Properties

Label 9801.2.a.bp.1.2
Level $9801$
Weight $2$
Character 9801.1
Self dual yes
Analytic conductor $78.261$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [9801,2,Mod(1,9801)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9801, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("9801.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9801 = 3^{4} \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9801.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: \(78.2613790211\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.5144904.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 9x^{3} + 4x^{2} + 20x + 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.52027\) of defining polynomial
Character \(\chi\) \(=\) 9801.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.52027 q^{2} +0.311228 q^{4} +1.08767 q^{5} -2.25617 q^{7} +2.56739 q^{8} +O(q^{10})\) \(q-1.52027 q^{2} +0.311228 q^{4} +1.08767 q^{5} -2.25617 q^{7} +2.56739 q^{8} -1.65355 q^{10} -6.66038 q^{13} +3.42999 q^{14} -4.52559 q^{16} -5.92449 q^{17} -2.94494 q^{19} +0.338512 q^{20} -1.36367 q^{23} -3.81698 q^{25} +10.1256 q^{26} -0.702181 q^{28} +4.90971 q^{29} +1.12796 q^{31} +1.74535 q^{32} +9.00683 q^{34} -2.45396 q^{35} +0.220942 q^{37} +4.47711 q^{38} +2.79247 q^{40} -4.79628 q^{41} -7.34915 q^{43} +2.07315 q^{46} -9.42999 q^{47} -1.90971 q^{49} +5.80285 q^{50} -2.07289 q^{52} +2.06189 q^{53} -5.79247 q^{56} -7.46410 q^{58} +8.60794 q^{59} -12.5676 q^{61} -1.71480 q^{62} +6.39779 q^{64} -7.24427 q^{65} -2.90971 q^{67} -1.84386 q^{68} +3.73068 q^{70} -8.76192 q^{71} +1.17644 q^{73} -0.335892 q^{74} -0.916546 q^{76} -13.9650 q^{79} -4.92233 q^{80} +7.29164 q^{82} -15.8901 q^{83} -6.44386 q^{85} +11.1727 q^{86} +0.635716 q^{89} +15.0269 q^{91} -0.424412 q^{92} +14.3361 q^{94} -3.20311 q^{95} +4.77493 q^{97} +2.90329 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + q^{2} + 9 q^{4} - 4 q^{5} - 3 q^{7} + 12 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + q^{2} + 9 q^{4} - 4 q^{5} - 3 q^{7} + 12 q^{8} + 5 q^{10} - 13 q^{14} + 9 q^{16} + 4 q^{17} + q^{19} + 7 q^{20} + q^{23} + 15 q^{25} - 7 q^{26} - q^{28} + 3 q^{29} + 24 q^{31} + 20 q^{32} + 25 q^{34} - 26 q^{35} - 13 q^{37} + 33 q^{40} - 17 q^{41} + 4 q^{43} - 32 q^{46} - 17 q^{47} + 12 q^{49} + 47 q^{50} - 13 q^{52} + 6 q^{53} - 48 q^{56} + 15 q^{58} + 25 q^{59} - 17 q^{61} + 49 q^{62} + 2 q^{64} - 26 q^{65} + 7 q^{67} + 2 q^{68} - 23 q^{70} + 6 q^{71} + 15 q^{73} + 24 q^{74} + 37 q^{76} - 19 q^{79} + 52 q^{80} - 11 q^{82} - 3 q^{83} + 5 q^{85} + 6 q^{86} + 3 q^{89} + 20 q^{91} - 40 q^{92} + q^{94} - 15 q^{95} + 4 q^{97} - 12 q^{98}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.52027 −1.07499 −0.537497 0.843265i \(-0.680630\pi\)
−0.537497 + 0.843265i \(0.680630\pi\)
\(3\) 0 0
\(4\) 0.311228 0.155614
\(5\) 1.08767 0.486419 0.243210 0.969974i \(-0.421800\pi\)
0.243210 + 0.969974i \(0.421800\pi\)
\(6\) 0 0
\(7\) −2.25617 −0.852751 −0.426375 0.904546i \(-0.640210\pi\)
−0.426375 + 0.904546i \(0.640210\pi\)
\(8\) 2.56739 0.907711
\(9\) 0 0
\(10\) −1.65355 −0.522898
\(11\) 0 0
\(12\) 0 0
\(13\) −6.66038 −1.84726 −0.923629 0.383289i \(-0.874791\pi\)
−0.923629 + 0.383289i \(0.874791\pi\)
\(14\) 3.42999 0.916702
\(15\) 0 0
\(16\) −4.52559 −1.13140
\(17\) −5.92449 −1.43690 −0.718449 0.695579i \(-0.755148\pi\)
−0.718449 + 0.695579i \(0.755148\pi\)
\(18\) 0 0
\(19\) −2.94494 −0.675615 −0.337808 0.941215i \(-0.609685\pi\)
−0.337808 + 0.941215i \(0.609685\pi\)
\(20\) 0.338512 0.0756935
\(21\) 0 0
\(22\) 0 0
\(23\) −1.36367 −0.284345 −0.142172 0.989842i \(-0.545409\pi\)
−0.142172 + 0.989842i \(0.545409\pi\)
\(24\) 0 0
\(25\) −3.81698 −0.763397
\(26\) 10.1256 1.98579
\(27\) 0 0
\(28\) −0.702181 −0.132700
\(29\) 4.90971 0.911711 0.455856 0.890054i \(-0.349333\pi\)
0.455856 + 0.890054i \(0.349333\pi\)
\(30\) 0 0
\(31\) 1.12796 0.202587 0.101293 0.994857i \(-0.467702\pi\)
0.101293 + 0.994857i \(0.467702\pi\)
\(32\) 1.74535 0.308536
\(33\) 0 0
\(34\) 9.00683 1.54466
\(35\) −2.45396 −0.414794
\(36\) 0 0
\(37\) 0.220942 0.0363227 0.0181613 0.999835i \(-0.494219\pi\)
0.0181613 + 0.999835i \(0.494219\pi\)
\(38\) 4.47711 0.726283
\(39\) 0 0
\(40\) 2.79247 0.441528
\(41\) −4.79628 −0.749052 −0.374526 0.927216i \(-0.622195\pi\)
−0.374526 + 0.927216i \(0.622195\pi\)
\(42\) 0 0
\(43\) −7.34915 −1.12074 −0.560368 0.828244i \(-0.689340\pi\)
−0.560368 + 0.828244i \(0.689340\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 2.07315 0.305669
\(47\) −9.42999 −1.37550 −0.687752 0.725945i \(-0.741403\pi\)
