Properties

Label 117.12.b.b.64.6
Level $117$
Weight $12$
Character 117.64
Analytic conductor $89.896$
Analytic rank $0$
Dimension $12$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [117,12,Mod(64,117)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(117, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("117.64");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 117 = 3^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 117.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: \(89.8961521255\)
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: \( x^{12} + 18433 x^{10} + 121088056 x^{8} + 340607607312 x^{6} + 380893885719552 x^{4} + \cdots + 14\!\cdots\!56 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{9}\cdot 3^{6}\cdot 13^{4} \)
Twist minimal: no (minimal twist has level 13)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 64.6
Root \(-3.38741i\) of defining polynomial
Character \(\chi\) \(=\) 117.64
Dual form 117.12.b.b.64.7

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.38741i q^{2} +2036.53 q^{4} -9091.88i q^{5} -45027.4i q^{7} -13836.0i q^{8} +O(q^{10})\) \(q-3.38741i q^{2} +2036.53 q^{4} -9091.88i q^{5} -45027.4i q^{7} -13836.0i q^{8} -30797.9 q^{10} -810639. i q^{11} +(1.11466e6 - 741417. i) q^{13} -152526. q^{14} +4.12394e6 q^{16} +851428. q^{17} -7.56605e6i q^{19} -1.85158e7i q^{20} -2.74596e6 q^{22} -3.69275e6 q^{23} -3.38341e7 q^{25} +(-2.51148e6 - 3.77580e6i) q^{26} -9.16995e7i q^{28} +9.33625e7 q^{29} -2.61508e8i q^{31} -4.23055e7i q^{32} -2.88413e6i q^{34} -4.09384e8 q^{35} +5.38221e8i q^{37} -2.56293e7 q^{38} -1.25795e8 q^{40} +6.71292e8i q^{41} -1.09294e9 q^{43} -1.65089e9i q^{44} +1.25088e7i q^{46} +1.79698e9i q^{47} -5.01432e7 q^{49} +1.14610e8i q^{50} +(2.27003e9 - 1.50991e9i) q^{52} +5.82497e9 q^{53} -7.37023e9 q^{55} -6.22997e8 q^{56} -3.16257e8i q^{58} +4.74847e9i q^{59} +6.50495e9 q^{61} -8.85834e8 q^{62} +8.30252e9 q^{64} +(-6.74087e9 - 1.01343e10i) q^{65} +8.11940e9i q^{67} +1.73395e9 q^{68} +1.38675e9i q^{70} +3.16640e9i q^{71} -8.22937e9i q^{73} +1.82317e9 q^{74} -1.54085e10i q^{76} -3.65010e10 q^{77} -3.75500e9 q^{79} -3.74943e10i q^{80} +2.27394e9 q^{82} +6.60119e9i q^{83} -7.74108e9i q^{85} +3.70222e9i q^{86} -1.12160e10 q^{88} -5.68420e10i q^{89} +(-3.33841e10 - 5.01902e10i) q^{91} -7.52038e9 q^{92} +6.08709e9 q^{94} -6.87896e10 q^{95} +1.62006e11i q^{97} +1.69856e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12290 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12290 q^{4} - 333446 q^{10} + 3208868 q^{13} + 5367450 q^{14} + 19025698 q^{16} - 12198768 q^{17} - 111171128 q^{22} - 5810592 q^{23} + 6102388 q^{25} + 64543986 q^{26} + 244463112 q^{29} + 562027560 q^{35} - 3171817788 q^{38} + 4092185498 q^{40} + 2294519976 q^{43} - 3573617796 q^{49} - 5597650396 q^{52} + 4602062760 q^{53} - 6178744976 q^{55} - 20017912662 q^{56} - 13775649944 q^{61} - 239765256 q^{62} - 3560815378 q^{64} + 7598401512 q^{65} - 40844682210 q^{68} - 19351803414 q^{74} + 80478036048 q^{77} + 18046097296 q^{79} - 255687836096 q^{82} + 239343029120 q^{88} + 104793638664 q^{91} + 135236877012 q^{92} - 78363161402 q^{94} - 145093149648 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/117\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(92\)
\(\chi(n)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 3.38741i 0.0748518i −0.999299 0.0374259i \(-0.988084\pi\)
0.999299 0.0374259i \(-0.0119158\pi\)
\(3\) 0 0
\(4\) 2036.53 0.994397
\(5\) 9091.88i 1.30112i −0.759453 0.650562i \(-0.774533\pi\)
0.759453 0.650562i \(-0.225467\pi\)
\(6\) 0 0
\(7\) 45027.4i 1.01260i −0.862357 0.506300i \(-0.831013\pi\)
0.862357 0.506300i \(-0.168987\pi\)
\(8\) 13836.0i 0.149284i
\(9\) 0 0
\(10\) −30797.9 −0.0973915
\(11\) 810639.i 1.51764i −0.651303 0.758818i \(-0.725777\pi\)
0.651303 0.758818i \(-0.274223\pi\)
\(12\) 0 0
\(13\) 1.11466e6 741417.i 0.832632 0.553827i
\(14\) −152526. −0.0757950
\(15\) 0 0
\(16\) 4.12394e6 0.983223
\(17\) 851428. 0.145438 0.0727192 0.997352i \(-0.476832\pi\)
0.0727192 + 0.997352i \(0.476832\pi\)
\(18\) 0 0
\(19\) 7.56605e6i 0.701011i −0.936561 0.350505i \(-0.886010\pi\)
0.936561 0.350505i \(-0.113990\pi\)
\(20\) 1.85158e7i 1.29383i
\(21\) 0 0
\(22\) −2.74596e6 −0.113598
\(23\) −3.69275e6 −0.119632 −0.0598159 0.998209i \(-0.519051\pi\)
−0.0598159 + 0.998209i \(0.519051\pi\)
\(24\) 0 0
\(25\) −3.38341e7 −0.692923
\(26\) −2.51148e6 3.77580e6i −0.0414550 0.0623240i
\(27\) 0 0
\(28\) 9.16995e7i 1.00693i
\(29\) 9.33625e7 0.845247 0.422623 0.906305i \(-0.361109\pi\)
0.422623 + 0.906305i \(0.361109\pi\)
\(30\) 0 0
\(31\) 2.61508e8i 1.64057i −0.571953 0.820286i \(-0.693814\pi\)
0.571953 0.820286i \(-0.306186\pi\)
\(32\) 4.23055e7i 0.222880i
\(33\) 0 0
\(34\) 2.88413e6i 0.0108863i
\(35\) −4.09384e8 −1.31752
\(36\) 0 0
\(37\) 5.38221e8i 1.27600i 0.770036 + 0.638000i \(0.220238\pi\)
−0.770036 + 0.638000i \(0.779762\pi\)
\(38\) −2.56293e7 −0.0524719
\(39\) 0 0
\(40\) −1.25795e8 −0.194237
\(41\) 6.71292e8i 0.904899i 0.891790 + 0.452449i \(0.149450\pi\)
−0.891790 + 0.452449i \(0.850550\pi\)
\(42\) 0 0
\(43\) −1.09294e9 −1.13375 −0.566876 0.823803i \(-0.691848\pi\)
