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Properties

Label 91.7.b.b.90.1
Level $91$
Weight $7$
Character 91.90
Self dual yes
Analytic conductor $20.935$
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] = [91,7,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, 7, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("91.90");
 
S:= CuspForms(chi, 7);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 91 = 7 \cdot 13 \)
Weight: \( k \) \(=\) \( 7 \)
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: \(20.9349216094\)
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)
 
\(f(q)\) \(=\) \(q+64.0000 q^{4} +198.000 q^{5} +343.000 q^{7} +729.000 q^{9} +O(q^{10})\) \(q+64.0000 q^{4} +198.000 q^{5} +343.000 q^{7} +729.000 q^{9} -2197.00 q^{13} +4096.00 q^{16} -11450.0 q^{19} +12672.0 q^{20} -19710.0 q^{23} +23579.0 q^{25} +21952.0 q^{28} +47322.0 q^{29} -7810.00 q^{31} +67914.0 q^{35} +46656.0 q^{36} -124290. q^{41} +27610.0 q^{43} +144342. q^{45} -5346.00 q^{47} +117649. q^{49} -140608. q^{52} -123030. q^{53} +210870. q^{59} +250047. q^{63} +262144. q^{64} -435006. q^{65} -439234. q^{73} -732800. q^{76} -953678. q^{79} +811008. q^{80} +531441. q^{81} -733626. q^{83} +571230. q^{89} -753571. q^{91} -1.26144e6 q^{92} -2.26710e6 q^{95} +1.44965e6 q^{97} +O(q^{100})\)

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 \(\pi\)
\(4\) 64.0000 1.00000
\(5\) 198.000 1.58400 0.792000 − 0.610521i \(-0.209040\pi\)
0.792000 + 0.610521i \(0.209040\pi\)
\(6\) 0 0
\(7\) 343.000 1.00000
\(8\) 0 0
\(9\) 729.000 1.00000
\(10\) 0 0
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −2197.00 −1.00000
\(14\) 0 0
\(15\) 0 0
\(16\) 4096.00 1.00000
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) 0 0
\(19\) −11450.0 −1.66934 −0.834670 − 0.550751i \(-0.814341\pi\)
−0.834670 + 0.550751i \(0.814341\pi\)
\(20\) 12672.0 1.58400
\(21\) 0 0
\(22\) 0 0
\(23\) −19710.0 −1.61996 −0.809978 − 0.586461i \(-0.800521\pi\)
−0.809978 + 0.586461i \(0.800521\pi\)
\(24\) 0 0
\(25\) 23579.0 1.50906
\(26\) 0 0
\(27\) 0 0
\(28\) 21952.0 1.00000
\(29\) 47322.0 1.94030 0.970150 − 0.242504i \(-0.0779687\pi\)
0.970150 + 0.242504i \(0.0779687\pi\)
\(30\) 0 0
\(31\) −7810.00 −0.262160 −0.131080 − 0.991372i \(-0.541844\pi\)
−0.131080 + 0.991372i \(0.541844\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 67914.0 1.58400
\(36\) 46656.0 1.00000
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −124290. −1.80337 −0.901685 − 0.432394i \(-0.857669\pi\)
−0.901685 + 0.432394i \(0.857669\pi\)
\(42\) 0 0
\(43\) 27610.0 0.347265 0.173633 − 0.984811i \(-0.444450\pi\)
0.173633 + 0.984811i \(0.444450\pi\)
\(44\) 0 0
\(45\) 144342. 1.58400
\(46\) 0 0
\(47\) −5346.00 −0.0514915 −0.0257457 − 0.999669i \(-0.508196\pi\)
−0.0257457 + 0.999669i \(0.508196\pi\)
\(48\) 0 0
\(49\) 117649. 1.00000
\(50\) 0 0
\(51\) 0 0
\(52\) −140608. −1.00000
\(53\) −123030. −0.826387 −0.413193 − 0.910643i \(-0.635587\pi\)
−0.413193 + 0.910643i \(0.635587\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 210870. 1.02674 0.513368 − 0.858169i \(-0.328398\pi\)
0.513368 + 0.858169i \(0.328398\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) 250047. 1.00000
\(64\) 262144. 1.00000
\(65\) −435006. −1.58400
\(66\) 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
\(73\) −439234. −1.12909 −0.564543 − 0.825403i \(-0.690948\pi\)
