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Properties

Label 9025.2.a.j.1.1
Level $9025$
Weight $2$
Character 9025.1
Self dual yes
Analytic conductor $72.065$
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] = [9025,2,Mod(1,9025)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(9025, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("9025.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 9025 = 5^{2} \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 9025.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: \(72.0649878242\)
Analytic rank: \(1\)
Dimension: \(1\)
Coefficient field: \(\mathbb{Q}\)
Coefficient ring: \(\mathbb{Z}\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Character \(\chi\) \(=\) 9025.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{2} +2.00000 q^{4} +4.00000 q^{7} -3.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{2} +2.00000 q^{4} +4.00000 q^{7} -3.00000 q^{9} -1.00000 q^{11} -2.00000 q^{13} +8.00000 q^{14} -4.00000 q^{16} +2.00000 q^{17} -6.00000 q^{18} -2.00000 q^{22} -6.00000 q^{23} -4.00000 q^{26} +8.00000 q^{28} -9.00000 q^{29} +7.00000 q^{31} -8.00000 q^{32} +4.00000 q^{34} -6.00000 q^{36} +2.00000 q^{37} -2.00000 q^{41} -2.00000 q^{43} -2.00000 q^{44} -12.0000 q^{46} -6.00000 q^{47} +9.00000 q^{49} -4.00000 q^{52} +4.00000 q^{53} -18.0000 q^{58} -9.00000 q^{59} -7.00000 q^{61} +14.0000 q^{62} -12.0000 q^{63} -8.00000 q^{64} -10.0000 q^{67} +4.00000 q^{68} -1.00000 q^{71} -10.0000 q^{73} +4.00000 q^{74} -4.00000 q^{77} -1.00000 q^{79} +9.00000 q^{81} -4.00000 q^{82} +6.00000 q^{83} -4.00000 q^{86} +11.0000 q^{89} -8.00000 q^{91} -12.0000 q^{92} -12.0000 q^{94} -6.00000 q^{97} +18.0000 q^{98} +3.00000 q^{99} +O(q^{100})\)

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 \(-0.250000\pi\)
0.707107 + 0.707107i \(0.250000\pi\)
\(3\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(4\) 2.00000 1.00000
\(5\) 0 0
\(6\) 0 0
\(7\) 4.00000 1.51186 0.755929 − 0.654654i \(-0.227186\pi\)
0.755929 + 0.654654i \(0.227186\pi\)
\(8\) 0 0
\(9\) −3.00000 −1.00000
\(10\) 0 0
\(11\) −1.00000 −0.301511 −0.150756 − 0.988571i \(-0.548171\pi\)
−0.150756 + 0.988571i \(0.548171\pi\)
\(12\) 0 0
\(13\) −2.00000 −0.554700 −0.277350 − 0.960769i \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) 8.00000 2.13809
\(15\) 0 0
\(16\) −4.00000 −1.00000
\(17\) 2.00000 0.485071 0.242536 − 0.970143i \(-0.422021\pi\)
0.242536 + 0.970143i \(0.422021\pi\)
\(18\) −6.00000 −1.41421
\(19\) 0 0
\(20\) 0 0
\(21\) 0 0
\(22\) −2.00000 −0.426401
\(23\) −6.00000 −1.25109 −0.625543 − 0.780189i \(-0.715123\pi\)
−0.625543 + 0.780189i \(0.715123\pi\)
\(24\) 0 0
\(25\) 0 0
\(26\) −4.00000 −0.784465
\(27\) 0 0
\(28\) 8.00000 1.51186
\(29\) −9.00000 −1.67126 −0.835629 − 0.549294i \(-0.814897\pi\)
−0.835629 + 0.549294i \(0.814897\pi\)
\(30\) 0 0
\(31\) 7.00000 1.25724 0.628619 − 0.777714i \(-0.283621\pi\)
0.628619 + 0.777714i \(0.283621\pi\)
\(32\) −8.00000 −1.41421
\(33\) 0 0
\(34\) 4.00000 0.685994
\(35\) 0 0
\(36\) −6.00000 −1.00000
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) −2.00000 −0.312348 −0.156174 − 0.987730i \(-0.549916\pi\)
−0.156174 + 0.987730i \(0.549916\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) −2.00000 −0.301511
\(45\) 0 0
\(46\) −12.0000 −1.76930
\(47\) −6.00000 −0.875190 −0.437595 − 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) 9.00000 1.28571
\(50\) 0 0
\(51\) 0 0
\(52\) −4.00000 −0.554700
\(53\) 4.00000 0.549442 0.274721 − 0.961524i \(-0.411414\pi\)
0.274721 + 0.961524i \(0.411414\pi\)
\(54\) 0 0
\(55\) 0 0
\(56\) 0 0
\(57\) 0 0
\(58\) −18.0000 −2.36352
\(59\) −9.00000 −1.17170 −0.585850 − 0.810419i \(-0.699239\pi\)
−0.585850 + 0.810419i \(0.699239\pi\)
\(60\) 0 0
\(61\) −7.00000 −0.896258 −0.448129 − 0.893969i \(-0.647910\pi\)
−0.448129 + 0.893969i \(0.647910\pi\)
\(62\) 14.0000 1.77800
\(63\) −12.0000 −1.51186
\(64\) −8.00000 −1.00000
\(65\) 0 0
\(66\) 0 0
\(67\) −10.0000 −1.22169 −0.610847 − 0.791748i \(-0.709171\pi\)
−0.610847 + 0.791748i \(0.709171\pi\)
\(68\) 4.00000 0.485071
\(69\) 0 0
\(70\) 0 0
\(71\) −1.00000 −0.118678 −0.0593391 − 0.998238i \(-0.518899\pi\)
−0.0593391 + 0.998238i \(0.518899\pi\)
\(72\) 0 0
\(73\) −10.0000 −1.17041 −0.585206 − 0.810885i \(-0.698986\pi\)
−0.585206 + 0.810885i \(0.698986\pi\)
