Properties

Label 1776.1.z.c.221.2
Level $1776$
Weight $1$
Character 1776.221
Analytic conductor $0.886$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -111
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [1776,1,Mod(221,1776)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1776, base_ring=CyclotomicField(4))
 
chi = DirichletCharacter(H, H._module([0, 3, 2, 2]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1776.221");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1776 = 2^{4} \cdot 3 \cdot 37 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1776.z (of order \(4\), degree \(2\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.886339462436\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(8\) over \(\Q(i)\)
Coefficient field: \(\Q(\zeta_{32})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{16}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{16} + \cdots)\)

Embedding invariants

Embedding label 221.2
Root \(0.980785 + 0.195090i\) of defining polynomial
Character \(\chi\) \(=\) 1776.221
Dual form 1776.1.z.c.1109.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.831470 - 0.555570i) q^{2} +(0.707107 + 0.707107i) q^{3} +(0.382683 + 0.923880i) q^{4} +(-0.785695 + 0.785695i) q^{5} +(-0.195090 - 0.980785i) q^{6} -0.765367i q^{7} +(0.195090 - 0.980785i) q^{8} +1.00000i q^{9} +O(q^{10})\) \(q+(-0.831470 - 0.555570i) q^{2} +(0.707107 + 0.707107i) q^{3} +(0.382683 + 0.923880i) q^{4} +(-0.785695 + 0.785695i) q^{5} +(-0.195090 - 0.980785i) q^{6} -0.765367i q^{7} +(0.195090 - 0.980785i) q^{8} +1.00000i q^{9} +(1.08979 - 0.216773i) q^{10} +(-0.382683 + 0.923880i) q^{12} +(-0.425215 + 0.636379i) q^{14} -1.11114 q^{15} +(-0.707107 + 0.707107i) q^{16} +1.66294 q^{17} +(0.555570 - 0.831470i) q^{18} +(-1.02656 - 0.425215i) q^{20} +(0.541196 - 0.541196i) q^{21} +1.96157i q^{23} +(0.831470 - 0.555570i) q^{24} -0.234633i q^{25} +(-0.707107 + 0.707107i) q^{27} +(0.707107 - 0.292893i) q^{28} +(-0.275899 - 0.275899i) q^{29} +(0.923880 + 0.617317i) q^{30} +(0.980785 - 0.195090i) q^{32} +(-1.38268 - 0.923880i) q^{34} +(0.601345 + 0.601345i) q^{35} +(-0.923880 + 0.382683i) q^{36} +(-0.707107 + 0.707107i) q^{37} +(0.617317 + 0.923880i) q^{40} +(-0.750661 + 0.149316i) q^{42} +(-0.785695 - 0.785695i) q^{45} +(1.08979 - 1.63099i) q^{46} -1.00000 q^{48} +0.414214 q^{49} +(-0.130355 + 0.195090i) q^{50} +(1.17588 + 1.17588i) q^{51} +(0.980785 - 0.195090i) q^{54} +(-0.750661 - 0.149316i) q^{56} +(0.0761205 + 0.382683i) q^{58} +(-0.275899 + 0.275899i) q^{59} +(-0.425215 - 1.02656i) q^{60} +0.765367 q^{63} +(-0.923880 - 0.382683i) q^{64} +(-1.30656 - 1.30656i) q^{67} +(0.636379 + 1.53636i) q^{68} +(-1.38704 + 1.38704i) q^{69} +(-0.165911 - 0.834089i) q^{70} +(0.980785 + 0.195090i) q^{72} +1.84776i q^{73} +(0.980785 - 0.195090i) q^{74} +(0.165911 - 0.165911i) q^{75} -1.11114i q^{80} -1.00000 q^{81} +(0.707107 + 0.292893i) q^{84} +(-1.30656 + 1.30656i) q^{85} -0.390181i q^{87} +0.390181i q^{89} +(0.216773 + 1.08979i) q^{90} +(-1.81225 + 0.750661i) q^{92} +(0.831470 + 0.555570i) q^{96} +(-0.344406 - 0.230125i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{34} + 16 q^{40} - 16 q^{48} - 16 q^{49} + 16 q^{58} - 16 q^{81}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1776\mathbb{Z}\right)^\times\).

