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Properties

Label 1003.2.a.b.1.1
Level $1003$
Weight $2$
Character 1003.1
Self dual yes
Analytic conductor $8.009$
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] = [1003,2,Mod(1,1003)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1003, 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("1003.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1003 = 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1003.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: \(8.00899532273\)
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 \(\chi\) \(=\) 1003.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.00000 q^{3} -2.00000 q^{4} -2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} +O(q^{10})\) \(q+2.00000 q^{3} -2.00000 q^{4} -2.00000 q^{5} +2.00000 q^{7} +1.00000 q^{9} -5.00000 q^{11} -4.00000 q^{12} +2.00000 q^{13} -4.00000 q^{15} +4.00000 q^{16} -1.00000 q^{17} -5.00000 q^{19} +4.00000 q^{20} +4.00000 q^{21} -5.00000 q^{23} -1.00000 q^{25} -4.00000 q^{27} -4.00000 q^{28} -6.00000 q^{29} +8.00000 q^{31} -10.0000 q^{33} -4.00000 q^{35} -2.00000 q^{36} +2.00000 q^{37} +4.00000 q^{39} -6.00000 q^{41} -2.00000 q^{43} +10.0000 q^{44} -2.00000 q^{45} +6.00000 q^{47} +8.00000 q^{48} -3.00000 q^{49} -2.00000 q^{51} -4.00000 q^{52} -11.0000 q^{53} +10.0000 q^{55} -10.0000 q^{57} -1.00000 q^{59} +8.00000 q^{60} -1.00000 q^{61} +2.00000 q^{63} -8.00000 q^{64} -4.00000 q^{65} -8.00000 q^{67} +2.00000 q^{68} -10.0000 q^{69} -6.00000 q^{71} +13.0000 q^{73} -2.00000 q^{75} +10.0000 q^{76} -10.0000 q^{77} +4.00000 q^{79} -8.00000 q^{80} -11.0000 q^{81} +6.00000 q^{83} -8.00000 q^{84} +2.00000 q^{85} -12.0000 q^{87} +10.0000 q^{89} +4.00000 q^{91} +10.0000 q^{92} +16.0000 q^{93} +10.0000 q^{95} +3.00000 q^{97} -5.00000 q^{99} +O(q^{100})\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(3\) 2.00000 1.15470 0.577350 − 0.816497i \(-0.304087\pi\)
0.577350 + 0.816497i \(0.304087\pi\)
\(4\) −2.00000 −1.00000
\(5\) −2.00000 −0.894427 −0.447214 − 0.894427i \(-0.647584\pi\)
−0.447214 + 0.894427i \(0.647584\pi\)
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 − 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 0 0
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −5.00000 −1.50756 −0.753778 − 0.657129i \(-0.771771\pi\)
−0.753778 + 0.657129i \(0.771771\pi\)
\(12\) −4.00000 −1.15470
\(13\) 2.00000 0.554700 0.277350 − 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) 0 0
\(15\) −4.00000 −1.03280
\(16\) 4.00000 1.00000
\(17\) −1.00000 −0.242536
\(18\) 0 0
\(19\) −5.00000 −1.14708 −0.573539 − 0.819178i \(-0.694430\pi\)
−0.573539 + 0.819178i \(0.694430\pi\)
\(20\) 4.00000 0.894427
\(21\) 4.00000 0.872872
\(22\) 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
\(27\) −4.00000 −0.769800
\(28\) −4.00000 −0.755929
\(29\) −6.00000 −1.11417 −0.557086 − 0.830455i \(-0.688081\pi\)
−0.557086 + 0.830455i \(0.688081\pi\)
\(30\) 0 0
\(31\) 8.00000 1.43684 0.718421 − 0.695608i \(-0.244865\pi\)
0.718421 + 0.695608i \(0.244865\pi\)
\(32\) 0 0
\(33\) −10.0000 −1.74078
\(34\) 0 0
\(35\) −4.00000 −0.676123
\(36\) −2.00000 −0.333333
\(37\) 2.00000 0.328798 0.164399 − 0.986394i \(-0.447432\pi\)
0.164399 + 0.986394i \(0.447432\pi\)
\(38\) 0 0
\(39\) 4.00000 0.640513
\(40\) 0 0
\(41\) −6.00000 −0.937043 −0.468521 − 0.883452i \(-0.655213\pi\)
−0.468521 + 0.883452i \(0.655213\pi\)
\(42\) 0 0
\(43\) −2.00000 −0.304997 −0.152499 − 0.988304i \(-0.548732\pi\)
−0.152499 + 0.988304i \(0.548732\pi\)
\(44\) 10.0000 1.50756
\(45\) −2.00000 −0.298142
\(46\) 0 0
\(47\) 6.00000 0.875190 0.437595 − 0.899172i \(-0.355830\pi\)
0.437595 + 0.899172i \(0.355830\pi\)
\(48\) 8.00000 1.15470
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −2.00000 −0.280056
\(52\) −4.00000 −0.554700
\(53\) −11.0000 −1.51097 −0.755483 − 0.655168i \(-0.772598\pi\)
−0.755483 + 0.655168i \(0.772598\pi\)
\(54\) 0 0
\(55\) 10.0000 1.34840
\(56\) 0 0
\(57\) −10.0000 −1.32453
\(58\) 0 0
\(59\) −1.00000 −0.130189
\(60\) 8.00000 1.03280
\(61\) −1.00000 −0.128037 −0.0640184 − 0.997949i \(-0.520392\pi\)
−0.0640184 + 0.997949i \(0.520392\pi\)
\(62\) 0 0
\(63\) 2.00000 0.251976
\(64\) −8.00000 −1.00000
\(65\) −4.00000 −0.496139
\(66\) 0 0
\(67\) −8.00000 −0.977356 −0.488678 − 0.872464i \(-0.662521\pi\)
−0.488678 + 0.872464i \(0.662521\pi\)
\(68\) 2.00000 0.242536
\(69\) −10.0000 −1.20386
\(70\) 0 0
