Properties

Label 225.12.a.d.1.1
Level $225$
Weight $12$
Character 225.1
Self dual yes
Analytic conductor $172.877$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [225,12,Mod(1,225)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(225, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 12, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("225.1");
 
S:= CuspForms(chi, 12);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 225 = 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 225.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: \(172.877215626\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 225.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2048.00 q^{4} +76885.0 q^{7} +O(q^{10})\) \(q-2048.00 q^{4} +76885.0 q^{7} +2.24862e6 q^{13} +4.19430e6 q^{16} +2.05819e7 q^{19} -1.57460e8 q^{28} +2.96477e8 q^{31} +7.82920e8 q^{37} -1.76836e9 q^{43} +3.93398e9 q^{49} -4.60516e9 q^{52} -1.29773e10 q^{61} -8.58993e9 q^{64} -5.95129e9 q^{67} -1.98055e10 q^{73} -4.21517e10 q^{76} +3.28858e10 q^{79} +1.72885e11 q^{91} +5.29686e10 q^{97} +O(q^{100})\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2048.00 −1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 76885.0 1.72903 0.864515 0.502607i \(-0.167626\pi\)
0.864515 + 0.502607i \(0.167626\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 2.24862e6 1.67968 0.839840 0.542834i \(-0.182649\pi\)
0.839840 + 0.542834i \(0.182649\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.19430e6 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 2.05819e7 1.90696 0.953478 0.301463i \(-0.0974748\pi\)
0.953478 + 0.301463i \(0.0974748\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) −1.57460e8 −1.72903
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 2.96477e8 1.85995 0.929976 0.367621i \(-0.119828\pi\)
0.929976 + 0.367621i \(0.119828\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.82920e8 1.85613 0.928064 0.372422i \(-0.121472\pi\)
0.928064 + 0.372422i \(0.121472\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −1.76836e9 −1.83440 −0.917199 0.398429i \(-0.869556\pi\)
−0.917199 + 0.398429i \(0.869556\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 3.93398e9 1.98954
\(50\) 0 0
\(51\) 0 0
\(52\) −4.60516e9 −1.67968
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 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
\(61\) −1.29773e10 −1.96730 −0.983649 0.180098i \(-0.942359\pi\)
−0.983649 + 0.180098i \(0.942359\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.58993e9 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −5.95129e9 −0.538517 −0.269259 0.963068i \(-0.586779\pi\)
−0.269259 + 0.963068i \(0.586779\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) −1.98055e10 −1.11818 −0.559088 0.829108i \(-0.688849\pi\)
−0.559088 + 0.829108i \(0.688849\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −4.21517e10 −1.90696
\(77\) 0 0
\(78\) 0 0
\(79\) 3.28858e10 1.20243 0.601215 0.799087i \(-0.294684\pi\)
0.601215 + 0.799087i \(0.294684\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 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
\(91\) 1.72885e11 2.90422
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 5.29686e10 0.626287 0.313144 0.949706i \(-0.398618\pi\)
0.313144 + 0.949706i \(0.398618\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 7.30681e10 0.621045 0.310523 0.950566i \(-0.399496\pi\)
0.310523 + 0.950566i \(0.399496\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 2.57951e9 0.0160580 0.00802899 0.999968i \(-0.497444\pi\)
0.00802899 + 0.999968i \(0.497444\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 3.22479e11 1.72903
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −2.85312e11 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −6.07185e11 −1.85995
\(125\) 0 0
\(126\) 0 0
\(127\) −3.94910e11 −1.06066 −0.530331 0.847791i \(-0.677932\pi\)
−0.530331 + 0.847791i \(0.677932\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 1.58244e12 3.29718
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 1.14947e12 1.87896 0.939479 0.342607i \(-0.111310\pi\)
0.939479 + 0.342607i \(0.111310\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −1.60342e12 −1.85613
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −1.31670e12 −1.36494 −0.682470 0.730913i \(-0.739094\pi\)
−0.682470 + 0.730913i \(0.739094\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.73122e12 1.44845 0.724225 0.689564i \(-0.242198\pi\)
0.724225 + 0.689564i \(0.242198\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.85523e12 −1.26289 −0.631445 0.775420i \(-0.717538\pi\)
−0.631445 + 0.775420i \(0.717538\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 3.26411e12 1.82133
\(170\) 0 0
\(171\) 0 0
\(172\) 3.62160e12 1.83440
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 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
