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Properties

Label 76.5.c.a.37.2
Level $76$
Weight $5$
Character 76.37
Self dual yes
Analytic conductor $7.856$
Analytic rank $0$
Dimension $2$
CM discriminant -19
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [76,5,Mod(37,76)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(76, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("76.37");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 76 = 2^{2} \cdot 19 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 76.c (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: yes
Analytic conductor: \(7.85611719437\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{57}) \)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x - 14 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 37.2
Root \(-3.27492\) of defining polynomial
Character \(\chi\) \(=\) 76.37

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+18.4743 q^{5} -20.1238 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q+18.4743 q^{5} -20.1238 q^{7} +81.0000 q^{9} +59.8762 q^{11} +572.866 q^{17} +361.000 q^{19} -158.000 q^{23} -283.702 q^{25} -371.771 q^{35} -2726.10 q^{43} +1496.41 q^{45} -4284.04 q^{47} -1996.03 q^{49} +1106.17 q^{55} +4248.75 q^{61} -1630.02 q^{63} +8130.81 q^{73} -1204.94 q^{77} +6561.00 q^{81} -5678.00 q^{83} +10583.3 q^{85} +6669.21 q^{95} +4849.98 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 31 q^{5} + 73 q^{7} + 162 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 31 q^{5} + 73 q^{7} + 162 q^{9} + 233 q^{11} + 353 q^{17} + 722 q^{19} - 316 q^{23} + 1539 q^{25} - 4979 q^{35} - 3527 q^{43} - 2511 q^{45} - 1207 q^{47} + 4275 q^{49} - 7459 q^{55} - 3167 q^{61} + 5913 q^{63} + 10033 q^{73} + 14917 q^{77} + 13122 q^{81} - 11356 q^{83} + 21461 q^{85} - 11191 q^{95} + 18873 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/76\mathbb{Z}\right)^\times\).

\(n\) \(21\) \(39\)
\(\chi(n)\) \(-1\) \(1\)

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
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 0 0
\(5\) 18.4743 0.738970 0.369485 − 0.929237i \(-0.379534\pi\)
0.369485 + 0.929237i \(0.379534\pi\)
\(6\) 0 0
\(7\) −20.1238 −0.410689 −0.205344 − 0.978690i \(-0.565831\pi\)
−0.205344 + 0.978690i \(0.565831\pi\)
\(8\) 0 0
\(9\) 81.0000 1.00000
\(10\) 0 0
\(11\) 59.8762 0.494845 0.247422 − 0.968908i \(-0.420416\pi\)
0.247422 + 0.968908i \(0.420416\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 572.866 1.98224 0.991118 − 0.132984i \(-0.0424559\pi\)
0.991118 + 0.132984i \(0.0424559\pi\)
\(18\) 0 0
\(19\) 361.000 1.00000
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) −158.000 −0.298677 −0.149338 − 0.988786i \(-0.547714\pi\)
−0.149338 + 0.988786i \(0.547714\pi\)
\(24\) 0 0
\(25\) −283.702 −0.453923
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −371.771 −0.303487
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −2726.10 −1.47437 −0.737183 − 0.675693i \(-0.763845\pi\)
−0.737183 + 0.675693i \(0.763845\pi\)
\(44\) 0 0
\(45\) 1496.41 0.738970
\(46\) 0 0
\(47\) −4284.04 −1.93936 −0.969680 − 0.244380i \(-0.921416\pi\)
−0.969680 + 0.244380i \(0.921416\pi\)
\(48\) 0 0
\(49\) −1996.03 −0.831335
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 1106.17 0.365676
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 4248.75 1.14183 0.570915 − 0.821009i \(-0.306589\pi\)
0.570915 + 0.821009i \(0.306589\pi\)
\(62\) 0 0
\(63\) −1630.02 −0.410689
\(64\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) 8130.81 1.52577 0.762883 − 0.646537i \(-0.223783\pi\)
0.762883 + 0.646537i \(0.223783\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1204.94 −0.203227
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) −5678.00 −0.824213 −0.412106 − 0.911136i \(-0.635207\pi\)
−0.412106 + 0.911136i \(0.635207\pi\)
\(84\) 0 0
\(85\) 10583.3 1.46481
\(86\) 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
\(95\) 6669.21 0.738970
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 4849.98 0.494845
\(100\) 0 0
\(101\) −9998.00 −0.980100 −0.490050 − 0.871694i \(-0.663021\pi\)
