Properties

Label 75.22.b.d.49.4
Level $75$
Weight $22$
Character 75.49
Analytic conductor $209.608$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [75,22,Mod(49,75)]
 
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, 1]))
 
N = Newforms(chi, 22, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.49");
 
S:= CuspForms(chi, 22);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 22 \)
Character orbit: \([\chi]\) \(=\) 75.b (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(209.608008215\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{649})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 325x^{2} + 26244 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{4}\cdot 7^{2} \)
Twist minimal: no (minimal twist has level 3)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.4
Root \(12.2377i\) of defining polynomial
Character \(\chi\) \(=\) 75.49
Dual form 75.22.b.d.49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1937.96i q^{2} -59049.0i q^{3} -1.65852e6 q^{4} +1.14434e8 q^{6} +4.78205e7i q^{7} +8.50053e8i q^{8} -3.48678e9 q^{9} +O(q^{10})\) \(q+1937.96i q^{2} -59049.0i q^{3} -1.65852e6 q^{4} +1.14434e8 q^{6} +4.78205e7i q^{7} +8.50053e8i q^{8} -3.48678e9 q^{9} +1.60111e11 q^{11} +9.79338e10i q^{12} +7.86968e11i q^{13} -9.26741e10 q^{14} -5.12553e12 q^{16} -2.97477e12i q^{17} -6.75723e12i q^{18} +2.99456e13 q^{19} +2.82376e12 q^{21} +3.10287e14i q^{22} +1.91401e14i q^{23} +5.01948e13 q^{24} -1.52511e15 q^{26} +2.05891e14i q^{27} -7.93112e13i q^{28} -9.68170e14 q^{29} +2.80459e15 q^{31} -8.15035e15i q^{32} -9.45437e15i q^{33} +5.76496e15 q^{34} +5.78290e15 q^{36} +3.05038e16i q^{37} +5.80331e16i q^{38} +4.64697e16 q^{39} -2.22806e16 q^{41} +5.47231e15i q^{42} -1.63711e17i q^{43} -2.65546e17 q^{44} -3.70927e17 q^{46} +4.08678e17i q^{47} +3.02657e17i q^{48} +5.56259e17 q^{49} -1.75657e17 q^{51} -1.30520e18i q^{52} -4.34009e17i q^{53} -3.99008e17 q^{54} -4.06500e16 q^{56} -1.76826e18i q^{57} -1.87627e18i q^{58} +5.14341e18 q^{59} +1.98980e18 q^{61} +5.43517e18i q^{62} -1.66740e17i q^{63} +5.04601e18 q^{64} +1.83222e19 q^{66} -1.36361e19i q^{67} +4.93370e18i q^{68} +1.13020e19 q^{69} +7.35641e18 q^{71} -2.96395e18i q^{72} -6.81650e19i q^{73} -5.91149e19 q^{74} -4.96652e19 q^{76} +7.65658e18i q^{77} +9.00562e19i q^{78} +2.12282e19 q^{79} +1.21577e19 q^{81} -4.31789e19i q^{82} -1.10803e20i q^{83} -4.68325e18 q^{84} +3.17266e20 q^{86} +5.71695e19i q^{87} +1.36102e20i q^{88} +7.67205e18 q^{89} -3.76333e19 q^{91} -3.17442e20i q^{92} -1.65608e20i q^{93} -7.92000e20 q^{94} -4.81270e20 q^{96} +4.63755e20i q^{97} +1.07801e21i q^{98} -5.58271e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 2358472 q^{4} + 78653268 q^{6} - 13947137604 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 2358472 q^{4} + 78653268 q^{6} - 13947137604 q^{9} + 439738245936 q^{11} + 1422595413792 q^{14} - 16577472881120 q^{16} - 23921170023248 q^{19} + 80294371034976 q^{21} - 286630441537104 q^{24} - 49\!\cdots\!32 q^{26}+ \cdots - 15\!\cdots\!36 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1937.96i 1.33822i 0.743162 + 0.669112i \(0.233325\pi\)
−0.743162 + 0.669112i \(0.766675\pi\)
\(3\) − 59049.0i − 0.577350i
\(4\) −1.65852e6 −0.790843
\(5\) 0 0
\(6\) 1.14434e8 0.772624
\(7\) 4.78205e7i 0.0639860i 0.999488 + 0.0319930i \(0.0101854\pi\)
−0.999488 + 0.0319930i \(0.989815\pi\)
\(8\) 8.50053e8i 0.279899i
\(9\) −3.48678e9 −0.333333
\(10\) 0 0
\(11\) 1.60111e11 1.86122 0.930609 0.366016i \(-0.119278\pi\)
0.930609 + 0.366016i \(0.119278\pi\)
\(12\) 9.79338e10i 0.456593i
\(13\) 7.86968e11i 1.58326i 0.611001 + 0.791630i \(0.290767\pi\)
−0.611001 + 0.791630i \(0.709233\pi\)
\(14\) −9.26741e10 −0.0856276
\(15\) 0 0
\(16\) −5.12553e12 −1.16541
\(17\) − 2.97477e12i − 0.357881i −0.983860 0.178941i \(-0.942733\pi\)
0.983860 0.178941i \(-0.0572670\pi\)
\(18\) − 6.75723e12i − 0.446075i
\(19\) 2.99456e13 1.12052 0.560260 0.828317i \(-0.310701\pi\)
0.560260 + 0.828317i \(0.310701\pi\)
\(20\) 0 0
\(21\) 2.82376e12 0.0369423
\(22\) 3.10287e14i 2.49073i
\(23\) 1.91401e14i 0.963390i 0.876339 + 0.481695i \(0.159979\pi\)
−0.876339 + 0.481695i \(0.840021\pi\)
\(24\) 5.01948e13 0.161600
\(25\) 0 0
\(26\) −1.52511e15 −2.11876
\(27\) 2.05891e14i 0.192450i
\(28\) − 7.93112e13i − 0.0506029i
\(29\) −9.68170e14 −0.427339 −0.213670 0.976906i \(-0.568542\pi\)
−0.213670 + 0.976906i \(0.568542\pi\)
\(30\) 0 0
\(31\) 2.80459e15 0.614570 0.307285 0.951618i \(-0.400580\pi\)
0.307285 + 0.951618i \(0.400580\pi\)
\(32\) − 8.15035e15i − 1.27968i
\(33\) − 9.45437e15i − 1.07457i
\(34\) 5.76496e15 0.478925
\(35\) 0 0
\(36\) 5.78290e15 0.263614
\(37\) 3.05038e16i 1.04288i 0.853287 + 0.521441i \(0.174605\pi\)
−0.853287 + 0.521441i \(0.825395\pi\)
\(38\) 5.80331e16i 1.49951i
\(39\) 4.64697e16 0.914095
\(40\) 0 0
\(41\) −2.22806e16 −0.259237 −0.129618 0.991564i \(-0.541375\pi\)
−0.129618 + 0.991564i \(0.541375\pi\)
\(42\) 5.47231e15i 0.0494371i
\(43\) − 1.63711e17i − 1.15521i −0.816317 0.577604i \(-0.803988\pi\)
0.816317 0.577604i \(-0.196012\pi\)
\(44\) −2.65546e17 −1.47193
\(45\) 0 0
\(46\) −3.70927e17 −1.28923
\(47\) 4.08678e17i 1.13332i 0.823951 + 0.566662i \(0.191765\pi\)
−0.823951 + 0.566662i \(0.808235\pi\)
\(48\) 3.02657e17i 0.672850i
\(49\) 5.56259e17 0.995906
\(50\) 0 0
\(51\) −1.75657e17 −0.206623
\(52\) − 1.30520e18i − 1.25211i
