Properties

Label 4600.2.e.l.4049.2
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 4049.2
Root \(-0.618034i\) of defining polynomial
Character \(\chi\) \(=\) 4600.4049
Dual form 4600.2.e.l.4049.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.23607i q^{3} -0.236068i q^{7} +1.47214 q^{9} +1.00000 q^{11} +2.23607i q^{13} +2.47214i q^{17} -1.00000 q^{19} -0.291796 q^{21} -1.00000i q^{23} -5.52786i q^{27} +6.23607 q^{29} -8.47214 q^{31} -1.23607i q^{33} -6.76393i q^{37} +2.76393 q^{39} +11.9443 q^{41} +11.4721i q^{43} +1.70820i q^{47} +6.94427 q^{49} +3.05573 q^{51} -3.23607i q^{53} +1.23607i q^{57} +1.23607 q^{59} -2.76393 q^{61} -0.347524i q^{63} +4.94427i q^{67} -1.23607 q^{69} +10.0000 q^{71} -0.527864i q^{73} -0.236068i q^{77} +7.18034 q^{79} -2.41641 q^{81} +9.00000i q^{83} -7.70820i q^{87} -2.00000 q^{89} +0.527864 q^{91} +10.4721i q^{93} +16.1803i q^{97} +1.47214 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4 q - 12 q^{9} + 4 q^{11} - 4 q^{19} - 28 q^{21} + 16 q^{29} - 16 q^{31} + 20 q^{39} + 12 q^{41} - 8 q^{49} + 48 q^{51} - 4 q^{59} - 20 q^{61} + 4 q^{69} + 40 q^{71} - 16 q^{79} + 44 q^{81} - 8 q^{89} + 20 q^{91}+ \cdots - 12 q^{99}+O(q^{100}) \) 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
\(3\) − 1.23607i − 0.713644i −0.934172 0.356822i \(-0.883860\pi\)
0.934172 0.356822i \(-0.116140\pi\)
\(4\) 0 0
\(5\) 0 0
\(6\) 0 0
\(7\) − 0.236068i − 0.0892253i −0.999004 0.0446127i \(-0.985795\pi\)
0.999004 0.0446127i \(-0.0142054\pi\)
\(8\) 0 0
\(9\) 1.47214 0.490712
\(10\) 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 \(0.100358\pi\)
−0.950708 + 0.310087i \(0.899642\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) 0 0
\(17\) 2.47214i 0.599581i 0.954005 + 0.299791i \(0.0969168\pi\)
−0.954005 + 0.299791i \(0.903083\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 0 0
\(21\) −0.291796 −0.0636751
\(22\) 0 0
\(23\) − 1.00000i − 0.208514i
\(24\) 0 0
\(25\) 0 0
\(26\) 0 0
\(27\) − 5.52786i − 1.06384i
\(28\) 0 0
\(29\) 6.23607 1.15801 0.579004 0.815324i \(-0.303441\pi\)
0.579004 + 0.815324i \(0.303441\pi\)
\(30\) 0 0
\(31\) −8.47214 −1.52164 −0.760820 0.648963i \(-0.775203\pi\)
−0.760820 + 0.648963i \(0.775203\pi\)
\(32\) 0 0
\(33\) − 1.23607i − 0.215172i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) − 6.76393i − 1.11198i −0.831188 0.555992i \(-0.812339\pi\)
0.831188 0.555992i \(-0.187661\pi\)
\(38\) 0 0
\(39\) 2.76393 0.442583
\(40\) 0 0
\(41\) 11.9443 1.86538 0.932691 0.360677i \(-0.117454\pi\)
0.932691 + 0.360677i \(0.117454\pi\)
\(42\) 0 0
\(43\) 11.4721i 1.74948i 0.484589 + 0.874742i \(0.338969\pi\)
−0.484589 + 0.874742i \(0.661031\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) 0 0
\(47\) 1.70820i 0.249167i 0.992209 + 0.124584i \(0.0397595\pi\)
−0.992209 + 0.124584i \(0.960241\pi\)
\(48\) 0 0
\(49\) 6.94427 0.992039
\(50\) 0 0
\(51\) 3.05573 0.427888
\(52\) 0 0
\(53\) − 3.23607i − 0.444508i −0.974989 0.222254i \(-0.928659\pi\)
0.974989 0.222254i \(-0.0713414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 1.23607i 0.163721i
\(58\) 0 0
\(59\) 1.23607 0.160922 0.0804612 0.996758i \(-0.474361\pi\)
0.0804612 + 0.996758i \(0.474361\pi\)
\(60\) 0 0
\(61\) −2.76393 −0.353885 −0.176943 0.984221i \(-0.556621\pi\)
−0.176943 + 0.984221i \(0.556621\pi\)
\(62\) 0 0
\(63\) − 0.347524i − 0.0437839i
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 4.94427i 0.604039i 0.953302 + 0.302019i \(0.0976608\pi\)
−0.953302 + 0.302019i \(0.902339\pi\)
\(68\) 0 0
\(69\) −1.23607 −0.148805
\(70\) 0 0
\(71\) 10.0000 1.18678 0.593391 0.804914i \(-0.297789\pi\)
0.593391 + 0.804914i \(0.297789\pi\)
\(72\) 0 0
\(73\) − 0.527864i − 0.0617818i −0.999523 0.0308909i \(-0.990166\pi\)
0.999523 0.0308909i \(-0.00983445\pi\)
\(74\) 0 0
\(75\) 0 0
\(76\) 0 0
\(77\) − 0.236068i − 0.0269024i
\(78\) 0 0
\(79\) 7.18034 0.807851 0.403926 0.914792i \(-0.367645\pi\)
0.403926 + 0.914792i \(0.367645\pi\)
\(80\) 0 0
\(81\) −2.41641 −0.268490
