Properties

Label 2548.1.co.b.155.1
Level $2548$
Weight $1$
Character 2548.155
Analytic conductor $1.272$
Analytic rank $0$
Dimension $6$
Projective image $D_{7}$
CM discriminant -52
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2548,1,Mod(155,2548)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2548, base_ring=CyclotomicField(14))
 
chi = DirichletCharacter(H, H._module([7, 12, 7]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("2548.155");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2548 = 2^{2} \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 2548.co (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: \(1.27161765219\)
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: yes
Projective image: \(D_{7}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{7} - \cdots)\)

Embedding invariants

Embedding label 155.1
Root \(0.222521 + 0.974928i\) of defining polynomial
Character \(\chi\) \(=\) 2548.155
Dual form 2548.1.co.b.1611.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.623490 - 0.781831i) q^{2} +(-0.222521 + 0.974928i) q^{4} +(0.222521 + 0.974928i) q^{7} +(0.900969 - 0.433884i) q^{8} +(0.623490 - 0.781831i) q^{9} +O(q^{10})\) \(q+(-0.623490 - 0.781831i) q^{2} +(-0.222521 + 0.974928i) q^{4} +(0.222521 + 0.974928i) q^{7} +(0.900969 - 0.433884i) q^{8} +(0.623490 - 0.781831i) q^{9} +(1.12349 + 1.40881i) q^{11} +(0.623490 + 0.781831i) q^{13} +(0.623490 - 0.781831i) q^{14} +(-0.900969 - 0.433884i) q^{16} +(-0.277479 - 1.21572i) q^{17} -1.00000 q^{18} -1.24698 q^{19} +(0.400969 - 1.75676i) q^{22} +(0.623490 - 0.781831i) q^{25} +(0.222521 - 0.974928i) q^{26} -1.00000 q^{28} +(0.400969 + 1.75676i) q^{29} -1.24698 q^{31} +(0.222521 + 0.974928i) q^{32} +(-0.777479 + 0.974928i) q^{34} +(0.623490 + 0.781831i) q^{36} +(0.777479 + 0.974928i) q^{38} +(-1.62349 + 0.781831i) q^{44} +(1.12349 + 1.40881i) q^{47} +(-0.900969 + 0.433884i) q^{49} -1.00000 q^{50} +(-0.900969 + 0.433884i) q^{52} +(-0.445042 + 1.94986i) q^{53} +(0.623490 + 0.781831i) q^{56} +(1.12349 - 1.40881i) q^{58} +(-0.400969 - 0.193096i) q^{59} +(-0.445042 - 1.94986i) q^{61} +(0.777479 + 0.974928i) q^{62} +(0.900969 + 0.433884i) q^{63} +(0.623490 - 0.781831i) q^{64} +0.445042 q^{67} +1.24698 q^{68} +(0.277479 - 1.21572i) q^{71} +(0.222521 - 0.974928i) q^{72} +(0.277479 - 1.21572i) q^{76} +(-1.12349 + 1.40881i) q^{77} +(-0.222521 - 0.974928i) q^{81} +(0.277479 - 0.347948i) q^{83} +(1.62349 + 0.781831i) q^{88} +(-0.623490 + 0.781831i) q^{91} +(0.400969 - 1.75676i) q^{94} +(0.900969 + 0.433884i) q^{98} +1.80194 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + q^{2} - q^{4} + q^{7} + q^{8} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + q^{2} - q^{4} + q^{7} + q^{8} - q^{9} + 2 q^{11} - q^{13} - q^{14} - q^{16} - 2 q^{17} - 6 q^{18} + 2 q^{19} - 2 q^{22} - q^{25} + q^{26} - 6 q^{28} - 2 q^{29} + 2 q^{31} + q^{32} - 5 q^{34} - q^{36} + 5 q^{38} - 5 q^{44} + 2 q^{47} - q^{49} - 6 q^{50} - q^{52} - 2 q^{53} - q^{56} + 2 q^{58} + 2 q^{59} - 2 q^{61} + 5 q^{62} + q^{63} - q^{64} + 2 q^{67} - 2 q^{68} + 2 q^{71} + q^{72} + 2 q^{76} - 2 q^{77} - q^{81} + 2 q^{83} + 5 q^{88} + q^{91} - 2 q^{94} + q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2548\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(885\) \(1275\)
\(\chi(n)\) \(-1\) \(e\left(\frac{6}{7}\right)\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.623490 0.781831i −0.623490 0.781831i
\(3\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(4\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(5\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(6\) 0 0
\(7\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(8\) 0.900969 0.433884i 0.900969 0.433884i
\(9\) 0.623490 0.781831i 0.623490 0.781831i
\(10\) 0 0
\(11\) 1.12349 + 1.40881i 1.12349 + 1.40881i 0.900969 + 0.433884i \(0.142857\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(12\) 0 0
\(13\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(14\) 0.623490 0.781831i 0.623490 0.781831i
\(15\) 0 0
\(16\) −0.900969 0.433884i −0.900969 0.433884i
\(17\) −0.277479 1.21572i −0.277479 1.21572i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(18\) −1.00000 −1.00000
\(19\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.400969 1.75676i 0.400969 1.75676i
\(23\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(24\) 0 0
