Properties

Label 100.10.c.c.49.1
Level $100$
Weight $10$
Character 100.49
Analytic conductor $51.504$
Analytic rank $0$
Dimension $4$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [100,10,Mod(49,100)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(100, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 10, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("100.49");
 
S:= CuspForms(chi, 10);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 100 = 2^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 10 \)
Character orbit: \([\chi]\) \(=\) 100.c (of order \(2\), degree \(1\), not 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: \(51.5035836164\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{79})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 39x^{2} + 400 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{12}\cdot 5^{2} \)
Twist minimal: no (minimal twist has level 20)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 49.1
Root \(-4.44410 + 0.500000i\) of defining polynomial
Character \(\chi\) \(=\) 100.49
Dual form 100.10.c.c.49.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-272.211i q^{3} +10002.6i q^{7} -54415.9 q^{9} +O(q^{10})\) \(q-272.211i q^{3} +10002.6i q^{7} -54415.9 q^{9} +47093.7 q^{11} +9362.93i q^{13} -108521. i q^{17} +665173. q^{19} +2.72281e6 q^{21} +576426. i q^{23} +9.45468e6i q^{27} +2.61067e6 q^{29} +3.87896e6 q^{31} -1.28194e7i q^{33} -1.41599e7i q^{37} +2.54869e6 q^{39} +4.62193e6 q^{41} +8.31227e6i q^{43} -2.51923e7i q^{47} -5.96977e7 q^{49} -2.95405e7 q^{51} +3.49333e7i q^{53} -1.81068e8i q^{57} -6.71868e6 q^{59} -4.75982e6 q^{61} -5.44299e8i q^{63} +1.38772e8i q^{67} +1.56909e8 q^{69} +3.54359e8 q^{71} +2.41596e8i q^{73} +4.71058e8i q^{77} -2.61173e8 q^{79} +1.50260e9 q^{81} -6.55898e8i q^{83} -7.10654e8i q^{87} +1.00221e9 q^{89} -9.36534e7 q^{91} -1.05590e9i q^{93} -1.24869e9i q^{97} -2.56264e9 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 69764 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 69764 q^{9} + 205440 q^{11} - 274544 q^{19} + 5680624 q^{21} + 13787496 q^{29} + 583664 q^{31} + 951056 q^{39} + 59546904 q^{41} - 223875828 q^{49} - 54013392 q^{51} + 185861712 q^{59} + 391347848 q^{61} + 344148528 q^{69} + 622414032 q^{71} - 1084523552 q^{79} + 3762476756 q^{81} + 924583704 q^{89} + 3080075536 q^{91} - 2952090240 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/100\mathbb{Z}\right)^\times\).

\(n\) \(51\) \(77\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) − 272.211i − 1.94026i −0.242583 0.970131i \(-0.577995\pi\)
0.242583 0.970131i \(-0.422005\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 10002.6i 1.57460i 0.616570 + 0.787300i \(0.288522\pi\)
−0.616570 + 0.787300i \(0.711478\pi\)
\(8\) 0 0
\(9\) −54415.9 −2.76461
\(10\) 0 0
\(11\) 47093.7 0.969830 0.484915 0.874561i \(-0.338851\pi\)
0.484915 + 0.874561i \(0.338851\pi\)
\(12\) 0 0
\(13\) 9362.93i 0.0909216i 0.998966 + 0.0454608i \(0.0144756\pi\)
−0.998966 + 0.0454608i \(0.985524\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 108521.i − 0.315131i −0.987509 0.157566i \(-0.949635\pi\)
0.987509 0.157566i \(-0.0503646\pi\)
\(18\) 0 0
\(19\) 665173. 1.17096 0.585482 0.810685i \(-0.300905\pi\)
0.585482 + 0.810685i \(0.300905\pi\)
\(20\) 0 0
\(21\) 2.72281e6 3.05514
\(22\) 0 0
\(23\) 576426.i 0.429505i 0.976669 + 0.214752i \(0.0688945\pi\)
−0.976669 + 0.214752i \(0.931106\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 9.45468e6i 3.42381i
\(28\) 0 0
\(29\) 2.61067e6 0.685427 0.342714 0.939440i \(-0.388654\pi\)
0.342714 + 0.939440i \(0.388654\pi\)
\(30\) 0 0
\(31\) 3.87896e6 0.754375 0.377188 0.926137i \(-0.376891\pi\)
0.377188 + 0.926137i \(0.376891\pi\)
\(32\) 0 0
\(33\) − 1.28194e7i − 1.88172i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 1.41599e7i − 1.24209i −0.783774 0.621046i \(-0.786708\pi\)
0.783774 0.621046i \(-0.213292\pi\)
\(38\) 0 0
\(39\) 2.54869e6 0.176412
\(40\) 0 0
\(41\) 4.62193e6 0.255444 0.127722 0.991810i \(-0.459233\pi\)
0.127722 + 0.991810i \(0.459233\pi\)
\(42\) 0 0
\(43\) 8.31227e6i 0.370776i 0.982665 + 0.185388i \(0.0593542\pi\)
−0.982665 + 0.185388i \(0.940646\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 2.51923e7i − 0.753056i −0.926405 0.376528i \(-0.877118\pi\)
0.926405 0.376528i \(-0.122882\pi\)
\(48\) 0 0
\(49\) −5.96977e7 −1.47937
\(50\) 0 0
\(51\) −2.95405e7 −0.611437
\(52\) 0 0
