Properties

Label 49.26.a.e.1.8
Level $49$
Weight $26$
Character 49.1
Self dual yes
Analytic conductor $194.038$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [49,26,Mod(1,49)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(49, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 26, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("49.1");
 
S:= CuspForms(chi, 26);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 49 = 7^{2} \)
Weight: \( k \) \(=\) \( 26 \)
Character orbit: \([\chi]\) \(=\) 49.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: \(194.038422177\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 6 x^{11} - 1893235651143 x^{10} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: multiple of \( 2^{48}\cdot 3^{20}\cdot 5^{5}\cdot 7^{15} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.8
Root \(-764338.\) of defining polynomial
Character \(\chi\) \(=\) 49.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+300.357 q^{2} +1.52968e6 q^{3} -3.34642e7 q^{4} +9.44412e8 q^{5} +4.59451e8 q^{6} -2.01295e10 q^{8} +1.49264e12 q^{9} +O(q^{10})\) \(q+300.357 q^{2} +1.52968e6 q^{3} -3.34642e7 q^{4} +9.44412e8 q^{5} +4.59451e8 q^{6} -2.01295e10 q^{8} +1.49264e12 q^{9} +2.83661e11 q^{10} -2.47255e12 q^{11} -5.11896e13 q^{12} -6.66288e13 q^{13} +1.44465e15 q^{15} +1.11683e15 q^{16} -2.96239e15 q^{17} +4.48325e14 q^{18} +5.82342e15 q^{19} -3.16040e16 q^{20} -7.42646e14 q^{22} -1.71029e17 q^{23} -3.07918e16 q^{24} +5.93891e17 q^{25} -2.00124e16 q^{26} +9.87186e17 q^{27} +3.51070e18 q^{29} +4.33911e17 q^{30} +5.01950e18 q^{31} +1.01088e18 q^{32} -3.78221e18 q^{33} -8.89773e17 q^{34} -4.99501e19 q^{36} -1.07367e19 q^{37} +1.74910e18 q^{38} -1.01921e20 q^{39} -1.90106e19 q^{40} +2.09028e20 q^{41} +2.88141e20 q^{43} +8.27418e19 q^{44} +1.40967e21 q^{45} -5.13698e19 q^{46} -6.92528e20 q^{47} +1.70839e21 q^{48} +1.78379e20 q^{50} -4.53151e21 q^{51} +2.22968e21 q^{52} +1.61704e21 q^{53} +2.96508e20 q^{54} -2.33510e21 q^{55} +8.90799e21 q^{57} +1.05446e21 q^{58} +1.36756e22 q^{59} -4.83441e22 q^{60} +2.85683e22 q^{61} +1.50764e21 q^{62} -3.71709e22 q^{64} -6.29250e22 q^{65} -1.13601e21 q^{66} +5.59142e22 q^{67} +9.91340e22 q^{68} -2.61621e23 q^{69} +2.07670e23 q^{71} -3.00461e22 q^{72} -3.73651e22 q^{73} -3.22485e21 q^{74} +9.08465e23 q^{75} -1.94876e23 q^{76} -3.06126e22 q^{78} +4.44457e23 q^{79} +1.05474e24 q^{80} +2.45383e23 q^{81} +6.27831e22 q^{82} -1.41422e22 q^{83} -2.79771e24 q^{85} +8.65451e22 q^{86} +5.37026e24 q^{87} +4.97712e22 q^{88} +2.95320e24 q^{89} +4.23404e23 q^{90} +5.72336e24 q^{92} +7.67825e24 q^{93} -2.08005e23 q^{94} +5.49971e24 q^{95} +1.54633e24 q^{96} +1.91879e23 q^{97} -3.69063e24 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 8460 q^{2} + 161848296 q^{4} - 61395028560 q^{8} + 4977863358780 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 8460 q^{2} + 161848296 q^{4} - 61395028560 q^{8} + 4977863358780 q^{9} + 39593455677648 q^{11} + 357546272706144 q^{15} + 18\!\cdots\!56 q^{16}+ \cdots - 14\!\cdots\!88 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 300.357 0.0518516 0.0259258 0.999664i \(-0.491747\pi\)
0.0259258 + 0.999664i \(0.491747\pi\)
\(3\) 1.52968e6 1.66183 0.830913 0.556402i \(-0.187818\pi\)
0.830913 + 0.556402i \(0.187818\pi\)
\(4\) −3.34642e7 −0.997311
\(5\) 9.44412e8 1.72996 0.864981 0.501805i \(-0.167330\pi\)
0.864981 + 0.501805i \(0.167330\pi\)
\(6\) 4.59451e8 0.0861685
\(7\) 0 0
\(8\) −2.01295e10 −0.103564
\(9\) 1.49264e12 1.76167
\(10\) 2.83661e11 0.0897014
\(11\) −2.47255e12 −0.237540 −0.118770 0.992922i \(-0.537895\pi\)
−0.118770 + 0.992922i \(0.537895\pi\)
\(12\) −5.11896e13 −1.65736
\(13\) −6.66288e13 −0.793177 −0.396588 0.917997i \(-0.629806\pi\)
−0.396588 + 0.917997i \(0.629806\pi\)
\(14\) 0 0
\(15\) 1.44465e15 2.87490
\(16\) 1.11683e15 0.991941
\(17\) −2.96239e15 −1.23319 −0.616596 0.787280i \(-0.711489\pi\)
−0.616596 + 0.787280i \(0.711489\pi\)
\(18\) 4.48325e14 0.0913454
\(19\) 5.82342e15 0.603612 0.301806 0.953369i \(-0.402410\pi\)
0.301806 + 0.953369i \(0.402410\pi\)
\(20\) −3.16040e16 −1.72531
\(21\) 0 0
\(22\) −7.42646e14 −0.0123168
\(23\) −1.71029e17 −1.62732 −0.813659 0.581342i \(-0.802528\pi\)
−0.813659 + 0.581342i \(0.802528\pi\)
\(24\) −3.07918e16 −0.172105
\(25\) 5.93891e17 1.99277
\(26\) −2.00124e16 −0.0411275
\(27\) 9.87186e17 1.26576
\(28\) 0 0
\(29\) 3.51070e18 1.84255 0.921273 0.388916i \(-0.127150\pi\)
0.921273 + 0.388916i \(0.127150\pi\)
\(30\) 4.33911e17 0.149068
\(31\) 5.01950e18 1.14456 0.572281 0.820057i \(-0.306059\pi\)
0.572281 + 0.820057i \(0.306059\pi\)
\(32\) 1.01088e18 0.154998
\(33\) −3.78221e18 −0.394750
\(34\) −8.89773e17 −0.0639430
\(35\) 0 0
\(36\) −4.99501e19 −1.75693
\(37\) −1.07367e19 −0.268133 −0.134067 0.990972i \(-0.542804\pi\)
−0.134067 + 0.990972i \(0.542804\pi\)
\(38\) 1.74910e18 0.0312983
\(39\) −1.01921e20 −1.31812
\(40\) −1.90106e19 −0.179162
\(41\) 2.09028e20 1.44680 0.723398 0.690432i \(-0.242579\pi\)
0.723398 + 0.690432i \(0.242579\pi\)
\(42\) 0 0
\(43\) 2.88141e20 1.09964 0.549818 0.835284i \(-0.314697\pi\)
0.549818 + 0.835284i \(0.314697\pi\)
\(44\) 8.27418e19 0.236901
\(45\) 1.40967e21 3.04762
\(46\) −5.13698e19 −0.0843791
\(47\) −6.92528e20 −0.869388 −0.434694 0.900578i \(-0.643144\pi\)
−0.434694 + 0.900578i \(0.643144\pi\)
