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Properties

Label 672.3.e.c.335.1
Level $672$
Weight $3$
Character 672.335
Self dual yes
Analytic conductor $18.311$
Analytic rank $0$
Dimension $1$
CM discriminant -168
Inner twists $2$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [672,3,Mod(335,672)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(672, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("672.335");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 672 = 2^{5} \cdot 3 \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 672.e (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: yes
Analytic conductor: \(18.3106737650\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 168)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 335.1
Character \(\chi\) \(=\) 672.335

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.00000 q^{3} -7.00000 q^{7} +9.00000 q^{9} +O(q^{10})\) \(q+3.00000 q^{3} -7.00000 q^{7} +9.00000 q^{9} +2.00000 q^{13} +22.0000 q^{17} -21.0000 q^{21} +38.0000 q^{23} +25.0000 q^{25} +27.0000 q^{27} -26.0000 q^{29} +34.0000 q^{31} +6.00000 q^{39} -26.0000 q^{41} +82.0000 q^{43} +49.0000 q^{49} +66.0000 q^{51} +22.0000 q^{53} -106.000 q^{59} -94.0000 q^{61} -63.0000 q^{63} +34.0000 q^{67} +114.000 q^{69} -58.0000 q^{71} +75.0000 q^{75} +81.0000 q^{81} -58.0000 q^{83} -78.0000 q^{87} -122.000 q^{89} -14.0000 q^{91} +102.000 q^{93} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/672\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(421\) \(449\) \(577\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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
\(3\) 3.00000 1.00000
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) −7.00000 −1.00000
\(8\) 0 0
\(9\) 9.00000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 2.00000 0.153846 0.0769231 − 0.997037i \(-0.475490\pi\)
0.0769231 + 0.997037i \(0.475490\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 22.0000 1.29412 0.647059 − 0.762440i \(-0.275999\pi\)
0.647059 + 0.762440i \(0.275999\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −21.0000 −1.00000
\(22\) 0 0
\(23\) 38.0000 1.65217 0.826087 − 0.563543i \(-0.190562\pi\)
0.826087 + 0.563543i \(0.190562\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 27.0000 1.00000
\(28\) 0 0
\(29\) −26.0000 −0.896552 −0.448276 − 0.893895i \(-0.647962\pi\)
−0.448276 + 0.893895i \(0.647962\pi\)
\(30\) 0 0
\(31\) 34.0000 1.09677 0.548387 − 0.836225i \(-0.315242\pi\)
0.548387 + 0.836225i \(0.315242\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 6.00000 0.153846
\(40\) 0 0
\(41\) −26.0000 −0.634146 −0.317073 − 0.948401i \(-0.602700\pi\)
−0.317073 + 0.948401i \(0.602700\pi\)
\(42\) 0 0
\(43\) 82.0000 1.90698 0.953488 − 0.301430i \(-0.0974639\pi\)
0.953488 + 0.301430i \(0.0974639\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 49.0000 1.00000
\(50\) 0 0
\(51\) 66.0000 1.29412
\(52\) 0 0
\(53\) 22.0000 0.415094 0.207547 − 0.978225i \(-0.433452\pi\)
0.207547 + 0.978225i \(0.433452\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −106.000 −1.79661 −0.898305 − 0.439372i \(-0.855201\pi\)
−0.898305 + 0.439372i \(0.855201\pi\)
\(60\) 0 0
\(61\) −94.0000 −1.54098 −0.770492 − 0.637450i \(-0.779989\pi\)
−0.770492 + 0.637450i \(0.779989\pi\)
\(62\) 0 0
\(63\) −63.0000 −1.00000
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 34.0000 0.507463 0.253731 − 0.967275i \(-0.418342\pi\)
0.253731 + 0.967275i \(0.418342\pi\)
\(68\) 0 0
\(69\) 114.000 1.65217
\(70\) 0 0
\(71\) −58.0000 −0.816901 −0.408451 − 0.912780i \(-0.633931\pi\)
−0.408451 + 0.912780i \(0.633931\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 75.0000 1.00000
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 81.0000 1.00000
\(82\) 0 0
\(83\) −58.0000 −0.698795 −0.349398 − 0.936975i \(-0.613614\pi\)
−0.349398 + 0.936975i \(0.613614\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) −78.0000 −0.896552
\(88\) 0 0
\(89\) −122.000 −1.37079 −0.685393 − 0.728173i \(-0.740370\pi\)
−0.685393 + 0.728173i \(0.740370\pi\)
\(90\) 0 0
\(91\) −14.0000 −0.153846
\(92\) 0 0
\(93\) 102.000 1.09677
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 178.000 1.72816 0.864078 − 0.503359i \(-0.167902\pi\)
0.864078 + 0.503359i \(0.167902\pi\)
\(104\) 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
\(115\) 0 0
\(116\) 0 0