\(53\) 3.49333e7i 0.608133i 0.952651 + 0.304066i \(0.0983444\pi\)
−0.952651 + 0.304066i \(0.901656\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 1.81068e8i − 2.27198i
\(58\) 0 0
\(59\) −6.71868e6 −0.0721855 −0.0360928 0.999348i \(-0.511491\pi\)
−0.0360928 + 0.999348i \(0.511491\pi\)
\(60\) 0 0
\(61\) −4.75982e6 −0.0440156 −0.0220078 0.999758i \(-0.507006\pi\)
−0.0220078 + 0.999758i \(0.507006\pi\)
\(62\) 0 0
\(63\) − 5.44299e8i − 4.35316i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 1.38772e8i 0.841330i 0.907216 + 0.420665i \(0.138203\pi\)
−0.907216 + 0.420665i \(0.861797\pi\)
\(68\) 0 0
\(69\) 1.56909e8 0.833352
\(70\) 0 0
\(71\) 3.54359e8 1.65494 0.827468 0.561513i \(-0.189781\pi\)
0.827468 + 0.561513i \(0.189781\pi\)
\(72\) 0 0
\(73\) 2.41596e8i 0.995719i 0.867258 + 0.497859i \(0.165880\pi\)
−0.867258 + 0.497859i \(0.834120\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.71058e8i 1.52709i
\(78\) 0 0
\(79\) −2.61173e8 −0.754409 −0.377204 0.926130i \(-0.623115\pi\)
−0.377204 + 0.926130i \(0.623115\pi\)
\(80\) 0 0
\(81\) 1.50260e9 3.87847
\(82\) 0 0
\(83\) − 6.55898e8i − 1.51700i −0.651674 0.758499i \(-0.725933\pi\)
0.651674 0.758499i \(-0.274067\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) − 7.10654e8i − 1.32991i
\(88\) 0 0
\(89\) 1.00221e9 1.69318 0.846590 0.532245i \(-0.178652\pi\)
0.846590 + 0.532245i \(0.178652\pi\)
\(90\) 0 0
\(91\) −9.36534e7 −0.143165
\(92\) 0 0
\(93\) − 1.05590e9i − 1.46368i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 1.24869e9i − 1.43213i −0.698034 0.716065i \(-0.745942\pi\)
0.698034 0.716065i \(-0.254058\pi\)
\(98\) 0 0
\(99\) −2.56264e9 −2.68120
\(100\) 0 0
\(101\) −9.39590e7 −0.0898446 −0.0449223 0.998990i \(-0.514304\pi\)
−0.0449223 + 0.998990i \(0.514304\pi\)
\(102\) 0 0
\(103\) − 3.58300e8i − 0.313675i −0.987624 0.156837i \(-0.949870\pi\)
0.987624 0.156837i \(-0.0501299\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 1.02197e9i 0.753719i 0.926270 + 0.376860i \(0.122996\pi\)
−0.926270 + 0.376860i \(0.877004\pi\)
\(108\) 0 0
\(109\) 8.94205e8 0.606761 0.303380 0.952869i \(-0.401885\pi\)
0.303380 + 0.952869i \(0.401885\pi\)
\(110\) 0 0
\(111\) −3.85449e9 −2.40998
\(112\) 0 0
\(113\) 1.11113e9i 0.641077i 0.947235 + 0.320539i \(0.103864\pi\)
−0.947235 + 0.320539i \(0.896136\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) − 5.09492e8i − 0.251363i
\(118\) 0 0
\(119\) 1.08548e9 0.496206
\(120\) 0 0
\(121\) −1.40134e8 −0.0594306
\(122\) 0 0
\(123\) − 1.25814e9i − 0.495628i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 5.51824e9i − 1.88228i −0.338020 0.941139i \(-0.609757\pi\)
0.338020 0.941139i \(-0.390243\pi\)
\(128\) 0 0
\(129\) 2.26269e9 0.719402
\(130\) 0 0
\(131\) 4.66527e9 1.38406 0.692031 0.721867i \(-0.256716\pi\)
0.692031 + 0.721867i \(0.256716\pi\)
\(132\) 0 0
\(133\) 6.65344e9i 1.84380i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 6.97325e9i 1.69119i 0.533825 + 0.845595i \(0.320754\pi\)
−0.533825 + 0.845595i \(0.679246\pi\)
\(138\) 0 0
\(139\) −6.27736e9 −1.42630 −0.713149 0.701013i \(-0.752732\pi\)
−0.713149 + 0.701013i \(0.752732\pi\)
\(140\) 0 0
\(141\) −6.85762e9 −1.46113
\(142\) 0 0
\(143\) 4.40935e8i 0.0881784i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.62504e10i 2.87036i
\(148\) 0 0
\(149\) 1.62156e9 0.269523 0.134761 0.990878i \(-0.456973\pi\)
0.134761 + 0.990878i \(0.456973\pi\)
\(150\) 0 0
\(151\) 1.17677e10 1.84202 0.921011 0.389537i \(-0.127365\pi\)
0.921011 + 0.389537i \(0.127365\pi\)
\(152\) 0 0
\(153\) 5.90524e9i 0.871217i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.71288e9i 0.224998i 0.993652 + 0.112499i \(0.0358855\pi\)
−0.993652 + 0.112499i \(0.964114\pi\)
\(158\) 0 0
\(159\) 9.50923e9 1.17994
\(160\) 0 0
\(161\) −5.76574e9 −0.676298
\(162\) 0 0
\(163\) 9.98636e9i 1.10806i 0.832497 + 0.554030i \(0.186911\pi\)
−0.832497 + 0.554030i \(0.813089\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 1.56153e10i − 1.55355i −0.629777 0.776776i \(-0.716854\pi\)
0.629777 0.776776i \(-0.283146\pi\)
\(168\) 0 0
\(169\) 1.05168e10 0.991733
\(170\) 0 0
\(171\) −3.61960e10 −3.23726
\(172\) 0 0
\(173\) 4.61727e8i 0.0391902i 0.999808 + 0.0195951i \(0.00623772\pi\)
−0.999808 + 0.0195951i \(0.993762\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 1.82890e9i 0.140059i
\(178\) 0 0
\(179\) 1.56860e10 1.14202 0.571011 0.820943i \(-0.306551\pi\)
0.571011 + 0.820943i \(0.306551\pi\)
\(180\) 0 0
\(181\) 5.26075e8 0.0364329 0.0182165 0.999834i \(-0.494201\pi\)
