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Properties

Label 91.5.b.a.90.1
Level $91$
Weight $5$
Character 91.90
Self dual yes
Analytic conductor $9.407$
Analytic rank $0$
Dimension $1$
CM discriminant -91
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] = [91,5,Mod(90,91)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(91, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 5, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.90");
 
S:= CuspForms(chi, 5);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 5 \)
Character orbit: \([\chi]\) \(=\) 91.b (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: \(9.40666664063\)
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 90.1
Character \(\chi\) \(=\) 91.90

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+16.0000 q^{4} -41.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} +O(q^{10})\) \(q+16.0000 q^{4} -41.0000 q^{5} +49.0000 q^{7} +81.0000 q^{9} +169.000 q^{13} +256.000 q^{16} -97.0000 q^{19} -656.000 q^{20} +967.000 q^{23} +1056.00 q^{25} +784.000 q^{28} -593.000 q^{29} +1103.00 q^{31} -2009.00 q^{35} +1296.00 q^{36} -2462.00 q^{41} -3673.00 q^{43} -3321.00 q^{45} +2143.00 q^{47} +2401.00 q^{49} +2704.00 q^{52} -5393.00 q^{53} +1138.00 q^{59} +3969.00 q^{63} +4096.00 q^{64} -6929.00 q^{65} -9817.00 q^{73} -1552.00 q^{76} -7993.00 q^{79} -10496.0 q^{80} +6561.00 q^{81} +11503.0 q^{83} +11383.0 q^{89} +8281.00 q^{91} +15472.0 q^{92} +3977.00 q^{95} -1657.00 q^{97} +O(q^{100})\)

Display 100 coefficients 10 coefficients

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/91\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(66\)
\(\chi(n)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(3\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(4\) 16.0000 1.00000
\(5\) −41.0000 −1.64000 −0.820000 − 0.572364i \(-0.806027\pi\)
−0.820000 + 0.572364i \(0.806027\pi\)
\(6\) 0 0
\(7\) 49.0000 1.00000
\(8\) 0 0
\(9\) 81.0000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 169.000 1.00000
\(14\) 0 0
\(15\) 0 0
\(16\) 256.000 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −97.0000 −0.268698 −0.134349 − 0.990934i \(-0.542894\pi\)
−0.134349 + 0.990934i \(0.542894\pi\)
\(20\) −656.000 −1.64000
\(21\) 0 0
\(22\) 0 0
\(23\) 967.000 1.82798 0.913989 − 0.405740i \(-0.132986\pi\)
0.913989 + 0.405740i \(0.132986\pi\)
\(24\) 0 0
\(25\) 1056.00 1.68960
\(26\) 0 0
\(27\) 0 0
\(28\) 784.000 1.00000
\(29\) −593.000 −0.705113 −0.352556 − 0.935791i \(-0.614688\pi\)
−0.352556 + 0.935791i \(0.614688\pi\)
\(30\) 0 0
\(31\) 1103.00 1.14776 0.573881 − 0.818938i \(-0.305437\pi\)
0.573881 + 0.818938i \(0.305437\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −2009.00 −1.64000
\(36\) 1296.00 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2462.00 −1.46460 −0.732302 − 0.680980i \(-0.761554\pi\)
−0.732302 + 0.680980i \(0.761554\pi\)
\(42\) 0 0
\(43\) −3673.00 −1.98648 −0.993240 − 0.116082i \(-0.962966\pi\)
−0.993240 + 0.116082i \(0.962966\pi\)
\(44\) 0 0
\(45\) −3321.00 −1.64000
\(46\) 0 0
\(47\) 2143.00 0.970122 0.485061 − 0.874480i \(-0.338797\pi\)
0.485061 + 0.874480i \(0.338797\pi\)
\(48\) 0 0
\(49\) 2401.00 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) 2704.00 1.00000
\(53\) −5393.00 −1.91990 −0.959950 − 0.280171i \(-0.909609\pi\)
−0.959950 + 0.280171i \(0.909609\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 1138.00 0.326918 0.163459 − 0.986550i \(-0.447735\pi\)
0.163459 + 0.986550i \(0.447735\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 3969.00 1.00000
\(64\) 4096.00 1.00000
\(65\) −6929.00 −1.64000
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0 0
\(73\) −9817.00 −1.84218 −0.921092 − 0.389345i \(-0.872702\pi\)
−0.921092 + 0.389345i \(0.872702\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −1552.00 −0.268698
\(77\) 0 0
\(78\) 0 0
\(79\) −7993.00 −1.28072 −0.640362 − 0.768073i \(-0.721216\pi\)
−0.640362 + 0.768073i \(0.721216\pi\)
