Properties

Label 3549.1.da.a.2480.1
Level $3549$
Weight $1$
Character 3549.2480
Analytic conductor $1.771$
Analytic rank $0$
Dimension $24$
Projective image $D_{78}$
CM discriminant -3
Inner twists $4$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3549,1,Mod(95,3549)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3549, base_ring=CyclotomicField(78))
 
chi = DirichletCharacter(H, H._module([39, 52, 37]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3549.95");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3549.da (of order \(78\), 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: \(1.77118172983\)
Analytic rank: \(0\)
Dimension: \(24\)
Coefficient field: \(\Q(\zeta_{39})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{24} - x^{23} + x^{21} - x^{20} + x^{18} - x^{17} + x^{15} - x^{14} + x^{12} - x^{10} + x^{9} - x^{7} + x^{6} - x^{4} + x^{3} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Projective image: \(D_{78}\)
Projective field: Galois closure of \(\mathbb{Q}[x]/(x^{78} - \cdots)\)

Embedding invariants

Embedding label 2480.1
Root \(-0.632445 + 0.774605i\) of defining polynomial
Character \(\chi\) \(=\) 3549.2480
Dual form 3549.1.da.a.1733.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-0.428693 - 0.903450i) q^{3} +(0.748511 - 0.663123i) q^{4} +(0.919979 + 0.391967i) q^{7} +(-0.632445 + 0.774605i) q^{9} +O(q^{10})\) \(q+(-0.428693 - 0.903450i) q^{3} +(0.748511 - 0.663123i) q^{4} +(0.919979 + 0.391967i) q^{7} +(-0.632445 + 0.774605i) q^{9} +(-0.919979 - 0.391967i) q^{12} +(0.500000 + 0.866025i) q^{13} +(0.120537 - 0.992709i) q^{16} +(-0.277840 + 0.160411i) q^{19} +(-0.0402659 - 0.999189i) q^{21} +(-0.692724 - 0.721202i) q^{25} +(0.970942 + 0.239316i) q^{27} +(0.948536 - 0.316668i) q^{28} +(1.27388 - 0.368985i) q^{31} +(0.0402659 + 0.999189i) q^{36} +(0.478243 - 1.94031i) q^{37} +(0.568065 - 0.822984i) q^{39} +(-0.0224054 + 0.0773523i) q^{43} +(-0.948536 + 0.316668i) q^{48} +(0.692724 + 0.721202i) q^{49} +(0.948536 + 0.316668i) q^{52} +(0.264032 + 0.182248i) q^{57} +(1.60339 + 1.01392i) q^{61} +(-0.885456 + 0.464723i) q^{63} +(-0.568065 - 0.822984i) q^{64} +(-0.151567 - 0.453999i) q^{67} +(-0.192732 - 1.18593i) q^{73} +(-0.354605 + 0.935016i) q^{75} +(-0.101594 + 0.304312i) q^{76} +(-1.07766 + 0.359776i) q^{79} +(-0.200026 - 0.979791i) q^{81} +(-0.692724 - 0.721202i) q^{84} +(0.120537 + 0.992709i) q^{91} +(-0.879463 - 0.992709i) q^{93} +(-1.04052 + 1.38468i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q - q^{3} + 2 q^{4} - q^{7} + q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q - q^{3} + 2 q^{4} - q^{7} + q^{9} + q^{12} + 12 q^{13} - 2 q^{16} + 3 q^{19} + q^{21} - q^{25} + 2 q^{27} + q^{28} - q^{36} - 2 q^{39} - q^{43} - q^{48} + q^{49} + q^{52} + q^{61} + 2 q^{63} + 2 q^{64} + 3 q^{73} - 2 q^{75} - 3 q^{76} - 2 q^{79} + q^{81} - q^{84} - 2 q^{91} - 26 q^{93} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3549\mathbb{Z}\right)^\times\).

\(n\) \(1184\) \(1522\) \(3382\)
\(\chi(n)\) \(-1\) \(e\left(\frac{1}{3}\right)\) \(e\left(\frac{17}{78}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 0.935016 0.354605i \(-0.115385\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(3\) −0.428693 0.903450i −0.428693 0.903450i
\(4\) 0.748511 0.663123i 0.748511 0.663123i
\(5\) 0 0 0.391967 0.919979i \(-0.371795\pi\)
−0.391967 + 0.919979i \(0.628205\pi\)
\(6\) 0 0
\(7\) 0.919979 + 0.391967i 0.919979 + 0.391967i
\(8\) 0 0
\(9\) −0.632445 + 0.774605i −0.632445 + 0.774605i
\(10\) 0 0
\(11\) 0 0 0.774605 0.632445i \(-0.217949\pi\)
−0.774605 + 0.632445i \(0.782051\pi\)
\(12\) −0.919979 0.391967i −0.919979 0.391967i
\(13\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(14\) 0 0
\(15\) 0 0
\(16\) 0.120537 0.992709i 0.120537 0.992709i
\(17\) 0 0 −0.885456 0.464723i \(-0.846154\pi\)
0.885456 + 0.464723i \(0.153846\pi\)
\(18\) 0 0
\(19\) −0.277840 + 0.160411i −0.277840 + 0.160411i −0.632445 0.774605i \(-0.717949\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(20\) 0 0
\(21\) −0.0402659 0.999189i −0.0402659 0.999189i
\(22\) 0 0
\(23\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(24\) 0 0
\(25\) −0.692724 0.721202i −0.692724 0.721202i
\(26\) 0 0
\(27\) 0.970942 + 0.239316i 0.970942 + 0.239316i
\(28\) 0.948536 0.316668i 0.948536 0.316668i
\(29\) 0 0 0.632445 0.774605i \(-0.282051\pi\)
−0.632445 + 0.774605i \(0.717949\pi\)
\(30\) 0 0
\(31\) 1.27388 0.368985i 1.27388 0.368985i 0.428693 0.903450i \(-0.358974\pi\)
0.845190 + 0.534466i \(0.179487\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.0402659 + 0.999189i 0.0402659 + 0.999189i
