Properties

Label 2576.2.a.v.1.2
Level $2576$
Weight $2$
Character 2576.1
Self dual yes
Analytic conductor $20.569$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [2576,2,Mod(1,2576)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(2576, 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("2576.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 2576 = 2^{4} \cdot 7 \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2576.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: \(20.5694635607\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.148.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 3x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 161)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(0.311108\) of defining polynomial
Character \(\chi\) \(=\) 2576.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.688892 q^{3} +0.688892 q^{5} +1.00000 q^{7} -2.52543 q^{9} +O(q^{10})\) \(q-0.688892 q^{3} +0.688892 q^{5} +1.00000 q^{7} -2.52543 q^{9} +0.903212 q^{11} +0.622216 q^{13} -0.474572 q^{15} +1.31111 q^{17} -7.05086 q^{19} -0.688892 q^{21} -1.00000 q^{23} -4.52543 q^{25} +3.80642 q^{27} +1.52543 q^{29} +3.54617 q^{31} -0.622216 q^{33} +0.688892 q^{35} +2.42864 q^{37} -0.428639 q^{39} -4.75557 q^{41} -6.23506 q^{43} -1.73975 q^{45} -9.73975 q^{47} +1.00000 q^{49} -0.903212 q^{51} -6.85728 q^{53} +0.622216 q^{55} +4.85728 q^{57} -9.97481 q^{59} +7.87310 q^{61} -2.52543 q^{63} +0.428639 q^{65} +5.76049 q^{67} +0.688892 q^{69} +7.61285 q^{71} -1.86665 q^{73} +3.11753 q^{75} +0.903212 q^{77} -7.65878 q^{79} +4.95407 q^{81} +11.0923 q^{83} +0.903212 q^{85} -1.05086 q^{87} -2.98418 q^{89} +0.622216 q^{91} -2.44293 q^{93} -4.85728 q^{95} -2.49532 q^{97} -2.28100 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - 2 q^{3} + 2 q^{5} + 3 q^{7} - q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q - 2 q^{3} + 2 q^{5} + 3 q^{7} - q^{9} - 4 q^{11} + 2 q^{13} - 8 q^{15} + 4 q^{17} - 8 q^{19} - 2 q^{21} - 3 q^{23} - 7 q^{25} - 2 q^{27} - 2 q^{29} - 16 q^{31} - 2 q^{33} + 2 q^{35} - 6 q^{37} + 12 q^{39} - 14 q^{41} + 8 q^{43} + 8 q^{45} - 16 q^{47} + 3 q^{49} + 4 q^{51} + 6 q^{53} + 2 q^{55} - 12 q^{57} + 10 q^{59} + 10 q^{61} - q^{63} - 12 q^{65} - 16 q^{67} + 2 q^{69} - 4 q^{71} - 6 q^{73} - 4 q^{75} - 4 q^{77} - 16 q^{79} - 5 q^{81} - 20 q^{83} - 4 q^{85} + 10 q^{87} + 4 q^{89} + 2 q^{91} + 12 q^{93} + 12 q^{95} + 6 q^{97}+O(q^{100}) \) Copy content Toggle raw display

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
\(3\) −0.688892 −0.397732 −0.198866 0.980027i \(-0.563726\pi\)
−0.198866 + 0.980027i \(0.563726\pi\)
\(4\) 0 0
\(5\) 0.688892 0.308082 0.154041 0.988064i \(-0.450771\pi\)
0.154041 + 0.988064i \(0.450771\pi\)
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) 0 0
\(9\) −2.52543 −0.841809
\(10\) 0 0
\(11\) 0.903212 0.272329 0.136164 0.990686i \(-0.456522\pi\)
0.136164 + 0.990686i \(0.456522\pi\)
\(12\) 0 0
\(13\) 0.622216 0.172572 0.0862858 0.996270i \(-0.472500\pi\)
0.0862858 + 0.996270i \(0.472500\pi\)
\(14\) 0 0
\(15\) −0.474572 −0.122534
\(16\) 0 0
\(17\) 1.31111 0.317990 0.158995 0.987279i \(-0.449175\pi\)
0.158995 + 0.987279i \(0.449175\pi\)
\(18\) 0 0
\(19\) −7.05086 −1.61758 −0.808789 0.588100i \(-0.799876\pi\)
−0.808789 + 0.588100i \(0.799876\pi\)
\(20\) 0 0
\(21\) −0.688892 −0.150329
\(22\) 0 0
\(23\) −1.00000 −0.208514
\(24\) 0 0
\(25\) −4.52543 −0.905086
\(26\) 0 0
\(27\) 3.80642 0.732547
\(28\) 0 0
\(29\) 1.52543 0.283265 0.141632 0.989919i \(-0.454765\pi\)
0.141632 + 0.989919i \(0.454765\pi\)
\(30\) 0 0
\(31\) 3.54617 0.636911 0.318456 0.947938i \(-0.396836\pi\)
0.318456 + 0.947938i \(0.396836\pi\)
\(32\) 0 0
\(33\) −0.622216 −0.108314
\(34\) 0 0
\(35\) 0.688892 0.116444
\(36\) 0 0
\(37\) 2.42864 0.399266 0.199633 0.979871i \(-0.436025\pi\)
0.199633 + 0.979871i \(0.436025\pi\)
\(38\) 0 0
\(39\) −0.428639 −0.0686372
\(40\) 0 0
\(41\) −4.75557 −0.742695 −0.371348 0.928494i \(-0.621104\pi\)
−0.371348 + 0.928494i \(0.621104\pi\)
\(42\) 0 0
\(43\) −6.23506 −0.950838 −0.475419 0.879759i \(-0.657704\pi\)
−0.475419 + 0.879759i \(0.657704\pi\)
\(44\) 0 0
\(45\) −1.73975 −0.259346
\(46\) 0 0
\(47\) −9.73975 −1.42069 −0.710344 0.703855i \(-0.751461\pi\)
−0.710344 + 0.703855i \(0.751461\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) 0 0