0.0182165 + 0.999834i \(0.494201\pi\)
\(182\) 0 0
\(183\) 1.29568e9i 0.0854017i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 5.11063e9i − 0.305624i
\(188\) 0 0
\(189\) −9.45710e10 −5.39113
\(190\) 0 0
\(191\) −2.60641e10 −1.41708 −0.708538 0.705673i \(-0.750645\pi\)
−0.708538 + 0.705673i \(0.750645\pi\)
\(192\) 0 0
\(193\) 4.83514e9i 0.250843i 0.992104 + 0.125421i \(0.0400283\pi\)
−0.992104 + 0.125421i \(0.959972\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 1.09275e10i − 0.516919i −0.966022 0.258460i \(-0.916785\pi\)
0.966022 0.258460i \(-0.0832149\pi\)
\(198\) 0 0
\(199\) 2.24042e10 1.01272 0.506361 0.862322i \(-0.330990\pi\)
0.506361 + 0.862322i \(0.330990\pi\)
\(200\) 0 0
\(201\) 3.77754e10 1.63240
\(202\) 0 0
\(203\) 2.61134e10i 1.07927i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 3.13667e10i − 1.18741i
\(208\) 0 0
\(209\) 3.13255e10 1.13564
\(210\) 0 0
\(211\) 3.17129e10 1.10145 0.550725 0.834687i \(-0.314351\pi\)
0.550725 + 0.834687i \(0.314351\pi\)
\(212\) 0 0
\(213\) − 9.64605e10i − 3.21101i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 3.87995e10i 1.18784i
\(218\) 0 0
\(219\) 6.57651e10 1.93195
\(220\) 0 0
\(221\) 1.01607e9 0.0286522
\(222\) 0 0
\(223\) 1.52662e10i 0.413388i 0.978406 + 0.206694i \(0.0662705\pi\)
−0.978406 + 0.206694i \(0.933730\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 7.53481e9i 0.188346i 0.995556 + 0.0941729i \(0.0300206\pi\)
−0.995556 + 0.0941729i \(0.969979\pi\)
\(228\) 0 0
\(229\) 1.28244e10 0.308160 0.154080 0.988058i \(-0.450759\pi\)
0.154080 + 0.988058i \(0.450759\pi\)
\(230\) 0 0
\(231\) 1.28227e11 2.96296
\(232\) 0 0
\(233\) 2.06272e10i 0.458500i 0.973368 + 0.229250i \(0.0736274\pi\)
−0.973368 + 0.229250i \(0.926373\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 7.10943e10i 1.46375i
\(238\) 0 0
\(239\) −3.96664e10 −0.786380 −0.393190 0.919457i \(-0.628628\pi\)
−0.393190 + 0.919457i \(0.628628\pi\)
\(240\) 0 0
\(241\) −3.75642e9 −0.0717294 −0.0358647 0.999357i \(-0.511419\pi\)
−0.0358647 + 0.999357i \(0.511419\pi\)
\(242\) 0 0
\(243\) − 2.22928e11i − 4.10144i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 6.22797e9i 0.106466i
\(248\) 0 0
\(249\) −1.78543e11 −2.94337
\(250\) 0 0
\(251\) 7.74656e10 1.23191 0.615953 0.787783i \(-0.288771\pi\)
0.615953 + 0.787783i \(0.288771\pi\)
\(252\) 0 0
\(253\) 2.71460e10i 0.416546i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) − 3.31990e10i − 0.474707i −0.971423 0.237354i \(-0.923720\pi\)
0.971423 0.237354i \(-0.0762800\pi\)
\(258\) 0 0
\(259\) 1.41636e11 1.95580
\(260\) 0 0
\(261\) −1.42062e11 −1.89494
\(262\) 0 0
\(263\) 1.00754e11i 1.29856i 0.760549 + 0.649281i \(0.224930\pi\)
−0.760549 + 0.649281i \(0.775070\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 2.72812e11i − 3.28521i
\(268\) 0 0
\(269\) 7.49946e10 0.873262 0.436631 0.899641i \(-0.356172\pi\)
0.436631 + 0.899641i \(0.356172\pi\)
\(270\) 0 0
\(271\) −5.80768e10 −0.654095 −0.327048 0.945008i \(-0.606054\pi\)
−0.327048 + 0.945008i \(0.606054\pi\)
\(272\) 0 0
\(273\) 2.54935e10i 0.277778i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.44538e11i − 1.47511i −0.675289 0.737554i \(-0.735981\pi\)
0.675289 0.737554i \(-0.264019\pi\)
\(278\) 0 0
\(279\) −2.11077e11 −2.08556
\(280\) 0 0
\(281\) 1.73877e11 1.66366 0.831828 0.555034i \(-0.187295\pi\)
0.831828 + 0.555034i \(0.187295\pi\)
\(282\) 0 0
\(283\) − 3.86637e8i − 0.00358315i −0.999998 0.00179157i \(-0.999430\pi\)
0.999998 0.00179157i \(-0.000570276\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 4.62311e10i 0.402222i
\(288\) 0 0
\(289\) 1.06811e11 0.900692
\(290\) 0 0
\(291\) −3.39908e11 −2.77870
\(292\) 0 0
\(293\) 6.25206e10i 0.495586i 0.968813 + 0.247793i \(0.0797053\pi\)
−0.968813 + 0.247793i \(0.920295\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 4.45255e11i 3.32051i
\(298\) 0 0
\(299\) −5.39703e9 −0.0390512
\(300\) 0 0
\(301\) −8.31440e10 −0.583824
\(302\) 0 0
\(303\) 2.55767e10i 0.174322i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.57636e10i 0.101282i 0.998717 + 0.0506410i \(0.0161264\pi\)
−0.998717 + 0.0506410i \(0.983874\pi\)
\(308\) 0 0
\(309\) −9.75334e10 −0.608611
\(310\) 0 0
\(311\) −2.01301e10 −0.122018 −0.0610091 0.998137i \(-0.519432\pi\)
−0.0610091 + 0.998137i \(0.519432\pi\)
\(312\) 0 0
\(313\) − 6.03790e10i − 0.355579i −0.984069 0.177790i \(-0.943105\pi\)
