Properties

Label 935.1.y.a.824.2
Level $935$
Weight $1$
Character 935.824
Analytic conductor $0.467$
Analytic rank $0$
Dimension $16$
Projective image $D_{16}$
CM discriminant -55
Inner twists $8$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [935,1,Mod(219,935)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(935, base_ring=CyclotomicField(8))
 
chi = DirichletCharacter(H, H._module([4, 4, 3]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("935.219");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 935 = 5 \cdot 11 \cdot 17 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 935.y (of order \(8\), degree \(4\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(0.466625786812\)
Analytic rank: \(0\)
Dimension: \(16\)
Relative dimension: \(4\) over \(\Q(\zeta_{8})\)
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: \(\Z[a_1, \ldots, a_{17}]\)
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 824.2
Root \(-0.980785 + 0.195090i\) of defining polynomial
Character \(\chi\) \(=\) 935.824
Dual form 935.1.y.a.219.2

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.275899 + 0.275899i) q^{2} +0.847759i q^{4} +(-0.923880 - 0.382683i) q^{5} +(1.81225 - 0.750661i) q^{7} +(-0.509796 - 0.509796i) q^{8} +(-0.707107 - 0.707107i) q^{9} +O(q^{10})\) \(q+(-0.275899 + 0.275899i) q^{2} +0.847759i q^{4} +(-0.923880 - 0.382683i) q^{5} +(1.81225 - 0.750661i) q^{7} +(-0.509796 - 0.509796i) q^{8} +(-0.707107 - 0.707107i) q^{9} +(0.360480 - 0.149316i) q^{10} +(-0.382683 - 0.923880i) q^{11} -1.66294i q^{13} +(-0.292893 + 0.707107i) q^{14} -0.566454 q^{16} +(0.831470 + 0.555570i) q^{17} +0.390181 q^{18} +(0.324423 - 0.783227i) q^{20} +(0.360480 + 0.149316i) q^{22} +(0.707107 + 0.707107i) q^{25} +(0.458804 + 0.458804i) q^{26} +(0.636379 + 1.53636i) q^{28} +(-0.541196 + 1.30656i) q^{31} +(0.666080 - 0.666080i) q^{32} +(-0.382683 + 0.0761205i) q^{34} -1.96157 q^{35} +(0.599456 - 0.599456i) q^{36} +(0.275899 + 0.666080i) q^{40} +(0.275899 + 0.275899i) q^{43} +(0.783227 - 0.324423i) q^{44} +(0.382683 + 0.923880i) q^{45} +(2.01367 - 2.01367i) q^{49} -0.390181 q^{50} +1.40977 q^{52} +1.00000i q^{55} +(-1.30656 - 0.541196i) q^{56} +(-0.211164 - 0.509796i) q^{62} +(-1.81225 - 0.750661i) q^{63} -0.198912i q^{64} +(-0.636379 + 1.53636i) q^{65} +(-0.470990 + 0.704886i) q^{68} +(0.541196 - 0.541196i) q^{70} +(-0.292893 + 0.707107i) q^{71} +0.720960i q^{72} +(-1.02656 - 0.425215i) q^{73} +(-1.38704 - 1.38704i) q^{77} +(0.523336 + 0.216773i) q^{80} +1.00000i q^{81} +(0.785695 - 0.785695i) q^{83} +(-0.555570 - 0.831470i) q^{85} -0.152241 q^{86} +(-0.275899 + 0.666080i) q^{88} +0.765367i q^{89} +(-0.360480 - 0.149316i) q^{90} +(-1.24830 - 3.01367i) q^{91} +1.11114i q^{98} +(-0.382683 + 0.923880i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{14} - 16 q^{16} + 16 q^{26} + 16 q^{44} - 16 q^{71} - 16 q^{80} - 32 q^{86}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/935\mathbb{Z}\right)^\times\).

