Properties

Label 1305.2.f.g.289.3
Level $1305$
Weight $2$
Character 1305.289
Analytic conductor $10.420$
Analytic rank $0$
Dimension $4$
CM discriminant -435
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1305,2,Mod(289,1305)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1305, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1305.289");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1305 = 3^{2} \cdot 5 \cdot 29 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1305.f (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: no
Analytic conductor: \(10.4204774638\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\sqrt{-5}, \sqrt{-29})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 17x^{2} + 36 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 289.3
Root \(3.81062i\) of defining polynomial
Character \(\chi\) \(=\) 1305.289
Dual form 1305.2.f.g.289.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{4} +2.23607i q^{5} -5.38516i q^{11} +4.00000 q^{16} -4.47214i q^{20} -2.23607i q^{23} -5.00000 q^{25} +5.38516i q^{29} +12.0416 q^{37} -5.38516i q^{41} +12.0416 q^{43} +10.7703i q^{44} +7.00000 q^{49} +11.1803i q^{53} +12.0416 q^{55} -8.00000 q^{64} +12.0416 q^{73} +8.94427i q^{80} -15.6525i q^{83} +10.7703i q^{89} +4.47214i q^{92} +12.0416 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4} + 16 q^{16} - 20 q^{25} + 28 q^{49} - 32 q^{64}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1305\mathbb{Z}\right)^\times\).

\(n\) \(146\) \(262\) \(901\)
\(\chi(n)\) \(1\) \(-1\) \(-1\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) − 5.38516i − 1.62369i −0.583874 0.811844i \(-0.698464\pi\)
0.583874 0.811844i \(-0.301536\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) − 4.47214i − 1.00000i
\(21\) 0 0
\(22\) 0 0
\(23\) − 2.23607i − 0.466252i −0.972446 0.233126i \(-0.925104\pi\)
0.972446 0.233126i \(-0.0748955\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 5.38516i 1.00000i
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 12.0416 1.97963 0.989813 0.142374i \(-0.0454735\pi\)
0.989813 + 0.142374i \(0.0454735\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) − 5.38516i − 0.841021i −0.907288 0.420511i \(-0.861851\pi\)
0.907288 0.420511i \(-0.138149\pi\)
\(42\) 0 0
\(43\) 12.0416 1.83633 0.918163 0.396203i \(-0.129672\pi\)
0.918163 + 0.396203i \(0.129672\pi\)
\(44\) 10.7703i 1.62369i
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 7.00000 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 11.1803i 1.53574i 0.640607 + 0.767869i \(0.278683\pi\)
−0.640607 + 0.767869i \(0.721317\pi\)
\(54\) 0 0
\(55\) 12.0416 1.62369
\(56\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 12.0416 1.40936 0.704681 0.709524i \(-0.251090\pi\)
0.704681 + 0.709524i \(0.251090\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 8.94427i 1.00000i
\(81\) 0 0
\(82\) 0 0
\(83\) − 15.6525i − 1.71808i −0.511906 0.859041i \(-0.671061\pi\)
0.511906 0.859041i \(-0.328939\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.7703i 1.14165i 0.821071 + 0.570826i \(0.193377\pi\)
−0.821071 + 0.570826i \(0.806623\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 4.47214i 0.466252i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 12.0416 1.22264 0.611319 0.791384i \(-0.290639\pi\)
0.611319 + 0.791384i \(0.290639\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 10.0000 1.00000
\(101\) − 5.38516i − 0.535844i −0.963441 0.267922i \(-0.913663\pi\)
0.963441 0.267922i \(-0.0863369\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 8.94427i − 0.864675i −0.901712 0.432338i \(-0.857689\pi\)
0.901712 0.432338i \(-0.142311\pi\)
\(108\) 0 0
\(109\) −1.00000 −0.0957826 −0.0478913 0.998853i \(-0.515250\pi\)
−0.0478913 + 0.998853i \(0.515250\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(114\) 0 0
\(115\) 5.00000 0.466252
\(116\) − 10.7703i − 1.00000i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −18.0000 −1.63636
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 11.1803i − 1.00000i
\(126\) 0 0
\(127\) 12.0416 1.06852 0.534259 0.845321i \(-0.320591\pi\)
0.534259 + 0.845321i \(0.320591\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) − 21.5407i − 1.88202i −0.338384 0.941008i \(-0.609880\pi\)
