Properties

Label 4608.2.d.q.2305.7
Level $4608$
Weight $2$
Character 4608.2305
Analytic conductor $36.795$
Analytic rank $0$
Dimension $8$
CM discriminant -24
Inner twists $8$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4608,2,Mod(2305,4608)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4608, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4608.2305");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4608 = 2^{9} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4608.d (of order \(2\), degree \(1\), not 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: no
Analytic conductor: \(36.7950652514\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{18} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 2305.7
Root \(-0.965926 - 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 4608.2305
Dual form 4608.2.d.q.2305.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+4.44949i q^{5} -4.87832 q^{7} +O(q^{10})\) \(q+4.44949i q^{5} -4.87832 q^{7} +3.46410i q^{11} -14.7980 q^{25} +5.34847i q^{29} -10.5352 q^{31} -21.7060i q^{35} +16.7980 q^{49} +12.4495i q^{53} -15.4135 q^{55} -11.3137i q^{59} +9.79796 q^{73} -16.8990i q^{77} -3.32124 q^{79} +17.3205i q^{83} -2.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 40 q^{25} + 56 q^{49} - 16 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4608\mathbb{Z}\right)^\times\).

\(n\) \(2053\) \(3583\) \(4097\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 0 0
\(4\) 0 0
\(5\) 4.44949i 1.98987i 0.100509 + 0.994936i \(0.467953\pi\)
−0.100509 + 0.994936i \(0.532047\pi\)
\(6\) 0 0
\(7\) −4.87832 −1.84383 −0.921915 0.387392i \(-0.873376\pi\)
−0.921915 + 0.387392i \(0.873376\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 3.46410i 1.04447i 0.852803 + 0.522233i \(0.174901\pi\)
−0.852803 + 0.522233i \(0.825099\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −14.7980 −2.95959
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.34847i 0.993186i 0.867984 + 0.496593i \(0.165416\pi\)
−0.867984 + 0.496593i \(0.834584\pi\)
\(30\) 0 0
\(31\) −10.5352 −1.89217 −0.946086 0.323915i \(-0.895001\pi\)
−0.946086 + 0.323915i \(0.895001\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 21.7060i − 3.66899i
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 16.7980 2.39971
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 12.4495i 1.71007i 0.518571 + 0.855034i \(0.326464\pi\)
−0.518571 + 0.855034i \(0.673536\pi\)
\(54\) 0 0
\(55\) −15.4135 −2.07835
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 11.3137i − 1.47292i −0.676481 0.736460i \(-0.736496\pi\)
0.676481 0.736460i \(-0.263504\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 9.79796 1.14676 0.573382 0.819288i \(-0.305631\pi\)
0.573382 + 0.819288i \(0.305631\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 16.8990i − 1.92582i
\(78\) 0 0
\(79\) −3.32124 −0.373668 −0.186834 0.982391i \(-0.559823\pi\)
−0.186834 + 0.982391i \(0.559823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 17.3205i 1.90117i 0.310460 + 0.950586i \(0.399517\pi\)
−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 \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 11.5505i 1.14932i 0.818393 + 0.574659i \(0.194865\pi\)
−0.818393 + 0.574659i \(0.805135\pi\)
\(102\) 0 0
\(103\) 20.2918 1.99941 0.999705 0.0242790i \(-0.00772901\pi\)
0.999705 + 0.0242790i \(0.00772901\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 11.3137i − 1.09374i −0.837218 0.546869i \(-0.815820\pi\)
0.837218 0.546869i \(-0.184180\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 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
\(121\) −1.00000 −0.0909091
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 43.5959i − 3.89934i
\(126\) 0 0
\(127\) −12.0922 −1.07301 −0.536507 0.843896i \(-0.680256\pi\)
−0.536507 + 0.843896i \(0.680256\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 22.6274i − 1.97697i −0.151330 0.988483i \(-0.548356\pi\)
0.151330 0.988483i \(-0.451644\pi\)
\(132\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −23.7980 −1.97631
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) − 15.1464i − 1.24084i −0.784268 0.620422i \(-0.786961\pi\)
0.784268 0.620422i \(-0.213039\pi\)
\(150\) 0 0
\(151\) 18.7347 1.52461 0.762305 0.647218i \(-0.224068\pi\)
0.762305 + 0.647218i \(0.224068\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) − 46.8761i − 3.76518i
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 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
