Properties

Label 2240.2.n.j.1119.7
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 1119.7
Root \(-0.141174 - 0.862555i\) of defining polynomial
Character \(\chi\) \(=\) 2240.1119
Dual form 2240.2.n.j.1119.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.17325 q^{3} +2.23607i q^{5} -2.64575i q^{7} -1.62348 q^{9} +O(q^{10})\) \(q-1.17325 q^{3} +2.23607i q^{5} -2.64575i q^{7} -1.62348 q^{9} +0.359964 q^{11} +5.64539i q^{13} -2.62348i q^{15} -2.03214 q^{17} +3.10414i q^{21} -5.00000 q^{25} +5.42451 q^{27} -10.7523i q^{29} -0.422329 q^{33} +5.91608 q^{35} -6.62348i q^{39} -3.63020i q^{45} +5.71383i q^{47} -7.00000 q^{49} +2.38421 q^{51} +0.804903i q^{55} +4.29531i q^{63} -12.6235 q^{65} -12.0000i q^{71} -10.5830 q^{73} +5.86627 q^{75} -0.952374i q^{77} -15.8704i q^{79} -1.49390 q^{81} -8.94427 q^{83} -4.54399i q^{85} +12.6151i q^{87} +14.9363 q^{91} +15.0696 q^{97} -0.584392 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q + 56 q^{9}+O(q^{10}) \) 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
\(3\) −1.17325 −0.677378 −0.338689 0.940898i \(-0.609984\pi\)
−0.338689 + 0.940898i \(0.609984\pi\)
\(4\) 0 0
\(5\) 2.23607i 1.00000i
\(6\) 0 0
\(7\) − 2.64575i − 1.00000i
\(8\) 0 0
\(9\) −1.62348 −0.541158
\(10\) 0 0
\(11\) 0.359964 0.108533 0.0542666 0.998526i \(-0.482718\pi\)
0.0542666 + 0.998526i \(0.482718\pi\)
\(12\) 0 0
\(13\) 5.64539i 1.56575i 0.622179 + 0.782875i \(0.286247\pi\)
−0.622179 + 0.782875i \(0.713753\pi\)
\(14\) 0 0
\(15\) − 2.62348i − 0.677378i
\(16\) 0 0
\(17\) −2.03214 −0.492865 −0.246433 0.969160i \(-0.579258\pi\)
−0.246433 + 0.969160i \(0.579258\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) 3.10414i 0.677378i
\(22\) 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
\(27\) 5.42451 1.04395
\(28\) 0 0
\(29\) − 10.7523i − 1.99665i −0.0578882 0.998323i \(-0.518437\pi\)
0.0578882 0.998323i \(-0.481563\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(32\) 0 0
\(33\) −0.422329 −0.0735180
\(34\) 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
\(39\) − 6.62348i − 1.06060i
\(40\) 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
\(45\) − 3.63020i − 0.541158i
\(46\) 0 0
\(47\) 5.71383i 0.833448i 0.909033 + 0.416724i \(0.136822\pi\)
−0.909033 + 0.416724i \(0.863178\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) 0 0
\(51\) 2.38421 0.333856
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) 0 0
\(55\) 0.804903i 0.108533i
\(56\) 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
\(63\) 4.29531i 0.541158i
\(64\) 0 0
\(65\) −12.6235 −1.56575
\(66\) 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 \(-0.747758\pi\)
0.702109 0.712069i \(-0.252242\pi\)
\(72\) 0 0
\(73\) −10.5830 −1.23865 −0.619324 0.785136i \(-0.712593\pi\)
−0.619324 + 0.785136i \(0.712593\pi\)
\(74\) 0 0
\(75\) 5.86627 0.677378
\(76\) 0 0
\(77\) − 0.952374i − 0.108533i
\(78\) 0 0
\(79\) − 15.8704i − 1.78556i −0.450490 0.892781i \(-0.648751\pi\)
0.450490 0.892781i \(-0.351249\pi\)
\(80\) 0 0
\(81\) −1.49390 −0.165989
\(82\) 0 0
\(83\) −8.94427 −0.981761 −0.490881 0.871227i \(-0.663325\pi\)
−0.490881 + 0.871227i \(0.663325\pi\)
\(84\) 0 0
\(85\) − 4.54399i − 0.492865i
\(86\) 0 0
\(87\) 12.6151i 1.35249i
\(88\) 0 0
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 14.9363 1.56575
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 15.0696 1.53009 0.765043 0.643979i \(-0.222718\pi\)
0.765043 + 0.643979i \(0.222718\pi\)
\(98\) 0 0
\(99\) −0.584392 −0.0587336
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) − 18.7513i − 1.84762i −0.382851 0.923810i \(-0.625058\pi\)
0.382851 0.923810i \(-0.374942\pi\)
\(104\) 0 0
\(105\) −6.94106 −0.677378
