Properties

Label 91.8.a.c.1.1
Level $91$
Weight $8$
Character 91.1
Self dual yes
Analytic conductor $28.427$
Analytic rank $1$
Dimension $10$
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] = [91,8,Mod(1,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, 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("91.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 91.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: \(28.4270373191\)
Analytic rank: \(1\)
Dimension: \(10\)
Coefficient field: \(\mathbb{Q}[x]/(x^{10} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{10} - 2 x^{9} - 957 x^{8} + 1224 x^{7} + 310102 x^{6} - 241884 x^{5} - 40367312 x^{4} + 11067840 x^{3} + 1840757376 x^{2} + 541859072 x - 4516262912 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2^{6}\cdot 3^{2} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-20.0501\) of defining polynomial
Character \(\chi\) \(=\) 91.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-22.0501 q^{2} -47.1151 q^{3} +358.208 q^{4} -467.623 q^{5} +1038.89 q^{6} +343.000 q^{7} -5076.12 q^{8} +32.8291 q^{9} +O(q^{10})\) \(q-22.0501 q^{2} -47.1151 q^{3} +358.208 q^{4} -467.623 q^{5} +1038.89 q^{6} +343.000 q^{7} -5076.12 q^{8} +32.8291 q^{9} +10311.1 q^{10} -4968.96 q^{11} -16877.0 q^{12} -2197.00 q^{13} -7563.19 q^{14} +22032.1 q^{15} +66078.5 q^{16} +20459.5 q^{17} -723.885 q^{18} +11244.3 q^{19} -167506. q^{20} -16160.5 q^{21} +109566. q^{22} +62996.4 q^{23} +239162. q^{24} +140546. q^{25} +48444.1 q^{26} +101494. q^{27} +122865. q^{28} -15545.6 q^{29} -485810. q^{30} +50810.4 q^{31} -807295. q^{32} +234113. q^{33} -451134. q^{34} -160395. q^{35} +11759.6 q^{36} +268555. q^{37} -247937. q^{38} +103512. q^{39} +2.37371e6 q^{40} -524959. q^{41} +356340. q^{42} -1.02709e6 q^{43} -1.77992e6 q^{44} -15351.6 q^{45} -1.38908e6 q^{46} +937764. q^{47} -3.11329e6 q^{48} +117649. q^{49} -3.09906e6 q^{50} -963950. q^{51} -786983. q^{52} +1.65322e6 q^{53} -2.23795e6 q^{54} +2.32360e6 q^{55} -1.74111e6 q^{56} -529774. q^{57} +342782. q^{58} -2.20262e6 q^{59} +7.89207e6 q^{60} +230413. q^{61} -1.12038e6 q^{62} +11260.4 q^{63} +9.34292e6 q^{64} +1.02737e6 q^{65} -5.16222e6 q^{66} +2.00255e6 q^{67} +7.32876e6 q^{68} -2.96808e6 q^{69} +3.53672e6 q^{70} +2.93841e6 q^{71} -166644. q^{72} +1.33302e6 q^{73} -5.92168e6 q^{74} -6.62183e6 q^{75} +4.02779e6 q^{76} -1.70435e6 q^{77} -2.28245e6 q^{78} -2.92073e6 q^{79} -3.08998e7 q^{80} -4.85369e6 q^{81} +1.15754e7 q^{82} -7.49486e6 q^{83} -5.78881e6 q^{84} -9.56732e6 q^{85} +2.26474e7 q^{86} +732431. q^{87} +2.52231e7 q^{88} -1.80466e6 q^{89} +338505. q^{90} -753571. q^{91} +2.25658e7 q^{92} -2.39394e6 q^{93} -2.06778e7 q^{94} -5.25807e6 q^{95} +3.80358e7 q^{96} -5.26831e6 q^{97} -2.59418e6 q^{98} -163126. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 10 q - 18 q^{2} - 80 q^{3} + 670 q^{4} - 927 q^{5} - 1419 q^{6} + 3430 q^{7} - 4878 q^{8} + 3612 q^{9} + 9420 q^{10} + 876 q^{11} - 8765 q^{12} - 21970 q^{13} - 6174 q^{14} - 5320 q^{15} + 41370 q^{16} + 6294 q^{17} - 16027 q^{18} - 97401 q^{19} - 166650 q^{20} - 27440 q^{21} + 74171 q^{22} - 15255 q^{23} + 196187 q^{24} + 162145 q^{25} + 39546 q^{26} - 181820 q^{27} + 229810 q^{28} - 340533 q^{29} - 325020 q^{30} - 148675 q^{31} - 642762 q^{32} - 624400 q^{33} - 1161518 q^{34} - 317961 q^{35} - 773917 q^{36} - 621782 q^{37} - 805092 q^{38} + 175760 q^{39} - 350478 q^{40} - 2043336 q^{41} - 486717 q^{42} - 1801391 q^{43} - 3953667 q^{44} - 1908807 q^{45} - 2707731 q^{46} - 1624701 q^{47} - 6068625 q^{48} + 1176490 q^{49} - 6891516 q^{50} + 1811700 q^{51} - 1471990 q^{52} - 199965 q^{53} - 2895913 q^{54} + 739086 q^{55} - 1673154 q^{56} + 2159088 q^{57} + 2071092 q^{58} - 8098908 q^{59} + 8096436 q^{60} + 2271618 q^{61} - 8910225 q^{62} + 1238916 q^{63} + 8099930 q^{64} + 2036619 q^{65} - 5999191 q^{66} + 1970272 q^{67} - 1766238 q^{68} - 4622962 q^{69} + 3231060 q^{70} - 7145820 q^{71} + 984975 q^{72} + 1409431 q^{73} - 5498643 q^{74} - 8857892 q^{75} - 2749534 q^{76} + 300468 q^{77} + 3117543 q^{78} - 9011055 q^{79} - 23850522 q^{80} + 11613490 q^{81} + 27962597 q^{82} - 15006567 q^{83} - 3006395 q^{84} - 9416628 q^{85} + 38357850 q^{86} - 15828996 q^{87} + 42205269 q^{88} - 11472777 q^{89} + 53425712 q^{90} - 7535710 q^{91} + 16755837 q^{92} + 36339848 q^{93} + 5133371 q^{94} + 29637939 q^{95} + 65329611 q^{96} + 3228571 q^{97} - 2117682 q^{98} + 19367194 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −22.0501 −1.94897 −0.974487 0.224443i \(-0.927944\pi\)
−0.974487 + 0.224443i \(0.927944\pi\)
\(3\) −47.1151 −1.00748 −0.503739 0.863856i \(-0.668043\pi\)
−0.503739 + 0.863856i \(0.668043\pi\)
\(4\) 358.208 2.79850
\(5\) −467.623 −1.67302 −0.836509 0.547953i \(-0.815407\pi\)
−0.836509 + 0.547953i \(0.815407\pi\)
\(6\) 1038.89 1.96355
\(7\) 343.000 0.377964
\(8\) −5076.12 −3.50523
\(9\) 32.8291 0.0150110
\(10\) 10311.1 3.26067
\(11\) −4968.96 −1.12562 −0.562810 0.826587i \(-0.690280\pi\)
−0.562810 + 0.826587i \(0.690280\pi\)
\(12\) −16877.0 −2.81943
\(13\) −2197.00 −0.277350
\(14\) −7563.19 −0.736643
\(15\) 22032.1 1.68553
\(16\) 66078.5 4.03311
\(17\) 20459.5 1.01001 0.505003 0.863118i \(-0.331492\pi\)
0.505003 + 0.863118i \(0.331492\pi\)
\(18\) −723.885 −0.0292561
\(19\) 11244.3 0.376091 0.188046 0.982160i \(-0.439785\pi\)
0.188046 + 0.982160i \(0.439785\pi\)
\(20\) −167506. −4.68194
\(21\) −16160.5 −0.380791
\(22\) 109566. 2.19380
\(23\) 62996.4 1.07961 0.539806 0.841789i \(-0.318497\pi\)
0.539806 + 0.841789i \(0.318497\pi\)
\(24\) 239162. 3.53144
\(25\) 140546. 1.79899
\(26\) 48444.1 0.540548
\(27\) 101494. 0.992354
