Properties

Label 961.1.h.a.115.1
Level $961$
Weight $1$
Character 961.115
Analytic conductor $0.480$
Analytic rank $0$
Dimension $8$
Projective image $D_{3}$
CM discriminant -31
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [961,1,Mod(115,961)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(961, base_ring=CyclotomicField(30))
 
chi = DirichletCharacter(H, H._module([17]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("961.115");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 961 = 31^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 961.h (of order \(30\), degree \(8\), 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: \(0.479601477140\)
Analytic rank: \(0\)
Dimension: \(8\)
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: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 31)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.31.1
Artin image: $S_3\times C_{15}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{45} - \cdots)\)

Embedding invariants

Embedding label 115.1
Root \(-0.104528 + 0.994522i\) of defining polynomial
Character \(\chi\) \(=\) 961.115
Dual form 961.1.h.a.117.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.309017 + 0.951057i) q^{2} +(0.500000 + 0.866025i) q^{5} +(-0.913545 + 0.406737i) q^{7} +(-0.809017 + 0.587785i) q^{8} +(0.913545 + 0.406737i) q^{9} +O(q^{10})\) \(q+(-0.309017 + 0.951057i) q^{2} +(0.500000 + 0.866025i) q^{5} +(-0.913545 + 0.406737i) q^{7} +(-0.809017 + 0.587785i) q^{8} +(0.913545 + 0.406737i) q^{9} +(-0.978148 + 0.207912i) q^{10} +(-0.104528 - 0.994522i) q^{14} +(-0.309017 - 0.951057i) q^{16} +(-0.669131 + 0.743145i) q^{18} +(-0.669131 - 0.743145i) q^{19} +(-0.809017 - 0.587785i) q^{35} +(0.913545 - 0.406737i) q^{38} +(-0.913545 - 0.406737i) q^{40} +(0.978148 - 0.207912i) q^{41} +(0.104528 + 0.994522i) q^{45} +(0.618034 + 1.90211i) q^{47} +(0.500000 - 0.866025i) q^{56} +(0.978148 + 0.207912i) q^{59} -1.00000 q^{63} +(0.309017 - 0.951057i) q^{64} +(-1.00000 - 1.73205i) q^{67} +(0.809017 - 0.587785i) q^{70} +(-0.913545 - 0.406737i) q^{71} +(-0.978148 + 0.207912i) q^{72} +(0.669131 - 0.743145i) q^{80} +(0.669131 + 0.743145i) q^{81} +(-0.104528 + 0.994522i) q^{82} +(-0.978148 - 0.207912i) q^{90} -2.00000 q^{94} +(0.309017 - 0.951057i) q^{95} +(0.809017 + 0.587785i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q + 2 q^{2} + 4 q^{5} - q^{7} - 2 q^{8} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 2 q^{2} + 4 q^{5} - q^{7} - 2 q^{8} + q^{9} + q^{10} + q^{14} + 2 q^{16} - q^{18} - q^{19} - 2 q^{35} + q^{38} - q^{40} - q^{41} - q^{45} - 4 q^{47} + 4 q^{56} - q^{59} - 8 q^{63} - 2 q^{64} - 8 q^{67} + 2 q^{70} - q^{71} + q^{72} + q^{80} + q^{81} + q^{82} + q^{90} - 16 q^{94} - 2 q^{95} + 2 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/961\mathbb{Z}\right)^\times\).

