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Properties

Label 300.7.g.d.101.1
Level $300$
Weight $7$
Character 300.101
Self dual yes
Analytic conductor $69.016$
Analytic rank $0$
Dimension $1$
CM discriminant -3
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [300,7,Mod(101,300)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(300, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 0]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("300.101");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 300 = 2^{2} \cdot 3 \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 300.g (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: \(69.0162250860\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 101.1
Character \(\chi\) \(=\) 300.101

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+27.0000 q^{3} +397.000 q^{7} +729.000 q^{9} +O(q^{10})\) \(q+27.0000 q^{3} +397.000 q^{7} +729.000 q^{9} +4033.00 q^{13} -2269.00 q^{19} +10719.0 q^{21} +19683.0 q^{27} -59221.0 q^{31} +89206.0 q^{37} +108891. q^{39} -42587.0 q^{43} +39960.0 q^{49} -61263.0 q^{57} +357839. q^{61} +289413. q^{63} +585397. q^{67} -638066. q^{73} -204622. q^{79} +531441. q^{81} +1.60110e6 q^{91} -1.59897e6 q^{93} -1.60826e6 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/300\mathbb{Z}\right)^\times\).

\(n\) \(101\) \(151\) \(277\)
\(\chi(n)\) \(-1\) \(1\) \(1\)

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
\(3\) 27.0000 1.00000
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 397.000 1.15743 0.578717 − 0.815528i \(-0.303554\pi\)
0.578717 + 0.815528i \(0.303554\pi\)
\(8\) 0 0
\(9\) 729.000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 4033.00 1.83569 0.917843 − 0.396945i \(-0.129930\pi\)
0.917843 + 0.396945i \(0.129930\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −2269.00 −0.330806 −0.165403 − 0.986226i \(-0.552893\pi\)
−0.165403 + 0.986226i \(0.552893\pi\)
\(20\) 0 0
\(21\) 10719.0 1.15743
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 19683.0 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) −59221.0 −1.98788 −0.993941 − 0.109914i \(-0.964943\pi\)
−0.993941 + 0.109914i \(0.964943\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 89206.0 1.76112 0.880560 − 0.473935i \(-0.157167\pi\)
0.880560 + 0.473935i \(0.157167\pi\)
\(38\) 0 0
\(39\) 108891. 1.83569
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −42587.0 −0.535638 −0.267819 − 0.963469i \(-0.586303\pi\)
−0.267819 + 0.963469i \(0.586303\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 39960.0 0.339654
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −61263.0 −0.330806
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 357839. 1.57652 0.788258 − 0.615345i \(-0.210983\pi\)
0.788258 + 0.615345i \(0.210983\pi\)
\(62\) 0 0
\(63\) 289413. 1.15743
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 585397. 1.94637 0.973187 − 0.230017i \(-0.0738783\pi\)
0.973187 + 0.230017i \(0.0738783\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −638066. −1.64020 −0.820100 − 0.572220i \(-0.806082\pi\)
−0.820100 + 0.572220i \(0.806082\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −204622. −0.415022 −0.207511 − 0.978233i \(-0.566536\pi\)
−0.207511 + 0.978233i \(0.566536\pi\)
\(80\) 0 0
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 1.60110e6 2.12468
\(92\) 0 0
\(93\) −1.59897e6 −1.98788
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.60826e6 −1.76215 −0.881073 − 0.472980i \(-0.843178\pi\)
−0.881073 + 0.472980i \(0.843178\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) −1.12695e6 −1.03132 −0.515658 − 0.856795i \(-0.672452\pi\)
−0.515658 + 0.856795i \(0.672452\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 2.30731e6 1.78167 0.890834 − 0.454329i \(-0.150121\pi\)
0.890834 + 0.454329i \(0.150121\pi\)
\(110\) 0 0
\(111\) 2.40856e6 1.76112
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 2.94006e6 1.83569
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.77156e6 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.95237e6 1.92951 0.964753 − 0.263158i \(-0.0847641\pi\)