\(28\) 122865. 1.05773
\(29\) −15545.6 −0.118362 −0.0591812 0.998247i \(-0.518849\pi\)
−0.0591812 + 0.998247i \(0.518849\pi\)
\(30\) −485810. −3.28505
\(31\) 50810.4 0.306328 0.153164 0.988201i \(-0.451054\pi\)
0.153164 + 0.988201i \(0.451054\pi\)
\(32\) −807295. −4.35519
\(33\) 234113. 1.13404
\(34\) −451134. −1.96847
\(35\) −160395. −0.632341
\(36\) 11759.6 0.0420083
\(37\) 268555. 0.871622 0.435811 0.900038i \(-0.356462\pi\)
0.435811 + 0.900038i \(0.356462\pi\)
\(38\) −247937. −0.732992
\(39\) 103512. 0.279424
\(40\) 2.37371e6 5.86432
\(41\) −524959. −1.18955 −0.594773 0.803893i \(-0.702758\pi\)
−0.594773 + 0.803893i \(0.702758\pi\)
\(42\) 356340. 0.742151
\(43\) −1.02709e6 −1.97000 −0.985002 0.172542i \(-0.944802\pi\)
−0.985002 + 0.172542i \(0.944802\pi\)
\(44\) −1.77992e6 −3.15005
\(45\) −15351.6 −0.0251137
\(46\) −1.38908e6 −2.10414
\(47\) 937764. 1.31750 0.658751 0.752361i \(-0.271085\pi\)
0.658751 + 0.752361i \(0.271085\pi\)
\(48\) −3.11329e6 −4.06327
\(49\) 117649. 0.142857
\(50\) −3.09906e6 −3.50618
\(51\) −963950. −1.01756
\(52\) −786983. −0.776165
\(53\) 1.65322e6 1.52533 0.762666 0.646793i \(-0.223890\pi\)
0.762666 + 0.646793i \(0.223890\pi\)
\(54\) −2.23795e6 −1.93407
\(55\) 2.32360e6 1.88318
\(56\) −1.74111e6 −1.32485
\(57\) −529774. −0.378904
\(58\) 342782. 0.230685
\(59\) −2.20262e6 −1.39623 −0.698116 0.715984i \(-0.745978\pi\)
−0.698116 + 0.715984i \(0.745978\pi\)
\(60\) 7.89207e6 4.71695
\(61\) 230413. 0.129973 0.0649864 0.997886i \(-0.479300\pi\)
0.0649864 + 0.997886i \(0.479300\pi\)
\(62\) −1.12038e6 −0.597025
\(63\) 11260.4 0.00567363
\(64\) 9.34292e6 4.45505
\(65\) 1.02737e6 0.464012
\(66\) −5.16222e6 −2.21021
\(67\) 2.00255e6 0.813434 0.406717 0.913554i \(-0.366674\pi\)
0.406717 + 0.913554i \(0.366674\pi\)
\(68\) 7.32876e6 2.82650
\(69\) −2.96808e6 −1.08769
\(70\) 3.53672e6 1.23242
\(71\) 2.93841e6 0.974335 0.487167 0.873309i \(-0.338030\pi\)
0.487167 + 0.873309i \(0.338030\pi\)
\(72\) −166644. −0.0526171
\(73\) 1.33302e6 0.401057 0.200528 0.979688i \(-0.435734\pi\)
0.200528 + 0.979688i \(0.435734\pi\)
\(74\) −5.92168e6 −1.69877
\(75\) −6.62183e6 −1.81244
\(76\) 4.02779e6 1.05249
\(77\) −1.70435e6 −0.425444
\(78\) −2.28245e6 −0.544590
\(79\) −2.92073e6 −0.666494 −0.333247 0.942840i \(-0.608144\pi\)
−0.333247 + 0.942840i \(0.608144\pi\)
\(80\) −3.08998e7 −6.74746
\(81\) −4.85369e6 −1.01479
\(82\) 1.15754e7 2.31840
\(83\) −7.49486e6 −1.43877 −0.719383 0.694614i \(-0.755575\pi\)
−0.719383 + 0.694614i \(0.755575\pi\)
\(84\) −5.78881e6 −1.06564
\(85\) −9.56732e6 −1.68976
\(86\) 2.26474e7 3.83949
\(87\) 732431. 0.119248
\(88\) 2.52231e7 3.94556
\(89\) −1.80466e6 −0.271350 −0.135675 0.990753i \(-0.543320\pi\)
−0.135675 + 0.990753i \(0.543320\pi\)
\(90\) 338505. 0.0489459
\(91\) −753571. −0.104828
\(92\) 2.25658e7 3.02130
\(93\) −2.39394e6 −0.308619
\(94\) −2.06778e7 −2.56778
\(95\) −5.25807e6 −0.629207
\(96\) 3.80358e7 4.38776
\(97\) −5.26831e6 −0.586098 −0.293049 0.956097i \(-0.594670\pi\)
−0.293049 + 0.956097i \(0.594670\pi\)
\(98\) −2.59418e6 −0.278425
\(99\) −163126. −0.0168967
\(100\) 5.03447e7 5.03447
\(101\) 8.54858e6 0.825599 0.412799 0.910822i \(-0.364551\pi\)
0.412799 + 0.910822i \(0.364551\pi\)
\(102\) 2.12552e7 1.98319
\(103\) −1.48721e7 −1.34104 −0.670519 0.741893i \(-0.733928\pi\)
−0.670519 + 0.741893i \(0.733928\pi\)
\(104\) 1.11522e7 0.972177
\(105\) 7.55700e6 0.637070
\(106\) −3.64536e7 −2.97283
\(107\) 1.38028e7 1.08924 0.544619 0.838684i \(-0.316674\pi\)
0.544619 + 0.838684i \(0.316674\pi\)
\(108\) 3.63559e7 2.77711
\(109\) −1.08168e7 −0.800031 −0.400016 0.916508i \(-0.630995\pi\)
−0.400016 + 0.916508i \(0.630995\pi\)
\(110\) −5.12357e7 −3.67027
\(111\) −1.26530e7 −0.878139
\(112\) 2.26649e7 1.52437
\(113\) 3.73413e6 0.243453 0.121726 0.992564i \(-0.461157\pi\)
0.121726 + 0.992564i \(0.461157\pi\)
\(114\) 1.16816e7 0.738473
\(115\) −2.94585e7 −1.80621
\(116\) −5.56856e6 −0.331238
\(117\) −72125.5 −0.00416330
\(118\) 4.85681e7 2.72122
\(119\) 7.01761e6 0.381746
\(120\) −1.11837e8 −5.90817
\(121\) 5.20344e6 0.267018
\(122\) −5.08063e6 −0.253313
\(123\) 2.47335e7 1.19844
\(124\) 1.82007e7 0.857259
\(125\) −2.91895e7 −1.33672
\(126\) −248293. −0.0110578
\(127\) 1.10183e7 0.477309 0.238655 0.971105i \(-0.423294\pi\)
0.238655 + 0.971105i \(0.423294\pi\)
\(128\) −1.02679e8 −4.32759
\(129\) 4.83912e7 1.98474
\(130\) −2.26536e7 −0.904347
\(131\) −9.49471e6 −0.369005 −0.184502 0.982832i \(-0.559067\pi\)
−0.184502 + 0.982832i \(0.559067\pi\)
\(132\) 8.38612e7 3.17360
\(133\) 3.85678e6 0.142149
\(134\) −4.41565e7 −1.58536
\(135\) −4.74608e7 −1.66023
\(136\) −1.03855e8 −3.54030
\(137\) 3.15294e7 1.04759 0.523797 0.851843i \(-0.324515\pi\)
0.523797 + 0.851843i \(0.324515\pi\)
\(138\) 6.54465e7 2.11987
\(139\) −4.38808e7 −1.38587 −0.692936 0.720999i \(-0.743683\pi\)
−0.692936 + 0.720999i \(0.743683\pi\)
\(140\) −5.74547e7 −1.76961
\(141\) −4.41828e7 −1.32735
\(142\) −6.47923e7 −1.89895
\(143\) 1.09168e7 0.312191
\(144\) 2.16929e6 0.0605410
\(145\) 7.26947e6 0.198023
\(146\) −2.93932e7 −0.781650
\(147\) −5.54304e6 −0.143925
\(148\) 9.61988e7 2.43923
\(149\) −2.35141e6 −0.0582338 −0.0291169 0.999576i \(-0.509270\pi\)
−0.0291169 + 0.999576i \(0.509270\pi\)
\(150\) 1.46012e8 3.53240
\(151\) 6.63014e7 1.56712 0.783562 0.621314i \(-0.213401\pi\)
0.783562 + 0.621314i \(0.213401\pi\)
\(152\) −5.70772e7 −1.31829
\(153\) 671666. 0.0151612
\(154\) 3.75812e7 0.829180
\(155\) −2.37601e7 −0.512492
\(156\) 3.70788e7 0.781968
\(157\) −1.89167e7 −0.390119 −0.195059 0.980791i \(-0.562490\pi\)
−0.195059 + 0.980791i \(0.562490\pi\)
\(158\) 6.44024e7 1.29898