\(51\) −0.903212 −0.126475
\(52\) 0 0
\(53\) −6.85728 −0.941920 −0.470960 0.882155i \(-0.656092\pi\)
−0.470960 + 0.882155i \(0.656092\pi\)
\(54\) 0 0
\(55\) 0.622216 0.0838995
\(56\) 0 0
\(57\) 4.85728 0.643362
\(58\) 0 0
\(59\) −9.97481 −1.29861 −0.649305 0.760528i \(-0.724940\pi\)
−0.649305 + 0.760528i \(0.724940\pi\)
\(60\) 0 0
\(61\) 7.87310 1.00805 0.504024 0.863690i \(-0.331852\pi\)
0.504024 + 0.863690i \(0.331852\pi\)
\(62\) 0 0
\(63\) −2.52543 −0.318174
\(64\) 0 0
\(65\) 0.428639 0.0531662
\(66\) 0 0
\(67\) 5.76049 0.703756 0.351878 0.936046i \(-0.385543\pi\)
0.351878 + 0.936046i \(0.385543\pi\)
\(68\) 0 0
\(69\) 0.688892 0.0829329
\(70\) 0 0
\(71\) 7.61285 0.903479 0.451739 0.892150i \(-0.350804\pi\)
0.451739 + 0.892150i \(0.350804\pi\)
\(72\) 0 0
\(73\) −1.86665 −0.218474 −0.109237 0.994016i \(-0.534841\pi\)
−0.109237 + 0.994016i \(0.534841\pi\)
\(74\) 0 0
\(75\) 3.11753 0.359982
\(76\) 0 0
\(77\) 0.903212 0.102931
\(78\) 0 0
\(79\) −7.65878 −0.861680 −0.430840 0.902428i \(-0.641783\pi\)
−0.430840 + 0.902428i \(0.641783\pi\)
\(80\) 0 0
\(81\) 4.95407 0.550452
\(82\) 0 0
\(83\) 11.0923 1.21754 0.608771 0.793346i \(-0.291663\pi\)
0.608771 + 0.793346i \(0.291663\pi\)
\(84\) 0 0
\(85\) 0.903212 0.0979671
\(86\) 0 0
\(87\) −1.05086 −0.112663
\(88\) 0 0
\(89\) −2.98418 −0.316322 −0.158161 0.987413i \(-0.550557\pi\)
−0.158161 + 0.987413i \(0.550557\pi\)
\(90\) 0 0
\(91\) 0.622216 0.0652259
\(92\) 0 0
\(93\) −2.44293 −0.253320
\(94\) 0 0
\(95\) −4.85728 −0.498346
\(96\) 0 0
\(97\) −2.49532 −0.253361 −0.126680 0.991944i \(-0.540432\pi\)
−0.126680 + 0.991944i \(0.540432\pi\)
\(98\) 0 0
\(99\) −2.28100 −0.229249
\(100\) 0 0
\(101\) −10.4286 −1.03769 −0.518844 0.854869i \(-0.673638\pi\)
−0.518844 + 0.854869i \(0.673638\pi\)
\(102\) 0 0
\(103\) −9.24443 −0.910881 −0.455440 0.890266i \(-0.650518\pi\)
−0.455440 + 0.890266i \(0.650518\pi\)
\(104\) 0 0
\(105\) −0.474572 −0.0463135
\(106\) 0 0
\(107\) −17.8479 −1.72542 −0.862711 0.505697i \(-0.831235\pi\)
−0.862711 + 0.505697i \(0.831235\pi\)
\(108\) 0 0
\(109\) 10.9906 1.05271 0.526356 0.850264i \(-0.323558\pi\)
0.526356 + 0.850264i \(0.323558\pi\)
\(110\) 0 0
\(111\) −1.67307 −0.158801
\(112\) 0 0
\(113\) −12.3684 −1.16352 −0.581761 0.813360i \(-0.697636\pi\)
−0.581761 + 0.813360i \(0.697636\pi\)
\(114\) 0 0
\(115\) −0.688892 −0.0642395
\(116\) 0 0
\(117\) −1.57136 −0.145272
\(118\) 0 0
\(119\) 1.31111 0.120189
\(120\) 0 0
\(121\) −10.1842 −0.925837
\(122\) 0 0
\(123\) 3.27607 0.295394
\(124\) 0 0
\(125\) −6.56199 −0.586922
\(126\) 0 0
\(127\) −8.42864 −0.747921 −0.373960 0.927445i \(-0.622000\pi\)
−0.373960 + 0.927445i \(0.622000\pi\)
\(128\) 0 0
\(129\) 4.29529 0.378179
\(130\) 0 0
\(131\) 17.2192 1.50445 0.752226 0.658905i \(-0.228980\pi\)
0.752226 + 0.658905i \(0.228980\pi\)
\(132\) 0 0
\(133\) −7.05086 −0.611387
\(134\) 0 0
\(135\) 2.62222 0.225684
\(136\) 0 0
\(137\) −5.18421 −0.442917 −0.221458 0.975170i \(-0.571082\pi\)
−0.221458 + 0.975170i \(0.571082\pi\)
\(138\) 0 0
\(139\) −11.8731 −1.00706 −0.503532 0.863977i \(-0.667966\pi\)
−0.503532 + 0.863977i \(0.667966\pi\)
\(140\) 0 0
\(141\) 6.70964 0.565053
\(142\) 0 0
\(143\) 0.561993 0.0469962
\(144\) 0 0
\(145\) 1.05086 0.0872688
\(146\) 0 0
\(147\) −0.688892 −0.0568189
\(148\) 0 0
\(149\) −17.6128 −1.44290 −0.721450 0.692466i \(-0.756524\pi\)
−0.721450 + 0.692466i \(0.756524\pi\)
\(150\) 0 0
\(151\) 8.04149 0.654407 0.327203 0.944954i \(-0.393894\pi\)
0.327203 + 0.944954i \(0.393894\pi\)
\(152\) 0 0
\(153\) −3.31111 −0.267687
\(154\) 0 0
\(155\) 2.44293 0.196221
\(156\) 0 0
\(157\) −9.31111 −0.743107 −0.371554 0.928411i \(-0.621175\pi\)
−0.371554 + 0.928411i \(0.621175\pi\)
\(158\) 0 0
\(159\) 4.72393 0.374632
\(160\) 0 0
\(161\) −1.00000 −0.0788110
\(162\) 0 0
\(163\) −1.53972 −0.120600 −0.0603000 0.998180i \(-0.519206\pi\)
−0.0603000 + 0.998180i \(0.519206\pi\)
\(164\) 0 0
\(165\) −0.428639 −0.0333695
\(166\) 0 0
\(167\) 14.3017 1.10670 0.553351 0.832948i \(-0.313349\pi\)
0.553351 + 0.832948i \(0.313349\pi\)
\(168\) 0 0
\(169\) −12.6128 −0.970219
\(170\) 0 0
\(171\) 17.8064 1.36169
\(172\) 0 0
\(173\) −2.56199 −0.194785 −0.0973923 0.995246i \(-0.531050\pi\)
−0.0973923 + 0.995246i \(0.531050\pi\)
\(174\) 0 0
\(175\) −4.52543 −0.342090
\(176\) 0 0
\(177\) 6.87157 0.516499
\(178\) 0 0