0.964753 + 0.263158i \(0.0847641\pi\)
\(128\) 0 0
\(129\) −1.14985e6 −0.535638
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −900793. −0.382887
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 1.26454e6 0.470855 0.235428 − 0.971892i \(-0.424351\pi\)
0.235428 + 0.971892i \(0.424351\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 1.07892e6 0.339654
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 6.87078e6 1.99561 0.997804 − 0.0662392i \(-0.0211000\pi\)
0.997804 + 0.0662392i \(0.0211000\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 6.24406e6 1.61350 0.806748 − 0.590896i \(-0.201226\pi\)
0.806748 + 0.590896i \(0.201226\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −5.61935e6 −1.29755 −0.648773 − 0.760982i \(-0.724718\pi\)
−0.648773 + 0.760982i \(0.724718\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 1.14383e7 2.36974
\(170\) 0 0
\(171\) −1.65410e6 −0.330806
\(172\) 0 0
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.03194e7 −1.74028 −0.870139 − 0.492806i \(-0.835971\pi\)
−0.870139 + 0.492806i \(0.835971\pi\)
\(182\) 0 0
\(183\) 9.66165e6 1.57652
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 7.81415e6 1.15743
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) −1.17442e7 −1.63362 −0.816811 − 0.576905i \(-0.804260\pi\)
−0.816811 + 0.576905i \(0.804260\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) −1.50164e7 −1.90549 −0.952747 − 0.303766i \(-0.901756\pi\)
−0.952747 + 0.303766i \(0.901756\pi\)
\(200\) 0 0
\(201\) 1.58057e7 1.94637
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) −3.09834e6 −0.329824 −0.164912 − 0.986308i \(-0.552734\pi\)
−0.164912 + 0.986308i \(0.552734\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −2.35107e7 −2.30084
\(218\) 0 0
\(219\) −1.72278e7 −1.64020
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.03524e7 0.933527 0.466764 − 0.884382i \(-0.345420\pi\)
0.466764 + 0.884382i \(0.345420\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) −1.84625e7 −1.53739 −0.768696 − 0.639614i \(-0.779094\pi\)
−0.768696 + 0.639614i \(0.779094\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −5.52479e6 −0.415022
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −2.11882e7 −1.51371 −0.756854 − 0.653584i \(-0.773265\pi\)
−0.756854 + 0.653584i \(0.773265\pi\)
\(242\) 0 0
\(243\) 1.43489e7 1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −9.15088e6 −0.607256
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 3.54148e7 2.03838
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 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
\(271\) −3.91457e7 −1.96687 −0.983436 − 0.181258i \(-0.941983\pi\)
−0.983436 + 0.181258i \(0.941983\pi\)
\(272\) 0 0
\(273\) 4.32297e7 2.12468
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.20769e7 −1.97973 −0.989863 − 0.142029i \(-0.954637\pi\)
−0.989863 + 0.142029i \(0.954637\pi\)
\(278\) 0 0
\(279\) −4.31721e7 −1.98788
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −4.43316e7 −1.95593 −0.977966 − 0.208762i \(-0.933057\pi\)
−0.977966 + 0.208762i \(0.933057\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.41376e7 1.00000
\(290\) 0 0
\(291\) −4.34231e7 −1.76215
\(292\) 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
\(301\) −1.69070e7 −0.619966
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 1.30191e7 0.449951 0.224975 − 0.974364i \(-0.427770\pi\)
0.224975 + 0.974364i \(0.427770\pi\)
\(308\) 0 0
\(309\) −3.04275e7 −1.03132
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 6.05168e7 1.97353 0.986763 − 0.162168i \(-0.0518488\pi\)
0.986763 + 0.162168i \(0.0518488\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 6.22974e7 1.78167
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −7.15453e7 −1.97286 −0.986432 − 0.164169i \(-0.947506\pi\)
−0.986432 + 0.164169i \(0.947506\pi\)
\(332\) 0 0
\(333\) 6.50312e7 1.76112
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 2.23317e7 0.583489 0.291745 − 0.956496i \(-0.405764\pi\)
0.291745 + 0.956496i \(0.405764\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −3.08425e7 −0.764307