0.984069 0.177790i \(-0.0568946\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 1.24292e11i 0.691313i 0.938361 + 0.345657i \(0.112344\pi\)
−0.938361 + 0.345657i \(0.887656\pi\)
\(318\) 0 0
\(319\) 1.22946e11 0.664748
\(320\) 0 0
\(321\) 2.78191e11 1.46241
\(322\) 0 0
\(323\) − 7.21850e10i − 0.369008i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 2.43412e11i − 1.17727i
\(328\) 0 0
\(329\) 2.51988e11 1.18576
\(330\) 0 0
\(331\) −1.14638e11 −0.524930 −0.262465 0.964941i \(-0.584535\pi\)
−0.262465 + 0.964941i \(0.584535\pi\)
\(332\) 0 0
\(333\) 7.70526e11i 3.43390i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 3.40785e11i 1.43928i 0.694345 + 0.719642i \(0.255694\pi\)
−0.694345 + 0.719642i \(0.744306\pi\)
\(338\) 0 0
\(339\) 3.02461e11 1.24386
\(340\) 0 0
\(341\) 1.82674e11 0.731615
\(342\) 0 0
\(343\) − 1.93491e11i − 0.754809i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 3.38240e11i 1.25240i 0.779664 + 0.626198i \(0.215390\pi\)
−0.779664 + 0.626198i \(0.784610\pi\)
\(348\) 0 0
\(349\) −2.34359e11 −0.845603 −0.422802 0.906222i \(-0.638953\pi\)
−0.422802 + 0.906222i \(0.638953\pi\)
\(350\) 0 0
\(351\) −8.85235e10 −0.311298
\(352\) 0 0
\(353\) 3.56202e11i 1.22098i 0.792022 + 0.610492i \(0.209028\pi\)
−0.792022 + 0.610492i \(0.790972\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 2.95481e11i − 0.962769i
\(358\) 0 0
\(359\) −3.37815e11 −1.07338 −0.536691 0.843779i \(-0.680326\pi\)
−0.536691 + 0.843779i \(0.680326\pi\)
\(360\) 0 0
\(361\) 1.19768e11 0.371157
\(362\) 0 0
\(363\) 3.81461e10i 0.115311i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 1.36383e11i − 0.392431i −0.980561 0.196215i \(-0.937135\pi\)
0.980561 0.196215i \(-0.0628652\pi\)
\(368\) 0 0
\(369\) −2.51506e11 −0.706204
\(370\) 0 0
\(371\) −3.49423e11 −0.957566
\(372\) 0 0
\(373\) 5.08929e11i 1.36134i 0.732589 + 0.680672i \(0.238312\pi\)
−0.732589 + 0.680672i \(0.761688\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 2.44436e10i 0.0623201i
\(378\) 0 0
\(379\) −9.70726e10 −0.241669 −0.120834 0.992673i \(-0.538557\pi\)
−0.120834 + 0.992673i \(0.538557\pi\)
\(380\) 0 0
\(381\) −1.50213e12 −3.65211
\(382\) 0 0
\(383\) − 7.13792e11i − 1.69503i −0.530772 0.847514i \(-0.678098\pi\)
0.530772 0.847514i \(-0.321902\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 4.52320e11i − 1.02505i
\(388\) 0 0
\(389\) −2.48149e11 −0.549465 −0.274732 0.961521i \(-0.588589\pi\)
−0.274732 + 0.961521i \(0.588589\pi\)
\(390\) 0 0
\(391\) 6.25540e10 0.135350
\(392\) 0 0
\(393\) − 1.26994e12i − 2.68544i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 7.04445e11i − 1.42328i −0.702545 0.711640i \(-0.747953\pi\)
0.702545 0.711640i \(-0.252047\pi\)
\(398\) 0 0
\(399\) 1.81114e12 3.57745
\(400\) 0 0
\(401\) −2.82127e11 −0.544873 −0.272437 0.962174i \(-0.587830\pi\)
−0.272437 + 0.962174i \(0.587830\pi\)
\(402\) 0 0
\(403\) 3.63184e10i 0.0685890i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 6.66844e11i − 1.20462i
\(408\) 0 0
\(409\) −9.15011e11 −1.61686 −0.808428 0.588595i \(-0.799681\pi\)
−0.808428 + 0.588595i \(0.799681\pi\)
\(410\) 0 0
\(411\) 1.89820e12 3.28135
\(412\) 0 0
\(413\) − 6.72041e10i − 0.113663i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.70877e12i 2.76739i
\(418\) 0 0
\(419\) −1.19010e12 −1.88635 −0.943174 0.332299i \(-0.892176\pi\)
−0.943174 + 0.332299i \(0.892176\pi\)
\(420\) 0 0
\(421\) 7.44534e11 1.15509 0.577544 0.816360i \(-0.304011\pi\)
0.577544 + 0.816360i \(0.304011\pi\)
\(422\) 0 0
\(423\) 1.37086e12i 2.08191i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 4.76104e10i − 0.0693069i
\(428\) 0 0
\(429\) 1.20027e11 0.171089
\(430\) 0 0
\(431\) −3.47758e11 −0.485433 −0.242716 0.970097i \(-0.578038\pi\)
−0.242716 + 0.970097i \(0.578038\pi\)
\(432\) 0 0
\(433\) − 8.13744e11i − 1.11248i −0.831022 0.556240i \(-0.812244\pi\)
0.831022 0.556240i \(-0.187756\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 3.83423e11i 0.502935i
\(438\) 0 0
\(439\) 1.09730e12 1.41005 0.705023 0.709184i \(-0.250937\pi\)
0.705023 + 0.709184i \(0.250937\pi\)
\(440\) 0 0
\(441\) 3.24851e12 4.08987
\(442\) 0 0
\(443\) 6.59236e10i 0.0813251i 0.999173 + 0.0406625i \(0.0129469\pi\)
−0.999173 + 0.0406625i \(0.987053\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 4.41407e11i − 0.522945i
\(448\) 0 0
\(449\) −2.15220e11 −0.249905 −0.124952 0.992163i \(-0.539878\pi\)
−0.124952 + 0.992163i \(0.539878\pi\)
\(450\) 0 0
\(451\) 2.17664e11 0.247737
\(452\) 0 0
