Properties

Label 75.9.c.a.26.1
Level $75$
Weight $9$
Character 75.26
Self dual yes
Analytic conductor $30.553$
Analytic rank $0$
Dimension $1$
CM discriminant -3
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] = [75,9,Mod(26,75)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(75, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0]))
 
N = Newforms(chi, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("75.26");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 75 = 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 75.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: \(30.5533957546\)
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 26.1
Character \(\chi\) \(=\) 75.26

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-81.0000 q^{3} +256.000 q^{4} +4273.00 q^{7} +6561.00 q^{9} +O(q^{10})\) \(q-81.0000 q^{3} +256.000 q^{4} +4273.00 q^{7} +6561.00 q^{9} -20736.0 q^{12} -56447.0 q^{13} +65536.0 q^{16} +157967. q^{19} -346113. q^{21} -531441. q^{27} +1.09389e6 q^{28} +1.22597e6 q^{31} +1.67962e6 q^{36} -503522. q^{37} +4.57221e6 q^{39} +6.83707e6 q^{43} -5.30842e6 q^{48} +1.24937e7 q^{49} -1.44504e7 q^{52} -1.27953e7 q^{57} -307393. q^{61} +2.80352e7 q^{63} +1.67772e7 q^{64} +3.18748e7 q^{67} -1.61693e7 q^{73} +4.04396e7 q^{76} -1.88870e7 q^{79} +4.30467e7 q^{81} -8.86049e7 q^{84} -2.41198e8 q^{91} -9.93033e7 q^{93} +8.21325e7 q^{97} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/75\mathbb{Z}\right)^\times\).

\(n\) \(26\) \(52\)
\(\chi(n)\) \(-1\) \(1\)

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 \(\pi\)
\(3\) −81.0000 −1.00000
\(4\) 256.000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 4273.00 1.77968 0.889838 0.456277i \(-0.150818\pi\)
0.889838 + 0.456277i \(0.150818\pi\)
\(8\) 0 0
\(9\) 6561.00 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) −20736.0 −1.00000
\(13\) −56447.0 −1.97637 −0.988183 0.153277i \(-0.951017\pi\)
−0.988183 + 0.153277i \(0.951017\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 65536.0 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 157967. 1.21214 0.606069 0.795412i \(-0.292746\pi\)
0.606069 + 0.795412i \(0.292746\pi\)
\(20\) 0 0
\(21\) −346113. −1.77968
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) −531441. −1.00000
\(28\) 1.09389e6 1.77968
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.22597e6 1.32749 0.663746 0.747958i \(-0.268966\pi\)
0.663746 + 0.747958i \(0.268966\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 1.67962e6 1.00000
\(37\) −503522. −0.268665 −0.134333 0.990936i \(-0.542889\pi\)
−0.134333 + 0.990936i \(0.542889\pi\)
\(38\) 0 0
\(39\) 4.57221e6 1.97637
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 6.83707e6 1.99985 0.999923 0.0124389i \(-0.00395953\pi\)
0.999923 + 0.0124389i \(0.00395953\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −5.30842e6 −1.00000
\(49\) 1.24937e7 2.16724
\(50\) 0 0
\(51\) 0 0
\(52\) −1.44504e7 −1.97637
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −1.27953e7 −1.21214
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −307393. −0.0222011 −0.0111006 0.999938i \(-0.503533\pi\)
−0.0111006 + 0.999938i \(0.503533\pi\)
\(62\) 0 0
\(63\) 2.80352e7 1.77968
\(64\) 1.67772e7 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 3.18748e7 1.58179 0.790895 0.611952i \(-0.209616\pi\)
0.790895 + 0.611952i \(0.209616\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −1.61693e7 −0.569376 −0.284688 0.958620i \(-0.591890\pi\)
−0.284688 + 0.958620i \(0.591890\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 4.04396e7 1.21214
\(77\) 0 0
\(78\) 0 0
\(79\) −1.88870e7 −0.484904 −0.242452 0.970163i \(-0.577952\pi\)
−0.242452 + 0.970163i \(0.577952\pi\)
\(80\) 0 0
\(81\) 4.30467e7 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) −8.86049e7 −1.77968