\(179\) 19.7748 1.47804 0.739018 0.673685i \(-0.235290\pi\)
0.739018 + 0.673685i \(0.235290\pi\)
\(180\) 0 0
\(181\) 12.3620 0.918857 0.459429 0.888215i \(-0.348054\pi\)
0.459429 + 0.888215i \(0.348054\pi\)
\(182\) 0 0
\(183\) −5.42372 −0.400933
\(184\) 0 0
\(185\) 1.67307 0.123007
\(186\) 0 0
\(187\) 1.18421 0.0865979
\(188\) 0 0
\(189\) 3.80642 0.276877
\(190\) 0 0
\(191\) 4.72393 0.341811 0.170906 0.985287i \(-0.445331\pi\)
0.170906 + 0.985287i \(0.445331\pi\)
\(192\) 0 0
\(193\) −0.714082 −0.0514007 −0.0257004 0.999670i \(-0.508182\pi\)
−0.0257004 + 0.999670i \(0.508182\pi\)
\(194\) 0 0
\(195\) −0.295286 −0.0211459
\(196\) 0 0
\(197\) −16.1017 −1.14720 −0.573600 0.819136i \(-0.694453\pi\)
−0.573600 + 0.819136i \(0.694453\pi\)
\(198\) 0 0
\(199\) −19.6128 −1.39032 −0.695159 0.718856i \(-0.744666\pi\)
−0.695159 + 0.718856i \(0.744666\pi\)
\(200\) 0 0
\(201\) −3.96836 −0.279906
\(202\) 0 0
\(203\) 1.52543 0.107064
\(204\) 0 0
\(205\) −3.27607 −0.228811
\(206\) 0 0
\(207\) 2.52543 0.175529
\(208\) 0 0
\(209\) −6.36842 −0.440513
\(210\) 0 0
\(211\) −21.2859 −1.46538 −0.732692 0.680561i \(-0.761736\pi\)
−0.732692 + 0.680561i \(0.761736\pi\)
\(212\) 0 0
\(213\) −5.24443 −0.359343
\(214\) 0 0
\(215\) −4.29529 −0.292936
\(216\) 0 0
\(217\) 3.54617 0.240730
\(218\) 0 0
\(219\) 1.28592 0.0868943
\(220\) 0 0
\(221\) 0.815792 0.0548761
\(222\) 0 0
\(223\) −13.8129 −0.924979 −0.462489 0.886625i \(-0.653044\pi\)
−0.462489 + 0.886625i \(0.653044\pi\)
\(224\) 0 0
\(225\) 11.4286 0.761909
\(226\) 0 0
\(227\) 16.8573 1.11886 0.559428 0.828879i \(-0.311021\pi\)
0.559428 + 0.828879i \(0.311021\pi\)
\(228\) 0 0
\(229\) 23.0257 1.52158 0.760789 0.648999i \(-0.224812\pi\)
0.760789 + 0.648999i \(0.224812\pi\)
\(230\) 0 0
\(231\) −0.622216 −0.0409388
\(232\) 0 0
\(233\) −1.61729 −0.105952 −0.0529762 0.998596i \(-0.516871\pi\)
−0.0529762 + 0.998596i \(0.516871\pi\)
\(234\) 0 0
\(235\) −6.70964 −0.437688
\(236\) 0 0
\(237\) 5.27607 0.342718
\(238\) 0 0
\(239\) −21.2859 −1.37687 −0.688436 0.725297i \(-0.741702\pi\)
−0.688436 + 0.725297i \(0.741702\pi\)
\(240\) 0 0
\(241\) −28.8736 −1.85991 −0.929955 0.367673i \(-0.880155\pi\)
−0.929955 + 0.367673i \(0.880155\pi\)
\(242\) 0 0
\(243\) −14.8321 −0.951479
\(244\) 0 0
\(245\) 0.688892 0.0440117
\(246\) 0 0
\(247\) −4.38715 −0.279148
\(248\) 0 0
\(249\) −7.64143 −0.484256
\(250\) 0 0
\(251\) −1.41927 −0.0895836 −0.0447918 0.998996i \(-0.514262\pi\)
−0.0447918 + 0.998996i \(0.514262\pi\)
\(252\) 0 0
\(253\) −0.903212 −0.0567844
\(254\) 0 0
\(255\) −0.622216 −0.0389647
\(256\) 0 0
\(257\) 18.8573 1.17628 0.588142 0.808757i \(-0.299859\pi\)
0.588142 + 0.808757i \(0.299859\pi\)
\(258\) 0 0
\(259\) 2.42864 0.150908
\(260\) 0 0
\(261\) −3.85236 −0.238455
\(262\) 0 0
\(263\) −2.44293 −0.150637 −0.0753187 0.997160i \(-0.523997\pi\)
−0.0753187 + 0.997160i \(0.523997\pi\)
\(264\) 0 0
\(265\) −4.72393 −0.290188
\(266\) 0 0
\(267\) 2.05578 0.125812
\(268\) 0 0
\(269\) 16.6637 1.01600 0.508002 0.861356i \(-0.330384\pi\)
0.508002 + 0.861356i \(0.330384\pi\)
\(270\) 0 0
\(271\) −32.6385 −1.98265 −0.991324 0.131439i \(-0.958040\pi\)
−0.991324 + 0.131439i \(0.958040\pi\)
\(272\) 0 0
\(273\) −0.428639 −0.0259424
\(274\) 0 0
\(275\) −4.08742 −0.246481
\(276\) 0 0
\(277\) −6.76986 −0.406761 −0.203381 0.979100i \(-0.565193\pi\)
−0.203381 + 0.979100i \(0.565193\pi\)
\(278\) 0 0
\(279\) −8.95560 −0.536158
\(280\) 0 0
\(281\) 27.1941 1.62226 0.811131 0.584865i \(-0.198853\pi\)
0.811131 + 0.584865i \(0.198853\pi\)
\(282\) 0 0
\(283\) 23.2257 1.38062 0.690312 0.723512i \(-0.257473\pi\)
0.690312 + 0.723512i \(0.257473\pi\)
\(284\) 0 0
\(285\) 3.34614 0.198208
\(286\) 0 0
\(287\) −4.75557 −0.280712
\(288\) 0 0
\(289\) −15.2810 −0.898882
\(290\) 0 0
\(291\) 1.71900 0.100770
\(292\) 0 0
\(293\) 5.25088 0.306760 0.153380 0.988167i \(-0.450984\pi\)
0.153380 + 0.988167i \(0.450984\pi\)
\(294\) 0 0
\(295\) −6.87157 −0.400078
\(296\) 0 0
\(297\) 3.43801 0.199493
\(298\) 0 0
\(299\) −0.622216 −0.0359837
\(300\) 0 0
\(301\) −6.23506 −0.359383
\(302\) 0 0
\(303\) 7.18421 0.412722
\(304\) 0 0
\(305\) 5.42372 0.310561
\(306\) 0 0
\(307\) 11.9333 0.681071 0.340535 0.940232i \(-0.389392\pi\)
0.340535 + 0.940232i \(0.389392\pi\)
\(308\) 0 0
\(309\) 6.36842 0.362287
\(310\) 0 0
\(311\) −0.0350337 −0.00198658 −0.000993289 1.00000i \(-0.500316\pi\)