\(74\) 4.00000 0.464991
\(75\) 0 0
\(76\) 0 0
\(77\) −4.00000 −0.455842
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) 9.00000 1.00000
\(82\) −4.00000 −0.441726
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) 0 0
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) 0 0
\(89\) 11.0000 1.16600 0.582999 − 0.812473i \(-0.301879\pi\)
0.582999 + 0.812473i \(0.301879\pi\)
\(90\) 0 0
\(91\) −8.00000 −0.838628
\(92\) −12.0000 −1.25109
\(93\) 0 0
\(94\) −12.0000 −1.23771
\(95\) 0 0
\(96\) 0 0
\(97\) −6.00000 −0.609208 −0.304604 − 0.952479i \(-0.598524\pi\)
−0.304604 + 0.952479i \(0.598524\pi\)
\(98\) 18.0000 1.81827
\(99\) 3.00000 0.301511
\(100\) 0 0
\(101\) 15.0000 1.49256 0.746278 − 0.665635i \(-0.231839\pi\)
0.746278 + 0.665635i \(0.231839\pi\)
\(102\) 0 0
\(103\) 16.0000 1.57653 0.788263 − 0.615338i \(-0.210980\pi\)
0.788263 + 0.615338i \(0.210980\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) 8.00000 0.777029
\(107\) −10.0000 −0.966736 −0.483368 − 0.875417i \(-0.660587\pi\)
−0.483368 + 0.875417i \(0.660587\pi\)
\(108\) 0 0
\(109\) −15.0000 −1.43674 −0.718370 − 0.695662i \(-0.755111\pi\)
−0.718370 + 0.695662i \(0.755111\pi\)
\(110\) 0 0
\(111\) 0 0
\(112\) −16.0000 −1.51186
\(113\) −12.0000 −1.12887 −0.564433 − 0.825479i \(-0.690905\pi\)
−0.564433 + 0.825479i \(0.690905\pi\)
\(114\) 0 0
\(115\) 0 0
\(116\) −18.0000 −1.67126
\(117\) 6.00000 0.554700
\(118\) −18.0000 −1.65703
\(119\) 8.00000 0.733359
\(120\) 0 0
\(121\) −10.0000 −0.909091
\(122\) −14.0000 −1.26750
\(123\) 0 0
\(124\) 14.0000 1.25724
\(125\) 0 0
\(126\) −24.0000 −2.13809
\(127\) −6.00000 −0.532414 −0.266207 − 0.963916i \(-0.585770\pi\)
−0.266207 + 0.963916i \(0.585770\pi\)
\(128\) 0 0
\(129\) 0 0
\(130\) 0 0
\(131\) 12.0000 1.04844 0.524222 − 0.851581i \(-0.324356\pi\)
0.524222 + 0.851581i \(0.324356\pi\)
\(132\) 0 0
\(133\) 0 0
\(134\) −20.0000 −1.72774
\(135\) 0 0
\(136\) 0 0
\(137\) −12.0000 −1.02523 −0.512615 − 0.858619i \(-0.671323\pi\)
−0.512615 + 0.858619i \(0.671323\pi\)
\(138\) 0 0
\(139\) 20.0000 1.69638 0.848189 − 0.529694i \(-0.177693\pi\)
0.848189 + 0.529694i \(0.177693\pi\)
\(140\) 0 0
\(141\) 0 0
\(142\) −2.00000 −0.167836
\(143\) 2.00000 0.167248
\(144\) 12.0000 1.00000
\(145\) 0 0
\(146\) −20.0000 −1.65521
\(147\) 0 0
\(148\) 4.00000 0.328798
\(149\) −1.00000 −0.0819232 −0.0409616 − 0.999161i \(-0.513042\pi\)
−0.0409616 + 0.999161i \(0.513042\pi\)
\(150\) 0 0
\(151\) −9.00000 −0.732410 −0.366205 − 0.930534i \(-0.619343\pi\)
−0.366205 + 0.930534i \(0.619343\pi\)
\(152\) 0 0
\(153\) −6.00000 −0.485071
\(154\) −8.00000 −0.644658
\(155\) 0 0
\(156\) 0 0
\(157\) −4.00000 −0.319235 −0.159617 − 0.987179i \(-0.551026\pi\)
−0.159617 + 0.987179i \(0.551026\pi\)
\(158\) −2.00000 −0.159111
\(159\) 0 0
\(160\) 0 0
\(161\) −24.0000 −1.89146
\(162\) 18.0000 1.41421
\(163\) −4.00000 −0.313304 −0.156652 − 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) −4.00000 −0.312348
\(165\) 0 0
\(166\) 12.0000 0.931381
\(167\) −12.0000 −0.928588 −0.464294 − 0.885681i \(-0.653692\pi\)
−0.464294 + 0.885681i \(0.653692\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) 0 0
\(172\) −4.00000 −0.304997
\(173\) −24.0000 −1.82469 −0.912343 − 0.409426i \(-0.865729\pi\)
−0.912343 + 0.409426i \(0.865729\pi\)
\(174\) 0 0
\(175\) 0 0
\(176\) 4.00000 0.301511
\(177\) 0 0
\(178\) 22.0000 1.64897
\(179\) −15.0000 −1.12115 −0.560576 − 0.828103i \(-0.689420\pi\)
−0.560576 + 0.828103i \(0.689420\pi\)
\(180\) 0 0
\(181\) 6.00000 0.445976 0.222988 − 0.974821i \(-0.428419\pi\)
0.222988 + 0.974821i \(0.428419\pi\)
\(182\) −16.0000 −1.18600
\(183\) 0 0
\(184\) 0 0
\(185\) 0 0
\(186\) 0 0
\(187\) −2.00000 −0.146254
\(188\) −12.0000 −0.875190
\(189\) 0 0
\(190\) 0 0
\(191\) −3.00000 −0.217072 −0.108536 − 0.994092i \(-0.534616\pi\)
−0.108536 + 0.994092i \(0.534616\pi\)
\(192\) 0 0
\(193\) 16.0000 1.15171 0.575853 − 0.817554i \(-0.304670\pi\)
0.575853 + 0.817554i \(0.304670\pi\)
\(194\) −12.0000 −0.861550
\(195\) 0 0
\(196\) 18.0000 1.28571
\(197\) 18.0000 1.28245 0.641223 − 0.767354i \(-0.278427\pi\)
0.641223 + 0.767354i \(0.278427\pi\)
\(198\) 6.00000 0.426401
\(199\) 13.0000 0.921546 0.460773 − 0.887518i \(-0.347572\pi\)
0.460773 + 0.887518i \(0.347572\pi\)
\(200\) 0 0
\(201\) 0 0
\(202\) 30.0000 2.11079
\(203\) −36.0000 −2.52670
\(204\) 0 0
