Properties

Label 2151.1.d.e.955.2
Level $2151$
Weight $1$
Character 2151.955
Self dual yes
Analytic conductor $1.073$
Analytic rank $0$
Dimension $4$
Projective image $D_{15}$
CM discriminant -239
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] = [2151,1,Mod(955,2151)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2151, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2151.955");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2151.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(1.07348884217\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{15})^+\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} - x^{3} - 4x^{2} + 4x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 239)
Projective image: \(D_{15}\)
Projective field: Galois closure of 15.1.44543599279432079.1

Embedding invariants

Embedding label 955.2
Root \(1.33826\) of defining polynomial
Character \(\chi\) \(=\) 2151.955

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.33826 q^{2} +0.790943 q^{4} +1.95630 q^{5} +0.279773 q^{8} +O(q^{10})\) \(q-1.33826 q^{2} +0.790943 q^{4} +1.95630 q^{5} +0.279773 q^{8} -2.61803 q^{10} -1.82709 q^{11} -1.16535 q^{16} +1.00000 q^{17} +1.54732 q^{20} +2.44512 q^{22} +2.82709 q^{25} +0.209057 q^{29} +1.33826 q^{31} +1.27977 q^{32} -1.33826 q^{34} +0.547318 q^{40} -1.44512 q^{44} +1.00000 q^{49} -3.78339 q^{50} -3.57433 q^{55} -0.279773 q^{58} -1.00000 q^{61} -1.79094 q^{62} -0.547318 q^{64} -1.00000 q^{67} +0.790943 q^{68} +1.61803 q^{71} -2.27977 q^{80} +0.209057 q^{83} +1.95630 q^{85} -0.511170 q^{88} -1.33826 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - q^{2} + 5 q^{4} - q^{5} + q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - q^{2} + 5 q^{4} - q^{5} + q^{8} - 6 q^{10} - q^{11} + 6 q^{16} + 4 q^{17} - q^{22} + 5 q^{25} - q^{29} + q^{31} + 5 q^{32} - q^{34} - 4 q^{40} + 5 q^{44} + 4 q^{49} - q^{55} - q^{58} - 4 q^{61} - 9 q^{62} + 4 q^{64} - 4 q^{67} + 5 q^{68} + 2 q^{71} - 9 q^{80} - q^{83} - q^{85} - 4 q^{88} - q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2151\mathbb{Z}\right)^\times\).

\(n\) \(479\) \(1441\)
\(\chi(n)\) \(1\) \(-1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(3\) 0 0
\(4\) 0.790943 0.790943
\(5\) 1.95630 1.95630 0.978148 0.207912i \(-0.0666667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0.279773 0.279773
\(9\) 0 0
\(10\) −2.61803 −2.61803
\(11\) −1.82709 −1.82709 −0.913545 0.406737i \(-0.866667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −1.16535 −1.16535
\(17\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 1.54732 1.54732
\(21\) 0 0
\(22\) 2.44512 2.44512
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 2.82709 2.82709
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0.209057 0.209057 0.104528 0.994522i \(-0.466667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(30\) 0 0
\(31\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(32\) 1.27977 1.27977
\(33\) 0 0
\(34\) −1.33826 −1.33826
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.547318 0.547318
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) −1.44512 −1.44512
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 1.00000 1.00000
\(50\) −3.78339 −3.78339
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) −3.57433 −3.57433
\(56\) 0 0
\(57\) 0 0
\(58\) −0.279773 −0.279773
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(62\) −1.79094 −1.79094
\(63\) 0 0
\(64\) −0.547318 −0.547318
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0.790943 0.790943
\(69\) 0 0
\(70\) 0 0
\(71\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(72\) 0 0
\(73\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) −2.27977 −2.27977
\(81\) 0 0
\(82\) 0 0
\(83\) 0.209057 0.209057 0.104528 0.994522i \(-0.466667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(84\) 0 0
\(85\) 1.95630 1.95630
\(86\) 0 0
\(87\) 0 0
\(88\) −0.511170 −0.511170
\(89\) 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
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −1.33826 −1.33826
\(99\) 0 0
\(100\) 2.23607 2.23607
\(101\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(110\) 4.78339 4.78339
\(111\) 0 0
\(112\) 0 0
\(113\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.165352 0.165352
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.33826 2.33826
\(122\) 1.33826 1.33826
\(123\) 0 0
\(124\) 1.05849 1.05849
\(125\) 3.57433 3.57433