\(n\) \(496\) \(562\) \(596\)
\(\chi(n)\) \(e\left(\frac{5}{8}\right)\) \(-1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.275899 + 0.275899i −0.275899 + 0.275899i −0.831470 0.555570i \(-0.812500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(3\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(4\) 0.847759i 0.847759i
\(5\) −0.923880 0.382683i −0.923880 0.382683i
\(6\) 0 0
\(7\) 1.81225 0.750661i 1.81225 0.750661i 0.831470 0.555570i \(-0.187500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(8\) −0.509796 0.509796i −0.509796 0.509796i
\(9\) −0.707107 0.707107i −0.707107 0.707107i
\(10\) 0.360480 0.149316i 0.360480 0.149316i
\(11\) −0.382683 0.923880i −0.382683 0.923880i
\(12\) 0 0
\(13\) 1.66294i 1.66294i −0.555570 0.831470i \(-0.687500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(14\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(15\) 0 0
\(16\) −0.566454 −0.566454
\(17\) 0.831470 + 0.555570i 0.831470 + 0.555570i
\(18\) 0.390181 0.390181
\(19\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(20\) 0.324423 0.783227i 0.324423 0.783227i
\(21\) 0 0
\(22\) 0.360480 + 0.149316i 0.360480 + 0.149316i
\(23\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(24\) 0 0
\(25\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(26\) 0.458804 + 0.458804i 0.458804 + 0.458804i
\(27\) 0 0
\(28\) 0.636379 + 1.53636i 0.636379 + 1.53636i
\(29\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(30\) 0 0
\(31\) −0.541196 + 1.30656i −0.541196 + 1.30656i 0.382683 + 0.923880i \(0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(32\) 0.666080 0.666080i 0.666080 0.666080i
\(33\) 0 0
\(34\) −0.382683 + 0.0761205i −0.382683 + 0.0761205i
\(35\) −1.96157 −1.96157
\(36\) 0.599456 0.599456i 0.599456 0.599456i
\(37\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.275899 + 0.666080i 0.275899 + 0.666080i
\(41\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(42\) 0 0
\(43\) 0.275899 + 0.275899i 0.275899 + 0.275899i 0.831470 0.555570i \(-0.187500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(44\) 0.783227 0.324423i 0.783227 0.324423i
\(45\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(46\) 0 0
\(47\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(48\) 0 0
\(49\) 2.01367 2.01367i 2.01367 2.01367i
\(50\) −0.390181 −0.390181
\(51\) 0 0
\(52\) 1.40977 1.40977
\(53\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(54\) 0 0
\(55\) 1.00000i 1.00000i
\(56\) −1.30656 0.541196i −1.30656 0.541196i
\(57\) 0 0
\(58\) 0 0
\(59\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(60\) 0 0
\(61\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(62\) −0.211164 0.509796i −0.211164 0.509796i
\(63\) −1.81225 0.750661i −1.81225 0.750661i
\(64\) 0.198912i 0.198912i
\(65\) −0.636379 + 1.53636i −0.636379 + 1.53636i
\(66\) 0 0
\(67\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(68\) −0.470990 + 0.704886i −0.470990 + 0.704886i
\(69\) 0 0
\(70\) 0.541196 0.541196i 0.541196 0.541196i
\(71\) −0.292893 + 0.707107i −0.292893 + 0.707107i 0.707107 + 0.707107i \(0.250000\pi\)
−1.00000 \(\pi\)
\(72\) 0.720960i 0.720960i
\(73\) −1.02656 0.425215i −1.02656 0.425215i −0.195090 0.980785i \(-0.562500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) −1.38704 1.38704i −1.38704 1.38704i
\(78\) 0 0
\(79\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(80\) 0.523336 + 0.216773i 0.523336 + 0.216773i
\(81\) 1.00000i 1.00000i
\(82\) 0 0
\(83\) 0.785695 0.785695i 0.785695 0.785695i −0.195090 0.980785i \(-0.562500\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(84\) 0 0
\(85\) −0.555570 0.831470i −0.555570 0.831470i
\(86\) −0.152241 −0.152241
\(87\) 0 0
\(88\) −0.275899 + 0.666080i −0.275899 + 0.666080i
\(89\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(90\) −0.360480 0.149316i −0.360480 0.149316i
\(91\) −1.24830 3.01367i −1.24830 3.01367i
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(98\) 1.11114i 1.11114i
\(99\) −0.382683 + 0.923880i −0.382683 + 0.923880i
\(100\) −0.599456 + 0.599456i −0.599456 + 0.599456i
\(101\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(102\) 0 0
\(103\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(104\) −0.847759 + 0.847759i −0.847759 + 0.847759i
\(105\) 0 0
\(106\) 0 0
\(107\) 1.02656 + 0.425215i 1.02656 + 0.425215i 0.831470 0.555570i \(-0.187500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(108\) 0 0
\(109\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(110\) −0.275899 0.275899i −0.275899 0.275899i
\(111\) 0 0
\(112\) −1.02656 + 0.425215i −1.02656 + 0.425215i
\(113\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −1.17588 + 1.17588i −1.17588 + 1.17588i
