Properties

Label 675.4.a.k.1.2
Level $675$
Weight $4$
Character 675.1
Self dual yes
Analytic conductor $39.826$
Analytic rank $1$
Dimension $2$
CM discriminant -15
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [675,4,Mod(1,675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(675, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("675.1");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 675 = 3^{3} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 675.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: \(39.8262892539\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{10})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 135)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.2
Root \(-0.618034\) of defining polynomial
Character \(\chi\) \(=\) 675.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.38197 q^{2} +3.43769 q^{4} +15.4296 q^{8} +O(q^{10})\) \(q-3.38197 q^{2} +3.43769 q^{4} +15.4296 q^{8} -79.6838 q^{16} -87.3181 q^{17} +102.125 q^{19} -121.807 q^{23} +337.371 q^{31} +146.051 q^{32} +295.307 q^{34} -345.382 q^{38} +411.945 q^{46} -545.601 q^{47} -343.000 q^{49} +706.813 q^{53} +943.735 q^{61} -1140.98 q^{62} +143.530 q^{64} -300.173 q^{68} +351.073 q^{76} -1339.35 q^{79} -1346.04 q^{83} -418.734 q^{92} +1845.20 q^{94} +1160.01 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 9 q^{2} + 27 q^{4} - 72 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 9 q^{2} + 27 q^{4} - 72 q^{8} + 223 q^{16} - 36 q^{17} + 164 q^{19} - 342 q^{23} + 232 q^{31} - 855 q^{32} + 7 q^{34} - 693 q^{38} + 1649 q^{46} - 686 q^{49} + 792 q^{53} + 358 q^{61} - 549 q^{62} + 3346 q^{64} + 909 q^{68} + 1809 q^{76} - 304 q^{79} - 1422 q^{83} - 5607 q^{92} - 1220 q^{94} + 3087 q^{98}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −3.38197 −1.19571 −0.597853 0.801606i \(-0.703979\pi\)
−0.597853 + 0.801606i \(0.703979\pi\)
\(3\) 0 0
\(4\) 3.43769 0.429712
\(5\) 0 0
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 15.4296 0.681897
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −79.6838 −1.24506
\(17\) −87.3181 −1.24575 −0.622875 0.782321i \(-0.714036\pi\)
−0.622875 + 0.782321i \(0.714036\pi\)
\(18\) 0 0
\(19\) 102.125 1.23310 0.616552 0.787314i \(-0.288529\pi\)
0.616552 + 0.787314i \(0.288529\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −121.807 −1.10428 −0.552139 0.833752i \(-0.686188\pi\)
−0.552139 + 0.833752i \(0.686188\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 337.371 1.95463 0.977316 0.211788i \(-0.0679286\pi\)
0.977316 + 0.211788i \(0.0679286\pi\)
\(32\) 146.051 0.806828
\(33\) 0 0
\(34\) 295.307 1.48955
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) −345.382 −1.47443
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 411.945 1.32039
\(47\) −545.601 −1.69328 −0.846639 0.532168i \(-0.821377\pi\)
−0.846639 + 0.532168i \(0.821377\pi\)
\(48\) 0 0
\(49\) −343.000 −1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 706.813 1.83185 0.915927 0.401344i \(-0.131457\pi\)
0.915927 + 0.401344i \(0.131457\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 943.735 1.98087 0.990434 0.137989i \(-0.0440639\pi\)
0.990434 + 0.137989i \(0.0440639\pi\)
\(62\) −1140.98 −2.33716
\(63\) 0 0
\(64\) 143.530 0.280331
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) −300.173 −0.535313
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 351.073 0.529880
\(77\) 0 0
\(78\) 0 0
\(79\) −1339.35 −1.90745 −0.953727 0.300674i \(-0.902789\pi\)
−0.953727 + 0.300674i \(0.902789\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −1346.04 −1.78009 −0.890045 0.455873i \(-0.849327\pi\)
−0.890045 + 0.455873i \(0.849327\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) −418.734 −0.474522
\(93\) 0 0
\(94\) 1845.20 2.02466
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 1160.01 1.19571
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −2390.42 −2.19036
\(107\) 17.8885 0.0161622 0.00808108 0.999967i \(-0.497428\pi\)
0.00808108 + 0.999967i \(0.497428\pi\)
