Properties

Label 4600.2.e.l.4049.4
Level $4600$
Weight $2$
Character 4600.4049
Analytic conductor $36.731$
Analytic rank $0$
Dimension $4$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [4600,2,Mod(4049,4600)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(4600, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 1, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("4600.4049");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 4600 = 2^{3} \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4600.e (of order \(2\), degree \(1\), not 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: \(36.7311849298\)
Analytic rank: \(0\)
Dimension: \(4\)
Coefficient field: \(\Q(i, \sqrt{5})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{4} + 3x^{2} + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 4049.4
Root \(1.61803i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.l.4049.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+3.23607i q^{3} +4.23607i q^{7} -7.47214 q^{9} +1.00000 q^{11} -2.23607i q^{13} -6.47214i q^{17} -1.00000 q^{19} -13.7082 q^{21} -1.00000i q^{23} -14.4721i q^{27} +1.76393 q^{29} +0.472136 q^{31} +3.23607i q^{33} -11.2361i q^{37} +7.23607 q^{39} -5.94427 q^{41} +2.52786i q^{43} -11.7082i q^{47} -10.9443 q^{49} +20.9443 q^{51} +1.23607i q^{53} -3.23607i q^{57} -3.23607 q^{59} -7.23607 q^{61} -31.6525i q^{63} -12.9443i q^{67} +3.23607 q^{69} +10.0000 q^{71} -9.47214i q^{73} +4.23607i q^{77} -15.1803 q^{79} +24.4164 q^{81} +9.00000i q^{83} +5.70820i q^{87} -2.00000 q^{89} +9.47214 q^{91} +1.52786i q^{93} -6.18034i q^{97} -7.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 4 q^{11} - 4 q^{19} - 28 q^{21} + 16 q^{29} - 16 q^{31} + 20 q^{39} + 12 q^{41} - 8 q^{49} + 48 q^{51} - 4 q^{59} - 20 q^{61} + 4 q^{69} + 40 q^{71} - 16 q^{79} + 44 q^{81} - 8 q^{89} + 20 q^{91}+ \cdots - 12 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/4600\mathbb{Z}\right)^\times\).

\(n\) \(1151\) \(1201\) \(2301\) \(2577\)
\(\chi(n)\) \(1\) \(1\) \(1\) \(-1\)

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
\(3\) 3.23607i 1.86834i 0.356822 + 0.934172i \(0.383860\pi\)
−0.356822 + 0.934172i \(0.616140\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) 4.23607i 1.60108i 0.599277 + 0.800542i \(0.295455\pi\)
−0.599277 + 0.800542i \(0.704545\pi\)
\(8\) 0 0
\(9\) −7.47214 −2.49071
\(10\) 0 0
\(11\) 1.00000 0.301511 0.150756 0.988571i \(-0.451829\pi\)
0.150756 + 0.988571i \(0.451829\pi\)
\(12\) 0 0
\(13\) − 2.23607i − 0.620174i −0.950708 0.310087i \(-0.899642\pi\)
0.950708 0.310087i \(-0.100358\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) − 6.47214i − 1.56972i −0.619671 0.784862i \(-0.712734\pi\)
0.619671 0.784862i \(-0.287266\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −13.7082 −2.99138
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 14.4721i − 2.78516i
\(28\) 0 0
\(29\) 1.76393 0.327554 0.163777 0.986497i \(-0.447632\pi\)
0.163777 + 0.986497i \(0.447632\pi\)
\(30\) 0 0
\(31\) 0.472136 0.0847981 0.0423991 0.999101i \(-0.486500\pi\)
0.0423991 + 0.999101i \(0.486500\pi\)
\(32\) 0 0
\(33\) 3.23607i 0.563327i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 11.2361i − 1.84720i −0.383360 0.923599i \(-0.625233\pi\)
0.383360 0.923599i \(-0.374767\pi\)
\(38\) 0 0
\(39\) 7.23607 1.15870
\(40\) 0 0
\(41\) −5.94427 −0.928339 −0.464170 0.885746i \(-0.653647\pi\)
−0.464170 + 0.885746i \(0.653647\pi\)
\(42\) 0 0
\(43\) 2.52786i 0.385496i 0.981248 + 0.192748i \(0.0617399\pi\)
−0.981248 + 0.192748i \(0.938260\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) − 11.7082i − 1.70782i −0.520423 0.853909i \(-0.674226\pi\)
0.520423 0.853909i \(-0.325774\pi\)
\(48\) 0 0
\(49\) −10.9443 −1.56347
\(50\) 0 0
\(51\) 20.9443 2.93278
\(52\) 0 0
\(53\) 1.23607i 0.169787i 0.996390 + 0.0848935i \(0.0270550\pi\)
−0.996390 + 0.0848935i \(0.972945\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) − 3.23607i − 0.428628i
\(58\) 0 0
\(59\) −3.23607 −0.421300 −0.210650 0.977562i \(-0.567558\pi\)
−0.210650 + 0.977562i \(0.567558\pi\)
\(60\) 0 0
\(61\) −7.23607 −0.926484 −0.463242 0.886232i \(-0.653314\pi\)
−0.463242 + 0.886232i \(0.653314\pi\)
