Properties

Label 3720.2.a.o.1.1
Level $3720$
Weight $2$
Character 3720.1
Self dual yes
Analytic conductor $29.704$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3720,2,Mod(1,3720)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3720, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3720.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3720 = 2^{3} \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3720.a (trivial)

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: yes
Analytic conductor: \(29.7043495519\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.48119\) of defining polynomial
Character \(\chi\) \(=\) 3720.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -1.00000 q^{5} -2.48119 q^{7} +1.00000 q^{9} -0.387873 q^{11} +2.48119 q^{13} -1.00000 q^{15} -0.418190 q^{17} -4.00000 q^{19} -2.48119 q^{21} +5.11871 q^{23} +1.00000 q^{25} +1.00000 q^{27} -4.24965 q^{29} -1.00000 q^{31} -0.387873 q^{33} +2.48119 q^{35} +3.83146 q^{37} +2.48119 q^{39} +4.31265 q^{41} -6.00000 q^{43} -1.00000 q^{45} -10.8568 q^{47} -0.843675 q^{49} -0.418190 q^{51} -8.85685 q^{53} +0.387873 q^{55} -4.00000 q^{57} -3.86177 q^{59} +12.2374 q^{61} -2.48119 q^{63} -2.48119 q^{65} -14.2193 q^{67} +5.11871 q^{69} +7.59991 q^{71} -4.60720 q^{73} +1.00000 q^{75} +0.962389 q^{77} -5.38058 q^{79} +1.00000 q^{81} -5.89446 q^{83} +0.418190 q^{85} -4.24965 q^{87} -5.93700 q^{89} -6.15633 q^{91} -1.00000 q^{93} +4.00000 q^{95} -12.3127 q^{97} -0.387873 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + 3 q^{3} - 3 q^{5} - 2 q^{7} + 3 q^{9} - 2 q^{11} + 2 q^{13} - 3 q^{15} - 12 q^{19} - 2 q^{21} - 6 q^{23} + 3 q^{25} + 3 q^{27} + 4 q^{29} - 3 q^{31} - 2 q^{33} + 2 q^{35} - 4 q^{37} + 2 q^{39} - 8 q^{41}+ \cdots - 2 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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
\(7\) −2.48119 −0.937803 −0.468902 0.883250i \(-0.655350\pi\)
−0.468902 + 0.883250i \(0.655350\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.387873 −0.116948 −0.0584741 0.998289i \(-0.518623\pi\)
−0.0584741 + 0.998289i \(0.518623\pi\)
\(12\) 0 0
\(13\) 2.48119 0.688159 0.344080 0.938940i \(-0.388191\pi\)
0.344080 + 0.938940i \(0.388191\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 0 0
\(17\) −0.418190 −0.101426 −0.0507130 0.998713i \(-0.516149\pi\)
−0.0507130 + 0.998713i \(0.516149\pi\)
\(18\) 0 0
\(19\) −4.00000 −0.917663 −0.458831 0.888523i \(-0.651732\pi\)
−0.458831 + 0.888523i \(0.651732\pi\)
\(20\) 0 0
\(21\) −2.48119 −0.541441
\(22\) 0 0
\(23\) 5.11871 1.06733 0.533663 0.845697i \(-0.320815\pi\)
0.533663 + 0.845697i \(0.320815\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −4.24965 −0.789140 −0.394570 0.918866i \(-0.629106\pi\)
−0.394570 + 0.918866i \(0.629106\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) 0 0
\(33\) −0.387873 −0.0675200
\(34\) 0 0
\(35\) 2.48119 0.419398
\(36\) 0 0
\(37\) 3.83146 0.629887 0.314944 0.949110i \(-0.398014\pi\)
0.314944 + 0.949110i \(0.398014\pi\)
\(38\) 0 0
\(39\) 2.48119 0.397309
\(40\) 0 0
\(41\) 4.31265 0.673523 0.336761 0.941590i \(-0.390668\pi\)
0.336761 + 0.941590i \(0.390668\pi\)
\(42\) 0 0
\(43\) −6.00000 −0.914991 −0.457496 0.889212i \(-0.651253\pi\)
−0.457496 + 0.889212i \(0.651253\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −10.8568 −1.58363 −0.791817 0.610758i \(-0.790865\pi\)
−0.791817 + 0.610758i \(0.790865\pi\)
\(48\) 0 0
\(49\) −0.843675 −0.120525
\(50\) 0 0
\(51\) −0.418190 −0.0585584
\(52\) 0 0
\(53\) −8.85685 −1.21658 −0.608291 0.793714i \(-0.708145\pi\)
−0.608291 + 0.793714i \(0.708145\pi\)
\(54\) 0 0
\(55\) 0.387873 0.0523008
\(56\) 0 0
\(57\) −4.00000 −0.529813
\(58\) 0 0
\(59\) −3.86177 −0.502760 −0.251380 0.967888i \(-0.580884\pi\)
−0.251380 + 0.967888i \(0.580884\pi\)
\(60\) 0 0
\(61\) 12.2374 1.56684 0.783421 0.621491i \(-0.213473\pi\)
0.783421 + 0.621491i \(0.213473\pi\)
\(62\) 0 0
\(63\) −2.48119 −0.312601
\(64\) 0 0
\(65\) −2.48119 −0.307754
\(66\) 0 0
\(67\) −14.2193 −1.73717 −0.868584 0.495542i \(-0.834969\pi\)
−0.868584 + 0.495542i \(0.834969\pi\)
\(68\) 0 0
\(69\) 5.11871 0.616221
\(70\) 0 0
