.166926\pi\)
\(182\) 0 0
\(183\) −1.57047e7 −0.189431
\(184\) 0 0
\(185\) 7.05062e6 + 1.22120e7i 0.0818703 + 0.141803i
\(186\) 0 0
\(187\) −1.93424e6 3.35021e6i −0.0216305 0.0374651i
\(188\) 0 0
\(189\) −8.86803e7 −0.955456
\(190\) 0 0
\(191\) −6.74668e7 −0.700605 −0.350303 0.936637i \(-0.613921\pi\)
−0.350303 + 0.936637i \(0.613921\pi\)
\(192\) 0 0
\(193\) −3.34728e6 5.79766e6i −0.0335152 0.0580500i 0.848781 0.528744i \(-0.177337\pi\)
−0.882296 + 0.470694i \(0.844004\pi\)
\(194\) 0 0
\(195\) 8.02060e6 + 1.38921e7i 0.0774615 + 0.134167i
\(196\) 0 0
\(197\) 1.11025e8 1.03464 0.517318 0.855793i \(-0.326931\pi\)
0.517318 + 0.855793i \(0.326931\pi\)
\(198\) 0 0
\(199\) −1.52124e7 + 2.63487e7i −0.136840 + 0.237014i −0.926299 0.376790i \(-0.877028\pi\)
0.789459 + 0.613803i \(0.210361\pi\)
\(200\) 0 0
\(201\) −3.06131e7 −0.265901
\(202\) 0 0
\(203\) −8.68848e7 + 1.50489e8i −0.728966 + 1.26261i
\(204\) 0 0
\(205\) 1.07105e7 1.85511e7i 0.0868301 0.150394i
\(206\) 0 0
\(207\) −2.37644e7 4.11612e7i −0.186222 0.322546i
\(208\) 0 0
\(209\) 2.60227e6 + 6.10252e7i 0.0197169 + 0.462378i
\(210\) 0 0
\(211\) 9.83274e7 + 1.70308e8i 0.720587 + 1.24809i 0.960765 + 0.277364i \(0.0894607\pi\)
−0.240178 + 0.970729i \(0.577206\pi\)
\(212\) 0 0
\(213\) −3.40387e7 + 5.89568e7i −0.241348 + 0.418028i
\(214\) 0 0
\(215\) 8.22239e6 1.42416e7i 0.0564240 0.0977292i
\(216\) 0 0
\(217\) −2.95940e8 −1.96605
\(218\) 0 0
\(219\) 1.66108e7 2.87708e7i 0.106865 0.185096i
\(220\) 0 0
\(221\) −1.63430e7 −0.101850
\(222\) 0 0
\(223\) −8.95579e7 1.55119e8i −0.540800 0.936693i −0.998858 0.0477711i \(-0.984788\pi\)
0.458058 0.888922i \(-0.348545\pi\)
\(224\) 0 0
\(225\) −6.19837e7 1.07359e8i −0.362776 0.628347i
\(226\) 0 0
\(227\) −4.13714e7 −0.234752 −0.117376 0.993088i \(-0.537448\pi\)
−0.117376 + 0.993088i \(0.537448\pi\)
\(228\) 0 0
\(229\) 3.16259e7 0.174028 0.0870140 0.996207i \(-0.472268\pi\)
0.0870140 + 0.996207i \(0.472268\pi\)
\(230\) 0 0
\(231\) −2.19009e7 3.79335e7i −0.116902 0.202479i
\(232\) 0 0
\(233\) 7.96025e7 + 1.37876e8i 0.412269 + 0.714071i 0.995138 0.0984954i \(-0.0314030\pi\)
−0.582868 + 0.812567i \(0.698070\pi\)
\(234\) 0 0
\(235\) 1.57068e8 0.789498
\(236\) 0 0
\(237\) −7.11696e6 + 1.23269e7i −0.0347276 + 0.0601500i
\(238\) 0 0
\(239\) 1.28680e8 0.609702 0.304851 0.952400i \(-0.401393\pi\)
0.304851 + 0.952400i \(0.401393\pi\)
\(240\) 0 0
\(241\) 2.10493e8 3.64584e8i 0.968673 1.67779i 0.269269 0.963065i \(-0.413218\pi\)
0.699404 0.714726i \(-0.253449\pi\)
\(242\) 0 0
\(243\) 9.50327e7 1.64601e8i 0.424865 0.735888i
\(244\) 0 0
\(245\) 6.68566e7 + 1.15799e8i 0.290444 + 0.503064i
\(246\) 0 0
\(247\) 2.28767e8 + 1.19384e8i 0.965950 + 0.504090i
\(248\) 0 0
\(249\) −5.97133e7 1.03426e8i −0.245117 0.424555i
\(250\) 0 0
\(251\) −5.06004e7 + 8.76424e7i −0.201974 + 0.349829i −0.949164 0.314781i \(-0.898069\pi\)
0.747190 + 0.664610i \(0.231402\pi\)
\(252\) 0 0
\(253\) 2.49080e7 4.31419e7i 0.0966978 0.167486i
\(254\) 0 0
\(255\) 3.51929e6 0.0132912
\(256\) 0 0
\(257\) −6.08914e6 + 1.05467e7i −0.0223764 + 0.0387570i −0.876997 0.480496i \(-0.840457\pi\)
0.854620 + 0.519253i \(0.173790\pi\)
\(258\) 0 0
\(259\) −1.62669e8 −0.581774
\(260\) 0 0
\(261\) −1.21810e8 2.10982e8i −0.424075 0.734519i
\(262\) 0 0
\(263\) 2.46088e8 + 4.26238e8i 0.834154 + 1.44480i 0.894718 + 0.446632i \(0.147377\pi\)
−0.0605637 + 0.998164i \(0.519290\pi\)
\(264\) 0 0
\(265\) 5.62912e7 0.185815
\(266\) 0 0
\(267\) −7.19297e7 −0.231270
\(268\) 0 0
\(269\) 1.59482e8 + 2.76231e8i 0.499550 + 0.865245i 1.00000 0.000519964i \(-0.000165510\pi\)
−0.500450 + 0.865765i \(0.666832\pi\)
\(270\) 0 0
\(271\) −2.28060e8 3.95012e8i −0.696077 1.20564i −0.969816 0.243836i \(-0.921594\pi\)
0.273740 0.961804i \(-0.411739\pi\)
\(272\) 0 0
\(273\) −1.85048e8 −0.550445
\(274\) 0 0
\(275\) 6.49664e7 1.12525e8i 0.188375 0.326276i
\(276\) 0 0
\(277\) 4.31963e8 1.22114 0.610572 0.791961i \(-0.290940\pi\)
0.610572 + 0.791961i \(0.290940\pi\)
\(278\) 0 0
\(279\) 2.07450e8 3.59314e8i 0.571872 0.990512i
\(280\) 0 0
\(281\) −1.42756e7 + 2.47261e7i −0.0383816 + 0.0664789i −0.884578 0.466392i \(-0.845554\pi\)
