Properties

Label 1840.2.m.b.1839.2
Level $1840$
Weight $2$
Character 1840.1839
Analytic conductor $14.692$
Analytic rank $0$
Dimension $4$
CM discriminant -115
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1840,2,Mod(1839,1840)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1840, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1840.1839");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1840 = 2^{4} \cdot 5 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1840.m (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(14.6924739719\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{5}, \sqrt{-23})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 9x^{2} + 49 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{4} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1839.2
Root \(1.11803 - 2.39792i\) of defining polynomial
Character \(\chi\) \(=\) 1840.1839
Dual form 1840.2.m.b.1839.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.23607 q^{5} +4.79583i q^{7} -3.00000 q^{9} +O(q^{10})\) \(q-2.23607 q^{5} +4.79583i q^{7} -3.00000 q^{9} -6.70820 q^{17} -4.79583i q^{23} +5.00000 q^{25} -1.00000 q^{29} -10.7238i q^{31} -10.7238i q^{35} +11.1803 q^{37} +7.00000 q^{41} +9.59166i q^{43} +6.70820 q^{45} -16.0000 q^{49} +2.23607 q^{53} -10.7238i q^{59} -14.3875i q^{63} +4.79583i q^{67} -10.7238i q^{71} +9.00000 q^{81} -14.3875i q^{83} +15.0000 q^{85} +4.47214 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 12 q^{9} + 20 q^{25} - 4 q^{29} + 28 q^{41} - 64 q^{49} + 36 q^{81} + 60 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1840\mathbb{Z}\right)^\times\).

\(n\) \(737\) \(1151\) \(1201\) \(1381\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 0 0
\(5\) −2.23607 −1.00000
\(6\) 0 0
\(7\) 4.79583i 1.81265i 0.422577 + 0.906327i \(0.361126\pi\)
−0.422577 + 0.906327i \(0.638874\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) −6.70820 −1.62698 −0.813489 0.581580i \(-0.802435\pi\)
−0.813489 + 0.581580i \(0.802435\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) − 4.79583i − 1.00000i
\(24\) 0 0
\(25\) 5.00000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −1.00000 −0.185695 −0.0928477 0.995680i \(-0.529597\pi\)
−0.0928477 + 0.995680i \(0.529597\pi\)
\(30\) 0 0
\(31\) − 10.7238i − 1.92605i −0.269408 0.963026i \(-0.586828\pi\)
0.269408 0.963026i \(-0.413172\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) − 10.7238i − 1.81265i
\(36\) 0 0
\(37\) 11.1803 1.83804 0.919018 0.394215i \(-0.128983\pi\)
0.919018 + 0.394215i \(0.128983\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 7.00000 1.09322 0.546608 0.837389i \(-0.315919\pi\)
0.546608 + 0.837389i \(0.315919\pi\)
\(42\) 0 0
\(43\) 9.59166i 1.46271i 0.681994 + 0.731357i \(0.261113\pi\)
−0.681994 + 0.731357i \(0.738887\pi\)
\(44\) 0 0
\(45\) 6.70820 1.00000
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −16.0000 −2.28571
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 2.23607 0.307148 0.153574 0.988137i \(-0.450922\pi\)
0.153574 + 0.988137i \(0.450922\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) − 10.7238i − 1.39612i −0.716039 0.698060i \(-0.754047\pi\)
0.716039 0.698060i \(-0.245953\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) − 14.3875i − 1.81265i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.79583i 0.585904i 0.956127 + 0.292952i \(0.0946376\pi\)
−0.956127 + 0.292952i \(0.905362\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) − 10.7238i − 1.27268i −0.771408 0.636341i \(-0.780447\pi\)
0.771408 0.636341i \(-0.219553\pi\)
\(72\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) 0 0
\(83\) − 14.3875i − 1.57923i −0.613601 0.789616i \(-0.710280\pi\)
0.613601 0.789616i \(-0.289720\pi\)
\(84\) 0 0
\(85\) 15.0000 1.62698
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 4.47214 0.454077 0.227038 0.973886i \(-0.427096\pi\)
0.227038 + 0.973886i \(0.427096\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) −17.0000 −1.69156 −0.845782 0.533529i \(-0.820865\pi\)
−0.845782 + 0.533529i \(0.820865\pi\)
\(102\) 0 0
\(103\) − 9.59166i − 0.945095i −0.881305 0.472547i \(-0.843335\pi\)
0.881305 0.472547i \(-0.156665\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 4.79583i 0.463631i 0.972760 + 0.231815i \(0.0744665\pi\)
