Properties

Label 3549.1.ds.a.2363.1
Level $3549$
Weight $1$
Character 3549.2363
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(179,3549)]
 
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, 5]))
 
N = Newforms(chi, 1, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3549.179");
 
S:= CuspForms(chi, 1);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3549 = 3 \cdot 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 3549.ds (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: \(\Z[a_1, \ldots, a_{4}]\)
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 2363.1
Root \(-0.632445 - 0.774605i\) of defining polynomial
Character \(\chi\) \(=\) 3549.2363
Dual form 3549.1.ds.a.1577.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(0.748511 + 0.663123i) q^{3} +(-0.692724 - 0.721202i) q^{4} +(-0.632445 - 0.774605i) q^{7} +(0.120537 + 0.992709i) q^{9} +O(q^{10})\) \(q+(0.748511 + 0.663123i) q^{3} +(-0.692724 - 0.721202i) q^{4} +(-0.632445 - 0.774605i) q^{7} +(0.120537 + 0.992709i) q^{9} +(-0.0402659 - 0.999189i) q^{12} +(0.500000 + 0.866025i) q^{13} +(-0.0402659 + 0.999189i) q^{16} +0.783933i q^{19} +(0.0402659 - 0.999189i) q^{21} +(0.996757 - 0.0804666i) q^{25} +(-0.568065 + 0.822984i) q^{27} +(-0.120537 + 0.992709i) q^{28} +(0.432420 - 0.205186i) q^{31} +(0.632445 - 0.774605i) q^{36} +(1.39963 - 0.664135i) q^{37} +(-0.200026 + 0.979791i) q^{39} +(1.26079 - 0.101781i) q^{43} +(-0.692724 + 0.721202i) q^{48} +(-0.200026 + 0.979791i) q^{49} +(0.278217 - 0.960518i) q^{52} +(-0.519844 + 0.586782i) q^{57} +(0.491287 - 1.29542i) q^{61} +(0.692724 - 0.721202i) q^{63} +(0.748511 - 0.663123i) q^{64} +(-0.393906 + 1.59814i) q^{67} +(1.20051 - 1.59759i) q^{73} +(0.799443 + 0.600742i) q^{75} +(0.565375 - 0.543050i) q^{76} +(0.416498 + 1.43792i) q^{79} +(-0.970942 + 0.239316i) q^{81} +(-0.748511 + 0.663123i) q^{84} +(0.354605 - 0.935016i) q^{91} +(0.459734 + 0.133164i) q^{93} +(-1.73065 - 0.0697427i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 24 q + 2 q^{3} - q^{4} + q^{7} - 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 24 q + 2 q^{3} - q^{4} + q^{7} - 2 q^{9} + q^{12} + 12 q^{13} + q^{16} - q^{21} - q^{25} + 2 q^{27} + 2 q^{28} - q^{36} + 3 q^{37} + q^{39} - q^{43} - q^{48} + q^{49} + q^{52} - 2 q^{61} + q^{63} + 2 q^{64} - 3 q^{73} + q^{75} - 3 q^{76} - 2 q^{79} - 2 q^{81} - 2 q^{84} + 2 q^{91} + 13 q^{93} + 3 q^{97}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(1184\) \(1522\) \(3382\)
\(\chi(n)\) \(-1\) \(e\left(\frac{2}{3}\right)\) \(e\left(\frac{23}{78}\right)\)

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 0.391967 0.919979i \(-0.371795\pi\)
−0.391967 + 0.919979i \(0.628205\pi\)
\(3\) 0.748511 + 0.663123i 0.748511 + 0.663123i
\(4\) −0.692724 0.721202i −0.692724 0.721202i
\(5\) 0 0 0.999189 0.0402659i \(-0.0128205\pi\)
−0.999189 + 0.0402659i \(0.987179\pi\)
\(6\) 0 0
\(7\) −0.632445 0.774605i −0.632445 0.774605i
\(8\) 0 0
\(9\) 0.120537 + 0.992709i 0.120537 + 0.992709i
\(10\) 0 0
\(11\) 0 0 −0.992709 0.120537i \(-0.961538\pi\)
0.992709 + 0.120537i \(0.0384615\pi\)
\(12\) −0.0402659 0.999189i −0.0402659 0.999189i
\(13\) 0.500000 + 0.866025i 0.500000 + 0.866025i
\(14\) 0 0
\(15\) 0 0
\(16\) −0.0402659 + 0.999189i −0.0402659 + 0.999189i
\(17\) 0 0 0.632445 0.774605i \(-0.282051\pi\)
−0.632445 + 0.774605i \(0.717949\pi\)
\(18\) 0 0
\(19\) 0.783933i 0.783933i 0.919979 + 0.391967i \(0.128205\pi\)
−0.919979 + 0.391967i \(0.871795\pi\)
\(20\) 0 0
\(21\) 0.0402659 0.999189i 0.0402659 0.999189i
\(22\) 0 0
\(23\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(24\) 0 0
\(25\) 0.996757 0.0804666i 0.996757 0.0804666i
\(26\) 0 0
\(27\) −0.568065 + 0.822984i −0.568065 + 0.822984i
\(28\) −0.120537 + 0.992709i −0.120537 + 0.992709i
\(29\) 0 0 0.799443 0.600742i \(-0.205128\pi\)
−0.799443 + 0.600742i \(0.794872\pi\)
\(30\) 0 0
\(31\) 0.432420 0.205186i 0.432420 0.205186i −0.200026 0.979791i \(-0.564103\pi\)
0.632445 + 0.774605i \(0.282051\pi\)
\(32\) 0 0
\(33\) 0 0
\(34\) 0 0
\(35\) 0 0
\(36\) 0.632445 0.774605i 0.632445 0.774605i
\(37\) 1.39963 0.664135i 1.39963 0.664135i 0.428693 0.903450i \(-0.358974\pi\)
0.970942 + 0.239316i \(0.0769231\pi\)
\(38\) 0 0
\(39\) −0.200026 + 0.979791i −0.200026 + 0.979791i
\(40\) 0 0
\(41\) 0 0 −0.979791 0.200026i \(-0.935897\pi\)
0.979791 + 0.200026i \(0.0641026\pi\)
\(42\) 0 0
\(43\) 1.26079 0.101781i 1.26079 0.101781i 0.568065 0.822984i \(-0.307692\pi\)
0.692724 + 0.721202i \(0.256410\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 0 0 0.960518 0.278217i \(-0.0897436\pi\)
−0.960518 + 0.278217i \(0.910256\pi\)
\(48\) −0.692724 + 0.721202i −0.692724 + 0.721202i
\(49\) −0.200026 + 0.979791i −0.200026 + 0.979791i
