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Properties

Label 91.9.b.a.90.1
Level $91$
Weight $9$
Character 91.90
Self dual yes
Analytic conductor $37.071$
Analytic rank $0$
Dimension $1$
CM discriminant -91
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [91,9,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, 9, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.90");
 
S:= CuspForms(chi, 9);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 9 \)
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: \(37.0714535156\)
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)
 
\(f(q)\) \(=\) \(q+256.000 q^{4} -431.000 q^{5} -2401.00 q^{7} +6561.00 q^{9} +O(q^{10})\) \(q+256.000 q^{4} -431.000 q^{5} -2401.00 q^{7} +6561.00 q^{9} -28561.0 q^{13} +65536.0 q^{16} +251233. q^{19} -110336. q^{20} +375407. q^{23} -204864. q^{25} -614656. q^{28} -1.06291e6 q^{29} +630433. q^{31} +1.03483e6 q^{35} +1.67962e6 q^{36} -409922. q^{41} +6.65333e6 q^{43} -2.82779e6 q^{45} +5.16691e6 q^{47} +5.76480e6 q^{49} -7.31162e6 q^{52} +1.33035e7 q^{53} +2.29397e7 q^{59} -1.57530e7 q^{63} +1.67772e7 q^{64} +1.23098e7 q^{65} -3.95770e7 q^{73} +6.43156e7 q^{76} -1.40121e7 q^{79} -2.82460e7 q^{80} +4.30467e7 q^{81} -3.74024e7 q^{83} -4.08821e6 q^{89} +6.85750e7 q^{91} +9.61042e7 q^{92} -1.08281e8 q^{95} +1.74313e8 q^{97} +O(q^{100})\)

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 \(\pi\)
\(4\) 256.000 1.00000
\(5\) −431.000 −0.689600 −0.344800 − 0.938676i \(-0.612053\pi\)
−0.344800 + 0.938676i \(0.612053\pi\)
\(6\) 0 0
\(7\) −2401.00 −1.00000
\(8\) 0 0
\(9\) 6561.00 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −28561.0 −1.00000
\(14\) 0 0
\(15\) 0 0
\(16\) 65536.0 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) 251233. 1.92780 0.963901 − 0.266262i \(-0.0857887\pi\)
0.963901 + 0.266262i \(0.0857887\pi\)
\(20\) −110336. −0.689600
\(21\) 0 0
\(22\) 0 0
\(23\) 375407. 1.34150 0.670751 − 0.741683i \(-0.265972\pi\)
0.670751 + 0.741683i \(0.265972\pi\)
\(24\) 0 0
\(25\) −204864. −0.524452
\(26\) 0 0
\(27\) 0 0
\(28\) −614656. −1.00000
\(29\) −1.06291e6 −1.50282 −0.751408 − 0.659838i \(-0.770625\pi\)
−0.751408 + 0.659838i \(0.770625\pi\)
\(30\) 0 0
\(31\) 630433. 0.682641 0.341320 − 0.939947i \(-0.389126\pi\)
0.341320 + 0.939947i \(0.389126\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 1.03483e6 0.689600
\(36\) 1.67962e6 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −409922. −0.145066 −0.0725330 − 0.997366i \(-0.523108\pi\)
−0.0725330 + 0.997366i \(0.523108\pi\)
\(42\) 0 0
\(43\) 6.65333e6 1.94610 0.973050 − 0.230595i \(-0.0740673\pi\)
0.973050 + 0.230595i \(0.0740673\pi\)
\(44\) 0 0
\(45\) −2.82779e6 −0.689600
\(46\) 0 0
\(47\) 5.16691e6 1.05886 0.529431 − 0.848353i \(-0.322405\pi\)
0.529431 + 0.848353i \(0.322405\pi\)
\(48\) 0 0
\(49\) 5.76480e6 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −7.31162e6 −1.00000
\(53\) 1.33035e7 1.68602 0.843009 − 0.537900i \(-0.180782\pi\)
0.843009 + 0.537900i \(0.180782\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 2.29397e7 1.89312 0.946562 − 0.322521i \(-0.104530\pi\)
0.946562 + 0.322521i \(0.104530\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −1.57530e7 −1.00000
\(64\) 1.67772e7 1.00000
\(65\) 1.23098e7 0.689600
\(66\) 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
\(73\) −3.95770e7 −1.39364 −0.696821 − 0.717245i \(-0.745403\pi\)
