Properties

Label 225.12.a.a.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 15)
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)
 
\(f(q)\) \(=\) \(q-56.0000 q^{2} +1088.00 q^{4} -27984.0 q^{7} +53760.0 q^{8} +O(q^{10})\) \(q-56.0000 q^{2} +1088.00 q^{4} -27984.0 q^{7} +53760.0 q^{8} +112028. q^{11} +1.09692e6 q^{13} +1.56710e6 q^{14} -5.23878e6 q^{16} -249566. q^{17} -1.37124e7 q^{19} -6.27357e6 q^{22} +4.13957e7 q^{23} -6.14276e7 q^{26} -3.04466e7 q^{28} +4.53385e6 q^{29} -2.65339e8 q^{31} +1.83271e8 q^{32} +1.39757e7 q^{34} +2.12137e8 q^{37} +7.67896e8 q^{38} +1.26697e9 q^{41} -1.41295e7 q^{43} +1.21886e8 q^{44} -2.31816e9 q^{46} -2.65727e9 q^{47} -1.19422e9 q^{49} +1.19345e9 q^{52} +2.40270e9 q^{53} -1.50442e9 q^{56} -2.53896e8 q^{58} -7.49874e9 q^{59} -4.06483e9 q^{61} +1.48590e10 q^{62} +4.65830e8 q^{64} -6.87151e9 q^{67} -2.71528e8 q^{68} +1.32837e10 q^{71} +2.88758e10 q^{73} -1.18797e10 q^{74} -1.49191e10 q^{76} -3.13499e9 q^{77} +2.71003e10 q^{79} -7.09503e10 q^{82} -3.43653e10 q^{83} +7.91255e8 q^{86} +6.02263e9 q^{88} +6.35004e10 q^{89} -3.06963e10 q^{91} +4.50386e10 q^{92} +1.48807e11 q^{94} -1.96345e10 q^{97} +6.68765e10 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −56.0000 −1.23744 −0.618718 0.785613i \(-0.712348\pi\)
−0.618718 + 0.785613i \(0.712348\pi\)
\(3\) 0 0
\(4\) 1088.00 0.531250
\(5\) 0 0
\(6\) 0 0
\(7\) −27984.0 −0.629319 −0.314659 0.949205i \(-0.601890\pi\)
−0.314659 + 0.949205i \(0.601890\pi\)
\(8\) 53760.0 0.580049
\(9\) 0 0
\(10\) 0 0
\(11\) 112028. 0.209733 0.104867 0.994486i \(-0.466558\pi\)
0.104867 + 0.994486i \(0.466558\pi\)
\(12\) 0 0
\(13\) 1.09692e6 0.819384 0.409692 0.912224i \(-0.365636\pi\)
0.409692 + 0.912224i \(0.365636\pi\)
\(14\) 1.56710e6 0.778742
\(15\) 0 0
\(16\) −5.23878e6 −1.24902
\(17\) −249566. −0.0426301 −0.0213150 0.999773i \(-0.506785\pi\)
−0.0213150 + 0.999773i \(0.506785\pi\)
\(18\) 0 0
\(19\) −1.37124e7 −1.27048 −0.635242 0.772313i \(-0.719100\pi\)
−0.635242 + 0.772313i \(0.719100\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −6.27357e6 −0.259531
\(23\) 4.13957e7 1.34107 0.670537 0.741877i \(-0.266064\pi\)
0.670537 + 0.741877i \(0.266064\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −6.14276e7 −1.01394
\(27\) 0 0
\(28\) −3.04466e7 −0.334326
\(29\) 4.53385e6 0.0410467 0.0205233 0.999789i \(-0.493467\pi\)
0.0205233 + 0.999789i \(0.493467\pi\)
\(30\) 0 0
\(31\) −2.65339e8 −1.66461 −0.832304 0.554320i \(-0.812978\pi\)
−0.832304 + 0.554320i \(0.812978\pi\)
\(32\) 1.83271e8 0.965539
\(33\) 0 0
\(34\) 1.39757e7 0.0527521
\(35\) 0 0
\(36\) 0 0
\(37\) 2.12137e8 0.502929 0.251465 0.967866i \(-0.419088\pi\)
0.251465 + 0.967866i \(0.419088\pi\)
\(38\) 7.67896e8 1.57214
\(39\) 0 0
\(40\) 0 0
\(41\) 1.26697e9 1.70787 0.853936 0.520379i \(-0.174209\pi\)
0.853936 + 0.520379i \(0.174209\pi\)
\(42\) 0 0
\(43\) −1.41295e7 −0.0146572 −0.00732861 0.999973i \(-0.502333\pi\)
−0.00732861 + 0.999973i \(0.502333\pi\)
\(44\) 1.21886e8 0.111421
\(45\) 0 0
\(46\) −2.31816e9 −1.65949
\(47\) −2.65727e9 −1.69004 −0.845022 0.534732i \(-0.820413\pi\)
−0.845022 + 0.534732i \(0.820413\pi\)
\(48\) 0 0
\(49\) −1.19422e9 −0.603958
\(50\) 0 0
\(51\) 0 0
\(52\) 1.19345e9 0.435298
\(53\) 2.40270e9 0.789191 0.394596 0.918855i \(-0.370885\pi\)
0.394596 + 0.918855i \(0.370885\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.50442e9 −0.365035
\(57\) 0 0
\(58\) −2.53896e8 −0.0507927
\(59\) −7.49874e9 −1.36553 −0.682766 0.730637i \(-0.739223\pi\)
−0.682766 + 0.730637i \(0.739223\pi\)
\(60\) 0 0
\(61\) −4.06483e9 −0.616209 −0.308105 0.951352i \(-0.599695\pi\)
−0.308105 + 0.951352i \(0.599695\pi\)
\(62\) 1.48590e10 2.05985
\(63\) 0 0
\(64\) 4.65830e8 0.0542297
\(65\) 0 0
\(66\) 0 0
\(67\) −6.87151e9 −0.621786 −0.310893 0.950445i \(-0.600628\pi\)
−0.310893 + 0.950445i \(0.600628\pi\)
\(68\) −2.71528e8 −0.0226472
\(69\) 0 0
\(70\) 0 0
\(71\) 1.32837e10 0.873774 0.436887 0.899516i \(-0.356081\pi\)
0.436887 + 0.899516i \(0.356081\pi\)
\(72\) 0 0
\(73\) 2.88758e10 1.63027 0.815134 0.579273i \(-0.196663\pi\)
0.815134 + 0.579273i \(0.196663\pi\)
\(74\) −1.18797e10 −0.622343
\(75\) 0 0
\(76\) −1.49191e10 −0.674945
\(77\) −3.13499e9 −0.131989
\(78\) 0 0
\(79\) 2.71003e10 0.990889 0.495445 0.868640i \(-0.335005\pi\)
0.495445 + 0.868640i \(0.335005\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −7.09503e10 −2.11338
\(83\) −3.43653e10 −0.957613 −0.478807 0.877920i \(-0.658930\pi\)
