Properties

Label 3364.1.j.c.1031.1
Level $3364$
Weight $1$
Character 3364.1031
Analytic conductor $1.679$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3364,1,Mod(571,3364)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3364, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 12]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3364.571");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3364 = 2^{2} \cdot 29^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3364.j (of order \(14\), degree \(6\), not 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: \(1.67885470250\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: \(\Q(\zeta_{14})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 116)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.38068692544.1

Embedding invariants

Embedding label 1031.1
Root \(0.900969 - 0.433884i\) of defining polynomial
Character \(\chi\) \(=\) 3364.1031
Dual form 3364.1.j.c.571.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.900969 + 0.433884i) q^{2} +(0.623490 + 0.781831i) q^{4} +(-1.12349 - 0.541044i) q^{5} +(0.222521 + 0.974928i) q^{8} +(-0.222521 - 0.974928i) q^{9} +O(q^{10})\) \(q+(0.900969 + 0.433884i) q^{2} +(0.623490 + 0.781831i) q^{4} +(-1.12349 - 0.541044i) q^{5} +(0.222521 + 0.974928i) q^{8} +(-0.222521 - 0.974928i) q^{9} +(-0.777479 - 0.974928i) q^{10} +(0.400969 - 1.75676i) q^{13} +(-0.222521 + 0.974928i) q^{16} +0.445042 q^{17} +(0.222521 - 0.974928i) q^{18} +(-0.277479 - 1.21572i) q^{20} +(0.346011 + 0.433884i) q^{25} +(1.12349 - 1.40881i) q^{26} +(-0.623490 + 0.781831i) q^{32} +(0.400969 + 0.193096i) q^{34} +(0.623490 - 0.781831i) q^{36} +(-0.400969 - 1.75676i) q^{37} +(0.277479 - 1.21572i) q^{40} +1.80194 q^{41} +(-0.277479 + 1.21572i) q^{45} +(-0.222521 - 0.974928i) q^{49} +(0.123490 + 0.541044i) q^{50} +(1.62349 - 0.781831i) q^{52} +(0.400969 + 0.193096i) q^{53} +(0.277479 - 0.347948i) q^{61} +(-0.900969 + 0.433884i) q^{64} +(-1.40097 + 1.75676i) q^{65} +(0.277479 + 0.347948i) q^{68} +(0.900969 - 0.433884i) q^{72} +(-0.400969 + 0.193096i) q^{73} +(0.400969 - 1.75676i) q^{74} +(0.777479 - 0.974928i) q^{80} +(-0.900969 + 0.433884i) q^{81} +(1.62349 + 0.781831i) q^{82} +(-0.500000 - 0.240787i) q^{85} +(1.12349 + 0.541044i) q^{89} +(-0.777479 + 0.974928i) q^{90} +(1.12349 + 1.40881i) q^{97} +(0.222521 - 0.974928i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - q^{4} - 2 q^{5} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - q^{4} - 2 q^{5} + q^{8} - q^{9} - 5 q^{10} - 2 q^{13} - q^{16} + 2 q^{17} + q^{18} - 2 q^{20} - 3 q^{25} + 2 q^{26} + q^{32} - 2 q^{34} - q^{36} + 2 q^{37} + 2 q^{40} + 2 q^{41} - 2 q^{45} - q^{49} - 4 q^{50} + 5 q^{52} - 2 q^{53} + 2 q^{61} - q^{64} - 4 q^{65} + 2 q^{68} + q^{72} + 2 q^{73} - 2 q^{74} + 5 q^{80} - q^{81} + 5 q^{82} - 3 q^{85} + 2 q^{89} - 5 q^{90} + 2 q^{97} + q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3364\mathbb{Z}\right)^\times\).