−0.566876 + 0.823803i \(0.691848\pi\)
\(44\) 1.65089e9i 1.50913i
\(45\) 0 0
\(46\) 1.25088e7i 0.00895466i
\(47\) 1.79698e9i 1.14289i 0.820641 + 0.571445i \(0.193617\pi\)
−0.820641 + 0.571445i \(0.806383\pi\)
\(48\) 0 0
\(49\) −5.01432e7 −0.0253591
\(50\) 1.14610e8i 0.0518665i
\(51\) 0 0
\(52\) 2.27003e9 1.50991e9i 0.827967 0.550724i
\(53\) 5.82497e9 1.91327 0.956636 0.291287i \(-0.0940835\pi\)
0.956636 + 0.291287i \(0.0940835\pi\)
\(54\) 0 0
\(55\) −7.37023e9 −1.97463
\(56\) −6.22997e8 −0.151165
\(57\) 0 0
\(58\) 3.16257e8i 0.0632683i
\(59\) 4.74847e9i 0.864704i 0.901705 + 0.432352i \(0.142316\pi\)
−0.901705 + 0.432352i \(0.857684\pi\)
\(60\) 0 0
\(61\) 6.50495e9 0.986120 0.493060 0.869995i \(-0.335878\pi\)
0.493060 + 0.869995i \(0.335878\pi\)
\(62\) −8.85834e8 −0.122800
\(63\) 0 0
\(64\) 8.30252e9 0.966540
\(65\) −6.74087e9 1.01343e10i −0.720597 1.08336i
\(66\) 0 0
\(67\) 8.11940e9i 0.734704i 0.930082 + 0.367352i \(0.119736\pi\)
−0.930082 + 0.367352i \(0.880264\pi\)
\(68\) 1.73395e9 0.144623
\(69\) 0 0
\(70\) 1.38675e9i 0.0986186i
\(71\) 3.16640e9i 0.208279i 0.994563 + 0.104139i \(0.0332088\pi\)
−0.994563 + 0.104139i \(0.966791\pi\)
\(72\) 0 0
\(73\) 8.22937e9i 0.464613i −0.972643 0.232306i \(-0.925373\pi\)
0.972643 0.232306i \(-0.0746272\pi\)
\(74\) 1.82317e9 0.0955110
\(75\) 0 0
\(76\) 1.54085e10i 0.697083i
\(77\) −3.65010e10 −1.53676
\(78\) 0 0
\(79\) −3.75500e9 −0.137297 −0.0686485 0.997641i \(-0.521869\pi\)
−0.0686485 + 0.997641i \(0.521869\pi\)
\(80\) 3.74943e10i 1.27929i
\(81\) 0 0
\(82\) 2.27394e9 0.0677333
\(83\) 6.60119e9i 0.183947i 0.995761 + 0.0919735i \(0.0293175\pi\)
−0.995761 + 0.0919735i \(0.970682\pi\)
\(84\) 0 0
\(85\) 7.74108e9i 0.189233i
\(86\) 3.70222e9i 0.0848634i
\(87\) 0 0
\(88\) −1.12160e10 −0.226559
\(89\) 5.68420e10i 1.07901i −0.841983 0.539503i \(-0.818612\pi\)
0.841983 0.539503i \(-0.181388\pi\)
\(90\) 0 0
\(91\) −3.33841e10 5.01902e10i −0.560805 0.843123i
\(92\) −7.52038e9 −0.118962
\(93\) 0 0
\(94\) 6.08709e9 0.0855474
\(95\) −6.87896e10 −0.912101
\(96\) 0 0
\(97\) 1.62006e11i 1.91551i 0.287578 + 0.957757i \(0.407150\pi\)
−0.287578 + 0.957757i \(0.592850\pi\)
\(98\) 1.69856e8i 0.00189818i
\(99\) 0 0
\(100\) −6.89040e10 −0.689040
\(101\) −1.31282e11 −1.24290 −0.621450 0.783454i \(-0.713456\pi\)
−0.621450 + 0.783454i \(0.713456\pi\)
\(102\) 0 0
\(103\) −5.50452e10 −0.467859 −0.233929 0.972254i \(-0.575158\pi\)
−0.233929 + 0.972254i \(0.575158\pi\)
\(104\) −1.02582e10 1.54223e10i −0.0826777 0.124299i
\(105\) 0 0
\(106\) 1.97316e10i 0.143212i
\(107\) −1.85072e11 −1.27564 −0.637822 0.770184i \(-0.720164\pi\)
−0.637822 + 0.770184i \(0.720164\pi\)
\(108\) 0 0
\(109\) 2.74414e10i 0.170829i −0.996346 0.0854143i \(-0.972779\pi\)
0.996346 0.0854143i \(-0.0272214\pi\)
\(110\) 2.49660e10i 0.147805i
\(111\) 0 0
\(112\) 1.85690e11i 0.995612i
\(113\) 1.64428e11 0.839547 0.419773 0.907629i \(-0.362110\pi\)
0.419773 + 0.907629i \(0.362110\pi\)
\(114\) 0 0
\(115\) 3.35740e10i 0.155656i
\(116\) 1.90135e11 0.840511
\(117\) 0 0
\(118\) 1.60850e10 0.0647247
\(119\) 3.83376e10i 0.147271i
\(120\) 0 0
\(121\) −3.71823e11 −1.30322
\(122\) 2.20349e10i 0.0738129i
\(123\) 0 0
\(124\) 5.32567e11i 1.63138i
\(125\) 1.36324e11i 0.399546i
\(126\) 0 0
\(127\) 5.48728e11 1.47379 0.736897 0.676005i \(-0.236290\pi\)
0.736897 + 0.676005i \(0.236290\pi\)
\(128\) 1.14766e11i 0.295228i
\(129\) 0 0
\(130\) −3.43291e10 + 2.28341e10i −0.0810912 + 0.0539380i
\(131\) −3.22742e11 −0.730910 −0.365455 0.930829i \(-0.619087\pi\)
−0.365455 + 0.930829i \(0.619087\pi\)
\(132\) 0 0
\(133\) −3.40680e11 −0.709843
\(134\) 2.75037e10 0.0549940
\(135\) 0 0
\(136\) 1.17803e10i 0.0217117i
\(137\) 1.94892e11i 0.345009i −0.985009 0.172505i \(-0.944814\pi\)
0.985009 0.172505i \(-0.0551859\pi\)
\(138\) 0 0
\(139\) −6.64756e11 −1.08663 −0.543314 0.839530i \(-0.682831\pi\)
−0.543314 + 0.839530i \(0.682831\pi\)
\(140\) −8.33721e11 −1.31014
\(141\) 0 0
\(142\) 1.07259e10 0.0155901
\(143\) −6.01021e11 9.03584e11i −0.840508 1.26363i
\(144\) 0 0
\(145\) 8.48840e11i 1.09977i
\(146\) −2.78762e10 −0.0347771
\(147\) 0 0
\(148\) 1.09610e12i 1.26885i
\(149\) 4.29539e11i 0.479157i −0.970877 0.239578i \(-0.922991\pi\)
0.970877 0.239578i \(-0.0770092\pi\)
\(150\) 0 0
\(151\) 1.20700e12i 1.25122i 0.780136 + 0.625610i \(0.215150\pi\)
−0.780136 + 0.625610i \(0.784850\pi\)
\(152\) −1.04684e11 −0.104650
\(153\) 0 0
\(154\) 1.23644e11i 0.115029i
\(155\) −2.37760e12 −2.13459
\(156\) 0 0
\(157\) −9.61496e10 −0.0804450 −0.0402225 0.999191i \(-0.512807\pi\)
−0.0402225 + 0.999191i \(0.512807\pi\)
\(158\) 1.27197e10i 0.0102769i
\(159\) 0 0
\(160\) −3.84636e11 −0.289995
\(161\) 1.66275e11i 0.121139i
\(162\) 0 0
\(163\) 1.50626e12i 1.02534i −0.858586 0.512670i \(-0.828656\pi\)
0.858586 0.512670i \(-0.171344\pi\)
\(164\) 1.36710e12i 0.899829i
\(165\) 0 0
\(166\) 2.23609e10 0.0137688
\(167\) 1.03474e12i 0.616441i −0.951315 0.308221i \(-0.900266\pi\)
0.951315 0.308221i \(-0.0997335\pi\)
\(168\) 0 0
\(169\) 6.92762e11 1.65285e12i 0.386551 0.922268i
\(170\) −2.62222e10 −0.0141645
\(171\) 0 0
\(172\) −2.22579e12 −1.12740
\(173\) 1.11263e11 0.0545881 0.0272941 0.999627i \(-0.491311\pi\)
0.0272941 + 0.999627i \(0.491311\pi\)
