Properties

Label 75.22.a.e.1.1
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
Analytic rank $1$
Dimension $3$
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] = [75,22,Mod(1,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.1");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(209.608008215\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: \(\mathbb{Q}[x]/(x^{3} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 6395796x - 2792983104 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3\cdot 5\cdot 7 \)
Twist minimal: no (minimal twist has level 15)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2272.56\) of defining polynomial
Character \(\chi\) \(=\) 75.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2540.56 q^{2} -59049.0 q^{3} +4.35728e6 q^{4} +1.50017e8 q^{6} -4.55015e8 q^{7} -5.74199e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q-2540.56 q^{2} -59049.0 q^{3} +4.35728e6 q^{4} +1.50017e8 q^{6} -4.55015e8 q^{7} -5.74199e9 q^{8} +3.48678e9 q^{9} -1.12317e10 q^{11} -2.57293e11 q^{12} -8.05046e11 q^{13} +1.15599e12 q^{14} +5.44998e12 q^{16} +3.31190e11 q^{17} -8.85838e12 q^{18} +1.86365e13 q^{19} +2.68682e13 q^{21} +2.85348e13 q^{22} -4.27137e13 q^{23} +3.39059e14 q^{24} +2.04527e15 q^{26} -2.05891e14 q^{27} -1.98263e15 q^{28} +3.70518e15 q^{29} -7.15562e15 q^{31} -1.80416e15 q^{32} +6.63222e14 q^{33} -8.41408e14 q^{34} +1.51929e16 q^{36} +2.94893e16 q^{37} -4.73471e16 q^{38} +4.75372e16 q^{39} -4.25039e16 q^{41} -6.82602e16 q^{42} +1.81773e17 q^{43} -4.89398e16 q^{44} +1.08517e17 q^{46} -3.00481e17 q^{47} -3.21816e17 q^{48} -3.51507e17 q^{49} -1.95564e16 q^{51} -3.50781e18 q^{52} -2.29068e18 q^{53} +5.23078e17 q^{54} +2.61269e18 q^{56} -1.10047e18 q^{57} -9.41323e18 q^{58} -6.29616e18 q^{59} +7.85242e18 q^{61} +1.81793e19 q^{62} -1.58654e18 q^{63} -6.84587e18 q^{64} -1.68495e18 q^{66} +9.53526e18 q^{67} +1.44309e18 q^{68} +2.52220e18 q^{69} +1.28098e19 q^{71} -2.00211e19 q^{72} -7.89120e18 q^{73} -7.49193e19 q^{74} +8.12044e19 q^{76} +5.11060e18 q^{77} -1.20771e20 q^{78} -7.72058e19 q^{79} +1.21577e19 q^{81} +1.07984e20 q^{82} +2.36090e20 q^{83} +1.17072e20 q^{84} -4.61806e20 q^{86} -2.18787e20 q^{87} +6.44925e19 q^{88} +1.56566e20 q^{89} +3.66308e20 q^{91} -1.86116e20 q^{92} +4.22532e20 q^{93} +7.63389e20 q^{94} +1.06534e20 q^{96} +1.41907e21 q^{97} +8.93025e20 q^{98} -3.91626e19 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 803 q^{2} - 177147 q^{3} + 6715073 q^{4} + 47416347 q^{6} + 1577598316 q^{7} + 1424191017 q^{8} + 10460353203 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 803 q^{2} - 177147 q^{3} + 6715073 q^{4} + 47416347 q^{6} + 1577598316 q^{7} + 1424191017 q^{8} + 10460353203 q^{9} + 84497282000 q^{11} - 396518345577 q^{12} - 1065489966310 q^{13} + 1543881561348 q^{14} + 9712801855841 q^{16} - 13851876239906 q^{17} - 2799887874003 q^{18} + 26858848298644 q^{19} - 93155602961484 q^{21} + 93991312008688 q^{22} - 75776598293952 q^{23} - 84097055362833 q^{24} + 16\!\cdots\!86 q^{26}+ \cdots + 29\!\cdots\!00 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2540.56 −1.75434 −0.877171 0.480179i \(-0.840572\pi\)
−0.877171 + 0.480179i \(0.840572\pi\)
\(3\) −59049.0 −0.577350
\(4\) 4.35728e6 2.07771
\(5\) 0 0
\(6\) 1.50017e8 1.01287
\(7\) −4.55015e8 −0.608830 −0.304415 0.952539i \(-0.598461\pi\)
−0.304415 + 0.952539i \(0.598461\pi\)
\(8\) −5.74199e9 −1.89068
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) −1.12317e10 −0.130564 −0.0652820 0.997867i \(-0.520795\pi\)
−0.0652820 + 0.997867i \(0.520795\pi\)
\(12\) −2.57293e11 −1.19957
\(13\) −8.05046e11 −1.61963 −0.809814 0.586686i \(-0.800432\pi\)
−0.809814 + 0.586686i \(0.800432\pi\)
\(14\) 1.15599e12 1.06810
\(15\) 0 0
\(16\) 5.44998e12 1.23918
\(17\) 3.31190e11 0.0398441 0.0199220 0.999802i \(-0.493658\pi\)
0.0199220 + 0.999802i \(0.493658\pi\)
\(18\) −8.85838e12 −0.584781
\(19\) 1.86365e13 0.697351 0.348675 0.937244i \(-0.386632\pi\)
0.348675 + 0.937244i \(0.386632\pi\)
\(20\) 0 0
\(21\) 2.68682e13 0.351508
\(22\) 2.85348e13 0.229054
\(23\) −4.27137e13 −0.214993 −0.107497 0.994205i \(-0.534284\pi\)
−0.107497 + 0.994205i \(0.534284\pi\)
\(24\) 3.39059e14 1.09158
\(25\) 0 0
\(26\) 2.04527e15 2.84138
\(27\) −2.05891e14 −0.192450
\(28\) −1.98263e15 −1.26498
\(29\) 3.70518e15 1.63542 0.817712 0.575627i \(-0.195242\pi\)
0.817712 + 0.575627i \(0.195242\pi\)
\(30\) 0 0
\(31\) −7.15562e15 −1.56801 −0.784007 0.620752i \(-0.786827\pi\)
−0.784007 + 0.620752i \(0.786827\pi\)
\(32\) −1.80416e15 −0.283269
\(33\) 6.63222e14 0.0753811
\(34\) −8.41408e14 −0.0699001
\(35\) 0 0
\(36\) 1.51929e16 0.692571
\(37\) 2.94893e16 1.00820 0.504100 0.863645i \(-0.331824\pi\)
0.504100 + 0.863645i \(0.331824\pi\)
\(38\) −4.73471e16 −1.22339
\(39\) 4.75372e16 0.935093
\(40\) 0 0
\(41\) −4.25039e16 −0.494536 −0.247268 0.968947i \(-0.579533\pi\)
−0.247268 + 0.968947i \(0.579533\pi\)
\(42\) −6.82602e16 −0.616666
\(43\) 1.81773e17 1.28266 0.641330 0.767265i \(-0.278383\pi\)
0.641330 + 0.767265i \(0.278383\pi\)
\(44\) −4.89398e16 −0.271275
\(45\) 0 0
\(46\) 1.08517e17 0.377172
\(47\) −3.00481e17 −0.833276 −0.416638 0.909072i \(-0.636792\pi\)