−0.000993289 1.00000i \(0.500316\pi\)
\(312\) 0 0
\(313\) −24.3116 −1.37417 −0.687086 0.726576i \(-0.741110\pi\)
−0.687086 + 0.726576i \(0.741110\pi\)
\(314\) 0 0
\(315\) −1.73975 −0.0980237
\(316\) 0 0
\(317\) 4.84299 0.272009 0.136005 0.990708i \(-0.456574\pi\)
0.136005 + 0.990708i \(0.456574\pi\)
\(318\) 0 0
\(319\) 1.37778 0.0771411
\(320\) 0 0
\(321\) 12.2953 0.686256
\(322\) 0 0
\(323\) −9.24443 −0.514374
\(324\) 0 0
\(325\) −2.81579 −0.156192
\(326\) 0 0
\(327\) −7.57136 −0.418697
\(328\) 0 0
\(329\) −9.73975 −0.536970
\(330\) 0 0
\(331\) 17.5812 0.966350 0.483175 0.875524i \(-0.339483\pi\)
0.483175 + 0.875524i \(0.339483\pi\)
\(332\) 0 0
\(333\) −6.13335 −0.336106
\(334\) 0 0
\(335\) 3.96836 0.216815
\(336\) 0 0
\(337\) −17.8666 −0.973258 −0.486629 0.873609i \(-0.661774\pi\)
−0.486629 + 0.873609i \(0.661774\pi\)
\(338\) 0 0
\(339\) 8.52051 0.462770
\(340\) 0 0
\(341\) 3.20294 0.173449
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) 0 0
\(345\) 0.474572 0.0255501
\(346\) 0 0
\(347\) 31.1842 1.67406 0.837028 0.547160i \(-0.184291\pi\)
0.837028 + 0.547160i \(0.184291\pi\)
\(348\) 0 0
\(349\) 35.9813 1.92603 0.963016 0.269443i \(-0.0868394\pi\)
0.963016 + 0.269443i \(0.0868394\pi\)
\(350\) 0 0
\(351\) 2.36842 0.126417
\(352\) 0 0
\(353\) 29.2672 1.55773 0.778867 0.627189i \(-0.215794\pi\)
0.778867 + 0.627189i \(0.215794\pi\)
\(354\) 0 0
\(355\) 5.24443 0.278346
\(356\) 0 0
\(357\) −0.903212 −0.0478030
\(358\) 0 0
\(359\) −24.2208 −1.27832 −0.639162 0.769072i \(-0.720719\pi\)
−0.639162 + 0.769072i \(0.720719\pi\)
\(360\) 0 0
\(361\) 30.7146 1.61656
\(362\) 0 0
\(363\) 7.01582 0.368235
\(364\) 0 0
\(365\) −1.28592 −0.0673080
\(366\) 0 0
\(367\) −23.6543 −1.23475 −0.617373 0.786670i \(-0.711803\pi\)
−0.617373 + 0.786670i \(0.711803\pi\)
\(368\) 0 0
\(369\) 12.0098 0.625208
\(370\) 0 0
\(371\) −6.85728 −0.356012
\(372\) 0 0
\(373\) −32.8385 −1.70032 −0.850158 0.526528i \(-0.823493\pi\)
−0.850158 + 0.526528i \(0.823493\pi\)
\(374\) 0 0
\(375\) 4.52051 0.233438
\(376\) 0 0
\(377\) 0.949145 0.0488834
\(378\) 0 0
\(379\) 4.63651 0.238161 0.119081 0.992885i \(-0.462005\pi\)
0.119081 + 0.992885i \(0.462005\pi\)
\(380\) 0 0
\(381\) 5.80642 0.297472
\(382\) 0 0
\(383\) 1.28592 0.0657074 0.0328537 0.999460i \(-0.489540\pi\)
0.0328537 + 0.999460i \(0.489540\pi\)
\(384\) 0 0
\(385\) 0.622216 0.0317110
\(386\) 0 0
\(387\) 15.7462 0.800424
\(388\) 0 0
\(389\) 24.8256 1.25871 0.629355 0.777118i \(-0.283319\pi\)
0.629355 + 0.777118i \(0.283319\pi\)
\(390\) 0 0
\(391\) −1.31111 −0.0663056
\(392\) 0 0
\(393\) −11.8622 −0.598369
\(394\) 0 0
\(395\) −5.27607 −0.265468
\(396\) 0 0
\(397\) −5.09234 −0.255577 −0.127789 0.991801i \(-0.540788\pi\)
−0.127789 + 0.991801i \(0.540788\pi\)
\(398\) 0 0
\(399\) 4.85728 0.243168
\(400\) 0 0
\(401\) 20.6222 1.02982 0.514912 0.857243i \(-0.327825\pi\)
0.514912 + 0.857243i \(0.327825\pi\)
\(402\) 0 0
\(403\) 2.20648 0.109913
\(404\) 0 0
\(405\) 3.41282 0.169584
\(406\) 0 0
\(407\) 2.19358 0.108732
\(408\) 0 0
\(409\) 37.3876 1.84870 0.924350 0.381547i \(-0.124608\pi\)
0.924350 + 0.381547i \(0.124608\pi\)
\(410\) 0 0
\(411\) 3.57136 0.176162
\(412\) 0 0
\(413\) −9.97481 −0.490828
\(414\) 0 0
\(415\) 7.64143 0.375103
\(416\) 0 0
\(417\) 8.17929 0.400541
\(418\) 0 0
\(419\) −20.7654 −1.01446 −0.507228 0.861812i \(-0.669330\pi\)
−0.507228 + 0.861812i \(0.669330\pi\)
\(420\) 0 0
\(421\) 2.59057 0.126257 0.0631284 0.998005i \(-0.479892\pi\)
0.0631284 + 0.998005i \(0.479892\pi\)
\(422\) 0 0
\(423\) 24.5970 1.19595
\(424\) 0 0
\(425\) −5.93332 −0.287808
\(426\) 0 0
\(427\) 7.87310 0.381006
\(428\) 0 0
\(429\) −0.387152 −0.0186919
\(430\) 0 0
\(431\) −11.3461 −0.546524 −0.273262 0.961940i \(-0.588103\pi\)
−0.273262 + 0.961940i \(0.588103\pi\)
\(432\) 0 0
\(433\) 33.0257 1.58711 0.793556 0.608497i \(-0.208227\pi\)
0.793556 + 0.608497i \(0.208227\pi\)
\(434\) 0 0
\(435\) −0.723926 −0.0347096
\(436\) 0 0
\(437\) 7.05086 0.337288
\(438\) 0 0
\(439\) 14.8321 0.707897 0.353949 0.935265i \(-0.384839\pi\)
0.353949 + 0.935265i \(0.384839\pi\)
\(440\) 0 0
\(441\) −2.52543 −0.120258
\(442\) 0 0
\(443\) 2.58073 0.122614 0.0613071 0.998119i \(-0.480473\pi\)
0.0613071 + 0.998119i \(0.480473\pi\)
\(444\) 0 0
\(445\) −2.05578 −0.0974532
\(446\) 0 0
\(447\) 12.1334 0.573888