\(50\) 0 0
\(51\) 0 0
\(52\) 0.278217 0.960518i 0.278217 0.960518i
\(53\) 0 0 −0.987050 0.160411i \(-0.948718\pi\)
0.987050 + 0.160411i \(0.0512821\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) −0.519844 + 0.586782i −0.519844 + 0.586782i
\(58\) 0 0
\(59\) 0 0 0.999189 0.0402659i \(-0.0128205\pi\)
−0.999189 + 0.0402659i \(0.987179\pi\)
\(60\) 0 0
\(61\) 0.491287 1.29542i 0.491287 1.29542i −0.428693 0.903450i \(-0.641026\pi\)
0.919979 0.391967i \(-0.128205\pi\)
\(62\) 0 0
\(63\) 0.692724 0.721202i 0.692724 0.721202i
\(64\) 0.748511 0.663123i 0.748511 0.663123i
\(65\) 0 0
\(66\) 0 0
\(67\) −0.393906 + 1.59814i −0.393906 + 1.59814i 0.354605 + 0.935016i \(0.384615\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(68\) 0 0
\(69\) 0 0
\(70\) 0 0
\(71\) 0 0 −0.316668 0.948536i \(-0.602564\pi\)
0.316668 + 0.948536i \(0.397436\pi\)
\(72\) 0 0
\(73\) 1.20051 1.59759i 1.20051 1.59759i 0.568065 0.822984i \(-0.307692\pi\)
0.632445 0.774605i \(-0.282051\pi\)
\(74\) 0 0
\(75\) 0.799443 + 0.600742i 0.799443 + 0.600742i
\(76\) 0.565375 0.543050i 0.565375 0.543050i
\(77\) 0 0
\(78\) 0 0
\(79\) 0.416498 + 1.43792i 0.416498 + 1.43792i 0.845190 + 0.534466i \(0.179487\pi\)
−0.428693 + 0.903450i \(0.641026\pi\)
\(80\) 0 0
\(81\) −0.970942 + 0.239316i −0.970942 + 0.239316i
\(82\) 0 0
\(83\) 0 0 −0.663123 0.748511i \(-0.730769\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(84\) −0.748511 + 0.663123i −0.748511 + 0.663123i
\(85\) 0 0
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) 0 0 −0.866025 0.500000i \(-0.833333\pi\)
0.866025 + 0.500000i \(0.166667\pi\)
\(90\) 0 0
\(91\) 0.354605 0.935016i 0.354605 0.935016i
\(92\) 0 0
\(93\) 0.459734 + 0.133164i 0.459734 + 0.133164i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) −1.73065 0.0697427i −1.73065 0.0697427i −0.845190 0.534466i \(-0.820513\pi\)
−0.885456 + 0.464723i \(0.846154\pi\)
\(98\) 0 0
\(99\) 0 0
\(100\) −0.748511 0.663123i −0.748511 0.663123i
\(101\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(102\) 0 0
\(103\) 0.0161084 0.0789044i 0.0161084 0.0789044i −0.970942 0.239316i \(-0.923077\pi\)
0.987050 + 0.160411i \(0.0512821\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 0 0 0.845190 0.534466i \(-0.179487\pi\)
−0.845190 + 0.534466i \(0.820513\pi\)
\(108\) 0.987050 0.160411i 0.987050 0.160411i
\(109\) −0.0860133 1.06547i −0.0860133 1.06547i −0.885456 0.464723i \(-0.846154\pi\)
0.799443 0.600742i \(-0.205128\pi\)
\(110\) 0 0
\(111\) 1.48804 + 0.431017i 1.48804 + 0.431017i
\(112\) 0.799443 0.600742i 0.799443 0.600742i
\(113\) 0 0 −0.200026 0.979791i \(-0.564103\pi\)
0.200026 + 0.979791i \(0.435897\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) −0.799443 + 0.600742i −0.799443 + 0.600742i
\(118\) 0 0
\(119\) 0 0
\(120\) 0 0
\(121\) 0.970942 + 0.239316i 0.970942 + 0.239316i
\(122\) 0 0
\(123\) 0 0
\(124\) −0.447528 0.169725i −0.447528 0.169725i
\(125\) 0 0
\(126\) 0 0
\(127\) 0.470293 + 1.62364i 0.470293 + 1.62364i 0.748511 + 0.663123i \(0.230769\pi\)
−0.278217 + 0.960518i \(0.589744\pi\)
\(128\) 0 0
\(129\) 1.01121 + 0.759873i 1.01121 + 0.759873i
\(130\) 0 0
\(131\) 0 0 0.799443 0.600742i \(-0.205128\pi\)
−0.799443 + 0.600742i \(0.794872\pi\)
\(132\) 0 0
\(133\) 0.607239 0.495795i 0.607239 0.495795i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) 0 0 −0.903450 0.428693i \(-0.858974\pi\)
0.903450 + 0.428693i \(0.141026\pi\)
\(138\) 0 0
\(139\) −0.0457473 + 1.13521i −0.0457473 + 1.13521i 0.799443 + 0.600742i \(0.205128\pi\)
−0.845190 + 0.534466i \(0.820513\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) 0 0
\(143\) 0 0
\(144\) −0.996757 + 0.0804666i −0.996757 + 0.0804666i
\(145\) 0 0
\(146\) 0 0
\(147\) −0.799443 + 0.600742i −0.799443 + 0.600742i
\(148\) −1.44854 0.549357i −1.44854 0.549357i
\(149\) 0 0 0.239316 0.970942i \(-0.423077\pi\)
−0.239316 + 0.970942i \(0.576923\pi\)
\(150\) 0 0
\(151\) −1.58098 0.457937i −1.58098 0.457937i −0.632445 0.774605i \(-0.717949\pi\)
−0.948536 + 0.316668i \(0.897436\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) 0 0
\(156\) 0.845190 0.534466i 0.845190 0.534466i
\(157\) −0.470293 + 0.297395i −0.470293 + 0.297395i −0.748511 0.663123i \(-0.769231\pi\)
0.278217 + 0.960518i \(0.410256\pi\)
\(158\) 0 0
\(159\) 0 0
\(160\) 0 0
\(161\) 0 0
\(162\) 0 0
\(163\) −0.132445 0.0914204i −0.132445 0.0914204i 0.500000 0.866025i \(-0.333333\pi\)
−0.632445 + 0.774605i \(0.717949\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 0 0 −0.600742 0.799443i \(-0.705128\pi\)
0.600742 + 0.799443i \(0.294872\pi\)
\(168\) 0 0
\(169\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(170\) 0 0
\(171\) −0.778217 + 0.0944927i −0.778217 + 0.0944927i
