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Properties

Label 368.7.f.a.321.1
Level $368$
Weight $7$
Character 368.321
Self dual yes
Analytic conductor $84.660$
Analytic rank $0$
Dimension $1$
CM discriminant -23
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] = [368,7,Mod(321,368)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(368, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1]))
 
N = Newforms(chi, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("368.321");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 368 = 2^{4} \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 368.f (of order \(2\), degree \(1\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(84.6599027721\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 23)
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 321.1
Character \(\chi\) \(=\) 368.321

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+38.0000 q^{3} +715.000 q^{9} +O(q^{10})\) \(q+38.0000 q^{3} +715.000 q^{9} +1082.00 q^{13} +12167.0 q^{23} +15625.0 q^{25} -532.000 q^{27} +30746.0 q^{29} -58754.0 q^{31} +41116.0 q^{39} +43634.0 q^{41} +205342. q^{47} +117649. q^{49} +253942. q^{59} +462346. q^{69} -667154. q^{71} +725042. q^{73} +593750. q^{75} -541451. q^{81} +1.16835e6 q^{87} -2.23265e6 q^{93} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/368\mathbb{Z}\right)^\times\).

\(n\) \(47\) \(97\) \(277\)
\(\chi(n)\) \(1\) \(-1\) \(1\)

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
\(3\) 38.0000 1.40741 0.703704 − 0.710494i \(-0.251528\pi\)
0.703704 + 0.710494i \(0.251528\pi\)
\(4\) 0 0
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) 0 0
\(9\) 715.000 0.980796
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 1082.00 0.492490 0.246245 − 0.969208i \(-0.420803\pi\)
0.246245 + 0.969208i \(0.420803\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 12167.0 1.00000
\(24\) 0 0
\(25\) 15625.0 1.00000
\(26\) 0 0
\(27\) −532.000 −0.0270284
\(28\) 0 0
\(29\) 30746.0 1.26065 0.630325 − 0.776331i \(-0.282922\pi\)
0.630325 + 0.776331i \(0.282922\pi\)
\(30\) 0 0
\(31\) −58754.0 −1.97221 −0.986103 − 0.166134i \(-0.946872\pi\)
−0.986103 + 0.166134i \(0.946872\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 41116.0 0.693134
\(40\) 0 0
\(41\) 43634.0 0.633102 0.316551 − 0.948576i \(-0.397475\pi\)
0.316551 + 0.948576i \(0.397475\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 205342. 1.97781 0.988904 − 0.148555i \(-0.0474621\pi\)
0.988904 + 0.148555i \(0.0474621\pi\)
\(48\) 0 0
\(49\) 117649. 1.00000
\(50\) 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
\(57\) 0 0
\(58\) 0 0
\(59\) 253942. 1.23646 0.618228 − 0.785999i \(-0.287851\pi\)
0.618228 + 0.785999i \(0.287851\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 462346. 1.40741
\(70\) 0 0
\(71\) −667154. −1.86402 −0.932011 − 0.362430i \(-0.881947\pi\)
−0.932011 + 0.362430i \(0.881947\pi\)
\(72\) 0 0
\(73\) 725042. 1.86378 0.931890 − 0.362741i \(-0.118159\pi\)
0.931890 + 0.362741i \(0.118159\pi\)
\(74\) 0 0
\(75\) 593750. 1.40741
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −541451. −1.01884
\(82\) 0 0
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 1.16835e6 1.77425
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −2.23265e6 −2.77570
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 505802. 0.490926 0.245463 − 0.969406i \(-0.421060\pi\)
0.245463 + 0.969406i \(0.421060\pi\)
\(102\) 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
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 773630. 0.483032
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.77156e6 1.00000
\(122\) 0 0
\(123\) 1.65809e6 0.891032
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) −2.70490e6 −1.32050 −0.660252 − 0.751044i \(-0.729551\pi\)
−0.660252 + 0.751044i \(0.729551\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) −3.32143e6 −1.47745 −0.738723 − 0.674009i \(-0.764571\pi\)
