Properties

Label 1911.1.w.e.1598.4
Level $1911$
Weight $1$
Character 1911.1598
Analytic conductor $0.954$
Analytic rank $0$
Dimension $8$
Projective image $D_{8}$
CM discriminant -39
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1911,1,Mod(116,1911)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1911, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1911.116");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1911 = 3 \cdot 7^{2} \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1911.w (of order \(6\), degree \(2\), 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: \(0.953713239142\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: 8.0.339738624.2
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} + 4x^{6} + 14x^{4} + 8x^{2} + 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{8}\)
Projective field: Galois closure of 8.2.90724673403.2

Embedding invariants

Embedding label 1598.4
Root \(-0.382683 + 0.662827i\) of defining polynomial
Character \(\chi\) \(=\) 1911.1598
Dual form 1911.1.w.e.116.4

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.923880 - 1.60021i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-1.20711 - 2.09077i) q^{4} +(-0.382683 + 0.662827i) q^{5} -1.84776 q^{6} -2.61313 q^{8} +(-0.500000 + 0.866025i) q^{9} +O(q^{10})\) \(q+(0.923880 - 1.60021i) q^{2} +(-0.500000 - 0.866025i) q^{3} +(-1.20711 - 2.09077i) q^{4} +(-0.382683 + 0.662827i) q^{5} -1.84776 q^{6} -2.61313 q^{8} +(-0.500000 + 0.866025i) q^{9} +(0.707107 + 1.22474i) q^{10} +(-0.923880 - 1.60021i) q^{11} +(-1.20711 + 2.09077i) q^{12} -1.00000 q^{13} +0.765367 q^{15} +(-1.20711 + 2.09077i) q^{16} +(0.923880 + 1.60021i) q^{18} +1.84776 q^{20} -3.41421 q^{22} +(1.30656 + 2.26303i) q^{24} +(0.207107 + 0.358719i) q^{25} +(-0.923880 + 1.60021i) q^{26} +1.00000 q^{27} +(0.707107 - 1.22474i) q^{30} +(0.923880 + 1.60021i) q^{32} +(-0.923880 + 1.60021i) q^{33} +2.41421 q^{36} +(0.500000 + 0.866025i) q^{39} +(1.00000 - 1.73205i) q^{40} -0.765367 q^{41} +(-2.23044 + 3.86324i) q^{44} +(-0.382683 - 0.662827i) q^{45} +(0.382683 - 0.662827i) q^{47} +2.41421 q^{48} +0.765367 q^{50} +(1.20711 + 2.09077i) q^{52} +(0.923880 - 1.60021i) q^{54} +1.41421 q^{55} +(-0.923880 - 1.60021i) q^{59} +(-0.923880 - 1.60021i) q^{60} +1.00000 q^{64} +(0.382683 - 0.662827i) q^{65} +(1.70711 + 2.95680i) q^{66} -0.765367 q^{71} +(1.30656 - 2.26303i) q^{72} +(0.207107 - 0.358719i) q^{75} +1.84776 q^{78} +(0.707107 - 1.22474i) q^{79} +(-0.923880 - 1.60021i) q^{80} +(-0.500000 - 0.866025i) q^{81} +(-0.707107 + 1.22474i) q^{82} -1.84776 q^{83} +(2.41421 + 4.18154i) q^{88} +(0.923880 - 1.60021i) q^{89} -1.41421 q^{90} +(-0.707107 - 1.22474i) q^{94} +(0.923880 - 1.60021i) q^{96} +1.84776 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q - 4 q^{3} - 4 q^{4} - 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{3} - 4 q^{4} - 4 q^{9} - 4 q^{12} - 8 q^{13} - 4 q^{16} - 16 q^{22} - 4 q^{25} + 8 q^{27} + 8 q^{36} + 4 q^{39} + 8 q^{40} + 8 q^{48} + 4 q^{52} + 8 q^{64} + 8 q^{66} - 4 q^{75} - 4 q^{81} + 8 q^{88}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1911\mathbb{Z}\right)^\times\).