−0.416638 + 0.909072i \(0.636792\pi\)
\(48\) −3.21816e17 −0.715442
\(49\) −3.51507e17 −0.629326
\(50\) 0 0
\(51\) −1.95564e16 −0.0230040
\(52\) −3.50781e18 −3.36512
\(53\) −2.29068e18 −1.79915 −0.899575 0.436766i \(-0.856124\pi\)
−0.899575 + 0.436766i \(0.856124\pi\)
\(54\) 5.23078e17 0.337623
\(55\) 0 0
\(56\) 2.61269e18 1.15110
\(57\) −1.10047e18 −0.402616
\(58\) −9.41323e18 −2.86909
\(59\) −6.29616e18 −1.60372 −0.801862 0.597509i \(-0.796157\pi\)
−0.801862 + 0.597509i \(0.796157\pi\)
\(60\) 0 0
\(61\) 7.85242e18 1.40942 0.704710 0.709496i \(-0.251077\pi\)
0.704710 + 0.709496i \(0.251077\pi\)
\(62\) 1.81793e19 2.75083
\(63\) −1.58654e18 −0.202943
\(64\) −6.84587e18 −0.742230
\(65\) 0 0
\(66\) −1.68495e18 −0.132244
\(67\) 9.53526e18 0.639068 0.319534 0.947575i \(-0.396474\pi\)
0.319534 + 0.947575i \(0.396474\pi\)
\(68\) 1.44309e18 0.0827846
\(69\) 2.52220e18 0.124127
\(70\) 0 0
\(71\) 1.28098e19 0.467015 0.233507 0.972355i \(-0.424980\pi\)
0.233507 + 0.972355i \(0.424980\pi\)
\(72\) −2.00211e19 −0.630226
\(73\) −7.89120e18 −0.214908 −0.107454 0.994210i \(-0.534270\pi\)
−0.107454 + 0.994210i \(0.534270\pi\)
\(74\) −7.49193e19 −1.76873
\(75\) 0 0
\(76\) 8.12044e19 1.44890
\(77\) 5.11060e18 0.0794913
\(78\) −1.20771e20 −1.64047
\(79\) −7.72058e19 −0.917414 −0.458707 0.888587i \(-0.651687\pi\)
−0.458707 + 0.888587i \(0.651687\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) 1.07984e20 0.867586
\(83\) 2.36090e20 1.67016 0.835081 0.550127i \(-0.185421\pi\)
0.835081 + 0.550127i \(0.185421\pi\)
\(84\) 1.17072e20 0.730334
\(85\) 0 0
\(86\) −4.61806e20 −2.25022
\(87\) −2.18787e20 −0.944213
\(88\) 6.44925e19 0.246854
\(89\) 1.56566e20 0.532234 0.266117 0.963941i \(-0.414259\pi\)
0.266117 + 0.963941i \(0.414259\pi\)
\(90\) 0 0
\(91\) 3.66308e20 0.986079
\(92\) −1.86116e20 −0.446695
\(93\) 4.22532e20 0.905293
\(94\) 7.63389e20 1.46185
\(95\) 0 0
\(96\) 1.06534e20 0.163546
\(97\) 1.41907e21 1.95389 0.976946 0.213488i \(-0.0684824\pi\)
0.976946 + 0.213488i \(0.0684824\pi\)
\(98\) 8.93025e20 1.10405
\(99\) −3.91626e19 −0.0435213
\(100\) 0 0
\(101\) −1.38972e21 −1.25185 −0.625925 0.779883i \(-0.715278\pi\)
−0.625925 + 0.779883i \(0.715278\pi\)
\(102\) 4.96843e19 0.0403568
\(103\) 1.44366e21 1.05846 0.529228 0.848479i \(-0.322482\pi\)
0.529228 + 0.848479i \(0.322482\pi\)
\(104\) 4.62257e21 3.06220
\(105\) 0 0
\(106\) 5.81960e21 3.15632
\(107\) −7.63760e20 −0.375342 −0.187671 0.982232i \(-0.560094\pi\)
−0.187671 + 0.982232i \(0.560094\pi\)
\(108\) −8.97126e20 −0.399856
\(109\) 5.56666e20 0.225225 0.112612 0.993639i \(-0.464078\pi\)
0.112612 + 0.993639i \(0.464078\pi\)
\(110\) 0 0
\(111\) −1.74131e21 −0.582084
\(112\) −2.47982e21 −0.754451
\(113\) 2.01218e21 0.557625 0.278812 0.960346i \(-0.410059\pi\)
0.278812 + 0.960346i \(0.410059\pi\)
\(114\) 2.79580e21 0.706325
\(115\) 0 0
\(116\) 1.61445e22 3.39794
\(117\) −2.80702e21 −0.539876
\(118\) 1.59958e22 2.81348
\(119\) −1.50696e20 −0.0242583
\(120\) 0 0
\(121\) −7.27410e21 −0.982953
\(122\) −1.99495e22 −2.47260
\(123\) 2.50981e21 0.285521
\(124\) −3.11791e22 −3.25788
\(125\) 0 0
\(126\) 4.03069e21 0.356032
\(127\) 1.13891e22 0.925875 0.462937 0.886391i \(-0.346796\pi\)
0.462937 + 0.886391i \(0.346796\pi\)
\(128\) 2.11759e22 1.58540
\(129\) −1.07335e22 −0.740544
\(130\) 0 0
\(131\) 1.80367e22 1.05879 0.529394 0.848376i \(-0.322419\pi\)
0.529394 + 0.848376i \(0.322419\pi\)
\(132\) 2.88985e21 0.156620
\(133\) −8.47988e21 −0.424568
\(134\) −2.42249e22 −1.12114
\(135\) 0 0
\(136\) −1.90169e21 −0.0753323
\(137\) −1.76881e22 −0.648805 −0.324403 0.945919i \(-0.605163\pi\)
−0.324403 + 0.945919i \(0.605163\pi\)
\(138\) −6.40780e21 −0.217760
\(139\) 4.69444e22 1.47887 0.739433 0.673230i \(-0.235094\pi\)
0.739433 + 0.673230i \(0.235094\pi\)
\(140\) 0 0
\(141\) 1.77431e22 0.481092
\(142\) −3.25441e22 −0.819304
\(143\) 9.04205e21 0.211465
\(144\) 1.90029e22 0.413061
\(145\) 0 0
\(146\) 2.00480e22 0.377022
\(147\) 2.07562e22 0.363341
\(148\) 1.28493e23 2.09475
\(149\) −1.89144e22 −0.287300 −0.143650 0.989629i \(-0.545884\pi\)
−0.143650 + 0.989629i \(0.545884\pi\)
\(150\) 0 0
\(151\) 1.07037e23 1.41344 0.706721 0.707493i \(-0.250174\pi\)
0.706721 + 0.707493i \(0.250174\pi\)
\(152\) −1.07011e23 −1.31847
\(153\) 1.15479e21 0.0132814
\(154\) −1.29838e22 −0.139455
\(155\) 0 0
\(156\) 2.07133e23 1.94286
\(157\) −9.13669e22 −0.801387 −0.400694 0.916212i \(-0.631231\pi\)
−0.400694 + 0.916212i \(0.631231\pi\)
\(158\) 1.96146e23 1.60946
\(159\) 1.35262e23 1.03874
\(160\) 0 0
\(161\) 1.94354e22 0.130895
\(162\) −3.08873e22 −0.194927
\(163\) 1.52600e23 0.902786 0.451393 0.892325i \(-0.350927\pi\)
0.451393 + 0.892325i \(0.350927\pi\)
\(164\) −1.85201e23 −1.02751
\(165\) 0 0
\(166\) −5.99802e23 −2.93003
\(167\) −1.39482e23 −0.639726 −0.319863 0.947464i \(-0.603637\pi\)
−0.319863 + 0.947464i \(0.603637\pi\)
\(168\) −1.54277e23 −0.664589
\(169\) 4.01034e23 1.62320
\(170\) 0 0
\(171\) 6.49814e22 0.232450
\(172\) 7.92038e23 2.66500
\(173\) 1.32428e23 0.419272 0.209636 0.977779i \(-0.432772\pi\)
0.209636 + 0.977779i \(0.432772\pi\)
\(174\) 5.55842e23 1.65647
\(175\) 0 0
\(176\) −6.12127e22 −0.161792
\(177\) 3.71782e23 0.925910