−0.564543 + 0.825403i \(0.690948\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −732800. −1.66934
\(77\) 0 0
\(78\) 0 0
\(79\) −953678. −1.93429 −0.967143 − 0.254235i \(-0.918176\pi\)
−0.967143 + 0.254235i \(0.918176\pi\)
\(80\) 811008. 1.58400
\(81\) 531441. 1.00000
\(82\) 0 0
\(83\) −733626. −1.28304 −0.641520 − 0.767106i \(-0.721696\pi\)
−0.641520 + 0.767106i \(0.721696\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 571230. 0.810291 0.405145 − 0.914252i \(-0.367221\pi\)
0.405145 + 0.914252i \(0.367221\pi\)
\(90\) 0 0
\(91\) −753571. −1.00000
\(92\) −1.26144e6 −1.61996
\(93\) 0 0
\(94\) 0 0
\(95\) −2.26710e6 −2.64423
\(96\) 0 0
\(97\) 1.44965e6 1.58835 0.794176 − 0.607688i \(-0.207903\pi\)
0.794176 + 0.607688i \(0.207903\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 1.50906e6 1.50906
\(101\) 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
\(107\) 1.77705e6 1.45060 0.725301 − 0.688432i \(-0.241701\pi\)
0.725301 + 0.688432i \(0.241701\pi\)
\(108\) 0 0
\(109\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 1.40493e6 1.00000
\(113\) −2.71107e6 −1.87891 −0.939454 − 0.342676i \(-0.888667\pi\)
−0.939454 + 0.342676i \(0.888667\pi\)
\(114\) 0 0
\(115\) −3.90258e6 −2.56601
\(116\) 3.02861e6 1.94030
\(117\) −1.60161e6 −1.00000
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 1.77156e6 1.00000
\(122\) 0 0
\(123\) 0 0
\(124\) −499840. −0.262160
\(125\) 1.57489e6 0.806345
\(126\) 0 0
\(127\) 3.99157e6 1.94864 0.974322 − 0.225158i \(-0.0722899\pi\)
0.974322 + 0.225158i \(0.0722899\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) −3.92735e6 −1.66934
\(134\) 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 \(\pi\)
\(140\) 4.34650e6 1.58400
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 2.98598e6 1.00000
\(145\) 9.36976e6 3.07344
\(146\) 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
\(155\) −1.54638e6 −0.415261
\(156\) 0 0
\(157\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) −6.76053e6 −1.61996
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −7.95456e6 −1.80337
\(165\) 0 0
\(166\) 0 0
\(167\) −752274. −0.161520 −0.0807601 − 0.996734i \(-0.525735\pi\)
−0.0807601 + 0.996734i \(0.525735\pi\)
\(168\) 0 0
\(169\) 4.82681e6 1.00000
\(170\) 0 0
\(171\) −8.34705e6 −1.66934
\(172\) 1.76704e6 0.347265
\(173\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(174\) 0 0
\(175\) 8.08760e6 1.50906
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) −6.63068e6 −1.15611 −0.578055 − 0.815998i \(-0.696188\pi\)
−0.578055 + 0.815998i \(0.696188\pi\)
\(180\) 9.23789e6 1.58400
\(181\) 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
\(188\) −342144. −0.0514915
\(189\) 0 0
\(190\) 0 0
\(191\) −1.96414e6 −0.281886 −0.140943 − 0.990018i \(-0.545013\pi\)
−0.140943 + 0.990018i \(0.545013\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 7.52954e6 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
\(203\) 1.62314e7 1.94030
\(204\) 0 0
\(205\) −2.46094e7 −2.85654
\(206\) 0 0
\(207\) −1.43686e7 −1.61996
\(208\) −8.99891e6 −1.00000
\(209\) 0 0
\(210\) 0 0
\(211\) −9.54626e6 −1.01622 −0.508108 − 0.861293i \(-0.669655\pi\)
−0.508108 + 0.861293i \(0.669655\pi\)
\(212\) −7.87392e6 −0.826387
\(213\) 0 0
\(214\) 0 0
\(215\) 5.46678e6 0.550068
\(216\) 0 0
\(217\) −2.67883e6 −0.262160
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.62801e7 −1.46805 −0.734026 − 0.679121i \(-0.762361\pi\)
