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Properties

Label 115.7.c.a.114.1
Level $115$
Weight $7$
Character 115.114
Self dual yes
Analytic conductor $26.456$
Analytic rank $0$
Dimension $1$
CM discriminant -115
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] = [115,7,Mod(114,115)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(115, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("115.114");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 115 = 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 115.c (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: \(26.4562196163\)
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 114.1
Character \(\chi\) \(=\) 115.114

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+64.0000 q^{4} -125.000 q^{5} -594.000 q^{7} +729.000 q^{9} +4096.00 q^{16} +8206.00 q^{17} -8000.00 q^{20} +12167.0 q^{23} +15625.0 q^{25} -38016.0 q^{28} -41382.0 q^{29} +3922.00 q^{31} +74250.0 q^{35} +46656.0 q^{36} +76806.0 q^{37} +130482. q^{41} -33066.0 q^{43} -91125.0 q^{45} +235187. q^{49} +179174. q^{53} -31302.0 q^{59} -433026. q^{63} +262144. q^{64} +127206. q^{67} +525184. q^{68} -388638. q^{71} -512000. q^{80} +531441. q^{81} -778426. q^{83} -1.02575e6 q^{85} +778688. q^{92} +356526. q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/115\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(51\)
\(\chi(n)\) \(-1\) \(-1\)

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 \(\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 64.0000 1.00000
\(5\) −125.000 −1.00000
\(6\) 0 0
\(7\) −594.000 −1.73178 −0.865889 − 0.500236i \(-0.833247\pi\)
−0.865889 + 0.500236i \(0.833247\pi\)
\(8\) 0 0
\(9\) 729.000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4096.00 1.00000
\(17\) 8206.00 1.67026 0.835131 − 0.550051i \(-0.185392\pi\)
0.835131 + 0.550051i \(0.185392\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −8000.00 −1.00000
\(21\) 0 0
\(22\) 0 0
\(23\) 12167.0 1.00000
\(24\) 0 0
\(25\) 15625.0 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) −38016.0 −1.73178
\(29\) −41382.0 −1.69675 −0.848374 − 0.529397i \(-0.822418\pi\)
−0.848374 + 0.529397i \(0.822418\pi\)
\(30\) 0 0
\(31\) 3922.00 0.131650 0.0658252 − 0.997831i \(-0.479032\pi\)
0.0658252 + 0.997831i \(0.479032\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 74250.0 1.73178
\(36\) 46656.0 1.00000
\(37\) 76806.0 1.51632 0.758158 − 0.652070i \(-0.226099\pi\)
0.758158 + 0.652070i \(0.226099\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 130482. 1.89321 0.946606 − 0.322394i \(-0.104488\pi\)
0.946606 + 0.322394i \(0.104488\pi\)
\(42\) 0 0
\(43\) −33066.0 −0.415888 −0.207944 − 0.978141i \(-0.566677\pi\)
−0.207944 + 0.978141i \(0.566677\pi\)
\(44\) 0 0
\(45\) −91125.0 −1.00000
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 235187. 1.99906
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 179174. 1.20350 0.601752 − 0.798683i \(-0.294470\pi\)
0.601752 + 0.798683i \(0.294470\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −31302.0 −0.152411 −0.0762055 − 0.997092i \(-0.524280\pi\)
−0.0762055 + 0.997092i \(0.524280\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −433026. −1.73178
\(64\) 262144. 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 127206. 0.422944 0.211472 − 0.977384i \(-0.432174\pi\)
0.211472 + 0.977384i \(0.432174\pi\)
\(68\) 525184. 1.67026
\(69\) 0 0
\(70\) 0 0
\(71\) −388638. −1.08585 −0.542925 − 0.839781i \(-0.682683\pi\)
−0.542925 + 0.839781i \(0.682683\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −512000. −1.00000
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) −778426. −1.36139 −0.680696 − 0.732566i \(-0.738322\pi\)
−0.680696 + 0.732566i \(0.738322\pi\)
\(84\) 0 0
\(85\) −1.02575e6 −1.67026
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 778688. 1.00000
