Properties

Label 1155.2.a.h.1.1
Level $1155$
Weight $2$
Character 1155.1
Self dual yes
Analytic conductor $9.223$
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] = [1155,2,Mod(1,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, 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("1155.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.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: \(9.22272143346\)
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 \(\chi\) \(=\) 1155.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} -1.00000 q^{15} +4.00000 q^{16} +3.00000 q^{17} -1.00000 q^{19} +2.00000 q^{20} +1.00000 q^{21} -3.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} -2.00000 q^{28} -9.00000 q^{29} -10.0000 q^{31} -1.00000 q^{33} -1.00000 q^{35} -2.00000 q^{36} -4.00000 q^{37} -4.00000 q^{39} +6.00000 q^{41} -1.00000 q^{43} +2.00000 q^{44} -1.00000 q^{45} -6.00000 q^{47} +4.00000 q^{48} +1.00000 q^{49} +3.00000 q^{51} +8.00000 q^{52} -9.00000 q^{53} +1.00000 q^{55} -1.00000 q^{57} +15.0000 q^{59} +2.00000 q^{60} -13.0000 q^{61} +1.00000 q^{63} -8.00000 q^{64} +4.00000 q^{65} +2.00000 q^{67} -6.00000 q^{68} -3.00000 q^{69} -6.00000 q^{71} -16.0000 q^{73} +1.00000 q^{75} +2.00000 q^{76} -1.00000 q^{77} +8.00000 q^{79} -4.00000 q^{80} +1.00000 q^{81} -3.00000 q^{83} -2.00000 q^{84} -3.00000 q^{85} -9.00000 q^{87} -9.00000 q^{89} -4.00000 q^{91} +6.00000 q^{92} -10.0000 q^{93} +1.00000 q^{95} +17.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 1.00000 0.577350
\(4\) −2.00000 −1.00000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −1.00000 −0.301511
\(12\) −2.00000 −0.577350
\(13\) −4.00000 −1.10940 −0.554700 0.832050i \(-0.687167\pi\)
−0.554700 + 0.832050i \(0.687167\pi\)
\(14\) 0 0
\(15\) −1.00000 −0.258199
\(16\) 4.00000 1.00000
\(17\) 3.00000 0.727607 0.363803 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) −1.00000 −0.229416 −0.114708 0.993399i \(-0.536593\pi\)
−0.114708 + 0.993399i \(0.536593\pi\)
\(20\) 2.00000 0.447214
\(21\) 1.00000 0.218218
\(22\) 0 0
\(23\) −3.00000 −0.625543 −0.312772 0.949828i \(-0.601257\pi\)
−0.312772 + 0.949828i \(0.601257\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) −2.00000 −0.377964
\(29\) −9.00000 −1.67126 −0.835629 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) −10.0000 −1.79605 −0.898027 0.439941i \(-0.854999\pi\)
−0.898027 + 0.439941i \(0.854999\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) −2.00000 −0.333333
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) −1.00000 −0.152499 −0.0762493 0.997089i \(-0.524294\pi\)
−0.0762493 + 0.997089i \(0.524294\pi\)
\(44\) 2.00000 0.301511
\(45\) −1.00000 −0.149071
\(46\) 0 0
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 4.00000 0.577350
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) 3.00000 0.420084
\(52\) 8.00000 1.10940
\(53\) −9.00000 −1.23625 −0.618123 0.786082i \(-0.712106\pi\)
−0.618123 + 0.786082i \(0.712106\pi\)
\(54\) 0 0
\(55\) 1.00000 0.134840
\(56\) 0 0
\(57\) −1.00000 −0.132453
\(58\) 0 0
\(59\) 15.0000 1.95283 0.976417 0.215894i \(-0.0692665\pi\)
0.976417 + 0.215894i \(0.0692665\pi\)
\(60\) 2.00000 0.258199
\(61\) −13.0000 −1.66448 −0.832240 0.554416i \(-0.812942\pi\)
−0.832240 + 0.554416i \(0.812942\pi\)
\(62\) 0 0
\(63\) 1.00000 0.125988
\(64\) −8.00000 −1.00000
\(65\) 4.00000 0.496139
\(66\) 0 0
\(67\) 2.00000 0.244339 0.122169 0.992509i \(-0.461015\pi\)
0.122169 + 0.992509i \(0.461015\pi\)
\(68\) −6.00000 −0.727607
\(69\) −3.00000 −0.361158
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −16.0000 −1.87266 −0.936329 0.351123i \(-0.885800\pi\)
−0.936329 + 0.351123i \(0.885800\pi\)
\(74\) 0 0
\(75\) 1.00000 0.115470
\(76\) 2.00000 0.229416
\(77\) −1.00000 −0.113961
\(78\) 0 0
\(79\) 8.00000 0.900070 0.450035 0.893011i \(-0.351411\pi\)
0.450035 + 0.893011i \(0.351411\pi\)
\(80\) −4.00000 −0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) −3.00000 −0.329293 −0.164646 0.986353i \(-0.552648\pi\)
−0.164646 + 0.986353i \(0.552648\pi\)
\(84\) −2.00000 −0.218218
\(85\) −3.00000 −0.325396
\(86\) 0 0
\(87\) −9.00000 −0.964901
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) −4.00000 −0.419314