\(178\) −3.97765e23 −0.933721
\(179\) 4.79594e23 1.06149 0.530746 0.847531i \(-0.321912\pi\)
0.530746 + 0.847531i \(0.321912\pi\)
\(180\) 0 0
\(181\) 2.92105e22 0.0575326 0.0287663 0.999586i \(-0.490842\pi\)
0.0287663 + 0.999586i \(0.490842\pi\)
\(182\) −9.30626e23 −1.72992
\(183\) −4.63677e23 −0.813728
\(184\) 2.45262e23 0.406483
\(185\) 0 0
\(186\) −1.07347e24 −1.58819
\(187\) −3.71984e21 −0.00520220
\(188\) −1.30928e24 −1.73131
\(189\) 9.36835e22 0.117169
\(190\) 0 0
\(191\) −8.53666e23 −0.955956 −0.477978 0.878372i \(-0.658630\pi\)
−0.477978 + 0.878372i \(0.658630\pi\)
\(192\) 4.04242e23 0.428527
\(193\) −4.47860e23 −0.449563 −0.224781 0.974409i \(-0.572167\pi\)
−0.224781 + 0.974409i \(0.572167\pi\)
\(194\) −3.60523e24 −3.42779
\(195\) 0 0
\(196\) −1.53162e24 −1.30756
\(197\) −9.31605e23 −0.753940 −0.376970 0.926226i \(-0.623034\pi\)
−0.376970 + 0.926226i \(0.623034\pi\)
\(198\) 9.94948e22 0.0763512
\(199\) −1.91931e24 −1.39698 −0.698488 0.715622i \(-0.746143\pi\)
−0.698488 + 0.715622i \(0.746143\pi\)
\(200\) 0 0
\(201\) −5.63048e23 −0.368966
\(202\) 3.53066e24 2.19617
\(203\) −1.68591e24 −0.995696
\(204\) −8.52130e22 −0.0477957
\(205\) 0 0
\(206\) −3.66769e24 −1.85689
\(207\) −1.48934e23 −0.0716645
\(208\) −4.38748e24 −2.00701
\(209\) −2.09320e23 −0.0910488
\(210\) 0 0
\(211\) 1.64026e24 0.645575 0.322788 0.946471i \(-0.395380\pi\)
0.322788 + 0.946471i \(0.395380\pi\)
\(212\) −9.98113e24 −3.73812
\(213\) −7.56408e23 −0.269631
\(214\) 1.94038e24 0.658478
\(215\) 0 0
\(216\) 1.18223e24 0.363861
\(217\) 3.25592e24 0.954654
\(218\) −1.41424e24 −0.395121
\(219\) 4.65967e23 0.124077
\(220\) 0 0
\(221\) −2.66623e23 −0.0645326
\(222\) 4.42391e24 1.02117
\(223\) −6.69527e23 −0.147424 −0.0737119 0.997280i \(-0.523485\pi\)
−0.0737119 + 0.997280i \(0.523485\pi\)
\(224\) 8.20919e23 0.172463
\(225\) 0 0
\(226\) −5.11205e24 −0.978265
\(227\) −4.72465e24 −0.863174 −0.431587 0.902071i \(-0.642046\pi\)
−0.431587 + 0.902071i \(0.642046\pi\)
\(228\) −4.79504e24 −0.836520
\(229\) 9.75963e24 1.62615 0.813076 0.582158i \(-0.197792\pi\)
0.813076 + 0.582158i \(0.197792\pi\)
\(230\) 0 0
\(231\) −3.01776e23 −0.0458943
\(232\) −2.12751e25 −3.09206
\(233\) 9.76904e24 1.35711 0.678555 0.734550i \(-0.262607\pi\)
0.678555 + 0.734550i \(0.262607\pi\)
\(234\) 7.13140e24 0.947127
\(235\) 0 0
\(236\) −2.74341e25 −3.33208
\(237\) 4.55893e24 0.529669
\(238\) 3.82853e23 0.0425573
\(239\) −7.90678e23 −0.0841050 −0.0420525 0.999115i \(-0.513390\pi\)
−0.0420525 + 0.999115i \(0.513390\pi\)
\(240\) 0 0
\(241\) −3.04969e24 −0.297219 −0.148610 0.988896i \(-0.547480\pi\)
−0.148610 + 0.988896i \(0.547480\pi\)
\(242\) 1.84803e25 1.72444
\(243\) −7.17898e23 −0.0641500
\(244\) 3.42152e25 2.92837
\(245\) 0 0
\(246\) −6.37632e24 −0.500901
\(247\) −1.50032e25 −1.12945
\(248\) 4.10875e25 2.96461
\(249\) −1.39409e25 −0.964268
\(250\) 0 0
\(251\) 1.91006e25 1.21471 0.607357 0.794429i \(-0.292230\pi\)
0.607357 + 0.794429i \(0.292230\pi\)
\(252\) −6.91300e24 −0.421658
\(253\) 4.79749e23 0.0280704
\(254\) −2.89348e25 −1.62430
\(255\) 0 0
\(256\) −3.94418e25 −2.03909
\(257\) −2.69227e25 −1.33605 −0.668023 0.744141i \(-0.732859\pi\)
−0.668023 + 0.744141i \(0.732859\pi\)
\(258\) 2.72692e25 1.29917
\(259\) −1.34181e25 −0.613822
\(260\) 0 0
\(261\) 1.29192e25 0.545141
\(262\) −4.58234e25 −1.85748
\(263\) −1.23729e25 −0.481879 −0.240939 0.970540i \(-0.577455\pi\)
−0.240939 + 0.970540i \(0.577455\pi\)
\(264\) −3.80822e24 −0.142521
\(265\) 0 0
\(266\) 2.15436e25 0.744838
\(267\) −9.24508e24 −0.307286
\(268\) 4.15478e25 1.32780
\(269\) 4.72661e25 1.45262 0.726308 0.687369i \(-0.241235\pi\)
0.726308 + 0.687369i \(0.241235\pi\)
\(270\) 0 0
\(271\) 4.40314e25 1.25194 0.625972 0.779846i \(-0.284703\pi\)
0.625972 + 0.779846i \(0.284703\pi\)
\(272\) 1.80498e24 0.0493740
\(273\) −2.16301e25 −0.569313
\(274\) 4.49376e25 1.13823
\(275\) 0 0
\(276\) 1.09900e25 0.257899
\(277\) 3.46903e25 0.783738 0.391869 0.920021i \(-0.371829\pi\)
0.391869 + 0.920021i \(0.371829\pi\)
\(278\) −1.19265e26 −2.59444
\(279\) −2.49501e25 −0.522671
\(280\) 0 0
\(281\) 3.44630e25 0.669788 0.334894 0.942256i \(-0.391300\pi\)
0.334894 + 0.942256i \(0.391300\pi\)
\(282\) −4.50773e25 −0.844000
\(283\) 5.31368e25 0.958601 0.479301 0.877651i \(-0.340890\pi\)
0.479301 + 0.877651i \(0.340890\pi\)
\(284\) 5.58160e25 0.970323
\(285\) 0 0
\(286\) −2.29719e25 −0.370982
\(287\) 1.93399e25 0.301089
\(288\) −6.29071e24 −0.0944232
\(289\) −6.89822e25 −0.998412
\(290\) 0 0
\(291\) −8.37946e25 −1.12808
\(292\) −3.43842e25 −0.446518
\(293\) −8.33915e25 −1.04475 −0.522374 0.852716i \(-0.674954\pi\)
−0.522374 + 0.852716i \(0.674954\pi\)
\(294\) −5.27322e25 −0.637425
\(295\) 0 0
\(296\) −1.69327e26 −1.90618
\(297\) 2.31251e24 0.0251270
\(298\) 4.80530e25 0.504022
\(299\) 3.43865e25 0.348209
\(300\) 0 0
\(301\) −8.27096e25 −0.780922
\(302\) −2.71935e26 −2.47966
\(303\) 8.20615e25 0.722756
\(304\) 1.01568e26 0.864144
\(305\) 0 0
\(306\) −2.93381e24 −0.0233000
\(307\) −1.20325e26 −0.923429 −0.461715 0.887029i \(-0.652766\pi\)
−0.461715 + 0.887029i \(0.652766\pi\)
\(308\) 2.22683e25 0.165160
\(309\) −8.52465e25 −0.611100
\(310\) 0 0
\(311\) 5.17519e25 0.346691 0.173345 0.984861i \(-0.444542\pi\)
