Properties

Label 35.8.a.a.1.2
Level $35$
Weight $8$
Character 35.1
Self dual yes
Analytic conductor $10.933$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [35,8,Mod(1,35)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(35, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("35.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 35 = 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 35.a (trivial)

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: yes
Analytic conductor: \(10.9334758919\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{11}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 11 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.31662\) of defining polynomial
Character \(\chi\) \(=\) 35.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+14.6332 q^{2} -54.7995 q^{3} +86.1320 q^{4} +125.000 q^{5} -801.895 q^{6} -343.000 q^{7} -612.665 q^{8} +815.985 q^{9} +O(q^{10})\) \(q+14.6332 q^{2} -54.7995 q^{3} +86.1320 q^{4} +125.000 q^{5} -801.895 q^{6} -343.000 q^{7} -612.665 q^{8} +815.985 q^{9} +1829.16 q^{10} -6473.63 q^{11} -4719.99 q^{12} -11681.7 q^{13} -5019.20 q^{14} -6849.94 q^{15} -19990.2 q^{16} +13460.5 q^{17} +11940.5 q^{18} +34955.5 q^{19} +10766.5 q^{20} +18796.2 q^{21} -94730.3 q^{22} +77831.4 q^{23} +33573.7 q^{24} +15625.0 q^{25} -170941. q^{26} +75130.9 q^{27} -29543.3 q^{28} -221135. q^{29} -100237. q^{30} -23222.3 q^{31} -214100. q^{32} +354752. q^{33} +196971. q^{34} -42875.0 q^{35} +70282.4 q^{36} -422392. q^{37} +511512. q^{38} +640151. q^{39} -76583.1 q^{40} +191818. q^{41} +275050. q^{42} +310754. q^{43} -557587. q^{44} +101998. q^{45} +1.13893e6 q^{46} -240747. q^{47} +1.09545e6 q^{48} +117649. q^{49} +228645. q^{50} -737628. q^{51} -1.00617e6 q^{52} -1.06654e6 q^{53} +1.09941e6 q^{54} -809204. q^{55} +210144. q^{56} -1.91554e6 q^{57} -3.23592e6 q^{58} +451838. q^{59} -589999. q^{60} -831659. q^{61} -339818. q^{62} -279883. q^{63} -574238. q^{64} -1.46021e6 q^{65} +5.19117e6 q^{66} +2.26405e6 q^{67} +1.15938e6 q^{68} -4.26512e6 q^{69} -627401. q^{70} -2.22036e6 q^{71} -499925. q^{72} +4.99377e6 q^{73} -6.18096e6 q^{74} -856242. q^{75} +3.01078e6 q^{76} +2.22046e6 q^{77} +9.36749e6 q^{78} -2.72773e6 q^{79} -2.49877e6 q^{80} -5.90170e6 q^{81} +2.80693e6 q^{82} -6.38392e6 q^{83} +1.61896e6 q^{84} +1.68256e6 q^{85} +4.54734e6 q^{86} +1.21181e7 q^{87} +3.96617e6 q^{88} -7.32978e6 q^{89} +1.49256e6 q^{90} +4.00682e6 q^{91} +6.70378e6 q^{92} +1.27257e6 q^{93} -3.52291e6 q^{94} +4.36943e6 q^{95} +1.17326e7 q^{96} -2.38676e6 q^{97} +1.72159e6 q^{98} -5.28239e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 16 q^{2} - 30 q^{3} - 40 q^{4} + 250 q^{5} - 768 q^{6} - 686 q^{7} - 960 q^{8} - 756 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 16 q^{2} - 30 q^{3} - 40 q^{4} + 250 q^{5} - 768 q^{6} - 686 q^{7} - 960 q^{8} - 756 q^{9} + 2000 q^{10} - 7906 q^{11} - 7848 q^{12} - 17818 q^{13} - 5488 q^{14} - 3750 q^{15} - 4320 q^{16} - 2398 q^{17} + 9792 q^{18} - 3612 q^{19} - 5000 q^{20} + 10290 q^{21} - 96688 q^{22} + 13844 q^{23} + 24960 q^{24} + 31250 q^{25} - 179328 q^{26} - 18090 q^{27} + 13720 q^{28} - 126898 q^{29} - 96000 q^{30} + 252768 q^{31} - 148224 q^{32} + 319230 q^{33} + 175296 q^{34} - 85750 q^{35} + 268560 q^{36} - 265860 q^{37} + 458800 q^{38} + 487974 q^{39} - 120000 q^{40} - 111920 q^{41} + 263424 q^{42} + 947572 q^{43} - 376920 q^{44} - 94500 q^{45} + 1051472 q^{46} + 271274 q^{47} + 1484064 q^{48} + 235298 q^{49} + 250000 q^{50} - 1130910 q^{51} - 232184 q^{52} - 1267792 q^{53} + 972000 q^{54} - 988250 q^{55} + 329280 q^{56} - 2871996 q^{57} - 3107120 q^{58} - 1360120 q^{59} - 981000 q^{60} - 1813680 q^{61} + 37392 q^{62} + 259308 q^{63} - 2489984 q^{64} - 2227250 q^{65} + 5142624 q^{66} - 2189312 q^{67} + 3159640 q^{68} - 5851980 q^{69} - 686000 q^{70} - 1494928 q^{71} + 46080 q^{72} + 7169788 q^{73} - 5967024 q^{74} - 468750 q^{75} + 7875376 q^{76} + 2711758 q^{77} + 9159504 q^{78} - 7942974 q^{79} - 540000 q^{80} - 4775598 q^{81} + 2391792 q^{82} - 304712 q^{83} + 2691864 q^{84} - 299750 q^{85} + 5417712 q^{86} + 14455086 q^{87} + 4463680 q^{88} - 17943528 q^{89} + 1224000 q^{90} + 6111574 q^{91} + 14774640 q^{92} + 8116992 q^{93} - 2823104 q^{94} - 451500 q^{95} + 13366272 q^{96} + 4258074 q^{97} + 1882384 q^{98} - 3030732 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.6332 1.29341 0.646704 0.762741i \(-0.276147\pi\)
0.646704 + 0.762741i \(0.276147\pi\)
\(3\) −54.7995 −1.17180 −0.585898 0.810385i \(-0.699258\pi\)
−0.585898 + 0.810385i \(0.699258\pi\)
\(4\) 86.1320 0.672906
\(5\) 125.000 0.447214
\(6\) −801.895 −1.51561
\(7\) −343.000 −0.377964
\(8\) −612.665 −0.423066
\(9\) 815.985 0.373107
\(10\) 1829.16 0.578430
\(11\) −6473.63 −1.46647 −0.733236 0.679974i \(-0.761991\pi\)
−0.733236 + 0.679974i \(0.761991\pi\)
\(12\) −4719.99 −0.788509
\(13\) −11681.7 −1.47470 −0.737351 0.675510i \(-0.763924\pi\)
−0.737351 + 0.675510i \(0.763924\pi\)
\(14\) −5019.20 −0.488863
\(15\) −6849.94 −0.524043
\(16\) −19990.2 −1.22010
\(17\) 13460.5 0.664491 0.332246 0.943193i \(-0.392194\pi\)
0.332246 + 0.943193i \(0.392194\pi\)
\(18\) 11940.5 0.482580
\(19\) 34955.5 1.16917 0.584585 0.811333i \(-0.301257\pi\)
0.584585 + 0.811333i \(0.301257\pi\)
\(20\) 10766.5 0.300933
\(21\) 18796.2 0.442897
\(22\) −94730.3 −1.89675
\(23\) 77831.4 1.33385 0.666926 0.745124i \(-0.267610\pi\)
0.666926 + 0.745124i \(0.267610\pi\)
\(24\) 33573.7 0.495747
\(25\) 15625.0 0.200000
\(26\) −170941. −1.90739
\(27\) 75130.9 0.734591
\(28\) −29543.3 −0.254335
\(29\) −221135. −1.68370 −0.841848 0.539715i \(-0.818532\pi\)
−0.841848 + 0.539715i \(0.818532\pi\)