\(181\) 1.27601e12 0.488228 0.244114 0.969747i \(-0.421503\pi\)
0.244114 + 0.969747i \(0.421503\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) 1.87475e12 0.503939 0.251969 0.967735i \(-0.418922\pi\)
0.251969 + 0.967735i \(0.418922\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −8.05678e12 −1.98954
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 8.39365e12 1.90660 0.953299 0.302028i \(-0.0976636\pi\)
0.953299 + 0.302028i \(0.0976636\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 9.43137e12 1.67968
\(209\) 0 0
\(210\) 0 0
\(211\) −1.12332e13 −1.84906 −0.924531 0.381106i \(-0.875543\pi\)
−0.924531 + 0.381106i \(0.875543\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.27946e13 3.21591
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.58316e13 1.92241 0.961207 0.275829i \(-0.0889524\pi\)
0.961207 + 0.275829i \(0.0889524\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −1.90357e13 −1.99744 −0.998720 0.0505740i \(-0.983895\pi\)
−0.998720 + 0.0505740i \(0.983895\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 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
\(241\) −2.27780e12 −0.180477 −0.0902386 0.995920i \(-0.528763\pi\)
−0.0902386 + 0.995920i \(0.528763\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 2.65775e13 1.96730
\(245\) 0 0
\(246\) 0 0
\(247\) 4.62808e13 3.20308
\(248\) 0 0
\(249\) 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
\(256\) 1.75922e13 1.00000
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 6.01948e13 3.20930
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.21882e13 0.538517
\(269\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) −4.72629e13 −1.96422 −0.982109 0.188315i \(-0.939697\pi\)
−0.982109 + 0.188315i \(0.939697\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 5.40745e13 1.99230 0.996148 0.0876833i \(-0.0279464\pi\)
0.996148 + 0.0876833i \(0.0279464\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) −1.27932e13 −0.418941 −0.209470 0.977815i \(-0.567174\pi\)
−0.209470 + 0.977815i \(0.567174\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −3.42719e13 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 4.05617e13 1.11818
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −1.35960e14 −3.17173
\(302\) 0 0
\(303\) 0 0
\(304\) 8.63267e13 1.90696
\(305\) 0 0
\(306\) 0 0
\(307\) −8.17725e13 −1.71138 −0.855689 0.517490i \(-0.826866\pi\)
−0.855689 + 0.517490i \(0.826866\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 5.35920e12 0.100834 0.0504169 0.998728i \(-0.483945\pi\)
0.0504169 + 0.998728i \(0.483945\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −6.73502e13 −1.20243
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 9.80000e13 1.35573 0.677863 0.735188i \(-0.262906\pi\)
0.677863 + 0.735188i \(0.262906\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.59216e14 1.99536 0.997682 0.0680538i \(-0.0216789\pi\)
0.997682 + 0.0680538i \(0.0216789\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.50437e14 1.71095
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 1.70909e14 1.76695 0.883475 0.468478i \(-0.155197\pi\)
0.883475 + 0.468478i \(0.155197\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 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
\(361\) 3.07124e14 2.63648
\(362\) 0 0
\(363\) 0 0
\(364\) −3.54068e14 −2.90422
\(365\) 0 0
\(366\) 0 0
\(367\) 2.63013e13 0.206212 0.103106 0.994670i \(-0.467122\pi\)
0.103106 + 0.994670i \(0.467122\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −2.77182e14 −1.98777 −0.993884 0.110426i \(-0.964779\pi\)
−0.993884 + 0.110426i \(0.964779\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −5.50919e13 −0.361886 −0.180943 0.983494i \(-0.557915\pi\)
−0.180943 + 0.983494i \(0.557915\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −1.08480e14 −0.626287
\(389\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −2.73891e14 −1.39389 −0.696947 0.717123i \(-0.745459\pi\)
−0.696947 + 0.717123i \(0.745459\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 6.66663e14 3.12412
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −8.33592e13 −0.360143 −0.180072 0.983654i \(-0.557633\pi\)
−0.180072 + 0.983654i \(0.557633\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −1.49644e14 −0.621045
\(413\) 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
\(421\) −3.20990e13 −0.118288 −0.0591438 0.998249i \(-0.518837\pi\)
−0.0591438 + 0.998249i \(0.518837\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −9.97759e14 −3.40152
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) −6.33443e14 −1.99997 −0.999986 0.00523115i \(-0.998335\pi\)
−0.999986 + 0.00523115i \(0.998335\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −5.28283e12 −0.0160580
\(437\) 0 0
\(438\) 0 0
\(439\) −4.50030e14 −1.31731 −0.658654 0.752446i \(-0.728874\pi\)
−0.658654 + 0.752446i \(0.728874\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −6.60437e14 −1.72903