\(n\) \(1683\) \(2525\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{7}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(3\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(4\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(5\) −1.12349 0.541044i −1.12349 0.541044i −0.222521 0.974928i \(-0.571429\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(6\) 0 0
\(7\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(8\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(9\) −0.222521 0.974928i −0.222521 0.974928i
\(10\) −0.777479 0.974928i −0.777479 0.974928i
\(11\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(12\) 0 0
\(13\) 0.400969 1.75676i 0.400969 1.75676i −0.222521 0.974928i \(-0.571429\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(17\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(18\) 0.222521 0.974928i 0.222521 0.974928i
\(19\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(20\) −0.277479 1.21572i −0.277479 1.21572i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(24\) 0 0
\(25\) 0.346011 + 0.433884i 0.346011 + 0.433884i
\(26\) 1.12349 1.40881i 1.12349 1.40881i
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0
\(30\) 0 0
\(31\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(32\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(33\) 0 0
\(34\) 0.400969 + 0.193096i 0.400969 + 0.193096i
\(35\) 0 0
\(36\) 0.623490 0.781831i 0.623490 0.781831i
\(37\) −0.400969 1.75676i −0.400969 1.75676i −0.623490 0.781831i \(-0.714286\pi\)
0.222521 0.974928i \(-0.428571\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.277479 1.21572i 0.277479 1.21572i
\(41\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(42\) 0 0
\(43\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(44\) 0 0
\(45\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(46\) 0 0
\(47\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(48\) 0 0
\(49\) −0.222521 0.974928i −0.222521 0.974928i
\(50\) 0.123490 + 0.541044i 0.123490 + 0.541044i
\(51\) 0 0
\(52\) 1.62349 0.781831i 1.62349 0.781831i
\(53\) 0.400969 + 0.193096i 0.400969 + 0.193096i 0.623490 0.781831i \(-0.285714\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(60\) 0 0
\(61\) 0.277479 0.347948i 0.277479 0.347948i −0.623490 0.781831i \(-0.714286\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(65\) −1.40097 + 1.75676i −1.40097 + 1.75676i
\(66\) 0 0
\(67\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(68\) 0.277479 + 0.347948i 0.277479 + 0.347948i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(72\) 0.900969 0.433884i 0.900969 0.433884i
\(73\) −0.400969 + 0.193096i −0.400969 + 0.193096i −0.623490 0.781831i \(-0.714286\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(74\) 0.400969 1.75676i 0.400969 1.75676i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(80\) 0.777479 0.974928i 0.777479 0.974928i
\(81\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(82\) 1.62349 + 0.781831i 1.62349 + 0.781831i
\(83\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(84\) 0 0
\(85\) −0.500000 0.240787i −0.500000 0.240787i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 1.12349 + 0.541044i 1.12349 + 0.541044i 0.900969 0.433884i \(-0.142857\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(90\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.12349 + 1.40881i 1.12349 + 1.40881i 0.900969 + 0.433884i \(0.142857\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(98\) 0.222521 0.974928i 0.222521 0.974928i
\(99\) 0 0
\(100\) −0.123490 + 0.541044i −0.123490 + 0.541044i
\(101\) 1.12349 0.541044i 1.12349 0.541044i 0.222521 0.974928i \(-0.428571\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(102\) 0 0
\(103\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(104\) 1.80194 1.80194
\(105\) 0 0
\(106\) 0.277479 + 0.347948i 0.277479 + 0.347948i
\(107\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(108\) 0 0
\(109\) −1.12349 + 1.40881i −1.12349 + 1.40881i −0.222521 + 0.974928i \(0.571429\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.777479 + 0.974928i −0.777479 + 0.974928i 0.222521 + 0.974928i \(0.428571\pi\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.80194 −1.80194
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.900969 0.433884i −0.900969 0.433884i
\(122\) 0.400969 0.193096i 0.400969 0.193096i
\(123\) 0 0
\(124\) 0 0
\(125\) 0.123490 + 0.541044i 0.123490 + 0.541044i
\(126\) 0 0