\(25\) 0.623490 0.781831i 0.623490 0.781831i
\(26\) 0.222521 0.974928i 0.222521 0.974928i
\(27\) 0 0
\(28\) −1.00000 −1.00000
\(29\) 0.400969 + 1.75676i 0.400969 + 1.75676i 0.623490 + 0.781831i \(0.285714\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(30\) 0 0
\(31\) −1.24698 −1.24698 −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(32\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(33\) 0 0
\(34\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(35\) 0 0
\(36\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(37\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(38\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(39\) 0 0
\(40\) 0 0
\(41\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(42\) 0 0
\(43\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(44\) −1.62349 + 0.781831i −1.62349 + 0.781831i
\(45\) 0 0
\(46\) 0 0
\(47\) 1.12349 + 1.40881i 1.12349 + 1.40881i 0.900969 + 0.433884i \(0.142857\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(48\) 0 0
\(49\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(50\) −1.00000 −1.00000
\(51\) 0 0
\(52\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(53\) −0.445042 + 1.94986i −0.445042 + 1.94986i −0.222521 + 0.974928i \(0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(57\) 0 0
\(58\) 1.12349 1.40881i 1.12349 1.40881i
\(59\) −0.400969 0.193096i −0.400969 0.193096i 0.222521 0.974928i \(-0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(60\) 0 0
\(61\) −0.445042 1.94986i −0.445042 1.94986i −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(62\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(63\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(64\) 0.623490 0.781831i 0.623490 0.781831i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(68\) 1.24698 1.24698
\(69\) 0 0
\(70\) 0 0
\(71\) 0.277479 1.21572i 0.277479 1.21572i −0.623490 0.781831i \(-0.714286\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(72\) 0.222521 0.974928i 0.222521 0.974928i
\(73\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0.277479 1.21572i 0.277479 1.21572i
\(77\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 0 0
\(81\) −0.222521 0.974928i −0.222521 0.974928i
\(82\) 0 0
\(83\) 0.277479 0.347948i 0.277479 0.347948i −0.623490 0.781831i \(-0.714286\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 1.62349 + 0.781831i 1.62349 + 0.781831i
\(89\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(90\) 0 0
\(91\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(92\) 0 0
\(93\) 0 0
\(94\) 0.400969 1.75676i 0.400969 1.75676i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(99\) 1.80194 1.80194
\(100\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(101\) 1.62349 0.781831i 1.62349 0.781831i 0.623490 0.781831i \(-0.285714\pi\)
1.00000 \(0\)
\(102\) 0 0
\(103\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(104\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(105\) 0 0
\(106\) 1.80194 0.867767i 1.80194 0.867767i
\(107\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(108\) 0 0
\(109\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.222521 0.974928i 0.222521 0.974928i
\(113\) −0.277479 + 0.347948i −0.277479 + 0.347948i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −1.80194 −1.80194
\(117\) 1.00000 1.00000
\(118\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i
\(119\) 1.12349 0.541044i 1.12349 0.541044i
\(120\) 0 0
\(121\) −0.500000 + 2.19064i −0.500000 + 2.19064i
\(122\) −1.24698 + 1.56366i −1.24698 + 1.56366i
\(123\) 0 0
\(124\) 0.277479 1.21572i 0.277479 1.21572i
\(125\) 0 0
\(126\) −0.222521 0.974928i −0.222521 0.974928i
\(127\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(128\) −1.00000 −1.00000
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(132\) 0 0
\(133\) −0.277479 1.21572i −0.277479 1.21572i
\(134\) −0.277479 0.347948i −0.277479 0.347948i
\(135\) 0 0
\(136\) −0.777479 0.974928i −0.777479 0.974928i
\(137\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(138\) 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
\(142\) −1.12349 + 0.541044i −1.12349 + 0.541044i
\(143\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(144\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(150\) 0 0
