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Properties

Label 2304.3.h.e.2177.4
Level $2304$
Weight $3$
Character 2304.2177
Analytic conductor $62.779$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2304,3,Mod(2177,2304)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2304, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("2304.2177");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2304 = 2^{8} \cdot 3^{2} \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 2304.h (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: \(62.7794529086\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{29}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: no (minimal twist has level 288)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 2177.4
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 2304.2177
Dual form 2304.3.h.e.2177.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+9.89949 q^{5} +O(q^{10})\) \(q+9.89949 q^{5} +24.0000i q^{13} +9.89949i q^{17} +73.0000 q^{25} -57.9828 q^{29} +70.0000i q^{37} +43.8406i q^{41} -49.0000 q^{49} +24.0416 q^{53} +22.0000i q^{61} +237.588i q^{65} -96.0000 q^{73} +98.0000i q^{85} -57.9828i q^{89} +144.000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q+O(q^{10}) \) Copy content Toggle raw display \( 4 q + 292 q^{25} - 196 q^{49} - 384 q^{73} + 576 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2304\mathbb{Z}\right)^\times\).

\(n\) \(1279\) \(1793\) \(2053\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 0 0
\(4\) 0 0
\(5\) 9.89949 1.97990 0.989949 − 0.141421i \(-0.0451672\pi\)
0.989949 + 0.141421i \(0.0451672\pi\)
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 24.0000i 1.84615i 0.384615 + 0.923077i \(0.374334\pi\)
−0.384615 + 0.923077i \(0.625666\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 9.89949i 0.582323i 0.956674 + 0.291162i \(0.0940417\pi\)
−0.956674 + 0.291162i \(0.905958\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 73.0000 2.92000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) −57.9828 −1.99941 −0.999703 − 0.0243830i \(-0.992238\pi\)
−0.999703 + 0.0243830i \(0.992238\pi\)
\(30\) 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
\(37\) 70.0000i 1.89189i 0.324324 + 0.945946i \(0.394863\pi\)
−0.324324 + 0.945946i \(0.605137\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 43.8406i 1.06928i 0.845079 + 0.534642i \(0.179553\pi\)
−0.845079 + 0.534642i \(0.820447\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 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
\(53\) 24.0416 0.453616 0.226808 − 0.973940i \(-0.427171\pi\)
0.226808 + 0.973940i \(0.427171\pi\)
\(54\) 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
\(61\) 22.0000i 0.360656i 0.983607 + 0.180328i \(0.0577159\pi\)
−0.983607 + 0.180328i \(0.942284\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 237.588i 3.65520i
\(66\) 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
\(73\) −96.0000 −1.31507 −0.657534 − 0.753425i \(-0.728401\pi\)
−0.657534 + 0.753425i \(0.728401\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 98.0000i 1.15294i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) − 57.9828i − 0.651492i −0.945457 − 0.325746i \(-0.894385\pi\)
0.945457 − 0.325746i \(-0.105615\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 144.000 1.48454 0.742268 − 0.670103i \(-0.233750\pi\)
0.742268 + 0.670103i \(0.233750\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 111.723 1.10617 0.553084 − 0.833126i \(-0.313451\pi\)
0.553084 + 0.833126i \(0.313451\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) − 120.000i − 1.10092i −0.834862 − 0.550459i \(-0.814453\pi\)
0.834862 − 0.550459i \(-0.185547\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 179.605i − 1.58943i −0.606985 − 0.794713i \(-0.707621\pi\)
0.606985 − 0.794713i \(-0.292379\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −121.000 −1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 475.176 3.80141
\(126\) 0 0
\(127\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 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
\(137\) 24.0416i 0.175486i 0.996143 + 0.0877432i \(0.0279655\pi\)
−0.996143 + 0.0877432i \(0.972035\pi\)
\(138\) 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
\(145\) −574.000 −3.95862
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 125.865 0.844732 0.422366 − 0.906425i \(-0.361200\pi\)
0.422366 + 0.906425i \(0.361200\pi\)
\(150\) 0 0
\(151\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 170.000i 1.08280i 0.840764 + 0.541401i \(0.182106\pi\)
−0.840764 + 0.541401i \(0.817894\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −407.000 −2.40828
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 159.806 0.923735 0.461867 − 0.886949i \(-0.347180\pi\)
