Properties

Label 1075.1.c.c.601.4
Level $1075$
Weight $1$
Character 1075.601
Analytic conductor $0.536$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -215
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1075,1,Mod(601,1075)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1075, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1075.601");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1075 = 5^{2} \cdot 43 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1075.c (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: \(0.536494888580\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.0.153664.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} + 5x^{4} + 6x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 215)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.9938375.1
Artin image: $C_4\times D_7$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{28} - \cdots)\)

Embedding invariants

Embedding label 601.4
Root \(-0.445042i\) of defining polynomial
Character \(\chi\) \(=\) 1075.601
Dual form 1075.1.c.c.601.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.445042i q^{2} -1.80194i q^{3} +0.801938 q^{4} +0.801938 q^{6} -1.24698i q^{7} +0.801938i q^{8} -2.24698 q^{9} +O(q^{10})\) \(q+0.445042i q^{2} -1.80194i q^{3} +0.801938 q^{4} +0.801938 q^{6} -1.24698i q^{7} +0.801938i q^{8} -2.24698 q^{9} -0.445042 q^{11} -1.44504i q^{12} +0.554958 q^{14} +0.445042 q^{16} -1.00000i q^{18} -2.24698 q^{21} -0.198062i q^{22} +1.44504 q^{24} +2.24698i q^{27} -1.00000i q^{28} +1.24698 q^{31} +1.00000i q^{32} +0.801938i q^{33} -1.80194 q^{36} +0.445042i q^{37} -1.80194 q^{41} -1.00000i q^{42} +1.00000i q^{43} -0.356896 q^{44} -0.801938i q^{48} -0.554958 q^{49} -1.00000 q^{54} +1.00000 q^{56} +1.80194 q^{59} +0.554958i q^{62} +2.80194i q^{63} -0.356896 q^{66} -1.80194i q^{72} -1.80194i q^{73} -0.198062 q^{74} +0.554958i q^{77} +0.445042 q^{79} +1.80194 q^{81} -0.801938i q^{82} -1.80194 q^{84} -0.445042 q^{86} -0.356896i q^{88} -2.24698i q^{93} +1.80194 q^{96} -0.246980i q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q - 4 q^{4} - 4 q^{6} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q - 4 q^{4} - 4 q^{6} - 4 q^{9} - 2 q^{11} + 4 q^{14} + 2 q^{16} - 4 q^{21} + 8 q^{24} - 2 q^{31} - 2 q^{36} - 2 q^{41} + 6 q^{44} - 4 q^{49} - 6 q^{54} + 6 q^{56} + 2 q^{59} + 6 q^{66} - 10 q^{74} + 2 q^{79} + 2 q^{81} - 2 q^{84} - 2 q^{86} + 2 q^{96} + 6 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1075\mathbb{Z}\right)^\times\).