\(n\) \(638\) \(1471\) \(1522\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(3\) −0.500000 0.866025i −0.500000 0.866025i
\(4\) −1.20711 2.09077i −1.20711 2.09077i
\(5\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(6\) −1.84776 −1.84776
\(7\) 0 0
\(8\) −2.61313 −2.61313
\(9\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(10\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(11\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(12\) −1.20711 + 2.09077i −1.20711 + 2.09077i
\(13\) −1.00000 −1.00000
\(14\) 0 0
\(15\) 0.765367 0.765367
\(16\) −1.20711 + 2.09077i −1.20711 + 2.09077i
\(17\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(18\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(19\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(20\) 1.84776 1.84776
\(21\) 0 0
\(22\) −3.41421 −3.41421
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 1.30656 + 2.26303i 1.30656 + 2.26303i
\(25\) 0.207107 + 0.358719i 0.207107 + 0.358719i
\(26\) −0.923880 + 1.60021i −0.923880 + 1.60021i
\(27\) 1.00000 1.00000
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0.707107 1.22474i 0.707107 1.22474i
\(31\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(32\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(33\) −0.923880 + 1.60021i −0.923880 + 1.60021i
\(34\) 0 0
\(35\) 0 0
\(36\) 2.41421 2.41421
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0 0
\(39\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(40\) 1.00000 1.73205i 1.00000 1.73205i
\(41\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(42\) 0 0
\(43\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(44\) −2.23044 + 3.86324i −2.23044 + 3.86324i
\(45\) −0.382683 0.662827i −0.382683 0.662827i
\(46\) 0 0
\(47\) 0.382683 0.662827i 0.382683 0.662827i −0.608761 0.793353i \(-0.708333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(48\) 2.41421 2.41421
\(49\) 0 0
\(50\) 0.765367 0.765367
\(51\) 0 0
\(52\) 1.20711 + 2.09077i 1.20711 + 2.09077i
\(53\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) 0.923880 1.60021i 0.923880 1.60021i
\(55\) 1.41421 1.41421
\(56\) 0 0
\(57\) 0 0
\(58\) 0 0
\(59\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(60\) −0.923880 1.60021i −0.923880 1.60021i
\(61\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 1.00000 1.00000
\(65\) 0.382683 0.662827i 0.382683 0.662827i
\(66\) 1.70711 + 2.95680i 1.70711 + 2.95680i
\(67\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(72\) 1.30656 2.26303i 1.30656 2.26303i
\(73\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(74\) 0 0
\(75\) 0.207107 0.358719i 0.207107 0.358719i
\(76\) 0 0
\(77\) 0 0
\(78\) 1.84776 1.84776
\(79\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(80\) −0.923880 1.60021i −0.923880 1.60021i
\(81\) −0.500000 0.866025i −0.500000 0.866025i
\(82\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(83\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 2.41421 + 4.18154i 2.41421 + 4.18154i
\(89\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(90\) −1.41421 −1.41421
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −0.707107 1.22474i −0.707107 1.22474i
\(95\) 0 0
\(96\) 0.923880 1.60021i 0.923880 1.60021i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) 0 0
\(99\) 1.84776 1.84776
\(100\) 0.500000 0.866025i 0.500000 0.866025i
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(104\) 2.61313 2.61313
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) −1.20711 2.09077i −1.20711 2.09077i
\(109\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(110\) 1.30656 2.26303i 1.30656 2.26303i
\(111\) 0 0
\(112\) 0 0
\(113\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.500000 0.866025i 0.500000 0.866025i
\(118\) −3.41421 −3.41421
\(119\) 0 0
\(120\) −2.00000 −2.00000
\(121\) −1.20711 + 2.09077i −1.20711 + 2.09077i
