Properties

Label 6080.2.a.bn.1.1
Level $6080$
Weight $2$
Character 6080.1
Self dual yes
Analytic conductor $48.549$
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] = [6080,2,Mod(1,6080)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6080, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6080.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6080 = 2^{6} \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6080.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: \(48.5490444289\)
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: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 3040)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(2.17009\) of defining polynomial
Character \(\chi\) \(=\) 6080.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-3.17009 q^{3} +1.00000 q^{5} +2.63090 q^{7} +7.04945 q^{9} +O(q^{10})\) \(q-3.17009 q^{3} +1.00000 q^{5} +2.63090 q^{7} +7.04945 q^{9} -0.630898 q^{11} +5.95774 q^{13} -3.17009 q^{15} +3.26180 q^{17} -1.00000 q^{19} -8.34017 q^{21} -6.04945 q^{23} +1.00000 q^{25} -12.8371 q^{27} -8.34017 q^{29} +1.41855 q^{31} +2.00000 q^{33} +2.63090 q^{35} -6.29791 q^{37} -18.8865 q^{39} -11.1278 q^{43} +7.04945 q^{45} +4.78765 q^{47} -0.0783777 q^{49} -10.3402 q^{51} -11.3763 q^{53} -0.630898 q^{55} +3.17009 q^{57} -0.241276 q^{59} -4.63090 q^{61} +18.5464 q^{63} +5.95774 q^{65} -0.489741 q^{67} +19.1773 q^{69} -9.41855 q^{71} -2.18342 q^{73} -3.17009 q^{75} -1.65983 q^{77} -13.9155 q^{79} +19.5464 q^{81} +1.86603 q^{83} +3.26180 q^{85} +26.4391 q^{87} +12.3402 q^{89} +15.6742 q^{91} -4.49693 q^{93} -1.00000 q^{95} -17.7165 q^{97} -4.44748 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 4 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 4 q^{3} + 3 q^{5} + 4 q^{7} + 3 q^{9} + 2 q^{11} + 2 q^{13} - 4 q^{15} + 2 q^{17} - 3 q^{19} - 14 q^{21} + 3 q^{25} - 10 q^{27} - 14 q^{29} - 10 q^{31} + 6 q^{33} + 4 q^{35} + 8 q^{37} - 10 q^{39} - 12 q^{43} + 3 q^{45} + 4 q^{47} + 3 q^{49} - 20 q^{51} - 4 q^{53} + 2 q^{55} + 4 q^{57} - 26 q^{59} - 10 q^{61} + 20 q^{63} + 2 q^{65} - 18 q^{67} + 18 q^{69} - 14 q^{71} - 2 q^{73} - 4 q^{75} - 16 q^{77} - 10 q^{79} + 23 q^{81} - 8 q^{83} + 2 q^{85} + 32 q^{87} + 26 q^{89} - 10 q^{91} + 4 q^{93} - 3 q^{95} - 12 q^{97} - 14 q^{99}+O(q^{100}) \) 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
\(3\) −3.17009 −1.83025 −0.915125 0.403170i \(-0.867908\pi\)
−0.915125 + 0.403170i \(0.867908\pi\)
\(4\) 0 0
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.63090 0.994386 0.497193 0.867640i \(-0.334364\pi\)
0.497193 + 0.867640i \(0.334364\pi\)
\(8\) 0 0
\(9\) 7.04945 2.34982
\(10\) 0 0
\(11\) −0.630898 −0.190223 −0.0951114 0.995467i \(-0.530321\pi\)
−0.0951114 + 0.995467i \(0.530321\pi\)
\(12\) 0 0
\(13\) 5.95774 1.65238 0.826190 0.563392i \(-0.190504\pi\)
0.826190 + 0.563392i \(0.190504\pi\)
\(14\) 0 0
\(15\) −3.17009 −0.818513
\(16\) 0 0
\(17\) 3.26180 0.791102 0.395551 0.918444i \(-0.370554\pi\)
0.395551 + 0.918444i \(0.370554\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416
\(20\) 0 0
\(21\) −8.34017 −1.81997
\(22\) 0 0
\(23\) −6.04945 −1.26140 −0.630699 0.776028i \(-0.717232\pi\)
−0.630699 + 0.776028i \(0.717232\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) −12.8371 −2.47050
\(28\) 0 0
\(29\) −8.34017 −1.54873 −0.774366 0.632738i \(-0.781931\pi\)
−0.774366 + 0.632738i \(0.781931\pi\)
\(30\) 0 0
\(31\) 1.41855 0.254779 0.127390 0.991853i \(-0.459340\pi\)
0.127390 + 0.991853i \(0.459340\pi\)
\(32\) 0 0
\(33\) 2.00000 0.348155
\(34\) 0 0
\(35\) 2.63090 0.444703
\(36\) 0 0
\(37\) −6.29791 −1.03537 −0.517685 0.855571i \(-0.673206\pi\)
−0.517685 + 0.855571i \(0.673206\pi\)
\(38\) 0 0
\(39\) −18.8865 −3.02427
\(40\) 0 0
\(41\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(42\) 0 0
\(43\) −11.1278 −1.69698 −0.848489 0.529213i \(-0.822487\pi\)
−0.848489 + 0.529213i \(0.822487\pi\)
\(44\) 0 0
\(45\) 7.04945 1.05087
\(46\) 0 0
\(47\) 4.78765 0.698351 0.349175 0.937057i \(-0.386462\pi\)
0.349175 + 0.937057i \(0.386462\pi\)
\(48\) 0 0
\(49\) −0.0783777 −0.0111968
\(50\) 0 0
\(51\) −10.3402 −1.44791
\(52\) 0 0
\(53\) −11.3763 −1.56265 −0.781327 0.624122i \(-0.785457\pi\)
−0.781327 + 0.624122i \(0.785457\pi\)