0.338384 0.941008i \(-0.390120\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 0 0
\(139\) 11.0000 0.933008 0.466504 0.884519i \(-0.345513\pi\)
0.466504 + 0.884519i \(0.345513\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −12.0416 −1.00000
\(146\) 0 0
\(147\) 0 0
\(148\) −24.0832 −1.97963
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) −13.0000 −1.05792 −0.528962 0.848645i \(-0.677419\pi\)
−0.528962 + 0.848645i \(0.677419\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −24.0832 −1.92205 −0.961024 0.276465i \(-0.910837\pi\)
−0.961024 + 0.276465i \(0.910837\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 12.0416 0.943170 0.471585 0.881820i \(-0.343682\pi\)
0.471585 + 0.881820i \(0.343682\pi\)
\(164\) 10.7703i 0.841021i
\(165\) 0 0
\(166\) 0 0
\(167\) 17.8885i 1.38426i 0.721774 + 0.692129i \(0.243327\pi\)
−0.721774 + 0.692129i \(0.756673\pi\)
\(168\) 0 0
\(169\) 13.0000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) −24.0832 −1.83633
\(173\) 24.5967i 1.87006i 0.354574 + 0.935028i \(0.384626\pi\)
−0.354574 + 0.935028i \(0.615374\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) − 21.5407i − 1.62369i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 17.0000 1.26360 0.631800 0.775131i \(-0.282316\pi\)
0.631800 + 0.775131i \(0.282316\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 26.9258i 1.97963i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) − 5.38516i − 0.389657i −0.980837 0.194828i \(-0.937585\pi\)
0.980837 0.194828i \(-0.0624150\pi\)
\(192\) 0 0
\(193\) −24.0832 −1.73355 −0.866773 0.498703i \(-0.833810\pi\)
−0.866773 + 0.498703i \(0.833810\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) − 2.23607i − 0.159313i −0.996822 0.0796566i \(-0.974618\pi\)
0.996822 0.0796566i \(-0.0253824\pi\)
\(198\) 0 0
\(199\) −19.0000 −1.34687 −0.673437 0.739244i \(-0.735183\pi\)
−0.673437 + 0.739244i \(0.735183\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 12.0416 0.841021
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) − 22.3607i − 1.53574i
\(213\) 0 0
\(214\) 0 0
\(215\) 26.9258i 1.83633i
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −24.0832 −1.62369
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 11.1803i 0.742065i 0.928620 + 0.371033i \(0.120996\pi\)
−0.928620 + 0.371033i \(0.879004\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) − 29.0689i − 1.90437i −0.305528 0.952183i \(-0.598833\pi\)
0.305528 0.952183i \(-0.401167\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 23.0000 1.48156 0.740780 0.671748i \(-0.234456\pi\)
0.740780 + 0.671748i \(0.234456\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 15.6525i 1.00000i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) − 21.5407i − 1.35963i −0.733382 0.679817i \(-0.762059\pi\)
0.733382 0.679817i \(-0.237941\pi\)
\(252\) 0 0
\(253\) −12.0416 −0.757049
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) − 15.6525i − 0.976375i −0.872739 0.488187i \(-0.837658\pi\)
0.872739 0.488187i \(-0.162342\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −25.0000 −1.53574
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 10.7703i 0.656679i 0.944560 + 0.328339i \(0.106489\pi\)
−0.944560 + 0.328339i \(0.893511\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 26.9258i 1.62369i
\(276\) 0 0
\(277\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(282\) 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
\(289\) −17.0000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −24.0832 −1.40936
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 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
\(305\) 0 0
\(306\) 0 0
\(307\) 12.0416 0.687250 0.343625 0.939107i \(-0.388345\pi\)
0.343625 + 0.939107i \(0.388345\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) − 5.38516i − 0.305365i −0.988275 0.152682i \(-0.951209\pi\)
0.988275 0.152682i \(-0.0487911\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 29.0000 1.62369
\(320\) − 17.8885i − 1.00000i
\(321\) 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 \(\pi\)
\(332\) 31.3050i 1.71808i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −24.0832 −1.31189 −0.655947 0.754807i \(-0.727731\pi\)
−0.655947 + 0.754807i \(0.727731\pi\)
