Properties

Label 3528.1.dh.a.181.1
Level $3528$
Weight $1$
Character 3528.181
Analytic conductor $1.761$
Analytic rank $0$
Dimension $12$
Projective image $D_{14}$
CM discriminant -24
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3528,1,Mod(181,3528)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3528, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([0, 7, 0, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3528.181");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3528 = 2^{3} \cdot 3^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3528.dh (of order \(14\), degree \(6\), 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: \(1.76070136457\)
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: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{14}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{14} + \cdots)\)

Embedding invariants

Embedding label 181.1
Root \(-0.974928 - 0.222521i\) of defining polynomial
Character \(\chi\) \(=\) 3528.181
Dual form 3528.1.dh.a.1189.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.974928 - 0.222521i) q^{2} +(0.900969 + 0.433884i) q^{4} +(1.21572 - 1.52446i) q^{5} +(0.623490 + 0.781831i) q^{7} +(-0.781831 - 0.623490i) q^{8} +O(q^{10})\) \(q+(-0.974928 - 0.222521i) q^{2} +(0.900969 + 0.433884i) q^{4} +(1.21572 - 1.52446i) q^{5} +(0.623490 + 0.781831i) q^{7} +(-0.781831 - 0.623490i) q^{8} +(-1.52446 + 1.21572i) q^{10} +(1.75676 + 0.400969i) q^{11} +(-0.433884 - 0.900969i) q^{14} +(0.623490 + 0.781831i) q^{16} +(1.75676 - 0.846011i) q^{20} +(-1.62349 - 0.781831i) q^{22} +(-0.623490 - 2.73169i) q^{25} +(0.222521 + 0.974928i) q^{28} +(-0.193096 - 0.400969i) q^{29} +1.56366i q^{31} +(-0.433884 - 0.900969i) q^{32} +1.94986 q^{35} +(-1.90097 + 0.433884i) q^{40} +(1.40881 + 1.12349i) q^{44} +(-0.222521 + 0.974928i) q^{49} +2.80194i q^{50} +(-0.867767 + 1.80194i) q^{53} +(2.74698 - 2.19064i) q^{55} -1.00000i q^{56} +(0.0990311 + 0.433884i) q^{58} +(0.347948 - 1.52446i) q^{62} +(0.222521 + 0.974928i) q^{64} +(-1.90097 - 0.433884i) q^{70} +(-1.52446 + 0.347948i) q^{73} +(0.781831 + 1.62349i) q^{77} -1.24698 q^{79} +1.94986 q^{80} +(-0.433884 - 1.90097i) q^{83} +(-1.12349 - 1.40881i) q^{88} -0.867767i q^{97} +(0.433884 - 0.900969i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 2 q^{4} - 2 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 2 q^{4} - 2 q^{7} - 2 q^{16} - 10 q^{22} + 2 q^{25} + 2 q^{28} - 14 q^{40} - 2 q^{49} + 14 q^{55} + 10 q^{58} + 2 q^{64} - 14 q^{70} + 4 q^{79} - 4 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3528\mathbb{Z}\right)^\times\).

