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Properties

Label 435.3.b.c.434.1
Level $435$
Weight $3$
Character 435.434
Self dual yes
Analytic conductor $11.853$
Analytic rank $0$
Dimension $1$
CM discriminant -435
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [435,3,Mod(434,435)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(435, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("435.434");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 435 = 3 \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 435.b (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(11.8528914997\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 434.1
Character \(\chi\) \(=\) 435.434

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} +4.00000 q^{4} -5.00000 q^{5} +9.00000 q^{9} +7.00000 q^{11} +12.0000 q^{12} -15.0000 q^{15} +16.0000 q^{16} -20.0000 q^{20} +41.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} +29.0000 q^{29} +21.0000 q^{33} +36.0000 q^{36} -71.0000 q^{37} -53.0000 q^{41} -59.0000 q^{43} +28.0000 q^{44} -45.0000 q^{45} +48.0000 q^{48} +49.0000 q^{49} -19.0000 q^{53} -35.0000 q^{55} -60.0000 q^{60} +64.0000 q^{64} +123.000 q^{69} +1.00000 q^{73} +75.0000 q^{75} -80.0000 q^{80} +81.0000 q^{81} -79.0000 q^{83} +87.0000 q^{87} -62.0000 q^{89} +164.000 q^{92} +49.0000 q^{97} +63.0000 q^{99} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/435\mathbb{Z}\right)^\times\).

\(n\) \(31\) \(146\) \(262\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\)

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 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 3.00000 1.00000
\(4\) 4.00000 1.00000
\(5\) −5.00000 −1.00000
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) 7.00000 0.636364 0.318182 − 0.948030i \(-0.396928\pi\)
0.318182 + 0.948030i \(0.396928\pi\)
\(12\) 12.0000 1.00000
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −15.0000 −1.00000
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −20.0000 −1.00000
\(21\) 0 0
\(22\) 0 0
\(23\) 41.0000 1.78261 0.891304 − 0.453406i \(-0.149791\pi\)
0.891304 + 0.453406i \(0.149791\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 27.0000 1.00000
\(28\) 0 0
\(29\) 29.0000 1.00000
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 21.0000 0.636364
\(34\) 0 0
\(35\) 0 0
\(36\) 36.0000 1.00000
\(37\) −71.0000 −1.91892 −0.959459 − 0.281847i \(-0.909053\pi\)
−0.959459 + 0.281847i \(0.909053\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −53.0000 −1.29268 −0.646341 − 0.763048i \(-0.723702\pi\)
−0.646341 + 0.763048i \(0.723702\pi\)
\(42\) 0 0
\(43\) −59.0000 −1.37209 −0.686047 − 0.727558i \(-0.740655\pi\)
−0.686047 + 0.727558i \(0.740655\pi\)
\(44\) 28.0000 0.636364
\(45\) −45.0000 −1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 48.0000 1.00000
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −19.0000 −0.358491 −0.179245 − 0.983804i \(-0.557366\pi\)
−0.179245 + 0.983804i \(0.557366\pi\)
\(54\) 0 0
\(55\) −35.0000 −0.636364
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) −60.0000 −1.00000
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 123.000 1.78261
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 1.00000 0.0136986 0.00684932 − 0.999977i \(-0.497820\pi\)
0.00684932 + 0.999977i \(0.497820\pi\)
\(74\) 0 0
\(75\) 75.0000 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −80.0000 −1.00000
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) −79.0000 −0.951807 −0.475904 − 0.879497i \(-0.657879\pi\)
−0.475904 + 0.879497i \(0.657879\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 87.0000 1.00000
\(88\) 0 0
\(89\) −62.0000 −0.696629 −0.348315 − 0.937378i \(-0.613246\pi\)
−0.348315 + 0.937378i \(0.613246\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 164.000 1.78261
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 49.0000 0.505155 0.252577 − 0.967577i \(-0.418722\pi\)
0.252577 + 0.967577i \(0.418722\pi\)
\(98\) 0 0
\(99\) 63.0000 0.636364
\(100\) 100.000 1.00000
