Properties

Label 453.1.t.a.299.1
Level $453$
Weight $1$
Character 453.299
Analytic conductor $0.226$
Analytic rank $0$
Dimension $20$
Projective image $D_{25}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [453,1,Mod(20,453)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(453, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([25, 34]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("453.20");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 453 = 3 \cdot 151 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 453.t (of order \(50\), degree \(20\), 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: \(0.226076450723\)
Analytic rank: \(0\)
Dimension: \(20\)
Coefficient field: \(\Q(\zeta_{50})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{20} - x^{15} + x^{10} - x^{5} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{25}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{25} - \cdots)\)

Embedding invariants

Embedding label 299.1
Root \(-0.876307 + 0.481754i\) of defining polynomial
Character \(\chi\) \(=\) 453.299
Dual form 453.1.t.a.50.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.728969 + 0.684547i) q^{3} +(0.309017 - 0.951057i) q^{4} +(-0.456288 + 0.718995i) q^{7} +(0.0627905 + 0.998027i) q^{9} +O(q^{10})\) \(q+(0.728969 + 0.684547i) q^{3} +(0.309017 - 0.951057i) q^{4} +(-0.456288 + 0.718995i) q^{7} +(0.0627905 + 0.998027i) q^{9} +(0.876307 - 0.481754i) q^{12} +(-0.0534698 + 0.113629i) q^{13} +(-0.809017 - 0.587785i) q^{16} +(0.598617 - 1.84235i) q^{19} +(-0.824805 + 0.211774i) q^{21} +(-0.992115 + 0.125333i) q^{25} +(-0.637424 + 0.770513i) q^{27} +(0.542804 + 0.656137i) q^{28} +(-0.996398 - 0.394502i) q^{31} +(0.968583 + 0.248690i) q^{36} +(-0.620759 + 1.31918i) q^{37} +(-0.116762 + 0.0462295i) q^{39} +(-0.200808 - 0.316423i) q^{43} +(-0.187381 - 0.982287i) q^{48} +(0.117025 + 0.248690i) q^{49} +(0.0915446 + 0.0859661i) q^{52} +(1.69755 - 0.933237i) q^{57} +(-1.17950 + 1.10762i) q^{61} +(-0.746226 - 0.410241i) q^{63} +(-0.809017 + 0.587785i) q^{64} +(0.110048 - 1.74915i) q^{67} +(0.331159 - 0.521823i) q^{73} +(-0.809017 - 0.587785i) q^{75} +(-1.56720 - 1.13864i) q^{76} +(1.84489 - 0.730444i) q^{79} +(-0.992115 + 0.125333i) q^{81} +(-0.0534698 + 0.849878i) q^{84} +(-0.0573011 - 0.0902921i) q^{91} +(-0.456288 - 0.969661i) q^{93} +(-0.101597 + 1.61484i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q - 5 q^{4}+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 5 q^{4} - 5 q^{16} - 5 q^{31} - 5 q^{37} - 5 q^{39} - 5 q^{63} - 5 q^{64} - 5 q^{75} - 5 q^{91}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/453\mathbb{Z}\right)^\times\).

