Properties

Label 216.3.h.b.53.1
Level $216$
Weight $3$
Character 216.53
Self dual yes
Analytic conductor $5.886$
Analytic rank $0$
Dimension $2$
CM discriminant -24
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 216 = 2^{3} \cdot 3^{3} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 216.h (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual: yes
Analytic conductor: \(5.88557371018\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{8})^+\)
Defining polynomial: \(x^{2} - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 2\cdot 3 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 53.1
Root \(-1.41421\) of defining polynomial
Character \(\chi\) \(=\) 216.53

$q$-expansion

\(f(q)\) \(=\) \(q+2.00000 q^{2} +4.00000 q^{4} -7.48528 q^{5} +13.4853 q^{7} +8.00000 q^{8} +O(q^{10})\) \(q+2.00000 q^{2} +4.00000 q^{4} -7.48528 q^{5} +13.4853 q^{7} +8.00000 q^{8} -14.9706 q^{10} +11.9706 q^{11} +26.9706 q^{14} +16.0000 q^{16} -29.9411 q^{20} +23.9411 q^{22} +31.0294 q^{25} +53.9411 q^{28} -50.0000 q^{29} -61.4264 q^{31} +32.0000 q^{32} -100.941 q^{35} -59.8823 q^{40} +47.8823 q^{44} +132.853 q^{49} +62.0589 q^{50} -4.57359 q^{53} -89.6030 q^{55} +107.882 q^{56} -100.000 q^{58} +10.0000 q^{59} -122.853 q^{62} +64.0000 q^{64} -201.882 q^{70} -143.794 q^{73} +161.426 q^{77} -58.0000 q^{79} -119.765 q^{80} -17.8528 q^{83} +95.7645 q^{88} +128.941 q^{97} +265.706 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q + 4q^{2} + 8q^{4} + 2q^{5} + 10q^{7} + 16q^{8} + O(q^{10}) \) \( 2q + 4q^{2} + 8q^{4} + 2q^{5} + 10q^{7} + 16q^{8} + 4q^{10} - 10q^{11} + 20q^{14} + 32q^{16} + 8q^{20} - 20q^{22} + 96q^{25} + 40q^{28} - 100q^{29} - 38q^{31} + 64q^{32} - 134q^{35} + 16q^{40} - 40q^{44} + 96q^{49} + 192q^{50} - 94q^{53} - 298q^{55} + 80q^{56} - 200q^{58} + 20q^{59} - 76q^{62} + 128q^{64} - 268q^{70} - 50q^{73} + 238q^{77} - 116q^{79} + 32q^{80} + 134q^{83} - 80q^{88} + 190q^{97} + 192q^{98} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/216\mathbb{Z}\right)^\times\).

\(n\) \(55\) \(109\) \(137\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000 1.00000
\(3\) 0 0
\(4\) 4.00000 1.00000
\(5\) −7.48528 −1.49706 −0.748528 0.663103i \(-0.769239\pi\)
−0.748528 + 0.663103i \(0.769239\pi\)
\(6\) 0 0
\(7\) 13.4853 1.92647 0.963234 0.268662i \(-0.0865816\pi\)
0.963234 + 0.268662i \(0.0865816\pi\)
\(8\) 8.00000 1.00000
\(9\) 0 0
\(10\) −14.9706 −1.49706
\(11\) 11.9706 1.08823 0.544116 0.839010i \(-0.316865\pi\)
0.544116 + 0.839010i \(0.316865\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 26.9706 1.92647
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −29.9411 −1.49706
\(21\) 0 0
\(22\) 23.9411 1.08823
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 31.0294 1.24118
\(26\) 0 0
\(27\) 0 0
\(28\) 53.9411 1.92647
\(29\) −50.0000 −1.72414 −0.862069 0.506791i \(-0.830832\pi\)
−0.862069 + 0.506791i \(0.830832\pi\)
\(30\) 0 0
\(31\) −61.4264 −1.98150 −0.990748 0.135711i \(-0.956668\pi\)
−0.990748 + 0.135711i \(0.956668\pi\)
\(32\) 32.0000 1.00000
\(33\) 0 0
\(34\) 0 0
\(35\) −100.941 −2.88403
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −59.8823 −1.49706
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 47.8823 1.08823
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 132.853 2.71128
\(50\) 62.0589 1.24118
\(51\) 0 0
\(52\) 0 0
\(53\) −4.57359 −0.0862942 −0.0431471 0.999069i \(-0.513738\pi\)
