Properties

Label 27.9.b.b.26.2
Level $27$
Weight $9$
Character 27.26
Analytic conductor $10.999$
Analytic rank $0$
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] = [27,9,Mod(26,27)]
 
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([1]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("27.26");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 27.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{-6}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2^{2}\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 26.2
Root \(2.44949i\) of defining polynomial
Character \(\chi\) \(=\) 27.26
Dual form 27.9.b.b.26.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+29.3939i q^{2} -608.000 q^{4} -823.029i q^{5} +1967.00 q^{7} -10346.6i q^{8} +O(q^{10})\) \(q+29.3939i q^{2} -608.000 q^{4} -823.029i q^{5} +1967.00 q^{7} -10346.6i q^{8} +24192.0 q^{10} -12580.6i q^{11} -45505.0 q^{13} +57817.8i q^{14} +148480. q^{16} -59610.8i q^{17} +152399. q^{19} +500401. i q^{20} +369792. q^{22} -131332. i q^{23} -286751. q^{25} -1.33757e6i q^{26} -1.19594e6 q^{28} -588583. i q^{29} -164350. q^{31} +1.71566e6i q^{32} +1.75219e6 q^{34} -1.61890e6i q^{35} -663937. q^{37} +4.47960e6i q^{38} -8.51558e6 q^{40} -938017. i q^{41} +575330. q^{43} +7.64899e6i q^{44} +3.86035e6 q^{46} -9.23426e6i q^{47} -1.89571e6 q^{49} -8.42872e6i q^{50} +2.76670e7 q^{52} +1.03765e7i q^{53} -1.03542e7 q^{55} -2.03519e7i q^{56} +1.73007e7 q^{58} -5.03987e6i q^{59} -1.92130e7 q^{61} -4.83088e6i q^{62} -1.24191e7 q^{64} +3.74519e7i q^{65} -598033. q^{67} +3.62434e7i q^{68} +4.75857e7 q^{70} +2.92721e7i q^{71} +1.28502e7 q^{73} -1.95157e7i q^{74} -9.26586e7 q^{76} -2.47460e7i q^{77} -2.35847e7 q^{79} -1.22203e8i q^{80} +2.75720e7 q^{82} +3.34509e7i q^{83} -4.90614e7 q^{85} +1.69112e7i q^{86} -1.30167e8 q^{88} -2.82848e7i q^{89} -8.95083e7 q^{91} +7.98498e7i q^{92} +2.71431e8 q^{94} -1.25429e8i q^{95} +1.36490e8 q^{97} -5.57223e7i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 1216 q^{4} + 3934 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 1216 q^{4} + 3934 q^{7} + 48384 q^{10} - 91010 q^{13} + 296960 q^{16} + 304798 q^{19} + 739584 q^{22} - 573502 q^{25} - 2391872 q^{28} - 328700 q^{31} + 3504384 q^{34} - 1327874 q^{37} - 17031168 q^{40} + 1150660 q^{43} + 7720704 q^{46} - 3791424 q^{49} + 55334080 q^{52} - 20708352 q^{55} + 34601472 q^{58} - 38425922 q^{61} - 24838144 q^{64} - 1196066 q^{67} + 95171328 q^{70} + 25700350 q^{73} - 185317184 q^{76} - 47169314 q^{79} + 55143936 q^{82} - 98122752 q^{85} - 260333568 q^{88} - 179016670 q^{91} + 542861568 q^{94} + 272979262 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 29.3939i 1.83712i 0.395285 + 0.918559i \(0.370646\pi\)
−0.395285 + 0.918559i \(0.629354\pi\)
\(3\) 0 0
\(4\) −608.000 −2.37500
\(5\) − 823.029i − 1.31685i −0.752648 0.658423i \(-0.771224\pi\)
0.752648 0.658423i \(-0.228776\pi\)
\(6\) 0 0
\(7\) 1967.00 0.819242 0.409621 0.912256i \(-0.365661\pi\)
0.409621 + 0.912256i \(0.365661\pi\)
\(8\) − 10346.6i − 2.52604i
\(9\) 0 0
\(10\) 24192.0 2.41920
\(11\) − 12580.6i − 0.859270i −0.903003 0.429635i \(-0.858642\pi\)
0.903003 0.429635i \(-0.141358\pi\)
\(12\) 0 0
\(13\) −45505.0 −1.59326 −0.796628 0.604470i \(-0.793385\pi\)
−0.796628 + 0.604470i \(0.793385\pi\)
\(14\) 57817.8i 1.50504i
\(15\) 0 0
\(16\) 148480. 2.26562
\(17\) − 59610.8i − 0.713722i −0.934157 0.356861i \(-0.883847\pi\)
0.934157 0.356861i \(-0.116153\pi\)
\(18\) 0 0
\(19\) 152399. 1.16941 0.584706 0.811245i \(-0.301210\pi\)
0.584706 + 0.811245i \(0.301210\pi\)
\(20\) 500401.i 3.12751i
\(21\) 0 0
\(22\) 369792. 1.57858
\(23\) − 131332.i − 0.469309i −0.972079 0.234654i \(-0.924604\pi\)
0.972079 0.234654i \(-0.0753958\pi\)
\(24\) 0 0
\(25\) −286751. −0.734083
\(26\) − 1.33757e6i − 2.92700i
\(27\) 0 0
\(28\) −1.19594e6 −1.94570
\(29\) − 588583.i − 0.832177i −0.909324 0.416089i \(-0.863401\pi\)
0.909324 0.416089i \(-0.136599\pi\)
\(30\) 0 0
\(31\) −164350. −0.177960 −0.0889801 0.996033i \(-0.528361\pi\)
−0.0889801 + 0.996033i \(0.528361\pi\)
\(32\) 1.71566e6i 1.63618i
\(33\) 0 0
\(34\) 1.75219e6 1.31119
\(35\) − 1.61890e6i − 1.07882i
\(36\) 0 0
\(37\) −663937. −0.354258 −0.177129 0.984188i \(-0.556681\pi\)
−0.177129 + 0.984188i \(0.556681\pi\)
\(38\) 4.47960e6i 2.14835i
\(39\) 0 0
\(40\) −8.51558e6 −3.32640
\(41\) − 938017.i − 0.331952i −0.986130 0.165976i \(-0.946923\pi\)
0.986130 0.165976i \(-0.0530774\pi\)
\(42\) 0 0
\(43\) 575330. 0.168284 0.0841421 0.996454i \(-0.473185\pi\)
0.0841421 + 0.996454i \(0.473185\pi\)
\(44\) 7.64899e6i 2.04077i
\(45\) 0 0
\(46\) 3.86035e6 0.862175
\(47\) − 9.23426e6i − 1.89239i −0.323596 0.946195i \(-0.604892\pi\)
0.323596 0.946195i \(-0.395108\pi\)
\(48\) 0 0
\(49\) −1.89571e6 −0.328843
\(50\) − 8.42872e6i − 1.34860i
\(51\) 0 0
\(52\) 2.76670e7 3.78398
\(53\) 1.03765e7i 1.31507i 0.753425 + 0.657533i \(0.228400\pi\)
−0.753425 + 0.657533i \(0.771600\pi\)
\(54\) 0 0