\(453\) − 3.20329e12i − 3.57400i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 7.36259e11i − 0.789601i −0.918767 0.394801i \(-0.870814\pi\)
0.918767 0.394801i \(-0.129186\pi\)
\(458\) 0 0
\(459\) 1.02603e12 1.07895
\(460\) 0 0
\(461\) −1.23730e12 −1.27591 −0.637957 0.770072i \(-0.720220\pi\)
−0.637957 + 0.770072i \(0.720220\pi\)
\(462\) 0 0
\(463\) − 7.36969e11i − 0.745306i −0.927971 0.372653i \(-0.878448\pi\)
0.927971 0.372653i \(-0.121552\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 1.22439e12i 1.19123i 0.803271 + 0.595614i \(0.203091\pi\)
−0.803271 + 0.595614i \(0.796909\pi\)
\(468\) 0 0
\(469\) −1.38808e12 −1.32476
\(470\) 0 0
\(471\) 4.66265e11 0.436555
\(472\) 0 0
\(473\) 3.91455e11i 0.359590i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 1.90093e12i − 1.68125i
\(478\) 0 0
\(479\) −1.11970e12 −0.971833 −0.485916 0.874005i \(-0.661514\pi\)
−0.485916 + 0.874005i \(0.661514\pi\)
\(480\) 0 0
\(481\) 1.32579e11 0.112933
\(482\) 0 0
\(483\) 1.56950e12i 1.31220i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 2.25337e11i 0.181532i 0.995872 + 0.0907659i \(0.0289315\pi\)
−0.995872 + 0.0907659i \(0.971068\pi\)
\(488\) 0 0
\(489\) 2.71840e12 2.14992
\(490\) 0 0
\(491\) −1.60716e11 −0.124794 −0.0623970 0.998051i \(-0.519874\pi\)
−0.0623970 + 0.998051i \(0.519874\pi\)
\(492\) 0 0
\(493\) − 2.83312e11i − 0.216000i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 3.54450e12i 2.60586i
\(498\) 0 0
\(499\) −8.05187e11 −0.581359 −0.290680 0.956820i \(-0.593881\pi\)
−0.290680 + 0.956820i \(0.593881\pi\)
\(500\) 0 0
\(501\) −4.25065e12 −3.01430
\(502\) 0 0
\(503\) − 1.68493e12i − 1.17361i −0.809727 0.586807i \(-0.800385\pi\)
0.809727 0.586807i \(-0.199615\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 2.86280e12i − 1.92422i
\(508\) 0 0
\(509\) −1.50208e12 −0.991890 −0.495945 0.868354i \(-0.665178\pi\)
−0.495945 + 0.868354i \(0.665178\pi\)
\(510\) 0 0
\(511\) −2.41658e12 −1.56786
\(512\) 0 0
\(513\) 6.28900e12i 4.00916i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 1.18640e12i − 0.730336i
\(518\) 0 0
\(519\) 1.25687e11 0.0760393
\(520\) 0 0
\(521\) 9.24724e11 0.549848 0.274924 0.961466i \(-0.411347\pi\)
0.274924 + 0.961466i \(0.411347\pi\)
\(522\) 0 0
\(523\) − 3.81782e10i − 0.0223130i −0.999938 0.0111565i \(-0.996449\pi\)
0.999938 0.0111565i \(-0.00355130\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 4.20947e11i − 0.237727i
\(528\) 0 0
\(529\) 1.46889e12 0.815526
\(530\) 0 0
\(531\) 3.65603e11 0.199565
\(532\) 0 0
\(533\) 4.32748e10i 0.0232254i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 4.26991e12i − 2.21582i
\(538\) 0 0
\(539\) −2.81139e12 −1.43473
\(540\) 0 0
\(541\) −1.05812e12 −0.531064 −0.265532 0.964102i \(-0.585548\pi\)
−0.265532 + 0.964102i \(0.585548\pi\)
\(542\) 0 0
\(543\) − 1.43203e11i − 0.0706894i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.05156e12i 0.502215i 0.967959 + 0.251108i \(0.0807948\pi\)
−0.967959 + 0.251108i \(0.919205\pi\)
\(548\) 0 0
\(549\) 2.59010e11 0.121686
\(550\) 0 0
\(551\) 1.73655e12 0.802611
\(552\) 0 0
\(553\) − 2.61240e12i − 1.18789i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 3.92684e11i 0.172860i 0.996258 + 0.0864301i \(0.0275459\pi\)
−0.996258 + 0.0864301i \(0.972454\pi\)
\(558\) 0 0
\(559\) −7.78272e10 −0.0337115
\(560\) 0 0
\(561\) −1.39117e12 −0.592990
\(562\) 0 0
\(563\) 2.36287e12i 0.991180i 0.868557 + 0.495590i \(0.165048\pi\)
−0.868557 + 0.495590i \(0.834952\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.50299e13i 6.10705i
\(568\) 0 0
\(569\) 2.50603e12 1.00226 0.501130 0.865372i \(-0.332918\pi\)
0.501130 + 0.865372i \(0.332918\pi\)
\(570\) 0 0
\(571\) −2.99195e12 −1.17785 −0.588927 0.808187i \(-0.700449\pi\)
−0.588927 + 0.808187i \(0.700449\pi\)
\(572\) 0 0
\(573\) 7.09494e12i 2.74950i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 3.06622e11i 0.115163i 0.998341 + 0.0575815i \(0.0183389\pi\)
−0.998341 + 0.0575815i \(0.981661\pi\)
\(578\) 0 0
\(579\) 1.31618e12 0.486700
\(580\) 0 0
\(581\) 6.56067e12 2.38867
\(582\) 0 0
\(583\) 1.64514e12i 0.589785i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 2.80823e12i 0.976249i 0.872774 + 0.488125i \(0.162319\pi\)
−0.872774 + 0.488125i \(0.837681\pi\)
\(588\) 0 0
\(589\) 2.58018e12 0.883346
\(590\) 0 0
\(591\) −2.97459e12 −1.00296
\(592\) 0 0
\(593\) − 1.34954e12i − 0.448166i −0.974570 0.224083i \(-0.928061\pi\)
0.974570 0.224083i \(-0.0719387\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 6.09867e12i − 1.96494i
