Properties

Label 404.1.o.a.371.1
Level $404$
Weight $1$
Character 404.371
Analytic conductor $0.202$
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] = [404,1,Mod(19,404)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(404, base_ring=CyclotomicField(50))
 
chi = DirichletCharacter(H, H._module([25, 48]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("404.19");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 404 = 2^{2} \cdot 101 \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 404.o (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: \(0.201622265104\)
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 371.1
Root \(-0.968583 - 0.248690i\) of defining polynomial
Character \(\chi\) \(=\) 404.371
Dual form 404.1.o.a.355.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.535827 + 0.844328i) q^{2} +(-0.425779 + 0.904827i) q^{4} +(0.781202 - 1.23098i) q^{5} +(-0.992115 + 0.125333i) q^{8} +(0.876307 + 0.481754i) q^{9} +O(q^{10})\) \(q+(0.535827 + 0.844328i) q^{2} +(-0.425779 + 0.904827i) q^{4} +(0.781202 - 1.23098i) q^{5} +(-0.992115 + 0.125333i) q^{8} +(0.876307 + 0.481754i) q^{9} +1.45794 q^{10} +(-1.80113 - 0.713118i) q^{13} +(-0.637424 - 0.770513i) q^{16} +(-0.500000 + 1.53884i) q^{17} +(0.0627905 + 0.998027i) q^{18} +(0.781202 + 1.23098i) q^{20} +(-0.479249 - 1.01846i) q^{25} +(-0.362989 - 1.90285i) q^{26} +(-0.574633 - 0.227513i) q^{29} +(0.309017 - 0.951057i) q^{32} +(-1.56720 + 0.402389i) q^{34} +(-0.809017 + 0.587785i) q^{36} +(-0.824805 + 0.211774i) q^{37} +(-0.620759 + 1.31918i) q^{40} +(-0.393950 - 1.21245i) q^{41} +(1.27760 - 0.702367i) q^{45} +(0.728969 - 0.684547i) q^{49} +(0.603116 - 0.950360i) q^{50} +(1.41213 - 1.32608i) q^{52} +(0.791759 + 1.68257i) q^{53} +(-0.115808 - 0.607087i) q^{58} +(0.844844 - 1.79538i) q^{61} +(0.968583 - 0.248690i) q^{64} +(-2.28488 + 1.66006i) q^{65} +(-1.17950 - 1.10762i) q^{68} +(-0.929776 - 0.368125i) q^{72} +(-0.0235315 + 0.374023i) q^{73} +(-0.620759 - 0.582932i) q^{74} +(-1.44644 + 0.182728i) q^{80} +(0.535827 + 0.844328i) q^{81} +(0.812619 - 0.982287i) q^{82} +(1.50368 + 1.81763i) q^{85} +(-1.11716 + 1.35041i) q^{89} +(1.27760 + 0.702367i) q^{90} +(0.791759 - 1.68257i) q^{97} +(0.968583 + 0.248690i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20 q+O(q^{10}) \) Copy content Toggle raw display \( 20 q - 5 q^{13} - 10 q^{17} - 5 q^{32} - 5 q^{36} - 5 q^{40} - 5 q^{45} - 5 q^{50} - 5 q^{61} - 5 q^{74} - 5 q^{80} + 20 q^{82} - 5 q^{90}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/404\mathbb{Z}\right)^\times\).

\(n\) \(203\) \(305\)
\(\chi(n)\) \(-1\) \(e\left(\frac{8}{25}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.535827 + 0.844328i 0.535827 + 0.844328i
\(3\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(4\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(5\) 0.781202 1.23098i 0.781202 1.23098i −0.187381 0.982287i \(-0.560000\pi\)
0.968583 0.248690i \(-0.0800000\pi\)
\(6\) 0 0
\(7\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(8\) −0.992115 + 0.125333i −0.992115 + 0.125333i
\(9\) 0.876307 + 0.481754i 0.876307 + 0.481754i
\(10\) 1.45794 1.45794
\(11\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(12\) 0 0
\(13\) −1.80113 0.713118i −1.80113 0.713118i −0.992115 0.125333i \(-0.960000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −0.637424 0.770513i −0.637424 0.770513i
\(17\) −0.500000 + 1.53884i −0.500000 + 1.53884i 0.309017 + 0.951057i \(0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(18\) 0.0627905 + 0.998027i 0.0627905 + 0.998027i
\(19\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(20\) 0.781202 + 1.23098i 0.781202 + 1.23098i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(24\) 0 0
\(25\) −0.479249 1.01846i −0.479249 1.01846i
\(26\) −0.362989 1.90285i −0.362989 1.90285i
\(27\) 0 0
\(28\) 0 0
\(29\) −0.574633 0.227513i −0.574633 0.227513i 0.0627905 0.998027i \(-0.480000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(30\) 0 0
\(31\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(32\) 0.309017 0.951057i 0.309017 0.951057i
\(33\) 0 0
\(34\) −1.56720 + 0.402389i −1.56720 + 0.402389i
\(35\) 0 0
\(36\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(37\) −0.824805 + 0.211774i −0.824805 + 0.211774i −0.637424 0.770513i \(-0.720000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) −0.620759 + 1.31918i −0.620759 + 1.31918i