\(122\) 0 0
\(123\) 0.382683 + 0.662827i 0.382683 + 0.662827i
\(124\) 0 0
\(125\) −1.08239 −1.08239
\(126\) 0 0
\(127\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) −0.707107 1.22474i −0.707107 1.22474i
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) 4.46088 4.46088
\(133\) 0 0
\(134\) 0 0
\(135\) −0.382683 + 0.662827i −0.382683 + 0.662827i
\(136\) 0 0
\(137\) −0.382683 0.662827i −0.382683 0.662827i 0.608761 0.793353i \(-0.291667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(138\) 0 0
\(139\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(140\) 0 0
\(141\) −0.765367 −0.765367
\(142\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(143\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(144\) −1.20711 2.09077i −1.20711 2.09077i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(150\) −0.382683 0.662827i −0.382683 0.662827i
\(151\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 1.20711 2.09077i 1.20711 2.09077i
\(157\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(158\) −1.30656 2.26303i −1.30656 2.26303i
\(159\) 0 0
\(160\) −1.41421 −1.41421
\(161\) 0 0
\(162\) −1.84776 −1.84776
\(163\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(164\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(165\) −0.707107 1.22474i −0.707107 1.22474i
\(166\) −1.70711 + 2.95680i −1.70711 + 2.95680i
\(167\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.46088 4.46088
\(177\) −0.923880 + 1.60021i −0.923880 + 1.60021i
\(178\) −1.70711 2.95680i −1.70711 2.95680i
\(179\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) −0.923880 + 1.60021i −0.923880 + 1.60021i
\(181\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −1.84776 −1.84776
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) −0.500000 0.866025i −0.500000 0.866025i
\(193\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(194\) 0 0
\(195\) −0.765367 −0.765367
\(196\) 0 0
\(197\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(198\) 1.70711 2.95680i 1.70711 2.95680i
\(199\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(200\) −0.541196 0.937379i −0.541196 0.937379i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0.292893 0.507306i 0.292893 0.507306i
\(206\) −1.30656 2.26303i −1.30656 2.26303i
\(207\) 0 0
\(208\) 1.20711 2.09077i 1.20711 2.09077i
\(209\) 0 0
\(210\) 0 0
\(211\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(212\) 0 0
\(213\) 0.382683 + 0.662827i 0.382683 + 0.662827i
\(214\) 0 0
\(215\) 0 0
\(216\) −2.61313 −2.61313
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) −1.70711 2.95680i −1.70711 2.95680i
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(224\) 0 0
\(225\) −0.414214 −0.414214
\(226\) 0 0
\(227\) 0.382683 + 0.662827i 0.382683 + 0.662827i 0.991445 0.130526i \(-0.0416667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(228\) 0 0
\(229\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(234\) −0.923880 1.60021i −0.923880 1.60021i
\(235\) 0.292893 + 0.507306i 0.292893 + 0.507306i
\(236\) −2.23044 + 3.86324i −2.23044 + 3.86324i
\(237\) −1.41421 −1.41421
\(238\) 0 0
\(239\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(240\) −0.923880 + 1.60021i −0.923880 + 1.60021i
\(241\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(242\) 2.23044 + 3.86324i 2.23044 + 3.86324i
\(243\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(244\) 0 0
\(245\) 0 0
\(246\) 1.41421 1.41421
\(247\) 0 0
\(248\) 0 0
\(249\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(250\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 1.30656 2.26303i 1.30656 2.26303i
\(255\) 0 0
\(256\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(257\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.84776 −1.84776
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 2.41421 4.18154i 2.41421 4.18154i