\(n\) \(785\) \(1081\) \(1765\) \(2647\)
\(\chi(n)\) \(1\) \(e\left(\frac{3}{14}\right)\) \(-1\) \(1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.974928 0.222521i −0.974928 0.222521i
\(3\) 0 0
\(4\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(5\) 1.21572 1.52446i 1.21572 1.52446i 0.433884 0.900969i \(-0.357143\pi\)
0.781831 0.623490i \(-0.214286\pi\)
\(6\) 0 0
\(7\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(8\) −0.781831 0.623490i −0.781831 0.623490i
\(9\) 0 0
\(10\) −1.52446 + 1.21572i −1.52446 + 1.21572i
\(11\) 1.75676 + 0.400969i 1.75676 + 0.400969i 0.974928 0.222521i \(-0.0714286\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(12\) 0 0
\(13\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(14\) −0.433884 0.900969i −0.433884 0.900969i
\(15\) 0 0
\(16\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(17\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(18\) 0 0
\(19\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(20\) 1.75676 0.846011i 1.75676 0.846011i
\(21\) 0 0
\(22\) −1.62349 0.781831i −1.62349 0.781831i
\(23\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(24\) 0 0
\(25\) −0.623490 2.73169i −0.623490 2.73169i
\(26\) 0 0
\(27\) 0 0
\(28\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(29\) −0.193096 0.400969i −0.193096 0.400969i 0.781831 0.623490i \(-0.214286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(30\) 0 0
\(31\) 1.56366i 1.56366i 0.623490 + 0.781831i \(0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(32\) −0.433884 0.900969i −0.433884 0.900969i
\(33\) 0 0
\(34\) 0 0
\(35\) 1.94986 1.94986
\(36\) 0 0
\(37\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −1.90097 + 0.433884i −1.90097 + 0.433884i
\(41\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(42\) 0 0
\(43\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(44\) 1.40881 + 1.12349i 1.40881 + 1.12349i
\(45\) 0 0
\(46\) 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
\(50\) 2.80194i 2.80194i
\(51\) 0 0
\(52\) 0 0
\(53\) −0.867767 + 1.80194i −0.867767 + 1.80194i −0.433884 + 0.900969i \(0.642857\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(54\) 0 0
\(55\) 2.74698 2.19064i 2.74698 2.19064i
\(56\) 1.00000i 1.00000i
\(57\) 0 0
\(58\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i
\(59\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(60\) 0 0
\(61\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(62\) 0.347948 1.52446i 0.347948 1.52446i
\(63\) 0 0
\(64\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) −1.90097 0.433884i −1.90097 0.433884i
\(71\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(72\) 0 0
\(73\) −1.52446 + 0.347948i −1.52446 + 0.347948i −0.900969 0.433884i \(-0.857143\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0.781831 + 1.62349i 0.781831 + 1.62349i
\(78\) 0 0
\(79\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(80\) 1.94986 1.94986
\(81\) 0 0
\(82\) 0 0
\(83\) −0.433884 1.90097i −0.433884 1.90097i −0.433884 0.900969i \(-0.642857\pi\)
1.00000i \(-0.5\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) −1.12349 1.40881i −1.12349 1.40881i
\(89\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.867767i 0.867767i −0.900969 0.433884i \(-0.857143\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(98\) 0.433884 0.900969i 0.433884 0.900969i
\(99\) 0 0
\(100\) 0.623490 2.73169i 0.623490 2.73169i
\(101\) −0.541044 + 0.678448i −0.541044 + 0.678448i −0.974928 0.222521i \(-0.928571\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(102\) 0 0
\(103\) 0.678448 + 0.541044i 0.678448 + 0.541044i 0.900969 0.433884i \(-0.142857\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 1.24698 1.56366i 1.24698 1.56366i
\(107\) −1.21572 + 0.277479i −1.21572 + 0.277479i −0.781831 0.623490i \(-0.785714\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(108\) 0 0
\(109\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(110\) −3.16557 + 1.52446i −3.16557 + 1.52446i