\(48\) 1.70839e21 1.64844
\(49\) 0 0
\(50\) 1.78379e20 0.103328
\(51\) −4.53151e21 −2.04935
\(52\) 2.22968e21 0.791044
\(53\) 1.61704e21 0.452139 0.226070 0.974111i \(-0.427412\pi\)
0.226070 + 0.974111i \(0.427412\pi\)
\(54\) 2.96508e20 0.0656318
\(55\) −2.33510e21 −0.410934
\(56\) 0 0
\(57\) 8.90799e21 1.00310
\(58\) 1.05446e21 0.0955391
\(59\) 1.36756e22 1.00068 0.500340 0.865829i \(-0.333208\pi\)
0.500340 + 0.865829i \(0.333208\pi\)
\(60\) −4.83441e22 −2.86717
\(61\) 2.85683e22 1.37804 0.689020 0.724742i \(-0.258041\pi\)
0.689020 + 0.724742i \(0.258041\pi\)
\(62\) 1.50764e21 0.0593475
\(63\) 0 0
\(64\) −3.71709e22 −0.983905
\(65\) −6.29250e22 −1.37217
\(66\) −1.13601e21 −0.0204684
\(67\) 5.59142e22 0.834809 0.417404 0.908721i \(-0.362940\pi\)
0.417404 + 0.908721i \(0.362940\pi\)
\(68\) 9.91340e22 1.22988
\(69\) −2.61621e23 −2.70432
\(70\) 0 0
\(71\) 2.07670e23 1.50191 0.750957 0.660351i \(-0.229593\pi\)
0.750957 + 0.660351i \(0.229593\pi\)
\(72\) −3.00461e22 −0.182445
\(73\) −3.73651e22 −0.190955 −0.0954775 0.995432i \(-0.530438\pi\)
−0.0954775 + 0.995432i \(0.530438\pi\)
\(74\) −3.22485e21 −0.0139031
\(75\) 9.08465e23 3.31164
\(76\) −1.94876e23 −0.601989
\(77\) 0 0
\(78\) −3.06126e22 −0.0683468
\(79\) 4.44457e23 0.846235 0.423117 0.906075i \(-0.360936\pi\)
0.423117 + 0.906075i \(0.360936\pi\)
\(80\) 1.05474e24 1.71602
\(81\) 2.45383e23 0.341808
\(82\) 6.27831e22 0.0750187
\(83\) −1.41422e22 −0.0145225 −0.00726125 0.999974i \(-0.502311\pi\)
−0.00726125 + 0.999974i \(0.502311\pi\)
\(84\) 0 0
\(85\) −2.79771e24 −2.13337
\(86\) 8.65451e22 0.0570180
\(87\) 5.37026e24 3.06199
\(88\) 4.97712e22 0.0246005
\(89\) 2.95320e24 1.26741 0.633705 0.773575i \(-0.281533\pi\)
0.633705 + 0.773575i \(0.281533\pi\)
\(90\) 4.23404e23 0.158024
\(91\) 0 0
\(92\) 5.72336e24 1.62294
\(93\) 7.67825e24 1.90206
\(94\) −2.08005e23 −0.0450792
\(95\) 5.49971e24 1.04423
\(96\) 1.54633e24 0.257579
\(97\) 1.91879e23 0.0280790 0.0140395 0.999901i \(-0.495531\pi\)
0.0140395 + 0.999901i \(0.495531\pi\)
\(98\) 0 0
\(99\) −3.69063e24 −0.418466
\(100\) −1.98741e25 −1.98741
\(101\) −3.36187e24 −0.296868 −0.148434 0.988922i \(-0.547423\pi\)
−0.148434 + 0.988922i \(0.547423\pi\)
\(102\) −1.36107e24 −0.106262
\(103\) −1.48667e25 −1.02742 −0.513712 0.857962i \(-0.671730\pi\)
−0.513712 + 0.857962i \(0.671730\pi\)
\(104\) 1.34120e24 0.0821445
\(105\) 0 0
\(106\) 4.85689e23 0.0234442
\(107\) 4.59102e24 0.197066 0.0985330 0.995134i \(-0.468585\pi\)
0.0985330 + 0.995134i \(0.468585\pi\)
\(108\) −3.30354e25 −1.26236
\(109\) −2.07503e25 −0.706633 −0.353317 0.935504i \(-0.614946\pi\)
−0.353317 + 0.935504i \(0.614946\pi\)
\(110\) −7.01364e23 −0.0213076
\(111\) −1.64238e25 −0.445591
\(112\) 0 0
\(113\) 2.42749e25 0.526838 0.263419 0.964682i \(-0.415150\pi\)
0.263419 + 0.964682i \(0.415150\pi\)
\(114\) 2.67557e24 0.0520123
\(115\) −1.61522e26 −2.81520
\(116\) −1.17483e26 −1.83759
\(117\) −9.94529e25 −1.39731
\(118\) 4.10755e24 0.0518869
\(119\) 0 0
\(120\) −2.90801e25 −0.297736
\(121\) −1.02234e26 −0.943575
\(122\) 8.58069e24 0.0714536
\(123\) 3.19747e26 2.40432
\(124\) −1.67974e26 −1.14149
\(125\) 2.79421e26 1.71745
\(126\) 0 0
\(127\) 1.57593e26 0.794311 0.397156 0.917751i \(-0.369997\pi\)
0.397156 + 0.917751i \(0.369997\pi\)
\(128\) −4.50841e25 −0.206015
\(129\) 4.40764e26 1.82741
\(130\) −1.89000e25 −0.0711490
\(131\) −8.74455e25 −0.299121 −0.149560 0.988753i \(-0.547786\pi\)
−0.149560 + 0.988753i \(0.547786\pi\)
\(132\) 1.26569e26 0.393688
\(133\) 0 0
\(134\) 1.67942e25 0.0432862
\(135\) 9.32310e26 2.18972
\(136\) 5.96314e25 0.127714
\(137\) −6.03372e26 −1.17917 −0.589587 0.807705i \(-0.700710\pi\)
−0.589587 + 0.807705i \(0.700710\pi\)
\(138\) −7.85795e25 −0.140224
\(139\) −5.46912e26 −0.891729 −0.445864 0.895101i \(-0.647104\pi\)
−0.445864 + 0.895101i \(0.647104\pi\)
\(140\) 0 0
\(141\) −1.05935e27 −1.44477
\(142\) 6.23752e25 0.0778767
\(143\) 1.64743e26 0.188411
\(144\) 1.66702e27 1.74747
\(145\) 3.31555e27 3.18754
\(146\) −1.12229e25 −0.00990133
\(147\) 0 0
\(148\) 3.59296e26 0.267412
\(149\) −1.57289e27 −1.07615 −0.538073 0.842898i \(-0.680848\pi\)
−0.538073 + 0.842898i \(0.680848\pi\)
\(150\) 2.72864e26 0.171714
\(151\) −1.68207e27 −0.974165 −0.487082 0.873356i \(-0.661939\pi\)
−0.487082 + 0.873356i \(0.661939\pi\)
\(152\) −1.17223e26 −0.0625124
\(153\) −4.42178e27 −2.17247
\(154\) 0 0
\(155\) 4.74048e27 1.98005
\(156\) 3.41070e27 1.31458
\(157\) 2.58732e27 0.920672 0.460336 0.887745i \(-0.347729\pi\)
0.460336 + 0.887745i \(0.347729\pi\)
\(158\) 1.33496e26 0.0438787
\(159\) 2.47356e27 0.751377
\(160\) 9.54688e26 0.268140
\(161\) 0 0
\(162\) 7.37025e25 0.0177233
\(163\) −6.00179e27 −1.33640 −0.668199 0.743983i \(-0.732935\pi\)
−0.668199 + 0.743983i \(0.732935\pi\)
\(164\) −6.99497e27 −1.44291
\(165\) −3.57197e27 −0.682902
\(166\) −4.24771e24 −0.000753016 0
\(167\) 4.11051e27 0.675989 0.337995 0.941148i \(-0.390251\pi\)
0.337995 + 0.941148i \(0.390251\pi\)
\(168\) 0 0
\(169\) −2.61702e27 −0.370871
\(170\) −8.40313e26 −0.110619
\(171\) 8.69228e27 1.06336
\(172\) −9.64241e27 −1.09668
\(173\) −5.87547e27 −0.621536 −0.310768 0.950486i \(-0.600586\pi\)
−0.310768 + 0.950486i \(0.600586\pi\)
\(174\) 1.61299e27 0.158769
\(175\) 0 0
\(176\) −2.76141e27 −0.235625
\(177\) 2.09193e28 1.66296
\(178\) 8.87013e26 0.0657173