−0.696821 + 0.717245i \(0.745403\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 6.43156e7 1.92780
\(77\) 0 0
\(78\) 0 0
\(79\) −1.40121e7 −0.359745 −0.179873 − 0.983690i \(-0.557569\pi\)
−0.179873 + 0.983690i \(0.557569\pi\)
\(80\) −2.82460e7 −0.689600
\(81\) 4.30467e7 1.00000
\(82\) 0 0
\(83\) −3.74024e7 −0.788110 −0.394055 − 0.919087i \(-0.628928\pi\)
−0.394055 + 0.919087i \(0.628928\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −4.08821e6 −0.0651588 −0.0325794 − 0.999469i \(-0.510372\pi\)
−0.0325794 + 0.999469i \(0.510372\pi\)
\(90\) 0 0
\(91\) 6.85750e7 1.00000
\(92\) 9.61042e7 1.34150
\(93\) 0 0
\(94\) 0 0
\(95\) −1.08281e8 −1.32941
\(96\) 0 0
\(97\) 1.74313e8 1.96899 0.984493 − 0.175424i \(-0.0561296\pi\)
0.984493 + 0.175424i \(0.0561296\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −5.24452e7 −0.524452
\(101\) 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
\(107\) −2.62001e8 −1.99879 −0.999396 − 0.0347576i \(-0.988934\pi\)
−0.999396 + 0.0347576i \(0.988934\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −1.57352e8 −1.00000
\(113\) −2.72615e8 −1.67200 −0.835999 − 0.548731i \(-0.815111\pi\)
−0.835999 + 0.548731i \(0.815111\pi\)
\(114\) 0 0
\(115\) −1.61800e8 −0.925099
\(116\) −2.72106e8 −1.50282
\(117\) −1.87389e8 −1.00000
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.14359e8 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) 1.61391e8 0.682641
\(125\) 2.56656e8 1.05126
\(126\) 0 0
\(127\) −1.13944e8 −0.438004 −0.219002 − 0.975724i \(-0.570280\pi\)
−0.219002 + 0.975724i \(0.570280\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −6.03210e8 −1.92780
\(134\) 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 \(\pi\)
\(140\) 2.64917e8 0.689600
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 4.29982e8 1.00000
\(145\) 4.58116e8 1.03634
\(146\) 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
\(155\) −2.71717e8 −0.470749
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −9.01352e8 −1.34150
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −1.04940e8 −0.145066
\(165\) 0 0
\(166\) 0 0
\(167\) −1.54656e9 −1.98839 −0.994195 − 0.107589i \(-0.965687\pi\)
−0.994195 + 0.107589i \(0.965687\pi\)
\(168\) 0 0
\(169\) 8.15731e8 1.00000
\(170\) 0 0
\(171\) 1.64834e9 1.92780
\(172\) 1.70325e9 1.94610
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 4.91878e8 0.524452
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −2.00131e9 −1.94941 −0.974703 − 0.223504i \(-0.928251\pi\)
−0.974703 + 0.223504i \(0.928251\pi\)
\(180\) −7.23914e8 −0.689600
\(181\) 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
\(188\) 1.32273e9 1.05886
\(189\) 0 0
\(190\) 0 0
\(191\) 2.61455e9 1.96455 0.982277 − 0.187436i \(-0.0600178\pi\)
0.982277 + 0.187436i \(0.0600178\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.47579e9 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
\(203\) 2.55205e9 1.50282
\(204\) 0 0
\(205\) 1.76676e8 0.100038
\(206\) 0 0
\(207\) 2.46305e9 1.34150
\(208\) −1.87177e9 −1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) 7.37195e8 0.371922 0.185961 − 0.982557i \(-0.440460\pi\)
0.185961 + 0.982557i \(0.440460\pi\)
\(212\) 3.40569e9 1.68602
\(213\) 0 0
\(214\) 0 0
\(215\) −2.86758e9 −1.34203
\(216\) 0 0
\(217\) −1.51367e9 −0.682641
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −2.24798e9 −0.909018 −0.454509 − 0.890742i \(-0.650185\pi\)
−0.454509 + 0.890742i \(0.650185\pi\)