\(106\) 0 0
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 0 0
\(109\) − 18.4004i − 1.76244i −0.472708 0.881219i \(-0.656723\pi\)
0.472708 0.881219i \(-0.343277\pi\)
\(110\) 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
\(119\) 5.37652i 0.492865i
\(120\) 0 0
\(121\) −10.8704 −0.988221
\(122\) 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
\(135\) 12.1296i 1.04395i
\(136\) 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
\(141\) − 6.70377i − 0.564560i
\(142\) 0 0
\(143\) 2.03214i 0.169936i
\(144\) 0 0
\(145\) 24.0428 1.99665
\(146\) 0 0
\(147\) 8.21278 0.677378
\(148\) 0 0
\(149\) 23.6643i 1.93866i 0.245770 + 0.969328i \(0.420959\pi\)
−0.245770 + 0.969328i \(0.579041\pi\)
\(150\) 0 0
\(151\) − 23.8704i − 1.94255i −0.237964 0.971274i \(-0.576480\pi\)
0.237964 0.971274i \(-0.423520\pi\)
\(152\) 0 0
\(153\) 3.29912 0.266718
\(154\) 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
\(165\) − 0.944356i − 0.0735180i
\(166\) 0 0
\(167\) − 25.2700i − 1.95545i −0.209881 0.977727i \(-0.567308\pi\)
0.209881 0.977727i \(-0.432692\pi\)
\(168\) 0 0
\(169\) −18.8704 −1.45157
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 15.0314i 1.14282i 0.820666 + 0.571409i \(0.193603\pi\)
−0.820666 + 0.571409i \(0.806397\pi\)
\(174\) 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 \(-0.645798\pi\)
−0.442189 + 0.896922i \(0.645798\pi\)
\(180\) 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
\(187\) −0.731495 −0.0534922
\(188\) 0 0
\(189\) − 14.3519i − 1.04395i
\(190\) 0 0
\(191\) − 18.6235i − 1.34755i −0.738938 0.673774i \(-0.764672\pi\)
0.738938 0.673774i \(-0.235328\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 14.8105 1.06060
\(196\) 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
\(203\) −28.4478 −1.99665
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 28.7927 1.98217 0.991086 0.133226i \(-0.0425335\pi\)
0.991086 + 0.133226i \(0.0425335\pi\)
\(212\) 0 0
\(213\) 14.0790i 0.964680i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 12.4166 0.839033
\(220\) 0 0
\(221\) − 11.4722i − 0.771703i
\(222\) 0 0
\(223\) − 20.3611i − 1.36348i −0.731594 0.681740i \(-0.761223\pi\)
0.731594 0.681740i \(-0.238777\pi\)
\(224\) 0 0
\(225\) 8.11738 0.541158
\(226\) 0 0
\(227\) −21.4083 −1.42092 −0.710460 0.703738i \(-0.751513\pi\)
−0.710460 + 0.703738i \(0.751513\pi\)
\(228\) 0 0
\(229\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(230\) 0 0
\(231\) 1.11738i 0.0735180i
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0 0
\(235\) −12.7765 −0.833448
\(236\) 0 0
\(237\) 18.6200i 1.20950i
\(238\) 0 0
\(239\) 21.1174i 1.36597i 0.730433 + 0.682985i \(0.239318\pi\)
−0.730433 + 0.682985i \(0.760682\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) −14.5208 −0.931510
\(244\) 0 0
\(245\) − 15.6525i − 1.00000i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 10.4939 0.665024
\(250\) 0 0
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 5.33126i 0.333856i
\(256\) 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
\(261\) 17.4560i 1.08050i
\(262\) 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
\(273\) −17.5241 −1.06060
\(274\) 0 0
\(275\) −1.79982 −0.108533
\(276\) 0 0
\(277\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 11.3765 0.678667 0.339333 0.940666i \(-0.389799\pi\)
0.339333 + 0.940666i \(0.389799\pi\)
\(282\) 0 0
\(283\) −19.0618 −1.13311 −0.566553 0.824025i \(-0.691723\pi\)
−0.566553 + 0.824025i \(0.691723\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −12.8704 −0.757084
\(290\) 0 0
\(291\) −17.6805 −1.03645
\(292\) 0 0
\(293\) 32.9200i 1.92320i 0.274446 + 0.961602i \(0.411505\pi\)