\(127\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(128\) −1.00000 −1.00000
\(129\) 0 0
\(130\) −2.02446 + 0.974928i −2.02446 + 0.974928i
\(131\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i
\(137\) 0.277479 + 1.21572i 0.277479 + 1.21572i 0.900969 + 0.433884i \(0.142857\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(138\) 0 0
\(139\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 1.00000 1.00000
\(145\) 0 0
\(146\) −0.445042 −0.445042
\(147\) 0 0
\(148\) 1.12349 1.40881i 1.12349 1.40881i
\(149\) −0.277479 0.347948i −0.277479 0.347948i 0.623490 0.781831i \(-0.285714\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(150\) 0 0
\(151\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(152\) 0 0
\(153\) −0.0990311 0.433884i −0.0990311 0.433884i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 1.12349 0.541044i 1.12349 0.541044i
\(161\) 0 0
\(162\) −1.00000 −1.00000
\(163\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(164\) 1.12349 + 1.40881i 1.12349 + 1.40881i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(168\) 0 0
\(169\) −2.02446 0.974928i −2.02446 0.974928i
\(170\) −0.346011 0.433884i −0.346011 0.433884i
\(171\) 0 0
\(172\) 0 0
\(173\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(179\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(180\) −1.12349 + 0.541044i −1.12349 + 0.541044i
\(181\) 0.777479 0.974928i 0.777479 0.974928i −0.222521 0.974928i \(-0.571429\pi\)
1.00000 \(0\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.500000 + 2.19064i −0.500000 + 2.19064i
\(186\) 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
\(193\) −0.777479 0.974928i −0.777479 0.974928i 0.222521 0.974928i \(-0.428571\pi\)
−1.00000 \(\pi\)
\(194\) 0.400969 + 1.75676i 0.400969 + 1.75676i
\(195\) 0 0
\(196\) 0.623490 0.781831i 0.623490 0.781831i
\(197\) −1.12349 + 0.541044i −1.12349 + 0.541044i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(198\) 0 0
\(199\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(200\) −0.346011 + 0.433884i −0.346011 + 0.433884i
\(201\) 0 0
\(202\) 1.24698 1.24698
\(203\) 0 0
\(204\) 0 0
\(205\) −2.02446 0.974928i −2.02446 0.974928i
\(206\) 0 0
\(207\) 0 0
\(208\) 1.62349 + 0.781831i 1.62349 + 0.781831i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(212\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −1.62349 + 0.781831i −1.62349 + 0.781831i
\(219\) 0 0
\(220\) 0 0
\(221\) 0.178448 0.781831i 0.178448 0.781831i
\(222\) 0 0
\(223\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(224\) 0 0
\(225\) 0.346011 0.433884i 0.346011 0.433884i
\(226\) −1.12349 + 0.541044i −1.12349 + 0.541044i
\(227\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(228\) 0 0
\(229\) 0.277479 0.347948i 0.277479 0.347948i −0.623490 0.781831i \(-0.714286\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(234\) −1.62349 0.781831i −1.62349 0.781831i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(240\) 0 0
\(241\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i 1.00000 \(0\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(242\) −0.623490 0.781831i −0.623490 0.781831i
\(243\) 0 0
\(244\) 0.445042 0.445042
\(245\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.123490 + 0.541044i −0.123490 + 0.541044i
\(251\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.900969 0.433884i −0.900969 0.433884i
\(257\) −1.12349 1.40881i −1.12349 1.40881i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.24698 −2.24698
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(264\) 0 0
\(265\) −0.346011 0.433884i −0.346011 0.433884i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.277479 + 1.21572i 0.277479 + 1.21572i 0.900969 + 0.433884i \(0.142857\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(270\) 0 0
\(271\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(272\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i
\(273\) 0 0
\(274\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(275\) 0 0
\(276\) 0 0
\(277\) 0.0990311 0.433884i 0.0990311 0.433884i −0.900969 0.433884i \(-0.857143\pi\)
1.00000 \(0\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.400969 + 1.75676i 0.400969 + 1.75676i 0.623490 + 0.781831i \(0.285714\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(282\) 0 0
\(283\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(289\) −0.801938 −0.801938
\(290\) 0 0
\(291\) 0 0
\(292\) −0.400969 0.193096i −0.400969 0.193096i
