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Properties

Label 2016.3.e.b.1007.7
Level $2016$
Weight $3$
Character 2016.1007
Analytic conductor $54.932$
Analytic rank $0$
Dimension $8$
CM discriminant -7
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2016,3,Mod(1007,2016)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2016, 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("2016.1007");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2016 = 2^{5} \cdot 3^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2016.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: no
Analytic conductor: \(54.9320212950\)
Analytic rank: \(0\)
Dimension: \(8\)
Coefficient field: 8.0.157351936.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + x^{4} + 16 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{23}]\)
Coefficient ring index: \( 2^{16}\cdot 3^{4} \)
Twist minimal: no (minimal twist has level 504)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1007.7
Root \(-0.581861 - 1.28897i\) of defining polynomial
Character \(\chi\) \(=\) 2016.1007
Dual form 2016.3.e.b.1007.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.00000i q^{7} +O(q^{10})\) \(q+7.00000i q^{7} +10.7240i q^{11} +42.6612 q^{23} +25.0000 q^{25} -53.1504 q^{29} +38.0000i q^{37} +63.4980 q^{43} -49.0000 q^{49} -70.5905 q^{53} -118.000 q^{67} -20.7437 q^{71} -75.0679 q^{77} +126.996i q^{79} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 200 q^{25} - 392 q^{49} - 944 q^{67}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2016\mathbb{Z}\right)^\times\).

\(n\) \(127\) \(577\) \(1765\) \(1793\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 7.00000i 1.00000i
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 10.7240i 0.974908i 0.873149 + 0.487454i \(0.162074\pi\)
−0.873149 + 0.487454i \(0.837926\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 42.6612 1.85483 0.927417 − 0.374029i \(-0.122024\pi\)
0.927417 + 0.374029i \(0.122024\pi\)
\(24\) 0 0
\(25\) 25.0000 1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −53.1504 −1.83277 −0.916386 − 0.400295i \(-0.868907\pi\)
−0.916386 + 0.400295i \(0.868907\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 38.0000i 1.02703i 0.858082 + 0.513514i \(0.171656\pi\)
−0.858082 + 0.513514i \(0.828344\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 63.4980 1.47670 0.738349 − 0.674419i \(-0.235606\pi\)
0.738349 + 0.674419i \(0.235606\pi\)
\(44\) 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
\(51\) 0 0
\(52\) 0 0
\(53\) −70.5905 −1.33190 −0.665948 − 0.745998i \(-0.731973\pi\)
−0.665948 + 0.745998i \(0.731973\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) −118.000 −1.76119 −0.880597 − 0.473866i \(-0.842858\pi\)
−0.880597 + 0.473866i \(0.842858\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −20.7437 −0.292164 −0.146082 − 0.989272i \(-0.546666\pi\)
−0.146082 + 0.989272i \(0.546666\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −75.0679 −0.974908
\(78\) 0 0
\(79\) 126.996i 1.60755i 0.594937 + 0.803773i \(0.297177\pi\)
−0.594937 + 0.803773i \(0.702823\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 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
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 56.6887i − 0.529801i −0.964276 − 0.264901i \(-0.914661\pi\)
0.964276 − 0.264901i \(-0.0853391\pi\)
\(108\) 0 0
\(109\) 106.000i 0.972477i 0.873826 + 0.486239i \(0.161631\pi\)
−0.873826 + 0.486239i \(0.838369\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 186.911i − 1.65408i −0.562144 − 0.827040i \(-0.690023\pi\)
0.562144 − 0.827040i \(-0.309977\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.99606 0.0495542
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 253.992i 1.99994i 0.00787402 + 0.999969i \(0.497494\pi\)
−0.00787402 + 0.999969i \(0.502506\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 26.6297i 0.194377i 0.995266 + 0.0971887i \(0.0309850\pi\)
−0.995266 + 0.0971887i \(0.969015\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −296.155 −1.98762 −0.993808 − 0.111111i \(-0.964559\pi\)
−0.993808 + 0.111111i \(0.964559\pi\)
\(150\) 0 0
\(151\) 274.000i 1.81457i 0.420517 + 0.907285i \(0.361849\pi\)
−0.420517 + 0.907285i \(0.638151\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 298.628i 1.85483i
\(162\) 0 0
\(163\) −74.0000 −0.453988 −0.226994 − 0.973896i \(-0.572890\pi\)
−0.226994 + 0.973896i \(0.572890\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 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
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 175.000i 1.00000i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 316.664i 1.76907i 0.466473 + 0.884535i \(0.345524\pi\)
