Properties

Label 500.1.p.a.471.1
Level $500$
Weight $1$
Character 500.471
Analytic conductor $0.250$
Analytic rank $0$
Dimension $20$
Projective image $D_{25}$
CM discriminant -4
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [500,1,Mod(11,500)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(500, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([25, 38]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("500.11");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 500 = 2^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 500.p (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: \(0.249532506317\)
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: \(\Z[a_1, a_2]\)
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 471.1
Root \(-0.728969 - 0.684547i\) of defining polynomial
Character \(\chi\) \(=\) 500.471
Dual form 500.1.p.a.431.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.929776 - 0.368125i) q^{2} +(0.728969 + 0.684547i) q^{4} +(0.876307 + 0.481754i) q^{5} +(-0.425779 - 0.904827i) q^{8} +(0.535827 - 0.844328i) q^{9} +O(q^{10})\) \(q+(-0.929776 - 0.368125i) q^{2} +(0.728969 + 0.684547i) q^{4} +(0.876307 + 0.481754i) q^{5} +(-0.425779 - 0.904827i) q^{8} +(0.535827 - 0.844328i) q^{9} +(-0.637424 - 0.770513i) q^{10} +(-1.06320 + 1.67534i) q^{13} +(0.0627905 + 0.998027i) q^{16} +(0.781202 - 0.733597i) q^{17} +(-0.809017 + 0.587785i) q^{18} +(0.309017 + 0.951057i) q^{20} +(0.535827 + 0.844328i) q^{25} +(1.60528 - 1.16630i) q^{26} +(-0.0235315 - 0.123357i) q^{29} +(0.309017 - 0.951057i) q^{32} +(-0.996398 + 0.394502i) q^{34} +(0.968583 - 0.248690i) q^{36} +(-0.0800484 - 1.27233i) q^{37} +(0.0627905 - 0.998027i) q^{40} +(-0.929324 - 1.12336i) q^{41} +(0.876307 - 0.481754i) q^{45} +(0.309017 + 0.951057i) q^{49} +(-0.187381 - 0.982287i) q^{50} +(-1.92189 + 0.493458i) q^{52} +(0.371808 + 0.0469702i) q^{53} +(-0.0235315 + 0.123357i) q^{58} +(-1.11716 + 1.35041i) q^{61} +(-0.637424 + 0.770513i) q^{64} +(-1.73879 + 0.955910i) q^{65} +1.07165 q^{68} +(-0.992115 - 0.125333i) q^{72} +(-0.824805 + 0.211774i) q^{73} +(-0.393950 + 1.21245i) q^{74} +(-0.425779 + 0.904827i) q^{80} +(-0.425779 - 0.904827i) q^{81} +(0.450527 + 1.38658i) q^{82} +(1.03799 - 0.266509i) q^{85} +(-1.80113 + 0.462452i) q^{89} +(-0.992115 + 0.125333i) q^{90} +(-0.362989 - 1.90285i) q^{97} +(0.0627905 - 0.998027i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 5 q^{18} - 5 q^{20} - 5 q^{32} - 5 q^{34} - 5 q^{37} - 5 q^{49} - 5 q^{53} - 5 q^{65} - 5 q^{85} - 5 q^{89}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/500\mathbb{Z}\right)^\times\).

\(n\) \(251\) \(377\)
\(\chi(n)\) \(-1\) \(e\left(\frac{3}{25}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.929776 0.368125i −0.929776 0.368125i
\(3\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(4\) 0.728969 + 0.684547i 0.728969 + 0.684547i
\(5\) 0.876307 + 0.481754i 0.876307 + 0.481754i
\(6\) 0 0
\(7\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(8\) −0.425779 0.904827i −0.425779 0.904827i
\(9\) 0.535827 0.844328i 0.535827 0.844328i
\(10\) −0.637424 0.770513i −0.637424 0.770513i
\(11\) 0 0 −0.929776 0.368125i \(-0.880000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(12\) 0 0
\(13\) −1.06320 + 1.67534i −1.06320 + 1.67534i −0.425779 + 0.904827i \(0.640000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0.0627905 + 0.998027i 0.0627905 + 0.998027i
\(17\) 0.781202 0.733597i 0.781202 0.733597i −0.187381 0.982287i \(-0.560000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(18\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(19\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(20\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(24\) 0 0
\(25\) 0.535827 + 0.844328i 0.535827 + 0.844328i
\(26\) 1.60528 1.16630i 1.60528 1.16630i
\(27\) 0 0
\(28\) 0 0
\(29\) −0.0235315 0.123357i −0.0235315 0.123357i 0.968583 0.248690i \(-0.0800000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(30\) 0 0
\(31\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(32\) 0.309017 0.951057i 0.309017 0.951057i
\(33\) 0 0
\(34\) −0.996398 + 0.394502i −0.996398 + 0.394502i
\(35\) 0 0
\(36\) 0.968583 0.248690i 0.968583 0.248690i
