Properties

Label 3234.2.e.a.2155.14
Level $3234$
Weight $2$
Character 3234.2155
Analytic conductor $25.824$
Analytic rank $0$
Dimension $16$
CM no
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [3234,2,Mod(2155,3234)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(3234, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 1, 1]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("3234.2155");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 3234 = 2 \cdot 3 \cdot 7^{2} \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3234.e (of order \(2\), degree \(1\), minimal)

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: no
Analytic conductor: \(25.8236200137\)
Analytic rank: \(0\)
Dimension: \(16\)
Coefficient field: \(\mathbb{Q}[x]/(x^{16} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{16} - 8 x^{15} + 74 x^{14} - 378 x^{13} + 1878 x^{12} - 6718 x^{11} + 22086 x^{10} - 56904 x^{9} + \cdots + 13417 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{6} \)
Twist minimal: no (minimal twist has level 462)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 2155.14
Root \(0.500000 - 0.921602i\) of defining polynomial
Character \(\chi\) \(=\) 3234.2155
Dual form 3234.2.e.a.2155.3

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.78763i q^{5} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} +O(q^{10})\) \(q+1.00000i q^{2} +1.00000i q^{3} -1.00000 q^{4} +1.78763i q^{5} -1.00000 q^{6} -1.00000i q^{8} -1.00000 q^{9} -1.78763 q^{10} +(-3.15519 + 1.02213i) q^{11} -1.00000i q^{12} +6.37742 q^{13} -1.78763 q^{15} +1.00000 q^{16} -0.106095 q^{17} -1.00000i q^{18} +4.15162 q^{19} -1.78763i q^{20} +(-1.02213 - 3.15519i) q^{22} +7.95866 q^{23} +1.00000 q^{24} +1.80439 q^{25} +6.37742i q^{26} -1.00000i q^{27} +7.65230i q^{29} -1.78763i q^{30} -1.27872i q^{31} +1.00000i q^{32} +(-1.02213 - 3.15519i) q^{33} -0.106095i q^{34} +1.00000 q^{36} +4.52184 q^{37} +4.15162i q^{38} +6.37742i q^{39} +1.78763 q^{40} +0.0321383 q^{41} +6.87723i q^{43} +(3.15519 - 1.02213i) q^{44} -1.78763i q^{45} +7.95866i q^{46} +9.88198i q^{47} +1.00000i q^{48} +1.80439i q^{50} -0.106095i q^{51} -6.37742 q^{52} -0.627441 q^{53} +1.00000 q^{54} +(-1.82718 - 5.64031i) q^{55} +4.15162i q^{57} -7.65230 q^{58} -12.4549i q^{59} +1.78763 q^{60} -9.95024 q^{61} +1.27872 q^{62} -1.00000 q^{64} +11.4004i q^{65} +(3.15519 - 1.02213i) q^{66} +10.8722 q^{67} +0.106095 q^{68} +7.95866i q^{69} -8.42785 q^{71} +1.00000i q^{72} -0.125008 q^{73} +4.52184i q^{74} +1.80439i q^{75} -4.15162 q^{76} -6.37742 q^{78} -10.1691i q^{79} +1.78763i q^{80} +1.00000 q^{81} +0.0321383i q^{82} -12.0657 q^{83} -0.189659i q^{85} -6.87723 q^{86} -7.65230 q^{87} +(1.02213 + 3.15519i) q^{88} -5.36009i q^{89} +1.78763 q^{90} -7.95866 q^{92} +1.27872 q^{93} -9.88198 q^{94} +7.42156i q^{95} -1.00000 q^{96} +15.7498i q^{97} +(3.15519 - 1.02213i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 16 q - 16 q^{4} - 16 q^{6} - 16 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 16 q - 16 q^{4} - 16 q^{6} - 16 q^{9} - 4 q^{10} + 8 q^{11} - 4 q^{15} + 16 q^{16} + 20 q^{19} + 2 q^{22} + 8 q^{23} + 16 q^{24} - 20 q^{25} + 2 q^{33} + 16 q^{36} - 28 q^{37} + 4 q^{40} + 32 q^{41} - 8 q^{44} + 16 q^{54} - 14 q^{55} + 4 q^{60} - 56 q^{61} - 8 q^{62} - 16 q^{64} - 8 q^{66} + 32 q^{67} - 56 q^{71} + 88 q^{73} - 20 q^{76} + 16 q^{81} + 8 q^{83} + 24 q^{86} - 2 q^{88} + 4 q^{90} - 8 q^{92} - 8 q^{93} - 28 q^{94} - 16 q^{96} - 8 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/3234\mathbb{Z}\right)^\times\).