−0.738723 + 0.674009i \(0.764571\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 5.20149e6 1.93680 0.968398 − 0.249411i \(-0.0802372\pi\)
0.968398 + 0.249411i \(0.0802372\pi\)
\(140\) 0 0
\(141\) 7.80300e6 2.78358
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 4.47066e6 1.40741
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) −6.18955e6 −1.79775 −0.898873 − 0.438208i \(-0.855613\pi\)
−0.898873 + 0.438208i \(0.855613\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 5.98500e6 1.38198 0.690989 − 0.722865i \(-0.257175\pi\)
0.690989 + 0.722865i \(0.257175\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 6.07493e6 1.30434 0.652171 − 0.758072i \(-0.273858\pi\)
0.652171 + 0.758072i \(0.273858\pi\)
\(168\) 0 0
\(169\) −3.65608e6 −0.757454
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 1.96467e6 0.379446 0.189723 − 0.981838i \(-0.439241\pi\)
0.189723 + 0.981838i \(0.439241\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 9.64980e6 1.74020
\(178\) 0 0
\(179\) 3.91917e6 0.683338 0.341669 − 0.939820i \(-0.389008\pi\)
0.341669 + 0.939820i \(0.389008\pi\)
\(180\) 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
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 3.99168e6 0.555244 0.277622 − 0.960690i \(-0.410454\pi\)
0.277622 + 0.960690i \(0.410454\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.49813e7 1.95952 0.979760 − 0.200177i \(-0.0641517\pi\)
0.979760 + 0.200177i \(0.0641517\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 8.69940e6 0.980796
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 1.26968e7 1.35160 0.675800 − 0.737085i \(-0.263798\pi\)
0.675800 + 0.737085i \(0.263798\pi\)
\(212\) 0 0
\(213\) −2.53519e7 −2.62344
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 2.75516e7 2.62310
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.24647e6 −0.112400 −0.0561999 − 0.998420i \(-0.517898\pi\)
−0.0561999 + 0.998420i \(0.517898\pi\)
\(224\) 0 0
\(225\) 1.11719e7 0.980796
\(226\) 0 0
\(227\) 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 0
\(233\) 7.80984e6 0.617411 0.308706 − 0.951158i \(-0.400104\pi\)
0.308706 + 0.951158i \(0.400104\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.69836e7 −1.97654 −0.988271 − 0.152712i \(-0.951199\pi\)
−0.988271 + 0.152712i \(0.951199\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −2.01873e7 −1.40689
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 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
\(257\) −2.31777e7 −1.36543 −0.682716 − 0.730684i \(-0.739201\pi\)
−0.682716 + 0.730684i \(0.739201\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 2.19834e7 1.23644
\(262\) 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
\(269\) −3.79262e7 −1.94842 −0.974210 − 0.225643i \(-0.927552\pi\)
−0.974210 + 0.225643i \(0.927552\pi\)
\(270\) 0 0
\(271\) −3.96187e7 −1.99064 −0.995320 − 0.0966371i \(-0.969191\pi\)
−0.995320 + 0.0966371i \(0.969191\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −3.91296e7 −1.84105 −0.920527 − 0.390680i \(-0.872240\pi\)
−0.920527 + 0.390680i \(0.872240\pi\)
\(278\) 0 0
\(279\) −4.20091e7 −1.93433
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 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
\(289\) 2.41376e7 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 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
\(299\) 1.31647e7 0.492490
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 1.92205e7 0.690934
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) −5.07075e7 −1.75250 −0.876248 − 0.481860i \(-0.839961\pi\)
−0.876248 + 0.481860i \(0.839961\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 2.58677e7 0.859958 0.429979 − 0.902839i \(-0.358521\pi\)
0.429979 + 0.902839i \(0.358521\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) −6.13691e7 −1.92651 −0.963257 − 0.268582i \(-0.913445\pi\)