\(30\) −100237. −0.677802
\(31\) −23222.3 −0.140004 −0.0700018 0.997547i \(-0.522301\pi\)
−0.0700018 + 0.997547i \(0.522301\pi\)
\(32\) −214100. −1.15503
\(33\) 354752. 1.71841
\(34\) 196971. 0.859459
\(35\) −42875.0 −0.169031
\(36\) 70282.4 0.251066
\(37\) −422392. −1.37091 −0.685456 0.728114i \(-0.740397\pi\)
−0.685456 + 0.728114i \(0.740397\pi\)
\(38\) 511512. 1.51221
\(39\) 640151. 1.72805
\(40\) −76583.1 −0.189201
\(41\) 191818. 0.434657 0.217329 0.976099i \(-0.430266\pi\)
0.217329 + 0.976099i \(0.430266\pi\)
\(42\) 275050. 0.572847
\(43\) 310754. 0.596042 0.298021 0.954559i \(-0.403673\pi\)
0.298021 + 0.954559i \(0.403673\pi\)
\(44\) −557587. −0.986798
\(45\) 101998. 0.166859
\(46\) 1.13893e6 1.72522
\(47\) −240747. −0.338235 −0.169117 0.985596i \(-0.554092\pi\)
−0.169117 + 0.985596i \(0.554092\pi\)
\(48\) 1.09545e6 1.42971
\(49\) 117649. 0.142857
\(50\) 228645. 0.258682
\(51\) −737628. −0.778649
\(52\) −1.00617e6 −0.992336
\(53\) −1.06654e6 −0.984040 −0.492020 0.870584i \(-0.663741\pi\)
−0.492020 + 0.870584i \(0.663741\pi\)
\(54\) 1.09941e6 0.950126
\(55\) −809204. −0.655826
\(56\) 210144. 0.159904
\(57\) −1.91554e6 −1.37003
\(58\) −3.23592e6 −2.17771
\(59\) 451838. 0.286418 0.143209 0.989692i \(-0.454258\pi\)
0.143209 + 0.989692i \(0.454258\pi\)
\(60\) −589999. −0.352632
\(61\) −831659. −0.469127 −0.234564 0.972101i \(-0.575366\pi\)
−0.234564 + 0.972101i \(0.575366\pi\)
\(62\) −339818. −0.181082
\(63\) −279883. −0.141021
\(64\) −574238. −0.273818
\(65\) −1.46021e6 −0.659507
\(66\) 5.19117e6 2.22260
\(67\) 2.26405e6 0.919654 0.459827 0.888009i \(-0.347912\pi\)
0.459827 + 0.888009i \(0.347912\pi\)
\(68\) 1.15938e6 0.447140
\(69\) −4.26512e6 −1.56300
\(70\) −627401. −0.218626
\(71\) −2.22036e6 −0.736241 −0.368120 0.929778i \(-0.619999\pi\)
−0.368120 + 0.929778i \(0.619999\pi\)
\(72\) −499925. −0.157849
\(73\) 4.99377e6 1.50244 0.751222 0.660049i \(-0.229465\pi\)
0.751222 + 0.660049i \(0.229465\pi\)
\(74\) −6.18096e6 −1.77315
\(75\) −856242. −0.234359
\(76\) 3.01078e6 0.786742
\(77\) 2.22046e6 0.554274
\(78\) 9.36749e6 2.23508
\(79\) −2.72773e6 −0.622452 −0.311226 0.950336i \(-0.600740\pi\)
−0.311226 + 0.950336i \(0.600740\pi\)
\(80\) −2.49877e6 −0.545647
\(81\) −5.90170e6 −1.23390
\(82\) 2.80693e6 0.562189
\(83\) −6.38392e6 −1.22550 −0.612751 0.790276i \(-0.709937\pi\)
−0.612751 + 0.790276i \(0.709937\pi\)
\(84\) 1.61896e6 0.298028
\(85\) 1.68256e6 0.297170
\(86\) 4.54734e6 0.770926
\(87\) 1.21181e7 1.97295
\(88\) 3.96617e6 0.620414
\(89\) −7.32978e6 −1.10211 −0.551056 0.834468i \(-0.685775\pi\)
−0.551056 + 0.834468i \(0.685775\pi\)
\(90\) 1.49256e6 0.215816
\(91\) 4.00682e6 0.557385
\(92\) 6.70378e6 0.897557
\(93\) 1.27257e6 0.164056
\(94\) −3.52291e6 −0.437476
\(95\) 4.36943e6 0.522869
\(96\) 1.17326e7 1.35346
\(97\) −2.38676e6 −0.265526 −0.132763 0.991148i \(-0.542385\pi\)
−0.132763 + 0.991148i \(0.542385\pi\)
\(98\) 1.72159e6 0.184773
\(99\) −5.28239e6 −0.547151
\(100\) 1.34581e6 0.134581
\(101\) −1.92113e7 −1.85538 −0.927688 0.373356i \(-0.878207\pi\)
−0.927688 + 0.373356i \(0.878207\pi\)
\(102\) −1.07939e7 −1.00711
\(103\) −4.45359e6 −0.401587 −0.200793 0.979634i \(-0.564352\pi\)
−0.200793 + 0.979634i \(0.564352\pi\)
\(104\) 7.15697e6 0.623896
\(105\) 2.34953e6 0.198070
\(106\) −1.56070e7 −1.27277
\(107\) 7.61385e6 0.600843 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(108\) 6.47118e6 0.494311
\(109\) −1.99698e7 −1.47700 −0.738502 0.674251i \(-0.764467\pi\)
−0.738502 + 0.674251i \(0.764467\pi\)
\(110\) −1.18413e7 −0.848251
\(111\) 2.31469e7 1.60643
\(112\) 6.85663e6 0.461156
\(113\) 2.57944e7 1.68171 0.840855 0.541261i \(-0.182053\pi\)
0.840855 + 0.541261i \(0.182053\pi\)
\(114\) −2.80306e7 −1.77201
\(115\) 9.72893e6 0.596517
\(116\) −1.90468e7 −1.13297
\(117\) −9.53209e6 −0.550222
\(118\) 6.61185e6 0.370456
\(119\) −4.61695e6 −0.251154
\(120\) 4.19672e6 0.221705
\(121\) 2.24208e7 1.15054
\(122\) −1.21699e7 −0.606773
\(123\) −1.05116e7 −0.509330
\(124\) −2.00018e6 −0.0942094
\(125\) 1.95312e6 0.0894427
\(126\) −4.09560e6 −0.182398
\(127\) −6.75687e6 −0.292707 −0.146353 0.989232i \(-0.546754\pi\)
−0.146353 + 0.989232i \(0.546754\pi\)
\(128\) 1.90018e7 0.800868
\(129\) −1.70292e7 −0.698440
\(130\) −2.13677e7 −0.853012
\(131\) −2.21063e6 −0.0859144 −0.0429572 0.999077i \(-0.513678\pi\)
−0.0429572 + 0.999077i \(0.513678\pi\)
\(132\) 3.05555e7 1.15633
\(133\) −1.19897e7 −0.441905
\(134\) 3.31304e7 1.18949
\(135\) 9.39137e6 0.328519
\(136\) −8.24677e6 −0.281124
\(137\) −7.10581e6 −0.236097 −0.118049 0.993008i \(-0.537664\pi\)
−0.118049 + 0.993008i \(0.537664\pi\)
\(138\) −6.24126e7 −2.02160
\(139\) 9.21848e6 0.291144 0.145572 0.989348i \(-0.453498\pi\)
0.145572 + 0.989348i \(0.453498\pi\)
\(140\) −3.69291e6 −0.113742
\(141\) 1.31928e7 0.396342
\(142\) −3.24911e7 −0.952260
\(143\) 7.56230e7 2.16261
\(144\) −1.63117e7 −0.455229
\(145\) −2.76418e7 −0.752972
\(146\) 7.30751e7 1.94328
\(147\) −6.44711e6 −0.167399
\(148\) −3.63814e7 −0.922495
\(149\) 1.33298e7 0.330119 0.165059 0.986284i \(-0.447218\pi\)
0.165059 + 0.986284i \(0.447218\pi\)
\(150\) −1.25296e7 −0.303122
\(151\) −6.41939e7 −1.51731 −0.758656 0.651492i \(-0.774143\pi\)
−0.758656 + 0.651492i \(0.774143\pi\)
\(152\) −2.14160e7 −0.494636
\(153\) 1.09836e7 0.247926
\(154\) 3.24925e7 0.716903
\(155\) −2.90279e6 −0.0626116
\(156\) 5.51375e7 1.16282
\(157\) 7.36596e7 1.51908 0.759540 0.650461i \(-0.225424\pi\)
0.759540 + 0.650461i \(0.225424\pi\)
\(158\) −3.99155e7 −0.805085
\(159\) 5.84460e7 1.15309
\(160\) −2.67625e7 −0.516544
\(161\) −2.66962e7 −0.504149
\(162\) −8.63610e7 −1.59593
\(163\) −3.50642e7 −0.634172 −0.317086 0.948397i \(-0.602704\pi\)