\(n\) \(302\) \(476\)
\(\chi(n)\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(3\) − 1.80194i − 1.80194i −0.433884 0.900969i \(-0.642857\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(4\) 0.801938 0.801938
\(5\) 0 0
\(6\) 0.801938 0.801938
\(7\) − 1.24698i − 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(8\) 0.801938i 0.801938i
\(9\) −2.24698 −2.24698
\(10\) 0 0
\(11\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(12\) − 1.44504i − 1.44504i
\(13\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(14\) 0.554958 0.554958
\(15\) 0 0
\(16\) 0.445042 0.445042
\(17\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(18\) − 1.00000i − 1.00000i
\(19\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(20\) 0 0
\(21\) −2.24698 −2.24698
\(22\) − 0.198062i − 0.198062i
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) 1.44504 1.44504
\(25\) 0 0
\(26\) 0 0
\(27\) 2.24698i 2.24698i
\(28\) − 1.00000i − 1.00000i
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(32\) 1.00000i 1.00000i
\(33\) 0.801938i 0.801938i
\(34\) 0 0
\(35\) 0 0
\(36\) −1.80194 −1.80194
\(37\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(42\) − 1.00000i − 1.00000i
\(43\) 1.00000i 1.00000i
\(44\) −0.356896 −0.356896
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) − 0.801938i − 0.801938i
\(49\) −0.554958 −0.554958
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(54\) −1.00000 −1.00000
\(55\) 0 0
\(56\) 1.00000 1.00000
\(57\) 0 0
\(58\) 0 0
\(59\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0.554958i 0.554958i
\(63\) 2.80194i 2.80194i
\(64\) 0 0
\(65\) 0 0
\(66\) −0.356896 −0.356896
\(67\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) − 1.80194i − 1.80194i
\(73\) − 1.80194i − 1.80194i −0.433884 0.900969i \(-0.642857\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(74\) −0.198062 −0.198062
\(75\) 0 0
\(76\) 0 0
\(77\) 0.554958i 0.554958i
\(78\) 0 0
\(79\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(80\) 0 0
\(81\) 1.80194 1.80194
\(82\) − 0.801938i − 0.801938i
\(83\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(84\) −1.80194 −1.80194
\(85\) 0 0
\(86\) −0.445042 −0.445042
\(87\) 0 0
\(88\) − 0.356896i − 0.356896i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) − 2.24698i − 2.24698i
\(94\) 0 0
\(95\) 0 0
\(96\) 1.80194 1.80194
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) − 0.246980i − 0.246980i
\(99\) 1.00000 1.00000
\(100\) 0 0
\(101\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(108\) 1.80194i 1.80194i
\(109\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(110\) 0 0
\(111\) 0.801938 0.801938
\(112\) − 0.554958i − 0.554958i
\(113\) 1.24698i 1.24698i 0.781831 + 0.623490i \(0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) 0.801938i 0.801938i
\(119\) 0 0
\(120\) 0 0
\(121\) −0.801938 −0.801938
\(122\) 0 0
\(123\) 3.24698i 3.24698i
\(124\) 1.00000 1.00000
\(125\) 0 0
\(126\) −1.24698 −1.24698
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 1.00000i 1.00000i
\(129\) 1.80194 1.80194
\(130\) 0 0
\(131\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(132\) 0.643104i 0.643104i
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 1.80194i 1.80194i 0.433884 + 0.900969i \(0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(138\) 0 0
\(139\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −1.00000 −1.00000
\(145\) 0 0
\(146\) 0.801938 0.801938
\(147\) 1.00000i 1.00000i
\(148\) 0.356896i 0.356896i
\(149\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(150\) 0 0
\(151\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −0.246980 −0.246980
\(155\) 0 0
\(156\) 0 0
\(157\) 1.80194i 1.80194i 0.433884 + 0.900969i \(0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(158\) 0.198062i 0.198062i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0.801938i 0.801938i
\(163\) − 0.445042i − 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(164\) −1.44504 −1.44504
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) − 1.80194i − 1.80194i
\(169\) −1.00000 −1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0.801938i 0.801938i
\(173\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −0.198062 −0.198062
\(177\) − 3.24698i − 3.24698i
\(178\) 0 0
\(179\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(180\) 0 0
