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Properties

Label 9405.2.a.b.1.1
Level $9405$
Weight $2$
Character 9405.1
Self dual yes
Analytic conductor $75.099$
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] = [9405,2,Mod(1,9405)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9405, 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("9405.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9405 = 3^{2} \cdot 5 \cdot 11 \cdot 19 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9405.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: \(75.0993031010\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9405.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} +O(q^{10})\) \(q-2.00000 q^{2} +2.00000 q^{4} +1.00000 q^{5} -2.00000 q^{10} -1.00000 q^{11} -1.00000 q^{13} -4.00000 q^{16} -3.00000 q^{17} +1.00000 q^{19} +2.00000 q^{20} +2.00000 q^{22} +4.00000 q^{23} +1.00000 q^{25} +2.00000 q^{26} -6.00000 q^{29} -8.00000 q^{31} +8.00000 q^{32} +6.00000 q^{34} +4.00000 q^{37} -2.00000 q^{38} +8.00000 q^{41} +4.00000 q^{43} -2.00000 q^{44} -8.00000 q^{46} +6.00000 q^{47} -7.00000 q^{49} -2.00000 q^{50} -2.00000 q^{52} -5.00000 q^{53} -1.00000 q^{55} +12.0000 q^{58} +1.00000 q^{59} +2.00000 q^{61} +16.0000 q^{62} -8.00000 q^{64} -1.00000 q^{65} -4.00000 q^{67} -6.00000 q^{68} +11.0000 q^{71} -4.00000 q^{73} -8.00000 q^{74} +2.00000 q^{76} +11.0000 q^{79} -4.00000 q^{80} -16.0000 q^{82} -1.00000 q^{83} -3.00000 q^{85} -8.00000 q^{86} +13.0000 q^{89} +8.00000 q^{92} -12.0000 q^{94} +1.00000 q^{95} -8.00000 q^{97} +14.0000 q^{98} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.00000 −1.41421 −0.707107 − 0.707107i \(-0.750000\pi\)
−0.707107 + 0.707107i \(0.750000\pi\)
\(3\) 0 0
\(4\) 2.00000 1.00000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(8\) 0 0
\(9\) 0 0
\(10\) −2.00000 −0.632456
\(11\) −1.00000 −0.301511
\(12\) 0 0
\(13\) −1.00000 −0.277350 −0.138675 − 0.990338i \(-0.544284\pi\)
−0.138675 + 0.990338i \(0.544284\pi\)
\(14\) 0 0
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) −3.00000 −0.727607 −0.363803 − 0.931476i \(-0.618522\pi\)
−0.363803 + 0.931476i \(0.618522\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416
\(20\) 2.00000 0.447214
\(21\) 0 0
\(22\) 2.00000 0.426401
\(23\) 4.00000 0.834058 0.417029 − 0.908893i \(-0.363071\pi\)
0.417029 + 0.908893i \(0.363071\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 2.00000 0.392232
\(27\) 0 0
\(28\) 0 0
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) −8.00000 −1.43684 −0.718421 − 0.695608i \(-0.755135\pi\)
−0.718421 + 0.695608i \(0.755135\pi\)
\(32\) 8.00000 1.41421
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 0 0
\(36\) 0 0
\(37\) 4.00000 0.657596 0.328798 − 0.944400i \(-0.393356\pi\)
0.328798 + 0.944400i \(0.393356\pi\)
\(38\) −2.00000 −0.324443
\(39\) 0 0
\(40\) 0 0
\(41\) 8.00000 1.24939 0.624695 − 0.780869i \(-0.285223\pi\)
0.624695 + 0.780869i \(0.285223\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −8.00000 −1.17954
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 0 0
\(49\) −7.00000 −1.00000
\(50\) −2.00000 −0.282843
\(51\) 0 0
\(52\) −2.00000 −0.277350
\(53\) −5.00000 −0.686803 −0.343401 − 0.939189i \(-0.611579\pi\)
−0.343401 + 0.939189i \(0.611579\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 0 0
\(58\) 12.0000 1.57568
\(59\) 1.00000 0.130189 0.0650945 − 0.997879i \(-0.479265\pi\)
0.0650945 + 0.997879i \(0.479265\pi\)
\(60\) 0 0
\(61\) 2.00000 0.256074 0.128037 − 0.991769i \(-0.459132\pi\)
0.128037 + 0.991769i \(0.459132\pi\)
\(62\) 16.0000 2.03200
\(63\) 0 0
\(64\) −8.00000 −1.00000
\(65\) −1.00000 −0.124035
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 − 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −6.00000 −0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) 11.0000 1.30546 0.652730 − 0.757591i \(-0.273624\pi\)
0.652730 + 0.757591i \(0.273624\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 − 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) −8.00000 −0.929981
\(75\) 0 0
\(76\) 2.00000 0.229416
\(77\) 0 0
\(78\) 0 0
\(79\) 11.0000 1.23760 0.618798 − 0.785550i \(-0.287620\pi\)
0.618798 + 0.785550i \(0.287620\pi\)
