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Properties

Label 7605.2.a.r.1.1
Level $7605$
Weight $2$
Character 7605.1
Self dual yes
Analytic conductor $60.726$
Analytic rank $1$
Dimension $1$
CM no
Inner twists $1$

Related objects

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [7605,2,Mod(1,7605)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(7605, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("7605.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 7605 = 3^{2} \cdot 5 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 7605.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: \(60.7262307372\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 585)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 7605.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +4.00000 q^{7} -3.00000 q^{8} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{4} +1.00000 q^{5} +4.00000 q^{7} -3.00000 q^{8} +1.00000 q^{10} +4.00000 q^{14} -1.00000 q^{16} -4.00000 q^{17} +4.00000 q^{19} -1.00000 q^{20} -8.00000 q^{23} +1.00000 q^{25} -4.00000 q^{28} -8.00000 q^{29} -4.00000 q^{31} +5.00000 q^{32} -4.00000 q^{34} +4.00000 q^{35} +4.00000 q^{37} +4.00000 q^{38} -3.00000 q^{40} -2.00000 q^{41} -8.00000 q^{43} -8.00000 q^{46} -8.00000 q^{47} +9.00000 q^{49} +1.00000 q^{50} -12.0000 q^{53} -12.0000 q^{56} -8.00000 q^{58} -2.00000 q^{61} -4.00000 q^{62} +7.00000 q^{64} +12.0000 q^{67} +4.00000 q^{68} +4.00000 q^{70} -4.00000 q^{71} +4.00000 q^{73} +4.00000 q^{74} -4.00000 q^{76} -8.00000 q^{79} -1.00000 q^{80} -2.00000 q^{82} -12.0000 q^{83} -4.00000 q^{85} -8.00000 q^{86} +18.0000 q^{89} +8.00000 q^{92} -8.00000 q^{94} +4.00000 q^{95} -4.00000 q^{97} +9.00000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107 0.353553 − 0.935414i \(-0.384973\pi\)
0.353553 + 0.935414i \(0.384973\pi\)
\(3\) 0 0
\(4\) −1.00000 −0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) −3.00000 −1.06066
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(12\) 0 0
\(13\) 0 0
\(14\) 4.00000 1.06904
\(15\) 0 0
\(16\) −1.00000 −0.250000
\(17\) −4.00000 −0.970143 −0.485071 − 0.874475i \(-0.661206\pi\)
−0.485071 + 0.874475i \(0.661206\pi\)
\(18\) 0 0
\(19\) 4.00000 0.917663 0.458831 − 0.888523i \(-0.348268\pi\)
0.458831 + 0.888523i \(0.348268\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 0 0
\(23\) −8.00000 −1.66812 −0.834058 − 0.551677i \(-0.813988\pi\)
−0.834058 + 0.551677i \(0.813988\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 0 0
\(28\) −4.00000 −0.755929
\(29\) −8.00000 −1.48556 −0.742781 − 0.669534i \(-0.766494\pi\)
−0.742781 + 0.669534i \(0.766494\pi\)
\(30\) 0 0
\(31\) −4.00000 −0.718421 −0.359211 − 0.933257i \(-0.616954\pi\)
−0.359211 + 0.933257i \(0.616954\pi\)
\(32\) 5.00000 0.883883
\(33\) 0 0
\(34\) −4.00000 −0.685994
\(35\) 4.00000 0.676123
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) 4.00000 0.648886
\(39\) 0 0
\(40\) −3.00000 −0.474342
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −8.00000 −1.21999 −0.609994 − 0.792406i \(-0.708828\pi\)
−0.609994 + 0.792406i \(0.708828\pi\)
\(44\) 0 0
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) −8.00000 −1.16692 −0.583460 − 0.812142i \(-0.698301\pi\)
−0.583460 + 0.812142i \(0.698301\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 1.00000 0.141421
\(51\) 0 0
\(52\) 0 0
\(53\) −12.0000 −1.64833 −0.824163 − 0.566352i \(-0.808354\pi\)
−0.824163 + 0.566352i \(0.808354\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) −12.0000 −1.60357
\(57\) 0 0
\(58\) −8.00000 −1.05045
\(59\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 − 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −4.00000 −0.508001
\(63\) 0 0
\(64\) 7.00000 0.875000
\(65\) 0 0
\(66\) 0 0
\(67\) 12.0000 1.46603 0.733017 − 0.680211i \(-0.238112\pi\)
0.733017 + 0.680211i \(0.238112\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 4.00000 0.478091
\(71\) −4.00000 −0.474713 −0.237356 − 0.971423i \(-0.576281\pi\)
−0.237356 + 0.971423i \(0.576281\pi\)
\(72\) 0 0
\(73\) 4.00000 0.468165 0.234082 − 0.972217i \(-0.424791\pi\)
0.234082 + 0.972217i \(0.424791\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) −4.00000 −0.458831