\(53\) − 4.34009e17i − 0.340881i −0.985368 0.170440i \(-0.945481\pi\)
0.985368 0.170440i \(-0.0545190\pi\)
\(54\) −3.99008e17 −0.257541
\(55\) 0 0
\(56\) −4.06500e16 −0.0179096
\(57\) − 1.76826e18i − 0.646932i
\(58\) − 1.87627e18i − 0.571875i
\(59\) 5.14341e18 1.31010 0.655051 0.755584i \(-0.272647\pi\)
0.655051 + 0.755584i \(0.272647\pi\)
\(60\) 0 0
\(61\) 1.98980e18 0.357147 0.178573 0.983927i \(-0.442852\pi\)
0.178573 + 0.983927i \(0.442852\pi\)
\(62\) 5.43517e18i 0.822432i
\(63\) − 1.66740e17i − 0.0213287i
\(64\) 5.04601e18 0.547089
\(65\) 0 0
\(66\) 1.83222e19 1.43802
\(67\) − 1.36361e19i − 0.913913i −0.889489 0.456956i \(-0.848940\pi\)
0.889489 0.456956i \(-0.151060\pi\)
\(68\) 4.93370e18i 0.283028i
\(69\) 1.13020e19 0.556213
\(70\) 0 0
\(71\) 7.35641e18 0.268196 0.134098 0.990968i \(-0.457186\pi\)
0.134098 + 0.990968i \(0.457186\pi\)
\(72\) − 2.96395e18i − 0.0932996i
\(73\) − 6.81650e19i − 1.85640i −0.372081 0.928200i \(-0.621356\pi\)
0.372081 0.928200i \(-0.378644\pi\)
\(74\) −5.91149e19 −1.39561
\(75\) 0 0
\(76\) −4.96652e19 −0.886155
\(77\) 7.65658e18i 0.119092i
\(78\) 9.00562e19i 1.22326i
\(79\) 2.12282e19 0.252249 0.126124 0.992014i \(-0.459746\pi\)
0.126124 + 0.992014i \(0.459746\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) − 4.31789e19i − 0.346917i
\(83\) − 1.10803e20i − 0.783849i −0.919997 0.391924i \(-0.871809\pi\)
0.919997 0.391924i \(-0.128191\pi\)
\(84\) −4.68325e18 −0.0292156
\(85\) 0 0
\(86\) 3.17266e20 1.54593
\(87\) 5.71695e19i 0.246724i
\(88\) 1.36102e20i 0.520952i
\(89\) 7.67205e18 0.0260805 0.0130403 0.999915i \(-0.495849\pi\)
0.0130403 + 0.999915i \(0.495849\pi\)
\(90\) 0 0
\(91\) −3.76333e19 −0.101306
\(92\) − 3.17442e20i − 0.761890i
\(93\) − 1.65608e20i − 0.354822i
\(94\) −7.92000e20 −1.51664
\(95\) 0 0
\(96\) −4.81270e20 −0.738824
\(97\) 4.63755e20i 0.638535i 0.947665 + 0.319268i \(0.103437\pi\)
−0.947665 + 0.319268i \(0.896563\pi\)
\(98\) 1.07801e21i 1.33274i
\(99\) −5.58271e20 −0.620406
\(100\) 0 0
\(101\) 1.75642e20 0.158217 0.0791086 0.996866i \(-0.474793\pi\)
0.0791086 + 0.996866i \(0.474793\pi\)
\(102\) − 3.40415e20i − 0.276508i
\(103\) 1.29018e21i 0.945933i 0.881080 + 0.472967i \(0.156817\pi\)
−0.881080 + 0.472967i \(0.843183\pi\)
\(104\) −6.68965e20 −0.443152
\(105\) 0 0
\(106\) 8.41091e20 0.456175
\(107\) 5.76913e19i 0.0283518i 0.999900 + 0.0141759i \(0.00451248\pi\)
−0.999900 + 0.0141759i \(0.995488\pi\)
\(108\) − 3.41474e20i − 0.152198i
\(109\) 5.74941e20 0.232619 0.116309 0.993213i \(-0.462894\pi\)
0.116309 + 0.993213i \(0.462894\pi\)
\(110\) 0 0
\(111\) 1.80122e21 0.602109
\(112\) − 2.45106e20i − 0.0745700i
\(113\) 4.56755e21i 1.26578i 0.774240 + 0.632892i \(0.218132\pi\)
−0.774240 + 0.632892i \(0.781868\pi\)
\(114\) 3.42680e21 0.865740
\(115\) 0 0
\(116\) 1.60573e21 0.337958
\(117\) − 2.74399e21i − 0.527753i
\(118\) 9.96770e21i 1.75321i
\(119\) 1.42255e20 0.0228994
\(120\) 0 0
\(121\) 1.82352e22 2.46413
\(122\) 3.85615e21i 0.477942i
\(123\) 1.31565e21i 0.149671i
\(124\) −4.65146e21 −0.486028
\(125\) 0 0
\(126\) 3.23134e20 0.0285425
\(127\) 1.84938e22i 1.50344i 0.659480 + 0.751722i \(0.270776\pi\)
−0.659480 + 0.751722i \(0.729224\pi\)
\(128\) − 7.31359e21i − 0.547553i
\(129\) −9.66700e21 −0.666960
\(130\) 0 0
\(131\) −1.25296e22 −0.735509 −0.367754 0.929923i \(-0.619873\pi\)
−0.367754 + 0.929923i \(0.619873\pi\)
\(132\) 1.56802e22i 0.849820i
\(133\) 1.43201e21i 0.0716976i
\(134\) 2.64261e22 1.22302
\(135\) 0 0
\(136\) 2.52871e21 0.100171
\(137\) 3.84075e22i 1.40880i 0.709801 + 0.704402i \(0.248785\pi\)
−0.709801 + 0.704402i \(0.751215\pi\)
\(138\) 2.19028e22i 0.744338i
\(139\) 9.58856e21 0.302063 0.151032 0.988529i \(-0.451740\pi\)
0.151032 + 0.988529i \(0.451740\pi\)
\(140\) 0 0
\(141\) 2.41320e22 0.654324
\(142\) 1.42564e22i 0.358907i
\(143\) 1.26002e23i 2.94679i
\(144\) 1.78716e22 0.388470
\(145\) 0 0
\(146\) 1.32101e23 2.48428
\(147\) − 3.28465e22i − 0.574986i
\(148\) − 5.05911e22i − 0.824757i
\(149\) −3.44428e22 −0.523170 −0.261585 0.965180i \(-0.584245\pi\)
−0.261585 + 0.965180i \(0.584245\pi\)
\(150\) 0 0
\(151\) −3.63152e22 −0.479546 −0.239773 0.970829i \(-0.577073\pi\)
−0.239773 + 0.970829i \(0.577073\pi\)
\(152\) 2.54553e22i 0.313632i
\(153\) 1.03724e22i 0.119294i
\(154\) −1.48381e22 −0.159372
\(155\) 0 0
\(156\) −7.70708e22 −0.722906
\(157\) − 9.41986e22i − 0.826225i −0.910680 0.413112i \(-0.864442\pi\)
0.910680 0.413112i \(-0.135558\pi\)
\(158\) 4.11393e22i 0.337565i
\(159\) −2.56278e22 −0.196808
\(160\) 0 0
\(161\) −9.15290e21 −0.0616435
\(162\) 2.35610e22i 0.148692i
\(163\) 1.80771e23i 1.06945i 0.845027 + 0.534723i \(0.179584\pi\)
−0.845027 + 0.534723i \(0.820416\pi\)
\(164\) 3.69528e22 0.205016
\(165\) 0 0
\(166\) 2.14732e23 1.04897
\(167\) 3.71934e22i 0.170586i 0.996356 + 0.0852929i \(0.0271826\pi\)
−0.996356 + 0.0852929i \(0.972817\pi\)
\(168\) 2.40034e21i 0.0103401i
\(169\) −3.72255e23 −1.50671
\(170\) 0 0
\(171\) −1.04414e23 −0.373507
\(172\) 2.71518e23i 0.913589i
\(173\) − 2.58398e23i − 0.818096i −0.912513 0.409048i \(-0.865861\pi\)
0.912513 0.409048i \(-0.134139\pi\)
\(174\) −1.10792e23 −0.330172
\(175\) 0 0
\(176\) −8.20652e23 −2.16908
\(177\) − 3.03713e23i − 0.756388i
\(178\) 1.48681e22i 0.0349016i
\(179\) 3.35184e23 0.741868 0.370934 0.928659i \(-0.379038\pi\)
0.370934 + 0.928659i \(0.379038\pi\)
\(180\) 0 0