\(37\) 0.478243 1.94031i 0.478243 1.94031i 0.200026 0.979791i \(-0.435897\pi\)
0.278217 0.960518i \(-0.410256\pi\)
\(38\) 0 0
\(39\) 0.568065 0.822984i 0.568065 0.822984i
\(40\) 0 0
\(41\) 0 0 0.0804666 0.996757i \(-0.474359\pi\)
−0.0804666 + 0.996757i \(0.525641\pi\)
\(42\) 0 0
\(43\) −0.0224054 + 0.0773523i −0.0224054 + 0.0773523i −0.970942 0.239316i \(-0.923077\pi\)
0.948536 + 0.316668i \(0.102564\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 −0.316668 0.948536i \(-0.602564\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(48\) −0.948536 + 0.316668i −0.948536 + 0.316668i
\(49\) 0.692724 + 0.721202i 0.692724 + 0.721202i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.948536 + 0.316668i 0.948536 + 0.316668i
\(53\) 0 0 −0.0402659 0.999189i \(-0.512821\pi\)
0.0402659 + 0.999189i \(0.487179\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0.264032 + 0.182248i 0.264032 + 0.182248i
\(58\) 0 0
\(59\) 0 0 0.992709 0.120537i \(-0.0384615\pi\)
−0.992709 + 0.120537i \(0.961538\pi\)
\(60\) 0 0
\(61\) 1.60339 + 1.01392i 1.60339 + 1.01392i 0.970942 + 0.239316i \(0.0769231\pi\)
0.632445 + 0.774605i \(0.282051\pi\)
\(62\) 0 0
\(63\) −0.885456 + 0.464723i −0.885456 + 0.464723i
\(64\) −0.568065 0.822984i −0.568065 0.822984i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.151567 0.453999i −0.151567 0.453999i 0.845190 0.534466i \(-0.179487\pi\)
−0.996757 + 0.0804666i \(0.974359\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 0.903450 0.428693i \(-0.141026\pi\)
−0.903450 + 0.428693i \(0.858974\pi\)
\(72\) 0 0
\(73\) −0.192732 1.18593i −0.192732 1.18593i −0.885456 0.464723i \(-0.846154\pi\)
0.692724 0.721202i \(-0.256410\pi\)
\(74\) 0 0
\(75\) −0.354605 + 0.935016i −0.354605 + 0.935016i
\(76\) −0.101594 + 0.304312i −0.101594 + 0.304312i
\(77\) 0 0
\(78\) 0 0
\(79\) −1.07766 + 0.359776i −1.07766 + 0.359776i −0.799443 0.600742i \(-0.794872\pi\)
−0.278217 + 0.960518i \(0.589744\pi\)
\(80\) 0 0
\(81\) −0.200026 0.979791i −0.200026 0.979791i
\(82\) 0 0
\(83\) 0 0 0.822984 0.568065i \(-0.192308\pi\)
−0.822984 + 0.568065i \(0.807692\pi\)
\(84\) −0.692724 0.721202i −0.692724 0.721202i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(90\) 0 0
\(91\) 0.120537 + 0.992709i 0.120537 + 0.992709i
\(92\) 0 0
\(93\) −0.879463 0.992709i −0.879463 0.992709i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.04052 + 1.38468i −1.04052 + 1.38468i −0.120537 + 0.992709i \(0.538462\pi\)
−0.919979 + 0.391967i \(0.871795\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.996757 0.0804666i −0.996757 0.0804666i
\(101\) 0 0 0.278217 0.960518i \(-0.410256\pi\)
−0.278217 + 0.960518i \(0.589744\pi\)
\(102\) 0 0
\(103\) 0.685430 + 1.44451i 0.685430 + 1.44451i 0.885456 + 0.464723i \(0.153846\pi\)
−0.200026 + 0.979791i \(0.564103\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(108\) 0.885456 0.464723i 0.885456 0.464723i
\(109\) 0.565375 0.543050i 0.565375 0.543050i −0.354605 0.935016i \(-0.615385\pi\)
0.919979 + 0.391967i \(0.128205\pi\)
\(110\) 0 0
\(111\) −1.95799 + 0.399727i −1.95799 + 0.399727i
\(112\) 0.500000 0.866025i 0.500000 0.866025i
\(113\) 0 0 0.996757 0.0804666i \(-0.0256410\pi\)
−0.996757 + 0.0804666i \(0.974359\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.987050 0.160411i −0.987050 0.160411i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.200026 0.979791i 0.200026 0.979791i
\(122\) 0 0
\(123\) 0 0
\(124\) 0.708833 1.12093i 0.708833 1.12093i
\(125\) 0 0
\(126\) 0 0
\(127\) −0.368039 + 1.80277i −0.368039 + 1.80277i 0.200026 + 0.979791i \(0.435897\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(128\) 0 0
\(129\) 0.0794890 0.0129182i 0.0794890 0.0129182i
\(130\) 0 0
\(131\) 0 0 −0.987050 0.160411i \(-0.948718\pi\)
0.987050 + 0.160411i \(0.0512821\pi\)
\(132\) 0 0
\(133\) −0.318483 + 0.0386709i −0.318483 + 0.0386709i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.239316 0.970942i \(-0.576923\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(138\) 0 0
\(139\) −1.55242 1.16657i −1.55242 1.16657i −0.919979 0.391967i \(-0.871795\pi\)
−0.632445 0.774605i \(-0.717949\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) 0.692724 + 0.721202i 0.692724 + 0.721202i
\(145\) 0 0
\(146\) 0 0
\(147\) 0.354605 0.935016i 0.354605 0.935016i
\(148\) −0.928693 1.76948i −0.928693 1.76948i
\(149\) 0 0 −0.979791 0.200026i \(-0.935897\pi\)
0.979791 + 0.200026i \(0.0641026\pi\)
\(150\) 0 0
\(151\) 0.151567 0.453999i 0.151567 0.453999i −0.845190 0.534466i \(-0.820513\pi\)
0.996757 + 0.0804666i \(0.0256410\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) −0.120537 0.992709i −0.120537 0.992709i