\(62\) 0 0
\(63\) − 31.6525i − 3.98784i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) − 12.9443i − 1.58139i −0.612207 0.790697i \(-0.709718\pi\)
0.612207 0.790697i \(-0.290282\pi\)
\(68\) 0 0
\(69\) 3.23607 0.389577
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) − 9.47214i − 1.10863i −0.832307 0.554315i \(-0.812980\pi\)
0.832307 0.554315i \(-0.187020\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) 4.23607i 0.482745i
\(78\) 0 0
\(79\) −15.1803 −1.70792 −0.853961 0.520337i \(-0.825806\pi\)
−0.853961 + 0.520337i \(0.825806\pi\)
\(80\) 0 0
\(81\) 24.4164 2.71293
\(82\) 0 0
\(83\) 9.00000i 0.987878i 0.869496 + 0.493939i \(0.164443\pi\)
−0.869496 + 0.493939i \(0.835557\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) 0 0
\(87\) 5.70820i 0.611984i
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 9.47214 0.992950
\(92\) 0 0
\(93\) 1.52786i 0.158432i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) − 6.18034i − 0.627518i −0.949503 0.313759i \(-0.898412\pi\)
0.949503 0.313759i \(-0.101588\pi\)
\(98\) 0 0
\(99\) −7.47214 −0.750978
\(100\) 0 0
\(101\) −18.9443 −1.88503 −0.942513 0.334170i \(-0.891544\pi\)
−0.942513 + 0.334170i \(0.891544\pi\)
\(102\) 0 0
\(103\) − 2.23607i − 0.220326i −0.993914 0.110163i \(-0.964863\pi\)
0.993914 0.110163i \(-0.0351373\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) 19.4164i 1.87705i 0.345204 + 0.938527i \(0.387810\pi\)
−0.345204 + 0.938527i \(0.612190\pi\)
\(108\) 0 0
\(109\) 12.7639 1.22256 0.611281 0.791413i \(-0.290654\pi\)
0.611281 + 0.791413i \(0.290654\pi\)
\(110\) 0 0
\(111\) 36.3607 3.45120
\(112\) 0 0
\(113\) 6.76393i 0.636297i 0.948041 + 0.318149i \(0.103061\pi\)
−0.948041 + 0.318149i \(0.896939\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 16.7082i 1.54467i
\(118\) 0 0
\(119\) 27.4164 2.51326
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) − 19.2361i − 1.73446i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) 4.18034i 0.370945i 0.982649 + 0.185473i \(0.0593816\pi\)
−0.982649 + 0.185473i \(0.940618\pi\)
\(128\) 0 0
\(129\) −8.18034 −0.720239
\(130\) 0 0
\(131\) −14.9443 −1.30569 −0.652844 0.757493i \(-0.726424\pi\)
−0.652844 + 0.757493i \(0.726424\pi\)
\(132\) 0 0
\(133\) − 4.23607i − 0.367314i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 11.5279i − 0.984892i −0.870343 0.492446i \(-0.836103\pi\)
0.870343 0.492446i \(-0.163897\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 37.8885 3.19079
\(142\) 0 0
\(143\) − 2.23607i − 0.186989i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 35.4164i − 2.92110i
\(148\) 0 0
\(149\) −10.9443 −0.896590 −0.448295 0.893886i \(-0.647969\pi\)
−0.448295 + 0.893886i \(0.647969\pi\)
\(150\) 0 0
\(151\) 11.5279 0.938124 0.469062 0.883165i \(-0.344592\pi\)
0.469062 + 0.883165i \(0.344592\pi\)
\(152\) 0 0
\(153\) 48.3607i 3.90973i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 0.291796i − 0.0232879i −0.999932 0.0116439i \(-0.996294\pi\)
0.999932 0.0116439i \(-0.00370646\pi\)
\(158\) 0 0
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) 4.23607 0.333849
\(162\) 0 0
\(163\) − 11.7082i − 0.917057i −0.888680 0.458529i \(-0.848377\pi\)
0.888680 0.458529i \(-0.151623\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) 5.41641i 0.419134i 0.977794 + 0.209567i \(0.0672055\pi\)
−0.977794 + 0.209567i \(0.932795\pi\)
\(168\) 0 0
\(169\) 8.00000 0.615385
\(170\) 0 0
\(171\) 7.47214 0.571409
\(172\) 0 0
\(173\) 14.2361i 1.08235i 0.840911 + 0.541174i \(0.182020\pi\)
−0.840911 + 0.541174i \(0.817980\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 10.4721i − 0.787134i
\(178\) 0 0
\(179\) 15.7082 1.17409 0.587043 0.809556i \(-0.300292\pi\)
0.587043 + 0.809556i \(0.300292\pi\)
\(180\) 0 0
\(181\) −0.944272 −0.0701872 −0.0350936 0.999384i \(-0.511173\pi\)
−0.0350936 + 0.999384i \(0.511173\pi\)
\(182\) 0 0
\(183\) − 23.4164i − 1.73099i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) − 6.47214i − 0.473289i
\(188\) 0 0
\(189\) 61.3050 4.45928
\(190\) 0 0
\(191\) −19.1803 −1.38784 −0.693920 0.720052i \(-0.744118\pi\)
−0.693920 + 0.720052i \(0.744118\pi\)
\(192\) 0 0
\(193\) 2.94427i 0.211933i 0.994370 + 0.105967i \(0.0337937\pi\)