\(181\) −1.92267e23 −0.378687 −0.189343 0.981911i \(-0.560636\pi\)
−0.189343 + 0.981911i \(0.560636\pi\)
\(182\) − 7.29316e22i − 0.135571i
\(183\) − 1.17496e23i − 0.206199i
\(184\) −1.62701e23 −0.269652
\(185\) 0 0
\(186\) 3.20941e23 0.474831
\(187\) − 4.76292e23i − 0.666095i
\(188\) − 6.77800e23i − 0.896281i
\(189\) −9.84583e21 −0.0123141
\(190\) 0 0
\(191\) −4.60821e23 −0.516038 −0.258019 0.966140i \(-0.583070\pi\)
−0.258019 + 0.966140i \(0.583070\pi\)
\(192\) − 2.97962e23i − 0.315862i
\(193\) − 1.29600e24i − 1.30093i −0.759536 0.650465i \(-0.774574\pi\)
0.759536 0.650465i \(-0.225426\pi\)
\(194\) −8.98736e23 −0.854503
\(195\) 0 0
\(196\) −9.22566e23 −0.787605
\(197\) 2.05711e24i 1.66480i 0.554172 + 0.832402i \(0.313035\pi\)
−0.554172 + 0.832402i \(0.686965\pi\)
\(198\) − 1.08190e24i − 0.830242i
\(199\) −5.34557e22 −0.0389078 −0.0194539 0.999811i \(-0.506193\pi\)
−0.0194539 + 0.999811i \(0.506193\pi\)
\(200\) 0 0
\(201\) −8.05198e23 −0.527648
\(202\) 3.40386e23i 0.211730i
\(203\) − 4.62984e22i − 0.0273437i
\(204\) 2.91330e23 0.163406
\(205\) 0 0
\(206\) −2.50032e24 −1.26587
\(207\) − 6.67374e23i − 0.321130i
\(208\) − 4.03363e24i − 1.84515i
\(209\) 4.79460e24 2.08553
\(210\) 0 0
\(211\) −3.16142e24 −1.24427 −0.622137 0.782908i \(-0.713735\pi\)
−0.622137 + 0.782908i \(0.713735\pi\)
\(212\) 7.19813e23i 0.269583i
\(213\) − 4.34388e23i − 0.154843i
\(214\) −1.11803e23 −0.0379410
\(215\) 0 0
\(216\) −1.75018e23 −0.0538665
\(217\) 1.34117e23i 0.0393239i
\(218\) 1.11421e24i 0.311296i
\(219\) −4.02508e24 −1.07179
\(220\) 0 0
\(221\) 2.34105e24 0.566619
\(222\) 3.49068e24i 0.805756i
\(223\) 3.10623e24i 0.683963i 0.939707 + 0.341981i \(0.111098\pi\)
−0.939707 + 0.341981i \(0.888902\pi\)
\(224\) 3.89754e23 0.0818817
\(225\) 0 0
\(226\) −8.85171e24 −1.69390
\(227\) − 2.04882e24i − 0.374310i −0.982330 0.187155i \(-0.940073\pi\)
0.982330 0.187155i \(-0.0599267\pi\)
\(228\) 2.93268e24i 0.511622i
\(229\) −4.12622e24 −0.687512 −0.343756 0.939059i \(-0.611699\pi\)
−0.343756 + 0.939059i \(0.611699\pi\)
\(230\) 0 0
\(231\) 4.52113e23 0.0687577
\(232\) − 8.22995e23i − 0.119612i
\(233\) − 3.43270e24i − 0.476869i −0.971159 0.238434i \(-0.923366\pi\)
0.971159 0.238434i \(-0.0766342\pi\)
\(234\) 5.31773e24 0.706252
\(235\) 0 0
\(236\) −8.53044e24 −1.03609
\(237\) − 1.25351e24i − 0.145636i
\(238\) 2.75684e23i 0.0306445i
\(239\) −8.29149e24 −0.881972 −0.440986 0.897514i \(-0.645371\pi\)
−0.440986 + 0.897514i \(0.645371\pi\)
\(240\) 0 0
\(241\) −1.30526e25 −1.27209 −0.636045 0.771652i \(-0.719431\pi\)
−0.636045 + 0.771652i \(0.719431\pi\)
\(242\) 3.53389e25i 3.29756i
\(243\) − 7.17898e23i − 0.0641500i
\(244\) −3.30012e24 −0.282447
\(245\) 0 0
\(246\) −2.54967e24 −0.200293
\(247\) 2.35662e25i 1.77407i
\(248\) 2.38405e24i 0.172017i
\(249\) −6.54282e24 −0.452555
\(250\) 0 0
\(251\) −1.77630e25 −1.12965 −0.564823 0.825212i \(-0.691056\pi\)
−0.564823 + 0.825212i \(0.691056\pi\)
\(252\) 2.76541e23i 0.0168676i
\(253\) 3.06453e25i 1.79308i
\(254\) −3.58402e25 −2.01194
\(255\) 0 0
\(256\) 2.47557e25 1.27984
\(257\) − 5.70058e24i − 0.282893i −0.989946 0.141446i \(-0.954825\pi\)
0.989946 0.141446i \(-0.0451752\pi\)
\(258\) − 1.87342e25i − 0.892542i
\(259\) −1.45871e24 −0.0667299
\(260\) 0 0
\(261\) 3.37580e24 0.142446
\(262\) − 2.42818e25i − 0.984275i
\(263\) − 9.73572e23i − 0.0379169i −0.999820 0.0189584i \(-0.993965\pi\)
0.999820 0.0189584i \(-0.00603502\pi\)
\(264\) 8.03672e24 0.300772
\(265\) 0 0
\(266\) −2.77518e24 −0.0959474
\(267\) − 4.53027e23i − 0.0150576i
\(268\) 2.26157e25i 0.722761i
\(269\) 5.78587e25 1.77816 0.889078 0.457756i \(-0.151347\pi\)
0.889078 + 0.457756i \(0.151347\pi\)
\(270\) 0 0
\(271\) 4.29033e25 1.21987 0.609934 0.792452i \(-0.291196\pi\)
0.609934 + 0.792452i \(0.291196\pi\)
\(272\) 1.52473e25i 0.417079i
\(273\) 2.22221e24i 0.0584893i
\(274\) −7.44321e25 −1.88530
\(275\) 0 0
\(276\) −1.87446e25 −0.439877
\(277\) 6.39000e25i 1.44366i 0.692073 + 0.721828i \(0.256698\pi\)
−0.692073 + 0.721828i \(0.743302\pi\)
\(278\) 1.85822e25i 0.404228i
\(279\) −9.77899e24 −0.204857
\(280\) 0 0
\(281\) 3.23104e25 0.627951 0.313976 0.949431i \(-0.398339\pi\)
0.313976 + 0.949431i \(0.398339\pi\)
\(282\) 4.67668e25i 0.875632i
\(283\) 1.00627e25i 0.181534i 0.995872 + 0.0907668i \(0.0289318\pi\)
−0.995872 + 0.0907668i \(0.971068\pi\)
\(284\) −1.22007e25 −0.212101
\(285\) 0 0
\(286\) −2.44186e26 −3.94346
\(287\) − 1.06547e24i − 0.0165875i
\(288\) 2.84185e25i 0.426560i
\(289\) 6.02427e25 0.871921
\(290\) 0 0
\(291\) 2.73842e25 0.368659
\(292\) 1.13053e26i 1.46812i
\(293\) − 4.93011e25i − 0.617656i −0.951118 0.308828i \(-0.900063\pi\)
0.951118 0.308828i \(-0.0999367\pi\)
\(294\) 6.36551e25 0.769461
\(295\) 0 0
\(296\) −2.59298e25 −0.291902
\(297\) 3.29654e25i 0.358191i
\(298\) − 6.67487e25i − 0.700119i
\(299\) −1.50627e26 −1.52530
\(300\) 0 0
\(301\) 7.82877e24 0.0739172
\(302\) − 7.03772e25i − 0.641740i
\(303\) − 1.03715e25i − 0.0913467i
\(304\) −1.53487e26 −1.30587
\(305\) 0 0
\(306\) −2.01012e25 −0.159642
\(307\) − 1.49828e25i − 0.114985i −0.998346 0.0574925i \(-0.981689\pi\)
0.998346 0.0574925i \(-0.0183105\pi\)
\(308\) − 1.26986e25i − 0.0941830i
\(309\) 7.61840e25 0.546135
\(310\) 0 0
\(311\) −1.48794e26 −0.996786 −0.498393 0.866951i \(-0.666076\pi\)
−0.498393 + 0.866951i \(0.666076\pi\)
\(312\) 3.95017e25i 0.255854i