0.846196 + 0.532871i \(0.178887\pi\)
\(282\) 0 0
\(283\) −8.79098e7 1.52264e8i −0.230560 0.399342i 0.727413 0.686200i \(-0.240723\pi\)
−0.957973 + 0.286858i \(0.907389\pi\)
\(284\) 0 0
\(285\) −4.92624e7 2.57081e7i −0.126055 0.0657829i
\(286\) 0 0
\(287\) 1.23554e8 + 2.14001e8i 0.308509 + 0.534354i
\(288\) 0 0
\(289\) 2.03377e8 3.52259e8i 0.495631 0.858458i
\(290\) 0 0
\(291\) 1.24798e8 2.16156e8i 0.296880 0.514211i
\(292\) 0 0
\(293\) 4.25093e8 0.987296 0.493648 0.869662i \(-0.335663\pi\)
0.493648 + 0.869662i \(0.335663\pi\)
\(294\) 0 0
\(295\) 1.45120e8 2.51355e8i 0.329117 0.570048i
\(296\) 0 0
\(297\) 1.30310e8 0.288623
\(298\) 0 0
\(299\) −1.05228e8 1.82260e8i −0.227657 0.394313i
\(300\) 0 0
\(301\) 9.48516e7 + 1.64288e8i 0.200476 + 0.347234i
\(302\) 0 0
\(303\) −2.56527e8 −0.529766
\(304\) 0 0
\(305\) −1.22740e8 −0.247707
\(306\) 0 0
\(307\) −2.70722e8 4.68904e8i −0.533997 0.924910i −0.999211 0.0397119i \(-0.987356\pi\)
0.465214 0.885198i \(-0.345977\pi\)
\(308\) 0 0
\(309\) 1.46598e8 + 2.53915e8i 0.282665 + 0.489590i
\(310\) 0 0
\(311\) −3.28809e7 −0.0619845 −0.0309922 0.999520i \(-0.509867\pi\)
−0.0309922 + 0.999520i \(0.509867\pi\)
\(312\) 0 0
\(313\) −4.15658e8 + 7.19940e8i −0.766180 + 1.32706i 0.173441 + 0.984844i \(0.444511\pi\)
−0.939621 + 0.342218i \(0.888822\pi\)
\(314\) 0 0
\(315\) −3.26616e8 −0.588777
\(316\) 0 0
\(317\) −3.48372e8 + 6.03398e8i −0.614237 + 1.06389i 0.376281 + 0.926506i \(0.377203\pi\)
−0.990518 + 0.137384i \(0.956131\pi\)
\(318\) 0 0
\(319\) 1.27672e8 2.21134e8i 0.220205 0.381407i
\(320\) 0 0
\(321\) −2.42498e7 4.20019e7i −0.0409205 0.0708764i
\(322\) 0 0
\(323\) 4.77775e7 3.03694e7i 0.0788886 0.0501449i
\(324\) 0 0
\(325\) −2.74461e8 4.75380e8i −0.443494 0.768155i
\(326\) 0 0
\(327\) 1.17261e7 2.03102e7i 0.0185454 0.0321216i
\(328\) 0 0
\(329\) −9.05952e8 + 1.56916e9i −1.40255 + 2.42930i
\(330\) 0 0
\(331\) 6.03206e7 0.0914256 0.0457128 0.998955i \(-0.485444\pi\)
0.0457128 + 0.998955i \(0.485444\pi\)
\(332\) 0 0
\(333\) 1.14029e8 1.97504e8i 0.169223 0.293103i
\(334\) 0 0
\(335\) −2.39256e8 −0.347701
\(336\) 0 0
\(337\) 4.50854e8 + 7.80902e8i 0.641699 + 1.11146i 0.985053 + 0.172249i \(0.0551035\pi\)
−0.343354 + 0.939206i \(0.611563\pi\)
\(338\) 0 0
\(339\) 1.48567e8 + 2.57325e8i 0.207120 + 0.358743i
\(340\) 0 0
\(341\) 4.34865e8 0.593901
\(342\) 0 0
\(343\) −3.97496e8 −0.531868
\(344\) 0 0
\(345\) 2.26596e7 + 3.92476e7i 0.0297088 + 0.0514572i
\(346\) 0 0
\(347\) −2.55975e8 4.43363e8i −0.328886 0.569647i 0.653405 0.757008i \(-0.273340\pi\)
−0.982291 + 0.187361i \(0.940006\pi\)
\(348\) 0 0
\(349\) −9.66879e8 −1.21754 −0.608770 0.793347i \(-0.708337\pi\)
−0.608770 + 0.793347i \(0.708337\pi\)
\(350\) 0 0
\(351\) 2.75258e8 4.76761e8i 0.339754 0.588472i
\(352\) 0 0
\(353\) 6.94120e8 0.839891 0.419945 0.907549i \(-0.362049\pi\)
0.419945 + 0.907549i \(0.362049\pi\)
\(354\) 0 0
\(355\) −2.66030e8 + 4.60777e8i −0.315595 + 0.546627i
\(356\) 0 0
\(357\) −2.02988e7 + 3.51586e7i −0.0236120 + 0.0408972i
\(358\) 0 0
\(359\) 4.15769e8 + 7.20133e8i 0.474266 + 0.821452i 0.999566 0.0294647i \(-0.00938028\pi\)
−0.525300 + 0.850917i \(0.676047\pi\)
\(360\) 0 0
\(361\) −8.90627e8 + 7.60955e7i −0.996370 + 0.0851302i
\(362\) 0 0
\(363\) −1.18073e8 2.04509e8i −0.129562 0.224409i
\(364\) 0 0
\(365\) 1.29822e8 2.24858e8i 0.139741 0.242038i
\(366\) 0 0
\(367\) −1.72325e8 + 2.98476e8i −0.181977 + 0.315194i −0.942554 0.334054i \(-0.891583\pi\)
0.760576 + 0.649248i \(0.224916\pi\)
\(368\) 0 0
\(369\) −3.46438e8 −0.358949
\(370\) 0 0
\(371\) −3.24681e8 + 5.62364e8i −0.330102 + 0.571754i
\(372\) 0 0
\(373\) 1.50373e9 1.50033 0.750167 0.661248i \(-0.229973\pi\)
0.750167 + 0.661248i \(0.229973\pi\)
\(374\) 0 0
\(375\) 1.31702e8 + 2.28115e8i 0.128969 + 0.223380i
\(376\) 0 0
\(377\) −5.39369e8 9.34215e8i −0.518432 0.897950i
\(378\) 0 0
\(379\) −1.55525e9 −1.46745 −0.733727 0.679445i \(-0.762221\pi\)
−0.733727 + 0.679445i \(0.762221\pi\)
\(380\) 0 0
\(381\) −1.47383e8 −0.136524
\(382\) 0 0
\(383\) −2.01073e8 3.48268e8i −0.182876 0.316751i 0.759983 0.649943i \(-0.225207\pi\)