−0.0431471 + 0.999069i \(0.513738\pi\)
\(54\) 0 0
\(55\) −89.6030 −1.62915
\(56\) 107.882 1.92647
\(57\) 0 0
\(58\) −100.000 −1.72414
\(59\) 10.0000 0.169492 0.0847458 0.996403i \(-0.472992\pi\)
0.0847458 + 0.996403i \(0.472992\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) −122.853 −1.98150
\(63\) 0 0
\(64\) 64.0000 1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −201.882 −2.88403
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −143.794 −1.96978 −0.984890 0.173181i \(-0.944595\pi\)
−0.984890 + 0.173181i \(0.944595\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 161.426 2.09645
\(78\) 0 0
\(79\) −58.0000 −0.734177 −0.367089 0.930186i \(-0.619645\pi\)
−0.367089 + 0.930186i \(0.619645\pi\)
\(80\) −119.765 −1.49706
\(81\) 0 0
\(82\) 0 0
\(83\) −17.8528 −0.215094 −0.107547 0.994200i \(-0.534300\pi\)
−0.107547 + 0.994200i \(0.534300\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 95.7645 1.08823
\(89\) 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
\(97\) 128.941 1.32929 0.664645 0.747159i \(-0.268583\pi\)
0.664645 + 0.747159i \(0.268583\pi\)
\(98\) 265.706 2.71128
\(99\) 0 0
\(100\) 124.118 1.24118
\(101\) −154.397 −1.52868 −0.764341 0.644812i \(-0.776936\pi\)
−0.764341 + 0.644812i \(0.776936\pi\)
\(102\) 0 0
\(103\) −10.0000 −0.0970874 −0.0485437 0.998821i \(-0.515458\pi\)
−0.0485437 + 0.998821i \(0.515458\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −9.14719 −0.0862942
\(107\) 212.706 1.98790 0.993952 0.109820i \(-0.0350273\pi\)
0.993952 + 0.109820i \(0.0350273\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) −179.206 −1.62915
\(111\) 0 0
\(112\) 215.765 1.92647
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −200.000 −1.72414
\(117\) 0 0
\(118\) 20.0000 0.169492
\(119\) 0 0
\(120\) 0 0
\(121\) 22.2944 0.184251
\(122\) 0 0
\(123\) 0 0
\(124\) −245.706 −1.98150
\(125\) −45.1320 −0.361056
\(126\) 0 0
\(127\) −208.338 −1.64046 −0.820229 0.572036i \(-0.806154\pi\)
−0.820229 + 0.572036i \(0.806154\pi\)
\(128\) 128.000 1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) −57.1177 −0.436013 −0.218007 0.975947i \(-0.569955\pi\)
−0.218007 + 0.975947i \(0.569955\pi\)
\(132\) 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 \(\pi\)
\(140\) −403.765 −2.88403
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 374.264 2.58113
\(146\) −287.588 −1.96978
\(147\) 0 0
\(148\) 0 0
\(149\) 85.6030 0.574517 0.287258 0.957853i \(-0.407256\pi\)
0.287258 + 0.957853i \(0.407256\pi\)
\(150\) 0 0
\(151\) 106.574 0.705785 0.352893 0.935664i \(-0.385198\pi\)
0.352893 + 0.935664i \(0.385198\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 322.853 2.09645
\(155\) 459.794 2.96641
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) −116.000 −0.734177
\(159\) 0 0
\(160\) −239.529 −1.49706
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −35.7056 −0.215094
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 169.000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 273.985 1.58373 0.791864 0.610698i \(-0.209111\pi\)
0.791864 + 0.610698i \(0.209111\pi\)
\(174\) 0 0
\(175\) 418.441 2.39109
\(176\) 191.529 1.08823
\(177\) 0 0
\(178\) 0 0
\(179\) −122.588 −0.684848 −0.342424 0.939545i \(-0.611248\pi\)
−0.342424 + 0.939545i \(0.611248\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −365.617 −1.89439 −0.947195 0.320658i \(-0.896096\pi\)
−0.947195 + 0.320658i \(0.896096\pi\)
\(194\) 257.882 1.32929