\(80\) −10496.0 −1.64000
\(81\) 6561.00 1.00000
\(82\) 0 0
\(83\) 11503.0 1.66976 0.834882 − 0.550429i \(-0.185536\pi\)
0.834882 + 0.550429i \(0.185536\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 11383.0 1.43707 0.718533 − 0.695493i \(-0.244814\pi\)
0.718533 + 0.695493i \(0.244814\pi\)
\(90\) 0 0
\(91\) 8281.00 1.00000
\(92\) 15472.0 1.82798
\(93\) 0 0
\(94\) 0 0
\(95\) 3977.00 0.440665
\(96\) 0 0
\(97\) −1657.00 −0.176108 −0.0880540 − 0.996116i \(-0.528065\pi\)
−0.0880540 + 0.996116i \(0.528065\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 16896.0 1.68960
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) −398.000 −0.0347629 −0.0173814 − 0.999849i \(-0.505533\pi\)
−0.0173814 + 0.999849i \(0.505533\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 12544.0 1.00000
\(113\) −7313.00 −0.572715 −0.286358 − 0.958123i \(-0.592445\pi\)
−0.286358 + 0.958123i \(0.592445\pi\)
\(114\) 0 0
\(115\) −39647.0 −2.99788
\(116\) −9488.00 −0.705113
\(117\) 13689.0 1.00000
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 14641.0 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 17648.0 1.14776
\(125\) −17671.0 −1.13094
\(126\) 0 0
\(127\) −20158.0 −1.24980 −0.624899 − 0.780705i \(-0.714860\pi\)
−0.624899 + 0.780705i \(0.714860\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −4753.00 −0.268698
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) −32144.0 −1.64000
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 20736.0 1.00000
\(145\) 24313.0 1.15639
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −45223.0 −1.88233
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 47383.0 1.82798
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −39392.0 −1.46460
\(165\) 0 0
\(166\) 0 0
\(167\) −55697.0 −1.99710 −0.998548 − 0.0538726i \(-0.982843\pi\)
−0.998548 + 0.0538726i \(0.982843\pi\)
\(168\) 0 0
\(169\) 28561.0 1.00000
\(170\) 0 0
\(171\) −7857.00 −0.268698
\(172\) −58768.0 −1.98648
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 51744.0 1.68960
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 7207.00 0.224931 0.112465 − 0.993656i \(-0.464125\pi\)
0.112465 + 0.993656i \(0.464125\pi\)
\(180\) −53136.0 −1.64000
\(181\) 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
\(188\) 34288.0 0.970122
\(189\) 0 0
\(190\) 0 0
\(191\) −72638.0 −1.99112 −0.995559 − 0.0941362i \(-0.969991\pi\)
−0.995559 + 0.0941362i \(0.969991\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 38416.0 1.00000
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −29057.0 −0.705113
\(204\) 0 0
\(205\) 100942. 2.40195
\(206\) 0 0
\(207\) 78327.0 1.82798
\(208\) 43264.0 1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) 68567.0 1.54010 0.770052 − 0.637981i \(-0.220230\pi\)
0.770052 + 0.637981i \(0.220230\pi\)
\(212\) −86288.0 −1.91990
\(213\) 0 0
\(214\) 0 0
\(215\) 150593. 3.25783
\(216\) 0 0
\(217\) 54047.0 1.14776
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −84817.0 −1.70558 −0.852792 − 0.522250i \(-0.825093\pi\)
−0.852792 + 0.522250i \(0.825093\pi\)
\(224\) 0 0
\(225\) 85536.0 1.68960
\(226\) 0 0
\(227\) −42542.0 −0.825593 −0.412797 − 0.910823i \(-0.635448\pi\)
−0.412797 + 0.910823i \(0.635448\pi\)
\(228\) 0 0
\(229\) −104782. −1.99809 −0.999047 − 0.0436577i \(-0.986099\pi\)
−0.999047 + 0.0436577i \(0.986099\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 32047.0 0.590304 0.295152 − 0.955450i \(-0.404630\pi\)
0.295152 + 0.955450i \(0.404630\pi\)
\(234\) 0 0
\(235\) −87863.0 −1.59100
\(236\) 18208.0 0.326918
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 17063.0 0.293779 0.146890 − 0.989153i \(-0.453074\pi\)
0.146890 + 0.989153i \(0.453074\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −98441.0 −1.64000
\(246\) 0 0
\(247\) −16393.0 −0.268698
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 63504.0 1.00000
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 65536.0 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −110864. −1.64000