−0.734026 + 0.679121i \(0.762361\pi\)
\(224\) 0 0
\(225\) 1.71891e7 1.50906
\(226\) 0 0
\(227\) −2.31415e7 −1.97840 −0.989198 − 0.146584i \(-0.953172\pi\)
−0.989198 + 0.146584i \(0.953172\pi\)
\(228\) 0 0
\(229\) 1.57223e6 0.130921 0.0654605 − 0.997855i \(-0.479148\pi\)
0.0654605 + 0.997855i \(0.479148\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −8.34075e6 −0.659382 −0.329691 − 0.944089i \(-0.606945\pi\)
−0.329691 + 0.944089i \(0.606945\pi\)
\(234\) 0 0
\(235\) −1.05851e6 −0.0815625
\(236\) 1.34957e7 1.02674
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −1.49716e7 −1.06959 −0.534794 − 0.844983i \(-0.679611\pi\)
−0.534794 + 0.844983i \(0.679611\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 2.32945e7 1.58400
\(246\) 0 0
\(247\) 2.51556e7 1.66934
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 1.60030e7 1.00000
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 1.67772e7 1.00000
\(257\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.78404e7 −1.58400
\(261\) 3.44977e7 1.94030
\(262\) 0 0
\(263\) −7.95267e6 −0.437165 −0.218583 − 0.975818i \(-0.570143\pi\)
−0.218583 + 0.975818i \(0.570143\pi\)
\(264\) 0 0
\(265\) −2.43599e7 −1.30900
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) 3.95047e7 1.98491 0.992454 − 0.122614i \(-0.0391278\pi\)
0.992454 + 0.122614i \(0.0391278\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 4.23899e7 1.99445 0.997226 − 0.0744393i \(-0.0237167\pi\)
0.997226 + 0.0744393i \(0.0237167\pi\)
\(278\) 0 0
\(279\) −5.69349e6 −0.262160
\(280\) 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
\(287\) −4.26315e7 −1.80337
\(288\) 0 0
\(289\) 2.41376e7 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) −2.81110e7 −1.12909
\(293\) −4.94402e7 −1.96552 −0.982759 − 0.184889i \(-0.940808\pi\)
−0.982759 + 0.184889i \(0.940808\pi\)
\(294\) 0 0
\(295\) 4.17523e7 1.62635
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 4.33029e7 1.61996
\(300\) 0 0
\(301\) 9.47023e6 0.347265
\(302\) 0 0
\(303\) 0 0
\(304\) −4.68992e7 −1.66934
\(305\) 0 0
\(306\) 0 0
\(307\) 3.95214e7 1.36590 0.682948 − 0.730467i \(-0.260698\pi\)
0.682948 + 0.730467i \(0.260698\pi\)
\(308\) 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
\(315\) 4.95093e7 1.58400
\(316\) −6.10354e7 −1.93429
\(317\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 5.19045e7 1.58400
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 3.40122e7 1.00000
\(325\) −5.18031e7 −1.50906
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) −1.83368e6 −0.0514915
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) −4.69521e7 −1.28304
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 4.63539e7 1.21115 0.605573 − 0.795790i \(-0.292944\pi\)
0.605573 + 0.795790i \(0.292944\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 4.03536e7 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −8.32679e7 −1.99292 −0.996459 − 0.0840853i \(-0.973203\pi\)
−0.996459 + 0.0840853i \(0.973203\pi\)
\(348\) 0 0
\(349\) 8.49792e7 1.99911 0.999554 − 0.0298592i \(-0.00950591\pi\)
0.999554 + 0.0298592i \(0.00950591\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.24687e7 0.283462 0.141731 − 0.989905i \(-0.454733\pi\)
0.141731 + 0.989905i \(0.454733\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 3.65587e7 0.810291
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 8.40566e7 1.78669
\(362\) 0 0
\(363\) 0 0
