Properties

Label 2240.2.n.j.1119.4
Level $2240$
Weight $2$
Character 2240.1119
Analytic conductor $17.886$
Analytic rank $0$
Dimension $16$
CM discriminant -35
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2240,2,Mod(1119,2240)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2240, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2240.1119");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2240 = 2^{6} \cdot 5 \cdot 7 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2240.n (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: \(17.8864900528\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: 16.0.968265199641600000000.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 9x^{14} + 44x^{12} + 261x^{10} + 1029x^{8} + 1044x^{6} + 704x^{4} + 576x^{2} + 256 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{24} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 1119.4
Root \(0.369600 - 2.25820i\) of defining polynomial
Character \(\chi\) \(=\) 2240.1119
Dual form 2240.2.n.j.1119.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.40932 q^{3} +2.23607i q^{5} +2.64575i q^{7} +8.62348 q^{9} +O(q^{10})\) \(q-3.40932 q^{3} +2.23607i q^{5} +2.64575i q^{7} +8.62348 q^{9} -5.55612 q^{11} +1.06281i q^{13} -7.62348i q^{15} -5.90512 q^{17} -9.02022i q^{21} -5.00000 q^{25} -19.1722 q^{27} +4.83619i q^{29} +18.9426 q^{33} -5.91608 q^{35} -3.62348i q^{39} +19.2827i q^{45} +13.6511i q^{47} -7.00000 q^{49} +20.1324 q^{51} -12.4239i q^{55} +22.8156i q^{63} -2.37652 q^{65} +12.0000i q^{71} -10.5830 q^{73} +17.0466 q^{75} -14.7001i q^{77} -14.8704i q^{79} +39.4939 q^{81} +8.94427 q^{83} -13.2042i q^{85} -16.4881i q^{87} -2.81194 q^{91} +3.45065 q^{97} -47.9130 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 56 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q + 56 q^{9} - 80 q^{25} - 112 q^{49} - 120 q^{65} + 304 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2240\mathbb{Z}\right)^\times\).

\(n\) \(897\) \(1471\) \(1541\) \(1921\)
\(\chi(n)\) \(-1\) \(-1\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −3.40932 −1.96837 −0.984186 0.177136i \(-0.943317\pi\)
−0.984186 + 0.177136i \(0.943317\pi\)
\(4\) 0 0
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) 2.64575i 1.00000i
\(8\) 0 0
\(9\) 8.62348 2.87449
\(10\) 0 0
\(11\) −5.55612 −1.67523 −0.837616 0.546259i \(-0.816051\pi\)
−0.837616 + 0.546259i \(0.816051\pi\)
\(12\) 0 0
\(13\) 1.06281i 0.294772i 0.989079 + 0.147386i \(0.0470859\pi\)
−0.989079 + 0.147386i \(0.952914\pi\)
\(14\) 0 0
\(15\) − 7.62348i − 1.96837i
\(16\) 0 0
\(17\) −5.90512 −1.43220 −0.716101 0.697997i \(-0.754075\pi\)
−0.716101 + 0.697997i \(0.754075\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) − 9.02022i − 1.96837i
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −19.1722 −3.68970
\(28\) 0 0
\(29\) 4.83619i 0.898058i 0.893517 + 0.449029i \(0.148230\pi\)
−0.893517 + 0.449029i \(0.851770\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) 18.9426 3.29748
\(34\) 0 0
\(35\) −5.91608 −1.00000
\(36\) 0 0
\(37\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(38\) 0 0
\(39\) − 3.62348i − 0.580220i
\(40\) 0 0
\(41\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(44\) 0 0
\(45\) 19.2827i 2.87449i
\(46\) 0 0
\(47\) 13.6511i 1.99122i 0.0936230 + 0.995608i \(0.470155\pi\)
−0.0936230 + 0.995608i \(0.529845\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 20.1324 2.81911