\(n\) \(199\) \(1079\) \(2059\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000i 0.707107i
\(3\) 1.00000i 0.577350i
\(4\) −1.00000 −0.500000
\(5\) 1.78763i 0.799451i 0.916635 + 0.399726i \(0.130895\pi\)
−0.916635 + 0.399726i \(0.869105\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0 0
\(8\) 1.00000i 0.353553i
\(9\) −1.00000 −0.333333
\(10\) −1.78763 −0.565297
\(11\) −3.15519 + 1.02213i −0.951327 + 0.308183i
\(12\) 1.00000i 0.288675i
\(13\) 6.37742 1.76878 0.884389 0.466751i \(-0.154576\pi\)
0.884389 + 0.466751i \(0.154576\pi\)
\(14\) 0 0
\(15\) −1.78763 −0.461563
\(16\) 1.00000 0.250000
\(17\) −0.106095 −0.0257319 −0.0128659 0.999917i \(-0.504095\pi\)
−0.0128659 + 0.999917i \(0.504095\pi\)
\(18\) 1.00000i 0.235702i
\(19\) 4.15162 0.952448 0.476224 0.879324i \(-0.342005\pi\)
0.476224 + 0.879324i \(0.342005\pi\)
\(20\) 1.78763i 0.399726i
\(21\) 0 0
\(22\) −1.02213 3.15519i −0.217919 0.672690i
\(23\) 7.95866 1.65950 0.829748 0.558138i \(-0.188484\pi\)
0.829748 + 0.558138i \(0.188484\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.80439 0.360878
\(26\) 6.37742i 1.25071i
\(27\) 1.00000i 0.192450i
\(28\) 0 0
\(29\) 7.65230i 1.42100i 0.703699 + 0.710498i \(0.251530\pi\)
−0.703699 + 0.710498i \(0.748470\pi\)
\(30\) 1.78763i 0.326375i
\(31\) 1.27872i 0.229665i −0.993385 0.114833i \(-0.963367\pi\)
0.993385 0.114833i \(-0.0366331\pi\)
\(32\) 1.00000i 0.176777i
\(33\) −1.02213 3.15519i −0.177930 0.549249i
\(34\) 0.106095i 0.0181952i
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 4.52184 0.743386 0.371693 0.928356i \(-0.378777\pi\)
0.371693 + 0.928356i \(0.378777\pi\)
\(38\) 4.15162i 0.673482i
\(39\) 6.37742i 1.02120i
\(40\) 1.78763 0.282649
\(41\) 0.0321383 0.00501915 0.00250958 0.999997i \(-0.499201\pi\)
0.00250958 + 0.999997i \(0.499201\pi\)
\(42\) 0 0
\(43\) 6.87723i 1.04877i 0.851482 + 0.524384i \(0.175704\pi\)
−0.851482 + 0.524384i \(0.824296\pi\)
\(44\) 3.15519 1.02213i 0.475663 0.154092i
\(45\) 1.78763i 0.266484i
\(46\) 7.95866i 1.17344i
\(47\) 9.88198i 1.44144i 0.693229 + 0.720718i \(0.256188\pi\)
−0.693229 + 0.720718i \(0.743812\pi\)
\(48\) 1.00000i 0.144338i
\(49\) 0 0
\(50\) 1.80439i 0.255179i
\(51\) 0.106095i 0.0148563i
\(52\) −6.37742 −0.884389
\(53\) −0.627441 −0.0861856 −0.0430928 0.999071i \(-0.513721\pi\)
−0.0430928 + 0.999071i \(0.513721\pi\)