\(205\) 0 0
\(206\) 32.0000 2.22955
\(207\) 18.0000 1.25109
\(208\) 8.00000 0.554700
\(209\) 0 0
\(210\) 0 0
\(211\) −5.00000 −0.344214 −0.172107 − 0.985078i \(-0.555058\pi\)
−0.172107 + 0.985078i \(0.555058\pi\)
\(212\) 8.00000 0.549442
\(213\) 0 0
\(214\) −20.0000 −1.36717
\(215\) 0 0
\(216\) 0 0
\(217\) 28.0000 1.90076
\(218\) −30.0000 −2.03186
\(219\) 0 0
\(220\) 0 0
\(221\) −4.00000 −0.269069
\(222\) 0 0
\(223\) −2.00000 −0.133930 −0.0669650 − 0.997755i \(-0.521332\pi\)
−0.0669650 + 0.997755i \(0.521332\pi\)
\(224\) −32.0000 −2.13809
\(225\) 0 0
\(226\) −24.0000 −1.59646
\(227\) 14.0000 0.929213 0.464606 − 0.885517i \(-0.346196\pi\)
0.464606 + 0.885517i \(0.346196\pi\)
\(228\) 0 0
\(229\) 17.0000 1.12339 0.561696 − 0.827344i \(-0.310149\pi\)
0.561696 + 0.827344i \(0.310149\pi\)
\(230\) 0 0
\(231\) 0 0
\(232\) 0 0
\(233\) 8.00000 0.524097 0.262049 − 0.965055i \(-0.415602\pi\)
0.262049 + 0.965055i \(0.415602\pi\)
\(234\) 12.0000 0.784465
\(235\) 0 0
\(236\) −18.0000 −1.17170
\(237\) 0 0
\(238\) 16.0000 1.03713
\(239\) −19.0000 −1.22901 −0.614504 − 0.788914i \(-0.710644\pi\)
−0.614504 + 0.788914i \(0.710644\pi\)
\(240\) 0 0
\(241\) −1.00000 −0.0644157 −0.0322078 − 0.999481i \(-0.510254\pi\)
−0.0322078 + 0.999481i \(0.510254\pi\)
\(242\) −20.0000 −1.28565
\(243\) 0 0
\(244\) −14.0000 −0.896258
\(245\) 0 0
\(246\) 0 0
\(247\) 0 0
\(248\) 0 0
\(249\) 0 0
\(250\) 0 0
\(251\) 13.0000 0.820553 0.410276 − 0.911961i \(-0.365432\pi\)
0.410276 + 0.911961i \(0.365432\pi\)
\(252\) −24.0000 −1.51186
\(253\) 6.00000 0.377217
\(254\) −12.0000 −0.752947
\(255\) 0 0
\(256\) 16.0000 1.00000
\(257\) −6.00000 −0.374270 −0.187135 − 0.982334i \(-0.559920\pi\)
−0.187135 + 0.982334i \(0.559920\pi\)
\(258\) 0 0
\(259\) 8.00000 0.497096
\(260\) 0 0
\(261\) 27.0000 1.67126
\(262\) 24.0000 1.48272
\(263\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(264\) 0 0
\(265\) 0 0
\(266\) 0 0
\(267\) 0 0
\(268\) −20.0000 −1.22169
\(269\) −3.00000 −0.182913 −0.0914566 − 0.995809i \(-0.529152\pi\)
−0.0914566 + 0.995809i \(0.529152\pi\)
\(270\) 0 0
\(271\) 3.00000 0.182237 0.0911185 − 0.995840i \(-0.470956\pi\)
0.0911185 + 0.995840i \(0.470956\pi\)
\(272\) −8.00000 −0.485071
\(273\) 0 0
\(274\) −24.0000 −1.44989
\(275\) 0 0
\(276\) 0 0
\(277\) 28.0000 1.68236 0.841178 − 0.540758i \(-0.181862\pi\)
0.841178 + 0.540758i \(0.181862\pi\)
\(278\) 40.0000 2.39904
\(279\) −21.0000 −1.25724
\(280\) 0 0
\(281\) −10.0000 −0.596550 −0.298275 − 0.954480i \(-0.596411\pi\)
−0.298275 + 0.954480i \(0.596411\pi\)
\(282\) 0 0
\(283\) −14.0000 −0.832214 −0.416107 − 0.909316i \(-0.636606\pi\)
−0.416107 + 0.909316i \(0.636606\pi\)
\(284\) −2.00000 −0.118678
\(285\) 0 0
\(286\) 4.00000 0.236525
\(287\) −8.00000 −0.472225
\(288\) 24.0000 1.41421
\(289\) −13.0000 −0.764706
\(290\) 0 0
\(291\) 0 0
\(292\) −20.0000 −1.17041
\(293\) 4.00000 0.233682 0.116841 − 0.993151i \(-0.462723\pi\)
0.116841 + 0.993151i \(0.462723\pi\)
\(294\) 0 0
\(295\) 0 0
\(296\) 0 0
\(297\) 0 0
\(298\) −2.00000 −0.115857
\(299\) 12.0000 0.693978
\(300\) 0 0
\(301\) −8.00000 −0.461112
\(302\) −18.0000 −1.03578
\(303\) 0 0
\(304\) 0 0
\(305\) 0 0
\(306\) −12.0000 −0.685994
\(307\) 16.0000 0.913168 0.456584 − 0.889680i \(-0.349073\pi\)
0.456584 + 0.889680i \(0.349073\pi\)
\(308\) −8.00000 −0.455842
\(309\) 0 0
\(310\) 0 0
\(311\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) −30.0000 −1.69570 −0.847850 − 0.530236i \(-0.822103\pi\)
−0.847850 + 0.530236i \(0.822103\pi\)
\(314\) −8.00000 −0.451466
\(315\) 0 0
\(316\) −2.00000 −0.112509
\(317\) −2.00000 −0.112331 −0.0561656 − 0.998421i \(-0.517887\pi\)
−0.0561656 + 0.998421i \(0.517887\pi\)
\(318\) 0 0
\(319\) 9.00000 0.503903
\(320\) 0 0
\(321\) 0 0
\(322\) −48.0000 −2.67494
\(323\) 0 0
\(324\) 18.0000 1.00000
\(325\) 0 0
\(326\) −8.00000 −0.443079
\(327\) 0 0
\(328\) 0 0
\(329\) −24.0000 −1.32316
\(330\) 0 0
\(331\) −20.0000 −1.09930 −0.549650 − 0.835395i \(-0.685239\pi\)
−0.549650 + 0.835395i \(0.685239\pi\)
\(332\) 12.0000 0.658586
\(333\) −6.00000 −0.328798
\(334\) −24.0000 −1.31322
\(335\) 0 0
\(336\) 0 0
\(337\) 34.0000 1.85210 0.926049 − 0.377403i \(-0.123183\pi\)
0.926049 + 0.377403i \(0.123183\pi\)
\(338\) −18.0000 −0.979071
\(339\) 0 0
\(340\) 0 0
\(341\) −7.00000 −0.379071
\(342\) 0 0
\(343\) 8.00000 0.431959
\(344\) 0 0
