Properties

Label 1400.1.cf.a.507.1
Level $1400$
Weight $1$
Character 1400.507
Analytic conductor $0.699$
Analytic rank $0$
Dimension $8$
Projective image $D_{6}$
CM discriminant -40
Inner twists $16$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1400,1,Mod(243,1400)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1400, base_ring=CyclotomicField(12))
 
chi = DirichletCharacter(H, H._module([6, 6, 9, 10]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1400.243");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1400 = 2^{3} \cdot 5^{2} \cdot 7 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1400.cf (of order \(12\), degree \(4\), 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: \(0.698691017686\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(2\) over \(\Q(\zeta_{12})\)
Coefficient field: \(\Q(\zeta_{24})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{8} - x^{4} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{6}\)
Projective field: Galois closure of 6.0.26891200.1

Embedding invariants

Embedding label 507.1
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 1400.507
Dual form 1400.1.cf.a.243.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(0.707107 - 0.707107i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-0.866025 + 0.500000i) q^{9} +O(q^{10})\) \(q+(-0.965926 - 0.258819i) q^{2} +(0.866025 + 0.500000i) q^{4} +(0.707107 - 0.707107i) q^{7} +(-0.707107 - 0.707107i) q^{8} +(-0.866025 + 0.500000i) q^{9} +(-0.500000 + 0.866025i) q^{11} +(-1.22474 + 1.22474i) q^{13} +(-0.866025 + 0.500000i) q^{14} +(0.500000 + 0.866025i) q^{16} +(0.965926 - 0.258819i) q^{18} +(0.866025 + 1.50000i) q^{19} +(0.707107 - 0.707107i) q^{22} +(-0.258819 + 0.965926i) q^{23} +(1.50000 - 0.866025i) q^{26} +(0.965926 - 0.258819i) q^{28} +(-0.258819 - 0.965926i) q^{32} -1.00000 q^{36} +(0.965926 + 0.258819i) q^{37} +(-0.448288 - 1.67303i) q^{38} -1.73205i q^{41} +(-0.866025 + 0.500000i) q^{44} +(0.500000 - 0.866025i) q^{46} +(-0.448288 + 1.67303i) q^{47} -1.00000i q^{49} +(-1.67303 + 0.448288i) q^{52} +(0.965926 - 0.258819i) q^{53} -1.00000 q^{56} +(-0.258819 + 0.965926i) q^{63} +1.00000i q^{64} +(0.965926 + 0.258819i) q^{72} +(-0.866025 - 0.500000i) q^{74} +1.73205i q^{76} +(0.258819 + 0.965926i) q^{77} +(0.500000 - 0.866025i) q^{81} +(-0.448288 + 1.67303i) q^{82} +(0.965926 - 0.258819i) q^{88} +1.73205i q^{91} +(-0.707107 + 0.707107i) q^{92} +(0.866025 - 1.50000i) q^{94} +(-0.258819 + 0.965926i) q^{98} -1.00000i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q - 4 q^{11} + 4 q^{16} + 12 q^{26} - 8 q^{36} + 4 q^{46} - 8 q^{56} + 4 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1400\mathbb{Z}\right)^\times\).