\(265\) 0 0
\(266\) 0 0
\(267\) −1.84776 −1.84776
\(268\) 0 0
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(271\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) −1.41421 −1.41421
\(275\) 0.382683 0.662827i 0.382683 0.662827i
\(276\) 0 0
\(277\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(278\) 1.30656 2.26303i 1.30656 2.26303i
\(279\) 0 0
\(280\) 0 0
\(281\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(282\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(283\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(284\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(285\) 0 0
\(286\) 3.41421 3.41421
\(287\) 0 0
\(288\) −1.84776 −1.84776
\(289\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(294\) 0 0
\(295\) 1.41421 1.41421
\(296\) 0 0
\(297\) −0.923880 1.60021i −0.923880 1.60021i
\(298\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(299\) 0 0
\(300\) −1.00000 −1.00000
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) −1.41421 −1.41421
\(310\) 0 0
\(311\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(312\) −1.30656 2.26303i −1.30656 2.26303i
\(313\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(314\) −2.61313 −2.61313
\(315\) 0 0
\(316\) −3.41421 −3.41421
\(317\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.382683 + 0.662827i −0.382683 + 0.662827i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −1.20711 + 2.09077i −1.20711 + 2.09077i
\(325\) −0.207107 0.358719i −0.207107 0.358719i
\(326\) 0 0
\(327\) 0 0
\(328\) 2.00000 2.00000
\(329\) 0 0
\(330\) −2.61313 −2.61313
\(331\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 2.23044 + 3.86324i 2.23044 + 3.86324i
\(333\) 0 0
\(334\) 1.70711 2.95680i 1.70711 2.95680i
\(335\) 0 0
\(336\) 0 0
\(337\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(338\) 0.923880 1.60021i 0.923880 1.60021i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) −1.00000 −1.00000
\(352\) 1.70711 2.95680i 1.70711 2.95680i
\(353\) 0.923880 + 1.60021i 0.923880 + 1.60021i 0.793353 + 0.608761i \(0.208333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(354\) 1.70711 + 2.95680i 1.70711 + 2.95680i
\(355\) 0.292893 0.507306i 0.292893 0.507306i
\(356\) −4.46088 −4.46088
\(357\) 0 0
\(358\) 0 0
\(359\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(360\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(361\) −0.500000 0.866025i −0.500000 0.866025i
\(362\) −1.30656 + 2.26303i −1.30656 + 2.26303i
\(363\) 2.41421 2.41421
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(368\) 0 0
\(369\) 0.382683 0.662827i 0.382683 0.662827i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(374\) 0 0
\(375\) 0.541196 + 0.937379i 0.541196 + 0.937379i
\(376\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(377\) 0 0
\(378\) 0 0
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) −0.707107 1.22474i −0.707107 1.22474i
\(382\) 0 0
\(383\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(390\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 1.70711 2.95680i 1.70711 2.95680i
\(395\) 0.541196 + 0.937379i 0.541196 + 0.937379i
\(396\) −2.23044 3.86324i −2.23044 3.86324i
\(397\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(398\) 3.69552 3.69552
\(399\) 0 0
\(400\) −1.00000 −1.00000
\(401\) −0.923880 + 1.60021i −0.923880 + 1.60021i −0.130526 + 0.991445i \(0.541667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0.765367 0.765367
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(410\) −0.541196 0.937379i −0.541196 0.937379i
\(411\) −0.382683 + 0.662827i −0.382683 + 0.662827i
\(412\) −3.41421 −3.41421
\(413\) 0 0
\(414\) 0 0
\(415\) 0.707107 1.22474i 0.707107 1.22474i
\(416\) −0.923880 1.60021i −0.923880 1.60021i
\(417\) −0.707107 1.22474i −0.707107 1.22474i
