Properties

Label 3481.1.d.a.672.1
Level $3481$
Weight $1$
Character 3481.672
Analytic conductor $1.737$
Analytic rank $0$
Dimension $28$
Projective image $D_{3}$
CM discriminant -59
Inner twists $56$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3481,1,Mod(506,3481)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3481, base_ring=CyclotomicField(58))
 
chi = DirichletCharacter(H, H._module([41]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3481.506");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3481 = 59^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3481.d (of order \(58\), degree \(28\), 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: \(1.73724530898\)
Analytic rank: \(0\)
Dimension: \(28\)
Coefficient field: \(\Q(\zeta_{58})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{28} - x^{27} + x^{26} - x^{25} + x^{24} - x^{23} + x^{22} - x^{21} + x^{20} - x^{19} + x^{18} - x^{17} + x^{16} - x^{15} + x^{14} - x^{13} + x^{12} - x^{11} + x^{10} - x^{9} + x^{8} - x^{7} + x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 59)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.59.1
Artin image: $S_3\times C_{29}$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{87} - \cdots)\)

Embedding invariants

Embedding label 672.1
Root \(0.856857 + 0.515554i\) of defining polynomial
Character \(\chi\) \(=\) 3481.672
Dual form 3481.1.d.a.1611.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.976621 + 0.214970i) q^{3} +(0.0541389 - 0.998533i) q^{4} +(0.161782 - 0.986827i) q^{5} +(-0.468408 + 0.883512i) q^{7} +O(q^{10})\) \(q+(-0.976621 + 0.214970i) q^{3} +(0.0541389 - 0.998533i) q^{4} +(0.161782 - 0.986827i) q^{5} +(-0.468408 + 0.883512i) q^{7} +(0.161782 + 0.986827i) q^{12} +(0.0541389 + 0.998533i) q^{15} +(-0.994138 - 0.108119i) q^{16} +(0.936817 + 1.76702i) q^{17} +(0.856857 - 0.515554i) q^{19} +(-0.976621 - 0.214970i) q^{20} +(0.267528 - 0.963550i) q^{21} +(0.796093 - 0.605174i) q^{27} +(0.856857 + 0.515554i) q^{28} +(0.725995 + 0.687699i) q^{29} +(0.796093 + 0.605174i) q^{35} +(0.370138 - 0.928977i) q^{41} +(0.994138 - 0.108119i) q^{48} +(-1.29477 - 1.52432i) q^{51} +(0.561187 - 0.827689i) q^{53} +(-0.725995 + 0.687699i) q^{57} +1.00000 q^{60} +(-0.161782 + 0.986827i) q^{64} +(1.81515 - 0.839778i) q^{68} +(-0.323564 - 1.97365i) q^{71} +(-0.468408 - 0.883512i) q^{76} +(-0.976621 - 0.214970i) q^{79} +(-0.267528 + 0.963550i) q^{80} +(-0.647386 + 0.762162i) q^{81} +(-0.947653 - 0.319302i) q^{84} +(1.89531 - 0.638603i) q^{85} +(-0.856857 - 0.515554i) q^{87} +(-0.370138 - 0.928977i) q^{95} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 28 q + q^{3} - q^{4} + q^{5} + q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 28 q + q^{3} - q^{4} + q^{5} + q^{7} + q^{12} - q^{15} - q^{16} - 2 q^{17} + q^{19} + q^{20} - q^{21} - q^{27} + q^{28} + q^{29} - q^{35} + q^{41} + q^{48} + 2 q^{51} + q^{53} - q^{57} + 28 q^{60} - q^{64} - 2 q^{68} - 2 q^{71} + q^{76} + q^{79} + q^{80} + q^{81} - q^{84} + 2 q^{85} - q^{87} - q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3481\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{15}{58}\right)\)

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 0 0.725995 0.687699i \(-0.241379\pi\)
−0.725995 + 0.687699i \(0.758621\pi\)
\(3\) −0.976621 + 0.214970i −0.976621 + 0.214970i −0.674480 0.738293i \(-0.735632\pi\)
−0.302140 + 0.953263i \(0.597701\pi\)
\(4\) 0.0541389 0.998533i 0.0541389 0.998533i
\(5\) 0.161782 0.986827i 0.161782 0.986827i −0.773726 0.633521i \(-0.781609\pi\)
0.935508 0.353306i \(-0.114943\pi\)
\(6\) 0 0
\(7\) −0.468408 + 0.883512i −0.468408 + 0.883512i 0.530940 + 0.847410i \(0.321839\pi\)
−0.999348 + 0.0361024i \(0.988506\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) 0 0
\(11\) 0 0 0.994138 0.108119i \(-0.0344828\pi\)
−0.994138 + 0.108119i \(0.965517\pi\)
\(12\) 0.161782 + 0.986827i 0.161782 + 0.986827i
\(13\) 0 0 −0.907575 0.419889i \(-0.862069\pi\)
0.907575 + 0.419889i \(0.137931\pi\)
\(14\) 0 0
\(15\) 0.0541389 + 0.998533i 0.0541389 + 0.998533i
\(16\) −0.994138 0.108119i −0.994138 0.108119i
\(17\) 0.936817 + 1.76702i 0.936817 + 1.76702i 0.468408 + 0.883512i \(0.344828\pi\)
0.468408 + 0.883512i \(0.344828\pi\)
\(18\) 0 0
\(19\) 0.856857 0.515554i 0.856857 0.515554i −0.0180541 0.999837i \(-0.505747\pi\)
0.874911 + 0.484283i \(0.160920\pi\)
\(20\) −0.976621 0.214970i −0.976621 0.214970i
\(21\) 0.267528 0.963550i 0.267528 0.963550i
\(22\) 0 0
\(23\) 0 0 −0.370138 0.928977i \(-0.620690\pi\)
0.370138 + 0.928977i \(0.379310\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) 0.796093 0.605174i 0.796093 0.605174i
\(28\) 0.856857 + 0.515554i 0.856857 + 0.515554i
\(29\) 0.725995 + 0.687699i 0.725995 + 0.687699i 0.958563 0.284881i \(-0.0919540\pi\)
−0.232567 + 0.972580i \(0.574713\pi\)
\(30\) 0 0
\(31\) 0 0 −0.856857 0.515554i \(-0.827586\pi\)
0.856857 + 0.515554i \(0.172414\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0.796093 + 0.605174i 0.796093 + 0.605174i
\(36\) 0 0
\(37\) 0 0 0.647386 0.762162i \(-0.275862\pi\)
−0.647386 + 0.762162i \(0.724138\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 0.370138 0.928977i 0.370138 0.928977i −0.619448 0.785037i \(-0.712644\pi\)
0.989587 0.143939i \(-0.0459770\pi\)
\(42\) 0 0
\(43\) 0 0 −0.994138 0.108119i \(-0.965517\pi\)
