Properties

Label 1287.1.ba.d.571.1
Level $1287$
Weight $1$
Character 1287.571
Analytic conductor $0.642$
Analytic rank $0$
Dimension $8$
Projective image $D_{15}$
CM discriminant -143
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1287,1,Mod(142,1287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1287, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([4, 3, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1287.142");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1287.ba (of order \(6\), degree \(2\), 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: \(0.642296671259\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{15})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{7} + x^{5} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{15}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{15} + \cdots)\)

Embedding invariants

Embedding label 571.1
Root \(-0.978148 - 0.207912i\) of defining polynomial
Character \(\chi\) \(=\) 1287.571
Dual form 1287.1.ba.d.142.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.978148 - 1.69420i) q^{2} +(0.669131 - 0.743145i) q^{3} +(-1.41355 + 2.44833i) q^{4} +(-1.91355 - 0.406737i) q^{6} +(-0.809017 - 1.40126i) q^{7} +3.57433 q^{8} +(-0.104528 - 0.994522i) q^{9} +O(q^{10})\) \(q+(-0.978148 - 1.69420i) q^{2} +(0.669131 - 0.743145i) q^{3} +(-1.41355 + 2.44833i) q^{4} +(-1.91355 - 0.406737i) q^{6} +(-0.809017 - 1.40126i) q^{7} +3.57433 q^{8} +(-0.104528 - 0.994522i) q^{9} +(0.500000 + 0.866025i) q^{11} +(0.873619 + 2.68872i) q^{12} +(0.500000 - 0.866025i) q^{13} +(-1.58268 + 2.74128i) q^{14} +(-2.08268 - 3.60730i) q^{16} +(-1.58268 + 1.14988i) q^{18} -1.33826 q^{19} +(-1.58268 - 0.336408i) q^{21} +(0.978148 - 1.69420i) q^{22} +(0.104528 - 0.181049i) q^{23} +(2.39169 - 2.65624i) q^{24} +(-0.500000 - 0.866025i) q^{25} -1.95630 q^{26} +(-0.809017 - 0.587785i) q^{27} +4.57433 q^{28} +(-2.28716 + 3.96149i) q^{32} +(0.978148 + 0.207912i) q^{33} +(2.58268 + 1.14988i) q^{36} +(1.30902 + 2.26728i) q^{38} +(-0.309017 - 0.951057i) q^{39} +(-0.104528 + 0.181049i) q^{41} +(0.978148 + 3.01043i) q^{42} -2.82709 q^{44} -0.408977 q^{46} +(-4.07433 - 0.866025i) q^{48} +(-0.809017 + 1.40126i) q^{49} +(-0.978148 + 1.69420i) q^{50} +(1.41355 + 2.44833i) q^{52} +0.618034 q^{53} +(-0.204489 + 1.94558i) q^{54} +(-2.89169 - 5.00856i) q^{56} +(-0.895472 + 0.994522i) q^{57} +(-1.30902 + 0.951057i) q^{63} +4.78339 q^{64} +(-0.604528 - 1.86055i) q^{66} +(-0.0646021 - 0.198825i) q^{69} +(-0.373619 - 3.55475i) q^{72} -0.618034 q^{73} +(-0.978148 - 0.207912i) q^{75} +(1.89169 - 3.27651i) q^{76} +(0.809017 - 1.40126i) q^{77} +(-1.30902 + 1.45381i) q^{78} +(-0.978148 + 0.207912i) q^{81} +0.408977 q^{82} +(0.913545 + 1.58231i) q^{83} +(3.06082 - 3.39939i) q^{84} +(1.78716 + 3.09546i) q^{88} -1.61803 q^{91} +(0.295511 + 0.511841i) q^{92} +(1.41355 + 4.35045i) q^{96} +3.16535 q^{98} +(0.809017 - 0.587785i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + q^{2} + q^{3} - 5 q^{4} - 9 q^{6} - 2 q^{7} + 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + q^{2} + q^{3} - 5 q^{4} - 9 q^{6} - 2 q^{7} + 2 q^{8} + q^{9} + 4 q^{11} + 4 q^{13} - 2 q^{14} - 6 q^{16} - 2 q^{18} - 2 q^{19} - 2 q^{21} - q^{22} - q^{23} + 4 q^{24} - 4 q^{25} + 2 q^{26} - 2 q^{27} + 10 q^{28} - 5 q^{32} - q^{33} + 10 q^{36} + 6 q^{38} + 2 q^{39} + q^{41} - q^{42} - 10 q^{44} + 2 q^{46} - 6 q^{48} - 2 q^{49} + q^{50} + 5 q^{52} - 4 q^{53} + q^{54} - 8 q^{56} - 9 q^{57} - 6 q^{63} + 8 q^{64} - 3 q^{66} + 2 q^{69} + 4 q^{72} + 4 q^{73} + q^{75} + 2 q^{77} - 6 q^{78} + q^{81} - 2 q^{82} + q^{83} + 5 q^{84} + q^{88} - 4 q^{91} + 5 q^{92} + 5 q^{96} + 4 q^{98} + 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1287\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(937\) \(1145\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.978148 1.69420i −0.978148 1.69420i −0.669131 0.743145i \(-0.733333\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(3\) 0.669131 0.743145i 0.669131 0.743145i
\(4\) −1.41355 + 2.44833i −1.41355 + 2.44833i
\(5\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) −1.91355 0.406737i −1.91355 0.406737i
\(7\) −0.809017 1.40126i −0.809017 1.40126i −0.913545 0.406737i \(-0.866667\pi\)
0.104528 0.994522i \(-0.466667\pi\)
\(8\) 3.57433 3.57433
\(9\) −0.104528 0.994522i −0.104528 0.994522i
\(10\) 0 0
\(11\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(12\) 0.873619 + 2.68872i 0.873619 + 2.68872i
\(13\) 0.500000 0.866025i 0.500000 0.866025i
\(14\) −1.58268 + 2.74128i −1.58268 + 2.74128i
\(15\) 0 0
\(16\) −2.08268 3.60730i −2.08268 3.60730i
\(17\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(18\) −1.58268 + 1.14988i −1.58268 + 1.14988i
