Properties

Label 1155.2.a.i.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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{3} -2.00000 q^{4} +1.00000 q^{5} -1.00000 q^{7} +1.00000 q^{9} -1.00000 q^{11} -2.00000 q^{12} -4.00000 q^{13} +1.00000 q^{15} +4.00000 q^{16} -5.00000 q^{17} +1.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} -5.00000 q^{23} +1.00000 q^{25} +1.00000 q^{27} +2.00000 q^{28} +3.00000 q^{29} -6.00000 q^{31} -1.00000 q^{33} -1.00000 q^{35} -2.00000 q^{36} -12.0000 q^{37} -4.00000 q^{39} -2.00000 q^{41} +13.0000 q^{43} +2.00000 q^{44} +1.00000 q^{45} -6.00000 q^{47} +4.00000 q^{48} +1.00000 q^{49} -5.00000 q^{51} +8.00000 q^{52} +1.00000 q^{53} -1.00000 q^{55} +1.00000 q^{57} -11.0000 q^{59} -2.00000 q^{60} +5.00000 q^{61} -1.00000 q^{63} -8.00000 q^{64} -4.00000 q^{65} -10.0000 q^{67} +10.0000 q^{68} -5.00000 q^{69} -6.00000 q^{71} -4.00000 q^{73} +1.00000 q^{75} -2.00000 q^{76} +1.00000 q^{77} -8.00000 q^{79} +4.00000 q^{80} +1.00000 q^{81} +5.00000 q^{83} +2.00000 q^{84} -5.00000 q^{85} +3.00000 q^{87} +13.0000 q^{89} +4.00000 q^{91} +10.0000 q^{92} -6.00000 q^{93} +1.00000 q^{95} -19.0000 q^{97} -1.00000 q^{99} +O(q^{100})\)

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
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) 0 0
\(19\) 1.00000 0.229416 0.114708 0.993399i \(-0.463407\pi\)
0.114708 + 0.993399i \(0.463407\pi\)
\(20\) −2.00000 −0.447214
\(21\) −1.00000 −0.218218
\(22\) 0 0
\(23\) −5.00000 −1.04257 −0.521286 0.853382i \(-0.674548\pi\)
−0.521286 + 0.853382i \(0.674548\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 2.00000 0.377964
\(29\) 3.00000 0.557086 0.278543 0.960424i \(-0.410149\pi\)
0.278543 + 0.960424i \(0.410149\pi\)
\(30\) 0 0
\(31\) −6.00000 −1.07763 −0.538816 0.842424i \(-0.681128\pi\)
−0.538816 + 0.842424i \(0.681128\pi\)
\(32\) 0 0
\(33\) −1.00000 −0.174078
\(34\) 0 0
\(35\) −1.00000 −0.169031
\(36\) −2.00000 −0.333333
\(37\) −12.0000 −1.97279 −0.986394 0.164399i \(-0.947432\pi\)
−0.986394 + 0.164399i \(0.947432\pi\)
\(38\) 0 0
\(39\) −4.00000 −0.640513
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) 13.0000 1.98248 0.991241 0.132068i \(-0.0421616\pi\)
0.991241 + 0.132068i \(0.0421616\pi\)
\(44\) 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
\(51\) −5.00000 −0.700140
\(52\) 8.00000 1.10940
\(53\) 1.00000 0.137361 0.0686803 0.997639i \(-0.478121\pi\)
0.0686803 + 0.997639i \(0.478121\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) 0 0
\(57\) 1.00000 0.132453
\(58\) 0 0
\(59\) −11.0000 −1.43208 −0.716039 0.698060i \(-0.754047\pi\)
−0.716039 + 0.698060i \(0.754047\pi\)
\(60\) −2.00000 −0.258199
\(61\) 5.00000 0.640184 0.320092 0.947386i \(-0.396286\pi\)
0.320092 + 0.947386i \(0.396286\pi\)
\(62\) 0 0
\(63\) −1.00000 −0.125988
\(64\) −8.00000 −1.00000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 10.0000 1.21268
\(69\) −5.00000 −0.601929
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 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 \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 4.00000 0.447214
\(81\) 1.00000 0.111111
\(82\) 0 0
\(83\) 5.00000 0.548821 0.274411 0.961613i \(-0.411517\pi\)
0.274411 + 0.961613i \(0.411517\pi\)
\(84\) 2.00000 0.218218
\(85\) −5.00000 −0.542326
\(86\) 0 0
\(87\) 3.00000 0.321634
\(88\) 0 0
\(89\) 13.0000 1.37800 0.688999 0.724763i \(-0.258051\pi\)
0.688999 + 0.724763i \(0.258051\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 10.0000 1.04257
\(93\) −6.00000 −0.622171
\(94\) 0 0
\(95\) 1.00000 0.102598
\(96\) 0 0
\(97\) −19.0000 −1.92916 −0.964579 0.263795i \(-0.915026\pi\)
−0.964579 + 0.263795i \(0.915026\pi\)
\(98\) 0 0
\(99\) −1.00000 −0.100504
\(100\) −2.00000 −0.200000
\(101\) 2.00000 0.199007 0.0995037 0.995037i \(-0.468274\pi\)
0.0995037 + 0.995037i \(0.468274\pi\)
\(102\) 0 0
\(103\) 9.00000 0.886796 0.443398 0.896325i \(-0.353773\pi\)
0.443398 + 0.896325i \(0.353773\pi\)
\(104\) 0 0
\(105\) −1.00000 −0.0975900
\(106\) 0 0
\(107\) 18.0000 1.74013 0.870063 0.492941i \(-0.164078\pi\)
0.870063 + 0.492941i \(0.164078\pi\)
\(108\) −2.00000 −0.192450
\(109\) −10.0000 −0.957826 −0.478913 0.877862i \(-0.658969\pi\)