−0.478807 + 0.877920i \(0.658930\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 7.91255e8 0.0181374
\(87\) 0 0
\(88\) 6.02263e9 0.121655
\(89\) 6.35004e10 1.20540 0.602700 0.797968i \(-0.294091\pi\)
0.602700 + 0.797968i \(0.294091\pi\)
\(90\) 0 0
\(91\) −3.06963e10 −0.515653
\(92\) 4.50386e10 0.712445
\(93\) 0 0
\(94\) 1.48807e11 2.09132
\(95\) 0 0
\(96\) 0 0
\(97\) −1.96345e10 −0.232153 −0.116077 0.993240i \(-0.537032\pi\)
−0.116077 + 0.993240i \(0.537032\pi\)
\(98\) 6.68765e10 0.747360
\(99\) 0 0
\(100\) 0 0
\(101\) 1.21293e11 1.14834 0.574169 0.818737i \(-0.305325\pi\)
0.574169 + 0.818737i \(0.305325\pi\)
\(102\) 0 0
\(103\) −1.32353e11 −1.12494 −0.562469 0.826818i \(-0.690149\pi\)
−0.562469 + 0.826818i \(0.690149\pi\)
\(104\) 5.89705e10 0.475282
\(105\) 0 0
\(106\) −1.34551e11 −0.976574
\(107\) 1.93758e10 0.133552 0.0667759 0.997768i \(-0.478729\pi\)
0.0667759 + 0.997768i \(0.478729\pi\)
\(108\) 0 0
\(109\) 2.32688e11 1.44853 0.724265 0.689522i \(-0.242179\pi\)
0.724265 + 0.689522i \(0.242179\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.46602e11 0.786034
\(113\) 3.19485e11 1.63124 0.815621 0.578586i \(-0.196395\pi\)
0.815621 + 0.578586i \(0.196395\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 4.93283e9 0.0218061
\(117\) 0 0
\(118\) 4.19929e11 1.68976
\(119\) 6.98385e9 0.0268279
\(120\) 0 0
\(121\) −2.72761e11 −0.956012
\(122\) 2.27630e11 0.762520
\(123\) 0 0
\(124\) −2.88689e11 −0.884323
\(125\) 0 0
\(126\) 0 0
\(127\) 5.61435e11 1.50792 0.753962 0.656918i \(-0.228140\pi\)
0.753962 + 0.656918i \(0.228140\pi\)
\(128\) −4.01426e11 −1.03264
\(129\) 0 0
\(130\) 0 0
\(131\) −4.57850e11 −1.03689 −0.518443 0.855112i \(-0.673488\pi\)
−0.518443 + 0.855112i \(0.673488\pi\)
\(132\) 0 0
\(133\) 3.83728e11 0.799539
\(134\) 3.84805e11 0.769421
\(135\) 0 0
\(136\) −1.34167e10 −0.0247275
\(137\) −4.20737e10 −0.0744814 −0.0372407 0.999306i \(-0.511857\pi\)
−0.0372407 + 0.999306i \(0.511857\pi\)
\(138\) 0 0
\(139\) 8.13653e11 1.33002 0.665009 0.746835i \(-0.268428\pi\)
0.665009 + 0.746835i \(0.268428\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −7.43889e11 −1.08124
\(143\) 1.22886e11 0.171852
\(144\) 0 0
\(145\) 0 0
\(146\) −1.61705e12 −2.01735
\(147\) 0 0
\(148\) 2.30805e11 0.267181
\(149\) 1.56119e11 0.174153 0.0870763 0.996202i \(-0.472248\pi\)
0.0870763 + 0.996202i \(0.472248\pi\)
\(150\) 0 0
\(151\) 5.50139e11 0.570295 0.285147 0.958484i \(-0.407957\pi\)
0.285147 + 0.958484i \(0.407957\pi\)
\(152\) −7.37180e11 −0.736943
\(153\) 0 0
\(154\) 1.75560e11 0.163328
\(155\) 0 0
\(156\) 0 0
\(157\) 1.87489e12 1.56866 0.784328 0.620346i \(-0.213008\pi\)
0.784328 + 0.620346i \(0.213008\pi\)
\(158\) −1.51762e12 −1.22616
\(159\) 0 0
\(160\) 0 0
\(161\) −1.15842e12 −0.843962
\(162\) 0 0
\(163\) 2.17731e12 1.48214 0.741068 0.671430i \(-0.234320\pi\)
0.741068 + 0.671430i \(0.234320\pi\)
\(164\) 1.37846e12 0.907307
\(165\) 0 0
\(166\) 1.92445e12 1.18499
\(167\) −1.18014e12 −0.703058 −0.351529 0.936177i \(-0.614338\pi\)
−0.351529 + 0.936177i \(0.614338\pi\)
\(168\) 0 0
\(169\) −5.88923e11 −0.328610
\(170\) 0 0
\(171\) 0 0
\(172\) −1.53729e10 −0.00778665
\(173\) 4.99181e11 0.244909 0.122454 0.992474i \(-0.460923\pi\)
0.122454 + 0.992474i \(0.460923\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −5.86890e11 −0.261961
\(177\) 0 0
\(178\) −3.55602e12 −1.49161
\(179\) −3.21203e12 −1.30643 −0.653217 0.757171i \(-0.726581\pi\)
−0.653217 + 0.757171i \(0.726581\pi\)
\(180\) 0 0
\(181\) −2.93245e12 −1.12201 −0.561007 0.827811i \(-0.689586\pi\)
−0.561007 + 0.827811i \(0.689586\pi\)
\(182\) 1.71899e12 0.638089
\(183\) 0 0
\(184\) 2.22543e12 0.777887
\(185\) 0 0
\(186\) 0 0
\(187\) −2.79584e10 −0.00894094
\(188\) −2.89111e12 −0.897836
\(189\) 0 0
\(190\) 0 0
\(191\) −2.28642e12 −0.650839 −0.325419 0.945570i \(-0.605505\pi\)
−0.325419 + 0.945570i \(0.605505\pi\)
\(192\) 0 0
\(193\) −5.86617e12 −1.57685 −0.788423 0.615133i \(-0.789102\pi\)
−0.788423 + 0.615133i \(0.789102\pi\)
\(194\) 1.09953e12 0.287275
\(195\) 0 0
\(196\) −1.29931e12 −0.320853
\(197\) 5.41588e11 0.130048 0.0650242 0.997884i \(-0.479288\pi\)
0.0650242 + 0.997884i \(0.479288\pi\)
\(198\) 0 0
\(199\) −6.12034e12 −1.39022 −0.695110 0.718903i \(-0.744644\pi\)
−0.695110 + 0.718903i \(0.744644\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) −6.79243e12 −1.42100
\(203\) −1.26875e11 −0.0258314
\(204\) 0 0
\(205\) 0 0
\(206\) 7.41176e12 1.39204
\(207\) 0 0
\(208\) −5.74654e12 −1.02343
\(209\) −1.53617e12 −0.266463
\(210\) 0 0