\(n\) \(3\)
\(\chi(n)\) \(e\left(\frac{17}{30}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(3\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(4\) 0 0
\(5\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(6\) 0 0
\(7\) −0.913545 + 0.406737i −0.913545 + 0.406737i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(8\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(9\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(10\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(11\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(12\) 0 0
\(13\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(14\) −0.104528 0.994522i −0.104528 0.994522i
\(15\) 0 0
\(16\) −0.309017 0.951057i −0.309017 0.951057i
\(17\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(18\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(19\) −0.669131 0.743145i −0.669131 0.743145i 0.309017 0.951057i \(-0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0 0
\(28\) 0 0
\(29\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(30\) 0 0
\(31\) 0 0
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) −0.809017 0.587785i −0.809017 0.587785i
\(36\) 0 0
\(37\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(38\) 0.913545 0.406737i 0.913545 0.406737i
\(39\) 0 0
\(40\) −0.913545 0.406737i −0.913545 0.406737i
\(41\) 0.978148 0.207912i 0.978148 0.207912i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(42\) 0 0
\(43\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(44\) 0 0
\(45\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(46\) 0 0
\(47\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) 0 0
\(52\) 0 0
\(53\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0.500000 0.866025i 0.500000 0.866025i
\(57\) 0 0
\(58\) 0 0
\(59\) 0.978148 + 0.207912i 0.978148 + 0.207912i 0.669131 0.743145i \(-0.266667\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(60\) 0 0
\(61\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(62\) 0 0
\(63\) −1.00000 −1.00000
\(64\) 0.309017 0.951057i 0.309017 0.951057i
\(65\) 0 0
\(66\) 0 0
\(67\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0.809017 0.587785i 0.809017 0.587785i
\(71\) −0.913545 0.406737i −0.913545 0.406737i −0.104528 0.994522i \(-0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(72\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(73\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(80\) 0.669131 0.743145i 0.669131 0.743145i
\(81\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(82\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(83\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(90\) −0.978148 0.207912i −0.978148 0.207912i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) −2.00000 −2.00000
\(95\) 0.309017 0.951057i 0.309017 0.951057i
\(96\) 0 0
\(97\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(102\) 0 0
\(103\) 0.978148 0.207912i 0.978148 0.207912i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0.104528 + 0.994522i 0.104528 + 0.994522i 0.913545 + 0.406737i \(0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(108\) 0 0
\(109\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(113\) 0.104528 0.994522i 0.104528 0.994522i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0 0
\(118\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(119\) 0 0
\(120\) 0 0
\(121\) −0.978148 0.207912i −0.978148 0.207912i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 1.00000
\(126\) 0.309017 0.951057i 0.309017 0.951057i
\(127\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(128\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(129\) 0 0
\(130\) 0 0
\(131\) 1.82709 0.813473i 1.82709 0.813473i 0.913545 0.406737i \(-0.133333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(132\) 0 0
\(133\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(134\) 1.95630 0.415823i 1.95630 0.415823i
