Properties

Label 225.12.a.f.1.1
Level $225$
Weight $12$
Character 225.1
Self dual yes
Analytic conductor $172.877$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,12,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 225.a (trivial)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(172.877215626\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+78.0000 q^{2} +4036.00 q^{4} +27760.0 q^{7} +155064. q^{8} +O(q^{10})\) \(q+78.0000 q^{2} +4036.00 q^{4} +27760.0 q^{7} +155064. q^{8} -637836. q^{11} -766214. q^{13} +2.16528e6 q^{14} +3.82926e6 q^{16} +3.08435e6 q^{17} -1.95114e7 q^{19} -4.97512e7 q^{22} +1.53124e7 q^{23} -5.97647e7 q^{26} +1.12039e8 q^{28} -1.07513e7 q^{29} -5.09374e7 q^{31} -1.88885e7 q^{32} +2.40580e8 q^{34} -6.64741e8 q^{37} -1.52189e9 q^{38} -8.98833e8 q^{41} +9.57947e8 q^{43} -2.57431e9 q^{44} +1.19436e9 q^{46} -1.55574e9 q^{47} -1.20671e9 q^{49} -3.09244e9 q^{52} +3.79242e9 q^{53} +4.30458e9 q^{56} -8.38598e8 q^{58} -5.55307e8 q^{59} +4.95042e9 q^{61} -3.97312e9 q^{62} -9.31563e9 q^{64} -5.29240e9 q^{67} +1.24485e10 q^{68} +1.48311e10 q^{71} -1.39710e10 q^{73} -5.18498e10 q^{74} -7.87480e10 q^{76} -1.77063e10 q^{77} +3.72054e9 q^{79} -7.01090e10 q^{82} +8.76845e9 q^{83} +7.47199e10 q^{86} -9.89054e10 q^{88} +2.54728e10 q^{89} -2.12701e10 q^{91} +6.18007e10 q^{92} -1.21348e11 q^{94} +3.90925e10 q^{97} -9.41233e10 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 78.0000 1.72357 0.861786 0.507271i \(-0.169346\pi\)
0.861786 + 0.507271i \(0.169346\pi\)
\(3\) 0 0
\(4\) 4036.00 1.97070
\(5\) 0 0
\(6\) 0 0
\(7\) 27760.0 0.624281 0.312141 0.950036i \(-0.398954\pi\)
0.312141 + 0.950036i \(0.398954\pi\)
\(8\) 155064. 1.67308
\(9\) 0 0
\(10\) 0 0
\(11\) −637836. −1.19412 −0.597062 0.802195i \(-0.703665\pi\)
−0.597062 + 0.802195i \(0.703665\pi\)
\(12\) 0 0
\(13\) −766214. −0.572350 −0.286175 0.958177i \(-0.592384\pi\)
−0.286175 + 0.958177i \(0.592384\pi\)
\(14\) 2.16528e6 1.07599
\(15\) 0 0
\(16\) 3.82926e6 0.912968
\(17\) 3.08435e6 0.526860 0.263430 0.964679i \(-0.415146\pi\)
0.263430 + 0.964679i \(0.415146\pi\)
\(18\) 0 0
\(19\) −1.95114e7 −1.80777 −0.903886 0.427773i \(-0.859298\pi\)
−0.903886 + 0.427773i \(0.859298\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −4.97512e7 −2.05816
\(23\) 1.53124e7 0.496066 0.248033 0.968752i \(-0.420216\pi\)
0.248033 + 0.968752i \(0.420216\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −5.97647e7 −0.986487
\(27\) 0 0
\(28\) 1.12039e8 1.23027
\(29\) −1.07513e7 −0.0973353 −0.0486677 0.998815i \(-0.515498\pi\)
−0.0486677 + 0.998815i \(0.515498\pi\)
\(30\) 0 0
\(31\) −5.09374e7 −0.319556 −0.159778 0.987153i \(-0.551078\pi\)
−0.159778 + 0.987153i \(0.551078\pi\)
\(32\) −1.88885e7 −0.0995112
\(33\) 0 0
\(34\) 2.40580e8 0.908081
\(35\) 0 0
\(36\) 0 0
\(37\) −6.64741e8 −1.57595 −0.787976 0.615706i \(-0.788871\pi\)
−0.787976 + 0.615706i \(0.788871\pi\)
\(38\) −1.52189e9 −3.11583
\(39\) 0 0
\(40\) 0 0
\(41\) −8.98833e8 −1.21162 −0.605812 0.795608i \(-0.707152\pi\)
−0.605812 + 0.795608i \(0.707152\pi\)
\(42\) 0 0
\(43\) 9.57947e8 0.993722 0.496861 0.867830i \(-0.334486\pi\)
0.496861 + 0.867830i \(0.334486\pi\)
\(44\) −2.57431e9 −2.35326
\(45\) 0 0
\(46\) 1.19436e9 0.855005
\(47\) −1.55574e9 −0.989462 −0.494731 0.869046i \(-0.664733\pi\)
−0.494731 + 0.869046i \(0.664733\pi\)
\(48\) 0 0
\(49\) −1.20671e9 −0.610273
\(50\) 0 0
\(51\) 0 0
\(52\) −3.09244e9 −1.12793
\(53\) 3.79242e9 1.24566 0.622829 0.782358i \(-0.285983\pi\)
0.622829 + 0.782358i \(0.285983\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 4.30458e9 1.04447
\(57\) 0 0
\(58\) −8.38598e8 −0.167765
\(59\) −5.55307e8 −0.101122 −0.0505612 0.998721i \(-0.516101\pi\)
−0.0505612 + 0.998721i \(0.516101\pi\)
\(60\) 0 0
\(61\) 4.95042e9 0.750461 0.375230 0.926932i \(-0.377564\pi\)
0.375230 + 0.926932i \(0.377564\pi\)
\(62\) −3.97312e9 −0.550779
\(63\) 0 0
\(64\) −9.31563e9 −1.08448
\(65\) 0 0
\(66\) 0 0
\(67\) −5.29240e9 −0.478896 −0.239448 0.970909i \(-0.576966\pi\)
−0.239448 + 0.970909i \(0.576966\pi\)
\(68\) 1.24485e10 1.03828
\(69\) 0 0
\(70\) 0 0
\(71\) 1.48311e10 0.975556 0.487778 0.872968i \(-0.337808\pi\)
0.487778 + 0.872968i \(0.337808\pi\)
\(72\) 0 0
\(73\) −1.39710e10 −0.788773 −0.394386 0.918945i \(-0.629043\pi\)
−0.394386 + 0.918945i \(0.629043\pi\)
\(74\) −5.18498e10 −2.71627
\(75\) 0 0
\(76\) −7.87480e10 −3.56258
\(77\) −1.77063e10 −0.745469
\(78\) 0 0
\(79\) 3.72054e9 0.136037 0.0680185 0.997684i \(-0.478332\pi\)
0.0680185 + 0.997684i \(0.478332\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.01090e10 −2.08832