\(338\) 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
\(347\) 24.5967i 1.32042i 0.751080 + 0.660211i \(0.229533\pi\)
−0.751080 + 0.660211i \(0.770467\pi\)
\(348\) 0 0
\(349\) −31.0000 −1.65939 −0.829696 0.558216i \(-0.811486\pi\)
−0.829696 + 0.558216i \(0.811486\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 4.47214i 0.238028i 0.992893 + 0.119014i \(0.0379733\pi\)
−0.992893 + 0.119014i \(0.962027\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) − 21.5407i − 1.14165i
\(357\) 0 0
\(358\) 0 0
\(359\) − 37.6962i − 1.98953i −0.102204 0.994763i \(-0.532589\pi\)
0.102204 0.994763i \(-0.467411\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 26.9258i 1.40936i
\(366\) 0 0
\(367\) 12.0416 0.628566 0.314283 0.949329i \(-0.398236\pi\)
0.314283 + 0.949329i \(0.398236\pi\)
\(368\) − 8.94427i − 0.466252i
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 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
\(383\) 38.0132i 1.94238i 0.238303 + 0.971191i \(0.423409\pi\)
−0.238303 + 0.971191i \(0.576591\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) −24.0832 −1.22264
\(389\) − 37.6962i − 1.91127i −0.294552 0.955635i \(-0.595170\pi\)
0.294552 0.955635i \(-0.404830\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −20.0000 −1.00000
\(401\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 10.7703i 0.535844i
\(405\) 0 0
\(406\) 0 0
\(407\) − 64.8460i − 3.21430i
\(408\) 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
\(415\) 35.0000 1.71808
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 17.8885i 0.864675i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 12.0416 0.578682 0.289341 0.957226i \(-0.406564\pi\)
0.289341 + 0.957226i \(0.406564\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) 0 0
\(438\) 0 0
\(439\) −4.00000 −0.190910 −0.0954548 0.995434i \(-0.530431\pi\)
−0.0954548 + 0.995434i \(0.530431\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −24.0832 −1.14165
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 37.6962i − 1.77899i −0.456943 0.889496i \(-0.651056\pi\)
0.456943 0.889496i \(-0.348944\pi\)
\(450\) 0 0
\(451\) −29.0000 −1.36556
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −10.0000 −0.466252
\(461\) − 5.38516i − 0.250812i −0.992105 0.125406i \(-0.959977\pi\)
0.992105 0.125406i \(-0.0400234\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 21.5407i 1.00000i
\(465\) 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
\(473\) − 64.8460i − 2.98162i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 43.0813i 1.96844i 0.176961 + 0.984218i \(0.443373\pi\)
−0.176961 + 0.984218i \(0.556627\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 36.0000 1.63636
\(485\) 26.9258i 1.22264i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) − 21.5407i − 0.972116i −0.873926 0.486058i \(-0.838434\pi\)
0.873926 0.486058i \(-0.161566\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −16.0000 −0.716258 −0.358129 0.933672i \(-0.616585\pi\)
−0.358129 + 0.933672i \(0.616585\pi\)
\(500\) 22.3607i 1.00000i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 12.0416 0.535844
\(506\) 0 0
\(507\) 0 0
\(508\) −24.0832 −1.06852
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 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 \(\pi\)
\(524\) 43.0813i 1.88202i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 18.0000 0.782609
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 20.0000 0.864675
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) − 37.6962i − 1.62369i
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) − 2.23607i − 0.0957826i
\(546\) 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
\(556\) −22.0000 −0.933008
\(557\) 38.0132i 1.61067i 0.592821 + 0.805335i \(0.298014\pi\)
−0.592821 + 0.805335i \(0.701986\pi\)
\(558\) 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
\(569\) 10.7703i 0.451516i 0.974183 + 0.225758i \(0.0724858\pi\)
−0.974183 + 0.225758i \(0.927514\pi\)
\(570\) 0 0
\(571\) −43.0000 −1.79949 −0.899747 0.436412i \(-0.856249\pi\)
−0.899747 + 0.436412i \(0.856249\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 11.1803i 0.466252i
\(576\) 0 0
\(577\) −24.0832 −1.00260 −0.501298 0.865275i \(-0.667144\pi\)
