Properties

Label 4560.2.d.k.2431.7
Level $4560$
Weight $2$
Character 4560.2431
Analytic conductor $36.412$
Analytic rank $0$
Dimension $12$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4560,2,Mod(2431,4560)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4560, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 0, 0, 0, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4560.2431");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4560 = 2^{4} \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4560.d (of order \(2\), degree \(1\), 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: \(36.4117833217\)
Analytic rank: \(0\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 4 x^{11} + 8 x^{10} + 4 x^{9} + 36 x^{8} - 128 x^{7} + 232 x^{6} + 104 x^{5} + 324 x^{4} - 784 x^{3} + 800 x^{2} - 320 x + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{19}]\)
Coefficient ring index: \( 2^{11}\cdot 3^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2431.7
Root \(0.531065 + 0.531065i\) of defining polynomial
Character \(\chi\) \(=\) 4560.2431
Dual form 4560.2.d.k.2431.6

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} +0.549104i q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -1.00000 q^{5} +0.549104i q^{7} +1.00000 q^{9} -1.98504i q^{11} -2.38877i q^{13} -1.00000 q^{15} -4.49807 q^{17} +(-4.14448 + 1.35030i) q^{19} +0.549104i q^{21} -2.88909i q^{23} +1.00000 q^{25} +1.00000 q^{27} +3.54699i q^{29} +0.0281611 q^{31} -1.98504i q^{33} -0.549104i q^{35} -3.37396i q^{37} -2.38877i q^{39} +4.40793i q^{41} +6.83806i q^{43} -1.00000 q^{45} +10.4981i q^{47} +6.69848 q^{49} -4.49807 q^{51} +12.4775i q^{53} +1.98504i q^{55} +(-4.14448 + 1.35030i) q^{57} +3.79880 q^{59} -10.8231 q^{61} +0.549104i q^{63} +2.38877i q^{65} -4.92887 q^{67} -2.88909i q^{69} -15.1763 q^{71} -7.76272 q^{73} +1.00000 q^{75} +1.09000 q^{77} +2.04703 q^{79} +1.00000 q^{81} +7.95317i q^{83} +4.49807 q^{85} +3.54699i q^{87} -0.0323659i q^{89} +1.31168 q^{91} +0.0281611 q^{93} +(4.14448 - 1.35030i) q^{95} +0.596131i q^{97} -1.98504i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 12 q^{3} - 12 q^{5} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q + 12 q^{3} - 12 q^{5} + 12 q^{9} - 12 q^{15} - 8 q^{17} + 8 q^{19} + 12 q^{25} + 12 q^{27} - 12 q^{45} - 20 q^{49} - 8 q^{51} + 8 q^{57} + 8 q^{59} - 8 q^{67} + 32 q^{71} - 24 q^{73} + 12 q^{75} + 56 q^{77} + 16 q^{79} + 12 q^{81} + 8 q^{85} - 16 q^{91} - 8 q^{95}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4560\mathbb{Z}\right)^\times\).

\(n\) \(1141\) \(1711\) \(1921\) \(2737\) \(3041\)
\(\chi(n)\) \(1\) \(-1\) \(-1\) \(1\) \(1\)

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
\(3\) 1.00000 0.577350
\(4\) 0 0
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 0.549104i 0.207542i 0.994601 + 0.103771i \(0.0330909\pi\)
−0.994601 + 0.103771i \(0.966909\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 1.98504i 0.598513i −0.954173 0.299257i \(-0.903261\pi\)
0.954173 0.299257i \(-0.0967387\pi\)
\(12\) 0 0
\(13\) 2.38877i 0.662525i −0.943539 0.331262i \(-0.892526\pi\)
0.943539 0.331262i \(-0.107474\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −4.49807 −1.09094 −0.545471 0.838130i \(-0.683649\pi\)
−0.545471 + 0.838130i \(0.683649\pi\)
\(18\) 0 0
\(19\) −4.14448 + 1.35030i −0.950808 + 0.309780i
\(20\) 0 0
\(21\) 0.549104i 0.119824i
\(22\) 0 0
\(23\) 2.88909i 0.602418i −0.953558 0.301209i \(-0.902610\pi\)
0.953558 0.301209i \(-0.0973901\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) 3.54699i 0.658660i 0.944215 + 0.329330i \(0.106823\pi\)
−0.944215 + 0.329330i \(0.893177\pi\)
\(30\) 0 0
\(31\) 0.0281611 0.00505789 0.00252895 0.999997i \(-0.499195\pi\)
0.00252895 + 0.999997i \(0.499195\pi\)
\(32\) 0 0
\(33\) 1.98504i 0.345552i
\(34\) 0 0
\(35\) 0.549104i 0.0928155i
\(36\) 0 0
\(37\) 3.37396i 0.554675i −0.960773 0.277338i \(-0.910548\pi\)
0.960773 0.277338i \(-0.0894521\pi\)
\(38\) 0 0
\(39\) 2.38877i 0.382509i
\(40\) 0 0
\(41\) 4.40793i 0.688402i 0.938896 + 0.344201i \(0.111850\pi\)
−0.938896 + 0.344201i \(0.888150\pi\)
\(42\) 0 0
\(43\) 6.83806i 1.04279i 0.853314 + 0.521397i \(0.174589\pi\)
−0.853314 + 0.521397i \(0.825411\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) 10.4981i 1.53130i 0.643257 + 0.765650i \(0.277583\pi\)
−0.643257 + 0.765650i \(0.722417\pi\)
\(48\) 0 0
\(49\) 6.69848 0.956926
\(50\) 0 0
