Properties

Label 3675.1.u.d.1451.2
Level $3675$
Weight $1$
Character 3675.1451
Analytic conductor $1.834$
Analytic rank $0$
Dimension $8$
Projective image $S_{4}$
CM/RM no
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3675,1,Mod(851,3675)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3675, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([3, 0, 4]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3675.851");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3675 = 3 \cdot 5^{2} \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3675.u (of order \(6\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(1.83406392143\)
Analytic rank: \(0\)
Dimension: \(8\)
Relative dimension: \(4\) over \(\Q(\zeta_{6})\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(S_{4}\)
Projective field: Galois closure of 4.2.77175.1

Embedding invariants

Embedding label 1451.2
Root \(-0.965926 + 0.258819i\) of defining polynomial
Character \(\chi\) \(=\) 3675.1451
Dual form 3675.1.u.d.851.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.965926 + 0.258819i) q^{3} +(-0.707107 - 0.707107i) q^{6} +1.00000i q^{8} +(0.866025 + 0.500000i) q^{9} +O(q^{10})\) \(q+(-0.866025 - 0.500000i) q^{2} +(0.965926 + 0.258819i) q^{3} +(-0.707107 - 0.707107i) q^{6} +1.00000i q^{8} +(0.866025 + 0.500000i) q^{9} +(0.866025 - 0.500000i) q^{11} -1.41421 q^{13} +(0.500000 - 0.866025i) q^{16} +(1.22474 - 0.707107i) q^{17} +(-0.500000 - 0.866025i) q^{18} -1.00000 q^{22} +(0.866025 + 0.500000i) q^{23} +(-0.258819 + 0.965926i) q^{24} +(1.22474 + 0.707107i) q^{26} +(0.707107 + 0.707107i) q^{27} +1.00000i q^{29} +(0.965926 - 0.258819i) q^{33} -1.41421 q^{34} +(-0.500000 + 0.866025i) q^{37} +(-1.36603 - 0.366025i) q^{39} -1.41421i q^{41} -1.00000 q^{43} +(-0.500000 - 0.866025i) q^{46} +(0.707107 - 0.707107i) q^{48} +(1.36603 - 0.366025i) q^{51} +(-0.258819 - 0.965926i) q^{54} +(0.500000 - 0.866025i) q^{58} +(1.22474 - 0.707107i) q^{59} +(0.707107 - 1.22474i) q^{61} -1.00000 q^{64} +(-0.965926 - 0.258819i) q^{66} +(0.500000 + 0.866025i) q^{67} +(0.707107 + 0.707107i) q^{69} +1.00000i q^{71} +(-0.500000 + 0.866025i) q^{72} +(0.866025 - 0.500000i) q^{74} +(1.00000 + 1.00000i) q^{78} +(0.500000 - 0.866025i) q^{79} +(0.500000 + 0.866025i) q^{81} +(-0.707107 + 1.22474i) q^{82} +(0.866025 + 0.500000i) q^{86} +(-0.258819 + 0.965926i) q^{87} +(0.500000 + 0.866025i) q^{88} +(1.22474 + 0.707107i) q^{89} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 8 q+O(q^{10}) \) Copy content Toggle raw display \( 8 q + 4 q^{16} - 4 q^{18} - 8 q^{22} - 4 q^{37} - 4 q^{39} - 8 q^{43} - 4 q^{46} + 4 q^{51} + 4 q^{58} - 8 q^{64} + 4 q^{67} - 4 q^{72} + 8 q^{78} + 4 q^{79} + 4 q^{81} + 4 q^{88} + 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3675\mathbb{Z}\right)^\times\).