\(71\) 7.59991 0.901943 0.450972 0.892538i \(-0.351078\pi\)
0.450972 + 0.892538i \(0.351078\pi\)
\(72\) 0 0
\(73\) −4.60720 −0.539232 −0.269616 0.962968i \(-0.586897\pi\)
−0.269616 + 0.962968i \(0.586897\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 0 0
\(77\) 0.962389 0.109674
\(78\) 0 0
\(79\) −5.38058 −0.605362 −0.302681 0.953092i \(-0.597882\pi\)
−0.302681 + 0.953092i \(0.597882\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −5.89446 −0.647001 −0.323501 0.946228i \(-0.604860\pi\)
−0.323501 + 0.946228i \(0.604860\pi\)
\(84\) 0 0
\(85\) 0.418190 0.0453591
\(86\) 0 0
\(87\) −4.24965 −0.455610
\(88\) 0 0
\(89\) −5.93700 −0.629320 −0.314660 0.949204i \(-0.601891\pi\)
−0.314660 + 0.949204i \(0.601891\pi\)
\(90\) 0 0
\(91\) −6.15633 −0.645358
\(92\) 0 0
\(93\) −1.00000 −0.103695
\(94\) 0 0
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −12.3127 −1.25016 −0.625080 0.780561i \(-0.714934\pi\)
−0.625080 + 0.780561i \(0.714934\pi\)
\(98\) 0 0
\(99\) −0.387873 −0.0389827
\(100\) 0 0
\(101\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(102\) 0 0
\(103\) 7.05571 0.695220 0.347610 0.937639i \(-0.386993\pi\)
0.347610 + 0.937639i \(0.386993\pi\)
\(104\) 0 0
\(105\) 2.48119 0.242140
\(106\) 0 0
\(107\) 11.0435 1.06761 0.533807 0.845606i \(-0.320761\pi\)
0.533807 + 0.845606i \(0.320761\pi\)
\(108\) 0 0
\(109\) −20.5950 −1.97264 −0.986321 0.164837i \(-0.947290\pi\)
−0.986321 + 0.164837i \(0.947290\pi\)
\(110\) 0 0
\(111\) 3.83146 0.363666
\(112\) 0 0
\(113\) 12.0508 1.13364 0.566821 0.823841i \(-0.308173\pi\)
0.566821 + 0.823841i \(0.308173\pi\)
\(114\) 0 0
\(115\) −5.11871 −0.477323
\(116\) 0 0
\(117\) 2.48119 0.229386
\(118\) 0 0
\(119\) 1.03761 0.0951177
\(120\) 0 0
\(121\) −10.8496 −0.986323
\(122\) 0 0
\(123\) 4.31265 0.388859
\(124\) 0 0
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 3.66291 0.325031 0.162515 0.986706i \(-0.448039\pi\)
0.162515 + 0.986706i \(0.448039\pi\)
\(128\) 0 0
\(129\) −6.00000 −0.528271
\(130\) 0 0
\(131\) −6.24965 −0.546034 −0.273017 0.962009i \(-0.588022\pi\)
−0.273017 + 0.962009i \(0.588022\pi\)
\(132\) 0 0
\(133\) 9.92478 0.860587
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) −8.73084 −0.745926 −0.372963 0.927846i \(-0.621658\pi\)
−0.372963 + 0.927846i \(0.621658\pi\)
\(138\) 0 0
\(139\) −11.0376 −0.936198 −0.468099 0.883676i \(-0.655061\pi\)
−0.468099 + 0.883676i \(0.655061\pi\)
\(140\) 0 0
\(141\) −10.8568 −0.914312
\(142\) 0 0
\(143\) −0.962389 −0.0804790
\(144\) 0 0
\(145\) 4.24965 0.352914
\(146\) 0 0
\(147\) −0.843675 −0.0695851
\(148\) 0 0
\(149\) −7.53690 −0.617447 −0.308724 0.951152i \(-0.599902\pi\)
−0.308724 + 0.951152i \(0.599902\pi\)
\(150\) 0 0
\(151\) −18.9683 −1.54362 −0.771808 0.635856i \(-0.780647\pi\)
−0.771808 + 0.635856i \(0.780647\pi\)
\(152\) 0 0
\(153\) −0.418190 −0.0338087
\(154\) 0 0
\(155\) 1.00000 0.0803219
\(156\) 0 0
\(157\) 4.77575 0.381146 0.190573 0.981673i \(-0.438965\pi\)
0.190573 + 0.981673i \(0.438965\pi\)
\(158\) 0 0
\(159\) −8.85685 −0.702394
\(160\) 0 0
\(161\) −12.7005 −1.00094
\(162\) 0 0
\(163\) 1.56959 0.122940 0.0614699 0.998109i \(-0.480421\pi\)
0.0614699 + 0.998109i \(0.480421\pi\)
\(164\) 0 0
\(165\) 0.387873 0.0301959
\(166\) 0 0
\(167\) −3.87399 −0.299779 −0.149889 0.988703i \(-0.547892\pi\)
−0.149889 + 0.988703i \(0.547892\pi\)
\(168\) 0 0
\(169\) −6.84367 −0.526437
\(170\) 0 0
\(171\) −4.00000 −0.305888
\(172\) 0 0
\(173\) −8.05079 −0.612090 −0.306045 0.952017i \(-0.599006\pi\)
−0.306045 + 0.952017i \(0.599006\pi\)
\(174\) 0 0
\(175\) −2.48119 −0.187561
\(176\) 0 0
\(177\) −3.86177 −0.290269
\(178\) 0 0
\(179\) −10.3879 −0.776426 −0.388213 0.921570i \(-0.626908\pi\)
−0.388213 + 0.921570i \(0.626908\pi\)
\(180\) 0 0
\(181\) 2.31265 0.171898 0.0859490 0.996300i \(-0.472608\pi\)
0.0859490 + 0.996300i \(0.472608\pi\)
\(182\) 0 0
\(183\) 12.2374 0.904617
\(184\) 0 0
\(185\) −3.83146 −0.281694
\(186\) 0 0
\(187\) 0.162205 0.0118616
\(188\) 0 0
\(189\) −2.48119 −0.180480
\(190\) 0 0
\(191\) 6.18901 0.447821 0.223911 0.974610i \(-0.428118\pi\)
0.223911 + 0.974610i \(0.428118\pi\)
\(192\) 0 0
\(193\) 1.55149 0.111679 0.0558394 0.998440i \(-0.482217\pi\)
