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Properties

Label 441.8.a.f.1.1
Level $441$
Weight $8$
Character 441.1
Self dual yes
Analytic conductor $137.762$
Analytic rank $0$
Dimension $1$
CM discriminant -7
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [441,8,Mod(1,441)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(441, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 8, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("441.1");
 
S:= CuspForms(chi, 8);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 441 = 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 441.a (trivial)

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: \(137.761796238\)
Analytic rank: \(0\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 49)
Fricke sign: \(1\)
Sato-Tate group: $N(\mathrm{U}(1))$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 441.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+13.0000 q^{2} +41.0000 q^{4} -1131.00 q^{8} +O(q^{10})\) \(q+13.0000 q^{2} +41.0000 q^{4} -1131.00 q^{8} -8684.00 q^{11} -19951.0 q^{16} -112892. q^{22} +67976.0 q^{23} -78125.0 q^{25} +253622. q^{29} -114595. q^{32} -278382. q^{37} +1.01420e6 q^{43} -356044. q^{44} +883688. q^{46} -1.01562e6 q^{50} +1.80461e6 q^{53} +3.29709e6 q^{58} +1.06399e6 q^{64} -4.86309e6 q^{67} +3.72174e6 q^{71} -3.61897e6 q^{74} +1.10175e6 q^{79} +1.31847e7 q^{86} +9.82160e6 q^{88} +2.78702e6 q^{92} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 13.0000 1.14905 0.574524 − 0.818488i \(-0.305187\pi\)
0.574524 + 0.818488i \(0.305187\pi\)
\(3\) 0 0
\(4\) 41.0000 0.320312
\(5\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) −1131.00 −0.780994
\(9\) 0 0
\(10\) 0 0
\(11\) −8684.00 −1.96719 −0.983593 − 0.180402i \(-0.942260\pi\)
−0.983593 + 0.180402i \(0.942260\pi\)
\(12\) 0 0
\(13\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −19951.0 −1.21771
\(17\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) 0 0
\(19\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −112892. −2.26039
\(23\) 67976.0 1.16495 0.582476 − 0.812848i \(-0.302084\pi\)
0.582476 + 0.812848i \(0.302084\pi\)
\(24\) 0 0
\(25\) −78125.0 −1.00000
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 253622. 1.93105 0.965526 − 0.260307i \(-0.0838238\pi\)
0.965526 + 0.260307i \(0.0838238\pi\)
\(30\) 0 0
\(31\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) −114595. −0.618217
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) −278382. −0.903514 −0.451757 − 0.892141i \(-0.649203\pi\)
−0.451757 + 0.892141i \(0.649203\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) 1.01420e6 1.94530 0.972648 − 0.232284i \(-0.0746198\pi\)
0.972648 + 0.232284i \(0.0746198\pi\)
\(44\) −356044. −0.630114
\(45\) 0 0
\(46\) 883688. 1.33859
\(47\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) −1.01562e6 −1.14905
\(51\) 0 0
\(52\) 0 0
\(53\) 1.80461e6 1.66501 0.832507 − 0.554015i \(-0.186905\pi\)
0.832507 + 0.554015i \(0.186905\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 3.29709e6 2.21887
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.06399e6 0.507351
\(65\) 0 0
\(66\) 0 0
\(67\) −4.86309e6 −1.97538 −0.987690 − 0.156424i \(-0.950003\pi\)
−0.987690 + 0.156424i \(0.950003\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 3.72174e6 1.23408 0.617039 − 0.786933i \(-0.288332\pi\)
0.617039 + 0.786933i \(0.288332\pi\)
\(72\) 0 0
\(73\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −3.61897e6 −1.03818
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 1.10175e6 0.251414 0.125707 − 0.992067i \(-0.459880\pi\)
0.125707 + 0.992067i \(0.459880\pi\)
\(80\) 0 0
\(81\) 0 0
\(82\) 0 0
\(83\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 1.31847e7 2.23524
\(87\) 0 0