0.994138 + 0.108119i \(0.0344828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.161782 0.986827i \(-0.551724\pi\)
0.161782 + 0.986827i \(0.448276\pi\)
\(48\) 0.994138 0.108119i 0.994138 0.108119i
\(49\) 0 0
\(50\) 0 0
\(51\) −1.29477 1.52432i −1.29477 1.52432i
\(52\) 0 0
\(53\) 0.561187 0.827689i 0.561187 0.827689i −0.436206 0.899847i \(-0.643678\pi\)
0.997393 + 0.0721578i \(0.0229885\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.725995 + 0.687699i −0.725995 + 0.687699i
\(58\) 0 0
\(59\) 0 0
\(60\) 1.00000 1.00000
\(61\) 0 0 0.725995 0.687699i \(-0.241379\pi\)
−0.725995 + 0.687699i \(0.758621\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.161782 + 0.986827i −0.161782 + 0.986827i
\(65\) 0 0
\(66\) 0 0
\(67\) 0 0 −0.647386 0.762162i \(-0.724138\pi\)
0.647386 + 0.762162i \(0.275862\pi\)
\(68\) 1.81515 0.839778i 1.81515 0.839778i
\(69\) 0 0
\(70\) 0 0
\(71\) −0.323564 1.97365i −0.323564 1.97365i −0.161782 0.986827i \(-0.551724\pi\)
−0.161782 0.986827i \(-0.551724\pi\)
\(72\) 0 0
\(73\) 0 0 −0.267528 0.963550i \(-0.586207\pi\)
0.267528 + 0.963550i \(0.413793\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) −0.468408 0.883512i −0.468408 0.883512i
\(77\) 0 0
\(78\) 0 0
\(79\) −0.976621 0.214970i −0.976621 0.214970i −0.302140 0.953263i \(-0.597701\pi\)
−0.674480 + 0.738293i \(0.735632\pi\)
\(80\) −0.267528 + 0.963550i −0.267528 + 0.963550i
\(81\) −0.647386 + 0.762162i −0.647386 + 0.762162i
\(82\) 0 0
\(83\) 0 0 −0.796093 0.605174i \(-0.793103\pi\)
0.796093 + 0.605174i \(0.206897\pi\)
\(84\) −0.947653 0.319302i −0.947653 0.319302i
\(85\) 1.89531 0.638603i 1.89531 0.638603i
\(86\) 0 0
\(87\) −0.856857 0.515554i −0.856857 0.515554i
\(88\) 0 0
\(89\) 0 0 −0.725995 0.687699i \(-0.758621\pi\)
0.725995 + 0.687699i \(0.241379\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) −0.370138 0.928977i −0.370138 0.928977i
\(96\) 0 0
\(97\) 0 0 0.267528 0.963550i \(-0.413793\pi\)
−0.267528 + 0.963550i \(0.586207\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) 0 0
\(101\) 0 0 −0.468408 0.883512i \(-0.655172\pi\)
0.468408 + 0.883512i \(0.344828\pi\)
\(102\) 0 0
\(103\) 0 0 −0.0541389 0.998533i \(-0.517241\pi\)
0.0541389 + 0.998533i \(0.482759\pi\)
\(104\) 0 0
\(105\) −0.907575 0.419889i −0.907575 0.419889i
\(106\) 0 0
\(107\) 0.994138 0.108119i 0.994138 0.108119i 0.403435 0.915008i \(-0.367816\pi\)
0.590703 + 0.806889i \(0.298851\pi\)
\(108\) −0.561187 0.827689i −0.561187 0.827689i
\(109\) 0 0 0.907575 0.419889i \(-0.137931\pi\)
−0.907575 + 0.419889i \(0.862069\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0.561187 0.827689i 0.561187 0.827689i
\(113\) 0 0 0.161782 0.986827i \(-0.448276\pi\)
−0.161782 + 0.986827i \(0.551724\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.725995 0.687699i 0.725995 0.687699i
\(117\) 0 0
\(118\) 0 0
\(119\) −2.00000 −2.00000
\(120\) 0 0
\(121\) 0.976621 0.214970i 0.976621 0.214970i
\(122\) 0 0
\(123\) −0.161782 + 0.986827i −0.161782 + 0.986827i
\(124\) 0 0
\(125\) 0.468408 0.883512i 0.468408 0.883512i
\(126\) 0 0
\(127\) −0.907575 + 0.419889i −0.907575 + 0.419889i −0.817422 0.576039i \(-0.804598\pi\)
−0.0901531 + 0.995928i \(0.528736\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 −0.907575 0.419889i \(-0.862069\pi\)
0.907575 + 0.419889i \(0.137931\pi\)
\(132\) 0 0
\(133\) 0.0541389 + 0.998533i 0.0541389 + 0.998533i
\(134\) 0 0
\(135\) −0.468408 0.883512i −0.468408 0.883512i
\(136\) 0 0
\(137\) 0.856857 0.515554i 0.856857 0.515554i −0.0180541 0.999837i \(-0.505747\pi\)
0.874911 + 0.484283i \(0.160920\pi\)
\(138\) 0 0
\(139\) 0.535057 1.92710i 0.535057 1.92710i 0.267528 0.963550i \(-0.413793\pi\)
0.267528 0.963550i \(-0.413793\pi\)
\(140\) 0.647386 0.762162i 0.647386 0.762162i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0 0
\(145\) 0.796093 0.605174i 0.796093 0.605174i
\(146\) 0 0
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 −0.856857 0.515554i \(-0.827586\pi\)
0.856857 + 0.515554i \(0.172414\pi\)
\(150\) 0 0
\(151\) 0 0 0.947653 0.319302i \(-0.103448\pi\)
−0.947653 + 0.319302i \(0.896552\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0 0 −0.976621 0.214970i \(-0.931034\pi\)
0.976621 + 0.214970i \(0.0689655\pi\)
\(158\) 0 0
\(159\) −0.370138 + 0.928977i −0.370138 + 0.928977i
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 0.535057 + 1.92710i 0.535057 + 1.92710i 0.267528 + 0.963550i \(0.413793\pi\)
0.267528 + 0.963550i \(0.413793\pi\)
\(164\) −0.907575 0.419889i −0.907575 0.419889i
\(165\) 0 0
\(166\) 0 0
\(167\) 0.561187 + 0.827689i 0.561187 + 0.827689i 0.997393 0.0721578i \(-0.0229885\pi\)
−0.436206 + 0.899847i \(0.643678\pi\)
\(168\) 0 0
\(169\) 0.647386 + 0.762162i 0.647386 + 0.762162i
\(170\) 0 0
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 0.0541389 0.998533i \(-0.482759\pi\)
−0.0541389 + 0.998533i \(0.517241\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.725995 0.687699i \(-0.241379\pi\)
−0.725995 + 0.687699i \(0.758621\pi\)
\(180\) 0 0
\(181\) −0.0541389 + 0.998533i −0.0541389 + 0.998533i 0.837686 + 0.546152i \(0.183908\pi\)
−0.891825 + 0.452381i \(0.850575\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.161782 + 0.986827i 0.161782 + 0.986827i
\(190\) 0 0
\(191\) 0 0 −0.267528 0.963550i \(-0.586207\pi\)
0.267528 + 0.963550i \(0.413793\pi\)
\(192\) −0.0541389 0.998533i −0.0541389 0.998533i