\(54\) 0 0
\(55\) −0.630898 −0.0850702
\(56\) 0 0
\(57\) 3.17009 0.419888
\(58\) 0 0
\(59\) −0.241276 −0.0314115 −0.0157057 0.999877i \(-0.505000\pi\)
−0.0157057 + 0.999877i \(0.505000\pi\)
\(60\) 0 0
\(61\) −4.63090 −0.592926 −0.296463 0.955044i \(-0.595807\pi\)
−0.296463 + 0.955044i \(0.595807\pi\)
\(62\) 0 0
\(63\) 18.5464 2.33662
\(64\) 0 0
\(65\) 5.95774 0.738967
\(66\) 0 0
\(67\) −0.489741 −0.0598313 −0.0299157 0.999552i \(-0.509524\pi\)
−0.0299157 + 0.999552i \(0.509524\pi\)
\(68\) 0 0
\(69\) 19.1773 2.30867
\(70\) 0 0
\(71\) −9.41855 −1.11778 −0.558888 0.829243i \(-0.688772\pi\)
−0.558888 + 0.829243i \(0.688772\pi\)
\(72\) 0 0
\(73\) −2.18342 −0.255550 −0.127775 0.991803i \(-0.540783\pi\)
−0.127775 + 0.991803i \(0.540783\pi\)
\(74\) 0 0
\(75\) −3.17009 −0.366050
\(76\) 0 0
\(77\) −1.65983 −0.189155
\(78\) 0 0
\(79\) −13.9155 −1.56561 −0.782807 0.622265i \(-0.786213\pi\)
−0.782807 + 0.622265i \(0.786213\pi\)
\(80\) 0 0
\(81\) 19.5464 2.17182
\(82\) 0 0
\(83\) 1.86603 0.204823 0.102412 0.994742i \(-0.467344\pi\)
0.102412 + 0.994742i \(0.467344\pi\)
\(84\) 0 0
\(85\) 3.26180 0.353791
\(86\) 0 0
\(87\) 26.4391 2.83457
\(88\) 0 0
\(89\) 12.3402 1.30806 0.654028 0.756470i \(-0.273078\pi\)
0.654028 + 0.756470i \(0.273078\pi\)
\(90\) 0 0
\(91\) 15.6742 1.64310
\(92\) 0 0
\(93\) −4.49693 −0.466310
\(94\) 0 0
\(95\) −1.00000 −0.102598
\(96\) 0 0
\(97\) −17.7165 −1.79883 −0.899417 0.437091i \(-0.856009\pi\)
−0.899417 + 0.437091i \(0.856009\pi\)
\(98\) 0 0
\(99\) −4.44748 −0.446989
\(100\) 0 0
\(101\) 15.2846 1.52087 0.760436 0.649412i \(-0.224985\pi\)
0.760436 + 0.649412i \(0.224985\pi\)
\(102\) 0 0
\(103\) −6.82991 −0.672971 −0.336486 0.941689i \(-0.609238\pi\)
−0.336486 + 0.941689i \(0.609238\pi\)
\(104\) 0 0
\(105\) −8.34017 −0.813918
\(106\) 0 0
\(107\) 8.68753 0.839856 0.419928 0.907558i \(-0.362055\pi\)
0.419928 + 0.907558i \(0.362055\pi\)
\(108\) 0 0
\(109\) −3.60197 −0.345006 −0.172503 0.985009i \(-0.555185\pi\)
−0.172503 + 0.985009i \(0.555185\pi\)
\(110\) 0 0
\(111\) 19.9649 1.89499
\(112\) 0 0
\(113\) −7.61757 −0.716600 −0.358300 0.933606i \(-0.616644\pi\)
−0.358300 + 0.933606i \(0.616644\pi\)
\(114\) 0 0
\(115\) −6.04945 −0.564114
\(116\) 0 0
\(117\) 41.9988 3.88279
\(118\) 0 0
\(119\) 8.58145 0.786660
\(120\) 0 0
\(121\) −10.6020 −0.963815
\(122\) 0 0
\(123\) 0 0
\(124\) 0 0
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −10.4052 −0.923314 −0.461657 0.887059i \(-0.652745\pi\)
−0.461657 + 0.887059i \(0.652745\pi\)
\(128\) 0 0
\(129\) 35.2762 3.10589
\(130\) 0 0
\(131\) 6.34017 0.553943 0.276972 0.960878i \(-0.410669\pi\)
0.276972 + 0.960878i \(0.410669\pi\)
\(132\) 0 0
\(133\) −2.63090 −0.228128
\(134\) 0 0
\(135\) −12.8371 −1.10484
\(136\) 0 0
\(137\) −8.15676 −0.696879 −0.348439 0.937331i \(-0.613288\pi\)
−0.348439 + 0.937331i \(0.613288\pi\)
\(138\) 0 0
\(139\) −10.2062 −0.865679 −0.432839 0.901471i \(-0.642488\pi\)
−0.432839 + 0.901471i \(0.642488\pi\)
\(140\) 0 0
\(141\) −15.1773 −1.27816
\(142\) 0 0
\(143\) −3.75872 −0.314320
\(144\) 0 0
\(145\) −8.34017 −0.692614
\(146\) 0 0
\(147\) 0.248464 0.0204930
\(148\) 0 0
\(149\) 18.5730 1.52156 0.760781 0.649008i \(-0.224816\pi\)
0.760781 + 0.649008i \(0.224816\pi\)
\(150\) 0 0
\(151\) 13.7587 1.11967 0.559835 0.828604i \(-0.310865\pi\)
0.559835 + 0.828604i \(0.310865\pi\)
\(152\) 0 0
\(153\) 22.9939 1.85894
\(154\) 0 0
\(155\) 1.41855 0.113941
\(156\) 0 0
\(157\) 3.84324 0.306724 0.153362 0.988170i \(-0.450990\pi\)
0.153362 + 0.988170i \(0.450990\pi\)
\(158\) 0 0
\(159\) 36.0638 2.86005
\(160\) 0 0
\(161\) −15.9155 −1.25432
\(162\) 0 0
\(163\) −24.3896 −1.91034 −0.955171 0.296054i \(-0.904329\pi\)
−0.955171 + 0.296054i \(0.904329\pi\)
\(164\) 0 0
\(165\) 2.00000 0.155700
\(166\) 0 0
\(167\) 21.2423 1.64378 0.821890 0.569646i \(-0.192920\pi\)
0.821890 + 0.569646i \(0.192920\pi\)
\(168\) 0 0
\(169\) 22.4947 1.73036
\(170\) 0 0
\(171\) −7.04945 −0.539085
\(172\) 0 0
\(173\) −15.2195 −1.15712 −0.578560 0.815640i \(-0.696385\pi\)
−0.578560 + 0.815640i \(0.696385\pi\)
\(174\) 0 0
\(175\) 2.63090 0.198877
\(176\) 0 0
\(177\) 0.764867 0.0574909
\(178\) 0 0
\(179\) 1.91548 0.143170 0.0715848 0.997435i \(-0.477194\pi\)
0.0715848 + 0.997435i \(0.477194\pi\)
\(180\) 0 0