\(224\) 0 0
\(225\) −1.34411e9 −0.524452
\(226\) 0 0
\(227\) 3.50065e9 1.31840 0.659198 − 0.751970i \(-0.270896\pi\)
0.659198 + 0.751970i \(0.270896\pi\)
\(228\) 0 0
\(229\) −5.47915e9 −1.99238 −0.996188 − 0.0872322i \(-0.972198\pi\)
−0.996188 + 0.0872322i \(0.972198\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −4.86758e9 −1.65154 −0.825771 − 0.564006i \(-0.809260\pi\)
−0.825771 + 0.564006i \(0.809260\pi\)
\(234\) 0 0
\(235\) −2.22694e9 −0.730192
\(236\) 5.87256e9 1.89312
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 6.45566e9 1.91369 0.956847 − 0.290593i \(-0.0938525\pi\)
0.956847 + 0.290593i \(0.0938525\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) −2.48463e9 −0.689600
\(246\) 0 0
\(247\) −7.17547e9 −1.92780
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) −4.03276e9 −1.00000
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 4.29497e9 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 3.15131e9 0.689600
\(261\) −6.97377e9 −1.50282
\(262\) 0 0
\(263\) −6.97921e9 −1.45876 −0.729379 − 0.684109i \(-0.760191\pi\)
−0.729379 + 0.684109i \(0.760191\pi\)
\(264\) 0 0
\(265\) −5.73380e9 −1.16268
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 6.84558e9 1.26921 0.634605 − 0.772837i \(-0.281163\pi\)
0.634605 + 0.772837i \(0.281163\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −4.84695e9 −0.823283 −0.411642 − 0.911346i \(-0.635044\pi\)
−0.411642 + 0.911346i \(0.635044\pi\)
\(278\) 0 0
\(279\) 4.13627e9 0.682641
\(280\) 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
\(287\) 9.84223e8 0.145066
\(288\) 0 0
\(289\) 6.97576e9 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −1.01317e10 −1.39364
\(293\) 4.01191e9 0.544353 0.272176 − 0.962247i \(-0.412257\pi\)
0.272176 + 0.962247i \(0.412257\pi\)
\(294\) 0 0
\(295\) −9.88700e9 −1.30550
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) −1.07220e10 −1.34150
\(300\) 0 0
\(301\) −1.59746e10 −1.94610
\(302\) 0 0
\(303\) 0 0
\(304\) 1.64648e10 1.92780
\(305\) 0 0
\(306\) 0 0
\(307\) 1.77479e10 1.99799 0.998996 − 0.0447956i \(-0.0142637\pi\)
0.998996 + 0.0447956i \(0.0142637\pi\)
\(308\) 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
\(315\) 6.78953e9 0.689600
\(316\) −3.58710e9 −0.359745
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −7.23098e9 −0.689600
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.10200e10 1.00000
\(325\) 5.85112e9 0.524452
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.24058e10 −1.05886
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −9.57501e9 −0.788110
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.66867e10 1.29375 0.646877 − 0.762594i \(-0.276075\pi\)
0.646877 + 0.762594i \(0.276075\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.38413e10 −1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.15943e10 −0.799699 −0.399849 − 0.916581i \(-0.630938\pi\)
−0.399849 + 0.916581i \(0.630938\pi\)
\(348\) 0 0
\(349\) 1.58466e10 1.06816 0.534078 − 0.845435i \(-0.320659\pi\)
0.534078 + 0.845435i \(0.320659\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 2.03181e10 1.30853 0.654266 − 0.756264i \(-0.272978\pi\)
0.654266 + 0.756264i \(0.272978\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.04658e9 −0.0651588
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 4.61345e10 2.71642
\(362\) 0 0
\(363\) 0 0
\(364\) 1.75552e10 1.00000
\(365\) 1.70577e10 0.961056