0.461867 + 0.886949i \(0.347180\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) − 360.000i − 1.98895i −0.104972 − 0.994475i \(-0.533475\pi\)
0.104972 − 0.994475i \(-0.466525\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 692.965i 3.74575i
\(186\) 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
\(193\) 190.000 0.984456 0.492228 − 0.870466i \(-0.336183\pi\)
0.492228 + 0.870466i \(0.336183\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 315.370 1.60086 0.800431 − 0.599426i \(-0.204604\pi\)
0.800431 + 0.599426i \(0.204604\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 434.000i 2.11707i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 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
\(221\) −237.588 −1.07506
\(222\) 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
\(229\) − 120.000i − 0.524017i −0.965066 − 0.262009i \(-0.915615\pi\)
0.965066 − 0.262009i \(-0.0843849\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 145.664i − 0.625167i −0.949890 − 0.312584i \(-0.898806\pi\)
0.949890 − 0.312584i \(-0.101194\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −240.000 −0.995851 −0.497925 − 0.867220i \(-0.665905\pi\)
−0.497925 + 0.867220i \(0.665905\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −485.075 −1.97990
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 315.370i 1.22712i 0.789648 + 0.613560i \(0.210263\pi\)
−0.789648 + 0.613560i \(0.789737\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 238.000 0.898113
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 465.276 1.72965 0.864826 − 0.502072i \(-0.167429\pi\)
0.864826 + 0.502072i \(0.167429\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 504.000i 1.81949i 0.415162 + 0.909747i \(0.363725\pi\)
−0.415162 + 0.909747i \(0.636275\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 552.958i 1.96782i 0.178664 + 0.983910i \(0.442822\pi\)
−0.178664 + 0.983910i \(0.557178\pi\)
\(282\) 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
\(289\) 191.000 0.660900
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 499.217 1.70381 0.851907 − 0.523693i \(-0.175446\pi\)
0.851907 + 0.523693i \(0.175446\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 217.789i 0.714062i
\(306\) 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
\(313\) −50.0000 −0.159744 −0.0798722 − 0.996805i \(-0.525451\pi\)
−0.0798722 + 0.996805i \(0.525451\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 329.512 1.03947 0.519735 − 0.854328i \(-0.326031\pi\)
0.519735 + 0.854328i \(0.326031\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1752.00i 5.39077i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 576.000 1.70920 0.854599 − 0.519288i \(-0.173803\pi\)
0.854599 + 0.519288i \(0.173803\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 598.000i 1.71347i 0.515759 + 0.856734i \(0.327510\pi\)
−0.515759 + 0.856734i \(0.672490\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 702.864i 1.99112i 0.0941474 + 0.995558i \(0.469988\pi\)
−0.0941474 + 0.995558i \(0.530012\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 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
\(365\) −950.352 −2.60370
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) − 550.000i − 1.47453i −0.675603 − 0.737265i \(-0.736117\pi\)
0.675603 − 0.737265i \(-0.263883\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 1391.59i − 3.69121i
\(378\) 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
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −213.546 −0.548962 −0.274481 − 0.961593i \(-0.588506\pi\)
−0.274481 + 0.961593i \(0.588506\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 650.000i − 1.63728i −0.574307 − 0.818640i \(-0.694729\pi\)
0.574307 − 0.818640i \(-0.305271\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 620.840i 1.54823i 0.633046 + 0.774114i \(0.281805\pi\)
−0.633046 + 0.774114i \(0.718195\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −240.000 −0.586797 −0.293399 − 0.955990i \(-0.594786\pi\)
−0.293399 + 0.955990i \(0.594786\pi\)
\(410\) 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
\(421\) − 840.000i − 1.99525i −0.0688836 − 0.997625i \(-0.521944\pi\)
0.0688836 − 0.997625i \(-0.478056\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 722.663i 1.70038i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −290.000 −0.669746 −0.334873 − 0.942263i \(-0.608693\pi\)
−0.334873 + 0.942263i \(0.608693\pi\)
\(434\) 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
\(443\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) − 574.000i − 1.28989i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 892.369i − 1.98746i −0.111814 − 0.993729i \(-0.535666\pi\)
0.111814 − 0.993729i \(-0.464334\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −336.000 −0.735230 −0.367615 − 0.929978i \(-0.619826\pi\)
