Properties

Label 197.14.a.b.1.13
Level $197$
Weight $14$
Character 197.1
Self dual yes
Analytic conductor $211.245$
Analytic rank $0$
Dimension $109$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [197,14,Mod(1,197)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(197, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 14, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("197.1");
 
S:= CuspForms(chi, 14);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 197 \)
Weight: \( k \) \(=\) \( 14 \)
Character orbit: \([\chi]\) \(=\) 197.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: \(211.244930035\)
Analytic rank: \(0\)
Dimension: \(109\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.13
Character \(\chi\) \(=\) 197.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-145.155 q^{2} -2025.07 q^{3} +12878.0 q^{4} -10162.8 q^{5} +293949. q^{6} +345875. q^{7} -680200. q^{8} +2.50658e6 q^{9} +O(q^{10})\) \(q-145.155 q^{2} -2025.07 q^{3} +12878.0 q^{4} -10162.8 q^{5} +293949. q^{6} +345875. q^{7} -680200. q^{8} +2.50658e6 q^{9} +1.47518e6 q^{10} +1.29462e6 q^{11} -2.60789e7 q^{12} -2.93098e7 q^{13} -5.02055e7 q^{14} +2.05804e7 q^{15} -6.76221e6 q^{16} -7.66490e7 q^{17} -3.63843e8 q^{18} +1.13939e8 q^{19} -1.30877e8 q^{20} -7.00421e8 q^{21} -1.87921e8 q^{22} -9.76784e8 q^{23} +1.37745e9 q^{24} -1.11742e9 q^{25} +4.25447e9 q^{26} -1.84739e9 q^{27} +4.45418e9 q^{28} +4.14550e9 q^{29} -2.98735e9 q^{30} -5.83659e9 q^{31} +6.55377e9 q^{32} -2.62170e9 q^{33} +1.11260e10 q^{34} -3.51506e9 q^{35} +3.22798e10 q^{36} +3.87665e9 q^{37} -1.65388e10 q^{38} +5.93545e10 q^{39} +6.91274e9 q^{40} -3.31987e10 q^{41} +1.01670e11 q^{42} -4.34571e10 q^{43} +1.66722e10 q^{44} -2.54739e10 q^{45} +1.41785e11 q^{46} +1.16895e11 q^{47} +1.36939e10 q^{48} +2.27405e10 q^{49} +1.62199e11 q^{50} +1.55219e11 q^{51} -3.77453e11 q^{52} +3.23420e10 q^{53} +2.68158e11 q^{54} -1.31570e10 q^{55} -2.35264e11 q^{56} -2.30734e11 q^{57} -6.01741e11 q^{58} -4.58259e11 q^{59} +2.65034e11 q^{60} -7.04452e11 q^{61} +8.47211e11 q^{62} +8.66964e11 q^{63} -8.95917e11 q^{64} +2.97870e11 q^{65} +3.80553e11 q^{66} +7.66795e11 q^{67} -9.87087e11 q^{68} +1.97806e12 q^{69} +5.10229e11 q^{70} -1.10301e12 q^{71} -1.70498e12 q^{72} +2.36793e12 q^{73} -5.62715e11 q^{74} +2.26285e12 q^{75} +1.46731e12 q^{76} +4.47777e11 q^{77} -8.61560e12 q^{78} +7.79158e11 q^{79} +6.87229e10 q^{80} -2.55213e11 q^{81} +4.81896e12 q^{82} -6.47070e11 q^{83} -9.02003e12 q^{84} +7.78968e11 q^{85} +6.30802e12 q^{86} -8.39492e12 q^{87} -8.80601e11 q^{88} +1.00322e12 q^{89} +3.69767e12 q^{90} -1.01375e13 q^{91} -1.25790e13 q^{92} +1.18195e13 q^{93} -1.69679e13 q^{94} -1.15794e12 q^{95} -1.32718e13 q^{96} +1.52626e13 q^{97} -3.30090e12 q^{98} +3.24507e12 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 109 q + 192 q^{2} + 8018 q^{3} + 471040 q^{4} + 88496 q^{5} + 383232 q^{6} + 1680731 q^{7} + 1820859 q^{8} + 59521391 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 109 q + 192 q^{2} + 8018 q^{3} + 471040 q^{4} + 88496 q^{5} + 383232 q^{6} + 1680731 q^{7} + 1820859 q^{8} + 59521391 q^{9} + 16373653 q^{10} + 21199298 q^{11} + 63225856 q^{12} + 59695238 q^{13} + 37888529 q^{14} + 87246239 q^{15} + 2130706432 q^{16} + 228353715 q^{17} + 400647337 q^{18} + 1139301305 q^{19} + 1109969259 q^{20} + 539982398 q^{21} + 1613315649 q^{22} + 920306804 q^{23} + 5542439613 q^{24} + 31241700999 q^{25} + 1864366110 q^{26} + 17825460755 q^{27} + 20413389070 q^{28} + 7185436621 q^{29} + 2050251883 q^{30} + 28475592572 q^{31} + 8334714660 q^{32} + 19623425846 q^{33} + 37845014194 q^{34} + 25255003636 q^{35} + 287968706746 q^{36} + 71523920490 q^{37} + 67778214914 q^{38} + 44951568463 q^{39} + 169184871486 q^{40} + 69139231052 q^{41} + 58715177635 q^{42} + 247544146139 q^{43} + 63861560722 q^{44} + 257443045479 q^{45} + 160530477869 q^{46} + 308496573061 q^{47} + 412228130018 q^{48} + 1736616239908 q^{49} + 1680360028531 q^{50} + 756579032995 q^{51} + 928015404666 q^{52} + 342783723680 q^{53} - 597894730601 q^{54} + 59276330527 q^{55} - 3822929869144 q^{56} - 562905761941 q^{57} + 62740419347 q^{58} + 827401964151 q^{59} - 2247133283907 q^{60} + 988213134514 q^{61} + 1937380192071 q^{62} + 1788190111357 q^{63} + 11682175668457 q^{64} + 2494670804291 q^{65} + 11819807890512 q^{66} + 8038740399790 q^{67} + 10126245189885 q^{68} + 5225665164579 q^{69} + 11464042631319 q^{70} + 4867145119603 q^{71} + 18133468947055 q^{72} + 9684156738615 q^{73} + 16996786880941 q^{74} + 16718732018262 q^{75} + 21454522032798 q^{76} + 6593100920650 q^{77} + 33749579076633 q^{78} + 7591753073823 q^{79} + 24349241260570 q^{80} + 38778649605417 q^{81} + 25555033184251 q^{82} + 16945724819556 q^{83} + 21855489402730 q^{84} + 15544906794766 q^{85} + 18664144286914 q^{86} + 19049540636401 q^{87} + 17318749473003 q^{88} + 11289674998576 q^{89} + 20983303956671 q^{90} + 47242561944227 q^{91} - 25046698097386 q^{92} - 5411884145985 q^{93} + 18338784709341 q^{94} + 6784117894603 q^{95} - 36827486682955 q^{96} + 45969533477736 q^{97} - 42983409526150 q^{98} + 12084396239183 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −145.155 −1.60375 −0.801877 0.597490i \(-0.796165\pi\)
−0.801877 + 0.597490i \(0.796165\pi\)
\(3\) −2025.07 −1.60381 −0.801903 0.597455i \(-0.796179\pi\)
−0.801903 + 0.597455i \(0.796179\pi\)
\(4\) 12878.0 1.57202
\(5\) −10162.8 −0.290876 −0.145438 0.989367i \(-0.546459\pi\)
−0.145438 + 0.989367i \(0.546459\pi\)
\(6\) 293949. 2.57211
\(7\) 345875. 1.11117 0.555587 0.831459i \(-0.312494\pi\)
0.555587 + 0.831459i \(0.312494\pi\)
\(8\) −680200. −0.917385
\(9\) 2.50658e6 1.57219
\(10\) 1.47518e6 0.466494
\(11\) 1.29462e6 0.220338 0.110169 0.993913i \(-0.464861\pi\)
0.110169 + 0.993913i \(0.464861\pi\)
\(12\) −2.60789e7 −2.52122
