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Properties

Label 207.7.d.a.91.1
Level $207$
Weight $7$
Character 207.91
Self dual yes
Analytic conductor $47.621$
Analytic rank $0$
Dimension $1$
CM discriminant -23
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [207,7,Mod(91,207)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("207.91"); S:= CuspForms(chi, 7); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(207, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 7, names="a")
 
Level: \( N \) \(=\) \( 207 = 3^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 7 \)
Character orbit: \([\chi]\) \(=\) 207.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [1,7] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(47.6211953093\)
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 91.1
Character \(\chi\) \(=\) 207.91

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+7.00000 q^{2} -15.0000 q^{4} -553.000 q^{8} +1082.00 q^{13} -2911.00 q^{16} +12167.0 q^{23} +15625.0 q^{25} +7574.00 q^{26} -30746.0 q^{29} +58754.0 q^{31} +15015.0 q^{32} -43634.0 q^{41} +85169.0 q^{46} +205342. q^{47} +117649. q^{49} +109375. q^{50} -16230.0 q^{52} -215222. q^{58} +253942. q^{59} +411278. q^{62} +291409. q^{64} -667154. q^{71} +725042. q^{73} -305438. q^{82} -182505. q^{92} +1.43739e6 q^{94} +823543. q^{98} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/207\mathbb{Z}\right)^\times\).

\(n\) \(28\) \(47\)
\(\chi(n)\) \(-1\) \(1\)