\(117\) 18.0000 0.153846
\(118\) 0 0
\(119\) −154.000 −1.29412
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 0 0
\(123\) −78.0000 −0.634146
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) 0 0
\(129\) 246.000 1.90698
\(130\) 0 0
\(131\) 38.0000 0.290076 0.145038 − 0.989426i \(-0.453670\pi\)
0.145038 + 0.989426i \(0.453670\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 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
\(147\) 147.000 1.00000
\(148\) 0 0
\(149\) 214.000 1.43624 0.718121 − 0.695918i \(-0.245003\pi\)
0.718121 + 0.695918i \(0.245003\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 198.000 1.29412
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −286.000 −1.82166 −0.910828 − 0.412786i \(-0.864556\pi\)
−0.910828 + 0.412786i \(0.864556\pi\)
\(158\) 0 0
\(159\) 66.0000 0.415094
\(160\) 0 0
\(161\) −266.000 −1.65217
\(162\) 0 0
\(163\) −158.000 −0.969325 −0.484663 − 0.874701i \(-0.661058\pi\)
−0.484663 + 0.874701i \(0.661058\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −165.000 −0.976331
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) −175.000 −1.00000
\(176\) 0 0
\(177\) −318.000 −1.79661
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 338.000 1.86740 0.933702 − 0.358052i \(-0.116559\pi\)
0.933702 + 0.358052i \(0.116559\pi\)
\(182\) 0 0
\(183\) −282.000 −1.54098
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −189.000 −1.00000
\(190\) 0 0
\(191\) 374.000 1.95812 0.979058 − 0.203583i \(-0.0652588\pi\)
0.979058 + 0.203583i \(0.0652588\pi\)
\(192\) 0 0
\(193\) −286.000 −1.48187 −0.740933 − 0.671579i \(-0.765616\pi\)
−0.740933 + 0.671579i \(0.765616\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −362.000 −1.83756 −0.918782 − 0.394766i \(-0.870826\pi\)
−0.918782 + 0.394766i \(0.870826\pi\)
\(198\) 0 0
\(199\) −302.000 −1.51759 −0.758794 − 0.651331i \(-0.774211\pi\)
−0.758794 + 0.651331i \(0.774211\pi\)
\(200\) 0 0
\(201\) 102.000 0.507463
\(202\) 0 0
\(203\) 182.000 0.896552
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 342.000 1.65217
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −254.000 −1.20379 −0.601896 − 0.798575i \(-0.705588\pi\)
−0.601896 + 0.798575i \(0.705588\pi\)
\(212\) 0 0
\(213\) −174.000 −0.816901
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −238.000 −1.09677
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 44.0000 0.199095
\(222\) 0 0
\(223\) 418.000 1.87444 0.937220 − 0.348739i \(-0.113390\pi\)
0.937220 + 0.348739i \(0.113390\pi\)
\(224\) 0 0
\(225\) 225.000 1.00000
\(226\) 0 0
\(227\) −442.000 −1.94714 −0.973568 − 0.228396i \(-0.926652\pi\)
−0.973568 + 0.228396i \(0.926652\pi\)
\(228\) 0 0
\(229\) 242.000 1.05677 0.528384 − 0.849005i \(-0.322798\pi\)
0.528384 + 0.849005i \(0.322798\pi\)
\(230\) 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
\(239\) 278.000 1.16318 0.581590 − 0.813482i \(-0.302431\pi\)
0.581590 + 0.813482i \(0.302431\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 243.000 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −174.000 −0.698795
\(250\) 0 0
\(251\) −394.000 −1.56972 −0.784861 − 0.619672i \(-0.787265\pi\)
−0.784861 + 0.619672i \(0.787265\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −458.000 −1.78210 −0.891051 − 0.453904i \(-0.850031\pi\)
−0.891051 + 0.453904i \(0.850031\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) −234.000 −0.896552
\(262\) 0 0
\(263\) −442.000 −1.68061 −0.840304 − 0.542115i \(-0.817624\pi\)
−0.840304 + 0.542115i \(0.817624\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −366.000 −1.37079
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −158.000 −0.583026 −0.291513 − 0.956567i \(-0.594159\pi\)
−0.291513 + 0.956567i \(0.594159\pi\)
\(272\) 0 0
\(273\) −42.0000 −0.153846
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 306.000 1.09677
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 182.000 0.634146
\(288\) 0 0
\(289\) 195.000 0.674740
\(290\) 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
\(299\) 76.0000 0.254181
\(300\) 0 0
\(301\) −574.000 −1.90698
\(302\) 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
\(309\) 534.000 1.72816
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −122.000 −0.384858 −0.192429 − 0.981311i \(-0.561637\pi\)
−0.192429 + 0.981311i \(0.561637\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 50.0000 0.153846