−0.466473 + 0.884535i \(0.654476\pi\)
\(180\) 0 0
\(181\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 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
\(191\) −374.526 −1.96087 −0.980435 − 0.196842i \(-0.936931\pi\)
−0.980435 + 0.196842i \(0.936931\pi\)
\(192\) 0 0
\(193\) 380.988 1.97403 0.987016 − 0.160622i \(-0.0513499\pi\)
0.987016 + 0.160622i \(0.0513499\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −4.83793 −0.0245580 −0.0122790 − 0.999925i \(-0.503909\pi\)
−0.0122790 + 0.999925i \(0.503909\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) − 372.053i − 1.83277i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 317.490 1.50469 0.752346 − 0.658768i \(-0.228922\pi\)
0.752346 + 0.658768i \(0.228922\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 341.994i − 1.46778i −0.679266 − 0.733892i \(-0.737702\pi\)
0.679266 − 0.733892i \(-0.262298\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −142.355 −0.595627 −0.297814 − 0.954624i \(-0.596257\pi\)
−0.297814 + 0.954624i \(0.596257\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 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
\(253\) 457.498i 1.80829i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) −266.000 −1.02703
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −471.872 −1.79419 −0.897095 − 0.441837i \(-0.854327\pi\)
−0.897095 + 0.441837i \(0.854327\pi\)
\(264\) 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
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 268.100i 0.974908i
\(276\) 0 0
\(277\) 317.490i 1.14617i 0.819495 + 0.573087i \(0.194254\pi\)
−0.819495 + 0.573087i \(0.805746\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 308.522i 1.09794i 0.835841 + 0.548972i \(0.184981\pi\)
−0.835841 + 0.548972i \(0.815019\pi\)
\(282\) 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
\(290\) 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
\(301\) 444.486i 1.47670i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 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
\(317\) 623.542 1.96701 0.983505 − 0.180879i \(-0.0578941\pi\)
0.983505 + 0.180879i \(0.0578941\pi\)
\(318\) 0 0
\(319\) − 569.984i − 1.78678i
\(320\) 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
\(331\) −190.494 −0.575511 −0.287755 − 0.957704i \(-0.592909\pi\)
−0.287755 + 0.957704i \(0.592909\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −634.980 −1.88421 −0.942107 − 0.335312i \(-0.891158\pi\)
−0.942107 + 0.335312i \(0.891158\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 343.000i − 1.00000i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 374.652i − 1.07969i −0.841765 − 0.539844i \(-0.818483\pi\)
0.841765 − 0.539844i \(-0.181517\pi\)
\(348\) 0 0
\(349\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 252.915 0.704499 0.352249 − 0.935906i \(-0.385417\pi\)
0.352249 + 0.935906i \(0.385417\pi\)
\(360\) 0 0
\(361\) 361.000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) − 494.134i − 1.33190i
\(372\) 0 0
\(373\) 698.478i 1.87260i 0.351206 + 0.936298i \(0.385772\pi\)
−0.351206 + 0.936298i \(0.614228\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 444.486 1.17279 0.586393 − 0.810026i \(-0.300547\pi\)
0.586393 + 0.810026i \(0.300547\pi\)
\(380\) 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
\(387\) 0 0
\(388\) 0 0
\(389\) −186.567 −0.479607 −0.239804 − 0.970821i \(-0.577083\pi\)
−0.239804 + 0.970821i \(0.577083\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) − 759.181i − 1.89322i −0.322381 − 0.946610i \(-0.604483\pi\)
0.322381 − 0.946610i \(-0.395517\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −407.512 −1.00126
\(408\) 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
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) − 825.474i − 1.96075i −0.197150 − 0.980373i \(-0.563168\pi\)
0.197150 − 0.980373i \(-0.436832\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 484.114 1.12323 0.561617 − 0.827397i \(-0.310179\pi\)
0.561617 + 0.827397i \(0.310179\pi\)
\(432\) 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
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 867.486i 1.95821i 0.203361 + 0.979104i \(0.434813\pi\)
−0.203361 + 0.979104i \(0.565187\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 572.287i 1.27458i 0.770624 + 0.637291i \(0.219945\pi\)
−0.770624 + 0.637291i \(0.780055\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 253.992 0.555781 0.277891 − 0.960613i \(-0.410365\pi\)
0.277891 + 0.960613i \(0.410365\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) − 634.980i − 1.37145i −0.727862 − 0.685724i \(-0.759486\pi\)