−0.478913 + 0.877862i \(0.658969\pi\)
\(110\) 0 0
\(111\) −12.0000 −1.13899
\(112\) −4.00000 −0.377964
\(113\) 9.00000 0.846649 0.423324 0.905978i \(-0.360863\pi\)
0.423324 + 0.905978i \(0.360863\pi\)
\(114\) 0 0
\(115\) −5.00000 −0.466252
\(116\) −6.00000 −0.557086
\(117\) −4.00000 −0.369800
\(118\) 0 0
\(119\) 5.00000 0.458349
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 0 0
\(123\) −2.00000 −0.180334
\(124\) 12.0000 1.07763
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) −19.0000 −1.68598 −0.842989 0.537931i \(-0.819206\pi\)
−0.842989 + 0.537931i \(0.819206\pi\)
\(128\) 0 0
\(129\) 13.0000 1.14459
\(130\) 0 0
\(131\) 4.00000 0.349482 0.174741 0.984614i \(-0.444091\pi\)
0.174741 + 0.984614i \(0.444091\pi\)
\(132\) 2.00000 0.174078
\(133\) −1.00000 −0.0867110
\(134\) 0 0
\(135\) 1.00000 0.0860663
\(136\) 0 0
\(137\) −10.0000 −0.854358 −0.427179 0.904167i \(-0.640493\pi\)
−0.427179 + 0.904167i \(0.640493\pi\)
\(138\) 0 0
\(139\) 4.00000 0.339276 0.169638 0.985506i \(-0.445740\pi\)
0.169638 + 0.985506i \(0.445740\pi\)
\(140\) 2.00000 0.169031
\(141\) −6.00000 −0.505291
\(142\) 0 0
\(143\) 4.00000 0.334497
\(144\) 4.00000 0.333333
\(145\) 3.00000 0.249136
\(146\) 0 0
\(147\) 1.00000 0.0824786
\(148\) 24.0000 1.97279
\(149\) 6.00000 0.491539 0.245770 0.969328i \(-0.420959\pi\)
0.245770 + 0.969328i \(0.420959\pi\)
\(150\) 0 0
\(151\) 6.00000 0.488273 0.244137 0.969741i \(-0.421495\pi\)
0.244137 + 0.969741i \(0.421495\pi\)
\(152\) 0 0
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) −6.00000 −0.481932
\(156\) 8.00000 0.640513
\(157\) 21.0000 1.67598 0.837991 0.545684i \(-0.183730\pi\)
0.837991 + 0.545684i \(0.183730\pi\)
\(158\) 0 0
\(159\) 1.00000 0.0793052
\(160\) 0 0
\(161\) 5.00000 0.394055
\(162\) 0 0
\(163\) 4.00000 0.313304 0.156652 0.987654i \(-0.449930\pi\)
0.156652 + 0.987654i \(0.449930\pi\)
\(164\) 4.00000 0.312348
\(165\) −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
\(172\) −26.0000 −1.98248
\(173\) 2.00000 0.152057 0.0760286 0.997106i \(-0.475776\pi\)
0.0760286 + 0.997106i \(0.475776\pi\)
\(174\) 0 0
\(175\) −1.00000 −0.0755929
\(176\) −4.00000 −0.301511
\(177\) −11.0000 −0.826811
\(178\) 0 0
\(179\) 22.0000 1.64436 0.822179 0.569230i \(-0.192758\pi\)
0.822179 + 0.569230i \(0.192758\pi\)
\(180\) −2.00000 −0.149071
\(181\) −2.00000 −0.148659 −0.0743294 0.997234i \(-0.523682\pi\)
−0.0743294 + 0.997234i \(0.523682\pi\)
\(182\) 0 0
\(183\) 5.00000 0.369611
\(184\) 0 0
\(185\) −12.0000 −0.882258
\(186\) 0 0
\(187\) 5.00000 0.365636
\(188\) 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
\(193\) −14.0000 −1.00774 −0.503871 0.863779i \(-0.668091\pi\)
−0.503871 + 0.863779i \(0.668091\pi\)
\(194\) 0 0
\(195\) −4.00000 −0.286446
\(196\) −2.00000 −0.142857
\(197\) −24.0000 −1.70993 −0.854965 0.518686i \(-0.826421\pi\)
−0.854965 + 0.518686i \(0.826421\pi\)
\(198\) 0 0
\(199\) −8.00000 −0.567105 −0.283552 0.958957i \(-0.591513\pi\)
−0.283552 + 0.958957i \(0.591513\pi\)
\(200\) 0 0
\(201\) −10.0000 −0.705346
\(202\) 0 0
\(203\) −3.00000 −0.210559
\(204\) 10.0000 0.700140
\(205\) −2.00000 −0.139686
\(206\) 0 0
\(207\) −5.00000 −0.347524
\(208\) −16.0000 −1.10940
\(209\) −1.00000 −0.0691714
\(210\) 0 0
\(211\) 26.0000 1.78991 0.894957 0.446153i \(-0.147206\pi\)
0.894957 + 0.446153i \(0.147206\pi\)
\(212\) −2.00000 −0.137361
\(213\) −6.00000 −0.411113
\(214\) 0 0
\(215\) 13.0000 0.886593
\(216\) 0 0
\(217\) 6.00000 0.407307
\(218\) 0 0
\(219\) −4.00000 −0.270295
\(220\) 2.00000 0.134840
\(221\) 20.0000 1.34535
\(222\) 0 0
\(223\) −13.0000 −0.870544 −0.435272 0.900299i \(-0.643348\pi\)
−0.435272 + 0.900299i \(0.643348\pi\)
\(224\) 0 0
\(225\) 1.00000 0.0666667
\(226\) 0 0
\(227\) −3.00000 −0.199117 −0.0995585 0.995032i \(-0.531743\pi\)
−0.0995585 + 0.995032i \(0.531743\pi\)
\(228\) −2.00000 −0.132453
\(229\) −14.0000 −0.925146 −0.462573 0.886581i \(-0.653074\pi\)
−0.462573 + 0.886581i \(0.653074\pi\)
\(230\) 0 0
\(231\) 1.00000 0.0657952
\(232\) 0 0
\(233\) 20.0000 1.31024 0.655122 0.755523i \(-0.272617\pi\)
0.655122 + 0.755523i \(0.272617\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 22.0000 1.43208