\(37\) −0.0800484 1.27233i −0.0800484 1.27233i −0.809017 0.587785i \(-0.800000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.0627905 0.998027i 0.0627905 0.998027i
\(41\) −0.929324 1.12336i −0.929324 1.12336i −0.992115 0.125333i \(-0.960000\pi\)
0.0627905 0.998027i \(-0.480000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(44\) 0 0
\(45\) 0.876307 0.481754i 0.876307 0.481754i
\(46\) 0 0
\(47\) 0 0 0.425779 0.904827i \(-0.360000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(48\) 0 0
\(49\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(50\) −0.187381 0.982287i −0.187381 0.982287i
\(51\) 0 0
\(52\) −1.92189 + 0.493458i −1.92189 + 0.493458i
\(53\) 0.371808 + 0.0469702i 0.371808 + 0.0469702i 0.309017 0.951057i \(-0.400000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −0.0235315 + 0.123357i −0.0235315 + 0.123357i
\(59\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(60\) 0 0
\(61\) −1.11716 + 1.35041i −1.11716 + 1.35041i −0.187381 + 0.982287i \(0.560000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) −0.637424 + 0.770513i −0.637424 + 0.770513i
\(65\) −1.73879 + 0.955910i −1.73879 + 0.955910i
\(66\) 0 0
\(67\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(68\) 1.07165 1.07165
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.425779 0.904827i \(-0.360000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(72\) −0.992115 0.125333i −0.992115 0.125333i
\(73\) −0.824805 + 0.211774i −0.824805 + 0.211774i −0.637424 0.770513i \(-0.720000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(74\) −0.393950 + 1.21245i −0.393950 + 1.21245i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(80\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(81\) −0.425779 0.904827i −0.425779 0.904827i
\(82\) 0.450527 + 1.38658i 0.450527 + 1.38658i
\(83\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(84\) 0 0
\(85\) 1.03799 0.266509i 1.03799 0.266509i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.80113 + 0.462452i −1.80113 + 0.462452i −0.992115 0.125333i \(-0.960000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(90\) −0.992115 + 0.125333i −0.992115 + 0.125333i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −0.362989 1.90285i −0.362989 1.90285i −0.425779 0.904827i \(-0.640000\pi\)
0.0627905 0.998027i \(-0.480000\pi\)
\(98\) 0.0627905 0.998027i 0.0627905 0.998027i
\(99\) 0 0
\(100\) −0.187381 + 0.982287i −0.187381 + 0.982287i
\(101\) −1.56720 1.13864i −1.56720 1.13864i −0.929776 0.368125i \(-0.880000\pi\)
−0.637424 0.770513i \(-0.720000\pi\)
\(102\) 0 0
\(103\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(104\) 1.96858 + 0.248690i 1.96858 + 0.248690i
\(105\) 0 0
\(106\) −0.328407 0.180543i −0.328407 0.180543i
\(107\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(108\) 0 0
\(109\) −0.0534698 0.849878i −0.0534698 0.849878i −0.929776 0.368125i \(-0.880000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −1.80113 0.462452i −1.80113 0.462452i −0.809017 0.587785i \(-0.800000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.0672897 0.106032i 0.0672897 0.106032i
\(117\) 0.844844 + 1.79538i 0.844844 + 1.79538i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.728969 + 0.684547i 0.728969 + 0.684547i
\(122\) 1.53583 0.844328i 1.53583 0.844328i
\(123\) 0 0
\(124\) 0 0
\(125\) 0.0627905 + 0.998027i 0.0627905 + 0.998027i
\(126\) 0 0
\(127\) 0 0 −0.929776 0.368125i \(-0.880000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(128\) 0.876307 0.481754i 0.876307 0.481754i
\(129\) 0 0
\(130\) 1.96858 0.248690i 1.96858 0.248690i
\(131\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) −0.996398 0.394502i −0.996398 0.394502i
\(137\) 1.69755 + 0.435857i 1.69755 + 0.435857i 0.968583 0.248690i \(-0.0800000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.876307 + 0.481754i 0.876307 + 0.481754i
\(145\) 0.0388067 0.119435i 0.0388067 0.119435i
\(146\) 0.844844 + 0.106729i 0.844844 + 0.106729i
\(147\) 0 0
\(148\) 0.812619 0.982287i 0.812619 0.982287i
\(149\) 0.303189 + 0.220280i 0.303189 + 0.220280i 0.728969 0.684547i \(-0.240000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(150\) 0 0
\(151\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(152\) 0 0