0.173345 + 0.984861i \(0.444542\pi\)
\(312\) −2.72958e26 −1.76796
\(313\) 1.55997e26 0.977017 0.488508 0.872559i \(-0.337541\pi\)
0.488508 + 0.872559i \(0.337541\pi\)
\(314\) 2.32123e26 1.40591
\(315\) 0 0
\(316\) −3.36408e26 −1.90612
\(317\) 3.05641e26 1.67529 0.837643 0.546219i \(-0.183933\pi\)
0.837643 + 0.546219i \(0.183933\pi\)
\(318\) −3.43642e26 −1.82230
\(319\) −4.16156e25 −0.213527
\(320\) 0 0
\(321\) 4.50993e25 0.216704
\(322\) −4.93767e25 −0.229634
\(323\) 6.17222e24 0.0277853
\(324\) 5.29744e25 0.230857
\(325\) 0 0
\(326\) −3.87689e26 −1.58379
\(327\) −3.28706e25 −0.130034
\(328\) 2.44057e26 0.935009
\(329\) 1.36723e26 0.507324
\(330\) 0 0
\(331\) −4.13740e26 −1.44057 −0.720283 0.693680i \(-0.755988\pi\)
−0.720283 + 0.693680i \(0.755988\pi\)
\(332\) 1.02871e27 3.47012
\(333\) 1.02823e26 0.336067
\(334\) 3.54361e26 1.12230
\(335\) 0 0
\(336\) 1.46431e26 0.435583
\(337\) 5.50567e26 1.58743 0.793717 0.608287i \(-0.208143\pi\)
0.793717 + 0.608287i \(0.208143\pi\)
\(338\) −1.01885e27 −2.84764
\(339\) −1.18817e26 −0.321945
\(340\) 0 0
\(341\) 8.03700e25 0.204726
\(342\) −1.65089e26 −0.407797
\(343\) 4.14088e26 0.991983
\(344\) −1.04374e27 −2.42510
\(345\) 0 0
\(346\) −3.36441e26 −0.735547
\(347\) −5.09948e26 −1.08160 −0.540800 0.841151i \(-0.681878\pi\)
−0.540800 + 0.841151i \(0.681878\pi\)
\(348\) −9.53318e26 −1.96180
\(349\) 1.73631e26 0.346706 0.173353 0.984860i \(-0.444540\pi\)
0.173353 + 0.984860i \(0.444540\pi\)
\(350\) 0 0
\(351\) 1.65752e26 0.311698
\(352\) 2.02638e25 0.0369848
\(353\) 2.48553e26 0.440337 0.220168 0.975462i \(-0.429339\pi\)
0.220168 + 0.975462i \(0.429339\pi\)
\(354\) −9.44533e26 −1.62436
\(355\) 0 0
\(356\) 6.82203e26 1.10583
\(357\) 8.89847e24 0.0140055
\(358\) −1.21844e27 −1.86222
\(359\) −9.77934e26 −1.45150 −0.725751 0.687957i \(-0.758508\pi\)
−0.725751 + 0.687957i \(0.758508\pi\)
\(360\) 0 0
\(361\) −3.66891e26 −0.513702
\(362\) −7.42110e25 −0.100932
\(363\) 4.29528e26 0.567508
\(364\) 1.59611e27 2.04879
\(365\) 0 0
\(366\) 1.17800e27 1.42756
\(367\) −1.01737e27 −1.19808 −0.599040 0.800719i \(-0.704451\pi\)
−0.599040 + 0.800719i \(0.704451\pi\)
\(368\) −2.32789e26 −0.266416
\(369\) −1.48202e26 −0.164845
\(370\) 0 0
\(371\) 1.04229e27 1.09538
\(372\) 1.84109e27 1.88094
\(373\) −2.95914e26 −0.293916 −0.146958 0.989143i \(-0.546948\pi\)
−0.146958 + 0.989143i \(0.546948\pi\)
\(374\) 9.45046e24 0.00912643
\(375\) 0 0
\(376\) 1.72536e27 1.57546
\(377\) −2.98284e27 −2.64878
\(378\) −2.38008e26 −0.205555
\(379\) 1.34431e27 1.12924 0.564621 0.825350i \(-0.309022\pi\)
0.564621 + 0.825350i \(0.309022\pi\)
\(380\) 0 0
\(381\) −6.72518e26 −0.534554
\(382\) 2.16879e27 1.67707
\(383\) −1.85260e27 −1.39378 −0.696890 0.717178i \(-0.745434\pi\)
−0.696890 + 0.717178i \(0.745434\pi\)
\(384\) −1.25042e27 −0.915328
\(385\) 0 0
\(386\) 1.13781e27 0.788686
\(387\) 6.33805e26 0.427553
\(388\) 6.18329e27 4.05963
\(389\) −7.11190e26 −0.454480 −0.227240 0.973839i \(-0.572970\pi\)
−0.227240 + 0.973839i \(0.572970\pi\)
\(390\) 0 0
\(391\) −1.41464e25 −0.00856621
\(392\) 2.01835e27 1.18985
\(393\) −1.06505e27 −0.611292
\(394\) 2.36680e27 1.32267
\(395\) 0 0
\(396\) −1.70642e26 −0.0904248
\(397\) 1.18258e27 0.610279 0.305140 0.952308i \(-0.401297\pi\)
0.305140 + 0.952308i \(0.401297\pi\)
\(398\) 4.87613e27 2.45077
\(399\) 5.00728e26 0.245125
\(400\) 0 0
\(401\) 2.57138e27 1.19440 0.597200 0.802092i \(-0.296280\pi\)
0.597200 + 0.802092i \(0.296280\pi\)
\(402\) 1.43046e27 0.647293
\(403\) 5.76061e27 2.53960
\(404\) −6.05539e27 −2.60099
\(405\) 0 0
\(406\) 4.28316e27 1.74679
\(407\) −3.31216e26 −0.131634
\(408\) 1.12293e26 0.0434931
\(409\) 3.03164e27 1.14441 0.572206 0.820110i \(-0.306088\pi\)
0.572206 + 0.820110i \(0.306088\pi\)
\(410\) 0 0
\(411\) 1.04446e27 0.374588
\(412\) 6.29042e27 2.19917
\(413\) 2.86485e27 0.976396
\(414\) 3.78374e26 0.125724
\(415\) 0 0
\(416\) 1.45243e27 0.458791
\(417\) −2.77202e27 −0.853823
\(418\) 5.31789e26 0.159731
\(419\) −3.03379e27 −0.888666 −0.444333 0.895862i \(-0.646559\pi\)
−0.444333 + 0.895862i \(0.646559\pi\)
\(420\) 0 0
\(421\) 2.04419e27 0.569586 0.284793 0.958589i \(-0.408075\pi\)
0.284793 + 0.958589i \(0.408075\pi\)
\(422\) −4.16717e27 −1.13256
\(423\) −1.04771e27 −0.277759
\(424\) 1.31531e28 3.40162
\(425\) 0 0
\(426\) 1.92170e27 0.473025
\(427\) −3.57297e27 −0.858097
\(428\) −3.32792e27 −0.779854
\(429\) −5.33924e26 −0.122089
\(430\) 0 0
\(431\) −2.51063e27 −0.546729 −0.273365 0.961910i \(-0.588137\pi\)
−0.273365 + 0.961910i \(0.588137\pi\)
\(432\) −1.12210e27 −0.238481
\(433\) −1.97726e27 −0.410149 −0.205075 0.978746i \(-0.565744\pi\)
−0.205075 + 0.978746i \(0.565744\pi\)
\(434\) −8.27184e27 −1.67479
\(435\) 0 0
\(436\) 2.42555e27 0.467953
\(437\) −7.96034e26 −0.149926
\(438\) −1.18382e27 −0.217674
\(439\) −9.05989e27 −1.62647 −0.813234 0.581937i \(-0.802295\pi\)
−0.813234 + 0.581937i \(0.802295\pi\)
\(440\) 0 0
\(441\) −1.22563e27 −0.209775
\(442\) 6.77372e26 0.113212
\(443\) 1.58685e27 0.258998 0.129499 0.991580i \(-0.458663\pi\)
0.129499 + 0.991580i \(0.458663\pi\)
\(444\) −7.58740e27 −1.20940
\(445\) 0 0
\(446\) 1.70097e27 0.258632
\(447\) 1.11687e27 0.165873