−0.490050 + 0.871694i \(0.663021\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) −2918.93 −0.220713
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) −11528.2 −0.814083
\(120\) 0 0
\(121\) −11055.8 −0.755128
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −16787.6 −1.07441
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −33665.5 −1.96175 −0.980873 − 0.194651i \(-0.937643\pi\)
−0.980873 + 0.194651i \(0.937643\pi\)
\(132\) 0 0
\(133\) −7264.68 −0.410689
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 8396.39 0.447354 0.223677 − 0.974663i \(-0.428194\pi\)
0.223677 + 0.974663i \(0.428194\pi\)
\(138\) 0 0
\(139\) 33381.1 1.72771 0.863856 − 0.503738i \(-0.168042\pi\)
0.863856 + 0.503738i \(0.168042\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −30212.3 −1.36085 −0.680427 − 0.732816i \(-0.738206\pi\)
−0.680427 + 0.732816i \(0.738206\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 46402.2 1.98224
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −49198.0 −1.99594 −0.997972 − 0.0636620i \(-0.979722\pi\)
−0.997972 + 0.0636620i \(0.979722\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 3179.55 0.122663
\(162\) 0 0
\(163\) 9362.00 0.352366 0.176183 − 0.984357i \(-0.443625\pi\)
0.176183 + 0.984357i \(0.443625\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 28561.0 1.00000
\(170\) 0 0
\(171\) 29241.0 1.00000
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 5709.15 0.186421
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 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
\(187\) 34301.1 0.980900
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 61997.2 1.69944 0.849719 − 0.527235i \(-0.176771\pi\)
0.849719 + 0.527235i \(0.176771\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −69518.0 −1.79129 −0.895643 − 0.444774i \(-0.853284\pi\)
−0.895643 + 0.444774i \(0.853284\pi\)
\(198\) 0 0
\(199\) 78104.5 1.97229 0.986143 − 0.165900i \(-0.0530529\pi\)
0.986143 + 0.165900i \(0.0530529\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −12798.0 −0.298677
\(208\) 0 0
\(209\) 21615.3 0.494845
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) −50362.7 −1.08951
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −22979.9 −0.453923
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 59034.7 1.12574 0.562868 − 0.826547i \(-0.309698\pi\)
0.562868 + 0.826547i \(0.309698\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −66110.9 −1.21776 −0.608879 − 0.793263i \(-0.708380\pi\)
−0.608879 + 0.793263i \(0.708380\pi\)
\(234\) 0 0
\(235\) −79144.5 −1.43313
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −105335. −1.84406 −0.922032 − 0.387113i \(-0.873472\pi\)
−0.922032 + 0.387113i \(0.873472\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −36875.2 −0.614331
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 74357.6 1.18026 0.590130 − 0.807308i \(-0.299076\pi\)
0.590130 + 0.807308i \(0.299076\pi\)
\(252\) 0 0
\(253\) −9460.45 −0.147799
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 104877. 1.51625 0.758124 − 0.652111i \(-0.226116\pi\)
0.758124 + 0.652111i \(0.226116\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −126718. −1.72544 −0.862720 − 0.505682i \(-0.831241\pi\)
−0.862720 + 0.505682i \(0.831241\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) −16987.0 −0.224622
\(276\) 0 0
\(277\) −133030. −1.73376 −0.866880 − 0.498517i \(-0.833878\pi\)
−0.866880 + 0.498517i \(0.833878\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −136595. −1.70554 −0.852771 − 0.522286i \(-0.825079\pi\)
−0.852771 + 0.522286i \(0.825079\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 244655. 2.92926
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 54859.5 0.605506
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 78492.4 0.843778
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −5413.87 −0.0559741 −0.0279871 − 0.999608i \(-0.508910\pi\)
−0.0279871 + 0.999608i \(0.508910\pi\)
\(312\) 0 0
\(313\) 152162. 1.55316 0.776582 − 0.630016i \(-0.216952\pi\)
0.776582 + 0.630016i \(0.216952\pi\)
\(314\) 0 0
\(315\) −30113.5 −0.303487