\(19\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(20\) 0 0
\(21\) −1.58268 0.336408i −1.58268 0.336408i
\(22\) 0.978148 1.69420i 0.978148 1.69420i
\(23\) 0.104528 0.181049i 0.104528 0.181049i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(24\) 2.39169 2.65624i 2.39169 2.65624i
\(25\) −0.500000 0.866025i −0.500000 0.866025i
\(26\) −1.95630 −1.95630
\(27\) −0.809017 0.587785i −0.809017 0.587785i
\(28\) 4.57433 4.57433
\(29\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(30\) 0 0
\(31\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(32\) −2.28716 + 3.96149i −2.28716 + 3.96149i
\(33\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(34\) 0 0
\(35\) 0 0
\(36\) 2.58268 + 1.14988i 2.58268 + 1.14988i
\(37\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(38\) 1.30902 + 2.26728i 1.30902 + 2.26728i
\(39\) −0.309017 0.951057i −0.309017 0.951057i
\(40\) 0 0
\(41\) −0.104528 + 0.181049i −0.104528 + 0.181049i −0.913545 0.406737i \(-0.866667\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(42\) 0.978148 + 3.01043i 0.978148 + 3.01043i
\(43\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(44\) −2.82709 −2.82709
\(45\) 0 0
\(46\) −0.408977 −0.408977
\(47\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(48\) −4.07433 0.866025i −4.07433 0.866025i
\(49\) −0.809017 + 1.40126i −0.809017 + 1.40126i
\(50\) −0.978148 + 1.69420i −0.978148 + 1.69420i
\(51\) 0 0
\(52\) 1.41355 + 2.44833i 1.41355 + 2.44833i
\(53\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(54\) −0.204489 + 1.94558i −0.204489 + 1.94558i
\(55\) 0 0
\(56\) −2.89169 5.00856i −2.89169 5.00856i
\(57\) −0.895472 + 0.994522i −0.895472 + 0.994522i
\(58\) 0 0
\(59\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(60\) 0 0
\(61\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(62\) 0 0
\(63\) −1.30902 + 0.951057i −1.30902 + 0.951057i
\(64\) 4.78339 4.78339
\(65\) 0 0
\(66\) −0.604528 1.86055i −0.604528 1.86055i
\(67\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) −0.0646021 0.198825i −0.0646021 0.198825i
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) −0.373619 3.55475i −0.373619 3.55475i
\(73\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(74\) 0 0
\(75\) −0.978148 0.207912i −0.978148 0.207912i
\(76\) 1.89169 3.27651i 1.89169 3.27651i
\(77\) 0.809017 1.40126i 0.809017 1.40126i
\(78\) −1.30902 + 1.45381i −1.30902 + 1.45381i
\(79\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(80\) 0 0
\(81\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(82\) 0.408977 0.408977
\(83\) 0.913545 + 1.58231i 0.913545 + 1.58231i 0.809017 + 0.587785i \(0.200000\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(84\) 3.06082 3.39939i 3.06082 3.39939i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 1.78716 + 3.09546i 1.78716 + 3.09546i
\(89\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(90\) 0 0
\(91\) −1.61803 −1.61803
\(92\) 0.295511 + 0.511841i 0.295511 + 0.511841i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 1.41355 + 4.35045i 1.41355 + 4.35045i
\(97\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(98\) 3.16535 3.16535
\(99\) 0.809017 0.587785i 0.809017 0.587785i
\(100\) 2.82709 2.82709
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) 0 0
\(103\) 0.104528 0.181049i 0.104528 0.181049i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(104\) 1.78716 3.09546i 1.78716 3.09546i
\(105\) 0 0
\(106\) −0.604528 1.04707i −0.604528 1.04707i
\(107\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(108\) 2.58268 1.14988i 2.58268 1.14988i
\(109\) 1.95630 1.95630 0.978148 0.207912i \(-0.0666667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −3.36984 + 5.83674i −3.36984 + 5.83674i
\(113\) 0.978148 1.69420i 0.978148 1.69420i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(114\) 2.56082 + 0.544320i 2.56082 + 0.544320i
\(115\) 0 0
\(116\) 0 0
\(117\) −0.913545 0.406737i −0.913545 0.406737i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(122\) 0 0
\(123\) 0.0646021 + 0.198825i 0.0646021 + 0.198825i
\(124\) 0 0
\(125\) 0 0
\(126\) 2.89169 + 1.28746i 2.89169 + 1.28746i
\(127\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(128\) −2.39169 4.14253i −2.39169 4.14253i
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(132\) −1.89169 + 2.10094i −1.89169 + 2.10094i
\(133\) 1.08268 + 1.87525i 1.08268 + 1.87525i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) −0.273659 + 0.303929i −0.273659 + 0.303929i
\(139\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 1.00000 1.00000
\(144\) −3.36984 + 2.44833i −3.36984 + 2.44833i
\(145\) 0 0
\(146\) 0.604528 + 1.04707i 0.604528 + 1.04707i