\(237\) −8.00000 −0.519656
\(238\) 0 0
\(239\) 25.0000 1.61712 0.808558 0.588417i \(-0.200249\pi\)
0.808558 + 0.588417i \(0.200249\pi\)
\(240\) 4.00000 0.258199
\(241\) 2.00000 0.128831 0.0644157 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −10.0000 −0.640184
\(245\) 1.00000 0.0638877
\(246\) 0 0
\(247\) −4.00000 −0.254514
\(248\) 0 0
\(249\) 5.00000 0.316862
\(250\) 0 0
\(251\) −28.0000 −1.76734 −0.883672 0.468106i \(-0.844936\pi\)
−0.883672 + 0.468106i \(0.844936\pi\)
\(252\) 2.00000 0.125988
\(253\) 5.00000 0.314347
\(254\) 0 0
\(255\) −5.00000 −0.313112
\(256\) 16.0000 1.00000
\(257\) 8.00000 0.499026 0.249513 0.968371i \(-0.419729\pi\)
0.249513 + 0.968371i \(0.419729\pi\)
\(258\) 0 0
\(259\) 12.0000 0.745644
\(260\) 8.00000 0.496139
\(261\) 3.00000 0.185695
\(262\) 0 0
\(263\) 4.00000 0.246651 0.123325 0.992366i \(-0.460644\pi\)
0.123325 + 0.992366i \(0.460644\pi\)
\(264\) 0 0
\(265\) 1.00000 0.0614295
\(266\) 0 0
\(267\) 13.0000 0.795587
\(268\) 20.0000 1.22169
\(269\) 17.0000 1.03651 0.518254 0.855227i \(-0.326582\pi\)
0.518254 + 0.855227i \(0.326582\pi\)
\(270\) 0 0
\(271\) −5.00000 −0.303728 −0.151864 0.988401i \(-0.548528\pi\)
−0.151864 + 0.988401i \(0.548528\pi\)
\(272\) −20.0000 −1.21268
\(273\) 4.00000 0.242091
\(274\) 0 0
\(275\) −1.00000 −0.0603023
\(276\) 10.0000 0.601929
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 0 0
\(279\) −6.00000 −0.359211
\(280\) 0 0
\(281\) −18.0000 −1.07379 −0.536895 0.843649i \(-0.680403\pi\)
−0.536895 + 0.843649i \(0.680403\pi\)
\(282\) 0 0
\(283\) −28.0000 −1.66443 −0.832214 0.554455i \(-0.812927\pi\)
−0.832214 + 0.554455i \(0.812927\pi\)
\(284\) 12.0000 0.712069
\(285\) 1.00000 0.0592349
\(286\) 0 0
\(287\) 2.00000 0.118056
\(288\) 0 0
\(289\) 8.00000 0.470588
\(290\) 0 0
\(291\) −19.0000 −1.11380
\(292\) 8.00000 0.468165
\(293\) −13.0000 −0.759468 −0.379734 0.925096i \(-0.623985\pi\)
−0.379734 + 0.925096i \(0.623985\pi\)
\(294\) 0 0
\(295\) −11.0000 −0.640445
\(296\) 0 0
\(297\) −1.00000 −0.0580259
\(298\) 0 0
\(299\) 20.0000 1.15663
\(300\) −2.00000 −0.115470
\(301\) −13.0000 −0.749308
\(302\) 0 0
\(303\) 2.00000 0.114897
\(304\) 4.00000 0.229416
\(305\) 5.00000 0.286299
\(306\) 0 0
\(307\) 14.0000 0.799022 0.399511 0.916728i \(-0.369180\pi\)
0.399511 + 0.916728i \(0.369180\pi\)
\(308\) −2.00000 −0.113961
\(309\) 9.00000 0.511992
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) −9.00000 −0.508710 −0.254355 0.967111i \(-0.581863\pi\)
−0.254355 + 0.967111i \(0.581863\pi\)
\(314\) 0 0
\(315\) −1.00000 −0.0563436
\(316\) 16.0000 0.900070
\(317\) 2.00000 0.112331 0.0561656 0.998421i \(-0.482113\pi\)
0.0561656 + 0.998421i \(0.482113\pi\)
\(318\) 0 0
\(319\) −3.00000 −0.167968
\(320\) −8.00000 −0.447214
\(321\) 18.0000 1.00466
\(322\) 0 0
\(323\) −5.00000 −0.278207
\(324\) −2.00000 −0.111111
\(325\) −4.00000 −0.221880
\(326\) 0 0
\(327\) −10.0000 −0.553001
\(328\) 0 0
\(329\) 6.00000 0.330791
\(330\) 0 0
\(331\) 25.0000 1.37412 0.687062 0.726599i \(-0.258900\pi\)
0.687062 + 0.726599i \(0.258900\pi\)
\(332\) −10.0000 −0.548821
\(333\) −12.0000 −0.657596
\(334\) 0 0
\(335\) −10.0000 −0.546358
\(336\) −4.00000 −0.218218
\(337\) 17.0000 0.926049 0.463025 0.886345i \(-0.346764\pi\)
0.463025 + 0.886345i \(0.346764\pi\)
\(338\) 0 0
\(339\) 9.00000 0.488813
\(340\) 10.0000 0.542326
\(341\) 6.00000 0.324918
\(342\) 0 0
\(343\) −1.00000 −0.0539949
\(344\) 0 0
\(345\) −5.00000 −0.269191
\(346\) 0 0
\(347\) 28.0000 1.50312 0.751559 0.659665i \(-0.229302\pi\)
0.751559 + 0.659665i \(0.229302\pi\)
\(348\) −6.00000 −0.321634
\(349\) 1.00000 0.0535288 0.0267644 0.999642i \(-0.491480\pi\)
0.0267644 + 0.999642i \(0.491480\pi\)
\(350\) 0 0
\(351\) −4.00000 −0.213504
\(352\) 0 0
\(353\) −26.0000 −1.38384 −0.691920 0.721974i \(-0.743235\pi\)
−0.691920 + 0.721974i \(0.743235\pi\)
\(354\) 0 0
\(355\) −6.00000 −0.318447
\(356\) −26.0000 −1.37800
\(357\) 5.00000 0.264628
\(358\) 0 0
\(359\) 1.00000 0.0527780 0.0263890 0.999652i \(-0.491599\pi\)
0.0263890 + 0.999652i \(0.491599\pi\)
\(360\) 0 0
\(361\) −18.0000 −0.947368
\(362\) 0 0
\(363\) 1.00000 0.0524864
\(364\) −8.00000 −0.419314