\(118\) 0 0
\(119\) 1.92388 + 0.382683i 1.92388 + 0.382683i
\(120\) 0 0
\(121\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(122\) 0 0
\(123\) 0 0
\(124\) −1.10765 0.458804i −1.10765 0.458804i
\(125\) −0.382683 0.923880i −0.382683 0.923880i
\(126\) 0.707107 0.292893i 0.707107 0.292893i
\(127\) 1.17588 + 1.17588i 1.17588 + 1.17588i 0.980785 + 0.195090i \(0.0625000\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(128\) 0.720960 + 0.720960i 0.720960 + 0.720960i
\(129\) 0 0
\(130\) −0.248303 0.599456i −0.248303 0.599456i
\(131\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −0.140652 0.707107i −0.140652 0.707107i
\(137\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(138\) 0 0
\(139\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(140\) 1.66294i 1.66294i
\(141\) 0 0
\(142\) −0.114281 0.275899i −0.114281 0.275899i
\(143\) −1.53636 + 0.636379i −1.53636 + 0.636379i
\(144\) 0.400544 + 0.400544i 0.400544 + 0.400544i
\(145\) 0 0
\(146\) 0.400544 0.165911i 0.400544 0.165911i
\(147\) 0 0
\(148\) 0 0
\(149\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(150\) 0 0
\(151\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(152\) 0 0
\(153\) −0.195090 0.980785i −0.195090 0.980785i
\(154\) 0.765367 0.765367
\(155\) 1.00000 1.00000i 1.00000 1.00000i
\(156\) 0 0
\(157\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −0.870275 + 0.360480i −0.870275 + 0.360480i
\(161\) 0 0
\(162\) −0.275899 0.275899i −0.275899 0.275899i
\(163\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0.433546i 0.433546i
\(167\) 0.750661 1.81225i 0.750661 1.81225i 0.195090 0.980785i \(-0.437500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(168\) 0 0
\(169\) −1.76537 −1.76537
\(170\) 0.382683 + 0.0761205i 0.382683 + 0.0761205i
\(171\) 0 0
\(172\) −0.233896 + 0.233896i −0.233896 + 0.233896i
\(173\) −0.750661 + 1.81225i −0.750661 + 1.81225i −0.195090 + 0.980785i \(0.562500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(174\) 0 0
\(175\) 1.81225 + 0.750661i 1.81225 + 0.750661i
\(176\) 0.216773 + 0.523336i 0.216773 + 0.523336i
\(177\) 0 0
\(178\) −0.211164 0.211164i −0.211164 0.211164i
\(179\) 0.541196 + 0.541196i 0.541196 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(180\) −0.783227 + 0.324423i −0.783227 + 0.324423i
\(181\) 0.541196 + 1.30656i 0.541196 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(182\) 1.17588 + 0.487064i 1.17588 + 0.487064i
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0.195090 0.980785i 0.195090 0.980785i
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(192\) 0 0
\(193\) 0.149316 + 0.360480i 0.149316 + 0.360480i 0.980785 0.195090i \(-0.0625000\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 1.70711 + 1.70711i 1.70711 + 1.70711i
\(197\) −1.02656 + 0.425215i −1.02656 + 0.425215i −0.831470 0.555570i \(-0.812500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(198\) −0.149316 0.360480i −0.149316 0.360480i
\(199\) −1.70711 0.707107i −1.70711 0.707107i −0.707107 0.707107i \(-0.750000\pi\)
−1.00000 \(\pi\)
\(200\) 0.720960i 0.720960i
\(201\) 0 0
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.941979i 0.941979i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) −0.400544 + 0.165911i −0.400544 + 0.165911i
\(215\) −0.149316 0.360480i −0.149316 0.360480i
\(216\) 0 0
\(217\) 2.77408i 2.77408i
\(218\) 0 0
\(219\) 0 0
\(220\) −0.847759 −0.847759
\(221\) 0.923880 1.38268i 0.923880 1.38268i
\(222\) 0 0
\(223\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(224\) 0.707107 1.70711i 0.707107 1.70711i
\(225\) 1.00000i 1.00000i
\(226\) 0 0
\(227\) −0.425215 1.02656i −0.425215 1.02656i −0.980785 0.195090i \(-0.937500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(228\) 0 0
\(229\) −1.00000 1.00000i −1.00000 1.00000i 1.00000i \(-0.5\pi\)
−1.00000 \(\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 1.53636 + 0.636379i 1.53636 + 0.636379i 0.980785 0.195090i \(-0.0625000\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(234\) 0.648847i 0.648847i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) −0.636379 + 0.425215i −0.636379 + 0.425215i
\(239\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(240\) 0 0
\(241\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(242\) 0.390181i 0.390181i
\(243\) 0 0
\(244\) 0 0
\(245\) −2.63099 + 1.08979i −2.63099 + 1.08979i
\(246\) 0 0
\(247\) 0 0
\(248\) 0.941979 0.390181i 0.941979 0.390181i
\(249\) 0 0
\(250\) 0.360480 + 0.149316i 0.360480 + 0.149316i
\(251\) 1.41421i 1.41421i 0.707107 + 0.707107i \(0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(252\) 0.636379 1.53636i 0.636379 1.53636i
\(253\) 0 0