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 356526. 0.390639 0.195320 − 0.980740i \(-0.437426\pi\)
0.195320 + 0.980740i \(0.437426\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.00000e6 1.00000
\(101\) −2.00396e6 −1.94502 −0.972511 − 0.232857i \(-0.925193\pi\)
−0.972511 + 0.232857i \(0.925193\pi\)
\(102\) 0 0
\(103\) 2.14673e6 1.96457 0.982283 − 0.187404i \(-0.0600075\pi\)
0.982283 + 0.187404i \(0.0600075\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −407594. −0.332718 −0.166359 − 0.986065i \(-0.553201\pi\)
−0.166359 + 0.986065i \(0.553201\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −2.43302e6 −1.73178
\(113\) 720974. 0.499671 0.249836 − 0.968288i \(-0.419623\pi\)
0.249836 + 0.968288i \(0.419623\pi\)
\(114\) 0 0
\(115\) −1.52088e6 −1.00000
\(116\) −2.64845e6 −1.69675
\(117\) 0 0
\(118\) 0 0
\(119\) −4.87436e6 −2.89252
\(120\) 0 0
\(121\) 1.77156e6 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 251008. 0.131650
\(125\) −1.95312e6 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 2.43124e6 1.08147 0.540735 − 0.841193i \(-0.318146\pi\)
0.540735 + 0.841193i \(0.318146\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −4.46227e6 −1.73538 −0.867690 − 0.497106i \(-0.834396\pi\)
−0.867690 + 0.497106i \(0.834396\pi\)
\(138\) 0 0
\(139\) −5.11722e6 −1.90542 −0.952708 − 0.303887i \(-0.901715\pi\)
−0.952708 + 0.303887i \(0.901715\pi\)
\(140\) 4.75200e6 1.73178
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 2.98598e6 1.00000
\(145\) 5.17275e6 1.69675
\(146\) 0 0
\(147\) 0 0
\(148\) 4.91558e6 1.51632
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 6.86336e6 1.99345 0.996727 − 0.0808455i \(-0.0257620\pi\)
0.996727 + 0.0808455i \(0.0257620\pi\)
\(152\) 0 0
\(153\) 5.98217e6 1.67026
\(154\) 0 0
\(155\) −490250. −0.131650
\(156\) 0 0
\(157\) −3.12359e6 −0.807153 −0.403576 − 0.914946i \(-0.632233\pi\)
−0.403576 + 0.914946i \(0.632233\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −7.22720e6 −1.73178
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 8.35085e6 1.89321
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 4.82681e6 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −2.11622e6 −0.415888
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −9.28125e6 −1.73178
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 8.74334e6 1.52447 0.762234 − 0.647302i \(-0.224103\pi\)
0.762234 + 0.647302i \(0.224103\pi\)
\(180\) −5.83200e6 −1.00000
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −9.60075e6 −1.51632
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 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
\(195\) 0 0
\(196\) 1.50520e7 1.99906
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 2.45809e7 2.93839
\(204\) 0 0
\(205\) −1.63102e7 −1.89321
\(206\) 0 0
\(207\) 8.86974e6 1.00000
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −1.20694e7 −1.28481 −0.642404 − 0.766366i \(-0.722063\pi\)
−0.642404 + 0.766366i \(0.722063\pi\)
\(212\) 1.14671e7 1.20350
\(213\) 0 0
\(214\) 0 0
\(215\) 4.13325e6 0.415888
\(216\) 0 0
\(217\) −2.32967e6 −0.227989
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 1.13906e7 1.00000
\(226\) 0 0
\(227\) −5.50191e6 −0.470366 −0.235183 − 0.971951i \(-0.575569\pi\)
−0.235183 + 0.971951i \(0.575569\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −2.00333e6 −0.152411
\(237\) 0 0
\(238\) 0 0
\(239\) −1.43726e7 −1.05279 −0.526396 − 0.850240i \(-0.676457\pi\)
−0.526396 + 0.850240i \(0.676457\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.93984e7 −1.99906
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −2.77137e7 −1.73178
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.67772e7 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −4.56228e7 −2.62592
\(260\) 0 0
\(261\) −3.01675e7 −1.69675