\(179\) 2.78322e27 0.192258 0.0961292 0.995369i \(-0.469354\pi\)
0.0961292 + 0.995369i \(0.469354\pi\)
\(180\) −4.71735e28 −3.03943
\(181\) 2.86445e28 1.72210 0.861051 0.508519i \(-0.169807\pi\)
0.861051 + 0.508519i \(0.169807\pi\)
\(182\) 0 0
\(183\) 4.37005e28 2.29006
\(184\) 3.44274e27 0.168531
\(185\) −1.01399e28 −0.463860
\(186\) 2.30621e27 0.0986252
\(187\) 7.32464e27 0.292932
\(188\) 2.31749e28 0.867051
\(189\) 0 0
\(190\) 1.65187e27 0.0541449
\(191\) 3.65037e28 1.12052 0.560261 0.828316i \(-0.310701\pi\)
0.560261 + 0.828316i \(0.310701\pi\)
\(192\) −5.68596e28 −1.63508
\(193\) 4.22298e28 1.13803 0.569014 0.822328i \(-0.307325\pi\)
0.569014 + 0.822328i \(0.307325\pi\)
\(194\) 5.76322e25 0.00145594
\(195\) −9.62553e28 −2.28030
\(196\) 0 0
\(197\) −3.69294e28 −0.770096 −0.385048 0.922896i \(-0.625815\pi\)
−0.385048 + 0.922896i \(0.625815\pi\)
\(198\) −1.10850e27 −0.0216982
\(199\) −4.31066e28 −0.792283 −0.396142 0.918189i \(-0.629651\pi\)
−0.396142 + 0.918189i \(0.629651\pi\)
\(200\) −1.19547e28 −0.206379
\(201\) 8.55310e28 1.38731
\(202\) −1.00976e27 −0.0153931
\(203\) 0 0
\(204\) 1.51644e29 2.04384
\(205\) 1.97409e29 2.50290
\(206\) −4.46532e27 −0.0532737
\(207\) −2.55285e29 −2.86680
\(208\) −7.44128e28 −0.786785
\(209\) −1.43987e28 −0.143382
\(210\) 0 0
\(211\) 1.08925e29 0.962934 0.481467 0.876464i \(-0.340104\pi\)
0.481467 + 0.876464i \(0.340104\pi\)
\(212\) −5.41129e28 −0.450923
\(213\) 3.17670e29 2.49592
\(214\) 1.37894e27 0.0102182
\(215\) 2.72124e29 1.90233
\(216\) −1.98716e28 −0.131087
\(217\) 0 0
\(218\) −6.23251e27 −0.0366401
\(219\) −5.71568e28 −0.317334
\(220\) 7.81424e28 0.409830
\(221\) 1.97380e29 0.978138
\(222\) −4.93300e27 −0.0231046
\(223\) 3.47042e28 0.153663 0.0768317 0.997044i \(-0.475520\pi\)
0.0768317 + 0.997044i \(0.475520\pi\)
\(224\) 0 0
\(225\) 8.86467e29 3.51060
\(226\) 7.29113e27 0.0273174
\(227\) 2.93853e29 1.04185 0.520927 0.853601i \(-0.325586\pi\)
0.520927 + 0.853601i \(0.325586\pi\)
\(228\) −2.98099e29 −1.00040
\(229\) 2.73707e29 0.869646 0.434823 0.900516i \(-0.356811\pi\)
0.434823 + 0.900516i \(0.356811\pi\)
\(230\) −4.85143e28 −0.145973
\(231\) 0 0
\(232\) −7.06686e28 −0.190821
\(233\) 5.86808e29 1.50157 0.750787 0.660544i \(-0.229674\pi\)
0.750787 + 0.660544i \(0.229674\pi\)
\(234\) −2.98714e28 −0.0724531
\(235\) −6.54031e29 −1.50401
\(236\) −4.57642e29 −0.997989
\(237\) 6.79878e29 1.40630
\(238\) 0 0
\(239\) 6.68799e29 1.24544 0.622719 0.782446i \(-0.286028\pi\)
0.622719 + 0.782446i \(0.286028\pi\)
\(240\) 1.61343e30 2.85173
\(241\) −4.01605e29 −0.673885 −0.336943 0.941525i \(-0.609393\pi\)
−0.336943 + 0.941525i \(0.609393\pi\)
\(242\) −3.07066e28 −0.0489259
\(243\) −4.61073e29 −0.697736
\(244\) −9.56017e29 −1.37433
\(245\) 0 0
\(246\) 9.60382e28 0.124668
\(247\) −3.88007e29 −0.478771
\(248\) −1.01040e29 −0.118535
\(249\) −2.16331e28 −0.0241339
\(250\) 8.39261e28 0.0890526
\(251\) 1.15661e29 0.116752 0.0583762 0.998295i \(-0.481408\pi\)
0.0583762 + 0.998295i \(0.481408\pi\)
\(252\) 0 0
\(253\) 4.22878e29 0.386553
\(254\) 4.73342e28 0.0411863
\(255\) −4.27962e30 −3.54530
\(256\) 1.23371e30 0.973222
\(257\) −1.30924e30 −0.983682 −0.491841 0.870685i \(-0.663676\pi\)
−0.491841 + 0.870685i \(0.663676\pi\)
\(258\) 1.32387e29 0.0947540
\(259\) 0 0
\(260\) 2.10574e30 1.36848
\(261\) 5.24021e30 3.24596
\(262\) −2.62649e28 −0.0155099
\(263\) −2.36399e30 −1.33107 −0.665533 0.746369i \(-0.731796\pi\)
−0.665533 + 0.746369i \(0.731796\pi\)
\(264\) 7.61341e28 0.0408818
\(265\) 1.52715e30 0.782183
\(266\) 0 0
\(267\) 4.51746e30 2.10622
\(268\) −1.87112e30 −0.832564
\(269\) 9.03280e29 0.383636 0.191818 0.981431i \(-0.438562\pi\)
0.191818 + 0.981431i \(0.438562\pi\)
\(270\) 2.80026e29 0.113541
\(271\) −2.30713e30 −0.893215 −0.446608 0.894730i \(-0.647368\pi\)
−0.446608 + 0.894730i \(0.647368\pi\)
\(272\) −3.30847e30 −1.22325
\(273\) 0 0
\(274\) −1.81227e29 −0.0611422
\(275\) −1.46842e30 −0.473361
\(276\) 8.75493e30 2.69705
\(277\) −2.38991e30 −0.703694 −0.351847 0.936057i \(-0.614446\pi\)
−0.351847 + 0.936057i \(0.614446\pi\)
\(278\) −1.64269e29 −0.0462376
\(279\) 7.49232e30 2.01634
\(280\) 0 0
\(281\) −4.32085e30 −1.06351 −0.531753 0.846899i \(-0.678467\pi\)
−0.531753 + 0.846899i \(0.678467\pi\)
\(282\) −3.18182e29 −0.0749139
\(283\) 1.53040e30 0.344725 0.172363 0.985034i \(-0.444860\pi\)
0.172363 + 0.985034i \(0.444860\pi\)
\(284\) −6.94952e30 −1.49788
\(285\) 8.41281e30 1.73532
\(286\) 4.94816e28 0.00976941
\(287\) 0 0
\(288\) 1.50888e30 0.273055
\(289\) 3.00511e30 0.520760
\(290\) 9.95847e29 0.165279
\(291\) 2.93514e29 0.0466624
\(292\) 1.25040e30 0.190442
\(293\) −2.06559e30 −0.301438 −0.150719 0.988577i \(-0.548159\pi\)
−0.150719 + 0.988577i \(0.548159\pi\)
\(294\) 0 0
\(295\) 1.29154e31 1.73114
\(296\) 2.16125e29 0.0277689
\(297\) −2.44086e30 −0.300668
\(298\) −4.72430e29 −0.0558000
\(299\) 1.13955e31 1.29075
\(300\) −3.04011e31 −3.30273
\(301\) 0 0
\(302\) −5.05221e29 −0.0505120
\(303\) −5.14260e30 −0.493344
\(304\) 6.50375e30 0.598748
\(305\) 2.69803e31 2.38396
\(306\) −1.32811e30 −0.112646
\(307\) 1.61945e31 1.31867 0.659337 0.751848i \(-0.270837\pi\)
0.659337 + 0.751848i \(0.270837\pi\)
\(308\) 0 0
\(309\) −2.27414e31 −1.70740
\(310\) 1.42384e30 0.102669
\(311\) −1.32005e31 −0.914294 −0.457147 0.889391i \(-0.651129\pi\)