\(345\) 0 0
\(346\) −48.0000 −2.58050
\(347\) 12.0000 0.644194 0.322097 − 0.946707i \(-0.395612\pi\)
0.322097 + 0.946707i \(0.395612\pi\)
\(348\) 0 0
\(349\) 14.0000 0.749403 0.374701 − 0.927146i \(-0.377745\pi\)
0.374701 + 0.927146i \(0.377745\pi\)
\(350\) 0 0
\(351\) 0 0
\(352\) 8.00000 0.426401
\(353\) 8.00000 0.425797 0.212899 − 0.977074i \(-0.431710\pi\)
0.212899 + 0.977074i \(0.431710\pi\)
\(354\) 0 0
\(355\) 0 0
\(356\) 22.0000 1.16600
\(357\) 0 0
\(358\) −30.0000 −1.58555
\(359\) −20.0000 −1.05556 −0.527780 − 0.849381i \(-0.676975\pi\)
−0.527780 + 0.849381i \(0.676975\pi\)
\(360\) 0 0
\(361\) 0 0
\(362\) 12.0000 0.630706
\(363\) 0 0
\(364\) −16.0000 −0.838628
\(365\) 0 0
\(366\) 0 0
\(367\) 16.0000 0.835193 0.417597 − 0.908633i \(-0.362873\pi\)
0.417597 + 0.908633i \(0.362873\pi\)
\(368\) 24.0000 1.25109
\(369\) 6.00000 0.312348
\(370\) 0 0
\(371\) 16.0000 0.830679
\(372\) 0 0
\(373\) −12.0000 −0.621336 −0.310668 − 0.950518i \(-0.600553\pi\)
−0.310668 + 0.950518i \(0.600553\pi\)
\(374\) −4.00000 −0.206835
\(375\) 0 0
\(376\) 0 0
\(377\) 18.0000 0.927047
\(378\) 0 0
\(379\) 29.0000 1.48963 0.744815 − 0.667271i \(-0.232538\pi\)
0.744815 + 0.667271i \(0.232538\pi\)
\(380\) 0 0
\(381\) 0 0
\(382\) −6.00000 −0.306987
\(383\) 6.00000 0.306586 0.153293 − 0.988181i \(-0.451012\pi\)
0.153293 + 0.988181i \(0.451012\pi\)
\(384\) 0 0
\(385\) 0 0
\(386\) 32.0000 1.62876
\(387\) 6.00000 0.304997
\(388\) −12.0000 −0.609208
\(389\) −33.0000 −1.67317 −0.836583 − 0.547840i \(-0.815450\pi\)
−0.836583 + 0.547840i \(0.815450\pi\)
\(390\) 0 0
\(391\) −12.0000 −0.606866
\(392\) 0 0
\(393\) 0 0
\(394\) 36.0000 1.81365
\(395\) 0 0
\(396\) 6.00000 0.301511
\(397\) 8.00000 0.401508 0.200754 − 0.979642i \(-0.435661\pi\)
0.200754 + 0.979642i \(0.435661\pi\)
\(398\) 26.0000 1.30326
\(399\) 0 0
\(400\) 0 0
\(401\) −3.00000 −0.149813 −0.0749064 − 0.997191i \(-0.523866\pi\)
−0.0749064 + 0.997191i \(0.523866\pi\)
\(402\) 0 0
\(403\) −14.0000 −0.697390
\(404\) 30.0000 1.49256
\(405\) 0 0
\(406\) −72.0000 −3.57330
\(407\) −2.00000 −0.0991363
\(408\) 0 0
\(409\) 5.00000 0.247234 0.123617 − 0.992330i \(-0.460551\pi\)
0.123617 + 0.992330i \(0.460551\pi\)
\(410\) 0 0
\(411\) 0 0
\(412\) 32.0000 1.57653
\(413\) −36.0000 −1.77144
\(414\) 36.0000 1.76930
\(415\) 0 0
\(416\) 16.0000 0.784465
\(417\) 0 0
\(418\) 0 0
\(419\) 9.00000 0.439679 0.219839 − 0.975536i \(-0.429447\pi\)
0.219839 + 0.975536i \(0.429447\pi\)
\(420\) 0 0
\(421\) 15.0000 0.731055 0.365528 − 0.930800i \(-0.380889\pi\)
0.365528 + 0.930800i \(0.380889\pi\)
\(422\) −10.0000 −0.486792
\(423\) 18.0000 0.875190
\(424\) 0 0
\(425\) 0 0
\(426\) 0 0
\(427\) −28.0000 −1.35501
\(428\) −20.0000 −0.966736
\(429\) 0 0
\(430\) 0 0
\(431\) −21.0000 −1.01153 −0.505767 − 0.862670i \(-0.668791\pi\)
−0.505767 + 0.862670i \(0.668791\pi\)
\(432\) 0 0
\(433\) −4.00000 −0.192228 −0.0961139 − 0.995370i \(-0.530641\pi\)
−0.0961139 + 0.995370i \(0.530641\pi\)
\(434\) 56.0000 2.68809
\(435\) 0 0
\(436\) −30.0000 −1.43674
\(437\) 0 0
\(438\) 0 0
\(439\) 13.0000 0.620456 0.310228 − 0.950662i \(-0.399595\pi\)
0.310228 + 0.950662i \(0.399595\pi\)
\(440\) 0 0
\(441\) −27.0000 −1.28571
\(442\) −8.00000 −0.380521
\(443\) −4.00000 −0.190046 −0.0950229 − 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 0 0
\(445\) 0 0
\(446\) −4.00000 −0.189405
\(447\) 0 0
\(448\) −32.0000 −1.51186
\(449\) −31.0000 −1.46298 −0.731490 − 0.681852i \(-0.761175\pi\)
−0.731490 + 0.681852i \(0.761175\pi\)
\(450\) 0 0
\(451\) 2.00000 0.0941763
\(452\) −24.0000 −1.12887
\(453\) 0 0
\(454\) 28.0000 1.31411
\(455\) 0 0
\(456\) 0 0
\(457\) −34.0000 −1.59045 −0.795226 − 0.606313i \(-0.792648\pi\)
−0.795226 + 0.606313i \(0.792648\pi\)
\(458\) 34.0000 1.58872
\(459\) 0 0
\(460\) 0 0
\(461\) −25.0000 −1.16437 −0.582183 − 0.813058i \(-0.697801\pi\)
−0.582183 + 0.813058i \(0.697801\pi\)
\(462\) 0 0
\(463\) 4.00000 0.185896 0.0929479 − 0.995671i \(-0.470371\pi\)
0.0929479 + 0.995671i \(0.470371\pi\)
\(464\) 36.0000 1.67126
\(465\) 0 0
\(466\) 16.0000 0.741186
\(467\) −10.0000 −0.462745 −0.231372 − 0.972865i \(-0.574322\pi\)
−0.231372 + 0.972865i \(0.574322\pi\)
\(468\) 12.0000 0.554700
\(469\) −40.0000 −1.84703
\(470\) 0 0
\(471\) 0 0
\(472\) 0 0
\(473\) 2.00000 0.0919601
\(474\) 0 0
\(475\) 0 0
\(476\) 16.0000 0.733359
