Properties

Label 75.22.a.j.1.6
Level $75$
Weight $22$
Character 75.1
Self dual yes
Analytic conductor $209.608$
Analytic rank $1$
Dimension $6$
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] = [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: \(6\)
Coefficient field: \(\mathbb{Q}[x]/(x^{6} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} - 530373x^{4} - 23850203x^{3} + 59940607490x^{2} + 2888057920800x - 684850637484000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18}\cdot 3^{6}\cdot 5^{7}\cdot 7^{2} \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.6
Root \(-571.187\) 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)
 
\(f(q)\) \(=\) \(q+2300.75 q^{2} -59049.0 q^{3} +3.19629e6 q^{4} -1.35857e8 q^{6} -2.10374e8 q^{7} +2.52885e9 q^{8} +3.48678e9 q^{9} +O(q^{10})\) \(q+2300.75 q^{2} -59049.0 q^{3} +3.19629e6 q^{4} -1.35857e8 q^{6} -2.10374e8 q^{7} +2.52885e9 q^{8} +3.48678e9 q^{9} +6.30187e10 q^{11} -1.88738e11 q^{12} -3.66488e11 q^{13} -4.84019e11 q^{14} -8.84863e11 q^{16} +5.08634e12 q^{17} +8.02222e12 q^{18} +7.36866e12 q^{19} +1.24224e13 q^{21} +1.44990e14 q^{22} +5.08468e12 q^{23} -1.49326e14 q^{24} -8.43196e14 q^{26} -2.05891e14 q^{27} -6.72418e14 q^{28} +1.15873e15 q^{29} -2.75579e15 q^{31} -7.33923e15 q^{32} -3.72119e15 q^{33} +1.17024e16 q^{34} +1.11448e16 q^{36} -1.13252e16 q^{37} +1.69534e16 q^{38} +2.16407e16 q^{39} -3.82078e16 q^{41} +2.85808e16 q^{42} +1.60836e17 q^{43} +2.01426e17 q^{44} +1.16986e16 q^{46} -7.69010e16 q^{47} +5.22503e16 q^{48} -5.14289e17 q^{49} -3.00343e17 q^{51} -1.17140e18 q^{52} +1.26003e18 q^{53} -4.73704e17 q^{54} -5.32005e17 q^{56} -4.35112e17 q^{57} +2.66595e18 q^{58} -4.45998e16 q^{59} -7.65725e18 q^{61} -6.34039e18 q^{62} -7.33530e17 q^{63} -1.50300e19 q^{64} -8.56153e18 q^{66} -4.35494e18 q^{67} +1.62574e19 q^{68} -3.00245e17 q^{69} -4.76688e19 q^{71} +8.81756e18 q^{72} -8.98402e18 q^{73} -2.60564e19 q^{74} +2.35524e19 q^{76} -1.32575e19 q^{77} +4.97899e19 q^{78} -9.52432e19 q^{79} +1.21577e19 q^{81} -8.79065e19 q^{82} +1.50799e20 q^{83} +3.97056e19 q^{84} +3.70043e20 q^{86} -6.84219e19 q^{87} +1.59365e20 q^{88} +7.17533e19 q^{89} +7.70996e19 q^{91} +1.62521e19 q^{92} +1.62727e20 q^{93} -1.76930e20 q^{94} +4.33374e20 q^{96} +2.27790e20 q^{97} -1.18325e21 q^{98} +2.19733e20 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 92 q^{2} - 354294 q^{3} + 4390448 q^{4} - 5432508 q^{6} + 1327143454 q^{7} - 4252271424 q^{8} + 20920706406 q^{9} - 78310112516 q^{11} - 259251563952 q^{12} + 116029746338 q^{13} - 313825730148 q^{14} + 1996701612800 q^{16} + 5382900513068 q^{17} + 320784164892 q^{18} + 6969638997622 q^{19} - 78366493815246 q^{21} - 19701817271864 q^{22} + 295754895311052 q^{23} + 251092375315776 q^{24} - 274314326263628 q^{26} - 12\!\cdots\!94 q^{27}+ \cdots - 27\!\cdots\!16 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2300.75 1.58875 0.794373 0.607431i \(-0.207800\pi\)
0.794373 + 0.607431i \(0.207800\pi\)
\(3\) −59049.0 −0.577350
\(4\) 3.19629e6 1.52411
\(5\) 0 0
\(6\) −1.35857e8 −0.917263
\(7\) −2.10374e8 −0.281490 −0.140745 0.990046i \(-0.544950\pi\)
−0.140745 + 0.990046i \(0.544950\pi\)
\(8\) 2.52885e9 0.832681
\(9\) 3.48678e9 0.333333
\(10\) 0 0
\(11\) 6.30187e10 0.732565 0.366283 0.930504i \(-0.380630\pi\)
0.366283 + 0.930504i \(0.380630\pi\)
\(12\) −1.88738e11 −0.879946
\(13\) −3.66488e11 −0.737317 −0.368659 0.929565i \(-0.620183\pi\)
−0.368659 + 0.929565i \(0.620183\pi\)
\(14\) −4.84019e11 −0.447216
\(15\) 0 0
\(16\) −8.84863e11 −0.201195
\(17\) 5.08634e12 0.611916 0.305958 0.952045i \(-0.401023\pi\)
0.305958 + 0.952045i \(0.401023\pi\)
\(18\) 8.02222e12 0.529582
\(19\) 7.36866e12 0.275725 0.137862 0.990451i \(-0.455977\pi\)
0.137862 + 0.990451i \(0.455977\pi\)
\(20\) 0 0
\(21\) 1.24224e13 0.162518
\(22\) 1.44990e14 1.16386
\(23\) 5.08468e12 0.0255930 0.0127965 0.999918i \(-0.495927\pi\)
0.0127965 + 0.999918i \(0.495927\pi\)
\(24\) −1.49326e14 −0.480748
\(25\) 0 0
\(26\) −8.43196e14 −1.17141
\(27\) −2.05891e14 −0.192450
\(28\) −6.72418e14 −0.429023
\(29\) 1.15873e15 0.511450 0.255725 0.966750i \(-0.417686\pi\)
0.255725 + 0.966750i \(0.417686\pi\)
\(30\) 0 0
\(31\) −2.75579e15 −0.603878 −0.301939 0.953327i \(-0.597634\pi\)
−0.301939 + 0.953327i \(0.597634\pi\)
\(32\) −7.33923e15 −1.15233
\(33\) −3.72119e15 −0.422947
\(34\) 1.17024e16 0.972179
\(35\) 0 0
\(36\) 1.11448e16 0.508037
\(37\) −1.13252e16 −0.387192 −0.193596 0.981081i \(-0.562015\pi\)
−0.193596 + 0.981081i \(0.562015\pi\)
\(38\) 1.69534e16 0.438056
\(39\) 2.16407e16 0.425690
\(40\) 0 0
\(41\) −3.82078e16 −0.444551 −0.222275 0.974984i \(-0.571348\pi\)
−0.222275 + 0.974984i \(0.571348\pi\)
\(42\) 2.85808e16 0.258200
\(43\) 1.60836e17 1.13492 0.567459 0.823402i \(-0.307927\pi\)
0.567459 + 0.823402i \(0.307927\pi\)
\(44\) 2.01426e17 1.11651
\(45\) 0 0
\(46\) 1.16986e16 0.0406608
\(47\) −7.69010e16 −0.213258 −0.106629 0.994299i \(-0.534006\pi\)
−0.106629 + 0.994299i \(0.534006\pi\)
\(48\) 5.22503e16 0.116160
\(49\) −5.14289e17 −0.920763
\(50\) 0 0
\(51\) −3.00343e17 −0.353290
\(52\) −1.17140e18 −1.12375
\(53\) 1.26003e18 0.989653 0.494826 0.868992i \(-0.335232\pi\)
