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Properties

Label 416.7.h.b.207.1
Level $416$
Weight $7$
Character 416.207
Self dual yes
Analytic conductor $95.702$
Analytic rank $0$
Dimension $1$
CM discriminant -104
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [416,7,Mod(207,416)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("416.207"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(416, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 416 = 2^{5} \cdot 13 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 416.h (of order \(2\), degree \(1\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,0,50,0,218] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(95.7024987859\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 104)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 207.1
Character \(\chi\) \(=\) 416.207

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+50.0000 q^{3} +218.000 q^{5} -614.000 q^{7} +1771.00 q^{9} +2197.00 q^{13} +10900.0 q^{15} +3170.00 q^{17} -30700.0 q^{21} +31899.0 q^{25} +52100.0 q^{27} -27830.0 q^{31} -133852. q^{35} -13894.0 q^{37} +109850. q^{39} +111490. q^{43} +386078. q^{45} +128554. q^{47} +259347. q^{49} +158500. q^{51} -1.08739e6 q^{63} +478946. q^{65} -317990. q^{71} +1.59495e6 q^{75} +1.31394e6 q^{81} +691060. q^{85} -1.34896e6 q^{91} -1.39150e6 q^{93} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/416\mathbb{Z}\right)^\times\).

\(n\) \(261\) \(287\) \(353\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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
\(3\) 50.0000 1.85185 0.925926 − 0.377705i \(-0.123287\pi\)
0.925926 + 0.377705i \(0.123287\pi\)
\(4\) 0 0
\(5\) 218.000 1.74400 0.872000 − 0.489506i \(-0.162823\pi\)
0.872000 + 0.489506i \(0.162823\pi\)
\(6\) 0 0
\(7\) −614.000 −1.79009 −0.895044 − 0.445978i \(-0.852856\pi\)
−0.895044 + 0.445978i \(0.852856\pi\)
\(8\) 0 0
\(9\) 1771.00 2.42936
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 2197.00 1.00000
\(14\) 0 0
\(15\) 10900.0 3.22963
\(16\) 0 0
\(17\) 3170.00 0.645227 0.322613 − 0.946531i \(-0.395439\pi\)
0.322613 + 0.946531i \(0.395439\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −30700.0 −3.31498
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 31899.0 2.04154
\(26\) 0 0
\(27\) 52100.0 2.64695
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −27830.0 −0.934175 −0.467087 − 0.884211i \(-0.654697\pi\)
−0.467087 + 0.884211i \(0.654697\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −133852. −3.12191
\(36\) 0 0
\(37\) −13894.0 −0.274298 −0.137149 − 0.990550i \(-0.543794\pi\)
−0.137149 + 0.990550i \(0.543794\pi\)
\(38\) 0 0
\(39\) 109850. 1.85185
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 111490. 1.40227 0.701133 − 0.713030i \(-0.252678\pi\)
0.701133 + 0.713030i \(0.252678\pi\)
\(44\) 0 0
\(45\) 386078. 4.23680
\(46\) 0 0
\(47\) 128554. 1.23820 0.619102 − 0.785311i \(-0.287497\pi\)
0.619102 + 0.785311i \(0.287497\pi\)
\(48\) 0 0
\(49\) 259347. 2.20441
\(50\) 0 0
\(51\) 158500. 1.19486
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −1.08739e6 −4.34876
\(64\) 0 0
\(65\) 478946. 1.74400
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −317990. −0.888461 −0.444231 − 0.895913i \(-0.646523\pi\)
−0.444231 + 0.895913i \(0.646523\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 1.59495e6 3.78062
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 1.31394e6 2.47241
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 691060. 1.12528
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −1.34896e6 −1.79009
\(92\) 0 0
\(93\) −1.39150e6 −1.72995