\(151\) 0.277479 1.21572i 0.277479 1.21572i −0.623490 0.781831i \(-0.714286\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(152\) −1.12349 + 0.541044i −1.12349 + 0.541044i
\(153\) −1.12349 0.541044i −1.12349 0.541044i
\(154\) 1.80194 1.80194
\(155\) 0 0
\(156\) 0 0
\(157\) 1.62349 + 0.781831i 1.62349 + 0.781831i 1.00000 \(0\)
0.623490 + 0.781831i \(0.285714\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(163\) −0.400969 0.193096i −0.400969 0.193096i 0.222521 0.974928i \(-0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) −0.445042 −0.445042
\(167\) −0.0990311 0.433884i −0.0990311 0.433884i 0.900969 0.433884i \(-0.142857\pi\)
−1.00000 \(\pi\)
\(168\) 0 0
\(169\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(170\) 0 0
\(171\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(172\) 0 0
\(173\) 0.0990311 0.433884i 0.0990311 0.433884i −0.900969 0.433884i \(-0.857143\pi\)
1.00000 \(0\)
\(174\) 0 0
\(175\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(176\) −0.400969 1.75676i −0.400969 1.75676i
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(180\) 0 0
\(181\) −0.277479 + 0.347948i −0.277479 + 0.347948i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(182\) 1.00000 1.00000
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 1.40097 1.75676i 1.40097 1.75676i
\(188\) −1.62349 + 0.781831i −1.62349 + 0.781831i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(192\) 0 0
\(193\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −0.222521 0.974928i −0.222521 0.974928i
\(197\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(198\) −1.12349 1.40881i −1.12349 1.40881i
\(199\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(200\) 0.222521 0.974928i 0.222521 0.974928i
\(201\) 0 0
\(202\) −1.62349 0.781831i −1.62349 0.781831i
\(203\) −1.62349 + 0.781831i −1.62349 + 0.781831i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −0.222521 0.974928i −0.222521 0.974928i
\(209\) −1.40097 1.75676i −1.40097 1.75676i
\(210\) 0 0
\(211\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(212\) −1.80194 0.867767i −1.80194 0.867767i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −0.277479 1.21572i −0.277479 1.21572i
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0.777479 0.974928i 0.777479 0.974928i
\(222\) 0 0
\(223\) −0.400969 + 1.75676i −0.400969 + 1.75676i 0.222521 + 0.974928i \(0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(224\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(225\) −0.222521 0.974928i −0.222521 0.974928i
\(226\) 0.445042 0.445042
\(227\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(228\) 0 0
\(229\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 1.12349 + 1.40881i 1.12349 + 1.40881i
\(233\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i 1.00000 \(0\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(234\) −0.623490 0.781831i −0.623490 0.781831i
\(235\) 0 0
\(236\) 0.277479 0.347948i 0.277479 0.347948i
\(237\) 0 0
\(238\) −1.12349 0.541044i −1.12349 0.541044i
\(239\) −0.400969 0.193096i −0.400969 0.193096i 0.222521 0.974928i \(-0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(240\) 0 0
\(241\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(242\) 2.02446 0.974928i 2.02446 0.974928i
\(243\) 0 0
\(244\) 2.00000 2.00000
\(245\) 0 0
\(246\) 0 0
\(247\) −0.777479 0.974928i −0.777479 0.974928i
\(248\) −1.12349 + 0.541044i −1.12349 + 0.541044i
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(252\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.623490 + 0.781831i 0.623490 + 0.781831i
\(257\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i 1.00000 \(0\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 1.62349 + 0.781831i 1.62349 + 0.781831i
\(262\) 0 0
\(263\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(267\) 0 0
\(268\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i
\(269\) −0.277479 + 0.347948i −0.277479 + 0.347948i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(270\) 0 0
\(271\) 0.445042 1.94986i 0.445042 1.94986i 0.222521 0.974928i \(-0.428571\pi\)
0.222521 0.974928i \(-0.428571\pi\)
\(272\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(273\) 0 0
\(274\) 0 0
\(275\) 1.80194 1.80194
\(276\) 0 0
\(277\) −0.277479 1.21572i −0.277479 1.21572i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(278\) 0 0