−0.994370 + 0.105967i \(0.966206\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 8.23607i 0.586796i 0.955990 + 0.293398i \(0.0947860\pi\)
−0.955990 + 0.293398i \(0.905214\pi\)
\(198\) 0 0
\(199\) −12.7082 −0.900861 −0.450430 0.892812i \(-0.648729\pi\)
−0.450430 + 0.892812i \(0.648729\pi\)
\(200\) 0 0
\(201\) 41.8885 2.95459
\(202\) 0 0
\(203\) 7.47214i 0.524441i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) 7.47214i 0.519349i
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −14.9443 −1.02881 −0.514403 0.857549i \(-0.671986\pi\)
−0.514403 + 0.857549i \(0.671986\pi\)
\(212\) 0 0
\(213\) 32.3607i 2.21732i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 0 0
\(219\) 30.6525 2.07130
\(220\) 0 0
\(221\) −14.4721 −0.973501
\(222\) 0 0
\(223\) 13.2361i 0.886353i 0.896434 + 0.443176i \(0.146148\pi\)
−0.896434 + 0.443176i \(0.853852\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) − 12.9443i − 0.859142i −0.903033 0.429571i \(-0.858665\pi\)
0.903033 0.429571i \(-0.141335\pi\)
\(228\) 0 0
\(229\) −6.94427 −0.458890 −0.229445 0.973322i \(-0.573691\pi\)
−0.229445 + 0.973322i \(0.573691\pi\)
\(230\) 0 0
\(231\) −13.7082 −0.901934
\(232\) 0 0
\(233\) 4.52786i 0.296630i 0.988940 + 0.148315i \(0.0473850\pi\)
−0.988940 + 0.148315i \(0.952615\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 49.1246i − 3.19099i
\(238\) 0 0
\(239\) −6.76393 −0.437522 −0.218761 0.975778i \(-0.570202\pi\)
−0.218761 + 0.975778i \(0.570202\pi\)
\(240\) 0 0
\(241\) 21.2361 1.36794 0.683968 0.729512i \(-0.260253\pi\)
0.683968 + 0.729512i \(0.260253\pi\)
\(242\) 0 0
\(243\) 35.5967i 2.28353i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) 2.23607i 0.142278i
\(248\) 0 0
\(249\) −29.1246 −1.84570
\(250\) 0 0
\(251\) −5.52786 −0.348916 −0.174458 0.984665i \(-0.555817\pi\)
−0.174458 + 0.984665i \(0.555817\pi\)
\(252\) 0 0
\(253\) − 1.00000i − 0.0628695i
\(254\) 0 0
\(255\) 0 0
\(256\) 0 0
\(257\) 14.0000i 0.873296i 0.899632 + 0.436648i \(0.143834\pi\)
−0.899632 + 0.436648i \(0.856166\pi\)
\(258\) 0 0
\(259\) 47.5967 2.95752
\(260\) 0 0
\(261\) −13.1803 −0.815843
\(262\) 0 0
\(263\) 8.00000i 0.493301i 0.969104 + 0.246651i \(0.0793300\pi\)
−0.969104 + 0.246651i \(0.920670\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) − 6.47214i − 0.396088i
\(268\) 0 0
\(269\) 25.1803 1.53527 0.767636 0.640886i \(-0.221433\pi\)
0.767636 + 0.640886i \(0.221433\pi\)
\(270\) 0 0
\(271\) −32.0689 −1.94805 −0.974023 0.226449i \(-0.927288\pi\)
−0.974023 + 0.226449i \(0.927288\pi\)
\(272\) 0 0
\(273\) 30.6525i 1.85517i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 14.2361i − 0.855362i −0.903930 0.427681i \(-0.859331\pi\)
0.903930 0.427681i \(-0.140669\pi\)
\(278\) 0 0
\(279\) −3.52786 −0.211208
\(280\) 0 0
\(281\) −22.1803 −1.32317 −0.661584 0.749871i \(-0.730116\pi\)
−0.661584 + 0.749871i \(0.730116\pi\)
\(282\) 0 0
\(283\) 28.3607i 1.68587i 0.538017 + 0.842934i \(0.319173\pi\)
−0.538017 + 0.842934i \(0.680827\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 25.1803i − 1.48635i
\(288\) 0 0
\(289\) −24.8885 −1.46403
\(290\) 0 0
\(291\) 20.0000 1.17242
\(292\) 0 0
\(293\) − 14.4721i − 0.845471i −0.906253 0.422736i \(-0.861070\pi\)
0.906253 0.422736i \(-0.138930\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 14.4721i − 0.839759i
\(298\) 0 0
\(299\) −2.23607 −0.129315
\(300\) 0 0
\(301\) −10.7082 −0.617211
\(302\) 0 0
\(303\) − 61.3050i − 3.52188i
\(304\) 0 0
\(305\) 0 0
\(306\) 0 0
\(307\) − 24.0000i − 1.36975i −0.728659 0.684876i \(-0.759856\pi\)
0.728659 0.684876i \(-0.240144\pi\)
\(308\) 0 0
\(309\) 7.23607 0.411646
\(310\) 0 0
\(311\) −3.23607 −0.183501 −0.0917503 0.995782i \(-0.529246\pi\)
−0.0917503 + 0.995782i \(0.529246\pi\)
\(312\) 0 0
\(313\) − 10.6525i − 0.602114i −0.953606 0.301057i \(-0.902661\pi\)
0.953606 0.301057i \(-0.0973394\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) 22.1246i 1.24264i 0.783556 + 0.621321i \(0.213404\pi\)
−0.783556 + 0.621321i \(0.786596\pi\)
\(318\) 0 0
\(319\) 1.76393 0.0987612
\(320\) 0 0
\(321\) −62.8328 −3.50699
\(322\) 0 0
\(323\) 6.47214i 0.360119i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) 41.3050i 2.28417i