\(126\) 0 0
\(127\) 1.82709 1.82709 0.913545 0.406737i \(-0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(128\) −0.547318 −0.547318
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.33826 1.33826
\(135\) 0 0
\(136\) 0.279773 0.279773
\(137\) 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
\(142\) −2.16535 −2.16535
\(143\) 0 0
\(144\) 0 0
\(145\) 0.408977 0.408977
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 2.61803 2.61803
\(156\) 0 0
\(157\) −0.209057 −0.209057 −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 2.50361 2.50361
\(161\) 0 0
\(162\) 0 0
\(163\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −0.279773 −0.279773
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) −2.61803 −2.61803
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 2.12920 2.12920
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −1.82709 −1.82709
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.790943 0.790943
\(197\) −1.82709 −1.82709 −0.913545 0.406737i \(-0.866667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0.790943 0.790943
\(201\) 0 0
\(202\) 0.827091 0.827091
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −0.209057 −0.209057 −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 2.16535 2.16535
\(219\) 0 0
\(220\) −2.82709 −2.82709
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.827091 0.827091
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.0584884 0.0584884
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −1.00000 −1.00000
\(240\) 0 0
\(241\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(242\) −3.12920 −3.12920
\(243\) 0 0
\(244\) −0.790943 −0.790943
\(245\) 1.95630 1.95630
\(246\) 0 0
\(247\) 0 0
\(248\) 0.374409 0.374409
\(249\) 0 0
\(250\) −4.78339 −4.78339
\(251\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −2.44512 −2.44512
\(255\) 0 0
\(256\) 1.27977 1.27977
\(257\) −1.82709 −1.82709 −0.913545 0.406737i \(-0.866667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0.209057 0.209057 0.104528 0.994522i \(-0.466667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −0.790943 −0.790943
\(269\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(270\) 0 0
\(271\) 1.82709 1.82709 0.913545 0.406737i \(-0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(272\) −1.16535 −1.16535
\(273\) 0 0
\(274\) 0 0
\(275\) −5.16535 −5.16535
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(284\) 1.27977 1.27977
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0 0
\(290\) −0.547318 −0.547318
\(291\) 0 0
\(292\) 0 0
\(293\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(294\) 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
\(305\) −1.95630 −1.95630
\(306\) 0 0
\(307\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −3.50361 −3.50361
\(311\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0.279773 0.279773
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) −0.381966 −0.381966
\(320\) −1.07072 −1.07072
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 2.61803 2.61803
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0.165352 0.165352
\(333\) 0 0
\(334\) 0 0
\(335\) −1.95630 −1.95630
\(336\) 0 0
\(337\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(338\) −1.33826 −1.33826
\(339\) 0 0
\(340\) 1.54732 1.54732
\(341\) −2.44512 −2.44512
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(348\) 0 0
\(349\) 1.82709 1.82709 0.913545 0.406737i \(-0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −2.33826 −2.33826
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 3.16535 3.16535
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −0.209057 −0.209057 −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(374\) 2.44512 2.44512
\(375\) 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
\(383\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.61803 2.61803
\(387\) 0 0
\(388\) 0 0
\(389\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.279773 0.279773
\(393\) 0 0
\(394\) 2.44512 2.44512
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −3.29456 −3.29456
\(401\) 0.209057 0.209057 0.104528 0.994522i \(-0.466667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.488830 −0.488830
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0.408977 0.408977
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0.209057 0.209057 0.104528 0.994522i \(-0.466667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(420\) 0 0