−0.317086 + 0.948397i \(0.602704\pi\)
\(164\) 1.65217e7 0.292483
\(165\) 4.43440e7 0.768495
\(166\) −9.34175e7 −1.58508
\(167\) 2.56950e6 0.0426915 0.0213458 0.999772i \(-0.493205\pi\)
0.0213458 + 0.999772i \(0.493205\pi\)
\(168\) −1.15158e7 −0.187375
\(169\) 7.37136e7 1.17475
\(170\) 2.46213e7 0.384362
\(171\) 2.85231e7 0.436225
\(172\) 2.67659e7 0.401081
\(173\) 8.03463e7 1.17979 0.589895 0.807480i \(-0.299169\pi\)
0.589895 + 0.807480i \(0.299169\pi\)
\(174\) 1.77327e8 2.55183
\(175\) −5.35938e6 −0.0755929
\(176\) 1.29409e8 1.78925
\(177\) −2.47605e7 −0.335624
\(178\) −1.07259e8 −1.42548
\(179\) 9.99074e7 1.30200 0.651001 0.759077i \(-0.274349\pi\)
0.651001 + 0.759077i \(0.274349\pi\)
\(180\) 8.78530e6 0.112280
\(181\) −1.07414e8 −1.34644 −0.673221 0.739442i \(-0.735090\pi\)
−0.673221 + 0.739442i \(0.735090\pi\)
\(182\) 5.86328e7 0.720927
\(183\) 4.55745e7 0.549722
\(184\) −4.76846e7 −0.564307
\(185\) −5.27990e7 −0.613090
\(186\) 1.86218e7 0.212191
\(187\) −8.71382e7 −0.974458
\(188\) −2.07360e7 −0.227600
\(189\) −2.57699e7 −0.277649
\(190\) 6.39390e7 0.676283
\(191\) −2.21085e7 −0.229584 −0.114792 0.993390i \(-0.536620\pi\)
−0.114792 + 0.993390i \(0.536620\pi\)
\(192\) 3.14679e7 0.320859
\(193\) −1.49793e8 −1.49983 −0.749913 0.661537i \(-0.769905\pi\)
−0.749913 + 0.661537i \(0.769905\pi\)
\(194\) −3.49261e7 −0.343434
\(195\) 8.00189e7 0.772808
\(196\) 1.01333e7 0.0961295
\(197\) −5.70107e7 −0.531281 −0.265641 0.964072i \(-0.585583\pi\)
−0.265641 + 0.964072i \(0.585583\pi\)
\(198\) −7.72985e7 −0.707690
\(199\) −6.84161e7 −0.615421 −0.307711 0.951480i \(-0.599563\pi\)
−0.307711 + 0.951480i \(0.599563\pi\)
\(200\) −9.57289e6 −0.0846132
\(201\) −1.24069e8 −1.07765
\(202\) −2.81124e8 −2.39976
\(203\) 7.58492e7 0.636377
\(204\) −6.35333e7 −0.523958
\(205\) 2.39773e7 0.194385
\(206\) −6.51704e7 −0.519416
\(207\) 6.35093e7 0.497669
\(208\) 2.33519e8 1.79929
\(209\) −2.26289e8 −1.71455
\(210\) 3.43812e7 0.256185
\(211\) −1.36201e8 −0.998141 −0.499071 0.866561i \(-0.666325\pi\)
−0.499071 + 0.866561i \(0.666325\pi\)
\(212\) −9.18635e7 −0.662167
\(213\) 1.21675e8 0.862724
\(214\) 1.11415e8 0.777135
\(215\) 3.88442e7 0.266558
\(216\) −4.60301e7 −0.310780
\(217\) 7.96525e6 0.0529164
\(218\) −2.92224e8 −1.91037
\(219\) −2.73656e8 −1.76056
\(220\) −6.96984e7 −0.441310
\(221\) −1.57241e8 −0.979927
\(222\) 3.38714e8 2.07777
\(223\) −1.47728e8 −0.892066 −0.446033 0.895017i \(-0.647164\pi\)
−0.446033 + 0.895017i \(0.647164\pi\)
\(224\) 7.34363e7 0.436559
\(225\) 1.27498e7 0.0746214
\(226\) 3.77456e8 2.17514
\(227\) 3.22427e8 1.82954 0.914768 0.403980i \(-0.132373\pi\)
0.914768 + 0.403980i \(0.132373\pi\)
\(228\) −1.64989e8 −0.921901
\(229\) 3.10033e8 1.70602 0.853010 0.521895i \(-0.174775\pi\)
0.853010 + 0.521895i \(0.174775\pi\)
\(230\) 1.42366e8 0.771540
\(231\) −1.21680e8 −0.649497
\(232\) 1.35481e8 0.712315
\(233\) 1.80410e8 0.934361 0.467181 0.884162i \(-0.345270\pi\)
0.467181 + 0.884162i \(0.345270\pi\)
\(234\) −1.39485e8 −0.711661
\(235\) −3.00934e7 −0.151263
\(236\) 3.89177e7 0.192733
\(237\) 1.49478e8 0.729387
\(238\) −6.75609e7 −0.324845
\(239\) 3.66489e8 1.73647 0.868237 0.496150i \(-0.165253\pi\)
0.868237 + 0.496150i \(0.165253\pi\)
\(240\) 1.36931e8 0.639387
\(241\) 2.19729e8 1.01118 0.505589 0.862775i \(-0.331275\pi\)
0.505589 + 0.862775i \(0.331275\pi\)
\(242\) 3.28089e8 1.48812
\(243\) 1.59099e8 0.711286
\(244\) −7.16324e7 −0.315679
\(245\) 1.47061e7 0.0638877
\(246\) −1.53818e8 −0.658771
\(247\) −4.08339e8 −1.72418
\(248\) 1.42275e7 0.0592308
\(249\) 3.49836e8 1.43604
\(250\) 2.85806e7 0.115686
\(251\) −1.29875e8 −0.518404 −0.259202 0.965823i \(-0.583460\pi\)
−0.259202 + 0.965823i \(0.583460\pi\)
\(252\) −2.41069e7 −0.0948940
\(253\) −5.03852e8 −1.95606
\(254\) −9.88750e7 −0.378589
\(255\) −9.22035e7 −0.348222
\(256\) 3.51561e8 1.30967
\(257\) −2.94635e8 −1.08273 −0.541363 0.840789i \(-0.682091\pi\)
−0.541363 + 0.840789i \(0.682091\pi\)
\(258\) −2.49192e8 −0.903369
\(259\) 1.44880e8 0.518156
\(260\) −1.25771e8 −0.443786
\(261\) −1.80443e8 −0.628199
\(262\) −3.23486e7 −0.111122
\(263\) 3.13955e8 1.06420 0.532098 0.846683i \(-0.321404\pi\)
0.532098 + 0.846683i \(0.321404\pi\)
\(264\) −2.17344e8 −0.726999
\(265\) −1.33318e8 −0.440076
\(266\) −1.75449e8 −0.571563
\(267\) 4.01668e8 1.29145
\(268\) 1.95007e8 0.618841
\(269\) −2.94745e8 −0.923238 −0.461619 0.887078i \(-0.652731\pi\)
−0.461619 + 0.887078i \(0.652731\pi\)
\(270\) 1.37426e8 0.424909
\(271\) −8.47290e7 −0.258607 −0.129303 0.991605i \(-0.541274\pi\)
−0.129303 + 0.991605i \(0.541274\pi\)
\(272\) −2.69077e8 −0.810748
\(273\) −2.19572e8 −0.653142
\(274\) −1.03981e8 −0.305371
\(275\) −1.01151e8 −0.293294
\(276\) −3.67364e8 −1.05175
\(277\) 1.96909e8 0.556655 0.278327 0.960486i \(-0.410220\pi\)
0.278327 + 0.960486i \(0.410220\pi\)
\(278\) 1.34896e8 0.376568
\(279\) −1.89491e7 −0.0522364
\(280\) 2.62680e7 0.0715112
\(281\) 3.32330e8 0.893505 0.446753 0.894658i \(-0.352580\pi\)
0.446753 + 0.894658i \(0.352580\pi\)
\(282\) 1.93054e8 0.512632
\(283\) 4.40437e8 1.15513 0.577566 0.816344i \(-0.304003\pi\)
0.577566 + 0.816344i \(0.304003\pi\)
\(284\) −1.91244e8 −0.495421
\(285\) −2.39443e8 −0.612696
\(286\) 1.10661e9 2.79714
\(287\) −6.57937e7 −0.164285
\(288\) −1.74702e8 −0.430948
\(289\) −2.29154e8 −0.558451
\(290\) −4.04490e8 −0.973900
\(291\) 1.30793e8 0.311143
\(292\) 4.30123e8 1.01100
\(293\) −3.05058e8 −0.708510 −0.354255 0.935149i \(-0.615265\pi\)
−0.354255 + 0.935149i \(0.615265\pi\)
\(294\) −9.43421e7 −0.216516
\(295\) 5.64797e7 0.128090
\(296\) 2.58785e8 0.579986
\(297\) −4.86370e8 −1.07726
\(298\) 1.95058e8 0.426979