\(328\) 0 0
\(329\) 49.5967 2.73436
\(330\) 0 0
\(331\) 30.9443 1.70085 0.850426 0.526095i \(-0.176345\pi\)
0.850426 + 0.526095i \(0.176345\pi\)
\(332\) 0 0
\(333\) 83.9574i 4.60084i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) 24.6525i 1.34291i 0.741047 + 0.671453i \(0.234329\pi\)
−0.741047 + 0.671453i \(0.765671\pi\)
\(338\) 0 0
\(339\) −21.8885 −1.18882
\(340\) 0 0
\(341\) 0.472136 0.0255676
\(342\) 0 0
\(343\) − 16.7082i − 0.902158i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 3.05573i − 0.164040i −0.996631 0.0820200i \(-0.973863\pi\)
0.996631 0.0820200i \(-0.0261372\pi\)
\(348\) 0 0
\(349\) 5.18034 0.277297 0.138649 0.990342i \(-0.455724\pi\)
0.138649 + 0.990342i \(0.455724\pi\)
\(350\) 0 0
\(351\) −32.3607 −1.72729
\(352\) 0 0
\(353\) − 4.88854i − 0.260191i −0.991501 0.130095i \(-0.958472\pi\)
0.991501 0.130095i \(-0.0415284\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) 88.7214i 4.69563i
\(358\) 0 0
\(359\) −19.7639 −1.04310 −0.521550 0.853221i \(-0.674646\pi\)
−0.521550 + 0.853221i \(0.674646\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) − 32.3607i − 1.69850i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) 23.6525i 1.23465i 0.786709 + 0.617325i \(0.211783\pi\)
−0.786709 + 0.617325i \(0.788217\pi\)
\(368\) 0 0
\(369\) 44.4164 2.31223
\(370\) 0 0
\(371\) −5.23607 −0.271843
\(372\) 0 0
\(373\) − 12.0000i − 0.621336i −0.950518 0.310668i \(-0.899447\pi\)
0.950518 0.310668i \(-0.100553\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 0 0
\(377\) − 3.94427i − 0.203140i
\(378\) 0 0
\(379\) −3.05573 −0.156962 −0.0784811 0.996916i \(-0.525007\pi\)
−0.0784811 + 0.996916i \(0.525007\pi\)
\(380\) 0 0
\(381\) −13.5279 −0.693053
\(382\) 0 0
\(383\) − 0.236068i − 0.0120625i −0.999982 0.00603126i \(-0.998080\pi\)
0.999982 0.00603126i \(-0.00191982\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) − 18.8885i − 0.960159i
\(388\) 0 0
\(389\) −6.18034 −0.313356 −0.156678 0.987650i \(-0.550078\pi\)
−0.156678 + 0.987650i \(0.550078\pi\)
\(390\) 0 0
\(391\) −6.47214 −0.327310
\(392\) 0 0
\(393\) − 48.3607i − 2.43947i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) − 22.3607i − 1.12225i −0.827731 0.561125i \(-0.810369\pi\)
0.827731 0.561125i \(-0.189631\pi\)
\(398\) 0 0
\(399\) 13.7082 0.686269
\(400\) 0 0
\(401\) −13.1246 −0.655412 −0.327706 0.944780i \(-0.606276\pi\)
−0.327706 + 0.944780i \(0.606276\pi\)
\(402\) 0 0
\(403\) − 1.05573i − 0.0525896i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 11.2361i − 0.556951i
\(408\) 0 0
\(409\) 26.4164 1.30621 0.653104 0.757269i \(-0.273467\pi\)
0.653104 + 0.757269i \(0.273467\pi\)
\(410\) 0 0
\(411\) 37.3050 1.84012
\(412\) 0 0
\(413\) − 13.7082i − 0.674537i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) − 64.7214i − 3.16942i
\(418\) 0 0
\(419\) 36.7771 1.79668 0.898339 0.439303i \(-0.144774\pi\)
0.898339 + 0.439303i \(0.144774\pi\)
\(420\) 0 0
\(421\) 15.8885 0.774360 0.387180 0.922004i \(-0.373449\pi\)
0.387180 + 0.922004i \(0.373449\pi\)
\(422\) 0 0
\(423\) 87.4853i 4.25368i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) − 30.6525i − 1.48338i
\(428\) 0 0
\(429\) 7.23607 0.349361
\(430\) 0 0
\(431\) 12.9443 0.623504 0.311752 0.950164i \(-0.399084\pi\)
0.311752 + 0.950164i \(0.399084\pi\)
\(432\) 0 0
\(433\) − 29.1246i − 1.39964i −0.714319 0.699820i \(-0.753264\pi\)
0.714319 0.699820i \(-0.246736\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.00000i 0.0478365i
\(438\) 0 0
\(439\) −12.3607 −0.589943 −0.294972 0.955506i \(-0.595310\pi\)
−0.294972 + 0.955506i \(0.595310\pi\)
\(440\) 0 0
\(441\) 81.7771 3.89415
\(442\) 0 0
\(443\) − 10.0000i − 0.475114i −0.971374 0.237557i \(-0.923653\pi\)
0.971374 0.237557i \(-0.0763467\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) 0 0
\(447\) − 35.4164i − 1.67514i
\(448\) 0 0
\(449\) 36.8328 1.73825 0.869124 0.494594i \(-0.164683\pi\)
0.869124 + 0.494594i \(0.164683\pi\)
\(450\) 0 0
\(451\) −5.94427 −0.279905
\(452\) 0 0
\(453\) 37.3050i 1.75274i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) 20.6525i 0.966082i 0.875598 + 0.483041i \(0.160468\pi\)
−0.875598 + 0.483041i \(0.839532\pi\)
\(458\) 0 0