−0.942859 + 0.333193i \(0.891874\pi\)
\(384\) 0 0
\(385\) −1.71167e8 2.96469e8i −0.152864 0.264769i
\(386\) 0 0
\(387\) −2.65959e8 −0.233253
\(388\) 0 0
\(389\) −4.47194e8 + 7.74563e8i −0.385188 + 0.667165i −0.991795 0.127837i \(-0.959197\pi\)
0.606607 + 0.795001i \(0.292530\pi\)
\(390\) 0 0
\(391\) −4.61719e7 −0.0390624
\(392\) 0 0
\(393\) −2.01199e8 + 3.48486e8i −0.167206 + 0.289609i
\(394\) 0 0
\(395\) −5.56226e7 + 9.63411e7i −0.0454110 + 0.0786542i
\(396\) 0 0
\(397\) 8.01015e7 + 1.38740e8i 0.0642501 + 0.111284i 0.896361 0.443325i \(-0.146201\pi\)
−0.832111 + 0.554609i \(0.812868\pi\)
\(398\) 0 0
\(399\) 5.40971e8 3.43864e8i 0.426352 0.271008i
\(400\) 0 0
\(401\) 1.81408e8 + 3.14208e8i 0.140492 + 0.243339i 0.927682 0.373371i \(-0.121798\pi\)
−0.787190 + 0.616711i \(0.788465\pi\)
\(402\) 0 0
\(403\) 9.18578e8 1.59102e9i 0.699115 1.21090i
\(404\) 0 0
\(405\) 1.97613e8 3.42276e8i 0.147817 0.256026i
\(406\) 0 0
\(407\) 2.39032e8 0.175742
\(408\) 0 0
\(409\) 8.24946e8 1.42885e9i 0.596203 1.03265i −0.397173 0.917744i \(-0.630009\pi\)
0.993376 0.114910i \(-0.0366579\pi\)
\(410\) 0 0
\(411\) 7.88319e8 0.560087
\(412\) 0 0
\(413\) 1.67407e9 + 2.89958e9i 1.16936 + 2.02539i
\(414\) 0 0
\(415\) −4.66689e8 8.08330e8i −0.320523 0.555162i
\(416\) 0 0
\(417\) 1.43119e8 0.0966542
\(418\) 0 0
\(419\) 2.44293e9 1.62242 0.811209 0.584756i \(-0.198810\pi\)
0.811209 + 0.584756i \(0.198810\pi\)
\(420\) 0 0
\(421\) −3.56978e7 6.18304e7i −0.0233160 0.0403845i 0.854132 0.520056i \(-0.174089\pi\)
−0.877448 + 0.479672i \(0.840756\pi\)
\(422\) 0 0
\(423\) −1.27012e9 2.19992e9i −0.815933 1.41324i
\(424\) 0 0
\(425\) −1.20428e8 −0.0760968
\(426\) 0 0
\(427\) 7.07952e8 1.22621e9i 0.440054 0.762196i
\(428\) 0 0
\(429\) 2.71916e8 0.166278
\(430\) 0 0
\(431\) −1.03682e9 + 1.79582e9i −0.623782 + 1.08042i 0.364993 + 0.931010i \(0.381071\pi\)
−0.988775 + 0.149412i \(0.952262\pi\)
\(432\) 0 0
\(433\) 7.75885e8 1.34387e9i 0.459293 0.795518i −0.539631 0.841902i \(-0.681436\pi\)
0.998924 + 0.0463833i \(0.0147696\pi\)
\(434\) 0 0
\(435\) 1.16147e8 + 2.01173e8i 0.0676544 + 0.117181i
\(436\) 0 0
\(437\) 6.46307e8 + 3.37282e8i 0.370471 + 0.193334i
\(438\) 0 0
\(439\) 7.87754e7 + 1.36443e8i 0.0444391 + 0.0769707i 0.887389 0.461021i \(-0.152517\pi\)
−0.842950 + 0.537991i \(0.819183\pi\)
\(440\) 0 0
\(441\) 1.08126e9 1.87280e9i 0.600338 1.03982i
\(442\) 0 0
\(443\) 8.59312e7 1.48837e8i 0.0469611 0.0813390i −0.841589 0.540118i \(-0.818380\pi\)
0.888550 + 0.458779i \(0.151713\pi\)
\(444\) 0 0
\(445\) −5.62167e8 −0.302416
\(446\) 0 0
\(447\) 3.87155e8 6.70571e8i 0.205026 0.355115i
\(448\) 0 0
\(449\) −2.76795e9 −1.44310 −0.721548 0.692364i \(-0.756569\pi\)
−0.721548 + 0.692364i \(0.756569\pi\)
\(450\) 0 0
\(451\) −1.81554e8 3.14461e8i −0.0931941 0.161417i
\(452\) 0 0
\(453\) 1.85583e8 + 3.21440e8i 0.0937984 + 0.162464i
\(454\) 0 0
\(455\) −1.44624e9 −0.719781
\(456\) 0 0
\(457\) −1.24275e9 −0.609084 −0.304542 0.952499i \(-0.598503\pi\)
−0.304542 + 0.952499i \(0.598503\pi\)
\(458\) 0 0
\(459\) −6.03890e7 1.04597e8i −0.0291483 0.0504863i
\(460\) 0 0
\(461\) 1.96118e9 + 3.39687e9i 0.932319 + 1.61482i 0.779346 + 0.626593i \(0.215551\pi\)
0.152973 + 0.988230i \(0.451115\pi\)
\(462\) 0 0
\(463\) 1.77320e9 0.830278 0.415139 0.909758i \(-0.363733\pi\)
0.415139 + 0.909758i \(0.363733\pi\)
\(464\) 0 0
\(465\) −1.97805e8 + 3.42609e8i −0.0912332 + 0.158021i
\(466\) 0 0
\(467\) 2.87562e9 1.30654 0.653269 0.757126i \(-0.273397\pi\)
0.653269 + 0.757126i \(0.273397\pi\)
\(468\) 0 0
\(469\) 1.38000e9 2.39023e9i 0.617696 1.06988i
\(470\) 0 0
\(471\) 2.32861e8 4.03328e8i 0.102689 0.177863i
\(472\) 0 0
\(473\) −1.39379e8 2.41411e8i −0.0605595 0.104892i
\(474\) 0 0
\(475\) 1.68573e9 + 8.79717e8i 0.721708 + 0.376630i
\(476\) 0 0
\(477\) −4.55195e8 7.88421e8i −0.192036 0.332617i
\(478\) 0 0
\(479\) −2.93157e8 + 5.07764e8i −0.121878 + 0.211099i −0.920508 0.390723i \(-0.872225\pi\)
0.798630 + 0.601822i \(0.205558\pi\)
\(480\) 0 0
\(481\) 5.04913e8 8.74535e8i 0.206875 0.358319i
\(482\) 0 0
\(483\) −5.22792e8 −0.211112
\(484\) 0 0
\(485\) 9.75355e8 1.68936e9i 0.388210 0.672399i