\(41\) −0.393950 1.21245i −0.393950 1.21245i −0.929776 0.368125i \(-0.880000\pi\)
0.535827 0.844328i \(-0.320000\pi\)
\(42\) 0 0
\(43\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(44\) 0 0
\(45\) 1.27760 0.702367i 1.27760 0.702367i
\(46\) 0 0
\(47\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(48\) 0 0
\(49\) 0.728969 0.684547i 0.728969 0.684547i
\(50\) 0.603116 0.950360i 0.603116 0.950360i
\(51\) 0 0
\(52\) 1.41213 1.32608i 1.41213 1.32608i
\(53\) 0.791759 + 1.68257i 0.791759 + 1.68257i 0.728969 + 0.684547i \(0.240000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −0.115808 0.607087i −0.115808 0.607087i
\(59\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(60\) 0 0
\(61\) 0.844844 1.79538i 0.844844 1.79538i 0.309017 0.951057i \(-0.400000\pi\)
0.535827 0.844328i \(-0.320000\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.968583 0.248690i 0.968583 0.248690i
\(65\) −2.28488 + 1.66006i −2.28488 + 1.66006i
\(66\) 0 0
\(67\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(68\) −1.17950 1.10762i −1.17950 1.10762i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(72\) −0.929776 0.368125i −0.929776 0.368125i
\(73\) −0.0235315 + 0.374023i −0.0235315 + 0.374023i 0.968583 + 0.248690i \(0.0800000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(74\) −0.620759 0.582932i −0.620759 0.582932i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(80\) −1.44644 + 0.182728i −1.44644 + 0.182728i
\(81\) 0.535827 + 0.844328i 0.535827 + 0.844328i
\(82\) 0.812619 0.982287i 0.812619 0.982287i
\(83\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(84\) 0 0
\(85\) 1.50368 + 1.81763i 1.50368 + 1.81763i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.11716 + 1.35041i −1.11716 + 1.35041i −0.187381 + 0.982287i \(0.560000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(90\) 1.27760 + 0.702367i 1.27760 + 0.702367i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 0.791759 1.68257i 0.791759 1.68257i 0.0627905 0.998027i \(-0.480000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(98\) 0.968583 + 0.248690i 0.968583 + 0.248690i
\(99\) 0 0
\(100\) 1.12558 1.12558
\(101\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(102\) 0 0
\(103\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(104\) 1.87631 + 0.481754i 1.87631 + 0.481754i
\(105\) 0 0
\(106\) −0.996398 + 1.57007i −0.996398 + 1.57007i
\(107\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(108\) 0 0
\(109\) −1.06320 + 0.134314i −1.06320 + 0.134314i −0.637424 0.770513i \(-0.720000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.683098 + 0.825723i −0.683098 + 0.825723i −0.992115 0.125333i \(-0.960000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0.450527 0.423073i 0.450527 0.423073i
\(117\) −1.23480 1.49261i −1.23480 1.49261i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.535827 + 0.844328i 0.535827 + 0.844328i
\(122\) 1.96858 0.248690i 1.96858 0.248690i
\(123\) 0 0
\(124\) 0 0
\(125\) −0.181646 0.0229472i −0.181646 0.0229472i
\(126\) 0 0
\(127\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(128\) 0.728969 + 0.684547i 0.728969 + 0.684547i
\(129\) 0 0
\(130\) −2.62594 1.03968i −2.62594 1.03968i
\(131\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0.303189 1.58937i 0.303189 1.58937i
\(137\) 0.303189 0.220280i 0.303189 0.220280i −0.425779 0.904827i \(-0.640000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(138\) 0 0
\(139\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.187381 0.982287i −0.187381 0.982287i
\(145\) −0.728969 + 0.529627i −0.728969 + 0.529627i
\(146\) −0.328407 + 0.180543i −0.328407 + 0.180543i
\(147\) 0 0
\(148\) 0.159566 0.836475i 0.159566 0.836475i
\(149\) 0.159566 + 0.339095i 0.159566 + 0.339095i 0.968583 0.248690i \(-0.0800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(150\) 0 0
\(151\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(152\) 0 0
\(153\) −1.17950 + 1.10762i −1.17950 + 1.10762i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 0.939097 0.516273i 0.939097 0.516273i 0.0627905 0.998027i \(-0.480000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) −0.929324 1.12336i −0.929324 1.12336i
\(161\) 0 0
\(162\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(163\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(164\) 1.26480 + 0.159781i 1.26480 + 0.159781i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(168\) 0 0
