Properties

Label 27.10.a.a.1.2
Level $27$
Weight $10$
Character 27.1
Self dual yes
Analytic conductor $13.906$
Analytic rank $1$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [27,10,Mod(1,27)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(27, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.1");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 27.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: \(13.9059675764\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{14}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(3.74166\) of defining polynomial
Character \(\chi\) \(=\) 27.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+22.4499 q^{2} -8.00000 q^{4} -493.899 q^{5} -763.000 q^{7} -11674.0 q^{8} -11088.0 q^{10} -56888.2 q^{11} -73015.0 q^{13} -17129.3 q^{14} -257984. q^{16} +168240. q^{17} -598129. q^{19} +3951.19 q^{20} -1.27714e6 q^{22} +2.40039e6 q^{23} -1.70919e6 q^{25} -1.63918e6 q^{26} +6104.00 q^{28} -4.65378e6 q^{29} -1.82410e6 q^{31} +185347. q^{32} +3.77698e6 q^{34} +376845. q^{35} +1.42722e7 q^{37} -1.34280e7 q^{38} +5.76576e6 q^{40} +2.98243e7 q^{41} +7.75690e6 q^{43} +455105. q^{44} +5.38887e7 q^{46} +3.09903e7 q^{47} -3.97714e7 q^{49} -3.83712e7 q^{50} +584120. q^{52} +9.90123e6 q^{53} +2.80970e7 q^{55} +8.90724e6 q^{56} -1.04477e8 q^{58} -1.25210e8 q^{59} -1.52766e8 q^{61} -4.09509e7 q^{62} +1.36249e8 q^{64} +3.60620e7 q^{65} -3.20752e8 q^{67} -1.34592e6 q^{68} +8.46014e6 q^{70} +1.09152e8 q^{71} +6.64640e7 q^{73} +3.20410e8 q^{74} +4.78503e6 q^{76} +4.34057e7 q^{77} -1.15357e8 q^{79} +1.27418e8 q^{80} +6.69554e8 q^{82} -6.14639e8 q^{83} -8.30935e7 q^{85} +1.74142e8 q^{86} +6.64111e8 q^{88} -3.37797e8 q^{89} +5.57104e7 q^{91} -1.92031e7 q^{92} +6.95731e8 q^{94} +2.95415e8 q^{95} -2.35307e8 q^{97} -8.92867e8 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 16 q^{4} - 1526 q^{7} - 22176 q^{10} - 146030 q^{13} - 515968 q^{16} - 1196258 q^{19} - 2554272 q^{22} - 3418378 q^{25} + 12208 q^{28} - 3648200 q^{31} + 7553952 q^{34} + 28544350 q^{37} + 11531520 q^{40}+ \cdots - 470613686 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 22.4499 0.992157 0.496078 0.868278i \(-0.334773\pi\)
0.496078 + 0.868278i \(0.334773\pi\)
\(3\) 0 0
\(4\) −8.00000 −0.0156250
\(5\) −493.899 −0.353405 −0.176703 0.984264i \(-0.556543\pi\)
−0.176703 + 0.984264i \(0.556543\pi\)
\(6\) 0 0
\(7\) −763.000 −0.120111 −0.0600556 0.998195i \(-0.519128\pi\)
−0.0600556 + 0.998195i \(0.519128\pi\)
\(8\) −11674.0 −1.00766
\(9\) 0 0
\(10\) −11088.0 −0.350633
\(11\) −56888.2 −1.17153 −0.585767 0.810480i \(-0.699207\pi\)
−0.585767 + 0.810480i \(0.699207\pi\)
\(12\) 0 0
\(13\) −73015.0 −0.709034 −0.354517 0.935050i \(-0.615355\pi\)
−0.354517 + 0.935050i \(0.615355\pi\)
\(14\) −17129.3 −0.119169
\(15\) 0 0
\(16\) −257984. −0.984131
\(17\) 168240. 0.488550 0.244275 0.969706i \(-0.421450\pi\)
0.244275 + 0.969706i \(0.421450\pi\)
\(18\) 0 0
\(19\) −598129. −1.05294 −0.526470 0.850194i \(-0.676485\pi\)
−0.526470 + 0.850194i \(0.676485\pi\)
\(20\) 3951.19 0.00552196
\(21\) 0 0
\(22\) −1.27714e6 −1.16235
\(23\) 2.40039e6 1.78857 0.894287 0.447493i \(-0.147683\pi\)
0.894287 + 0.447493i \(0.147683\pi\)
\(24\) 0 0
\(25\) −1.70919e6 −0.875105
\(26\) −1.63918e6 −0.703473
\(27\) 0 0
\(28\) 6104.00 0.00187674
\(29\) −4.65378e6 −1.22184 −0.610921 0.791691i \(-0.709201\pi\)
−0.610921 + 0.791691i \(0.709201\pi\)
\(30\) 0 0
\(31\) −1.82410e6 −0.354749 −0.177374 0.984143i \(-0.556760\pi\)
−0.177374 + 0.984143i \(0.556760\pi\)
\(32\) 185347. 0.0312471
\(33\) 0 0
\(34\) 3.77698e6 0.484718
\(35\) 376845. 0.0424479
\(36\) 0 0
\(37\) 1.42722e7 1.25194 0.625968 0.779848i \(-0.284704\pi\)
0.625968 + 0.779848i \(0.284704\pi\)
\(38\) −1.34280e7 −1.04468
\(39\) 0 0
\(40\) 5.76576e6 0.356112
\(41\) 2.98243e7 1.64833 0.824163 0.566353i \(-0.191646\pi\)
0.824163 + 0.566353i \(0.191646\pi\)
\(42\) 0 0
\(43\) 7.75690e6 0.346003 0.173002 0.984922i \(-0.444653\pi\)
0.173002 + 0.984922i \(0.444653\pi\)
\(44\) 455105. 0.0183052
\(45\) 0 0
\(46\) 5.38887e7 1.77455
\(47\) 3.09903e7 0.926372 0.463186 0.886261i \(-0.346706\pi\)
0.463186 + 0.886261i \(0.346706\pi\)
\(48\) 0 0
\(49\) −3.97714e7 −0.985573
\(50\) −3.83712e7 −0.868241
\(51\) 0 0
\(52\) 584120. 0.0110787
\(53\) 9.90123e6 0.172365 0.0861823 0.996279i \(-0.472533\pi\)
0.0861823 + 0.996279i \(0.472533\pi\)
\(54\) 0 0
\(55\) 2.80970e7 0.414026
\(56\) 8.90724e6 0.121031
\(57\) 0 0
\(58\) −1.04477e8 −1.21226
\(59\) −1.25210e8 −1.34525 −0.672627 0.739982i \(-0.734834\pi\)
−0.672627 + 0.739982i \(0.734834\pi\)
\(60\) 0 0
\(61\) −1.52766e8 −1.41268 −0.706340 0.707873i \(-0.749655\pi\)
−0.706340 + 0.707873i \(0.749655\pi\)
\(62\) −4.09509e7 −0.351966
\(63\) 0 0
\(64\) 1.36249e8 1.01513
\(65\) 3.60620e7 0.250576
\(66\) 0 0