0.494826 + 0.868992i \(0.335232\pi\)
\(54\) −4.73704e17 −0.305754
\(55\) 0 0
\(56\) −5.32005e17 −0.234391
\(57\) −4.35112e17 −0.159190
\(58\) 2.66595e18 0.812564
\(59\) −4.45998e16 −0.0113602 −0.00568012 0.999984i \(-0.501808\pi\)
−0.00568012 + 0.999984i \(0.501808\pi\)
\(60\) 0 0
\(61\) −7.65725e18 −1.37439 −0.687195 0.726473i \(-0.741158\pi\)
−0.687195 + 0.726473i \(0.741158\pi\)
\(62\) −6.34039e18 −0.959408
\(63\) −7.33530e17 −0.0938301
\(64\) −1.50300e19 −1.62956
\(65\) 0 0
\(66\) −8.56153e18 −0.671954
\(67\) −4.35494e18 −0.291875 −0.145938 0.989294i \(-0.546620\pi\)
−0.145938 + 0.989294i \(0.546620\pi\)
\(68\) 1.62574e19 0.932629
\(69\) −3.00245e17 −0.0147761
\(70\) 0 0
\(71\) −4.76688e19 −1.73789 −0.868943 0.494912i \(-0.835200\pi\)
−0.868943 + 0.494912i \(0.835200\pi\)
\(72\) 8.81756e18 0.277560
\(73\) −8.98402e18 −0.244670 −0.122335 0.992489i \(-0.539038\pi\)
−0.122335 + 0.992489i \(0.539038\pi\)
\(74\) −2.60564e19 −0.615150
\(75\) 0 0
\(76\) 2.35524e19 0.420235
\(77\) −1.32575e19 −0.206210
\(78\) 4.97899e19 0.676313
\(79\) −9.52432e19 −1.13175 −0.565874 0.824492i \(-0.691461\pi\)
−0.565874 + 0.824492i \(0.691461\pi\)
\(80\) 0 0
\(81\) 1.21577e19 0.111111
\(82\) −8.79065e19 −0.706278
\(83\) 1.50799e20 1.06679 0.533396 0.845866i \(-0.320916\pi\)
0.533396 + 0.845866i \(0.320916\pi\)
\(84\) 3.97056e19 0.247696
\(85\) 0 0
\(86\) 3.70043e20 1.80310
\(87\) −6.84219e19 −0.295286
\(88\) 1.59365e20 0.609993
\(89\) 7.17533e19 0.243920 0.121960 0.992535i \(-0.461082\pi\)
0.121960 + 0.992535i \(0.461082\pi\)
\(90\) 0 0
\(91\) 7.70996e19 0.207548
\(92\) 1.62521e19 0.0390066
\(93\) 1.62727e20 0.348649
\(94\) −1.76930e20 −0.338812
\(95\) 0 0
\(96\) 4.33374e20 0.665297
\(97\) 2.27790e20 0.313639 0.156820 0.987627i \(-0.449876\pi\)
0.156820 + 0.987627i \(0.449876\pi\)
\(98\) −1.18325e21 −1.46286
\(99\) 2.19733e20 0.244188
\(100\) 0 0
\(101\) 1.49001e21 1.34220 0.671098 0.741369i \(-0.265823\pi\)
0.671098 + 0.741369i \(0.265823\pi\)
\(102\) −6.91015e20 −0.561288
\(103\) −1.97182e21 −1.44569 −0.722847 0.691008i \(-0.757167\pi\)
−0.722847 + 0.691008i \(0.757167\pi\)
\(104\) −9.26793e20 −0.613950
\(105\) 0 0
\(106\) 2.89900e21 1.57231
\(107\) −1.05324e21 −0.517603 −0.258801 0.965931i \(-0.583327\pi\)
−0.258801 + 0.965931i \(0.583327\pi\)
\(108\) −6.58089e20 −0.293315
\(109\) −3.54520e21 −1.43438 −0.717188 0.696880i \(-0.754571\pi\)
−0.717188 + 0.696880i \(0.754571\pi\)
\(110\) 0 0
\(111\) 6.68740e20 0.223546
\(112\) 1.86152e20 0.0566343
\(113\) 2.70245e20 0.0748919 0.0374459 0.999299i \(-0.488078\pi\)
0.0374459 + 0.999299i \(0.488078\pi\)
\(114\) −1.00108e21 −0.252912
\(115\) 0 0
\(116\) 3.70364e21 0.779507
\(117\) −1.27786e21 −0.245772
\(118\) −1.02613e20 −0.0180485
\(119\) −1.07004e21 −0.172248
\(120\) 0 0
\(121\) −3.42889e21 −0.463348
\(122\) −1.76174e22 −2.18355
\(123\) 2.25613e21 0.256661
\(124\) −8.80833e21 −0.920377
\(125\) 0 0
\(126\) −1.68767e21 −0.149072
\(127\) −2.21609e22 −1.80156 −0.900780 0.434277i \(-0.857004\pi\)
−0.900780 + 0.434277i \(0.857004\pi\)
\(128\) −1.91889e22 −1.43663
\(129\) −9.49720e21 −0.655245
\(130\) 0 0
\(131\) 9.46139e19 0.00555400 0.00277700 0.999996i \(-0.499116\pi\)
0.00277700 + 0.999996i \(0.499116\pi\)
\(132\) −1.18940e22 −0.644618
\(133\) −1.55018e21 −0.0776138
\(134\) −1.00196e22 −0.463715
\(135\) 0 0
\(136\) 1.28626e22 0.509531
\(137\) −4.94865e22 −1.81518 −0.907592 0.419854i \(-0.862081\pi\)
−0.907592 + 0.419854i \(0.862081\pi\)
\(138\) −6.90789e20 −0.0234755
\(139\) −1.77582e22 −0.559428 −0.279714 0.960083i \(-0.590240\pi\)
−0.279714 + 0.960083i \(0.590240\pi\)
\(140\) 0 0
\(141\) 4.54093e21 0.123124
\(142\) −1.09674e23 −2.76106
\(143\) −2.30956e22 −0.540133
\(144\) −3.08533e21 −0.0670649
\(145\) 0 0
\(146\) −2.06700e22 −0.388718
\(147\) 3.03682e22 0.531603
\(148\) −3.61986e22 −0.590125
\(149\) −3.44662e22 −0.523525 −0.261762 0.965132i \(-0.584304\pi\)
−0.261762 + 0.965132i \(0.584304\pi\)
\(150\) 0 0
\(151\) 6.39694e22 0.844723 0.422361 0.906428i \(-0.361201\pi\)
0.422361 + 0.906428i \(0.361201\pi\)
\(152\) 1.86342e22 0.229591
\(153\) 1.77350e22 0.203972
\(154\) −3.05022e22 −0.327615
\(155\) 0 0
\(156\) 6.91702e22 0.648800
\(157\) 4.97870e22 0.436686 0.218343 0.975872i \(-0.429935\pi\)
0.218343 + 0.975872i \(0.429935\pi\)
\(158\) −2.19131e23 −1.79806
\(159\) −7.44033e22 −0.571376
\(160\) 0 0
\(161\) −1.06969e21 −0.00720418
\(162\) 2.79717e22 0.176527
\(163\) 1.16432e23 0.688814 0.344407 0.938821i \(-0.388080\pi\)
0.344407 + 0.938821i \(0.388080\pi\)
\(164\) −1.22123e23 −0.677545
\(165\) 0 0
\(166\) 3.46952e23 1.69486
\(167\) −5.65142e22 −0.259200 −0.129600 0.991566i \(-0.541369\pi\)
−0.129600 + 0.991566i \(0.541369\pi\)
\(168\) 3.14144e22 0.135326
\(169\) −1.12751e23 −0.456363
\(170\) 0 0
\(171\) 2.56929e22 0.0919082
\(172\) 5.14079e23 1.72974
\(173\) −3.02320e23 −0.957157 −0.478579 0.878045i \(-0.658848\pi\)
−0.478579 + 0.878045i \(0.658848\pi\)
\(174\) −1.57422e23 −0.469134
\(175\) 0 0
\(176\) −5.57629e22 −0.147388
\(177\) 2.63358e21 0.00655883
\(178\) 1.65086e23 0.387526
\(179\) 8.48756e23 1.87856 0.939282 0.343147i \(-0.111493\pi\)
0.939282 + 0.343147i \(0.111493\pi\)
\(180\) 0 0