\(n\) \(351\) \(701\) \(801\) \(1177\)
\(\chi(n)\) \(-1\) \(-1\) \(e\left(\frac{1}{6}\right)\) \(e\left(\frac{1}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.965926 0.258819i −0.965926 0.258819i
\(3\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(4\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(5\) 0 0
\(6\) 0 0
\(7\) 0.707107 0.707107i 0.707107 0.707107i
\(8\) −0.707107 0.707107i −0.707107 0.707107i
\(9\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(10\) 0 0
\(11\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(12\) 0 0
\(13\) −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(14\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(15\) 0 0
\(16\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(17\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(18\) 0.965926 0.258819i 0.965926 0.258819i
\(19\) 0.866025 + 1.50000i 0.866025 + 1.50000i 0.866025 + 0.500000i \(0.166667\pi\)
1.00000i \(0.5\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0.707107 0.707107i 0.707107 0.707107i
\(23\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 1.50000 0.866025i 1.50000 0.866025i
\(27\) 0 0
\(28\) 0.965926 0.258819i 0.965926 0.258819i
\(29\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(30\) 0 0
\(31\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(32\) −0.258819 0.965926i −0.258819 0.965926i
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) −1.00000 −1.00000
\(37\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(38\) −0.448288 1.67303i −0.448288 1.67303i
\(39\) 0 0
\(40\) 0 0
\(41\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(42\) 0 0
\(43\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(44\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(45\) 0 0
\(46\) 0.500000 0.866025i 0.500000 0.866025i
\(47\) −0.448288 + 1.67303i −0.448288 + 1.67303i 0.258819 + 0.965926i \(0.416667\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(48\) 0 0
\(49\) 1.00000i 1.00000i
\(50\) 0 0
\(51\) 0 0
\(52\) −1.67303 + 0.448288i −1.67303 + 0.448288i
\(53\) 0.965926 0.258819i 0.965926 0.258819i 0.258819 0.965926i \(-0.416667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −1.00000 −1.00000
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(60\) 0 0
\(61\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(62\) 0 0
\(63\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(64\) 1.00000i 1.00000i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(72\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(73\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(74\) −0.866025 0.500000i −0.866025 0.500000i
\(75\) 0 0
\(76\) 1.73205i 1.73205i
\(77\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(78\) 0 0
\(79\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(80\) 0 0
\(81\) 0.500000 0.866025i 0.500000 0.866025i
\(82\) −0.448288 + 1.67303i −0.448288 + 1.67303i
\(83\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0.965926 0.258819i 0.965926 0.258819i
\(89\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(90\) 0 0
\(91\) 1.73205i 1.73205i
\(92\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(93\) 0 0
\(94\) 0.866025 1.50000i 0.866025 1.50000i
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(98\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(99\) 1.00000i 1.00000i
\(100\) 0 0
\(101\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(102\) 0 0
\(103\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(104\) 1.73205 1.73205
\(105\) 0 0
\(106\) −1.00000 −1.00000
\(107\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(108\) 0 0
\(109\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(113\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 0.448288 1.67303i 0.448288 1.67303i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 0 0
\(126\) 0.500000 0.866025i 0.500000 0.866025i
\(127\) 0.707107 0.707107i 0.707107 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(128\) 0.258819 0.965926i 0.258819 0.965926i
\(129\) 0 0
\(130\) 0 0
\(131\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(132\) 0 0
\(133\) 1.67303 + 0.448288i 1.67303 + 0.448288i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(138\) 0 0
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) −0.448288 1.67303i −0.448288 1.67303i
\(144\) −0.866025 0.500000i −0.866025 0.500000i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(149\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(152\) 0.448288 1.67303i 0.448288 1.67303i
\(153\) 0 0
\(154\) 1.00000i 1.00000i
\(155\) 0 0
\(156\) 0 0
\(157\) −1.67303 + 0.448288i −1.67303 + 0.448288i −0.965926 0.258819i \(-0.916667\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(162\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(163\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(164\) 0.866025 1.50000i 0.866025 1.50000i