\(299\) −9.09203e8 −1.96703
\(300\) −7.37498e7 −0.157702
\(301\) −1.06589e8 −0.225283
\(302\) −9.39366e8 −1.96250
\(303\) 1.05277e9 2.17412
\(304\) −6.98766e8 −1.42651
\(305\) −1.03957e8 −0.209800
\(306\) 1.60725e8 0.320670
\(307\) −2.41616e8 −0.476587 −0.238293 0.971193i \(-0.576588\pi\)
−0.238293 + 0.971193i \(0.576588\pi\)
\(308\) 1.91252e8 0.372975
\(309\) 2.44054e8 0.470578
\(310\) −4.24772e7 −0.0809823
\(311\) −6.71768e7 −0.126636 −0.0633181 0.997993i \(-0.520168\pi\)
−0.0633181 + 0.997993i \(0.520168\pi\)
\(312\) −3.92198e8 −0.731079
\(313\) −6.75084e8 −1.24438 −0.622190 0.782867i \(-0.713757\pi\)
−0.622190 + 0.782867i \(0.713757\pi\)
\(314\) 1.07788e9 1.96479
\(315\) −3.49854e7 −0.0630666
\(316\) −2.34944e8 −0.418852
\(317\) −4.65149e8 −0.820135 −0.410067 0.912055i \(-0.634495\pi\)
−0.410067 + 0.912055i \(0.634495\pi\)
\(318\) 8.55255e8 1.49142
\(319\) 1.43154e9 2.46909
\(320\) −7.17797e7 −0.122455
\(321\) −4.17235e8 −0.704066
\(322\) −3.90652e8 −0.652070
\(323\) 4.70517e8 0.776903
\(324\) −5.08325e8 −0.830298
\(325\) −1.82527e8 −0.294940
\(326\) −5.13103e8 −0.820244
\(327\) 1.09434e9 1.73075
\(328\) −1.17520e8 −0.183889
\(329\) 8.25762e7 0.127841
\(330\) 6.48897e8 0.993978
\(331\) 2.69779e8 0.408893 0.204447 0.978878i \(-0.434460\pi\)
0.204447 + 0.978878i \(0.434460\pi\)
\(332\) −5.49860e8 −0.824648
\(333\) −3.44665e8 −0.511497
\(334\) 3.76002e7 0.0552176
\(335\) 2.83006e8 0.411282
\(336\) −3.75740e8 −0.540381
\(337\) −6.29093e8 −0.895385 −0.447693 0.894187i \(-0.647754\pi\)
−0.447693 + 0.894187i \(0.647754\pi\)
\(338\) 1.07867e9 1.51943
\(339\) −1.41352e9 −1.97062
\(340\) 1.44922e8 0.199967
\(341\) 1.50333e8 0.205312
\(342\) 4.17386e8 0.564218
\(343\) −4.03536e7 −0.0539949
\(344\) −1.90388e8 −0.252165
\(345\) −5.33140e8 −0.698996
\(346\) 1.17573e9 1.52595
\(347\) −7.61715e8 −0.978676 −0.489338 0.872094i \(-0.662762\pi\)
−0.489338 + 0.872094i \(0.662762\pi\)
\(348\) 1.04375e9 1.32761
\(349\) −3.31639e8 −0.417616 −0.208808 0.977957i \(-0.566958\pi\)
−0.208808 + 0.977957i \(0.566958\pi\)
\(350\) −7.84251e7 −0.0977725
\(351\) −8.77657e8 −1.08330
\(352\) 1.38601e9 1.69381
\(353\) 7.57419e8 0.916484 0.458242 0.888828i \(-0.348479\pi\)
0.458242 + 0.888828i \(0.348479\pi\)
\(354\) −3.62326e8 −0.434099
\(355\) −2.77545e8 −0.329257
\(356\) −6.31329e8 −0.741619
\(357\) 2.53006e8 0.294302
\(358\) 1.46197e9 1.68402
\(359\) 1.46796e9 1.67449 0.837246 0.546827i \(-0.184164\pi\)
0.837246 + 0.546827i \(0.184164\pi\)
\(360\) −6.24907e7 −0.0705922
\(361\) 3.28013e8 0.366958
\(362\) −1.57182e9 −1.74150
\(363\) −1.22865e9 −1.34820
\(364\) 3.45116e8 0.375068
\(365\) 6.24221e8 0.671914
\(366\) 6.66903e8 0.711015
\(367\) 1.64615e9 1.73835 0.869177 0.494502i \(-0.164649\pi\)
0.869177 + 0.494502i \(0.164649\pi\)
\(368\) −1.55586e9 −1.62744
\(369\) 1.56521e8 0.162174
\(370\) −7.72620e8 −0.792976
\(371\) 3.65824e8 0.371932
\(372\) 1.09609e8 0.110394
\(373\) −1.16387e9 −1.16124 −0.580622 0.814173i \(-0.697191\pi\)
−0.580622 + 0.814173i \(0.697191\pi\)
\(374\) −1.27512e9 −1.26037
\(375\) −1.07030e8 −0.104809
\(376\) 1.47497e8 0.143096
\(377\) 2.58323e9 2.48295
\(378\) −3.77098e8 −0.359114
\(379\) 4.07762e8 0.384742 0.192371 0.981322i \(-0.438382\pi\)
0.192371 + 0.981322i \(0.438382\pi\)
\(380\) 3.76348e8 0.351841
\(381\) 3.70273e8 0.342993
\(382\) −3.23519e8 −0.296946
\(383\) 7.10345e8 0.646061 0.323031 0.946389i \(-0.395298\pi\)
0.323031 + 0.946389i \(0.395298\pi\)
\(384\) −1.04129e9 −0.938454
\(385\) 2.77557e8 0.247879
\(386\) −2.19196e9 −1.93989
\(387\) 2.53571e8 0.222388
\(388\) −2.05576e8 −0.178674
\(389\) 1.95091e8 0.168040 0.0840202 0.996464i \(-0.473224\pi\)
0.0840202 + 0.996464i \(0.473224\pi\)
\(390\) 1.17094e9 0.999556
\(391\) 1.04765e9 0.886333
\(392\) −7.20794e7 −0.0604380
\(393\) 1.21141e8 0.100674
\(394\) −8.34252e8 −0.687164
\(395\) −3.40966e8 −0.278369
\(396\) −4.54983e8 −0.368181
\(397\) 1.58231e9 1.26919 0.634593 0.772846i \(-0.281168\pi\)
0.634593 + 0.772846i \(0.281168\pi\)
\(398\) −1.00115e9 −0.795991
\(399\) 6.57031e8 0.517822
\(400\) −3.12346e8 −0.244021
\(401\) −2.13647e9 −1.65460 −0.827298 0.561763i \(-0.810123\pi\)
−0.827298 + 0.561763i \(0.810123\pi\)
\(402\) −1.81553e9 −1.39384
\(403\) 2.71276e8 0.206464
\(404\) −1.65471e9 −1.24849
\(405\) −7.37712e8 −0.551816
\(406\) 1.10992e9 0.823096
\(407\) 2.73441e9 2.01040
\(408\) 4.51919e8 0.329420
\(409\) −4.05635e8 −0.293159 −0.146580 0.989199i \(-0.546826\pi\)
−0.146580 + 0.989199i \(0.546826\pi\)
\(410\) 3.50866e8 0.251419
\(411\) 3.89395e8 0.276658
\(412\) −3.83596e8 −0.270230
\(413\) −1.54980e8 −0.108256
\(414\) 9.29347e8 0.643690
\(415\) −7.97990e8 −0.548061
\(416\) 2.50105e9 1.70332
\(417\) −5.05168e8 −0.341161
\(418\) −3.31134e9 −2.21762
\(419\) 1.07475e9 0.713771 0.356886 0.934148i \(-0.383839\pi\)
0.356886 + 0.934148i \(0.383839\pi\)
\(420\) 2.02370e8 0.133282
\(421\) −8.32900e8 −0.544009 −0.272004 0.962296i \(-0.587686\pi\)
−0.272004 + 0.962296i \(0.587686\pi\)
\(422\) −1.99306e9 −1.29100
\(423\) −1.96446e8 −0.126198
\(424\) 6.53434e8 0.416314
\(425\) 2.10320e8 0.132898
\(426\) 1.78050e9 1.11586
\(427\) 2.85259e8 0.177313
\(428\) 6.55796e8 0.404311
\(429\) −4.14411e9 −2.53414
\(430\) 5.68418e8 0.344769
\(431\) 8.26292e6 0.00497122 0.00248561 0.999997i \(-0.499209\pi\)
0.00248561 + 0.999997i \(0.499209\pi\)
\(432\) −1.50188e9 −0.896277
\(433\) −3.10619e9 −1.83874 −0.919369 0.393396i \(-0.871300\pi\)
−0.919369 + 0.393396i \(0.871300\pi\)
\(434\) 1.16558e8 0.0684426
\(435\) 1.51476e9 0.882330
\(436\) −1.72004e9 −0.993885
\(437\) 2.72063e9 1.55950
\(438\) −4.00448e9 −2.27712
\(439\) −1.28295e9 −0.723742 −0.361871 0.932228i \(-0.617862\pi\)