\(153\) −0.200808 1.05267i −0.200808 1.05267i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) −0.574633 + 1.76854i −0.574633 + 1.76854i 0.0627905 + 0.998027i \(0.480000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0.728969 0.684547i 0.728969 0.684547i
\(161\) 0 0
\(162\) 0.0627905 + 0.998027i 0.0627905 + 0.998027i
\(163\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(164\) 0.0915446 1.45506i 0.0915446 1.45506i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(168\) 0 0
\(169\) −1.25058 2.65763i −1.25058 2.65763i
\(170\) −1.06320 0.134314i −1.06320 0.134314i
\(171\) 0 0
\(172\) 0 0
\(173\) 1.03799 + 1.63560i 1.03799 + 1.63560i 0.728969 + 0.684547i \(0.240000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 1.84489 + 0.233064i 1.84489 + 0.233064i
\(179\) 0 0 0.425779 0.904827i \(-0.360000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(180\) 0.968583 + 0.248690i 0.968583 + 0.248690i
\(181\) 0.348445 1.82662i 0.348445 1.82662i −0.187381 0.982287i \(-0.560000\pi\)
0.535827 0.844328i \(-0.320000\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 0.542804 1.15352i 0.542804 1.15352i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(192\) 0 0
\(193\) 1.75261 1.75261 0.876307 0.481754i \(-0.160000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(194\) −0.362989 + 1.90285i −0.362989 + 1.90285i
\(195\) 0 0
\(196\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(197\) −1.44644 0.182728i −1.44644 0.182728i −0.637424 0.770513i \(-0.720000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(198\) 0 0
\(199\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(200\) 0.535827 0.844328i 0.535827 0.844328i
\(201\) 0 0
\(202\) 1.03799 + 1.63560i 1.03799 + 1.63560i
\(203\) 0 0
\(204\) 0 0
\(205\) −0.273190 1.43211i −0.273190 1.43211i
\(206\) 0 0
\(207\) 0 0
\(208\) −1.73879 0.955910i −1.73879 0.955910i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(212\) 0.238883 + 0.288760i 0.238883 + 0.288760i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.263146 + 0.809880i −0.263146 + 0.809880i
\(219\) 0 0
\(220\) 0 0
\(221\) 0.398449 + 2.08874i 0.398449 + 2.08874i
\(222\) 0 0
\(223\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(224\) 0 0
\(225\) 1.00000 1.00000
\(226\) 1.50441 + 1.09302i 1.50441 + 1.09302i
\(227\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(228\) 0 0
\(229\) 0.371808 + 0.0469702i 0.371808 + 0.0469702i 0.309017 0.951057i \(-0.400000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −0.101597 + 0.0738147i −0.101597 + 0.0738147i
\(233\) 1.06279 0.998027i 1.06279 0.998027i 0.0627905 0.998027i \(-0.480000\pi\)
1.00000 \(0\)
\(234\) −0.124591 1.98031i −0.124591 1.98031i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.929776 0.368125i \(-0.880000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(240\) 0 0
\(241\) −0.683098 + 1.07639i −0.683098 + 1.07639i 0.309017 + 0.951057i \(0.400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(242\) −0.425779 0.904827i −0.425779 0.904827i
\(243\) 0 0
\(244\) −1.73879 + 0.219661i −1.73879 + 0.219661i
\(245\) −0.187381 + 0.982287i −0.187381 + 0.982287i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0.309017 0.951057i 0.309017 0.951057i
\(251\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.992115 + 0.125333i −0.992115 + 0.125333i
\(257\) 0.688925 + 0.500534i 0.688925 + 0.500534i 0.876307 0.481754i \(-0.160000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −1.92189 0.493458i −1.92189 0.493458i
\(261\) −0.116762 0.0462295i −0.116762 0.0462295i
\(262\) 0 0
\(263\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(264\) 0 0
\(265\) 0.303189 + 0.220280i 0.303189 + 0.220280i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 1.53583 + 0.844328i 1.53583 + 0.844328i 1.00000 \(0\)
0.535827 + 0.844328i \(0.320000\pi\)
\(270\) 0 0
\(271\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(272\) 0.781202 + 0.733597i 0.781202 + 0.733597i
\(273\) 0 0
\(274\) −1.41789 1.03016i −1.41789 1.03016i
\(275\) 0 0
\(276\) 0 0
\(277\) 0.00788530 0.125333i 0.00788530 0.125333i −0.992115 0.125333i \(-0.960000\pi\)
1.00000 \(0\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.273190 + 0.256543i −0.273190 + 0.256543i −0.809017 0.587785i \(-0.800000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(282\) 0 0
\(283\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) −0.637424 0.770513i −0.637424 0.770513i