\(165\) 0 0
\(166\) 0 0
\(167\) 1.22474 + 1.22474i 1.22474 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(168\) 0 0
\(169\) 2.00000i 2.00000i
\(170\) 0 0
\(171\) −1.50000 0.866025i −1.50000 0.866025i
\(172\) 0 0
\(173\) −1.67303 0.448288i −1.67303 0.448288i −0.707107 0.707107i \(-0.750000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) −1.00000 −1.00000
\(177\) 0 0
\(178\) 0 0
\(179\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0.448288 1.67303i 0.448288 1.67303i
\(183\) 0 0
\(184\) 0.866025 0.500000i 0.866025 0.500000i
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) −1.22474 + 1.22474i −1.22474 + 1.22474i
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) 0 0
\(193\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.500000 0.866025i 0.500000 0.866025i
\(197\) −0.707107 + 0.707107i −0.707107 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(198\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(199\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −0.258819 0.965926i −0.258819 0.965926i
\(208\) −1.67303 0.448288i −1.67303 0.448288i
\(209\) −1.73205 −1.73205
\(210\) 0 0
\(211\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(212\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0 0
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) −0.866025 0.500000i −0.866025 0.500000i
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(228\) 0 0
\(229\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(234\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −1.50000 0.866025i −1.50000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
−1.00000 \(\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) −2.89778 0.776457i −2.89778 0.776457i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(252\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(253\) −0.707107 0.707107i −0.707107 0.707107i
\(254\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(255\) 0 0
\(256\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(257\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(258\) 0 0
\(259\) 0.866025 0.500000i 0.866025 0.500000i
\(260\) 0 0
\(261\) 0 0
\(262\) 1.67303 0.448288i 1.67303 0.448288i
\(263\) 1.93185 0.517638i 1.93185 0.517638i 0.965926 0.258819i \(-0.0833333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) −1.50000 0.866025i −1.50000 0.866025i
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(270\) 0 0
\(271\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(272\) 0 0
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) −0.517638 1.93185i −0.517638 1.93185i −0.258819 0.965926i \(-0.583333\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(282\) 0 0
\(283\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 1.73205i 1.73205i
\(287\) −1.22474 1.22474i −1.22474 1.22474i
\(288\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(289\) 0.866025 0.500000i 0.866025 0.500000i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.22474 1.22474i 1.22474 1.22474i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.500000 0.866025i −0.500000 0.866025i
\(297\) 0 0
\(298\) 0 0
\(299\) −0.866025 1.50000i −0.866025 1.50000i
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(305\) 0 0
\(306\) 0 0
\(307\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(308\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(312\) 0 0
\(313\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(314\) 1.73205 1.73205
\(315\) 0 0
\(316\) 0 0
\(317\) 1.93185 + 0.517638i 1.93185 + 0.517638i 0.965926 + 0.258819i \(0.0833333\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) −0.258819 0.965926i −0.258819 0.965926i
\(323\) 0 0
\(324\) 0.866025 0.500000i 0.866025 0.500000i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) −1.22474 + 1.22474i −1.22474 + 1.22474i
\(329\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(330\) 0 0
\(331\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(332\) 0 0
\(333\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(334\) −0.866025 1.50000i −0.866025 1.50000i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(338\) −0.517638 + 1.93185i −0.517638 + 1.93185i
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(343\) −0.707107 0.707107i −0.707107 0.707107i
\(344\) 0 0
\(345\) 0 0
\(346\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(347\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(353\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) −0.707107 0.707107i −0.707107 0.707107i
\(359\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(360\) 0 0
\(361\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(362\) 0 0
\(363\) 0 0
\(364\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(365\) 0 0
\(366\) 0 0