\(147\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(148\) 0 0
\(149\) 0.913545 1.58231i 0.913545 1.58231i 0.104528 0.994522i \(-0.466667\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(150\) 0.604528 + 1.86055i 0.604528 + 1.86055i
\(151\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(152\) −4.78339 −4.78339
\(153\) 0 0
\(154\) −3.16535 −3.16535
\(155\) 0 0
\(156\) 2.76531 + 0.587785i 2.76531 + 0.587785i
\(157\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(158\) 0 0
\(159\) 0.413545 0.459289i 0.413545 0.459289i
\(160\) 0 0
\(161\) −0.338261 −0.338261
\(162\) 1.30902 + 1.45381i 1.30902 + 1.45381i
\(163\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(164\) −0.295511 0.511841i −0.295511 0.511841i
\(165\) 0 0
\(166\) 1.78716 3.09546i 1.78716 3.09546i
\(167\) 0.309017 0.535233i 0.309017 0.535233i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(168\) −5.65701 1.20243i −5.65701 1.20243i
\(169\) −0.500000 0.866025i −0.500000 0.866025i
\(170\) 0 0
\(171\) 0.139886 + 1.33093i 0.139886 + 1.33093i
\(172\) 0 0
\(173\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(174\) 0 0
\(175\) −0.809017 + 1.40126i −0.809017 + 1.40126i
\(176\) 2.08268 3.60730i 2.08268 3.60730i
\(177\) 0 0
\(178\) 0 0
\(179\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(180\) 0 0
\(181\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(182\) 1.58268 + 2.74128i 1.58268 + 2.74128i
\(183\) 0 0
\(184\) 0.373619 0.647127i 0.373619 0.647127i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.169131 + 1.60917i −0.169131 + 1.60917i
\(190\) 0 0
\(191\) 0.978148 + 1.69420i 0.978148 + 1.69420i 0.669131 + 0.743145i \(0.266667\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(192\) 3.20071 3.55475i 3.20071 3.55475i
\(193\) 0.913545 1.58231i 0.913545 1.58231i 0.104528 0.994522i \(-0.466667\pi\)
0.809017 0.587785i \(-0.200000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) −2.28716 3.96149i −2.28716 3.96149i
\(197\) −0.618034 −0.618034 −0.309017 0.951057i \(-0.600000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(198\) −1.78716 0.795697i −1.78716 0.795697i
\(199\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(200\) −1.78716 3.09546i −1.78716 3.09546i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) −0.408977 −0.408977
\(207\) −0.190983 0.0850311i −0.190983 0.0850311i
\(208\) −4.16535 −4.16535
\(209\) −0.669131 1.15897i −0.669131 1.15897i
\(210\) 0 0
\(211\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(212\) −0.873619 + 1.51315i −0.873619 + 1.51315i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) −2.89169 2.10094i −2.89169 2.10094i
\(217\) 0 0
\(218\) −1.91355 3.31436i −1.91355 3.31436i
\(219\) −0.413545 + 0.459289i −0.413545 + 0.459289i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(224\) 7.40142 7.40142
\(225\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(226\) −3.82709 −3.82709
\(227\) 0.669131 + 1.15897i 0.669131 + 1.15897i 0.978148 + 0.207912i \(0.0666667\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(228\) −1.16913 3.59821i −1.16913 3.59821i
\(229\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(230\) 0 0
\(231\) −0.500000 1.53884i −0.500000 1.53884i
\(232\) 0 0
\(233\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(234\) 0.204489 + 1.94558i 0.204489 + 1.94558i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0.669131 1.15897i 0.669131 1.15897i −0.309017 0.951057i \(-0.600000\pi\)
0.978148 0.207912i \(-0.0666667\pi\)
\(240\) 0 0
\(241\) −0.104528 0.181049i −0.104528 0.181049i 0.809017 0.587785i \(-0.200000\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(242\) 1.95630 1.95630
\(243\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(244\) 0 0
\(245\) 0 0
\(246\) 0.273659 0.303929i 0.273659 0.303929i
\(247\) −0.669131 + 1.15897i −0.669131 + 1.15897i
\(248\) 0 0
\(249\) 1.78716 + 0.379874i 1.78716 + 0.379874i
\(250\) 0 0
\(251\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(252\) −0.478148 4.54927i −0.478148 4.54927i
\(253\) 0.209057 0.209057
\(254\) 0 0
\(255\) 0 0
\(256\) −2.28716 + 3.96149i −2.28716 + 3.96149i
\(257\) −0.913545 + 1.58231i −0.913545 + 1.58231i −0.104528 + 0.994522i \(0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(264\) 3.49622 + 0.743145i 3.49622 + 0.743145i
\(265\) 0 0
\(266\) 2.11803 3.66854i 2.11803 3.66854i
\(267\) 0 0
\(268\) 0 0
\(269\) 1.82709 1.82709 0.913545 0.406737i \(-0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(270\) 0 0
\(271\) 1.95630 1.95630 0.978148 0.207912i \(-0.0666667\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(272\) 0 0
\(273\) −1.08268 + 1.20243i −1.08268 + 1.20243i
\(274\) 0 0
\(275\) 0.500000 0.866025i 0.500000 0.866025i