\(289\) 0.00932071 0.148149i 0.00932071 0.148149i
\(290\) −0.0800484 + 0.0967619i −0.0800484 + 0.0967619i
\(291\) 0 0
\(292\) −0.746226 0.410241i −0.746226 0.410241i
\(293\) 0.0388067 + 0.119435i 0.0388067 + 0.119435i 0.968583 0.248690i \(-0.0800000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.11716 + 0.614163i −1.11716 + 0.614163i
\(297\) 0 0
\(298\) −0.200808 0.316423i −0.200808 0.316423i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −1.62954 + 0.645180i −1.62954 + 0.645180i
\(306\) −0.200808 + 1.05267i −0.200808 + 1.05267i
\(307\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(312\) 0 0
\(313\) 1.50441 0.595638i 1.50441 0.595638i 0.535827 0.844328i \(-0.320000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(314\) 1.18532 1.43281i 1.18532 1.43281i
\(315\) 0 0
\(316\) 0 0
\(317\) −0.115808 + 0.607087i −0.115808 + 0.607087i 0.876307 + 0.481754i \(0.160000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) −0.929776 + 0.368125i −0.929776 + 0.368125i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.309017 0.951057i 0.309017 0.951057i
\(325\) −1.98423 −1.98423
\(326\) 0 0
\(327\) 0 0
\(328\) −0.620759 + 1.31918i −0.620759 + 1.31918i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(332\) 0 0
\(333\) −1.11716 0.614163i −1.11716 0.614163i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.393950 0.476203i −0.393950 0.476203i 0.535827 0.844328i \(-0.320000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(338\) 0.184426 + 2.93137i 0.184426 + 2.93137i
\(339\) 0 0
\(340\) 0.939097 + 0.516273i 0.939097 + 0.516273i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −0.362989 1.90285i −0.362989 1.90285i
\(347\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(348\) 0 0
\(349\) 1.03137 0.749337i 1.03137 0.749337i 0.0627905 0.998027i \(-0.480000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −1.17950 1.10762i −1.17950 1.10762i −0.992115 0.125333i \(-0.960000\pi\)
−0.187381 0.982287i \(-0.560000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −1.62954 0.895846i −1.62954 0.895846i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(360\) −0.809017 0.587785i −0.809017 0.587785i
\(361\) 0.535827 + 0.844328i 0.535827 + 0.844328i
\(362\) −0.996398 + 1.57007i −0.996398 + 1.57007i
\(363\) 0 0
\(364\) 0 0
\(365\) −0.824805 0.211774i −0.824805 0.211774i
\(366\) 0 0
\(367\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(368\) 0 0
\(369\) −1.44644 + 0.182728i −1.44644 + 0.182728i
\(370\) −0.929324 + 0.872693i −0.929324 + 0.872693i
\(371\) 0 0
\(372\) 0 0
\(373\) 1.50441 + 0.595638i 1.50441 + 0.595638i 0.968583 0.248690i \(-0.0800000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.231683 + 0.0917299i 0.231683 + 0.0917299i
\(378\) 0 0
\(379\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −1.62954 0.645180i −1.62954 0.645180i
\(387\) 0 0
\(388\) 1.03799 1.63560i 1.03799 1.63560i
\(389\) 0.781202 + 1.23098i 0.781202 + 1.23098i 0.968583 + 0.248690i \(0.0800000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0.728969 0.684547i 0.728969 0.684547i
\(393\) 0 0
\(394\) 1.27760 + 0.702367i 1.27760 + 0.702367i
\(395\) 0 0
\(396\) 0 0
\(397\) 0.450527 + 0.423073i 0.450527 + 0.423073i 0.876307 0.481754i \(-0.160000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(401\) −1.41789 + 1.03016i −1.41789 + 1.03016i −0.425779 + 0.904827i \(0.640000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.362989 1.90285i −0.362989 1.90285i
\(405\) 0.0627905 0.998027i 0.0627905 0.998027i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.18532 0.469303i 1.18532 0.469303i 0.309017 0.951057i \(-0.400000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(410\) −0.273190 + 1.43211i −0.273190 + 1.43211i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 1.26480 + 1.52888i 1.26480 + 1.52888i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(420\) 0 0
\(421\) 1.69755 0.933237i 1.69755 0.933237i 0.728969 0.684547i \(-0.240000\pi\)
0.968583 0.248690i \(-0.0800000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −0.115808 0.356420i −0.115808 0.356420i
\(425\) 1.03799 + 0.266509i 1.03799 + 0.266509i
\(426\) 0 0
\(427\) 0 0
\(428\) 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 0