\(181\) −3.06671e23 −0.604016 −0.302008 0.953305i \(-0.597657\pi\)
−0.302008 + 0.953305i \(0.597657\pi\)
\(182\) 1.77387e23 0.329740
\(183\) 4.52153e23 0.793504
\(184\) 1.28584e22 0.0213108
\(185\) 0 0
\(186\) 3.74394e23 0.553914
\(187\) 3.20535e23 0.448268
\(188\) −2.45798e23 −0.325028
\(189\) 4.33142e22 0.0541728
\(190\) 0 0
\(191\) −1.59483e24 −1.78593 −0.892964 0.450128i \(-0.851378\pi\)
−0.892964 + 0.450128i \(0.851378\pi\)
\(192\) 8.87509e23 0.940827
\(193\) 1.20399e24 1.20857 0.604283 0.796770i \(-0.293460\pi\)
0.604283 + 0.796770i \(0.293460\pi\)
\(194\) 5.24087e23 0.498293
\(195\) 0 0
\(196\) −1.64382e24 −1.40335
\(197\) −2.54274e23 −0.205782 −0.102891 0.994693i \(-0.532809\pi\)
−0.102891 + 0.994693i \(0.532809\pi\)
\(198\) 5.05550e23 0.387953
\(199\) −1.83859e24 −1.33822 −0.669111 0.743162i \(-0.733325\pi\)
−0.669111 + 0.743162i \(0.733325\pi\)
\(200\) 0 0
\(201\) 2.57155e23 0.168514
\(202\) 3.42815e24 2.13241
\(203\) −2.43767e23 −0.143968
\(204\) −9.59986e23 −0.538453
\(205\) 0 0
\(206\) −4.53667e24 −2.29684
\(207\) 1.77292e22 0.00853100
\(208\) 3.24292e23 0.148344
\(209\) 4.64363e23 0.201986
\(210\) 0 0
\(211\) −2.45445e24 −0.966025 −0.483013 0.875613i \(-0.660457\pi\)
−0.483013 + 0.875613i \(0.660457\pi\)
\(212\) 4.02741e24 1.50834
\(213\) 2.81479e24 1.00337
\(214\) −2.42323e24 −0.822339
\(215\) 0 0
\(216\) −5.20668e23 −0.160249
\(217\) 5.79748e23 0.169986
\(218\) −8.15662e24 −2.27886
\(219\) 5.30497e23 0.141260
\(220\) 0 0
\(221\) −1.86408e24 −0.451176
\(222\) 1.53860e24 0.355157
\(223\) −2.05271e24 −0.451988 −0.225994 0.974129i \(-0.572563\pi\)
−0.225994 + 0.974129i \(0.572563\pi\)
\(224\) 1.54399e24 0.324369
\(225\) 0 0
\(226\) 6.21767e23 0.118984
\(227\) 5.12767e24 0.936804 0.468402 0.883516i \(-0.344830\pi\)
0.468402 + 0.883516i \(0.344830\pi\)
\(228\) −1.39075e24 −0.242623
\(229\) 2.29982e24 0.383196 0.191598 0.981473i \(-0.438633\pi\)
0.191598 + 0.981473i \(0.438633\pi\)
\(230\) 0 0
\(231\) 7.82843e23 0.119055
\(232\) 2.93026e24 0.425875
\(233\) −1.35634e25 −1.88422 −0.942109 0.335308i \(-0.891160\pi\)
−0.942109 + 0.335308i \(0.891160\pi\)
\(234\) −2.94004e24 −0.390470
\(235\) 0 0
\(236\) −1.42554e23 −0.0173143
\(237\) 5.62402e24 0.653415
\(238\) −2.46188e24 −0.273659
\(239\) 1.88596e24 0.200610 0.100305 0.994957i \(-0.468018\pi\)
0.100305 + 0.994957i \(0.468018\pi\)
\(240\) 0 0
\(241\) 1.04662e25 1.02002 0.510010 0.860168i \(-0.329642\pi\)
0.510010 + 0.860168i \(0.329642\pi\)
\(242\) −7.88903e24 −0.736143
\(243\) −7.17898e23 −0.0641500
\(244\) −2.44748e25 −2.09472
\(245\) 0 0
\(246\) 5.19079e24 0.407770
\(247\) −2.70052e24 −0.203297
\(248\) −6.96899e24 −0.502837
\(249\) −8.90456e24 −0.615913
\(250\) 0 0
\(251\) 1.72531e25 1.09722 0.548610 0.836078i \(-0.315157\pi\)
0.548610 + 0.836078i \(0.315157\pi\)
\(252\) −2.34458e24 −0.143008
\(253\) 3.20430e23 0.0187485
\(254\) −5.09867e25 −2.86222
\(255\) 0 0
\(256\) −1.26285e25 −0.652878
\(257\) 8.63659e24 0.428593 0.214296 0.976769i \(-0.431254\pi\)
0.214296 + 0.976769i \(0.431254\pi\)
\(258\) −2.18507e25 −1.04102
\(259\) 2.38253e24 0.108991
\(260\) 0 0
\(261\) 4.04024e24 0.170483
\(262\) 2.17683e23 0.00882390
\(263\) 6.94089e24 0.270321 0.135161 0.990824i \(-0.456845\pi\)
0.135161 + 0.990824i \(0.456845\pi\)
\(264\) −9.41034e24 −0.352179
\(265\) 0 0
\(266\) −3.56657e24 −0.123309
\(267\) −4.23696e24 −0.140827
\(268\) −1.39197e25 −0.444850
\(269\) −2.37580e25 −0.730149 −0.365075 0.930978i \(-0.618957\pi\)
−0.365075 + 0.930978i \(0.618957\pi\)
\(270\) 0 0
\(271\) −4.58603e25 −1.30394 −0.651972 0.758243i \(-0.726058\pi\)
−0.651972 + 0.758243i \(0.726058\pi\)
\(272\) −4.50072e24 −0.123114
\(273\) −4.55266e24 −0.119828
\(274\) −1.13856e26 −2.88386
\(275\) 0 0
\(276\) −9.59672e23 −0.0225205
\(277\) −6.09487e25 −1.37698 −0.688489 0.725247i \(-0.741726\pi\)
−0.688489 + 0.725247i \(0.741726\pi\)
\(278\) −4.08572e25 −0.888788
\(279\) −9.60886e24 −0.201293
\(280\) 0 0
\(281\) 3.41139e25 0.663003 0.331501 0.943455i \(-0.392445\pi\)
0.331501 + 0.943455i \(0.392445\pi\)
\(282\) 1.04475e25 0.195613
\(283\) −4.80459e25 −0.866760 −0.433380 0.901211i \(-0.642679\pi\)
−0.433380 + 0.901211i \(0.642679\pi\)
\(284\) −1.52363e26 −2.64873
\(285\) 0 0
\(286\) −5.31371e25 −0.858133
\(287\) 8.03794e24 0.125137
\(288\) −2.55903e25 −0.384109
\(289\) −4.32211e25 −0.625559
\(290\) 0 0
\(291\) −1.34507e25 −0.181080
\(292\) −2.87156e25 −0.372904
\(293\) 1.15610e26 1.44840 0.724198 0.689593i \(-0.242210\pi\)
0.724198 + 0.689593i \(0.242210\pi\)
\(294\) 6.98697e25 0.844582
\(295\) 0 0
\(296\) −2.86397e25 −0.322408
\(297\) −1.29750e25 −0.140982
\(298\) −7.92980e25 −0.831748
\(299\) −1.86347e24 −0.0188702
\(300\) 0 0
\(301\) −3.38358e25 −0.319468
\(302\) 1.47177e26 1.34205
\(303\) −8.79838e25 −0.774917
\(304\) −6.52025e24 −0.0554743
\(305\) 0 0
\(306\) 4.08037e25 0.324060
\(307\) 1.37622e26 1.05618 0.528088 0.849190i \(-0.322909\pi\)
0.528088 + 0.849190i \(0.322909\pi\)
\(308\) −4.23749e25 −0.314287
\(309\) 1.16434e26 0.834672
\(310\) 0 0
\(311\) 1.64151e26 1.09967 0.549833 0.835275i \(-0.314691\pi\)
0.549833 + 0.835275i \(0.314691\pi\)
\(312\) 5.47262e25 0.354464
\(313\) 8.22532e25 0.515154 0.257577 0.966258i \(-0.417076\pi\)