0.0558394 + 0.998440i \(0.482217\pi\)
\(194\) 0 0
\(195\) −2.48119 −0.177682
\(196\) 0 0
\(197\) 15.1793 1.08148 0.540742 0.841189i \(-0.318144\pi\)
0.540742 + 0.841189i \(0.318144\pi\)
\(198\) 0 0
\(199\) 0.730841 0.0518079 0.0259040 0.999664i \(-0.491754\pi\)
0.0259040 + 0.999664i \(0.491754\pi\)
\(200\) 0 0
\(201\) −14.2193 −1.00295
\(202\) 0 0
\(203\) 10.5442 0.740058
\(204\) 0 0
\(205\) −4.31265 −0.301209
\(206\) 0 0
\(207\) 5.11871 0.355775
\(208\) 0 0
\(209\) 1.55149 0.107319
\(210\) 0 0
\(211\) 5.27504 0.363149 0.181574 0.983377i \(-0.441881\pi\)
0.181574 + 0.983377i \(0.441881\pi\)
\(212\) 0 0
\(213\) 7.59991 0.520737
\(214\) 0 0
\(215\) 6.00000 0.409197
\(216\) 0 0
\(217\) 2.48119 0.168434
\(218\) 0 0
\(219\) −4.60720 −0.311326
\(220\) 0 0
\(221\) −1.03761 −0.0697973
\(222\) 0 0
\(223\) −29.5125 −1.97630 −0.988150 0.153488i \(-0.950949\pi\)
−0.988150 + 0.153488i \(0.950949\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −6.67021 −0.442717 −0.221359 0.975192i \(-0.571049\pi\)
−0.221359 + 0.975192i \(0.571049\pi\)
\(228\) 0 0
\(229\) 7.27504 0.480748 0.240374 0.970680i \(-0.422730\pi\)
0.240374 + 0.970680i \(0.422730\pi\)
\(230\) 0 0
\(231\) 0.962389 0.0633205
\(232\) 0 0
\(233\) 23.0943 1.51296 0.756478 0.654019i \(-0.226918\pi\)
0.756478 + 0.654019i \(0.226918\pi\)
\(234\) 0 0
\(235\) 10.8568 0.708223
\(236\) 0 0
\(237\) −5.38058 −0.349506
\(238\) 0 0
\(239\) 12.8872 0.833601 0.416801 0.908998i \(-0.363151\pi\)
0.416801 + 0.908998i \(0.363151\pi\)
\(240\) 0 0
\(241\) 20.3634 1.31172 0.655862 0.754881i \(-0.272305\pi\)
0.655862 + 0.754881i \(0.272305\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 0 0
\(245\) 0.843675 0.0539004
\(246\) 0 0
\(247\) −9.92478 −0.631498
\(248\) 0 0
\(249\) −5.89446 −0.373546
\(250\) 0 0
\(251\) −18.2374 −1.15114 −0.575568 0.817754i \(-0.695219\pi\)
−0.575568 + 0.817754i \(0.695219\pi\)
\(252\) 0 0
\(253\) −1.98541 −0.124822
\(254\) 0 0
\(255\) 0.418190 0.0261881
\(256\) 0 0
\(257\) −5.16950 −0.322464 −0.161232 0.986916i \(-0.551547\pi\)
−0.161232 + 0.986916i \(0.551547\pi\)
\(258\) 0 0
\(259\) −9.50659 −0.590711
\(260\) 0 0
\(261\) −4.24965 −0.263047
\(262\) 0 0
\(263\) −9.58769 −0.591202 −0.295601 0.955311i \(-0.595520\pi\)
−0.295601 + 0.955311i \(0.595520\pi\)
\(264\) 0 0
\(265\) 8.85685 0.544072
\(266\) 0 0
\(267\) −5.93700 −0.363338
\(268\) 0 0
\(269\) 27.4495 1.67362 0.836812 0.547491i \(-0.184417\pi\)
0.836812 + 0.547491i \(0.184417\pi\)
\(270\) 0 0
\(271\) −10.3634 −0.629534 −0.314767 0.949169i \(-0.601926\pi\)
−0.314767 + 0.949169i \(0.601926\pi\)
\(272\) 0 0
\(273\) −6.15633 −0.372598
\(274\) 0 0
\(275\) −0.387873 −0.0233896
\(276\) 0 0
\(277\) 14.8691 0.893396 0.446698 0.894685i \(-0.352600\pi\)
0.446698 + 0.894685i \(0.352600\pi\)
\(278\) 0 0
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 2.71037 0.161687 0.0808436 0.996727i \(-0.474239\pi\)
0.0808436 + 0.996727i \(0.474239\pi\)
\(282\) 0 0
\(283\) 17.3077 1.02884 0.514419 0.857539i \(-0.328008\pi\)
0.514419 + 0.857539i \(0.328008\pi\)
\(284\) 0 0
\(285\) 4.00000 0.236940
\(286\) 0 0
\(287\) −10.7005 −0.631632
\(288\) 0 0
\(289\) −16.8251 −0.989713
\(290\) 0 0
\(291\) −12.3127 −0.721780
\(292\) 0 0
\(293\) 18.1563 1.06070 0.530352 0.847778i \(-0.322060\pi\)
0.530352 + 0.847778i \(0.322060\pi\)
\(294\) 0 0
\(295\) 3.86177 0.224841
\(296\) 0 0
\(297\) −0.387873 −0.0225067
\(298\) 0 0
\(299\) 12.7005 0.734490
\(300\) 0 0
\(301\) 14.8872 0.858082
\(302\) 0 0
\(303\) 0 0
\(304\) 0 0
\(305\) −12.2374 −0.700713
\(306\) 0 0
\(307\) −14.9199 −0.851521 −0.425760 0.904836i \(-0.639993\pi\)
−0.425760 + 0.904836i \(0.639993\pi\)
\(308\) 0 0
\(309\) 7.05571 0.401385
\(310\) 0 0
\(311\) 5.86177 0.332391 0.166195 0.986093i \(-0.446852\pi\)
0.166195 + 0.986093i \(0.446852\pi\)
\(312\) 0 0
\(313\) −22.5828 −1.27645 −0.638227 0.769849i \(-0.720332\pi\)
−0.638227 + 0.769849i \(0.720332\pi\)
\(314\) 0 0
\(315\) 2.48119 0.139799
\(316\) 0 0
\(317\) 29.7948 1.67344 0.836721 0.547629i \(-0.184469\pi\)
0.836721 + 0.547629i \(0.184469\pi\)
\(318\) 0 0
\(319\) 1.64832 0.0922884
\(320\) 0 0
\(321\) 11.0435 0.616388
\(322\) 0 0
\(323\) 1.67276 0.0930749
\(324\) 0 0