\(448\) 3.11497e27 0.451892
\(449\) −1.13467e28 −1.60799 −0.803994 0.594638i \(-0.797296\pi\)
−0.803994 + 0.594638i \(0.797296\pi\)
\(450\) 0 0
\(451\) 4.77392e26 0.0645686
\(452\) 8.76762e27 1.15859
\(453\) −6.32045e27 −0.816051
\(454\) 1.20033e28 1.51430
\(455\) 0 0
\(456\) 6.31887e27 0.761217
\(457\) −1.26834e28 −1.49319 −0.746594 0.665280i \(-0.768312\pi\)
−0.746594 + 0.665280i \(0.768312\pi\)
\(458\) −2.47949e28 −2.85282
\(459\) −6.81891e25 −0.00766799
\(460\) 0 0
\(461\) 3.19162e27 0.342888 0.171444 0.985194i \(-0.445157\pi\)
0.171444 + 0.985194i \(0.445157\pi\)
\(462\) 7.66679e26 0.0805143
\(463\) −7.25637e26 −0.0744936 −0.0372468 0.999306i \(-0.511859\pi\)
−0.0372468 + 0.999306i \(0.511859\pi\)
\(464\) 2.01932e28 2.02659
\(465\) 0 0
\(466\) −2.48188e28 −2.38083
\(467\) 8.56374e27 0.803223 0.401611 0.915810i \(-0.368450\pi\)
0.401611 + 0.915810i \(0.368450\pi\)
\(468\) −1.22310e28 −1.12171
\(469\) −4.33869e27 −0.389084
\(470\) 0 0
\(471\) 5.39512e27 0.462681
\(472\) 3.61525e28 3.03213
\(473\) −2.04163e27 −0.167469
\(474\) −1.15822e28 −0.929221
\(475\) 0 0
\(476\) −6.56627e26 −0.0504018
\(477\) −7.98710e27 −0.599717
\(478\) 2.00876e27 0.147549
\(479\) 1.42126e28 1.02130 0.510648 0.859790i \(-0.329406\pi\)
0.510648 + 0.859790i \(0.329406\pi\)
\(480\) 0 0
\(481\) −2.37402e28 −1.63291
\(482\) 7.74791e27 0.521424
\(483\) −1.14764e27 −0.0755720
\(484\) −3.16953e28 −2.04230
\(485\) 0 0
\(486\) 1.82386e27 0.112541
\(487\) −1.18992e28 −0.718561 −0.359281 0.933230i \(-0.616978\pi\)
−0.359281 + 0.933230i \(0.616978\pi\)
\(488\) −4.50885e28 −2.66476
\(489\) −9.01088e27 −0.521224
\(490\) 0 0
\(491\) −1.56760e28 −0.868718 −0.434359 0.900740i \(-0.643025\pi\)
−0.434359 + 0.900740i \(0.643025\pi\)
\(492\) 1.09360e28 0.593230
\(493\) 1.22712e27 0.0651620
\(494\) 3.81166e28 1.98144
\(495\) 0 0
\(496\) −3.89980e28 −1.94305
\(497\) −5.82866e27 −0.284333
\(498\) 3.54177e28 1.69166
\(499\) 4.89071e27 0.228726 0.114363 0.993439i \(-0.463517\pi\)
0.114363 + 0.993439i \(0.463517\pi\)
\(500\) 0 0
\(501\) 8.23625e27 0.369346
\(502\) −4.85263e28 −2.13102
\(503\) −1.30763e27 −0.0562367 −0.0281183 0.999605i \(-0.508952\pi\)
−0.0281183 + 0.999605i \(0.508952\pi\)
\(504\) 9.10989e27 0.383701
\(505\) 0 0
\(506\) −1.21883e27 −0.0492450
\(507\) −2.36807e28 −0.937153
\(508\) 4.96257e28 1.92370
\(509\) −3.01844e28 −1.14616 −0.573080 0.819499i \(-0.694252\pi\)
−0.573080 + 0.819499i \(0.694252\pi\)
\(510\) 0 0
\(511\) 3.59061e27 0.130843
\(512\) 5.57951e28 1.99187
\(513\) −3.83709e27 −0.134205
\(514\) 6.83988e28 2.34388
\(515\) 0 0
\(516\) −4.67690e28 −1.53864
\(517\) 3.37492e27 0.108796
\(518\) 3.40894e28 1.07685
\(519\) −7.81975e27 −0.242067
\(520\) 0 0
\(521\) −2.89002e28 −0.859221 −0.429610 0.903014i \(-0.641349\pi\)
−0.429610 + 0.903014i \(0.641349\pi\)
\(522\) −3.28219e28 −0.956364
\(523\) 2.23003e28 0.636859 0.318429 0.947947i \(-0.396845\pi\)
0.318429 + 0.947947i \(0.396845\pi\)
\(524\) 7.85912e28 2.19986
\(525\) 0 0
\(526\) 3.14342e28 0.845380
\(527\) −2.36987e27 −0.0624760
\(528\) 3.61455e27 0.0934109
\(529\) −3.76471e28 −0.953778
\(530\) 0 0
\(531\) −2.19534e28 −0.534575
\(532\) −3.69492e28 −0.882131
\(533\) 3.42176e28 0.800965
\(534\) 2.34877e28 0.539084
\(535\) 0 0
\(536\) −5.47514e28 −1.20827
\(537\) −2.83195e28 −0.612852
\(538\) −1.20082e29 −2.54838
\(539\) 3.94803e27 0.0821672
\(540\) 0 0
\(541\) 4.31280e28 0.863352 0.431676 0.902029i \(-0.357922\pi\)
0.431676 + 0.902029i \(0.357922\pi\)
\(542\) −1.11864e29 −2.19634
\(543\) −1.72485e27 −0.0332165
\(544\) −5.97519e26 −0.0112866
\(545\) 0 0
\(546\) 5.49526e28 0.998769
\(547\) −1.23562e28 −0.220301 −0.110150 0.993915i \(-0.535133\pi\)
−0.110150 + 0.993915i \(0.535133\pi\)
\(548\) −7.70719e28 −1.34803
\(549\) 2.73797e28 0.469806
\(550\) 0 0
\(551\) 6.90516e28 1.14046
\(552\) −1.44825e28 −0.234683
\(553\) 3.51298e28 0.558550
\(554\) −8.81328e28 −1.37494
\(555\) 0 0
\(556\) 2.04550e29 3.07266
\(557\) −6.98412e28 −1.02951 −0.514757 0.857336i \(-0.672118\pi\)
−0.514757 + 0.857336i \(0.672118\pi\)
\(558\) 6.33872e28 0.916944
\(559\) −1.46336e29 −2.07743
\(560\) 0 0
\(561\) 2.19653e26 0.00300349
\(562\) −8.75553e28 −1.17504
\(563\) 1.18199e29 1.55695 0.778474 0.627676i \(-0.215994\pi\)
0.778474 + 0.627676i \(0.215994\pi\)
\(564\) 7.73116e28 0.999572
\(565\) 0 0
\(566\) −1.34997e29 −1.68171
\(567\) −5.53192e27 −0.0676478
\(568\) −7.35539e28 −0.882975
\(569\) −8.18956e28 −0.965121 −0.482560 0.875863i \(-0.660293\pi\)
−0.482560 + 0.875863i \(0.660293\pi\)
\(570\) 0 0
\(571\) −1.32446e29 −1.50439 −0.752193 0.658943i \(-0.771004\pi\)
−0.752193 + 0.658943i \(0.771004\pi\)
\(572\) 3.93988e28 0.439364
\(573\) 5.04081e28 0.551921
\(574\) −4.91341e28 −0.528212
\(575\) 0 0
\(576\) −2.38701e28 −0.247410
\(577\) −6.37145e28 −0.648474 −0.324237 0.945976i \(-0.605107\pi\)
−0.324237 + 0.945976i \(0.605107\pi\)
\(578\) 1.75253e29 1.75156
\(579\) 2.64457e28 0.259555
\(580\) 0 0
\(581\) −1.07425e29 −1.01685
\(582\) 2.12885e29 1.97904
\(583\) 2.57283e28 0.234904
\(584\) 4.53112e28 0.406322
\(585\) 0 0
\(586\) 2.11861e29 1.83285
\(587\) −8.14731e28 −0.692332 −0.346166 0.938173i \(-0.612517\pi\)
−0.346166 + 0.938173i \(0.612517\pi\)