\(195\) 0 0
\(196\) 531.411 2.71128
\(197\) 393.985 1.99992 0.999962 0.00876993i \(-0.00279159\pi\)
0.999962 + 0.00876993i \(0.00279159\pi\)
\(198\) 0 0
\(199\) −195.985 −0.984848 −0.492424 0.870355i \(-0.663889\pi\)
−0.492424 + 0.870355i \(0.663889\pi\)
\(200\) 248.235 1.24118
\(201\) 0 0
\(202\) −308.794 −1.52868
\(203\) −674.264 −3.32150
\(204\) 0 0
\(205\) 0 0
\(206\) −20.0000 −0.0970874
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) −18.2944 −0.0862942
\(213\) 0 0
\(214\) 425.411 1.98790
\(215\) 0 0
\(216\) 0 0
\(217\) −828.352 −3.81729
\(218\) 0 0
\(219\) 0 0
\(220\) −358.412 −1.62915
\(221\) 0 0
\(222\) 0 0
\(223\) 230.000 1.03139 0.515695 0.856772i \(-0.327534\pi\)
0.515695 + 0.856772i \(0.327534\pi\)
\(224\) 431.529 1.92647
\(225\) 0 0
\(226\) 0 0
\(227\) 346.000 1.52423 0.762115 0.647442i \(-0.224161\pi\)
0.762115 + 0.647442i \(0.224161\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −400.000 −1.72414
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 40.0000 0.169492
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −382.000 −1.58506 −0.792531 0.609831i \(-0.791237\pi\)
−0.792531 + 0.609831i \(0.791237\pi\)
\(242\) 44.5887 0.184251
\(243\) 0 0
\(244\) 0 0
\(245\) −994.441 −4.05894
\(246\) 0 0
\(247\) 0 0
\(248\) −491.411 −1.98150
\(249\) 0 0
\(250\) −90.2641 −0.361056
\(251\) −470.000 −1.87251 −0.936255 0.351321i \(-0.885733\pi\)
−0.936255 + 0.351321i \(0.885733\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −416.676 −1.64046
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −114.235 −0.436013
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 34.2346 0.129187
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 430.000 1.59851 0.799257 0.600990i \(-0.205227\pi\)
0.799257 + 0.600990i \(0.205227\pi\)
\(270\) 0 0
\(271\) 495.690 1.82912 0.914558 0.404455i \(-0.132539\pi\)
0.914558 + 0.404455i \(0.132539\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 371.440 1.35069
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −807.529 −2.88403
\(281\) 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
\(287\) 0 0
\(288\) 0 0
\(289\) 289.000 1.00000
\(290\) 748.528 2.58113
\(291\) 0 0
\(292\) −575.176 −1.96978
\(293\) −386.000 −1.31741 −0.658703 0.752403i \(-0.728895\pi\)
−0.658703 + 0.752403i \(0.728895\pi\)
\(294\) 0 0
\(295\) −74.8528 −0.253738
\(296\) 0 0
\(297\) 0 0
\(298\) 171.206 0.574517
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 213.147 0.705785
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 645.706 2.09645
\(309\) 0 0
\(310\) 919.588 2.96641
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 23.4996 0.0750785 0.0375392 0.999295i \(-0.488048\pi\)
0.0375392 + 0.999295i \(0.488048\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −232.000 −0.734177
\(317\) −633.690 −1.99902 −0.999512 0.0312439i \(-0.990053\pi\)
−0.999512 + 0.0312439i \(0.990053\pi\)
\(318\) 0 0
\(319\) −598.528 −1.87626
\(320\) −479.058 −1.49706
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −71.4113 −0.215094
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −190.000 −0.563798 −0.281899 0.959444i \(-0.590964\pi\)
−0.281899 + 0.959444i \(0.590964\pi\)
\(338\) 338.000 1.00000
\(339\) 0 0
\(340\) 0 0
\(341\) −735.309 −2.15633
\(342\) 0 0
\(343\) 1130.78 3.29673
\(344\) 0 0
\(345\) 0 0
\(346\) 547.970 1.58373
\(347\) −646.970 −1.86447 −0.932233 0.361859i \(-0.882142\pi\)