\(364\) −4.82285e7 −1.00000
\(365\) −8.69683e7 −1.78847
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) −8.07322e7 −1.61996
\(369\) −9.06074e7 −1.80337
\(370\) 0 0
\(371\) −4.21993e7 −0.826387
\(372\) 0 0
\(373\) −6.15662e7 −1.18636 −0.593179 − 0.805070i \(-0.702128\pi\)
−0.593179 + 0.805070i \(0.702128\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) −1.03966e8 −1.94030
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) −1.45094e8 −2.64423
\(381\) 0 0
\(382\) 0 0
\(383\) 2.87569e7 0.511854 0.255927 − 0.966696i \(-0.417619\pi\)
0.255927 + 0.966696i \(0.417619\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 2.01277e7 0.347265
\(388\) 9.27773e7 1.58835
\(389\) −3.87884e6 −0.0658951 −0.0329475 − 0.999457i \(-0.510489\pi\)
−0.0329475 + 0.999457i \(0.510489\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −1.88828e8 −3.06391
\(396\) 0 0
\(397\) 1.07102e8 1.71169 0.855844 − 0.517234i \(-0.173038\pi\)
0.855844 + 0.517234i \(0.173038\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 9.65796e7 1.50906
\(401\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 1.71586e7 0.262160
\(404\) 0 0
\(405\) 1.05225e8 1.58400
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −5.14774e7 −0.752396 −0.376198 − 0.926539i \(-0.622769\pi\)
−0.376198 + 0.926539i \(0.622769\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 7.23284e7 1.02674
\(414\) 0 0
\(415\) −1.45258e8 −2.03234
\(416\) 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
\(423\) −3.89723e6 −0.0514915
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 1.13731e8 1.45060
\(429\) 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
\(437\) 2.25680e8 2.70426
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 8.57661e7 1.00000
\(442\) 0 0
\(443\) 3.44060e7 0.395752 0.197876 − 0.980227i \(-0.436596\pi\)
0.197876 + 0.980227i \(0.436596\pi\)
\(444\) 0 0
\(445\) 1.13104e8 1.28350
\(446\) 0 0
\(447\) 0 0
\(448\) 8.99154e7 1.00000
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) −1.73508e8 −1.87891
\(453\) 0 0
\(454\) 0 0
\(455\) −1.49207e8 −1.58400
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) −2.49765e8 −2.56601
\(461\) 1.03823e8 1.05972 0.529861 − 0.848085i \(-0.322244\pi\)
0.529861 + 0.848085i \(0.322244\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 1.93831e8 1.94030
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(468\) −1.02503e8 −1.00000
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −2.69980e8 −2.51913
\(476\) 0 0
\(477\) −8.96889e7 −0.826387
\(478\) 0 0
\(479\) 1.90710e8 1.73527 0.867636 − 0.497200i \(-0.165638\pi\)
0.867636 + 0.497200i \(0.165638\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 1.13380e8 1.00000
\(485\) 2.87030e8 2.51595
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −2.10935e8 −1.78199 −0.890993 − 0.454016i \(-0.849991\pi\)
−0.890993 + 0.454016i \(0.849991\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −3.19898e7 −0.262160
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 1.00793e8 0.806345
\(501\) 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
\(508\) 2.55460e8 1.94864
\(509\) 2.33946e8 1.77403 0.887017 − 0.461736i \(-0.152773\pi\)
0.887017 + 0.461736i \(0.152773\pi\)
\(510\) 0 0
\(511\) −1.50657e8 −1.12909
\(512\) 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
\(529\) 2.40448e8 1.62426
\(530\) 0 0
\(531\) 1.53724e8 1.02674
\(532\) −2.51350e8 −1.66934