\(588\) 9.04404e28 0.754919
\(589\) −1.33356e29 −1.09346
\(590\) 0 0
\(591\) 5.50104e28 0.435287
\(592\) 1.60716e29 1.24934
\(593\) −1.55191e29 −1.18520 −0.592601 0.805496i \(-0.701899\pi\)
−0.592601 + 0.805496i \(0.701899\pi\)
\(594\) −5.87507e27 −0.0440814
\(595\) 0 0
\(596\) −8.24152e28 −0.596927
\(597\) 1.13334e29 0.806544
\(598\) −8.73609e28 −0.610878
\(599\) 2.80190e27 0.0192518 0.00962590 0.999954i \(-0.496936\pi\)
0.00962590 + 0.999954i \(0.496936\pi\)
\(600\) 0 0
\(601\) −9.09945e28 −0.603716 −0.301858 0.953353i \(-0.597607\pi\)
−0.301858 + 0.953353i \(0.597607\pi\)
\(602\) 2.10129e29 1.37000
\(603\) 3.32474e28 0.213023
\(604\) 4.66392e29 2.93673
\(605\) 0 0
\(606\) −2.08482e29 −1.26796
\(607\) 2.38304e28 0.142446 0.0712231 0.997460i \(-0.477310\pi\)
0.0712231 + 0.997460i \(0.477310\pi\)
\(608\) −3.36232e28 −0.197538
\(609\) 9.95515e28 0.574865
\(610\) 0 0
\(611\) 2.41901e29 1.34960
\(612\) 5.03174e27 0.0275949
\(613\) −2.31419e29 −1.24757 −0.623784 0.781597i \(-0.714405\pi\)
−0.623784 + 0.781597i \(0.714405\pi\)
\(614\) 3.05693e29 1.62001
\(615\) 0 0
\(616\) −2.93450e28 −0.150292
\(617\) −3.03072e29 −1.52599 −0.762995 0.646404i \(-0.776272\pi\)
−0.762995 + 0.646404i \(0.776272\pi\)
\(618\) 2.16574e29 1.07208
\(619\) −5.21875e28 −0.253989 −0.126994 0.991903i \(-0.540533\pi\)
−0.126994 + 0.991903i \(0.540533\pi\)
\(620\) 0 0
\(621\) 8.79438e27 0.0413755
\(622\) −1.31479e29 −0.608214
\(623\) −7.12400e28 −0.324040
\(624\) 2.59076e29 1.15875
\(625\) 0 0
\(626\) −3.96321e29 −1.71402
\(627\) 1.23601e28 0.0525671
\(628\) −3.98111e29 −1.66505
\(629\) 9.76657e27 0.0401708
\(630\) 0 0
\(631\) −2.18444e29 −0.869025 −0.434513 0.900666i \(-0.643079\pi\)
−0.434513 + 0.900666i \(0.643079\pi\)
\(632\) 4.43315e29 1.73454
\(633\) −9.68557e28 −0.372723
\(634\) −7.76498e29 −2.93902
\(635\) 0 0
\(636\) 5.89376e29 2.15820
\(637\) 2.82979e29 1.01927
\(638\) 1.05727e29 0.374600
\(639\) 4.46651e28 0.155672
\(640\) 0 0
\(641\) 2.14377e29 0.723050 0.361525 0.932362i \(-0.382256\pi\)
0.361525 + 0.932362i \(0.382256\pi\)
\(642\) −1.14577e29 −0.380173
\(643\) 1.80517e29 0.589255 0.294627 0.955612i \(-0.404804\pi\)
0.294627 + 0.955612i \(0.404804\pi\)
\(644\) 8.46855e28 0.271961
\(645\) 0 0
\(646\) −1.56809e28 −0.0487449
\(647\) −4.97248e29 −1.52082 −0.760411 0.649442i \(-0.775003\pi\)
−0.760411 + 0.649442i \(0.775003\pi\)
\(648\) −6.98092e28 −0.210075
\(649\) 7.07167e28 0.209388
\(650\) 0 0
\(651\) −1.92259e29 −0.551170
\(652\) 6.64921e29 1.87573
\(653\) −4.69947e29 −1.30455 −0.652275 0.757983i \(-0.726185\pi\)
−0.652275 + 0.757983i \(0.726185\pi\)
\(654\) 8.35096e28 0.228123
\(655\) 0 0
\(656\) −2.31645e29 −0.612820
\(657\) −2.75149e28 −0.0716360
\(658\) −3.47353e29 −0.890019
\(659\) −4.34544e29 −1.09581 −0.547907 0.836539i \(-0.684575\pi\)
−0.547907 + 0.836539i \(0.684575\pi\)
\(660\) 0 0
\(661\) 4.23242e29 1.03389 0.516944 0.856019i \(-0.327070\pi\)
0.516944 + 0.856019i \(0.327070\pi\)
\(662\) 1.05113e30 2.52725
\(663\) 1.57438e28 0.0372579
\(664\) −1.35563e30 −3.15774
\(665\) 0 0
\(666\) −2.61227e29 −0.589575
\(667\) −1.58262e29 −0.351606
\(668\) −6.07761e29 −1.32917
\(669\) 3.95349e28 0.0851151
\(670\) 0 0
\(671\) −8.81962e28 −0.184019
\(672\) −4.84744e28 −0.0995716
\(673\) −4.15920e28 −0.0841108 −0.0420554 0.999115i \(-0.513391\pi\)
−0.0420554 + 0.999115i \(0.513391\pi\)
\(674\) −1.39875e30 −2.78490
\(675\) 0 0
\(676\) 1.74742e30 3.37254
\(677\) 1.60090e29 0.304218 0.152109 0.988364i \(-0.451394\pi\)
0.152109 + 0.988364i \(0.451394\pi\)
\(678\) 3.01861e29 0.564801
\(679\) −6.45698e29 −1.18959
\(680\) 0 0
\(681\) 2.78986e29 0.498354
\(682\) −2.04185e29 −0.359159
\(683\) −1.83629e29 −0.318071 −0.159035 0.987273i \(-0.550838\pi\)
−0.159035 + 0.987273i \(0.550838\pi\)
\(684\) 2.83142e29 0.482965
\(685\) 0 0
\(686\) −1.05201e30 −1.74028
\(687\) −5.76297e29 −0.938859
\(688\) 9.90661e29 1.58945
\(689\) 1.84410e30 2.91396
\(690\) 0 0
\(691\) 9.13171e29 1.39969 0.699846 0.714293i \(-0.253252\pi\)
0.699846 + 0.714293i \(0.253252\pi\)
\(692\) 5.77027e29 0.871128
\(693\) 1.78196e28 0.0264971
\(694\) 1.29555e30 1.89750
\(695\) 0 0
\(696\) 1.25627e30 1.78520
\(697\) −1.40769e28 −0.0197043
\(698\) −4.41121e29 −0.608241
\(699\) −5.76852e29 −0.783527
\(700\) 0 0
\(701\) −2.77685e29 −0.366027 −0.183014 0.983110i \(-0.558585\pi\)
−0.183014 + 0.983110i \(0.558585\pi\)
\(702\) −4.21102e29 −0.546824
\(703\) 5.49577e29 0.703069
\(704\) 7.68909e28 0.0969085
\(705\) 0 0
\(706\) −6.31463e29 −0.772501
\(707\) 6.32343e29 0.762164
\(708\) 1.61996e30 1.92378
\(709\) −6.85586e29 −0.802189 −0.401094 0.916037i \(-0.631370\pi\)
−0.401094 + 0.916037i \(0.631370\pi\)
\(710\) 0 0
\(711\) −2.69200e29 −0.305805
\(712\) −8.99002e29 −1.00628
\(713\) 3.05643e29 0.337113
\(714\) −2.26071e28 −0.0245705
\(715\) 0 0
\(716\) 2.08972e30 2.20548
\(717\) 4.66888e28 0.0485581
\(718\) 2.48450e30 2.54643
\(719\) 1.48014e28 0.0149503 0.00747517 0.999972i \(-0.497621\pi\)
0.00747517 + 0.999972i \(0.497621\pi\)
\(720\) 0 0
\(721\) −6.56885e29 −0.644420
\(722\) 9.32107e29 0.901209
\(723\) 1.80081e29 0.171600
\(724\) 1.27278e29 0.119536
\(725\) 0 0
\(726\) −1.09124e30 −0.995603