\(51\) −4.49807 −0.629856
\(52\) 0 0
\(53\) 12.4775i 1.71391i 0.515391 + 0.856955i \(0.327647\pi\)
−0.515391 + 0.856955i \(0.672353\pi\)
\(54\) 0 0
\(55\) 1.98504i 0.267663i
\(56\) 0 0
\(57\) −4.14448 + 1.35030i −0.548949 + 0.178851i
\(58\) 0 0
\(59\) 3.79880 0.494562 0.247281 0.968944i \(-0.420463\pi\)
0.247281 + 0.968944i \(0.420463\pi\)
\(60\) 0 0
\(61\) −10.8231 −1.38576 −0.692878 0.721055i \(-0.743658\pi\)
−0.692878 + 0.721055i \(0.743658\pi\)
\(62\) 0 0
\(63\) 0.549104i 0.0691806i
\(64\) 0 0
\(65\) 2.38877i 0.296290i
\(66\) 0 0
\(67\) −4.92887 −0.602157 −0.301079 0.953599i \(-0.597347\pi\)
−0.301079 + 0.953599i \(0.597347\pi\)
\(68\) 0 0
\(69\) 2.88909i 0.347806i
\(70\) 0 0
\(71\) −15.1763 −1.80110 −0.900548 0.434757i \(-0.856834\pi\)
−0.900548 + 0.434757i \(0.856834\pi\)
\(72\) 0 0
\(73\) −7.76272 −0.908558 −0.454279 0.890859i \(-0.650103\pi\)
−0.454279 + 0.890859i \(0.650103\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 1.09000 0.124217
\(78\) 0 0
\(79\) 2.04703 0.230309 0.115154 0.993348i \(-0.463264\pi\)
0.115154 + 0.993348i \(0.463264\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 7.95317i 0.872974i 0.899711 + 0.436487i \(0.143777\pi\)
−0.899711 + 0.436487i \(0.856223\pi\)
\(84\) 0 0
\(85\) 4.49807 0.487884
\(86\) 0 0
\(87\) 3.54699i 0.380278i
\(88\) 0 0
\(89\) 0.0323659i 0.00343078i −0.999999 0.00171539i \(-0.999454\pi\)
0.999999 0.00171539i \(-0.000546026\pi\)
\(90\) 0 0
\(91\) 1.31168 0.137502
\(92\) 0 0
\(93\) 0.0281611 0.00292017
\(94\) 0 0
\(95\) 4.14448 1.35030i 0.425214 0.138538i
\(96\) 0 0
\(97\) 0.596131i 0.0605279i 0.999542 + 0.0302640i \(0.00963479\pi\)
−0.999542 + 0.0302640i \(0.990365\pi\)
\(98\) 0 0
\(99\) 1.98504i 0.199504i
\(100\) 0 0
\(101\) −13.8004 −1.37319 −0.686594 0.727041i \(-0.740895\pi\)
−0.686594 + 0.727041i \(0.740895\pi\)
\(102\) 0 0
\(103\) 10.5966 1.04411 0.522056 0.852911i \(-0.325165\pi\)
0.522056 + 0.852911i \(0.325165\pi\)
\(104\) 0 0
\(105\) 0.549104i 0.0535871i
\(106\) 0 0
\(107\) 1.48472 0.143533 0.0717665 0.997421i \(-0.477136\pi\)
0.0717665 + 0.997421i \(0.477136\pi\)
\(108\) 0 0
\(109\) 0.973949i 0.0932874i −0.998912 0.0466437i \(-0.985147\pi\)
0.998912 0.0466437i \(-0.0148525\pi\)
\(110\) 0 0
\(111\) 3.37396i 0.320242i
\(112\) 0 0
\(113\) 8.25885i 0.776927i −0.921464 0.388464i \(-0.873006\pi\)
0.921464 0.388464i \(-0.126994\pi\)
\(114\) 0 0
\(115\) 2.88909i 0.269409i
\(116\) 0 0
\(117\) 2.38877i 0.220842i
\(118\) 0 0
\(119\) 2.46991i 0.226416i
\(120\) 0 0
\(121\) 7.05960 0.641782
\(122\) 0 0
\(123\) 4.40793i 0.397449i
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −12.2186 −1.08423 −0.542113 0.840306i \(-0.682376\pi\)
−0.542113 + 0.840306i \(0.682376\pi\)
\(128\) 0 0
\(129\) 6.83806i 0.602058i
\(130\) 0 0
\(131\) 12.4849i 1.09081i 0.838173 + 0.545404i \(0.183624\pi\)
−0.838173 + 0.545404i \(0.816376\pi\)
\(132\) 0 0
\(133\) −0.741454 2.27575i −0.0642922 0.197333i
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 6.63266 0.566666 0.283333 0.959022i \(-0.408560\pi\)
0.283333 + 0.959022i \(0.408560\pi\)
\(138\) 0 0
\(139\) 4.43383i 0.376073i 0.982162 + 0.188036i \(0.0602123\pi\)
−0.982162 + 0.188036i \(0.939788\pi\)
\(140\) 0 0
\(141\) 10.4981i 0.884097i
\(142\) 0 0
\(143\) −4.74181 −0.396530
\(144\) 0 0
\(145\) 3.54699i 0.294562i
\(146\) 0 0
\(147\) 6.69848 0.552482
\(148\) 0 0
\(149\) −4.58255 −0.375417 −0.187709 0.982225i \(-0.560106\pi\)
−0.187709 + 0.982225i \(0.560106\pi\)
\(150\) 0 0
\(151\) 7.32098 0.595773 0.297886 0.954601i \(-0.403718\pi\)
0.297886 + 0.954601i \(0.403718\pi\)
\(152\) 0 0
\(153\) −4.49807 −0.363647
\(154\) 0 0
\(155\) −0.0281611 −0.00226196
\(156\) 0 0
\(157\) −10.4479 −0.833830 −0.416915 0.908946i \(-0.636889\pi\)
−0.416915 + 0.908946i \(0.636889\pi\)
\(158\) 0 0
\(159\) 12.4775i 0.989527i
\(160\) 0 0
\(161\) 1.58641 0.125027
\(162\) 0 0
\(163\) 12.6983i 0.994611i 0.867576 + 0.497305i \(0.165677\pi\)
−0.867576 + 0.497305i \(0.834323\pi\)
\(164\) 0 0
\(165\) 1.98504i 0.154536i
\(166\) 0 0
\(167\) −16.2673 −1.25880 −0.629399 0.777082i \(-0.716699\pi\)
−0.629399 + 0.777082i \(0.716699\pi\)
\(168\) 0 0
\(169\) 7.29380 0.561061
\(170\) 0 0
\(171\) −4.14448 + 1.35030i −0.316936 + 0.103260i
\(172\) 0 0
\(173\) 17.4612i 1.32755i −0.747932 0.663776i \(-0.768953\pi\)
0.747932 0.663776i \(-0.231047\pi\)
\(174\) 0 0
\(175\) 0.549104i 0.0415084i
\(176\) 0 0
\(177\) 3.79880 0.285536
\(178\) 0 0
