↑

Properties

Label 300.7.g.b.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 $2$

Related objects

Downloads

Learn more

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)
 
\(f(q)\) \(=\) \(q-27.0000 q^{3} +683.000 q^{7} +729.000 q^{9} +O(q^{10})\) \(q-27.0000 q^{3} +683.000 q^{7} +729.000 q^{9} +3527.00 q^{13} +12851.0 q^{19} -18441.0 q^{21} -19683.0 q^{27} +23939.0 q^{31} -89206.0 q^{37} -95229.0 q^{39} -153973. q^{43} +348840. q^{49} -346977. q^{57} +62999.0 q^{61} +497907. q^{63} +412523. q^{67} +638066. q^{73} -204622. q^{79} +531441. q^{81} +2.40894e6 q^{91} -646353. q^{93} -1.55182e6 q^{97} +O(q^{100})\)

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
\(7\) 683.000 1.99125 0.995627 − 0.0934196i \(-0.0297798\pi\)
0.995627 + 0.0934196i \(0.0297798\pi\)
\(8\) 0 0
\(9\) 729.000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 3527.00 1.60537 0.802685 − 0.596403i \(-0.203404\pi\)
0.802685 + 0.596403i \(0.203404\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 12851.0 1.87360 0.936798 − 0.349870i \(-0.113774\pi\)
0.936798 + 0.349870i \(0.113774\pi\)
\(20\) 0 0
\(21\) −18441.0 −1.99125
\(22\) 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
\(31\) 23939.0 0.803565 0.401782 − 0.915735i \(-0.368391\pi\)
0.401782 + 0.915735i \(0.368391\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −89206.0 −1.76112 −0.880560 − 0.473935i \(-0.842833\pi\)
−0.880560 + 0.473935i \(0.842833\pi\)
\(38\) 0 0
\(39\) −95229.0 −1.60537
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) −153973. −1.93660 −0.968298 − 0.249796i \(-0.919636\pi\)
−0.968298 + 0.249796i \(0.919636\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) 0 0
\(49\) 348840. 2.96509
\(50\) 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
\(57\) −346977. −1.87360
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 62999.0 0.277552 0.138776 − 0.990324i \(-0.455683\pi\)
0.138776 + 0.990324i \(0.455683\pi\)
\(62\) 0 0
\(63\) 497907. 1.99125
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 412523. 1.37159 0.685794 − 0.727796i \(-0.259455\pi\)
0.685794 + 0.727796i \(0.259455\pi\)
\(68\) 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 \(-0.193918\pi\)
0.820100 + 0.572220i \(0.193918\pi\)
\(74\) 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
\(91\) 2.40894e6 3.19670
\(92\) 0 0
\(93\) −646353. −0.803565
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.55182e6 −1.70030 −0.850150 − 0.526541i \(-0.823489\pi\)
−0.850150 + 0.526541i \(0.823489\pi\)
\(98\) 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 \(-0.327548\pi\)
0.515658 + 0.856795i \(0.327548\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) −134569. −0.103912 −0.0519560 − 0.998649i \(-0.516546\pi\)
−0.0519560 + 0.998649i \(0.516546\pi\)
\(110\) 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
\(117\) 2.57118e6 1.60537
\(118\) 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 \(-0.915236\pi\)
−0.964753 + 0.263158i \(0.915236\pi\)
\(128\) 0 0
\(129\) 4.15727e6 1.93660
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 8.77723e6 3.73081
\(134\) 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
\(147\) −9.41868e6 −2.96509
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −3.04038e6 −0.883074 −0.441537 − 0.897243i \(-0.645567\pi\)
−0.441537 + 0.897243i \(0.645567\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −838657. −0.216713 −0.108357 − 0.994112i \(-0.534559\pi\)
−0.108357 + 0.994112i \(0.534559\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −8.51785e6 −1.96683 −0.983416 − 0.181363i \(-0.941949\pi\)
−0.983416 + 0.181363i \(0.941949\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) 7.61292e6 1.57722
\(170\) 0 0
\(171\) 9.36838e6 1.87360
\(172\) 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
\(181\) 1.02211e7 1.72370 0.861852 − 0.507160i \(-0.169305\pi\)
0.861852 + 0.507160i \(0.169305\pi\)
\(182\) 0 0
\(183\) −1.70097e6 −0.277552
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −1.34435e7 −1.99125
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 1.31141e6 0.182417 0.0912086 − 0.995832i \(-0.470927\pi\)
0.0912086 + 0.995832i \(0.470927\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 3.36193e6 0.426609 0.213304 − 0.976986i \(-0.431577\pi\)