\(55\) −1.03542e7 −1.13153
\(56\) − 2.03519e7i − 2.06943i
\(57\) 0 0
\(58\) 1.73007e7 1.52881
\(59\) − 5.03987e6i − 0.415922i −0.978137 0.207961i \(-0.933317\pi\)
0.978137 0.207961i \(-0.0666827\pi\)
\(60\) 0 0
\(61\) −1.92130e7 −1.38763 −0.693817 0.720151i \(-0.744072\pi\)
−0.693817 + 0.720151i \(0.744072\pi\)
\(62\) − 4.83088e6i − 0.326934i
\(63\) 0 0
\(64\) −1.24191e7 −0.740234
\(65\) 3.74519e7i 2.09807i
\(66\) 0 0
\(67\) −598033. −0.0296774 −0.0148387 0.999890i \(-0.504723\pi\)
−0.0148387 + 0.999890i \(0.504723\pi\)
\(68\) 3.62434e7i 1.69509i
\(69\) 0 0
\(70\) 4.75857e7 1.98191
\(71\) 2.92721e7i 1.15191i 0.817480 + 0.575957i \(0.195370\pi\)
−0.817480 + 0.575957i \(0.804630\pi\)
\(72\) 0 0
\(73\) 1.28502e7 0.452499 0.226249 0.974069i \(-0.427354\pi\)
0.226249 + 0.974069i \(0.427354\pi\)
\(74\) − 1.95157e7i − 0.650814i
\(75\) 0 0
\(76\) −9.26586e7 −2.77735
\(77\) − 2.47460e7i − 0.703950i
\(78\) 0 0
\(79\) −2.35847e7 −0.605510 −0.302755 0.953068i \(-0.597906\pi\)
−0.302755 + 0.953068i \(0.597906\pi\)
\(80\) − 1.22203e8i − 2.98348i
\(81\) 0 0
\(82\) 2.75720e7 0.609835
\(83\) 3.34509e7i 0.704849i 0.935840 + 0.352424i \(0.114643\pi\)
−0.935840 + 0.352424i \(0.885357\pi\)
\(84\) 0 0
\(85\) −4.90614e7 −0.939862
\(86\) 1.69112e7i 0.309158i
\(87\) 0 0
\(88\) −1.30167e8 −2.17055
\(89\) − 2.82848e7i − 0.450809i −0.974265 0.225405i \(-0.927630\pi\)
0.974265 0.225405i \(-0.0723704\pi\)
\(90\) 0 0
\(91\) −8.95083e7 −1.30526
\(92\) 7.98498e7i 1.11461i
\(93\) 0 0
\(94\) 2.71431e8 3.47654
\(95\) − 1.25429e8i − 1.53994i
\(96\) 0 0
\(97\) 1.36490e8 1.54175 0.770873 0.636989i \(-0.219820\pi\)
0.770873 + 0.636989i \(0.219820\pi\)
\(98\) − 5.57223e7i − 0.604122i
\(99\) 0 0
\(100\) 1.74345e8 1.74345
\(101\) 4.02494e7i 0.386789i 0.981121 + 0.193394i \(0.0619497\pi\)
−0.981121 + 0.193394i \(0.938050\pi\)
\(102\) 0 0
\(103\) 3.35907e7 0.298449 0.149225 0.988803i \(-0.452322\pi\)
0.149225 + 0.988803i \(0.452322\pi\)
\(104\) 4.70824e8i 4.02462i
\(105\) 0 0
\(106\) −3.05006e8 −2.41593
\(107\) 1.95483e8i 1.49133i 0.666322 + 0.745664i \(0.267868\pi\)
−0.666322 + 0.745664i \(0.732132\pi\)
\(108\) 0 0
\(109\) −1.01182e8 −0.716800 −0.358400 0.933568i \(-0.616678\pi\)
−0.358400 + 0.933568i \(0.616678\pi\)
\(110\) − 3.04349e8i − 2.07875i
\(111\) 0 0
\(112\) 2.92060e8 1.85610
\(113\) − 3.14504e8i − 1.92891i −0.264238 0.964457i \(-0.585120\pi\)
0.264238 0.964457i \(-0.414880\pi\)
\(114\) 0 0
\(115\) −1.08090e8 −0.618007
\(116\) 3.57858e8i 1.97642i
\(117\) 0 0
\(118\) 1.48141e8 0.764097
\(119\) − 1.17254e8i − 0.584711i
\(120\) 0 0
\(121\) 5.60879e7 0.261654
\(122\) − 5.64743e8i − 2.54925i
\(123\) 0 0
\(124\) 9.99248e7 0.422656
\(125\) − 8.54913e7i − 0.350172i
\(126\) 0 0
\(127\) 4.30869e8 1.65627 0.828133 0.560532i \(-0.189403\pi\)
0.828133 + 0.560532i \(0.189403\pi\)
\(128\) 7.41647e7i 0.276285i
\(129\) 0 0
\(130\) −1.10086e9 −3.85441
\(131\) − 1.88933e8i − 0.641538i −0.947158 0.320769i \(-0.896059\pi\)
0.947158 0.320769i \(-0.103941\pi\)
\(132\) 0 0
\(133\) 2.99769e8 0.958032
\(134\) − 1.75785e7i − 0.0545209i
\(135\) 0 0
\(136\) −6.16772e8 −1.80289
\(137\) − 1.22998e8i − 0.349152i −0.984644 0.174576i \(-0.944144\pi\)
0.984644 0.174576i \(-0.0558555\pi\)
\(138\) 0 0
\(139\) 1.72072e8 0.460947 0.230474 0.973079i \(-0.425972\pi\)
0.230474 + 0.973079i \(0.425972\pi\)
\(140\) 9.84289e8i 2.56219i
\(141\) 0 0
\(142\) −8.60420e8 −2.11620
\(143\) 5.72479e8i 1.36904i
\(144\) 0 0
\(145\) −4.84421e8 −1.09585
\(146\) 3.77716e8i 0.831294i
\(147\) 0 0
\(148\) 4.03674e8 0.841363
\(149\) 7.31667e8i 1.48446i 0.670145 + 0.742230i \(0.266232\pi\)
−0.670145 + 0.742230i \(0.733768\pi\)
\(150\) 0 0
\(151\) 1.85952e8 0.357679 0.178840 0.983878i \(-0.442766\pi\)
0.178840 + 0.983878i \(0.442766\pi\)
\(152\) − 1.57682e9i − 2.95398i
\(153\) 0 0
\(154\) 7.27381e8 1.29324
\(155\) 1.35265e8i 0.234346i
\(156\) 0 0
\(157\) 9.74007e8 1.60311 0.801556 0.597920i \(-0.204006\pi\)
0.801556 + 0.597920i \(0.204006\pi\)
\(158\) − 6.93245e8i − 1.11239i
\(159\) 0 0
\(160\) 1.41204e9 2.15460
\(161\) − 2.58330e8i − 0.384477i
\(162\) 0 0
\(163\) −1.15499e9 −1.63616 −0.818082 0.575102i \(-0.804962\pi\)
−0.818082 + 0.575102i \(0.804962\pi\)
\(164\) 5.70315e8i 0.788386i
\(165\) 0 0
\(166\) −9.83253e8 −1.29489
\(167\) 1.14302e8i 0.146956i 0.997297 + 0.0734779i \(0.0234098\pi\)
−0.997297 + 0.0734779i \(0.976590\pi\)
\(168\) 0 0
\(169\) 1.25497e9 1.53847
\(170\) − 1.44210e9i − 1.72664i
\(171\) 0 0
\(172\) −3.49801e8 −0.399675
\(173\) 1.65322e9i 1.84564i 0.385235 + 0.922819i \(0.374121\pi\)
−0.385235 + 0.922819i \(0.625879\pi\)
\(174\) 0 0
\(175\) −5.64039e8 −0.601391
\(176\) − 1.86796e9i − 1.94678i
\(177\) 0 0
\(178\) 8.31400e8 0.828190
\(179\) − 4.17229e8i − 0.406408i −0.979136 0.203204i \(-0.934865\pi\)
0.979136 0.203204i \(-0.0651355\pi\)
\(180\) 0 0
\(181\) 1.16424e9 1.08475 0.542374 0.840137i \(-0.317526\pi\)
0.542374 + 0.840137i \(0.317526\pi\)
\(182\) − 2.63100e9i − 2.39792i
\(183\) 0 0