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −494.000 −1.49245 −0.746224 − 0.665695i \(-0.768135\pi\)
−0.746224 + 0.665695i \(0.768135\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.00000 0.00593472 0.00296736 − 0.999996i \(-0.499055\pi\)
0.00296736 + 0.999996i \(0.499055\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −343.000 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 674.000 1.93123 0.965616 − 0.259972i \(-0.0837135\pi\)
0.965616 + 0.259972i \(0.0837135\pi\)
\(350\) 0 0
\(351\) 54.0000 0.153846
\(352\) 0 0
\(353\) 694.000 1.96601 0.983003 − 0.183590i \(-0.0587720\pi\)
0.983003 + 0.183590i \(0.0587720\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) −462.000 −1.29412
\(358\) 0 0
\(359\) −634.000 −1.76602 −0.883008 − 0.469357i \(-0.844486\pi\)
−0.883008 + 0.469357i \(0.844486\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 363.000 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −638.000 −1.73842 −0.869210 − 0.494443i \(-0.835372\pi\)
−0.869210 + 0.494443i \(0.835372\pi\)
\(368\) 0 0
\(369\) −234.000 −0.634146
\(370\) 0 0
\(371\) −154.000 −0.415094
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −52.0000 −0.137931
\(378\) 0 0
\(379\) 754.000 1.98945 0.994723 − 0.102597i \(-0.0327154\pi\)
0.994723 + 0.102597i \(0.0327154\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 738.000 1.90698
\(388\) 0 0
\(389\) 22.0000 0.0565553 0.0282776 − 0.999600i \(-0.490998\pi\)
0.0282776 + 0.999600i \(0.490998\pi\)
\(390\) 0 0
\(391\) 836.000 2.13811
\(392\) 0 0
\(393\) 114.000 0.290076
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 578.000 1.45592 0.727960 − 0.685620i \(-0.240469\pi\)
0.727960 + 0.685620i \(0.240469\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 68.0000 0.168734
\(404\) 0 0
\(405\) 0 0
\(406\) 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
\(413\) 742.000 1.79661
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 614.000 1.46539 0.732697 − 0.680555i \(-0.238261\pi\)
0.732697 + 0.680555i \(0.238261\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 550.000 1.29412
\(426\) 0 0
\(427\) 658.000 1.54098
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −106.000 −0.245940 −0.122970 − 0.992410i \(-0.539242\pi\)
−0.122970 + 0.992410i \(0.539242\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −494.000 −1.12528 −0.562642 − 0.826700i \(-0.690215\pi\)
−0.562642 + 0.826700i \(0.690215\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 642.000 1.43624
\(448\) 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
\(457\) 242.000 0.529540 0.264770 − 0.964312i \(-0.414704\pi\)
0.264770 + 0.964312i \(0.414704\pi\)
\(458\) 0 0
\(459\) 594.000 1.29412
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 38.0000 0.0813704 0.0406852 − 0.999172i \(-0.487046\pi\)
0.0406852 + 0.999172i \(0.487046\pi\)
\(468\) 0 0
\(469\) −238.000 −0.507463
\(470\) 0 0
\(471\) −858.000 −1.82166
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 198.000 0.415094
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) −798.000 −1.65217
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) −474.000 −0.969325
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) −572.000 −1.16024
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 406.000 0.816901
\(498\) 0 0
\(499\) 514.000 1.03006 0.515030 − 0.857172i \(-0.327781\pi\)
0.515030 + 0.857172i \(0.327781\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −495.000 −0.976331
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −986.000 −1.89251 −0.946257 − 0.323415i \(-0.895169\pi\)
−0.946257 + 0.323415i \(0.895169\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) −525.000 −1.00000
\(526\) 0 0
\(527\) 748.000 1.41935
\(528\) 0 0
\(529\) 915.000 1.72968
\(530\) 0 0
\(531\) −954.000 −1.79661
\(532\) 0 0
\(533\) −52.0000 −0.0975610
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 1014.00 1.86740
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 418.000 0.764168 0.382084 − 0.924128i \(-0.375206\pi\)
0.382084 + 0.924128i \(0.375206\pi\)
\(548\) 0 0
\(549\) −846.000 −1.54098
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −986.000 −1.77020 −0.885099 − 0.465403i \(-0.845909\pi\)
−0.885099 + 0.465403i \(0.845909\pi\)
\(558\) 0 0
\(559\) 164.000 0.293381
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 902.000 1.60213 0.801066 − 0.598576i \(-0.204267\pi\)