\(477\) −12.0000 −0.549442
\(478\) −38.0000 −1.73808
\(479\) 15.0000 0.685367 0.342684 − 0.939451i \(-0.388664\pi\)
0.342684 + 0.939451i \(0.388664\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −2.00000 −0.0910975
\(483\) 0 0
\(484\) −20.0000 −0.909091
\(485\) 0 0
\(486\) 0 0
\(487\) −38.0000 −1.72194 −0.860972 − 0.508652i \(-0.830144\pi\)
−0.860972 + 0.508652i \(0.830144\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) 13.0000 0.586682 0.293341 − 0.956008i \(-0.405233\pi\)
0.293341 + 0.956008i \(0.405233\pi\)
\(492\) 0 0
\(493\) −18.0000 −0.810679
\(494\) 0 0
\(495\) 0 0
\(496\) −28.0000 −1.25724
\(497\) −4.00000 −0.179425
\(498\) 0 0
\(499\) 28.0000 1.25345 0.626726 − 0.779240i \(-0.284395\pi\)
0.626726 + 0.779240i \(0.284395\pi\)
\(500\) 0 0
\(501\) 0 0
\(502\) 26.0000 1.16044
\(503\) 26.0000 1.15928 0.579641 − 0.814872i \(-0.303193\pi\)
0.579641 + 0.814872i \(0.303193\pi\)
\(504\) 0 0
\(505\) 0 0
\(506\) 12.0000 0.533465
\(507\) 0 0
\(508\) −12.0000 −0.532414
\(509\) −18.0000 −0.797836 −0.398918 − 0.916987i \(-0.630614\pi\)
−0.398918 + 0.916987i \(0.630614\pi\)
\(510\) 0 0
\(511\) −40.0000 −1.76950
\(512\) 32.0000 1.41421
\(513\) 0 0
\(514\) −12.0000 −0.529297
\(515\) 0 0
\(516\) 0 0
\(517\) 6.00000 0.263880
\(518\) 16.0000 0.703000
\(519\) 0 0
\(520\) 0 0
\(521\) −15.0000 −0.657162 −0.328581 − 0.944476i \(-0.606570\pi\)
−0.328581 + 0.944476i \(0.606570\pi\)
\(522\) 54.0000 2.36352
\(523\) 16.0000 0.699631 0.349816 − 0.936819i \(-0.386244\pi\)
0.349816 + 0.936819i \(0.386244\pi\)
\(524\) 24.0000 1.04844
\(525\) 0 0
\(526\) 0 0
\(527\) 14.0000 0.609850
\(528\) 0 0
\(529\) 13.0000 0.565217
\(530\) 0 0
\(531\) 27.0000 1.17170
\(532\) 0 0
\(533\) 4.00000 0.173259
\(534\) 0 0
\(535\) 0 0
\(536\) 0 0
\(537\) 0 0
\(538\) −6.00000 −0.258678
\(539\) −9.00000 −0.387657
\(540\) 0 0
\(541\) 23.0000 0.988847 0.494424 − 0.869221i \(-0.335379\pi\)
0.494424 + 0.869221i \(0.335379\pi\)
\(542\) 6.00000 0.257722
\(543\) 0 0
\(544\) −16.0000 −0.685994
\(545\) 0 0
\(546\) 0 0
\(547\) 22.0000 0.940652 0.470326 − 0.882493i \(-0.344136\pi\)
0.470326 + 0.882493i \(0.344136\pi\)
\(548\) −24.0000 −1.02523
\(549\) 21.0000 0.896258
\(550\) 0 0
\(551\) 0 0
\(552\) 0 0
\(553\) −4.00000 −0.170097
\(554\) 56.0000 2.37921
\(555\) 0 0
\(556\) 40.0000 1.69638
\(557\) −14.0000 −0.593199 −0.296600 − 0.955002i \(-0.595853\pi\)
−0.296600 + 0.955002i \(0.595853\pi\)
\(558\) −42.0000 −1.77800
\(559\) 4.00000 0.169182
\(560\) 0 0
\(561\) 0 0
\(562\) −20.0000 −0.843649
\(563\) −24.0000 −1.01148 −0.505740 − 0.862686i \(-0.668780\pi\)
−0.505740 + 0.862686i \(0.668780\pi\)
\(564\) 0 0
\(565\) 0 0
\(566\) −28.0000 −1.17693
\(567\) 36.0000 1.51186
\(568\) 0 0
\(569\) −15.0000 −0.628833 −0.314416 − 0.949285i \(-0.601809\pi\)
−0.314416 + 0.949285i \(0.601809\pi\)
\(570\) 0 0
\(571\) 13.0000 0.544033 0.272017 − 0.962293i \(-0.412309\pi\)
0.272017 + 0.962293i \(0.412309\pi\)
\(572\) 4.00000 0.167248
\(573\) 0 0
\(574\) −16.0000 −0.667827
\(575\) 0 0
\(576\) 24.0000 1.00000
\(577\) −6.00000 −0.249783 −0.124892 − 0.992170i \(-0.539858\pi\)
−0.124892 + 0.992170i \(0.539858\pi\)
\(578\) −26.0000 −1.08146
\(579\) 0 0
\(580\) 0 0
\(581\) 24.0000 0.995688
\(582\) 0 0
\(583\) −4.00000 −0.165663
\(584\) 0 0
\(585\) 0 0
\(586\) 8.00000 0.330477
\(587\) 24.0000 0.990586 0.495293 − 0.868726i \(-0.335061\pi\)
0.495293 + 0.868726i \(0.335061\pi\)
\(588\) 0 0
\(589\) 0 0
\(590\) 0 0
\(591\) 0 0
\(592\) −8.00000 −0.328798
\(593\) 32.0000 1.31408 0.657041 − 0.753855i \(-0.271808\pi\)
0.657041 + 0.753855i \(0.271808\pi\)
\(594\) 0 0
\(595\) 0 0
\(596\) −2.00000 −0.0819232
\(597\) 0 0
\(598\) 24.0000 0.981433
\(599\) 12.0000 0.490307 0.245153 − 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −17.0000 −0.693444 −0.346722 − 0.937968i \(-0.612705\pi\)
−0.346722 + 0.937968i \(0.612705\pi\)
\(602\) −16.0000 −0.652111
\(603\) 30.0000 1.22169
\(604\) −18.0000 −0.732410
\(605\) 0 0
\(606\) 0 0
\(607\) −14.0000 −0.568242 −0.284121 − 0.958788i \(-0.591702\pi\)
−0.284121 + 0.958788i \(0.591702\pi\)
\(608\) 0 0
\(609\) 0 0
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) −12.0000 −0.485071
\(613\) −24.0000 −0.969351 −0.484675 − 0.874694i \(-0.661062\pi\)
−0.484675 + 0.874694i \(0.661062\pi\)
\(614\) 32.0000 1.29141
\(615\) 0 0
\(616\) 0 0