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 5.69263e7 1.33917 0.669586 − 0.742734i \(-0.266471\pi\)
0.669586 + 0.742734i \(0.266471\pi\)
\(350\) 0 0
\(351\) 7.93815e7 1.83569
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 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
\(361\) −4.18975e7 −0.890567
\(362\) 0 0
\(363\) 4.78321e7 1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −9.02833e7 −1.82646 −0.913228 − 0.407449i \(-0.866418\pi\)
−0.913228 + 0.407449i \(0.866418\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −5.49950e7 −1.05973 −0.529866 − 0.848081i \(-0.677758\pi\)
−0.529866 + 0.848081i \(0.677758\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −7.16333e7 −1.31582 −0.657911 − 0.753096i \(-0.728560\pi\)
−0.657911 + 0.753096i \(0.728560\pi\)
\(380\) 0 0
\(381\) 1.06714e8 1.92951
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −3.10459e7 −0.535638
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −4.56049e7 −0.728854 −0.364427 − 0.931232i \(-0.618735\pi\)
−0.364427 + 0.931232i \(0.618735\pi\)
\(398\) 0 0
\(399\) −2.43214e7 −0.382887
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −2.38838e8 −3.64913
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.36835e8 1.99999 0.999994 − 0.00355979i \(-0.00113312\pi\)
0.999994 + 0.00355979i \(0.00113312\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 3.41425e7 0.470855
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 1.44474e8 1.93617 0.968086 − 0.250620i \(-0.0806343\pi\)
0.968086 + 0.250620i \(0.0806343\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.42062e8 1.82471
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) −3.78287e7 −0.465970 −0.232985 − 0.972480i \(-0.574849\pi\)
−0.232985 + 0.972480i \(0.574849\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.65775e8 −1.95942 −0.979708 − 0.200431i \(-0.935766\pi\)
−0.979708 + 0.200431i \(0.935766\pi\)
\(440\) 0 0
\(441\) 2.91308e7 0.339654
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 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 0
\(453\) 1.85511e8 1.99561
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 2.93439e7 0.307446 0.153723 − 0.988114i \(-0.450874\pi\)
0.153723 + 0.988114i \(0.450874\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(462\) 0 0
\(463\) −1.92743e8 −1.94194 −0.970968 − 0.239209i \(-0.923112\pi\)
−0.970968 + 0.239209i \(0.923112\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 2.32403e8 2.25280
\(470\) 0 0
\(471\) 1.68590e8 1.61350
\(472\) 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
\(481\) 3.59768e8 3.23286
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 1.20514e7 0.104340 0.0521700 − 0.998638i \(-0.483386\pi\)
0.0521700 + 0.998638i \(0.483386\pi\)
\(488\) 0 0
\(489\) −1.51722e8 −1.29755
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 2.24255e8 1.80485 0.902424 − 0.430849i \(-0.141786\pi\)
0.902424 + 0.430849i \(0.141786\pi\)
\(500\) 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
\(507\) 3.08834e8 2.36974
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −2.53312e8 −1.89842
\(512\) 0 0
\(513\) −4.46607e7 −0.330806
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 1.44953e8 1.01326 0.506632 − 0.862162i \(-0.330890\pi\)
0.506632 + 0.862162i \(0.330890\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.48036e8 1.00000
\(530\) 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
\(539\) 0 0
\(540\) 0 0
\(541\) −3.13113e8 −1.97746 −0.988732 − 0.149696i \(-0.952171\pi\)
−0.988732 + 0.149696i \(0.952171\pi\)
\(542\) 0 0
\(543\) −2.78624e8 −1.74028
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 3.24645e8 1.98357 0.991783 − 0.127929i \(-0.0408330\pi\)
0.991783 + 0.127929i \(0.0408330\pi\)
\(548\) 0 0
\(549\) 2.60865e8 1.57652
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −8.12349e7 −0.480361
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −1.71753e8 −0.983263
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 2.10982e8 1.15743