−0.361871 + 0.932228i \(0.617862\pi\)
\(440\) 4.95771e8 0.277458
\(441\) 9.59998e7 0.0533010
\(442\) −2.30095e9 −1.26745
\(443\) −1.73263e9 −0.946878 −0.473439 0.880827i \(-0.656988\pi\)
−0.473439 + 0.880827i \(0.656988\pi\)
\(444\) 1.99368e9 1.08098
\(445\) −9.16223e8 −0.492880
\(446\) −2.16175e9 −1.15381
\(447\) −7.30464e8 −0.386832
\(448\) 1.96964e8 0.103493
\(449\) 1.86903e9 0.974439 0.487219 0.873280i \(-0.338011\pi\)
0.487219 + 0.873280i \(0.338011\pi\)
\(450\) 1.86570e8 0.0965160
\(451\) −1.24176e9 −0.637413
\(452\) 2.22172e9 1.13163
\(453\) 3.51780e9 1.77798
\(454\) 4.71816e9 2.36634
\(455\) 5.00853e8 0.249270
\(456\) 1.17359e9 0.579613
\(457\) 1.76868e9 0.866849 0.433425 0.901190i \(-0.357305\pi\)
0.433425 + 0.901190i \(0.357305\pi\)
\(458\) 4.53679e9 2.20658
\(459\) 1.01130e9 0.488129
\(460\) 8.37972e8 0.401400
\(461\) −2.55825e8 −0.121616 −0.0608078 0.998149i \(-0.519368\pi\)
−0.0608078 + 0.998149i \(0.519368\pi\)
\(462\) −1.78057e9 −0.840065
\(463\) −4.19121e9 −1.96249 −0.981243 0.192777i \(-0.938250\pi\)
−0.981243 + 0.192777i \(0.938250\pi\)
\(464\) 4.42052e9 2.05428
\(465\) 1.59071e8 0.0733680
\(466\) 2.63998e9 1.20851
\(467\) −2.94239e9 −1.33688 −0.668438 0.743768i \(-0.733037\pi\)
−0.668438 + 0.743768i \(0.733037\pi\)
\(468\) −8.21018e8 −0.370247
\(469\) −7.76569e8 −0.347596
\(470\) −4.40364e8 −0.195645
\(471\) −4.03651e9 −1.78005
\(472\) −2.76825e8 −0.121174
\(473\) −2.01171e9 −0.874079
\(474\) 2.18735e9 0.943396
\(475\) 5.46179e8 0.233834
\(476\) −3.97667e8 −0.169003
\(477\) −8.70283e8 −0.367152
\(478\) 5.36292e9 2.24597
\(479\) 4.57933e9 1.90383 0.951914 0.306366i \(-0.0991132\pi\)
0.951914 + 0.306366i \(0.0991132\pi\)
\(480\) 1.46657e9 0.605284
\(481\) 4.93425e9 2.02169
\(482\) 3.21535e9 1.30787
\(483\) 1.46294e9 0.590760
\(484\) 1.93115e9 0.774206
\(485\) −2.98345e8 −0.118747
\(486\) 2.32813e9 0.919984
\(487\) −1.40131e9 −0.549771 −0.274886 0.961477i \(-0.588640\pi\)
−0.274886 + 0.961477i \(0.588640\pi\)
\(488\) 5.09528e8 0.198472
\(489\) 1.92150e9 0.743121
\(490\) 2.15198e8 0.0826329
\(491\) −4.79712e9 −1.82892 −0.914462 0.404672i \(-0.867386\pi\)
−0.914462 + 0.404672i \(0.867386\pi\)
\(492\) −9.05381e8 −0.342731
\(493\) −2.97658e9 −1.11880
\(494\) −5.97533e9 −2.23007
\(495\) −6.60299e8 −0.244693
\(496\) 4.64218e8 0.170819
\(497\) 7.61585e8 0.278273
\(498\) 5.11923e9 1.85739
\(499\) −2.01049e9 −0.724351 −0.362176 0.932110i \(-0.617966\pi\)
−0.362176 + 0.932110i \(0.617966\pi\)
\(500\) 1.68227e8 0.0601866
\(501\) −1.40807e8 −0.0500258
\(502\) −1.90050e9 −0.670509
\(503\) 1.68618e9 0.590766 0.295383 0.955379i \(-0.404553\pi\)
0.295383 + 0.955379i \(0.404553\pi\)
\(504\) 1.71474e8 0.0596613
\(505\) −2.40141e9 −0.829749
\(506\) −7.37300e9 −2.52998
\(507\) −4.03947e9 −1.37656
\(508\) −5.81983e8 −0.196964
\(509\) 1.63477e9 0.549470 0.274735 0.961520i \(-0.411410\pi\)
0.274735 + 0.961520i \(0.411410\pi\)
\(510\) −1.34924e9 −0.450394
\(511\) −1.71286e9 −0.567871
\(512\) 2.71225e9 0.893068
\(513\) 2.62624e9 0.858862
\(514\) −4.31147e9 −1.40041
\(515\) −5.56698e8 −0.179595
\(516\) −1.46676e9 −0.469985
\(517\) 1.55851e9 0.496012
\(518\) 2.12007e9 0.670187
\(519\) −4.40294e9 −1.38247
\(520\) 8.94621e8 0.279015
\(521\) 3.28595e9 1.01796 0.508978 0.860779i \(-0.330023\pi\)
0.508978 + 0.860779i \(0.330023\pi\)
\(522\) −2.64046e9 −0.812518
\(523\) 1.29734e9 0.396549 0.198274 0.980147i \(-0.436466\pi\)
0.198274 + 0.980147i \(0.436466\pi\)
\(524\) −1.90406e8 −0.0578123
\(525\) 2.93691e8 0.0885795
\(526\) 4.59418e9 1.37644
\(527\) −3.12583e8 −0.0930313
\(528\) −7.09155e9 −2.09663
\(529\) 2.65291e9 0.779161
\(530\) −1.95087e9 −0.569198
\(531\) 3.68693e8 0.106865
\(532\) −1.03270e9 −0.297360
\(533\) −2.24076e9 −0.640990
\(534\) 5.87771e9 1.67038
\(535\) 9.51731e8 0.268705
\(536\) −1.38710e9 −0.389074
\(537\) −5.47487e9 −1.52568
\(538\) −4.31308e9 −1.19412
\(539\) −7.61617e8 −0.209496
\(540\) 8.08897e8 0.221063
\(541\) 1.23726e9 0.335948 0.167974 0.985791i \(-0.446278\pi\)
0.167974 + 0.985791i \(0.446278\pi\)
\(542\) −1.23986e9 −0.334484
\(543\) 5.88625e9 1.57775
\(544\) −2.88189e9 −0.767505
\(545\) −2.49623e9 −0.660536
\(546\) −3.21305e9 −0.844779
\(547\) −4.27288e9 −1.11626 −0.558130 0.829754i \(-0.688481\pi\)
−0.558130 + 0.829754i \(0.688481\pi\)
\(548\) −6.12037e8 −0.158871
\(549\) −6.78621e8 −0.175035
\(550\) −1.48016e9 −0.379350
\(551\) −7.72986e9 −1.96853
\(552\) 2.61309e9 0.661253
\(553\) 9.35610e8 0.235265
\(554\) 2.88141e9 0.719982
\(555\) 2.89336e9 0.718417
\(556\) 7.94006e8 0.195912
\(557\) −2.41921e9 −0.593171 −0.296586 0.955006i \(-0.595848\pi\)
−0.296586 + 0.955006i \(0.595848\pi\)
\(558\) −2.77286e8 −0.0675630
\(559\) −3.63013e9 −0.878985
\(560\) 8.57079e8 0.206235
\(561\) 4.77513e9 1.14187
\(562\) 4.86306e9 1.15567
\(563\) −3.03839e9 −0.717570 −0.358785 0.933420i \(-0.616809\pi\)
−0.358785 + 0.933420i \(0.616809\pi\)
\(564\) 1.13632e9 0.266701
\(565\) 3.22430e9 0.752084
\(566\) 6.44503e9 1.49406
\(567\) 2.02428e9 0.466370
\(568\) 1.36034e9 0.311478
\(569\) 2.85539e9 0.649788 0.324894 0.945750i \(-0.394671\pi\)
0.324894 + 0.945750i \(0.394671\pi\)
\(570\) −3.50383e9 −0.792466
\(571\) 1.87867e9 0.422304 0.211152 0.977453i \(-0.432279\pi\)
0.211152 + 0.977453i \(0.432279\pi\)
\(572\) 6.51356e9 1.45523
\(573\) 1.21153e9 0.269026
\(574\) −9.62776e8 −0.212488
\(575\) 1.21612e9 0.266770
\(576\) −4.68569e8 −0.102163
\(577\) 2.19182e9 0.474996 0.237498 0.971388i \(-0.423673\pi\)
0.237498 + 0.971388i \(0.423673\pi\)
\(578\) −3.35327e9 −0.722306
\(579\) 8.20858e9 1.75749
\(580\) −2.38085e9 −0.506679
\(581\) 2.18969e9 0.463196
\(582\) 1.91393e9 0.402435
\(583\) 6.90441e9 1.44307