\(71\) −6.00000 −0.712069 −0.356034 − 0.934473i \(-0.615871\pi\)
−0.356034 + 0.934473i \(0.615871\pi\)
\(72\) 0 0
\(73\) 13.0000 1.52153 0.760767 − 0.649025i \(-0.224823\pi\)
0.760767 + 0.649025i \(0.224823\pi\)
\(74\) 0 0
\(75\) −2.00000 −0.230940
\(76\) 10.0000 1.14708
\(77\) −10.0000 −1.13961
\(78\) 0 0
\(79\) 4.00000 0.450035 0.225018 − 0.974355i \(-0.427756\pi\)
0.225018 + 0.974355i \(0.427756\pi\)
\(80\) −8.00000 −0.894427
\(81\) −11.0000 −1.22222
\(82\) 0 0
\(83\) 6.00000 0.658586 0.329293 − 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) −8.00000 −0.872872
\(85\) 2.00000 0.216930
\(86\) 0 0
\(87\) −12.0000 −1.28654
\(88\) 0 0
\(89\) 10.0000 1.06000 0.529999 − 0.847998i \(-0.322192\pi\)
0.529999 + 0.847998i \(0.322192\pi\)
\(90\) 0 0
\(91\) 4.00000 0.419314
\(92\) 10.0000 1.04257
\(93\) 16.0000 1.65912
\(94\) 0 0
\(95\) 10.0000 1.02598
\(96\) 0 0
\(97\) 3.00000 0.304604 0.152302 − 0.988334i \(-0.451331\pi\)
0.152302 + 0.988334i \(0.451331\pi\)
\(98\) 0 0
\(99\) −5.00000 −0.502519
\(100\) 2.00000 0.200000
\(101\) −2.00000 −0.199007 −0.0995037 − 0.995037i \(-0.531726\pi\)
−0.0995037 + 0.995037i \(0.531726\pi\)
\(102\) 0 0
\(103\) −6.00000 −0.591198 −0.295599 − 0.955312i \(-0.595519\pi\)
−0.295599 + 0.955312i \(0.595519\pi\)
\(104\) 0 0
\(105\) −8.00000 −0.780720
\(106\) 0 0
\(107\) 6.00000 0.580042 0.290021 − 0.957020i \(-0.406338\pi\)
0.290021 + 0.957020i \(0.406338\pi\)
\(108\) 8.00000 0.769800
\(109\) −7.00000 −0.670478 −0.335239 − 0.942133i \(-0.608817\pi\)
−0.335239 + 0.942133i \(0.608817\pi\)
\(110\) 0 0
\(111\) 4.00000 0.379663
\(112\) 8.00000 0.755929
\(113\) 6.00000 0.564433 0.282216 − 0.959351i \(-0.408930\pi\)
0.282216 + 0.959351i \(0.408930\pi\)
\(114\) 0 0
\(115\) 10.0000 0.932505
\(116\) 12.0000 1.11417
\(117\) 2.00000 0.184900
\(118\) 0 0
\(119\) −2.00000 −0.183340
\(120\) 0 0
\(121\) 14.0000 1.27273
\(122\) 0 0
\(123\) −12.0000 −1.08200
\(124\) −16.0000 −1.43684
\(125\) 12.0000 1.07331
\(126\) 0 0
\(127\) 16.0000 1.41977 0.709885 − 0.704317i \(-0.248747\pi\)
0.709885 + 0.704317i \(0.248747\pi\)
\(128\) 0 0
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −4.00000 −0.349482 −0.174741 − 0.984614i \(-0.555909\pi\)
−0.174741 + 0.984614i \(0.555909\pi\)
\(132\) 20.0000 1.74078
\(133\) −10.0000 −0.867110
\(134\) 0 0
\(135\) 8.00000 0.688530
\(136\) 0 0
\(137\) 9.00000 0.768922 0.384461 − 0.923141i \(-0.374387\pi\)
0.384461 + 0.923141i \(0.374387\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 − 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 8.00000 0.676123
\(141\) 12.0000 1.01058
\(142\) 0 0
\(143\) −10.0000 −0.836242
\(144\) 4.00000 0.333333
\(145\) 12.0000 0.996546
\(146\) 0 0
\(147\) −6.00000 −0.494872
\(148\) −4.00000 −0.328798
\(149\) −4.00000 −0.327693 −0.163846 − 0.986486i \(-0.552390\pi\)
−0.163846 + 0.986486i \(0.552390\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 − 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 0 0
\(153\) −1.00000 −0.0808452
\(154\) 0 0
\(155\) −16.0000 −1.28515
\(156\) −8.00000 −0.640513
\(157\) 10.0000 0.798087 0.399043 − 0.916932i \(-0.369342\pi\)
0.399043 + 0.916932i \(0.369342\pi\)
\(158\) 0 0
\(159\) −22.0000 −1.74471
\(160\) 0 0
\(161\) −10.0000 −0.788110
\(162\) 0 0
\(163\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(164\) 12.0000 0.937043
\(165\) 20.0000 1.55700
\(166\) 0 0
\(167\) 12.0000 0.928588 0.464294 − 0.885681i \(-0.346308\pi\)
0.464294 + 0.885681i \(0.346308\pi\)
\(168\) 0 0
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −5.00000 −0.382360
\(172\) 4.00000 0.304997
\(173\) 1.00000 0.0760286 0.0380143 − 0.999277i \(-0.487897\pi\)
0.0380143 + 0.999277i \(0.487897\pi\)
\(174\) 0 0
\(175\) −2.00000 −0.151186
\(176\) −20.0000 −1.50756
\(177\) −2.00000 −0.150329
\(178\) 0 0
\(179\) −18.0000 −1.34538 −0.672692 − 0.739923i \(-0.734862\pi\)
−0.672692 + 0.739923i \(0.734862\pi\)
\(180\) 4.00000 0.298142
\(181\) 22.0000 1.63525 0.817624 − 0.575753i \(-0.195291\pi\)
0.817624 + 0.575753i \(0.195291\pi\)
\(182\) 0 0
\(183\) −2.00000 −0.147844
\(184\) 0 0
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) 5.00000 0.365636
\(188\) −12.0000 −0.875190
\(189\) −8.00000 −0.581914
\(190\) 0 0
\(191\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(192\) −16.0000 −1.15470
\(193\) −8.00000 −0.575853 −0.287926 − 0.957653i \(-0.592966\pi\)
−0.287926 + 0.957653i \(0.592966\pi\)
\(194\) 0 0
\(195\) −8.00000 −0.572892
\(196\) 6.00000 0.428571