\(533\) 2.73065e8 1.80337
\(534\) 0 0
\(535\) 3.51856e8 2.29775
\(536\) 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
\(547\) −2.26051e8 −1.38116 −0.690579 − 0.723257i \(-0.742644\pi\)
−0.690579 + 0.723257i \(0.742644\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) −5.41837e8 −3.23902
\(552\) 0 0
\(553\) −3.27112e8 −1.93429
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) −6.06592e7 −0.347265
\(560\) 2.78176e8 1.58400
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) −5.36792e8 −2.97619
\(566\) 0 0
\(567\) 1.82284e8 1.00000
\(568\) 0 0
\(569\) −3.65530e8 −1.98420 −0.992101 − 0.125444i \(-0.959964\pi\)
−0.992101 + 0.125444i \(0.959964\pi\)
\(570\) 0 0
\(571\) −3.46206e8 −1.85963 −0.929815 − 0.368027i \(-0.880033\pi\)
−0.929815 + 0.368027i \(0.880033\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −4.64742e8 −2.44460
\(576\) 1.91103e8 1.00000
\(577\) 1.42711e8 0.742898 0.371449 − 0.928453i \(-0.378861\pi\)
0.371449 + 0.928453i \(0.378861\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 5.99664e8 3.07344
\(581\) −2.51634e8 −1.28304
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −3.17119e8 −1.58400
\(586\) 0 0
\(587\) −2.65657e8 −1.31343 −0.656716 − 0.754138i \(-0.728055\pi\)
−0.656716 + 0.754138i \(0.728055\pi\)
\(588\) 0 0
\(589\) 8.94245e7 0.437634
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 3.31871e7 0.159149 0.0795747 − 0.996829i \(-0.474644\pi\)
0.0795747 + 0.996829i \(0.474644\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −8.99575e7 −0.418559 −0.209280 − 0.977856i \(-0.567112\pi\)
−0.209280 + 0.977856i \(0.567112\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 3.50769e8 1.58400
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 1.17452e7 0.0514915
\(612\) 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
\(619\) −9.12405e7 −0.384694 −0.192347 − 0.981327i \(-0.561610\pi\)
−0.192347 + 0.981327i \(0.561610\pi\)
\(620\) −9.89683e7 −0.415261
\(621\) 0 0
\(622\) 0 0
\(623\) 1.95932e8 0.810291
\(624\) 0 0
\(625\) −5.65933e7 −0.231806
\(626\) 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
\(635\) 7.90331e8 3.08665
\(636\) 0 0
\(637\) −2.58475e8 −1.00000
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 2.44868e8 0.929732 0.464866 − 0.885381i \(-0.346103\pi\)
0.464866 + 0.885381i \(0.346103\pi\)
\(642\) 0 0
\(643\) 5.25613e8 1.97712 0.988561 − 0.150822i \(-0.0481922\pi\)
0.988561 + 0.150822i \(0.0481922\pi\)
\(644\) −4.32674e8 −1.61996
\(645\) 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
\(653\) 1.88509e8 0.677006 0.338503 − 0.940965i \(-0.390079\pi\)
0.338503 + 0.940965i \(0.390079\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −5.09092e8 −1.80337
\(657\) −3.20202e8 −1.12909
\(658\) 0 0
\(659\) 3.70353e8 1.29408 0.647038 − 0.762458i \(-0.276007\pi\)
0.647038 + 0.762458i \(0.276007\pi\)
\(660\) 0 0
\(661\) 1.70221e8 0.589397 0.294699 − 0.955590i \(-0.404781\pi\)
0.294699 + 0.955590i \(0.404781\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) −7.77615e8 −2.64423
\(666\) 0 0
\(667\) −9.32717e8 −3.14320
\(668\) −4.81455e7 −0.161520
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 2.71379e8 0.890288 0.445144 − 0.895459i \(-0.353152\pi\)
0.445144 + 0.895459i \(0.353152\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 3.08916e8 1.00000
\(677\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 4.97229e8 1.58835