\(80\) −4.00000 −0.447214
\(81\) 0 0
\(82\) −16.0000 −1.76690
\(83\) −1.00000 −0.109764 −0.0548821 − 0.998493i \(-0.517478\pi\)
−0.0548821 + 0.998493i \(0.517478\pi\)
\(84\) 0 0
\(85\) −3.00000 −0.325396
\(86\) −8.00000 −0.862662
\(87\) 0 0
\(88\) 0 0
\(89\) 13.0000 1.37800 0.688999 − 0.724763i \(-0.258051\pi\)
0.688999 + 0.724763i \(0.258051\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 8.00000 0.834058
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −8.00000 −0.812277 −0.406138 − 0.913812i \(-0.633125\pi\)
−0.406138 + 0.913812i \(0.633125\pi\)
\(98\) 14.0000 1.41421
\(99\) 0 0
\(100\) 2.00000 0.200000
\(101\) −10.0000 −0.995037 −0.497519 − 0.867453i \(-0.665755\pi\)
−0.497519 + 0.867453i \(0.665755\pi\)
\(102\) 0 0
\(103\) −16.0000 −1.57653 −0.788263 − 0.615338i \(-0.789020\pi\)
−0.788263 + 0.615338i \(0.789020\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 10.0000 0.971286
\(107\) 18.0000 1.74013 0.870063 − 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) 0 0
\(109\) 6.00000 0.574696 0.287348 − 0.957826i \(-0.407226\pi\)
0.287348 + 0.957826i \(0.407226\pi\)
\(110\) 2.00000 0.190693
\(111\) 0 0
\(112\) 0 0
\(113\) −9.00000 −0.846649 −0.423324 − 0.905978i \(-0.639137\pi\)
−0.423324 + 0.905978i \(0.639137\pi\)
\(114\) 0 0
\(115\) 4.00000 0.373002
\(116\) −12.0000 −1.11417
\(117\) 0 0
\(118\) −2.00000 −0.184115
\(119\) 0 0
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −4.00000 −0.362143
\(123\) 0 0
\(124\) −16.0000 −1.43684
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −11.0000 −0.976092 −0.488046 − 0.872818i \(-0.662290\pi\)
−0.488046 + 0.872818i \(0.662290\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 2.00000 0.175412
\(131\) 3.00000 0.262111 0.131056 − 0.991375i \(-0.458163\pi\)
0.131056 + 0.991375i \(0.458163\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) 8.00000 0.691095
\(135\) 0 0
\(136\) 0 0
\(137\) 6.00000 0.512615 0.256307 − 0.966595i \(-0.417494\pi\)
0.256307 + 0.966595i \(0.417494\pi\)
\(138\) 0 0
\(139\) −12.0000 −1.01783 −0.508913 − 0.860818i \(-0.669953\pi\)
−0.508913 + 0.860818i \(0.669953\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −22.0000 −1.84620
\(143\) 1.00000 0.0836242
\(144\) 0 0
\(145\) −6.00000 −0.498273
\(146\) 8.00000 0.662085
\(147\) 0 0
\(148\) 8.00000 0.657596
\(149\) −7.00000 −0.573462 −0.286731 − 0.958011i \(-0.592569\pi\)
−0.286731 + 0.958011i \(0.592569\pi\)
\(150\) 0 0
\(151\) 8.00000 0.651031 0.325515 − 0.945537i \(-0.394462\pi\)
0.325515 + 0.945537i \(0.394462\pi\)
\(152\) 0 0
\(153\) 0 0
\(154\) 0 0
\(155\) −8.00000 −0.642575
\(156\) 0 0
\(157\) −17.0000 −1.35675 −0.678374 − 0.734717i \(-0.737315\pi\)
−0.678374 + 0.734717i \(0.737315\pi\)
\(158\) −22.0000 −1.75023
\(159\) 0 0
\(160\) 8.00000 0.632456
\(161\) 0 0
\(162\) 0 0
\(163\) 1.00000 0.0783260 0.0391630 − 0.999233i \(-0.487531\pi\)
0.0391630 + 0.999233i \(0.487531\pi\)
\(164\) 16.0000 1.24939
\(165\) 0 0
\(166\) 2.00000 0.155230
\(167\) −8.00000 −0.619059 −0.309529 − 0.950890i \(-0.600171\pi\)
−0.309529 + 0.950890i \(0.600171\pi\)
\(168\) 0 0
\(169\) −12.0000 −0.923077
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) 8.00000 0.609994
\(173\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) −26.0000 −1.94878
\(179\) −5.00000 −0.373718 −0.186859 − 0.982387i \(-0.559831\pi\)
−0.186859 + 0.982387i \(0.559831\pi\)
\(180\) 0 0
\(181\) 2.00000 0.148659 0.0743294 − 0.997234i \(-0.476318\pi\)
0.0743294 + 0.997234i \(0.476318\pi\)
\(182\) 0 0
\(183\) 0 0
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) 3.00000 0.219382
\(188\) 12.0000 0.875190
\(189\) 0 0
\(190\) −2.00000 −0.145095
\(191\) 14.0000 1.01300 0.506502 − 0.862239i \(-0.330938\pi\)
0.506502 + 0.862239i \(0.330938\pi\)
\(192\) 0 0
\(193\) 1.00000 0.0719816 0.0359908 − 0.999352i \(-0.488541\pi\)
0.0359908 + 0.999352i \(0.488541\pi\)
\(194\) 16.0000 1.14873
\(195\) 0 0
\(196\) −14.0000 −1.00000
\(197\) 5.00000 0.356235 0.178118 − 0.984009i \(-0.442999\pi\)
0.178118 + 0.984009i \(0.442999\pi\)
\(198\) 0 0
\(199\) −21.0000 −1.48865 −0.744325 − 0.667817i \(-0.767229\pi\)
−0.744325 + 0.667817i \(0.767229\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 20.0000 1.40720