\(83\) 8.76845e9 0.244339 0.122170 0.992509i \(-0.461015\pi\)
0.122170 + 0.992509i \(0.461015\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.47199e10 1.71275
\(87\) 0 0
\(88\) −9.89054e10 −1.99786
\(89\) 2.54728e10 0.483539 0.241769 0.970334i \(-0.422272\pi\)
0.241769 + 0.970334i \(0.422272\pi\)
\(90\) 0 0
\(91\) −2.12701e10 −0.357307
\(92\) 6.18007e10 0.977598
\(93\) 0 0
\(94\) −1.21348e11 −1.70541
\(95\) 0 0
\(96\) 0 0
\(97\) 3.90925e10 0.462220 0.231110 0.972928i \(-0.425764\pi\)
0.231110 + 0.972928i \(0.425764\pi\)
\(98\) −9.41233e10 −1.05185
\(99\) 0 0
\(100\) 0 0
\(101\) −9.31078e9 −0.0881492 −0.0440746 0.999028i \(-0.514034\pi\)
−0.0440746 + 0.999028i \(0.514034\pi\)
\(102\) 0 0
\(103\) −4.85751e10 −0.412865 −0.206433 0.978461i \(-0.566185\pi\)
−0.206433 + 0.978461i \(0.566185\pi\)
\(104\) −1.18812e11 −0.957586
\(105\) 0 0
\(106\) 2.95809e11 2.14698
\(107\) 2.25596e11 1.55496 0.777482 0.628906i \(-0.216497\pi\)
0.777482 + 0.628906i \(0.216497\pi\)
\(108\) 0 0
\(109\) −6.94512e10 −0.432348 −0.216174 0.976355i \(-0.569358\pi\)
−0.216174 + 0.976355i \(0.569358\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.06300e11 0.569949
\(113\) −3.59665e11 −1.83640 −0.918198 0.396122i \(-0.870356\pi\)
−0.918198 + 0.396122i \(0.870356\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −4.33921e10 −0.191819
\(117\) 0 0
\(118\) −4.33139e10 −0.174292
\(119\) 8.56217e10 0.328909
\(120\) 0 0
\(121\) 1.21523e11 0.425931
\(122\) 3.86133e11 1.29347
\(123\) 0 0
\(124\) −2.05583e11 −0.629751
\(125\) 0 0
\(126\) 0 0
\(127\) −2.50273e11 −0.672192 −0.336096 0.941828i \(-0.609107\pi\)
−0.336096 + 0.941828i \(0.609107\pi\)
\(128\) −6.87936e11 −1.76967
\(129\) 0 0
\(130\) 0 0
\(131\) 9.78918e10 0.221694 0.110847 0.993837i \(-0.464644\pi\)
0.110847 + 0.993837i \(0.464644\pi\)
\(132\) 0 0
\(133\) −5.41637e11 −1.12856
\(134\) −4.12807e11 −0.825412
\(135\) 0 0
\(136\) 4.78272e11 0.881477
\(137\) −1.55015e11 −0.274416 −0.137208 0.990542i \(-0.543813\pi\)
−0.137208 + 0.990542i \(0.543813\pi\)
\(138\) 0 0
\(139\) −1.12627e12 −1.84102 −0.920512 0.390715i \(-0.872228\pi\)
−0.920512 + 0.390715i \(0.872228\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 1.15682e12 1.68144
\(143\) 4.88719e11 0.683457
\(144\) 0 0
\(145\) 0 0
\(146\) −1.08974e12 −1.35951
\(147\) 0 0
\(148\) −2.68289e12 −3.10573
\(149\) 1.38458e12 1.54452 0.772261 0.635306i \(-0.219126\pi\)
0.772261 + 0.635306i \(0.219126\pi\)
\(150\) 0 0
\(151\) −6.98601e11 −0.724196 −0.362098 0.932140i \(-0.617939\pi\)
−0.362098 + 0.932140i \(0.617939\pi\)
\(152\) −3.02552e12 −3.02454
\(153\) 0 0
\(154\) −1.38109e12 −1.28487
\(155\) 0 0
\(156\) 0 0
\(157\) −2.13127e12 −1.78316 −0.891580 0.452863i \(-0.850403\pi\)
−0.891580 + 0.452863i \(0.850403\pi\)
\(158\) 2.90202e11 0.234470
\(159\) 0 0
\(160\) 0 0
\(161\) 4.25071e11 0.309684
\(162\) 0 0
\(163\) 1.63564e11 0.111342 0.0556708 0.998449i \(-0.482270\pi\)
0.0556708 + 0.998449i \(0.482270\pi\)
\(164\) −3.62769e12 −2.38775
\(165\) 0 0
\(166\) 6.83939e11 0.421137
\(167\) −8.80943e9 −0.00524816 −0.00262408 0.999997i \(-0.500835\pi\)
−0.00262408 + 0.999997i \(0.500835\pi\)
\(168\) 0 0
\(169\) −1.20508e12 −0.672416
\(170\) 0 0
\(171\) 0 0
\(172\) 3.86627e12 1.95833
\(173\) −7.30852e11 −0.358571 −0.179286 0.983797i \(-0.557379\pi\)
−0.179286 + 0.983797i \(0.557379\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −2.44244e12 −1.09020
\(177\) 0 0
\(178\) 1.98688e12 0.833414
\(179\) −3.92371e12 −1.59590 −0.797950 0.602724i \(-0.794082\pi\)
−0.797950 + 0.602724i \(0.794082\pi\)
\(180\) 0 0
\(181\) 2.27931e12 0.872110 0.436055 0.899920i \(-0.356375\pi\)
0.436055 + 0.899920i \(0.356375\pi\)
\(182\) −1.65907e12 −0.615845
\(183\) 0 0
\(184\) 2.37440e12 0.829956
\(185\) 0 0
\(186\) 0 0
\(187\) −1.96731e12 −0.629136
\(188\) −6.27897e12 −1.94994
\(189\) 0 0
\(190\) 0 0
\(191\) −3.38709e12 −0.964147 −0.482073 0.876131i \(-0.660116\pi\)
−0.482073 + 0.876131i \(0.660116\pi\)
\(192\) 0 0
\(193\) 4.92921e12 1.32499 0.662494 0.749067i \(-0.269498\pi\)
0.662494 + 0.749067i \(0.269498\pi\)
\(194\) 3.04921e12 0.796670
\(195\) 0 0
\(196\) −4.87028e12 −1.20267
\(197\) 6.35785e12 1.52667 0.763337 0.646001i \(-0.223560\pi\)
0.763337 + 0.646001i \(0.223560\pi\)
\(198\) 0 0
\(199\) −3.78554e12 −0.859875 −0.429938 0.902859i \(-0.641464\pi\)
−0.429938 + 0.902859i \(0.641464\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −7.26241e11 −0.151932
\(203\) −2.98455e11 −0.0607646
\(204\) 0 0
\(205\) 0 0
\(206\) −3.78885e12 −0.711604
\(207\) 0 0
\(208\) −2.93404e12 −0.522537