\(293\) 1.12349 1.40881i 1.12349 1.40881i 0.222521 0.974928i \(-0.428571\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 1.62349 0.781831i 1.62349 0.781831i
\(297\) 0 0
\(298\) −0.0990311 0.433884i −0.0990311 0.433884i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.500000 + 0.240787i −0.500000 + 0.240787i
\(306\) 0.0990311 0.433884i 0.0990311 0.433884i
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(312\) 0 0
\(313\) 0.400969 0.193096i 0.400969 0.193096i −0.222521 0.974928i \(-0.571429\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(314\) −1.12349 0.541044i −1.12349 0.541044i
\(315\) 0 0
\(316\) 0 0
\(317\) −0.400969 0.193096i −0.400969 0.193096i 0.222521 0.974928i \(-0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.24698 1.24698
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.900969 0.433884i −0.900969 0.433884i
\(325\) 0.900969 0.433884i 0.900969 0.433884i
\(326\) 0 0
\(327\) 0 0
\(328\) 0.400969 + 1.75676i 0.400969 + 1.75676i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −1.62349 + 0.781831i −1.62349 + 0.781831i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.400969 + 1.75676i −0.400969 + 1.75676i 0.222521 + 0.974928i \(0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(338\) −1.40097 1.75676i −1.40097 1.75676i
\(339\) 0 0
\(340\) −0.123490 0.541044i −0.123490 0.541044i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −0.400969 0.193096i −0.400969 0.193096i
\(347\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(348\) 0 0
\(349\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 1.62349 + 0.781831i 1.62349 + 0.781831i 1.00000 \(0\)
0.623490 + 0.781831i \(0.285714\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(360\) −1.24698 −1.24698
\(361\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(362\) 1.12349 0.541044i 1.12349 0.541044i
\(363\) 0 0
\(364\) 0 0
\(365\) 0.554958 0.554958
\(366\) 0 0
\(367\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(368\) 0 0
\(369\) −0.400969 1.75676i −0.400969 1.75676i
\(370\) −1.40097 + 1.75676i −1.40097 + 1.75676i
\(371\) 0 0
\(372\) 0 0
\(373\) 0.777479 + 0.974928i 0.777479 + 0.974928i 1.00000 \(0\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.277479 1.21572i −0.277479 1.21572i
\(387\) 0 0
\(388\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(389\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.900969 0.433884i 0.900969 0.433884i
\(393\) 0 0
\(394\) −1.24698 −1.24698
\(395\) 0 0
\(396\) 0 0
\(397\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i 1.00000 \(0\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.500000 + 0.240787i −0.500000 + 0.240787i
\(401\) 1.62349 + 0.781831i 1.62349 + 0.781831i 1.00000 \(0\)
0.623490 + 0.781831i \(0.285714\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(405\) 1.24698 1.24698
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.777479 + 0.974928i −0.777479 + 0.974928i 0.222521 + 0.974928i \(0.428571\pi\)
−1.00000 \(\pi\)
\(410\) −1.40097 1.75676i −1.40097 1.75676i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.12349 + 1.40881i 1.12349 + 1.40881i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(420\) 0 0
\(421\) 1.80194 0.867767i 1.80194 0.867767i 0.900969 0.433884i \(-0.142857\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i
\(425\) 0.153989 + 0.193096i 0.153989 + 0.193096i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(432\) 0 0
\(433\) 1.80194 + 0.867767i 1.80194 + 0.867767i 0.900969 + 0.433884i \(0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −1.80194 −1.80194
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(440\) 0 0
\(441\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(442\) 0.500000 0.626980i 0.500000 0.626980i
\(443\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(444\) 0 0
\(445\) −0.969501 1.21572i −0.969501 1.21572i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.400969 + 0.193096i −0.400969 + 0.193096i −0.623490 0.781831i \(-0.714286\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(450\) 0.500000 0.240787i 0.500000 0.240787i
\(451\) 0 0
\(452\) −1.24698 −1.24698
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.777479 0.974928i 0.777479 0.974928i −0.222521 0.974928i \(-0.571429\pi\)
1.00000 \(0\)
\(458\) 0.400969 0.193096i 0.400969 0.193096i
\(459\) 0 0
\(460\) 0 0
\(461\) −1.24698 + 1.56366i −1.24698 + 1.56366i −0.623490 + 0.781831i \(0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(467\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(468\) −1.12349 1.40881i −1.12349 1.40881i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.0990311 0.433884i 0.0990311 0.433884i