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −1.30656 + 2.26303i −1.30656 + 2.26303i
\(423\) 0.382683 + 0.662827i 0.382683 + 0.662827i
\(424\) 0 0
\(425\) 0 0
\(426\) 1.41421 1.41421
\(427\) 0 0
\(428\) 0 0
\(429\) 0.923880 1.60021i 0.923880 1.60021i
\(430\) 0 0
\(431\) 0.382683 + 0.662827i 0.382683 + 0.662827i 0.991445 0.130526i \(-0.0416667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(432\) −1.20711 + 2.09077i −1.20711 + 2.09077i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(440\) −3.69552 −3.69552
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(444\) 0 0
\(445\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(446\) 0 0
\(447\) 0.765367 0.765367
\(448\) 0 0
\(449\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(450\) −0.382683 + 0.662827i −0.382683 + 0.662827i
\(451\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(452\) 0 0
\(453\) 0 0
\(454\) 1.41421 1.41421
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(468\) −2.41421 −2.41421
\(469\) 0 0
\(470\) 1.08239 1.08239
\(471\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(472\) 2.41421 + 4.18154i 2.41421 + 4.18154i
\(473\) 0 0
\(474\) −1.30656 + 2.26303i −1.30656 + 2.26303i
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) −1.70711 + 2.95680i −1.70711 + 2.95680i
\(479\) −0.382683 0.662827i −0.382683 0.662827i 0.608761 0.793353i \(-0.291667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(480\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 5.82843 5.82843
\(485\) 0 0
\(486\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(487\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(492\) 0.923880 1.60021i 0.923880 1.60021i
\(493\) 0 0
\(494\) 0 0
\(495\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(496\) 0 0
\(497\) 0 0
\(498\) 3.41421 3.41421
\(499\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) 1.30656 + 2.26303i 1.30656 + 2.26303i
\(501\) −0.923880 1.60021i −0.923880 1.60021i
\(502\) 0 0
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.500000 0.866025i −0.500000 0.866025i
\(508\) −1.70711 2.95680i −1.70711 2.95680i
\(509\) 0.382683 0.662827i 0.382683 0.662827i −0.608761 0.793353i \(-0.708333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.84776 1.84776
\(513\) 0 0
\(514\) 0 0
\(515\) 0.541196 + 0.937379i 0.541196 + 0.937379i
\(516\) 0 0
\(517\) −1.41421 −1.41421
\(518\) 0 0
\(519\) 0 0
\(520\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(521\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) −2.23044 3.86324i −2.23044 3.86324i
\(529\) −0.500000 0.866025i −0.500000 0.866025i
\(530\) 0 0
\(531\) 1.84776 1.84776
\(532\) 0 0
\(533\) 0.765367 0.765367
\(534\) −1.70711 + 2.95680i −1.70711 + 2.95680i
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 1.84776 1.84776
\(541\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(542\) 0 0
\(543\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(548\) −0.923880 + 1.60021i −0.923880 + 1.60021i
\(549\) 0 0
\(550\) −0.707107 1.22474i −0.707107 1.22474i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) −1.70711 2.95680i −1.70711 2.95680i
\(557\) 0.382683 + 0.662827i 0.382683 + 0.662827i 0.991445 0.130526i \(-0.0416667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.707107 1.22474i 0.707107 1.22474i
\(563\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(564\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(565\) 0 0
\(566\) 0 0
\(567\) 0 0
\(568\) 2.00000 2.00000
\(569\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(570\) 0 0
\(571\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(572\) 2.23044 3.86324i 2.23044 3.86324i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(577\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(578\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0.382683 + 0.662827i 0.382683 + 0.662827i