\(159\) −7.78914e7 −1.53674
\(160\) 3.77510e8 7.28632
\(161\) 2.16078e7 0.408055
\(162\) 1.07024e8 1.97779
\(163\) −3.18530e7 −0.576094 −0.288047 0.957616i \(-0.593006\pi\)
−0.288047 + 0.957616i \(0.593006\pi\)
\(164\) −1.88044e8 −3.32895
\(165\) −1.09477e8 −1.89726
\(166\) 1.65263e8 2.80412
\(167\) −3.39660e7 −0.564335 −0.282167 0.959365i \(-0.591053\pi\)
−0.282167 + 0.959365i \(0.591053\pi\)
\(168\) 8.20325e7 1.33476
\(169\) 4.82681e6 0.0769231
\(170\) 2.10961e8 3.29329
\(171\) 369139. 0.00564551
\(172\) −3.67911e8 −5.51306
\(173\) 8.91771e6 0.130946 0.0654730 0.997854i \(-0.479144\pi\)
0.0654730 + 0.997854i \(0.479144\pi\)
\(174\) −1.61502e7 −0.232410
\(175\) 4.82073e7 0.679954
\(176\) −3.28342e8 −4.53975
\(177\) 1.03777e8 1.40667
\(178\) 3.97929e7 0.528854
\(179\) −5.18357e7 −0.675528 −0.337764 0.941231i \(-0.609671\pi\)
−0.337764 + 0.941231i \(0.609671\pi\)
\(180\) −5.49908e6 −0.0702807
\(181\) 7.62815e7 0.956190 0.478095 0.878308i \(-0.341327\pi\)
0.478095 + 0.878308i \(0.341327\pi\)
\(182\) 1.66163e7 0.204308
\(183\) −1.08559e7 −0.130945
\(184\) −3.19777e8 −3.78430
\(185\) −1.25583e8 −1.45824
\(186\) 5.27866e7 0.601490
\(187\) −1.01662e8 −1.13688
\(188\) 3.35915e8 3.68703
\(189\) 3.48124e7 0.375075
\(190\) 1.15941e8 1.22631
\(191\) 9.33400e7 0.969284 0.484642 0.874712i \(-0.338950\pi\)
0.484642 + 0.874712i \(0.338950\pi\)
\(192\) −4.40192e8 −4.48836
\(193\) 1.59630e8 1.59832 0.799161 0.601117i \(-0.205277\pi\)
0.799161 + 0.601117i \(0.205277\pi\)
\(194\) 1.16167e8 1.14229
\(195\) −4.84045e7 −0.467481
\(196\) 4.21428e7 0.399786
\(197\) −2.50462e7 −0.233405 −0.116702 0.993167i \(-0.537232\pi\)
−0.116702 + 0.993167i \(0.537232\pi\)
\(198\) 3.59696e6 0.0329312
\(199\) −8.14028e7 −0.732240 −0.366120 0.930568i \(-0.619314\pi\)
−0.366120 + 0.930568i \(0.619314\pi\)
\(200\) −7.13428e8 −6.30587
\(201\) −9.43504e7 −0.819516
\(202\) −1.88497e8 −1.60907
\(203\) −5.33214e6 −0.0447368
\(204\) −3.45295e8 −2.84764
\(205\) 2.45483e8 1.99013
\(206\) 3.27931e8 2.61365
\(207\) 2.06811e6 0.0162061
\(208\) −1.45174e8 −1.11858
\(209\) −5.58723e7 −0.423336
\(210\) −1.66633e8 −1.24163
\(211\) 2.41270e8 1.76813 0.884067 0.467361i \(-0.154795\pi\)
0.884067 + 0.467361i \(0.154795\pi\)
\(212\) 5.92196e8 4.26864
\(213\) −1.38443e8 −0.981620
\(214\) −3.04353e8 −2.12290
\(215\) 4.80289e8 3.29585
\(216\) −5.15195e8 −3.47843
\(217\) 1.74280e7 0.115781
\(218\) 2.38512e8 1.55924
\(219\) −6.28052e7 −0.404056
\(220\) 8.32333e8 5.27008
\(221\) −4.49495e7 −0.280125
\(222\) 2.79000e8 1.71147
\(223\) 1.53450e8 0.926618 0.463309 0.886197i \(-0.346662\pi\)
0.463309 + 0.886197i \(0.346662\pi\)
\(224\) −2.76902e8 −1.64611
\(225\) 4.61399e6 0.0270046
\(226\) −8.23380e7 −0.474483
\(227\) −1.33687e7 −0.0758575 −0.0379288 0.999280i \(-0.512076\pi\)
−0.0379288 + 0.999280i \(0.512076\pi\)
\(228\) −1.89769e8 −1.06036
\(229\) 3.89793e7 0.214492 0.107246 0.994233i \(-0.465797\pi\)
0.107246 + 0.994233i \(0.465797\pi\)
\(230\) 6.49564e8 3.52026
\(231\) 8.03008e7 0.428625
\(232\) 7.89113e7 0.414888
\(233\) −2.83726e8 −1.46945 −0.734723 0.678367i \(-0.762688\pi\)
−0.734723 + 0.678367i \(0.762688\pi\)
\(234\) 1.59038e6 0.00811417
\(235\) −4.38520e8 −2.20420
\(236\) −7.88997e8 −3.90736
\(237\) 1.37610e8 0.671478
\(238\) −1.54739e8 −0.744013
\(239\) 1.19229e8 0.564922 0.282461 0.959279i \(-0.408849\pi\)
0.282461 + 0.959279i \(0.408849\pi\)
\(240\) 1.45585e9 6.79792
\(241\) −3.04631e8 −1.40189 −0.700946 0.713215i \(-0.747238\pi\)
−0.700946 + 0.713215i \(0.747238\pi\)
\(242\) −1.14736e8 −0.520412
\(243\) 6.71468e6 0.0300195
\(244\) 8.25357e7 0.363729
\(245\) −5.50153e7 −0.239003
\(246\) −5.45376e8 −2.33573
\(247\) −2.47036e7 −0.104309
\(248\) −2.57920e8 −1.07375
\(249\) 3.53121e8 1.44952
\(250\) 6.43631e8 2.60524
\(251\) 1.75280e8 0.699640 0.349820 0.936817i \(-0.386243\pi\)
0.349820 + 0.936817i \(0.386243\pi\)
\(252\) 4.03356e6 0.0158777
\(253\) −3.13027e8 −1.21523
\(254\) −2.42954e8 −0.930264
\(255\) 4.50765e8 1.70239
\(256\) 1.06819e9 3.97931
\(257\) 2.16263e8 0.794724 0.397362 0.917662i \(-0.369926\pi\)
0.397362 + 0.917662i \(0.369926\pi\)
\(258\) −1.06703e9 −3.86820
\(259\) 9.21145e7 0.329442
\(260\) 3.68011e8 1.29854
\(261\) −510347. −0.00177674
\(262\) 2.09359e8 0.719181
\(263\) 5.74623e8 1.94777 0.973885 0.227040i \(-0.0729049\pi\)
0.973885 + 0.227040i \(0.0729049\pi\)
\(264\) −1.18839e9 −3.97506
\(265\) −7.73082e8 −2.55191
\(266\) −8.50425e7 −0.277045
\(267\) 8.50265e7 0.273379
\(268\) 7.17331e8 2.27640
\(269\) −4.09722e8 −1.28338 −0.641692 0.766963i \(-0.721767\pi\)
−0.641692 + 0.766963i \(0.721767\pi\)
\(270\) 1.04652e9 3.23574
\(271\) −4.71283e8 −1.43843 −0.719215 0.694787i \(-0.755498\pi\)
−0.719215 + 0.694787i \(0.755498\pi\)
\(272\) 1.35193e9 4.07346
\(273\) 3.55045e7 0.105612
\(274\) −6.95226e8 −2.04173
\(275\) −6.98368e8 −2.02498
\(276\) −1.06319e9 −3.04389
\(277\) −3.25552e8 −0.920326 −0.460163 0.887834i \(-0.652209\pi\)
−0.460163 + 0.887834i \(0.652209\pi\)
\(278\) 9.67578e8 2.70103
\(279\) 1.66806e6 0.00459829
\(280\) 8.14182e8 2.21650
\(281\) −2.59883e7 −0.0698724 −0.0349362 0.999390i \(-0.511123\pi\)
−0.0349362 + 0.999390i \(0.511123\pi\)
\(282\) 9.74237e8 2.58698
\(283\) 1.86622e8 0.489453 0.244727 0.969592i \(-0.421302\pi\)
0.244727 + 0.969592i \(0.421302\pi\)
\(284\) 1.05256e9 2.72668
\(285\) 2.47734e8 0.633912
\(286\) −2.40717e8 −0.608451
\(287\) −1.80061e8 −0.449606
\(288\) −2.65028e7 −0.0653758
\(289\) 8.25224e6 0.0201108
\(290\) −1.60293e8 −0.385941
\(291\) 2.48217e8 0.590481
\(292\) 4.77498e8 1.12236
\(293\) −7.58493e8 −1.76163 −0.880815 0.473460i \(-0.843005\pi\)
−0.880815 + 0.473460i \(0.843005\pi\)