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) −11.0000 −0.574195 −0.287098 0.957901i \(-0.592690\pi\)
−0.287098 + 0.957901i \(0.592690\pi\)
\(368\) −20.0000 −1.04257
\(369\) −2.00000 −0.104116
\(370\) 0 0
\(371\) −1.00000 −0.0519174
\(372\) 12.0000 0.622171
\(373\) −11.0000 −0.569558 −0.284779 0.958593i \(-0.591920\pi\)
−0.284779 + 0.958593i \(0.591920\pi\)
\(374\) 0 0
\(375\) 1.00000 0.0516398
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 13.0000 0.667765 0.333883 0.942615i \(-0.391641\pi\)
0.333883 + 0.942615i \(0.391641\pi\)
\(380\) −2.00000 −0.102598
\(381\) −19.0000 −0.973399
\(382\) 0 0
\(383\) 24.0000 1.22634 0.613171 0.789950i \(-0.289894\pi\)
0.613171 + 0.789950i \(0.289894\pi\)
\(384\) 0 0
\(385\) 1.00000 0.0509647
\(386\) 0 0
\(387\) 13.0000 0.660827
\(388\) 38.0000 1.92916
\(389\) −12.0000 −0.608424 −0.304212 0.952604i \(-0.598393\pi\)
−0.304212 + 0.952604i \(0.598393\pi\)
\(390\) 0 0
\(391\) 25.0000 1.26430
\(392\) 0 0
\(393\) 4.00000 0.201773
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 2.00000 0.100504
\(397\) −38.0000 −1.90717 −0.953583 0.301131i \(-0.902636\pi\)
−0.953583 + 0.301131i \(0.902636\pi\)
\(398\) 0 0
\(399\) −1.00000 −0.0500626
\(400\) 4.00000 0.200000
\(401\) 22.0000 1.09863 0.549314 0.835616i \(-0.314889\pi\)
0.549314 + 0.835616i \(0.314889\pi\)
\(402\) 0 0
\(403\) 24.0000 1.19553
\(404\) −4.00000 −0.199007
\(405\) 1.00000 0.0496904
\(406\) 0 0
\(407\) 12.0000 0.594818
\(408\) 0 0
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) −10.0000 −0.493264
\(412\) −18.0000 −0.886796
\(413\) 11.0000 0.541275
\(414\) 0 0
\(415\) 5.00000 0.245440
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) 0 0
\(419\) −33.0000 −1.61216 −0.806078 0.591810i \(-0.798414\pi\)
−0.806078 + 0.591810i \(0.798414\pi\)
\(420\) 2.00000 0.0975900
\(421\) 1.00000 0.0487370 0.0243685 0.999703i \(-0.492242\pi\)
0.0243685 + 0.999703i \(0.492242\pi\)
\(422\) 0 0
\(423\) −6.00000 −0.291730
\(424\) 0 0
\(425\) −5.00000 −0.242536
\(426\) 0 0
\(427\) −5.00000 −0.241967
\(428\) −36.0000 −1.74013
\(429\) 4.00000 0.193122
\(430\) 0 0
\(431\) 36.0000 1.73406 0.867029 0.498257i \(-0.166026\pi\)
0.867029 + 0.498257i \(0.166026\pi\)
\(432\) 4.00000 0.192450
\(433\) 10.0000 0.480569 0.240285 0.970702i \(-0.422759\pi\)
0.240285 + 0.970702i \(0.422759\pi\)
\(434\) 0 0
\(435\) 3.00000 0.143839
\(436\) 20.0000 0.957826
\(437\) −5.00000 −0.239182
\(438\) 0 0
\(439\) 21.0000 1.00228 0.501138 0.865368i \(-0.332915\pi\)
0.501138 + 0.865368i \(0.332915\pi\)
\(440\) 0 0
\(441\) 1.00000 0.0476190
\(442\) 0 0
\(443\) 24.0000 1.14027 0.570137 0.821549i \(-0.306890\pi\)
0.570137 + 0.821549i \(0.306890\pi\)
\(444\) 24.0000 1.13899
\(445\) 13.0000 0.616259
\(446\) 0 0
\(447\) 6.00000 0.283790
\(448\) 8.00000 0.377964
\(449\) −6.00000 −0.283158 −0.141579 0.989927i \(-0.545218\pi\)
−0.141579 + 0.989927i \(0.545218\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) −18.0000 −0.846649
\(453\) 6.00000 0.281905
\(454\) 0 0
\(455\) 4.00000 0.187523
\(456\) 0 0
\(457\) −27.0000 −1.26301 −0.631503 0.775373i \(-0.717562\pi\)
−0.631503 + 0.775373i \(0.717562\pi\)
\(458\) 0 0
\(459\) −5.00000 −0.233380
\(460\) 10.0000 0.466252
\(461\) −10.0000 −0.465746 −0.232873 0.972507i \(-0.574813\pi\)
−0.232873 + 0.972507i \(0.574813\pi\)
\(462\) 0 0
\(463\) −22.0000 −1.02243 −0.511213 0.859454i \(-0.670804\pi\)
−0.511213 + 0.859454i \(0.670804\pi\)
\(464\) 12.0000 0.557086
\(465\) −6.00000 −0.278243
\(466\) 0 0
\(467\) −32.0000 −1.48078 −0.740392 0.672176i \(-0.765360\pi\)
−0.740392 + 0.672176i \(0.765360\pi\)
\(468\) 8.00000 0.369800
\(469\) 10.0000 0.461757
\(470\) 0 0
\(471\) 21.0000 0.967629
\(472\) 0 0
\(473\) −13.0000 −0.597741
\(474\) 0 0
\(475\) 1.00000 0.0458831
\(476\) −10.0000 −0.458349
\(477\) 1.00000 0.0457869
\(478\) 0 0
\(479\) 28.0000 1.27935 0.639676 0.768644i \(-0.279068\pi\)
0.639676 + 0.768644i \(0.279068\pi\)
\(480\) 0 0
\(481\) 48.0000 2.18861
\(482\) 0 0
\(483\) 5.00000 0.227508
\(484\) −2.00000 −0.0909091
\(485\) −19.0000 −0.862746
\(486\) 0 0
\(487\) −34.0000 −1.54069 −0.770344 0.637629i \(-0.779915\pi\)
−0.770344 + 0.637629i \(0.779915\pi\)