\(209\) 1.24451e13 2.15870
\(210\) 0 0
\(211\) 1.79494e11 0.0295458 0.0147729 0.999891i \(-0.495297\pi\)
0.0147729 + 0.999891i \(0.495297\pi\)
\(212\) 1.53062e13 2.45482
\(213\) 0 0
\(214\) 1.75965e13 2.68009
\(215\) 0 0
\(216\) 0 0
\(217\) −1.41402e12 −0.199493
\(218\) −5.41719e12 −0.745184
\(219\) 0 0
\(220\) 0 0
\(221\) −2.36328e12 −0.301548
\(222\) 0 0
\(223\) −2.22568e12 −0.270263 −0.135132 0.990828i \(-0.543146\pi\)
−0.135132 + 0.990828i \(0.543146\pi\)
\(224\) −5.24344e11 −0.0621230
\(225\) 0 0
\(226\) −2.80538e13 −3.16516
\(227\) −4.15848e11 −0.0457923 −0.0228961 0.999738i \(-0.507289\pi\)
−0.0228961 + 0.999738i \(0.507289\pi\)
\(228\) 0 0
\(229\) 1.81248e13 1.90186 0.950928 0.309414i \(-0.100133\pi\)
0.950928 + 0.309414i \(0.100133\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.66713e12 −0.162850
\(233\) −1.87641e10 −0.00179007 −0.000895033 1.00000i \(-0.500285\pi\)
−0.000895033 1.00000i \(0.500285\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.24122e12 −0.199282
\(237\) 0 0
\(238\) 6.67849e12 0.566898
\(239\) 1.76252e13 1.46200 0.730999 0.682379i \(-0.239054\pi\)
0.730999 + 0.682379i \(0.239054\pi\)
\(240\) 0 0
\(241\) −8.90117e11 −0.0705267 −0.0352633 0.999378i \(-0.511227\pi\)
−0.0352633 + 0.999378i \(0.511227\pi\)
\(242\) 9.47880e12 0.734123
\(243\) 0 0
\(244\) 1.99799e13 1.47894
\(245\) 0 0
\(246\) 0 0
\(247\) 1.49499e13 1.03468
\(248\) −7.89856e12 −0.534643
\(249\) 0 0
\(250\) 0 0
\(251\) 2.42280e13 1.53502 0.767508 0.641040i \(-0.221497\pi\)
0.767508 + 0.641040i \(0.221497\pi\)
\(252\) 0 0
\(253\) −9.76677e12 −0.592364
\(254\) −1.95213e13 −1.15857
\(255\) 0 0
\(256\) −3.45806e13 −1.96568
\(257\) 7.80492e12 0.434246 0.217123 0.976144i \(-0.430333\pi\)
0.217123 + 0.976144i \(0.430333\pi\)
\(258\) 0 0
\(259\) −1.84532e13 −0.983837
\(260\) 0 0
\(261\) 0 0
\(262\) 7.63556e12 0.382106
\(263\) 1.65956e13 0.813272 0.406636 0.913590i \(-0.366702\pi\)
0.406636 + 0.913590i \(0.366702\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −4.22477e13 −1.94515
\(267\) 0 0
\(268\) −2.13601e13 −0.943762
\(269\) −2.85236e13 −1.23471 −0.617357 0.786683i \(-0.711797\pi\)
−0.617357 + 0.786683i \(0.711797\pi\)
\(270\) 0 0
\(271\) 2.33800e13 0.971658 0.485829 0.874054i \(-0.338518\pi\)
0.485829 + 0.874054i \(0.338518\pi\)
\(272\) 1.18108e13 0.481006
\(273\) 0 0
\(274\) −1.20911e13 −0.472976
\(275\) 0 0
\(276\) 0 0
\(277\) 3.03641e13 1.11872 0.559361 0.828924i \(-0.311047\pi\)
0.559361 + 0.828924i \(0.311047\pi\)
\(278\) −8.78487e13 −3.17314
\(279\) 0 0
\(280\) 0 0
\(281\) −1.59749e13 −0.543942 −0.271971 0.962306i \(-0.587675\pi\)
−0.271971 + 0.962306i \(0.587675\pi\)
\(282\) 0 0
\(283\) −2.87045e13 −0.939993 −0.469997 0.882668i \(-0.655745\pi\)
−0.469997 + 0.882668i \(0.655745\pi\)
\(284\) 5.98583e13 1.92253
\(285\) 0 0
\(286\) 3.81201e13 1.17799
\(287\) −2.49516e13 −0.756394
\(288\) 0 0
\(289\) −2.47587e13 −0.722419
\(290\) 0 0
\(291\) 0 0
\(292\) −5.63870e13 −1.55444
\(293\) 4.81754e13 1.30333 0.651663 0.758508i \(-0.274071\pi\)
0.651663 + 0.758508i \(0.274071\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.03077e14 −2.63669
\(297\) 0 0
\(298\) 1.07997e14 2.66209
\(299\) −1.17325e13 −0.283923
\(300\) 0 0
\(301\) 2.65926e13 0.620362
\(302\) −5.44909e13 −1.24820
\(303\) 0 0
\(304\) −7.47143e13 −1.65044
\(305\) 0 0
\(306\) 0 0
\(307\) −2.57350e13 −0.538597 −0.269298 0.963057i \(-0.586792\pi\)
−0.269298 + 0.963057i \(0.586792\pi\)
\(308\) −7.14627e13 −1.46910
\(309\) 0 0
\(310\) 0 0
\(311\) −3.46043e13 −0.674446 −0.337223 0.941425i \(-0.609488\pi\)
−0.337223 + 0.941425i \(0.609488\pi\)
\(312\) 0 0
\(313\) −1.26066e13 −0.237194 −0.118597 0.992942i \(-0.537840\pi\)
−0.118597 + 0.992942i \(0.537840\pi\)
\(314\) −1.66239e14 −3.07341
\(315\) 0 0
\(316\) 1.50161e13 0.268089
\(317\) −8.18243e13 −1.43568 −0.717838 0.696210i \(-0.754868\pi\)
−0.717838 + 0.696210i \(0.754868\pi\)
\(318\) 0 0
\(319\) 6.85754e12 0.116230
\(320\) 0 0
\(321\) 0 0
\(322\) 3.31555e13 0.533764
\(323\) −6.01801e13 −0.952443
\(324\) 0 0
\(325\) 0 0
\(326\) 1.27580e13 0.191905
\(327\) 0 0
\(328\) −1.39377e14 −2.02714
\(329\) −4.31874e13 −0.617703
\(330\) 0 0
\(331\) −3.40115e13 −0.470513 −0.235256 0.971933i \(-0.575593\pi\)
−0.235256 + 0.971933i \(0.575593\pi\)
\(332\) 3.53895e13 0.481520
\(333\) 0 0
\(334\) −6.87135e11 −0.00904558
\(335\) 0 0
\(336\) 0 0
\(337\) 5.99439e13 0.751244 0.375622 0.926773i \(-0.377429\pi\)
0.375622 + 0.926773i \(0.377429\pi\)
\(338\) −9.39960e13 −1.15896
\(339\) 0 0
\(340\) 0 0
\(341\) 3.24897e13 0.381590
\(342\) 0 0
\(343\) −8.83888e13 −1.00526