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) − 12.4239i − 1.67523i
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(62\) 0 0
\(63\) 22.8156i 2.87449i
\(64\) 0 0
\(65\) −2.37652 −0.294772
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 12.0000i 1.42414i 0.702109 + 0.712069i \(0.252242\pi\)
−0.702109 + 0.712069i \(0.747758\pi\)
\(72\) 0 0
\(73\) −10.5830 −1.23865 −0.619324 0.785136i \(-0.712593\pi\)
−0.619324 + 0.785136i \(0.712593\pi\)
\(74\) 0 0
\(75\) 17.0466 1.96837
\(76\) 0 0
\(77\) − 14.7001i − 1.67523i
\(78\) 0 0
\(79\) − 14.8704i − 1.67305i −0.547926 0.836527i \(-0.684582\pi\)
0.547926 0.836527i \(-0.315418\pi\)
\(80\) 0 0
\(81\) 39.4939 4.38821
\(82\) 0 0
\(83\) 8.94427 0.981761 0.490881 0.871227i \(-0.336675\pi\)
0.490881 + 0.871227i \(0.336675\pi\)
\(84\) 0 0
\(85\) − 13.2042i − 1.43220i
\(86\) 0 0
\(87\) − 16.4881i − 1.76771i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −2.81194 −0.294772
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 3.45065 0.350361 0.175180 0.984536i \(-0.443949\pi\)
0.175180 + 0.984536i \(0.443949\pi\)
\(98\) 0 0
\(99\) −47.9130 −4.81544
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) − 16.1055i − 1.58693i −0.608618 0.793463i \(-0.708276\pi\)
0.608618 0.793463i \(-0.291724\pi\)
\(104\) 0 0
\(105\) 20.1698 1.96837
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) 0.652160i 0.0624656i 0.999512 + 0.0312328i \(0.00994332\pi\)
−0.999512 + 0.0312328i \(0.990057\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 9.16515i 0.847319i
\(118\) 0 0
\(119\) − 15.6235i − 1.43220i
\(120\) 0 0
\(121\) 19.8704 1.80640
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) − 11.1803i − 1.00000i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) − 42.8704i − 3.68970i
\(136\) 0 0
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) − 46.5409i − 3.91945i
\(142\) 0 0
\(143\) − 5.90512i − 0.493811i
\(144\) 0 0
\(145\) −10.8140 −0.898058
\(146\) 0 0
\(147\) 23.8653 1.96837
\(148\) 0 0
\(149\) 23.6643i 1.93866i 0.245770 + 0.969328i \(0.420959\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) − 6.87043i − 0.559107i −0.960130 0.279554i \(-0.909814\pi\)
0.960130 0.279554i \(-0.0901865\pi\)
\(152\) 0 0
\(153\) −50.9226 −4.11685
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 13.4164i − 1.07075i −0.844616 0.535373i \(-0.820171\pi\)
0.844616 0.535373i \(-0.179829\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 42.3569i 3.29748i
\(166\) 0 0
\(167\) − 17.3328i − 1.34125i −0.741796 0.670625i \(-0.766026\pi\)
0.741796 0.670625i \(-0.233974\pi\)
\(168\) 0 0
\(169\) 11.8704 0.913110
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) − 26.2118i − 1.99284i −0.0845218 0.996422i \(-0.526936\pi\)
0.0845218 0.996422i \(-0.473064\pi\)
\(174\) 0 0
\(175\) − 13.2288i − 1.00000i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 11.8322 0.884377 0.442189 0.896922i \(-0.354202\pi\)
0.442189 + 0.896922i \(0.354202\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 32.8095 2.39927
\(188\) 0 0
\(189\) − 50.7250i − 3.68970i
\(190\) 0 0
\(191\) 8.37652i 0.606104i 0.952974 + 0.303052i \(0.0980056\pi\)
−0.952974 + 0.303052i \(0.901994\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 8.10234 0.580220
\(196\) 0 0
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −12.7954 −0.898058
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 11.0445 0.760332 0.380166 0.924918i \(-0.375867\pi\)
0.380166 + 0.924918i \(0.375867\pi\)