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 2.46027e10 1.34150
\(369\) −2.68950e9 −0.145066
\(370\) 0 0
\(371\) −3.19417e10 −1.68602
\(372\) 0 0
\(373\) 1.22909e10 0.634960 0.317480 − 0.948265i \(-0.397163\pi\)
0.317480 + 0.948265i \(0.397163\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 3.03579e10 1.50282
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) −2.77200e10 −1.32941
\(381\) 0 0
\(382\) 0 0
\(383\) −4.04984e10 −1.88210 −0.941050 − 0.338268i \(-0.890159\pi\)
−0.941050 + 0.338268i \(0.890159\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 4.36525e10 1.94610
\(388\) 4.46241e10 1.96899
\(389\) −2.11339e10 −0.922957 −0.461478 − 0.887152i \(-0.652681\pi\)
−0.461478 + 0.887152i \(0.652681\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 6.03922e9 0.248080
\(396\) 0 0
\(397\) −3.71922e10 −1.49724 −0.748618 − 0.663001i \(-0.769282\pi\)
−0.748618 + 0.663001i \(0.769282\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.34260e10 −0.524452
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) −1.80058e10 −0.682641
\(404\) 0 0
\(405\) −1.85531e10 −0.689600
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 5.23042e8 0.0186914 0.00934572 − 0.999956i \(-0.497025\pi\)
0.00934572 + 0.999956i \(0.497025\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) −5.50782e10 −1.89312
\(414\) 0 0
\(415\) 1.61204e10 0.543481
\(416\) 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
\(423\) 3.39001e10 1.05886
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −6.70722e10 −1.99879
\(429\) 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
\(437\) 9.43146e10 2.58615
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 3.78229e10 1.00000
\(442\) 0 0
\(443\) −1.96542e10 −0.510316 −0.255158 − 0.966899i \(-0.582128\pi\)
−0.255158 + 0.966899i \(0.582128\pi\)
\(444\) 0 0
\(445\) 1.76202e9 0.0449335
\(446\) 0 0
\(447\) 0 0
\(448\) −4.02821e10 −1.00000
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −6.97894e10 −1.67200
\(453\) 0 0
\(454\) 0 0
\(455\) −2.95558e10 −0.689600
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −4.14209e10 −0.925099
\(461\) −1.97994e10 −0.438377 −0.219189 − 0.975683i \(-0.570341\pi\)
−0.219189 + 0.975683i \(0.570341\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) −6.96591e10 −1.50282
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −4.79715e10 −1.00000
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −5.14686e10 −1.01104
\(476\) 0 0
\(477\) 8.72842e10 1.68602
\(478\) 0 0
\(479\) 9.87808e10 1.87642 0.938211 − 0.346064i \(-0.112482\pi\)
0.938211 + 0.346064i \(0.112482\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 5.48759e10 1.00000
\(485\) −7.51289e10 −1.35781
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −1.06192e11 −1.82712 −0.913561 − 0.406701i \(-0.866679\pi\)
−0.913561 + 0.406701i \(0.866679\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 4.13161e10 0.682641
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 6.57039e10 1.05126
\(501\) 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
\(508\) −2.91697e10 −0.438004
\(509\) −1.07683e11 −1.60427 −0.802134 − 0.597143i \(-0.796302\pi\)
−0.802134 + 0.597143i \(0.796302\pi\)
\(510\) 0 0
\(511\) 9.50244e10 1.39364
\(512\) 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
\(529\) 6.26194e10 0.799625
\(530\) 0 0
\(531\) 1.50507e11 1.89312
\(532\) −1.54422e11 −1.92780