−0.501298 + 0.865275i \(0.667144\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 24.0832 1.00000
\(581\) 0 0
\(582\) 0 0
\(583\) 60.2080 2.49356
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 44.7214i 1.84585i 0.384982 + 0.922924i \(0.374208\pi\)
−0.384982 + 0.922924i \(0.625792\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 48.1664 1.97963
\(593\) 31.3050i 1.28554i 0.766059 + 0.642770i \(0.222215\pi\)
−0.766059 + 0.642770i \(0.777785\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 43.0813i 1.76025i 0.474737 + 0.880127i \(0.342543\pi\)
−0.474737 + 0.880127i \(0.657457\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 26.0000 1.05792
\(605\) − 40.2492i − 1.63636i
\(606\) 0 0
\(607\) 48.1664 1.95501 0.977506 0.210905i \(-0.0676412\pi\)
0.977506 + 0.210905i \(0.0676412\pi\)
\(608\) 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
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 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
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 48.1664 1.92205
\(629\) 0 0
\(630\) 0 0
\(631\) −28.0000 −1.11466 −0.557331 0.830290i \(-0.688175\pi\)
−0.557331 + 0.830290i \(0.688175\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 26.9258i 1.06852i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) − 5.38516i − 0.212701i −0.994329 0.106351i \(-0.966083\pi\)
0.994329 0.106351i \(-0.0339166\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 42.4853i − 1.67027i −0.550046 0.835135i \(-0.685390\pi\)
0.550046 0.835135i \(-0.314610\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −24.0832 −0.943170
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 48.1664 1.88202
\(656\) − 21.5407i − 0.841021i
\(657\) 0 0
\(658\) 0 0
\(659\) − 37.6962i − 1.46843i −0.678915 0.734217i \(-0.737550\pi\)
0.678915 0.734217i \(-0.262450\pi\)
\(660\) 0 0
\(661\) 47.0000 1.82809 0.914044 0.405615i \(-0.132943\pi\)
0.914044 + 0.405615i \(0.132943\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 12.0416 0.466252
\(668\) − 35.7771i − 1.38426i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −26.0000 −1.00000
\(677\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 51.4296i 1.96790i 0.178451 + 0.983949i \(0.442891\pi\)
−0.178451 + 0.983949i \(0.557109\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 48.1664 1.83633
\(689\) 0 0
\(690\) 0 0
\(691\) 32.0000 1.21734 0.608669 0.793424i \(-0.291704\pi\)
0.608669 + 0.793424i \(0.291704\pi\)
\(692\) − 49.1935i − 1.87006i
\(693\) 0 0
\(694\) 0 0
\(695\) 24.5967i 0.933008i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 43.0813i 1.62369i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −49.0000 −1.84023 −0.920117 0.391644i \(-0.871906\pi\)
−0.920117 + 0.391644i \(0.871906\pi\)
\(710\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) −34.0000 −1.26360
\(725\) − 26.9258i − 1.00000i
\(726\) 0 0
\(727\) 48.1664 1.78639 0.893196 0.449667i \(-0.148458\pi\)
0.893196 + 0.449667i \(0.148458\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −24.0832 −0.889533 −0.444766 0.895647i \(-0.646713\pi\)
−0.444766 + 0.895647i \(0.646713\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) − 53.8516i − 1.97963i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) − 29.0689i − 1.05792i
\(756\) 0 0
\(757\) 12.0416 0.437659 0.218830 0.975763i \(-0.429776\pi\)
0.218830 + 0.975763i \(0.429776\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 10.7703i 0.389657i
\(765\) 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
\(772\) 48.1664 1.73355
\(773\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 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
\(784\) 28.0000 1.00000
\(785\) − 53.8516i − 1.92205i
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 4.47214i 0.159313i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 38.0000 1.34687
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) − 64.8460i − 2.28836i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) − 37.6962i − 1.32533i −0.748918 0.662663i \(-0.769426\pi\)
0.748918 0.662663i \(-0.230574\pi\)
\(810\) 0 0
\(811\) 53.0000 1.86108 0.930541 0.366188i \(-0.119337\pi\)
0.930541 + 0.366188i \(0.119337\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 26.9258i 0.943170i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −24.0832 −0.841021