\(67\) −3.20752e8 −1.94461 −0.972306 0.233712i \(-0.924913\pi\)
−0.972306 + 0.233712i \(0.924913\pi\)
\(68\) −1.34592e6 −0.00763359
\(69\) 0 0
\(70\) 8.46014e6 0.0421150
\(71\) 1.09152e8 0.509763 0.254881 0.966972i \(-0.417964\pi\)
0.254881 + 0.966972i \(0.417964\pi\)
\(72\) 0 0
\(73\) 6.64640e7 0.273926 0.136963 0.990576i \(-0.456266\pi\)
0.136963 + 0.990576i \(0.456266\pi\)
\(74\) 3.20410e8 1.24212
\(75\) 0 0
\(76\) 4.78503e6 0.0164522
\(77\) 4.34057e7 0.140714
\(78\) 0 0
\(79\) −1.15357e8 −0.333214 −0.166607 0.986023i \(-0.553281\pi\)
−0.166607 + 0.986023i \(0.553281\pi\)
\(80\) 1.27418e8 0.347797
\(81\) 0 0
\(82\) 6.69554e8 1.63540
\(83\) −6.14639e8 −1.42157 −0.710785 0.703409i \(-0.751660\pi\)
−0.710785 + 0.703409i \(0.751660\pi\)
\(84\) 0 0
\(85\) −8.30935e7 −0.172656
\(86\) 1.74142e8 0.343290
\(87\) 0 0
\(88\) 6.64111e8 1.18051
\(89\) −3.37797e8 −0.570691 −0.285345 0.958425i \(-0.592108\pi\)
−0.285345 + 0.958425i \(0.592108\pi\)
\(90\) 0 0
\(91\) 5.57104e7 0.0851629
\(92\) −1.92031e7 −0.0279465
\(93\) 0 0
\(94\) 6.95731e8 0.919106
\(95\) 2.95415e8 0.372114
\(96\) 0 0
\(97\) −2.35307e8 −0.269874 −0.134937 0.990854i \(-0.543083\pi\)
−0.134937 + 0.990854i \(0.543083\pi\)
\(98\) −8.92867e8 −0.977843
\(99\) 0 0
\(100\) 1.36735e7 0.0136735
\(101\) −9.16170e8 −0.876052 −0.438026 0.898962i \(-0.644322\pi\)
−0.438026 + 0.898962i \(0.644322\pi\)
\(102\) 0 0
\(103\) 1.00799e8 0.0882451 0.0441225 0.999026i \(-0.485951\pi\)
0.0441225 + 0.999026i \(0.485951\pi\)
\(104\) 8.52375e8 0.714465
\(105\) 0 0
\(106\) 2.22282e8 0.171013
\(107\) 1.50887e9 1.11282 0.556411 0.830907i \(-0.312178\pi\)
0.556411 + 0.830907i \(0.312178\pi\)
\(108\) 0 0
\(109\) −2.26733e9 −1.53849 −0.769247 0.638951i \(-0.779369\pi\)
−0.769247 + 0.638951i \(0.779369\pi\)
\(110\) 6.30776e8 0.410779
\(111\) 0 0
\(112\) 1.96842e8 0.118205
\(113\) 1.15962e9 0.669054 0.334527 0.942386i \(-0.391423\pi\)
0.334527 + 0.942386i \(0.391423\pi\)
\(114\) 0 0
\(115\) −1.18555e9 −0.632092
\(116\) 3.72303e7 0.0190913
\(117\) 0 0
\(118\) −2.81095e9 −1.33470
\(119\) −1.28367e8 −0.0586803
\(120\) 0 0
\(121\) 8.78315e8 0.372491
\(122\) −3.42960e9 −1.40160
\(123\) 0 0
\(124\) 1.45928e7 0.00554295
\(125\) 1.80881e9 0.662672
\(126\) 0 0
\(127\) −2.61195e9 −0.890938 −0.445469 0.895297i \(-0.646963\pi\)
−0.445469 + 0.895297i \(0.646963\pi\)
\(128\) 2.96388e9 0.975924
\(129\) 0 0
\(130\) 8.09590e8 0.248611
\(131\) −3.26886e9 −0.969786 −0.484893 0.874574i \(-0.661141\pi\)
−0.484893 + 0.874574i \(0.661141\pi\)
\(132\) 0 0
\(133\) 4.56372e8 0.126470
\(134\) −7.20087e9 −1.92936
\(135\) 0 0
\(136\) −1.96403e9 −0.492292
\(137\) −5.66884e7 −0.0137484 −0.00687419 0.999976i \(-0.502188\pi\)
−0.00687419 + 0.999976i \(0.502188\pi\)
\(138\) 0 0
\(139\) 6.79897e9 1.54482 0.772408 0.635127i \(-0.219052\pi\)
0.772408 + 0.635127i \(0.219052\pi\)
\(140\) −3.01476e6 −0.000663249 0
\(141\) 0 0
\(142\) 2.45045e9 0.505764
\(143\) 4.15369e9 0.830657
\(144\) 0 0
\(145\) 2.29850e9 0.431805
\(146\) 1.49211e9 0.271778
\(147\) 0 0
\(148\) −1.14177e8 −0.0195615
\(149\) 1.45949e9 0.242585 0.121292 0.992617i \(-0.461296\pi\)
0.121292 + 0.992617i \(0.461296\pi\)
\(150\) 0 0
\(151\) 4.95922e9 0.776277 0.388139 0.921601i \(-0.373118\pi\)
0.388139 + 0.921601i \(0.373118\pi\)
\(152\) 6.98254e9 1.06100
\(153\) 0 0
\(154\) 9.74455e8 0.139611
\(155\) 9.00921e8 0.125370
\(156\) 0 0
\(157\) 5.69095e9 0.747543 0.373772 0.927521i \(-0.378064\pi\)
0.373772 + 0.927521i \(0.378064\pi\)
\(158\) −2.58977e9 −0.330601
\(159\) 0 0
\(160\) −9.15425e7 −0.0110429
\(161\) −1.83150e9 −0.214828
\(162\) 0 0
\(163\) 7.88882e9 0.875322 0.437661 0.899140i \(-0.355807\pi\)
0.437661 + 0.899140i \(0.355807\pi\)
\(164\) −2.38594e8 −0.0257551
\(165\) 0 0
\(166\) −1.37986e10 −1.41042
\(167\) 9.25008e9 0.920283 0.460142 0.887846i \(-0.347799\pi\)
0.460142 + 0.887846i \(0.347799\pi\)
\(168\) 0 0
\(169\) −5.27331e9 −0.497271
\(170\) −1.86544e9 −0.171302
\(171\) 0 0
\(172\) −6.20552e7 −0.00540630
\(173\) −1.85425e10 −1.57384 −0.786922 0.617052i \(-0.788327\pi\)
−0.786922 + 0.617052i \(0.788327\pi\)
\(174\) 0 0
\(175\) 1.30411e9 0.105110
\(176\) 1.46762e10 1.15294
\(177\) 0 0
\(178\) −7.58352e9 −0.566215
\(179\) −1.83931e10 −1.33911 −0.669555 0.742762i \(-0.733515\pi\)
−0.669555 + 0.742762i \(0.733515\pi\)
\(180\) 0 0
\(181\) 1.17014e9 0.0810375 0.0405187 0.999179i \(-0.487099\pi\)
0.0405187 + 0.999179i \(0.487099\pi\)
\(182\) 1.25070e9 0.0844949
\(183\) 0 0
\(184\) −2.80221e10 −1.80227
\(185\) −7.04901e9 −0.442441
\(186\) 0 0
\(187\) −9.57086e9 −0.572353
\(188\) −2.47922e8 −0.0144746
\(189\) 0 0
\(190\) 6.63205e9 0.369196
\(191\) −1.03749e10 −0.564073 −0.282036 0.959404i \(-0.591010\pi\)
−0.282036 + 0.959404i \(0.591010\pi\)
\(192\) 0 0
\(193\) −1.17395e10 −0.609037 −0.304518 0.952506i \(-0.598496\pi\)