\(172\) −0.946784 0.838778i −0.946784 0.838778i
\(173\) 0 0 0.970942 0.239316i \(-0.0769231\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(174\) 0 0
\(175\) −0.692724 0.721202i −0.692724 0.721202i
\(176\) 0 0
\(177\) 0 0
\(178\) 0 0
\(179\) 0 0 0.354605 0.935016i \(-0.384615\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(180\) 0 0
\(181\) −1.76517 0.926432i −1.76517 0.926432i −0.919979 0.391967i \(-0.871795\pi\)
−0.845190 0.534466i \(-0.820513\pi\)
\(182\) 0 0
\(183\) 1.22675 0.643850i 1.22675 0.643850i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 0 0
\(188\) 0 0
\(189\) 0.996757 0.0804666i 0.996757 0.0804666i
\(190\) 0 0
\(191\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(192\) 1.00000 1.00000
\(193\) −0.299974 0.113765i −0.299974 0.113765i 0.200026 0.979791i \(-0.435897\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0.845190 0.534466i 0.845190 0.534466i
\(197\) 0 0 −0.903450 0.428693i \(-0.858974\pi\)
0.903450 + 0.428693i \(0.141026\pi\)
\(198\) 0 0
\(199\) −1.55512 0.983395i −1.55512 0.983395i −0.987050 0.160411i \(-0.948718\pi\)
−0.568065 0.822984i \(-0.692308\pi\)
\(200\) 0 0
\(201\) −1.35460 + 0.935016i −1.35460 + 0.935016i
\(202\) 0 0
\(203\) 0 0
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 0 0
\(208\) −0.885456 + 0.464723i −0.885456 + 0.464723i
\(209\) 0 0
\(210\) 0 0
\(211\) 0.554631 + 1.91481i 0.554631 + 1.91481i 0.354605 + 0.935016i \(0.384615\pi\)
0.200026 + 0.979791i \(0.435897\pi\)
\(212\) 0 0
\(213\) 0 0
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) −0.432420 0.205186i −0.432420 0.205186i
\(218\) 0 0
\(219\) 1.95799 0.399727i 1.95799 0.399727i
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 0 0 −0.0402659 0.999189i \(-0.512821\pi\)
0.0402659 + 0.999189i \(0.487179\pi\)
\(224\) 0 0
\(225\) 0.200026 + 0.979791i 0.200026 + 0.979791i
\(226\) 0 0
\(227\) 0 0 0.721202 0.692724i \(-0.243590\pi\)
−0.721202 + 0.692724i \(0.756410\pi\)
\(228\) 0.783297 0.0315658i 0.783297 0.0315658i
\(229\) −0.965727 0.458243i −0.965727 0.458243i −0.120537 0.992709i \(-0.538462\pi\)
−0.845190 + 0.534466i \(0.820513\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 0 0 −0.948536 0.316668i \(-0.897436\pi\)
0.948536 + 0.316668i \(0.102564\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) −0.641762 + 1.35248i −0.641762 + 1.35248i
\(238\) 0 0
\(239\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(240\) 0 0
\(241\) −0.255812 + 0.404534i −0.255812 + 0.404534i −0.948536 0.316668i \(-0.897436\pi\)
0.692724 + 0.721202i \(0.256410\pi\)
\(242\) 0 0
\(243\) −0.885456 0.464723i −0.885456 0.464723i
\(244\) −1.27458 + 0.543050i −1.27458 + 0.543050i
\(245\) 0 0
\(246\) 0 0
\(247\) −0.678906 + 0.391967i −0.678906 + 0.391967i
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 0 0 −0.428693 0.903450i \(-0.641026\pi\)
0.428693 + 0.903450i \(0.358974\pi\)
\(252\) −1.00000 −1.00000
\(253\) 0 0
\(254\) 0 0
\(255\) 0 0
\(256\) −0.996757 0.0804666i −0.996757 0.0804666i
\(257\) 0 0 0.278217 0.960518i \(-0.410256\pi\)
−0.278217 + 0.960518i \(0.589744\pi\)
\(258\) 0 0
\(259\) −1.39963 0.664135i −1.39963 0.664135i
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 0.120537 0.992709i \(-0.461538\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) 1.42545 0.822984i 1.42545 0.822984i
\(269\) 0 0 0.200026 0.979791i \(-0.435897\pi\)
−0.200026 + 0.979791i \(0.564103\pi\)
\(270\) 0 0
\(271\) 1.38546 + 1.33075i 1.38546 + 1.33075i 0.885456 + 0.464723i \(0.153846\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(272\) 0 0
\(273\) 0.885456 0.464723i 0.885456 0.464723i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) 1.87251 0.304312i 1.87251 0.304312i 0.885456 0.464723i \(-0.153846\pi\)
0.987050 + 0.160411i \(0.0512821\pi\)
\(278\) 0 0
\(279\) 0.255812 + 0.404534i 0.255812 + 0.404534i
\(280\) 0 0
\(281\) 0 0 0.663123 0.748511i \(-0.269231\pi\)
−0.663123 + 0.748511i \(0.730769\pi\)
\(282\) 0 0
\(283\) 0.0457473 0.0662764i 0.0457473 0.0662764i −0.799443 0.600742i \(-0.794872\pi\)
0.845190 + 0.534466i \(0.179487\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −0.200026 0.979791i −0.200026 0.979791i
\(290\) 0 0
\(291\) −1.24916 1.19983i −1.24916 1.19983i
\(292\) −1.98381 + 0.240878i −1.98381 + 0.240878i
\(293\) 0 0 0.774605 0.632445i \(-0.217949\pi\)
−0.774605 + 0.632445i \(0.782051\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) 0 0
\(299\) 0 0
\(300\) −0.120537 0.992709i −0.120537 0.992709i
\(301\) −0.876221 0.912242i −0.876221 0.912242i
\(302\) 0 0
\(303\) 0 0
\(304\) −0.783297 0.0315658i −0.783297 0.0315658i
\(305\) 0 0
\(306\) 0 0
\(307\) −1.53901 1.06230i −1.53901 1.06230i −0.970942 0.239316i \(-0.923077\pi\)