\(n\) \(152\) \(157\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{25}\right)\)

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 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(3\) 0.728969 + 0.684547i 0.728969 + 0.684547i
\(4\) 0.309017 0.951057i 0.309017 0.951057i
\(5\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(6\) 0 0
\(7\) −0.456288 + 0.718995i −0.456288 + 0.718995i −0.992115 0.125333i \(-0.960000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(8\) 0 0
\(9\) 0.0627905 + 0.998027i 0.0627905 + 0.998027i
\(10\) 0 0
\(11\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(12\) 0.876307 0.481754i 0.876307 0.481754i
\(13\) −0.0534698 + 0.113629i −0.0534698 + 0.113629i −0.929776 0.368125i \(-0.880000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.809017 0.587785i −0.809017 0.587785i
\(17\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(18\) 0 0
\(19\) 0.598617 1.84235i 0.598617 1.84235i 0.0627905 0.998027i \(-0.480000\pi\)
0.535827 0.844328i \(-0.320000\pi\)
\(20\) 0 0
\(21\) −0.824805 + 0.211774i −0.824805 + 0.211774i
\(22\) 0 0
\(23\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(24\) 0 0
\(25\) −0.992115 + 0.125333i −0.992115 + 0.125333i
\(26\) 0 0
\(27\) −0.637424 + 0.770513i −0.637424 + 0.770513i
\(28\) 0.542804 + 0.656137i 0.542804 + 0.656137i
\(29\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(30\) 0 0
\(31\) −0.996398 0.394502i −0.996398 0.394502i −0.187381 0.982287i \(-0.560000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.968583 + 0.248690i 0.968583 + 0.248690i
\(37\) −0.620759 + 1.31918i −0.620759 + 1.31918i 0.309017 + 0.951057i \(0.400000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(38\) 0 0
\(39\) −0.116762 + 0.0462295i −0.116762 + 0.0462295i
\(40\) 0 0
\(41\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(42\) 0 0
\(43\) −0.200808 0.316423i −0.200808 0.316423i 0.728969 0.684547i \(-0.240000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(48\) −0.187381 0.982287i −0.187381 0.982287i
\(49\) 0.117025 + 0.248690i 0.117025 + 0.248690i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.0915446 + 0.0859661i 0.0915446 + 0.0859661i
\(53\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.69755 0.933237i 1.69755 0.933237i
\(58\) 0 0
\(59\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(60\) 0 0
\(61\) −1.17950 + 1.10762i −1.17950 + 1.10762i −0.187381 + 0.982287i \(0.560000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(62\) 0 0
\(63\) −0.746226 0.410241i −0.746226 0.410241i
\(64\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(65\) 0 0
\(66\) 0 0
\(67\) 0.110048 1.74915i 0.110048 1.74915i −0.425779 0.904827i \(-0.640000\pi\)
0.535827 0.844328i \(-0.320000\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(72\) 0 0
\(73\) 0.331159 0.521823i 0.331159 0.521823i −0.637424 0.770513i \(-0.720000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(74\) 0 0
\(75\) −0.809017 0.587785i −0.809017 0.587785i
\(76\) −1.56720 1.13864i −1.56720 1.13864i
\(77\) 0 0
\(78\) 0 0
\(79\) 1.84489 0.730444i 1.84489 0.730444i 0.876307 0.481754i \(-0.160000\pi\)
0.968583 0.248690i \(-0.0800000\pi\)
\(80\) 0 0
\(81\) −0.992115 + 0.125333i −0.992115 + 0.125333i
\(82\) 0 0
\(83\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(84\) −0.0534698 + 0.849878i −0.0534698 + 0.849878i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(90\) 0 0
\(91\) −0.0573011 0.0902921i −0.0573011 0.0902921i
\(92\) 0 0
\(93\) −0.456288 0.969661i −0.456288 0.969661i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.101597 + 1.61484i −0.101597 + 1.61484i 0.535827 + 0.844328i \(0.320000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.187381 + 0.982287i −0.187381 + 0.982287i
\(101\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(102\) 0 0
\(103\) 0.371808 + 1.94908i 0.371808 + 1.94908i 0.309017 + 0.951057i \(0.400000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(108\) 0.535827 + 0.844328i 0.535827 + 0.844328i