\(n\) \(1177\) \(1226\) \(2551\)
\(\chi(n)\) \(1\) \(-1\) \(e\left(\frac{1}{3}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(3\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(4\) 0 0
\(5\) 0 0
\(6\) −0.707107 0.707107i −0.707107 0.707107i
\(7\) 0 0
\(8\) 1.00000i 1.00000i
\(9\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(10\) 0 0
\(11\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(12\) 0 0
\(13\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.500000 0.866025i 0.500000 0.866025i
\(17\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(18\) −0.500000 0.866025i −0.500000 0.866025i
\(19\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) −1.00000 −1.00000
\(23\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(24\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(25\) 0 0
\(26\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(27\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(28\) 0 0
\(29\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(30\) 0 0
\(31\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(32\) 0 0
\(33\) 0.965926 0.258819i 0.965926 0.258819i
\(34\) −1.41421 −1.41421
\(35\) 0 0
\(36\) 0 0
\(37\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(38\) 0 0
\(39\) −1.36603 0.366025i −1.36603 0.366025i
\(40\) 0 0
\(41\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(42\) 0 0
\(43\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −0.500000 0.866025i −0.500000 0.866025i
\(47\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(48\) 0.707107 0.707107i 0.707107 0.707107i
\(49\) 0 0
\(50\) 0 0
\(51\) 1.36603 0.366025i 1.36603 0.366025i
\(52\) 0 0
\(53\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(54\) −0.258819 0.965926i −0.258819 0.965926i
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 0.500000 0.866025i 0.500000 0.866025i
\(59\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(60\) 0 0
\(61\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −1.00000 −1.00000
\(65\) 0 0
\(66\) −0.965926 0.258819i −0.965926 0.258819i
\(67\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(68\) 0 0
\(69\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(70\) 0 0
\(71\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(72\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(73\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(74\) 0.866025 0.500000i 0.866025 0.500000i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(79\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(80\) 0 0
\(81\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(82\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(83\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(87\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(88\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(89\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(98\) 0 0
\(99\) 1.00000 1.00000
\(100\) 0 0
\(101\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(102\) −1.36603 0.366025i −1.36603 0.366025i
\(103\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(104\) 1.41421i 1.41421i
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(108\) 0 0
\(109\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(110\) 0 0
\(111\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(112\) 0 0
\(113\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.22474 0.707107i −1.22474 0.707107i
\(118\) −1.41421 −1.41421
\(119\) 0 0
\(120\) 0 0
\(121\) 0 0
\(122\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(123\) 0.366025 1.36603i 0.366025 1.36603i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(128\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(129\) −0.965926 0.258819i −0.965926 0.258819i
\(130\) 0 0
\(131\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 1.00000i 1.00000i
\(135\) 0 0
\(136\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(137\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(138\) −0.258819 0.965926i −0.258819 0.965926i