\(181\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 1.00000 1.00000
\(187\) 0 0
\(188\) 0 0
\(189\) 2.80194 2.80194
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.445042 −0.445042
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0.445042i 0.445042i
\(199\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0.554958i 0.554958i
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) −1.80194 −1.80194
\(217\) − 1.55496i − 1.55496i
\(218\) − 0.554958i − 0.554958i
\(219\) −3.24698 −3.24698
\(220\) 0 0
\(221\) 0 0
\(222\) 0.356896i 0.356896i
\(223\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 1.24698 1.24698
\(225\) 0 0
\(226\) −0.554958 −0.554958
\(227\) − 1.24698i − 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(228\) 0 0
\(229\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(230\) 0 0
\(231\) 1.00000 1.00000
\(232\) 0 0
\(233\) − 1.80194i − 1.80194i −0.433884 0.900969i \(-0.642857\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 1.44504 1.44504
\(237\) − 0.801938i − 0.801938i
\(238\) 0 0
\(239\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(240\) 0 0
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) − 0.356896i − 0.356896i
\(243\) − 1.00000i − 1.00000i
\(244\) 0 0
\(245\) 0 0
\(246\) −1.44504 −1.44504
\(247\) 0 0
\(248\) 1.00000i 1.00000i
\(249\) 0 0
\(250\) 0 0
\(251\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(252\) 2.24698i 2.24698i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.445042 −0.445042
\(257\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(258\) 0.801938i 0.801938i
\(259\) 0.554958 0.554958
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) − 0.445042i − 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(264\) −0.643104 −0.643104
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(270\) 0 0
\(271\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −0.801938 −0.801938
\(275\) 0 0
\(276\) 0 0
\(277\) − 1.24698i − 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(278\) 0.801938i 0.801938i
\(279\) −2.80194 −2.80194
\(280\) 0 0
\(281\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(282\) 0 0
\(283\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 2.24698i 2.24698i
\(288\) − 2.24698i − 2.24698i
\(289\) −1.00000 −1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) − 1.44504i − 1.44504i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) −0.445042 −0.445042
\(295\) 0 0
\(296\) −0.356896 −0.356896
\(297\) − 1.00000i − 1.00000i
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 1.24698 1.24698
\(302\) 0 0
\(303\) − 2.24698i − 2.24698i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(308\) 0.445042i 0.445042i
\(309\) 0 0
\(310\) 0 0
\(311\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(312\) 0 0
\(313\) 1.24698i 1.24698i 0.781831 + 0.623490i \(0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(314\) −0.801938 −0.801938
\(315\) 0 0
\(316\) 0.356896 0.356896
\(317\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 1.44504 1.44504
\(325\) 0 0
\(326\) 0.198062 0.198062
\(327\) 2.24698i 2.24698i
\(328\) − 1.44504i − 1.44504i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) − 1.00000i − 1.00000i
\(334\) 0 0
\(335\) 0 0
\(336\) −1.00000 −1.00000
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) − 0.445042i − 0.445042i
\(339\) 2.24698 2.24698
\(340\) 0 0
\(341\) −0.554958 −0.554958
\(342\) 0 0
\(343\) − 0.554958i − 0.554958i
\(344\) −0.801938 −0.801938
\(345\) 0 0
\(346\) 0 0
\(347\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) − 0.445042i − 0.445042i
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 1.44504 1.44504
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 1.00000 1.00000
\(362\) − 0.801938i − 0.801938i
\(363\) 1.44504i 1.44504i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(368\) 0 0
\(369\) 4.04892 4.04892
\(370\) 0 0
\(371\) 0 0
\(372\) − 1.80194i − 1.80194i
\(373\) 1.24698i 1.24698i 0.781831 + 0.623490i \(0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 1.24698i 1.24698i
\(379\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) − 1.80194i − 1.80194i −0.433884 0.900969i \(-0.642857\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(384\) 1.80194 1.80194
\(385\) 0 0
\(386\) 0 0
\(387\) − 2.24698i − 2.24698i
\(388\) 0 0