\(488\) 0 0
\(489\) 4.00000 0.180886
\(490\) 0 0
\(491\) −37.0000 −1.66979 −0.834893 0.550412i \(-0.814471\pi\)
−0.834893 + 0.550412i \(0.814471\pi\)
\(492\) 4.00000 0.180334
\(493\) −15.0000 −0.675566
\(494\) 0 0
\(495\) −1.00000 −0.0449467
\(496\) −24.0000 −1.07763
\(497\) 6.00000 0.269137
\(498\) 0 0
\(499\) −5.00000 −0.223831 −0.111915 0.993718i \(-0.535699\pi\)
−0.111915 + 0.993718i \(0.535699\pi\)
\(500\) −2.00000 −0.0894427
\(501\) 12.0000 0.536120
\(502\) 0 0
\(503\) −21.0000 −0.936344 −0.468172 0.883637i \(-0.655087\pi\)
−0.468172 + 0.883637i \(0.655087\pi\)
\(504\) 0 0
\(505\) 2.00000 0.0889988
\(506\) 0 0
\(507\) 3.00000 0.133235
\(508\) 38.0000 1.68598
\(509\) 11.0000 0.487566 0.243783 0.969830i \(-0.421611\pi\)
0.243783 + 0.969830i \(0.421611\pi\)
\(510\) 0 0
\(511\) 4.00000 0.176950
\(512\) 0 0
\(513\) 1.00000 0.0441511
\(514\) 0 0
\(515\) 9.00000 0.396587
\(516\) −26.0000 −1.14459
\(517\) 6.00000 0.263880
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −21.0000 −0.920027 −0.460013 0.887912i \(-0.652155\pi\)
−0.460013 + 0.887912i \(0.652155\pi\)
\(522\) 0 0
\(523\) −24.0000 −1.04945 −0.524723 0.851273i \(-0.675831\pi\)
−0.524723 + 0.851273i \(0.675831\pi\)
\(524\) −8.00000 −0.349482
\(525\) −1.00000 −0.0436436
\(526\) 0 0
\(527\) 30.0000 1.30682
\(528\) −4.00000 −0.174078
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) −11.0000 −0.477359
\(532\) 2.00000 0.0867110
\(533\) 8.00000 0.346518
\(534\) 0 0
\(535\) 18.0000 0.778208
\(536\) 0 0
\(537\) 22.0000 0.949370
\(538\) 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
\(543\) −2.00000 −0.0858282
\(544\) 0 0
\(545\) −10.0000 −0.428353
\(546\) 0 0
\(547\) 11.0000 0.470326 0.235163 0.971956i \(-0.424438\pi\)
0.235163 + 0.971956i \(0.424438\pi\)
\(548\) 20.0000 0.854358
\(549\) 5.00000 0.213395
\(550\) 0 0
\(551\) 3.00000 0.127804
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) −12.0000 −0.509372
\(556\) −8.00000 −0.339276
\(557\) −42.0000 −1.77960 −0.889799 0.456354i \(-0.849155\pi\)
−0.889799 + 0.456354i \(0.849155\pi\)
\(558\) 0 0
\(559\) −52.0000 −2.19937
\(560\) −4.00000 −0.169031
\(561\) 5.00000 0.211100
\(562\) 0 0
\(563\) −36.0000 −1.51722 −0.758610 0.651546i \(-0.774121\pi\)
−0.758610 + 0.651546i \(0.774121\pi\)
\(564\) 12.0000 0.505291
\(565\) 9.00000 0.378633
\(566\) 0 0
\(567\) −1.00000 −0.0419961
\(568\) 0 0
\(569\) −25.0000 −1.04805 −0.524027 0.851701i \(-0.675571\pi\)
−0.524027 + 0.851701i \(0.675571\pi\)
\(570\) 0 0
\(571\) −20.0000 −0.836974 −0.418487 0.908223i \(-0.637439\pi\)
−0.418487 + 0.908223i \(0.637439\pi\)
\(572\) −8.00000 −0.334497
\(573\) 12.0000 0.501307
\(574\) 0 0
\(575\) −5.00000 −0.208514
\(576\) −8.00000 −0.333333
\(577\) −18.0000 −0.749350 −0.374675 0.927156i \(-0.622246\pi\)
−0.374675 + 0.927156i \(0.622246\pi\)
\(578\) 0 0
\(579\) −14.0000 −0.581820
\(580\) −6.00000 −0.249136
\(581\) −5.00000 −0.207435
\(582\) 0 0
\(583\) −1.00000 −0.0414158
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) 8.00000 0.330195 0.165098 0.986277i \(-0.447206\pi\)
0.165098 + 0.986277i \(0.447206\pi\)
\(588\) −2.00000 −0.0824786
\(589\) −6.00000 −0.247226
\(590\) 0 0
\(591\) −24.0000 −0.987228
\(592\) −48.0000 −1.97279
\(593\) −26.0000 −1.06769 −0.533846 0.845582i \(-0.679254\pi\)
−0.533846 + 0.845582i \(0.679254\pi\)
\(594\) 0 0
\(595\) 5.00000 0.204980
\(596\) −12.0000 −0.491539
\(597\) −8.00000 −0.327418
\(598\) 0 0
\(599\) −34.0000 −1.38920 −0.694601 0.719395i \(-0.744419\pi\)
−0.694601 + 0.719395i \(0.744419\pi\)
\(600\) 0 0
\(601\) −13.0000 −0.530281 −0.265141 0.964210i \(-0.585418\pi\)
−0.265141 + 0.964210i \(0.585418\pi\)
\(602\) 0 0
\(603\) −10.0000 −0.407231
\(604\) −12.0000 −0.488273
\(605\) 1.00000 0.0406558
\(606\) 0 0
\(607\) −38.0000 −1.54237 −0.771186 0.636610i \(-0.780336\pi\)
−0.771186 + 0.636610i \(0.780336\pi\)
\(608\) 0 0
\(609\) −3.00000 −0.121566
\(610\) 0 0
\(611\) 24.0000 0.970936
\(612\) 10.0000 0.404226
\(613\) −26.0000 −1.05013 −0.525065 0.851062i \(-0.675959\pi\)
−0.525065 + 0.851062i \(0.675959\pi\)
\(614\) 0 0
\(615\) −2.00000 −0.0806478
\(616\) 0 0
\(617\) 26.0000 1.04672 0.523360 0.852111i \(-0.324678\pi\)