\(n\) \(223\) \(593\) \(1297\) \(1333\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(e\left(\frac{3}{4}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.831470 0.555570i −0.831470 0.555570i
\(3\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(4\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(5\) −0.785695 + 0.785695i −0.785695 + 0.785695i −0.980785 0.195090i \(-0.937500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(6\) −0.195090 0.980785i −0.195090 0.980785i
\(7\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(8\) 0.195090 0.980785i 0.195090 0.980785i
\(9\) 1.00000i 1.00000i
\(10\) 1.08979 0.216773i 1.08979 0.216773i
\(11\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(12\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(13\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(14\) −0.425215 + 0.636379i −0.425215 + 0.636379i
\(15\) −1.11114 −1.11114
\(16\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(17\) 1.66294 1.66294 0.831470 0.555570i \(-0.187500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(18\) 0.555570 0.831470i 0.555570 0.831470i
\(19\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(20\) −1.02656 0.425215i −1.02656 0.425215i
\(21\) 0.541196 0.541196i 0.541196 0.541196i
\(22\) 0 0
\(23\) 1.96157i 1.96157i 0.195090 + 0.980785i \(0.437500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(24\) 0.831470 0.555570i 0.831470 0.555570i
\(25\) 0.234633i 0.234633i
\(26\) 0 0
\(27\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(28\) 0.707107 0.292893i 0.707107 0.292893i
\(29\) −0.275899 0.275899i −0.275899 0.275899i 0.555570 0.831470i \(-0.312500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(30\) 0.923880 + 0.617317i 0.923880 + 0.617317i
\(31\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(32\) 0.980785 0.195090i 0.980785 0.195090i
\(33\) 0 0
\(34\) −1.38268 0.923880i −1.38268 0.923880i
\(35\) 0.601345 + 0.601345i 0.601345 + 0.601345i
\(36\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(37\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(38\) 0 0
\(39\) 0 0
\(40\) 0.617317 + 0.923880i 0.617317 + 0.923880i
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) −0.750661 + 0.149316i −0.750661 + 0.149316i
\(43\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(44\) 0 0
\(45\) −0.785695 0.785695i −0.785695 0.785695i
\(46\) 1.08979 1.63099i 1.08979 1.63099i
\(47\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(48\) −1.00000 −1.00000
\(49\) 0.414214 0.414214
\(50\) −0.130355 + 0.195090i −0.130355 + 0.195090i
\(51\) 1.17588 + 1.17588i 1.17588 + 1.17588i
\(52\) 0 0
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0.980785 0.195090i 0.980785 0.195090i
\(55\) 0 0
\(56\) −0.750661 0.149316i −0.750661 0.149316i
\(57\) 0 0
\(58\) 0.0761205 + 0.382683i 0.0761205 + 0.382683i
\(59\) −0.275899 + 0.275899i −0.275899 + 0.275899i −0.831470 0.555570i \(-0.812500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(60\) −0.425215 1.02656i −0.425215 1.02656i
\(61\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(62\) 0 0
\(63\) 0.765367 0.765367
\(64\) −0.923880 0.382683i −0.923880 0.382683i
\(65\) 0 0
\(66\) 0 0
\(67\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(68\) 0.636379 + 1.53636i 0.636379 + 1.53636i
\(69\) −1.38704 + 1.38704i −1.38704 + 1.38704i
\(70\) −0.165911 0.834089i −0.165911 0.834089i
\(71\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(72\) 0.980785 + 0.195090i 0.980785 + 0.195090i
\(73\) 1.84776i 1.84776i 0.382683 + 0.923880i \(0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(74\) 0.980785 0.195090i 0.980785 0.195090i
\(75\) 0.165911 0.165911i 0.165911 0.165911i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(80\) 1.11114i 1.11114i
\(81\) −1.00000 −1.00000
\(82\) 0 0
\(83\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(84\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(85\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(86\) 0 0
\(87\) 0.390181i 0.390181i
\(88\) 0 0
\(89\) 0.390181i 0.390181i 0.980785 + 0.195090i \(0.0625000\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(90\) 0.216773 + 1.08979i 0.216773 + 1.08979i
\(91\) 0 0
\(92\) −1.81225 + 0.750661i −1.81225 + 0.750661i
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(97\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(98\) −0.344406 0.230125i −0.344406 0.230125i
\(99\) 0 0
\(100\) 0.216773 0.0897902i 0.216773 0.0897902i
\(101\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(102\) −0.324423 1.63099i −0.324423 1.63099i
\(103\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(104\) 0 0
\(105\) 0.850430i 0.850430i
\(106\) 0 0
\(107\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(108\) −0.923880 0.382683i −0.923880 0.382683i
\(109\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(110\) 0 0
\(111\) −1.00000 −1.00000