\(212\) 0 0
\(213\) − 40.9119i − 2.80323i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 36.0809 2.43812
\(220\) 0 0
\(221\) − 6.27604i − 0.422172i
\(222\) 0 0
\(223\) 8.74216i 0.585418i 0.956202 + 0.292709i \(0.0945567\pi\)
−0.956202 + 0.292709i \(0.905443\pi\)
\(224\) 0 0
\(225\) −43.1174 −2.87449
\(226\) 0 0
\(227\) 7.66058 0.508450 0.254225 0.967145i \(-0.418180\pi\)
0.254225 + 0.967145i \(0.418180\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 50.1174i 3.29748i
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −30.5248 −1.99122
\(236\) 0 0
\(237\) 50.6981i 3.29319i
\(238\) 0 0
\(239\) 30.1174i 1.94813i 0.226266 + 0.974066i \(0.427348\pi\)
−0.226266 + 0.974066i \(0.572652\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −77.1307 −4.94794
\(244\) 0 0
\(245\) − 15.6525i − 1.00000i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) −30.4939 −1.93247
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 45.0175i 2.81911i
\(256\) 0 0
\(257\) −31.7490 −1.98045 −0.990225 0.139482i \(-0.955456\pi\)
−0.990225 + 0.139482i \(0.955456\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 41.7048i 2.58146i
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 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
\(273\) 9.58681 0.580220
\(274\) 0 0
\(275\) 27.7806 1.67523
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 21.6235 1.28995 0.644974 0.764204i \(-0.276868\pi\)
0.644974 + 0.764204i \(0.276868\pi\)
\(282\) 0 0
\(283\) 14.4792 0.860700 0.430350 0.902662i \(-0.358390\pi\)
0.430350 + 0.902662i \(0.358390\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 17.8704 1.05120
\(290\) 0 0
\(291\) −11.7644 −0.689641
\(292\) 0 0
\(293\) − 8.32322i − 0.486248i −0.969995 0.243124i \(-0.921828\pi\)
0.969995 0.243124i \(-0.0781721\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 106.523 6.18110
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 26.4326 1.50859 0.754295 0.656535i \(-0.227979\pi\)
0.754295 + 0.656535i \(0.227979\pi\)
\(308\) 0 0
\(309\) 54.9090i 3.12366i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −31.9801 −1.80762 −0.903810 0.427934i \(-0.859241\pi\)
−0.903810 + 0.427934i \(0.859241\pi\)
\(314\) 0 0
\(315\) −51.0172 −2.87449
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) − 26.8704i − 1.50446i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) − 5.31407i − 0.294772i
\(326\) 0 0
\(327\) − 2.22342i − 0.122956i
\(328\) 0 0
\(329\) −36.1174 −1.99122
\(330\) 0 0
\(331\) 35.4965 1.95106 0.975531 0.219860i \(-0.0705600\pi\)
0.975531 + 0.219860i \(0.0705600\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) − 18.5203i − 1.00000i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) − 20.3765i − 1.08762i
\(352\) 0 0
\(353\) 13.2685 0.706212 0.353106 0.935583i \(-0.385126\pi\)
0.353106 + 0.935583i \(0.385126\pi\)
\(354\) 0 0
\(355\) −26.8328 −1.42414
\(356\) 0 0
\(357\) 53.2655i 2.81911i
\(358\) 0 0
\(359\) − 36.0000i − 1.90001i −0.312239 0.950004i \(-0.601079\pi\)
0.312239 0.950004i \(-0.398921\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) −67.7447 −3.55567
\(364\) 0 0
\(365\) − 23.6643i − 1.23865i
\(366\) 0 0
\(367\) 19.7872i 1.03289i 0.856322 + 0.516443i \(0.172744\pi\)
−0.856322 + 0.516443i \(0.827256\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(374\) 0 0
\(375\) 38.1174i 1.96837i
\(376\) 0 0
\(377\) −5.13997 −0.264722
\(378\) 0 0
\(379\) −35.4965 −1.82333 −0.911666 0.410932i \(-0.865203\pi\)