\(727\) 9.70389e29 0.872639 0.436320 0.899792i \(-0.356282\pi\)
0.436320 + 0.899792i \(0.356282\pi\)
\(728\) −2.10334e30 −1.86436
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) 6.02015e28 0.0511064
\(732\) −2.02037e30 −1.69070
\(733\) 2.15653e30 1.77895 0.889473 0.456988i \(-0.151072\pi\)
0.889473 + 0.456988i \(0.151072\pi\)
\(734\) 2.58469e30 2.10184
\(735\) 0 0
\(736\) 7.70623e28 0.0609011
\(737\) −1.07097e29 −0.0834393
\(738\) 3.76516e29 0.289195
\(739\) 6.89652e29 0.522232 0.261116 0.965307i \(-0.415910\pi\)
0.261116 + 0.965307i \(0.415910\pi\)
\(740\) 0 0
\(741\) 8.85925e29 0.652088
\(742\) −2.64801e30 −1.92167
\(743\) 1.50941e30 1.08000 0.540002 0.841664i \(-0.318424\pi\)
0.540002 + 0.841664i \(0.318424\pi\)
\(744\) −2.42618e30 −1.71162
\(745\) 0 0
\(746\) 7.51787e29 0.515629
\(747\) 8.23197e29 0.556721
\(748\) −1.62084e28 −0.0108087
\(749\) 3.47522e29 0.228520
\(750\) 0 0
\(751\) −2.21031e30 −1.41330 −0.706649 0.707565i \(-0.749794\pi\)
−0.706649 + 0.707565i \(0.749794\pi\)
\(752\) −1.63761e30 −1.03258
\(753\) −1.12787e30 −0.701315
\(754\) 7.57808e30 4.64686
\(755\) 0 0
\(756\) 4.08206e29 0.243445
\(757\) −1.21005e30 −0.711700 −0.355850 0.934543i \(-0.615809\pi\)
−0.355850 + 0.934543i \(0.615809\pi\)
\(758\) −3.41529e30 −1.98108
\(759\) −2.83287e28 −0.0162064
\(760\) 0 0
\(761\) 9.67440e29 0.538375 0.269188 0.963088i \(-0.413245\pi\)
0.269188 + 0.963088i \(0.413245\pi\)
\(762\) 1.70857e30 0.937790
\(763\) −2.53291e29 −0.137124
\(764\) −3.71966e30 −1.98620
\(765\) 0 0
\(766\) 4.70664e30 2.44517
\(767\) 5.06870e30 2.59744
\(768\) 2.32900e30 1.17727
\(769\) 9.63350e29 0.480350 0.240175 0.970730i \(-0.422795\pi\)
0.240175 + 0.970730i \(0.422795\pi\)
\(770\) 0 0
\(771\) 1.58976e30 0.771366
\(772\) −1.95145e30 −0.934063
\(773\) −1.68443e30 −0.795368 −0.397684 0.917522i \(-0.630186\pi\)
−0.397684 + 0.917522i \(0.630186\pi\)
\(774\) −1.61022e30 −0.750074
\(775\) 0 0
\(776\) −8.14828e30 −3.69418
\(777\) 7.92324e29 0.354391
\(778\) 1.80682e30 0.797313
\(779\) −7.92123e29 −0.344865
\(780\) 0 0
\(781\) −1.43876e29 −0.0609753
\(782\) 3.59397e28 0.0150281
\(783\) −7.62864e29 −0.314738
\(784\) −1.91571e30 −0.779849
\(785\) 0 0
\(786\) 2.70582e30 1.07242
\(787\) 2.94830e30 1.15302 0.576510 0.817090i \(-0.304414\pi\)
0.576510 + 0.817090i \(0.304414\pi\)
\(788\) −4.05927e30 −1.56647
\(789\) 7.30610e29 0.278213
\(790\) 0 0
\(791\) −9.15570e29 −0.339499
\(792\) 2.24871e29 0.0822848
\(793\) −6.32156e30 −2.28274
\(794\) −3.00440e30 −1.07064
\(795\) 0 0
\(796\) −8.36299e30 −2.90252
\(797\) −2.94698e30 −1.00940 −0.504702 0.863294i \(-0.668398\pi\)
−0.504702 + 0.863294i \(0.668398\pi\)
\(798\) −1.27213e30 −0.430032
\(799\) −9.95162e28 −0.0332011
\(800\) 0 0
\(801\) 5.45913e29 0.177411
\(802\) −6.53274e30 −2.09539
\(803\) 8.86317e28 0.0280593
\(804\) −2.45336e30 −0.766606
\(805\) 0 0
\(806\) −1.46352e31 −4.45532
\(807\) −2.79102e30 −0.838668
\(808\) 7.97975e30 2.36685
\(809\) −6.41034e30 −1.87682 −0.938408 0.345529i \(-0.887699\pi\)
−0.938408 + 0.345529i \(0.887699\pi\)
\(810\) 0 0
\(811\) 8.93724e29 0.254967 0.127484 0.991841i \(-0.459310\pi\)
0.127484 + 0.991841i \(0.459310\pi\)
\(812\) −7.34600e30 −2.06877
\(813\) −2.60001e30 −0.722810
\(814\) 8.41473e29 0.230932
\(815\) 0 0
\(816\) −1.06582e29 −0.0285061
\(817\) 3.38762e30 0.894464
\(818\) −7.70204e30 −2.00769
\(819\) 1.27724e30 0.328693
\(820\) 0 0
\(821\) −2.68313e29 −0.0673036 −0.0336518 0.999434i \(-0.510714\pi\)
−0.0336518 + 0.999434i \(0.510714\pi\)
\(822\) −2.65352e30 −0.657155
\(823\) −1.38875e30 −0.339568 −0.169784 0.985481i \(-0.554307\pi\)
−0.169784 + 0.985481i \(0.554307\pi\)
\(824\) −8.28946e30 −2.00120
\(825\) 0 0
\(826\) −7.27831e30 −1.71293
\(827\) 2.46558e30 0.572943 0.286471 0.958089i \(-0.407518\pi\)
0.286471 + 0.958089i \(0.407518\pi\)
\(828\) −6.48946e29 −0.148898
\(829\) 8.57512e30 1.94275 0.971376 0.237547i \(-0.0763434\pi\)
0.971376 + 0.237547i \(0.0763434\pi\)
\(830\) 0 0
\(831\) −2.04843e30 −0.452491
\(832\) 5.51124e30 1.20214
\(833\) −1.16416e29 −0.0250749
\(834\) 7.04248e30 1.49790
\(835\) 0 0
\(836\) −9.12066e29 −0.189173
\(837\) 1.47328e30 0.301764
\(838\) 7.70753e30 1.55902
\(839\) 3.24689e30 0.648585 0.324293 0.945957i \(-0.394874\pi\)
0.324293 + 0.945957i \(0.394874\pi\)
\(840\) 0 0
\(841\) 8.59553e30 1.67461
\(842\) −5.19339e30 −0.999249
\(843\) −2.03501e30 −0.386702
\(844\) 7.14708e30 1.34132
\(845\) 0 0
\(846\) 2.66177e30 0.487284
\(847\) 3.30982e30 0.598452
\(848\) −1.24841e31 −2.22947
\(849\) −3.13768e30 −0.553449
\(850\) 0 0
\(851\) −1.25960e30 −0.216756
\(852\) −3.29588e30 −0.560216
\(853\) −2.10639e30 −0.353650 −0.176825 0.984242i \(-0.556583\pi\)
−0.176825 + 0.984242i \(0.556583\pi\)
\(854\) 9.07733e30 1.50540
\(855\) 0 0
\(856\) 4.38551e30 0.709651
\(857\) −6.52123e30 −1.04239 −0.521196 0.853437i \(-0.674514\pi\)
−0.521196 + 0.853437i \(0.674514\pi\)
\(858\) 1.35647e30 0.214187
\(859\) 5.13120e30 0.800369 0.400185 0.916435i \(-0.368946\pi\)
0.400185 + 0.916435i \(0.368946\pi\)
\(860\) 0 0
\(861\) −1.14200e30 −0.173834
\(862\) 6.37841e30 0.959150
\(863\) −8.76491e30 −1.30207 −0.651035 0.759048i \(-0.725665\pi\)
−0.651035 + 0.759048i \(0.725665\pi\)
\(864\) 3.71460e29 0.0545152
\(865\) 0 0