\(112\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(113\) 1.96157 1.96157 0.980785 0.195090i \(-0.0625000\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(114\) 0 0
\(115\) −1.54120 1.54120i −1.54120 1.54120i
\(116\) 0.149316 0.360480i 0.149316 0.360480i
\(117\) 0 0
\(118\) 0.382683 0.0761205i 0.382683 0.0761205i
\(119\) 1.27276i 1.27276i
\(120\) −0.216773 + 1.08979i −0.216773 + 1.08979i
\(121\) 1.00000i 1.00000i
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) −0.601345 0.601345i −0.601345 0.601345i
\(126\) −0.636379 0.425215i −0.636379 0.425215i
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(129\) 0 0
\(130\) 0 0
\(131\) −0.275899 0.275899i −0.275899 0.275899i 0.555570 0.831470i \(-0.312500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0.360480 + 1.81225i 0.360480 + 1.81225i
\(135\) 1.11114i 1.11114i
\(136\) 0.324423 1.63099i 0.324423 1.63099i
\(137\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(138\) 1.92388 0.382683i 1.92388 0.382683i
\(139\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(140\) −0.325446 + 0.785695i −0.325446 + 0.785695i
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.707107 0.707107i −0.707107 0.707107i
\(145\) 0.433546 0.433546
\(146\) 1.02656 1.53636i 1.02656 1.53636i
\(147\) 0.292893 + 0.292893i 0.292893 + 0.292893i
\(148\) −0.923880 0.382683i −0.923880 0.382683i
\(149\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(150\) −0.230125 + 0.0457747i −0.230125 + 0.0457747i
\(151\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(152\) 0 0
\(153\) 1.66294i 1.66294i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −0.617317 + 0.923880i −0.617317 + 0.923880i
\(161\) 1.50132 1.50132
\(162\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(163\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 1.66294i 1.66294i −0.555570 0.831470i \(-0.687500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(168\) −0.425215 0.636379i −0.425215 0.636379i
\(169\) 1.00000i 1.00000i
\(170\) 1.81225 0.360480i 1.81225 0.360480i
\(171\) 0 0
\(172\) 0 0
\(173\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(174\) −0.216773 + 0.324423i −0.216773 + 0.324423i
\(175\) −0.179580 −0.179580
\(176\) 0 0
\(177\) −0.390181 −0.390181
\(178\) 0.216773 0.324423i 0.216773 0.324423i
\(179\) −1.17588 1.17588i −1.17588 1.17588i −0.980785 0.195090i \(-0.937500\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(180\) 0.425215 1.02656i 0.425215 1.02656i
\(181\) 1.30656 1.30656i 1.30656 1.30656i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(185\) 1.11114i 1.11114i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(190\) 0 0
\(191\) 1.66294 1.66294 0.831470 0.555570i \(-0.187500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(192\) −0.382683 0.923880i −0.382683 0.923880i
\(193\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.158513 + 0.382683i 0.158513 + 0.382683i
\(197\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(198\) 0 0
\(199\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −0.230125 0.0457747i −0.230125 0.0457747i
\(201\) 1.84776i 1.84776i
\(202\) 0 0
\(203\) −0.211164 + 0.211164i −0.211164 + 0.211164i
\(204\) −0.636379 + 1.53636i −0.636379 + 1.53636i
\(205\) 0 0
\(206\) 0 0
\(207\) −1.96157 −1.96157
\(208\) 0 0
\(209\) 0 0
\(210\) 0.472474 0.707107i 0.472474 0.707107i
\(211\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(217\) 0 0
\(218\) 0 0
\(219\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(220\) 0 0
\(221\) 0 0
\(222\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(223\) 1.41421 1.41421 0.707107 0.707107i \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(224\) −0.149316 0.750661i −0.149316 0.750661i
\(225\) 0.234633 0.234633
\(226\) −1.63099 1.08979i −1.63099 1.08979i
\(227\) 0.785695 + 0.785695i 0.785695 + 0.785695i 0.980785 0.195090i \(-0.0625000\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(228\) 0 0
\(229\) 1.41421 1.41421i 1.41421 1.41421i 0.707107 0.707107i \(-0.250000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(230\) 0.425215 + 2.13770i 0.425215 + 2.13770i
\(231\) 0 0
\(232\) −0.324423 + 0.216773i −0.324423 + 0.216773i
\(233\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) −0.360480 0.149316i −0.360480 0.149316i
\(237\) 0 0
\(238\) −0.707107 + 1.05826i −0.707107 + 1.05826i
\(239\) −0.390181 −0.390181 −0.195090 0.980785i \(-0.562500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(240\) 0.785695 0.785695i 0.785695 0.785695i
\(241\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(242\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(243\) −0.707107 0.707107i −0.707107 0.707107i
\(244\) 0 0
\(245\) −0.325446 + 0.325446i −0.325446 + 0.325446i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0.165911 + 0.834089i 0.165911 + 0.834089i