−0.304518 + 0.952506i \(0.598496\pi\)
\(194\) −5.28263e9 −0.267758
\(195\) 0 0
\(196\) 3.18172e8 0.0153996
\(197\) 2.61163e10 1.23542 0.617708 0.786408i \(-0.288062\pi\)
0.617708 + 0.786408i \(0.288062\pi\)
\(198\) 0 0
\(199\) 2.31315e10 1.04560 0.522800 0.852455i \(-0.324887\pi\)
0.522800 + 0.852455i \(0.324887\pi\)
\(200\) 1.99530e10 0.881807
\(201\) 0 0
\(202\) −2.05680e10 −0.869181
\(203\) 3.55084e9 0.146757
\(204\) 0 0
\(205\) −1.47302e10 −0.582527
\(206\) 2.26294e9 0.0875530
\(207\) 0 0
\(208\) 1.88367e10 0.697782
\(209\) 3.40265e10 1.23355
\(210\) 0 0
\(211\) 2.57907e10 0.895759 0.447880 0.894094i \(-0.352179\pi\)
0.447880 + 0.894094i \(0.352179\pi\)
\(212\) −7.92099e7 −0.00269320
\(213\) 0 0
\(214\) 3.38741e10 1.10409
\(215\) −3.83113e9 −0.122279
\(216\) 0 0
\(217\) 1.39179e9 0.0426093
\(218\) −5.09015e10 −1.52643
\(219\) 0 0
\(220\) −2.24776e8 −0.00646916
\(221\) −1.22840e10 −0.346398
\(222\) 0 0
\(223\) −1.00337e9 −0.0271699 −0.0135850 0.999908i \(-0.504324\pi\)
−0.0135850 + 0.999908i \(0.504324\pi\)
\(224\) −1.41420e8 −0.00375313
\(225\) 0 0
\(226\) 2.60333e10 0.663806
\(227\) 5.27911e10 1.31961 0.659803 0.751438i \(-0.270640\pi\)
0.659803 + 0.751438i \(0.270640\pi\)
\(228\) 0 0
\(229\) 3.10909e10 0.747091 0.373546 0.927612i \(-0.378142\pi\)
0.373546 + 0.927612i \(0.378142\pi\)
\(230\) −2.66156e10 −0.627134
\(231\) 0 0
\(232\) 5.43281e10 1.23120
\(233\) −3.41332e10 −0.758710 −0.379355 0.925251i \(-0.623854\pi\)
−0.379355 + 0.925251i \(0.623854\pi\)
\(234\) 0 0
\(235\) −1.53061e10 −0.327385
\(236\) 1.00168e9 0.0210196
\(237\) 0 0
\(238\) −2.88183e9 −0.0582200
\(239\) −9.02670e10 −1.78953 −0.894764 0.446539i \(-0.852656\pi\)
−0.894764 + 0.446539i \(0.852656\pi\)
\(240\) 0 0
\(241\) −7.73393e10 −1.47681 −0.738403 0.674360i \(-0.764420\pi\)
−0.738403 + 0.674360i \(0.764420\pi\)
\(242\) 1.97181e10 0.369570
\(243\) 0 0
\(244\) 1.22213e9 0.0220731
\(245\) 1.96431e10 0.348307
\(246\) 0 0
\(247\) 4.36724e10 0.746570
\(248\) 2.12945e10 0.357466
\(249\) 0 0
\(250\) 4.06077e10 0.657474
\(251\) 4.53113e10 0.720568 0.360284 0.932843i \(-0.382680\pi\)
0.360284 + 0.932843i \(0.382680\pi\)
\(252\) 0 0
\(253\) −1.36554e11 −2.09538
\(254\) −5.86380e10 −0.883950
\(255\) 0 0
\(256\) −3.22043e9 −0.0468635
\(257\) 1.35887e11 1.94303 0.971514 0.236984i \(-0.0761588\pi\)
0.971514 + 0.236984i \(0.0761588\pi\)
\(258\) 0 0
\(259\) −1.08897e10 −0.150372
\(260\) −2.88496e8 −0.00391525
\(261\) 0 0
\(262\) −7.33858e10 −0.962179
\(263\) 4.81731e10 0.620875 0.310437 0.950594i \(-0.399525\pi\)
0.310437 + 0.950594i \(0.399525\pi\)
\(264\) 0 0
\(265\) −4.89021e9 −0.0609145
\(266\) 1.02455e10 0.125478
\(267\) 0 0
\(268\) 2.56602e9 0.0303846
\(269\) −1.33744e11 −1.55736 −0.778682 0.627418i \(-0.784112\pi\)
−0.778682 + 0.627418i \(0.784112\pi\)
\(270\) 0 0
\(271\) −7.03388e10 −0.792197 −0.396098 0.918208i \(-0.629636\pi\)
−0.396098 + 0.918208i \(0.629636\pi\)
\(272\) −4.34032e10 −0.480797
\(273\) 0 0
\(274\) −1.27265e9 −0.0136406
\(275\) 9.72326e10 1.02521
\(276\) 0 0
\(277\) −1.34327e11 −1.37090 −0.685449 0.728121i \(-0.740394\pi\)
−0.685449 + 0.728121i \(0.740394\pi\)
\(278\) 1.52637e11 1.53270
\(279\) 0 0
\(280\) −4.39927e9 −0.0427730
\(281\) −1.15851e11 −1.10847 −0.554233 0.832362i \(-0.686988\pi\)
−0.554233 + 0.832362i \(0.686988\pi\)
\(282\) 0 0
\(283\) 1.40288e11 1.30011 0.650055 0.759887i \(-0.274746\pi\)
0.650055 + 0.759887i \(0.274746\pi\)
\(284\) −8.73214e8 −0.00796504
\(285\) 0 0
\(286\) 9.32501e10 0.824142
\(287\) −2.27559e10 −0.197982
\(288\) 0 0
\(289\) −9.02832e10 −0.761319
\(290\) 5.16012e10 0.428419
\(291\) 0 0
\(292\) −5.31712e8 −0.00428010
\(293\) −1.59942e11 −1.26782 −0.633910 0.773407i \(-0.718551\pi\)
−0.633910 + 0.773407i \(0.718551\pi\)
\(294\) 0 0
\(295\) 6.18409e10 0.475420
\(296\) −1.66613e11 −1.26153
\(297\) 0 0
\(298\) 3.27655e10 0.240682
\(299\) −1.75265e11 −1.26816
\(300\) 0 0
\(301\) −5.91852e9 −0.0415589
\(302\) 1.11334e11 0.770189
\(303\) 0 0
\(304\) 1.54308e11 1.03623
\(305\) 7.54512e10 0.499249
\(306\) 0 0
\(307\) 1.98532e11 1.27558 0.637791 0.770210i \(-0.279849\pi\)
0.637791 + 0.770210i \(0.279849\pi\)
\(308\) −3.47245e8 −0.00219866
\(309\) 0 0
\(310\) 2.02256e10 0.124387
\(311\) −2.24811e10 −0.136269 −0.0681344 0.997676i \(-0.521705\pi\)
−0.0681344 + 0.997676i \(0.521705\pi\)
\(312\) 0 0
\(313\) −5.95978e10 −0.350978 −0.175489 0.984481i \(-0.556151\pi\)
−0.175489 + 0.984481i \(0.556151\pi\)
\(314\) 1.27761e11 0.741680
\(315\) 0 0
\(316\) 9.22860e8 0.00520647
\(317\) 3.02644e11 1.68332 0.841658 0.540012i \(-0.181580\pi\)
0.841658 + 0.540012i \(0.181580\pi\)
\(318\) 0 0
\(319\) 2.64745e11 1.43143
\(320\) −6.72931e10 −0.358753
\(321\) 0 0
\(322\) −4.11171e10 −0.213143
\(323\) −1.00629e11 −0.514414
\(324\) 0 0
\(325\) 1.24796e11 0.620479
\(326\) 1.77104e11 0.868456