\(486\) 0 0
\(487\) −3.21097e9 −1.25975 −0.629876 0.776695i \(-0.716894\pi\)
−0.629876 + 0.776695i \(0.716894\pi\)
\(488\) 0 0
\(489\) −9.67875e7 1.67641e8i −0.0374316 0.0648334i
\(490\) 0 0
\(491\) 1.30136e9 + 2.25402e9i 0.496149 + 0.859355i 0.999990 0.00444096i \(-0.00141361\pi\)
−0.503841 + 0.863796i \(0.668080\pi\)
\(492\) 0 0
\(493\) −2.36665e8 −0.0889549
\(494\) 0 0
\(495\) 4.79942e8 0.177857
\(496\) 0 0
\(497\) −3.06886e9 5.31541e9i −1.12132 1.94218i
\(498\) 0 0
\(499\) 2.17591e8 + 3.76879e8i 0.0783951 + 0.135784i 0.902558 0.430569i \(-0.141687\pi\)
−0.824163 + 0.566353i \(0.808354\pi\)
\(500\) 0 0
\(501\) −4.81547e8 −0.171083
\(502\) 0 0
\(503\) −2.27657e9 + 3.94313e9i −0.797614 + 1.38151i 0.123551 + 0.992338i \(0.460572\pi\)
−0.921166 + 0.389170i \(0.872762\pi\)
\(504\) 0 0
\(505\) −2.00489e9 −0.692740
\(506\) 0 0
\(507\) 9.05549e7 1.56846e8i 0.0308591 0.0534496i
\(508\) 0 0
\(509\) 4.76163e8 8.24739e8i 0.160046 0.277207i −0.774839 0.632158i \(-0.782169\pi\)
0.934885 + 0.354951i \(0.115503\pi\)
\(510\) 0 0
\(511\) 1.49759e9 + 2.59391e9i 0.496502 + 0.859967i
\(512\) 0 0
\(513\) 8.12453e7 + 1.90527e9i 0.0265697 + 0.623081i
\(514\) 0 0
\(515\) 1.14573e9 + 1.98447e9i 0.369622 + 0.640205i
\(516\) 0 0
\(517\) 1.33124e9 2.30578e9i 0.423682 0.733838i
\(518\) 0 0
\(519\) 5.35416e6 9.27368e6i 0.00168115 0.00291183i
\(520\) 0 0
\(521\) −4.55659e9 −1.41159 −0.705794 0.708417i \(-0.749410\pi\)
−0.705794 + 0.708417i \(0.749410\pi\)
\(522\) 0 0
\(523\) −2.64354e9 + 4.57875e9i −0.808035 + 1.39956i 0.106188 + 0.994346i \(0.466136\pi\)
−0.914223 + 0.405212i \(0.867198\pi\)
\(524\) 0 0
\(525\) −1.36357e9 −0.411264
\(526\) 0 0
\(527\) −2.01527e8 3.49056e8i −0.0599787 0.103886i
\(528\) 0 0
\(529\) 1.40513e9 + 2.43375e9i 0.412687 + 0.714794i
\(530\) 0 0
\(531\) −4.69402e9 −1.36055
\(532\) 0 0
\(533\) −1.53401e9 −0.438816
\(534\) 0 0
\(535\) −1.89524e8 3.28266e8i −0.0535090 0.0926803i
\(536\) 0 0
\(537\) −5.19195e8 8.99272e8i −0.144684 0.250600i
\(538\) 0 0
\(539\) 2.26659e9 0.623464
\(540\) 0 0
\(541\) −2.04906e9 + 3.54908e9i −0.556372 + 0.963664i 0.441424 + 0.897299i \(0.354474\pi\)
−0.997795 + 0.0663652i \(0.978860\pi\)
\(542\) 0 0
\(543\) −2.00491e6 −0.000537396
\(544\) 0 0
\(545\) 9.16453e7 1.58734e8i 0.0242506 0.0420032i
\(546\) 0 0
\(547\) 5.08604e8 8.80927e8i 0.132869 0.230136i −0.791912 0.610635i \(-0.790914\pi\)
0.924781 + 0.380499i \(0.124248\pi\)
\(548\) 0 0
\(549\) 9.92531e8 + 1.71911e9i 0.256000 + 0.443406i
\(550\) 0 0
\(551\) 3.31280e9 + 1.72882e9i 0.843655 + 0.440269i
\(552\) 0 0
\(553\) −6.41649e8 1.11137e9i −0.161347 0.279460i
\(554\) 0 0
\(555\) −1.08727e8 + 1.88321e8i −0.0269969 + 0.0467599i
\(556\) 0 0
\(557\) 3.21011e9 5.56008e9i 0.787094 1.36329i −0.140645 0.990060i \(-0.544918\pi\)
0.927740 0.373228i \(-0.121749\pi\)
\(558\) 0 0
\(559\) −1.17765e9 −0.285152
\(560\) 0 0
\(561\) 2.98279e7 5.16634e7i 0.00713268 0.0123542i
\(562\) 0 0
\(563\) −6.93235e9 −1.63720 −0.818599 0.574366i \(-0.805249\pi\)
−0.818599 + 0.574366i \(0.805249\pi\)
\(564\) 0 0
\(565\) 1.16112e9 + 2.01112e9i 0.270837 + 0.469104i
\(566\) 0 0
\(567\) 2.27962e9 + 3.94842e9i 0.525197 + 0.909667i
\(568\) 0 0
\(569\) −4.91652e9 −1.11883 −0.559416 0.828887i \(-0.688974\pi\)
−0.559416 + 0.828887i \(0.688974\pi\)
\(570\) 0 0
\(571\) −9.03802e8 −0.203164 −0.101582 0.994827i \(-0.532390\pi\)
−0.101582 + 0.994827i \(0.532390\pi\)
\(572\) 0 0
\(573\) −5.20202e8 9.01016e8i −0.115513 0.200074i
\(574\) 0 0
\(575\) −7.75399e8 1.34303e9i −0.170093 0.294611i
\(576\) 0 0
\(577\) −3.68572e9 −0.798742 −0.399371 0.916789i \(-0.630771\pi\)
−0.399371 + 0.916789i \(0.630771\pi\)
\(578\) 0 0
\(579\) 5.16183e7 8.94054e7i 0.0110517 0.0191421i
\(580\) 0 0
\(581\) 1.07672e10 2.27765
\(582\) 0 0
\(583\) 4.77099e8 8.26360e8i 0.0997169 0.172715i
\(584\) 0 0
\(585\) 1.01380e9 1.75595e9i 0.209366 0.362632i
\(586\) 0 0
\(587\) 9.40741e7 + 1.62941e8i 0.0191971 + 0.0332504i 0.875464 0.483283i \(-0.160556\pi\)
−0.856267 + 0.516533i \(0.827222\pi\)
\(588\) 0 0
\(589\) 2.71128e8 + 6.35817e9i 0.0546727 + 1.28212i
\(590\) 0 0