\(157\) −0.319818 + 0.240328i −0.319818 + 0.240328i −0.748511 0.663123i \(-0.769231\pi\)
0.428693 + 0.903450i \(0.358974\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −1.84519 + 0.534466i −1.84519 + 0.534466i −0.845190 + 0.534466i \(0.820513\pi\)
−1.00000 \(\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.774605 0.632445i \(-0.782051\pi\)
0.774605 + 0.632445i \(0.217949\pi\)
\(168\) 0 0
\(169\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(170\) 0 0
\(171\) 0.0514636 0.316668i 0.0514636 0.316668i
\(172\) 0.0345234 + 0.0727566i 0.0345234 + 0.0727566i
\(173\) 0 0 −0.948536 0.316668i \(-0.897436\pi\)
0.948536 + 0.316668i \(0.102564\pi\)
\(174\) 0 0
\(175\) −0.354605 0.935016i −0.354605 0.935016i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.0402659 0.999189i \(-0.487179\pi\)
−0.0402659 + 0.999189i \(0.512821\pi\)
\(180\) 0 0
\(181\) 0.0670708 + 0.552378i 0.0670708 + 0.552378i 0.987050 + 0.160411i \(0.0512821\pi\)
−0.919979 + 0.391967i \(0.871795\pi\)
\(182\) 0 0
\(183\) 0.228667 1.88324i 0.228667 1.88324i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.799443 + 0.600742i 0.799443 + 0.600742i
\(190\) 0 0
\(191\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(192\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(193\) −1.06806 0.0430415i −1.06806 0.0430415i −0.500000 0.866025i \(-0.666667\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.996757 + 0.0804666i 0.996757 + 0.0804666i
\(197\) 0 0 0.721202 0.692724i \(-0.243590\pi\)
−0.721202 + 0.692724i \(0.756410\pi\)
\(198\) 0 0
\(199\) −0.237952 + 1.95971i −0.237952 + 1.95971i 0.0402659 + 0.999189i \(0.487179\pi\)
−0.278217 + 0.960518i \(0.589744\pi\)
\(200\) 0 0
\(201\) −0.345190 + 0.331560i −0.345190 + 0.331560i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) 0.919979 0.391967i 0.919979 0.391967i
\(209\) 0 0
\(210\) 0 0
\(211\) 0.111301 0.545190i 0.111301 0.545190i −0.885456 0.464723i \(-0.846154\pi\)
0.996757 0.0804666i \(-0.0256410\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 1.31658 + 0.159861i 1.31658 + 0.159861i
\(218\) 0 0
\(219\) −0.988802 + 0.682521i −0.988802 + 0.682521i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 0.799443 0.600742i \(-0.205128\pi\)
−0.799443 + 0.600742i \(0.794872\pi\)
\(224\) 0 0
\(225\) 0.996757 0.0804666i 0.996757 0.0804666i
\(226\) 0 0
\(227\) 0 0 −0.663123 0.748511i \(-0.730769\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(228\) 0.318483 0.0386709i 0.318483 0.0386709i
\(229\) 0.752982 + 0.218104i 0.752982 + 0.218104i 0.632445 0.774605i \(-0.282051\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 0.996757 0.0804666i \(-0.0256410\pi\)
−0.996757 + 0.0804666i \(0.974359\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) 0.787025 + 0.819379i 0.787025 + 0.819379i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −1.31658 0.159861i −1.31658 0.159861i −0.568065 0.822984i \(-0.692308\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(242\) 0 0
\(243\) −0.799443 + 0.600742i −0.799443 + 0.600742i
\(244\) 1.87251 0.304312i 1.87251 0.304312i
\(245\) 0 0
\(246\) 0 0
\(247\) −0.277840 0.160411i −0.277840 0.160411i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 0.692724 0.721202i \(-0.256410\pi\)
−0.692724 + 0.721202i \(0.743590\pi\)
\(252\) −0.354605 + 0.935016i −0.354605 + 0.935016i
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.970942 0.239316i −0.970942 0.239316i
\(257\) 0 0 0.748511 0.663123i \(-0.230769\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(258\) 0 0
\(259\) 1.20051 1.59759i 1.20051 1.59759i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 −0.632445 0.774605i \(-0.717949\pi\)
0.632445 + 0.774605i \(0.282051\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −0.414507 0.239316i −0.414507 0.239316i
\(269\) 0 0 0.568065 0.822984i \(-0.307692\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(270\) 0 0
\(271\) 1.29944 1.46677i 1.29944 1.46677i 0.500000 0.866025i \(-0.333333\pi\)
0.799443 0.600742i \(-0.205128\pi\)
\(272\) 0 0
\(273\) 0.845190 0.534466i 0.845190 0.534466i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 0.759177 0.398447i 0.759177 0.398447i −0.0402659 0.999189i \(-0.512821\pi\)
0.799443 + 0.600742i \(0.205128\pi\)
\(278\) 0 0
\(279\) −0.519844 + 1.22012i −0.519844 + 1.22012i
\(280\) 0 0
\(281\) 0 0 −0.822984 0.568065i \(-0.807692\pi\)
0.822984 + 0.568065i \(0.192308\pi\)
\(282\) 0 0
\(283\) −1.10759 + 1.15312i −1.10759 + 1.15312i −0.120537 + 0.992709i \(0.538462\pi\)
−0.987050 + 0.160411i \(0.948718\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) 0.568065 + 0.822984i 0.568065 + 0.822984i