\(139\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0.500000 0.866025i 0.500000 0.866025i
\(143\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(144\) 0.866025 0.500000i 0.866025 0.500000i
\(145\) 0 0
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(150\) 0 0
\(151\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(152\) 0 0
\(153\) 1.41421 1.41421
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(158\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 1.00000i 1.00000i
\(163\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(168\) 0 0
\(169\) 1.00000 1.00000
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(174\) 0.707107 0.707107i 0.707107 0.707107i
\(175\) 0 0
\(176\) 1.00000i 1.00000i
\(177\) 1.36603 0.366025i 1.36603 0.366025i
\(178\) −0.707107 1.22474i −0.707107 1.22474i
\(179\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(180\) 0 0
\(181\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(182\) 0 0
\(183\) 1.00000 1.00000i 1.00000 1.00000i
\(184\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(185\) 0 0
\(186\) 0 0
\(187\) 0.707107 1.22474i 0.707107 1.22474i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(192\) −0.965926 0.258819i −0.965926 0.258819i
\(193\) 0.500000 + 0.866025i 0.500000 + 0.866025i 1.00000 \(0\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(198\) −0.866025 0.500000i −0.866025 0.500000i
\(199\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(200\) 0 0
\(201\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(208\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(214\) 0 0
\(215\) 0 0
\(216\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(217\) 0 0
\(218\) 1.00000i 1.00000i
\(219\) 0 0
\(220\) 0 0
\(221\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(222\) 0.965926 0.258819i 0.965926 0.258819i
\(223\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(227\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(228\) 0 0
\(229\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.00000 −1.00000
\(233\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(234\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(235\) 0 0
\(236\) 0 0
\(237\) 0.707107 0.707107i 0.707107 0.707107i
\(238\) 0 0
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(242\) 0 0
\(243\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(244\) 0 0
\(245\) 0 0
\(246\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(252\) 0 0
\(253\) 1.00000 1.00000
\(254\) −0.866025 0.500000i −0.866025 0.500000i
\(255\) 0 0
\(256\) 0 0
\(257\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(258\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(259\) 0 0
\(260\) 0 0
\(261\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(262\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(263\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(264\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(265\) 0 0
\(266\) 0 0
\(267\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(268\) 0 0
\(269\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(270\) 0 0
\(271\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(272\) 1.41421i 1.41421i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(282\) 0 0
\(283\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 1.41421 1.41421
\(287\) 0 0
\(288\) 0 0
\(289\) 0.500000 0.866025i 0.500000 0.866025i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −0.866025 0.500000i −0.866025 0.500000i
\(297\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(298\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(299\) −1.22474 0.707107i −1.22474 0.707107i
\(300\) 0 0
\(301\) 0 0
\(302\) 1.00000i 1.00000i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) −1.22474 0.707107i −1.22474 0.707107i
\(307\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(312\) 0.366025 1.36603i 0.366025 1.36603i
\(313\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(314\) 1.41421i 1.41421i
\(315\) 0 0
\(316\) 0 0