\(82\) 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
\(87\) − 7.70820i − 0.826406i
\(88\) 0 0
\(89\) −2.00000 −0.212000 −0.106000 0.994366i \(-0.533804\pi\)
−0.106000 + 0.994366i \(0.533804\pi\)
\(90\) 0 0
\(91\) 0.527864 0.0553352
\(92\) 0 0
\(93\) 10.4721i 1.08591i
\(94\) 0 0
\(95\) 0 0
\(96\) 0 0
\(97\) 16.1803i 1.64286i 0.570306 + 0.821432i \(0.306825\pi\)
−0.570306 + 0.821432i \(0.693175\pi\)
\(98\) 0 0
\(99\) 1.47214 0.147955
\(100\) 0 0
\(101\) −1.05573 −0.105049 −0.0525244 0.998620i \(-0.516727\pi\)
−0.0525244 + 0.998620i \(0.516727\pi\)
\(102\) 0 0
\(103\) 2.23607i 0.220326i 0.993914 + 0.110163i \(0.0351373\pi\)
−0.993914 + 0.110163i \(0.964863\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 0 0
\(107\) − 7.41641i − 0.716971i −0.933535 0.358486i \(-0.883293\pi\)
0.933535 0.358486i \(-0.116707\pi\)
\(108\) 0 0
\(109\) 17.2361 1.65092 0.825458 0.564464i \(-0.190917\pi\)
0.825458 + 0.564464i \(0.190917\pi\)
\(110\) 0 0
\(111\) −8.36068 −0.793561
\(112\) 0 0
\(113\) 11.2361i 1.05700i 0.848933 + 0.528500i \(0.177245\pi\)
−0.848933 + 0.528500i \(0.822755\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) 0 0
\(117\) 3.29180i 0.304327i
\(118\) 0 0
\(119\) 0.583592 0.0534978
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) 0 0
\(123\) − 14.7639i − 1.33122i
\(124\) 0 0
\(125\) 0 0
\(126\) 0 0
\(127\) − 18.1803i − 1.61324i −0.591067 0.806622i \(-0.701293\pi\)
0.591067 0.806622i \(-0.298707\pi\)
\(128\) 0 0
\(129\) 14.1803 1.24851
\(130\) 0 0
\(131\) 2.94427 0.257242 0.128621 0.991694i \(-0.458945\pi\)
0.128621 + 0.991694i \(0.458945\pi\)
\(132\) 0 0
\(133\) 0.236068i 0.0204697i
\(134\) 0 0
\(135\) 0 0
\(136\) 0 0
\(137\) − 20.4721i − 1.74905i −0.484978 0.874526i \(-0.661172\pi\)
0.484978 0.874526i \(-0.338828\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) 0 0
\(141\) 2.11146 0.177817
\(142\) 0 0
\(143\) 2.23607i 0.186989i
\(144\) 0 0
\(145\) 0 0
\(146\) 0 0
\(147\) − 8.58359i − 0.707963i
\(148\) 0 0
\(149\) 6.94427 0.568897 0.284448 0.958691i \(-0.408190\pi\)
0.284448 + 0.958691i \(0.408190\pi\)
\(150\) 0 0
\(151\) 20.4721 1.66600 0.832999 0.553274i \(-0.186622\pi\)
0.832999 + 0.553274i \(0.186622\pi\)
\(152\) 0 0
\(153\) 3.63932i 0.294222i
\(154\) 0 0
\(155\) 0 0
\(156\) 0 0
\(157\) − 13.7082i − 1.09403i −0.837122 0.547017i \(-0.815763\pi\)
0.837122 0.547017i \(-0.184237\pi\)
\(158\) 0 0
\(159\) −4.00000 −0.317221
\(160\) 0 0
\(161\) −0.236068 −0.0186048
\(162\) 0 0
\(163\) 1.70820i 0.133797i 0.997760 + 0.0668984i \(0.0213103\pi\)
−0.997760 + 0.0668984i \(0.978690\pi\)
\(164\) 0 0
\(165\) 0 0
\(166\) 0 0
\(167\) − 21.4164i − 1.65725i −0.559803 0.828626i \(-0.689123\pi\)
0.559803 0.828626i \(-0.310877\pi\)
\(168\) 0 0
\(169\) 8.00000 0.615385
\(170\) 0 0
\(171\) −1.47214 −0.112577
\(172\) 0 0
\(173\) 9.76393i 0.742338i 0.928565 + 0.371169i \(0.121043\pi\)
−0.928565 + 0.371169i \(0.878957\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 0 0
\(177\) − 1.52786i − 0.114841i
\(178\) 0 0
\(179\) 2.29180 0.171297 0.0856484 0.996325i \(-0.472704\pi\)
0.0856484 + 0.996325i \(0.472704\pi\)
\(180\) 0 0
\(181\) 16.9443 1.25946 0.629729 0.776815i \(-0.283166\pi\)
0.629729 + 0.776815i \(0.283166\pi\)
\(182\) 0 0
\(183\) 3.41641i 0.252548i
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) 2.47214i 0.180780i
\(188\) 0 0
\(189\) −1.30495 −0.0949213
\(190\) 0 0
\(191\) 3.18034 0.230121 0.115061 0.993358i \(-0.463294\pi\)
0.115061 + 0.993358i \(0.463294\pi\)
\(192\) 0 0
\(193\) − 14.9443i − 1.07571i −0.843037 0.537856i \(-0.819234\pi\)
0.843037 0.537856i \(-0.180766\pi\)
\(194\) 0 0
\(195\) 0 0
\(196\) 0 0
\(197\) 3.76393i 0.268169i 0.990970 + 0.134085i \(0.0428094\pi\)
−0.990970 + 0.134085i \(0.957191\pi\)
\(198\) 0 0
\(199\) 0.708204 0.0502032 0.0251016 0.999685i \(-0.492009\pi\)
0.0251016 + 0.999685i \(0.492009\pi\)
\(200\) 0 0
\(201\) 6.11146 0.431069
\(202\) 0 0
\(203\) − 1.47214i − 0.103324i
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) − 1.47214i − 0.102321i