\(433\) −0.273190 + 1.43211i −0.273190 + 1.43211i 0.535827 + 0.844328i \(0.320000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.542804 0.656137i 0.542804 0.656137i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(440\) 0 0
\(441\) 0.968583 + 0.248690i 0.968583 + 0.248690i
\(442\) 0.398449 2.08874i 0.398449 2.08874i
\(443\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(444\) 0 0
\(445\) −1.80113 0.462452i −1.80113 0.462452i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.574633 + 1.76854i −0.574633 + 1.76854i 0.0627905 + 0.998027i \(0.480000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(450\) −0.929776 0.368125i −0.929776 0.368125i
\(451\) 0 0
\(452\) −0.996398 1.57007i −0.996398 1.57007i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.190983 + 0.587785i 0.190983 + 0.587785i 1.00000 \(0\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(458\) −0.328407 0.180543i −0.328407 0.180543i
\(459\) 0 0
\(460\) 0 0
\(461\) 0.0672897 1.06954i 0.0672897 1.06954i −0.809017 0.587785i \(-0.800000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(464\) 0.121636 0.0312307i 0.121636 0.0312307i
\(465\) 0 0
\(466\) −1.35556 + 0.536702i −1.35556 + 0.536702i
\(467\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(468\) −0.613161 + 1.88711i −0.613161 + 1.88711i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.238883 0.288760i 0.238883 0.288760i
\(478\) 0 0
\(479\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(480\) 0 0
\(481\) 2.21670 + 1.21864i 2.21670 + 1.21864i
\(482\) 1.03137 0.749337i 1.03137 0.749337i
\(483\) 0 0
\(484\) 0.0627905 + 0.998027i 0.0627905 + 0.998027i
\(485\) 0.598617 1.84235i 0.598617 1.84235i
\(486\) 0 0
\(487\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(488\) 1.69755 + 0.435857i 1.69755 + 0.435857i
\(489\) 0 0
\(490\) 0.535827 0.844328i 0.535827 0.844328i
\(491\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(492\) 0 0
\(493\) −0.108877 0.0791038i −0.108877 0.0791038i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(500\) −0.637424 + 0.770513i −0.637424 + 0.770513i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(504\) 0 0
\(505\) −0.824805 1.75280i −0.824805 1.75280i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.574221 0.904827i 0.574221 0.904827i −0.425779 0.904827i \(-0.640000\pi\)
1.00000 \(0\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.968583 + 0.248690i 0.968583 + 0.248690i
\(513\) 0 0
\(514\) −0.456288 0.718995i −0.456288 0.718995i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 1.60528 + 1.16630i 1.60528 + 1.16630i
\(521\) 0.371808 + 0.0469702i 0.371808 + 0.0469702i 0.309017 0.951057i \(-0.400000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(522\) 0.0915446 + 0.0859661i 0.0915446 + 0.0859661i
\(523\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.187381 0.982287i −0.187381 0.982287i
\(530\) −0.200808 0.316423i −0.200808 0.316423i
\(531\) 0 0
\(532\) 0 0
\(533\) 2.87007 0.362574i 2.87007 0.362574i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −1.11716 1.35041i −1.11716 1.35041i
\(539\) 0 0
\(540\) 0 0
\(541\) −0.0800484 0.0967619i −0.0800484 0.0967619i 0.728969 0.684547i \(-0.240000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) −0.456288 0.969661i −0.456288 0.969661i
\(545\) 0.362576 0.770513i 0.362576 0.770513i
\(546\) 0 0
\(547\) 0 0 0.425779 0.904827i \(-0.360000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(548\) 0.939097 + 1.47978i 0.939097 + 1.47978i
\(549\) 0.541587 + 1.66683i 0.541587 + 1.66683i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −0.0534698 + 0.113629i −0.0534698 + 0.113629i
\(555\) 0 0
\(556\) 0 0
\(557\) −1.85955 −1.85955 −0.929776 0.368125i \(-0.880000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.348445 0.137959i 0.348445 0.137959i
\(563\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(564\) 0 0
\(565\) −1.35556 1.27295i −1.35556 1.27295i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) −0.200808 + 1.05267i −0.200808 + 1.05267i 0.728969 + 0.684547i \(0.240000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(570\) 0 0
\(571\) 0 0 0.425779 0.904827i \(-0.360000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(577\) 0.331159 + 0.521823i 0.331159 + 0.521823i 0.968583 0.248690i \(-0.0800000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(578\) −0.0632033 + 0.134314i −0.0632033 + 0.134314i