\(197\) 26.0000 1.85242 0.926212 − 0.377004i \(-0.123046\pi\)
0.926212 + 0.377004i \(0.123046\pi\)
\(198\) 0 0
\(199\) −24.0000 −1.70131 −0.850657 − 0.525720i \(-0.823796\pi\)
−0.850657 + 0.525720i \(0.823796\pi\)
\(200\) 0 0
\(201\) −16.0000 −1.12855
\(202\) 0 0
\(203\) −12.0000 −0.842235
\(204\) 4.00000 0.280056
\(205\) 12.0000 0.838116
\(206\) 0 0
\(207\) −5.00000 −0.347524
\(208\) 8.00000 0.554700
\(209\) 25.0000 1.72929
\(210\) 0 0
\(211\) −23.0000 −1.58339 −0.791693 − 0.610920i \(-0.790800\pi\)
−0.791693 + 0.610920i \(0.790800\pi\)
\(212\) 22.0000 1.51097
\(213\) −12.0000 −0.822226
\(214\) 0 0
\(215\) 4.00000 0.272798
\(216\) 0 0
\(217\) 16.0000 1.08615
\(218\) 0 0
\(219\) 26.0000 1.75692
\(220\) −20.0000 −1.34840
\(221\) −2.00000 −0.134535
\(222\) 0 0
\(223\) 9.00000 0.602685 0.301342 − 0.953516i \(-0.402565\pi\)
0.301342 + 0.953516i \(0.402565\pi\)
\(224\) 0 0
\(225\) −1.00000 −0.0666667
\(226\) 0 0
\(227\) −8.00000 −0.530979 −0.265489 − 0.964114i \(-0.585534\pi\)
−0.265489 + 0.964114i \(0.585534\pi\)
\(228\) 20.0000 1.32453
\(229\) −20.0000 −1.32164 −0.660819 − 0.750546i \(-0.729791\pi\)
−0.660819 + 0.750546i \(0.729791\pi\)
\(230\) 0 0
\(231\) −20.0000 −1.31590
\(232\) 0 0
\(233\) −25.0000 −1.63780 −0.818902 − 0.573933i \(-0.805417\pi\)
−0.818902 + 0.573933i \(0.805417\pi\)
\(234\) 0 0
\(235\) −12.0000 −0.782794
\(236\) 2.00000 0.130189
\(237\) 8.00000 0.519656
\(238\) 0 0
\(239\) −15.0000 −0.970269 −0.485135 − 0.874439i \(-0.661229\pi\)
−0.485135 + 0.874439i \(0.661229\pi\)
\(240\) −16.0000 −1.03280
\(241\) 2.00000 0.128831 0.0644157 − 0.997923i \(-0.479482\pi\)
0.0644157 + 0.997923i \(0.479482\pi\)
\(242\) 0 0
\(243\) −10.0000 −0.641500
\(244\) 2.00000 0.128037
\(245\) 6.00000 0.383326
\(246\) 0 0
\(247\) −10.0000 −0.636285
\(248\) 0 0
\(249\) 12.0000 0.760469
\(250\) 0 0
\(251\) 27.0000 1.70422 0.852112 − 0.523359i \(-0.175321\pi\)
0.852112 + 0.523359i \(0.175321\pi\)
\(252\) −4.00000 −0.251976
\(253\) 25.0000 1.57174
\(254\) 0 0
\(255\) 4.00000 0.250490
\(256\) 16.0000 1.00000
\(257\) 17.0000 1.06043 0.530215 − 0.847863i \(-0.322111\pi\)
0.530215 + 0.847863i \(0.322111\pi\)
\(258\) 0 0
\(259\) 4.00000 0.248548
\(260\) 8.00000 0.496139
\(261\) −6.00000 −0.371391
\(262\) 0 0
\(263\) 27.0000 1.66489 0.832446 − 0.554107i \(-0.186940\pi\)
0.832446 + 0.554107i \(0.186940\pi\)
\(264\) 0 0
\(265\) 22.0000 1.35145
\(266\) 0 0
\(267\) 20.0000 1.22398
\(268\) 16.0000 0.977356
\(269\) 5.00000 0.304855 0.152428 − 0.988315i \(-0.451291\pi\)
0.152428 + 0.988315i \(0.451291\pi\)
\(270\) 0 0
\(271\) −9.00000 −0.546711 −0.273356 − 0.961913i \(-0.588134\pi\)
−0.273356 + 0.961913i \(0.588134\pi\)
\(272\) −4.00000 −0.242536
\(273\) 8.00000 0.484182
\(274\) 0 0
\(275\) 5.00000 0.301511
\(276\) 20.0000 1.20386
\(277\) −26.0000 −1.56219 −0.781094 − 0.624413i \(-0.785338\pi\)
−0.781094 + 0.624413i \(0.785338\pi\)
\(278\) 0 0
\(279\) 8.00000 0.478947
\(280\) 0 0
\(281\) −3.00000 −0.178965 −0.0894825 − 0.995988i \(-0.528521\pi\)
−0.0894825 + 0.995988i \(0.528521\pi\)
\(282\) 0 0
\(283\) 16.0000 0.951101 0.475551 − 0.879688i \(-0.342249\pi\)
0.475551 + 0.879688i \(0.342249\pi\)
\(284\) 12.0000 0.712069
\(285\) 20.0000 1.18470
\(286\) 0 0
\(287\) −12.0000 −0.708338
\(288\) 0 0
\(289\) 1.00000 0.0588235
\(290\) 0 0
\(291\) 6.00000 0.351726
\(292\) −26.0000 −1.52153
\(293\) −11.0000 −0.642627 −0.321313 − 0.946973i \(-0.604124\pi\)
−0.321313 + 0.946973i \(0.604124\pi\)
\(294\) 0 0
\(295\) 2.00000 0.116445
\(296\) 0 0
\(297\) 20.0000 1.16052
\(298\) 0 0
\(299\) −10.0000 −0.578315
\(300\) 4.00000 0.230940
\(301\) −4.00000 −0.230556
\(302\) 0 0
\(303\) −4.00000 −0.229794
\(304\) −20.0000 −1.14708
\(305\) 2.00000 0.114520
\(306\) 0 0
\(307\) 25.0000 1.42683 0.713413 − 0.700744i \(-0.247149\pi\)
0.713413 + 0.700744i \(0.247149\pi\)
\(308\) 20.0000 1.13961
\(309\) −12.0000 −0.682656
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 − 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 0 0
\(313\) 6.00000 0.339140 0.169570 − 0.985518i \(-0.445762\pi\)
0.169570 + 0.985518i \(0.445762\pi\)
\(314\) 0 0
\(315\) −4.00000 −0.225374
\(316\) −8.00000 −0.450035
\(317\) −32.0000 −1.79730 −0.898650 − 0.438667i \(-0.855451\pi\)
−0.898650 + 0.438667i \(0.855451\pi\)
\(318\) 0 0
\(319\) 30.0000 1.67968
\(320\) 16.0000 0.894427
\(321\) 12.0000 0.669775
\(322\) 0 0
\(323\) 5.00000 0.278207
\(324\) 22.0000 1.22222
\(325\) −2.00000 −0.110940
\(326\) 0 0
\(327\) −14.0000 −0.774202
\(328\) 0 0