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) −6.69260e6 −5.78132
\(106\) 0 0
\(107\) 2.44717e6 1.99762 0.998810 − 0.0487741i \(-0.0155314\pi\)
0.998810 + 0.0487741i \(0.0155314\pi\)
\(108\) 0 0
\(109\) −2.58127e6 −1.99321 −0.996607 − 0.0823070i \(-0.973771\pi\)
−0.996607 + 0.0823070i \(0.973771\pi\)
\(110\) 0 0
\(111\) −694700. −0.507959
\(112\) 0 0
\(113\) 419330. 0.290617 0.145308 − 0.989386i \(-0.453583\pi\)
0.145308 + 0.989386i \(0.453583\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.89089e6 2.42936
\(118\) 0 0
\(119\) −1.94638e6 −1.15501
\(120\) 0 0
\(121\) 1.77156e6 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 3.54773e6 1.81644
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 5.57450e6 2.59679
\(130\) 0 0
\(131\) −3.97192e6 −1.76680 −0.883398 − 0.468624i \(-0.844750\pi\)
−0.883398 + 0.468624i \(0.844750\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 1.13578e7 4.61629
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −5.34784e6 −1.99129 −0.995643 − 0.0932421i \(-0.970277\pi\)
−0.995643 + 0.0932421i \(0.970277\pi\)
\(140\) 0 0
\(141\) 6.42770e6 2.29297
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.29674e7 4.08225
\(148\) 0 0
\(149\) −6.40951e6 −1.93761 −0.968804 − 0.247827i \(-0.920283\pi\)
−0.968804 + 0.247827i \(0.920283\pi\)
\(150\) 0 0
\(151\) −5.65031e6 −1.64112 −0.820562 − 0.571557i \(-0.806339\pi\)
−0.820562 + 0.571557i \(0.806339\pi\)
\(152\) 0 0
\(153\) 5.61407e6 1.56749
\(154\) 0 0
\(155\) −6.06694e6 −1.62920
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 8.80527e6 1.89057 0.945287 − 0.326241i \(-0.105782\pi\)
0.945287 + 0.326241i \(0.105782\pi\)
\(168\) 0 0
\(169\) 4.82681e6 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −1.95860e7 −3.65453
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 1.12106e7 1.95465 0.977325 − 0.211746i \(-0.0679149\pi\)
0.977325 + 0.211746i \(0.0679149\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −3.02889e6 −0.478375
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −3.19894e7 −4.73828
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 2.39473e7 3.22963
\(196\) 0 0
\(197\) 4.37695e6 0.572497 0.286249 − 0.958155i \(-0.407592\pi\)
0.286249 + 0.958155i \(0.407592\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.82029e7 −1.93773 −0.968863 − 0.247598i \(-0.920359\pi\)
−0.968863 + 0.247598i \(0.920359\pi\)
\(212\) 0 0
\(213\) −1.58995e7 −1.64530
\(214\) 0 0
\(215\) 2.43048e7 2.44555
\(216\) 0 0
\(217\) 1.70876e7 1.67225
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 6.96449e6 0.645227
\(222\) 0 0
\(223\) −3.08293e6 −0.278003 −0.139002 − 0.990292i \(-0.544389\pi\)
−0.139002 + 0.990292i \(0.544389\pi\)
\(224\) 0 0
\(225\) 5.64931e7 4.95962
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −7.53559e6 −0.627496 −0.313748 − 0.949506i \(-0.601585\pi\)
−0.313748 + 0.949506i \(0.601585\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −1.04145e7 −0.823325 −0.411662 − 0.911336i \(-0.635052\pi\)
−0.411662 + 0.911336i \(0.635052\pi\)
\(234\) 0 0
\(235\) 2.80248e7 2.15943
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 1.69138e7 1.23893 0.619465 − 0.785024i \(-0.287350\pi\)
0.619465 + 0.785024i \(0.287350\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 2.77162e7 1.93159
\(244\) 0 0
\(245\) 5.65376e7 3.84450
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.21784e7 0.770139 0.385070 − 0.922888i \(-0.374177\pi\)
0.385070 + 0.922888i \(0.374177\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 3.45530e7 2.08384
\(256\) 0 0
\(257\) −1.58527e7 −0.933906 −0.466953 − 0.884282i \(-0.654648\pi\)
−0.466953 + 0.884282i \(0.654648\pi\)
\(258\) 0 0
\(259\) 8.53092e6 0.491017