\(181\) 8.15676 0.606287 0.303143 0.952945i \(-0.401964\pi\)
0.303143 + 0.952945i \(0.401964\pi\)
\(182\) 0 0
\(183\) 14.6803 1.08520
\(184\) 0 0
\(185\) −6.29791 −0.463032
\(186\) 0 0
\(187\) −2.05786 −0.150486
\(188\) 0 0
\(189\) −33.7731 −2.45663
\(190\) 0 0
\(191\) 5.57531 0.403415 0.201707 0.979446i \(-0.435351\pi\)
0.201707 + 0.979446i \(0.435351\pi\)
\(192\) 0 0
\(193\) −14.8794 −1.07104 −0.535520 0.844523i \(-0.679884\pi\)
−0.535520 + 0.844523i \(0.679884\pi\)
\(194\) 0 0
\(195\) −18.8865 −1.35249
\(196\) 0 0
\(197\) 18.9627 1.35103 0.675517 0.737345i \(-0.263921\pi\)
0.675517 + 0.737345i \(0.263921\pi\)
\(198\) 0 0
\(199\) −15.2039 −1.07778 −0.538889 0.842377i \(-0.681156\pi\)
−0.538889 + 0.842377i \(0.681156\pi\)
\(200\) 0 0
\(201\) 1.55252 0.109506
\(202\) 0 0
\(203\) −21.9421 −1.54004
\(204\) 0 0
\(205\) 0 0
\(206\) 0 0
\(207\) −42.6453 −2.96405
\(208\) 0 0
\(209\) 0.630898 0.0436401
\(210\) 0 0
\(211\) 0.0845208 0.00581865 0.00290933 0.999996i \(-0.499074\pi\)
0.00290933 + 0.999996i \(0.499074\pi\)
\(212\) 0 0
\(213\) 29.8576 2.04581
\(214\) 0 0
\(215\) −11.1278 −0.758911
\(216\) 0 0
\(217\) 3.73206 0.253349
\(218\) 0 0
\(219\) 6.92162 0.467720
\(220\) 0 0
\(221\) 19.4329 1.30720
\(222\) 0 0
\(223\) 3.66701 0.245561 0.122781 0.992434i \(-0.460819\pi\)
0.122781 + 0.992434i \(0.460819\pi\)
\(224\) 0 0
\(225\) 7.04945 0.469963
\(226\) 0 0
\(227\) −28.7721 −1.90967 −0.954834 0.297139i \(-0.903967\pi\)
−0.954834 + 0.297139i \(0.903967\pi\)
\(228\) 0 0
\(229\) 6.38962 0.422238 0.211119 0.977460i \(-0.432289\pi\)
0.211119 + 0.977460i \(0.432289\pi\)
\(230\) 0 0
\(231\) 5.26180 0.346201
\(232\) 0 0
\(233\) −5.91548 −0.387536 −0.193768 0.981047i \(-0.562071\pi\)
−0.193768 + 0.981047i \(0.562071\pi\)
\(234\) 0 0
\(235\) 4.78765 0.312312
\(236\) 0 0
\(237\) 44.1133 2.86546
\(238\) 0 0
\(239\) 24.9627 1.61470 0.807350 0.590073i \(-0.200901\pi\)
0.807350 + 0.590073i \(0.200901\pi\)
\(240\) 0 0
\(241\) −21.9421 −1.41342 −0.706709 0.707505i \(-0.749821\pi\)
−0.706709 + 0.707505i \(0.749821\pi\)
\(242\) 0 0
\(243\) −23.4524 −1.50447
\(244\) 0 0
\(245\) −0.0783777 −0.00500737
\(246\) 0 0
\(247\) −5.95774 −0.379082
\(248\) 0 0
\(249\) −5.91548 −0.374878
\(250\) 0 0
\(251\) −7.05172 −0.445100 −0.222550 0.974921i \(-0.571438\pi\)
−0.222550 + 0.974921i \(0.571438\pi\)
\(252\) 0 0
\(253\) 3.81658 0.239946
\(254\) 0 0
\(255\) −10.3402 −0.647527
\(256\) 0 0
\(257\) 8.45467 0.527388 0.263694 0.964606i \(-0.415059\pi\)
0.263694 + 0.964606i \(0.415059\pi\)
\(258\) 0 0
\(259\) −16.5692 −1.02956
\(260\) 0 0
\(261\) −58.7936 −3.63923
\(262\) 0 0
\(263\) −28.9854 −1.78732 −0.893660 0.448745i \(-0.851871\pi\)
−0.893660 + 0.448745i \(0.851871\pi\)
\(264\) 0 0
\(265\) −11.3763 −0.698840
\(266\) 0 0
\(267\) −39.1194 −2.39407
\(268\) 0 0
\(269\) −6.34017 −0.386567 −0.193284 0.981143i \(-0.561914\pi\)
−0.193284 + 0.981143i \(0.561914\pi\)
\(270\) 0 0
\(271\) −21.5259 −1.30760 −0.653801 0.756666i \(-0.726827\pi\)
−0.653801 + 0.756666i \(0.726827\pi\)
\(272\) 0 0
\(273\) −49.6886 −3.00729
\(274\) 0 0
\(275\) −0.630898 −0.0380446
\(276\) 0 0
\(277\) 1.68649 0.101331 0.0506657 0.998716i \(-0.483866\pi\)
0.0506657 + 0.998716i \(0.483866\pi\)
\(278\) 0 0
\(279\) 10.0000 0.598684
\(280\) 0 0
\(281\) 17.6742 1.05435 0.527177 0.849755i \(-0.323250\pi\)
0.527177 + 0.849755i \(0.323250\pi\)
\(282\) 0 0
\(283\) 9.33791 0.555081 0.277540 0.960714i \(-0.410481\pi\)
0.277540 + 0.960714i \(0.410481\pi\)
\(284\) 0 0
\(285\) 3.17009 0.187780
\(286\) 0 0
\(287\) 0 0
\(288\) 0 0
\(289\) −6.36069 −0.374158
\(290\) 0 0
\(291\) 56.1627 3.29232
\(292\) 0 0
\(293\) −8.63809 −0.504642 −0.252321 0.967644i \(-0.581194\pi\)
−0.252321 + 0.967644i \(0.581194\pi\)
\(294\) 0 0
\(295\) −0.241276 −0.0140476
\(296\) 0 0
\(297\) 8.09890 0.469946
\(298\) 0 0
\(299\) −36.0410 −2.08431
\(300\) 0 0
\(301\) −29.2762 −1.68745
\(302\) 0 0
\(303\) −48.4534 −2.78358
\(304\) 0 0
\(305\) −4.63090 −0.265164
\(306\) 0 0
\(307\) 33.6814 1.92230 0.961149 0.276029i \(-0.0890186\pi\)
0.961149 + 0.276029i \(0.0890186\pi\)
\(308\) 0 0
\(309\) 21.6514 1.23171
\(310\) 0 0
\(311\) −12.3630 −0.701039 −0.350520 0.936555i \(-0.613995\pi\)
−0.350520 + 0.936555i \(0.613995\pi\)
\(312\) 0 0