−0.274446 + 0.961602i \(0.588495\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 1.95263 0.113303
\(298\) 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
\(307\) 33.1408 1.89145 0.945724 0.324971i \(-0.105355\pi\)
0.945724 + 0.324971i \(0.105355\pi\)
\(308\) 0 0
\(309\) 22.0000i 1.25154i
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 2.87679 0.162606 0.0813030 0.996689i \(-0.474092\pi\)
0.0813030 + 0.996689i \(0.474092\pi\)
\(314\) 0 0
\(315\) −9.60461 −0.541158
\(316\) 0 0
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) − 3.87043i − 0.216702i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) − 28.2269i − 1.56575i
\(326\) 0 0
\(327\) 21.5883i 1.19384i
\(328\) 0 0
\(329\) 15.1174 0.833448
\(330\) 0 0
\(331\) −35.4965 −1.95106 −0.975531 0.219860i \(-0.929440\pi\)
−0.975531 + 0.219860i \(0.929440\pi\)
\(332\) 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
\(351\) 30.6235i 1.63456i
\(352\) 0 0
\(353\) −37.0803 −1.97358 −0.986792 0.161993i \(-0.948208\pi\)
−0.986792 + 0.161993i \(0.948208\pi\)
\(354\) 0 0
\(355\) 26.8328 1.42414
\(356\) 0 0
\(357\) − 6.30803i − 0.333856i
\(358\) 0 0
\(359\) 36.0000i 1.90001i 0.312239 + 0.950004i \(0.398921\pi\)
−0.312239 + 0.950004i \(0.601079\pi\)
\(360\) 0 0
\(361\) 19.0000 1.00000
\(362\) 0 0
\(363\) 12.7538 0.669399
\(364\) 0 0
\(365\) − 23.6643i − 1.23865i
\(366\) 0 0
\(367\) 38.3075i 1.99964i 0.0190919 + 0.999818i \(0.493923\pi\)
−0.0190919 + 0.999818i \(0.506077\pi\)
\(368\) 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
\(375\) 13.1174i 0.677378i
\(376\) 0 0
\(377\) 60.7007 3.12625
\(378\) 0 0
\(379\) 35.4965 1.82333 0.911666 0.410932i \(-0.134797\pi\)
0.911666 + 0.410932i \(0.134797\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 15.8745i − 0.811149i −0.914062 0.405575i \(-0.867071\pi\)
0.914062 0.405575i \(-0.132929\pi\)
\(384\) 0 0
\(385\) 2.12957 0.108533
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 30.8170i 1.56248i 0.624230 + 0.781241i \(0.285413\pi\)
−0.624230 + 0.781241i \(0.714587\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 35.4874 1.78556
\(396\) 0 0
\(397\) − 19.7244i − 0.989941i −0.868910 0.494971i \(-0.835179\pi\)
0.868910 0.494971i \(-0.164821\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 39.1174 1.95343 0.976714 0.214544i \(-0.0688266\pi\)
0.976714 + 0.214544i \(0.0688266\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) − 3.34047i − 0.165989i
\(406\) 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
\(421\) 23.1688i 1.12918i 0.825372 + 0.564590i \(0.190966\pi\)
−0.825372 + 0.564590i \(0.809034\pi\)
\(422\) 0 0
\(423\) − 9.27626i − 0.451027i
\(424\) 0 0
\(425\) 10.1607 0.492865
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) − 2.38421i − 0.115111i
\(430\) 0 0
\(431\) 34.3643i 1.65527i 0.561266 + 0.827636i \(0.310315\pi\)
−0.561266 + 0.827636i \(0.689685\pi\)
\(432\) 0 0
\(433\) 10.5830 0.508587 0.254293 0.967127i \(-0.418157\pi\)
0.254293 + 0.967127i \(0.418157\pi\)
\(434\) 0 0
\(435\) −28.2083 −1.35249
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 11.3643 0.541158
\(442\) 0 0
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 27.7643i − 1.31320i
\(448\) 0 0
\(449\) 40.3643 1.90491 0.952455 0.304679i \(-0.0985491\pi\)
0.952455 + 0.304679i \(0.0985491\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 28.0061i 1.31584i
\(454\) 0 0
\(455\) 33.3986i 1.56575i
\(456\) 0 0
\(457\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(458\) 0 0
\(459\) −11.0233 −0.514525
\(460\) 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
\(467\) 14.3688 0.664908 0.332454 0.943119i \(-0.392123\pi\)
0.332454 + 0.943119i \(0.392123\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 15.7409i 0.725300i