\(179\) −5.91855 −0.442373 −0.221186 0.975232i \(-0.570993\pi\)
−0.221186 + 0.975232i \(0.570993\pi\)
\(180\) 0 0
\(181\) 23.8524i 1.77294i −0.462789 0.886469i \(-0.653151\pi\)
0.462789 0.886469i \(-0.346849\pi\)
\(182\) 0 0
\(183\) −10.8231 −0.800067
\(184\) 0 0
\(185\) 3.37396i 0.248058i
\(186\) 0 0
\(187\) 8.92887i 0.652944i
\(188\) 0 0
\(189\) 0.549104i 0.0399414i
\(190\) 0 0
\(191\) 0.779128i 0.0563757i 0.999603 + 0.0281879i \(0.00897367\pi\)
−0.999603 + 0.0281879i \(0.991026\pi\)
\(192\) 0 0
\(193\) 14.1438i 1.01809i −0.860740 0.509045i \(-0.829999\pi\)
0.860740 0.509045i \(-0.170001\pi\)
\(194\) 0 0
\(195\) 2.38877i 0.171063i
\(196\) 0 0
\(197\) −25.5679 −1.82164 −0.910820 0.412803i \(-0.864550\pi\)
−0.910820 + 0.412803i \(0.864550\pi\)
\(198\) 0 0
\(199\) 7.29620i 0.517214i −0.965983 0.258607i \(-0.916737\pi\)
0.965983 0.258607i \(-0.0832634\pi\)
\(200\) 0 0
\(201\) −4.92887 −0.347656
\(202\) 0 0
\(203\) −1.94767 −0.136700
\(204\) 0 0
\(205\) 4.40793i 0.307863i
\(206\) 0 0
\(207\) 2.88909i 0.200806i
\(208\) 0 0
\(209\) 2.68040 + 8.22697i 0.185407 + 0.569072i
\(210\) 0 0
\(211\) 11.7321 0.807674 0.403837 0.914831i \(-0.367676\pi\)
0.403837 + 0.914831i \(0.367676\pi\)
\(212\) 0 0
\(213\) −15.1763 −1.03986
\(214\) 0 0
\(215\) 6.83806i 0.466352i
\(216\) 0 0
\(217\) 0.0154634i 0.00104972i
\(218\) 0 0
\(219\) −7.76272 −0.524556
\(220\) 0 0
\(221\) 10.7448i 0.722776i
\(222\) 0 0
\(223\) −5.27596 −0.353304 −0.176652 0.984273i \(-0.556527\pi\)
−0.176652 + 0.984273i \(0.556527\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −4.83586 −0.320967 −0.160484 0.987038i \(-0.551305\pi\)
−0.160484 + 0.987038i \(0.551305\pi\)
\(228\) 0 0
\(229\) −18.1941 −1.20230 −0.601151 0.799135i \(-0.705291\pi\)
−0.601151 + 0.799135i \(0.705291\pi\)
\(230\) 0 0
\(231\) 1.09000 0.0717165
\(232\) 0 0
\(233\) −5.14794 −0.337253 −0.168626 0.985680i \(-0.553933\pi\)
−0.168626 + 0.985680i \(0.553933\pi\)
\(234\) 0 0
\(235\) 10.4981i 0.684819i
\(236\) 0 0
\(237\) 2.04703 0.132969
\(238\) 0 0
\(239\) 12.9516i 0.837767i 0.908040 + 0.418884i \(0.137578\pi\)
−0.908040 + 0.418884i \(0.862422\pi\)
\(240\) 0 0
\(241\) 8.60963i 0.554595i 0.960784 + 0.277298i \(0.0894388\pi\)
−0.960784 + 0.277298i \(0.910561\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) −6.69848 −0.427950
\(246\) 0 0
\(247\) 3.22555 + 9.90019i 0.205237 + 0.629934i
\(248\) 0 0
\(249\) 7.95317i 0.504012i
\(250\) 0 0
\(251\) 27.3239i 1.72467i 0.506338 + 0.862335i \(0.330999\pi\)
−0.506338 + 0.862335i \(0.669001\pi\)
\(252\) 0 0
\(253\) −5.73498 −0.360555
\(254\) 0 0
\(255\) 4.49807 0.281680
\(256\) 0 0
\(257\) 15.0604i 0.939441i 0.882815 + 0.469720i \(0.155645\pi\)
−0.882815 + 0.469720i \(0.844355\pi\)
\(258\) 0 0
\(259\) 1.85265 0.115118
\(260\) 0 0
\(261\) 3.54699i 0.219553i
\(262\) 0 0
\(263\) 28.5652i 1.76141i 0.473668 + 0.880703i \(0.342930\pi\)
−0.473668 + 0.880703i \(0.657070\pi\)
\(264\) 0 0
\(265\) 12.4775i 0.766484i
\(266\) 0 0
\(267\) 0.0323659i 0.00198076i
\(268\) 0 0
\(269\) 27.4252i 1.67214i −0.548621 0.836071i \(-0.684847\pi\)
0.548621 0.836071i \(-0.315153\pi\)
\(270\) 0 0
\(271\) 27.5383i 1.67284i 0.548093 + 0.836418i \(0.315354\pi\)
−0.548093 + 0.836418i \(0.684646\pi\)
\(272\) 0 0
\(273\) 1.31168 0.0793866
\(274\) 0 0
\(275\) 1.98504i 0.119703i
\(276\) 0 0
\(277\) −14.2412 −0.855672 −0.427836 0.903856i \(-0.640724\pi\)
−0.427836 + 0.903856i \(0.640724\pi\)
\(278\) 0 0
\(279\) 0.0281611 0.00168596
\(280\) 0 0
\(281\) 4.13443i 0.246640i −0.992367 0.123320i \(-0.960646\pi\)
0.992367 0.123320i \(-0.0393541\pi\)
\(282\) 0 0
\(283\) 22.4044i 1.33180i 0.746040 + 0.665901i \(0.231953\pi\)
−0.746040 + 0.665901i \(0.768047\pi\)
\(284\) 0 0
\(285\) 4.14448 1.35030i 0.245498 0.0799847i
\(286\) 0 0
\(287\) −2.42041 −0.142872
\(288\) 0 0
\(289\) 3.23263 0.190155
\(290\) 0 0
\(291\) 0.596131i 0.0349458i
\(292\) 0 0
\(293\) 9.64601i 0.563526i −0.959484 0.281763i \(-0.909081\pi\)
0.959484 0.281763i \(-0.0909192\pi\)
\(294\) 0 0
\(295\) −3.79880 −0.221175
\(296\) 0 0
\(297\) 1.98504i 0.115184i
\(298\) 0 0
\(299\) −6.90137 −0.399117
\(300\) 0 0
\(301\) −3.75481 −0.216423
\(302\) 0 0
\(303\) −13.8004 −0.792811
\(304\) 0 0
\(305\) 10.8231 0.619729
\(306\) 0 0
\(307\) 27.6037 1.57542 0.787712 0.616043i \(-0.211265\pi\)
0.787712 + 0.616043i \(0.211265\pi\)
\(308\) 0 0
\(309\) 10.5966 0.602818
\(310\) 0 0
\(311\) 6.71077i 0.380533i −0.981732 0.190266i \(-0.939065\pi\)