\(579\) 0 0
\(580\) 0.110048 0.0604991i 0.110048 0.0604991i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0.542804 + 0.656137i 0.542804 + 0.656137i
\(585\) −0.124591 + 1.98031i −0.124591 + 1.98031i
\(586\) 0.00788530 0.125333i 0.00788530 0.125333i
\(587\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.26480 0.159781i 1.26480 0.159781i
\(593\) 0.0388067 0.119435i 0.0388067 0.119435i −0.929776 0.368125i \(-0.880000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0.0702235 + 0.368125i 0.0702235 + 0.368125i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(600\) 0 0
\(601\) 1.60528 + 1.16630i 1.60528 + 1.16630i 0.876307 + 0.481754i \(0.160000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(606\) 0 0
\(607\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 1.75261 1.75261
\(611\) 0 0
\(612\) 0.574221 0.904827i 0.574221 0.904827i
\(613\) −1.23480 0.317042i −1.23480 0.317042i −0.425779 0.904827i \(-0.640000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.620759 1.31918i −0.620759 1.31918i −0.929776 0.368125i \(-0.880000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(618\) 0 0
\(619\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(626\) −1.61803 −1.61803
\(627\) 0 0
\(628\) −1.62954 + 0.895846i −1.62954 + 0.895846i
\(629\) −0.995914 0.935225i −0.995914 0.935225i
\(630\) 0 0
\(631\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.331159 0.521823i 0.331159 0.521823i
\(635\) 0 0
\(636\) 0 0
\(637\) −1.92189 0.493458i −1.92189 0.493458i
\(638\) 0 0
\(639\) 0 0
\(640\) 1.00000 1.00000
\(641\) −0.101597 1.61484i −0.101597 1.61484i −0.637424 0.770513i \(-0.720000\pi\)
0.535827 0.844328i \(-0.320000\pi\)
\(642\) 0 0
\(643\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(648\) −0.637424 + 0.770513i −0.637424 + 0.770513i
\(649\) 0 0
\(650\) 1.84489 + 0.730444i 1.84489 + 0.730444i
\(651\) 0 0
\(652\) 0 0
\(653\) −0.362989 1.90285i −0.362989 1.90285i −0.425779 0.904827i \(-0.640000\pi\)
0.0627905 0.998027i \(-0.480000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 1.06279 0.998027i 1.06279 0.998027i
\(657\) −0.263146 + 0.809880i −0.263146 + 0.809880i
\(658\) 0 0
\(659\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(660\) 0 0
\(661\) −1.56720 + 0.402389i −1.56720 + 0.402389i −0.929776 0.368125i \(-0.880000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0.812619 + 0.982287i 0.812619 + 0.982287i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 0.0672897 + 0.106032i 0.0672897 + 0.106032i 0.876307 0.481754i \(-0.160000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(674\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(675\) 0 0
\(676\) 0.907634 2.79341i 0.907634 2.79341i
\(677\) 0.598617 0.153699i 0.598617 0.153699i 0.0627905 0.998027i \(-0.480000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −0.683098 0.825723i −0.683098 0.825723i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(684\) 0 0
\(685\) 1.27760 + 1.19975i 1.27760 + 1.19975i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.473998 + 0.572965i −0.473998 + 0.572965i
\(690\) 0 0
\(691\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(692\) −0.362989 + 1.90285i −0.362989 + 1.90285i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) −1.55008 0.195821i −1.55008 0.195821i
\(698\) −1.23480 + 0.317042i −1.23480 + 0.317042i
\(699\) 0 0
\(700\) 0 0
\(701\) −0.613161 1.88711i −0.613161 1.88711i −0.425779 0.904827i \(-0.640000\pi\)
−0.187381 0.982287i \(-0.560000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0.688925 + 1.46404i 0.688925 + 1.46404i
\(707\) 0 0
\(708\) 0 0
\(709\) 1.26480 + 1.52888i 1.26480 + 1.52888i 0.728969 + 0.684547i \(0.240000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 1.18532 + 1.43281i 1.18532 + 1.43281i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(720\) 0.535827 + 0.844328i 0.535827 + 0.844328i
\(721\) 0 0
\(722\) −0.187381 0.982287i −0.187381 0.982287i
\(723\) 0 0
\(724\) 1.50441 1.09302i 1.50441 1.09302i
\(725\) 0.0915446 0.0859661i 0.0915446 0.0859661i
\(726\) 0 0
\(727\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(728\) 0 0