\(329\) 12.0000 0.661581
\(330\) 0 0
\(331\) −8.00000 −0.439720 −0.219860 − 0.975531i \(-0.570560\pi\)
−0.219860 + 0.975531i \(0.570560\pi\)
\(332\) −12.0000 −0.658586
\(333\) 2.00000 0.109599
\(334\) 0 0
\(335\) 16.0000 0.874173
\(336\) 16.0000 0.872872
\(337\) 27.0000 1.47078 0.735392 − 0.677642i \(-0.236998\pi\)
0.735392 + 0.677642i \(0.236998\pi\)
\(338\) 0 0
\(339\) 12.0000 0.651751
\(340\) −4.00000 −0.216930
\(341\) −40.0000 −2.16612
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 0 0
\(345\) 20.0000 1.07676
\(346\) 0 0
\(347\) −17.0000 −0.912608 −0.456304 − 0.889824i \(-0.650827\pi\)
−0.456304 + 0.889824i \(0.650827\pi\)
\(348\) 24.0000 1.28654
\(349\) −26.0000 −1.39175 −0.695874 − 0.718164i \(-0.744983\pi\)
−0.695874 + 0.718164i \(0.744983\pi\)
\(350\) 0 0
\(351\) −8.00000 −0.427008
\(352\) 0 0
\(353\) −4.00000 −0.212899 −0.106449 − 0.994318i \(-0.533948\pi\)
−0.106449 + 0.994318i \(0.533948\pi\)
\(354\) 0 0
\(355\) 12.0000 0.636894
\(356\) −20.0000 −1.06000
\(357\) −4.00000 −0.211702
\(358\) 0 0
\(359\) −25.0000 −1.31945 −0.659725 − 0.751507i \(-0.729327\pi\)
−0.659725 + 0.751507i \(0.729327\pi\)
\(360\) 0 0
\(361\) 6.00000 0.315789
\(362\) 0 0
\(363\) 28.0000 1.46962
\(364\) −8.00000 −0.419314
\(365\) −26.0000 −1.36090
\(366\) 0 0
\(367\) 13.0000 0.678594 0.339297 − 0.940679i \(-0.389811\pi\)
0.339297 + 0.940679i \(0.389811\pi\)
\(368\) −20.0000 −1.04257
\(369\) −6.00000 −0.312348
\(370\) 0 0
\(371\) −22.0000 −1.14218
\(372\) −32.0000 −1.65912
\(373\) −7.00000 −0.362446 −0.181223 − 0.983442i \(-0.558006\pi\)
−0.181223 + 0.983442i \(0.558006\pi\)
\(374\) 0 0
\(375\) 24.0000 1.23935
\(376\) 0 0
\(377\) −12.0000 −0.618031
\(378\) 0 0
\(379\) 34.0000 1.74646 0.873231 − 0.487306i \(-0.162020\pi\)
0.873231 + 0.487306i \(0.162020\pi\)
\(380\) −20.0000 −1.02598
\(381\) 32.0000 1.63941
\(382\) 0 0
\(383\) 1.00000 0.0510976 0.0255488 − 0.999674i \(-0.491867\pi\)
0.0255488 + 0.999674i \(0.491867\pi\)
\(384\) 0 0
\(385\) 20.0000 1.01929
\(386\) 0 0
\(387\) −2.00000 −0.101666
\(388\) −6.00000 −0.304604
\(389\) 6.00000 0.304212 0.152106 − 0.988364i \(-0.451394\pi\)
0.152106 + 0.988364i \(0.451394\pi\)
\(390\) 0 0
\(391\) 5.00000 0.252861
\(392\) 0 0
\(393\) −8.00000 −0.403547
\(394\) 0 0
\(395\) −8.00000 −0.402524
\(396\) 10.0000 0.502519
\(397\) −7.00000 −0.351320 −0.175660 − 0.984451i \(-0.556206\pi\)
−0.175660 + 0.984451i \(0.556206\pi\)
\(398\) 0 0
\(399\) −20.0000 −1.00125
\(400\) −4.00000 −0.200000
\(401\) 1.00000 0.0499376 0.0249688 − 0.999688i \(-0.492051\pi\)
0.0249688 + 0.999688i \(0.492051\pi\)
\(402\) 0 0
\(403\) 16.0000 0.797017
\(404\) 4.00000 0.199007
\(405\) 22.0000 1.09319
\(406\) 0 0
\(407\) −10.0000 −0.495682
\(408\) 0 0
\(409\) −22.0000 −1.08783 −0.543915 − 0.839140i \(-0.683059\pi\)
−0.543915 + 0.839140i \(0.683059\pi\)
\(410\) 0 0
\(411\) 18.0000 0.887875
\(412\) 12.0000 0.591198
\(413\) −2.00000 −0.0984136
\(414\) 0 0
\(415\) −12.0000 −0.589057
\(416\) 0 0
\(417\) −8.00000 −0.391762
\(418\) 0 0
\(419\) −17.0000 −0.830504 −0.415252 − 0.909706i \(-0.636307\pi\)
−0.415252 + 0.909706i \(0.636307\pi\)
\(420\) 16.0000 0.780720
\(421\) 10.0000 0.487370 0.243685 − 0.969854i \(-0.421644\pi\)
0.243685 + 0.969854i \(0.421644\pi\)
\(422\) 0 0
\(423\) 6.00000 0.291730
\(424\) 0 0
\(425\) 1.00000 0.0485071
\(426\) 0 0
\(427\) −2.00000 −0.0967868
\(428\) −12.0000 −0.580042
\(429\) −20.0000 −0.965609
\(430\) 0 0
\(431\) −17.0000 −0.818861 −0.409431 − 0.912341i \(-0.634273\pi\)
−0.409431 + 0.912341i \(0.634273\pi\)
\(432\) −16.0000 −0.769800
\(433\) 11.0000 0.528626 0.264313 − 0.964437i \(-0.414855\pi\)
0.264313 + 0.964437i \(0.414855\pi\)
\(434\) 0 0
\(435\) 24.0000 1.15071
\(436\) 14.0000 0.670478
\(437\) 25.0000 1.19591
\(438\) 0 0
\(439\) 18.0000 0.859093 0.429547 − 0.903045i \(-0.358673\pi\)
0.429547 + 0.903045i \(0.358673\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) 0 0
\(443\) −8.00000 −0.380091 −0.190046 − 0.981775i \(-0.560864\pi\)
−0.190046 + 0.981775i \(0.560864\pi\)
\(444\) −8.00000 −0.379663
\(445\) −20.0000 −0.948091
\(446\) 0 0
\(447\) −8.00000 −0.378387
\(448\) −16.0000 −0.755929
\(449\) −30.0000 −1.41579 −0.707894 − 0.706319i \(-0.750354\pi\)
−0.707894 + 0.706319i \(0.750354\pi\)
\(450\) 0 0
\(451\) 30.0000 1.41264
\(452\) −12.0000 −0.564433
\(453\) −16.0000 −0.751746
\(454\) 0 0
\(455\) −8.00000 −0.375046
\(456\) 0 0
\(457\) −14.0000 −0.654892 −0.327446 − 0.944870i \(-0.606188\pi\)
