Properties

Label 3675.1.bj.a.3074.1
Level $3675$
Weight $1$
Character 3675.3074
Analytic conductor $1.834$
Analytic rank $0$
Dimension $12$
Projective image $D_{7}$
CM discriminant -3
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3675,1,Mod(449,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 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("3675.449");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3675.bj (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: \(1.83406392143\)
Analytic rank: \(0\)
Dimension: \(12\)
Relative dimension: \(2\) over \(\Q(\zeta_{14})\)
Coefficient field: \(\Q(\zeta_{28})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 147)
Projective image: \(D_{7}\)
Projective field: Galois closure of 7.1.373714754427.1

Embedding invariants

Embedding label 3074.1
Root \(-0.433884 - 0.900969i\) of defining polynomial
Character \(\chi\) \(=\) 3675.3074
Dual form 3675.1.bj.a.2024.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.781831 + 0.623490i) q^{3} +(0.900969 - 0.433884i) q^{4} +(-0.974928 + 0.222521i) q^{7} +(0.222521 - 0.974928i) q^{9} +O(q^{10})\) \(q+(-0.781831 + 0.623490i) q^{3} +(0.900969 - 0.433884i) q^{4} +(-0.974928 + 0.222521i) q^{7} +(0.222521 - 0.974928i) q^{9} +(-0.433884 + 0.900969i) q^{12} +(1.21572 - 0.277479i) q^{13} +(0.623490 - 0.781831i) q^{16} +0.445042 q^{19} +(0.623490 - 0.781831i) q^{21} +(0.433884 + 0.900969i) q^{27} +(-0.781831 + 0.623490i) q^{28} -1.80194 q^{31} +(-0.222521 - 0.974928i) q^{36} +(0.781831 - 1.62349i) q^{37} +(-0.777479 + 0.974928i) q^{39} +(-0.347948 - 0.277479i) q^{43} +1.00000i q^{48} +(0.900969 - 0.433884i) q^{49} +(0.974928 - 0.777479i) q^{52} +(-0.347948 + 0.277479i) q^{57} +(1.62349 + 0.781831i) q^{61} +1.00000i q^{63} +(0.222521 - 0.974928i) q^{64} -1.24698i q^{67} +(1.75676 + 0.400969i) q^{73} +(0.400969 - 0.193096i) q^{76} +1.80194 q^{79} +(-0.900969 - 0.433884i) q^{81} +(0.222521 - 0.974928i) q^{84} +(-1.12349 + 0.541044i) q^{91} +(1.40881 - 1.12349i) q^{93} +0.445042i q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{4} + 2 q^{9} - 2 q^{16} + 4 q^{19} - 2 q^{21} - 4 q^{31} - 2 q^{36} - 10 q^{39} + 2 q^{49} + 10 q^{61} + 2 q^{64} - 4 q^{76} + 4 q^{79} - 2 q^{81} + 2 q^{84} - 4 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3675\mathbb{Z}\right)^\times\).