0.257577 + 0.966258i \(0.417076\pi\)
\(314\) 1.14547e26 0.693783
\(315\) 0 0
\(316\) −3.04425e26 −1.72491
\(317\) 2.64449e26 1.44950 0.724752 0.689010i \(-0.241954\pi\)
0.724752 + 0.689010i \(0.241954\pi\)
\(318\) −1.71183e26 −0.907772
\(319\) 7.30217e25 0.374671
\(320\) 0 0
\(321\) 6.21926e25 0.298838
\(322\) −2.46108e24 −0.0114456
\(323\) 3.74795e25 0.168720
\(324\) 3.88595e25 0.169346
\(325\) 0 0
\(326\) 2.67880e26 1.09435
\(327\) 2.09341e26 0.828137
\(328\) −9.66218e25 −0.370169
\(329\) 1.61780e25 0.0600299
\(330\) 0 0
\(331\) −5.13496e26 −1.78790 −0.893949 0.448168i \(-0.852077\pi\)
−0.893949 + 0.448168i \(0.852077\pi\)
\(332\) 4.81999e26 1.62591
\(333\) −3.94884e25 −0.129064
\(334\) −1.30025e26 −0.411802
\(335\) 0 0
\(336\) −1.09921e25 −0.0326978
\(337\) 7.57301e25 0.218351 0.109175 0.994023i \(-0.465179\pi\)
0.109175 + 0.994023i \(0.465179\pi\)
\(338\) −2.59412e26 −0.725045
\(339\) −1.59577e25 −0.0432388
\(340\) 0 0
\(341\) −1.73667e26 −0.442380
\(342\) 5.91130e25 0.146019
\(343\) 2.25697e26 0.540676
\(344\) 4.06730e26 0.945024
\(345\) 0 0
\(346\) −6.95563e26 −1.52068
\(347\) 2.39671e26 0.508343 0.254171 0.967159i \(-0.418197\pi\)
0.254171 + 0.967159i \(0.418197\pi\)
\(348\) −2.18696e26 −0.450049
\(349\) 2.95784e26 0.590619 0.295310 0.955402i \(-0.404577\pi\)
0.295310 + 0.955402i \(0.404577\pi\)
\(350\) 0 0
\(351\) 7.54566e25 0.141897
\(352\) −4.62509e26 −0.844155
\(353\) 4.45964e26 0.790069 0.395034 0.918666i \(-0.370733\pi\)
0.395034 + 0.918666i \(0.370733\pi\)
\(354\) 6.05920e24 0.0104203
\(355\) 0 0
\(356\) 2.29345e26 0.371761
\(357\) 6.31846e25 0.0994477
\(358\) 1.95277e27 2.98456
\(359\) −1.82800e26 −0.271321 −0.135661 0.990755i \(-0.543316\pi\)
−0.135661 + 0.990755i \(0.543316\pi\)
\(360\) 0 0
\(361\) −6.59912e26 −0.923976
\(362\) −7.05574e26 −0.959627
\(363\) 2.02473e26 0.267514
\(364\) 2.46433e26 0.316326
\(365\) 0 0
\(366\) 1.04029e27 1.26068
\(367\) −7.38077e26 −0.869176 −0.434588 0.900629i \(-0.643106\pi\)
−0.434588 + 0.900629i \(0.643106\pi\)
\(368\) −4.49925e24 −0.00514917
\(369\) −1.33222e26 −0.148184
\(370\) 0 0
\(371\) −2.65077e26 −0.278578
\(372\) 5.20123e26 0.531380
\(373\) 1.76554e27 1.75362 0.876808 0.480841i \(-0.159668\pi\)
0.876808 + 0.480841i \(0.159668\pi\)
\(374\) 7.37470e26 0.712184
\(375\) 0 0
\(376\) −1.94471e26 −0.177575
\(377\) −4.24660e26 −0.377101
\(378\) 9.96551e25 0.0860668
\(379\) −9.86553e26 −0.828722 −0.414361 0.910113i \(-0.635995\pi\)
−0.414361 + 0.910113i \(0.635995\pi\)
\(380\) 0 0
\(381\) 1.30858e27 1.04013
\(382\) −3.66930e27 −2.83739
\(383\) −1.49870e27 −1.12753 −0.563766 0.825934i \(-0.690648\pi\)
−0.563766 + 0.825934i \(0.690648\pi\)
\(384\) 1.13308e27 0.829438
\(385\) 0 0
\(386\) 2.77008e27 1.92010
\(387\) 5.60800e26 0.378306
\(388\) 7.28083e26 0.478022
\(389\) 1.22249e27 0.781224 0.390612 0.920556i \(-0.372264\pi\)
0.390612 + 0.920556i \(0.372264\pi\)
\(390\) 0 0
\(391\) 2.58624e25 0.0156608
\(392\) −1.30056e27 −0.766702
\(393\) −5.58685e24 −0.00320661
\(394\) −5.85021e26 −0.326935
\(395\) 0 0
\(396\) 7.02330e26 0.372170
\(397\) −3.53943e26 −0.182656 −0.0913279 0.995821i \(-0.529111\pi\)
−0.0913279 + 0.995821i \(0.529111\pi\)
\(398\) −4.23014e27 −2.12610
\(399\) 9.15364e25 0.0448103
\(400\) 0 0
\(401\) 1.38817e27 0.644804 0.322402 0.946603i \(-0.395510\pi\)
0.322402 + 0.946603i \(0.395510\pi\)
\(402\) 5.91649e26 0.267726
\(403\) 1.00996e27 0.445249
\(404\) 4.76252e27 2.04566
\(405\) 0 0
\(406\) −5.60847e26 −0.228729
\(407\) −7.13698e26 −0.283644
\(408\) −7.59524e26 −0.294178
\(409\) −4.10310e27 −1.54888 −0.774439 0.632648i \(-0.781968\pi\)
−0.774439 + 0.632648i \(0.781968\pi\)
\(410\) 0 0
\(411\) 2.92213e27 1.04800
\(412\) −6.30252e27 −2.20340
\(413\) 9.38266e24 0.00319779
\(414\) 4.07904e25 0.0135536
\(415\) 0 0
\(416\) 2.68974e27 0.849631
\(417\) 1.04860e27 0.322986
\(418\) 1.06838e27 0.320905
\(419\) 1.74062e27 0.509867 0.254934 0.966959i \(-0.417946\pi\)
0.254934 + 0.966959i \(0.417946\pi\)
\(420\) 0 0
\(421\) 3.18977e26 0.0888786 0.0444393 0.999012i \(-0.485850\pi\)
0.0444393 + 0.999012i \(0.485850\pi\)
\(422\) −5.64707e27 −1.53477
\(423\) −2.68137e26 −0.0710859
\(424\) 3.18642e27 0.824065
\(425\) 0 0
\(426\) 6.47613e27 1.59410
\(427\) 1.61089e27 0.386877
\(428\) −3.36646e27 −0.788884
\(429\) 1.36377e27 0.311846
\(430\) 0 0
\(431\) 4.01990e27 0.875395 0.437697 0.899122i \(-0.355794\pi\)
0.437697 + 0.899122i \(0.355794\pi\)
\(432\) 1.82185e26 0.0387199
\(433\) 5.98127e27 1.24071 0.620355 0.784321i \(-0.286988\pi\)
0.620355 + 0.784321i \(0.286988\pi\)
\(434\) 1.33386e27 0.270064
\(435\) 0 0
\(436\) −1.13315e28 −2.18615
\(437\) 3.74673e25 0.00705662
\(438\) 1.22054e27 0.224427
\(439\) −2.93253e27 −0.526460 −0.263230 0.964733i \(-0.584788\pi\)
−0.263230 + 0.964733i \(0.584788\pi\)
\(440\) 0 0
\(441\) −1.79321e27 −0.306921
\(442\) −4.28879e27 −0.716804
\(443\) 2.72621e27 0.444959 0.222480 0.974937i \(-0.428585\pi\)
0.222480 + 0.974937i \(0.428585\pi\)
\(444\) 2.13749e27 0.340709
\(445\) 0 0
\(446\) −4.72277e27 −0.718094
\(447\) 2.03519e27 0.302257
\(448\) 3.16194e27 0.458705
\(449\) 5.45985e27 0.773739 0.386869 0.922135i \(-0.373556\pi\)