\(77\) 0 0
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 − 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −2.00000 −0.220863
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) −4.00000 −0.433861
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) 18.0000 1.90800 0.953998 − 0.299813i \(-0.0969242\pi\)
0.953998 + 0.299813i \(0.0969242\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) −8.00000 −0.825137
\(95\) 4.00000 0.410391
\(96\) 0 0
\(97\) −4.00000 −0.406138 −0.203069 − 0.979164i \(-0.565092\pi\)
−0.203069 + 0.979164i \(0.565092\pi\)
\(98\) 9.00000 0.909137
\(99\) 0 0
\(100\) −1.00000 −0.100000
\(101\) 16.0000 1.59206 0.796030 − 0.605257i \(-0.206930\pi\)
0.796030 + 0.605257i \(0.206930\pi\)
\(102\) 0 0
\(103\) −4.00000 −0.394132 −0.197066 − 0.980390i \(-0.563141\pi\)
−0.197066 + 0.980390i \(0.563141\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −12.0000 −1.16554
\(107\) −8.00000 −0.773389 −0.386695 − 0.922208i \(-0.626383\pi\)
−0.386695 + 0.922208i \(0.626383\pi\)
\(108\) 0 0
\(109\) 8.00000 0.766261 0.383131 − 0.923694i \(-0.374846\pi\)
0.383131 + 0.923694i \(0.374846\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −4.00000 −0.377964
\(113\) −4.00000 −0.376288 −0.188144 − 0.982141i \(-0.560247\pi\)
−0.188144 + 0.982141i \(0.560247\pi\)
\(114\) 0 0
\(115\) −8.00000 −0.746004
\(116\) 8.00000 0.742781
\(117\) 0 0
\(118\) 0 0
\(119\) −16.0000 −1.46672
\(120\) 0 0
\(121\) −11.0000 −1.00000
\(122\) −2.00000 −0.181071
\(123\) 0 0
\(124\) 4.00000 0.359211
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −12.0000 −1.06483 −0.532414 − 0.846484i \(-0.678715\pi\)
−0.532414 + 0.846484i \(0.678715\pi\)
\(128\) −3.00000 −0.265165
\(129\) 0 0
\(130\) 0 0
\(131\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 16.0000 1.38738
\(134\) 12.0000 1.03664
\(135\) 0 0
\(136\) 12.0000 1.02899
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −20.0000 −1.69638 −0.848189 − 0.529694i \(-0.822307\pi\)
−0.848189 + 0.529694i \(0.822307\pi\)
\(140\) −4.00000 −0.338062
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) 0 0
\(144\) 0 0
\(145\) −8.00000 −0.664364
\(146\) 4.00000 0.331042
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) 10.0000 0.819232 0.409616 − 0.912258i \(-0.365663\pi\)
0.409616 + 0.912258i \(0.365663\pi\)
\(150\) 0 0
\(151\) 20.0000 1.62758 0.813788 − 0.581161i \(-0.197401\pi\)
0.813788 + 0.581161i \(0.197401\pi\)
\(152\) −12.0000 −0.973329
\(153\) 0 0
\(154\) 0 0
\(155\) −4.00000 −0.321288
\(156\) 0 0
\(157\) −22.0000 −1.75579 −0.877896 − 0.478852i \(-0.841053\pi\)
−0.877896 + 0.478852i \(0.841053\pi\)
\(158\) −8.00000 −0.636446
\(159\) 0 0
\(160\) 5.00000 0.395285
\(161\) −32.0000 −2.52195
\(162\) 0 0
\(163\) 12.0000 0.939913 0.469956 − 0.882690i \(-0.344270\pi\)
0.469956 + 0.882690i \(0.344270\pi\)
\(164\) 2.00000 0.156174
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 8.00000 0.619059 0.309529 − 0.950890i \(-0.399829\pi\)
0.309529 + 0.950890i \(0.399829\pi\)
\(168\) 0 0
\(169\) 0 0
\(170\) −4.00000 −0.306786
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 4.00000 0.304114 0.152057 − 0.988372i \(-0.451410\pi\)
0.152057 + 0.988372i \(0.451410\pi\)
\(174\) 0 0
\(175\) 4.00000 0.302372
\(176\) 0 0
\(177\) 0 0
\(178\) 18.0000 1.34916
\(179\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(180\) 0 0
\(181\) −6.00000 −0.445976 −0.222988 − 0.974821i \(-0.571581\pi\)
−0.222988 + 0.974821i \(0.571581\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 24.0000 1.76930
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 0 0
\(188\) 8.00000 0.583460
\(189\) 0 0
\(190\) 4.00000 0.290191
\(191\) 8.00000 0.578860 0.289430 − 0.957199i \(-0.406534\pi\)
0.289430 + 0.957199i \(0.406534\pi\)
\(192\) 0 0
\(193\) 4.00000 0.287926 0.143963 − 0.989583i \(-0.454015\pi\)
0.143963 + 0.989583i \(0.454015\pi\)
\(194\) −4.00000 −0.287183
\(195\) 0 0
\(196\) −9.00000 −0.642857
\(197\) −26.0000 −1.85242 −0.926212 − 0.377004i \(-0.876954\pi\)
−0.926212 + 0.377004i \(0.876954\pi\)
\(198\) 0 0
\(199\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(200\) −3.00000 −0.212132
