Properties

Label 891.2.d.a.890.1
Level $891$
Weight $2$
Character 891.890
Analytic conductor $7.115$
Analytic rank $0$
Dimension $4$
CM discriminant -11
Inner twists $4$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [891,2,Mod(890,891)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(891, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("891.890"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: \( N \) \(=\) \( 891 = 3^{4} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 891.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(7.11467082010\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-3}, \sqrt{-11})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 2x^{2} - 3x + 9 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 3^{2} \)
Twist minimal: no (minimal twist has level 99)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 890.1
Root \(-1.18614 - 1.26217i\) of defining polynomial
Character \(\chi\) \(=\) 891.890
Dual form 891.2.d.a.890.4

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} -4.25639i q^{5} -3.31662i q^{11} +4.00000 q^{16} +8.51278i q^{20} +3.31662i q^{23} -13.1168 q^{25} -11.1168 q^{31} -5.11684 q^{37} +6.63325i q^{44} -7.07568i q^{47} +7.00000 q^{49} -1.43710i q^{53} -14.1168 q^{55} +11.3321i q^{59} -8.00000 q^{64} -2.11684 q^{67} +5.69349i q^{71} -17.0256i q^{80} -16.5831i q^{89} -6.63325i q^{92} -17.1168 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 16 q^{16} - 18 q^{25} - 10 q^{31} + 14 q^{37} + 28 q^{49} - 22 q^{55} - 32 q^{64} + 26 q^{67} - 34 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/891\mathbb{Z}\right)^\times\).

\(n\) \(244\) \(650\)
\(\chi(n)\) \(-1\) \(-1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) − 4.25639i − 1.90351i −0.306851 0.951757i \(-0.599275\pi\)
0.306851 0.951757i \(-0.400725\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 3.31662i − 1.00000i
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 8.51278i 1.90351i
\(21\) 0 0
\(22\) 0 0
\(23\) 3.31662i 0.691564i 0.938315 + 0.345782i \(0.112386\pi\)
−0.938315 + 0.345782i \(0.887614\pi\)
\(24\) 0 0
\(25\) −13.1168 −2.62337
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) −11.1168 −1.99664 −0.998322 0.0579057i \(-0.981558\pi\)
−0.998322 + 0.0579057i \(0.981558\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −5.11684 −0.841204 −0.420602 0.907245i \(-0.638181\pi\)
−0.420602 + 0.907245i \(0.638181\pi\)
\(38\) 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 \(\pi\)
\(44\) 6.63325i 1.00000i
\(45\) 0 0
\(46\) 0 0
\(47\) − 7.07568i − 1.03209i −0.856560 0.516047i \(-0.827403\pi\)
0.856560 0.516047i \(-0.172597\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) − 1.43710i − 0.197400i −0.995117 0.0987002i \(-0.968532\pi\)
0.995117 0.0987002i \(-0.0314685\pi\)
\(54\) 0 0
\(55\) −14.1168 −1.90351
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 11.3321i 1.47531i 0.675178 + 0.737655i \(0.264067\pi\)
−0.675178 + 0.737655i \(0.735933\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −2.11684 −0.258614 −0.129307 0.991605i \(-0.541275\pi\)
−0.129307 + 0.991605i \(0.541275\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 5.69349i 0.675692i 0.941201 + 0.337846i \(0.109698\pi\)
−0.941201 + 0.337846i \(0.890302\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) − 17.0256i − 1.90351i
\(81\) 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
\(89\) − 16.5831i − 1.75781i −0.476999 0.878904i \(-0.658275\pi\)
0.476999 0.878904i \(-0.341725\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) − 6.63325i − 0.691564i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −17.1168 −1.73795 −0.868976 0.494854i \(-0.835222\pi\)
−0.868976 + 0.494854i \(0.835222\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 26.2337 2.62337
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 19.2337 1.89515 0.947576 0.319531i \(-0.103525\pi\)
0.947576 + 0.319531i \(0.103525\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 19.8448i − 1.86685i −0.358778 0.933423i \(-0.616806\pi\)