\(254\) −0.648847 −0.648847
\(255\) 0 0
\(256\) −0.198912 −0.198912
\(257\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.30246 0.539496i −1.30246 0.539496i
\(261\) 0 0
\(262\) 0 0
\(263\) 1.38704 + 1.38704i 1.38704 + 1.38704i 0.831470 + 0.555570i \(0.187500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.707107 + 1.70711i −0.707107 + 1.70711i 1.00000i \(0.5\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(270\) 0 0
\(271\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(272\) −0.470990 0.314705i −0.470990 0.314705i
\(273\) 0 0
\(274\) 0 0
\(275\) 0.382683 0.923880i 0.382683 0.923880i
\(276\) 0 0
\(277\) −1.53636 0.636379i −1.53636 0.636379i −0.555570 0.831470i \(-0.687500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(278\) 0 0
\(279\) 1.30656 0.541196i 1.30656 0.541196i
\(280\) 1.00000 + 1.00000i 1.00000 + 1.00000i
\(281\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(282\) 0 0
\(283\) −0.636379 1.53636i −0.636379 1.53636i −0.831470 0.555570i \(-0.812500\pi\)
0.195090 0.980785i \(-0.437500\pi\)
\(284\) −0.599456 0.248303i −0.599456 0.248303i
\(285\) 0 0
\(286\) 0.248303 0.599456i 0.248303 0.599456i
\(287\) 0 0
\(288\) −0.941979 −0.941979
\(289\) 0.382683 + 0.923880i 0.382683 + 0.923880i
\(290\) 0 0
\(291\) 0 0
\(292\) 0.360480 0.870275i 0.360480 0.870275i
\(293\) 1.66294i 1.66294i −0.555570 0.831470i \(-0.687500\pi\)
0.555570 0.831470i \(-0.312500\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0 0
\(301\) 0.707107 + 0.292893i 0.707107 + 0.292893i
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) 0.324423 + 0.216773i 0.324423 + 0.216773i
\(307\) 1.96157 1.96157 0.980785 0.195090i \(-0.0625000\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(308\) 1.17588 1.17588i 1.17588 1.17588i
\(309\) 0 0
\(310\) 0.551799i 0.551799i
\(311\) 0.707107 + 0.292893i 0.707107 + 0.292893i 0.707107 0.707107i \(-0.250000\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(314\) 0 0
\(315\) 1.38704 + 1.38704i 1.38704 + 1.38704i
\(316\) 0 0
\(317\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.0761205 + 0.183771i −0.0761205 + 0.183771i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.847759 −0.847759
\(325\) 1.17588 1.17588i 1.17588 1.17588i
\(326\) 0 0
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.30656 + 1.30656i 1.30656 + 1.30656i 0.923880 + 0.382683i \(0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(332\) 0.666080 + 0.666080i 0.666080 + 0.666080i
\(333\) 0 0
\(334\) 0.292893 + 0.707107i 0.292893 + 0.707107i
\(335\) 0 0
\(336\) 0 0
\(337\) −0.425215 + 1.02656i −0.425215 + 1.02656i 0.555570 + 0.831470i \(0.312500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(338\) 0.487064 0.487064i 0.487064 0.487064i
\(339\) 0 0
\(340\) 0.704886 0.470990i 0.704886 0.470990i
\(341\) 1.41421 1.41421
\(342\) 0 0
\(343\) 1.38704 3.34861i 1.38704 3.34861i
\(344\) 0.281305i 0.281305i
\(345\) 0 0
\(346\) −0.292893 0.707107i −0.292893 0.707107i
\(347\) 0.360480 0.149316i 0.360480 0.149316i −0.195090 0.980785i \(-0.562500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(348\) 0 0
\(349\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(350\) −0.707107 + 0.292893i −0.707107 + 0.292893i
\(351\) 0 0
\(352\) −0.870275 0.360480i −0.870275 0.360480i
\(353\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(354\) 0 0
\(355\) 0.541196 0.541196i 0.541196 0.541196i
\(356\) −0.648847 −0.648847
\(357\) 0 0
\(358\) −0.298631 −0.298631
\(359\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(360\) 0.275899 0.666080i 0.275899 0.666080i
\(361\) 1.00000i 1.00000i
\(362\) −0.509796 0.211164i −0.509796 0.211164i
\(363\) 0 0
\(364\) 2.55487 1.05826i 2.55487 1.05826i
\(365\) 0.785695 + 0.785695i 0.785695 + 0.785695i
\(366\) 0 0
\(367\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) 0 0
\(373\) 0.390181 0.390181 0.195090 0.980785i \(-0.437500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(374\) 0.216773 + 0.324423i 0.216773 + 0.324423i
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.707107 + 0.292893i 0.707107 + 0.292893i 0.707107 0.707107i \(-0.250000\pi\)
1.00000i \(0.5\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0.211164 + 0.211164i 0.211164 + 0.211164i
\(383\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(384\) 0 0
\(385\) 0.750661 + 1.81225i 0.750661 + 1.81225i
\(386\) −0.140652 0.0582601i −0.140652 0.0582601i
\(387\) 0.390181i 0.390181i
\(388\) 0 0
\(389\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −2.05312 −2.05312
\(393\) 0 0
\(394\) 0.165911 0.400544i 0.165911 0.400544i
\(395\) 0 0
\(396\) −0.783227 0.324423i −0.783227 0.324423i