\(201\) 0 0
\(202\) 16.0000 1.12576
\(203\) −32.0000 −2.24596
\(204\) 0 0
\(205\) −2.00000 −0.139686
\(206\) −4.00000 −0.278693
\(207\) 0 0
\(208\) 0 0
\(209\) 0 0
\(210\) 0 0
\(211\) 12.0000 0.826114 0.413057 − 0.910705i \(-0.364461\pi\)
0.413057 + 0.910705i \(0.364461\pi\)
\(212\) 12.0000 0.824163
\(213\) 0 0
\(214\) −8.00000 −0.546869
\(215\) −8.00000 −0.545595
\(216\) 0 0
\(217\) −16.0000 −1.08615
\(218\) 8.00000 0.541828
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 0 0
\(223\) 4.00000 0.267860 0.133930 − 0.990991i \(-0.457240\pi\)
0.133930 + 0.990991i \(0.457240\pi\)
\(224\) 20.0000 1.33631
\(225\) 0 0
\(226\) −4.00000 −0.266076
\(227\) 12.0000 0.796468 0.398234 − 0.917284i \(-0.369623\pi\)
0.398234 + 0.917284i \(0.369623\pi\)
\(228\) 0 0
\(229\) −16.0000 −1.05731 −0.528655 − 0.848837i \(-0.677303\pi\)
−0.528655 + 0.848837i \(0.677303\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 24.0000 1.57568
\(233\) −28.0000 −1.83434 −0.917170 − 0.398495i \(-0.869533\pi\)
−0.917170 + 0.398495i \(0.869533\pi\)
\(234\) 0 0
\(235\) −8.00000 −0.521862
\(236\) 0 0
\(237\) 0 0
\(238\) −16.0000 −1.03713
\(239\) 20.0000 1.29369 0.646846 − 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) −24.0000 −1.54598 −0.772988 − 0.634421i \(-0.781239\pi\)
−0.772988 + 0.634421i \(0.781239\pi\)
\(242\) −11.0000 −0.707107
\(243\) 0 0
\(244\) 2.00000 0.128037
\(245\) 9.00000 0.574989
\(246\) 0 0
\(247\) 0 0
\(248\) 12.0000 0.762001
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 24.0000 1.51487 0.757433 − 0.652913i \(-0.226453\pi\)
0.757433 + 0.652913i \(0.226453\pi\)
\(252\) 0 0
\(253\) 0 0
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) −17.0000 −1.06250
\(257\) 12.0000 0.748539 0.374270 − 0.927320i \(-0.377893\pi\)
0.374270 + 0.927320i \(0.377893\pi\)
\(258\) 0 0
\(259\) 16.0000 0.994192
\(260\) 0 0
\(261\) 0 0
\(262\) 0 0
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) −12.0000 −0.737154
\(266\) 16.0000 0.981023
\(267\) 0 0
\(268\) −12.0000 −0.733017
\(269\) −16.0000 −0.975537 −0.487769 − 0.872973i \(-0.662189\pi\)
−0.487769 + 0.872973i \(0.662189\pi\)
\(270\) 0 0
\(271\) −20.0000 −1.21491 −0.607457 − 0.794353i \(-0.707810\pi\)
−0.607457 + 0.794353i \(0.707810\pi\)
\(272\) 4.00000 0.242536
\(273\) 0 0
\(274\) 6.00000 0.362473
\(275\) 0 0
\(276\) 0 0
\(277\) 14.0000 0.841178 0.420589 − 0.907251i \(-0.361823\pi\)
0.420589 + 0.907251i \(0.361823\pi\)
\(278\) −20.0000 −1.19952
\(279\) 0 0
\(280\) −12.0000 −0.717137
\(281\) −18.0000 −1.07379 −0.536895 − 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 0 0
\(287\) −8.00000 −0.472225
\(288\) 0 0
\(289\) −1.00000 −0.0588235
\(290\) −8.00000 −0.469776
\(291\) 0 0
\(292\) −4.00000 −0.234082
\(293\) −6.00000 −0.350524 −0.175262 − 0.984522i \(-0.556077\pi\)
−0.175262 + 0.984522i \(0.556077\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) −12.0000 −0.697486
\(297\) 0 0
\(298\) 10.0000 0.579284
\(299\) 0 0
\(300\) 0 0
\(301\) −32.0000 −1.84445
\(302\) 20.0000 1.15087
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) −2.00000 −0.114520
\(306\) 0 0
\(307\) −28.0000 −1.59804 −0.799022 − 0.601302i \(-0.794649\pi\)
−0.799022 + 0.601302i \(0.794649\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) −4.00000 −0.227185
\(311\) −8.00000 −0.453638 −0.226819 − 0.973937i \(-0.572833\pi\)
−0.226819 + 0.973937i \(0.572833\pi\)
\(312\) 0 0
\(313\) −10.0000 −0.565233 −0.282617 − 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) −22.0000 −1.24153
\(315\) 0 0
\(316\) 8.00000 0.450035
\(317\) −2.00000 −0.112331 −0.0561656 − 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 0 0
\(320\) 7.00000 0.391312
\(321\) 0 0
\(322\) −32.0000 −1.78329
\(323\) −16.0000 −0.890264
\(324\) 0 0
\(325\) 0 0
\(326\) 12.0000 0.664619
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −32.0000 −1.76422
\(330\) 0 0
\(331\) −4.00000 −0.219860 −0.109930 − 0.993939i \(-0.535063\pi\)
−0.109930 + 0.993939i \(0.535063\pi\)
\(332\) 12.0000 0.658586
\(333\) 0 0
\(334\) 8.00000 0.437741
\(335\) 12.0000 0.655630
\(336\) 0 0
\(337\) −18.0000 −0.980522 −0.490261 − 0.871576i \(-0.663099\pi\)