\(598\) 0 0
\(599\) −3.38598e12 −1.07464 −0.537321 0.843378i \(-0.680564\pi\)
−0.537321 + 0.843378i \(0.680564\pi\)
\(600\) 0 0
\(601\) 4.62837e12 1.44708 0.723540 0.690282i \(-0.242514\pi\)
0.723540 + 0.690282i \(0.242514\pi\)
\(602\) 0 0
\(603\) − 7.55142e12i − 2.32595i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 4.78828e12i − 1.43163i −0.698291 0.715814i \(-0.746056\pi\)
0.698291 0.715814i \(-0.253944\pi\)
\(608\) 0 0
\(609\) 7.10837e12 2.09407
\(610\) 0 0
\(611\) 2.35874e11 0.0684690
\(612\) 0 0
\(613\) − 1.02749e12i − 0.293905i −0.989144 0.146953i \(-0.953054\pi\)
0.989144 0.146953i \(-0.0469465\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 4.53079e12i 1.25861i 0.777159 + 0.629304i \(0.216660\pi\)
−0.777159 + 0.629304i \(0.783340\pi\)
\(618\) 0 0
\(619\) 3.42902e12 0.938777 0.469388 0.882992i \(-0.344474\pi\)
0.469388 + 0.882992i \(0.344474\pi\)
\(620\) 0 0
\(621\) −5.44992e12 −1.47054
\(622\) 0 0
\(623\) 1.00247e13i 2.66608i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 8.52714e12i − 2.20343i
\(628\) 0 0
\(629\) −1.53664e12 −0.391422
\(630\) 0 0
\(631\) −3.52587e12 −0.885389 −0.442695 0.896672i \(-0.645977\pi\)
−0.442695 + 0.896672i \(0.645977\pi\)
\(632\) 0 0
\(633\) − 8.63260e12i − 2.13710i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 5.58946e11i − 0.134506i
\(638\) 0 0
\(639\) −1.92828e13 −4.57526
\(640\) 0 0
\(641\) −7.00458e12 −1.63878 −0.819390 0.573236i \(-0.805688\pi\)
−0.819390 + 0.573236i \(0.805688\pi\)
\(642\) 0 0
\(643\) − 4.31047e12i − 0.994432i −0.867627 0.497216i \(-0.834356\pi\)
0.867627 0.497216i \(-0.165644\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 2.84949e12i 0.639290i 0.947537 + 0.319645i \(0.103564\pi\)
−0.947537 + 0.319645i \(0.896436\pi\)
\(648\) 0 0
\(649\) −3.16407e11 −0.0700077
\(650\) 0 0
\(651\) 1.05617e13 2.30472
\(652\) 0 0
\(653\) 1.93213e12i 0.415841i 0.978146 + 0.207920i \(0.0666695\pi\)
−0.978146 + 0.207920i \(0.933331\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 1.31467e13i − 2.75278i
\(658\) 0 0
\(659\) −8.58675e12 −1.77355 −0.886777 0.462198i \(-0.847061\pi\)
−0.886777 + 0.462198i \(0.847061\pi\)
\(660\) 0 0
\(661\) 1.11478e12 0.227133 0.113567 0.993530i \(-0.463772\pi\)
0.113567 + 0.993530i \(0.463772\pi\)
\(662\) 0 0
\(663\) − 2.76586e11i − 0.0555928i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.50486e12i 0.294394i
\(668\) 0 0
\(669\) 4.15562e12 0.802081
\(670\) 0 0
\(671\) −2.24157e11 −0.0426876
\(672\) 0 0
\(673\) 7.57716e12i 1.42377i 0.702298 + 0.711883i \(0.252157\pi\)
−0.702298 + 0.711883i \(0.747843\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 5.08558e12i 0.930447i 0.885193 + 0.465223i \(0.154026\pi\)
−0.885193 + 0.465223i \(0.845974\pi\)
\(678\) 0 0
\(679\) 1.24901e13 2.25503
\(680\) 0 0
\(681\) 2.05106e12 0.365440
\(682\) 0 0
\(683\) − 1.78278e11i − 0.0313476i −0.999877 0.0156738i \(-0.995011\pi\)
0.999877 0.0156738i \(-0.00498933\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 3.49093e12i − 0.597911i
\(688\) 0 0
\(689\) −3.27078e11 −0.0552924
\(690\) 0 0
\(691\) −8.66612e11 −0.144602 −0.0723008 0.997383i \(-0.523034\pi\)
−0.0723008 + 0.997383i \(0.523034\pi\)
\(692\) 0 0
\(693\) − 2.56330e13i − 4.22182i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) − 5.01574e11i − 0.0804985i
\(698\) 0 0
\(699\) 5.61497e12 0.889610
\(700\) 0 0
\(701\) 5.18099e12 0.810367 0.405183 0.914235i \(-0.367208\pi\)
0.405183 + 0.914235i \(0.367208\pi\)
\(702\) 0 0
\(703\) − 9.41882e12i − 1.45445i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 9.39831e11i − 0.141469i
\(708\) 0 0
\(709\) 1.22351e13 1.81844 0.909218 0.416320i \(-0.136681\pi\)
0.909218 + 0.416320i \(0.136681\pi\)
\(710\) 0 0
\(711\) 1.42120e13 2.08565
\(712\) 0 0
\(713\) 2.23593e12i 0.324008i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 1.07976e13i 1.52578i
\(718\) 0 0
\(719\) 1.11881e13 1.56127 0.780634 0.624989i \(-0.214896\pi\)
0.780634 + 0.624989i \(0.214896\pi\)
\(720\) 0 0
\(721\) 3.58392e12 0.493913
\(722\) 0 0
\(723\) 1.02254e12i 0.139174i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 7.94265e12i − 1.05453i −0.849699 0.527267i \(-0.823217\pi\)
0.849699 0.527267i \(-0.176783\pi\)
\(728\) 0 0
\(729\) −3.11078e13 −4.07940
\(730\) 0 0
\(731\) 9.02052e11 0.116843
\(732\) 0 0
\(733\) 1.09012e13i 1.39478i 0.716690 + 0.697392i \(0.245656\pi\)
−0.716690 + 0.697392i \(0.754344\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 6.53530e12i 0.815947i