\(13\) −2.93098e7 −1.68415 −0.842077 0.539358i \(-0.818667\pi\)
−0.842077 + 0.539358i \(0.818667\pi\)
\(14\) −5.02055e7 −1.78205
\(15\) 2.05804e7 0.466509
\(16\) −6.76221e6 −0.100765
\(17\) −7.66490e7 −0.770173 −0.385087 0.922880i \(-0.625828\pi\)
−0.385087 + 0.922880i \(0.625828\pi\)
\(18\) −3.63843e8 −2.52141
\(19\) 1.13939e8 0.555614 0.277807 0.960637i \(-0.410392\pi\)
0.277807 + 0.960637i \(0.410392\pi\)
\(20\) −1.30877e8 −0.457265
\(21\) −7.00421e8 −1.78211
\(22\) −1.87921e8 −0.353368
\(23\) −9.76784e8 −1.37584 −0.687920 0.725787i \(-0.741476\pi\)
−0.687920 + 0.725787i \(0.741476\pi\)
\(24\) 1.37745e9 1.47131
\(25\) −1.11742e9 −0.915391
\(26\) 4.25447e9 2.70097
\(27\) −1.84739e9 −0.917685
\(28\) 4.45418e9 1.74679
\(29\) 4.14550e9 1.29417 0.647083 0.762420i \(-0.275989\pi\)
0.647083 + 0.762420i \(0.275989\pi\)
\(30\) −2.98735e9 −0.748165
\(31\) −5.83659e9 −1.18116 −0.590579 0.806980i \(-0.701101\pi\)
−0.590579 + 0.806980i \(0.701101\pi\)
\(32\) 6.55377e9 1.07899
\(33\) −2.62170e9 −0.353380
\(34\) 1.11260e10 1.23517
\(35\) −3.51506e9 −0.323214
\(36\) 3.22798e10 2.47152
\(37\) 3.87665e9 0.248396 0.124198 0.992257i \(-0.460364\pi\)
0.124198 + 0.992257i \(0.460364\pi\)
\(38\) −1.65388e10 −0.891068
\(39\) 5.93545e10 2.70105
\(40\) 6.91274e9 0.266846
\(41\) −3.31987e10 −1.09151 −0.545753 0.837946i \(-0.683756\pi\)
−0.545753 + 0.837946i \(0.683756\pi\)
\(42\) 1.01670e11 2.85806
\(43\) −4.34571e10 −1.04837 −0.524186 0.851604i \(-0.675631\pi\)
−0.524186 + 0.851604i \(0.675631\pi\)
\(44\) 1.66722e10 0.346377
\(45\) −2.54739e10 −0.457313
\(46\) 1.41785e11 2.20651
\(47\) 1.16895e11 1.58183 0.790916 0.611925i \(-0.209605\pi\)
0.790916 + 0.611925i \(0.209605\pi\)
\(48\) 1.36939e10 0.161607
\(49\) 2.27405e10 0.234707
\(50\) 1.62199e11 1.46806
\(51\) 1.55219e11 1.23521
\(52\) −3.77453e11 −2.64753
\(53\) 3.23420e10 0.200435 0.100218 0.994966i \(-0.468046\pi\)
0.100218 + 0.994966i \(0.468046\pi\)
\(54\) 2.68158e11 1.47174
\(55\) −1.31570e10 −0.0640912
\(56\) −2.35264e11 −1.01937
\(57\) −2.30734e11 −0.891098
\(58\) −6.01741e11 −2.07552
\(59\) −4.58259e11 −1.41440 −0.707202 0.707012i \(-0.750043\pi\)
−0.707202 + 0.707012i \(0.750043\pi\)
\(60\) 2.65034e11 0.733363
\(61\) −7.04452e11 −1.75068 −0.875342 0.483505i \(-0.839364\pi\)
−0.875342 + 0.483505i \(0.839364\pi\)
\(62\) 8.47211e11 1.89429
\(63\) 8.66964e11 1.74698
\(64\) −8.95917e11 −1.62966
\(65\) 2.97870e11 0.489880
\(66\) 3.80553e11 0.566734
\(67\) 7.66795e11 1.03560 0.517801 0.855501i \(-0.326751\pi\)
0.517801 + 0.855501i \(0.326751\pi\)
\(68\) −9.87087e11 −1.21073
\(69\) 1.97806e12 2.20658
\(70\) 5.10229e11 0.518356
\(71\) −1.10301e12 −1.02188 −0.510939 0.859617i \(-0.670702\pi\)
−0.510939 + 0.859617i \(0.670702\pi\)
\(72\) −1.70498e12 −1.44231
\(73\) 2.36793e12 1.83135 0.915674 0.401921i \(-0.131657\pi\)
0.915674 + 0.401921i \(0.131657\pi\)
\(74\) −5.62715e11 −0.398366
\(75\) 2.26285e12 1.46811
\(76\) 1.46731e12 0.873439
\(77\) 4.47777e11 0.244834
\(78\) −8.61560e12 −4.33182
\(79\) 7.79158e11 0.360620 0.180310 0.983610i \(-0.442290\pi\)
0.180310 + 0.983610i \(0.442290\pi\)
\(80\) 6.87229e10 0.0293101
\(81\) −2.55213e11 −0.100404
\(82\) 4.81896e12 1.75050
\(83\) −6.47070e11 −0.217242 −0.108621 0.994083i \(-0.534643\pi\)
−0.108621 + 0.994083i \(0.534643\pi\)
\(84\) −9.02003e12 −2.80151
\(85\) 7.78968e11 0.224025
\(86\) 6.30802e12 1.68133
\(87\) −8.39492e12 −2.07559
\(88\) −8.80601e11 −0.202135
\(89\) 1.00322e12 0.213974 0.106987 0.994260i \(-0.465880\pi\)
0.106987 + 0.994260i \(0.465880\pi\)
\(90\) 3.69767e12 0.733418
\(91\) −1.01375e13 −1.87139
\(92\) −1.25790e13 −2.16285
\(93\) 1.18195e13 1.89435
\(94\) −1.69679e13 −2.53687
\(95\) −1.15794e12 −0.161615
\(96\) −1.32718e13 −1.73049
\(97\) 1.52626e13 1.86043 0.930213 0.367020i \(-0.119622\pi\)
0.930213 + 0.367020i \(0.119622\pi\)
\(98\) −3.30090e12 −0.376411
\(99\) 3.24507e12 0.346414
\(100\) −1.43902e13 −1.43902
\(101\) −1.76318e13 −1.65275 −0.826374 0.563121i \(-0.809600\pi\)
−0.826374 + 0.563121i \(0.809600\pi\)
\(102\) −2.25309e13 −1.98097
\(103\) −4.93405e12 −0.407157 −0.203578 0.979059i \(-0.565257\pi\)
−0.203578 + 0.979059i \(0.565257\pi\)
\(104\) 1.99366e13 1.54502
\(105\) 7.11824e12 0.518372
\(106\) −4.69461e12 −0.321448
\(107\) −2.84116e13 −1.83021 −0.915105 0.403216i \(-0.867892\pi\)
−0.915105 + 0.403216i \(0.867892\pi\)
\(108\) −2.37907e13 −1.44262
\(109\) −1.38782e13 −0.792614 −0.396307 0.918118i \(-0.629708\pi\)
−0.396307 + 0.918118i \(0.629708\pi\)
\(110\) 1.90980e12 0.102787
\(111\) −7.85048e12 −0.398379
\(112\) −2.33888e12 −0.111967
\(113\) 1.35222e13 0.610993 0.305497 0.952193i \(-0.401178\pi\)
0.305497 + 0.952193i \(0.401178\pi\)
\(114\) 3.34922e13 1.42910
\(115\) 9.92686e12 0.400199
\(116\) 5.33858e13 2.03446
\(117\) −7.34675e13 −2.64781
\(118\) 6.65187e13 2.26835
\(119\) −2.65110e13 −0.855796
\(120\) −1.39988e13 −0.427968
\(121\) −3.28467e13 −0.951451
\(122\) 1.02255e14 2.80766
\(123\) 6.72296e13 1.75056
\(124\) −7.51637e13 −1.85681
\(125\) 2.37619e13 0.557142
\(126\) −1.25844e14 −2.80172
\(127\) −5.73033e13 −1.21187 −0.605934 0.795515i \(-0.707201\pi\)
−0.605934 + 0.795515i \(0.707201\pi\)
\(128\) 7.63585e13 1.53459
\(129\) 8.80036e13 1.68139
\(130\) −4.32374e13 −0.785647
\(131\) −1.03566e13 −0.179041 −0.0895205 0.995985i \(-0.528533\pi\)
−0.0895205 + 0.995985i \(0.528533\pi\)
\(132\) −3.37623e13 −0.555522
\(133\) 3.94086e13 0.617384
\(134\) −1.11304e14 −1.66085
\(135\) 1.87746e13 0.266933
\(136\) 5.21366e13 0.706545
\(137\) −5.90787e13 −0.763392 −0.381696 0.924288i \(-0.624660\pi\)
−0.381696 + 0.924288i \(0.624660\pi\)
\(138\) −2.87125e14 −3.53881
\(139\) 8.92735e13 1.04985 0.524924 0.851149i \(-0.324094\pi\)
0.524924 + 0.851149i \(0.324094\pi\)
\(140\) −4.52670e13 −0.508100
\(141\) −2.36721e14 −2.53695
\(142\) 1.60107e14 1.63884
\(143\) −3.79451e13 −0.371084