\(211\) 1.92054e12 0.316133 0.158067 0.987428i \(-0.449474\pi\)
0.158067 + 0.987428i \(0.449474\pi\)
\(212\) 2.61414e12 0.419258
\(213\) 0 0
\(214\) −1.08505e12 −0.165262
\(215\) 0 0
\(216\) 0 0
\(217\) 7.42525e12 1.04757
\(218\) −1.30305e13 −1.79246
\(219\) 0 0
\(220\) 0 0
\(221\) −2.73754e11 −0.0349304
\(222\) 0 0
\(223\) −8.97214e12 −1.08948 −0.544740 0.838605i \(-0.683372\pi\)
−0.544740 + 0.838605i \(0.683372\pi\)
\(224\) −5.12867e12 −0.607632
\(225\) 0 0
\(226\) −1.78911e13 −2.01856
\(227\) 1.08223e13 1.19173 0.595864 0.803086i \(-0.296810\pi\)
0.595864 + 0.803086i \(0.296810\pi\)
\(228\) 0 0
\(229\) −8.41898e12 −0.883414 −0.441707 0.897159i \(-0.645627\pi\)
−0.441707 + 0.897159i \(0.645627\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 2.43740e11 0.0238091
\(233\) −1.69297e13 −1.61507 −0.807537 0.589817i \(-0.799200\pi\)
−0.807537 + 0.589817i \(0.799200\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −8.15863e12 −0.725439
\(237\) 0 0
\(238\) −3.91096e11 −0.0331979
\(239\) −6.99835e12 −0.580506 −0.290253 0.956950i \(-0.593740\pi\)
−0.290253 + 0.956950i \(0.593740\pi\)
\(240\) 0 0
\(241\) −9.06165e12 −0.717982 −0.358991 0.933341i \(-0.616879\pi\)
−0.358991 + 0.933341i \(0.616879\pi\)
\(242\) 1.52746e13 1.18300
\(243\) 0 0
\(244\) −4.42253e12 −0.327361
\(245\) 0 0
\(246\) 0 0
\(247\) −1.50415e13 −1.04101
\(248\) −1.42646e13 −0.965553
\(249\) 0 0
\(250\) 0 0
\(251\) −2.59291e13 −1.64279 −0.821394 0.570362i \(-0.806803\pi\)
−0.821394 + 0.570362i \(0.806803\pi\)
\(252\) 0 0
\(253\) 4.63748e12 0.281267
\(254\) −3.14404e13 −1.86596
\(255\) 0 0
\(256\) 2.15259e13 1.22360
\(257\) 8.77582e12 0.488265 0.244133 0.969742i \(-0.421497\pi\)
0.244133 + 0.969742i \(0.421497\pi\)
\(258\) 0 0
\(259\) −5.93644e12 −0.316503
\(260\) 0 0
\(261\) 0 0
\(262\) 2.56396e13 1.28308
\(263\) 8.38157e12 0.410742 0.205371 0.978684i \(-0.434160\pi\)
0.205371 + 0.978684i \(0.434160\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −2.14888e13 −0.989380
\(267\) 0 0
\(268\) −7.47621e12 −0.330324
\(269\) 3.22500e13 1.39602 0.698011 0.716087i \(-0.254068\pi\)
0.698011 + 0.716087i \(0.254068\pi\)
\(270\) 0 0
\(271\) −2.89199e12 −0.120189 −0.0600947 0.998193i \(-0.519140\pi\)
−0.0600947 + 0.998193i \(0.519140\pi\)
\(272\) 1.30742e12 0.0532460
\(273\) 0 0
\(274\) 2.35613e12 0.0921660
\(275\) 0 0
\(276\) 0 0
\(277\) −9.73968e12 −0.358844 −0.179422 0.983772i \(-0.557423\pi\)
−0.179422 + 0.983772i \(0.557423\pi\)
\(278\) −4.55645e13 −1.64581
\(279\) 0 0
\(280\) 0 0
\(281\) 9.36930e12 0.319023 0.159512 0.987196i \(-0.449008\pi\)
0.159512 + 0.987196i \(0.449008\pi\)
\(282\) 0 0
\(283\) −4.70902e13 −1.54207 −0.771036 0.636791i \(-0.780261\pi\)
−0.771036 + 0.636791i \(0.780261\pi\)
\(284\) 1.44527e13 0.464193
\(285\) 0 0
\(286\) −6.88161e12 −0.212656
\(287\) −3.54549e13 −1.07480
\(288\) 0 0
\(289\) −3.42096e13 −0.998183
\(290\) 0 0
\(291\) 0 0
\(292\) 3.14169e13 0.866080
\(293\) −1.11851e13 −0.302600 −0.151300 0.988488i \(-0.548346\pi\)
−0.151300 + 0.988488i \(0.548346\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.14045e13 0.291723
\(297\) 0 0
\(298\) −8.74264e12 −0.215503
\(299\) 4.54079e13 1.09885
\(300\) 0 0
\(301\) 3.95401e11 0.00922406
\(302\) −3.08078e13 −0.705704
\(303\) 0 0
\(304\) 7.18364e13 1.58686
\(305\) 0 0
\(306\) 0 0
\(307\) −4.85767e13 −1.01664 −0.508320 0.861168i \(-0.669733\pi\)
−0.508320 + 0.861168i \(0.669733\pi\)
\(308\) −3.41087e12 −0.0701191
\(309\) 0 0
\(310\) 0 0
\(311\) −5.18050e13 −1.00969 −0.504846 0.863209i \(-0.668451\pi\)
−0.504846 + 0.863209i \(0.668451\pi\)
\(312\) 0 0
\(313\) −2.45673e13 −0.462236 −0.231118 0.972926i \(-0.574238\pi\)
−0.231118 + 0.972926i \(0.574238\pi\)
\(314\) −1.04994e14 −1.94111
\(315\) 0 0
\(316\) 2.94851e13 0.526410
\(317\) −1.74942e13 −0.306950 −0.153475 0.988153i \(-0.549046\pi\)
−0.153475 + 0.988153i \(0.549046\pi\)
\(318\) 0 0
\(319\) 5.07918e11 0.00860885
\(320\) 0 0
\(321\) 0 0
\(322\) 6.48714e13 1.04435
\(323\) 3.42215e12 0.0541609
\(324\) 0 0
\(325\) 0 0
\(326\) −1.21929e14 −1.83405
\(327\) 0 0
\(328\) 6.81123e13 0.990648
\(329\) 7.43611e13 1.06358
\(330\) 0 0
\(331\) −6.84453e13 −0.946868 −0.473434 0.880829i \(-0.656986\pi\)
−0.473434 + 0.880829i \(0.656986\pi\)
\(332\) −3.73894e13 −0.508732
\(333\) 0 0
\(334\) 6.60876e13 0.869990
\(335\) 0 0
\(336\) 0 0
\(337\) 7.69939e13 0.964921 0.482460 0.875918i \(-0.339743\pi\)
0.482460 + 0.875918i \(0.339743\pi\)
\(338\) 3.29797e13 0.406635
\(339\) 0 0
\(340\) 0 0
\(341\) −2.97254e13 −0.349123
\(342\) 0 0