\(325\) 2.48119 0.137632
\(326\) 0 0
\(327\) −20.5950 −1.13891
\(328\) 0 0
\(329\) 26.9380 1.48514
\(330\) 0 0
\(331\) −7.91890 −0.435262 −0.217631 0.976031i \(-0.569833\pi\)
−0.217631 + 0.976031i \(0.569833\pi\)
\(332\) 0 0
\(333\) 3.83146 0.209962
\(334\) 0 0
\(335\) 14.2193 0.776885
\(336\) 0 0
\(337\) 13.7562 0.749350 0.374675 0.927156i \(-0.377754\pi\)
0.374675 + 0.927156i \(0.377754\pi\)
\(338\) 0 0
\(339\) 12.0508 0.654509
\(340\) 0 0
\(341\) 0.387873 0.0210045
\(342\) 0 0
\(343\) 19.4617 1.05083
\(344\) 0 0
\(345\) −5.11871 −0.275582
\(346\) 0 0
\(347\) −35.1998 −1.88963 −0.944813 0.327611i \(-0.893756\pi\)
−0.944813 + 0.327611i \(0.893756\pi\)
\(348\) 0 0
\(349\) −18.2071 −0.974604 −0.487302 0.873233i \(-0.662019\pi\)
−0.487302 + 0.873233i \(0.662019\pi\)
\(350\) 0 0
\(351\) 2.48119 0.132436
\(352\) 0 0
\(353\) 0.478825 0.0254853 0.0127426 0.999919i \(-0.495944\pi\)
0.0127426 + 0.999919i \(0.495944\pi\)
\(354\) 0 0
\(355\) −7.59991 −0.403361
\(356\) 0 0
\(357\) 1.03761 0.0549162
\(358\) 0 0
\(359\) 20.9502 1.10571 0.552854 0.833278i \(-0.313539\pi\)
0.552854 + 0.833278i \(0.313539\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) 0 0
\(363\) −10.8496 −0.569454
\(364\) 0 0
\(365\) 4.60720 0.241152
\(366\) 0 0
\(367\) 20.8773 1.08979 0.544894 0.838505i \(-0.316570\pi\)
0.544894 + 0.838505i \(0.316570\pi\)
\(368\) 0 0
\(369\) 4.31265 0.224508
\(370\) 0 0
\(371\) 21.9756 1.14091
\(372\) 0 0
\(373\) 8.20123 0.424644 0.212322 0.977200i \(-0.431897\pi\)
0.212322 + 0.977200i \(0.431897\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) −10.5442 −0.543054
\(378\) 0 0
\(379\) −34.9234 −1.79389 −0.896946 0.442139i \(-0.854220\pi\)
−0.896946 + 0.442139i \(0.854220\pi\)
\(380\) 0 0
\(381\) 3.66291 0.187657
\(382\) 0 0
\(383\) 27.7586 1.41840 0.709199 0.705008i \(-0.249057\pi\)
0.709199 + 0.705008i \(0.249057\pi\)
\(384\) 0 0
\(385\) −0.962389 −0.0490479
\(386\) 0 0
\(387\) −6.00000 −0.304997
\(388\) 0 0
\(389\) 16.1989 0.821315 0.410657 0.911790i \(-0.365299\pi\)
0.410657 + 0.911790i \(0.365299\pi\)
\(390\) 0 0
\(391\) −2.14060 −0.108255
\(392\) 0 0
\(393\) −6.24965 −0.315253
\(394\) 0 0
\(395\) 5.38058 0.270726
\(396\) 0 0
\(397\) 28.2374 1.41720 0.708598 0.705612i \(-0.249328\pi\)
0.708598 + 0.705612i \(0.249328\pi\)
\(398\) 0 0
\(399\) 9.92478 0.496860
\(400\) 0 0
\(401\) −7.83734 −0.391378 −0.195689 0.980666i \(-0.562694\pi\)
−0.195689 + 0.980666i \(0.562694\pi\)
\(402\) 0 0
\(403\) −2.48119 −0.123597
\(404\) 0 0
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) −1.48612 −0.0736642
\(408\) 0 0
\(409\) 17.0132 0.841247 0.420624 0.907235i \(-0.361811\pi\)
0.420624 + 0.907235i \(0.361811\pi\)
\(410\) 0 0
\(411\) −8.73084 −0.430661
\(412\) 0 0
\(413\) 9.58181 0.471490
\(414\) 0 0
\(415\) 5.89446 0.289348
\(416\) 0 0
\(417\) −11.0376 −0.540514
\(418\) 0 0
\(419\) −18.1382 −0.886110 −0.443055 0.896494i \(-0.646105\pi\)
−0.443055 + 0.896494i \(0.646105\pi\)
\(420\) 0 0
\(421\) −27.1695 −1.32416 −0.662080 0.749433i \(-0.730326\pi\)
−0.662080 + 0.749433i \(0.730326\pi\)
\(422\) 0 0
\(423\) −10.8568 −0.527878
\(424\) 0 0
\(425\) −0.418190 −0.0202852
\(426\) 0 0
\(427\) −30.3634 −1.46939
\(428\) 0 0
\(429\) −0.962389 −0.0464646
\(430\) 0 0
\(431\) 14.6375 0.705065 0.352532 0.935800i \(-0.385321\pi\)
0.352532 + 0.935800i \(0.385321\pi\)
\(432\) 0 0
\(433\) 23.2424 1.11696 0.558478 0.829519i \(-0.311385\pi\)
0.558478 + 0.829519i \(0.311385\pi\)
\(434\) 0 0
\(435\) 4.24965 0.203755
\(436\) 0 0
\(437\) −20.4749 −0.979445
\(438\) 0 0
\(439\) −26.5501 −1.26717 −0.633583 0.773675i \(-0.718417\pi\)
−0.633583 + 0.773675i \(0.718417\pi\)
\(440\) 0 0
\(441\) −0.843675 −0.0401750
\(442\) 0 0
\(443\) 14.6702 0.697003 0.348501 0.937308i \(-0.386691\pi\)
0.348501 + 0.937308i \(0.386691\pi\)
\(444\) 0 0
\(445\) 5.93700 0.281441
\(446\) 0 0
\(447\) −7.53690 −0.356483
\(448\) 0 0
\(449\) −10.6883 −0.504412 −0.252206 0.967674i \(-0.581156\pi\)
−0.252206 + 0.967674i \(0.581156\pi\)
\(450\) 0 0
\(451\) −1.67276 −0.0787672
\(452\) 0 0
\(453\) −18.9683 −0.891207
\(454\) 0 0
\(455\) 6.15633 0.288613
\(456\) 0 0
\(457\) −17.2931 −0.808939 −0.404469 0.914552i \(-0.632544\pi\)