\(203\) 0 0
\(204\) 0 0
\(205\) 8.00000 0.558744
\(206\) 32.0000 2.22955
\(207\) 0 0
\(208\) 4.00000 0.277350
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 − 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) −10.0000 −0.686803
\(213\) 0 0
\(214\) −36.0000 −2.46091
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 0 0
\(218\) −12.0000 −0.812743
\(219\) 0 0
\(220\) −2.00000 −0.134840
\(221\) 3.00000 0.201802
\(222\) 0 0
\(223\) −22.0000 −1.47323 −0.736614 − 0.676313i \(-0.763577\pi\)
−0.736614 + 0.676313i \(0.763577\pi\)
\(224\) 0 0
\(225\) 0 0
\(226\) 18.0000 1.19734
\(227\) −20.0000 −1.32745 −0.663723 − 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −1.00000 −0.0660819 −0.0330409 − 0.999454i \(-0.510519\pi\)
−0.0330409 + 0.999454i \(0.510519\pi\)
\(230\) −8.00000 −0.527504
\(231\) 0 0
\(232\) 0 0
\(233\) −6.00000 −0.393073 −0.196537 − 0.980497i \(-0.562969\pi\)
−0.196537 + 0.980497i \(0.562969\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) 2.00000 0.130189
\(237\) 0 0
\(238\) 0 0
\(239\) 20.0000 1.29369 0.646846 − 0.762620i \(-0.276088\pi\)
0.646846 + 0.762620i \(0.276088\pi\)
\(240\) 0 0
\(241\) 1.00000 0.0644157 0.0322078 − 0.999481i \(-0.489746\pi\)
0.0322078 + 0.999481i \(0.489746\pi\)
\(242\) −2.00000 −0.128565
\(243\) 0 0
\(244\) 4.00000 0.256074
\(245\) −7.00000 −0.447214
\(246\) 0 0
\(247\) −1.00000 −0.0636285
\(248\) 0 0
\(249\) 0 0
\(250\) −2.00000 −0.126491
\(251\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(252\) 0 0
\(253\) −4.00000 −0.251478
\(254\) 22.0000 1.38040
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −18.0000 −1.12281 −0.561405 − 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 0 0
\(259\) 0 0
\(260\) −2.00000 −0.124035
\(261\) 0 0
\(262\) −6.00000 −0.370681
\(263\) 3.00000 0.184988 0.0924940 − 0.995713i \(-0.470516\pi\)
0.0924940 + 0.995713i \(0.470516\pi\)
\(264\) 0 0
\(265\) −5.00000 −0.307148
\(266\) 0 0
\(267\) 0 0
\(268\) −8.00000 −0.488678
\(269\) −3.00000 −0.182913 −0.0914566 − 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) 12.0000 0.728948 0.364474 − 0.931214i \(-0.381249\pi\)
0.364474 + 0.931214i \(0.381249\pi\)
\(272\) 12.0000 0.727607
\(273\) 0 0
\(274\) −12.0000 −0.724947
\(275\) −1.00000 −0.0603023
\(276\) 0 0
\(277\) 4.00000 0.240337 0.120168 − 0.992754i \(-0.461657\pi\)
0.120168 + 0.992754i \(0.461657\pi\)
\(278\) 24.0000 1.43942
\(279\) 0 0
\(280\) 0 0
\(281\) 12.0000 0.715860 0.357930 − 0.933748i \(-0.383483\pi\)
0.357930 + 0.933748i \(0.383483\pi\)
\(282\) 0 0
\(283\) −26.0000 −1.54554 −0.772770 − 0.634686i \(-0.781129\pi\)
−0.772770 + 0.634686i \(0.781129\pi\)
\(284\) 22.0000 1.30546
\(285\) 0 0
\(286\) −2.00000 −0.118262
\(287\) 0 0
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 12.0000 0.704664
\(291\) 0 0
\(292\) −8.00000 −0.468165
\(293\) −14.0000 −0.817889 −0.408944 − 0.912559i \(-0.634103\pi\)
−0.408944 + 0.912559i \(0.634103\pi\)
\(294\) 0 0
\(295\) 1.00000 0.0582223
\(296\) 0 0
\(297\) 0 0
\(298\) 14.0000 0.810998
\(299\) −4.00000 −0.231326
\(300\) 0 0
\(301\) 0 0
\(302\) −16.0000 −0.920697
\(303\) 0 0
\(304\) −4.00000 −0.229416
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) −7.00000 −0.399511 −0.199756 − 0.979846i \(-0.564015\pi\)
−0.199756 + 0.979846i \(0.564015\pi\)
\(308\) 0 0
\(309\) 0 0
\(310\) 16.0000 0.908739
\(311\) −6.00000 −0.340229 −0.170114 − 0.985424i \(-0.554414\pi\)
−0.170114 + 0.985424i \(0.554414\pi\)
\(312\) 0 0
\(313\) −1.00000 −0.0565233 −0.0282617 − 0.999601i \(-0.508997\pi\)
−0.0282617 + 0.999601i \(0.508997\pi\)
\(314\) 34.0000 1.91873
\(315\) 0 0
\(316\) 22.0000 1.23760
\(317\) 18.0000 1.01098 0.505490 − 0.862832i \(-0.331312\pi\)
0.505490 + 0.862832i \(0.331312\pi\)
\(318\) 0 0
\(319\) 6.00000 0.335936
\(320\) −8.00000 −0.447214
\(321\) 0 0
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) 0 0
\(325\) −1.00000 −0.0554700
\(326\) −2.00000 −0.110770
\(327\) 0 0
\(328\) 0 0
\(329\) 0 0
\(330\) 0 0
\(331\) 10.0000 0.549650 0.274825 − 0.961494i \(-0.411380\pi\)
0.274825 + 0.961494i \(0.411380\pi\)
\(332\) −2.00000 −0.109764
\(333\) 0 0
\(334\) 16.0000 0.875481
\(335\) −4.00000 −0.218543
\(336\) 0 0
\(337\) 6.00000 0.326841 0.163420 − 0.986557i \(-0.447747\pi\)