−0.490261 + 0.871576i \(0.663099\pi\)
\(338\) 0 0
\(339\) 0 0
\(340\) 4.00000 0.216930
\(341\) 0 0
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 24.0000 1.29399
\(345\) 0 0
\(346\) 4.00000 0.215041
\(347\) 32.0000 1.71785 0.858925 − 0.512101i \(-0.171133\pi\)
0.858925 + 0.512101i \(0.171133\pi\)
\(348\) 0 0
\(349\) 16.0000 0.856460 0.428230 − 0.903670i \(-0.359137\pi\)
0.428230 + 0.903670i \(0.359137\pi\)
\(350\) 4.00000 0.213809
\(351\) 0 0
\(352\) 0 0
\(353\) −18.0000 −0.958043 −0.479022 − 0.877803i \(-0.659008\pi\)
−0.479022 + 0.877803i \(0.659008\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) −18.0000 −0.953998
\(357\) 0 0
\(358\) 0 0
\(359\) −28.0000 −1.47778 −0.738892 − 0.673824i \(-0.764651\pi\)
−0.738892 + 0.673824i \(0.764651\pi\)
\(360\) 0 0
\(361\) −3.00000 −0.157895
\(362\) −6.00000 −0.315353
\(363\) 0 0
\(364\) 0 0
\(365\) 4.00000 0.209370
\(366\) 0 0
\(367\) 12.0000 0.626395 0.313197 − 0.949688i \(-0.398600\pi\)
0.313197 + 0.949688i \(0.398600\pi\)
\(368\) 8.00000 0.417029
\(369\) 0 0
\(370\) 4.00000 0.207950
\(371\) −48.0000 −2.49204
\(372\) 0 0
\(373\) −14.0000 −0.724893 −0.362446 − 0.932005i \(-0.618058\pi\)
−0.362446 + 0.932005i \(0.618058\pi\)
\(374\) 0 0
\(375\) 0 0
\(376\) 24.0000 1.23771
\(377\) 0 0
\(378\) 0 0
\(379\) −12.0000 −0.616399 −0.308199 − 0.951322i \(-0.599726\pi\)
−0.308199 + 0.951322i \(0.599726\pi\)
\(380\) −4.00000 −0.205196
\(381\) 0 0
\(382\) 8.00000 0.409316
\(383\) 16.0000 0.817562 0.408781 − 0.912633i \(-0.365954\pi\)
0.408781 + 0.912633i \(0.365954\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 4.00000 0.203595
\(387\) 0 0
\(388\) 4.00000 0.203069
\(389\) 16.0000 0.811232 0.405616 − 0.914044i \(-0.367057\pi\)
0.405616 + 0.914044i \(0.367057\pi\)
\(390\) 0 0
\(391\) 32.0000 1.61831
\(392\) −27.0000 −1.36371
\(393\) 0 0
\(394\) −26.0000 −1.30986
\(395\) −8.00000 −0.402524
\(396\) 0 0
\(397\) 20.0000 1.00377 0.501886 − 0.864934i \(-0.332640\pi\)
0.501886 + 0.864934i \(0.332640\pi\)
\(398\) 0 0
\(399\) 0 0
\(400\) −1.00000 −0.0500000
\(401\) 10.0000 0.499376 0.249688 − 0.968326i \(-0.419672\pi\)
0.249688 + 0.968326i \(0.419672\pi\)
\(402\) 0 0
\(403\) 0 0
\(404\) −16.0000 −0.796030
\(405\) 0 0
\(406\) −32.0000 −1.58813
\(407\) 0 0
\(408\) 0 0
\(409\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(410\) −2.00000 −0.0987730
\(411\) 0 0
\(412\) 4.00000 0.197066
\(413\) 0 0
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) 0 0
\(418\) 0 0
\(419\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(420\) 0 0
\(421\) 16.0000 0.779792 0.389896 − 0.920859i \(-0.372511\pi\)
0.389896 + 0.920859i \(0.372511\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) 36.0000 1.74831
\(425\) −4.00000 −0.194029
\(426\) 0 0
\(427\) −8.00000 −0.387147
\(428\) 8.00000 0.386695
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) −12.0000 −0.578020 −0.289010 − 0.957326i \(-0.593326\pi\)
−0.289010 + 0.957326i \(0.593326\pi\)
\(432\) 0 0
\(433\) −2.00000 −0.0961139 −0.0480569 − 0.998845i \(-0.515303\pi\)
−0.0480569 + 0.998845i \(0.515303\pi\)
\(434\) −16.0000 −0.768025
\(435\) 0 0
\(436\) −8.00000 −0.383131
\(437\) −32.0000 −1.53077
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 − 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 − 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 0 0
\(445\) 18.0000 0.853282
\(446\) 4.00000 0.189405
\(447\) 0 0
\(448\) 28.0000 1.32288
\(449\) −6.00000 −0.283158 −0.141579 − 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 0 0
\(452\) 4.00000 0.188144
\(453\) 0 0
\(454\) 12.0000 0.563188
\(455\) 0 0
\(456\) 0 0
\(457\) 28.0000 1.30978 0.654892 − 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) −16.0000 −0.747631
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −34.0000 −1.58354 −0.791769 − 0.610821i \(-0.790840\pi\)
−0.791769 + 0.610821i \(0.790840\pi\)
\(462\) 0 0
\(463\) 20.0000 0.929479 0.464739 − 0.885448i \(-0.346148\pi\)
0.464739 + 0.885448i \(0.346148\pi\)
\(464\) 8.00000 0.371391
\(465\) 0 0
\(466\) −28.0000 −1.29707
\(467\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(468\) 0 0
\(469\) 48.0000 2.21643