−0.404469 + 0.914552i \(0.632544\pi\)
\(458\) 0 0
\(459\) −0.418190 −0.0195195
\(460\) 0 0
\(461\) −8.52610 −0.397100 −0.198550 0.980091i \(-0.563623\pi\)
−0.198550 + 0.980091i \(0.563623\pi\)
\(462\) 0 0
\(463\) −9.53690 −0.443218 −0.221609 0.975136i \(-0.571131\pi\)
−0.221609 + 0.975136i \(0.571131\pi\)
\(464\) 0 0
\(465\) 1.00000 0.0463739
\(466\) 0 0
\(467\) 4.15633 0.192332 0.0961659 0.995365i \(-0.469342\pi\)
0.0961659 + 0.995365i \(0.469342\pi\)
\(468\) 0 0
\(469\) 35.2809 1.62912
\(470\) 0 0
\(471\) 4.77575 0.220055
\(472\) 0 0
\(473\) 2.32724 0.107007
\(474\) 0 0
\(475\) −4.00000 −0.183533
\(476\) 0 0
\(477\) −8.85685 −0.405527
\(478\) 0 0
\(479\) 12.5867 0.575103 0.287551 0.957765i \(-0.407159\pi\)
0.287551 + 0.957765i \(0.407159\pi\)
\(480\) 0 0
\(481\) 9.50659 0.433463
\(482\) 0 0
\(483\) −12.7005 −0.577894
\(484\) 0 0
\(485\) 12.3127 0.559089
\(486\) 0 0
\(487\) −39.3503 −1.78313 −0.891565 0.452892i \(-0.850392\pi\)
−0.891565 + 0.452892i \(0.850392\pi\)
\(488\) 0 0
\(489\) 1.56959 0.0709794
\(490\) 0 0
\(491\) −35.0494 −1.58176 −0.790878 0.611974i \(-0.790376\pi\)
−0.790878 + 0.611974i \(0.790376\pi\)
\(492\) 0 0
\(493\) 1.77716 0.0800393
\(494\) 0 0
\(495\) 0.387873 0.0174336
\(496\) 0 0
\(497\) −18.8568 −0.845845
\(498\) 0 0
\(499\) −23.2243 −1.03966 −0.519830 0.854270i \(-0.674005\pi\)
−0.519830 + 0.854270i \(0.674005\pi\)
\(500\) 0 0
\(501\) −3.87399 −0.173077
\(502\) 0 0
\(503\) 0.367405 0.0163818 0.00819089 0.999966i \(-0.497393\pi\)
0.00819089 + 0.999966i \(0.497393\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 0 0
\(507\) −6.84367 −0.303938
\(508\) 0 0
\(509\) −0.0484156 −0.00214598 −0.00107299 0.999999i \(-0.500342\pi\)
−0.00107299 + 0.999999i \(0.500342\pi\)
\(510\) 0 0
\(511\) 11.4314 0.505694
\(512\) 0 0
\(513\) −4.00000 −0.176604
\(514\) 0 0
\(515\) −7.05571 −0.310912
\(516\) 0 0
\(517\) 4.21108 0.185203
\(518\) 0 0
\(519\) −8.05079 −0.353390
\(520\) 0 0
\(521\) −4.93795 −0.216336 −0.108168 0.994133i \(-0.534498\pi\)
−0.108168 + 0.994133i \(0.534498\pi\)
\(522\) 0 0
\(523\) 18.5745 0.812207 0.406103 0.913827i \(-0.366887\pi\)
0.406103 + 0.913827i \(0.366887\pi\)
\(524\) 0 0
\(525\) −2.48119 −0.108288
\(526\) 0 0
\(527\) 0.418190 0.0182167
\(528\) 0 0
\(529\) 3.20123 0.139184
\(530\) 0 0
\(531\) −3.86177 −0.167587
\(532\) 0 0
\(533\) 10.7005 0.463491
\(534\) 0 0
\(535\) −11.0435 −0.477452
\(536\) 0 0
\(537\) −10.3879 −0.448270
\(538\) 0 0
\(539\) 0.327239 0.0140952
\(540\) 0 0
\(541\) −14.2922 −0.614469 −0.307234 0.951634i \(-0.599404\pi\)
−0.307234 + 0.951634i \(0.599404\pi\)
\(542\) 0 0
\(543\) 2.31265 0.0992453
\(544\) 0 0
\(545\) 20.5950 0.882192
\(546\) 0 0
\(547\) 11.3439 0.485031 0.242516 0.970148i \(-0.422027\pi\)
0.242516 + 0.970148i \(0.422027\pi\)
\(548\) 0 0
\(549\) 12.2374 0.522281
\(550\) 0 0
\(551\) 16.9986 0.724164
\(552\) 0 0
\(553\) 13.3503 0.567711
\(554\) 0 0
\(555\) −3.83146 −0.162636
\(556\) 0 0
\(557\) 25.5672 1.08332 0.541659 0.840598i \(-0.317796\pi\)
0.541659 + 0.840598i \(0.317796\pi\)
\(558\) 0 0
\(559\) −14.8872 −0.629660
\(560\) 0 0
\(561\) 0.162205 0.00684829
\(562\) 0 0
\(563\) 38.6820 1.63025 0.815125 0.579285i \(-0.196668\pi\)
0.815125 + 0.579285i \(0.196668\pi\)
\(564\) 0 0
\(565\) −12.0508 −0.506980
\(566\) 0 0
\(567\) −2.48119 −0.104200
\(568\) 0 0
\(569\) 22.0122 0.922800 0.461400 0.887192i \(-0.347347\pi\)
0.461400 + 0.887192i \(0.347347\pi\)
\(570\) 0 0
\(571\) 4.18664 0.175206 0.0876028 0.996155i \(-0.472079\pi\)
0.0876028 + 0.996155i \(0.472079\pi\)
\(572\) 0 0
\(573\) 6.18901 0.258550
\(574\) 0 0
\(575\) 5.11871 0.213465
\(576\) 0 0
\(577\) 39.8251 1.65794 0.828971 0.559292i \(-0.188927\pi\)
0.828971 + 0.559292i \(0.188927\pi\)
\(578\) 0 0
\(579\) 1.55149 0.0644778
\(580\) 0 0
\(581\) 14.6253 0.606760
\(582\) 0 0
\(583\) 3.43533 0.142277
\(584\) 0 0
\(585\) −2.48119 −0.102585
\(586\) 0 0
\(587\) −9.85940 −0.406941 −0.203471 0.979081i \(-0.565222\pi\)
−0.203471 + 0.979081i \(0.565222\pi\)
\(588\) 0 0
\(589\) 4.00000 0.164817
\(590\) 0 0
\(591\) 15.1793 0.624395
\(592\) 0 0
\(593\) 12.1114 0.497356 0.248678 0.968586i \(-0.420004\pi\)
0.248678 + 0.968586i \(0.420004\pi\)