\(591\) 8.56053e8 + 1.48273e9i 0.170586 + 0.295464i
\(592\) 0 0
\(593\) 8.27618e8 1.43348e9i 0.162982 0.282293i −0.772955 0.634461i \(-0.781222\pi\)
0.935937 + 0.352168i \(0.114556\pi\)
\(594\) 0 0
\(595\) −1.58646e8 + 2.74782e8i −0.0308758 + 0.0534785i
\(596\) 0 0
\(597\) −4.69181e8 −0.0902464
\(598\) 0 0
\(599\) −1.80344e9 + 3.12365e9i −0.342852 + 0.593838i −0.984961 0.172776i \(-0.944726\pi\)
0.642109 + 0.766614i \(0.278060\pi\)
\(600\) 0 0
\(601\) −2.35303e8 −0.0442146 −0.0221073 0.999756i \(-0.507038\pi\)
−0.0221073 + 0.999756i \(0.507038\pi\)
\(602\) 0 0
\(603\) 1.93473e9 + 3.35105e9i 0.359343 + 0.622401i
\(604\) 0 0
\(605\) −9.22803e8 1.59834e9i −0.169420 0.293444i
\(606\) 0 0
\(607\) −2.73499e9 −0.496358 −0.248179 0.968714i \(-0.579832\pi\)
−0.248179 + 0.968714i \(0.579832\pi\)
\(608\) 0 0
\(609\) −2.67969e9 −0.480756
\(610\) 0 0
\(611\) −5.62404e9 9.74112e9i −0.997479 1.72768i
\(612\) 0 0
\(613\) 2.32909e8 + 4.03410e8i 0.0408390 + 0.0707352i 0.885722 0.464215i \(-0.153664\pi\)
−0.844883 + 0.534950i \(0.820330\pi\)
\(614\) 0 0
\(615\) 3.30332e8 0.0572647
\(616\) 0 0
\(617\) 1.09918e9 1.90383e9i 0.188396 0.326311i −0.756320 0.654202i \(-0.773005\pi\)
0.944715 + 0.327891i \(0.106338\pi\)
\(618\) 0 0
\(619\) 1.05418e10 1.78647 0.893236 0.449587i \(-0.148429\pi\)
0.893236 + 0.449587i \(0.148429\pi\)
\(620\) 0 0
\(621\) 7.77651e8 1.34693e9i 0.130306 0.225697i
\(622\) 0 0
\(623\) 3.24251e9 5.61620e9i 0.537246 0.930538i
\(624\) 0 0
\(625\) −1.45503e9 2.52018e9i −0.238392 0.412907i
\(626\) 0 0
\(627\) −7.94923e8 + 5.05287e8i −0.128792 + 0.0818656i
\(628\) 0 0
\(629\) −1.10773e8 1.91865e8i −0.0177483 0.0307410i
\(630\) 0 0
\(631\) 5.25684e9 9.10511e9i 0.832955 1.44272i −0.0627288 0.998031i \(-0.519980\pi\)
0.895684 0.444691i \(-0.146686\pi\)
\(632\) 0 0
\(633\) −1.51630e9 + 2.62631e9i −0.237615 + 0.411561i
\(634\) 0 0
\(635\) −1.15187e9 −0.178524
\(636\) 0 0
\(637\) 4.78777e9 8.29267e9i 0.733915 1.27118i
\(638\) 0 0
\(639\) 8.60492e9 1.30465
\(640\) 0 0
\(641\) 1.69373e9 + 2.93362e9i 0.254004 + 0.439948i 0.964624 0.263628i \(-0.0849191\pi\)
−0.710621 + 0.703575i \(0.751586\pi\)
\(642\) 0 0
\(643\) 1.55153e9 + 2.68732e9i 0.230155 + 0.398640i 0.957854 0.287257i \(-0.0927433\pi\)
−0.727699 + 0.685897i \(0.759410\pi\)
\(644\) 0 0
\(645\) 2.53594e8 0.0372118
\(646\) 0 0
\(647\) −4.87239e8 −0.0707257 −0.0353628 0.999375i \(-0.511259\pi\)
−0.0353628 + 0.999375i \(0.511259\pi\)
\(648\) 0 0
\(649\) −2.45995e9 4.26075e9i −0.353239 0.611828i
\(650\) 0 0
\(651\) −2.28184e9 3.95226e9i −0.324154 0.561451i
\(652\) 0 0
\(653\) 1.15984e10 1.63005 0.815024 0.579427i \(-0.196724\pi\)
0.815024 + 0.579427i \(0.196724\pi\)
\(654\) 0 0
\(655\) −1.57247e9 + 2.72359e9i −0.218644 + 0.378702i
\(656\) 0 0
\(657\) −4.19918e9 −0.577678
\(658\) 0 0
\(659\) −5.16391e9 + 8.94416e9i −0.702878 + 1.21742i 0.264574 + 0.964365i \(0.414769\pi\)
−0.967452 + 0.253055i \(0.918565\pi\)
\(660\) 0 0
\(661\) 5.26311e9 9.11597e9i 0.708822 1.22772i −0.256472 0.966552i \(-0.582560\pi\)
0.965294 0.261164i \(-0.0841064\pi\)
\(662\) 0 0
\(663\) −1.26013e8 2.18260e8i −0.0167926 0.0290856i
\(664\) 0 0
\(665\) 4.22796e9 2.68747e9i 0.557513 0.354379i
\(666\) 0 0
\(667\) −1.52381e9 2.63932e9i −0.198834 0.344391i
\(668\) 0 0
\(669\) 1.38107e9 2.39208e9i 0.178330 0.308876i
\(670\) 0 0
\(671\) −1.04029e9 + 1.80184e9i −0.132931 + 0.230243i
\(672\) 0 0
\(673\) −4.39297e9 −0.555527 −0.277764 0.960649i \(-0.589593\pi\)
−0.277764 + 0.960649i \(0.589593\pi\)
\(674\) 0 0
\(675\) 2.02831e9 3.51314e9i 0.253847 0.439676i
\(676\) 0 0
\(677\) −3.03159e9 −0.375500 −0.187750 0.982217i \(-0.560120\pi\)
−0.187750 + 0.982217i \(0.560120\pi\)
\(678\) 0 0
\(679\) 1.12515e10 + 1.94881e10i 1.37932 + 2.38905i
\(680\) 0 0
\(681\) −3.18993e8 5.52512e8i −0.0387049 0.0670389i
\(682\) 0 0
\(683\) −2.40218e9 −0.288492 −0.144246 0.989542i \(-0.546076\pi\)
−0.144246 + 0.989542i \(0.546076\pi\)
\(684\) 0 0
\(685\) 6.16110e9 0.732389
\(686\) 0 0
\(687\) 2.43851e8 + 4.22362e8i 0.0286930 + 0.0496977i
\(688\) 0 0
\(689\) −2.01558e9 3.49109e9i −0.234765 0.406624i
\(690\) 0 0