\(459\) −93.6656 −4.37194
\(460\) 0 0
\(461\) 33.6525 1.56735 0.783676 0.621170i \(-0.213342\pi\)
0.783676 + 0.621170i \(0.213342\pi\)
\(462\) 0 0
\(463\) − 26.4721i − 1.23026i −0.788424 0.615132i \(-0.789103\pi\)
0.788424 0.615132i \(-0.210897\pi\)
\(464\) 0 0
\(465\) 0 0
\(466\) 0 0
\(467\) 17.0000i 0.786666i 0.919396 + 0.393333i \(0.128678\pi\)
−0.919396 + 0.393333i \(0.871322\pi\)
\(468\) 0 0
\(469\) 54.8328 2.53194
\(470\) 0 0
\(471\) 0.944272 0.0435098
\(472\) 0 0
\(473\) 2.52786i 0.116231i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 9.23607i − 0.422891i
\(478\) 0 0
\(479\) 12.1246 0.553988 0.276994 0.960872i \(-0.410662\pi\)
0.276994 + 0.960872i \(0.410662\pi\)
\(480\) 0 0
\(481\) −25.1246 −1.14558
\(482\) 0 0
\(483\) 13.7082i 0.623745i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) 39.1246i 1.77291i 0.462819 + 0.886453i \(0.346838\pi\)
−0.462819 + 0.886453i \(0.653162\pi\)
\(488\) 0 0
\(489\) 37.8885 1.71338
\(490\) 0 0
\(491\) −3.23607 −0.146042 −0.0730209 0.997330i \(-0.523264\pi\)
−0.0730209 + 0.997330i \(0.523264\pi\)
\(492\) 0 0
\(493\) − 11.4164i − 0.514169i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) 42.3607i 1.90014i
\(498\) 0 0
\(499\) 42.5410 1.90440 0.952199 0.305479i \(-0.0988166\pi\)
0.952199 + 0.305479i \(0.0988166\pi\)
\(500\) 0 0
\(501\) −17.5279 −0.783087
\(502\) 0 0
\(503\) 16.5967i 0.740012i 0.929029 + 0.370006i \(0.120644\pi\)
−0.929029 + 0.370006i \(0.879356\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) 25.8885i 1.14975i
\(508\) 0 0
\(509\) −30.3607 −1.34571 −0.672857 0.739773i \(-0.734933\pi\)
−0.672857 + 0.739773i \(0.734933\pi\)
\(510\) 0 0
\(511\) 40.1246 1.77501
\(512\) 0 0
\(513\) 14.4721i 0.638960i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) − 11.7082i − 0.514926i
\(518\) 0 0
\(519\) −46.0689 −2.02220
\(520\) 0 0
\(521\) 9.34752 0.409522 0.204761 0.978812i \(-0.434358\pi\)
0.204761 + 0.978812i \(0.434358\pi\)
\(522\) 0 0
\(523\) − 18.8885i − 0.825938i −0.910745 0.412969i \(-0.864492\pi\)
0.910745 0.412969i \(-0.135508\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 3.05573i − 0.133110i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 24.1803 1.04934
\(532\) 0 0
\(533\) 13.2918i 0.575732i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 50.8328i 2.19360i
\(538\) 0 0
\(539\) −10.9443 −0.471403
\(540\) 0 0
\(541\) 8.70820 0.374395 0.187197 0.982322i \(-0.440060\pi\)
0.187197 + 0.982322i \(0.440060\pi\)
\(542\) 0 0
\(543\) − 3.05573i − 0.131134i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 15.7082i 0.671634i 0.941927 + 0.335817i \(0.109012\pi\)
−0.941927 + 0.335817i \(0.890988\pi\)
\(548\) 0 0
\(549\) 54.0689 2.30760
\(550\) 0 0
\(551\) −1.76393 −0.0751460
\(552\) 0 0
\(553\) − 64.3050i − 2.73452i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 36.1803i − 1.53301i −0.642238 0.766505i \(-0.721994\pi\)
0.642238 0.766505i \(-0.278006\pi\)
\(558\) 0 0
\(559\) 5.65248 0.239074
\(560\) 0 0
\(561\) 20.9443 0.884268
\(562\) 0 0
\(563\) − 39.3607i − 1.65885i −0.558614 0.829427i \(-0.688667\pi\)
0.558614 0.829427i \(-0.311333\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 103.430i 4.34363i
\(568\) 0 0
\(569\) −12.5836 −0.527532 −0.263766 0.964587i \(-0.584965\pi\)
−0.263766 + 0.964587i \(0.584965\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) 0 0
\(573\) − 62.0689i − 2.59296i
\(574\) 0 0
\(575\) 0 0
\(576\) 0 0
\(577\) − 9.00000i − 0.374675i −0.982296 0.187337i \(-0.940014\pi\)
0.982296 0.187337i \(-0.0599858\pi\)
\(578\) 0 0
\(579\) −9.52786 −0.395965
\(580\) 0 0
\(581\) −38.1246 −1.58168
\(582\) 0 0
\(583\) 1.23607i 0.0511927i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) − 27.5967i − 1.13904i −0.821978 0.569520i \(-0.807129\pi\)
0.821978 0.569520i \(-0.192871\pi\)
\(588\) 0 0
\(589\) −0.472136 −0.0194540
\(590\) 0 0
\(591\) −26.6525 −1.09634
\(592\) 0 0
\(593\) 44.8885i 1.84335i 0.387961 + 0.921676i \(0.373180\pi\)
−0.387961 + 0.921676i \(0.626820\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 41.1246i − 1.68312i
\(598\) 0 0
\(599\) −16.8328 −0.687770 −0.343885 0.939012i \(-0.611743\pi\)
−0.343885 + 0.939012i \(0.611743\pi\)
\(600\) 0 0