\(344\) 1.48543e14 1.66257
\(345\) 0 0
\(346\) −5.70064e13 −0.618024
\(347\) 9.78685e13 1.04431 0.522157 0.852850i \(-0.325128\pi\)
0.522157 + 0.852850i \(0.325128\pi\)
\(348\) 0 0
\(349\) −1.42790e14 −1.47624 −0.738120 0.674670i \(-0.764286\pi\)
−0.738120 + 0.674670i \(0.764286\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 1.20478e13 0.118829
\(353\) 1.44246e14 1.40069 0.700346 0.713804i \(-0.253029\pi\)
0.700346 + 0.713804i \(0.253029\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 1.02808e14 0.952911
\(357\) 0 0
\(358\) −3.06050e14 −2.75065
\(359\) 1.24349e14 1.10059 0.550293 0.834972i \(-0.314516\pi\)
0.550293 + 0.834972i \(0.314516\pi\)
\(360\) 0 0
\(361\) 2.64205e14 2.26804
\(362\) 1.77786e14 1.50314
\(363\) 0 0
\(364\) −8.58461e13 −0.704147
\(365\) 0 0
\(366\) 0 0
\(367\) 1.60110e14 1.25532 0.627659 0.778488i \(-0.284013\pi\)
0.627659 + 0.778488i \(0.284013\pi\)
\(368\) 5.86351e13 0.452892
\(369\) 0 0
\(370\) 0 0
\(371\) 1.05277e14 0.777641
\(372\) 0 0
\(373\) 3.47258e13 0.249031 0.124516 0.992218i \(-0.460262\pi\)
0.124516 + 0.992218i \(0.460262\pi\)
\(374\) −1.53450e14 −1.08436
\(375\) 0 0
\(376\) −2.41239e14 −1.65545
\(377\) 8.23777e12 0.0557099
\(378\) 0 0
\(379\) −1.46500e14 −0.962327 −0.481163 0.876631i \(-0.659786\pi\)
−0.481163 + 0.876631i \(0.659786\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −2.64193e14 −1.66178
\(383\) 6.43419e13 0.398933 0.199467 0.979905i \(-0.436079\pi\)
0.199467 + 0.979905i \(0.436079\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.84478e14 2.28371
\(387\) 0 0
\(388\) 1.57777e14 0.910899
\(389\) 3.98900e13 0.227061 0.113530 0.993535i \(-0.463784\pi\)
0.113530 + 0.993535i \(0.463784\pi\)
\(390\) 0 0
\(391\) 4.72287e13 0.261357
\(392\) −1.87117e14 −1.02103
\(393\) 0 0
\(394\) 4.95912e14 2.63133
\(395\) 0 0
\(396\) 0 0
\(397\) −1.06552e14 −0.542268 −0.271134 0.962542i \(-0.587399\pi\)
−0.271134 + 0.962542i \(0.587399\pi\)
\(398\) −2.95272e14 −1.48206
\(399\) 0 0
\(400\) 0 0
\(401\) −3.41445e13 −0.164447 −0.0822236 0.996614i \(-0.526202\pi\)
−0.0822236 + 0.996614i \(0.526202\pi\)
\(402\) 0 0
\(403\) 3.90289e13 0.182898
\(404\) −3.75783e13 −0.173716
\(405\) 0 0
\(406\) −2.32795e13 −0.104732
\(407\) 4.23996e14 1.88188
\(408\) 0 0
\(409\) 5.33349e13 0.230427 0.115213 0.993341i \(-0.463245\pi\)
0.115213 + 0.993341i \(0.463245\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.96049e14 −0.813635
\(413\) −1.54153e13 −0.0631288
\(414\) 0 0
\(415\) 0 0
\(416\) 1.44726e13 0.0569553
\(417\) 0 0
\(418\) 9.70716e14 3.72068
\(419\) 1.01288e14 0.383159 0.191580 0.981477i \(-0.438639\pi\)
0.191580 + 0.981477i \(0.438639\pi\)
\(420\) 0 0
\(421\) −1.57928e14 −0.581981 −0.290991 0.956726i \(-0.593985\pi\)
−0.290991 + 0.956726i \(0.593985\pi\)
\(422\) 1.40005e13 0.0509244
\(423\) 0 0
\(424\) 5.88067e14 2.08408
\(425\) 0 0
\(426\) 0 0
\(427\) 1.37424e14 0.468499
\(428\) 9.10504e14 3.06437
\(429\) 0 0
\(430\) 0 0
\(431\) −5.13171e14 −1.66202 −0.831012 0.556254i \(-0.812238\pi\)
−0.831012 + 0.556254i \(0.812238\pi\)
\(432\) 0 0
\(433\) 7.49248e13 0.236560 0.118280 0.992980i \(-0.462262\pi\)
0.118280 + 0.992980i \(0.462262\pi\)
\(434\) −1.10294e14 −0.343841
\(435\) 0 0
\(436\) −2.80305e14 −0.852030
\(437\) −2.98766e14 −0.896773
\(438\) 0 0
\(439\) −3.68335e14 −1.07817 −0.539086 0.842250i \(-0.681230\pi\)
−0.539086 + 0.842250i \(0.681230\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −1.84335e14 −0.519740
\(443\) −1.11248e14 −0.309793 −0.154896 0.987931i \(-0.549504\pi\)
−0.154896 + 0.987931i \(0.549504\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −1.73603e14 −0.465818
\(447\) 0 0
\(448\) −2.58602e14 −0.677022
\(449\) 8.83314e13 0.228434 0.114217 0.993456i \(-0.463564\pi\)
0.114217 + 0.993456i \(0.463564\pi\)
\(450\) 0 0
\(451\) 5.73308e14 1.44683
\(452\) −1.45161e15 −3.61899
\(453\) 0 0
\(454\) −3.24361e13 −0.0789263
\(455\) 0 0
\(456\) 0 0
\(457\) 9.42094e12 0.0221083 0.0110541 0.999939i \(-0.496481\pi\)
0.0110541 + 0.999939i \(0.496481\pi\)
\(458\) 1.41373e15 3.27799
\(459\) 0 0
\(460\) 0 0
\(461\) 6.97134e14 1.55941 0.779706 0.626146i \(-0.215369\pi\)
0.779706 + 0.626146i \(0.215369\pi\)
\(462\) 0 0
\(463\) 1.87941e14 0.410513 0.205256 0.978708i \(-0.434197\pi\)
0.205256 + 0.978708i \(0.434197\pi\)
\(464\) −4.11694e13 −0.0888640
\(465\) 0 0
\(466\) −1.46360e12 −0.00308531
\(467\) 3.20007e14 0.666678 0.333339 0.942807i \(-0.391825\pi\)
0.333339 + 0.942807i \(0.391825\pi\)
\(468\) 0 0
\(469\) −1.46917e14 −0.298966
\(470\) 0 0
\(471\) 0 0
\(472\) −8.61081e13 −0.169185
\(473\) −6.11013e14 −1.18663