−0.687752 + 0.725945i \(0.741403\pi\)
\(48\) 0 0
\(49\) −1.90971 −0.272816
\(50\) 5.80285 0.820647
\(51\) 0 0
\(52\) −2.07289 −0.287459
\(53\) 2.06189 0.283223 0.141611 0.989922i \(-0.454772\pi\)
0.141611 + 0.989922i \(0.454772\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −5.79247 −0.774051
\(57\) 0 0
\(58\) −7.46410 −0.980085
\(59\) 8.60794 1.12066 0.560329 0.828270i \(-0.310675\pi\)
0.560329 + 0.828270i \(0.310675\pi\)
\(60\) 0 0
\(61\) −12.5676 −1.60912 −0.804561 0.593870i \(-0.797599\pi\)
−0.804561 + 0.593870i \(0.797599\pi\)
\(62\) −1.71480 −0.217780
\(63\) 0 0
\(64\) 6.39779 0.799723
\(65\) −7.24427 −0.898541
\(66\) 0 0
\(67\) −2.90971 −0.355478 −0.177739 0.984078i \(-0.556878\pi\)
−0.177739 + 0.984078i \(0.556878\pi\)
\(68\) −1.84386 −0.223601
\(69\) 0 0
\(70\) 3.73068 0.445902
\(71\) −8.76192 −1.03985 −0.519924 0.854212i \(-0.674040\pi\)
−0.519924 + 0.854212i \(0.674040\pi\)
\(72\) 0 0
\(73\) 1.17644 0.137692 0.0688460 0.997627i \(-0.478068\pi\)
0.0688460 + 0.997627i \(0.478068\pi\)
\(74\) −0.335892 −0.0390467
\(75\) 0 0
\(76\) −0.916546 −0.105135
\(77\) 0 0
\(78\) 0 0
\(79\) −13.9650 −1.57119 −0.785594 0.618742i \(-0.787643\pi\)
−0.785594 + 0.618742i \(0.787643\pi\)
\(80\) −4.92233 −0.550334
\(81\) 0 0
\(82\) 7.29164 0.805227
\(83\) −15.8901 −1.74417 −0.872084 0.489356i \(-0.837232\pi\)
−0.872084 + 0.489356i \(0.837232\pi\)
\(84\) 0 0
\(85\) −6.44386 −0.698935
\(86\) 11.1727 1.20478
\(87\) 0 0
\(88\) 0 0
\(89\) 0.635716 0.0673857 0.0336929 0.999432i \(-0.489273\pi\)
0.0336929 + 0.999432i \(0.489273\pi\)
\(90\) 0 0
\(91\) 15.0269 1.57525
\(92\) −0.424412 −0.0442480
\(93\) 0 0
\(94\) 14.3361 1.47866
\(95\) −3.20311 −0.328632
\(96\) 0 0
\(97\) 4.77493 0.484820 0.242410 0.970174i \(-0.422062\pi\)
0.242410 + 0.970174i \(0.422062\pi\)
\(98\) 2.90329 0.293276
\(99\) 0 0
\(100\) −1.18795 −0.118795
\(101\) 8.92449 0.888020 0.444010 0.896022i \(-0.353556\pi\)
0.444010 + 0.896022i \(0.353556\pi\)
\(102\) 0 0
\(103\) 2.93811 0.289500 0.144750 0.989468i \(-0.453762\pi\)
0.144750 + 0.989468i \(0.453762\pi\)
\(104\) −17.0998 −1.67678
\(105\) 0 0
\(106\) −3.13464 −0.304463
\(107\) 11.0619 1.06939 0.534697 0.845044i \(-0.320426\pi\)
0.534697 + 0.845044i \(0.320426\pi\)
\(108\) 0 0
\(109\) 15.3142 1.46683 0.733416 0.679780i \(-0.237925\pi\)
0.733416 + 0.679780i \(0.237925\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 10.2105 0.964800
\(113\) −12.8987 −1.21341 −0.606704 0.794927i \(-0.707509\pi\)
−0.606704 + 0.794927i \(0.707509\pi\)
\(114\) 0 0
\(115\) −1.48322 −0.138311
\(116\) 1.52804 0.141875
\(117\) 0 0
\(118\) −13.0864 −1.20470
\(119\) 13.3666 1.22532
\(120\) 0 0
\(121\) 0 0
\(122\) 19.1062 1.72980
\(123\) 0 0
\(124\) 0.351051 0.0315253
\(125\) −9.58993 −0.857750
\(126\) 0 0
\(127\) 20.7553 1.84174 0.920869 0.389871i \(-0.127480\pi\)
0.920869 + 0.389871i \(0.127480\pi\)
\(128\) −13.2171 −1.16823
\(129\) 0 0
\(130\) 11.0133 0.965927
\(131\) 10.5500 0.921758 0.460879 0.887463i \(-0.347534\pi\)
0.460879 + 0.887463i \(0.347534\pi\)
\(132\) 0 0
\(133\) 6.64427 0.576131
\(134\) 4.42356 0.382137
\(135\) 0 0
\(136\) −15.2105 −1.30429
\(137\) −11.9624 −1.02202 −0.511009 0.859575i \(-0.670728\pi\)
−0.511009 + 0.859575i \(0.670728\pi\)
\(138\) 0 0
\(139\) 12.6631 1.07407 0.537034 0.843560i \(-0.319545\pi\)
0.537034 + 0.843560i \(0.319545\pi\)
\(140\) −0.763739 −0.0645477
\(141\) 0 0
\(142\) 13.3205 1.11783
\(143\) 0 0
\(144\) 0 0
\(145\) 5.34013 0.443474
\(146\) −1.78851 −0.148018
\(147\) 0 0
\(148\) 0.0687633 0.00565231
\(149\) −13.3131 −1.09065 −0.545325 0.838225i \(-0.683594\pi\)
−0.545325 + 0.838225i \(0.683594\pi\)
\(150\) 0 0
\(151\) −8.26300 −0.672433 −0.336217 0.941785i \(-0.609147\pi\)
−0.336217 + 0.941785i \(0.609147\pi\)
\(152\) −7.56082 −0.613263
\(153\) 0 0
\(154\) 0 0
\(155\) 1.22684 0.0985421
\(156\) 0 0
\(157\) 22.3628 1.78475 0.892374 0.451298i \(-0.149039\pi\)
0.892374 + 0.451298i \(0.149039\pi\)
\(158\) 21.2306 1.68902
\(159\) 0 0
\(160\) 1.89835 0.150078
\(161\) 3.07666 0.242475
\(162\) 0 0
\(163\) 9.15930 0.717412 0.358706 0.933451i \(-0.383218\pi\)
0.358706 + 0.933451i \(0.383218\pi\)
\(164\) −1.49273 −0.116563
\(165\) 0 0
\(166\) 24.1573 1.87497
\(167\) −19.9346 −1.54259 −0.771294 0.636479i \(-0.780390\pi\)
−0.771294 + 0.636479i \(0.780390\pi\)
\(168\) 0 0
\(169\) 31.3607 2.41236
\(170\) 9.79642 0.751351
\(171\) 0 0
\(172\) −2.28726 −0.174402
\(173\) 23.3672 1.77658 0.888289 0.459285i \(-0.151894\pi\)
0.888289 + 0.459285i \(0.151894\pi\)
\(174\) 0 0