\(92\) 6.00000 0.625543
\(93\) −10.0000 −1.03695
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) 17.0000 1.72609 0.863044 0.505128i \(-0.168555\pi\)
0.863044 + 0.505128i \(0.168555\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −2.00000 −0.200000
\(101\) 18.0000 1.79107 0.895533 0.444994i \(-0.146794\pi\)
0.895533 + 0.444994i \(0.146794\pi\)
\(102\) 0 0
\(103\) −19.0000 −1.87213 −0.936063 0.351833i \(-0.885559\pi\)
−0.936063 + 0.351833i \(0.885559\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) −6.00000 −0.580042 −0.290021 0.957020i \(-0.593662\pi\)
−0.290021 + 0.957020i \(0.593662\pi\)
\(108\) −2.00000 −0.192450
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) 0 0
\(111\) −4.00000 −0.379663
\(112\) 4.00000 0.377964
\(113\) 15.0000 1.41108 0.705541 0.708669i \(-0.250704\pi\)
0.705541 + 0.708669i \(0.250704\pi\)
\(114\) 0 0
\(115\) 3.00000 0.279751
\(116\) 18.0000 1.67126
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 3.00000 0.275010
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) 6.00000 0.541002
\(124\) 20.0000 1.79605
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) −1.00000 −0.0887357 −0.0443678 0.999015i \(-0.514127\pi\)
−0.0443678 + 0.999015i \(0.514127\pi\)
\(128\) 0 0
\(129\) −1.00000 −0.0880451
\(130\) 0 0
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 2.00000 0.174078
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) −1.00000 −0.0860663
\(136\) 0 0
\(137\) 18.0000 1.53784 0.768922 0.639343i \(-0.220793\pi\)
0.768922 + 0.639343i \(0.220793\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 2.00000 0.169031
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 4.00000 0.333333
\(145\) 9.00000 0.747409
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 8.00000 0.657596
\(149\) −18.0000 −1.47462 −0.737309 0.675556i \(-0.763904\pi\)
−0.737309 + 0.675556i \(0.763904\pi\)
\(150\) 0 0
\(151\) 2.00000 0.162758 0.0813788 0.996683i \(-0.474068\pi\)
0.0813788 + 0.996683i \(0.474068\pi\)
\(152\) 0 0
\(153\) 3.00000 0.242536
\(154\) 0 0
\(155\) 10.0000 0.803219
\(156\) 8.00000 0.640513
\(157\) 17.0000 1.35675 0.678374 0.734717i \(-0.262685\pi\)
0.678374 + 0.734717i \(0.262685\pi\)
\(158\) 0 0
\(159\) −9.00000 −0.713746
\(160\) 0 0
\(161\) −3.00000 −0.236433
\(162\) 0 0
\(163\) −16.0000 −1.25322 −0.626608 0.779334i \(-0.715557\pi\)
−0.626608 + 0.779334i \(0.715557\pi\)
\(164\) −12.0000 −0.937043
\(165\) 1.00000 0.0778499
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) 3.00000 0.230769
\(170\) 0 0
\(171\) −1.00000 −0.0764719
\(172\) 2.00000 0.152499
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) −4.00000 −0.301511
\(177\) 15.0000 1.12747
\(178\) 0 0
\(179\) −6.00000 −0.448461 −0.224231 0.974536i \(-0.571987\pi\)
−0.224231 + 0.974536i \(0.571987\pi\)
\(180\) 2.00000 0.149071
\(181\) −10.0000 −0.743294 −0.371647 0.928374i \(-0.621207\pi\)
−0.371647 + 0.928374i \(0.621207\pi\)
\(182\) 0 0
\(183\) −13.0000 −0.960988
\(184\) 0 0
\(185\) 4.00000 0.294086
\(186\) 0 0
\(187\) −3.00000 −0.219382
\(188\) 12.0000 0.875190
\(189\) 1.00000 0.0727393
\(190\) 0 0
\(191\) 12.0000 0.868290 0.434145 0.900843i \(-0.357051\pi\)
0.434145 + 0.900843i \(0.357051\pi\)
\(192\) −8.00000 −0.577350
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) 0 0
\(195\) 4.00000 0.286446
\(196\) −2.00000 −0.142857
\(197\) 12.0000 0.854965 0.427482 0.904024i \(-0.359401\pi\)
0.427482 + 0.904024i \(0.359401\pi\)
\(198\) 0 0
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) 0 0
\(201\) 2.00000 0.141069
\(202\) 0 0
\(203\) −9.00000 −0.631676
\(204\) −6.00000 −0.420084
\(205\) −6.00000 −0.419058
\(206\) 0 0
\(207\) −3.00000 −0.208514
\(208\) −16.0000 −1.10940
\(209\) 1.00000 0.0691714
\(210\) 0 0
\(211\) −22.0000 −1.51454 −0.757271 0.653101i \(-0.773468\pi\)
−0.757271 + 0.653101i \(0.773468\pi\)
\(212\) 18.0000 1.23625
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) 1.00000 0.0681994
\(216\) 0 0
\(217\) −10.0000 −0.678844
\(218\) 0 0
\(219\) −16.0000 −1.08118
\(220\) −2.00000 −0.134840
\(221\) −12.0000 −0.807207
\(222\) 0 0
\(223\) 23.0000 1.54019 0.770097 0.637927i \(-0.220208\pi\)