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(138\) 0 0
\(139\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.669131 0.743145i 0.669131 0.743145i
\(143\) 0 0
\(144\) 0.104528 0.994522i 0.104528 0.994522i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(150\) 0 0
\(151\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(152\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(163\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(168\) 0 0
\(169\) −0.104528 0.994522i −0.104528 0.994522i
\(170\) 0 0
\(171\) −0.309017 0.951057i −0.309017 0.951057i
\(172\) 0 0
\(173\) 1.33826 1.48629i 1.33826 1.48629i 0.669131 0.743145i \(-0.266667\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(180\) 0 0
\(181\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(191\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) 0 0
\(193\) −0.913545 + 0.406737i −0.913545 + 0.406737i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(194\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(195\) 0 0
\(196\) 0 0
\(197\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(198\) 0 0
\(199\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(203\) 0 0
\(204\) 0 0
\(205\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(206\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(212\) 0 0
\(213\) 0 0
\(214\) −0.978148 0.207912i −0.978148 0.207912i
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 1.00000 1.00000
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(227\) −1.95630 + 0.415823i −1.95630 + 0.415823i −0.978148 + 0.207912i \(0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(228\) 0 0
\(229\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(234\) 0 0
\(235\) −1.33826 + 1.48629i −1.33826 + 1.48629i
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(240\) 0 0
\(241\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(242\) 0.500000 0.866025i 0.500000 0.866025i
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(251\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) −0.913545 0.406737i −0.913545 0.406737i −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
\(262\) 0.209057 + 1.98904i 0.209057 + 1.98904i
\(263\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(270\) 0 0
\(271\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 1.00000 1.00000
\(281\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(282\) 0 0
\(283\) −1.61803 1.17557i −1.61803 1.17557i −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(288\) 0 0
\(289\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) −0.209057 1.98904i −0.209057 1.98904i −0.104528 0.994522i \(-0.533333\pi\)
−0.104528 0.994522i \(-0.533333\pi\)
\(294\) 0 0
\(295\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(296\) 0 0
\(297\) 0 0
\(298\) −1.33826 1.48629i −1.33826 1.48629i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(305\) 0 0
\(306\) 0 0
\(307\) 0.978148 + 0.207912i 0.978148 + 0.207912i 0.669131 0.743145i \(-0.266667\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(312\) 0 0
\(313\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(314\) −0.809017 0.587785i −0.809017 0.587785i
\(315\) −0.500000 0.866025i −0.500000 0.866025i
\(316\) 0 0
\(317\) −0.913545 + 0.406737i −0.913545 + 0.406737i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.978148 0.207912i 0.978148 0.207912i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(327\) 0 0
\(328\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(329\) −1.33826 1.48629i −1.33826 1.48629i
\(330\) 0 0
\(331\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 1.00000 1.73205i 1.00000 1.73205i
\(336\) 0 0
\(337\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(338\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 1.00000 1.00000
\(343\) 0.309017 0.951057i 0.309017 0.951057i
\(344\) 0 0
\(345\) 0 0
\(346\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(347\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(348\) 0 0