−0.457147 + 0.889391i \(0.651129\pi\)
\(312\) 2.05162e30 0.136510
\(313\) 1.51942e31 0.971344 0.485672 0.874141i \(-0.338575\pi\)
0.485672 + 0.874141i \(0.338575\pi\)
\(314\) 7.77120e29 0.0477383
\(315\) 0 0
\(316\) −1.48734e31 −0.843960
\(317\) −1.02409e29 −0.00558593 −0.00279297 0.999996i \(-0.500889\pi\)
−0.00279297 + 0.999996i \(0.500889\pi\)
\(318\) 7.42950e29 0.0389601
\(319\) −8.68036e30 −0.437678
\(320\) −3.51046e31 −1.70212
\(321\) 7.02280e30 0.327490
\(322\) 0 0
\(323\) −1.72512e31 −0.744369
\(324\) −8.21156e30 −0.340889
\(325\) −3.95702e31 −1.58062
\(326\) −1.80268e30 −0.0692944
\(327\) −3.17414e31 −1.17430
\(328\) −4.20764e30 −0.149836
\(329\) 0 0
\(330\) −1.07286e30 −0.0354096
\(331\) 5.49413e30 0.174602 0.0873009 0.996182i \(-0.472176\pi\)
0.0873009 + 0.996182i \(0.472176\pi\)
\(332\) 4.73258e29 0.0144835
\(333\) −1.60261e31 −0.472362
\(334\) 1.23462e30 0.0350512
\(335\) 5.28060e31 1.44419
\(336\) 0 0
\(337\) −4.63930e31 −1.17782 −0.588910 0.808198i \(-0.700443\pi\)
−0.588910 + 0.808198i \(0.700443\pi\)
\(338\) −7.86039e29 −0.0192303
\(339\) 3.71329e31 0.875513
\(340\) 9.36233e31 2.12764
\(341\) −1.24110e31 −0.271879
\(342\) 2.61079e30 0.0551372
\(343\) 0 0
\(344\) −5.80014e30 −0.113883
\(345\) −2.47078e32 −4.67837
\(346\) −1.76474e30 −0.0322276
\(347\) 4.66143e30 0.0821110 0.0410555 0.999157i \(-0.486928\pi\)
0.0410555 + 0.999157i \(0.486928\pi\)
\(348\) −1.79711e32 −3.05376
\(349\) 1.05203e32 1.72469 0.862344 0.506322i \(-0.168995\pi\)
0.862344 + 0.506322i \(0.168995\pi\)
\(350\) 0 0
\(351\) −6.57750e31 −1.00397
\(352\) −2.49945e30 −0.0368181
\(353\) 3.18266e31 0.452487 0.226244 0.974071i \(-0.427355\pi\)
0.226244 + 0.974071i \(0.427355\pi\)
\(354\) 6.28325e30 0.0862270
\(355\) 1.96126e32 2.59825
\(356\) −9.88264e31 −1.26400
\(357\) 0 0
\(358\) 8.35960e29 0.00996891
\(359\) 8.23739e31 0.948656 0.474328 0.880348i \(-0.342691\pi\)
0.474328 + 0.880348i \(0.342691\pi\)
\(360\) −2.83759e31 −0.315623
\(361\) −5.91643e31 −0.635652
\(362\) 8.60357e30 0.0892938
\(363\) −1.56385e32 −1.56806
\(364\) 0 0
\(365\) −3.52881e31 −0.330345
\(366\) 1.31257e31 0.118744
\(367\) −9.43154e31 −0.824625 −0.412313 0.911042i \(-0.635279\pi\)
−0.412313 + 0.911042i \(0.635279\pi\)
\(368\) −1.91010e32 −1.61420
\(369\) 3.12004e32 2.54877
\(370\) −3.04559e30 −0.0240519
\(371\) 0 0
\(372\) −2.56947e32 −1.89695
\(373\) −1.43554e32 −1.02484 −0.512420 0.858735i \(-0.671251\pi\)
−0.512420 + 0.858735i \(0.671251\pi\)
\(374\) 2.20001e30 0.0151890
\(375\) 4.27426e32 2.85410
\(376\) 1.39402e31 0.0900372
\(377\) −2.33913e32 −1.46146
\(378\) 0 0
\(379\) −8.58402e31 −0.501996 −0.250998 0.967988i \(-0.580759\pi\)
−0.250998 + 0.967988i \(0.580759\pi\)
\(380\) −1.84043e32 −1.04142
\(381\) 2.41067e32 1.32001
\(382\) 1.09641e31 0.0581009
\(383\) −1.23830e31 −0.0635098 −0.0317549 0.999496i \(-0.510110\pi\)
−0.0317549 + 0.999496i \(0.510110\pi\)
\(384\) −6.89643e31 −0.342361
\(385\) 0 0
\(386\) 1.26840e31 0.0590086
\(387\) 4.30091e32 1.93720
\(388\) −6.42108e30 −0.0280035
\(389\) 1.04615e32 0.441798 0.220899 0.975297i \(-0.429101\pi\)
0.220899 + 0.975297i \(0.429101\pi\)
\(390\) −2.89110e31 −0.118237
\(391\) 5.06655e32 2.00679
\(392\) 0 0
\(393\) −1.33764e32 −0.497087
\(394\) −1.10920e31 −0.0399308
\(395\) 4.19750e32 1.46395
\(396\) 1.23504e32 0.417341
\(397\) −6.93650e31 −0.227122 −0.113561 0.993531i \(-0.536226\pi\)
−0.113561 + 0.993531i \(0.536226\pi\)
\(398\) −1.29474e31 −0.0410812
\(399\) 0 0
\(400\) 6.63273e32 1.97671
\(401\) −1.05318e32 −0.304226 −0.152113 0.988363i \(-0.548608\pi\)
−0.152113 + 0.988363i \(0.548608\pi\)
\(402\) 2.56898e31 0.0719342
\(403\) −3.34443e32 −0.907840
\(404\) 1.12502e32 0.296070
\(405\) 2.31743e32 0.591315
\(406\) 0 0
\(407\) 2.65471e31 0.0636922
\(408\) 9.12172e31 0.212239
\(409\) −5.60357e32 −1.26451 −0.632257 0.774759i \(-0.717871\pi\)
−0.632257 + 0.774759i \(0.717871\pi\)
\(410\) 5.92931e31 0.129780
\(411\) −9.22968e32 −1.95958
\(412\) 4.97503e32 1.02466
\(413\) 0 0
\(414\) −7.66767e31 −0.148648
\(415\) −1.33561e31 −0.0251234
\(416\) −6.73537e31 −0.122941
\(417\) −8.36603e32 −1.48190
\(418\) −4.32474e30 −0.00743458
\(419\) −8.52671e32 −1.42268 −0.711339 0.702849i \(-0.751911\pi\)
−0.711339 + 0.702849i \(0.751911\pi\)
\(420\) 0 0
\(421\) −4.69154e32 −0.737546 −0.368773 0.929519i \(-0.620222\pi\)
−0.368773 + 0.929519i \(0.620222\pi\)
\(422\) 3.27163e31 0.0499297
\(423\) −1.03370e33 −1.53157
\(424\) −3.25502e31 −0.0468253
\(425\) −1.75934e33 −2.45746
\(426\) 9.54143e31 0.129418
\(427\) 0 0
\(428\) −1.53635e32 −0.196536
\(429\) 2.52004e32 0.313106
\(430\) 8.17343e31 0.0986389
\(431\) −1.15764e33 −1.35708 −0.678542 0.734562i \(-0.737388\pi\)
−0.678542 + 0.734562i \(0.737388\pi\)
\(432\) 1.10252e33 1.25556
\(433\) 7.58667e32 0.839368 0.419684 0.907670i \(-0.362141\pi\)
0.419684 + 0.907670i \(0.362141\pi\)
\(434\) 0 0
\(435\) 5.07173e33 5.29713
\(436\) 6.94394e32 0.704733
\(437\) −9.95975e32 −0.982269
\(438\) −1.71674e31 −0.0164543
\(439\) 1.86519e33 1.73746 0.868732 0.495283i \(-0.164936\pi\)
0.868732 + 0.495283i \(0.164936\pi\)
\(440\) 4.70045e31 0.0425580
\(441\) 0 0
\(442\) 5.92845e31 0.0507181
\(443\) 8.79334e32 0.731320 0.365660 0.930749i \(-0.380843\pi\)
0.365660 + 0.930749i \(0.380843\pi\)
\(444\) 5.49609e32 0.444393
\(445\) 2.78903e33 2.19257