\(184\) −1.35884e9 −1.18549
\(185\) 5.46439e8i 0.466503i
\(186\) 0 0
\(187\) −7.49938e8 −0.613280
\(188\) 5.61443e9i 4.49443i
\(189\) 0 0
\(190\) 3.68684e9 2.82904
\(191\) − 1.61055e9i − 1.21016i −0.796166 0.605078i \(-0.793142\pi\)
0.796166 0.605078i \(-0.206858\pi\)
\(192\) 0 0
\(193\) −7.10279e8 −0.511917 −0.255959 0.966688i \(-0.582391\pi\)
−0.255959 + 0.966688i \(0.582391\pi\)
\(194\) 4.01196e9i 2.83237i
\(195\) 0 0
\(196\) 1.15259e9 0.781001
\(197\) − 1.41623e9i − 0.940303i −0.882586 0.470152i \(-0.844199\pi\)
0.882586 0.470152i \(-0.155801\pi\)
\(198\) 0 0
\(199\) −2.34324e9 −1.49419 −0.747093 0.664720i \(-0.768551\pi\)
−0.747093 + 0.664720i \(0.768551\pi\)
\(200\) 2.96691e9i 1.85432i
\(201\) 0 0
\(202\) −1.18309e9 −0.710576
\(203\) − 1.15774e9i − 0.681754i
\(204\) 0 0
\(205\) −7.72015e8 −0.437130
\(206\) 9.87362e8i 0.548286i
\(207\) 0 0
\(208\) −6.75658e9 −3.60972
\(209\) − 1.91727e9i − 1.00484i
\(210\) 0 0
\(211\) 1.06517e9 0.537389 0.268695 0.963225i \(-0.413408\pi\)
0.268695 + 0.963225i \(0.413408\pi\)
\(212\) − 6.30892e9i − 3.12328i
\(213\) 0 0
\(214\) −5.74600e9 −2.73975
\(215\) − 4.73513e8i − 0.221604i
\(216\) 0 0
\(217\) −3.23276e8 −0.145792
\(218\) − 2.97414e9i − 1.31685i
\(219\) 0 0
\(220\) 6.29534e9 2.68738
\(221\) 2.71259e9i 1.13714i
\(222\) 0 0
\(223\) 1.99586e9 0.807068 0.403534 0.914965i \(-0.367782\pi\)
0.403534 + 0.914965i \(0.367782\pi\)
\(224\) 3.37471e9i 1.34043i
\(225\) 0 0
\(226\) 9.24451e9 3.54364
\(227\) − 1.79923e9i − 0.677615i −0.940856 0.338807i \(-0.889977\pi\)
0.940856 0.338807i \(-0.110023\pi\)
\(228\) 0 0
\(229\) 1.59787e9 0.581032 0.290516 0.956870i \(-0.406173\pi\)
0.290516 + 0.956870i \(0.406173\pi\)
\(230\) − 3.17718e9i − 1.13535i
\(231\) 0 0
\(232\) −6.08986e9 −2.10211
\(233\) − 4.98411e9i − 1.69108i −0.533912 0.845540i \(-0.679279\pi\)
0.533912 0.845540i \(-0.320721\pi\)
\(234\) 0 0
\(235\) −7.60006e9 −2.49199
\(236\) 3.06424e9i 0.987814i
\(237\) 0 0
\(238\) 3.44656e9 1.07418
\(239\) 5.89754e9i 1.80751i 0.428055 + 0.903753i \(0.359199\pi\)
−0.428055 + 0.903753i \(0.640801\pi\)
\(240\) 0 0
\(241\) 2.26174e9 0.670463 0.335231 0.942136i \(-0.391185\pi\)
0.335231 + 0.942136i \(0.391185\pi\)
\(242\) 1.64864e9i 0.480689i
\(243\) 0 0
\(244\) 1.16815e10 3.29563
\(245\) 1.56023e9i 0.433035i
\(246\) 0 0
\(247\) −6.93492e9 −1.86317
\(248\) 1.70047e9i 0.449534i
\(249\) 0 0
\(250\) 2.51292e9 0.643307
\(251\) 3.31815e9i 0.835989i 0.908450 + 0.417994i \(0.137267\pi\)
−0.908450 + 0.417994i \(0.862733\pi\)
\(252\) 0 0
\(253\) −1.65223e9 −0.403263
\(254\) 1.26649e10i 3.04276i
\(255\) 0 0
\(256\) −5.35927e9 −1.24780
\(257\) − 3.05018e9i − 0.699185i −0.936902 0.349593i \(-0.886320\pi\)
0.936902 0.349593i \(-0.113680\pi\)
\(258\) 0 0
\(259\) −1.30596e9 −0.290223
\(260\) − 2.27708e10i − 4.98292i
\(261\) 0 0
\(262\) 5.55347e9 1.17858
\(263\) 1.97422e9i 0.412640i 0.978485 + 0.206320i \(0.0661489\pi\)
−0.978485 + 0.206320i \(0.933851\pi\)
\(264\) 0 0
\(265\) 8.54016e9 1.73174
\(266\) 8.81137e9i 1.76002i
\(267\) 0 0
\(268\) 3.63604e8 0.0704838
\(269\) 3.49334e9i 0.667163i 0.942721 + 0.333581i \(0.108257\pi\)
−0.942721 + 0.333581i \(0.891743\pi\)
\(270\) 0 0
\(271\) 1.43226e9 0.265549 0.132774 0.991146i \(-0.457611\pi\)
0.132774 + 0.991146i \(0.457611\pi\)
\(272\) − 8.85101e9i − 1.61703i
\(273\) 0 0
\(274\) 3.61538e9 0.641434
\(275\) 3.60749e9i 0.630775i
\(276\) 0 0
\(277\) 3.35095e9 0.569179 0.284589 0.958650i \(-0.408143\pi\)
0.284589 + 0.958650i \(0.408143\pi\)
\(278\) 5.05787e9i 0.846814i
\(279\) 0 0
\(280\) −1.67502e10 −2.72513
\(281\) − 4.29040e9i − 0.688133i −0.938945 0.344067i \(-0.888195\pi\)
0.938945 0.344067i \(-0.111805\pi\)
\(282\) 0 0
\(283\) −9.91145e7 −0.0154522 −0.00772612 0.999970i \(-0.502459\pi\)
−0.00772612 + 0.999970i \(0.502459\pi\)
\(284\) − 1.77974e10i − 2.73580i
\(285\) 0 0
\(286\) −1.68274e10 −2.51508
\(287\) − 1.84508e9i − 0.271949i
\(288\) 0 0
\(289\) 3.42231e9 0.490601
\(290\) − 1.42390e10i − 2.01320i
\(291\) 0 0
\(292\) −7.81291e9 −1.07469
\(293\) 9.02972e8i 0.122519i 0.998122 + 0.0612596i \(0.0195117\pi\)
−0.998122 + 0.0612596i \(0.980488\pi\)
\(294\) 0 0
\(295\) −4.14796e9 −0.547705
\(296\) 6.86952e9i 0.894869i
\(297\) 0 0
\(298\) −2.15065e10 −2.72713
\(299\) 5.97626e9i 0.747729i
\(300\) 0 0
\(301\) 1.13167e9 0.137865
\(302\) 5.46586e9i 0.657099i
\(303\) 0 0
\(304\) 2.26282e10 2.64945
\(305\) 1.58128e10i 1.82730i
\(306\) 0 0
\(307\) 3.36397e9 0.378703 0.189352 0.981909i \(-0.439361\pi\)
0.189352 + 0.981909i \(0.439361\pi\)
\(308\) 1.50456e10i 1.67188i
\(309\) 0 0
\(310\) −3.97596e9 −0.430521
\(311\) − 1.33023e10i − 1.42196i −0.703215 0.710978i \(-0.748253\pi\)
0.703215 0.710978i \(-0.251747\pi\)
\(312\) 0 0
\(313\) 1.31249e10 1.36748 0.683738 0.729728i \(-0.260353\pi\)
0.683738 + 0.729728i \(0.260353\pi\)
\(314\) 2.86299e10i 2.94510i
\(315\) 0 0
\(316\) 1.43395e10 1.43809
\(317\) 6.98520e9i 0.691739i 0.938283 + 0.345869i \(0.112416\pi\)
−0.938283 + 0.345869i \(0.887584\pi\)