\(88\) 9.82160e6 1.53636
\(89\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 2.78702e6 0.373149
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −3.20312e6 −0.320312
\(101\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 2.34599e7 1.91318
\(107\) −6.01802e6 −0.474909 −0.237455 − 0.971399i \(-0.576313\pi\)
−0.237455 + 0.971399i \(0.576313\pi\)
\(108\) 0 0
\(109\) −2.26341e7 −1.67406 −0.837030 − 0.547158i \(-0.815710\pi\)
−0.837030 + 0.547158i \(0.815710\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 1.87956e7 1.22541 0.612706 − 0.790311i \(-0.290081\pi\)
0.612706 + 0.790311i \(0.290081\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 1.03985e7 0.618540
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 5.59247e7 2.86982
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 3.17339e7 1.37471 0.687353 − 0.726323i \(-0.258772\pi\)
0.687353 + 0.726323i \(0.258772\pi\)
\(128\) 2.85001e7 1.20119
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −6.32202e7 −2.26981
\(135\) 0 0
\(136\) 0 0
\(137\) −3.12871e6 −0.103955 −0.0519773 − 0.998648i \(-0.516552\pi\)
−0.0519773 + 0.998648i \(0.516552\pi\)
\(138\) 0 0
\(139\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 4.83827e7 1.41801
\(143\) 0 0
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) −1.14137e7 −0.289407
\(149\) 8.07570e7 1.99999 0.999995 − 0.00300609i \(-0.000956869\pi\)
0.999995 + 0.00300609i \(0.000956869\pi\)
\(150\) 0 0
\(151\) −4.37642e6 −0.103443 −0.0517214 − 0.998662i \(-0.516471\pi\)
−0.0517214 + 0.998662i \(0.516471\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 1.43228e7 0.288887
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.77658e6 0.0321313 0.0160656 − 0.999871i \(-0.494886\pi\)
0.0160656 + 0.999871i \(0.494886\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −6.27485e7 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 4.15824e7 0.623103
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 1.73254e8 2.39547
\(177\) 0 0
\(178\) 0 0
\(179\) 1.33147e8 1.73519 0.867595 − 0.497271i \(-0.165664\pi\)
0.867595 + 0.497271i \(0.165664\pi\)
\(180\) 0 0
\(181\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) −7.68809e7 −0.909821
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 1.70419e8 1.76970 0.884851 − 0.465874i \(-0.154260\pi\)
0.884851 + 0.465874i \(0.154260\pi\)
\(192\) 0 0
\(193\) 1.94899e8 1.95146 0.975728 − 0.218987i \(-0.0702752\pi\)
0.975728 + 0.218987i \(0.0702752\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) −1.94567e8 −1.81317 −0.906585 − 0.422024i \(-0.861320\pi\)
−0.906585 + 0.422024i \(0.861320\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 8.83594e7 0.780994
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 4.37312e7 0.320482 0.160241 − 0.987078i \(-0.448773\pi\)
0.160241 + 0.987078i \(0.448773\pi\)
\(212\) 7.39890e7 0.533325
\(213\) 0 0
\(214\) −7.82343e7 −0.545693
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −2.94244e8 −1.92358
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 2.44343e8 1.40806
\(227\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −2.86846e8 −1.50814
\(233\) −2.33756e8 −1.21065 −0.605323 − 0.795980i \(-0.706956\pi\)
−0.605323 + 0.795980i \(0.706956\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) −2.60700e8 −1.23523 −0.617615 − 0.786481i \(-0.711901\pi\)
−0.617615 + 0.786481i \(0.711901\pi\)
\(240\) 0 0
\(241\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(242\) 7.27021e8 3.29756
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −5.90304e8 −2.29168
\(254\) 4.12540e8 1.57960
\(255\) 0 0
\(256\) 2.34310e8 0.872872
\(257\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −2.42562e8 −0.822200 −0.411100 − 0.911590i \(-0.634855\pi\)