\(478\) 0 0
\(479\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(480\) 0 0
\(481\) −3.24698 −3.24698
\(482\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i
\(483\) 0 0
\(484\) −0.222521 0.974928i −0.222521 0.974928i
\(485\) −0.500000 2.19064i −0.500000 2.19064i
\(486\) 0 0
\(487\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(488\) 0.400969 + 0.193096i 0.400969 + 0.193096i
\(489\) 0 0
\(490\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(491\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(500\) −0.346011 + 0.433884i −0.346011 + 0.433884i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(504\) 0 0
\(505\) −1.55496 −1.55496
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.400969 1.75676i 0.400969 1.75676i −0.222521 0.974928i \(-0.571429\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.623490 0.781831i −0.623490 0.781831i
\(513\) 0 0
\(514\) −0.400969 1.75676i −0.400969 1.75676i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −2.02446 0.974928i −2.02446 0.974928i
\(521\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.623490 0.781831i 0.623490 0.781831i
\(530\) −0.123490 0.541044i −0.123490 0.541044i
\(531\) 0 0
\(532\) 0 0
\(533\) 0.722521 3.16557i 0.722521 3.16557i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(539\) 0 0
\(540\) 0 0
\(541\) 1.12349 + 1.40881i 1.12349 + 1.40881i 0.900969 + 0.433884i \(0.142857\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.277479 + 0.347948i −0.277479 + 0.347948i
\(545\) 2.02446 0.974928i 2.02446 0.974928i
\(546\) 0 0
\(547\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(548\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(549\) −0.400969 0.193096i −0.400969 0.193096i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0.277479 0.347948i 0.277479 0.347948i
\(555\) 0 0
\(556\) 0 0
\(557\) 1.62349 0.781831i 1.62349 0.781831i 0.623490 0.781831i \(-0.285714\pi\)
1.00000 \(0\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(563\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(564\) 0 0
\(565\) 1.40097 0.674671i 1.40097 0.674671i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.400969 + 1.75676i −0.400969 + 1.75676i 0.222521 + 0.974928i \(0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(570\) 0 0
\(571\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(577\) 1.12349 1.40881i 1.12349 1.40881i 0.222521 0.974928i \(-0.428571\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(578\) −0.722521 0.347948i −0.722521 0.347948i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.277479 0.347948i −0.277479 0.347948i
\(585\) 2.02446 + 0.974928i 2.02446 + 0.974928i
\(586\) 1.62349 0.781831i 1.62349 0.781831i
\(587\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.80194 1.80194
\(593\) 0.0990311 0.433884i 0.0990311 0.433884i −0.900969 0.433884i \(-0.857143\pi\)
1.00000 \(0\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.0990311 0.433884i 0.0990311 0.433884i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(600\) 0 0
\(601\) 0.445042 + 1.94986i 0.445042 + 1.94986i 0.222521 + 0.974928i \(0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(606\) 0 0
\(607\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) −0.554958 −0.554958
\(611\) 0 0
\(612\) 0.277479 0.347948i 0.277479 0.347948i
\(613\) −1.12349 1.40881i −1.12349 1.40881i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0.277479 + 1.21572i 0.277479 + 1.21572i 0.900969 + 0.433884i \(0.142857\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(618\) 0 0
\(619\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.277479 1.21572i 0.277479 1.21572i
\(626\) 0.445042 0.445042
\(627\) 0 0
\(628\) −0.777479 0.974928i −0.777479 0.974928i
\(629\) −0.178448 0.781831i −0.178448 0.781831i
\(630\) 0 0
\(631\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −0.277479 0.347948i −0.277479 0.347948i
\(635\) 0 0
\(636\) 0 0
\(637\) −1.80194 −1.80194
\(638\) 0 0
\(639\) 0 0
\(640\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(641\) −1.24698 + 1.56366i −1.24698 + 1.56366i −0.623490 + 0.781831i \(0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(642\) 0 0
\(643\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(648\) −0.623490 0.781831i −0.623490 0.781831i
\(649\) 0 0
\(650\) 1.00000 1.00000
\(651\) 0 0
\(652\) 0 0
\(653\) −1.62349 + 0.781831i −1.62349 + 0.781831i −0.623490 + 0.781831i \(0.714286\pi\)