\(533\) 1.17078e10 0.145066
\(534\) 0 0
\(535\) 1.12922e11 1.37837
\(536\) 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
\(547\) 9.42605e10 1.05288 0.526442 − 0.850211i \(-0.323526\pi\)
0.526442 + 0.850211i \(0.323526\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −2.67039e11 −2.89713
\(552\) 0 0
\(553\) 3.36431e10 0.359745
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −1.90026e11 −1.94610
\(560\) 6.78187e10 0.689600
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 1.17497e11 1.15301
\(566\) 0 0
\(567\) −1.03355e11 −1.00000
\(568\) 0 0
\(569\) 2.06701e11 1.97194 0.985971 − 0.166916i \(-0.0533809\pi\)
0.985971 + 0.166916i \(0.0533809\pi\)
\(570\) 0 0
\(571\) 1.86322e11 1.75275 0.876374 − 0.481631i \(-0.159955\pi\)
0.876374 + 0.481631i \(0.159955\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −7.69074e10 −0.703553
\(576\) 1.10075e11 1.00000
\(577\) −1.93751e11 −1.74800 −0.873998 − 0.485929i \(-0.838481\pi\)
−0.873998 + 0.485929i \(0.838481\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 1.17278e11 1.03634
\(581\) 8.98031e10 0.788110
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 8.07645e10 0.689600
\(586\) 0 0
\(587\) 2.36453e11 1.99156 0.995778 − 0.0917946i \(-0.0292603\pi\)
0.995778 + 0.0917946i \(0.0292603\pi\)
\(588\) 0 0
\(589\) 1.58386e11 1.31600
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 1.45666e11 1.17798 0.588992 − 0.808139i \(-0.299525\pi\)
0.588992 + 0.808139i \(0.299525\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −6.18231e10 −0.480224 −0.240112 − 0.970745i \(-0.577184\pi\)
−0.240112 + 0.970745i \(0.577184\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −9.23887e10 −0.689600
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −1.47572e11 −1.05886
\(612\) 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
\(619\) 2.06849e11 1.40893 0.704466 − 0.709738i \(-0.251187\pi\)
0.704466 + 0.709738i \(0.251187\pi\)
\(620\) −6.95595e10 −0.470749
\(621\) 0 0
\(622\) 0 0
\(623\) 9.81579e9 0.0651588
\(624\) 0 0
\(625\) −3.05936e10 −0.200498
\(626\) 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
\(635\) 4.91100e10 0.302047
\(636\) 0 0
\(637\) −1.64648e11 −1.00000
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −3.10656e11 −1.84013 −0.920064 − 0.391769i \(-0.871863\pi\)
−0.920064 + 0.391769i \(0.871863\pi\)
\(642\) 0 0
\(643\) 1.08106e11 0.632420 0.316210 − 0.948689i \(-0.397590\pi\)
0.316210 + 0.948689i \(0.397590\pi\)
\(644\) −2.30746e11 −1.34150
\(645\) 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
\(653\) 3.25779e11 1.79172 0.895860 − 0.444337i \(-0.146561\pi\)
0.895860 + 0.444337i \(0.146561\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −2.68646e10 −0.145066
\(657\) −2.59665e11 −1.39364
\(658\) 0 0
\(659\) −3.74964e11 −1.98814 −0.994072 − 0.108724i \(-0.965323\pi\)
−0.994072 + 0.108724i \(0.965323\pi\)
\(660\) 0 0
\(661\) 4.75043e10 0.248844 0.124422 − 0.992229i \(-0.460292\pi\)
0.124422 + 0.992229i \(0.460292\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 2.59984e11 1.32941
\(666\) 0 0
\(667\) −3.99025e11 −2.01603
\(668\) −3.95920e11 −1.98839
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 3.74951e10 0.182774 0.0913870 − 0.995815i \(-0.470870\pi\)
0.0913870 + 0.995815i \(0.470870\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 2.08827e11 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) −4.18525e11 −1.96899