\(290\) 0 0
\(291\) 1.69705 + 0.346455i 1.69705 + 0.346455i
\(292\) −0.930676 0.759873i −0.930676 0.759873i
\(293\) 0 0 −0.999189 0.0402659i \(-0.987179\pi\)
0.999189 + 0.0402659i \(0.0128205\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) 0.354605 + 0.935016i 0.354605 + 0.935016i
\(301\) −0.0509320 + 0.0623804i −0.0509320 + 0.0623804i
\(302\) 0 0
\(303\) 0 0
\(304\) 0.125752 + 0.295150i 0.125752 + 0.295150i
\(305\) 0 0
\(306\) 0 0
\(307\) 0.222431 0.902438i 0.222431 0.902438i −0.748511 0.663123i \(-0.769231\pi\)
0.970942 0.239316i \(-0.0769231\pi\)
\(308\) 0 0
\(309\) 1.01121 1.23850i 1.01121 1.23850i
\(310\) 0 0
\(311\) 0 0 −0.278217 0.960518i \(-0.589744\pi\)
0.278217 + 0.960518i \(0.410256\pi\)
\(312\) 0 0
\(313\) −0.238540 + 0.823534i −0.238540 + 0.823534i 0.748511 + 0.663123i \(0.230769\pi\)
−0.987050 + 0.160411i \(0.948718\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) −0.568065 + 0.983917i −0.568065 + 0.983917i
\(317\) 0 0 −0.999189 0.0402659i \(-0.987179\pi\)
0.999189 + 0.0402659i \(0.0128205\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) −0.799443 0.600742i −0.799443 0.600742i
\(325\) 0.278217 0.960518i 0.278217 0.960518i
\(326\) 0 0
\(327\) −0.732990 0.277987i −0.732990 0.277987i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) −1.44124 + 0.0580798i −1.44124 + 0.0580798i −0.748511 0.663123i \(-0.769231\pi\)
−0.692724 + 0.721202i \(0.743590\pi\)
\(332\) 0 0
\(333\) 1.20051 + 1.59759i 1.20051 + 1.59759i
\(334\) 0 0
\(335\) 0 0
\(336\) −0.996757 0.0804666i −0.996757 0.0804666i
\(337\) 1.69038 1.69038 0.845190 0.534466i \(-0.179487\pi\)
0.845190 + 0.534466i \(0.179487\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.354605 + 0.935016i 0.354605 + 0.935016i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(348\) 0 0
\(349\) 0.277840 + 1.70962i 0.277840 + 1.70962i 0.632445 + 0.774605i \(0.282051\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(350\) 0 0
\(351\) 0.278217 + 0.960518i 0.278217 + 0.960518i
\(352\) 0 0
\(353\) 0 0 −0.534466 0.845190i \(-0.679487\pi\)
0.534466 + 0.845190i \(0.320513\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.999189 0.0402659i \(-0.987179\pi\)
0.999189 + 0.0402659i \(0.0128205\pi\)
\(360\) 0 0
\(361\) −0.448536 + 0.776888i −0.448536 + 0.776888i
\(362\) 0 0
\(363\) −0.970942 + 0.239316i −0.970942 + 0.239316i
\(364\) 0.748511 + 0.663123i 0.748511 + 0.663123i
\(365\) 0 0
\(366\) 0 0
\(367\) −1.96770 0.319782i −1.96770 0.319782i −0.996757 0.0804666i \(-0.974359\pi\)
−0.970942 0.239316i \(-0.923077\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −1.31658 0.159861i −1.31658 0.159861i
\(373\) 1.47764 0.240139i 1.47764 0.240139i 0.632445 0.774605i \(-0.282051\pi\)
0.845190 + 0.534466i \(0.179487\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −0.150475 + 1.86397i −0.150475 + 1.86397i 0.278217 + 0.960518i \(0.410256\pi\)
−0.428693 + 0.903450i \(0.641026\pi\)
\(380\) 0 0
\(381\) 1.78649 0.440331i 1.78649 0.440331i
\(382\) 0 0
\(383\) 0 0 0.534466 0.845190i \(-0.320513\pi\)
−0.534466 + 0.845190i \(0.679487\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) −0.0457473 0.0662764i −0.0457473 0.0662764i
\(388\) 0.139372 + 1.72643i 0.139372 + 1.72643i
\(389\) 0 0 −0.845190 0.534466i \(-0.820513\pi\)
0.845190 + 0.534466i \(0.179487\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) −0.380472 0.506316i −0.380472 0.506316i 0.568065 0.822984i \(-0.307692\pi\)
−0.948536 + 0.316668i \(0.897436\pi\)
\(398\) 0 0
\(399\) 0.171469 + 0.271156i 0.171469 + 0.271156i
\(400\) −0.799443 + 0.600742i −0.799443 + 0.600742i
\(401\) 0 0 −0.992709 0.120537i \(-0.961538\pi\)
0.992709 + 0.120537i \(0.0384615\pi\)
\(402\) 0 0
\(403\) 0.956491 + 0.918722i 0.956491 + 0.918722i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.27497 + 0.880052i 1.27497 + 0.880052i 0.996757 0.0804666i \(-0.0256410\pi\)
0.278217 + 0.960518i \(0.410256\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 1.47094 + 0.626710i 1.47094 + 0.626710i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −0.388427 + 1.90264i −0.388427 + 1.90264i
\(418\) 0 0
\(419\) 0 0 −0.948536 0.316668i \(-0.897436\pi\)
0.948536 + 0.316668i \(0.102564\pi\)
\(420\) 0 0
\(421\) 1.63397 1.12785i 1.63397 1.12785i 0.748511 0.663123i \(-0.230769\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 1.07766 + 1.56126i 1.07766 + 1.56126i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 0.979791 0.200026i \(-0.0641026\pi\)
−0.979791 + 0.200026i \(0.935897\pi\)
\(432\) 0.354605 0.935016i 0.354605 0.935016i