\(313\) 4.07223 0.230176 0.115088 0.993355i \(-0.463285\pi\)
0.115088 + 0.993355i \(0.463285\pi\)
\(314\) 0 0
\(315\) 18.5464 1.04497
\(316\) 0 0
\(317\) −3.13501 −0.176080 −0.0880400 0.996117i \(-0.528060\pi\)
−0.0880400 + 0.996117i \(0.528060\pi\)
\(318\) 0 0
\(319\) 5.26180 0.294604
\(320\) 0 0
\(321\) −27.5402 −1.53715
\(322\) 0 0
\(323\) −3.26180 −0.181491
\(324\) 0 0
\(325\) 5.95774 0.330476
\(326\) 0 0
\(327\) 11.4186 0.631447
\(328\) 0 0
\(329\) 12.5958 0.694430
\(330\) 0 0
\(331\) 29.1773 1.60373 0.801864 0.597507i \(-0.203842\pi\)
0.801864 + 0.597507i \(0.203842\pi\)
\(332\) 0 0
\(333\) −44.3968 −2.43293
\(334\) 0 0
\(335\) −0.489741 −0.0267574
\(336\) 0 0
\(337\) −28.4501 −1.54978 −0.774889 0.632098i \(-0.782194\pi\)
−0.774889 + 0.632098i \(0.782194\pi\)
\(338\) 0 0
\(339\) 24.1483 1.31156
\(340\) 0 0
\(341\) −0.894960 −0.0484648
\(342\) 0 0
\(343\) −18.6225 −1.00552
\(344\) 0 0
\(345\) 19.1773 1.03247
\(346\) 0 0
\(347\) 18.2472 0.979563 0.489782 0.871845i \(-0.337077\pi\)
0.489782 + 0.871845i \(0.337077\pi\)
\(348\) 0 0
\(349\) −2.08452 −0.111582 −0.0557909 0.998442i \(-0.517768\pi\)
−0.0557909 + 0.998442i \(0.517768\pi\)
\(350\) 0 0
\(351\) −76.4801 −4.08221
\(352\) 0 0
\(353\) 35.4596 1.88732 0.943662 0.330912i \(-0.107356\pi\)
0.943662 + 0.330912i \(0.107356\pi\)
\(354\) 0 0
\(355\) −9.41855 −0.499885
\(356\) 0 0
\(357\) −27.2039 −1.43978
\(358\) 0 0
\(359\) 18.5197 0.977433 0.488717 0.872443i \(-0.337465\pi\)
0.488717 + 0.872443i \(0.337465\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) 0 0
\(363\) 33.6092 1.76402
\(364\) 0 0
\(365\) −2.18342 −0.114285
\(366\) 0 0
\(367\) −24.8020 −1.29466 −0.647328 0.762212i \(-0.724113\pi\)
−0.647328 + 0.762212i \(0.724113\pi\)
\(368\) 0 0
\(369\) 0 0
\(370\) 0 0
\(371\) −29.9299 −1.55388
\(372\) 0 0
\(373\) −25.1206 −1.30070 −0.650349 0.759636i \(-0.725377\pi\)
−0.650349 + 0.759636i \(0.725377\pi\)
\(374\) 0 0
\(375\) −3.17009 −0.163703
\(376\) 0 0
\(377\) −49.6886 −2.55909
\(378\) 0 0
\(379\) −10.8904 −0.559404 −0.279702 0.960087i \(-0.590236\pi\)
−0.279702 + 0.960087i \(0.590236\pi\)
\(380\) 0 0
\(381\) 32.9854 1.68990
\(382\) 0 0
\(383\) 8.19061 0.418520 0.209260 0.977860i \(-0.432894\pi\)
0.209260 + 0.977860i \(0.432894\pi\)
\(384\) 0 0
\(385\) −1.65983 −0.0845926
\(386\) 0 0
\(387\) −78.4450 −3.98759
\(388\) 0 0
\(389\) 29.6020 1.50088 0.750440 0.660939i \(-0.229842\pi\)
0.750440 + 0.660939i \(0.229842\pi\)
\(390\) 0 0
\(391\) −19.7321 −0.997893
\(392\) 0 0
\(393\) −20.0989 −1.01386
\(394\) 0 0
\(395\) −13.9155 −0.700164
\(396\) 0 0
\(397\) 1.60197 0.0804005 0.0402002 0.999192i \(-0.487200\pi\)
0.0402002 + 0.999192i \(0.487200\pi\)
\(398\) 0 0
\(399\) 8.34017 0.417531
\(400\) 0 0
\(401\) −31.1461 −1.55536 −0.777680 0.628660i \(-0.783604\pi\)
−0.777680 + 0.628660i \(0.783604\pi\)
\(402\) 0 0
\(403\) 8.45136 0.420992
\(404\) 0 0
\(405\) 19.5464 0.971267
\(406\) 0 0
\(407\) 3.97334 0.196951
\(408\) 0 0
\(409\) −0.979481 −0.0484322 −0.0242161 0.999707i \(-0.507709\pi\)
−0.0242161 + 0.999707i \(0.507709\pi\)
\(410\) 0 0
\(411\) 25.8576 1.27546
\(412\) 0 0
\(413\) −0.634773 −0.0312352
\(414\) 0 0
\(415\) 1.86603 0.0915999
\(416\) 0 0
\(417\) 32.3545 1.58441
\(418\) 0 0
\(419\) 19.9155 0.972935 0.486467 0.873699i \(-0.338285\pi\)
0.486467 + 0.873699i \(0.338285\pi\)
\(420\) 0 0
\(421\) 3.05172 0.148732 0.0743658 0.997231i \(-0.476307\pi\)
0.0743658 + 0.997231i \(0.476307\pi\)
\(422\) 0 0
\(423\) 33.7503 1.64100
\(424\) 0 0
\(425\) 3.26180 0.158220
\(426\) 0 0
\(427\) −12.1834 −0.589597
\(428\) 0 0
\(429\) 11.9155 0.575285
\(430\) 0 0
\(431\) 16.3812 0.789055 0.394528 0.918884i \(-0.370908\pi\)
0.394528 + 0.918884i \(0.370908\pi\)
\(432\) 0 0
\(433\) 12.2134 0.586938 0.293469 0.955969i \(-0.405190\pi\)
0.293469 + 0.955969i \(0.405190\pi\)
\(434\) 0 0
\(435\) 26.4391 1.26766
\(436\) 0 0
\(437\) 6.04945 0.289384
\(438\) 0 0
\(439\) 25.6475 1.22409 0.612045 0.790823i \(-0.290347\pi\)
0.612045 + 0.790823i \(0.290347\pi\)
\(440\) 0 0
\(441\) −0.552520 −0.0263105
\(442\) 0 0
\(443\) 8.57304 0.407317 0.203659 0.979042i \(-0.434717\pi\)
0.203659 + 0.979042i \(0.434717\pi\)
\(444\) 0 0
\(445\) 12.3402 0.584980
\(446\) 0 0
\(447\) −58.8781 −2.78484
\(448\) 0 0