\(446\) 1.04236e31 0.00796771
\(447\) −2.40603e33 −1.78837
\(448\) 0 0
\(449\) −1.64616e33 −1.15716 −0.578578 0.815627i \(-0.696392\pi\)
−0.578578 + 0.815627i \(0.696392\pi\)
\(450\) 2.66256e32 0.182030
\(451\) −5.16832e32 −0.343671
\(452\) −8.12340e32 −0.525421
\(453\) −2.57303e33 −1.61889
\(454\) 8.82607e31 0.0540218
\(455\) 0 0
\(456\) −1.79313e32 −0.103885
\(457\) 1.49461e32 0.0842509 0.0421255 0.999112i \(-0.486587\pi\)
0.0421255 + 0.999112i \(0.486587\pi\)
\(458\) 8.22098e31 0.0450926
\(459\) −2.92443e33 −1.56093
\(460\) 5.40521e33 2.80763
\(461\) −1.53425e33 −0.775594 −0.387797 0.921745i \(-0.626764\pi\)
−0.387797 + 0.921745i \(0.626764\pi\)
\(462\) 0 0
\(463\) 1.72850e33 0.827765 0.413883 0.910330i \(-0.364172\pi\)
0.413883 + 0.910330i \(0.364172\pi\)
\(464\) 3.92084e33 1.82770
\(465\) 7.25143e33 3.29050
\(466\) 1.76252e32 0.0778591
\(467\) −5.52669e32 −0.237686 −0.118843 0.992913i \(-0.537919\pi\)
−0.118843 + 0.992913i \(0.537919\pi\)
\(468\) 3.32811e33 1.39356
\(469\) 0 0
\(470\) −1.96443e32 −0.0779853
\(471\) 3.95778e33 1.53000
\(472\) −2.75282e32 −0.103634
\(473\) −7.12442e32 −0.261207
\(474\) 2.04206e32 0.0729188
\(475\) 3.45848e33 1.20286
\(476\) 0 0
\(477\) 2.41366e33 0.796519
\(478\) 2.00878e32 0.0645780
\(479\) 5.94920e33 1.86322 0.931611 0.363456i \(-0.118403\pi\)
0.931611 + 0.363456i \(0.118403\pi\)
\(480\) 1.46037e33 0.445602
\(481\) 7.15375e32 0.212677
\(482\) −1.20625e32 −0.0349421
\(483\) 0 0
\(484\) 3.42117e33 0.941038
\(485\) 1.81213e32 0.0485755
\(486\) −1.38486e32 −0.0361788
\(487\) −1.29730e33 −0.330316 −0.165158 0.986267i \(-0.552813\pi\)
−0.165158 + 0.986267i \(0.552813\pi\)
\(488\) −5.75066e32 −0.142715
\(489\) −9.18084e33 −2.22086
\(490\) 0 0
\(491\) −1.96688e33 −0.452125 −0.226062 0.974113i \(-0.572585\pi\)
−0.226062 + 0.974113i \(0.572585\pi\)
\(492\) −1.07001e34 −2.39786
\(493\) −1.04000e34 −2.27221
\(494\) −1.16541e32 −0.0248251
\(495\) −3.48547e33 −0.723930
\(496\) 5.60591e33 1.13534
\(497\) 0 0
\(498\) −6.49765e30 −0.00125138
\(499\) −3.31607e33 −0.622827 −0.311413 0.950275i \(-0.600802\pi\)
−0.311413 + 0.950275i \(0.600802\pi\)
\(500\) −9.35061e33 −1.71283
\(501\) 6.28778e33 1.12338
\(502\) 3.47396e31 0.00605380
\(503\) −3.38649e33 −0.575640 −0.287820 0.957685i \(-0.592930\pi\)
−0.287820 + 0.957685i \(0.592930\pi\)
\(504\) 0 0
\(505\) −3.17499e33 −0.513571
\(506\) 1.27014e32 0.0200434
\(507\) −4.00321e33 −0.616323
\(508\) −5.27373e33 −0.792175
\(509\) 4.03391e33 0.591227 0.295613 0.955308i \(-0.404476\pi\)
0.295613 + 0.955308i \(0.404476\pi\)
\(510\) −1.28541e33 −0.183830
\(511\) 0 0
\(512\) 1.88332e33 0.256478
\(513\) 5.74880e33 0.764029
\(514\) −3.93239e32 −0.0510055
\(515\) −1.40403e34 −1.77741
\(516\) −1.47498e34 −1.82249
\(517\) 1.71231e33 0.206514
\(518\) 0 0
\(519\) −8.98760e33 −1.03288
\(520\) 1.26665e33 0.142107
\(521\) −7.20328e32 −0.0788967 −0.0394483 0.999222i \(-0.512560\pi\)
−0.0394483 + 0.999222i \(0.512560\pi\)
\(522\) 1.57393e33 0.168308
\(523\) −7.40980e33 −0.773634 −0.386817 0.922156i \(-0.626426\pi\)
−0.386817 + 0.922156i \(0.626426\pi\)
\(524\) 2.92630e33 0.298316
\(525\) 0 0
\(526\) −7.10041e32 −0.0690179
\(527\) −1.48697e34 −1.41146
\(528\) −4.22408e33 −0.391569
\(529\) 1.82053e34 1.64817
\(530\) 4.58690e32 0.0405575
\(531\) 2.04127e34 1.76287
\(532\) 0 0
\(533\) −1.39273e34 −1.14756
\(534\) 1.35685e33 0.109211
\(535\) 4.33581e33 0.340917
\(536\) −1.12552e33 −0.0864560
\(537\) 4.25745e33 0.319500
\(538\) 2.71306e32 0.0198921
\(539\) 0 0
\(540\) −3.11990e34 −2.18383
\(541\) −2.28059e34 −1.55985 −0.779923 0.625876i \(-0.784742\pi\)
−0.779923 + 0.625876i \(0.784742\pi\)
\(542\) −6.92963e32 −0.0463147
\(543\) 4.38170e34 2.86184
\(544\) −2.99462e33 −0.191142
\(545\) −1.95969e34 −1.22245
\(546\) 0 0
\(547\) 7.07649e33 0.421674 0.210837 0.977521i \(-0.432381\pi\)
0.210837 + 0.977521i \(0.432381\pi\)
\(548\) 2.01914e34 1.17600
\(549\) 4.26423e34 2.42765
\(550\) −4.41051e32 −0.0245446
\(551\) 2.04443e34 1.11218
\(552\) 5.26630e33 0.280070
\(553\) 0 0
\(554\) −7.17825e32 −0.0364877
\(555\) −1.55108e34 −0.770855
\(556\) 1.83020e34 0.889331
\(557\) −3.52550e34 −1.67506 −0.837530 0.546391i \(-0.816001\pi\)
−0.837530 + 0.546391i \(0.816001\pi\)
\(558\) 2.25037e33 0.104551
\(559\) −1.91985e34 −0.872206
\(560\) 0 0
\(561\) 1.12044e34 0.486802
\(562\) −1.29780e33 −0.0551446
\(563\) −5.41988e33 −0.225234 −0.112617 0.993638i \(-0.535923\pi\)
−0.112617 + 0.993638i \(0.535923\pi\)
\(564\) 3.54502e34 1.44089
\(565\) 2.29255e34 0.911409
\(566\) 4.59665e32 0.0178746
\(567\) 0 0
\(568\) −4.18030e33 −0.155544
\(569\) 3.14649e33 0.114531 0.0572655 0.998359i \(-0.481762\pi\)
0.0572655 + 0.998359i \(0.481762\pi\)
\(570\) 2.52685e33 0.0899794
\(571\) 1.08463e34 0.377860 0.188930 0.981991i \(-0.439498\pi\)
0.188930 + 0.981991i \(0.439498\pi\)
\(572\) −5.51299e33 −0.187904
\(573\) 5.58391e34 1.86211
\(574\) 0 0
\(575\) −1.01573e35 −3.24287
\(576\) −5.54828e34 −1.73331
\(577\) −2.14266e34 −0.655023 −0.327512 0.944847i \(-0.606210\pi\)
−0.327512 + 0.944847i \(0.606210\pi\)
\(578\) 9.02606e32 0.0270023
\(579\) 6.45982e34 1.89120
\(580\) −1.10952e35 −3.17897
\(581\) 0 0
\(582\) 8.81590e31 0.00241952
\(583\) −3.99820e33 −0.107401
\(584\) 7.52142e32 0.0197760
\(585\) −9.39245e34 −2.41730
\(586\) −6.20413e32 −0.0156301
