Properties

Label 20.3.d.c.19.2
Level $20$
Weight $3$
Character 20.19
Analytic conductor $0.545$
Analytic rank $0$
Dimension $2$
CM discriminant -4
Inner twists $4$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [20,3,Mod(19,20)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(20, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("20.19");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 20 = 2^{2} \cdot 5 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 20.d (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: \(0.544960528721\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(i)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

Embedding invariants

Embedding label 19.2
Root \(1.00000i\) of defining polynomial
Character \(\chi\) \(=\) 20.19
Dual form 20.3.d.c.19.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000i q^{2} -4.00000 q^{4} +(3.00000 - 4.00000i) q^{5} -8.00000i q^{8} -9.00000 q^{9} +O(q^{10})\) \(q+2.00000i q^{2} -4.00000 q^{4} +(3.00000 - 4.00000i) q^{5} -8.00000i q^{8} -9.00000 q^{9} +(8.00000 + 6.00000i) q^{10} +24.0000i q^{13} +16.0000 q^{16} -16.0000i q^{17} -18.0000i q^{18} +(-12.0000 + 16.0000i) q^{20} +(-7.00000 - 24.0000i) q^{25} -48.0000 q^{26} +42.0000 q^{29} +32.0000i q^{32} +32.0000 q^{34} +36.0000 q^{36} +24.0000i q^{37} +(-32.0000 - 24.0000i) q^{40} -18.0000 q^{41} +(-27.0000 + 36.0000i) q^{45} -49.0000 q^{49} +(48.0000 - 14.0000i) q^{50} -96.0000i q^{52} -56.0000i q^{53} +84.0000i q^{58} +22.0000 q^{61} -64.0000 q^{64} +(96.0000 + 72.0000i) q^{65} +64.0000i q^{68} +72.0000i q^{72} -96.0000i q^{73} -48.0000 q^{74} +(48.0000 - 64.0000i) q^{80} +81.0000 q^{81} -36.0000i q^{82} +(-64.0000 - 48.0000i) q^{85} -78.0000 q^{89} +(-72.0000 - 54.0000i) q^{90} +144.000i q^{97} -98.0000i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 8 q^{4} + 6 q^{5} - 18 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 8 q^{4} + 6 q^{5} - 18 q^{9} + 16 q^{10} + 32 q^{16} - 24 q^{20} - 14 q^{25} - 96 q^{26} + 84 q^{29} + 64 q^{34} + 72 q^{36} - 64 q^{40} - 36 q^{41} - 54 q^{45} - 98 q^{49} + 96 q^{50} + 44 q^{61} - 128 q^{64} + 192 q^{65} - 96 q^{74} + 96 q^{80} + 162 q^{81} - 128 q^{85} - 156 q^{89} - 144 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/20\mathbb{Z}\right)^\times\).

\(n\) \(11\) \(17\)
\(\chi(n)\) \(-1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.00000i 1.00000i
\(3\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) −4.00000 −1.00000
\(5\) 3.00000 4.00000i 0.600000 0.800000i
\(6\) 0 0
\(7\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 8.00000i 1.00000i
\(9\) −9.00000 −1.00000
\(10\) 8.00000 + 6.00000i 0.800000 + 0.600000i
\(11\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) 24.0000i 1.84615i 0.384615 + 0.923077i \(0.374334\pi\)
−0.384615 + 0.923077i \(0.625666\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 16.0000 1.00000
\(17\) 16.0000i 0.941176i −0.882353 0.470588i \(-0.844042\pi\)
0.882353 0.470588i \(-0.155958\pi\)
\(18\) 18.0000i 1.00000i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) −12.0000 + 16.0000i −0.600000 + 0.800000i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 0 0
\(25\) −7.00000 24.0000i −0.280000 0.960000i
\(26\) −48.0000 −1.84615
\(27\) 0 0
\(28\) 0 0
\(29\) 42.0000 1.44828 0.724138 0.689655i \(-0.242238\pi\)
0.724138 + 0.689655i \(0.242238\pi\)
\(30\) 0 0
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 32.0000i 1.00000i
\(33\) 0 0
\(34\) 32.0000 0.941176
\(35\) 0 0
\(36\) 36.0000 1.00000
\(37\) 24.0000i 0.648649i 0.945946 + 0.324324i \(0.105137\pi\)
−0.945946 + 0.324324i \(0.894863\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −32.0000 24.0000i −0.800000 0.600000i