\(317\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(318\) 0 0
\(319\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(328\) 1.41421 1.41421
\(329\) 0 0
\(330\) 0 0
\(331\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(332\) 0 0
\(333\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(334\) 0.707107 1.22474i 0.707107 1.22474i
\(335\) 0 0
\(336\) 0 0
\(337\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(338\) −0.866025 0.500000i −0.866025 0.500000i
\(339\) 0.258819 0.965926i 0.258819 0.965926i
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 1.00000i 1.00000i
\(345\) 0 0
\(346\) 0 0
\(347\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(348\) 0 0
\(349\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(350\) 0 0
\(351\) −1.00000 1.00000i −1.00000 1.00000i
\(352\) 0 0
\(353\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(354\) −1.36603 0.366025i −1.36603 0.366025i
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(360\) 0 0
\(361\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(362\) 0 0
\(363\) 0 0
\(364\) 0 0
\(365\) 0 0
\(366\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(367\) −0.707107 1.22474i −0.707107 1.22474i −0.965926 0.258819i \(-0.916667\pi\)
0.258819 0.965926i \(-0.416667\pi\)
\(368\) 0.866025 0.500000i 0.866025 0.500000i
\(369\) 0.707107 1.22474i 0.707107 1.22474i
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(374\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(375\) 0 0
\(376\) 0 0
\(377\) 1.41421i 1.41421i
\(378\) 0 0
\(379\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(380\) 0 0
\(381\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(382\) 0 0
\(383\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(384\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(385\) 0 0
\(386\) 1.00000i 1.00000i
\(387\) −0.866025 0.500000i −0.866025 0.500000i
\(388\) 0 0
\(389\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) 1.41421 1.41421
\(392\) 0 0
\(393\) −1.00000 1.00000i −1.00000 1.00000i
\(394\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(395\) 0 0
\(396\) 0 0
\(397\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(398\) 1.41421i 1.41421i
\(399\) 0 0
\(400\) 0 0
\(401\) −0.866025 0.500000i −0.866025 0.500000i 1.00000i \(-0.5\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(402\) 0.258819 0.965926i 0.258819 0.965926i
\(403\) 0 0
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 1.00000i 1.00000i
\(408\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(409\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 1.00000i 1.00000i
\(415\) 0 0
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(420\) 0 0
\(421\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0.707107 0.707107i 0.707107 0.707107i
\(427\) 0 0
\(428\) 0 0
\(429\) −1.36603 + 0.366025i −1.36603 + 0.366025i
\(430\) 0 0
\(431\) 1.73205 1.00000i 1.73205 1.00000i 0.866025 0.500000i \(-0.166667\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(432\) 0.965926 0.258819i 0.965926 0.258819i
\(433\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 2.00000 2.00000
\(443\) −1.73205 1.00000i −1.73205 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
−0.866025 0.500000i \(-0.833333\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) −0.707107 0.707107i −0.707107 0.707107i
\(448\) 0 0
\(449\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(450\) 0 0
\(451\) −0.707107 1.22474i −0.707107 1.22474i
\(452\) 0 0
\(453\) −0.258819 0.965926i −0.258819 0.965926i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(458\) 1.22474 0.707107i 1.22474 0.707107i
\(459\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(460\) 0 0
\(461\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 0 0
\(463\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(464\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(465\) 0 0
\(466\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(467\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) −0.366025 1.36603i −0.366025 1.36603i