\(144\) −1.69500e13 −0.158422
\(145\) −4.21299e13 −0.376442
\(146\) −3.43718e14 −2.93703
\(147\) −4.60511e13 −0.376424
\(148\) 4.99235e13 0.390485
\(149\) −3.00467e13 −0.224950 −0.112475 0.993655i \(-0.535878\pi\)
−0.112475 + 0.993655i \(0.535878\pi\)
\(150\) −3.28465e14 −2.35448
\(151\) −1.09032e14 −0.748521 −0.374260 0.927324i \(-0.622103\pi\)
−0.374260 + 0.927324i \(0.622103\pi\)
\(152\) −7.75012e13 −0.509712
\(153\) −1.92127e14 −1.21086
\(154\) −6.49972e13 −0.392654
\(155\) 5.93161e13 0.343571
\(156\) 7.64368e14 4.24612
\(157\) 2.21346e14 1.17957 0.589784 0.807561i \(-0.299213\pi\)
0.589784 + 0.807561i \(0.299213\pi\)
\(158\) −1.13099e14 −0.578345
\(159\) −6.54948e13 −0.321459
\(160\) −6.66046e13 −0.313852
\(161\) −3.37845e14 −1.52880
\(162\) 3.70455e13 0.161023
\(163\) 2.75363e14 1.14997 0.574985 0.818164i \(-0.305008\pi\)
0.574985 + 0.818164i \(0.305008\pi\)
\(164\) −4.27533e14 −1.71587
\(165\) 2.66438e13 0.102790
\(166\) 9.39255e13 0.348402
\(167\) −4.51190e14 −1.60954 −0.804771 0.593586i \(-0.797712\pi\)
−0.804771 + 0.593586i \(0.797712\pi\)
\(168\) 4.76426e14 1.63488
\(169\) 5.56192e14 1.83637
\(170\) −1.13071e14 −0.359281
\(171\) 2.85597e14 0.873533
\(172\) −5.59641e14 −1.64807
\(173\) −3.30467e14 −0.937193 −0.468596 0.883412i \(-0.655240\pi\)
−0.468596 + 0.883412i \(0.655240\pi\)
\(174\) 1.21857e15 3.32873
\(175\) −3.86488e14 −1.01716
\(176\) −8.75450e12 −0.0222023
\(177\) 9.28007e14 2.26843
\(178\) −1.45622e14 −0.343161
\(179\) −4.56742e13 −0.103783 −0.0518915 0.998653i \(-0.516525\pi\)
−0.0518915 + 0.998653i \(0.516525\pi\)
\(180\) −3.28053e14 −0.718908
\(181\) −6.46033e14 −1.36567 −0.682833 0.730575i \(-0.739252\pi\)
−0.682833 + 0.730575i \(0.739252\pi\)
\(182\) 1.47152e15 3.00124
\(183\) 1.42656e15 2.80776
\(184\) 6.64409e14 1.26217
\(185\) −3.93976e13 −0.0722526
\(186\) −1.71566e15 −3.03807
\(187\) −9.92314e13 −0.169699
\(188\) 1.50538e15 2.48668
\(189\) −6.38965e14 −1.01971
\(190\) 1.68081e14 0.259191
\(191\) 5.19900e14 0.774825 0.387412 0.921907i \(-0.373369\pi\)
0.387412 + 0.921907i \(0.373369\pi\)
\(192\) 1.81429e15 2.61366
\(193\) 5.43076e14 0.756376 0.378188 0.925729i \(-0.376547\pi\)
0.378188 + 0.925729i \(0.376547\pi\)
\(194\) −2.21545e15 −2.98366
\(195\) −6.03207e14 −0.785673
\(196\) 2.92852e14 0.368964
\(197\) 5.84517e13 0.0712470
\(198\) −4.71039e14 −0.555563
\(199\) −5.89769e14 −0.673189 −0.336594 0.941650i \(-0.609275\pi\)
−0.336594 + 0.941650i \(0.609275\pi\)
\(200\) 7.60069e14 0.839766
\(201\) −1.55281e15 −1.66091
\(202\) 2.55934e15 2.65060
\(203\) 1.43382e15 1.43804
\(204\) 1.99892e15 1.94178
\(205\) 3.37392e14 0.317493
\(206\) 7.16203e14 0.652979
\(207\) −2.44839e15 −2.16308
\(208\) 1.98199e14 0.169703
\(209\) 1.47508e14 0.122423
\(210\) −1.03325e15 −0.831341
\(211\) 7.79987e14 0.608487 0.304244 0.952594i \(-0.401596\pi\)
0.304244 + 0.952594i \(0.401596\pi\)
\(212\) 4.16501e14 0.315089
\(213\) 2.23367e15 1.63889
\(214\) 4.12409e15 2.93520
\(215\) 4.41646e14 0.304947
\(216\) 1.25659e15 0.841871
\(217\) −2.01873e15 −1.31247
\(218\) 2.01450e15 1.27116
\(219\) −4.79523e15 −2.93713
\(220\) −1.69436e14 −0.100753
\(221\) 2.24657e15 1.29709
\(222\) 1.13954e15 0.638902
\(223\) 1.43792e15 0.782985 0.391493 0.920181i \(-0.371959\pi\)
0.391493 + 0.920181i \(0.371959\pi\)
\(224\) 2.26678e15 1.19894
\(225\) −2.80091e15 −1.43917
\(226\) −1.96281e15 −0.979882
\(227\) −3.72324e15 −1.80615 −0.903073 0.429486i \(-0.858695\pi\)
−0.903073 + 0.429486i \(0.858695\pi\)
\(228\) −2.97140e15 −1.40083
\(229\) −2.18510e15 −1.00125 −0.500623 0.865666i \(-0.666896\pi\)
−0.500623 + 0.865666i \(0.666896\pi\)
\(230\) −1.44094e15 −0.641821
\(231\) −9.06780e14 −0.392667
\(232\) −2.81977e15 −1.18725
\(233\) −2.22110e15 −0.909401 −0.454700 0.890645i \(-0.650254\pi\)
−0.454700 + 0.890645i \(0.650254\pi\)
\(234\) 1.06642e16 4.24644
\(235\) −1.18798e15 −0.460117
\(236\) −5.90147e15 −2.22348
\(237\) −1.57785e15 −0.578364
\(238\) 3.84820e15 1.37249
\(239\) −4.73143e15 −1.64212 −0.821062 0.570838i \(-0.806618\pi\)
−0.821062 + 0.570838i \(0.806618\pi\)
\(240\) −1.39169e14 −0.0470077
\(241\) 1.43724e15 0.472519 0.236259 0.971690i \(-0.424078\pi\)
0.236259 + 0.971690i \(0.424078\pi\)
\(242\) 4.76786e15 1.52589
\(243\) 3.46216e15 1.07871
\(244\) −9.07195e15 −2.75212
\(245\) −2.31107e14 −0.0682706
\(246\) −9.75873e15 −2.80747
\(247\) −3.33953e15 −0.935740
\(248\) 3.97005e15 1.08358
\(249\) 1.31036e15 0.348414
\(250\) −3.44916e15 −0.893518
\(251\) 1.08501e15 0.273875 0.136938 0.990580i \(-0.456274\pi\)
0.136938 + 0.990580i \(0.456274\pi\)
\(252\) 1.11648e16 2.74629
\(253\) −1.26457e15 −0.303150
\(254\) 8.31787e15 1.94354
\(255\) −1.57746e15 −0.359293
\(256\) −3.74448e15 −0.831442
\(257\) 4.84146e15 1.04812 0.524060 0.851682i \(-0.324417\pi\)
0.524060 + 0.851682i \(0.324417\pi\)
\(258\) −1.27742e16 −2.69653
\(259\) 1.34083e15 0.276011
\(260\) 3.83598e15 0.770104
\(261\) 1.03910e16 2.03468
\(262\) 1.50331e15 0.287138
\(263\) −4.14252e15 −0.771885 −0.385942 0.922523i \(-0.626124\pi\)
−0.385942 + 0.922523i \(0.626124\pi\)
\(264\) 1.78328e15 0.324186
\(265\) −3.28685e14 −0.0583018
\(266\) −5.72036e15 −0.990132
\(267\) −2.03159e15 −0.343172
\(268\) 9.87480e15 1.62799
\(269\) 8.88717e15 1.43012 0.715062 0.699061i \(-0.246399\pi\)
0.715062 + 0.699061i \(0.246399\pi\)
\(270\) −2.72524e15 −0.428094
\(271\) 5.37246e15 0.823896 0.411948 0.911207i \(-0.364848\pi\)
0.411948 + 0.911207i \(0.364848\pi\)
\(272\) 5.18316e14 0.0776063
\(273\) 2.05292e16 3.00134
\(274\) 8.57558e15 1.22429
\(275\) −1.44664e15 −0.201696
\(276\) 2.54734e16 3.46879
\(277\) 9.67553e15 1.28693 0.643467 0.765474i \(-0.277495\pi\)
0.643467 + 0.765474i \(0.277495\pi\)
\(278\) −1.29585e16 −1.68370
\(279\) −1.46299e16 −1.85701
\(280\) 2.39094e15 0.296512
\(281\) −9.60785e15 −1.16422 −0.582110 0.813110i \(-0.697773\pi\)