\(729\) −0.992115 0.125333i −0.992115 0.125333i
\(730\) 0.688925 + 0.500534i 0.688925 + 0.500534i
\(731\) 0 0
\(732\) 0 0
\(733\) −1.17950 + 1.10762i −1.17950 + 1.10762i −0.187381 + 0.982287i \(0.560000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 1.41213 + 0.362574i 1.41213 + 0.362574i
\(739\) 0 0 −0.929776 0.368125i \(-0.880000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(740\) 1.18532 0.469303i 1.18532 0.469303i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(744\) 0 0
\(745\) 0.159566 + 0.339095i 0.159566 + 0.339095i
\(746\) −1.17950 1.10762i −1.17950 1.10762i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −0.181646 0.170577i −0.181646 0.170577i
\(755\) 0 0
\(756\) 0 0
\(757\) −1.41789 1.03016i −1.41789 1.03016i −0.992115 0.125333i \(-0.960000\pi\)
−0.425779 0.904827i \(-0.640000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.791759 + 0.313480i 0.791759 + 0.313480i 0.728969 0.684547i \(-0.240000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.331159 1.01920i 0.331159 1.01920i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.541587 + 0.297740i 0.541587 + 0.297740i 0.728969 0.684547i \(-0.240000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.27760 + 1.19975i 1.27760 + 1.19975i
\(773\) 0.542804 0.656137i 0.542804 0.656137i −0.425779 0.904827i \(-0.640000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) −1.56720 + 1.13864i −1.56720 + 1.13864i
\(777\) 0 0
\(778\) −0.273190 1.43211i −0.273190 1.43211i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.929776 + 0.368125i −0.929776 + 0.368125i
\(785\) −1.35556 + 1.27295i −1.35556 + 1.27295i
\(786\) 0 0
\(787\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(788\) −0.929324 1.12336i −0.929324 1.12336i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −1.07463 3.30738i −1.07463 3.30738i
\(794\) −0.263146 0.559214i −0.263146 0.559214i
\(795\) 0 0
\(796\) 0 0
\(797\) −0.620759 + 1.31918i −0.620759 + 1.31918i 0.309017 + 0.951057i \(0.400000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.968583 0.248690i 0.968583 0.248690i
\(801\) −0.574633 + 1.76854i −0.574633 + 1.76854i
\(802\) 1.69755 0.435857i 1.69755 0.435857i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) −0.362989 + 1.90285i −0.362989 + 1.90285i
\(809\) 1.03799 + 0.266509i 1.03799 + 0.266509i 0.728969 0.684547i \(-0.240000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(810\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(811\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) −1.27485 −1.27485
\(819\) 0 0
\(820\) 0.781202 1.23098i 0.781202 1.23098i
\(821\) 0.688925 1.46404i 0.688925 1.46404i −0.187381 0.982287i \(-0.560000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(822\) 0 0
\(823\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(828\) 0 0
\(829\) −1.73879 + 0.955910i −1.73879 + 0.955910i −0.809017 + 0.587785i \(0.800000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −0.613161 1.88711i −0.613161 1.88711i
\(833\) 0.939097 + 0.516273i 0.939097 + 0.516273i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(840\) 0 0
\(841\) 0.915113 0.362319i 0.915113 0.362319i
\(842\) −1.92189 + 0.242791i −1.92189 + 0.242791i
\(843\) 0 0
\(844\) 0 0
\(845\) 0.184426 2.93137i 0.184426 2.93137i
\(846\) 0 0
\(847\) 0 0
\(848\) −0.0235315 + 0.374023i −0.0235315 + 0.374023i
\(849\) 0 0
\(850\) −0.866986 0.629902i −0.866986 0.629902i
\(851\) 0 0
\(852\) 0 0
\(853\) −0.273190 0.256543i −0.273190 0.256543i 0.535827 0.844328i \(-0.320000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.500000 + 0.363271i −0.500000 + 0.363271i −0.809017 0.587785i \(-0.800000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(858\) 0 0
\(859\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(864\) 0 0
\(865\) 0.121636 + 1.93334i 0.121636 + 1.93334i
\(866\) 0.781202 1.23098i 0.781202 1.23098i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −0.746226 + 0.410241i −0.746226 + 0.410241i
\(873\) −1.80113 0.713118i −1.80113 0.713118i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.791759 + 0.313480i 0.791759 + 0.313480i 0.728969 0.684547i \(-0.240000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 1.60528 0.202793i 1.60528 0.202793i 0.728969 0.684547i \(-0.240000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(882\) −0.809017 0.587785i −0.809017 0.587785i