\(109\) 1.18532 1.43281i 1.18532 1.43281i 0.309017 0.951057i \(-0.400000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(110\) 0 0
\(111\) −1.35556 + 0.536702i −1.35556 + 0.536702i
\(112\) 0.791759 0.313480i 0.791759 0.313480i
\(113\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.116762 0.0462295i −0.116762 0.0462295i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.0627905 0.998027i 0.0627905 0.998027i
\(122\) 0 0
\(123\) 0 0
\(124\) −0.683098 + 0.825723i −0.683098 + 0.825723i
\(125\) 0 0
\(126\) 0 0
\(127\) −0.263146 + 0.559214i −0.263146 + 0.559214i −0.992115 0.125333i \(-0.960000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(128\) 0 0
\(129\) 0.0702235 0.368125i 0.0702235 0.368125i
\(130\) 0 0
\(131\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(132\) 0 0
\(133\) 1.05150 + 1.27105i 1.05150 + 1.27105i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(138\) 0 0
\(139\) 0.541587 0.297740i 0.541587 0.297740i −0.187381 0.982287i \(-0.560000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.535827 0.844328i 0.535827 0.844328i
\(145\) 0 0
\(146\) 0 0
\(147\) −0.0849327 + 0.261396i −0.0849327 + 0.261396i
\(148\) 1.06279 + 0.998027i 1.06279 + 0.998027i
\(149\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(150\) 0 0
\(151\) −0.809017 0.587785i −0.809017 0.587785i
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0.00788530 + 0.125333i 0.00788530 + 0.125333i
\(157\) 1.84489 + 0.233064i 1.84489 + 0.233064i 0.968583 0.248690i \(-0.0800000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) 1.27760 0.702367i 1.27760 0.702367i 0.309017 0.951057i \(-0.400000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(168\) 0 0
\(169\) 0.627371 + 0.758362i 0.627371 + 0.758362i
\(170\) 0 0
\(171\) 1.87631 + 0.481754i 1.87631 + 0.481754i
\(172\) −0.362989 + 0.0931997i −0.362989 + 0.0931997i
\(173\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(174\) 0 0
\(175\) 0.362576 0.770513i 0.362576 0.770513i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(180\) 0 0
\(181\) −0.0800484 + 1.27233i −0.0800484 + 1.27233i 0.728969 + 0.684547i \(0.240000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(182\) 0 0
\(183\) −1.61803 −1.61803
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) −0.263146 0.809880i −0.263146 0.809880i
\(190\) 0 0
\(191\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(192\) −0.992115 0.125333i −0.992115 0.125333i
\(193\) 1.18532 1.43281i 1.18532 1.43281i 0.309017 0.951057i \(-0.400000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.272681 0.0344476i 0.272681 0.0344476i
\(197\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(198\) 0 0
\(199\) −0.200808 1.05267i −0.200808 1.05267i −0.929776 0.368125i \(-0.880000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(200\) 0 0
\(201\) 1.27760 1.19975i 1.27760 1.19975i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.110048 0.0604991i 0.110048 0.0604991i
\(209\) 0 0
\(210\) 0 0
\(211\) 0.781202 + 1.23098i 0.781202 + 1.23098i 0.968583 + 0.248690i \(0.0800000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0.738289 0.536399i 0.738289 0.536399i
\(218\) 0 0
\(219\) 0.598617 0.153699i 0.598617 0.153699i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) −1.62954 + 0.645180i −1.62954 + 0.645180i −0.992115 0.125333i \(-0.960000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(224\) 0 0
\(225\) −0.187381 0.982287i −0.187381 0.982287i
\(226\) 0 0
\(227\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(228\) −0.362989 1.90285i −0.362989 1.90285i
\(229\) −0.866986 + 1.36615i −0.866986 + 1.36615i 0.0627905 + 0.998027i \(0.480000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 1.84489 + 0.730444i 1.84489 + 0.730444i
\(238\) 0 0
\(239\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(240\) 0 0
\(241\) −1.35556 + 1.27295i −1.35556 + 1.27295i −0.425779 + 0.904827i \(0.640000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(242\) 0 0
