Properties

Label 3380.1.bz.a.3047.1
Level $3380$
Weight $1$
Character 3380.3047
Analytic conductor $1.687$
Analytic rank $0$
Dimension $24$
Projective image $D_{52}$
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] = [3380,1,Mod(187,3380)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3380, base_ring=CyclotomicField(52))
 
chi = DirichletCharacter(H, H._module([26, 13, 31]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3380.187");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3380 = 2^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3380.bz (of order \(52\), degree \(24\), 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: \(1.68683974270\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{52})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{22} + x^{20} - x^{18} + x^{16} - x^{14} + x^{12} - x^{10} + x^{8} - x^{6} + x^{4} - x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{52}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{52} + \cdots)\)

Embedding invariants

Embedding label 3047.1
Root \(-0.663123 - 0.748511i\) of defining polynomial
Character \(\chi\) \(=\) 3380.3047
Dual form 3380.1.bz.a.203.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.568065 - 0.822984i) q^{2} +(-0.354605 + 0.935016i) q^{4} +(0.992709 - 0.120537i) q^{5} +(0.970942 - 0.239316i) q^{8} +(0.822984 - 0.568065i) q^{9} +O(q^{10})\) \(q+(-0.568065 - 0.822984i) q^{2} +(-0.354605 + 0.935016i) q^{4} +(0.992709 - 0.120537i) q^{5} +(0.970942 - 0.239316i) q^{8} +(0.822984 - 0.568065i) q^{9} +(-0.663123 - 0.748511i) q^{10} +(0.239316 - 0.970942i) q^{13} +(-0.748511 - 0.663123i) q^{16} +(0.943521 - 1.56077i) q^{17} +(-0.935016 - 0.354605i) q^{18} +(-0.239316 + 0.970942i) q^{20} +(0.970942 - 0.239316i) q^{25} +(-0.935016 + 0.354605i) q^{26} +(-1.45743 + 1.00599i) q^{29} +(-0.120537 + 0.992709i) q^{32} +(-1.82047 + 0.110118i) q^{34} +(0.239316 + 0.970942i) q^{36} +(-1.31658 + 0.159861i) q^{37} +(0.935016 - 0.354605i) q^{40} +(0.344186 - 0.107253i) q^{41} +(0.748511 - 0.663123i) q^{45} +(-0.885456 + 0.464723i) q^{49} +(-0.748511 - 0.663123i) q^{50} +(0.822984 + 0.568065i) q^{52} +(-0.535277 + 0.885456i) q^{53} +(1.65583 + 0.627974i) q^{58} +(-0.902438 - 0.222431i) q^{61} +(0.885456 - 0.464723i) q^{64} +(0.120537 - 0.992709i) q^{65} +(1.12477 + 1.43566i) q^{68} +(0.663123 - 0.748511i) q^{72} +(1.13613 - 1.64597i) q^{73} +(0.879463 + 0.992709i) q^{74} +(-0.822984 - 0.568065i) q^{80} +(0.354605 - 0.935016i) q^{81} +(-0.283788 - 0.222333i) q^{82} +(0.748511 - 1.66312i) q^{85} +(-1.41163 + 1.41163i) q^{89} +(-0.970942 - 0.239316i) q^{90} +(1.48611 + 1.31658i) q^{97} +(0.885456 + 0.464723i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{2} - 2 q^{4} + 2 q^{8}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 2 q^{2} - 2 q^{4} + 2 q^{8} - 2 q^{16} - 2 q^{17} + 2 q^{25} + 2 q^{32} + 2 q^{34} + 2 q^{41} + 2 q^{45} + 2 q^{49} - 2 q^{50} - 24 q^{53} - 2 q^{64} - 2 q^{65} - 2 q^{68} - 4 q^{73} + 26 q^{74} + 2 q^{81} - 2 q^{82} + 2 q^{85} - 2 q^{89} - 2 q^{90} - 2 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3380\mathbb{Z}\right)^\times\).

\(n\) \(677\) \(1691\) \(1861\)
\(\chi(n)\) \(e\left(\frac{1}{4}\right)\) \(-1\) \(e\left(\frac{3}{52}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.568065 0.822984i −0.568065 0.822984i
\(3\) 0 0 0.954721 0.297503i \(-0.0961538\pi\)
−0.954721 + 0.297503i \(0.903846\pi\)
\(4\) −0.354605 + 0.935016i −0.354605 + 0.935016i
\(5\) 0.992709 0.120537i 0.992709 0.120537i
\(6\) 0 0
\(7\) 0 0 −0.239316 0.970942i \(-0.576923\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(8\) 0.970942 0.239316i 0.970942 0.239316i
\(9\) 0.822984 0.568065i 0.822984 0.568065i
\(10\) −0.663123 0.748511i −0.663123 0.748511i
\(11\) 0 0 0.983620 0.180255i \(-0.0576923\pi\)
−0.983620 + 0.180255i \(0.942308\pi\)
\(12\) 0 0
\(13\) 0.239316 0.970942i 0.239316 0.970942i
\(14\) 0 0
\(15\) 0 0
\(16\) −0.748511 0.663123i −0.748511 0.663123i
\(17\) 0.943521 1.56077i 0.943521 1.56077i 0.120537 0.992709i \(-0.461538\pi\)
0.822984 0.568065i \(-0.192308\pi\)
\(18\) −0.935016 0.354605i −0.935016 0.354605i
\(19\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(20\) −0.239316 + 0.970942i −0.239316 + 0.970942i
\(21\) 0 0
\(22\) 0 0
\(23\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(24\) 0 0
\(25\) 0.970942 0.239316i 0.970942 0.239316i
\(26\) −0.935016 + 0.354605i −0.935016 + 0.354605i
\(27\) 0 0
\(28\) 0 0
\(29\) −1.45743 + 1.00599i −1.45743 + 1.00599i −0.464723 + 0.885456i \(0.653846\pi\)
−0.992709 + 0.120537i \(0.961538\pi\)
\(30\) 0 0
\(31\) 0 0 −0.787183 0.616719i \(-0.788462\pi\)
0.787183 + 0.616719i \(0.211538\pi\)
\(32\) −0.120537 + 0.992709i −0.120537 + 0.992709i
\(33\) 0 0
\(34\) −1.82047 + 0.110118i −1.82047 + 0.110118i
\(35\) 0 0