−0.963257 + 0.268582i \(0.913445\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 1.69062e7 0.492490
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −7.25286e7 −1.99998 −0.999989 − 0.00477828i \(-0.998479\pi\)
−0.999989 + 0.00477828i \(0.998479\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 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
\(347\) −708554. −0.0169584 −0.00847919 − 0.999964i \(-0.502699\pi\)
−0.00847919 + 0.999964i \(0.502699\pi\)
\(348\) 0 0
\(349\) 9.50019e6 0.223489 0.111744 − 0.993737i \(-0.464356\pi\)
0.111744 + 0.993737i \(0.464356\pi\)
\(350\) 0 0
\(351\) −575624. −0.0133112
\(352\) 0 0
\(353\) −7.62365e7 −1.73316 −0.866580 − 0.499038i \(-0.833687\pi\)
−0.866580 + 0.499038i \(0.833687\pi\)
\(354\) 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
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) 6.73193e7 1.40741
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 3.11983e7 0.620943
\(370\) 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
\(377\) 3.32672e7 0.620857
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −1.02786e8 −1.85849
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) −1.26214e8 −2.07937
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.02325e8 1.63535 0.817676 − 0.575679i \(-0.195262\pi\)
0.817676 + 0.575679i \(0.195262\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −6.35718e7 −0.971291
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.11816e8 −1.63431 −0.817153 − 0.576421i \(-0.804449\pi\)
−0.817153 + 0.576421i \(0.804449\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.97657e8 2.72586
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 1.46820e8 1.93983
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 1.16094e8 1.37220 0.686099 − 0.727509i \(-0.259322\pi\)
0.686099 + 0.727509i \(0.259322\pi\)
\(440\) 0 0
\(441\) 8.41190e7 0.980796
\(442\) 0 0
\(443\) −1.42821e8 −1.64279 −0.821393 − 0.570363i \(-0.806803\pi\)
−0.821393 + 0.570363i \(0.806803\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.58038e8 1.74591 0.872955 − 0.487801i \(-0.162201\pi\)
0.872955 + 0.487801i \(0.162201\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −2.35203e8 −2.53016
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.86172e8 −1.90026 −0.950129 − 0.311856i \(-0.899049\pi\)
−0.950129 + 0.311856i \(0.899049\pi\)
\(462\) 0 0
\(463\) 6.20833e7 0.625506 0.312753 − 0.949834i \(-0.398749\pi\)
0.312753 + 0.949834i \(0.398749\pi\)
\(464\) 0 0
\(465\) 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
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) −1.55985e8 −1.35050 −0.675252 − 0.737587i \(-0.735965\pi\)
−0.675252 + 0.737587i \(0.735965\pi\)
\(488\) 0 0
\(489\) 2.27430e8 1.94501
\(490\) 0 0
\(491\) −1.90755e8 −1.61151 −0.805754 − 0.592250i \(-0.798240\pi\)
−0.805754 + 0.592250i \(0.798240\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −757946. −0.00610010 −0.00305005 − 0.999995i \(-0.500971\pi\)
−0.00305005 + 0.999995i \(0.500971\pi\)
\(500\) 0 0
\(501\) 2.30847e8 1.83574
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −1.38931e8 −1.06605
\(508\) 0 0
\(509\) −2.00351e8 −1.51928 −0.759641 − 0.650343i \(-0.774625\pi\)
−0.759641 + 0.650343i \(0.774625\pi\)
\(510\) 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
\(519\) 7.46573e7 0.534036
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.48036e8 1.00000
\(530\) 0 0
\(531\) 1.81569e8 1.21271
\(532\) 0 0
\(533\) 4.72120e7 0.311796
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 1.48929e8 0.961735
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3.15808e8 −1.99449 −0.997245 − 0.0741740i \(-0.976368\pi\)
−0.997245 + 0.0741740i \(0.976368\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.27720e8 −0.780361 −0.390180 − 0.920738i \(-0.627587\pi\)