\(279\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(280\) 0 0
\(281\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(282\) 0 0
\(283\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(284\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(285\) 0 0
\(286\) 1.62349 0.781831i 1.62349 0.781831i
\(287\) 0 0
\(288\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(289\) −0.500000 + 0.240787i −0.500000 + 0.240787i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) −1.12349 + 0.541044i −1.12349 + 0.541044i
\(303\) 0 0
\(304\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(305\) 0 0
\(306\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(307\) −1.24698 1.56366i −1.24698 1.56366i −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 0.781831i \(-0.714286\pi\)
\(308\) −1.12349 1.40881i −1.12349 1.40881i
\(309\) 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
\(313\) −1.80194 −1.80194 −0.900969 0.433884i \(-0.857143\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(314\) −0.400969 1.75676i −0.400969 1.75676i
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(318\) 0 0
\(319\) −2.02446 + 2.53859i −2.02446 + 2.53859i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0.346011 + 1.51597i 0.346011 + 1.51597i
\(324\) 1.00000 1.00000
\(325\) 1.00000 1.00000
\(326\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i
\(327\) 0 0
\(328\) 0 0
\(329\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(330\) 0 0
\(331\) −0.400969 1.75676i −0.400969 1.75676i −0.623490 0.781831i \(-0.714286\pi\)
0.222521 0.974928i \(-0.428571\pi\)
\(332\) 0.277479 + 0.347948i 0.277479 + 0.347948i
\(333\) 0 0
\(334\) −0.277479 + 0.347948i −0.277479 + 0.347948i
\(335\) 0 0
\(336\) 0 0
\(337\) −1.12349 0.541044i −1.12349 0.541044i −0.222521 0.974928i \(-0.571429\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(338\) 0.900969 0.433884i 0.900969 0.433884i
\(339\) 0 0
\(340\) 0 0
\(341\) −1.40097 1.75676i −1.40097 1.75676i
\(342\) 1.24698 1.24698
\(343\) −0.623490 0.781831i −0.623490 0.781831i
\(344\) 0 0
\(345\) 0 0
\(346\) −0.400969 + 0.193096i −0.400969 + 0.193096i
\(347\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(348\) 0 0
\(349\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(350\) −0.222521 0.974928i −0.222521 0.974928i
\(351\) 0 0
\(352\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(353\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −1.62349 0.781831i −1.62349 0.781831i −0.623490 0.781831i \(-0.714286\pi\)
−1.00000 \(\pi\)
\(360\) 0 0
\(361\) 0.554958 0.554958
\(362\) 0.445042 0.445042
\(363\) 0 0
\(364\) −0.623490 0.781831i −0.623490 0.781831i
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −2.00000 −2.00000
\(372\) 0 0
\(373\) −0.445042 −0.445042 −0.222521 0.974928i \(-0.571429\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(374\) −2.24698 −2.24698
\(375\) 0 0
\(376\) 1.62349 + 0.781831i 1.62349 + 0.781831i
\(377\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(378\) 0 0
\(379\) 0.277479 + 0.347948i 0.277479 + 0.347948i 0.900969 0.433884i \(-0.142857\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0.277479 0.347948i 0.277479 0.347948i −0.623490 0.781831i \(-0.714286\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.12349 + 0.541044i −1.12349 + 0.541044i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(397\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.900969 + 0.433884i −0.900969 + 0.433884i
\(401\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(402\) 0 0
\(403\) −0.777479 0.974928i −0.777479 0.974928i
\(404\) 0.400969 + 1.75676i 0.400969 + 1.75676i
\(405\) 0 0
\(406\) 1.62349 + 0.781831i 1.62349 + 0.781831i
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.0990311 0.433884i 0.0990311 0.433884i
\(414\) 0 0
\(415\) 0 0
\(416\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(417\) 0 0
\(418\) −0.500000 + 2.19064i −0.500000 + 2.19064i
\(419\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(420\) 0 0
\(421\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(422\) 0 0
\(423\) 1.80194 1.80194
\(424\) 0.445042 + 1.94986i 0.445042 + 1.94986i
\(425\) −1.12349 0.541044i −1.12349 0.541044i
\(426\) 0 0
\(427\) 1.80194 0.867767i 1.80194 0.867767i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.12349 + 0.541044i 1.12349 + 0.541044i 0.900969 0.433884i \(-0.142857\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(432\) 0 0