−0.568065 0.822984i \(-0.692308\pi\)
\(308\) 0 0
\(309\) 0.0643806 0.0483789i 0.0643806 0.0483789i
\(310\) 0 0
\(311\) 0 0 −0.428693 0.903450i \(-0.641026\pi\)
0.428693 + 0.903450i \(0.358974\pi\)
\(312\) 0 0
\(313\) −0.813261 + 1.71391i −0.813261 + 1.71391i −0.120537 + 0.992709i \(0.538462\pi\)
−0.692724 + 0.721202i \(0.743590\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0.748511 1.29646i 0.748511 1.29646i
\(317\) 0 0 −0.160411 0.987050i \(-0.551282\pi\)
0.160411 + 0.987050i \(0.448718\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 0 0
\(321\) 0 0
\(322\) 0 0
\(323\) 0 0
\(324\) 0.845190 + 0.534466i 0.845190 + 0.534466i
\(325\) 0.568065 + 0.822984i 0.568065 + 0.822984i
\(326\) 0 0
\(327\) 0.642152 0.854550i 0.642152 0.854550i
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 1.68948 0.640736i 1.68948 0.640736i 0.692724 0.721202i \(-0.256410\pi\)
0.996757 + 0.0804666i \(0.0256410\pi\)
\(332\) 0 0
\(333\) 0.828000 + 1.30938i 0.828000 + 1.30938i
\(334\) 0 0
\(335\) 0 0
\(336\) 0.996757 + 0.0804666i 0.996757 + 0.0804666i
\(337\) −1.97410 −1.97410 −0.987050 0.160411i \(-0.948718\pi\)
−0.987050 + 0.160411i \(0.948718\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 0 0
\(341\) 0 0
\(342\) 0 0
\(343\) 0.885456 0.464723i 0.885456 0.464723i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) 0 0 0.799443 0.600742i \(-0.205128\pi\)
−0.799443 + 0.600742i \(0.794872\pi\)
\(348\) 0 0
\(349\) −0.678906 1.59345i −0.678906 1.59345i −0.799443 0.600742i \(-0.794872\pi\)
0.120537 0.992709i \(-0.461538\pi\)
\(350\) 0 0
\(351\) −0.996757 0.0804666i −0.996757 0.0804666i
\(352\) 0 0
\(353\) 0 0 0.935016 0.354605i \(-0.115385\pi\)
−0.935016 + 0.354605i \(0.884615\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 0 0
\(358\) 0 0
\(359\) 0 0 −0.160411 0.987050i \(-0.551282\pi\)
0.160411 + 0.987050i \(0.448718\pi\)
\(360\) 0 0
\(361\) 0.385449 0.385449
\(362\) 0 0
\(363\) 0.568065 + 0.822984i 0.568065 + 0.822984i
\(364\) −0.919979 + 0.391967i −0.919979 + 0.391967i
\(365\) 0 0
\(366\) 0 0
\(367\) −0.0482209 0.397135i −0.0482209 0.397135i −0.996757 0.0804666i \(-0.974359\pi\)
0.948536 0.316668i \(-0.102564\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) 0 0
\(372\) −0.222431 0.423807i −0.222431 0.423807i
\(373\) 0.234068 1.92773i 0.234068 1.92773i −0.120537 0.992709i \(-0.538462\pi\)
0.354605 0.935016i \(-0.384615\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) 0 0
\(378\) 0 0
\(379\) −1.94529 0.397135i −1.94529 0.397135i −0.996757 0.0804666i \(-0.974359\pi\)
−0.948536 0.316668i \(-0.897436\pi\)
\(380\) 0 0
\(381\) −0.724653 + 1.52717i −0.724653 + 1.52717i
\(382\) 0 0
\(383\) 0 0 −0.935016 0.354605i \(-0.884615\pi\)
0.935016 + 0.354605i \(0.115385\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 0.253011 + 1.23933i 0.253011 + 1.23933i
\(388\) 1.14856 + 1.29646i 1.14856 + 1.29646i
\(389\) 0 0 −0.632445 0.774605i \(-0.717949\pi\)
0.632445 + 0.774605i \(0.282051\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 0.670319 1.27719i 0.670319 1.27719i −0.278217 0.960518i \(-0.589744\pi\)
0.948536 0.316668i \(-0.102564\pi\)
\(398\) 0 0
\(399\) 0.783297 + 0.0315658i 0.783297 + 0.0315658i
\(400\) 0.0402659 + 0.999189i 0.0402659 + 0.999189i
\(401\) 0 0 −0.999189 0.0402659i \(-0.987179\pi\)
0.999189 + 0.0402659i \(0.0128205\pi\)
\(402\) 0 0
\(403\) 0.393906 + 0.271894i 0.393906 + 0.271894i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) 0 0
\(408\) 0 0
\(409\) 1.17720 + 0.240328i 1.17720 + 0.240328i 0.748511 0.663123i \(-0.230769\pi\)
0.428693 + 0.903450i \(0.358974\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) −0.0680647 + 0.0430415i −0.0680647 + 0.0430415i
\(413\) 0 0
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) −0.787025 + 0.819379i −0.787025 + 0.819379i
\(418\) 0 0
\(419\) 0 0 −0.692724 0.721202i \(-0.743590\pi\)
0.692724 + 0.721202i \(0.256410\pi\)
\(420\) 0 0
\(421\) 0.616337 + 0.695701i 0.616337 + 0.695701i 0.970942 0.239316i \(-0.0769231\pi\)
−0.354605 + 0.935016i \(0.615385\pi\)
\(422\) 0 0
\(423\) 0 0
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −1.31415 + 0.438727i −1.31415 + 0.438727i
\(428\) 0 0
\(429\) 0 0
\(430\) 0 0
\(431\) 0 0 −0.239316 0.970942i \(-0.576923\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(432\) −0.799443 0.600742i −0.799443 0.600742i
\(433\) 0.203753 0.128845i 0.203753 0.128845i −0.428693 0.903450i \(-0.641026\pi\)
0.632445 + 0.774605i \(0.282051\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) −0.708833 + 0.800107i −0.708833 + 0.800107i
\(437\) 0 0
\(438\) 0 0
\(439\) 1.83399 0.148055i 1.83399 0.148055i 0.885456 0.464723i \(-0.153846\pi\)
0.948536 + 0.316668i \(0.102564\pi\)
\(440\) 0 0