\(243\) −0.809017 0.587785i −0.809017 0.587785i
\(244\) 0.688925 + 1.46404i 0.688925 + 1.46404i
\(245\) 0 0
\(246\) 0 0
\(247\) 0.177337 + 0.166531i 0.177337 + 0.166531i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(252\) −0.620759 + 0.582932i −0.620759 + 0.582932i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(257\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(258\) 0 0
\(259\) −0.665239 1.04825i −0.665239 1.04825i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −1.62954 0.645180i −1.62954 0.645180i
\(269\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(270\) 0 0
\(271\) −1.62954 0.645180i −1.62954 0.645180i −0.637424 0.770513i \(-0.720000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(272\) 0 0
\(273\) 0.0200385 0.105045i 0.0200385 0.105045i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.371808 0.0469702i 0.371808 0.0469702i 0.0627905 0.998027i \(-0.480000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(278\) 0 0
\(279\) 0.331159 1.01920i 0.331159 1.01920i
\(280\) 0 0
\(281\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(282\) 0 0
\(283\) −0.613161 + 1.88711i −0.613161 + 1.88711i −0.187381 + 0.982287i \(0.560000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(290\) 0 0
\(291\) −1.17950 + 1.10762i −1.17950 + 1.10762i
\(292\) −0.393950 0.476203i −0.393950 0.476203i
\(293\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(301\) 0.319132 0.319132
\(302\) 0 0
\(303\) 0 0
\(304\) −1.56720 + 1.13864i −1.56720 + 1.13864i
\(305\) 0 0
\(306\) 0 0
\(307\) −0.124591 1.98031i −0.124591 1.98031i −0.187381 0.982287i \(-0.560000\pi\)
0.0627905 0.998027i \(-0.480000\pi\)
\(308\) 0 0
\(309\) −1.06320 + 1.67534i −1.06320 + 1.67534i
\(310\) 0 0
\(311\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(312\) 0 0
\(313\) 0.450527 0.423073i 0.450527 0.423073i −0.425779 0.904827i \(-0.640000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.124591 1.98031i −0.124591 1.98031i
\(317\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.187381 + 0.982287i −0.187381 + 0.982287i
\(325\) 0.0388067 0.119435i 0.0388067 0.119435i
\(326\) 0 0
\(327\) 1.84489 0.233064i 1.84489 0.233064i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 0.303189 1.58937i 0.303189 1.58937i −0.425779 0.904827i \(-0.640000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(332\) 0 0
\(333\) −1.35556 0.536702i −1.35556 0.536702i
\(334\) 0 0
\(335\) 0 0
\(336\) 0.791759 + 0.313480i 0.791759 + 0.313480i
\(337\) 1.27760 + 0.702367i 1.27760 + 0.702367i 0.968583 0.248690i \(-0.0800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) −1.07705 0.136063i −1.07705 0.136063i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(348\) 0 0
\(349\) 1.26480 + 0.159781i 1.26480 + 0.159781i 0.728969 0.684547i \(-0.240000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(350\) 0 0
\(351\) −0.0534698 0.113629i −0.0534698 0.113629i
\(352\) 0 0
\(353\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(360\) 0 0
\(361\) −2.22691 1.61795i −2.22691 1.61795i
\(362\) 0 0
\(363\) 0.728969 0.684547i 0.728969 0.684547i
\(364\) −0.103580 + 0.0265948i −0.103580 + 0.0265948i
\(365\) 0 0
\(366\) 0 0
\(367\) −0.996398 0.394502i −0.996398 0.394502i −0.187381 0.982287i \(-0.560000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −1.06320 + 0.134314i −1.06320 + 0.134314i
\(373\) 1.03799 1.63560i 1.03799 1.63560i 0.309017 0.951057i \(-0.400000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) 0.238883 + 1.25227i 0.238883 + 1.25227i 0.876307 + 0.481754i \(0.160000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(380\) 0 0
\(381\) −0.574633 + 0.227513i −0.574633 + 0.227513i
\(382\) 0 0
\(383\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.303189 0.220280i 0.303189 0.220280i
\(388\) 1.50441 + 0.595638i 1.50441 + 0.595638i
\(389\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −1.23480 0.317042i −1.23480 0.317042i −0.425779 0.904827i \(-0.640000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(398\) 0 0