\(174\) 0 0
\(175\) 1.52346e12i 0.701653i
\(176\) 3.34302e12i 1.49217i
\(177\) 0 0
\(178\) −1.92547e11 −0.0807656
\(179\) −2.06269e12 −0.838964 −0.419482 0.907764i \(-0.637788\pi\)
−0.419482 + 0.907764i \(0.637788\pi\)
\(180\) 0 0
\(181\) 3.46724e12 1.32663 0.663317 0.748338i \(-0.269148\pi\)
0.663317 + 0.748338i \(0.269148\pi\)
\(182\) −1.70015e11 + 1.13086e11i −0.0631093 + 0.0419773i
\(183\) 0 0
\(184\) 5.10927e10i 0.0178592i
\(185\) 4.89344e12 1.66023
\(186\) 0 0
\(187\) 6.90200e11i 0.220722i
\(188\) 3.65959e12i 1.13649i
\(189\) 0 0
\(190\) 2.33018e11i 0.0682725i
\(191\) 4.63569e12 1.31957 0.659783 0.751456i \(-0.270648\pi\)
0.659783 + 0.751456i \(0.270648\pi\)
\(192\) 0 0
\(193\) 1.13783e12i 0.305852i −0.988238 0.152926i \(-0.951130\pi\)
0.988238 0.152926i \(-0.0488696\pi\)
\(194\) 5.48779e11 0.143380
\(195\) 0 0
\(196\) −1.02118e11 −0.0252170
\(197\) 2.32298e12i 0.557803i 0.960320 + 0.278901i \(0.0899702\pi\)
−0.960320 + 0.278901i \(0.910030\pi\)
\(198\) 0 0
\(199\) 2.94354e11 0.0668618 0.0334309 0.999441i \(-0.489357\pi\)
0.0334309 + 0.999441i \(0.489357\pi\)
\(200\) 4.68127e11i 0.103442i
\(201\) 0 0
\(202\) 4.44704e11i 0.0930333i
\(203\) 4.20387e12i 0.855897i
\(204\) 0 0
\(205\) 6.10330e12 1.17739
\(206\) 1.86461e11i 0.0350201i
\(207\) 0 0
\(208\) 4.59678e12 3.05756e12i 0.818663 0.544536i
\(209\) −6.13333e12 −1.06388
\(210\) 0 0
\(211\) 8.00209e11 0.131719 0.0658597 0.997829i \(-0.479021\pi\)
0.0658597 + 0.997829i \(0.479021\pi\)
\(212\) 1.18627e13 1.90255
\(213\) 0 0
\(214\) 6.26914e11i 0.0954843i
\(215\) 9.93684e12i 1.47515i
\(216\) 0 0
\(217\) −1.17750e13 −1.66124
\(218\) −9.29552e10 −0.0127868
\(219\) 0 0
\(220\) −1.50097e13 −1.96357
\(221\) 9.49050e11 6.31263e11i 0.121097 0.0805477i
\(222\) 0 0
\(223\) 1.48493e13i 1.80314i 0.432631 + 0.901571i \(0.357585\pi\)
−0.432631 + 0.901571i \(0.642415\pi\)
\(224\) −1.90491e12 −0.225689
\(225\) 0 0
\(226\) 5.56985e11i 0.0628416i
\(227\) 1.57088e13i 1.72982i 0.501928 + 0.864910i \(0.332624\pi\)
−0.501928 + 0.864910i \(0.667376\pi\)
\(228\) 0 0
\(229\) 2.72068e12i 0.285485i 0.989760 + 0.142742i \(0.0455920\pi\)
−0.989760 + 0.142742i \(0.954408\pi\)
\(230\) 1.13729e11 0.0116511
\(231\) 0 0
\(232\) 1.29176e12i 0.126182i
\(233\) 8.42380e12 0.803620 0.401810 0.915723i \(-0.368381\pi\)
0.401810 + 0.915723i \(0.368381\pi\)
\(234\) 0 0
\(235\) 1.63379e13 1.48704
\(236\) 9.67037e12i 0.859859i
\(237\) 0 0
\(238\) −1.29865e11 −0.0110235
\(239\) 2.27825e13i 1.88979i −0.327374 0.944895i \(-0.606164\pi\)
0.327374 0.944895i \(-0.393836\pi\)
\(240\) 0 0
\(241\) 6.32011e12i 0.500761i 0.968147 + 0.250381i \(0.0805558\pi\)
−0.968147 + 0.250381i \(0.919444\pi\)
\(242\) 1.25952e12i 0.0975483i
\(243\) 0 0
\(244\) 1.32475e13 0.980595
\(245\) 4.55896e11i 0.0329953i
\(246\) 0 0
\(247\) −5.60960e12 8.43356e12i −0.388239 0.583684i
\(248\) −3.61821e12 −0.244912
\(249\) 0 0
\(250\) −4.61784e11 −0.0299067
\(251\) −5.09290e12 −0.322671 −0.161335 0.986900i \(-0.551580\pi\)
−0.161335 + 0.986900i \(0.551580\pi\)
\(252\) 0 0
\(253\) 2.99349e12i 0.181558i
\(254\) 1.85877e12i 0.110316i
\(255\) 0 0
\(256\) 1.66148e13 0.944442
\(257\) 2.23129e13 1.24144 0.620718 0.784034i \(-0.286841\pi\)
0.620718 + 0.784034i \(0.286841\pi\)
\(258\) 0 0
\(259\) 2.42347e13 1.29208
\(260\) −1.37280e13 2.06388e13i −0.716560 1.07729i
\(261\) 0 0
\(262\) 1.09326e12i 0.0547099i
\(263\) −1.02420e12 −0.0501911 −0.0250956 0.999685i \(-0.507989\pi\)
−0.0250956 + 0.999685i \(0.507989\pi\)
\(264\) 0 0
\(265\) 5.29599e13i 2.48940i
\(266\) 1.15402e12i 0.0531331i
\(267\) 0 0
\(268\) 1.65354e13i 0.730588i
\(269\) 3.39292e13 1.46871 0.734355 0.678766i \(-0.237485\pi\)
0.734355 + 0.678766i \(0.237485\pi\)
\(270\) 0 0
\(271\) 4.37089e13i 1.81651i 0.418414 + 0.908257i \(0.362586\pi\)
−0.418414 + 0.908257i \(0.637414\pi\)
\(272\) 3.51123e12 0.142998
\(273\) 0 0
\(274\) −6.60178e11 −0.0258246
\(275\) 2.74272e13i 1.05160i
\(276\) 0 0
\(277\) 1.69270e13 0.623651 0.311826 0.950139i \(-0.399060\pi\)
0.311826 + 0.950139i \(0.399060\pi\)
\(278\) 2.25180e12i 0.0813361i
\(279\) 0 0
\(280\) 5.66422e12i 0.196685i
\(281\) 4.31608e13i 1.46962i 0.678274 + 0.734809i \(0.262728\pi\)
−0.678274 + 0.734809i \(0.737272\pi\)
\(282\) 0 0
\(283\) −4.77458e13 −1.56354 −0.781772 0.623565i \(-0.785684\pi\)
−0.781772 + 0.623565i \(0.785684\pi\)
\(284\) 6.44846e12i 0.207112i
\(285\) 0 0
\(286\) −3.06081e12 + 2.03590e12i −0.0945852 + 0.0629136i
\(287\) 3.02265e13 0.916301
\(288\) 0 0
\(289\) −3.35470e13 −0.978848
\(290\) −2.87537e12 −0.0823198
\(291\) 0 0
\(292\) 1.67593e13i 0.462010i
\(293\) 1.64453e13i 0.444908i 0.974943 + 0.222454i \(0.0714067\pi\)
−0.974943 + 0.222454i \(0.928593\pi\)
\(294\) 0 0
\(295\) 4.31725e13 1.12509
\(296\) 7.44680e12 0.190487
\(297\) 0 0
\(298\) −1.45502e12 −0.0358658
\(299\) −4.11615e12 + 2.73787e12i −0.0996093 + 0.0662553i
\(300\) 0 0
\(301\) 4.92121e13i 1.14804i
\(302\) 4.08860e12 0.0936561
\(303\) 0 0
\(304\) 3.12019e13i 0.689250i
\(305\) 5.91422e13i 1.28306i
\(306\) 0 0
\(307\) 5.79282e12i 0.121235i 0.998161 + 0.0606176i \(0.0193070\pi\)
−0.998161 + 0.0606176i \(0.980693\pi\)
\(308\) −7.43352e13 −1.52815
\(309\) 0 0