\(441\) −0.996757 0.0804666i −0.996757 0.0804666i
\(442\) 0 0
\(443\) 0 0 −0.948536 0.316668i \(-0.897436\pi\)
0.948536 + 0.316668i \(0.102564\pi\)
\(444\) −0.719954 1.37176i −0.719954 1.37176i
\(445\) 0 0
\(446\) 0 0
\(447\) 0 0
\(448\) −0.987050 0.160411i −0.987050 0.160411i
\(449\) 0 0 −0.960518 0.278217i \(-0.910256\pi\)
0.960518 + 0.278217i \(0.0897436\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −0.879714 1.39116i −0.879714 1.39116i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 0.910663 0.185913i 0.910663 0.185913i 0.278217 0.960518i \(-0.410256\pi\)
0.632445 + 0.774605i \(0.282051\pi\)
\(458\) 0 0
\(459\) 0 0
\(460\) 0 0
\(461\) 0 0 0.391967 0.919979i \(-0.371795\pi\)
−0.391967 + 0.919979i \(0.628205\pi\)
\(462\) 0 0
\(463\) −1.19272 + 0.144823i −1.19272 + 0.144823i −0.692724 0.721202i \(-0.743590\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 0 0 0.919979 0.391967i \(-0.128205\pi\)
−0.919979 + 0.391967i \(0.871795\pi\)
\(468\) 0.987050 + 0.160411i 0.987050 + 0.160411i
\(469\) 1.48705 0.705614i 1.48705 0.705614i
\(470\) 0 0
\(471\) −0.549229 0.0892584i −0.549229 0.0892584i
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 0.0630804 + 0.781391i 0.0630804 + 0.781391i
\(476\) 0 0
\(477\) 0 0
\(478\) 0 0
\(479\) 0 0 0.822984 0.568065i \(-0.192308\pi\)
−0.822984 + 0.568065i \(0.807692\pi\)
\(480\) 0 0
\(481\) 1.27497 + 0.880052i 1.27497 + 0.880052i
\(482\) 0 0
\(483\) 0 0
\(484\) −0.500000 0.866025i −0.500000 0.866025i
\(485\) 0 0
\(486\) 0 0
\(487\) 1.84519 0.534466i 1.84519 0.534466i 0.845190 0.534466i \(-0.179487\pi\)
1.00000 \(0\)
\(488\) 0 0
\(489\) −0.0385138 0.156257i −0.0385138 0.156257i
\(490\) 0 0
\(491\) 0 0 −0.845190 0.534466i \(-0.820513\pi\)
0.845190 + 0.534466i \(0.179487\pi\)
\(492\) 0 0
\(493\) 0 0
\(494\) 0 0
\(495\) 0 0
\(496\) 0.187607 + 0.440331i 0.187607 + 0.440331i
\(497\) 0 0
\(498\) 0 0
\(499\) −1.54795 1.26386i −1.54795 1.26386i −0.799443 0.600742i \(-0.794872\pi\)
−0.748511 0.663123i \(-0.769231\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 0 0
\(503\) 0 0 0.278217 0.960518i \(-0.410256\pi\)
−0.278217 + 0.960518i \(0.589744\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −0.948536 + 0.316668i −0.948536 + 0.316668i
\(508\) 0.845190 1.46391i 0.845190 1.46391i
\(509\) 0 0 0.391967 0.919979i \(-0.371795\pi\)
−0.391967 + 0.919979i \(0.628205\pi\)
\(510\) 0 0
\(511\) −1.99676 + 0.0804666i −1.99676 + 0.0804666i
\(512\) 0 0
\(513\) −0.645164 0.445325i −0.645164 0.445325i
\(514\) 0 0
\(515\) 0 0
\(516\) −0.152466 1.25567i −0.152466 1.25567i
\(517\) 0 0
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 0 0 0.0402659 0.999189i \(-0.487179\pi\)
−0.0402659 + 0.999189i \(0.512821\pi\)
\(522\) 0 0
\(523\) −0.0643806 + 1.59759i −0.0643806 + 1.59759i 0.568065 + 0.822984i \(0.307692\pi\)
−0.632445 + 0.774605i \(0.717949\pi\)
\(524\) 0 0
\(525\) −0.0402659 0.999189i −0.0402659 0.999189i
\(526\) 0 0
\(527\) 0 0
\(528\) 0 0
\(529\) −0.500000 0.866025i −0.500000 0.866025i
\(530\) 0 0
\(531\) 0 0
\(532\) −0.778217 0.0944927i −0.778217 0.0944927i
\(533\) 0 0
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) 0 0
\(539\) 0 0
\(540\) 0 0
\(541\) 0.965727 1.52717i 0.965727 1.52717i 0.120537 0.992709i \(-0.461538\pi\)
0.845190 0.534466i \(-0.179487\pi\)
\(542\) 0 0
\(543\) −0.706910 1.86397i −0.706910 1.86397i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 0.0670708 0.552378i 0.0670708 0.552378i −0.919979 0.391967i \(-0.871795\pi\)
0.987050 0.160411i \(-0.0512821\pi\)
\(548\) 0 0
\(549\) 1.34519 + 0.331560i 1.34519 + 0.331560i
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) 0.850405 1.23202i 0.850405 1.23202i
\(554\) 0 0
\(555\) 0 0
\(556\) 0.850405 0.753393i 0.850405 0.753393i
\(557\) 0 0 −0.663123 0.748511i \(-0.730769\pi\)
0.663123 + 0.748511i \(0.269231\pi\)
\(558\) 0 0
\(559\) 0.718540 + 1.04098i 0.718540 + 1.04098i
\(560\) 0 0
\(561\) 0 0
\(562\) 0 0
\(563\) 0 0 −0.948536 0.316668i \(-0.897436\pi\)
0.948536 + 0.316668i \(0.102564\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.799443 + 0.600742i 0.799443 + 0.600742i
\(568\) 0 0
\(569\) 0 0 0.845190 0.534466i \(-0.179487\pi\)
−0.845190 + 0.534466i \(0.820513\pi\)
\(570\) 0 0
\(571\) −0.0800206 0.391967i −0.0800206 0.391967i 0.919979 0.391967i \(-0.128205\pi\)
−1.00000 \(\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) 0 0
\(575\) 0 0
\(576\) 0.748511 + 0.663123i 0.748511 + 0.663123i
\(577\) −0.139372 0.0804666i −0.139372 0.0804666i 0.428693 0.903450i \(-0.358974\pi\)
−0.568065 + 0.822984i \(0.692308\pi\)
\(578\) 0 0
\(579\) −0.149094 0.284074i −0.149094 0.284074i
\(580\) 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 \(0.833333\pi\)