−0.411100 + 0.911590i \(0.634855\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.99387e8 −0.632739
\(269\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(270\) 0 0
\(271\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −4.06732e7 −0.119449
\(275\) 6.78438e8 1.96719
\(276\) 0 0
\(277\) 5.97469e8 1.68903 0.844513 − 0.535535i \(-0.179890\pi\)
0.844513 + 0.535535i \(0.179890\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 5.23796e7 0.140828 0.0704142 − 0.997518i \(-0.477568\pi\)
0.0704142 + 0.997518i \(0.477568\pi\)
\(282\) 0 0
\(283\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 1.52592e8 0.395290
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −4.10339e8 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 3.14850e8 0.705639
\(297\) 0 0
\(298\) 1.04984e9 2.29809
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −5.68935e7 −0.118861
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 4.51718e7 0.0805310
\(317\) −5.84674e8 −1.03088 −0.515438 − 0.856927i \(-0.672371\pi\)
−0.515438 + 0.856927i \(0.672371\pi\)
\(318\) 0 0
\(319\) −2.20245e9 −3.79874
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 2.30955e7 0.0369204
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 6.88824e8 1.04402 0.522012 − 0.852938i \(-0.325182\pi\)
0.522012 + 0.852938i \(0.325182\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −5.61896e8 −0.799745 −0.399872 − 0.916571i \(-0.630946\pi\)
−0.399872 + 0.916571i \(0.630946\pi\)
\(338\) −8.15731e8 −1.14905
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) −1.14706e9 −1.51926
\(345\) 0 0
\(346\) 0 0
\(347\) 1.06448e9 1.36768 0.683839 − 0.729633i \(-0.260309\pi\)
0.683839 + 0.729633i \(0.260309\pi\)
\(348\) 0 0
\(349\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 9.95143e8 1.21615
\(353\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 1.73092e9 1.99382
\(359\) 1.74745e9 1.99331 0.996655 − 0.0817184i \(-0.0260408\pi\)
0.996655 + 0.0817184i \(0.0260408\pi\)
\(360\) 0 0
\(361\) −8.93872e8 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) −1.35619e9 −1.41858
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 1.78662e9 1.78258 0.891292 − 0.453430i \(-0.149800\pi\)
0.891292 + 0.453430i \(0.149800\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 1.72231e9 1.62508 0.812539 − 0.582906i \(-0.198085\pi\)
0.812539 + 0.582906i \(0.198085\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 2.21544e9 2.03347
\(383\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 2.53368e9 2.24232
\(387\) 0 0
\(388\) 0 0
\(389\) −7.53566e8 −0.649079 −0.324540 − 0.945872i \(-0.605209\pi\)
−0.324540 + 0.945872i \(0.605209\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −2.52938e9 −2.08342
\(395\) 0 0
\(396\) 0 0
\(397\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 1.55867e9 1.21771
\(401\) −1.87870e9 −1.45496 −0.727480 − 0.686129i \(-0.759309\pi\)
−0.727480 + 0.686129i \(0.759309\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 2.41747e9 1.77738
\(408\) 0 0
\(409\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) −3.04946e9 −1.99176 −0.995878 − 0.0906996i \(-0.971090\pi\)
−0.995878 + 0.0906996i \(0.971090\pi\)
\(422\) 5.68506e8 0.368249
\(423\) 0 0
\(424\) −2.04101e9 −1.30037
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −2.46739e8 −0.152119
\(429\) 0 0
\(430\) 0 0
\(431\) −4.10137e8 −0.246751 −0.123375 − 0.992360i \(-0.539372\pi\)
−0.123375 + 0.992360i \(0.539372\pi\)
\(432\) 0 0
\(433\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −9.27999e8 −0.536222
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 1.16533e9 0.636845 0.318423 − 0.947949i \(-0.396847\pi\)
0.318423 + 0.947949i \(0.396847\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 1.24479e9 0.648981 0.324491 − 0.945889i \(-0.394807\pi\)