\(316\) 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
\(323\) 206805. 1.98224
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 86211.1 0.796473
\(330\) 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
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 88484.9 0.752109
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −201682. −1.67497 −0.837487 − 0.546457i \(-0.815976\pi\)
−0.837487 + 0.546457i \(0.815976\pi\)
\(348\) 0 0
\(349\) −225949. −1.85506 −0.927532 − 0.373745i \(-0.878074\pi\)
−0.927532 + 0.373745i \(0.878074\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 10882.0 0.0873292 0.0436646 − 0.999046i \(-0.486097\pi\)
0.0436646 + 0.999046i \(0.486097\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 241539. 1.87412 0.937061 − 0.349165i \(-0.113535\pi\)
0.937061 + 0.349165i \(0.113535\pi\)
\(360\) 0 0
\(361\) 130321. 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 150211. 1.12750
\(366\) 0 0
\(367\) −266878. −1.98144 −0.990719 − 0.135923i \(-0.956600\pi\)
−0.990719 + 0.135923i \(0.956600\pi\)
\(368\) 0 0
\(369\) 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
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) −22260.3 −0.150179
\(386\) 0 0
\(387\) −220814. −1.47437
\(388\) 0 0
\(389\) 38543.1 0.254711 0.127355 − 0.991857i \(-0.459351\pi\)
0.127355 + 0.991857i \(0.459351\pi\)
\(390\) 0 0
\(391\) −90512.9 −0.592048
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −297079. −1.88491 −0.942456 − 0.334329i \(-0.891490\pi\)
−0.942456 + 0.334329i \(0.891490\pi\)
\(398\) 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
\(405\) 121210. 0.738970
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −104897. −0.609068
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 229522. 1.30736 0.653682 − 0.756770i \(-0.273224\pi\)
0.653682 + 0.756770i \(0.273224\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) −347008. −1.93936
\(424\) 0 0
\(425\) −162523. −0.899783
\(426\) 0 0
\(427\) −85500.8 −0.468937
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −57038.0 −0.298677
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) −161679. −0.831335
\(442\) 0 0
\(443\) 160753. 0.819126 0.409563 − 0.912282i \(-0.365681\pi\)
0.409563 + 0.912282i \(0.365681\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 374513. 1.79322 0.896612 − 0.442817i \(-0.146021\pi\)
0.896612 + 0.442817i \(0.146021\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −199550. −0.938967 −0.469484 − 0.882941i \(-0.655560\pi\)
−0.469484 + 0.882941i \(0.655560\pi\)
\(462\) 0 0
\(463\) 279914. 1.30576 0.652879 − 0.757462i \(-0.273561\pi\)
0.652879 + 0.757462i \(0.273561\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −51999.4 −0.238432 −0.119216 − 0.992868i \(-0.538038\pi\)
−0.119216 + 0.992868i \(0.538038\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −163229. −0.729583
\(474\) 0 0
\(475\) −102416. −0.453923
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 428482. 1.86750 0.933752 − 0.357921i \(-0.116514\pi\)
0.933752 + 0.357921i \(0.116514\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 360562. 1.49561 0.747803 − 0.663921i \(-0.231109\pi\)
0.747803 + 0.663921i \(0.231109\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 89599.7 0.365676
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 497221. 1.99687 0.998433 − 0.0559673i \(-0.0178243\pi\)
0.998433 + 0.0559673i \(0.0178243\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 358882. 1.41846 0.709228 − 0.704979i \(-0.249044\pi\)
0.709228 + 0.704979i \(0.249044\pi\)
\(504\) 0 0
\(505\) −184706. −0.724265
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −163622. −0.626615
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −256512. −0.959682
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 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
\(529\) −254877. −0.910792
\(530\) 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
\(539\) −119515. −0.411382
\(540\) 0 0
\(541\) 576412. 1.96942 0.984711 − 0.174196i \(-0.0557327\pi\)
0.984711 + 0.174196i \(0.0557327\pi\)
\(542\) 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