−0.911666 + 0.410932i \(0.865203\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 15.8745i 0.811149i 0.914062 + 0.405575i \(0.132929\pi\)
−0.914062 + 0.405575i \(0.867071\pi\)
\(384\) 0 0
\(385\) 32.8704 1.67523
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) − 36.7330i − 1.86244i −0.364459 0.931219i \(-0.618746\pi\)
0.364459 0.931219i \(-0.381254\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 33.2513 1.67305
\(396\) 0 0
\(397\) 39.8490i 1.99997i 0.00579782 + 0.999983i \(0.498154\pi\)
−0.00579782 + 0.999983i \(0.501846\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −12.1174 −0.605113 −0.302556 0.953131i \(-0.597840\pi\)
−0.302556 + 0.953131i \(0.597840\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 88.3110i 4.38821i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 20.0000i 0.981761i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) − 40.9171i − 1.99418i −0.0762630 0.997088i \(-0.524299\pi\)
0.0762630 0.997088i \(-0.475701\pi\)
\(422\) 0 0
\(423\) 117.720i 5.72373i
\(424\) 0 0
\(425\) 29.5256 1.43220
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 20.1324i 0.972004i
\(430\) 0 0
\(431\) 37.3643i 1.79978i 0.436121 + 0.899888i \(0.356352\pi\)
−0.436121 + 0.899888i \(0.643648\pi\)
\(432\) 0 0
\(433\) 10.5830 0.508587 0.254293 0.967127i \(-0.418157\pi\)
0.254293 + 0.967127i \(0.418157\pi\)
\(434\) 0 0
\(435\) 36.8686 1.76771
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) −60.3643 −2.87449
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 80.6793i − 3.81600i
\(448\) 0 0
\(449\) −31.3643 −1.48017 −0.740087 0.672511i \(-0.765216\pi\)
−0.740087 + 0.672511i \(0.765216\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 23.4235i 1.10053i
\(454\) 0 0
\(455\) − 6.28769i − 0.294772i
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) 113.214 5.28439
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) −28.1165 −1.30108 −0.650538 0.759473i \(-0.725457\pi\)
−0.650538 + 0.759473i \(0.725457\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 45.7409i 2.10763i
\(472\) 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
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 7.71590i 0.350361i
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −13.7886 −0.622273 −0.311136 0.950365i \(-0.600710\pi\)
−0.311136 + 0.950365i \(0.600710\pi\)
\(492\) 0 0
\(493\) − 28.5583i − 1.28620i
\(494\) 0 0
\(495\) − 107.137i − 4.81544i
\(496\) 0 0
\(497\) −31.7490 −1.42414
\(498\) 0 0
\(499\) −44.3812 −1.98677 −0.993387 0.114816i \(-0.963372\pi\)
−0.993387 + 0.114816i \(0.963372\pi\)
\(500\) 0 0
\(501\) 59.0930i 2.64008i
\(502\) 0 0
\(503\) − 21.0145i − 0.936989i −0.883466 0.468495i \(-0.844797\pi\)
0.883466 0.468495i \(-0.155203\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −40.4701 −1.79734
\(508\) 0 0
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) − 28.0000i − 1.23865i
\(512\) 0 0
\(513\) 0 0
\(514\) 0 0
\(515\) 36.0131 1.58693
\(516\) 0 0
\(517\) − 75.8470i − 3.33575i
\(518\) 0 0
\(519\) 89.3643i 3.92266i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 26.8328 1.17332 0.586659 0.809834i \(-0.300443\pi\)
0.586659 + 0.809834i \(0.300443\pi\)
\(524\) 0 0
\(525\) 45.1011i 1.96837i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −40.3396 −1.74078
\(538\) 0 0
\(539\) 38.8928 1.67523
\(540\) 0 0
\(541\) 28.3650i 1.21951i 0.792592 + 0.609753i \(0.208731\pi\)