\(448\) 0 0
\(449\) −42.1990 −1.99149 −0.995746 0.0921366i \(-0.970630\pi\)
−0.995746 + 0.0921366i \(0.970630\pi\)
\(450\) 0 0
\(451\) −4.29529 −0.202257
\(452\) 0 0
\(453\) −5.53972 −0.260279
\(454\) 0 0
\(455\) 0.428639 0.0200949
\(456\) 0 0
\(457\) −8.07313 −0.377645 −0.188823 0.982011i \(-0.560467\pi\)
−0.188823 + 0.982011i \(0.560467\pi\)
\(458\) 0 0
\(459\) 4.99063 0.232943
\(460\) 0 0
\(461\) 15.8064 0.736179 0.368089 0.929790i \(-0.380012\pi\)
0.368089 + 0.929790i \(0.380012\pi\)
\(462\) 0 0
\(463\) 20.6035 0.957525 0.478762 0.877944i \(-0.341086\pi\)
0.478762 + 0.877944i \(0.341086\pi\)
\(464\) 0 0
\(465\) −1.68292 −0.0780433
\(466\) 0 0
\(467\) 10.0731 0.466129 0.233064 0.972461i \(-0.425125\pi\)
0.233064 + 0.972461i \(0.425125\pi\)
\(468\) 0 0
\(469\) 5.76049 0.265995
\(470\) 0 0
\(471\) 6.41435 0.295558
\(472\) 0 0
\(473\) −5.63158 −0.258940
\(474\) 0 0
\(475\) 31.9081 1.46405
\(476\) 0 0
\(477\) 17.3176 0.792917
\(478\) 0 0
\(479\) 23.2543 1.06251 0.531257 0.847210i \(-0.321720\pi\)
0.531257 + 0.847210i \(0.321720\pi\)
\(480\) 0 0
\(481\) 1.51114 0.0689019
\(482\) 0 0
\(483\) 0.688892 0.0313457
\(484\) 0 0
\(485\) −1.71900 −0.0780559
\(486\) 0 0
\(487\) −9.51114 −0.430991 −0.215495 0.976505i \(-0.569137\pi\)
−0.215495 + 0.976505i \(0.569137\pi\)
\(488\) 0 0
\(489\) 1.06070 0.0479665
\(490\) 0 0
\(491\) 20.4701 0.923804 0.461902 0.886931i \(-0.347167\pi\)
0.461902 + 0.886931i \(0.347167\pi\)
\(492\) 0 0
\(493\) 2.00000 0.0900755
\(494\) 0 0
\(495\) −1.57136 −0.0706274
\(496\) 0 0
\(497\) 7.61285 0.341483
\(498\) 0 0
\(499\) −20.3368 −0.910399 −0.455200 0.890389i \(-0.650432\pi\)
−0.455200 + 0.890389i \(0.650432\pi\)
\(500\) 0 0
\(501\) −9.85236 −0.440171
\(502\) 0 0
\(503\) −6.27655 −0.279858 −0.139929 0.990162i \(-0.544687\pi\)
−0.139929 + 0.990162i \(0.544687\pi\)
\(504\) 0 0
\(505\) −7.18421 −0.319693
\(506\) 0 0
\(507\) 8.68889 0.385887
\(508\) 0 0
\(509\) −36.3684 −1.61200 −0.806001 0.591914i \(-0.798372\pi\)
−0.806001 + 0.591914i \(0.798372\pi\)
\(510\) 0 0
\(511\) −1.86665 −0.0825756
\(512\) 0 0
\(513\) −26.8385 −1.18495
\(514\) 0 0
\(515\) −6.36842 −0.280626
\(516\) 0 0
\(517\) −8.79706 −0.386894
\(518\) 0 0
\(519\) 1.76494 0.0774721
\(520\) 0 0
\(521\) 27.0988 1.18722 0.593610 0.804753i \(-0.297702\pi\)
0.593610 + 0.804753i \(0.297702\pi\)
\(522\) 0 0
\(523\) 21.1240 0.923687 0.461844 0.886961i \(-0.347188\pi\)
0.461844 + 0.886961i \(0.347188\pi\)
\(524\) 0 0
\(525\) 3.11753 0.136060
\(526\) 0 0
\(527\) 4.64941 0.202532
\(528\) 0 0
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) 25.1907 1.09318
\(532\) 0 0
\(533\) −2.95899 −0.128168
\(534\) 0 0
\(535\) −12.2953 −0.531572
\(536\) 0 0
\(537\) −13.6227 −0.587863
\(538\) 0 0
\(539\) 0.903212 0.0389041
\(540\) 0 0
\(541\) 5.61729 0.241506 0.120753 0.992683i \(-0.461469\pi\)
0.120753 + 0.992683i \(0.461469\pi\)
\(542\) 0 0
\(543\) −8.51606 −0.365459
\(544\) 0 0
\(545\) 7.57136 0.324321
\(546\) 0 0
\(547\) 18.7971 0.803704 0.401852 0.915705i \(-0.368367\pi\)
0.401852 + 0.915705i \(0.368367\pi\)
\(548\) 0 0
\(549\) −19.8829 −0.848583
\(550\) 0 0
\(551\) −10.7556 −0.458203
\(552\) 0 0
\(553\) −7.65878 −0.325684
\(554\) 0 0
\(555\) −1.15257 −0.0489237
\(556\) 0 0
\(557\) 28.8385 1.22193 0.610964 0.791658i \(-0.290782\pi\)
0.610964 + 0.791658i \(0.290782\pi\)
\(558\) 0 0
\(559\) −3.87955 −0.164088
\(560\) 0 0
\(561\) −0.815792 −0.0344428
\(562\) 0 0
\(563\) 24.2034 1.02005 0.510026 0.860159i \(-0.329636\pi\)
0.510026 + 0.860159i \(0.329636\pi\)
\(564\) 0 0
\(565\) −8.52051 −0.358460
\(566\) 0 0
\(567\) 4.95407 0.208051
\(568\) 0 0
\(569\) 2.69535 0.112995 0.0564974 0.998403i \(-0.482007\pi\)
0.0564974 + 0.998403i \(0.482007\pi\)
\(570\) 0 0
\(571\) −28.6035 −1.19702 −0.598509 0.801116i \(-0.704240\pi\)
−0.598509 + 0.801116i \(0.704240\pi\)
\(572\) 0 0
\(573\) −3.25428 −0.135949
\(574\) 0 0
\(575\) 4.52543 0.188723
\(576\) 0 0
\(577\) 47.1655 1.96352 0.981762 0.190113i \(-0.0608855\pi\)
0.981762 + 0.190113i \(0.0608855\pi\)
\(578\) 0 0
\(579\) 0.491925 0.0204437
\(580\) 0 0
\(581\) 11.0923 0.460188
\(582\) 0 0
\(583\) −6.19358 −0.256512
\(584\) 0 0
\(585\) −1.08250 −0.0447558
\(586\) 0 0
\(587\) 6.88247 0.284070 0.142035 0.989862i \(-0.454635\pi\)
0.142035 + 0.989862i \(0.454635\pi\)
\(588\) 0 0
\(589\) −25.0035 −1.03025