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 4.21975e11 1.92780
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 4.36032e11 1.94610
\(689\) −3.79961e11 −1.68602
\(690\) 0 0
\(691\) 3.71448e11 1.62924 0.814622 − 0.579992i \(-0.196944\pi\)
0.814622 + 0.579992i \(0.196944\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 1.25921e11 0.524452
\(701\) −4.34253e11 −1.79834 −0.899168 − 0.437603i \(-0.855828\pi\)
−0.899168 + 0.437603i \(0.855828\pi\)
\(702\) 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
\(711\) −9.19335e10 −0.359745
\(712\) 0 0
\(713\) 2.36669e11 0.915763
\(714\) 0 0
\(715\) 0 0
\(716\) −5.12335e11 −1.94941
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −1.85322e11 −0.689600
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 2.17753e11 0.788154
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 2.82430e11 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −5.33774e11 −1.84902 −0.924510 − 0.381157i \(-0.875526\pi\)
−0.924510 + 0.381157i \(0.875526\pi\)
\(734\) 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
\(747\) −2.45397e11 −0.788110
\(748\) 0 0
\(749\) 6.29064e11 1.99879
\(750\) 0 0
\(751\) −3.54793e11 −1.11536 −0.557680 − 0.830056i \(-0.688308\pi\)
−0.557680 + 0.830056i \(0.688308\pi\)
\(752\) 3.38619e11 1.05886
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −2.17105e11 −0.661128 −0.330564 − 0.943784i \(-0.607239\pi\)
−0.330564 + 0.943784i \(0.607239\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 1.77092e11 0.528031 0.264015 − 0.964518i \(-0.414953\pi\)
0.264015 + 0.964518i \(0.414953\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 6.69325e11 1.96455
\(765\) 0 0
\(766\) 0 0
\(767\) −6.55180e11 −1.89312
\(768\) 0 0
\(769\) −1.98335e11 −0.567145 −0.283572 − 0.958951i \(-0.591520\pi\)
−0.283572 + 0.958951i \(0.591520\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −5.73104e11 −1.60515 −0.802574 − 0.596552i \(-0.796537\pi\)
−0.802574 + 0.596552i \(0.796537\pi\)
\(774\) 0 0
\(775\) −1.29153e11 −0.358012
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −1.02986e11 −0.279659
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 3.77802e11 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −7.67233e11 −1.99999 −0.999997 − 0.00254129i \(-0.999191\pi\)
−0.999997 + 0.00254129i \(0.999191\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 6.54548e11 1.67200
\(792\) 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
\(801\) −2.68227e10 −0.0651588
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 3.88483e11 0.925099
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −8.70127e10 −0.203137 −0.101568 − 0.994829i \(-0.532386\pi\)
−0.101568 + 0.994829i \(0.532386\pi\)
\(810\) 0 0
\(811\) −7.04670e11 −1.62893 −0.814465 − 0.580213i \(-0.802969\pi\)
−0.814465 + 0.580213i \(0.802969\pi\)
\(812\) 6.53326e11 1.50282
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 1.67154e12 3.75169
\(818\) 0 0
\(819\) 4.49920e11 1.00000
\(820\) 4.52292e10 0.100038
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 7.78285e11 1.69644 0.848222 − 0.529641i \(-0.177673\pi\)
0.848222 + 0.529641i \(0.177673\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 6.30540e11 1.34150
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −4.79174e11 −1.00000
\(833\) 0 0
\(834\) 0 0
\(835\) 6.66569e11 1.37119
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −7.37316e11 −1.48801 −0.744005 − 0.668174i \(-0.767076\pi\)