−0.582110 + 0.813110i \(0.697773\pi\)
\(282\) 3.43612e16 4.06864
\(283\) −1.56977e16 −1.81645 −0.908225 0.418482i \(-0.862562\pi\)
−0.908225 + 0.418482i \(0.862562\pi\)
\(284\) −1.42045e16 −1.60642
\(285\) 2.34490e15 0.259199
\(286\) 5.50793e15 0.595127
\(287\) −1.14826e16 −1.21285
\(288\) 1.64276e16 1.69637
\(289\) −4.02951e15 −0.406833
\(290\) 6.11537e15 0.603720
\(291\) −3.09078e16 −2.98376
\(292\) 3.04943e16 2.87892
\(293\) 8.90199e14 0.0821954 0.0410977 0.999155i \(-0.486915\pi\)
0.0410977 + 0.999155i \(0.486915\pi\)
\(294\) 6.68455e15 0.603691
\(295\) 4.65720e15 0.411417
\(296\) −2.63689e15 −0.227875
\(297\) −2.39167e15 −0.202201
\(298\) 4.36143e15 0.360764
\(299\) 2.86294e16 2.31712
\(300\) 2.91411e16 2.30790
\(301\) −1.50307e16 −1.16492
\(302\) 1.58266e16 1.20044
\(303\) 3.57055e16 2.65069
\(304\) −7.70478e14 −0.0559863
\(305\) 7.15921e15 0.509232
\(306\) 2.78882e16 1.94192
\(307\) 1.77890e16 1.21270 0.606348 0.795200i \(-0.292634\pi\)
0.606348 + 0.795200i \(0.292634\pi\)
\(308\) 5.76648e15 0.384885
\(309\) 9.99179e15 0.653000
\(310\) −8.61004e15 −0.551003
\(311\) −4.48278e15 −0.280934 −0.140467 0.990085i \(-0.544860\pi\)
−0.140467 + 0.990085i \(0.544860\pi\)
\(312\) −4.03729e16 −2.47791
\(313\) −1.49198e16 −0.896862 −0.448431 0.893817i \(-0.648017\pi\)
−0.448431 + 0.893817i \(0.648017\pi\)
\(314\) −3.21294e16 −1.89174
\(315\) −8.81078e15 −0.508155
\(316\) 1.00340e16 0.566903
\(317\) −1.64023e16 −0.907859 −0.453930 0.891038i \(-0.649978\pi\)
−0.453930 + 0.891038i \(0.649978\pi\)
\(318\) 9.50691e15 0.515541
\(319\) 5.36685e15 0.285154
\(320\) 9.10503e15 0.474031
\(321\) 5.75354e16 2.93530
\(322\) 4.90400e16 2.45181
\(323\) −8.73330e15 −0.427919
\(324\) −3.28664e15 −0.157837
\(325\) 3.27514e16 1.54166
\(326\) −3.99704e16 −1.84427
\(327\) 2.81044e16 1.27120
\(328\) 2.25817e16 1.00133
\(329\) 4.04311e16 1.75769
\(330\) −3.86748e15 −0.164850
\(331\) −4.04451e16 −1.69038 −0.845188 0.534469i \(-0.820512\pi\)
−0.845188 + 0.534469i \(0.820512\pi\)
\(332\) −8.33298e15 −0.341509
\(333\) 9.71713e15 0.390527
\(334\) 6.54925e16 2.58131
\(335\) −7.79279e15 −0.301232
\(336\) 4.73639e15 0.179573
\(337\) −3.09233e16 −1.14998 −0.574991 0.818160i \(-0.694994\pi\)
−0.574991 + 0.818160i \(0.694994\pi\)
\(338\) −8.07341e16 −2.94509
\(339\) −2.73833e16 −0.979914
\(340\) 1.00316e16 0.352173
\(341\) −7.55617e15 −0.260255
\(342\) −4.14559e16 −1.40093
\(343\) −2.56461e16 −0.850374
\(344\) 2.95595e16 0.961762
\(345\) −2.01026e16 −0.641842
\(346\) 4.79690e16 1.50303
\(347\) −4.75672e16 −1.46274 −0.731368 0.681983i \(-0.761118\pi\)
−0.731368 + 0.681983i \(0.761118\pi\)
\(348\) −1.08110e17 −3.26288
\(349\) 1.75469e16 0.519797 0.259899 0.965636i \(-0.416311\pi\)
0.259899 + 0.965636i \(0.416311\pi\)
\(350\) 5.61007e16 1.63127
\(351\) 5.41466e16 1.54552
\(352\) 8.48465e15 0.237742
\(353\) 2.14820e16 0.590933 0.295467 0.955353i \(-0.404525\pi\)
0.295467 + 0.955353i \(0.404525\pi\)
\(354\) −1.34705e17 −3.63800
\(355\) 1.12096e16 0.297240
\(356\) 1.29195e16 0.336372
\(357\) 5.36865e16 1.37253
\(358\) 6.62985e15 0.166442
\(359\) 9.72533e15 0.239768 0.119884 0.992788i \(-0.461748\pi\)
0.119884 + 0.992788i \(0.461748\pi\)
\(360\) 1.73273e16 0.419533
\(361\) −2.90709e16 −0.691293
\(362\) 9.37751e16 2.19019
\(363\) 6.65168e16 1.52594
\(364\) −1.30551e17 −2.94186
\(365\) −2.40648e16 −0.532696
\(366\) −2.07073e17 −4.50295
\(367\) −2.51912e16 −0.538171 −0.269085 0.963116i \(-0.586721\pi\)
−0.269085 + 0.963116i \(0.586721\pi\)
\(368\) 6.60522e15 0.138636
\(369\) −8.32152e16 −1.71606
\(370\) 5.71876e15 0.115875
\(371\) 1.11863e16 0.222718
\(372\) 1.52212e17 2.97796
\(373\) 5.80195e16 1.11549 0.557746 0.830011i \(-0.311666\pi\)
0.557746 + 0.830011i \(0.311666\pi\)
\(374\) 1.44040e16 0.272155
\(375\) −4.81195e16 −0.893547
\(376\) −7.95121e16 −1.45115
\(377\) −1.21504e17 −2.17957
\(378\) 9.27491e16 1.63536
\(379\) 2.67049e16 0.462845 0.231423 0.972853i \(-0.425662\pi\)
0.231423 + 0.972853i \(0.425662\pi\)
\(380\) −1.49119e16 −0.254063
\(381\) 1.16043e17 1.94360
\(382\) −7.54662e16 −1.24263
\(383\) −8.05130e16 −1.30339 −0.651695 0.758481i \(-0.725942\pi\)
−0.651695 + 0.758481i \(0.725942\pi\)
\(384\) −1.54631e17 −2.46119
\(385\) −4.55067e15 −0.0712165
\(386\) −7.88302e16 −1.21304
\(387\) −1.08929e17 −1.64824
\(388\) 1.96552e17 2.92463
\(389\) −1.30815e17 −1.91419 −0.957094 0.289779i \(-0.906418\pi\)
−0.957094 + 0.289779i \(0.906418\pi\)
\(390\) 8.75587e16 1.26003
\(391\) 7.48695e16 1.05963
\(392\) −1.54681e16 −0.215316
\(393\) 2.09728e16 0.287147
\(394\) −8.48457e15 −0.114263
\(395\) −7.91842e15 −0.104896
\(396\) 4.17901e16 0.544572
\(397\) 1.86167e16 0.238651 0.119326 0.992855i \(-0.461927\pi\)
0.119326 + 0.992855i \(0.461927\pi\)
\(398\) 8.56080e16 1.07963
\(399\) −7.98052e16 −0.990164
\(400\) 7.55623e15 0.0922391
\(401\) −1.20070e17 −1.44210 −0.721052 0.692881i \(-0.756341\pi\)
−0.721052 + 0.692881i \(0.756341\pi\)
\(402\) 2.25399e17 2.66368
\(403\) 1.71069e17 1.98925
\(404\) −2.27062e17 −2.59816
\(405\) 2.59368e15 0.0292051
\(406\) −2.08127e17 −2.30626
\(407\) 5.01879e15 0.0547312
\(408\) −1.05580e17 −1.13316
\(409\) −1.76100e17 −1.86020 −0.930098 0.367313i \(-0.880278\pi\)
−0.930098 + 0.367313i \(0.880278\pi\)
\(410\) −4.89741e16 −0.509180
\(411\) 1.19638e17 1.22433
\(412\) −6.35408e16 −0.640060
\(413\) −1.58500e17 −1.57165
\(414\) 3.55396e17 3.46905
\(415\) 6.57604e15 0.0631905
\(416\) −1.92090e17 −1.81718
\(417\) −1.80785e17 −1.68375
\(418\) −2.14115e16 −0.196337
\(419\) 9.38626e16 0.847426 0.423713 0.905797i \(-0.360727\pi\)
0.423713 + 0.905797i \(0.360727\pi\)
\(420\) 9.16688e16 0.814894
\(421\) 1.10977e17 0.971407 0.485703 0.874124i \(-0.338563\pi\)
0.485703 + 0.874124i \(0.338563\pi\)
\(422\) −1.13219e17 −0.975863
\(423\) 2.93007e17 2.48694
\(424\) −2.19990e16 −0.183876