0.324491 + 0.945889i \(0.394807\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 7.70620e8 0.392515
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 3.40168e9 1.66720 0.833599 − 0.552370i \(-0.186276\pi\)
0.833599 + 0.552370i \(0.186276\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 3.75457e9 1.75803 0.879017 − 0.476791i \(-0.158200\pi\)
0.879017 + 0.476791i \(0.158200\pi\)
\(464\) −5.06001e9 −2.35147
\(465\) 0 0
\(466\) −3.03883e9 −1.39109
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) −8.80735e9 −3.82676
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −3.38909e9 −1.41934
\(479\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 2.29291e9 0.919239
\(485\) 0 0
\(486\) 0 0
\(487\) −3.93841e9 −1.54514 −0.772572 − 0.634927i \(-0.781030\pi\)
−0.772572 + 0.634927i \(0.781030\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −3.50996e9 −1.33819 −0.669095 − 0.743177i \(-0.733318\pi\)
−0.669095 + 0.743177i \(0.733318\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.51210e9 −0.544788 −0.272394 − 0.962186i \(-0.587815\pi\)
−0.272394 + 0.962186i \(0.587815\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −7.67395e9 −2.63325
\(507\) 0 0
\(508\) 1.30109e9 0.440336
\(509\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −6.01982e8 −0.198216
\(513\) 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 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) 0 0
\(523\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −3.15331e9 −0.944748
\(527\) 0 0
\(528\) 0 0
\(529\) 1.21591e9 0.357114
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 5.50016e9 1.54276
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −3.82386e9 −1.03827 −0.519136 − 0.854692i \(-0.673746\pi\)
−0.519136 + 0.854692i \(0.673746\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −5.76648e9 −1.50645 −0.753225 − 0.657763i \(-0.771503\pi\)
−0.753225 + 0.657763i \(0.771503\pi\)
\(548\) −1.28277e8 −0.0332979
\(549\) 0 0
\(550\) 8.81969e9 2.26039
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 7.76710e9 1.94077
\(555\) 0 0
\(556\) 0 0
\(557\) 9.88321e7 0.0242329 0.0121164 − 0.999927i \(-0.496143\pi\)
0.0121164 + 0.999927i \(0.496143\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 6.80935e8 0.161819
\(563\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) −4.20929e9 −0.963807
\(569\) −1.86126e9 −0.423559 −0.211780 − 0.977317i \(-0.567926\pi\)
−0.211780 + 0.977317i \(0.567926\pi\)
\(570\) 0 0
\(571\) −4.92469e9 −1.10701 −0.553506 − 0.832845i \(-0.686710\pi\)
−0.553506 + 0.832845i \(0.686710\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) −5.31063e9 −1.16495
\(576\) 0 0
\(577\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −5.33440e9 −1.14905
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −1.56712e10 −3.27539
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 5.55400e9 1.10022
\(593\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 3.31104e9 0.640622
\(597\) 0 0
\(598\) 0 0
\(599\) −1.01242e10 −1.92472 −0.962358 − 0.271784i \(-0.912386\pi\)
−0.962358 + 0.271784i \(0.912386\pi\)
\(600\) 0 0
\(601\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) −1.79433e8 −0.0331340
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.81409e9 −0.318087 −0.159044 − 0.987272i \(-0.550841\pi\)
−0.159044 + 0.987272i \(0.550841\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 7.64091e9 1.30963 0.654813 − 0.755791i \(-0.272747\pi\)
0.654813 + 0.755791i \(0.272747\pi\)
\(618\) 0 0
\(619\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 6.10352e9 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −9.66981e9 −1.53220 −0.766100 − 0.642722i \(-0.777805\pi\)
−0.766100 + 0.642722i \(0.777805\pi\)