\(472\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(473\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(474\) −0.965926 + 0.258819i −0.965926 + 0.258819i
\(475\) 0 0
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(480\) 0 0
\(481\) 0.707107 1.22474i 0.707107 1.22474i
\(482\) 0 0
\(483\) 0 0
\(484\) 0 0
\(485\) 0 0
\(486\) 0.258819 0.965926i 0.258819 0.965926i
\(487\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(488\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(489\) 0 0
\(490\) 0 0
\(491\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(492\) 0 0
\(493\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(500\) 0 0
\(501\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(502\) 0.707107 1.22474i 0.707107 1.22474i
\(503\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) −0.866025 0.500000i −0.866025 0.500000i
\(507\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(508\) 0 0
\(509\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 1.00000i 1.00000i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(522\) 0.866025 0.500000i 0.866025 0.500000i
\(523\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −1.00000 −1.00000
\(527\) 0 0
\(528\) 0.258819 0.965926i 0.258819 0.965926i
\(529\) 0 0
\(530\) 0 0
\(531\) 1.41421 1.41421
\(532\) 0 0
\(533\) 2.00000i 2.00000i
\(534\) −0.366025 1.36603i −0.366025 1.36603i
\(535\) 0 0
\(536\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.500000 + 0.866025i −0.500000 + 0.866025i 0.500000 + 0.866025i \(0.333333\pi\)
−1.00000 \(\pi\)
\(542\) 1.22474 0.707107i 1.22474 0.707107i
\(543\) 0 0
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(548\) 0 0
\(549\) 1.22474 0.707107i 1.22474 0.707107i
\(550\) 0 0
\(551\) 0 0
\(552\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(553\) 0 0
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(558\) 0 0
\(559\) 1.41421 1.41421
\(560\) 0 0
\(561\) 1.00000 1.00000i 1.00000 1.00000i
\(562\) 0.500000 0.866025i 0.500000 0.866025i
\(563\) 1.22474 0.707107i 1.22474 0.707107i 0.258819 0.965926i \(-0.416667\pi\)
0.965926 + 0.258819i \(0.0833333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 1.41421i 1.41421i
\(567\) 0 0
\(568\) −1.00000 −1.00000
\(569\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.866025 0.500000i −0.866025 0.500000i
\(577\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(578\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(579\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0.258819 0.965926i 0.258819 0.965926i
\(592\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(593\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(594\) −0.707107 0.707107i −0.707107 0.707107i
\(595\) 0 0
\(596\) 0 0
\(597\) −0.366025 1.36603i −0.366025 1.36603i
\(598\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(599\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(600\) 0 0
\(601\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(602\) 0 0
\(603\) 1.00000i 1.00000i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(614\) 1.22474 + 0.707107i 1.22474 + 0.707107i
\(615\) 0 0
\(616\) 0 0
\(617\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(618\) 0 0
\(619\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(620\) 0 0
\(621\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(622\) −1.41421 −1.41421
\(623\) 0 0
\(624\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(625\) 0 0
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 1.41421i 1.41421i
\(630\) 0 0
\(631\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(632\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(633\) 0 0
\(634\) −0.500000 0.866025i −0.500000 0.866025i
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 1.00000i 1.00000i
\(639\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(640\) 0 0
\(641\) −0.866025 + 0.500000i −0.866025 + 0.500000i −0.866025 0.500000i \(-0.833333\pi\)
1.00000i \(0.5\pi\)
\(642\) 0 0