\(389\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) − 0.445042i − 0.445042i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0.801938 0.801938
\(397\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.00000 1.00000
\(405\) 0 0
\(406\) 0 0
\(407\) − 0.198062i − 0.198062i
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 3.24698 3.24698
\(412\) 0 0
\(413\) − 2.24698i − 2.24698i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 3.24698i − 3.24698i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(432\) 1.00000i 1.00000i
\(433\) − 0.445042i − 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(434\) 0.692021 0.692021
\(435\) 0 0
\(436\) −1.00000 −1.00000
\(437\) 0 0
\(438\) − 1.44504i − 1.44504i
\(439\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(440\) 0 0
\(441\) 1.24698 1.24698
\(442\) 0 0
\(443\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(444\) 0.643104 0.643104
\(445\) 0 0
\(446\) −0.890084 −0.890084
\(447\) 0 0
\(448\) 0 0
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 0 0
\(451\) 0.801938 0.801938
\(452\) 1.00000i 1.00000i
\(453\) 0 0
\(454\) 0.554958 0.554958
\(455\) 0 0
\(456\) 0 0
\(457\) 1.80194i 1.80194i 0.433884 + 0.900969i \(0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(458\) 0.198062i 0.198062i
\(459\) 0 0
\(460\) 0 0
\(461\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(462\) 0.445042i 0.445042i
\(463\) − 1.80194i − 1.80194i −0.433884 0.900969i \(-0.642857\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0.801938 0.801938
\(467\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 3.24698 3.24698
\(472\) 1.44504i 1.44504i
\(473\) − 0.445042i − 0.445042i
\(474\) 0.356896 0.356896
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) − 0.554958i − 0.554958i
\(479\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.643104 −0.643104
\(485\) 0 0
\(486\) 0.445042 0.445042
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) −0.801938 −0.801938
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 2.60388i 2.60388i
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.554958 0.554958
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0.890084i 0.890084i
\(503\) 1.24698i 1.24698i 0.781831 + 0.623490i \(0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(504\) −2.24698 −2.24698
\(505\) 0 0
\(506\) 0 0
\(507\) 1.80194i 1.80194i
\(508\) 0 0
\(509\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(510\) 0 0
\(511\) −2.24698 −2.24698
\(512\) 0.801938i 0.801938i
\(513\) 0 0
\(514\) −0.198062 −0.198062
\(515\) 0 0
\(516\) 1.44504 1.44504
\(517\) 0 0
\(518\) 0.246980i 0.246980i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) − 0.445042i − 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0.198062 0.198062
\(527\) 0 0
\(528\) 0.356896i 0.356896i
\(529\) −1.00000 −1.00000
\(530\) 0 0
\(531\) −4.04892 −4.04892
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0.198062i 0.198062i
\(539\) 0.246980 0.246980
\(540\) 0 0
\(541\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(542\) − 0.801938i − 0.801938i
\(543\) 3.24698i 3.24698i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(548\) 1.44504i 1.44504i
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) − 0.554958i − 0.554958i
\(554\) 0.554958 0.554958
\(555\) 0 0
\(556\) 1.44504 1.44504
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) − 1.24698i − 1.24698i
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) − 0.198062i − 0.198062i
\(563\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) − 2.24698i − 2.24698i
\(568\) 0 0
\(569\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(570\) 0 0
\(571\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −1.00000 −1.00000
\(575\) 0 0
\(576\) 0 0
\(577\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(578\) − 0.445042i − 0.445042i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 1.44504 1.44504
\(585\) 0 0
\(586\) 0 0
\(587\) − 1.24698i − 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(588\) 0.801938i 0.801938i
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.198062i 0.198062i
\(593\) − 0.445042i − 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(594\) 0.445042 0.445042
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0.554958i 0.554958i
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 1.00000 1.00000