\(472\) 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
\(485\) 33.6967i 1.53009i
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −43.3690 −1.95722 −0.978609 0.205731i \(-0.934043\pi\)
−0.978609 + 0.205731i \(0.934043\pi\)
\(492\) 0 0
\(493\) 21.8501i 0.984077i
\(494\) 0 0
\(495\) − 1.30674i − 0.0587336i
\(496\) 0 0
\(497\) −31.7490 −1.42414
\(498\) 0 0
\(499\) −26.6329 −1.19225 −0.596127 0.802890i \(-0.703294\pi\)
−0.596127 + 0.802890i \(0.703294\pi\)
\(500\) 0 0
\(501\) 29.6482i 1.32458i
\(502\) 0 0
\(503\) − 44.8262i − 1.99870i −0.0360049 0.999352i \(-0.511463\pi\)
0.0360049 0.999352i \(-0.488537\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 22.1398 0.983263
\(508\) 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
\(515\) 41.9292 1.84762
\(516\) 0 0
\(517\) 2.05677i 0.0904567i
\(518\) 0 0
\(519\) − 17.6357i − 0.774120i
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −26.8328 −1.17332 −0.586659 0.809834i \(-0.699557\pi\)
−0.586659 + 0.809834i \(0.699557\pi\)
\(524\) 0 0
\(525\) − 15.5207i − 0.677378i
\(526\) 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
\(537\) 13.8821 0.599058
\(538\) 0 0
\(539\) −2.51975 −0.108533
\(540\) 0 0
\(541\) − 46.1132i − 1.98256i −0.131765 0.991281i \(-0.542065\pi\)
0.131765 0.991281i \(-0.457935\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 41.1445 1.76244
\(546\) 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
\(553\) −41.9892 −1.78556
\(554\) 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
\(561\) 0.858229 0.0362345
\(562\) 0 0
\(563\) −44.7214 −1.88478 −0.942390 0.334515i \(-0.891427\pi\)
−0.942390 + 0.334515i \(0.891427\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 3.95249i 0.165989i
\(568\) 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 \(-0.233526\pi\)
0.742741 + 0.669579i \(0.233526\pi\)
\(572\) 0 0
\(573\) 21.8501i 0.912800i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −11.8500 −0.493322 −0.246661 0.969102i \(-0.579333\pi\)
−0.246661 + 0.969102i \(0.579333\pi\)
\(578\) 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 \(-0.440906\pi\)
0.184585 + 0.982817i \(0.440906\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 24.8875 1.02201 0.511003 0.859579i \(-0.329274\pi\)
0.511003 + 0.859579i \(0.329274\pi\)
\(594\) 0 0
\(595\) −12.0223 −0.492865
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 45.1174i 1.84345i 0.387849 + 0.921723i \(0.373218\pi\)
−0.387849 + 0.921723i \(0.626782\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) − 24.3070i − 0.988221i
\(606\) 0 0
\(607\) − 39.9173i − 1.62019i −0.586296 0.810097i \(-0.699414\pi\)
0.586296 0.810097i \(-0.300586\pi\)
\(608\) 0 0
\(609\) 33.3765 1.35249
\(610\) 0 0
\(611\) −32.2568 −1.30497
\(612\) 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
\(631\) 8.12957i 0.323633i 0.986821 + 0.161817i \(0.0517353\pi\)
−0.986821 + 0.161817i \(0.948265\pi\)
\(632\) 0 0
\(633\) −33.7812 −1.34268
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) − 39.5177i − 1.56575i
\(638\) 0 0
\(639\) 19.4817i 0.770684i
\(640\) 0 0
\(641\) −18.0000 −0.710957 −0.355479 0.934684i \(-0.615682\pi\)
−0.355479 + 0.934684i \(0.615682\pi\)
\(642\) 0 0
\(643\) −40.1804 −1.58456 −0.792279 0.610158i \(-0.791106\pi\)
−0.792279 + 0.610158i \(0.791106\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 47.6235i 1.87227i 0.351636 + 0.936137i \(0.385626\pi\)
−0.351636 + 0.936137i \(0.614374\pi\)
\(648\) 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
\(657\) 17.1812 0.670304
\(658\) 0 0
\(659\) −41.2093 −1.60528 −0.802642 0.596461i \(-0.796573\pi\)
−0.802642 + 0.596461i \(0.796573\pi\)