\(36\) 0.239316 + 0.970942i 0.239316 + 0.970942i
\(37\) −1.31658 + 0.159861i −1.31658 + 0.159861i −0.748511 0.663123i \(-0.769231\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0.935016 0.354605i 0.935016 0.354605i
\(41\) 0.344186 0.107253i 0.344186 0.107253i −0.120537 0.992709i \(-0.538462\pi\)
0.464723 + 0.885456i \(0.346154\pi\)
\(42\) 0 0
\(43\) 0 0 0.787183 0.616719i \(-0.211538\pi\)
−0.787183 + 0.616719i \(0.788462\pi\)
\(44\) 0 0
\(45\) 0.748511 0.663123i 0.748511 0.663123i
\(46\) 0 0
\(47\) 0 0 0.935016 0.354605i \(-0.115385\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(48\) 0 0
\(49\) −0.885456 + 0.464723i −0.885456 + 0.464723i
\(50\) −0.748511 0.663123i −0.748511 0.663123i
\(51\) 0 0
\(52\) 0.822984 + 0.568065i 0.822984 + 0.568065i
\(53\) −0.535277 + 0.885456i −0.535277 + 0.885456i 0.464723 + 0.885456i \(0.346154\pi\)
−1.00000 \(\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) 1.65583 + 0.627974i 1.65583 + 0.627974i
\(59\) 0 0 0.0603785 0.998176i \(-0.480769\pi\)
−0.0603785 + 0.998176i \(0.519231\pi\)
\(60\) 0 0
\(61\) −0.902438 0.222431i −0.902438 0.222431i −0.239316 0.970942i \(-0.576923\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0.885456 0.464723i 0.885456 0.464723i
\(65\) 0.120537 0.992709i 0.120537 0.992709i
\(66\) 0 0
\(67\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(68\) 1.12477 + 1.43566i 1.12477 + 1.43566i
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.954721 0.297503i \(-0.0961538\pi\)
−0.954721 + 0.297503i \(0.903846\pi\)
\(72\) 0.663123 0.748511i 0.663123 0.748511i
\(73\) 1.13613 1.64597i 1.13613 1.64597i 0.568065 0.822984i \(-0.307692\pi\)
0.568065 0.822984i \(-0.307692\pi\)
\(74\) 0.879463 + 0.992709i 0.879463 + 0.992709i
\(75\) 0 0
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) 0 0 0.935016 0.354605i \(-0.115385\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(80\) −0.822984 0.568065i −0.822984 0.568065i
\(81\) 0.354605 0.935016i 0.354605 0.935016i
\(82\) −0.283788 0.222333i −0.283788 0.222333i
\(83\) 0 0 −0.464723 0.885456i \(-0.653846\pi\)
0.464723 + 0.885456i \(0.346154\pi\)
\(84\) 0 0
\(85\) 0.748511 1.66312i 0.748511 1.66312i
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −1.41163 + 1.41163i −1.41163 + 1.41163i −0.663123 + 0.748511i \(0.730769\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(90\) −0.970942 0.239316i −0.970942 0.239316i
\(91\) 0 0
\(92\) 0 0
\(93\) 0 0
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 1.48611 + 1.31658i 1.48611 + 1.31658i 0.822984 + 0.568065i \(0.192308\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(98\) 0.885456 + 0.464723i 0.885456 + 0.464723i
\(99\) 0 0
\(100\) −0.120537 + 0.992709i −0.120537 + 0.992709i
\(101\) 1.97094 + 0.239316i 1.97094 + 0.239316i 1.00000 \(0\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(102\) 0 0
\(103\) 0 0 −0.297503 0.954721i \(-0.596154\pi\)
0.297503 + 0.954721i \(0.403846\pi\)
\(104\) 1.00000i 1.00000i
\(105\) 0 0
\(106\) 1.03279 0.0624722i 1.03279 0.0624722i
\(107\) 0 0 −0.998176 0.0603785i \(-0.980769\pi\)
0.998176 + 0.0603785i \(0.0192308\pi\)
\(108\) 0 0
\(109\) −0.646140 + 0.506219i −0.646140 + 0.506219i −0.885456 0.464723i \(-0.846154\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) 0 0
\(113\) −0.0359256 + 0.115289i −0.0359256 + 0.115289i −0.970942 0.239316i \(-0.923077\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −0.423807 1.71945i −0.423807 1.71945i
\(117\) −0.354605 0.935016i −0.354605 0.935016i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.935016 0.354605i 0.935016 0.354605i
\(122\) 0.329586 + 0.869047i 0.329586 + 0.869047i
\(123\) 0 0
\(124\) 0 0
\(125\) 0.935016 0.354605i 0.935016 0.354605i
\(126\) 0 0
\(127\) 0 0 −0.911900 0.410413i \(-0.865385\pi\)
0.911900 + 0.410413i \(0.134615\pi\)
\(128\) −0.885456 0.464723i −0.885456 0.464723i
\(129\) 0 0
\(130\) −0.885456 + 0.464723i −0.885456 + 0.464723i
\(131\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 0 0
\(135\) 0 0
\(136\) 0.542586 1.74122i 0.542586 1.74122i
\(137\) −1.48611 0.180446i −1.48611 0.180446i −0.663123 0.748511i \(-0.730769\pi\)
−0.822984 + 0.568065i \(0.807692\pi\)
\(138\) 0 0
\(139\) 0 0 0.663123 0.748511i \(-0.269231\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.992709 0.120537i −0.992709 0.120537i
\(145\) −1.32555 + 1.17433i −1.32555 + 1.17433i
\(146\) −2.00000 −2.00000
\(147\) 0 0
\(148\) 0.317391 1.28771i 0.317391 1.28771i
\(149\) −0.506219 + 1.12477i −0.506219 + 1.12477i 0.464723 + 0.885456i \(0.346154\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(150\) 0 0
\(151\) 0 0 0.911900 0.410413i \(-0.134615\pi\)
−0.911900 + 0.410413i \(0.865385\pi\)
\(152\) 0 0
\(153\) −0.110118 1.82047i −0.110118 1.82047i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) 1.57149 + 0.0950579i 1.57149 + 0.0950579i 0.822984 0.568065i \(-0.192308\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 1.00000i 1.00000i