0.386869 + 0.922135i \(0.373556\pi\)
\(450\) 0 0
\(451\) −2.40780e27 −0.325662
\(452\) 8.63784e26 0.114144
\(453\) −3.77733e27 −0.487701
\(454\) 1.17975e28 1.48834
\(455\) 0 0
\(456\) −1.10033e27 −0.132554
\(457\) −3.99132e27 −0.469891 −0.234945 0.972009i \(-0.575491\pi\)
−0.234945 + 0.972009i \(0.575491\pi\)
\(458\) 5.29131e27 0.608801
\(459\) −1.04723e27 −0.117763
\(460\) 0 0
\(461\) 4.71199e27 0.506226 0.253113 0.967437i \(-0.418546\pi\)
0.253113 + 0.967437i \(0.418546\pi\)
\(462\) 1.80113e27 0.189149
\(463\) 1.54367e28 1.58473 0.792364 0.610049i \(-0.208850\pi\)
0.792364 + 0.610049i \(0.208850\pi\)
\(464\) −1.02532e27 −0.102901
\(465\) 0 0
\(466\) −3.12060e28 −2.99354
\(467\) 1.83848e28 1.72438 0.862188 0.506589i \(-0.169094\pi\)
0.862188 + 0.506589i \(0.169094\pi\)
\(468\) −4.08443e27 −0.374585
\(469\) 9.16168e26 0.0821600
\(470\) 0 0
\(471\) −2.93987e27 −0.252121
\(472\) −1.12786e26 −0.00945944
\(473\) 1.01357e28 0.831401
\(474\) 1.29395e28 1.03811
\(475\) 0 0
\(476\) −3.42015e27 −0.262526
\(477\) 4.39344e27 0.329884
\(478\) 4.33911e27 0.318719
\(479\) −1.87722e28 −1.34894 −0.674469 0.738303i \(-0.735628\pi\)
−0.674469 + 0.738303i \(0.735628\pi\)
\(480\) 0 0
\(481\) 4.15054e27 0.285484
\(482\) 2.40800e28 1.62055
\(483\) 6.31639e25 0.00415934
\(484\) −1.09598e28 −0.706195
\(485\) 0 0
\(486\) −1.65170e27 −0.101918
\(487\) 7.96543e27 0.481011 0.240506 0.970648i \(-0.422687\pi\)
0.240506 + 0.970648i \(0.422687\pi\)
\(488\) −1.93641e28 −1.14443
\(489\) −6.87518e27 −0.397687
\(490\) 0 0
\(491\) 2.38795e28 1.32333 0.661666 0.749799i \(-0.269850\pi\)
0.661666 + 0.749799i \(0.269850\pi\)
\(492\) 7.21126e27 0.391181
\(493\) 5.89370e27 0.312965
\(494\) −6.21323e27 −0.322986
\(495\) 0 0
\(496\) 2.43850e27 0.121497
\(497\) 1.00283e28 0.489198
\(498\) −2.04872e28 −0.978528
\(499\) −1.85870e27 −0.0869270 −0.0434635 0.999055i \(-0.513839\pi\)
−0.0434635 + 0.999055i \(0.513839\pi\)
\(500\) 0 0
\(501\) 3.33711e27 0.149649
\(502\) 3.96951e28 1.74320
\(503\) 3.05087e28 1.31208 0.656039 0.754727i \(-0.272231\pi\)
0.656039 + 0.754727i \(0.272231\pi\)
\(504\) −1.85499e27 −0.0781305
\(505\) 0 0
\(506\) 7.37229e26 0.0297867
\(507\) 6.65785e27 0.263482
\(508\) −7.08328e28 −2.74578
\(509\) −2.93108e28 −1.11299 −0.556494 0.830851i \(-0.687854\pi\)
−0.556494 + 0.830851i \(0.687854\pi\)
\(510\) 0 0
\(511\) 1.89001e27 0.0688722
\(512\) 1.11870e28 0.399373
\(513\) −1.51714e27 −0.0530632
\(514\) 1.98706e28 0.680924
\(515\) 0 0
\(516\) −3.03559e28 −0.998667
\(517\) −4.84620e27 −0.156225
\(518\) 5.48160e27 0.173159
\(519\) 1.78517e28 0.552615
\(520\) 0 0
\(521\) 1.39401e28 0.414448 0.207224 0.978293i \(-0.433557\pi\)
0.207224 + 0.978293i \(0.433557\pi\)
\(522\) 9.29558e27 0.270855
\(523\) −1.92103e28 −0.548613 −0.274307 0.961642i \(-0.588448\pi\)
−0.274307 + 0.961642i \(0.588448\pi\)
\(524\) 3.02414e26 0.00846492
\(525\) 0 0
\(526\) 1.59693e28 0.429472
\(527\) −1.40169e28 −0.369522
\(528\) 3.29274e27 0.0850946
\(529\) −3.94457e28 −0.999345
\(530\) 0 0
\(531\) −1.55510e26 −0.00378674
\(532\) −4.95482e27 −0.118292
\(533\) 1.40027e28 0.327775
\(534\) −9.74819e27 −0.223739
\(535\) 0 0
\(536\) −1.10130e28 −0.243039
\(537\) −5.01182e28 −1.08459
\(538\) −5.46613e28 −1.16002
\(539\) −3.24098e28 −0.674519
\(540\) 0 0
\(541\) 2.24623e27 0.0449659 0.0224830 0.999747i \(-0.492843\pi\)
0.0224830 + 0.999747i \(0.492843\pi\)
\(542\) −1.05513e29 −2.07164
\(543\) 1.81086e28 0.348729
\(544\) −3.73299e28 −0.705128
\(545\) 0 0
\(546\) −1.04745e28 −0.190376
\(547\) −1.02940e28 −0.183535 −0.0917675 0.995780i \(-0.529252\pi\)
−0.0917675 + 0.995780i \(0.529252\pi\)
\(548\) −1.58173e29 −2.76654
\(549\) −2.66992e28 −0.458130
\(550\) 0 0
\(551\) 8.53829e27 0.141019
\(552\) −7.59276e26 −0.0123038
\(553\) 2.00367e28 0.318576
\(554\) −1.40228e29 −2.18767
\(555\) 0 0
\(556\) −5.67605e28 −0.852630
\(557\) −7.27319e28 −1.07213 −0.536063 0.844178i \(-0.680089\pi\)
−0.536063 + 0.844178i \(0.680089\pi\)
\(558\) −2.21076e28 −0.319803
\(559\) −5.89444e28 −0.836794
\(560\) 0 0
\(561\) −1.89273e28 −0.258808
\(562\) 7.84876e28 1.05334
\(563\) 9.50226e28 1.25167 0.625834 0.779957i \(-0.284759\pi\)
0.625834 + 0.779957i \(0.284759\pi\)
\(564\) 1.45141e28 0.187655
\(565\) 0 0
\(566\) −1.10542e29 −1.37706
\(567\) −2.55766e27 −0.0312767
\(568\) −1.20547e29 −1.44710
\(569\) −9.95783e28 −1.17351 −0.586754 0.809765i \(-0.699594\pi\)
−0.586754 + 0.809765i \(0.699594\pi\)
\(570\) 0 0
\(571\) 9.60678e28 1.09119 0.545593 0.838050i \(-0.316304\pi\)
0.545593 + 0.838050i \(0.316304\pi\)
\(572\) −7.38203e28 −0.823223
\(573\) 9.41731e28 1.03111
\(574\) 1.84933e28 0.198810
\(575\) 0 0
\(576\) −5.24065e28 −0.543187
\(577\) 9.48715e28 0.965583 0.482792 0.875735i \(-0.339623\pi\)
0.482792 + 0.875735i \(0.339623\pi\)
\(578\) −9.94408e28 −0.993853
\(579\) −7.10943e28 −0.697766
\(580\) 0 0
\(581\) −3.17243e28 −0.300292
\(582\) −3.09468e28 −0.287690
\(583\) 7.94052e28 0.724985
\(584\) −2.27192e28 −0.203732
\(585\) 0 0
\(586\) 2.65991e29 2.30113
\(587\) −7.25220e28 −0.616268 −0.308134 0.951343i \(-0.599705\pi\)
−0.308134 + 0.951343i \(0.599705\pi\)
\(588\) 9.70658e28 0.810222
\(589\) −2.03065e28 −0.166504