\(108\) 0 0
\(109\) 250.227 0.219885 0.109942 0.993938i \(-0.464933\pi\)
0.109942 + 0.993938i \(0.464933\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1426.61 −1.18765 −0.593824 0.804595i \(-0.702383\pi\)
−0.593824 + 0.804595i \(0.702383\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −1331.00 −1.00000
\(122\) −3191.68 −2.36853
\(123\) 0 0
\(124\) 1159.78 0.839928
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1653.82 −1.14202
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −1347.28 −0.849473
\(137\) −3173.05 −1.97877 −0.989387 0.145306i \(-0.953583\pi\)
−0.989387 + 0.145306i \(0.953583\pi\)
\(138\) 0 0
\(139\) −1604.00 −0.978773 −0.489387 0.872067i \(-0.662779\pi\)
−0.489387 + 0.872067i \(0.662779\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −3112.00 −1.67716 −0.838579 0.544779i \(-0.816613\pi\)
−0.838579 + 0.544779i \(0.816613\pi\)
\(152\) 1575.74 0.840850
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 4529.64 2.28075
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 4552.27 2.12846
\(167\) 55.7692 0.0258416 0.0129208 0.999917i \(-0.495887\pi\)
0.0129208 + 0.999917i \(0.495887\pi\)
\(168\) 0 0
\(169\) −2197.00 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −4136.43 −1.81784 −0.908921 0.416968i \(-0.863093\pi\)
−0.908921 + 0.416968i \(0.863093\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −4714.17 −1.93592 −0.967960 0.251103i \(-0.919207\pi\)
−0.967960 + 0.251103i \(0.919207\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −1879.42 −0.753004
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −1875.61 −0.727621
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −1179.13 −0.429712
\(197\) 2971.21 1.07457 0.537283 0.843402i \(-0.319451\pi\)
0.537283 + 0.843402i \(0.319451\pi\)
\(198\) 0 0
\(199\) −5456.00 −1.94355 −0.971773 0.235919i \(-0.924190\pi\)
−0.971773 + 0.235919i \(0.924190\pi\)
\(200\) 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
\(211\) 5957.80 1.94385 0.971924 0.235295i \(-0.0756056\pi\)
0.971924 + 0.235295i \(0.0756056\pi\)
\(212\) 2429.81 0.787169
\(213\) 0 0
\(214\) −60.4984 −0.0193252
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −846.261 −0.262918
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 4824.75 1.42008
\(227\) −1275.88 −0.373054 −0.186527 0.982450i \(-0.559723\pi\)
−0.186527 + 0.982450i \(0.559723\pi\)
\(228\) 0 0
\(229\) 5854.13 1.68931 0.844655 0.535311i \(-0.179805\pi\)
0.844655 + 0.535311i \(0.179805\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 4449.78 1.25114 0.625568 0.780170i \(-0.284867\pi\)
0.625568 + 0.780170i \(0.284867\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −5640.38 −1.50759 −0.753794 0.657111i \(-0.771778\pi\)
−0.753794 + 0.657111i \(0.771778\pi\)
\(242\) 4501.40 1.19571
\(243\) 0 0
\(244\) 3244.27 0.851202
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 5205.48 1.33286
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4444.94 1.08519
\(257\) −1066.63 −0.258890 −0.129445 0.991587i \(-0.541320\pi\)
−0.129445 + 0.991587i \(0.541320\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 6341.49 1.48682 0.743409 0.668837i \(-0.233208\pi\)
0.743409 + 0.668837i \(0.233208\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 1675.28 0.375520 0.187760 0.982215i \(-0.439877\pi\)
0.187760 + 0.982215i \(0.439877\pi\)
\(272\) 6957.84 1.55103
\(273\) 0 0
\(274\) 10731.1 2.36603
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 5424.67 1.17032
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2711.45 0.551893
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −9824.90 −1.95897 −0.979483 0.201529i \(-0.935409\pi\)
−0.979483 + 0.201529i \(0.935409\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 10524.7 2.00539
\(303\) 0 0
\(304\) −8137.68 −1.53529
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −4604.28 −0.819656
\(317\) 650.333 0.115225 0.0576125 0.998339i \(-0.481651\pi\)