\(449\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −4.41639e14 −1.03640 −0.518202 0.855258i \(-0.673398\pi\)
−0.518202 + 0.855258i \(0.673398\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −7.53453e14 −1.64574 −0.822869 0.568231i \(-0.807628\pi\)
−0.822869 + 0.568231i \(0.807628\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) −4.57565e14 −0.931113
\(470\) 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
\(481\) 1.76049e15 3.11770
\(482\) 0 0
\(483\) 0 0
\(484\) 5.84318e14 1.00000
\(485\) 0 0
\(486\) 0 0
\(487\) −4.03932e14 −0.668188 −0.334094 0.942540i \(-0.608430\pi\)
−0.334094 + 0.942540i \(0.608430\pi\)
\(488\) 0 0
\(489\) 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
\(496\) 1.24351e15 1.85995
\(497\) 0 0
\(498\) 0 0
\(499\) −5.44758e14 −0.788225 −0.394113 0.919062i \(-0.628948\pi\)
−0.394113 + 0.919062i \(0.628948\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 8.08775e14 1.06066
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −1.52275e15 −1.93336
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) −2.85954e14 −0.319548 −0.159774 0.987154i \(-0.551077\pi\)
−0.159774 + 0.987154i \(0.551077\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −9.52810e14 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −3.24084e15 −3.29718
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 1.78270e15 1.65384 0.826921 0.562319i \(-0.190091\pi\)
0.826921 + 0.562319i \(0.190091\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.19604e15 1.91739 0.958695 0.284436i \(-0.0918062\pi\)
0.958695 + 0.284436i \(0.0918062\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 2.52843e15 2.07904
\(554\) 0 0
\(555\) 0 0
\(556\) −2.35412e15 −1.87896
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −3.97636e15 −3.08120
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 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
\(571\) −1.21874e15 −0.840261 −0.420131 0.907464i \(-0.638016\pi\)
−0.420131 + 0.907464i \(0.638016\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.32758e15 0.864157 0.432078 0.901836i \(-0.357780\pi\)
0.432078 + 0.901836i \(0.357780\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 6.10206e15 3.54685
\(590\) 0 0
\(591\) 0 0
\(592\) 3.28380e15 1.85613
\(593\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 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
\(601\) 3.04058e15 1.58178 0.790892 0.611956i \(-0.209617\pi\)
0.790892 + 0.611956i \(0.209617\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2.69660e15 1.36494
\(605\) 0 0
\(606\) 0 0
\(607\) −4.03360e15 −1.98680 −0.993401 0.114689i \(-0.963413\pi\)
−0.993401 + 0.114689i \(0.963413\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −3.37012e15 −1.57258 −0.786289 0.617858i \(-0.788000\pi\)
−0.786289 + 0.617858i \(0.788000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 1.56531e15 0.692312 0.346156 0.938177i \(-0.387487\pi\)
0.346156 + 0.938177i \(0.387487\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) −3.54553e15 −1.44845
\(629\) 0 0
\(630\) 0 0
\(631\) −4.79425e15 −1.90791 −0.953957 0.299943i \(-0.903032\pi\)
−0.953957 + 0.299943i \(0.903032\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 8.84600e15 3.34180
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 3.01389e15 1.08135 0.540676 0.841231i \(-0.318168\pi\)
0.540676 + 0.841231i \(0.318168\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 3.79951e15 1.26289
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 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
\(661\) −1.77508e15 −0.547155 −0.273578 0.961850i \(-0.588207\pi\)
−0.273578 + 0.961850i \(0.588207\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 7.06267e15 1.97190 0.985952 0.167026i \(-0.0534164\pi\)
0.985952 + 0.167026i \(0.0534164\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −6.68490e15 −1.82133
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 4.07249e15 1.08287
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −7.41703e15 −1.83440
\(689\) 0 0
\(690\) 0 0
\(691\) −8.23279e15 −1.98801 −0.994003 0.109351i \(-0.965123\pi\)
−0.994003 + 0.109351i \(0.965123\pi\)
\(692\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 1.61140e16 3.53955
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.99833e15 1.04778 0.523891 0.851785i \(-0.324480\pi\)
0.523891 + 0.851785i \(0.324480\pi\)
\(710\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 5.61784e15 1.07381
\(722\) 0 0
\(723\) 0 0
\(724\) −2.61327e15 −0.488228
\(725\) 0 0
\(726\) 0 0
\(727\) 9.33970e15 1.70566 0.852832 0.522185i \(-0.174883\pi\)
0.852832 + 0.522185i \(0.174883\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 4.23742e15 0.739655 0.369827 0.929100i \(-0.379417\pi\)
0.369827 + 0.929100i \(0.379417\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.72182e15 −0.287372 −0.143686 0.989623i \(-0.545895\pi\)