\(193\) 0.994138 + 0.108119i 0.994138 + 0.108119i 0.590703 0.806889i \(-0.298851\pi\)
0.403435 + 0.915008i \(0.367816\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 1.95324 + 0.429941i 1.95324 + 0.429941i 0.976621 + 0.214970i \(0.0689655\pi\)
0.976621 + 0.214970i \(0.0689655\pi\)
\(198\) 0 0
\(199\) −0.647386 + 0.762162i −0.647386 + 0.762162i −0.983745 0.179572i \(-0.942529\pi\)
0.336359 + 0.941734i \(0.390805\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 0 0
\(203\) −0.947653 + 0.319302i −0.947653 + 0.319302i
\(204\) −1.59219 + 1.21035i −1.59219 + 1.21035i
\(205\) −0.856857 0.515554i −0.856857 0.515554i
\(206\) 0 0
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.947653 0.319302i \(-0.896552\pi\)
0.947653 + 0.319302i \(0.103448\pi\)
\(212\) −0.796093 0.605174i −0.796093 0.605174i
\(213\) 0.740276 + 1.85795i 0.740276 + 1.85795i
\(214\) 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
\(223\) 1.81515 + 0.839778i 1.81515 + 0.839778i 0.907575 + 0.419889i \(0.137931\pi\)
0.907575 + 0.419889i \(0.137931\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 0 0 0.907575 0.419889i \(-0.137931\pi\)
−0.907575 + 0.419889i \(0.862069\pi\)
\(228\) 0.647386 + 0.762162i 0.647386 + 0.762162i
\(229\) 0 0 0.468408 0.883512i \(-0.344828\pi\)
−0.468408 + 0.883512i \(0.655172\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.976621 0.214970i \(-0.0689655\pi\)
−0.976621 + 0.214970i \(0.931034\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.00000 1.00000
\(238\) 0 0
\(239\) −0.976621 + 0.214970i −0.976621 + 0.214970i −0.674480 0.738293i \(-0.735632\pi\)
−0.302140 + 0.953263i \(0.597701\pi\)
\(240\) 0.0541389 0.998533i 0.0541389 0.998533i
\(241\) 0.161782 0.986827i 0.161782 0.986827i −0.773726 0.633521i \(-0.781609\pi\)
0.935508 0.353306i \(-0.114943\pi\)
\(242\) 0 0
\(243\) 0 0
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) −0.0541389 0.998533i −0.0541389 0.998533i −0.891825 0.452381i \(-0.850575\pi\)
0.837686 0.546152i \(-0.183908\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) −1.71371 + 1.03111i −1.71371 + 1.03111i
\(256\) 0.976621 + 0.214970i 0.976621 + 0.214970i
\(257\) −0.267528 + 0.963550i −0.267528 + 0.963550i 0.700695 + 0.713461i \(0.252874\pi\)
−0.968223 + 0.250089i \(0.919540\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) −0.796093 + 0.605174i −0.796093 + 0.605174i −0.922143 0.386850i \(-0.873563\pi\)
0.126050 + 0.992024i \(0.459770\pi\)
\(264\) 0 0
\(265\) −0.725995 0.687699i −0.725995 0.687699i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0 0 0.947653 0.319302i \(-0.103448\pi\)
−0.947653 + 0.319302i \(0.896552\pi\)
\(270\) 0 0
\(271\) −0.796093 0.605174i −0.796093 0.605174i 0.126050 0.992024i \(-0.459770\pi\)
−0.922143 + 0.386850i \(0.873563\pi\)
\(272\) −0.740276 1.85795i −0.740276 1.85795i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.370138 0.928977i 0.370138 0.928977i −0.619448 0.785037i \(-0.712644\pi\)
0.989587 0.143939i \(-0.0459770\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.267528 0.963550i −0.267528 0.963550i −0.968223 0.250089i \(-0.919540\pi\)
0.700695 0.713461i \(-0.252874\pi\)
\(282\) 0 0
\(283\) 0 0 −0.161782 0.986827i \(-0.551724\pi\)
0.161782 + 0.986827i \(0.448276\pi\)
\(284\) −1.98828 + 0.216238i −1.98828 + 0.216238i
\(285\) 0.561187 + 0.827689i 0.561187 + 0.827689i
\(286\) 0 0
\(287\) 0.647386 + 0.762162i 0.647386 + 0.762162i
\(288\) 0 0
\(289\) −1.68356 + 2.48307i −1.68356 + 2.48307i
\(290\) 0 0
\(291\) 0 0
\(292\) 0 0
\(293\) 0.725995 0.687699i 0.725995 0.687699i −0.232567 0.972580i \(-0.574713\pi\)
0.958563 + 0.284881i \(0.0919540\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) −0.907575 + 0.419889i −0.907575 + 0.419889i
\(305\) 0 0
\(306\) 0 0
\(307\) 0.161782 + 0.986827i 0.161782 + 0.986827i 0.935508 + 0.353306i \(0.114943\pi\)
−0.773726 + 0.633521i \(0.781609\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0.994138 + 0.108119i 0.994138 + 0.108119i 0.590703 0.806889i \(-0.298851\pi\)
0.403435 + 0.915008i \(0.367816\pi\)
\(312\) 0 0
\(313\) 0 0 0.370138 0.928977i \(-0.379310\pi\)
−0.370138 + 0.928977i \(0.620690\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.267528 + 0.963550i −0.267528 + 0.963550i
\(317\) 1.29477 1.52432i 1.29477 1.52432i 0.647386 0.762162i \(-0.275862\pi\)
0.647386 0.762162i \(-0.275862\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.947653 + 0.319302i 0.947653 + 0.319302i
\(321\) −0.947653 + 0.319302i −0.947653 + 0.319302i
\(322\) 0 0
\(323\) 1.71371 + 1.03111i 1.71371 + 1.03111i
\(324\) 0.725995 + 0.687699i 0.725995 + 0.687699i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.370138 + 0.928977i 0.370138 + 0.928977i 0.989587 + 0.143939i \(0.0459770\pi\)
−0.619448 + 0.785037i \(0.712644\pi\)
\(332\) 0 0
\(333\) 0 0
\(334\) 0 0
\(335\) 0 0
\(336\) −0.370138 + 0.928977i −0.370138 + 0.928977i
\(337\) 0 0 −0.468408 0.883512i \(-0.655172\pi\)
0.468408 + 0.883512i \(0.344828\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) −0.535057 1.92710i −0.535057 1.92710i
\(341\) 0 0
\(342\) 0 0
\(343\) −0.994138 + 0.108119i −0.994138 + 0.108119i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.468408 0.883512i \(-0.344828\pi\)
−0.468408 + 0.883512i \(0.655172\pi\)
\(348\) −0.561187 + 0.827689i −0.561187 + 0.827689i
\(349\) 0 0 0.161782 0.986827i \(-0.448276\pi\)