\(399\) −0.103580 + 1.64636i −0.103580 + 1.64636i
\(400\) 0.876307 + 0.481754i 0.876307 + 0.481754i
\(401\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(402\) 0 0
\(403\) 0.0981041 0.0921259i 0.0981041 0.0921259i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) −0.746226 1.58581i −0.746226 1.58581i −0.809017 0.587785i \(-0.800000\pi\)
0.0627905 0.998027i \(-0.480000\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.96858 + 0.248690i 1.96858 + 0.248690i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 0.598617 + 0.153699i 0.598617 + 0.153699i
\(418\) 0 0
\(419\) 0 0 −0.929776 0.368125i \(-0.880000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(420\) 0 0
\(421\) 1.07165 1.07165 0.535827 0.844328i \(-0.320000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −0.258183 1.35345i −0.258183 1.35345i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(432\) 0.968583 0.248690i 0.968583 0.248690i
\(433\) −1.56720 0.402389i −1.56720 0.402389i −0.637424 0.770513i \(-0.720000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.996398 1.57007i −0.996398 1.57007i
\(437\) 0 0
\(438\) 0 0
\(439\) 0.00788530 + 0.125333i 0.00788530 + 0.125333i 1.00000 \(0\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(440\) 0 0
\(441\) −0.240851 + 0.132409i −0.240851 + 0.132409i
\(442\) 0 0
\(443\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(444\) 0.0915446 + 1.45506i 0.0915446 + 1.45506i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.0534698 0.849878i −0.0534698 0.849878i
\(449\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −0.187381 0.982287i −0.187381 0.982287i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.115808 + 0.356420i −0.115808 + 0.356420i −0.992115 0.125333i \(-0.960000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(462\) 0 0
\(463\) −0.929324 1.12336i −0.929324 1.12336i −0.992115 0.125333i \(-0.960000\pi\)
0.0627905 0.998027i \(-0.480000\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(468\) −0.0800484 + 0.0967619i −0.0800484 + 0.0967619i
\(469\) 1.20742 + 0.877242i 1.20742 + 0.877242i
\(470\) 0 0
\(471\) 1.18532 + 1.43281i 1.18532 + 1.43281i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) −0.362989 + 1.90285i −0.362989 + 1.90285i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(480\) 0 0
\(481\) −0.116705 0.141073i −0.116705 0.141073i
\(482\) 0 0
\(483\) 0 0
\(484\) −0.929776 0.368125i −0.929776 0.368125i
\(485\) 0 0
\(486\) 0 0
\(487\) −0.116762 0.0462295i −0.116762 0.0462295i 0.309017 0.951057i \(-0.400000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(488\) 0 0
\(489\) 1.41213 + 0.362574i 1.41213 + 0.362574i
\(490\) 0 0
\(491\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.574221 + 0.904827i 0.574221 + 0.904827i
\(497\) 0 0
\(498\) 0 0
\(499\) 0.598617 + 1.84235i 0.598617 + 1.84235i 0.535827 + 0.844328i \(0.320000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.0618003 + 0.982287i −0.0618003 + 0.982287i
\(508\) 0.450527 + 0.423073i 0.450527 + 0.423073i
\(509\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(510\) 0 0
\(511\) 0.224084 + 0.476203i 0.224084 + 0.476203i
\(512\) 0 0
\(513\) 1.03799 + 1.63560i 1.03799 + 1.63560i
\(514\) 0 0
\(515\) 0 0
\(516\) −0.328407 0.180543i −0.328407 0.180543i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(522\) 0 0
\(523\) 1.26480 0.159781i 1.26480 0.159781i 0.535827 0.844328i \(-0.320000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(524\) 0 0
\(525\) 0.791759 0.313480i 0.791759 0.313480i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.809017 0.587785i −0.809017 0.587785i
\(530\) 0 0
\(531\) 0 0
\(532\) 1.53377 0.607262i 1.53377 0.607262i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) −0.328407 0.180543i −0.328407 0.180543i 0.309017 0.951057i \(-0.400000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(542\) 0 0