\(175\) 8.61175 0.650987
\(176\) 0 0
\(177\) 0 0
\(178\) −0.966461 −0.0724393
\(179\) 10.8595 0.811678 0.405839 0.913945i \(-0.366979\pi\)
0.405839 + 0.913945i \(0.366979\pi\)
\(180\) 0 0
\(181\) 12.4309 0.923980 0.461990 0.886885i \(-0.347136\pi\)
0.461990 + 0.886885i \(0.347136\pi\)
\(182\) −22.8450 −1.69339
\(183\) 0 0
\(184\) −3.50108 −0.258103
\(185\) 0.240311 0.0176680
\(186\) 0 0
\(187\) 0 0
\(188\) −2.93487 −0.214048
\(189\) 0 0
\(190\) 4.86960 0.353278
\(191\) 2.34494 0.169674 0.0848370 0.996395i \(-0.472963\pi\)
0.0848370 + 0.996395i \(0.472963\pi\)
\(192\) 0 0
\(193\) 4.17644 0.300627 0.150313 0.988638i \(-0.451972\pi\)
0.150313 + 0.988638i \(0.451972\pi\)
\(194\) −7.25919 −0.521179
\(195\) 0 0
\(196\) −0.594356 −0.0424540
\(197\) −15.4170 −1.09841 −0.549207 0.835686i \(-0.685070\pi\)
−0.549207 + 0.835686i \(0.685070\pi\)
\(198\) 0 0
\(199\) −5.76678 −0.408796 −0.204398 0.978888i \(-0.565524\pi\)
−0.204398 + 0.978888i \(0.565524\pi\)
\(200\) −9.79970 −0.692943
\(201\) 0 0
\(202\) −13.5676 −0.954616
\(203\) −11.0771 −0.777462
\(204\) 0 0
\(205\) −5.21675 −0.364353
\(206\) −4.46672 −0.311211
\(207\) 0 0
\(208\) 30.1422 2.08998
\(209\) 0 0
\(210\) 0 0
\(211\) −10.6959 −0.736333 −0.368167 0.929760i \(-0.620014\pi\)
−0.368167 + 0.929760i \(0.620014\pi\)
\(212\) 0.641718 0.0440734
\(213\) 0 0
\(214\) −16.8171 −1.14959
\(215\) −7.99342 −0.545147
\(216\) 0 0
\(217\) −2.54486 −0.172756
\(218\) −23.2817 −1.57684
\(219\) 0 0
\(220\) 0 0
\(221\) 39.4593 2.65432
\(222\) 0 0
\(223\) −21.5772 −1.44491 −0.722457 0.691415i \(-0.756987\pi\)
−0.722457 + 0.691415i \(0.756987\pi\)
\(224\) −3.93779 −0.263105
\(225\) 0 0
\(226\) 19.6096 1.30441
\(227\) 4.17914 0.277379 0.138690 0.990336i \(-0.455711\pi\)
0.138690 + 0.990336i \(0.455711\pi\)
\(228\) 0 0
\(229\) −19.5701 −1.29323 −0.646614 0.762817i \(-0.723816\pi\)
−0.646614 + 0.762817i \(0.723816\pi\)
\(230\) 2.25489 0.148683
\(231\) 0 0
\(232\) 12.6052 0.827570
\(233\) −10.4641 −0.685526 −0.342763 0.939422i \(-0.611363\pi\)
−0.342763 + 0.939422i \(0.611363\pi\)
\(234\) 0 0
\(235\) −10.2567 −0.669072
\(236\) 2.67903 0.174390
\(237\) 0 0
\(238\) −20.3209 −1.31721
\(239\) −13.8412 −0.895316 −0.447658 0.894205i \(-0.647742\pi\)
−0.447658 + 0.894205i \(0.647742\pi\)
\(240\) 0 0
\(241\) −2.13234 −0.137356 −0.0686781 0.997639i \(-0.521878\pi\)
−0.0686781 + 0.997639i \(0.521878\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) −3.91140 −0.250402
\(245\) −2.07713 −0.132703
\(246\) 0 0
\(247\) 19.6144 1.24804
\(248\) 2.89591 0.183890
\(249\) 0 0
\(250\) 14.5793 0.922076
\(251\) 15.5736 0.982996 0.491498 0.870879i \(-0.336450\pi\)
0.491498 + 0.870879i \(0.336450\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −31.5538 −1.97986
\(255\) 0 0
\(256\) 7.29797 0.456123
\(257\) 6.94630 0.433298 0.216649 0.976250i \(-0.430487\pi\)
0.216649 + 0.976250i \(0.430487\pi\)
\(258\) 0 0
\(259\) −0.498482 −0.0309742
\(260\) −2.25462 −0.139825
\(261\) 0 0
\(262\) −16.0389 −0.990885
\(263\) −28.7587 −1.77334 −0.886668 0.462406i \(-0.846986\pi\)
−0.886668 + 0.462406i \(0.846986\pi\)
\(264\) 0 0
\(265\) 2.24265 0.137765
\(266\) −10.1011 −0.619338
\(267\) 0 0
\(268\) −0.905583 −0.0553173
\(269\) −7.91233 −0.482424 −0.241212 0.970473i \(-0.577545\pi\)
−0.241212 + 0.970473i \(0.577545\pi\)
\(270\) 0 0
\(271\) 14.4421 0.877297 0.438649 0.898659i \(-0.355457\pi\)
0.438649 + 0.898659i \(0.355457\pi\)
\(272\) 26.8118 1.62570
\(273\) 0 0
\(274\) 18.1861 1.09866
\(275\) 0 0
\(276\) 0 0
\(277\) −13.5300 −0.812940 −0.406470 0.913664i \(-0.633240\pi\)
−0.406470 + 0.913664i \(0.633240\pi\)
\(278\) −19.2513 −1.15462
\(279\) 0 0
\(280\) −6.30027 −0.376513
\(281\) −5.25347 −0.313395 −0.156698 0.987647i \(-0.550085\pi\)
−0.156698 + 0.987647i \(0.550085\pi\)
\(282\) 0 0
\(283\) −17.8919 −1.06356 −0.531781 0.846882i \(-0.678477\pi\)
−0.531781 + 0.846882i \(0.678477\pi\)
\(284\) −2.72695 −0.161815
\(285\) 0 0
\(286\) 0 0
\(287\) 10.8212 0.638755
\(288\) 0 0
\(289\) 18.0995 1.06468
\(290\) −8.11845 −0.476732
\(291\) 0 0
\(292\) 0.366141 0.0214268
\(293\) −9.22950 −0.539193 −0.269596 0.962973i \(-0.586890\pi\)
−0.269596 + 0.962973i \(0.586890\pi\)
\(294\) 0 0
\(295\) 9.36256 0.545109
\(296\) 0.567245 0.0329705
\(297\) 0 0
\(298\) 20.2395 1.17244
\(299\) 9.08256 0.525258
\(300\) 0 0
\(301\) 16.5809 0.955708
\(302\) 12.5620 0.722862
\(303\) 0 0
\(304\) 13.3276 0.764390
\(305\) −13.6694 −0.782708
\(306\) 0 0
\(307\) −30.2755 −1.72791 −0.863957 0.503566i \(-0.832021\pi\)