\(n\) \(1177\) \(1226\) \(2551\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{2}{7}\right)\)

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 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(3\) −0.781831 + 0.623490i −0.781831 + 0.623490i
\(4\) 0.900969 0.433884i 0.900969 0.433884i
\(5\) 0 0
\(6\) 0 0
\(7\) −0.974928 + 0.222521i −0.974928 + 0.222521i
\(8\) 0 0
\(9\) 0.222521 0.974928i 0.222521 0.974928i
\(10\) 0 0
\(11\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(12\) −0.433884 + 0.900969i −0.433884 + 0.900969i
\(13\) 1.21572 0.277479i 1.21572 0.277479i 0.433884 0.900969i \(-0.357143\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.623490 0.781831i 0.623490 0.781831i
\(17\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(18\) 0 0
\(19\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(20\) 0 0
\(21\) 0.623490 0.781831i 0.623490 0.781831i
\(22\) 0 0
\(23\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.433884 + 0.900969i 0.433884 + 0.900969i
\(28\) −0.781831 + 0.623490i −0.781831 + 0.623490i
\(29\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(30\) 0 0
\(31\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −0.222521 0.974928i −0.222521 0.974928i
\(37\) 0.781831 1.62349i 0.781831 1.62349i 1.00000i \(-0.5\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(38\) 0 0
\(39\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(40\) 0 0
\(41\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(42\) 0 0
\(43\) −0.347948 0.277479i −0.347948 0.277479i 0.433884 0.900969i \(-0.357143\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(48\) 1.00000i 1.00000i
\(49\) 0.900969 0.433884i 0.900969 0.433884i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.974928 0.777479i 0.974928 0.777479i
\(53\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.347948 + 0.277479i −0.347948 + 0.277479i
\(58\) 0 0
\(59\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(60\) 0 0
\(61\) 1.62349 + 0.781831i 1.62349 + 0.781831i 1.00000 \(0\)
0.623490 + 0.781831i \(0.285714\pi\)
\(62\) 0 0
\(63\) 1.00000i 1.00000i
\(64\) 0.222521 0.974928i 0.222521 0.974928i
\(65\) 0 0
\(66\) 0 0
\(67\) 1.24698i 1.24698i −0.781831 0.623490i \(-0.785714\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(72\) 0 0
\(73\) 1.75676 + 0.400969i 1.75676 + 0.400969i 0.974928 0.222521i \(-0.0714286\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0.400969 0.193096i 0.400969 0.193096i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(80\) 0 0
\(81\) −0.900969 0.433884i −0.900969 0.433884i
\(82\) 0 0
\(83\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(84\) 0.222521 0.974928i 0.222521 0.974928i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(90\) 0 0
\(91\) −1.12349 + 0.541044i −1.12349 + 0.541044i
\(92\) 0 0
\(93\) 1.40881 1.12349i 1.40881 1.12349i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(102\) 0 0
\(103\) 0.347948 0.277479i 0.347948 0.277479i −0.433884 0.900969i \(-0.642857\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(108\) 0.781831 + 0.623490i 0.781831 + 0.623490i
\(109\) 0.277479 + 1.21572i 0.277479 + 1.21572i 0.900969 + 0.433884i \(0.142857\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(110\) 0 0
\(111\) 0.400969 + 1.75676i 0.400969 + 1.75676i
\(112\) −0.433884 + 0.900969i −0.433884 + 0.900969i
\(113\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 1.24698i 1.24698i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(122\) 0 0
\(123\) 0 0
\(124\) −1.62349 + 0.781831i −1.62349 + 0.781831i
\(125\) 0 0
\(126\) 0 0
\(127\) 0.193096 0.400969i 0.193096 0.400969i −0.781831 0.623490i \(-0.785714\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(128\) 0 0
\(129\) 0.445042 0.445042
\(130\) 0 0
\(131\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(132\) 0 0
\(133\) −0.433884 + 0.0990311i −0.433884 + 0.0990311i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(138\) 0 0