\(470\) −8.00000 −0.369012
\(471\) 0 0
\(472\) 0 0
\(473\) 0 0
\(474\) 0 0
\(475\) 4.00000 0.183533
\(476\) 16.0000 0.733359
\(477\) 0 0
\(478\) 20.0000 0.914779
\(479\) 12.0000 0.548294 0.274147 − 0.961688i \(-0.411605\pi\)
0.274147 + 0.961688i \(0.411605\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −24.0000 −1.09317
\(483\) 0 0
\(484\) 11.0000 0.500000
\(485\) −4.00000 −0.181631
\(486\) 0 0
\(487\) −12.0000 −0.543772 −0.271886 − 0.962329i \(-0.587647\pi\)
−0.271886 + 0.962329i \(0.587647\pi\)
\(488\) 6.00000 0.271607
\(489\) 0 0
\(490\) 9.00000 0.406579
\(491\) −40.0000 −1.80517 −0.902587 − 0.430507i \(-0.858335\pi\)
−0.902587 + 0.430507i \(0.858335\pi\)
\(492\) 0 0
\(493\) 32.0000 1.44121
\(494\) 0 0
\(495\) 0 0
\(496\) 4.00000 0.179605
\(497\) −16.0000 −0.717698
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 24.0000 1.07117
\(503\) 16.0000 0.713405 0.356702 − 0.934218i \(-0.383901\pi\)
0.356702 + 0.934218i \(0.383901\pi\)
\(504\) 0 0
\(505\) 16.0000 0.711991
\(506\) 0 0
\(507\) 0 0
\(508\) 12.0000 0.532414
\(509\) −18.0000 −0.797836 −0.398918 − 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) 16.0000 0.707798
\(512\) −11.0000 −0.486136
\(513\) 0 0
\(514\) 12.0000 0.529297
\(515\) −4.00000 −0.176261
\(516\) 0 0
\(517\) 0 0
\(518\) 16.0000 0.703000
\(519\) 0 0
\(520\) 0 0
\(521\) −16.0000 −0.700973 −0.350486 − 0.936568i \(-0.613984\pi\)
−0.350486 + 0.936568i \(0.613984\pi\)
\(522\) 0 0
\(523\) −16.0000 −0.699631 −0.349816 − 0.936819i \(-0.613756\pi\)
−0.349816 + 0.936819i \(0.613756\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 0 0
\(527\) 16.0000 0.696971
\(528\) 0 0
\(529\) 41.0000 1.78261
\(530\) −12.0000 −0.521247
\(531\) 0 0
\(532\) −16.0000 −0.693688
\(533\) 0 0
\(534\) 0 0
\(535\) −8.00000 −0.345870
\(536\) −36.0000 −1.55496
\(537\) 0 0
\(538\) −16.0000 −0.689809
\(539\) 0 0
\(540\) 0 0
\(541\) −8.00000 −0.343947 −0.171973 − 0.985102i \(-0.555014\pi\)
−0.171973 + 0.985102i \(0.555014\pi\)
\(542\) −20.0000 −0.859074
\(543\) 0 0
\(544\) −20.0000 −0.857493
\(545\) 8.00000 0.342682
\(546\) 0 0
\(547\) −16.0000 −0.684111 −0.342055 − 0.939680i \(-0.611123\pi\)
−0.342055 + 0.939680i \(0.611123\pi\)
\(548\) −6.00000 −0.256307
\(549\) 0 0
\(550\) 0 0
\(551\) −32.0000 −1.36325
\(552\) 0 0
\(553\) −32.0000 −1.36078
\(554\) 14.0000 0.594803
\(555\) 0 0
\(556\) 20.0000 0.848189
\(557\) 18.0000 0.762684 0.381342 − 0.924434i \(-0.375462\pi\)
0.381342 + 0.924434i \(0.375462\pi\)
\(558\) 0 0
\(559\) 0 0
\(560\) −4.00000 −0.169031
\(561\) 0 0
\(562\) −18.0000 −0.759284
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) −4.00000 −0.168281
\(566\) 16.0000 0.672530
\(567\) 0 0
\(568\) 12.0000 0.503509
\(569\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 − 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) 0 0
\(573\) 0 0
\(574\) −8.00000 −0.333914
\(575\) −8.00000 −0.333623
\(576\) 0 0
\(577\) −4.00000 −0.166522 −0.0832611 − 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 0 0
\(580\) 8.00000 0.332182
\(581\) −48.0000 −1.99138
\(582\) 0 0
\(583\) 0 0
\(584\) −12.0000 −0.496564
\(585\) 0 0
\(586\) −6.00000 −0.247858
\(587\) −20.0000 −0.825488 −0.412744 − 0.910847i \(-0.635430\pi\)
−0.412744 + 0.910847i \(0.635430\pi\)
\(588\) 0 0
\(589\) −16.0000 −0.659269
\(590\) 0 0
\(591\) 0 0
\(592\) −4.00000 −0.164399
\(593\) −14.0000 −0.574911 −0.287456 − 0.957794i \(-0.592809\pi\)
−0.287456 + 0.957794i \(0.592809\pi\)
\(594\) 0 0
\(595\) −16.0000 −0.655936
\(596\) −10.0000 −0.409616
\(597\) 0 0
\(598\) 0 0
\(599\) −16.0000 −0.653742 −0.326871 − 0.945069i \(-0.605994\pi\)
−0.326871 + 0.945069i \(0.605994\pi\)
\(600\) 0 0
\(601\) −10.0000 −0.407909 −0.203954 − 0.978980i \(-0.565379\pi\)
−0.203954 + 0.978980i \(0.565379\pi\)
\(602\) −32.0000 −1.30422
\(603\) 0 0
\(604\) −20.0000 −0.813788
\(605\) −11.0000 −0.447214
\(606\) 0 0
\(607\) −20.0000 −0.811775 −0.405887 − 0.913923i \(-0.633038\pi\)
−0.405887 + 0.913923i \(0.633038\pi\)
\(608\) 20.0000 0.811107
\(609\) 0 0
\(610\) −2.00000 −0.0809776
\(611\) 0 0
\(612\) 0 0
\(613\) 44.0000 1.77714 0.888572 − 0.458738i \(-0.151698\pi\)
0.888572 + 0.458738i \(0.151698\pi\)