\(294\) 1.22225e8 0.280507
\(295\) 1.03000e9 2.33592
\(296\) −1.36322e9 −3.05524
\(297\) −5.04320e8 −1.11701
\(298\) 5.18488e7 0.113496
\(299\) −1.38403e8 −0.299431
\(300\) −2.37199e9 −5.07212
\(301\) −3.52290e8 −0.744592
\(302\) −1.46195e9 −3.05428
\(303\) −4.02767e8 −0.831772
\(304\) 7.43003e8 1.51682
\(305\) −1.07746e8 −0.217447
\(306\) −1.48103e7 −0.0295488
\(307\) 3.91537e8 0.772304 0.386152 0.922435i \(-0.373804\pi\)
0.386152 + 0.922435i \(0.373804\pi\)
\(308\) −6.10514e8 −1.19061
\(309\) 7.00698e8 1.35106
\(310\) 5.23913e8 0.998834
\(311\) −3.63348e8 −0.684955 −0.342477 0.939526i \(-0.611266\pi\)
−0.342477 + 0.939526i \(0.611266\pi\)
\(312\) −5.25438e8 −0.979446
\(313\) 5.35465e8 0.987020 0.493510 0.869740i \(-0.335714\pi\)
0.493510 + 0.869740i \(0.335714\pi\)
\(314\) 4.17116e8 0.760331
\(315\) −5.26561e6 −0.00949208
\(316\) −1.04623e9 −1.86518
\(317\) −3.64638e8 −0.642917 −0.321459 0.946924i \(-0.604173\pi\)
−0.321459 + 0.946924i \(0.604173\pi\)
\(318\) 1.71752e9 2.99506
\(319\) 7.72455e7 0.133231
\(320\) −4.36896e9 −7.45338
\(321\) −6.50318e8 −1.09738
\(322\) −4.76454e8 −0.795289
\(323\) 2.30052e8 0.379854
\(324\) −1.73863e9 −2.83988
\(325\) −3.08779e8 −0.498950
\(326\) 7.02363e8 1.12279
\(327\) 5.09635e8 0.806014
\(328\) 2.66475e9 4.16964
\(329\) 3.21653e8 0.497969
\(330\) 2.41397e9 3.69772
\(331\) 6.19834e8 0.939458 0.469729 0.882811i \(-0.344352\pi\)
0.469729 + 0.882811i \(0.344352\pi\)
\(332\) −2.68472e9 −4.02639
\(333\) 8.81643e6 0.0130839
\(334\) 7.48954e8 1.09987
\(335\) −9.36439e8 −1.36089
\(336\) −1.06786e9 −1.53577
\(337\) −6.79975e8 −0.967806 −0.483903 0.875122i \(-0.660781\pi\)
−0.483903 + 0.875122i \(0.660781\pi\)
\(338\) −1.06432e8 −0.149921
\(339\) −1.75934e8 −0.245273
\(340\) −3.42709e9 −4.72879
\(341\) −2.52475e8 −0.344809
\(342\) −8.13955e6 −0.0110030
\(343\) 4.03536e7 0.0539949
\(344\) 5.21361e9 6.90533
\(345\) 1.38794e9 1.81972
\(346\) −1.96637e8 −0.255210
\(347\) 8.63591e8 1.10957 0.554785 0.831994i \(-0.312800\pi\)
0.554785 + 0.831994i \(0.312800\pi\)
\(348\) 2.62363e8 0.333714
\(349\) 2.13960e8 0.269429 0.134714 0.990884i \(-0.456988\pi\)
0.134714 + 0.990884i \(0.456988\pi\)
\(350\) −1.06298e9 −1.32521
\(351\) −2.22982e8 −0.275230
\(352\) 4.01142e9 4.90229
\(353\) −1.13948e9 −1.37878 −0.689389 0.724391i \(-0.742121\pi\)
−0.689389 + 0.724391i \(0.742121\pi\)
\(354\) −2.28829e9 −2.74157
\(355\) −1.37407e9 −1.63008
\(356\) −6.46443e8 −0.759373
\(357\) −3.30635e8 −0.384601
\(358\) 1.14298e9 1.31659
\(359\) −8.30408e8 −0.947242 −0.473621 0.880729i \(-0.657053\pi\)
−0.473621 + 0.880729i \(0.657053\pi\)
\(360\) 7.79267e7 0.0880293
\(361\) −7.67438e8 −0.858555
\(362\) −1.68202e9 −1.86359
\(363\) −2.45160e8 −0.269015
\(364\) −2.69935e8 −0.293363
\(365\) −6.23349e8 −0.670975
\(366\) 2.39374e8 0.255208
\(367\) 9.59695e8 1.01345 0.506725 0.862108i \(-0.330856\pi\)
0.506725 + 0.862108i \(0.330856\pi\)
\(368\) 4.16270e9 4.35420
\(369\) −1.72339e7 −0.0178563
\(370\) 2.76911e9 2.84207
\(371\) 5.67053e8 0.576521
\(372\) −8.57527e8 −0.863670
\(373\) −1.25855e9 −1.25571 −0.627855 0.778330i \(-0.716067\pi\)
−0.627855 + 0.778330i \(0.716067\pi\)
\(374\) 2.24167e9 2.21575
\(375\) 1.37526e9 1.34672
\(376\) −4.76020e9 −4.61815
\(377\) 3.41536e7 0.0328278
\(378\) −7.67618e8 −0.731011
\(379\) −4.65618e8 −0.439332 −0.219666 0.975575i \(-0.570497\pi\)
−0.219666 + 0.975575i \(0.570497\pi\)
\(380\) −1.88348e9 −1.76084
\(381\) −5.19126e8 −0.480878
\(382\) −2.05816e9 −1.88911
\(383\) −7.97037e8 −0.724908 −0.362454 0.932002i \(-0.618061\pi\)
−0.362454 + 0.932002i \(0.618061\pi\)
\(384\) 4.83772e9 4.35995
\(385\) 7.96995e8 0.711775
\(386\) −3.51987e9 −3.11509
\(387\) −3.37183e7 −0.0295718
\(388\) −1.88715e9 −1.64020
\(389\) 1.01828e9 0.877092 0.438546 0.898709i \(-0.355494\pi\)
0.438546 + 0.898709i \(0.355494\pi\)
\(390\) 1.06732e9 0.911109
\(391\) 1.28887e9 1.09041
\(392\) −5.97200e8 −0.500748
\(393\) 4.47344e8 0.371764
\(394\) 5.52271e8 0.454900
\(395\) 1.36580e9 1.11506
\(396\) −5.84332e7 −0.0472854
\(397\) 8.55854e8 0.686488 0.343244 0.939246i \(-0.388474\pi\)
0.343244 + 0.939246i \(0.388474\pi\)
\(398\) 1.79494e9 1.42712
\(399\) −1.81712e8 −0.143212
\(400\) 9.28706e9 7.25552
\(401\) 5.97175e8 0.462483 0.231242 0.972896i \(-0.425721\pi\)
0.231242 + 0.972896i \(0.425721\pi\)
\(402\) 2.08044e9 1.59722
\(403\) −1.11630e8 −0.0849601
\(404\) 3.06217e9 2.31044
\(405\) 2.26969e9 1.69775
\(406\) 1.17574e8 0.0871909
\(407\) −1.33444e9 −0.981114
\(408\) 4.89313e9 3.56678
\(409\) 2.47188e9 1.78647 0.893234 0.449592i \(-0.148431\pi\)
0.893234 + 0.449592i \(0.148431\pi\)
\(410\) −5.41292e9 −3.87872
\(411\) −1.48551e9 −1.05543
\(412\) −5.32729e9 −3.75289
\(413\) −7.55499e8 −0.527726
\(414\) −4.56021e7 −0.0315852
\(415\) 3.50477e9 2.40708
\(416\) 1.77363e9 1.20791
\(417\) 2.06745e9 1.39623
\(418\) 1.23199e9 0.825070
\(419\) 4.84975e7 0.0322085 0.0161042 0.999870i \(-0.494874\pi\)
0.0161042 + 0.999870i \(0.494874\pi\)
\(420\) 2.70698e9 1.78284
\(421\) −1.16612e9 −0.761652 −0.380826 0.924647i \(-0.624360\pi\)
−0.380826 + 0.924647i \(0.624360\pi\)
\(422\) −5.32004e9 −3.44605
\(423\) 3.07859e7 0.0197770
\(424\) −8.39193e9 −5.34664
\(425\) 2.87550e9 1.81699
\(426\) 3.05269e9 1.91315
\(427\) 7.90316e7 0.0491251
\(428\) 4.94426e9 3.04823
\(429\) −5.14346e8 −0.314525
\(430\) −1.05904e10 −6.42353
\(431\) 1.98606e8 0.119487 0.0597436 0.998214i \(-0.480972\pi\)
0.0597436 + 0.998214i \(0.480972\pi\)
\(432\) 6.70656e9 4.00227
\(433\) −1.72363e9 −1.02032 −0.510161 0.860079i \(-0.670414\pi\)
−0.510161 + 0.860079i \(0.670414\pi\)
\(434\) −3.84289e8 −0.225654
\(435\) −3.42501e8 −0.199503