0.770097 + 0.637927i \(0.220208\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) 21.0000 1.39382 0.696909 0.717159i \(-0.254558\pi\)
0.696909 + 0.717159i \(0.254558\pi\)
\(228\) 2.00000 0.132453
\(229\) −22.0000 −1.45380 −0.726900 0.686743i \(-0.759040\pi\)
−0.726900 + 0.686743i \(0.759040\pi\)
\(230\) 0 0
\(231\) −1.00000 −0.0657952
\(232\) 0 0
\(233\) 24.0000 1.57229 0.786146 0.618041i \(-0.212073\pi\)
0.786146 + 0.618041i \(0.212073\pi\)
\(234\) 0 0
\(235\) 6.00000 0.391397
\(236\) −30.0000 −1.95283
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) 21.0000 1.35838 0.679189 0.733964i \(-0.262332\pi\)
0.679189 + 0.733964i \(0.262332\pi\)
\(240\) −4.00000 −0.258199
\(241\) −10.0000 −0.644157 −0.322078 0.946713i \(-0.604381\pi\)
−0.322078 + 0.946713i \(0.604381\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) 26.0000 1.66448
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) 4.00000 0.254514
\(248\) 0 0
\(249\) −3.00000 −0.190117
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −2.00000 −0.125988
\(253\) 3.00000 0.188608
\(254\) 0 0
\(255\) −3.00000 −0.187867
\(256\) 16.0000 1.00000
\(257\) −24.0000 −1.49708 −0.748539 0.663090i \(-0.769245\pi\)
−0.748539 + 0.663090i \(0.769245\pi\)
\(258\) 0 0
\(259\) −4.00000 −0.248548
\(260\) −8.00000 −0.496139
\(261\) −9.00000 −0.557086
\(262\) 0 0
\(263\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 9.00000 0.552866
\(266\) 0 0
\(267\) −9.00000 −0.550791
\(268\) −4.00000 −0.244339
\(269\) 3.00000 0.182913 0.0914566 0.995809i \(-0.470848\pi\)
0.0914566 + 0.995809i \(0.470848\pi\)
\(270\) 0 0
\(271\) 29.0000 1.76162 0.880812 0.473466i \(-0.156997\pi\)
0.880812 + 0.473466i \(0.156997\pi\)
\(272\) 12.0000 0.727607
\(273\) −4.00000 −0.242091
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 6.00000 0.361158
\(277\) −10.0000 −0.600842 −0.300421 0.953807i \(-0.597127\pi\)
−0.300421 + 0.953807i \(0.597127\pi\)
\(278\) 0 0
\(279\) −10.0000 −0.598684
\(280\) 0 0
\(281\) 6.00000 0.357930 0.178965 0.983855i \(-0.442725\pi\)
0.178965 + 0.983855i \(0.442725\pi\)
\(282\) 0 0
\(283\) 32.0000 1.90220 0.951101 0.308879i \(-0.0999539\pi\)
0.951101 + 0.308879i \(0.0999539\pi\)
\(284\) 12.0000 0.712069
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 6.00000 0.354169
\(288\) 0 0
\(289\) −8.00000 −0.470588
\(290\) 0 0
\(291\) 17.0000 0.996558
\(292\) 32.0000 1.87266
\(293\) 3.00000 0.175262 0.0876309 0.996153i \(-0.472070\pi\)
0.0876309 + 0.996153i \(0.472070\pi\)
\(294\) 0 0
\(295\) −15.0000 −0.873334
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 12.0000 0.693978
\(300\) −2.00000 −0.115470
\(301\) −1.00000 −0.0576390
\(302\) 0 0
\(303\) 18.0000 1.03407
\(304\) −4.00000 −0.229416
\(305\) 13.0000 0.744378
\(306\) 0 0
\(307\) −10.0000 −0.570730 −0.285365 0.958419i \(-0.592115\pi\)
−0.285365 + 0.958419i \(0.592115\pi\)
\(308\) 2.00000 0.113961
\(309\) −19.0000 −1.08087
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 11.0000 0.621757 0.310878 0.950450i \(-0.399377\pi\)
0.310878 + 0.950450i \(0.399377\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) −16.0000 −0.900070
\(317\) −18.0000 −1.01098 −0.505490 0.862832i \(-0.668688\pi\)
−0.505490 + 0.862832i \(0.668688\pi\)
\(318\) 0 0
\(319\) 9.00000 0.503903
\(320\) 8.00000 0.447214
\(321\) −6.00000 −0.334887
\(322\) 0 0
\(323\) −3.00000 −0.166924
\(324\) −2.00000 −0.111111
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) 2.00000 0.110600
\(328\) 0 0
\(329\) −6.00000 −0.330791
\(330\) 0 0
\(331\) −31.0000 −1.70391 −0.851957 0.523612i \(-0.824584\pi\)
−0.851957 + 0.523612i \(0.824584\pi\)
\(332\) 6.00000 0.329293
\(333\) −4.00000 −0.219199
\(334\) 0 0
\(335\) −2.00000 −0.109272
\(336\) 4.00000 0.218218
\(337\) −13.0000 −0.708155 −0.354078 0.935216i \(-0.615205\pi\)
−0.354078 + 0.935216i \(0.615205\pi\)
\(338\) 0 0
\(339\) 15.0000 0.814688
\(340\) 6.00000 0.325396
\(341\) 10.0000 0.541530
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 3.00000 0.161515
\(346\) 0 0
\(347\) −12.0000 −0.644194 −0.322097 0.946707i \(-0.604388\pi\)
−0.322097 + 0.946707i \(0.604388\pi\)
\(348\) 18.0000 0.964901