−0.744005 + 0.668174i \(0.767076\pi\)
\(840\) 0 0
\(841\) 6.29538e11 1.25846
\(842\) 0 0
\(843\) 0 0
\(844\) 1.88722e11 0.371922
\(845\) −3.51580e11 −0.689600
\(846\) 0 0
\(847\) −5.14676e11 −1.00000
\(848\) 8.71857e11 1.68602
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 8.54577e11 1.61419 0.807096 − 0.590420i \(-0.201038\pi\)
0.807096 + 0.590420i \(0.201038\pi\)
\(854\) 0 0
\(855\) −7.10434e11 −1.32941
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) −7.34101e11 −1.34203
\(861\) 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
\(868\) −3.87499e11 −0.682641
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.14367e12 1.96899
\(874\) 0 0
\(875\) −6.16230e11 −1.05126
\(876\) 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
\(883\) −6.96390e11 −1.14554 −0.572769 − 0.819717i \(-0.694131\pi\)
−0.572769 + 0.819717i \(0.694131\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 2.73580e11 0.438004
\(890\) 0 0
\(891\) 0 0
\(892\) −5.75482e11 −0.909018
\(893\) 1.29810e12 2.04128
\(894\) 0 0
\(895\) 8.62565e11 1.34431
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −6.70095e11 −1.02588
\(900\) −3.44093e11 −0.524452
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −9.24120e11 −1.36552 −0.682762 − 0.730641i \(-0.739221\pi\)
−0.682762 + 0.730641i \(0.739221\pi\)
\(908\) 8.96167e11 1.31840
\(909\) 0 0
\(910\) 0 0
\(911\) 7.38419e11 1.07209 0.536043 − 0.844191i \(-0.319918\pi\)
0.536043 + 0.844191i \(0.319918\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) −1.40266e12 −1.99238
\(917\) 0 0
\(918\) 0 0
\(919\) −1.28314e12 −1.79892 −0.899458 − 0.437008i \(-0.856038\pi\)
−0.899458 + 0.437008i \(0.856038\pi\)
\(920\) 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
\(929\) −1.32581e12 −1.78000 −0.889999 − 0.455963i \(-0.849295\pi\)
−0.889999 + 0.455963i \(0.849295\pi\)
\(930\) 0 0
\(931\) 1.44831e12 1.92780
\(932\) −1.24610e12 −1.65154
\(933\) 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
\(940\) −5.70097e11 −0.730192
\(941\) −1.47570e12 −1.88208 −0.941041 − 0.338294i \(-0.890150\pi\)
−0.941041 + 0.338294i \(0.890150\pi\)
\(942\) 0 0
\(943\) −1.53888e11 −0.194606
\(944\) 1.50337e12 1.89312
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 1.13036e12 1.39364
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 5.34226e11 0.647670 0.323835 − 0.946114i \(-0.395028\pi\)
0.323835 + 0.946114i \(0.395028\pi\)
\(954\) 0 0
\(955\) −1.12687e12 −1.35476
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −4.55445e11 −0.534002
\(962\) 0 0
\(963\) −1.71899e12 −1.99879
\(964\) 1.65265e12 1.91369
\(965\) 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
\(980\) −6.36065e11 −0.689600
\(981\) 0 0
\(982\) 0 0
\(983\) 2.41541e11 0.258689 0.129344 − 0.991600i \(-0.458713\pi\)
0.129344 + 0.991600i \(0.458713\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −1.83692e12 −1.92780
\(989\) 2.49771e12 2.61069
\(990\) 0 0
\(991\) −1.50156e12 −1.55686 −0.778428 − 0.627734i \(-0.783983\pi\)
−0.778428 + 0.627734i \(0.783983\pi\)
\(992\) 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 91.9.b.a.90.1 ✓ 1
7.6 odd 2 91.9.b.b.90.1 yes 1
13.12 even 2 91.9.b.b.90.1 yes 1
91.90 odd 2 CM 91.9.b.a.90.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.9.b.a.90.1 ✓ 1 1.1 even 1 trivial
91.9.b.a.90.1 ✓ 1 91.90 odd 2 CM
91.9.b.b.90.1 yes 1 7.6 odd 2
91.9.b.b.90.1 yes 1 13.12 even 2