\(586\) 1.70711 2.95680i 1.70711 2.95680i
\(587\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 1.30656 2.26303i 1.30656 2.26303i
\(591\) −0.923880 1.60021i −0.923880 1.60021i
\(592\) 0 0
\(593\) 0.923880 1.60021i 0.923880 1.60021i 0.130526 0.991445i \(-0.458333\pi\)
0.793353 0.608761i \(-0.208333\pi\)
\(594\) −3.41421 −3.41421
\(595\) 0 0
\(596\) 1.84776 1.84776
\(597\) 1.00000 1.73205i 1.00000 1.73205i
\(598\) 0 0
\(599\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(600\) −0.541196 + 0.937379i −0.541196 + 0.937379i
\(601\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.923880 1.60021i −0.923880 1.60021i
\(606\) 0 0
\(607\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) −0.382683 + 0.662827i −0.382683 + 0.662827i
\(612\) 0 0
\(613\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(614\) 0 0
\(615\) −0.585786 −0.585786
\(616\) 0 0
\(617\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(618\) −1.30656 + 2.26303i −1.30656 + 2.26303i
\(619\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −2.41421 −2.41421
\(625\) 0.207107 0.358719i 0.207107 0.358719i
\(626\) 0 0
\(627\) 0 0
\(628\) −1.70711 + 2.95680i −1.70711 + 2.95680i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) −1.84776 + 3.20041i −1.84776 + 3.20041i
\(633\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(634\) −1.70711 2.95680i −1.70711 2.95680i
\(635\) −0.541196 + 0.937379i −0.541196 + 0.937379i
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0.382683 0.662827i 0.382683 0.662827i
\(640\) 0 0
\(641\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(648\) 1.30656 + 2.26303i 1.30656 + 2.26303i
\(649\) −1.70711 + 2.95680i −1.70711 + 2.95680i
\(650\) −0.765367 −0.765367
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0.923880 1.60021i 0.923880 1.60021i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) −1.70711 + 2.95680i −1.70711 + 2.95680i
\(661\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 4.82843 4.82843
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) −2.23044 3.86324i −2.23044 3.86324i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(674\) 1.30656 2.26303i 1.30656 2.26303i
\(675\) 0.207107 + 0.358719i 0.207107 + 0.358719i
\(676\) −1.20711 2.09077i −1.20711 2.09077i
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0.382683 0.662827i 0.382683 0.662827i
\(682\) 0 0
\(683\) −0.382683 0.662827i −0.382683 0.662827i 0.608761 0.793353i \(-0.291667\pi\)
−0.991445 + 0.130526i \(0.958333\pi\)
\(684\) 0 0
\(685\) 0.585786 0.585786
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −0.541196 + 0.937379i −0.541196 + 0.937379i
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) −0.923880 + 1.60021i −0.923880 + 1.60021i
\(703\) 0 0
\(704\) −0.923880 1.60021i −0.923880 1.60021i
\(705\) 0.292893 0.507306i 0.292893 0.507306i
\(706\) 3.41421 3.41421
\(707\) 0 0
\(708\) 4.46088 4.46088
\(709\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) −0.541196 0.937379i −0.541196 0.937379i
\(711\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(712\) −2.41421 + 4.18154i −2.41421 + 4.18154i
\(713\) 0 0
\(714\) 0 0
\(715\) −1.41421 −1.41421
\(716\) 0 0
\(717\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(718\) −1.70711 2.95680i −1.70711 2.95680i
\(719\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 1.84776 1.84776
\(721\) 0 0
\(722\) −1.84776 −1.84776
\(723\) 0 0
\(724\) 1.70711 + 2.95680i 1.70711 + 2.95680i
\(725\) 0 0
\(726\) 2.23044 3.86324i 2.23044 3.86324i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 1.00000 1.00000
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(734\) 2.61313 2.61313
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −0.707107 1.22474i −0.707107 1.22474i
\(739\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(744\) 0 0
\(745\) −0.292893 0.507306i −0.292893 0.507306i