\(821\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(822\) 0 0
\(823\) 48.1664 1.67897 0.839487 0.543379i \(-0.182856\pi\)
0.839487 + 0.543379i \(0.182856\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) −40.0000 −1.38426
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 43.0813i 1.48733i 0.668551 + 0.743666i \(0.266915\pi\)
−0.668551 + 0.743666i \(0.733085\pi\)
\(840\) 0 0
\(841\) −29.0000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 29.0689i 1.00000i
\(846\) 0 0
\(847\) 0 0
\(848\) 44.7214i 1.53574i
\(849\) 0 0
\(850\) 0 0
\(851\) − 26.9258i − 0.923005i
\(852\) 0 0
\(853\) 12.0416 0.412296 0.206148 0.978521i \(-0.433907\pi\)
0.206148 + 0.978521i \(0.433907\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 51.4296i 1.75680i 0.477926 + 0.878400i \(0.341389\pi\)
−0.477926 + 0.878400i \(0.658611\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) − 53.8516i − 1.83633i
\(861\) 0 0
\(862\) 0 0
\(863\) 17.8885i 0.608933i 0.952523 + 0.304467i \(0.0984782\pi\)
−0.952523 + 0.304467i \(0.901522\pi\)
\(864\) 0 0
\(865\) −55.0000 −1.87006
\(866\) 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
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 48.1664 1.62369
\(881\) 59.2368i 1.99574i 0.0652421 + 0.997869i \(0.479218\pi\)
−0.0652421 + 0.997869i \(0.520782\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 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
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 38.0132i 1.26360i
\(906\) 0 0
\(907\) −60.2080 −1.99917 −0.999586 0.0287559i \(-0.990845\pi\)
−0.999586 + 0.0287559i \(0.990845\pi\)
\(908\) − 22.3607i − 0.742065i
\(909\) 0 0
\(910\) 0 0
\(911\) 59.2368i 1.96260i 0.192476 + 0.981302i \(0.438348\pi\)
−0.192476 + 0.981302i \(0.561652\pi\)
\(912\) 0 0
\(913\) −84.2912 −2.78963
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 44.0000 1.45143 0.725713 0.687998i \(-0.241510\pi\)
0.725713 + 0.687998i \(0.241510\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −60.2080 −1.97963
\(926\) 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
\(932\) 58.1378i 1.90437i
\(933\) 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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) −12.0416 −0.392128
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 49.1935i − 1.59353i −0.604287 0.796767i \(-0.706542\pi\)
0.604287 0.796767i \(-0.293458\pi\)
\(954\) 0 0
\(955\) 12.0416 0.389657
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) −46.0000 −1.48156
\(965\) − 53.8516i − 1.73355i
\(966\) 0 0
\(967\) −60.2080 −1.93616 −0.968079 0.250645i \(-0.919357\pi\)
−0.968079 + 0.250645i \(0.919357\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 59.2368i 1.90100i 0.310725 + 0.950500i \(0.399428\pi\)
−0.310725 + 0.950500i \(0.600572\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 55.9017i − 1.78845i −0.447614 0.894227i \(-0.647726\pi\)
0.447614 0.894227i \(-0.352274\pi\)
\(978\) 0 0
\(979\) 58.0000 1.85369
\(980\) − 31.3050i − 1.00000i
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 5.00000 0.159313
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) − 26.9258i − 0.856191i
\(990\) 0 0
\(991\) −7.00000 −0.222362 −0.111181 0.993800i \(-0.535463\pi\)
−0.111181 + 0.993800i \(0.535463\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) − 42.4853i − 1.34687i
\(996\) 0 0
\(997\) −60.2080 −1.90681 −0.953403 0.301700i \(-0.902446\pi\)
−0.953403 + 0.301700i \(0.902446\pi\)
\(998\) 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 1305.2.f.g.289.3 yes 4
3.2 odd 2 inner 1305.2.f.g.289.2 yes 4
5.4 even 2 inner 1305.2.f.g.289.1 4
15.14 odd 2 inner 1305.2.f.g.289.4 yes 4
29.28 even 2 inner 1305.2.f.g.289.4 yes 4
87.86 odd 2 inner 1305.2.f.g.289.1 4
145.144 even 2 inner 1305.2.f.g.289.2 yes 4
435.434 odd 2 CM 1305.2.f.g.289.3 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1305.2.f.g.289.1 4 5.4 even 2 inner
1305.2.f.g.289.1 4 87.86 odd 2 inner
1305.2.f.g.289.2 yes 4 3.2 odd 2 inner
1305.2.f.g.289.2 yes 4 145.144 even 2 inner
1305.2.f.g.289.3 yes 4 1.1 even 1 trivial
1305.2.f.g.289.3 yes 4 435.434 odd 2 CM
1305.2.f.g.289.4 yes 4 15.14 odd 2 inner
1305.2.f.g.289.4 yes 4 29.28 even 2 inner