\(276\) 0.578108 + 0.122881i 0.578108 + 0.122881i
\(277\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.309017 + 0.535233i 0.309017 + 0.535233i 0.978148 0.207912i \(-0.0666667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(282\) 0 0
\(283\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) −0.978148 1.69420i −0.978148 1.69420i
\(287\) 0.338261 0.338261
\(288\) 4.17886 + 1.86055i 4.17886 + 1.86055i
\(289\) 1.00000 1.00000
\(290\) 0 0
\(291\) 0 0
\(292\) 0.873619 1.51315i 0.873619 1.51315i
\(293\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(294\) 2.11803 2.35232i 2.11803 2.35232i
\(295\) 0 0
\(296\) 0 0
\(297\) 0.104528 0.994522i 0.104528 0.994522i
\(298\) −3.57433 −3.57433
\(299\) −0.104528 0.181049i −0.104528 0.181049i
\(300\) 1.89169 2.10094i 1.89169 2.10094i
\(301\) 0 0
\(302\) −0.978148 + 1.69420i −0.978148 + 1.69420i
\(303\) 0 0
\(304\) 2.78716 + 4.82751i 2.78716 + 4.82751i
\(305\) 0 0
\(306\) 0 0
\(307\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(308\) 2.28716 + 3.96149i 2.28716 + 3.96149i
\(309\) −0.0646021 0.198825i −0.0646021 0.198825i
\(310\) 0 0
\(311\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(312\) −1.10453 3.39939i −1.10453 3.39939i
\(313\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(314\) −3.16535 −3.16535
\(315\) 0 0
\(316\) 0 0
\(317\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(318\) −1.18264 0.251377i −1.18264 0.251377i
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0.330869 + 0.573083i 0.330869 + 0.573083i
\(323\) 0 0
\(324\) 0.873619 2.68872i 0.873619 2.68872i
\(325\) −1.00000 −1.00000
\(326\) 0 0
\(327\) 1.30902 1.45381i 1.30902 1.45381i
\(328\) −0.373619 + 0.647127i −0.373619 + 0.647127i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(332\) −5.16535 −5.16535
\(333\) 0 0
\(334\) −1.20906 −1.20906
\(335\) 0 0
\(336\) 2.08268 + 6.40982i 2.08268 + 6.40982i
\(337\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(338\) −0.978148 + 1.69420i −0.978148 + 1.69420i
\(339\) −0.604528 1.86055i −0.604528 1.86055i
\(340\) 0 0
\(341\) 0 0
\(342\) 2.11803 1.53884i 2.11803 1.53884i
\(343\) 1.00000 1.00000
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) 0.913545 + 1.58231i 0.913545 + 1.58231i 0.809017 + 0.587785i \(0.200000\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(350\) 3.16535 3.16535
\(351\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(352\) −4.57433 −4.57433
\(353\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0.978148 + 1.69420i 0.978148 + 1.69420i
\(359\) 0.209057 0.209057 0.104528 0.994522i \(-0.466667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(360\) 0 0
\(361\) 0.790943 0.790943
\(362\) −1.30902 2.26728i −1.30902 2.26728i
\(363\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(364\) 2.28716 3.96149i 2.28716 3.96149i
\(365\) 0 0
\(366\) 0 0
\(367\) −0.309017 0.535233i −0.309017 0.535233i 0.669131 0.743145i \(-0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(368\) −0.870796 −0.870796
\(369\) 0.190983 + 0.0850311i 0.190983 + 0.0850311i
\(370\) 0 0
\(371\) −0.500000 0.866025i −0.500000 0.866025i
\(372\) 0 0
\(373\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 2.89169 1.28746i 2.89169 1.28746i
\(379\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 1.91355 3.31436i 1.91355 3.31436i
\(383\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(384\) −4.67886 0.994522i −4.67886 0.994522i
\(385\) 0 0
\(386\) −3.57433 −3.57433
\(387\) 0 0
\(388\) 0 0
\(389\) −0.669131 1.15897i −0.669131 1.15897i −0.978148 0.207912i \(-0.933333\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.89169 + 5.00856i −2.89169 + 5.00856i
\(393\) 0 0
\(394\) 0.604528 + 1.04707i 0.604528 + 1.04707i
\(395\) 0 0
\(396\) 0.295511 + 2.81160i 0.295511 + 2.81160i
\(397\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(398\) 1.91355 + 3.31436i 1.91355 + 3.31436i
\(399\) 2.11803 + 0.450202i 2.11803 + 0.450202i
\(400\) −2.08268 + 3.60730i −2.08268 + 3.60730i
\(401\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0.295511 + 0.511841i 0.295511 + 0.511841i
\(413\) 0 0
\(414\) 0.0427497 + 0.406737i 0.0427497 + 0.406737i
\(415\) 0 0
\(416\) 2.28716 + 3.96149i 2.28716 + 3.96149i
\(417\) 0 0
\(418\) −1.30902 + 2.26728i −1.30902 + 2.26728i
\(419\) −0.913545 + 1.58231i −0.913545 + 1.58231i −0.104528 + 0.994522i \(0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(420\) 0 0
\(421\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 2.20906 2.20906
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0.669131 0.743145i 0.669131 0.743145i
\(430\) 0 0
\(431\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(432\) −0.435398 + 4.14253i −0.435398 + 4.14253i