−0.161782 + 0.986827i \(0.551724\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) −2.00000 −2.00000
\(356\) 0 0
\(357\) 1.95324 0.429941i 1.95324 0.429941i
\(358\) 0 0
\(359\) 0.161782 0.986827i 0.161782 0.986827i −0.773726 0.633521i \(-0.781609\pi\)
0.935508 0.353306i \(-0.114943\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 0 0
\(363\) −0.907575 + 0.419889i −0.907575 + 0.419889i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 0 0 −0.907575 0.419889i \(-0.862069\pi\)
0.907575 + 0.419889i \(0.137931\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0.468408 + 0.883512i 0.468408 + 0.883512i
\(372\) 0 0
\(373\) −1.71371 + 1.03111i −1.71371 + 1.03111i −0.856857 + 0.515554i \(0.827586\pi\)
−0.856857 + 0.515554i \(0.827586\pi\)
\(374\) 0 0
\(375\) −0.267528 + 0.963550i −0.267528 + 0.963550i
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.947653 + 0.319302i 0.947653 + 0.319302i 0.750350 0.661041i \(-0.229885\pi\)
0.197303 + 0.980342i \(0.436782\pi\)
\(380\) −0.947653 + 0.319302i −0.947653 + 0.319302i
\(381\) 0.796093 0.605174i 0.796093 0.605174i
\(382\) 0 0
\(383\) −1.45199 1.37540i −1.45199 1.37540i −0.725995 0.687699i \(-0.758621\pi\)
−0.725995 0.687699i \(-0.758621\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) 1.59219 + 1.21035i 1.59219 + 1.21035i 0.796093 + 0.605174i \(0.206897\pi\)
0.796093 + 0.605174i \(0.206897\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −0.370138 + 0.928977i −0.370138 + 0.928977i
\(396\) 0 0
\(397\) 0 0 −0.994138 0.108119i \(-0.965517\pi\)
0.994138 + 0.108119i \(0.0344828\pi\)
\(398\) 0 0
\(399\) −0.267528 0.963550i −0.267528 0.963550i
\(400\) 0 0
\(401\) 0 0 −0.161782 0.986827i \(-0.551724\pi\)
0.161782 + 0.986827i \(0.448276\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) 0 0
\(405\) 0.647386 + 0.762162i 0.647386 + 0.762162i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 0.0541389 0.998533i \(-0.482759\pi\)
−0.0541389 + 0.998533i \(0.517241\pi\)
\(410\) 0 0
\(411\) −0.725995 + 0.687699i −0.725995 + 0.687699i
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −0.108278 + 1.99707i −0.108278 + 1.99707i
\(418\) 0 0
\(419\) 0 0 0.561187 0.827689i \(-0.310345\pi\)
−0.561187 + 0.827689i \(0.689655\pi\)
\(420\) −0.468408 + 0.883512i −0.468408 + 0.883512i
\(421\) 0 0 −0.647386 0.762162i \(-0.724138\pi\)
0.647386 + 0.762162i \(0.275862\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0 0
\(428\) −0.0541389 0.998533i −0.0541389 0.998533i
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.370138 0.928977i \(-0.379310\pi\)
−0.370138 + 0.928977i \(0.620690\pi\)
\(432\) −0.856857 + 0.515554i −0.856857 + 0.515554i
\(433\) −0.976621 0.214970i −0.976621 0.214970i −0.302140 0.953263i \(-0.597701\pi\)
−0.674480 + 0.738293i \(0.735632\pi\)
\(434\) 0 0
\(435\) −0.647386 + 0.762162i −0.647386 + 0.762162i
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) −1.89531 + 0.638603i −1.89531 + 0.638603i −0.947653 + 0.319302i \(0.896552\pi\)
−0.947653 + 0.319302i \(0.896552\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 0 0 −0.725995 0.687699i \(-0.758621\pi\)
0.725995 + 0.687699i \(0.241379\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.796093 0.605174i −0.796093 0.605174i
\(449\) 0.370138 + 0.928977i 0.370138 + 0.928977i 0.989587 + 0.143939i \(0.0459770\pi\)
−0.619448 + 0.785037i \(0.712644\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 −0.0541389 0.998533i \(-0.517241\pi\)
0.0541389 + 0.998533i \(0.482759\pi\)
\(458\) 0 0
\(459\) 1.81515 + 0.839778i 1.81515 + 0.839778i
\(460\) 0 0
\(461\) −1.98828 + 0.216238i −1.98828 + 0.216238i −0.994138 + 0.108119i \(0.965517\pi\)
−0.994138 + 0.108119i \(0.965517\pi\)
\(462\) 0 0
\(463\) 0 0 0.907575 0.419889i \(-0.137931\pi\)
−0.907575 + 0.419889i \(0.862069\pi\)
\(464\) −0.647386 0.762162i −0.647386 0.762162i
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.161782 0.986827i \(-0.448276\pi\)
−0.161782 + 0.986827i \(0.551724\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) −0.108278 + 1.99707i −0.108278 + 1.99707i
\(477\) 0 0
\(478\) 0 0
\(479\) 0.936817 1.76702i 0.936817 1.76702i 0.468408 0.883512i \(-0.344828\pi\)
0.468408 0.883512i \(-0.344828\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.161782 0.986827i −0.161782 0.986827i
\(485\) 0 0
\(486\) 0 0
\(487\) −0.0541389 0.998533i −0.0541389 0.998533i −0.891825 0.452381i \(-0.850575\pi\)
0.837686 0.546152i \(-0.183908\pi\)
\(488\) 0 0
\(489\) −0.936817 1.76702i −0.936817 1.76702i
\(490\) 0 0
\(491\) 0.856857 0.515554i 0.856857 0.515554i −0.0180541 0.999837i \(-0.505747\pi\)
0.874911 + 0.484283i \(0.160920\pi\)
\(492\) 0.976621 + 0.214970i 0.976621 + 0.214970i
\(493\) −0.535057 + 1.92710i −0.535057 + 1.92710i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 1.89531 + 0.638603i 1.89531 + 0.638603i
\(498\) 0 0
\(499\) −0.796093 + 0.605174i −0.796093 + 0.605174i −0.922143 0.386850i \(-0.873563\pi\)
0.126050 + 0.992024i \(0.459770\pi\)
\(500\) −0.856857 0.515554i −0.856857 0.515554i
\(501\) −0.725995 0.687699i −0.725995 0.687699i
\(502\) 0 0
\(503\) 0 0 −0.856857 0.515554i \(-0.827586\pi\)
0.856857 + 0.515554i \(0.172414\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.796093 0.605174i −0.796093 0.605174i
\(508\) 0.370138 + 0.928977i 0.370138 + 0.928977i
\(509\) 0 0 0.647386 0.762162i \(-0.275862\pi\)
−0.647386 + 0.762162i \(0.724138\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0 0