\(449\) −0.282314 −0.0133232 −0.00666161 0.999978i \(-0.502120\pi\)
−0.00666161 + 0.999978i \(0.502120\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 0 0
\(453\) −43.6163 −2.04927
\(454\) 0 0
\(455\) 15.6742 0.734818
\(456\) 0 0
\(457\) −26.8781 −1.25731 −0.628653 0.777686i \(-0.716393\pi\)
−0.628653 + 0.777686i \(0.716393\pi\)
\(458\) 0 0
\(459\) −41.8720 −1.95442
\(460\) 0 0
\(461\) −39.8720 −1.85702 −0.928512 0.371302i \(-0.878911\pi\)
−0.928512 + 0.371302i \(0.878911\pi\)
\(462\) 0 0
\(463\) 6.63090 0.308164 0.154082 0.988058i \(-0.450758\pi\)
0.154082 + 0.988058i \(0.450758\pi\)
\(464\) 0 0
\(465\) −4.49693 −0.208540
\(466\) 0 0
\(467\) −14.6309 −0.677037 −0.338519 0.940960i \(-0.609926\pi\)
−0.338519 + 0.940960i \(0.609926\pi\)
\(468\) 0 0
\(469\) −1.28846 −0.0594954
\(470\) 0 0
\(471\) −12.1834 −0.561382
\(472\) 0 0
\(473\) 7.02052 0.322804
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) 0 0
\(477\) −80.1966 −3.67195
\(478\) 0 0
\(479\) 35.2534 1.61077 0.805384 0.592753i \(-0.201959\pi\)
0.805384 + 0.592753i \(0.201959\pi\)
\(480\) 0 0
\(481\) −37.5213 −1.71083
\(482\) 0 0
\(483\) 50.4534 2.29571
\(484\) 0 0
\(485\) −17.7165 −0.804463
\(486\) 0 0
\(487\) 2.79872 0.126822 0.0634110 0.997987i \(-0.479802\pi\)
0.0634110 + 0.997987i \(0.479802\pi\)
\(488\) 0 0
\(489\) 77.3172 3.49641
\(490\) 0 0
\(491\) −15.3028 −0.690607 −0.345304 0.938491i \(-0.612224\pi\)
−0.345304 + 0.938491i \(0.612224\pi\)
\(492\) 0 0
\(493\) −27.2039 −1.22520
\(494\) 0 0
\(495\) −4.44748 −0.199899
\(496\) 0 0
\(497\) −24.7792 −1.11150
\(498\) 0 0
\(499\) 16.6765 0.746541 0.373271 0.927722i \(-0.378236\pi\)
0.373271 + 0.927722i \(0.378236\pi\)
\(500\) 0 0
\(501\) −67.3400 −3.00853
\(502\) 0 0
\(503\) 15.9383 0.710652 0.355326 0.934742i \(-0.384370\pi\)
0.355326 + 0.934742i \(0.384370\pi\)
\(504\) 0 0
\(505\) 15.2846 0.680155
\(506\) 0 0
\(507\) −71.3100 −3.16699
\(508\) 0 0
\(509\) −5.71769 −0.253432 −0.126716 0.991939i \(-0.540444\pi\)
−0.126716 + 0.991939i \(0.540444\pi\)
\(510\) 0 0
\(511\) −5.74435 −0.254115
\(512\) 0 0
\(513\) 12.8371 0.566772
\(514\) 0 0
\(515\) −6.82991 −0.300962
\(516\) 0 0
\(517\) −3.02052 −0.132842
\(518\) 0 0
\(519\) 48.2472 2.11782
\(520\) 0 0
\(521\) −10.8950 −0.477317 −0.238658 0.971104i \(-0.576708\pi\)
−0.238658 + 0.971104i \(0.576708\pi\)
\(522\) 0 0
\(523\) −6.67316 −0.291797 −0.145898 0.989300i \(-0.546607\pi\)
−0.145898 + 0.989300i \(0.546607\pi\)
\(524\) 0 0
\(525\) −8.34017 −0.363995
\(526\) 0 0
\(527\) 4.62702 0.201556
\(528\) 0 0
\(529\) 13.5958 0.591123
\(530\) 0 0
\(531\) −1.70086 −0.0738112
\(532\) 0 0
\(533\) 0 0
\(534\) 0 0
\(535\) 8.68753 0.375595
\(536\) 0 0
\(537\) −6.07223 −0.262036
\(538\) 0 0
\(539\) 0.0494483 0.00212989
\(540\) 0 0
\(541\) 27.7503 1.19308 0.596540 0.802584i \(-0.296542\pi\)
0.596540 + 0.802584i \(0.296542\pi\)
\(542\) 0 0
\(543\) −25.8576 −1.10966
\(544\) 0 0
\(545\) −3.60197 −0.154291
\(546\) 0 0
\(547\) −33.8381 −1.44681 −0.723407 0.690421i \(-0.757425\pi\)
−0.723407 + 0.690421i \(0.757425\pi\)
\(548\) 0 0
\(549\) −32.6453 −1.39327
\(550\) 0 0
\(551\) 8.34017 0.355303
\(552\) 0 0
\(553\) −36.6102 −1.55682
\(554\) 0 0
\(555\) 19.9649 0.847464
\(556\) 0 0
\(557\) −18.5646 −0.786609 −0.393304 0.919408i \(-0.628668\pi\)
−0.393304 + 0.919408i \(0.628668\pi\)
\(558\) 0 0
\(559\) −66.2967 −2.80405
\(560\) 0 0
\(561\) 6.52359 0.275426
\(562\) 0 0
\(563\) 27.8504 1.17376 0.586878 0.809675i \(-0.300357\pi\)
0.586878 + 0.809675i \(0.300357\pi\)
\(564\) 0 0
\(565\) −7.61757 −0.320473
\(566\) 0 0
\(567\) 51.4245 2.15963
\(568\) 0 0
\(569\) −46.8515 −1.96412 −0.982058 0.188579i \(-0.939612\pi\)
−0.982058 + 0.188579i \(0.939612\pi\)
\(570\) 0 0
\(571\) 27.2001 1.13829 0.569144 0.822238i \(-0.307275\pi\)
0.569144 + 0.822238i \(0.307275\pi\)
\(572\) 0 0
\(573\) −17.6742 −0.738350
\(574\) 0 0
\(575\) −6.04945 −0.252279
\(576\) 0 0
\(577\) −42.4801 −1.76847 −0.884235 0.467042i \(-0.845320\pi\)
−0.884235 + 0.467042i \(0.845320\pi\)
\(578\) 0 0
\(579\) 47.1689 1.96027
\(580\) 0 0
\(581\) 4.90934 0.203674
\(582\) 0 0
\(583\) 7.17727 0.297252
\(584\) 0 0
\(585\) 41.9988 1.73644
\(586\) 0 0
\(587\) −9.49920 −0.392074 −0.196037 0.980597i \(-0.562807\pi\)
−0.196037 + 0.980597i \(0.562807\pi\)
\(588\) 0 0