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −1.63301e7 −0.820503 −0.410252 − 0.911972i \(-0.634559\pi\)
−0.410252 + 0.911972i \(0.634559\pi\)
\(272\) 0 0
\(273\) −6.74479e7 −3.31498
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) −4.92869e7 −2.26944
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −3.82458e7 −1.68742 −0.843712 − 0.536796i \(-0.819635\pi\)
−0.843712 + 0.536796i \(0.819635\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −1.40887e7 −0.583682
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 4.91523e7 1.95407 0.977037 − 0.213069i \(-0.0683458\pi\)
0.977037 + 0.213069i \(0.0683458\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −6.84549e7 −2.51018
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −6.13286e7 −2.00000 −1.00000 0.000361172i \(-0.999885\pi\)
−1.00000 0.000361172i \(0.999885\pi\)
\(314\) 0 0
\(315\) −2.37052e8 −7.58423
\(316\) 0 0
\(317\) 4.54692e7 1.42738 0.713690 − 0.700461i \(-0.247022\pi\)
0.713690 + 0.700461i \(0.247022\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 1.22358e8 3.69930
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 7.00821e7 2.04154
\(326\) 0 0
\(327\) −1.29064e8 −3.69114
\(328\) 0 0
\(329\) −7.89322e7 −2.21649
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −2.46063e7 −0.666366
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −3.55087e7 −0.927779 −0.463890 − 0.885893i \(-0.653547\pi\)
−0.463890 + 0.885893i \(0.653547\pi\)
\(338\) 0 0
\(339\) 2.09665e7 0.538179
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −8.70026e7 −2.15600
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.81636e7 −0.434723 −0.217361 − 0.976091i \(-0.569745\pi\)
−0.217361 + 0.976091i \(0.569745\pi\)
\(348\) 0 0
\(349\) −5.97550e7 −1.40572 −0.702859 − 0.711329i \(-0.748094\pi\)
−0.702859 + 0.711329i \(0.748094\pi\)
\(350\) 0 0
\(351\) 1.14464e8 2.64695
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −6.93218e7 −1.54948
\(356\) 0 0
\(357\) −9.73190e7 −2.13891
\(358\) 0 0
\(359\) −8.10358e7 −1.75143 −0.875716 − 0.482826i \(-0.839610\pi\)
−0.875716 + 0.482826i \(0.839610\pi\)
\(360\) 0 0
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) 8.85780e7 1.85185
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 1.77387e8 3.36378
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 6.77127e6 0.120524 0.0602621 − 0.998183i \(-0.480806\pi\)
0.0602621 + 0.998183i \(0.480806\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 1.97449e8 3.40660
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −1.98596e8 −3.27184
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.09696e8 −1.75315 −0.876577 − 0.481261i \(-0.840179\pi\)
−0.876577 + 0.481261i \(0.840179\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −6.11425e7 −0.934175
\(404\) 0 0
\(405\) 2.86439e8 4.31189
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −2.67392e8 −3.68757
\(418\) 0 0
\(419\) −1.45495e8 −1.97791 −0.988955 − 0.148219i \(-0.952646\pi\)
−0.988955 + 0.148219i \(0.952646\pi\)
\(420\) 0 0
\(421\) 1.45559e8 1.95072 0.975358 − 0.220627i \(-0.0708105\pi\)
0.975358 + 0.220627i \(0.0708105\pi\)
\(422\) 0 0
\(423\) 2.27669e8 3.00804
\(424\) 0 0
\(425\) 1.01120e8 1.31725
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.47692e8 1.84469 0.922347 − 0.386363i \(-0.126269\pi\)
0.922347 + 0.386363i \(0.126269\pi\)
\(432\) 0 0
\(433\) 5.54331e7 0.682819 0.341409 − 0.939915i \(-0.389096\pi\)
0.341409 + 0.939915i \(0.389096\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 4.59304e8 5.35530
\(442\) 0 0
\(443\) −8.60423e7 −0.989693 −0.494847 − 0.868980i \(-0.664776\pi\)