\(738\) 0 0
\(739\) −1.54854e12 −0.190996 −0.0954978 0.995430i \(-0.530444\pi\)
−0.0954978 + 0.995430i \(0.530444\pi\)
\(740\) 0 0
\(741\) 1.69532e12 0.206572
\(742\) 0 0
\(743\) 5.96254e12i 0.717764i 0.933383 + 0.358882i \(0.116842\pi\)
−0.933383 + 0.358882i \(0.883158\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 3.56913e13i 4.19392i
\(748\) 0 0
\(749\) −1.02223e13 −1.18681
\(750\) 0 0
\(751\) 9.92051e12 1.13803 0.569016 0.822326i \(-0.307324\pi\)
0.569016 + 0.822326i \(0.307324\pi\)
\(752\) 0 0
\(753\) − 2.10870e13i − 2.39022i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 1.23733e13i − 1.36947i −0.728791 0.684737i \(-0.759917\pi\)
0.728791 0.684737i \(-0.240083\pi\)
\(758\) 0 0
\(759\) 7.38944e12 0.808209
\(760\) 0 0
\(761\) −8.52217e12 −0.921127 −0.460563 0.887627i \(-0.652353\pi\)
−0.460563 + 0.887627i \(0.652353\pi\)
\(762\) 0 0
\(763\) 8.94434e12i 0.955406i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) − 6.29066e10i − 0.00656322i
\(768\) 0 0
\(769\) −2.99595e12 −0.308934 −0.154467 0.987998i \(-0.549366\pi\)
−0.154467 + 0.987998i \(0.549366\pi\)
\(770\) 0 0
\(771\) −9.03713e12 −0.921056
\(772\) 0 0
\(773\) 6.44491e11i 0.0649245i 0.999473 + 0.0324623i \(0.0103349\pi\)
−0.999473 + 0.0324623i \(0.989665\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) − 3.85548e13i − 3.79476i
\(778\) 0 0
\(779\) 3.07438e12 0.299116
\(780\) 0 0
\(781\) 1.66881e13 1.60501
\(782\) 0 0
\(783\) 2.46831e13i 2.34677i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 8.05096e12i − 0.748103i −0.927408 0.374052i \(-0.877968\pi\)
0.927408 0.374052i \(-0.122032\pi\)
\(788\) 0 0
\(789\) 2.74264e13 2.51955
\(790\) 0 0
\(791\) −1.11141e13 −1.00944
\(792\) 0 0
\(793\) − 4.45659e10i − 0.00400196i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 1.11381e13i 0.977799i 0.872340 + 0.488899i \(0.162601\pi\)
−0.872340 + 0.488899i \(0.837399\pi\)
\(798\) 0 0
\(799\) −2.73388e12 −0.237312
\(800\) 0 0
\(801\) −5.45361e13 −4.68099
\(802\) 0 0
\(803\) 1.13776e13i 0.965678i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 2.04144e13i − 1.69436i
\(808\) 0 0
\(809\) −1.69716e13 −1.39301 −0.696506 0.717550i \(-0.745263\pi\)
−0.696506 + 0.717550i \(0.745263\pi\)
\(810\) 0 0
\(811\) −1.17583e13 −0.954444 −0.477222 0.878783i \(-0.658356\pi\)
−0.477222 + 0.878783i \(0.658356\pi\)
\(812\) 0 0
\(813\) 1.58092e13i 1.26912i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 5.52910e12i 0.434165i
\(818\) 0 0
\(819\) 5.09623e12 0.395796
\(820\) 0 0
\(821\) 1.99657e13 1.53370 0.766851 0.641825i \(-0.221823\pi\)
0.766851 + 0.641825i \(0.221823\pi\)
\(822\) 0 0
\(823\) 5.52813e11i 0.0420029i 0.999779 + 0.0210014i \(0.00668546\pi\)
−0.999779 + 0.0210014i \(0.993315\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.57036e13i 1.16741i 0.811965 + 0.583707i \(0.198398\pi\)
−0.811965 + 0.583707i \(0.801602\pi\)
\(828\) 0 0
\(829\) −1.05928e13 −0.778958 −0.389479 0.921035i \(-0.627345\pi\)
−0.389479 + 0.921035i \(0.627345\pi\)
\(830\) 0 0
\(831\) −3.93449e13 −2.86209
\(832\) 0 0
\(833\) 6.47843e12i 0.466195i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.66743e13i 2.58284i
\(838\) 0 0
\(839\) 6.55506e11 0.0456718 0.0228359 0.999739i \(-0.492730\pi\)
0.0228359 + 0.999739i \(0.492730\pi\)
\(840\) 0 0
\(841\) −7.69153e12 −0.530189
\(842\) 0 0
\(843\) − 4.73312e13i − 3.22793i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 1.40170e12i − 0.0935794i
\(848\) 0 0
\(849\) −1.05247e11 −0.00695224
\(850\) 0 0
\(851\) 8.16215e12 0.533484
\(852\) 0 0
\(853\) 2.08805e13i 1.35043i 0.737622 + 0.675213i \(0.235949\pi\)
−0.737622 + 0.675213i \(0.764051\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 2.19074e13i 1.38732i 0.720303 + 0.693660i \(0.244003\pi\)
−0.720303 + 0.693660i \(0.755997\pi\)
\(858\) 0 0
\(859\) −1.47238e13 −0.922680 −0.461340 0.887223i \(-0.652631\pi\)
−0.461340 + 0.887223i \(0.652631\pi\)
\(860\) 0 0
\(861\) 1.25846e13 0.780416
\(862\) 0 0
\(863\) − 2.74805e12i − 0.168646i −0.996438 0.0843230i \(-0.973127\pi\)
0.996438 0.0843230i \(-0.0268727\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 2.90752e13i − 1.74758i
\(868\) 0 0
\(869\) −1.22996e13 −0.731648
\(870\) 0 0
\(871\) −1.29932e12 −0.0764950
\(872\) 0 0
\(873\) 6.79486e13i 3.95928i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) − 6.68274e12i − 0.381466i −0.981642 0.190733i \(-0.938913\pi\)
0.981642 0.190733i \(-0.0610865\pi\)
\(878\) 0 0