−0.367615 + 0.929978i \(0.619826\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −906.511 −1.96640 −0.983201 − 0.182529i \(-0.941572\pi\)
−0.983201 + 0.182529i \(0.941572\pi\)
\(462\) 0 0
\(463\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 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
\(481\) −1680.00 −3.49272
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 1425.53 2.93923
\(486\) 0 0
\(487\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) − 574.000i − 1.16430i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 1106.00 2.19010
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 960.251 1.88654 0.943272 − 0.332021i \(-0.107730\pi\)
0.943272 + 0.332021i \(0.107730\pi\)
\(510\) 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
\(521\) 227.688i 0.437022i 0.975835 + 0.218511i \(0.0701199\pi\)
−0.975835 + 0.218511i \(0.929880\pi\)
\(522\) 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
\(529\) 529.000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) −1052.17 −1.97406
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 840.000i 1.55268i 0.630314 + 0.776340i \(0.282926\pi\)
−0.630314 + 0.776340i \(0.717074\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 1187.94i − 2.17971i
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 519.016 0.931807 0.465903 − 0.884836i \(-0.345729\pi\)
0.465903 + 0.884836i \(0.345729\pi\)
\(558\) 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
\(565\) − 1778.00i − 3.14690i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) − 1062.07i − 1.86656i −0.359146 − 0.933282i \(-0.616932\pi\)
0.359146 − 0.933282i \(-0.383068\pi\)
\(570\) 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
\(577\) −1150.00 −1.99307 −0.996534 − 0.0831889i \(-0.973490\pi\)
−0.996534 + 0.0831889i \(0.973490\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) − 1178.04i − 1.98658i −0.115665 − 0.993288i \(-0.536900\pi\)
0.115665 − 0.993288i \(-0.463100\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1102.00 −1.83361 −0.916805 − 0.399334i \(-0.869241\pi\)
−0.916805 + 0.399334i \(0.869241\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −1197.84 −1.97990
\(606\) 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
\(613\) − 70.0000i − 0.114192i −0.998369 − 0.0570962i \(-0.981816\pi\)
0.998369 − 0.0570962i \(-0.0181842\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 1008.33i − 1.63425i −0.576458 − 0.817127i \(-0.695566\pi\)
0.576458 − 0.817127i \(-0.304434\pi\)
\(618\) 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
\(625\) 2879.00 4.60640
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) −692.965 −1.10169
\(630\) 0 0
\(631\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 1176.00i − 1.84615i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 1144.10i − 1.78487i −0.451180 − 0.892433i \(-0.648997\pi\)
0.451180 − 0.892433i \(-0.351003\pi\)
\(642\) 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
\(653\) 363.453 0.556589 0.278295 − 0.960496i \(-0.410231\pi\)
0.278295 + 0.960496i \(0.410231\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) − 1178.00i − 1.78215i −0.453858 − 0.891074i \(-0.649953\pi\)
0.453858 − 0.891074i \(-0.350047\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 770.000 1.14413 0.572065 − 0.820208i \(-0.306142\pi\)
0.572065 + 0.820208i \(0.306142\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 1028.13 1.51866 0.759330 − 0.650705i \(-0.225527\pi\)
0.759330 + 0.650705i \(0.225527\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 238.000i 0.347445i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 576.999i 0.837444i
\(690\) 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
\(697\) −434.000 −0.622669
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 552.958 0.788812 0.394406 − 0.918936i \(-0.370950\pi\)
0.394406 + 0.918936i \(0.370950\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1320.00i 1.86178i 0.365303 + 0.930889i \(0.380965\pi\)
−0.365303 + 0.930889i \(0.619035\pi\)
\(710\) 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
\(725\) −4232.74 −5.83826
\(726\) 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
\(733\) 216.000i 0.294679i 0.989086 + 0.147340i \(0.0470711\pi\)
−0.989086 + 0.147340i \(0.952929\pi\)
\(734\) 0 0
\(735\) 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
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 1246.00 1.67248
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 936.000i 1.23646i 0.785997 + 0.618230i \(0.212150\pi\)
−0.785997 + 0.618230i \(0.787850\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1129.96i 1.48483i 0.669940 + 0.742416i \(0.266320\pi\)