\(421\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(422\) 0.279773 0.279773
\(423\) 0 0
\(424\) 0 0
\(425\) 2.82709 2.82709
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.95630 1.95630 0.978148 0.207912i \(-0.0666667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.27977 −1.27977
\(437\) 0 0
\(438\) 0 0
\(439\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(440\) −1.00000 −1.00000
\(441\) 0 0
\(442\) 0 0
\(443\) 1.95630 1.95630 0.978148 0.207912i \(-0.0666667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −0.488830 −0.488830
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −0.243625 −0.243625
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 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
\(478\) 1.33826 1.33826
\(479\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −0.827091 −0.827091
\(483\) 0 0
\(484\) 1.84943 1.84943
\(485\) 0 0
\(486\) 0 0
\(487\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(488\) −0.279773 −0.279773
\(489\) 0 0
\(490\) −2.61803 −2.61803
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0.209057 0.209057
\(494\) 0 0
\(495\) 0 0
\(496\) −1.55955 −1.55955
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 2.82709 2.82709
\(501\) 0 0
\(502\) −1.33826 −1.33826
\(503\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(504\) 0 0
\(505\) −1.20906 −1.20906
\(506\) 0 0
\(507\) 0 0
\(508\) 1.44512 1.44512
\(509\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.16535 −1.16535
\(513\) 0 0
\(514\) 2.44512 2.44512
\(515\) 0 0
\(516\) 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
\(523\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −0.279773 −0.279773
\(527\) 1.33826 1.33826
\(528\) 0 0
\(529\) 1.00000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) −0.279773 −0.279773
\(537\) 0 0
\(538\) 1.79094 1.79094
\(539\) −1.82709 −1.82709
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) −2.44512 −2.44512
\(543\) 0 0
\(544\) 1.27977 1.27977
\(545\) −3.16535 −3.16535
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 6.91259 6.91259
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) −1.20906 −1.20906
\(566\) −1.79094 −1.79094
\(567\) 0 0
\(568\) 0.452682 0.452682
\(569\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(570\) 0 0
\(571\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0.323478 0.323478
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −2.16535 −2.16535
\(587\) 1.95630 1.95630 0.978148 0.207912i \(-0.0666667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 4.57433 4.57433
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 2.61803 2.61803
\(611\) 0 0
\(612\) 0 0
\(613\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(614\) 2.61803 2.61803
\(615\) 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 \(\pi\)
\(620\) 2.07072 2.07072
\(621\) 0 0
\(622\) −1.33826 −1.33826
\(623\) 0 0
\(624\) 0 0
\(625\) 4.16535 4.16535
\(626\) 0 0
\(627\) 0 0
\(628\) −0.165352 −0.165352
\(629\) 0 0
\(630\) 0 0
\(631\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 3.57433 3.57433
\(636\) 0 0
\(637\) 0 0
\(638\) 0.511170 0.511170
\(639\) 0 0
\(640\) −1.07072 −1.07072
\(641\) −1.82709 −1.82709 −0.913545 0.406737i \(-0.866667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(642\) 0 0
\(643\) 1.82709 1.82709 0.913545 0.406737i \(-0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −1.54732 −1.54732
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 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
\(661\) 1.82709 1.82709 0.913545 0.406737i \(-0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0.0584884 0.0584884
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 2.61803 2.61803
\(671\) 1.82709 1.82709
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 2.61803 2.61803
\(675\) 0 0
\(676\) 0.790943 0.790943
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.547318 0.547318
\(681\) 0 0
\(682\) 3.27222 3.27222
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.79094 1.79094
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −2.44512 −2.44512
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.00000 1.00000
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) −4.23607 −4.23607
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0.827091 0.827091
\(719\) 1.95630 1.95630 0.978148 0.207912i \(-0.0666667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.33826 −1.33826