\(313\) 1.10961e26i 0.694951i 0.937689 + 0.347476i \(0.112961\pi\)
−0.937689 + 0.347476i \(0.887039\pi\)
\(314\) 1.82553e26 1.10567
\(315\) 0 0
\(316\) −3.52074e25 −0.199489
\(317\) 7.14805e25i 0.391801i 0.980624 + 0.195900i \(0.0627629\pi\)
−0.980624 + 0.195900i \(0.937237\pi\)
\(318\) − 4.96656e25i − 0.263373i
\(319\) −1.55014e26 −0.795371
\(320\) 0 0
\(321\) 3.40661e24 0.0163689
\(322\) − 1.77379e25i − 0.0824928i
\(323\) − 8.90810e25i − 0.401013i
\(324\) −2.01637e25 −0.0878715
\(325\) 0 0
\(326\) −3.50326e26 −1.43116
\(327\) − 3.39497e25i − 0.134302i
\(328\) − 1.89397e25i − 0.0725601i
\(329\) −1.95432e25 −0.0725168
\(330\) 0 0
\(331\) −5.05668e26 −1.76065 −0.880323 0.474375i \(-0.842674\pi\)
−0.880323 + 0.474375i \(0.842674\pi\)
\(332\) 1.83769e26i 0.619901i
\(333\) − 1.06360e26i − 0.347628i
\(334\) −7.20791e25 −0.228282
\(335\) 0 0
\(336\) −1.44732e25 −0.0430530
\(337\) 3.50546e26i 1.01072i 0.862909 + 0.505360i \(0.168640\pi\)
−0.862909 + 0.505360i \(0.831360\pi\)
\(338\) − 7.21413e26i − 2.01632i
\(339\) 2.69709e26 0.730801
\(340\) 0 0
\(341\) 4.49044e26 1.14385
\(342\) − 2.02349e26i − 0.499835i
\(343\) 5.33106e25i 0.127710i
\(344\) 1.39163e26 0.323341
\(345\) 0 0
\(346\) 5.00763e26 1.09480
\(347\) − 3.51664e26i − 0.745879i −0.927856 0.372940i \(-0.878350\pi\)
0.927856 0.372940i \(-0.121650\pi\)
\(348\) − 9.48166e25i − 0.195120i
\(349\) −5.74285e26 −1.14673 −0.573364 0.819300i \(-0.694362\pi\)
−0.573364 + 0.819300i \(0.694362\pi\)
\(350\) 0 0
\(351\) −1.62030e26 −0.304698
\(352\) − 1.30496e27i − 2.38176i
\(353\) − 1.85373e25i − 0.0328406i −0.999865 0.0164203i \(-0.994773\pi\)
0.999865 0.0164203i \(-0.00522698\pi\)
\(354\) 5.88583e26 1.01222
\(355\) 0 0
\(356\) −1.27242e25 −0.0206256
\(357\) − 8.40001e24i − 0.0132210i
\(358\) 6.49572e26i 0.992785i
\(359\) 3.41606e26 0.507030 0.253515 0.967331i \(-0.418413\pi\)
0.253515 + 0.967331i \(0.418413\pi\)
\(360\) 0 0
\(361\) 1.82527e26 0.255565
\(362\) − 3.72605e26i − 0.506768i
\(363\) − 1.07677e27i − 1.42267i
\(364\) 6.24154e25 0.0801175
\(365\) 0 0
\(366\) 2.27702e26 0.275940
\(367\) 5.79162e26i 0.682034i 0.940057 + 0.341017i \(0.110771\pi\)
−0.940057 + 0.341017i \(0.889229\pi\)
\(368\) − 9.81031e26i − 1.12274i
\(369\) 7.76878e25 0.0864123
\(370\) 0 0
\(371\) 2.07546e25 0.0218116
\(372\) 2.74664e26i 0.280609i
\(373\) 1.74513e27i 1.73335i 0.498877 + 0.866673i \(0.333746\pi\)
−0.498877 + 0.866673i \(0.666254\pi\)
\(374\) 9.23032e26 0.891384
\(375\) 0 0
\(376\) −3.47398e26 −0.317216
\(377\) − 7.61919e26i − 0.676588i
\(378\) − 1.90808e25i − 0.0164790i
\(379\) −1.08818e27 −0.914093 −0.457046 0.889443i \(-0.651093\pi\)
−0.457046 + 0.889443i \(0.651093\pi\)
\(380\) 0 0
\(381\) 1.09204e27 0.868014
\(382\) − 8.93050e26i − 0.690575i
\(383\) 2.35974e27i 1.77532i 0.460498 + 0.887661i \(0.347671\pi\)
−0.460498 + 0.887661i \(0.652329\pi\)
\(384\) −4.31860e26 −0.316130
\(385\) 0 0
\(386\) 2.51159e27 1.74094
\(387\) 5.70827e26i 0.385070i
\(388\) − 7.69145e26i − 0.504981i
\(389\) 2.72162e27 1.73923 0.869616 0.493729i \(-0.164367\pi\)
0.869616 + 0.493729i \(0.164367\pi\)
\(390\) 0 0
\(391\) 5.69373e26 0.344779
\(392\) 4.72850e26i 0.278753i
\(393\) 7.39859e26i 0.424646i
\(394\) −3.98660e27 −2.22788
\(395\) 0 0
\(396\) 9.25903e26 0.490644
\(397\) − 2.20086e27i − 1.13577i −0.823106 0.567887i \(-0.807761\pi\)
0.823106 0.567887i \(-0.192239\pi\)
\(398\) − 1.03595e26i − 0.0520673i
\(399\) 8.45589e25 0.0413946
\(400\) 0 0
\(401\) 2.61699e27 1.21559 0.607793 0.794096i \(-0.292055\pi\)
0.607793 + 0.794096i \(0.292055\pi\)
\(402\) − 1.56044e27i − 0.706111i
\(403\) 2.20712e27i 0.973023i
\(404\) −2.91305e26 −0.125125
\(405\) 0 0
\(406\) 8.97242e25 0.0365920
\(407\) 4.88398e27i 1.94103i
\(408\) − 1.49318e26i − 0.0578335i
\(409\) 8.92474e26 0.336900 0.168450 0.985710i \(-0.446124\pi\)
0.168450 + 0.985710i \(0.446124\pi\)
\(410\) 0 0
\(411\) 2.26793e27 0.813373
\(412\) − 2.13979e27i − 0.748085i
\(413\) 2.45961e26i 0.0838282i
\(414\) 1.29334e27 0.429744
\(415\) 0 0
\(416\) 6.41407e27 2.02607
\(417\) − 5.66195e26i − 0.174396i
\(418\) 9.29172e27i 2.79091i
\(419\) 2.48238e27 0.727144 0.363572 0.931566i \(-0.381557\pi\)
0.363572 + 0.931566i \(0.381557\pi\)
\(420\) 0 0
\(421\) 4.09797e27 1.14184 0.570922 0.821004i \(-0.306586\pi\)
0.570922 + 0.821004i \(0.306586\pi\)
\(422\) − 6.12669e27i − 1.66512i
\(423\) − 1.42497e27i − 0.377774i
\(424\) 3.68931e26 0.0954121
\(425\) 0 0
\(426\) 8.41825e26 0.207215
\(427\) 9.51534e25i 0.0228524i
\(428\) − 9.56820e25i − 0.0224218i
\(429\) 7.44029e27 1.70133
\(430\) 0 0
\(431\) −4.89793e27 −1.06660 −0.533299 0.845927i \(-0.679048\pi\)
−0.533299 + 0.845927i \(0.679048\pi\)
\(432\) − 1.05530e27i − 0.224283i
\(433\) 2.06118e27i 0.427557i 0.976882 + 0.213778i \(0.0685770\pi\)
−0.976882 + 0.213778i \(0.931423\pi\)
\(434\) −2.59913e26 −0.0526242
\(435\) 0 0
\(436\) −9.53549e26 −0.183965
\(437\) 5.73161e27i 1.07950i
\(438\) − 7.80042e27i − 1.43430i
\(439\) −1.00358e28 −1.80167 −0.900835 0.434161i \(-0.857045\pi\)
−0.900835 + 0.434161i \(0.857045\pi\)
\(440\) 0 0
\(441\) −1.93956e27 −0.331969
\(442\) 4.53684e27i 0.758263i
\(443\) − 4.23345e27i − 0.690964i −0.938425 0.345482i \(-0.887715\pi\)
0.938425 0.345482i \(-0.112285\pi\)
\(444\) −2.98735e27 −0.476173
\(445\) 0 0
\(446\) −6.01973e27 −0.915295
\(447\) 2.03382e27i 0.302053i
\(448\) 2.41303e26i 0.0350061i