\(111\) 0 0
\(112\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(113\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.445042i 0.445042i
\(117\) 0 0
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 2.02446 + 0.974928i 2.02446 + 0.974928i
\(122\) 0 0
\(123\) 0 0
\(124\) −0.678448 + 1.40881i −0.678448 + 1.40881i
\(125\) −3.16557 1.52446i −3.16557 1.52446i
\(126\) 0 0
\(127\) −1.12349 + 0.541044i −1.12349 + 0.541044i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(128\) 1.00000i 1.00000i
\(129\) 0 0
\(130\) 0 0
\(131\) −0.974928 1.22252i −0.974928 1.22252i −0.974928 0.222521i \(-0.928571\pi\)
1.00000i \(-0.5\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(138\) 0 0
\(139\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(140\) 1.75676 + 0.846011i 1.75676 + 0.846011i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) −0.846011 0.193096i −0.846011 0.193096i
\(146\) 1.56366 1.56366
\(147\) 0 0
\(148\) 0 0
\(149\) −1.21572 0.277479i −1.21572 0.277479i −0.433884 0.900969i \(-0.642857\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(150\) 0 0
\(151\) 1.80194 + 0.867767i 1.80194 + 0.867767i 0.900969 + 0.433884i \(0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) −0.400969 1.75676i −0.400969 1.75676i
\(155\) 2.38374 + 1.90097i 2.38374 + 1.90097i
\(156\) 0 0
\(157\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(158\) 1.21572 + 0.277479i 1.21572 + 0.277479i
\(159\) 0 0
\(160\) −1.90097 0.433884i −1.90097 0.433884i
\(161\) 0 0
\(162\) 0 0
\(163\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 1.94986i 1.94986i
\(167\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(168\) 0 0
\(169\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) −1.75676 0.846011i −1.75676 0.846011i −0.974928 0.222521i \(-0.928571\pi\)
−0.781831 0.623490i \(-0.785714\pi\)
\(174\) 0 0
\(175\) 1.74698 2.19064i 1.74698 2.19064i
\(176\) 0.781831 + 1.62349i 0.781831 + 1.62349i
\(177\) 0 0
\(178\) 0 0
\(179\) 0.193096 + 0.400969i 0.193096 + 0.400969i 0.974928 0.222521i \(-0.0714286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(180\) 0 0
\(181\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(192\) 0 0
\(193\) 0.777479 0.974928i 0.777479 0.974928i −0.222521 0.974928i \(-0.571429\pi\)
1.00000 \(0\)
\(194\) −0.193096 + 0.846011i −0.193096 + 0.846011i
\(195\) 0 0
\(196\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(197\) 1.80194i 1.80194i −0.433884 0.900969i \(-0.642857\pi\)
0.433884 0.900969i \(-0.357143\pi\)
\(198\) 0 0
\(199\) −1.22252 0.974928i −1.22252 0.974928i −0.222521 0.974928i \(-0.571429\pi\)
−1.00000 \(\pi\)
\(200\) −1.21572 + 2.52446i −1.21572 + 2.52446i
\(201\) 0 0
\(202\) 0.678448 0.541044i 0.678448 0.541044i
\(203\) 0.193096 0.400969i 0.193096 0.400969i
\(204\) 0 0
\(205\) 0 0
\(206\) −0.541044 0.678448i −0.541044 0.678448i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(212\) −1.56366 + 1.24698i −1.56366 + 1.24698i
\(213\) 0 0
\(214\) 1.24698 1.24698
\(215\) 0 0
\(216\) 0 0
\(217\) −1.22252 + 0.974928i −1.22252 + 0.974928i
\(218\) 0 0
\(219\) 0 0
\(220\) 3.42543 0.781831i 3.42543 0.781831i
\(221\) 0 0
\(222\) 0 0
\(223\) 0.376510 0.781831i 0.376510 0.781831i −0.623490 0.781831i \(-0.714286\pi\)
1.00000 \(0\)
\(224\) 0.433884 0.900969i 0.433884 0.900969i
\(225\) 0 0
\(226\) 0 0
\(227\) 0.867767 0.867767 0.433884 0.900969i \(-0.357143\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(228\) 0 0
\(229\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i
\(233\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(240\) 0 0
\(241\) 0.846011 1.75676i 0.846011 1.75676i 0.222521 0.974928i \(-0.428571\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(242\) −1.75676 1.40097i −1.75676 1.40097i
\(243\) 0 0
\(244\) 0 0
\(245\) 1.21572 + 1.52446i 1.21572 + 1.52446i
\(246\) 0 0
\(247\) 0 0
\(248\) 0.974928 1.22252i 0.974928 1.22252i
\(249\) 0 0
\(250\) 2.74698 + 2.19064i 2.74698 + 2.19064i