\(262\) 0 0
\(263\) 3.37330e7 1.85433 0.927166 − 0.374651i \(-0.122238\pi\)
0.927166 + 0.374651i \(0.122238\pi\)
\(264\) 0 0
\(265\) −2.23968e7 −1.20350
\(266\) 0 0
\(267\) 0 0
\(268\) 8.14118e6 0.422944
\(269\) −1.61391e7 −0.829132 −0.414566 − 0.910019i \(-0.636067\pi\)
−0.414566 + 0.910019i \(0.636067\pi\)
\(270\) 0 0
\(271\) −1.12039e7 −0.562940 −0.281470 − 0.959570i \(-0.590822\pi\)
−0.281470 + 0.959570i \(0.590822\pi\)
\(272\) 3.36118e7 1.67026
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 2.85914e6 0.131650
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 4.17687e7 1.84286 0.921428 − 0.388548i \(-0.127023\pi\)
0.921428 + 0.388548i \(0.127023\pi\)
\(284\) −2.48728e7 −1.08585
\(285\) 0 0
\(286\) 0 0
\(287\) −7.75063e7 −3.27862
\(288\) 0 0
\(289\) 4.32009e7 1.78978
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.90075e7 0.755652 0.377826 − 0.925877i \(-0.376672\pi\)
0.377826 + 0.925877i \(0.376672\pi\)
\(294\) 0 0
\(295\) 3.91275e6 0.152411
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.96412e7 0.720226
\(302\) 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
\(311\) −4.27549e7 −1.42136 −0.710681 − 0.703515i \(-0.751613\pi\)
−0.710681 + 0.703515i \(0.751613\pi\)
\(312\) 0 0
\(313\) −2.14959e7 −0.701008 −0.350504 − 0.936561i \(-0.613990\pi\)
−0.350504 + 0.936561i \(0.613990\pi\)
\(314\) 0 0
\(315\) 5.41282e7 1.73178
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −3.27680e7 −1.00000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 3.40122e7 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 7.07036e7 1.94966 0.974828 − 0.222960i \(-0.0715720\pi\)
0.974828 + 0.222960i \(0.0715720\pi\)
\(332\) −4.98193e7 −1.36139
\(333\) 5.59916e7 1.51632
\(334\) 0 0
\(335\) −1.59008e7 −0.422944
\(336\) 0 0
\(337\) 7.56037e7 1.97539 0.987696 − 0.156383i \(-0.0499835\pi\)
0.987696 + 0.156383i \(0.0499835\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −6.56480e7 −1.67026
\(341\) 0 0
\(342\) 0 0
\(343\) −6.98176e7 −1.73014
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 8.48681e7 1.99649 0.998247 − 0.0591865i \(-0.0188507\pi\)
0.998247 + 0.0591865i \(0.0188507\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 4.85798e7 1.08585
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −7.21338e7 −1.45929 −0.729643 − 0.683828i \(-0.760314\pi\)
−0.729643 + 0.683828i \(0.760314\pi\)
\(368\) 4.98360e7 1.00000
\(369\) 9.51214e7 1.89321
\(370\) 0 0
\(371\) −1.06429e8 −2.08420
\(372\) 0 0
\(373\) 7.96342e7 1.53452 0.767261 − 0.641335i \(-0.221619\pi\)
0.767261 + 0.641335i \(0.221619\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 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
\(383\) −1.11818e8 −1.99029 −0.995146 − 0.0984144i \(-0.968623\pi\)
−0.995146 + 0.0984144i \(0.968623\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −2.41051e7 −0.415888
\(388\) 2.28177e7 0.390639
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 9.98424e7 1.67026
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 6.40000e7 1.00000
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −1.28253e8 −1.94502
\(405\) −6.64301e7 −1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5.36670e6 −0.0784400 −0.0392200 − 0.999231i \(-0.512487\pi\)
−0.0392200 + 0.999231i \(0.512487\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.37391e8 1.96457
\(413\) 1.85934e7 0.263942
\(414\) 0 0
\(415\) 9.73032e7 1.36139
\(416\) 0 0
\(417\) 0 0
\(418\) 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
\(425\) 1.28219e8 1.67026
\(426\) 0 0
\(427\) 0 0
\(428\) −2.60860e7 −0.332718
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −1.29024e8 −1.58931 −0.794654 − 0.607063i \(-0.792347\pi\)
−0.794654 + 0.607063i \(0.792347\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.68638e8 −1.99324 −0.996622 − 0.0821217i \(-0.973830\pi\)