−0.327446 + 0.944870i \(0.606188\pi\)
\(458\) 0 0
\(459\) 4.00000 0.186704
\(460\) −20.0000 −0.932505
\(461\) −15.0000 −0.698620 −0.349310 − 0.937007i \(-0.613584\pi\)
−0.349310 + 0.937007i \(0.613584\pi\)
\(462\) 0 0
\(463\) −26.0000 −1.20832 −0.604161 − 0.796862i \(-0.706492\pi\)
−0.604161 + 0.796862i \(0.706492\pi\)
\(464\) −24.0000 −1.11417
\(465\) −32.0000 −1.48396
\(466\) 0 0
\(467\) 20.0000 0.925490 0.462745 − 0.886492i \(-0.346865\pi\)
0.462745 + 0.886492i \(0.346865\pi\)
\(468\) −4.00000 −0.184900
\(469\) −16.0000 −0.738811
\(470\) 0 0
\(471\) 20.0000 0.921551
\(472\) 0 0
\(473\) 10.0000 0.459800
\(474\) 0 0
\(475\) 5.00000 0.229416
\(476\) 4.00000 0.183340
\(477\) −11.0000 −0.503655
\(478\) 0 0
\(479\) −12.0000 −0.548294 −0.274147 − 0.961688i \(-0.588395\pi\)
−0.274147 + 0.961688i \(0.588395\pi\)
\(480\) 0 0
\(481\) 4.00000 0.182384
\(482\) 0 0
\(483\) −20.0000 −0.910032
\(484\) −28.0000 −1.27273
\(485\) −6.00000 −0.272446
\(486\) 0 0
\(487\) −24.0000 −1.08754 −0.543772 − 0.839233i \(-0.683004\pi\)
−0.543772 + 0.839233i \(0.683004\pi\)
\(488\) 0 0
\(489\) 0 0
\(490\) 0 0
\(491\) −15.0000 −0.676941 −0.338470 − 0.940977i \(-0.609909\pi\)
−0.338470 + 0.940977i \(0.609909\pi\)
\(492\) 24.0000 1.08200
\(493\) 6.00000 0.270226
\(494\) 0 0
\(495\) 10.0000 0.449467
\(496\) 32.0000 1.43684
\(497\) −12.0000 −0.538274
\(498\) 0 0
\(499\) −20.0000 −0.895323 −0.447661 − 0.894203i \(-0.647743\pi\)
−0.447661 + 0.894203i \(0.647743\pi\)
\(500\) −24.0000 −1.07331
\(501\) 24.0000 1.07224
\(502\) 0 0
\(503\) 1.00000 0.0445878 0.0222939 − 0.999751i \(-0.492903\pi\)
0.0222939 + 0.999751i \(0.492903\pi\)
\(504\) 0 0
\(505\) 4.00000 0.177998
\(506\) 0 0
\(507\) −18.0000 −0.799408
\(508\) −32.0000 −1.41977
\(509\) −36.0000 −1.59567 −0.797836 − 0.602875i \(-0.794022\pi\)
−0.797836 + 0.602875i \(0.794022\pi\)
\(510\) 0 0
\(511\) 26.0000 1.15017
\(512\) 0 0
\(513\) 20.0000 0.883022
\(514\) 0 0
\(515\) 12.0000 0.528783
\(516\) 8.00000 0.352180
\(517\) −30.0000 −1.31940
\(518\) 0 0
\(519\) 2.00000 0.0877903
\(520\) 0 0
\(521\) −42.0000 −1.84005 −0.920027 − 0.391856i \(-0.871833\pi\)
−0.920027 + 0.391856i \(0.871833\pi\)
\(522\) 0 0
\(523\) 33.0000 1.44299 0.721495 − 0.692420i \(-0.243455\pi\)
0.721495 + 0.692420i \(0.243455\pi\)
\(524\) 8.00000 0.349482
\(525\) −4.00000 −0.174574
\(526\) 0 0
\(527\) −8.00000 −0.348485
\(528\) −40.0000 −1.74078
\(529\) 2.00000 0.0869565
\(530\) 0 0
\(531\) −1.00000 −0.0433963
\(532\) 20.0000 0.867110
\(533\) −12.0000 −0.519778
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 0 0
\(537\) −36.0000 −1.55351
\(538\) 0 0
\(539\) 15.0000 0.646096
\(540\) −16.0000 −0.688530
\(541\) −14.0000 −0.601907 −0.300954 − 0.953639i \(-0.597305\pi\)
−0.300954 + 0.953639i \(0.597305\pi\)
\(542\) 0 0
\(543\) 44.0000 1.88822
\(544\) 0 0
\(545\) 14.0000 0.599694
\(546\) 0 0
\(547\) 28.0000 1.19719 0.598597 − 0.801050i \(-0.295725\pi\)
0.598597 + 0.801050i \(0.295725\pi\)
\(548\) −18.0000 −0.768922
\(549\) −1.00000 −0.0426790
\(550\) 0 0
\(551\) 30.0000 1.27804
\(552\) 0 0
\(553\) 8.00000 0.340195
\(554\) 0 0
\(555\) −8.00000 −0.339581
\(556\) 8.00000 0.339276
\(557\) −39.0000 −1.65248 −0.826242 − 0.563316i \(-0.809525\pi\)
−0.826242 + 0.563316i \(0.809525\pi\)
\(558\) 0 0
\(559\) −4.00000 −0.169182
\(560\) −16.0000 −0.676123
\(561\) 10.0000 0.422200
\(562\) 0 0
\(563\) −28.0000 −1.18006 −0.590030 − 0.807382i \(-0.700884\pi\)
−0.590030 + 0.807382i \(0.700884\pi\)
\(564\) −24.0000 −1.01058
\(565\) −12.0000 −0.504844
\(566\) 0 0
\(567\) −22.0000 −0.923913
\(568\) 0 0
\(569\) 10.0000 0.419222 0.209611 − 0.977785i \(-0.432780\pi\)
0.209611 + 0.977785i \(0.432780\pi\)
\(570\) 0 0
\(571\) −28.0000 −1.17176 −0.585882 − 0.810397i \(-0.699252\pi\)
−0.585882 + 0.810397i \(0.699252\pi\)
\(572\) 20.0000 0.836242
\(573\) 0 0
\(574\) 0 0
\(575\) 5.00000 0.208514
\(576\) −8.00000 −0.333333
\(577\) −25.0000 −1.04076 −0.520382 − 0.853934i \(-0.674210\pi\)
−0.520382 + 0.853934i \(0.674210\pi\)
\(578\) 0 0
\(579\) −16.0000 −0.664937
\(580\) −24.0000 −0.996546
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 55.0000 2.27787
\(584\) 0 0
\(585\) −4.00000 −0.165380
\(586\) 0 0
\(587\) −28.0000 −1.15568 −0.577842 − 0.816149i \(-0.696105\pi\)
−0.577842 + 0.816149i \(0.696105\pi\)
\(588\) 12.0000 0.494872
\(589\) −40.0000 −1.64817
\(590\) 0 0
\(591\) 52.0000 2.13899
\(592\) 8.00000 0.328798
\(593\) 39.0000 1.60154 0.800769 − 0.598973i \(-0.204424\pi\)