\(343\) 8.87526e13 1.00940
\(344\) −7.59605e11 −0.00850190
\(345\) 0 0
\(346\) −2.79541e13 −0.303059
\(347\) 1.29519e14 1.38204 0.691020 0.722836i \(-0.257162\pi\)
0.691020 + 0.722836i \(0.257162\pi\)
\(348\) 0 0
\(349\) 3.64763e13 0.377112 0.188556 0.982062i \(-0.439619\pi\)
0.188556 + 0.982062i \(0.439619\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 2.05315e13 0.202505
\(353\) 1.03351e14 1.00358 0.501790 0.864989i \(-0.332675\pi\)
0.501790 + 0.864989i \(0.332675\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 6.90884e13 0.640369
\(357\) 0 0
\(358\) 1.79874e14 1.61663
\(359\) 1.52982e14 1.35400 0.677002 0.735981i \(-0.263279\pi\)
0.677002 + 0.735981i \(0.263279\pi\)
\(360\) 0 0
\(361\) 7.15402e13 0.614130
\(362\) 1.64217e14 1.38842
\(363\) 0 0
\(364\) −3.33975e13 −0.273941
\(365\) 0 0
\(366\) 0 0
\(367\) −1.49106e14 −1.16905 −0.584523 0.811377i \(-0.698718\pi\)
−0.584523 + 0.811377i \(0.698718\pi\)
\(368\) −2.16863e14 −1.67503
\(369\) 0 0
\(370\) 0 0
\(371\) −6.72371e13 −0.496653
\(372\) 0 0
\(373\) 5.61185e13 0.402446 0.201223 0.979545i \(-0.435508\pi\)
0.201223 + 0.979545i \(0.435508\pi\)
\(374\) 1.56567e12 0.0110638
\(375\) 0 0
\(376\) −1.42855e14 −0.980308
\(377\) 4.97328e12 0.0336330
\(378\) 0 0
\(379\) 7.57417e13 0.497530 0.248765 0.968564i \(-0.419975\pi\)
0.248765 + 0.968564i \(0.419975\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.28040e14 0.805372
\(383\) 2.02071e13 0.125289 0.0626443 0.998036i \(-0.480047\pi\)
0.0626443 + 0.998036i \(0.480047\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 3.28505e14 1.95125
\(387\) 0 0
\(388\) −2.13623e13 −0.123332
\(389\) −1.96148e14 −1.11651 −0.558253 0.829671i \(-0.688528\pi\)
−0.558253 + 0.829671i \(0.688528\pi\)
\(390\) 0 0
\(391\) −1.03310e13 −0.0571701
\(392\) −6.42014e13 −0.350325
\(393\) 0 0
\(394\) −3.03289e13 −0.160927
\(395\) 0 0
\(396\) 0 0
\(397\) −3.12645e14 −1.59112 −0.795562 0.605872i \(-0.792824\pi\)
−0.795562 + 0.605872i \(0.792824\pi\)
\(398\) 3.42739e14 1.72031
\(399\) 0 0
\(400\) 0 0
\(401\) 1.29292e14 0.622696 0.311348 0.950296i \(-0.399219\pi\)
0.311348 + 0.950296i \(0.399219\pi\)
\(402\) 0 0
\(403\) −2.91056e14 −1.36395
\(404\) 1.31967e14 0.610055
\(405\) 0 0
\(406\) 7.10501e12 0.0319648
\(407\) 2.37653e13 0.105481
\(408\) 0 0
\(409\) 1.70213e14 0.735385 0.367692 0.929948i \(-0.380148\pi\)
0.367692 + 0.929948i \(0.380148\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.44000e14 −0.597624
\(413\) 2.09845e14 0.859355
\(414\) 0 0
\(415\) 0 0
\(416\) 2.01034e14 0.791147
\(417\) 0 0
\(418\) 8.60258e13 0.329731
\(419\) −7.47406e13 −0.282735 −0.141367 0.989957i \(-0.545150\pi\)
−0.141367 + 0.989957i \(0.545150\pi\)
\(420\) 0 0
\(421\) −1.85677e14 −0.684239 −0.342119 0.939656i \(-0.611145\pi\)
−0.342119 + 0.939656i \(0.611145\pi\)
\(422\) −1.07550e14 −0.391195
\(423\) 0 0
\(424\) 1.29169e14 0.457769
\(425\) 0 0
\(426\) 0 0
\(427\) 1.13750e14 0.387792
\(428\) 2.10809e13 0.0709494
\(429\) 0 0
\(430\) 0 0
\(431\) 3.42728e14 1.11001 0.555003 0.831849i \(-0.312717\pi\)
0.555003 + 0.831849i \(0.312717\pi\)
\(432\) 0 0
\(433\) −4.77829e14 −1.50865 −0.754326 0.656500i \(-0.772036\pi\)
−0.754326 + 0.656500i \(0.772036\pi\)
\(434\) −4.15814e14 −1.29630
\(435\) 0 0
\(436\) 2.53164e14 0.769532
\(437\) −5.67636e14 −1.70381
\(438\) 0 0
\(439\) −5.50255e14 −1.61068 −0.805339 0.592814i \(-0.798017\pi\)
−0.805339 + 0.592814i \(0.798017\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 1.53302e13 0.0432242
\(443\) 2.60772e13 0.0726173 0.0363087 0.999341i \(-0.488440\pi\)
0.0363087 + 0.999341i \(0.488440\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 5.02440e14 1.34816
\(447\) 0 0
\(448\) −1.30358e13 −0.0341278
\(449\) 1.54203e14 0.398784 0.199392 0.979920i \(-0.436103\pi\)
0.199392 + 0.979920i \(0.436103\pi\)
\(450\) 0 0
\(451\) 1.41936e14 0.358197
\(452\) 3.47599e14 0.866598
\(453\) 0 0
\(454\) −6.06048e14 −1.47469
\(455\) 0 0
\(456\) 0 0
\(457\) −7.76543e13 −0.182233 −0.0911163 0.995840i \(-0.529044\pi\)
−0.0911163 + 0.995840i \(0.529044\pi\)
\(458\) 4.71463e14 1.09317
\(459\) 0 0
\(460\) 0 0
\(461\) −6.59986e14 −1.47632 −0.738158 0.674627i \(-0.764304\pi\)
−0.738158 + 0.674627i \(0.764304\pi\)
\(462\) 0 0
\(463\) 2.96908e14 0.648525 0.324262 0.945967i \(-0.394884\pi\)
0.324262 + 0.945967i \(0.394884\pi\)
\(464\) −2.37519e13 −0.0512683
\(465\) 0 0
\(466\) 9.48065e14 1.99855
\(467\) −8.37123e14 −1.74400 −0.872000 0.489507i \(-0.837177\pi\)
−0.872000 + 0.489507i \(0.837177\pi\)