−0.996622 + 0.0821217i \(0.973830\pi\)
\(440\) 0 0
\(441\) 1.71451e8 1.99906
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −1.55714e8 −1.73178
\(449\) 6.48794e6 0.0716750 0.0358375 − 0.999358i \(-0.488590\pi\)
0.0358375 + 0.999358i \(0.488590\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.61423e7 0.499671
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.49095e8 −1.56212 −0.781058 − 0.624458i \(-0.785320\pi\)
−0.781058 + 0.624458i \(0.785320\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −9.73360e7 −1.00000
\(461\) −1.88338e8 −1.92236 −0.961180 − 0.275922i \(-0.911017\pi\)
−0.961180 + 0.275922i \(0.911017\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −1.69501e8 −1.69675
\(465\) 0 0
\(466\) 0 0
\(467\) 1.93951e8 1.90433 0.952163 − 0.305589i \(-0.0988535\pi\)
0.952163 + 0.305589i \(0.0988535\pi\)
\(468\) 0 0
\(469\) −7.55604e7 −0.732446
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −3.11959e8 −2.89252
\(477\) 1.30618e8 1.20350
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.13380e8 1.00000
\(485\) −4.45658e7 −0.390639
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.46627e7 0.208351 0.104176 − 0.994559i \(-0.466780\pi\)
0.104176 + 0.994559i \(0.466780\pi\)
\(492\) 0 0
\(493\) −3.39581e8 −2.83402
\(494\) 0 0
\(495\) 0 0
\(496\) 1.60645e7 0.131650
\(497\) 2.30851e8 1.88045
\(498\) 0 0
\(499\) 2.75885e7 0.222037 0.111019 − 0.993818i \(-0.464589\pi\)
0.111019 + 0.993818i \(0.464589\pi\)
\(500\) −1.25000e8 −1.00000
\(501\) 0 0
\(502\) 0 0
\(503\) 9.63802e7 0.757328 0.378664 − 0.925534i \(-0.376384\pi\)
0.378664 + 0.925534i \(0.376384\pi\)
\(504\) 0 0
\(505\) 2.50495e8 1.94502
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 8.34815e7 0.633048 0.316524 − 0.948584i \(-0.397484\pi\)
0.316524 + 0.948584i \(0.397484\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −2.68342e8 −1.96457
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −8.54107e7 −0.597045 −0.298522 − 0.954403i \(-0.596494\pi\)
−0.298522 + 0.954403i \(0.596494\pi\)
\(524\) 1.55599e8 1.08147
\(525\) 0 0
\(526\) 0 0
\(527\) 3.21839e7 0.219891
\(528\) 0 0
\(529\) 1.48036e8 1.00000
\(530\) 0 0
\(531\) −2.28192e7 −0.152411
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 5.09492e7 0.332718
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 2.30037e8 1.45280 0.726400 − 0.687272i \(-0.241192\pi\)
0.726400 + 0.687272i \(0.241192\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) −2.85586e8 −1.73538
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −3.27502e8 −1.90542
\(557\) −2.83152e8 −1.63853 −0.819263 − 0.573418i \(-0.805617\pi\)
−0.819263 + 0.573418i \(0.805617\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 3.04128e8 1.73178
\(561\) 0 0
\(562\) 0 0
\(563\) −950906. −0.00532859 −0.00266430 − 0.999996i \(-0.500848\pi\)
−0.00266430 + 0.999996i \(0.500848\pi\)
\(564\) 0 0
\(565\) −9.01218e7 −0.499671
\(566\) 0 0
\(567\) −3.15676e8 −1.73178
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.90109e8 1.00000
\(576\) 1.91103e8 1.00000
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 3.31056e8 1.69675
\(581\) 4.62385e8 2.35763
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 3.14597e8 1.51632
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 6.09296e8 2.89252
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −3.92438e8 −1.82596 −0.912979 − 0.408007i \(-0.866224\pi\)
−0.912979 + 0.408007i \(0.866224\pi\)
\(600\) 0 0
\(601\) −1.76304e8 −0.812156 −0.406078 − 0.913838i \(-0.633104\pi\)
−0.406078 + 0.913838i \(0.633104\pi\)
\(602\) 0 0
\(603\) 9.27332e7 0.422944
\(604\) 4.39255e8 1.99345
\(605\) −2.21445e8 −1.00000
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 3.82859e8 1.67026
\(613\) −2.62431e8 −1.13929 −0.569643 − 0.821892i \(-0.692919\pi\)