\(41\) −18.0000 −0.439024 −0.219512 0.975610i \(-0.570447\pi\)
−0.219512 + 0.975610i \(0.570447\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) 0 0
\(45\) −27.0000 + 36.0000i −0.600000 + 0.800000i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) −49.0000 −1.00000
\(50\) 48.0000 14.0000i 0.960000 0.280000i
\(51\) 0 0
\(52\) 96.0000i 1.84615i
\(53\) 56.0000i 1.05660i −0.849057 0.528302i \(-0.822829\pi\)
0.849057 0.528302i \(-0.177171\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 84.0000i 1.44828i
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 22.0000 0.360656 0.180328 0.983607i \(-0.442284\pi\)
0.180328 + 0.983607i \(0.442284\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −64.0000 −1.00000
\(65\) 96.0000 + 72.0000i 1.47692 + 1.10769i
\(66\) 0 0
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 64.0000i 0.941176i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 72.0000i 1.00000i
\(73\) 96.0000i 1.31507i −0.753425 0.657534i \(-0.771599\pi\)
0.753425 0.657534i \(-0.228401\pi\)
\(74\) −48.0000 −0.648649
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 48.0000 64.0000i 0.600000 0.800000i
\(81\) 81.0000 1.00000
\(82\) 36.0000i 0.439024i
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) 0 0
\(85\) −64.0000 48.0000i −0.752941 0.564706i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −78.0000 −0.876404 −0.438202 0.898876i \(-0.644385\pi\)
−0.438202 + 0.898876i \(0.644385\pi\)
\(90\) −72.0000 54.0000i −0.800000 0.600000i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 144.000i 1.48454i 0.670103 + 0.742268i \(0.266250\pi\)
−0.670103 + 0.742268i \(0.733750\pi\)
\(98\) 98.0000i 1.00000i
\(99\) 0 0
\(100\) 28.0000 + 96.0000i 0.280000 + 0.960000i
\(101\) −198.000 −1.96040 −0.980198 0.198020i \(-0.936549\pi\)
−0.980198 + 0.198020i \(0.936549\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 192.000 1.84615
\(105\) 0 0
\(106\) 112.000 1.05660
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 0 0
\(109\) 182.000 1.66972 0.834862 0.550459i \(-0.185547\pi\)
0.834862 + 0.550459i \(0.185547\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) 224.000i 1.98230i 0.132743 + 0.991150i \(0.457621\pi\)
−0.132743 + 0.991150i \(0.542379\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −168.000 −1.44828
\(117\) 216.000i 1.84615i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 121.000 1.00000
\(122\) 44.0000i 0.360656i
\(123\) 0 0
\(124\) 0 0
\(125\) −117.000 44.0000i −0.936000 0.352000i
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 128.000i 1.00000i
\(129\) 0 0
\(130\) −144.000 + 192.000i −1.10769 + 1.47692i
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −128.000 −0.941176
\(137\) 176.000i 1.28467i −0.766423 0.642336i \(-0.777965\pi\)
0.766423 0.642336i \(-0.222035\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −144.000 −1.00000
\(145\) 126.000 168.000i 0.868966 1.15862i
\(146\) 192.000 1.31507
\(147\) 0 0
\(148\) 96.0000i 0.648649i
\(149\) 102.000 0.684564 0.342282 0.939597i \(-0.388800\pi\)
0.342282 + 0.939597i \(0.388800\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 144.000i 0.941176i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 264.000i 1.68153i 0.541401 + 0.840764i \(0.317894\pi\)
−0.541401 + 0.840764i \(0.682106\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 128.000 + 96.0000i 0.800000 + 0.600000i
\(161\) 0 0
\(162\) 162.000i 1.00000i
\(163\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 72.0000 0.439024
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) −407.000 −2.40828
\(170\) 96.0000 128.000i 0.564706 0.752941i
\(171\) 0 0