\(101\) −173.000 −1.71287 −0.856436 − 0.516254i \(-0.827326\pi\)
−0.856436 + 0.516254i \(0.827326\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 134.000 1.25234 0.626168 − 0.779688i \(-0.284622\pi\)
0.626168 + 0.779688i \(0.284622\pi\)
\(108\) 108.000 1.00000
\(109\) −217.000 −1.99083 −0.995413 − 0.0956727i \(-0.969500\pi\)
−0.995413 + 0.0956727i \(0.969500\pi\)
\(110\) 0 0
\(111\) −213.000 −1.91892
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) −205.000 −1.78261
\(116\) 116.000 1.00000
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −72.0000 −0.595041
\(122\) 0 0
\(123\) −159.000 −1.29268
\(124\) 0 0
\(125\) −125.000 −1.00000
\(126\) 0 0
\(127\) 109.000 0.858268 0.429134 − 0.903241i \(-0.358819\pi\)
0.429134 + 0.903241i \(0.358819\pi\)
\(128\) 0 0
\(129\) −177.000 −1.37209
\(130\) 0 0
\(131\) 202.000 1.54198 0.770992 − 0.636844i \(-0.219761\pi\)
0.770992 + 0.636844i \(0.219761\pi\)
\(132\) 84.0000 0.636364
\(133\) 0 0
\(134\) 0 0
\(135\) −135.000 −1.00000
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) −157.000 −1.12950 −0.564748 − 0.825263i \(-0.691027\pi\)
−0.564748 + 0.825263i \(0.691027\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 144.000 1.00000
\(145\) −145.000 −1.00000
\(146\) 0 0
\(147\) 147.000 1.00000
\(148\) −284.000 −1.91892
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −133.000 −0.880795 −0.440397 − 0.897803i \(-0.645162\pi\)
−0.440397 + 0.897803i \(0.645162\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −266.000 −1.69427 −0.847134 − 0.531380i \(-0.821674\pi\)
−0.847134 + 0.531380i \(0.821674\pi\)
\(158\) 0 0
\(159\) −57.0000 −0.358491
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 181.000 1.11043 0.555215 − 0.831707i \(-0.312636\pi\)
0.555215 + 0.831707i \(0.312636\pi\)
\(164\) −212.000 −1.29268
\(165\) −105.000 −0.636364
\(166\) 0 0
\(167\) 14.0000 0.0838323 0.0419162 − 0.999121i \(-0.486654\pi\)
0.0419162 + 0.999121i \(0.486654\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −236.000 −1.37209
\(173\) −259.000 −1.49711 −0.748555 − 0.663073i \(-0.769252\pi\)
−0.748555 + 0.663073i \(0.769252\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 112.000 0.636364
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) −180.000 −1.00000
\(181\) −73.0000 −0.403315 −0.201657 − 0.979456i \(-0.564633\pi\)
−0.201657 + 0.979456i \(0.564633\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 355.000 1.91892
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) −353.000 −1.84817 −0.924084 − 0.382190i \(-0.875170\pi\)
−0.924084 + 0.382190i \(0.875170\pi\)
\(192\) 192.000 1.00000
\(193\) −194.000 −1.00518 −0.502591 − 0.864525i \(-0.667620\pi\)
−0.502591 + 0.864525i \(0.667620\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 196.000 1.00000
\(197\) 389.000 1.97462 0.987310 − 0.158807i \(-0.0507648\pi\)
0.987310 + 0.158807i \(0.0507648\pi\)
\(198\) 0 0
\(199\) −37.0000 −0.185930 −0.0929648 − 0.995669i \(-0.529634\pi\)
−0.0929648 + 0.995669i \(0.529634\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 265.000 1.29268
\(206\) 0 0
\(207\) 369.000 1.78261
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −76.0000 −0.358491
\(213\) 0 0
\(214\) 0 0
\(215\) 295.000 1.37209
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 3.00000 0.0136986
\(220\) −140.000 −0.636364
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 225.000 1.00000
\(226\) 0 0
\(227\) 329.000 1.44934 0.724670 − 0.689096i \(-0.241992\pi\)
0.724670 + 0.689096i \(0.241992\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −379.000 −1.62661 −0.813305 − 0.581838i \(-0.802334\pi\)
−0.813305 + 0.581838i \(0.802334\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) −240.000 −1.00000
\(241\) 47.0000 0.195021 0.0975104 − 0.995235i \(-0.468912\pi\)
0.0975104 + 0.995235i \(0.468912\pi\)
\(242\) 0 0
\(243\) 243.000 1.00000
\(244\) 0 0
\(245\) −245.000 −1.00000
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −237.000 −0.951807
\(250\) 0 0
\(251\) −38.0000 −0.151394 −0.0756972 − 0.997131i \(-0.524118\pi\)