\(261\) −48033.0 −0.705113
\(262\) 0 0
\(263\) 50887.0 0.735691 0.367845 − 0.929887i \(-0.380096\pi\)
0.367845 + 0.929887i \(0.380096\pi\)
\(264\) 0 0
\(265\) 221113. 3.14864
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −62782.0 −0.854863 −0.427432 − 0.904048i \(-0.640581\pi\)
−0.427432 + 0.904048i \(0.640581\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −83233.0 −1.08477 −0.542383 − 0.840131i \(-0.682478\pi\)
−0.542383 + 0.840131i \(0.682478\pi\)
\(278\) 0 0
\(279\) 89343.0 1.14776
\(280\) 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
\(287\) −120638. −1.46460
\(288\) 0 0
\(289\) 83521.0 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −157072. −1.84218
\(293\) −103577. −1.20650 −0.603251 − 0.797551i \(-0.706128\pi\)
−0.603251 + 0.797551i \(0.706128\pi\)
\(294\) 0 0
\(295\) −46658.0 −0.536145
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 163423. 1.82798
\(300\) 0 0
\(301\) −179977. −1.98648
\(302\) 0 0
\(303\) 0 0
\(304\) −24832.0 −0.268698
\(305\) 0 0
\(306\) 0 0
\(307\) 4223.00 0.0448068 0.0224034 − 0.999749i \(-0.492868\pi\)
0.0224034 + 0.999749i \(0.492868\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(314\) 0 0
\(315\) −162729. −1.64000
\(316\) −127888. −1.28072
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −167936. −1.64000
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 104976. 1.00000
\(325\) 178464. 1.68960
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 105007. 0.970122
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 184048. 1.66976
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −206113. −1.81487 −0.907435 − 0.420192i \(-0.861963\pi\)
−0.907435 + 0.420192i \(0.861963\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 117649. 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −131918. −1.09558 −0.547791 − 0.836615i \(-0.684531\pi\)
−0.547791 + 0.836615i \(0.684531\pi\)
\(348\) 0 0
\(349\) −117577. −0.965320 −0.482660 − 0.875808i \(-0.660329\pi\)
−0.482660 + 0.875808i \(0.660329\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 103618. 0.831545 0.415773 − 0.909469i \(-0.363511\pi\)
0.415773 + 0.909469i \(0.363511\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 182128. 1.43707
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) −120912. −0.927801
\(362\) 0 0
\(363\) 0 0
\(364\) 132496. 1.00000
\(365\) 402497. 3.02118
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 247552. 1.82798
\(369\) −199422. −1.46460
\(370\) 0 0
\(371\) −264257. −1.91990
\(372\) 0 0
\(373\) 225842. 1.62326 0.811628 − 0.584175i \(-0.198582\pi\)
0.811628 + 0.584175i \(0.198582\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −100217. −0.705113
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 63632.0 0.440665
\(381\) 0 0
\(382\) 0 0
\(383\) −289022. −1.97030 −0.985152 − 0.171683i \(-0.945080\pi\)
−0.985152 + 0.171683i \(0.945080\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −297513. −1.98648
\(388\) −26512.0 −0.176108
\(389\) 157042. 1.03781 0.518904 − 0.854833i \(-0.326340\pi\)
0.518904 + 0.854833i \(0.326340\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 327713. 2.10039
\(396\) 0 0
\(397\) 294743. 1.87009 0.935045 − 0.354529i \(-0.115359\pi\)
0.935045 + 0.354529i \(0.115359\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 270336. 1.68960
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 186407. 1.14776
\(404\) 0 0
\(405\) −269001. −1.64000
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 235463. 1.40759 0.703795 − 0.710403i \(-0.251487\pi\)
0.703795 + 0.710403i \(0.251487\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 55762.0 0.326918
\(414\) 0 0
\(415\) −471623. −2.73841
\(416\) 0 0
\(417\) 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 \(\pi\)
\(422\) 0 0
\(423\) 173583. 0.970122
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −6368.00 −0.0347629
\(429\) 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