\(601\) −14.9443 −0.609590 −0.304795 0.952418i \(-0.598588\pi\)
−0.304795 + 0.952418i \(0.598588\pi\)
\(602\) 0 0
\(603\) 96.7214i 3.93880i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) − 23.1246i − 0.938599i −0.883039 0.469300i \(-0.844506\pi\)
0.883039 0.469300i \(-0.155494\pi\)
\(608\) 0 0
\(609\) −24.1803 −0.979837
\(610\) 0 0
\(611\) −26.1803 −1.05914
\(612\) 0 0
\(613\) − 18.0689i − 0.729795i −0.931048 0.364898i \(-0.881104\pi\)
0.931048 0.364898i \(-0.118896\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) − 25.3050i − 1.01874i −0.860548 0.509369i \(-0.829879\pi\)
0.860548 0.509369i \(-0.170121\pi\)
\(618\) 0 0
\(619\) 9.52786 0.382957 0.191479 0.981497i \(-0.438672\pi\)
0.191479 + 0.981497i \(0.438672\pi\)
\(620\) 0 0
\(621\) −14.4721 −0.580747
\(622\) 0 0
\(623\) − 8.47214i − 0.339429i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) − 3.23607i − 0.129236i
\(628\) 0 0
\(629\) −72.7214 −2.89959
\(630\) 0 0
\(631\) −24.1246 −0.960386 −0.480193 0.877163i \(-0.659433\pi\)
−0.480193 + 0.877163i \(0.659433\pi\)
\(632\) 0 0
\(633\) − 48.3607i − 1.92216i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 24.4721i 0.969621i
\(638\) 0 0
\(639\) −74.7214 −2.95593
\(640\) 0 0
\(641\) 27.7771 1.09713 0.548564 0.836108i \(-0.315175\pi\)
0.548564 + 0.836108i \(0.315175\pi\)
\(642\) 0 0
\(643\) 1.94427i 0.0766746i 0.999265 + 0.0383373i \(0.0122061\pi\)
−0.999265 + 0.0383373i \(0.987794\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) − 10.1803i − 0.400230i −0.979772 0.200115i \(-0.935868\pi\)
0.979772 0.200115i \(-0.0641317\pi\)
\(648\) 0 0
\(649\) −3.23607 −0.127027
\(650\) 0 0
\(651\) −6.47214 −0.253663
\(652\) 0 0
\(653\) − 39.5410i − 1.54736i −0.633577 0.773680i \(-0.718414\pi\)
0.633577 0.773680i \(-0.281586\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) 70.7771i 2.76128i
\(658\) 0 0
\(659\) −13.4721 −0.524800 −0.262400 0.964959i \(-0.584514\pi\)
−0.262400 + 0.964959i \(0.584514\pi\)
\(660\) 0 0
\(661\) −21.8197 −0.848686 −0.424343 0.905501i \(-0.639495\pi\)
−0.424343 + 0.905501i \(0.639495\pi\)
\(662\) 0 0
\(663\) − 46.8328i − 1.81884i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 1.76393i − 0.0682997i
\(668\) 0 0
\(669\) −42.8328 −1.65601
\(670\) 0 0
\(671\) −7.23607 −0.279345
\(672\) 0 0
\(673\) 48.3050i 1.86202i 0.364995 + 0.931010i \(0.381071\pi\)
−0.364995 + 0.931010i \(0.618929\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 17.7082i 0.680582i 0.940320 + 0.340291i \(0.110526\pi\)
−0.940320 + 0.340291i \(0.889474\pi\)
\(678\) 0 0
\(679\) 26.1803 1.00471
\(680\) 0 0
\(681\) 41.8885 1.60517
\(682\) 0 0
\(683\) 41.7771i 1.59856i 0.600962 + 0.799278i \(0.294784\pi\)
−0.600962 + 0.799278i \(0.705216\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 22.4721i − 0.857365i
\(688\) 0 0
\(689\) 2.76393 0.105297
\(690\) 0 0
\(691\) −42.4721 −1.61572 −0.807858 0.589377i \(-0.799373\pi\)
−0.807858 + 0.589377i \(0.799373\pi\)
\(692\) 0 0
\(693\) − 31.6525i − 1.20238i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 38.4721i 1.45724i
\(698\) 0 0
\(699\) −14.6525 −0.554208
\(700\) 0 0
\(701\) −37.5279 −1.41741 −0.708704 0.705506i \(-0.750720\pi\)
−0.708704 + 0.705506i \(0.750720\pi\)
\(702\) 0 0
\(703\) 11.2361i 0.423776i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) − 80.2492i − 3.01808i
\(708\) 0 0
\(709\) −9.41641 −0.353641 −0.176820 0.984243i \(-0.556581\pi\)
−0.176820 + 0.984243i \(0.556581\pi\)
\(710\) 0 0
\(711\) 113.430 4.25394
\(712\) 0 0
\(713\) − 0.472136i − 0.0176816i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) − 21.8885i − 0.817443i
\(718\) 0 0
\(719\) −37.7082 −1.40628 −0.703139 0.711052i \(-0.748219\pi\)
−0.703139 + 0.711052i \(0.748219\pi\)
\(720\) 0 0
\(721\) 9.47214 0.352761
\(722\) 0 0
\(723\) 68.7214i 2.55577i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) − 44.7214i − 1.65862i −0.558786 0.829312i \(-0.688733\pi\)
0.558786 0.829312i \(-0.311267\pi\)
\(728\) 0 0
\(729\) −41.9443 −1.55349
\(730\) 0 0
\(731\) 16.3607 0.605122
\(732\) 0 0
\(733\) − 39.5279i − 1.45999i −0.683450 0.729997i \(-0.739521\pi\)
0.683450 0.729997i \(-0.260479\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) − 12.9443i − 0.476808i