\(54\) 1.00000 0.136083
\(55\) −1.82718 5.64031i −0.246378 0.760539i
\(56\) 0 0
\(57\) 4.15162i 0.549896i
\(58\) −7.65230 −1.00480
\(59\) 12.4549i 1.62150i −0.585395 0.810748i \(-0.699061\pi\)
0.585395 0.810748i \(-0.300939\pi\)
\(60\) 1.78763 0.230782
\(61\) −9.95024 −1.27400 −0.636999 0.770865i \(-0.719824\pi\)
−0.636999 + 0.770865i \(0.719824\pi\)
\(62\) 1.27872 0.162398
\(63\) 0 0
\(64\) −1.00000 −0.125000
\(65\) 11.4004i 1.41405i
\(66\) 3.15519 1.02213i 0.388378 0.125815i
\(67\) 10.8722 1.32825 0.664124 0.747623i \(-0.268805\pi\)
0.664124 + 0.747623i \(0.268805\pi\)
\(68\) 0.106095 0.0128659
\(69\) 7.95866i 0.958110i
\(70\) 0 0
\(71\) −8.42785 −1.00020 −0.500101 0.865967i \(-0.666704\pi\)
−0.500101 + 0.865967i \(0.666704\pi\)
\(72\) 1.00000i 0.117851i
\(73\) −0.125008 −0.0146311 −0.00731556 0.999973i \(-0.502329\pi\)
−0.00731556 + 0.999973i \(0.502329\pi\)
\(74\) 4.52184i 0.525654i
\(75\) 1.80439i 0.208353i
\(76\) −4.15162 −0.476224
\(77\) 0 0
\(78\) −6.37742 −0.722100
\(79\) 10.1691i 1.14412i −0.820213 0.572059i \(-0.806145\pi\)
0.820213 0.572059i \(-0.193855\pi\)
\(80\) 1.78763i 0.199863i
\(81\) 1.00000 0.111111
\(82\) 0.0321383i 0.00354908i
\(83\) −12.0657 −1.32438 −0.662191 0.749335i \(-0.730373\pi\)
−0.662191 + 0.749335i \(0.730373\pi\)
\(84\) 0 0
\(85\) 0.189659i 0.0205714i
\(86\) −6.87723 −0.741591
\(87\) −7.65230 −0.820412
\(88\) 1.02213 + 3.15519i 0.108959 + 0.336345i
\(89\) 5.36009i 0.568169i −0.958799 0.284084i \(-0.908310\pi\)
0.958799 0.284084i \(-0.0916896\pi\)
\(90\) 1.78763 0.188432
\(91\) 0 0
\(92\) −7.95866 −0.829748
\(93\) 1.27872 0.132597
\(94\) −9.88198 −1.01925
\(95\) 7.42156i 0.761436i
\(96\) −1.00000 −0.102062
\(97\) 15.7498i 1.59915i 0.600569 + 0.799573i \(0.294941\pi\)
−0.600569 + 0.799573i \(0.705059\pi\)
\(98\) 0 0
\(99\) 3.15519 1.02213i 0.317109 0.102728i
\(100\) −1.80439 −0.180439
\(101\) −7.57300 −0.753541 −0.376771 0.926307i \(-0.622966\pi\)
−0.376771 + 0.926307i \(0.622966\pi\)
\(102\) 0.106095 0.0105050
\(103\) 5.23715i 0.516032i −0.966141 0.258016i \(-0.916931\pi\)
0.966141 0.258016i \(-0.0830687\pi\)
\(104\) 6.37742i 0.625357i
\(105\) 0 0
\(106\) 0.627441i 0.0609424i
\(107\) 4.76790i 0.460930i −0.973081 0.230465i \(-0.925975\pi\)
0.973081 0.230465i \(-0.0740247\pi\)
\(108\) 1.00000i 0.0962250i