\(474\) 0 0
\(475\) 0 0
\(476\) 3.45569e14 0.648181
\(477\) 0 0
\(478\) 1.37477e15 2.51986
\(479\) 1.50382e14 0.272491 0.136245 0.990675i \(-0.456496\pi\)
0.136245 + 0.990675i \(0.456496\pi\)
\(480\) 0 0
\(481\) 5.09334e14 0.901996
\(482\) −6.94291e13 −0.121558
\(483\) 0 0
\(484\) 4.90467e14 0.839384
\(485\) 0 0
\(486\) 0 0
\(487\) −1.76546e14 −0.292045 −0.146022 0.989281i \(-0.546647\pi\)
−0.146022 + 0.989281i \(0.546647\pi\)
\(488\) 7.67632e14 1.25558
\(489\) 0 0
\(490\) 0 0
\(491\) 8.60958e14 1.36155 0.680775 0.732492i \(-0.261643\pi\)
0.680775 + 0.732492i \(0.261643\pi\)
\(492\) 0 0
\(493\) −3.31607e13 −0.0512821
\(494\) 1.16609e15 1.78334
\(495\) 0 0
\(496\) −1.95053e14 −0.291745
\(497\) 4.11711e14 0.609021
\(498\) 0 0
\(499\) 6.01209e14 0.869907 0.434953 0.900453i \(-0.356765\pi\)
0.434953 + 0.900453i \(0.356765\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.88979e15 2.64571
\(503\) 1.09203e15 1.51221 0.756103 0.654453i \(-0.227101\pi\)
0.756103 + 0.654453i \(0.227101\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −7.61808e14 −1.02098
\(507\) 0 0
\(508\) −1.01010e15 −1.32469
\(509\) −8.76371e14 −1.13695 −0.568474 0.822702i \(-0.692466\pi\)
−0.568474 + 0.822702i \(0.692466\pi\)
\(510\) 0 0
\(511\) −3.87835e14 −0.492416
\(512\) −1.28839e15 −1.61832
\(513\) 0 0
\(514\) 6.08784e14 0.748455
\(515\) 0 0
\(516\) 0 0
\(517\) 9.92308e14 1.18154
\(518\) −1.43935e15 −1.69571
\(519\) 0 0
\(520\) 0 0
\(521\) 3.71989e14 0.424544 0.212272 0.977211i \(-0.431914\pi\)
0.212272 + 0.977211i \(0.431914\pi\)
\(522\) 0 0
\(523\) 9.73507e14 1.08788 0.543939 0.839125i \(-0.316932\pi\)
0.543939 + 0.839125i \(0.316932\pi\)
\(524\) 3.95091e14 0.436893
\(525\) 0 0
\(526\) 1.29445e15 1.40173
\(527\) −1.57109e14 −0.168361
\(528\) 0 0
\(529\) −7.18341e14 −0.753919
\(530\) 0 0
\(531\) 0 0
\(532\) −2.18605e15 −2.22405
\(533\) 6.88699e14 0.693473
\(534\) 0 0
\(535\) 0 0
\(536\) −8.20661e14 −0.801230
\(537\) 0 0
\(538\) −2.22484e15 −2.12812
\(539\) 7.69683e14 0.728741
\(540\) 0 0
\(541\) 1.74606e15 1.61985 0.809925 0.586533i \(-0.199508\pi\)
0.809925 + 0.586533i \(0.199508\pi\)
\(542\) 1.82364e15 1.67472
\(543\) 0 0
\(544\) −5.82588e13 −0.0524285
\(545\) 0 0
\(546\) 0 0
\(547\) −1.74624e14 −0.152467 −0.0762333 0.997090i \(-0.524289\pi\)
−0.0762333 + 0.997090i \(0.524289\pi\)
\(548\) −6.25639e14 −0.540793
\(549\) 0 0
\(550\) 0 0
\(551\) 2.09772e14 0.175960
\(552\) 0 0
\(553\) 1.03282e14 0.0849254
\(554\) 2.36840e15 1.92820
\(555\) 0 0
\(556\) −4.54561e15 −3.62811
\(557\) 1.58365e15 1.25157 0.625785 0.779995i \(-0.284779\pi\)
0.625785 + 0.779995i \(0.284779\pi\)
\(558\) 0 0
\(559\) −7.33993e14 −0.568757
\(560\) 0 0
\(561\) 0 0
\(562\) −1.24604e15 −0.937523
\(563\) −9.75798e14 −0.727049 −0.363525 0.931585i \(-0.618427\pi\)
−0.363525 + 0.931585i \(0.618427\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −2.23895e15 −1.62015
\(567\) 0 0
\(568\) 2.29977e15 1.63218
\(569\) −1.92427e15 −1.35254 −0.676269 0.736655i \(-0.736404\pi\)
−0.676269 + 0.736655i \(0.736404\pi\)
\(570\) 0 0
\(571\) 1.61132e15 1.11092 0.555461 0.831543i \(-0.312542\pi\)
0.555461 + 0.831543i \(0.312542\pi\)
\(572\) 1.97247e15 1.34689
\(573\) 0 0
\(574\) −1.94623e15 −1.30370
\(575\) 0 0
\(576\) 0 0
\(577\) 2.53250e15 1.64848 0.824239 0.566242i \(-0.191603\pi\)
0.824239 + 0.566242i \(0.191603\pi\)
\(578\) −1.93118e15 −1.24514
\(579\) 0 0
\(580\) 0 0
\(581\) 2.43412e14 0.152536
\(582\) 0 0
\(583\) −2.41894e15 −1.48747
\(584\) −2.16640e15 −1.31968
\(585\) 0 0
\(586\) 3.75768e15 2.24638
\(587\) −3.11508e15 −1.84484 −0.922421 0.386186i \(-0.873792\pi\)
−0.922421 + 0.386186i \(0.873792\pi\)
\(588\) 0 0
\(589\) 9.93860e14 0.577685
\(590\) 0 0
\(591\) 0 0
\(592\) −2.54547e15 −1.43879
\(593\) −2.20723e15 −1.23608 −0.618040 0.786146i \(-0.712073\pi\)
−0.618040 + 0.786146i \(0.712073\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 5.58817e15 3.04379
\(597\) 0 0
\(598\) −9.15138e14 −0.489362
\(599\) −3.41159e15 −1.80763 −0.903813 0.427927i \(-0.859244\pi\)
−0.903813 + 0.427927i \(0.859244\pi\)
\(600\) 0 0
\(601\) −1.57545e15 −0.819588 −0.409794 0.912178i \(-0.634399\pi\)
−0.409794 + 0.912178i \(0.634399\pi\)
\(602\) 2.07422e15 1.06924
\(603\) 0 0
\(604\) −2.81956e15 −1.42718
\(605\) 0 0
\(606\) 0 0
\(607\) 5.05592e14 0.249036 0.124518 0.992217i \(-0.460262\pi\)
0.124518 + 0.992217i \(0.460262\pi\)
\(608\) 3.68541e14 0.179894
\(609\) 0 0
\(610\) 0 0
\(611\) 1.19203e15 0.566319
\(612\) 0 0
\(613\) 1.98375e15 0.925665 0.462832 0.886446i \(-0.346833\pi\)
0.462832 + 0.886446i \(0.346833\pi\)