0.800769 + 0.598973i \(0.204424\pi\)
\(594\) 0 0
\(595\) 4.00000 0.163984
\(596\) 8.00000 0.327693
\(597\) −48.0000 −1.96451
\(598\) 0 0
\(599\) −24.0000 −0.980613 −0.490307 − 0.871550i \(-0.663115\pi\)
−0.490307 + 0.871550i \(0.663115\pi\)
\(600\) 0 0
\(601\) 38.0000 1.55005 0.775026 − 0.631929i \(-0.217737\pi\)
0.775026 + 0.631929i \(0.217737\pi\)
\(602\) 0 0
\(603\) −8.00000 −0.325785
\(604\) 16.0000 0.651031
\(605\) −28.0000 −1.13836
\(606\) 0 0
\(607\) −16.0000 −0.649420 −0.324710 − 0.945814i \(-0.605267\pi\)
−0.324710 + 0.945814i \(0.605267\pi\)
\(608\) 0 0
\(609\) −24.0000 −0.972529
\(610\) 0 0
\(611\) 12.0000 0.485468
\(612\) 2.00000 0.0808452
\(613\) −16.0000 −0.646234 −0.323117 − 0.946359i \(-0.604731\pi\)
−0.323117 + 0.946359i \(0.604731\pi\)
\(614\) 0 0
\(615\) 24.0000 0.967773
\(616\) 0 0
\(617\) 20.0000 0.805170 0.402585 − 0.915383i \(-0.368112\pi\)
0.402585 + 0.915383i \(0.368112\pi\)
\(618\) 0 0
\(619\) −2.00000 −0.0803868 −0.0401934 − 0.999192i \(-0.512797\pi\)
−0.0401934 + 0.999192i \(0.512797\pi\)
\(620\) 32.0000 1.28515
\(621\) 20.0000 0.802572
\(622\) 0 0
\(623\) 20.0000 0.801283
\(624\) 16.0000 0.640513
\(625\) −19.0000 −0.760000
\(626\) 0 0
\(627\) 50.0000 1.99681
\(628\) −20.0000 −0.798087
\(629\) −2.00000 −0.0797452
\(630\) 0 0
\(631\) −15.0000 −0.597141 −0.298570 − 0.954388i \(-0.596510\pi\)
−0.298570 + 0.954388i \(0.596510\pi\)
\(632\) 0 0
\(633\) −46.0000 −1.82834
\(634\) 0 0
\(635\) −32.0000 −1.26988
\(636\) 44.0000 1.74471
\(637\) −6.00000 −0.237729
\(638\) 0 0
\(639\) −6.00000 −0.237356
\(640\) 0 0
\(641\) 24.0000 0.947943 0.473972 − 0.880540i \(-0.342820\pi\)
0.473972 + 0.880540i \(0.342820\pi\)
\(642\) 0 0
\(643\) 6.00000 0.236617 0.118308 − 0.992977i \(-0.462253\pi\)
0.118308 + 0.992977i \(0.462253\pi\)
\(644\) 20.0000 0.788110
\(645\) 8.00000 0.315000
\(646\) 0 0
\(647\) 17.0000 0.668339 0.334169 − 0.942513i \(-0.391544\pi\)
0.334169 + 0.942513i \(0.391544\pi\)
\(648\) 0 0
\(649\) 5.00000 0.196267
\(650\) 0 0
\(651\) 32.0000 1.25418
\(652\) 0 0
\(653\) 48.0000 1.87839 0.939193 − 0.343391i \(-0.111576\pi\)
0.939193 + 0.343391i \(0.111576\pi\)
\(654\) 0 0
\(655\) 8.00000 0.312586
\(656\) −24.0000 −0.937043
\(657\) 13.0000 0.507178
\(658\) 0 0
\(659\) −6.00000 −0.233727 −0.116863 − 0.993148i \(-0.537284\pi\)
−0.116863 + 0.993148i \(0.537284\pi\)
\(660\) −40.0000 −1.55700
\(661\) 22.0000 0.855701 0.427850 − 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 0 0
\(663\) −4.00000 −0.155347
\(664\) 0 0
\(665\) 20.0000 0.775567
\(666\) 0 0
\(667\) 30.0000 1.16160
\(668\) −24.0000 −0.928588
\(669\) 18.0000 0.695920
\(670\) 0 0
\(671\) 5.00000 0.193023
\(672\) 0 0
\(673\) 37.0000 1.42625 0.713123 − 0.701039i \(-0.247280\pi\)
0.713123 + 0.701039i \(0.247280\pi\)
\(674\) 0 0
\(675\) 4.00000 0.153960
\(676\) 18.0000 0.692308
\(677\) −16.0000 −0.614930 −0.307465 − 0.951559i \(-0.599481\pi\)
−0.307465 + 0.951559i \(0.599481\pi\)
\(678\) 0 0
\(679\) 6.00000 0.230259
\(680\) 0 0
\(681\) −16.0000 −0.613121
\(682\) 0 0
\(683\) 36.0000 1.37750 0.688751 − 0.724998i \(-0.258159\pi\)
0.688751 + 0.724998i \(0.258159\pi\)
\(684\) 10.0000 0.382360
\(685\) −18.0000 −0.687745
\(686\) 0 0
\(687\) −40.0000 −1.52610
\(688\) −8.00000 −0.304997
\(689\) −22.0000 −0.838133
\(690\) 0 0
\(691\) 39.0000 1.48363 0.741815 − 0.670605i \(-0.233965\pi\)
0.741815 + 0.670605i \(0.233965\pi\)
\(692\) −2.00000 −0.0760286
\(693\) −10.0000 −0.379869
\(694\) 0 0
\(695\) 8.00000 0.303457
\(696\) 0 0
\(697\) 6.00000 0.227266
\(698\) 0 0
\(699\) −50.0000 −1.89117
\(700\) 4.00000 0.151186
\(701\) 20.0000 0.755390 0.377695 − 0.925930i \(-0.376717\pi\)
0.377695 + 0.925930i \(0.376717\pi\)
\(702\) 0 0
\(703\) −10.0000 −0.377157
\(704\) 40.0000 1.50756
\(705\) −24.0000 −0.903892
\(706\) 0 0
\(707\) −4.00000 −0.150435
\(708\) 4.00000 0.150329
\(709\) −20.0000 −0.751116 −0.375558 − 0.926799i \(-0.622549\pi\)
−0.375558 + 0.926799i \(0.622549\pi\)
\(710\) 0 0
\(711\) 4.00000 0.150012
\(712\) 0 0
\(713\) −40.0000 −1.49801
\(714\) 0 0
\(715\) 20.0000 0.747958
\(716\) 36.0000 1.34538
\(717\) −30.0000 −1.12037
\(718\) 0 0
\(719\) 0 0 − 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(720\) −8.00000 −0.298142
\(721\) −12.0000 −0.446903
\(722\) 0 0
\(723\) 4.00000 0.148762
\(724\) −44.0000 −1.63525
\(725\) 6.00000 0.222834
\(726\) 0 0
\(727\) 33.0000 1.22390 0.611951 − 0.790896i \(-0.290385\pi\)
0.611951 + 0.790896i \(0.290385\pi\)
\(728\) 0 0
\(729\) 13.0000 0.481481