−0.143686 + 0.989623i \(0.545895\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 1.20840e16 1.84582 0.922911 0.385013i \(-0.125803\pi\)
0.922911 + 0.385013i \(0.125803\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.28671e16 1.88127 0.940636 0.339416i \(-0.110229\pi\)
0.940636 + 0.339416i \(0.110229\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 1.98325e14 0.0277647
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −9.03163e15 −1.21108 −0.605538 0.795817i \(-0.707042\pi\)
−0.605538 + 0.795817i \(0.707042\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −3.83949e15 −0.503939
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 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
\(783\) 0 0
\(784\) 1.65003e16 1.98954
\(785\) 0 0
\(786\) 0 0
\(787\) 1.58074e16 1.86637 0.933187 0.359391i \(-0.117015\pi\)
0.933187 + 0.359391i \(0.117015\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.91809e16 −3.30443
\(794\) 0 0
\(795\) 0 0
\(796\) −1.71902e16 −1.90660
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 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
\(811\) 1.99246e16 1.99423 0.997113 0.0759375i \(-0.0241950\pi\)
0.997113 + 0.0759375i \(0.0241950\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −3.63962e16 −3.49812
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 1.77442e16 1.63816 0.819081 0.573678i \(-0.194484\pi\)
0.819081 + 0.573678i \(0.194484\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −1.54331e16 −1.36900 −0.684501 0.729012i \(-0.739980\pi\)
−0.684501 + 0.729012i \(0.739980\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.93155e16 −1.67968
\(833\) 0 0
\(834\) 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
\(841\) −1.22005e16 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 2.30057e16 1.84906
\(845\) 0 0
\(846\) 0 0
\(847\) −2.19362e16 −1.72903
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1.86296e16 1.41248 0.706241 0.707972i \(-0.250390\pi\)
0.706241 + 0.707972i \(0.250390\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) −2.70738e16 −1.97509 −0.987545 0.157335i \(-0.949710\pi\)
−0.987545 + 0.157335i \(0.949710\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) −4.66834e16 −3.21591
\(869\) 0 0
\(870\) 0 0
\(871\) −1.33822e16 −0.904537
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 3.06270e16 1.99346 0.996728 0.0808282i \(-0.0257565\pi\)
0.996728 + 0.0808282i \(0.0257565\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) −1.35313e16 −0.848311 −0.424155 0.905589i \(-0.639429\pi\)
−0.424155 + 0.905589i \(0.639429\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −3.03626e16 −1.83392
\(890\) 0 0
\(891\) 0 0
\(892\) −3.24230e16 −1.92241
\(893\) 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
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −3.66352e16 −1.98179 −0.990896 0.134631i \(-0.957015\pi\)
−0.990896 + 0.134631i \(0.957015\pi\)
\(908\) 0 0
\(909\) 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
\(916\) 3.89851e16 1.99744
\(917\) 0 0
\(918\) 0 0
\(919\) 7.17141e15 0.360885 0.180443 0.983586i \(-0.442247\pi\)
0.180443 + 0.983586i \(0.442247\pi\)
\(920\) 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
\(931\) 8.09687e16 3.79397
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.76725e16 0.799338 0.399669 0.916660i \(-0.369125\pi\)
0.399669 + 0.916660i \(0.369125\pi\)
\(938\) 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
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) −4.45350e16 −1.87818
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 6.24901e16 2.45942
\(962\) 0 0
\(963\) 0 0
\(964\) 4.66494e15 0.180477
\(965\) 0 0
\(966\) 0 0
\(967\) −2.33345e16 −0.887470 −0.443735 0.896158i \(-0.646347\pi\)
−0.443735 + 0.896158i \(0.646347\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 8.83771e16 3.24877
\(974\) 0 0
\(975\) 0 0
\(976\) −5.44307e16 −1.96730
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −9.47830e16 −3.20308
\(989\) 0 0
\(990\) 0 0
\(991\) −8.04799e15 −0.267474 −0.133737 0.991017i \(-0.542698\pi\)
−0.133737 + 0.991017i \(0.542698\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −6.12213e16 −1.96825 −0.984123 0.177489i \(-0.943203\pi\)
−0.984123 + 0.177489i \(0.943203\pi\)
\(998\) 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 225.12.a.d.1.1 yes 1
3.2 odd 2 CM 225.12.a.d.1.1 yes 1
5.2 odd 4 225.12.b.e.199.2 2
5.3 odd 4 225.12.b.e.199.1 2
5.4 even 2 225.12.a.c.1.1 1
15.2 even 4 225.12.b.e.199.2 2
15.8 even 4 225.12.b.e.199.1 2
15.14 odd 2 225.12.a.c.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
225.12.a.c.1.1 1 5.4 even 2
225.12.a.c.1.1 1 15.14 odd 2
225.12.a.d.1.1 yes 1 1.1 even 1 trivial
225.12.a.d.1.1 yes 1 3.2 odd 2 CM
225.12.b.e.199.1 2 5.3 odd 4
225.12.b.e.199.1 2 15.8 even 4
225.12.b.e.199.2 2 5.2 odd 4
225.12.b.e.199.2 2 15.2 even 4