\(513\) 0.370138 0.928977i 0.370138 0.928977i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) −1.12237 1.65538i −1.12237 1.65538i −0.561187 0.827689i \(-0.689655\pi\)
−0.561187 0.827689i \(-0.689655\pi\)
\(522\) 0 0
\(523\) −0.647386 0.762162i −0.647386 0.762162i 0.336359 0.941734i \(-0.390805\pi\)
−0.983745 + 0.179572i \(0.942529\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.725995 + 0.687699i −0.725995 + 0.687699i
\(530\) 0 0
\(531\) 0 0
\(532\) 1.00000 1.00000
\(533\) 0 0
\(534\) 0 0
\(535\) 0.0541389 0.998533i 0.0541389 0.998533i
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) −0.907575 + 0.419889i −0.907575 + 0.419889i
\(541\) 0 0 −0.561187 0.827689i \(-0.689655\pi\)
0.561187 + 0.827689i \(0.310345\pi\)
\(542\) 0 0
\(543\) −0.161782 0.986827i −0.161782 0.986827i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.98828 0.216238i −1.98828 0.216238i −0.994138 0.108119i \(-0.965517\pi\)
−0.994138 0.108119i \(-0.965517\pi\)
\(548\) −0.468408 0.883512i −0.468408 0.883512i
\(549\) 0 0
\(550\) 0 0
\(551\) 0.976621 + 0.214970i 0.976621 + 0.214970i
\(552\) 0 0
\(553\) 0.647386 0.762162i 0.647386 0.762162i
\(554\) 0 0
\(555\) 0 0
\(556\) −1.89531 0.638603i −1.89531 0.638603i
\(557\) 0.947653 0.319302i 0.947653 0.319302i 0.197303 0.980342i \(-0.436782\pi\)
0.750350 + 0.661041i \(0.229885\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −0.725995 0.687699i −0.725995 0.687699i
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 0.796093 0.605174i \(-0.206897\pi\)
−0.796093 + 0.605174i \(0.793103\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) −0.370138 0.928977i −0.370138 0.928977i
\(568\) 0 0
\(569\) 0 0 0.267528 0.963550i \(-0.413793\pi\)
−0.267528 + 0.963550i \(0.586207\pi\)
\(570\) 0 0
\(571\) 0 0 0.856857 0.515554i \(-0.172414\pi\)
−0.856857 + 0.515554i \(0.827586\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) −0.907575 0.419889i −0.907575 0.419889i −0.0901531 0.995928i \(-0.528736\pi\)
−0.817422 + 0.576039i \(0.804598\pi\)
\(578\) 0 0
\(579\) −0.994138 + 0.108119i −0.994138 + 0.108119i
\(580\) −0.561187 0.827689i −0.561187 0.827689i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.976621 0.214970i \(-0.0689655\pi\)
−0.976621 + 0.214970i \(0.931034\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) −2.00000 −2.00000
\(592\) 0 0
\(593\) −0.976621 + 0.214970i −0.976621 + 0.214970i −0.674480 0.738293i \(-0.735632\pi\)
−0.302140 + 0.953263i \(0.597701\pi\)
\(594\) 0 0
\(595\) −0.323564 + 1.97365i −0.323564 + 1.97365i
\(596\) 0 0
\(597\) 0.468408 0.883512i 0.468408 0.883512i
\(598\) 0 0
\(599\) −0.907575 + 0.419889i −0.907575 + 0.419889i −0.817422 0.576039i \(-0.804598\pi\)
−0.0901531 + 0.995928i \(0.528736\pi\)
\(600\) 0 0
\(601\) 0 0 0.994138 0.108119i \(-0.0344828\pi\)
−0.994138 + 0.108119i \(0.965517\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) −0.0541389 0.998533i −0.0541389 0.998533i
\(606\) 0 0
\(607\) −0.468408 0.883512i −0.468408 0.883512i −0.999348 0.0361024i \(-0.988506\pi\)
0.530940 0.847410i \(-0.321839\pi\)
\(608\) 0 0
\(609\) 0.856857 0.515554i 0.856857 0.515554i
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0 0 −0.370138 0.928977i \(-0.620690\pi\)
0.370138 + 0.928977i \(0.379310\pi\)
\(614\) 0 0
\(615\) 0.947653 + 0.319302i 0.947653 + 0.319302i
\(616\) 0 0
\(617\) −0.796093 + 0.605174i −0.796093 + 0.605174i −0.922143 0.386850i \(-0.873563\pi\)
0.126050 + 0.992024i \(0.459770\pi\)
\(618\) 0 0
\(619\) 0.725995 + 0.687699i 0.725995 + 0.687699i 0.958563 0.284881i \(-0.0919540\pi\)
−0.232567 + 0.972580i \(0.574713\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.796093 0.605174i −0.796093 0.605174i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) 0 0
\(631\) −0.740276 + 1.85795i −0.740276 + 1.85795i −0.370138 + 0.928977i \(0.620690\pi\)
−0.370138 + 0.928977i \(0.620690\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) 0.267528 + 0.963550i 0.267528 + 0.963550i
\(636\) 0.907575 + 0.419889i 0.907575 + 0.419889i
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 1.29477 + 1.52432i 1.29477 + 1.52432i 0.647386 + 0.762162i \(0.275862\pi\)
0.647386 + 0.762162i \(0.275862\pi\)
\(642\) 0 0
\(643\) 0.561187 0.827689i 0.561187 0.827689i −0.436206 0.899847i \(-0.643678\pi\)
0.997393 + 0.0721578i \(0.0229885\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0.725995 0.687699i 0.725995 0.687699i −0.232567 0.972580i \(-0.574713\pi\)
0.958563 + 0.284881i \(0.0919540\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 1.95324 0.429941i 1.95324 0.429941i
\(653\) −0.0541389 + 0.998533i −0.0541389 + 0.998533i 0.837686 + 0.546152i \(0.183908\pi\)
−0.891825 + 0.452381i \(0.850575\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.468408 + 0.883512i −0.468408 + 0.883512i
\(657\) 0 0
\(658\) 0 0
\(659\) 0 0 −0.561187 0.827689i \(-0.689655\pi\)
0.561187 + 0.827689i \(0.310345\pi\)
\(660\) 0 0
\(661\) 0.161782 + 0.986827i 0.161782 + 0.986827i 0.935508 + 0.353306i \(0.114943\pi\)
−0.773726 + 0.633521i \(0.781609\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0.994138 + 0.108119i 0.994138 + 0.108119i
\(666\) 0 0
\(667\) 0 0
\(668\) 0.856857 0.515554i 0.856857 0.515554i
\(669\) −1.95324 0.429941i −1.95324 0.429941i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0 0 −0.796093 0.605174i \(-0.793103\pi\)