\(449\) 6.74432e27 0.955766 0.477883 0.878424i \(-0.341404\pi\)
0.477883 + 0.878424i \(0.341404\pi\)
\(450\) 0 0
\(451\) −3.56737e27 −0.482496
\(452\) − 7.57537e27i − 1.00104i
\(453\) 2.14438e27i 0.276866i
\(454\) 3.97052e27 0.500911
\(455\) 0 0
\(456\) 1.50311e27 0.181076
\(457\) − 1.13821e28i − 1.34000i −0.742362 0.669998i \(-0.766295\pi\)
0.742362 0.669998i \(-0.233705\pi\)
\(458\) − 7.99644e27i − 0.920045i
\(459\) 6.12478e26 0.0688743
\(460\) 0 0
\(461\) 6.24175e27 0.670574 0.335287 0.942116i \(-0.391167\pi\)
0.335287 + 0.942116i \(0.391167\pi\)
\(462\) 8.76175e26i 0.0920132i
\(463\) 2.31855e27i 0.238021i 0.992893 + 0.119011i \(0.0379723\pi\)
−0.992893 + 0.119011i \(0.962028\pi\)
\(464\) 4.96238e27 0.498025
\(465\) 0 0
\(466\) 6.65242e27 0.638157
\(467\) − 9.47846e26i − 0.0889018i −0.999012 0.0444509i \(-0.985846\pi\)
0.999012 0.0444509i \(-0.0141538\pi\)
\(468\) 4.55096e27i 0.417370i
\(469\) 6.52086e26 0.0584776
\(470\) 0 0
\(471\) −5.56234e27 −0.477021
\(472\) 4.37217e27i 0.366696i
\(473\) − 2.62120e28i − 2.15009i
\(474\) 2.42924e27 0.194893
\(475\) 0 0
\(476\) −2.35932e26 −0.0181098
\(477\) 1.51330e27i 0.113627i
\(478\) − 1.60685e28i − 1.18028i
\(479\) 1.76996e28 1.27186 0.635932 0.771745i \(-0.280616\pi\)
0.635932 + 0.771745i \(0.280616\pi\)
\(480\) 0 0
\(481\) −2.40055e28 −1.65115
\(482\) − 2.52954e28i − 1.70234i
\(483\) 5.40470e26i 0.0355899i
\(484\) −3.02434e28 −1.94874
\(485\) 0 0
\(486\) 1.39125e27 0.0858471
\(487\) 1.99209e28i 1.20297i 0.798885 + 0.601484i \(0.205424\pi\)
−0.798885 + 0.601484i \(0.794576\pi\)
\(488\) 1.69144e27i 0.0999649i
\(489\) 1.06743e28 0.617445
\(490\) 0 0
\(491\) −5.71237e27 −0.316563 −0.158282 0.987394i \(-0.550595\pi\)
−0.158282 + 0.987394i \(0.550595\pi\)
\(492\) − 2.18203e27i − 0.118366i
\(493\) 2.88008e27i 0.152937i
\(494\) −4.56702e28 −2.37411
\(495\) 0 0
\(496\) −1.43750e28 −0.716226
\(497\) 3.51787e26i 0.0171608i
\(498\) − 1.26797e28i − 0.605620i
\(499\) 1.17032e28 0.547332 0.273666 0.961825i \(-0.411764\pi\)
0.273666 + 0.961825i \(0.411764\pi\)
\(500\) 0 0
\(501\) 2.19623e27 0.0984878
\(502\) − 3.44239e28i − 1.51172i
\(503\) 3.15184e28i 1.35550i 0.735292 + 0.677750i \(0.237045\pi\)
−0.735292 + 0.677750i \(0.762955\pi\)
\(504\) 1.41738e26 0.00596987
\(505\) 0 0
\(506\) −5.93893e28 −2.39954
\(507\) 2.19813e28i 0.869899i
\(508\) − 3.06723e28i − 1.18899i
\(509\) −4.58029e28 −1.73923 −0.869613 0.493733i \(-0.835632\pi\)
−0.869613 + 0.493733i \(0.835632\pi\)
\(510\) 0 0
\(511\) 3.25969e27 0.118784
\(512\) 3.26377e28i 1.16516i
\(513\) 6.16552e27i 0.215644i
\(514\) 1.10475e28 0.378574
\(515\) 0 0
\(516\) 1.60329e28 0.527461
\(517\) 6.54337e28i 2.10936i
\(518\) − 2.82691e27i − 0.0892996i
\(519\) −1.52581e28 −0.472328
\(520\) 0 0
\(521\) −5.19044e27 −0.154315 −0.0771575 0.997019i \(-0.524584\pi\)
−0.0771575 + 0.997019i \(0.524584\pi\)
\(522\) 6.54215e27i 0.190625i
\(523\) 2.08282e27i 0.0594817i 0.999558 + 0.0297408i \(0.00946820\pi\)
−0.999558 + 0.0297408i \(0.990532\pi\)
\(524\) 2.07805e28 0.581672
\(525\) 0 0
\(526\) 1.88674e27 0.0507413
\(527\) − 8.34299e27i − 0.219943i
\(528\) 4.84587e28i 1.25232i
\(529\) 2.83724e27 0.0718805
\(530\) 0 0
\(531\) −1.79340e28 −0.436701
\(532\) − 2.37502e27i − 0.0567016i
\(533\) − 1.75342e28i − 0.410439i
\(534\) 8.77946e26 0.0201504
\(535\) 0 0
\(536\) 1.15914e28 0.255803
\(537\) − 1.97923e28i − 0.428318i
\(538\) 1.12128e29i 2.37957i
\(539\) 8.90630e28 1.85360
\(540\) 0 0
\(541\) −1.25240e28 −0.250710 −0.125355 0.992112i \(-0.540007\pi\)
−0.125355 + 0.992112i \(0.540007\pi\)
\(542\) 8.31446e28i 1.63246i
\(543\) 1.13532e28i 0.218635i
\(544\) −2.42454e28 −0.457974
\(545\) 0 0
\(546\) −4.30654e27 −0.0782718
\(547\) 5.63104e28i 1.00397i 0.864875 + 0.501987i \(0.167397\pi\)
−0.864875 + 0.501987i \(0.832603\pi\)
\(548\) − 6.36996e28i − 1.11414i
\(549\) −6.93801e27 −0.119049
\(550\) 0 0
\(551\) −2.89924e28 −0.478842
\(552\) 9.60733e27i 0.155683i
\(553\) 1.01514e27i 0.0161404i
\(554\) −1.23835e29 −1.93193
\(555\) 0 0
\(556\) −1.59028e28 −0.238885
\(557\) 4.80914e28i 0.708904i 0.935074 + 0.354452i \(0.115333\pi\)
−0.935074 + 0.354452i \(0.884667\pi\)
\(558\) − 1.89513e28i − 0.274144i
\(559\) 1.28836e29 1.82899
\(560\) 0 0
\(561\) −2.81246e28 −0.384570
\(562\) 6.26161e28i 0.840339i
\(563\) 1.81812e28i 0.239488i 0.992805 + 0.119744i \(0.0382074\pi\)
−0.992805 + 0.119744i \(0.961793\pi\)
\(564\) −4.00234e28 −0.517468
\(565\) 0 0
\(566\) −1.95011e28 −0.242933
\(567\) 5.81386e26i 0.00710956i
\(568\) 6.25333e27i 0.0750679i
\(569\) −1.75144e28 −0.206403 −0.103201 0.994660i \(-0.532909\pi\)
−0.103201 + 0.994660i \(0.532909\pi\)
\(570\) 0 0
\(571\) 1.67794e29 1.90589 0.952946 0.303140i \(-0.0980350\pi\)
0.952946 + 0.303140i \(0.0980350\pi\)
\(572\) − 2.08977e29i − 2.33045i
\(573\) 2.72110e28i 0.297935i
\(574\) 2.06484e27 0.0221978
\(575\) 0 0
\(576\) −1.75943e28 −0.182363
\(577\) 7.49346e28i 0.762669i 0.924437 + 0.381335i \(0.124535\pi\)
−0.924437 + 0.381335i \(0.875465\pi\)
\(578\) 1.16748e29i 1.16683i
\(579\) −7.65276e28 −0.751092
\(580\) 0 0
\(581\) 5.29867e27 0.0501554
\(582\) 5.30694e28i 0.493348i
\(583\) − 6.94895e28i − 0.634453i
\(584\) 5.79439e28 0.519604
\(585\) 0 0
\(586\) 9.55433e28 0.826562
\(587\) 1.69908e29i 1.44382i 0.691985 + 0.721912i \(0.256736\pi\)
−0.691985 + 0.721912i \(0.743264\pi\)
\(588\) 5.44766e28i 0.454724i
\(589\) 8.39849e28 0.688638
\(590\) 0 0