\(109\) 8.50535i 0.814665i 0.913280 + 0.407332i \(0.133541\pi\)
−0.913280 + 0.407332i \(0.866459\pi\)
\(110\) 5.64031 1.82718i 0.537783 0.174215i
\(111\) 4.52184i 0.429194i
\(112\) 0 0
\(113\) 10.3003 0.968969 0.484484 0.874800i \(-0.339007\pi\)
0.484484 + 0.874800i \(0.339007\pi\)
\(114\) −4.15162 −0.388835
\(115\) 14.2271i 1.32669i
\(116\) 7.65230i 0.710498i
\(117\) −6.37742 −0.589593
\(118\) 12.4549 1.14657
\(119\) 0 0
\(120\) 1.78763i 0.163187i
\(121\) 8.91051 6.45003i 0.810046 0.586366i
\(122\) 9.95024i 0.900852i
\(123\) 0.0321383i 0.00289781i
\(124\) 1.27872i 0.114833i
\(125\) 12.1637i 1.08796i
\(126\) 0 0
\(127\) 3.18422i 0.282553i −0.989970 0.141277i \(-0.954879\pi\)
0.989970 0.141277i \(-0.0451207\pi\)
\(128\) 1.00000i 0.0883883i
\(129\) −6.87723 −0.605506
\(130\) −11.4004 −0.999885
\(131\) −1.08620 −0.0949016 −0.0474508 0.998874i \(-0.515110\pi\)
−0.0474508 + 0.998874i \(0.515110\pi\)
\(132\) 1.02213 + 3.15519i 0.0889649 + 0.274624i
\(133\) 0 0
\(134\) 10.8722i 0.939213i
\(135\) 1.78763 0.153854
\(136\) 0.106095i 0.00909758i
\(137\) −7.58196 −0.647771 −0.323885 0.946096i \(-0.604989\pi\)
−0.323885 + 0.946096i \(0.604989\pi\)
\(138\) −7.95866 −0.677486
\(139\) −1.34643 −0.114203 −0.0571013 0.998368i \(-0.518186\pi\)
−0.0571013 + 0.998368i \(0.518186\pi\)
\(140\) 0 0
\(141\) −9.88198 −0.832213
\(142\) 8.42785i 0.707249i
\(143\) −20.1220 + 6.51854i −1.68269 + 0.545108i
\(144\) −1.00000 −0.0833333
\(145\) −13.6795 −1.13602
\(146\) 0.125008i 0.0103458i
\(147\) 0 0
\(148\) −4.52184 −0.371693
\(149\) 0.490157i 0.0401552i 0.999798 + 0.0200776i \(0.00639134\pi\)
−0.999798 + 0.0200776i \(0.993609\pi\)
\(150\) −1.80439 −0.147328
\(151\) 3.37323i 0.274509i 0.990536 + 0.137255i \(0.0438278\pi\)
−0.990536 + 0.137255i \(0.956172\pi\)
\(152\) 4.15162i 0.336741i
\(153\) 0.106095 0.00857728
\(154\) 0 0
\(155\) 2.28588 0.183606
\(156\) 6.37742i 0.510602i
\(157\) 4.06299i 0.324262i −0.986769 0.162131i \(-0.948163\pi\)
0.986769 0.162131i \(-0.0518367\pi\)
\(158\) 10.1691 0.809013
\(159\) 0.627441i 0.0497593i
\(160\) −1.78763 −0.141324
\(161\) 0 0
\(162\) 1.00000i 0.0785674i
\(163\) 20.7414 1.62459 0.812297 0.583244i \(-0.198217\pi\)
0.812297 + 0.583244i \(0.198217\pi\)
\(164\) −0.0321383 −0.00250958
\(165\) 5.64031 1.82718i 0.439098 0.142246i
\(166\) 12.0657i 0.936480i