\(349\) −1.00000 −0.0535288 −0.0267644 0.999642i \(-0.508520\pi\)
−0.0267644 + 0.999642i \(0.508520\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) −6.00000 −0.319348 −0.159674 0.987170i \(-0.551044\pi\)
−0.159674 + 0.987170i \(0.551044\pi\)
\(354\) 0 0
\(355\) 6.00000 0.318447
\(356\) 18.0000 0.953998
\(357\) 3.00000 0.158777
\(358\) 0 0
\(359\) −27.0000 −1.42501 −0.712503 0.701669i \(-0.752438\pi\)
−0.712503 + 0.701669i \(0.752438\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) 8.00000 0.419314
\(365\) 16.0000 0.837478
\(366\) 0 0
\(367\) 17.0000 0.887393 0.443696 0.896177i \(-0.353667\pi\)
0.443696 + 0.896177i \(0.353667\pi\)
\(368\) −12.0000 −0.625543
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) −9.00000 −0.467257
\(372\) 20.0000 1.03695
\(373\) 23.0000 1.19089 0.595447 0.803394i \(-0.296975\pi\)
0.595447 + 0.803394i \(0.296975\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) 0 0
\(377\) 36.0000 1.85409
\(378\) 0 0
\(379\) −19.0000 −0.975964 −0.487982 0.872854i \(-0.662267\pi\)
−0.487982 + 0.872854i \(0.662267\pi\)
\(380\) −2.00000 −0.102598
\(381\) −1.00000 −0.0512316
\(382\) 0 0
\(383\) −12.0000 −0.613171 −0.306586 0.951843i \(-0.599187\pi\)
−0.306586 + 0.951843i \(0.599187\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 0 0
\(387\) −1.00000 −0.0508329
\(388\) −34.0000 −1.72609
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) −9.00000 −0.455150
\(392\) 0 0
\(393\) 0 0
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 2.00000 0.100504
\(397\) 2.00000 0.100377 0.0501886 0.998740i \(-0.484018\pi\)
0.0501886 + 0.998740i \(0.484018\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 4.00000 0.200000
\(401\) 6.00000 0.299626 0.149813 0.988714i \(-0.452133\pi\)
0.149813 + 0.988714i \(0.452133\pi\)
\(402\) 0 0
\(403\) 40.0000 1.99254
\(404\) −36.0000 −1.79107
\(405\) −1.00000 −0.0496904
\(406\) 0 0
\(407\) 4.00000 0.198273
\(408\) 0 0
\(409\) 14.0000 0.692255 0.346128 0.938187i \(-0.387496\pi\)
0.346128 + 0.938187i \(0.387496\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 38.0000 1.87213
\(413\) 15.0000 0.738102
\(414\) 0 0
\(415\) 3.00000 0.147264
\(416\) 0 0
\(417\) −4.00000 −0.195881
\(418\) 0 0
\(419\) −3.00000 −0.146560 −0.0732798 0.997311i \(-0.523347\pi\)
−0.0732798 + 0.997311i \(0.523347\pi\)
\(420\) 2.00000 0.0975900
\(421\) 17.0000 0.828529 0.414265 0.910156i \(-0.364039\pi\)
0.414265 + 0.910156i \(0.364039\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) 3.00000 0.145521
\(426\) 0 0
\(427\) −13.0000 −0.629114
\(428\) 12.0000 0.580042
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 12.0000 0.578020 0.289010 0.957326i \(-0.406674\pi\)
0.289010 + 0.957326i \(0.406674\pi\)
\(432\) 4.00000 0.192450
\(433\) 2.00000 0.0961139 0.0480569 0.998845i \(-0.484697\pi\)
0.0480569 + 0.998845i \(0.484697\pi\)
\(434\) 0 0
\(435\) 9.00000 0.431517
\(436\) −4.00000 −0.191565
\(437\) 3.00000 0.143509
\(438\) 0 0
\(439\) 35.0000 1.67046 0.835229 0.549902i \(-0.185335\pi\)
0.835229 + 0.549902i \(0.185335\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) −24.0000 −1.14027 −0.570137 0.821549i \(-0.693110\pi\)
−0.570137 + 0.821549i \(0.693110\pi\)
\(444\) 8.00000 0.379663
\(445\) 9.00000 0.426641
\(446\) 0 0
\(447\) −18.0000 −0.851371
\(448\) −8.00000 −0.377964
\(449\) −18.0000 −0.849473 −0.424736 0.905317i \(-0.639633\pi\)
−0.424736 + 0.905317i \(0.639633\pi\)
\(450\) 0 0
\(451\) −6.00000 −0.282529
\(452\) −30.0000 −1.41108
\(453\) 2.00000 0.0939682
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −1.00000 −0.0467780 −0.0233890 0.999726i \(-0.507446\pi\)
−0.0233890 + 0.999726i \(0.507446\pi\)
\(458\) 0 0
\(459\) 3.00000 0.140028
\(460\) −6.00000 −0.279751
\(461\) −6.00000 −0.279448 −0.139724 0.990190i \(-0.544622\pi\)
−0.139724 + 0.990190i \(0.544622\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) −36.0000 −1.67126
\(465\) 10.0000 0.463739
\(466\) 0 0
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 8.00000 0.369800
\(469\) 2.00000 0.0923514
\(470\) 0 0
\(471\) 17.0000 0.783319
\(472\) 0 0
\(473\) 1.00000 0.0459800
\(474\) 0 0
\(475\) −1.00000 −0.0458831
\(476\) −6.00000 −0.275010
\(477\) −9.00000 −0.412082