\(691\) −1.44559e10 −1.66676 −0.833378 0.552703i \(-0.813596\pi\)
−0.833378 + 0.552703i \(0.813596\pi\)
\(692\) 0 0
\(693\) −2.76825e9 + 4.79475e9i −0.315965 + 0.547268i
\(694\) 0 0
\(695\) 1.11855e9 0.126388
\(696\) 0 0
\(697\) −1.68274e8 + 2.91458e8i −0.0188235 + 0.0326033i
\(698\) 0 0
\(699\) −1.22755e9 + 2.12617e9i −0.135946 + 0.235466i
\(700\) 0 0
\(701\) 7.07010e9 + 1.22458e10i 0.775198 + 1.34268i 0.934683 + 0.355482i \(0.115683\pi\)
−0.159485 + 0.987200i \(0.550984\pi\)
\(702\) 0 0
\(703\) 1.49030e8 + 3.49488e9i 0.0161782 + 0.379392i
\(704\) 0 0
\(705\) 1.21107e9 + 2.09764e9i 0.130169 + 0.225460i
\(706\) 0 0
\(707\) 1.15640e10 2.00294e10i 1.23066 2.13157i
\(708\) 0 0
\(709\) −8.20648e9 + 1.42140e10i −0.864759 + 1.49781i 0.00252638 + 0.999997i \(0.499196\pi\)
−0.867286 + 0.497810i \(0.834138\pi\)
\(710\) 0 0
\(711\) 1.79915e9 0.187726
\(712\) 0 0
\(713\) 2.59514e9 4.49492e9i 0.268132 0.464418i
\(714\) 0 0
\(715\) 2.12516e9 0.217430
\(716\) 0 0
\(717\) 9.92182e8 + 1.71851e9i 0.100525 + 0.174115i
\(718\) 0 0
\(719\) −5.85186e9 1.01357e10i −0.587141 1.01696i −0.994605 0.103737i \(-0.966920\pi\)
0.407463 0.913222i \(-0.366413\pi\)
\(720\) 0 0
\(721\) −2.64338e10 −2.62656
\(722\) 0 0
\(723\) 6.49200e9 0.638843
\(724\) 0 0
\(725\) −3.97449e9 6.88402e9i −0.387345 0.670902i
\(726\) 0 0
\(727\) −4.91258e9 8.50884e9i −0.474176 0.821296i 0.525387 0.850863i \(-0.323920\pi\)
−0.999563 + 0.0295671i \(0.990587\pi\)
\(728\) 0 0
\(729\) −4.24078e9 −0.405415
\(730\) 0 0
\(731\) −1.29183e8 + 2.23752e8i −0.0122319 + 0.0211863i
\(732\) 0 0
\(733\) 7.76094e9 0.727865 0.363932 0.931425i \(-0.381434\pi\)
0.363932 + 0.931425i \(0.381434\pi\)
\(734\) 0 0
\(735\) −1.03099e9 + 1.78573e9i −0.0957745 + 0.165886i
\(736\) 0 0
\(737\) −2.02783e9 + 3.51230e9i −0.186593 + 0.323188i
\(738\) 0 0
\(739\) 6.90874e9 + 1.19663e10i 0.629714 + 1.09070i 0.987609 + 0.156934i \(0.0501611\pi\)
−0.357896 + 0.933762i \(0.616506\pi\)
\(740\) 0 0
\(741\) 1.69533e8 + 3.97569e9i 0.0153070 + 0.358962i
\(742\) 0 0
\(743\) −7.75066e9 1.34245e10i −0.693231 1.20071i −0.970773 0.239998i \(-0.922853\pi\)
0.277542 0.960713i \(-0.410480\pi\)
\(744\) 0 0
\(745\) 3.02581e9 5.24085e9i 0.268098 0.464360i
\(746\) 0 0
\(747\) −7.54770e9 + 1.30730e10i −0.662510 + 1.14750i
\(748\) 0 0
\(749\) 4.37262e9 0.380238
\(750\) 0 0
\(751\) 6.96809e9 1.20691e10i 0.600308 1.03976i −0.392466 0.919766i \(-0.628378\pi\)
0.992774 0.119997i \(-0.0382886\pi\)
\(752\) 0 0
\(753\) −1.56061e9 −0.133203
\(754\) 0 0
\(755\) 1.45043e9 + 2.51221e9i 0.122654 + 0.212443i
\(756\) 0 0
\(757\) 9.93533e9 + 1.72085e10i 0.832428 + 1.44181i 0.896107 + 0.443837i \(0.146383\pi\)
−0.0636791 + 0.997970i \(0.520283\pi\)
\(758\) 0 0
\(759\) 7.68210e8 0.0637726
\(760\) 0 0
\(761\) −8.10861e9 −0.666961 −0.333480 0.942757i \(-0.608223\pi\)
−0.333480 + 0.942757i \(0.608223\pi\)
\(762\) 0 0
\(763\) 1.05720e9 + 1.83112e9i 0.0861630 + 0.149239i
\(764\) 0 0
\(765\) −2.22417e8 3.85238e8i −0.0179620 0.0311110i
\(766\) 0 0
\(767\) −2.07849e10 −1.66327
\(768\) 0 0
\(769\) −1.21009e10 + 2.09594e10i −0.959568 + 1.66202i −0.236017 + 0.971749i \(0.575842\pi\)
−0.723551 + 0.690271i \(0.757491\pi\)
\(770\) 0 0
\(771\) −1.87801e8 −0.0147573
\(772\) 0 0
\(773\) −1.12978e10 + 1.95684e10i −0.879763 + 1.52379i −0.0281627 + 0.999603i \(0.508966\pi\)
−0.851601 + 0.524191i \(0.824368\pi\)
\(774\) 0 0
\(775\) 6.76879e9 1.17239e10i 0.522342 0.904724i
\(776\) 0 0
\(777\) −1.25425e9 2.17243e9i −0.0959206 0.166139i
\(778\) 0 0
\(779\) 4.48454e9 2.85056e9i 0.339889 0.216048i
\(780\) 0 0
\(781\) 4.50949e9 + 7.81067e9i 0.338726 + 0.586691i
\(782\) 0 0
\(783\) 3.98604e9 6.90402e9i 0.296739 0.513968i
\(784\) 0 0
\(785\) 1.81993e9 3.15221e9i 0.134280 0.232579i
\(786\) 0 0
\(787\) −1.39366e10 −1.01916 −0.509582 0.860422i \(-0.670200\pi\)
−0.509582 + 0.860422i \(0.670200\pi\)
\(788\) 0 0
\(789\) −3.79492e9 + 6.57300e9i −0.275064 + 0.476424i
\(790\) 0 0
\(791\) −2.67889e10 −1.92459
\(792\) 0 0
\(793\) 4.39487e9 + 7.61214e9i 0.312961 + 0.542064i
\(794\) 0 0
\(795\) 4.34032e8 + 7.51766e8i 0.0306364 + 0.0530637i
\(796\) 0 0