\(327\) 0 0
\(328\) −3.48168e11 −1.66095
\(329\) −2.36456e10 −0.111268
\(330\) 0 0
\(331\) 4.50482e10 0.206277 0.103139 0.994667i \(-0.467111\pi\)
0.103139 + 0.994667i \(0.467111\pi\)
\(332\) 4.91711e9 0.0222120
\(333\) 0 0
\(334\) 2.07664e11 0.913065
\(335\) 1.58419e11 0.687236
\(336\) 0 0
\(337\) −1.02480e11 −0.432818 −0.216409 0.976303i \(-0.569435\pi\)
−0.216409 + 0.976303i \(0.569435\pi\)
\(338\) −1.18385e11 −0.493371
\(339\) 0 0
\(340\) 6.64748e8 0.00269775
\(341\) 1.03770e11 0.415600
\(342\) 0 0
\(343\) 6.11354e10 0.238490
\(344\) −9.05539e10 −0.348654
\(345\) 0 0
\(346\) −4.16279e11 −1.56150
\(347\) −1.90947e11 −0.707017 −0.353509 0.935431i \(-0.615012\pi\)
−0.353509 + 0.935431i \(0.615012\pi\)
\(348\) 0 0
\(349\) −2.17959e11 −0.786431 −0.393216 0.919446i \(-0.628637\pi\)
−0.393216 + 0.919446i \(0.628637\pi\)
\(350\) 2.92772e10 0.104285
\(351\) 0 0
\(352\) −1.05440e10 −0.0366071
\(353\) −3.13329e11 −1.07403 −0.537013 0.843574i \(-0.680447\pi\)
−0.537013 + 0.843574i \(0.680447\pi\)
\(354\) 0 0
\(355\) −5.39099e10 −0.180153
\(356\) 2.70238e9 0.00891704
\(357\) 0 0
\(358\) −4.12924e11 −1.32861
\(359\) 2.37983e11 0.756173 0.378087 0.925770i \(-0.376582\pi\)
0.378087 + 0.925770i \(0.376582\pi\)
\(360\) 0 0
\(361\) 3.50706e10 0.108683
\(362\) 2.62697e10 0.0804019
\(363\) 0 0
\(364\) −4.45684e8 −0.00133067
\(365\) −3.28265e10 −0.0968069
\(366\) 0 0
\(367\) 3.25345e11 0.936153 0.468077 0.883688i \(-0.344947\pi\)
0.468077 + 0.883688i \(0.344947\pi\)
\(368\) −6.19263e11 −1.76019
\(369\) 0 0
\(370\) −1.58250e11 −0.438971
\(371\) −7.55464e9 −0.0207029
\(372\) 0 0
\(373\) −5.13899e11 −1.37464 −0.687319 0.726356i \(-0.741212\pi\)
−0.687319 + 0.726356i \(0.741212\pi\)
\(374\) −2.14865e11 −0.567863
\(375\) 0 0
\(376\) −3.61780e11 −0.933467
\(377\) 3.39796e11 0.866328
\(378\) 0 0
\(379\) 6.49235e11 1.61631 0.808156 0.588968i \(-0.200466\pi\)
0.808156 + 0.588968i \(0.200466\pi\)
\(380\) −2.36332e9 −0.00581429
\(381\) 0 0
\(382\) −2.32917e11 −0.559649
\(383\) 5.68193e11 1.34928 0.674638 0.738148i \(-0.264300\pi\)
0.674638 + 0.738148i \(0.264300\pi\)
\(384\) 0 0
\(385\) −2.14380e10 −0.0497292
\(386\) −2.63552e11 −0.604260
\(387\) 0 0
\(388\) 1.88245e9 0.00421679
\(389\) −1.85758e11 −0.411315 −0.205657 0.978624i \(-0.565933\pi\)
−0.205657 + 0.978624i \(0.565933\pi\)
\(390\) 0 0
\(391\) 4.03842e11 0.873808
\(392\) 4.64291e11 0.993122
\(393\) 0 0
\(394\) 5.86309e11 1.22573
\(395\) 5.69749e10 0.117760
\(396\) 0 0
\(397\) −3.00165e11 −0.606461 −0.303230 0.952917i \(-0.598065\pi\)
−0.303230 + 0.952917i \(0.598065\pi\)
\(398\) 5.19302e11 1.03740
\(399\) 0 0
\(400\) 4.40943e11 0.861218
\(401\) 6.13010e11 1.18391 0.591954 0.805972i \(-0.298357\pi\)
0.591954 + 0.805972i \(0.298357\pi\)
\(402\) 0 0
\(403\) 1.33187e11 0.251529
\(404\) 7.32936e9 0.0136883
\(405\) 0 0
\(406\) 7.97161e10 0.145606
\(407\) −8.11918e11 −1.46669
\(408\) 0 0
\(409\) 1.78447e11 0.315322 0.157661 0.987493i \(-0.449605\pi\)
0.157661 + 0.987493i \(0.449605\pi\)
\(410\) −3.30692e11 −0.577958
\(411\) 0 0
\(412\) −8.06395e8 −0.00137883
\(413\) 9.55350e10 0.161580
\(414\) 0 0
\(415\) 3.03569e11 0.502391
\(416\) −1.35331e10 −0.0221553
\(417\) 0 0
\(418\) 7.63892e11 1.22388
\(419\) 1.13405e11 0.179751 0.0898753 0.995953i \(-0.471353\pi\)
0.0898753 + 0.995953i \(0.471353\pi\)
\(420\) 0 0
\(421\) 1.70461e11 0.264457 0.132229 0.991219i \(-0.457787\pi\)
0.132229 + 0.991219i \(0.457787\pi\)
\(422\) 5.78999e11 0.888734
\(423\) 0 0
\(424\) −1.15587e11 −0.173685
\(425\) −2.87554e11 −0.427532
\(426\) 0 0
\(427\) 1.16561e11 0.169679
\(428\) −1.20710e10 −0.0173878
\(429\) 0 0
\(430\) −8.60086e10 −0.121320
\(431\) −6.06017e11 −0.845936 −0.422968 0.906145i \(-0.639012\pi\)
−0.422968 + 0.906145i \(0.639012\pi\)
\(432\) 0 0
\(433\) −8.94608e11 −1.22303 −0.611515 0.791233i \(-0.709440\pi\)
−0.611515 + 0.791233i \(0.709440\pi\)
\(434\) 3.12456e10 0.0422751
\(435\) 0 0
\(436\) 1.81387e10 0.0240390
\(437\) −1.43574e12 −1.88326
\(438\) 0 0
\(439\) 8.16164e11 1.04879 0.524393 0.851476i \(-0.324292\pi\)
0.524393 + 0.851476i \(0.324292\pi\)
\(440\) −3.28003e11 −0.417197
\(441\) 0 0
\(442\) −2.75776e11 −0.343681
\(443\) 2.55777e11 0.315533 0.157767 0.987476i \(-0.449571\pi\)
0.157767 + 0.987476i \(0.449571\pi\)
\(444\) 0 0
\(445\) 1.66838e11 0.201685
\(446\) −2.25256e10 −0.0269568
\(447\) 0 0
\(448\) −1.03958e11 −0.121929
\(449\) −2.65080e11 −0.307800 −0.153900 0.988086i \(-0.549183\pi\)
−0.153900 + 0.988086i \(0.549183\pi\)
\(450\) 0 0
\(451\) −1.69665e12 −1.93107
\(452\) −9.27693e9 −0.0104540
\(453\) 0 0
\(454\) 1.18516e12 1.30926
\(455\) −2.75153e10 −0.0300970
\(456\) 0 0
\(457\) −1.25565e12 −1.34662 −0.673312 0.739358i \(-0.735129\pi\)
−0.673312 + 0.739358i \(0.735129\pi\)
\(458\) 6.97989e11 0.741232
\(459\) 0 0
\(460\) 9.48441e9 0.00987643