\(437\) −93799.0 −0.491174
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 194481. 1.00000
\(442\) 0 0
\(443\) 239527. 1.22053 0.610263 − 0.792199i \(-0.291064\pi\)
0.610263 + 0.792199i \(0.291064\pi\)
\(444\) 0 0
\(445\) −466703. −2.35679
\(446\) 0 0
\(447\) 0 0
\(448\) 200704. 1.00000
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −117008. −0.572715
\(453\) 0 0
\(454\) 0 0
\(455\) −339521. −1.64000
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −634352. −2.99788
\(461\) 331858. 1.56153 0.780765 − 0.624825i \(-0.214830\pi\)
0.780765 + 0.624825i \(0.214830\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −151808. −0.705113
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) 219024. 1.00000
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −102432. −0.453992
\(476\) 0 0
\(477\) −436833. −1.91990
\(478\) 0 0
\(479\) −80657.0 −0.351537 −0.175768 − 0.984432i \(-0.556241\pi\)
−0.175768 + 0.984432i \(0.556241\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 234256. 1.00000
\(485\) 67937.0 0.288817
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −100238. −0.415786 −0.207893 − 0.978152i \(-0.566661\pi\)
−0.207893 + 0.978152i \(0.566661\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 282368. 1.14776
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −282736. −1.13094
\(501\) 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
\(508\) −322528. −1.24980
\(509\) 491863. 1.89849 0.949246 − 0.314536i \(-0.101849\pi\)
0.949246 + 0.314536i \(0.101849\pi\)
\(510\) 0 0
\(511\) −481033. −1.84218
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−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
\(529\) 655248. 2.34150
\(530\) 0 0
\(531\) 92178.0 0.326918
\(532\) −76048.0 −0.268698
\(533\) −416078. −1.46460
\(534\) 0 0
\(535\) 16318.0 0.0570111
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −522793. −1.74725 −0.873625 − 0.486600i \(-0.838237\pi\)
−0.873625 + 0.486600i \(0.838237\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 57521.0 0.189462
\(552\) 0 0
\(553\) −391657. −1.28072
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −620737. −1.98648
\(560\) −514304. −1.64000
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 299833. 0.939253
\(566\) 0 0
\(567\) 321489. 1.00000
\(568\) 0 0
\(569\) 645247. 1.99297 0.996487 − 0.0837523i \(-0.0266904\pi\)
0.996487 + 0.0837523i \(0.0266904\pi\)
\(570\) 0 0
\(571\) 631607. 1.93720 0.968601 − 0.248622i \(-0.0799777\pi\)
0.968601 + 0.248622i \(0.0799777\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 1.02115e6 3.08855
\(576\) 331776. 1.00000
\(577\) −644542. −1.93597 −0.967987 − 0.251000i \(-0.919241\pi\)
−0.967987 + 0.251000i \(0.919241\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 389008. 1.15639
\(581\) 563647. 1.66976
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −561249. −1.64000
\(586\) 0 0
\(587\) 31663.0 0.0918916 0.0459458 − 0.998944i \(-0.485370\pi\)
0.0459458 + 0.998944i \(0.485370\pi\)
\(588\) 0 0
\(589\) −106991. −0.308402
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 318823. 0.906651 0.453326 − 0.891345i \(-0.350237\pi\)
0.453326 + 0.891345i \(0.350237\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 442327. 1.23279 0.616396 − 0.787436i \(-0.288592\pi\)
0.616396 + 0.787436i \(0.288592\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −600281. −1.64000
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 362167. 0.970122
\(612\) 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
\(619\) 294578. 0.768810 0.384405 − 0.923165i \(-0.374407\pi\)
0.384405 + 0.923165i \(0.374407\pi\)
\(620\) −723568. −1.88233
\(621\) 0 0
\(622\) 0 0
\(623\) 557767. 1.43707
\(624\) 0 0
\(625\) 64511.0 0.165148
\(626\) 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
\(634\) 0 0
\(635\) 826478. 2.04967
\(636\) 0 0
\(637\) 405769. 1.00000
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 164287. 0.399841 0.199920 − 0.979812i \(-0.435932\pi\)