\(208\) 0 0
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 2.94427 0.202692 0.101346 0.994851i \(-0.467685\pi\)
0.101346 + 0.994851i \(0.467685\pi\)
\(212\) 0 0
\(213\) − 12.3607i − 0.846940i
\(214\) 0 0
\(215\) 0 0
\(216\) 0 0
\(217\) 2.00000i 0.135769i
\(218\) 0 0
\(219\) −0.652476 −0.0440902
\(220\) 0 0
\(221\) −5.52786 −0.371844
\(222\) 0 0
\(223\) 8.76393i 0.586876i 0.955978 + 0.293438i \(0.0947995\pi\)
−0.955978 + 0.293438i \(0.905201\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 0 0
\(227\) 4.94427i 0.328163i 0.986447 + 0.164081i \(0.0524660\pi\)
−0.986447 + 0.164081i \(0.947534\pi\)
\(228\) 0 0
\(229\) 10.9443 0.723218 0.361609 0.932330i \(-0.382228\pi\)
0.361609 + 0.932330i \(0.382228\pi\)
\(230\) 0 0
\(231\) −0.291796 −0.0191988
\(232\) 0 0
\(233\) 13.4721i 0.882589i 0.897362 + 0.441294i \(0.145481\pi\)
−0.897362 + 0.441294i \(0.854519\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 0 0
\(237\) − 8.87539i − 0.576518i
\(238\) 0 0
\(239\) −11.2361 −0.726801 −0.363400 0.931633i \(-0.618384\pi\)
−0.363400 + 0.931633i \(0.618384\pi\)
\(240\) 0 0
\(241\) 16.7639 1.07986 0.539930 0.841710i \(-0.318451\pi\)
0.539930 + 0.841710i \(0.318451\pi\)
\(242\) 0 0
\(243\) − 13.5967i − 0.872232i
\(244\) 0 0
\(245\) 0 0
\(246\) 0 0
\(247\) − 2.23607i − 0.142278i
\(248\) 0 0
\(249\) 11.1246 0.704994
\(250\) 0 0
\(251\) −14.4721 −0.913473 −0.456737 0.889602i \(-0.650982\pi\)
−0.456737 + 0.889602i \(0.650982\pi\)
\(252\) 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
\(259\) −1.59675 −0.0992171
\(260\) 0 0
\(261\) 9.18034 0.568249
\(262\) 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
\(267\) 2.47214i 0.151292i
\(268\) 0 0
\(269\) 2.81966 0.171918 0.0859589 0.996299i \(-0.472605\pi\)
0.0859589 + 0.996299i \(0.472605\pi\)
\(270\) 0 0
\(271\) 26.0689 1.58357 0.791786 0.610799i \(-0.209152\pi\)
0.791786 + 0.610799i \(0.209152\pi\)
\(272\) 0 0
\(273\) − 0.652476i − 0.0394896i
\(274\) 0 0
\(275\) 0 0
\(276\) 0 0
\(277\) − 9.76393i − 0.586658i −0.956012 0.293329i \(-0.905237\pi\)
0.956012 0.293329i \(-0.0947631\pi\)
\(278\) 0 0
\(279\) −12.4721 −0.746687
\(280\) 0 0
\(281\) 0.180340 0.0107582 0.00537909 0.999986i \(-0.498288\pi\)
0.00537909 + 0.999986i \(0.498288\pi\)
\(282\) 0 0
\(283\) − 16.3607i − 0.972541i −0.873808 0.486271i \(-0.838357\pi\)
0.873808 0.486271i \(-0.161643\pi\)
\(284\) 0 0
\(285\) 0 0
\(286\) 0 0
\(287\) − 2.81966i − 0.166439i
\(288\) 0 0
\(289\) 10.8885 0.640503
\(290\) 0 0
\(291\) 20.0000 1.17242
\(292\) 0 0
\(293\) − 5.52786i − 0.322941i −0.986878 0.161471i \(-0.948376\pi\)
0.986878 0.161471i \(-0.0516237\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) − 5.52786i − 0.320759i
\(298\) 0 0
\(299\) 2.23607 0.129315
\(300\) 0 0
\(301\) 2.70820 0.156098
\(302\) 0 0
\(303\) 1.30495i 0.0749675i
\(304\) 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
\(309\) 2.76393 0.157235
\(310\) 0 0
\(311\) 1.23607 0.0700910 0.0350455 0.999386i \(-0.488842\pi\)
0.0350455 + 0.999386i \(0.488842\pi\)
\(312\) 0 0
\(313\) 20.6525i 1.16735i 0.811988 + 0.583673i \(0.198385\pi\)
−0.811988 + 0.583673i \(0.801615\pi\)
\(314\) 0 0
\(315\) 0 0
\(316\) 0 0
\(317\) − 18.1246i − 1.01798i −0.860773 0.508990i \(-0.830019\pi\)
0.860773 0.508990i \(-0.169981\pi\)
\(318\) 0 0
\(319\) 6.23607 0.349153
\(320\) 0 0
\(321\) −9.16718 −0.511662
\(322\) 0 0
\(323\) − 2.47214i − 0.137553i
\(324\) 0 0
\(325\) 0 0
\(326\) 0 0
\(327\) − 21.3050i − 1.17817i
\(328\) 0 0
\(329\) 0.403252 0.0222320
\(330\) 0 0
\(331\) 13.0557 0.717608 0.358804 0.933413i \(-0.383185\pi\)
0.358804 + 0.933413i \(0.383185\pi\)
\(332\) 0 0
\(333\) − 9.95743i − 0.545664i
\(334\) 0 0
\(335\) 0 0
\(336\) 0 0
\(337\) − 6.65248i − 0.362383i −0.983448 0.181192i \(-0.942005\pi\)
0.983448 0.181192i \(-0.0579955\pi\)
\(338\) 0 0
\(339\) 13.8885 0.754322
\(340\) 0 0
\(341\) −8.47214 −0.458792
\(342\) 0 0
\(343\) − 3.29180i − 0.177740i
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) − 20.9443i − 1.12435i −0.827019 0.562174i \(-0.809965\pi\)