\(590\) 0 0
\(591\) 11.0923 0.456278
\(592\) 0 0
\(593\) 7.41927 0.304673 0.152336 0.988329i \(-0.451320\pi\)
0.152336 + 0.988329i \(0.451320\pi\)
\(594\) 0 0
\(595\) 0.903212 0.0370281
\(596\) 0 0
\(597\) 13.5111 0.552974
\(598\) 0 0
\(599\) 2.71408 0.110894 0.0554472 0.998462i \(-0.482342\pi\)
0.0554472 + 0.998462i \(0.482342\pi\)
\(600\) 0 0
\(601\) 28.4099 1.15886 0.579432 0.815021i \(-0.303274\pi\)
0.579432 + 0.815021i \(0.303274\pi\)
\(602\) 0 0
\(603\) −14.5477 −0.592428
\(604\) 0 0
\(605\) −7.01582 −0.285234
\(606\) 0 0
\(607\) 22.8508 0.927486 0.463743 0.885970i \(-0.346506\pi\)
0.463743 + 0.885970i \(0.346506\pi\)
\(608\) 0 0
\(609\) −1.05086 −0.0425828
\(610\) 0 0
\(611\) −6.06022 −0.245170
\(612\) 0 0
\(613\) −33.3590 −1.34736 −0.673680 0.739024i \(-0.735287\pi\)
−0.673680 + 0.739024i \(0.735287\pi\)
\(614\) 0 0
\(615\) 2.25686 0.0910055
\(616\) 0 0
\(617\) 13.7748 0.554552 0.277276 0.960790i \(-0.410568\pi\)
0.277276 + 0.960790i \(0.410568\pi\)
\(618\) 0 0
\(619\) −28.6035 −1.14967 −0.574835 0.818269i \(-0.694934\pi\)
−0.574835 + 0.818269i \(0.694934\pi\)
\(620\) 0 0
\(621\) −3.80642 −0.152747
\(622\) 0 0
\(623\) −2.98418 −0.119559
\(624\) 0 0
\(625\) 18.1066 0.724265
\(626\) 0 0
\(627\) 4.38715 0.175206
\(628\) 0 0
\(629\) 3.18421 0.126963
\(630\) 0 0
\(631\) −30.4143 −1.21078 −0.605388 0.795930i \(-0.706982\pi\)
−0.605388 + 0.795930i \(0.706982\pi\)
\(632\) 0 0
\(633\) 14.6637 0.582830
\(634\) 0 0
\(635\) −5.80642 −0.230421
\(636\) 0 0
\(637\) 0.622216 0.0246531
\(638\) 0 0
\(639\) −19.2257 −0.760557
\(640\) 0 0
\(641\) −5.30465 −0.209521 −0.104761 0.994497i \(-0.533408\pi\)
−0.104761 + 0.994497i \(0.533408\pi\)
\(642\) 0 0
\(643\) 50.5718 1.99436 0.997179 0.0750565i \(-0.0239137\pi\)
0.997179 + 0.0750565i \(0.0239137\pi\)
\(644\) 0 0
\(645\) 2.95899 0.116510
\(646\) 0 0
\(647\) 14.3936 0.565871 0.282936 0.959139i \(-0.408692\pi\)
0.282936 + 0.959139i \(0.408692\pi\)
\(648\) 0 0
\(649\) −9.00937 −0.353649
\(650\) 0 0
\(651\) −2.44293 −0.0957460
\(652\) 0 0
\(653\) 7.98126 0.312331 0.156166 0.987731i \(-0.450087\pi\)
0.156166 + 0.987731i \(0.450087\pi\)
\(654\) 0 0
\(655\) 11.8622 0.463495
\(656\) 0 0
\(657\) 4.71408 0.183914
\(658\) 0 0
\(659\) 17.5111 0.682137 0.341069 0.940038i \(-0.389211\pi\)
0.341069 + 0.940038i \(0.389211\pi\)
\(660\) 0 0
\(661\) −28.7719 −1.11910 −0.559548 0.828798i \(-0.689025\pi\)
−0.559548 + 0.828798i \(0.689025\pi\)
\(662\) 0 0
\(663\) −0.561993 −0.0218260
\(664\) 0 0
\(665\) −4.85728 −0.188357
\(666\) 0 0
\(667\) −1.52543 −0.0590648
\(668\) 0 0
\(669\) 9.51558 0.367894
\(670\) 0 0
\(671\) 7.11108 0.274520
\(672\) 0 0
\(673\) −39.0651 −1.50585 −0.752925 0.658106i \(-0.771358\pi\)
−0.752925 + 0.658106i \(0.771358\pi\)
\(674\) 0 0
\(675\) −17.2257 −0.663017
\(676\) 0 0
\(677\) −10.3749 −0.398739 −0.199369 0.979924i \(-0.563889\pi\)
−0.199369 + 0.979924i \(0.563889\pi\)
\(678\) 0 0
\(679\) −2.49532 −0.0957614
\(680\) 0 0
\(681\) −11.6128 −0.445005
\(682\) 0 0
\(683\) −1.41927 −0.0543069 −0.0271535 0.999631i \(-0.508644\pi\)
−0.0271535 + 0.999631i \(0.508644\pi\)
\(684\) 0 0
\(685\) −3.57136 −0.136455
\(686\) 0 0
\(687\) −15.8622 −0.605181
\(688\) 0 0
\(689\) −4.26671 −0.162549
\(690\) 0 0
\(691\) 10.0350 0.381751 0.190875 0.981614i \(-0.438867\pi\)
0.190875 + 0.981614i \(0.438867\pi\)
\(692\) 0 0
\(693\) −2.28100 −0.0866479
\(694\) 0 0
\(695\) −8.17929 −0.310258
\(696\) 0 0
\(697\) −6.23506 −0.236170
\(698\) 0 0
\(699\) 1.11414 0.0421407
\(700\) 0 0
\(701\) 39.8765 1.50611 0.753057 0.657955i \(-0.228578\pi\)
0.753057 + 0.657955i \(0.228578\pi\)
\(702\) 0 0
\(703\) −17.1240 −0.645843
\(704\) 0 0
\(705\) 4.62222 0.174083
\(706\) 0 0
\(707\) −10.4286 −0.392209
\(708\) 0 0
\(709\) 11.3778 0.427302 0.213651 0.976910i \(-0.431465\pi\)
0.213651 + 0.976910i \(0.431465\pi\)
\(710\) 0 0
\(711\) 19.3417 0.725370
\(712\) 0 0
\(713\) −3.54617 −0.132805
\(714\) 0 0
\(715\) 0.387152 0.0144787
\(716\) 0 0
\(717\) 14.6637 0.547626
\(718\) 0 0
\(719\) 36.5783 1.36414 0.682070 0.731287i \(-0.261080\pi\)
0.682070 + 0.731287i \(0.261080\pi\)
\(720\) 0 0
\(721\) −9.24443 −0.344281
\(722\) 0 0
\(723\) 19.8908 0.739746
\(724\) 0 0
\(725\) −6.90321 −0.256379
\(726\) 0 0
\(727\) −16.3872 −0.607766 −0.303883 0.952709i \(-0.598283\pi\)
−0.303883 + 0.952709i \(0.598283\pi\)
\(728\) 0 0