−1.00000 \(1.00000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(657\) 0.277479 + 0.347948i 0.277479 + 0.347948i
\(658\) 0 0
\(659\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(660\) 0 0
\(661\) 1.62349 0.781831i 1.62349 0.781831i 0.623490 0.781831i \(-0.285714\pi\)
1.00000 \(0\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −1.80194 −1.80194
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.12349 + 0.541044i −1.12349 + 0.541044i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(674\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(675\) 0 0
\(676\) −0.500000 2.19064i −0.500000 2.19064i
\(677\) −0.777479 0.974928i −0.777479 0.974928i 0.222521 0.974928i \(-0.428571\pi\)
−1.00000 \(\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.123490 0.541044i 0.123490 0.541044i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(684\) 0 0
\(685\) 0.346011 1.51597i 0.346011 1.51597i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.500000 0.626980i 0.500000 0.626980i
\(690\) 0 0
\(691\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(692\) −0.277479 0.347948i −0.277479 0.347948i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.801938 0.801938
\(698\) −0.400969 0.193096i −0.400969 0.193096i
\(699\) 0 0
\(700\) 0 0
\(701\) 0.400969 + 0.193096i 0.400969 + 0.193096i 0.623490 0.781831i \(-0.285714\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 1.12349 + 1.40881i 1.12349 + 1.40881i
\(707\) 0 0
\(708\) 0 0
\(709\) −0.277479 + 1.21572i −0.277479 + 1.21572i 0.623490 + 0.781831i \(0.285714\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(720\) −1.12349 0.541044i −1.12349 0.541044i
\(721\) 0 0
\(722\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(723\) 0 0
\(724\) 1.24698 1.24698
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(728\) 0 0
\(729\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(730\) 0.500000 + 0.240787i 0.500000 + 0.240787i
\(731\) 0 0
\(732\) 0 0
\(733\) 0.445042 + 1.94986i 0.445042 + 1.94986i 0.222521 + 0.974928i \(0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0.400969 1.75676i 0.400969 1.75676i
\(739\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(740\) −2.02446 + 0.974928i −2.02446 + 0.974928i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(744\) 0 0
\(745\) 0.123490 + 0.541044i 0.123490 + 0.541044i
\(746\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) −0.777479 + 0.974928i −0.777479 + 0.974928i 0.222521 + 0.974928i \(0.428571\pi\)
−1.00000 \(\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.777479 0.974928i 0.777479 0.974928i −0.222521 0.974928i \(-0.571429\pi\)
1.00000 \(0\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −0.123490 + 0.541044i −0.123490 + 0.541044i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.12349 0.541044i 1.12349 0.541044i 0.222521 0.974928i \(-0.428571\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.277479 1.21572i 0.277479 1.21572i
\(773\) −0.777479 0.974928i −0.777479 0.974928i 0.222521 0.974928i \(-0.428571\pi\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(777\) 0 0
\(778\) 0.400969 + 0.193096i 0.400969 + 0.193096i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) 1.40097 + 0.674671i 1.40097 + 0.674671i
\(786\) 0 0
\(787\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(788\) −1.12349 0.541044i −1.12349 0.541044i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.500000 0.626980i −0.500000 0.626980i
\(794\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i
\(795\) 0 0
\(796\) 0 0
\(797\) −1.62349 + 0.781831i −1.62349 + 0.781831i −0.623490 + 0.781831i \(0.714286\pi\)
−1.00000 \(1.00000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.554958 −0.554958
\(801\) 0.277479 1.21572i 0.277479 1.21572i
\(802\) 1.12349 + 1.40881i 1.12349 + 1.40881i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(809\) 1.12349 1.40881i 1.12349 1.40881i 0.222521 0.974928i \(-0.428571\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(810\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(811\) 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
\(818\) −1.12349 + 0.541044i −1.12349 + 0.541044i
\(819\) 0 0
\(820\) −0.500000 2.19064i −0.500000 2.19064i
\(821\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i 1.00000 \(0\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(822\) 0 0
\(823\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(828\) 0 0