−0.932233 + 0.361859i \(0.882142\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 836.881 2.39109
\(351\) 0 0
\(352\) 383.058 1.08823
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −245.176 −0.684848
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 1076.34 2.94887
\(366\) 0 0
\(367\) −193.780 −0.528010 −0.264005 0.964521i \(-0.585044\pi\)
−0.264005 + 0.964521i \(0.585044\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −61.6762 −0.166243
\(372\) 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
\(385\) −1208.32 −3.13850
\(386\) −731.235 −1.89439
\(387\) 0 0
\(388\) 515.765 1.32929
\(389\) 558.367 1.43539 0.717695 0.696358i \(-0.245197\pi\)
0.717695 + 0.696358i \(0.245197\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 1062.82 2.71128
\(393\) 0 0
\(394\) 787.970 1.99992
\(395\) 434.146 1.09910
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) −391.970 −0.984848
\(399\) 0 0
\(400\) 496.471 1.24118
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −617.588 −1.52868
\(405\) 0 0
\(406\) −1348.53 −3.32150
\(407\) 0 0
\(408\) 0 0
\(409\) 698.411 1.70761 0.853803 0.520595i \(-0.174290\pi\)
0.853803 + 0.520595i \(0.174290\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −40.0000 −0.0970874
\(413\) 134.853 0.326520
\(414\) 0 0
\(415\) 133.633 0.322008
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 730.000 1.74224 0.871122 0.491067i \(-0.163393\pi\)
0.871122 + 0.491067i \(0.163393\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −36.5887 −0.0862942
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 850.823 1.98790
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −140.176 −0.323732 −0.161866 0.986813i \(-0.551751\pi\)
−0.161866 + 0.986813i \(0.551751\pi\)
\(434\) −1656.70 −3.81729
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 582.249 1.32631 0.663154 0.748483i \(-0.269218\pi\)
0.663154 + 0.748483i \(0.269218\pi\)
\(440\) −716.824 −1.62915
\(441\) 0 0
\(442\) 0 0
\(443\) −86.0000 −0.194131 −0.0970655 0.995278i \(-0.530946\pi\)
−0.0970655 + 0.995278i \(0.530946\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 460.000 1.03139
\(447\) 0 0
\(448\) 863.058 1.92647
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 692.000 1.52423
\(455\) 0 0
\(456\) 0 0
\(457\) 379.881 0.831250 0.415625 0.909536i \(-0.363563\pi\)
0.415625 + 0.909536i \(0.363563\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 918.367 1.99212 0.996059 0.0886899i \(-0.0282680\pi\)
0.996059 + 0.0886899i \(0.0282680\pi\)
\(462\) 0 0
\(463\) 897.161 1.93771 0.968856 0.247625i \(-0.0796501\pi\)
0.968856 + 0.247625i \(0.0796501\pi\)
\(464\) −800.000 −1.72414
\(465\) 0 0
\(466\) 0 0
\(467\) −910.970 −1.95068 −0.975342 0.220698i \(-0.929167\pi\)
−0.975342 + 0.220698i \(0.929167\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 80.0000 0.169492
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −764.000 −1.58506
\(483\) 0 0
\(484\) 89.1775 0.184251
\(485\) −965.161 −1.99002
\(486\) 0 0
\(487\) −970.000 −1.99179 −0.995893 0.0905356i \(-0.971142\pi\)
−0.995893 + 0.0905356i \(0.971142\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −1988.88 −4.05894
\(491\) −511.705 −1.04217 −0.521084 0.853505i \(-0.674472\pi\)
−0.521084 + 0.853505i \(0.674472\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −982.823 −1.98150
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −180.528 −0.361056
\(501\) 0 0