\(436\) −3.87467e9 −2.23889
\(437\) 7.08347e8 0.406033
\(438\) 1.38486e9 0.787495
\(439\) −8.39890e8 −0.473801 −0.236901 0.971534i \(-0.576132\pi\)
−0.236901 + 0.971534i \(0.576132\pi\)
\(440\) −1.17949e10 −6.60099
\(441\) 3.86231e6 0.00214443
\(442\) 9.91142e8 0.545957
\(443\) −2.24667e8 −0.122780 −0.0613899 0.998114i \(-0.519553\pi\)
−0.0613899 + 0.998114i \(0.519553\pi\)
\(444\) −4.53241e9 −2.45747
\(445\) 8.43898e8 0.453973
\(446\) −3.38360e9 −1.80595
\(447\) 1.10787e8 0.0586693
\(448\) 3.20462e9 1.68385
\(449\) −1.76424e9 −0.919805 −0.459902 0.887970i \(-0.652116\pi\)
−0.459902 + 0.887970i \(0.652116\pi\)
\(450\) −1.01739e8 −0.0526313
\(451\) 2.60850e9 1.33898
\(452\) 1.33760e9 0.681303
\(453\) −3.12379e9 −1.57884
\(454\) 2.94782e8 0.147844
\(455\) 3.52387e8 0.175380
\(456\) 2.68920e9 1.32815
\(457\) −3.98300e9 −1.95211 −0.976054 0.217527i \(-0.930201\pi\)
−0.976054 + 0.217527i \(0.930201\pi\)
\(458\) −8.59499e8 −0.418039
\(459\) 2.07651e9 1.00228
\(460\) −1.05523e10 −5.05469
\(461\) −2.43458e9 −1.15737 −0.578684 0.815552i \(-0.696434\pi\)
−0.578684 + 0.815552i \(0.696434\pi\)
\(462\) −1.77064e9 −0.835380
\(463\) 1.46969e8 0.0688166 0.0344083 0.999408i \(-0.489045\pi\)
0.0344083 + 0.999408i \(0.489045\pi\)
\(464\) −1.02723e9 −0.477369
\(465\) 1.11946e9 0.516324
\(466\) 6.25620e9 2.86391
\(467\) 1.70476e9 0.774559 0.387280 0.921962i \(-0.373415\pi\)
0.387280 + 0.921962i \(0.373415\pi\)
\(468\) −2.58359e7 −0.0116510
\(469\) 6.86875e8 0.307449
\(470\) 9.66942e9 4.29594
\(471\) 8.91262e8 0.393036
\(472\) 1.11808e10 4.89412
\(473\) 5.10355e9 2.21747
\(474\) −3.03432e9 −1.30869
\(475\) 1.58034e9 0.676584
\(476\) 2.51376e9 1.06832
\(477\) 5.42736e7 0.0228968
\(478\) −2.62901e9 −1.10102
\(479\) 1.54581e9 0.642660 0.321330 0.946967i \(-0.395870\pi\)
0.321330 + 0.946967i \(0.395870\pi\)
\(480\) −1.77864e10 −7.34080
\(481\) −5.90016e8 −0.241744
\(482\) 6.71715e9 2.73225
\(483\) −1.01805e9 −0.411107
\(484\) 1.86391e9 0.747252
\(485\) 2.46358e9 0.980552
\(486\) −1.48060e8 −0.0585072
\(487\) 1.57113e9 0.616396 0.308198 0.951322i \(-0.400274\pi\)
0.308198 + 0.951322i \(0.400274\pi\)
\(488\) −1.16960e9 −0.455585
\(489\) 1.50076e9 0.580402
\(490\) 1.21310e9 0.465810
\(491\) −4.44730e9 −1.69555 −0.847776 0.530354i \(-0.822059\pi\)
−0.847776 + 0.530354i \(0.822059\pi\)
\(492\) 8.85973e9 3.35384
\(493\) −3.18055e8 −0.119547
\(494\) 5.44718e8 0.203295
\(495\) 7.62816e7 0.0282684
\(496\) 3.35747e9 1.23545
\(497\) 1.00787e9 0.368264
\(498\) −7.78636e9 −2.82509
\(499\) 6.30533e7 0.0227173 0.0113586 0.999935i \(-0.496384\pi\)
0.0113586 + 0.999935i \(0.496384\pi\)
\(500\) −1.04559e10 −3.74082
\(501\) 1.60031e9 0.568554
\(502\) −3.86495e9 −1.36358
\(503\) −1.20044e9 −0.420584 −0.210292 0.977639i \(-0.567441\pi\)
−0.210292 + 0.977639i \(0.567441\pi\)
\(504\) −5.71590e7 −0.0198874
\(505\) −3.99751e9 −1.38124
\(506\) 6.90228e9 2.36846
\(507\) −2.27415e8 −0.0774983
\(508\) 3.94683e9 1.33575
\(509\) −2.58483e9 −0.868800 −0.434400 0.900720i \(-0.643040\pi\)
−0.434400 + 0.900720i \(0.643040\pi\)
\(510\) −9.93943e9 −3.31792
\(511\) 4.57225e8 0.151585
\(512\) −1.04108e10 −3.42798
\(513\) 1.14122e9 0.373216
\(514\) −4.76863e9 −1.54890
\(515\) 6.95451e9 2.24358
\(516\) 1.73341e10 5.55428
\(517\) −4.65972e9 −1.48301
\(518\) −2.03114e9 −0.642074
\(519\) −4.20159e8 −0.131925
\(520\) −5.21504e9 −1.62647
\(521\) 2.64704e9 0.820028 0.410014 0.912079i \(-0.365524\pi\)
0.410014 + 0.912079i \(0.365524\pi\)
\(522\) 1.12532e7 0.00346282
\(523\) 3.04754e9 0.931522 0.465761 0.884911i \(-0.345781\pi\)
0.465761 + 0.884911i \(0.345781\pi\)
\(524\) −3.40108e9 −1.03266
\(525\) −2.27129e9 −0.685038
\(526\) −1.26705e10 −3.79616
\(527\) 1.03956e9 0.309393
\(528\) 1.54698e10 4.57369
\(529\) 5.63716e8 0.165564
\(530\) 1.70465e10 4.97360
\(531\) −7.23100e7 −0.0209589
\(532\) 1.38153e9 0.397805
\(533\) 1.15333e9 0.329921
\(534\) −1.87485e9 −0.532808
\(535\) −6.45448e9 −1.82231
\(536\) −1.01652e10 −2.85127
\(537\) 2.44224e9 0.680580
\(538\) 9.03442e9 2.50128
\(539\) −5.84594e8 −0.160803
\(540\) −1.70009e10 −4.64615
\(541\) 4.88226e9 1.32566 0.662828 0.748772i \(-0.269356\pi\)
0.662828 + 0.748772i \(0.269356\pi\)
\(542\) 1.03918e10 2.80346
\(543\) −3.59401e9 −0.963340
\(544\) −1.65169e10 −4.39877
\(545\) 5.05819e9 1.33847
\(546\) −7.82880e8 −0.205836
\(547\) −5.03886e9 −1.31637 −0.658183 0.752858i \(-0.728675\pi\)
−0.658183 + 0.752858i \(0.728675\pi\)
\(548\) 1.12941e10 2.93169
\(549\) 7.56424e6 0.00195102
\(550\) 1.53991e10 3.94663
\(551\) −1.74799e8 −0.0445151
\(552\) 1.50663e10 3.81259
\(553\) −1.00181e9 −0.251911
\(554\) 7.17847e9 1.79369
\(555\) 5.91683e9 1.46914
\(556\) −1.57185e10 −3.87836
\(557\) 4.09225e9 1.00339 0.501694 0.865045i \(-0.332710\pi\)
0.501694 + 0.865045i \(0.332710\pi\)
\(558\) −3.67809e7 −0.00896195
\(559\) 2.25651e9 0.546381
\(560\) −1.05986e10 −2.55030
\(561\) 4.78983e9 1.14538
\(562\) 5.73046e8 0.136180
\(563\) −2.85540e9 −0.674353 −0.337176 0.941442i \(-0.609472\pi\)
−0.337176 + 0.941442i \(0.609472\pi\)
\(564\) −1.58266e10 −3.71460
\(565\) −1.74616e9 −0.407301
\(566\) −4.11505e9 −0.953932
\(567\) −1.66482e9 −0.383553
\(568\) −1.49157e10 −3.41527
\(569\) −2.69662e9 −0.613659 −0.306830 0.951764i \(-0.599268\pi\)
−0.306830 + 0.951764i \(0.599268\pi\)
\(570\) −5.46257e9 −1.23548
\(571\) −5.34341e9 −1.20114 −0.600568 0.799574i \(-0.705059\pi\)
−0.600568 + 0.799574i \(0.705059\pi\)
\(572\) 3.91049e9 0.873666
\(573\) −4.39772e9 −0.976532
\(574\) 3.97036e9 0.876271
\(575\) 8.85388e9 1.94221
\(576\) 3.06719e8 0.0668748
\(577\) 1.51880e9 0.329143 0.164571 0.986365i \(-0.447376\pi\)
0.164571 + 0.986365i \(0.447376\pi\)