0.523360 + 0.852111i \(0.324678\pi\)
\(618\) 0 0
\(619\) 10.0000 0.401934 0.200967 0.979598i \(-0.435592\pi\)
0.200967 + 0.979598i \(0.435592\pi\)
\(620\) 12.0000 0.481932
\(621\) −5.00000 −0.200643
\(622\) 0 0
\(623\) −13.0000 −0.520834
\(624\) −16.0000 −0.640513
\(625\) 1.00000 0.0400000
\(626\) 0 0
\(627\) −1.00000 −0.0399362
\(628\) −42.0000 −1.67598
\(629\) 60.0000 2.39236
\(630\) 0 0
\(631\) 13.0000 0.517522 0.258761 0.965941i \(-0.416686\pi\)
0.258761 + 0.965941i \(0.416686\pi\)
\(632\) 0 0
\(633\) 26.0000 1.03341
\(634\) 0 0
\(635\) −19.0000 −0.753992
\(636\) −2.00000 −0.0793052
\(637\) −4.00000 −0.158486
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) −34.0000 −1.34292 −0.671460 0.741041i \(-0.734332\pi\)
−0.671460 + 0.741041i \(0.734332\pi\)
\(642\) 0 0
\(643\) 13.0000 0.512670 0.256335 0.966588i \(-0.417485\pi\)
0.256335 + 0.966588i \(0.417485\pi\)
\(644\) −10.0000 −0.394055
\(645\) 13.0000 0.511875
\(646\) 0 0
\(647\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(648\) 0 0
\(649\) 11.0000 0.431788
\(650\) 0 0
\(651\) 6.00000 0.235159
\(652\) −8.00000 −0.313304
\(653\) 11.0000 0.430463 0.215232 0.976563i \(-0.430949\pi\)
0.215232 + 0.976563i \(0.430949\pi\)
\(654\) 0 0
\(655\) 4.00000 0.156293
\(656\) −8.00000 −0.312348
\(657\) −4.00000 −0.156055
\(658\) 0 0
\(659\) −37.0000 −1.44132 −0.720658 0.693291i \(-0.756160\pi\)
−0.720658 + 0.693291i \(0.756160\pi\)
\(660\) 2.00000 0.0778499
\(661\) 14.0000 0.544537 0.272268 0.962221i \(-0.412226\pi\)
0.272268 + 0.962221i \(0.412226\pi\)
\(662\) 0 0
\(663\) 20.0000 0.776736
\(664\) 0 0
\(665\) −1.00000 −0.0387783
\(666\) 0 0
\(667\) −15.0000 −0.580802
\(668\) −24.0000 −0.928588
\(669\) −13.0000 −0.502609
\(670\) 0 0
\(671\) −5.00000 −0.193023
\(672\) 0 0
\(673\) 11.0000 0.424019 0.212009 0.977268i \(-0.431999\pi\)
0.212009 + 0.977268i \(0.431999\pi\)
\(674\) 0 0
\(675\) 1.00000 0.0384900
\(676\) −6.00000 −0.230769
\(677\) 31.0000 1.19143 0.595713 0.803197i \(-0.296869\pi\)
0.595713 + 0.803197i \(0.296869\pi\)
\(678\) 0 0
\(679\) 19.0000 0.729153
\(680\) 0 0
\(681\) −3.00000 −0.114960
\(682\) 0 0
\(683\) 12.0000 0.459167 0.229584 0.973289i \(-0.426264\pi\)
0.229584 + 0.973289i \(0.426264\pi\)
\(684\) −2.00000 −0.0764719
\(685\) −10.0000 −0.382080
\(686\) 0 0
\(687\) −14.0000 −0.534133
\(688\) 52.0000 1.98248
\(689\) −4.00000 −0.152388
\(690\) 0 0
\(691\) 22.0000 0.836919 0.418460 0.908235i \(-0.362570\pi\)
0.418460 + 0.908235i \(0.362570\pi\)
\(692\) −4.00000 −0.152057
\(693\) 1.00000 0.0379869
\(694\) 0 0
\(695\) 4.00000 0.151729
\(696\) 0 0
\(697\) 10.0000 0.378777
\(698\) 0 0
\(699\) 20.0000 0.756469
\(700\) 2.00000 0.0755929
\(701\) −7.00000 −0.264386 −0.132193 0.991224i \(-0.542202\pi\)
−0.132193 + 0.991224i \(0.542202\pi\)
\(702\) 0 0
\(703\) −12.0000 −0.452589
\(704\) 8.00000 0.301511
\(705\) −6.00000 −0.225973
\(706\) 0 0
\(707\) −2.00000 −0.0752177
\(708\) 22.0000 0.826811
\(709\) 25.0000 0.938895 0.469447 0.882960i \(-0.344453\pi\)
0.469447 + 0.882960i \(0.344453\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) 0 0
\(713\) 30.0000 1.12351
\(714\) 0 0
\(715\) 4.00000 0.149592
\(716\) −44.0000 −1.64436
\(717\) 25.0000 0.933642
\(718\) 0 0
\(719\) −39.0000 −1.45445 −0.727227 0.686397i \(-0.759191\pi\)
−0.727227 + 0.686397i \(0.759191\pi\)
\(720\) 4.00000 0.149071
\(721\) −9.00000 −0.335178
\(722\) 0 0
\(723\) 2.00000 0.0743808
\(724\) 4.00000 0.148659
\(725\) 3.00000 0.111417
\(726\) 0 0
\(727\) −5.00000 −0.185440 −0.0927199 0.995692i \(-0.529556\pi\)
−0.0927199 + 0.995692i \(0.529556\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −65.0000 −2.40411
\(732\) −10.0000 −0.369611
\(733\) −46.0000 −1.69905 −0.849524 0.527549i \(-0.823111\pi\)
−0.849524 + 0.527549i \(0.823111\pi\)
\(734\) 0 0
\(735\) 1.00000 0.0368856
\(736\) 0 0
\(737\) 10.0000 0.368355
\(738\) 0 0
\(739\) −18.0000 −0.662141 −0.331070 0.943606i \(-0.607410\pi\)
−0.331070 + 0.943606i \(0.607410\pi\)
\(740\) 24.0000 0.882258
\(741\) −4.00000 −0.146944
\(742\) 0 0
\(743\) 18.0000 0.660356 0.330178 0.943919i \(-0.392891\pi\)
0.330178 + 0.943919i \(0.392891\pi\)
\(744\) 0 0
\(745\) 6.00000 0.219823