−0.863957 + 0.503566i \(0.832021\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −1.86513 −0.105932
\(311\) 7.17472 0.406841 0.203420 0.979091i \(-0.434794\pi\)
0.203420 + 0.979091i \(0.434794\pi\)
\(312\) 0 0
\(313\) −29.7516 −1.68166 −0.840831 0.541297i \(-0.817933\pi\)
−0.840831 + 0.541297i \(0.817933\pi\)
\(314\) −33.9976 −1.91859
\(315\) 0 0
\(316\) −4.34630 −0.244499
\(317\) −29.2162 −1.64095 −0.820473 0.571685i \(-0.806290\pi\)
−0.820473 + 0.571685i \(0.806290\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 6.95865 0.389001
\(321\) 0 0
\(322\) −4.67737 −0.260660
\(323\) 17.4472 0.970791
\(324\) 0 0
\(325\) 25.4226 1.41019
\(326\) −13.9246 −0.771214
\(327\) 0 0
\(328\) −12.3139 −0.679923
\(329\) 21.2756 1.17296
\(330\) 0 0
\(331\) 30.5439 1.67885 0.839423 0.543478i \(-0.182893\pi\)
0.839423 + 0.543478i \(0.182893\pi\)
\(332\) −4.94545 −0.271417
\(333\) 0 0
\(334\) 30.3061 1.65827
\(335\) −3.16480 −0.172911
\(336\) 0 0
\(337\) −23.1138 −1.25909 −0.629544 0.776965i \(-0.716758\pi\)
−0.629544 + 0.776965i \(0.716758\pi\)
\(338\) −47.6767 −2.59327
\(339\) 0 0
\(340\) −2.00551 −0.108764
\(341\) 0 0
\(342\) 0 0
\(343\) 20.1018 1.08539
\(344\) −18.8682 −1.01730
\(345\) 0 0
\(346\) −35.5246 −1.90981
\(347\) 25.5935 1.37393 0.686966 0.726690i \(-0.258942\pi\)
0.686966 + 0.726690i \(0.258942\pi\)
\(348\) 0 0
\(349\) 2.43088 0.130122 0.0650611 0.997881i \(-0.479276\pi\)
0.0650611 + 0.997881i \(0.479276\pi\)
\(350\) −13.0922 −0.699808
\(351\) 0 0
\(352\) 0 0
\(353\) 0.825819 0.0439539 0.0219770 0.999758i \(-0.493004\pi\)
0.0219770 + 0.999758i \(0.493004\pi\)
\(354\) 0 0
\(355\) −9.53004 −0.505802
\(356\) 0.197852 0.0104861
\(357\) 0 0
\(358\) −16.5094 −0.872549
\(359\) −29.7329 −1.56924 −0.784622 0.619975i \(-0.787143\pi\)
−0.784622 + 0.619975i \(0.787143\pi\)
\(360\) 0 0
\(361\) −10.3273 −0.543544
\(362\) −18.8983 −0.993274
\(363\) 0 0
\(364\) 4.67679 0.245131
\(365\) 1.27957 0.0669760
\(366\) 0 0
\(367\) 36.3364 1.89674 0.948372 0.317159i \(-0.102729\pi\)
0.948372 + 0.317159i \(0.102729\pi\)
\(368\) 6.17141 0.321707
\(369\) 0 0
\(370\) −0.365338 −0.0189930
\(371\) −4.65197 −0.241518
\(372\) 0 0
\(373\) −0.309250 −0.0160123 −0.00800617 0.999968i \(-0.502548\pi\)
−0.00800617 + 0.999968i \(0.502548\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −24.2105 −1.24856
\(377\) −32.7006 −1.68416
\(378\) 0 0
\(379\) −16.7314 −0.859434 −0.429717 0.902964i \(-0.641387\pi\)
−0.429717 + 0.902964i \(0.641387\pi\)
\(380\) −0.996896 −0.0511397
\(381\) 0 0
\(382\) −3.56495 −0.182399
\(383\) 12.3894 0.633071 0.316535 0.948581i \(-0.397480\pi\)
0.316535 + 0.948581i \(0.397480\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −6.34933 −0.323172
\(387\) 0 0
\(388\) 1.48609 0.0754447
\(389\) 21.8529 1.10799 0.553994 0.832521i \(-0.313103\pi\)
0.553994 + 0.832521i \(0.313103\pi\)
\(390\) 0 0
\(391\) 8.07904 0.408575
\(392\) −4.90299 −0.247638
\(393\) 0 0
\(394\) 23.4380 1.18079
\(395\) −15.1893 −0.764256
\(396\) 0 0
\(397\) 21.9721 1.10275 0.551374 0.834258i \(-0.314104\pi\)
0.551374 + 0.834258i \(0.314104\pi\)
\(398\) 8.76707 0.439453
\(399\) 0 0
\(400\) 17.2741 0.863705
\(401\) −16.2137 −0.809675 −0.404837 0.914389i \(-0.632672\pi\)
−0.404837 + 0.914389i \(0.632672\pi\)
\(402\) 0 0
\(403\) −7.51261 −0.374230
\(404\) 2.77755 0.138188
\(405\) 0 0
\(406\) 16.8403 0.835768
\(407\) 0 0
\(408\) 0 0
\(409\) −9.02007 −0.446014 −0.223007 0.974817i \(-0.571587\pi\)
−0.223007 + 0.974817i \(0.571587\pi\)
\(410\) 7.93087 0.391678
\(411\) 0 0
\(412\) 0.914420 0.0450502
\(413\) −19.4209 −0.955642
\(414\) 0 0
\(415\) −17.2832 −0.848397
\(416\) −11.6247 −0.569946
\(417\) 0 0
\(418\) 0 0
\(419\) 22.7915 1.11344 0.556718 0.830702i \(-0.312060\pi\)
0.556718 + 0.830702i \(0.312060\pi\)
\(420\) 0 0
\(421\) 20.5745 1.00274 0.501370 0.865233i \(-0.332829\pi\)
0.501370 + 0.865233i \(0.332829\pi\)
\(422\) 16.2606 0.791554
\(423\) 0 0
\(424\) 5.29369 0.257084
\(425\) 22.6137 1.09692
\(426\) 0 0
\(427\) 28.3547 1.37218
\(428\) 3.44277 0.166412
\(429\) 0 0
\(430\) 12.1522 0.586030
\(431\) 5.19653 0.250308 0.125154 0.992137i \(-0.460057\pi\)
0.125154 + 0.992137i \(0.460057\pi\)
\(432\) 0 0
\(433\) 3.76824 0.181090 0.0905451 0.995892i \(-0.471139\pi\)
0.0905451 + 0.995892i \(0.471139\pi\)
\(434\) 3.86887 0.185712
\(435\) 0 0
\(436\) 4.76620 0.228259
\(437\) 4.01592 0.192108
\(438\) 0 0
\(439\) 11.1730 0.533257 0.266628 0.963799i \(-0.414090\pi\)
0.266628 + 0.963799i \(0.414090\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −59.9889 −2.85338
\(443\) 16.1963 0.769509 0.384755 0.923019i \(-0.374286\pi\)