0.163420 + 0.986557i \(0.447747\pi\)
\(338\) 24.0000 1.30543
\(339\) 0 0
\(340\) −6.00000 −0.325396
\(341\) 8.00000 0.433224
\(342\) 0 0
\(343\) 0 0
\(344\) 0 0
\(345\) 0 0
\(346\) 0 0
\(347\) −1.00000 −0.0536828 −0.0268414 − 0.999640i \(-0.508545\pi\)
−0.0268414 + 0.999640i \(0.508545\pi\)
\(348\) 0 0
\(349\) 8.00000 0.428230 0.214115 − 0.976808i \(-0.431313\pi\)
0.214115 + 0.976808i \(0.431313\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) −8.00000 −0.426401
\(353\) −12.0000 −0.638696 −0.319348 − 0.947638i \(-0.603464\pi\)
−0.319348 + 0.947638i \(0.603464\pi\)
\(354\) 0 0
\(355\) 11.0000 0.583819
\(356\) 26.0000 1.37800
\(357\) 0 0
\(358\) 10.0000 0.528516
\(359\) −11.0000 −0.580558 −0.290279 − 0.956942i \(-0.593748\pi\)
−0.290279 + 0.956942i \(0.593748\pi\)
\(360\) 0 0
\(361\) 1.00000 0.0526316
\(362\) −4.00000 −0.210235
\(363\) 0 0
\(364\) 0 0
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) −11.0000 −0.574195 −0.287098 − 0.957901i \(-0.592690\pi\)
−0.287098 + 0.957901i \(0.592690\pi\)
\(368\) −16.0000 −0.834058
\(369\) 0 0
\(370\) −8.00000 −0.415900
\(371\) 0 0
\(372\) 0 0
\(373\) 6.00000 0.310668 0.155334 − 0.987862i \(-0.450355\pi\)
0.155334 + 0.987862i \(0.450355\pi\)
\(374\) −6.00000 −0.310253
\(375\) 0 0
\(376\) 0 0
\(377\) 6.00000 0.309016
\(378\) 0 0
\(379\) 8.00000 0.410932 0.205466 − 0.978664i \(-0.434129\pi\)
0.205466 + 0.978664i \(0.434129\pi\)
\(380\) 2.00000 0.102598
\(381\) 0 0
\(382\) −28.0000 −1.43260
\(383\) 15.0000 0.766464 0.383232 − 0.923652i \(-0.374811\pi\)
0.383232 + 0.923652i \(0.374811\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) −2.00000 −0.101797
\(387\) 0 0
\(388\) −16.0000 −0.812277
\(389\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) −10.0000 −0.503793
\(395\) 11.0000 0.553470
\(396\) 0 0
\(397\) 3.00000 0.150566 0.0752828 − 0.997162i \(-0.476014\pi\)
0.0752828 + 0.997162i \(0.476014\pi\)
\(398\) 42.0000 2.10527
\(399\) 0 0
\(400\) −4.00000 −0.200000
\(401\) 14.0000 0.699127 0.349563 − 0.936913i \(-0.386330\pi\)
0.349563 + 0.936913i \(0.386330\pi\)
\(402\) 0 0
\(403\) 8.00000 0.398508
\(404\) −20.0000 −0.995037
\(405\) 0 0
\(406\) 0 0
\(407\) −4.00000 −0.198273
\(408\) 0 0
\(409\) 21.0000 1.03838 0.519192 − 0.854658i \(-0.326233\pi\)
0.519192 + 0.854658i \(0.326233\pi\)
\(410\) −16.0000 −0.790184
\(411\) 0 0
\(412\) −32.0000 −1.57653
\(413\) 0 0
\(414\) 0 0
\(415\) −1.00000 −0.0490881
\(416\) −8.00000 −0.392232
\(417\) 0 0
\(418\) 2.00000 0.0978232
\(419\) 14.0000 0.683945 0.341972 − 0.939710i \(-0.388905\pi\)
0.341972 + 0.939710i \(0.388905\pi\)
\(420\) 0 0
\(421\) −18.0000 −0.877266 −0.438633 − 0.898666i \(-0.644537\pi\)
−0.438633 + 0.898666i \(0.644537\pi\)
\(422\) 10.0000 0.486792
\(423\) 0 0
\(424\) 0 0
\(425\) −3.00000 −0.145521
\(426\) 0 0
\(427\) 0 0
\(428\) 36.0000 1.74013
\(429\) 0 0
\(430\) −8.00000 −0.385794
\(431\) 12.0000 0.578020 0.289010 − 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 0 0
\(433\) −16.0000 −0.768911 −0.384455 − 0.923144i \(-0.625611\pi\)
−0.384455 + 0.923144i \(0.625611\pi\)
\(434\) 0 0
\(435\) 0 0
\(436\) 12.0000 0.574696
\(437\) 4.00000 0.191346
\(438\) 0 0
\(439\) 1.00000 0.0477274 0.0238637 − 0.999715i \(-0.492403\pi\)
0.0238637 + 0.999715i \(0.492403\pi\)
\(440\) 0 0
\(441\) 0 0
\(442\) −6.00000 −0.285391
\(443\) −16.0000 −0.760183 −0.380091 − 0.924949i \(-0.624107\pi\)
−0.380091 + 0.924949i \(0.624107\pi\)
\(444\) 0 0
\(445\) 13.0000 0.616259
\(446\) 44.0000 2.08346
\(447\) 0 0
\(448\) 0 0
\(449\) −13.0000 −0.613508 −0.306754 − 0.951789i \(-0.599243\pi\)
−0.306754 + 0.951789i \(0.599243\pi\)
\(450\) 0 0
\(451\) −8.00000 −0.376705
\(452\) −18.0000 −0.846649
\(453\) 0 0
\(454\) 40.0000 1.87729
\(455\) 0 0
\(456\) 0 0
\(457\) 28.0000 1.30978 0.654892 − 0.755722i \(-0.272714\pi\)
0.654892 + 0.755722i \(0.272714\pi\)
\(458\) 2.00000 0.0934539
\(459\) 0 0
\(460\) 8.00000 0.373002
\(461\) −19.0000 −0.884918 −0.442459 − 0.896789i \(-0.645894\pi\)
−0.442459 + 0.896789i \(0.645894\pi\)
\(462\) 0 0
\(463\) −15.0000 −0.697109 −0.348555 − 0.937288i \(-0.613327\pi\)
−0.348555 + 0.937288i \(0.613327\pi\)
\(464\) 24.0000 1.11417
\(465\) 0 0
\(466\) 12.0000 0.555889