−0.792592 + 0.609753i \(0.791269\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) −1.45827 −0.0624656
\(546\) 0 0
\(547\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 39.3434 1.67305
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) −111.858 −4.72266
\(562\) 0 0
\(563\) 44.7214 1.88478 0.942390 0.334515i \(-0.108573\pi\)
0.942390 + 0.334515i \(0.108573\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 104.491i 4.38821i
\(568\) 0 0
\(569\) −6.00000 −0.251533 −0.125767 0.992060i \(-0.540139\pi\)
−0.125767 + 0.992060i \(0.540139\pi\)
\(570\) 0 0
\(571\) −35.4965 −1.48548 −0.742741 0.669579i \(-0.766474\pi\)
−0.742741 + 0.669579i \(0.766474\pi\)
\(572\) 0 0
\(573\) − 28.5583i − 1.19304i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 46.2448 1.92519 0.962597 0.270936i \(-0.0873333\pi\)
0.962597 + 0.270936i \(0.0873333\pi\)
\(578\) 0 0
\(579\) 0 0
\(580\) 0 0
\(581\) 23.6643i 0.981761i
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) −20.4939 −0.847319
\(586\) 0 0
\(587\) −8.94427 −0.369170 −0.184585 0.982817i \(-0.559094\pi\)
−0.184585 + 0.982817i \(0.559094\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −48.6992 −1.99984 −0.999919 0.0127518i \(-0.995941\pi\)
−0.999919 + 0.0127518i \(0.995941\pi\)
\(594\) 0 0
\(595\) 34.9352 1.43220
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 6.11738i 0.249949i 0.992160 + 0.124975i \(0.0398849\pi\)
−0.992160 + 0.124975i \(0.960115\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 44.4316i 1.80640i
\(606\) 0 0
\(607\) 5.06046i 0.205398i 0.994712 + 0.102699i \(0.0327478\pi\)
−0.994712 + 0.102699i \(0.967252\pi\)
\(608\) 0 0
\(609\) 43.6235 1.76771
\(610\) 0 0
\(611\) −14.5086 −0.586954
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(618\) 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
\(625\) 25.0000 1.00000
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) − 38.8704i − 1.54741i −0.633548 0.773704i \(-0.718402\pi\)
0.633548 0.773704i \(-0.281598\pi\)
\(632\) 0 0
\(633\) −37.6541 −1.49662
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 7.43970i − 0.294772i
\(638\) 0 0
\(639\) 103.482i 4.09367i
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −46.8886 −1.84910 −0.924552 0.381055i \(-0.875561\pi\)
−0.924552 + 0.381055i \(0.875561\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 47.6235i − 1.87227i −0.351636 0.936137i \(-0.614374\pi\)
0.351636 0.936137i \(-0.385626\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) −91.2623 −3.56048
\(658\) 0 0
\(659\) −47.1253 −1.83574 −0.917871 0.396878i \(-0.870093\pi\)
−0.917871 + 0.396878i \(0.870093\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 21.3971i 0.830993i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) − 29.8048i − 1.15232i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) 95.8612 3.68970
\(676\) 0 0
\(677\) − 32.5886i − 1.25248i −0.779629 0.626242i \(-0.784592\pi\)
0.779629 0.626242i \(-0.215408\pi\)
\(678\) 0 0
\(679\) 9.12957i 0.350361i
\(680\) 0 0
\(681\) −26.1174 −1.00082
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) − 126.766i − 4.81544i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 49.2851i 1.86147i 0.365690 + 0.930737i \(0.380833\pi\)
−0.365690 + 0.930737i \(0.619167\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 104.069 3.91945
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 25.7563i − 0.967299i −0.875262 0.483650i \(-0.839311\pi\)
0.875262 0.483650i \(-0.160689\pi\)
\(710\) 0 0