\(588\) 0.987050 + 0.160411i 0.987050 + 0.160411i
\(589\) 0.160852 + 0.338988i 0.160852 + 0.338988i
\(590\) 0 0
\(591\) 0 0
\(592\) 0.607239 + 1.42524i 0.607239 + 1.42524i
\(593\) 0 0 0.316668 0.948536i \(-0.397436\pi\)
−0.316668 + 0.948536i \(0.602564\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) −0.511909 1.76731i −0.511909 1.76731i
\(598\) 0 0
\(599\) 0 0 0.278217 0.960518i \(-0.410256\pi\)
−0.278217 + 0.960518i \(0.589744\pi\)
\(600\) 0 0
\(601\) −1.16367 + 0.495795i −1.16367 + 0.495795i −0.885456 0.464723i \(-0.846154\pi\)
−0.278217 + 0.960518i \(0.589744\pi\)
\(602\) 0 0
\(603\) −1.63397 0.198399i −1.63397 0.198399i
\(604\) 0.764919 + 1.45743i 0.764919 + 1.45743i
\(605\) 0 0
\(606\) 0 0
\(607\) −1.47764 1.30907i −1.47764 1.30907i −0.845190 0.534466i \(-0.820513\pi\)
−0.632445 0.774605i \(-0.717949\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 0 0
\(612\) 0 0
\(613\) 0.149094 + 0.284074i 0.149094 + 0.284074i 0.948536 0.316668i \(-0.102564\pi\)
−0.799443 + 0.600742i \(0.794872\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 0 0 −0.534466 0.845190i \(-0.679487\pi\)
0.534466 + 0.845190i \(0.320513\pi\)
\(618\) 0 0
\(619\) −0.490585 1.46948i −0.490585 1.46948i −0.845190 0.534466i \(-0.820513\pi\)
0.354605 0.935016i \(-0.384615\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 0 0
\(624\) −0.970942 0.239316i −0.970942 0.239316i
\(625\) 0.987050 0.160411i 0.987050 0.160411i
\(626\) 0 0
\(627\) 0 0
\(628\) 0.540266 + 0.133164i 0.540266 + 0.133164i
\(629\) 0 0
\(630\) 0 0
\(631\) 1.20003 0.979791i 1.20003 0.979791i 0.200026 0.979791i \(-0.435897\pi\)
1.00000 \(0\)
\(632\) 0 0
\(633\) −0.854605 + 1.80104i −0.854605 + 1.80104i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) −0.948536 + 0.316668i −0.948536 + 0.316668i
\(638\) 0 0
\(639\) 0 0
\(640\) 0 0
\(641\) 0 0 0.919979 0.391967i \(-0.128205\pi\)
−0.919979 + 0.391967i \(0.871795\pi\)
\(642\) 0 0
\(643\) −0.314339 + 0.0641728i −0.314339 + 0.0641728i −0.354605 0.935016i \(-0.615385\pi\)
0.0402659 + 0.999189i \(0.487179\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 0 0 −0.120537 0.992709i \(-0.538462\pi\)
0.120537 + 0.992709i \(0.461538\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) −0.187607 0.440331i −0.187607 0.440331i
\(652\) 0.0258155 + 0.158849i 0.0258155 + 0.158849i
\(653\) 0 0 0.500000 0.866025i \(-0.333333\pi\)
−0.500000 + 0.866025i \(0.666667\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 1.73065 + 0.999189i 1.73065 + 0.999189i
\(658\) 0 0
\(659\) 0 0 −0.987050 0.160411i \(-0.948718\pi\)
0.987050 + 0.160411i \(0.0512821\pi\)
\(660\) 0 0
\(661\) −1.85640 + 0.704039i −1.85640 + 0.704039i −0.885456 + 0.464723i \(0.846154\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(662\) 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
\(673\) −0.319818 + 1.56657i −0.319818 + 1.56657i 0.428693 + 0.903450i \(0.358974\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(674\) 0 0
\(675\) −0.500000 + 0.866025i −0.500000 + 0.866025i
\(676\) 0.970942 0.239316i 0.970942 0.239316i
\(677\) 0 0 −0.500000 0.866025i \(-0.666667\pi\)
0.500000 + 0.866025i \(0.333333\pi\)
\(678\) 0 0
\(679\) 1.04052 + 1.38468i 1.04052 + 1.38468i
\(680\) 0 0
\(681\) 0 0
\(682\) 0 0
\(683\) 0 0 −0.774605 0.632445i \(-0.782051\pi\)
0.774605 + 0.632445i \(0.217949\pi\)
\(684\) 0.607239 + 0.495795i 0.607239 + 0.495795i
\(685\) 0 0
\(686\) 0 0
\(687\) −0.418986 0.983395i −0.418986 0.983395i
\(688\) 0.0509320 + 1.26386i 0.0509320 + 1.26386i
\(689\) 0 0
\(690\) 0 0
\(691\) 0.231378 1.42373i 0.231378 1.42373i −0.568065 0.822984i \(-0.692308\pi\)
0.799443 0.600742i \(-0.205128\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 0 0
\(700\) −0.0402659 + 0.999189i −0.0402659 + 0.999189i
\(701\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(702\) 0 0
\(703\) 0.520637 + 1.09722i 0.520637 + 1.09722i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) −0.106718 0.120460i −0.106718 0.120460i 0.692724 0.721202i \(-0.256410\pi\)
−0.799443 + 0.600742i \(0.794872\pi\)
\(710\) 0 0
\(711\) −1.37723 + 0.586782i −1.37723 + 0.586782i
\(712\) 0 0
\(713\) 0 0
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) 0 0
\(719\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(720\) 0 0
\(721\) −0.0713074 + 0.0374250i −0.0713074 + 0.0374250i
\(722\) 0 0
\(723\) −0.459734 + 0.133164i −0.459734 + 0.133164i
\(724\) 0.554631 + 1.91481i 0.554631 + 1.91481i
\(725\) 0 0
\(726\) 0 0
\(727\) −0.251489 + 0.663123i −0.251489 + 0.663123i 0.748511 + 0.663123i \(0.230769\pi\)
−1.00000 \(\pi\)
\(728\) 0 0
\(729\) −0.354605 0.935016i −0.354605 0.935016i
\(730\) 0 0
\(731\) 0 0
\(732\) −1.31415 0.438727i −1.31415 0.438727i