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −5.34211e8 −1.66934
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 1.13091e8 0.347265
\(689\) 2.70297e8 0.826387
\(690\) 0 0
\(691\) 6.26098e8 1.89761 0.948807 − 0.315856i \(-0.102292\pi\)
0.948807 + 0.315856i \(0.102292\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 5.17606e8 1.50906
\(701\) 6.21641e8 1.80462 0.902309 − 0.431090i \(-0.141871\pi\)
0.902309 + 0.431090i \(0.141871\pi\)
\(702\) 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
\(711\) −6.95231e8 −1.93429
\(712\) 0 0
\(713\) 1.53935e8 0.424687
\(714\) 0 0
\(715\) 0 0
\(716\) −4.24363e8 −1.15611
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 5.91225e8 1.58400
\(721\) 0 0
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 1.11581e9 2.92802
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 3.87420e8 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −7.54032e8 −1.91460 −0.957299 − 0.289099i \(-0.906644\pi\)
−0.957299 + 0.289099i \(0.906644\pi\)
\(734\) 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
\(747\) −5.34813e8 −1.28304
\(748\) 0 0
\(749\) 6.09528e8 1.45060
\(750\) 0 0
\(751\) 8.46028e8 1.99740 0.998700 − 0.0509697i \(-0.0162312\pi\)
0.998700 + 0.0509697i \(0.0162312\pi\)
\(752\) −2.18972e7 −0.0514915
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −8.59105e8 −1.98043 −0.990213 − 0.139566i \(-0.955429\pi\)
−0.990213 + 0.139566i \(0.955429\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 8.65173e8 1.96313 0.981565 − 0.191129i \(-0.0612147\pi\)
0.981565 + 0.191129i \(0.0612147\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) −1.25705e8 −0.281886
\(765\) 0 0
\(766\) 0 0
\(767\) −4.63281e8 −1.02674
\(768\) 0 0
\(769\) −7.46346e8 −1.64120 −0.820600 − 0.571504i \(-0.806360\pi\)
−0.820600 + 0.571504i \(0.806360\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) −4.26089e8 −0.922491 −0.461246 − 0.887272i \(-0.652597\pi\)
−0.461246 + 0.887272i \(0.652597\pi\)
\(774\) 0 0
\(775\) −1.84152e8 −0.395614
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 1.42312e9 3.01044
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 4.81890e8 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −1.85811e6 −0.00381194 −0.00190597 − 0.999998i \(-0.500607\pi\)
−0.00190597 + 0.999998i \(0.500607\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −9.29897e8 −1.87891
\(792\) 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
\(801\) 4.16427e8 0.810291
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) −1.33858e9 −2.56601
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.00639e9 1.90073 0.950364 − 0.311139i \(-0.100710\pi\)
0.950364 + 0.311139i \(0.100710\pi\)
\(810\) 0 0
\(811\) 4.77666e8 0.895492 0.447746 − 0.894161i \(-0.352227\pi\)
0.447746 + 0.894161i \(0.352227\pi\)
\(812\) 1.03881e9 1.94030
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −3.16134e8 −0.579703
\(818\) 0 0
\(819\) −5.49353e8 −1.00000
\(820\) −1.57500e9 −2.85654
\(821\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) −1.01861e9 −1.82729 −0.913645 − 0.406512i \(-0.866745\pi\)
−0.913645 + 0.406512i \(0.866745\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) −9.19590e8 −1.61996
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −5.75930e8 −1.00000
\(833\) 0 0
\(834\) 0 0
\(835\) −1.48950e8 −0.255848
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.00772e9 −1.70629 −0.853144 − 0.521676i \(-0.825307\pi\)