\(310\) 8.05389e12i 0.159778i
\(311\) −6.88588e13 −1.34208 −0.671039 0.741422i \(-0.734152\pi\)
−0.671039 + 0.741422i \(0.734152\pi\)
\(312\) 0 0
\(313\) −2.94167e13 −0.553477 −0.276738 0.960945i \(-0.589254\pi\)
−0.276738 + 0.960945i \(0.589254\pi\)
\(314\) 3.25698e11i 0.00602146i
\(315\) 0 0
\(316\) −7.64715e12 −0.136528
\(317\) 2.48320e13i 0.435698i −0.975983 0.217849i \(-0.930096\pi\)
0.975983 0.217849i \(-0.0699040\pi\)
\(318\) 0 0
\(319\) 7.56832e13i 1.28278i
\(320\) 7.54855e13i 1.25759i
\(321\) 0 0
\(322\) 5.63241e11 0.00906749
\(323\) 6.44195e12i 0.101954i
\(324\) 0 0
\(325\) −3.77134e13 + 2.50852e13i −0.576949 + 0.383759i
\(326\) −5.10231e12 −0.0767486
\(327\) 0 0
\(328\) 9.28796e12 0.135087
\(329\) 8.09132e13 1.15729
\(330\) 0 0
\(331\) 3.37169e13i 0.466438i −0.972424 0.233219i \(-0.925074\pi\)
0.972424 0.233219i \(-0.0749259\pi\)
\(332\) 1.34435e13i 0.182916i
\(333\) 0 0
\(334\) −3.50509e12 −0.0461418
\(335\) 7.38206e13 0.955941
\(336\) 0 0
\(337\) −8.98229e13 −1.12570 −0.562850 0.826559i \(-0.690295\pi\)
−0.562850 + 0.826559i \(0.690295\pi\)
\(338\) −5.59888e12 2.34667e12i −0.0690335 0.0289341i
\(339\) 0 0
\(340\) 1.57649e13i 0.188173i
\(341\) −2.11988e14 −2.48979
\(342\) 0 0
\(343\) 8.67761e13i 0.986922i
\(344\) 1.51218e13i 0.169251i
\(345\) 0 0
\(346\) 3.76894e11i 0.00408602i
\(347\) −8.75394e13 −0.934096 −0.467048 0.884232i \(-0.654682\pi\)
−0.467048 + 0.884232i \(0.654682\pi\)
\(348\) 0 0
\(349\) 3.87065e13i 0.400169i 0.979779 + 0.200085i \(0.0641217\pi\)
−0.979779 + 0.200085i \(0.935878\pi\)
\(350\) 5.16059e12 0.0525201
\(351\) 0 0
\(352\) −3.42945e13 −0.338251
\(353\) 3.24210e13i 0.314822i −0.987533 0.157411i \(-0.949685\pi\)
0.987533 0.157411i \(-0.0503148\pi\)
\(354\) 0 0
\(355\) 2.87885e13 0.270996
\(356\) 1.15760e14i 1.07296i
\(357\) 0 0
\(358\) 6.98719e12i 0.0627980i
\(359\) 1.45534e14i 1.28809i 0.764990 + 0.644043i \(0.222744\pi\)
−0.764990 + 0.644043i \(0.777256\pi\)
\(360\) 0 0
\(361\) 5.92451e13 0.508584
\(362\) 1.17449e13i 0.0993011i
\(363\) 0 0
\(364\) −6.79876e13 1.02214e14i −0.557663 0.838399i
\(365\) −7.48205e13 −0.604518
\(366\) 0 0
\(367\) 6.95668e13 0.545430 0.272715 0.962095i \(-0.412078\pi\)
0.272715 + 0.962095i \(0.412078\pi\)
\(368\) −1.52287e13 −0.117625
\(369\) 0 0
\(370\) 1.65761e13i 0.124272i
\(371\) 2.62284e14i 1.93738i
\(372\) 0 0
\(373\) 1.20982e14 0.867604 0.433802 0.901008i \(-0.357172\pi\)
0.433802 + 0.901008i \(0.357172\pi\)
\(374\) −2.33799e12 −0.0165215
\(375\) 0 0
\(376\) 2.48629e13 0.170615
\(377\) 1.04067e14 6.92206e13i 0.703779 0.468120i
\(378\) 0 0
\(379\) 3.22918e13i 0.212117i −0.994360 0.106059i \(-0.966177\pi\)
0.994360 0.106059i \(-0.0338231\pi\)
\(380\) −1.40092e14 −0.906991
\(381\) 0 0
\(382\) 1.57030e13i 0.0987719i
\(383\) 6.60313e13i 0.409408i 0.978824 + 0.204704i \(0.0656232\pi\)
−0.978824 + 0.204704i \(0.934377\pi\)
\(384\) 0 0
\(385\) 3.31862e14i 1.99951i
\(386\) −3.85428e12 −0.0228936
\(387\) 0 0
\(388\) 3.29929e14i 1.90478i
\(389\) −1.13990e14 −0.648851 −0.324426 0.945911i \(-0.605171\pi\)
−0.324426 + 0.945911i \(0.605171\pi\)
\(390\) 0 0
\(391\) −3.14411e12 −0.0173991
\(392\) 6.93779e11i 0.00378572i
\(393\) 0 0
\(394\) 7.86887e12 0.0417525
\(395\) 3.41400e13i 0.178640i
\(396\) 0 0
\(397\) 2.68777e14i 1.36787i −0.729543 0.683935i \(-0.760267\pi\)
0.729543 0.683935i \(-0.239733\pi\)
\(398\) 9.97097e11i 0.00500473i
\(399\) 0 0
\(400\) −1.39530e14 −0.681297
\(401\) 3.43265e14i 1.65324i 0.562762 + 0.826619i \(0.309739\pi\)
−0.562762 + 0.826619i \(0.690261\pi\)
\(402\) 0 0
\(403\) −1.93886e14 2.91492e14i −0.908593 1.36599i
\(404\) −2.67358e14 −1.23594
\(405\) 0 0
\(406\) −1.42402e13 −0.0640655
\(407\) 4.36303e14 1.93650
\(408\) 0 0
\(409\) 2.66850e14i 1.15289i 0.817135 + 0.576446i \(0.195561\pi\)
−0.817135 + 0.576446i \(0.804439\pi\)
\(410\) 2.06744e13i 0.0881294i
\(411\) 0 0
\(412\) −1.12101e14 −0.465237
\(413\) 2.13811e14 0.875599
\(414\) 0 0
\(415\) 6.00172e13 0.239338
\(416\) −3.13660e13 4.71561e13i −0.123437 0.185577i
\(417\) 0 0
\(418\) 2.07761e13i 0.0796333i
\(419\) 2.13925e14 0.809253 0.404626 0.914482i \(-0.367402\pi\)
0.404626 + 0.914482i \(0.367402\pi\)
\(420\) 0 0
\(421\) 1.24393e14i 0.458398i 0.973380 + 0.229199i \(0.0736107\pi\)
−0.973380 + 0.229199i \(0.926389\pi\)
\(422\) 2.71063e12i 0.00985944i
\(423\) 0 0
\(424\) 8.05940e13i 0.285621i
\(425\) −2.88073e13 −0.100777
\(426\) 0 0
\(427\) 2.92901e14i 0.998545i
\(428\) −3.76903e14 −1.26850
\(429\) 0 0
\(430\) 3.36601e13 0.110418
\(431\) 3.23913e14i 1.04907i −0.851389 0.524535i \(-0.824239\pi\)
0.851389 0.524535i \(-0.175761\pi\)
\(432\) 0 0
\(433\) −2.35022e13 −0.0742036 −0.0371018 0.999311i \(-0.511813\pi\)
−0.0371018 + 0.999311i \(0.511813\pi\)
\(434\) 3.98868e13i 0.124347i
\(435\) 0 0
\(436\) 5.58851e13i 0.169871i
\(437\) 2.79395e13i 0.0838632i
\(438\) 0 0
\(439\) 4.02014e14 1.17676 0.588379 0.808586i \(-0.299767\pi\)
0.588379 + 0.808586i \(0.299767\pi\)
\(440\) 1.01974e14i 0.294781i
\(441\) 0 0
\(442\) −2.13835e12 3.21482e12i −0.00602914 0.00906430i
\(443\) 3.08170e14 0.858164 0.429082 0.903265i \(-0.358837\pi\)
0.429082 + 0.903265i \(0.358837\pi\)
\(444\) 0 0
\(445\) −5.16800e14 −1.40392
\(446\) 5.03007e13 0.134969