\(467\) −30.0000 −1.38823 −0.694117 − 0.719862i \(-0.744205\pi\)
−0.694117 + 0.719862i \(0.744205\pi\)
\(468\) 0 0
\(469\) 0 0
\(470\) −12.0000 −0.553519
\(471\) 0 0
\(472\) 0 0
\(473\) −4.00000 −0.183920
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) 0 0
\(477\) 0 0
\(478\) −40.0000 −1.82956
\(479\) −9.00000 −0.411220 −0.205610 − 0.978634i \(-0.565918\pi\)
−0.205610 + 0.978634i \(0.565918\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) 2.00000 0.0909091
\(485\) −8.00000 −0.363261
\(486\) 0 0
\(487\) −14.0000 −0.634401 −0.317200 − 0.948359i \(-0.602743\pi\)
−0.317200 + 0.948359i \(0.602743\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 14.0000 0.632456
\(491\) −43.0000 −1.94056 −0.970281 − 0.241979i \(-0.922203\pi\)
−0.970281 + 0.241979i \(0.922203\pi\)
\(492\) 0 0
\(493\) 18.0000 0.810679
\(494\) 2.00000 0.0899843
\(495\) 0 0
\(496\) 32.0000 1.43684
\(497\) 0 0
\(498\) 0 0
\(499\) −27.0000 −1.20869 −0.604343 − 0.796724i \(-0.706564\pi\)
−0.604343 + 0.796724i \(0.706564\pi\)
\(500\) 2.00000 0.0894427
\(501\) 0 0
\(502\) 0 0
\(503\) −16.0000 −0.713405 −0.356702 − 0.934218i \(-0.616099\pi\)
−0.356702 + 0.934218i \(0.616099\pi\)
\(504\) 0 0
\(505\) −10.0000 −0.444994
\(506\) 8.00000 0.355643
\(507\) 0 0
\(508\) −22.0000 −0.976092
\(509\) −10.0000 −0.443242 −0.221621 − 0.975133i \(-0.571135\pi\)
−0.221621 + 0.975133i \(0.571135\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −32.0000 −1.41421
\(513\) 0 0
\(514\) 36.0000 1.58789
\(515\) −16.0000 −0.705044
\(516\) 0 0
\(517\) −6.00000 −0.263880
\(518\) 0 0
\(519\) 0 0
\(520\) 0 0
\(521\) 27.0000 1.18289 0.591446 − 0.806345i \(-0.298557\pi\)
0.591446 + 0.806345i \(0.298557\pi\)
\(522\) 0 0
\(523\) 31.0000 1.35554 0.677768 − 0.735276i \(-0.262948\pi\)
0.677768 + 0.735276i \(0.262948\pi\)
\(524\) 6.00000 0.262111
\(525\) 0 0
\(526\) −6.00000 −0.261612
\(527\) 24.0000 1.04546
\(528\) 0 0
\(529\) −7.00000 −0.304348
\(530\) 10.0000 0.434372
\(531\) 0 0
\(532\) 0 0
\(533\) −8.00000 −0.346518
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 0 0
\(537\) 0 0
\(538\) 6.00000 0.258678
\(539\) 7.00000 0.301511
\(540\) 0 0
\(541\) 20.0000 0.859867 0.429934 − 0.902861i \(-0.358537\pi\)
0.429934 + 0.902861i \(0.358537\pi\)
\(542\) −24.0000 −1.03089
\(543\) 0 0
\(544\) −24.0000 −1.02899
\(545\) 6.00000 0.257012
\(546\) 0 0
\(547\) 36.0000 1.53925 0.769624 − 0.638497i \(-0.220443\pi\)
0.769624 + 0.638497i \(0.220443\pi\)
\(548\) 12.0000 0.512615
\(549\) 0 0
\(550\) 2.00000 0.0852803
\(551\) −6.00000 −0.255609
\(552\) 0 0
\(553\) 0 0
\(554\) −8.00000 −0.339887
\(555\) 0 0
\(556\) −24.0000 −1.01783
\(557\) 22.0000 0.932170 0.466085 − 0.884740i \(-0.345664\pi\)
0.466085 + 0.884740i \(0.345664\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) −24.0000 −1.01238
\(563\) −6.00000 −0.252870 −0.126435 − 0.991975i \(-0.540353\pi\)
−0.126435 + 0.991975i \(0.540353\pi\)
\(564\) 0 0
\(565\) −9.00000 −0.378633
\(566\) 52.0000 2.18572
\(567\) 0 0
\(568\) 0 0
\(569\) −10.0000 −0.419222 −0.209611 − 0.977785i \(-0.567220\pi\)
−0.209611 + 0.977785i \(0.567220\pi\)
\(570\) 0 0
\(571\) −6.00000 −0.251092 −0.125546 − 0.992088i \(-0.540068\pi\)
−0.125546 + 0.992088i \(0.540068\pi\)
\(572\) 2.00000 0.0836242
\(573\) 0 0
\(574\) 0 0
\(575\) 4.00000 0.166812
\(576\) 0 0
\(577\) 5.00000 0.208153 0.104076 − 0.994569i \(-0.466811\pi\)
0.104076 + 0.994569i \(0.466811\pi\)
\(578\) 16.0000 0.665512
\(579\) 0 0
\(580\) −12.0000 −0.498273
\(581\) 0 0
\(582\) 0 0
\(583\) 5.00000 0.207079
\(584\) 0 0
\(585\) 0 0
\(586\) 28.0000 1.15667
\(587\) −18.0000 −0.742940 −0.371470 − 0.928445i \(-0.621146\pi\)
−0.371470 + 0.928445i \(0.621146\pi\)
\(588\) 0 0
\(589\) −8.00000 −0.329634
\(590\) −2.00000 −0.0823387
\(591\) 0 0
\(592\) −16.0000 −0.657596
\(593\) 1.00000 0.0410651 0.0205325 − 0.999789i \(-0.493464\pi\)
0.0205325 + 0.999789i \(0.493464\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −14.0000 −0.573462
\(597\) 0 0
\(598\) 8.00000 0.327144
\(599\) 21.0000 0.858037 0.429018 − 0.903296i \(-0.358860\pi\)
0.429018 + 0.903296i \(0.358860\pi\)
\(600\) 0 0
\(601\) −22.0000 −0.897399 −0.448699 − 0.893683i \(-0.648113\pi\)
−0.448699 + 0.893683i \(0.648113\pi\)
\(602\) 0 0