−0.669940 + 0.742416i \(0.733680\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 962.000 1.25098 0.625488 − 0.780234i \(-0.284900\pi\)
0.625488 + 0.780234i \(0.284900\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1333.60 −1.72523 −0.862615 − 0.505860i \(-0.831175\pi\)
−0.862615 + 0.505860i \(0.831175\pi\)
\(774\) 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
\(785\) 1682.91i 2.14384i
\(786\) 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
\(793\) −528.000 −0.665826
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 24.0416 0.0301652 0.0150826 − 0.999886i \(-0.495199\pi\)
0.0150826 + 0.999886i \(0.495199\pi\)
\(798\) 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
\(809\) 1469.37i 1.81628i 0.418670 + 0.908138i \(0.362496\pi\)
−0.418670 + 0.908138i \(0.637504\pi\)
\(810\) 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
\(821\) 383.252 0.466811 0.233406 − 0.972379i \(-0.425013\pi\)
0.233406 + 0.972379i \(0.425013\pi\)
\(822\) 0 0
\(823\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) − 1080.00i − 1.30277i −0.758745 − 0.651387i \(-0.774187\pi\)
0.758745 − 0.651387i \(-0.225813\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) − 485.075i − 0.582323i
\(834\) 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
\(841\) 2521.00 2.99762
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) −4029.09 −4.76816
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) − 410.000i − 0.480657i −0.970692 − 0.240328i \(-0.922745\pi\)
0.970692 − 0.240328i \(-0.0772551\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 838.629i − 0.978563i −0.872126 − 0.489282i \(-0.837259\pi\)
0.872126 − 0.489282i \(-0.162741\pi\)
\(858\) 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
\(865\) 1582.00 1.82890
\(866\) 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
\(875\) 0 0
\(876\) 0 0
\(877\) − 1610.00i − 1.83580i −0.396807 − 0.917902i \(-0.629882\pi\)
0.396807 − 0.917902i \(-0.370118\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) − 1653.22i − 1.87652i −0.345929 − 0.938261i \(-0.612436\pi\)
0.345929 − 0.938261i \(-0.387564\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 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
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 238.000i 0.264151i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) − 3563.82i − 3.93792i
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 5110.00i 5.52432i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) − 1483.51i − 1.59689i −0.602068 − 0.798445i \(-0.705657\pi\)
0.602068 − 0.798445i \(-0.294343\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −430.000 −0.458911 −0.229456 − 0.973319i \(-0.573695\pi\)
−0.229456 + 0.973319i \(0.573695\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −227.688 −0.241964 −0.120982 − 0.992655i \(-0.538604\pi\)
−0.120982 + 0.992655i \(0.538604\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) − 2304.00i − 2.42782i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 159.806i 0.167687i 0.996479 + 0.0838437i \(0.0267197\pi\)
−0.996479 + 0.0838437i \(0.973280\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −961.000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 1880.90 1.94912
\(966\) 0 0
\(967\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 1687.16i 1.72687i 0.504456 + 0.863437i \(0.331693\pi\)
−0.504456 + 0.863437i \(0.668307\pi\)
\(978\) 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
\(985\) 3122.00 3.16954
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1850.00i 1.85557i 0.373119 + 0.927783i \(0.378288\pi\)
−0.373119 + 0.927783i \(0.621712\pi\)
\(998\) 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 2304.3.h.e.2177.4 4
3.2 odd 2 inner 2304.3.h.e.2177.2 4
4.3 odd 2 CM 2304.3.h.e.2177.4 4
8.3 odd 2 inner 2304.3.h.e.2177.1 4
8.5 even 2 inner 2304.3.h.e.2177.1 4
12.11 even 2 inner 2304.3.h.e.2177.2 4
16.3 odd 4 288.3.e.b.161.1 ✓ 2
16.5 even 4 576.3.e.e.449.2 2
16.11 odd 4 576.3.e.e.449.2 2
16.13 even 4 288.3.e.b.161.1 ✓ 2
24.5 odd 2 inner 2304.3.h.e.2177.3 4
24.11 even 2 inner 2304.3.h.e.2177.3 4
48.5 odd 4 576.3.e.e.449.1 2
48.11 even 4 576.3.e.e.449.1 2
48.29 odd 4 288.3.e.b.161.2 yes 2
48.35 even 4 288.3.e.b.161.2 yes 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
288.3.e.b.161.1 ✓ 2 16.3 odd 4
288.3.e.b.161.1 ✓ 2 16.13 even 4
288.3.e.b.161.2 yes 2 48.29 odd 4
288.3.e.b.161.2 yes 2 48.35 even 4
576.3.e.e.449.1 2 48.5 odd 4
576.3.e.e.449.1 2 48.11 even 4
576.3.e.e.449.2 2 16.5 even 4
576.3.e.e.449.2 2 16.11 odd 4
2304.3.h.e.2177.1 4 8.3 odd 2 inner
2304.3.h.e.2177.1 4 8.5 even 2 inner
2304.3.h.e.2177.2 4 3.2 odd 2 inner
2304.3.h.e.2177.2 4 12.11 even 2 inner
2304.3.h.e.2177.3 4 24.5 odd 2 inner
2304.3.h.e.2177.3 4 24.11 even 2 inner
2304.3.h.e.2177.4 4 1.1 even 1 trivial
2304.3.h.e.2177.4 4 4.3 odd 2 CM