0.358778 0.933423i \(-0.383194\pi\)
\(114\) 0 0
\(115\) 14.1168 1.31640
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 22.2337 1.99664
\(125\) 34.5484i 3.09011i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 14.2063i − 1.21372i −0.794808 0.606861i \(-0.792428\pi\)
0.794808 0.606861i \(-0.207572\pi\)
\(138\) 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
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 10.2337 0.841204
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 47.3176i 3.80064i
\(156\) 0 0
\(157\) −20.1168 −1.60550 −0.802749 0.596316i \(-0.796630\pi\)
−0.802749 + 0.596316i \(0.796630\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 25.2337 1.97645 0.988227 0.152992i \(-0.0488907\pi\)
0.988227 + 0.152992i \(0.0488907\pi\)
\(164\) 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
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 13.2665i − 1.00000i
\(177\) 0 0
\(178\) 0 0
\(179\) − 9.89497i − 0.739585i −0.929114 0.369792i \(-0.879429\pi\)
0.929114 0.369792i \(-0.120571\pi\)
\(180\) 0 0
\(181\) 3.88316 0.288633 0.144316 0.989532i \(-0.453902\pi\)
0.144316 + 0.989532i \(0.453902\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 21.7793i 1.60124i
\(186\) 0 0
\(187\) 0 0
\(188\) 14.1514i 1.03209i
\(189\) 0 0
\(190\) 0 0
\(191\) 1.38219i 0.100012i 0.998749 + 0.0500060i \(0.0159241\pi\)
−0.998749 + 0.0500060i \(0.984076\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 7.23369 0.512783 0.256391 0.966573i \(-0.417466\pi\)
0.256391 + 0.966573i \(0.417466\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 2.87419i 0.197400i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 28.2337 1.90351
\(221\) 0 0
\(222\) 0 0
\(223\) −1.00000 −0.0669650 −0.0334825 0.999439i \(-0.510660\pi\)
−0.0334825 + 0.999439i \(0.510660\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 5.00000 0.330409 0.165205 0.986259i \(-0.447172\pi\)
0.165205 + 0.986259i \(0.447172\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) −30.1168 −1.96461
\(236\) − 22.6641i − 1.47531i
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) − 29.7947i − 1.90351i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 16.5831i − 1.04672i −0.852112 0.523359i \(-0.824679\pi\)
0.852112 0.523359i \(-0.175321\pi\)
\(252\) 0 0
\(253\) 11.0000 0.691564
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) − 26.5330i − 1.65508i −0.561405 0.827541i \(-0.689739\pi\)
0.561405 0.827541i \(-0.310261\pi\)
\(258\) 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
\(265\) −6.11684 −0.375755
\(266\) 0 0
\(267\) 0 0
\(268\) 4.23369 0.258614
\(269\) − 32.6140i − 1.98851i −0.107031 0.994256i \(-0.534134\pi\)
0.107031 0.994256i \(-0.465866\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 43.5036i 2.62337i
\(276\) 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 \(\pi\)
\(284\) − 11.3870i − 0.675692i
\(285\) 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
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 48.2337 2.80827
\(296\) 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
\(311\) − 26.9754i − 1.52964i −0.644246 0.764818i \(-0.722829\pi\)
0.644246 0.764818i \(-0.277171\pi\)
\(312\) 0 0
\(313\) −19.0000 −1.07394 −0.536972 0.843600i \(-0.680432\pi\)
−0.536972 + 0.843600i \(0.680432\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 23.2164i 1.30396i 0.758236 + 0.651981i \(0.226062\pi\)
−0.758236 + 0.651981i \(0.773938\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 34.0511i 1.90351i
\(321\) 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
\(331\) −26.1168 −1.43551 −0.717756 0.696295i \(-0.754831\pi\)
−0.717756 + 0.696295i \(0.754831\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 9.01011i 0.492275i
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 36.8704i 1.99664i
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) − 36.4829i − 1.94179i −0.239511 0.970894i \(-0.576987\pi\)
0.239511 0.970894i \(-0.423013\pi\)