\(397\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(398\) 0.666080 0.275899i 0.666080 0.275899i
\(399\) 0 0
\(400\) −0.400544 0.400544i −0.400544 0.400544i
\(401\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(402\) 0 0
\(403\) 2.17273 + 0.899976i 2.17273 + 0.899976i
\(404\) 0 0
\(405\) 0.382683 0.923880i 0.382683 0.923880i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) −1.02656 + 0.425215i −1.02656 + 0.425215i
\(416\) −1.10765 1.10765i −1.10765 1.10765i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(420\) 0 0
\(421\) 0.765367i 0.765367i 0.923880 + 0.382683i \(0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0.195090 + 0.980785i 0.195090 + 0.980785i
\(426\) 0 0
\(427\) 0 0
\(428\) −0.360480 + 0.870275i −0.360480 + 0.870275i
\(429\) 0 0
\(430\) 0.140652 + 0.0582601i 0.140652 + 0.0582601i
\(431\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(432\) 0 0
\(433\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(434\) −0.765367 0.765367i −0.765367 0.765367i
\(435\) 0 0
\(436\) 0 0
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(440\) 0.509796 0.509796i 0.509796 0.509796i
\(441\) −2.84776 −2.84776
\(442\) 0.126584 + 0.636379i 0.126584 + 0.636379i
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) 0.292893 0.707107i 0.292893 0.707107i
\(446\) 0 0
\(447\) 0 0
\(448\) −0.149316 0.360480i −0.149316 0.360480i
\(449\) −1.30656 + 0.541196i −1.30656 + 0.541196i −0.923880 0.382683i \(-0.875000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(450\) 0.275899 + 0.275899i 0.275899 + 0.275899i
\(451\) 0 0
\(452\) 0 0
\(453\) 0 0
\(454\) 0.400544 + 0.165911i 0.400544 + 0.165911i
\(455\) 3.26197i 3.26197i
\(456\) 0 0
\(457\) −1.38704 + 1.38704i −1.38704 + 1.38704i −0.555570 + 0.831470i \(0.687500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(458\) 0.551799 0.551799
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(462\) 0 0
\(463\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) −0.599456 + 0.248303i −0.599456 + 0.248303i
\(467\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(468\) −0.996859 0.996859i −0.996859 0.996859i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0.149316 0.360480i 0.149316 0.360480i
\(474\) 0 0
\(475\) 0 0
\(476\) −0.324423 + 1.63099i −0.324423 + 1.63099i
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) 0 0
\(483\) 0 0
\(484\) −0.599456 0.599456i −0.599456 0.599456i
\(485\) 0 0
\(486\) 0 0
\(487\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0.425215 1.02656i 0.425215 1.02656i
\(491\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0.707107 0.707107i 0.707107 0.707107i
\(496\) 0.306563 0.740108i 0.306563 0.740108i
\(497\) 1.50132i 1.50132i
\(498\) 0 0
\(499\) 0.707107 + 1.70711i 0.707107 + 1.70711i 0.707107 + 0.707107i \(0.250000\pi\)
1.00000i \(0.5\pi\)
\(500\) 0.783227 0.324423i 0.783227 0.324423i
\(501\) 0 0
\(502\) −0.390181 0.390181i −0.390181 0.390181i
\(503\) 0.360480 0.149316i 0.360480 0.149316i −0.195090 0.980785i \(-0.562500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(504\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) −0.996859 + 0.996859i −0.996859 + 0.996859i
\(509\) 2.00000 2.00000 1.00000 \(0\)
1.00000 \(0\)
\(510\) 0 0
\(511\) −2.17958 −2.17958
\(512\) −0.666080 + 0.666080i −0.666080 + 0.666080i
\(513\) 0 0
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 1.10765 0.458804i 1.10765 0.458804i
\(521\) −0.541196 1.30656i −0.541196 1.30656i −0.923880 0.382683i \(-0.875000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(522\) 0 0
\(523\) 1.11114i 1.11114i −0.831470 0.555570i \(-0.812500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) −0.765367 −0.765367
\(527\) −1.17588 + 0.785695i −1.17588 + 0.785695i
\(528\) 0 0
\(529\) −0.707107 + 0.707107i −0.707107 + 0.707107i
\(530\) 0 0
\(531\) 0 0
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) −0.785695 0.785695i −0.785695 0.785695i
\(536\) 0 0
\(537\) 0 0
\(538\) −0.275899 0.666080i −0.275899 0.666080i
\(539\) −2.63099 1.08979i −2.63099 1.08979i
\(540\) 0 0
\(541\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 0.923880 0.183771i 0.923880 0.183771i
\(545\) 0 0
\(546\) 0 0
\(547\) −0.636379 + 1.53636i −0.636379 + 1.53636i 0.195090 + 0.980785i \(0.437500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(548\) 0 0
\(549\) 0 0
\(550\) 0.149316 + 0.360480i 0.149316 + 0.360480i
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) 0.599456 0.248303i 0.599456 0.248303i
\(555\) 0 0
\(556\) 0 0
\(557\) 1.96157i 1.96157i −0.195090 0.980785i \(-0.562500\pi\)
0.195090 0.980785i \(-0.437500\pi\)
\(558\) −0.211164 + 0.509796i −0.211164 + 0.509796i
\(559\) 0.458804 0.458804i 0.458804 0.458804i