\(738\) 0 0
\(739\) −9.05573 −0.333120 −0.166560 0.986031i \(-0.553266\pi\)
−0.166560 + 0.986031i \(0.553266\pi\)
\(740\) 0 0
\(741\) −7.23607 −0.265824
\(742\) 0 0
\(743\) 39.1803i 1.43739i 0.695327 + 0.718694i \(0.255260\pi\)
−0.695327 + 0.718694i \(0.744740\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) − 67.2492i − 2.46052i
\(748\) 0 0
\(749\) −82.2492 −3.00532
\(750\) 0 0
\(751\) −22.5967 −0.824567 −0.412284 0.911056i \(-0.635269\pi\)
−0.412284 + 0.911056i \(0.635269\pi\)
\(752\) 0 0
\(753\) − 17.8885i − 0.651895i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 16.7639i 0.609295i 0.952465 + 0.304648i \(0.0985387\pi\)
−0.952465 + 0.304648i \(0.901461\pi\)
\(758\) 0 0
\(759\) 3.23607 0.117462
\(760\) 0 0
\(761\) −52.8885 −1.91721 −0.958604 0.284742i \(-0.908092\pi\)
−0.958604 + 0.284742i \(0.908092\pi\)
\(762\) 0 0
\(763\) 54.0689i 1.95743i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 7.23607i 0.261279i
\(768\) 0 0
\(769\) 46.0689 1.66129 0.830643 0.556805i \(-0.187973\pi\)
0.830643 + 0.556805i \(0.187973\pi\)
\(770\) 0 0
\(771\) −45.3050 −1.63162
\(772\) 0 0
\(773\) − 4.76393i − 0.171347i −0.996323 0.0856734i \(-0.972696\pi\)
0.996323 0.0856734i \(-0.0273041\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 154.026i 5.52566i
\(778\) 0 0
\(779\) 5.94427 0.212976
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) 0 0
\(783\) − 25.5279i − 0.912291i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) − 39.7214i − 1.41591i −0.706256 0.707957i \(-0.749617\pi\)
0.706256 0.707957i \(-0.250383\pi\)
\(788\) 0 0
\(789\) −25.8885 −0.921657
\(790\) 0 0
\(791\) −28.6525 −1.01876
\(792\) 0 0
\(793\) 16.1803i 0.574581i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 15.2361i − 0.539689i −0.962904 0.269845i \(-0.913028\pi\)
0.962904 0.269845i \(-0.0869723\pi\)
\(798\) 0 0
\(799\) −75.7771 −2.68080
\(800\) 0 0
\(801\) 14.9443 0.528030
\(802\) 0 0
\(803\) − 9.47214i − 0.334264i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) 81.4853i 2.86842i
\(808\) 0 0
\(809\) 9.11146 0.320342 0.160171 0.987089i \(-0.448795\pi\)
0.160171 + 0.987089i \(0.448795\pi\)
\(810\) 0 0
\(811\) 15.4164 0.541343 0.270672 0.962672i \(-0.412754\pi\)
0.270672 + 0.962672i \(0.412754\pi\)
\(812\) 0 0
\(813\) − 103.777i − 3.63962i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 2.52786i − 0.0884388i
\(818\) 0 0
\(819\) −70.7771 −2.47315
\(820\) 0 0
\(821\) −22.2361 −0.776044 −0.388022 0.921650i \(-0.626842\pi\)
−0.388022 + 0.921650i \(0.626842\pi\)
\(822\) 0 0
\(823\) 8.94427i 0.311778i 0.987775 + 0.155889i \(0.0498242\pi\)
−0.987775 + 0.155889i \(0.950176\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) − 3.00000i − 0.104320i −0.998639 0.0521601i \(-0.983389\pi\)
0.998639 0.0521601i \(-0.0166106\pi\)
\(828\) 0 0
\(829\) 4.23607 0.147125 0.0735624 0.997291i \(-0.476563\pi\)
0.0735624 + 0.997291i \(0.476563\pi\)
\(830\) 0 0
\(831\) 46.0689 1.59811
\(832\) 0 0
\(833\) 70.8328i 2.45421i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) − 6.83282i − 0.236177i
\(838\) 0 0
\(839\) −8.59675 −0.296793 −0.148396 0.988928i \(-0.547411\pi\)
−0.148396 + 0.988928i \(0.547411\pi\)
\(840\) 0 0
\(841\) −25.8885 −0.892708
\(842\) 0 0
\(843\) − 71.7771i − 2.47213i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) − 42.3607i − 1.45553i
\(848\) 0 0
\(849\) −91.7771 −3.14978
\(850\) 0 0
\(851\) −11.2361 −0.385167
\(852\) 0 0
\(853\) 28.2361i 0.966785i 0.875404 + 0.483392i \(0.160596\pi\)
−0.875404 + 0.483392i \(0.839404\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) 35.3050i 1.20599i 0.797743 + 0.602997i \(0.206027\pi\)
−0.797743 + 0.602997i \(0.793973\pi\)
\(858\) 0 0
\(859\) 21.7082 0.740674 0.370337 0.928897i \(-0.379242\pi\)
0.370337 + 0.928897i \(0.379242\pi\)
\(860\) 0 0
\(861\) 81.4853 2.77701
\(862\) 0 0
\(863\) − 34.5410i − 1.17579i −0.808937 0.587895i \(-0.799957\pi\)
0.808937 0.587895i \(-0.200043\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 80.5410i − 2.73532i
\(868\) 0 0
\(869\) −15.1803 −0.514958
\(870\) 0 0
\(871\) −28.9443 −0.980739
\(872\) 0 0
\(873\) 46.1803i 1.56297i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 2.58359i 0.0872417i 0.999048 + 0.0436209i \(0.0138894\pi\)