−0.390180 + 0.920738i \(0.627587\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 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
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.90109e8 1.00000
\(576\) 0 0
\(577\) 9.00812e7 0.468929 0.234464 − 0.972125i \(-0.424666\pi\)
0.234464 + 0.972125i \(0.424666\pi\)
\(578\) 0 0
\(579\) 1.51684e8 0.781455
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 3.99910e8 1.97719 0.988594 − 0.150604i \(-0.0481217\pi\)
0.988594 + 0.150604i \(0.0481217\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 5.69288e8 2.75784
\(592\) 0 0
\(593\) 2.78321e8 1.33470 0.667348 − 0.744746i \(-0.267429\pi\)
0.667348 + 0.744746i \(0.267429\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.62397e8 −0.755611 −0.377806 − 0.925885i \(-0.623321\pi\)
−0.377806 + 0.925885i \(0.623321\pi\)
\(600\) 0 0
\(601\) −4.34057e8 −1.99951 −0.999755 − 0.0221248i \(-0.992957\pi\)
−0.999755 + 0.0221248i \(0.992957\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 3.69151e8 1.65059 0.825294 − 0.564704i \(-0.191010\pi\)
0.825294 + 0.564704i \(0.191010\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.22180e8 0.974050
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) −6.47284e6 −0.0270284
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 2.44141e8 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 4.82480e8 1.90225
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 1.27296e8 0.492490
\(638\) 0 0
\(639\) −4.77015e8 −1.82822
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −2.76738e8 −1.02178 −0.510888 − 0.859648i \(-0.670683\pi\)
−0.510888 + 0.859648i \(0.670683\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −7.17455e7 −0.257665 −0.128832 − 0.991666i \(-0.541123\pi\)
−0.128832 + 0.991666i \(0.541123\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 5.18405e8 1.82799
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 3.74087e8 1.26065
\(668\) 0 0
\(669\) −4.73657e7 −0.158192
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 5.18377e8 1.70059 0.850297 − 0.526304i \(-0.176422\pi\)
0.850297 + 0.526304i \(0.176422\pi\)
\(674\) 0 0
\(675\) −8.31250e6 −0.0270284
\(676\) 0 0
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −6.13389e8 −1.92519 −0.962595 − 0.270945i \(-0.912664\pi\)
−0.962595 + 0.270945i \(0.912664\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 6.23542e8 1.88987 0.944934 − 0.327261i \(-0.106125\pi\)
0.944934 + 0.327261i \(0.106125\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 2.96774e8 0.868949
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) −7.14860e8 −1.97221
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) −1.02538e9 −2.78180
\(718\) 0 0
\(719\) −6.96357e8 −1.87346 −0.936732 − 0.350047i \(-0.886166\pi\)
−0.936732 + 0.350047i \(0.886166\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 4.80406e8 1.26065
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −3.72400e8 −0.961229
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −1.26013e8 −0.312236 −0.156118 − 0.987738i \(-0.549898\pi\)
−0.156118 + 0.987738i \(0.549898\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 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
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 6.49422e8 1.47358 0.736789 − 0.676123i \(-0.236341\pi\)
0.736789 + 0.676123i \(0.236341\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.74765e8 0.608942
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −8.80751e8 −1.92172
\(772\) 0 0
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) −9.18031e8 −1.97221
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) −1.63569e7 −0.0340734
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 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
\(807\) −1.44120e9 −2.74222
\(808\) 0 0
\(809\) −2.83551e8 −0.535531 −0.267766 − 0.963484i \(-0.586285\pi\)