\(543\) −0.929324 + 0.872693i −0.929324 + 0.872693i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) −1.11716 + 0.614163i −1.11716 + 0.614163i −0.929776 0.368125i \(-0.880000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(548\) 0 0
\(549\) −1.17950 1.10762i −1.17950 1.10762i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −0.316616 + 1.65976i −0.316616 + 1.65976i
\(554\) 0 0
\(555\) 0 0
\(556\) −0.115808 0.607087i −0.115808 0.607087i
\(557\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(558\) 0 0
\(559\) 0.0466920 0.00589857i 0.0466920 0.00589857i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.362576 0.770513i 0.362576 0.770513i
\(568\) 0 0
\(569\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(570\) 0 0
\(571\) 1.07165 1.07165 0.535827 0.844328i \(-0.320000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) −0.637424 0.770513i −0.637424 0.770513i
\(577\) −0.0800484 + 0.0967619i −0.0800484 + 0.0967619i −0.809017 0.587785i \(-0.800000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(578\) 0 0
\(579\) 1.84489 0.233064i 1.84489 0.233064i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(588\) 0.222357 + 0.161552i 0.222357 + 0.161552i
\(589\) −1.32327 + 1.59956i −1.32327 + 1.59956i
\(590\) 0 0
\(591\) 0 0
\(592\) 1.27760 0.702367i 1.27760 0.702367i
\(593\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 0.574221 0.904827i 0.574221 0.904827i
\(598\) 0 0
\(599\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(600\) 0 0
\(601\) −0.273190 0.256543i −0.273190 0.256543i 0.535827 0.844328i \(-0.320000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(602\) 0 0
\(603\) 1.75261 1.75261
\(604\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(605\) 0 0
\(606\) 0 0
\(607\) −1.35556 1.27295i −1.35556 1.27295i −0.929776 0.368125i \(-0.880000\pi\)
−0.425779 0.904827i \(-0.640000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0.00788530 + 0.125333i 0.00788530 + 0.125333i 1.00000 \(0\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.425779 0.904827i \(-0.360000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(618\) 0 0
\(619\) 0.238883 0.288760i 0.238883 0.288760i −0.637424 0.770513i \(-0.720000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0.121636 + 0.0312307i 0.121636 + 0.0312307i
\(625\) 0.968583 0.248690i 0.968583 0.248690i
\(626\) 0 0
\(627\) 0 0
\(628\) 0.791759 1.68257i 0.791759 1.68257i
\(629\) 0 0
\(630\) 0 0
\(631\) 0.238883 0.288760i 0.238883 0.288760i −0.637424 0.770513i \(-0.720000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(632\) 0 0
\(633\) −0.273190 + 1.43211i −0.273190 + 1.43211i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.0345157 −0.0345157
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.425779 0.904827i \(-0.360000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(642\) 0 0
\(643\) 1.72897 0.684547i 1.72897 0.684547i 0.728969 0.684547i \(-0.240000\pi\)
1.00000 \(0\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0.905380 + 0.114376i 0.905380 + 0.114376i
\(652\) −0.273190 1.43211i −0.273190 1.43211i
\(653\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 0.541587 + 0.297740i 0.541587 + 0.297740i
\(658\) 0 0
\(659\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(660\) 0 0
\(661\) 0.541587 0.297740i 0.541587 0.297740i −0.187381 0.982287i \(-0.560000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) −1.62954 0.645180i −1.62954 0.645180i
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −1.41789 + 0.779494i −1.41789 + 0.779494i −0.992115 0.125333i \(-0.960000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(674\) 0 0
\(675\) 0.535827 0.844328i 0.535827 0.844328i
\(676\) 0.915113 0.362319i 0.915113 0.362319i
\(677\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(678\) 0 0
\(679\) −1.11470 0.809880i −1.11470 0.809880i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(684\) 1.03799 1.63560i 1.03799 1.63560i
\(685\) 0 0
\(686\) 0 0
\(687\) −1.56720 + 0.402389i −1.56720 + 0.402389i