\(549\) 344149. 1.14183
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −94987.4 −0.306165 −0.153083 − 0.988213i \(-0.548920\pi\)
−0.153083 + 0.988213i \(0.548920\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −132032. −0.410689
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −442318. −1.35663 −0.678317 − 0.734770i \(-0.737290\pi\)
−0.678317 + 0.734770i \(0.737290\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 44824.9 0.135576
\(576\) 0 0
\(577\) −179948. −0.540500 −0.270250 − 0.962790i \(-0.587106\pi\)
−0.270250 + 0.962790i \(0.587106\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 114263. 0.338495
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 38758.7 0.112485 0.0562423 − 0.998417i \(-0.482088\pi\)
0.0562423 + 0.998417i \(0.482088\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −702398. −1.99744 −0.998720 − 0.0505740i \(-0.983895\pi\)
−0.998720 + 0.0505740i \(0.983895\pi\)
\(594\) 0 0
\(595\) −212975. −0.601583
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −204248. −0.558017
\(606\) 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
\(613\) 28243.5 0.0751620 0.0375810 − 0.999294i \(-0.488035\pi\)
0.0375810 + 0.999294i \(0.488035\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −761325. −1.99986 −0.999930 − 0.0118526i \(-0.996227\pi\)
−0.999930 + 0.0118526i \(0.996227\pi\)
\(618\) 0 0
\(619\) −328078. −0.856241 −0.428120 − 0.903722i \(-0.640824\pi\)
−0.428120 + 0.903722i \(0.640824\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −132825. −0.340031
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 506384. 1.27181 0.635903 − 0.771769i \(-0.280628\pi\)
0.635903 + 0.771769i \(0.280628\pi\)
\(632\) 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
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 410564. 0.993023 0.496511 − 0.868030i \(-0.334614\pi\)
0.496511 + 0.868030i \(0.334614\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −795843. −1.90116 −0.950580 − 0.310480i \(-0.899510\pi\)
−0.950580 + 0.310480i \(0.899510\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 762372. 1.78789 0.893945 − 0.448178i \(-0.147927\pi\)
0.893945 + 0.448178i \(0.147927\pi\)
\(654\) 0 0
\(655\) −621945. −1.44967
\(656\) 0 0
\(657\) 658595. 1.52577
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −134209. −0.303487
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 254399. 0.565029
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 155117. 0.330581
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 278468. 0.583202 0.291601 − 0.956540i \(-0.405812\pi\)
0.291601 + 0.956540i \(0.405812\pi\)
\(692\) 0 0
\(693\) −97599.7 −0.203227
\(694\) 0 0
\(695\) 616692. 1.27673
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 222802. 0.453402 0.226701 − 0.973964i \(-0.427206\pi\)
0.226701 + 0.973964i \(0.427206\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 201197. 0.402516
\(708\) 0 0
\(709\) 731762. 1.45572 0.727859 − 0.685727i \(-0.240515\pi\)
0.727859 + 0.685727i \(0.240515\pi\)
\(710\) 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
\(719\) −837359. −1.61977 −0.809886 − 0.586588i \(-0.800471\pi\)
−0.809886 + 0.586588i \(0.800471\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 631766. 1.19533 0.597664 − 0.801747i \(-0.296096\pi\)
0.597664 + 0.801747i \(0.296096\pi\)
\(728\) 0 0
\(729\) 531441. 1.00000
\(730\) 0 0
\(731\) −1.56169e6 −2.92254
\(732\) 0 0
\(733\) 538322. 1.00192 0.500961 − 0.865470i \(-0.332980\pi\)
0.500961 + 0.865470i \(0.332980\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.04695e6 1.91707 0.958536 − 0.284970i \(-0.0919838\pi\)
0.958536 + 0.284970i \(0.0919838\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −558150. −1.00563
\(746\) 0 0
\(747\) −459918. −0.824213
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.14504e6 1.99815 0.999076 − 0.0429669i \(-0.0136810\pi\)
0.999076 + 0.0429669i \(0.0136810\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −238330. −0.411537 −0.205769 − 0.978601i \(-0.565969\pi\)
−0.205769 + 0.978601i \(0.565969\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 857246. 1.46481