\(879\) 1.70188e13 0.961567
\(880\) 0 0
\(881\) 1.42384e13 0.796289 0.398144 0.917323i \(-0.369654\pi\)
0.398144 + 0.917323i \(0.369654\pi\)
\(882\) 0 0
\(883\) − 2.46098e13i − 1.36234i −0.732125 0.681170i \(-0.761471\pi\)
0.732125 0.681170i \(-0.238529\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 9.12625e12i 0.495035i 0.968883 + 0.247518i \(0.0796148\pi\)
−0.968883 + 0.247518i \(0.920385\pi\)
\(888\) 0 0
\(889\) 5.51966e13 2.96384
\(890\) 0 0
\(891\) 7.07630e13 3.76146
\(892\) 0 0
\(893\) − 1.67572e13i − 0.881802i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 1.46913e12i 0.0757696i
\(898\) 0 0
\(899\) 1.01267e13 0.517069
\(900\) 0 0
\(901\) 3.79098e12 0.191642
\(902\) 0 0
\(903\) 2.26327e13i 1.13277i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 3.32446e13i − 1.63113i −0.578668 0.815564i \(-0.696427\pi\)
0.578668 0.815564i \(-0.303573\pi\)
\(908\) 0 0
\(909\) 5.11286e12 0.248386
\(910\) 0 0
\(911\) −1.42660e13 −0.686232 −0.343116 0.939293i \(-0.611482\pi\)
−0.343116 + 0.939293i \(0.611482\pi\)
\(912\) 0 0
\(913\) − 3.08887e13i − 1.47123i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 4.66647e13i 2.17935i
\(918\) 0 0
\(919\) −3.23171e13 −1.49456 −0.747278 0.664512i \(-0.768640\pi\)
−0.747278 + 0.664512i \(0.768640\pi\)
\(920\) 0 0
\(921\) 4.29102e12 0.196514
\(922\) 0 0
\(923\) 3.31784e12i 0.150469i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1.94972e13i 0.867190i
\(928\) 0 0
\(929\) 3.49040e12 0.153746 0.0768732 0.997041i \(-0.475506\pi\)
0.0768732 + 0.997041i \(0.475506\pi\)
\(930\) 0 0
\(931\) −3.97093e13 −1.73228
\(932\) 0 0
\(933\) 5.47964e12i 0.236747i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.48063e13i 0.627507i 0.949504 + 0.313754i \(0.101587\pi\)
−0.949504 + 0.313754i \(0.898413\pi\)
\(938\) 0 0
\(939\) −1.64358e13 −0.689916
\(940\) 0 0
\(941\) 3.90935e13 1.62537 0.812684 0.582705i \(-0.198006\pi\)
0.812684 + 0.582705i \(0.198006\pi\)
\(942\) 0 0
\(943\) 2.66420e12i 0.109714i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 2.30013e13i − 0.929346i −0.885482 0.464673i \(-0.846172\pi\)
0.885482 0.464673i \(-0.153828\pi\)
\(948\) 0 0
\(949\) −2.26205e12 −0.0905323
\(950\) 0 0
\(951\) 3.38335e13 1.34133
\(952\) 0 0
\(953\) − 3.32845e13i − 1.30714i −0.756864 0.653572i \(-0.773270\pi\)
0.756864 0.653572i \(-0.226730\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 3.34673e13i − 1.28978i
\(958\) 0 0
\(959\) −6.97504e13 −2.66295
\(960\) 0 0
\(961\) −1.13933e13 −0.430918
\(962\) 0 0
\(963\) − 5.56112e13i − 2.08374i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 5.77848e12i 0.212518i 0.994339 + 0.106259i \(0.0338872\pi\)
−0.994339 + 0.106259i \(0.966113\pi\)
\(968\) 0 0
\(969\) −1.96495e13 −0.715971
\(970\) 0 0
\(971\) −4.46200e13 −1.61080 −0.805402 0.592728i \(-0.798051\pi\)
−0.805402 + 0.592728i \(0.798051\pi\)
\(972\) 0 0
\(973\) − 6.27897e13i − 2.24585i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 3.90697e13i − 1.37188i −0.727660 0.685938i \(-0.759392\pi\)
0.727660 0.685938i \(-0.240608\pi\)
\(978\) 0 0
\(979\) 4.71977e13 1.64210
\(980\) 0 0
\(981\) −4.86589e13 −1.67746
\(982\) 0 0
\(983\) 1.96137e12i 0.0669990i 0.999439 + 0.0334995i \(0.0106652\pi\)
−0.999439 + 0.0334995i \(0.989335\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 6.85938e13i − 2.30069i
\(988\) 0 0
\(989\) −4.79141e12 −0.159250
\(990\) 0 0
\(991\) −4.33785e13 −1.42871 −0.714354 0.699785i \(-0.753279\pi\)
−0.714354 + 0.699785i \(0.753279\pi\)
\(992\) 0 0
\(993\) 3.12057e13i 1.01850i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 2.79360e12i 0.0895440i 0.998997 + 0.0447720i \(0.0142561\pi\)
−0.998997 + 0.0447720i \(0.985744\pi\)
\(998\) 0 0
\(999\) 1.33878e14 4.25269
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 100.10.c.c.49.1 4
4.3 odd 2 400.10.c.l.49.4 4
5.2 odd 4 20.10.a.b.1.1 2
5.3 odd 4 100.10.a.c.1.2 2
5.4 even 2 inner 100.10.c.c.49.4 4
15.2 even 4 180.10.a.e.1.1 2
20.3 even 4 400.10.a.l.1.1 2
20.7 even 4 80.10.a.j.1.2 2
20.19 odd 2 400.10.c.l.49.1 4
40.27 even 4 320.10.a.l.1.1 2
40.37 odd 4 320.10.a.t.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.10.a.b.1.1 2 5.2 odd 4
80.10.a.j.1.2 2 20.7 even 4
100.10.a.c.1.2 2 5.3 odd 4
100.10.c.c.49.1 4 1.1 even 1 trivial
100.10.c.c.49.4 4 5.4 even 2 inner
180.10.a.e.1.1 2 15.2 even 4
320.10.a.l.1.1 2 40.27 even 4
320.10.a.t.1.2 2 40.37 odd 4
400.10.a.l.1.1 2 20.3 even 4
400.10.c.l.49.1 4 20.19 odd 2
400.10.c.l.49.4 4 4.3 odd 2