\(730\) 0 0
\(731\) 2.00000 0.0739727
\(732\) 4.00000 0.147844
\(733\) 9.00000 0.332423 0.166211 − 0.986090i \(-0.446847\pi\)
0.166211 + 0.986090i \(0.446847\pi\)
\(734\) 0 0
\(735\) 12.0000 0.442627
\(736\) 0 0
\(737\) 40.0000 1.47342
\(738\) 0 0
\(739\) −50.0000 −1.83928 −0.919640 − 0.392763i \(-0.871519\pi\)
−0.919640 + 0.392763i \(0.871519\pi\)
\(740\) 8.00000 0.294086
\(741\) −20.0000 −0.734718
\(742\) 0 0
\(743\) −24.0000 −0.880475 −0.440237 − 0.897881i \(-0.645106\pi\)
−0.440237 + 0.897881i \(0.645106\pi\)
\(744\) 0 0
\(745\) 8.00000 0.293097
\(746\) 0 0
\(747\) 6.00000 0.219529
\(748\) −10.0000 −0.365636
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) −45.0000 −1.64207 −0.821037 − 0.570875i \(-0.806604\pi\)
−0.821037 + 0.570875i \(0.806604\pi\)
\(752\) 24.0000 0.875190
\(753\) 54.0000 1.96787
\(754\) 0 0
\(755\) 16.0000 0.582300
\(756\) 16.0000 0.581914
\(757\) 1.00000 0.0363456 0.0181728 − 0.999835i \(-0.494215\pi\)
0.0181728 + 0.999835i \(0.494215\pi\)
\(758\) 0 0
\(759\) 50.0000 1.81489
\(760\) 0 0
\(761\) −45.0000 −1.63125 −0.815624 − 0.578582i \(-0.803606\pi\)
−0.815624 + 0.578582i \(0.803606\pi\)
\(762\) 0 0
\(763\) −14.0000 −0.506834
\(764\) 0 0
\(765\) 2.00000 0.0723102
\(766\) 0 0
\(767\) −2.00000 −0.0722158
\(768\) 32.0000 1.15470
\(769\) 14.0000 0.504853 0.252426 − 0.967616i \(-0.418771\pi\)
0.252426 + 0.967616i \(0.418771\pi\)
\(770\) 0 0
\(771\) 34.0000 1.22448
\(772\) 16.0000 0.575853
\(773\) −46.0000 −1.65451 −0.827253 − 0.561830i \(-0.810097\pi\)
−0.827253 + 0.561830i \(0.810097\pi\)
\(774\) 0 0
\(775\) −8.00000 −0.287368
\(776\) 0 0
\(777\) 8.00000 0.286998
\(778\) 0 0
\(779\) 30.0000 1.07486
\(780\) 16.0000 0.572892
\(781\) 30.0000 1.07348
\(782\) 0 0
\(783\) 24.0000 0.857690
\(784\) −12.0000 −0.428571
\(785\) −20.0000 −0.713831
\(786\) 0 0
\(787\) 18.0000 0.641631 0.320815 − 0.947142i \(-0.396043\pi\)
0.320815 + 0.947142i \(0.396043\pi\)
\(788\) −52.0000 −1.85242
\(789\) 54.0000 1.92245
\(790\) 0 0
\(791\) 12.0000 0.426671
\(792\) 0 0
\(793\) −2.00000 −0.0710221
\(794\) 0 0
\(795\) 44.0000 1.56052
\(796\) 48.0000 1.70131
\(797\) −12.0000 −0.425062 −0.212531 − 0.977154i \(-0.568171\pi\)
−0.212531 + 0.977154i \(0.568171\pi\)
\(798\) 0 0
\(799\) −6.00000 −0.212265
\(800\) 0 0
\(801\) 10.0000 0.353333
\(802\) 0 0
\(803\) −65.0000 −2.29380
\(804\) 32.0000 1.12855
\(805\) 20.0000 0.704907
\(806\) 0 0
\(807\) 10.0000 0.352017
\(808\) 0 0
\(809\) −50.0000 −1.75791 −0.878953 − 0.476908i \(-0.841757\pi\)
−0.878953 + 0.476908i \(0.841757\pi\)
\(810\) 0 0
\(811\) −53.0000 −1.86108 −0.930541 − 0.366188i \(-0.880663\pi\)
−0.930541 + 0.366188i \(0.880663\pi\)
\(812\) 24.0000 0.842235
\(813\) −18.0000 −0.631288
\(814\) 0 0
\(815\) 0 0
\(816\) −8.00000 −0.280056
\(817\) 10.0000 0.349856
\(818\) 0 0
\(819\) 4.00000 0.139771
\(820\) −24.0000 −0.838116
\(821\) −13.0000 −0.453703 −0.226852 − 0.973929i \(-0.572843\pi\)
−0.226852 + 0.973929i \(0.572843\pi\)
\(822\) 0 0
\(823\) 31.0000 1.08059 0.540296 − 0.841475i \(-0.318312\pi\)
0.540296 + 0.841475i \(0.318312\pi\)
\(824\) 0 0
\(825\) 10.0000 0.348155
\(826\) 0 0
\(827\) −4.00000 −0.139094 −0.0695468 − 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 10.0000 0.347524
\(829\) −55.0000 −1.91023 −0.955114 − 0.296237i \(-0.904268\pi\)
−0.955114 + 0.296237i \(0.904268\pi\)
\(830\) 0 0
\(831\) −52.0000 −1.80386
\(832\) −16.0000 −0.554700
\(833\) 3.00000 0.103944
\(834\) 0 0
\(835\) −24.0000 −0.830554
\(836\) −50.0000 −1.72929
\(837\) −32.0000 −1.10608
\(838\) 0 0
\(839\) 35.0000 1.20833 0.604167 − 0.796858i \(-0.293506\pi\)
0.604167 + 0.796858i \(0.293506\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) 0 0
\(843\) −6.00000 −0.206651
\(844\) 46.0000 1.58339
\(845\) 18.0000 0.619219
\(846\) 0 0
\(847\) 28.0000 0.962091
\(848\) −44.0000 −1.51097
\(849\) 32.0000 1.09824
\(850\) 0 0
\(851\) −10.0000 −0.342796
\(852\) 24.0000 0.822226
\(853\) 34.0000 1.16414 0.582069 − 0.813139i \(-0.302243\pi\)
0.582069 + 0.813139i \(0.302243\pi\)
\(854\) 0 0
\(855\) 10.0000 0.341993
\(856\) 0 0
\(857\) 3.00000 0.102478 0.0512390 − 0.998686i \(-0.483683\pi\)
0.0512390 + 0.998686i \(0.483683\pi\)
\(858\) 0 0
\(859\) 58.0000 1.97893 0.989467 − 0.144757i \(-0.0462401\pi\)
0.989467 + 0.144757i \(0.0462401\pi\)
\(860\) −8.00000 −0.272798
\(861\) −24.0000 −0.817918
\(862\) 0 0
\(863\) −32.0000 −1.08929 −0.544646 − 0.838666i \(-0.683336\pi\)
−0.544646 + 0.838666i \(0.683336\pi\)
\(864\) 0 0