\(746\) −1.30656 2.26303i −1.30656 2.26303i
\(747\) 0.923880 1.60021i 0.923880 1.60021i
\(748\) 0 0
\(749\) 0 0
\(750\) 2.00000 2.00000
\(751\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(752\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.382683 0.662827i 0.382683 0.662827i −0.608761 0.793353i \(-0.708333\pi\)
0.991445 + 0.130526i \(0.0416667\pi\)
\(762\) −2.61313 −2.61313
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(767\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(768\) 0.500000 0.866025i 0.500000 0.866025i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0.923880 + 1.60021i 0.923880 + 1.60021i 0.793353 + 0.608761i \(0.208333\pi\)
0.130526 + 0.991445i \(0.458333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(781\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) 1.08239 1.08239
\(786\) 0 0
\(787\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(788\) −2.23044 3.86324i −2.23044 3.86324i
\(789\) 0 0
\(790\) 2.00000 2.00000
\(791\) 0 0
\(792\) −4.82843 −4.82843
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 2.41421 4.18154i 2.41421 4.18154i
\(797\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.382683 + 0.662827i −0.382683 + 0.662827i
\(801\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(802\) 1.70711 + 2.95680i 1.70711 + 2.95680i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(810\) 0.707107 1.22474i 0.707107 1.22474i
\(811\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −1.41421 −1.41421
\(821\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(822\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(823\) 1.00000 + 1.73205i 1.00000 + 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(824\) −1.84776 + 3.20041i −1.84776 + 3.20041i
\(825\) −0.765367 −0.765367
\(826\) 0 0
\(827\) 0.765367 0.765367 0.382683 0.923880i \(-0.375000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(828\) 0 0
\(829\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(830\) −1.30656 2.26303i −1.30656 2.26303i
\(831\) 0 0
\(832\) −1.00000 −1.00000
\(833\) 0 0
\(834\) −2.61313 −2.61313
\(835\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(840\) 0 0
\(841\) 1.00000 1.00000
\(842\) 0 0
\(843\) −0.382683 0.662827i −0.382683 0.662827i
\(844\) 1.70711 + 2.95680i 1.70711 + 2.95680i
\(845\) −0.382683 + 0.662827i −0.382683 + 0.662827i
\(846\) 1.41421 1.41421
\(847\) 0 0
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0.923880 1.60021i 0.923880 1.60021i
\(853\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(858\) −1.70711 2.95680i −1.70711 2.95680i
\(859\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.41421 1.41421
\(863\) −0.923880 + 1.60021i −0.923880 + 1.60021i −0.130526 + 0.991445i \(0.541667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(864\) 0.923880 + 1.60021i 0.923880 + 1.60021i
\(865\) 0 0
\(866\) 0 0
\(867\) 1.00000 1.00000
\(868\) 0 0
\(869\) −2.61313 −2.61313
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(878\) −1.30656 2.26303i −1.30656 2.26303i
\(879\) −0.923880 1.60021i −0.923880 1.60021i
\(880\) −1.70711 + 2.95680i −1.70711 + 2.95680i
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(884\) 0 0
\(885\) −0.707107 1.22474i −0.707107 1.22474i
\(886\) 0 0
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 2.61313 2.61313
\(891\) −0.923880 + 1.60021i −0.923880 + 1.60021i
\(892\) 0 0
\(893\) 0 0
\(894\) 0.707107 1.22474i 0.707107 1.22474i
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −1.70711 + 2.95680i −1.70711 + 2.95680i
\(899\) 0 0
\(900\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(901\) 0 0
\(902\) 2.61313 2.61313
\(903\) 0 0
\(904\) 0 0
\(905\) 0.541196 0.937379i 0.541196 0.937379i
\(906\) 0 0
\(907\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(908\) 0.923880 1.60021i 0.923880 1.60021i