\(587\) 1.24448e34 0.306911 0.153455 0.988156i \(-0.450960\pi\)
0.153455 + 0.988156i \(0.450960\pi\)
\(588\) 0 0
\(589\) 2.92307e34 0.690872
\(590\) 3.87922e33 0.0897623
\(591\) −5.64904e34 −1.27977
\(592\) −1.19911e34 −0.265972
\(593\) −3.18871e34 −0.692519 −0.346260 0.938139i \(-0.612548\pi\)
−0.346260 + 0.938139i \(0.612548\pi\)
\(594\) −7.33130e32 −0.0155902
\(595\) 0 0
\(596\) 5.26357e34 1.07325
\(597\) −6.59395e34 −1.31664
\(598\) 3.42271e33 0.0669276
\(599\) 8.22623e34 1.57531 0.787655 0.616117i \(-0.211295\pi\)
0.787655 + 0.616117i \(0.211295\pi\)
\(600\) −1.82870e34 −0.342966
\(601\) −4.72278e34 −0.867495 −0.433747 0.901034i \(-0.642809\pi\)
−0.433747 + 0.901034i \(0.642809\pi\)
\(602\) 0 0
\(603\) 8.34598e34 1.47066
\(604\) 5.62892e34 0.971545
\(605\) −9.65506e34 −1.63235
\(606\) −1.54461e33 −0.0255807
\(607\) −8.65747e34 −1.40453 −0.702267 0.711913i \(-0.747829\pi\)
−0.702267 + 0.711913i \(0.747829\pi\)
\(608\) 5.88678e33 0.0935585
\(609\) 0 0
\(610\) 8.10371e33 0.123612
\(611\) 4.61423e34 0.689579
\(612\) 1.47972e35 2.16663
\(613\) 1.01940e35 1.46247 0.731235 0.682125i \(-0.238944\pi\)
0.731235 + 0.682125i \(0.238944\pi\)
\(614\) 4.86412e33 0.0683754
\(615\) 3.01973e35 4.15939
\(616\) 0 0
\(617\) 2.76112e34 0.365192 0.182596 0.983188i \(-0.441550\pi\)
0.182596 + 0.983188i \(0.441550\pi\)
\(618\) −6.83053e33 −0.0885316
\(619\) −8.47803e34 −1.07687 −0.538433 0.842668i \(-0.680984\pi\)
−0.538433 + 0.842668i \(0.680984\pi\)
\(620\) −1.58636e35 −1.97473
\(621\) −1.68838e35 −2.05980
\(622\) −3.96486e33 −0.0474077
\(623\) 0 0
\(624\) −1.13828e35 −1.30750
\(625\) 8.68954e34 0.978355
\(626\) 4.56368e33 0.0503658
\(627\) −2.20254e34 −0.238276
\(628\) −8.65827e34 −0.918196
\(629\) 3.18063e34 0.330659
\(630\) 0 0
\(631\) −8.38341e34 −0.837633 −0.418817 0.908071i \(-0.637555\pi\)
−0.418817 + 0.908071i \(0.637555\pi\)
\(632\) −8.94670e33 −0.0876394
\(633\) 1.66621e35 1.60023
\(634\) −3.07591e31 −0.000289640 0
\(635\) 1.48833e35 1.37413
\(636\) −8.27756e34 −0.749357
\(637\) 0 0
\(638\) −2.60721e33 −0.0226943
\(639\) 3.09977e35 2.64587
\(640\) −4.25779e34 −0.356398
\(641\) −2.39743e34 −0.196798 −0.0983990 0.995147i \(-0.531372\pi\)
−0.0983990 + 0.995147i \(0.531372\pi\)
\(642\) 2.10935e33 0.0169809
\(643\) −1.85782e35 −1.46678 −0.733391 0.679807i \(-0.762063\pi\)
−0.733391 + 0.679807i \(0.762063\pi\)
\(644\) 0 0
\(645\) 4.16263e35 3.16134
\(646\) −5.18152e33 −0.0385968
\(647\) −2.25609e35 −1.64836 −0.824181 0.566326i \(-0.808364\pi\)
−0.824181 + 0.566326i \(0.808364\pi\)
\(648\) −4.93944e33 −0.0353990
\(649\) −3.38135e34 −0.237701
\(650\) −1.18852e34 −0.0819576
\(651\) 0 0
\(652\) 2.00845e35 1.33280
\(653\) 1.67096e35 1.08780 0.543902 0.839149i \(-0.316946\pi\)
0.543902 + 0.839149i \(0.316946\pi\)
\(654\) −9.53376e33 −0.0608895
\(655\) −8.25846e34 −0.517467
\(656\) 2.33448e35 1.43514
\(657\) −5.57728e34 −0.336399
\(658\) 0 0
\(659\) 2.08497e34 0.121069 0.0605344 0.998166i \(-0.480720\pi\)
0.0605344 + 0.998166i \(0.480720\pi\)
\(660\) 1.19533e35 0.681066
\(661\) 1.06866e35 0.597475 0.298738 0.954335i \(-0.403434\pi\)
0.298738 + 0.954335i \(0.403434\pi\)
\(662\) 1.65020e33 0.00905339
\(663\) 3.01929e35 1.62550
\(664\) 2.84676e32 0.00150401
\(665\) 0 0
\(666\) −4.81354e33 −0.0244927
\(667\) −6.00432e35 −2.99841
\(668\) −1.37555e35 −0.674172
\(669\) 5.30864e34 0.255362
\(670\) 1.58607e34 0.0748835
\(671\) −7.06365e34 −0.327339
\(672\) 0 0
\(673\) −2.13809e34 −0.0954635 −0.0477317 0.998860i \(-0.515199\pi\)
−0.0477317 + 0.998860i \(0.515199\pi\)
\(674\) −1.39345e34 −0.0610719
\(675\) 5.86281e35 2.52237
\(676\) 8.75764e34 0.369874
\(677\) 2.21336e35 0.917684 0.458842 0.888518i \(-0.348264\pi\)
0.458842 + 0.888518i \(0.348264\pi\)
\(678\) 1.11531e34 0.0453968
\(679\) 0 0
\(680\) 5.63166e34 0.220940
\(681\) 4.49502e35 1.73138
\(682\) −3.72772e33 −0.0140974
\(683\) −3.91218e35 −1.45265 −0.726323 0.687354i \(-0.758772\pi\)
−0.726323 + 0.687354i \(0.758772\pi\)
\(684\) −2.90880e35 −1.06051
\(685\) −5.69832e35 −2.03993
\(686\) 0 0
\(687\) 4.18685e35 1.44520
\(688\) 3.21804e35 1.09078
\(689\) −1.07741e35 −0.358626
\(690\) −7.42115e34 −0.242581
\(691\) 3.80918e35 1.22280 0.611400 0.791322i \(-0.290607\pi\)
0.611400 + 0.791322i \(0.290607\pi\)
\(692\) 1.96618e35 0.619865
\(693\) 0 0
\(694\) 1.40009e33 0.00425759
\(695\) −5.16511e35 −1.54266
\(696\) −1.08101e35 −0.317112
\(697\) −6.19223e35 −1.78417
\(698\) 3.15985e34 0.0894280
\(699\) 8.97630e35 2.49536
\(700\) 0 0
\(701\) 8.68030e34 0.232841 0.116421 0.993200i \(-0.462858\pi\)
0.116421 + 0.993200i \(0.462858\pi\)
\(702\) −1.97560e34 −0.0520576
\(703\) −6.25245e34 −0.161848
\(704\) 9.19067e34 0.233716
\(705\) −1.00046e36 −2.49940
\(706\) 9.55934e33 0.0234622
\(707\) 0 0
\(708\) −7.00047e35 −1.65848
\(709\) 4.90839e35 1.14251 0.571256 0.820772i \(-0.306456\pi\)
0.571256 + 0.820772i \(0.306456\pi\)
\(710\) 5.89079e34 0.134724
\(711\) 6.63415e35 1.49079
\(712\) −5.94464e34 −0.131258
\(713\) −8.58482e35 −1.86257
\(714\) 0 0
\(715\) 1.55585e35 0.325944
\(716\) −9.31384e34 −0.191741
\(717\) 1.02305e36 2.06970
\(718\) 2.47416e34 0.0491894
\(719\) −5.04396e35 −0.985509 −0.492755 0.870168i \(-0.664010\pi\)
−0.492755 + 0.870168i \(0.664010\pi\)
\(720\) 1.57436e36 3.02306
\(721\) 0 0
\(722\) −1.77704e34 −0.0329596
\(723\) −6.14328e35 −1.11988
\(724\) −9.58565e35 −1.71747