−0.853144 + 0.521676i \(0.825307\pi\)
\(840\) 0 0
\(841\) 1.64455e9 2.76477
\(842\) 0 0
\(843\) 0 0
\(844\) −6.10961e8 −1.01622
\(845\) 9.55708e8 1.58400
\(846\) 0 0
\(847\) 6.07645e8 1.00000
\(848\) −5.03931e8 −0.826387
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −3.80695e8 −0.613380 −0.306690 − 0.951809i \(-0.599222\pi\)
−0.306690 + 0.951809i \(0.599222\pi\)
\(854\) 0 0
\(855\) −1.65272e9 −2.64423
\(856\) 0 0
\(857\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 3.49874e8 0.550068
\(861\) 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
\(868\) −1.71445e8 −0.262160
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 1.05679e9 1.58835
\(874\) 0 0
\(875\) 5.40188e8 0.806345
\(876\) 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
\(883\) 8.90402e7 0.129331 0.0646657 − 0.997907i \(-0.479402\pi\)
0.0646657 + 0.997907i \(0.479402\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 1.36911e9 1.94864
\(890\) 0 0
\(891\) 0 0
\(892\) −1.04192e9 −1.46805
\(893\) 6.12117e7 0.0859568
\(894\) 0 0
\(895\) −1.31287e9 −1.83128
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) −3.69585e8 −0.508669
\(900\) 1.10010e9 1.50906
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −1.47053e9 −1.97085 −0.985423 − 0.170121i \(-0.945584\pi\)
−0.985423 + 0.170121i \(0.945584\pi\)
\(908\) −1.48105e9 −1.97840
\(909\) 0 0
\(910\) 0 0
\(911\) 1.03490e9 1.36881 0.684407 − 0.729100i \(-0.260061\pi\)
0.684407 + 0.729100i \(0.260061\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.00623e8 0.130921
\(917\) 0 0
\(918\) 0 0
\(919\) 6.69876e8 0.863074 0.431537 − 0.902095i \(-0.357971\pi\)
0.431537 + 0.902095i \(0.357971\pi\)
\(920\) 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
\(929\) 1.50349e9 1.87523 0.937615 − 0.347675i \(-0.113029\pi\)
0.937615 + 0.347675i \(0.113029\pi\)
\(930\) 0 0
\(931\) −1.34708e9 −1.66934
\(932\) −5.33808e8 −0.659382
\(933\) 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
\(940\) −6.77445e7 −0.0815625
\(941\) −1.61097e9 −1.93338 −0.966691 − 0.255947i \(-0.917613\pi\)
−0.966691 + 0.255947i \(0.917613\pi\)
\(942\) 0 0
\(943\) 2.44976e9 2.92138
\(944\) 8.63724e8 1.02674
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) 9.64997e8 1.12909
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.03382e9 1.19445 0.597223 − 0.802075i \(-0.296271\pi\)
0.597223 + 0.802075i \(0.296271\pi\)
\(954\) 0 0
\(955\) −3.88900e8 −0.446507
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −8.26508e8 −0.931272
\(962\) 0 0
\(963\) 1.29547e9 1.45060
\(964\) −9.58180e8 −1.06959
\(965\) 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
\(980\) 1.49085e9 1.58400
\(981\) 0 0
\(982\) 0 0
\(983\) −5.53086e8 −0.582280 −0.291140 − 0.956680i \(-0.594035\pi\)
−0.291140 + 0.956680i \(0.594035\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 1.60996e9 1.66934
\(989\) −5.44193e8 −0.562554
\(990\) 0 0
\(991\) −5.31220e8 −0.545825 −0.272913 − 0.962039i \(-0.587987\pi\)
−0.272913 + 0.962039i \(0.587987\pi\)
\(992\) 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 91.7.b.b.90.1 yes 1
7.6 odd 2 91.7.b.a.90.1 ✓ 1
13.12 even 2 91.7.b.a.90.1 ✓ 1
91.90 odd 2 CM 91.7.b.b.90.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.7.b.a.90.1 ✓ 1 7.6 odd 2
91.7.b.a.90.1 ✓ 1 13.12 even 2
91.7.b.b.90.1 yes 1 1.1 even 1 trivial
91.7.b.b.90.1 yes 1 91.90 odd 2 CM