\(433\) 0.400969 0.193096i 0.400969 0.193096i −0.222521 0.974928i \(-0.571429\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(434\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(435\) 0 0
\(436\) 0 0
\(437\) 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
\(441\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(442\) −1.24698 −1.24698
\(443\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 1.62349 0.781831i 1.62349 0.781831i
\(447\) 0 0
\(448\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(449\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(450\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(451\) 0 0
\(452\) −0.277479 0.347948i −0.277479 0.347948i
\(453\) 0 0
\(454\) −1.12349 1.40881i −1.12349 1.40881i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(462\) 0 0
\(463\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i 0.900969 + 0.433884i \(0.142857\pi\)
−1.00000 \(\pi\)
\(464\) 0.400969 1.75676i 0.400969 1.75676i
\(465\) 0 0
\(466\) 0.277479 0.347948i 0.277479 0.347948i
\(467\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(468\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(469\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i
\(470\) 0 0
\(471\) 0 0
\(472\) −0.445042 −0.445042
\(473\) 0 0
\(474\) 0 0
\(475\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(476\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(477\) 1.24698 + 1.56366i 1.24698 + 1.56366i
\(478\) 0.0990311 + 0.433884i 0.0990311 + 0.433884i
\(479\) −1.24698 1.56366i −1.24698 1.56366i −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 0.781831i \(-0.714286\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −2.02446 0.974928i −2.02446 0.974928i
\(485\) 0 0
\(486\) 0 0
\(487\) 1.12349 0.541044i 1.12349 0.541044i 0.222521 0.974928i \(-0.428571\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(488\) −1.24698 1.56366i −1.24698 1.56366i
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0 0
\(493\) 2.02446 0.974928i 2.02446 0.974928i
\(494\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(495\) 0 0
\(496\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(497\) 1.24698 1.24698
\(498\) 0 0
\(499\) 1.12349 1.40881i 1.12349 1.40881i 0.222521 0.974928i \(-0.428571\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(504\) 1.00000 1.00000
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.222521 0.974928i 0.222521 0.974928i
\(513\) 0 0
\(514\) 0.277479 0.347948i 0.277479 0.347948i
\(515\) 0 0
\(516\) 0 0
\(517\) −0.722521 + 3.16557i −0.722521 + 3.16557i
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(522\) −0.400969 1.75676i −0.400969 1.75676i
\(523\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0.346011 + 1.51597i 0.346011 + 1.51597i
\(528\) 0 0
\(529\) −0.900969 0.433884i −0.900969 0.433884i
\(530\) 0 0
\(531\) −0.400969 + 0.193096i −0.400969 + 0.193096i
\(532\) 1.24698 1.24698
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0.400969 0.193096i 0.400969 0.193096i
\(537\) 0 0
\(538\) 0.445042 0.445042
\(539\) −1.62349 0.781831i −1.62349 0.781831i
\(540\) 0 0
\(541\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(542\) −1.80194 + 0.867767i −1.80194 + 0.867767i
\(543\) 0 0
\(544\) 1.12349 0.541044i 1.12349 0.541044i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(548\) 0 0
\(549\) −1.80194 0.867767i −1.80194 0.867767i
\(550\) −1.12349 1.40881i −1.12349 1.40881i
\(551\) −0.500000 2.19064i −0.500000 2.19064i
\(552\) 0 0
\(553\) 0 0
\(554\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 1.24698 1.24698
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.900969 0.433884i 0.900969 0.433884i
\(568\) −0.277479 1.21572i −0.277479 1.21572i
\(569\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(570\) 0 0
\(571\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(572\) −1.62349 0.781831i −1.62349 0.781831i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.222521 0.974928i −0.222521 0.974928i
\(577\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(578\) 0.500000 + 0.240787i 0.500000 + 0.240787i
\(579\) 0 0
\(580\) 0 0
\(581\) 0.400969 + 0.193096i 0.400969 + 0.193096i
\(582\) 0 0
\(583\) −3.24698 + 1.56366i −3.24698 + 1.56366i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 1.55496 1.55496