0.199920 + 0.979812i \(0.435932\pi\)
\(642\) 0 0
\(643\) −483502. −1.16944 −0.584718 − 0.811237i \(-0.698795\pi\)
−0.584718 + 0.811237i \(0.698795\pi\)
\(644\) 758128. 1.82798
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −830318. −1.94723 −0.973617 − 0.228189i \(-0.926720\pi\)
−0.973617 + 0.228189i \(0.926720\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −630272. −1.46460
\(657\) −795177. −1.84218
\(658\) 0 0
\(659\) 47287.0 0.108886 0.0544429 − 0.998517i \(-0.482662\pi\)
0.0544429 + 0.998517i \(0.482662\pi\)
\(660\) 0 0
\(661\) 578183. 1.32331 0.661656 − 0.749807i \(-0.269854\pi\)
0.661656 + 0.749807i \(0.269854\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 194873. 0.440665
\(666\) 0 0
\(667\) −573431. −1.28893
\(668\) −891152. −1.99710
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 669167. 1.47742 0.738711 − 0.674023i \(-0.235435\pi\)
0.738711 + 0.674023i \(0.235435\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 456976. 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −81193.0 −0.176108
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −125712. −0.268698
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −940288. −1.98648
\(689\) −911417. −1.91990
\(690\) 0 0
\(691\) −290737. −0.608898 −0.304449 − 0.952529i \(-0.598472\pi\)
−0.304449 + 0.952529i \(0.598472\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 827904. 1.68960
\(701\) −220673. −0.449069 −0.224535 − 0.974466i \(-0.572086\pi\)
−0.224535 + 0.974466i \(0.572086\pi\)
\(702\) 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
\(711\) −647433. −1.28072
\(712\) 0 0
\(713\) 1.06660e6 2.09808
\(714\) 0 0
\(715\) 0 0
\(716\) 115312. 0.224931
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −850176. −1.64000
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) −626208. −1.19136
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 531441. 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 1.05410e6 1.96189 0.980946 − 0.194281i \(-0.0622372\pi\)
0.980946 + 0.194281i \(0.0622372\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 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
\(747\) 931743. 1.66976
\(748\) 0 0
\(749\) −19502.0 −0.0347629
\(750\) 0 0
\(751\) −530473. −0.940553 −0.470277 − 0.882519i \(-0.655846\pi\)
−0.470277 + 0.882519i \(0.655846\pi\)
\(752\) 548608. 0.970122
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −663073. −1.15710 −0.578548 − 0.815648i \(-0.696381\pi\)
−0.578548 + 0.815648i \(0.696381\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −702617. −1.21325 −0.606624 − 0.794989i \(-0.707476\pi\)
−0.606624 + 0.794989i \(0.707476\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −1.16221e6 −1.99112
\(765\) 0 0
\(766\) 0 0
\(767\) 192322. 0.326918
\(768\) 0 0
\(769\) −947497. −1.60223 −0.801116 − 0.598510i \(-0.795760\pi\)
−0.801116 + 0.598510i \(0.795760\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −1.13454e6 −1.89872 −0.949361 − 0.314186i \(-0.898268\pi\)
−0.949361 + 0.314186i \(0.898268\pi\)
\(774\) 0 0
\(775\) 1.16477e6 1.93926
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 238814. 0.393536
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 614656. 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −1.23874e6 −2.00000 −0.999999 − 0.00127065i \(-0.999596\pi\)
−0.999999 + 0.00127065i \(0.999596\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −358337. −0.572715
\(792\) 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
\(801\) 922023. 1.43707
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −1.94270e6 −2.99788
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −877313. −1.34047 −0.670236 − 0.742148i \(-0.733807\pi\)
−0.670236 + 0.742148i \(0.733807\pi\)
\(810\) 0 0
\(811\) −1.25294e6 −1.90497 −0.952487 − 0.304578i \(-0.901485\pi\)
−0.952487 + 0.304578i \(0.901485\pi\)
\(812\) −464912. −0.705113