\(729\) −4.64449 −0.172018
\(730\) 0 0
\(731\) −8.17484 −0.302357
\(732\) 0 0
\(733\) 47.9976 1.77283 0.886415 0.462891i \(-0.153188\pi\)
0.886415 + 0.462891i \(0.153188\pi\)
\(734\) 0 0
\(735\) −0.474572 −0.0175049
\(736\) 0 0
\(737\) 5.20294 0.191653
\(738\) 0 0
\(739\) −22.4514 −0.825888 −0.412944 0.910756i \(-0.635499\pi\)
−0.412944 + 0.910756i \(0.635499\pi\)
\(740\) 0 0
\(741\) 3.02227 0.111026
\(742\) 0 0
\(743\) 46.5892 1.70919 0.854596 0.519294i \(-0.173805\pi\)
0.854596 + 0.519294i \(0.173805\pi\)
\(744\) 0 0
\(745\) −12.1334 −0.444532
\(746\) 0 0
\(747\) −28.0129 −1.02494
\(748\) 0 0
\(749\) −17.8479 −0.652148
\(750\) 0 0
\(751\) −40.8113 −1.48923 −0.744614 0.667496i \(-0.767366\pi\)
−0.744614 + 0.667496i \(0.767366\pi\)
\(752\) 0 0
\(753\) 0.977725 0.0356303
\(754\) 0 0
\(755\) 5.53972 0.201611
\(756\) 0 0
\(757\) −1.30465 −0.0474185 −0.0237092 0.999719i \(-0.507548\pi\)
−0.0237092 + 0.999719i \(0.507548\pi\)
\(758\) 0 0
\(759\) 0.622216 0.0225850
\(760\) 0 0
\(761\) −17.4795 −0.633631 −0.316816 0.948487i \(-0.602614\pi\)
−0.316816 + 0.948487i \(0.602614\pi\)
\(762\) 0 0
\(763\) 10.9906 0.397888
\(764\) 0 0
\(765\) −2.28100 −0.0824696
\(766\) 0 0
\(767\) −6.20648 −0.224103
\(768\) 0 0
\(769\) 0.361963 0.0130527 0.00652636 0.999979i \(-0.497923\pi\)
0.00652636 + 0.999979i \(0.497923\pi\)
\(770\) 0 0
\(771\) −12.9906 −0.467846
\(772\) 0 0
\(773\) 14.5970 0.525019 0.262509 0.964929i \(-0.415450\pi\)
0.262509 + 0.964929i \(0.415450\pi\)
\(774\) 0 0
\(775\) −16.0479 −0.576459
\(776\) 0 0
\(777\) −1.67307 −0.0600211
\(778\) 0 0
\(779\) 33.5308 1.20137
\(780\) 0 0
\(781\) 6.87601 0.246043
\(782\) 0 0
\(783\) 5.80642 0.207505
\(784\) 0 0
\(785\) −6.41435 −0.228938
\(786\) 0 0
\(787\) 7.00048 0.249540 0.124770 0.992186i \(-0.460181\pi\)
0.124770 + 0.992186i \(0.460181\pi\)
\(788\) 0 0
\(789\) 1.68292 0.0599134
\(790\) 0 0
\(791\) −12.3684 −0.439770
\(792\) 0 0
\(793\) 4.89877 0.173960
\(794\) 0 0
\(795\) 3.25428 0.115417
\(796\) 0 0
\(797\) 14.5841 0.516596 0.258298 0.966065i \(-0.416838\pi\)
0.258298 + 0.966065i \(0.416838\pi\)
\(798\) 0 0
\(799\) −12.7699 −0.451765
\(800\) 0 0
\(801\) 7.53633 0.266283
\(802\) 0 0
\(803\) −1.68598 −0.0594969
\(804\) 0 0
\(805\) −0.688892 −0.0242803
\(806\) 0 0
\(807\) −11.4795 −0.404097
\(808\) 0 0
\(809\) 24.0602 0.845912 0.422956 0.906150i \(-0.360992\pi\)
0.422956 + 0.906150i \(0.360992\pi\)
\(810\) 0 0
\(811\) 19.7081 0.692045 0.346023 0.938226i \(-0.387532\pi\)
0.346023 + 0.938226i \(0.387532\pi\)
\(812\) 0 0
\(813\) 22.4844 0.788563
\(814\) 0 0
\(815\) −1.06070 −0.0371547
\(816\) 0 0
\(817\) 43.9625 1.53805
\(818\) 0 0
\(819\) −1.57136 −0.0549078
\(820\) 0 0
\(821\) −24.7797 −0.864818 −0.432409 0.901678i \(-0.642336\pi\)
−0.432409 + 0.901678i \(0.642336\pi\)
\(822\) 0 0
\(823\) 52.2578 1.82159 0.910796 0.412856i \(-0.135469\pi\)
0.910796 + 0.412856i \(0.135469\pi\)
\(824\) 0 0
\(825\) 2.81579 0.0980333
\(826\) 0 0
\(827\) −30.8385 −1.07236 −0.536181 0.844103i \(-0.680133\pi\)
−0.536181 + 0.844103i \(0.680133\pi\)
\(828\) 0 0
\(829\) −26.3497 −0.915162 −0.457581 0.889168i \(-0.651284\pi\)
−0.457581 + 0.889168i \(0.651284\pi\)
\(830\) 0 0
\(831\) 4.66370 0.161782
\(832\) 0 0
\(833\) 1.31111 0.0454272
\(834\) 0 0
\(835\) 9.85236 0.340955
\(836\) 0 0
\(837\) 13.4982 0.466567
\(838\) 0 0
\(839\) −22.1432 −0.764468 −0.382234 0.924066i \(-0.624845\pi\)
−0.382234 + 0.924066i \(0.624845\pi\)
\(840\) 0 0
\(841\) −26.6731 −0.919761
\(842\) 0 0
\(843\) −18.7338 −0.645225
\(844\) 0 0
\(845\) −8.68889 −0.298907
\(846\) 0 0
\(847\) −10.1842 −0.349934
\(848\) 0 0
\(849\) −16.0000 −0.549119
\(850\) 0 0
\(851\) −2.42864 −0.0832527
\(852\) 0 0
\(853\) 31.8479 1.09045 0.545226 0.838289i \(-0.316444\pi\)
0.545226 + 0.838289i \(0.316444\pi\)
\(854\) 0 0
\(855\) 12.2667 0.419513
\(856\) 0 0
\(857\) −35.5308 −1.21371 −0.606855 0.794813i \(-0.707569\pi\)
−0.606855 + 0.794813i \(0.707569\pi\)
\(858\) 0 0
\(859\) −1.53326 −0.0523143 −0.0261571 0.999658i \(-0.508327\pi\)
−0.0261571 + 0.999658i \(0.508327\pi\)
\(860\) 0 0
\(861\) 3.27607 0.111648
\(862\) 0 0
\(863\) −48.4068 −1.64779 −0.823894 0.566744i \(-0.808203\pi\)
−0.823894 + 0.566744i \(0.808203\pi\)
\(864\) 0 0
\(865\) −1.76494 −0.0600096
\(866\) 0 0
\(867\) 10.5270 0.357514
\(868\) 0 0
\(869\) −6.91750 −0.234660