\(169\) 2.00657 + 1.88429i 2.00657 + 1.88429i
\(170\) −0.728969 + 2.24353i −0.728969 + 2.24353i
\(171\) 0 0
\(172\) 0 0
\(173\) 0.791759 + 0.313480i 0.791759 + 0.313480i 0.728969 0.684547i \(-0.240000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) −1.73879 0.219661i −1.73879 0.219661i
\(179\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(180\) 0.0915446 + 1.45506i 0.0915446 + 1.45506i
\(181\) 1.96858 0.248690i 1.96858 0.248690i 0.968583 0.248690i \(-0.0800000\pi\)
1.00000 \(0\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −0.383650 + 1.18075i −0.383650 + 1.18075i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(192\) 0 0
\(193\) −1.11716 0.614163i −1.11716 0.614163i −0.187381 0.982287i \(-0.560000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(194\) 1.84489 0.233064i 1.84489 0.233064i
\(195\) 0 0
\(196\) 0.309017 + 0.951057i 0.309017 + 0.951057i
\(197\) 0.331159 0.521823i 0.331159 0.521823i −0.637424 0.770513i \(-0.720000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(198\) 0 0
\(199\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(200\) 0.603116 + 0.950360i 0.603116 + 0.950360i
\(201\) 0 0
\(202\) −0.992115 + 0.125333i −0.992115 + 0.125333i
\(203\) 0 0
\(204\) 0 0
\(205\) −1.80026 0.462227i −1.80026 0.462227i
\(206\) 0 0
\(207\) 0 0
\(208\) 0.598617 + 1.84235i 0.598617 + 1.84235i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.876307 0.481754i \(-0.840000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(212\) −1.85955 −1.85955
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) −0.683098 0.825723i −0.683098 0.825723i
\(219\) 0 0
\(220\) 0 0
\(221\) 1.99794 2.41510i 1.99794 2.41510i
\(222\) 0 0
\(223\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(224\) 0 0
\(225\) 0.0706758 1.12336i 0.0706758 1.12336i
\(226\) −1.06320 0.134314i −1.06320 0.134314i
\(227\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(228\) 0 0
\(229\) −1.35556 1.27295i −1.35556 1.27295i −0.929776 0.368125i \(-0.880000\pi\)
−0.425779 0.904827i \(-0.640000\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0.598617 + 0.153699i 0.598617 + 0.153699i
\(233\) 1.84489 0.730444i 1.84489 0.730444i 0.876307 0.481754i \(-0.160000\pi\)
0.968583 0.248690i \(-0.0800000\pi\)
\(234\) 0.598617 1.84235i 0.598617 1.84235i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(240\) 0 0
\(241\) −0.101597 0.0738147i −0.101597 0.0738147i 0.535827 0.844328i \(-0.320000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(242\) −0.425779 + 0.904827i −0.425779 + 0.904827i
\(243\) 0 0
\(244\) 1.26480 + 1.52888i 1.26480 + 1.52888i
\(245\) −0.273190 1.43211i −0.273190 1.43211i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.0779556 0.165664i −0.0779556 0.165664i
\(251\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.187381 + 0.982287i −0.187381 + 0.982287i
\(257\) −1.73879 + 0.955910i −1.73879 + 0.955910i −0.809017 + 0.587785i \(0.800000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −0.529215 2.77424i −0.529215 2.77424i
\(261\) −0.393950 0.476203i −0.393950 0.476203i
\(262\) 0 0
\(263\) 0 0 0.425779 0.904827i \(-0.360000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(264\) 0 0
\(265\) 2.68973 + 0.339792i 2.68973 + 0.339792i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) 0.121636 0.0312307i 0.121636 0.0312307i −0.187381 0.982287i \(-0.560000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(270\) 0 0
\(271\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(272\) 1.50441 0.595638i 1.50441 0.595638i
\(273\) 0 0
\(274\) 0.348445 + 0.137959i 0.348445 + 0.137959i
\(275\) 0 0
\(276\) 0 0
\(277\) 0.238883 + 1.25227i 0.238883 + 1.25227i 0.876307 + 0.481754i \(0.160000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) −0.0534698 0.849878i −0.0534698 0.849878i −0.929776 0.368125i \(-0.880000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(282\) 0 0
\(283\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.728969 0.684547i 0.728969 0.684547i
\(289\) −1.30902 0.951057i −1.30902 0.951057i
\(290\) −0.837780 0.331700i −0.837780 0.331700i
\(291\) 0 0
\(292\) −0.328407 0.180543i −0.328407 0.180543i
\(293\) 1.75261 1.75261 0.876307 0.481754i \(-0.160000\pi\)
0.876307 + 0.481754i \(0.160000\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0.791759 0.313480i 0.791759 0.313480i
\(297\) 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