Coefficient data

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



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 7.00000 0.875000 0.437500 − 0.899218i \(-0.355864\pi\)
0.437500 + 0.899218i \(0.355864\pi\)
\(3\) 0 0
\(4\) −15.0000 −0.234375
\(5\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(6\) 0 0
\(7\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(8\) −553.000 −1.08008
\(9\) 0 0
\(10\) 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
\(16\) −2911.00 −0.710693
\(17\) 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
\(26\) 7574.00 0.430929
\(27\) 0 0
\(28\) 0 0
\(29\) −30746.0 −1.26065 −0.630325 − 0.776331i \(-0.717078\pi\)
−0.630325 + 0.776331i \(0.717078\pi\)
\(30\) 0 0
\(31\) 58754.0 1.97221 0.986103 − 0.166134i \(-0.0531284\pi\)
0.986103 + 0.166134i \(0.0531284\pi\)
\(32\) 15015.0 0.458221
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −43634.0 −0.633102 −0.316551 − 0.948576i \(-0.602525\pi\)
−0.316551 + 0.948576i \(0.602525\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 85169.0 0.875000
\(47\) 205342. 1.97781 0.988904 − 0.148555i \(-0.0474621\pi\)
0.988904 + 0.148555i \(0.0474621\pi\)
\(48\) 0 0
\(49\) 117649. 1.00000
\(50\) 109375. 0.875000
\(51\) 0 0
\(52\) −16230.0 −0.115427
\(53\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −215222. −1.10307
\(59\) 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 \(\pi\)
\(62\) 411278. 1.72568
\(63\) 0 0
\(64\) 291409. 1.11164
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −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
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) −305438. −0.553964
\(83\) 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
\(91\) 0 0
\(92\) −182505. −0.234375
\(93\) 0 0
\(94\) 1.43739e6 1.73058
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 823543. 0.875000
\(99\) 0 0
\(100\) −234375. −0.234375
\(101\) −505802. −0.490926 −0.245463 − 0.969406i \(-0.578940\pi\)
−0.245463 + 0.969406i \(0.578940\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −598346. −0.531927
\(105\) 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
\(116\) 461190. 0.295465
\(117\) 0 0
\(118\) 1.77759e6 1.08190
\(119\) 0 0
\(120\) 0 0
\(121\) 1.77156e6 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −881310. −0.462236
\(125\) 0 0
\(126\) 0 0
\(127\) 2.70490e6 1.32050 0.660252 − 0.751044i \(-0.270449\pi\)
0.660252 + 0.751044i \(0.270449\pi\)
\(128\) 1.07890e6 0.514461
\(129\) 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 \(-0.919763\pi\)
−0.968398 + 0.249411i \(0.919763\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −4.67008e6 −1.63102
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 5.07529e6 1.63081
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 6.18955e6 1.79775 0.898873 − 0.438208i \(-0.144387\pi\)
0.898873 + 0.438208i \(0.144387\pi\)
\(152\) 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 \(-0.742825\pi\)
−0.690989 + 0.722865i \(0.742825\pi\)
\(164\) 654510. 0.148383
\(165\) 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 \(-0.560759\pi\)
−0.189723 + 0.981838i \(0.560759\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 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
\(184\) −6.72835e6 −1.08008
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −3.08013e6 −0.463549
\(189\) 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
\(196\) −1.76474e6 −0.234375
\(197\) −1.49813e7 −1.95952 −0.979760 − 0.200177i \(-0.935848\pi\)
−0.979760 + 0.200177i \(0.935848\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −8.64062e6 −1.08008
\(201\) 0 0
\(202\) −3.54061e6 −0.429561
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −3.14970e6 −0.350009
\(209\) 0 0
\(210\) 0 0
\(211\) −1.26968e7 −1.35160 −0.675800 − 0.737085i \(-0.736202\pi\)
−0.675800 + 0.737085i \(0.736202\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 1.24647e6 0.112400 0.0561999 − 0.998420i \(-0.482102\pi\)
0.0561999 + 0.998420i \(0.482102\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 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
\(232\) 1.70025e7 1.36160
\(233\) −7.80984e6 −0.617411 −0.308706 − 0.951158i \(-0.599896\pi\)
−0.308706 + 0.951158i \(0.599896\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −3.80913e6 −0.289794
\(237\) 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 \(\pi\)
\(242\) 1.24009e7 0.875000
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) −3.24910e7 −2.13014
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.89343e7 1.15544
\(255\) 0 0
\(256\) −1.10979e7 −0.661484
\(257\) 2.31777e7 1.36543 0.682716 − 0.730684i \(-0.260799\pi\)
0.682716 + 0.730684i \(0.260799\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) −2.32500e7 −1.29277
\(263\) 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 \(-0.0724481\pi\)
0.974210 + 0.225643i \(0.0724481\pi\)
\(270\) 0 0
\(271\) 3.96187e7 1.99064 0.995320 − 0.0966371i \(-0.0308086\pi\)
0.995320 + 0.0966371i \(0.0308086\pi\)
\(272\) 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 \(0.872240\pi\)
\(278\) −3.64105e7 −1.69470
\(279\) 0 0
\(280\) 0 0
\(281\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 1.00073e7 0.436880
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 2.41376e7 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −1.08756e7 −0.436823
\(293\) 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
\(302\) 4.33269e7 1.57303
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 5.07075e7 1.75250 0.876248 − 0.481860i \(-0.160039\pi\)