\(590\) 0 0
\(591\) 1.50146e28 0.118808
\(592\) 1.00212e28 0.0779010
\(593\) −5.96548e28 −0.455587 −0.227793 0.973710i \(-0.573151\pi\)
−0.227793 + 0.973710i \(0.573151\pi\)
\(594\) −2.98522e28 −0.223985
\(595\) 0 0
\(596\) −1.10164e29 −0.797910
\(597\) 1.08567e29 0.772623
\(598\) −4.28738e27 −0.0299799
\(599\) 1.36596e29 0.938548 0.469274 0.883053i \(-0.344516\pi\)
0.469274 + 0.883053i \(0.344516\pi\)
\(600\) 0 0
\(601\) 2.91002e29 1.93069 0.965347 0.260968i \(-0.0840418\pi\)
0.965347 + 0.260968i \(0.0840418\pi\)
\(602\) −7.78476e28 −0.507554
\(603\) −1.51847e28 −0.0972917
\(604\) 2.04465e29 1.28745
\(605\) 0 0
\(606\) −2.02429e29 −1.23115
\(607\) 1.53630e29 0.918322 0.459161 0.888353i \(-0.348150\pi\)
0.459161 + 0.888353i \(0.348150\pi\)
\(608\) −5.40803e28 −0.317725
\(609\) 1.43942e28 0.0831201
\(610\) 0 0
\(611\) 2.81833e28 0.157238
\(612\) 5.66862e28 0.310876
\(613\) −6.03634e28 −0.325416 −0.162708 0.986674i \(-0.552023\pi\)
−0.162708 + 0.986674i \(0.552023\pi\)
\(614\) 3.16634e29 1.67799
\(615\) 0 0
\(616\) −3.35263e28 −0.171707
\(617\) 6.56975e28 0.330791 0.165396 0.986227i \(-0.447110\pi\)
0.165396 + 0.986227i \(0.447110\pi\)
\(618\) 2.67886e29 1.32608
\(619\) 2.24352e29 1.09189 0.545945 0.837821i \(-0.316171\pi\)
0.545945 + 0.837821i \(0.316171\pi\)
\(620\) 0 0
\(621\) −1.04689e27 −0.00492538
\(622\) 3.77671e29 1.74709
\(623\) −1.50951e28 −0.0686610
\(624\) −1.91491e28 −0.0856466
\(625\) 0 0
\(626\) 1.89244e29 0.818449
\(627\) −2.74202e28 −0.116617
\(628\) 1.59134e29 0.665558
\(629\) −5.76037e28 −0.236929
\(630\) 0 0
\(631\) 1.40053e29 0.557166 0.278583 0.960412i \(-0.410135\pi\)
0.278583 + 0.960412i \(0.410135\pi\)
\(632\) −2.40856e29 −0.942384
\(633\) 1.44933e29 0.557735
\(634\) 6.08430e29 2.30289
\(635\) 0 0
\(636\) −2.37815e29 −0.870842
\(637\) 1.88480e29 0.678894
\(638\) 1.68005e29 0.595256
\(639\) −1.66211e29 −0.579295
\(640\) 0 0
\(641\) −2.34559e29 −0.791121 −0.395560 0.918440i \(-0.629450\pi\)
−0.395560 + 0.918440i \(0.629450\pi\)
\(642\) 1.43090e29 0.474777
\(643\) 1.42258e29 0.464368 0.232184 0.972672i \(-0.425413\pi\)
0.232184 + 0.972672i \(0.425413\pi\)
\(644\) −3.41903e27 −0.0109800
\(645\) 0 0
\(646\) 8.62310e28 0.268054
\(647\) 1.58157e29 0.483720 0.241860 0.970311i \(-0.422243\pi\)
0.241860 + 0.970311i \(0.422243\pi\)
\(648\) 3.07449e28 0.0925201
\(649\) −2.81062e27 −0.00832211
\(650\) 0 0
\(651\) −3.42336e28 −0.0981413
\(652\) 3.72150e29 1.04983
\(653\) 4.13878e29 1.14890 0.574452 0.818539i \(-0.305215\pi\)
0.574452 + 0.818539i \(0.305215\pi\)
\(654\) 4.81640e29 1.31570
\(655\) 0 0
\(656\) 3.38087e28 0.0894412
\(657\) −3.13253e28 −0.0815566
\(658\) 3.72215e28 0.0953723
\(659\) −7.78856e28 −0.196409 −0.0982043 0.995166i \(-0.531310\pi\)
−0.0982043 + 0.995166i \(0.531310\pi\)
\(660\) 0 0
\(661\) −9.22514e28 −0.225350 −0.112675 0.993632i \(-0.535942\pi\)
−0.112675 + 0.993632i \(0.535942\pi\)
\(662\) −1.18142e30 −2.84052
\(663\) 1.10072e29 0.260487
\(664\) 3.81349e29 0.888297
\(665\) 0 0
\(666\) −9.08530e28 −0.205050
\(667\) 5.89177e27 0.0130895
\(668\) −1.80636e29 −0.395049
\(669\) 1.21211e29 0.260955
\(670\) 0 0
\(671\) −4.82550e29 −1.00683
\(672\) −9.11709e28 −0.187274
\(673\) 3.42657e27 0.00692949 0.00346475 0.999994i \(-0.498897\pi\)
0.00346475 + 0.999994i \(0.498897\pi\)
\(674\) 1.74236e29 0.346904
\(675\) 0 0
\(676\) −3.60386e29 −0.695549
\(677\) −1.60981e29 −0.305910 −0.152955 0.988233i \(-0.548879\pi\)
−0.152955 + 0.988233i \(0.548879\pi\)
\(678\) −3.67147e28 −0.0686955
\(679\) −4.79211e28 −0.0882864
\(680\) 0 0
\(681\) −3.02784e29 −0.540864
\(682\) −3.99563e29 −0.702829
\(683\) 2.93386e28 0.0508186 0.0254093 0.999677i \(-0.491911\pi\)
0.0254093 + 0.999677i \(0.491911\pi\)
\(684\) 8.21221e28 0.140078
\(685\) 0 0
\(686\) 5.19272e29 0.858997
\(687\) −1.35802e29 −0.221238
\(688\) −1.42318e29 −0.228339
\(689\) −4.61784e29 −0.729688
\(690\) 0 0
\(691\) −7.95216e29 −1.21889 −0.609446 0.792827i \(-0.708608\pi\)
−0.609446 + 0.792827i \(0.708608\pi\)
\(692\) −9.66305e29 −1.45881
\(693\) −4.62261e28 −0.0687366
\(694\) 5.51424e29 0.807627
\(695\) 0 0
\(696\) −1.73029e29 −0.245879
\(697\) −1.94338e29 −0.272028
\(698\) 6.80525e29 0.938344
\(699\) 8.00904e29 1.08785
\(700\) 0 0
\(701\) 1.10069e30 1.45086 0.725430 0.688296i \(-0.241641\pi\)
0.725430 + 0.688296i \(0.241641\pi\)
\(702\) 1.73607e29 0.225438
\(703\) −8.34514e28 −0.106759
\(704\) −9.47174e29 −1.19376
\(705\) 0 0
\(706\) 1.02605e30 1.25522
\(707\) −3.13461e29 −0.377815
\(708\) 8.41769e27 0.00999640
\(709\) 9.95317e29 1.16460 0.582299 0.812975i \(-0.302153\pi\)
0.582299 + 0.812975i \(0.302153\pi\)
\(710\) 0 0
\(711\) −3.32093e29 −0.377249
\(712\) 1.81454e29 0.203107
\(713\) −1.40123e28 −0.0154550
\(714\) 1.45372e29 0.157997
\(715\) 0 0
\(716\) 2.71287e30 2.86314
\(717\) −1.11364e29 −0.115822
\(718\) −4.20577e29 −0.431061
\(719\) −1.52067e29 −0.153597 −0.0767983 0.997047i \(-0.524470\pi\)
−0.0767983 + 0.997047i \(0.524470\pi\)
\(720\) 0 0
\(721\) 4.14820e29 0.406949
\(722\) −1.51829e30 −1.46796
\(723\) −6.18017e29 −0.588909
\(724\) −9.80212e29 −0.920587
\(725\) 0 0
\(726\) 4.65839e29 0.425012
\(727\) −8.18896e29 −0.736406 −0.368203 0.929745i \(-0.620027\pi\)