\(478\) 0 0
\(479\) 24.0000 1.09659 0.548294 0.836286i \(-0.315277\pi\)
0.548294 + 0.836286i \(0.315277\pi\)
\(480\) 0 0
\(481\) 16.0000 0.729537
\(482\) 0 0
\(483\) −3.00000 −0.136505
\(484\) −2.00000 −0.0909091
\(485\) −17.0000 −0.771930
\(486\) 0 0
\(487\) 26.0000 1.17817 0.589086 0.808070i \(-0.299488\pi\)
0.589086 + 0.808070i \(0.299488\pi\)
\(488\) 0 0
\(489\) −16.0000 −0.723545
\(490\) 0 0
\(491\) −9.00000 −0.406164 −0.203082 0.979162i \(-0.565096\pi\)
−0.203082 + 0.979162i \(0.565096\pi\)
\(492\) −12.0000 −0.541002
\(493\) −27.0000 −1.21602
\(494\) 0 0
\(495\) 1.00000 0.0449467
\(496\) −40.0000 −1.79605
\(497\) −6.00000 −0.269137
\(498\) 0 0
\(499\) −13.0000 −0.581960 −0.290980 0.956729i \(-0.593981\pi\)
−0.290980 + 0.956729i \(0.593981\pi\)
\(500\) 2.00000 0.0894427
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) 27.0000 1.20387 0.601935 0.798545i \(-0.294397\pi\)
0.601935 + 0.798545i \(0.294397\pi\)
\(504\) 0 0
\(505\) −18.0000 −0.800989
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 2.00000 0.0887357
\(509\) −15.0000 −0.664863 −0.332432 0.943127i \(-0.607869\pi\)
−0.332432 + 0.943127i \(0.607869\pi\)
\(510\) 0 0
\(511\) −16.0000 −0.707798
\(512\) 0 0
\(513\) −1.00000 −0.0441511
\(514\) 0 0
\(515\) 19.0000 0.837240
\(516\) 2.00000 0.0880451
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) 18.0000 0.790112
\(520\) 0 0
\(521\) −15.0000 −0.657162 −0.328581 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 0 0
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) 0 0
\(525\) 1.00000 0.0436436
\(526\) 0 0
\(527\) −30.0000 −1.30682
\(528\) −4.00000 −0.174078
\(529\) −14.0000 −0.608696
\(530\) 0 0
\(531\) 15.0000 0.650945
\(532\) 2.00000 0.0867110
\(533\) −24.0000 −1.03956
\(534\) 0 0
\(535\) 6.00000 0.259403
\(536\) 0 0
\(537\) −6.00000 −0.258919
\(538\) 0 0
\(539\) −1.00000 −0.0430730
\(540\) 2.00000 0.0860663
\(541\) −10.0000 −0.429934 −0.214967 0.976621i \(-0.568964\pi\)
−0.214967 + 0.976621i \(0.568964\pi\)
\(542\) 0 0
\(543\) −10.0000 −0.429141
\(544\) 0 0
\(545\) −2.00000 −0.0856706
\(546\) 0 0
\(547\) 17.0000 0.726868 0.363434 0.931620i \(-0.381604\pi\)
0.363434 + 0.931620i \(0.381604\pi\)
\(548\) −36.0000 −1.53784
\(549\) −13.0000 −0.554826
\(550\) 0 0
\(551\) 9.00000 0.383413
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) 4.00000 0.169791
\(556\) 8.00000 0.339276
\(557\) −18.0000 −0.762684 −0.381342 0.924434i \(-0.624538\pi\)
−0.381342 + 0.924434i \(0.624538\pi\)
\(558\) 0 0
\(559\) 4.00000 0.169182
\(560\) −4.00000 −0.169031
\(561\) −3.00000 −0.126660
\(562\) 0 0
\(563\) −12.0000 −0.505740 −0.252870 0.967500i \(-0.581374\pi\)
−0.252870 + 0.967500i \(0.581374\pi\)
\(564\) 12.0000 0.505291
\(565\) −15.0000 −0.631055
\(566\) 0 0
\(567\) 1.00000 0.0419961
\(568\) 0 0
\(569\) −45.0000 −1.88650 −0.943249 0.332086i \(-0.892248\pi\)
−0.943249 + 0.332086i \(0.892248\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) −8.00000 −0.334497
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −3.00000 −0.125109
\(576\) −8.00000 −0.333333
\(577\) 14.0000 0.582828 0.291414 0.956597i \(-0.405874\pi\)
0.291414 + 0.956597i \(0.405874\pi\)
\(578\) 0 0
\(579\) −10.0000 −0.415586
\(580\) −18.0000 −0.747409
\(581\) −3.00000 −0.124461
\(582\) 0 0
\(583\) 9.00000 0.372742
\(584\) 0 0
\(585\) 4.00000 0.165380
\(586\) 0 0
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) −2.00000 −0.0824786
\(589\) 10.0000 0.412043
\(590\) 0 0
\(591\) 12.0000 0.493614
\(592\) −16.0000 −0.657596
\(593\) −42.0000 −1.72473 −0.862367 0.506284i \(-0.831019\pi\)
−0.862367 + 0.506284i \(0.831019\pi\)
\(594\) 0 0
\(595\) −3.00000 −0.122988
\(596\) 36.0000 1.47462
\(597\) −16.0000 −0.654836
\(598\) 0 0
\(599\) −30.0000 −1.22577 −0.612883 0.790173i \(-0.709990\pi\)
−0.612883 + 0.790173i \(0.709990\pi\)
\(600\) 0 0
\(601\) 5.00000 0.203954 0.101977 0.994787i \(-0.467483\pi\)
0.101977 + 0.994787i \(0.467483\pi\)
\(602\) 0 0
\(603\) 2.00000 0.0814463
\(604\) −4.00000 −0.162758
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 14.0000 0.568242 0.284121 0.958788i \(-0.408298\pi\)
0.284121 + 0.958788i \(0.408298\pi\)
\(608\) 0 0
\(609\) −9.00000 −0.364698