0.981732 0.190266i \(-0.0609352\pi\)
\(312\) 0 0
\(313\) 1.30165 0.0735734 0.0367867 0.999323i \(-0.488288\pi\)
0.0367867 + 0.999323i \(0.488288\pi\)
\(314\) 0 0
\(315\) 0.549104i 0.0309385i
\(316\) 0 0
\(317\) 21.4736i 1.20608i 0.797712 + 0.603039i \(0.206044\pi\)
−0.797712 + 0.603039i \(0.793956\pi\)
\(318\) 0 0
\(319\) 7.04094 0.394217
\(320\) 0 0
\(321\) 1.48472 0.0828688
\(322\) 0 0
\(323\) 18.6422 6.07373i 1.03728 0.337952i
\(324\) 0 0
\(325\) 2.38877i 0.132505i
\(326\) 0 0
\(327\) 0.973949i 0.0538595i
\(328\) 0 0
\(329\) −5.76453 −0.317809
\(330\) 0 0
\(331\) −29.9657 −1.64706 −0.823532 0.567270i \(-0.808000\pi\)
−0.823532 + 0.567270i \(0.808000\pi\)
\(332\) 0 0
\(333\) 3.37396i 0.184892i
\(334\) 0 0
\(335\) 4.92887 0.269293
\(336\) 0 0
\(337\) 29.3254i 1.59746i −0.601693 0.798728i \(-0.705507\pi\)
0.601693 0.798728i \(-0.294493\pi\)
\(338\) 0 0
\(339\) 8.25885i 0.448559i
\(340\) 0 0
\(341\) 0.0559011i 0.00302722i
\(342\) 0 0
\(343\) 7.52189i 0.406144i
\(344\) 0 0
\(345\) 2.88909i 0.155544i
\(346\) 0 0
\(347\) 6.22343i 0.334091i −0.985949 0.167046i \(-0.946577\pi\)
0.985949 0.167046i \(-0.0534228\pi\)
\(348\) 0 0
\(349\) 7.83180 0.419227 0.209613 0.977784i \(-0.432779\pi\)
0.209613 + 0.977784i \(0.432779\pi\)
\(350\) 0 0
\(351\) 2.38877i 0.127503i
\(352\) 0 0
\(353\) −4.10154 −0.218303 −0.109151 0.994025i \(-0.534813\pi\)
−0.109151 + 0.994025i \(0.534813\pi\)
\(354\) 0 0
\(355\) 15.1763 0.805474
\(356\) 0 0
\(357\) 2.46991i 0.130721i
\(358\) 0 0
\(359\) 8.67706i 0.457958i −0.973431 0.228979i \(-0.926461\pi\)
0.973431 0.228979i \(-0.0735387\pi\)
\(360\) 0 0
\(361\) 15.3534 11.1926i 0.808073 0.589082i
\(362\) 0 0
\(363\) 7.05960 0.370533
\(364\) 0 0
\(365\) 7.76272 0.406320
\(366\) 0 0
\(367\) 6.58967i 0.343978i −0.985099 0.171989i \(-0.944981\pi\)
0.985099 0.171989i \(-0.0550193\pi\)
\(368\) 0 0
\(369\) 4.40793i 0.229467i
\(370\) 0 0
\(371\) −6.85142 −0.355708
\(372\) 0 0
\(373\) 21.1789i 1.09660i −0.836281 0.548302i \(-0.815275\pi\)
0.836281 0.548302i \(-0.184725\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 8.47294 0.436379
\(378\) 0 0
\(379\) 23.3582 1.19983 0.599914 0.800064i \(-0.295201\pi\)
0.599914 + 0.800064i \(0.295201\pi\)
\(380\) 0 0
\(381\) −12.2186 −0.625978
\(382\) 0 0
\(383\) −8.77542 −0.448403 −0.224201 0.974543i \(-0.571977\pi\)
−0.224201 + 0.974543i \(0.571977\pi\)
\(384\) 0 0
\(385\) −1.09000 −0.0555514
\(386\) 0 0
\(387\) 6.83806i 0.347598i
\(388\) 0 0
\(389\) 9.15133 0.463991 0.231995 0.972717i \(-0.425475\pi\)
0.231995 + 0.972717i \(0.425475\pi\)
\(390\) 0 0
\(391\) 12.9953i 0.657203i
\(392\) 0 0
\(393\) 12.4849i 0.629778i
\(394\) 0 0
\(395\) −2.04703 −0.102997
\(396\) 0 0
\(397\) 17.7505 0.890873 0.445436 0.895314i \(-0.353049\pi\)
0.445436 + 0.895314i \(0.353049\pi\)
\(398\) 0 0
\(399\) −0.741454 2.27575i −0.0371191 0.113930i
\(400\) 0 0
\(401\) 33.1362i 1.65474i 0.561655 + 0.827372i \(0.310165\pi\)
−0.561655 + 0.827372i \(0.689835\pi\)
\(402\) 0 0
\(403\) 0.0672704i 0.00335098i
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −6.69746 −0.331981
\(408\) 0 0
\(409\) 10.8447i 0.536235i 0.963386 + 0.268117i \(0.0864015\pi\)
−0.963386 + 0.268117i \(0.913599\pi\)
\(410\) 0 0
\(411\) 6.63266 0.327165
\(412\) 0 0
\(413\) 2.08594i 0.102642i
\(414\) 0 0
\(415\) 7.95317i 0.390406i
\(416\) 0 0
\(417\) 4.43383i 0.217126i
\(418\) 0 0
\(419\) 36.6151i 1.78876i −0.447305 0.894381i \(-0.647616\pi\)
0.447305 0.894381i \(-0.352384\pi\)
\(420\) 0 0
\(421\) 31.2843i 1.52470i −0.647163 0.762352i \(-0.724045\pi\)
0.647163 0.762352i \(-0.275955\pi\)
\(422\) 0 0
\(423\) 10.4981i 0.510434i
\(424\) 0 0
\(425\) −4.49807 −0.218188
\(426\) 0 0
\(427\) 5.94301i 0.287602i
\(428\) 0 0
\(429\) −4.74181 −0.228937
\(430\) 0 0
\(431\) −2.11974 −0.102104 −0.0510522 0.998696i \(-0.516257\pi\)
−0.0510522 + 0.998696i \(0.516257\pi\)
\(432\) 0 0
\(433\) 4.68020i 0.224916i −0.993656 0.112458i \(-0.964128\pi\)
0.993656 0.112458i \(-0.0358724\pi\)
\(434\) 0 0
\(435\) 3.54699i 0.170065i
\(436\) 0 0
\(437\) 3.90114 + 11.9738i 0.186617 + 0.572784i
\(438\) 0 0
\(439\) 10.9361 0.521952 0.260976 0.965345i \(-0.415956\pi\)
0.260976 + 0.965345i \(0.415956\pi\)
\(440\) 0 0
\(441\) 6.69848 0.318975
\(442\) 0 0
\(443\) 16.6930i 0.793107i 0.918012 + 0.396554i \(0.129794\pi\)
−0.918012 + 0.396554i \(0.870206\pi\)
\(444\) 0 0
\(445\) 0.0323659i 0.00153429i
\(446\) 0 0
\(447\) −4.58255 −0.216747