\(318\) 0 0
\(319\) −7.40472e9 −0.715065
\(320\) 1.02213e10i 0.974774i
\(321\) 0 0
\(322\) 7.59331e9 0.706330
\(323\) − 9.08462e9i − 0.834635i
\(324\) 0 0
\(325\) 1.30486e10 1.16958
\(326\) − 3.39495e10i − 3.00582i
\(327\) 0 0
\(328\) −9.70533e9 −0.838523
\(329\) − 1.81638e10i − 1.55033i
\(330\) 0 0
\(331\) −1.18733e10 −0.989140 −0.494570 0.869138i \(-0.664674\pi\)
−0.494570 + 0.869138i \(0.664674\pi\)
\(332\) − 2.03382e10i − 1.67402i
\(333\) 0 0
\(334\) −3.35977e9 −0.269975
\(335\) 4.92198e8i 0.0390806i
\(336\) 0 0
\(337\) 9.15419e9 0.709741 0.354871 0.934915i \(-0.384525\pi\)
0.354871 + 0.934915i \(0.384525\pi\)
\(338\) 3.68886e10i 2.82634i
\(339\) 0 0
\(340\) 2.98293e10 2.23217
\(341\) 2.06762e9i 0.152916i
\(342\) 0 0
\(343\) −1.50682e10 −1.08864
\(344\) − 5.95274e9i − 0.425092i
\(345\) 0 0
\(346\) −4.85946e10 −3.39065
\(347\) − 2.05081e10i − 1.41451i −0.706957 0.707257i \(-0.749933\pi\)
0.706957 0.707257i \(-0.250067\pi\)
\(348\) 0 0
\(349\) 1.34799e10 0.908625 0.454312 0.890842i \(-0.349885\pi\)
0.454312 + 0.890842i \(0.349885\pi\)
\(350\) − 1.65793e10i − 1.10483i
\(351\) 0 0
\(352\) 2.15840e10 1.40592
\(353\) − 9.78259e9i − 0.630021i −0.949088 0.315011i \(-0.897992\pi\)
0.949088 0.315011i \(-0.102008\pi\)
\(354\) 0 0
\(355\) 2.40917e10 1.51689
\(356\) 1.71972e10i 1.07067i
\(357\) 0 0
\(358\) 1.22640e10 0.746619
\(359\) − 1.67338e10i − 1.00743i −0.863868 0.503717i \(-0.831965\pi\)
0.863868 0.503717i \(-0.168035\pi\)
\(360\) 0 0
\(361\) 6.24189e9 0.367525
\(362\) 3.42216e10i 1.99281i
\(363\) 0 0
\(364\) 5.44211e10 3.10000
\(365\) − 1.05761e10i − 0.595871i
\(366\) 0 0
\(367\) −1.49879e10 −0.826184 −0.413092 0.910689i \(-0.635551\pi\)
−0.413092 + 0.910689i \(0.635551\pi\)
\(368\) − 1.95002e10i − 1.06328i
\(369\) 0 0
\(370\) −1.60620e10 −0.857022
\(371\) 2.04106e10i 1.07736i
\(372\) 0 0
\(373\) −1.97964e10 −1.02271 −0.511353 0.859371i \(-0.670855\pi\)
−0.511353 + 0.859371i \(0.670855\pi\)
\(374\) − 2.20436e10i − 1.12667i
\(375\) 0 0
\(376\) −9.55436e10 −4.78025
\(377\) 2.67835e10i 1.32587i
\(378\) 0 0
\(379\) −5.34899e9 −0.259248 −0.129624 0.991563i \(-0.541377\pi\)
−0.129624 + 0.991563i \(0.541377\pi\)
\(380\) 7.62607e10i 3.65735i
\(381\) 0 0
\(382\) 4.73404e10 2.22320
\(383\) 1.44938e10i 0.673577i 0.941580 + 0.336788i \(0.109341\pi\)
−0.941580 + 0.336788i \(0.890659\pi\)
\(384\) 0 0
\(385\) −2.03667e10 −0.926994
\(386\) − 2.08778e10i − 0.940452i
\(387\) 0 0
\(388\) −8.29857e10 −3.66165
\(389\) − 1.80507e10i − 0.788308i −0.919044 0.394154i \(-0.871038\pi\)
0.919044 0.394154i \(-0.128962\pi\)
\(390\) 0 0
\(391\) −7.82879e9 −0.334956
\(392\) 1.96143e10i 0.830668i
\(393\) 0 0
\(394\) 4.16284e10 1.72745
\(395\) 1.94108e10i 0.797363i
\(396\) 0 0
\(397\) 3.82640e10 1.54038 0.770191 0.637814i \(-0.220161\pi\)
0.770191 + 0.637814i \(0.220161\pi\)
\(398\) − 6.88769e10i − 2.74499i
\(399\) 0 0
\(400\) −4.25768e10 −1.66316
\(401\) 4.26602e10i 1.64985i 0.565241 + 0.824926i \(0.308783\pi\)
−0.565241 + 0.824926i \(0.691217\pi\)
\(402\) 0 0
\(403\) 7.47875e9 0.283536
\(404\) − 2.44716e10i − 0.918623i
\(405\) 0 0
\(406\) 3.40305e10 1.25246
\(407\) 8.35271e9i 0.304404i
\(408\) 0 0
\(409\) −3.53953e10 −1.26489 −0.632444 0.774606i \(-0.717948\pi\)
−0.632444 + 0.774606i \(0.717948\pi\)
\(410\) − 2.26925e10i − 0.803059i
\(411\) 0 0
\(412\) −2.04232e10 −0.708817
\(413\) − 9.91343e9i − 0.340741i
\(414\) 0 0
\(415\) 2.75311e10 0.928177
\(416\) − 7.80712e10i − 2.60686i
\(417\) 0 0
\(418\) 5.63559e10 1.84601
\(419\) 1.45926e10i 0.473453i 0.971576 + 0.236726i \(0.0760745\pi\)
−0.971576 + 0.236726i \(0.923926\pi\)
\(420\) 0 0
\(421\) −2.65099e10 −0.843879 −0.421939 0.906624i \(-0.638650\pi\)
−0.421939 + 0.906624i \(0.638650\pi\)
\(422\) 3.13095e10i 0.987247i
\(423\) 0 0
\(424\) 1.07362e11 3.32191
\(425\) 1.70935e10i 0.523931i
\(426\) 0 0
\(427\) −3.77919e10 −1.13681
\(428\) − 1.18854e11i − 3.54191i
\(429\) 0 0
\(430\) 1.39184e10 0.407113
\(431\) 4.61676e9i 0.133791i 0.997760 + 0.0668957i \(0.0213095\pi\)
−0.997760 + 0.0668957i \(0.978691\pi\)
\(432\) 0 0
\(433\) −1.64494e10 −0.467948 −0.233974 0.972243i \(-0.575173\pi\)
−0.233974 + 0.972243i \(0.575173\pi\)
\(434\) − 9.50235e9i − 0.267838i
\(435\) 0 0
\(436\) 6.15188e10 1.70240
\(437\) − 2.00148e10i − 0.548816i
\(438\) 0 0
\(439\) −2.61475e10 −0.703999 −0.351999 0.936000i \(-0.614498\pi\)
−0.351999 + 0.936000i \(0.614498\pi\)
\(440\) 1.07131e11i 2.85828i
\(441\) 0 0
\(442\) −7.97335e10 −2.08906
\(443\) − 4.31455e10i − 1.12027i −0.828403 0.560133i \(-0.810750\pi\)
0.828403 0.560133i \(-0.189250\pi\)
\(444\) 0 0
\(445\) −2.32792e10 −0.593646
\(446\) 5.86660e10i 1.48268i
\(447\) 0 0
\(448\) −2.44283e10 −0.606431
\(449\) − 2.18287e10i − 0.537084i −0.963268 0.268542i \(-0.913458\pi\)
0.963268 0.268542i \(-0.0865418\pi\)
\(450\) 0 0
\(451\) −1.18008e10 −0.285237
\(452\) 1.91219e11i 4.58117i
\(453\) 0 0
\(454\) 5.28863e10 1.24486
\(455\) 7.36679e10i 1.71883i
\(456\) 0 0