0.213304 + 0.976986i \(0.431577\pi\)
\(200\) 0 0
\(201\) −1.11381e7 −1.37159
\(202\) 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
\(211\) −1.44988e7 −1.54342 −0.771712 − 0.635972i \(-0.780599\pi\)
−0.771712 + 0.635972i \(0.780599\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.63503e7 1.60010
\(218\) 0 0
\(219\) −1.72278e7 −1.64020
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 2.21631e7 1.99856 0.999279 − 0.0379619i \(-0.0120866\pi\)
0.999279 + 0.0379619i \(0.0120866\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(228\) 0 0
\(229\) 2.25354e7 1.87654 0.938270 − 0.345903i \(-0.112428\pi\)
0.938270 + 0.345903i \(0.112428\pi\)
\(230\) 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
\(241\) −5.25168e6 −0.375187 −0.187593 − 0.982247i \(-0.560069\pi\)
−0.187593 + 0.982247i \(0.560069\pi\)
\(242\) 0 0
\(243\) −1.43489e7 −1.00000
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 4.53255e7 3.00782
\(248\) 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
\(259\) −6.09277e7 −3.50684
\(260\) 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
\(273\) −6.50414e7 −3.19670
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −1.58100e7 −0.743861 −0.371931 − 0.928261i \(-0.621304\pi\)
−0.371931 + 0.928261i \(0.621304\pi\)
\(278\) 0 0
\(279\) 1.74515e7 0.803565
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) −3.03612e7 −1.33955 −0.669776 − 0.742563i \(-0.733610\pi\)
−0.669776 + 0.742563i \(0.733610\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.41376e7 1.00000
\(290\) 0 0
\(291\) 4.18991e7 1.70030
\(292\) 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
\(301\) −1.05164e8 −3.85626
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −4.23216e7 −1.46267 −0.731337 − 0.682017i \(-0.761103\pi\)
−0.731337 + 0.682017i \(0.761103\pi\)
\(308\) 0 0
\(309\) −3.04275e7 −1.03132
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 2.16453e7 0.705879 0.352940 − 0.935646i \(-0.385182\pi\)
0.352940 + 0.935646i \(0.385182\pi\)
\(314\) 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
\(327\) 3.63336e6 0.103912
\(328\) 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
\(337\) 7.45723e7 1.94844 0.974222 − 0.225590i \(-0.0724309\pi\)
0.974222 + 0.225590i \(0.0724309\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 1.57903e8 3.91299
\(344\) 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
\(351\) −6.94219e7 −1.60537
\(352\) 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
\(361\) 1.18102e8 2.51036
\(362\) 0 0
\(363\) −4.78321e7 −1.00000
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) −1.02572e7 −0.207505 −0.103753 − 0.994603i \(-0.533085\pi\)
−0.103753 + 0.994603i \(0.533085\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.03727e8 −1.99879 −0.999393 − 0.0348373i \(-0.988909\pi\)
−0.999393 + 0.0348373i \(0.988909\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.06828e8 1.96231 0.981155 − 0.193220i \(-0.0618930\pi\)
0.981155 + 0.193220i \(0.0618930\pi\)
\(380\) 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
\(387\) −1.12246e8 −1.93660
\(388\) 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
\(397\) 7.81205e7 1.24851 0.624257 − 0.781219i \(-0.285402\pi\)
0.624257 + 0.781219i \(0.285402\pi\)
\(398\) 0 0
\(399\) −2.36985e8 −3.73081
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 8.44329e7 1.29002
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −6.79956e7 −0.993828 −0.496914 − 0.867800i \(-0.665534\pi\)
−0.496914 + 0.867800i \(0.665534\pi\)
\(410\) 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
\(427\) 4.30283e7 0.552676
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 1.17829e8 1.45140 0.725700 − 0.688011i \(-0.241516\pi\)
0.725700 + 0.688011i \(0.241516\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 1.12259e8 1.32686 0.663432 − 0.748237i \(-0.269099\pi\)
0.663432 + 0.748237i \(0.269099\pi\)
\(440\) 0 0
\(441\) 2.54304e8 2.96509
\(442\) 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
\(453\) 8.20903e7 0.883074
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −2.93439e7 −0.307446 −0.153723 − 0.988114i \(-0.549126\pi\)
−0.153723 + 0.988114i \(0.549126\pi\)
\(458\) 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 \(-0.0768880\pi\)