\(461\) 1.49720e12 1.54393 0.771963 0.635667i \(-0.219275\pi\)
0.771963 + 0.635667i \(0.219275\pi\)
\(462\) 0 0
\(463\) −1.88248e12 −1.90377 −0.951886 0.306453i \(-0.900858\pi\)
−0.951886 + 0.306453i \(0.900858\pi\)
\(464\) 1.20060e12 1.20245
\(465\) 0 0
\(466\) −7.66289e11 −0.752759
\(467\) 1.43018e11 0.139145 0.0695723 0.997577i \(-0.477837\pi\)
0.0695723 + 0.997577i \(0.477837\pi\)
\(468\) 0 0
\(469\) 2.44734e11 0.233570
\(470\) −3.43621e11 −0.324817
\(471\) 0 0
\(472\) 1.46170e12 1.35556
\(473\) −4.41276e11 −0.405355
\(474\) 0 0
\(475\) 1.02232e12 0.921433
\(476\) 1.02694e9 0.000916879 0
\(477\) 0 0
\(478\) −2.02649e12 −1.77549
\(479\) 3.78756e11 0.328738 0.164369 0.986399i \(-0.447441\pi\)
0.164369 + 0.986399i \(0.447441\pi\)
\(480\) 0 0
\(481\) −1.04208e12 −0.887666
\(482\) −1.73626e12 −1.46522
\(483\) 0 0
\(484\) −7.02652e9 −0.00582018
\(485\) 1.16218e11 0.0953750
\(486\) 0 0
\(487\) 9.66599e9 0.00778692 0.00389346 0.999992i \(-0.498761\pi\)
0.00389346 + 0.999992i \(0.498761\pi\)
\(488\) 1.78339e12 1.42350
\(489\) 0 0
\(490\) 4.40986e11 0.345575
\(491\) −6.90819e11 −0.536411 −0.268205 0.963362i \(-0.586431\pi\)
−0.268205 + 0.963362i \(0.586431\pi\)
\(492\) 0 0
\(493\) −7.82952e11 −0.596931
\(494\) 9.80443e11 0.740715
\(495\) 0 0
\(496\) 4.70589e11 0.349119
\(497\) −8.32828e10 −0.0612282
\(498\) 0 0
\(499\) 7.97646e11 0.575914 0.287957 0.957643i \(-0.407024\pi\)
0.287957 + 0.957643i \(0.407024\pi\)
\(500\) −1.44705e10 −0.0103542
\(501\) 0 0
\(502\) 1.01724e12 0.714916
\(503\) 4.71233e11 0.328231 0.164116 0.986441i \(-0.447523\pi\)
0.164116 + 0.986441i \(0.447523\pi\)
\(504\) 0 0
\(505\) 4.52495e11 0.309601
\(506\) −3.06563e12 −2.07894
\(507\) 0 0
\(508\) 2.08956e10 0.0139209
\(509\) 2.09988e12 1.38664 0.693320 0.720630i \(-0.256147\pi\)
0.693320 + 0.720630i \(0.256147\pi\)
\(510\) 0 0
\(511\) −5.07120e10 −0.0329016
\(512\) −1.58981e12 −1.02242
\(513\) 0 0
\(514\) 3.05066e12 1.92779
\(515\) −4.97847e10 −0.0311863
\(516\) 0 0
\(517\) −1.76298e12 −1.08528
\(518\) −2.44472e11 −0.149192
\(519\) 0 0
\(520\) −4.20987e11 −0.252495
\(521\) −2.49095e12 −1.48114 −0.740569 0.671980i \(-0.765444\pi\)
−0.740569 + 0.671980i \(0.765444\pi\)
\(522\) 0 0
\(523\) 9.27462e11 0.542049 0.271024 0.962572i \(-0.412638\pi\)
0.271024 + 0.962572i \(0.412638\pi\)
\(524\) 2.61509e10 0.0151529
\(525\) 0 0
\(526\) 1.08148e12 0.616005
\(527\) −3.06886e11 −0.173312
\(528\) 0 0
\(529\) 3.96073e12 2.19900
\(530\) −1.09785e11 −0.0604368
\(531\) 0 0
\(532\) −3.65098e9 −0.00197609
\(533\) −2.17762e12 −1.16872
\(534\) 0 0
\(535\) −7.45230e11 −0.393277
\(536\) 3.74445e12 1.95951
\(537\) 0 0
\(538\) −3.00255e12 −1.54515
\(539\) 2.26252e12 1.15463
\(540\) 0 0
\(541\) 1.99506e12 1.00131 0.500655 0.865647i \(-0.333093\pi\)
0.500655 + 0.865647i \(0.333093\pi\)
\(542\) −1.57910e12 −0.785984
\(543\) 0 0
\(544\) 3.11827e10 0.0152658
\(545\) 1.11983e12 0.543712
\(546\) 0 0
\(547\) −1.49828e11 −0.0715567 −0.0357784 0.999360i \(-0.511391\pi\)
−0.0357784 + 0.999360i \(0.511391\pi\)
\(548\) 4.53507e8 0.000214818 0
\(549\) 0 0
\(550\) 2.18287e12 1.01717
\(551\) 2.78356e12 1.28653
\(552\) 0 0
\(553\) 8.80177e10 0.0400228
\(554\) −3.01564e12 −1.36015
\(555\) 0 0
\(556\) −5.43918e10 −0.0241378
\(557\) 6.75741e11 0.297462 0.148731 0.988878i \(-0.452481\pi\)
0.148731 + 0.988878i \(0.452481\pi\)
\(558\) 0 0
\(559\) −5.66370e11 −0.245328
\(560\) −9.72199e10 −0.0417743
\(561\) 0 0
\(562\) −2.60086e12 −1.09977
\(563\) 1.25366e12 0.525885 0.262943 0.964811i \(-0.415307\pi\)
0.262943 + 0.964811i \(0.415307\pi\)
\(564\) 0 0
\(565\) −5.72733e11 −0.236447
\(566\) 3.14945e12 1.28991
\(567\) 0 0
\(568\) −1.27423e12 −0.513667
\(569\) 3.15164e12 1.26047 0.630233 0.776406i \(-0.282959\pi\)
0.630233 + 0.776406i \(0.282959\pi\)
\(570\) 0 0
\(571\) −4.23310e12 −1.66647 −0.833233 0.552923i \(-0.813513\pi\)
−0.833233 + 0.552923i \(0.813513\pi\)
\(572\) −3.32295e10 −0.0129790
\(573\) 0 0
\(574\) −5.10870e11 −0.196429
\(575\) −4.10273e12 −1.56519
\(576\) 0 0
\(577\) 4.23367e12 1.59011 0.795053 0.606540i \(-0.207443\pi\)
0.795053 + 0.606540i \(0.207443\pi\)
\(578\) −2.02685e12 −0.755348
\(579\) 0 0
\(580\) −1.83880e10 −0.00674696
\(581\) 4.68969e11 0.170747
\(582\) 0 0
\(583\) −5.63263e11 −0.201931
\(584\) −7.75899e11 −0.276024
\(585\) 0 0
\(586\) −3.59068e12 −1.25788
\(587\) −1.39753e12 −0.485837 −0.242919 0.970047i \(-0.578105\pi\)
−0.242919 + 0.970047i \(0.578105\pi\)
\(588\) 0 0
\(589\) 1.09105e12 0.373529
\(590\) 1.38833e12 0.471691
\(591\) 0 0
\(592\) −3.68199e12 −1.23207
\(593\) 6.32131e11 0.209923 0.104962 0.994476i \(-0.466528\pi\)
0.104962 + 0.994476i \(0.466528\pi\)
\(594\) 0 0
\(595\) 6.34003e10 0.0207379
\(596\) −1.16759e10 −0.00379039
\(597\) 0 0
\(598\) −3.93468e12 −1.25821
\(599\) −5.15452e12 −1.63594 −0.817970 0.575261i \(-0.804900\pi\)