0.727862 − 0.685724i \(-0.240514\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) − 826.000i − 1.76119i
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 680.952i 1.43965i
\(474\) 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
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 398.000i 0.817248i 0.912703 + 0.408624i \(0.133991\pi\)
−0.912703 + 0.408624i \(0.866009\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 509.947i 1.03859i 0.854596 + 0.519294i \(0.173805\pi\)
−0.854596 + 0.519294i \(0.826195\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 145.206i − 0.292164i
\(498\) 0 0
\(499\) −952.470 −1.90876 −0.954379 − 0.298597i \(-0.903481\pi\)
−0.954379 + 0.298597i \(0.903481\pi\)
\(500\) 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
\(507\) 0 0
\(508\) 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
\(521\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 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
\(529\) 1290.98 2.44041
\(530\) 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
\(539\) − 525.475i − 0.974908i
\(540\) 0 0
\(541\) − 1079.47i − 1.99532i −0.0683919 − 0.997659i \(-0.521787\pi\)
0.0683919 − 0.997659i \(-0.478213\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 842.000 1.53931 0.769653 − 0.638463i \(-0.220429\pi\)
0.769653 + 0.638463i \(0.220429\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −888.972 −1.60755
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1052.75 1.89004 0.945021 − 0.327009i \(-0.106041\pi\)
0.945021 + 0.327009i \(0.106041\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 1120.98i − 1.97009i −0.172306 − 0.985044i \(-0.555122\pi\)
0.172306 − 0.985044i \(-0.444878\pi\)
\(570\) 0 0
\(571\) 1126.00 1.97198 0.985989 − 0.166807i \(-0.0533458\pi\)
0.985989 + 0.166807i \(0.0533458\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1066.53 1.85483
\(576\) 0 0
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) − 757.012i − 1.29848i
\(584\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 715.095 1.19381 0.596907 − 0.802310i \(-0.296396\pi\)
0.596907 + 0.802310i \(0.296396\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 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
\(613\) − 1206.46i − 1.96813i −0.177814 − 0.984064i \(-0.556903\pi\)
0.177814 − 0.984064i \(-0.443097\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 383.699i 0.621879i 0.950430 + 0.310939i \(0.100644\pi\)
−0.950430 + 0.310939i \(0.899356\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 625.000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 761.976i 1.20757i 0.797147 + 0.603785i \(0.206341\pi\)
−0.797147 + 0.603785i \(0.793659\pi\)
\(632\) 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
\(641\) 1278.19i 1.99406i 0.0770186 + 0.997030i \(0.475460\pi\)
−0.0770186 + 0.997030i \(0.524540\pi\)
\(642\) 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
\(651\) 0 0
\(652\) 0 0
\(653\) 470.120 0.719938 0.359969 − 0.932964i \(-0.382787\pi\)
0.359969 + 0.932964i \(0.382787\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) − 1260.16i − 1.91223i −0.292999 − 0.956113i \(-0.594653\pi\)
0.292999 − 0.956113i \(-0.405347\pi\)
\(660\) 0 0
\(661\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −2267.46 −3.39949
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1269.96 −1.88701 −0.943507 − 0.331352i \(-0.892495\pi\)
−0.943507 + 0.331352i \(0.892495\pi\)
\(674\) 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
\(683\) 1140.68i 1.67010i 0.550178 + 0.835048i \(0.314560\pi\)
−0.550178 + 0.835048i \(0.685440\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 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
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 913.702 1.30343 0.651713 − 0.758465i \(-0.274051\pi\)
0.651713 + 0.758465i \(0.274051\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 1382.00i − 1.94922i −0.223900 − 0.974612i \(-0.571879\pi\)
0.223900 − 0.974612i \(-0.428121\pi\)
\(710\) 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
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −1328.76 −1.83277
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 1265.43i − 1.71700i
\(738\) 0 0
\(739\) 1226.00 1.65900 0.829499 − 0.558508i \(-0.188626\pi\)
0.829499 + 0.558508i \(0.188626\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 967.054 1.30155 0.650777 − 0.759269i \(-0.274443\pi\)
0.650777 + 0.759269i \(0.274443\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 396.821 0.529801
\(750\) 0 0
\(751\) − 802.000i − 1.06791i −0.845513 − 0.533955i \(-0.820705\pi\)
0.845513 − 0.533955i \(-0.179295\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) − 1402.00i − 1.85205i −0.377465 − 0.926024i \(-0.623204\pi\)