0.970968 + 0.239209i \(0.0768880\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 2.81753e8 2.73118
\(470\) 0 0
\(471\) 2.26437e7 0.216713
\(472\) 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
\(481\) −3.14630e8 −2.82725
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.93756e8 −1.67752 −0.838761 − 0.544500i \(-0.816720\pi\)
−0.838761 + 0.544500i \(0.816720\pi\)
\(488\) 0 0
\(489\) 2.29982e8 1.96683
\(490\) 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
\(499\) −2.04851e8 −1.64868 −0.824338 − 0.566097i \(-0.808453\pi\)
−0.824338 + 0.566097i \(0.808453\pi\)
\(500\) 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
\(507\) −2.05549e8 −1.57722
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 4.35799e8 3.26606
\(512\) 0 0
\(513\) −2.52946e8 −1.87360
\(514\) 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
\(523\) 2.86103e8 1.99994 0.999971 − 0.00767504i \(-0.00244307\pi\)
0.999971 + 0.00767504i \(0.00244307\pi\)
\(524\) 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
\(541\) 1.15502e8 0.729452 0.364726 − 0.931115i \(-0.381163\pi\)
0.364726 + 0.931115i \(0.381163\pi\)
\(542\) 0 0
\(543\) −2.75970e8 −1.72370
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −3.24645e8 −1.98357 −0.991783 − 0.127929i \(-0.959167\pi\)
−0.991783 + 0.127929i \(0.959167\pi\)
\(548\) 0 0
\(549\) 4.59263e7 0.277552
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.39757e8 −0.826414
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −5.43063e8 −3.10896
\(560\) 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
\(567\) 3.62974e8 1.99125
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) −3.71943e8 −1.99787 −0.998937 − 0.0460873i \(-0.985325\pi\)
−0.998937 + 0.0460873i \(0.985325\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −2.91790e8 −1.51895 −0.759475 − 0.650537i \(-0.774544\pi\)
−0.759475 + 0.650537i \(0.774544\pi\)
\(578\) 0 0
\(579\) −3.54080e7 −0.182417
\(580\) 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
\(589\) 3.07640e8 1.50556
\(590\) 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
\(597\) −9.07721e7 −0.426609
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) −2.91337e8 −1.34206 −0.671031 − 0.741429i \(-0.734148\pi\)
−0.671031 + 0.741429i \(0.734148\pi\)
\(602\) 0 0
\(603\) 3.00729e8 1.37159
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.60399e8 1.61145 0.805727 − 0.592287i \(-0.201775\pi\)
0.805727 + 0.592287i \(0.201775\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.58281e8 −0.687142 −0.343571 − 0.939127i \(-0.611637\pi\)
−0.343571 + 0.939127i \(0.611637\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 2.38163e8 1.00416 0.502079 − 0.864822i \(-0.332569\pi\)
0.502079 + 0.864822i \(0.332569\pi\)
\(620\) 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
\(631\) 1.97599e8 0.786498 0.393249 − 0.919432i \(-0.371351\pi\)
0.393249 + 0.919432i \(0.371351\pi\)
\(632\) 0 0
\(633\) 3.91468e8 1.54342
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.23036e9 4.76007
\(638\) 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 \(-0.735544\pi\)
−0.674277 + 0.738479i \(0.735544\pi\)
\(644\) 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
\(651\) −4.41459e8 −1.60010
\(652\) 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
\(669\) −5.98405e8 −1.99856
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −3.12399e7 −0.102486 −0.0512430 − 0.998686i \(-0.516318\pi\)
−0.0512430 + 0.998686i \(0.516318\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −1.05989e9 −3.38573
\(680\) 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
\(687\) −6.08454e8 −1.87654
\(688\) 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
\(703\) −1.14639e9 −3.29963
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 4.47139e7 0.125460 0.0627298 − 0.998031i \(-0.480019\pi\)
0.0627298 + 0.998031i \(0.480019\pi\)
\(710\) 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
\(721\) 7.69704e8 2.05361
\(722\) 0 0
\(723\) 1.41795e8 0.375187
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −2.57552e7 −0.0670287 −0.0335144 − 0.999438i \(-0.510670\pi\)
−0.0335144 + 0.999438i \(0.510670\pi\)
\(728\) 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 \(-0.752409\pi\)
−0.712438 + 0.701735i \(0.752409\pi\)
\(734\) 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