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 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
\(601\) −1.12349 1.40881i −1.12349 1.40881i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(602\) 0 0
\(603\) 0.277479 0.347948i 0.277479 0.347948i
\(604\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) −0.277479 1.21572i −0.277479 1.21572i
\(609\) 0 0
\(610\) 0 0
\(611\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(612\) 0.777479 0.974928i 0.777479 0.974928i
\(613\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(614\) −0.445042 + 1.94986i −0.445042 + 1.94986i
\(615\) 0 0
\(616\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(617\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(618\) 0 0
\(619\) 0.445042 0.445042 0.222521 0.974928i \(-0.428571\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.222521 0.974928i −0.222521 0.974928i
\(626\) 1.12349 + 1.40881i 1.12349 + 1.40881i
\(627\) 0 0
\(628\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(629\) 0 0
\(630\) 0 0
\(631\) −0.400969 0.193096i −0.400969 0.193096i 0.222521 0.974928i \(-0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.900969 0.433884i −0.900969 0.433884i
\(638\) 3.24698 3.24698
\(639\) −0.777479 0.974928i −0.777479 0.974928i
\(640\) 0 0
\(641\) 0.400969 1.75676i 0.400969 1.75676i −0.222521 0.974928i \(-0.571429\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(642\) 0 0
\(643\) −0.400969 0.193096i −0.400969 0.193096i 0.222521 0.974928i \(-0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0.969501 1.21572i 0.969501 1.21572i
\(647\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(648\) −0.623490 0.781831i −0.623490 0.781831i
\(649\) −0.178448 0.781831i −0.178448 0.781831i
\(650\) −0.623490 0.781831i −0.623490 0.781831i
\(651\) 0 0
\(652\) 0.277479 0.347948i 0.277479 0.347948i
\(653\) 0.400969 + 0.193096i 0.400969 + 0.193096i 0.623490 0.781831i \(-0.285714\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0 0
\(658\) 1.80194 1.80194
\(659\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(660\) 0 0
\(661\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(662\) −1.12349 + 1.40881i −1.12349 + 1.40881i
\(663\) 0 0
\(664\) 0.0990311 0.433884i 0.0990311 0.433884i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0.445042 0.445042
\(669\) 0 0
\(670\) 0 0
\(671\) 2.24698 2.81762i 2.24698 2.81762i
\(672\) 0 0
\(673\) 0.777479 + 0.974928i 0.777479 + 0.974928i 1.00000 \(0\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(674\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(675\) 0 0
\(676\) −0.900969 0.433884i −0.900969 0.433884i
\(677\) −1.12349 + 1.40881i −1.12349 + 1.40881i −0.222521 + 0.974928i \(0.571429\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −0.500000 + 2.19064i −0.500000 + 2.19064i
\(683\) 1.12349 0.541044i 1.12349 0.541044i 0.222521 0.974928i \(-0.428571\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(684\) −0.777479 0.974928i −0.777479 0.974928i
\(685\) 0 0
\(686\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(687\) 0 0
\(688\) 0 0
\(689\) −1.80194 + 0.867767i −1.80194 + 0.867767i
\(690\) 0 0
\(691\) −1.62349 + 0.781831i −1.62349 + 0.781831i −0.623490 + 0.781831i \(0.714286\pi\)
−1.00000 \(1.00000\pi\)
\(692\) 0.400969 + 0.193096i 0.400969 + 0.193096i
\(693\) 0.400969 + 1.75676i 0.400969 + 1.75676i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.623490 + 0.781831i −0.623490 + 0.781831i
\(701\) 0.777479 0.974928i 0.777479 0.974928i −0.222521 0.974928i \(-0.571429\pi\)
1.00000 \(0\)
\(702\) 0 0
\(703\) 0 0
\(704\) 1.80194 1.80194
\(705\) 0 0
\(706\) 0 0
\(707\) 1.12349 + 1.40881i 1.12349 + 1.40881i
\(708\) 0 0
\(709\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0.400969 + 1.75676i 0.400969 + 1.75676i
\(719\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −0.346011 0.433884i −0.346011 0.433884i
\(723\) 0 0
\(724\) −0.277479 0.347948i −0.277479 0.347948i
\(725\) 1.62349 + 0.781831i 1.62349 + 0.781831i
\(726\) 0 0
\(727\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(728\) −0.222521 + 0.974928i −0.222521 + 0.974928i
\(729\) −0.900969 0.433884i −0.900969 0.433884i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0.500000 + 0.626980i 0.500000 + 0.626980i
\(738\) 0 0