\(578\) −1.81963e8 −0.0391954
\(579\) −7.52098e9 −1.61027
\(580\) 2.60398e9 0.554166
\(581\) −2.57074e9 −0.543802
\(582\) −5.47321e9 −1.15083
\(583\) −8.21477e9 −1.71694
\(584\) −6.76656e9 −1.40580
\(585\) 3.37275e7 0.00696528
\(586\) 1.67249e10 3.43337
\(587\) −4.06252e9 −0.829014 −0.414507 0.910046i \(-0.636046\pi\)
−0.414507 + 0.910046i \(0.636046\pi\)
\(588\) −1.98556e9 −0.402775
\(589\) 5.71325e8 0.115207
\(590\) −2.27115e10 −4.55265
\(591\) 1.18005e9 0.235150
\(592\) 1.77457e10 3.51535
\(593\) −1.42420e9 −0.280466 −0.140233 0.990119i \(-0.544785\pi\)
−0.140233 + 0.990119i \(0.544785\pi\)
\(594\) 1.11203e10 2.17703
\(595\) −3.28159e9 −0.638668
\(596\) −8.42293e8 −0.162968
\(597\) 3.83530e9 0.737716
\(598\) 3.05180e9 0.583583
\(599\) −5.04245e8 −0.0958623 −0.0479312 0.998851i \(-0.515263\pi\)
−0.0479312 + 0.998851i \(0.515263\pi\)
\(600\) 3.36132e10 6.35303
\(601\) −3.01576e9 −0.566678 −0.283339 0.959020i \(-0.591442\pi\)
−0.283339 + 0.959020i \(0.591442\pi\)
\(602\) 7.76805e9 1.45119
\(603\) 6.57419e7 0.0122105
\(604\) 2.37497e10 4.38560
\(605\) −2.43324e9 −0.446727
\(606\) 8.88106e9 1.62110
\(607\) −3.64713e9 −0.661898 −0.330949 0.943649i \(-0.607369\pi\)
−0.330949 + 0.943649i \(0.607369\pi\)
\(608\) −9.07744e9 −1.63795
\(609\) 2.51224e8 0.0450713
\(610\) 2.37582e9 0.423798
\(611\) −2.06027e9 −0.365409
\(612\) 2.40596e8 0.0424286
\(613\) −4.88911e9 −0.857271 −0.428635 0.903478i \(-0.641005\pi\)
−0.428635 + 0.903478i \(0.641005\pi\)
\(614\) −8.63344e9 −1.50520
\(615\) −1.15659e10 −2.00501
\(616\) 8.65151e9 1.49128
\(617\) −5.88695e8 −0.100900 −0.0504501 0.998727i \(-0.516066\pi\)
−0.0504501 + 0.998727i \(0.516066\pi\)
\(618\) −1.54505e10 −2.63319
\(619\) −6.16760e9 −1.04520 −0.522600 0.852578i \(-0.675038\pi\)
−0.522600 + 0.852578i \(0.675038\pi\)
\(620\) −8.51106e9 −1.43421
\(621\) 6.39375e9 1.07136
\(622\) 8.01188e9 1.33496
\(623\) −6.18997e8 −0.102561
\(624\) 6.83990e9 1.12695
\(625\) 2.66950e9 0.437371
\(626\) −1.18071e10 −1.92368
\(627\) 2.63243e9 0.426501
\(628\) −6.77612e9 −1.09175
\(629\) 5.49451e9 0.880342
\(630\) 1.16107e8 0.0184998
\(631\) 1.00481e10 1.59213 0.796067 0.605209i \(-0.206910\pi\)
0.796067 + 0.605209i \(0.206910\pi\)
\(632\) 1.48260e10 2.33622
\(633\) −1.13675e10 −1.78135
\(634\) 8.04032e9 1.25303
\(635\) −5.15239e9 −0.798547
\(636\) −2.79013e10 −4.30056
\(637\) −2.58475e8 −0.0396214
\(638\) −1.70327e9 −0.259664
\(639\) 9.64653e7 0.0146257
\(640\) 4.80149e10 7.24013
\(641\) 8.25313e8 0.123770 0.0618850 0.998083i \(-0.480289\pi\)
0.0618850 + 0.998083i \(0.480289\pi\)
\(642\) 1.43396e10 2.13877
\(643\) −1.11163e10 −1.64900 −0.824502 0.565859i \(-0.808545\pi\)
−0.824502 + 0.565859i \(0.808545\pi\)
\(644\) 7.74007e9 1.14194
\(645\) −2.26288e10 −3.32050
\(646\) −5.07267e9 −0.740326
\(647\) 3.35721e9 0.487319 0.243659 0.969861i \(-0.421652\pi\)
0.243659 + 0.969861i \(0.421652\pi\)
\(648\) 2.46379e10 3.55706
\(649\) 1.09447e10 1.57163
\(650\) 6.80863e9 0.972440
\(651\) −8.21120e8 −0.116647
\(652\) −1.14100e10 −1.61220
\(653\) −3.12328e9 −0.438949 −0.219475 0.975618i \(-0.570434\pi\)
−0.219475 + 0.975618i \(0.570434\pi\)
\(654\) −1.12375e10 −1.57090
\(655\) 4.43994e9 0.617351
\(656\) −3.46885e10 −4.79757
\(657\) 4.37617e7 0.00602027
\(658\) −7.09249e9 −0.970529
\(659\) −1.85958e8 −0.0253113 −0.0126557 0.999920i \(-0.504029\pi\)
−0.0126557 + 0.999920i \(0.504029\pi\)
\(660\) −3.92154e10 −5.30949
\(661\) −7.98250e9 −1.07506 −0.537532 0.843244i \(-0.680643\pi\)
−0.537532 + 0.843244i \(0.680643\pi\)
\(662\) −1.36674e10 −1.83098
\(663\) 2.11780e9 0.282220
\(664\) 3.80448e10 5.04321
\(665\) −1.80352e9 −0.237818
\(666\) −1.94403e8 −0.0255002
\(667\) −9.79315e8 −0.127786
\(668\) −1.21669e10 −1.57929
\(669\) −7.22982e9 −0.933547
\(670\) 2.06486e10 2.65234
\(671\) −1.14491e9 −0.146300
\(672\) 1.30463e10 1.65842
\(673\) −9.52399e9 −1.20439 −0.602194 0.798350i \(-0.705706\pi\)
−0.602194 + 0.798350i \(0.705706\pi\)
\(674\) 1.49935e10 1.88623
\(675\) 1.42646e10 1.78523
\(676\) 1.72900e9 0.215269
\(677\) −4.95260e9 −0.613442 −0.306721 0.951800i \(-0.599232\pi\)
−0.306721 + 0.951800i \(0.599232\pi\)
\(678\) 3.87936e9 0.478031
\(679\) −1.80703e9 −0.221524
\(680\) 4.85649e10 5.92299
\(681\) 6.29867e8 0.0764247
\(682\) 5.56711e9 0.672023
\(683\) 1.44209e10 1.73189 0.865943 0.500142i \(-0.166719\pi\)
0.865943 + 0.500142i \(0.166719\pi\)
\(684\) 1.32228e8 0.0157990
\(685\) −1.47438e10 −1.75264
\(686\) −8.89802e8 −0.105235
\(687\) −1.83651e9 −0.216095
\(688\) −6.78683e10 −7.94524
\(689\) −3.63212e9 −0.423051
\(690\) −3.06043e10 −3.54658
\(691\) 1.55754e10 1.79584 0.897919 0.440161i \(-0.145079\pi\)
0.897919 + 0.440161i \(0.145079\pi\)
\(692\) 3.19440e9 0.366453
\(693\) −5.59524e7 −0.00638634
\(694\) −1.90423e10 −2.16252
\(695\) 2.05197e10 2.31859
\(696\) −3.71791e9 −0.417990
\(697\) −1.07404e10 −1.20145
\(698\) −4.71785e9 −0.525110
\(699\) 1.33678e10 1.48043
\(700\) 1.72682e10 1.90285
\(701\) −3.18364e9 −0.349069 −0.174534 0.984651i \(-0.555842\pi\)
−0.174534 + 0.984651i \(0.555842\pi\)
\(702\) 4.91678e9 0.536415
\(703\) 3.01971e9 0.327809
\(704\) −4.64246e10 −5.01469
\(705\) 2.06609e10 2.22069
\(706\) 2.51256e10 2.68720
\(707\) 2.93216e9 0.312047
\(708\) 3.71736e10 3.93658
\(709\) −9.62913e9 −1.01467 −0.507335 0.861749i \(-0.669369\pi\)
−0.507335 + 0.861749i \(0.669369\pi\)
\(710\) 3.02984e10 3.17698
\(711\) −9.58848e7 −0.0100047
\(712\) 9.16066e9 0.951145
\(713\) 3.20087e9 0.330716
\(714\) 7.29054e9 0.749577
\(715\) −5.10495e9 −0.522300
\(716\) −1.85680e10 −1.89047
\(717\) −5.61747e9 −0.569146
\(718\) 1.83106e10 1.84615
\(719\) −3.39050e8 −0.0340183 −0.0170092 0.999855i \(-0.505414\pi\)