\(688\) −0.0235315 + 0.374023i −0.0235315 + 0.374023i
\(689\) 0 0
\(690\) 0 0
\(691\) 0.303189 0.220280i 0.303189 0.220280i −0.425779 0.904827i \(-0.640000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.620759 0.582932i −0.620759 0.582932i
\(701\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(702\) 0 0
\(703\) 2.05880 + 1.93334i 2.05880 + 1.93334i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.613161 1.88711i −0.613161 1.88711i −0.425779 0.904827i \(-0.640000\pi\)
−0.187381 0.982287i \(-0.560000\pi\)
\(710\) 0 0
\(711\) 0.844844 + 1.79538i 0.844844 + 1.79538i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(720\) 0 0
\(721\) −1.57103 0.622015i −1.57103 0.622015i
\(722\) 0 0
\(723\) −1.85955 −1.85955
\(724\) 1.18532 + 0.469303i 1.18532 + 0.469303i
\(725\) 0 0
\(726\) 0 0
\(727\) 1.03137 + 1.24672i 1.03137 + 1.24672i 0.968583 + 0.248690i \(0.0800000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(728\) 0 0
\(729\) −0.187381 0.982287i −0.187381 0.982287i
\(730\) 0 0
\(731\) 0 0
\(732\) −0.500000 + 1.53884i −0.500000 + 1.53884i
\(733\) −0.273190 + 1.43211i −0.273190 + 1.43211i 0.535827 + 0.844328i \(0.320000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −0.866986 0.629902i −0.866986 0.629902i 0.0627905 0.998027i \(-0.480000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(740\) 0 0
\(741\) 0.0152751 + 0.242791i 0.0152751 + 0.242791i
\(742\) 0 0
\(743\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −0.574633 + 1.76854i −0.574633 + 1.76854i 0.0627905 + 0.998027i \(0.480000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −0.851559 −0.851559
\(757\) −1.41789 + 1.03016i −1.41789 + 1.03016i −0.425779 + 0.904827i \(0.640000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(762\) 0 0
\(763\) 0.489334 + 1.50602i 0.489334 + 1.50602i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(769\) 0.121636 + 1.93334i 0.121636 + 1.93334i 0.309017 + 0.951057i \(0.400000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.996398 1.57007i −0.996398 1.57007i
\(773\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(774\) 0 0
\(775\) 1.03799 + 0.266509i 1.03799 + 0.266509i
\(776\) 0 0
\(777\) 0.232637 1.21953i 0.232637 1.21953i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.0515014 0.269980i 0.0515014 0.269980i
\(785\) 0 0
\(786\) 0 0
\(787\) −1.27485 −1.27485 −0.637424 0.770513i \(-0.720000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.0627905 0.193249i −0.0627905 0.193249i
\(794\) 0 0
\(795\) 0 0
\(796\) −1.06320 0.134314i −1.06320 0.134314i
\(797\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −0.746226 1.58581i −0.746226 1.58581i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(810\) 0 0
\(811\) 1.03799 + 0.266509i 1.03799 + 0.266509i 0.728969 0.684547i \(-0.240000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(812\) 0 0
\(813\) −0.746226 1.58581i −0.746226 1.58581i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −0.703170 + 0.180543i −0.703170 + 0.180543i
\(818\) 0 0
\(819\) 0.0865160 0.0628575i 0.0865160 0.0628575i
\(820\) 0 0
\(821\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(822\) 0 0
\(823\) −0.824805 + 0.211774i −0.824805 + 0.211774i −0.637424 0.770513i \(-0.720000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(828\) 0 0
\(829\) −0.273190 1.43211i −0.273190 1.43211i −0.809017 0.587785i \(-0.800000\pi\)
0.535827 0.844328i \(-0.320000\pi\)
\(830\) 0 0
\(831\) 0.303189 + 0.220280i 0.303189 + 0.220280i
\(832\) −0.0235315 0.123357i −0.0235315 0.123357i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0.939097 0.516273i 0.939097 0.516273i
\(838\) 0 0
\(839\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(840\) 0 0
\(841\) −0.929776 0.368125i −0.929776 0.368125i
\(842\) 0 0
\(843\) 0 0
\(844\) 1.41213 0.362574i 1.41213 0.362574i
\(845\) 0 0
\(846\) 0 0
\(847\) 0.688925 + 0.500534i 0.688925 + 0.500534i
\(848\) 0 0