\(448\) 0 0
\(449\) 22.4864i 1.06120i −0.847622 0.530600i \(-0.821967\pi\)
0.847622 0.530600i \(-0.178033\pi\)
\(450\) 0 0
\(451\) 8.74993 0.412018
\(452\) 0 0
\(453\) 7.32098 0.343969
\(454\) 0 0
\(455\) −1.31168 −0.0614926
\(456\) 0 0
\(457\) 28.8834 1.35111 0.675555 0.737310i \(-0.263904\pi\)
0.675555 + 0.737310i \(0.263904\pi\)
\(458\) 0 0
\(459\) −4.49807 −0.209952
\(460\) 0 0
\(461\) −16.2365 −0.756208 −0.378104 0.925763i \(-0.623424\pi\)
−0.378104 + 0.925763i \(0.623424\pi\)
\(462\) 0 0
\(463\) 21.2699i 0.988498i 0.869320 + 0.494249i \(0.164557\pi\)
−0.869320 + 0.494249i \(0.835443\pi\)
\(464\) 0 0
\(465\) −0.0281611 −0.00130594
\(466\) 0 0
\(467\) 2.74688i 0.127111i −0.997978 0.0635553i \(-0.979756\pi\)
0.997978 0.0635553i \(-0.0202439\pi\)
\(468\) 0 0
\(469\) 2.70646i 0.124973i
\(470\) 0 0
\(471\) −10.4479 −0.481412
\(472\) 0 0
\(473\) 13.5739 0.624126
\(474\) 0 0
\(475\) −4.14448 + 1.35030i −0.190162 + 0.0619559i
\(476\) 0 0
\(477\) 12.4775i 0.571303i
\(478\) 0 0
\(479\) 22.8836i 1.04558i −0.852462 0.522790i \(-0.824891\pi\)
0.852462 0.522790i \(-0.175109\pi\)
\(480\) 0 0
\(481\) −8.05960 −0.367486
\(482\) 0 0
\(483\) 1.58641 0.0721843
\(484\) 0 0
\(485\) 0.596131i 0.0270689i
\(486\) 0 0
\(487\) 26.9321 1.22041 0.610205 0.792244i \(-0.291087\pi\)
0.610205 + 0.792244i \(0.291087\pi\)
\(488\) 0 0
\(489\) 12.6983i 0.574239i
\(490\) 0 0
\(491\) 5.99912i 0.270737i 0.990795 + 0.135368i \(0.0432217\pi\)
−0.990795 + 0.135368i \(0.956778\pi\)
\(492\) 0 0
\(493\) 15.9546i 0.718560i
\(494\) 0 0
\(495\) 1.98504i 0.0892211i
\(496\) 0 0
\(497\) 8.33337i 0.373803i
\(498\) 0 0
\(499\) 10.8066i 0.483768i −0.970305 0.241884i \(-0.922235\pi\)
0.970305 0.241884i \(-0.0777653\pi\)
\(500\) 0 0
\(501\) −16.2673 −0.726767
\(502\) 0 0
\(503\) 21.9373i 0.978135i 0.872246 + 0.489068i \(0.162663\pi\)
−0.872246 + 0.489068i \(0.837337\pi\)
\(504\) 0 0
\(505\) 13.8004 0.614109
\(506\) 0 0
\(507\) 7.29380 0.323929
\(508\) 0 0
\(509\) 17.1858i 0.761748i 0.924627 + 0.380874i \(0.124377\pi\)
−0.924627 + 0.380874i \(0.875623\pi\)
\(510\) 0 0
\(511\) 4.26254i 0.188564i
\(512\) 0 0
\(513\) −4.14448 + 1.35030i −0.182983 + 0.0596171i
\(514\) 0 0
\(515\) −10.5966 −0.466941
\(516\) 0 0
\(517\) 20.8391 0.916504
\(518\) 0 0
\(519\) 17.4612i 0.766462i
\(520\) 0 0
\(521\) 4.41552i 0.193447i 0.995311 + 0.0967237i \(0.0308363\pi\)
−0.995311 + 0.0967237i \(0.969164\pi\)
\(522\) 0 0
\(523\) −12.4144 −0.542842 −0.271421 0.962461i \(-0.587494\pi\)
−0.271421 + 0.962461i \(0.587494\pi\)
\(524\) 0 0
\(525\) 0.549104i 0.0239649i
\(526\) 0 0
\(527\) −0.126671 −0.00551787
\(528\) 0 0
\(529\) 14.6531 0.637093
\(530\) 0 0
\(531\) 3.79880 0.164854
\(532\) 0 0
\(533\) 10.5295 0.456084
\(534\) 0 0
\(535\) −1.48472 −0.0641899
\(536\) 0 0
\(537\) −5.91855 −0.255404
\(538\) 0 0
\(539\) 13.2968i 0.572733i
\(540\) 0 0
\(541\) −20.9115 −0.899057 −0.449528 0.893266i \(-0.648408\pi\)
−0.449528 + 0.893266i \(0.648408\pi\)
\(542\) 0 0
\(543\) 23.8524i 1.02361i
\(544\) 0 0
\(545\) 0.973949i 0.0417194i
\(546\) 0 0
\(547\) −39.4357 −1.68615 −0.843075 0.537796i \(-0.819257\pi\)
−0.843075 + 0.537796i \(0.819257\pi\)
\(548\) 0 0
\(549\) −10.8231 −0.461919
\(550\) 0 0
\(551\) −4.78950 14.7004i −0.204039 0.626260i
\(552\) 0 0
\(553\) 1.12403i 0.0477987i
\(554\) 0 0
\(555\) 3.37396i 0.143217i
\(556\) 0 0
\(557\) −25.9645 −1.10015 −0.550075 0.835115i \(-0.685401\pi\)
−0.550075 + 0.835115i \(0.685401\pi\)
\(558\) 0 0
\(559\) 16.3345 0.690877
\(560\) 0 0
\(561\) 8.92887i 0.376977i
\(562\) 0 0
\(563\) −32.9955 −1.39060 −0.695298 0.718721i \(-0.744728\pi\)
−0.695298 + 0.718721i \(0.744728\pi\)
\(564\) 0 0
\(565\) 8.25885i 0.347452i
\(566\) 0 0
\(567\) 0.549104i 0.0230602i
\(568\) 0 0
\(569\) 1.61692i 0.0677849i 0.999425 + 0.0338924i \(0.0107904\pi\)
−0.999425 + 0.0338924i \(0.989210\pi\)
\(570\) 0 0
\(571\) 4.46476i 0.186844i −0.995627 0.0934221i \(-0.970219\pi\)
0.995627 0.0934221i \(-0.0297806\pi\)
\(572\) 0 0
\(573\) 0.779128i 0.0325485i
\(574\) 0 0
\(575\) 2.88909i 0.120484i
\(576\) 0 0
\(577\) −33.5216 −1.39552 −0.697761 0.716331i \(-0.745820\pi\)
−0.697761 + 0.716331i \(0.745820\pi\)
\(578\) 0 0
\(579\) 14.1438i 0.587795i
\(580\) 0 0
\(581\) −4.36712 −0.181179
\(582\) 0 0
\(583\) 24.7683 1.02580
\(584\) 0 0
\(585\) 2.38877i 0.0987633i
\(586\) 0 0
\(587\) 8.24563i 0.340333i 0.985415 + 0.170167i \(0.0544306\pi\)
−0.985415 + 0.170167i \(0.945569\pi\)
\(588\) 0 0
