Properties

Label 1764.2.e.d.1079.4
Level $1764$
Weight $2$
Character 1764.1079
Analytic conductor $14.086$
Analytic rank $0$
Dimension $4$
CM discriminant -4
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1764,2,Mod(1079,1764)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1764, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1764.1079");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1764 = 2^{2} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1764.e (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: no
Analytic conductor: \(14.0856109166\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(\zeta_{8})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1079.4
Root \(-0.707107 - 0.707107i\) of defining polynomial
Character \(\chi\) \(=\) 1764.1079
Dual form 1764.2.e.d.1079.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.41421i q^{2} -2.00000 q^{4} +4.00000i q^{5} -2.82843i q^{8} +O(q^{10})\) \(q+1.41421i q^{2} -2.00000 q^{4} +4.00000i q^{5} -2.82843i q^{8} -5.65685 q^{10} +1.41421 q^{13} +4.00000 q^{16} +8.00000i q^{17} -8.00000i q^{20} -11.0000 q^{25} +2.00000i q^{26} +9.89949i q^{29} +5.65685i q^{32} -11.3137 q^{34} -12.0000 q^{37} +11.3137 q^{40} -10.0000i q^{41} -15.5563i q^{50} -2.82843 q^{52} -12.7279i q^{53} -14.0000 q^{58} -1.41421 q^{61} -8.00000 q^{64} +5.65685i q^{65} -16.0000i q^{68} +15.5563 q^{73} -16.9706i q^{74} +16.0000i q^{80} +14.1421 q^{82} -32.0000 q^{85} +10.0000i q^{89} +7.07107 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 8 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 4 q - 8 q^{4} + 16 q^{16} - 44 q^{25} - 48 q^{37} - 56 q^{58} - 32 q^{64} - 128 q^{85}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1764\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(883\) \(1081\)
\(\chi(n)\) \(-1\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.41421i 1.00000i
\(3\) 0 0
\(4\) −2.00000 −1.00000
\(5\) 4.00000i 1.78885i 0.447214 + 0.894427i \(0.352416\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) − 2.82843i − 1.00000i
\(9\) 0 0
\(10\) −5.65685 −1.78885
\(11\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 1.41421 0.392232 0.196116 0.980581i \(-0.437167\pi\)
0.196116 + 0.980581i \(0.437167\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 4.00000 1.00000
\(17\) 8.00000i 1.94029i 0.242536 + 0.970143i \(0.422021\pi\)
−0.242536 + 0.970143i \(0.577979\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) − 8.00000i − 1.78885i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −11.0000 −2.20000
\(26\) 2.00000i 0.392232i
\(27\) 0 0
\(28\) 0 0
\(29\) 9.89949i 1.83829i 0.393919 + 0.919145i \(0.371119\pi\)
−0.393919 + 0.919145i \(0.628881\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 5.65685i 1.00000i
\(33\) 0 0
\(34\) −11.3137 −1.94029
\(35\) 0 0
\(36\) 0 0
\(37\) −12.0000 −1.97279 −0.986394 0.164399i \(-0.947432\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 11.3137 1.78885
\(41\) − 10.0000i − 1.56174i −0.624695 0.780869i \(-0.714777\pi\)
0.624695 0.780869i \(-0.285223\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) − 15.5563i − 2.20000i
\(51\) 0 0
\(52\) −2.82843 −0.392232
\(53\) − 12.7279i − 1.74831i −0.485643 0.874157i \(-0.661414\pi\)
0.485643 0.874157i \(-0.338586\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −14.0000 −1.83829
\(59\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −1.41421 −0.181071 −0.0905357 0.995893i \(-0.528858\pi\)
−0.0905357 + 0.995893i \(0.528858\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) 5.65685i 0.701646i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) − 16.0000i − 1.94029i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0 0
\(73\) 15.5563 1.82073 0.910366 0.413803i \(-0.135800\pi\)
0.910366 + 0.413803i \(0.135800\pi\)
\(74\) − 16.9706i − 1.97279i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 16.0000i 1.78885i
\(81\) 0 0