\(167\) −3.42870 −0.265321 −0.132660 0.991162i \(-0.542352\pi\)
−0.132660 + 0.991162i \(0.542352\pi\)
\(168\) 0 0
\(169\) 27.6715 2.12857
\(170\) 0.189659 0.0145461
\(171\) −4.15162 −0.317483
\(172\) 6.87723i 0.524384i
\(173\) 21.1524 1.60819 0.804094 0.594502i \(-0.202651\pi\)
0.804094 + 0.594502i \(0.202651\pi\)
\(174\) 7.65230i 0.580119i
\(175\) 0 0
\(176\) −3.15519 + 1.02213i −0.237832 + 0.0770458i
\(177\) 12.4549 0.936171
\(178\) 5.36009 0.401756
\(179\) −14.0866 −1.05288 −0.526441 0.850212i \(-0.676474\pi\)
−0.526441 + 0.850212i \(0.676474\pi\)
\(180\) 1.78763i 0.133242i
\(181\) 12.4579i 0.925992i −0.886360 0.462996i \(-0.846775\pi\)
0.886360 0.462996i \(-0.153225\pi\)
\(182\) 0 0
\(183\) 9.95024i 0.735543i
\(184\) 7.95866i 0.586720i
\(185\) 8.08337i 0.594301i
\(186\) 1.27872i 0.0937604i
\(187\) 0.334751 0.108443i 0.0244794 0.00793013i
\(188\) 9.88198i 0.720718i
\(189\) 0 0
\(190\) −7.42156 −0.538416
\(191\) −27.2293 −1.97024 −0.985122 0.171854i \(-0.945024\pi\)
−0.985122 + 0.171854i \(0.945024\pi\)
\(192\) 1.00000i 0.0721688i
\(193\) 26.6381i 1.91745i 0.284337 + 0.958724i \(0.408227\pi\)
−0.284337 + 0.958724i \(0.591773\pi\)
\(194\) −15.7498 −1.13077
\(195\) −11.4004 −0.816403
\(196\) 0 0
\(197\) 11.7764i 0.839031i 0.907748 + 0.419515i \(0.137800\pi\)
−0.907748 + 0.419515i \(0.862200\pi\)
\(198\) 1.02213 + 3.15519i 0.0726395 + 0.224230i
\(199\) 21.4008i 1.51706i −0.651637 0.758531i \(-0.725917\pi\)
0.651637 0.758531i \(-0.274083\pi\)
\(200\) 1.80439i 0.127590i
\(201\) 10.8722i 0.766864i
\(202\) 7.57300i 0.532834i
\(203\) 0 0
\(204\) 0.106095i 0.00742815i
\(205\) 0.0574512i 0.00401257i
\(206\) 5.23715 0.364889
\(207\) −7.95866 −0.553165
\(208\) 6.37742 0.442194
\(209\) −13.0992 + 4.24349i −0.906089 + 0.293529i
\(210\) 0 0
\(211\) 25.2654i 1.73934i 0.493630 + 0.869672i \(0.335670\pi\)
−0.493630 + 0.869672i \(0.664330\pi\)
\(212\) 0.627441 0.0430928
\(213\) 8.42785i 0.577467i
\(214\) 4.76790 0.325927
\(215\) −12.2939 −0.838439
\(216\) −1.00000 −0.0680414
\(217\) 0 0
\(218\) −8.50535 −0.576055
\(219\) 0.125008i 0.00844728i
\(220\) 1.82718 + 5.64031i 0.123189 + 0.380270i
\(221\) −0.676613 −0.0455139
\(222\) −4.52184 −0.303486
\(223\) 9.80133i 0.656346i −0.944618 0.328173i \(-0.893567\pi\)
0.944618 0.328173i \(-0.106433\pi\)
\(224\) 0 0
\(225\) −1.80439 −0.120293
\(226\) 10.3003i 0.685164i