\(305\) −1.55008 2.44254i −1.55008 2.44254i
\(306\) −1.56720 0.402389i −1.56720 0.402389i
\(307\) 0 0 0.425779 0.904827i \(-0.360000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(312\) 0 0
\(313\) −1.61803 −1.61803 −0.809017 0.587785i \(-0.800000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(314\) 0.939097 + 0.516273i 0.939097 + 0.516273i
\(315\) 0 0
\(316\) 0 0
\(317\) −1.41789 1.03016i −1.41789 1.03016i −0.992115 0.125333i \(-0.960000\pi\)
−0.425779 0.904827i \(-0.640000\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.450527 1.38658i 0.450527 1.38658i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.992115 + 0.125333i −0.992115 + 0.125333i
\(325\) 0.136911 + 2.17614i 0.136911 + 2.17614i
\(326\) 0 0
\(327\) 0 0
\(328\) 0.542804 + 1.15352i 0.542804 + 1.15352i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.0627905 0.998027i \(-0.480000\pi\)
−0.0627905 + 0.998027i \(0.520000\pi\)
\(332\) 0 0
\(333\) −0.824805 0.211774i −0.824805 0.211774i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) −0.362989 + 0.0931997i −0.362989 + 0.0931997i −0.425779 0.904827i \(-0.640000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(338\) −0.515788 + 2.70386i −0.515788 + 2.70386i
\(339\) 0 0
\(340\) −2.28488 + 0.586657i −2.28488 + 0.586657i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0.159566 + 0.836475i 0.159566 + 0.836475i
\(347\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(348\) 0 0
\(349\) −0.746226 + 0.410241i −0.746226 + 0.410241i −0.809017 0.587785i \(-0.800000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) 0.331159 0.521823i 0.331159 0.521823i −0.637424 0.770513i \(-0.720000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −0.746226 1.58581i −0.746226 1.58581i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(360\) −1.17950 + 0.856954i −1.17950 + 0.856954i
\(361\) −0.187381 0.982287i −0.187381 0.982287i
\(362\) 1.26480 + 1.52888i 1.26480 + 1.52888i
\(363\) 0 0
\(364\) 0 0
\(365\) 0.442031 + 0.321154i 0.442031 + 0.321154i
\(366\) 0 0
\(367\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(368\) 0 0
\(369\) 0.238883 1.25227i 0.238883 1.25227i
\(370\) −1.20251 + 0.308753i −1.20251 + 0.308753i
\(371\) 0 0
\(372\) 0 0
\(373\) 0.791759 0.313480i 0.791759 0.313480i 0.0627905 0.998027i \(-0.480000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.872746 + 0.819564i 0.872746 + 0.819564i
\(378\) 0 0
\(379\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 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
\(386\) −0.0800484 1.27233i −0.0800484 1.27233i
\(387\) 0 0
\(388\) 1.18532 + 1.43281i 1.18532 + 1.43281i
\(389\) −1.17950 + 1.10762i −1.17950 + 1.10762i −0.187381 + 0.982287i \(0.560000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.637424 + 0.770513i −0.637424 + 0.770513i
\(393\) 0 0
\(394\) 0.618034 0.618034
\(395\) 0 0
\(396\) 0 0
\(397\) −1.80113 + 0.713118i −1.80113 + 0.713118i −0.809017 + 0.587785i \(0.800000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.479249 + 1.01846i −0.479249 + 1.01846i
\(401\) 1.87631 + 0.481754i 1.87631 + 0.481754i 1.00000 \(0\)
0.876307 + 0.481754i \(0.160000\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.637424 0.770513i −0.637424 0.770513i
\(405\) 1.45794 1.45794
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.574221 0.904827i 0.574221 0.904827i −0.425779 0.904827i \(-0.640000\pi\)
1.00000 \(0\)
\(410\) −0.574354 1.76768i −0.574354 1.76768i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) −1.23480 + 1.49261i −1.23480 + 1.49261i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(420\) 0 0
\(421\) 0.0388067 0.119435i 0.0388067 0.119435i −0.929776 0.368125i \(-0.880000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) −0.996398 1.57007i −0.996398 1.57007i
\(425\) 1.80687 0.228260i 1.80687 0.228260i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(432\) 0 0
\(433\) −0.116762 0.0462295i −0.116762 0.0462295i 0.309017 0.951057i \(-0.400000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.331159 1.01920i 0.331159 1.01920i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(440\) 0 0
\(441\) 0.968583 0.248690i 0.968583 0.248690i
\(442\) 3.10969 + 0.392845i 3.10969 + 0.392845i
\(443\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(444\) 0 0