\(746\) 0 0
\(747\) 5.00000 0.182940
\(748\) −10.0000 −0.365636
\(749\) −18.0000 −0.657706
\(750\) 0 0
\(751\) 5.00000 0.182453 0.0912263 0.995830i \(-0.470921\pi\)
0.0912263 + 0.995830i \(0.470921\pi\)
\(752\) −24.0000 −0.875190
\(753\) −28.0000 −1.02038
\(754\) 0 0
\(755\) 6.00000 0.218362
\(756\) 2.00000 0.0727393
\(757\) 10.0000 0.363456 0.181728 0.983349i \(-0.441831\pi\)
0.181728 + 0.983349i \(0.441831\pi\)
\(758\) 0 0
\(759\) 5.00000 0.181489
\(760\) 0 0
\(761\) −34.0000 −1.23250 −0.616250 0.787551i \(-0.711349\pi\)
−0.616250 + 0.787551i \(0.711349\pi\)
\(762\) 0 0
\(763\) 10.0000 0.362024
\(764\) −24.0000 −0.868290
\(765\) −5.00000 −0.180775
\(766\) 0 0
\(767\) 44.0000 1.58875
\(768\) 16.0000 0.577350
\(769\) 5.00000 0.180305 0.0901523 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 8.00000 0.288113
\(772\) 28.0000 1.00774
\(773\) −34.0000 −1.22290 −0.611448 0.791285i \(-0.709412\pi\)
−0.611448 + 0.791285i \(0.709412\pi\)
\(774\) 0 0
\(775\) −6.00000 −0.215526
\(776\) 0 0
\(777\) 12.0000 0.430498
\(778\) 0 0
\(779\) −2.00000 −0.0716574
\(780\) 8.00000 0.286446
\(781\) 6.00000 0.214697
\(782\) 0 0
\(783\) 3.00000 0.107211
\(784\) 4.00000 0.142857
\(785\) 21.0000 0.749522
\(786\) 0 0
\(787\) −36.0000 −1.28326 −0.641631 0.767014i \(-0.721742\pi\)
−0.641631 + 0.767014i \(0.721742\pi\)
\(788\) 48.0000 1.70993
\(789\) 4.00000 0.142404
\(790\) 0 0
\(791\) −9.00000 −0.320003
\(792\) 0 0
\(793\) −20.0000 −0.710221
\(794\) 0 0
\(795\) 1.00000 0.0354663
\(796\) 16.0000 0.567105
\(797\) −26.0000 −0.920967 −0.460484 0.887668i \(-0.652324\pi\)
−0.460484 + 0.887668i \(0.652324\pi\)
\(798\) 0 0
\(799\) 30.0000 1.06132
\(800\) 0 0
\(801\) 13.0000 0.459332
\(802\) 0 0
\(803\) 4.00000 0.141157
\(804\) 20.0000 0.705346
\(805\) 5.00000 0.176227
\(806\) 0 0
\(807\) 17.0000 0.598428
\(808\) 0 0
\(809\) 30.0000 1.05474 0.527372 0.849635i \(-0.323177\pi\)
0.527372 + 0.849635i \(0.323177\pi\)
\(810\) 0 0
\(811\) 20.0000 0.702295 0.351147 0.936320i \(-0.385792\pi\)
0.351147 + 0.936320i \(0.385792\pi\)
\(812\) 6.00000 0.210559
\(813\) −5.00000 −0.175358
\(814\) 0 0
\(815\) 4.00000 0.140114
\(816\) −20.0000 −0.700140
\(817\) 13.0000 0.454812
\(818\) 0 0
\(819\) 4.00000 0.139771
\(820\) 4.00000 0.139686
\(821\) −3.00000 −0.104701 −0.0523504 0.998629i \(-0.516671\pi\)
−0.0523504 + 0.998629i \(0.516671\pi\)
\(822\) 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
\(827\) 20.0000 0.695468 0.347734 0.937593i \(-0.386951\pi\)
0.347734 + 0.937593i \(0.386951\pi\)
\(828\) 10.0000 0.347524
\(829\) 6.00000 0.208389 0.104194 0.994557i \(-0.466774\pi\)
0.104194 + 0.994557i \(0.466774\pi\)
\(830\) 0 0
\(831\) 10.0000 0.346896
\(832\) 32.0000 1.10940
\(833\) −5.00000 −0.173240
\(834\) 0 0
\(835\) 12.0000 0.415277
\(836\) 2.00000 0.0691714
\(837\) −6.00000 −0.207390
\(838\) 0 0
\(839\) −39.0000 −1.34643 −0.673215 0.739447i \(-0.735087\pi\)
−0.673215 + 0.739447i \(0.735087\pi\)
\(840\) 0 0
\(841\) −20.0000 −0.689655
\(842\) 0 0
\(843\) −18.0000 −0.619953
\(844\) −52.0000 −1.78991
\(845\) 3.00000 0.103203
\(846\) 0 0
\(847\) −1.00000 −0.0343604
\(848\) 4.00000 0.137361
\(849\) −28.0000 −0.960958
\(850\) 0 0
\(851\) 60.0000 2.05677
\(852\) 12.0000 0.411113
\(853\) 22.0000 0.753266 0.376633 0.926363i \(-0.377082\pi\)
0.376633 + 0.926363i \(0.377082\pi\)
\(854\) 0 0
\(855\) 1.00000 0.0341993
\(856\) 0 0
\(857\) 42.0000 1.43469 0.717346 0.696717i \(-0.245357\pi\)
0.717346 + 0.696717i \(0.245357\pi\)
\(858\) 0 0
\(859\) 34.0000 1.16007 0.580033 0.814593i \(-0.303040\pi\)
0.580033 + 0.814593i \(0.303040\pi\)
\(860\) −26.0000 −0.886593
\(861\) 2.00000 0.0681598
\(862\) 0 0
\(863\) 3.00000 0.102121 0.0510606 0.998696i \(-0.483740\pi\)
0.0510606 + 0.998696i \(0.483740\pi\)
\(864\) 0 0
\(865\) 2.00000 0.0680020
\(866\) 0 0
\(867\) 8.00000 0.271694
\(868\) −12.0000 −0.407307
\(869\) 8.00000 0.271381
\(870\) 0 0
\(871\) 40.0000 1.35535
\(872\) 0 0
\(873\) −19.0000 −0.643053
\(874\) 0 0
\(875\) −1.00000 −0.0338062
\(876\) 8.00000 0.270295
\(877\) 11.0000 0.371444 0.185722 0.982602i \(-0.440538\pi\)
0.185722 + 0.982602i \(0.440538\pi\)
\(878\) 0 0
\(879\) −13.0000 −0.438479