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 2.00833e8 1.07876 0.539381 − 0.842062i \(-0.318658\pi\)
0.539381 + 0.842062i \(0.318658\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −3.62346e8 −1.88624 −0.943119 − 0.332456i \(-0.892123\pi\)
−0.943119 + 0.332456i \(0.892123\pi\)
\(578\) 0 0
\(579\) −3.17094e8 −1.63362
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 1.34372e8 0.657604
\(590\) 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
\(597\) −4.05444e8 −1.90549
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −1.33106e8 −0.613162 −0.306581 − 0.951845i \(-0.599185\pi\)
−0.306581 + 0.951845i \(0.599185\pi\)
\(602\) 0 0
\(603\) 4.26754e8 1.94637
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −3.60399e8 −1.61145 −0.805727 − 0.592287i \(-0.798225\pi\)
−0.805727 + 0.592287i \(0.798225\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.58281e8 0.687142 0.343571 − 0.939127i \(-0.388363\pi\)
0.343571 + 0.939127i \(0.388363\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) −4.74352e8 −1.99999 −0.999997 − 0.00240251i \(-0.999235\pi\)
−0.999997 + 0.00240251i \(0.999235\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 3.01300e8 1.19925 0.599627 − 0.800280i \(-0.295316\pi\)
0.599627 + 0.800280i \(0.295316\pi\)
\(632\) 0 0
\(633\) −8.36552e7 −0.329824
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.61159e8 0.623498
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 3.58510e8 1.34855 0.674277 − 0.738479i \(-0.264456\pi\)
0.674277 + 0.738479i \(0.264456\pi\)
\(644\) 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
\(651\) −6.34790e8 −2.30084
\(652\) 0 0
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −4.65150e8 −1.64020
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1.58097e8 −0.547419 −0.273710 − 0.961812i \(-0.588251\pi\)
−0.273710 + 0.961812i \(0.588251\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 2.79515e8 0.933527
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 3.12399e7 0.102486 0.0512430 − 0.998686i \(-0.483682\pi\)
0.0512430 + 0.998686i \(0.483682\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −6.38480e8 −2.03957
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −4.98488e8 −1.53739
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −4.01570e8 −1.21710 −0.608551 − 0.793515i \(-0.708249\pi\)
−0.608551 + 0.793515i \(0.708249\pi\)
\(692\) 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
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) −2.02408e8 −0.582589
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −6.38446e8 −1.79137 −0.895685 − 0.444690i \(-0.853314\pi\)
−0.895685 + 0.444690i \(0.853314\pi\)
\(710\) 0 0
\(711\) −1.49169e8 −0.415022
\(712\) 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
\(721\) −4.47398e8 −1.19368
\(722\) 0 0
\(723\) −5.72080e8 −1.51371
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −6.78028e8 −1.76459 −0.882296 − 0.470695i \(-0.844003\pi\)
−0.882296 + 0.470695i \(0.844003\pi\)
\(728\) 0 0
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 5.61163e8 1.42488 0.712438 − 0.701735i \(-0.247591\pi\)
0.712438 + 0.701735i \(0.247591\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.77287e8 0.439281 0.219641 − 0.975581i \(-0.429512\pi\)
0.219641 + 0.975581i \(0.429512\pi\)
\(740\) 0 0
\(741\) −2.47074e8 −0.607256
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −2.97133e8 −0.701506 −0.350753 − 0.936468i \(-0.614074\pi\)
−0.350753 + 0.936468i \(0.614074\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 2.85001e8 0.656991 0.328495 − 0.944506i \(-0.393458\pi\)
0.328495 + 0.944506i \(0.393458\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 9.16002e8 2.06216
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 6.13281e8 1.34859 0.674296 − 0.738461i \(-0.264447\pi\)
0.674296 + 0.738461i \(0.264447\pi\)
\(770\) 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
\(777\) 9.56199e8 2.03838
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 7.60176e8 1.55952 0.779759 − 0.626080i \(-0.215342\pi\)
0.779759 + 0.626080i \(0.215342\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.44316e9 2.89399