\(433\) −0.652458 0.277987i −0.652458 0.277987i 0.0402659 0.999189i \(-0.487179\pi\)
−0.692724 + 0.721202i \(0.743590\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0.0630804 0.781391i 0.0630804 0.781391i
\(437\) 0 0
\(438\) 0 0
\(439\) −1.91674 + 0.472433i −1.91674 + 0.472433i −0.919979 + 0.391967i \(0.871795\pi\)
−0.996757 + 0.0804666i \(0.974359\pi\)
\(440\) 0 0
\(441\) −0.996757 + 0.0804666i −0.996757 + 0.0804666i
\(442\) 0 0
\(443\) 0 0 0.996757 0.0804666i \(-0.0256410\pi\)
−0.996757 + 0.0804666i \(0.974359\pi\)
\(444\) −1.20051 + 1.59759i −1.20051 + 1.59759i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.200026 0.979791i −0.200026 0.979791i
\(449\) 0 0 0.979791 0.200026i \(-0.0641026\pi\)
−0.979791 + 0.200026i \(0.935897\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −0.475142 + 0.0576926i −0.475142 + 0.0576926i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) −1.63397 + 1.12785i −1.63397 + 1.12785i −0.748511 + 0.663123i \(0.769231\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.160411 0.987050i \(-0.448718\pi\)
−0.160411 + 0.987050i \(0.551282\pi\)
\(462\) 0 0
\(463\) −1.44854 + 0.549357i −1.44854 + 0.549357i −0.948536 0.316668i \(-0.897436\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 −0.632445 0.774605i \(-0.717949\pi\)
0.632445 + 0.774605i \(0.282051\pi\)
\(468\) −0.845190 + 0.534466i −0.845190 + 0.534466i
\(469\) 0.0385138 0.477079i 0.0385138 0.477079i
\(470\) 0 0
\(471\) 0.354228 + 0.185913i 0.354228 + 0.185913i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.308156 + 0.0892584i 0.308156 + 0.0892584i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 −0.960518 0.278217i \(-0.910256\pi\)
0.960518 + 0.278217i \(0.0897436\pi\)
\(480\) 0 0
\(481\) 1.91948 0.555984i 1.91948 0.555984i
\(482\) 0 0
\(483\) 0 0
\(484\) −0.500000 0.866025i −0.500000 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) −1.29944 + 1.46677i −1.29944 + 1.46677i −0.500000 + 0.866025i \(0.666667\pi\)
−0.799443 + 0.600742i \(0.794872\pi\)
\(488\) 0 0
\(489\) 1.27388 + 1.43792i 1.27388 + 1.43792i
\(490\) 0 0
\(491\) 0 0 0.919979 0.391967i \(-0.128205\pi\)
−0.919979 + 0.391967i \(0.871795\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) −0.212745 1.30907i −0.212745 1.30907i
\(497\) 0 0
\(498\) 0 0
\(499\) −0.642152 1.01548i −0.642152 1.01548i −0.996757 0.0804666i \(-0.974359\pi\)
0.354605 0.935016i \(-0.384615\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 −0.200026 0.979791i \(-0.564103\pi\)
0.200026 + 0.979791i \(0.435897\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 0.996757 + 0.0804666i 0.996757 + 0.0804666i
\(508\) 0.919979 + 1.59345i 0.919979 + 1.59345i
\(509\) 0 0 0.935016 0.354605i \(-0.115385\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(510\) 0 0
\(511\) 0.287534 1.16657i 0.287534 1.16657i
\(512\) 0 0
\(513\) −0.308156 + 0.0892584i −0.308156 + 0.0892584i
\(514\) 0 0
\(515\) 0 0
\(516\) 0.0509320 0.0623804i 0.0509320 0.0623804i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.919979 0.391967i \(-0.128205\pi\)
−0.919979 + 0.391967i \(0.871795\pi\)
\(522\) 0 0
\(523\) −0.152466 + 1.25567i −0.152466 + 1.25567i 0.692724 + 0.721202i \(0.256410\pi\)
−0.845190 + 0.534466i \(0.820513\pi\)
\(524\) 0 0
\(525\) −0.692724 + 0.721202i −0.692724 + 0.721202i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) 1.00000 1.00000
\(530\) 0 0
\(531\) 0 0
\(532\) −0.212745 + 0.240139i −0.212745 + 0.240139i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.866514 1.15312i 0.866514 1.15312i −0.120537 0.992709i \(-0.538462\pi\)
0.987050 0.160411i \(-0.0512821\pi\)
\(542\) 0 0
\(543\) 0.470293 0.297395i 0.470293 0.297395i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.141860 0.374055i 0.141860 0.374055i −0.845190 0.534466i \(-0.820513\pi\)
0.987050 + 0.160411i \(0.0512821\pi\)
\(548\) 0 0
\(549\) −1.79944 + 0.600742i −1.79944 + 0.600742i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −1.13245 0.0914204i −1.13245 0.0914204i
\(554\) 0 0
\(555\) 0 0
\(556\) −1.93559 + 0.156257i −1.93559 + 0.156257i
\(557\) 0 0 −0.0804666 0.996757i \(-0.525641\pi\)
0.0804666 + 0.996757i \(0.474359\pi\)
\(558\) 0 0
\(559\) −0.0781918 + 0.0192725i −0.0781918 + 0.0192725i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.200026 0.979791i 0.200026 0.979791i
\(568\) 0 0
\(569\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(570\) 0 0
\(571\) 0.854605 1.80104i 0.854605 1.80104i 0.354605 0.935016i \(-0.384615\pi\)
0.500000 0.866025i \(-0.333333\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.996757 + 0.0804666i 0.996757 + 0.0804666i