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) −6.00000 −0.242536
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) 0 0
\(615\) −6.00000 −0.241943
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) −22.0000 −0.884255 −0.442127 0.896952i \(-0.645776\pi\)
−0.442127 + 0.896952i \(0.645776\pi\)
\(620\) −20.0000 −0.803219
\(621\) −3.00000 −0.120386
\(622\) 0 0
\(623\) −9.00000 −0.360577
\(624\) −16.0000 −0.640513
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) 1.00000 0.0399362
\(628\) −34.0000 −1.35675
\(629\) −12.0000 −0.478471
\(630\) 0 0
\(631\) −43.0000 −1.71180 −0.855901 0.517139i \(-0.826997\pi\)
−0.855901 + 0.517139i \(0.826997\pi\)
\(632\) 0 0
\(633\) −22.0000 −0.874421
\(634\) 0 0
\(635\) 1.00000 0.0396838
\(636\) 18.0000 0.713746
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −6.00000 −0.236986 −0.118493 0.992955i \(-0.537806\pi\)
−0.118493 + 0.992955i \(0.537806\pi\)
\(642\) 0 0
\(643\) 17.0000 0.670415 0.335207 0.942144i \(-0.391194\pi\)
0.335207 + 0.942144i \(0.391194\pi\)
\(644\) 6.00000 0.236433
\(645\) 1.00000 0.0393750
\(646\) 0 0
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 0 0
\(649\) −15.0000 −0.588802
\(650\) 0 0
\(651\) −10.0000 −0.391931
\(652\) 32.0000 1.25322
\(653\) −3.00000 −0.117399 −0.0586995 0.998276i \(-0.518695\pi\)
−0.0586995 + 0.998276i \(0.518695\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 24.0000 0.937043
\(657\) −16.0000 −0.624219
\(658\) 0 0
\(659\) 39.0000 1.51922 0.759612 0.650376i \(-0.225389\pi\)
0.759612 + 0.650376i \(0.225389\pi\)
\(660\) −2.00000 −0.0778499
\(661\) −22.0000 −0.855701 −0.427850 0.903850i \(-0.640729\pi\)
−0.427850 + 0.903850i \(0.640729\pi\)
\(662\) 0 0
\(663\) −12.0000 −0.466041
\(664\) 0 0
\(665\) 1.00000 0.0387783
\(666\) 0 0
\(667\) 27.0000 1.04544
\(668\) −24.0000 −0.928588
\(669\) 23.0000 0.889231
\(670\) 0 0
\(671\) 13.0000 0.501859
\(672\) 0 0
\(673\) 41.0000 1.58043 0.790217 0.612827i \(-0.209968\pi\)
0.790217 + 0.612827i \(0.209968\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) −6.00000 −0.230769
\(677\) 15.0000 0.576497 0.288248 0.957556i \(-0.406927\pi\)
0.288248 + 0.957556i \(0.406927\pi\)
\(678\) 0 0
\(679\) 17.0000 0.652400
\(680\) 0 0
\(681\) 21.0000 0.804722
\(682\) 0 0
\(683\) −12.0000 −0.459167 −0.229584 0.973289i \(-0.573736\pi\)
−0.229584 + 0.973289i \(0.573736\pi\)
\(684\) 2.00000 0.0764719
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) −22.0000 −0.839352
\(688\) −4.00000 −0.152499
\(689\) 36.0000 1.37149
\(690\) 0 0
\(691\) −10.0000 −0.380418 −0.190209 0.981744i \(-0.560917\pi\)
−0.190209 + 0.981744i \(0.560917\pi\)
\(692\) −36.0000 −1.36851
\(693\) −1.00000 −0.0379869
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) 18.0000 0.681799
\(698\) 0 0
\(699\) 24.0000 0.907763
\(700\) −2.00000 −0.0755929
\(701\) −51.0000 −1.92624 −0.963122 0.269066i \(-0.913285\pi\)
−0.963122 + 0.269066i \(0.913285\pi\)
\(702\) 0 0
\(703\) 4.00000 0.150863
\(704\) 8.00000 0.301511
\(705\) 6.00000 0.225973
\(706\) 0 0
\(707\) 18.0000 0.676960
\(708\) −30.0000 −1.12747
\(709\) −31.0000 −1.16423 −0.582115 0.813107i \(-0.697775\pi\)
−0.582115 + 0.813107i \(0.697775\pi\)
\(710\) 0 0
\(711\) 8.00000 0.300023
\(712\) 0 0
\(713\) 30.0000 1.12351
\(714\) 0 0
\(715\) −4.00000 −0.149592
\(716\) 12.0000 0.448461
\(717\) 21.0000 0.784259
\(718\) 0 0
\(719\) 51.0000 1.90198 0.950990 0.309223i \(-0.100069\pi\)
0.950990 + 0.309223i \(0.100069\pi\)
\(720\) −4.00000 −0.149071
\(721\) −19.0000 −0.707597
\(722\) 0 0
\(723\) −10.0000 −0.371904
\(724\) 20.0000 0.743294
\(725\) −9.00000 −0.334252
\(726\) 0 0
\(727\) 23.0000 0.853023 0.426511 0.904482i \(-0.359742\pi\)
0.426511 + 0.904482i \(0.359742\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −3.00000 −0.110959
\(732\) 26.0000 0.960988
\(733\) −22.0000 −0.812589 −0.406294 0.913742i \(-0.633179\pi\)
−0.406294 + 0.913742i \(0.633179\pi\)
\(734\) 0 0
\(735\) −1.00000 −0.0368856
\(736\) 0 0
\(737\) −2.00000 −0.0736709
\(738\) 0 0
\(739\) −22.0000 −0.809283 −0.404642 0.914475i \(-0.632604\pi\)
−0.404642 + 0.914475i \(0.632604\pi\)
\(740\) −8.00000 −0.294086
\(741\) 4.00000 0.146944
\(742\) 0 0