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 356281. 0.533763
\(818\) 0 0
\(819\) 670761. 1.00000
\(820\) 1.61507e6 2.40195
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 1.30224e6 1.92261 0.961307 − 0.275480i \(-0.0888367\pi\)
0.961307 + 0.275480i \(0.0888367\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 1.25323e6 1.82798
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 692224. 1.00000
\(833\) 0 0
\(834\) 0 0
\(835\) 2.28358e6 3.27524
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 1.31466e6 1.86762 0.933811 − 0.357767i \(-0.116462\pi\)
0.933811 + 0.357767i \(0.116462\pi\)
\(840\) 0 0
\(841\) −355632. −0.502816
\(842\) 0 0
\(843\) 0 0
\(844\) 1.09707e6 1.54010
\(845\) −1.17100e6 −1.64000
\(846\) 0 0
\(847\) 717409. 1.00000
\(848\) −1.38061e6 −1.91990
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 451943. 0.621134 0.310567 − 0.950551i \(-0.399481\pi\)
0.310567 + 0.950551i \(0.399481\pi\)
\(854\) 0 0
\(855\) 322137. 0.440665
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 2.40949e6 3.25783
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 864752. 1.14776
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −134217. −0.176108
\(874\) 0 0
\(875\) −865879. −1.13094
\(876\) 0 0
\(877\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 720722. 0.924371 0.462186 − 0.886783i \(-0.347065\pi\)
0.462186 + 0.886783i \(0.347065\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) −987742. −1.24980
\(890\) 0 0
\(891\) 0 0
\(892\) −1.35707e6 −1.70558
\(893\) −207871. −0.260670
\(894\) 0 0
\(895\) −295487. −0.368886
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −654079. −0.809302
\(900\) 1.36858e6 1.68960
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −655273. −0.796540 −0.398270 − 0.917268i \(-0.630389\pi\)
−0.398270 + 0.917268i \(0.630389\pi\)
\(908\) −680672. −0.825593
\(909\) 0 0
\(910\) 0 0
\(911\) −1.45463e6 −1.75274 −0.876368 − 0.481641i \(-0.840041\pi\)
−0.876368 + 0.481641i \(0.840041\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −1.67651e6 −1.99809
\(917\) 0 0
\(918\) 0 0
\(919\) 378722. 0.448425 0.224212 − 0.974540i \(-0.428019\pi\)
0.224212 + 0.974540i \(0.428019\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 1.67794e6 1.94422 0.972111 − 0.234522i \(-0.0753525\pi\)
0.972111 + 0.234522i \(0.0753525\pi\)
\(930\) 0 0
\(931\) −232897. −0.268698
\(932\) 512752. 0.590304
\(933\) 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
\(940\) −1.40581e6 −1.59100
\(941\) 1.74466e6 1.97030 0.985150 − 0.171697i \(-0.0549249\pi\)
0.985150 + 0.171697i \(0.0549249\pi\)
\(942\) 0 0
\(943\) −2.38075e6 −2.67726
\(944\) 291328. 0.326918
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) −1.65907e6 −1.84218
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.47781e6 1.62717 0.813583 − 0.581449i \(-0.197514\pi\)
0.813583 + 0.581449i \(0.197514\pi\)
\(954\) 0 0
\(955\) 2.97816e6 3.26543
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 293088. 0.317359
\(962\) 0 0
\(963\) −32238.0 −0.0347629
\(964\) 273008. 0.293779
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 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
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.57506e6 −1.64000
\(981\) 0 0
\(982\) 0 0
\(983\) 1.27510e6 1.31959 0.659794 − 0.751447i \(-0.270644\pi\)
0.659794 + 0.751447i \(0.270644\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −262288. −0.268698
\(989\) −3.55179e6 −3.63124
\(990\) 0 0
\(991\) 653762. 0.665691 0.332845 − 0.942981i \(-0.391991\pi\)
0.332845 + 0.942981i \(0.391991\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(998\) 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 91.5.b.a.90.1 ✓ 1
7.6 odd 2 91.5.b.b.90.1 yes 1
13.12 even 2 91.5.b.b.90.1 yes 1
91.90 odd 2 CM 91.5.b.a.90.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.5.b.a.90.1 ✓ 1 1.1 even 1 trivial
91.5.b.a.90.1 ✓ 1 91.90 odd 2 CM
91.5.b.b.90.1 yes 1 7.6 odd 2
91.5.b.b.90.1 yes 1 13.12 even 2