\(354\) 0 0
\(355\) 24.2337 1.28619
\(356\) 33.1662i 1.75781i
\(357\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) 9.88316 0.515897 0.257948 0.966159i \(-0.416954\pi\)
0.257948 + 0.966159i \(0.416954\pi\)
\(368\) 13.2665i 0.691564i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 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
\(379\) −25.0000 −1.28416 −0.642082 0.766636i \(-0.721929\pi\)
−0.642082 + 0.766636i \(0.721929\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 35.4333i − 1.81056i −0.424818 0.905279i \(-0.639662\pi\)
0.424818 0.905279i \(-0.360338\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 34.2337 1.73795
\(389\) 31.2318i 1.58352i 0.610835 + 0.791758i \(0.290834\pi\)
−0.610835 + 0.791758i \(0.709166\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −35.4674 −1.78006 −0.890028 0.455905i \(-0.849316\pi\)
−0.890028 + 0.455905i \(0.849316\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −52.4674 −2.62337
\(401\) − 12.7143i − 0.634920i −0.948272 0.317460i \(-0.897170\pi\)
0.948272 0.317460i \(-0.102830\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 16.9707i 0.841204i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −38.4674 −1.89515
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 4.20149i 0.205256i 0.994720 + 0.102628i \(0.0327251\pi\)
−0.994720 + 0.102628i \(0.967275\pi\)
\(420\) 0 0
\(421\) −29.4674 −1.43615 −0.718076 0.695965i \(-0.754977\pi\)
−0.718076 + 0.695965i \(0.754977\pi\)
\(422\) 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
\(433\) 29.0000 1.39365 0.696826 0.717241i \(-0.254595\pi\)
0.696826 + 0.717241i \(0.254595\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0.0549029i 0.00260851i 0.999999 + 0.00130426i \(0.000415158\pi\)
−0.999999 + 0.00130426i \(0.999585\pi\)
\(444\) 0 0
\(445\) −70.5842 −3.34601
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 25.4834i − 1.20264i −0.799009 0.601319i \(-0.794642\pi\)
0.799009 0.601319i \(-0.205358\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 39.6897i 1.86685i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −28.2337 −1.31640
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −31.0000 −1.44069 −0.720346 0.693615i \(-0.756017\pi\)
−0.720346 + 0.693615i \(0.756017\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) − 24.1561i − 1.11781i −0.829231 0.558906i \(-0.811221\pi\)
0.829231 0.558906i \(-0.188779\pi\)
\(468\) 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
\(484\) 22.0000 1.00000
\(485\) 72.8559i 3.30822i
\(486\) 0 0
\(487\) 12.8832 0.583792 0.291896 0.956450i \(-0.405714\pi\)
0.291896 + 0.956450i \(0.405714\pi\)
\(488\) 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
\(496\) −44.4674 −1.99664
\(497\) 0 0
\(498\) 0 0
\(499\) 37.2337 1.66681 0.833404 0.552664i \(-0.186389\pi\)
0.833404 + 0.552664i \(0.186389\pi\)
\(500\) − 69.0969i − 3.09011i
\(501\) 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
\(508\) 0 0
\(509\) 3.31662i 0.147007i 0.997295 + 0.0735034i \(0.0234180\pi\)
−0.997295 + 0.0735034i \(0.976582\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) − 81.8661i − 3.60745i
\(516\) 0 0
\(517\) −23.4674 −1.03209
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) − 8.56768i − 0.375357i −0.982231 0.187678i \(-0.939904\pi\)
0.982231 0.187678i \(-0.0600963\pi\)
\(522\) 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
\(529\) 12.0000 0.521739
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 23.2164i − 1.00000i
\(540\) 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 \(\pi\)
\(548\) 28.4125i 1.21372i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) −84.4674 −3.55357
\(566\) 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
\(575\) − 43.5036i − 1.81423i
\(576\) 0 0
\(577\) −32.1168 −1.33704 −0.668521 0.743693i \(-0.733072\pi\)
−0.668521 + 0.743693i \(0.733072\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −4.76631 −0.197400