\(794\) 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
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 5.82267e8 1.09159 0.545795 − 0.837918i \(-0.316228\pi\)
0.545795 + 0.837918i \(0.316228\pi\)
\(812\) 0 0
\(813\) −1.05693e9 −1.96687
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 9.66299e7 0.177193
\(818\) 0 0
\(819\) 1.16720e9 2.12468
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −75347.0 −0.000135166 0 −6.75828e−5 − 1.00000i \(-0.500022\pi\)
−6.75828e−5 1.00000i \(0.500022\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −8.48197e8 −1.48879 −0.744395 − 0.667740i \(-0.767262\pi\)
−0.744395 + 0.667740i \(0.767262\pi\)
\(830\) 0 0
\(831\) −1.13608e9 −1.97973
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −1.16565e9 −1.98788
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 5.94823e8 1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 7.03310e8 1.15743
\(848\) 0 0
\(849\) −1.19695e9 −1.95593
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −2.34104e8 −0.377191 −0.188596 − 0.982055i \(-0.560394\pi\)
−0.188596 + 0.982055i \(0.560394\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) −2.87739e8 −0.453962 −0.226981 − 0.973899i \(-0.572886\pi\)
−0.226981 + 0.973899i \(0.572886\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 6.51714e8 1.00000
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 2.36091e9 3.57293
\(872\) 0 0
\(873\) −1.17242e9 −1.76215
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −4.67258e6 −0.00692721 −0.00346360 − 0.999994i \(-0.501103\pi\)
−0.00346360 + 0.999994i \(0.501103\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.33842e9 −1.94406 −0.972029 − 0.234859i \(-0.924537\pi\)
−0.972029 + 0.234859i \(0.924537\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.56909e9 2.23328
\(890\) 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
\(903\) −4.56490e8 −0.619966
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −5.18340e8 −0.694693 −0.347347 − 0.937737i \(-0.612917\pi\)
−0.347347 + 0.937737i \(0.612917\pi\)
\(908\) 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
\(919\) 9.05179e8 1.16624 0.583120 − 0.812386i \(-0.301832\pi\)
0.583120 + 0.812386i \(0.301832\pi\)
\(920\) 0 0
\(921\) 3.51515e8 0.449951
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −8.21544e8 −1.03132
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) −9.06692e7 −0.112360
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.41345e9 1.71816 0.859078 − 0.511845i \(-0.171038\pi\)
0.859078 + 0.511845i \(0.171038\pi\)
\(938\) 0 0
\(939\) 1.63395e9 1.97353
\(940\) 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
\(949\) −2.57332e9 −3.01089
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.61962e9 2.95168
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −6.93538e8 −0.766992 −0.383496 − 0.923542i \(-0.625280\pi\)
−0.383496 + 0.923542i \(0.625280\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 5.02022e8 0.544984
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.68203e9 1.78167
\(982\) 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
\(989\) 0 0
\(990\) 0 0
\(991\) 1.61391e9 1.65828 0.829138 − 0.559043i \(-0.188832\pi\)
0.829138 + 0.559043i \(0.188832\pi\)
\(992\) 0 0
\(993\) −1.93172e9 −1.97286
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.14627e9 1.15664 0.578322 − 0.815808i \(-0.303708\pi\)
0.578322 + 0.815808i \(0.303708\pi\)
\(998\) 0 0
\(999\) 1.75584e9 1.76112
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 300.7.g.d.101.1 yes 1
3.2 odd 2 CM 300.7.g.d.101.1 yes 1
5.2 odd 4 300.7.b.b.149.1 2
5.3 odd 4 300.7.b.b.149.2 2
5.4 even 2 300.7.g.a.101.1 ✓ 1
15.2 even 4 300.7.b.b.149.1 2
15.8 even 4 300.7.b.b.149.2 2
15.14 odd 2 300.7.g.a.101.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.7.b.b.149.1 2 5.2 odd 4
300.7.b.b.149.1 2 15.2 even 4
300.7.b.b.149.2 2 5.3 odd 4
300.7.b.b.149.2 2 15.8 even 4
300.7.g.a.101.1 ✓ 1 5.4 even 2
300.7.g.a.101.1 ✓ 1 15.14 odd 2
300.7.g.d.101.1 yes 1 1.1 even 1 trivial
300.7.g.d.101.1 yes 1 3.2 odd 2 CM