−0.569643 + 0.821892i \(0.692919\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −4.69502e8 −1.99886 −0.999429 − 0.0337908i \(-0.989242\pi\)
−0.999429 + 0.0337908i \(0.989242\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) −3.13760e7 −0.131650
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.44141e8 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) −1.99910e8 −0.807153
\(629\) 6.30270e8 2.53265
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −2.83317e8 −1.08585
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 1.93195e8 0.726715 0.363357 − 0.931650i \(-0.381630\pi\)
0.363357 + 0.931650i \(0.381630\pi\)
\(644\) −4.62541e8 −1.73178
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) −3.03905e8 −1.08147
\(656\) 5.34454e8 1.89321
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −5.03495e8 −1.69675
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 3.08916e8 1.00000
\(677\) −5.67996e8 −1.83054 −0.915270 − 0.402840i \(-0.868023\pi\)
−0.915270 + 0.402840i \(0.868023\pi\)
\(678\) 0 0
\(679\) −2.11776e8 −0.676501
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 5.57784e8 1.73538
\(686\) 0 0
\(687\) 0 0
\(688\) −1.35438e8 −0.415888
\(689\) 0 0
\(690\) 0 0
\(691\) −5.36935e8 −1.62737 −0.813687 − 0.581303i \(-0.802543\pi\)
−0.813687 + 0.581303i \(0.802543\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 6.39653e8 1.90542
\(696\) 0 0
\(697\) 1.07074e9 3.16216
\(698\) 0 0
\(699\) 0 0
\(700\) −5.94000e8 −1.73178
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 1.19035e9 3.36835
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 4.77190e7 0.131650
\(714\) 0 0
\(715\) 0 0
\(716\) 5.59574e8 1.52447
\(717\) 0 0
\(718\) 0 0
\(719\) −7.38742e8 −1.98749 −0.993747 − 0.111654i \(-0.964385\pi\)
−0.993747 + 0.111654i \(0.964385\pi\)
\(720\) −3.73248e8 −1.00000
\(721\) −1.27516e9 −3.40219
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −6.46594e8 −1.69675
\(726\) 0 0
\(727\) 2.40089e8 0.624841 0.312420 − 0.949944i \(-0.398860\pi\)
0.312420 + 0.949944i \(0.398860\pi\)
\(728\) 0 0
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) −2.71340e8 −0.694642
\(732\) 0 0
\(733\) 5.07465e8 1.28853 0.644264 − 0.764803i \(-0.277164\pi\)
0.644264 + 0.764803i \(0.277164\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −4.37605e8 −1.08430 −0.542149 − 0.840282i \(-0.682389\pi\)
−0.542149 + 0.840282i \(0.682389\pi\)
\(740\) −6.14448e8 −1.51632
\(741\) 0 0
\(742\) 0 0
\(743\) −4.00202e8 −0.975692 −0.487846 − 0.872930i \(-0.662217\pi\)
−0.487846 + 0.872930i \(0.662217\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −5.67473e8 −1.36139
\(748\) 0 0
\(749\) 2.42111e8 0.576194
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −8.57920e8 −1.99345
\(756\) 0 0
\(757\) −1.38049e8 −0.318234 −0.159117 − 0.987260i \(-0.550865\pi\)
−0.159117 + 0.987260i \(0.550865\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.26162e8 −0.286269 −0.143134 − 0.989703i \(-0.545718\pi\)
−0.143134 + 0.989703i \(0.545718\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −7.47772e8 −1.67026
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 1.32246e8 0.286315 0.143158 − 0.989700i \(-0.454274\pi\)
0.143158 + 0.989700i \(0.454274\pi\)
\(774\) 0 0
\(775\) 6.12812e7 0.131650
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 9.63326e8 1.99906
\(785\) 3.90449e8 0.807153
\(786\) 0 0
\(787\) −8.49163e8 −1.74208 −0.871038 − 0.491216i \(-0.836553\pi\)
−0.871038 + 0.491216i \(0.836553\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −4.28259e8 −0.865320
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −5.84876e8 −1.15528 −0.577642 − 0.816290i \(-0.696027\pi\)
−0.577642 + 0.816290i \(0.696027\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 9.03400e8 1.73178
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 4.81926e8 0.910196 0.455098 − 0.890441i \(-0.349604\pi\)