\(367\) 1.67303 0.448288i 1.67303 0.448288i 0.707107 0.707107i \(-0.250000\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(368\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(369\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(370\) 0 0
\(371\) 0.500000 0.866025i 0.500000 0.866025i
\(372\) 0 0
\(373\) 0.517638 1.93185i 0.517638 1.93185i 0.258819 0.965926i \(-0.416667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 1.50000 0.866025i 1.50000 0.866025i
\(377\) 0 0
\(378\) 0 0
\(379\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 1.67303 + 0.448288i 1.67303 + 0.448288i 0.965926 0.258819i \(-0.0833333\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(393\) 0 0
\(394\) 0.866025 0.500000i 0.866025 0.500000i
\(395\) 0 0
\(396\) 0.500000 0.866025i 0.500000 0.866025i
\(397\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0 0
\(401\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(408\) 0 0
\(409\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 1.00000i 1.00000i
\(415\) 0 0
\(416\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(417\) 0 0
\(418\) 1.67303 + 0.448288i 1.67303 + 0.448288i
\(419\) −1.73205 −1.73205 −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(420\) 0 0
\(421\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(422\) −0.965926 0.258819i −0.965926 0.258819i
\(423\) −0.448288 1.67303i −0.448288 1.67303i
\(424\) −0.866025 0.500000i −0.866025 0.500000i
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(432\) 0 0
\(433\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) −1.67303 + 0.448288i −1.67303 + 0.448288i
\(438\) 0 0
\(439\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(440\) 0 0
\(441\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(442\) 0 0
\(443\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(449\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(450\) 0 0
\(451\) 1.50000 + 0.866025i 1.50000 + 0.866025i
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(462\) 0 0
\(463\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.258819 0.965926i \(-0.416667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(468\) 1.22474 1.22474i 1.22474 1.22474i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(478\) 0 0
\(479\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(480\) 0 0
\(481\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(482\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 0.517638 + 1.93185i 0.517638 + 1.93185i 0.258819 + 0.965926i \(0.416667\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(492\) 0 0
\(493\) 0 0
\(494\) 2.59808 + 1.50000i 2.59808 + 1.50000i
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) −1.73205 + 1.00000i −1.73205 + 1.00000i −0.866025 + 0.500000i \(0.833333\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) −0.448288 + 1.67303i −0.448288 + 1.67303i
\(503\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(504\) 0.866025 0.500000i 0.866025 0.500000i
\(505\) 0 0
\(506\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(507\) 0 0
\(508\) 0.965926 0.258819i 0.965926 0.258819i
\(509\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.707107 0.707107i 0.707107 0.707107i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) −1.22474 1.22474i −1.22474 1.22474i
\(518\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(519\) 0 0
\(520\) 0 0
\(521\) 1.50000 + 0.866025i 1.50000 + 0.866025i 1.00000 \(0\)
0.500000 + 0.866025i \(0.333333\pi\)
\(522\) 0 0
\(523\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(524\) −1.73205 −1.73205
\(525\) 0 0
\(526\) −2.00000 −2.00000
\(527\) 0 0
\(528\) 0 0
\(529\) 0 0
\(530\) 0 0
\(531\) 0 0
\(532\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(533\) 2.12132 + 2.12132i 2.12132 + 2.12132i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(540\) 0 0
\(541\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 2.00000i 2.00000i
\(555\) 0 0
\(556\) 0 0
\(557\) 0.258819 + 0.965926i 0.258819 + 0.965926i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) −0.965926 0.258819i −0.965926 0.258819i
\(563\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.258819 0.965926i −0.258819 0.965926i
\(568\) 0 0
\(569\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(570\) 0 0
\(571\) 1.00000 1.73205i 1.00000 1.73205i 0.500000 0.866025i \(-0.333333\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(572\) 0.448288 1.67303i 0.448288 1.67303i
\(573\) 0 0
\(574\) 0.866025 + 1.50000i 0.866025 + 1.50000i
\(575\) 0 0
\(576\) −0.500000 0.866025i −0.500000 0.866025i
\(577\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(578\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(579\) 0 0
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(584\) 0 0
\(585\) 0 0