\(251\) 0.974928 + 1.22252i 0.974928 + 1.22252i 0.974928 + 0.222521i \(0.0714286\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.21572 0.277479i 1.21572 0.277479i
\(255\) 0 0
\(256\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(257\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0.678448 + 1.40881i 0.678448 + 1.40881i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 1.69202 + 3.51352i 1.69202 + 3.51352i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.433884 1.90097i −0.433884 1.90097i −0.433884 0.900969i \(-0.642857\pi\)
1.00000i \(-0.5\pi\)
\(270\) 0 0
\(271\) 0.376510 0.781831i 0.376510 0.781831i −0.623490 0.781831i \(-0.714286\pi\)
1.00000 \(0\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 5.04892i 5.04892i
\(276\) 0 0
\(277\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) −1.52446 1.21572i −1.52446 1.21572i
\(281\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(282\) 0 0
\(283\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.623490 0.781831i 0.623490 0.781831i
\(290\) 0.781831 + 0.376510i 0.781831 + 0.376510i
\(291\) 0 0
\(292\) −1.52446 0.347948i −1.52446 0.347948i
\(293\) 0.867767 0.867767 0.433884 0.900969i \(-0.357143\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −1.56366 1.24698i −1.56366 1.24698i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(308\) 1.80194i 1.80194i
\(309\) 0 0
\(310\) −1.90097 2.38374i −1.90097 2.38374i
\(311\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(312\) 0 0
\(313\) 1.56366i 1.56366i −0.623490 0.781831i \(-0.714286\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −1.12349 0.541044i −1.12349 0.541044i
\(317\) −0.781831 + 1.62349i −0.781831 + 1.62349i 1.00000i \(0.5\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(318\) 0 0
\(319\) −0.178448 0.781831i −0.178448 0.781831i
\(320\) 1.75676 + 0.846011i 1.75676 + 0.846011i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(332\) 0.433884 1.90097i 0.433884 1.90097i
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 1.24698 + 1.56366i 1.24698 + 1.56366i 0.623490 + 0.781831i \(0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(338\) −0.781831 0.623490i −0.781831 0.623490i
\(339\) 0 0
\(340\) 0 0
\(341\) −0.626980 + 2.74698i −0.626980 + 2.74698i
\(342\) 0 0
\(343\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(344\) 0 0
\(345\) 0 0
\(346\) 1.52446 + 1.21572i 1.52446 + 1.21572i
\(347\) 0.193096 0.400969i 0.193096 0.400969i −0.781831 0.623490i \(-0.785714\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(348\) 0 0
\(349\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(350\) −2.19064 + 1.74698i −2.19064 + 1.74698i
\(351\) 0 0
\(352\) −0.400969 1.75676i −0.400969 1.75676i
\(353\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −0.0990311 0.433884i −0.0990311 0.433884i
\(359\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(360\) 0 0
\(361\) −1.00000 −1.00000
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) −1.32288 + 2.74698i −1.32288 + 2.74698i
\(366\) 0 0
\(367\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −1.94986 + 0.445042i −1.94986 + 0.445042i
\(372\) 0 0
\(373\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(384\) 0 0
\(385\) 3.42543 + 0.781831i 3.42543 + 0.781831i
\(386\) −0.974928 + 0.777479i −0.974928 + 0.777479i
\(387\) 0 0
\(388\) 0.376510 0.781831i 0.376510 0.781831i
\(389\) 0.347948 + 0.277479i 0.347948 + 0.277479i 0.781831 0.623490i \(-0.214286\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.781831 0.623490i 0.781831 0.623490i
\(393\) 0 0
\(394\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(395\) −1.51597 + 1.90097i −1.51597 + 1.90097i
\(396\) 0 0
\(397\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(398\) 0.974928 + 1.22252i 0.974928 + 1.22252i
\(399\) 0 0
\(400\) 1.74698 2.19064i 1.74698 2.19064i
\(401\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.781831 + 0.376510i −0.781831 + 0.376510i
\(405\) 0 0
\(406\) −0.277479 + 0.347948i −0.277479 + 0.347948i