\(139\) −0.777479 0.974928i −0.777479 0.974928i 0.222521 0.974928i \(-0.428571\pi\)
−1.00000 \(\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.623490 0.781831i −0.623490 0.781831i
\(145\) 0 0
\(146\) 0 0
\(147\) −0.433884 + 0.900969i −0.433884 + 0.900969i
\(148\) 1.80194i 1.80194i
\(149\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(150\) 0 0
\(151\) 0.400969 0.193096i 0.400969 0.193096i −0.222521 0.974928i \(-0.571429\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(157\) −0.974928 0.777479i −0.974928 0.777479i 1.00000i \(-0.5\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.974928 + 0.777479i 0.974928 + 0.777479i 0.974928 0.222521i \(-0.0714286\pi\)
1.00000i \(0.5\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(168\) 0 0
\(169\) 0.500000 0.240787i 0.500000 0.240787i
\(170\) 0 0
\(171\) 0.0990311 0.433884i 0.0990311 0.433884i
\(172\) −0.433884 0.0990311i −0.433884 0.0990311i
\(173\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(180\) 0 0
\(181\) −0.277479 + 1.21572i −0.277479 + 1.21572i 0.623490 + 0.781831i \(0.285714\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(182\) 0 0
\(183\) −1.75676 + 0.400969i −1.75676 + 0.400969i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.623490 0.781831i −0.623490 0.781831i
\(190\) 0 0
\(191\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(192\) 0.433884 + 0.900969i 0.433884 + 0.900969i
\(193\) −0.974928 + 0.777479i −0.974928 + 0.777479i −0.974928 0.222521i \(-0.928571\pi\)
1.00000i \(0.5\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.623490 0.781831i 0.623490 0.781831i
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 0 0
\(199\) 0.277479 + 0.347948i 0.277479 + 0.347948i 0.900969 0.433884i \(-0.142857\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(200\) 0 0
\(201\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.541044 1.12349i 0.541044 1.12349i
\(209\) 0 0
\(210\) 0 0
\(211\) 0.400969 1.75676i 0.400969 1.75676i −0.222521 0.974928i \(-0.571429\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.75676 0.400969i 1.75676 0.400969i
\(218\) 0 0
\(219\) −1.62349 + 0.781831i −1.62349 + 0.781831i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0.781831 + 1.62349i 0.781831 + 1.62349i 0.781831 + 0.623490i \(0.214286\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(228\) −0.193096 + 0.400969i −0.193096 + 0.400969i
\(229\) 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
\(233\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −1.40881 + 1.12349i −1.40881 + 1.12349i
\(238\) 0 0
\(239\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(240\) 0 0
\(241\) −1.12349 + 0.541044i −1.12349 + 0.541044i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(242\) 0 0
\(243\) 0.974928 0.222521i 0.974928 0.222521i
\(244\) 1.80194 1.80194
\(245\) 0 0
\(246\) 0 0
\(247\) 0.541044 0.123490i 0.541044 0.123490i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(252\) 0.433884 + 0.900969i 0.433884 + 0.900969i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.222521 0.974928i −0.222521 0.974928i
\(257\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(258\) 0 0
\(259\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −0.541044 1.12349i −0.541044 1.12349i
\(269\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(270\) 0 0
\(271\) −1.12349 + 0.541044i −1.12349 + 0.541044i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(272\) 0 0
\(273\) 0.541044 1.12349i 0.541044 1.12349i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.541044 + 1.12349i −0.541044 + 1.12349i 0.433884 + 0.900969i \(0.357143\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(278\) 0 0
\(279\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(280\) 0 0
\(281\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(282\) 0 0
\(283\) −1.75676 + 0.400969i −1.75676 + 0.400969i −0.974928 0.222521i \(-0.928571\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.623490 0.781831i −0.623490 0.781831i
\(290\) 0 0