\(172\) 0 0
\(173\) 104.000i 0.601156i 0.953757 + 0.300578i \(0.0971796\pi\)
−0.953757 + 0.300578i \(0.902820\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 156.000i 0.876404i
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 108.000 144.000i 0.600000 0.800000i
\(181\) −38.0000 −0.209945 −0.104972 0.994475i \(-0.533475\pi\)
−0.104972 + 0.994475i \(0.533475\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 96.0000 + 72.0000i 0.518919 + 0.389189i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 336.000i 1.74093i −0.492228 0.870466i \(-0.663817\pi\)
0.492228 0.870466i \(-0.336183\pi\)
\(194\) −288.000 −1.48454
\(195\) 0 0
\(196\) 196.000 1.00000
\(197\) 56.0000i 0.284264i −0.989848 0.142132i \(-0.954604\pi\)
0.989848 0.142132i \(-0.0453957\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) −192.000 + 56.0000i −0.960000 + 0.280000i
\(201\) 0 0
\(202\) 396.000i 1.96040i
\(203\) 0 0
\(204\) 0 0
\(205\) −54.0000 + 72.0000i −0.263415 + 0.351220i
\(206\) 0 0
\(207\) 0 0
\(208\) 384.000i 1.84615i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 224.000i 1.05660i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 364.000i 1.66972i
\(219\) 0 0
\(220\) 0 0
\(221\) 384.000 1.73756
\(222\) 0 0
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 63.0000 + 216.000i 0.280000 + 0.960000i
\(226\) −448.000 −1.98230
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) 0 0
\(229\) 442.000 1.93013 0.965066 0.262009i \(-0.0843849\pi\)
0.965066 + 0.262009i \(0.0843849\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 336.000i 1.44828i
\(233\) 416.000i 1.78541i −0.450644 0.892704i \(-0.648806\pi\)
0.450644 0.892704i \(-0.351194\pi\)
\(234\) 432.000 1.84615
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) −418.000 −1.73444 −0.867220 0.497925i \(-0.834095\pi\)
−0.867220 + 0.497925i \(0.834095\pi\)
\(242\) 242.000i 1.00000i
\(243\) 0 0
\(244\) −88.0000 −0.360656
\(245\) −147.000 + 196.000i −0.600000 + 0.800000i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 88.0000 234.000i 0.352000 0.936000i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 256.000 1.00000
\(257\) 64.0000i 0.249027i 0.992218 + 0.124514i \(0.0397370\pi\)
−0.992218 + 0.124514i \(0.960263\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −384.000 288.000i −1.47692 1.10769i
\(261\) −378.000 −1.44828
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −224.000 168.000i −0.845283 0.633962i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −138.000 −0.513011 −0.256506 0.966543i \(-0.582571\pi\)
−0.256506 + 0.966543i \(0.582571\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) 256.000i 0.941176i
\(273\) 0 0
\(274\) 352.000 1.28467
\(275\) 0 0
\(276\) 0 0
\(277\) 504.000i 1.81949i 0.415162 + 0.909747i \(0.363725\pi\)
−0.415162 + 0.909747i \(0.636275\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 462.000 1.64413 0.822064 0.569395i \(-0.192822\pi\)
0.822064 + 0.569395i \(0.192822\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 288.000i 1.00000i
\(289\) 33.0000 0.114187
\(290\) 336.000 + 252.000i 1.15862 + 0.868966i
\(291\) 0 0
\(292\) 384.000i 1.31507i
\(293\) 136.000i 0.464164i −0.972696 0.232082i \(-0.925446\pi\)
0.972696 0.232082i \(-0.0745537\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 192.000 0.648649
\(297\) 0 0
\(298\) 204.000i 0.684564i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 66.0000 88.0000i 0.216393 0.288525i
\(306\) −288.000 −0.941176
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(312\) 0 0
\(313\) 624.000i 1.99361i 0.0798722 + 0.996805i \(0.474549\pi\)
−0.0798722 + 0.996805i \(0.525451\pi\)
\(314\) −528.000 −1.68153
\(315\) 0 0