\(591\) 1.21471e29 0.961175
\(592\) − 1.56348e29i − 1.21539i
\(593\) 1.26592e29i 0.966787i 0.875403 + 0.483394i \(0.160596\pi\)
−0.875403 + 0.483394i \(0.839404\pi\)
\(594\) −6.38854e28 −0.479340
\(595\) 0 0
\(596\) 5.71241e28 0.413746
\(597\) 3.15650e27i 0.0224634i
\(598\) − 2.91907e29i − 2.04119i
\(599\) −1.73983e29 −1.19544 −0.597718 0.801706i \(-0.703926\pi\)
−0.597718 + 0.801706i \(0.703926\pi\)
\(600\) 0 0
\(601\) −2.58691e29 −1.71633 −0.858164 0.513376i \(-0.828395\pi\)
−0.858164 + 0.513376i \(0.828395\pi\)
\(602\) 1.51718e28i 0.0989178i
\(603\) 4.75461e28i 0.304638i
\(604\) 6.02294e28 0.379246
\(605\) 0 0
\(606\) 2.00994e28 0.122242
\(607\) − 1.61646e29i − 0.966238i −0.875555 0.483119i \(-0.839504\pi\)
0.875555 0.483119i \(-0.160496\pi\)
\(608\) − 2.44067e29i − 1.43391i
\(609\) −2.73387e27 −0.0157869
\(610\) 0 0
\(611\) −3.21617e29 −1.79434
\(612\) − 1.72028e28i − 0.0943427i
\(613\) − 8.86426e28i − 0.477868i −0.971036 0.238934i \(-0.923202\pi\)
0.971036 0.238934i \(-0.0767979\pi\)
\(614\) 2.90361e28 0.153876
\(615\) 0 0
\(616\) −6.50850e27 −0.0333337
\(617\) − 7.14976e28i − 0.359996i −0.983667 0.179998i \(-0.942391\pi\)
0.983667 0.179998i \(-0.0576091\pi\)
\(618\) 1.47641e29i 0.730851i
\(619\) 3.79378e28 0.184638 0.0923188 0.995730i \(-0.470572\pi\)
0.0923188 + 0.995730i \(0.470572\pi\)
\(620\) 0 0
\(621\) −3.94078e28 −0.185404
\(622\) − 2.88356e29i − 1.33392i
\(623\) 3.66882e26i 0.00166879i
\(624\) −2.38182e29 −1.06530
\(625\) 0 0
\(626\) −2.15037e29 −0.930000
\(627\) − 2.83116e29i − 1.20408i
\(628\) 1.56230e29i 0.653414i
\(629\) 9.07416e28 0.373228
\(630\) 0 0
\(631\) 4.22002e29 1.67883 0.839415 0.543491i \(-0.182898\pi\)
0.839415 + 0.543491i \(0.182898\pi\)
\(632\) 1.80451e28i 0.0706041i
\(633\) 1.86679e29i 0.718382i
\(634\) −1.38526e29 −0.524317
\(635\) 0 0
\(636\) 4.25042e28 0.155644
\(637\) 4.37758e29i 1.57678i
\(638\) − 3.00411e29i − 1.06438i
\(639\) −2.56502e28 −0.0893988
\(640\) 0 0
\(641\) −2.45898e29 −0.829364 −0.414682 0.909966i \(-0.636107\pi\)
−0.414682 + 0.909966i \(0.636107\pi\)
\(642\) 6.60186e27i 0.0219053i
\(643\) − 3.81795e29i − 1.24628i −0.782111 0.623139i \(-0.785857\pi\)
0.782111 0.623139i \(-0.214143\pi\)
\(644\) 1.51803e28 0.0487503
\(645\) 0 0
\(646\) 1.72635e29 0.536646
\(647\) − 3.01566e29i − 0.922332i −0.887314 0.461166i \(-0.847431\pi\)
0.887314 0.461166i \(-0.152569\pi\)
\(648\) 1.03347e28i 0.0310999i
\(649\) 8.23515e29 2.43839
\(650\) 0 0
\(651\) 7.91947e27 0.0227037
\(652\) − 2.99812e29i − 0.845764i
\(653\) − 2.27685e29i − 0.632042i −0.948752 0.316021i \(-0.897653\pi\)
0.948752 0.316021i \(-0.102347\pi\)
\(654\) 6.57929e28 0.179727
\(655\) 0 0
\(656\) 1.14200e29 0.302117
\(657\) 2.37677e29i 0.618800i
\(658\) − 3.78738e28i − 0.0970437i
\(659\) 4.45804e29 1.12421 0.562104 0.827066i \(-0.309992\pi\)
0.562104 + 0.827066i \(0.309992\pi\)
\(660\) 0 0
\(661\) 4.54039e29 1.10912 0.554559 0.832144i \(-0.312887\pi\)
0.554559 + 0.832144i \(0.312887\pi\)
\(662\) − 9.79963e29i − 2.35614i
\(663\) − 1.38236e29i − 0.327138i
\(664\) 9.41886e28 0.219398
\(665\) 0 0
\(666\) 2.06121e29 0.465204
\(667\) − 1.85309e29i − 0.411694i
\(668\) − 6.16859e28i − 0.134907i
\(669\) 1.83420e29 0.394886
\(670\) 0 0
\(671\) 3.18588e29 0.664727
\(672\) − 2.30146e28i − 0.0472744i
\(673\) − 5.05512e28i − 0.102229i −0.998693 0.0511144i \(-0.983723\pi\)
0.998693 0.0511144i \(-0.0162773\pi\)
\(674\) −6.79342e29 −1.35257
\(675\) 0 0
\(676\) 6.17391e29 1.19157
\(677\) − 3.44673e29i − 0.654977i −0.944855 0.327489i \(-0.893798\pi\)
0.944855 0.327489i \(-0.106202\pi\)
\(678\) 5.22685e29i 0.977975i
\(679\) −2.21770e28 −0.0408573
\(680\) 0 0
\(681\) −1.20981e29 −0.216108
\(682\) 8.70228e29i 1.53072i
\(683\) − 3.07293e29i − 0.532273i −0.963935 0.266137i \(-0.914253\pi\)
0.963935 0.266137i \(-0.0857472\pi\)
\(684\) 1.73172e29 0.295385
\(685\) 0 0
\(686\) −1.03314e29 −0.170905
\(687\) 2.43649e29i 0.396935i
\(688\) 8.39108e29i 1.34629i
\(689\) 3.41552e29 0.539703
\(690\) 0 0
\(691\) 7.85532e28 0.120405 0.0602025 0.998186i \(-0.480825\pi\)
0.0602025 + 0.998186i \(0.480825\pi\)
\(692\) 4.28557e29i 0.646986i
\(693\) − 2.66968e28i − 0.0396973i
\(694\) 6.81509e29 0.998153
\(695\) 0 0
\(696\) −4.85971e28 −0.0690578
\(697\) 6.62797e28i 0.0927761i
\(698\) − 1.11294e30i − 1.53458i
\(699\) −2.02698e29 −0.275320
\(700\) 0 0
\(701\) 7.75849e29 1.02268 0.511338 0.859380i \(-0.329150\pi\)
0.511338 + 0.859380i \(0.329150\pi\)
\(702\) − 3.14006e29i − 0.407755i
\(703\) 9.13453e29i 1.16857i
\(704\) 8.07920e29 1.01825
\(705\) 0 0
\(706\) 3.59244e28 0.0439481
\(707\) 8.39929e27i 0.0101237i
\(708\) 5.03714e29i 0.598184i
\(709\) −1.53428e30 −1.79523 −0.897613 0.440785i \(-0.854700\pi\)
−0.897613 + 0.440785i \(0.854700\pi\)
\(710\) 0 0
\(711\) −7.40182e28 −0.0840829
\(712\) 6.52165e27i 0.00729991i
\(713\) 5.36801e29i 0.592070i
\(714\) 1.62788e28 0.0176926
\(715\) 0 0
\(716\) −5.55909e29 −0.586701
\(717\) 4.89604e29i 0.509206i
\(718\) 6.62017e29i 0.678520i
\(719\) −5.56901e29 −0.562503 −0.281252 0.959634i \(-0.590750\pi\)
−0.281252 + 0.959634i \(0.590750\pi\)
\(720\) 0 0
\(721\) −6.16973e28 −0.0605265
\(722\) 3.53729e29i 0.342003i
\(723\) 7.70743e29i 0.734442i
\(724\) 3.18879e29 0.299482
\(725\) 0 0
\(726\) 2.08673e30 1.90384
\(727\) − 1.43096e29i − 0.128681i −0.997928 0.0643407i \(-0.979506\pi\)
0.997928 0.0643407i \(-0.0204945\pi\)