\(173\) − 1.75255i − 0.133244i −0.997778 0.0666220i \(-0.978778\pi\)
0.997778 0.0666220i \(-0.0212222\pi\)
\(174\) 0 0
\(175\) 72.1891 5.45698
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) − 11.3137i − 0.845626i −0.906217 0.422813i \(-0.861043\pi\)
0.906217 0.422813i \(-0.138957\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 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
\(193\) −9.79796 −0.705273 −0.352636 0.935760i \(-0.614715\pi\)
−0.352636 + 0.935760i \(0.614715\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) − 7.14643i − 0.509162i −0.967051 0.254581i \(-0.918062\pi\)
0.967051 0.254581i \(-0.0819375\pi\)
\(198\) 0 0
\(199\) −7.42101 −0.526062 −0.263031 0.964787i \(-0.584722\pi\)
−0.263031 + 0.964787i \(0.584722\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 26.0915i − 1.83127i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 51.3939 3.48884
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −7.99247 −0.535215 −0.267608 0.963528i \(-0.586233\pi\)
−0.267608 + 0.963528i \(0.586233\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 10.3923i 0.689761i 0.938647 + 0.344881i \(0.112081\pi\)
−0.938647 + 0.344881i \(0.887919\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
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
\(241\) −29.3939 −1.89343 −0.946713 0.322078i \(-0.895619\pi\)
−0.946713 + 0.322078i \(0.895619\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 74.7423i 4.77511i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 31.1769i 1.96787i 0.178529 + 0.983935i \(0.442866\pi\)
−0.178529 + 0.983935i \(0.557134\pi\)
\(252\) 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
\(265\) −55.3939 −3.40282
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) − 29.3485i − 1.78941i −0.446660 0.894704i \(-0.647387\pi\)
0.446660 0.894704i \(-0.352613\pi\)
\(270\) 0 0
\(271\) −1.76416 −0.107165 −0.0535825 0.998563i \(-0.517064\pi\)
−0.0535825 + 0.998563i \(0.517064\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) − 51.2616i − 3.09119i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 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
\(293\) 32.0454i 1.87211i 0.351850 + 0.936056i \(0.385553\pi\)
−0.351850 + 0.936056i \(0.614447\pi\)
\(294\) 0 0
\(295\) 50.3402 2.93092
\(296\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 9.79796 0.553813 0.276907 0.960897i \(-0.410691\pi\)
0.276907 + 0.960897i \(0.410691\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 9.75255i − 0.547758i −0.961764 0.273879i \(-0.911693\pi\)
0.961764 0.273879i \(-0.0883068\pi\)
\(318\) 0 0
\(319\) −18.5276 −1.03735
\(320\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 29.3939 1.60119 0.800593 0.599208i \(-0.204518\pi\)
0.800593 + 0.599208i \(0.204518\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) − 36.4949i − 1.97631i
\(342\) 0 0
\(343\) −47.7975 −2.58082
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 24.2487i 1.30174i 0.759190 + 0.650870i \(0.225596\pi\)
−0.759190 + 0.650870i \(0.774404\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 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
\(365\) 43.5959i 2.28191i
\(366\) 0 0
\(367\) 23.4060 1.22178 0.610890 0.791715i \(-0.290812\pi\)
0.610890 + 0.791715i \(0.290812\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 60.7325i − 3.15308i
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 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
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 75.1918 3.83213
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 39.1464i 1.98480i 0.123043 + 0.992401i \(0.460735\pi\)
−0.123043 + 0.992401i \(0.539265\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) − 14.7778i − 0.743552i
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 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 \(-0.579522\pi\)
−0.247234 + 0.968956i \(0.579522\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 55.1918i 2.71581i
\(414\) 0 0
\(415\) −77.0674 −3.78309
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 10.3923i 0.507697i 0.967244 + 0.253849i \(0.0816965\pi\)
−0.967244 + 0.253849i \(0.918303\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 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
\(439\) −41.3621 −1.97411 −0.987054 0.160391i \(-0.948725\pi\)