0.0576125 + 0.998339i \(0.481651\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) −8917.33 −1.53614
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 5852.00 0.971767 0.485884 0.874023i \(-0.338498\pi\)
0.485884 + 0.874023i \(0.338498\pi\)
\(332\) −4627.29 −0.764925
\(333\) 0 0
\(334\) −188.610 −0.0308990
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) 7430.18 1.19571
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 13989.3 2.17360
\(347\) 8550.72 1.32284 0.661422 0.750014i \(-0.269953\pi\)
0.661422 + 0.750014i \(0.269953\pi\)
\(348\) 0 0
\(349\) 8974.04 1.37642 0.688208 0.725513i \(-0.258398\pi\)
0.688208 + 0.725513i \(0.258398\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −2473.09 −0.372888 −0.186444 0.982466i \(-0.559696\pi\)
−0.186444 + 0.982466i \(0.559696\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 3570.44 0.520548
\(362\) 15943.2 2.31479
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 9706.01 1.37489
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −8418.38 −1.15464
\(377\) 0 0
\(378\) 0 0
\(379\) −9933.39 −1.34629 −0.673145 0.739510i \(-0.735057\pi\)
−0.673145 + 0.739510i \(0.735057\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) −14779.7 −1.97182 −0.985911 0.167269i \(-0.946505\pi\)
−0.985911 + 0.167269i \(0.946505\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 10635.9 1.37566
\(392\) −5292.34 −0.681897
\(393\) 0 0
\(394\) −10048.5 −1.28487
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 18452.0 2.32391
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −14327.4 −1.73213 −0.866067 0.499929i \(-0.833360\pi\)
−0.866067 + 0.499929i \(0.833360\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −17164.0 −1.98699 −0.993493 0.113890i \(-0.963669\pi\)
−0.993493 + 0.113890i \(0.963669\pi\)
\(422\) −20149.1 −2.32427
\(423\) 0 0
\(424\) 10905.8 1.24914
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 61.4953 0.00694507
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 860.206 0.0944871
\(437\) −12439.4 −1.36169
\(438\) 0 0
\(439\) 2349.99 0.255488 0.127744 0.991807i \(-0.459226\pi\)
0.127744 + 0.991807i \(0.459226\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 4965.77 0.532575 0.266288 0.963894i \(-0.414203\pi\)
0.266288 + 0.963894i \(0.414203\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −4904.25 −0.510347
\(453\) 0 0
\(454\) 4314.99 0.446063
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) −19798.5 −2.01992
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −15049.0 −1.49599
\(467\) 19237.5 1.90622 0.953111 0.302620i \(-0.0978613\pi\)
0.953111 + 0.302620i \(0.0978613\pi\)
\(468\) 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
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 19075.6 1.80263
\(483\) 0 0
\(484\) −4575.57 −0.429712
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 14561.4 1.35075
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −26883.0 −2.43363
\(497\) 0 0
\(498\) 0 0
\(499\) 18.3109 0.00164271 0.000821353 1.00000i \(-0.499739\pi\)
0.000821353 1.00000i \(0.499739\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −15470.1 −1.37133 −0.685663 0.727919i \(-0.740488\pi\)
−0.685663 + 0.727919i \(0.740488\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1802.04 −0.155547
\(513\) 0 0
\(514\) 3607.32 0.309556
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −21446.7 −1.77780
\(527\) −29458.6 −2.43498
\(528\) 0 0
\(529\) 2669.82 0.219432
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3238.00 −0.257324 −0.128662 0.991688i \(-0.541068\pi\)
−0.128662 + 0.991688i \(0.541068\pi\)
\(542\) −5665.73 −0.449011
\(543\) 0 0
\(544\) −12752.9 −1.00511
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) −10908.0 −0.850302
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −5514.06 −0.420590
\(557\) −26184.4 −1.99186 −0.995931 0.0901226i \(-0.971274\pi\)
−0.995931 + 0.0901226i \(0.971274\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 14632.8 1.09538 0.547691 0.836681i \(-0.315507\pi\)