\(447\) 0 0
\(448\) 3.73841e14i 0.978719i
\(449\) 1.99777e14i 0.516643i 0.966059 + 0.258322i \(0.0831694\pi\)
−0.966059 + 0.258322i \(0.916831\pi\)
\(450\) 0 0
\(451\) 5.44175e14 1.37331
\(452\) 3.34862e14 0.834843
\(453\) 0 0
\(454\) 5.32121e13 0.129480
\(455\) −4.56323e14 + 3.03524e14i −1.09701 + 0.729677i
\(456\) 0 0
\(457\) 4.57271e13i 0.107309i −0.998560 0.0536543i \(-0.982913\pi\)
0.998560 0.0536543i \(-0.0170869\pi\)
\(458\) 9.21606e12 0.0213690
\(459\) 0 0
\(460\) 6.83744e13i 0.154784i
\(461\) 3.40488e14i 0.761635i −0.924650 0.380817i \(-0.875643\pi\)
0.924650 0.380817i \(-0.124357\pi\)
\(462\) 0 0
\(463\) 3.63954e14i 0.794971i 0.917608 + 0.397486i \(0.130117\pi\)
−0.917608 + 0.397486i \(0.869883\pi\)
\(464\) 3.85021e14 0.831066
\(465\) 0 0
\(466\) 2.85349e13i 0.0601524i
\(467\) 1.10274e14 0.229738 0.114869 0.993381i \(-0.463355\pi\)
0.114869 + 0.993381i \(0.463355\pi\)
\(468\) 0 0
\(469\) 3.65596e14 0.743962
\(470\) 5.53431e13i 0.111308i
\(471\) 0 0
\(472\) 6.56995e13 0.129087
\(473\) 8.85976e14i 1.72062i
\(474\) 0 0
\(475\) 2.55991e14i 0.485746i
\(476\) 7.80755e13i 0.146446i
\(477\) 0 0
\(478\) −7.71737e13 −0.141454
\(479\) 1.66407e14i 0.301526i 0.988570 + 0.150763i \(0.0481731\pi\)
−0.988570 + 0.150763i \(0.951827\pi\)
\(480\) 0 0
\(481\) 3.99046e14 + 5.99932e14i 0.706684 + 1.06244i
\(482\) 2.14088e13 0.0374829
\(483\) 0 0
\(484\) −7.57228e14 −1.29592
\(485\) 1.47293e15 2.49232
\(486\) 0 0
\(487\) 4.21283e13i 0.0696891i 0.999393 + 0.0348446i \(0.0110936\pi\)
−0.999393 + 0.0348446i \(0.988906\pi\)
\(488\) 9.00021e13i 0.147212i
\(489\) 0 0
\(490\) 1.54431e12 0.00246976
\(491\) −3.56448e13 −0.0563700 −0.0281850 0.999603i \(-0.508973\pi\)
−0.0281850 + 0.999603i \(0.508973\pi\)
\(492\) 0 0
\(493\) 7.94914e13 0.122931
\(494\) −2.85679e13 + 1.90020e13i −0.0436898 + 0.0290604i
\(495\) 0 0
\(496\) 1.07844e15i 1.61305i
\(497\) 1.42575e14 0.210903
\(498\) 0 0
\(499\) 2.70054e14i 0.390748i 0.980729 + 0.195374i \(0.0625921\pi\)
−0.980729 + 0.195374i \(0.937408\pi\)
\(500\) 2.77627e14i 0.397307i
\(501\) 0 0
\(502\) 1.72517e13i 0.0241525i
\(503\) 4.93264e14 0.683054 0.341527 0.939872i \(-0.389056\pi\)
0.341527 + 0.939872i \(0.389056\pi\)
\(504\) 0 0
\(505\) 1.19360e15i 1.61717i
\(506\) 1.01402e13 0.0135899
\(507\) 0 0
\(508\) 1.11750e15 1.46554
\(509\) 7.49682e14i 0.972589i 0.873795 + 0.486294i \(0.161652\pi\)
−0.873795 + 0.486294i \(0.838348\pi\)
\(510\) 0 0
\(511\) −3.70548e14 −0.470467
\(512\) 2.91321e14i 0.365921i
\(513\) 0 0
\(514\) 7.55830e13i 0.0929238i
\(515\) 5.00464e14i 0.608742i
\(516\) 0 0
\(517\) 1.45670e15 1.73449
\(518\) 8.20928e13i 0.0967145i
\(519\) 0 0
\(520\) −1.40218e14 + 9.32664e13i −0.161728 + 0.107574i
\(521\) −1.28499e15 −1.46654 −0.733270 0.679938i \(-0.762007\pi\)
−0.733270 + 0.679938i \(0.762007\pi\)
\(522\) 0 0
\(523\) 1.71007e14 0.191097 0.0955486 0.995425i \(-0.469539\pi\)
0.0955486 + 0.995425i \(0.469539\pi\)
\(524\) −6.57273e14 −0.726815
\(525\) 0 0
\(526\) 3.46937e12i 0.00375690i
\(527\) 2.22655e14i 0.238602i
\(528\) 0 0
\(529\) −9.39173e14 −0.985688
\(530\) −1.79397e14 −0.186336
\(531\) 0 0
\(532\) −6.93803e14 −0.705866
\(533\) 4.97707e14 + 7.48260e14i 0.501157 + 0.753447i
\(534\) 0 0
\(535\) 1.68265e15i 1.65977i
\(536\) 1.12340e14 0.109680
\(537\) 0 0
\(538\) 1.14932e14i 0.109936i
\(539\) 4.06480e13i 0.0384859i
\(540\) 0 0
\(541\) 1.59371e15i 1.47851i −0.673424 0.739256i \(-0.735177\pi\)
0.673424 0.739256i \(-0.264823\pi\)
\(542\) 1.48060e14 0.135969
\(543\) 0 0
\(544\) 3.60201e13i 0.0324153i
\(545\) −2.49494e14 −0.222269
\(546\) 0 0
\(547\) 1.12365e15 0.981073 0.490537 0.871421i \(-0.336801\pi\)
0.490537 + 0.871421i \(0.336801\pi\)
\(548\) 3.96902e14i 0.343076i
\(549\) 0 0
\(550\) 9.29072e13 0.0787145
\(551\) 7.06386e14i 0.592527i
\(552\) 0 0
\(553\) 1.69078e14i 0.139027i
\(554\) 5.73387e13i 0.0466814i
\(555\) 0 0
\(556\) −1.35379e15 −1.08054
\(557\) 5.45822e14i 0.431368i −0.976463 0.215684i \(-0.930802\pi\)
0.976463 0.215684i \(-0.0691981\pi\)
\(558\) 0 0
\(559\) −1.21825e15 + 8.10321e14i −0.943998 + 0.627903i
\(560\) −1.68827e15 −1.29541
\(561\) 0 0
\(562\) 1.46203e14 0.110004
\(563\) 1.07491e15 0.800897 0.400448 0.916319i \(-0.368854\pi\)
0.400448 + 0.916319i \(0.368854\pi\)
\(564\) 0 0
\(565\) 1.49496e15i 1.09235i
\(566\) 1.61735e14i 0.117034i
\(567\) 0 0
\(568\) 4.38102e13 0.0310928
\(569\) 3.71485e14 0.261110 0.130555 0.991441i \(-0.458324\pi\)
0.130555 + 0.991441i \(0.458324\pi\)
\(570\) 0 0
\(571\) 5.89713e14 0.406576 0.203288 0.979119i \(-0.434837\pi\)
0.203288 + 0.979119i \(0.434837\pi\)
\(572\) −1.22400e15 1.84017e15i −0.835799 1.25655i
\(573\) 0 0
\(574\) 1.02390e14i 0.0685868i
\(575\) 1.24941e14 0.0828956
\(576\) 0 0
\(577\) 4.47961e14i 0.291590i −0.989315 0.145795i \(-0.953426\pi\)
0.989315 0.145795i \(-0.0465740\pi\)
\(578\) 1.13637e14i 0.0732686i
\(579\) 0 0
\(580\) 1.72869e15i 1.09361i
\(581\) 2.97235e14 0.186265
\(582\) 0 0
\(583\) 4.72195e15i 2.90365i
\(584\) −1.13861e14 −0.0693594
\(585\) 0 0
\(586\) 5.57070e13 0.0333022
\(587\) 2.90281e14i 0.171913i −0.996299 0.0859567i \(-0.972605\pi\)
0.996299 0.0859567i \(-0.0273947\pi\)
\(588\) 0 0
\(589\) −1.97858e15 −1.15006