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.376510 + 0.781831i 0.376510 + 0.781831i
\(413\) 0 0
\(414\) 0 0
\(415\) −3.42543 1.64960i −3.42543 1.64960i
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 1.75676 + 0.846011i 1.75676 + 0.846011i 0.974928 + 0.222521i \(0.0714286\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(420\) 0 0
\(421\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 1.80194 0.867767i 1.80194 0.867767i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −1.21572 0.277479i −1.21572 0.277479i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(432\) 0 0
\(433\) −1.52446 1.21572i −1.52446 1.21572i −0.900969 0.433884i \(-0.857143\pi\)
−0.623490 0.781831i \(-0.714286\pi\)
\(434\) 1.40881 0.678448i 1.40881 0.678448i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −0.846011 0.193096i −0.846011 0.193096i −0.222521 0.974928i \(-0.571429\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(440\) −3.51352 −3.51352
\(441\) 0 0
\(442\) 0 0
\(443\) 1.75676 + 0.400969i 1.75676 + 0.400969i 0.974928 0.222521i \(-0.0714286\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −0.541044 + 0.678448i −0.541044 + 0.678448i
\(447\) 0 0
\(448\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(449\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) −0.846011 0.193096i −0.846011 0.193096i
\(455\) 0 0
\(456\) 0 0
\(457\) −0.277479 0.347948i −0.277479 0.347948i 0.623490 0.781831i \(-0.285714\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) −1.40881 + 0.678448i −1.40881 + 0.678448i −0.974928 0.222521i \(-0.928571\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(462\) 0 0
\(463\) −0.400969 0.193096i −0.400969 0.193096i 0.222521 0.974928i \(-0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(464\) 0.193096 0.400969i 0.193096 0.400969i
\(465\) 0 0
\(466\) 0 0
\(467\) −0.781831 0.376510i −0.781831 0.376510i 1.00000i \(-0.5\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(468\) 0 0
\(469\) 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 0
\(478\) 0 0
\(479\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −1.21572 + 1.52446i −1.21572 + 1.52446i
\(483\) 0 0
\(484\) 1.40097 + 1.75676i 1.40097 + 1.75676i
\(485\) −1.32288 1.05496i −1.32288 1.05496i
\(486\) 0 0
\(487\) −0.777479 + 0.974928i −0.777479 + 0.974928i 0.222521 + 0.974928i \(0.428571\pi\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) −0.846011 1.75676i −0.846011 1.75676i
\(491\) 0.445042i 0.445042i −0.974928 0.222521i \(-0.928571\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −1.22252 + 0.974928i −1.22252 + 0.974928i
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(500\) −2.19064 2.74698i −2.19064 2.74698i
\(501\) 0 0
\(502\) −0.678448 1.40881i −0.678448 1.40881i
\(503\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(504\) 0 0
\(505\) 0.376510 + 1.64960i 0.376510 + 1.64960i
\(506\) 0 0
\(507\) 0 0
\(508\) −1.24698 −1.24698
\(509\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(510\) 0 0
\(511\) −1.22252 0.974928i −1.22252 0.974928i
\(512\) 0.433884 0.900969i 0.433884 0.900969i
\(513\) 0 0
\(514\) 0 0
\(515\) 1.64960 0.376510i 1.64960 0.376510i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(522\) 0 0
\(523\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(524\) −0.347948 1.52446i −0.347948 1.52446i
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.623490 0.781831i −0.623490 0.781831i
\(530\) −0.867767 3.80194i −0.867767 3.80194i
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −1.05496 + 2.19064i −1.05496 + 2.19064i
\(536\) 0 0
\(537\) 0 0
\(538\) 1.94986i 1.94986i
\(539\) −0.781831 + 1.62349i −0.781831 + 1.62349i
\(540\) 0 0
\(541\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(542\) −0.541044 + 0.678448i −0.541044 + 0.678448i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −1.12349 + 4.92233i −1.12349 + 4.92233i
\(551\) 0 0
\(552\) 0 0