\(594\) 0 0
\(595\) −1.03761 −0.0425379
\(596\) 0 0
\(597\) 0.730841 0.0299113
\(598\) 0 0
\(599\) 19.8764 0.812126 0.406063 0.913845i \(-0.366901\pi\)
0.406063 + 0.913845i \(0.366901\pi\)
\(600\) 0 0
\(601\) −28.2130 −1.15083 −0.575416 0.817861i \(-0.695160\pi\)
−0.575416 + 0.817861i \(0.695160\pi\)
\(602\) 0 0
\(603\) −14.2193 −0.579056
\(604\) 0 0
\(605\) 10.8496 0.441097
\(606\) 0 0
\(607\) −23.7464 −0.963836 −0.481918 0.876216i \(-0.660060\pi\)
−0.481918 + 0.876216i \(0.660060\pi\)
\(608\) 0 0
\(609\) 10.5442 0.427272
\(610\) 0 0
\(611\) −26.9380 −1.08979
\(612\) 0 0
\(613\) −18.9560 −0.765628 −0.382814 0.923826i \(-0.625045\pi\)
−0.382814 + 0.923826i \(0.625045\pi\)
\(614\) 0 0
\(615\) −4.31265 −0.173903
\(616\) 0 0
\(617\) 47.7255 1.92135 0.960677 0.277667i \(-0.0895613\pi\)
0.960677 + 0.277667i \(0.0895613\pi\)
\(618\) 0 0
\(619\) −15.9307 −0.640307 −0.320154 0.947366i \(-0.603735\pi\)
−0.320154 + 0.947366i \(0.603735\pi\)
\(620\) 0 0
\(621\) 5.11871 0.205407
\(622\) 0 0
\(623\) 14.7308 0.590179
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.55149 0.0619606
\(628\) 0 0
\(629\) −1.60228 −0.0638870
\(630\) 0 0
\(631\) 9.92478 0.395099 0.197550 0.980293i \(-0.436702\pi\)
0.197550 + 0.980293i \(0.436702\pi\)
\(632\) 0 0
\(633\) 5.27504 0.209664
\(634\) 0 0
\(635\) −3.66291 −0.145358
\(636\) 0 0
\(637\) −2.09332 −0.0829404
\(638\) 0 0
\(639\) 7.59991 0.300648
\(640\) 0 0
\(641\) 20.6639 0.816174 0.408087 0.912943i \(-0.366196\pi\)
0.408087 + 0.912943i \(0.366196\pi\)
\(642\) 0 0
\(643\) 18.5745 0.732507 0.366254 0.930515i \(-0.380640\pi\)
0.366254 + 0.930515i \(0.380640\pi\)
\(644\) 0 0
\(645\) 6.00000 0.236250
\(646\) 0 0
\(647\) 11.7889 0.463470 0.231735 0.972779i \(-0.425560\pi\)
0.231735 + 0.972779i \(0.425560\pi\)
\(648\) 0 0
\(649\) 1.49788 0.0587969
\(650\) 0 0
\(651\) 2.48119 0.0972457
\(652\) 0 0
\(653\) −10.2579 −0.401422 −0.200711 0.979650i \(-0.564325\pi\)
−0.200711 + 0.979650i \(0.564325\pi\)
\(654\) 0 0
\(655\) 6.24965 0.244194
\(656\) 0 0
\(657\) −4.60720 −0.179744
\(658\) 0 0
\(659\) 22.5477 0.878334 0.439167 0.898405i \(-0.355274\pi\)
0.439167 + 0.898405i \(0.355274\pi\)
\(660\) 0 0
\(661\) −26.3780 −1.02599 −0.512993 0.858393i \(-0.671463\pi\)
−0.512993 + 0.858393i \(0.671463\pi\)
\(662\) 0 0
\(663\) −1.03761 −0.0402975
\(664\) 0 0
\(665\) −9.92478 −0.384866
\(666\) 0 0
\(667\) −21.7527 −0.842269
\(668\) 0 0
\(669\) −29.5125 −1.14102
\(670\) 0 0
\(671\) −4.74657 −0.183239
\(672\) 0 0
\(673\) 21.6156 0.833222 0.416611 0.909085i \(-0.363218\pi\)
0.416611 + 0.909085i \(0.363218\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) 0 0
\(677\) 9.31994 0.358195 0.179097 0.983831i \(-0.442682\pi\)
0.179097 + 0.983831i \(0.442682\pi\)
\(678\) 0 0
\(679\) 30.5501 1.17240
\(680\) 0 0
\(681\) −6.67021 −0.255603
\(682\) 0 0
\(683\) −41.5329 −1.58921 −0.794607 0.607124i \(-0.792323\pi\)
−0.794607 + 0.607124i \(0.792323\pi\)
\(684\) 0 0
\(685\) 8.73084 0.333588
\(686\) 0 0
\(687\) 7.27504 0.277560
\(688\) 0 0
\(689\) −21.9756 −0.837202
\(690\) 0 0
\(691\) −8.80018 −0.334775 −0.167387 0.985891i \(-0.553533\pi\)
−0.167387 + 0.985891i \(0.553533\pi\)
\(692\) 0 0
\(693\) 0.962389 0.0365581
\(694\) 0 0
\(695\) 11.0376 0.418680
\(696\) 0 0
\(697\) −1.80351 −0.0683128
\(698\) 0 0
\(699\) 23.0943 0.873506
\(700\) 0 0
\(701\) −24.3488 −0.919643 −0.459822 0.888011i \(-0.652087\pi\)
−0.459822 + 0.888011i \(0.652087\pi\)
\(702\) 0 0
\(703\) −15.3258 −0.578024
\(704\) 0 0
\(705\) 10.8568 0.408893
\(706\) 0 0
\(707\) 0 0
\(708\) 0 0
\(709\) 3.40105 0.127729 0.0638645 0.997959i \(-0.479657\pi\)
0.0638645 + 0.997959i \(0.479657\pi\)
\(710\) 0 0
\(711\) −5.38058 −0.201787
\(712\) 0 0
\(713\) −5.11871 −0.191697
\(714\) 0 0
\(715\) 0.962389 0.0359913
\(716\) 0 0
\(717\) 12.8872 0.481280
\(718\) 0 0
\(719\) −13.6385 −0.508629 −0.254315 0.967122i \(-0.581850\pi\)
−0.254315 + 0.967122i \(0.581850\pi\)
\(720\) 0 0
\(721\) −17.5066 −0.651979
\(722\) 0 0
\(723\) 20.3634 0.757324
\(724\) 0 0
\(725\) −4.24965 −0.157828
\(726\) 0 0
\(727\) −8.41582 −0.312126 −0.156063 0.987747i \(-0.549880\pi\)