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 1.70711 + 2.95680i 1.70711 + 2.95680i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 1.70711 2.95680i 1.70711 2.95680i
\(923\) 0.765367 0.765367
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(928\) 0 0
\(929\) −0.923880 + 1.60021i −0.923880 + 1.60021i −0.130526 + 0.991445i \(0.541667\pi\)
−0.793353 + 0.608761i \(0.791667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −1.30656 + 2.26303i −1.30656 + 2.26303i
\(937\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0.707107 1.22474i 0.707107 1.22474i
\(941\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(942\) 1.30656 + 2.26303i 1.30656 + 2.26303i
\(943\) 0 0
\(944\) 4.46088 4.46088
\(945\) 0 0
\(946\) 0 0
\(947\) −0.382683 + 0.662827i −0.382683 + 0.662827i −0.991445 0.130526i \(-0.958333\pi\)
0.608761 + 0.793353i \(0.291667\pi\)
\(948\) 1.70711 + 2.95680i 1.70711 + 2.95680i
\(949\) 0 0
\(950\) 0 0
\(951\) −1.84776 −1.84776
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 2.23044 + 3.86324i 2.23044 + 3.86324i
\(957\) 0 0
\(958\) −1.41421 −1.41421
\(959\) 0 0
\(960\) 0.765367 0.765367
\(961\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(968\) 3.15432 5.46345i 3.15432 5.46345i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(972\) 2.41421 2.41421
\(973\) 0 0
\(974\) 0 0
\(975\) −0.207107 + 0.358719i −0.207107 + 0.358719i
\(976\) 0 0
\(977\) −0.923880 1.60021i −0.923880 1.60021i −0.793353 0.608761i \(-0.791667\pi\)
−0.130526 0.991445i \(-0.541667\pi\)
\(978\) 0 0
\(979\) −3.41421 −3.41421
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0.382683 + 0.662827i 0.382683 + 0.662827i 0.991445 0.130526i \(-0.0416667\pi\)
−0.608761 + 0.793353i \(0.708333\pi\)
\(984\) −1.00000 1.73205i −1.00000 1.73205i
\(985\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 1.30656 + 2.26303i 1.30656 + 2.26303i
\(991\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) −1.53073 −1.53073
\(996\) 2.23044 3.86324i 2.23044 3.86324i
\(997\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(998\) 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 1911.1.w.e.1598.4 8
3.2 odd 2 inner 1911.1.w.e.1598.1 8
7.2 even 3 1911.1.h.e.1520.1 yes 4
7.3 odd 6 1911.1.w.f.116.4 8
7.4 even 3 inner 1911.1.w.e.116.4 8
7.5 odd 6 1911.1.h.d.1520.1 4
7.6 odd 2 1911.1.w.f.1598.4 8
13.12 even 2 inner 1911.1.w.e.1598.1 8
21.2 odd 6 1911.1.h.e.1520.4 yes 4
21.5 even 6 1911.1.h.d.1520.4 yes 4
21.11 odd 6 inner 1911.1.w.e.116.1 8
21.17 even 6 1911.1.w.f.116.1 8
21.20 even 2 1911.1.w.f.1598.1 8
39.38 odd 2 CM 1911.1.w.e.1598.4 8
91.12 odd 6 1911.1.h.d.1520.4 yes 4
91.25 even 6 inner 1911.1.w.e.116.1 8
91.38 odd 6 1911.1.w.f.116.1 8
91.51 even 6 1911.1.h.e.1520.4 yes 4
91.90 odd 2 1911.1.w.f.1598.1 8
273.38 even 6 1911.1.w.f.116.4 8
273.116 odd 6 inner 1911.1.w.e.116.4 8
273.194 even 6 1911.1.h.d.1520.1 4
273.233 odd 6 1911.1.h.e.1520.1 yes 4
273.272 even 2 1911.1.w.f.1598.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1911.1.h.d.1520.1 4 7.5 odd 6
1911.1.h.d.1520.1 4 273.194 even 6
1911.1.h.d.1520.4 yes 4 21.5 even 6
1911.1.h.d.1520.4 yes 4 91.12 odd 6
1911.1.h.e.1520.1 yes 4 7.2 even 3
1911.1.h.e.1520.1 yes 4 273.233 odd 6
1911.1.h.e.1520.4 yes 4 21.2 odd 6
1911.1.h.e.1520.4 yes 4 91.51 even 6
1911.1.w.e.116.1 8 21.11 odd 6 inner
1911.1.w.e.116.1 8 91.25 even 6 inner
1911.1.w.e.116.4 8 7.4 even 3 inner
1911.1.w.e.116.4 8 273.116 odd 6 inner
1911.1.w.e.1598.1 8 3.2 odd 2 inner
1911.1.w.e.1598.1 8 13.12 even 2 inner
1911.1.w.e.1598.4 8 1.1 even 1 trivial
1911.1.w.e.1598.4 8 39.38 odd 2 CM
1911.1.w.f.116.1 8 21.17 even 6
1911.1.w.f.116.1 8 91.38 odd 6
1911.1.w.f.116.4 8 7.3 odd 6
1911.1.w.f.116.4 8 273.38 even 6
1911.1.w.f.1598.1 8 21.20 even 2
1911.1.w.f.1598.1 8 91.90 odd 2
1911.1.w.f.1598.4 8 7.6 odd 2
1911.1.w.f.1598.4 8 273.272 even 2