0.827019 0.562174i \(-0.190035\pi\)
\(348\) 0 0
\(349\) −17.1803 −0.919643 −0.459821 0.888011i \(-0.652087\pi\)
−0.459821 + 0.888011i \(0.652087\pi\)
\(350\) 0 0
\(351\) 12.3607 0.659764
\(352\) 0 0
\(353\) 30.8885i 1.64403i 0.569465 + 0.822016i \(0.307150\pi\)
−0.569465 + 0.822016i \(0.692850\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 0 0
\(357\) − 0.721360i − 0.0381784i
\(358\) 0 0
\(359\) −24.2361 −1.27913 −0.639565 0.768737i \(-0.720886\pi\)
−0.639565 + 0.768737i \(0.720886\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 12.3607i 0.648767i
\(364\) 0 0
\(365\) 0 0
\(366\) 0 0
\(367\) − 7.65248i − 0.399456i −0.979851 0.199728i \(-0.935994\pi\)
0.979851 0.199728i \(-0.0640059\pi\)
\(368\) 0 0
\(369\) 17.5836 0.915365
\(370\) 0 0
\(371\) −0.763932 −0.0396614
\(372\) 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
\(377\) 13.9443i 0.718167i
\(378\) 0 0
\(379\) −20.9443 −1.07583 −0.537917 0.842997i \(-0.680789\pi\)
−0.537917 + 0.842997i \(0.680789\pi\)
\(380\) 0 0
\(381\) −22.4721 −1.15128
\(382\) 0 0
\(383\) 4.23607i 0.216453i 0.994126 + 0.108226i \(0.0345172\pi\)
−0.994126 + 0.108226i \(0.965483\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 0 0
\(387\) 16.8885i 0.858493i
\(388\) 0 0
\(389\) 16.1803 0.820376 0.410188 0.912001i \(-0.365463\pi\)
0.410188 + 0.912001i \(0.365463\pi\)
\(390\) 0 0
\(391\) 2.47214 0.125021
\(392\) 0 0
\(393\) − 3.63932i − 0.183579i
\(394\) 0 0
\(395\) 0 0
\(396\) 0 0
\(397\) 22.3607i 1.12225i 0.827731 + 0.561125i \(0.189631\pi\)
−0.827731 + 0.561125i \(0.810369\pi\)
\(398\) 0 0
\(399\) 0.291796 0.0146081
\(400\) 0 0
\(401\) 27.1246 1.35454 0.677269 0.735735i \(-0.263163\pi\)
0.677269 + 0.735735i \(0.263163\pi\)
\(402\) 0 0
\(403\) − 18.9443i − 0.943681i
\(404\) 0 0
\(405\) 0 0
\(406\) 0 0
\(407\) − 6.76393i − 0.335276i
\(408\) 0 0
\(409\) −0.416408 −0.0205900 −0.0102950 0.999947i \(-0.503277\pi\)
−0.0102950 + 0.999947i \(0.503277\pi\)
\(410\) 0 0
\(411\) −25.3050 −1.24820
\(412\) 0 0
\(413\) − 0.291796i − 0.0143583i
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 24.7214i 1.21061i
\(418\) 0 0
\(419\) −34.7771 −1.69897 −0.849486 0.527611i \(-0.823088\pi\)
−0.849486 + 0.527611i \(0.823088\pi\)
\(420\) 0 0
\(421\) −19.8885 −0.969308 −0.484654 0.874706i \(-0.661055\pi\)
−0.484654 + 0.874706i \(0.661055\pi\)
\(422\) 0 0
\(423\) 2.51471i 0.122269i
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) 0.652476i 0.0315755i
\(428\) 0 0
\(429\) 2.76393 0.133444
\(430\) 0 0
\(431\) −4.94427 −0.238157 −0.119079 0.992885i \(-0.537994\pi\)
−0.119079 + 0.992885i \(0.537994\pi\)
\(432\) 0 0
\(433\) 11.1246i 0.534615i 0.963611 + 0.267307i \(0.0861339\pi\)
−0.963611 + 0.267307i \(0.913866\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 0 0
\(437\) 1.00000i 0.0478365i
\(438\) 0 0
\(439\) 32.3607 1.54449 0.772245 0.635324i \(-0.219134\pi\)
0.772245 + 0.635324i \(0.219134\pi\)
\(440\) 0 0
\(441\) 10.2229 0.486805
\(442\) 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
\(447\) − 8.58359i − 0.405990i
\(448\) 0 0
\(449\) −16.8328 −0.794390 −0.397195 0.917734i \(-0.630016\pi\)
−0.397195 + 0.917734i \(0.630016\pi\)
\(450\) 0 0
\(451\) 11.9443 0.562434
\(452\) 0 0
\(453\) − 25.3050i − 1.18893i
\(454\) 0 0
\(455\) 0 0
\(456\) 0 0
\(457\) − 10.6525i − 0.498302i −0.968465 0.249151i \(-0.919848\pi\)
0.968465 0.249151i \(-0.0801515\pi\)
\(458\) 0 0
\(459\) 13.6656 0.637857
\(460\) 0 0
\(461\) 2.34752 0.109335 0.0546676 0.998505i \(-0.482590\pi\)
0.0546676 + 0.998505i \(0.482590\pi\)
\(462\) 0 0
\(463\) − 17.5279i − 0.814589i −0.913297 0.407294i \(-0.866472\pi\)
0.913297 0.407294i \(-0.133528\pi\)
\(464\) 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
\(469\) 1.16718 0.0538956
\(470\) 0 0
\(471\) −16.9443 −0.780751
\(472\) 0 0
\(473\) 11.4721i 0.527489i
\(474\) 0 0
\(475\) 0 0
\(476\) 0 0
\(477\) − 4.76393i − 0.218125i
\(478\) 0 0
\(479\) −28.1246 −1.28505 −0.642523 0.766266i \(-0.722112\pi\)