\(797\) −9.99801e9 −0.699535 −0.349767 0.936837i \(-0.613739\pi\)
−0.349767 + 0.936837i \(0.613739\pi\)
\(798\) 0 0
\(799\) −2.46772e9 −0.171152
\(800\) 0 0
\(801\) 4.54592e9 + 7.87377e9i 0.312542 + 0.541338i
\(802\) 0 0
\(803\) −2.20062e9 3.81159e9i −0.149983 0.259778i
\(804\) 0 0
\(805\) −4.08588e9 −0.276058
\(806\) 0 0
\(807\) −2.45937e9 + 4.25975e9i −0.164727 + 0.285316i
\(808\) 0 0
\(809\) 1.63818e10 1.08778 0.543892 0.839155i \(-0.316950\pi\)
0.543892 + 0.839155i \(0.316950\pi\)
\(810\) 0 0
\(811\) 2.38271e9 4.12697e9i 0.156855 0.271681i −0.776878 0.629651i \(-0.783198\pi\)
0.933733 + 0.357971i \(0.116531\pi\)
\(812\) 0 0
\(813\) 3.51691e9 6.09146e9i 0.229532 0.397562i
\(814\) 0 0
\(815\) −7.56442e8 1.31020e9i −0.0489468 0.0847783i
\(816\) 0 0
\(817\) 3.44277e9 2.18837e9i 0.220867 0.140392i
\(818\) 0 0
\(819\) 1.16949e10 + 2.02562e10i 0.743881 + 1.28844i
\(820\) 0 0
\(821\) 1.12396e10 1.94676e10i 0.708843 1.22775i −0.256443 0.966559i \(-0.582551\pi\)
0.965286 0.261193i \(-0.0841160\pi\)
\(822\) 0 0
\(823\) −7.56189e9 + 1.30976e10i −0.472859 + 0.819015i −0.999517 0.0310617i \(-0.990111\pi\)
0.526659 + 0.850077i \(0.323444\pi\)
\(824\) 0 0
\(825\) 2.00369e9 0.124234
\(826\) 0 0
\(827\) 1.60761e10 2.78447e10i 0.988353 1.71188i 0.362385 0.932028i \(-0.381962\pi\)
0.625968 0.779849i \(-0.284704\pi\)
\(828\) 0 0
\(829\) 8.77112e9 0.534705 0.267353 0.963599i \(-0.413851\pi\)
0.267353 + 0.963599i \(0.413851\pi\)
\(830\) 0 0
\(831\) 3.33064e9 + 5.76884e9i 0.201337 + 0.348726i
\(832\) 0 0
\(833\) −1.05039e9 1.81933e9i −0.0629642 0.109057i
\(834\) 0 0
\(835\) −3.76353e9 −0.223714
\(836\) 0 0
\(837\) 1.35769e10 0.800317
\(838\) 0 0
\(839\) −1.19267e10 2.06577e10i −0.697195 1.20758i −0.969435 0.245347i \(-0.921098\pi\)
0.272241 0.962229i \(-0.412235\pi\)
\(840\) 0 0
\(841\) 8.14276e8 + 1.41037e9i 0.0472047 + 0.0817610i
\(842\) 0 0
\(843\) −4.40288e8 −0.0253128
\(844\) 0 0
\(845\) 7.07731e8 1.22583e9i 0.0403525 0.0698925i
\(846\) 0 0
\(847\) 2.12905e10 1.20391
\(848\) 0 0
\(849\) 1.35565e9 2.34806e9i 0.0760277 0.131684i
\(850\) 0 0
\(851\) 1.42647e9 2.47071e9i 0.0793429 0.137426i
\(852\) 0 0
\(853\) 6.70793e9 + 1.16185e10i 0.370056 + 0.640955i 0.989574 0.144027i \(-0.0460053\pi\)
−0.619518 + 0.784982i \(0.712672\pi\)
\(854\) 0 0
\(855\) 2.99233e8 + 7.01724e9i 0.0163730 + 0.383959i
\(856\) 0 0
\(857\) −7.79764e9 1.35059e10i −0.423185 0.732978i 0.573064 0.819511i \(-0.305755\pi\)
−0.996249 + 0.0865326i \(0.972421\pi\)
\(858\) 0 0
\(859\) 1.76013e9 3.04863e9i 0.0947475 0.164108i −0.814756 0.579804i \(-0.803129\pi\)
0.909503 + 0.415697i \(0.136462\pi\)
\(860\) 0 0
\(861\) −1.90531e9 + 3.30010e9i −0.101732 + 0.176204i
\(862\) 0 0
\(863\) −1.79917e10 −0.952870 −0.476435 0.879210i \(-0.658071\pi\)
−0.476435 + 0.879210i \(0.658071\pi\)
\(864\) 0 0
\(865\) 4.18455e7 7.24785e7i 0.00219833 0.00380761i
\(866\) 0 0
\(867\) 6.27252e9 0.326870
\(868\) 0 0
\(869\) 9.42864e8 + 1.63309e9i 0.0487393 + 0.0844190i
\(870\) 0 0
\(871\) 8.56688e9 + 1.48383e10i 0.439298 + 0.760886i
\(872\) 0 0
\(873\) −3.15486e10 −1.60483
\(874\) 0 0
\(875\) −2.37480e10 −1.19839
\(876\) 0 0
\(877\) −1.47815e10 2.56023e10i −0.739980 1.28168i −0.952504 0.304527i \(-0.901502\pi\)
0.212524 0.977156i \(-0.431832\pi\)
\(878\) 0 0
\(879\) 3.27767e9 + 5.67710e9i 0.162781 + 0.281945i
\(880\) 0 0
\(881\) 2.49389e10 1.22874 0.614372 0.789016i \(-0.289409\pi\)
0.614372 + 0.789016i \(0.289409\pi\)
\(882\) 0 0
\(883\) −7.51697e8 + 1.30198e9i −0.0367435 + 0.0636416i −0.883812 0.467842i \(-0.845032\pi\)
0.847069 + 0.531483i \(0.178365\pi\)
\(884\) 0 0
\(885\) 4.47578e9 0.217054
\(886\) 0 0
\(887\) 1.36244e10 2.35982e10i 0.655519 1.13539i −0.326245 0.945285i \(-0.605783\pi\)
0.981763 0.190107i \(-0.0608834\pi\)
\(888\) 0 0
\(889\) 6.64385e9 1.15075e10i 0.317149 0.549319i
\(890\) 0 0
\(891\) −3.34976e9 5.80196e9i −0.158651 0.274791i
\(892\) 0 0
\(893\) 3.45428e10 + 1.80265e10i 1.62322 + 0.847092i
\(894\) 0 0
\(895\) −4.05777e9 7.02826e9i −0.189194 0.327693i
\(896\) 0 0
\(897\) 1.62271e9 2.81062e9i 0.0750703 0.130026i
\(898\) 0 0
\(899\) 1.33020e10 2.30398e10i 0.610603 1.05759i