\(433\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −2.76531 + 4.78966i −2.76531 + 4.78966i
\(437\) −0.139886 + 0.242290i −0.139886 + 0.242290i
\(438\) 1.18264 + 0.251377i 1.18264 + 0.251377i
\(439\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) 1.47815 + 0.658114i 1.47815 + 0.658114i
\(442\) 0 0
\(443\) −0.913545 1.58231i −0.913545 1.58231i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −0.564602 1.73767i −0.564602 1.73767i
\(448\) −3.86984 6.70276i −3.86984 6.70276i
\(449\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(450\) 1.78716 + 0.795697i 1.78716 + 0.795697i
\(451\) −0.209057 −0.209057
\(452\) 2.76531 + 4.78966i 2.76531 + 4.78966i
\(453\) −0.978148 0.207912i −0.978148 0.207912i
\(454\) 1.30902 2.26728i 1.30902 2.26728i
\(455\) 0 0
\(456\) −3.20071 + 3.55475i −3.20071 + 3.55475i
\(457\) −0.978148 1.69420i −0.978148 1.69420i −0.669131 0.743145i \(-0.733333\pi\)
−0.309017 0.951057i \(-0.600000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0.669131 + 1.15897i 0.669131 + 1.15897i 0.978148 + 0.207912i \(0.0666667\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(462\) −2.11803 + 2.35232i −2.11803 + 2.35232i
\(463\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(468\) 2.28716 1.66172i 2.28716 1.66172i
\(469\) 0 0
\(470\) 0 0
\(471\) −0.500000 1.53884i −0.500000 1.53884i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.669131 + 1.15897i 0.669131 + 1.15897i
\(476\) 0 0
\(477\) −0.0646021 0.614648i −0.0646021 0.614648i
\(478\) −2.61803 −2.61803
\(479\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −0.204489 + 0.354185i −0.204489 + 0.354185i
\(483\) −0.226341 + 0.251377i −0.226341 + 0.251377i
\(484\) −1.41355 2.44833i −1.41355 2.44833i
\(485\) 0 0
\(486\) 1.95630 1.95630
\(487\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(492\) −0.578108 0.122881i −0.578108 0.122881i
\(493\) 0 0
\(494\) 2.61803 2.61803
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) −1.10453 3.39939i −1.10453 3.39939i
\(499\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(500\) 0 0
\(501\) −0.190983 0.587785i −0.190983 0.587785i
\(502\) 1.58268 + 2.74128i 1.58268 + 2.74128i
\(503\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(504\) −4.67886 + 3.39939i −4.67886 + 3.39939i
\(505\) 0 0
\(506\) −0.204489 0.354185i −0.204489 0.354185i
\(507\) −0.978148 0.207912i −0.978148 0.207912i
\(508\) 0 0
\(509\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(510\) 0 0
\(511\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(512\) 4.16535 4.16535
\(513\) 1.08268 + 0.786610i 1.08268 + 0.786610i
\(514\) 3.57433 3.57433
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −0.209057 −0.209057 −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(522\) 0 0
\(523\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(524\) 0 0
\(525\) 0.500000 + 1.53884i 0.500000 + 1.53884i
\(526\) 0 0
\(527\) 0 0
\(528\) −1.28716 3.96149i −1.28716 3.96149i
\(529\) 0.478148 + 0.828176i 0.478148 + 0.828176i
\(530\) 0 0
\(531\) 0 0
\(532\) −6.12165 −6.12165
\(533\) 0.104528 + 0.181049i 0.104528 + 0.181049i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(538\) −1.78716 3.09546i −1.78716 3.09546i
\(539\) −1.61803 −1.61803
\(540\) 0 0
\(541\) −1.82709 −1.82709 −0.913545 0.406737i \(-0.866667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(542\) −1.91355 3.31436i −1.91355 3.31436i
\(543\) 0.895472 0.994522i 0.895472 0.994522i
\(544\) 0 0
\(545\) 0 0
\(546\) 3.09618 + 0.658114i 3.09618 + 0.658114i
\(547\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) −1.95630 −1.95630
\(551\) 0 0
\(552\) −0.230909 0.710666i −0.230909 0.710666i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 1.61803 1.61803 0.809017 0.587785i \(-0.200000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.604528 1.04707i 0.604528 1.04707i
\(563\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 1.08268 + 1.20243i 1.08268 + 1.20243i
\(568\) 0 0
\(569\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(570\) 0 0
\(571\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(572\) −1.41355 + 2.44833i −1.41355 + 2.44833i
\(573\) 1.91355 + 0.406737i 1.91355 + 0.406737i
\(574\) −0.330869 0.573083i −0.330869 0.573083i
\(575\) −0.209057 −0.209057
\(576\) −0.500000 4.75718i −0.500000 4.75718i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −0.978148 1.69420i −0.978148 1.69420i
\(579\) −0.564602 1.73767i −0.564602 1.73767i
\(580\) 0 0
\(581\) 1.47815 2.56023i 1.47815 2.56023i
\(582\) 0 0
\(583\) 0.309017 + 0.535233i 0.309017 + 0.535233i
\(584\) −2.20906 −2.20906
\(585\) 0 0
\(586\) 1.95630 1.95630
\(587\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(588\) −4.47437 0.951057i −4.47437 0.951057i