\(711\) − 128.235i − 4.80918i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 13.2042 0.493811
\(716\) 0 0
\(717\) − 102.680i − 3.83465i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) 42.6113 1.58693
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) − 24.1809i − 0.898058i
\(726\) 0 0
\(727\) − 5.29150i − 0.196251i −0.995174 0.0981255i \(-0.968715\pi\)
0.995174 0.0981255i \(-0.0312847\pi\)
\(728\) 0 0
\(729\) 144.482 5.35117
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) − 33.4722i − 1.23632i −0.786051 0.618161i \(-0.787878\pi\)
0.786051 0.618161i \(-0.212122\pi\)
\(734\) 0 0
\(735\) 53.3643i 1.96837i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −36.1486 −1.32975 −0.664875 0.746955i \(-0.731515\pi\)
−0.664875 + 0.746955i \(0.731515\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) −52.9150 −1.93866
\(746\) 0 0
\(747\) 77.1307 2.82207
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 52.6113i − 1.91981i −0.280321 0.959906i \(-0.590441\pi\)
0.280321 0.959906i \(-0.409559\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 15.3627 0.559107
\(756\) 0 0
\(757\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) −1.72545 −0.0624656
\(764\) 0 0
\(765\) − 113.866i − 4.11685i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 108.243 3.89826
\(772\) 0 0
\(773\) 49.2351i 1.77086i 0.464770 + 0.885431i \(0.346137\pi\)
−0.464770 + 0.885431i \(0.653863\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) − 66.6734i − 2.38576i
\(782\) 0 0
\(783\) − 92.7206i − 3.31356i
\(784\) 0 0
\(785\) 30.0000 1.07075
\(786\) 0 0
\(787\) −22.1814 −0.790681 −0.395340 0.918535i \(-0.629373\pi\)
−0.395340 + 0.918535i \(0.629373\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 21.0770i 0.746585i 0.927714 + 0.373293i \(0.121771\pi\)
−0.927714 + 0.373293i \(0.878229\pi\)
\(798\) 0 0
\(799\) − 80.6113i − 2.85182i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 58.8004 2.07502
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) −55.3643 −1.94651 −0.973253 0.229736i \(-0.926214\pi\)
−0.973253 + 0.229736i \(0.926214\pi\)
\(810\) 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
\(818\) 0 0
\(819\) −24.2487 −0.847319
\(820\) 0 0
\(821\) 46.4054i 1.61956i 0.586734 + 0.809780i \(0.300414\pi\)
−0.586734 + 0.809780i \(0.699586\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) −94.7129 −3.29748
\(826\) 0 0
\(827\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 41.3358 1.43220
\(834\) 0 0
\(835\) 38.7573 1.34125
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) 5.61128 0.193492
\(842\) 0 0
\(843\) −73.7214 −2.53910
\(844\) 0 0
\(845\) 26.5431i 0.913110i
\(846\) 0 0
\(847\) 52.5722i 1.80640i
\(848\) 0 0
\(849\) −49.3643 −1.69418
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 40.2492i 1.37811i 0.724710 + 0.689054i \(0.241974\pi\)
−0.724710 + 0.689054i \(0.758026\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 31.7490 1.08453 0.542263 0.840209i \(-0.317568\pi\)
0.542263 + 0.840209i \(0.317568\pi\)
\(858\) 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
\(865\) 58.6113 1.99284
\(866\) 0 0
\(867\) −60.9260 −2.06916
\(868\) 0 0
\(869\) 82.6218i 2.80275i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 29.7566 1.00711
\(874\) 0 0
\(875\) 29.5804 1.00000
\(876\) 0 0
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0 0
\(879\) 28.3765i 0.957116i
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 47.6235i 1.59904i 0.600639 + 0.799521i \(0.294913\pi\)