−0.368203 + 0.929745i \(0.620027\pi\)
\(728\) 1.94973e29 0.172821
\(729\) 4.23912e28 0.0370370
\(730\) 0 0
\(731\) 8.18067e29 0.694475
\(732\) 1.44521e30 1.20939
\(733\) −4.95962e29 −0.409125 −0.204563 0.978853i \(-0.565577\pi\)
−0.204563 + 0.978853i \(0.565577\pi\)
\(734\) −1.69813e30 −1.38090
\(735\) 0 0
\(736\) −3.73177e28 −0.0294915
\(737\) −2.74443e29 −0.213818
\(738\) −3.06511e29 −0.235426
\(739\) −7.82517e29 −0.592553 −0.296277 0.955102i \(-0.595745\pi\)
−0.296277 + 0.955102i \(0.595745\pi\)
\(740\) 0 0
\(741\) 1.59463e29 0.117373
\(742\) −6.09876e29 −0.442589
\(743\) −3.93104e29 −0.281271 −0.140636 0.990061i \(-0.544915\pi\)
−0.140636 + 0.990061i \(0.544915\pi\)
\(744\) 4.11512e29 0.290313
\(745\) 0 0
\(746\) 4.06206e30 2.78605
\(747\) 5.25805e29 0.355597
\(748\) 1.02452e30 0.683211
\(749\) 2.21574e29 0.145700
\(750\) 0 0
\(751\) −1.84532e30 −1.17992 −0.589959 0.807433i \(-0.700856\pi\)
−0.589959 + 0.807433i \(0.700856\pi\)
\(752\) 6.80469e28 0.0429063
\(753\) −1.01878e30 −0.633480
\(754\) −9.77037e29 −0.599117
\(755\) 0 0
\(756\) 1.38445e29 0.0825654
\(757\) −1.84934e30 −1.08770 −0.543852 0.839181i \(-0.683035\pi\)
−0.543852 + 0.839181i \(0.683035\pi\)
\(758\) −2.26981e30 −1.31663
\(759\) −1.89211e28 −0.0108245
\(760\) 0 0
\(761\) −2.72378e30 −1.51577 −0.757885 0.652389i \(-0.773767\pi\)
−0.757885 + 0.652389i \(0.773767\pi\)
\(762\) 3.01071e30 1.65250
\(763\) 7.45820e29 0.403763
\(764\) −5.09755e30 −2.72195
\(765\) 0 0
\(766\) −3.44814e30 −1.79136
\(767\) 1.63453e28 0.00837609
\(768\) 7.45700e29 0.376939
\(769\) −3.33728e29 −0.166405 −0.0832026 0.996533i \(-0.526515\pi\)
−0.0832026 + 0.996533i \(0.526515\pi\)
\(770\) 0 0
\(771\) −5.09982e29 −0.247448
\(772\) 3.84830e30 1.84199
\(773\) −3.35597e30 −1.58465 −0.792325 0.610100i \(-0.791129\pi\)
−0.792325 + 0.610100i \(0.791129\pi\)
\(774\) 1.29026e30 0.601032
\(775\) 0 0
\(776\) 5.76046e29 0.261161
\(777\) −1.40686e29 −0.0629259
\(778\) 2.81265e30 1.24117
\(779\) −2.81540e29 −0.122574
\(780\) 0 0
\(781\) −3.00402e30 −1.27311
\(782\) 5.95029e28 0.0248810
\(783\) −2.38572e29 −0.0984286
\(784\) 4.55075e29 0.185253
\(785\) 0 0
\(786\) −1.28539e28 −0.00509448
\(787\) 1.64029e29 0.0641484 0.0320742 0.999485i \(-0.489789\pi\)
0.0320742 + 0.999485i \(0.489789\pi\)
\(788\) −8.12735e29 −0.313634
\(789\) −4.09853e29 −0.156070
\(790\) 0 0
\(791\) −5.68527e28 −0.0210813
\(792\) 5.55671e29 0.203331
\(793\) 2.80629e30 1.01336
\(794\) −8.14335e29 −0.290194
\(795\) 0 0
\(796\) −5.87669e30 −2.03960
\(797\) −1.33943e30 −0.458784 −0.229392 0.973334i \(-0.573674\pi\)
−0.229392 + 0.973334i \(0.573674\pi\)
\(798\) 2.10602e29 0.0711922
\(799\) −3.91145e29 −0.130496
\(800\) 0 0
\(801\) 2.50188e29 0.0813066
\(802\) 3.19384e30 1.02443
\(803\) −5.66161e29 −0.179237
\(804\) 8.21943e29 0.256835
\(805\) 0 0
\(806\) 2.32368e30 0.707388
\(807\) 1.40289e30 0.421552
\(808\) 3.76802e30 1.11762
\(809\) 5.46468e30 1.59995 0.799973 0.600036i \(-0.204847\pi\)
0.799973 + 0.600036i \(0.204847\pi\)
\(810\) 0 0
\(811\) −6.64567e30 −1.89592 −0.947959 0.318393i \(-0.896857\pi\)
−0.947959 + 0.318393i \(0.896857\pi\)
\(812\) −7.79151e29 −0.219424
\(813\) 2.70800e30 0.752833
\(814\) −1.64204e30 −0.450638
\(815\) 0 0
\(816\) 2.65763e29 0.0710800
\(817\) 1.18515e30 0.312925
\(818\) −9.44020e30 −2.46077
\(819\) 2.68830e29 0.0691825
\(820\) 0 0
\(821\) −3.94037e30 −0.988403 −0.494201 0.869347i \(-0.664539\pi\)
−0.494201 + 0.869347i \(0.664539\pi\)
\(822\) 6.72308e30 1.66500
\(823\) −2.57770e29 −0.0630280 −0.0315140 0.999503i \(-0.510033\pi\)
−0.0315140 + 0.999503i \(0.510033\pi\)
\(824\) −4.98644e30 −1.20380
\(825\) 0 0
\(826\) 2.15872e28 0.00508048
\(827\) −9.53726e29 −0.221624 −0.110812 0.993841i \(-0.535345\pi\)
−0.110812 + 0.993841i \(0.535345\pi\)
\(828\) 5.66677e28 0.0130022
\(829\) −2.35934e29 −0.0534526 −0.0267263 0.999643i \(-0.508508\pi\)
−0.0267263 + 0.999643i \(0.508508\pi\)
\(830\) 0 0
\(831\) 3.59896e30 0.794998
\(832\) 5.50833e30 1.20150
\(833\) −2.61585e30 −0.563430
\(834\) 2.41258e30 0.513142
\(835\) 0 0
\(836\) 1.48424e30 0.307850
\(837\) 5.67394e29 0.116216
\(838\) 4.00473e30 0.810049
\(839\) 1.50684e30 0.301000 0.150500 0.988610i \(-0.451912\pi\)
0.150500 + 0.988610i \(0.451912\pi\)
\(840\) 0 0
\(841\) −3.79019e30 −0.738419
\(842\) 7.33886e29 0.141206
\(843\) −2.01439e30 −0.382785
\(844\) −7.84514e30 −1.47233
\(845\) 0 0
\(846\) −6.16916e29 −0.112937
\(847\) 7.21351e29 0.130428
\(848\) −1.11495e30 −0.199113
\(849\) 2.83706e30 0.500424
\(850\) 0 0
\(851\) −5.75849e28 −0.00990942
\(852\) 8.99691e30 1.52925
\(853\) −8.74672e30 −1.46852 −0.734261 0.678867i \(-0.762471\pi\)
−0.734261 + 0.678867i \(0.762471\pi\)
\(854\) 3.70625e30 0.614649
\(855\) 0 0
\(856\) −2.66348e30 −0.430998
\(857\) −1.79796e30 −0.287397 −0.143699 0.989622i \(-0.545900\pi\)
−0.143699 + 0.989622i \(0.545900\pi\)
\(858\) 3.13769e30 0.495444
\(859\) 2.30872e30 0.360116 0.180058 0.983656i \(-0.442371\pi\)
0.180058 + 0.983656i \(0.442371\pi\)
\(860\) 0 0
\(861\) −4.74632e29 −0.0722477
\(862\) 9.24878e30 1.39078
\(863\) −7.08890e30 −1.05309 −0.526545 0.850147i \(-0.676513\pi\)
−0.526545 + 0.850147i \(0.676513\pi\)