\(617\) −18.0000 −0.724653 −0.362326 − 0.932051i \(-0.618017\pi\)
−0.362326 + 0.932051i \(0.618017\pi\)
\(618\) 0 0
\(619\) 4.00000 0.160774 0.0803868 − 0.996764i \(-0.474384\pi\)
0.0803868 + 0.996764i \(0.474384\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 44.0000 1.76282
\(624\) 0 0
\(625\) 0 0
\(626\) −60.0000 −2.39808
\(627\) 0 0
\(628\) −8.00000 −0.319235
\(629\) 4.00000 0.159490
\(630\) 0 0
\(631\) 1.00000 0.0398094 0.0199047 − 0.999802i \(-0.493664\pi\)
0.0199047 + 0.999802i \(0.493664\pi\)
\(632\) 0 0
\(633\) 0 0
\(634\) −4.00000 −0.158860
\(635\) 0 0
\(636\) 0 0
\(637\) −18.0000 −0.713186
\(638\) 18.0000 0.712627
\(639\) 3.00000 0.118678
\(640\) 0 0
\(641\) −21.0000 −0.829450 −0.414725 − 0.909947i \(-0.636122\pi\)
−0.414725 + 0.909947i \(0.636122\pi\)
\(642\) 0 0
\(643\) 46.0000 1.81406 0.907031 − 0.421063i \(-0.138343\pi\)
0.907031 + 0.421063i \(0.138343\pi\)
\(644\) −48.0000 −1.89146
\(645\) 0 0
\(646\) 0 0
\(647\) 6.00000 0.235884 0.117942 − 0.993020i \(-0.462370\pi\)
0.117942 + 0.993020i \(0.462370\pi\)
\(648\) 0 0
\(649\) 9.00000 0.353281
\(650\) 0 0
\(651\) 0 0
\(652\) −8.00000 −0.313304
\(653\) −10.0000 −0.391330 −0.195665 − 0.980671i \(-0.562687\pi\)
−0.195665 + 0.980671i \(0.562687\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) 8.00000 0.312348
\(657\) 30.0000 1.17041
\(658\) −48.0000 −1.87123
\(659\) −20.0000 −0.779089 −0.389545 − 0.921008i \(-0.627368\pi\)
−0.389545 + 0.921008i \(0.627368\pi\)
\(660\) 0 0
\(661\) 15.0000 0.583432 0.291716 − 0.956505i \(-0.405774\pi\)
0.291716 + 0.956505i \(0.405774\pi\)
\(662\) −40.0000 −1.55464
\(663\) 0 0
\(664\) 0 0
\(665\) 0 0
\(666\) −12.0000 −0.464991
\(667\) 54.0000 2.09089
\(668\) −24.0000 −0.928588
\(669\) 0 0
\(670\) 0 0
\(671\) 7.00000 0.270232
\(672\) 0 0
\(673\) −20.0000 −0.770943 −0.385472 − 0.922720i \(-0.625961\pi\)
−0.385472 + 0.922720i \(0.625961\pi\)
\(674\) 68.0000 2.61926
\(675\) 0 0
\(676\) −18.0000 −0.692308
\(677\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(678\) 0 0
\(679\) −24.0000 −0.921035
\(680\) 0 0
\(681\) 0 0
\(682\) −14.0000 −0.536088
\(683\) 30.0000 1.14792 0.573959 − 0.818884i \(-0.305407\pi\)
0.573959 + 0.818884i \(0.305407\pi\)
\(684\) 0 0
\(685\) 0 0
\(686\) 16.0000 0.610883
\(687\) 0 0
\(688\) 8.00000 0.304997
\(689\) −8.00000 −0.304776
\(690\) 0 0
\(691\) 3.00000 0.114125 0.0570627 − 0.998371i \(-0.481827\pi\)
0.0570627 + 0.998371i \(0.481827\pi\)
\(692\) −48.0000 −1.82469
\(693\) 12.0000 0.455842
\(694\) 24.0000 0.911028
\(695\) 0 0
\(696\) 0 0
\(697\) −4.00000 −0.151511
\(698\) 28.0000 1.05982
\(699\) 0 0
\(700\) 0 0
\(701\) 6.00000 0.226617 0.113308 − 0.993560i \(-0.463855\pi\)
0.113308 + 0.993560i \(0.463855\pi\)
\(702\) 0 0
\(703\) 0 0
\(704\) 8.00000 0.301511
\(705\) 0 0
\(706\) 16.0000 0.602168
\(707\) 60.0000 2.25653
\(708\) 0 0
\(709\) 25.0000 0.938895 0.469447 − 0.882960i \(-0.344453\pi\)
0.469447 + 0.882960i \(0.344453\pi\)
\(710\) 0 0
\(711\) 3.00000 0.112509
\(712\) 0 0
\(713\) −42.0000 −1.57291
\(714\) 0 0
\(715\) 0 0
\(716\) −30.0000 −1.12115
\(717\) 0 0
\(718\) −40.0000 −1.49279
\(719\) −21.0000 −0.783168 −0.391584 − 0.920142i \(-0.628073\pi\)
−0.391584 + 0.920142i \(0.628073\pi\)
\(720\) 0 0
\(721\) 64.0000 2.38348
\(722\) 0 0
\(723\) 0 0
\(724\) 12.0000 0.445976
\(725\) 0 0
\(726\) 0 0
\(727\) −8.00000 −0.296704 −0.148352 − 0.988935i \(-0.547397\pi\)
−0.148352 + 0.988935i \(0.547397\pi\)
\(728\) 0 0
\(729\) −27.0000 −1.00000
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 0 0
\(733\) 6.00000 0.221615 0.110808 − 0.993842i \(-0.464656\pi\)
0.110808 + 0.993842i \(0.464656\pi\)
\(734\) 32.0000 1.18114
\(735\) 0 0
\(736\) 48.0000 1.76930
\(737\) 10.0000 0.368355
\(738\) 12.0000 0.441726
\(739\) −29.0000 −1.06678 −0.533391 − 0.845869i \(-0.679083\pi\)
−0.533391 + 0.845869i \(0.679083\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 32.0000 1.17476
\(743\) 50.0000 1.83432 0.917161 − 0.398517i \(-0.130475\pi\)
0.917161 + 0.398517i \(0.130475\pi\)
\(744\) 0 0
\(745\) 0 0
\(746\) −24.0000 −0.878702
\(747\) −18.0000 −0.658586
\(748\) −4.00000 −0.146254
\(749\) −40.0000 −1.46157
\(750\) 0 0
\(751\) −41.0000 −1.49611 −0.748056 − 0.663636i \(-0.769012\pi\)
−0.748056 + 0.663636i \(0.769012\pi\)
\(752\) 24.0000 0.875190
\(753\) 0 0
\(754\) 36.0000 1.31104
\(755\) 0 0
\(756\) 0 0