\(82\) 14.1421 1.56174
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −32.0000 −3.47089
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 10.0000i 1.06000i 0.847998 + 0.529999i \(0.177808\pi\)
−0.847998 + 0.529999i \(0.822192\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 7.07107 0.717958 0.358979 0.933346i \(-0.383125\pi\)
0.358979 + 0.933346i \(0.383125\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 22.0000 2.20000
\(101\) − 2.00000i − 0.199007i −0.995037 0.0995037i \(-0.968274\pi\)
0.995037 0.0995037i \(-0.0317255\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) − 4.00000i − 0.392232i
\(105\) 0 0
\(106\) 18.0000 1.74831
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) − 21.2132i − 1.99557i −0.0665190 0.997785i \(-0.521189\pi\)
0.0665190 0.997785i \(-0.478811\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) − 19.7990i − 1.83829i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) − 2.00000i − 0.181071i
\(123\) 0 0
\(124\) 0 0
\(125\) − 24.0000i − 2.14663i
\(126\) 0 0
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) − 11.3137i − 1.00000i
\(129\) 0 0
\(130\) −8.00000 −0.701646
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 22.6274 1.94029
\(137\) 9.89949i 0.845771i 0.906183 + 0.422885i \(0.138983\pi\)
−0.906183 + 0.422885i \(0.861017\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −39.5980 −3.28843
\(146\) 22.0000i 1.82073i
\(147\) 0 0
\(148\) 24.0000 1.97279
\(149\) 4.24264i 0.347571i 0.984784 + 0.173785i \(0.0555999\pi\)
−0.984784 + 0.173785i \(0.944400\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −7.07107 −0.564333 −0.282166 0.959366i \(-0.591053\pi\)
−0.282166 + 0.959366i \(0.591053\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −22.6274 −1.78885
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 20.0000i 1.56174i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −11.0000 −0.846154
\(170\) − 45.2548i − 3.47089i
\(171\) 0 0
\(172\) 0 0
\(173\) 26.0000i 1.97674i 0.152057 + 0.988372i \(0.451410\pi\)
−0.152057 + 0.988372i \(0.548590\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −14.1421 −1.06000
\(179\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −26.8701 −1.99724 −0.998618 0.0525588i \(-0.983262\pi\)
−0.998618 + 0.0525588i \(0.983262\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) − 48.0000i − 3.52903i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) 0 0
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 10.0000i 0.717958i
\(195\) 0 0
\(196\) 0 0
\(197\) 21.2132i 1.51138i 0.654931 + 0.755689i \(0.272698\pi\)
−0.654931 + 0.755689i \(0.727302\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 31.1127i 2.20000i
\(201\) 0 0
\(202\) 2.82843 0.199007
\(203\) 0 0
\(204\) 0 0
\(205\) 40.0000 2.79372
\(206\) 0 0
\(207\) 0 0
\(208\) 5.65685 0.392232
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 25.4558i 1.74831i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 8.48528i 0.574696i
\(219\) 0 0
\(220\) 0 0
\(221\) 11.3137i 0.761042i
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 30.0000 1.99557
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) −18.3848 −1.21490 −0.607450 0.794358i \(-0.707808\pi\)
−0.607450 + 0.794358i \(0.707808\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 28.0000 1.83829
\(233\) 7.07107i 0.463241i 0.972806 + 0.231621i \(0.0744028\pi\)
−0.972806 + 0.231621i \(0.925597\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) 26.8701 1.73085 0.865426 0.501036i \(-0.167048\pi\)
0.865426 + 0.501036i \(0.167048\pi\)
\(242\) − 15.5563i − 1.00000i
\(243\) 0 0
\(244\) 2.82843 0.181071
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 33.9411 2.14663
\(251\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) 2.00000i 0.124757i 0.998053 + 0.0623783i \(0.0198685\pi\)
−0.998053 + 0.0623783i \(0.980131\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) − 11.3137i − 0.701646i