\(603\) 0 0
\(604\) 16.0000 0.651031
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −32.0000 −1.29884 −0.649420 − 0.760430i \(-0.724988\pi\)
−0.649420 + 0.760430i \(0.724988\pi\)
\(608\) 8.00000 0.324443
\(609\) 0 0
\(610\) −4.00000 −0.161955
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 34.0000 1.37325 0.686624 − 0.727013i \(-0.259092\pi\)
0.686624 + 0.727013i \(0.259092\pi\)
\(614\) 14.0000 0.564994
\(615\) 0 0
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 − 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 0 0
\(619\) 7.00000 0.281354 0.140677 − 0.990056i \(-0.455072\pi\)
0.140677 + 0.990056i \(0.455072\pi\)
\(620\) −16.0000 −0.642575
\(621\) 0 0
\(622\) 12.0000 0.481156
\(623\) 0 0
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) 2.00000 0.0799361
\(627\) 0 0
\(628\) −34.0000 −1.35675
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) 4.00000 0.159237 0.0796187 − 0.996825i \(-0.474630\pi\)
0.0796187 + 0.996825i \(0.474630\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −36.0000 −1.42974
\(635\) −11.0000 −0.436522
\(636\) 0 0
\(637\) 7.00000 0.277350
\(638\) −12.0000 −0.475085
\(639\) 0 0
\(640\) 0 0
\(641\) −38.0000 −1.50091 −0.750455 − 0.660922i \(-0.770166\pi\)
−0.750455 + 0.660922i \(0.770166\pi\)
\(642\) 0 0
\(643\) −16.0000 −0.630978 −0.315489 − 0.948929i \(-0.602169\pi\)
−0.315489 + 0.948929i \(0.602169\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 6.00000 0.236067
\(647\) 20.0000 0.786281 0.393141 − 0.919478i \(-0.371389\pi\)
0.393141 + 0.919478i \(0.371389\pi\)
\(648\) 0 0
\(649\) −1.00000 −0.0392534
\(650\) 2.00000 0.0784465
\(651\) 0 0
\(652\) 2.00000 0.0783260
\(653\) −16.0000 −0.626128 −0.313064 − 0.949732i \(-0.601356\pi\)
−0.313064 + 0.949732i \(0.601356\pi\)
\(654\) 0 0
\(655\) 3.00000 0.117220
\(656\) −32.0000 −1.24939
\(657\) 0 0
\(658\) 0 0
\(659\) 12.0000 0.467454 0.233727 − 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −16.0000 −0.622328 −0.311164 − 0.950356i \(-0.600719\pi\)
−0.311164 + 0.950356i \(0.600719\pi\)
\(662\) −20.0000 −0.777322
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) 0 0
\(667\) −24.0000 −0.929284
\(668\) −16.0000 −0.619059
\(669\) 0 0
\(670\) 8.00000 0.309067
\(671\) −2.00000 −0.0772091
\(672\) 0 0
\(673\) −5.00000 −0.192736 −0.0963679 − 0.995346i \(-0.530723\pi\)
−0.0963679 + 0.995346i \(0.530723\pi\)
\(674\) −12.0000 −0.462223
\(675\) 0 0
\(676\) −24.0000 −0.923077
\(677\) −38.0000 −1.46046 −0.730229 − 0.683202i \(-0.760587\pi\)
−0.730229 + 0.683202i \(0.760587\pi\)
\(678\) 0 0
\(679\) 0 0
\(680\) 0 0
\(681\) 0 0
\(682\) −16.0000 −0.612672
\(683\) −1.00000 −0.0382639 −0.0191320 − 0.999817i \(-0.506090\pi\)
−0.0191320 + 0.999817i \(0.506090\pi\)
\(684\) 0 0
\(685\) 6.00000 0.229248
\(686\) 0 0
\(687\) 0 0
\(688\) −16.0000 −0.609994
\(689\) 5.00000 0.190485
\(690\) 0 0
\(691\) −4.00000 −0.152167 −0.0760836 − 0.997101i \(-0.524242\pi\)
−0.0760836 + 0.997101i \(0.524242\pi\)
\(692\) 0 0
\(693\) 0 0
\(694\) 2.00000 0.0759190
\(695\) −12.0000 −0.455186
\(696\) 0 0
\(697\) −24.0000 −0.909065
\(698\) −16.0000 −0.605609
\(699\) 0 0
\(700\) 0 0
\(701\) −11.0000 −0.415464 −0.207732 − 0.978186i \(-0.566608\pi\)
−0.207732 + 0.978186i \(0.566608\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) 24.0000 0.903252
\(707\) 0 0
\(708\) 0 0
\(709\) −25.0000 −0.938895 −0.469447 − 0.882960i \(-0.655547\pi\)
−0.469447 + 0.882960i \(0.655547\pi\)
\(710\) −22.0000 −0.825645
\(711\) 0 0
\(712\) 0 0
\(713\) −32.0000 −1.19841
\(714\) 0 0
\(715\) 1.00000 0.0373979
\(716\) −10.0000 −0.373718
\(717\) 0 0
\(718\) 22.0000 0.821033
\(719\) −30.0000 −1.11881 −0.559406 − 0.828894i \(-0.688971\pi\)
−0.559406 + 0.828894i \(0.688971\pi\)
\(720\) 0 0
\(721\) 0 0
\(722\) −2.00000 −0.0744323
\(723\) 0 0
\(724\) 4.00000 0.148659
\(725\) −6.00000 −0.222834
\(726\) 0 0
\(727\) 4.00000 0.148352 0.0741759 − 0.997245i \(-0.476367\pi\)
0.0741759 + 0.997245i \(0.476367\pi\)
\(728\) 0 0
\(729\) 0 0
\(730\) 8.00000 0.296093
\(731\) −12.0000 −0.443836
\(732\) 0 0
\(733\) 6.00000 0.221615 0.110808 − 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 22.0000 0.812035
\(735\) 0 0
\(736\) 32.0000 1.17954
\(737\) 4.00000 0.147342