\(614\) −28.0000 −1.12999
\(615\) 0 0
\(616\) 0 0
\(617\) 38.0000 1.52982 0.764911 − 0.644136i \(-0.222783\pi\)
0.764911 + 0.644136i \(0.222783\pi\)
\(618\) 0 0
\(619\) −28.0000 −1.12542 −0.562708 − 0.826656i \(-0.690240\pi\)
−0.562708 + 0.826656i \(0.690240\pi\)
\(620\) 4.00000 0.160644
\(621\) 0 0
\(622\) −8.00000 −0.320771
\(623\) 72.0000 2.88462
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −10.0000 −0.399680
\(627\) 0 0
\(628\) 22.0000 0.877896
\(629\) −16.0000 −0.637962
\(630\) 0 0
\(631\) 20.0000 0.796187 0.398094 − 0.917345i \(-0.369672\pi\)
0.398094 + 0.917345i \(0.369672\pi\)
\(632\) 24.0000 0.954669
\(633\) 0 0
\(634\) −2.00000 −0.0794301
\(635\) −12.0000 −0.476205
\(636\) 0 0
\(637\) 0 0
\(638\) 0 0
\(639\) 0 0
\(640\) −3.00000 −0.118585
\(641\) −24.0000 −0.947943 −0.473972 − 0.880540i \(-0.657180\pi\)
−0.473972 + 0.880540i \(0.657180\pi\)
\(642\) 0 0
\(643\) −4.00000 −0.157745 −0.0788723 − 0.996885i \(-0.525132\pi\)
−0.0788723 + 0.996885i \(0.525132\pi\)
\(644\) 32.0000 1.26098
\(645\) 0 0
\(646\) −16.0000 −0.629512
\(647\) −32.0000 −1.25805 −0.629025 − 0.777385i \(-0.716546\pi\)
−0.629025 + 0.777385i \(0.716546\pi\)
\(648\) 0 0
\(649\) 0 0
\(650\) 0 0
\(651\) 0 0
\(652\) −12.0000 −0.469956
\(653\) −4.00000 −0.156532 −0.0782660 − 0.996933i \(-0.524938\pi\)
−0.0782660 + 0.996933i \(0.524938\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 2.00000 0.0780869
\(657\) 0 0
\(658\) −32.0000 −1.24749
\(659\) 24.0000 0.934907 0.467454 − 0.884018i \(-0.345171\pi\)
0.467454 + 0.884018i \(0.345171\pi\)
\(660\) 0 0
\(661\) 32.0000 1.24466 0.622328 − 0.782757i \(-0.286187\pi\)
0.622328 + 0.782757i \(0.286187\pi\)
\(662\) −4.00000 −0.155464
\(663\) 0 0
\(664\) 36.0000 1.39707
\(665\) 16.0000 0.620453
\(666\) 0 0
\(667\) 64.0000 2.47809
\(668\) −8.00000 −0.309529
\(669\) 0 0
\(670\) 12.0000 0.463600
\(671\) 0 0
\(672\) 0 0
\(673\) 50.0000 1.92736 0.963679 − 0.267063i \(-0.0860531\pi\)
0.963679 + 0.267063i \(0.0860531\pi\)
\(674\) −18.0000 −0.693334
\(675\) 0 0
\(676\) 0 0
\(677\) 36.0000 1.38359 0.691796 − 0.722093i \(-0.256820\pi\)
0.691796 + 0.722093i \(0.256820\pi\)
\(678\) 0 0
\(679\) −16.0000 −0.614024
\(680\) 12.0000 0.460179
\(681\) 0 0
\(682\) 0 0
\(683\) −20.0000 −0.765279 −0.382639 − 0.923898i \(-0.624985\pi\)
−0.382639 + 0.923898i \(0.624985\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 8.00000 0.305441
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) 0 0
\(690\) 0 0
\(691\) 28.0000 1.06517 0.532585 − 0.846376i \(-0.321221\pi\)
0.532585 + 0.846376i \(0.321221\pi\)
\(692\) −4.00000 −0.152057
\(693\) 0 0
\(694\) 32.0000 1.21470
\(695\) −20.0000 −0.758643
\(696\) 0 0
\(697\) 8.00000 0.303022
\(698\) 16.0000 0.605609
\(699\) 0 0
\(700\) −4.00000 −0.151186
\(701\) −24.0000 −0.906467 −0.453234 − 0.891392i \(-0.649730\pi\)
−0.453234 + 0.891392i \(0.649730\pi\)
\(702\) 0 0
\(703\) 16.0000 0.603451
\(704\) 0 0
\(705\) 0 0
\(706\) −18.0000 −0.677439
\(707\) 64.0000 2.40697
\(708\) 0 0
\(709\) 16.0000 0.600893 0.300446 − 0.953799i \(-0.402864\pi\)
0.300446 + 0.953799i \(0.402864\pi\)
\(710\) −4.00000 −0.150117
\(711\) 0 0
\(712\) −54.0000 −2.02374
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 0 0
\(716\) 0 0
\(717\) 0 0
\(718\) −28.0000 −1.04495
\(719\) 24.0000 0.895049 0.447524 − 0.894272i \(-0.352306\pi\)
0.447524 + 0.894272i \(0.352306\pi\)
\(720\) 0 0
\(721\) −16.0000 −0.595871
\(722\) −3.00000 −0.111648
\(723\) 0 0
\(724\) 6.00000 0.222988
\(725\) −8.00000 −0.297113
\(726\) 0 0
\(727\) −44.0000 −1.63187 −0.815935 − 0.578144i \(-0.803777\pi\)
−0.815935 + 0.578144i \(0.803777\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 4.00000 0.148047
\(731\) 32.0000 1.18356
\(732\) 0 0
\(733\) 36.0000 1.32969 0.664845 − 0.746981i \(-0.268498\pi\)
0.664845 + 0.746981i \(0.268498\pi\)
\(734\) 12.0000 0.442928
\(735\) 0 0
\(736\) −40.0000 −1.47442
\(737\) 0 0
\(738\) 0 0
\(739\) 36.0000 1.32428 0.662141 − 0.749380i \(-0.269648\pi\)
0.662141 + 0.749380i \(0.269648\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) −48.0000 −1.76214
\(743\) −32.0000 −1.17397 −0.586983 − 0.809599i \(-0.699684\pi\)
−0.586983 + 0.809599i \(0.699684\pi\)