\(643\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(648\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(649\) 0.707107 1.22474i 0.707107 1.22474i
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0.258819 0.965926i 0.258819 0.965926i
\(655\) 0 0
\(656\) −1.22474 0.707107i −1.22474 0.707107i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(660\) 0 0
\(661\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(662\) 0.866025 0.500000i 0.866025 0.500000i
\(663\) −1.93185 + 0.517638i −1.93185 + 0.517638i
\(664\) 0 0
\(665\) 0 0
\(666\) 1.00000 1.00000
\(667\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 1.41421i 1.41421i
\(672\) 0 0
\(673\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(678\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(688\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(689\) 0 0
\(690\) 0 0
\(691\) −0.707107 + 1.22474i −0.707107 + 1.22474i 0.258819 + 0.965926i \(0.416667\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −1.00000 −1.00000
\(695\) 0 0
\(696\) −0.965926 0.258819i −0.965926 0.258819i
\(697\) −1.00000 1.73205i −1.00000 1.73205i
\(698\) 0 0
\(699\) −0.707107 0.707107i −0.707107 0.707107i
\(700\) 0 0
\(701\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(702\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(703\) 0 0
\(704\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.00000 + 1.73205i −1.00000 + 1.73205i −0.500000 + 0.866025i \(0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 0.866025 0.500000i 0.866025 0.500000i
\(712\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) −0.500000 0.866025i −0.500000 0.866025i
\(719\) −1.22474 0.707107i −1.22474 0.707107i −0.258819 0.965926i \(-0.583333\pi\)
−0.965926 + 0.258819i \(0.916667\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 1.00000i 1.00000i
\(723\) 0 0
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(728\) 0 0
\(729\) 1.00000i 1.00000i
\(730\) 0 0
\(731\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(732\) 0 0
\(733\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(734\) 1.41421i 1.41421i
\(735\) 0 0
\(736\) 0 0
\(737\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(738\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(739\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.866025 0.500000i 0.866025 0.500000i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(752\) 0 0
\(753\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(754\) −0.707107 + 1.22474i −0.707107 + 1.22474i
\(755\) 0 0
\(756\) 0 0
\(757\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(758\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(759\) 0.965926 + 0.258819i 0.965926 + 0.258819i
\(760\) 0 0
\(761\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(762\) −0.707107 0.707107i −0.707107 0.707107i
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(767\) −1.73205 + 1.00000i −1.73205 + 1.00000i
\(768\) 0 0
\(769\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0 0
\(773\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(774\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 1.00000 1.00000
\(779\) 0 0
\(780\) 0 0
\(781\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(782\) −1.22474 0.707107i −1.22474 0.707107i
\(783\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(784\) 0 0
\(785\) 0 0
\(786\) 0.366025 + 1.36603i 0.366025 + 1.36603i
\(787\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(788\) 0 0
\(789\) 0.965926 0.258819i 0.965926 0.258819i
\(790\) 0 0
\(791\) 0 0
\(792\) 1.00000i 1.00000i
\(793\) −1.00000 + 1.73205i −1.00000 + 1.73205i
\(794\) 1.22474 0.707107i 1.22474 0.707107i
\(795\) 0 0
\(796\) 0 0
\(797\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0.707107 + 1.22474i 0.707107 + 1.22474i
\(802\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(803\) 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 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(810\) 0 0
\(811\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(812\) 0 0