−0.0170092 + 0.999855i \(0.505414\pi\)
\(720\) −1.01441e9 −0.101286
\(721\) −5.10112e9 −0.506864
\(722\) 1.69221e10 1.67330
\(723\) 1.43527e10 1.41237
\(724\) 2.73247e10 2.67590
\(725\) −2.18487e9 −0.212933
\(726\) 5.40581e9 0.524304
\(727\) 9.26879e8 0.0894648 0.0447324 0.998999i \(-0.485756\pi\)
0.0447324 + 0.998999i \(0.485756\pi\)
\(728\) 3.82522e9 0.367448
\(729\) 1.02987e10 0.984542
\(730\) 1.37449e10 1.30771
\(731\) −2.10137e10 −1.98971
\(732\) −3.88868e9 −0.366449
\(733\) −6.82501e9 −0.640088 −0.320044 0.947403i \(-0.603698\pi\)
−0.320044 + 0.947403i \(0.603698\pi\)
\(734\) −2.11614e10 −1.97519
\(735\) 2.59205e9 0.240790
\(736\) −5.08567e10 −4.70192
\(737\) −9.95061e9 −0.915616
\(738\) 3.80010e8 0.0348015
\(739\) 1.27514e10 1.16225 0.581127 0.813813i \(-0.302612\pi\)
0.581127 + 0.813813i \(0.302612\pi\)
\(740\) −4.49847e10 −4.08088
\(741\) 1.16391e9 0.105089
\(742\) −1.25036e10 −1.12362
\(743\) 8.91329e9 0.797218 0.398609 0.917121i \(-0.369493\pi\)
0.398609 + 0.917121i \(0.369493\pi\)
\(744\) 1.21519e10 1.08178
\(745\) 1.09957e9 0.0974263
\(746\) 2.77512e10 2.44735
\(747\) −2.46049e8 −0.0215973
\(748\) −3.64163e10 −3.18156
\(749\) 4.73435e9 0.411693
\(750\) −3.03247e10 −2.62472
\(751\) −7.84865e9 −0.676169 −0.338084 0.941116i \(-0.609779\pi\)
−0.338084 + 0.941116i \(0.609779\pi\)
\(752\) 6.19660e10 5.31363
\(753\) −8.25834e9 −0.704872
\(754\) −7.53092e8 −0.0639806
\(755\) −3.10040e10 −2.62183
\(756\) 1.24701e10 1.04965
\(757\) 1.18026e10 0.988873 0.494437 0.869214i \(-0.335374\pi\)
0.494437 + 0.869214i \(0.335374\pi\)
\(758\) 1.02669e10 0.856247
\(759\) 1.47483e10 1.22432
\(760\) 2.66906e10 2.20552
\(761\) 1.54333e10 1.26944 0.634718 0.772744i \(-0.281116\pi\)
0.634718 + 0.772744i \(0.281116\pi\)
\(762\) 1.14468e10 0.937220
\(763\) −3.71017e9 −0.302383
\(764\) 3.34352e10 2.71254
\(765\) −3.14086e8 −0.0253650
\(766\) 1.75748e10 1.41283
\(767\) 4.83916e9 0.387245
\(768\) −5.03277e10 −4.00906
\(769\) −1.08288e10 −0.858692 −0.429346 0.903140i \(-0.641256\pi\)
−0.429346 + 0.903140i \(0.641256\pi\)
\(770\) −1.75738e10 −1.38723
\(771\) −1.01892e10 −0.800666
\(772\) 5.71808e10 4.47291
\(773\) −1.34170e10 −1.04478 −0.522392 0.852705i \(-0.674960\pi\)
−0.522392 + 0.852705i \(0.674960\pi\)
\(774\) 7.43492e8 0.0576346
\(775\) 7.14120e9 0.551080
\(776\) 2.67426e10 2.05441
\(777\) −4.33998e9 −0.331905
\(778\) −2.24533e10 −1.70943
\(779\) −5.90277e9 −0.447378
\(780\) −1.73389e10 −1.30825
\(781\) −1.46009e10 −1.09673
\(782\) −2.84198e10 −2.12519
\(783\) −1.57778e9 −0.117458
\(784\) 7.77406e9 0.576158
\(785\) 8.84588e9 0.652675
\(786\) −9.86398e9 −0.724559
\(787\) 1.38342e9 0.101168 0.0505840 0.998720i \(-0.483892\pi\)
0.0505840 + 0.998720i \(0.483892\pi\)
\(788\) −8.97174e9 −0.653183
\(789\) −2.70734e10 −1.96234
\(790\) −3.01160e10 −2.17322
\(791\) 1.28081e9 0.0920165
\(792\) 8.28050e8 0.0592268
\(793\) −5.06217e8 −0.0360479
\(794\) −1.88717e10 −1.33795
\(795\) 3.64238e10 2.57099
\(796\) −2.91592e10 −2.04918
\(797\) −7.26613e9 −0.508392 −0.254196 0.967153i \(-0.581811\pi\)
−0.254196 + 0.967153i \(0.581811\pi\)
\(798\) 4.00678e9 0.279117
\(799\) 1.91862e10 1.33068
\(800\) −1.13462e11 −7.83494
\(801\) −5.92452e7 −0.00407323
\(802\) −1.31678e10 −0.901368
\(803\) −6.62372e9 −0.451437
\(804\) −3.37971e10 −2.29342
\(805\) −1.01043e10 −0.682684
\(806\) 2.46147e9 0.165585
\(807\) 1.93041e10 1.29298
\(808\) −4.33936e10 −2.89392
\(809\) −1.49759e10 −0.994427 −0.497213 0.867628i \(-0.665643\pi\)
−0.497213 + 0.867628i \(0.665643\pi\)
\(810\) −5.00471e10 −3.30888
\(811\) −6.06531e9 −0.399282 −0.199641 0.979869i \(-0.563978\pi\)
−0.199641 + 0.979869i \(0.563978\pi\)
\(812\) −1.91001e9 −0.125196
\(813\) 2.22045e10 1.44919
\(814\) 2.94246e10 1.91217
\(815\) 1.48952e10 0.963816
\(816\) −6.36964e10 −4.10392
\(817\) −1.15488e10 −0.740901
\(818\) −5.45052e10 −3.48178
\(819\) −2.47390e7 −0.00157358
\(820\) 8.79339e10 5.56939
\(821\) 3.43518e9 0.216645 0.108323 0.994116i \(-0.465452\pi\)
0.108323 + 0.994116i \(0.465452\pi\)
\(822\) 3.27556e10 2.05700
\(823\) −1.57126e8 −0.00982537 −0.00491269 0.999988i \(-0.501564\pi\)
−0.00491269 + 0.999988i \(0.501564\pi\)
\(824\) 7.54924e10 4.70065
\(825\) 3.29036e10 2.04012
\(826\) 1.66588e10 1.02852
\(827\) 2.08809e10 1.28375 0.641874 0.766810i \(-0.278157\pi\)
0.641874 + 0.766810i \(0.278157\pi\)
\(828\) 7.40815e8 0.0453527
\(829\) 2.58551e9 0.157618 0.0788088 0.996890i \(-0.474888\pi\)
0.0788088 + 0.996890i \(0.474888\pi\)
\(830\) −7.72805e10 −4.69134
\(831\) 1.53384e10 0.927208
\(832\) −2.05264e10 −1.23561
\(833\) 2.40704e9 0.144286
\(834\) −4.55875e10 −2.72123
\(835\) 1.58833e10 0.944142
\(836\) −2.00139e10 −1.18471
\(837\) 5.15695e9 0.303986
\(838\) −1.06938e9 −0.0627735
\(839\) −7.67324e9 −0.448551 −0.224276 0.974526i \(-0.572002\pi\)
−0.224276 + 0.974526i \(0.572002\pi\)
\(840\) −3.83602e10 −2.23308
\(841\) −1.70082e10 −0.985990
\(842\) 2.57131e10 1.48444
\(843\) 1.22444e9 0.0703949
\(844\) 8.64249e10 4.94812
\(845\) −2.25713e9 −0.128694
\(846\) −6.78834e8 −0.0385449
\(847\) 1.78478e9 0.100924
\(848\) 1.09242e11 6.15183
\(849\) −8.79272e9 −0.493113
\(850\) −6.34051e10 −3.54126
\(851\) 1.69180e10 0.941014
\(852\) −4.95915e10 −2.74707
\(853\) −2.43199e10 −1.34165 −0.670825 0.741616i \(-0.734060\pi\)
−0.670825 + 0.741616i \(0.734060\pi\)
\(854\) −1.74266e9 −0.0957435
\(855\) −1.72618e8 −0.00944504
\(856\) −7.00645e10 −3.81803
\(857\) −1.57922e10 −0.857055 −0.428528 0.903529i \(-0.640968\pi\)
−0.428528 + 0.903529i \(0.640968\pi\)
\(858\) 1.13414e10 0.613001
\(859\) −1.62472e10 −0.874588 −0.437294 0.899319i \(-0.644063\pi\)