\(161\) 0 0
\(162\) −0.970942 + 0.239316i −0.970942 + 0.239316i
\(163\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(164\) −0.0217671 + 0.359852i −0.0217671 + 0.359852i
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 0.822984 0.568065i \(-0.192308\pi\)
−0.822984 + 0.568065i \(0.807692\pi\)
\(168\) 0 0
\(169\) −0.885456 0.464723i −0.885456 0.464723i
\(170\) −1.79393 + 0.328749i −1.79393 + 0.328749i
\(171\) 0 0
\(172\) 0 0
\(173\) 1.74122 0.783659i 1.74122 0.783659i 0.748511 0.663123i \(-0.230769\pi\)
0.992709 0.120537i \(-0.0384615\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) 0 0
\(178\) 1.96365 + 0.359852i 1.96365 + 0.359852i
\(179\) 0 0 −0.970942 0.239316i \(-0.923077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(180\) 0.354605 + 0.935016i 0.354605 + 0.935016i
\(181\) −1.28771 1.45352i −1.28771 1.45352i −0.822984 0.568065i \(-0.807692\pi\)
−0.464723 0.885456i \(-0.653846\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) −1.28771 + 0.317391i −1.28771 + 0.317391i
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0 0
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 0 0
\(193\) 1.92773 + 0.475142i 1.92773 + 0.475142i 0.992709 + 0.120537i \(0.0384615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(194\) 0.239316 1.97094i 0.239316 1.97094i
\(195\) 0 0
\(196\) −0.120537 0.992709i −0.120537 0.992709i
\(197\) −0.112032 + 0.922670i −0.112032 + 0.922670i 0.822984 + 0.568065i \(0.192308\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(198\) 0 0
\(199\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(200\) 0.885456 0.464723i 0.885456 0.464723i
\(201\) 0 0
\(202\) −0.922670 1.75800i −0.922670 1.75800i
\(203\) 0 0
\(204\) 0 0
\(205\) 0.328749 0.147958i 0.328749 0.147958i
\(206\) 0 0
\(207\) 0 0
\(208\) −0.822984 + 0.568065i −0.822984 + 0.568065i
\(209\) 0 0
\(210\) 0 0
\(211\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(212\) −0.638104 0.814480i −0.638104 0.814480i
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 0 0
\(218\) 0.783659 + 0.244198i 0.783659 + 0.244198i
\(219\) 0 0
\(220\) 0 0
\(221\) −1.28962 1.28962i −1.28962 1.28962i
\(222\) 0 0
\(223\) 0 0 −0.663123 0.748511i \(-0.730769\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(224\) 0 0
\(225\) 0.663123 0.748511i 0.663123 0.748511i
\(226\) 0.115289 0.0359256i 0.115289 0.0359256i
\(227\) 0 0 0.354605 0.935016i \(-0.384615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(228\) 0 0
\(229\) 1.05555 + 1.34731i 1.05555 + 1.34731i 0.935016 + 0.354605i \(0.115385\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) −1.17433 + 1.32555i −1.17433 + 1.32555i
\(233\) −1.50308 0.468379i −1.50308 0.468379i −0.568065 0.822984i \(-0.692308\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(234\) −0.568065 + 0.822984i −0.568065 + 0.822984i
\(235\) 0 0
\(236\) 0 0
\(237\) 0 0
\(238\) 0 0
\(239\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(240\) 0 0
\(241\) 0.115289 + 1.90596i 0.115289 + 1.90596i 0.354605 + 0.935016i \(0.384615\pi\)
−0.239316 + 0.970942i \(0.576923\pi\)
\(242\) −0.822984 0.568065i −0.822984 0.568065i
\(243\) 0 0
\(244\) 0.527986 0.764919i 0.527986 0.764919i
\(245\) −0.822984 + 0.568065i −0.822984 + 0.568065i
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) −0.822984 0.568065i −0.822984 0.568065i
\(251\) 0 0 0.992709 0.120537i \(-0.0384615\pi\)
−0.992709 + 0.120537i \(0.961538\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) 0.120537 + 0.992709i 0.120537 + 0.992709i
\(257\) −0.424644 0.943521i −0.424644 0.943521i −0.992709 0.120537i \(-0.961538\pi\)
0.568065 0.822984i \(-0.307692\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) 0.885456 + 0.464723i 0.885456 + 0.464723i
\(261\) −0.627974 + 1.65583i −0.627974 + 1.65583i
\(262\) 0 0
\(263\) 0 0 −0.180255 0.983620i \(-0.557692\pi\)
0.180255 + 0.983620i \(0.442308\pi\)
\(264\) 0 0
\(265\) −0.424644 + 0.943521i −0.424644 + 0.943521i
\(266\) 0 0
\(267\) 0 0
\(268\) 0 0
\(269\) −0.616337 + 1.17433i −0.616337 + 1.17433i 0.354605 + 0.935016i \(0.384615\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(270\) 0 0
\(271\) 0 0 0.410413 0.911900i \(-0.365385\pi\)
−0.410413 + 0.911900i \(0.634615\pi\)
\(272\) −1.74122 + 0.542586i −1.74122 + 0.542586i
\(273\) 0 0
\(274\) 0.695701 + 1.32555i 0.695701 + 1.32555i
\(275\) 0 0
\(276\) 0 0
\(277\) 1.21026 0.731626i 1.21026 0.731626i 0.239316 0.970942i \(-0.423077\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(278\) 0 0
\(279\) 0 0
\(280\) 0 0
\(281\) 0.509195 + 1.63406i 0.509195 + 1.63406i 0.748511 + 0.663123i \(0.230769\pi\)
−0.239316 + 0.970942i \(0.576923\pi\)
\(282\) 0 0
\(283\) 0 0 0.616719 0.787183i \(-0.288462\pi\)