\(425\) 8.56492e16 0.705010
\(426\) −3.24228e17 −2.62838
\(427\) −2.43652e17 −1.94531
\(428\) −3.65885e17 −2.87713
\(429\) 7.68415e16 0.595146
\(430\) −6.41072e16 −0.489059
\(431\) −1.35233e17 −1.01620 −0.508102 0.861297i \(-0.669653\pi\)
−0.508102 + 0.861297i \(0.669653\pi\)
\(432\) 1.24924e16 0.0924703
\(433\) 9.96386e16 0.726534 0.363267 0.931685i \(-0.381661\pi\)
0.363267 + 0.931685i \(0.381661\pi\)
\(434\) 2.93029e17 2.10488
\(435\) 8.53159e16 0.603740
\(436\) −1.78724e17 −1.24601
\(437\) −1.11294e17 −0.764436
\(438\) 6.96052e17 4.71043
\(439\) 1.63741e17 1.09178 0.545892 0.837855i \(-0.316191\pi\)
0.545892 + 0.837855i \(0.316191\pi\)
\(440\) 8.94938e15 0.0587963
\(441\) 5.70009e16 0.369004
\(442\) −3.26101e17 −2.08021
\(443\) 7.00534e16 0.440357 0.220178 0.975460i \(-0.429336\pi\)
0.220178 + 0.975460i \(0.429336\pi\)
\(444\) −1.01099e17 −0.626262
\(445\) −1.01955e16 −0.0622399
\(446\) −2.08722e17 −1.25571
\(447\) 6.08465e16 0.360776
\(448\) −3.09875e17 −1.81084
\(449\) 7.90873e16 0.455518 0.227759 0.973718i \(-0.426860\pi\)
0.227759 + 0.973718i \(0.426860\pi\)
\(450\) 4.06566e17 2.30807
\(451\) −4.29797e16 −0.240501
\(452\) 1.74139e17 0.960496
\(453\) 2.20797e17 1.20048
\(454\) 5.40448e17 2.89661
\(455\) 1.03026e17 0.544342
\(456\) 1.56945e17 0.817480
\(457\) 6.98717e16 0.358795 0.179397 0.983777i \(-0.442585\pi\)
0.179397 + 0.983777i \(0.442585\pi\)
\(458\) 3.17179e17 1.60575
\(459\) 1.41600e17 0.706776
\(460\) 1.27838e17 0.629123
\(461\) 1.57968e17 0.766502 0.383251 0.923644i \(-0.374804\pi\)
0.383251 + 0.923644i \(0.374804\pi\)
\(462\) 1.31624e17 0.629740
\(463\) −2.34071e17 −1.10426 −0.552131 0.833758i \(-0.686185\pi\)
−0.552131 + 0.833758i \(0.686185\pi\)
\(464\) −2.80327e16 −0.130406
\(465\) −1.20119e17 −0.551021
\(466\) 3.22405e17 1.45845
\(467\) −5.92394e16 −0.264272 −0.132136 0.991232i \(-0.542184\pi\)
−0.132136 + 0.991232i \(0.542184\pi\)
\(468\) −9.46116e17 −4.16243
\(469\) 2.65215e17 1.15073
\(470\) 1.72442e17 0.737915
\(471\) −4.48240e17 −1.89180
\(472\) 3.11708e17 1.29755
\(473\) −5.62605e16 −0.230997
\(474\) 2.29033e17 0.927553
\(475\) −1.27318e17 −0.508604
\(476\) −3.41409e17 −1.34533
\(477\) 8.10679e16 0.315123
\(478\) 6.86792e17 2.63356
\(479\) −2.37177e17 −0.897205 −0.448603 0.893731i \(-0.648078\pi\)
−0.448603 + 0.893731i \(0.648078\pi\)
\(480\) 1.34879e17 0.503357
\(481\) −1.13624e17 −0.418337
\(482\) −2.08623e17 −0.757804
\(483\) 6.84160e17 2.45189
\(484\) −4.23000e17 −1.49570
\(485\) −1.55111e17 −0.541154
\(486\) −5.02550e17 −1.72999
\(487\) 1.18832e17 0.403641 0.201821 0.979423i \(-0.435314\pi\)
0.201821 + 0.979423i \(0.435314\pi\)
\(488\) 4.79168e17 1.60605
\(489\) −5.57630e17 −1.84433
\(490\) 3.35464e16 0.109489
\(491\) −1.43052e17 −0.460750 −0.230375 0.973102i \(-0.573995\pi\)
−0.230375 + 0.973102i \(0.573995\pi\)
\(492\) 8.65784e17 2.75193
\(493\) −3.17748e17 −0.996732
\(494\) 4.84750e17 1.50070
\(495\) −3.29790e16 −0.100764
\(496\) 3.94682e16 0.119019
\(497\) −3.81503e17 −1.13548
\(498\) −1.90206e17 −0.558770
\(499\) 6.60855e15 0.0191625 0.00958126 0.999954i \(-0.496950\pi\)
0.00958126 + 0.999954i \(0.496950\pi\)
\(500\) 3.06006e17 0.875840
\(501\) 9.13691e17 2.58139
\(502\) −1.57494e17 −0.439228
\(503\) −6.41119e17 −1.76501 −0.882504 0.470304i \(-0.844144\pi\)
−0.882504 + 0.470304i \(0.844144\pi\)
\(504\) −5.89709e17 −1.60265
\(505\) 1.79188e17 0.480745
\(506\) 1.83558e17 0.486178
\(507\) −1.12633e18 −2.94518
\(508\) −7.37953e17 −1.90509
\(509\) −5.47621e17 −1.39577 −0.697887 0.716208i \(-0.745876\pi\)
−0.697887 + 0.716208i \(0.745876\pi\)
\(510\) 2.28977e17 0.576217
\(511\) 8.19009e17 2.03495
\(512\) −8.19983e16 −0.201164
\(513\) −2.10489e17 −0.509879
\(514\) −7.02762e17 −1.68092
\(515\) 5.01438e16 0.118432
\(516\) 1.13331e18 2.64318
\(517\) 1.51335e17 0.348538
\(518\) −1.94629e17 −0.442654
\(519\) 6.69219e17 1.50308
\(520\) −2.02611e17 −0.449409
\(521\) −1.31923e15 −0.00288985 −0.00144492 0.999999i \(-0.500460\pi\)
−0.00144492 + 0.999999i \(0.500460\pi\)
\(522\) −1.50831e18 −3.26312
\(523\) −5.26578e17 −1.12513 −0.562564 0.826754i \(-0.690185\pi\)
−0.562564 + 0.826754i \(0.690185\pi\)
\(524\) −1.33372e17 −0.281457
\(525\) 7.82665e17 1.63132
\(526\) 6.01309e17 1.23791
\(527\) 4.47369e17 0.909696
\(528\) 1.77285e16 0.0356082
\(529\) 4.50071e17 0.892934
\(530\) 4.77104e16 0.0935017
\(531\) −1.14867e18 −2.22371
\(532\) 5.07505e17 0.970543
\(533\) 9.73048e17 1.83826
\(534\) 2.94895e17 0.550364
\(535\) 2.88741e17 0.532365
\(536\) −5.21574e17 −0.950046
\(537\) 9.24935e16 0.166448
\(538\) −1.29002e18 −2.29356
\(539\) 2.94403e16 0.0517149
\(540\) 2.41780e17 0.419625
\(541\) −7.09136e16 −0.121604 −0.0608020 0.998150i \(-0.519366\pi\)
−0.0608020 + 0.998150i \(0.519366\pi\)
\(542\) −7.79840e17 −1.32133
\(543\) 1.30826e18 2.19026
\(544\) −5.02340e17 −0.831007
\(545\) 1.41042e17 0.230553
\(546\) −2.97992e18 −4.81341
\(547\) −2.39261e17 −0.381903 −0.190952 0.981599i \(-0.561157\pi\)
−0.190952 + 0.981599i \(0.561157\pi\)
\(548\) −7.60817e17 −1.20007
\(549\) −1.76577e18 −2.75241
\(550\) 2.09987e17 0.323470
\(551\) 4.72333e17 0.719057
\(552\) −1.34547e18 −2.02428
\(553\) 2.69491e17 0.400711
\(554\) −1.40445e18 −2.06392
\(555\) 7.97828e16 0.115879
\(556\) 1.14967e18 1.65039
\(557\) 2.71586e17 0.385344 0.192672 0.981263i \(-0.438285\pi\)
0.192672 + 0.981263i \(0.438285\pi\)
\(558\) 2.12360e18 2.97818
\(559\) 1.27372e18 1.76562
\(560\) 2.37695e16 0.0325686
\(561\) 2.00950e17 0.272164
\(562\) 1.39463e18 1.86712
\(563\) 6.09127e17 0.806127 0.403064 0.915172i \(-0.367945\pi\)
0.403064 + 0.915172i \(0.367945\pi\)
\(564\) −3.04849e18 −3.98815
\(565\) −1.37423e17 −0.177723
\(566\) 2.27860e18 2.91314
\(567\) −8.82719e16 −0.111566
\(568\) 7.50265e17 0.937456
\(569\) 5.16902e17 0.638525 0.319263 0.947666i \(-0.396565\pi\)