\(502\) −940.000 −1.87251
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 1155.70 2.28852
\(506\) 0 0
\(507\) 0 0
\(508\) −833.352 −1.64046
\(509\) 394.691 0.775425 0.387713 0.921780i \(-0.373265\pi\)
0.387713 + 0.921780i \(0.373265\pi\)
\(510\) 0 0
\(511\) −1939.10 −3.79472
\(512\) 512.000 1.00000
\(513\) 0 0
\(514\) 0 0
\(515\) 74.8528 0.145345
\(516\) 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
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) −228.471 −0.436013
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 529.000 1.00000
\(530\) 68.4693 0.129187
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −1592.16 −2.97600
\(536\) 0 0
\(537\) 0 0
\(538\) 860.000 1.59851
\(539\) 1590.32 2.95051
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 991.381 1.82912
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 742.880 1.35069
\(551\) 0 0
\(552\) 0 0
\(553\) −782.146 −1.41437
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1008.54 1.81067 0.905335 0.424698i \(-0.139620\pi\)
0.905335 + 0.424698i \(0.139620\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −1615.06 −2.88403
\(561\) 0 0
\(562\) 0 0
\(563\) 1096.38 1.94739 0.973695 0.227854i \(-0.0731709\pi\)
0.973695 + 0.227854i \(0.0731709\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 290.000 0.502600 0.251300 0.967909i \(-0.419142\pi\)
0.251300 + 0.967909i \(0.419142\pi\)
\(578\) 578.000 1.00000
\(579\) 0 0
\(580\) 1497.06 2.58113
\(581\) −240.750 −0.414372
\(582\) 0 0
\(583\) −54.7485 −0.0939082
\(584\) −1150.35 −1.96978
\(585\) 0 0
\(586\) −772.000 −1.31741
\(587\) 241.823 0.411963 0.205982 0.978556i \(-0.433961\pi\)
0.205982 + 0.978556i \(0.433961\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −149.706 −0.253738
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 342.412 0.574517
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 683.853 1.13786 0.568929 0.822387i \(-0.307358\pi\)
0.568929 + 0.822387i \(0.307358\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 426.294 0.705785
\(605\) −166.880 −0.275834
\(606\) 0 0
\(607\) −730.000 −1.20264 −0.601318 0.799010i \(-0.705357\pi\)
−0.601318 + 0.799010i \(0.705357\pi\)
\(608\) 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
\(616\) 1291.41 2.09645
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 1839.18 2.96641
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −437.910 −0.700656
\(626\) 46.9991 0.0750785
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −406.868 −0.644799 −0.322399 0.946604i \(-0.604489\pi\)
−0.322399 + 0.946604i \(0.604489\pi\)
\(632\) −464.000 −0.734177
\(633\) 0 0
\(634\) −1267.38 −1.99902
\(635\) 1559.47 2.45586
\(636\) 0 0
\(637\) 0 0
\(638\) −1197.06 −1.87626
\(639\) 0 0
\(640\) −958.116 −1.49706
\(641\) 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
\(649\) 119.706 0.184446
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1224.25 −1.87481 −0.937403 0.348245i \(-0.886778\pi\)
−0.937403 + 0.348245i \(0.886778\pi\)
\(654\) 0 0
\(655\) 427.542 0.652737
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −1315.35 −1.99598 −0.997991 0.0633633i \(-0.979817\pi\)
−0.997991 + 0.0633633i \(0.979817\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −142.823 −0.215094
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1249.00 1.85587 0.927933 0.372746i \(-0.121584\pi\)
0.927933 + 0.372746i \(0.121584\pi\)
\(674\) −380.000 −0.563798