\(553\) −0.777479 0.974928i −0.777479 0.974928i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0.445042i 0.445042i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 1.21572 + 1.52446i 1.21572 + 1.52446i
\(561\) 0 0
\(562\) 0 0
\(563\) 0.347948 + 1.52446i 0.347948 + 1.52446i 0.781831 + 0.623490i \(0.214286\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) 1.52446 + 0.347948i 1.52446 + 0.347948i 0.900969 0.433884i \(-0.142857\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(578\) −0.781831 + 0.623490i −0.781831 + 0.623490i
\(579\) 0 0
\(580\) −0.678448 0.541044i −0.678448 0.541044i
\(581\) 1.21572 1.52446i 1.21572 1.52446i
\(582\) 0 0
\(583\) −2.24698 + 2.81762i −2.24698 + 2.81762i
\(584\) 1.40881 + 0.678448i 1.40881 + 0.678448i
\(585\) 0 0
\(586\) −0.846011 0.193096i −0.846011 0.193096i
\(587\) −0.867767 −0.867767 −0.433884 0.900969i \(-0.642857\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.974928 0.777479i −0.974928 0.777479i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(600\) 0 0
\(601\) −1.52446 0.347948i −1.52446 0.347948i −0.623490 0.781831i \(-0.714286\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 1.24698 + 1.56366i 1.24698 + 1.56366i
\(605\) 3.94740 1.90097i 3.94740 1.90097i
\(606\) 0 0
\(607\) 0.867767i 0.867767i −0.900969 0.433884i \(-0.857143\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0.400969 1.75676i 0.400969 1.75676i
\(617\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(620\) 1.32288 + 2.74698i 1.32288 + 2.74698i
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −3.64795 + 1.75676i −3.64795 + 1.75676i
\(626\) −0.347948 + 1.52446i −0.347948 + 1.52446i
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) 1.12349 + 1.40881i 1.12349 + 1.40881i 0.900969 + 0.433884i \(0.142857\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(632\) 0.974928 + 0.777479i 0.974928 + 0.777479i
\(633\) 0 0
\(634\) 1.12349 1.40881i 1.12349 1.40881i
\(635\) −0.541044 + 2.37047i −0.541044 + 2.37047i
\(636\) 0 0
\(637\) 0 0
\(638\) 0.801938i 0.801938i
\(639\) 0 0
\(640\) −1.52446 1.21572i −1.52446 1.21572i
\(641\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(642\) 0 0
\(643\) 0 0 0.781831 0.623490i \(-0.214286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −0.347948 + 0.277479i −0.347948 + 0.277479i −0.781831 0.623490i \(-0.785714\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(654\) 0 0
\(655\) −3.04892 −3.04892
\(656\) 0 0
\(657\) 0 0
\(658\) 0 0
\(659\) −0.781831 + 1.62349i −0.781831 + 1.62349i 1.00000i \(0.5\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(660\) 0 0
\(661\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) −0.846011 + 1.75676i −0.846011 + 1.75676i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.400969 + 1.75676i −0.400969 + 1.75676i 0.222521 + 0.974928i \(0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(674\) −0.867767 1.80194i −0.867767 1.80194i
\(675\) 0 0
\(676\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(677\) 0.193096 + 0.846011i 0.193096 + 0.846011i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(678\) 0 0
\(679\) 0.678448 0.541044i 0.678448 0.541044i
\(680\) 0 0
\(681\) 0 0
\(682\) 1.22252 2.53859i 1.22252 2.53859i
\(683\) −0.347948 0.277479i −0.347948 0.277479i 0.433884 0.900969i \(-0.357143\pi\)
−0.781831 + 0.623490i \(0.785714\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.974928 0.222521i 0.974928 0.222521i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(692\) −1.21572 1.52446i −1.21572 1.52446i
\(693\) 0 0
\(694\) −0.277479 + 0.347948i −0.277479 + 0.347948i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 2.52446 1.21572i 2.52446 1.21572i
\(701\) 1.21572 0.277479i 1.21572 0.277479i 0.433884 0.900969i \(-0.357143\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.80194i 1.80194i
\(705\) 0 0
\(706\) 0 0
\(707\) −0.867767 −0.867767
\(708\) 0 0
\(709\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.445042i 0.445042i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(720\) 0 0