−0.494847 + 0.868980i \(0.664776\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −3.20476e8 −3.58816
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.82516e8 −3.03912
\(454\) 0 0
\(455\) −2.94073e8 −3.12191
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 1.65157e8 1.70789
\(460\) 0 0
\(461\) −1.72671e8 −1.76245 −0.881226 − 0.472696i \(-0.843281\pi\)
−0.881226 + 0.472696i \(0.843281\pi\)
\(462\) 0 0
\(463\) 1.05552e8 1.06347 0.531733 − 0.846912i \(-0.321541\pi\)
0.531733 + 0.846912i \(0.321541\pi\)
\(464\) 0 0
\(465\) −3.03347e8 −3.01704
\(466\) 0 0
\(467\) −2.87454e7 −0.282239 −0.141120 − 0.989993i \(-0.545070\pi\)
−0.141120 + 0.989993i \(0.545070\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.01712e7 −0.0925476 −0.0462738 − 0.998929i \(-0.514735\pi\)
−0.0462738 + 0.998929i \(0.514735\pi\)
\(480\) 0 0
\(481\) −3.05251e7 −0.274298
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.97305e8 1.70825 0.854126 − 0.520066i \(-0.174093\pi\)
0.854126 + 0.520066i \(0.174093\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 7.99411e7 0.675345 0.337672 − 0.941264i \(-0.390360\pi\)
0.337672 + 0.941264i \(0.390360\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.95246e8 1.59042
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 4.40264e8 3.50106
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 2.41340e8 1.85185
\(508\) 0 0
\(509\) 2.61109e8 1.98001 0.990006 − 0.141024i \(-0.0450394\pi\)
0.990006 + 0.141024i \(0.0450394\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −2.81764e8 −1.99238 −0.996191 − 0.0872021i \(-0.972207\pi\)
−0.996191 + 0.0872021i \(0.972207\pi\)
\(522\) 0 0
\(523\) 8.89336e7 0.621671 0.310836 − 0.950464i \(-0.399391\pi\)
0.310836 + 0.950464i \(0.399391\pi\)
\(524\) 0 0
\(525\) −9.79299e8 −6.76764
\(526\) 0 0
\(527\) −8.82211e7 −0.602755
\(528\) 0 0
\(529\) 1.48036e8 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 5.33483e8 3.48385
\(536\) 0 0
\(537\) 5.60529e8 3.61972
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3.94438e7 −0.249108 −0.124554 − 0.992213i \(-0.539750\pi\)
−0.124554 + 0.992213i \(0.539750\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −5.62717e8 −3.47617
\(546\) 0 0
\(547\) −3.05993e8 −1.86960 −0.934801 − 0.355171i \(-0.884423\pi\)
−0.934801 + 0.355171i \(0.884423\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) −1.51445e8 −0.885880
\(556\) 0 0
\(557\) −1.62805e8 −0.942111 −0.471055 − 0.882104i \(-0.656127\pi\)
−0.471055 + 0.882104i \(0.656127\pi\)
\(558\) 0 0
\(559\) 2.44944e8 1.40227
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −3.35941e8 −1.88251 −0.941255 − 0.337697i \(-0.890352\pi\)
−0.941255 + 0.337697i \(0.890352\pi\)
\(564\) 0 0
\(565\) 9.14139e7 0.506836
\(566\) 0 0
\(567\) −8.06760e8 −4.42583
\(568\) 0 0
\(569\) −3.45458e8 −1.87524 −0.937622 − 0.347656i \(-0.886978\pi\)
−0.937622 + 0.347656i \(0.886978\pi\)
\(570\) 0 0
\(571\) −6.07173e7 −0.326140 −0.163070 − 0.986615i \(-0.552140\pi\)
−0.163070 + 0.986615i \(0.552140\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 8.48213e8 4.23680
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 2.18848e8 1.06018
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) −4.24311e8 −2.01434
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −3.37535e8 −1.55487 −0.777437 − 0.628961i \(-0.783480\pi\)
−0.777437 + 0.628961i \(0.783480\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.86200e8 1.74400
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.82433e8 1.23820
\(612\) 0 0
\(613\) −8.30155e7 −0.360394 −0.180197 − 0.983631i \(-0.557674\pi\)
−0.180197 + 0.983631i \(0.557674\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.74984e8 1.12633