−0.600639 + 0.799521i \(0.705087\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −219.433 −7.35127
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 26.4575i 0.884377i
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) − 12.0000i − 0.397578i −0.980042 0.198789i \(-0.936299\pi\)
0.980042 0.198789i \(-0.0637008\pi\)
\(912\) 0 0
\(913\) −49.6954 −1.64468
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 60.6113i − 1.99938i −0.0248659 0.999691i \(-0.507916\pi\)
0.0248659 0.999691i \(-0.492084\pi\)
\(920\) 0 0
\(921\) −90.1174 −2.96947
\(922\) 0 0
\(923\) −12.7538 −0.419795
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) − 138.886i − 4.56161i
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 73.3643i 2.39927i
\(936\) 0 0
\(937\) 24.6167 0.804191 0.402096 0.915598i \(-0.368282\pi\)
0.402096 + 0.915598i \(0.368282\pi\)
\(938\) 0 0
\(939\) 109.030 3.55807
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 113.424 3.68970
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) − 11.2478i − 0.365118i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) −18.7305 −0.606104
\(956\) 0 0
\(957\) 91.6099i 2.96133i
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −31.0000 −1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 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
\(975\) 18.1174i 0.580220i
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 5.62388i 0.179557i
\(982\) 0 0
\(983\) − 2.60600i − 0.0831184i −0.999136 0.0415592i \(-0.986767\pi\)
0.999136 0.0415592i \(-0.0132325\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 123.136 3.91945
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 52.0000i 1.65183i 0.563791 + 0.825917i \(0.309342\pi\)
−0.563791 + 0.825917i \(0.690658\pi\)
\(992\) 0 0
\(993\) −121.019 −3.84042
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 46.2259i 1.46399i 0.681310 + 0.731995i \(0.261411\pi\)
−0.681310 + 0.731995i \(0.738589\pi\)
\(998\) 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 2240.2.n.j.1119.4 yes 16
4.3 odd 2 inner 2240.2.n.j.1119.15 yes 16
5.4 even 2 inner 2240.2.n.j.1119.13 yes 16
7.6 odd 2 inner 2240.2.n.j.1119.13 yes 16
8.3 odd 2 inner 2240.2.n.j.1119.1 16
8.5 even 2 inner 2240.2.n.j.1119.14 yes 16
20.19 odd 2 inner 2240.2.n.j.1119.2 yes 16
28.27 even 2 inner 2240.2.n.j.1119.2 yes 16
35.34 odd 2 CM 2240.2.n.j.1119.4 yes 16
40.19 odd 2 inner 2240.2.n.j.1119.16 yes 16
40.29 even 2 inner 2240.2.n.j.1119.3 yes 16
56.13 odd 2 inner 2240.2.n.j.1119.3 yes 16
56.27 even 2 inner 2240.2.n.j.1119.16 yes 16
140.139 even 2 inner 2240.2.n.j.1119.15 yes 16
280.69 odd 2 inner 2240.2.n.j.1119.14 yes 16
280.139 even 2 inner 2240.2.n.j.1119.1 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2240.2.n.j.1119.1 16 8.3 odd 2 inner
2240.2.n.j.1119.1 16 280.139 even 2 inner
2240.2.n.j.1119.2 yes 16 20.19 odd 2 inner
2240.2.n.j.1119.2 yes 16 28.27 even 2 inner
2240.2.n.j.1119.3 yes 16 40.29 even 2 inner
2240.2.n.j.1119.3 yes 16 56.13 odd 2 inner
2240.2.n.j.1119.4 yes 16 1.1 even 1 trivial
2240.2.n.j.1119.4 yes 16 35.34 odd 2 CM
2240.2.n.j.1119.13 yes 16 5.4 even 2 inner
2240.2.n.j.1119.13 yes 16 7.6 odd 2 inner
2240.2.n.j.1119.14 yes 16 8.5 even 2 inner
2240.2.n.j.1119.14 yes 16 280.69 odd 2 inner
2240.2.n.j.1119.15 yes 16 4.3 odd 2 inner
2240.2.n.j.1119.15 yes 16 140.139 even 2 inner
2240.2.n.j.1119.16 yes 16 40.19 odd 2 inner
2240.2.n.j.1119.16 yes 16 56.27 even 2 inner