\(586\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(587\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(593\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0 0
\(598\) 0.448288 + 1.67303i 0.448288 + 1.67303i
\(599\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) −0.448288 + 1.67303i −0.448288 + 1.67303i 0.258819 + 0.965926i \(0.416667\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(608\) 1.22474 1.22474i 1.22474 1.22474i
\(609\) 0 0
\(610\) 0 0
\(611\) −1.50000 2.59808i −1.50000 2.59808i
\(612\) 0 0
\(613\) 0.965926 0.258819i 0.965926 0.258819i 0.258819 0.965926i \(-0.416667\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0.500000 0.866025i 0.500000 0.866025i
\(617\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(618\) 0 0
\(619\) −0.866025 + 1.50000i −0.866025 + 1.50000i 1.00000i \(0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) −1.67303 0.448288i −1.67303 0.448288i
\(629\) 0 0
\(630\) 0 0
\(631\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −1.73205 1.00000i −1.73205 1.00000i
\(635\) 0 0
\(636\) 0 0
\(637\) 1.22474 + 1.22474i 1.22474 + 1.22474i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(644\) 1.00000i 1.00000i
\(645\) 0 0
\(646\) 0 0
\(647\) 1.67303 0.448288i 1.67303 0.448288i 0.707107 0.707107i \(-0.250000\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(648\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0.258819 0.965926i 0.258819 0.965926i −0.707107 0.707107i \(-0.750000\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.50000 0.866025i 1.50000 0.866025i
\(657\) 0 0
\(658\) −0.448288 1.67303i −0.448288 1.67303i
\(659\) 2.00000i 2.00000i 1.00000i \(-0.5\pi\)
1.00000i \(-0.5\pi\)
\(660\) 0 0
\(661\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(662\) −0.258819 0.965926i −0.258819 0.965926i
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 1.00000 1.00000
\(667\) 0 0
\(668\) 0.448288 + 1.67303i 0.448288 + 1.67303i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 1.00000 1.73205i 1.00000 1.73205i
\(677\) 0.448288 1.67303i 0.448288 1.67303i −0.258819 0.965926i \(-0.583333\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(684\) −0.866025 1.50000i −0.866025 1.50000i
\(685\) 0 0
\(686\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(687\) 0 0
\(688\) 0 0
\(689\) −0.866025 + 1.50000i −0.866025 + 1.50000i
\(690\) 0 0
\(691\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(692\) −1.22474 1.22474i −1.22474 1.22474i
\(693\) −0.707107 0.707107i −0.707107 0.707107i
\(694\) 0 0
\(695\) 0 0
\(696\) 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 0
\(703\) 0.448288 + 1.67303i 0.448288 + 1.67303i
\(704\) −0.866025 0.500000i −0.866025 0.500000i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.41421 1.41421i 1.41421 1.41421i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) −1.22474 1.22474i −1.22474 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 0.965926i \(-0.583333\pi\)
\(728\) 1.22474 1.22474i 1.22474 1.22474i
\(729\) 1.00000i 1.00000i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.67303 0.448288i −1.67303 0.448288i −0.707107 0.707107i \(-0.750000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(734\) −1.73205 −1.73205
\(735\) 0 0
\(736\) 1.00000 1.00000
\(737\) 0 0
\(738\) −0.448288 1.67303i −0.448288 1.67303i
\(739\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(743\) −0.707107 0.707107i −0.707107 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(752\) −1.67303 + 0.448288i −1.67303 + 0.448288i
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(758\) 0.258819 0.965926i 0.258819 0.965926i
\(759\) 0 0
\(760\) 0 0
\(761\) −1.50000 + 0.866025i −1.50000 + 0.866025i −0.500000 + 0.866025i \(0.666667\pi\)
−1.00000 \(\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) −1.50000 0.866025i −1.50000 0.866025i
\(767\) 0 0
\(768\) 0 0
\(769\) 1.73205 1.73205 0.866025 0.500000i \(-0.166667\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0.448288 + 1.67303i 0.448288 + 1.67303i 0.707107 + 0.707107i \(0.250000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 2.59808 1.50000i 2.59808 1.50000i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.866025 0.500000i 0.866025 0.500000i
\(785\) 0 0
\(786\) 0 0
\(787\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(788\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(810\) 0 0
\(811\) 1.73205i 1.73205i −0.500000 0.866025i \(-0.666667\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0.866025 0.500000i 0.866025 0.500000i
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −0.866025 1.50000i −0.866025 1.50000i
\(820\) 0 0
\(821\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(822\) 0 0
\(823\) −1.93185 + 0.517638i −1.93185 + 0.517638i −0.965926 + 0.258819i \(0.916667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(828\) 0.258819 0.965926i 0.258819 0.965926i