\(251\) −0.785695 + 0.785695i −0.785695 + 0.785695i −0.980785 0.195090i \(-0.937500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(252\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(253\) 0 0
\(254\) 0 0
\(255\) −1.84776 −1.84776
\(256\) 1.00000i 1.00000i
\(257\) −1.96157 −1.96157 −0.980785 0.195090i \(-0.937500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(258\) 0 0
\(259\) 0.541196 + 0.541196i 0.541196 + 0.541196i
\(260\) 0 0
\(261\) 0.275899 0.275899i 0.275899 0.275899i
\(262\) 0.0761205 + 0.382683i 0.0761205 + 0.382683i
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) −0.275899 + 0.275899i −0.275899 + 0.275899i
\(268\) 0.707107 1.70711i 0.707107 1.70711i
\(269\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(270\) −0.617317 + 0.923880i −0.617317 + 0.923880i
\(271\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(272\) −1.17588 + 1.17588i −1.17588 + 1.17588i
\(273\) 0 0
\(274\) 0 0
\(275\) 0 0
\(276\) −1.81225 0.750661i −1.81225 0.750661i
\(277\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(278\) −1.81225 + 0.360480i −1.81225 + 0.360480i
\(279\) 0 0
\(280\) 0.707107 0.472474i 0.707107 0.472474i
\(281\) 1.96157i 1.96157i −0.195090 0.980785i \(-0.562500\pi\)
0.195090 0.980785i \(-0.437500\pi\)
\(282\) 0 0
\(283\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(289\) 1.76537 1.76537
\(290\) −0.360480 0.240865i −0.360480 0.240865i
\(291\) 0 0
\(292\) −1.70711 + 0.707107i −1.70711 + 0.707107i
\(293\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(294\) −0.0808091 0.406255i −0.0808091 0.406255i
\(295\) 0.433546i 0.433546i
\(296\) 0.555570 + 0.831470i 0.555570 + 0.831470i
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0.216773 + 0.0897902i 0.216773 + 0.0897902i
\(301\) 0 0
\(302\) −0.785695 + 1.17588i −0.785695 + 1.17588i
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0.923880 1.38268i 0.923880 1.38268i
\(307\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 1.96157i 1.96157i −0.195090 0.980785i \(-0.562500\pi\)
0.195090 0.980785i \(-0.437500\pi\)
\(312\) 0 0
\(313\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(314\) −0.360480 1.81225i −0.360480 1.81225i
\(315\) −0.601345 + 0.601345i −0.601345 + 0.601345i
\(316\) 0 0
\(317\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 1.02656 0.425215i 1.02656 0.425215i
\(321\) 0 0
\(322\) −1.24830 0.834089i −1.24830 0.834089i
\(323\) 0 0
\(324\) −0.382683 0.923880i −0.382683 0.923880i
\(325\) 0 0
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(332\) 0 0
\(333\) −0.707107 0.707107i −0.707107 0.707107i
\(334\) −0.923880 + 1.38268i −0.923880 + 1.38268i
\(335\) 2.05312 2.05312
\(336\) 0.765367i 0.765367i
\(337\) −1.41421 −1.41421 −0.707107 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(338\) 0.555570 0.831470i 0.555570 0.831470i
\(339\) 1.38704 + 1.38704i 1.38704 + 1.38704i
\(340\) −1.70711 0.707107i −1.70711 0.707107i
\(341\) 0 0
\(342\) 0 0
\(343\) 1.08239i 1.08239i
\(344\) 0 0
\(345\) 2.17958i 2.17958i
\(346\) 0 0
\(347\) 1.38704 1.38704i 1.38704 1.38704i 0.555570 0.831470i \(-0.312500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(348\) 0.360480 0.149316i 0.360480 0.149316i
\(349\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(350\) 0.149316 + 0.0997695i 0.149316 + 0.0997695i
\(351\) 0 0
\(352\) 0 0
\(353\) −1.11114 −1.11114 −0.555570 0.831470i \(-0.687500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(354\) 0.324423 + 0.216773i 0.324423 + 0.216773i
\(355\) 0 0
\(356\) −0.360480 + 0.149316i −0.360480 + 0.149316i
\(357\) 0.899976 0.899976i 0.899976 0.899976i
\(358\) 0.324423 + 1.63099i 0.324423 + 1.63099i
\(359\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(360\) −0.923880 + 0.617317i −0.923880 + 0.617317i
\(361\) 1.00000i 1.00000i
\(362\) −1.81225 + 0.360480i −1.81225 + 0.360480i
\(363\) 0.707107 0.707107i 0.707107 0.707107i
\(364\) 0 0
\(365\) −1.45177 1.45177i −1.45177 1.45177i
\(366\) 0 0
\(367\) −2.00000 −2.00000 −1.00000 \(\pi\)
−1.00000 \(\pi\)
\(368\) −1.38704 1.38704i −1.38704 1.38704i
\(369\) 0 0
\(370\) −0.617317 + 0.923880i −0.617317 + 0.923880i
\(371\) 0 0
\(372\) 0 0
\(373\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(374\) 0 0
\(375\) 0.850430i 0.850430i
\(376\) 0 0
\(377\) 0 0
\(378\) −0.149316 0.750661i −0.149316 0.750661i
\(379\) 0.541196 0.541196i 0.541196 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −1.38268 0.923880i −1.38268 0.923880i
\(383\) −1.66294 −1.66294 −0.831470 0.555570i \(-0.812500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(384\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(385\) 0 0
\(386\) 0 0
\(387\) 0 0
\(388\) 0 0
\(389\) −1.17588 + 1.17588i −1.17588 + 1.17588i −0.195090 + 0.980785i \(0.562500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(390\) 0 0
\(391\) 3.26197i 3.26197i
\(392\) 0.0808091 0.406255i 0.0808091 0.406255i