\(560\) 1.11114 1.11114
\(561\) 0 0
\(562\) 0 0
\(563\) −1.17588 + 1.17588i −1.17588 + 1.17588i −0.195090 + 0.980785i \(0.562500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0.599456 + 0.248303i 0.599456 + 0.248303i
\(567\) 0.750661 + 1.81225i 0.750661 + 1.81225i
\(568\) 0.509796 0.211164i 0.509796 0.211164i
\(569\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(570\) 0 0
\(571\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(572\) −0.539496 1.30246i −0.539496 1.30246i
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.140652 + 0.140652i −0.140652 + 0.140652i
\(577\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(578\) −0.360480 0.149316i −0.360480 0.149316i
\(579\) 0 0
\(580\) 0 0
\(581\) 0.834089 2.01367i 0.834089 2.01367i
\(582\) 0 0
\(583\) 0 0
\(584\) 0.306563 + 0.740108i 0.306563 + 0.740108i
\(585\) 1.53636 0.636379i 1.53636 0.636379i
\(586\) 0.458804 + 0.458804i 0.458804 + 0.458804i
\(587\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0 0
\(593\) −0.275899 + 0.275899i −0.275899 + 0.275899i −0.831470 0.555570i \(-0.812500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(594\) 0 0
\(595\) −1.63099 1.08979i −1.63099 1.08979i
\(596\) 0 0
\(597\) 0 0
\(598\) 0 0
\(599\) 1.84776i 1.84776i −0.382683 0.923880i \(-0.625000\pi\)
0.382683 0.923880i \(-0.375000\pi\)
\(600\) 0 0
\(601\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(602\) −0.275899 + 0.114281i −0.275899 + 0.114281i
\(603\) 0 0
\(604\) 0 0
\(605\) 0.923880 0.382683i 0.923880 0.382683i
\(606\) 0 0
\(607\) −1.02656 0.425215i −1.02656 0.425215i −0.195090 0.980785i \(-0.562500\pi\)
−0.831470 + 0.555570i \(0.812500\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0.831470 0.165390i 0.831470 0.165390i
\(613\) −1.11114 −1.11114 −0.555570 0.831470i \(-0.687500\pi\)
−0.555570 + 0.831470i \(0.687500\pi\)
\(614\) −0.541196 + 0.541196i −0.541196 + 0.541196i
\(615\) 0 0
\(616\) 1.41421i 1.41421i
\(617\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(618\) 0 0
\(619\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(620\) 0.847759 + 0.847759i 0.847759 + 0.847759i
\(621\) 0 0
\(622\) −0.275899 + 0.114281i −0.275899 + 0.114281i
\(623\) 0.574531 + 1.38704i 0.574531 + 1.38704i
\(624\) 0 0
\(625\) 1.00000i 1.00000i
\(626\) 0 0
\(627\) 0 0
\(628\) 0 0
\(629\) 0 0
\(630\) −0.765367 −0.765367
\(631\) 1.00000 1.00000i 1.00000 1.00000i 1.00000i \(-0.5\pi\)
1.00000 \(0\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0 0
\(635\) −0.636379 1.53636i −0.636379 1.53636i
\(636\) 0 0
\(637\) −3.34861 3.34861i −3.34861 3.34861i
\(638\) 0 0
\(639\) 0.707107 0.292893i 0.707107 0.292893i
\(640\) −0.390181 0.941979i −0.390181 0.941979i
\(641\) 1.30656 + 0.541196i 1.30656 + 0.541196i 0.923880 0.382683i \(-0.125000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(642\) 0 0
\(643\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(648\) 0.509796 0.509796i 0.509796 0.509796i
\(649\) 0 0
\(650\) 0.648847i 0.648847i
\(651\) 0 0
\(652\) 0 0
\(653\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.425215 + 1.02656i 0.425215 + 1.02656i
\(658\) 0 0
\(659\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(660\) 0 0
\(661\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(662\) −0.720960 −0.720960
\(663\) 0 0
\(664\) −0.801088 −0.801088
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 1.53636 + 0.636379i 1.53636 + 0.636379i
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.360480 0.149316i 0.360480 0.149316i −0.195090 0.980785i \(-0.562500\pi\)
0.555570 + 0.831470i \(0.312500\pi\)
\(674\) −0.165911 0.400544i −0.165911 0.400544i
\(675\) 0 0
\(676\) 1.49661i 1.49661i
\(677\) −0.149316 + 0.360480i −0.149316 + 0.360480i −0.980785 0.195090i \(-0.937500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −0.140652 + 0.707107i −0.140652 + 0.707107i
\(681\) 0 0
\(682\) −0.390181 + 0.390181i −0.390181 + 0.390181i
\(683\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0.541196 + 1.30656i 0.541196 + 1.30656i
\(687\) 0 0
\(688\) −0.156284 0.156284i −0.156284 0.156284i
\(689\) 0 0
\(690\) 0 0
\(691\) −0.292893 0.707107i −0.292893 0.707107i 0.707107 0.707107i \(-0.250000\pi\)
−1.00000 \(\pi\)
\(692\) −1.53636 0.636379i −1.53636 0.636379i
\(693\) 1.96157i 1.96157i
\(694\) −0.0582601 + 0.140652i −0.0582601 + 0.140652i
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.636379 + 1.53636i −0.636379 + 1.53636i
\(701\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) −0.183771 + 0.0761205i −0.183771 + 0.0761205i
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.30656 0.541196i −1.30656 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(710\) 0.298631i 0.298631i