\(589\) −0.116713 + 0.0380259i −0.00480908 + 0.00156683i
\(590\) 0 0
\(591\) −25.5679 −1.05172
\(592\) 0 0
\(593\) −10.4320 −0.428392 −0.214196 0.976791i \(-0.568713\pi\)
−0.214196 + 0.976791i \(0.568713\pi\)
\(594\) 0 0
\(595\) 2.46991i 0.101256i
\(596\) 0 0
\(597\) 7.29620i 0.298614i
\(598\) 0 0
\(599\) 26.7797 1.09419 0.547094 0.837071i \(-0.315734\pi\)
0.547094 + 0.837071i \(0.315734\pi\)
\(600\) 0 0
\(601\) 19.8979i 0.811652i 0.913950 + 0.405826i \(0.133016\pi\)
−0.913950 + 0.405826i \(0.866984\pi\)
\(602\) 0 0
\(603\) −4.92887 −0.200719
\(604\) 0 0
\(605\) −7.05960 −0.287013
\(606\) 0 0
\(607\) −10.1488 −0.411928 −0.205964 0.978560i \(-0.566033\pi\)
−0.205964 + 0.978560i \(0.566033\pi\)
\(608\) 0 0
\(609\) −1.94767 −0.0789235
\(610\) 0 0
\(611\) 25.0774 1.01452
\(612\) 0 0
\(613\) 11.9759 0.483701 0.241851 0.970313i \(-0.422246\pi\)
0.241851 + 0.970313i \(0.422246\pi\)
\(614\) 0 0
\(615\) 4.40793i 0.177745i
\(616\) 0 0
\(617\) 2.74054 0.110330 0.0551650 0.998477i \(-0.482432\pi\)
0.0551650 + 0.998477i \(0.482432\pi\)
\(618\) 0 0
\(619\) 25.1098i 1.00925i −0.863339 0.504625i \(-0.831631\pi\)
0.863339 0.504625i \(-0.168369\pi\)
\(620\) 0 0
\(621\) 2.88909i 0.115935i
\(622\) 0 0
\(623\) 0.0177723 0.000712031
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 2.68040 + 8.22697i 0.107045 + 0.328554i
\(628\) 0 0
\(629\) 15.1763i 0.605119i
\(630\) 0 0
\(631\) 24.7053i 0.983501i −0.870736 0.491750i \(-0.836357\pi\)
0.870736 0.491750i \(-0.163643\pi\)
\(632\) 0 0
\(633\) 11.7321 0.466311
\(634\) 0 0
\(635\) 12.2186 0.484881
\(636\) 0 0
\(637\) 16.0011i 0.633987i
\(638\) 0 0
\(639\) −15.1763 −0.600365
\(640\) 0 0
\(641\) 21.8782i 0.864137i −0.901841 0.432068i \(-0.857784\pi\)
0.901841 0.432068i \(-0.142216\pi\)
\(642\) 0 0
\(643\) 32.1896i 1.26943i −0.772745 0.634717i \(-0.781117\pi\)
0.772745 0.634717i \(-0.218883\pi\)
\(644\) 0 0
\(645\) 6.83806i 0.269248i
\(646\) 0 0
\(647\) 5.96906i 0.234668i 0.993093 + 0.117334i \(0.0374348\pi\)
−0.993093 + 0.117334i \(0.962565\pi\)
\(648\) 0 0
\(649\) 7.54079i 0.296002i
\(650\) 0 0
\(651\) 0.0154634i 0.000606058i
\(652\) 0 0
\(653\) −15.1388 −0.592429 −0.296214 0.955121i \(-0.595724\pi\)
−0.296214 + 0.955121i \(0.595724\pi\)
\(654\) 0 0
\(655\) 12.4849i 0.487824i
\(656\) 0 0
\(657\) −7.76272 −0.302853
\(658\) 0 0
\(659\) 25.6974 1.00103 0.500515 0.865728i \(-0.333144\pi\)
0.500515 + 0.865728i \(0.333144\pi\)
\(660\) 0 0
\(661\) 40.2383i 1.56509i 0.622595 + 0.782545i \(0.286079\pi\)
−0.622595 + 0.782545i \(0.713921\pi\)
\(662\) 0 0
\(663\) 10.7448i 0.417295i
\(664\) 0 0
\(665\) 0.741454 + 2.27575i 0.0287524 + 0.0882498i
\(666\) 0 0
\(667\) 10.2476 0.396789
\(668\) 0 0
\(669\) −5.27596 −0.203980
\(670\) 0 0
\(671\) 21.4843i 0.829394i
\(672\) 0 0
\(673\) 38.5749i 1.48695i −0.668763 0.743476i \(-0.733176\pi\)
0.668763 0.743476i \(-0.266824\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 2.86548i 0.110129i −0.998483 0.0550646i \(-0.982464\pi\)
0.998483 0.0550646i \(-0.0175365\pi\)
\(678\) 0 0
\(679\) −0.327338 −0.0125621
\(680\) 0 0
\(681\) −4.83586 −0.185311
\(682\) 0 0
\(683\) −3.62407 −0.138671 −0.0693355 0.997593i \(-0.522088\pi\)
−0.0693355 + 0.997593i \(0.522088\pi\)
\(684\) 0 0
\(685\) −6.63266 −0.253421
\(686\) 0 0
\(687\) −18.1941 −0.694150
\(688\) 0 0
\(689\) 29.8057 1.13551
\(690\) 0 0
\(691\) 29.2507i 1.11275i 0.830932 + 0.556375i \(0.187808\pi\)
−0.830932 + 0.556375i \(0.812192\pi\)
\(692\) 0 0
\(693\) 1.09000 0.0414055
\(694\) 0 0
\(695\) 4.43383i 0.168185i
\(696\) 0 0
\(697\) 19.8272i 0.751007i
\(698\) 0 0
\(699\) −5.14794 −0.194713
\(700\) 0 0
\(701\) −21.9657 −0.829635 −0.414817 0.909905i \(-0.636155\pi\)
−0.414817 + 0.909905i \(0.636155\pi\)
\(702\) 0 0
\(703\) 4.55585 + 13.9833i 0.171827 + 0.527390i
\(704\) 0 0
\(705\) 10.4981i 0.395380i
\(706\) 0 0
\(707\) 7.57784i 0.284994i
\(708\) 0 0
\(709\) 51.9631 1.95151 0.975757 0.218855i \(-0.0702321\pi\)
0.975757 + 0.218855i \(0.0702321\pi\)
\(710\) 0 0
\(711\) 2.04703 0.0767695
\(712\) 0 0
\(713\) 0.0813602i 0.00304696i
\(714\) 0 0
\(715\) 4.74181 0.177334
\(716\) 0 0
\(717\) 12.9516i 0.483685i
\(718\) 0 0
\(719\) 28.1730i 1.05068i −0.850894 0.525338i \(-0.823939\pi\)
0.850894 0.525338i \(-0.176061\pi\)
\(720\) 0 0
\(721\) 5.81863i 0.216697i
\(722\) 0 0
\(723\) 8.60963i 0.320196i
\(724\) 0 0
\(725\) 3.54699i 0.131732i
\(726\) 0 0
\(727\) 23.6145i 0.875814i 0.899020 + 0.437907i \(0.144280\pi\)