\(607\) 1.80194i 1.80194i 0.433884 + 0.900969i \(0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) −0.445042 −0.445042
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) − 0.801938i − 0.801938i
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) −0.554958 −0.554958
\(627\) 0 0
\(628\) 1.44504i 1.44504i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0.356896i 0.356896i
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 1.24698i − 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(648\) 1.44504i 1.44504i
\(649\) −0.801938 −0.801938
\(650\) 0 0
\(651\) −2.80194 −2.80194
\(652\) − 0.356896i − 0.356896i
\(653\) 1.24698i 1.24698i 0.781831 + 0.623490i \(0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(654\) −1.00000 −1.00000
\(655\) 0 0
\(656\) −0.801938 −0.801938
\(657\) 4.04892i 4.04892i
\(658\) 0 0
\(659\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(660\) 0 0
\(661\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.445042 0.445042
\(667\) 0 0
\(668\) 0 0
\(669\) 3.60388 3.60388
\(670\) 0 0
\(671\) 0 0
\(672\) − 2.24698i − 2.24698i
\(673\) − 0.445042i − 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) −0.801938 −0.801938
\(677\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(678\) 1.00000i 1.00000i
\(679\) 0 0
\(680\) 0 0
\(681\) −2.24698 −2.24698
\(682\) − 0.246980i − 0.246980i
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.246980 0.246980
\(687\) − 0.801938i − 0.801938i
\(688\) 0.445042i 0.445042i
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(692\) 0 0
\(693\) − 1.24698i − 1.24698i
\(694\) −0.198062 −0.198062
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) −3.24698 −3.24698
\(700\) 0 0
\(701\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 1.55496i − 1.55496i
\(708\) − 2.60388i − 2.60388i
\(709\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(710\) 0 0
\(711\) −1.00000 −1.00000
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 2.24698i 2.24698i
\(718\) − 0.890084i − 0.890084i
\(719\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0.445042i 0.445042i
\(723\) 0 0
\(724\) −1.44504 −1.44504
\(725\) 0 0
\(726\) −0.643104 −0.643104
\(727\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 2.00000i 2.00000i 1.00000i \(0.5\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 1.80194i 1.80194i
\(739\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) − 1.80194i − 1.80194i −0.433884 0.900969i \(-0.642857\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(744\) 1.80194 1.80194
\(745\) 0 0
\(746\) −0.554958 −0.554958
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) − 3.60388i − 3.60388i
\(754\) 0 0
\(755\) 0 0
\(756\) 2.24698 2.24698
\(757\) − 1.24698i − 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(758\) 0.198062i 0.198062i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 1.55496i 1.55496i
\(764\) 0 0
\(765\) 0 0
\(766\) 0.801938 0.801938
\(767\) 0 0
\(768\) 0.801938i 0.801938i
\(769\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(770\) 0 0
\(771\) 0.801938 0.801938
\(772\) 0 0
\(773\) 1.24698i 1.24698i 0.781831 + 0.623490i \(0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(774\) 1.00000 1.00000
\(775\) 0 0
\(776\) 0 0
\(777\) − 1.00000i − 1.00000i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.246980 −0.246980
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) −0.801938 −0.801938
\(790\) 0 0
\(791\) 1.55496 1.55496
\(792\) 0.801938i 0.801938i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0.554958i 0.554958i
\(803\) 0.801938i 0.801938i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 0.801938i − 0.801938i
\(808\) 1.00000i 1.00000i
\(809\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(810\) 0 0
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 3.24698i 3.24698i
\(814\) 0.0881460 0.0881460
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(822\) 1.44504i 1.44504i
\(823\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 1.00000 1.00000
\(827\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(828\) 0 0
\(829\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(830\) 0 0
\(831\) −2.24698 −2.24698
\(832\) 0 0
\(833\) 0 0
\(834\) 1.44504 1.44504
\(835\) 0 0
\(836\) 0 0
\(837\) 2.80194i 2.80194i