\(723\) 0 0
\(724\) 0 0
\(725\) 0.591023 0.591023
\(726\) 0 0
\(727\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(734\) 0.279773 0.279773
\(735\) 0 0
\(736\) 0 0
\(737\) 1.82709 1.82709
\(738\) 0 0
\(739\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.16535 2.16535
\(747\) 0 0
\(748\) −1.44512 −1.44512
\(749\) 0 0
\(750\) 0 0
\(751\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 2.67652 2.67652
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.54732 −1.54732
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 3.78339 3.78339
\(776\) 0 0
\(777\) 0 0
\(778\) 1.79094 1.79094
\(779\) 0 0
\(780\) 0 0
\(781\) −2.95630 −2.95630
\(782\) 0 0
\(783\) 0 0
\(784\) −1.16535 −1.16535
\(785\) −0.408977 −0.408977
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) −1.44512 −1.44512
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 3.61803 3.61803
\(801\) 0 0
\(802\) −0.279773 −0.279773
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −0.172909 −0.172909
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) −3.82709 −3.82709
\(816\) 0 0
\(817\) 0 0
\(818\) 2.61803 2.61803
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.209057 0.209057 0.104528 0.994522i \(-0.466667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) −0.547318 −0.547318
\(831\) 0 0
\(832\) 0 0
\(833\) 1.00000 1.00000
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −0.279773 −0.279773
\(839\) −1.82709 −1.82709 −0.913545 0.406737i \(-0.866667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(840\) 0 0
\(841\) −0.956295 −0.956295
\(842\) −0.827091 −0.827091
\(843\) 0 0
\(844\) −0.165352 −0.165352
\(845\) 1.95630 1.95630
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) −3.78339 −3.78339
\(851\) 0 0
\(852\) 0 0
\(853\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 1.82709 1.82709 0.913545 0.406737i \(-0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −2.61803 −2.61803
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −0.452682 −0.452682
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −0.209057 −0.209057 −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(878\) −0.827091 −0.827091
\(879\) 0 0
\(880\) 4.16535 4.16535
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −2.61803 −2.61803
\(887\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(888\) 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
\(899\) 0.279773 0.279773
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −0.172909 −0.172909
\(905\) 0 0
\(906\) 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
\(913\) −0.381966 −0.381966
\(914\) 2.16535 2.16535
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0.267545 0.267545
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) −3.57433 −3.57433
\(936\) 0 0
\(937\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(938\) 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
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 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
\(956\) −0.790943 −0.790943
\(957\) 0 0
\(958\) 1.79094 1.79094
\(959\) 0 0
\(960\) 0 0
\(961\) 0.790943 0.790943
\(962\) 0 0
\(963\) 0 0
\(964\) 0.488830 0.488830
\(965\) −3.82709 −3.82709
\(966\) 0 0
\(967\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(968\) 0.654182 0.654182
\(969\) 0 0
\(970\) 0 0
\(971\) 1.95630 1.95630 0.978148 0.207912i \(-0.0666667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1.33826 1.33826
\(975\) 0 0
\(976\) 1.16535 1.16535
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.54732 1.54732
\(981\) 0 0
\(982\) 0 0
\(983\) −1.82709 −1.82709 −0.913545 0.406737i \(-0.866667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(984\) 0 0
\(985\) −3.57433 −3.57433
\(986\) −0.279773 −0.279773
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 1.71267 1.71267
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 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 2151.1.d.e.955.2 4
3.2 odd 2 239.1.b.c.238.3 4
12.11 even 2 3824.1.h.c.3345.1 4
239.238 odd 2 CM 2151.1.d.e.955.2 4
717.716 even 2 239.1.b.c.238.3 4
2868.2867 odd 2 3824.1.h.c.3345.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
239.1.b.c.238.3 4 3.2 odd 2
239.1.b.c.238.3 4 717.716 even 2
2151.1.d.e.955.2 4 1.1 even 1 trivial
2151.1.d.e.955.2 4 239.238 odd 2 CM
3824.1.h.c.3345.1 4 12.11 even 2
3824.1.h.c.3345.1 4 2868.2867 odd 2