\(866\) 5.02335e30 0.719542
\(867\) 4.07333e30 0.576434
\(868\) 1.41869e31 1.98350
\(869\) 8.67154e29 0.119781
\(870\) 0 0
\(871\) −7.67632e30 −1.03505
\(872\) −3.19637e30 −0.425828
\(873\) 4.94799e30 0.651297
\(874\) 2.02237e30 0.263021
\(875\) 0 0
\(876\) 2.03035e30 0.257797
\(877\) 2.89130e30 0.362741 0.181371 0.983415i \(-0.441947\pi\)
0.181371 + 0.983415i \(0.441947\pi\)
\(878\) 2.30172e31 2.85338
\(879\) 4.92418e30 0.603186
\(880\) 0 0
\(881\) 6.24707e30 0.747187 0.373594 0.927593i \(-0.378125\pi\)
0.373594 + 0.927593i \(0.378125\pi\)
\(882\) 3.11378e30 0.368017
\(883\) −4.71833e30 −0.551063 −0.275531 0.961292i \(-0.588854\pi\)
−0.275531 + 0.961292i \(0.588854\pi\)
\(884\) −1.16175e30 −0.134080
\(885\) 0 0
\(886\) −4.03148e30 −0.454370
\(887\) 1.75208e31 1.95145 0.975724 0.219005i \(-0.0702813\pi\)
0.975724 + 0.219005i \(0.0702813\pi\)
\(888\) 9.99861e30 1.10053
\(889\) −5.18223e30 −0.563700
\(890\) 0 0
\(891\) −1.36552e29 −0.0145071
\(892\) −2.91732e30 −0.306304
\(893\) −5.59990e30 −0.581086
\(894\) −2.83748e30 −0.290997
\(895\) 0 0
\(896\) −9.63536e30 −0.965237
\(897\) −2.03049e30 −0.201039
\(898\) 2.88269e31 2.82096
\(899\) −2.65129e31 −2.56437
\(900\) 0 0
\(901\) −7.58650e29 −0.0716855
\(902\) −1.21284e30 −0.113275
\(903\) 4.88392e30 0.450866
\(904\) −1.15539e31 −1.05429
\(905\) 0 0
\(906\) 1.60575e31 1.43163
\(907\) −1.53542e31 −1.35316 −0.676582 0.736367i \(-0.736540\pi\)
−0.676582 + 0.736367i \(0.736540\pi\)
\(908\) −2.05866e31 −1.79343
\(909\) −4.84565e30 −0.417283
\(910\) 0 0
\(911\) 1.68310e31 1.41634 0.708168 0.706044i \(-0.249522\pi\)
0.708168 + 0.706044i \(0.249522\pi\)
\(912\) −5.99752e30 −0.498914
\(913\) −2.65170e30 −0.218063
\(914\) 3.22228e31 2.61956
\(915\) 0 0
\(916\) 4.25255e31 3.37868
\(917\) −8.20699e30 −0.644623
\(918\) 1.73238e29 0.0134523
\(919\) −2.16596e31 −1.66279 −0.831394 0.555683i \(-0.812457\pi\)
−0.831394 + 0.555683i \(0.812457\pi\)
\(920\) 0 0
\(921\) 7.10508e30 0.533142
\(922\) −8.10850e30 −0.601542
\(923\) −1.03125e31 −0.756391
\(924\) −1.31492e30 −0.0953553
\(925\) 0 0
\(926\) 1.84352e30 0.130687
\(927\) 5.03372e30 0.352819
\(928\) −6.68473e30 −0.463266
\(929\) 2.21057e31 1.51474 0.757372 0.652983i \(-0.226483\pi\)
0.757372 + 0.652983i \(0.226483\pi\)
\(930\) 0 0
\(931\) −6.55086e30 −0.438861
\(932\) 4.25665e31 2.81969
\(933\) −3.05590e30 −0.200162
\(934\) −2.17567e31 −1.40913
\(935\) 0 0
\(936\) 1.61179e31 1.02073
\(937\) −8.91926e30 −0.558551 −0.279276 0.960211i \(-0.590094\pi\)
−0.279276 + 0.960211i \(0.590094\pi\)
\(938\) 1.10227e31 0.682586
\(939\) −9.21149e30 −0.564081
\(940\) 0 0
\(941\) −1.27960e31 −0.766270 −0.383135 0.923692i \(-0.625156\pi\)
−0.383135 + 0.923692i \(0.625156\pi\)
\(942\) −1.37066e31 −0.811701
\(943\) 1.81550e30 0.106322
\(944\) −3.43139e31 −1.98731
\(945\) 0 0
\(946\) 5.18687e30 0.293798
\(947\) 1.21493e31 0.680578 0.340289 0.940321i \(-0.389475\pi\)
0.340289 + 0.940321i \(0.389475\pi\)
\(948\) 1.98645e31 1.10050
\(949\) 6.35278e30 0.348071
\(950\) 0 0
\(951\) −1.80478e31 −0.967226
\(952\) 8.65298e29 0.0458646
\(953\) −3.78850e30 −0.198606 −0.0993030 0.995057i \(-0.531661\pi\)
−0.0993030 + 0.995057i \(0.531661\pi\)
\(954\) 2.02917e31 1.05211
\(955\) 0 0
\(956\) −3.44521e30 −0.174746
\(957\) 2.45736e30 0.123280
\(958\) −3.61080e31 −1.79170
\(959\) 8.04834e30 0.395012
\(960\) 0 0
\(961\) 3.03775e31 1.45867
\(962\) 6.03135e31 2.86468
\(963\) −2.66307e30 −0.125114
\(964\) −1.32884e31 −0.617536
\(965\) 0 0
\(966\) 2.91565e30 0.132579
\(967\) −2.64473e31 −1.18961 −0.594804 0.803871i \(-0.702770\pi\)
−0.594804 + 0.803871i \(0.702770\pi\)
\(968\) 4.17678e31 1.85845
\(969\) −3.64463e29 −0.0160418
\(970\) 0 0
\(971\) 1.46560e31 0.631269 0.315634 0.948881i \(-0.397783\pi\)
0.315634 + 0.948881i \(0.397783\pi\)
\(972\) −3.12808e30 −0.133285
\(973\) −2.13604e31 −0.900378
\(974\) 3.02306e31 1.26060
\(975\) 0 0
\(976\) 4.27955e31 1.74653
\(977\) 2.70997e30 0.109414 0.0547068 0.998502i \(-0.482578\pi\)
0.0547068 + 0.998502i \(0.482578\pi\)
\(978\) 2.28927e31 0.914404
\(979\) −1.75851e30 −0.0694906
\(980\) 0 0
\(981\) 1.94097e30 0.0750749
\(982\) 3.98257e31 1.52403
\(983\) 4.42442e31 1.67511 0.837556 0.546351i \(-0.183983\pi\)
0.837556 + 0.546351i \(0.183983\pi\)
\(984\) −1.44113e31 −0.539828
\(985\) 0 0
\(986\) −3.11757e30 −0.114316
\(987\) −8.07337e30 −0.292904
\(988\) −6.53733e31 −2.34667
\(989\) −7.76422e30 −0.275763
\(990\) 0 0
\(991\) −1.26640e31 −0.440349 −0.220175 0.975460i \(-0.570663\pi\)
−0.220175 + 0.975460i \(0.570663\pi\)
\(992\) 1.29099e31 0.444170
\(993\) 2.44309e31 0.831712
\(994\) 1.48081e31 0.498817
\(995\) 0 0
\(996\) −6.07445e31 −2.00347
\(997\) −3.47593e31 −1.13441 −0.567207 0.823575i \(-0.691976\pi\)
−0.567207 + 0.823575i \(0.691976\pi\)
\(998\) −1.24251e31 −0.401264
\(999\) −6.07159e30 −0.194028
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 75.22.a.e.1.1 3
5.2 odd 4 75.22.b.e.49.1 6
5.3 odd 4 75.22.b.e.49.6 6
5.4 even 2 15.22.a.d.1.3 3
15.14 odd 2 45.22.a.b.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.22.a.d.1.3 3 5.4 even 2
45.22.a.b.1.1 3 15.14 odd 2
75.22.a.e.1.1 3 1.1 even 1 trivial
75.22.b.e.49.1 6 5.2 odd 4
75.22.b.e.49.6 6 5.3 odd 4