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −4.40440e7 −0.176984
\(630\) 0 0
\(631\) 4.93350e8 1.96366 0.981832 − 0.189754i \(-0.0607689\pi\)
0.981832 + 0.189754i \(0.0607689\pi\)
\(632\) 0 0
\(633\) −9.10143e8 −3.58838
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 5.69785e8 2.20441
\(638\) 0 0
\(639\) −5.63160e8 −2.15839
\(640\) 0 0
\(641\) 4.15127e8 1.57618 0.788091 − 0.615558i \(-0.211069\pi\)
0.788091 + 0.615558i \(0.211069\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 1.21524e9 4.52880
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 8.54381e8 3.09677
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −8.65878e8 −3.08129
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 4.36753e8 1.52609 0.763044 − 0.646346i \(-0.223704\pi\)
0.763044 + 0.646346i \(0.223704\pi\)
\(660\) 0 0
\(661\) −3.83714e8 −1.32863 −0.664314 − 0.747454i \(-0.731276\pi\)
−0.664314 + 0.747454i \(0.731276\pi\)
\(662\) 0 0
\(663\) 3.48224e8 1.19486
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1.54147e8 −0.514821
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 6.07111e8 1.99170 0.995848 − 0.0910264i \(-0.0290148\pi\)
0.995848 + 0.0910264i \(0.0290148\pi\)
\(674\) 0 0
\(675\) 1.66194e9 5.40385
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −3.76780e8 −1.16203
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.16583e9 −3.47280
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −5.20726e8 −1.52468
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 1.40124e9 3.99894
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −2.61192e8 −0.732859 −0.366430 − 0.930446i \(-0.619420\pi\)
−0.366430 + 0.930446i \(0.619420\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 8.45689e8 2.29431
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 4.27945e8 1.10460
\(730\) 0 0
\(731\) 3.53423e8 0.904780
\(732\) 0 0
\(733\) −6.63862e8 −1.68564 −0.842821 − 0.538193i \(-0.819107\pi\)
−0.842821 + 0.538193i \(0.819107\pi\)
\(734\) 0 0
\(735\) 2.82688e9 7.11944
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 3.66280e8 0.892989 0.446495 − 0.894786i \(-0.352672\pi\)
0.446495 + 0.894786i \(0.352672\pi\)
\(744\) 0 0
\(745\) −1.39727e9 −3.37919
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −1.50256e9 −3.57591
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 6.08920e8 1.42618
\(754\) 0 0
\(755\) −1.23177e9 −2.86212
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.58490e9 3.56803
\(764\) 0 0
\(765\) 1.22387e9 2.73369
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −7.92634e8 −1.72946
\(772\) 0 0
\(773\) −7.84772e8 −1.69905 −0.849523 − 0.527552i \(-0.823110\pi\)
−0.849523 + 0.527552i \(0.823110\pi\)
\(774\) 0 0
\(775\) −8.87749e8 −1.90715
\(776\) 0 0
\(777\) 4.26546e8 0.909290
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −2.57469e8 −0.520229
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 4.07516e8 0.798922
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −5.63710e8 −1.06466 −0.532329 − 0.846538i \(-0.678683\pi\)
−0.532329 + 0.846538i \(0.678683\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) −8.16504e8 −1.51945
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −2.38900e9 −4.34876
\(820\) 0 0
\(821\) 7.52144e8 1.35916 0.679581 − 0.733600i \(-0.262162\pi\)
0.679581 + 0.733600i \(0.262162\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 8.22130e8 1.42235
\(834\) 0 0
\(835\) 1.91955e9 3.29716
\(836\) 0 0
\(837\) −1.44994e9 −2.47272
\(838\) 0 0
\(839\) −7.83672e8 −1.32693 −0.663466 − 0.748207i \(-0.730915\pi\)
−0.663466 + 0.748207i \(0.730915\pi\)
\(840\) 0 0
\(841\) 5.94823e8 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.05224e9 1.74400