\(589\) 0 0
\(590\) 0 0
\(591\) −0.413545 + 0.459289i −0.413545 + 0.459289i
\(592\) 0 0
\(593\) −1.82709 −1.82709 −0.913545 0.406737i \(-0.866667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(594\) −1.78716 + 0.795697i −1.78716 + 0.795697i
\(595\) 0 0
\(596\) 2.58268 + 4.47333i 2.58268 + 4.47333i
\(597\) −1.30902 + 1.45381i −1.30902 + 1.45381i
\(598\) −0.204489 + 0.354185i −0.204489 + 0.354185i
\(599\) −0.309017 + 0.535233i −0.309017 + 0.535233i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(600\) −3.49622 0.743145i −3.49622 0.743145i
\(601\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 2.82709 2.82709
\(605\) 0 0
\(606\) 0 0
\(607\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(608\) 3.06082 5.30150i 3.06082 5.30150i
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(614\) −0.978148 1.69420i −0.978148 1.69420i
\(615\) 0 0
\(616\) 2.89169 5.00856i 2.89169 5.00856i
\(617\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(618\) −0.273659 + 0.303929i −0.273659 + 0.303929i
\(619\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(620\) 0 0
\(621\) −0.190983 + 0.0850311i −0.190983 + 0.0850311i
\(622\) 1.20906 1.20906
\(623\) 0 0
\(624\) −2.78716 + 3.09546i −2.78716 + 3.09546i
\(625\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(626\) 1.58268 2.74128i 1.58268 2.74128i
\(627\) −1.30902 0.278240i −1.30902 0.278240i
\(628\) 2.28716 + 3.96149i 2.28716 + 3.96149i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0 0
\(636\) 0.539926 + 1.66172i 0.539926 + 1.66172i
\(637\) 0.809017 + 1.40126i 0.809017 + 1.40126i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(642\) 0 0
\(643\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(644\) 0.478148 0.828176i 0.478148 0.828176i
\(645\) 0 0
\(646\) 0 0
\(647\) 1.33826 1.33826 0.669131 0.743145i \(-0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(648\) −3.49622 + 0.743145i −3.49622 + 0.743145i
\(649\) 0 0
\(650\) 0.978148 + 1.69420i 0.978148 + 1.69420i
\(651\) 0 0
\(652\) 0 0
\(653\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) −3.74346 0.795697i −3.74346 0.795697i
\(655\) 0 0
\(656\) 0.870796 0.870796
\(657\) 0.0646021 + 0.614648i 0.0646021 + 0.614648i
\(658\) 0 0
\(659\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(660\) 0 0
\(661\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 3.26531 + 5.65569i 3.26531 + 5.65569i
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0.873619 + 1.51315i 0.873619 + 1.51315i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 4.95252 5.50033i 4.95252 5.50033i
\(673\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(674\) 0 0
\(675\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(676\) 2.82709 2.82709
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) −2.56082 + 2.84408i −2.56082 + 2.84408i
\(679\) 0 0
\(680\) 0 0
\(681\) 1.30902 + 0.278240i 1.30902 + 0.278240i
\(682\) 0 0
\(683\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(684\) −3.45630 1.53884i −3.45630 1.53884i
\(685\) 0 0
\(686\) −0.978148 1.69420i −0.978148 1.69420i
\(687\) 0 0
\(688\) 0 0
\(689\) 0.309017 0.535233i 0.309017 0.535233i
\(690\) 0 0
\(691\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(692\) 0 0
\(693\) −1.47815 0.658114i −1.47815 0.658114i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 1.78716 3.09546i 1.78716 3.09546i
\(699\) 0 0
\(700\) −2.28716 3.96149i −2.28716 3.96149i
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 1.58268 + 1.14988i 1.58268 + 1.14988i
\(703\) 0 0
\(704\) 2.39169 + 4.14253i 2.39169 + 4.14253i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 1.41355 2.44833i 1.41355 2.44833i
\(717\) −0.413545 1.27276i −0.413545 1.27276i
\(718\) −0.204489 0.354185i −0.204489 0.354185i
\(719\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0 0
\(721\) −0.338261 −0.338261
\(722\) −0.773659 1.34002i −0.773659 1.34002i
\(723\) −0.204489 0.0434654i −0.204489 0.0434654i
\(724\) −1.89169 + 3.27651i −1.89169 + 3.27651i
\(725\) 0 0
\(726\) 1.30902 1.45381i 1.30902 1.45381i
\(727\) −0.913545 1.58231i −0.913545 1.58231i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(728\) −5.78339 −5.78339
\(729\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −0.104528 + 0.181049i −0.104528 + 0.181049i −0.913545 0.406737i \(-0.866667\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(734\) −0.604528 + 1.04707i −0.604528 + 1.04707i
\(735\) 0 0
\(736\) 0.478148 + 0.828176i 0.478148 + 0.828176i
\(737\) 0 0
\(738\) −0.0427497 0.406737i −0.0427497 0.406737i
\(739\) −1.82709 −1.82709 −0.913545 0.406737i \(-0.866667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(740\) 0 0