−0.972760 + 0.231815i \(0.925534\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −15.6525 −1.47246 −0.736231 0.676731i \(-0.763396\pi\)
−0.736231 + 0.676731i \(0.763396\pi\)
\(114\) 0 0
\(115\) 10.7238i 1.00000i
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) − 32.1714i − 2.94915i
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −11.1803 −1.00000
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 21.4476i 1.87389i 0.349482 + 0.936943i \(0.386358\pi\)
−0.349482 + 0.936943i \(0.613642\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) −13.4164 −1.14624 −0.573121 0.819471i \(-0.694267\pi\)
−0.573121 + 0.819471i \(0.694267\pi\)
\(138\) 0 0
\(139\) − 10.7238i − 0.909581i −0.890598 0.454791i \(-0.849714\pi\)
0.890598 0.454791i \(-0.150286\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 2.23607 0.185695
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) − 21.4476i − 1.74538i −0.488273 0.872691i \(-0.662373\pi\)
0.488273 0.872691i \(-0.337627\pi\)
\(152\) 0 0
\(153\) 20.1246 1.62698
\(154\) 0 0
\(155\) 23.9792i 1.92605i
\(156\) 0 0
\(157\) −24.5967 −1.96303 −0.981517 0.191375i \(-0.938705\pi\)
−0.981517 + 0.191375i \(0.938705\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 23.0000 1.81265
\(162\) 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
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 23.9792i 1.81265i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 21.4476i 1.60307i 0.597948 + 0.801535i \(0.295983\pi\)
−0.597948 + 0.801535i \(0.704017\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −25.0000 −1.83804
\(186\) 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
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 4.79583i − 0.336601i
\(204\) 0 0
\(205\) −15.6525 −1.09322
\(206\) 0 0
\(207\) 14.3875i 1.00000i
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) − 10.7238i − 0.738257i −0.929378 0.369129i \(-0.879656\pi\)
0.929378 0.369129i \(-0.120344\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) − 21.4476i − 1.46271i
\(216\) 0 0
\(217\) 51.4296 3.49127
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) −15.0000 −1.00000
\(226\) 0 0
\(227\) − 28.7750i − 1.90986i −0.296826 0.954932i \(-0.595928\pi\)
0.296826 0.954932i \(-0.404072\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) − 10.7238i − 0.693665i −0.937927 0.346833i \(-0.887257\pi\)
0.937927 0.346833i \(-0.112743\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 35.7771 2.28571
\(246\) 0 0
\(247\) 0 0
\(248\) 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
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 53.6190i 3.33172i
\(260\) 0 0
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) − 14.3875i − 0.887171i −0.896232 0.443585i \(-0.853706\pi\)
0.896232 0.443585i \(-0.146294\pi\)
\(264\) 0 0
\(265\) −5.00000 −0.307148
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 31.0000 1.89010 0.945052 0.326921i \(-0.106011\pi\)
0.945052 + 0.326921i \(0.106011\pi\)
\(270\) 0 0
\(271\) 32.1714i 1.95427i 0.212610 + 0.977137i \(0.431804\pi\)
−0.212610 + 0.977137i \(0.568196\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 32.1714i 1.92605i
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) − 33.5708i − 1.99558i −0.0664602 0.997789i \(-0.521171\pi\)
0.0664602 0.997789i \(-0.478829\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 33.5708i 1.98162i
\(288\) 0 0
\(289\) 28.0000 1.64706
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −6.70820 −0.391897 −0.195949 0.980614i \(-0.562779\pi\)
−0.195949 + 0.980614i \(0.562779\pi\)
\(294\) 0 0
\(295\) 23.9792i 1.39612i
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) −46.0000 −2.65140
\(302\) 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
\(311\) − 21.4476i − 1.21618i −0.793867 0.608091i \(-0.791935\pi\)
0.793867 0.608091i \(-0.208065\pi\)
\(312\) 0 0
\(313\) 11.1803 0.631950 0.315975 0.948767i \(-0.397668\pi\)
0.315975 + 0.948767i \(0.397668\pi\)
\(314\) 0 0
\(315\) 32.1714i 1.81265i
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 32.1714i 1.76830i 0.467202 + 0.884150i \(0.345262\pi\)
−0.467202 + 0.884150i \(0.654738\pi\)
\(332\) 0 0
\(333\) −33.5410 −1.83804