\(584\) −3.05951e9 −0.635633
\(585\) −1.19151e9 −0.246067
\(586\) −4.46399e9 −0.916392
\(587\) −4.70415e9 −0.959949 −0.479974 0.877283i \(-0.659354\pi\)
−0.479974 + 0.877283i \(0.659354\pi\)
\(588\) −5.55302e8 −0.112644
\(589\) −8.11747e8 −0.163688
\(590\) 8.26482e8 0.165673
\(591\) 3.12416e9 0.622554
\(592\) 8.44368e9 1.67265
\(593\) 3.66996e9 0.722719 0.361359 0.932427i \(-0.382313\pi\)
0.361359 + 0.932427i \(0.382313\pi\)
\(594\) −7.11718e9 −1.39333
\(595\) −5.77118e8 −0.112320
\(596\) 1.14812e9 0.222139
\(597\) 3.74917e9 0.721149
\(598\) −1.33046e10 −2.54418
\(599\) −5.46935e9 −1.03978 −0.519890 0.854233i \(-0.674027\pi\)
−0.519890 + 0.854233i \(0.674027\pi\)
\(600\) 5.24590e8 0.0991494
\(601\) 2.74417e9 0.515645 0.257822 0.966192i \(-0.416995\pi\)
0.257822 + 0.966192i \(0.416995\pi\)
\(602\) −1.55974e9 −0.291383
\(603\) 1.84743e9 0.343129
\(604\) −5.52915e9 −1.02101
\(605\) 2.80260e9 0.514537
\(606\) 1.54054e10 2.81203
\(607\) −2.26388e9 −0.410859 −0.205430 0.978672i \(-0.565859\pi\)
−0.205430 + 0.978672i \(0.565859\pi\)
\(608\) −7.48397e9 −1.35042
\(609\) −4.15650e9 −0.745705
\(610\) −1.52123e9 −0.271357
\(611\) 2.81233e9 0.498795
\(612\) 9.46035e8 0.166831
\(613\) −9.92731e9 −1.74068 −0.870342 0.492448i \(-0.836102\pi\)
−0.870342 + 0.492448i \(0.836102\pi\)
\(614\) −3.53563e9 −0.616422
\(615\) −1.31394e9 −0.227779
\(616\) −1.36040e9 −0.234495
\(617\) 2.27156e9 0.389338 0.194669 0.980869i \(-0.437637\pi\)
0.194669 + 0.980869i \(0.437637\pi\)
\(618\) 3.57131e9 0.608650
\(619\) 6.90552e8 0.117025 0.0585126 0.998287i \(-0.481364\pi\)
0.0585126 + 0.998287i \(0.481364\pi\)
\(620\) −2.50023e8 −0.0421317
\(621\) 5.84755e9 0.979836
\(622\) −9.83015e8 −0.163792
\(623\) 2.51412e9 0.416560
\(624\) −1.27967e10 −2.10840
\(625\) 2.44141e8 0.0400000
\(626\) −9.87867e9 −1.60949
\(627\) 1.24005e10 2.00911
\(628\) 6.34445e9 1.02220
\(629\) −5.68560e9 −0.910959
\(630\) −5.11949e8 −0.0815709
\(631\) −1.00992e10 −1.60023 −0.800116 0.599846i \(-0.795229\pi\)
−0.800116 + 0.599846i \(0.795229\pi\)
\(632\) 1.67118e9 0.263338
\(633\) 7.46375e9 1.16962
\(634\) −6.80665e9 −1.06077
\(635\) −8.44609e8 −0.130902
\(636\) 5.03407e9 0.775925
\(637\) −1.37434e9 −0.210672
\(638\) 2.09482e10 3.19355
\(639\) −1.81178e9 −0.274697
\(640\) 2.37523e9 0.358159
\(641\) 8.50418e8 0.127535 0.0637675 0.997965i \(-0.479688\pi\)
0.0637675 + 0.997965i \(0.479688\pi\)
\(642\) −6.10550e9 −0.910644
\(643\) 1.56624e9 0.232339 0.116169 0.993229i \(-0.462938\pi\)
0.116169 + 0.993229i \(0.462938\pi\)
\(644\) −2.29940e9 −0.339245
\(645\) −2.12865e9 −0.312352
\(646\) 6.88520e9 1.00485
\(647\) 1.06053e10 1.53942 0.769712 0.638391i \(-0.220400\pi\)
0.769712 + 0.638391i \(0.220400\pi\)
\(648\) 3.61576e9 0.522020
\(649\) −2.92503e9 −0.420024
\(650\) −2.67096e9 −0.381478
\(651\) −4.36492e8 −0.0620073
\(652\) −3.02015e9 −0.426739
\(653\) −1.83644e9 −0.258096 −0.129048 0.991638i \(-0.541192\pi\)
−0.129048 + 0.991638i \(0.541192\pi\)
\(654\) 1.60137e10 2.23856
\(655\) −2.76328e8 −0.0384221
\(656\) −3.83448e9 −0.530327
\(657\) 4.07484e9 0.560573
\(658\) 1.20836e9 0.165350
\(659\) −5.03583e9 −0.685444 −0.342722 0.939437i \(-0.611349\pi\)
−0.342722 + 0.939437i \(0.611349\pi\)
\(660\) 3.81944e9 0.517125
\(661\) −1.27453e10 −1.71651 −0.858254 0.513225i \(-0.828451\pi\)
−0.858254 + 0.513225i \(0.828451\pi\)
\(662\) 3.94774e9 0.528866
\(663\) 8.61674e9 1.14827
\(664\) 3.91121e9 0.518469
\(665\) −1.49872e9 −0.197626
\(666\) −5.04357e9 −0.661574
\(667\) −1.72112e10 −2.24580
\(668\) 2.21316e8 0.0287274
\(669\) 8.09544e9 1.04532
\(670\) 4.14130e9 0.531955
\(671\) 5.38386e9 0.687962
\(672\) −4.02427e9 −0.511558
\(673\) 1.02193e10 1.29231 0.646155 0.763206i \(-0.276376\pi\)
0.646155 + 0.763206i \(0.276376\pi\)
\(674\) −9.20567e9 −1.15810
\(675\) 1.17392e9 0.146918
\(676\) 6.34910e9 0.790494
\(677\) 1.04119e10 1.28965 0.644823 0.764332i \(-0.276931\pi\)
0.644823 + 0.764332i \(0.276931\pi\)
\(678\) −2.06844e10 −2.54882
\(679\) 8.18659e8 0.100360
\(680\) −1.03085e9 −0.125722
\(681\) −1.76688e10 −2.14384
\(682\) 2.19986e9 0.265552
\(683\) 6.56705e9 0.788675 0.394338 0.918966i \(-0.370974\pi\)
0.394338 + 0.918966i \(0.370974\pi\)
\(684\) 2.45675e9 0.293539
\(685\) −8.88226e8 −0.105586
\(686\) −5.90504e8 −0.0698375
\(687\) −1.69897e10 −1.99911
\(688\) −6.21203e9 −0.727233
\(689\) 1.24590e10 1.45117
\(690\) −7.80158e9 −0.904088
\(691\) 4.44242e9 0.512208 0.256104 0.966649i \(-0.417561\pi\)
0.256104 + 0.966649i \(0.417561\pi\)
\(692\) 6.92039e9 0.793888
\(693\) 1.81186e9 0.206804
\(694\) −1.11464e10 −1.26583
\(695\) 1.15231e9 0.130203
\(696\) −7.42431e9 −0.834688
\(697\) 2.58197e9 0.288826
\(698\) −4.85296e9 −0.540148
\(699\) −9.88638e9 −1.09488
\(700\) −4.61614e8 −0.0508669
\(701\) −7.92343e9 −0.868761 −0.434380 0.900729i \(-0.643033\pi\)
−0.434380 + 0.900729i \(0.643033\pi\)
\(702\) −1.28430e10 −1.40115
\(703\) −1.47649e10 −1.60283
\(704\) 3.71741e9 0.401546
\(705\) 1.64910e9 0.177250
\(706\) 1.10835e10 1.18539
\(707\) 6.58948e9 0.701266
\(708\) −2.13267e9 −0.225843
\(709\) 9.27216e9 0.977055 0.488528 0.872548i \(-0.337534\pi\)
0.488528 + 0.872548i \(0.337534\pi\)
\(710\) −4.06139e9 −0.425864
\(711\) −2.22578e9 −0.232241
\(712\) 4.49070e9 0.466266
\(713\) −1.80743e9 −0.186744
\(714\) 3.70230e9 0.380652
\(715\) 9.45288e9 0.967148
\(716\) 8.60522e9 0.876126
\(717\) −2.00834e10 −2.03479
\(718\) 2.14810e10 2.16580
\(719\) −5.48244e9 −0.550076 −0.275038 0.961433i \(-0.588690\pi\)
−0.275038 + 0.961433i \(0.588690\pi\)
\(720\) −2.03896e9 −0.203585
\(721\) 1.52758e9 0.151786
\(722\) 4.79990e9 0.474626
\(723\) −1.20410e10 −1.18489
\(724\) −9.25181e9 −0.906029
\(725\) −3.45523e9 −0.336739