−0.817970 + 0.575261i \(0.804900\pi\)
\(600\) 0 0
\(601\) −7.37044e11 −0.230440 −0.115220 0.993340i \(-0.536757\pi\)
−0.115220 + 0.993340i \(0.536757\pi\)
\(602\) −1.32870e11 −0.0412329
\(603\) 0 0
\(604\) −3.96737e10 −0.0121293
\(605\) −4.33799e11 −0.131640
\(606\) 0 0
\(607\) 1.87389e12 0.560267 0.280134 0.959961i \(-0.409621\pi\)
0.280134 + 0.959961i \(0.409621\pi\)
\(608\) −1.10861e11 −0.0329013
\(609\) 0 0
\(610\) 1.69387e12 0.495333
\(611\) −2.26276e12 −0.656829
\(612\) 0 0
\(613\) −1.62540e12 −0.464931 −0.232465 0.972605i \(-0.574679\pi\)
−0.232465 + 0.972605i \(0.574679\pi\)
\(614\) 4.45703e12 1.26558
\(615\) 0 0
\(616\) −5.06716e11 −0.141792
\(617\) −4.48555e12 −1.24604 −0.623021 0.782205i \(-0.714095\pi\)
−0.623021 + 0.782205i \(0.714095\pi\)
\(618\) 0 0
\(619\) 2.64489e12 0.724102 0.362051 0.932158i \(-0.382077\pi\)
0.362051 + 0.932158i \(0.382077\pi\)
\(620\) −7.20737e9 −0.00195891
\(621\) 0 0
\(622\) −5.04700e11 −0.135200
\(623\) 2.57739e11 0.0685463
\(624\) 0 0
\(625\) 2.44489e12 0.640913
\(626\) −1.33797e12 −0.348226
\(627\) 0 0
\(628\) −4.55276e10 −0.0116804
\(629\) 2.40115e12 0.611633
\(630\) 0 0
\(631\) −2.53561e12 −0.636723 −0.318361 0.947969i \(-0.603133\pi\)
−0.318361 + 0.947969i \(0.603133\pi\)
\(632\) 1.34668e12 0.335767
\(633\) 0 0
\(634\) 6.79434e12 1.67011
\(635\) 1.29004e12 0.314862
\(636\) 0 0
\(637\) 2.90391e12 0.698805
\(638\) 5.94351e12 1.42020
\(639\) 0 0
\(640\) −1.46386e12 −0.344897
\(641\) −7.11969e11 −0.166571 −0.0832856 0.996526i \(-0.526541\pi\)
−0.0832856 + 0.996526i \(0.526541\pi\)
\(642\) 0 0
\(643\) −5.97690e11 −0.137888 −0.0689441 0.997621i \(-0.521963\pi\)
−0.0689441 + 0.997621i \(0.521963\pi\)
\(644\) 1.46520e10 0.00335668
\(645\) 0 0
\(646\) −2.25912e12 −0.510379
\(647\) 3.90064e12 0.875119 0.437559 0.899189i \(-0.355843\pi\)
0.437559 + 0.899189i \(0.355843\pi\)
\(648\) 0 0
\(649\) 7.12295e12 1.57601
\(650\) 2.80167e12 0.615612
\(651\) 0 0
\(652\) −6.31106e10 −0.0136769
\(653\) 2.73627e11 0.0588912 0.0294456 0.999566i \(-0.490626\pi\)
0.0294456 + 0.999566i \(0.490626\pi\)
\(654\) 0 0
\(655\) 1.61449e12 0.342727
\(656\) −7.69419e12 −1.62217
\(657\) 0 0
\(658\) −5.30843e11 −0.110395
\(659\) −1.93239e12 −0.399126 −0.199563 0.979885i \(-0.563952\pi\)
−0.199563 + 0.979885i \(0.563952\pi\)
\(660\) 0 0
\(661\) 8.99819e12 1.83336 0.916682 0.399617i \(-0.130857\pi\)
0.916682 + 0.399617i \(0.130857\pi\)
\(662\) 1.01133e12 0.204660
\(663\) 0 0
\(664\) 7.17528e12 1.43246
\(665\) −2.25402e11 −0.0446951
\(666\) 0 0
\(667\) −1.11709e13 −2.18536
\(668\) −7.40007e10 −0.0143794
\(669\) 0 0
\(670\) 3.55650e12 0.681846
\(671\) 8.69060e12 1.65500
\(672\) 0 0
\(673\) −3.80149e12 −0.714309 −0.357155 0.934045i \(-0.616253\pi\)
−0.357155 + 0.934045i \(0.616253\pi\)
\(674\) −2.30068e12 −0.429424
\(675\) 0 0
\(676\) 4.21865e10 0.00776986
\(677\) −2.51015e12 −0.459252 −0.229626 0.973279i \(-0.573750\pi\)
−0.229626 + 0.973279i \(0.573750\pi\)
\(678\) 0 0
\(679\) 1.79539e11 0.0324149
\(680\) 9.70031e11 0.173978
\(681\) 0 0
\(682\) 2.32962e12 0.412341
\(683\) −1.00278e12 −0.176325 −0.0881623 0.996106i \(-0.528099\pi\)
−0.0881623 + 0.996106i \(0.528099\pi\)
\(684\) 0 0
\(685\) 2.79983e10 0.00485875
\(686\) 1.37249e12 0.236619
\(687\) 0 0
\(688\) −2.00116e12 −0.340513
\(689\) −7.22939e11 −0.122212
\(690\) 0 0
\(691\) 1.14248e12 0.190632 0.0953161 0.995447i \(-0.469614\pi\)
0.0953161 + 0.995447i \(0.469614\pi\)
\(692\) 1.48340e11 0.0245913
\(693\) 0 0
\(694\) −4.28675e12 −0.701472
\(695\) −3.35801e12 −0.545946
\(696\) 0 0
\(697\) 5.01764e12 0.805289
\(698\) −4.89317e12 −0.780263
\(699\) 0 0
\(700\) −1.04329e10 −0.00164234
\(701\) −5.29555e12 −0.828285 −0.414143 0.910212i \(-0.635919\pi\)
−0.414143 + 0.910212i \(0.635919\pi\)
\(702\) 0 0
\(703\) −8.53660e12 −1.31821
\(704\) −7.75095e12 −1.18926
\(705\) 0 0
\(706\) −7.03422e12 −1.06560
\(707\) 6.99037e11 0.105224
\(708\) 0 0
\(709\) −1.01497e13 −1.50849 −0.754246 0.656592i \(-0.771998\pi\)
−0.754246 + 0.656592i \(0.771998\pi\)
\(710\) −1.21027e12 −0.178740
\(711\) 0 0
\(712\) 3.94343e12 0.575062
\(713\) −4.37856e12 −0.634495
\(714\) 0 0
\(715\) −2.05150e12 −0.293559
\(716\) 1.47145e11 0.0209236
\(717\) 0 0
\(718\) 5.34271e12 0.750242
\(719\) 5.75277e12 0.802780 0.401390 0.915907i \(-0.368527\pi\)
0.401390 + 0.915907i \(0.368527\pi\)
\(720\) 0 0
\(721\) −7.69100e10 −0.0105992
\(722\) 7.87333e11 0.107830
\(723\) 0 0
\(724\) −9.36116e9 −0.00126621
\(725\) 7.95420e12 1.06924
\(726\) 0 0
\(727\) −1.10169e13 −1.46270 −0.731351 0.682001i \(-0.761110\pi\)
−0.731351 + 0.682001i \(0.761110\pi\)
\(728\) −6.50362e11 −0.0858152
\(729\) 0 0
\(730\) −7.36953e11 −0.0960476
\(731\) 1.30502e12 0.169040
\(732\) 0 0
\(733\) −1.75672e12 −0.224768 −0.112384 0.993665i \(-0.535849\pi\)
−0.112384 + 0.993665i \(0.535849\pi\)