\(725\) 2.08497e36 3.67177
\(726\) −4.69713e34 −0.0813064
\(727\) 2.86199e34 0.0486954 0.0243477 0.999704i \(-0.492249\pi\)
0.0243477 + 0.999704i \(0.492249\pi\)
\(728\) 0 0
\(729\) −9.13206e35 −1.50132
\(730\) −1.05990e34 −0.0171289
\(731\) −8.53585e35 −1.35606
\(732\) −1.46240e36 −2.28391
\(733\) −3.05062e35 −0.468370 −0.234185 0.972192i \(-0.575242\pi\)
−0.234185 + 0.972192i \(0.575242\pi\)
\(734\) −2.83283e34 −0.0427582
\(735\) 0 0
\(736\) −1.72890e35 −0.252231
\(737\) −1.38250e35 −0.198300
\(738\) 9.37127e34 0.132158
\(739\) 5.77113e35 0.800212 0.400106 0.916469i \(-0.368973\pi\)
0.400106 + 0.916469i \(0.368973\pi\)
\(740\) 3.39324e35 0.462613
\(741\) −5.93528e35 −0.795635
\(742\) 0 0
\(743\) 4.31674e35 0.559494 0.279747 0.960074i \(-0.409749\pi\)
0.279747 + 0.960074i \(0.409749\pi\)
\(744\) −1.54559e35 −0.196985
\(745\) −1.48546e36 −1.86169
\(746\) −4.31175e34 −0.0531396
\(747\) −2.11093e34 −0.0255838
\(748\) −2.45113e35 −0.292144
\(749\) 0 0
\(750\) 1.28380e35 0.147990
\(751\) 1.13779e36 1.28992 0.644959 0.764217i \(-0.276874\pi\)
0.644959 + 0.764217i \(0.276874\pi\)
\(752\) −7.73433e35 −0.862382
\(753\) 1.76925e35 0.194022
\(754\) −7.02575e34 −0.0757794
\(755\) −1.58857e36 −1.68527
\(756\) 0 0
\(757\) −1.93410e36 −1.98509 −0.992546 0.121872i \(-0.961110\pi\)
−0.992546 + 0.121872i \(0.961110\pi\)
\(758\) −2.57827e34 −0.0260293
\(759\) 6.46869e35 0.642383
\(760\) −1.10706e35 −0.108144
\(761\) −5.50323e35 −0.528822 −0.264411 0.964410i \(-0.585178\pi\)
−0.264411 + 0.964410i \(0.585178\pi\)
\(762\) 7.24063e34 0.0684446
\(763\) 0 0
\(764\) −1.22157e36 −1.11751
\(765\) −4.17599e36 −3.75830
\(766\) −3.71931e33 −0.00329309
\(767\) −9.11186e35 −0.793715
\(768\) 1.88718e36 1.61733
\(769\) −3.37946e35 −0.284949 −0.142475 0.989798i \(-0.545506\pi\)
−0.142475 + 0.989798i \(0.545506\pi\)
\(770\) 0 0
\(771\) −2.00272e36 −1.63471
\(772\) −1.41319e36 −1.13497
\(773\) 1.97789e36 1.56300 0.781499 0.623907i \(-0.214456\pi\)
0.781499 + 0.623907i \(0.214456\pi\)
\(774\) 1.29181e35 0.100447
\(775\) 2.98104e36 2.28085
\(776\) −3.86243e33 −0.00290797
\(777\) 0 0
\(778\) 3.14218e34 0.0229080
\(779\) 1.21726e36 0.873303
\(780\) 3.22111e36 2.27417
\(781\) −5.13474e35 −0.356764
\(782\) 1.52177e35 0.104056
\(783\) 3.46571e36 2.33222
\(784\) 0 0
\(785\) 2.44350e36 1.59273
\(786\) −4.01769e34 −0.0257748
\(787\) −3.56804e35 −0.225292 −0.112646 0.993635i \(-0.535933\pi\)
−0.112646 + 0.993635i \(0.535933\pi\)
\(788\) 1.23582e36 0.768026
\(789\) −3.61616e36 −2.21200
\(790\) 1.26075e35 0.0759084
\(791\) 0 0
\(792\) 7.42905e34 0.0433380
\(793\) −1.90347e36 −1.09303
\(794\) −2.08343e34 −0.0117767
\(795\) 2.33606e36 1.29985
\(796\) 1.44253e36 0.790153
\(797\) −2.08564e36 −1.12463 −0.562317 0.826922i \(-0.690090\pi\)
−0.562317 + 0.826922i \(0.690090\pi\)
\(798\) 0 0
\(799\) 2.05154e36 1.07212
\(800\) 6.00353e35 0.308874
\(801\) 4.40806e36 2.23276
\(802\) −3.16328e34 −0.0157746
\(803\) 9.23871e34 0.0453594
\(804\) −2.86223e36 −1.38358
\(805\) 0 0
\(806\) −1.00452e35 −0.0470730
\(807\) 1.38173e36 0.637536
\(808\) 6.76728e34 0.0307448
\(809\) 1.27866e36 0.572003 0.286001 0.958229i \(-0.407674\pi\)
0.286001 + 0.958229i \(0.407674\pi\)
\(810\) 6.96056e34 0.0306606
\(811\) 3.41904e36 1.48301 0.741505 0.670947i \(-0.234112\pi\)
0.741505 + 0.670947i \(0.234112\pi\)
\(812\) 0 0
\(813\) −3.52918e36 −1.48437
\(814\) 7.97359e33 0.00330255
\(815\) −5.66816e36 −2.31192
\(816\) −5.06092e36 −2.03284
\(817\) 1.67797e36 0.663754
\(818\) −1.68307e35 −0.0655671
\(819\) 0 0
\(820\) −6.60614e36 −2.49617
\(821\) −5.12217e36 −1.90618 −0.953092 0.302681i \(-0.902118\pi\)
−0.953092 + 0.302681i \(0.902118\pi\)
\(822\) −2.77220e35 −0.101608
\(823\) 5.26462e35 0.190050 0.0950252 0.995475i \(-0.469707\pi\)
0.0950252 + 0.995475i \(0.469707\pi\)
\(824\) 2.99260e35 0.106404
\(825\) −2.24622e36 −0.786644
\(826\) 0 0
\(827\) 2.25397e36 0.765825 0.382913 0.923785i \(-0.374921\pi\)
0.382913 + 0.923785i \(0.374921\pi\)
\(828\) 8.54293e36 2.85909
\(829\) −3.48267e36 −1.14810 −0.574052 0.818819i \(-0.694629\pi\)
−0.574052 + 0.818819i \(0.694629\pi\)
\(830\) −4.01159e33 −0.00130269
\(831\) −3.65580e36 −1.16942
\(832\) 2.47665e36 0.780410
\(833\) 0 0
\(834\) −2.51279e35 −0.0768389
\(835\) 3.88202e36 1.16944
\(836\) 4.81840e35 0.142996
\(837\) 4.95518e36 1.44874
\(838\) −2.56106e35 −0.0737682
\(839\) 1.38679e36 0.393536 0.196768 0.980450i \(-0.436955\pi\)
0.196768 + 0.980450i \(0.436955\pi\)
\(840\) 0 0
\(841\) 8.69464e36 2.39498
\(842\) −1.40914e35 −0.0382430
\(843\) −6.60953e36 −1.76736
\(844\) −3.64509e36 −0.960345
\(845\) −2.47154e36 −0.641592
\(846\) −3.10478e35 −0.0794147
\(847\) 0 0
\(848\) 1.80595e36 0.448496
\(849\) 2.34102e36 0.572874
\(850\) −5.28428e35 −0.127424
\(851\) 1.83629e36 0.436338
\(852\) −1.06306e37 −2.48921
\(853\) 1.28825e36 0.297261 0.148631 0.988893i \(-0.452513\pi\)
0.148631 + 0.988893i \(0.452513\pi\)
\(854\) 0 0
\(855\) 8.20909e36 1.83958
\(856\) −9.24149e34 −0.0204089
\(857\) −1.05441e36 −0.229482 −0.114741 0.993395i \(-0.536604\pi\)
−0.114741 + 0.993395i \(0.536604\pi\)
\(858\) 7.56912e34 0.0162351
\(859\) 7.57439e36 1.60115 0.800576 0.599231i \(-0.204527\pi\)
0.800576 + 0.599231i \(0.204527\pi\)
\(860\) −9.10641e36 −1.89722
\(861\) 0 0
\(862\) −3.47704e35 −0.0703670