−0.999048 + 0.0436209i \(0.986111\pi\)
\(878\) 0 0
\(879\) 46.8328 1.57963
\(880\) 0 0
\(881\) 37.7082 1.27042 0.635211 0.772339i \(-0.280913\pi\)
0.635211 + 0.772339i \(0.280913\pi\)
\(882\) 0 0
\(883\) − 3.05573i − 0.102833i −0.998677 0.0514167i \(-0.983626\pi\)
0.998677 0.0514167i \(-0.0163737\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) 12.3607i 0.415031i 0.978232 + 0.207516i \(0.0665377\pi\)
−0.978232 + 0.207516i \(0.933462\pi\)
\(888\) 0 0
\(889\) −17.7082 −0.593914
\(890\) 0 0
\(891\) 24.4164 0.817980
\(892\) 0 0
\(893\) 11.7082i 0.391800i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 7.23607i − 0.241605i
\(898\) 0 0
\(899\) 0.832816 0.0277760
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) − 34.6525i − 1.15316i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 5.58359i 0.185400i 0.995694 + 0.0927001i \(0.0295498\pi\)
−0.995694 + 0.0927001i \(0.970450\pi\)
\(908\) 0 0
\(909\) 141.554 4.69506
\(910\) 0 0
\(911\) −31.0689 −1.02936 −0.514679 0.857383i \(-0.672089\pi\)
−0.514679 + 0.857383i \(0.672089\pi\)
\(912\) 0 0
\(913\) 9.00000i 0.297857i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 63.3050i − 2.09051i
\(918\) 0 0
\(919\) −43.7771 −1.44407 −0.722036 0.691855i \(-0.756794\pi\)
−0.722036 + 0.691855i \(0.756794\pi\)
\(920\) 0 0
\(921\) 77.6656 2.55917
\(922\) 0 0
\(923\) − 22.3607i − 0.736011i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 16.7082i 0.548769i
\(928\) 0 0
\(929\) −31.0000 −1.01708 −0.508539 0.861039i \(-0.669814\pi\)
−0.508539 + 0.861039i \(0.669814\pi\)
\(930\) 0 0
\(931\) 10.9443 0.358684
\(932\) 0 0
\(933\) − 10.4721i − 0.342842i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) − 7.12461i − 0.232751i −0.993205 0.116375i \(-0.962872\pi\)
0.993205 0.116375i \(-0.0371276\pi\)
\(938\) 0 0
\(939\) 34.4721 1.12496
\(940\) 0 0
\(941\) −13.2361 −0.431483 −0.215742 0.976450i \(-0.569217\pi\)
−0.215742 + 0.976450i \(0.569217\pi\)
\(942\) 0 0
\(943\) 5.94427i 0.193572i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) − 45.1935i − 1.46859i −0.678830 0.734296i \(-0.737513\pi\)
0.678830 0.734296i \(-0.262487\pi\)
\(948\) 0 0
\(949\) −21.1803 −0.687543
\(950\) 0 0
\(951\) −71.5967 −2.32168
\(952\) 0 0
\(953\) 5.34752i 0.173223i 0.996242 + 0.0866116i \(0.0276039\pi\)
−0.996242 + 0.0866116i \(0.972396\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) 5.70820i 0.184520i
\(958\) 0 0
\(959\) 48.8328 1.57689
\(960\) 0 0
\(961\) −30.7771 −0.992809
\(962\) 0 0
\(963\) − 145.082i − 4.67520i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) − 5.88854i − 0.189363i −0.995508 0.0946814i \(-0.969817\pi\)
0.995508 0.0946814i \(-0.0301832\pi\)
\(968\) 0 0
\(969\) −20.9443 −0.672827
\(970\) 0 0
\(971\) 37.2492 1.19538 0.597692 0.801726i \(-0.296084\pi\)
0.597692 + 0.801726i \(0.296084\pi\)
\(972\) 0 0
\(973\) − 84.7214i − 2.71604i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) − 22.5410i − 0.721151i −0.932730 0.360576i \(-0.882580\pi\)
0.932730 0.360576i \(-0.117420\pi\)
\(978\) 0 0
\(979\) −2.00000 −0.0639203
\(980\) 0 0
\(981\) −95.3738 −3.04505
\(982\) 0 0
\(983\) 27.6525i 0.881977i 0.897513 + 0.440989i \(0.145372\pi\)
−0.897513 + 0.440989i \(0.854628\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) 160.498i 5.10872i
\(988\) 0 0
\(989\) 2.52786 0.0803814
\(990\) 0 0
\(991\) −41.5967 −1.32136 −0.660682 0.750666i \(-0.729733\pi\)
−0.660682 + 0.750666i \(0.729733\pi\)
\(992\) 0 0
\(993\) 100.138i 3.17778i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) 2.12461i 0.0672871i 0.999434 + 0.0336436i \(0.0107111\pi\)
−0.999434 + 0.0336436i \(0.989289\pi\)
\(998\) 0 0
\(999\) −162.610 −5.14475
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 4600.2.e.l.4049.4 4
5.2 odd 4 4600.2.a.u.1.2 yes 2
5.3 odd 4 4600.2.a.q.1.1 2
5.4 even 2 inner 4600.2.e.l.4049.1 4
20.3 even 4 9200.2.a.bz.1.2 2
20.7 even 4 9200.2.a.bn.1.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.q.1.1 2 5.3 odd 4
4600.2.a.u.1.2 yes 2 5.2 odd 4
4600.2.e.l.4049.1 4 5.4 even 2 inner
4600.2.e.l.4049.4 4 1.1 even 1 trivial
9200.2.a.bn.1.1 2 20.7 even 4
9200.2.a.bz.1.2 2 20.3 even 4