\(457\) 2.46905e10 0.566064 0.283032 0.959110i \(-0.408660\pi\)
0.283032 + 0.959110i \(0.408660\pi\)
\(458\) 4.69676e10i 1.06742i
\(459\) 0 0
\(460\) 6.57186e10 1.46777
\(461\) − 2.61732e10i − 0.579500i −0.957102 0.289750i \(-0.906428\pi\)
0.957102 0.289750i \(-0.0935722\pi\)
\(462\) 0 0
\(463\) 7.04102e10 1.53219 0.766093 0.642729i \(-0.222198\pi\)
0.766093 + 0.642729i \(0.222198\pi\)
\(464\) − 8.73928e10i − 1.88540i
\(465\) 0 0
\(466\) 1.46502e11 3.10671
\(467\) 1.07220e10i 0.225428i 0.993627 + 0.112714i \(0.0359543\pi\)
−0.993627 + 0.112714i \(0.964046\pi\)
\(468\) 0 0
\(469\) −1.17633e9 −0.0243130
\(470\) − 2.23395e11i − 4.57807i
\(471\) 0 0
\(472\) −5.21458e10 −1.05063
\(473\) − 7.23798e9i − 0.144602i
\(474\) 0 0
\(475\) −4.37006e10 −0.858445
\(476\) 7.12907e10i 1.38869i
\(477\) 0 0
\(478\) −1.73352e11 −3.32060
\(479\) − 1.40034e10i − 0.266005i −0.991116 0.133003i \(-0.957538\pi\)
0.991116 0.133003i \(-0.0424619\pi\)
\(480\) 0 0
\(481\) 3.02125e10 0.564424
\(482\) 6.64814e10i 1.23172i
\(483\) 0 0
\(484\) −3.41014e10 −0.621429
\(485\) − 1.12335e11i − 2.03024i
\(486\) 0 0
\(487\) 1.72271e9 0.0306264 0.0153132 0.999883i \(-0.495125\pi\)
0.0153132 + 0.999883i \(0.495125\pi\)
\(488\) 1.98790e11i 3.50521i
\(489\) 0 0
\(490\) −4.58611e10 −0.795536
\(491\) − 3.70419e10i − 0.637334i −0.947867 0.318667i \(-0.896765\pi\)
0.947867 0.318667i \(-0.103235\pi\)
\(492\) 0 0
\(493\) −3.50859e10 −0.593943
\(494\) − 2.03844e11i − 3.42287i
\(495\) 0 0
\(496\) −2.44027e10 −0.403191
\(497\) 5.75782e10i 0.943696i
\(498\) 0 0
\(499\) −1.33497e10 −0.215312 −0.107656 0.994188i \(-0.534335\pi\)
−0.107656 + 0.994188i \(0.534335\pi\)
\(500\) 5.19787e10i 0.831659i
\(501\) 0 0
\(502\) −9.75332e10 −1.53581
\(503\) − 5.91175e10i − 0.923516i −0.887006 0.461758i \(-0.847219\pi\)
0.887006 0.461758i \(-0.152781\pi\)
\(504\) 0 0
\(505\) 3.31264e10 0.509341
\(506\) − 4.85655e10i − 0.740842i
\(507\) 0 0
\(508\) −2.61968e11 −3.93363
\(509\) 3.32051e10i 0.494691i 0.968927 + 0.247345i \(0.0795583\pi\)
−0.968927 + 0.247345i \(0.920442\pi\)
\(510\) 0 0
\(511\) 2.52763e10 0.370706
\(512\) − 1.38544e11i − 2.01607i
\(513\) 0 0
\(514\) 8.96565e10 1.28449
\(515\) − 2.76461e10i − 0.393012i
\(516\) 0 0
\(517\) −1.16172e11 −1.62608
\(518\) − 3.83873e10i − 0.533174i
\(519\) 0 0
\(520\) 3.87502e11 5.29981
\(521\) − 2.53746e10i − 0.344388i −0.985063 0.172194i \(-0.944914\pi\)
0.985063 0.172194i \(-0.0550856\pi\)
\(522\) 0 0
\(523\) 1.27775e11 1.70781 0.853903 0.520432i \(-0.174229\pi\)
0.853903 + 0.520432i \(0.174229\pi\)
\(524\) 1.14871e11i 1.52365i
\(525\) 0 0
\(526\) −5.80299e10 −0.758069
\(527\) 9.79703e9i 0.127014i
\(528\) 0 0
\(529\) 6.10629e10 0.779749
\(530\) 2.51029e11i 3.18141i
\(531\) 0 0
\(532\) −1.82259e11 −2.27533
\(533\) 4.26845e10i 0.528885i
\(534\) 0 0
\(535\) 1.60888e11 1.96385
\(536\) 6.18763e9i 0.0749662i
\(537\) 0 0
\(538\) −1.02683e11 −1.22566
\(539\) 2.38492e10i 0.282565i
\(540\) 0 0
\(541\) 6.16835e9 0.0720078 0.0360039 0.999352i \(-0.488537\pi\)
0.0360039 + 0.999352i \(0.488537\pi\)
\(542\) 4.20997e10i 0.487845i
\(543\) 0 0
\(544\) 1.02272e11 1.16778
\(545\) 8.32758e10i 0.943915i
\(546\) 0 0
\(547\) 1.02309e11 1.14279 0.571394 0.820676i \(-0.306403\pi\)
0.571394 + 0.820676i \(0.306403\pi\)
\(548\) 7.47826e10i 0.829237i
\(549\) 0 0
\(550\) −1.06038e11 −1.15881
\(551\) − 8.96995e10i − 0.973158i
\(552\) 0 0
\(553\) −4.63910e10 −0.496059
\(554\) 9.84974e10i 1.04565i
\(555\) 0 0
\(556\) −1.04620e11 −1.09475
\(557\) 1.00553e11i 1.04466i 0.852743 + 0.522331i \(0.174937\pi\)
−0.852743 + 0.522331i \(0.825063\pi\)
\(558\) 0 0
\(559\) −2.61804e10 −0.268120
\(560\) − 2.40374e11i − 2.44419i
\(561\) 0 0
\(562\) 1.26112e11 1.26418
\(563\) − 1.29004e11i − 1.28402i −0.766697 0.642009i \(-0.778101\pi\)
0.766697 0.642009i \(-0.221899\pi\)
\(564\) 0 0
\(565\) −2.58846e11 −2.54008
\(566\) − 2.91336e9i − 0.0283876i
\(567\) 0 0
\(568\) 3.02868e11 2.90978
\(569\) 1.38131e10i 0.131778i 0.997827 + 0.0658889i \(0.0209883\pi\)
−0.997827 + 0.0658889i \(0.979012\pi\)
\(570\) 0 0
\(571\) 7.83161e10 0.736727 0.368363 0.929682i \(-0.379918\pi\)
0.368363 + 0.929682i \(0.379918\pi\)
\(572\) − 3.48067e11i − 3.25147i
\(573\) 0 0
\(574\) 5.42341e10 0.499602
\(575\) 3.76595e10i 0.344511i
\(576\) 0 0
\(577\) −6.44834e10 −0.581761 −0.290880 0.956759i \(-0.593948\pi\)
−0.290880 + 0.956759i \(0.593948\pi\)
\(578\) 1.00595e11i 0.901291i
\(579\) 0 0
\(580\) 2.94528e11 2.60264
\(581\) 6.57980e10i 0.577442i
\(582\) 0 0
\(583\) 1.30542e11 1.13000
\(584\) − 1.32956e11i − 1.14303i
\(585\) 0 0
\(586\) −2.65418e10 −0.225082
\(587\) 1.10003e11i 0.926512i 0.886225 + 0.463256i \(0.153319\pi\)
−0.886225 + 0.463256i \(0.846681\pi\)
\(588\) 0 0
\(589\) −2.50468e10 −0.208109
\(590\) − 1.21925e11i − 1.00620i
\(591\) 0 0
\(592\) −9.85814e10 −0.802616
\(593\) − 9.33048e9i − 0.0754545i −0.999288 0.0377273i \(-0.987988\pi\)
0.999288 0.0377273i \(-0.0120118\pi\)
\(594\) 0 0
\(595\) −9.65037e10 −0.769974