\(334\) 0 0
\(335\) − 10.7238i − 0.585904i
\(336\) 0 0
\(337\) −31.3050 −1.70529 −0.852645 0.522491i \(-0.825003\pi\)
−0.852645 + 0.522491i \(0.825003\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 43.1625i − 2.33056i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 19.0000 1.01705 0.508523 0.861048i \(-0.330192\pi\)
0.508523 + 0.861048i \(0.330192\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 23.9792i 1.27268i
\(356\) 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
\(365\) 0 0
\(366\) 0 0
\(367\) 4.79583i 0.250340i 0.992135 + 0.125170i \(0.0399477\pi\)
−0.992135 + 0.125170i \(0.960052\pi\)
\(368\) 0 0
\(369\) −21.0000 −1.09322
\(370\) 0 0
\(371\) 10.7238i 0.556752i
\(372\) 0 0
\(373\) −4.47214 −0.231558 −0.115779 0.993275i \(-0.536937\pi\)
−0.115779 + 0.993275i \(0.536937\pi\)
\(374\) 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
\(383\) − 33.5708i − 1.71539i −0.514160 0.857694i \(-0.671896\pi\)
0.514160 0.857694i \(-0.328104\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 28.7750i − 1.46271i
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 32.1714i 1.62698i
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −20.1246 −1.00000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 39.0000 1.92843 0.964213 0.265129i \(-0.0854146\pi\)
0.964213 + 0.265129i \(0.0854146\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 51.4296 2.53068
\(414\) 0 0
\(415\) 32.1714i 1.57923i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) −33.5410 −1.62698
\(426\) 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
\(433\) −24.5967 −1.18204 −0.591022 0.806655i \(-0.701275\pi\)
−0.591022 + 0.806655i \(0.701275\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) − 21.4476i − 1.02364i −0.859093 0.511819i \(-0.828972\pi\)
0.859093 0.511819i \(-0.171028\pi\)
\(440\) 0 0
\(441\) 48.0000 2.28571
\(442\) 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
\(448\) 0 0
\(449\) −41.0000 −1.93491 −0.967455 0.253044i \(-0.918568\pi\)
−0.967455 + 0.253044i \(0.918568\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −42.4853 −1.98738 −0.993689 0.112170i \(-0.964220\pi\)
−0.993689 + 0.112170i \(0.964220\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −2.00000 −0.0931493 −0.0465746 0.998915i \(-0.514831\pi\)
−0.0465746 + 0.998915i \(0.514831\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 43.1625i 1.99732i 0.0517364 + 0.998661i \(0.483524\pi\)
−0.0517364 + 0.998661i \(0.516476\pi\)
\(468\) 0 0
\(469\) −23.0000 −1.06204
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −6.70820 −0.307148
\(478\) 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
\(485\) −10.0000 −0.454077
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 10.7238i − 0.483959i −0.970281 0.241979i \(-0.922203\pi\)
0.970281 0.241979i \(-0.0777966\pi\)
\(492\) 0 0
\(493\) 6.70820 0.302122
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 51.4296 2.30693
\(498\) 0 0
\(499\) 32.1714i 1.44019i 0.693875 + 0.720095i \(0.255902\pi\)
−0.693875 + 0.720095i \(0.744098\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) − 14.3875i − 0.641507i −0.947163 0.320753i \(-0.896064\pi\)
0.947163 0.320753i \(-0.103936\pi\)
\(504\) 0 0
\(505\) 38.0132 1.69156
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 14.0000 0.620539 0.310270 0.950649i \(-0.399581\pi\)
0.310270 + 0.950649i \(0.399581\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 21.4476i 0.945095i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 9.59166i 0.419414i 0.977764 + 0.209707i \(0.0672510\pi\)
−0.977764 + 0.209707i \(0.932749\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 71.9375i 3.13365i
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 32.1714i 1.39612i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) − 10.7238i − 0.463631i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −18.0000 −0.773880 −0.386940 0.922105i \(-0.626468\pi\)
−0.386940 + 0.922105i \(0.626468\pi\)
\(542\) 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
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 46.9574 1.98965 0.994825 0.101603i \(-0.0323971\pi\)
0.994825 + 0.101603i \(0.0323971\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) − 33.5708i − 1.41484i −0.706793 0.707421i \(-0.749859\pi\)