0.876248 + 0.481860i \(0.160039\pi\)
\(308\) 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 \(-0.0865549\pi\)
0.963257 + 0.268582i \(0.0865549\pi\)
\(318\) 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
\(326\) −4.18950e7 −1.20923
\(327\) 0 0
\(328\) 2.41296e7 0.683799
\(329\) 0 0
\(330\) 0 0
\(331\) 7.25286e7 1.99998 0.999989 − 0.00477828i \(-0.00152098\pi\)
0.999989 + 0.00477828i \(0.00152098\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 4.25245e7 1.14130
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) −2.55926e7 −0.662772
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −1.37527e7 −0.332016
\(347\) −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
\(351\) 0 0
\(352\) 0 0
\(353\) 7.62365e7 1.73316 0.866580 − 0.499038i \(-0.166313\pi\)
0.866580 + 0.499038i \(0.166313\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 2.74342e7 0.597921
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 4.70459e7 1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −3.54181e7 −0.710693
\(369\) 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
\(376\) −1.13554e8 −2.13619
\(377\) −3.32672e7 −0.620857
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.79418e7 0.485839
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −6.50599e7 −1.08008
\(393\) 0 0
\(394\) −1.04869e8 −1.71458
\(395\) 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
\(400\) −4.54844e7 −0.710693
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 6.35718e7 0.971291
\(404\) 7.58703e6 0.115061
\(405\) 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
\(416\) 1.62462e7 0.225669
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −8.88779e7 −1.18265
\(423\) 0 0
\(424\) 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 \(-0.740678\pi\)
−0.686099 + 0.727509i \(0.740678\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 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
\(446\) 8.72526e6 0.0983499
\(447\) 0 0
\(448\) 0 0
\(449\) −1.58038e8 −1.74591 −0.872955 − 0.487801i \(-0.837799\pi\)
−0.872955 + 0.487801i \(0.837799\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 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
\(460\) 0 0
\(461\) 1.86172e8 1.90026 0.950129 − 0.311856i \(-0.100951\pi\)
0.950129 + 0.311856i \(0.100951\pi\)
\(462\) 0 0
\(463\) −6.20833e7 −0.625506 −0.312753 − 0.949834i \(-0.601251\pi\)
−0.312753 + 0.949834i \(0.601251\pi\)
\(464\) 8.95016e7 0.895936
\(465\) 0 0
\(466\) −5.46689e7 −0.540235
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) −1.40430e8 −1.33547
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −1.88885e8 −1.72947
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −2.65734e7 −0.234375
\(485\) 0 0
\(486\) 0 0
\(487\) 1.55985e8 1.35050 0.675252 − 0.737587i \(-0.264035\pi\)
0.675252 + 0.737587i \(0.264035\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 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
\(496\) −1.71033e8 −1.40163
\(497\) 0 0
\(498\) 0 0
\(499\) 757946. 0.00610010 0.00305005 − 0.999995i \(-0.499029\pi\)
0.00305005 + 0.999995i \(0.499029\pi\)
\(500\) 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
\(507\) 0 0
\(508\) −4.05735e7 −0.309493
\(509\) 2.00351e8 1.51928 0.759641 − 0.650343i \(-0.225375\pi\)
0.759641 + 0.650343i \(0.225375\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −1.46735e8 −1.09326
\(513\) 0 0
\(514\) 1.62244e8 1.19475
\(515\) 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
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 4.98215e7 0.346277
\(525\) 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
\(533\) −4.72120e7 −0.311796
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 2.65483e8 1.70487
\(539\) 0 0
\(540\) 0 0
\(541\) −3.15808e8 −1.99449 −0.997245 − 0.0741740i \(-0.976368\pi\)
−0.997245 + 0.0741740i \(0.976368\pi\)
\(542\) 2.77331e8 1.74181
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 1.27720e8 0.780361 0.390180 − 0.920738i \(-0.372413\pi\)
0.390180 + 0.920738i \(0.372413\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −2.73907e8 −1.61092
\(555\) 0 0
\(556\) 7.80224e7 0.453936
\(557\) 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
\(568\) 3.68936e8 2.01329
\(569\) 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 \(0.424666\pi\)
\(578\) 1.68963e8 0.875000
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −4.00948e8 −2.01303
\(585\) 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
\(591\) 0 0
\(592\) 0 0
\(593\) −2.78321e8 −1.33470 −0.667348 − 0.744746i \(-0.732571\pi\)
−0.667348 + 0.744746i \(0.732571\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 9.21529e7 0.430929
\(599\) −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
\(604\) −9.28433e7 −0.421347
\(605\) 0 0
\(606\) 0 0
\(607\) −3.69151e8 −1.65059 −0.825294 − 0.564704i \(-0.808990\pi\)
−0.825294 + 0.564704i \(0.808990\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 2.22180e8 0.974050
\(612\) 0 0
\(613\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(614\) 3.54953e8 1.53343
\(615\) 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
\(621\) 0 0
\(622\) 1.81074e8 0.752463
\(623\) 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
\(633\) 0 0
\(634\) 4.29584e8 1.68570
\(635\) 0 0
\(636\) 0 0
\(637\) 1.27296e8 0.492490
\(638\) 0 0
\(639\) 0 0
\(640\) 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
\(650\) 1.18344e8 0.430929
\(651\) 0 0
\(652\) 8.97750e7 0.323901
\(653\) 7.17455e7 0.257665 0.128832 − 0.991666i \(-0.458877\pi\)