\(596\) − 4.44854e11i − 3.52559i
\(597\) 0 0
\(598\) −1.75665e11 −1.37367
\(599\) 8.12693e10i 0.631276i 0.948880 + 0.315638i \(0.102219\pi\)
−0.948880 + 0.315638i \(0.897781\pi\)
\(600\) 0 0
\(601\) 1.80583e11 1.38414 0.692069 0.721831i \(-0.256699\pi\)
0.692069 + 0.721831i \(0.256699\pi\)
\(602\) 3.32643e10i 0.253275i
\(603\) 0 0
\(604\) −1.13059e11 −0.849488
\(605\) − 4.61619e10i − 0.344558i
\(606\) 0 0
\(607\) 1.15750e11 0.852644 0.426322 0.904571i \(-0.359809\pi\)
0.426322 + 0.904571i \(0.359809\pi\)
\(608\) 2.61465e11i 1.91337i
\(609\) 0 0
\(610\) −4.64800e11 −3.35696
\(611\) 4.20205e11i 3.01506i
\(612\) 0 0
\(613\) 7.01257e10 0.496632 0.248316 0.968679i \(-0.420123\pi\)
0.248316 + 0.968679i \(0.420123\pi\)
\(614\) 9.88802e10i 0.695722i
\(615\) 0 0
\(616\) −2.56038e11 −1.77820
\(617\) − 1.79037e10i − 0.123538i −0.998090 0.0617691i \(-0.980326\pi\)
0.998090 0.0617691i \(-0.0196742\pi\)
\(618\) 0 0
\(619\) 1.35992e11 0.926300 0.463150 0.886280i \(-0.346719\pi\)
0.463150 + 0.886280i \(0.346719\pi\)
\(620\) − 8.22410e10i − 0.556572i
\(621\) 0 0
\(622\) 3.91007e11 2.61230
\(623\) − 5.56362e10i − 0.369322i
\(624\) 0 0
\(625\) −1.82374e11 −1.19521
\(626\) 3.85793e11i 2.51221i
\(627\) 0 0
\(628\) −5.92197e11 −3.80739
\(629\) 3.95778e10i 0.252842i
\(630\) 0 0
\(631\) −2.50988e11 −1.58320 −0.791600 0.611040i \(-0.790752\pi\)
−0.791600 + 0.611040i \(0.790752\pi\)
\(632\) 2.44022e11i 1.52954i
\(633\) 0 0
\(634\) −2.05322e11 −1.27080
\(635\) − 3.54617e11i − 2.18105i
\(636\) 0 0
\(637\) 8.62644e10 0.523931
\(638\) − 2.17653e11i − 1.31366i
\(639\) 0 0
\(640\) 6.10397e10 0.363825
\(641\) 2.57286e10i 0.152400i 0.997093 + 0.0761999i \(0.0242787\pi\)
−0.997093 + 0.0761999i \(0.975721\pi\)
\(642\) 0 0
\(643\) 4.29706e10 0.251378 0.125689 0.992070i \(-0.459886\pi\)
0.125689 + 0.992070i \(0.459886\pi\)
\(644\) 1.57064e11i 0.913134i
\(645\) 0 0
\(646\) 2.67032e11 1.53332
\(647\) − 2.20975e11i − 1.26103i −0.776177 0.630516i \(-0.782843\pi\)
0.776177 0.630516i \(-0.217157\pi\)
\(648\) 0 0
\(649\) −6.34045e10 −0.357389
\(650\) 3.83549e11i 2.14866i
\(651\) 0 0
\(652\) 7.02232e11 3.88589
\(653\) − 2.99011e11i − 1.64450i −0.569125 0.822251i \(-0.692718\pi\)
0.569125 0.822251i \(-0.307282\pi\)
\(654\) 0 0
\(655\) −1.55497e11 −0.844806
\(656\) − 1.39277e11i − 0.752079i
\(657\) 0 0
\(658\) 5.33904e11 2.84813
\(659\) 2.99806e11i 1.58964i 0.606846 + 0.794820i \(0.292435\pi\)
−0.606846 + 0.794820i \(0.707565\pi\)
\(660\) 0 0
\(661\) −2.11448e11 −1.10764 −0.553820 0.832637i \(-0.686830\pi\)
−0.553820 + 0.832637i \(0.686830\pi\)
\(662\) − 3.49001e11i − 1.81717i
\(663\) 0 0
\(664\) 3.46105e11 1.78047
\(665\) − 2.46718e11i − 1.26158i
\(666\) 0 0
\(667\) −7.72997e10 −0.390548
\(668\) − 6.94954e10i − 0.349020i
\(669\) 0 0
\(670\) −1.44676e10 −0.0717956
\(671\) 2.41710e11i 1.19235i
\(672\) 0 0
\(673\) −1.90166e11 −0.926985 −0.463493 0.886101i \(-0.653404\pi\)
−0.463493 + 0.886101i \(0.653404\pi\)
\(674\) 2.69077e11i 1.30388i
\(675\) 0 0
\(676\) −7.63024e11 −3.65386
\(677\) − 1.14815e11i − 0.546570i −0.961933 0.273285i \(-0.911890\pi\)
0.961933 0.273285i \(-0.0881102\pi\)
\(678\) 0 0
\(679\) 2.68475e11 1.26306
\(680\) 5.07621e11i 2.37413i
\(681\) 0 0
\(682\) −6.07753e10 −0.280925
\(683\) − 3.07050e11i − 1.41100i −0.708710 0.705500i \(-0.750723\pi\)
0.708710 0.705500i \(-0.249277\pi\)
\(684\) 0 0
\(685\) −1.01231e11 −0.459780
\(686\) − 4.42914e11i − 1.99997i
\(687\) 0 0
\(688\) 8.54250e10 0.381269
\(689\) − 4.72183e11i − 2.09524i
\(690\) 0 0
\(691\) −2.20597e11 −0.967582 −0.483791 0.875184i \(-0.660741\pi\)
−0.483791 + 0.875184i \(0.660741\pi\)
\(692\) − 1.00516e12i − 4.38339i
\(693\) 0 0
\(694\) 6.02812e11 2.59863
\(695\) − 1.41620e11i − 0.606996i
\(696\) 0 0
\(697\) −5.59160e10 −0.236922
\(698\) 3.96226e11i 1.66925i
\(699\) 0 0
\(700\) 3.42936e11 1.42830
\(701\) 4.11156e11i 1.70269i 0.524609 + 0.851343i \(0.324212\pi\)
−0.524609 + 0.851343i \(0.675788\pi\)
\(702\) 0 0
\(703\) −1.01183e11 −0.414274
\(704\) 1.56239e11i 0.636062i
\(705\) 0 0
\(706\) 2.87548e11 1.15742
\(707\) 7.91705e10i 0.316874i
\(708\) 0 0
\(709\) 1.41043e11 0.558169 0.279084 0.960267i \(-0.409969\pi\)
0.279084 + 0.960267i \(0.409969\pi\)
\(710\) 7.08150e11i 2.78671i
\(711\) 0 0
\(712\) −2.92653e11 −1.13876
\(713\) 2.15844e10i 0.0835183i
\(714\) 0 0
\(715\) 4.71167e11 1.80281
\(716\) 2.53675e11i 0.965219i
\(717\) 0 0
\(718\) 4.91871e11 1.85078
\(719\) 1.87514e11i 0.701645i 0.936442 + 0.350823i \(0.114098\pi\)
−0.936442 + 0.350823i \(0.885902\pi\)
\(720\) 0 0
\(721\) 6.60730e10 0.244502
\(722\) 1.83473e11i 0.675187i
\(723\) 0 0
\(724\) −7.07859e11 −2.57628
\(725\) 1.68777e11i 0.610887i
\(726\) 0 0
\(727\) 6.74621e10 0.241503 0.120751 0.992683i \(-0.461470\pi\)
0.120751 + 0.992683i \(0.461470\pi\)
\(728\) 9.26111e11i 3.29714i
\(729\) 0 0
\(730\) 3.10871e11 1.09469
\(731\) − 3.42959e10i − 0.120108i
\(732\) 0 0
\(733\) −5.51348e11 −1.90990 −0.954948 0.296773i \(-0.904090\pi\)