\(227\) 23.2907 1.54586 0.772931 0.634491i \(-0.218790\pi\)
0.772931 + 0.634491i \(0.218790\pi\)
\(228\) 4.15162i 0.274948i
\(229\) 6.53063i 0.431556i −0.976442 0.215778i \(-0.930771\pi\)
0.976442 0.215778i \(-0.0692288\pi\)
\(230\) −14.2271 −0.938109
\(231\) 0 0
\(232\) 7.65230 0.502398
\(233\) 10.8677i 0.711964i 0.934493 + 0.355982i \(0.115853\pi\)
−0.934493 + 0.355982i \(0.884147\pi\)
\(234\) 6.37742i 0.416905i
\(235\) −17.6653 −1.15236
\(236\) 12.4549i 0.810748i
\(237\) 10.1691 0.660557
\(238\) 0 0
\(239\) 7.69148i 0.497521i −0.968565 0.248760i \(-0.919977\pi\)
0.968565 0.248760i \(-0.0800231\pi\)
\(240\) −1.78763 −0.115391
\(241\) −22.8135 −1.46955 −0.734773 0.678313i \(-0.762711\pi\)
−0.734773 + 0.678313i \(0.762711\pi\)
\(242\) 6.45003 + 8.91051i 0.414624 + 0.572789i
\(243\) 1.00000i 0.0641500i
\(244\) 9.95024 0.636999
\(245\) 0 0
\(246\) −0.0321383 −0.00204906
\(247\) 26.4766 1.68467
\(248\) −1.27872 −0.0811988
\(249\) 12.0657i 0.764632i
\(250\) −12.1637 −0.769301
\(251\) 18.1111i 1.14317i −0.820544 0.571583i \(-0.806330\pi\)
0.820544 0.571583i \(-0.193670\pi\)
\(252\) 0 0
\(253\) −25.1111 + 8.13478i −1.57872 + 0.511429i
\(254\) 3.18422 0.199795
\(255\) 0.189659 0.0118769
\(256\) 1.00000 0.0625000
\(257\) 3.94247i 0.245924i −0.992411 0.122962i \(-0.960761\pi\)
0.992411 0.122962i \(-0.0392394\pi\)
\(258\) 6.87723i 0.428158i
\(259\) 0 0
\(260\) 11.4004i 0.707026i
\(261\) 7.65230i 0.473665i
\(262\) 1.08620i 0.0671056i
\(263\) 2.06715i 0.127466i 0.997967 + 0.0637329i \(0.0203006\pi\)
−0.997967 + 0.0637329i \(0.979699\pi\)
\(264\) −3.15519 + 1.02213i −0.194189 + 0.0629077i
\(265\) 1.12163i 0.0689012i
\(266\) 0 0
\(267\) 5.36009 0.328032
\(268\) −10.8722 −0.664124
\(269\) 2.80480i 0.171011i −0.996338 0.0855057i \(-0.972749\pi\)
0.996338 0.0855057i \(-0.0272506\pi\)
\(270\) 1.78763i 0.108792i
\(271\) −9.28092 −0.563776 −0.281888 0.959447i \(-0.590961\pi\)
−0.281888 + 0.959447i \(0.590961\pi\)
\(272\) −0.106095 −0.00643296
\(273\) 0 0
\(274\) 7.58196i 0.458043i
\(275\) −5.69320 + 1.84432i −0.343313 + 0.111217i
\(276\) 7.95866i 0.479055i
\(277\) 5.23096i 0.314298i −0.987575 0.157149i \(-0.949770\pi\)
0.987575 0.157149i \(-0.0502303\pi\)
\(278\) 1.34643i 0.0807534i
\(279\) 1.27872i 0.0765550i
\(280\) 0 0
\(281\) 6.87281i 0.409998i 0.978762 + 0.204999i \(0.0657190\pi\)