\(632\) −1.24608e9 −0.196353
\(633\) 0 0
\(634\) −7.60076e9 −1.18453
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) −2.86319e10 −4.36493
\(639\) 0 0
\(640\) 0 0
\(641\) 1.32357e10 1.98492 0.992461 − 0.122558i \(-0.0391097\pi\)
0.992461 + 0.122558i \(0.0391097\pi\)
\(642\) 0 0
\(643\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 7.28398e7 0.0102921
\(653\) −1.57247e9 −0.220997 −0.110499 − 0.993876i \(-0.535245\pi\)
−0.110499 + 0.993876i \(0.535245\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) 1.17145e10 1.59450 0.797252 − 0.603647i \(-0.206286\pi\)
0.797252 + 0.603647i \(0.206286\pi\)
\(660\) 0 0
\(661\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 8.95472e9 1.19963
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 1.72402e10 2.24958
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.45874e10 1.84470 0.922349 − 0.386357i \(-0.126267\pi\)
0.922349 + 0.386357i \(0.126267\pi\)
\(674\) −7.30465e9 −0.918945
\(675\) 0 0
\(676\) −2.57269e9 −0.320312
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 1.26307e10 1.51689 0.758444 − 0.651738i \(-0.225960\pi\)
0.758444 + 0.651738i \(0.225960\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) −2.02344e10 −2.36881
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 1.38382e10 1.57153
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 4.76773e9 0.522755 0.261378 − 0.965237i \(-0.415823\pi\)
0.261378 + 0.965237i \(0.415823\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −9.23972e9 −0.998055
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 1.34870e10 1.42120 0.710599 − 0.703597i \(-0.248424\pi\)
0.710599 + 0.703597i \(0.248424\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 5.45905e9 0.555803
\(717\) 0 0
\(718\) 2.27169e10 2.29041
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −1.16203e10 −1.14905
\(723\) 0 0
\(724\) 0 0
\(725\) −1.98142e10 −1.93105
\(726\) 0 0
\(727\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) −7.78971e9 −0.720193
\(737\) 4.22311e10 3.88594
\(738\) 0 0
\(739\) 1.05824e10 0.964557 0.482278 − 0.876018i \(-0.339809\pi\)
0.482278 + 0.876018i \(0.339809\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.10452e10 0.987902 0.493951 − 0.869490i \(-0.335552\pi\)
0.493951 + 0.869490i \(0.335552\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 2.32260e10 2.04828
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 2.15179e10 1.85379 0.926894 − 0.375324i \(-0.122468\pi\)
0.926894 + 0.375324i \(0.122468\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −5.11830e9 −0.428835 −0.214417 − 0.976742i \(-0.568785\pi\)
−0.214417 + 0.976742i \(0.568785\pi\)
\(758\) 2.23900e10 1.86729
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 6.98716e9 0.566858
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 7.99085e9 0.625076
\(773\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −9.79636e9 −0.745824
\(779\) 0 0
\(780\) 0 0
\(781\) −3.23196e10 −2.42766
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) −7.97726e9 −0.580781
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 8.95273e9 0.618217
\(801\) 0 0
\(802\) −2.44230e10 −1.67182
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 2.76440e10 1.83561 0.917806 − 0.397030i \(-0.129959\pi\)
0.917806 + 0.397030i \(0.129959\pi\)
\(810\) 0 0
\(811\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 3.14271e10 2.04230
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 1.18646e10 0.748261 0.374130 − 0.927376i \(-0.377941\pi\)
0.374130 + 0.927376i \(0.377941\pi\)
\(822\) 0 0
\(823\) 2.69117e10 1.68284 0.841419 − 0.540383i \(-0.181721\pi\)
0.841419 + 0.540383i \(0.181721\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −6.00744e9 −0.369335 −0.184667 − 0.982801i \(-0.559121\pi\)