\(726\) −1.79791e10 −1.74377
\(727\) −9.22691e9 −0.890606 −0.445303 0.895380i \(-0.646904\pi\)
−0.445303 + 0.895380i \(0.646904\pi\)
\(728\) −2.45484e9 −0.235811
\(729\) 4.18848e9 0.400415
\(730\) 9.13438e9 0.869059
\(731\) 4.18290e9 0.396065
\(732\) 3.92542e9 0.369911
\(733\) 4.96865e9 0.465988 0.232994 0.972478i \(-0.425148\pi\)
0.232994 + 0.972478i \(0.425148\pi\)
\(734\) 2.40885e10 2.24840
\(735\) −8.05888e8 −0.0748633
\(736\) −1.66637e10 −1.54063
\(737\) −1.46566e10 −1.34865
\(738\) 2.29041e9 0.209757
\(739\) −1.96084e9 −0.178726 −0.0893628 0.995999i \(-0.528483\pi\)
−0.0893628 + 0.995999i \(0.528483\pi\)
\(740\) −4.54768e9 −0.412552
\(741\) 2.23768e10 2.02038
\(742\) 5.35320e9 0.481060
\(743\) 9.56947e9 0.855908 0.427954 0.903801i \(-0.359235\pi\)
0.427954 + 0.903801i \(0.359235\pi\)
\(744\) −7.79660e8 −0.0694064
\(745\) 1.66622e9 0.147634
\(746\) −1.70312e10 −1.50196
\(747\) −5.20918e9 −0.457244
\(748\) −7.50539e9 −0.655719
\(749\) −2.61155e9 −0.227097
\(750\) −1.56620e9 −0.135560
\(751\) 8.11719e9 0.699304 0.349652 0.936880i \(-0.386300\pi\)
0.349652 + 0.936880i \(0.386300\pi\)
\(752\) 4.81257e9 0.412681
\(753\) 7.11710e9 0.607464
\(754\) 3.78010e10 3.21147
\(755\) −8.02424e9 −0.678562
\(756\) −2.21961e9 −0.186832
\(757\) −9.11117e9 −0.763376 −0.381688 0.924291i \(-0.624657\pi\)
−0.381688 + 0.924291i \(0.624657\pi\)
\(758\) 5.96689e9 0.497629
\(759\) 2.76109e10 2.29210
\(760\) −2.67700e9 −0.221208
\(761\) 1.71359e10 1.40948 0.704742 0.709464i \(-0.251063\pi\)
0.704742 + 0.709464i \(0.251063\pi\)
\(762\) 5.41830e9 0.443630
\(763\) 6.84965e9 0.558255
\(764\) −1.90425e9 −0.154489
\(765\) 1.37294e9 0.110876
\(766\) 1.03947e10 0.835621
\(767\) −5.27823e9 −0.422381
\(768\) −1.92654e10 −1.53466
\(769\) −1.82316e10 −1.44572 −0.722858 0.690997i \(-0.757172\pi\)
−0.722858 + 0.690997i \(0.757172\pi\)
\(770\) 4.06156e9 0.320609
\(771\) 1.61458e10 1.26873
\(772\) −1.29020e10 −1.00924
\(773\) −1.45965e10 −1.13663 −0.568316 0.822810i \(-0.692405\pi\)
−0.568316 + 0.822810i \(0.692405\pi\)
\(774\) 3.71056e9 0.287638
\(775\) −3.62849e8 −0.0280007
\(776\) 1.46228e9 0.112335
\(777\) −7.93937e9 −0.607173
\(778\) 2.85481e9 0.217345
\(779\) 6.70510e9 0.508188
\(780\) 6.89219e9 0.520027
\(781\) 1.43738e10 1.07968
\(782\) 1.53305e10 1.14639
\(783\) −1.66141e10 −1.23683
\(784\) −2.35182e9 −0.174300
\(785\) 9.20745e9 0.679353
\(786\) 1.77269e9 0.130213
\(787\) 1.42358e10 1.04104 0.520522 0.853848i \(-0.325737\pi\)
0.520522 + 0.853848i \(0.325737\pi\)
\(788\) −4.91045e9 −0.357503
\(789\) −1.72046e10 −1.24702
\(790\) −4.98944e9 −0.360045
\(791\) −8.84748e9 −0.635627
\(792\) 3.23633e9 0.231481
\(793\) 9.71519e9 0.691823
\(794\) 2.31544e10 1.64158
\(795\) 7.30575e9 0.515680
\(796\) −5.89281e9 −0.414121
\(797\) 1.03325e10 0.722935 0.361467 0.932385i \(-0.382276\pi\)
0.361467 + 0.932385i \(0.382276\pi\)
\(798\) 9.61450e9 0.669756
\(799\) −3.24057e9 −0.224754
\(800\) −3.34531e9 −0.231005
\(801\) −5.98099e9 −0.411206
\(802\) −3.12635e10 −2.14007
\(803\) −3.23278e10 −2.20329
\(804\) −1.06863e10 −0.725155
\(805\) −3.33702e9 −0.225462
\(806\) 3.96965e9 0.267042
\(807\) 1.61519e10 1.08185
\(808\) 1.17701e10 0.784947
\(809\) −1.58064e10 −1.04957 −0.524787 0.851233i \(-0.675855\pi\)
−0.524787 + 0.851233i \(0.675855\pi\)
\(810\) −1.07951e10 −0.713724
\(811\) 2.87547e9 0.189294 0.0946469 0.995511i \(-0.469828\pi\)
0.0946469 + 0.995511i \(0.469828\pi\)
\(812\) 6.53304e9 0.428222
\(813\) 4.64311e9 0.303034
\(814\) 4.00133e10 2.60027
\(815\) −4.38303e9 −0.283611
\(816\) 1.47453e10 0.950032
\(817\) 1.08626e10 0.696875
\(818\) −5.93575e9 −0.379175
\(819\) 3.26951e9 0.207964
\(820\) 2.06521e9 0.130803
\(821\) −1.42014e10 −0.895633 −0.447817 0.894125i \(-0.647798\pi\)
−0.447817 + 0.894125i \(0.647798\pi\)
\(822\) 5.69811e9 0.357832
\(823\) −2.79354e10 −1.74685 −0.873424 0.486961i \(-0.838105\pi\)
−0.873424 + 0.486961i \(0.838105\pi\)
\(824\) 2.72856e9 0.169898
\(825\) 5.54300e9 0.343681
\(826\) −2.26787e9 −0.140019
\(827\) −1.48560e8 −0.00913341 −0.00456670 0.999990i \(-0.501454\pi\)
−0.00456670 + 0.999990i \(0.501454\pi\)
\(828\) 5.47018e9 0.334885
\(829\) 3.98861e9 0.243154 0.121577 0.992582i \(-0.461205\pi\)
0.121577 + 0.992582i \(0.461205\pi\)
\(830\) −1.16772e10 −0.708868
\(831\) −1.07905e10 −0.652286
\(832\) 6.70807e9 0.403800
\(833\) 1.58361e9 0.0949273
\(834\) −7.39225e9 −0.441261
\(835\) 3.21188e8 0.0190922
\(836\) −1.94907e10 −1.15373
\(837\) −1.74471e9 −0.102845
\(838\) 1.57271e10 0.923198
\(839\) 2.43693e9 0.142455 0.0712273 0.997460i \(-0.477308\pi\)
0.0712273 + 0.997460i \(0.477308\pi\)
\(840\) −1.43947e9 −0.0837966
\(841\) 3.16506e10 1.83483
\(842\) −1.21880e10 −0.703625
\(843\) −1.82115e10 −1.04701
\(844\) −1.17313e10 −0.671655
\(845\) 9.21419e9 0.525362
\(846\) −2.87464e9 −0.163225
\(847\) −7.69033e9 −0.434863
\(848\) 2.13204e10 1.20063
\(849\) −2.41357e10 −1.35358
\(850\) 3.07767e9 0.171892
\(851\) −3.28754e10 −1.82859
\(852\) 1.04801e10 0.580533
\(853\) 1.54310e9 0.0851282 0.0425641 0.999094i \(-0.486447\pi\)
0.0425641 + 0.999094i \(0.486447\pi\)
\(854\) 4.17427e9 0.229339
\(855\) 3.56539e9 0.195086
\(856\) −4.66474e9 −0.254196
\(857\) −1.29972e10 −0.705369 −0.352684 0.935742i \(-0.614731\pi\)
−0.352684 + 0.935742i \(0.614731\pi\)
\(858\) −6.06417e10 −3.27768
\(859\) −1.98316e10 −1.06754 −0.533768 0.845631i \(-0.679224\pi\)
−0.533768 + 0.845631i \(0.679224\pi\)
\(860\) 3.34573e9 0.179369
\(861\) 3.60546e9 0.192509
\(862\) 1.20913e8 0.00642982
\(863\) 4.94264e9 0.261771 0.130886 0.991397i \(-0.458218\pi\)
0.130886 + 0.991397i \(0.458218\pi\)
\(864\) −1.60855e10 −0.848472
\(865\) 1.00433e10 0.527618