\(863\) −3.22715e36 −0.643701 −0.321850 0.946791i \(-0.604305\pi\)
−0.321850 + 0.946791i \(0.604305\pi\)
\(864\) 9.97927e35 0.196190
\(865\) −5.54886e36 −1.07523
\(866\) 2.27871e35 0.0435226
\(867\) 4.59687e36 0.865413
\(868\) 0 0
\(869\) −1.09894e36 −0.201014
\(870\) 1.52333e36 0.274665
\(871\) −3.72549e36 −0.662151
\(872\) 4.17694e35 0.0731817
\(873\) 2.86407e35 0.0494658
\(874\) −2.99148e35 −0.0509323
\(875\) 0 0
\(876\) 1.91271e36 0.316481
\(877\) 5.54904e36 0.905154 0.452577 0.891725i \(-0.350505\pi\)
0.452577 + 0.891725i \(0.350505\pi\)
\(878\) 5.60221e35 0.0900903
\(879\) −3.15969e36 −0.500938
\(880\) −2.60791e36 −0.407623
\(881\) 9.25222e36 1.42576 0.712881 0.701285i \(-0.247390\pi\)
0.712881 + 0.701285i \(0.247390\pi\)
\(882\) 0 0
\(883\) 2.33411e36 0.349633 0.174817 0.984601i \(-0.444067\pi\)
0.174817 + 0.984601i \(0.444067\pi\)
\(884\) −6.60518e36 −0.975508
\(885\) 1.97564e37 2.87685
\(886\) 2.64114e35 0.0379201
\(887\) 1.00103e37 1.41710 0.708550 0.705661i \(-0.249350\pi\)
0.708550 + 0.705661i \(0.249350\pi\)
\(888\) 3.30603e35 0.0461471
\(889\) 0 0
\(890\) 8.37706e35 0.113688
\(891\) −6.06722e35 −0.0811929
\(892\) −1.16135e36 −0.153250
\(893\) −4.03288e36 −0.524774
\(894\) −7.22668e35 −0.0927299
\(895\) 2.62851e36 0.332600
\(896\) 0 0
\(897\) 1.74315e37 2.14500
\(898\) −4.94434e35 −0.0600004
\(899\) 1.76220e37 2.10891
\(900\) −2.96649e37 −3.50116
\(901\) −4.79030e36 −0.557574
\(902\) −1.55234e35 −0.0178199
\(903\) 0 0
\(904\) −4.88642e35 −0.0545614
\(905\) 2.70522e37 2.97917
\(906\) −7.72829e35 −0.0839423
\(907\) 1.19643e37 1.28173 0.640865 0.767654i \(-0.278576\pi\)
0.640865 + 0.767654i \(0.278576\pi\)
\(908\) −9.83355e36 −1.03905
\(909\) −5.01807e36 −0.522984
\(910\) 0 0
\(911\) −6.18003e35 −0.0626629 −0.0313315 0.999509i \(-0.509975\pi\)
−0.0313315 + 0.999509i \(0.509975\pi\)
\(912\) 9.94868e36 0.995016
\(913\) 3.49673e34 0.00344967
\(914\) 4.48916e34 0.00436855
\(915\) 4.12713e37 3.96172
\(916\) −9.15940e36 −0.867308
\(917\) 0 0
\(918\) −8.78372e35 −0.0809366
\(919\) −1.87864e37 −1.70765 −0.853826 0.520558i \(-0.825724\pi\)
−0.853826 + 0.520558i \(0.825724\pi\)
\(920\) 3.25136e36 0.291553
\(921\) 2.47724e37 2.19141
\(922\) −4.60822e35 −0.0402158
\(923\) −1.38368e37 −1.19128
\(924\) 0 0
\(925\) −6.37645e36 −0.534327
\(926\) 5.19167e35 0.0429210
\(927\) −2.21907e37 −1.80998
\(928\) 3.54890e36 0.285590
\(929\) 1.07023e37 0.849732 0.424866 0.905256i \(-0.360321\pi\)
0.424866 + 0.905256i \(0.360321\pi\)
\(930\) 2.17802e36 0.170618
\(931\) 0 0
\(932\) −1.96371e37 −1.49754
\(933\) −2.01926e37 −1.51940
\(934\) −1.65998e35 −0.0123244
\(935\) 6.91748e36 0.506761
\(936\) 2.00194e36 0.144711
\(937\) −1.91721e37 −1.36749 −0.683746 0.729720i \(-0.739650\pi\)
−0.683746 + 0.729720i \(0.739650\pi\)
\(938\) 0 0
\(939\) 2.32423e37 1.61421
\(940\) 2.18867e37 1.49997
\(941\) −2.06934e37 −1.39946 −0.699732 0.714405i \(-0.746697\pi\)
−0.699732 + 0.714405i \(0.746697\pi\)
\(942\) 1.18875e36 0.0793329
\(943\) −3.57500e37 −2.35440
\(944\) 1.52732e37 0.992615
\(945\) 0 0
\(946\) −2.13987e35 −0.0135440
\(947\) 7.48084e36 0.467278 0.233639 0.972323i \(-0.424937\pi\)
0.233639 + 0.972323i \(0.424937\pi\)
\(948\) −2.27516e37 −1.40251
\(949\) 2.48959e36 0.151461
\(950\) 1.03878e36 0.0623702
\(951\) −1.56653e35 −0.00928286
\(952\) 0 0
\(953\) 2.03734e37 1.17599 0.587994 0.808865i \(-0.299918\pi\)
0.587994 + 0.808865i \(0.299918\pi\)
\(954\) 7.24959e35 0.0413008
\(955\) 3.44746e37 1.93846
\(956\) −2.23808e37 −1.24209
\(957\) −1.32782e37 −0.727345
\(958\) 1.78688e36 0.0966112
\(959\) 0 0
\(960\) −5.36989e37 −2.82862
\(961\) 5.96261e36 0.310023
\(962\) 2.14868e35 0.0110276
\(963\) 6.85274e36 0.347165
\(964\) 1.34394e37 0.672074
\(965\) 3.98823e37 1.96874
\(966\) 0 0
\(967\) −1.36949e37 −0.658763 −0.329382 0.944197i \(-0.606840\pi\)
−0.329382 + 0.944197i \(0.606840\pi\)
\(968\) 2.05791e36 0.0977203
\(969\) −2.63889e37 −1.23701
\(970\) 5.44286e34 0.00251872
\(971\) −3.88125e37 −1.77309 −0.886545 0.462642i \(-0.846901\pi\)
−0.886545 + 0.462642i \(0.846901\pi\)
\(972\) 1.54294e37 0.695860
\(973\) 0 0
\(974\) −3.89654e35 −0.0171274
\(975\) −6.05299e37 −2.62671
\(976\) 3.19059e37 1.36693
\(977\) −2.41102e35 −0.0101981 −0.00509905 0.999987i \(-0.501623\pi\)
−0.00509905 + 0.999987i \(0.501623\pi\)
\(978\) −2.75753e36 −0.115155
\(979\) −7.30192e36 −0.301060
\(980\) 0 0
\(981\) −3.09728e37 −1.24485
\(982\) −5.90765e35 −0.0234434
\(983\) −2.16610e37 −0.848710 −0.424355 0.905496i \(-0.639499\pi\)
−0.424355 + 0.905496i \(0.639499\pi\)
\(984\) −6.43635e36 −0.249001
\(985\) −3.48766e37 −1.33224
\(986\) −3.12373e36 −0.117818
\(987\) 0 0
\(988\) 1.29844e37 0.477484
\(989\) −4.92805e37 −1.78946
\(990\) −1.04689e36 −0.0375370
\(991\) 7.62118e36 0.269837 0.134918 0.990857i \(-0.456923\pi\)
0.134918 + 0.990857i \(0.456923\pi\)
\(992\) 5.07412e36 0.177405
\(993\) 8.40427e36 0.290158
\(994\) 0 0
\(995\) −4.07104e37 −1.37062
\(996\) 7.23935e35 0.0240690
\(997\) 3.72156e37 1.22190 0.610949 0.791670i \(-0.290788\pi\)
0.610949 + 0.791670i \(0.290788\pi\)
\(998\) −9.96005e35 −0.0322946
\(999\) −1.05991e37 −0.339392
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 49.26.a.e.1.8 yes 12
7.6 odd 2 inner 49.26.a.e.1.7 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.26.a.e.1.7 12 7.6 odd 2 inner
49.26.a.e.1.8 yes 12 1.1 even 1 trivial