\(728\) − 3.19902e28i − 0.0283556i
\(729\) −4.23912e28 −0.0370370
\(730\) 0 0
\(731\) −4.87003e29 −0.413428
\(732\) 1.94869e29i 0.163071i
\(733\) 6.81141e29i 0.561882i 0.959725 + 0.280941i \(0.0906465\pi\)
−0.959725 + 0.280941i \(0.909353\pi\)
\(734\) −1.12239e30 −0.912714
\(735\) 0 0
\(736\) 1.55999e30 1.23283
\(737\) − 2.18328e30i − 1.70099i
\(738\) 1.50555e29i 0.115639i
\(739\) −2.16993e30 −1.64315 −0.821577 0.570097i \(-0.806906\pi\)
−0.821577 + 0.570097i \(0.806906\pi\)
\(740\) 0 0
\(741\) 1.39156e30 1.02426
\(742\) 4.02214e28i 0.0291888i
\(743\) 1.40097e29i 0.100241i 0.998743 + 0.0501206i \(0.0159606\pi\)
−0.998743 + 0.0501206i \(0.984039\pi\)
\(744\) 1.40776e29 0.0993143
\(745\) 0 0
\(746\) −3.38199e30 −2.31960
\(747\) 3.86347e29i 0.261283i
\(748\) 7.89939e29i 0.526777i
\(749\) −2.75883e27 −0.00181412
\(750\) 0 0
\(751\) 8.06099e29 0.515429 0.257715 0.966221i \(-0.417031\pi\)
0.257715 + 0.966221i \(0.417031\pi\)
\(752\) − 2.09469e30i − 1.32079i
\(753\) 1.04889e30i 0.652202i
\(754\) 1.47656e30 0.905427
\(755\) 0 0
\(756\) 1.63295e28 0.00973853
\(757\) − 1.54959e30i − 0.911401i −0.890133 0.455700i \(-0.849389\pi\)
0.890133 0.455700i \(-0.150611\pi\)
\(758\) − 2.10885e30i − 1.22326i
\(759\) 1.80958e30 1.03523
\(760\) 0 0
\(761\) 2.04061e29 0.113559 0.0567793 0.998387i \(-0.481917\pi\)
0.0567793 + 0.998387i \(0.481917\pi\)
\(762\) 2.11633e30i 1.16160i
\(763\) 2.74940e28i 0.0148843i
\(764\) 7.64280e29 0.408105
\(765\) 0 0
\(766\) −4.57307e30 −2.37578
\(767\) 4.04770e30i 2.07423i
\(768\) − 1.46180e30i − 0.738915i
\(769\) −1.27880e30 −0.637641 −0.318821 0.947815i \(-0.603287\pi\)
−0.318821 + 0.947815i \(0.603287\pi\)
\(770\) 0 0
\(771\) −3.36614e29 −0.163328
\(772\) 2.14944e30i 1.02883i
\(773\) − 3.43774e30i − 1.62326i −0.584172 0.811630i \(-0.698581\pi\)
0.584172 0.811630i \(-0.301419\pi\)
\(774\) −1.10624e30 −0.515309
\(775\) 0 0
\(776\) −3.94216e29 −0.178725
\(777\) 8.61352e28i 0.0385265i
\(778\) 5.27439e30i 2.32748i
\(779\) −6.67206e29 −0.290480
\(780\) 0 0
\(781\) 1.17784e30 0.499172
\(782\) 1.10342e30i 0.461392i
\(783\) − 1.99338e29i − 0.0822414i
\(784\) −2.85112e30 −1.16064
\(785\) 0 0
\(786\) −1.43381e30 −0.568271
\(787\) 1.06317e30i 0.415784i 0.978152 + 0.207892i \(0.0666602\pi\)
−0.978152 + 0.207892i \(0.933340\pi\)
\(788\) − 3.41176e30i − 1.31660i
\(789\) −5.74885e28 −0.0218913
\(790\) 0 0
\(791\) −2.18423e29 −0.0809925
\(792\) − 4.74560e29i − 0.173651i
\(793\) 1.56591e30i 0.565456i
\(794\) 4.26517e30 1.51992
\(795\) 0 0
\(796\) 8.86572e28 0.0307699
\(797\) − 2.59470e30i − 0.888742i −0.895843 0.444371i \(-0.853427\pi\)
0.895843 0.444371i \(-0.146573\pi\)
\(798\) 1.63871e29i 0.0553953i
\(799\) 1.21572e30 0.405595
\(800\) 0 0
\(801\) −2.67508e28 −0.00869351
\(802\) 5.07160e30i 1.62672i
\(803\) − 1.09139e31i − 3.45516i
\(804\) 1.33544e30 0.417286
\(805\) 0 0
\(806\) −4.27730e30 −1.30212
\(807\) − 3.41650e30i − 1.02662i
\(808\) 1.49305e29i 0.0442848i
\(809\) 4.54632e30 1.33107 0.665534 0.746367i \(-0.268204\pi\)
0.665534 + 0.746367i \(0.268204\pi\)
\(810\) 0 0
\(811\) −5.33885e30 −1.52310 −0.761550 0.648106i \(-0.775561\pi\)
−0.761550 + 0.648106i \(0.775561\pi\)
\(812\) 7.67867e28i 0.0216246i
\(813\) − 2.53339e30i − 0.704291i
\(814\) −9.46493e30 −2.59753
\(815\) 0 0
\(816\) 9.00335e29 0.240801
\(817\) − 4.90243e30i − 1.29443i
\(818\) 1.72957e30i 0.450847i
\(819\) 1.31219e29 0.0337688
\(820\) 0 0
\(821\) −6.77652e30 −1.69982 −0.849911 0.526926i \(-0.823344\pi\)
−0.849911 + 0.526926i \(0.823344\pi\)
\(822\) 4.39514e30i 1.08848i
\(823\) − 2.32096e30i − 0.567506i −0.958897 0.283753i \(-0.908420\pi\)
0.958897 0.283753i \(-0.0915795\pi\)
\(824\) −1.09672e30 −0.264766
\(825\) 0 0
\(826\) −4.76661e29 −0.112181
\(827\) 3.93890e30i 0.915307i 0.889131 + 0.457654i \(0.151310\pi\)
−0.889131 + 0.457654i \(0.848690\pi\)
\(828\) 1.10685e30i 0.253963i
\(829\) −3.15777e30 −0.715415 −0.357708 0.933834i \(-0.616442\pi\)
−0.357708 + 0.933834i \(0.616442\pi\)
\(830\) 0 0
\(831\) 3.77323e30 0.833495
\(832\) 3.97105e30i 0.866184i
\(833\) − 1.65474e30i − 0.356416i
\(834\) 1.09726e30 0.233381
\(835\) 0 0
\(836\) −7.95194e30 −1.64933
\(837\) 5.77440e29i 0.118274i
\(838\) 4.81074e30i 0.973082i
\(839\) 1.71221e28 0.00342024 0.00171012 0.999999i \(-0.499456\pi\)
0.00171012 + 0.999999i \(0.499456\pi\)
\(840\) 0 0
\(841\) −4.19549e30 −0.817381
\(842\) 7.94168e30i 1.52804i
\(843\) − 1.90790e30i − 0.362548i
\(844\) 5.24327e30 0.984026
\(845\) 0 0
\(846\) 2.76153e30 0.505547
\(847\) 8.72016e29i 0.157670i
\(848\) 2.22453e30i 0.397266i
\(849\) 5.94193e29 0.104809
\(850\) 0 0
\(851\) −5.83845e30 −1.00470
\(852\) 7.20441e29i 0.122457i
\(853\) − 2.27930e30i − 0.382680i −0.981524 0.191340i \(-0.938717\pi\)
0.981524 0.191340i \(-0.0612833\pi\)
\(854\) −1.84403e29 −0.0305816
\(855\) 0 0
\(856\) −4.90406e28 −0.00793563
\(857\) − 1.06629e31i − 1.70441i −0.523204 0.852207i \(-0.675263\pi\)
0.523204 0.852207i \(-0.324737\pi\)
\(858\) 1.44190e31i 2.27676i
\(859\) 8.93547e29 0.139376 0.0696882 0.997569i \(-0.477800\pi\)
0.0696882 + 0.997569i \(0.477800\pi\)
\(860\) 0 0
\(861\) −6.29150e28 −0.00957682
\(862\) − 9.49196e30i − 1.42735i
\(863\) − 2.82417e30i − 0.419543i −0.977750 0.209772i \(-0.932728\pi\)
0.977750 0.209772i \(-0.0672721\pi\)
\(864\) 1.67809e30 0.246275
\(865\) 0 0
\(866\) −3.99448e30 −0.572167
\(867\) − 3.55727e30i − 0.503404i