−0.987054 + 0.160391i \(0.948725\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) − 31.1769i − 1.48126i −0.671913 0.740630i \(-0.734527\pi\)
0.671913 0.740630i \(-0.265473\pi\)
\(444\) 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
\(461\) − 40.9444i − 1.90697i −0.301440 0.953485i \(-0.597467\pi\)
0.301440 0.953485i \(-0.402533\pi\)
\(462\) 0 0
\(463\) −5.86393 −0.272520 −0.136260 0.990673i \(-0.543508\pi\)
−0.136260 + 0.990673i \(0.543508\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 17.3205i − 0.801498i −0.916188 0.400749i \(-0.868750\pi\)
0.916188 0.400749i \(-0.131250\pi\)
\(468\) 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
\(485\) − 8.89898i − 0.404082i
\(486\) 0 0
\(487\) −32.5911 −1.47684 −0.738422 0.674338i \(-0.764429\pi\)
−0.738422 + 0.674338i \(0.764429\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 22.6274i − 1.02116i −0.859830 0.510581i \(-0.829431\pi\)
0.859830 0.510581i \(-0.170569\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −51.3939 −2.28700
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) − 33.8434i − 1.50008i −0.661392 0.750040i \(-0.730034\pi\)
0.661392 0.750040i \(-0.269966\pi\)
\(510\) 0 0
\(511\) −47.7975 −2.11444
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 90.2882i 3.97857i
\(516\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 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
\(535\) 50.3402 2.17640
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 58.1898i 2.50641i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 16.2020 0.688981
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 41.8434i − 1.77296i −0.462767 0.886480i \(-0.653143\pi\)
0.462767 0.886480i \(-0.346857\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 38.1051i 1.60594i 0.596020 + 0.802970i \(0.296748\pi\)
−0.596020 + 0.802970i \(0.703252\pi\)
\(564\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −29.3939 −1.22368 −0.611842 0.790980i \(-0.709571\pi\)
−0.611842 + 0.790980i \(0.709571\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) − 84.4949i − 3.50544i
\(582\) 0 0
\(583\) −43.1263 −1.78611
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 45.2548i 1.86787i 0.357447 + 0.933933i \(0.383647\pi\)
−0.357447 + 0.933933i \(0.616353\pi\)
\(588\) 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
\(601\) −48.9898 −1.99834 −0.999168 0.0407909i \(-0.987012\pi\)
−0.999168 + 0.0407909i \(0.987012\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 4.44949i − 0.180897i
\(606\) 0 0
\(607\) 15.6206 0.634019 0.317010 0.948422i \(-0.397321\pi\)
0.317010 + 0.948422i \(0.397321\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 119.990 4.79959
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −38.8194 −1.54538 −0.772689 0.634785i \(-0.781089\pi\)
−0.772689 + 0.634785i \(0.781089\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) − 53.8043i − 2.13516i
\(636\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 39.1918 1.53841
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 46.2474i 1.80980i 0.425622 + 0.904901i \(0.360055\pi\)
−0.425622 + 0.904901i \(0.639945\pi\)
\(654\) 0 0
\(655\) 100.680 3.93391
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 45.2548i 1.76288i 0.472298 + 0.881439i \(0.343425\pi\)
−0.472298 + 0.881439i \(0.656575\pi\)
\(660\) 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
\(667\) 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
\(677\) − 34.7423i − 1.33526i −0.744495 0.667628i \(-0.767310\pi\)
0.744495 0.667628i \(-0.232690\pi\)
\(678\) 0 0
\(679\) 9.75663 0.374425
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) − 51.9615i − 1.98825i −0.108227 0.994126i \(-0.534517\pi\)
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 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 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
\(701\) − 0.944387i − 0.0356690i −0.999841 0.0178345i \(-0.994323\pi\)
0.999841 0.0178345i \(-0.00567720\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 56.3470i − 2.11915i
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 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
\(721\) −98.9898 −3.68657
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 79.1464i − 2.93942i
\(726\) 0 0
\(727\) −36.6909 −1.36079 −0.680395 0.732845i \(-0.738192\pi\)
−0.680395 + 0.732845i \(0.738192\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 67.3939 2.46912
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 55.1918i 2.01667i
\(750\) 0 0