0.384755 + 0.923019i \(0.374286\pi\)
\(444\) 0 0
\(445\) 0.691446 0.0327777
\(446\) 32.8032 1.55328
\(447\) 0 0
\(448\) −14.4345 −0.681964
\(449\) 3.27004 0.154323 0.0771614 0.997019i \(-0.475414\pi\)
0.0771614 + 0.997019i \(0.475414\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −4.01444 −0.188823
\(453\) 0 0
\(454\) −6.35343 −0.298181
\(455\) 16.3443 0.766231
\(456\) 0 0
\(457\) 15.2626 0.713954 0.356977 0.934113i \(-0.383807\pi\)
0.356977 + 0.934113i \(0.383807\pi\)
\(458\) 29.7519 1.39021
\(459\) 0 0
\(460\) −0.461618 −0.0215231
\(461\) 7.82454 0.364425 0.182213 0.983259i \(-0.441674\pi\)
0.182213 + 0.983259i \(0.441674\pi\)
\(462\) 0 0
\(463\) 16.8395 0.782599 0.391299 0.920263i \(-0.372026\pi\)
0.391299 + 0.920263i \(0.372026\pi\)
\(464\) −22.2194 −1.03151
\(465\) 0 0
\(466\) 15.9083 0.736937
\(467\) −38.0169 −1.75921 −0.879606 0.475704i \(-0.842193\pi\)
−0.879606 + 0.475704i \(0.842193\pi\)
\(468\) 0 0
\(469\) 6.56480 0.303134
\(470\) 15.5929 0.719249
\(471\) 0 0
\(472\) 22.1000 1.01723
\(473\) 0 0
\(474\) 0 0
\(475\) 11.2408 0.515762
\(476\) 4.16006 0.190676
\(477\) 0 0
\(478\) 21.0425 0.962460
\(479\) −15.5460 −0.710317 −0.355158 0.934806i \(-0.615573\pi\)
−0.355158 + 0.934806i \(0.615573\pi\)
\(480\) 0 0
\(481\) −1.47156 −0.0670973
\(482\) 3.24174 0.147657
\(483\) 0 0
\(484\) 0 0
\(485\) 5.19353 0.235826
\(486\) 0 0
\(487\) −21.1203 −0.957053 −0.478526 0.878073i \(-0.658829\pi\)
−0.478526 + 0.878073i \(0.658829\pi\)
\(488\) −32.2661 −1.46062
\(489\) 0 0
\(490\) 3.15781 0.142655
\(491\) 41.3904 1.86792 0.933960 0.357377i \(-0.116329\pi\)
0.933960 + 0.357377i \(0.116329\pi\)
\(492\) 0 0
\(493\) −29.0875 −1.31004
\(494\) −29.8192 −1.34163
\(495\) 0 0
\(496\) −5.10467 −0.229206
\(497\) 19.7683 0.886732
\(498\) 0 0
\(499\) 7.07932 0.316914 0.158457 0.987366i \(-0.449348\pi\)
0.158457 + 0.987366i \(0.449348\pi\)
\(500\) −2.98465 −0.133478
\(501\) 0 0
\(502\) −23.6761 −1.05672
\(503\) 44.1749 1.96966 0.984830 0.173524i \(-0.0555154\pi\)
0.984830 + 0.173524i \(0.0555154\pi\)
\(504\) 0 0
\(505\) 9.70686 0.431950
\(506\) 0 0
\(507\) 0 0
\(508\) 6.45964 0.286600
\(509\) 25.8703 1.14668 0.573341 0.819317i \(-0.305647\pi\)
0.573341 + 0.819317i \(0.305647\pi\)
\(510\) 0 0
\(511\) −2.65424 −0.117417
\(512\) 15.3392 0.677905
\(513\) 0 0
\(514\) −10.5603 −0.465793
\(515\) 3.19568 0.140818
\(516\) 0 0
\(517\) 0 0
\(518\) 0.757828 0.0332971
\(519\) 0 0
\(520\) −18.5989 −0.815615
\(521\) −25.5813 −1.12074 −0.560369 0.828243i \(-0.689341\pi\)
−0.560369 + 0.828243i \(0.689341\pi\)
\(522\) 0 0
\(523\) 7.58921 0.331853 0.165926 0.986138i \(-0.446939\pi\)
0.165926 + 0.986138i \(0.446939\pi\)
\(524\) 3.28345 0.143438
\(525\) 0 0
\(526\) 43.7210 1.90633
\(527\) −6.68256 −0.291097
\(528\) 0 0
\(529\) −21.1404 −0.919148
\(530\) −3.40944 −0.148097
\(531\) 0 0
\(532\) 2.06788 0.0896540
\(533\) 31.9450 1.38369
\(534\) 0 0
\(535\) 12.0316 0.520173
\(536\) −7.47038 −0.322671
\(537\) 0 0
\(538\) 12.0289 0.518603
\(539\) 0 0
\(540\) 0 0
\(541\) 31.6585 1.36111 0.680553 0.732699i \(-0.261740\pi\)
0.680553 + 0.732699i \(0.261740\pi\)
\(542\) −21.9560 −0.943090
\(543\) 0 0
\(544\) −10.3403 −0.443336
\(545\) 16.6567 0.713495
\(546\) 0 0
\(547\) −30.8160 −1.31760 −0.658798 0.752320i \(-0.728935\pi\)
−0.658798 + 0.752320i \(0.728935\pi\)
\(548\) −3.72303 −0.159040
\(549\) 0 0
\(550\) 0 0
\(551\) −14.4588 −0.615966
\(552\) 0 0
\(553\) 31.5074 1.33983
\(554\) 20.5693 0.873906
\(555\) 0 0
\(556\) 3.94110 0.167140
\(557\) 4.98818 0.211356 0.105678 0.994400i \(-0.466299\pi\)
0.105678 + 0.994400i \(0.466299\pi\)
\(558\) 0 0
\(559\) 48.9481 2.07029
\(560\) 11.1056 0.469297
\(561\) 0 0
\(562\) 7.98670 0.336898
\(563\) −3.03235 −0.127798 −0.0638992 0.997956i \(-0.520354\pi\)
−0.0638992 + 0.997956i \(0.520354\pi\)
\(564\) 0 0
\(565\) −14.0295 −0.590225
\(566\) 27.2005 1.14332
\(567\) 0 0
\(568\) −22.4953 −0.943882
\(569\) 47.5101 1.99173 0.995863 0.0908639i \(-0.0289628\pi\)
0.995863 + 0.0908639i \(0.0289628\pi\)
\(570\) 0 0
\(571\) −8.47272 −0.354572 −0.177286 0.984159i \(-0.556732\pi\)
−0.177286 + 0.984159i \(0.556732\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −16.4512 −0.686658
\(575\) 5.20510 0.217068
\(576\) 0 0
\(577\) −35.2635 −1.46804 −0.734020 0.679128i \(-0.762358\pi\)
−0.734020 + 0.679128i \(0.762358\pi\)
\(578\) −27.5162 −1.14452
\(579\) 0 0
\(580\) 1.66200 0.0690106
\(581\) 35.8508 1.48734
\(582\) 0 0
\(583\) 0 0
\(584\) 3.02039 0.124984
\(585\) 0 0
\(586\) 14.0313 0.579630
\(587\) −25.3219 −1.04515 −0.522574 0.852594i \(-0.675028\pi\)