−0.978762 + 0.204999i \(0.934281\pi\)
\(282\) 9.88198i 0.588463i
\(283\) −12.0108 −0.713970 −0.356985 0.934110i \(-0.616195\pi\)
−0.356985 + 0.934110i \(0.616195\pi\)
\(284\) 8.42785 0.500101
\(285\) −7.42156 −0.439615
\(286\) −6.51854 20.1220i −0.385449 1.18984i
\(287\) 0 0
\(288\) 1.00000i 0.0589256i
\(289\) −16.9887 −0.999338
\(290\) 13.6795i 0.803285i
\(291\) −15.7498 −0.923268
\(292\) 0.125008 0.00731556
\(293\) −14.1147 −0.824592 −0.412296 0.911050i \(-0.635273\pi\)
−0.412296 + 0.911050i \(0.635273\pi\)
\(294\) 0 0
\(295\) 22.2648 1.29631
\(296\) 4.52184i 0.262827i
\(297\) 1.02213 + 3.15519i 0.0593099 + 0.183083i
\(298\) −0.490157 −0.0283940
\(299\) 50.7557 2.93528
\(300\) 1.80439i 0.104176i
\(301\) 0 0
\(302\) −3.37323 −0.194107
\(303\) 7.57300i 0.435057i
\(304\) 4.15162 0.238112
\(305\) 17.7873i 1.01850i
\(306\) 0.106095i 0.00606506i
\(307\) 21.0753 1.20283 0.601417 0.798936i \(-0.294603\pi\)
0.601417 + 0.798936i \(0.294603\pi\)
\(308\) 0 0
\(309\) 5.23715 0.297931
\(310\) 2.28588i 0.129829i
\(311\) 13.9495i 0.791003i 0.918465 + 0.395502i \(0.129429\pi\)
−0.918465 + 0.395502i \(0.870571\pi\)
\(312\) 6.37742 0.361050
\(313\) 3.98407i 0.225193i 0.993641 + 0.112596i \(0.0359167\pi\)
−0.993641 + 0.112596i \(0.964083\pi\)
\(314\) 4.06299 0.229288
\(315\) 0 0
\(316\) 10.1691i 0.572059i
\(317\) 12.2985 0.690755 0.345377 0.938464i \(-0.387751\pi\)
0.345377 + 0.938464i \(0.387751\pi\)
\(318\) 0.627441 0.0351851
\(319\) −7.82163 24.1445i −0.437927 1.35183i
\(320\) 1.78763i 0.0999314i
\(321\) 4.76790 0.266118
\(322\) 0 0
\(323\) −0.440467 −0.0245082
\(324\) −1.00000 −0.0555556
\(325\) 11.5073 0.638313
\(326\) 20.7414i 1.14876i
\(327\) −8.50535 −0.470347
\(328\) 0.0321383i 0.00177454i
\(329\) 0 0
\(330\) 1.82718 + 5.64031i 0.100583 + 0.310489i
\(331\) −31.0132 −1.70464 −0.852319 0.523022i \(-0.824804\pi\)
−0.852319 + 0.523022i \(0.824804\pi\)
\(332\) 12.0657 0.662191
\(333\) −4.52184 −0.247795
\(334\) 3.42870i 0.187610i
\(335\) 19.4354i 1.06187i
\(336\) 0 0
\(337\) 13.6161i 0.741718i −0.928689 0.370859i \(-0.879063\pi\)
0.928689 0.370859i \(-0.120937\pi\)
\(338\) 27.6715i 1.50513i
\(339\) 10.3003i 0.559434i
\(340\) 0.189659i 0.0102857i
\(341\) 1.30702 + 4.03461i 0.0707789 + 0.218487i
\(342\) 4.15162i 0.224494i
\(343\) 0 0
\(344\) 6.87723 0.370795
