Properties

Label 2304.2.a.z.1.3
Level $2304$
Weight $2$
Character 2304.1
Self dual yes
Analytic conductor $18.398$
Analytic rank $0$
Dimension $4$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,2,Mod(1,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2304.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2304.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: \(18.3975326257\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{24})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - 4x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{5} \)
Twist minimal: no (minimal twist has level 1152)
Fricke sign: \(-1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.3
Root \(1.93185\) of defining polynomial
Character \(\chi\) \(=\) 2304.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.46410 q^{5} -4.89898 q^{7} +O(q^{10})\) \(q+3.46410 q^{5} -4.89898 q^{7} +5.65685 q^{11} +7.00000 q^{25} +10.3923 q^{29} -4.89898 q^{31} -16.9706 q^{35} +17.0000 q^{49} -3.46410 q^{53} +19.5959 q^{55} +11.3137 q^{59} +14.0000 q^{73} -27.7128 q^{77} +14.6969 q^{79} -5.65685 q^{83} +2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 28 q^{25} + 68 q^{49} + 56 q^{73} + 8 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 3.46410 1.54919 0.774597 0.632456i \(-0.217953\pi\)
0.774597 + 0.632456i \(0.217953\pi\)
\(6\) 0 0
\(7\) −4.89898 −1.85164 −0.925820 0.377964i \(-0.876624\pi\)
−0.925820 + 0.377964i \(0.876624\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 5.65685 1.70561 0.852803 0.522233i \(-0.174901\pi\)
0.852803 + 0.522233i \(0.174901\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) 7.00000 1.40000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 10.3923 1.92980 0.964901 0.262613i \(-0.0845842\pi\)
0.964901 + 0.262613i \(0.0845842\pi\)
\(30\) 0 0
\(31\) −4.89898 −0.879883 −0.439941 0.898027i \(-0.645001\pi\)
−0.439941 + 0.898027i \(0.645001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −16.9706 −2.86855
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 17.0000 2.42857
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) −3.46410 −0.475831 −0.237915 0.971286i \(-0.576464\pi\)
−0.237915 + 0.971286i \(0.576464\pi\)
\(54\) 0 0
\(55\) 19.5959 2.64231
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.3137 1.47292 0.736460 0.676481i \(-0.236496\pi\)
0.736460 + 0.676481i \(0.236496\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 14.0000 1.63858 0.819288 0.573382i \(-0.194369\pi\)
0.819288 + 0.573382i \(0.194369\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −27.7128 −3.15817
\(78\) 0 0
\(79\) 14.6969 1.65353 0.826767 0.562544i \(-0.190177\pi\)
0.826767 + 0.562544i \(0.190177\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) −5.65685 −0.620920 −0.310460 0.950586i \(-0.600483\pi\)
−0.310460 + 0.950586i \(0.600483\pi\)
\(84\) 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
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −3.46410 −0.344691 −0.172345 0.985037i \(-0.555135\pi\)
−0.172345 + 0.985037i \(0.555135\pi\)
\(102\) 0 0
\(103\) −14.6969 −1.44813 −0.724066 0.689730i \(-0.757729\pi\)
−0.724066 + 0.689730i \(0.757729\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 11.3137 1.09374 0.546869 0.837218i \(-0.315820\pi\)
0.546869 + 0.837218i \(0.315820\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 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
\(121\) 21.0000 1.90909
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 6.92820 0.619677
\(126\) 0 0
\(127\) 4.89898 0.434714 0.217357 0.976092i \(-0.430256\pi\)
0.217357 + 0.976092i \(0.430256\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 22.6274 1.97697 0.988483 0.151330i \(-0.0483556\pi\)
0.988483 + 0.151330i \(0.0483556\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 36.0000 2.98964
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −24.2487 −1.98653 −0.993266 0.115857i \(-0.963039\pi\)
−0.993266 + 0.115857i \(0.963039\pi\)
\(150\) 0 0
\(151\) −24.4949 −1.99337 −0.996683 0.0813788i \(-0.974068\pi\)
−0.996683 + 0.0813788i \(0.974068\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −16.9706 −1.36311
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 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
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −13.0000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 17.3205 1.31685 0.658427 0.752645i \(-0.271222\pi\)
0.658427 + 0.752645i \(0.271222\pi\)
\(174\) 0 0
\(175\) −34.2929 −2.59230
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −11.3137 −0.845626 −0.422813 0.906217i \(-0.638957\pi\)