\(349\) −1.61803 + 1.17557i −1.61803 + 1.17557i −0.809017 + 0.587785i \(0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(354\) 0 0
\(355\) −0.104528 0.994522i −0.104528 0.994522i
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) −0.669131 + 0.743145i −0.669131 + 0.743145i −0.978148 0.207912i \(-0.933333\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(360\) −0.669131 0.743145i −0.669131 0.743145i
\(361\) 0 0
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(368\) 0 0
\(369\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) −1.61803 1.17557i −1.61803 1.17557i
\(377\) 0 0
\(378\) 0 0
\(379\) 1.82709 0.813473i 1.82709 0.813473i 0.913545 0.406737i \(-0.133333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(383\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.104528 0.994522i −0.104528 0.994522i
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(410\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(411\) 0 0
\(412\) 0 0
\(413\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(420\) 0 0
\(421\) −0.669131 + 0.743145i −0.669131 + 0.743145i −0.978148 0.207912i \(-0.933333\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(422\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(423\) −0.209057 + 1.98904i −0.209057 + 1.98904i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) −1.95630 0.415823i −1.95630 0.415823i −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 0.207912i \(-0.933333\pi\)
\(432\) 0 0
\(433\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) −0.913545 0.406737i −0.913545 0.406737i −0.104528 0.994522i \(-0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(449\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0.209057 1.98904i 0.209057 1.98904i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(462\) 0 0
\(463\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 1.00000 1.00000
\(467\) −0.309017 + 0.951057i −0.309017 + 0.951057i 0.669131 + 0.743145i \(0.266667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(468\) 0 0
\(469\) 1.61803 + 1.17557i 1.61803 + 1.17557i
\(470\) −1.00000 1.73205i −1.00000 1.73205i
\(471\) 0 0
\(472\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0.104528 + 0.994522i 0.104528 + 0.994522i 0.913545 + 0.406737i \(0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(486\) 0 0
\(487\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.00000 1.00000
\(498\) 0 0
\(499\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) −0.913545 + 0.406737i −0.913545 + 0.406737i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(504\) 0.809017 0.587785i 0.809017 0.587785i
\(505\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(513\) 0 0
\(514\) 0.669131 0.743145i 0.669131 0.743145i
\(515\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(522\) 0 0
\(523\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 0.309017 0.951057i 0.309017 0.951057i
\(530\) 0 0
\(531\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(536\) 1.82709 + 0.813473i 1.82709 + 0.813473i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.104528 + 0.994522i 0.104528 + 0.994522i 0.913545 + 0.406737i \(0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0.669131 0.743145i 0.669131 0.743145i
\(546\) 0 0
\(547\) 0.104528 0.994522i 0.104528 0.994522i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(561\) 0 0
\(562\) −0.809017 0.587785i −0.809017 0.587785i
\(563\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(564\) 0 0
\(565\) 0.913545 0.406737i 0.913545 0.406737i
\(566\) 1.61803 1.17557i 1.61803 1.17557i
\(567\) −0.913545 0.406737i −0.913545 0.406737i
\(568\) 0.978148 0.207912i 0.978148 0.207912i
\(569\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(570\) 0 0
\(571\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −0.309017 0.951057i −0.309017 0.951057i
\(575\) 0 0