\(866\) −4.54536e10 −2.37824
\(867\) 1.25575e10 0.654391
\(868\) 6.86063e8 0.0356078
\(869\) 1.76583e10 0.912809
\(870\) 2.21658e10 1.14121
\(871\) −2.64480e10 −1.35621
\(872\) 1.22348e10 0.624870
\(873\) −1.94756e9 −0.0990697
\(874\) 3.98117e10 2.01707
\(875\) −6.69922e8 −0.0338062
\(876\) −2.35705e10 −1.18469
\(877\) −7.37011e9 −0.368957 −0.184478 0.982837i \(-0.559060\pi\)
−0.184478 + 0.982837i \(0.559060\pi\)
\(878\) −1.87737e10 −0.936094
\(879\) 1.67170e10 0.830229
\(880\) 1.61761e10 0.800176
\(881\) 9.74385e9 0.480081 0.240041 0.970763i \(-0.422839\pi\)
0.240041 + 0.970763i \(0.422839\pi\)
\(882\) 1.40479e9 0.0689400
\(883\) −2.79540e10 −1.36641 −0.683206 0.730226i \(-0.739415\pi\)
−0.683206 + 0.730226i \(0.739415\pi\)
\(884\) −1.35435e10 −0.659399
\(885\) −3.09506e9 −0.150095
\(886\) −2.53541e10 −1.22470
\(887\) −2.08176e10 −1.00161 −0.500804 0.865561i \(-0.666962\pi\)
−0.500804 + 0.865561i \(0.666962\pi\)
\(888\) −1.41813e10 −0.679626
\(889\) 2.31761e9 0.110633
\(890\) −1.34073e10 −0.637495
\(891\) 3.82054e10 1.80948
\(892\) −1.27241e10 −0.600276
\(893\) −8.41542e9 −0.395454
\(894\) −1.06891e10 −0.500332
\(895\) 1.24884e10 0.582273
\(896\) −6.51763e9 −0.302700
\(897\) 4.98239e10 2.30496
\(898\) 2.73500e10 1.26035
\(899\) 5.13526e9 0.235724
\(900\) 1.09816e9 0.0502132
\(901\) −1.43562e10 −0.653886
\(902\) −1.81710e10 −0.824435
\(903\) 5.84100e9 0.263986
\(904\) −1.58033e10 −0.711474
\(905\) −1.34268e10 −0.602147
\(906\) 5.14768e10 2.29965
\(907\) 3.61602e10 1.60918 0.804591 0.593830i \(-0.202385\pi\)
0.804591 + 0.593830i \(0.202385\pi\)
\(908\) 2.77713e10 1.23111
\(909\) −1.56761e10 −0.692254
\(910\) 7.32910e9 0.322408
\(911\) 1.79811e10 0.787954 0.393977 0.919120i \(-0.371099\pi\)
0.393977 + 0.919120i \(0.371099\pi\)
\(912\) 3.82920e10 1.67158
\(913\) 4.13272e10 1.79717
\(914\) 2.58816e10 1.12119
\(915\) 5.69681e9 0.245843
\(916\) 2.67038e10 1.14799
\(917\) 7.58245e8 0.0324726
\(918\) 1.47986e10 0.631351
\(919\) −1.01373e10 −0.430841 −0.215421 0.976521i \(-0.569112\pi\)
−0.215421 + 0.976521i \(0.569112\pi\)
\(920\) −5.96057e9 −0.252366
\(921\) 1.32405e10 0.558463
\(922\) −3.74355e9 −0.157299
\(923\) 2.59376e10 1.08574
\(924\) −1.04805e10 −0.437050
\(925\) −6.59987e9 −0.274182
\(926\) −6.13311e10 −2.53830
\(927\) −3.63406e9 −0.149835
\(928\) 4.73449e10 1.94471
\(929\) −2.39605e10 −0.980485 −0.490242 0.871586i \(-0.663092\pi\)
−0.490242 + 0.871586i \(0.663092\pi\)
\(930\) 2.32773e9 0.0948948
\(931\) 4.11248e9 0.167024
\(932\) 1.55391e10 0.628738
\(933\) 3.68125e9 0.148392
\(934\) −4.30567e10 −1.72913
\(935\) −1.08923e10 −0.435791
\(936\) 5.83998e9 0.232780
\(937\) 1.16627e10 0.463138 0.231569 0.972818i \(-0.425614\pi\)
0.231569 + 0.972818i \(0.425614\pi\)
\(938\) −1.13637e10 −0.449584
\(939\) 3.69943e10 1.45816
\(940\) −2.59200e9 −0.101786
\(941\) 3.03134e10 1.18596 0.592982 0.805216i \(-0.297951\pi\)
0.592982 + 0.805216i \(0.297951\pi\)
\(942\) −5.90672e10 −2.30233
\(943\) 1.49295e10 0.579768
\(944\) −9.03231e9 −0.349460
\(945\) −3.22124e9 −0.124169
\(946\) −2.94378e10 −1.13054
\(947\) 2.84339e10 1.08796 0.543979 0.839099i \(-0.316917\pi\)
0.543979 + 0.839099i \(0.316917\pi\)
\(948\) 1.28748e10 0.490809
\(949\) −5.83357e10 −2.21566
\(950\) 7.99238e9 0.302443
\(951\) 2.54900e10 0.961031
\(952\) 2.82864e9 0.106255
\(953\) −4.09127e10 −1.53120 −0.765602 0.643315i \(-0.777559\pi\)
−0.765602 + 0.643315i \(0.777559\pi\)
\(954\) −1.27351e10 −0.474878
\(955\) −2.76356e9 −0.102673
\(956\) 3.15664e10 1.16848
\(957\) −7.84479e10 −2.89327
\(958\) 6.70105e10 2.46243
\(959\) 2.43729e9 0.0892365
\(960\) 3.93349e9 0.143492
\(961\) −2.69733e10 −0.980399
\(962\) 7.22042e10 2.61487
\(963\) 6.21278e9 0.224179
\(964\) 1.89257e10 0.680428
\(965\) −1.87241e10 −0.670742
\(966\) 2.14075e10 0.764094
\(967\) −4.11324e10 −1.46282 −0.731411 0.681937i \(-0.761138\pi\)
−0.731411 + 0.681937i \(0.761138\pi\)
\(968\) −1.37364e10 −0.486754
\(969\) −2.57841e10 −0.910372
\(970\) −4.36576e9 −0.153588
\(971\) 2.85539e10 1.00092 0.500459 0.865760i \(-0.333165\pi\)
0.500459 + 0.865760i \(0.333165\pi\)
\(972\) 1.37035e10 0.478629
\(973\) −3.16194e9 −0.110042
\(974\) −2.05057e10 −0.711079
\(975\) 1.00024e10 0.345610
\(976\) 1.66250e10 0.572384
\(977\) 3.64395e10 1.25009 0.625045 0.780589i \(-0.285081\pi\)
0.625045 + 0.780589i \(0.285081\pi\)
\(978\) 2.81178e10 0.961159
\(979\) 4.74503e10 1.61622
\(980\) 1.26667e9 0.0429904
\(981\) −1.62951e10 −0.551081
\(982\) −7.01975e10 −2.36555
\(983\) −3.71636e10 −1.24790 −0.623951 0.781463i \(-0.714474\pi\)
−0.623951 + 0.781463i \(0.714474\pi\)
\(984\) 6.44006e9 0.215480
\(985\) −7.12634e9 −0.237596
\(986\) −4.35570e10 −1.44707
\(987\) −4.52513e9 −0.149803
\(988\) −3.51711e10 −1.16021
\(989\) 2.41864e10 0.795032
\(990\) −9.66231e9 −0.316489
\(991\) 1.80657e10 0.589655 0.294827 0.955551i \(-0.404738\pi\)
0.294827 + 0.955551i \(0.404738\pi\)
\(992\) 4.97190e9 0.161708
\(993\) −1.47838e10 −0.479140
\(994\) 1.11445e10 0.359921
\(995\) −8.55201e9 −0.275225
\(996\) 3.01321e10 0.966320
\(997\) −2.62408e10 −0.838580 −0.419290 0.907852i \(-0.637721\pi\)
−0.419290 + 0.907852i \(0.637721\pi\)
\(998\) −2.94199e10 −0.936882
\(999\) −3.17347e10 −1.00706
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 35.8.a.a.1.2 2
3.2 odd 2 315.8.a.c.1.1 2
4.3 odd 2 560.8.a.i.1.2 2
5.2 odd 4 175.8.b.c.99.4 4
5.3 odd 4 175.8.b.c.99.1 4
5.4 even 2 175.8.a.b.1.1 2
7.6 odd 2 245.8.a.b.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
35.8.a.a.1.2 2 1.1 even 1 trivial
175.8.a.b.1.1 2 5.4 even 2
175.8.b.c.99.1 4 5.3 odd 4
175.8.b.c.99.4 4 5.2 odd 4
245.8.a.b.1.2 2 7.6 odd 2
315.8.a.c.1.1 2 3.2 odd 2
560.8.a.i.1.2 2 4.3 odd 2