\(829\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0.400969 + 1.75676i 0.400969 + 1.75676i
\(833\) −0.0990311 0.433884i −0.0990311 0.433884i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 2.00000 2.00000
\(843\) 0 0
\(844\) 0 0
\(845\) 1.74698 + 2.19064i 1.74698 + 2.19064i
\(846\) 0 0
\(847\) 0 0
\(848\) −0.277479 + 0.347948i −0.277479 + 0.347948i
\(849\) 0 0
\(850\) 0.0549581 + 0.240787i 0.0549581 + 0.240787i
\(851\) 0 0
\(852\) 0 0
\(853\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.445042 + 1.94986i −0.445042 + 1.94986i −0.222521 + 0.974928i \(0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(858\) 0 0
\(859\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(864\) 0 0
\(865\) 0.500000 + 0.240787i 0.500000 + 0.240787i
\(866\) 1.24698 + 1.56366i 1.24698 + 1.56366i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −1.62349 0.781831i −1.62349 0.781831i
\(873\) 1.12349 1.40881i 1.12349 1.40881i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 1.24698 1.56366i 1.24698 1.56366i 0.623490 0.781831i \(-0.285714\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i 0.900969 + 0.433884i \(0.142857\pi\)
−1.00000 \(\pi\)
\(882\) −1.00000 −1.00000
\(883\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(884\) 0.722521 0.347948i 0.722521 0.347948i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −0.346011 1.51597i −0.346011 1.51597i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −0.445042 −0.445042
\(899\) 0 0
\(900\) 0.554958 0.554958
\(901\) 0.178448 + 0.0859360i 0.178448 + 0.0859360i
\(902\) 0 0
\(903\) 0 0
\(904\) −1.12349 0.541044i −1.12349 0.541044i
\(905\) −1.40097 + 0.674671i −1.40097 + 0.674671i
\(906\) 0 0
\(907\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(908\) 0 0
\(909\) −0.777479 0.974928i −0.777479 0.974928i
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.12349 0.541044i 1.12349 0.541044i
\(915\) 0 0
\(916\) 0.445042 0.445042
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.80194 + 0.867767i −1.80194 + 0.867767i
\(923\) 0 0
\(924\) 0 0
\(925\) 0.623490 0.781831i 0.623490 0.781831i
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −0.400969 1.75676i −0.400969 1.75676i
\(937\) −0.277479 1.21572i −0.277479 1.21572i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −0.445042 + 1.94986i −0.445042 + 1.94986i −0.222521 + 0.974928i \(0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(948\) 0 0
\(949\) 0.178448 + 0.781831i 0.178448 + 0.781831i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −1.12349 1.40881i −1.12349 1.40881i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(954\) 0.277479 0.347948i 0.277479 0.347948i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(962\) −2.92543 1.40881i −2.92543 1.40881i
\(963\) 0 0
\(964\) −0.277479 + 0.347948i −0.277479 + 0.347948i
\(965\) 0.346011 + 1.51597i 0.346011 + 1.51597i
\(966\) 0 0
\(967\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(968\) 0.222521 0.974928i 0.222521 0.974928i
\(969\) 0 0
\(970\) 0.500000 2.19064i 0.500000 2.19064i
\(971\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0.277479 + 0.347948i 0.277479 + 0.347948i
\(977\) −0.277479 1.21572i −0.277479 1.21572i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −1.12349 + 0.541044i −1.12349 + 0.541044i
\(981\) 1.62349 + 0.781831i 1.62349 + 0.781831i
\(982\) 0 0
\(983\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(984\) 0 0
\(985\) 1.55496 1.55496
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.277479 1.21572i 0.277479 1.21572i −0.623490 0.781831i \(-0.714286\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(998\) 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 3364.1.j.c.1031.1 6
4.3 odd 2 CM 3364.1.j.c.1031.1 6
29.2 odd 28 3364.1.h.c.267.2 12
29.3 odd 28 3364.1.d.a.3363.4 6
29.4 even 14 3364.1.j.b.1619.1 6
29.5 even 14 3364.1.j.b.1415.1 6
29.6 even 14 116.1.j.a.83.1 yes 6
29.7 even 7 3364.1.b.b.1683.3 3
29.8 odd 28 3364.1.h.d.1111.2 12
29.9 even 14 3364.1.j.a.571.1 6
29.10 odd 28 3364.1.h.c.63.2 12
29.11 odd 28 3364.1.h.e.2719.2 12
29.12 odd 4 3364.1.h.d.651.1 12
29.13 even 14 116.1.j.a.7.1 6
29.14 odd 28 3364.1.h.e.2759.1 12
29.15 odd 28 3364.1.h.e.2759.2 12
29.16 even 7 3364.1.j.d.2327.1 6
29.17 odd 4 3364.1.h.d.651.2 12
29.18 odd 28 3364.1.h.e.2719.1 12
29.19 odd 28 3364.1.h.c.63.1 12
29.20 even 7 inner 3364.1.j.c.571.1 6
29.21 odd 28 3364.1.h.d.1111.1 12
29.22 even 14 3364.1.b.c.1683.3 3
29.23 even 7 3364.1.j.d.2287.1 6
29.24 even 7 3364.1.j.e.1415.1 6