\(757\) −26.0000 −0.944986 −0.472493 − 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) 58.0000 2.10665
\(759\) 0 0
\(760\) 0 0
\(761\) −18.0000 −0.652499 −0.326250 − 0.945284i \(-0.605785\pi\)
−0.326250 + 0.945284i \(0.605785\pi\)
\(762\) 0 0
\(763\) −60.0000 −2.17215
\(764\) −6.00000 −0.217072
\(765\) 0 0
\(766\) 12.0000 0.433578
\(767\) 18.0000 0.649942
\(768\) 0 0
\(769\) 5.00000 0.180305 0.0901523 − 0.995928i \(-0.471265\pi\)
0.0901523 + 0.995928i \(0.471265\pi\)
\(770\) 0 0
\(771\) 0 0
\(772\) 32.0000 1.15171
\(773\) 34.0000 1.22290 0.611448 − 0.791285i \(-0.290588\pi\)
0.611448 + 0.791285i \(0.290588\pi\)
\(774\) 12.0000 0.431331
\(775\) 0 0
\(776\) 0 0
\(777\) 0 0
\(778\) −66.0000 −2.36621
\(779\) 0 0
\(780\) 0 0
\(781\) 1.00000 0.0357828
\(782\) −24.0000 −0.858238
\(783\) 0 0
\(784\) −36.0000 −1.28571
\(785\) 0 0
\(786\) 0 0
\(787\) 22.0000 0.784215 0.392108 − 0.919919i \(-0.371746\pi\)
0.392108 + 0.919919i \(0.371746\pi\)
\(788\) 36.0000 1.28245
\(789\) 0 0
\(790\) 0 0
\(791\) −48.0000 −1.70668
\(792\) 0 0
\(793\) 14.0000 0.497155
\(794\) 16.0000 0.567819
\(795\) 0 0
\(796\) 26.0000 0.921546
\(797\) 24.0000 0.850124 0.425062 − 0.905164i \(-0.360252\pi\)
0.425062 + 0.905164i \(0.360252\pi\)
\(798\) 0 0
\(799\) −12.0000 −0.424529
\(800\) 0 0
\(801\) −33.0000 −1.16600
\(802\) −6.00000 −0.211867
\(803\) 10.0000 0.352892
\(804\) 0 0
\(805\) 0 0
\(806\) −28.0000 −0.986258
\(807\) 0 0
\(808\) 0 0
\(809\) −43.0000 −1.51180 −0.755900 − 0.654687i \(-0.772800\pi\)
−0.755900 + 0.654687i \(0.772800\pi\)
\(810\) 0 0
\(811\) −9.00000 −0.316033 −0.158016 − 0.987436i \(-0.550510\pi\)
−0.158016 + 0.987436i \(0.550510\pi\)
\(812\) −72.0000 −2.52670
\(813\) 0 0
\(814\) −4.00000 −0.140200
\(815\) 0 0
\(816\) 0 0
\(817\) 0 0
\(818\) 10.0000 0.349642
\(819\) 24.0000 0.838628
\(820\) 0 0
\(821\) 21.0000 0.732905 0.366453 − 0.930437i \(-0.380572\pi\)
0.366453 + 0.930437i \(0.380572\pi\)
\(822\) 0 0
\(823\) 30.0000 1.04573 0.522867 − 0.852414i \(-0.324862\pi\)
0.522867 + 0.852414i \(0.324862\pi\)
\(824\) 0 0
\(825\) 0 0
\(826\) −72.0000 −2.50520
\(827\) 6.00000 0.208640 0.104320 − 0.994544i \(-0.466733\pi\)
0.104320 + 0.994544i \(0.466733\pi\)
\(828\) 36.0000 1.25109
\(829\) −2.00000 −0.0694629 −0.0347314 − 0.999397i \(-0.511058\pi\)
−0.0347314 + 0.999397i \(0.511058\pi\)
\(830\) 0 0
\(831\) 0 0
\(832\) 16.0000 0.554700
\(833\) 18.0000 0.623663
\(834\) 0 0
\(835\) 0 0
\(836\) 0 0
\(837\) 0 0
\(838\) 18.0000 0.621800
\(839\) −48.0000 −1.65714 −0.828572 − 0.559883i \(-0.810846\pi\)
−0.828572 + 0.559883i \(0.810846\pi\)
\(840\) 0 0
\(841\) 52.0000 1.79310
\(842\) 30.0000 1.03387
\(843\) 0 0
\(844\) −10.0000 −0.344214
\(845\) 0 0
\(846\) 36.0000 1.23771
\(847\) −40.0000 −1.37442
\(848\) −16.0000 −0.549442
\(849\) 0 0
\(850\) 0 0
\(851\) −12.0000 −0.411355
\(852\) 0 0
\(853\) 16.0000 0.547830 0.273915 − 0.961754i \(-0.411681\pi\)
0.273915 + 0.961754i \(0.411681\pi\)
\(854\) −56.0000 −1.91628
\(855\) 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
\(859\) 1.00000 0.0341196 0.0170598 − 0.999854i \(-0.494569\pi\)
0.0170598 + 0.999854i \(0.494569\pi\)
\(860\) 0 0
\(861\) 0 0
\(862\) −42.0000 −1.43053
\(863\) 20.0000 0.680808 0.340404 − 0.940279i \(-0.389436\pi\)
0.340404 + 0.940279i \(0.389436\pi\)
\(864\) 0 0
\(865\) 0 0
\(866\) −8.00000 −0.271851
\(867\) 0 0
\(868\) 56.0000 1.90076
\(869\) 1.00000 0.0339227
\(870\) 0 0
\(871\) 20.0000 0.677674
\(872\) 0 0
\(873\) 18.0000 0.609208
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −58.0000 −1.95852 −0.979260 − 0.202606i \(-0.935059\pi\)
−0.979260 + 0.202606i \(0.935059\pi\)
\(878\) 26.0000 0.877457
\(879\) 0 0
\(880\) 0 0
\(881\) 37.0000 1.24656 0.623281 − 0.781998i \(-0.285799\pi\)
0.623281 + 0.781998i \(0.285799\pi\)
\(882\) −54.0000 −1.81827
\(883\) 8.00000 0.269221 0.134611 − 0.990899i \(-0.457022\pi\)
0.134611 + 0.990899i \(0.457022\pi\)
\(884\) −8.00000 −0.269069
\(885\) 0 0
\(886\) −8.00000 −0.268765
\(887\) −16.0000 −0.537227 −0.268614 − 0.963248i \(-0.586566\pi\)
−0.268614 + 0.963248i \(0.586566\pi\)
\(888\) 0 0
\(889\) −24.0000 −0.804934
\(890\) 0 0
\(891\) −9.00000 −0.301511
\(892\) −4.00000 −0.133930
\(893\) 0 0
\(894\) 0 0
\(895\) 0 0
\(896\) 0 0
\(897\) 0 0
\(898\) −62.0000 −2.06897
\(899\) −63.0000 −2.10117
\(900\) 0 0
\(901\) 8.00000 0.266519