\(734\) 7.30398e12 0.928811
\(735\) 0 0
\(736\) 4.44905e11 0.0558878
\(737\) 1.82470e13 2.27818
\(738\) 0 0
\(739\) −4.52901e12 −0.558603 −0.279301 0.960204i \(-0.590103\pi\)
−0.279301 + 0.960204i \(0.590103\pi\)
\(740\) 5.63921e10 0.00691314
\(741\) 0 0
\(742\) −1.69601e11 −0.0205405
\(743\) −2.25595e12 −0.271569 −0.135784 0.990738i \(-0.543355\pi\)
−0.135784 + 0.990738i \(0.543355\pi\)
\(744\) 0 0
\(745\) −7.20841e11 −0.0857307
\(746\) −1.15370e13 −1.36386
\(747\) 0 0
\(748\) 7.65669e10 0.00894301
\(749\) −1.15127e12 −0.133662
\(750\) 0 0
\(751\) −9.29242e12 −1.06598 −0.532990 0.846122i \(-0.678932\pi\)
−0.532990 + 0.846122i \(0.678932\pi\)
\(752\) −7.99500e12 −0.911671
\(753\) 0 0
\(754\) 7.62840e12 0.859533
\(755\) −2.44935e12 −0.274340
\(756\) 0 0
\(757\) 7.28505e12 0.806308 0.403154 0.915132i \(-0.367914\pi\)
0.403154 + 0.915132i \(0.367914\pi\)
\(758\) 1.45753e13 1.60364
\(759\) 0 0
\(760\) −3.44867e12 −0.374965
\(761\) 2.78674e12 0.301207 0.150604 0.988594i \(-0.451878\pi\)
0.150604 + 0.988594i \(0.451878\pi\)
\(762\) 0 0
\(763\) 1.72998e12 0.184790
\(764\) 8.29995e10 0.00881364
\(765\) 0 0
\(766\) 1.27559e13 1.33869
\(767\) 9.14219e12 0.953830
\(768\) 0 0
\(769\) 4.63000e12 0.477433 0.238716 0.971089i \(-0.423273\pi\)
0.238716 + 0.971089i \(0.423273\pi\)
\(770\) −4.81282e11 −0.0493391
\(771\) 0 0
\(772\) 9.39164e10 0.00951620
\(773\) −8.49811e12 −0.856080 −0.428040 0.903760i \(-0.640796\pi\)
−0.428040 + 0.903760i \(0.640796\pi\)
\(774\) 0 0
\(775\) 3.11773e12 0.310442
\(776\) 2.74697e12 0.271941
\(777\) 0 0
\(778\) −4.17026e12 −0.408089
\(779\) −1.78388e13 −1.73559
\(780\) 0 0
\(781\) −6.20944e12 −0.597204
\(782\) 9.06623e12 0.866954
\(783\) 0 0
\(784\) 1.02604e13 0.969933
\(785\) −2.81075e12 −0.264186
\(786\) 0 0
\(787\) 3.57079e12 0.331801 0.165900 0.986143i \(-0.446947\pi\)
0.165900 + 0.986143i \(0.446947\pi\)
\(788\) −2.08930e11 −0.0193034
\(789\) 0 0
\(790\) 1.27908e12 0.116836
\(791\) −8.84787e11 −0.0803608
\(792\) 0 0
\(793\) 1.11542e13 1.00164
\(794\) −6.73868e12 −0.601704
\(795\) 0 0
\(796\) −1.85052e11 −0.0163375
\(797\) −9.06883e12 −0.796139 −0.398070 0.917355i \(-0.630320\pi\)
−0.398070 + 0.917355i \(0.630320\pi\)
\(798\) 0 0
\(799\) 5.21381e12 0.452579
\(800\) −3.16793e11 −0.0273445
\(801\) 0 0
\(802\) 1.37620e13 1.17462
\(803\) −3.78101e12 −0.320914
\(804\) 0 0
\(805\) 9.04576e11 0.0759213
\(806\) 2.99003e12 0.249556
\(807\) 0 0
\(808\) 1.06953e13 0.882762
\(809\) −1.10403e13 −0.906173 −0.453087 0.891467i \(-0.649677\pi\)
−0.453087 + 0.891467i \(0.649677\pi\)
\(810\) 0 0
\(811\) −1.04507e12 −0.0848302 −0.0424151 0.999100i \(-0.513505\pi\)
−0.0424151 + 0.999100i \(0.513505\pi\)
\(812\) −2.84067e10 −0.00229308
\(813\) 0 0
\(814\) −1.82275e13 −1.45518
\(815\) −3.89628e12 −0.309343
\(816\) 0 0
\(817\) −4.63963e12 −0.364321
\(818\) 4.00612e12 0.312849
\(819\) 0 0
\(820\) 1.17841e11 0.00910198
\(821\) 1.04318e13 0.801340 0.400670 0.916223i \(-0.368777\pi\)
0.400670 + 0.916223i \(0.368777\pi\)
\(822\) 0 0
\(823\) 7.03298e12 0.534367 0.267184 0.963646i \(-0.413907\pi\)
0.267184 + 0.963646i \(0.413907\pi\)
\(824\) −1.17673e12 −0.0889210
\(825\) 0 0
\(826\) 2.14476e12 0.160313
\(827\) 1.69223e13 1.25801 0.629006 0.777400i \(-0.283462\pi\)
0.629006 + 0.777400i \(0.283462\pi\)
\(828\) 0 0
\(829\) 2.27155e12 0.167042 0.0835212 0.996506i \(-0.473383\pi\)
0.0835212 + 0.996506i \(0.473383\pi\)
\(830\) 6.81511e12 0.498450
\(831\) 0 0
\(832\) −9.94821e12 −0.719764
\(833\) −6.69114e12 −0.481502
\(834\) 0 0
\(835\) −4.56860e12 −0.325233
\(836\) −2.72212e11 −0.0192743
\(837\) 0 0
\(838\) 2.54594e12 0.178341
\(839\) −1.79457e13 −1.25035 −0.625175 0.780484i \(-0.714973\pi\)
−0.625175 + 0.780484i \(0.714973\pi\)
\(840\) 0 0
\(841\) 7.15056e12 0.492899
\(842\) 3.82684e12 0.262383
\(843\) 0 0
\(844\) −2.06325e11 −0.0139962
\(845\) 2.60448e12 0.175738
\(846\) 0 0
\(847\) −6.70154e11 −0.0447404
\(848\) −2.55436e12 −0.169629
\(849\) 0 0
\(850\) −6.45557e12 −0.424179
\(851\) 3.42588e13 2.23918
\(852\) 0 0
\(853\) −2.20102e13 −1.42349 −0.711744 0.702439i \(-0.752095\pi\)
−0.711744 + 0.702439i \(0.752095\pi\)
\(854\) 2.61678e12 0.168348
\(855\) 0 0
\(856\) −1.76145e13 −1.12135
\(857\) 2.34048e13 1.48215 0.741074 0.671424i \(-0.234317\pi\)
0.741074 + 0.671424i \(0.234317\pi\)
\(858\) 0 0
\(859\) 4.26808e10 0.00267463 0.00133732 0.999999i \(-0.499574\pi\)
0.00133732 + 0.999999i \(0.499574\pi\)
\(860\) 3.06490e10 0.00191062
\(861\) 0 0
\(862\) −1.36051e13 −0.839301
\(863\) −3.10119e13 −1.90318 −0.951590 0.307370i \(-0.900551\pi\)
−0.951590 + 0.307370i \(0.900551\pi\)
\(864\) 0 0
\(865\) 9.15814e12 0.556205
\(866\) −2.00839e13 −1.21344
\(867\) 0 0
\(868\) −1.11343e10 −0.000665770 0
\(869\) 6.56247e12 0.390372
\(870\) 0 0
\(871\) 2.34197e13 1.37880