0.796093 + 0.605174i \(0.206897\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0.796093 0.605174i 0.796093 0.605174i
\(677\) −1.71371 1.03111i −1.71371 1.03111i −0.856857 0.515554i \(-0.827586\pi\)
−0.856857 0.515554i \(-0.827586\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.947653 0.319302i \(-0.896552\pi\)
0.947653 + 0.319302i \(0.103448\pi\)
\(684\) 0 0
\(685\) −0.370138 0.928977i −0.370138 0.928977i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0 0
\(690\) 0 0
\(691\) 0 0 −0.468408 0.883512i \(-0.655172\pi\)
0.468408 + 0.883512i \(0.344828\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) −1.81515 0.839778i −1.81515 0.839778i
\(696\) 0 0
\(697\) 1.98828 0.216238i 1.98828 0.216238i
\(698\) 0 0
\(699\) 0 0
\(700\) 0 0
\(701\) 0 0 0.468408 0.883512i \(-0.344828\pi\)
−0.468408 + 0.883512i \(0.655172\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.00000 −1.00000 −0.500000 0.866025i \(-0.666667\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0.907575 0.419889i 0.907575 0.419889i
\(718\) 0 0
\(719\) 0 0 0.994138 0.108119i \(-0.0344828\pi\)
−0.994138 + 0.108119i \(0.965517\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) 0 0
\(723\) 0.0541389 + 0.998533i 0.0541389 + 0.998533i
\(724\) 0.994138 + 0.108119i 0.994138 + 0.108119i
\(725\) 0 0
\(726\) 0 0
\(727\) −1.71371 + 1.03111i −1.71371 + 1.03111i −0.856857 + 0.515554i \(0.827586\pi\)
−0.856857 + 0.515554i \(0.827586\pi\)
\(728\) 0 0
\(729\) 0.267528 0.963550i 0.267528 0.963550i
\(730\) 0 0
\(731\) 0 0
\(732\) 0 0
\(733\) −1.89531 0.638603i −1.89531 0.638603i −0.947653 0.319302i \(-0.896552\pi\)
−0.947653 0.319302i \(-0.896552\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.856857 0.515554i \(-0.827586\pi\)
0.856857 + 0.515554i \(0.172414\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) 1.59219 + 1.21035i 1.59219 + 1.21035i 0.796093 + 0.605174i \(0.206897\pi\)
0.796093 + 0.605174i \(0.206897\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) −0.370138 + 0.928977i −0.370138 + 0.928977i
\(750\) 0 0
\(751\) 0 0 −0.994138 0.108119i \(-0.965517\pi\)
0.994138 + 0.108119i \(0.0344828\pi\)
\(752\) 0 0
\(753\) 0.267528 + 0.963550i 0.267528 + 0.963550i
\(754\) 0 0
\(755\) 0 0
\(756\) 0.994138 0.108119i 0.994138 0.108119i
\(757\) 0.561187 + 0.827689i 0.561187 + 0.827689i 0.997393 0.0721578i \(-0.0229885\pi\)
−0.436206 + 0.899847i \(0.643678\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.561187 0.827689i 0.561187 0.827689i −0.436206 0.899847i \(-0.643678\pi\)
0.997393 + 0.0721578i \(0.0229885\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −1.00000 −1.00000
\(769\) 0 0 0.725995 0.687699i \(-0.241379\pi\)
−0.725995 + 0.687699i \(0.758621\pi\)
\(770\) 0 0
\(771\) 0.0541389 0.998533i 0.0541389 0.998533i
\(772\) 0.161782 0.986827i 0.161782 0.986827i
\(773\) 0 0 0.561187 0.827689i \(-0.310345\pi\)
−0.561187 + 0.827689i \(0.689655\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) −0.161782 0.986827i −0.161782 0.986827i
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0.994138 + 0.108119i 0.994138 + 0.108119i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 1.95324 + 0.429941i 1.95324 + 0.429941i 0.976621 + 0.214970i \(0.0689655\pi\)
0.976621 + 0.214970i \(0.0689655\pi\)
\(788\) 0.535057 1.92710i 0.535057 1.92710i
\(789\) 0.647386 0.762162i 0.647386 0.762162i
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 0 0
\(794\) 0 0
\(795\) 0.856857 + 0.515554i 0.856857 + 0.515554i
\(796\) 0.725995 + 0.687699i 0.725995 + 0.687699i
\(797\) 0 0 −0.725995 0.687699i \(-0.758621\pi\)
0.725995 + 0.687699i \(0.241379\pi\)
\(798\) 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 0
\(809\) 0 0 −0.468408 0.883512i \(-0.655172\pi\)
0.468408 + 0.883512i \(0.344828\pi\)
\(810\) 0 0
\(811\) 0 0 −0.0541389 0.998533i \(-0.517241\pi\)
0.0541389 + 0.998533i \(0.482759\pi\)
\(812\) 0.267528 + 0.963550i 0.267528 + 0.963550i
\(813\) 0.907575 + 0.419889i 0.907575 + 0.419889i
\(814\) 0 0
\(815\) 1.98828 0.216238i 1.98828 0.216238i
\(816\) 1.12237 + 1.65538i 1.12237 + 1.65538i
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) −0.561187 + 0.827689i −0.561187 + 0.827689i
\(821\) 0 0 0.161782 0.986827i \(-0.448276\pi\)
−0.161782 + 0.986827i \(0.551724\pi\)
\(822\) 0 0
\(823\) 0 0 0.976621 0.214970i \(-0.0689655\pi\)
−0.976621 + 0.214970i \(0.931034\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(828\) 0 0
\(829\) −0.976621 + 0.214970i −0.976621 + 0.214970i −0.674480 0.738293i \(-0.735632\pi\)
−0.302140 + 0.953263i \(0.597701\pi\)
\(830\) 0 0
\(831\) −0.161782 + 0.986827i −0.161782 + 0.986827i
\(832\) 0 0
\(833\) 0 0
\(834\) 0 0
\(835\) 0.907575 0.419889i 0.907575 0.419889i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.907575 0.419889i \(-0.862069\pi\)
0.907575 + 0.419889i \(0.137931\pi\)
\(840\) 0 0
\(841\) 0 0
\(842\) 0 0
\(843\) 0.468408 + 0.883512i 0.468408 + 0.883512i
\(844\) 0 0
\(845\) 0.856857 0.515554i 0.856857 0.515554i
\(846\) 0 0
\(847\) −0.267528 + 0.963550i −0.267528 + 0.963550i
\(848\) −0.647386 + 0.762162i −0.647386 + 0.762162i
\(849\) 0 0
\(850\) 0 0
\(851\) 0 0
\(852\) 1.89531 0.638603i 1.89531 0.638603i
\(853\) −0.796093 + 0.605174i −0.796093 + 0.605174i −0.922143 0.386850i \(-0.873563\pi\)
0.126050 + 0.992024i \(0.459770\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.856857 0.515554i \(-0.827586\pi\)