\(733\) 1.83224 + 0.374055i 1.83224 + 0.374055i 0.987050 0.160411i \(-0.0512821\pi\)
0.845190 + 0.534466i \(0.179487\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 0 0
\(738\) 0 0
\(739\) −0.670319 1.27719i −0.670319 1.27719i −0.948536 0.316668i \(-0.897436\pi\)
0.278217 0.960518i \(-0.410256\pi\)
\(740\) 0 0
\(741\) −0.768090 0.156807i −0.768090 0.156807i
\(742\) 0 0
\(743\) 0 0 0.960518 0.278217i \(-0.0897436\pi\)
−0.960518 + 0.278217i \(0.910256\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 0 0
\(748\) 0 0
\(749\) 0 0
\(750\) 0 0
\(751\) −1.35136 1.01548i −1.35136 1.01548i −0.996757 0.0804666i \(-0.974359\pi\)
−0.354605 0.935016i \(-0.615385\pi\)
\(752\) 0 0
\(753\) 0 0
\(754\) 0 0
\(755\) 0 0
\(756\) −0.748511 0.663123i −0.748511 0.663123i
\(757\) 0.491287 + 0.511484i 0.491287 + 0.511484i 0.919979 0.391967i \(-0.128205\pi\)
−0.428693 + 0.903450i \(0.641026\pi\)
\(758\) 0 0
\(759\) 0 0
\(760\) 0 0
\(761\) 0 0 −0.992709 0.120537i \(-0.961538\pi\)
0.992709 + 0.120537i \(0.0384615\pi\)
\(762\) 0 0
\(763\) −0.770916 + 0.740475i −0.770916 + 0.740475i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 0 0
\(768\) −0.692724 0.721202i −0.692724 0.721202i
\(769\) −1.30314 + 1.25168i −1.30314 + 1.25168i −0.354605 + 0.935016i \(0.615385\pi\)
−0.948536 + 0.316668i \(0.897436\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 0.125752 + 0.295150i 0.125752 + 0.295150i
\(773\) 0 0 0.534466 0.845190i \(-0.320513\pi\)
−0.534466 + 0.845190i \(0.679487\pi\)
\(774\) 0 0
\(775\) 0.414507 0.239316i 0.414507 0.239316i
\(776\) 0 0
\(777\) −0.607239 1.42524i −0.607239 1.42524i
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 0 0
\(782\) 0 0
\(783\) 0 0
\(784\) −0.970942 0.239316i −0.970942 0.239316i
\(785\) 0 0
\(786\) 0 0
\(787\) −1.44124 + 1.38433i −1.44124 + 1.38433i −0.692724 + 0.721202i \(0.743590\pi\)
−0.748511 + 0.663123i \(0.769231\pi\)
\(788\) 0 0
\(789\) 0 0
\(790\) 0 0
\(791\) 0 0
\(792\) 0 0
\(793\) 1.36751 0.222242i 1.36751 0.222242i
\(794\) 0 0
\(795\) 0 0
\(796\) 0.368039 + 1.80277i 0.368039 + 1.80277i
\(797\) 0 0 0.692724 0.721202i \(-0.256410\pi\)
−0.692724 + 0.721202i \(0.743590\pi\)
\(798\) 0 0
\(799\) 0 0
\(800\) 0 0
\(801\) 0 0
\(802\) 0 0
\(803\) 0 0
\(804\) 1.61270 + 0.329236i 1.61270 + 0.329236i
\(805\) 0 0
\(806\) 0 0
\(807\) 0 0
\(808\) 0 0
\(809\) 0 0 −0.354605 0.935016i \(-0.615385\pi\)
0.354605 + 0.935016i \(0.384615\pi\)
\(810\) 0 0
\(811\) −0.364312 0.694138i −0.364312 0.694138i 0.632445 0.774605i \(-0.282051\pi\)
−0.996757 + 0.0804666i \(0.974359\pi\)
\(812\) 0 0
\(813\) 0.154579 + 1.91481i 0.154579 + 1.91481i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) 0.0797898 + 0.988374i 0.0797898 + 0.988374i
\(818\) 0 0
\(819\) 0.970942 + 0.239316i 0.970942 + 0.239316i
\(820\) 0 0
\(821\) 0 0 0.774605 0.632445i \(-0.217949\pi\)
−0.774605 + 0.632445i \(0.782051\pi\)
\(822\) 0 0
\(823\) −0.120537 0.208776i −0.120537 0.208776i 0.799443 0.600742i \(-0.205128\pi\)
−0.919979 + 0.391967i \(0.871795\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) 0 0 −0.239316 0.970942i \(-0.576923\pi\)
0.239316 + 0.970942i \(0.423077\pi\)
\(828\) 0 0
\(829\) 1.49676 0.785559i 1.49676 0.785559i 0.500000 0.866025i \(-0.333333\pi\)
0.996757 + 0.0804666i \(0.0256410\pi\)
\(830\) 0 0
\(831\) 1.60339 + 1.01392i 1.60339 + 1.01392i
\(832\) 0.948536 + 0.316668i 0.948536 + 0.316668i
\(833\) 0 0
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) −0.0767779 + 0.472433i −0.0767779 + 0.472433i
\(838\) 0 0
\(839\) 0 0 0.903450 0.428693i \(-0.141026\pi\)
−0.903450 + 0.428693i \(0.858974\pi\)
\(840\) 0 0
\(841\) 0.278217 0.960518i 0.278217 0.960518i
\(842\) 0 0
\(843\) 0 0
\(844\) 0.996757 1.72643i 0.996757 1.72643i
\(845\) 0 0
\(846\) 0 0
\(847\) −0.428693 0.903450i −0.428693 0.903450i
\(848\) 0 0
\(849\) 0.0781918 0.0192725i 0.0781918 0.0192725i
\(850\) 0 0
\(851\) 0 0
\(852\) 0 0
\(853\) −0.869047 + 0.329586i −0.869047 + 0.329586i −0.748511 0.663123i \(-0.769231\pi\)
−0.120537 + 0.992709i \(0.538462\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 0 0 0.845190 0.534466i \(-0.179487\pi\)
−0.845190 + 0.534466i \(0.820513\pi\)
\(858\) 0 0
\(859\) −0.338119 0.213814i −0.338119 0.213814i 0.354605 0.935016i \(-0.384615\pi\)
−0.692724 + 0.721202i \(0.743590\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 0 0 0.721202 0.692724i \(-0.243590\pi\)
−0.721202 + 0.692724i \(0.756410\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) 0.500000 0.866025i 0.500000 0.866025i
\(868\) 0.151567 + 0.453999i 0.151567 + 0.453999i
\(869\) 0 0
\(870\) 0 0
\(871\) −1.58098 + 0.457937i −1.58098 + 0.457937i
\(872\) 0 0
\(873\) −0.139372 1.72643i −0.139372 1.72643i
\(874\) 0 0
\(875\) 0 0