0.801066 + 0.598576i \(0.204267\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −567.000 −1.00000
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −974.000 −1.70578 −0.852890 − 0.522091i \(-0.825152\pi\)
−0.852890 + 0.522091i \(0.825152\pi\)
\(572\) 0 0
\(573\) 1122.00 1.95812
\(574\) 0 0
\(575\) 950.000 1.65217
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) −858.000 −1.48187
\(580\) 0 0
\(581\) 406.000 0.698795
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 278.000 0.473595 0.236797 − 0.971559i \(-0.423902\pi\)
0.236797 + 0.971559i \(0.423902\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −1086.00 −1.83756
\(592\) 0 0
\(593\) 214.000 0.360877 0.180438 − 0.983586i \(-0.442248\pi\)
0.180438 + 0.983586i \(0.442248\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −906.000 −1.51759
\(598\) 0 0
\(599\) −1114.00 −1.85977 −0.929883 − 0.367855i \(-0.880092\pi\)
−0.929883 + 0.367855i \(0.880092\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 306.000 0.507463
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 1186.00 1.95387 0.976936 − 0.213533i \(-0.0684972\pi\)
0.976936 + 0.213533i \(0.0684972\pi\)
\(608\) 0 0
\(609\) 546.000 0.896552
\(610\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 1026.00 1.65217
\(622\) 0 0
\(623\) 854.000 1.37079
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) −762.000 −1.20379
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 98.0000 0.153846
\(638\) 0 0
\(639\) −522.000 −0.816901
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −714.000 −1.09677
\(652\) 0 0
\(653\) −794.000 −1.21593 −0.607963 − 0.793965i \(-0.708013\pi\)
−0.607963 + 0.793965i \(0.708013\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −622.000 −0.940998 −0.470499 − 0.882400i \(-0.655926\pi\)
−0.470499 + 0.882400i \(0.655926\pi\)
\(662\) 0 0
\(663\) 132.000 0.199095
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −988.000 −1.48126
\(668\) 0 0
\(669\) 1254.00 1.87444
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1342.00 −1.99406 −0.997028 − 0.0770370i \(-0.975454\pi\)
−0.997028 + 0.0770370i \(0.975454\pi\)
\(674\) 0 0
\(675\) 675.000 1.00000
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) −1326.00 −1.94714
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 726.000 1.05677
\(688\) 0 0
\(689\) 44.0000 0.0638607
\(690\) 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
\(697\) −572.000 −0.820660
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 646.000 0.921541 0.460770 − 0.887519i \(-0.347573\pi\)
0.460770 + 0.887519i \(0.347573\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 1292.00 1.81206
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 834.000 1.16318
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −1246.00 −1.72816
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −650.000 −0.896552
\(726\) 0 0
\(727\) 82.0000 0.112792 0.0563961 − 0.998408i \(-0.482039\pi\)
0.0563961 + 0.998408i \(0.482039\pi\)
\(728\) 0 0
\(729\) 729.000 1.00000
\(730\) 0 0
\(731\) 1804.00 2.46785
\(732\) 0 0
\(733\) −1438.00 −1.96180 −0.980900 − 0.194511i \(-0.937688\pi\)
−0.980900 + 0.194511i \(0.937688\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 34.0000 0.0460081 0.0230041 − 0.999735i \(-0.492677\pi\)
0.0230041 + 0.999735i \(0.492677\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1402.00 −1.88694 −0.943472 − 0.331451i \(-0.892462\pi\)
−0.943472 + 0.331451i \(0.892462\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) −522.000 −0.698795
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) −1182.00 −1.56972
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1222.00 1.60578 0.802891 − 0.596126i \(-0.203294\pi\)
0.802891 + 0.596126i \(0.203294\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) −212.000 −0.276402
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −1374.00 −1.78210
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 850.000 1.09677
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −702.000 −0.896552
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) −1326.00 −1.68061
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −188.000 −0.237074
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) −1098.00 −1.37079
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) −474.000 −0.583026