\(576\) 0.669131 0.743145i 0.669131 0.743145i
\(577\) 1.33826 + 1.48629i 1.33826 + 1.48629i 0.669131 + 0.743145i \(0.266667\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(578\) 0.104528 0.994522i 0.104528 0.994522i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 1.95630 + 0.415823i 1.95630 + 0.415823i
\(587\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) −1.00000 −1.00000
\(591\) 0 0
\(592\) 0 0
\(593\) 0.809017 + 0.587785i 0.809017 + 0.587785i 0.913545 0.406737i \(-0.133333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0.978148 0.207912i 0.978148 0.207912i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(600\) 0 0
\(601\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(602\) 0 0
\(603\) −0.209057 1.98904i −0.209057 1.98904i
\(604\) 0 0
\(605\) −0.309017 0.951057i −0.309017 0.951057i
\(606\) 0 0
\(607\) 1.33826 1.48629i 1.33826 1.48629i 0.669131 0.743145i \(-0.266667\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 0.913545 0.406737i \(-0.133333\pi\)
−0.913545 + 0.406737i \(0.866667\pi\)
\(614\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(615\) 0 0
\(616\) 0 0
\(617\) −1.95630 0.415823i −1.95630 0.415823i −0.978148 0.207912i \(-0.933333\pi\)
−0.978148 0.207912i \(-0.933333\pi\)
\(618\) 0 0
\(619\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0.309017 0.951057i 0.309017 0.951057i
\(623\) 0 0
\(624\) 0 0
\(625\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0.978148 0.207912i 0.978148 0.207912i
\(631\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −0.104528 0.994522i −0.104528 0.994522i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) −0.669131 0.743145i −0.669131 0.743145i
\(640\) −0.104528 + 0.994522i −0.104528 + 0.994522i
\(641\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(642\) 0 0
\(643\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(648\) −0.978148 0.207912i −0.978148 0.207912i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.618034 1.90211i 0.618034 1.90211i 0.309017 0.951057i \(-0.400000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(654\) 0 0
\(655\) 1.61803 + 1.17557i 1.61803 + 1.17557i
\(656\) −0.500000 0.866025i −0.500000 0.866025i
\(657\) 0 0
\(658\) 1.82709 0.813473i 1.82709 0.813473i
\(659\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(660\) 0 0
\(661\) 0.978148 0.207912i 0.978148 0.207912i 0.309017 0.951057i \(-0.400000\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.104528 + 0.994522i 0.104528 + 0.994522i
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 1.33826 + 1.48629i 1.33826 + 1.48629i
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) −0.978148 0.207912i −0.978148 0.207912i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) −0.913545 0.406737i −0.913545 0.406737i −0.104528 0.994522i \(-0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) −0.618034 1.90211i −0.618034 1.90211i
\(699\) 0 0
\(700\) 0 0
\(701\) −0.669131 0.743145i −0.669131 0.743145i 0.309017 0.951057i \(-0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(708\) 0 0
\(709\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(710\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) −0.500000 0.866025i −0.500000 0.866025i
\(719\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(720\) 0.913545 0.406737i 0.913545 0.406737i
\(721\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(722\) 0 0
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 0.104528 + 0.994522i 0.104528 + 0.994522i 0.913545 + 0.406737i \(0.133333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(728\) 0 0
\(729\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) 0.104528 0.994522i 0.104528 0.994522i −0.809017 0.587785i \(-0.800000\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(739\) 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
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) −2.00000 −2.00000
\(746\) 0.309017 0.951057i 0.309017 0.951057i