\(291\) −0.277479 0.347948i −0.277479 0.347948i
\(292\) 1.75676 0.400969i 1.75676 0.400969i
\(293\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0.400969 + 0.193096i 0.400969 + 0.193096i
\(302\) 0 0
\(303\) 0 0
\(304\) 0.277479 0.347948i 0.277479 0.347948i
\(305\) 0 0
\(306\) 0 0
\(307\) 0.433884 0.0990311i 0.433884 0.0990311i 1.00000i \(-0.5\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(308\) 0 0
\(309\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i
\(310\) 0 0
\(311\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(312\) 0 0
\(313\) 1.80194i 1.80194i −0.433884 0.900969i \(-0.642857\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 1.62349 0.781831i 1.62349 0.781831i
\(317\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.00000 −1.00000
\(325\) 0 0
\(326\) 0 0
\(327\) −0.974928 0.777479i −0.974928 0.777479i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.400969 + 0.193096i 0.400969 + 0.193096i 0.623490 0.781831i \(-0.285714\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(332\) 0 0
\(333\) −1.40881 1.12349i −1.40881 1.12349i
\(334\) 0 0
\(335\) 0 0
\(336\) −0.222521 0.974928i −0.222521 0.974928i
\(337\) −0.974928 0.777479i −0.974928 0.777479i 1.00000i \(-0.5\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −0.781831 + 0.623490i −0.781831 + 0.623490i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(348\) 0 0
\(349\) 0.277479 0.347948i 0.277479 0.347948i −0.623490 0.781831i \(-0.714286\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(350\) 0 0
\(351\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(352\) 0 0
\(353\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(360\) 0 0
\(361\) −0.801938 −0.801938
\(362\) 0 0
\(363\) 0.433884 0.900969i 0.433884 0.900969i
\(364\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(365\) 0 0
\(366\) 0 0
\(367\) 1.21572 + 0.277479i 1.21572 + 0.277479i 0.781831 0.623490i \(-0.214286\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0.781831 1.62349i 0.781831 1.62349i
\(373\) 0.445042i 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.0990311 0.433884i −0.0990311 0.433884i 0.900969 0.433884i \(-0.142857\pi\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i
\(382\) 0 0
\(383\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.347948 + 0.277479i −0.347948 + 0.277479i
\(388\) 0.193096 + 0.400969i 0.193096 + 0.400969i
\(389\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.40881 + 1.12349i −1.40881 + 1.12349i −0.433884 + 0.900969i \(0.642857\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(398\) 0 0
\(399\) 0.277479 0.347948i 0.277479 0.347948i
\(400\) 0 0
\(401\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(402\) 0 0
\(403\) −2.19064 + 0.500000i −2.19064 + 0.500000i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −1.62349 0.781831i −1.62349 0.781831i −0.623490 0.781831i \(-0.714286\pi\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.193096 0.400969i 0.193096 0.400969i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 1.21572 + 0.277479i 1.21572 + 0.277479i
\(418\) 0 0
\(419\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(420\) 0 0
\(421\) −1.12349 0.541044i −1.12349 0.541044i −0.222521 0.974928i \(-0.571429\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.75676 0.400969i −1.75676 0.400969i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(432\) 0.974928 + 0.222521i 0.974928 + 0.222521i
\(433\) 0.347948 0.277479i 0.347948 0.277479i −0.433884 0.900969i \(-0.642857\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(437\) 0 0
\(438\) 0 0
\(439\) 0.277479 + 1.21572i 0.277479 + 1.21572i 0.900969 + 0.433884i \(0.142857\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(440\) 0 0
\(441\) −0.222521 0.974928i −0.222521 0.974928i
\(442\) 0 0
\(443\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(444\) 1.12349 + 1.40881i 1.12349 + 1.40881i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 1.00000i 1.00000i