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −2.41198e8 −3.51729
\(92\) 0 0
\(93\) −9.93033e7 −1.32749
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 8.21325e7 0.927744 0.463872 0.885902i \(-0.346460\pi\)
0.463872 + 0.885902i \(0.346460\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −4.44490e7 −0.394923 −0.197462 0.980311i \(-0.563270\pi\)
−0.197462 + 0.980311i \(0.563270\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) −1.36049e8 −1.00000
\(109\) −2.71340e8 −1.92224 −0.961122 0.276125i \(-0.910950\pi\)
−0.961122 + 0.276125i \(0.910950\pi\)
\(110\) 0 0
\(111\) 4.07853e7 0.268665
\(112\) 2.80035e8 1.77968
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −3.70349e8 −1.97637
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.14359e8 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 3.13848e8 1.32749
\(125\) 0 0
\(126\) 0 0
\(127\) 4.00562e8 1.53977 0.769883 0.638185i \(-0.220314\pi\)
0.769883 + 0.638185i \(0.220314\pi\)
\(128\) 0 0
\(129\) −5.53803e8 −1.99985
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 6.74993e8 2.15721
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 7.09431e8 1.90043 0.950213 0.311602i \(-0.100865\pi\)
0.950213 + 0.311602i \(0.100865\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 4.29982e8 1.00000
\(145\) 0 0
\(146\) 0 0
\(147\) −1.01199e9 −2.16724
\(148\) −1.28902e8 −0.268665
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −1.00464e9 −1.93243 −0.966214 0.257740i \(-0.917022\pi\)
−0.966214 + 0.257740i \(0.917022\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.17048e9 1.97637
\(157\) −1.03379e9 −1.70151 −0.850757 0.525559i \(-0.823856\pi\)
−0.850757 + 0.525559i \(0.823856\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 9.68184e7 0.137154 0.0685768 0.997646i \(-0.478154\pi\)
0.0685768 + 0.997646i \(0.478154\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 2.37053e9 2.90602
\(170\) 0 0
\(171\) 1.03642e9 1.21214
\(172\) 1.75029e9 1.99985
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.09376e9 −1.01908 −0.509539 0.860448i \(-0.670184\pi\)
−0.509539 + 0.860448i \(0.670184\pi\)
\(182\) 0 0
\(183\) 2.48988e7 0.0222011
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −2.27085e9 −1.77968
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) −1.35895e9 −1.00000
\(193\) 2.47302e9 1.78237 0.891185 0.453641i \(-0.149875\pi\)
0.891185 + 0.453641i \(0.149875\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 3.19839e9 2.16724
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −3.13036e9 −1.99610 −0.998050 0.0624175i \(-0.980119\pi\)
−0.998050 + 0.0624175i \(0.980119\pi\)
\(200\) 0 0
\(201\) −2.58186e9 −1.58179
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −3.69931e9 −1.97637
\(209\) 0 0
\(210\) 0 0
\(211\) −3.33759e9 −1.68385 −0.841924 0.539596i \(-0.818577\pi\)
−0.841924 + 0.539596i \(0.818577\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 5.23856e9 2.36251
\(218\) 0 0
\(219\) 1.30971e9 0.569376
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.10425e9 −0.446527 −0.223263 0.974758i \(-0.571671\pi\)
−0.223263 + 0.974758i \(0.571671\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) −3.27560e9 −1.21214
\(229\) −3.75229e9 −1.36444 −0.682220 0.731147i \(-0.738986\pi\)
−0.682220 + 0.731147i \(0.738986\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.52985e9 0.484904
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −5.19544e8 −0.154012 −0.0770059 0.997031i \(-0.524536\pi\)
−0.0770059 + 0.997031i \(0.524536\pi\)
\(242\) 0 0
\(243\) −3.48678e9 −1.00000
\(244\) −7.86926e7 −0.0222011
\(245\) 0 0
\(246\) 0 0
\(247\) −8.91676e9 −2.39563
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 7.17700e9 1.77968
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4.29497e9 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) −2.15155e9 −0.478137