\(747\) 0 0
\(748\) 0 0
\(749\) −0.500000 0.866025i −0.500000 0.866025i
\(750\) 0 0
\(751\) −0.913545 + 0.406737i −0.913545 + 0.406737i −0.809017 0.587785i \(-0.800000\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(752\) 1.61803 1.17557i 1.61803 1.17557i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(758\) 0.209057 + 1.98904i 0.209057 + 1.98904i
\(759\) 0 0
\(760\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(761\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(762\) 0 0
\(763\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.00000 −1.00000
\(777\) 0 0
\(778\) 0 0
\(779\) −0.809017 0.587785i −0.809017 0.587785i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0 0
\(785\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(786\) 0 0
\(787\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(792\) 0 0
\(793\) 0 0
\(794\) 0.669131 + 0.743145i 0.669131 + 0.743145i
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(809\) 0 0 −0.978148 0.207912i \(-0.933333\pi\)
0.978148 + 0.207912i \(0.0666667\pi\)
\(810\) −0.809017 0.587785i −0.809017 0.587785i
\(811\) −1.00000 1.73205i −1.00000 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 0.866025i \(-0.666667\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0.913545 + 0.406737i 0.913545 + 0.406737i
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(822\) 0 0
\(823\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(824\) −0.669131 + 0.743145i −0.669131 + 0.743145i
\(825\) 0 0
\(826\) 0.104528 0.994522i 0.104528 0.994522i
\(827\) 0 0 0.978148 0.207912i \(-0.0666667\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(828\) 0 0
\(829\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 1.00000 1.00000
\(839\) 0.618034 1.90211i 0.618034 1.90211i 0.309017 0.951057i \(-0.400000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(840\) 0 0
\(841\) −0.809017 0.587785i −0.809017 0.587785i
\(842\) −0.500000 0.866025i −0.500000 0.866025i
\(843\) 0 0
\(844\) 0 0
\(845\) 0.809017 0.587785i 0.809017 0.587785i
\(846\) −1.82709 0.813473i −1.82709 0.813473i
\(847\) 0.978148 0.207912i 0.978148 0.207912i
\(848\) 0 0
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.618034 + 1.90211i 0.618034 + 1.90211i 0.309017 + 0.951057i \(0.400000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(854\) 0 0
\(855\) 0.669131 0.743145i 0.669131 0.743145i
\(856\) −0.669131 0.743145i −0.669131 0.743145i
\(857\) −0.209057 + 1.98904i −0.209057 + 1.98904i −0.104528 + 0.994522i \(0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(858\) 0 0
\(859\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 1.00000 1.73205i 1.00000 1.73205i
\(863\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(864\) 0 0
\(865\) 1.95630 + 0.415823i 1.95630 + 0.415823i
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0.809017 + 0.587785i 0.809017 + 0.587785i
\(873\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(874\) 0 0
\(875\) −0.913545 + 0.406737i −0.913545 + 0.406737i
\(876\) 0 0
\(877\) −0.913545 0.406737i −0.913545 0.406737i −0.104528 0.994522i \(-0.533333\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(878\) −0.978148 + 0.207912i −0.978148 + 0.207912i
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.669131 0.743145i \(-0.266667\pi\)
−0.669131 + 0.743145i \(0.733333\pi\)
\(882\) 0 0
\(883\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0.669131 0.743145i 0.669131 0.743145i
\(887\) −0.669131 0.743145i −0.669131 0.743145i 0.309017 0.951057i \(-0.400000\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 1.00000 1.73205i 1.00000 1.73205i
\(894\) 0 0
\(895\) 0 0
\(896\) −0.978148 0.207912i −0.978148 0.207912i
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(905\) 0 0
\(906\) 0 0
\(907\) 0.809017 0.587785i 0.809017 0.587785i −0.104528 0.994522i \(-0.533333\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(908\) 0 0
\(909\) 0.978148 0.207912i 0.978148 0.207912i