\(675\) 0 0
\(676\) 676.000 1.00000
\(677\) −1346.00 −1.98818 −0.994092 0.108545i \(-0.965381\pi\)
−0.994092 + 0.108545i \(0.965381\pi\)
\(678\) 0 0
\(679\) 1738.81 2.56084
\(680\) 0 0
\(681\) 0 0
\(682\) −1470.62 −2.15633
\(683\) −1334.00 −1.95315 −0.976574 0.215182i \(-0.930965\pi\)
−0.976574 + 0.215182i \(0.930965\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 2261.56 3.29673
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 1095.94 1.58373
\(693\) 0 0
\(694\) −1293.94 −1.86447
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1673.76 2.39109
\(701\) 1238.40 1.76661 0.883306 0.468797i \(-0.155312\pi\)
0.883306 + 0.468797i \(0.155312\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 766.116 1.08823
\(705\) 0 0
\(706\) 0 0
\(707\) −2082.09 −2.94496
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) −490.352 −0.684848
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) −134.853 −0.187036
\(722\) 722.000 1.00000
\(723\) 0 0
\(724\) 0 0
\(725\) −1551.47 −2.13996
\(726\) 0 0
\(727\) 631.662 0.868861 0.434430 0.900705i \(-0.356950\pi\)
0.434430 + 0.900705i \(0.356950\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 2152.68 2.94887
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) −387.559 −0.528010
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −123.352 −0.166243
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −640.763 −0.860084
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 2868.40 3.82963
\(750\) 0 0
\(751\) 1167.69 1.55485 0.777424 0.628977i \(-0.216526\pi\)
0.777424 + 0.628977i \(0.216526\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) −797.733 −1.05660
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1534.09 1.99491 0.997456 0.0712917i \(-0.0227121\pi\)
0.997456 + 0.0712917i \(0.0227121\pi\)
\(770\) −2416.64 −3.13850
\(771\) 0 0
\(772\) −1462.47 −1.89439
\(773\) −1154.00 −1.49288 −0.746442 0.665450i \(-0.768240\pi\)
−0.746442 + 0.665450i \(0.768240\pi\)
\(774\) 0 0
\(775\) −1906.03 −2.45939
\(776\) 1031.53 1.32929
\(777\) 0 0
\(778\) 1116.73 1.43539
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 2125.65 2.71128
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 1575.94 1.99992
\(789\) 0 0
\(790\) 868.293 1.09910
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −783.939 −0.984848
\(797\) 159.102 0.199626 0.0998129 0.995006i \(-0.468176\pi\)
0.0998129 + 0.995006i \(0.468176\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 992.942 1.24118
\(801\) 0 0
\(802\) 0 0
\(803\) −1721.29 −2.14358
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −1235.18 −1.52868
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) −2697.06 −3.32150
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1396.82 1.70761
\(819\) 0 0
\(820\) 0 0
\(821\) 670.000 0.816078 0.408039 0.912965i \(-0.366213\pi\)
0.408039 + 0.912965i \(0.366213\pi\)
\(822\) 0 0
\(823\) −1601.13 −1.94548 −0.972740 0.231898i \(-0.925507\pi\)
−0.972740 + 0.231898i \(0.925507\pi\)
\(824\) −80.0000 −0.0970874
\(825\) 0 0
\(826\) 269.706 0.326520
\(827\) 1546.00 1.86941 0.934704 0.355428i \(-0.115665\pi\)
0.934704 + 0.355428i \(0.115665\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 267.267 0.322008
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 1460.00 1.74224
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1659.00 1.97265
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −1265.01 −1.49706