\(449\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −0.193096 + 0.400969i −0.193096 + 0.400969i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 1.40881 + 1.12349i 1.40881 + 1.12349i 0.974928 + 0.222521i \(0.0714286\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(462\) 0 0
\(463\) −0.541044 1.12349i −0.541044 1.12349i −0.974928 0.222521i \(-0.928571\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(468\) −0.541044 1.12349i −0.541044 1.12349i
\(469\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(470\) 0 0
\(471\) 1.24698 1.24698
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(480\) 0 0
\(481\) 0.500000 2.19064i 0.500000 2.19064i
\(482\) 0 0
\(483\) 0 0
\(484\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(485\) 0 0
\(486\) 0 0
\(487\) −1.40881 + 1.12349i −1.40881 + 1.12349i −0.433884 + 0.900969i \(0.642857\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(488\) 0 0
\(489\) −1.24698 −1.24698
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(497\) 0 0
\(498\) 0 0
\(499\) −0.400969 + 1.75676i −0.400969 + 1.75676i 0.222521 + 0.974928i \(0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.240787 + 0.500000i −0.240787 + 0.500000i
\(508\) 0.445042i 0.445042i
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) −1.80194 −1.80194
\(512\) 0 0
\(513\) 0.193096 + 0.400969i 0.193096 + 0.400969i
\(514\) 0 0
\(515\) 0 0
\(516\) 0.400969 0.193096i 0.400969 0.193096i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) −1.40881 1.12349i −1.40881 1.12349i −0.974928 0.222521i \(-0.928571\pi\)
−0.433884 0.900969i \(-0.642857\pi\)
\(524\) 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 0
\(531\) 0 0
\(532\) −0.347948 + 0.277479i −0.347948 + 0.277479i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i 1.00000 \(0\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(542\) 0 0
\(543\) −0.541044 1.12349i −0.541044 1.12349i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.40881 + 1.12349i −1.40881 + 1.12349i −0.433884 + 0.900969i \(0.642857\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(548\) 0 0
\(549\) 1.12349 1.40881i 1.12349 1.40881i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.75676 + 0.400969i −1.75676 + 0.400969i
\(554\) 0 0
\(555\) 0 0
\(556\) −1.12349 0.541044i −1.12349 0.541044i
\(557\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) −0.500000 0.240787i −0.500000 0.240787i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.974928 + 0.222521i 0.974928 + 0.222521i
\(568\) 0 0
\(569\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(570\) 0 0
\(571\) 1.62349 + 0.781831i 1.62349 + 0.781831i 1.00000 \(0\)
0.623490 + 0.781831i \(0.285714\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.900969 0.433884i −0.900969 0.433884i
\(577\) −1.21572 + 0.277479i −1.21572 + 0.277479i −0.781831 0.623490i \(-0.785714\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(578\) 0 0
\(579\) 0.277479 1.21572i 0.277479 1.21572i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(588\) 1.00000i 1.00000i
\(589\) −0.801938 −0.801938
\(590\) 0 0
\(591\) 0 0
\(592\) −0.781831 1.62349i −0.781831 1.62349i
\(593\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.433884 0.0990311i −0.433884 0.0990311i
\(598\) 0 0
\(599\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(600\) 0 0
\(601\) −0.445042 1.94986i −0.445042 1.94986i −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(602\) 0 0
\(603\) −1.21572 0.277479i −1.21572 0.277479i
\(604\) 0.277479 0.347948i 0.277479 0.347948i
\(605\) 0 0
\(606\) 0 0
\(607\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 1.75676 + 0.400969i 1.75676 + 0.400969i 0.974928 0.222521i \(-0.0714286\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(618\) 0 0
\(619\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) −1.21572 0.277479i −1.21572 0.277479i
\(629\) 0 0
\(630\) 0 0
\(631\) −1.12349 + 1.40881i −1.12349 + 1.40881i −0.222521 + 0.974928i \(0.571429\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(632\) 0 0