\(660\) 0 0
\(661\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(662\) 0 0
\(663\) 13.4598i 0.522735i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 23.8887i 0.923592i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(674\) 0 0
\(675\) −27.1226 −1.04395
\(676\) 0 0
\(677\) − 18.8409i − 0.724115i −0.932156 0.362058i \(-0.882074\pi\)
0.932156 0.362058i \(-0.117926\pi\)
\(678\) 0 0
\(679\) − 39.8704i − 1.53009i
\(680\) 0 0
\(681\) 25.1174 0.962500
\(682\) 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
\(693\) 1.54616i 0.0587336i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) − 7.87256i − 0.297342i −0.988887 0.148671i \(-0.952500\pi\)
0.988887 0.148671i \(-0.0474996\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 14.9901 0.564560
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) − 27.4884i − 1.03235i −0.856484 0.516174i \(-0.827356\pi\)
0.856484 0.516174i \(-0.172644\pi\)
\(710\) 0 0
\(711\) 25.7652i 0.966272i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) −4.54399 −0.169936
\(716\) 0 0
\(717\) − 24.7760i − 0.925278i
\(718\) 0 0
\(719\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) 0 0
\(721\) −49.6113 −1.84762
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 53.7613i 1.99665i
\(726\) 0 0
\(727\) 5.29150i 0.196251i 0.995174 + 0.0981255i \(0.0312847\pi\)
−0.995174 + 0.0981255i \(0.968715\pi\)
\(728\) 0 0
\(729\) 21.5183 0.796974
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 53.5968i 1.97964i 0.142318 + 0.989821i \(0.454545\pi\)
−0.142318 + 0.989821i \(0.545455\pi\)
\(734\) 0 0
\(735\) 18.3643i 0.677378i
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 17.0961 0.628889 0.314445 0.949276i \(-0.398182\pi\)
0.314445 + 0.949276i \(0.398182\pi\)
\(740\) 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
\(747\) 14.5208 0.531288
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) − 39.6113i − 1.44544i −0.691143 0.722718i \(-0.742893\pi\)
0.691143 0.722718i \(-0.257107\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 53.3759 1.94255
\(756\) 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
\(763\) −48.6829 −1.76244
\(764\) 0 0
\(765\) 7.37706i 0.266718i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 37.2497 1.34151
\(772\) 0 0
\(773\) − 46.9990i − 1.69044i −0.534421 0.845218i \(-0.679470\pi\)
0.534421 0.845218i \(-0.320530\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) − 4.31956i − 0.154566i
\(782\) 0 0
\(783\) − 58.3258i − 2.08439i
\(784\) 0 0
\(785\) 30.0000 1.07075
\(786\) 0 0
\(787\) −55.7224 −1.98629 −0.993145 0.116892i \(-0.962707\pi\)
−0.993145 + 0.116892i \(0.962707\pi\)
\(788\) 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
\(797\) 34.8247i 1.23355i 0.787138 + 0.616777i \(0.211562\pi\)
−0.787138 + 0.616777i \(0.788438\pi\)
\(798\) 0 0
\(799\) − 11.6113i − 0.410778i
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) −3.80950 −0.134434
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 16.3643 0.575339 0.287670 0.957730i \(-0.407120\pi\)
0.287670 + 0.957730i \(0.407120\pi\)
\(810\) 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
\(821\) − 52.3215i − 1.82603i −0.407923 0.913016i \(-0.633747\pi\)
0.407923 0.913016i \(-0.366253\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 2.11164 0.0735180
\(826\) 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
\(833\) 14.2249 0.492865
\(834\) 0 0
\(835\) 56.5055 1.95545
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0 0
\(841\) −86.6113 −2.98660
\(842\) 0 0
\(843\) −13.3476 −0.459714
\(844\) 0 0
\(845\) − 42.1956i − 1.45157i
\(846\) 0 0
\(847\) 28.7604i 0.988221i
\(848\) 0 0
\(849\) 22.3643 0.767542