0.377465 − 0.926024i \(-0.376796\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) −742.000 −0.972477
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 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
\(781\) − 222.455i − 0.284833i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1308.38 1.65408
\(792\) 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
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 1014.43i − 1.25393i −0.779048 − 0.626964i \(-0.784297\pi\)
0.779048 − 0.626964i \(-0.215703\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 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
\(821\) −1641.99 −1.99999 −0.999997 − 0.00264006i \(-0.999160\pi\)
−0.999997 + 0.00264006i \(0.999160\pi\)
\(822\) 0 0
\(823\) − 1523.95i − 1.85170i −0.377886 − 0.925852i \(-0.623349\pi\)
0.377886 − 0.925852i \(-0.376651\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 953.026i − 1.15239i −0.817312 − 0.576195i \(-0.804537\pi\)
0.817312 − 0.576195i \(-0.195463\pi\)
\(828\) 0 0
\(829\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1983.96 2.35905
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 41.9724i 0.0495542i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 1621.12i 1.90496i
\(852\) 0 0
\(853\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 1504.48 1.74331 0.871655 − 0.490119i \(-0.163047\pi\)
0.871655 + 0.490119i \(0.163047\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −1361.90 −1.56721
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1587.45i 1.81009i 0.425314 + 0.905046i \(0.360164\pi\)
−0.425314 + 0.905046i \(0.639836\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 698.478 0.791029 0.395514 − 0.918460i \(-0.370566\pi\)
0.395514 + 0.918460i \(0.370566\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −1777.94 −1.99994
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 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
\(907\) 317.490 0.350044 0.175022 − 0.984564i \(-0.444000\pi\)
0.175022 + 0.984564i \(0.444000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −448.798 −0.492643 −0.246321 − 0.969188i \(-0.579222\pi\)
−0.246321 + 0.969188i \(0.579222\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 466.000i 0.507073i 0.967326 + 0.253536i \(0.0815938\pi\)
−0.967326 + 0.253536i \(0.918406\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 950.000i 1.02703i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 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
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 233.253i − 0.246307i −0.992388 − 0.123154i \(-0.960699\pi\)
0.992388 − 0.123154i \(-0.0393008\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1899.03i 1.99268i 0.0854655 + 0.996341i \(0.472762\pi\)
−0.0854655 + 0.996341i \(0.527238\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) −186.408 −0.194377
\(960\) 0 0
\(961\) −961.000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 1904.94i − 1.96995i −0.172699 − 0.984975i \(-0.555249\pi\)
0.172699 − 0.984975i \(-0.444751\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1262.38i 1.29210i 0.763296 + 0.646048i \(0.223580\pi\)
−0.763296 + 0.646048i \(0.776420\pi\)
\(978\) 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
\(989\) 2708.90 2.73903
\(990\) 0 0
\(991\) 1406.00i 1.41877i 0.704822 + 0.709384i \(0.251027\pi\)
−0.704822 + 0.709384i \(0.748973\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) 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 2016.3.e.b.1007.7 8
3.2 odd 2 inner 2016.3.e.b.1007.6 8
4.3 odd 2 504.3.e.b.251.3 ✓ 8
7.6 odd 2 CM 2016.3.e.b.1007.7 8
8.3 odd 2 inner 2016.3.e.b.1007.3 8
8.5 even 2 504.3.e.b.251.5 yes 8
12.11 even 2 504.3.e.b.251.6 yes 8
21.20 even 2 inner 2016.3.e.b.1007.6 8
24.5 odd 2 504.3.e.b.251.4 yes 8
24.11 even 2 inner 2016.3.e.b.1007.2 8
28.27 even 2 504.3.e.b.251.3 ✓ 8
56.13 odd 2 504.3.e.b.251.5 yes 8
56.27 even 2 inner 2016.3.e.b.1007.3 8
84.83 odd 2 504.3.e.b.251.6 yes 8
168.83 odd 2 inner 2016.3.e.b.1007.2 8
168.125 even 2 504.3.e.b.251.4 yes 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
504.3.e.b.251.3 ✓ 8 4.3 odd 2
504.3.e.b.251.3 ✓ 8 28.27 even 2
504.3.e.b.251.4 yes 8 24.5 odd 2
504.3.e.b.251.4 yes 8 168.125 even 2
504.3.e.b.251.5 yes 8 8.5 even 2
504.3.e.b.251.5 yes 8 56.13 odd 2
504.3.e.b.251.6 yes 8 12.11 even 2
504.3.e.b.251.6 yes 8 84.83 odd 2
2016.3.e.b.1007.2 8 24.11 even 2 inner
2016.3.e.b.1007.2 8 168.83 odd 2 inner
2016.3.e.b.1007.3 8 8.3 odd 2 inner
2016.3.e.b.1007.3 8 56.27 even 2 inner
2016.3.e.b.1007.6 8 3.2 odd 2 inner
2016.3.e.b.1007.6 8 21.20 even 2 inner
2016.3.e.b.1007.7 8 1.1 even 1 trivial
2016.3.e.b.1007.7 8 7.6 odd 2 CM