\(614\) −2.00733e15 −0.928310
\(615\) 0 0
\(616\) −2.74561e15 −1.24723
\(617\) −1.93099e15 −0.869382 −0.434691 0.900580i \(-0.643142\pi\)
−0.434691 + 0.900580i \(0.643142\pi\)
\(618\) 0 0
\(619\) 9.47689e14 0.419148 0.209574 0.977793i \(-0.432792\pi\)
0.209574 + 0.977793i \(0.432792\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) −2.69913e15 −1.16246
\(623\) 7.07124e14 0.301864
\(624\) 0 0
\(625\) 0 0
\(626\) −9.83314e14 −0.408821
\(627\) 0 0
\(628\) −8.60181e15 −3.51408
\(629\) −2.05030e15 −0.830306
\(630\) 0 0
\(631\) 1.00593e15 0.400318 0.200159 0.979763i \(-0.435854\pi\)
0.200159 + 0.979763i \(0.435854\pi\)
\(632\) 5.76922e14 0.227601
\(633\) 0 0
\(634\) −6.38230e15 −2.47449
\(635\) 0 0
\(636\) 0 0
\(637\) 9.24597e14 0.349290
\(638\) 5.34888e14 0.200332
\(639\) 0 0
\(640\) 0 0
\(641\) −1.13144e15 −0.412966 −0.206483 0.978450i \(-0.566202\pi\)
−0.206483 + 0.978450i \(0.566202\pi\)
\(642\) 0 0
\(643\) 8.68306e14 0.311539 0.155769 0.987793i \(-0.450214\pi\)
0.155769 + 0.987793i \(0.450214\pi\)
\(644\) 1.71559e15 0.610296
\(645\) 0 0
\(646\) −4.69405e15 −1.64160
\(647\) −1.37636e15 −0.477265 −0.238633 0.971110i \(-0.576699\pi\)
−0.238633 + 0.971110i \(0.576699\pi\)
\(648\) 0 0
\(649\) 3.54195e14 0.120753
\(650\) 0 0
\(651\) 0 0
\(652\) 6.60146e14 0.219421
\(653\) 2.55276e15 0.841370 0.420685 0.907207i \(-0.361790\pi\)
0.420685 + 0.907207i \(0.361790\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −3.44187e15 −1.10617
\(657\) 0 0
\(658\) −3.36862e15 −1.06466
\(659\) −9.26792e14 −0.290478 −0.145239 0.989397i \(-0.546395\pi\)
−0.145239 + 0.989397i \(0.546395\pi\)
\(660\) 0 0
\(661\) −1.90332e15 −0.586683 −0.293342 0.956008i \(-0.594767\pi\)
−0.293342 + 0.956008i \(0.594767\pi\)
\(662\) −2.65289e15 −0.810963
\(663\) 0 0
\(664\) 1.35967e15 0.408799
\(665\) 0 0
\(666\) 0 0
\(667\) −1.64627e14 −0.0482847
\(668\) −3.55548e13 −0.0103426
\(669\) 0 0
\(670\) 0 0
\(671\) −3.15756e15 −0.896143
\(672\) 0 0
\(673\) −4.92990e15 −1.37643 −0.688217 0.725505i \(-0.741606\pi\)
−0.688217 + 0.725505i \(0.741606\pi\)
\(674\) 4.67563e15 1.29482
\(675\) 0 0
\(676\) −4.86369e15 −1.32513
\(677\) 4.30293e14 0.116286 0.0581429 0.998308i \(-0.481482\pi\)
0.0581429 + 0.998308i \(0.481482\pi\)
\(678\) 0 0
\(679\) 1.08521e15 0.288555
\(680\) 0 0
\(681\) 0 0
\(682\) 2.53420e15 0.657698
\(683\) −3.81244e15 −0.981498 −0.490749 0.871301i \(-0.663277\pi\)
−0.490749 + 0.871301i \(0.663277\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −6.89433e15 −1.73264
\(687\) 0 0
\(688\) 3.66823e15 0.907236
\(689\) −2.90580e15 −0.712952
\(690\) 0 0
\(691\) 4.03729e15 0.974901 0.487450 0.873151i \(-0.337927\pi\)
0.487450 + 0.873151i \(0.337927\pi\)
\(692\) −2.94972e15 −0.706638
\(693\) 0 0
\(694\) 7.63374e15 1.79995
\(695\) 0 0
\(696\) 0 0
\(697\) −2.77232e15 −0.638356
\(698\) −1.11376e16 −2.54441
\(699\) 0 0
\(700\) 0 0
\(701\) 4.71267e15 1.05152 0.525761 0.850632i \(-0.323781\pi\)
0.525761 + 0.850632i \(0.323781\pi\)
\(702\) 0 0
\(703\) 1.29700e16 2.84896
\(704\) 5.94185e15 1.29501
\(705\) 0 0
\(706\) 1.12512e16 2.41419
\(707\) −2.58467e14 −0.0550299
\(708\) 0 0
\(709\) 8.66706e14 0.181684 0.0908422 0.995865i \(-0.471044\pi\)
0.0908422 + 0.995865i \(0.471044\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 3.94991e15 0.808997
\(713\) −7.79972e14 −0.158521
\(714\) 0 0
\(715\) 0 0
\(716\) −1.58361e16 −3.14505
\(717\) 0 0
\(718\) 9.69924e15 1.89694
\(719\) −2.62666e13 −0.00509795 −0.00254897 0.999997i \(-0.500811\pi\)
−0.00254897 + 0.999997i \(0.500811\pi\)
\(720\) 0 0
\(721\) −1.34844e15 −0.257744
\(722\) 2.06080e16 3.90913
\(723\) 0 0
\(724\) 9.19929e15 1.71867
\(725\) 0 0
\(726\) 0 0
\(727\) 5.68018e15 1.03734 0.518672 0.854973i \(-0.326427\pi\)
0.518672 + 0.854973i \(0.326427\pi\)
\(728\) −3.29823e15 −0.597803
\(729\) 0 0
\(730\) 0 0
\(731\) 2.95465e15 0.523552
\(732\) 0 0
\(733\) 1.62312e15 0.283320 0.141660 0.989915i \(-0.454756\pi\)
0.141660 + 0.989915i \(0.454756\pi\)
\(734\) 1.24885e16 2.16363
\(735\) 0 0
\(736\) −2.89227e14 −0.0493641
\(737\) 3.37568e15 0.571861
\(738\) 0 0
\(739\) 3.64794e15 0.608840 0.304420 0.952538i \(-0.401537\pi\)
0.304420 + 0.952538i \(0.401537\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 8.21164e15 1.34032
\(743\) 6.76045e15 1.09531 0.547655 0.836704i \(-0.315521\pi\)
0.547655 + 0.836704i \(0.315521\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.70861e15 0.429224
\(747\) 0 0
\(748\) −7.94007e15 −1.23984
\(749\) 6.26254e15 0.970734
\(750\) 0 0
\(751\) 2.64833e13 0.00404532 0.00202266 0.999998i \(-0.499356\pi\)