−0.616719 + 0.787183i \(0.711538\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0.464723 + 0.885456i 0.464723 + 0.885456i
\(289\) −1.08106 2.05979i −1.08106 2.05979i
\(290\) 1.71945 + 0.423807i 1.71945 + 0.423807i
\(291\) 0 0
\(292\) 1.13613 + 1.64597i 1.13613 + 1.64597i
\(293\) −0.234068 0.0576926i −0.234068 0.0576926i 0.120537 0.992709i \(-0.461538\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −1.24006 + 0.470293i −1.24006 + 0.470293i
\(297\) 0 0
\(298\) 1.21323 0.222333i 1.21323 0.222333i
\(299\) 0 0
\(300\) 0 0
\(301\) 0 0
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −0.922670 0.112032i −0.922670 0.112032i
\(306\) −1.43566 + 1.12477i −1.43566 + 1.12477i
\(307\) 0 0 0.992709 0.120537i \(-0.0384615\pi\)
−0.992709 + 0.120537i \(0.961538\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 0.992709 0.120537i \(-0.0384615\pi\)
−0.992709 + 0.120537i \(0.961538\pi\)
\(312\) 0 0
\(313\) −0.222333 0.283788i −0.222333 0.283788i 0.663123 0.748511i \(-0.269231\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(314\) −0.814480 1.34731i −0.814480 1.34731i
\(315\) 0 0
\(316\) 0 0
\(317\) −1.88546 0.464723i −1.88546 0.464723i −0.885456 0.464723i \(-0.846154\pi\)
−1.00000 \(\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0.822984 0.568065i 0.822984 0.568065i
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.748511 + 0.663123i 0.748511 + 0.663123i
\(325\) 1.00000i 1.00000i
\(326\) 0 0
\(327\) 0 0
\(328\) 0.308518 0.186505i 0.308518 0.186505i
\(329\) 0 0
\(330\) 0 0
\(331\) 0 0 0.517338 0.855781i \(-0.326923\pi\)
−0.517338 + 0.855781i \(0.673077\pi\)
\(332\) 0 0
\(333\) −0.992709 + 0.879463i −0.992709 + 0.879463i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 0.872172 0.872172i 0.872172 0.872172i −0.120537 0.992709i \(-0.538462\pi\)
0.992709 + 0.120537i \(0.0384615\pi\)
\(338\) 0.120537 + 0.992709i 0.120537 + 0.992709i
\(339\) 0 0
\(340\) 1.28962 + 1.28962i 1.28962 + 1.28962i
\(341\) 0 0
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) −1.63406 0.987826i −1.63406 0.987826i
\(347\) 0 0 0.180255 0.983620i \(-0.442308\pi\)
−0.180255 + 0.983620i \(0.557692\pi\)
\(348\) 0 0
\(349\) −0.0217671 0.118779i −0.0217671 0.118779i 0.970942 0.239316i \(-0.0769231\pi\)
−0.992709 + 0.120537i \(0.961538\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 0 0
\(353\) −0.114544 0.464723i −0.114544 0.464723i 0.885456 0.464723i \(-0.153846\pi\)
−1.00000 \(\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) −0.819328 1.82047i −0.819328 1.82047i
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 0.855781 0.517338i \(-0.173077\pi\)
−0.855781 + 0.517338i \(0.826923\pi\)
\(360\) 0.568065 0.822984i 0.568065 0.822984i
\(361\) 1.00000i 1.00000i
\(362\) −0.464723 + 1.88546i −0.464723 + 1.88546i
\(363\) 0 0
\(364\) 0 0
\(365\) 0.929446 1.77091i 0.929446 1.77091i
\(366\) 0 0
\(367\) 0 0 0.180255 0.983620i \(-0.442308\pi\)
−0.180255 + 0.983620i \(0.557692\pi\)
\(368\) 0 0
\(369\) 0.222333 0.283788i 0.222333 0.283788i
\(370\) 0.992709 + 0.879463i 0.992709 + 0.879463i
\(371\) 0 0
\(372\) 0 0
\(373\) 0.354605 + 1.93502i 0.354605 + 1.93502i 0.354605 + 0.935016i \(0.384615\pi\)
1.00000i \(0.500000\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0.627974 + 1.65583i 0.627974 + 1.65583i
\(378\) 0 0
\(379\) 0 0 −0.297503 0.954721i \(-0.596154\pi\)
0.297503 + 0.954721i \(0.403846\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) 0 0
\(383\) 0 0 0.239316 0.970942i \(-0.423077\pi\)
−0.239316 + 0.970942i \(0.576923\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −0.704039 1.85640i −0.704039 1.85640i
\(387\) 0 0
\(388\) −1.75800 + 0.922670i −1.75800 + 0.922670i
\(389\) 1.45352 + 0.358261i 1.45352 + 0.358261i 0.885456 0.464723i \(-0.153846\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) −0.748511 + 0.663123i −0.748511 + 0.663123i
\(393\) 0 0
\(394\) 0.822984 0.431935i 0.822984 0.431935i
\(395\) 0 0
\(396\) 0 0
\(397\) 1.09148 + 1.23202i 1.09148 + 1.23202i 0.970942 + 0.239316i \(0.0769231\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −0.885456 0.464723i −0.885456 0.464723i
\(401\) 0.0950579 + 1.57149i 0.0950579 + 1.57149i 0.663123 + 0.748511i \(0.269231\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −0.922670 + 1.75800i −0.922670 + 1.75800i
\(405\) 0.239316 0.970942i 0.239316 0.970942i
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 0.420733 + 1.35018i 0.420733 + 1.35018i 0.885456 + 0.464723i \(0.153846\pi\)
−0.464723 + 0.885456i \(0.653846\pi\)
\(410\) −0.308518 0.186505i −0.308518 0.186505i
\(411\) 0 0
\(412\) 0 0
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0.935016 + 0.354605i 0.935016 + 0.354605i
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 −0.935016 0.354605i \(-0.884615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(420\) 0 0
\(421\) 0.177016 0.568065i 0.177016 0.568065i −0.822984 0.568065i \(-0.807692\pi\)