\(633\) 0.781831 + 1.62349i 0.781831 + 1.62349i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0.974928 0.777479i 0.974928 0.777479i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(642\) 0 0
\(643\) 0.974928 + 0.777479i 0.974928 + 0.777479i 0.974928 0.222521i \(-0.0714286\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(652\) 1.21572 + 0.277479i 1.21572 + 0.277479i
\(653\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.781831 1.62349i 0.781831 1.62349i
\(658\) 0 0
\(659\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(660\) 0 0
\(661\) −0.445042 + 1.94986i −0.445042 + 1.94986i −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 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1.62349 0.781831i −1.62349 0.781831i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.21572 0.277479i 1.21572 0.277479i 0.433884 0.900969i \(-0.357143\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0.346011 0.433884i 0.346011 0.433884i
\(677\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(678\) 0 0
\(679\) −0.0990311 0.433884i −0.0990311 0.433884i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(684\) −0.0990311 0.433884i −0.0990311 0.433884i
\(685\) 0 0
\(686\) 0 0
\(687\) 0.445042i 0.445042i
\(688\) −0.433884 + 0.0990311i −0.433884 + 0.0990311i
\(689\) 0 0
\(690\) 0 0
\(691\) 1.24698 + 1.56366i 1.24698 + 1.56366i 0.623490 + 0.781831i \(0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(702\) 0 0
\(703\) 0.347948 0.722521i 0.347948 0.722521i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.400969 + 0.193096i −0.400969 + 0.193096i −0.623490 0.781831i \(-0.714286\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(710\) 0 0
\(711\) 0.400969 1.75676i 0.400969 1.75676i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(720\) 0 0
\(721\) −0.277479 + 0.347948i −0.277479 + 0.347948i
\(722\) 0 0
\(723\) 0.541044 1.12349i 0.541044 1.12349i
\(724\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(725\) 0 0
\(726\) 0 0
\(727\) 1.56366 1.24698i 1.56366 1.24698i 0.781831 0.623490i \(-0.214286\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(728\) 0 0
\(729\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(730\) 0 0
\(731\) 0 0
\(732\) −1.40881 + 1.12349i −1.40881 + 1.12349i
\(733\) −0.433884 + 0.0990311i −0.433884 + 0.0990311i −0.433884 0.900969i \(-0.642857\pi\)
1.00000i \(0.5\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 1.12349 0.541044i 1.12349 0.541044i 0.222521 0.974928i \(-0.428571\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(740\) 0 0
\(741\) −0.346011 + 0.433884i −0.346011 + 0.433884i
\(742\) 0 0
\(743\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.277479 + 0.347948i −0.277479 + 0.347948i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −0.900969 0.433884i −0.900969 0.433884i
\(757\) 0.541044 + 1.12349i 0.541044 + 1.12349i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(762\) 0 0
\(763\) −0.541044 1.12349i −0.541044 1.12349i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0.781831 + 0.623490i 0.781831 + 0.623490i
\(769\) 0.445042 1.94986i 0.445042 1.94986i 0.222521 0.974928i \(-0.428571\pi\)
0.222521 0.974928i \(-0.428571\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.541044 + 1.12349i −0.541044 + 1.12349i
\(773\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) −0.781831 1.62349i −0.781831 1.62349i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.222521 0.974928i 0.222521 0.974928i
\(785\) 0 0
\(786\) 0 0
\(787\) −1.40881 + 1.12349i −1.40881 + 1.12349i −0.433884 + 0.900969i \(0.642857\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 2.19064 + 0.500000i 2.19064 + 0.500000i
\(794\) 0 0
\(795\) 0 0
\(796\) 0.400969 + 0.193096i 0.400969 + 0.193096i
\(797\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(810\) 0 0
\(811\) −1.12349 + 0.541044i −1.12349 + 0.541044i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(812\) 0 0