−0.522574 + 0.852594i \(0.675028\pi\)
\(588\) 0 0
\(589\) −3.32176 −0.136871
\(590\) −14.2336 −0.585990
\(591\) 0 0
\(592\) −0.999894 −0.0410954
\(593\) 39.8626 1.63696 0.818480 0.574535i \(-0.194817\pi\)
0.818480 + 0.574535i \(0.194817\pi\)
\(594\) 0 0
\(595\) 14.5384 0.596017
\(596\) −4.14340 −0.169720
\(597\) 0 0
\(598\) −13.8080 −0.564649
\(599\) −31.9030 −1.30352 −0.651761 0.758425i \(-0.725969\pi\)
−0.651761 + 0.758425i \(0.725969\pi\)
\(600\) 0 0
\(601\) −18.0780 −0.737417 −0.368709 0.929545i \(-0.620200\pi\)
−0.368709 + 0.929545i \(0.620200\pi\)
\(602\) −25.2075 −1.02738
\(603\) 0 0
\(604\) −2.57167 −0.104640
\(605\) 0 0
\(606\) 0 0
\(607\) 1.54758 0.0628145 0.0314072 0.999507i \(-0.490001\pi\)
0.0314072 + 0.999507i \(0.490001\pi\)
\(608\) −5.13993 −0.208452
\(609\) 0 0
\(610\) 20.7812 0.841407
\(611\) 62.8073 2.54091
\(612\) 0 0
\(613\) −10.7046 −0.432355 −0.216177 0.976354i \(-0.569359\pi\)
−0.216177 + 0.976354i \(0.569359\pi\)
\(614\) 46.0270 1.85750
\(615\) 0 0
\(616\) 0 0
\(617\) 32.0693 1.29106 0.645530 0.763735i \(-0.276636\pi\)
0.645530 + 0.763735i \(0.276636\pi\)
\(618\) 0 0
\(619\) 14.2834 0.574098 0.287049 0.957916i \(-0.407326\pi\)
0.287049 + 0.957916i \(0.407326\pi\)
\(620\) 0.381826 0.0153345
\(621\) 0 0
\(622\) −10.9075 −0.437352
\(623\) −1.43428 −0.0574632
\(624\) 0 0
\(625\) 8.65427 0.346171
\(626\) 45.2306 1.80778
\(627\) 0 0
\(628\) 6.95993 0.277731
\(629\) −1.30897 −0.0521920
\(630\) 0 0
\(631\) −20.6442 −0.821833 −0.410917 0.911673i \(-0.634791\pi\)
−0.410917 + 0.911673i \(0.634791\pi\)
\(632\) −35.8537 −1.42618
\(633\) 0 0
\(634\) 44.4166 1.76401
\(635\) 22.5749 0.895857
\(636\) 0 0
\(637\) 12.7194 0.503962
\(638\) 0 0
\(639\) 0 0
\(640\) −14.3758 −0.568252
\(641\) −11.7046 −0.462304 −0.231152 0.972918i \(-0.574249\pi\)
−0.231152 + 0.972918i \(0.574249\pi\)
\(642\) 0 0
\(643\) 18.4574 0.727888 0.363944 0.931421i \(-0.381430\pi\)
0.363944 + 0.931421i \(0.381430\pi\)
\(644\) 0.957543 0.0377325
\(645\) 0 0
\(646\) −26.5246 −1.04360
\(647\) −26.2873 −1.03346 −0.516730 0.856148i \(-0.672851\pi\)
−0.516730 + 0.856148i \(0.672851\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) −38.6492 −1.51595
\(651\) 0 0
\(652\) 2.85063 0.111639
\(653\) −27.9054 −1.09202 −0.546011 0.837778i \(-0.683854\pi\)
−0.546011 + 0.837778i \(0.683854\pi\)
\(654\) 0 0
\(655\) 11.4749 0.448361
\(656\) 21.7060 0.847477
\(657\) 0 0
\(658\) −32.3447 −1.26093
\(659\) −45.9712 −1.79078 −0.895392 0.445278i \(-0.853105\pi\)
−0.895392 + 0.445278i \(0.853105\pi\)
\(660\) 0 0
\(661\) −12.1654 −0.473181 −0.236590 0.971609i \(-0.576030\pi\)
−0.236590 + 0.971609i \(0.576030\pi\)
\(662\) −46.4351 −1.80475
\(663\) 0 0
\(664\) −40.7962 −1.58320
\(665\) 7.22675 0.280241
\(666\) 0 0
\(667\) −6.69523 −0.259240
\(668\) −6.20421 −0.240048
\(669\) 0 0
\(670\) 4.81135 0.185879
\(671\) 0 0
\(672\) 0 0
\(673\) 38.4548 1.48232 0.741161 0.671327i \(-0.234275\pi\)
0.741161 + 0.671327i \(0.234275\pi\)
\(674\) 35.1392 1.35351
\(675\) 0 0
\(676\) 9.76030 0.375396
\(677\) −2.09086 −0.0803583 −0.0401791 0.999192i \(-0.512793\pi\)
−0.0401791 + 0.999192i \(0.512793\pi\)
\(678\) 0 0
\(679\) −10.7730 −0.413431
\(680\) −16.5439 −0.634431
\(681\) 0 0
\(682\) 0 0
\(683\) −41.6302 −1.59294 −0.796468 0.604681i \(-0.793301\pi\)
−0.796468 + 0.604681i \(0.793301\pi\)
\(684\) 0 0
\(685\) −13.0111 −0.497129
\(686\) −30.5602 −1.16679
\(687\) 0 0
\(688\) 33.2593 1.26800
\(689\) −13.7330 −0.523185
\(690\) 0 0
\(691\) 20.5196 0.780603 0.390301 0.920687i \(-0.372371\pi\)
0.390301 + 0.920687i \(0.372371\pi\)
\(692\) 7.27253 0.276460
\(693\) 0 0
\(694\) −38.9091 −1.47697
\(695\) 13.7732 0.522447
\(696\) 0 0
\(697\) 28.4155 1.07631
\(698\) −3.69560 −0.139881
\(699\) 0 0
\(700\) 2.68021 0.101303
\(701\) 12.6814 0.478972 0.239486 0.970900i \(-0.423021\pi\)
0.239486 + 0.970900i \(0.423021\pi\)
\(702\) 0 0
\(703\) −0.650661 −0.0245401
\(704\) 0 0
\(705\) 0 0
\(706\) −1.25547 −0.0472502
\(707\) −20.1351 −0.757259
\(708\) 0 0
\(709\) 19.5544 0.734380 0.367190 0.930146i \(-0.380320\pi\)
0.367190 + 0.930146i \(0.380320\pi\)
\(710\) 14.4883 0.543735
\(711\) 0 0
\(712\) 1.63213 0.0611667
\(713\) −1.53816 −0.0576045
\(714\) 0 0
\(715\) 0 0
\(716\) 3.37978 0.126308
\(717\) 0 0
\(718\) 45.2021 1.68693
\(719\) −32.9171 −1.22760 −0.613800 0.789462i \(-0.710360\pi\)
−0.613800 + 0.789462i \(0.710360\pi\)
\(720\) 0 0
\(721\) −6.62886 −0.246872
\(722\) 15.7004 0.584307
\(723\) 0 0
\(724\) 3.86883 0.143784
\(725\) −18.7403 −0.695997
\(726\) 0 0