−0.642523 + 0.766266i \(0.722112\pi\)
\(480\) 0 0
\(481\) 15.1246 0.689623
\(482\) 0 0
\(483\) 0.291796i 0.0132772i
\(484\) 0 0
\(485\) 0 0
\(486\) 0 0
\(487\) − 1.12461i − 0.0509610i −0.999675 0.0254805i \(-0.991888\pi\)
0.999675 0.0254805i \(-0.00811158\pi\)
\(488\) 0 0
\(489\) 2.11146 0.0954833
\(490\) 0 0
\(491\) 1.23607 0.0557830 0.0278915 0.999611i \(-0.491121\pi\)
0.0278915 + 0.999611i \(0.491121\pi\)
\(492\) 0 0
\(493\) 15.4164i 0.694320i
\(494\) 0 0
\(495\) 0 0
\(496\) 0 0
\(497\) − 2.36068i − 0.105891i
\(498\) 0 0
\(499\) −24.5410 −1.09861 −0.549303 0.835623i \(-0.685107\pi\)
−0.549303 + 0.835623i \(0.685107\pi\)
\(500\) 0 0
\(501\) −26.4721 −1.18269
\(502\) 0 0
\(503\) − 32.5967i − 1.45342i −0.686946 0.726709i \(-0.741049\pi\)
0.686946 0.726709i \(-0.258951\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) − 9.88854i − 0.439166i
\(508\) 0 0
\(509\) 14.3607 0.636526 0.318263 0.948002i \(-0.396901\pi\)
0.318263 + 0.948002i \(0.396901\pi\)
\(510\) 0 0
\(511\) −0.124612 −0.00551250
\(512\) 0 0
\(513\) 5.52786i 0.244061i
\(514\) 0 0
\(515\) 0 0
\(516\) 0 0
\(517\) 1.70820i 0.0751267i
\(518\) 0 0
\(519\) 12.0689 0.529765
\(520\) 0 0
\(521\) 40.6525 1.78102 0.890509 0.454966i \(-0.150349\pi\)
0.890509 + 0.454966i \(0.150349\pi\)
\(522\) 0 0
\(523\) 16.8885i 0.738484i 0.929333 + 0.369242i \(0.120383\pi\)
−0.929333 + 0.369242i \(0.879617\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) − 20.9443i − 0.912347i
\(528\) 0 0
\(529\) −1.00000 −0.0434783
\(530\) 0 0
\(531\) 1.81966 0.0789665
\(532\) 0 0
\(533\) 26.7082i 1.15686i
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) − 2.83282i − 0.122245i
\(538\) 0 0
\(539\) 6.94427 0.299111
\(540\) 0 0
\(541\) −4.70820 −0.202421 −0.101211 0.994865i \(-0.532272\pi\)
−0.101211 + 0.994865i \(0.532272\pi\)
\(542\) 0 0
\(543\) − 20.9443i − 0.898805i
\(544\) 0 0
\(545\) 0 0
\(546\) 0 0
\(547\) 2.29180i 0.0979901i 0.998799 + 0.0489951i \(0.0156019\pi\)
−0.998799 + 0.0489951i \(0.984398\pi\)
\(548\) 0 0
\(549\) −4.06888 −0.173656
\(550\) 0 0
\(551\) −6.23607 −0.265665
\(552\) 0 0
\(553\) − 1.69505i − 0.0720808i
\(554\) 0 0
\(555\) 0 0
\(556\) 0 0
\(557\) − 13.8197i − 0.585558i −0.956180 0.292779i \(-0.905420\pi\)
0.956180 0.292779i \(-0.0945800\pi\)
\(558\) 0 0
\(559\) −25.6525 −1.08498
\(560\) 0 0
\(561\) 3.05573 0.129013
\(562\) 0 0
\(563\) 5.36068i 0.225926i 0.993599 + 0.112963i \(0.0360341\pi\)
−0.993599 + 0.112963i \(0.963966\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) 0 0
\(567\) 0.570437i 0.0239561i
\(568\) 0 0
\(569\) −39.4164 −1.65242 −0.826211 0.563361i \(-0.809508\pi\)
−0.826211 + 0.563361i \(0.809508\pi\)
\(570\) 0 0
\(571\) 24.0000 1.00437 0.502184 0.864761i \(-0.332530\pi\)
0.502184 + 0.864761i \(0.332530\pi\)
\(572\) 0 0
\(573\) − 3.93112i − 0.164225i
\(574\) 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
\(579\) −18.4721 −0.767676
\(580\) 0 0
\(581\) 2.12461 0.0881437
\(582\) 0 0
\(583\) − 3.23607i − 0.134024i
\(584\) 0 0
\(585\) 0 0
\(586\) 0 0
\(587\) 21.5967i 0.891393i 0.895184 + 0.445697i \(0.147044\pi\)
−0.895184 + 0.445697i \(0.852956\pi\)
\(588\) 0 0
\(589\) 8.47214 0.349088
\(590\) 0 0
\(591\) 4.65248 0.191377
\(592\) 0 0
\(593\) 9.11146i 0.374163i 0.982344 + 0.187081i \(0.0599028\pi\)
−0.982344 + 0.187081i \(0.940097\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) 0 0
\(597\) − 0.875388i − 0.0358273i
\(598\) 0 0
\(599\) 36.8328 1.50495 0.752474 0.658622i \(-0.228860\pi\)
0.752474 + 0.658622i \(0.228860\pi\)
\(600\) 0 0
\(601\) 2.94427 0.120099 0.0600497 0.998195i \(-0.480874\pi\)
0.0600497 + 0.998195i \(0.480874\pi\)
\(602\) 0 0
\(603\) 7.27864i 0.296409i
\(604\) 0 0
\(605\) 0 0
\(606\) 0 0
\(607\) 17.1246i 0.695067i 0.937668 + 0.347533i \(0.112981\pi\)
−0.937668 + 0.347533i \(0.887019\pi\)
\(608\) 0 0
\(609\) −1.81966 −0.0737363
\(610\) 0 0
\(611\) −3.81966 −0.154527
\(612\) 0 0
\(613\) 40.0689i 1.61837i 0.587556 + 0.809183i \(0.300090\pi\)