−0.0756972 + 0.997131i \(0.524118\pi\)
\(252\) 0 0
\(253\) 287.000 1.13439
\(254\) 0 0
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 269.000 1.04669 0.523346 − 0.852120i \(-0.324683\pi\)
0.523346 + 0.852120i \(0.324683\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 261.000 1.00000
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 95.0000 0.358491
\(266\) 0 0
\(267\) −186.000 −0.696629
\(268\) 0 0
\(269\) −422.000 −1.56877 −0.784387 − 0.620272i \(-0.787022\pi\)
−0.784387 + 0.620272i \(0.787022\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 175.000 0.636364
\(276\) 492.000 1.78261
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 0 0
\(291\) 147.000 0.505155
\(292\) 4.00000 0.0136986
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 189.000 0.636364
\(298\) 0 0
\(299\) 0 0
\(300\) 300.000 1.00000
\(301\) 0 0
\(302\) 0 0
\(303\) −519.000 −1.71287
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 469.000 1.52769 0.763844 − 0.645401i \(-0.223310\pi\)
0.763844 + 0.645401i \(0.223310\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −593.000 −1.90675 −0.953376 − 0.301784i \(-0.902418\pi\)
−0.953376 + 0.301784i \(0.902418\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 203.000 0.636364
\(320\) −320.000 −1.00000
\(321\) 402.000 1.25234
\(322\) 0 0
\(323\) 0 0
\(324\) 324.000 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) −651.000 −1.99083
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −316.000 −0.951807
\(333\) −639.000 −1.91892
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 94.0000 0.278932 0.139466 − 0.990227i \(-0.455461\pi\)
0.139466 + 0.990227i \(0.455461\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) −615.000 −1.78261
\(346\) 0 0
\(347\) 89.0000 0.256484 0.128242 − 0.991743i \(-0.459067\pi\)
0.128242 + 0.991743i \(0.459067\pi\)
\(348\) 348.000 1.00000
\(349\) 263.000 0.753582 0.376791 − 0.926298i \(-0.377028\pi\)
0.376791 + 0.926298i \(0.377028\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 686.000 1.94334 0.971671 − 0.236336i \(-0.0759466\pi\)
0.971671 + 0.236336i \(0.0759466\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −248.000 −0.696629
\(357\) 0 0
\(358\) 0 0
\(359\) 703.000 1.95822 0.979109 − 0.203338i \(-0.0651790\pi\)
0.979109 + 0.203338i \(0.0651790\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) −216.000 −0.595041
\(364\) 0 0
\(365\) −5.00000 −0.0136986
\(366\) 0 0
\(367\) 589.000 1.60490 0.802452 − 0.596716i \(-0.203528\pi\)
0.802452 + 0.596716i \(0.203528\pi\)
\(368\) 656.000 1.78261
\(369\) −477.000 −1.29268
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) −375.000 −1.00000
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 327.000 0.858268
\(382\) 0 0
\(383\) −679.000 −1.77285 −0.886423 − 0.462876i \(-0.846817\pi\)
−0.886423 + 0.462876i \(0.846817\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −531.000 −1.37209
\(388\) 196.000 0.505155
\(389\) 643.000 1.65296 0.826478 − 0.562969i \(-0.190341\pi\)
0.826478 + 0.562969i \(0.190341\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 606.000 1.54198
\(394\) 0 0
\(395\) 0 0
\(396\) 252.000 0.636364
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 400.000 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −692.000 −1.71287
\(405\) −405.000 −1.00000
\(406\) 0 0
\(407\) −497.000 −1.22113
\(408\) 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
\(415\) 395.000 0.951807
\(416\) 0 0
\(417\) −471.000 −1.12950
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 536.000 1.25234
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 432.000 1.00000
\(433\) 721.000 1.66513 0.832564 − 0.553930i \(-0.186872\pi\)
0.832564 + 0.553930i \(0.186872\pi\)
\(434\) 0 0
\(435\) −435.000 −1.00000
\(436\) −868.000 −1.99083
\(437\) 0 0
\(438\) 0 0
\(439\) −862.000 −1.96355 −0.981777 − 0.190038i \(-0.939139\pi\)
−0.981777 + 0.190038i \(0.939139\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) −852.000 −1.91892
\(445\) 310.000 0.696629