\(468\) 0 0
\(469\) 1.92292e14 0.391302
\(470\) 0 0
\(471\) 0 0
\(472\) −4.03132e14 −0.792075
\(473\) −1.58291e12 −0.00307410
\(474\) 0 0
\(475\) 0 0
\(476\) 7.59843e12 0.0142523
\(477\) 0 0
\(478\) 3.91907e14 0.718340
\(479\) −1.21855e14 −0.220800 −0.110400 0.993887i \(-0.535213\pi\)
−0.110400 + 0.993887i \(0.535213\pi\)
\(480\) 0 0
\(481\) 2.32698e14 0.412092
\(482\) 5.07452e14 0.888457
\(483\) 0 0
\(484\) −2.96764e14 −0.507881
\(485\) 0 0
\(486\) 0 0
\(487\) −9.45308e13 −0.156374 −0.0781870 0.996939i \(-0.524913\pi\)
−0.0781870 + 0.996939i \(0.524913\pi\)
\(488\) −2.18525e14 −0.357431
\(489\) 0 0
\(490\) 0 0
\(491\) 2.09824e14 0.331823 0.165911 0.986141i \(-0.446943\pi\)
0.165911 + 0.986141i \(0.446943\pi\)
\(492\) 0 0
\(493\) −1.13149e12 −0.00174982
\(494\) 8.42321e14 1.28819
\(495\) 0 0
\(496\) 1.39005e15 2.07913
\(497\) −3.71732e14 −0.549882
\(498\) 0 0
\(499\) −2.64936e14 −0.383343 −0.191671 0.981459i \(-0.561391\pi\)
−0.191671 + 0.981459i \(0.561391\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 1.45203e15 2.03285
\(503\) −3.64546e14 −0.504811 −0.252405 0.967622i \(-0.581222\pi\)
−0.252405 + 0.967622i \(0.581222\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −2.59699e14 −0.348051
\(507\) 0 0
\(508\) 6.10842e14 0.801085
\(509\) −2.36671e14 −0.307042 −0.153521 0.988145i \(-0.549061\pi\)
−0.153521 + 0.988145i \(0.549061\pi\)
\(510\) 0 0
\(511\) −8.08062e14 −1.02596
\(512\) −3.83327e14 −0.481487
\(513\) 0 0
\(514\) −4.91446e14 −0.604197
\(515\) 0 0
\(516\) 0 0
\(517\) −2.97689e14 −0.354458
\(518\) 3.32441e14 0.391652
\(519\) 0 0
\(520\) 0 0
\(521\) −2.57734e14 −0.294147 −0.147073 0.989126i \(-0.546985\pi\)
−0.147073 + 0.989126i \(0.546985\pi\)
\(522\) 0 0
\(523\) 1.48029e15 1.65420 0.827101 0.562053i \(-0.189988\pi\)
0.827101 + 0.562053i \(0.189988\pi\)
\(524\) −4.98141e14 −0.550846
\(525\) 0 0
\(526\) −4.69368e14 −0.508267
\(527\) 6.62196e13 0.0709624
\(528\) 0 0
\(529\) 7.60797e14 0.798477
\(530\) 0 0
\(531\) 0 0
\(532\) 4.17496e14 0.424755
\(533\) 1.38977e15 1.39940
\(534\) 0 0
\(535\) 0 0
\(536\) −3.69413e14 −0.360666
\(537\) 0 0
\(538\) −1.80600e15 −1.72749
\(539\) −1.33786e14 −0.126670
\(540\) 0 0
\(541\) −1.21529e15 −1.12744 −0.563720 0.825966i \(-0.690630\pi\)
−0.563720 + 0.825966i \(0.690630\pi\)
\(542\) 1.61951e14 0.148727
\(543\) 0 0
\(544\) −4.57383e13 −0.0411610
\(545\) 0 0
\(546\) 0 0
\(547\) 4.93440e14 0.430828 0.215414 0.976523i \(-0.430890\pi\)
0.215414 + 0.976523i \(0.430890\pi\)
\(548\) −4.57762e13 −0.0395682
\(549\) 0 0
\(550\) 0 0
\(551\) −6.21701e13 −0.0521492
\(552\) 0 0
\(553\) −7.58375e14 −0.623585
\(554\) 5.45422e14 0.444047
\(555\) 0 0
\(556\) 8.85254e14 0.706572
\(557\) 5.15728e14 0.407584 0.203792 0.979014i \(-0.434673\pi\)
0.203792 + 0.979014i \(0.434673\pi\)
\(558\) 0 0
\(559\) −1.54990e13 −0.0120099
\(560\) 0 0
\(561\) 0 0
\(562\) −5.24681e14 −0.394771
\(563\) 2.86863e14 0.213737 0.106868 0.994273i \(-0.465918\pi\)
0.106868 + 0.994273i \(0.465918\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 2.63705e15 1.90822
\(567\) 0 0
\(568\) 7.14134e14 0.506832
\(569\) 5.24421e14 0.368606 0.184303 0.982869i \(-0.440997\pi\)
0.184303 + 0.982869i \(0.440997\pi\)
\(570\) 0 0
\(571\) −1.33323e15 −0.919194 −0.459597 0.888127i \(-0.652006\pi\)
−0.459597 + 0.888127i \(0.652006\pi\)
\(572\) 1.33700e14 0.0912963
\(573\) 0 0
\(574\) 1.98547e15 1.32999
\(575\) 0 0
\(576\) 0 0
\(577\) −1.21802e15 −0.792844 −0.396422 0.918068i \(-0.629748\pi\)
−0.396422 + 0.918068i \(0.629748\pi\)
\(578\) 1.91574e15 1.23519
\(579\) 0 0
\(580\) 0 0
\(581\) 9.61677e14 0.602644
\(582\) 0 0
\(583\) 2.69170e14 0.165519
\(584\) 1.55237e15 0.945634
\(585\) 0 0
\(586\) 6.26367e14 0.374448
\(587\) 7.51065e14 0.444803 0.222402 0.974955i \(-0.428610\pi\)
0.222402 + 0.974955i \(0.428610\pi\)
\(588\) 0 0
\(589\) 3.63844e15 2.11486
\(590\) 0 0
\(591\) 0 0
\(592\) −1.11134e15 −0.628170
\(593\) −1.95401e15 −1.09427 −0.547137 0.837043i \(-0.684282\pi\)
−0.547137 + 0.837043i \(0.684282\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 1.69857e14 0.0925186
\(597\) 0 0
\(598\) −2.54284e15 −1.35976
\(599\) −2.08899e15 −1.10685 −0.553425 0.832899i \(-0.686679\pi\)
−0.553425 + 0.832899i \(0.686679\pi\)
\(600\) 0 0
\(601\) −2.78205e15 −1.44729 −0.723645 0.690173i \(-0.757535\pi\)
−0.723645 + 0.690173i \(0.757535\pi\)
\(602\) −2.21425e13 −0.0114142
\(603\) 0 0
\(604\) 5.98551e14 0.302969
\(605\) 0 0
\(606\) 0 0
\(607\) −2.48334e15 −1.22320 −0.611601 0.791166i \(-0.709474\pi\)