−0.587556 + 0.809183i \(0.699910\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 37.3050i 1.50184i 0.660393 + 0.750920i \(0.270390\pi\)
−0.660393 + 0.750920i \(0.729610\pi\)
\(618\) 0 0
\(619\) 18.4721 0.742458 0.371229 0.928541i \(-0.378937\pi\)
0.371229 + 0.928541i \(0.378937\pi\)
\(620\) 0 0
\(621\) −5.52786 −0.221826
\(622\) 0 0
\(623\) 0.472136i 0.0189157i
\(624\) 0 0
\(625\) 0 0
\(626\) 0 0
\(627\) 1.23607i 0.0493638i
\(628\) 0 0
\(629\) 16.7214 0.666724
\(630\) 0 0
\(631\) 16.1246 0.641911 0.320955 0.947094i \(-0.395996\pi\)
0.320955 + 0.947094i \(0.395996\pi\)
\(632\) 0 0
\(633\) − 3.63932i − 0.144650i
\(634\) 0 0
\(635\) 0 0
\(636\) 0 0
\(637\) 15.5279i 0.615236i
\(638\) 0 0
\(639\) 14.7214 0.582368
\(640\) 0 0
\(641\) −43.7771 −1.72909 −0.864546 0.502555i \(-0.832394\pi\)
−0.864546 + 0.502555i \(0.832394\pi\)
\(642\) 0 0
\(643\) − 15.9443i − 0.628781i −0.949294 0.314390i \(-0.898200\pi\)
0.949294 0.314390i \(-0.101800\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 0 0
\(647\) 12.1803i 0.478859i 0.970914 + 0.239429i \(0.0769603\pi\)
−0.970914 + 0.239429i \(0.923040\pi\)
\(648\) 0 0
\(649\) 1.23607 0.0485199
\(650\) 0 0
\(651\) 2.47214 0.0968906
\(652\) 0 0
\(653\) 27.5410i 1.07776i 0.842381 + 0.538882i \(0.181153\pi\)
−0.842381 + 0.538882i \(0.818847\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 0 0
\(657\) − 0.777088i − 0.0303171i
\(658\) 0 0
\(659\) −4.52786 −0.176381 −0.0881903 0.996104i \(-0.528108\pi\)
−0.0881903 + 0.996104i \(0.528108\pi\)
\(660\) 0 0
\(661\) −44.1803 −1.71842 −0.859208 0.511626i \(-0.829043\pi\)
−0.859208 + 0.511626i \(0.829043\pi\)
\(662\) 0 0
\(663\) 6.83282i 0.265365i
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) − 6.23607i − 0.241462i
\(668\) 0 0
\(669\) 10.8328 0.418821
\(670\) 0 0
\(671\) −2.76393 −0.106700
\(672\) 0 0
\(673\) − 14.3050i − 0.551415i −0.961242 0.275708i \(-0.911088\pi\)
0.961242 0.275708i \(-0.0889122\pi\)
\(674\) 0 0
\(675\) 0 0
\(676\) 0 0
\(677\) 4.29180i 0.164947i 0.996593 + 0.0824736i \(0.0262820\pi\)
−0.996593 + 0.0824736i \(0.973718\pi\)
\(678\) 0 0
\(679\) 3.81966 0.146585
\(680\) 0 0
\(681\) 6.11146 0.234192
\(682\) 0 0
\(683\) − 29.7771i − 1.13939i −0.821857 0.569694i \(-0.807062\pi\)
0.821857 0.569694i \(-0.192938\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 0 0
\(687\) − 13.5279i − 0.516120i
\(688\) 0 0
\(689\) 7.23607 0.275672
\(690\) 0 0
\(691\) −33.5279 −1.27546 −0.637730 0.770260i \(-0.720126\pi\)
−0.637730 + 0.770260i \(0.720126\pi\)
\(692\) 0 0
\(693\) − 0.347524i − 0.0132014i
\(694\) 0 0
\(695\) 0 0
\(696\) 0 0
\(697\) 29.5279i 1.11845i
\(698\) 0 0
\(699\) 16.6525 0.629854
\(700\) 0 0
\(701\) −46.4721 −1.75523 −0.877614 0.479368i \(-0.840866\pi\)
−0.877614 + 0.479368i \(0.840866\pi\)
\(702\) 0 0
\(703\) 6.76393i 0.255107i
\(704\) 0 0
\(705\) 0 0
\(706\) 0 0
\(707\) 0.249224i 0.00937302i
\(708\) 0 0
\(709\) 17.4164 0.654087 0.327043 0.945009i \(-0.393948\pi\)
0.327043 + 0.945009i \(0.393948\pi\)
\(710\) 0 0
\(711\) 10.5704 0.396422
\(712\) 0 0
\(713\) 8.47214i 0.317284i
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 13.8885i 0.518677i
\(718\) 0 0
\(719\) −24.2918 −0.905931 −0.452966 0.891528i \(-0.649634\pi\)
−0.452966 + 0.891528i \(0.649634\pi\)
\(720\) 0 0
\(721\) 0.527864 0.0196587
\(722\) 0 0
\(723\) − 20.7214i − 0.770636i
\(724\) 0 0
\(725\) 0 0
\(726\) 0 0
\(727\) 44.7214i 1.65862i 0.558786 + 0.829312i \(0.311267\pi\)
−0.558786 + 0.829312i \(0.688733\pi\)
\(728\) 0 0
\(729\) −24.0557 −0.890953
\(730\) 0 0
\(731\) −28.3607 −1.04896
\(732\) 0 0
\(733\) − 48.4721i − 1.79036i −0.445706 0.895180i \(-0.647047\pi\)
0.445706 0.895180i \(-0.352953\pi\)
\(734\) 0 0
\(735\) 0 0
\(736\) 0 0
\(737\) 4.94427i 0.182125i
\(738\) 0 0
\(739\) −26.9443 −0.991161 −0.495581 0.868562i \(-0.665045\pi\)
−0.495581 + 0.868562i \(0.665045\pi\)
\(740\) 0 0
\(741\) −2.76393 −0.101536
\(742\) 0 0
\(743\) 16.8197i 0.617053i 0.951216 + 0.308527i \(0.0998359\pi\)