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 38.2526i − 1.57885i −0.613845 0.789427i \(-0.710378\pi\)
0.613845 0.789427i \(-0.289622\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −20.4674 −0.841204
\(593\) 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
\(599\) 33.1662i 1.35514i 0.735460 + 0.677568i \(0.236966\pi\)
−0.735460 + 0.677568i \(0.763034\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 46.8203i 1.90351i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 49.6396i 1.99841i 0.0398207 + 0.999207i \(0.487321\pi\)
−0.0398207 + 0.999207i \(0.512679\pi\)
\(618\) 0 0
\(619\) 43.5842 1.75180 0.875899 0.482495i \(-0.160269\pi\)
0.875899 + 0.482495i \(0.160269\pi\)
\(620\) − 94.6352i − 3.80064i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 81.4674 3.25870
\(626\) 0 0
\(627\) 0 0
\(628\) 40.2337 1.60550
\(629\) 0 0
\(630\) 0 0
\(631\) 46.5842 1.85449 0.927244 0.374457i \(-0.122171\pi\)
0.927244 + 0.374457i \(0.122171\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 23.2164i 0.916992i 0.888697 + 0.458496i \(0.151612\pi\)
−0.888697 + 0.458496i \(0.848388\pi\)
\(642\) 0 0
\(643\) 41.0000 1.61688 0.808441 0.588577i \(-0.200312\pi\)
0.808441 + 0.588577i \(0.200312\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 43.1161i 1.69507i 0.530740 + 0.847535i \(0.321914\pi\)
−0.530740 + 0.847535i \(0.678086\pi\)
\(648\) 0 0
\(649\) 37.5842 1.47531
\(650\) 0 0
\(651\) 0 0
\(652\) −50.4674 −1.97645
\(653\) 42.5090i 1.66351i 0.555147 + 0.831753i \(0.312662\pi\)
−0.555147 + 0.831753i \(0.687338\pi\)
\(654\) 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
\(661\) 49.5842 1.92860 0.964301 0.264807i \(-0.0853084\pi\)
0.964301 + 0.264807i \(0.0853084\pi\)
\(662\) 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
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 44.0010i 1.68365i 0.539750 + 0.841825i \(0.318519\pi\)
−0.539750 + 0.841825i \(0.681481\pi\)
\(684\) 0 0
\(685\) −60.4674 −2.31034
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 34.5842 1.31565 0.657823 0.753173i \(-0.271478\pi\)
0.657823 + 0.753173i \(0.271478\pi\)
\(692\) 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
\(703\) 0 0
\(704\) 26.5330i 1.00000i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 52.5842 1.97484 0.987421 0.158114i \(-0.0505412\pi\)
0.987421 + 0.158114i \(0.0505412\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) − 36.8704i − 1.38081i
\(714\) 0 0
\(715\) 0 0
\(716\) 19.7899i 0.739585i
\(717\) 0 0
\(718\) 0 0
\(719\) 52.4589i 1.95639i 0.207700 + 0.978193i \(0.433402\pi\)
−0.207700 + 0.978193i \(0.566598\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −7.76631 −0.288633
\(725\) 0 0
\(726\) 0 0
\(727\) −35.1168 −1.30241 −0.651206 0.758901i \(-0.725737\pi\)
−0.651206 + 0.758901i \(0.725737\pi\)
\(728\) 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
\(737\) 7.02078i 0.258614i
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) − 43.5586i − 1.60124i
\(741\) 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
\(751\) 31.5842 1.15252 0.576262 0.817265i \(-0.304511\pi\)
0.576262 + 0.817265i \(0.304511\pi\)
\(752\) − 28.3027i − 1.03209i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −53.4674 −1.94330 −0.971652 0.236414i \(-0.924028\pi\)
−0.971652 + 0.236414i \(0.924028\pi\)
\(758\) 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
\(764\) − 2.76439i − 0.100012i
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 13.2665i 0.477163i 0.971123 + 0.238581i \(0.0766824\pi\)
−0.971123 + 0.238581i \(0.923318\pi\)
\(774\) 0 0
\(775\) 145.818 5.23793
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 18.8832 0.675692
\(782\) 0 0
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) 85.6251i 3.05609i
\(786\) 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
\(796\) −14.4674 −0.512783
\(797\) 25.5932i 0.906559i 0.891368 + 0.453279i \(0.149746\pi\)
−0.891368 + 0.453279i \(0.850254\pi\)
\(798\) 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