\(316\) 0 0
\(317\) 616.000i 1.94322i −0.236593 0.971609i \(-0.576031\pi\)
0.236593 0.971609i \(-0.423969\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −192.000 + 256.000i −0.600000 + 0.800000i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −324.000 −1.00000
\(325\) 576.000 168.000i 1.77231 0.516923i
\(326\) 0 0
\(327\) 0 0
\(328\) 144.000i 0.439024i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) 216.000i 0.648649i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 576.000i 1.70920i −0.519288 0.854599i \(-0.673803\pi\)
0.519288 0.854599i \(-0.326197\pi\)
\(338\) 814.000i 2.40828i
\(339\) 0 0
\(340\) 256.000 + 192.000i 0.752941 + 0.564706i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −208.000 −0.601156
\(347\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(348\) 0 0
\(349\) −598.000 −1.71347 −0.856734 0.515759i \(-0.827510\pi\)
−0.856734 + 0.515759i \(0.827510\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 544.000i 1.54108i 0.637394 + 0.770538i \(0.280012\pi\)
−0.637394 + 0.770538i \(0.719988\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 312.000 0.876404
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(360\) 288.000 + 216.000i 0.800000 + 0.600000i
\(361\) 361.000 1.00000
\(362\) 76.0000i 0.209945i
\(363\) 0 0
\(364\) 0 0
\(365\) −384.000 288.000i −1.05205 0.789041i
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 162.000 0.439024
\(370\) −144.000 + 192.000i −0.389189 + 0.518919i
\(371\) 0 0
\(372\) 0 0
\(373\) 504.000i 1.35121i 0.737265 + 0.675603i \(0.236117\pi\)
−0.737265 + 0.675603i \(0.763883\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 1008.00i 2.67374i
\(378\) 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
\(386\) 672.000 1.74093
\(387\) 0 0
\(388\) 576.000i 1.48454i
\(389\) −378.000 −0.971722 −0.485861 0.874036i \(-0.661494\pi\)
−0.485861 + 0.874036i \(0.661494\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 392.000i 1.00000i
\(393\) 0 0
\(394\) 112.000 0.284264
\(395\) 0 0
\(396\) 0 0
\(397\) 456.000i 1.14861i −0.818640 0.574307i \(-0.805271\pi\)
0.818640 0.574307i \(-0.194729\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −112.000 384.000i −0.280000 0.960000i
\(401\) −798.000 −1.99002 −0.995012 0.0997506i \(-0.968195\pi\)
−0.995012 + 0.0997506i \(0.968195\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 792.000 1.96040
\(405\) 243.000 324.000i 0.600000 0.800000i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 782.000 1.91198 0.955990 0.293399i \(-0.0947863\pi\)
0.955990 + 0.293399i \(0.0947863\pi\)
\(410\) −144.000 108.000i −0.351220 0.263415i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −768.000 −1.84615
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −58.0000 −0.137767 −0.0688836 0.997625i \(-0.521944\pi\)
−0.0688836 + 0.997625i \(0.521944\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −448.000 −1.05660
\(425\) −384.000 + 112.000i −0.903529 + 0.263529i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(432\) 0 0
\(433\) 816.000i 1.88453i −0.334873 0.942263i \(-0.608693\pi\)
0.334873 0.942263i \(-0.391307\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −728.000 −1.66972
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(440\) 0 0
\(441\) 441.000 1.00000
\(442\) 768.000i 1.73756i
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0 0
\(445\) −234.000 + 312.000i −0.525843 + 0.701124i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 702.000 1.56347 0.781737 0.623608i \(-0.214334\pi\)
0.781737 + 0.623608i \(0.214334\pi\)
\(450\) −432.000 + 126.000i −0.960000 + 0.280000i
\(451\) 0 0