\(260\) 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
\(268\) 8.15996e9 1.58179
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −2.98709e9 −0.553824 −0.276912 0.960895i \(-0.589311\pi\)
−0.276912 + 0.960895i \(0.589311\pi\)
\(272\) 0 0
\(273\) 1.95370e10 3.51729
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.19798e9 −0.713053 −0.356526 0.934285i \(-0.616039\pi\)
−0.356526 + 0.934285i \(0.616039\pi\)
\(278\) 0 0
\(279\) 8.04357e9 1.32749
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −1.17255e10 −1.82803 −0.914016 0.405678i \(-0.867036\pi\)
−0.914016 + 0.405678i \(0.867036\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 6.97576e9 1.00000
\(290\) 0 0
\(291\) −6.65273e9 −0.927744
\(292\) −4.13934e9 −0.569376
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 2.92148e10 3.55907
\(302\) 0 0
\(303\) 0 0
\(304\) 1.03525e10 1.21214
\(305\) 0 0
\(306\) 0 0
\(307\) −1.63938e10 −1.84555 −0.922775 0.385338i \(-0.874085\pi\)
−0.922775 + 0.385338i \(0.874085\pi\)
\(308\) 0 0
\(309\) 3.60037e9 0.394923
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) −7.32851e9 −0.763551 −0.381776 0.924255i \(-0.624687\pi\)
−0.381776 + 0.924255i \(0.624687\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −4.83508e9 −0.484904
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.10200e10 1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) 2.19786e10 1.92224
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.62495e10 −1.35372 −0.676858 0.736113i \(-0.736659\pi\)
−0.676858 + 0.736113i \(0.736659\pi\)
\(332\) 0 0
\(333\) −3.30361e9 −0.268665
\(334\) 0 0
\(335\) 0 0
\(336\) −2.26829e10 −1.77968
\(337\) −1.18646e10 −0.919886 −0.459943 0.887948i \(-0.652130\pi\)
−0.459943 + 0.887948i \(0.652130\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 2.87527e10 2.07731
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −2.96004e10 −1.99524 −0.997621 0.0689403i \(-0.978038\pi\)
−0.997621 + 0.0689403i \(0.978038\pi\)
\(350\) 0 0
\(351\) 2.99983e10 1.97637
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 7.97001e9 0.469278
\(362\) 0 0
\(363\) −1.73631e10 −1.00000
\(364\) −6.17467e10 −3.51729
\(365\) 0 0
\(366\) 0 0
\(367\) 3.54815e10 1.95586 0.977931 0.208929i \(-0.0669978\pi\)
0.977931 + 0.208929i \(0.0669978\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −2.54217e10 −1.32749
\(373\) −3.07802e10 −1.59014 −0.795070 0.606517i \(-0.792566\pi\)
−0.795070 + 0.606517i \(0.792566\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 3.73548e10 1.81046 0.905230 0.424921i \(-0.139698\pi\)
0.905230 + 0.424921i \(0.139698\pi\)
\(380\) 0 0
\(381\) −3.24455e10 −1.53977
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.48580e10 1.99985
\(388\) 2.10259e10 0.927744
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 3.29221e10 1.32533 0.662667 0.748914i \(-0.269425\pi\)
0.662667 + 0.748914i \(0.269425\pi\)
\(398\) 0 0
\(399\) −5.46744e10 −2.15721
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −6.92022e10 −2.62361
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 4.27011e10 1.52597 0.762985 0.646416i \(-0.223733\pi\)
0.762985 + 0.646416i \(0.223733\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.13789e10 −0.394923
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −5.74639e10 −1.90043
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −1.16089e10 −0.369541 −0.184771 0.982782i \(-0.559154\pi\)
−0.184771 + 0.982782i \(0.559154\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.31349e9 −0.0395108
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) −3.48285e10 −1.00000
\(433\) 9.74052e9 0.277096 0.138548 0.990356i \(-0.455756\pi\)
0.138548 + 0.990356i \(0.455756\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −6.94631e10 −1.92224
\(437\) 0 0
\(438\) 0 0
\(439\) −7.01153e9 −0.188779 −0.0943897 0.995535i \(-0.530090\pi\)