0.319263 + 0.947666i \(0.396565\pi\)
\(570\) −3.40375e17 −0.415691
\(571\) −1.84006e17 −0.222176 −0.111088 0.993811i \(-0.535434\pi\)
−0.111088 + 0.993811i \(0.535434\pi\)
\(572\) −4.88658e17 −0.583353
\(573\) −1.05283e18 −1.24267
\(574\) 1.66676e18 1.94511
\(575\) 1.09148e18 1.25943
\(576\) −2.24569e18 −2.56214
\(577\) 5.01932e16 0.0566243 0.0283121 0.999599i \(-0.490987\pi\)
0.0283121 + 0.999599i \(0.490987\pi\)
\(578\) 5.84904e17 0.652460
\(579\) −1.09977e18 −1.21308
\(580\) −5.42549e17 −0.591776
\(581\) −2.23805e17 −0.241393
\(582\) 4.48643e18 4.78522
\(583\) 4.18706e16 0.0441636
\(584\) −1.61067e18 −1.68005
\(585\) 7.46636e17 0.770186
\(586\) −1.29217e17 −0.131821
\(587\) 1.04740e18 1.05674 0.528368 0.849016i \(-0.322804\pi\)
0.528368 + 0.849016i \(0.322804\pi\)
\(588\) −5.93046e17 −0.591747
\(589\) −6.65014e17 −0.656269
\(590\) −6.76016e17 −0.659810
\(591\) −1.18369e17 −0.114266
\(592\) −2.62147e16 −0.0250296
\(593\) 1.41371e18 1.33507 0.667534 0.744579i \(-0.267350\pi\)
0.667534 + 0.744579i \(0.267350\pi\)
\(594\) 3.47163e17 0.324281
\(595\) 2.69426e17 0.248931
\(596\) −3.86941e17 −0.353626
\(597\) 1.19432e18 1.07966
\(598\) −4.15570e18 −3.71610
\(599\) −2.05968e17 −0.182191 −0.0910954 0.995842i \(-0.529037\pi\)
−0.0910954 + 0.995842i \(0.529037\pi\)
\(600\) −1.53919e18 −1.34682
\(601\) −9.31064e17 −0.805927 −0.402963 0.915216i \(-0.632020\pi\)
−0.402963 + 0.915216i \(0.632020\pi\)
\(602\) 2.18179e18 1.86825
\(603\) 1.92204e18 1.62817
\(604\) −1.40412e18 −1.17669
\(605\) 3.33814e17 0.276755
\(606\) −5.18284e18 −4.25105
\(607\) 2.63045e17 0.213454 0.106727 0.994288i \(-0.465963\pi\)
0.106727 + 0.994288i \(0.465963\pi\)
\(608\) 7.46729e17 0.599501
\(609\) −2.90359e18 −2.30634
\(610\) −1.03920e18 −0.816683
\(611\) −3.42618e18 −2.66405
\(612\) −2.47422e18 −1.90350
\(613\) −1.16532e18 −0.887060 −0.443530 0.896259i \(-0.646274\pi\)
−0.443530 + 0.896259i \(0.646274\pi\)
\(614\) −2.58216e18 −1.94486
\(615\) −6.83241e17 −0.509197
\(616\) −3.04578e17 −0.224607
\(617\) −4.22470e17 −0.308278 −0.154139 0.988049i \(-0.549260\pi\)
−0.154139 + 0.988049i \(0.549260\pi\)
\(618\) −1.45036e18 −1.04725
\(619\) 2.34248e18 1.67373 0.836866 0.547407i \(-0.184385\pi\)
0.836866 + 0.547407i \(0.184385\pi\)
\(620\) 7.63874e17 0.540102
\(621\) 1.80450e18 1.26259
\(622\) 6.50698e17 0.450549
\(623\) 3.46988e17 0.237762
\(624\) −4.01367e17 −0.272171
\(625\) 1.12255e18 0.753332
\(626\) 2.16569e18 1.43835
\(627\) −2.98713e17 −0.196343
\(628\) 2.85049e18 1.85431
\(629\) −2.97141e17 −0.191308
\(630\) 1.27893e18 0.814954
\(631\) −1.03646e18 −0.653678 −0.326839 0.945080i \(-0.605983\pi\)
−0.326839 + 0.945080i \(0.605983\pi\)
\(632\) −5.29983e17 −0.330827
\(633\) −1.57953e18 −0.975895
\(634\) 2.38087e18 1.45598
\(635\) 5.82362e17 0.352504
\(636\) −8.43443e17 −0.505341
\(637\) −6.66520e17 −0.395282
\(638\) −7.79026e17 −0.457317
\(639\) −2.76478e18 −1.60659
\(640\) −7.76016e17 −0.446376
\(641\) 2.48877e18 1.41712 0.708561 0.705650i \(-0.249345\pi\)
0.708561 + 0.705650i \(0.249345\pi\)
\(642\) −8.35156e18 −4.70750
\(643\) 1.68962e18 0.942797 0.471399 0.881920i \(-0.343749\pi\)
0.471399 + 0.881920i \(0.343749\pi\)
\(644\) −4.35078e18 −2.40330
\(645\) −8.94363e17 −0.489075
\(646\) 1.26768e18 0.686277
\(647\) 3.24216e18 1.73763 0.868813 0.495141i \(-0.164884\pi\)
0.868813 + 0.495141i \(0.164884\pi\)
\(648\) 1.73596e17 0.0921090
\(649\) −5.93272e17 −0.311647
\(650\) −4.75404e18 −2.47244
\(651\) 4.08807e18 2.10495
\(652\) 3.54614e18 1.80778
\(653\) −2.68182e18 −1.35361 −0.676806 0.736161i \(-0.736637\pi\)
−0.676806 + 0.736161i \(0.736637\pi\)
\(654\) −4.07949e18 −2.03869
\(655\) 1.05252e17 0.0520788
\(656\) 2.24496e17 0.109985
\(657\) 5.93542e18 2.87923
\(658\) −5.86878e18 −2.81890
\(659\) 6.03146e17 0.286858 0.143429 0.989661i \(-0.454187\pi\)
0.143429 + 0.989661i \(0.454187\pi\)
\(660\) 3.43119e17 0.161588
\(661\) 2.93438e18 1.36838 0.684190 0.729304i \(-0.260156\pi\)
0.684190 + 0.729304i \(0.260156\pi\)
\(662\) 5.87081e18 2.71095
\(663\) −4.54946e18 −2.08028
\(664\) 4.40137e17 0.199295
\(665\) −4.00502e17 −0.179582
\(666\) −1.41049e18 −0.626308
\(667\) −4.04926e18 −1.78056
\(668\) −5.81043e18 −2.53024
\(669\) −2.91189e18 −1.25576
\(670\) 1.13116e18 0.483102
\(671\) −9.11999e17 −0.385743
\(672\) −4.59039e18 −1.92287
\(673\) 3.11819e18 1.29361 0.646807 0.762654i \(-0.276104\pi\)
0.646807 + 0.762654i \(0.276104\pi\)
\(674\) 4.48867e18 1.84429
\(675\) 2.06431e18 0.840041
\(676\) 7.16265e18 2.88682
\(677\) 2.72429e18 1.08749 0.543747 0.839249i \(-0.317005\pi\)
0.543747 + 0.839249i \(0.317005\pi\)
\(678\) 3.97483e18 1.57154
\(679\) 5.27895e18 2.06726
\(680\) −5.29854e17 −0.205517
\(681\) 7.53982e18 2.89671
\(682\) 1.09682e18 0.417384
\(683\) 3.95022e18 1.48897 0.744487 0.667637i \(-0.232694\pi\)
0.744487 + 0.667637i \(0.232694\pi\)
\(684\) 3.67793e18 1.37321
\(685\) 6.00405e17 0.222053
\(686\) 3.72267e18 1.36379
\(687\) 4.42498e18 1.60580
\(688\) 2.93866e17 0.105639
\(689\) −9.47939e17 −0.337563
\(690\) 2.91799e18 1.02936
\(691\) 3.49814e18 1.22245 0.611223 0.791458i \(-0.290678\pi\)
0.611223 + 0.791458i \(0.290678\pi\)
\(692\) −4.25577e18 −1.47329
\(693\) 1.12239e18 0.384926
\(694\) 6.90462e18 2.34587
\(695\) −9.07269e17 −0.305376
\(696\) 5.71023e18 1.90412
\(697\) 2.54465e18 0.840648
\(698\) −2.54702e18 −0.833626
\(699\) 4.49789e18 1.45850
\(700\) −4.97720e18 −1.59900
\(701\) −2.04271e18 −0.650190 −0.325095 0.945681i \(-0.605396\pi\)
−0.325095 + 0.945681i \(0.605396\pi\)
\(702\) −7.85966e18 −2.47864
\(703\) 4.41701e17 0.138013
\(704\) −1.15987e18 −0.359078
\(705\) 2.40575e18 0.737939
\(706\) −3.11822e18 −0.947711
\(707\) −6.09839e18 −1.83649
\(708\) 1.19509e19 3.56602
\(709\) 6.55708e17 0.193870 0.0969348 0.995291i \(-0.469096\pi\)
0.0969348 + 0.995291i \(0.469096\pi\)
\(710\) −1.62714e18 −0.476700