0.00202266 + 0.999998i \(0.499356\pi\)
\(752\) −5.95734e15 −0.903347
\(753\) 0 0
\(754\) 6.42546e14 0.0960200
\(755\) 0 0
\(756\) 0 0
\(757\) −7.37364e15 −1.07809 −0.539045 0.842277i \(-0.681215\pi\)
−0.539045 + 0.842277i \(0.681215\pi\)
\(758\) −1.14270e16 −1.65864
\(759\) 0 0
\(760\) 0 0
\(761\) −6.20983e15 −0.881991 −0.440996 0.897509i \(-0.645375\pi\)
−0.440996 + 0.897509i \(0.645375\pi\)
\(762\) 0 0
\(763\) −1.92796e15 −0.269907
\(764\) −1.36703e16 −1.90005
\(765\) 0 0
\(766\) 5.01866e15 0.687591
\(767\) 4.25484e14 0.0578774
\(768\) 0 0
\(769\) −8.86195e15 −1.18832 −0.594162 0.804346i \(-0.702516\pi\)
−0.594162 + 0.804346i \(0.702516\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.98943e16 2.61116
\(773\) −2.95047e15 −0.384507 −0.192254 0.981345i \(-0.561580\pi\)
−0.192254 + 0.981345i \(0.561580\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 6.06184e15 0.773330
\(777\) 0 0
\(778\) 3.11142e15 0.391355
\(779\) 1.75375e16 2.19034
\(780\) 0 0
\(781\) −9.45980e15 −1.16493
\(782\) 3.68384e15 0.450468
\(783\) 0 0
\(784\) −4.62081e15 −0.557160
\(785\) 0 0
\(786\) 0 0
\(787\) −1.35545e16 −1.60038 −0.800190 0.599746i \(-0.795268\pi\)
−0.800190 + 0.599746i \(0.795268\pi\)
\(788\) 2.56603e16 3.00862
\(789\) 0 0
\(790\) 0 0
\(791\) −9.98429e15 −1.14643
\(792\) 0 0
\(793\) −3.79308e15 −0.429526
\(794\) −8.31105e15 −0.934638
\(795\) 0 0
\(796\) −1.52784e16 −1.69456
\(797\) −1.59922e16 −1.76152 −0.880762 0.473558i \(-0.842969\pi\)
−0.880762 + 0.473558i \(0.842969\pi\)
\(798\) 0 0
\(799\) −4.79846e15 −0.521308
\(800\) 0 0
\(801\) 0 0
\(802\) −2.66327e15 −0.283437
\(803\) 8.91121e15 0.941892
\(804\) 0 0
\(805\) 0 0
\(806\) 3.04426e15 0.315238
\(807\) 0 0
\(808\) −1.44377e15 −0.147481
\(809\) −9.45859e15 −0.959643 −0.479821 0.877366i \(-0.659299\pi\)
−0.479821 + 0.877366i \(0.659299\pi\)
\(810\) 0 0
\(811\) −1.07996e16 −1.08092 −0.540461 0.841369i \(-0.681750\pi\)
−0.540461 + 0.841369i \(0.681750\pi\)
\(812\) −1.20456e15 −0.119749
\(813\) 0 0
\(814\) 3.30717e16 3.24356
\(815\) 0 0
\(816\) 0 0
\(817\) −1.86909e16 −1.79642
\(818\) 4.16012e15 0.397158
\(819\) 0 0
\(820\) 0 0
\(821\) 1.10411e16 1.03306 0.516529 0.856270i \(-0.327224\pi\)
0.516529 + 0.856270i \(0.327224\pi\)
\(822\) 0 0
\(823\) −1.26104e16 −1.16420 −0.582101 0.813117i \(-0.697769\pi\)
−0.582101 + 0.813117i \(0.697769\pi\)
\(824\) −7.53224e15 −0.690756
\(825\) 0 0
\(826\) −1.20239e15 −0.108807
\(827\) −7.86172e14 −0.0706704 −0.0353352 0.999376i \(-0.511250\pi\)
−0.0353352 + 0.999376i \(0.511250\pi\)
\(828\) 0 0
\(829\) −5.92589e15 −0.525659 −0.262829 0.964842i \(-0.584656\pi\)
−0.262829 + 0.964842i \(0.584656\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 7.13777e15 0.620704
\(833\) −3.72192e15 −0.321528
\(834\) 0 0
\(835\) 0 0
\(836\) 5.02283e16 4.25416
\(837\) 0 0
\(838\) 7.90044e15 0.660403
\(839\) −2.10209e16 −1.74566 −0.872831 0.488022i \(-0.837718\pi\)
−0.872831 + 0.488022i \(0.837718\pi\)
\(840\) 0 0
\(841\) −1.20849e16 −0.990526
\(842\) −1.23184e16 −1.00309
\(843\) 0 0
\(844\) 7.24437e14 0.0582260
\(845\) 0 0
\(846\) 0 0
\(847\) 3.37348e15 0.265901
\(848\) 1.45222e16 1.13725
\(849\) 0 0
\(850\) 0 0
\(851\) −1.01788e16 −0.781775
\(852\) 0 0
\(853\) 1.98378e16 1.50409 0.752046 0.659110i \(-0.229067\pi\)
0.752046 + 0.659110i \(0.229067\pi\)
\(854\) 1.07190e16 0.807491
\(855\) 0 0
\(856\) 3.49818e16 2.60157
\(857\) 7.96639e15 0.588663 0.294332 0.955703i \(-0.404903\pi\)
0.294332 + 0.955703i \(0.404903\pi\)
\(858\) 0 0
\(859\) −1.59749e16 −1.16540 −0.582701 0.812687i \(-0.698004\pi\)
−0.582701 + 0.812687i \(0.698004\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −4.00274e16 −2.86462
\(863\) −4.02926e15 −0.286527 −0.143264 0.989685i \(-0.545760\pi\)
−0.143264 + 0.989685i \(0.545760\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 5.84413e15 0.407729
\(867\) 0 0
\(868\) −5.70699e15 −0.393142
\(869\) −2.37310e15 −0.162445
\(870\) 0 0
\(871\) 4.05511e15 0.274096
\(872\) −1.07694e16 −0.723352
\(873\) 0 0
\(874\) −2.33037e16 −1.54565
\(875\) 0 0
\(876\) 0 0
\(877\) −9.76958e15 −0.635884 −0.317942 0.948110i \(-0.602992\pi\)
−0.317942 + 0.948110i \(0.602992\pi\)
\(878\) −2.87301e16 −1.85831
\(879\) 0 0
\(880\) 0 0
\(881\) −1.62583e16 −1.03207 −0.516033 0.856569i \(-0.672592\pi\)
−0.516033 + 0.856569i \(0.672592\pi\)
\(882\) 0 0
\(883\) −1.66061e16 −1.04108 −0.520540 0.853837i \(-0.674269\pi\)
−0.520540 + 0.853837i \(0.674269\pi\)
\(884\) −9.53818e15 −0.594262
\(885\) 0 0
\(886\) −8.67733e15 −0.533950
\(887\) −1.91314e16 −1.16995 −0.584975 0.811051i \(-0.698896\pi\)