−0.0943897 + 0.995535i \(0.530090\pi\)
\(440\) 0 0
\(441\) 8.19713e10 2.16724
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 1.04410e10 0.268665
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 7.16890e10 1.77968
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 8.13760e10 1.93243
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.72608e10 0.624991 0.312495 0.949919i \(-0.398835\pi\)
0.312495 + 0.949919i \(0.398835\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 6.87853e10 1.49683 0.748413 0.663232i \(-0.230816\pi\)
0.748413 + 0.663232i \(0.230816\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −9.48093e10 −1.97637
\(469\) 1.36201e11 2.81507
\(470\) 0 0
\(471\) 8.37374e10 1.70151
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 2.84223e10 0.530981
\(482\) 0 0
\(483\) 0 0
\(484\) 5.48759e10 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −1.17841e10 −0.209498 −0.104749 0.994499i \(-0.533404\pi\)
−0.104749 + 0.994499i \(0.533404\pi\)
\(488\) 0 0
\(489\) −7.84229e9 −0.137154
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 8.03450e10 1.32749
\(497\) 0 0
\(498\) 0 0
\(499\) −5.04931e10 −0.814386 −0.407193 0.913342i \(-0.633492\pi\)
−0.407193 + 0.913342i \(0.633492\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.92013e11 −2.90602
\(508\) 1.02544e11 1.53977
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −6.90913e10 −1.01330
\(512\) 0 0
\(513\) −8.39501e10 −1.21214
\(514\) 0 0
\(515\) 0 0
\(516\) −1.41774e11 −1.99985
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −1.15606e11 −1.54516 −0.772582 0.634915i \(-0.781035\pi\)
−0.772582 + 0.634915i \(0.781035\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 7.83110e10 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 1.72798e11 2.15721
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −1.69433e11 −1.97792 −0.988962 0.148171i \(-0.952661\pi\)
−0.988962 + 0.148171i \(0.952661\pi\)
\(542\) 0 0
\(543\) 8.85944e10 1.01908
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 6.18266e10 0.690599 0.345300 0.938492i \(-0.387777\pi\)
0.345300 + 0.938492i \(0.387777\pi\)
\(548\) 0 0
\(549\) −2.01681e9 −0.0222011
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8.07043e10 −0.862971
\(554\) 0 0
\(555\) 0 0
\(556\) 1.81614e11 1.90043
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −3.85932e11 −3.95243
\(560\) 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
\(567\) 1.83939e11 1.77968
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 2.39863e10 0.225641 0.112821 0.993615i \(-0.464011\pi\)
0.112821 + 0.993615i \(0.464011\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 1.10075e11 1.00000
\(577\) −2.14996e11 −1.93966 −0.969832 0.243775i \(-0.921614\pi\)
−0.969832 + 0.243775i \(0.921614\pi\)
\(578\) 0 0
\(579\) −2.00314e11 −1.78237
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) −2.59070e11 −2.16724
\(589\) 1.93662e11 1.60910
\(590\) 0 0
\(591\) 0 0
\(592\) −3.29988e10 −0.268665
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 2.53559e11 1.99610
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.88317e11 −1.44341 −0.721707 0.692199i \(-0.756642\pi\)
−0.721707 + 0.692199i \(0.756642\pi\)
\(602\) 0 0
\(603\) 2.09131e11 1.58179
\(604\) −2.57188e11 −1.93243
\(605\) 0 0
\(606\) 0 0
\(607\) 2.65989e11 1.95933 0.979666 0.200634i \(-0.0643001\pi\)
0.979666 + 0.200634i \(0.0643001\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.57998e10 0.111894 0.0559472 0.998434i \(-0.482182\pi\)
0.0559472 + 0.998434i \(0.482182\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 5.00608e10 0.340985 0.170493 0.985359i \(-0.445464\pi\)
0.170493 + 0.985359i \(0.445464\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 2.99644e11 1.97637
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) −2.64651e11 −1.70151
\(629\) 0 0
\(630\) 0 0