\(910\) 0 0
\(911\) 0 0 −0.669131 0.743145i \(-0.733333\pi\)
0.669131 + 0.743145i \(0.266667\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −1.33826 + 1.48629i −1.33826 + 1.48629i
\(918\) 0 0
\(919\) −0.209057 + 1.98904i −0.209057 + 1.98904i −0.104528 + 0.994522i \(0.533333\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0.978148 + 0.207912i 0.978148 + 0.207912i
\(928\) 0 0
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 0 0
\(934\) −0.809017 0.587785i −0.809017 0.587785i
\(935\) 0 0
\(936\) 0 0
\(937\) 1.82709 0.813473i 1.82709 0.813473i 0.913545 0.406737i \(-0.133333\pi\)
0.913545 0.406737i \(-0.133333\pi\)
\(938\) −1.61803 + 1.17557i −1.61803 + 1.17557i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.104528 0.994522i \(-0.466667\pi\)
−0.104528 + 0.994522i \(0.533333\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) −0.104528 0.994522i −0.104528 0.994522i
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.104528 0.994522i \(-0.533333\pi\)
0.104528 + 0.994522i \(0.466667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(954\) 0 0
\(955\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(956\) 0 0
\(957\) 0 0
\(958\) −0.978148 0.207912i −0.978148 0.207912i
\(959\) 0 0
\(960\) 0 0
\(961\) 0 0
\(962\) 0 0
\(963\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(964\) 0 0
\(965\) −0.809017 0.587785i −0.809017 0.587785i
\(966\) 0 0
\(967\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(968\) 0.913545 0.406737i 0.913545 0.406737i
\(969\) 0 0
\(970\) −0.913545 0.406737i −0.913545 0.406737i
\(971\) −1.95630 + 0.415823i −1.95630 + 0.415823i −0.978148 + 0.207912i \(0.933333\pi\)
−0.978148 + 0.207912i \(0.933333\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −0.309017 0.951057i −0.309017 0.951057i −0.978148 0.207912i \(-0.933333\pi\)
0.669131 0.743145i \(-0.266667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.104528 0.994522i 0.104528 0.994522i
\(982\) 0 0
\(983\) 0 0 −0.913545 0.406737i \(-0.866667\pi\)
0.913545 + 0.406737i \(0.133333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) −0.309017 + 0.951057i −0.309017 + 0.951057i
\(995\) 0 0
\(996\) 0 0
\(997\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(998\) 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 961.1.h.a.115.1 8
31.2 even 5 961.1.e.a.522.1 2
31.3 odd 30 inner 961.1.h.a.229.1 8
31.4 even 5 inner 961.1.h.a.414.1 8
31.5 even 3 961.1.f.a.333.1 4
31.6 odd 6 inner 961.1.h.a.513.1 8
31.7 even 15 inner 961.1.h.a.117.1 8
31.8 even 5 inner 961.1.h.a.726.1 8
31.9 even 15 961.1.f.a.573.1 4
31.10 even 15 31.1.b.a.30.1 1
31.11 odd 30 961.1.f.a.430.1 4
31.12 odd 30 961.1.e.a.440.1 2
31.13 odd 30 961.1.f.a.587.1 4
31.14 even 15 inner 961.1.h.a.623.1 8
31.15 odd 10 inner 961.1.h.a.145.1 8
31.16 even 5 inner 961.1.h.a.145.1 8
31.17 odd 30 inner 961.1.h.a.623.1 8
31.18 even 15 961.1.f.a.587.1 4
31.19 even 15 961.1.e.a.440.1 2
31.20 even 15 961.1.f.a.430.1 4
31.21 odd 30 31.1.b.a.30.1 1
31.22 odd 30 961.1.f.a.573.1 4
31.23 odd 10 inner 961.1.h.a.726.1 8
31.24 odd 30 inner 961.1.h.a.117.1 8
31.25 even 3 inner 961.1.h.a.513.1 8
31.26 odd 6 961.1.f.a.333.1 4
31.27 odd 10 inner 961.1.h.a.414.1 8
31.28 even 15 inner 961.1.h.a.229.1 8
31.29 odd 10 961.1.e.a.522.1 2
31.30 odd 2 CM 961.1.h.a.115.1 8
93.41 odd 30 279.1.d.b.154.1 1
93.83 even 30 279.1.d.b.154.1 1
124.83 even 30 496.1.e.a.433.1 1
124.103 odd 30 496.1.e.a.433.1 1
155.52 even 60 775.1.c.a.774.1 2
155.72 odd 60 775.1.c.a.774.1 2
155.83 even 60 775.1.c.a.774.2 2
155.103 odd 60 775.1.c.a.774.2 2
155.114 odd 30 775.1.d.b.526.1 1
155.134 even 30 775.1.d.b.526.1 1
217.10 odd 30 1519.1.n.a.30.1 2
217.41 odd 30 1519.1.c.a.1177.1 1
217.52 even 30 1519.1.n.a.30.1 2
217.72 even 15 1519.1.n.b.557.1 2
217.83 even 30 1519.1.c.a.1177.1 1
217.103 odd 30 1519.1.n.a.557.1 2
217.114 odd 30 1519.1.n.b.557.1 2
217.145 even 30 1519.1.n.a.557.1 2