\(849\) −1.73879 + 0.955910i −1.73879 + 0.955910i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −0.746226 0.410241i −0.746226 0.410241i 0.0627905 0.998027i \(-0.480000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(858\) 0 0
\(859\) −1.92189 0.242791i −1.92189 0.242791i −0.929776 0.368125i \(-0.880000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) −0.929776 + 0.368125i −0.929776 + 0.368125i
\(868\) −0.282001 0.867911i −0.282001 0.867911i
\(869\) 0 0
\(870\) 0 0
\(871\) 0.192871 + 0.106032i 0.192871 + 0.106032i
\(872\) 0 0
\(873\) −1.61803 −1.61803
\(874\) 0 0
\(875\) 0 0
\(876\) 0.0388067 0.616814i 0.0388067 0.616814i
\(877\) 0.238883 1.25227i 0.238883 1.25227i −0.637424 0.770513i \(-0.720000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(882\) 0 0
\(883\) −0.574633 + 1.76854i −0.574633 + 1.76854i 0.0627905 + 0.998027i \(0.480000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(888\) 0 0
\(889\) −0.282001 0.444363i −0.282001 0.444363i
\(890\) 0 0
\(891\) 0 0
\(892\) 0.110048 + 1.74915i 0.110048 + 1.74915i
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) −0.992115 0.125333i −0.992115 0.125333i
\(901\) 0 0
\(902\) 0 0
\(903\) 0.232637 + 0.218461i 0.232637 + 0.218461i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.93717 1.93717 0.968583 0.248690i \(-0.0800000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(912\) −1.92189 0.242791i −1.92189 0.242791i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 1.03137 + 1.24672i 1.03137 + 1.24672i
\(917\) 0 0
\(918\) 0 0
\(919\) 0.542804 1.15352i 0.542804 1.15352i −0.425779 0.904827i \(-0.640000\pi\)
0.968583 0.248690i \(-0.0800000\pi\)
\(920\) 0 0
\(921\) 1.26480 1.52888i 1.26480 1.52888i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 0.450527 1.38658i 0.450527 1.38658i
\(926\) 0 0
\(927\) −1.92189 + 0.493458i −1.92189 + 0.493458i
\(928\) 0 0
\(929\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(930\) 0 0
\(931\) 0.528228 0.0667307i 0.528228 0.0667307i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −1.80113 0.713118i −1.80113 0.713118i −0.992115 0.125333i \(-0.960000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(938\) 0 0
\(939\) 0.618034 0.618034
\(940\) 0 0
\(941\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(948\) 1.26480 1.52888i 1.26480 1.52888i
\(949\) 0.0415873 + 0.0655311i 0.0415873 + 0.0655311i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.108209 + 0.101615i 0.108209 + 0.101615i
\(962\) 0 0
\(963\) 0 0
\(964\) 0.791759 + 1.68257i 0.791759 + 1.68257i
\(965\) 0 0
\(966\) 0 0
\(967\) −1.44644 + 1.35830i −1.44644 + 1.35830i −0.637424 + 0.770513i \(0.720000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.929776 0.368125i \(-0.880000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(972\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(973\) −0.0330462 + 0.525254i −0.0330462 + 0.525254i
\(974\) 0 0
\(975\) 0.110048 0.0604991i 0.110048 0.0604991i
\(976\) 1.60528 0.202793i 1.60528 0.202793i
\(977\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.50441 + 1.09302i 1.50441 + 1.09302i
\(982\) 0 0
\(983\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0.213180 0.117197i 0.213180 0.117197i
\(989\) 0 0
\(990\) 0 0
\(991\) 1.60528 1.16630i 1.60528 1.16630i 0.728969 0.684547i \(-0.240000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(992\) 0 0
\(993\) 1.30902 0.951057i 1.30902 0.951057i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.574221 + 0.904827i 0.574221 + 0.904827i 1.00000 \(0\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(998\) 0 0
\(999\) −0.620759 1.31918i −0.620759 1.31918i
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 453.1.t.a.299.1 yes 20
3.2 odd 2 CM 453.1.t.a.299.1 yes 20
151.50 even 25 inner 453.1.t.a.50.1 20
453.50 odd 50 inner 453.1.t.a.50.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
453.1.t.a.50.1 20 151.50 even 25 inner
453.1.t.a.50.1 20 453.50 odd 50 inner
453.1.t.a.299.1 yes 20 1.1 even 1 trivial
453.1.t.a.299.1 yes 20 3.2 odd 2 CM