\(631\) −1.12984e11 −0.712686 −0.356343 0.934355i \(-0.615976\pi\)
−0.356343 + 0.934355i \(0.615976\pi\)
\(632\) 0 0
\(633\) 2.70345e11 1.68385
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −7.05233e11 −4.28327
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) −1.88544e11 −1.10298 −0.551490 0.834181i \(-0.685941\pi\)
−0.551490 + 0.834181i \(0.685941\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −4.24323e11 −2.36251
\(652\) 2.47855e10 0.137154
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −1.06087e11 −0.569376
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 3.56009e11 1.86490 0.932449 0.361302i \(-0.117668\pi\)
0.932449 + 0.361302i \(0.117668\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 8.94441e10 0.446527
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.28934e11 1.11597 0.557983 0.829853i \(-0.311576\pi\)
0.557983 + 0.829853i \(0.311576\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 6.06856e11 2.90602
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 3.50952e11 1.65108
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 2.65324e11 1.21214
\(685\) 0 0
\(686\) 0 0
\(687\) 3.03936e11 1.36444
\(688\) 4.48074e11 1.99985
\(689\) 0 0
\(690\) 0 0
\(691\) 1.55801e11 0.683374 0.341687 0.939814i \(-0.389002\pi\)
0.341687 + 0.939814i \(0.389002\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −7.95399e10 −0.325659
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.29057e11 0.510735 0.255367 0.966844i \(-0.417804\pi\)
0.255367 + 0.966844i \(0.417804\pi\)
\(710\) 0 0
\(711\) −1.23918e11 −0.484904
\(712\) 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 \(\pi\)
\(720\) 0 0
\(721\) −1.89930e11 −0.702835
\(722\) 0 0
\(723\) 4.20830e10 0.154012
\(724\) −2.80002e11 −1.01908
\(725\) 0 0
\(726\) 0 0
\(727\) −4.11506e11 −1.47312 −0.736560 0.676372i \(-0.763551\pi\)
−0.736560 + 0.676372i \(0.763551\pi\)
\(728\) 0 0
\(729\) 2.82430e11 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 6.37410e9 0.0222011
\(733\) 5.77330e11 1.99990 0.999949 0.0100913i \(-0.00321221\pi\)
0.999949 + 0.0100913i \(0.00321221\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −4.35662e11 −1.46074 −0.730368 0.683054i \(-0.760651\pi\)
−0.730368 + 0.683054i \(0.760651\pi\)
\(740\) 0 0
\(741\) 7.22258e11 2.39563
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −5.35831e11 −1.68449 −0.842244 0.539097i \(-0.818766\pi\)
−0.842244 + 0.539097i \(0.818766\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −5.81337e11 −1.77968
\(757\) −6.36051e11 −1.93691 −0.968453 0.249197i \(-0.919833\pi\)
−0.968453 + 0.249197i \(0.919833\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −1.15944e12 −3.42097
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −3.47892e11 −1.00000
\(769\) −5.07613e11 −1.45153 −0.725767 0.687941i \(-0.758515\pi\)
−0.725767 + 0.687941i \(0.758515\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 6.33092e11 1.78237
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.74276e11 0.478137
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 8.18789e11 2.16724
\(785\) 0 0
\(786\) 0 0
\(787\) 6.52899e11 1.70195 0.850976 0.525205i \(-0.176011\pi\)
0.850976 + 0.525205i \(0.176011\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.73514e10 0.0438775
\(794\) 0 0
\(795\) 0 0
\(796\) −8.01373e11 −1.99610
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −6.60957e11 −1.58179
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 7.00891e11 1.62019 0.810097 0.586296i \(-0.199415\pi\)
0.810097 + 0.586296i \(0.199415\pi\)
\(812\) 0 0
\(813\) 2.41954e11 0.553824
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.08003e12 2.42409
\(818\) 0 0
\(819\) −1.58250e12 −3.51729
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −7.02937e11 −1.53220 −0.766102 0.642719i \(-0.777806\pi\)