\(741\) −1.22379e9 −3.00782
\(742\) 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
\(757\) −5.67163e8 −1.30744 −0.653718 − 0.756738i \(-0.726792\pi\)
−0.653718 + 0.756738i \(0.726792\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −9.19106e7 −0.206915
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 2.75017e8 0.604756 0.302378 − 0.953188i \(-0.402220\pi\)
0.302378 + 0.953188i \(0.402220\pi\)
\(770\) 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
\(777\) 1.64505e9 3.50684
\(778\) 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
\(787\) −1.48497e8 −0.304644 −0.152322 − 0.988331i \(-0.548675\pi\)
−0.152322 + 0.988331i \(0.548675\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.22197e8 0.445574
\(794\) 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
\(811\) −1.06528e9 −1.99711 −0.998556 − 0.0537135i \(-0.982894\pi\)
−0.998556 + 0.0537135i \(0.982894\pi\)
\(812\) 0 0
\(813\) 1.05693e9 1.96687
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −1.97871e9 −3.62840
\(818\) 0 0
\(819\) 1.75612e9 3.19670
\(820\) 0 0
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 9.65480e8 1.73198 0.865992 − 0.500059i \(-0.166688\pi\)
0.865992 + 0.500059i \(0.166688\pi\)
\(824\) 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
\(831\) 4.26869e8 0.743861
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −4.71191e8 −0.803565
\(838\) 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
\(847\) 1.20998e9 1.99125
\(848\) 0 0
\(849\) 8.19753e8 1.33955
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −1.17276e9 −1.88956 −0.944782 − 0.327699i \(-0.893727\pi\)
−0.944782 + 0.327699i \(0.893727\pi\)
\(854\) 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
\(871\) 1.45497e9 2.20191
\(872\) 0 0
\(873\) −1.13127e9 −1.70030
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −1.17064e9 −1.73550 −0.867752 − 0.496997i \(-0.834436\pi\)
−0.867752 + 0.496997i \(0.834436\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −3.89149e8 −0.565241 −0.282620 − 0.959232i \(-0.591204\pi\)
−0.282620 + 0.959232i \(0.591204\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −2.69947e9 −3.84213
\(890\) 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
\(903\) 2.83942e9 3.85626
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.18340e8 0.694693 0.347347 − 0.937737i \(-0.387083\pi\)
0.347347 + 0.937737i \(0.387083\pi\)
\(908\) 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
\(919\) 6.39528e8 0.823974 0.411987 − 0.911190i \(-0.364835\pi\)
0.411987 + 0.911190i \(0.364835\pi\)
\(920\) 0 0
\(921\) 1.14268e9 1.46267
\(922\) 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
\(931\) 4.48294e9 5.55538
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −2.25938e7 −0.0274644 −0.0137322 − 0.999906i \(-0.504371\pi\)
−0.0137322 + 0.999906i \(0.504371\pi\)
\(938\) 0 0
\(939\) −5.84423e8 −0.705879
\(940\) 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
\(949\) 2.25046e9 2.63313
\(950\) 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
\(961\) −3.14428e8 −0.354284
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 6.93538e8 0.766992 0.383496 − 0.923542i \(-0.374720\pi\)
0.383496 + 0.923542i \(0.374720\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 8.63679e8 0.937592
\(974\) 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
\(981\) −9.81008e7 −0.103912
\(982\) 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
\(991\) −1.74933e9 −1.79743 −0.898715 − 0.438533i \(-0.855498\pi\)
−0.898715 + 0.438533i \(0.855498\pi\)
\(992\) 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 \(-0.696292\pi\)
−0.578322 + 0.815808i \(0.696292\pi\)
\(998\) 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.b.101.1 ✓ 1
3.2 odd 2 CM 300.7.g.b.101.1 ✓ 1
5.2 odd 4 300.7.b.a.149.2 2
5.3 odd 4 300.7.b.a.149.1 2
5.4 even 2 300.7.g.c.101.1 yes 1
15.2 even 4 300.7.b.a.149.2 2
15.8 even 4 300.7.b.a.149.1 2
15.14 odd 2 300.7.g.c.101.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
300.7.b.a.149.1 2 5.3 odd 4
300.7.b.a.149.1 2 15.8 even 4
300.7.b.a.149.2 2 5.2 odd 4
300.7.b.a.149.2 2 15.2 even 4
300.7.g.b.101.1 ✓ 1 1.1 even 1 trivial
300.7.g.b.101.1 ✓ 1 3.2 odd 2 CM
300.7.g.c.101.1 yes 1 5.4 even 2
300.7.g.c.101.1 yes 1 15.14 odd 2