\(829\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.22474 1.22474i −1.22474 1.22474i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) −1.50000 0.866025i −1.50000 0.866025i
\(837\) 0 0
\(838\) 1.67303 + 0.448288i 1.67303 + 0.448288i
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) −1.00000 −1.00000
\(842\) 0 0
\(843\) 0 0
\(844\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(845\) 0 0
\(846\) 1.73205i 1.73205i
\(847\) 0 0
\(848\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(849\) 0 0
\(850\) 0 0
\(851\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(852\) 0 0
\(853\) −1.22474 + 1.22474i −1.22474 + 1.22474i −0.258819 + 0.965926i \(0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.965926 0.258819i \(-0.0833333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(858\) 0 0
\(859\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) −0.258819 + 0.965926i −0.258819 + 0.965926i 0.707107 + 0.707107i \(0.250000\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 1.73205 1.73205
\(875\) 0 0
\(876\) 0 0
\(877\) 0.965926 + 0.258819i 0.965926 + 0.258819i 0.707107 0.707107i \(-0.250000\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.73205i 1.73205i 0.500000 + 0.866025i \(0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(882\) −0.258819 0.965926i −0.258819 0.965926i
\(883\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(888\) 0 0
\(889\) 1.00000i 1.00000i
\(890\) 0 0
\(891\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(892\) 0 0
\(893\) −2.89778 + 0.776457i −2.89778 + 0.776457i
\(894\) 0 0
\(895\) 0 0
\(896\) −0.500000 0.866025i −0.500000 0.866025i
\(897\) 0 0
\(898\) 0.258819 0.965926i 0.258819 0.965926i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) −1.22474 1.22474i −1.22474 1.22474i
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) −0.448288 + 1.67303i −0.448288 + 1.67303i
\(918\) 0 0
\(919\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(927\) 0 0
\(928\) 0 0
\(929\) −0.866025 1.50000i −0.866025 1.50000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(-0.5\pi\)
\(930\) 0 0
\(931\) 1.50000 0.866025i 1.50000 0.866025i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(937\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(942\) 0 0
\(943\) 1.67303 + 0.448288i 1.67303 + 0.448288i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.965926 0.258819i \(-0.916667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(954\) 0.866025 0.500000i 0.866025 0.500000i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(962\) 1.67303 0.448288i 1.67303 0.448288i
\(963\) 0 0
\(964\) −0.866025 1.50000i −0.866025 1.50000i
\(965\) 0 0
\(966\) 0 0
\(967\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 1.50000 0.866025i 1.50000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
1.00000 \(0\)
\(972\) 0 0
\(973\) 0 0
\(974\) 2.00000i 2.00000i
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) −1.93185 0.517638i −1.93185 0.517638i
\(983\) 0.448288 + 1.67303i 0.448288 + 1.67303i 0.707107 + 0.707107i \(0.250000\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −2.12132 2.12132i −2.12132 2.12132i
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 −0.258819 0.965926i \(-0.583333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(998\) 1.93185 0.517638i 1.93185 0.517638i
\(999\) 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 1400.1.cf.a.507.1 yes 8
5.2 odd 4 inner 1400.1.cf.a.843.2 yes 8
5.3 odd 4 inner 1400.1.cf.a.843.1 yes 8
5.4 even 2 inner 1400.1.cf.a.507.2 yes 8
7.5 odd 6 inner 1400.1.cf.a.1307.2 yes 8
8.3 odd 2 inner 1400.1.cf.a.507.2 yes 8
35.12 even 12 inner 1400.1.cf.a.243.1 8
35.19 odd 6 inner 1400.1.cf.a.1307.1 yes 8
35.33 even 12 inner 1400.1.cf.a.243.2 yes 8
40.3 even 4 inner 1400.1.cf.a.843.2 yes 8
40.19 odd 2 CM 1400.1.cf.a.507.1 yes 8
40.27 even 4 inner 1400.1.cf.a.843.1 yes 8
56.19 even 6 inner 1400.1.cf.a.1307.1 yes 8
280.19 even 6 inner 1400.1.cf.a.1307.2 yes 8
280.187 odd 12 inner 1400.1.cf.a.243.2 yes 8
280.243 odd 12 inner 1400.1.cf.a.243.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1400.1.cf.a.243.1 8 35.12 even 12 inner
1400.1.cf.a.243.1 8 280.243 odd 12 inner
1400.1.cf.a.243.2 yes 8 35.33 even 12 inner
1400.1.cf.a.243.2 yes 8 280.187 odd 12 inner
1400.1.cf.a.507.1 yes 8 1.1 even 1 trivial
1400.1.cf.a.507.1 yes 8 40.19 odd 2 CM
1400.1.cf.a.507.2 yes 8 5.4 even 2 inner
1400.1.cf.a.507.2 yes 8 8.3 odd 2 inner
1400.1.cf.a.843.1 yes 8 5.3 odd 4 inner
1400.1.cf.a.843.1 yes 8 40.27 even 4 inner
1400.1.cf.a.843.2 yes 8 5.2 odd 4 inner
1400.1.cf.a.843.2 yes 8 40.3 even 4 inner
1400.1.cf.a.1307.1 yes 8 35.19 odd 6 inner
1400.1.cf.a.1307.1 yes 8 56.19 even 6 inner
1400.1.cf.a.1307.2 yes 8 7.5 odd 6 inner
1400.1.cf.a.1307.2 yes 8 280.19 even 6 inner