−0.611601 + 0.791166i \(0.709474\pi\)
\(608\) −2.51309e15 −1.22670
\(609\) 0 0
\(610\) 0 0
\(611\) −2.91482e15 −1.38479
\(612\) 0 0
\(613\) 3.54510e15 1.65423 0.827116 0.562032i \(-0.189980\pi\)
0.827116 + 0.562032i \(0.189980\pi\)
\(614\) 2.72030e15 1.25803
\(615\) 0 0
\(616\) −1.68537e14 −0.0765600
\(617\) 1.21585e14 0.0547410 0.0273705 0.999625i \(-0.491287\pi\)
0.0273705 + 0.999625i \(0.491287\pi\)
\(618\) 0 0
\(619\) −1.33849e15 −0.591994 −0.295997 0.955189i \(-0.595652\pi\)
−0.295997 + 0.955189i \(0.595652\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 2.90108e15 1.24943
\(623\) −1.77700e15 −0.758581
\(624\) 0 0
\(625\) 0 0
\(626\) 1.37577e15 0.571988
\(627\) 0 0
\(628\) 2.03988e15 0.833348
\(629\) −5.29422e13 −0.0214399
\(630\) 0 0
\(631\) −2.23141e15 −0.888010 −0.444005 0.896024i \(-0.646443\pi\)
−0.444005 + 0.896024i \(0.646443\pi\)
\(632\) 1.45691e15 0.574764
\(633\) 0 0
\(634\) 9.79674e14 0.379831
\(635\) 0 0
\(636\) 0 0
\(637\) −1.30997e15 −0.494873
\(638\) −2.84434e13 −0.0106529
\(639\) 0 0
\(640\) 0 0
\(641\) −2.53917e15 −0.926773 −0.463386 0.886156i \(-0.653366\pi\)
−0.463386 + 0.886156i \(0.653366\pi\)
\(642\) 0 0
\(643\) 1.40215e15 0.503076 0.251538 0.967847i \(-0.419064\pi\)
0.251538 + 0.967847i \(0.419064\pi\)
\(644\) −1.26036e15 −0.448355
\(645\) 0 0
\(646\) −1.91641e14 −0.0670207
\(647\) 1.45024e15 0.502882 0.251441 0.967873i \(-0.419096\pi\)
0.251441 + 0.967873i \(0.419096\pi\)
\(648\) 0 0
\(649\) −8.40069e14 −0.286397
\(650\) 0 0
\(651\) 0 0
\(652\) 2.36891e15 0.787384
\(653\) 6.74967e14 0.222464 0.111232 0.993794i \(-0.464520\pi\)
0.111232 + 0.993794i \(0.464520\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −6.63738e15 −2.13317
\(657\) 0 0
\(658\) −4.16422e15 −1.31611
\(659\) −3.55334e15 −1.11370 −0.556849 0.830614i \(-0.687990\pi\)
−0.556849 + 0.830614i \(0.687990\pi\)
\(660\) 0 0
\(661\) 4.08998e15 1.26070 0.630351 0.776310i \(-0.282911\pi\)
0.630351 + 0.776310i \(0.282911\pi\)
\(662\) 3.83294e15 1.17169
\(663\) 0 0
\(664\) −1.84748e15 −0.555462
\(665\) 0 0
\(666\) 0 0
\(667\) 1.87682e14 0.0550466
\(668\) −1.28399e15 −0.373500
\(669\) 0 0
\(670\) 0 0
\(671\) −4.55375e14 −0.129239
\(672\) 0 0
\(673\) 2.31714e14 0.0646948 0.0323474 0.999477i \(-0.489702\pi\)
0.0323474 + 0.999477i \(0.489702\pi\)
\(674\) −4.31166e15 −1.19403
\(675\) 0 0
\(676\) −6.40748e14 −0.174574
\(677\) −4.21442e15 −1.13894 −0.569469 0.822013i \(-0.692851\pi\)
−0.569469 + 0.822013i \(0.692851\pi\)
\(678\) 0 0
\(679\) 5.49452e14 0.146099
\(680\) 0 0
\(681\) 0 0
\(682\) 1.66462e15 0.432018
\(683\) −9.30314e14 −0.239506 −0.119753 0.992804i \(-0.538210\pi\)
−0.119753 + 0.992804i \(0.538210\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −4.97015e15 −1.24907
\(687\) 0 0
\(688\) 7.40216e13 0.0183072
\(689\) 2.63557e15 0.646650
\(690\) 0 0
\(691\) −3.13309e15 −0.756561 −0.378280 0.925691i \(-0.623485\pi\)
−0.378280 + 0.925691i \(0.623485\pi\)
\(692\) 5.43109e14 0.130108
\(693\) 0 0
\(694\) −7.25305e15 −1.71019
\(695\) 0 0
\(696\) 0 0
\(697\) −3.16193e14 −0.0728067
\(698\) −2.04267e15 −0.466653
\(699\) 0 0
\(700\) 0 0
\(701\) 6.21787e15 1.38737 0.693686 0.720278i \(-0.255986\pi\)
0.693686 + 0.720278i \(0.255986\pi\)
\(702\) 0 0
\(703\) −2.90891e15 −0.638964
\(704\) 5.21860e13 0.0113738
\(705\) 0 0
\(706\) −5.78764e15 −1.24187
\(707\) −3.39428e15 −0.722670
\(708\) 0 0
\(709\) −1.38076e15 −0.289444 −0.144722 0.989472i \(-0.546229\pi\)
−0.144722 + 0.989472i \(0.546229\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 3.41378e15 0.699191
\(713\) −1.09839e16 −2.23236
\(714\) 0 0
\(715\) 0 0
\(716\) −3.49469e15 −0.694043
\(717\) 0 0
\(718\) −8.56698e15 −1.67550
\(719\) 1.22219e15 0.237209 0.118604 0.992942i \(-0.462158\pi\)
0.118604 + 0.992942i \(0.462158\pi\)
\(720\) 0 0
\(721\) 3.70376e15 0.707945
\(722\) −4.00625e15 −0.759948
\(723\) 0 0
\(724\) −3.19051e15 −0.596070
\(725\) 0 0
\(726\) 0 0
\(727\) 7.75213e15 1.41573 0.707867 0.706346i \(-0.249658\pi\)
0.707867 + 0.706346i \(0.249658\pi\)
\(728\) −1.65023e15 −0.299104
\(729\) 0 0
\(730\) 0 0
\(731\) 3.52625e12 0.000624839 0
\(732\) 0 0
\(733\) −8.29426e15 −1.44779 −0.723895 0.689910i \(-0.757650\pi\)
−0.723895 + 0.689910i \(0.757650\pi\)
\(734\) 8.34994e15 1.44662
\(735\) 0 0
\(736\) 7.58665e15 1.29486
\(737\) −7.69802e14 −0.130409
\(738\) 0 0
\(739\) −4.08381e14 −0.0681587 −0.0340793 0.999419i \(-0.510850\pi\)
−0.0340793 + 0.999419i \(0.510850\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 3.76528e15 0.614576