\(876\) −1.64463 1.13521i −1.64463 1.13521i
\(877\) −1.53901 + 1.06230i −1.53901 + 1.06230i −0.568065 + 0.822984i \(0.692308\pi\)
−0.970942 + 0.239316i \(0.923077\pi\)
\(878\) 0 0
\(879\) 0 0
\(880\) 0 0
\(881\) 0 0 0.632445 0.774605i \(-0.282051\pi\)
−0.632445 + 0.774605i \(0.717949\pi\)
\(882\) 0 0
\(883\) 0.152466 + 1.25567i 0.152466 + 1.25567i 0.845190 + 0.534466i \(0.179487\pi\)
−0.692724 + 0.721202i \(0.743590\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 0 0 0.692724 0.721202i \(-0.256410\pi\)
−0.692724 + 0.721202i \(0.743590\pi\)
\(888\) 0 0
\(889\) 0.960245 1.39116i 0.960245 1.39116i
\(890\) 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
\(900\) 0.568065 0.822984i 0.568065 0.822984i
\(901\) 0 0
\(902\) 0 0
\(903\) −0.0509320 1.26386i −0.0509320 1.26386i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) −0.759177 0.398447i −0.759177 0.398447i 0.0402659 0.999189i \(-0.487179\pi\)
−0.799443 + 0.600742i \(0.794872\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) 0 0 −0.748511 0.663123i \(-0.769231\pi\)
0.748511 + 0.663123i \(0.230769\pi\)
\(912\) −0.565375 0.543050i −0.565375 0.543050i
\(913\) 0 0
\(914\) 0 0
\(915\) 0 0
\(916\) 0.338496 + 1.01392i 0.338496 + 1.01392i
\(917\) 0 0
\(918\) 0 0
\(919\) −0.759177 0.398447i −0.759177 0.398447i 0.0402659 0.999189i \(-0.487179\pi\)
−0.799443 + 0.600742i \(0.794872\pi\)
\(920\) 0 0
\(921\) −0.447528 1.81569i −0.447528 1.81569i
\(922\) 0 0
\(923\) 0 0
\(924\) 0 0
\(925\) 1.34166 0.774605i 1.34166 0.774605i
\(926\) 0 0
\(927\) 0.0802707 + 0.00648012i 0.0802707 + 0.00648012i
\(928\) 0 0
\(929\) 0 0 −0.992709 0.120537i \(-0.961538\pi\)
0.992709 + 0.120537i \(0.0384615\pi\)
\(930\) 0 0
\(931\) −0.768090 0.156807i −0.768090 0.156807i
\(932\) 0 0
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 1.55242 0.382638i 1.55242 0.382638i 0.632445 0.774605i \(-0.282051\pi\)
0.919979 + 0.391967i \(0.128205\pi\)
\(938\) 0 0
\(939\) −1.74527 + 0.743589i −1.74527 + 0.743589i
\(940\) 0 0
\(941\) 0 0 0.600742 0.799443i \(-0.294872\pi\)
−0.600742 + 0.799443i \(0.705128\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 0 0 −0.721202 0.692724i \(-0.756410\pi\)
0.721202 + 0.692724i \(0.243590\pi\)
\(948\) 1.41998 0.474059i 1.41998 0.474059i
\(949\) 1.98381 + 0.240878i 1.98381 + 0.240878i
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) 0 0 −0.632445 0.774605i \(-0.717949\pi\)
0.632445 + 0.774605i \(0.282051\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 0 0
\(958\) 0 0
\(959\) 0 0
\(960\) 0 0
\(961\) −0.487560 + 0.597152i −0.487560 + 0.597152i
\(962\) 0 0
\(963\) 0 0
\(964\) 0.468959 0.0957386i 0.468959 0.0957386i
\(965\) 0 0
\(966\) 0 0
\(967\) −0.345190 + 1.40049i −0.345190 + 1.40049i 0.500000 + 0.866025i \(0.333333\pi\)
−0.845190 + 0.534466i \(0.820513\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) 0 0 −0.568065 0.822984i \(-0.692308\pi\)
0.568065 + 0.822984i \(0.307692\pi\)
\(972\) 0.278217 + 0.960518i 0.278217 + 0.960518i
\(973\) 0.908271 0.682521i 0.908271 0.682521i
\(974\) 0 0
\(975\) −0.120537 + 0.992709i −0.120537 + 0.992709i
\(976\) 1.27458 + 0.543050i 1.27458 + 0.543050i
\(977\) 0 0 −0.822984 0.568065i \(-0.807692\pi\)
0.822984 + 0.568065i \(0.192308\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) 0 0
\(981\) 1.04733 0.213814i 1.04733 0.213814i
\(982\) 0 0
\(983\) 0 0 0.903450 0.428693i \(-0.141026\pi\)
−0.903450 + 0.428693i \(0.858974\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 0 0
\(988\) 0.752982 + 0.218104i 0.752982 + 0.218104i
\(989\) 0 0
\(990\) 0 0
\(991\) −1.83996 −1.83996 −0.919979 0.391967i \(-0.871795\pi\)
−0.919979 + 0.391967i \(0.871795\pi\)
\(992\) 0 0
\(993\) 1.68948 + 0.640736i 1.68948 + 0.640736i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) −0.542249 + 0.664135i −0.542249 + 0.664135i −0.970942 0.239316i \(-0.923077\pi\)
0.428693 + 0.903450i \(0.358974\pi\)
\(998\) 0 0
\(999\) −0.248511 + 1.52915i −0.248511 + 1.52915i
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.ds.a.2363.1 yes 24
3.2 odd 2 CM 3549.1.ds.a.2363.1 yes 24
7.2 even 3 3549.1.da.a.842.1 24
21.2 odd 6 3549.1.da.a.842.1 24
169.56 even 78 3549.1.da.a.3098.1 yes 24
507.56 odd 78 3549.1.da.a.3098.1 yes 24
1183.394 even 78 inner 3549.1.ds.a.1577.1 yes 24
3549.1577 odd 78 inner 3549.1.ds.a.1577.1 yes 24
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3549.1.da.a.842.1 24 7.2 even 3
3549.1.da.a.842.1 24 21.2 odd 6
3549.1.da.a.3098.1 yes 24 169.56 even 78
3549.1.da.a.3098.1 yes 24 507.56 odd 78
3549.1.ds.a.1577.1 yes 24 1183.394 even 78 inner
3549.1.ds.a.1577.1 yes 24 3549.1577 odd 78 inner
3549.1.ds.a.2363.1 yes 24 1.1 even 1 trivial
3549.1.ds.a.2363.1 yes 24 3.2 odd 2 CM