\(813\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(814\) 0.500000 0.866025i 0.500000 0.866025i
\(815\) 0 0
\(816\) 0.366025 1.36603i 0.366025 1.36603i
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(822\) 0 0
\(823\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(828\) 0 0
\(829\) 0.707107 + 1.22474i 0.707107 + 1.22474i 0.965926 + 0.258819i \(0.0833333\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 1.41421 1.41421
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(843\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 0 0
\(848\) 0 0
\(849\) −0.366025 1.36603i −0.366025 1.36603i
\(850\) 0 0
\(851\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(852\) 0 0
\(853\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −1.22474 + 0.707107i −1.22474 + 0.707107i −0.965926 0.258819i \(-0.916667\pi\)
−0.258819 + 0.965926i \(0.583333\pi\)
\(858\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(859\) 0.707107 1.22474i 0.707107 1.22474i −0.258819 0.965926i \(-0.583333\pi\)
0.965926 0.258819i \(-0.0833333\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −2.00000 −2.00000
\(863\) 0.866025 + 0.500000i 0.866025 + 0.500000i 0.866025 0.500000i \(-0.166667\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.707107 0.707107i 0.707107 0.707107i
\(868\) 0 0
\(869\) 1.00000i 1.00000i
\(870\) 0 0
\(871\) −0.707107 1.22474i −0.707107 1.22474i
\(872\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(873\) 0 0
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(878\) 1.22474 0.707107i 1.22474 0.707107i
\(879\) 0.366025 1.36603i 0.366025 1.36603i
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0 0
\(883\) 1.00000 1.00000 0.500000 0.866025i \(-0.333333\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 1.00000 + 1.73205i 1.00000 + 1.73205i
\(887\) 1.22474 + 0.707107i 1.22474 + 0.707107i 0.965926 0.258819i \(-0.0833333\pi\)
0.258819 + 0.965926i \(0.416667\pi\)
\(888\) −0.707107 0.707107i −0.707107 0.707107i
\(889\) 0 0
\(890\) 0 0
\(891\) 0.866025 + 0.500000i 0.866025 + 0.500000i
\(892\) 0 0
\(893\) 0 0
\(894\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(895\) 0 0
\(896\) 0 0
\(897\) −1.00000 1.00000i −1.00000 1.00000i
\(898\) 0.500000 0.866025i 0.500000 0.866025i
\(899\) 0 0
\(900\) 0 0
\(901\) 0 0
\(902\) 1.41421i 1.41421i
\(903\) 0 0
\(904\) 1.00000 1.00000
\(905\) 0 0
\(906\) −0.258819 + 0.965926i −0.258819 + 0.965926i
\(907\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 1.00000i 1.00000i −0.866025 0.500000i \(-0.833333\pi\)
0.866025 0.500000i \(-0.166667\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.866025 + 0.500000i −0.866025 + 0.500000i
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) −1.00000 1.00000i −1.00000 1.00000i
\(919\) 0.500000 0.866025i 0.500000 0.866025i −0.500000 0.866025i \(-0.666667\pi\)
1.00000 \(0\)
\(920\) 0 0
\(921\) −1.36603 0.366025i −1.36603 0.366025i
\(922\) 0.707107 1.22474i 0.707107 1.22474i
\(923\) 1.41421i 1.41421i
\(924\) 0 0
\(925\) 0 0
\(926\) 1.73205 + 1.00000i 1.73205 + 1.00000i
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.36603 0.366025i 1.36603 0.366025i
\(934\) −0.707107 1.22474i −0.707107 1.22474i
\(935\) 0 0
\(936\) 0.707107 1.22474i 0.707107 1.22474i
\(937\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(942\) −0.366025 + 1.36603i −0.366025 + 1.36603i
\(943\) 0.707107 1.22474i 0.707107 1.22474i
\(944\) 1.41421i 1.41421i
\(945\) 0 0
\(946\) 1.00000 1.00000
\(947\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(952\) 0 0
\(953\) 1.00000i 1.00000i 0.866025 + 0.500000i \(0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0.258819 + 0.965926i 0.258819 + 0.965926i
\(958\) 1.41421 1.41421
\(959\) 0 0
\(960\) 0 0
\(961\) 0.500000 0.866025i 0.500000 0.866025i
\(962\) −1.22474 + 0.707107i −1.22474 + 0.707107i
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 1.00000i 1.00000i
\(975\) 0 0
\(976\) −0.707107 1.22474i −0.707107 1.22474i