\(711\) 0 0
\(712\) 0.390181 0.390181i 0.390181 0.390181i
\(713\) 0 0
\(714\) 0 0
\(715\) 1.66294 1.66294
\(716\) −0.458804 + 0.458804i −0.458804 + 0.458804i
\(717\) 0 0
\(718\) 0 0
\(719\) −1.30656 0.541196i −1.30656 0.541196i −0.382683 0.923880i \(-0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(720\) −0.216773 0.523336i −0.216773 0.523336i
\(721\) 0 0
\(722\) 0.275899 + 0.275899i 0.275899 + 0.275899i
\(723\) 0 0
\(724\) −1.10765 + 0.458804i −1.10765 + 0.458804i
\(725\) 0 0
\(726\) 0 0
\(727\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(728\) −0.899976 + 2.17273i −0.899976 + 2.17273i
\(729\) 0.707107 0.707107i 0.707107 0.707107i
\(730\) −0.433546 −0.433546
\(731\) 0.0761205 + 0.382683i 0.0761205 + 0.382683i
\(732\) 0 0
\(733\) −0.785695 + 0.785695i −0.785695 + 0.785695i −0.980785 0.195090i \(-0.937500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 0 0
\(743\) −1.53636 0.636379i −1.53636 0.636379i −0.555570 0.831470i \(-0.687500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −0.107651 + 0.107651i −0.107651 + 0.107651i
\(747\) −1.11114 −1.11114
\(748\) 0.831470 + 0.165390i 0.831470 + 0.165390i
\(749\) 2.17958 2.17958
\(750\) 0 0
\(751\) −0.707107 + 1.70711i −0.707107 + 1.70711i 1.00000i \(0.5\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(758\) −0.275899 + 0.114281i −0.275899 + 0.114281i
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0.648847 0.648847
\(765\) −0.195090 + 0.980785i −0.195090 + 0.980785i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(770\) −0.707107 0.292893i −0.707107 0.292893i
\(771\) 0 0
\(772\) −0.305600 + 0.126584i −0.305600 + 0.126584i
\(773\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(774\) 0.107651 + 0.107651i 0.107651 + 0.107651i
\(775\) −1.30656 + 0.541196i −1.30656 + 0.541196i
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0.765367 0.765367
\(782\) 0 0
\(783\) 0 0
\(784\) −1.14065 + 1.14065i −1.14065 + 1.14065i
\(785\) 0 0
\(786\) 0 0
\(787\) 0.360480 + 0.149316i 0.360480 + 0.149316i 0.555570 0.831470i \(-0.312500\pi\)
−0.195090 + 0.980785i \(0.562500\pi\)
\(788\) −0.360480 0.870275i −0.360480 0.870275i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0.666080 0.275899i 0.666080 0.275899i
\(793\) 0 0
\(794\) 0 0
\(795\) 0 0
\(796\) 0.599456 1.44722i 0.599456 1.44722i
\(797\) 0 0 0.707107 0.707107i \(-0.250000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.941979 0.941979
\(801\) 0.541196 0.541196i 0.541196 0.541196i
\(802\) 0 0
\(803\) 1.11114i 1.11114i
\(804\) 0 0
\(805\) 0 0
\(806\) −0.847759 + 0.351153i −0.847759 + 0.351153i
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(810\) 0.149316 + 0.360480i 0.149316 + 0.360480i
\(811\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0 0
\(819\) −1.24830 + 3.01367i −1.24830 + 3.01367i
\(820\) 0 0
\(821\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(822\) 0 0
\(823\) 0 0 0.923880 0.382683i \(-0.125000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0.750661 + 1.81225i 0.750661 + 1.81225i 0.555570 + 0.831470i \(0.312500\pi\)
0.195090 + 0.980785i \(0.437500\pi\)
\(828\) 0 0
\(829\) 0.765367i 0.765367i −0.923880 0.382683i \(-0.875000\pi\)
0.923880 0.382683i \(-0.125000\pi\)
\(830\) 0.165911 0.400544i 0.165911 0.400544i
\(831\) 0 0
\(832\) −0.330779 −0.330779
\(833\) 2.79304 0.555570i 2.79304 0.555570i
\(834\) 0 0
\(835\) −1.38704 + 1.38704i −1.38704 + 1.38704i
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) −0.707107 1.70711i −0.707107 1.70711i −0.707107 0.707107i \(-0.750000\pi\)
1.00000i \(-0.5\pi\)
\(840\) 0 0
\(841\) 0.707107 + 0.707107i 0.707107 + 0.707107i
\(842\) −0.211164 0.211164i −0.211164 0.211164i
\(843\) 0 0
\(844\) 0 0
\(845\) 1.63099 + 0.675577i 1.63099 + 0.675577i
\(846\) 0 0
\(847\) −0.750661 + 1.81225i −0.750661 + 1.81225i
\(848\) 0 0
\(849\) 0 0
\(850\) −0.324423 0.216773i −0.324423 0.216773i
\(851\) 0 0
\(852\) 0 0
\(853\) −0.149316 + 0.360480i −0.149316 + 0.360480i −0.980785 0.195090i \(-0.937500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) −0.306563 0.740108i −0.306563 0.740108i
\(857\) −1.81225 + 0.750661i −1.81225 + 0.750661i −0.831470 + 0.555570i \(0.812500\pi\)
−0.980785 + 0.195090i \(0.937500\pi\)
\(858\) 0 0
\(859\) −0.541196 0.541196i −0.541196 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(860\) 0.305600 0.126584i 0.305600 0.126584i
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(864\) 0 0
\(865\) 1.38704 1.38704i 1.38704 1.38704i
\(866\) 0 0
\(867\) 0 0
\(868\) −2.35175 −2.35175
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 0 0
\(873\) 0 0
\(874\) 0 0
\(875\) −1.38704 1.38704i −1.38704 1.38704i
\(876\) 0 0