\(445\) 0.789600 + 2.43014i 0.789600 + 2.43014i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) 0.110048 0.0604991i 0.110048 0.0604991i −0.425779 0.904827i \(-0.640000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(450\) 0.986354 0.542253i 0.986354 0.542253i
\(451\) 0 0
\(452\) −0.456288 0.969661i −0.456288 0.969661i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −0.824805 1.75280i −0.824805 1.75280i −0.637424 0.770513i \(-0.720000\pi\)
−0.187381 0.982287i \(-0.560000\pi\)
\(458\) 0.348445 1.82662i 0.348445 1.82662i
\(459\) 0 0
\(460\) 0 0
\(461\) −1.41789 + 1.03016i −1.41789 + 1.03016i −0.425779 + 0.904827i \(0.640000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(462\) 0 0
\(463\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(464\) 0.190983 + 0.587785i 0.190983 + 0.587785i
\(465\) 0 0
\(466\) 1.60528 + 1.16630i 1.60528 + 1.16630i
\(467\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(468\) 1.87631 0.481754i 1.87631 0.481754i
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) −0.116762 + 1.85588i −0.116762 + 1.85588i
\(478\) 0 0
\(479\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(480\) 0 0
\(481\) 1.63660 + 0.206751i 1.63660 + 0.206751i
\(482\) 0.00788530 0.125333i 0.00788530 0.125333i
\(483\) 0 0
\(484\) −0.992115 + 0.125333i −0.992115 + 0.125333i
\(485\) −1.45269 2.28907i −1.45269 2.28907i
\(486\) 0 0
\(487\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(488\) −0.613161 + 1.88711i −0.613161 + 1.88711i
\(489\) 0 0
\(490\) 1.06279 0.998027i 1.06279 0.998027i
\(491\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(492\) 0 0
\(493\) 0.637424 0.770513i 0.637424 0.770513i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(500\) 0.0981041 0.154587i 0.0981041 0.154587i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(504\) 0 0
\(505\) 0.781202 + 1.23098i 0.781202 + 1.23098i
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) −0.263146 + 0.559214i −0.263146 + 0.559214i −0.992115 0.125333i \(-0.960000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −0.929776 + 0.368125i −0.929776 + 0.368125i
\(513\) 0 0
\(514\) −1.73879 0.955910i −1.73879 0.955910i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 2.05880 1.93334i 2.05880 1.93334i
\(521\) 1.26480 + 1.52888i 1.26480 + 1.52888i 0.728969 + 0.684547i \(0.240000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(522\) 0.190983 0.587785i 0.190983 0.587785i
\(523\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.992115 0.125333i −0.992115 0.125333i
\(530\) 1.15434 + 2.45309i 1.15434 + 2.45309i
\(531\) 0 0
\(532\) 0 0
\(533\) −0.155067 + 2.46472i −0.155067 + 2.46472i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0.0915446 + 0.0859661i 0.0915446 + 0.0859661i
\(539\) 0 0
\(540\) 0 0
\(541\) −1.41789 + 1.03016i −1.41789 + 1.03016i −0.425779 + 0.904827i \(0.640000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.30902 + 0.951057i 1.30902 + 0.951057i
\(545\) −0.665239 + 1.41371i −0.665239 + 1.41371i
\(546\) 0 0
\(547\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(548\) 0.0702235 + 0.368125i 0.0702235 + 0.368125i
\(549\) 1.60528 1.16630i 1.60528 1.16630i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −0.929324 + 0.872693i −0.929324 + 0.872693i
\(555\) 0 0
\(556\) 0 0
\(557\) −1.35556 + 1.27295i −1.35556 + 1.27295i −0.425779 + 0.904827i \(0.640000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 0.688925 0.500534i 0.688925 0.500534i
\(563\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(564\) 0 0
\(565\) 0.482809 + 1.48593i 0.482809 + 1.48593i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 0.598617 0.153699i 0.598617 0.153699i 0.0627905 0.998027i \(-0.480000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(570\) 0 0
\(571\) 0 0 0.187381 0.982287i \(-0.440000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.968583 + 0.248690i 0.968583 + 0.248690i
\(577\) 1.18532 + 0.469303i 1.18532 + 0.469303i 0.876307 0.481754i \(-0.160000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(578\) 0.101597 1.61484i 0.101597 1.61484i
\(579\) 0 0
\(580\) −0.168841 0.885095i −0.168841 0.885095i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) −0.0235315 0.374023i −0.0235315 0.374023i
\(585\) −2.80200 + 0.353975i −2.80200 + 0.353975i
\(586\) 0.939097 + 1.47978i 0.939097 + 1.47978i
\(587\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 0.688925 + 0.500534i 0.688925 + 0.500534i