0.856857 + 0.515554i \(0.172414\pi\)
\(858\) 0 0
\(859\) 0 0 0.947653 0.319302i \(-0.103448\pi\)
−0.947653 + 0.319302i \(0.896552\pi\)
\(860\) 0 0
\(861\) −0.796093 0.605174i −0.796093 0.605174i
\(862\) 0 0
\(863\) 0 0 0.647386 0.762162i \(-0.275862\pi\)
−0.647386 + 0.762162i \(0.724138\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 1.11041 2.78693i 1.11041 2.78693i
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) 0.561187 + 0.827689i 0.561187 + 0.827689i
\(876\) 0 0
\(877\) −0.647386 0.762162i −0.647386 0.762162i 0.336359 0.941734i \(-0.390805\pi\)
−0.983745 + 0.179572i \(0.942529\pi\)
\(878\) 0 0
\(879\) −0.561187 + 0.827689i −0.561187 + 0.827689i
\(880\) 0 0
\(881\) 0 0 0.0541389 0.998533i \(-0.482759\pi\)
−0.0541389 + 0.998533i \(0.517241\pi\)
\(882\) 0 0
\(883\) 0.725995 0.687699i 0.725995 0.687699i −0.232567 0.972580i \(-0.574713\pi\)
0.958563 + 0.284881i \(0.0919540\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.725995 0.687699i \(-0.241379\pi\)
−0.725995 + 0.687699i \(0.758621\pi\)
\(888\) 0 0
\(889\) 0.0541389 0.998533i 0.0541389 0.998533i
\(890\) 0 0
\(891\) 0 0
\(892\) 0.936817 1.76702i 0.936817 1.76702i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0 0
\(901\) 1.98828 + 0.216238i 1.98828 + 0.216238i
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 0.976621 + 0.214970i 0.976621 + 0.214970i
\(906\) 0 0
\(907\) −0.647386 + 0.762162i −0.647386 + 0.762162i −0.983745 0.179572i \(-0.942529\pi\)
0.336359 + 0.941734i \(0.390805\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0.947653 0.319302i 0.947653 0.319302i 0.197303 0.980342i \(-0.436782\pi\)
0.750350 + 0.661041i \(0.229885\pi\)
\(912\) 0.796093 0.605174i 0.796093 0.605174i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.947653 0.319302i \(-0.896552\pi\)
0.947653 + 0.319302i \(0.103448\pi\)
\(920\) 0 0
\(921\) −0.370138 0.928977i −0.370138 0.928977i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0 0 −0.0541389 0.998533i \(-0.517241\pi\)
0.0541389 + 0.998533i \(0.482759\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) −0.994138 + 0.108119i −0.994138 + 0.108119i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 0 0 0.468408 0.883512i \(-0.344828\pi\)
−0.468408 + 0.883512i \(0.655172\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 0.976621 0.214970i \(-0.0689655\pi\)
−0.976621 + 0.214970i \(0.931034\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 1.00000 1.00000
\(946\) 0 0
\(947\) −0.976621 + 0.214970i −0.976621 + 0.214970i −0.674480 0.738293i \(-0.735632\pi\)
−0.302140 + 0.953263i \(0.597701\pi\)
\(948\) 0.0541389 0.998533i 0.0541389 0.998533i
\(949\) 0 0
\(950\) 0 0
\(951\) −0.936817 + 1.76702i −0.936817 + 1.76702i
\(952\) 0 0
\(953\) 1.81515 0.839778i 1.81515 0.839778i 0.907575 0.419889i \(-0.137931\pi\)
0.907575 0.419889i \(-0.137931\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0.161782 + 0.986827i 0.161782 + 0.986827i
\(957\) 0 0
\(958\) 0 0
\(959\) 0.0541389 + 0.998533i 0.0541389 + 0.998533i
\(960\) −0.994138 0.108119i −0.994138 0.108119i
\(961\) 0.468408 + 0.883512i 0.468408 + 0.883512i
\(962\) 0 0
\(963\) 0 0
\(964\) −0.976621 0.214970i −0.976621 0.214970i
\(965\) 0.267528 0.963550i 0.267528 0.963550i
\(966\) 0 0
\(967\) 0 0 −0.370138 0.928977i \(-0.620690\pi\)
0.370138 + 0.928977i \(0.379310\pi\)
\(968\) 0 0
\(969\) −1.89531 0.638603i −1.89531 0.638603i
\(970\) 0 0
\(971\) −0.796093 + 0.605174i −0.796093 + 0.605174i −0.922143 0.386850i \(-0.873563\pi\)
0.126050 + 0.992024i \(0.459770\pi\)
\(972\) 0 0
\(973\) 1.45199 + 1.37540i 1.45199 + 1.37540i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 0.947653 0.319302i \(-0.103448\pi\)
−0.947653 + 0.319302i \(0.896552\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 −0.976621 0.214970i \(-0.931034\pi\)
0.976621 + 0.214970i \(0.0689655\pi\)
\(984\) 0 0
\(985\) 0.740276 1.85795i 0.740276 1.85795i
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 −0.161782 0.986827i \(-0.551724\pi\)
0.161782 + 0.986827i \(0.448276\pi\)
\(992\) 0 0
\(993\) −0.561187 0.827689i −0.561187 0.827689i
\(994\) 0 0
\(995\) 0.647386 + 0.762162i 0.647386 + 0.762162i
\(996\) 0 0
\(997\) 0.561187 0.827689i 0.561187 0.827689i −0.436206 0.899847i \(-0.643678\pi\)
0.997393 + 0.0721578i \(0.0229885\pi\)
\(998\) 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 3481.1.d.a.672.1 28
59.2 odd 58 inner 3481.1.d.a.2511.1 28
59.3 even 29 inner 3481.1.d.a.1505.1 28
59.4 even 29 inner 3481.1.d.a.2374.1 28
59.5 even 29 inner 3481.1.d.a.1106.1 28
59.6 odd 58 inner 3481.1.d.a.946.1 28
59.7 even 29 inner 3481.1.d.a.3428.1 28
59.8 odd 58 inner 3481.1.d.a.3183.1 28
59.9 even 29 inner 3481.1.d.a.506.1 28
59.10 odd 58 59.1.b.a.58.1 1
59.11 odd 58 inner 3481.1.d.a.1839.1 28
59.12 even 29 inner 3481.1.d.a.893.1 28
59.13 odd 58 inner 3481.1.d.a.2117.1 28
59.14 odd 58 inner 3481.1.d.a.1558.1 28
59.15 even 29 inner 3481.1.d.a.809.1 28
59.16 even 29 inner 3481.1.d.a.1404.1 28
59.17 even 29 inner 3481.1.d.a.806.1 28
59.18 odd 58 inner 3481.1.d.a.1611.1 28
59.19 even 29 inner 3481.1.d.a.1105.1 28
59.20 even 29 inner 3481.1.d.a.3182.1 28
59.21 even 29 inner 3481.1.d.a.3181.1 28
59.22 even 29 inner 3481.1.d.a.3344.1 28
59.23 odd 58 inner 3481.1.d.a.1311.1 28
59.24 odd 58 inner 3481.1.d.a.2451.1 28
59.25 even 29 inner 3481.1.d.a.805.1 28
59.26 even 29 inner 3481.1.d.a.2922.1 28