\(721\) 0.867767i 0.867767i
\(722\) 0.974928 + 0.222521i 0.974928 + 0.222521i
\(723\) 0 0
\(724\) 0 0
\(725\) −0.974928 + 0.777479i −0.974928 + 0.777479i
\(726\) 0 0
\(727\) 1.52446 + 1.21572i 1.52446 + 1.21572i 0.900969 + 0.433884i \(0.142857\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 1.90097 2.38374i 1.90097 2.38374i
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.974928 0.222521i \(-0.928571\pi\)
0.974928 + 0.222521i \(0.0714286\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 2.00000 2.00000
\(743\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(744\) 0 0
\(745\) −1.90097 + 1.51597i −1.90097 + 1.51597i
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −0.974928 0.777479i −0.974928 0.777479i
\(750\) 0 0
\(751\) 1.12349 + 1.40881i 1.12349 + 1.40881i 0.900969 + 0.433884i \(0.142857\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 3.51352 1.69202i 3.51352 1.69202i
\(756\) 0 0
\(757\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 1.52446 0.347948i 1.52446 0.347948i 0.623490 0.781831i \(-0.285714\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(770\) −3.16557 1.52446i −3.16557 1.52446i
\(771\) 0 0
\(772\) 1.12349 0.541044i 1.12349 0.541044i
\(773\) 0.193096 0.846011i 0.193096 0.846011i −0.781831 0.623490i \(-0.785714\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(774\) 0 0
\(775\) 4.27144 0.974928i 4.27144 0.974928i
\(776\) −0.541044 + 0.678448i −0.541044 + 0.678448i
\(777\) 0 0
\(778\) −0.277479 0.347948i −0.277479 0.347948i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(788\) 0.781831 1.62349i 0.781831 1.62349i
\(789\) 0 0
\(790\) 1.90097 1.51597i 1.90097 1.51597i
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) −0.678448 1.40881i −0.678448 1.40881i
\(797\) 0.193096 0.846011i 0.193096 0.846011i −0.781831 0.623490i \(-0.785714\pi\)
0.974928 0.222521i \(-0.0714286\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −2.19064 + 1.74698i −2.19064 + 1.74698i
\(801\) 0 0
\(802\) 0 0
\(803\) −2.81762 −2.81762
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0.846011 0.193096i 0.846011 0.193096i
\(809\) 0 0 0.974928 0.222521i \(-0.0714286\pi\)
−0.974928 + 0.222521i \(0.928571\pi\)
\(810\) 0 0
\(811\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(812\) 0.347948 0.277479i 0.347948 0.277479i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0.541044 + 1.12349i 0.541044 + 1.12349i 0.974928 + 0.222521i \(0.0714286\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(822\) 0 0
\(823\) −0.777479 0.974928i −0.777479 0.974928i 0.222521 0.974928i \(-0.428571\pi\)
−1.00000 \(\pi\)
\(824\) −0.193096 0.846011i −0.193096 0.846011i
\(825\) 0 0
\(826\) 0 0
\(827\) −0.347948 + 0.277479i −0.347948 + 0.277479i −0.781831 0.623490i \(-0.785714\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(828\) 0 0
\(829\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(830\) 2.97247 + 2.37047i 2.97247 + 2.37047i
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) −1.52446 1.21572i −1.52446 1.21572i
\(839\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(840\) 0 0
\(841\) 0.500000 0.626980i 0.500000 0.626980i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 1.75676 0.846011i 1.75676 0.846011i
\(846\) 0 0
\(847\) 0.500000 + 2.19064i 0.500000 + 2.19064i
\(848\) −1.94986 + 0.445042i −1.94986 + 0.445042i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(857\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(858\) 0 0
\(859\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) −3.42543 + 1.64960i −3.42543 + 1.64960i
\(866\) 1.21572 + 1.52446i 1.21572 + 1.52446i
\(867\) 0 0
\(868\) −1.52446 + 0.347948i −1.52446 + 0.347948i
\(869\) −2.19064 0.500000i −2.19064 0.500000i
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.781831 3.42543i −0.781831 3.42543i
\(876\) 0 0
\(877\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(878\) 0.781831 + 0.376510i 0.781831 + 0.376510i
\(879\) 0 0