\(593\) −0.116762 0.0462295i −0.116762 0.0462295i 0.309017 0.951057i \(-0.400000\pi\)
−0.425779 + 0.904827i \(0.640000\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.374763 −0.374763
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(600\) 0 0
\(601\) −0.866986 + 1.36615i −0.866986 + 1.36615i 0.0627905 + 0.998027i \(0.480000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 1.45794 1.45794
\(606\) 0 0
\(607\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 1.23173 2.61756i 1.23173 2.61756i
\(611\) 0 0
\(612\) −0.500000 1.53884i −0.500000 1.53884i
\(613\) −0.116762 + 0.0462295i −0.116762 + 0.0462295i −0.425779 0.904827i \(-0.640000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −1.62954 0.895846i −1.62954 0.895846i −0.992115 0.125333i \(-0.960000\pi\)
−0.637424 0.770513i \(-0.720000\pi\)
\(618\) 0 0
\(619\) 0 0 −0.929776 0.368125i \(-0.880000\pi\)
0.929776 + 0.368125i \(0.120000\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.547323 0.661600i 0.547323 0.661600i
\(626\) −0.866986 1.36615i −0.866986 1.36615i
\(627\) 0 0
\(628\) 0.0672897 + 1.06954i 0.0672897 + 1.06954i
\(629\) 0.0865160 1.37513i 0.0865160 1.37513i
\(630\) 0 0
\(631\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.110048 1.74915i 0.110048 1.74915i
\(635\) 0 0
\(636\) 0 0
\(637\) −1.80113 + 0.713118i −1.80113 + 0.713118i
\(638\) 0 0
\(639\) 0 0
\(640\) 1.41213 0.362574i 1.41213 0.362574i
\(641\) 0.371808 1.94908i 0.371808 1.94908i 0.0627905 0.998027i \(-0.480000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(642\) 0 0
\(643\) 0 0 0.968583 0.248690i \(-0.0800000\pi\)
−0.968583 + 0.248690i \(0.920000\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.309017 0.951057i \(-0.600000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(648\) −0.637424 0.770513i −0.637424 0.770513i
\(649\) 0 0
\(650\) −1.76401 + 1.28163i −1.76401 + 1.28163i
\(651\) 0 0
\(652\) 0 0
\(653\) 0.303189 1.58937i 0.303189 1.58937i −0.425779 0.904827i \(-0.640000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.683098 + 1.07639i −0.683098 + 1.07639i
\(657\) −0.200808 + 0.316423i −0.200808 + 0.316423i
\(658\) 0 0
\(659\) 0 0 −0.425779 0.904827i \(-0.640000\pi\)
0.425779 + 0.904827i \(0.360000\pi\)
\(660\) 0 0
\(661\) −1.11716 + 0.614163i −1.11716 + 0.614163i −0.929776 0.368125i \(-0.880000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −0.263146 0.809880i −0.263146 0.809880i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.824805 + 0.211774i −0.824805 + 0.211774i −0.637424 0.770513i \(-0.720000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(674\) −0.273190 0.256543i −0.273190 0.256543i
\(675\) 0 0
\(676\) −2.55932 + 1.01330i −2.55932 + 1.01330i
\(677\) 0.121636 + 0.0312307i 0.121636 + 0.0312307i 0.309017 0.951057i \(-0.400000\pi\)
−0.187381 + 0.982287i \(0.560000\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) −1.71963 1.61484i −1.71963 1.61484i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.992115 0.125333i \(-0.960000\pi\)
0.992115 + 0.125333i \(0.0400000\pi\)
\(684\) 0 0
\(685\) −0.0343075 0.545302i −0.0343075 0.545302i
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) −0.226188 3.59516i −0.226188 3.59516i
\(690\) 0 0
\(691\) 0 0 −0.637424 0.770513i \(-0.720000\pi\)
0.637424 + 0.770513i \(0.280000\pi\)
\(692\) −0.620759 + 0.582932i −0.620759 + 0.582932i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 2.06275 2.06275
\(698\) −0.746226 0.410241i −0.746226 0.410241i
\(699\) 0 0
\(700\) 0 0
\(701\) 0.331159 + 1.01920i 0.331159 + 1.01920i 0.968583 + 0.248690i \(0.0800000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) 0.618034 0.618034
\(707\) 0 0
\(708\) 0 0
\(709\) 0.939097 + 1.47978i 0.939097 + 1.47978i 0.876307 + 0.481754i \(0.160000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) 0.939097 1.47978i 0.939097 1.47978i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(720\) −1.35556 0.536702i −1.35556 0.536702i
\(721\) 0 0
\(722\) 0.728969 0.684547i 0.728969 0.684547i
\(723\) 0 0
\(724\) −0.613161 + 1.88711i −0.613161 + 1.88711i
\(725\) 0.0436801 + 0.694275i 0.0436801 + 0.694275i
\(726\) 0 0
\(727\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(728\) 0 0
\(729\) 0.0627905 + 0.998027i 0.0627905 + 0.998027i
\(730\) −0.0343075 + 0.545302i −0.0343075 + 0.545302i
\(731\) 0 0
\(732\) 0 0
\(733\) −0.200808 1.05267i −0.200808 1.05267i −0.929776 0.368125i \(-0.880000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 1.18532 0.469303i 1.18532 0.469303i