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 523.000 1.16481 0.582405 − 0.812899i \(-0.302112\pi\)
0.582405 + 0.812899i \(0.302112\pi\)
\(450\) 0 0
\(451\) −371.000 −0.822616
\(452\) 0 0
\(453\) −399.000 −0.880795
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −820.000 −1.78261
\(461\) −893.000 −1.93709 −0.968547 − 0.248832i \(-0.919953\pi\)
−0.968547 + 0.248832i \(0.919953\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 464.000 1.00000
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −798.000 −1.69427
\(472\) 0 0
\(473\) −413.000 −0.873150
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −171.000 −0.358491
\(478\) 0 0
\(479\) 898.000 1.87474 0.937370 − 0.348337i \(-0.113253\pi\)
0.937370 + 0.348337i \(0.113253\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −288.000 −0.595041
\(485\) −245.000 −0.505155
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 543.000 1.11043
\(490\) 0 0
\(491\) −518.000 −1.05499 −0.527495 − 0.849558i \(-0.676869\pi\)
−0.527495 + 0.849558i \(0.676869\pi\)
\(492\) −636.000 −1.29268
\(493\) 0 0
\(494\) 0 0
\(495\) −315.000 −0.636364
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −742.000 −1.48697 −0.743487 − 0.668750i \(-0.766829\pi\)
−0.743487 + 0.668750i \(0.766829\pi\)
\(500\) −500.000 −1.00000
\(501\) 42.0000 0.0838323
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 865.000 1.71287
\(506\) 0 0
\(507\) 507.000 1.00000
\(508\) 436.000 0.858268
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) −708.000 −1.37209
\(517\) 0 0
\(518\) 0 0
\(519\) −777.000 −1.49711
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 808.000 1.54198
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 336.000 0.636364
\(529\) 1152.00 2.17769
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −670.000 −1.25234
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 343.000 0.636364
\(540\) −540.000 −1.00000
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) −219.000 −0.403315
\(544\) 0 0
\(545\) 1085.00 1.99083
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 1065.00 1.91892
\(556\) −628.000 −1.12950
\(557\) −331.000 −0.594255 −0.297127 − 0.954838i \(-0.596029\pi\)
−0.297127 + 0.954838i \(0.596029\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −1022.00 −1.79613 −0.898067 − 0.439859i \(-0.855028\pi\)
−0.898067 + 0.439859i \(0.855028\pi\)
\(570\) 0 0
\(571\) 707.000 1.23818 0.619089 − 0.785321i \(-0.287502\pi\)
0.619089 + 0.785321i \(0.287502\pi\)
\(572\) 0 0
\(573\) −1059.00 −1.84817
\(574\) 0 0
\(575\) 1025.00 1.78261
\(576\) 576.000 1.00000
\(577\) 574.000 0.994801 0.497400 − 0.867521i \(-0.334288\pi\)
0.497400 + 0.867521i \(0.334288\pi\)
\(578\) 0 0
\(579\) −582.000 −1.00518
\(580\) −580.000 −1.00000
\(581\) 0 0
\(582\) 0 0
\(583\) −133.000 −0.228130
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −826.000 −1.40716 −0.703578 − 0.710619i \(-0.748415\pi\)
−0.703578 + 0.710619i \(0.748415\pi\)
\(588\) 588.000 1.00000
\(589\) 0 0
\(590\) 0 0
\(591\) 1167.00 1.97462
\(592\) −1136.00 −1.91892
\(593\) 206.000 0.347386 0.173693 − 0.984800i \(-0.444430\pi\)
0.173693 + 0.984800i \(0.444430\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −111.000 −0.185930
\(598\) 0 0
\(599\) 658.000 1.09850 0.549249 − 0.835659i \(-0.314914\pi\)
0.549249 + 0.835659i \(0.314914\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −532.000 −0.880795
\(605\) 360.000 0.595041
\(606\) 0 0
\(607\) −1106.00 −1.82208 −0.911038 − 0.412323i \(-0.864718\pi\)
−0.911038 + 0.412323i \(0.864718\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 795.000 1.29268
\(616\) 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
\(621\) 1107.00 1.78261
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −1064.00 −1.69427
\(629\) 0 0
\(630\) 0 0
\(631\) −478.000 −0.757528 −0.378764 − 0.925493i \(-0.623651\pi\)
−0.378764 + 0.925493i \(0.623651\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −545.000 −0.858268
\(636\) −228.000 −0.358491
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −1253.00 −1.95476 −0.977379 − 0.211495i \(-0.932167\pi\)