\(393\) 0.390181i 0.390181i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) 0.165911 + 0.165911i 0.165911 + 0.165911i
\(401\) 1.11114 1.11114 0.555570 0.831470i \(-0.312500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(402\) −1.02656 + 1.53636i −1.02656 + 1.53636i
\(403\) 0 0
\(404\) 0 0
\(405\) 0.785695 0.785695i 0.785695 0.785695i
\(406\) 0.292893 0.0582601i 0.292893 0.0582601i
\(407\) 0 0
\(408\) 1.38268 0.923880i 1.38268 0.923880i
\(409\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0.211164 + 0.211164i 0.211164 + 0.211164i
\(414\) 1.63099 + 1.08979i 1.63099 + 1.08979i
\(415\) 0 0
\(416\) 0 0
\(417\) 1.84776 1.84776
\(418\) 0 0
\(419\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(420\) −0.785695 + 0.325446i −0.785695 + 0.325446i
\(421\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(422\) −0.275899 1.38704i −0.275899 1.38704i
\(423\) 0 0
\(424\) 0 0
\(425\) 0.390181i 0.390181i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 1.96157 1.96157 0.980785 0.195090i \(-0.0625000\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(432\) 1.00000i 1.00000i
\(433\) −0.765367 −0.765367 −0.382683 0.923880i \(-0.625000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(434\) 0 0
\(435\) 0.306563 + 0.306563i 0.306563 + 0.306563i
\(436\) 0 0
\(437\) 0 0
\(438\) 1.81225 0.360480i 1.81225 0.360480i
\(439\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(440\) 0 0
\(441\) 0.414214i 0.414214i
\(442\) 0 0
\(443\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(444\) −0.382683 0.923880i −0.382683 0.923880i
\(445\) −0.306563 0.306563i −0.306563 0.306563i
\(446\) −1.17588 0.785695i −1.17588 0.785695i
\(447\) 0 0
\(448\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(449\) 1.66294 1.66294 0.831470 0.555570i \(-0.187500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(450\) −0.195090 0.130355i −0.195090 0.130355i
\(451\) 0 0
\(452\) 0.750661 + 1.81225i 0.750661 + 1.81225i
\(453\) 1.00000 1.00000i 1.00000 1.00000i
\(454\) −0.216773 1.08979i −0.216773 1.08979i
\(455\) 0 0
\(456\) 0 0
\(457\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(458\) −1.96157 + 0.390181i −1.96157 + 0.390181i
\(459\) −1.17588 + 1.17588i −1.17588 + 1.17588i
\(460\) 0.834089 2.01367i 0.834089 2.01367i
\(461\) 1.17588 + 1.17588i 1.17588 + 1.17588i 0.980785 + 0.195090i \(0.0625000\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(464\) 0.390181 0.390181
\(465\) 0 0
\(466\) 0 0
\(467\) 0.275899 + 0.275899i 0.275899 + 0.275899i 0.831470 0.555570i \(-0.187500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(468\) 0 0
\(469\) −1.00000 + 1.00000i −1.00000 + 1.00000i
\(470\) 0 0
\(471\) 1.84776i 1.84776i
\(472\) 0.216773 + 0.324423i 0.216773 + 0.324423i
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 1.17588 0.487064i 1.17588 0.487064i
\(477\) 0 0
\(478\) 0.324423 + 0.216773i 0.324423 + 0.216773i
\(479\) −1.11114 −1.11114 −0.555570 0.831470i \(-0.687500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(480\) −1.08979 + 0.216773i −1.08979 + 0.216773i
\(481\) 0 0
\(482\) 0 0
\(483\) 1.06159 + 1.06159i 1.06159 + 1.06159i
\(484\) 0.923880 0.382683i 0.923880 0.382683i
\(485\) 0 0
\(486\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(487\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0.451406 0.0897902i 0.451406 0.0897902i
\(491\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(492\) 0 0
\(493\) −0.458804 0.458804i −0.458804 0.458804i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(500\) 0.325446 0.785695i 0.325446 0.785695i
\(501\) 1.17588 1.17588i 1.17588 1.17588i
\(502\) 1.08979 0.216773i 1.08979 0.216773i
\(503\) 1.11114i 1.11114i −0.831470 0.555570i \(-0.812500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(504\) 0.149316 0.750661i 0.149316 0.750661i
\(505\) 0 0
\(506\) 0 0
\(507\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(508\) 0 0
\(509\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(510\) 1.53636 + 1.02656i 1.53636 + 1.02656i
\(511\) 1.41421 1.41421
\(512\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(513\) 0 0
\(514\) 1.63099 + 1.08979i 1.63099 + 1.08979i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) −0.149316 0.750661i −0.149316 0.750661i
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(522\) −0.382683 + 0.0761205i −0.382683 + 0.0761205i
\(523\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(524\) 0.149316 0.360480i 0.149316 0.360480i
\(525\) −0.126983 0.126983i −0.126983 0.126983i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −2.84776 −2.84776
\(530\) 0 0
\(531\) −0.275899 0.275899i −0.275899 0.275899i
\(532\) 0 0
\(533\) 0 0
\(534\) 0.382683 0.0761205i 0.382683 0.0761205i
\(535\) 0 0
\(536\) −1.53636 + 1.02656i −1.53636 + 1.02656i
\(537\) 1.66294i 1.66294i
\(538\) 0 0
\(539\) 0 0
\(540\) 1.02656 0.425215i 1.02656 0.425215i
\(541\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(542\) 1.53636 + 1.02656i 1.53636 + 1.02656i