\(880\) 3.42543 + 0.781831i 3.42543 + 0.781831i
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1.62349 0.781831i −1.62349 0.781831i
\(887\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(888\) 0 0
\(889\) −1.12349 0.541044i −1.12349 0.541044i
\(890\) 0 0
\(891\) 0 0
\(892\) 0.678448 0.541044i 0.678448 0.541044i
\(893\) 0 0
\(894\) 0 0
\(895\) 0.846011 + 0.193096i 0.846011 + 0.193096i
\(896\) 0.781831 0.623490i 0.781831 0.623490i
\(897\) 0 0
\(898\) 0 0
\(899\) 0.626980 0.301938i 0.626980 0.301938i
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(908\) 0.781831 + 0.376510i 0.781831 + 0.376510i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(912\) 0 0
\(913\) 3.51352i 3.51352i
\(914\) 0.193096 + 0.400969i 0.193096 + 0.400969i
\(915\) 0 0
\(916\) 0 0
\(917\) 0.347948 1.52446i 0.347948 1.52446i
\(918\) 0 0
\(919\) 1.62349 0.781831i 1.62349 0.781831i 0.623490 0.781831i \(-0.285714\pi\)
1.00000 \(0\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.52446 0.347948i 1.52446 0.347948i
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0.347948 + 0.277479i 0.347948 + 0.277479i
\(927\) 0 0
\(928\) −0.277479 + 0.347948i −0.277479 + 0.347948i
\(929\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0.678448 + 0.541044i 0.678448 + 0.541044i
\(935\) 0 0
\(936\) 0 0
\(937\) 1.22252 0.974928i 1.22252 0.974928i 0.222521 0.974928i \(-0.428571\pi\)
1.00000 \(0\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.541044 + 0.678448i 0.541044 + 0.678448i 0.974928 0.222521i \(-0.0714286\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −0.974928 + 0.777479i −0.974928 + 0.777479i −0.974928 0.222521i \(-0.928571\pi\)
1.00000i \(0.5\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.433884 0.900969i \(-0.357143\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −1.44504 −1.44504
\(962\) 0 0
\(963\) 0 0
\(964\) 1.52446 1.21572i 1.52446 1.21572i
\(965\) −0.541044 2.37047i −0.541044 2.37047i
\(966\) 0 0
\(967\) −0.277479 + 1.21572i −0.277479 + 1.21572i 0.623490 + 0.781831i \(0.285714\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(968\) −0.974928 2.02446i −0.974928 2.02446i
\(969\) 0 0
\(970\) 1.05496 + 1.32288i 1.05496 + 1.32288i
\(971\) 0.347948 + 1.52446i 0.347948 + 1.52446i 0.781831 + 0.623490i \(0.214286\pi\)
−0.433884 + 0.900969i \(0.642857\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0.974928 0.777479i 0.974928 0.777479i
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.781831 0.623490i \(-0.785714\pi\)
0.781831 + 0.623490i \(0.214286\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0.433884 + 1.90097i 0.433884 + 1.90097i
\(981\) 0 0
\(982\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i
\(983\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(984\) 0 0
\(985\) −2.74698 2.19064i −2.74698 2.19064i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0.400969 1.75676i 0.400969 1.75676i −0.222521 0.974928i \(-0.571429\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(992\) 1.40881 0.678448i 1.40881 0.678448i
\(993\) 0 0
\(994\) 0 0
\(995\) −2.97247 + 0.678448i −2.97247 + 0.678448i
\(996\) 0 0
\(997\) 0 0 −0.433884 0.900969i \(-0.642857\pi\)
0.433884 + 0.900969i \(0.357143\pi\)
\(998\) 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 3528.1.dh.a.181.1 12
3.2 odd 2 inner 3528.1.dh.a.181.2 yes 12
8.5 even 2 inner 3528.1.dh.a.181.2 yes 12
24.5 odd 2 CM 3528.1.dh.a.181.1 12
49.13 odd 14 inner 3528.1.dh.a.1189.2 yes 12
147.62 even 14 inner 3528.1.dh.a.1189.1 yes 12
392.13 odd 14 inner 3528.1.dh.a.1189.1 yes 12
1176.797 even 14 inner 3528.1.dh.a.1189.2 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3528.1.dh.a.181.1 12 1.1 even 1 trivial
3528.1.dh.a.181.1 12 24.5 odd 2 CM
3528.1.dh.a.181.2 yes 12 3.2 odd 2 inner
3528.1.dh.a.181.2 yes 12 8.5 even 2 inner
3528.1.dh.a.1189.1 yes 12 147.62 even 14 inner
3528.1.dh.a.1189.1 yes 12 392.13 odd 14 inner
3528.1.dh.a.1189.2 yes 12 49.13 odd 14 inner
3528.1.dh.a.1189.2 yes 12 1176.797 even 14 inner