0.455098 + 0.890441i \(0.349604\pi\)
\(810\) 0 0
\(811\) −9.55985e8 −1.79221 −0.896104 − 0.443844i \(-0.853614\pi\)
−0.896104 + 0.443844i \(0.853614\pi\)
\(812\) 1.57318e9 2.93839
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −1.04386e9 −1.89321
\(821\) 3.92618e8 0.709481 0.354740 − 0.934965i \(-0.384569\pi\)
0.354740 + 0.934965i \(0.384569\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −4.19694e8 −0.742021 −0.371011 − 0.928629i \(-0.620989\pi\)
−0.371011 + 0.928629i \(0.620989\pi\)
\(828\) 5.67664e8 1.00000
\(829\) 7.06632e8 1.24031 0.620154 − 0.784480i \(-0.287070\pi\)
0.620154 + 0.784480i \(0.287070\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1.92994e9 3.33895
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.11765e9 1.87896
\(842\) 0 0
\(843\) 0 0
\(844\) −7.72441e8 −1.28481
\(845\) −6.03351e8 −1.00000
\(846\) 0 0
\(847\) −1.05231e9 −1.73178
\(848\) 7.33897e8 1.20350
\(849\) 0 0
\(850\) 0 0
\(851\) 9.34499e8 1.51632
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1.19331e9 −1.88266 −0.941331 − 0.337484i \(-0.890424\pi\)
−0.941331 + 0.337484i \(0.890424\pi\)
\(860\) 2.64528e8 0.415888
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) −1.49099e8 −0.227989
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 2.59907e8 0.390639
\(874\) 0 0
\(875\) 1.16016e9 1.73178
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 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
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −1.09292e9 −1.52447
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −1.62300e8 −0.223378
\(900\) 7.29000e8 1.00000
\(901\) 1.47030e9 2.01017
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.48159e9 −1.98566 −0.992831 − 0.119525i \(-0.961863\pi\)
−0.992831 + 0.119525i \(0.961863\pi\)
\(908\) −3.52122e8 −0.470366
\(909\) −1.46089e9 −1.94502
\(910\) 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
\(917\) −1.44416e9 −1.87287
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.20009e9 1.51632
\(926\) 0 0
\(927\) 1.56497e9 1.96457
\(928\) 0 0
\(929\) 1.58127e9 1.97223 0.986116 − 0.166060i \(-0.0531046\pi\)
0.986116 + 0.166060i \(0.0531046\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 5.51002e8 0.669783 0.334892 − 0.942257i \(-0.391300\pi\)
0.334892 + 0.942257i \(0.391300\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 1.58757e9 1.89321
\(944\) −1.28213e8 −0.152411
\(945\) 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
\(953\) −1.05139e9 −1.21475 −0.607375 − 0.794415i \(-0.707777\pi\)
−0.607375 + 0.794415i \(0.707777\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −9.19848e8 −1.05279
\(957\) 0 0
\(958\) 0 0
\(959\) 2.65059e9 3.00529
\(960\) 0 0
\(961\) −8.72122e8 −0.982668
\(962\) 0 0
\(963\) −2.97136e8 −0.332718
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 3.03963e9 3.29976
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.80419e9 −1.93463 −0.967317 − 0.253570i \(-0.918395\pi\)
−0.967317 + 0.253570i \(0.918395\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.88150e9 −1.99906
\(981\) 0 0
\(982\) 0 0
\(983\) 1.52023e9 1.60047 0.800237 − 0.599684i \(-0.204707\pi\)
0.800237 + 0.599684i \(0.204707\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) −4.02314e8 −0.415888
\(990\) 0 0
\(991\) 1.91887e9 1.97163 0.985815 − 0.167837i \(-0.0536784\pi\)
0.985815 + 0.167837i \(0.0536784\pi\)
\(992\) 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
\(999\) 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 115.7.c.a.114.1 ✓ 1
5.4 even 2 115.7.c.b.114.1 yes 1
23.22 odd 2 115.7.c.b.114.1 yes 1
115.114 odd 2 CM 115.7.c.a.114.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
115.7.c.a.114.1 ✓ 1 1.1 even 1 trivial
115.7.c.a.114.1 ✓ 1 115.114 odd 2 CM
115.7.c.b.114.1 yes 1 5.4 even 2
115.7.c.b.114.1 yes 1 23.22 odd 2