0.547691 + 0.836681i \(0.315507\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −20150.2 −1.47681 −0.738407 0.674355i \(-0.764422\pi\)
−0.738407 + 0.674355i \(0.764422\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −9170.04 −0.659902
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 33227.5 2.34235
\(587\) −23301.7 −1.63844 −0.819220 0.573480i \(-0.805593\pi\)
−0.819220 + 0.573480i \(0.805593\pi\)
\(588\) 0 0
\(589\) 34453.9 2.41027
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −20132.8 −1.39419 −0.697094 0.716980i \(-0.745524\pi\)
−0.697094 + 0.716980i \(0.745524\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 29441.1 1.99821 0.999107 0.0422630i \(-0.0134567\pi\)
0.999107 + 0.0422630i \(0.0134567\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −10698.1 −0.720695
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 14915.4 0.994903
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −24813.5 −1.61905 −0.809527 0.587083i \(-0.800276\pi\)
−0.809527 + 0.587083i \(0.800276\pi\)
\(618\) 0 0
\(619\) −3476.00 −0.225706 −0.112853 0.993612i \(-0.535999\pi\)
−0.112853 + 0.993612i \(0.535999\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 722.379 0.0455744 0.0227872 0.999740i \(-0.492746\pi\)
0.0227872 + 0.999740i \(0.492746\pi\)
\(632\) −20665.6 −1.30069
\(633\) 0 0
\(634\) −2199.41 −0.137775
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 30158.1 1.83677
\(647\) 29732.3 1.80664 0.903320 0.428967i \(-0.141122\pi\)
0.903320 + 0.428967i \(0.141122\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −9864.90 −0.591184 −0.295592 0.955314i \(-0.595517\pi\)
−0.295592 + 0.955314i \(0.595517\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 32978.0 1.94054 0.970269 0.242029i \(-0.0778130\pi\)
0.970269 + 0.242029i \(0.0778130\pi\)
\(662\) −19791.3 −1.16195
\(663\) 0 0
\(664\) −20768.9 −1.21384
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 191.718 0.0111045
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −7552.61 −0.429712
\(677\) 32901.5 1.86781 0.933905 0.357521i \(-0.116378\pi\)
0.933905 + 0.357521i \(0.116378\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5944.71 0.333042 0.166521 0.986038i \(-0.446747\pi\)
0.166521 + 0.986038i \(0.446747\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 29909.6 1.64662 0.823311 0.567591i \(-0.192124\pi\)
0.823311 + 0.567591i \(0.192124\pi\)
\(692\) −14219.8 −0.781148
\(693\) 0 0
\(694\) −28918.3 −1.58173
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −30349.9 −1.64579
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 8363.91 0.445864
\(707\) 0 0
\(708\) 0 0
\(709\) 37726.0 1.99835 0.999175 0.0406201i \(-0.0129334\pi\)
0.999175 + 0.0406201i \(0.0129334\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −41093.9 −2.15846
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −12075.1 −0.622422
\(723\) 0 0
\(724\) −16205.9 −0.831888
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −17790.0 −0.890963
\(737\) 0 0
\(738\) 0 0
\(739\) 38352.2 1.90908 0.954539 0.298085i \(-0.0963478\pi\)
0.954539 + 0.298085i \(0.0963478\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −10241.2 −0.505670 −0.252835 0.967509i \(-0.581363\pi\)
−0.252835 + 0.967509i \(0.581363\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −23503.8 −1.14203 −0.571015 0.820940i \(-0.693450\pi\)
−0.571015 + 0.820940i \(0.693450\pi\)
\(752\) 43475.5 2.10823
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 33594.4 1.60977
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 49984.5 2.35772
\(767\) 0 0
\(768\) 0 0
\(769\) −31750.8 −1.48890 −0.744449 0.667679i \(-0.767288\pi\)
−0.744449 + 0.667679i \(0.767288\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 41284.7 1.92097 0.960484 0.278335i \(-0.0897827\pi\)
0.960484 + 0.278335i \(0.0897827\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) −35970.3 −1.64488
\(783\) 0 0
\(784\) 27331.5 1.24506
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 10214.1 0.461754