\(590\) 1.46243e14i 0.0842148i
\(591\) 0 0
\(592\) 2.21959e15i 1.25459i
\(593\) 1.53135e15i 0.857581i −0.903404 0.428790i \(-0.858940\pi\)
0.903404 0.428790i \(-0.141060\pi\)
\(594\) 0 0
\(595\) −3.48561e14 −0.191618
\(596\) 8.74767e14i 0.476472i
\(597\) 0 0
\(598\) 9.27427e12 + 1.39431e13i 0.00495933 + 0.00745594i
\(599\) 2.34047e14 0.124009 0.0620047 0.998076i \(-0.480251\pi\)
0.0620047 + 0.998076i \(0.480251\pi\)
\(600\) 0 0
\(601\) −2.19750e15 −1.14319 −0.571597 0.820535i \(-0.693676\pi\)
−0.571597 + 0.820535i \(0.693676\pi\)
\(602\) 1.66701e14 0.0859327
\(603\) 0 0
\(604\) 2.45808e15i 1.24421i
\(605\) 3.38057e15i 1.69565i
\(606\) 0 0
\(607\) 3.02645e15 1.49072 0.745360 0.666662i \(-0.232277\pi\)
0.745360 + 0.666662i \(0.232277\pi\)
\(608\) −3.20086e14 −0.156241
\(609\) 0 0
\(610\) −2.00339e14 −0.0960397
\(611\) 1.33231e15 + 2.00301e15i 0.632963 + 0.951606i
\(612\) 0 0
\(613\) 1.31524e15i 0.613724i 0.951754 + 0.306862i \(0.0992791\pi\)
−0.951754 + 0.306862i \(0.900721\pi\)
\(614\) 1.96226e13 0.00907468
\(615\) 0 0
\(616\) 5.05026e14i 0.229414i
\(617\) 7.90565e14i 0.355934i −0.984036 0.177967i \(-0.943048\pi\)
0.984036 0.177967i \(-0.0569520\pi\)
\(618\) 0 0
\(619\) 5.75601e13i 0.0254579i 0.999919 + 0.0127290i \(0.00405186\pi\)
−0.999919 + 0.0127290i \(0.995948\pi\)
\(620\) −4.84204e15 −2.12263
\(621\) 0 0
\(622\) 2.33253e14i 0.100457i
\(623\) −2.55945e15 −1.09260
\(624\) 0 0
\(625\) −2.89150e15 −1.21278
\(626\) 9.96462e13i 0.0414288i
\(627\) 0 0
\(628\) −1.95811e14 −0.0799943
\(629\) 4.58256e14i 0.185579i
\(630\) 0 0
\(631\) 3.34887e15i 1.33271i 0.745633 + 0.666356i \(0.232147\pi\)
−0.745633 + 0.666356i \(0.767853\pi\)
\(632\) 5.19540e13i 0.0204963i
\(633\) 0 0
\(634\) −8.41160e13 −0.0326128
\(635\) 4.98897e15i 1.91759i
\(636\) 0 0
\(637\) −5.58925e13 + 3.71771e13i −0.0211148 + 0.0140446i
\(638\) −2.56370e14 −0.0960182
\(639\) 0 0
\(640\) −1.04344e15 −0.384128
\(641\) 5.29783e14 0.193365 0.0966827 0.995315i \(-0.469177\pi\)
0.0966827 + 0.995315i \(0.469177\pi\)
\(642\) 0 0
\(643\) 1.01506e15i 0.364194i −0.983281 0.182097i \(-0.941712\pi\)
0.983281 0.182097i \(-0.0582885\pi\)
\(644\) 3.38623e14i 0.120460i
\(645\) 0 0
\(646\) −2.18215e13 −0.00763143
\(647\) −1.92293e14 −0.0666793 −0.0333396 0.999444i \(-0.510614\pi\)
−0.0333396 + 0.999444i \(0.510614\pi\)
\(648\) 0 0
\(649\) 3.84929e15 1.31231
\(650\) 8.49737e13 + 1.27751e14i 0.0287251 + 0.0431857i
\(651\) 0 0
\(652\) 3.06753e15i 1.01959i
\(653\) 2.34263e15 0.772114 0.386057 0.922475i \(-0.373837\pi\)
0.386057 + 0.922475i \(0.373837\pi\)
\(654\) 0 0
\(655\) 2.93433e15i 0.951004i
\(656\) 2.76836e15i 0.889717i
\(657\) 0 0
\(658\) 2.74086e14i 0.0866253i
\(659\) −5.91488e15 −1.85386 −0.926928 0.375239i \(-0.877560\pi\)
−0.926928 + 0.375239i \(0.877560\pi\)
\(660\) 0 0
\(661\) 4.81720e15i 1.48486i −0.669922 0.742432i \(-0.733672\pi\)
0.669922 0.742432i \(-0.266328\pi\)
\(662\) −1.14213e14 −0.0349138
\(663\) 0 0
\(664\) 9.13337e13 0.0274604
\(665\) 3.09742e15i 0.923594i
\(666\) 0 0
\(667\) −3.44764e14 −0.101118
\(668\) 2.10728e15i 0.612988i
\(669\) 0 0
\(670\) 2.50060e14i 0.0715540i
\(671\) 5.27316e15i 1.49657i
\(672\) 0 0
\(673\) −1.84391e15 −0.514821 −0.257410 0.966302i \(-0.582869\pi\)
−0.257410 + 0.966302i \(0.582869\pi\)
\(674\) 3.04267e14i 0.0842607i
\(675\) 0 0
\(676\) 1.41083e15 3.36608e15i 0.384385 0.917101i
\(677\) 2.85026e15 0.770276 0.385138 0.922859i \(-0.374154\pi\)
0.385138 + 0.922859i \(0.374154\pi\)
\(678\) 0 0
\(679\) 7.29470e15 1.93965
\(680\) −1.07105e14 −0.0282495
\(681\) 0 0
\(682\) 7.18091e14i 0.186365i
\(683\) 4.69918e15i 1.20978i −0.796308 0.604892i \(-0.793216\pi\)
0.796308 0.604892i \(-0.206784\pi\)
\(684\) 0 0
\(685\) −1.77193e15 −0.448899
\(686\) −2.93946e14 −0.0738729
\(687\) 0 0
\(688\) −4.50720e15 −1.11473
\(689\) 6.49285e15 4.31873e15i 1.59305 1.05962i
\(690\) 0 0
\(691\) 2.34559e15i 0.566399i −0.959061 0.283199i \(-0.908604\pi\)
0.959061 0.283199i \(-0.0913958\pi\)
\(692\) 2.26590e14 0.0542823
\(693\) 0 0
\(694\) 2.96532e14i 0.0699188i
\(695\) 6.04388e15i 1.41384i
\(696\) 0 0
\(697\) 5.71556e14i 0.131607i
\(698\) 1.31115e14 0.0299534
\(699\) 0 0
\(700\) 3.10257e15i 0.697722i
\(701\) −2.09935e15 −0.468421 −0.234210 0.972186i \(-0.575250\pi\)
−0.234210 + 0.972186i \(0.575250\pi\)
\(702\) 0 0
\(703\) 4.07221e15 0.894490
\(704\) 6.73034e15i 1.46686i
\(705\) 0 0
\(706\) −1.09823e14 −0.0235650
\(707\) 5.91127e15i 1.25856i
\(708\) 0 0
\(709\) 6.10587e15i 1.27995i −0.768396 0.639975i \(-0.778945\pi\)
0.768396 0.639975i \(-0.221055\pi\)
\(710\) 9.75185e13i 0.0202846i
\(711\) 0 0
\(712\) −7.86463e14 −0.161079
\(713\) 9.65683e14i 0.196265i
\(714\) 0 0
\(715\) −8.21528e15 + 5.46441e15i −1.64414 + 1.09360i
\(716\) −4.20073e15 −0.834263
\(717\) 0 0
\(718\) 4.92983e14 0.0964156
\(719\) −5.91981e15 −1.14894 −0.574472 0.818524i \(-0.694793\pi\)
−0.574472 + 0.818524i \(0.694793\pi\)
\(720\) 0 0
\(721\) 2.47854e15i 0.473754i
\(722\) 2.00687e14i 0.0380685i
\(723\) 0 0
\(724\) 7.06112e15 1.31920
\(725\) −3.15884e15 −0.585690
\(726\) 0 0
\(727\) 2.61053e15 0.476748 0.238374 0.971173i \(-0.423386\pi\)
0.238374 + 0.971173i \(0.423386\pi\)
\(728\) −6.94429e14 + 4.61901e14i −0.125865 + 0.0837194i