\(883\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(884\) −1.13939 + 1.79538i −1.13939 + 1.79538i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.50441 + 1.09302i 1.50441 + 1.09302i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 1.18532 1.43281i 1.18532 1.43281i
\(899\) 0 0
\(900\) 0.728969 + 0.684547i 0.728969 + 0.684547i
\(901\) 0.324914 0.236064i 0.324914 0.236064i
\(902\) 0 0
\(903\) 0 0
\(904\) 0.348445 + 1.82662i 0.348445 + 1.82662i
\(905\) 1.18532 1.43281i 1.18532 1.43281i
\(906\) 0 0
\(907\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(908\) 0 0
\(909\) −1.80113 + 0.713118i −1.80113 + 0.713118i
\(910\) 0 0
\(911\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 0.0388067 0.616814i 0.0388067 0.616814i
\(915\) 0 0
\(916\) 0.238883 + 0.288760i 0.238883 + 0.288760i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −0.456288 + 0.969661i −0.456288 + 0.969661i
\(923\) 0 0
\(924\) 0 0
\(925\) 1.03137 0.749337i 1.03137 0.749337i
\(926\) 0 0
\(927\) 0 0
\(928\) −0.124591 0.0157395i −0.124591 0.0157395i
\(929\) −0.824805 + 1.75280i −0.824805 + 1.75280i −0.187381 + 0.982287i \(0.560000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 1.45794 1.45794
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 1.26480 1.52888i 1.26480 1.52888i
\(937\) 1.18532 0.469303i 1.18532 0.469303i 0.309017 0.951057i \(-0.400000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.121636 + 0.0312307i 0.121636 + 0.0312307i 0.309017 0.951057i \(-0.400000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(942\) 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 \(0.0400000\pi\)
\(948\) 0 0
\(949\) 0.522142 1.60699i 0.522142 1.60699i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.456288 + 0.969661i −0.456288 + 0.969661i 0.535827 + 0.844328i \(0.320000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(954\) −0.328407 + 0.180543i −0.328407 + 0.180543i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.0627905 0.998027i 0.0627905 0.998027i
\(962\) −1.61242 1.94908i −1.61242 1.94908i
\(963\) 0 0
\(964\) −1.23480 + 0.317042i −1.23480 + 0.317042i
\(965\) 1.53583 + 0.844328i 1.53583 + 0.844328i
\(966\) 0 0
\(967\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(968\) 0.309017 0.951057i 0.309017 0.951057i
\(969\) 0 0
\(970\) −1.23480 + 1.49261i −1.23480 + 1.49261i
\(971\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −1.41789 1.03016i −1.41789 1.03016i
\(977\) 0.542804 0.656137i 0.542804 0.656137i −0.425779 0.904827i \(-0.640000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(981\) −0.746226 0.410241i −0.746226 0.410241i
\(982\) 0 0
\(983\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(984\) 0 0
\(985\) −1.17950 0.856954i −1.17950 0.856954i
\(986\) 0.0721112 + 0.113629i 0.0721112 + 0.113629i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 0.541587 0.297740i 0.541587 0.297740i −0.187381 0.982287i \(-0.560000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(998\) 0 0
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 500.1.p.a.471.1 yes 20
4.3 odd 2 CM 500.1.p.a.471.1 yes 20
5.2 odd 4 2500.1.n.a.2399.2 40
5.3 odd 4 2500.1.n.a.2399.1 40
5.4 even 2 2500.1.p.a.1351.1 20
20.3 even 4 2500.1.n.a.2399.1 40
20.7 even 4 2500.1.n.a.2399.2 40
20.19 odd 2 2500.1.p.a.1351.1 20
125.56 even 25 inner 500.1.p.a.431.1 20
125.58 odd 100 2500.1.n.a.99.2 40
125.67 odd 100 2500.1.n.a.99.1 40
125.69 even 50 2500.1.p.a.1151.1 20
500.67 even 100 2500.1.n.a.99.1 40
500.183 even 100 2500.1.n.a.99.2 40
500.319 odd 50 2500.1.p.a.1151.1 20
500.431 odd 50 inner 500.1.p.a.431.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
500.1.p.a.431.1 20 125.56 even 25 inner
500.1.p.a.431.1 20 500.431 odd 50 inner
500.1.p.a.471.1 yes 20 1.1 even 1 trivial
500.1.p.a.471.1 yes 20 4.3 odd 2 CM
2500.1.n.a.99.1 40 125.67 odd 100
2500.1.n.a.99.1 40 500.67 even 100
2500.1.n.a.99.2 40 125.58 odd 100
2500.1.n.a.99.2 40 500.183 even 100
2500.1.n.a.2399.1 40 5.3 odd 4
2500.1.n.a.2399.1 40 20.3 even 4
2500.1.n.a.2399.2 40 5.2 odd 4
2500.1.n.a.2399.2 40 20.7 even 4
2500.1.p.a.1151.1 20 125.69 even 50
2500.1.p.a.1151.1 20 500.319 odd 50
2500.1.p.a.1351.1 20 5.4 even 2
2500.1.p.a.1351.1 20 20.19 odd 2