\(589\) −1.41855 −0.0584504
\(590\) 0 0
\(591\) −60.1133 −2.47273
\(592\) 0 0
\(593\) −22.1978 −0.911554 −0.455777 0.890094i \(-0.650639\pi\)
−0.455777 + 0.890094i \(0.650639\pi\)
\(594\) 0 0
\(595\) 8.58145 0.351805
\(596\) 0 0
\(597\) 48.1978 1.97260
\(598\) 0 0
\(599\) 12.7070 0.519194 0.259597 0.965717i \(-0.416410\pi\)
0.259597 + 0.965717i \(0.416410\pi\)
\(600\) 0 0
\(601\) 27.4186 1.11843 0.559213 0.829024i \(-0.311103\pi\)
0.559213 + 0.829024i \(0.311103\pi\)
\(602\) 0 0
\(603\) −3.45240 −0.140593
\(604\) 0 0
\(605\) −10.6020 −0.431031
\(606\) 0 0
\(607\) 15.1578 0.615236 0.307618 0.951510i \(-0.400468\pi\)
0.307618 + 0.951510i \(0.400468\pi\)
\(608\) 0 0
\(609\) 69.5585 2.81865
\(610\) 0 0
\(611\) 28.5236 1.15394
\(612\) 0 0
\(613\) −27.2306 −1.09983 −0.549917 0.835219i \(-0.685341\pi\)
−0.549917 + 0.835219i \(0.685341\pi\)
\(614\) 0 0
\(615\) 0 0
\(616\) 0 0
\(617\) 39.0082 1.57041 0.785206 0.619234i \(-0.212557\pi\)
0.785206 + 0.619234i \(0.212557\pi\)
\(618\) 0 0
\(619\) −17.8927 −0.719168 −0.359584 0.933113i \(-0.617081\pi\)
−0.359584 + 0.933113i \(0.617081\pi\)
\(620\) 0 0
\(621\) 77.6574 3.11628
\(622\) 0 0
\(623\) 32.4657 1.30071
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −2.00000 −0.0798723
\(628\) 0 0
\(629\) −20.5425 −0.819083
\(630\) 0 0
\(631\) 42.1894 1.67953 0.839766 0.542948i \(-0.182692\pi\)
0.839766 + 0.542948i \(0.182692\pi\)
\(632\) 0 0
\(633\) −0.267938 −0.0106496
\(634\) 0 0
\(635\) −10.4052 −0.412919
\(636\) 0 0
\(637\) −0.466954 −0.0185014
\(638\) 0 0
\(639\) −66.3956 −2.62657
\(640\) 0 0
\(641\) −11.3028 −0.446435 −0.223218 0.974769i \(-0.571656\pi\)
−0.223218 + 0.974769i \(0.571656\pi\)
\(642\) 0 0
\(643\) 29.4680 1.16210 0.581052 0.813866i \(-0.302641\pi\)
0.581052 + 0.813866i \(0.302641\pi\)
\(644\) 0 0
\(645\) 35.2762 1.38900
\(646\) 0 0
\(647\) −37.7503 −1.48412 −0.742059 0.670335i \(-0.766151\pi\)
−0.742059 + 0.670335i \(0.766151\pi\)
\(648\) 0 0
\(649\) 0.152221 0.00597518
\(650\) 0 0
\(651\) −11.8310 −0.463692
\(652\) 0 0
\(653\) 9.13624 0.357529 0.178764 0.983892i \(-0.442790\pi\)
0.178764 + 0.983892i \(0.442790\pi\)
\(654\) 0 0
\(655\) 6.34017 0.247731
\(656\) 0 0
\(657\) −15.3919 −0.600495
\(658\) 0 0
\(659\) 24.8059 0.966301 0.483150 0.875537i \(-0.339492\pi\)
0.483150 + 0.875537i \(0.339492\pi\)
\(660\) 0 0
\(661\) 30.0866 1.17023 0.585117 0.810949i \(-0.301049\pi\)
0.585117 + 0.810949i \(0.301049\pi\)
\(662\) 0 0
\(663\) −61.6041 −2.39250
\(664\) 0 0
\(665\) −2.63090 −0.102022
\(666\) 0 0
\(667\) 50.4534 1.95357
\(668\) 0 0
\(669\) −11.6248 −0.449439
\(670\) 0 0
\(671\) 2.92162 0.112788
\(672\) 0 0
\(673\) 23.0894 0.890033 0.445016 0.895522i \(-0.353198\pi\)
0.445016 + 0.895522i \(0.353198\pi\)
\(674\) 0 0
\(675\) −12.8371 −0.494100
\(676\) 0 0
\(677\) −6.96388 −0.267644 −0.133822 0.991005i \(-0.542725\pi\)
−0.133822 + 0.991005i \(0.542725\pi\)
\(678\) 0 0
\(679\) −46.6102 −1.78874
\(680\) 0 0
\(681\) 91.2099 3.49517
\(682\) 0 0
\(683\) −41.5103 −1.58835 −0.794173 0.607692i \(-0.792096\pi\)
−0.794173 + 0.607692i \(0.792096\pi\)
\(684\) 0 0
\(685\) −8.15676 −0.311654
\(686\) 0 0
\(687\) −20.2557 −0.772801
\(688\) 0 0
\(689\) −67.7770 −2.58210
\(690\) 0 0
\(691\) −11.4680 −0.436263 −0.218132 0.975919i \(-0.569996\pi\)
−0.218132 + 0.975919i \(0.569996\pi\)
\(692\) 0 0
\(693\) −11.7009 −0.444479
\(694\) 0 0
\(695\) −10.2062 −0.387143
\(696\) 0 0
\(697\) 0 0
\(698\) 0 0
\(699\) 18.7526 0.709288
\(700\) 0 0
\(701\) 2.16063 0.0816059 0.0408030 0.999167i \(-0.487008\pi\)
0.0408030 + 0.999167i \(0.487008\pi\)
\(702\) 0 0
\(703\) 6.29791 0.237530
\(704\) 0 0
\(705\) −15.1773 −0.571609
\(706\) 0 0
\(707\) 40.2122 1.51233
\(708\) 0 0
\(709\) 24.4924 0.919831 0.459916 0.887963i \(-0.347880\pi\)
0.459916 + 0.887963i \(0.347880\pi\)
\(710\) 0 0
\(711\) −98.0965 −3.67890
\(712\) 0 0
\(713\) −8.58145 −0.321378
\(714\) 0 0
\(715\) −3.75872 −0.140568
\(716\) 0 0
\(717\) −79.1338 −2.95531
\(718\) 0 0
\(719\) 43.4680 1.62108 0.810541 0.585681i \(-0.199173\pi\)
0.810541 + 0.585681i \(0.199173\pi\)
\(720\) 0 0
\(721\) −17.9688 −0.669193
\(722\) 0 0
\(723\) 69.5585 2.58691
\(724\) 0 0
\(725\) −8.34017 −0.309746
\(726\) 0 0
\(727\) 30.7442 1.14024 0.570119 0.821562i \(-0.306897\pi\)