\(738\) 0 0
\(739\) −26.0000 −0.956425 −0.478213 − 0.878244i \(-0.658715\pi\)
−0.478213 + 0.878244i \(0.658715\pi\)
\(740\) 8.00000 0.294086
\(741\) 0 0
\(742\) 0 0
\(743\) −50.0000 −1.83432 −0.917161 − 0.398517i \(-0.869525\pi\)
−0.917161 + 0.398517i \(0.869525\pi\)
\(744\) 0 0
\(745\) −7.00000 −0.256460
\(746\) −12.0000 −0.439351
\(747\) 0 0
\(748\) 6.00000 0.219382
\(749\) 0 0
\(750\) 0 0
\(751\) 40.0000 1.45962 0.729810 − 0.683650i \(-0.239608\pi\)
0.729810 + 0.683650i \(0.239608\pi\)
\(752\) −24.0000 −0.875190
\(753\) 0 0
\(754\) −12.0000 −0.437014
\(755\) 8.00000 0.291150
\(756\) 0 0
\(757\) 27.0000 0.981332 0.490666 − 0.871348i \(-0.336754\pi\)
0.490666 + 0.871348i \(0.336754\pi\)
\(758\) −16.0000 −0.581146
\(759\) 0 0
\(760\) 0 0
\(761\) 22.0000 0.797499 0.398750 − 0.917060i \(-0.369444\pi\)
0.398750 + 0.917060i \(0.369444\pi\)
\(762\) 0 0
\(763\) 0 0
\(764\) 28.0000 1.01300
\(765\) 0 0
\(766\) −30.0000 −1.08394
\(767\) −1.00000 −0.0361079
\(768\) 0 0
\(769\) −16.0000 −0.576975 −0.288487 − 0.957484i \(-0.593152\pi\)
−0.288487 + 0.957484i \(0.593152\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 2.00000 0.0719816
\(773\) −43.0000 −1.54660 −0.773301 − 0.634039i \(-0.781396\pi\)
−0.773301 + 0.634039i \(0.781396\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 0 0
\(778\) 0 0
\(779\) 8.00000 0.286630
\(780\) 0 0
\(781\) −11.0000 −0.393611
\(782\) 24.0000 0.858238
\(783\) 0 0
\(784\) 28.0000 1.00000
\(785\) −17.0000 −0.606756
\(786\) 0 0
\(787\) −23.0000 −0.819861 −0.409931 − 0.912117i \(-0.634447\pi\)
−0.409931 + 0.912117i \(0.634447\pi\)
\(788\) 10.0000 0.356235
\(789\) 0 0
\(790\) −22.0000 −0.782725
\(791\) 0 0
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) −6.00000 −0.212932
\(795\) 0 0
\(796\) −42.0000 −1.48865
\(797\) −2.00000 −0.0708436 −0.0354218 − 0.999372i \(-0.511277\pi\)
−0.0354218 + 0.999372i \(0.511277\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 8.00000 0.282843
\(801\) 0 0
\(802\) −28.0000 −0.988714
\(803\) 4.00000 0.141157
\(804\) 0 0
\(805\) 0 0
\(806\) −16.0000 −0.563576
\(807\) 0 0
\(808\) 0 0
\(809\) 27.0000 0.949269 0.474635 − 0.880183i \(-0.342580\pi\)
0.474635 + 0.880183i \(0.342580\pi\)
\(810\) 0 0
\(811\) 45.0000 1.58016 0.790082 − 0.613001i \(-0.210038\pi\)
0.790082 + 0.613001i \(0.210038\pi\)
\(812\) 0 0
\(813\) 0 0
\(814\) 8.00000 0.280400
\(815\) 1.00000 0.0350285
\(816\) 0 0
\(817\) 4.00000 0.139942
\(818\) −42.0000 −1.46850
\(819\) 0 0
\(820\) 16.0000 0.558744
\(821\) 27.0000 0.942306 0.471153 − 0.882051i \(-0.343838\pi\)
0.471153 + 0.882051i \(0.343838\pi\)
\(822\) 0 0
\(823\) −24.0000 −0.836587 −0.418294 − 0.908312i \(-0.637372\pi\)
−0.418294 + 0.908312i \(0.637372\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) 0 0
\(827\) −48.0000 −1.66912 −0.834562 − 0.550914i \(-0.814279\pi\)
−0.834562 + 0.550914i \(0.814279\pi\)
\(828\) 0 0
\(829\) −38.0000 −1.31979 −0.659897 − 0.751356i \(-0.729400\pi\)
−0.659897 + 0.751356i \(0.729400\pi\)
\(830\) 2.00000 0.0694210
\(831\) 0 0
\(832\) 8.00000 0.277350
\(833\) 21.0000 0.727607
\(834\) 0 0
\(835\) −8.00000 −0.276851
\(836\) −2.00000 −0.0691714
\(837\) 0 0
\(838\) −28.0000 −0.967244
\(839\) 16.0000 0.552381 0.276191 − 0.961103i \(-0.410928\pi\)
0.276191 + 0.961103i \(0.410928\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 36.0000 1.24064
\(843\) 0 0
\(844\) −10.0000 −0.344214
\(845\) −12.0000 −0.412813
\(846\) 0 0
\(847\) 0 0
\(848\) 20.0000 0.686803
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 16.0000 0.548473
\(852\) 0 0
\(853\) 26.0000 0.890223 0.445112 − 0.895475i \(-0.353164\pi\)
0.445112 + 0.895475i \(0.353164\pi\)
\(854\) 0 0
\(855\) 0 0
\(856\) 0 0
\(857\) −18.0000 −0.614868 −0.307434 − 0.951569i \(-0.599470\pi\)
−0.307434 + 0.951569i \(0.599470\pi\)
\(858\) 0 0
\(859\) −41.0000 −1.39890 −0.699451 − 0.714681i \(-0.746572\pi\)
−0.699451 + 0.714681i \(0.746572\pi\)
\(860\) 8.00000 0.272798
\(861\) 0 0
\(862\) −24.0000 −0.817443
\(863\) 4.00000 0.136162 0.0680808 − 0.997680i \(-0.478312\pi\)
0.0680808 + 0.997680i \(0.478312\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) 32.0000 1.08740
\(867\) 0 0
\(868\) 0 0
\(869\) −11.0000 −0.373149
\(870\) 0 0