0.128832 + 0.991666i \(0.458877\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.27019e8 0.449941
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(662\) 5.07700e8 1.74998
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −3.74087e8 −1.26065
\(668\) −9.11239e7 −0.305705
\(669\) 0 0
\(670\) 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
\(675\) 0 0
\(676\) 5.48413e7 0.177528
\(677\) 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 \(-0.893875\pi\)
−0.944934 + 0.327261i \(0.893875\pi\)
\(692\) 2.94700e7 0.0889327
\(693\) 0 0
\(694\) −4.95988e6 −0.0148386
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 6.65013e7 0.195553
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 5.33655e8 1.51652
\(707\) 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
\(716\) −5.87876e7 −0.160157
\(717\) 0 0
\(718\) 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
\(722\) 3.29321e8 0.875000
\(723\) 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
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 1.82688e8 0.458221
\(737\) 0 0
\(738\) 0 0
\(739\) 1.26013e8 0.312236 0.156118 − 0.987738i \(-0.450102\pi\)
0.156118 + 0.987738i \(0.450102\pi\)
\(740\) 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 \(\pi\)
\(752\) −5.97751e8 −1.40562
\(753\) 0 0
\(754\) −2.32870e8 −0.543250
\(755\) 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 \(-0.763659\pi\)
−0.736789 + 0.676123i \(0.763659\pi\)
\(762\) 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
\(771\) 0 0
\(772\) −5.98752e7 −0.130135
\(773\) 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
\(783\) 0 0
\(784\) −3.42476e8 −0.710693
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) 2.24719e8 0.459262
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 7.16276e8 1.43093
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 2.34609e8 0.458221
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 4.45003e8 0.849880
\(807\) 0 0
\(808\) 2.79709e8 0.530239
\(809\) 2.83551e8 0.535531 0.267766 − 0.963484i \(-0.413715\pi\)
0.267766 + 0.963484i \(0.413715\pi\)
\(810\) 0 0
\(811\) −1.05551e9 −1.97878 −0.989392 − 0.145268i \(-0.953596\pi\)
−0.989392 + 0.145268i \(0.953596\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −7.82711e8 −1.43002
\(819\) 0 0
\(820\) 0 0
\(821\) 5.08386e8 0.918679 0.459340 − 0.888261i \(-0.348086\pi\)
0.459340 + 0.888261i \(0.348086\pi\)
\(822\) 0 0
\(823\) −8.05686e8 −1.44533 −0.722664 − 0.691199i \(-0.757083\pi\)
−0.722664 + 0.691199i \(0.757083\pi\)
\(824\) 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
\(831\) 0 0
\(832\) 3.15305e8 0.547470
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 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
\(844\) 1.90453e8 0.316781
\(845\) 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 \(-0.366513\pi\)
0.407177 + 0.913349i \(0.366513\pi\)
\(858\) 0 0
\(859\) 1.26724e9 1.99931 0.999654 − 0.0262855i \(-0.00836789\pi\)
0.999654 + 0.0262855i \(0.00836789\pi\)
\(860\) 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
\(867\) 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
\(877\) −1.86924e8 −0.277119 −0.138560 − 0.990354i \(-0.544247\pi\)
−0.138560 + 0.990354i \(0.544247\pi\)
\(878\) −8.12659e8 −1.20067
\(879\) 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 \(-0.965287\pi\)
−0.994060 + 0.108838i \(0.965287\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −9.99747e8 −1.43744
\(887\) 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
\(892\) −1.86970e7 −0.0263437
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −1.10626e9 −1.52767
\(899\) −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
\(909\) 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
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.30321e9 1.66273
\(923\) −7.21861e8 −0.918012
\(924\) 0 0
\(925\) 0 0
\(926\) −4.34583e8 −0.547318
\(927\) 0 0
\(928\) −4.61651e8 −0.577657
\(929\) −8.42564e8 −1.05089 −0.525443 − 0.850829i \(-0.676100\pi\)
−0.525443 + 0.850829i \(0.676100\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.17148e8 0.144706
\(933\) 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 1.00000 \(0\)
−1.00000 \(\pi\)
\(942\) 0 0
\(943\) −5.30895e8 −0.633102
\(944\) −7.39225e8 −0.878741
\(945\) 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
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 4.04754e8 0.463252
\(957\) 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 \(-0.909593\pi\)
−0.959936 + 0.280219i \(0.909593\pi\)
\(968\) −9.79673e8 −1.08008
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1.09189e9 1.18169
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −1.33529e9 −1.41007
\(983\) 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 \(-0.906529\pi\)
−0.957194 + 0.289446i \(0.906529\pi\)
\(992\) 8.82191e8 0.903707
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.64078e9 1.65564 0.827819 − 0.560995i \(-0.189581\pi\)
0.827819 + 0.560995i \(0.189581\pi\)
\(998\) 5.30562e6 0.00533758
\(999\) 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 207.7.d.a.91.1 1
3.2 odd 2 23.7.b.a.22.1 ✓ 1
12.11 even 2 368.7.f.a.321.1 1
23.22 odd 2 CM 207.7.d.a.91.1 1
69.68 even 2 23.7.b.a.22.1 ✓ 1
276.275 odd 2 368.7.f.a.321.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
23.7.b.a.22.1 ✓ 1 3.2 odd 2
23.7.b.a.22.1 ✓ 1 69.68 even 2
207.7.d.a.91.1 1 1.1 even 1 trivial
207.7.d.a.91.1 1 23.22 odd 2 CM
368.7.f.a.321.1 1 12.11 even 2
368.7.f.a.321.1 1 276.275 odd 2