\(744\) 0 0
\(745\) 10.0000 0.366372
\(746\) −14.0000 −0.512576
\(747\) 0 0
\(748\) 0 0
\(749\) −32.0000 −1.16925
\(750\) 0 0
\(751\) −16.0000 −0.583848 −0.291924 − 0.956441i \(-0.594295\pi\)
−0.291924 + 0.956441i \(0.594295\pi\)
\(752\) 8.00000 0.291730
\(753\) 0 0
\(754\) 0 0
\(755\) 20.0000 0.727875
\(756\) 0 0
\(757\) 2.00000 0.0726912 0.0363456 − 0.999339i \(-0.488428\pi\)
0.0363456 + 0.999339i \(0.488428\pi\)
\(758\) −12.0000 −0.435860
\(759\) 0 0
\(760\) −12.0000 −0.435286
\(761\) −34.0000 −1.23250 −0.616250 − 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) 0 0
\(763\) 32.0000 1.15848
\(764\) −8.00000 −0.289430
\(765\) 0 0
\(766\) 16.0000 0.578103
\(767\) 0 0
\(768\) 0 0
\(769\) −40.0000 −1.44244 −0.721218 − 0.692708i \(-0.756418\pi\)
−0.721218 + 0.692708i \(0.756418\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) −4.00000 −0.143963
\(773\) −6.00000 −0.215805 −0.107903 − 0.994161i \(-0.534413\pi\)
−0.107903 + 0.994161i \(0.534413\pi\)
\(774\) 0 0
\(775\) −4.00000 −0.143684
\(776\) 12.0000 0.430775
\(777\) 0 0
\(778\) 16.0000 0.573628
\(779\) −8.00000 −0.286630
\(780\) 0 0
\(781\) 0 0
\(782\) 32.0000 1.14432
\(783\) 0 0
\(784\) −9.00000 −0.321429
\(785\) −22.0000 −0.785214
\(786\) 0 0
\(787\) −44.0000 −1.56843 −0.784215 − 0.620489i \(-0.786934\pi\)
−0.784215 + 0.620489i \(0.786934\pi\)
\(788\) 26.0000 0.926212
\(789\) 0 0
\(790\) −8.00000 −0.284627
\(791\) −16.0000 −0.568895
\(792\) 0 0
\(793\) 0 0
\(794\) 20.0000 0.709773
\(795\) 0 0
\(796\) 0 0
\(797\) 28.0000 0.991811 0.495905 − 0.868377i \(-0.334836\pi\)
0.495905 + 0.868377i \(0.334836\pi\)
\(798\) 0 0
\(799\) 32.0000 1.13208
\(800\) 5.00000 0.176777
\(801\) 0 0
\(802\) 10.0000 0.353112
\(803\) 0 0
\(804\) 0 0
\(805\) −32.0000 −1.12785
\(806\) 0 0
\(807\) 0 0
\(808\) −48.0000 −1.68863
\(809\) 24.0000 0.843795 0.421898 − 0.906644i \(-0.361364\pi\)
0.421898 + 0.906644i \(0.361364\pi\)
\(810\) 0 0
\(811\) −28.0000 −0.983213 −0.491606 − 0.870817i \(-0.663590\pi\)
−0.491606 + 0.870817i \(0.663590\pi\)
\(812\) 32.0000 1.12298
\(813\) 0 0
\(814\) 0 0
\(815\) 12.0000 0.420342
\(816\) 0 0
\(817\) −32.0000 −1.11954
\(818\) 0 0
\(819\) 0 0
\(820\) 2.00000 0.0698430
\(821\) 42.0000 1.46581 0.732905 − 0.680331i \(-0.238164\pi\)
0.732905 + 0.680331i \(0.238164\pi\)
\(822\) 0 0
\(823\) 12.0000 0.418294 0.209147 − 0.977884i \(-0.432931\pi\)
0.209147 + 0.977884i \(0.432931\pi\)
\(824\) 12.0000 0.418040
\(825\) 0 0
\(826\) 0 0
\(827\) −44.0000 −1.53003 −0.765015 − 0.644013i \(-0.777268\pi\)
−0.765015 + 0.644013i \(0.777268\pi\)
\(828\) 0 0
\(829\) 14.0000 0.486240 0.243120 − 0.969996i \(-0.421829\pi\)
0.243120 + 0.969996i \(0.421829\pi\)
\(830\) −12.0000 −0.416526
\(831\) 0 0
\(832\) 0 0
\(833\) −36.0000 −1.24733
\(834\) 0 0
\(835\) 8.00000 0.276851
\(836\) 0 0
\(837\) 0 0
\(838\) 0 0
\(839\) 36.0000 1.24286 0.621429 − 0.783470i \(-0.286552\pi\)
0.621429 + 0.783470i \(0.286552\pi\)
\(840\) 0 0
\(841\) 35.0000 1.20690
\(842\) 16.0000 0.551396
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) 0 0
\(846\) 0 0
\(847\) −44.0000 −1.51186
\(848\) 12.0000 0.412082
\(849\) 0 0
\(850\) −4.00000 −0.137199
\(851\) −32.0000 −1.09695
\(852\) 0 0
\(853\) 12.0000 0.410872 0.205436 − 0.978671i \(-0.434139\pi\)
0.205436 + 0.978671i \(0.434139\pi\)
\(854\) −8.00000 −0.273754
\(855\) 0 0
\(856\) 24.0000 0.820303
\(857\) −28.0000 −0.956462 −0.478231 − 0.878234i \(-0.658722\pi\)
−0.478231 + 0.878234i \(0.658722\pi\)
\(858\) 0 0
\(859\) 44.0000 1.50126 0.750630 − 0.660722i \(-0.229750\pi\)
0.750630 + 0.660722i \(0.229750\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −12.0000 −0.408722
\(863\) 48.0000 1.63394 0.816970 − 0.576681i \(-0.195652\pi\)
0.816970 + 0.576681i \(0.195652\pi\)
\(864\) 0 0
\(865\) 4.00000 0.136004
\(866\) −2.00000 −0.0679628
\(867\) 0 0
\(868\) 16.0000 0.543075
\(869\) 0 0
\(870\) 0 0
\(871\) 0 0
\(872\) −24.0000 −0.812743
\(873\) 0 0
\(874\) −32.0000 −1.08242
\(875\) 4.00000 0.135225
\(876\) 0 0
\(877\) 52.0000 1.75592 0.877958 − 0.478738i \(-0.158906\pi\)
0.877958 + 0.478738i \(0.158906\pi\)
\(878\) 32.0000 1.07995
\(879\) 0 0
\(880\) 0 0