\(452\) 896.000i 1.98230i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 336.000i 0.735230i −0.929978 0.367615i \(-0.880174\pi\)
0.929978 0.367615i \(-0.119826\pi\)
\(458\) 884.000i 1.93013i
\(459\) 0 0
\(460\) 0 0
\(461\) 522.000 1.13232 0.566161 0.824295i \(-0.308428\pi\)
0.566161 + 0.824295i \(0.308428\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 672.000 1.44828
\(465\) 0 0
\(466\) 832.000 1.78541
\(467\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 864.000i 1.84615i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 504.000i 1.05660i
\(478\) 0 0
\(479\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) −576.000 −1.19751
\(482\) 836.000i 1.73444i
\(483\) 0 0
\(484\) −484.000 −1.00000
\(485\) 576.000 + 432.000i 1.18763 + 0.890722i
\(486\) 0 0
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 176.000i 0.360656i
\(489\) 0 0
\(490\) −392.000 294.000i −0.800000 0.600000i
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 672.000i 1.36308i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 468.000 + 176.000i 0.936000 + 0.352000i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(504\) 0 0
\(505\) −594.000 + 792.000i −1.17624 + 1.56832i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −918.000 −1.80354 −0.901768 0.432220i \(-0.857730\pi\)
−0.901768 + 0.432220i \(0.857730\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 512.000i 1.00000i
\(513\) 0 0
\(514\) −128.000 −0.249027
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 576.000 768.000i 1.10769 1.47692i
\(521\) −558.000 −1.07102 −0.535509 0.844530i \(-0.679880\pi\)
−0.535509 + 0.844530i \(0.679880\pi\)
\(522\) 756.000i 1.44828i
\(523\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −529.000 −1.00000
\(530\) 336.000 448.000i 0.633962 0.845283i
\(531\) 0 0
\(532\) 0 0
\(533\) 432.000i 0.810507i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 276.000i 0.513011i
\(539\) 0 0
\(540\) 0 0
\(541\) 682.000 1.26063 0.630314 0.776340i \(-0.282926\pi\)
0.630314 + 0.776340i \(0.282926\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 512.000 0.941176
\(545\) 546.000 728.000i 1.00183 1.33578i
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 704.000i 1.28467i
\(549\) −198.000 −0.360656
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −1008.00 −1.81949
\(555\) 0 0
\(556\) 0 0
\(557\) 1064.00i 1.91023i 0.296230 + 0.955117i \(0.404271\pi\)
−0.296230 + 0.955117i \(0.595729\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 924.000i 1.64413i
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 896.000 + 672.000i 1.58584 + 1.18938i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 462.000 0.811951 0.405975 0.913884i \(-0.366932\pi\)
0.405975 + 0.913884i \(0.366932\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 576.000 1.00000
\(577\) 96.0000i 0.166378i −0.996534 0.0831889i \(-0.973490\pi\)
0.996534 0.0831889i \(-0.0265105\pi\)
\(578\) 66.0000i 0.114187i
\(579\) 0 0
\(580\) −504.000 + 672.000i −0.868966 + 1.15862i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −768.000 −1.31507
\(585\) −864.000 648.000i −1.47692 1.10769i
\(586\) 272.000 0.464164
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 384.000i 0.648649i
\(593\) 736.000i 1.24115i −0.784148 0.620573i \(-0.786900\pi\)
0.784148 0.620573i \(-0.213100\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −408.000 −0.684564
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(600\) 0 0
\(601\) 1102.00 1.83361 0.916805 0.399334i \(-0.130759\pi\)
0.916805 + 0.399334i \(0.130759\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 363.000 484.000i 0.600000 0.800000i
\(606\) 0 0
\(607\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 176.000 + 132.000i 0.288525 + 0.216393i