29.25 even 7 3364.1.j.e.1619.1 6
29.26 odd 28 3364.1.d.a.3363.1 6
29.27 odd 28 3364.1.h.c.267.1 12
29.28 even 2 3364.1.j.a.1031.1 6
87.35 odd 14 1044.1.bb.a.199.1 6
87.71 odd 14 1044.1.bb.a.703.1 6
116.3 even 28 3364.1.d.a.3363.4 6
116.7 odd 14 3364.1.b.b.1683.3 3
116.11 even 28 3364.1.h.e.2719.2 12
116.15 even 28 3364.1.h.e.2759.2 12
116.19 even 28 3364.1.h.c.63.1 12
116.23 odd 14 3364.1.j.d.2287.1 6
116.27 even 28 3364.1.h.c.267.1 12
116.31 even 28 3364.1.h.c.267.2 12
116.35 odd 14 116.1.j.a.83.1 yes 6
116.39 even 28 3364.1.h.c.63.2 12
116.43 even 28 3364.1.h.e.2759.1 12
116.47 even 28 3364.1.h.e.2719.1 12
116.51 odd 14 3364.1.b.c.1683.3 3
116.55 even 28 3364.1.d.a.3363.1 6
116.63 odd 14 3364.1.j.b.1415.1 6
116.67 odd 14 3364.1.j.a.571.1 6
116.71 odd 14 116.1.j.a.7.1 6
116.75 even 4 3364.1.h.d.651.2 12
116.79 even 28 3364.1.h.d.1111.1 12
116.83 odd 14 3364.1.j.e.1619.1 6
116.91 odd 14 3364.1.j.b.1619.1 6
116.95 even 28 3364.1.h.d.1111.2 12
116.99 even 4 3364.1.h.d.651.1 12
116.103 odd 14 3364.1.j.d.2327.1 6
116.107 odd 14 inner 3364.1.j.c.571.1 6
116.111 odd 14 3364.1.j.e.1415.1 6
116.115 odd 2 3364.1.j.a.1031.1 6
145.13 odd 28 2900.1.bd.a.1399.1 12
145.42 odd 28 2900.1.bd.a.1399.2 12
145.64 even 14 2900.1.bj.a.2751.1 6
145.93 odd 28 2900.1.bd.a.199.2 12
145.122 odd 28 2900.1.bd.a.199.1 12
145.129 even 14 2900.1.bj.a.1051.1 6
232.13 even 14 1856.1.bh.a.703.1 6
232.35 odd 14 1856.1.bh.a.895.1 6
232.93 even 14 1856.1.bh.a.895.1 6
232.187 odd 14 1856.1.bh.a.703.1 6
348.35 even 14 1044.1.bb.a.199.1 6
348.71 even 14 1044.1.bb.a.703.1 6
580.187 even 28 2900.1.bd.a.1399.2 12
580.267 even 28 2900.1.bd.a.199.1 12
580.303 even 28 2900.1.bd.a.1399.1 12
580.383 even 28 2900.1.bd.a.199.2 12
580.419 odd 14 2900.1.bj.a.1051.1 6
580.499 odd 14 2900.1.bj.a.2751.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
116.1.j.a.7.1 6 29.13 even 14
116.1.j.a.7.1 6 116.71 odd 14
116.1.j.a.83.1 yes 6 29.6 even 14
116.1.j.a.83.1 yes 6 116.35 odd 14
1044.1.bb.a.199.1 6 87.35 odd 14
1044.1.bb.a.199.1 6 348.35 even 14
1044.1.bb.a.703.1 6 87.71 odd 14
1044.1.bb.a.703.1 6 348.71 even 14
1856.1.bh.a.703.1 6 232.13 even 14
1856.1.bh.a.703.1 6 232.187 odd 14
1856.1.bh.a.895.1 6 232.35 odd 14
1856.1.bh.a.895.1 6 232.93 even 14
2900.1.bd.a.199.1 12 145.122 odd 28
2900.1.bd.a.199.1 12 580.267 even 28
2900.1.bd.a.199.2 12 145.93 odd 28
2900.1.bd.a.199.2 12 580.383 even 28
2900.1.bd.a.1399.1 12 145.13 odd 28
2900.1.bd.a.1399.1 12 580.303 even 28
2900.1.bd.a.1399.2 12 145.42 odd 28
2900.1.bd.a.1399.2 12 580.187 even 28
2900.1.bj.a.1051.1 6 145.129 even 14
2900.1.bj.a.1051.1 6 580.419 odd 14
2900.1.bj.a.2751.1 6 145.64 even 14
2900.1.bj.a.2751.1 6 580.499 odd 14
3364.1.b.b.1683.3 3 29.7 even 7
3364.1.b.b.1683.3 3 116.7 odd 14
3364.1.b.c.1683.3 3 29.22 even 14
3364.1.b.c.1683.3 3 116.51 odd 14
3364.1.d.a.3363.1 6 29.26 odd 28
3364.1.d.a.3363.1 6 116.55 even 28
3364.1.d.a.3363.4 6 29.3 odd 28
3364.1.d.a.3363.4 6 116.3 even 28
3364.1.h.c.63.1 12 29.19 odd 28
3364.1.h.c.63.1 12 116.19 even 28
3364.1.h.c.63.2 12 29.10 odd 28
3364.1.h.c.63.2 12 116.39 even 28
3364.1.h.c.267.1 12 29.27 odd 28
3364.1.h.c.267.1 12 116.27 even 28
3364.1.h.c.267.2 12 29.2 odd 28
3364.1.h.c.267.2 12 116.31 even 28
3364.1.h.d.651.1 12 29.12 odd 4
3364.1.h.d.651.1 12 116.99 even 4
3364.1.h.d.651.2 12 29.17 odd 4
3364.1.h.d.651.2 12 116.75 even 4
3364.1.h.d.1111.1 12 29.21 odd 28
3364.1.h.d.1111.1 12 116.79 even 28
3364.1.h.d.1111.2 12 29.8 odd 28
3364.1.h.d.1111.2 12 116.95 even 28
3364.1.h.e.2719.1 12 29.18 odd 28
3364.1.h.e.2719.1 12 116.47 even 28
3364.1.h.e.2719.2 12 29.11 odd 28
3364.1.h.e.2719.2 12 116.11 even 28
3364.1.h.e.2759.1 12 29.14 odd 28
3364.1.h.e.2759.1 12 116.43 even 28
3364.1.h.e.2759.2 12 29.15 odd 28
3364.1.h.e.2759.2 12 116.15 even 28
3364.1.j.a.571.1 6 29.9 even 14
3364.1.j.a.571.1 6 116.67 odd 14
3364.1.j.a.1031.1 6 29.28 even 2
3364.1.j.a.1031.1 6 116.115 odd 2
3364.1.j.b.1415.1 6 29.5 even 14
3364.1.j.b.1415.1 6 116.63 odd 14
3364.1.j.b.1619.1 6 29.4 even 14
3364.1.j.b.1619.1 6 116.91 odd 14
3364.1.j.c.571.1 6 29.20 even 7 inner
3364.1.j.c.571.1 6 116.107 odd 14 inner
3364.1.j.c.1031.1 6 1.1 even 1 trivial
3364.1.j.c.1031.1 6 4.3 odd 2 CM
3364.1.j.d.2287.1 6 29.23 even 7
3364.1.j.d.2287.1 6 116.23 odd 14
3364.1.j.d.2327.1 6 29.16 even 7
3364.1.j.d.2327.1 6 116.103 odd 14
3364.1.j.e.1415.1 6 29.24 even 7
3364.1.j.e.1415.1 6 116.111 odd 14
3364.1.j.e.1619.1 6 29.25 even 7
3364.1.j.e.1619.1 6 116.83 odd 14