−0.422813 + 0.906217i \(0.638957\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 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
\(193\) −26.0000 −1.87152 −0.935760 0.352636i \(-0.885285\pi\)
−0.935760 + 0.352636i \(0.885285\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 24.2487 1.72765 0.863825 0.503793i \(-0.168062\pi\)
0.863825 + 0.503793i \(0.168062\pi\)
\(198\) 0 0
\(199\) 24.4949 1.73640 0.868199 0.496217i \(-0.165278\pi\)
0.868199 + 0.496217i \(0.165278\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −50.9117 −3.57330
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 24.0000 1.62923
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −14.6969 −0.984180 −0.492090 0.870544i \(-0.663767\pi\)
−0.492090 + 0.870544i \(0.663767\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 28.2843 1.87729 0.938647 0.344881i \(-0.112081\pi\)
0.938647 + 0.344881i \(0.112081\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 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
\(241\) 10.0000 0.644157 0.322078 0.946713i \(-0.395619\pi\)
0.322078 + 0.946713i \(0.395619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 58.8897 3.76233
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 5.65685 0.357057 0.178529 0.983935i \(-0.442866\pi\)
0.178529 + 0.983935i \(0.442866\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.3923 0.633630 0.316815 0.948487i \(-0.397387\pi\)
0.316815 + 0.948487i \(0.397387\pi\)
\(270\) 0 0
\(271\) 24.4949 1.48796 0.743980 0.668202i \(-0.232936\pi\)
0.743980 + 0.668202i \(0.232936\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 39.5980 2.38785
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 0 0
\(283\) 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
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −31.1769 −1.82137 −0.910687 0.413096i \(-0.864447\pi\)
−0.910687 + 0.413096i \(0.864447\pi\)
\(294\) 0 0
\(295\) 39.1918 2.28184
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 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
\(313\) −34.0000 −1.92179 −0.960897 0.276907i \(-0.910691\pi\)
−0.960897 + 0.276907i \(0.910691\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −17.3205 −0.972817 −0.486408 0.873732i \(-0.661693\pi\)
−0.486408 + 0.873732i \(0.661693\pi\)
\(318\) 0 0
\(319\) 58.7878 3.29148
\(320\) 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
\(331\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 22.0000 1.19842 0.599208 0.800593i \(-0.295482\pi\)
0.599208 + 0.800593i \(0.295482\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) −27.7128 −1.50073
\(342\) 0 0
\(343\) −48.9898 −2.64520
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −28.2843 −1.51838 −0.759190 0.650870i \(-0.774404\pi\)
−0.759190 + 0.650870i \(0.774404\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 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
\(361\) −19.0000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 48.4974 2.53847
\(366\) 0 0
\(367\) −4.89898 −0.255725 −0.127862 0.991792i \(-0.540812\pi\)
−0.127862 + 0.991792i \(0.540812\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 16.9706 0.881068
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) −96.0000 −4.89261
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −24.2487 −1.22946 −0.614729 0.788738i \(-0.710735\pi\)
−0.614729 + 0.788738i \(0.710735\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 50.9117 2.56165
\(396\) 0 0
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 10.0000 0.494468 0.247234 0.968956i \(-0.420478\pi\)
0.247234 + 0.968956i \(0.420478\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −55.4256 −2.72732
\(414\) 0 0
\(415\) −19.5959 −0.961926
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −39.5980 −1.93449 −0.967244 0.253849i \(-0.918303\pi\)
−0.967244 + 0.253849i \(0.918303\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 14.0000 0.672797 0.336399 0.941720i \(-0.390791\pi\)
0.336399 + 0.941720i \(0.390791\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 24.4949 1.16908 0.584539 0.811366i \(-0.301275\pi\)
0.584539 + 0.811366i \(0.301275\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −28.2843 −1.34383 −0.671913 0.740630i \(-0.734527\pi\)
−0.671913 + 0.740630i \(0.734527\pi\)
\(444\) 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
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −38.0000 −1.77757 −0.888783 0.458329i \(-0.848448\pi\)
−0.888783 + 0.458329i \(0.848448\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −38.1051 −1.77473 −0.887366 0.461065i \(-0.847467\pi\)