1.00000 \(0\)
\(422\) 0 0
\(423\) 0 0
\(424\) −0.307819 + 0.987826i −0.307819 + 0.987826i
\(425\) 0.542586 1.74122i 0.542586 1.74122i
\(426\) 0 0
\(427\) 0 0
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.410413 0.911900i \(-0.634615\pi\)
0.410413 + 0.911900i \(0.365385\pi\)
\(432\) 0 0
\(433\) −0.120537 0.00729113i −0.120537 0.00729113i 1.00000i \(-0.5\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.244198 0.783659i −0.244198 0.783659i
\(437\) 0 0
\(438\) 0 0
\(439\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(440\) 0 0
\(441\) −0.464723 + 0.885456i −0.464723 + 0.885456i
\(442\) −0.328749 + 1.79393i −0.328749 + 1.79393i
\(443\) 0 0 0.297503 0.954721i \(-0.403846\pi\)
−0.297503 + 0.954721i \(0.596154\pi\)
\(444\) 0 0
\(445\) −1.23119 + 1.57149i −1.23119 + 1.57149i
\(446\) 0 0
\(447\) 0 0
\(448\) 0 0
\(449\) −0.807380 1.79393i −0.807380 1.79393i −0.568065 0.822984i \(-0.692308\pi\)
−0.239316 0.970942i \(-0.576923\pi\)
\(450\) −0.992709 0.120537i −0.992709 0.120537i
\(451\) 0 0
\(452\) −0.0950579 0.0744731i −0.0950579 0.0744731i
\(453\) 0 0
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.423807 0.222431i 0.423807 0.222431i −0.239316 0.970942i \(-0.576923\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(458\) 0.509195 1.63406i 0.509195 1.63406i
\(459\) 0 0
\(460\) 0 0
\(461\) −1.96365 0.359852i −1.96365 0.359852i −0.992709 0.120537i \(-0.961538\pi\)
−0.970942 0.239316i \(-0.923077\pi\)
\(462\) 0 0
\(463\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(464\) 1.75800 + 0.213460i 1.75800 + 0.213460i
\(465\) 0 0
\(466\) 0.468379 + 1.50308i 0.468379 + 1.50308i
\(467\) 0 0 −0.180255 0.983620i \(-0.557692\pi\)
0.180255 + 0.983620i \(0.442308\pi\)
\(468\) 1.00000 1.00000
\(469\) 0 0
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) 0.0624722 + 1.03279i 0.0624722 + 1.03279i
\(478\) 0 0
\(479\) 0 0 −0.616719 0.787183i \(-0.711538\pi\)
0.616719 + 0.787183i \(0.288462\pi\)
\(480\) 0 0
\(481\) −0.159861 + 1.31658i −0.159861 + 1.31658i
\(482\) 1.50308 1.17759i 1.50308 1.17759i
\(483\) 0 0
\(484\) 1.00000i 1.00000i
\(485\) 1.63397 + 1.12785i 1.63397 + 1.12785i
\(486\) 0 0
\(487\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(488\) −0.929446 −0.929446
\(489\) 0 0
\(490\) 0.935016 + 0.354605i 0.935016 + 0.354605i
\(491\) 0 0 0.663123 0.748511i \(-0.269231\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(492\) 0 0
\(493\) 0.195010 + 3.22390i 0.195010 + 3.22390i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 0 0
\(498\) 0 0
\(499\) 0 0 −0.855781 0.517338i \(-0.826923\pi\)
0.855781 + 0.517338i \(0.173077\pi\)
\(500\) 1.00000i 1.00000i
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.410413 0.911900i \(-0.634615\pi\)
0.410413 + 0.911900i \(0.365385\pi\)
\(504\) 0 0
\(505\) 1.98542 1.98542
\(506\) 0 0
\(507\) 0 0
\(508\) 0 0
\(509\) 0.344186 1.87816i 0.344186 1.87816i −0.120537 0.992709i \(-0.538462\pi\)
0.464723 0.885456i \(-0.346154\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) 0.748511 0.663123i 0.748511 0.663123i
\(513\) 0 0
\(514\) −0.535277 + 0.885456i −0.535277 + 0.885456i
\(515\) 0 0
\(516\) 0 0
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) −0.120537 0.992709i −0.120537 0.992709i
\(521\) 1.23202 + 1.09148i 1.23202 + 1.09148i 0.992709 + 0.120537i \(0.0384615\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(522\) 1.71945 0.423807i 1.71945 0.423807i
\(523\) 0 0 0.998176 0.0603785i \(-0.0192308\pi\)
−0.998176 + 0.0603785i \(0.980769\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000i 1.00000i
\(530\) 1.01773 0.186505i 1.01773 0.186505i
\(531\) 0 0
\(532\) 0 0
\(533\) −0.0217671 0.359852i −0.0217671 0.359852i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 1.31658 0.159861i 1.31658 0.159861i
\(539\) 0 0
\(540\) 0 0
\(541\) −0.103342 1.70844i −0.103342 1.70844i −0.568065 0.822984i \(-0.692308\pi\)
0.464723 0.885456i \(-0.346154\pi\)
\(542\) 0 0
\(543\) 0 0
\(544\) 1.43566 + 1.12477i 1.43566 + 1.12477i
\(545\) −0.580411 + 0.580411i −0.580411 + 0.580411i
\(546\) 0 0
\(547\) 0 0 0.983620 0.180255i \(-0.0576923\pi\)
−0.983620 + 0.180255i \(0.942308\pi\)
\(548\) 0.695701 1.32555i 0.695701 1.32555i
\(549\) −0.869047 + 0.329586i −0.869047 + 0.329586i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0 0
\(554\) −1.28962 0.580411i −1.28962 0.580411i
\(555\) 0 0
\(556\) 0 0
\(557\) 0 0 0.885456 0.464723i \(-0.153846\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) 0 0
\(561\) 0 0
\(562\) 1.05555 1.34731i 1.05555 1.34731i
\(563\) 0 0 −0.954721 0.297503i \(-0.903846\pi\)
0.954721 + 0.297503i \(0.0961538\pi\)
\(564\) 0 0
\(565\) −0.0217671 + 0.118779i −0.0217671 + 0.118779i
\(566\) 0 0
\(567\) 0 0
\(568\) 0 0
\(569\) 1.39974 1.24006i 1.39974 1.24006i 0.464723 0.885456i \(-0.346154\pi\)