−0.977379 + 0.211495i \(0.932167\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 885.000 1.37209
\(646\) 0 0
\(647\) −511.000 −0.789799 −0.394900 − 0.918724i \(-0.629221\pi\)
−0.394900 + 0.918724i \(0.629221\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 724.000 1.11043
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −1010.00 −1.54198
\(656\) −848.000 −1.29268
\(657\) 9.00000 0.0136986
\(658\) 0 0
\(659\) 103.000 0.156297 0.0781487 − 0.996942i \(-0.475099\pi\)
0.0781487 + 0.996942i \(0.475099\pi\)
\(660\) −420.000 −0.636364
\(661\) 887.000 1.34191 0.670953 − 0.741500i \(-0.265885\pi\)
0.670953 + 0.741500i \(0.265885\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1189.00 1.78261
\(668\) 56.0000 0.0838323
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 675.000 1.00000
\(676\) 676.000 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 987.000 1.44934
\(682\) 0 0
\(683\) −1279.00 −1.87262 −0.936310 − 0.351174i \(-0.885783\pi\)
−0.936310 + 0.351174i \(0.885783\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −944.000 −1.37209
\(689\) 0 0
\(690\) 0 0
\(691\) −358.000 −0.518090 −0.259045 − 0.965865i \(-0.583408\pi\)
−0.259045 + 0.965865i \(0.583408\pi\)
\(692\) −1036.00 −1.49711
\(693\) 0 0
\(694\) 0 0
\(695\) 785.000 1.12950
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −1137.00 −1.62661
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 448.000 0.636364
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 983.000 1.38646 0.693230 − 0.720717i \(-0.256187\pi\)
0.693230 + 0.720717i \(0.256187\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −720.000 −1.00000
\(721\) 0 0
\(722\) 0 0
\(723\) 141.000 0.195021
\(724\) −292.000 −0.403315
\(725\) 725.000 1.00000
\(726\) 0 0
\(727\) −866.000 −1.19120 −0.595598 − 0.803282i \(-0.703085\pi\)
−0.595598 + 0.803282i \(0.703085\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 886.000 1.20873 0.604366 − 0.796707i \(-0.293427\pi\)
0.604366 + 0.796707i \(0.293427\pi\)
\(734\) 0 0
\(735\) −735.000 −1.00000
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 1420.00 1.91892
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −711.000 −0.951807
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −114.000 −0.151394
\(754\) 0 0
\(755\) 665.000 0.880795
\(756\) 0 0
\(757\) 1369.00 1.80845 0.904227 − 0.427052i \(-0.140448\pi\)
0.904227 + 0.427052i \(0.140448\pi\)
\(758\) 0 0
\(759\) 861.000 1.13439
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −1412.00 −1.84817
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 768.000 1.00000
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 807.000 1.04669
\(772\) −776.000 −1.00518
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 783.000 1.00000
\(784\) 784.000 1.00000
\(785\) 1330.00 1.69427
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1556.00 1.97462
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 285.000 0.358491
\(796\) −148.000 −0.185930
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −558.000 −0.696629
\(802\) 0 0
\(803\) 7.00000 0.00871731
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) −1266.00 −1.56877
\(808\) 0 0
\(809\) −197.000 −0.243511 −0.121755 − 0.992560i \(-0.538852\pi\)
−0.121755 + 0.992560i \(0.538852\pi\)
\(810\) 0 0
\(811\) 1187.00 1.46363 0.731813 − 0.681506i \(-0.238675\pi\)
0.731813 + 0.681506i \(0.238675\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −905.000 −1.11043
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 1060.00 1.29268
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −674.000 −0.818955 −0.409478 − 0.912320i \(-0.634289\pi\)
−0.409478 + 0.912320i \(0.634289\pi\)
\(824\) 0 0
\(825\) 525.000 0.636364
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 1476.00 1.78261
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −70.0000 −0.0838323
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 178.000 0.212157 0.106079 − 0.994358i \(-0.466170\pi\)
0.106079 + 0.994358i \(0.466170\pi\)
\(840\) 0 0