\(872\) 2.64688e13 1.55028
\(873\) 0 0
\(874\) −3.22324e13 −1.86849
\(875\) −1.38012e12 −0.0795943
\(876\) 0 0
\(877\) −1.64046e13 −0.936416 −0.468208 0.883618i \(-0.655100\pi\)
−0.468208 + 0.883618i \(0.655100\pi\)
\(878\) 1.83228e13 1.04056
\(879\) 0 0
\(880\) −7.24857e12 −0.407456
\(881\) −2.25741e13 −1.26246 −0.631231 0.775595i \(-0.717450\pi\)
−0.631231 + 0.775595i \(0.717450\pi\)
\(882\) 0 0
\(883\) −7.75871e12 −0.429503 −0.214752 0.976669i \(-0.568894\pi\)
−0.214752 + 0.976669i \(0.568894\pi\)
\(884\) 9.82723e10 0.00541247
\(885\) 0 0
\(886\) 5.74219e12 0.313058
\(887\) −4.44036e12 −0.240858 −0.120429 0.992722i \(-0.538427\pi\)
−0.120429 + 0.992722i \(0.538427\pi\)
\(888\) 0 0
\(889\) 1.99291e12 0.107012
\(890\) 3.74549e12 0.200103
\(891\) 0 0
\(892\) 8.02694e9 0.000424530 0
\(893\) −1.85362e13 −0.975414
\(894\) 0 0
\(895\) 9.08433e12 0.473249
\(896\) −2.26144e12 −0.117219
\(897\) 0 0
\(898\) −5.95104e12 −0.305386
\(899\) 8.48897e12 0.433447
\(900\) 0 0
\(901\) 1.66578e12 0.0842087
\(902\) −3.80897e13 −1.91592
\(903\) 0 0
\(904\) −1.35373e13 −0.674178
\(905\) −5.77933e11 −0.0286391
\(906\) 0 0
\(907\) 1.71069e13 0.839342 0.419671 0.907676i \(-0.362145\pi\)
0.419671 + 0.907676i \(0.362145\pi\)
\(908\) −4.22329e11 −0.0206189
\(909\) 0 0
\(910\) −6.17717e11 −0.0298609
\(911\) 1.88236e13 0.905460 0.452730 0.891648i \(-0.350450\pi\)
0.452730 + 0.891648i \(0.350450\pi\)
\(912\) 0 0
\(913\) 3.49657e13 1.66542
\(914\) −2.81893e13 −1.33606
\(915\) 0 0
\(916\) −2.48727e11 −0.0116733
\(917\) 2.49414e12 0.116482
\(918\) 0 0
\(919\) −4.20992e13 −1.94695 −0.973473 0.228800i \(-0.926520\pi\)
−0.973473 + 0.228800i \(0.926520\pi\)
\(920\) 1.38401e13 0.636933
\(921\) 0 0
\(922\) 3.36121e13 1.53182
\(923\) −7.96971e12 −0.361439
\(924\) 0 0
\(925\) −2.43938e13 −1.09558
\(926\) −4.22615e13 −1.88884
\(927\) 0 0
\(928\) −8.62564e11 −0.0381791
\(929\) 1.23689e13 0.544831 0.272415 0.962180i \(-0.412177\pi\)
0.272415 + 0.962180i \(0.412177\pi\)
\(930\) 0 0
\(931\) 2.37885e13 1.03775
\(932\) 2.73066e11 0.0118548
\(933\) 0 0
\(934\) 3.21076e12 0.138053
\(935\) 4.72703e12 0.202272
\(936\) 0 0
\(937\) 2.10110e13 0.890467 0.445234 0.895414i \(-0.353121\pi\)
0.445234 + 0.895414i \(0.353121\pi\)
\(938\) 5.49426e12 0.231738
\(939\) 0 0
\(940\) 1.22449e11 0.00511539
\(941\) 1.49975e13 0.623543 0.311771 0.950157i \(-0.399078\pi\)
0.311771 + 0.950157i \(0.399078\pi\)
\(942\) 0 0
\(943\) 7.15900e13 2.94815
\(944\) 3.23021e13 1.32391
\(945\) 0 0
\(946\) −9.90662e12 −0.402175
\(947\) 1.30832e12 0.0528615 0.0264307 0.999651i \(-0.491586\pi\)
0.0264307 + 0.999651i \(0.491586\pi\)
\(948\) 0 0
\(949\) −4.85287e12 −0.194223
\(950\) 2.29509e13 0.914206
\(951\) 0 0
\(952\) 1.49855e12 0.0591297
\(953\) 4.18691e13 1.64428 0.822140 0.569285i \(-0.192780\pi\)
0.822140 + 0.569285i \(0.192780\pi\)
\(954\) 0 0
\(955\) 5.12417e12 0.199346
\(956\) 7.22136e11 0.0279614
\(957\) 0 0
\(958\) 8.50306e12 0.326160
\(959\) 4.32532e10 0.00165133
\(960\) 0 0
\(961\) −2.31123e13 −0.874153
\(962\) −2.33947e13 −0.880703
\(963\) 0 0
\(964\) 6.18714e11 0.0230751
\(965\) 5.79815e12 0.215237
\(966\) 0 0
\(967\) −1.10706e12 −0.0407148 −0.0203574 0.999793i \(-0.506480\pi\)
−0.0203574 + 0.999793i \(0.506480\pi\)
\(968\) −1.02534e13 −0.375344
\(969\) 0 0
\(970\) 2.60908e12 0.0946270
\(971\) −2.21624e13 −0.800075 −0.400038 0.916499i \(-0.631003\pi\)
−0.400038 + 0.916499i \(0.631003\pi\)
\(972\) 0 0
\(973\) −5.18762e12 −0.185550
\(974\) 2.17001e11 0.00772585
\(975\) 0 0
\(976\) 3.94113e13 1.39026
\(977\) −2.19683e13 −0.771384 −0.385692 0.922628i \(-0.626037\pi\)
−0.385692 + 0.922628i \(0.626037\pi\)
\(978\) 0 0
\(979\) 1.92166e13 0.668583
\(980\) −1.57145e11 −0.00544229
\(981\) 0 0
\(982\) −1.55088e13 −0.532204
\(983\) 4.07229e13 1.39107 0.695533 0.718494i \(-0.255168\pi\)
0.695533 + 0.718494i \(0.255168\pi\)
\(984\) 0 0
\(985\) −1.28988e13 −0.436602
\(986\) −1.75772e13 −0.592249
\(987\) 0 0
\(988\) −3.49379e11 −0.0116652
\(989\) 1.86196e13 0.618853
\(990\) 0 0
\(991\) 1.69138e13 0.557070 0.278535 0.960426i \(-0.410151\pi\)
0.278535 + 0.960426i \(0.410151\pi\)
\(992\) −3.38091e11 −0.0110849
\(993\) 0 0
\(994\) −1.86969e12 −0.0607480
\(995\) −1.14246e13 −0.369521
\(996\) 0 0
\(997\) −4.00661e13 −1.28425 −0.642125 0.766600i \(-0.721947\pi\)
−0.642125 + 0.766600i \(0.721947\pi\)
\(998\) 1.79071e13 0.571397
\(999\) 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 27.10.a.a.1.2 yes 2
3.2 odd 2 inner 27.10.a.a.1.1 2
9.2 odd 6 81.10.c.f.28.2 4
9.4 even 3 81.10.c.f.55.1 4
9.5 odd 6 81.10.c.f.55.2 4
9.7 even 3 81.10.c.f.28.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.10.a.a.1.1 2 3.2 odd 2 inner
27.10.a.a.1.2 yes 2 1.1 even 1 trivial
81.10.c.f.28.1 4 9.7 even 3
81.10.c.f.28.2 4 9.2 odd 6
81.10.c.f.55.1 4 9.4 even 3
81.10.c.f.55.2 4 9.5 odd 6