\(729\) 0 0
\(730\) 2.53447e14i 0.0452493i
\(731\) −9.30556e14 −0.164891
\(732\) 0 0
\(733\) 9.10402e15i 1.58914i 0.607175 + 0.794568i \(0.292303\pi\)
−0.607175 + 0.794568i \(0.707697\pi\)
\(734\) 2.35651e14i 0.0408264i
\(735\) 0 0
\(736\) 1.56224e14i 0.0266636i
\(737\) 6.58190e15 1.11501
\(738\) 0 0
\(739\) 2.00804e15i 0.335141i −0.985860 0.167570i \(-0.946408\pi\)
0.985860 0.167570i \(-0.0535921\pi\)
\(740\) 9.96561e15 1.65093
\(741\) 0 0
\(742\) −8.88461e14 −0.145016
\(743\) 8.48953e15i 1.37545i 0.725971 + 0.687725i \(0.241391\pi\)
−0.725971 + 0.687725i \(0.758609\pi\)
\(744\) 0 0
\(745\) −3.90531e15 −0.623442
\(746\) 4.09815e14i 0.0649417i
\(747\) 0 0
\(748\) 1.40561e15i 0.219486i
\(749\) 8.33331e15i 1.29172i
\(750\) 0 0
\(751\) −5.75001e15 −0.878312 −0.439156 0.898411i \(-0.644722\pi\)
−0.439156 + 0.898411i \(0.644722\pi\)
\(752\) 7.41062e15i 1.12372i
\(753\) 0 0
\(754\) −2.34478e14 3.52518e14i −0.0350397 0.0526792i
\(755\) 1.09739e16 1.62799
\(756\) 0 0
\(757\) 4.57782e15 0.669317 0.334658 0.942339i \(-0.391379\pi\)
0.334658 + 0.942339i \(0.391379\pi\)
\(758\) −1.09385e14 −0.0158774
\(759\) 0 0
\(760\) 9.51770e14i 0.136162i
\(761\) 3.71156e15i 0.527158i 0.964638 + 0.263579i \(0.0849029\pi\)
−0.964638 + 0.263579i \(0.915097\pi\)
\(762\) 0 0
\(763\) −1.23562e15 −0.172981
\(764\) 9.44070e15 1.31217
\(765\) 0 0
\(766\) 2.23675e14 0.0306450
\(767\) 3.52059e15 + 5.29291e15i 0.478896 + 0.719980i
\(768\) 0 0
\(769\) 2.37640e15i 0.318658i −0.987226 0.159329i \(-0.949067\pi\)
0.987226 0.159329i \(-0.0509331\pi\)
\(770\) 1.12415e15 0.149667
\(771\) 0 0
\(772\) 2.31721e15i 0.304138i
\(773\) 4.48843e15i 0.584934i −0.956276 0.292467i \(-0.905524\pi\)
0.956276 0.292467i \(-0.0944762\pi\)
\(774\) 0 0
\(775\) 8.84788e15i 1.13679i
\(776\) 2.24150e15 0.285956
\(777\) 0 0
\(778\) 3.86132e14i 0.0485677i
\(779\) 5.07903e15 0.634344
\(780\) 0 0
\(781\) 2.56681e15 0.316091
\(782\) 1.06504e13i 0.00130235i
\(783\) 0 0
\(784\) −2.06788e14 −0.0249337
\(785\) 8.74180e14i 0.104669i
\(786\) 0 0
\(787\) 1.79432e15i 0.211856i 0.994374 + 0.105928i \(0.0337812\pi\)
−0.994374 + 0.105928i \(0.966219\pi\)
\(788\) 4.73080e15i 0.554677i
\(789\) 0 0
\(790\) 1.15646e14 0.0133716
\(791\) 7.40378e15i 0.850125i
\(792\) 0 0
\(793\) 7.25079e15 4.82288e15i 0.821074 0.546140i
\(794\) −9.10458e14 −0.102388
\(795\) 0 0
\(796\) 5.99459e14 0.0664872
\(797\) −1.29926e15 −0.143111 −0.0715556 0.997437i \(-0.522796\pi\)
−0.0715556 + 0.997437i \(0.522796\pi\)
\(798\) 0 0
\(799\) 1.53000e15i 0.166220i
\(800\) 1.43137e15i 0.154439i
\(801\) 0 0
\(802\) 1.16278e15 0.123748
\(803\) −6.67105e15 −0.705113
\(804\) 0 0
\(805\) 1.51175e15 0.157617
\(806\) −9.87401e14 + 6.56772e14i −0.102247 + 0.0680099i
\(807\) 0 0
\(808\) 1.81641e15i 0.185545i
\(809\) 1.71565e16 1.74065 0.870326 0.492476i \(-0.163908\pi\)
0.870326 + 0.492476i \(0.163908\pi\)
\(810\) 0 0
\(811\) 8.99602e15i 0.900401i −0.892928 0.450200i \(-0.851353\pi\)
0.892928 0.450200i \(-0.148647\pi\)
\(812\) 8.56130e15i 0.851102i
\(813\) 0 0
\(814\) 1.47793e15i 0.144951i
\(815\) −1.36947e16 −1.33409
\(816\) 0 0
\(817\) 8.26921e15i 0.794772i
\(818\) 9.03929e14 0.0862961
\(819\) 0 0
\(820\) 1.24295e16 1.17079
\(821\) 1.60428e16i 1.50104i −0.660848 0.750520i \(-0.729803\pi\)
0.660848 0.750520i \(-0.270197\pi\)
\(822\) 0 0
\(823\) −5.49308e15 −0.507128 −0.253564 0.967319i \(-0.581603\pi\)
−0.253564 + 0.967319i \(0.581603\pi\)
\(824\) 7.61603e14i 0.0698440i
\(825\) 0 0
\(826\) 7.24266e14i 0.0655402i
\(827\) 8.14813e15i 0.732450i 0.930526 + 0.366225i \(0.119350\pi\)
−0.930526 + 0.366225i \(0.880650\pi\)
\(828\) 0 0
\(829\) 1.98376e16 1.75971 0.879853 0.475246i \(-0.157641\pi\)
0.879853 + 0.475246i \(0.157641\pi\)
\(830\) 2.03303e14i 0.0179149i
\(831\) 0 0
\(832\) 9.25446e15 6.15563e15i 0.804772 0.535296i
\(833\) −4.26934e13 −0.00368819
\(834\) 0 0
\(835\) −9.40775e15 −0.802066
\(836\) −1.24907e16 −1.05792
\(837\) 0 0
\(838\) 7.24651e14i 0.0605741i
\(839\) 2.34908e15i 0.195077i −0.995232 0.0975387i \(-0.968903\pi\)
0.995232 0.0975387i \(-0.0310970\pi\)
\(840\) 0 0
\(841\) −3.48395e15 −0.285558
\(842\) 4.21368e14 0.0343119
\(843\) 0 0
\(844\) 1.62965e15 0.130981
\(845\) −1.50275e16 6.29850e15i −1.19998 0.502951i
\(846\) 0 0
\(847\) 1.67423e16i 1.31964i
\(848\) 2.40218e16 1.88117
\(849\) 0 0
\(850\) 9.75821e13i 0.00754338i
\(851\) 1.98751e15i 0.152650i
\(852\) 0 0
\(853\) 1.56183e16i 1.18417i −0.805875 0.592086i \(-0.798305\pi\)
0.805875 0.592086i \(-0.201695\pi\)
\(854\) −9.92175e14 −0.0747429
\(855\) 0 0
\(856\) 2.56064e15i 0.190434i
\(857\) −1.24481e16 −0.919834 −0.459917 0.887962i \(-0.652121\pi\)
−0.459917 + 0.887962i \(0.652121\pi\)
\(858\) 0 0
\(859\) −1.17701e16 −0.858651 −0.429326 0.903150i \(-0.641249\pi\)
−0.429326 + 0.903150i \(0.641249\pi\)
\(860\) 2.02366e16i 1.46689i
\(861\) 0 0
\(862\) −1.09723e15 −0.0785247
\(863\) 4.09282e15i 0.291047i −0.989355 0.145524i \(-0.953513\pi\)
0.989355 0.145524i \(-0.0464867\pi\)
\(864\) 0 0
\(865\) 1.01159e15i 0.0710259i
\(866\) 7.96115e13i 0.00555428i
\(867\) 0 0
\(868\) −2.39801e16 −1.65194
\(869\) 3.04395e15i 0.208367i
\(870\) 0 0
\(871\) 6.01986e15 + 9.05035e15i 0.406899 + 0.611738i