−0.437294 + 0.899319i \(0.644063\pi\)
\(860\) 1.72043e11 9.22345
\(861\) 8.48358e9 0.452968
\(862\) −4.37928e9 −0.232877
\(863\) −1.85124e10 −0.980449 −0.490225 0.871596i \(-0.663085\pi\)
−0.490225 + 0.871596i \(0.663085\pi\)
\(864\) −8.19355e10 −4.32189
\(865\) −4.17012e9 −0.219075
\(866\) 3.80063e10 1.98858
\(867\) −3.88805e8 −0.0202612
\(868\) 6.24284e9 0.324014
\(869\) 1.45130e10 0.750218
\(870\) 7.55220e9 0.388827
\(871\) −4.39961e9 −0.225606
\(872\) 5.49075e10 2.80430
\(873\) −1.72954e8 −0.00879792
\(874\) −1.56192e10 −0.791348
\(875\) −1.00120e10 −0.505233
\(876\) −2.24973e10 −1.13075
\(877\) 1.60267e10 0.802314 0.401157 0.916009i \(-0.368608\pi\)
0.401157 + 0.916009i \(0.368608\pi\)
\(878\) 1.85197e10 0.923427
\(879\) 3.57364e10 1.77480
\(880\) 1.53540e11 7.59507
\(881\) 2.20272e10 1.08529 0.542643 0.839964i \(-0.317424\pi\)
0.542643 + 0.839964i \(0.317424\pi\)
\(882\) −8.51644e7 −0.00417944
\(883\) −2.75424e10 −1.34629 −0.673146 0.739510i \(-0.735057\pi\)
−0.673146 + 0.739510i \(0.735057\pi\)
\(884\) −1.61013e10 −0.783930
\(885\) −4.85283e10 −2.35339
\(886\) 4.95394e9 0.239295
\(887\) 5.33447e9 0.256660 0.128330 0.991732i \(-0.459038\pi\)
0.128330 + 0.991732i \(0.459038\pi\)
\(888\) 6.42282e10 3.07808
\(889\) 3.77926e9 0.180406
\(890\) −1.86081e10 −0.884782
\(891\) 2.41178e10 1.14226
\(892\) 5.49671e10 2.59314
\(893\) 1.05445e10 0.495501
\(894\) −2.44286e9 −0.114345
\(895\) 2.42396e10 1.13017
\(896\) −3.52188e10 −1.63567
\(897\) 6.52087e9 0.301670
\(898\) 3.89017e10 1.79268
\(899\) −7.89877e8 −0.0362577
\(900\) 1.65277e9 0.0755725
\(901\) 3.38240e10 1.54059
\(902\) −5.75178e10 −2.60963
\(903\) 1.65982e10 0.750159
\(904\) −1.89549e10 −0.853359
\(905\) −3.56710e10 −1.59972
\(906\) 6.88800e10 3.07712
\(907\) 1.78222e10 0.793115 0.396558 0.918010i \(-0.370205\pi\)
0.396558 + 0.918010i \(0.370205\pi\)
\(908\) −4.78878e9 −0.212287
\(909\) 2.80642e8 0.0123931
\(910\) −7.77018e9 −0.341811
\(911\) −1.58172e8 −0.00693129 −0.00346565 0.999994i \(-0.501103\pi\)
−0.00346565 + 0.999994i \(0.501103\pi\)
\(912\) −3.50066e10 −1.52816
\(913\) 3.72417e10 1.61950
\(914\) 8.78257e10 3.80461
\(915\) 5.07647e9 0.219073
\(916\) 1.39627e10 0.600255
\(917\) −3.25668e9 −0.139471
\(918\) −4.57874e10 −1.95342
\(919\) 1.68755e10 0.717220 0.358610 0.933487i \(-0.383251\pi\)
0.358610 + 0.933487i \(0.383251\pi\)
\(920\) 1.49535e11 6.33119
\(921\) −1.84473e10 −0.778079
\(922\) 5.36829e10 2.25568
\(923\) −6.45569e9 −0.270232
\(924\) 2.87644e10 1.19951
\(925\) 3.77444e10 1.56804
\(926\) −3.24069e9 −0.134122
\(927\) −4.88236e8 −0.0201303
\(928\) 1.25499e10 0.515492
\(929\) 1.42088e10 0.581438 0.290719 0.956808i \(-0.406105\pi\)
0.290719 + 0.956808i \(0.406105\pi\)
\(930\) −2.46842e10 −1.00630
\(931\) 1.32288e9 0.0537273
\(932\) −1.01633e11 −4.11225
\(933\) 1.71192e10 0.690076
\(934\) −3.75902e10 −1.50960
\(935\) 4.75397e10 1.90202
\(936\) 3.66118e8 0.0145934
\(937\) −3.87897e10 −1.54038 −0.770189 0.637815i \(-0.779838\pi\)
−0.770189 + 0.637815i \(0.779838\pi\)
\(938\) −1.51457e10 −0.599210
\(939\) −2.52285e10 −0.994401
\(940\) −1.57081e11 −6.16847
\(941\) 3.60809e10 1.41161 0.705804 0.708407i \(-0.250586\pi\)
0.705804 + 0.708407i \(0.250586\pi\)
\(942\) −1.96524e10 −0.766017
\(943\) −3.30705e10 −1.28425
\(944\) −1.45546e11 −5.63116
\(945\) −1.62791e10 −0.627507
\(946\) −1.12534e11 −4.32180
\(947\) 2.13252e9 0.0815957 0.0407978 0.999167i \(-0.487010\pi\)
0.0407978 + 0.999167i \(0.487010\pi\)
\(948\) 4.92931e10 1.87913
\(949\) −2.92864e9 −0.111233
\(950\) −3.48466e10 −1.31864
\(951\) 1.71800e10 0.647724
\(952\) −3.56222e10 −1.33811
\(953\) 3.44469e10 1.28922 0.644608 0.764514i \(-0.277021\pi\)
0.644608 + 0.764514i \(0.277021\pi\)
\(954\) −1.19674e9 −0.0446252
\(955\) −4.36479e10 −1.62163
\(956\) 4.27087e10 1.58093
\(957\) −3.63942e9 −0.134227
\(958\) −3.40852e10 −1.25253
\(959\) 1.08146e10 0.395953
\(960\) 2.05844e11 7.50911
\(961\) −2.49309e10 −0.906163
\(962\) 1.30099e10 0.471154
\(963\) 4.53132e8 0.0163506
\(964\) −1.09121e11 −3.92319
\(965\) −7.46467e10 −2.67402
\(966\) 2.24481e10 0.801236
\(967\) −3.00671e10 −1.06930 −0.534650 0.845074i \(-0.679556\pi\)
−0.534650 + 0.845074i \(0.679556\pi\)
\(968\) −2.64133e10 −0.935962
\(969\) −1.08389e10 −0.382695
\(970\) −5.43223e10 −1.91107
\(971\) −1.27577e10 −0.447205 −0.223602 0.974680i \(-0.571782\pi\)
−0.223602 + 0.974680i \(0.571782\pi\)
\(972\) 2.40525e9 0.0840096
\(973\) −1.50511e10 −0.523810
\(974\) −3.46436e10 −1.20134
\(975\) 1.45482e10 0.502680
\(976\) 1.52253e10 0.524194
\(977\) 6.08745e9 0.208835 0.104418 0.994534i \(-0.466702\pi\)
0.104418 + 0.994534i \(0.466702\pi\)
\(978\) −3.30919e10 −1.13119
\(979\) 8.96728e9 0.305437
\(980\) −1.97069e10 −0.668849
\(981\) −3.55106e8 −0.0120093
\(982\) 9.80636e10 3.30459
\(983\) 2.52717e10 0.848589 0.424294 0.905524i \(-0.360522\pi\)
0.424294 + 0.905524i \(0.360522\pi\)
\(984\) −1.25550e11 −4.20082
\(985\) 1.17122e10 0.390490
\(986\) 7.01315e9 0.232994
\(987\) −1.51547e10 −0.501692
\(988\) −8.84905e9 −0.291909
\(989\) −6.47027e10 −2.12684
\(990\) −1.68202e9 −0.0550945
\(991\) −1.17673e10 −0.384079 −0.192039 0.981387i \(-0.561510\pi\)
−0.192039 + 0.981387i \(0.561510\pi\)
\(992\) −4.10190e10 −1.33412
\(993\) −2.92035e10 −0.946483
\(994\) −2.22238e10 −0.717737
\(995\) 3.80658e10 1.22505
\(996\) 1.26491e11 4.05650
\(997\) −3.40723e10 −1.08885 −0.544425 0.838810i \(-0.683252\pi\)
−0.544425 + 0.838810i \(0.683252\pi\)
\(998\) −1.39033e9 −0.0442754
\(999\) 2.72567e10 0.864957
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 91.8.a.c.1.1 10
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.8.a.c.1.1 10 1.1 even 1 trivial