\(611\) 0 0
\(612\) 576.000i 0.941176i
\(613\) 1224.00i 1.99674i 0.0570962 + 0.998369i \(0.481816\pi\)
−0.0570962 + 0.998369i \(0.518184\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 1216.00i 1.97083i −0.170178 0.985413i \(-0.554434\pi\)
0.170178 0.985413i \(-0.445566\pi\)
\(618\) 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
\(625\) −527.000 + 336.000i −0.843200 + 0.537600i
\(626\) −1248.00 −1.99361
\(627\) 0 0
\(628\) 1056.00i 1.68153i
\(629\) 384.000 0.610493
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 1232.00 1.94322
\(635\) 0 0
\(636\) 0 0
\(637\) 1176.00i 1.84615i
\(638\) 0 0
\(639\) 0 0
\(640\) −512.000 384.000i −0.800000 0.600000i
\(641\) −1218.00 −1.90016 −0.950078 0.312012i \(-0.898997\pi\)
−0.950078 + 0.312012i \(0.898997\pi\)
\(642\) 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 \(0.5\pi\)
\(648\) 648.000i 1.00000i
\(649\) 0 0
\(650\) 336.000 + 1152.00i 0.516923 + 1.77231i
\(651\) 0 0
\(652\) 0 0
\(653\) 1144.00i 1.75191i 0.482389 + 0.875957i \(0.339769\pi\)
−0.482389 + 0.875957i \(0.660231\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −288.000 −0.439024
\(657\) 864.000i 1.31507i
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) −1178.00 −1.78215 −0.891074 0.453858i \(-0.850047\pi\)
−0.891074 + 0.453858i \(0.850047\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 432.000 0.648649
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1104.00i 1.64042i 0.572065 + 0.820208i \(0.306142\pi\)
−0.572065 + 0.820208i \(0.693858\pi\)
\(674\) 1152.00 1.70920
\(675\) 0 0
\(676\) 1628.00 2.40828
\(677\) 104.000i 0.153619i 0.997046 + 0.0768095i \(0.0244733\pi\)
−0.997046 + 0.0768095i \(0.975527\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −384.000 + 512.000i −0.564706 + 0.752941i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −704.000 528.000i −1.02774 0.770803i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 1344.00 1.95065
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 416.000i 0.601156i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 288.000i 0.413199i
\(698\) 1196.00i 1.71347i
\(699\) 0 0
\(700\) 0 0
\(701\) 1302.00 1.85735 0.928673 0.370899i \(-0.120950\pi\)
0.928673 + 0.370899i \(0.120950\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −1088.00 −1.54108
\(707\) 0 0
\(708\) 0 0
\(709\) −518.000 −0.730606 −0.365303 0.930889i \(-0.619035\pi\)
−0.365303 + 0.930889i \(0.619035\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 624.000i 0.876404i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) −432.000 + 576.000i −0.600000 + 0.800000i
\(721\) 0 0
\(722\) 722.000i 1.00000i
\(723\) 0 0
\(724\) 152.000 0.209945
\(725\) −294.000 1008.00i −0.405517 1.39034i
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) −729.000 −1.00000
\(730\) 576.000 768.000i 0.789041 1.05205i
\(731\) 0 0
\(732\) 0 0
\(733\) 216.000i 0.294679i −0.989086 0.147340i \(-0.952929\pi\)
0.989086 0.147340i \(-0.0470711\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 324.000i 0.439024i
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) −384.000 288.000i −0.518919 0.389189i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 306.000 408.000i 0.410738 0.547651i
\(746\) −1008.00 −1.35121
\(747\) 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
\(754\) −2016.00 −2.67374
\(755\) 0 0
\(756\) 0 0
\(757\) 936.000i 1.23646i −0.785997 0.618230i \(-0.787850\pi\)
0.785997 0.618230i \(-0.212150\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −78.0000 −0.102497 −0.0512484 0.998686i \(-0.516320\pi\)