\(741\) 0.413545 + 1.27276i 0.413545 + 1.27276i
\(742\) −0.978148 + 1.69420i −0.978148 + 1.69420i
\(743\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 1.47815 1.07394i 1.47815 1.07394i
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.669131 + 1.15897i −0.669131 + 1.15897i 0.309017 + 0.951057i \(0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(752\) 0 0
\(753\) −1.08268 + 1.20243i −1.08268 + 1.20243i
\(754\) 0 0
\(755\) 0 0
\(756\) −3.70071 2.68872i −3.70071 2.68872i
\(757\) 1.82709 1.82709 0.913545 0.406737i \(-0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(758\) 0 0
\(759\) 0.139886 0.155360i 0.139886 0.155360i
\(760\) 0 0
\(761\) −0.809017 + 1.40126i −0.809017 + 1.40126i 0.104528 + 0.994522i \(0.466667\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(762\) 0 0
\(763\) −1.58268 2.74128i −1.58268 2.74128i
\(764\) −5.53062 −5.53062
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 1.41355 + 4.35045i 1.41355 + 4.35045i
\(769\) −0.104528 + 0.181049i −0.104528 + 0.181049i −0.913545 0.406737i \(-0.866667\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(770\) 0 0
\(771\) 0.564602 + 1.73767i 0.564602 + 1.73767i
\(772\) 2.58268 + 4.47333i 2.58268 + 4.47333i
\(773\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −1.30902 + 2.26728i −1.30902 + 2.26728i
\(779\) 0.139886 0.242290i 0.139886 0.242290i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 6.73968 6.73968
\(785\) 0 0
\(786\) 0 0
\(787\) −0.104528 + 0.181049i −0.104528 + 0.181049i −0.913545 0.406737i \(-0.866667\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(788\) 0.873619 1.51315i 0.873619 1.51315i
\(789\) 0 0
\(790\) 0 0
\(791\) −3.16535 −3.16535
\(792\) 2.89169 2.10094i 2.89169 2.10094i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 2.76531 4.78966i 2.76531 4.78966i
\(797\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(798\) −1.30902 4.02874i −1.30902 4.02874i
\(799\) 0 0
\(800\) 4.57433 4.57433
\(801\) 0 0
\(802\) 0 0
\(803\) −0.309017 0.535233i −0.309017 0.535233i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 1.22256 1.35779i 1.22256 1.35779i
\(808\) 0 0
\(809\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(810\) 0 0
\(811\) 0.209057 0.209057 0.104528 0.994522i \(-0.466667\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(812\) 0 0
\(813\) 1.30902 1.45381i 1.30902 1.45381i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1.95630 1.95630
\(819\) 0.169131 + 1.60917i 0.169131 + 1.60917i
\(820\) 0 0
\(821\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(822\) 0 0
\(823\) 0.809017 1.40126i 0.809017 1.40126i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(824\) 0.373619 0.647127i 0.373619 0.647127i
\(825\) −0.309017 0.951057i −0.309017 0.951057i
\(826\) 0 0
\(827\) −1.33826 −1.33826 −0.669131 0.743145i \(-0.733333\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(828\) 0.478148 0.347395i 0.478148 0.347395i
\(829\) 1.82709 1.82709 0.913545 0.406737i \(-0.133333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 2.39169 4.14253i 2.39169 4.14253i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 3.78339 3.78339
\(837\) 0 0
\(838\) 3.57433 3.57433
\(839\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(840\) 0 0
\(841\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(842\) 0 0
\(843\) 0.604528 + 0.128496i 0.604528 + 0.128496i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 1.61803 1.61803
\(848\) −1.28716 2.22943i −1.28716 2.22943i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.669131 + 1.15897i 0.669131 + 1.15897i 0.978148 + 0.207912i \(0.0666667\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(858\) −1.91355 0.406737i −1.91355 0.406737i
\(859\) 0.978148 1.69420i 0.978148 1.69420i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(860\) 0 0
\(861\) 0.226341 0.251377i 0.226341 0.251377i
\(862\) −1.58268 2.74128i −1.58268 2.74128i
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) 4.17886 1.86055i 4.17886 1.86055i
\(865\) 0 0
\(866\) −1.30902 2.26728i −1.30902 2.26728i
\(867\) 0.669131 0.743145i 0.669131 0.743145i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 6.99244 6.99244
\(873\) 0 0
\(874\) 0.547318 0.547318
\(875\) 0 0
\(876\) −0.539926 1.66172i −0.539926 1.66172i
\(877\) 0.309017 0.535233i 0.309017 0.535233i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(878\) 0 0
\(879\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(880\) 0 0
\(881\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(882\) −0.330869 3.14801i −0.330869 3.14801i
\(883\) 0.618034 0.618034 0.309017 0.951057i \(-0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) −1.78716 + 3.09546i −1.78716 + 3.09546i