\(727\) 5.34049 0.198068 0.0990339 0.995084i \(-0.468425\pi\)
0.0990339 + 0.995084i \(0.468425\pi\)
\(728\) 38.5800 1.42987
\(729\) 0 0
\(730\) −1.94530 −0.0719988
\(731\) 43.5400 1.61038
\(732\) 0 0
\(733\) −1.98487 −0.0733128 −0.0366564 0.999328i \(-0.511671\pi\)
−0.0366564 + 0.999328i \(0.511671\pi\)
\(734\) −55.2412 −2.03899
\(735\) 0 0
\(736\) −2.38007 −0.0877307
\(737\) 0 0
\(738\) 0 0
\(739\) −6.52951 −0.240192 −0.120096 0.992762i \(-0.538320\pi\)
−0.120096 + 0.992762i \(0.538320\pi\)
\(740\) 0.0747915 0.00274939
\(741\) 0 0
\(742\) 7.07227 0.259631
\(743\) 7.07132 0.259421 0.129711 0.991552i \(-0.458595\pi\)
0.129711 + 0.991552i \(0.458595\pi\)
\(744\) 0 0
\(745\) −14.4802 −0.530513
\(746\) 0.470144 0.0172132
\(747\) 0 0
\(748\) 0 0
\(749\) −24.9575 −0.911926
\(750\) 0 0
\(751\) −24.6750 −0.900403 −0.450202 0.892927i \(-0.648648\pi\)
−0.450202 + 0.892927i \(0.648648\pi\)
\(752\) 42.6763 1.55624
\(753\) 0 0
\(754\) 49.7138 1.81047
\(755\) −8.98738 −0.327084
\(756\) 0 0
\(757\) 4.53023 0.164654 0.0823271 0.996605i \(-0.473765\pi\)
0.0823271 + 0.996605i \(0.473765\pi\)
\(758\) 25.4363 0.923888
\(759\) 0 0
\(760\) −8.22364 −0.298303
\(761\) 33.9151 1.22942 0.614710 0.788753i \(-0.289273\pi\)
0.614710 + 0.788753i \(0.289273\pi\)
\(762\) 0 0
\(763\) −34.5513 −1.25084
\(764\) 0.729810 0.0264036
\(765\) 0 0
\(766\) −18.8353 −0.680548
\(767\) −57.3321 −2.07014
\(768\) 0 0
\(769\) 2.78366 0.100381 0.0501906 0.998740i \(-0.484017\pi\)
0.0501906 + 0.998740i \(0.484017\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.29982 0.0467817
\(773\) −7.15532 −0.257359 −0.128680 0.991686i \(-0.541074\pi\)
−0.128680 + 0.991686i \(0.541074\pi\)
\(774\) 0 0
\(775\) −4.30539 −0.154654
\(776\) 12.2591 0.440077
\(777\) 0 0
\(778\) −33.2224 −1.19108
\(779\) 14.1247 0.506071
\(780\) 0 0
\(781\) 0 0
\(782\) −12.2823 −0.439216
\(783\) 0 0
\(784\) 8.64259 0.308664
\(785\) 24.3233 0.868135
\(786\) 0 0
\(787\) 20.0978 0.716411 0.358205 0.933643i \(-0.383389\pi\)
0.358205 + 0.933643i \(0.383389\pi\)
\(788\) −4.79819 −0.170928
\(789\) 0 0
\(790\) 23.0919 0.821571
\(791\) 29.1016 1.03474
\(792\) 0 0
\(793\) 83.7053 2.97246
\(794\) −33.4036 −1.18545
\(795\) 0 0
\(796\) −1.79478 −0.0636143
\(797\) 21.8979 0.775663 0.387832 0.921730i \(-0.373224\pi\)
0.387832 + 0.921730i \(0.373224\pi\)
\(798\) 0 0
\(799\) 55.8678 1.97646
\(800\) −6.66195 −0.235536
\(801\) 0 0
\(802\) 24.6493 0.870396
\(803\) 0 0
\(804\) 0 0
\(805\) 3.34638 0.117945
\(806\) 11.4212 0.402295
\(807\) 0 0
\(808\) 22.9127 0.806065
\(809\) −5.33571 −0.187593 −0.0937967 0.995591i \(-0.529900\pi\)
−0.0937967 + 0.995591i \(0.529900\pi\)
\(810\) 0 0
\(811\) 7.04918 0.247530 0.123765 0.992312i \(-0.460503\pi\)
0.123765 + 0.992312i \(0.460503\pi\)
\(812\) −3.44751 −0.120984
\(813\) 0 0
\(814\) 0 0
\(815\) 9.96226 0.348963
\(816\) 0 0
\(817\) 21.6428 0.757186
\(818\) 13.7130 0.479462
\(819\) 0 0
\(820\) −1.62360 −0.0566984
\(821\) 2.35318 0.0821264 0.0410632 0.999157i \(-0.486926\pi\)
0.0410632 + 0.999157i \(0.486926\pi\)
\(822\) 0 0
\(823\) −48.7811 −1.70040 −0.850201 0.526457i \(-0.823520\pi\)
−0.850201 + 0.526457i \(0.823520\pi\)
\(824\) 7.54328 0.262783
\(825\) 0 0
\(826\) 29.5251 1.02731
\(827\) 21.2670 0.739527 0.369764 0.929126i \(-0.379439\pi\)
0.369764 + 0.929126i \(0.379439\pi\)
\(828\) 0 0
\(829\) 24.7732 0.860408 0.430204 0.902732i \(-0.358442\pi\)
0.430204 + 0.902732i \(0.358442\pi\)
\(830\) 26.2751 0.912022
\(831\) 0 0
\(832\) −42.6117 −1.47729
\(833\) 11.3141 0.392010
\(834\) 0 0
\(835\) −21.6822 −0.750344
\(836\) 0 0
\(837\) 0 0
\(838\) −34.6492 −1.19694
\(839\) −3.50888 −0.121140 −0.0605701 0.998164i \(-0.519292\pi\)
−0.0605701 + 0.998164i \(0.519292\pi\)
\(840\) 0 0
\(841\) −4.89470 −0.168783
\(842\) −31.2789 −1.07794
\(843\) 0 0
\(844\) −3.32885 −0.114584
\(845\) 34.1099 1.17342
\(846\) 0 0
\(847\) 0 0
\(848\) −9.33129 −0.320438
\(849\) 0 0
\(850\) −34.3789 −1.17919
\(851\) −0.301292 −0.0103282
\(852\) 0 0
\(853\) 2.29563 0.0786008 0.0393004 0.999227i \(-0.487487\pi\)
0.0393004 + 0.999227i \(0.487487\pi\)
\(854\) −43.1069 −1.47509
\(855\) 0 0
\(856\) 28.4002 0.970700
\(857\) −15.8536 −0.541548 −0.270774 0.962643i \(-0.587280\pi\)
−0.270774 + 0.962643i \(0.587280\pi\)
\(858\) 0 0
\(859\) 4.97552 0.169763 0.0848813 0.996391i \(-0.472949\pi\)
0.0848813 + 0.996391i \(0.472949\pi\)
\(860\) −2.48777 −0.0848324
\(861\) 0 0
\(862\) −7.90014 −0.269080
\(863\) 40.3432 1.37330 0.686650 0.726988i \(-0.259081\pi\)
0.686650 + 0.726988i \(0.259081\pi\)
\(864\) 0 0
\(865\) 25.4158 0.864161