\(543\) 1.84776 1.84776
\(544\) 1.63099 0.324423i 1.63099 0.324423i
\(545\) 0 0
\(546\) 0 0
\(547\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0 0
\(551\) 0 0
\(552\) 1.08979 + 1.63099i 1.08979 + 1.63099i
\(553\) 0 0
\(554\) 0 0
\(555\) 0.785695 0.785695i 0.785695 0.785695i
\(556\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(557\) 1.38704 + 1.38704i 1.38704 + 1.38704i 0.831470 + 0.555570i \(0.187500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −0.850430 −0.850430
\(561\) 0 0
\(562\) −1.08979 + 1.63099i −1.08979 + 1.63099i
\(563\) −1.17588 1.17588i −1.17588 1.17588i −0.980785 0.195090i \(-0.937500\pi\)
−0.195090 0.980785i \(-0.562500\pi\)
\(564\) 0 0
\(565\) −1.54120 + 1.54120i −1.54120 + 1.54120i
\(566\) 0 0
\(567\) 0.765367i 0.765367i
\(568\) 0 0
\(569\) 1.96157i 1.96157i −0.195090 0.980785i \(-0.562500\pi\)
0.195090 0.980785i \(-0.437500\pi\)
\(570\) 0 0
\(571\) −0.541196 + 0.541196i −0.541196 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(572\) 0 0
\(573\) 1.17588 + 1.17588i 1.17588 + 1.17588i
\(574\) 0 0
\(575\) 0.460249 0.460249
\(576\) 0.382683 0.923880i 0.382683 0.923880i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −1.46785 0.980785i −1.46785 0.980785i
\(579\) 0 0
\(580\) 0.165911 + 0.400544i 0.165911 + 0.400544i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 1.81225 + 0.360480i 1.81225 + 0.360480i
\(585\) 0 0
\(586\) 0 0
\(587\) −1.17588 + 1.17588i −1.17588 + 1.17588i −0.195090 + 0.980785i \(0.562500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(588\) −0.158513 + 0.382683i −0.158513 + 0.382683i
\(589\) 0 0
\(590\) −0.240865 + 0.360480i −0.240865 + 0.360480i
\(591\) 0 0
\(592\) 1.00000i 1.00000i
\(593\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(594\) 0 0
\(595\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(600\) −0.130355 0.195090i −0.130355 0.195090i
\(601\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(602\) 0 0
\(603\) 1.30656 1.30656i 1.30656 1.30656i
\(604\) 1.30656 0.541196i 1.30656 0.541196i
\(605\) 0.785695 + 0.785695i 0.785695 + 0.785695i
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) −0.298631 −0.298631
\(610\) 0 0
\(611\) 0 0
\(612\) −1.53636 + 0.636379i −1.53636 + 0.636379i
\(613\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(614\) 0.275899 + 1.38704i 0.275899 + 1.38704i
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(618\) 0 0
\(619\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(620\) 0 0
\(621\) −1.38704 1.38704i −1.38704 1.38704i
\(622\) −1.08979 + 1.63099i −1.08979 + 1.63099i
\(623\) 0.298631 0.298631
\(624\) 0 0
\(625\) 1.17958 1.17958
\(626\) 0 0
\(627\) 0 0
\(628\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(629\) −1.17588 + 1.17588i −1.17588 + 1.17588i
\(630\) 0.834089 0.165911i 0.834089 0.165911i
\(631\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(632\) 0 0
\(633\) 1.41421i 1.41421i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −1.08979 0.216773i −1.08979 0.216773i
\(641\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(642\) 0 0
\(643\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(644\) 0.574531 + 1.38704i 0.574531 + 1.38704i
\(645\) 0 0
\(646\) 0 0
\(647\) 1.11114i 1.11114i 0.831470 + 0.555570i \(0.187500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(648\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) 0 0
\(653\) −1.38704 1.38704i −1.38704 1.38704i −0.831470 0.555570i \(-0.812500\pi\)
−0.555570 0.831470i \(-0.687500\pi\)
\(654\) 0 0
\(655\) 0.433546 0.433546
\(656\) 0 0
\(657\) −1.84776 −1.84776
\(658\) 0 0
\(659\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(660\) 0 0
\(661\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(667\) 0.541196 0.541196i 0.541196 0.541196i
\(668\) 1.53636 0.636379i 1.53636 0.636379i
\(669\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(670\) −1.70711 1.14065i −1.70711 1.14065i
\(671\) 0 0
\(672\) 0.425215 0.636379i 0.425215 0.636379i
\(673\) −1.84776 −1.84776 −0.923880 0.382683i \(-0.875000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(674\) 1.17588 + 0.785695i 1.17588 + 0.785695i
\(675\) 0.165911 + 0.165911i 0.165911 + 0.165911i
\(676\) −0.923880 + 0.382683i −0.923880 + 0.382683i
\(677\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(678\) −0.382683 1.92388i −0.382683 1.92388i
\(679\) 0 0
\(680\) 1.02656 + 1.53636i 1.02656 + 1.53636i
\(681\) 1.11114i 1.11114i
\(682\) 0 0
\(683\) 0.275899 0.275899i 0.275899 0.275899i −0.555570 0.831470i \(-0.687500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) −0.601345 + 0.899976i −0.601345 + 0.899976i
\(687\) 2.00000 2.00000
\(688\) 0 0
\(689\) 0 0
\(690\) −1.21091 + 1.81225i −1.21091 + 1.81225i
\(691\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) −1.92388 + 0.382683i −1.92388 + 0.382683i
\(695\) 2.05312i 2.05312i