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 50.9117 3.12748
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 20.0000i 1.21942i 0.792624 + 0.609711i \(0.208714\pi\)
−0.792624 + 0.609711i \(0.791286\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 32.0000i 1.94029i
\(273\) 0 0
\(274\) −14.0000 −0.845771
\(275\) 0 0
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 15.5563i 0.928014i 0.885832 + 0.464007i \(0.153589\pi\)
−0.885832 + 0.464007i \(0.846411\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −47.0000 −2.76471
\(290\) − 56.0000i − 3.28843i
\(291\) 0 0
\(292\) −31.1127 −1.82073
\(293\) 34.0000i 1.98630i 0.116841 + 0.993151i \(0.462723\pi\)
−0.116841 + 0.993151i \(0.537277\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 33.9411i 1.97279i
\(297\) 0 0
\(298\) −6.00000 −0.347571
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) − 5.65685i − 0.323911i
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 35.3553 1.99840 0.999201 0.0399680i \(-0.0127256\pi\)
0.999201 + 0.0399680i \(0.0127256\pi\)
\(314\) − 10.0000i − 0.564333i
\(315\) 0 0
\(316\) 0 0
\(317\) 4.24264i 0.238290i 0.992877 + 0.119145i \(0.0380154\pi\)
−0.992877 + 0.119145i \(0.961985\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) − 32.0000i − 1.78885i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) −15.5563 −0.862911
\(326\) 0 0
\(327\) 0 0
\(328\) −28.2843 −1.56174
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 18.0000 0.980522 0.490261 0.871576i \(-0.336901\pi\)
0.490261 + 0.871576i \(0.336901\pi\)
\(338\) − 15.5563i − 0.846154i
\(339\) 0 0
\(340\) 64.0000 3.47089
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −36.7696 −1.97674
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) 18.3848 0.984115 0.492057 0.870563i \(-0.336245\pi\)
0.492057 + 0.870563i \(0.336245\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 16.0000i 0.851594i 0.904819 + 0.425797i \(0.140006\pi\)
−0.904819 + 0.425797i \(0.859994\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) − 20.0000i − 1.06000i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) − 38.0000i − 1.99724i
\(363\) 0 0
\(364\) 0 0
\(365\) 62.2254i 3.25703i
\(366\) 0 0
\(367\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 67.8823 3.52903
\(371\) 0 0
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 14.0000i 0.721037i
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) − 19.7990i − 1.00774i
\(387\) 0 0
\(388\) −14.1421 −0.717958
\(389\) 9.89949i 0.501924i 0.967997 + 0.250962i \(0.0807470\pi\)
−0.967997 + 0.250962i \(0.919253\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) −30.0000 −1.51138
\(395\) 0 0
\(396\) 0 0
\(397\) −18.3848 −0.922705 −0.461353 0.887217i \(-0.652636\pi\)
−0.461353 + 0.887217i \(0.652636\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −44.0000 −2.20000
\(401\) − 26.8701i − 1.34183i −0.741536 0.670913i \(-0.765902\pi\)
0.741536 0.670913i \(-0.234098\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 4.00000i 0.199007i
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 24.0416 1.18878 0.594391 0.804176i \(-0.297393\pi\)
0.594391 + 0.804176i \(0.297393\pi\)
\(410\) 56.5685i 2.79372i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 8.00000i 0.392232i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 28.0000 1.36464 0.682318 0.731055i \(-0.260972\pi\)
0.682318 + 0.731055i \(0.260972\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −36.0000 −1.74831
\(425\) − 88.0000i − 4.26863i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(432\) 0 0
\(433\) 7.07107 0.339814 0.169907 0.985460i \(-0.445653\pi\)
0.169907 + 0.985460i \(0.445653\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −12.0000 −0.574696
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −16.0000 −0.761042
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −40.0000 −1.89618