0.706793 0.707421i \(-0.250141\pi\)
\(564\) 0 0
\(565\) 35.0000 1.47246
\(566\) 0 0
\(567\) 43.1625i 1.81265i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) − 23.9792i − 1.00000i
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 69.0000 2.86260
\(582\) 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
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 71.9375i 2.94915i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) − 21.4476i − 0.876326i −0.898896 0.438163i \(-0.855629\pi\)
0.898896 0.438163i \(-0.144371\pi\)
\(600\) 0 0
\(601\) −37.0000 −1.50926 −0.754631 0.656150i \(-0.772184\pi\)
−0.754631 + 0.656150i \(0.772184\pi\)
\(602\) 0 0
\(603\) − 14.3875i − 0.585904i
\(604\) 0 0
\(605\) 24.5967 1.00000
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 31.3050 1.26440 0.632198 0.774807i \(-0.282153\pi\)
0.632198 + 0.774807i \(0.282153\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −24.5967 −0.990228 −0.495114 0.868828i \(-0.664874\pi\)
−0.495114 + 0.868828i \(0.664874\pi\)
\(618\) 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
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −75.0000 −2.99045
\(630\) 0 0
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 32.1714i 1.27268i
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) − 33.5708i − 1.32390i −0.749546 0.661952i \(-0.769728\pi\)
0.749546 0.661952i \(-0.230272\pi\)
\(644\) 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
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) − 47.9583i − 1.87389i
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 4.79583i 0.185695i
\(668\) 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
\(676\) 0 0
\(677\) 29.0689 1.11721 0.558604 0.829435i \(-0.311337\pi\)
0.558604 + 0.829435i \(0.311337\pi\)
\(678\) 0 0
\(679\) 21.4476i 0.823084i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 30.0000 1.14624
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 21.4476i 0.815906i 0.913003 + 0.407953i \(0.133757\pi\)
−0.913003 + 0.407953i \(0.866243\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 23.9792i 0.909581i
\(696\) 0 0
\(697\) −46.9574 −1.77864
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 81.5291i − 3.06622i
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −51.4296 −1.92605
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) − 53.6190i − 1.99965i −0.0186469 0.999826i \(-0.505936\pi\)
0.0186469 0.999826i \(-0.494064\pi\)
\(720\) 0 0
\(721\) 46.0000 1.71313
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −5.00000 −0.185695
\(726\) 0 0
\(727\) − 52.7541i − 1.95654i −0.207328 0.978272i \(-0.566477\pi\)
0.207328 0.978272i \(-0.433523\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) − 64.3428i − 2.37981i
\(732\) 0 0
\(733\) −42.4853 −1.56923 −0.784615 0.619983i \(-0.787139\pi\)
−0.784615 + 0.619983i \(0.787139\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) − 53.6190i − 1.97241i −0.165535 0.986204i \(-0.552935\pi\)
0.165535 0.986204i \(-0.447065\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 9.59166i − 0.351884i −0.984401 0.175942i \(-0.943703\pi\)
0.984401 0.175942i \(-0.0562971\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 43.1625i 1.57923i
\(748\) 0 0
\(749\) −23.0000 −0.840402
\(750\) 0 0
\(751\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 47.9583i 1.74538i
\(756\) 0 0
\(757\) −15.6525 −0.568899 −0.284449 0.958691i \(-0.591811\pi\)
−0.284449 + 0.958691i \(0.591811\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −13.0000 −0.471250 −0.235625 0.971844i \(-0.575714\pi\)
−0.235625 + 0.971844i \(0.575714\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −45.0000 −1.62698
\(766\) 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
\(773\) −40.2492 −1.44766 −0.723832 0.689976i \(-0.757621\pi\)
−0.723832 + 0.689976i \(0.757621\pi\)
\(774\) 0 0
\(775\) − 53.6190i − 1.92605i
\(776\) 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
\(785\) 55.0000 1.96303
\(786\) 0 0
\(787\) 4.79583i 0.170953i 0.996340 + 0.0854765i \(0.0272412\pi\)
−0.996340 + 0.0854765i \(0.972759\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) − 75.0666i − 2.66906i