0.935016 0.354605i \(-0.115385\pi\)
\(570\) 0 0
\(571\) 0 0 −0.464723 0.885456i \(-0.653846\pi\)
0.464723 + 0.885456i \(0.346154\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.464723 0.885456i 0.464723 0.885456i
\(577\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(578\) −1.08106 + 2.05979i −1.08106 + 2.05979i
\(579\) 0 0
\(580\) −0.627974 1.65583i −0.627974 1.65583i
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0.709210 1.87003i 0.709210 1.87003i
\(585\) −0.464723 0.885456i −0.464723 0.885456i
\(586\) 0.0854858 + 0.225408i 0.0854858 + 0.225408i
\(587\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) 1.09148 + 0.753393i 1.09148 + 0.753393i
\(593\) −1.65583 + 0.869047i −1.65583 + 0.869047i −0.663123 + 0.748511i \(0.730769\pi\)
−0.992709 + 0.120537i \(0.961538\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −0.872172 0.872172i −0.872172 0.872172i
\(597\) 0 0
\(598\) 0 0
\(599\) 0 0 −0.935016 0.354605i \(-0.884615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(600\) 0 0
\(601\) 1.00599 1.45743i 1.00599 1.45743i 0.120537 0.992709i \(-0.461538\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 0 0
\(605\) 0.885456 0.464723i 0.885456 0.464723i
\(606\) 0 0
\(607\) 0 0 −0.297503 0.954721i \(-0.596154\pi\)
0.297503 + 0.954721i \(0.403846\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0.431935 + 0.822984i 0.431935 + 0.822984i
\(611\) 0 0
\(612\) 1.74122 + 0.542586i 1.74122 + 0.542586i
\(613\) −1.17433 + 1.32555i −1.17433 + 1.32555i −0.239316 + 0.970942i \(0.576923\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) −0.695701 + 0.616337i −0.695701 + 0.616337i −0.935016 0.354605i \(-0.884615\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(618\) 0 0
\(619\) 0 0 0.954721 0.297503i \(-0.0961538\pi\)
−0.954721 + 0.297503i \(0.903846\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) 0 0
\(625\) 0.885456 0.464723i 0.885456 0.464723i
\(626\) −0.107253 + 0.344186i −0.107253 + 0.344186i
\(627\) 0 0
\(628\) −0.646140 + 1.43566i −0.646140 + 1.43566i
\(629\) −0.992709 + 2.20571i −0.992709 + 2.20571i
\(630\) 0 0
\(631\) 0 0 −0.517338 0.855781i \(-0.673077\pi\)
0.517338 + 0.855781i \(0.326923\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) 0.688601 + 1.81569i 0.688601 + 1.81569i
\(635\) 0 0
\(636\) 0 0
\(637\) 0.239316 + 0.970942i 0.239316 + 0.970942i
\(638\) 0 0
\(639\) 0 0
\(640\) −0.935016 0.354605i −0.935016 0.354605i
\(641\) 0 0 0.568065 0.822984i \(-0.307692\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(642\) 0 0
\(643\) 0 0 −0.464723 0.885456i \(-0.653846\pi\)
0.464723 + 0.885456i \(0.346154\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.983620 0.180255i \(-0.942308\pi\)
0.983620 + 0.180255i \(0.0576923\pi\)
\(648\) 0.120537 0.992709i 0.120537 0.992709i
\(649\) 0 0
\(650\) −0.822984 + 0.568065i −0.822984 + 0.568065i
\(651\) 0 0
\(652\) 0 0
\(653\) −0.731626 + 0.731626i −0.731626 + 0.731626i −0.970942 0.239316i \(-0.923077\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −0.328749 0.147958i −0.328749 0.147958i
\(657\) 2.00000i 2.00000i
\(658\) 0 0
\(659\) 0 0 −0.239316 0.970942i \(-0.576923\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(660\) 0 0
\(661\) 1.56077 + 0.943521i 1.56077 + 0.943521i 0.992709 + 0.120537i \(0.0384615\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 1.28771 + 0.317391i 1.28771 + 0.317391i
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) −0.509195 1.63406i −0.509195 1.63406i −0.748511 0.663123i \(-0.769231\pi\)
0.239316 0.970942i \(-0.423077\pi\)
\(674\) −1.21323 0.222333i −1.21323 0.222333i
\(675\) 0 0
\(676\) 0.748511 0.663123i 0.748511 0.663123i
\(677\) 1.28962 + 1.28962i 1.28962 + 1.28962i 0.935016 + 0.354605i \(0.115385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0.328749 1.79393i 0.328749 1.79393i
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(684\) 0 0
\(685\) −1.49702 −1.49702
\(686\) 0 0
\(687\) 0 0
\(688\) 0 0
\(689\) 0.731626 + 0.731626i 0.731626 + 0.731626i
\(690\) 0 0
\(691\) 0 0 0.517338 0.855781i \(-0.326923\pi\)
−0.517338 + 0.855781i \(0.673077\pi\)
\(692\) 0.115289 + 1.90596i 0.115289 + 1.90596i
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0.157350 0.638392i 0.157350 0.638392i
\(698\) −0.0853881 + 0.0853881i −0.0853881 + 0.0853881i
\(699\) 0 0
\(700\) 0 0
\(701\) 0.475142 + 0.0576926i 0.475142 + 0.0576926i 0.354605 0.935016i \(-0.384615\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 0 0
\(705\) 0 0
\(706\) −0.317391 + 0.358261i −0.317391 + 0.358261i
\(707\) 0 0
\(708\) 0 0
\(709\) 0.244198 0.783659i 0.244198 0.783659i −0.748511 0.663123i \(-0.769231\pi\)
0.992709 0.120537i \(-0.0384615\pi\)
\(710\) 0 0
\(711\) 0 0
\(712\) −1.03279 + 1.70844i −1.03279 + 1.70844i
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(720\) −1.00000 −1.00000