−0.0512484 + 0.998686i \(0.516320\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 576.000 + 432.000i 0.752941 + 0.564706i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 962.000 1.25098 0.625488 0.780234i \(-0.284900\pi\)
0.625488 + 0.780234i \(0.284900\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1344.00i 1.74093i
\(773\) 1496.00i 1.93532i −0.252264 0.967658i \(-0.581175\pi\)
0.252264 0.967658i \(-0.418825\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1152.00 1.48454
\(777\) 0 0
\(778\) 756.000i 0.971722i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −784.000 −1.00000
\(785\) 1056.00 + 792.000i 1.34522 + 1.00892i
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 224.000i 0.284264i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 528.000i 0.665826i
\(794\) 912.000 1.14861
\(795\) 0 0
\(796\) 0 0
\(797\) 1144.00i 1.43538i 0.696361 + 0.717691i \(0.254801\pi\)
−0.696361 + 0.717691i \(0.745199\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 768.000 224.000i 0.960000 0.280000i
\(801\) 702.000 0.876404
\(802\) 1596.00i 1.99002i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 1584.00i 1.96040i
\(809\) −1518.00 −1.87639 −0.938195 0.346106i \(-0.887504\pi\)
−0.938195 + 0.346106i \(0.887504\pi\)
\(810\) 648.000 + 486.000i 0.800000 + 0.600000i
\(811\) 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
\(818\) 1564.00i 1.91198i
\(819\) 0 0
\(820\) 216.000 288.000i 0.263415 0.351220i
\(821\) −858.000 −1.04507 −0.522533 0.852619i \(-0.675013\pi\)
−0.522533 + 0.852619i \(0.675013\pi\)
\(822\) 0 0
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) −1258.00 −1.51749 −0.758745 0.651387i \(-0.774187\pi\)
−0.758745 + 0.651387i \(0.774187\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1536.00i 1.84615i
\(833\) 784.000i 0.941176i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 923.000 1.09750
\(842\) 116.000i 0.137767i
\(843\) 0 0
\(844\) 0 0
\(845\) −1221.00 + 1628.00i −1.44497 + 1.92663i
\(846\) 0 0
\(847\) 0 0
\(848\) 896.000i 1.05660i
\(849\) 0 0
\(850\) −224.000 768.000i −0.263529 0.903529i
\(851\) 0 0
\(852\) 0 0
\(853\) 1656.00i 1.94138i −0.240328 0.970692i \(-0.577255\pi\)
0.240328 0.970692i \(-0.422745\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 464.000i 0.541424i 0.962660 + 0.270712i \(0.0872590\pi\)
−0.962660 + 0.270712i \(0.912741\pi\)
\(858\) 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
\(865\) 416.000 + 312.000i 0.480925 + 0.360694i
\(866\) 1632.00 1.88453
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 1456.00i 1.66972i
\(873\) 1296.00i 1.48454i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 696.000i 0.793615i −0.917902 0.396807i \(-0.870118\pi\)
0.917902 0.396807i \(-0.129882\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −738.000 −0.837684 −0.418842 0.908059i \(-0.637564\pi\)
−0.418842 + 0.908059i \(0.637564\pi\)
\(882\) 882.000i 1.00000i
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) −1536.00 −1.73756
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −624.000 468.000i −0.701124 0.525843i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1404.00i 1.56347i
\(899\) 0 0
\(900\) −252.000 864.000i −0.280000 0.960000i
\(901\) −896.000 −0.994451
\(902\) 0 0
\(903\) 0 0
\(904\) 1792.00 1.98230
\(905\) −114.000 + 152.000i −0.125967 + 0.167956i
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) 0 0
\(909\) 1782.00 1.96040
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 672.000 0.735230
\(915\) 0 0
\(916\) −1768.00 −1.93013
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1044.00i 1.13232i
\(923\) 0 0
\(924\) 0 0
\(925\) 576.000 168.000i 0.622703 0.181622i
\(926\) 0 0
\(927\) 0 0
\(928\) 1344.00i 1.44828i
\(929\) −258.000 −0.277718 −0.138859 0.990312i \(-0.544343\pi\)