−0.951216 + 0.308527i \(0.900164\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) 0 0
\(747\) 13.2492i 0.484764i
\(748\) 0 0
\(749\) −1.75078 −0.0639720
\(750\) 0 0
\(751\) 26.5967 0.970529 0.485265 0.874367i \(-0.338723\pi\)
0.485265 + 0.874367i \(0.338723\pi\)
\(752\) 0 0
\(753\) 17.8885i 0.651895i
\(754\) 0 0
\(755\) 0 0
\(756\) 0 0
\(757\) 21.2361i 0.771838i 0.922533 + 0.385919i \(0.126116\pi\)
−0.922533 + 0.385919i \(0.873884\pi\)
\(758\) 0 0
\(759\) −1.23607 −0.0448664
\(760\) 0 0
\(761\) −17.1115 −0.620290 −0.310145 0.950689i \(-0.600378\pi\)
−0.310145 + 0.950689i \(0.600378\pi\)
\(762\) 0 0
\(763\) − 4.06888i − 0.147303i
\(764\) 0 0
\(765\) 0 0
\(766\) 0 0
\(767\) 2.76393i 0.0997998i
\(768\) 0 0
\(769\) −12.0689 −0.435215 −0.217608 0.976036i \(-0.569825\pi\)
−0.217608 + 0.976036i \(0.569825\pi\)
\(770\) 0 0
\(771\) 17.3050 0.623223
\(772\) 0 0
\(773\) − 9.23607i − 0.332198i −0.986109 0.166099i \(-0.946883\pi\)
0.986109 0.166099i \(-0.0531172\pi\)
\(774\) 0 0
\(775\) 0 0
\(776\) 0 0
\(777\) 1.97369i 0.0708057i
\(778\) 0 0
\(779\) −11.9443 −0.427948
\(780\) 0 0
\(781\) 10.0000 0.357828
\(782\) 0 0
\(783\) − 34.4721i − 1.23193i
\(784\) 0 0
\(785\) 0 0
\(786\) 0 0
\(787\) 49.7214i 1.77238i 0.463327 + 0.886188i \(0.346656\pi\)
−0.463327 + 0.886188i \(0.653344\pi\)
\(788\) 0 0
\(789\) 9.88854 0.352041
\(790\) 0 0
\(791\) 2.65248 0.0943112
\(792\) 0 0
\(793\) − 6.18034i − 0.219470i
\(794\) 0 0
\(795\) 0 0
\(796\) 0 0
\(797\) − 10.7639i − 0.381278i −0.981660 0.190639i \(-0.938944\pi\)
0.981660 0.190639i \(-0.0610560\pi\)
\(798\) 0 0
\(799\) −4.22291 −0.149396
\(800\) 0 0
\(801\) −2.94427 −0.104031
\(802\) 0 0
\(803\) − 0.527864i − 0.0186279i
\(804\) 0 0
\(805\) 0 0
\(806\) 0 0
\(807\) − 3.48529i − 0.122688i
\(808\) 0 0
\(809\) 44.8885 1.57820 0.789099 0.614267i \(-0.210548\pi\)
0.789099 + 0.614267i \(0.210548\pi\)
\(810\) 0 0
\(811\) −11.4164 −0.400884 −0.200442 0.979706i \(-0.564238\pi\)
−0.200442 + 0.979706i \(0.564238\pi\)
\(812\) 0 0
\(813\) − 32.2229i − 1.13011i
\(814\) 0 0
\(815\) 0 0
\(816\) 0 0
\(817\) − 11.4721i − 0.401359i
\(818\) 0 0
\(819\) 0.777088 0.0271536
\(820\) 0 0
\(821\) −17.7639 −0.619966 −0.309983 0.950742i \(-0.600323\pi\)
−0.309983 + 0.950742i \(0.600323\pi\)
\(822\) 0 0
\(823\) − 8.94427i − 0.311778i −0.987775 0.155889i \(-0.950176\pi\)
0.987775 0.155889i \(-0.0498242\pi\)
\(824\) 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
\(829\) −0.236068 −0.00819898 −0.00409949 0.999992i \(-0.501305\pi\)
−0.00409949 + 0.999992i \(0.501305\pi\)
\(830\) 0 0
\(831\) −12.0689 −0.418665
\(832\) 0 0
\(833\) 17.1672i 0.594808i
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 46.8328i 1.61878i
\(838\) 0 0
\(839\) 40.5967 1.40156 0.700778 0.713380i \(-0.252836\pi\)
0.700778 + 0.713380i \(0.252836\pi\)
\(840\) 0 0
\(841\) 9.88854 0.340984
\(842\) 0 0
\(843\) − 0.222912i − 0.00767751i
\(844\) 0 0
\(845\) 0 0
\(846\) 0 0
\(847\) 2.36068i 0.0811139i
\(848\) 0 0
\(849\) −20.2229 −0.694049
\(850\) 0 0
\(851\) −6.76393 −0.231865
\(852\) 0 0
\(853\) 23.7639i 0.813662i 0.913504 + 0.406831i \(0.133366\pi\)
−0.913504 + 0.406831i \(0.866634\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) − 27.3050i − 0.932719i −0.884595 0.466360i \(-0.845565\pi\)
0.884595 0.466360i \(-0.154435\pi\)
\(858\) 0 0
\(859\) 8.29180 0.282912 0.141456 0.989945i \(-0.454822\pi\)
0.141456 + 0.989945i \(0.454822\pi\)
\(860\) 0 0
\(861\) −3.48529 −0.118778
\(862\) 0 0
\(863\) 32.5410i 1.10771i 0.832613 + 0.553855i \(0.186844\pi\)
−0.832613 + 0.553855i \(0.813156\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 0 0
\(867\) − 13.4590i − 0.457091i
\(868\) 0 0
\(869\) 7.18034 0.243576
\(870\) 0 0
\(871\) −11.0557 −0.374609
\(872\) 0 0
\(873\) 23.8197i 0.806173i
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) 29.4164i 0.993322i 0.867945 + 0.496661i \(0.165441\pi\)
−0.867945 + 0.496661i \(0.834559\pi\)
\(878\) 0 0
\(879\) −6.83282 −0.230465
\(880\) 0 0
\(881\) 24.2918 0.818411 0.409206 0.912442i \(-0.365806\pi\)