\(813\) 0.541044 1.12349i 0.541044 1.12349i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −0.154851 0.123490i −0.154851 0.123490i
\(818\) 0 0
\(819\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(820\) 0 0
\(821\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(822\) 0 0
\(823\) −0.347948 0.277479i −0.347948 0.277479i 0.433884 0.900969i \(-0.357143\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(828\) 0 0
\(829\) −1.62349 + 0.781831i −1.62349 + 0.781831i −0.623490 + 0.781831i \(0.714286\pi\)
−1.00000 \(1.00000\pi\)
\(830\) 0 0
\(831\) −0.277479 1.21572i −0.277479 1.21572i
\(832\) 1.24698i 1.24698i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −0.781831 1.62349i −0.781831 1.62349i
\(838\) 0 0
\(839\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(840\) 0 0
\(841\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(842\) 0 0
\(843\) 0 0
\(844\) −0.400969 1.75676i −0.400969 1.75676i
\(845\) 0 0
\(846\) 0 0
\(847\) 0.781831 0.623490i 0.781831 0.623490i
\(848\) 0 0
\(849\) 1.12349 1.40881i 1.12349 1.40881i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −0.193096 + 0.400969i −0.193096 + 0.400969i −0.974928 0.222521i \(-0.928571\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(858\) 0 0
\(859\) −0.400969 + 0.193096i −0.400969 + 0.193096i −0.623490 0.781831i \(-0.714286\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.974928 + 0.222521i 0.974928 + 0.222521i
\(868\) 1.40881 1.12349i 1.40881 1.12349i
\(869\) 0 0
\(870\) 0 0
\(871\) −0.346011 1.51597i −0.346011 1.51597i
\(872\) 0 0
\(873\) 0.433884 + 0.0990311i 0.433884 + 0.0990311i
\(874\) 0 0
\(875\) 0 0
\(876\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(877\) −0.347948 + 0.277479i −0.347948 + 0.277479i −0.781831 0.623490i \(-0.785714\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.24698i 1.24698i 0.781831 + 0.623490i \(0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(888\) 0 0
\(889\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i
\(890\) 0 0
\(891\) 0 0
\(892\) 1.40881 + 1.12349i 1.40881 + 1.12349i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) −0.433884 + 0.0990311i −0.433884 + 0.0990311i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.21572 + 0.277479i 1.21572 + 0.277479i 0.781831 0.623490i \(-0.214286\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(912\) 0.445042i 0.445042i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0.0990311 0.433884i 0.0990311 0.433884i
\(917\) 0 0
\(918\) 0 0
\(919\) 1.12349 + 0.541044i 1.12349 + 0.541044i 0.900969 0.433884i \(-0.142857\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(920\) 0 0
\(921\) −0.277479 + 0.347948i −0.277479 + 0.347948i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) −0.193096 0.400969i −0.193096 0.400969i
\(928\) 0 0
\(929\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(930\) 0 0
\(931\) 0.400969 0.193096i 0.400969 0.193096i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0.347948 + 0.277479i 0.347948 + 0.277479i 0.781831 0.623490i \(-0.214286\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(938\) 0 0
\(939\) 1.12349 + 1.40881i 1.12349 + 1.40881i
\(940\) 0 0
\(941\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(948\) −0.781831 + 1.62349i −0.781831 + 1.62349i
\(949\) 2.24698 2.24698
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 2.24698 2.24698
\(962\) 0 0
\(963\) 0 0
\(964\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(965\) 0 0
\(966\) 0 0
\(967\) 0.433884 0.0990311i 0.433884 0.0990311i 1.00000i \(-0.5\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(972\) 0.781831 0.623490i 0.781831 0.623490i
\(973\) 0.974928 + 0.777479i 0.974928 + 0.777479i
\(974\) 0 0
\(975\) 0 0
\(976\) 1.62349 0.781831i 1.62349 0.781831i
\(977\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.24698 1.24698
\(982\) 0 0
\(983\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0.433884 0.346011i 0.433884 0.346011i
\(989\) 0 0