\(739\) 0 0 0.309017 0.951057i \(-0.400000\pi\)
−0.309017 + 0.951057i \(0.600000\pi\)
\(740\) −0.905028 0.849878i −0.905028 0.849878i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(744\) 0 0
\(745\) 0.542072 + 0.0684796i 0.542072 + 0.0684796i
\(746\) 0.688925 + 0.500534i 0.688925 + 0.500534i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 0.809017 0.587785i \(-0.200000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) −0.224339 + 1.17603i −0.224339 + 1.17603i
\(755\) 0 0
\(756\) 0 0
\(757\) 0.939097 1.47978i 0.939097 1.47978i 0.0627905 0.998027i \(-0.480000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0.159566 0.836475i 0.159566 0.836475i −0.809017 0.587785i \(-0.800000\pi\)
0.968583 0.248690i \(-0.0800000\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) 0.442031 + 2.31721i 0.442031 + 2.31721i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) −0.500000 0.363271i −0.500000 0.363271i 0.309017 0.951057i \(-0.400000\pi\)
−0.809017 + 0.587785i \(0.800000\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 1.03137 0.749337i 1.03137 0.749337i
\(773\) 0.0702235 0.368125i 0.0702235 0.368125i −0.929776 0.368125i \(-0.880000\pi\)
1.00000 \(0\)
\(774\) 0 0
\(775\) 0 0
\(776\) −0.574633 + 1.76854i −0.574633 + 1.76854i
\(777\) 0 0
\(778\) −1.56720 0.402389i −1.56720 0.402389i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.992115 0.125333i −0.992115 0.125333i
\(785\) 0.0981041 1.55932i 0.0981041 1.55932i
\(786\) 0 0
\(787\) 0 0 0.992115 0.125333i \(-0.0400000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(788\) 0.331159 + 0.521823i 0.331159 + 0.521823i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −2.80200 + 2.63125i −2.80200 + 2.63125i
\(794\) −1.56720 1.13864i −1.56720 1.13864i
\(795\) 0 0
\(796\) 0 0
\(797\) −1.73879 0.955910i −1.73879 0.955910i −0.929776 0.368125i \(-0.880000\pi\)
−0.809017 0.587785i \(-0.800000\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) −1.11671 + 0.141073i −1.11671 + 0.141073i
\(801\) −1.62954 + 0.645180i −1.62954 + 0.645180i
\(802\) 0.598617 + 1.84235i 0.598617 + 1.84235i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0.309017 0.951057i 0.309017 0.951057i
\(809\) −1.98423 −1.98423 −0.992115 0.125333i \(-0.960000\pi\)
−0.992115 + 0.125333i \(0.960000\pi\)
\(810\) 0.781202 + 1.23098i 0.781202 + 1.23098i
\(811\) 0 0 −0.968583 0.248690i \(-0.920000\pi\)
0.968583 + 0.248690i \(0.0800000\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 1.07165 1.07165
\(819\) 0 0
\(820\) 1.18475 1.43211i 1.18475 1.43211i
\(821\) 1.18532 + 0.469303i 1.18532 + 0.469303i 0.876307 0.481754i \(-0.160000\pi\)
0.309017 + 0.951057i \(0.400000\pi\)
\(822\) 0 0
\(823\) 0 0 0.728969 0.684547i \(-0.240000\pi\)
−0.728969 + 0.684547i \(0.760000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(828\) 0 0
\(829\) −1.92189 + 0.242791i −1.92189 + 0.242791i −0.992115 0.125333i \(-0.960000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −1.92189 0.242791i −1.92189 0.242791i
\(833\) 0.688925 + 1.46404i 0.688925 + 1.46404i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 0.929776 0.368125i \(-0.120000\pi\)
−0.929776 + 0.368125i \(0.880000\pi\)
\(840\) 0 0
\(841\) −0.450527 0.423073i −0.450527 0.423073i
\(842\) 0.121636 0.0312307i 0.121636 0.0312307i
\(843\) 0 0
\(844\) 0 0
\(845\) 3.88706 0.998027i 3.88706 0.998027i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.791759 1.68257i 0.791759 1.68257i
\(849\) 0 0
\(850\) 1.16089 + 1.40328i 1.16089 + 1.40328i
\(851\) 0 0
\(852\) 0 0
\(853\) 1.53583 0.844328i 1.53583 0.844328i 0.535827 0.844328i \(-0.320000\pi\)
1.00000 \(0\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.273190 + 0.256543i −0.273190 + 0.256543i −0.809017 0.587785i \(-0.800000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(858\) 0 0
\(859\) 0 0 0.535827 0.844328i \(-0.320000\pi\)
−0.535827 + 0.844328i \(0.680000\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.876307 0.481754i \(-0.160000\pi\)
−0.876307 + 0.481754i \(0.840000\pi\)
\(864\) 0 0
\(865\) 1.00441 0.729747i 1.00441 0.729747i
\(866\) −0.0235315 0.123357i −0.0235315 0.123357i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) 1.03799 0.266509i 1.03799 0.266509i
\(873\) 1.50441 1.09302i 1.50441 1.09302i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.541587 1.66683i 0.541587 1.66683i −0.187381 0.982287i \(-0.560000\pi\)