\(577\) −1.66367 + 0.960518i −1.66367 + 0.960518i −0.692724 + 0.721202i \(0.743590\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(578\) 0 0
\(579\) 0.418986 + 0.983395i 0.418986 + 0.983395i
\(580\) 0 0
\(581\) 0 0
\(582\) 0 0
\(583\) 0 0
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 0 0 0.866025 0.500000i \(-0.166667\pi\)
−0.866025 + 0.500000i \(0.833333\pi\)
\(588\) −0.354605 0.935016i −0.354605 0.935016i
\(589\) −0.294747 + 0.306864i −0.294747 + 0.306864i
\(590\) 0 0
\(591\) 0 0
\(592\) −1.86852 0.708635i −1.86852 0.708635i
\(593\) 0 0 −0.0804666 0.996757i \(-0.525641\pi\)
0.0804666 + 0.996757i \(0.474359\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) 1.87251 0.625134i 1.87251 0.625134i
\(598\) 0 0
\(599\) 0 0 −0.948536 0.316668i \(-0.897436\pi\)
0.948536 + 0.316668i \(0.102564\pi\)
\(600\) 0 0
\(601\) 0.0794890 0.0129182i 0.0794890 0.0129182i −0.120537 0.992709i \(-0.538462\pi\)
0.200026 + 0.979791i \(0.435897\pi\)
\(602\) 0 0
\(603\) 0.447528 + 0.169725i 0.447528 + 0.169725i
\(604\) −0.187607 0.440331i −0.187607 0.440331i
\(605\) 0 0
\(606\) 0 0
\(607\) 1.68490 + 0.136019i 1.68490 + 0.136019i 0.885456 0.464723i \(-0.153846\pi\)
0.799443 + 0.600742i \(0.205128\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) −0.642152 + 0.854550i −0.642152 + 0.854550i −0.996757 0.0804666i \(-0.974359\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 0.391967 0.919979i \(-0.371795\pi\)
−0.391967 + 0.919979i \(0.628205\pi\)
\(618\) 0 0
\(619\) 0.160803 1.99190i 0.160803 1.99190i 0.0402659 0.999189i \(-0.487179\pi\)
0.120537 0.992709i \(-0.461538\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −0.748511 0.663123i −0.748511 0.663123i
\(625\) −0.0402659 + 0.999189i −0.0402659 + 0.999189i
\(626\) 0 0
\(627\) 0 0
\(628\) −0.0800206 + 0.391967i −0.0800206 + 0.391967i
\(629\) 0 0
\(630\) 0 0
\(631\) 1.99676 + 0.0804666i 1.99676 + 0.0804666i 1.00000 \(0\)
0.996757 + 0.0804666i \(0.0256410\pi\)
\(632\) 0 0
\(633\) −0.540266 + 0.133164i −0.540266 + 0.133164i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.278217 + 0.960518i −0.278217 + 0.960518i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.354605 0.935016i \(-0.384615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(642\) 0 0
\(643\) 0.0860133 + 1.06547i 0.0860133 + 1.06547i 0.885456 + 0.464723i \(0.153846\pi\)
−0.799443 + 0.600742i \(0.794872\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.987050 0.160411i \(-0.948718\pi\)
0.987050 + 0.160411i \(0.0512821\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −0.419979 1.25799i −0.419979 1.25799i
\(652\) −1.02673 + 1.62364i −1.02673 + 1.62364i
\(653\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.04052 + 0.600742i 1.04052 + 0.600742i
\(658\) 0 0
\(659\) 0 0 0.845190 0.534466i \(-0.179487\pi\)
−0.845190 + 0.534466i \(0.820513\pi\)
\(660\) 0 0
\(661\) −0.999468 1.58053i −0.999468 1.58053i −0.799443 0.600742i \(-0.794872\pi\)
−0.200026 0.979791i \(-0.564103\pi\)
\(662\) 0 0
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) 0 0
\(668\) 0 0
\(669\) 0 0
\(670\) 0 0
\(671\) 0 0
\(672\) 0 0
\(673\) 1.26079 + 0.101781i 1.26079 + 0.101781i 0.692724 0.721202i \(-0.256410\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(674\) 0 0
\(675\) −0.500000 0.866025i −0.500000 0.866025i
\(676\) 0.200026 + 0.979791i 0.200026 + 0.979791i
\(677\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(678\) 0 0
\(679\) −1.50000 + 0.866025i −1.50000 + 0.866025i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 0.464723 0.885456i \(-0.346154\pi\)
−0.464723 + 0.885456i \(0.653846\pi\)
\(684\) −0.171469 0.271156i −0.171469 0.271156i
\(685\) 0 0
\(686\) 0 0
\(687\) −0.125752 0.773781i −0.125752 0.773781i
\(688\) 0.0740877 + 0.0315658i 0.0740877 + 0.0315658i
\(689\) 0 0
\(690\) 0 0
\(691\) 0.294326 0.560791i 0.294326 0.560791i −0.692724 0.721202i \(-0.743590\pi\)
0.987050 + 0.160411i \(0.0512821\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.885456 0.464723i −0.885456 0.464723i
\(701\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(702\) 0 0
\(703\) 0.178372 + 0.615812i 0.178372 + 0.615812i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −1.73556 0.823534i −1.73556 0.823534i −0.987050 0.160411i \(-0.948718\pi\)
−0.748511 0.663123i \(-0.769231\pi\)
\(710\) 0 0
\(711\) 0.402877 1.06230i 0.402877 1.06230i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.692724 0.721202i \(-0.743590\pi\)
0.692724 + 0.721202i \(0.256410\pi\)
\(720\) 0 0
\(721\) 0.0643806 + 1.59759i 0.0643806 + 1.59759i
\(722\) 0 0
\(723\) 0.419979 + 1.25799i 0.419979 + 1.25799i
\(724\) 0.416498 + 0.368985i 0.416498 + 0.368985i
\(725\) 0 0
\(726\) 0 0