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) − 38.1838i − 1.80200i −0.433816 0.901002i \(-0.642833\pi\)
0.433816 0.901002i \(-0.357167\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 42.4264i 1.99557i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 8.00000 0.374224 0.187112 0.982339i \(-0.440087\pi\)
0.187112 + 0.982339i \(0.440087\pi\)
\(458\) − 26.0000i − 1.21490i
\(459\) 0 0
\(460\) 0 0
\(461\) 38.0000i 1.76984i 0.465746 + 0.884918i \(0.345786\pi\)
−0.465746 + 0.884918i \(0.654214\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 39.5980i 1.83829i
\(465\) 0 0
\(466\) −10.0000 −0.463241
\(467\) 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
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(480\) 0 0
\(481\) −16.9706 −0.773791
\(482\) 38.0000i 1.73085i
\(483\) 0 0
\(484\) 22.0000 1.00000
\(485\) 28.2843i 1.28432i
\(486\) 0 0
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 4.00000i 0.181071i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(492\) 0 0
\(493\) −79.1960 −3.56681
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 48.0000i 2.14663i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) 8.00000 0.355995
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 44.0000i 1.95027i 0.221621 + 0.975133i \(0.428865\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 22.6274i 1.00000i
\(513\) 0 0
\(514\) −2.82843 −0.124757
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 16.0000 0.701646
\(521\) 40.0000i 1.75243i 0.481919 + 0.876216i \(0.339940\pi\)
−0.481919 + 0.876216i \(0.660060\pi\)
\(522\) 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
\(529\) −23.0000 −1.00000
\(530\) 72.0000i 3.12748i
\(531\) 0 0
\(532\) 0 0
\(533\) − 14.1421i − 0.612564i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −28.2843 −1.21942
\(539\) 0 0
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −45.2548 −1.94029
\(545\) 24.0000i 1.02805i
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) − 19.7990i − 0.845771i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 39.5980i 1.68236i
\(555\) 0 0
\(556\) 0 0
\(557\) 46.6690i 1.97743i 0.149805 + 0.988716i \(0.452135\pi\)
−0.149805 + 0.988716i \(0.547865\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −22.0000 −0.928014
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 84.8528 3.56978
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 9.89949i 0.415008i 0.978234 + 0.207504i \(0.0665341\pi\)
−0.978234 + 0.207504i \(0.933466\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −35.3553 −1.47186 −0.735931 0.677057i \(-0.763255\pi\)
−0.735931 + 0.677057i \(0.763255\pi\)
\(578\) − 66.4680i − 2.76471i
\(579\) 0 0
\(580\) 79.1960 3.28843
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) − 44.0000i − 1.82073i
\(585\) 0 0
\(586\) −48.0833 −1.98630
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −48.0000 −1.97279
\(593\) − 16.0000i − 0.657041i −0.944497 0.328521i \(-0.893450\pi\)
0.944497 0.328521i \(-0.106550\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) − 8.48528i − 0.347571i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) −41.0122 −1.67292 −0.836461 0.548026i \(-0.815379\pi\)
−0.836461 + 0.548026i \(0.815379\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 44.0000i − 1.78885i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 8.00000 0.323911
\(611\) 0 0
\(612\) 0 0
\(613\) 36.0000 1.45403 0.727013 0.686624i \(-0.240908\pi\)
0.727013 + 0.686624i \(0.240908\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 49.4975i − 1.99269i −0.0854011 0.996347i \(-0.527217\pi\)
0.0854011 0.996347i \(-0.472783\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 41.0000 1.64000
\(626\) 50.0000i 1.99840i
\(627\) 0 0
\(628\) 14.1421 0.564333
\(629\) − 96.0000i − 3.82777i
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −6.00000 −0.238290
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 45.2548 1.78885