−0.156063 + 0.987747i \(0.549880\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 2.50914 0.0928040
\(732\) 0 0
\(733\) 33.8496 1.25026 0.625131 0.780520i \(-0.285046\pi\)
0.625131 + 0.780520i \(0.285046\pi\)
\(734\) 0 0
\(735\) 0.843675 0.0311194
\(736\) 0 0
\(737\) 5.51530 0.203159
\(738\) 0 0
\(739\) −5.86811 −0.215862 −0.107931 0.994158i \(-0.534423\pi\)
−0.107931 + 0.994158i \(0.534423\pi\)
\(740\) 0 0
\(741\) −9.92478 −0.364596
\(742\) 0 0
\(743\) −22.9887 −0.843375 −0.421687 0.906741i \(-0.638562\pi\)
−0.421687 + 0.906741i \(0.638562\pi\)
\(744\) 0 0
\(745\) 7.53690 0.276131
\(746\) 0 0
\(747\) −5.89446 −0.215667
\(748\) 0 0
\(749\) −27.4010 −1.00121
\(750\) 0 0
\(751\) 37.2144 1.35797 0.678986 0.734151i \(-0.262419\pi\)
0.678986 + 0.734151i \(0.262419\pi\)
\(752\) 0 0
\(753\) −18.2374 −0.664609
\(754\) 0 0
\(755\) 18.9683 0.690326
\(756\) 0 0
\(757\) −8.53198 −0.310100 −0.155050 0.987907i \(-0.549554\pi\)
−0.155050 + 0.987907i \(0.549554\pi\)
\(758\) 0 0
\(759\) −1.98541 −0.0720659
\(760\) 0 0
\(761\) −19.2266 −0.696965 −0.348482 0.937315i \(-0.613303\pi\)
−0.348482 + 0.937315i \(0.613303\pi\)
\(762\) 0 0
\(763\) 51.1002 1.84995
\(764\) 0 0
\(765\) 0.418190 0.0151197
\(766\) 0 0
\(767\) −9.58181 −0.345979
\(768\) 0 0
\(769\) 42.2071 1.52203 0.761014 0.648736i \(-0.224702\pi\)
0.761014 + 0.648736i \(0.224702\pi\)
\(770\) 0 0
\(771\) −5.16950 −0.186175
\(772\) 0 0
\(773\) 0.609572 0.0219248 0.0109624 0.999940i \(-0.496510\pi\)
0.0109624 + 0.999940i \(0.496510\pi\)
\(774\) 0 0
\(775\) −1.00000 −0.0359211
\(776\) 0 0
\(777\) −9.50659 −0.341047
\(778\) 0 0
\(779\) −17.2506 −0.618067
\(780\) 0 0
\(781\) −2.94780 −0.105481
\(782\) 0 0
\(783\) −4.24965 −0.151870
\(784\) 0 0
\(785\) −4.77575 −0.170454
\(786\) 0 0
\(787\) 12.3225 0.439250 0.219625 0.975584i \(-0.429517\pi\)
0.219625 + 0.975584i \(0.429517\pi\)
\(788\) 0 0
\(789\) −9.58769 −0.341331
\(790\) 0 0
\(791\) −29.9003 −1.06313
\(792\) 0 0
\(793\) 30.3634 1.07824
\(794\) 0 0
\(795\) 8.85685 0.314120
\(796\) 0 0
\(797\) −23.0289 −0.815726 −0.407863 0.913043i \(-0.633726\pi\)
−0.407863 + 0.913043i \(0.633726\pi\)
\(798\) 0 0
\(799\) 4.54023 0.160622
\(800\) 0 0
\(801\) −5.93700 −0.209773
\(802\) 0 0
\(803\) 1.78701 0.0630622
\(804\) 0 0
\(805\) 12.7005 0.447635
\(806\) 0 0
\(807\) 27.4495 0.966267
\(808\) 0 0
\(809\) 17.4250 0.612631 0.306316 0.951930i \(-0.400904\pi\)
0.306316 + 0.951930i \(0.400904\pi\)
\(810\) 0 0
\(811\) −15.8134 −0.555282 −0.277641 0.960685i \(-0.589553\pi\)
−0.277641 + 0.960685i \(0.589553\pi\)
\(812\) 0 0
\(813\) −10.3634 −0.363462
\(814\) 0 0
\(815\) −1.56959 −0.0549804
\(816\) 0 0
\(817\) 24.0000 0.839654
\(818\) 0 0
\(819\) −6.15633 −0.215119
\(820\) 0 0
\(821\) −16.0729 −0.560946 −0.280473 0.959862i \(-0.590491\pi\)
−0.280473 + 0.959862i \(0.590491\pi\)
\(822\) 0 0
\(823\) 8.11616 0.282912 0.141456 0.989945i \(-0.454822\pi\)
0.141456 + 0.989945i \(0.454822\pi\)
\(824\) 0 0
\(825\) −0.387873 −0.0135040
\(826\) 0 0
\(827\) −36.5658 −1.27152 −0.635759 0.771888i \(-0.719313\pi\)
−0.635759 + 0.771888i \(0.719313\pi\)
\(828\) 0 0
\(829\) −40.7974 −1.41695 −0.708475 0.705736i \(-0.750617\pi\)
−0.708475 + 0.705736i \(0.750617\pi\)
\(830\) 0 0
\(831\) 14.8691 0.515802
\(832\) 0 0
\(833\) 0.352817 0.0122244
\(834\) 0 0
\(835\) 3.87399 0.134065
\(836\) 0 0
\(837\) −1.00000 −0.0345651
\(838\) 0 0
\(839\) 36.5477 1.26177 0.630884 0.775878i \(-0.282693\pi\)
0.630884 + 0.775878i \(0.282693\pi\)
\(840\) 0 0
\(841\) −10.9405 −0.377259
\(842\) 0 0
\(843\) 2.71037 0.0933502
\(844\) 0 0
\(845\) 6.84367 0.235430
\(846\) 0 0
\(847\) 26.9199 0.924977
\(848\) 0 0
\(849\) 17.3077 0.594000
\(850\) 0 0
\(851\) 19.6121 0.672295
\(852\) 0 0
\(853\) 50.3343 1.72341 0.861706 0.507408i \(-0.169396\pi\)
0.861706 + 0.507408i \(0.169396\pi\)
\(854\) 0 0
\(855\) 4.00000 0.136797
\(856\) 0 0
\(857\) 21.0982 0.720702 0.360351 0.932817i \(-0.382657\pi\)
0.360351 + 0.932817i \(0.382657\pi\)
\(858\) 0 0
\(859\) 11.0738 0.377833 0.188917 0.981993i \(-0.439502\pi\)
0.188917 + 0.981993i \(0.439502\pi\)
\(860\) 0 0
\(861\) −10.7005 −0.364673
\(862\) 0 0
\(863\) 33.1002 1.12674 0.563371 0.826204i \(-0.309504\pi\)
0.563371 + 0.826204i \(0.309504\pi\)