−0.899020 + 0.437907i \(0.855720\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 30.7581i 1.13763i
\(732\) 0 0
\(733\) −17.7108 −0.654164 −0.327082 0.944996i \(-0.606065\pi\)
−0.327082 + 0.944996i \(0.606065\pi\)
\(734\) 0 0
\(735\) −6.69848 −0.247077
\(736\) 0 0
\(737\) 9.78403i 0.360399i
\(738\) 0 0
\(739\) 5.44899i 0.200444i 0.994965 + 0.100222i \(0.0319553\pi\)
−0.994965 + 0.100222i \(0.968045\pi\)
\(740\) 0 0
\(741\) 3.22555 + 9.90019i 0.118493 + 0.363693i
\(742\) 0 0
\(743\) −3.92965 −0.144165 −0.0720825 0.997399i \(-0.522964\pi\)
−0.0720825 + 0.997399i \(0.522964\pi\)
\(744\) 0 0
\(745\) 4.58255 0.167892
\(746\) 0 0
\(747\) 7.95317i 0.290991i
\(748\) 0 0
\(749\) 0.815264i 0.0297891i
\(750\) 0 0
\(751\) 12.9639 0.473061 0.236530 0.971624i \(-0.423990\pi\)
0.236530 + 0.971624i \(0.423990\pi\)
\(752\) 0 0
\(753\) 27.3239i 0.995739i
\(754\) 0 0
\(755\) −7.32098 −0.266438
\(756\) 0 0
\(757\) 29.1638 1.05998 0.529989 0.848004i \(-0.322196\pi\)
0.529989 + 0.848004i \(0.322196\pi\)
\(758\) 0 0
\(759\) −5.73498 −0.208167
\(760\) 0 0
\(761\) 26.6750 0.966970 0.483485 0.875353i \(-0.339371\pi\)
0.483485 + 0.875353i \(0.339371\pi\)
\(762\) 0 0
\(763\) 0.534800 0.0193610
\(764\) 0 0
\(765\) 4.49807 0.162628
\(766\) 0 0
\(767\) 9.07445i 0.327660i
\(768\) 0 0
\(769\) 29.0447 1.04738 0.523690 0.851909i \(-0.324555\pi\)
0.523690 + 0.851909i \(0.324555\pi\)
\(770\) 0 0
\(771\) 15.0604i 0.542386i
\(772\) 0 0
\(773\) 11.6718i 0.419806i 0.977722 + 0.209903i \(0.0673149\pi\)
−0.977722 + 0.209903i \(0.932685\pi\)
\(774\) 0 0
\(775\) 0.0281611 0.00101158
\(776\) 0 0
\(777\) 1.85265 0.0664636
\(778\) 0 0
\(779\) −5.95201 18.2686i −0.213253 0.654539i
\(780\) 0 0
\(781\) 30.1256i 1.07798i
\(782\) 0 0
\(783\) 3.54699i 0.126759i
\(784\) 0 0
\(785\) 10.4479 0.372900
\(786\) 0 0
\(787\) 46.9721 1.67438 0.837188 0.546916i \(-0.184198\pi\)
0.837188 + 0.546916i \(0.184198\pi\)
\(788\) 0 0
\(789\) 28.5652i 1.01695i
\(790\) 0 0
\(791\) 4.53497 0.161245
\(792\) 0 0
\(793\) 25.8539i 0.918098i
\(794\) 0 0
\(795\) 12.4775i 0.442530i
\(796\) 0 0
\(797\) 22.4468i 0.795107i 0.917579 + 0.397554i \(0.130141\pi\)
−0.917579 + 0.397554i \(0.869859\pi\)
\(798\) 0 0
\(799\) 47.2211i 1.67056i
\(800\) 0 0
\(801\) 0.0323659i 0.00114359i
\(802\) 0 0
\(803\) 15.4094i 0.543784i
\(804\) 0 0
\(805\) −1.58641 −0.0559137
\(806\) 0 0
\(807\) 27.4252i 0.965412i
\(808\) 0 0
\(809\) 36.7277 1.29128 0.645639 0.763643i \(-0.276591\pi\)
0.645639 + 0.763643i \(0.276591\pi\)
\(810\) 0 0
\(811\) −9.26926 −0.325488 −0.162744 0.986668i \(-0.552034\pi\)
−0.162744 + 0.986668i \(0.552034\pi\)
\(812\) 0 0
\(813\) 27.5383i 0.965812i
\(814\) 0 0
\(815\) 12.6983i 0.444803i
\(816\) 0 0
\(817\) −9.23342 28.3402i −0.323036 0.991498i
\(818\) 0 0
\(819\) 1.31168 0.0458339
\(820\) 0 0
\(821\) 7.20869 0.251585 0.125792 0.992057i \(-0.459853\pi\)
0.125792 + 0.992057i \(0.459853\pi\)
\(822\) 0 0
\(823\) 12.7708i 0.445163i −0.974914 0.222582i \(-0.928552\pi\)
0.974914 0.222582i \(-0.0714484\pi\)
\(824\) 0 0
\(825\) 1.98504i 0.0691104i
\(826\) 0 0
\(827\) −3.41946 −0.118906 −0.0594532 0.998231i \(-0.518936\pi\)
−0.0594532 + 0.998231i \(0.518936\pi\)
\(828\) 0 0
\(829\) 18.4787i 0.641791i −0.947115 0.320895i \(-0.896016\pi\)
0.947115 0.320895i \(-0.103984\pi\)
\(830\) 0 0
\(831\) −14.2412 −0.494022
\(832\) 0 0
\(833\) −30.1303 −1.04395
\(834\) 0 0
\(835\) 16.2673 0.562952
\(836\) 0 0
\(837\) 0.0281611 0.000973391
\(838\) 0 0
\(839\) 20.4966 0.707622 0.353811 0.935317i \(-0.384886\pi\)
0.353811 + 0.935317i \(0.384886\pi\)
\(840\) 0 0
\(841\) 16.4188 0.566167
\(842\) 0 0
\(843\) 4.13443i 0.142397i
\(844\) 0 0
\(845\) −7.29380 −0.250914
\(846\) 0 0
\(847\) 3.87645i 0.133197i
\(848\) 0 0
\(849\) 22.4044i 0.768916i
\(850\) 0 0
\(851\) −9.74768 −0.334146
\(852\) 0 0
\(853\) −50.2946 −1.72206 −0.861028 0.508558i \(-0.830179\pi\)
−0.861028 + 0.508558i \(0.830179\pi\)
\(854\) 0 0
\(855\) 4.14448 1.35030i 0.141738 0.0461792i
\(856\) 0 0
\(857\) 41.0677i 1.40285i 0.712745 + 0.701423i \(0.247452\pi\)
−0.712745 + 0.701423i \(0.752548\pi\)
\(858\) 0 0
\(859\) 53.8104i 1.83599i −0.396596 0.917993i \(-0.629809\pi\)
0.396596 0.917993i \(-0.370191\pi\)
\(860\) 0 0
\(861\) −2.42041 −0.0824874
\(862\) 0 0
\(863\) 32.7392 1.11445 0.557227 0.830360i \(-0.311865\pi\)
0.557227 + 0.830360i \(0.311865\pi\)
\(864\) 0 0
\(865\) 17.4612i 0.593699i
\(866\) 0 0
\(867\) 3.23263 0.109786