\(739\) 0.277479 1.21572i 0.277479 1.21572i −0.623490 0.781831i \(-0.714286\pi\)
0.900969 0.433884i \(-0.142857\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 1.24698 + 1.56366i 1.24698 + 1.56366i
\(743\) −1.62349 + 0.781831i −1.62349 + 0.781831i −0.623490 + 0.781831i \(0.714286\pi\)
−1.00000 \(1.00000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.277479 + 0.347948i 0.277479 + 0.347948i
\(747\) −0.0990311 0.433884i −0.0990311 0.433884i
\(748\) 1.40097 + 1.75676i 1.40097 + 1.75676i
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(752\) −0.400969 1.75676i −0.400969 1.75676i
\(753\) 0 0
\(754\) 1.80194 1.80194
\(755\) 0 0
\(756\) 0 0
\(757\) 0.0990311 0.433884i 0.0990311 0.433884i −0.900969 0.433884i \(-0.857143\pi\)
1.00000 \(0\)
\(758\) 0.0990311 0.433884i 0.0990311 0.433884i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 0.222521 0.974928i \(-0.428571\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −0.445042 −0.445042
\(767\) −0.0990311 0.433884i −0.0990311 0.433884i
\(768\) 0 0
\(769\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(774\) 0 0
\(775\) −0.777479 + 0.974928i −0.777479 + 0.974928i
\(776\) 0 0
\(777\) 0 0
\(778\) 1.12349 + 0.541044i 1.12349 + 0.541044i
\(779\) 0 0
\(780\) 0 0
\(781\) 2.02446 0.974928i 2.02446 0.974928i
\(782\) 0 0
\(783\) 0 0
\(784\) 1.00000 1.00000
\(785\) 0 0
\(786\) 0 0
\(787\) −1.62349 + 0.781831i −1.62349 + 0.781831i −0.623490 + 0.781831i \(0.714286\pi\)
−1.00000 \(1.00000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) −0.400969 0.193096i −0.400969 0.193096i
\(792\) 1.62349 0.781831i 1.62349 0.781831i
\(793\) 1.24698 1.56366i 1.24698 1.56366i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) −1.12349 1.40881i −1.12349 1.40881i −0.900969 0.433884i \(-0.857143\pi\)
−0.222521 0.974928i \(-0.571429\pi\)
\(798\) 0 0
\(799\) 1.40097 1.75676i 1.40097 1.75676i
\(800\) 0.900969 + 0.433884i 0.900969 + 0.433884i
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) −0.277479 + 1.21572i −0.277479 + 1.21572i
\(807\) 0 0
\(808\) 1.12349 1.40881i 1.12349 1.40881i
\(809\) 1.24698 1.56366i 1.24698 1.56366i 0.623490 0.781831i \(-0.285714\pi\)
0.623490 0.781831i \(-0.285714\pi\)
\(810\) 0 0
\(811\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i 0.900969 + 0.433884i \(0.142857\pi\)
−1.00000 \(\pi\)
\(812\) −0.400969 1.75676i −0.400969 1.75676i
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0.222521 + 0.974928i 0.222521 + 0.974928i
\(820\) 0 0
\(821\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(822\) 0 0
\(823\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −0.400969 + 0.193096i −0.400969 + 0.193096i
\(827\) 1.12349 + 0.541044i 1.12349 + 0.541044i 0.900969 0.433884i \(-0.142857\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(828\) 0 0
\(829\) −0.277479 + 1.21572i −0.277479 + 1.21572i 0.623490 + 0.781831i \(0.285714\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.00000 1.00000
\(833\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(834\) 0 0
\(835\) 0 0
\(836\) 2.02446 0.974928i 2.02446 0.974928i
\(837\) 0 0
\(838\) 0 0
\(839\) −0.400969 0.193096i −0.400969 0.193096i 0.222521 0.974928i \(-0.428571\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(840\) 0 0
\(841\) −2.02446 + 0.974928i −2.02446 + 0.974928i
\(842\) 0 0
\(843\) 0 0
\(844\) 0 0
\(845\) 0 0
\(846\) −1.12349 1.40881i −1.12349 1.40881i
\(847\) −2.24698 −2.24698
\(848\) 1.24698 1.56366i 1.24698 1.56366i
\(849\) 0 0
\(850\) 0.277479 + 1.21572i 0.277479 + 1.21572i
\(851\) 0 0
\(852\) 0 0
\(853\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(854\) −1.80194 0.867767i −1.80194 0.867767i
\(855\) 0 0
\(856\) 0 0
\(857\) −0.277479 + 0.347948i −0.277479 + 0.347948i −0.900969 0.433884i \(-0.857143\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(858\) 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
\(862\) −0.277479 1.21572i −0.277479 1.21572i
\(863\) 1.80194 1.80194 0.900969 0.433884i \(-0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −0.400969 0.193096i −0.400969 0.193096i
\(867\) 0 0
\(868\) 1.24698 1.24698
\(869\) 0 0
\(870\) 0 0
\(871\) 0.277479 + 0.347948i 0.277479 + 0.347948i
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.24698 1.24698 0.623490 0.781831i \(-0.285714\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(882\) 0.900969 0.433884i 0.900969 0.433884i