\(727\) −1.56806 + 0.822984i −1.56806 + 0.822984i −0.568065 + 0.822984i \(0.692308\pi\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) 0.885456 + 0.464723i 0.885456 + 0.464723i
\(730\) 0 0
\(731\) 0 0
\(732\) −1.07766 1.56126i −1.07766 1.56126i
\(733\) −0.839709 + 0.398447i −0.839709 + 0.398447i −0.799443 0.600742i \(-0.794872\pi\)
−0.0402659 + 0.999189i \(0.512821\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) 0.380472 0.506316i 0.380472 0.506316i −0.568065 0.822984i \(-0.692308\pi\)
0.948536 + 0.316668i \(0.102564\pi\)
\(740\) 0 0
\(741\) −0.0258155 + 0.319782i −0.0258155 + 0.319782i
\(742\) 0 0
\(743\) 0 0 −0.979791 0.200026i \(-0.935897\pi\)
0.979791 + 0.200026i \(0.0641026\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) 0.652458 1.72039i 0.652458 1.72039i −0.0402659 0.999189i \(-0.512821\pi\)
0.692724 0.721202i \(-0.256410\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) 0.996757 0.0804666i 0.996757 0.0804666i
\(757\) −1.67977 0.560791i −1.67977 0.560791i −0.692724 0.721202i \(-0.743590\pi\)
−0.987050 + 0.160411i \(0.948718\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.160411 0.987050i \(-0.551282\pi\)
0.160411 + 0.987050i \(0.448718\pi\)
\(762\) 0 0
\(763\) 0.732990 0.277987i 0.732990 0.277987i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) 0.200026 + 0.979791i 0.200026 + 0.979791i
\(769\) 0.456763 1.36817i 0.456763 1.36817i −0.428693 0.903450i \(-0.641026\pi\)
0.885456 0.464723i \(-0.153846\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −0.828000 + 0.676041i −0.828000 + 0.676041i
\(773\) 0 0 −0.992709 0.120537i \(-0.961538\pi\)
0.992709 + 0.120537i \(0.0384615\pi\)
\(774\) 0 0
\(775\) −1.14856 0.663123i −1.14856 0.663123i
\(776\) 0 0
\(777\) −1.95799 0.399727i −1.95799 0.399727i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) 0.799443 0.600742i 0.799443 0.600742i
\(785\) 0 0
\(786\) 0 0
\(787\) −0.796732 0.899324i −0.796732 0.899324i 0.200026 0.979791i \(-0.435897\pi\)
−0.996757 + 0.0804666i \(0.974359\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) −0.0763874 + 1.89553i −0.0763874 + 1.89553i
\(794\) 0 0
\(795\) 0 0
\(796\) 1.12142 + 1.62465i 1.12142 + 1.62465i
\(797\) 0 0 0.948536 0.316668i \(-0.102564\pi\)
−0.948536 + 0.316668i \(0.897436\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) −0.0385138 + 0.477079i −0.0385138 + 0.477079i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 0.845190 0.534466i \(-0.179487\pi\)
−0.845190 + 0.534466i \(0.820513\pi\)
\(810\) 0 0
\(811\) 0.318483 + 0.0386709i 0.318483 + 0.0386709i 0.278217 0.960518i \(-0.410256\pi\)
0.0402659 + 0.999189i \(0.487179\pi\)
\(812\) 0 0
\(813\) −1.88221 0.545190i −1.88221 0.545190i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) −0.00618307 0.0250857i −0.00618307 0.0250857i
\(818\) 0 0
\(819\) −0.845190 0.534466i −0.845190 0.534466i
\(820\) 0 0
\(821\) 0 0 −0.464723 0.885456i \(-0.653846\pi\)
0.464723 + 0.885456i \(0.346154\pi\)
\(822\) 0 0
\(823\) −0.709210 −0.709210 −0.354605 0.935016i \(-0.615385\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.663123 0.748511i \(-0.730769\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(828\) 0 0
\(829\) 1.47094 + 1.10534i 1.47094 + 1.10534i 0.970942 + 0.239316i \(0.0769231\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(830\) 0 0
\(831\) −0.685430 0.515067i −0.685430 0.515067i
\(832\) 0.428693 0.903450i 0.428693 0.903450i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 1.32517 0.0534025i 1.32517 0.0534025i
\(838\) 0 0
\(839\) 0 0 −0.721202 0.692724i \(-0.756410\pi\)
0.721202 + 0.692724i \(0.243590\pi\)
\(840\) 0 0
\(841\) −0.200026 0.979791i −0.200026 0.979791i
\(842\) 0 0
\(843\) 0 0
\(844\) −0.278217 0.481887i −0.278217 0.481887i
\(845\) 0 0
\(846\) 0 0
\(847\) 0.568065 0.822984i 0.568065 0.822984i
\(848\) 0 0
\(849\) 1.51660 + 0.506316i 1.51660 + 0.506316i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) 0.922670 1.75800i 0.922670 1.75800i 0.354605 0.935016i \(-0.384615\pi\)
0.568065 0.822984i \(-0.307692\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.799443 0.600742i \(-0.205128\pi\)
−0.799443 + 0.600742i \(0.794872\pi\)
\(858\) 0 0
\(859\) 1.59370 + 1.19759i 1.59370 + 1.19759i 0.845190 + 0.534466i \(0.179487\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.979791 0.200026i \(-0.0641026\pi\)
−0.979791 + 0.200026i \(0.935897\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.500000 0.866025i 0.500000 0.866025i
\(868\) 1.09148 0.753393i 1.09148 0.753393i
\(869\) 0 0
\(870\) 0 0
\(871\) 0.317391 0.358261i 0.317391 0.358261i
\(872\) 0 0
\(873\) −0.414507 1.68172i −0.414507 1.68172i