\(743\) 1.87602e15 0.303948 0.151974 0.988384i \(-0.451437\pi\)
0.151974 + 0.988384i \(0.451437\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −3.14264e15 −0.498002
\(747\) 0 0
\(748\) −3.04187e13 −0.00474987
\(749\) −5.42214e14 −0.0840467
\(750\) 0 0
\(751\) 9.97525e15 1.52372 0.761858 0.647744i \(-0.224287\pi\)
0.761858 + 0.647744i \(0.224287\pi\)
\(752\) 1.39209e16 2.11090
\(753\) 0 0
\(754\) −2.78504e14 −0.0416187
\(755\) 0 0
\(756\) 0 0
\(757\) −1.09293e16 −1.59795 −0.798977 0.601361i \(-0.794625\pi\)
−0.798977 + 0.601361i \(0.794625\pi\)
\(758\) −4.24153e15 −0.615662
\(759\) 0 0
\(760\) 0 0
\(761\) 3.39091e15 0.481615 0.240808 0.970573i \(-0.422588\pi\)
0.240808 + 0.970573i \(0.422588\pi\)
\(762\) 0 0
\(763\) −6.51153e15 −0.911587
\(764\) −2.48763e15 −0.345758
\(765\) 0 0
\(766\) −1.13160e15 −0.155037
\(767\) −8.22553e15 −1.11890
\(768\) 0 0
\(769\) −6.40507e15 −0.858873 −0.429437 0.903097i \(-0.641288\pi\)
−0.429437 + 0.903097i \(0.641288\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −6.38239e15 −0.837699
\(773\) 6.92865e15 0.902945 0.451473 0.892285i \(-0.350899\pi\)
0.451473 + 0.892285i \(0.350899\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.05555e15 −0.134660
\(777\) 0 0
\(778\) 1.09843e16 1.38161
\(779\) −1.73732e16 −2.16982
\(780\) 0 0
\(781\) 1.48815e15 0.183259
\(782\) 5.78534e14 0.0707444
\(783\) 0 0
\(784\) 6.25627e15 0.754358
\(785\) 0 0
\(786\) 0 0
\(787\) 2.81869e15 0.332802 0.166401 0.986058i \(-0.446785\pi\)
0.166401 + 0.986058i \(0.446785\pi\)
\(788\) 5.89248e14 0.0690882
\(789\) 0 0
\(790\) 0 0
\(791\) −8.94046e15 −1.02657
\(792\) 0 0
\(793\) −4.45880e15 −0.504912
\(794\) 1.75081e16 1.96892
\(795\) 0 0
\(796\) −6.65893e15 −0.738555
\(797\) 3.10085e15 0.341554 0.170777 0.985310i \(-0.445372\pi\)
0.170777 + 0.985310i \(0.445372\pi\)
\(798\) 0 0
\(799\) 6.63165e14 0.0720467
\(800\) 0 0
\(801\) 0 0
\(802\) −7.24033e15 −0.770547
\(803\) 3.23490e15 0.341921
\(804\) 0 0
\(805\) 0 0
\(806\) 1.62991e16 1.68780
\(807\) 0 0
\(808\) 6.52073e15 0.666092
\(809\) −1.32956e16 −1.34894 −0.674468 0.738304i \(-0.735627\pi\)
−0.674468 + 0.738304i \(0.735627\pi\)
\(810\) 0 0
\(811\) 6.98127e15 0.698747 0.349373 0.936984i \(-0.386394\pi\)
0.349373 + 0.936984i \(0.386394\pi\)
\(812\) −1.38040e14 −0.0137230
\(813\) 0 0
\(814\) −1.33086e15 −0.130526
\(815\) 0 0
\(816\) 0 0
\(817\) 1.93750e14 0.0186218
\(818\) −9.53193e15 −0.909992
\(819\) 0 0
\(820\) 0 0
\(821\) −6.46007e15 −0.604436 −0.302218 0.953239i \(-0.597727\pi\)
−0.302218 + 0.953239i \(0.597727\pi\)
\(822\) 0 0
\(823\) 4.54918e15 0.419985 0.209993 0.977703i \(-0.432656\pi\)
0.209993 + 0.977703i \(0.432656\pi\)
\(824\) −7.11529e15 −0.652519
\(825\) 0 0
\(826\) −1.17513e16 −1.06340
\(827\) −3.58820e15 −0.322549 −0.161275 0.986910i \(-0.551560\pi\)
−0.161275 + 0.986910i \(0.551560\pi\)
\(828\) 0 0
\(829\) 1.35990e16 1.20630 0.603152 0.797626i \(-0.293911\pi\)
0.603152 + 0.797626i \(0.293911\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 5.10979e14 0.0444350
\(833\) 2.98037e14 0.0257468
\(834\) 0 0
\(835\) 0 0
\(836\) −1.67136e15 −0.141558
\(837\) 0 0
\(838\) 4.18547e15 0.349867
\(839\) 1.49622e16 1.24253 0.621263 0.783602i \(-0.286620\pi\)
0.621263 + 0.783602i \(0.286620\pi\)
\(840\) 0 0
\(841\) −1.21800e16 −0.998315
\(842\) 1.03979e16 0.846702
\(843\) 0 0
\(844\) 2.08955e15 0.167946
\(845\) 0 0
\(846\) 0 0
\(847\) 7.63295e15 0.601636
\(848\) −1.25872e16 −0.985718
\(849\) 0 0
\(850\) 0 0
\(851\) 8.78156e15 0.674465
\(852\) 0 0
\(853\) 9.31290e15 0.706098 0.353049 0.935605i \(-0.385145\pi\)
0.353049 + 0.935605i \(0.385145\pi\)
\(854\) −6.37001e15 −0.479868
\(855\) 0 0
\(856\) 1.04165e15 0.0774666
\(857\) −2.01081e16 −1.48585 −0.742927 0.669373i \(-0.766563\pi\)
−0.742927 + 0.669373i \(0.766563\pi\)
\(858\) 0 0
\(859\) −1.78636e16 −1.30319 −0.651594 0.758568i \(-0.725900\pi\)
−0.651594 + 0.758568i \(0.725900\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.91928e16 −1.37356
\(863\) 2.51194e16 1.78628 0.893141 0.449776i \(-0.148496\pi\)
0.893141 + 0.449776i \(0.148496\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 2.67584e16 1.86686
\(867\) 0 0
\(868\) 8.07867e15 0.556521
\(869\) 3.03599e15 0.207822
\(870\) 0 0
\(871\) −7.53752e15 −0.509481
\(872\) 1.25093e16 0.840218
\(873\) 0 0
\(874\) 3.17876e16 2.10836
\(875\) 0 0
\(876\) 0 0
\(877\) −1.05249e16 −0.685045 −0.342522 0.939510i \(-0.611281\pi\)
−0.342522 + 0.939510i \(0.611281\pi\)
\(878\) 3.08143e16 1.99311