\(868\) − 2.22435e29i − 0.0310990i
\(869\) 3.39886e30 0.469490
\(870\) 0 0
\(871\) 1.07312e31 1.44696
\(872\) 4.88730e29i 0.0651097i
\(873\) − 1.61701e30i − 0.212845i
\(874\) −1.11076e31 −1.44461
\(875\) 0 0
\(876\) 6.67566e30 0.847620
\(877\) 9.06575e30i 1.13739i 0.822550 + 0.568693i \(0.192551\pi\)
−0.822550 + 0.568693i \(0.807449\pi\)
\(878\) − 1.94490e31i − 2.41104i
\(879\) −2.91118e30 −0.356604
\(880\) 0 0
\(881\) 1.51630e31 1.81359 0.906794 0.421574i \(-0.138522\pi\)
0.906794 + 0.421574i \(0.138522\pi\)
\(882\) − 3.75877e30i − 0.444248i
\(883\) 1.16543e31i 1.36113i 0.732688 + 0.680565i \(0.238265\pi\)
−0.732688 + 0.680565i \(0.761735\pi\)
\(884\) −3.88267e30 −0.448107
\(885\) 0 0
\(886\) 8.20424e30 0.924664
\(887\) − 7.76908e30i − 0.865309i −0.901560 0.432655i \(-0.857577\pi\)
0.901560 0.432655i \(-0.142423\pi\)
\(888\) 1.53113e30i 0.168529i
\(889\) −8.84384e29 −0.0961994
\(890\) 0 0
\(891\) 1.94657e30 0.206802
\(892\) − 5.15174e30i − 0.540907i
\(893\) 1.22381e31i 1.26991i
\(894\) −3.94144e30 −0.404214
\(895\) 0 0
\(896\) 3.49740e29 0.0350357
\(897\) 8.89435e30i 0.880630i
\(898\) 1.30702e31i 1.27903i
\(899\) −2.71532e30 −0.262630
\(900\) 0 0
\(901\) −1.29108e30 −0.121995
\(902\) − 6.91340e30i − 0.645688i
\(903\) − 4.62281e29i − 0.0426761i
\(904\) −3.88266e30 −0.354292
\(905\) 0 0
\(906\) −4.15570e30 −0.370509
\(907\) 5.71533e30i 0.503692i 0.967767 + 0.251846i \(0.0810376\pi\)
−0.967767 + 0.251846i \(0.918962\pi\)
\(908\) 3.39800e30i 0.296021i
\(909\) −6.12425e29 −0.0527390
\(910\) 0 0
\(911\) 2.22259e30 0.187032 0.0935160 0.995618i \(-0.470189\pi\)
0.0935160 + 0.995618i \(0.470189\pi\)
\(912\) 9.06324e30i 0.753942i
\(913\) − 1.77408e31i − 1.45891i
\(914\) 2.20581e31 1.79322
\(915\) 0 0
\(916\) 6.84342e30 0.543714
\(917\) − 5.99171e29i − 0.0470623i
\(918\) 1.18695e30i 0.0921693i
\(919\) 9.94638e30 0.763576 0.381788 0.924250i \(-0.375308\pi\)
0.381788 + 0.924250i \(0.375308\pi\)
\(920\) 0 0
\(921\) −8.84722e29 −0.0663866
\(922\) 1.20962e31i 0.897378i
\(923\) 5.78926e30i 0.424624i
\(924\) −7.49838e29 −0.0543766
\(925\) 0 0
\(926\) −4.49324e30 −0.318526
\(927\) − 4.49859e30i − 0.315311i
\(928\) 7.89093e30i 0.546858i
\(929\) −6.57341e30 −0.450429 −0.225214 0.974309i \(-0.572308\pi\)
−0.225214 + 0.974309i \(0.572308\pi\)
\(930\) 0 0
\(931\) 1.66575e31 1.11593
\(932\) 5.69320e30i 0.377128i
\(933\) 8.78615e30i 0.575495i
\(934\) 1.83688e30 0.118970
\(935\) 0 0
\(936\) 2.33254e30 0.147717
\(937\) − 9.88672e30i − 0.619137i −0.950877 0.309568i \(-0.899815\pi\)
0.950877 0.309568i \(-0.100185\pi\)
\(938\) 1.26371e30i 0.0782561i
\(939\) 6.55213e30 0.401230
\(940\) 0 0
\(941\) 1.35253e31 0.809947 0.404973 0.914328i \(-0.367281\pi\)
0.404973 + 0.914328i \(0.367281\pi\)
\(942\) − 1.07796e31i − 0.638361i
\(943\) − 4.26454e30i − 0.249746i
\(944\) −2.63627e31 −1.52681
\(945\) 0 0
\(946\) 5.07976e31 2.87731
\(947\) − 2.26830e31i − 1.27065i −0.772245 0.635325i \(-0.780866\pi\)
0.772245 0.635325i \(-0.219134\pi\)
\(948\) 2.07896e30i 0.115175i
\(949\) 5.36437e31 2.93916
\(950\) 0 0
\(951\) 4.22085e30 0.226206
\(952\) 1.20924e29i 0.00640952i
\(953\) − 3.45875e31i − 1.81319i −0.422002 0.906595i \(-0.638673\pi\)
0.422002 0.906595i \(-0.361327\pi\)
\(954\) −2.93270e30 −0.152058
\(955\) 0 0
\(956\) 1.37516e31 0.697501
\(957\) 9.15344e30i 0.459207i
\(958\) 3.43011e31i 1.70204i
\(959\) −1.83667e30 −0.0901438
\(960\) 0 0
\(961\) −1.29598e31 −0.622304
\(962\) − 4.65216e31i − 2.20961i
\(963\) − 2.01157e29i − 0.00945059i
\(964\) 2.16480e31 1.00602
\(965\) 0 0
\(966\) −1.04741e30 −0.0476272
\(967\) − 1.92820e31i − 0.867311i −0.901079 0.433655i \(-0.857224\pi\)
0.901079 0.433655i \(-0.142776\pi\)
\(968\) 1.55009e31i 0.689707i
\(969\) −5.26015e30 −0.231525
\(970\) 0 0
\(971\) −2.95793e31 −1.27405 −0.637024 0.770844i \(-0.719835\pi\)
−0.637024 + 0.770844i \(0.719835\pi\)
\(972\) 1.19065e30i 0.0507326i
\(973\) 4.58530e29i 0.0193278i
\(974\) −3.86058e31 −1.60984
\(975\) 0 0
\(976\) −1.01988e31 −0.416222
\(977\) 4.86581e31i 1.96455i 0.187453 + 0.982274i \(0.439977\pi\)
−0.187453 + 0.982274i \(0.560023\pi\)
\(978\) 2.06864e31i 0.826280i
\(979\) 1.22838e30 0.0485415
\(980\) 0 0
\(981\) −2.00469e30 −0.0775396
\(982\) − 1.10703e31i − 0.423633i
\(983\) − 4.31738e31i − 1.63459i −0.576220 0.817294i \(-0.695473\pi\)
0.576220 0.817294i \(-0.304527\pi\)
\(984\) −1.11837e30 −0.0418926
\(985\) 0 0
\(986\) −5.58146e30 −0.204664
\(987\) 1.15401e30i 0.0418676i
\(988\) − 3.90850e31i − 1.40301i
\(989\) 3.13345e31 1.11292
\(990\) 0 0
\(991\) 3.44591e31 1.19820 0.599101 0.800673i \(-0.295525\pi\)
0.599101 + 0.800673i \(0.295525\pi\)
\(992\) − 2.28584e31i − 0.786453i
\(993\) 2.98592e31i 1.01651i
\(994\) −6.81748e29 −0.0229650
\(995\) 0 0
\(996\) 1.08514e31 0.357900
\(997\) 3.62353e31i 1.18259i 0.806457 + 0.591293i \(0.201382\pi\)
−0.806457 + 0.591293i \(0.798618\pi\)
\(998\) 2.26804e31i 0.732452i
\(999\) −6.28046e30 −0.200703
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.b.d.49.4 4
5.2 odd 4 75.22.a.d.1.1 2
5.3 odd 4 3.22.a.c.1.2 2
5.4 even 2 inner 75.22.b.d.49.1 4
15.8 even 4 9.22.a.e.1.1 2
20.3 even 4 48.22.a.g.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.22.a.c.1.2 2 5.3 odd 4
9.22.a.e.1.1 2 15.8 even 4
48.22.a.g.1.2 2 20.3 even 4
75.22.a.d.1.1 2 5.2 odd 4
75.22.b.d.49.1 4 5.4 even 2 inner
75.22.b.d.49.4 4 1.1 even 1 trivial