−0.887366 + 0.461065i \(0.847467\pi\)
\(462\) 0 0
\(463\) −34.2929 −1.59372 −0.796862 0.604161i \(-0.793508\pi\)
−0.796862 + 0.604161i \(0.793508\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −39.5980 −1.83238 −0.916188 0.400749i \(-0.868750\pi\)
−0.916188 + 0.400749i \(0.868750\pi\)
\(468\) 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
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 6.92820 0.314594
\(486\) 0 0
\(487\) −44.0908 −1.99795 −0.998973 0.0453143i \(-0.985571\pi\)
−0.998973 + 0.0453143i \(0.985571\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −22.6274 −1.02116 −0.510581 0.859830i \(-0.670569\pi\)
−0.510581 + 0.859830i \(0.670569\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −12.0000 −0.533993
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −45.0333 −1.99607 −0.998033 0.0626839i \(-0.980034\pi\)
−0.998033 + 0.0626839i \(0.980034\pi\)
\(510\) 0 0
\(511\) −68.5857 −3.03405
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) −50.9117 −2.24344
\(516\) 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
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 39.1918 1.69441
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 96.1665 4.14219
\(540\) 0 0
\(541\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −72.0000 −3.06175
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 45.0333 1.90812 0.954062 0.299611i \(-0.0968568\pi\)
0.954062 + 0.299611i \(0.0968568\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 28.2843 1.19204 0.596020 0.802970i \(-0.296748\pi\)
0.596020 + 0.802970i \(0.296748\pi\)
\(564\) 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
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −38.0000 −1.58196 −0.790980 0.611842i \(-0.790429\pi\)
−0.790980 + 0.611842i \(0.790429\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 27.7128 1.14972
\(582\) 0 0
\(583\) −19.5959 −0.811580
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.2548 1.86787 0.933933 0.357447i \(-0.116353\pi\)
0.933933 + 0.357447i \(0.116353\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 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
\(601\) 2.00000 0.0815817 0.0407909 0.999168i \(-0.487012\pi\)
0.0407909 + 0.999168i \(0.487012\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 72.7461 2.95755
\(606\) 0 0
\(607\) 44.0908 1.78959 0.894795 0.446476i \(-0.147321\pi\)
0.894795 + 0.446476i \(0.147321\pi\)
\(608\) 0 0
\(609\) 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
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −11.0000 −0.440000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −4.89898 −0.195025 −0.0975126 0.995234i \(-0.531089\pi\)
−0.0975126 + 0.995234i \(0.531089\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 16.9706 0.673456
\(636\) 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
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 64.0000 2.51222
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 17.3205 0.677804 0.338902 0.940822i \(-0.389945\pi\)
0.338902 + 0.940822i \(0.389945\pi\)
\(654\) 0 0
\(655\) 78.3837 3.06270
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −45.2548 −1.76288 −0.881439 0.472298i \(-0.843425\pi\)
−0.881439 + 0.472298i \(0.843425\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 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
\(673\) −34.0000 −1.31060 −0.655302 0.755367i \(-0.727459\pi\)
−0.655302 + 0.755367i \(0.727459\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −51.9615 −1.99704 −0.998522 0.0543526i \(-0.982690\pi\)
−0.998522 + 0.0543526i \(0.982690\pi\)
\(678\) 0 0
\(679\) −9.79796 −0.376011
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 5.65685 0.216454 0.108227 0.994126i \(-0.465483\pi\)
0.108227 + 0.994126i \(0.465483\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 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
\(701\) 38.1051 1.43921 0.719605 0.694383i \(-0.244323\pi\)
0.719605 + 0.694383i \(0.244323\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 16.9706 0.638244
\(708\) 0 0
\(709\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(710\) 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
\(721\) 72.0000 2.68142
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 72.7461 2.70172
\(726\) 0 0
\(727\) 53.8888 1.99862 0.999312 0.0370879i \(-0.0118082\pi\)
0.999312 + 0.0370879i \(0.0118082\pi\)
\(728\) 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
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −84.0000 −3.07752
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −55.4256 −2.02521
\(750\) 0 0