−0.954948 + 0.296773i \(0.904090\pi\)
\(734\) − 4.40553e11i − 1.51780i
\(735\) 0 0
\(736\) 2.25321e11 0.767875
\(737\) 7.52360e9i 0.0255009i
\(738\) 0 0
\(739\) −3.53894e11 −1.18658 −0.593288 0.804990i \(-0.702171\pi\)
−0.593288 + 0.804990i \(0.702171\pi\)
\(740\) − 3.32235e11i − 1.10795i
\(741\) 0 0
\(742\) −5.99946e11 −1.97923
\(743\) 4.61456e10i 0.151417i 0.997130 + 0.0757086i \(0.0241219\pi\)
−0.997130 + 0.0757086i \(0.975878\pi\)
\(744\) 0 0
\(745\) 6.02183e11 1.95480
\(746\) − 5.81893e11i − 1.87883i
\(747\) 0 0
\(748\) 4.55962e11 1.45654
\(749\) 3.84515e11i 1.22176i
\(750\) 0 0
\(751\) −2.78555e11 −0.875690 −0.437845 0.899050i \(-0.644258\pi\)
−0.437845 + 0.899050i \(0.644258\pi\)
\(752\) − 1.37110e12i − 4.28745i
\(753\) 0 0
\(754\) −7.87270e11 −2.43578
\(755\) − 1.53044e11i − 0.471008i
\(756\) 0 0
\(757\) −2.54100e11 −0.773786 −0.386893 0.922125i \(-0.626452\pi\)
−0.386893 + 0.922125i \(0.626452\pi\)
\(758\) − 1.57228e11i − 0.476268i
\(759\) 0 0
\(760\) −1.29777e12 −3.88993
\(761\) 5.04366e11i 1.50386i 0.659243 + 0.751930i \(0.270877\pi\)
−0.659243 + 0.751930i \(0.729123\pi\)
\(762\) 0 0
\(763\) −1.99025e11 −0.587233
\(764\) 9.79216e11i 2.87412i
\(765\) 0 0
\(766\) −4.26029e11 −1.23744
\(767\) 2.29339e11i 0.662670i
\(768\) 0 0
\(769\) 2.16447e11 0.618937 0.309468 0.950910i \(-0.399849\pi\)
0.309468 + 0.950910i \(0.399849\pi\)
\(770\) − 5.98655e11i − 1.70300i
\(771\) 0 0
\(772\) 4.31850e11 1.21580
\(773\) 4.05778e11i 1.13650i 0.822855 + 0.568252i \(0.192380\pi\)
−0.822855 + 0.568252i \(0.807620\pi\)
\(774\) 0 0
\(775\) 4.71275e10 0.130637
\(776\) − 1.41221e12i − 3.89451i
\(777\) 0 0
\(778\) 5.30581e11 1.44821
\(779\) − 1.42953e11i − 0.388189i
\(780\) 0 0
\(781\) 3.68260e11 0.989806
\(782\) − 2.30119e11i − 0.615354i
\(783\) 0 0
\(784\) −2.81475e11 −0.745034
\(785\) − 8.01636e11i − 2.11105i
\(786\) 0 0
\(787\) 6.28441e11 1.63820 0.819098 0.573653i \(-0.194474\pi\)
0.819098 + 0.573653i \(0.194474\pi\)
\(788\) 8.61066e11i 2.23322i
\(789\) 0 0
\(790\) −5.70560e11 −1.46485
\(791\) − 6.18630e11i − 1.58025i
\(792\) 0 0
\(793\) 8.74286e11 2.21086
\(794\) 1.12473e12i 2.82986i
\(795\) 0 0
\(796\) 1.42469e12 3.54869
\(797\) 3.23780e11i 0.802447i 0.915980 + 0.401224i \(0.131415\pi\)
−0.915980 + 0.401224i \(0.868585\pi\)
\(798\) 0 0
\(799\) −5.50462e11 −1.35064
\(800\) − 4.91968e11i − 1.20109i
\(801\) 0 0
\(802\) −1.25395e12 −3.03097
\(803\) − 1.61663e11i − 0.388819i
\(804\) 0 0
\(805\) −2.12613e11 −0.506297
\(806\) 2.19829e11i 0.520889i
\(807\) 0 0
\(808\) 4.16446e11 0.977042
\(809\) 4.65514e10i 0.108677i 0.998523 + 0.0543386i \(0.0173050\pi\)
−0.998523 + 0.0543386i \(0.982695\pi\)
\(810\) 0 0
\(811\) 9.90434e9 0.0228951 0.0114475 0.999934i \(-0.496356\pi\)
0.0114475 + 0.999934i \(0.496356\pi\)
\(812\) 7.03908e11i 1.61917i
\(813\) 0 0
\(814\) −2.45519e11 −0.559225
\(815\) 9.50587e11i 2.15457i
\(816\) 0 0
\(817\) 8.76797e10 0.196794
\(818\) − 1.04040e12i − 2.32375i
\(819\) 0 0
\(820\) 4.69385e11 1.03818
\(821\) 3.00799e11i 0.662069i 0.943619 + 0.331034i \(0.107398\pi\)
−0.943619 + 0.331034i \(0.892602\pi\)
\(822\) 0 0
\(823\) 3.64988e10 0.0795571 0.0397785 0.999209i \(-0.487335\pi\)
0.0397785 + 0.999209i \(0.487335\pi\)
\(824\) − 3.47551e11i − 0.753894i
\(825\) 0 0
\(826\) 2.91394e11 0.625980
\(827\) 3.63233e11i 0.776540i 0.921546 + 0.388270i \(0.126927\pi\)
−0.921546 + 0.388270i \(0.873073\pi\)
\(828\) 0 0
\(829\) −1.99283e11 −0.421941 −0.210971 0.977492i \(-0.567662\pi\)
−0.210971 + 0.977492i \(0.567662\pi\)
\(830\) 8.09245e11i 1.70517i
\(831\) 0 0
\(832\) 5.65130e11 1.17938
\(833\) 1.13005e11i 0.234702i
\(834\) 0 0
\(835\) 9.40735e10 0.193518
\(836\) 1.16570e12i 2.38650i
\(837\) 0 0
\(838\) −4.28933e11 −0.869788
\(839\) − 4.55688e11i − 0.919644i −0.888011 0.459822i \(-0.847913\pi\)
0.888011 0.459822i \(-0.152087\pi\)
\(840\) 0 0
\(841\) 1.53816e11 0.307481
\(842\) − 7.79229e11i − 1.55030i
\(843\) 0 0
\(844\) −6.47623e11 −1.27630
\(845\) − 1.03288e12i − 2.02592i
\(846\) 0 0
\(847\) 1.10325e11 0.214358
\(848\) 1.54070e12i 2.97945i
\(849\) 0 0
\(850\) −5.02443e11 −0.962523
\(851\) 8.71961e10i 0.166257i
\(852\) 0 0
\(853\) −3.80568e11 −0.718846 −0.359423 0.933175i \(-0.617027\pi\)
−0.359423 + 0.933175i \(0.617027\pi\)
\(854\) − 1.11085e12i − 2.08845i
\(855\) 0 0
\(856\) 2.02259e12 3.76715
\(857\) 2.04675e11i 0.379438i 0.981838 + 0.189719i \(0.0607577\pi\)
−0.981838 + 0.189719i \(0.939242\pi\)
\(858\) 0 0
\(859\) 2.40882e11 0.442417 0.221208 0.975227i \(-0.429000\pi\)
0.221208 + 0.975227i \(0.429000\pi\)
\(860\) 2.87896e11i 0.526310i
\(861\) 0 0
\(862\) −1.35704e11 −0.245790
\(863\) − 8.63484e11i − 1.55672i −0.627817 0.778361i \(-0.716051\pi\)
0.627817 0.778361i \(-0.283949\pi\)
\(864\) 0 0
\(865\) 1.36065e12 2.43042
\(866\) − 4.83510e11i − 0.859675i
\(867\) 0 0
\(868\) 1.96552e11 0.346257
\(869\) 2.96709e11i 0.520297i
\(870\) 0 0
\(871\) 2.72135e10 0.0472837
\(872\) 1.04690e12i 1.81066i
\(873\) 0 0
\(874\) 5.88314e11 1.00824