−0.267766 + 0.963484i \(0.586285\pi\)
\(810\) 0 0
\(811\) 1.05551e9 1.97878 0.989392 − 0.145268i \(-0.0464044\pi\)
0.989392 + 0.145268i \(0.0464044\pi\)
\(812\) 0 0
\(813\) −1.50551e9 −2.80164
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −5.08386e8 −0.918679 −0.459340 − 0.888261i \(-0.651914\pi\)
−0.459340 + 0.888261i \(0.651914\pi\)
\(822\) 0 0
\(823\) 8.05686e8 1.44533 0.722664 − 0.691199i \(-0.242917\pi\)
0.722664 + 0.691199i \(0.242917\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) −1.11478e9 −1.95671 −0.978357 − 0.206925i \(-0.933655\pi\)
−0.978357 + 0.206925i \(0.933655\pi\)
\(830\) 0 0
\(831\) −1.48693e9 −2.59111
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 3.12571e7 0.0533056
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 3.50493e8 0.589239
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −6.36633e8 −1.02575 −0.512876 − 0.858463i \(-0.671420\pi\)
−0.512876 + 0.858463i \(0.671420\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −5.12572e8 −0.814353 −0.407177 − 0.913349i \(-0.633487\pi\)
−0.407177 + 0.913349i \(0.633487\pi\)
\(858\) 0 0
\(859\) −1.26724e9 −1.99931 −0.999654 − 0.0262855i \(-0.991632\pi\)
−0.999654 + 0.0262855i \(0.991632\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −4.00781e8 −0.623555 −0.311777 − 0.950155i \(-0.600924\pi\)
−0.311777 + 0.950155i \(0.600924\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 9.17228e8 1.40741
\(868\) 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
\(877\) −1.86924e8 −0.277119 −0.138560 − 0.990354i \(-0.544247\pi\)
−0.138560 + 0.990354i \(0.544247\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.36875e9 1.98812 0.994060 − 0.108838i \(-0.0347129\pi\)
0.994060 + 0.108838i \(0.0347129\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 1.39031e9 1.99223 0.996117 − 0.0880391i \(-0.0280600\pi\)
0.996117 + 0.0880391i \(0.0280600\pi\)
\(888\) 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
\(897\) 5.00258e8 0.693134
\(898\) 0 0
\(899\) −1.80645e9 −2.48626
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 3.61648e8 0.481498
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) −1.92689e9 −2.46648
\(922\) 0 0
\(923\) −7.21861e8 −0.918012
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 8.42564e8 1.05089 0.525443 − 0.850829i \(-0.323900\pi\)
0.525443 + 0.850829i \(0.323900\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 9.82974e8 1.21031
\(934\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) 5.30895e8 0.633102
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 1.32946e9 1.56539 0.782697 − 0.622403i \(-0.213843\pi\)
0.782697 + 0.622403i \(0.213843\pi\)
\(948\) 0 0
\(949\) 7.84495e8 0.917892
\(950\) 0 0
\(951\) −2.33203e9 −2.71139
\(952\) 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
\(961\) 2.56453e9 2.88960
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 1.73601e9 1.91987 0.959936 − 0.280219i \(-0.0904071\pi\)
0.959936 + 0.280219i \(0.0904071\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 6.42438e8 0.693134
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 1.86316e9 1.91439 0.957194 − 0.289446i \(-0.0934710\pi\)
0.957194 + 0.289446i \(0.0934710\pi\)
\(992\) 0 0
\(993\) −2.75609e9 −2.81478
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.64078e9 1.65564 0.827819 − 0.560995i \(-0.189581\pi\)
0.827819 + 0.560995i \(0.189581\pi\)
\(998\) 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 368.7.f.a.321.1 1
4.3 odd 2 23.7.b.a.22.1 ✓ 1
12.11 even 2 207.7.d.a.91.1 1
23.22 odd 2 CM 368.7.f.a.321.1 1
92.91 even 2 23.7.b.a.22.1 ✓ 1
276.275 odd 2 207.7.d.a.91.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.7.b.a.22.1 ✓ 1 4.3 odd 2
23.7.b.a.22.1 ✓ 1 92.91 even 2
207.7.d.a.91.1 1 12.11 even 2
207.7.d.a.91.1 1 276.275 odd 2
368.7.f.a.321.1 1 1.1 even 1 trivial
368.7.f.a.321.1 1 23.22 odd 2 CM