\(879\) 0 0
\(880\) 0 0
\(881\) 2.10050e16 1.33338 0.666691 0.745334i \(-0.267710\pi\)
0.666691 + 0.745334i \(0.267710\pi\)
\(882\) 0 0
\(883\) 1.55181e16 0.972868 0.486434 0.873717i \(-0.338297\pi\)
0.486434 + 0.873717i \(0.338297\pi\)
\(884\) −2.97845e14 −0.0185568
\(885\) 0 0
\(886\) −1.46032e15 −0.0898593
\(887\) 7.61183e15 0.465489 0.232744 0.972538i \(-0.425229\pi\)
0.232744 + 0.972538i \(0.425229\pi\)
\(888\) 0 0
\(889\) −1.57112e16 −0.948964
\(890\) 0 0
\(891\) 0 0
\(892\) −9.76169e15 −0.578786
\(893\) 3.64376e16 2.14717
\(894\) 0 0
\(895\) 0 0
\(896\) 1.12335e16 0.649863
\(897\) 0 0
\(898\) −8.63537e15 −0.493470
\(899\) −1.20301e15 −0.0683266
\(900\) 0 0
\(901\) −5.99632e14 −0.0336433
\(902\) −7.94842e15 −0.443246
\(903\) 0 0
\(904\) 1.71755e16 0.946200
\(905\) 0 0
\(906\) 0 0
\(907\) 2.39103e16 1.29344 0.646719 0.762728i \(-0.276141\pi\)
0.646719 + 0.762728i \(0.276141\pi\)
\(908\) 1.17747e16 0.633105
\(909\) 0 0
\(910\) 0 0
\(911\) −2.44923e16 −1.29324 −0.646618 0.762814i \(-0.723817\pi\)
−0.646618 + 0.762814i \(0.723817\pi\)
\(912\) 0 0
\(913\) −3.84987e15 −0.200843
\(914\) 4.34864e15 0.225501
\(915\) 0 0
\(916\) −9.15985e15 −0.469314
\(917\) 1.28125e16 0.652532
\(918\) 0 0
\(919\) −3.21575e16 −1.61826 −0.809128 0.587632i \(-0.800060\pi\)
−0.809128 + 0.587632i \(0.800060\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 3.69592e16 1.82685
\(923\) 1.45712e16 0.715956
\(924\) 0 0
\(925\) 0 0
\(926\) −1.66268e16 −0.802508
\(927\) 0 0
\(928\) 8.30925e14 0.0396322
\(929\) 1.27739e16 0.605672 0.302836 0.953043i \(-0.402067\pi\)
0.302836 + 0.953043i \(0.402067\pi\)
\(930\) 0 0
\(931\) 1.63757e16 0.767319
\(932\) −1.84196e16 −0.858008
\(933\) 0 0
\(934\) 4.68789e16 2.15809
\(935\) 0 0
\(936\) 0 0
\(937\) −1.11554e15 −0.0504565 −0.0252283 0.999682i \(-0.508031\pi\)
−0.0252283 + 0.999682i \(0.508031\pi\)
\(938\) −1.07684e16 −0.484211
\(939\) 0 0
\(940\) 0 0
\(941\) 3.97991e16 1.75845 0.879226 0.476405i \(-0.158060\pi\)
0.879226 + 0.476405i \(0.158060\pi\)
\(942\) 0 0
\(943\) 5.24471e16 2.29038
\(944\) 3.92843e16 1.70558
\(945\) 0 0
\(946\) 8.86427e13 0.00380401
\(947\) −3.26979e16 −1.39507 −0.697534 0.716552i \(-0.745719\pi\)
−0.697534 + 0.716552i \(0.745719\pi\)
\(948\) 0 0
\(949\) 3.16745e16 1.33581
\(950\) 0 0
\(951\) 0 0
\(952\) 3.75452e14 0.0155615
\(953\) −2.74452e15 −0.113098 −0.0565491 0.998400i \(-0.518010\pi\)
−0.0565491 + 0.998400i \(0.518010\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −7.61420e15 −0.308394
\(957\) 0 0
\(958\) 6.82390e15 0.273226
\(959\) 1.17739e15 0.0468725
\(960\) 0 0
\(961\) 4.49963e16 1.77092
\(962\) −1.30311e16 −0.509938
\(963\) 0 0
\(964\) −9.85907e15 −0.381428
\(965\) 0 0
\(966\) 0 0
\(967\) −7.81335e15 −0.297161 −0.148580 0.988900i \(-0.547470\pi\)
−0.148580 + 0.988900i \(0.547470\pi\)
\(968\) −1.46637e16 −0.554533
\(969\) 0 0
\(970\) 0 0
\(971\) −4.79623e16 −1.78318 −0.891588 0.452848i \(-0.850408\pi\)
−0.891588 + 0.452848i \(0.850408\pi\)
\(972\) 0 0
\(973\) −2.27693e16 −0.837005
\(974\) 5.29373e15 0.193503
\(975\) 0 0
\(976\) 2.12948e16 0.769660
\(977\) 4.28901e16 1.54148 0.770738 0.637152i \(-0.219888\pi\)
0.770738 + 0.637152i \(0.219888\pi\)
\(978\) 0 0
\(979\) 7.11382e15 0.252812
\(980\) 0 0
\(981\) 0 0
\(982\) −1.17501e16 −0.410610
\(983\) −1.90984e16 −0.663671 −0.331836 0.943337i \(-0.607668\pi\)
−0.331836 + 0.943337i \(0.607668\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 6.33637e13 0.00216530
\(987\) 0 0
\(988\) −1.63651e16 −0.553039
\(989\) −5.84903e14 −0.0196564
\(990\) 0 0
\(991\) −7.64435e15 −0.254060 −0.127030 0.991899i \(-0.540544\pi\)
−0.127030 + 0.991899i \(0.540544\pi\)
\(992\) −4.86291e16 −1.60724
\(993\) 0 0
\(994\) 2.08170e16 0.680445
\(995\) 0 0
\(996\) 0 0
\(997\) −4.72891e16 −1.52033 −0.760164 0.649731i \(-0.774882\pi\)
−0.760164 + 0.649731i \(0.774882\pi\)
\(998\) 1.48364e16 0.474362
\(999\) 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.a.1.1 1
3.2 odd 2 75.12.a.b.1.1 1
5.2 odd 4 225.12.b.b.199.1 2
5.3 odd 4 225.12.b.b.199.2 2
5.4 even 2 45.12.a.b.1.1 1
15.2 even 4 75.12.b.b.49.2 2
15.8 even 4 75.12.b.b.49.1 2
15.14 odd 2 15.12.a.a.1.1 1
60.59 even 2 240.12.a.e.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
15.12.a.a.1.1 1 15.14 odd 2
45.12.a.b.1.1 1 5.4 even 2
75.12.a.b.1.1 1 3.2 odd 2
75.12.b.b.49.1 2 15.8 even 4
75.12.b.b.49.2 2 15.2 even 4
225.12.a.a.1.1 1 1.1 even 1 trivial
225.12.b.b.199.1 2 5.2 odd 4
225.12.b.b.199.2 2 5.3 odd 4
240.12.a.e.1.1 1 60.59 even 2