−0.766102 + 0.642719i \(0.777806\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 4.11968e11 0.872258 0.436129 0.899884i \(-0.356349\pi\)
0.436129 + 0.899884i \(0.356349\pi\)
\(830\) 0 0
\(831\) 3.40037e11 0.713053
\(832\) −9.47024e11 −1.97637
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −6.51529e11 −1.32749
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.00246e11 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) −8.54422e11 −1.68385
\(845\) 0 0
\(846\) 0 0
\(847\) 9.15955e11 1.77968
\(848\) 0 0
\(849\) 9.49762e11 1.82803
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −4.38996e11 −0.829210 −0.414605 0.910001i \(-0.636080\pi\)
−0.414605 + 0.910001i \(0.636080\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −8.02752e11 −1.47438 −0.737189 0.675686i \(-0.763847\pi\)
−0.737189 + 0.675686i \(0.763847\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −5.65036e11 −1.00000
\(868\) 1.34107e12 2.36251
\(869\) 0 0
\(870\) 0 0
\(871\) −1.79924e12 −3.12620
\(872\) 0 0
\(873\) 5.38871e11 0.927744
\(874\) 0 0
\(875\) 0 0
\(876\) 3.35286e11 0.569376
\(877\) 1.11362e12 1.88252 0.941262 0.337677i \(-0.109641\pi\)
0.941262 + 0.337677i \(0.109641\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.21517e12 1.99891 0.999457 0.0329612i \(-0.0104938\pi\)
0.999457 + 0.0329612i \(0.0104938\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.71160e12 2.74028
\(890\) 0 0
\(891\) 0 0
\(892\) −2.82688e11 −0.446527
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −2.36640e12 −3.55907
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.20490e12 −1.78042 −0.890212 0.455547i \(-0.849444\pi\)
−0.890212 + 0.455547i \(0.849444\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) −8.38555e11 −1.21214
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −9.60587e11 −1.36444
\(917\) 0 0
\(918\) 0 0
\(919\) 1.41417e12 1.98261 0.991307 0.131571i \(-0.0420022\pi\)
0.991307 + 0.131571i \(0.0420022\pi\)
\(920\) 0 0
\(921\) 1.32790e12 1.84555
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −2.91630e11 −0.394923
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 1.97360e12 2.62700
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.39864e11 −0.311176 −0.155588 0.987822i \(-0.549727\pi\)
−0.155588 + 0.987822i \(0.549727\pi\)
\(938\) 0 0
\(939\) 5.93609e11 0.763551
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 3.91642e11 0.484904
\(949\) 9.12707e11 1.12530
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 6.50104e11 0.762236
\(962\) 0 0
\(963\) 0 0
\(964\) −1.33003e11 −0.154012
\(965\) 0 0
\(966\) 0 0
\(967\) 1.51552e12 1.73322 0.866611 0.498984i \(-0.166293\pi\)
0.866611 + 0.498984i \(0.166293\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) −8.92617e11 −1.00000
\(973\) 3.03140e12 3.38214
\(974\) 0 0
\(975\) 0 0
\(976\) −2.01453e10 −0.0222011
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −1.78026e12 −1.92224
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −2.28269e12 −2.39563
\(989\) 0 0
\(990\) 0 0
\(991\) −1.11521e12 −1.15627 −0.578136 0.815940i \(-0.696220\pi\)
−0.578136 + 0.815940i \(0.696220\pi\)
\(992\) 0 0
\(993\) 1.31621e12 1.35372
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −5.81390e11 −0.588420 −0.294210 0.955741i \(-0.595056\pi\)
−0.294210 + 0.955741i \(0.595056\pi\)
\(998\) 0 0
\(999\) 2.67592e11 0.268665
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 75.9.c.a.26.1 1
3.2 odd 2 CM 75.9.c.a.26.1 1
5.2 odd 4 75.9.d.a.74.2 2
5.3 odd 4 75.9.d.a.74.1 2
5.4 even 2 75.9.c.b.26.1 yes 1
15.2 even 4 75.9.d.a.74.2 2
15.8 even 4 75.9.d.a.74.1 2
15.14 odd 2 75.9.c.b.26.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
75.9.c.a.26.1 1 1.1 even 1 trivial
75.9.c.a.26.1 1 3.2 odd 2 CM
75.9.c.b.26.1 yes 1 5.4 even 2
75.9.c.b.26.1 yes 1 15.14 odd 2
75.9.d.a.74.1 2 5.3 odd 4
75.9.d.a.74.1 2 15.8 even 4
75.9.d.a.74.2 2 5.2 odd 4
75.9.d.a.74.2 2 15.2 even 4