−0.138859 + 0.990312i \(0.544343\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1664.00i 1.78541i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −1728.00 −1.84615
\(937\) 1824.00i 1.94664i 0.229456 + 0.973319i \(0.426305\pi\)
−0.229456 + 0.973319i \(0.573695\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 1482.00 1.57492 0.787460 0.616366i \(-0.211396\pi\)
0.787460 + 0.616366i \(0.211396\pi\)
\(942\) 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
\(949\) 2304.00 2.42782
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1456.00i 1.52781i −0.645331 0.763903i \(-0.723280\pi\)
0.645331 0.763903i \(-0.276720\pi\)
\(954\) −1008.00 −1.05660
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 961.000 1.00000
\(962\) 1152.00i 1.19751i
\(963\) 0 0
\(964\) 1672.00 1.73444
\(965\) −1344.00 1008.00i −1.39275 1.04456i
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 968.000i 1.00000i
\(969\) 0 0
\(970\) −864.000 + 1152.00i −0.890722 + 1.18763i
\(971\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 352.000 0.360656
\(977\) 496.000i 0.507677i −0.967247 0.253838i \(-0.918307\pi\)
0.967247 0.253838i \(-0.0816931\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 588.000 784.000i 0.600000 0.800000i
\(981\) −1638.00 −1.66972
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) −224.000 168.000i −0.227411 0.170558i
\(986\) 1344.00 1.36308
\(987\) 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
\(997\) 744.000i 0.746239i 0.927783 + 0.373119i \(0.121712\pi\)
−0.927783 + 0.373119i \(0.878288\pi\)
\(998\) 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 20.3.d.c.19.2 yes 2
3.2 odd 2 180.3.f.c.19.1 2
4.3 odd 2 CM 20.3.d.c.19.2 yes 2
5.2 odd 4 100.3.b.a.51.1 1
5.3 odd 4 100.3.b.b.51.1 1
5.4 even 2 inner 20.3.d.c.19.1 2
8.3 odd 2 320.3.h.d.319.2 2
8.5 even 2 320.3.h.d.319.2 2
12.11 even 2 180.3.f.c.19.1 2
15.2 even 4 900.3.c.d.451.1 1
15.8 even 4 900.3.c.a.451.1 1
15.14 odd 2 180.3.f.c.19.2 2
16.3 odd 4 1280.3.e.a.639.1 2
16.5 even 4 1280.3.e.d.639.2 2
16.11 odd 4 1280.3.e.d.639.2 2
16.13 even 4 1280.3.e.a.639.1 2
20.3 even 4 100.3.b.b.51.1 1
20.7 even 4 100.3.b.a.51.1 1
20.19 odd 2 inner 20.3.d.c.19.1 2
40.3 even 4 1600.3.b.c.1151.1 1
40.13 odd 4 1600.3.b.c.1151.1 1
40.19 odd 2 320.3.h.d.319.1 2
40.27 even 4 1600.3.b.a.1151.1 1
40.29 even 2 320.3.h.d.319.1 2
40.37 odd 4 1600.3.b.a.1151.1 1
60.23 odd 4 900.3.c.a.451.1 1
60.47 odd 4 900.3.c.d.451.1 1
60.59 even 2 180.3.f.c.19.2 2
80.19 odd 4 1280.3.e.d.639.1 2
80.29 even 4 1280.3.e.d.639.1 2
80.59 odd 4 1280.3.e.a.639.2 2
80.69 even 4 1280.3.e.a.639.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
20.3.d.c.19.1 2 5.4 even 2 inner
20.3.d.c.19.1 2 20.19 odd 2 inner
20.3.d.c.19.2 yes 2 1.1 even 1 trivial
20.3.d.c.19.2 yes 2 4.3 odd 2 CM
100.3.b.a.51.1 1 5.2 odd 4
100.3.b.a.51.1 1 20.7 even 4
100.3.b.b.51.1 1 5.3 odd 4
100.3.b.b.51.1 1 20.3 even 4
180.3.f.c.19.1 2 3.2 odd 2
180.3.f.c.19.1 2 12.11 even 2
180.3.f.c.19.2 2 15.14 odd 2
180.3.f.c.19.2 2 60.59 even 2
320.3.h.d.319.1 2 40.19 odd 2
320.3.h.d.319.1 2 40.29 even 2
320.3.h.d.319.2 2 8.3 odd 2
320.3.h.d.319.2 2 8.5 even 2
900.3.c.a.451.1 1 15.8 even 4
900.3.c.a.451.1 1 60.23 odd 4
900.3.c.d.451.1 1 15.2 even 4
900.3.c.d.451.1 1 60.47 odd 4
1280.3.e.a.639.1 2 16.3 odd 4
1280.3.e.a.639.1 2 16.13 even 4
1280.3.e.a.639.2 2 80.59 odd 4
1280.3.e.a.639.2 2 80.69 even 4
1280.3.e.d.639.1 2 80.19 odd 4
1280.3.e.d.639.1 2 80.29 even 4
1280.3.e.d.639.2 2 16.5 even 4
1280.3.e.d.639.2 2 16.11 odd 4
1600.3.b.a.1151.1 1 40.27 even 4
1600.3.b.a.1151.1 1 40.37 odd 4
1600.3.b.c.1151.1 1 40.3 even 4
1600.3.b.c.1151.1 1 40.13 odd 4