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.04962e6 1.77493 0.887465 − 0.460875i \(-0.152464\pi\)
0.887465 + 0.460875i \(0.152464\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −908896. −1.47494
\(786\) 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
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) −2.45418e6 −3.84427
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 486842. 0.755018
\(804\) 0 0
\(805\) 58739.9 0.0906445
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −223470. −0.341446 −0.170723 − 0.985319i \(-0.554610\pi\)
−0.170723 + 0.985319i \(0.554610\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 172956. 0.260388
\(816\) 0 0
\(817\) −984124. −1.47437
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.17432e6 −1.74220 −0.871101 − 0.491104i \(-0.836594\pi\)
−0.871101 + 0.491104i \(0.836594\pi\)
\(822\) 0 0
\(823\) 143923. 0.212486 0.106243 − 0.994340i \(-0.466118\pi\)
0.106243 + 0.994340i \(0.466118\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) −1.14346e6 −1.64790
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 707281. 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 527643. 0.738970
\(846\) 0 0
\(847\) 222485. 0.310123
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −394318. −0.541937 −0.270968 − 0.962588i \(-0.587344\pi\)
−0.270968 + 0.962588i \(0.587344\pi\)
\(854\) 0 0
\(855\) 540206. 0.738970
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −1.47565e6 −1.99985 −0.999927 − 0.0120966i \(-0.996149\pi\)
−0.999927 + 0.0120966i \(0.996149\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 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
\(875\) 337829. 0.441247
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.41242e6 1.81975 0.909877 − 0.414879i \(-0.136176\pi\)
0.909877 + 0.414879i \(0.136176\pi\)
\(882\) 0 0
\(883\) −694207. −0.890365 −0.445182 − 0.895440i \(-0.646861\pi\)
−0.445182 + 0.895440i \(0.646861\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 392848. 0.494845
\(892\) 0 0
\(893\) −1.54654e6 −1.93936
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) −809838. −0.980100
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −339977. −0.407857
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 677477. 0.805667
\(918\) 0 0
\(919\) 1.41552e6 1.67604 0.838022 − 0.545636i \(-0.183712\pi\)
0.838022 + 0.545636i \(0.183712\pi\)
\(920\) 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
\(929\) −1.31392e6 −1.52243 −0.761214 − 0.648501i \(-0.775396\pi\)
−0.761214 + 0.648501i \(0.775396\pi\)
\(930\) 0 0
\(931\) −720568. −0.831335
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 633687. 0.724856
\(936\) 0 0
\(937\) 1.35678e6 1.54536 0.772682 − 0.634794i \(-0.218915\pi\)
0.772682 + 0.634794i \(0.218915\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.55528e6 1.73424 0.867120 − 0.498099i \(-0.165969\pi\)
0.867120 + 0.498099i \(0.165969\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 1.14535e6 1.25583
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −168967. −0.183723
\(960\) 0 0
\(961\) 923521. 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.33392e6 1.42652 0.713259 − 0.700900i \(-0.247218\pi\)
0.713259 + 0.700900i \(0.247218\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) −671754. −0.709553
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −1.28429e6 −1.32371
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 430724. 0.440359
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 1.44292e6 1.45746
\(996\) 0 0
\(997\) −486609. −0.489542 −0.244771 − 0.969581i \(-0.578713\pi\)
−0.244771 + 0.969581i \(0.578713\pi\)
\(998\) 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 76.5.c.a.37.2 ✓ 2
3.2 odd 2 684.5.h.b.37.1 2
4.3 odd 2 304.5.e.b.113.2 2
19.18 odd 2 CM 76.5.c.a.37.2 ✓ 2
57.56 even 2 684.5.h.b.37.1 2
76.75 even 2 304.5.e.b.113.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
76.5.c.a.37.2 ✓ 2 1.1 even 1 trivial
76.5.c.a.37.2 ✓ 2 19.18 odd 2 CM
304.5.e.b.113.2 2 4.3 odd 2
304.5.e.b.113.2 2 76.75 even 2
684.5.h.b.37.1 2 3.2 odd 2
684.5.h.b.37.1 2 57.56 even 2