\(846\) 0 0
\(847\) 300.646 0.354954
\(848\) −73.1775 −0.0862942
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1701.65 1.98790
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) −2050.85 −2.37093
\(866\) −280.352 −0.323732
\(867\) 0 0
\(868\) −3313.41 −3.81729
\(869\) −694.293 −0.798956
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −608.618 −0.695564
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 1164.50 1.32631
\(879\) 0 0
\(880\) −1433.65 −1.62915
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −172.000 −0.194131
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −2809.50 −3.16029
\(890\) 0 0
\(891\) 0 0
\(892\) 920.000 1.03139
\(893\) 0 0
\(894\) 0 0
\(895\) 917.605 1.02526
\(896\) 1726.12 1.92647
\(897\) 0 0
\(898\) 0 0
\(899\) 3071.32 3.41637
\(900\) 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 \(\pi\)
\(908\) 1384.00 1.52423
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) −213.708 −0.234073
\(914\) 759.763 0.831250
\(915\) 0 0
\(916\) 0 0
\(917\) −770.249 −0.839966
\(918\) 0 0
\(919\) 1153.92 1.25563 0.627815 0.778362i \(-0.283949\pi\)
0.627815 + 0.778362i \(0.283949\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1836.73 1.99212
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 1794.32 1.93771
\(927\) 0 0
\(928\) −1600.00 −1.72414
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −1821.94 −1.95068
\(935\) 0 0
\(936\) 0 0
\(937\) −1729.29 −1.84556 −0.922782 0.385323i \(-0.874090\pi\)
−0.922782 + 0.385323i \(0.874090\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −1801.75 −1.91472 −0.957358 0.288904i \(-0.906709\pi\)
−0.957358 + 0.288904i \(0.906709\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 160.000 0.169492
\(945\) 0 0
\(946\) 0 0
\(947\) 534.939 0.564878 0.282439 0.959285i \(-0.408857\pi\)
0.282439 + 0.959285i \(0.408857\pi\)
\(948\) 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
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2812.20 2.92633
\(962\) 0 0
\(963\) 0 0
\(964\) −1528.00 −1.58506
\(965\) 2736.75 2.83601
\(966\) 0 0
\(967\) −691.956 −0.715570 −0.357785 0.933804i \(-0.616468\pi\)
−0.357785 + 0.933804i \(0.616468\pi\)
\(968\) 178.355 0.184251
\(969\) 0 0
\(970\) −1930.32 −1.99002
\(971\) −1151.68 −1.18607 −0.593036 0.805176i \(-0.702071\pi\)
−0.593036 + 0.805176i \(0.702071\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −1940.00 −1.99179
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −3977.76 −4.05894
\(981\) 0 0
\(982\) −1023.41 −1.04217
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) −2949.09 −2.99400
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 539.366 0.544264 0.272132 0.962260i \(-0.412271\pi\)
0.272132 + 0.962260i \(0.412271\pi\)
\(992\) −1965.65 −1.98150
\(993\) 0 0
\(994\) 0 0
\(995\) 1467.00 1.47437
\(996\) 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 216.3.h.b.53.1 yes 2
3.2 odd 2 216.3.h.a.53.2 2
4.3 odd 2 864.3.h.b.593.1 2
8.3 odd 2 864.3.h.a.593.2 2
8.5 even 2 216.3.h.a.53.2 2
12.11 even 2 864.3.h.a.593.2 2
24.5 odd 2 CM 216.3.h.b.53.1 yes 2
24.11 even 2 864.3.h.b.593.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
216.3.h.a.53.2 2 3.2 odd 2
216.3.h.a.53.2 2 8.5 even 2
216.3.h.b.53.1 yes 2 1.1 even 1 trivial
216.3.h.b.53.1 yes 2 24.5 odd 2 CM
864.3.h.a.593.2 2 8.3 odd 2
864.3.h.a.593.2 2 12.11 even 2
864.3.h.b.593.1 2 4.3 odd 2
864.3.h.b.593.1 2 24.11 even 2