\(880\) −4.00000 −0.134840
\(881\) −15.0000 −0.505363 −0.252681 0.967550i \(-0.581312\pi\)
−0.252681 + 0.967550i \(0.581312\pi\)
\(882\) 0 0
\(883\) 40.0000 1.34611 0.673054 0.739594i \(-0.264982\pi\)
0.673054 + 0.739594i \(0.264982\pi\)
\(884\) −40.0000 −1.34535
\(885\) −11.0000 −0.369761
\(886\) 0 0
\(887\) 33.0000 1.10803 0.554016 0.832506i \(-0.313095\pi\)
0.554016 + 0.832506i \(0.313095\pi\)
\(888\) 0 0
\(889\) 19.0000 0.637240
\(890\) 0 0
\(891\) −1.00000 −0.0335013
\(892\) 26.0000 0.870544
\(893\) −6.00000 −0.200782
\(894\) 0 0
\(895\) 22.0000 0.735379
\(896\) 0 0
\(897\) 20.0000 0.667781
\(898\) 0 0
\(899\) −18.0000 −0.600334
\(900\) −2.00000 −0.0666667
\(901\) −5.00000 −0.166574
\(902\) 0 0
\(903\) −13.0000 −0.432613
\(904\) 0 0
\(905\) −2.00000 −0.0664822
\(906\) 0 0
\(907\) 16.0000 0.531271 0.265636 0.964073i \(-0.414418\pi\)
0.265636 + 0.964073i \(0.414418\pi\)
\(908\) 6.00000 0.199117
\(909\) 2.00000 0.0663358
\(910\) 0 0
\(911\) 56.0000 1.85536 0.927681 0.373373i \(-0.121799\pi\)
0.927681 + 0.373373i \(0.121799\pi\)
\(912\) 4.00000 0.132453
\(913\) −5.00000 −0.165476
\(914\) 0 0
\(915\) 5.00000 0.165295
\(916\) 28.0000 0.925146
\(917\) −4.00000 −0.132092
\(918\) 0 0
\(919\) −50.0000 −1.64935 −0.824674 0.565608i \(-0.808641\pi\)
−0.824674 + 0.565608i \(0.808641\pi\)
\(920\) 0 0
\(921\) 14.0000 0.461316
\(922\) 0 0
\(923\) 24.0000 0.789970
\(924\) −2.00000 −0.0657952
\(925\) −12.0000 −0.394558
\(926\) 0 0
\(927\) 9.00000 0.295599
\(928\) 0 0
\(929\) 26.0000 0.853032 0.426516 0.904480i \(-0.359741\pi\)
0.426516 + 0.904480i \(0.359741\pi\)
\(930\) 0 0
\(931\) 1.00000 0.0327737
\(932\) −40.0000 −1.31024
\(933\) 8.00000 0.261908
\(934\) 0 0
\(935\) 5.00000 0.163517
\(936\) 0 0
\(937\) 22.0000 0.718709 0.359354 0.933201i \(-0.382997\pi\)
0.359354 + 0.933201i \(0.382997\pi\)
\(938\) 0 0
\(939\) −9.00000 −0.293704
\(940\) 12.0000 0.391397
\(941\) −12.0000 −0.391189 −0.195594 0.980685i \(-0.562664\pi\)
−0.195594 + 0.980685i \(0.562664\pi\)
\(942\) 0 0
\(943\) 10.0000 0.325645
\(944\) −44.0000 −1.43208
\(945\) −1.00000 −0.0325300
\(946\) 0 0
\(947\) 47.0000 1.52729 0.763647 0.645634i \(-0.223407\pi\)
0.763647 + 0.645634i \(0.223407\pi\)
\(948\) 16.0000 0.519656
\(949\) 16.0000 0.519382
\(950\) 0 0
\(951\) 2.00000 0.0648544
\(952\) 0 0
\(953\) 58.0000 1.87880 0.939402 0.342817i \(-0.111381\pi\)
0.939402 + 0.342817i \(0.111381\pi\)
\(954\) 0 0
\(955\) 12.0000 0.388311
\(956\) −50.0000 −1.61712
\(957\) −3.00000 −0.0969762
\(958\) 0 0
\(959\) 10.0000 0.322917
\(960\) −8.00000 −0.258199
\(961\) 5.00000 0.161290
\(962\) 0 0
\(963\) 18.0000 0.580042
\(964\) −4.00000 −0.128831
\(965\) −14.0000 −0.450676
\(966\) 0 0
\(967\) 1.00000 0.0321578 0.0160789 0.999871i \(-0.494882\pi\)
0.0160789 + 0.999871i \(0.494882\pi\)
\(968\) 0 0
\(969\) −5.00000 −0.160623
\(970\) 0 0
\(971\) 57.0000 1.82922 0.914609 0.404341i \(-0.132499\pi\)
0.914609 + 0.404341i \(0.132499\pi\)
\(972\) −2.00000 −0.0641500
\(973\) −4.00000 −0.128234
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) 20.0000 0.640184
\(977\) −3.00000 −0.0959785 −0.0479893 0.998848i \(-0.515281\pi\)
−0.0479893 + 0.998848i \(0.515281\pi\)
\(978\) 0 0
\(979\) −13.0000 −0.415482
\(980\) −2.00000 −0.0638877
\(981\) −10.0000 −0.319275
\(982\) 0 0
\(983\) −6.00000 −0.191370 −0.0956851 0.995412i \(-0.530504\pi\)
−0.0956851 + 0.995412i \(0.530504\pi\)
\(984\) 0 0
\(985\) −24.0000 −0.764704
\(986\) 0 0
\(987\) 6.00000 0.190982
\(988\) 8.00000 0.254514
\(989\) −65.0000 −2.06688
\(990\) 0 0
\(991\) −25.0000 −0.794151 −0.397076 0.917786i \(-0.629975\pi\)
−0.397076 + 0.917786i \(0.629975\pi\)
\(992\) 0 0
\(993\) 25.0000 0.793351
\(994\) 0 0
\(995\) −8.00000 −0.253617
\(996\) −10.0000 −0.316862
\(997\) −26.0000 −0.823428 −0.411714 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 0 0
\(999\) −12.0000 −0.379663
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.i.1.1 1
3.2 odd 2 3465.2.a.h.1.1 1
5.4 even 2 5775.2.a.m.1.1 1
7.6 odd 2 8085.2.a.k.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.i.1.1 1 1.1 even 1 trivial
3465.2.a.h.1.1 1 3.2 odd 2
5775.2.a.m.1.1 1 5.4 even 2
8085.2.a.k.1.1 1 7.6 odd 2