\(900\) 0 0
\(901\) −8.84398e8 −0.0402820
\(902\) 0 0
\(903\) −1.46270e9 + 2.53348e9i −0.0661073 + 0.114501i
\(904\) 0 0
\(905\) −1.56693e7 −0.000702717
\(906\) 0 0
\(907\) −6.83931e9 1.18460e10i −0.304359 0.527166i 0.672759 0.739862i \(-0.265109\pi\)
−0.977118 + 0.212696i \(0.931776\pi\)
\(908\) 0 0
\(909\) 1.62124e10 + 2.80807e10i 0.715934 + 1.24003i
\(910\) 0 0
\(911\) 2.39180e10 1.04812 0.524059 0.851682i \(-0.324417\pi\)
0.524059 + 0.851682i \(0.324417\pi\)
\(912\) 0 0
\(913\) −1.58218e10 −0.688031
\(914\) 0 0
\(915\) −9.46386e8 1.63919e9i −0.0408408 0.0707384i
\(916\) 0 0
\(917\) −1.81396e10 3.14187e10i −0.776847 1.34554i
\(918\) 0 0
\(919\) 3.00107e10 1.27547 0.637737 0.770254i \(-0.279871\pi\)
0.637737 + 0.770254i \(0.279871\pi\)
\(920\) 0 0
\(921\) 4.17479e9 7.23095e9i 0.176086 0.304991i
\(922\) 0 0
\(923\) 3.81021e10 1.59494
\(924\) 0 0
\(925\) 3.72059e9 6.44425e9i 0.154567 0.267717i
\(926\) 0 0
\(927\) 1.85298e10 3.20945e10i 0.763997 1.32328i
\(928\) 0 0
\(929\) 1.26478e10 + 2.19066e10i 0.517559 + 0.896439i 0.999792 + 0.0203955i \(0.00649255\pi\)
−0.482233 + 0.876043i \(0.660174\pi\)
\(930\) 0 0
\(931\) 1.41316e9 + 3.31398e10i 0.0573942 + 1.34594i
\(932\) 0 0
\(933\) −2.53528e8 4.39123e8i −0.0102197 0.0177011i
\(934\) 0 0
\(935\) 2.33120e8 4.03776e8i 0.00932693 0.0161547i
\(936\) 0 0
\(937\) 2.71637e9 4.70490e9i 0.107870 0.186836i −0.807037 0.590501i \(-0.798930\pi\)
0.914907 + 0.403664i \(0.132264\pi\)
\(938\) 0 0
\(939\) −1.28197e10 −0.505298
\(940\) 0 0
\(941\) −1.42404e10 + 2.46651e10i −0.557132 + 0.964980i 0.440603 + 0.897702i \(0.354765\pi\)
−0.997734 + 0.0672779i \(0.978569\pi\)
\(942\) 0 0
\(943\) −4.33384e9 −0.168299
\(944\) 0 0
\(945\) −5.34398e9 9.25605e9i −0.205994 0.356791i
\(946\) 0 0
\(947\) −1.86941e10 3.23791e10i −0.715285 1.23891i −0.962850 0.270039i \(-0.912964\pi\)
0.247565 0.968871i \(-0.420370\pi\)
\(948\) 0 0
\(949\) −1.85938e10 −0.706212
\(950\) 0 0
\(951\) −1.07445e10 −0.405091
\(952\) 0 0
\(953\) −9.95647e9 1.72451e10i −0.372632 0.645418i 0.617338 0.786698i \(-0.288211\pi\)
−0.989970 + 0.141281i \(0.954878\pi\)
\(954\) 0 0
\(955\) −4.06563e9 7.04189e9i −0.151049 0.261624i
\(956\) 0 0
\(957\) 3.93764e9 0.145226
\(958\) 0 0
\(959\) −3.55365e10 + 6.15511e10i −1.30110 + 2.25357i
\(960\) 0 0
\(961\) 1.77957e10 0.646818
\(962\) 0 0
\(963\) −3.06515e9 + 5.30900e9i −0.110601 + 0.191567i
\(964\) 0 0
\(965\) 4.03422e8 6.98748e8i 0.0144515 0.0250308i
\(966\) 0 0
\(967\) −1.79745e10 3.11328e10i −0.639241 1.10720i −0.985600 0.169096i \(-0.945915\pi\)
0.346358 0.938102i \(-0.387418\pi\)
\(968\) 0 0
\(969\) 7.73968e8 + 4.03903e8i 0.0273269 + 0.0142608i
\(970\) 0 0
\(971\) 1.09527e10 + 1.89706e10i 0.383930 + 0.664987i 0.991620 0.129187i \(-0.0412369\pi\)
−0.607690 + 0.794175i \(0.707904\pi\)
\(972\) 0 0
\(973\) −6.45164e9 + 1.11746e10i −0.224530 + 0.388898i
\(974\) 0 0
\(975\) 4.23245e9 7.33081e9i 0.146243 0.253300i
\(976\) 0 0
\(977\) −2.41093e10 −0.827091 −0.413546 0.910483i \(-0.635710\pi\)
−0.413546 + 0.910483i \(0.635710\pi\)
\(978\) 0 0
\(979\) −4.76467e9 + 8.25265e9i −0.162291 + 0.281096i
\(980\) 0 0
\(981\) −2.96433e9 −0.100250
\(982\) 0 0
\(983\) 2.51995e10 + 4.36468e10i 0.846163 + 1.46560i 0.884607 + 0.466337i \(0.154426\pi\)
−0.0384442 + 0.999261i \(0.512240\pi\)
\(984\) 0 0
\(985\) 6.69048e9 + 1.15883e10i 0.223065 + 0.386359i
\(986\) 0 0
\(987\) −2.79413e10 −0.924989
\(988\) 0 0
\(989\) −3.32708e9 −0.109364
\(990\) 0 0
\(991\) 5.46607e9 + 9.46751e9i 0.178409 + 0.309014i 0.941336 0.337471i \(-0.109572\pi\)
−0.762927 + 0.646485i \(0.776238\pi\)
\(992\) 0 0
\(993\) 4.65101e8 + 8.05578e8i 0.0150739 + 0.0261087i
\(994\) 0 0
\(995\) −3.66688e9 −0.118009
\(996\) 0 0
\(997\) 7.06226e9 1.22322e10i 0.225689 0.390905i −0.730837 0.682552i \(-0.760870\pi\)
0.956526 + 0.291647i \(0.0942034\pi\)
\(998\) 0 0
\(999\) 7.46279e9 0.236822
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 76.8.e.a.45.7 22
19.11 even 3 inner 76.8.e.a.49.7 yes 22
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.8.e.a.45.7 22 1.1 even 1 trivial
76.8.e.a.49.7 yes 22 19.11 even 3 inner