\(641\) 41.0122i 1.61988i 0.586510 + 0.809942i \(0.300502\pi\)
−0.586510 + 0.809942i \(0.699498\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 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
\(650\) − 22.0000i − 0.862911i
\(651\) 0 0
\(652\) 0 0
\(653\) − 49.4975i − 1.93699i −0.249041 0.968493i \(-0.580115\pi\)
0.249041 0.968493i \(-0.419885\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) − 40.0000i − 1.56174i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 26.8701 1.04512 0.522562 0.852601i \(-0.324976\pi\)
0.522562 + 0.852601i \(0.324976\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 24.0000 0.925132 0.462566 0.886585i \(-0.346929\pi\)
0.462566 + 0.886585i \(0.346929\pi\)
\(674\) 25.4558i 0.980522i
\(675\) 0 0
\(676\) 22.0000 0.846154
\(677\) 2.00000i 0.0768662i 0.999261 + 0.0384331i \(0.0122367\pi\)
−0.999261 + 0.0384331i \(0.987763\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 90.5097i 3.47089i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −39.5980 −1.51296
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) − 18.0000i − 0.685745i
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) − 52.0000i − 1.97674i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 80.0000 3.03022
\(698\) 26.0000i 0.984115i
\(699\) 0 0
\(700\) 0 0
\(701\) − 43.8406i − 1.65584i −0.560848 0.827919i \(-0.689525\pi\)
0.560848 0.827919i \(-0.310475\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −22.6274 −0.851594
\(707\) 0 0
\(708\) 0 0
\(709\) 30.0000 1.12667 0.563337 0.826227i \(-0.309517\pi\)
0.563337 + 0.826227i \(0.309517\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 28.2843 1.06000
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 26.8701i 1.00000i
\(723\) 0 0
\(724\) 53.7401 1.99724
\(725\) − 108.894i − 4.04424i
\(726\) 0 0
\(727\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) −88.0000 −3.25703
\(731\) 0 0
\(732\) 0 0
\(733\) 41.0122 1.51482 0.757410 0.652940i \(-0.226464\pi\)
0.757410 + 0.652940i \(0.226464\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 96.0000i 3.52903i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −16.9706 −0.621753
\(746\) − 19.7990i − 0.724893i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −19.7990 −0.721037
\(755\) 0 0
\(756\) 0 0
\(757\) 18.0000 0.654221 0.327111 0.944986i \(-0.393925\pi\)
0.327111 + 0.944986i \(0.393925\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) − 38.0000i − 1.37750i −0.724999 0.688749i \(-0.758160\pi\)
0.724999 0.688749i \(-0.241840\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −52.3259 −1.88692 −0.943460 0.331486i \(-0.892450\pi\)
−0.943460 + 0.331486i \(0.892450\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 28.0000 1.00774
\(773\) − 34.0000i − 1.22290i −0.791285 0.611448i \(-0.790588\pi\)
0.791285 0.611448i \(-0.209412\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) − 20.0000i − 0.717958i
\(777\) 0 0
\(778\) −14.0000 −0.501924
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) − 28.2843i − 1.00951i
\(786\) 0 0
\(787\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(788\) − 42.4264i − 1.51138i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) − 26.0000i − 0.922705i
\(795\) 0 0
\(796\) 0 0
\(797\) 22.0000i 0.779280i 0.920967 + 0.389640i \(0.127401\pi\)
−0.920967 + 0.389640i \(0.872599\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) − 62.2254i − 2.20000i
\(801\) 0 0
\(802\) 38.0000 1.34183
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −5.65685 −0.199007
\(809\) − 46.6690i − 1.64080i −0.571793 0.820398i \(-0.693752\pi\)
0.571793 0.820398i \(-0.306248\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 34.0000i 1.18878i
\(819\) 0 0
\(820\) −80.0000 −2.79372
\(821\) 55.1543i 1.92490i 0.271460 + 0.962450i \(0.412493\pi\)
−0.271460 + 0.962450i \(0.587507\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 52.3259 1.81735 0.908677 0.417500i \(-0.137094\pi\)