\(696\) −0.382683 0.0761205i −0.382683 0.0761205i
\(697\) 0 0
\(698\) 0.149316 + 0.750661i 0.149316 + 0.750661i
\(699\) 0 0
\(700\) −0.0687225 0.165911i −0.0687225 0.165911i
\(701\) 0.275899 + 0.275899i 0.275899 + 0.275899i 0.831470 0.555570i \(-0.187500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0.923880 + 0.617317i 0.923880 + 0.617317i
\(707\) 0 0
\(708\) −0.149316 0.360480i −0.149316 0.360480i
\(709\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.382683 + 0.0761205i 0.382683 + 0.0761205i
\(713\) 0 0
\(714\) −1.24830 + 0.248303i −1.24830 + 0.248303i
\(715\) 0 0
\(716\) 0.636379 1.53636i 0.636379 1.53636i
\(717\) −0.275899 0.275899i −0.275899 0.275899i
\(718\) 0 0
\(719\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(720\) 1.11114 1.11114
\(721\) 0 0
\(722\) 0.555570 0.831470i 0.555570 0.831470i
\(723\) 0 0
\(724\) 1.70711 + 0.707107i 1.70711 + 0.707107i
\(725\) −0.0647351 + 0.0647351i −0.0647351 + 0.0647351i
\(726\) −0.980785 + 0.195090i −0.980785 + 0.195090i
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) 0 0
\(729\) 1.00000i 1.00000i
\(730\) 0.400544 + 2.01367i 0.400544 + 2.01367i
\(731\) 0 0
\(732\) 0 0
\(733\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(734\) 1.66294 + 1.11114i 1.66294 + 1.11114i
\(735\) −0.460249 −0.460249
\(736\) 0.382683 + 1.92388i 0.382683 + 1.92388i
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(740\) 1.02656 0.425215i 1.02656 0.425215i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0.750661 0.149316i 0.750661 0.149316i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) −0.472474 + 0.707107i −0.472474 + 0.707107i
\(751\) 1.84776 1.84776 0.923880 0.382683i \(-0.125000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(752\) 0 0
\(753\) −1.11114 −1.11114
\(754\) 0 0
\(755\) 1.11114 + 1.11114i 1.11114 + 1.11114i
\(756\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(757\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(758\) −0.750661 + 0.149316i −0.750661 + 0.149316i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0.636379 + 1.53636i 0.636379 + 1.53636i
\(765\) −1.30656 1.30656i −1.30656 1.30656i
\(766\) 1.38268 + 0.923880i 1.38268 + 0.923880i
\(767\) 0 0
\(768\) 0.707107 0.707107i 0.707107 0.707107i
\(769\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(770\) 0 0
\(771\) −1.38704 1.38704i −1.38704 1.38704i
\(772\) 0 0
\(773\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 0.765367i 0.765367i
\(778\) 1.63099 0.324423i 1.63099 0.324423i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 1.81225 2.71223i 1.81225 2.71223i
\(783\) 0.390181 0.390181
\(784\) −0.292893 + 0.292893i −0.292893 + 0.292893i
\(785\) −2.05312 −2.05312
\(786\) −0.216773 + 0.324423i −0.216773 + 0.324423i
\(787\) 1.00000 + 1.00000i 1.00000 + 1.00000i 1.00000 \(0\)
1.00000i \(0.5\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 1.50132i 1.50132i
\(792\) 0 0
\(793\) 0 0
\(794\) −0.360480 1.81225i −0.360480 1.81225i
\(795\) 0 0
\(796\) 0 0
\(797\) 1.17588 + 1.17588i 1.17588 + 1.17588i 0.980785 + 0.195090i \(0.0625000\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −0.0457747 0.230125i −0.0457747 0.230125i
\(801\) −0.390181 −0.390181
\(802\) −0.923880 0.617317i −0.923880 0.617317i
\(803\) 0 0
\(804\) 1.70711 0.707107i 1.70711 0.707107i
\(805\) −1.17958 + 1.17958i −1.17958 + 1.17958i
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 1.96157i 1.96157i −0.195090 0.980785i \(-0.562500\pi\)
0.195090 0.980785i \(-0.437500\pi\)
\(810\) −1.08979 + 0.216773i −1.08979 + 0.216773i
\(811\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(812\) −0.275899 0.114281i −0.275899 0.114281i
\(813\) −1.30656 1.30656i −1.30656 1.30656i
\(814\) 0 0
\(815\) 0 0
\(816\) −1.66294 −1.66294
\(817\) 0 0
\(818\) 0 0
\(819\) 0 0
\(820\) 0 0
\(821\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(822\) 0 0
\(823\) 1.41421i 1.41421i −0.707107 0.707107i \(-0.750000\pi\)
0.707107 0.707107i \(-0.250000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −0.0582601 0.292893i −0.0582601 0.292893i
\(827\) 0.785695 0.785695i 0.785695 0.785695i −0.195090 0.980785i \(-0.562500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(828\) −0.750661 1.81225i −0.750661 1.81225i
\(829\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 0 0
\(833\) 0.688812 0.688812
\(834\) −1.53636 1.02656i −1.53636 1.02656i
\(835\) 1.30656 + 1.30656i 1.30656 + 1.30656i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(840\) 0.834089 + 0.165911i 0.834089 + 0.165911i
\(841\) 0.847759i 0.847759i
\(842\) 0 0
\(843\) 1.38704 1.38704i 1.38704 1.38704i
\(844\) −0.541196 + 1.30656i −0.541196 + 1.30656i
\(845\) −0.785695 0.785695i −0.785695 0.785695i
\(846\) 0 0
\(847\) −0.765367 −0.765367
\(848\) 0 0
\(849\) 0 0
\(850\) −0.216773 + 0.324423i −0.216773 + 0.324423i
\(851\) −1.38704 1.38704i −1.38704 1.38704i