\(871\) 4.00000 0.135535
\(872\) 0 0
\(873\) 0 0
\(874\) −8.00000 −0.270604
\(875\) 0 0
\(876\) 0 0
\(877\) −7.00000 −0.236373 −0.118187 − 0.992991i \(-0.537708\pi\)
−0.118187 + 0.992991i \(0.537708\pi\)
\(878\) −2.00000 −0.0674967
\(879\) 0 0
\(880\) 4.00000 0.134840
\(881\) 48.0000 1.61716 0.808581 − 0.588386i \(-0.200236\pi\)
0.808581 + 0.588386i \(0.200236\pi\)
\(882\) 0 0
\(883\) 56.0000 1.88455 0.942275 − 0.334840i \(-0.108682\pi\)
0.942275 + 0.334840i \(0.108682\pi\)
\(884\) 6.00000 0.201802
\(885\) 0 0
\(886\) 32.0000 1.07506
\(887\) −34.0000 −1.14161 −0.570804 − 0.821086i \(-0.693368\pi\)
−0.570804 + 0.821086i \(0.693368\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) −26.0000 −0.871522
\(891\) 0 0
\(892\) −44.0000 −1.47323
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) −5.00000 −0.167132
\(896\) 0 0
\(897\) 0 0
\(898\) 26.0000 0.867631
\(899\) 48.0000 1.60089
\(900\) 0 0
\(901\) 15.0000 0.499722
\(902\) 16.0000 0.532742
\(903\) 0 0
\(904\) 0 0
\(905\) 2.00000 0.0664822
\(906\) 0 0
\(907\) −34.0000 −1.12895 −0.564476 − 0.825450i \(-0.690922\pi\)
−0.564476 + 0.825450i \(0.690922\pi\)
\(908\) −40.0000 −1.32745
\(909\) 0 0
\(910\) 0 0
\(911\) −16.0000 −0.530104 −0.265052 − 0.964234i \(-0.585389\pi\)
−0.265052 + 0.964234i \(0.585389\pi\)
\(912\) 0 0
\(913\) 1.00000 0.0330952
\(914\) −56.0000 −1.85232
\(915\) 0 0
\(916\) −2.00000 −0.0660819
\(917\) 0 0
\(918\) 0 0
\(919\) 36.0000 1.18753 0.593765 − 0.804638i \(-0.297641\pi\)
0.593765 + 0.804638i \(0.297641\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) 38.0000 1.25146
\(923\) −11.0000 −0.362069
\(924\) 0 0
\(925\) 4.00000 0.131519
\(926\) 30.0000 0.985861
\(927\) 0 0
\(928\) −48.0000 −1.57568
\(929\) −50.0000 −1.64045 −0.820223 − 0.572043i \(-0.806151\pi\)
−0.820223 + 0.572043i \(0.806151\pi\)
\(930\) 0 0
\(931\) −7.00000 −0.229416
\(932\) −12.0000 −0.393073
\(933\) 0 0
\(934\) 60.0000 1.96326
\(935\) 3.00000 0.0981105
\(936\) 0 0
\(937\) −42.0000 −1.37208 −0.686040 − 0.727564i \(-0.740653\pi\)
−0.686040 + 0.727564i \(0.740653\pi\)
\(938\) 0 0
\(939\) 0 0
\(940\) 12.0000 0.391397
\(941\) 40.0000 1.30396 0.651981 − 0.758235i \(-0.273938\pi\)
0.651981 + 0.758235i \(0.273938\pi\)
\(942\) 0 0
\(943\) 32.0000 1.04206
\(944\) −4.00000 −0.130189
\(945\) 0 0
\(946\) 8.00000 0.260102
\(947\) −42.0000 −1.36482 −0.682408 − 0.730971i \(-0.739067\pi\)
−0.682408 + 0.730971i \(0.739067\pi\)
\(948\) 0 0
\(949\) 4.00000 0.129845
\(950\) −2.00000 −0.0648886
\(951\) 0 0
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 − 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) 14.0000 0.453029
\(956\) 40.0000 1.29369
\(957\) 0 0
\(958\) 18.0000 0.581554
\(959\) 0 0
\(960\) 0 0
\(961\) 33.0000 1.06452
\(962\) 8.00000 0.257930
\(963\) 0 0
\(964\) 2.00000 0.0644157
\(965\) 1.00000 0.0321911
\(966\) 0 0
\(967\) −46.0000 −1.47926 −0.739630 − 0.673014i \(-0.765000\pi\)
−0.739630 + 0.673014i \(0.765000\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 16.0000 0.513729
\(971\) −15.0000 −0.481373 −0.240686 − 0.970603i \(-0.577373\pi\)
−0.240686 + 0.970603i \(0.577373\pi\)
\(972\) 0 0
\(973\) 0 0
\(974\) 28.0000 0.897178
\(975\) 0 0
\(976\) −8.00000 −0.256074
\(977\) −50.0000 −1.59964 −0.799821 − 0.600239i \(-0.795072\pi\)
−0.799821 + 0.600239i \(0.795072\pi\)
\(978\) 0 0
\(979\) −13.0000 −0.415482
\(980\) −14.0000 −0.447214
\(981\) 0 0
\(982\) 86.0000 2.74437
\(983\) 32.0000 1.02064 0.510321 − 0.859984i \(-0.329527\pi\)
0.510321 + 0.859984i \(0.329527\pi\)
\(984\) 0 0
\(985\) 5.00000 0.159313
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) −2.00000 −0.0636285
\(989\) 16.0000 0.508770
\(990\) 0 0
\(991\) −10.0000 −0.317660 −0.158830 − 0.987306i \(-0.550772\pi\)
−0.158830 + 0.987306i \(0.550772\pi\)
\(992\) −64.0000 −2.03200
\(993\) 0 0
\(994\) 0 0
\(995\) −21.0000 −0.665745
\(996\) 0 0
\(997\) 2.00000 0.0633406 0.0316703 − 0.999498i \(-0.489917\pi\)
0.0316703 + 0.999498i \(0.489917\pi\)
\(998\) 54.0000 1.70934
\(999\) 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 9405.2.a.b.1.1 ✓ 1
3.2 odd 2 9405.2.a.m.1.1 yes 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
9405.2.a.b.1.1 ✓ 1 1.1 even 1 trivial
9405.2.a.m.1.1 yes 1 3.2 odd 2