−0.184667 + 0.982801i \(0.559121\pi\)
\(828\) 0 0
\(829\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 4.70742e10 2.72896
\(842\) −3.96430e10 −2.28862
\(843\) 0 0
\(844\) 1.79298e9 0.102654
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) −3.60038e10 −2.02751
\(849\) 0 0
\(850\) 0 0
\(851\) −1.89233e10 −1.05255
\(852\) 0 0
\(853\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 6.80638e9 0.370901
\(857\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −5.33179e9 −0.283529
\(863\) 3.08505e10 1.63390 0.816949 − 0.576710i \(-0.195664\pi\)
0.816949 + 0.576710i \(0.195664\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) −9.56761e9 −0.494578
\(870\) 0 0
\(871\) 0 0
\(872\) 2.55992e10 1.30743
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −2.72959e10 −1.36647 −0.683233 − 0.730200i \(-0.739427\pi\)
−0.683233 + 0.730200i \(0.739427\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(882\) 0 0
\(883\) 4.04711e10 1.97825 0.989127 − 0.147064i \(-0.0469822\pi\)
0.989127 + 0.147064i \(0.0469822\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.51492e10 0.731766
\(887\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1.61822e10 0.745711
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) −2.12578e10 −0.957039
\(905\) 0 0
\(906\) 0 0
\(907\) 3.66883e10 1.63268 0.816341 − 0.577570i \(-0.195999\pi\)
0.816341 + 0.577570i \(0.195999\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 4.35058e10 1.90648 0.953242 − 0.302207i \(-0.0977235\pi\)
0.953242 + 0.302207i \(0.0977235\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 4.42219e10 1.91569
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 5.24885e9 0.223079 0.111540 − 0.993760i \(-0.464422\pi\)
0.111540 + 0.993760i \(0.464422\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 2.17486e10 0.903514
\(926\) 4.88095e10 2.02007
\(927\) 0 0
\(928\) −2.90638e10 −1.19381
\(929\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −9.58400e9 −0.387785
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) −1.14496e11 −4.39713
\(947\) −3.83891e10 −1.46887 −0.734435 − 0.678679i \(-0.762553\pi\)
−0.734435 + 0.678679i \(0.762553\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 4.11396e10 1.53970 0.769848 − 0.638227i \(-0.220332\pi\)
0.769848 + 0.638227i \(0.220332\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.06887e10 −0.395659
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −2.75126e10 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −5.53494e10 −1.96843 −0.984216 − 0.176969i \(-0.943371\pi\)
−0.984216 + 0.176969i \(0.943371\pi\)
\(968\) −6.32508e10 −2.24131
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −5.11993e10 −1.77545
\(975\) 0 0
\(976\) 0 0
\(977\) −2.76883e10 −0.949871 −0.474935 − 0.880021i \(-0.657529\pi\)
−0.474935 + 0.880021i \(0.657529\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −4.56295e10 −1.53764
\(983\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 6.89415e10 2.26618
\(990\) 0 0
\(991\) 2.40869e10 0.786183 0.393091 − 0.919499i \(-0.371406\pi\)
0.393091 + 0.919499i \(0.371406\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(998\) −1.96573e10 −0.625988
\(999\) 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 441.8.a.f.1.1 1
3.2 odd 2 49.8.a.a.1.1 ✓ 1
7.6 odd 2 CM 441.8.a.f.1.1 1
21.2 odd 6 49.8.c.c.18.1 2
21.5 even 6 49.8.c.c.18.1 2
21.11 odd 6 49.8.c.c.30.1 2
21.17 even 6 49.8.c.c.30.1 2
21.20 even 2 49.8.a.a.1.1 ✓ 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
49.8.a.a.1.1 ✓ 1 3.2 odd 2
49.8.a.a.1.1 ✓ 1 21.20 even 2
49.8.c.c.18.1 2 21.2 odd 6
49.8.c.c.18.1 2 21.5 even 6
49.8.c.c.30.1 2 21.11 odd 6
49.8.c.c.30.1 2 21.17 even 6
441.8.a.f.1.1 1 1.1 even 1 trivial
441.8.a.f.1.1 1 7.6 odd 2 CM