217.165 even 15 1519.1.n.b.30.1 2
217.207 odd 30 1519.1.n.b.30.1 2
248.21 odd 30 1984.1.e.a.1921.1 1
248.83 even 30 1984.1.e.b.1921.1 1
248.165 even 30 1984.1.e.a.1921.1 1
248.227 odd 30 1984.1.e.b.1921.1 1
279.41 odd 30 2511.1.m.a.1270.1 2
279.52 odd 30 2511.1.m.e.433.1 2
279.83 even 30 2511.1.m.a.433.1 2
279.103 even 15 2511.1.m.e.1270.1 2
279.176 even 30 2511.1.m.a.1270.1 2
279.196 even 15 2511.1.m.e.433.1 2
279.227 odd 30 2511.1.m.a.433.1 2
279.238 odd 30 2511.1.m.e.1270.1 2
341.10 odd 30 3751.1.d.b.1332.1 1
341.21 even 30 3751.1.d.b.1332.1 1
341.41 odd 30 3751.1.t.a.2913.1 4
341.52 even 30 3751.1.t.a.2913.1 4
341.72 odd 30 3751.1.t.a.2665.1 4
341.83 even 30 3751.1.t.a.2665.1 4
341.103 even 15 3751.1.t.c.3657.1 4
341.114 odd 30 3751.1.t.c.3657.1 4
341.134 odd 30 3751.1.t.a.2138.1 4
341.145 even 30 3751.1.t.a.2138.1 4
341.196 even 15 3751.1.t.c.2138.1 4
341.207 odd 30 3751.1.t.c.2138.1 4
341.227 odd 30 3751.1.t.a.3657.1 4
341.238 even 30 3751.1.t.a.3657.1 4
341.258 even 15 3751.1.t.c.2665.1 4
341.269 odd 30 3751.1.t.c.2665.1 4
341.289 even 15 3751.1.t.c.2913.1 4
341.300 odd 30 3751.1.t.c.2913.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
31.1.b.a.30.1 1 31.10 even 15
31.1.b.a.30.1 1 31.21 odd 30
279.1.d.b.154.1 1 93.41 odd 30
279.1.d.b.154.1 1 93.83 even 30
496.1.e.a.433.1 1 124.83 even 30
496.1.e.a.433.1 1 124.103 odd 30
775.1.c.a.774.1 2 155.52 even 60
775.1.c.a.774.1 2 155.72 odd 60
775.1.c.a.774.2 2 155.83 even 60
775.1.c.a.774.2 2 155.103 odd 60
775.1.d.b.526.1 1 155.114 odd 30
775.1.d.b.526.1 1 155.134 even 30
961.1.e.a.440.1 2 31.12 odd 30
961.1.e.a.440.1 2 31.19 even 15
961.1.e.a.522.1 2 31.2 even 5
961.1.e.a.522.1 2 31.29 odd 10
961.1.f.a.333.1 4 31.5 even 3
961.1.f.a.333.1 4 31.26 odd 6
961.1.f.a.430.1 4 31.11 odd 30
961.1.f.a.430.1 4 31.20 even 15
961.1.f.a.573.1 4 31.9 even 15
961.1.f.a.573.1 4 31.22 odd 30
961.1.f.a.587.1 4 31.13 odd 30
961.1.f.a.587.1 4 31.18 even 15
961.1.h.a.115.1 8 1.1 even 1 trivial
961.1.h.a.115.1 8 31.30 odd 2 CM
961.1.h.a.117.1 8 31.7 even 15 inner
961.1.h.a.117.1 8 31.24 odd 30 inner
961.1.h.a.145.1 8 31.15 odd 10 inner
961.1.h.a.145.1 8 31.16 even 5 inner
961.1.h.a.229.1 8 31.3 odd 30 inner
961.1.h.a.229.1 8 31.28 even 15 inner
961.1.h.a.414.1 8 31.4 even 5 inner
961.1.h.a.414.1 8 31.27 odd 10 inner
961.1.h.a.513.1 8 31.6 odd 6 inner
961.1.h.a.513.1 8 31.25 even 3 inner
961.1.h.a.623.1 8 31.14 even 15 inner
961.1.h.a.623.1 8 31.17 odd 30 inner
961.1.h.a.726.1 8 31.8 even 5 inner
961.1.h.a.726.1 8 31.23 odd 10 inner
1519.1.c.a.1177.1 1 217.41 odd 30
1519.1.c.a.1177.1 1 217.83 even 30
1519.1.n.a.30.1 2 217.10 odd 30
1519.1.n.a.30.1 2 217.52 even 30
1519.1.n.a.557.1 2 217.103 odd 30
1519.1.n.a.557.1 2 217.145 even 30
1519.1.n.b.30.1 2 217.165 even 15
1519.1.n.b.30.1 2 217.207 odd 30
1519.1.n.b.557.1 2 217.72 even 15
1519.1.n.b.557.1 2 217.114 odd 30
1984.1.e.a.1921.1 1 248.21 odd 30
1984.1.e.a.1921.1 1 248.165 even 30
1984.1.e.b.1921.1 1 248.83 even 30
1984.1.e.b.1921.1 1 248.227 odd 30
2511.1.m.a.433.1 2 279.83 even 30
2511.1.m.a.433.1 2 279.227 odd 30
2511.1.m.a.1270.1 2 279.41 odd 30
2511.1.m.a.1270.1 2 279.176 even 30
2511.1.m.e.433.1 2 279.52 odd 30
2511.1.m.e.433.1 2 279.196 even 15
2511.1.m.e.1270.1 2 279.103 even 15
2511.1.m.e.1270.1 2 279.238 odd 30
3751.1.d.b.1332.1 1 341.10 odd 30
3751.1.d.b.1332.1 1 341.21 even 30
3751.1.t.a.2138.1 4 341.134 odd 30
3751.1.t.a.2138.1 4 341.145 even 30
3751.1.t.a.2665.1 4 341.72 odd 30
3751.1.t.a.2665.1 4 341.83 even 30
3751.1.t.a.2913.1 4 341.41 odd 30
3751.1.t.a.2913.1 4 341.52 even 30
3751.1.t.a.3657.1 4 341.227 odd 30
3751.1.t.a.3657.1 4 341.238 even 30
3751.1.t.c.2138.1 4 341.196 even 15
3751.1.t.c.2138.1 4 341.207 odd 30
3751.1.t.c.2665.1 4 341.258 even 15
3751.1.t.c.2665.1 4 341.269 odd 30
3751.1.t.c.2913.1 4 341.289 even 15
3751.1.t.c.2913.1 4 341.300 odd 30
3751.1.t.c.3657.1 4 341.103 even 15
3751.1.t.c.3657.1 4 341.114 odd 30