\(850\) 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
\(865\) −33.6113 −1.14282
\(866\) 0 0
\(867\) 15.1003 0.512832
\(868\) 0 0
\(869\) − 5.71278i − 0.193793i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) −24.4651 −0.828019
\(874\) 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
\(879\) − 38.6235i − 1.30274i
\(880\) 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 \(-0.705087\pi\)
0.600639 0.799521i \(-0.294913\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.537750 −0.0180153
\(892\) 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 \(0.0637008\pi\)
−0.980042 + 0.198789i \(0.936299\pi\)
\(912\) 0 0
\(913\) −3.21961 −0.106554
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) − 31.6113i − 1.04276i −0.853325 0.521380i \(-0.825417\pi\)
0.853325 0.521380i \(-0.174583\pi\)
\(920\) 0 0
\(921\) −38.8826 −1.28123
\(922\) 0 0
\(923\) 67.7447 2.22984
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 30.4423i 0.999855i
\(928\) 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
\(935\) − 1.63567i − 0.0534922i
\(936\) 0 0
\(937\) 36.2356 1.18377 0.591883 0.806024i \(-0.298385\pi\)
0.591883 + 0.806024i \(0.298385\pi\)
\(938\) 0 0
\(939\) −3.37521 −0.110146
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 32.0918 1.04395
\(946\) 0 0
\(947\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(948\) 0 0
\(949\) − 59.7452i − 1.93941i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 41.6434 1.34755
\(956\) 0 0
\(957\) 4.54099i 0.146789i
\(958\) 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
\(975\) 33.1174i 1.06060i
\(976\) 0 0
\(977\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 29.8726i 0.953759i
\(982\) 0 0
\(983\) 52.9548i 1.68900i 0.535559 + 0.844498i \(0.320101\pi\)
−0.535559 + 0.844498i \(0.679899\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) −17.7365 −0.564560
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) − 52.0000i − 1.65183i −0.563791 0.825917i \(-0.690658\pi\)
0.563791 0.825917i \(-0.309342\pi\)
\(992\) 0 0
\(993\) 41.6464 1.32161
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 14.1479i 0.448069i 0.974581 + 0.224034i \(0.0719228\pi\)
−0.974581 + 0.224034i \(0.928077\pi\)
\(998\) 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.1119.7 yes 16
4.3 odd 2 inner 2240.2.n.j.1119.12 yes 16
5.4 even 2 inner 2240.2.n.j.1119.10 yes 16
7.6 odd 2 inner 2240.2.n.j.1119.10 yes 16
8.3 odd 2 inner 2240.2.n.j.1119.6 yes 16
8.5 even 2 inner 2240.2.n.j.1119.9 yes 16
20.19 odd 2 inner 2240.2.n.j.1119.5 16
28.27 even 2 inner 2240.2.n.j.1119.5 16
35.34 odd 2 CM 2240.2.n.j.1119.7 yes 16
40.19 odd 2 inner 2240.2.n.j.1119.11 yes 16
40.29 even 2 inner 2240.2.n.j.1119.8 yes 16
56.13 odd 2 inner 2240.2.n.j.1119.8 yes 16
56.27 even 2 inner 2240.2.n.j.1119.11 yes 16
140.139 even 2 inner 2240.2.n.j.1119.12 yes 16
280.69 odd 2 inner 2240.2.n.j.1119.9 yes 16
280.139 even 2 inner 2240.2.n.j.1119.6 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2240.2.n.j.1119.5 16 20.19 odd 2 inner
2240.2.n.j.1119.5 16 28.27 even 2 inner
2240.2.n.j.1119.6 yes 16 8.3 odd 2 inner
2240.2.n.j.1119.6 yes 16 280.139 even 2 inner
2240.2.n.j.1119.7 yes 16 1.1 even 1 trivial
2240.2.n.j.1119.7 yes 16 35.34 odd 2 CM
2240.2.n.j.1119.8 yes 16 40.29 even 2 inner
2240.2.n.j.1119.8 yes 16 56.13 odd 2 inner
2240.2.n.j.1119.9 yes 16 8.5 even 2 inner
2240.2.n.j.1119.9 yes 16 280.69 odd 2 inner
2240.2.n.j.1119.10 yes 16 5.4 even 2 inner
2240.2.n.j.1119.10 yes 16 7.6 odd 2 inner
2240.2.n.j.1119.11 yes 16 40.19 odd 2 inner
2240.2.n.j.1119.11 yes 16 56.27 even 2 inner
2240.2.n.j.1119.12 yes 16 4.3 odd 2 inner
2240.2.n.j.1119.12 yes 16 140.139 even 2 inner