\(711\) 1.95302e18 0.566963
\(712\) −6.82389e17 −0.196296
\(713\) 5.70109e18 1.62508
\(714\) −7.79288e18 −2.20120
\(715\) 3.85629e17 0.107939
\(716\) −5.88194e17 −0.163149
\(717\) 9.58148e18 2.63365
\(718\) −1.41168e18 −0.384528
\(719\) 3.13690e18 0.846765 0.423382 0.905951i \(-0.360843\pi\)
0.423382 + 0.905951i \(0.360843\pi\)
\(720\) 1.72260e17 0.0460811
\(721\) −1.70656e18 −0.452422
\(722\) 4.21979e18 1.10866
\(723\) −2.91052e18 −0.757829
\(724\) −8.31963e18 −2.14686
\(725\) −4.63227e18 −1.18467
\(726\) −9.65525e18 −2.44723
\(727\) −3.16136e18 −0.794146 −0.397073 0.917787i \(-0.629974\pi\)
−0.397073 + 0.917787i \(0.629974\pi\)
\(728\) 6.89555e18 1.71678
\(729\) −6.60422e18 −1.62964
\(730\) 3.49314e18 0.854313
\(731\) 3.33094e18 0.807429
\(732\) 1.83713e19 4.41386
\(733\) −3.49551e18 −0.832405 −0.416202 0.909272i \(-0.636639\pi\)
−0.416202 + 0.909272i \(0.636639\pi\)
\(734\) 3.65664e18 0.863093
\(735\) 4.68008e17 0.109493
\(736\) −6.40162e18 −1.48451
\(737\) 9.92709e17 0.228183
\(738\) 1.20791e19 2.75213
\(739\) 6.65779e18 1.50363 0.751816 0.659374i \(-0.229178\pi\)
0.751816 + 0.659374i \(0.229178\pi\)
\(740\) −5.07363e17 −0.113583
\(741\) 6.76278e18 1.50075
\(742\) −1.62375e18 −0.357185
\(743\) 6.25592e18 1.36416 0.682078 0.731279i \(-0.261076\pi\)
0.682078 + 0.731279i \(0.261076\pi\)
\(744\) −8.03962e18 −1.73785
\(745\) 3.05358e17 0.0654325
\(746\) −8.42183e18 −1.78897
\(747\) −1.62193e18 −0.341546
\(748\) −1.27790e18 −0.266771
\(749\) −9.82685e18 −2.03368
\(750\) 6.98479e18 1.43303
\(751\) −2.91968e18 −0.593849 −0.296925 0.954901i \(-0.595961\pi\)
−0.296925 + 0.954901i \(0.595961\pi\)
\(752\) −7.90469e17 −0.159393
\(753\) −2.19721e18 −0.439242
\(754\) 1.76369e19 3.49550
\(755\) 1.10807e18 0.217727
\(756\) −8.22861e18 −1.60300
\(757\) −1.42277e17 −0.0274796 −0.0137398 0.999906i \(-0.504374\pi\)
−0.0137398 + 0.999906i \(0.504374\pi\)
\(758\) −3.87635e18 −0.742289
\(759\) 2.56083e18 0.486194
\(760\) 7.87629e17 0.148263
\(761\) −1.21558e16 −0.00226874 −0.00113437 0.999999i \(-0.500361\pi\)
−0.00113437 + 0.999999i \(0.500361\pi\)
\(762\) −1.68443e19 −3.11706
\(763\) −4.80013e18 −0.880732
\(764\) 6.69529e18 1.21804
\(765\) 1.95255e18 0.352211
\(766\) 1.16869e19 2.09031
\(767\) 1.34315e19 2.38207
\(768\) 7.58283e18 1.33347
\(769\) −2.25480e18 −0.393175 −0.196588 0.980486i \(-0.562986\pi\)
−0.196588 + 0.980486i \(0.562986\pi\)
\(770\) 6.60553e17 0.114214
\(771\) −9.80428e18 −1.68098
\(772\) 6.99374e18 1.18904
\(773\) −1.60398e17 −0.0270415 −0.0135208 0.999909i \(-0.504304\pi\)
−0.0135208 + 0.999909i \(0.504304\pi\)
\(774\) 1.58116e19 2.64338
\(775\) 6.52192e18 1.08122
\(776\) −1.03816e19 −1.70673
\(777\) −2.71528e18 −0.442668
\(778\) 1.89884e19 3.06988
\(779\) −3.78262e18 −0.606456
\(780\) −7.76812e18 −1.23510
\(781\) −1.42798e18 −0.225159
\(782\) −1.08677e19 −1.69939
\(783\) −7.65834e18 −1.18764
\(784\) −1.53776e17 −0.0236501
\(785\) −2.24949e18 −0.343108
\(786\) −3.04431e18 −0.460513
\(787\) 1.55309e17 0.0233002 0.0116501 0.999932i \(-0.496292\pi\)
0.0116501 + 0.999932i \(0.496292\pi\)
\(788\) 7.52743e17 0.112002
\(789\) 8.38889e18 1.23795
\(790\) 1.14940e18 0.168227
\(791\) 4.67698e18 0.678919
\(792\) −2.20730e18 −0.317795
\(793\) 2.06474e19 2.94842
\(794\) −2.70231e18 −0.382738
\(795\) 6.65611e17 0.0935048
\(796\) −7.59506e18 −1.05827
\(797\) 8.03018e18 1.10980 0.554902 0.831916i \(-0.312756\pi\)
0.554902 + 0.831916i \(0.312756\pi\)
\(798\) 1.15841e19 1.58798
\(799\) −8.95989e18 −1.21828
\(800\) −7.32331e18 −0.987695
\(801\) 2.51465e18 0.336408
\(802\) 1.74288e19 2.31278
\(803\) 3.06558e18 0.403516
\(804\) −1.99972e19 −2.61098
\(805\) 3.43345e18 0.444691
\(806\) −2.48316e19 −3.19027
\(807\) −1.79971e19 −2.29364
\(808\) 1.19931e19 1.51621
\(809\) −1.04069e19 −1.30513 −0.652567 0.757731i \(-0.726308\pi\)
−0.652567 + 0.757731i \(0.726308\pi\)
\(810\) −3.76486e17 −0.0468378
\(811\) 4.35231e18 0.537136 0.268568 0.963261i \(-0.413450\pi\)
0.268568 + 0.963261i \(0.413450\pi\)
\(812\) 1.84648e19 2.26064
\(813\) −1.08796e19 −1.32137
\(814\) −7.28503e17 −0.0877754
\(815\) −2.79846e18 −0.334499
\(816\) −1.04963e18 −0.124465
\(817\) −4.95145e18 −0.582491
\(818\) 2.55619e19 2.98329
\(819\) −2.54106e19 −2.94218
\(820\) 4.34494e18 0.499107
\(821\) −4.07100e18 −0.463949 −0.231975 0.972722i \(-0.574519\pi\)
−0.231975 + 0.972722i \(0.574519\pi\)
\(822\) −1.73661e19 −1.96353
\(823\) −1.12410e19 −1.26097 −0.630484 0.776202i \(-0.717144\pi\)
−0.630484 + 0.776202i \(0.717144\pi\)
\(824\) 3.35614e18 0.373519
\(825\) 2.92954e18 0.323481
\(826\) 2.30072e19 2.52054
\(827\) −5.36960e18 −0.583655 −0.291828 0.956471i \(-0.594263\pi\)
−0.291828 + 0.956471i \(0.594263\pi\)
\(828\) −3.15304e19 −3.40042
\(829\) −1.15117e19 −1.23179 −0.615893 0.787830i \(-0.711205\pi\)
−0.615893 + 0.787830i \(0.711205\pi\)
\(830\) −9.54546e17 −0.101342
\(831\) −1.95936e19 −2.06399
\(832\) 2.62592e19 2.74460
\(833\) −1.74303e18 −0.180765
\(834\) 2.62419e19 2.70032
\(835\) 4.58535e18 0.468178
\(836\) 1.89961e18 0.192452
\(837\) 1.07824e19 1.08393
\(838\) −1.36246e19 −1.35906
\(839\) −4.36647e18 −0.432193 −0.216097 0.976372i \(-0.569333\pi\)
−0.216097 + 0.976372i \(0.569333\pi\)
\(840\) −4.84182e18 −0.475547
\(841\) 6.92453e18 0.674864
\(842\) −1.61090e19 −1.55790
\(843\) 1.94566e19 1.86718
\(844\) 1.00447e19 0.956557
\(845\) −5.65246e18 −0.534157
\(846\) −4.25315e19 −3.98844
\(847\) −1.13608e19 −1.05723
\(848\) −2.18703e17 −0.0201968
\(849\) 3.17889e19 2.91323
\(850\) −1.24324e19 −1.13066
\(851\) −3.78665e18 −0.341753
\(852\) 2.87652e19 2.57638
\(853\) −6.93088e17 −0.0616056 −0.0308028 0.999525i \(-0.509806\pi\)
−0.0308028 + 0.999525i \(0.509806\pi\)
\(854\) 3.53674e19 3.11980
\(855\) −2.90247e18 −0.254090
\(856\) 1.93256e19 1.67901
\(857\) 9.79130e18 0.844239 0.422119 0.906540i \(-0.361286\pi\)