\(870\) 0 0
\(871\) 3.58427 0.121448
\(872\) 0 0
\(873\) 6.30174 0.213282
\(874\) 0 0
\(875\) −6.56199 −0.221836
\(876\) 0 0
\(877\) −26.6780 −0.900852 −0.450426 0.892814i \(-0.648728\pi\)
−0.450426 + 0.892814i \(0.648728\pi\)
\(878\) 0 0
\(879\) −3.61729 −0.122008
\(880\) 0 0
\(881\) 16.3749 0.551683 0.275842 0.961203i \(-0.411043\pi\)
0.275842 + 0.961203i \(0.411043\pi\)
\(882\) 0 0
\(883\) 16.0415 0.539839 0.269920 0.962883i \(-0.413003\pi\)
0.269920 + 0.962883i \(0.413003\pi\)
\(884\) 0 0
\(885\) 4.73377 0.159124
\(886\) 0 0
\(887\) −28.8637 −0.969149 −0.484575 0.874750i \(-0.661026\pi\)
−0.484575 + 0.874750i \(0.661026\pi\)
\(888\) 0 0
\(889\) −8.42864 −0.282687
\(890\) 0 0
\(891\) 4.47457 0.149904
\(892\) 0 0
\(893\) 68.6735 2.29807
\(894\) 0 0
\(895\) 13.6227 0.455356
\(896\) 0 0
\(897\) 0.428639 0.0143119
\(898\) 0 0
\(899\) 5.40943 0.180414
\(900\) 0 0
\(901\) −8.99063 −0.299521
\(902\) 0 0
\(903\) 4.29529 0.142938
\(904\) 0 0
\(905\) 8.51606 0.283083
\(906\) 0 0
\(907\) 0.387152 0.0128552 0.00642759 0.999979i \(-0.497954\pi\)
0.00642759 + 0.999979i \(0.497954\pi\)
\(908\) 0 0
\(909\) 26.3368 0.873536
\(910\) 0 0
\(911\) −23.3921 −0.775014 −0.387507 0.921867i \(-0.626664\pi\)
−0.387507 + 0.921867i \(0.626664\pi\)
\(912\) 0 0
\(913\) 10.0187 0.331572
\(914\) 0 0
\(915\) −3.73636 −0.123520
\(916\) 0 0
\(917\) 17.2192 0.568629
\(918\) 0 0
\(919\) −19.4839 −0.642716 −0.321358 0.946958i \(-0.604139\pi\)
−0.321358 + 0.946958i \(0.604139\pi\)
\(920\) 0 0
\(921\) −8.22077 −0.270884
\(922\) 0 0
\(923\) 4.73683 0.155915
\(924\) 0 0
\(925\) −10.9906 −0.361370
\(926\) 0 0
\(927\) 23.3461 0.766788
\(928\) 0 0
\(929\) −44.6548 −1.46508 −0.732538 0.680726i \(-0.761665\pi\)
−0.732538 + 0.680726i \(0.761665\pi\)
\(930\) 0 0
\(931\) −7.05086 −0.231082
\(932\) 0 0
\(933\) 0.0241344 0.000790126 0
\(934\) 0 0
\(935\) 0.815792 0.0266792
\(936\) 0 0
\(937\) −28.5368 −0.932257 −0.466128 0.884717i \(-0.654352\pi\)
−0.466128 + 0.884717i \(0.654352\pi\)
\(938\) 0 0
\(939\) 16.7481 0.546552
\(940\) 0 0
\(941\) −37.2607 −1.21467 −0.607333 0.794448i \(-0.707760\pi\)
−0.607333 + 0.794448i \(0.707760\pi\)
\(942\) 0 0
\(943\) 4.75557 0.154863
\(944\) 0 0
\(945\) 2.62222 0.0853007
\(946\) 0 0
\(947\) 20.9906 0.682104 0.341052 0.940044i \(-0.389217\pi\)
0.341052 + 0.940044i \(0.389217\pi\)
\(948\) 0 0
\(949\) −1.16146 −0.0377025
\(950\) 0 0
\(951\) −3.33630 −0.108187
\(952\) 0 0
\(953\) −14.8859 −0.482200 −0.241100 0.970500i \(-0.577508\pi\)
−0.241100 + 0.970500i \(0.577508\pi\)
\(954\) 0 0
\(955\) 3.25428 0.105306
\(956\) 0 0
\(957\) −0.949145 −0.0306815
\(958\) 0 0
\(959\) −5.18421 −0.167407
\(960\) 0 0
\(961\) −18.4247 −0.594344
\(962\) 0 0
\(963\) 45.0736 1.45248
\(964\) 0 0
\(965\) −0.491925 −0.0158356
\(966\) 0 0
\(967\) −29.5397 −0.949933 −0.474967 0.880004i \(-0.657540\pi\)
−0.474967 + 0.880004i \(0.657540\pi\)
\(968\) 0 0
\(969\) 6.36842 0.204583
\(970\) 0 0
\(971\) −20.5491 −0.659452 −0.329726 0.944077i \(-0.606956\pi\)
−0.329726 + 0.944077i \(0.606956\pi\)
\(972\) 0 0
\(973\) −11.8731 −0.380634
\(974\) 0 0
\(975\) 1.93978 0.0621226
\(976\) 0 0
\(977\) 37.2128 1.19054 0.595271 0.803525i \(-0.297045\pi\)
0.595271 + 0.803525i \(0.297045\pi\)
\(978\) 0 0
\(979\) −2.69535 −0.0861436
\(980\) 0 0
\(981\) −27.7560 −0.886182
\(982\) 0 0
\(983\) 33.2859 1.06166 0.530828 0.847480i \(-0.321881\pi\)
0.530828 + 0.847480i \(0.321881\pi\)
\(984\) 0 0
\(985\) −11.0923 −0.353431
\(986\) 0 0
\(987\) 6.70964 0.213570
\(988\) 0 0
\(989\) 6.23506 0.198263
\(990\) 0 0
\(991\) −12.7368 −0.404599 −0.202299 0.979324i \(-0.564841\pi\)
−0.202299 + 0.979324i \(0.564841\pi\)
\(992\) 0 0
\(993\) −12.1116 −0.384349
\(994\) 0 0
\(995\) −13.5111 −0.428332
\(996\) 0 0
\(997\) −13.0509 −0.413325 −0.206662 0.978412i \(-0.566260\pi\)
−0.206662 + 0.978412i \(0.566260\pi\)
\(998\) 0 0
\(999\) 9.24443 0.292481
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 2576.2.a.v.1.2 3
4.3 odd 2 161.2.a.c.1.3 3
12.11 even 2 1449.2.a.m.1.1 3
20.19 odd 2 4025.2.a.j.1.1 3
28.27 even 2 1127.2.a.f.1.3 3
92.91 even 2 3703.2.a.c.1.3 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
161.2.a.c.1.3 3 4.3 odd 2
1127.2.a.f.1.3 3 28.27 even 2
1449.2.a.m.1.1 3 12.11 even 2
2576.2.a.v.1.2 3 1.1 even 1 trivial
3703.2.a.c.1.3 3 92.91 even 2
4025.2.a.j.1.1 3 20.19 odd 2