\(881\) 24.0000 0.808581 0.404290 − 0.914631i \(-0.367519\pi\)
0.404290 + 0.914631i \(0.367519\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 − 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 24.0000 0.806296
\(887\) 40.0000 1.34307 0.671534 − 0.740973i \(-0.265636\pi\)
0.671534 + 0.740973i \(0.265636\pi\)
\(888\) 0 0
\(889\) −48.0000 −1.60987
\(890\) 18.0000 0.603361
\(891\) 0 0
\(892\) −4.00000 −0.133930
\(893\) −32.0000 −1.07084
\(894\) 0 0
\(895\) 0 0
\(896\) −12.0000 −0.400892
\(897\) 0 0
\(898\) −6.00000 −0.200223
\(899\) 32.0000 1.06726
\(900\) 0 0
\(901\) 48.0000 1.59911
\(902\) 0 0
\(903\) 0 0
\(904\) 12.0000 0.399114
\(905\) −6.00000 −0.199447
\(906\) 0 0
\(907\) −40.0000 −1.32818 −0.664089 − 0.747653i \(-0.731180\pi\)
−0.664089 + 0.747653i \(0.731180\pi\)
\(908\) −12.0000 −0.398234
\(909\) 0 0
\(910\) 0 0
\(911\) −8.00000 −0.265052 −0.132526 − 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(912\) 0 0
\(913\) 0 0
\(914\) 28.0000 0.926158
\(915\) 0 0
\(916\) 16.0000 0.528655
\(917\) 0 0
\(918\) 0 0
\(919\) −8.00000 −0.263896 −0.131948 − 0.991257i \(-0.542123\pi\)
−0.131948 + 0.991257i \(0.542123\pi\)
\(920\) 24.0000 0.791257
\(921\) 0 0
\(922\) −34.0000 −1.11973
\(923\) 0 0
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) 20.0000 0.657241
\(927\) 0 0
\(928\) −40.0000 −1.31306
\(929\) −26.0000 −0.853032 −0.426516 − 0.904480i \(-0.640259\pi\)
−0.426516 + 0.904480i \(0.640259\pi\)
\(930\) 0 0
\(931\) 36.0000 1.17985
\(932\) 28.0000 0.917170
\(933\) 0 0
\(934\) 0 0
\(935\) 0 0
\(936\) 0 0
\(937\) −22.0000 −0.718709 −0.359354 − 0.933201i \(-0.617003\pi\)
−0.359354 + 0.933201i \(0.617003\pi\)
\(938\) 48.0000 1.56726
\(939\) 0 0
\(940\) 8.00000 0.260931
\(941\) 2.00000 0.0651981 0.0325991 − 0.999469i \(-0.489622\pi\)
0.0325991 + 0.999469i \(0.489622\pi\)
\(942\) 0 0
\(943\) 16.0000 0.521032
\(944\) 0 0
\(945\) 0 0
\(946\) 0 0
\(947\) −12.0000 −0.389948 −0.194974 − 0.980808i \(-0.562462\pi\)
−0.194974 + 0.980808i \(0.562462\pi\)
\(948\) 0 0
\(949\) 0 0
\(950\) 4.00000 0.129777
\(951\) 0 0
\(952\) 48.0000 1.55569
\(953\) 12.0000 0.388718 0.194359 − 0.980930i \(-0.437737\pi\)
0.194359 + 0.980930i \(0.437737\pi\)
\(954\) 0 0
\(955\) 8.00000 0.258874
\(956\) −20.0000 −0.646846
\(957\) 0 0
\(958\) 12.0000 0.387702
\(959\) 24.0000 0.775000
\(960\) 0 0
\(961\) −15.0000 −0.483871
\(962\) 0 0
\(963\) 0 0
\(964\) 24.0000 0.772988
\(965\) 4.00000 0.128765
\(966\) 0 0
\(967\) 20.0000 0.643157 0.321578 − 0.946883i \(-0.395787\pi\)
0.321578 + 0.946883i \(0.395787\pi\)
\(968\) 33.0000 1.06066
\(969\) 0 0
\(970\) −4.00000 −0.128432
\(971\) 24.0000 0.770197 0.385098 − 0.922876i \(-0.374168\pi\)
0.385098 + 0.922876i \(0.374168\pi\)
\(972\) 0 0
\(973\) −80.0000 −2.56468
\(974\) −12.0000 −0.384505
\(975\) 0 0
\(976\) 2.00000 0.0640184
\(977\) −14.0000 −0.447900 −0.223950 − 0.974601i \(-0.571895\pi\)
−0.223950 + 0.974601i \(0.571895\pi\)
\(978\) 0 0
\(979\) 0 0
\(980\) −9.00000 −0.287494
\(981\) 0 0
\(982\) −40.0000 −1.27645
\(983\) 40.0000 1.27580 0.637901 − 0.770118i \(-0.279803\pi\)
0.637901 + 0.770118i \(0.279803\pi\)
\(984\) 0 0
\(985\) −26.0000 −0.828429
\(986\) 32.0000 1.01909
\(987\) 0 0
\(988\) 0 0
\(989\) 64.0000 2.03508
\(990\) 0 0
\(991\) −8.00000 −0.254128 −0.127064 − 0.991894i \(-0.540555\pi\)
−0.127064 + 0.991894i \(0.540555\pi\)
\(992\) −20.0000 −0.635001
\(993\) 0 0
\(994\) −16.0000 −0.507489
\(995\) 0 0
\(996\) 0 0
\(997\) 30.0000 0.950110 0.475055 − 0.879956i \(-0.342428\pi\)
0.475055 + 0.879956i \(0.342428\pi\)
\(998\) −20.0000 −0.633089
\(999\) 0 0
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 7605.2.a.r.1.1 1
3.2 odd 2 7605.2.a.e.1.1 1
13.5 odd 4 585.2.b.c.181.1 yes 2
13.8 odd 4 585.2.b.c.181.2 yes 2
13.12 even 2 7605.2.a.b.1.1 1
39.5 even 4 585.2.b.b.181.2 yes 2
39.8 even 4 585.2.b.b.181.1 ✓ 2
39.38 odd 2 7605.2.a.o.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
585.2.b.b.181.1 ✓ 2 39.8 even 4
585.2.b.b.181.2 yes 2 39.5 even 4
585.2.b.c.181.1 yes 2 13.5 odd 4
585.2.b.c.181.2 yes 2 13.8 odd 4
7605.2.a.b.1.1 1 13.12 even 2
7605.2.a.e.1.1 1 3.2 odd 2
7605.2.a.o.1.1 1 39.38 odd 2
7605.2.a.r.1.1 1 1.1 even 1 trivial