0.570119 + 0.821562i \(0.306897\pi\)
\(728\) 0 0
\(729\) 15.7070 0.581741
\(730\) 0 0
\(731\) −36.2967 −1.34248
\(732\) 0 0
\(733\) −12.6004 −0.465405 −0.232702 0.972548i \(-0.574757\pi\)
−0.232702 + 0.972548i \(0.574757\pi\)
\(734\) 0 0
\(735\) 0.248464 0.00916474
\(736\) 0 0
\(737\) 0.308976 0.0113813
\(738\) 0 0
\(739\) 52.1978 1.92013 0.960063 0.279782i \(-0.0902623\pi\)
0.960063 + 0.279782i \(0.0902623\pi\)
\(740\) 0 0
\(741\) 18.8865 0.693815
\(742\) 0 0
\(743\) −19.0322 −0.698225 −0.349113 0.937081i \(-0.613517\pi\)
−0.349113 + 0.937081i \(0.613517\pi\)
\(744\) 0 0
\(745\) 18.5730 0.680463
\(746\) 0 0
\(747\) 13.1545 0.481298
\(748\) 0 0
\(749\) 22.8560 0.835141
\(750\) 0 0
\(751\) −31.3874 −1.14534 −0.572670 0.819786i \(-0.694093\pi\)
−0.572670 + 0.819786i \(0.694093\pi\)
\(752\) 0 0
\(753\) 22.3545 0.814645
\(754\) 0 0
\(755\) 13.7587 0.500731
\(756\) 0 0
\(757\) 33.0661 1.20181 0.600904 0.799321i \(-0.294807\pi\)
0.600904 + 0.799321i \(0.294807\pi\)
\(758\) 0 0
\(759\) −12.0989 −0.439162
\(760\) 0 0
\(761\) −18.1217 −0.656910 −0.328455 0.944520i \(-0.606528\pi\)
−0.328455 + 0.944520i \(0.606528\pi\)
\(762\) 0 0
\(763\) −9.47641 −0.343069
\(764\) 0 0
\(765\) 22.9939 0.831345
\(766\) 0 0
\(767\) −1.43746 −0.0519037
\(768\) 0 0
\(769\) −12.0806 −0.435639 −0.217820 0.975989i \(-0.569894\pi\)
−0.217820 + 0.975989i \(0.569894\pi\)
\(770\) 0 0
\(771\) −26.8020 −0.965251
\(772\) 0 0
\(773\) 15.5864 0.560603 0.280301 0.959912i \(-0.409566\pi\)
0.280301 + 0.959912i \(0.409566\pi\)
\(774\) 0 0
\(775\) 1.41855 0.0509558
\(776\) 0 0
\(777\) 52.5257 1.88435
\(778\) 0 0
\(779\) 0 0
\(780\) 0 0
\(781\) 5.94214 0.212627
\(782\) 0 0
\(783\) 107.064 3.82614
\(784\) 0 0
\(785\) 3.84324 0.137171
\(786\) 0 0
\(787\) −24.4775 −0.872527 −0.436264 0.899819i \(-0.643698\pi\)
−0.436264 + 0.899819i \(0.643698\pi\)
\(788\) 0 0
\(789\) 91.8864 3.27124
\(790\) 0 0
\(791\) −20.0410 −0.712577
\(792\) 0 0
\(793\) −27.5897 −0.979738
\(794\) 0 0
\(795\) 36.0638 1.27905
\(796\) 0 0
\(797\) −22.2134 −0.786839 −0.393419 0.919359i \(-0.628708\pi\)
−0.393419 + 0.919359i \(0.628708\pi\)
\(798\) 0 0
\(799\) 15.6163 0.552467
\(800\) 0 0
\(801\) 86.9914 3.07369
\(802\) 0 0
\(803\) 1.37751 0.0486114
\(804\) 0 0
\(805\) −15.9155 −0.560947
\(806\) 0 0
\(807\) 20.0989 0.707515
\(808\) 0 0
\(809\) 8.55479 0.300770 0.150385 0.988627i \(-0.451949\pi\)
0.150385 + 0.988627i \(0.451949\pi\)
\(810\) 0 0
\(811\) −13.3028 −0.467126 −0.233563 0.972342i \(-0.575038\pi\)
−0.233563 + 0.972342i \(0.575038\pi\)
\(812\) 0 0
\(813\) 68.2388 2.39324
\(814\) 0 0
\(815\) −24.3896 −0.854331
\(816\) 0 0
\(817\) 11.1278 0.389313
\(818\) 0 0
\(819\) 110.494 3.86099
\(820\) 0 0
\(821\) 19.1317 0.667701 0.333850 0.942626i \(-0.391652\pi\)
0.333850 + 0.942626i \(0.391652\pi\)
\(822\) 0 0
\(823\) 40.3728 1.40731 0.703654 0.710543i \(-0.251551\pi\)
0.703654 + 0.710543i \(0.251551\pi\)
\(824\) 0 0
\(825\) 2.00000 0.0696311
\(826\) 0 0
\(827\) −4.46308 −0.155196 −0.0775982 0.996985i \(-0.524725\pi\)
−0.0775982 + 0.996985i \(0.524725\pi\)
\(828\) 0 0
\(829\) −27.4329 −0.952785 −0.476392 0.879233i \(-0.658056\pi\)
−0.476392 + 0.879233i \(0.658056\pi\)
\(830\) 0 0
\(831\) −5.34632 −0.185462
\(832\) 0 0
\(833\) −0.255652 −0.00885782
\(834\) 0 0
\(835\) 21.2423 0.735121
\(836\) 0 0
\(837\) −18.2101 −0.629432
\(838\) 0 0
\(839\) −25.8987 −0.894121 −0.447060 0.894504i \(-0.647529\pi\)
−0.447060 + 0.894504i \(0.647529\pi\)
\(840\) 0 0
\(841\) 40.5585 1.39857
\(842\) 0 0
\(843\) −56.0288 −1.92973
\(844\) 0 0
\(845\) 22.4947 0.773840
\(846\) 0 0
\(847\) −27.8927 −0.958404
\(848\) 0 0
\(849\) −29.6020 −1.01594
\(850\) 0 0
\(851\) 38.0989 1.30601
\(852\) 0 0
\(853\) 2.58145 0.0883871 0.0441936 0.999023i \(-0.485928\pi\)
0.0441936 + 0.999023i \(0.485928\pi\)
\(854\) 0 0
\(855\) −7.04945 −0.241086
\(856\) 0 0
\(857\) 38.9783 1.33147 0.665736 0.746187i \(-0.268118\pi\)
0.665736 + 0.746187i \(0.268118\pi\)
\(858\) 0 0
\(859\) 1.87444 0.0639551 0.0319776 0.999489i \(-0.489819\pi\)
0.0319776 + 0.999489i \(0.489819\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) 0 0
\(863\) 3.00104 0.102157 0.0510784 0.998695i \(-0.483734\pi\)
0.0510784 + 0.998695i \(0.483734\pi\)
\(864\) 0 0
\(865\) −15.2195 −0.517480
\(866\) 0 0