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −18756.1 −0.835164
\(797\) −39665.3 −1.76288 −0.881441 0.472294i \(-0.843426\pi\)
−0.881441 + 0.472294i \(0.843426\pi\)
\(798\) 0 0
\(799\) 47640.8 2.10940
\(800\) 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
\(809\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) −14092.0 −0.610157 −0.305078 0.952327i \(-0.598683\pi\)
−0.305078 + 0.952327i \(0.598683\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 48454.7 2.07112
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 39151.1 1.64621 0.823107 0.567887i \(-0.192239\pi\)
0.823107 + 0.567887i \(0.192239\pi\)
\(828\) 0 0
\(829\) −45254.0 −1.89594 −0.947971 0.318356i \(-0.896869\pi\)
−0.947971 + 0.318356i \(0.896869\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 29950.1 1.24575
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −24389.0 −1.00000
\(842\) 58048.0 2.37585
\(843\) 0 0
\(844\) 20481.1 0.835294
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −56321.6 −2.28077
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 276.012 0.0110209
\(857\) 8378.73 0.333969 0.166985 0.985959i \(-0.446597\pi\)
0.166985 + 0.985959i \(0.446597\pi\)
\(858\) 0 0
\(859\) 22021.7 0.874703 0.437352 0.899291i \(-0.355917\pi\)
0.437352 + 0.899291i \(0.355917\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 50691.4 1.99949 0.999743 0.0226629i \(-0.00721446\pi\)
0.999743 + 0.0226629i \(0.00721446\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 3860.90 0.149939
\(873\) 0 0
\(874\) 42069.8 1.62818
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) −7947.60 −0.305488
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −16794.1 −0.636803
\(887\) 52699.6 1.99490 0.997451 0.0713489i \(-0.0227304\pi\)
0.997451 + 0.0713489i \(0.0227304\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) −55719.2 −2.08799
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) −61717.6 −2.28203
\(902\) 0 0
\(903\) 0 0
\(904\) −22012.0 −0.809854
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) −4386.09 −0.160306
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 20124.7 0.725917
\(917\) 0 0
\(918\) 0 0
\(919\) −21224.0 −0.761823 −0.380911 0.924612i \(-0.624390\pi\)
−0.380911 + 0.924612i \(0.624390\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) −35028.7 −1.23310
\(932\) 15297.0 0.537627
\(933\) 0 0
\(934\) −65060.6 −2.27928
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 9428.49 0.323532 0.161766 0.986829i \(-0.448281\pi\)
0.161766 + 0.986829i \(0.448281\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 30289.8 1.02957 0.514786 0.857319i \(-0.327871\pi\)
0.514786 + 0.857319i \(0.327871\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 84028.0 2.82058
\(962\) 0 0
\(963\) 0 0
\(964\) −19389.9 −0.647828
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −20536.7 −0.681897
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −75200.4 −2.46630
\(977\) 13018.4 0.426300 0.213150 0.977019i \(-0.431628\pi\)
0.213150 + 0.977019i \(0.431628\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 21586.1 0.700397 0.350198 0.936676i \(-0.386114\pi\)
0.350198 + 0.936676i \(0.386114\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 32733.6 1.04926 0.524630 0.851330i \(-0.324204\pi\)
0.524630 + 0.851330i \(0.324204\pi\)
\(992\) 49273.5 1.57705
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) −61.9270 −0.00196419
\(999\) 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 675.4.a.k.1.2 2
3.2 odd 2 675.4.a.o.1.1 2
5.2 odd 4 135.4.b.a.109.2 4
5.3 odd 4 135.4.b.a.109.3 yes 4
5.4 even 2 675.4.a.o.1.1 2
15.2 even 4 135.4.b.a.109.3 yes 4
15.8 even 4 135.4.b.a.109.2 4
15.14 odd 2 CM 675.4.a.k.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.b.a.109.2 4 5.2 odd 4
135.4.b.a.109.2 4 15.8 even 4
135.4.b.a.109.3 yes 4 5.3 odd 4
135.4.b.a.109.3 yes 4 15.2 even 4
675.4.a.k.1.2 2 1.1 even 1 trivial
675.4.a.k.1.2 2 15.14 odd 2 CM
675.4.a.o.1.1 2 3.2 odd 2
675.4.a.o.1.1 2 5.4 even 2