59.27 even 29 inner 3481.1.d.a.2076.1 28
59.28 even 29 inner 3481.1.d.a.2869.1 28
59.29 even 29 inner 3481.1.d.a.1702.1 28
59.30 odd 58 inner 3481.1.d.a.1702.1 28
59.31 odd 58 inner 3481.1.d.a.2869.1 28
59.32 odd 58 inner 3481.1.d.a.2076.1 28
59.33 odd 58 inner 3481.1.d.a.2922.1 28
59.34 odd 58 inner 3481.1.d.a.805.1 28
59.35 even 29 inner 3481.1.d.a.2451.1 28
59.36 even 29 inner 3481.1.d.a.1311.1 28
59.37 odd 58 inner 3481.1.d.a.3344.1 28
59.38 odd 58 inner 3481.1.d.a.3181.1 28
59.39 odd 58 inner 3481.1.d.a.3182.1 28
59.40 odd 58 inner 3481.1.d.a.1105.1 28
59.41 even 29 inner 3481.1.d.a.1611.1 28
59.42 odd 58 inner 3481.1.d.a.806.1 28
59.43 odd 58 inner 3481.1.d.a.1404.1 28
59.44 odd 58 inner 3481.1.d.a.809.1 28
59.45 even 29 inner 3481.1.d.a.1558.1 28
59.46 even 29 inner 3481.1.d.a.2117.1 28
59.47 odd 58 inner 3481.1.d.a.893.1 28
59.48 even 29 inner 3481.1.d.a.1839.1 28
59.49 even 29 59.1.b.a.58.1 1
59.50 odd 58 inner 3481.1.d.a.506.1 28
59.51 even 29 inner 3481.1.d.a.3183.1 28
59.52 odd 58 inner 3481.1.d.a.3428.1 28
59.53 even 29 inner 3481.1.d.a.946.1 28
59.54 odd 58 inner 3481.1.d.a.1106.1 28
59.55 odd 58 inner 3481.1.d.a.2374.1 28
59.56 odd 58 inner 3481.1.d.a.1505.1 28
59.57 even 29 inner 3481.1.d.a.2511.1 28
59.58 odd 2 CM 3481.1.d.a.672.1 28
177.128 even 58 531.1.c.a.235.1 1
177.167 odd 58 531.1.c.a.235.1 1
236.167 odd 58 944.1.h.a.353.1 1
236.187 even 58 944.1.h.a.353.1 1
295.49 even 58 1475.1.c.b.176.1 1
295.69 odd 58 1475.1.c.b.176.1 1
295.108 odd 116 1475.1.d.a.1474.1 2
295.128 even 116 1475.1.d.a.1474.1 2
295.167 odd 116 1475.1.d.a.1474.2 2
295.187 even 116 1475.1.d.a.1474.2 2
413.10 even 174 2891.1.g.b.471.1 2
413.69 even 58 2891.1.c.e.589.1 1
413.108 odd 174 2891.1.g.b.471.1 2
413.128 odd 174 2891.1.g.d.2713.1 2
413.167 odd 58 2891.1.c.e.589.1 1
413.187 even 174 2891.1.g.b.2713.1 2
413.226 even 87 2891.1.g.d.2713.1 2
413.285 odd 174 2891.1.g.b.2713.1 2
413.305 odd 174 2891.1.g.d.471.1 2
413.403 even 87 2891.1.g.d.471.1 2
472.69 odd 58 3776.1.h.b.2241.1 1
472.187 even 58 3776.1.h.a.2241.1 1
472.285 even 58 3776.1.h.b.2241.1 1
472.403 odd 58 3776.1.h.a.2241.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
59.1.b.a.58.1 1 59.10 odd 58
59.1.b.a.58.1 1 59.49 even 29
531.1.c.a.235.1 1 177.128 even 58
531.1.c.a.235.1 1 177.167 odd 58
944.1.h.a.353.1 1 236.167 odd 58
944.1.h.a.353.1 1 236.187 even 58
1475.1.c.b.176.1 1 295.49 even 58
1475.1.c.b.176.1 1 295.69 odd 58
1475.1.d.a.1474.1 2 295.108 odd 116
1475.1.d.a.1474.1 2 295.128 even 116
1475.1.d.a.1474.2 2 295.167 odd 116
1475.1.d.a.1474.2 2 295.187 even 116
2891.1.c.e.589.1 1 413.69 even 58
2891.1.c.e.589.1 1 413.167 odd 58
2891.1.g.b.471.1 2 413.10 even 174
2891.1.g.b.471.1 2 413.108 odd 174
2891.1.g.b.2713.1 2 413.187 even 174
2891.1.g.b.2713.1 2 413.285 odd 174
2891.1.g.d.471.1 2 413.305 odd 174
2891.1.g.d.471.1 2 413.403 even 87
2891.1.g.d.2713.1 2 413.128 odd 174
2891.1.g.d.2713.1 2 413.226 even 87
3481.1.d.a.506.1 28 59.9 even 29 inner
3481.1.d.a.506.1 28 59.50 odd 58 inner
3481.1.d.a.672.1 28 1.1 even 1 trivial
3481.1.d.a.672.1 28 59.58 odd 2 CM
3481.1.d.a.805.1 28 59.25 even 29 inner
3481.1.d.a.805.1 28 59.34 odd 58 inner
3481.1.d.a.806.1 28 59.17 even 29 inner
3481.1.d.a.806.1 28 59.42 odd 58 inner
3481.1.d.a.809.1 28 59.15 even 29 inner
3481.1.d.a.809.1 28 59.44 odd 58 inner
3481.1.d.a.893.1 28 59.12 even 29 inner
3481.1.d.a.893.1 28 59.47 odd 58 inner
3481.1.d.a.946.1 28 59.6 odd 58 inner
3481.1.d.a.946.1 28 59.53 even 29 inner
3481.1.d.a.1105.1 28 59.19 even 29 inner
3481.1.d.a.1105.1 28 59.40 odd 58 inner
3481.1.d.a.1106.1 28 59.5 even 29 inner
3481.1.d.a.1106.1 28 59.54 odd 58 inner
3481.1.d.a.1311.1 28 59.23 odd 58 inner
3481.1.d.a.1311.1 28 59.36 even 29 inner
3481.1.d.a.1404.1 28 59.16 even 29 inner
3481.1.d.a.1404.1 28 59.43 odd 58 inner
3481.1.d.a.1505.1 28 59.3 even 29 inner
3481.1.d.a.1505.1 28 59.56 odd 58 inner
3481.1.d.a.1558.1 28 59.14 odd 58 inner
3481.1.d.a.1558.1 28 59.45 even 29 inner
3481.1.d.a.1611.1 28 59.18 odd 58 inner
3481.1.d.a.1611.1 28 59.41 even 29 inner
3481.1.d.a.1702.1 28 59.29 even 29 inner
3481.1.d.a.1702.1 28 59.30 odd 58 inner
3481.1.d.a.1839.1 28 59.11 odd 58 inner
3481.1.d.a.1839.1 28 59.48 even 29 inner
3481.1.d.a.2076.1 28 59.27 even 29 inner
3481.1.d.a.2076.1 28 59.32 odd 58 inner
3481.1.d.a.2117.1 28 59.13 odd 58 inner
3481.1.d.a.2117.1 28 59.46 even 29 inner
3481.1.d.a.2374.1 28 59.4 even 29 inner
3481.1.d.a.2374.1 28 59.55 odd 58 inner
3481.1.d.a.2451.1 28 59.24 odd 58 inner
3481.1.d.a.2451.1 28 59.35 even 29 inner
3481.1.d.a.2511.1 28 59.2 odd 58 inner
3481.1.d.a.2511.1 28 59.57 even 29 inner
3481.1.d.a.2869.1 28 59.28 even 29 inner
3481.1.d.a.2869.1 28 59.31 odd 58 inner
3481.1.d.a.2922.1 28 59.26 even 29 inner
3481.1.d.a.2922.1 28 59.33 odd 58 inner
3481.1.d.a.3181.1 28 59.21 even 29 inner
3481.1.d.a.3181.1 28 59.38 odd 58 inner
3481.1.d.a.3182.1 28 59.20 even 29 inner
3481.1.d.a.3182.1 28 59.39 odd 58 inner
3481.1.d.a.3183.1 28 59.8 odd 58 inner
3481.1.d.a.3183.1 28 59.51 even 29 inner
3481.1.d.a.3344.1 28 59.22 even 29 inner
3481.1.d.a.3344.1 28 59.37 odd 58 inner
3481.1.d.a.3428.1 28 59.7 even 29 inner
3481.1.d.a.3428.1 28 59.52 odd 58 inner
3776.1.h.a.2241.1 1 472.187 even 58
3776.1.h.a.2241.1 1 472.403 odd 58
3776.1.h.b.2241.1 1 472.69 odd 58
3776.1.h.b.2241.1 1 472.285 even 58