\(864\) 1.51108e30 0.221766
\(865\) 0 0
\(866\) 1.37614e31 1.97117
\(867\) 2.55216e30 0.361166
\(868\) 1.85305e30 0.259077
\(869\) −6.00210e30 −0.829079
\(870\) 0 0
\(871\) 1.59603e30 0.215205
\(872\) −8.96529e30 −1.19438
\(873\) 7.94253e29 0.104546
\(874\) 8.62028e28 0.0112112
\(875\) 0 0
\(876\) 1.69563e30 0.215296
\(877\) −6.15937e30 −0.772752 −0.386376 0.922341i \(-0.626273\pi\)
−0.386376 + 0.922341i \(0.626273\pi\)
\(878\) −6.74702e30 −0.836411
\(879\) −6.82668e30 −0.836231
\(880\) 0 0
\(881\) 4.62731e30 0.553454 0.276727 0.960949i \(-0.410750\pi\)
0.276727 + 0.960949i \(0.410750\pi\)
\(882\) −4.12573e30 −0.487619
\(883\) 1.04142e30 0.121629 0.0608147 0.998149i \(-0.480630\pi\)
0.0608147 + 0.998149i \(0.480630\pi\)
\(884\) −5.95816e30 −0.687643
\(885\) 0 0
\(886\) 6.27233e30 0.706927
\(887\) −1.43153e31 −1.59441 −0.797207 0.603705i \(-0.793690\pi\)
−0.797207 + 0.603705i \(0.793690\pi\)
\(888\) 1.69114e30 0.186142
\(889\) 4.66209e30 0.507121
\(890\) 0 0
\(891\) 7.66160e29 0.0813961
\(892\) −6.56107e30 −0.688880
\(893\) −5.66657e29 −0.0588004
\(894\) 4.68247e30 0.480210
\(895\) 0 0
\(896\) 4.03685e30 0.404397
\(897\) 1.10036e29 0.0108947
\(898\) 1.25618e31 1.22927
\(899\) −3.19322e30 −0.308853
\(900\) 0 0
\(901\) 6.40892e30 0.605585
\(902\) −5.53975e30 −0.517395
\(903\) 1.99797e30 0.184445
\(904\) 6.83410e29 0.0623610
\(905\) 0 0
\(906\) −8.69068e30 −0.774833
\(907\) −1.50854e31 −1.32947 −0.664736 0.747079i \(-0.731456\pi\)
−0.664736 + 0.747079i \(0.731456\pi\)
\(908\) 1.63895e31 1.42779
\(909\) 5.19536e30 0.447399
\(910\) 0 0
\(911\) −4.70909e30 −0.396272 −0.198136 0.980175i \(-0.563489\pi\)
−0.198136 + 0.980175i \(0.563489\pi\)
\(912\) 3.85014e29 0.0320281
\(913\) 9.50318e30 0.781495
\(914\) −9.18303e30 −0.746536
\(915\) 0 0
\(916\) 7.35090e30 0.584034
\(917\) −1.99043e28 −0.00156340
\(918\) −2.40942e30 −0.187096
\(919\) −3.23144e30 −0.248075 −0.124038 0.992278i \(-0.539584\pi\)
−0.124038 + 0.992278i \(0.539584\pi\)
\(920\) 0 0
\(921\) −8.12646e30 −0.609783
\(922\) 1.08411e31 0.804264
\(923\) 1.74700e31 1.28137
\(924\) 2.50220e30 0.181454
\(925\) 0 0
\(926\) 3.55160e31 2.51773
\(927\) −6.87531e30 −0.481898
\(928\) −8.50419e30 −0.589358
\(929\) −2.63040e31 −1.80242 −0.901210 0.433382i \(-0.857320\pi\)
−0.901210 + 0.433382i \(0.857320\pi\)
\(930\) 0 0
\(931\) −3.78962e30 −0.253877
\(932\) −4.33526e31 −2.87176
\(933\) −9.69298e30 −0.634892
\(934\) 4.22988e31 2.73959
\(935\) 0 0
\(936\) −3.23153e30 −0.204650
\(937\) −2.02013e31 −1.26507 −0.632535 0.774532i \(-0.717986\pi\)
−0.632535 + 0.774532i \(0.717986\pi\)
\(938\) 2.10787e30 0.130531
\(939\) −4.85697e30 −0.297424
\(940\) 0 0
\(941\) −3.09975e31 −1.85625 −0.928123 0.372275i \(-0.878578\pi\)
−0.928123 + 0.372275i \(0.878578\pi\)
\(942\) −6.76390e30 −0.400556
\(943\) −1.94274e29 −0.0113774
\(944\) 3.94648e28 0.00228562
\(945\) 0 0
\(946\) 2.33196e31 1.32088
\(947\) −2.18440e31 −1.22365 −0.611826 0.790992i \(-0.709565\pi\)
−0.611826 + 0.790992i \(0.709565\pi\)
\(948\) 1.79760e31 0.995877
\(949\) 3.29253e30 0.180399
\(950\) 0 0
\(951\) −1.56154e31 −0.836871
\(952\) −2.70596e30 −0.143428
\(953\) 1.26936e31 0.665439 0.332719 0.943026i \(-0.392034\pi\)
0.332719 + 0.943026i \(0.392034\pi\)
\(954\) 1.01082e31 0.524102
\(955\) 0 0
\(956\) 6.02807e30 0.305753
\(957\) −4.31186e30 −0.216316
\(958\) −4.31901e31 −2.14312
\(959\) 1.04107e31 0.510956
\(960\) 0 0
\(961\) −1.32311e31 −0.635332
\(962\) 9.54935e30 0.453561
\(963\) −3.67241e30 −0.172534
\(964\) 3.34530e31 1.55463
\(965\) 0 0
\(966\) 1.45324e29 0.00660813
\(967\) 1.90794e31 0.858195 0.429097 0.903258i \(-0.358832\pi\)
0.429097 + 0.903258i \(0.358832\pi\)
\(968\) −8.67116e30 −0.385821
\(969\) −2.21313e30 −0.0974108
\(970\) 0 0
\(971\) 2.37666e31 1.02368 0.511841 0.859080i \(-0.328964\pi\)
0.511841 + 0.859080i \(0.328964\pi\)
\(972\) −2.29461e30 −0.0977718
\(973\) 3.73587e30 0.157473
\(974\) 1.83265e31 0.764204
\(975\) 0 0
\(976\) 6.77562e30 0.276520
\(977\) 2.16504e31 0.874123 0.437062 0.899432i \(-0.356019\pi\)
0.437062 + 0.899432i \(0.356019\pi\)
\(978\) −1.58181e31 −0.631823
\(979\) 4.52180e30 0.178687
\(980\) 0 0
\(981\) −1.23614e31 −0.478125
\(982\) 5.49406e31 2.10244
\(983\) 3.62066e31 1.37080 0.685402 0.728165i \(-0.259627\pi\)
0.685402 + 0.728165i \(0.259627\pi\)
\(984\) 5.70542e30 0.213717
\(985\) 0 0
\(986\) 1.35599e31 0.497221
\(987\) −9.55294e29 −0.0346583
\(988\) −8.63167e30 −0.309847
\(989\) 8.17799e29 0.0290460
\(990\) 0 0
\(991\) 2.36223e30 0.0821387 0.0410694 0.999156i \(-0.486924\pi\)
0.0410694 + 0.999156i \(0.486924\pi\)
\(992\) 2.02254e31 0.695865
\(993\) 3.03214e31 1.03224
\(994\) 2.30726e31 0.777211
\(995\) 0 0
\(996\) −2.84616e31 −0.938720
\(997\) 2.58169e31 0.842570 0.421285 0.906928i \(-0.361579\pi\)
0.421285 + 0.906928i \(0.361579\pi\)
\(998\) −4.27641e30 −0.138105
\(999\) 2.33175e30 0.0745152
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.j.1.6 yes 6
5.2 odd 4 75.22.b.i.49.11 12
5.3 odd 4 75.22.b.i.49.2 12
5.4 even 2 75.22.a.i.1.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.22.a.i.1.1 6 5.4 even 2
75.22.a.j.1.6 yes 6 1.1 even 1 trivial
75.22.b.i.49.2 12 5.3 odd 4
75.22.b.i.49.11 12 5.2 odd 4