0.422119 + 0.906540i \(0.361286\pi\)
\(858\) −1.11539e19 −0.954467
\(859\) 8.47286e18 0.719572 0.359786 0.933035i \(-0.382850\pi\)
0.359786 + 0.933035i \(0.382850\pi\)
\(860\) 5.68752e18 0.479384
\(861\) 2.32530e19 1.94518
\(862\) 1.96298e19 1.62974
\(863\) 1.39686e18 0.115102 0.0575508 0.998343i \(-0.481671\pi\)
0.0575508 + 0.998343i \(0.481671\pi\)
\(864\) −1.21074e19 −0.990170
\(865\) 3.35847e18 0.272607
\(866\) −1.44631e19 −1.16518
\(867\) 8.16004e18 0.652481
\(868\) −2.59972e19 −2.06324
\(869\) 1.00871e18 0.0794584
\(870\) −1.23840e19 −0.968250
\(871\) −2.24746e19 −1.74411
\(872\) 9.43997e18 0.727132
\(873\) 3.82570e19 2.92495
\(874\) 1.61549e19 1.22597
\(875\) 8.21864e18 0.619081
\(876\) −6.17531e19 −4.61723
\(877\) 8.92989e18 0.662748 0.331374 0.943499i \(-0.392488\pi\)
0.331374 + 0.943499i \(0.392488\pi\)
\(878\) −2.37678e19 −1.75095
\(879\) −1.80271e18 −0.131825
\(880\) 8.89702e16 0.00645814
\(881\) 1.28545e19 0.926218 0.463109 0.886301i \(-0.346734\pi\)
0.463109 + 0.886301i \(0.346734\pi\)
\(882\) −8.27397e18 −0.591791
\(883\) −6.29105e18 −0.446661 −0.223331 0.974743i \(-0.571693\pi\)
−0.223331 + 0.974743i \(0.571693\pi\)
\(884\) 2.89314e19 2.03906
\(885\) −9.43115e18 −0.659832
\(886\) −1.01686e19 −0.706224
\(887\) 8.99934e18 0.620450 0.310225 0.950663i \(-0.399596\pi\)
0.310225 + 0.950663i \(0.399596\pi\)
\(888\) 5.33989e18 0.365467
\(889\) −1.98198e19 −1.34660
\(890\) 1.47993e18 0.0998175
\(891\) −3.30404e17 −0.0221228
\(892\) 1.85176e19 1.23087
\(893\) 1.33189e19 0.878889
\(894\) −8.83219e18 −0.578595
\(895\) 4.64178e17 0.0301880
\(896\) 2.64105e19 1.70520
\(897\) −5.79765e19 −3.71622
\(898\) −1.14799e19 −0.730539
\(899\) −2.41956e19 −1.52861
\(900\) −3.60701e19 −2.26241
\(901\) −2.47898e18 −0.154370
\(902\) 6.23873e18 0.385703
\(903\) 3.04383e19 1.86831
\(904\) −9.19777e18 −0.560516
\(905\) 6.56551e18 0.397240
\(906\) −3.20499e19 −1.92528
\(907\) 2.22298e19 1.32583 0.662915 0.748694i \(-0.269319\pi\)
0.662915 + 0.748694i \(0.269319\pi\)
\(908\) −4.79480e19 −2.83931
\(909\) −4.41955e19 −2.59844
\(910\) −1.49547e19 −0.872990
\(911\) 2.10175e18 0.121818 0.0609091 0.998143i \(-0.480600\pi\)
0.0609091 + 0.998143i \(0.480600\pi\)
\(912\) 1.56027e18 0.0897912
\(913\) −8.37711e17 −0.0478667
\(914\) −1.01422e19 −0.575418
\(915\) −1.44979e19 −0.816710
\(916\) −2.81398e19 −1.57398
\(917\) −3.58208e18 −0.198946
\(918\) −2.05540e19 −1.13349
\(919\) −2.10844e19 −1.15454 −0.577272 0.816552i \(-0.695883\pi\)
−0.577272 + 0.816552i \(0.695883\pi\)
\(920\) −6.75225e18 −0.367137
\(921\) −3.60239e19 −1.94493
\(922\) −2.29299e19 −1.22928
\(923\) 3.23290e19 1.72100
\(924\) −1.16775e19 −0.617281
\(925\) −4.33184e18 −0.227380
\(926\) 3.39767e19 1.77096
\(927\) −1.23676e19 −0.640129
\(928\) 2.71686e19 1.39639
\(929\) −2.72627e18 −0.139145 −0.0695723 0.997577i \(-0.522163\pi\)
−0.0695723 + 0.997577i \(0.522163\pi\)
\(930\) 1.74359e19 0.883702
\(931\) 2.59103e18 0.130406
\(932\) −2.86034e19 −1.42960
\(933\) 9.07793e18 0.450564
\(934\) 8.59890e18 0.423827
\(935\) 1.00847e18 0.0493614
\(936\) 4.99726e19 2.42906
\(937\) −3.00915e19 −1.45257 −0.726285 0.687394i \(-0.758755\pi\)
−0.726285 + 0.687394i \(0.758755\pi\)
\(938\) −3.84974e19 −1.84549
\(939\) 3.02137e19 1.43839
\(940\) −1.52989e19 −0.723316
\(941\) −1.54939e19 −0.727490 −0.363745 0.931499i \(-0.618502\pi\)
−0.363745 + 0.931499i \(0.618502\pi\)
\(942\) 6.50643e19 3.03398
\(943\) 3.24280e19 1.50174
\(944\) 3.09884e18 0.142522
\(945\) 6.49368e18 0.296609
\(946\) 8.16650e18 0.370462
\(947\) −7.49897e18 −0.337852 −0.168926 0.985629i \(-0.554030\pi\)
−0.168926 + 0.985629i \(0.554030\pi\)
\(948\) −2.03196e19 −0.909202
\(949\) −6.94038e19 −3.08427
\(950\) 1.84808e19 0.815676
\(951\) 3.32157e19 1.45603
\(952\) 1.80328e19 0.785095
\(953\) 6.28397e18 0.271725 0.135863 0.990728i \(-0.456619\pi\)
0.135863 + 0.990728i \(0.456619\pi\)
\(954\) −1.17674e19 −0.505379
\(955\) −5.28364e18 −0.225378
\(956\) −6.09315e19 −2.58146
\(957\) −1.08682e19 −0.457332
\(958\) 3.44275e19 1.43890
\(959\) −2.04338e19 −0.848261
\(960\) −1.84383e19 −0.760253
\(961\) 9.64822e18 0.395135
\(962\) 1.64931e19 0.670910
\(963\) −7.12160e19 −2.87744
\(964\) 1.85088e19 0.742811
\(965\) −5.51917e18 −0.220012
\(966\) −9.93094e19 −3.93223
\(967\) −4.43950e19 −1.74607 −0.873035 0.487657i \(-0.837852\pi\)
−0.873035 + 0.487657i \(0.837852\pi\)
\(968\) 2.23423e19 0.872847
\(969\) 1.76855e19 0.686300
\(970\) 2.25151e19 0.867877
\(971\) −2.37034e19 −0.907581 −0.453790 0.891108i \(-0.649929\pi\)
−0.453790 + 0.891108i \(0.649929\pi\)
\(972\) 4.45857e19 1.69576
\(973\) 3.08775e19 1.16656
\(974\) −1.72491e19 −0.647341
\(975\) −6.63239e19 −2.47252
\(976\) 4.76365e18 0.176407
\(977\) −2.43638e19 −0.896254 −0.448127 0.893970i \(-0.647909\pi\)
−0.448127 + 0.893970i \(0.647909\pi\)
\(978\) 8.09429e19 2.95785
\(979\) 1.29879e18 0.0471467
\(980\) −2.97620e18 −0.107323
\(981\) −3.47869e19 −1.24614
\(982\) 2.07648e19 0.738930
\(983\) −1.91129e19 −0.675662 −0.337831 0.941207i \(-0.609693\pi\)
−0.337831 + 0.941207i \(0.609693\pi\)
\(984\) −4.57296e19 −1.60594
\(985\) −5.94033e17 −0.0207241
\(986\) 4.61228e19 1.59851
\(987\) −8.18758e19 −2.81899
\(988\) −4.30065e19 −1.47101
\(989\) 4.24482e19 1.44239
\(990\) 4.78708e18 0.161600
\(991\) 2.98770e19 1.00198 0.500989 0.865453i \(-0.332970\pi\)
0.500989 + 0.865453i \(0.332970\pi\)
\(992\) −3.82516e19 −1.27445
\(993\) 8.19040e19 2.71104
\(994\) 5.53771e19 1.82104
\(995\) 5.99370e18 0.195815
\(996\) 1.68749e19 0.547715
\(997\) −2.20460e19 −0.710905 −0.355453 0.934694i \(-0.615673\pi\)
−0.355453 + 0.934694i \(0.615673\pi\)
\(998\) −9.59265e17 −0.0307320
\(999\) −7.16167e18 −0.227949
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 197.14.a.b.1.13 109
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
197.14.a.b.1.13 109 1.1 even 1 trivial