\(867\) 20.1639 0.684803
\(868\) 0 0
\(869\) 8.77924 0.297815
\(870\) 0 0
\(871\) −2.91775 −0.0988641
\(872\) 0 0
\(873\) −124.891 −4.22693
\(874\) 0 0
\(875\) 2.63090 0.0889406
\(876\) 0 0
\(877\) 4.40910 0.148885 0.0744423 0.997225i \(-0.476282\pi\)
0.0744423 + 0.997225i \(0.476282\pi\)
\(878\) 0 0
\(879\) 27.3835 0.923622
\(880\) 0 0
\(881\) 2.09049 0.0704303 0.0352151 0.999380i \(-0.488788\pi\)
0.0352151 + 0.999380i \(0.488788\pi\)
\(882\) 0 0
\(883\) −17.2534 −0.580623 −0.290311 0.956932i \(-0.593759\pi\)
−0.290311 + 0.956932i \(0.593759\pi\)
\(884\) 0 0
\(885\) 0.764867 0.0257107
\(886\) 0 0
\(887\) 47.9760 1.61088 0.805438 0.592680i \(-0.201930\pi\)
0.805438 + 0.592680i \(0.201930\pi\)
\(888\) 0 0
\(889\) −27.3751 −0.918130
\(890\) 0 0
\(891\) −12.3318 −0.413130
\(892\) 0 0
\(893\) −4.78765 −0.160213
\(894\) 0 0
\(895\) 1.91548 0.0640274
\(896\) 0 0
\(897\) 114.253 3.81480
\(898\) 0 0
\(899\) −11.8310 −0.394585
\(900\) 0 0
\(901\) −37.1071 −1.23622
\(902\) 0 0
\(903\) 92.8080 3.08846
\(904\) 0 0
\(905\) 8.15676 0.271140
\(906\) 0 0
\(907\) −7.99281 −0.265397 −0.132698 0.991156i \(-0.542364\pi\)
−0.132698 + 0.991156i \(0.542364\pi\)
\(908\) 0 0
\(909\) 107.748 3.57377
\(910\) 0 0
\(911\) 6.02666 0.199672 0.0998361 0.995004i \(-0.468168\pi\)
0.0998361 + 0.995004i \(0.468168\pi\)
\(912\) 0 0
\(913\) −1.17727 −0.0389621
\(914\) 0 0
\(915\) 14.6803 0.485317
\(916\) 0 0
\(917\) 16.6803 0.550834
\(918\) 0 0
\(919\) −42.0098 −1.38578 −0.692888 0.721045i \(-0.743662\pi\)
−0.692888 + 0.721045i \(0.743662\pi\)
\(920\) 0 0
\(921\) −106.773 −3.51829
\(922\) 0 0
\(923\) −56.1133 −1.84699
\(924\) 0 0
\(925\) −6.29791 −0.207074
\(926\) 0 0
\(927\) −48.1471 −1.58136
\(928\) 0 0
\(929\) −40.4079 −1.32574 −0.662870 0.748735i \(-0.730662\pi\)
−0.662870 + 0.748735i \(0.730662\pi\)
\(930\) 0 0
\(931\) 0.0783777 0.00256873
\(932\) 0 0
\(933\) 39.1917 1.28308
\(934\) 0 0
\(935\) −2.05786 −0.0672992
\(936\) 0 0
\(937\) −20.8227 −0.680249 −0.340124 0.940380i \(-0.610469\pi\)
−0.340124 + 0.940380i \(0.610469\pi\)
\(938\) 0 0
\(939\) −12.9093 −0.421280
\(940\) 0 0
\(941\) −38.3591 −1.25047 −0.625235 0.780436i \(-0.714997\pi\)
−0.625235 + 0.780436i \(0.714997\pi\)
\(942\) 0 0
\(943\) 0 0
\(944\) 0 0
\(945\) −33.7731 −1.09864
\(946\) 0 0
\(947\) −25.9337 −0.842733 −0.421367 0.906890i \(-0.638449\pi\)
−0.421367 + 0.906890i \(0.638449\pi\)
\(948\) 0 0
\(949\) −13.0082 −0.422265
\(950\) 0 0
\(951\) 9.93827 0.322270
\(952\) 0 0
\(953\) −38.8482 −1.25842 −0.629208 0.777237i \(-0.716621\pi\)
−0.629208 + 0.777237i \(0.716621\pi\)
\(954\) 0 0
\(955\) 5.57531 0.180413
\(956\) 0 0
\(957\) −16.6803 −0.539199
\(958\) 0 0
\(959\) −21.4596 −0.692966
\(960\) 0 0
\(961\) −28.9877 −0.935088
\(962\) 0 0
\(963\) 61.2423 1.97351
\(964\) 0 0
\(965\) −14.8794 −0.478984
\(966\) 0 0
\(967\) −10.0183 −0.322165 −0.161083 0.986941i \(-0.551499\pi\)
−0.161083 + 0.986941i \(0.551499\pi\)
\(968\) 0 0
\(969\) 10.3402 0.332174
\(970\) 0 0
\(971\) 23.8432 0.765166 0.382583 0.923921i \(-0.375035\pi\)
0.382583 + 0.923921i \(0.375035\pi\)
\(972\) 0 0
\(973\) −26.8515 −0.860819
\(974\) 0 0
\(975\) −18.8865 −0.604854
\(976\) 0 0
\(977\) −0.481330 −0.0153991 −0.00769956 0.999970i \(-0.502451\pi\)
−0.00769956 + 0.999970i \(0.502451\pi\)
\(978\) 0 0
\(979\) −7.78539 −0.248822
\(980\) 0 0
\(981\) −25.3919 −0.810701
\(982\) 0 0
\(983\) 1.05681 0.0337071 0.0168536 0.999858i \(-0.494635\pi\)
0.0168536 + 0.999858i \(0.494635\pi\)
\(984\) 0 0
\(985\) 18.9627 0.604201
\(986\) 0 0
\(987\) −39.9299 −1.27098
\(988\) 0 0
\(989\) 67.3172 2.14056
\(990\) 0 0
\(991\) −46.8059 −1.48684 −0.743419 0.668826i \(-0.766797\pi\)
−0.743419 + 0.668826i \(0.766797\pi\)
\(992\) 0 0
\(993\) −92.4945 −2.93522
\(994\) 0 0
\(995\) −15.2039 −0.481997
\(996\) 0 0
\(997\) 9.53427 0.301953 0.150977 0.988537i \(-0.451758\pi\)
0.150977 + 0.988537i \(0.451758\pi\)
\(998\) 0 0
\(999\) 80.8469 2.55788
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 6080.2.a.bn.1.1 3
4.3 odd 2 6080.2.a.cb.1.3 3
8.3 odd 2 3040.2.a.i.1.1 3
8.5 even 2 3040.2.a.o.1.3 yes 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3040.2.a.i.1.1 3 8.3 odd 2
3040.2.a.o.1.3 yes 3 8.5 even 2
6080.2.a.bn.1.1 3 1.1 even 1 trivial
6080.2.a.cb.1.3 3 4.3 odd 2