\(751\) 53.8888 1.96643 0.983215 0.182453i \(-0.0584036\pi\)
0.983215 + 0.182453i \(0.0584036\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −84.8528 −3.08811
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −26.0000 −0.937584 −0.468792 0.883309i \(-0.655311\pi\)
−0.468792 + 0.883309i \(0.655311\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 51.9615 1.86893 0.934463 0.356060i \(-0.115880\pi\)
0.934463 + 0.356060i \(0.115880\pi\)
\(774\) 0 0
\(775\) −34.2929 −1.23184
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 17.3205 0.613524 0.306762 0.951786i \(-0.400754\pi\)
0.306762 + 0.951786i \(0.400754\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 79.1960 2.79476
\(804\) 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
\(811\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −31.1769 −1.08808 −0.544041 0.839059i \(-0.683106\pi\)
−0.544041 + 0.839059i \(0.683106\pi\)
\(822\) 0 0
\(823\) 34.2929 1.19537 0.597687 0.801730i \(-0.296087\pi\)
0.597687 + 0.801730i \(0.296087\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −56.5685 −1.96708 −0.983540 0.180688i \(-0.942168\pi\)
−0.983540 + 0.180688i \(0.942168\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 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
\(841\) 79.0000 2.72414
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −45.0333 −1.54919
\(846\) 0 0
\(847\) −102.879 −3.53495
\(848\) 0 0
\(849\) 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
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 60.0000 2.04006
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 83.1384 2.82028
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −33.9411 −1.14742
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −39.1918 −1.31004
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −50.9117 −1.69800
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −32.0000 −1.05905
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −110.851 −3.66063
\(918\) 0 0
\(919\) 34.2929 1.13122 0.565608 0.824674i \(-0.308641\pi\)
0.565608 + 0.824674i \(0.308641\pi\)
\(920\) 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
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 58.0000 1.89478 0.947389 0.320085i \(-0.103712\pi\)
0.947389 + 0.320085i \(0.103712\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −38.1051 −1.24219 −0.621096 0.783735i \(-0.713312\pi\)
−0.621096 + 0.783735i \(0.713312\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 56.5685 1.83823 0.919115 0.393989i \(-0.128905\pi\)
0.919115 + 0.393989i \(0.128905\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 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
\(961\) −7.00000 −0.225806
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) −90.0666 −2.89935
\(966\) 0 0
\(967\) 4.89898 0.157541 0.0787703 0.996893i \(-0.474901\pi\)
0.0787703 + 0.996893i \(0.474901\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −62.2254 −1.99691 −0.998454 0.0555842i \(-0.982298\pi\)
−0.998454 + 0.0555842i \(0.982298\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 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
\(985\) 84.0000 2.67646
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) −24.4949 −0.778106 −0.389053 0.921215i \(-0.627198\pi\)
−0.389053 + 0.921215i \(0.627198\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 84.8528 2.69002
\(996\) 0 0
\(997\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 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 2304.2.a.z.1.3 4
3.2 odd 2 inner 2304.2.a.z.1.1 4
4.3 odd 2 inner 2304.2.a.z.1.4 4
8.3 odd 2 inner 2304.2.a.z.1.2 4
8.5 even 2 inner 2304.2.a.z.1.1 4
12.11 even 2 inner 2304.2.a.z.1.2 4
16.3 odd 4 1152.2.d.g.577.1 4
16.5 even 4 1152.2.d.g.577.4 yes 4
16.11 odd 4 1152.2.d.g.577.3 yes 4
16.13 even 4 1152.2.d.g.577.2 yes 4
24.5 odd 2 CM 2304.2.a.z.1.3 4
24.11 even 2 inner 2304.2.a.z.1.4 4
48.5 odd 4 1152.2.d.g.577.2 yes 4
48.11 even 4 1152.2.d.g.577.1 4
48.29 odd 4 1152.2.d.g.577.4 yes 4
48.35 even 4 1152.2.d.g.577.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1152.2.d.g.577.1 4 16.3 odd 4
1152.2.d.g.577.1 4 48.11 even 4
1152.2.d.g.577.2 yes 4 16.13 even 4
1152.2.d.g.577.2 yes 4 48.5 odd 4
1152.2.d.g.577.3 yes 4 16.11 odd 4
1152.2.d.g.577.3 yes 4 48.35 even 4
1152.2.d.g.577.4 yes 4 16.5 even 4
1152.2.d.g.577.4 yes 4 48.29 odd 4
2304.2.a.z.1.1 4 3.2 odd 2 inner
2304.2.a.z.1.1 4 8.5 even 2 inner
2304.2.a.z.1.2 4 8.3 odd 2 inner
2304.2.a.z.1.2 4 12.11 even 2 inner
2304.2.a.z.1.3 4 1.1 even 1 trivial
2304.2.a.z.1.3 4 24.5 odd 2 CM
2304.2.a.z.1.4 4 4.3 odd 2 inner
2304.2.a.z.1.4 4 24.11 even 2 inner