\(721\) 0 0
\(722\) 0.822984 0.568065i 0.822984 0.568065i
\(723\) 0 0
\(724\) 1.81569 0.688601i 1.81569 0.688601i
\(725\) −1.17433 + 1.32555i −1.17433 + 1.32555i
\(726\) 0 0
\(727\) 0 0 0.855781 0.517338i \(-0.173077\pi\)
−0.855781 + 0.517338i \(0.826923\pi\)
\(728\) 0 0
\(729\) −0.239316 0.970942i −0.239316 0.970942i
\(730\) −1.98542 + 0.241073i −1.98542 + 0.241073i
\(731\) 0 0
\(732\) 0 0
\(733\) −0.112032 + 0.213460i −0.112032 + 0.213460i −0.935016 0.354605i \(-0.884615\pi\)
0.822984 + 0.568065i \(0.192308\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) −0.359852 0.0217671i −0.359852 0.0217671i
\(739\) 0 0 0.998176 0.0603785i \(-0.0192308\pi\)
−0.998176 + 0.0603785i \(0.980769\pi\)
\(740\) 0.159861 1.31658i 0.159861 1.31658i
\(741\) 0 0
\(742\) 0 0
\(743\) 0 0 0.935016 0.354605i \(-0.115385\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(744\) 0 0
\(745\) −0.366951 + 1.17759i −0.366951 + 1.17759i
\(746\) 1.39105 1.39105i 1.39105 1.39105i
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0 0 −0.822984 0.568065i \(-0.807692\pi\)
0.822984 + 0.568065i \(0.192308\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 1.00599 1.45743i 1.00599 1.45743i
\(755\) 0 0
\(756\) 0 0
\(757\) −0.542586 + 0.244198i −0.542586 + 0.244198i −0.663123 0.748511i \(-0.730769\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) −1.01773 + 0.186505i −1.01773 + 0.186505i −0.663123 0.748511i \(-0.730769\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 0 0
\(765\) −0.328749 1.79393i −0.328749 1.79393i
\(766\) 0 0
\(767\) 0 0
\(768\) 0 0
\(769\) 0.542586 0.244198i 0.542586 0.244198i −0.120537 0.992709i \(-0.538462\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −1.12785 + 1.63397i −1.12785 + 1.63397i
\(773\) −1.24006 + 1.39974i −1.24006 + 1.39974i −0.354605 + 0.935016i \(0.615385\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 1.75800 + 0.922670i 1.75800 + 0.922670i
\(777\) 0 0
\(778\) −0.530851 1.39974i −0.530851 1.39974i
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.970942 + 0.239316i 0.970942 + 0.239316i
\(785\) 1.57149 0.0950579i 1.57149 0.0950579i
\(786\) 0 0
\(787\) 0 0 −0.935016 0.354605i \(-0.884615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(788\) −0.822984 0.431935i −0.822984 0.431935i
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.431935 + 0.822984i −0.431935 + 0.822984i
\(794\) 0.393906 1.59814i 0.393906 1.59814i
\(795\) 0 0
\(796\) 0 0
\(797\) −1.79393 0.807380i −1.79393 0.807380i −0.970942 0.239316i \(-0.923077\pi\)
−0.822984 0.568065i \(-0.807692\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0.120537 + 0.992709i 0.120537 + 0.992709i
\(801\) −0.359852 + 1.96365i −0.359852 + 1.96365i
\(802\) 1.23932 0.970942i 1.23932 0.970942i
\(803\) 0 0
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 1.97094 0.239316i 1.97094 0.239316i
\(809\) 0.0576926 + 0.234068i 0.0576926 + 0.234068i 0.992709 0.120537i \(-0.0384615\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(810\) −0.935016 + 0.354605i −0.935016 + 0.354605i
\(811\) 0 0 0.998176 0.0603785i \(-0.0192308\pi\)
−0.998176 + 0.0603785i \(0.980769\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 0.872172 1.11325i 0.872172 1.11325i
\(819\) 0 0
\(820\) 0.0217671 + 0.359852i 0.0217671 + 0.359852i
\(821\) −1.05555 + 0.638104i −1.05555 + 0.638104i −0.935016 0.354605i \(-0.884615\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(822\) 0 0
\(823\) 0 0 −0.707107 0.707107i \(-0.750000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.935016 0.354605i \(-0.884615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(828\) 0 0
\(829\) −0.358261 0.317391i −0.358261 0.317391i 0.464723 0.885456i \(-0.346154\pi\)
−0.822984 + 0.568065i \(0.807692\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) −0.239316 0.970942i −0.239316 0.970942i
\(833\) −0.110118 + 1.82047i −0.110118 + 1.82047i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 0 0 −0.787183 0.616719i \(-0.788462\pi\)
0.787183 + 0.616719i \(0.211538\pi\)
\(840\) 0 0
\(841\) 0.757482 1.99732i 0.757482 1.99732i
\(842\) −0.568065 + 0.177016i −0.568065 + 0.177016i
\(843\) 0 0
\(844\) 0 0
\(845\) −0.935016 0.354605i −0.935016 0.354605i
\(846\) 0 0
\(847\) 0 0
\(848\) 0.987826 0.307819i 0.987826 0.307819i
\(849\) 0 0
\(850\) −1.74122 + 0.542586i −1.74122 + 0.542586i
\(851\) 0 0
\(852\) 0 0
\(853\) −0.688601 + 0.169725i −0.688601 + 0.169725i −0.568065 0.822984i \(-0.692308\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −0.0359256 + 0.593921i −0.0359256 + 0.593921i 0.935016 + 0.354605i \(0.115385\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(858\) 0 0
\(859\) 0 0 0.663123 0.748511i \(-0.269231\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 −0.935016 0.354605i \(-0.884615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(864\) 0 0