\(868\) 0 0
\(869\) 4.06344i 0.137843i
\(870\) 0 0
\(871\) 11.7739i 0.398944i
\(872\) 0 0
\(873\) 0.596131i 0.0201760i
\(874\) 0 0
\(875\) 0.549104i 0.0185631i
\(876\) 0 0
\(877\) 22.2494i 0.751309i −0.926760 0.375655i \(-0.877418\pi\)
0.926760 0.375655i \(-0.122582\pi\)
\(878\) 0 0
\(879\) 9.64601i 0.325352i
\(880\) 0 0
\(881\) 24.4761 0.824622 0.412311 0.911043i \(-0.364722\pi\)
0.412311 + 0.911043i \(0.364722\pi\)
\(882\) 0 0
\(883\) 23.6715i 0.796611i 0.917253 + 0.398305i \(0.130402\pi\)
−0.917253 + 0.398305i \(0.869598\pi\)
\(884\) 0 0
\(885\) −3.79880 −0.127695
\(886\) 0 0
\(887\) −46.2726 −1.55368 −0.776840 0.629697i \(-0.783179\pi\)
−0.776840 + 0.629697i \(0.783179\pi\)
\(888\) 0 0
\(889\) 6.70929i 0.225022i
\(890\) 0 0
\(891\) 1.98504i 0.0665015i
\(892\) 0 0
\(893\) −14.1755 43.5090i −0.474366 1.45597i
\(894\) 0 0
\(895\) 5.91855 0.197835
\(896\) 0 0
\(897\) −6.90137 −0.230430
\(898\) 0 0
\(899\) 0.0998874i 0.00333143i
\(900\) 0 0
\(901\) 56.1245i 1.86978i
\(902\) 0 0
\(903\) −3.75481 −0.124952
\(904\) 0 0
\(905\) 23.8524i 0.792882i
\(906\) 0 0
\(907\) −12.9697 −0.430651 −0.215325 0.976542i \(-0.569081\pi\)
−0.215325 + 0.976542i \(0.569081\pi\)
\(908\) 0 0
\(909\) −13.8004 −0.457730
\(910\) 0 0
\(911\) 24.6057 0.815222 0.407611 0.913156i \(-0.366362\pi\)
0.407611 + 0.913156i \(0.366362\pi\)
\(912\) 0 0
\(913\) 15.7874 0.522487
\(914\) 0 0
\(915\) 10.8231 0.357801
\(916\) 0 0
\(917\) −6.85549 −0.226388
\(918\) 0 0
\(919\) 31.4360i 1.03698i −0.855084 0.518490i \(-0.826494\pi\)
0.855084 0.518490i \(-0.173506\pi\)
\(920\) 0 0
\(921\) 27.6037 0.909572
\(922\) 0 0
\(923\) 36.2526i 1.19327i
\(924\) 0 0
\(925\) 3.37396i 0.110935i
\(926\) 0 0
\(927\) 10.5966 0.348037
\(928\) 0 0
\(929\) 2.57967 0.0846361 0.0423180 0.999104i \(-0.486526\pi\)
0.0423180 + 0.999104i \(0.486526\pi\)
\(930\) 0 0
\(931\) −27.7617 + 9.04495i −0.909854 + 0.296436i
\(932\) 0 0
\(933\) 6.71077i 0.219701i
\(934\) 0 0
\(935\) 8.92887i 0.292005i
\(936\) 0 0
\(937\) 24.9943 0.816527 0.408264 0.912864i \(-0.366134\pi\)
0.408264 + 0.912864i \(0.366134\pi\)
\(938\) 0 0
\(939\) 1.30165 0.0424776
\(940\) 0 0
\(941\) 14.8521i 0.484164i −0.970256 0.242082i \(-0.922170\pi\)
0.970256 0.242082i \(-0.0778303\pi\)
\(942\) 0 0
\(943\) 12.7349 0.414706
\(944\) 0 0
\(945\) 0.549104i 0.0178624i
\(946\) 0 0
\(947\) 1.00685i 0.0327183i −0.999866 0.0163592i \(-0.994792\pi\)
0.999866 0.0163592i \(-0.00520752\pi\)
\(948\) 0 0
\(949\) 18.5433i 0.601942i
\(950\) 0 0
\(951\) 21.4736i 0.696329i
\(952\) 0 0
\(953\) 8.23298i 0.266692i 0.991070 + 0.133346i \(0.0425722\pi\)
−0.991070 + 0.133346i \(0.957428\pi\)
\(954\) 0 0
\(955\) 0.779128i 0.0252120i
\(956\) 0 0
\(957\) 7.04094 0.227601
\(958\) 0 0
\(959\) 3.64202i 0.117607i
\(960\) 0 0
\(961\) −30.9992 −0.999974
\(962\) 0 0
\(963\) 1.48472 0.0478443
\(964\) 0 0
\(965\) 14.1438i 0.455304i
\(966\) 0 0
\(967\) 8.69905i 0.279743i −0.990170 0.139871i \(-0.955331\pi\)
0.990170 0.139871i \(-0.0446689\pi\)
\(968\) 0 0
\(969\) 18.6422 6.07373i 0.598872 0.195116i
\(970\) 0 0
\(971\) −60.1366 −1.92988 −0.964938 0.262479i \(-0.915460\pi\)
−0.964938 + 0.262479i \(0.915460\pi\)
\(972\) 0 0
\(973\) −2.43463 −0.0780508
\(974\) 0 0
\(975\) 2.38877i 0.0765017i
\(976\) 0 0
\(977\) 35.3094i 1.12965i 0.825211 + 0.564824i \(0.191056\pi\)
−0.825211 + 0.564824i \(0.808944\pi\)
\(978\) 0 0
\(979\) −0.0642478 −0.00205337
\(980\) 0 0
\(981\) 0.973949i 0.0310958i
\(982\) 0 0
\(983\) −4.04148 −0.128903 −0.0644516 0.997921i \(-0.520530\pi\)
−0.0644516 + 0.997921i \(0.520530\pi\)
\(984\) 0 0
\(985\) 25.5679 0.814662
\(986\) 0 0
\(987\) −5.76453 −0.183487
\(988\) 0 0
\(989\) 19.7558 0.628198
\(990\) 0 0
\(991\) −2.28257 −0.0725082 −0.0362541 0.999343i \(-0.511543\pi\)
−0.0362541 + 0.999343i \(0.511543\pi\)
\(992\) 0 0
\(993\) −29.9657 −0.950933
\(994\) 0 0
\(995\) 7.29620i 0.231305i
\(996\) 0 0
\(997\) −7.58884 −0.240341 −0.120170 0.992753i \(-0.538344\pi\)
−0.120170 + 0.992753i \(0.538344\pi\)
\(998\) 0 0
\(999\) 3.37396i 0.106747i
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 4560.2.d.k.2431.7 yes 12
4.3 odd 2 4560.2.d.i.2431.6 12
19.18 odd 2 4560.2.d.i.2431.7 yes 12
76.75 even 2 inner 4560.2.d.k.2431.6 yes 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4560.2.d.i.2431.6 12 4.3 odd 2
4560.2.d.i.2431.7 yes 12 19.18 odd 2
4560.2.d.k.2431.6 yes 12 76.75 even 2 inner
4560.2.d.k.2431.7 yes 12 1.1 even 1 trivial