\(875\) − 1.68161e11i − 0.286876i
\(876\) 0 0
\(877\) −4.79159e11 −0.809993 −0.404997 0.914318i \(-0.632727\pi\)
−0.404997 + 0.914318i \(0.632727\pi\)
\(878\) − 7.68576e11i − 1.29333i
\(879\) 0 0
\(880\) −1.53739e12 −2.56362
\(881\) 6.43291e11i 1.06783i 0.845537 + 0.533917i \(0.179280\pi\)
−0.845537 + 0.533917i \(0.820720\pi\)
\(882\) 0 0
\(883\) −1.08619e12 −1.78674 −0.893371 0.449319i \(-0.851667\pi\)
−0.893371 + 0.449319i \(0.851667\pi\)
\(884\) − 1.64925e12i − 2.70071i
\(885\) 0 0
\(886\) 1.26821e12 2.05806
\(887\) − 1.68383e11i − 0.272022i −0.990707 0.136011i \(-0.956572\pi\)
0.990707 0.136011i \(-0.0434283\pi\)
\(888\) 0 0
\(889\) 8.47519e11 1.35688
\(890\) − 6.84266e11i − 1.09060i
\(891\) 0 0
\(892\) −1.21348e12 −1.91679
\(893\) − 1.40729e12i − 2.21299i
\(894\) 0 0
\(895\) −3.43391e11 −0.535177
\(896\) 1.45882e11i 0.226344i
\(897\) 0 0
\(898\) 6.41630e11 0.986687
\(899\) 9.67336e10i 0.148094i
\(900\) 0 0
\(901\) 6.18552e11 0.938592
\(902\) − 3.46871e11i − 0.524013i
\(903\) 0 0
\(904\) −3.25407e12 −4.87251
\(905\) − 9.58204e11i − 1.42845i
\(906\) 0 0
\(907\) −6.69645e11 −0.989500 −0.494750 0.869035i \(-0.664740\pi\)
−0.494750 + 0.869035i \(0.664740\pi\)
\(908\) 1.09393e12i 1.60933i
\(909\) 0 0
\(910\) −2.16539e12 −3.15769
\(911\) 1.19351e12i 1.73282i 0.499337 + 0.866408i \(0.333577\pi\)
−0.499337 + 0.866408i \(0.666423\pi\)
\(912\) 0 0
\(913\) 4.20832e11 0.605656
\(914\) 7.25750e11i 1.03993i
\(915\) 0 0
\(916\) −9.71506e11 −1.37995
\(917\) − 3.71631e11i − 0.525575i
\(918\) 0 0
\(919\) −7.65817e11 −1.07365 −0.536825 0.843693i \(-0.680377\pi\)
−0.536825 + 0.843693i \(0.680377\pi\)
\(920\) 1.11837e12i 1.56111i
\(921\) 0 0
\(922\) 7.69333e11 1.06461
\(923\) − 1.33203e12i − 1.83529i
\(924\) 0 0
\(925\) 1.90385e11 0.260055
\(926\) 2.06963e12i 2.81481i
\(927\) 0 0
\(928\) 1.00981e12 1.36159
\(929\) 4.26646e11i 0.572803i 0.958110 + 0.286401i \(0.0924591\pi\)
−0.958110 + 0.286401i \(0.907541\pi\)
\(930\) 0 0
\(931\) −2.88905e11 −0.384553
\(932\) 3.03034e12i 4.01631i
\(933\) 0 0
\(934\) −3.15160e11 −0.414137
\(935\) 6.17221e11i 0.807596i
\(936\) 0 0
\(937\) 1.35431e12 1.75695 0.878477 0.477784i \(-0.158560\pi\)
0.878477 + 0.477784i \(0.158560\pi\)
\(938\) − 3.45769e10i − 0.0446658i
\(939\) 0 0
\(940\) 4.62084e12 5.91847
\(941\) 1.88655e10i 0.0240608i 0.999928 + 0.0120304i \(0.00382948\pi\)
−0.999928 + 0.0120304i \(0.996171\pi\)
\(942\) 0 0
\(943\) −1.23192e11 −0.155788
\(944\) − 7.48321e11i − 0.942323i
\(945\) 0 0
\(946\) 2.12752e11 0.265650
\(947\) 6.03707e11i 0.750631i 0.926897 + 0.375316i \(0.122466\pi\)
−0.926897 + 0.375316i \(0.877534\pi\)
\(948\) 0 0
\(949\) −5.84747e11 −0.720947
\(950\) − 1.28453e12i − 1.57706i
\(951\) 0 0
\(952\) −1.21319e12 −1.47700
\(953\) − 6.59941e10i − 0.0800080i −0.999200 0.0400040i \(-0.987263\pi\)
0.999200 0.0400040i \(-0.0127371\pi\)
\(954\) 0 0
\(955\) −1.32553e12 −1.59359
\(956\) − 3.58571e12i − 4.29283i
\(957\) 0 0
\(958\) 4.11613e11 0.488683
\(959\) − 2.41937e11i − 0.286040i
\(960\) 0 0
\(961\) −8.25880e11 −0.968330
\(962\) 8.88061e11i 1.03691i
\(963\) 0 0
\(964\) −1.37514e12 −1.59235
\(965\) 5.84580e11i 0.674116i
\(966\) 0 0
\(967\) 5.39935e11 0.617498 0.308749 0.951144i \(-0.400090\pi\)
0.308749 + 0.951144i \(0.400090\pi\)
\(968\) − 5.80322e11i − 0.660948i
\(969\) 0 0
\(970\) 3.30196e12 3.72979
\(971\) 7.23031e11i 0.813355i 0.913572 + 0.406677i \(0.133313\pi\)
−0.913572 + 0.406677i \(0.866687\pi\)
\(972\) 0 0
\(973\) 3.38466e11 0.377627
\(974\) 5.06370e10i 0.0562642i
\(975\) 0 0
\(976\) −2.85274e12 −3.14386
\(977\) 4.79457e11i 0.526225i 0.964765 + 0.263112i \(0.0847490\pi\)
−0.964765 + 0.263112i \(0.915251\pi\)
\(978\) 0 0
\(979\) −3.55839e11 −0.387367
\(980\) − 9.48617e11i − 1.02846i
\(981\) 0 0
\(982\) 1.08880e12 1.17086
\(983\) − 9.92587e11i − 1.06305i −0.847042 0.531526i \(-0.821619\pi\)
0.847042 0.531526i \(-0.178381\pi\)
\(984\) 0 0
\(985\) −1.16560e12 −1.23823
\(986\) − 1.03131e12i − 1.09114i
\(987\) 0 0
\(988\) 4.21643e12 4.42504
\(989\) − 7.55591e10i − 0.0789772i
\(990\) 0 0
\(991\) 9.53660e11 0.988778 0.494389 0.869241i \(-0.335392\pi\)
0.494389 + 0.869241i \(0.335392\pi\)
\(992\) − 2.81969e11i − 0.291175i
\(993\) 0 0
\(994\) −1.69245e12 −1.73368
\(995\) 1.92855e12i 1.96761i
\(996\) 0 0
\(997\) −1.51632e12 −1.53465 −0.767324 0.641259i \(-0.778412\pi\)
−0.767324 + 0.641259i \(0.778412\pi\)
\(998\) − 3.92399e11i − 0.395553i
\(999\) 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 27.9.b.b.26.2 yes 2
3.2 odd 2 inner 27.9.b.b.26.1 2
4.3 odd 2 432.9.e.d.161.1 2
9.2 odd 6 81.9.d.e.53.1 4
9.4 even 3 81.9.d.e.26.1 4
9.5 odd 6 81.9.d.e.26.2 4
9.7 even 3 81.9.d.e.53.2 4
12.11 even 2 432.9.e.d.161.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.b.b.26.1 2 3.2 odd 2 inner
27.9.b.b.26.2 yes 2 1.1 even 1 trivial
81.9.d.e.26.1 4 9.4 even 3
81.9.d.e.26.2 4 9.5 odd 6
81.9.d.e.53.1 4 9.2 odd 6
81.9.d.e.53.2 4 9.7 even 3
432.9.e.d.161.1 2 4.3 odd 2
432.9.e.d.161.2 2 12.11 even 2