\(743\) 30.0000 1.10059 0.550297 0.834969i \(-0.314515\pi\)
0.550297 + 0.834969i \(0.314515\pi\)
\(744\) 0 0
\(745\) 18.0000 0.659469
\(746\) 0 0
\(747\) −3.00000 −0.109764
\(748\) 6.00000 0.219382
\(749\) −6.00000 −0.219235
\(750\) 0 0
\(751\) −19.0000 −0.693320 −0.346660 0.937991i \(-0.612684\pi\)
−0.346660 + 0.937991i \(0.612684\pi\)
\(752\) −24.0000 −0.875190
\(753\) 12.0000 0.437304
\(754\) 0 0
\(755\) −2.00000 −0.0727875
\(756\) −2.00000 −0.0727393
\(757\) −34.0000 −1.23575 −0.617876 0.786276i \(-0.712006\pi\)
−0.617876 + 0.786276i \(0.712006\pi\)
\(758\) 0 0
\(759\) 3.00000 0.108893
\(760\) 0 0
\(761\) −30.0000 −1.08750 −0.543750 0.839248i \(-0.682996\pi\)
−0.543750 + 0.839248i \(0.682996\pi\)
\(762\) 0 0
\(763\) 2.00000 0.0724049
\(764\) −24.0000 −0.868290
\(765\) −3.00000 −0.108465
\(766\) 0 0
\(767\) −60.0000 −2.16647
\(768\) 16.0000 0.577350
\(769\) −37.0000 −1.33425 −0.667127 0.744944i \(-0.732476\pi\)
−0.667127 + 0.744944i \(0.732476\pi\)
\(770\) 0 0
\(771\) −24.0000 −0.864339
\(772\) 20.0000 0.719816
\(773\) 6.00000 0.215805 0.107903 0.994161i \(-0.465587\pi\)
0.107903 + 0.994161i \(0.465587\pi\)
\(774\) 0 0
\(775\) −10.0000 −0.359211
\(776\) 0 0
\(777\) −4.00000 −0.143499
\(778\) 0 0
\(779\) −6.00000 −0.214972
\(780\) −8.00000 −0.286446
\(781\) 6.00000 0.214697
\(782\) 0 0
\(783\) −9.00000 −0.321634
\(784\) 4.00000 0.142857
\(785\) −17.0000 −0.606756
\(786\) 0 0
\(787\) 8.00000 0.285169 0.142585 0.989783i \(-0.454459\pi\)
0.142585 + 0.989783i \(0.454459\pi\)
\(788\) −24.0000 −0.854965
\(789\) 0 0
\(790\) 0 0
\(791\) 15.0000 0.533339
\(792\) 0 0
\(793\) 52.0000 1.84657
\(794\) 0 0
\(795\) 9.00000 0.319197
\(796\) 32.0000 1.13421
\(797\) −42.0000 −1.48772 −0.743858 0.668338i \(-0.767006\pi\)
−0.743858 + 0.668338i \(0.767006\pi\)
\(798\) 0 0
\(799\) −18.0000 −0.636794
\(800\) 0 0
\(801\) −9.00000 −0.317999
\(802\) 0 0
\(803\) 16.0000 0.564628
\(804\) −4.00000 −0.141069
\(805\) 3.00000 0.105736
\(806\) 0 0
\(807\) 3.00000 0.105605
\(808\) 0 0
\(809\) 6.00000 0.210949 0.105474 0.994422i \(-0.466364\pi\)
0.105474 + 0.994422i \(0.466364\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 18.0000 0.631676
\(813\) 29.0000 1.01707
\(814\) 0 0
\(815\) 16.0000 0.560456
\(816\) 12.0000 0.420084
\(817\) 1.00000 0.0349856
\(818\) 0 0
\(819\) −4.00000 −0.139771
\(820\) 12.0000 0.419058
\(821\) 9.00000 0.314102 0.157051 0.987590i \(-0.449801\pi\)
0.157051 + 0.987590i \(0.449801\pi\)
\(822\) 0 0
\(823\) −4.00000 −0.139431 −0.0697156 0.997567i \(-0.522209\pi\)
−0.0697156 + 0.997567i \(0.522209\pi\)
\(824\) 0 0
\(825\) −1.00000 −0.0348155
\(826\) 0 0
\(827\) 24.0000 0.834562 0.417281 0.908778i \(-0.362983\pi\)
0.417281 + 0.908778i \(0.362983\pi\)
\(828\) 6.00000 0.208514
\(829\) 50.0000 1.73657 0.868286 0.496064i \(-0.165222\pi\)
0.868286 + 0.496064i \(0.165222\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 32.0000 1.10940
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) −12.0000 −0.415277
\(836\) −2.00000 −0.0691714
\(837\) −10.0000 −0.345651
\(838\) 0 0
\(839\) 3.00000 0.103572 0.0517858 0.998658i \(-0.483509\pi\)
0.0517858 + 0.998658i \(0.483509\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 0 0
\(843\) 6.00000 0.206651
\(844\) 44.0000 1.51454
\(845\) −3.00000 −0.103203
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) −36.0000 −1.23625
\(849\) 32.0000 1.09824
\(850\) 0 0
\(851\) 12.0000 0.411355
\(852\) 12.0000 0.411113
\(853\) −46.0000 −1.57501 −0.787505 0.616308i \(-0.788628\pi\)
−0.787505 + 0.616308i \(0.788628\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 18.0000 0.614868 0.307434 0.951569i \(-0.400530\pi\)
0.307434 + 0.951569i \(0.400530\pi\)
\(858\) 0 0
\(859\) 50.0000 1.70598 0.852989 0.521929i \(-0.174787\pi\)
0.852989 + 0.521929i \(0.174787\pi\)
\(860\) −2.00000 −0.0681994
\(861\) 6.00000 0.204479
\(862\) 0 0
\(863\) −3.00000 −0.102121 −0.0510606 0.998696i \(-0.516260\pi\)
−0.0510606 + 0.998696i \(0.516260\pi\)
\(864\) 0 0
\(865\) −18.0000 −0.612018
\(866\) 0 0
\(867\) −8.00000 −0.271694
\(868\) 20.0000 0.678844
\(869\) −8.00000 −0.271381
\(870\) 0 0
\(871\) −8.00000 −0.271070
\(872\) 0 0
\(873\) 17.0000 0.575363