\(852\) 0 0
\(853\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0.390181i 0.390181i −0.980785 0.195090i \(-0.937500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(858\) 0 0
\(859\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −1.63099 1.08979i −1.63099 1.08979i
\(863\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(864\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(865\) 0 0
\(866\) 0.636379 + 0.425215i 0.636379 + 0.425215i
\(867\) 1.24830 + 1.24830i 1.24830 + 1.24830i
\(868\) 0 0
\(869\) 0 0
\(870\) −0.0845805 0.425215i −0.0845805 0.425215i
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −0.460249 + 0.460249i −0.460249 + 0.460249i
\(876\) −1.70711 0.707107i −1.70711 0.707107i
\(877\) −1.30656 1.30656i −1.30656 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 0.923880i \(-0.625000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(882\) 0.230125 0.344406i 0.230125 0.344406i
\(883\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(884\) 0 0
\(885\) 0.306563 0.306563i 0.306563 0.306563i
\(886\) 0 0
\(887\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(888\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(889\) 0 0
\(890\) 0.0845805 + 0.425215i 0.0845805 + 0.425215i
\(891\) 0 0
\(892\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(893\) 0 0
\(894\) 0 0
\(895\) 1.84776 1.84776
\(896\) 0.636379 0.425215i 0.636379 0.425215i
\(897\) 0 0
\(898\) −1.38268 0.923880i −1.38268 0.923880i
\(899\) 0 0
\(900\) 0.0897902 + 0.216773i 0.0897902 + 0.216773i
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0.382683 1.92388i 0.382683 1.92388i
\(905\) 2.05312i 2.05312i
\(906\) −1.38704 + 0.275899i −1.38704 + 0.275899i
\(907\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(908\) −0.425215 + 1.02656i −0.425215 + 1.02656i
\(909\) 0 0
\(910\) 0 0
\(911\) 1.11114 1.11114 0.555570 0.831470i \(-0.312500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.84776 + 0.765367i 1.84776 + 0.765367i
\(917\) −0.211164 + 0.211164i −0.211164 + 0.211164i
\(918\) 1.63099 0.324423i 1.63099 0.324423i
\(919\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(920\) −1.81225 + 1.21091i −1.81225 + 1.21091i
\(921\) 1.41421i 1.41421i
\(922\) −0.324423 1.63099i −0.324423 1.63099i
\(923\) 0 0
\(924\) 0 0
\(925\) 0.165911 + 0.165911i 0.165911 + 0.165911i
\(926\) 0 0
\(927\) 0 0
\(928\) −0.324423 0.216773i −0.324423 0.216773i
\(929\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0 0
\(933\) 1.38704 1.38704i 1.38704 1.38704i
\(934\) −0.0761205 0.382683i −0.0761205 0.382683i
\(935\) 0 0
\(936\) 0 0
\(937\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(938\) 1.38704 0.275899i 1.38704 0.275899i
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(942\) 1.02656 1.53636i 1.02656 1.53636i
\(943\) 0 0
\(944\) 0.390181i 0.390181i
\(945\) −0.850430 −0.850430
\(946\) 0 0
\(947\) 1.17588 + 1.17588i 1.17588 + 1.17588i 0.980785 + 0.195090i \(0.0625000\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 0 0
\(951\) 0 0
\(952\) −1.24830 0.248303i −1.24830 0.248303i
\(953\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(954\) 0 0
\(955\) −1.30656 + 1.30656i −1.30656 + 1.30656i
\(956\) −0.149316 0.360480i −0.149316 0.360480i
\(957\) 0 0
\(958\) 0.923880 + 0.617317i 0.923880 + 0.617317i
\(959\) 0 0
\(960\) 1.02656 + 0.425215i 1.02656 + 0.425215i
\(961\) 1.00000 1.00000
\(962\) 0 0
\(963\) 0 0
\(964\) 0 0
\(965\) 0 0
\(966\) −0.292893 1.47247i −0.292893 1.47247i
\(967\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(968\) −0.980785 0.195090i −0.980785 0.195090i
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(972\) 0.382683 0.923880i 0.382683 0.923880i
\(973\) −1.00000 1.00000i −1.00000 1.00000i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) −1.11114 −1.11114 −0.555570 0.831470i \(-0.687500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.425215 0.176130i −0.425215 0.176130i
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0.126584 + 0.636379i 0.126584 + 0.636379i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(998\) 0 0
\(999\) 1.00000i 1.00000i
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 1776.1.z.c.221.2 16
3.2 odd 2 inner 1776.1.z.c.221.7 yes 16
16.5 even 4 inner 1776.1.z.c.1109.2 yes 16
37.36 even 2 inner 1776.1.z.c.221.7 yes 16
48.5 odd 4 inner 1776.1.z.c.1109.7 yes 16
111.110 odd 2 CM 1776.1.z.c.221.2 16
592.517 even 4 inner 1776.1.z.c.1109.7 yes 16
1776.1109 odd 4 inner 1776.1.z.c.1109.2 yes 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1776.1.z.c.221.2 16 1.1 even 1 trivial
1776.1.z.c.221.2 16 111.110 odd 2 CM
1776.1.z.c.221.7 yes 16 3.2 odd 2 inner
1776.1.z.c.221.7 yes 16 37.36 even 2 inner
1776.1.z.c.1109.2 yes 16 16.5 even 4 inner
1776.1.z.c.1109.2 yes 16 1776.1109 odd 4 inner
1776.1.z.c.1109.7 yes 16 48.5 odd 4 inner
1776.1.z.c.1109.7 yes 16 592.517 even 4 inner