\(751\) −31.0340 −1.13245 −0.566224 0.824251i \(-0.691596\pi\)
−0.566224 + 0.824251i \(0.691596\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 83.3600i 3.03378i
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 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
\(769\) −48.9898 −1.76662 −0.883309 0.468792i \(-0.844689\pi\)
−0.883309 + 0.468792i \(0.844689\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 50.7423i 1.82508i 0.408993 + 0.912538i \(0.365880\pi\)
−0.408993 + 0.912538i \(0.634120\pi\)
\(774\) 0 0
\(775\) 155.899 5.60006
\(776\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 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
\(797\) 25.7526i 0.912202i 0.889928 + 0.456101i \(0.150754\pi\)
−0.889928 + 0.456101i \(0.849246\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 33.9411i 1.19776i
\(804\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 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
\(821\) − 56.0454i − 1.95600i −0.208609 0.977999i \(-0.566894\pi\)
0.208609 0.977999i \(-0.433106\pi\)
\(822\) 0 0
\(823\) 56.7756 1.97907 0.989537 0.144280i \(-0.0460866\pi\)
0.989537 + 0.144280i \(0.0460866\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 56.5685i 1.96708i 0.180688 + 0.983540i \(0.442168\pi\)
−0.180688 + 0.983540i \(0.557832\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 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
\(841\) 0.393877 0.0135820
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 57.8434i 1.98987i
\(846\) 0 0
\(847\) 4.87832 0.167621
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 7.79796 0.265139
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) − 11.5051i − 0.390284i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 212.675i 7.18971i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 58.9898 1.97845
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 50.3402 1.68269
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) − 56.3470i − 1.87928i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) −60.0000 −1.98571
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 110.384i 3.64519i
\(918\) 0 0
\(919\) −11.1066 −0.366374 −0.183187 0.983078i \(-0.558641\pi\)
−0.183187 + 0.983078i \(0.558641\pi\)
\(920\) 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
\(941\) 7.05561i 0.230006i 0.993365 + 0.115003i \(0.0366878\pi\)
−0.993365 + 0.115003i \(0.963312\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 56.5685i 1.83823i 0.393989 + 0.919115i \(0.371095\pi\)
−0.393989 + 0.919115i \(0.628905\pi\)
\(948\) 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
\(961\) 79.9898 2.58032
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) − 43.5959i − 1.40340i
\(966\) 0 0
\(967\) −40.3765 −1.29842 −0.649211 0.760609i \(-0.724901\pi\)
−0.649211 + 0.760609i \(0.724901\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 3.46410i 0.111168i 0.998454 + 0.0555842i \(0.0177021\pi\)
−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
\(985\) 31.7980 1.01317
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 58.3327 1.85300 0.926500 0.376296i \(-0.122802\pi\)
0.926500 + 0.376296i \(0.122802\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 33.0197i − 1.04680i
\(996\) 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 4608.2.d.q.2305.7 8
3.2 odd 2 inner 4608.2.d.q.2305.1 8
4.3 odd 2 inner 4608.2.d.q.2305.8 8
8.3 odd 2 inner 4608.2.d.q.2305.2 8
8.5 even 2 inner 4608.2.d.q.2305.1 8
12.11 even 2 inner 4608.2.d.q.2305.2 8
16.3 odd 4 4608.2.a.bc.1.3 yes 4
16.5 even 4 4608.2.a.s.1.2 yes 4
16.11 odd 4 4608.2.a.s.1.1 4
16.13 even 4 4608.2.a.bc.1.4 yes 4
24.5 odd 2 CM 4608.2.d.q.2305.7 8
24.11 even 2 inner 4608.2.d.q.2305.8 8
48.5 odd 4 4608.2.a.bc.1.4 yes 4
48.11 even 4 4608.2.a.bc.1.3 yes 4
48.29 odd 4 4608.2.a.s.1.2 yes 4
48.35 even 4 4608.2.a.s.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4608.2.a.s.1.1 4 16.11 odd 4
4608.2.a.s.1.1 4 48.35 even 4
4608.2.a.s.1.2 yes 4 16.5 even 4
4608.2.a.s.1.2 yes 4 48.29 odd 4
4608.2.a.bc.1.3 yes 4 16.3 odd 4
4608.2.a.bc.1.3 yes 4 48.11 even 4
4608.2.a.bc.1.4 yes 4 16.13 even 4
4608.2.a.bc.1.4 yes 4 48.5 odd 4
4608.2.d.q.2305.1 8 3.2 odd 2 inner
4608.2.d.q.2305.1 8 8.5 even 2 inner
4608.2.d.q.2305.2 8 8.3 odd 2 inner
4608.2.d.q.2305.2 8 12.11 even 2 inner
4608.2.d.q.2305.7 8 1.1 even 1 trivial
4608.2.d.q.2305.7 8 24.5 odd 2 CM
4608.2.d.q.2305.8 8 4.3 odd 2 inner
4608.2.d.q.2305.8 8 24.11 even 2 inner