\(874\) 0 0
\(875\) 0 0
\(876\) −0.287534 + 1.16657i −0.287534 + 1.16657i
\(877\) 0.670319 0.643850i 0.670319 0.643850i −0.278217 0.960518i \(-0.589744\pi\)
0.948536 + 0.316668i \(0.102564\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 −0.0402659 0.999189i \(-0.512821\pi\)
0.0402659 + 0.999189i \(0.487179\pi\)
\(882\) 0 0
\(883\) −0.0285570 0.0752986i −0.0285570 0.0752986i 0.919979 0.391967i \(-0.128205\pi\)
−0.948536 + 0.316668i \(0.897436\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(888\) 0 0
\(889\) −1.04522 + 1.51426i −1.04522 + 1.51426i
\(890\) 0 0
\(891\) 0 0
\(892\) 0 0
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) 0 0
\(899\) 0 0
\(900\) 0.692724 0.721202i 0.692724 0.721202i
\(901\) 0 0
\(902\) 0 0
\(903\) 0.0781918 + 0.0192725i 0.0781918 + 0.0192725i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 1.27458 + 0.543050i 1.27458 + 0.543050i 0.919979 0.391967i \(-0.128205\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 0.568065 0.822984i \(-0.307692\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(912\) 0.212745 0.240139i 0.212745 0.240139i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0.708245 0.336066i 0.708245 0.336066i
\(917\) 0 0
\(918\) 0 0
\(919\) −1.10759 + 0.832298i −1.10759 + 0.832298i −0.987050 0.160411i \(-0.948718\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(920\) 0 0
\(921\) −0.910663 + 0.185913i −0.910663 + 0.185913i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) −1.73065 + 0.999189i −1.73065 + 0.999189i
\(926\) 0 0
\(927\) −1.55242 0.382638i −1.55242 0.382638i
\(928\) 0 0
\(929\) 0 0 −0.160411 0.987050i \(-0.551282\pi\)
0.160411 + 0.987050i \(0.448718\pi\)
\(930\) 0 0
\(931\) −0.308156 0.0892584i −0.308156 0.0892584i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −0.946784 + 0.838778i −0.946784 + 0.838778i −0.987050 0.160411i \(-0.948718\pi\)
0.0402659 + 0.999189i \(0.487179\pi\)
\(938\) 0 0
\(939\) 0.846282 0.137534i 0.846282 0.137534i
\(940\) 0 0
\(941\) 0 0 −0.160411 0.987050i \(-0.551282\pi\)
0.160411 + 0.987050i \(0.448718\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 0.663123 0.748511i \(-0.269231\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(948\) 1.13245 + 0.0914204i 1.13245 + 0.0914204i
\(949\) 0.930676 0.759873i 0.930676 0.759873i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 0.0402659 0.999189i \(-0.487179\pi\)
−0.0402659 + 0.999189i \(0.512821\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) 0.641437 0.405620i 0.641437 0.405620i
\(962\) 0 0
\(963\) 0 0
\(964\) −1.09148 + 0.753393i −1.09148 + 0.753393i
\(965\) 0 0
\(966\) 0 0
\(967\) −0.419979 + 0.474059i −0.419979 + 0.474059i −0.919979 0.391967i \(-0.871795\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.692724 0.721202i \(-0.743590\pi\)
0.692724 + 0.721202i \(0.256410\pi\)
\(972\) −0.200026 + 0.979791i −0.200026 + 0.979791i
\(973\) −0.970942 1.68172i −0.970942 1.68172i
\(974\) 0 0
\(975\) −0.987050 + 0.160411i −0.987050 + 0.160411i
\(976\) 1.19979 1.46948i 1.19979 1.46948i
\(977\) 0 0 −0.721202 0.692724i \(-0.756410\pi\)
0.721202 + 0.692724i \(0.243590\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 0.0630804 + 0.781391i 0.0630804 + 0.781391i
\(982\) 0 0
\(983\) 0 0 0.960518 0.278217i \(-0.0897436\pi\)
−0.960518 + 0.278217i \(0.910256\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) −0.314339 + 0.0641728i −0.314339 + 0.0641728i
\(989\) 0 0
\(990\) 0 0
\(991\) −0.987050 + 1.70962i −0.987050 + 1.70962i −0.354605 + 0.935016i \(0.615385\pi\)
−0.632445 + 0.774605i \(0.717949\pi\)
\(992\) 0 0
\(993\) 0.670319 + 1.27719i 0.670319 + 1.27719i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 1.22675 + 0.643850i 1.22675 + 0.643850i 0.948536 0.316668i \(-0.102564\pi\)
0.278217 + 0.960518i \(0.410256\pi\)
\(998\) 0 0
\(999\) 0.928693 1.76948i 0.928693 1.76948i
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 3549.1.da.a.2480.1 yes 24
3.2 odd 2 CM 3549.1.da.a.2480.1 yes 24
7.4 even 3 3549.1.ds.a.452.1 yes 24
21.11 odd 6 3549.1.ds.a.452.1 yes 24
169.43 even 78 3549.1.ds.a.212.1 yes 24
507.212 odd 78 3549.1.ds.a.212.1 yes 24
1183.550 even 78 inner 3549.1.da.a.1733.1 24
3549.1733 odd 78 inner 3549.1.da.a.1733.1 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3549.1.da.a.1733.1 24 1183.550 even 78 inner
3549.1.da.a.1733.1 24 3549.1733 odd 78 inner
3549.1.da.a.2480.1 yes 24 1.1 even 1 trivial
3549.1.da.a.2480.1 yes 24 3.2 odd 2 CM
3549.1.ds.a.212.1 yes 24 169.43 even 78
3549.1.ds.a.212.1 yes 24 507.212 odd 78
3549.1.ds.a.452.1 yes 24 7.4 even 3
3549.1.ds.a.452.1 yes 24 21.11 odd 6