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) 0.801938i 0.801938i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.00000i 1.00000i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(858\) 0 0
\(859\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(860\) 0 0
\(861\) 4.04892 4.04892
\(862\) − 0.801938i − 0.801938i
\(863\) 1.24698i 1.24698i 0.781831 + 0.623490i \(0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(864\) −2.24698 −2.24698
\(865\) 0 0
\(866\) 0.198062 0.198062
\(867\) 1.80194i 1.80194i
\(868\) − 1.24698i − 1.24698i
\(869\) −0.198062 −0.198062
\(870\) 0 0
\(871\) 0 0
\(872\) − 1.00000i − 1.00000i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) −2.60388 −2.60388
\(877\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(878\) 0.198062i 0.198062i
\(879\) 0 0
\(880\) 0 0
\(881\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(882\) 0.554958i 0.554958i
\(883\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 2.00000i − 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(888\) 0.643104i 0.643104i
\(889\) 0 0
\(890\) 0 0
\(891\) −0.801938 −0.801938
\(892\) 1.60388i 1.60388i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 1.24698 1.24698
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0.356896i 0.356896i
\(903\) − 2.24698i − 2.24698i
\(904\) −1.00000 −1.00000
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(908\) − 1.00000i − 1.00000i
\(909\) −2.80194 −2.80194
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.801938 −0.801938
\(915\) 0 0
\(916\) 0.356896 0.356896
\(917\) 0 0
\(918\) 0 0
\(919\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) − 0.198062i − 0.198062i
\(923\) 0 0
\(924\) 0.801938 0.801938
\(925\) 0 0
\(926\) 0.801938 0.801938
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) − 1.44504i − 1.44504i
\(933\) 3.24698i 3.24698i
\(934\) 0.890084 0.890084
\(935\) 0 0
\(936\) 0 0
\(937\) − 1.24698i − 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(938\) 0 0
\(939\) 2.24698 2.24698
\(940\) 0 0
\(941\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(942\) 1.44504i 1.44504i
\(943\) 0 0
\(944\) 0.801938 0.801938
\(945\) 0 0
\(946\) 0.198062 0.198062
\(947\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(948\) − 0.643104i − 0.643104i
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.24698i 1.24698i 0.781831 + 0.623490i \(0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) −1.00000 −1.00000
\(957\) 0 0
\(958\) − 0.554958i − 0.554958i
\(959\) 2.24698 2.24698
\(960\) 0 0
\(961\) 0.554958 0.554958
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) − 0.643104i − 0.643104i
\(969\) 0 0
\(970\) 0 0
\(971\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(972\) − 0.801938i − 0.801938i
\(973\) − 2.24698i − 2.24698i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(978\) − 0.356896i − 0.356896i
\(979\) 0 0
\(980\) 0 0
\(981\) 2.80194 2.80194
\(982\) 0 0
\(983\) − 0.445042i − 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(984\) −2.60388 −2.60388
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 1.24698i 1.24698i
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(998\) 0 0
\(999\) −1.00000 −1.00000
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 1075.1.c.c.601.4 6
5.2 odd 4 215.1.d.a.214.2 3
5.3 odd 4 215.1.d.b.214.2 yes 3
5.4 even 2 inner 1075.1.c.c.601.3 6
15.2 even 4 1935.1.h.b.1504.2 3
15.8 even 4 1935.1.h.a.1504.2 3
20.3 even 4 3440.1.p.a.3009.1 3
20.7 even 4 3440.1.p.b.3009.3 3
43.42 odd 2 inner 1075.1.c.c.601.3 6
215.42 even 4 215.1.d.b.214.2 yes 3
215.128 even 4 215.1.d.a.214.2 3
215.214 odd 2 CM 1075.1.c.c.601.4 6
645.128 odd 4 1935.1.h.b.1504.2 3
645.257 odd 4 1935.1.h.a.1504.2 3
860.343 odd 4 3440.1.p.b.3009.3 3
860.687 odd 4 3440.1.p.a.3009.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
215.1.d.a.214.2 3 5.2 odd 4
215.1.d.a.214.2 3 215.128 even 4
215.1.d.b.214.2 yes 3 5.3 odd 4
215.1.d.b.214.2 yes 3 215.42 even 4
1075.1.c.c.601.3 6 5.4 even 2 inner
1075.1.c.c.601.3 6 43.42 odd 2 inner
1075.1.c.c.601.4 6 1.1 even 1 trivial
1075.1.c.c.601.4 6 215.214 odd 2 CM
1935.1.h.a.1504.2 3 15.8 even 4
1935.1.h.a.1504.2 3 645.257 odd 4
1935.1.h.b.1504.2 3 15.2 even 4
1935.1.h.b.1504.2 3 645.128 odd 4
3440.1.p.a.3009.1 3 20.3 even 4
3440.1.p.a.3009.1 3 860.687 odd 4
3440.1.p.b.3009.3 3 20.7 even 4
3440.1.p.b.3009.3 3 860.343 odd 4