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 0.777479 + 0.974928i 0.777479 + 0.974928i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 1.12349 1.40881i 1.12349 1.40881i
\(892\) −1.62349 0.781831i −1.62349 0.781831i
\(893\) −1.40097 1.75676i −1.40097 1.75676i
\(894\) 0 0
\(895\) 0 0
\(896\) −0.222521 0.974928i −0.222521 0.974928i
\(897\) 0 0
\(898\) 0 0
\(899\) −0.500000 2.19064i −0.500000 2.19064i
\(900\) 1.00000 1.00000
\(901\) 2.49396 2.49396
\(902\) 0 0
\(903\) 0 0
\(904\) −0.0990311 + 0.433884i −0.0990311 + 0.433884i
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(908\) −0.400969 + 1.75676i −0.400969 + 1.75676i
\(909\) 0.400969 1.75676i 0.400969 1.75676i
\(910\) 0 0
\(911\) 0 0 −0.222521 0.974928i \(-0.571429\pi\)
0.222521 + 0.974928i \(0.428571\pi\)
\(912\) 0 0
\(913\) 0.801938 0.801938
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 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
\(922\) 0 0
\(923\) 1.12349 0.541044i 1.12349 0.541044i
\(924\) 0 0
\(925\) 0 0
\(926\) 0.400969 0.193096i 0.400969 0.193096i
\(927\) 0 0
\(928\) −1.62349 + 0.781831i −1.62349 + 0.781831i
\(929\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(930\) 0 0
\(931\) 1.12349 0.541044i 1.12349 0.541044i
\(932\) −0.445042 −0.445042
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0.900969 0.433884i 0.900969 0.433884i
\(937\) 0.400969 + 0.193096i 0.400969 + 0.193096i 0.623490 0.781831i \(-0.285714\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(938\) 0.277479 0.347948i 0.277479 0.347948i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.900969 0.433884i \(-0.857143\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0.277479 + 0.347948i 0.277479 + 0.347948i
\(945\) 0 0
\(946\) 0 0
\(947\) 1.80194 + 0.867767i 1.80194 + 0.867767i 0.900969 + 0.433884i \(0.142857\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 1.24698 1.24698
\(951\) 0 0
\(952\) 0.777479 0.974928i 0.777479 0.974928i
\(953\) −0.277479 + 1.21572i −0.277479 + 1.21572i 0.623490 + 0.781831i \(0.285714\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(954\) 0.445042 1.94986i 0.445042 1.94986i
\(955\) 0 0
\(956\) 0.277479 0.347948i 0.277479 0.347948i
\(957\) 0 0
\(958\) −0.445042 + 1.94986i −0.445042 + 1.94986i
\(959\) 0 0
\(960\) 0 0
\(961\) 0.554958 0.554958
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) −1.24698 1.56366i −1.24698 1.56366i −0.623490 0.781831i \(-0.714286\pi\)
−0.623490 0.781831i \(-0.714286\pi\)
\(968\) 0.500000 + 2.19064i 0.500000 + 2.19064i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.623490 0.781831i \(-0.285714\pi\)
−0.623490 + 0.781831i \(0.714286\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) −1.12349 0.541044i −1.12349 0.541044i
\(975\) 0 0
\(976\) −0.445042 + 1.94986i −0.445042 + 1.94986i
\(977\) 0 0 0.900969 0.433884i \(-0.142857\pi\)
−0.900969 + 0.433884i \(0.857143\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 1.12349 0.541044i 1.12349 0.541044i 0.222521 0.974928i \(-0.428571\pi\)
0.900969 + 0.433884i \(0.142857\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −2.02446 0.974928i −2.02446 0.974928i
\(987\) 0 0
\(988\) 1.12349 0.541044i 1.12349 0.541044i
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.623490 0.781831i \(-0.714286\pi\)
0.623490 + 0.781831i \(0.285714\pi\)
\(992\) −0.277479 1.21572i −0.277479 1.21572i
\(993\) 0 0
\(994\) −0.777479 0.974928i −0.777479 0.974928i
\(995\) 0 0
\(996\) 0 0
\(997\) 0.400969 + 1.75676i 0.400969 + 1.75676i 0.623490 + 0.781831i \(0.285714\pi\)
−0.222521 + 0.974928i \(0.571429\pi\)
\(998\) −1.80194 −1.80194
\(999\) 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 2548.1.co.b.155.1 yes 6
4.3 odd 2 2548.1.co.a.155.1 6
13.12 even 2 2548.1.co.a.155.1 6
49.43 even 7 inner 2548.1.co.b.1611.1 yes 6
52.51 odd 2 CM 2548.1.co.b.155.1 yes 6
196.43 odd 14 2548.1.co.a.1611.1 yes 6
637.337 even 14 2548.1.co.a.1611.1 yes 6
2548.1611 odd 14 inner 2548.1.co.b.1611.1 yes 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2548.1.co.a.155.1 6 4.3 odd 2
2548.1.co.a.155.1 6 13.12 even 2
2548.1.co.a.1611.1 yes 6 196.43 odd 14
2548.1.co.a.1611.1 yes 6 637.337 even 14
2548.1.co.b.155.1 yes 6 1.1 even 1 trivial
2548.1.co.b.155.1 yes 6 52.51 odd 2 CM
2548.1.co.b.1611.1 yes 6 49.43 even 7 inner
2548.1.co.b.1611.1 yes 6 2548.1611 odd 14 inner