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) − 107.404i − 3.76221i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) −49.0000 −1.70803 −0.854016 0.520246i \(-0.825840\pi\)
−0.854016 + 0.520246i \(0.825840\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 28.5842 0.992771 0.496385 0.868102i \(-0.334660\pi\)
0.496385 + 0.868102i \(0.334660\pi\)
\(830\) 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
\(839\) − 36.4829i − 1.25953i −0.776786 0.629764i \(-0.783151\pi\)
0.776786 0.629764i \(-0.216849\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) − 55.3331i − 1.90351i
\(846\) 0 0
\(847\) 0 0
\(848\) − 5.74839i − 0.197400i
\(849\) 0 0
\(850\) 0 0
\(851\) − 16.9707i − 0.581746i
\(852\) 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
\(859\) 58.5842 1.99887 0.999434 0.0336436i \(-0.0107111\pi\)
0.999434 + 0.0336436i \(0.0107111\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) − 46.4327i − 1.58059i −0.612727 0.790295i \(-0.709928\pi\)
0.612727 0.790295i \(-0.290072\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) −56.4674 −1.90351
\(881\) − 41.0719i − 1.38375i −0.722019 0.691873i \(-0.756786\pi\)
0.722019 0.691873i \(-0.243214\pi\)
\(882\) 0 0
\(883\) −10.7663 −0.362315 −0.181158 0.983454i \(-0.557984\pi\)
−0.181158 + 0.983454i \(0.557984\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 2.00000 0.0669650
\(893\) 0 0
\(894\) 0 0
\(895\) −42.1168 −1.40781
\(896\) 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
\(905\) − 16.5282i − 0.549417i
\(906\) 0 0
\(907\) 8.00000 0.265636 0.132818 0.991140i \(-0.457597\pi\)
0.132818 + 0.991140i \(0.457597\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 55.2781i 1.83145i 0.401809 + 0.915723i \(0.368381\pi\)
−0.401809 + 0.915723i \(0.631619\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −10.0000 −0.330409
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 67.1168 2.20679
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 52.5138i − 1.72292i −0.507825 0.861460i \(-0.669550\pi\)
0.507825 0.861460i \(-0.330450\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 60.2337 1.96461
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 45.3283i 1.47531i
\(945\) 0 0
\(946\) 0 0
\(947\) − 60.9716i − 1.98131i −0.136385 0.990656i \(-0.543548\pi\)
0.136385 0.990656i \(-0.456452\pi\)
\(948\) 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
\(955\) 5.88316 0.190374
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 92.5842 2.98659
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 43.1161i 1.38366i 0.722059 + 0.691831i \(0.243196\pi\)
−0.722059 + 0.691831i \(0.756804\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 56.3826i − 1.80384i −0.431903 0.901920i \(-0.642158\pi\)
0.431903 0.901920i \(-0.357842\pi\)
\(978\) 0 0
\(979\) −55.0000 −1.75781
\(980\) 59.5894i 1.90351i
\(981\) 0 0
\(982\) 0 0
\(983\) 62.4087i 1.99053i 0.0972017 + 0.995265i \(0.469011\pi\)
−0.0972017 + 0.995265i \(0.530989\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 20.0000 0.635321 0.317660 0.948205i \(-0.397103\pi\)
0.317660 + 0.948205i \(0.397103\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 30.7894i − 0.976089i
\(996\) 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 891.2.d.a.890.1 4
3.2 odd 2 inner 891.2.d.a.890.4 4
9.2 odd 6 99.2.g.a.32.1 4
9.4 even 3 99.2.g.a.65.1 yes 4
9.5 odd 6 297.2.g.a.197.2 4
9.7 even 3 297.2.g.a.98.2 4
11.10 odd 2 CM 891.2.d.a.890.1 4
33.32 even 2 inner 891.2.d.a.890.4 4
99.32 even 6 297.2.g.a.197.2 4
99.43 odd 6 297.2.g.a.98.2 4
99.65 even 6 99.2.g.a.32.1 4
99.76 odd 6 99.2.g.a.65.1 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
99.2.g.a.32.1 4 9.2 odd 6
99.2.g.a.32.1 4 99.65 even 6
99.2.g.a.65.1 yes 4 9.4 even 3
99.2.g.a.65.1 yes 4 99.76 odd 6
297.2.g.a.98.2 4 9.7 even 3
297.2.g.a.98.2 4 99.43 odd 6
297.2.g.a.197.2 4 9.5 odd 6
297.2.g.a.197.2 4 99.32 even 6
891.2.d.a.890.1 4 1.1 even 1 trivial
891.2.d.a.890.1 4 11.10 odd 2 CM
891.2.d.a.890.4 4 3.2 odd 2 inner
891.2.d.a.890.4 4 33.32 even 2 inner