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −126.000 −0.153846
\(820\) 0 0
\(821\) 1558.00 1.89769 0.948843 − 0.315749i \(-0.102256\pi\)
0.948843 + 0.315749i \(0.102256\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −286.000 −0.344994 −0.172497 − 0.985010i \(-0.555184\pi\)
−0.172497 + 0.985010i \(0.555184\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 1078.00 1.29412
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 918.000 1.09677
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −165.000 −0.196195
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) −847.000 −1.00000
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 1682.00 1.97186 0.985932 − 0.167147i \(-0.0534554\pi\)
0.985932 + 0.167147i \(0.0534554\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −314.000 −0.366394 −0.183197 − 0.983076i \(-0.558645\pi\)
−0.183197 + 0.983076i \(0.558645\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 546.000 0.634146
\(862\) 0 0
\(863\) 374.000 0.433372 0.216686 − 0.976241i \(-0.430475\pi\)
0.216686 + 0.976241i \(0.430475\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 585.000 0.674740
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 68.0000 0.0780712
\(872\) 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
\(880\) 0 0
\(881\) −1706.00 −1.93644 −0.968218 − 0.250108i \(-0.919534\pi\)
−0.968218 + 0.250108i \(0.919534\pi\)
\(882\) 0 0
\(883\) −1598.00 −1.80974 −0.904870 − 0.425689i \(-0.860032\pi\)
−0.904870 + 0.425689i \(0.860032\pi\)
\(884\) 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
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 228.000 0.254181
\(898\) 0 0
\(899\) −884.000 −0.983315
\(900\) 0 0
\(901\) 484.000 0.537181
\(902\) 0 0
\(903\) −1722.00 −1.90698
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −302.000 −0.332966 −0.166483 − 0.986044i \(-0.553241\pi\)
−0.166483 + 0.986044i \(0.553241\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −1066.00 −1.17014 −0.585071 − 0.810982i \(-0.698934\pi\)
−0.585071 + 0.810982i \(0.698934\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −266.000 −0.290076
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) −116.000 −0.125677
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 1602.00 1.72816
\(928\) 0 0
\(929\) 886.000 0.953714 0.476857 − 0.878981i \(-0.341776\pi\)
0.476857 + 0.878981i \(0.341776\pi\)
\(930\) 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
\(940\) 0 0
\(941\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) −988.000 −1.04772
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) −366.000 −0.384858
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 195.000 0.202914
\(962\) 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
\(971\) −1642.00 −1.69104 −0.845520 − 0.533944i \(-0.820709\pi\)
−0.845520 + 0.533944i \(0.820709\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 150.000 0.153846
\(976\) 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
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 3116.00 3.15066
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −1482.00 −1.49245
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1394.00 1.39819 0.699097 − 0.715027i \(-0.253585\pi\)
0.699097 + 0.715027i \(0.253585\pi\)
\(998\) 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 672.3.e.c.335.1 1
3.2 odd 2 672.3.e.a.335.1 1
4.3 odd 2 168.3.e.c.83.1 yes 1
7.6 odd 2 672.3.e.b.335.1 1
8.3 odd 2 672.3.e.d.335.1 1
8.5 even 2 168.3.e.a.83.1 ✓ 1
12.11 even 2 168.3.e.b.83.1 yes 1
21.20 even 2 672.3.e.d.335.1 1
24.5 odd 2 168.3.e.d.83.1 yes 1
24.11 even 2 672.3.e.b.335.1 1
28.27 even 2 168.3.e.d.83.1 yes 1
56.13 odd 2 168.3.e.b.83.1 yes 1
56.27 even 2 672.3.e.a.335.1 1
84.83 odd 2 168.3.e.a.83.1 ✓ 1
168.83 odd 2 CM 672.3.e.c.335.1 1
168.125 even 2 168.3.e.c.83.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
168.3.e.a.83.1 ✓ 1 8.5 even 2
168.3.e.a.83.1 ✓ 1 84.83 odd 2
168.3.e.b.83.1 yes 1 12.11 even 2
168.3.e.b.83.1 yes 1 56.13 odd 2
168.3.e.c.83.1 yes 1 4.3 odd 2
168.3.e.c.83.1 yes 1 168.125 even 2
168.3.e.d.83.1 yes 1 24.5 odd 2
168.3.e.d.83.1 yes 1 28.27 even 2
672.3.e.a.335.1 1 3.2 odd 2
672.3.e.a.335.1 1 56.27 even 2
672.3.e.b.335.1 1 7.6 odd 2
672.3.e.b.335.1 1 24.11 even 2
672.3.e.c.335.1 1 1.1 even 1 trivial
672.3.e.c.335.1 1 168.83 odd 2 CM
672.3.e.d.335.1 1 8.3 odd 2
672.3.e.d.335.1 1 21.20 even 2