\(841\) 841.000 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −845.000 −1.00000
\(846\) 0 0
\(847\) 0 0
\(848\) −304.000 −0.358491
\(849\) 0 0
\(850\) 0 0
\(851\) −2911.00 −3.42068
\(852\) 0 0
\(853\) 1561.00 1.83001 0.915006 − 0.403441i \(-0.132186\pi\)
0.915006 + 0.403441i \(0.132186\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −931.000 −1.08635 −0.543174 − 0.839620i \(-0.682778\pi\)
−0.543174 + 0.839620i \(0.682778\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 1180.00 1.37209
\(861\) 0 0
\(862\) 0 0
\(863\) 1406.00 1.62920 0.814600 − 0.580023i \(-0.196956\pi\)
0.814600 + 0.580023i \(0.196956\pi\)
\(864\) 0 0
\(865\) 1295.00 1.49711
\(866\) 0 0
\(867\) 867.000 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 441.000 0.505155
\(874\) 0 0
\(875\) 0 0
\(876\) 12.0000 0.0136986
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −560.000 −0.636364
\(881\) 1747.00 1.98297 0.991487 − 0.130206i \(-0.0415639\pi\)
0.991487 + 0.130206i \(0.0415639\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 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
\(891\) 567.000 0.636364
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 900.000 1.00000
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 365.000 0.403315
\(906\) 0 0
\(907\) −1811.00 −1.99669 −0.998346 − 0.0574880i \(-0.981691\pi\)
−0.998346 + 0.0574880i \(0.981691\pi\)
\(908\) 1316.00 1.44934
\(909\) −1557.00 −1.71287
\(910\) 0 0
\(911\) 1687.00 1.85181 0.925906 − 0.377755i \(-0.123304\pi\)
0.925906 + 0.377755i \(0.123304\pi\)
\(912\) 0 0
\(913\) −553.000 −0.605696
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 98.0000 0.106638 0.0533188 − 0.998578i \(-0.483020\pi\)
0.0533188 + 0.998578i \(0.483020\pi\)
\(920\) 0 0
\(921\) 1407.00 1.52769
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1775.00 −1.91892
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −1516.00 −1.62661
\(933\) −1779.00 −1.90675
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) −2173.00 −2.30435
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −514.000 −0.539349 −0.269675 − 0.962951i \(-0.586916\pi\)
−0.269675 + 0.962951i \(0.586916\pi\)
\(954\) 0 0
\(955\) 1765.00 1.84817
\(956\) 0 0
\(957\) 609.000 0.636364
\(958\) 0 0
\(959\) 0 0
\(960\) −960.000 −1.00000
\(961\) 961.000 1.00000
\(962\) 0 0
\(963\) 1206.00 1.25234
\(964\) 188.000 0.195021
\(965\) 970.000 1.00518
\(966\) 0 0
\(967\) −1691.00 −1.74871 −0.874354 − 0.485289i \(-0.838714\pi\)
−0.874354 + 0.485289i \(0.838714\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1567.00 1.61380 0.806900 − 0.590688i \(-0.201144\pi\)
0.806900 + 0.590688i \(0.201144\pi\)
\(972\) 972.000 1.00000
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1171.00 −1.19857 −0.599284 − 0.800537i \(-0.704548\pi\)
−0.599284 + 0.800537i \(0.704548\pi\)
\(978\) 0 0
\(979\) −434.000 −0.443309
\(980\) −980.000 −1.00000
\(981\) −1953.00 −1.99083
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −1945.00 −1.97462
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −2419.00 −2.44590
\(990\) 0 0
\(991\) −1933.00 −1.95055 −0.975277 − 0.220984i \(-0.929073\pi\)
−0.975277 + 0.220984i \(0.929073\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 185.000 0.185930
\(996\) −948.000 −0.951807
\(997\) −1631.00 −1.63591 −0.817954 − 0.575284i \(-0.804892\pi\)
−0.817954 + 0.575284i \(0.804892\pi\)
\(998\) 0 0
\(999\) −1917.00 −1.91892
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 435.3.b.c.434.1 yes 1
3.2 odd 2 435.3.b.d.434.1 yes 1
5.4 even 2 435.3.b.b.434.1 yes 1
15.14 odd 2 435.3.b.a.434.1 ✓ 1
29.28 even 2 435.3.b.a.434.1 ✓ 1
87.86 odd 2 435.3.b.b.434.1 yes 1
145.144 even 2 435.3.b.d.434.1 yes 1
435.434 odd 2 CM 435.3.b.c.434.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
435.3.b.a.434.1 ✓ 1 15.14 odd 2
435.3.b.a.434.1 ✓ 1 29.28 even 2
435.3.b.b.434.1 yes 1 5.4 even 2
435.3.b.b.434.1 yes 1 87.86 odd 2
435.3.b.c.434.1 yes 1 1.1 even 1 trivial
435.3.b.c.434.1 yes 1 435.434 odd 2 CM
435.3.b.d.434.1 yes 1 3.2 odd 2
435.3.b.d.434.1 yes 1 145.144 even 2