0.908677 + 0.417500i \(0.137094\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −11.3137 −0.392232
\(833\) 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
\(841\) −69.0000 −2.37931
\(842\) 39.5980i 1.36464i
\(843\) 0 0
\(844\) 0 0
\(845\) − 44.0000i − 1.51365i
\(846\) 0 0
\(847\) 0 0
\(848\) − 50.9117i − 1.74831i
\(849\) 0 0
\(850\) 124.451 4.26863
\(851\) 0 0
\(852\) 0 0
\(853\) −57.9828 −1.98529 −0.992646 0.121054i \(-0.961372\pi\)
−0.992646 + 0.121054i \(0.961372\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 58.0000i 1.98124i 0.136637 + 0.990621i \(0.456370\pi\)
−0.136637 + 0.990621i \(0.543630\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −104.000 −3.53611
\(866\) 10.0000i 0.339814i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) − 16.9706i − 0.574696i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 12.0000 0.405211 0.202606 0.979260i \(-0.435059\pi\)
0.202606 + 0.979260i \(0.435059\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 32.0000i 1.07811i 0.842271 + 0.539054i \(0.181218\pi\)
−0.842271 + 0.539054i \(0.818782\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) − 22.6274i − 0.761042i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) − 56.5685i − 1.89618i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 54.0000 1.80200
\(899\) 0 0
\(900\) 0 0
\(901\) 101.823 3.39223
\(902\) 0 0
\(903\) 0 0
\(904\) −60.0000 −1.99557
\(905\) − 107.480i − 3.57276i
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 11.3137i 0.374224i
\(915\) 0 0
\(916\) 36.7696 1.21490
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −53.7401 −1.76984
\(923\) 0 0
\(924\) 0 0
\(925\) 132.000 4.34013
\(926\) 0 0
\(927\) 0 0
\(928\) −56.0000 −1.83829
\(929\) 46.0000i 1.50921i 0.656179 + 0.754606i \(0.272172\pi\)
−0.656179 + 0.754606i \(0.727828\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 14.1421i − 0.463241i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 60.8112 1.98661 0.993307 0.115501i \(-0.0368473\pi\)
0.993307 + 0.115501i \(0.0368473\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 20.0000i 0.651981i 0.945373 + 0.325991i \(0.105698\pi\)
−0.945373 + 0.325991i \(0.894302\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 22.0000 0.714150
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) − 21.2132i − 0.687163i −0.939123 0.343582i \(-0.888360\pi\)
0.939123 0.343582i \(-0.111640\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 31.0000 1.00000
\(962\) − 24.0000i − 0.773791i
\(963\) 0 0
\(964\) −53.7401 −1.73085
\(965\) − 56.0000i − 1.80270i
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 31.1127i 1.00000i
\(969\) 0 0
\(970\) −40.0000 −1.28432
\(971\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −5.65685 −0.181071
\(977\) − 49.4975i − 1.58356i −0.610803 0.791782i \(-0.709153\pi\)
0.610803 0.791782i \(-0.290847\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −84.8528 −2.70364
\(986\) − 112.000i − 3.56681i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 52.3259 1.65718 0.828589 0.559857i \(-0.189144\pi\)
0.828589 + 0.559857i \(0.189144\pi\)
\(998\) 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 1764.2.e.d.1079.4 yes 4
3.2 odd 2 inner 1764.2.e.d.1079.1 4
4.3 odd 2 CM 1764.2.e.d.1079.4 yes 4
7.6 odd 2 inner 1764.2.e.d.1079.3 yes 4
12.11 even 2 inner 1764.2.e.d.1079.1 4
21.20 even 2 inner 1764.2.e.d.1079.2 yes 4
28.27 even 2 inner 1764.2.e.d.1079.3 yes 4
84.83 odd 2 inner 1764.2.e.d.1079.2 yes 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1764.2.e.d.1079.1 4 3.2 odd 2 inner
1764.2.e.d.1079.1 4 12.11 even 2 inner
1764.2.e.d.1079.2 yes 4 21.20 even 2 inner
1764.2.e.d.1079.2 yes 4 84.83 odd 2 inner
1764.2.e.d.1079.3 yes 4 7.6 odd 2 inner
1764.2.e.d.1079.3 yes 4 28.27 even 2 inner
1764.2.e.d.1079.4 yes 4 1.1 even 1 trivial
1764.2.e.d.1079.4 yes 4 4.3 odd 2 CM