\(977\) 0.866025 0.500000i 0.866025 0.500000i 1.00000i \(-0.5\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(978\) 0 0
\(979\) 1.41421 1.41421
\(980\) 0 0
\(981\) 1.00000i 1.00000i
\(982\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(983\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(984\) 1.36603 + 0.366025i 1.36603 + 0.366025i
\(985\) 0 0
\(986\) 1.41421i 1.41421i
\(987\) 0 0
\(988\) 0 0
\(989\) −0.866025 0.500000i −0.866025 0.500000i
\(990\) 0 0
\(991\) −0.500000 0.866025i −0.500000 0.866025i 0.500000 0.866025i \(-0.333333\pi\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(998\) 0 0
\(999\) −0.965926 + 0.258819i −0.965926 + 0.258819i
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 3675.1.u.d.1451.2 8
3.2 odd 2 inner 3675.1.u.d.1451.3 8
5.2 odd 4 3675.1.p.c.2774.3 8
5.3 odd 4 3675.1.p.b.2774.2 8
5.4 even 2 3675.1.u.e.1451.3 8
7.2 even 3 3675.1.c.h.1226.3 yes 4
7.3 odd 6 inner 3675.1.u.d.851.4 8
7.4 even 3 inner 3675.1.u.d.851.3 8
7.5 odd 6 3675.1.c.h.1226.4 yes 4
7.6 odd 2 inner 3675.1.u.d.1451.1 8
15.2 even 4 3675.1.p.b.2774.4 8
15.8 even 4 3675.1.p.c.2774.1 8
15.14 odd 2 3675.1.u.e.1451.2 8
21.2 odd 6 3675.1.c.h.1226.1 yes 4
21.5 even 6 3675.1.c.h.1226.2 yes 4
21.11 odd 6 inner 3675.1.u.d.851.2 8
21.17 even 6 inner 3675.1.u.d.851.1 8
21.20 even 2 inner 3675.1.u.d.1451.4 8
35.2 odd 12 3675.1.f.c.2549.4 4
35.3 even 12 3675.1.p.b.2174.1 8
35.4 even 6 3675.1.u.e.851.2 8
35.9 even 6 3675.1.c.g.1226.2 yes 4
35.12 even 12 3675.1.f.c.2549.1 4
35.13 even 4 3675.1.p.b.2774.3 8
35.17 even 12 3675.1.p.c.2174.4 8
35.18 odd 12 3675.1.p.b.2174.4 8
35.19 odd 6 3675.1.c.g.1226.1 4
35.23 odd 12 3675.1.f.d.2549.1 4
35.24 odd 6 3675.1.u.e.851.1 8
35.27 even 4 3675.1.p.c.2774.2 8
35.32 odd 12 3675.1.p.c.2174.1 8
35.33 even 12 3675.1.f.d.2549.4 4
35.34 odd 2 3675.1.u.e.1451.4 8
105.2 even 12 3675.1.f.d.2549.2 4
105.17 odd 12 3675.1.p.b.2174.3 8
105.23 even 12 3675.1.f.c.2549.3 4
105.32 even 12 3675.1.p.b.2174.2 8
105.38 odd 12 3675.1.p.c.2174.2 8
105.44 odd 6 3675.1.c.g.1226.4 yes 4
105.47 odd 12 3675.1.f.d.2549.3 4
105.53 even 12 3675.1.p.c.2174.3 8
105.59 even 6 3675.1.u.e.851.4 8
105.62 odd 4 3675.1.p.b.2774.1 8
105.68 odd 12 3675.1.f.c.2549.2 4
105.74 odd 6 3675.1.u.e.851.3 8
105.83 odd 4 3675.1.p.c.2774.4 8
105.89 even 6 3675.1.c.g.1226.3 yes 4
105.104 even 2 3675.1.u.e.1451.1 8
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3675.1.c.g.1226.1 4 35.19 odd 6
3675.1.c.g.1226.2 yes 4 35.9 even 6
3675.1.c.g.1226.3 yes 4 105.89 even 6
3675.1.c.g.1226.4 yes 4 105.44 odd 6
3675.1.c.h.1226.1 yes 4 21.2 odd 6
3675.1.c.h.1226.2 yes 4 21.5 even 6
3675.1.c.h.1226.3 yes 4 7.2 even 3
3675.1.c.h.1226.4 yes 4 7.5 odd 6
3675.1.f.c.2549.1 4 35.12 even 12
3675.1.f.c.2549.2 4 105.68 odd 12
3675.1.f.c.2549.3 4 105.23 even 12
3675.1.f.c.2549.4 4 35.2 odd 12
3675.1.f.d.2549.1 4 35.23 odd 12
3675.1.f.d.2549.2 4 105.2 even 12
3675.1.f.d.2549.3 4 105.47 odd 12
3675.1.f.d.2549.4 4 35.33 even 12
3675.1.p.b.2174.1 8 35.3 even 12
3675.1.p.b.2174.2 8 105.32 even 12
3675.1.p.b.2174.3 8 105.17 odd 12
3675.1.p.b.2174.4 8 35.18 odd 12
3675.1.p.b.2774.1 8 105.62 odd 4
3675.1.p.b.2774.2 8 5.3 odd 4
3675.1.p.b.2774.3 8 35.13 even 4
3675.1.p.b.2774.4 8 15.2 even 4
3675.1.p.c.2174.1 8 35.32 odd 12
3675.1.p.c.2174.2 8 105.38 odd 12
3675.1.p.c.2174.3 8 105.53 even 12
3675.1.p.c.2174.4 8 35.17 even 12
3675.1.p.c.2774.1 8 15.8 even 4
3675.1.p.c.2774.2 8 35.27 even 4
3675.1.p.c.2774.3 8 5.2 odd 4
3675.1.p.c.2774.4 8 105.83 odd 4
3675.1.u.d.851.1 8 21.17 even 6 inner
3675.1.u.d.851.2 8 21.11 odd 6 inner
3675.1.u.d.851.3 8 7.4 even 3 inner
3675.1.u.d.851.4 8 7.3 odd 6 inner
3675.1.u.d.1451.1 8 7.6 odd 2 inner
3675.1.u.d.1451.2 8 1.1 even 1 trivial
3675.1.u.d.1451.3 8 3.2 odd 2 inner
3675.1.u.d.1451.4 8 21.20 even 2 inner
3675.1.u.e.851.1 8 35.24 odd 6
3675.1.u.e.851.2 8 35.4 even 6
3675.1.u.e.851.3 8 105.74 odd 6
3675.1.u.e.851.4 8 105.59 even 6
3675.1.u.e.1451.1 8 105.104 even 2
3675.1.u.e.1451.2 8 15.14 odd 2
3675.1.u.e.1451.3 8 5.4 even 2
3675.1.u.e.1451.4 8 35.34 odd 2