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 20.1246 0.712850 0.356425 0.934324i \(-0.383995\pi\)
0.356425 + 0.934324i \(0.383995\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −51.4296 −1.81265
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 19.0000 0.668004 0.334002 0.942572i \(-0.391601\pi\)
0.334002 + 0.942572i \(0.391601\pi\)
\(810\) 0 0
\(811\) 32.1714i 1.12969i 0.825197 + 0.564846i \(0.191064\pi\)
−0.825197 + 0.564846i \(0.808936\pi\)
\(812\) 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
\(821\) 38.0000 1.32621 0.663105 0.748527i \(-0.269238\pi\)
0.663105 + 0.748527i \(0.269238\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 43.1625i 1.50091i 0.660923 + 0.750453i \(0.270165\pi\)
−0.660923 + 0.750453i \(0.729835\pi\)
\(828\) 0 0
\(829\) −21.0000 −0.729360 −0.364680 0.931133i \(-0.618822\pi\)
−0.364680 + 0.931133i \(0.618822\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 107.331 3.71881
\(834\) 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
\(841\) −28.0000 −0.965517
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −29.0689 −1.00000
\(846\) 0 0
\(847\) − 52.7541i − 1.81265i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) − 53.6190i − 1.83804i
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 32.1714i 1.09767i 0.835929 + 0.548837i \(0.184929\pi\)
−0.835929 + 0.548837i \(0.815071\pi\)
\(860\) 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
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −13.4164 −0.454077
\(874\) 0 0
\(875\) − 53.6190i − 1.81265i
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 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
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) − 47.9583i − 1.60307i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 10.7238i 0.357659i
\(900\) 0 0
\(901\) −15.0000 −0.499722
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) − 52.7541i − 1.75167i −0.482608 0.875836i \(-0.660310\pi\)
0.482608 0.875836i \(-0.339690\pi\)
\(908\) 0 0
\(909\) 51.0000 1.69156
\(910\) 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
\(916\) 0 0
\(917\) −102.859 −3.39671
\(918\) 0 0
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 55.9017 1.83804
\(926\) 0 0
\(927\) 28.7750i 0.945095i
\(928\) 0 0
\(929\) −29.0000 −0.951459 −0.475730 0.879592i \(-0.657816\pi\)
−0.475730 + 0.879592i \(0.657816\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 58.1378 1.89928 0.949639 0.313346i \(-0.101450\pi\)
0.949639 + 0.313346i \(0.101450\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) − 33.5708i − 1.09322i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 22.3607 0.724333 0.362167 0.932113i \(-0.382037\pi\)
0.362167 + 0.932113i \(0.382037\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) − 64.3428i − 2.07774i
\(960\) 0 0
\(961\) −84.0000 −2.70968
\(962\) 0 0
\(963\) − 14.3875i − 0.463631i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 51.4296 1.64876
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −33.5410 −1.07307 −0.536536 0.843877i \(-0.680267\pi\)
−0.536536 + 0.843877i \(0.680267\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 62.3458i 1.98852i 0.106979 + 0.994261i \(0.465882\pi\)
−0.106979 + 0.994261i \(0.534118\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 46.0000 1.46271
\(990\) 0 0
\(991\) − 53.6190i − 1.70326i −0.524140 0.851632i \(-0.675613\pi\)
0.524140 0.851632i \(-0.324387\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 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 1840.2.m.b.1839.2 yes 4
4.3 odd 2 inner 1840.2.m.b.1839.1 4
5.4 even 2 inner 1840.2.m.b.1839.3 yes 4
20.19 odd 2 inner 1840.2.m.b.1839.4 yes 4
23.22 odd 2 inner 1840.2.m.b.1839.3 yes 4
92.91 even 2 inner 1840.2.m.b.1839.4 yes 4
115.114 odd 2 CM 1840.2.m.b.1839.2 yes 4
460.459 even 2 inner 1840.2.m.b.1839.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1840.2.m.b.1839.1 4 4.3 odd 2 inner
1840.2.m.b.1839.1 4 460.459 even 2 inner
1840.2.m.b.1839.2 yes 4 1.1 even 1 trivial
1840.2.m.b.1839.2 yes 4 115.114 odd 2 CM
1840.2.m.b.1839.3 yes 4 5.4 even 2 inner
1840.2.m.b.1839.3 yes 4 23.22 odd 2 inner
1840.2.m.b.1839.4 yes 4 20.19 odd 2 inner
1840.2.m.b.1839.4 yes 4 92.91 even 2 inner