−0.584975 + 0.811051i \(0.698896\pi\)
\(888\) 0 0
\(889\) −6.94758e15 −0.419637
\(890\) 0 0
\(891\) 0 0
\(892\) −8.98286e15 −0.532608
\(893\) 3.03547e16 1.78872
\(894\) 0 0
\(895\) 0 0
\(896\) −1.90971e16 −1.10477
\(897\) 0 0
\(898\) 6.88985e15 0.393722
\(899\) 5.47641e14 0.0311041
\(900\) 0 0
\(901\) 1.16972e16 0.656287
\(902\) 4.47180e16 2.49372
\(903\) 0 0
\(904\) −5.57710e16 −3.07243
\(905\) 0 0
\(906\) 0 0
\(907\) 1.07252e16 0.580182 0.290091 0.956999i \(-0.406314\pi\)
0.290091 + 0.956999i \(0.406314\pi\)
\(908\) −1.67836e15 −0.0902430
\(909\) 0 0
\(910\) 0 0
\(911\) 2.82249e16 1.49032 0.745162 0.666883i \(-0.232372\pi\)
0.745162 + 0.666883i \(0.232372\pi\)
\(912\) 0 0
\(913\) −5.59284e15 −0.291771
\(914\) 7.34833e14 0.0381052
\(915\) 0 0
\(916\) 7.31516e16 3.74799
\(917\) 2.71748e15 0.138399
\(918\) 0 0
\(919\) 1.83551e16 0.923679 0.461839 0.886964i \(-0.347190\pi\)
0.461839 + 0.886964i \(0.347190\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 5.43764e16 2.68776
\(923\) −1.13638e16 −0.558359
\(924\) 0 0
\(925\) 0 0
\(926\) 1.46594e16 0.707549
\(927\) 0 0
\(928\) 2.03075e14 0.00968596
\(929\) 1.10125e16 0.522155 0.261078 0.965318i \(-0.415922\pi\)
0.261078 + 0.965318i \(0.415922\pi\)
\(930\) 0 0
\(931\) 2.35446e16 1.10323
\(932\) −7.57317e13 −0.00352769
\(933\) 0 0
\(934\) 2.49605e16 1.14907
\(935\) 0 0
\(936\) 0 0
\(937\) −8.36357e15 −0.378289 −0.189145 0.981949i \(-0.560571\pi\)
−0.189145 + 0.981949i \(0.560571\pi\)
\(938\) −1.14595e16 −0.515289
\(939\) 0 0
\(940\) 0 0
\(941\) −2.22942e16 −0.985028 −0.492514 0.870305i \(-0.663922\pi\)
−0.492514 + 0.870305i \(0.663922\pi\)
\(942\) 0 0
\(943\) −1.37633e16 −0.601045
\(944\) −2.12642e15 −0.0923214
\(945\) 0 0
\(946\) −4.76590e16 −2.04524
\(947\) −3.78038e16 −1.61291 −0.806457 0.591293i \(-0.798618\pi\)
−0.806457 + 0.591293i \(0.798618\pi\)
\(948\) 0 0
\(949\) 1.07048e16 0.451454
\(950\) 0 0
\(951\) 0 0
\(952\) 1.32768e16 0.550290
\(953\) 4.79568e15 0.197624 0.0988119 0.995106i \(-0.468496\pi\)
0.0988119 + 0.995106i \(0.468496\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 7.11355e16 2.88116
\(957\) 0 0
\(958\) 1.17298e16 0.469657
\(959\) −4.30321e15 −0.171313
\(960\) 0 0
\(961\) −2.28139e16 −0.897884
\(962\) 3.97280e16 1.55466
\(963\) 0 0
\(964\) −3.59251e15 −0.138987
\(965\) 0 0
\(966\) 0 0
\(967\) −3.37420e16 −1.28329 −0.641645 0.767002i \(-0.721748\pi\)
−0.641645 + 0.767002i \(0.721748\pi\)
\(968\) 1.88439e16 0.712616
\(969\) 0 0
\(970\) 0 0
\(971\) −3.58587e16 −1.33318 −0.666590 0.745425i \(-0.732247\pi\)
−0.666590 + 0.745425i \(0.732247\pi\)
\(972\) 0 0
\(973\) −3.12651e16 −1.14932
\(974\) −1.37706e16 −0.503360
\(975\) 0 0
\(976\) 1.89565e16 0.685147
\(977\) −1.20023e16 −0.431366 −0.215683 0.976463i \(-0.569198\pi\)
−0.215683 + 0.976463i \(0.569198\pi\)
\(978\) 0 0
\(979\) −1.62474e16 −0.577405
\(980\) 0 0
\(981\) 0 0
\(982\) 6.71547e16 2.34673
\(983\) −4.42687e15 −0.153834 −0.0769170 0.997038i \(-0.524508\pi\)
−0.0769170 + 0.997038i \(0.524508\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.58653e15 −0.0883884
\(987\) 0 0
\(988\) 6.03378e16 2.03904
\(989\) 1.46684e16 0.492951
\(990\) 0 0
\(991\) −3.79167e16 −1.26016 −0.630080 0.776530i \(-0.716978\pi\)
−0.630080 + 0.776530i \(0.716978\pi\)
\(992\) 9.62130e14 0.0317994
\(993\) 0 0
\(994\) 3.21135e16 1.04969
\(995\) 0 0
\(996\) 0 0
\(997\) −5.76003e13 −0.00185183 −0.000925915 1.00000i \(-0.500295\pi\)
−0.000925915 1.00000i \(0.500295\pi\)
\(998\) 4.68943e16 1.49935
\(999\) 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 225.12.a.f.1.1 1
3.2 odd 2 75.12.a.a.1.1 1
5.2 odd 4 225.12.b.a.199.2 2
5.3 odd 4 225.12.b.a.199.1 2
5.4 even 2 9.12.a.a.1.1 1
15.2 even 4 75.12.b.a.49.1 2
15.8 even 4 75.12.b.a.49.2 2
15.14 odd 2 3.12.a.a.1.1 1
20.19 odd 2 144.12.a.l.1.1 1
45.4 even 6 81.12.c.e.55.1 2
45.14 odd 6 81.12.c.a.55.1 2
45.29 odd 6 81.12.c.a.28.1 2
45.34 even 6 81.12.c.e.28.1 2
60.59 even 2 48.12.a.f.1.1 1
105.104 even 2 147.12.a.c.1.1 1
120.29 odd 2 192.12.a.q.1.1 1
120.59 even 2 192.12.a.g.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3.12.a.a.1.1 1 15.14 odd 2
9.12.a.a.1.1 1 5.4 even 2
48.12.a.f.1.1 1 60.59 even 2
75.12.a.a.1.1 1 3.2 odd 2
75.12.b.a.49.1 2 15.2 even 4
75.12.b.a.49.2 2 15.8 even 4
81.12.c.a.28.1 2 45.29 odd 6
81.12.c.a.55.1 2 45.14 odd 6
81.12.c.e.28.1 2 45.34 even 6
81.12.c.e.55.1 2 45.4 even 6
144.12.a.l.1.1 1 20.19 odd 2
147.12.a.c.1.1 1 105.104 even 2
192.12.a.g.1.1 1 120.59 even 2
192.12.a.q.1.1 1 120.29 odd 2
225.12.a.f.1.1 1 1.1 even 1 trivial
225.12.b.a.199.1 2 5.3 odd 4
225.12.b.a.199.2 2 5.2 odd 4