\(887\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) −0.669131 0.743145i −0.669131 0.743145i
\(892\) 0 0
\(893\) 0 0
\(894\) −2.39169 + 2.65624i −2.39169 + 2.65624i
\(895\) 0 0
\(896\) −3.86984 + 6.70276i −3.86984 + 6.70276i
\(897\) −0.204489 0.0434654i −0.204489 0.0434654i
\(898\) 0 0
\(899\) 0 0
\(900\) −0.295511 2.81160i −0.295511 2.81160i
\(901\) 0 0
\(902\) 0.204489 + 0.354185i 0.204489 + 0.354185i
\(903\) 0 0
\(904\) 3.49622 6.05563i 3.49622 6.05563i
\(905\) 0 0
\(906\) 0.604528 + 1.86055i 0.604528 + 1.86055i
\(907\) 0.104528 + 0.181049i 0.104528 + 0.181049i 0.913545 0.406737i \(-0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(908\) −3.78339 −3.78339
\(909\) 0 0
\(910\) 0 0
\(911\) 0.809017 + 1.40126i 0.809017 + 1.40126i 0.913545 + 0.406737i \(0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(912\) 5.45252 + 1.15897i 5.45252 + 1.15897i
\(913\) −0.913545 + 1.58231i −0.913545 + 1.58231i
\(914\) −1.91355 + 3.31436i −1.91355 + 3.31436i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0.669131 0.743145i 0.669131 0.743145i
\(922\) 1.30902 2.26728i 1.30902 2.26728i
\(923\) 0 0
\(924\) 4.47437 + 0.951057i 4.47437 + 0.951057i
\(925\) 0 0
\(926\) 0 0
\(927\) −0.190983 0.0850311i −0.190983 0.0850311i
\(928\) 0 0
\(929\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(930\) 0 0
\(931\) 1.08268 1.87525i 1.08268 1.87525i
\(932\) 0 0
\(933\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(934\) −1.95630 3.38840i −1.95630 3.38840i
\(935\) 0 0
\(936\) −3.26531 1.45381i −3.26531 1.45381i
\(937\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(938\) 0 0
\(939\) 1.58268 + 0.336408i 1.58268 + 0.336408i
\(940\) 0 0
\(941\) 0.309017 0.535233i 0.309017 0.535233i −0.669131 0.743145i \(-0.733333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(942\) −2.11803 + 2.35232i −2.11803 + 2.35232i
\(943\) 0.0218524 + 0.0378495i 0.0218524 + 0.0378495i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(948\) 0 0
\(949\) −0.309017 + 0.535233i −0.309017 + 0.535233i
\(950\) 1.30902 2.26728i 1.30902 2.26728i
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(954\) −0.978148 + 0.710666i −0.978148 + 0.710666i
\(955\) 0 0
\(956\) 1.89169 + 3.27651i 1.89169 + 3.27651i
\(957\) 0 0
\(958\) −0.978148 + 1.69420i −0.978148 + 1.69420i
\(959\) 0 0
\(960\) 0 0
\(961\) −0.500000 0.866025i −0.500000 0.866025i
\(962\) 0 0
\(963\) 0 0
\(964\) 0.591023 0.591023
\(965\) 0 0
\(966\) 0.647278 + 0.137583i 0.647278 + 0.137583i
\(967\) −0.978148 + 1.69420i −0.978148 + 1.69420i −0.309017 + 0.951057i \(0.600000\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(968\) −1.78716 + 3.09546i −1.78716 + 3.09546i
\(969\) 0 0
\(970\) 0 0
\(971\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(972\) −1.41355 2.44833i −1.41355 2.44833i
\(973\) 0 0
\(974\) 0 0
\(975\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(976\) 0 0
\(977\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) −0.204489 1.94558i −0.204489 1.94558i
\(982\) 0 0
\(983\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(984\) 0.230909 + 0.710666i 0.230909 + 0.710666i
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −1.89169 3.27651i −1.89169 3.27651i
\(989\) 0 0
\(990\) 0 0
\(991\) −1.95630 −1.95630 −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) −3.45630 + 3.83860i −3.45630 + 3.83860i
\(997\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(998\) 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 1287.1.ba.d.571.1 yes 8
3.2 odd 2 3861.1.ba.c.3574.4 8
9.2 odd 6 3861.1.ba.c.2287.4 8
9.7 even 3 inner 1287.1.ba.d.142.1 yes 8
11.10 odd 2 1287.1.ba.c.571.4 yes 8
13.12 even 2 1287.1.ba.c.571.4 yes 8
33.32 even 2 3861.1.ba.d.3574.1 8
39.38 odd 2 3861.1.ba.d.3574.1 8
99.43 odd 6 1287.1.ba.c.142.4 8
99.65 even 6 3861.1.ba.d.2287.1 8
117.25 even 6 1287.1.ba.c.142.4 8
117.38 odd 6 3861.1.ba.d.2287.1 8
143.142 odd 2 CM 1287.1.ba.d.571.1 yes 8
429.428 even 2 3861.1.ba.c.3574.4 8
1287.142 odd 6 inner 1287.1.ba.d.142.1 yes 8
1287.857 even 6 3861.1.ba.c.2287.4 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1287.1.ba.c.142.4 8 99.43 odd 6
1287.1.ba.c.142.4 8 117.25 even 6
1287.1.ba.c.571.4 yes 8 11.10 odd 2
1287.1.ba.c.571.4 yes 8 13.12 even 2
1287.1.ba.d.142.1 yes 8 9.7 even 3 inner
1287.1.ba.d.142.1 yes 8 1287.142 odd 6 inner
1287.1.ba.d.571.1 yes 8 1.1 even 1 trivial
1287.1.ba.d.571.1 yes 8 143.142 odd 2 CM
3861.1.ba.c.2287.4 8 9.2 odd 6
3861.1.ba.c.2287.4 8 1287.857 even 6
3861.1.ba.c.3574.4 8 3.2 odd 2
3861.1.ba.c.3574.4 8 429.428 even 2
3861.1.ba.d.2287.1 8 99.65 even 6
3861.1.ba.d.2287.1 8 117.38 odd 6
3861.1.ba.d.3574.1 8 33.32 even 2
3861.1.ba.d.3574.1 8 39.38 odd 2