0.409206 + 0.912442i \(0.365806\pi\)
\(882\) 0 0
\(883\) − 20.9443i − 0.704831i −0.935844 0.352415i \(-0.885360\pi\)
0.935844 0.352415i \(-0.114640\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 0 0
\(887\) − 32.3607i − 1.08657i −0.839550 0.543283i \(-0.817181\pi\)
0.839550 0.543283i \(-0.182819\pi\)
\(888\) 0 0
\(889\) −4.29180 −0.143942
\(890\) 0 0
\(891\) −2.41641 −0.0809527
\(892\) 0 0
\(893\) − 1.70820i − 0.0571629i
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) − 2.76393i − 0.0922850i
\(898\) 0 0
\(899\) −52.8328 −1.76207
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 0 0
\(903\) − 3.34752i − 0.111399i
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 32.4164i 1.07637i 0.842827 + 0.538185i \(0.180890\pi\)
−0.842827 + 0.538185i \(0.819110\pi\)
\(908\) 0 0
\(909\) −1.55418 −0.0515487
\(910\) 0 0
\(911\) 27.0689 0.896832 0.448416 0.893825i \(-0.351988\pi\)
0.448416 + 0.893825i \(0.351988\pi\)
\(912\) 0 0
\(913\) 9.00000i 0.297857i
\(914\) 0 0
\(915\) 0 0
\(916\) 0 0
\(917\) − 0.695048i − 0.0229525i
\(918\) 0 0
\(919\) 27.7771 0.916282 0.458141 0.888880i \(-0.348516\pi\)
0.458141 + 0.888880i \(0.348516\pi\)
\(920\) 0 0
\(921\) −29.6656 −0.977516
\(922\) 0 0
\(923\) 22.3607i 0.736011i
\(924\) 0 0
\(925\) 0 0
\(926\) 0 0
\(927\) 3.29180i 0.108117i
\(928\) 0 0
\(929\) −31.0000 −1.01708 −0.508539 0.861039i \(-0.669814\pi\)
−0.508539 + 0.861039i \(0.669814\pi\)
\(930\) 0 0
\(931\) −6.94427 −0.227589
\(932\) 0 0
\(933\) − 1.52786i − 0.0500200i
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) 33.1246i 1.08213i 0.840980 + 0.541067i \(0.181979\pi\)
−0.840980 + 0.541067i \(0.818021\pi\)
\(938\) 0 0
\(939\) 25.5279 0.833070
\(940\) 0 0
\(941\) −8.76393 −0.285696 −0.142848 0.989745i \(-0.545626\pi\)
−0.142848 + 0.989745i \(0.545626\pi\)
\(942\) 0 0
\(943\) − 11.9443i − 0.388959i
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) 53.1935i 1.72856i 0.503014 + 0.864278i \(0.332224\pi\)
−0.503014 + 0.864278i \(0.667776\pi\)
\(948\) 0 0
\(949\) 1.18034 0.0383155
\(950\) 0 0
\(951\) −22.4033 −0.726475
\(952\) 0 0
\(953\) 36.6525i 1.18729i 0.804727 + 0.593645i \(0.202312\pi\)
−0.804727 + 0.593645i \(0.797688\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 0 0
\(957\) − 7.70820i − 0.249171i
\(958\) 0 0
\(959\) −4.83282 −0.156060
\(960\) 0 0
\(961\) 40.7771 1.31539
\(962\) 0 0
\(963\) − 10.9180i − 0.351826i
\(964\) 0 0
\(965\) 0 0
\(966\) 0 0
\(967\) 29.8885i 0.961151i 0.876953 + 0.480575i \(0.159572\pi\)
−0.876953 + 0.480575i \(0.840428\pi\)
\(968\) 0 0
\(969\) −3.05573 −0.0981641
\(970\) 0 0
\(971\) −43.2492 −1.38793 −0.693967 0.720007i \(-0.744139\pi\)
−0.693967 + 0.720007i \(0.744139\pi\)
\(972\) 0 0
\(973\) 4.72136i 0.151360i
\(974\) 0 0
\(975\) 0 0
\(976\) 0 0
\(977\) 44.5410i 1.42499i 0.701675 + 0.712497i \(0.252436\pi\)
−0.701675 + 0.712497i \(0.747564\pi\)
\(978\) 0 0
\(979\) −2.00000 −0.0639203
\(980\) 0 0
\(981\) 25.3738 0.810124
\(982\) 0 0
\(983\) − 3.65248i − 0.116496i −0.998302 0.0582479i \(-0.981449\pi\)
0.998302 0.0582479i \(-0.0185514\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) 0 0
\(987\) − 0.498447i − 0.0158657i
\(988\) 0 0
\(989\) 11.4721 0.364793
\(990\) 0 0
\(991\) 7.59675 0.241319 0.120659 0.992694i \(-0.461499\pi\)
0.120659 + 0.992694i \(0.461499\pi\)
\(992\) 0 0
\(993\) − 16.1378i − 0.512117i
\(994\) 0 0
\(995\) 0 0
\(996\) 0 0
\(997\) − 38.1246i − 1.20742i −0.797205 0.603709i \(-0.793689\pi\)
0.797205 0.603709i \(-0.206311\pi\)
\(998\) 0 0
\(999\) −37.3901 −1.18297
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.4049.2 4
5.2 odd 4 4600.2.a.u.1.1 yes 2
5.3 odd 4 4600.2.a.q.1.2 2
5.4 even 2 inner 4600.2.e.l.4049.3 4
20.3 even 4 9200.2.a.bz.1.1 2
20.7 even 4 9200.2.a.bn.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
4600.2.a.q.1.2 2 5.3 odd 4
4600.2.a.u.1.1 yes 2 5.2 odd 4
4600.2.e.l.4049.2 4 1.1 even 1 trivial
4600.2.e.l.4049.3 4 5.4 even 2 inner
9200.2.a.bn.1.2 2 20.7 even 4
9200.2.a.bz.1.1 2 20.3 even 4