\(877\) 1.02656 0.425215i 1.02656 0.425215i 0.195090 0.980785i \(-0.437500\pi\)
0.831470 + 0.555570i \(0.187500\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0.566454i 0.566454i
\(881\) 0.292893 0.707107i 0.292893 0.707107i −0.707107 0.707107i \(-0.750000\pi\)
1.00000 \(0\)
\(882\) 0.785695 0.785695i 0.785695 0.785695i
\(883\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(884\) 1.17218 + 0.783227i 1.17218 + 0.783227i
\(885\) 0 0
\(886\) 0 0
\(887\) 0.425215 1.02656i 0.425215 1.02656i −0.555570 0.831470i \(-0.687500\pi\)
0.980785 0.195090i \(-0.0625000\pi\)
\(888\) 0 0
\(889\) 3.01367 + 1.24830i 3.01367 + 1.24830i
\(890\) 0.114281 + 0.275899i 0.114281 + 0.275899i
\(891\) 0.923880 0.382683i 0.923880 0.382683i
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) −0.292893 0.707107i −0.292893 0.707107i
\(896\) 1.84776 + 0.765367i 1.84776 + 0.765367i
\(897\) 0 0
\(898\) 0.211164 0.509796i 0.211164 0.509796i
\(899\) 0 0
\(900\) 0.847759 0.847759
\(901\) 0 0
\(902\) 0 0
\(903\) 0 0
\(904\) 0 0
\(905\) 1.41421i 1.41421i
\(906\) 0 0
\(907\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(908\) 0.870275 0.360480i 0.870275 0.360480i
\(909\) 0 0
\(910\) −0.899976 0.899976i −0.899976 0.899976i
\(911\) 1.30656 0.541196i 1.30656 0.541196i 0.382683 0.923880i \(-0.375000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(912\) 0 0
\(913\) −1.02656 0.425215i −1.02656 0.425215i
\(914\) 0.765367i 0.765367i
\(915\) 0 0
\(916\) 0.847759 0.847759i 0.847759 0.847759i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0 0
\(923\) 1.17588 + 0.487064i 1.17588 + 0.487064i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 0 0
\(928\) 0 0
\(929\) 0.707107 + 1.70711i 0.707107 + 1.70711i 0.707107 + 0.707107i \(0.250000\pi\)
1.00000i \(0.5\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.539496 + 1.30246i −0.539496 + 1.30246i
\(933\) 0 0
\(934\) 0 0
\(935\) −0.555570 + 0.831470i −0.555570 + 0.831470i
\(936\) 1.19891 1.19891
\(937\) 1.38704 1.38704i 1.38704 1.38704i 0.555570 0.831470i \(-0.312500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0 0 −0.382683 0.923880i \(-0.625000\pi\)
0.382683 + 0.923880i \(0.375000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0.0582601 + 0.140652i 0.0582601 + 0.140652i
\(947\) 0 0 −0.923880 0.382683i \(-0.875000\pi\)
0.923880 + 0.382683i \(0.125000\pi\)
\(948\) 0 0
\(949\) −0.707107 + 1.70711i −0.707107 + 1.70711i
\(950\) 0 0
\(951\) 0 0
\(952\) −0.785695 1.17588i −0.785695 1.17588i
\(953\) 1.96157 1.96157 0.980785 0.195090i \(-0.0625000\pi\)
0.980785 + 0.195090i \(0.0625000\pi\)
\(954\) 0 0
\(955\) −0.292893 + 0.707107i −0.292893 + 0.707107i
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.707107 0.707107i −0.707107 0.707107i
\(962\) 0 0
\(963\) −0.425215 1.02656i −0.425215 1.02656i
\(964\) 0 0
\(965\) 0.390181i 0.390181i
\(966\) 0 0
\(967\) 1.38704 1.38704i 1.38704 1.38704i 0.555570 0.831470i \(-0.312500\pi\)
0.831470 0.555570i \(-0.187500\pi\)
\(968\) 0.720960 0.720960
\(969\) 0 0
\(970\) 0 0
\(971\) −1.30656 + 1.30656i −1.30656 + 1.30656i −0.382683 + 0.923880i \(0.625000\pi\)
−0.923880 + 0.382683i \(0.875000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(978\) 0 0
\(979\) 0.707107 0.292893i 0.707107 0.292893i
\(980\) −0.923880 2.23044i −0.923880 2.23044i
\(981\) 0 0
\(982\) 0 0
\(983\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(984\) 0 0
\(985\) 1.11114 1.11114
\(986\) 0 0
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0.390181i 0.390181i
\(991\) 0 0 0.382683 0.923880i \(-0.375000\pi\)
−0.382683 + 0.923880i \(0.625000\pi\)
\(992\) 0.509796 + 1.23076i 0.509796 + 1.23076i
\(993\) 0 0
\(994\) −0.414214 0.414214i −0.414214 0.414214i
\(995\) 1.30656 + 1.30656i 1.30656 + 1.30656i
\(996\) 0 0
\(997\) −0.636379 1.53636i −0.636379 1.53636i −0.831470 0.555570i \(-0.812500\pi\)
0.195090 0.980785i \(-0.437500\pi\)
\(998\) −0.666080 0.275899i −0.666080 0.275899i
\(999\) 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 935.1.y.a.824.2 yes 16
5.4 even 2 inner 935.1.y.a.824.3 yes 16
11.10 odd 2 inner 935.1.y.a.824.3 yes 16
17.15 even 8 inner 935.1.y.a.219.2 16
55.54 odd 2 CM 935.1.y.a.824.2 yes 16
85.49 even 8 inner 935.1.y.a.219.3 yes 16
187.32 odd 8 inner 935.1.y.a.219.3 yes 16
935.219 odd 8 inner 935.1.y.a.219.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
935.1.y.a.219.2 16 17.15 even 8 inner
935.1.y.a.219.2 16 935.219 odd 8 inner
935.1.y.a.219.3 yes 16 85.49 even 8 inner
935.1.y.a.219.3 yes 16 187.32 odd 8 inner
935.1.y.a.824.2 yes 16 1.1 even 1 trivial
935.1.y.a.824.2 yes 16 55.54 odd 2 CM
935.1.y.a.824.3 yes 16 5.4 even 2 inner
935.1.y.a.824.3 yes 16 11.10 odd 2 inner