0.728969 0.684547i \(-0.240000\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0.0672897 1.06954i 0.0672897 1.06954i −0.809017 0.587785i \(-0.800000\pi\)
0.876307 0.481754i \(-0.160000\pi\)
\(882\) 0.728969 + 0.684547i 0.728969 + 0.684547i
\(883\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(884\) 1.33456 + 2.83609i 1.33456 + 2.83609i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.0627905 0.998027i \(-0.520000\pi\)
0.0627905 + 0.998027i \(0.480000\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −1.62875 + 1.96882i −1.62875 + 1.96882i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0.110048 + 0.0604991i 0.110048 + 0.0604991i
\(899\) 0 0
\(900\) 0.986354 + 0.542253i 0.986354 + 0.542253i
\(901\) −2.98509 + 0.377105i −2.98509 + 0.377105i
\(902\) 0 0
\(903\) 0 0
\(904\) 0.574221 0.904827i 0.574221 0.904827i
\(905\) 1.23173 2.61756i 1.23173 2.61756i
\(906\) 0 0
\(907\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(908\) 0 0
\(909\) −0.809017 + 0.587785i −0.809017 + 0.587785i
\(910\) 0 0
\(911\) 0 0 −0.535827 0.844328i \(-0.680000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 1.03799 1.63560i 1.03799 1.63560i
\(915\) 0 0
\(916\) 1.72897 0.684547i 1.72897 0.684547i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −1.62954 0.645180i −1.62954 0.645180i
\(923\) 0 0
\(924\) 0 0
\(925\) 0.610970 + 0.738536i 0.610970 + 0.738536i
\(926\) 0 0
\(927\) 0 0
\(928\) −0.393950 + 0.476203i −0.393950 + 0.476203i
\(929\) 0.331159 + 0.521823i 0.331159 + 0.521823i 0.968583 0.248690i \(-0.0800000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) −0.124591 + 1.98031i −0.124591 + 1.98031i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 1.41213 + 1.32608i 1.41213 + 1.32608i
\(937\) 0.121636 1.93334i 0.121636 1.93334i −0.187381 0.982287i \(-0.560000\pi\)
0.309017 0.951057i \(-0.400000\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) −0.115808 + 0.356420i −0.115808 + 0.356420i −0.992115 0.125333i \(-0.960000\pi\)
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 \(0.0400000\pi\)
\(948\) 0 0
\(949\) 0.309106 0.656884i 0.309106 0.656884i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 1.30902 0.951057i 1.30902 0.951057i 0.309017 0.951057i \(-0.400000\pi\)
1.00000 \(0\)
\(954\) −1.62954 + 0.895846i −1.62954 + 0.895846i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.728969 0.684547i 0.728969 0.684547i
\(962\) 0.702370 + 1.49261i 0.702370 + 1.49261i
\(963\) 0 0
\(964\) 0.110048 0.0604991i 0.110048 0.0604991i
\(965\) −1.62875 + 0.895411i −1.62875 + 0.895411i
\(966\) 0 0
\(967\) 0 0 −0.187381 0.982287i \(-0.560000\pi\)
0.187381 + 0.982287i \(0.440000\pi\)
\(968\) −0.637424 0.770513i −0.637424 0.770513i
\(969\) 0 0
\(970\) 1.15434 2.45309i 1.15434 2.45309i
\(971\) 0 0 −0.809017 0.587785i \(-0.800000\pi\)
0.809017 + 0.587785i \(0.200000\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) −1.92189 + 0.493458i −1.92189 + 0.493458i
\(977\) 1.41213 + 1.32608i 1.41213 + 1.32608i 0.876307 + 0.481754i \(0.160000\pi\)
0.535827 + 0.844328i \(0.320000\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 1.41213 + 0.362574i 1.41213 + 0.362574i
\(981\) −0.996398 0.394502i −0.996398 0.394502i
\(982\) 0 0
\(983\) 0 0 −0.728969 0.684547i \(-0.760000\pi\)
0.728969 + 0.684547i \(0.240000\pi\)
\(984\) 0 0
\(985\) −0.383650 0.815299i −0.383650 0.815299i
\(986\) 0.992115 + 0.125333i 0.992115 + 0.125333i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 0.637424 0.770513i \(-0.280000\pi\)
−0.637424 + 0.770513i \(0.720000\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.72897 + 0.684547i 1.72897 + 0.684547i 1.00000 \(0\)
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 404.1.o.a.371.1 yes 20
3.2 odd 2 3636.1.bt.a.775.1 20
4.3 odd 2 CM 404.1.o.a.371.1 yes 20
12.11 even 2 3636.1.bt.a.775.1 20
101.52 even 25 inner 404.1.o.a.355.1 20
303.254 odd 50 3636.1.bt.a.1567.1 20
404.355 odd 50 inner 404.1.o.a.355.1 20
1212.1163 even 50 3636.1.bt.a.1567.1 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
404.1.o.a.355.1 20 101.52 even 25 inner
404.1.o.a.355.1 20 404.355 odd 50 inner
404.1.o.a.371.1 yes 20 1.1 even 1 trivial
404.1.o.a.371.1 yes 20 4.3 odd 2 CM
3636.1.bt.a.775.1 20 3.2 odd 2
3636.1.bt.a.775.1 20 12.11 even 2
3636.1.bt.a.1567.1 20 303.254 odd 50
3636.1.bt.a.1567.1 20 1212.1163 even 50