\(990\) 0 0
\(991\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i 1.00000 \(0\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(992\) 0 0
\(993\) −0.433884 + 0.0990311i −0.433884 + 0.0990311i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −0.867767 + 1.80194i −0.867767 + 1.80194i −0.433884 + 0.900969i \(0.642857\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(998\) 0 0
\(999\) 1.80194 1.80194
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 3675.1.bj.a.3074.1 12
3.2 odd 2 CM 3675.1.bj.a.3074.1 12
5.2 odd 4 147.1.l.a.134.1 yes 6
5.3 odd 4 3675.1.bm.a.1751.1 6
5.4 even 2 inner 3675.1.bj.a.3074.2 12
15.2 even 4 147.1.l.a.134.1 yes 6
15.8 even 4 3675.1.bm.a.1751.1 6
15.14 odd 2 inner 3675.1.bj.a.3074.2 12
20.7 even 4 2352.1.cj.a.1457.1 6
35.2 odd 12 1029.1.n.b.998.1 12
35.12 even 12 1029.1.n.a.998.1 12
35.17 even 12 1029.1.n.a.128.1 12
35.27 even 4 1029.1.l.a.932.1 6
35.32 odd 12 1029.1.n.b.128.1 12
45.2 even 12 3969.1.bt.a.134.1 12
45.7 odd 12 3969.1.bt.a.134.1 12
45.22 odd 12 3969.1.bt.a.2780.1 12
45.32 even 12 3969.1.bt.a.2780.1 12
49.15 even 7 inner 3675.1.bj.a.2024.2 12
60.47 odd 4 2352.1.cj.a.1457.1 6
105.2 even 12 1029.1.n.b.998.1 12
105.17 odd 12 1029.1.n.a.128.1 12
105.32 even 12 1029.1.n.b.128.1 12
105.47 odd 12 1029.1.n.a.998.1 12
105.62 odd 4 1029.1.l.a.932.1 6
147.113 odd 14 inner 3675.1.bj.a.2024.2 12
245.64 even 14 inner 3675.1.bj.a.2024.1 12
245.107 odd 84 1029.1.n.b.410.1 12
245.113 odd 28 3675.1.bm.a.701.1 6
245.122 even 84 1029.1.n.a.863.1 12
245.132 even 28 1029.1.l.a.785.1 6
245.162 odd 28 147.1.l.a.113.1 6
245.172 odd 84 1029.1.n.b.863.1 12
245.187 even 84 1029.1.n.a.410.1 12
735.107 even 84 1029.1.n.b.410.1 12
735.113 even 28 3675.1.bm.a.701.1 6
735.122 odd 84 1029.1.n.a.863.1 12
735.377 odd 28 1029.1.l.a.785.1 6
735.407 even 28 147.1.l.a.113.1 6
735.554 odd 14 inner 3675.1.bj.a.2024.1 12
735.662 even 84 1029.1.n.b.863.1 12
735.677 odd 84 1029.1.n.a.410.1 12
980.407 even 28 2352.1.cj.a.113.1 6
2205.407 even 84 3969.1.bt.a.701.1 12
2205.652 odd 84 3969.1.bt.a.3347.1 12
2205.1877 even 84 3969.1.bt.a.3347.1 12
2205.2122 odd 84 3969.1.bt.a.701.1 12
2940.407 odd 28 2352.1.cj.a.113.1 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
147.1.l.a.113.1 6 245.162 odd 28
147.1.l.a.113.1 6 735.407 even 28
147.1.l.a.134.1 yes 6 5.2 odd 4
147.1.l.a.134.1 yes 6 15.2 even 4
1029.1.l.a.785.1 6 245.132 even 28
1029.1.l.a.785.1 6 735.377 odd 28
1029.1.l.a.932.1 6 35.27 even 4
1029.1.l.a.932.1 6 105.62 odd 4
1029.1.n.a.128.1 12 35.17 even 12
1029.1.n.a.128.1 12 105.17 odd 12
1029.1.n.a.410.1 12 245.187 even 84
1029.1.n.a.410.1 12 735.677 odd 84
1029.1.n.a.863.1 12 245.122 even 84
1029.1.n.a.863.1 12 735.122 odd 84
1029.1.n.a.998.1 12 35.12 even 12
1029.1.n.a.998.1 12 105.47 odd 12
1029.1.n.b.128.1 12 35.32 odd 12
1029.1.n.b.128.1 12 105.32 even 12
1029.1.n.b.410.1 12 245.107 odd 84
1029.1.n.b.410.1 12 735.107 even 84
1029.1.n.b.863.1 12 245.172 odd 84
1029.1.n.b.863.1 12 735.662 even 84
1029.1.n.b.998.1 12 35.2 odd 12
1029.1.n.b.998.1 12 105.2 even 12
2352.1.cj.a.113.1 6 980.407 even 28
2352.1.cj.a.113.1 6 2940.407 odd 28
2352.1.cj.a.1457.1 6 20.7 even 4
2352.1.cj.a.1457.1 6 60.47 odd 4
3675.1.bj.a.2024.1 12 245.64 even 14 inner
3675.1.bj.a.2024.1 12 735.554 odd 14 inner
3675.1.bj.a.2024.2 12 49.15 even 7 inner
3675.1.bj.a.2024.2 12 147.113 odd 14 inner
3675.1.bj.a.3074.1 12 1.1 even 1 trivial
3675.1.bj.a.3074.1 12 3.2 odd 2 CM
3675.1.bj.a.3074.2 12 5.4 even 2 inner
3675.1.bj.a.3074.2 12 15.14 odd 2 inner
3675.1.bm.a.701.1 6 245.113 odd 28
3675.1.bm.a.701.1 6 735.113 even 28
3675.1.bm.a.1751.1 6 5.3 odd 4
3675.1.bm.a.1751.1 6 15.8 even 4
3969.1.bt.a.134.1 12 45.2 even 12
3969.1.bt.a.134.1 12 45.7 odd 12
3969.1.bt.a.701.1 12 2205.407 even 84
3969.1.bt.a.701.1 12 2205.2122 odd 84
3969.1.bt.a.2780.1 12 45.22 odd 12
3969.1.bt.a.2780.1 12 45.32 even 12
3969.1.bt.a.3347.1 12 2205.652 odd 84
3969.1.bt.a.3347.1 12 2205.1877 even 84