\(846\) 0 0
\(847\) −1.08774e9 −1.79009
\(848\) 0 0
\(849\) −1.91229e9 −3.12486
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.14583e9 −1.84617 −0.923086 − 0.384593i \(-0.874342\pi\)
−0.923086 + 0.384593i \(0.874342\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.10042e8 −0.174831 −0.0874153 − 0.996172i \(-0.527861\pi\)
−0.0874153 + 0.996172i \(0.527861\pi\)
\(858\) 0 0
\(859\) 1.25286e9 1.97661 0.988307 − 0.152475i \(-0.0487243\pi\)
0.988307 + 0.152475i \(0.0487243\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −5.47423e8 −0.851708 −0.425854 − 0.904792i \(-0.640026\pi\)
−0.425854 + 0.904792i \(0.640026\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −7.04433e8 −1.08089
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −2.17831e9 −3.25158
\(876\) 0 0
\(877\) 2.72943e8 0.404645 0.202322 − 0.979319i \(-0.435151\pi\)
0.202322 + 0.979319i \(0.435151\pi\)
\(878\) 0 0
\(879\) 2.45762e9 3.61866
\(880\) 0 0
\(881\) −3.77235e8 −0.551676 −0.275838 − 0.961204i \(-0.588955\pi\)
−0.275838 + 0.961204i \(0.588955\pi\)
\(882\) 0 0
\(883\) −1.25086e9 −1.81688 −0.908442 − 0.418011i \(-0.862727\pi\)
−0.908442 + 0.418011i \(0.862727\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 2.44391e9 3.40891
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −3.42274e9 −4.64848
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.36131e8 0.182446 0.0912232 − 0.995830i \(-0.470922\pi\)
0.0912232 + 0.995830i \(0.470922\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 2.43876e9 3.16272
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −6.98624e8 −0.888461
\(924\) 0 0
\(925\) −4.43205e8 −0.559989
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.61380e9 −1.96169 −0.980844 − 0.194793i \(-0.937597\pi\)
−0.980844 + 0.194793i \(0.937597\pi\)
\(938\) 0 0
\(939\) −3.06643e9 −3.70370
\(940\) 0 0
\(941\) 1.11296e9 1.33571 0.667854 − 0.744292i \(-0.267213\pi\)
0.667854 + 0.744292i \(0.267213\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −6.97369e9 −8.26356
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 2.27346e9 2.64330
\(952\) 0 0
\(953\) 1.68629e9 1.94829 0.974146 − 0.225921i \(-0.0725390\pi\)
0.974146 + 0.225921i \(0.0725390\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −1.12995e8 −0.127318
\(962\) 0 0
\(963\) 4.33394e9 4.85293
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.27382e9 −1.40873 −0.704365 − 0.709838i \(-0.748768\pi\)
−0.704365 + 0.709838i \(0.748768\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 7.85702e7 0.0858223 0.0429111 − 0.999079i \(-0.486337\pi\)
0.0429111 + 0.999079i \(0.486337\pi\)
\(972\) 0 0
\(973\) 3.28357e9 3.56458
\(974\) 0 0
\(975\) 3.50411e9 3.78062
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −4.57143e9 −4.84223
\(982\) 0 0
\(983\) 1.53648e9 1.61758 0.808792 − 0.588095i \(-0.200122\pi\)
0.808792 + 0.588095i \(0.200122\pi\)
\(984\) 0 0
\(985\) 9.54176e8 0.998435
\(986\) 0 0
\(987\) −3.94661e9 −4.10462
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 0 0
\(999\) −7.23877e8 −0.726053
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 416.7.h.b.207.1 1
4.3 odd 2 104.7.h.a.51.1 ✓ 1
8.3 odd 2 416.7.h.a.207.1 1
8.5 even 2 104.7.h.b.51.1 yes 1
13.12 even 2 416.7.h.a.207.1 1
52.51 odd 2 104.7.h.b.51.1 yes 1
104.51 odd 2 CM 416.7.h.b.207.1 1
104.77 even 2 104.7.h.a.51.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
104.7.h.a.51.1 ✓ 1 4.3 odd 2
104.7.h.a.51.1 ✓ 1 104.77 even 2
104.7.h.b.51.1 yes 1 8.5 even 2
104.7.h.b.51.1 yes 1 52.51 odd 2
416.7.h.a.207.1 1 8.3 odd 2
416.7.h.a.207.1 1 13.12 even 2
416.7.h.b.207.1 1 1.1 even 1 trivial
416.7.h.b.207.1 1 104.51 odd 2 CM