\(872\) −3.79678e14 −0.0255020
\(873\) 0 0
\(874\) 9.46426e13 0.00627731
\(875\) −6.13831e15 −0.404580
\(876\) 0 0
\(877\) 1.66555e16i 1.08408i −0.840354 0.542038i \(-0.817653\pi\)
0.840354 0.542038i \(-0.182347\pi\)
\(878\) 1.36179e15i 0.0880824i
\(879\) 0 0
\(880\) −3.03943e16 −1.94150
\(881\) −2.33295e16 −1.48094 −0.740472 0.672087i \(-0.765398\pi\)
−0.740472 + 0.672087i \(0.765398\pi\)
\(882\) 0 0
\(883\) 5.66211e14 0.0354972 0.0177486 0.999842i \(-0.494350\pi\)
0.0177486 + 0.999842i \(0.494350\pi\)
\(884\) 1.93277e15 1.28558e15i 0.120418 0.0800964i
\(885\) 0 0
\(886\) 1.04390e15i 0.0642352i
\(887\) −8.53270e15 −0.521803 −0.260901 0.965365i \(-0.584020\pi\)
−0.260901 + 0.965365i \(0.584020\pi\)
\(888\) 0 0
\(889\) 2.47078e16i 1.49236i
\(890\) 1.75061e15i 0.105086i
\(891\) 0 0
\(892\) 3.02410e16i 1.79304i
\(893\) 1.35960e16 0.801177
\(894\) 0 0
\(895\) 1.87538e16i 1.09160i
\(896\) −5.16760e15 −0.298948
\(897\) 0 0
\(898\) 6.76726e14 0.0386717
\(899\) 2.44150e16i 1.38669i
\(900\) 0 0
\(901\) 4.95954e15 0.278263
\(902\) 1.84334e15i 0.102795i
\(903\) 0 0
\(904\) 2.27502e15i 0.125331i
\(905\) 3.15237e16i 1.72612i
\(906\) 0 0
\(907\) 9.92538e15 0.536917 0.268458 0.963291i \(-0.413486\pi\)
0.268458 + 0.963291i \(0.413486\pi\)
\(908\) 3.19914e16i 1.72013i
\(909\) 0 0
\(910\) 1.02816e15 + 1.54575e15i 0.0546177 + 0.0821130i
\(911\) −1.06854e16 −0.564208 −0.282104 0.959384i \(-0.591032\pi\)
−0.282104 + 0.959384i \(0.591032\pi\)
\(912\) 0 0
\(913\) 5.35118e15 0.279165
\(914\) −1.54896e14 −0.00803225
\(915\) 0 0
\(916\) 5.54074e15i 0.283885i
\(917\) 1.45323e16i 0.740119i
\(918\) 0 0
\(919\) −6.16827e15 −0.310405 −0.155202 0.987883i \(-0.549603\pi\)
−0.155202 + 0.987883i \(0.549603\pi\)
\(920\) 4.64529e14 0.0232370
\(921\) 0 0
\(922\) −1.15337e15 −0.0570098
\(923\) 2.34762e15 + 3.52945e15i 0.115350 + 0.173420i
\(924\) 0 0
\(925\) 1.82102e16i 0.884170i
\(926\) 1.23286e15 0.0595051
\(927\) 0 0
\(928\) 3.94975e15i 0.188389i
\(929\) 1.71984e16i 0.815460i −0.913103 0.407730i \(-0.866320\pi\)
0.913103 0.407730i \(-0.133680\pi\)
\(930\) 0 0
\(931\) 3.79386e14i 0.0177770i
\(932\) 1.71553e16 0.799117
\(933\) 0 0
\(934\) 3.73544e14i 0.0171963i
\(935\) −6.27522e15 −0.287187
\(936\) 0 0
\(937\) −1.38540e16 −0.626626 −0.313313 0.949650i \(-0.601439\pi\)
−0.313313 + 0.949650i \(0.601439\pi\)
\(938\) 1.23842e15i 0.0556869i
\(939\) 0 0
\(940\) 3.32725e16 1.47871
\(941\) 3.01742e16i 1.33319i −0.745419 0.666597i \(-0.767750\pi\)
0.745419 0.666597i \(-0.232250\pi\)
\(942\) 0 0
\(943\) 2.47891e15i 0.108255i
\(944\) 1.95824e16i 0.850196i
\(945\) 0 0
\(946\) 3.00116e15 0.128792
\(947\) 1.47817e16i 0.630665i 0.948981 + 0.315333i \(0.102116\pi\)
−0.948981 + 0.315333i \(0.897884\pi\)
\(948\) 0 0
\(949\) −6.10140e15 9.17293e15i −0.257315 0.386851i
\(950\) 8.67145e14 0.0363590
\(951\) 0 0
\(952\) −5.30437e14 −0.0219852
\(953\) 2.28957e16 0.943501 0.471750 0.881732i \(-0.343622\pi\)
0.471750 + 0.881732i \(0.343622\pi\)
\(954\) 0 0
\(955\) 4.21471e16i 1.71692i
\(956\) 4.63972e16i 1.87920i
\(957\) 0 0
\(958\) 5.63687e14 0.0225698
\(959\) −8.77548e15 −0.349356
\(960\) 0 0
\(961\) −4.29779e16 −1.69148
\(962\) 2.03221e15 1.35173e15i 0.0795255 0.0528966i
\(963\) 0 0
\(964\) 1.28711e16i 0.497956i
\(965\) −1.03450e16 −0.397951
\(966\) 0 0
\(967\) 3.65782e16i 1.39116i −0.718448 0.695580i \(-0.755147\pi\)
0.718448 0.695580i \(-0.244853\pi\)
\(968\) 5.14453e15i 0.194550i
\(969\) 0 0
\(970\) 4.98943e15i 0.186555i
\(971\) −2.24568e16 −0.834915 −0.417458 0.908696i \(-0.637079\pi\)
−0.417458 + 0.908696i \(0.637079\pi\)
\(972\) 0 0
\(973\) 2.99323e16i 1.10032i
\(974\) 1.42706e14 0.00521636
\(975\) 0 0
\(976\) 2.68260e16 0.969575
\(977\) 2.77848e16i 0.998590i 0.866432 + 0.499295i \(0.166408\pi\)
−0.866432 + 0.499295i \(0.833592\pi\)
\(978\) 0 0
\(979\) −4.60783e16 −1.63754
\(980\) 9.28444e14i 0.0328105i
\(981\) 0 0
\(982\) 1.20744e14i 0.00421940i
\(983\) 6.91991e15i 0.240467i 0.992746 + 0.120234i \(0.0383644\pi\)
−0.992746 + 0.120234i \(0.961636\pi\)
\(984\) 0 0
\(985\) 2.11202e16 0.725770
\(986\) 2.69270e14i 0.00920163i
\(987\) 0 0
\(988\) −1.14241e16 1.71752e16i −0.386063 0.580413i
\(989\) 4.03594e15 0.135633
\(990\) 0 0
\(991\) 4.49174e16 1.49283 0.746414 0.665482i \(-0.231774\pi\)
0.746414 + 0.665482i \(0.231774\pi\)
\(992\) −1.10632e16 −0.365651
\(993\) 0 0
\(994\) 4.82959e14i 0.0157865i
\(995\) 2.67623e15i 0.0869954i
\(996\) 0 0
\(997\) −4.98182e16 −1.60164 −0.800819 0.598907i \(-0.795602\pi\)
−0.800819 + 0.598907i \(0.795602\pi\)
\(998\) 9.14782e14 0.0292482
\(999\) 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 117.12.b.b.64.6 12
3.2 odd 2 13.12.b.a.12.7 yes 12
12.11 even 2 208.12.f.b.129.4 12
13.12 even 2 inner 117.12.b.b.64.7 12
39.5 even 4 169.12.a.e.1.7 12
39.8 even 4 169.12.a.e.1.6 12
39.38 odd 2 13.12.b.a.12.6 12
156.155 even 2 208.12.f.b.129.3 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
13.12.b.a.12.6 12 39.38 odd 2
13.12.b.a.12.7 yes 12 3.2 odd 2
117.12.b.b.64.6 12 1.1 even 1 trivial
117.12.b.b.64.7 12 13.12 even 2 inner
169.12.a.e.1.6 12 39.8 even 4
169.12.a.e.1.7 12 39.5 even 4
208.12.f.b.129.3 12 156.155 even 2
208.12.f.b.129.4 12 12.11 even 2