\(864\) 0 0
\(865\) 8.05079 0.273735
\(866\) 0 0
\(867\) −16.8251 −0.571411
\(868\) 0 0
\(869\) 2.08698 0.0707960
\(870\) 0 0
\(871\) −35.2809 −1.19545
\(872\) 0 0
\(873\) −12.3127 −0.416720
\(874\) 0 0
\(875\) 2.48119 0.0838797
\(876\) 0 0
\(877\) 17.6775 0.596927 0.298463 0.954421i \(-0.403526\pi\)
0.298463 + 0.954421i \(0.403526\pi\)
\(878\) 0 0
\(879\) 18.1563 0.612398
\(880\) 0 0
\(881\) 42.4772 1.43109 0.715547 0.698565i \(-0.246178\pi\)
0.715547 + 0.698565i \(0.246178\pi\)
\(882\) 0 0
\(883\) −47.7499 −1.60691 −0.803456 0.595364i \(-0.797008\pi\)
−0.803456 + 0.595364i \(0.797008\pi\)
\(884\) 0 0
\(885\) 3.86177 0.129812
\(886\) 0 0
\(887\) −19.9452 −0.669696 −0.334848 0.942272i \(-0.608685\pi\)
−0.334848 + 0.942272i \(0.608685\pi\)
\(888\) 0 0
\(889\) −9.08840 −0.304815
\(890\) 0 0
\(891\) −0.387873 −0.0129942
\(892\) 0 0
\(893\) 43.4274 1.45324
\(894\) 0 0
\(895\) 10.3879 0.347228
\(896\) 0 0
\(897\) 12.7005 0.424058
\(898\) 0 0
\(899\) 4.24965 0.141734
\(900\) 0 0
\(901\) 3.70385 0.123393
\(902\) 0 0
\(903\) 14.8872 0.495414
\(904\) 0 0
\(905\) −2.31265 −0.0768751
\(906\) 0 0
\(907\) 9.79526 0.325246 0.162623 0.986688i \(-0.448005\pi\)
0.162623 + 0.986688i \(0.448005\pi\)
\(908\) 0 0
\(909\) 0 0
\(910\) 0 0
\(911\) −5.38646 −0.178461 −0.0892307 0.996011i \(-0.528441\pi\)
−0.0892307 + 0.996011i \(0.528441\pi\)
\(912\) 0 0
\(913\) 2.28630 0.0756656
\(914\) 0 0
\(915\) −12.2374 −0.404557
\(916\) 0 0
\(917\) 15.5066 0.512073
\(918\) 0 0
\(919\) 31.3865 1.03534 0.517672 0.855579i \(-0.326799\pi\)
0.517672 + 0.855579i \(0.326799\pi\)
\(920\) 0 0
\(921\) −14.9199 −0.491626
\(922\) 0 0
\(923\) 18.8568 0.620681
\(924\) 0 0
\(925\) 3.83146 0.125977
\(926\) 0 0
\(927\) 7.05571 0.231740
\(928\) 0 0
\(929\) 12.1236 0.397764 0.198882 0.980023i \(-0.436269\pi\)
0.198882 + 0.980023i \(0.436269\pi\)
\(930\) 0 0
\(931\) 3.37470 0.110601
\(932\) 0 0
\(933\) 5.86177 0.191906
\(934\) 0 0
\(935\) −0.162205 −0.00530466
\(936\) 0 0
\(937\) −43.4880 −1.42069 −0.710346 0.703853i \(-0.751462\pi\)
−0.710346 + 0.703853i \(0.751462\pi\)
\(938\) 0 0
\(939\) −22.5828 −0.736961
\(940\) 0 0
\(941\) 9.28726 0.302756 0.151378 0.988476i \(-0.451629\pi\)
0.151378 + 0.988476i \(0.451629\pi\)
\(942\) 0 0
\(943\) 22.0752 0.718868
\(944\) 0 0
\(945\) 2.48119 0.0807133
\(946\) 0 0
\(947\) −18.2071 −0.591652 −0.295826 0.955242i \(-0.595595\pi\)
−0.295826 + 0.955242i \(0.595595\pi\)
\(948\) 0 0
\(949\) −11.4314 −0.371078
\(950\) 0 0
\(951\) 29.7948 0.966163
\(952\) 0 0
\(953\) −47.5936 −1.54171 −0.770853 0.637013i \(-0.780170\pi\)
−0.770853 + 0.637013i \(0.780170\pi\)
\(954\) 0 0
\(955\) −6.18901 −0.200272
\(956\) 0 0
\(957\) 1.64832 0.0532827
\(958\) 0 0
\(959\) 21.6629 0.699532
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 0 0
\(963\) 11.0435 0.355872
\(964\) 0 0
\(965\) −1.55149 −0.0499443
\(966\) 0 0
\(967\) 32.0917 1.03200 0.516000 0.856589i \(-0.327420\pi\)
0.516000 + 0.856589i \(0.327420\pi\)
\(968\) 0 0
\(969\) 1.67276 0.0537368
\(970\) 0 0
\(971\) 48.0503 1.54201 0.771004 0.636830i \(-0.219755\pi\)
0.771004 + 0.636830i \(0.219755\pi\)
\(972\) 0 0
\(973\) 27.3865 0.877970
\(974\) 0 0
\(975\) 2.48119 0.0794618
\(976\) 0 0
\(977\) 12.3272 0.394383 0.197192 0.980365i \(-0.436818\pi\)
0.197192 + 0.980365i \(0.436818\pi\)
\(978\) 0 0
\(979\) 2.30280 0.0735978
\(980\) 0 0
\(981\) −20.5950 −0.657547
\(982\) 0 0
\(983\) 27.0132 0.861586 0.430793 0.902451i \(-0.358234\pi\)
0.430793 + 0.902451i \(0.358234\pi\)
\(984\) 0 0
\(985\) −15.1793 −0.483654
\(986\) 0 0
\(987\) 26.9380 0.857444
\(988\) 0 0
\(989\) −30.7123 −0.976594
\(990\) 0 0
\(991\) 26.9887 0.857325 0.428663 0.903465i \(-0.358985\pi\)
0.428663 + 0.903465i \(0.358985\pi\)
\(992\) 0 0
\(993\) −7.91890 −0.251299
\(994\) 0 0
\(995\) −0.730841 −0.0231692
\(996\) 0 0
\(997\) −4.32250 −0.136895 −0.0684475 0.997655i \(-0.521805\pi\)
−0.0684475 + 0.997655i \(0.521805\pi\)
\(998\) 0 0
\(999\) 3.83146 0.121222
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 3720.2.a.o.1.1 3
4.3 odd 2 7440.2.a.bo.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3720.2.a.o.1.1 3 1.1 even 1 trivial
7440.2.a.bo.1.3 3 4.3 odd 2