\(866\) −5.72876 −0.194671
\(867\) 0 0
\(868\) −0.792029 −0.0268832
\(869\) 0 0
\(870\) 0 0
\(871\) 19.3798 0.656660
\(872\) 39.3175 1.33146
\(873\) 0 0
\(874\) −6.10530 −0.206515
\(875\) 21.6365 0.731447
\(876\) 0 0
\(877\) −44.3405 −1.49727 −0.748637 0.662980i \(-0.769291\pi\)
−0.748637 + 0.662980i \(0.769291\pi\)
\(878\) −16.9860 −0.573248
\(879\) 0 0
\(880\) 0 0
\(881\) −48.6174 −1.63796 −0.818981 0.573820i \(-0.805461\pi\)
−0.818981 + 0.573820i \(0.805461\pi\)
\(882\) 0 0
\(883\) −11.8277 −0.398033 −0.199017 0.979996i \(-0.563775\pi\)
−0.199017 + 0.979996i \(0.563775\pi\)
\(884\) 12.2808 0.413049
\(885\) 0 0
\(886\) −24.6228 −0.827218
\(887\) −17.6043 −0.591094 −0.295547 0.955328i \(-0.595502\pi\)
−0.295547 + 0.955328i \(0.595502\pi\)
\(888\) 0 0
\(889\) −46.8275 −1.57054
\(890\) −1.05119 −0.0352359
\(891\) 0 0
\(892\) −6.71541 −0.224849
\(893\) 27.7707 0.929312
\(894\) 0 0
\(895\) 11.8115 0.394815
\(896\) 29.8199 0.996213
\(897\) 0 0
\(898\) −4.97135 −0.165896
\(899\) 5.53794 0.184701
\(900\) 0 0
\(901\) −12.2157 −0.406963
\(902\) 0 0
\(903\) 0 0
\(904\) −33.1161 −1.10142
\(905\) 13.5206 0.449442
\(906\) 0 0
\(907\) 31.4287 1.04357 0.521786 0.853076i \(-0.325266\pi\)
0.521786 + 0.853076i \(0.325266\pi\)
\(908\) 1.30066 0.0431641
\(909\) 0 0
\(910\) −24.8477 −0.823695
\(911\) −25.4380 −0.842798 −0.421399 0.906875i \(-0.638461\pi\)
−0.421399 + 0.906875i \(0.638461\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −23.2033 −0.767497
\(915\) 0 0
\(916\) −6.09075 −0.201244
\(917\) −23.8026 −0.786030
\(918\) 0 0
\(919\) −33.3699 −1.10077 −0.550386 0.834910i \(-0.685519\pi\)
−0.550386 + 0.834910i \(0.685519\pi\)
\(920\) −3.80800 −0.125546
\(921\) 0 0
\(922\) −11.8954 −0.391755
\(923\) 58.3577 1.92087
\(924\) 0 0
\(925\) −0.843332 −0.0277286
\(926\) −25.6007 −0.841290
\(927\) 0 0
\(928\) 8.56915 0.281296
\(929\) −14.4401 −0.473765 −0.236883 0.971538i \(-0.576126\pi\)
−0.236883 + 0.971538i \(0.576126\pi\)
\(930\) 0 0
\(931\) 5.62399 0.184319
\(932\) −3.25672 −0.106677
\(933\) 0 0
\(934\) 57.7960 1.89114
\(935\) 0 0
\(936\) 0 0
\(937\) 29.6541 0.968757 0.484379 0.874859i \(-0.339046\pi\)
0.484379 + 0.874859i \(0.339046\pi\)
\(938\) −9.98028 −0.325868
\(939\) 0 0
\(940\) −3.19216 −0.104117
\(941\) 43.1420 1.40639 0.703194 0.710998i \(-0.251756\pi\)
0.703194 + 0.710998i \(0.251756\pi\)
\(942\) 0 0
\(943\) 6.54054 0.212989
\(944\) −38.9560 −1.26791
\(945\) 0 0
\(946\) 0 0
\(947\) 37.0608 1.20432 0.602158 0.798377i \(-0.294308\pi\)
0.602158 + 0.798377i \(0.294308\pi\)
\(948\) 0 0
\(949\) −7.83554 −0.254352
\(950\) −17.0890 −0.554442
\(951\) 0 0
\(952\) 34.3174 1.11223
\(953\) 25.1122 0.813463 0.406731 0.913548i \(-0.366669\pi\)
0.406731 + 0.913548i \(0.366669\pi\)
\(954\) 0 0
\(955\) 2.55051 0.0825326
\(956\) −4.30778 −0.139323
\(957\) 0 0
\(958\) 23.6342 0.763587
\(959\) 26.9892 0.871526
\(960\) 0 0
\(961\) −29.7277 −0.958959
\(962\) 2.23717 0.0721292
\(963\) 0 0
\(964\) −0.663644 −0.0213745
\(965\) 4.54257 0.146231
\(966\) 0 0
\(967\) 7.41108 0.238324 0.119162 0.992875i \(-0.461979\pi\)
0.119162 + 0.992875i \(0.461979\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) −7.89557 −0.253512
\(971\) 2.88466 0.0925733 0.0462866 0.998928i \(-0.485261\pi\)
0.0462866 + 0.998928i \(0.485261\pi\)
\(972\) 0 0
\(973\) −28.5700 −0.915913
\(974\) 32.1086 1.02883
\(975\) 0 0
\(976\) 56.8761 1.82056
\(977\) −17.9632 −0.574693 −0.287346 0.957827i \(-0.592773\pi\)
−0.287346 + 0.957827i \(0.592773\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.646461 −0.0206504
\(981\) 0 0
\(982\) −62.9246 −2.00800
\(983\) −31.5601 −1.00661 −0.503305 0.864109i \(-0.667883\pi\)
−0.503305 + 0.864109i \(0.667883\pi\)
\(984\) 0 0
\(985\) −16.7685 −0.534290
\(986\) 44.2210 1.40828
\(987\) 0 0
\(988\) 6.10455 0.194211
\(989\) 10.0218 0.318675
\(990\) 0 0
\(991\) 28.4307 0.903132 0.451566 0.892238i \(-0.350866\pi\)
0.451566 + 0.892238i \(0.350866\pi\)
\(992\) 1.96867 0.0625054
\(993\) 0 0
\(994\) −30.0533 −0.953232
\(995\) −6.27233 −0.198846
\(996\) 0 0
\(997\) 41.4489 1.31270 0.656350 0.754457i \(-0.272099\pi\)
0.656350 + 0.754457i \(0.272099\pi\)
\(998\) −10.7625 −0.340681
\(999\) 0 0
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 9801.2.a.bp.1.2 yes 5
3.2 odd 2 9801.2.a.bo.1.4 yes 5
11.10 odd 2 9801.2.a.bn.1.4 5
33.32 even 2 9801.2.a.bq.1.2 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9801.2.a.bn.1.4 5 11.10 odd 2
9801.2.a.bo.1.4 yes 5 3.2 odd 2
9801.2.a.bp.1.2 yes 5 1.1 even 1 trivial
9801.2.a.bq.1.2 yes 5 33.32 even 2