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 32.0000 1.08118
\(877\) −31.0000 −1.04680 −0.523398 0.852088i \(-0.675336\pi\)
−0.523398 + 0.852088i \(0.675336\pi\)
\(878\) 0 0
\(879\) 3.00000 0.101187
\(880\) 4.00000 0.134840
\(881\) −45.0000 −1.51609 −0.758044 0.652203i \(-0.773845\pi\)
−0.758044 + 0.652203i \(0.773845\pi\)
\(882\) 0 0
\(883\) −16.0000 −0.538443 −0.269221 0.963078i \(-0.586766\pi\)
−0.269221 + 0.963078i \(0.586766\pi\)
\(884\) 24.0000 0.807207
\(885\) −15.0000 −0.504219
\(886\) 0 0
\(887\) 9.00000 0.302190 0.151095 0.988519i \(-0.451720\pi\)
0.151095 + 0.988519i \(0.451720\pi\)
\(888\) 0 0
\(889\) −1.00000 −0.0335389
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) −46.0000 −1.54019
\(893\) 6.00000 0.200782
\(894\) 0 0
\(895\) 6.00000 0.200558
\(896\) 0 0
\(897\) 12.0000 0.400668
\(898\) 0 0
\(899\) 90.0000 3.00167
\(900\) −2.00000 −0.0666667
\(901\) −27.0000 −0.899500
\(902\) 0 0
\(903\) −1.00000 −0.0332779
\(904\) 0 0
\(905\) 10.0000 0.332411
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) −42.0000 −1.39382
\(909\) 18.0000 0.597022
\(910\) 0 0
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) −4.00000 −0.132453
\(913\) 3.00000 0.0992855
\(914\) 0 0
\(915\) 13.0000 0.429767
\(916\) 44.0000 1.45380
\(917\) 0 0
\(918\) 0 0
\(919\) −34.0000 −1.12156 −0.560778 0.827966i \(-0.689498\pi\)
−0.560778 + 0.827966i \(0.689498\pi\)
\(920\) 0 0
\(921\) −10.0000 −0.329511
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) 2.00000 0.0657952
\(925\) −4.00000 −0.131519
\(926\) 0 0
\(927\) −19.0000 −0.624042
\(928\) 0 0
\(929\) −18.0000 −0.590561 −0.295280 0.955411i \(-0.595413\pi\)
−0.295280 + 0.955411i \(0.595413\pi\)
\(930\) 0 0
\(931\) −1.00000 −0.0327737
\(932\) −48.0000 −1.57229
\(933\) 0 0
\(934\) 0 0
\(935\) 3.00000 0.0981105
\(936\) 0 0
\(937\) 2.00000 0.0653372 0.0326686 0.999466i \(-0.489599\pi\)
0.0326686 + 0.999466i \(0.489599\pi\)
\(938\) 0 0
\(939\) 11.0000 0.358971
\(940\) −12.0000 −0.391397
\(941\) 24.0000 0.782378 0.391189 0.920310i \(-0.372064\pi\)
0.391189 + 0.920310i \(0.372064\pi\)
\(942\) 0 0
\(943\) −18.0000 −0.586161
\(944\) 60.0000 1.95283
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) 9.00000 0.292461 0.146230 0.989251i \(-0.453286\pi\)
0.146230 + 0.989251i \(0.453286\pi\)
\(948\) −16.0000 −0.519656
\(949\) 64.0000 2.07753
\(950\) 0 0
\(951\) −18.0000 −0.583690
\(952\) 0 0
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −12.0000 −0.388311
\(956\) −42.0000 −1.35838
\(957\) 9.00000 0.290929
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) 8.00000 0.258199
\(961\) 69.0000 2.22581
\(962\) 0 0
\(963\) −6.00000 −0.193347
\(964\) 20.0000 0.644157
\(965\) 10.0000 0.321911
\(966\) 0 0
\(967\) 11.0000 0.353736 0.176868 0.984235i \(-0.443403\pi\)
0.176868 + 0.984235i \(0.443403\pi\)
\(968\) 0 0
\(969\) −3.00000 −0.0963739
\(970\) 0 0
\(971\) 3.00000 0.0962746 0.0481373 0.998841i \(-0.484672\pi\)
0.0481373 + 0.998841i \(0.484672\pi\)
\(972\) −2.00000 −0.0641500
\(973\) −4.00000 −0.128234
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) −52.0000 −1.66448
\(977\) 3.00000 0.0959785 0.0479893 0.998848i \(-0.484719\pi\)
0.0479893 + 0.998848i \(0.484719\pi\)
\(978\) 0 0
\(979\) 9.00000 0.287641
\(980\) 2.00000 0.0638877
\(981\) 2.00000 0.0638551
\(982\) 0 0
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 0 0
\(985\) −12.0000 −0.382352
\(986\) 0 0
\(987\) −6.00000 −0.190982
\(988\) −8.00000 −0.254514
\(989\) 3.00000 0.0953945
\(990\) 0 0
\(991\) −1.00000 −0.0317660 −0.0158830 0.999874i \(-0.505056\pi\)
−0.0158830 + 0.999874i \(0.505056\pi\)
\(992\) 0 0
\(993\) −31.0000 −0.983755
\(994\) 0 0
\(995\) 16.0000 0.507234
\(996\) 6.00000 0.190117
\(997\) 14.0000 0.443384 0.221692 0.975117i \(-0.428842\pi\)
0.221692 + 0.975117i \(0.428842\pi\)
\(998\) 0 0
\(999\) −4.00000 −0.126554
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 1155.2.a.h.1.1 1
3.2 odd 2 3465.2.a.j.1.1 1
5.4 even 2 5775.2.a.k.1.1 1
7.6 odd 2 8085.2.a.m.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.h.1.1 1 1.1 even 1 trivial
3465.2.a.j.1.1 1 3.2 odd 2
5775.2.a.k.1.1 1 5.4 even 2
8085.2.a.m.1.1 1 7.6 odd 2