\(865\) 1.63406 0.987826i 1.63406 0.987826i
\(866\) 0.0624722 + 0.103342i 0.0624722 + 0.103342i
\(867\) 0 0
\(868\) 0 0
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −0.506219 + 0.646140i −0.506219 + 0.646140i
\(873\) 1.97094 + 0.239316i 1.97094 + 0.239316i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 0.198399 1.63397i 0.198399 1.63397i −0.464723 0.885456i \(-0.653846\pi\)
0.663123 0.748511i \(-0.269231\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) −0.271894 1.10312i −0.271894 1.10312i −0.935016 0.354605i \(-0.884615\pi\)
0.663123 0.748511i \(-0.269231\pi\)
\(882\) 0.992709 0.120537i 0.992709 0.120537i
\(883\) 0 0 0.180255 0.983620i \(-0.442308\pi\)
−0.180255 + 0.983620i \(0.557692\pi\)
\(884\) 1.66312 0.748511i 1.66312 0.748511i
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.410413 0.911900i \(-0.365385\pi\)
−0.410413 + 0.911900i \(0.634615\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 1.99271 + 0.120537i 1.99271 + 0.120537i
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −1.01773 + 1.68353i −1.01773 + 1.68353i
\(899\) 0 0
\(900\) 0.464723 + 0.885456i 0.464723 + 0.885456i
\(901\) 0.876952 + 1.67089i 0.876952 + 1.67089i
\(902\) 0 0
\(903\) 0 0
\(904\) −0.00729113 + 0.120537i −0.00729113 + 0.120537i
\(905\) −1.45352 1.28771i −1.45352 1.28771i
\(906\) 0 0
\(907\) 0 0 0.0603785 0.998176i \(-0.480769\pi\)
−0.0603785 + 0.998176i \(0.519231\pi\)
\(908\) 0 0
\(909\) 1.75800 0.922670i 1.75800 0.922670i
\(910\) 0 0
\(911\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) −0.423807 0.222431i −0.423807 0.222431i
\(915\) 0 0
\(916\) −1.63406 + 0.509195i −1.63406 + 0.509195i
\(917\) 0 0
\(918\) 0 0
\(919\) 0 0 −0.663123 0.748511i \(-0.730769\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 0.819328 + 1.82047i 0.819328 + 1.82047i
\(923\) 0 0
\(924\) 0 0
\(925\) −1.24006 + 0.470293i −1.24006 + 0.470293i
\(926\) 0 0
\(927\) 0 0
\(928\) −0.822984 1.56806i −0.822984 1.56806i
\(929\) −0.807380 + 0.147958i −0.807380 + 0.147958i −0.568065 0.822984i \(-0.692308\pi\)
−0.239316 + 0.970942i \(0.576923\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 0.970942 1.23932i 0.970942 1.23932i
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) −0.568065 0.822984i −0.568065 0.822984i
\(937\) −1.28962 + 0.580411i −1.28962 + 0.580411i −0.935016 0.354605i \(-0.884615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 0 0
\(941\) 0.107253 + 0.585260i 0.107253 + 0.585260i 0.992709 + 0.120537i \(0.0384615\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.935016 0.354605i \(-0.115385\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(948\) 0 0
\(949\) −1.32625 1.49702i −1.32625 1.49702i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −0.307819 0.509195i −0.307819 0.509195i 0.663123 0.748511i \(-0.269231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(954\) 0.814480 0.638104i 0.814480 0.638104i
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.239316 + 0.970942i 0.239316 + 0.970942i
\(962\) 1.17433 0.616337i 1.17433 0.616337i
\(963\) 0 0
\(964\) −1.82298 0.568065i −1.82298 0.568065i
\(965\) 1.97094 + 0.239316i 1.97094 + 0.239316i
\(966\) 0 0
\(967\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(968\) 0.822984 0.568065i 0.822984 0.568065i
\(969\) 0 0
\(970\) 1.98542i 1.98542i
\(971\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 0 0
\(975\) 0 0
\(976\) 0.527986 + 0.764919i 0.527986 + 0.764919i
\(977\) 0.0290582 + 0.239316i 0.0290582 + 0.239316i 1.00000 \(0\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −0.239316 0.970942i −0.239316 0.970942i
\(981\) −0.244198 + 0.783659i −0.244198 + 0.783659i
\(982\) 0 0
\(983\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(984\) 0 0
\(985\) 0.929446i 0.929446i
\(986\) 2.54244 1.99187i 2.54244 1.99187i
\(987\) 0 0
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(992\) 0 0
\(993\) 0 0
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −1.70844 1.03279i −1.70844 1.03279i −0.885456 0.464723i \(-0.846154\pi\)
−0.822984 0.568065i \(-0.807692\pi\)
\(998\) 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 3380.1.bz.a.3047.1 yes 24
4.3 odd 2 CM 3380.1.bz.a.3047.1 yes 24
5.3 odd 4 3380.1.cg.a.343.1 yes 24
20.3 even 4 3380.1.cg.a.343.1 yes 24
169.34 odd 52 3380.1.cg.a.2907.1 yes 24
676.203 even 52 3380.1.cg.a.2907.1 yes 24
845.203 even 52 inner 3380.1.bz.a.203.1 24
3380.203 odd 52 inner 3380.1.bz.a.203.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3380.1.bz.a.203.1 24 845.203 even 52 inner
3380.1.bz.a.203.1 24 3380.203 odd 52 inner
3380.1.bz.a.3047.1 yes 24 1.1 even 1 trivial
3380.1.bz.a.3047.1 yes 24 4.3 odd 2 CM
3380.1.cg.a.343.1 yes 24 5.3 odd 4
3380.1.cg.a.343.1 yes 24 20.3 even 4
3380.1.cg.a.2907.1 yes 24 169.34 odd 52
3380.1.cg.a.2907.1 yes 24 676.203 even 52