\(865\) −2.00000 −0.0680020
\(866\) 0 0
\(867\) 2.00000 0.0679236
\(868\) −32.0000 −1.08615
\(869\) −20.0000 −0.678454
\(870\) 0 0
\(871\) −16.0000 −0.542139
\(872\) 0 0
\(873\) 3.00000 0.101535
\(874\) 0 0
\(875\) 24.0000 0.811348
\(876\) −52.0000 −1.75692
\(877\) −36.0000 −1.21563 −0.607817 − 0.794077i \(-0.707955\pi\)
−0.607817 + 0.794077i \(0.707955\pi\)
\(878\) 0 0
\(879\) −22.0000 −0.742042
\(880\) 40.0000 1.34840
\(881\) 3.00000 0.101073 0.0505363 − 0.998722i \(-0.483907\pi\)
0.0505363 + 0.998722i \(0.483907\pi\)
\(882\) 0 0
\(883\) 17.0000 0.572096 0.286048 − 0.958215i \(-0.407658\pi\)
0.286048 + 0.958215i \(0.407658\pi\)
\(884\) 4.00000 0.134535
\(885\) 4.00000 0.134459
\(886\) 0 0
\(887\) 48.0000 1.61168 0.805841 − 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) 0 0
\(889\) 32.0000 1.07325
\(890\) 0 0
\(891\) 55.0000 1.84257
\(892\) −18.0000 −0.602685
\(893\) −30.0000 −1.00391
\(894\) 0 0
\(895\) 36.0000 1.20335
\(896\) 0 0
\(897\) −20.0000 −0.667781
\(898\) 0 0
\(899\) −48.0000 −1.60089
\(900\) 2.00000 0.0666667
\(901\) 11.0000 0.366463
\(902\) 0 0
\(903\) −8.00000 −0.266223
\(904\) 0 0
\(905\) −44.0000 −1.46261
\(906\) 0 0
\(907\) −28.0000 −0.929725 −0.464862 − 0.885383i \(-0.653896\pi\)
−0.464862 + 0.885383i \(0.653896\pi\)
\(908\) 16.0000 0.530979
\(909\) −2.00000 −0.0663358
\(910\) 0 0
\(911\) 26.0000 0.861418 0.430709 − 0.902491i \(-0.358263\pi\)
0.430709 + 0.902491i \(0.358263\pi\)
\(912\) −40.0000 −1.32453
\(913\) −30.0000 −0.992855
\(914\) 0 0
\(915\) 4.00000 0.132236
\(916\) 40.0000 1.32164
\(917\) −8.00000 −0.264183
\(918\) 0 0
\(919\) −10.0000 −0.329870 −0.164935 − 0.986304i \(-0.552741\pi\)
−0.164935 + 0.986304i \(0.552741\pi\)
\(920\) 0 0
\(921\) 50.0000 1.64756
\(922\) 0 0
\(923\) −12.0000 −0.394985
\(924\) 40.0000 1.31590
\(925\) −2.00000 −0.0657596
\(926\) 0 0
\(927\) −6.00000 −0.197066
\(928\) 0 0
\(929\) 29.0000 0.951459 0.475730 − 0.879592i \(-0.342184\pi\)
0.475730 + 0.879592i \(0.342184\pi\)
\(930\) 0 0
\(931\) 15.0000 0.491605
\(932\) 50.0000 1.63780
\(933\) 16.0000 0.523816
\(934\) 0 0
\(935\) −10.0000 −0.327035
\(936\) 0 0
\(937\) −8.00000 −0.261349 −0.130674 − 0.991425i \(-0.541714\pi\)
−0.130674 + 0.991425i \(0.541714\pi\)
\(938\) 0 0
\(939\) 12.0000 0.391605
\(940\) 24.0000 0.782794
\(941\) −6.00000 −0.195594 −0.0977972 − 0.995206i \(-0.531180\pi\)
−0.0977972 + 0.995206i \(0.531180\pi\)
\(942\) 0 0
\(943\) 30.0000 0.976934
\(944\) −4.00000 −0.130189
\(945\) 16.0000 0.520480
\(946\) 0 0
\(947\) 42.0000 1.36482 0.682408 − 0.730971i \(-0.260933\pi\)
0.682408 + 0.730971i \(0.260933\pi\)
\(948\) −16.0000 −0.519656
\(949\) 26.0000 0.843996
\(950\) 0 0
\(951\) −64.0000 −2.07534
\(952\) 0 0
\(953\) 3.00000 0.0971795 0.0485898 − 0.998819i \(-0.484527\pi\)
0.0485898 + 0.998819i \(0.484527\pi\)
\(954\) 0 0
\(955\) 0 0
\(956\) 30.0000 0.970269
\(957\) 60.0000 1.93952
\(958\) 0 0
\(959\) 18.0000 0.581250
\(960\) 32.0000 1.03280
\(961\) 33.0000 1.06452
\(962\) 0 0
\(963\) 6.00000 0.193347
\(964\) −4.00000 −0.128831
\(965\) 16.0000 0.515058
\(966\) 0 0
\(967\) −50.0000 −1.60789 −0.803946 − 0.594703i \(-0.797270\pi\)
−0.803946 + 0.594703i \(0.797270\pi\)
\(968\) 0 0
\(969\) 10.0000 0.321246
\(970\) 0 0
\(971\) 29.0000 0.930654 0.465327 − 0.885139i \(-0.345937\pi\)
0.465327 + 0.885139i \(0.345937\pi\)
\(972\) 20.0000 0.641500
\(973\) −8.00000 −0.256468
\(974\) 0 0
\(975\) −4.00000 −0.128103
\(976\) −4.00000 −0.128037
\(977\) 4.00000 0.127971 0.0639857 − 0.997951i \(-0.479619\pi\)
0.0639857 + 0.997951i \(0.479619\pi\)
\(978\) 0 0
\(979\) −50.0000 −1.59801
\(980\) −12.0000 −0.383326
\(981\) −7.00000 −0.223493
\(982\) 0 0
\(983\) 53.0000 1.69044 0.845219 − 0.534421i \(-0.179470\pi\)
0.845219 + 0.534421i \(0.179470\pi\)
\(984\) 0 0
\(985\) −52.0000 −1.65686
\(986\) 0 0
\(987\) 24.0000 0.763928
\(988\) 20.0000 0.636285
\(989\) 10.0000 0.317982
\(990\) 0 0
\(991\) −37.0000 −1.17534 −0.587672 − 0.809099i \(-0.699955\pi\)
−0.587672 + 0.809099i \(0.699955\pi\)
\(992\) 0 0
\(993\) −16.0000 −0.507745
\(994\) 0 0
\(995\) 48.0000 1.52170
\(996\) −24.0000 −0.760469
\(997\) −8.00000 −0.253363 −0.126681 − 0.991943i \(-0.540433\pi\)
−0.126681 + 0.991943i \(0.540433\pi\)
\(998\) 0 0
\(999\) −8.00000 −0.253109
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 1003.2.a.b.1.1 ✓ 1
3.2 odd 2 9027.2.a.d.1.1 1
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1003.2.a.b.1.1 ✓ 1 1.1 even 1 trivial
9027.2.a.d.1.1 1 3.2 odd 2