\(902\) 4.00000 0.133185
\(903\) 0 0
\(904\) 0 0
\(905\) 0 0
\(906\) 0 0
\(907\) 12.0000 0.398453 0.199227 − 0.979953i \(-0.436157\pi\)
0.199227 + 0.979953i \(0.436157\pi\)
\(908\) 28.0000 0.929213
\(909\) −45.0000 −1.49256
\(910\) 0 0
\(911\) −41.0000 −1.35839 −0.679195 − 0.733958i \(-0.737671\pi\)
−0.679195 + 0.733958i \(0.737671\pi\)
\(912\) 0 0
\(913\) −6.00000 −0.198571
\(914\) −68.0000 −2.24924
\(915\) 0 0
\(916\) 34.0000 1.12339
\(917\) 48.0000 1.58510
\(918\) 0 0
\(919\) 16.0000 0.527791 0.263896 − 0.964551i \(-0.414993\pi\)
0.263896 + 0.964551i \(0.414993\pi\)
\(920\) 0 0
\(921\) 0 0
\(922\) −50.0000 −1.64666
\(923\) 2.00000 0.0658308
\(924\) 0 0
\(925\) 0 0
\(926\) 8.00000 0.262896
\(927\) −48.0000 −1.57653
\(928\) 72.0000 2.36352
\(929\) 51.0000 1.67326 0.836628 − 0.547772i \(-0.184524\pi\)
0.836628 + 0.547772i \(0.184524\pi\)
\(930\) 0 0
\(931\) 0 0
\(932\) 16.0000 0.524097
\(933\) 0 0
\(934\) −20.0000 −0.654420
\(935\) 0 0
\(936\) 0 0
\(937\) 56.0000 1.82944 0.914720 − 0.404088i \(-0.132411\pi\)
0.914720 + 0.404088i \(0.132411\pi\)
\(938\) −80.0000 −2.61209
\(939\) 0 0
\(940\) 0 0
\(941\) −35.0000 −1.14097 −0.570484 − 0.821309i \(-0.693244\pi\)
−0.570484 + 0.821309i \(0.693244\pi\)
\(942\) 0 0
\(943\) 12.0000 0.390774
\(944\) 36.0000 1.17170
\(945\) 0 0
\(946\) 4.00000 0.130051
\(947\) 42.0000 1.36482 0.682408 − 0.730971i \(-0.260933\pi\)
0.682408 + 0.730971i \(0.260933\pi\)
\(948\) 0 0
\(949\) 20.0000 0.649227
\(950\) 0 0
\(951\) 0 0
\(952\) 0 0
\(953\) −24.0000 −0.777436 −0.388718 − 0.921357i \(-0.627082\pi\)
−0.388718 + 0.921357i \(0.627082\pi\)
\(954\) −24.0000 −0.777029
\(955\) 0 0
\(956\) −38.0000 −1.22901
\(957\) 0 0
\(958\) 30.0000 0.969256
\(959\) −48.0000 −1.55000
\(960\) 0 0
\(961\) 18.0000 0.580645
\(962\) −8.00000 −0.257930
\(963\) 30.0000 0.966736
\(964\) −2.00000 −0.0644157
\(965\) 0 0
\(966\) 0 0
\(967\) −22.0000 −0.707472 −0.353736 − 0.935345i \(-0.615089\pi\)
−0.353736 + 0.935345i \(0.615089\pi\)
\(968\) 0 0
\(969\) 0 0
\(970\) 0 0
\(971\) −36.0000 −1.15529 −0.577647 − 0.816286i \(-0.696029\pi\)
−0.577647 + 0.816286i \(0.696029\pi\)
\(972\) 0 0
\(973\) 80.0000 2.56468
\(974\) −76.0000 −2.43520
\(975\) 0 0
\(976\) 28.0000 0.896258
\(977\) −44.0000 −1.40768 −0.703842 − 0.710356i \(-0.748534\pi\)
−0.703842 + 0.710356i \(0.748534\pi\)
\(978\) 0 0
\(979\) −11.0000 −0.351562
\(980\) 0 0
\(981\) 45.0000 1.43674
\(982\) 26.0000 0.829693
\(983\) 36.0000 1.14822 0.574111 − 0.818778i \(-0.305348\pi\)
0.574111 + 0.818778i \(0.305348\pi\)
\(984\) 0 0
\(985\) 0 0
\(986\) −36.0000 −1.14647
\(987\) 0 0
\(988\) 0 0
\(989\) 12.0000 0.381578
\(990\) 0 0
\(991\) −44.0000 −1.39771 −0.698853 − 0.715265i \(-0.746306\pi\)
−0.698853 + 0.715265i \(0.746306\pi\)
\(992\) −56.0000 −1.77800
\(993\) 0 0
\(994\) −8.00000 −0.253745
\(995\) 0 0
\(996\) 0 0
\(997\) −26.0000 −0.823428 −0.411714 − 0.911313i \(-0.635070\pi\)
−0.411714 + 0.911313i \(0.635070\pi\)
\(998\) 56.0000 1.77265
\(999\) 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 9025.2.a.j.1.1 1
5.2 odd 4 1805.2.b.a.1084.2 2
5.3 odd 4 1805.2.b.a.1084.1 2
5.4 even 2 9025.2.a.a.1.1 1
19.8 odd 6 475.2.e.c.26.1 2
19.12 odd 6 475.2.e.c.201.1 2
19.18 odd 2 9025.2.a.b.1.1 1
95.8 even 12 95.2.i.a.64.2 yes 4
95.12 even 12 95.2.i.a.49.2 yes 4
95.18 even 4 1805.2.b.b.1084.2 2
95.27 even 12 95.2.i.a.64.1 yes 4
95.37 even 4 1805.2.b.b.1084.1 2
95.69 odd 6 475.2.e.a.201.1 2
95.84 odd 6 475.2.e.a.26.1 2
95.88 even 12 95.2.i.a.49.1 ✓ 4
95.94 odd 2 9025.2.a.i.1.1 1
285.8 odd 12 855.2.be.a.64.1 4
285.107 odd 12 855.2.be.a.334.1 4
285.122 odd 12 855.2.be.a.64.2 4
285.278 odd 12 855.2.be.a.334.2 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.i.a.49.1 ✓ 4 95.88 even 12
95.2.i.a.49.2 yes 4 95.12 even 12
95.2.i.a.64.1 yes 4 95.27 even 12
95.2.i.a.64.2 yes 4 95.8 even 12
475.2.e.a.26.1 2 95.84 odd 6
475.2.e.a.201.1 2 95.69 odd 6
475.2.e.c.26.1 2 19.8 odd 6
475.2.e.c.201.1 2 19.12 odd 6
855.2.be.a.64.1 4 285.8 odd 12
855.2.be.a.64.2 4 285.122 odd 12
855.2.be.a.334.1 4 285.107 odd 12
855.2.be.a.334.2 4 285.278 odd 12
1805.2.b.a.1084.1 2 5.3 odd 4
1805.2.b.a.1084.2 2 5.2 odd 4
1805.2.b.b.1084.1 2 95.37 even 4
1805.2.b.b.1084.2 2 95.18 even 4
9025.2.a.a.1.1 1 5.4 even 2
9025.2.a.b.1.1 1 19.18 odd 2
9025.2.a.i.1.1 1 95.94 odd 2
9025.2.a.j.1.1 1 1.1 even 1 trivial