\(345\) −14.2271 −0.765963
\(346\) 21.1524i 1.13716i
\(347\) 27.9100i 1.49829i 0.662407 + 0.749144i \(0.269535\pi\)
−0.662407 + 0.749144i \(0.730465\pi\)
\(348\) 7.65230 0.410206
\(349\) −27.8147 −1.48888 −0.744442 0.667687i \(-0.767285\pi\)
−0.744442 + 0.667687i \(0.767285\pi\)
\(350\) 0 0
\(351\) 6.37742i 0.340401i
\(352\) −1.02213 3.15519i −0.0544796 0.168172i
\(353\) 21.5149i 1.14512i 0.819862 + 0.572562i \(0.194050\pi\)
−0.819862 + 0.572562i \(0.805950\pi\)
\(354\) 12.4549i 0.661973i
\(355\) 15.0659i 0.799612i
\(356\) 5.36009i 0.284084i
\(357\) 0 0
\(358\) 14.0866i 0.744500i
\(359\) 20.8942i 1.10276i 0.834256 + 0.551378i \(0.185898\pi\)
−0.834256 + 0.551378i \(0.814102\pi\)
\(360\) −1.78763 −0.0942162
\(361\) −1.76402 −0.0928431
\(362\) 12.4579 0.654775
\(363\) 6.45003 + 8.91051i 0.338539 + 0.467680i
\(364\) 0 0
\(365\) 0.223468i 0.0116969i
\(366\) 9.95024 0.520107
\(367\) 5.34315i 0.278910i −0.990228 0.139455i \(-0.955465\pi\)
0.990228 0.139455i \(-0.0445351\pi\)
\(368\) 7.95866 0.414874
\(369\) −0.0321383 −0.00167305
\(370\) −8.08337 −0.420234
\(371\) 0 0
\(372\) −1.27872 −0.0662986
\(373\) 25.3735i 1.31379i 0.753982 + 0.656895i \(0.228131\pi\)
−0.753982 + 0.656895i \(0.771869\pi\)
\(374\) 0.108443 + 0.334751i 0.00560745 + 0.0173096i
\(375\) −12.1637 −0.628131
\(376\) 9.88198 0.509624
\(377\) 48.8019i 2.51343i
\(378\) 0 0
\(379\) 22.7532 1.16875 0.584377 0.811483i \(-0.301339\pi\)
0.584377 + 0.811483i \(0.301339\pi\)
\(380\) 7.42156i 0.380718i
\(381\) 3.18422 0.163132
\(382\) 27.2293i 1.39317i
\(383\) 16.5480i 0.845565i −0.906231 0.422783i \(-0.861053\pi\)
0.906231 0.422783i \(-0.138947\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −26.6381 −1.35584
\(387\) 6.87723i 0.349589i
\(388\) 15.7498i 0.799573i
\(389\) 7.39812 0.375100 0.187550 0.982255i \(-0.439945\pi\)
0.187550 + 0.982255i \(0.439945\pi\)
\(390\) 11.4004i 0.577284i
\(391\) −0.844376 −0.0427019
\(392\) 0 0
\(393\) 1.08620i 0.0547915i
\(394\) −11.7764 −0.593284
\(395\) 18.1786 0.914666
\(396\) −3.15519 + 1.02213i −0.158554 + 0.0513639i
\(397\) 3.86738i 0.194098i 0.995280 + 0.0970492i \(0.0309404\pi\)
−0.995280 + 0.0970492i \(0.969060\pi\)
\(398\) 21.4008 1.07272
\(399\) 0 0
\(400\) 1.80439 0.0902194
\(401\) 3.13361 0.156485 0.0782426 0.996934i \(-0.475069\pi\)
0.0782426 + 0.996934i \(0.475069\pi\)
\(402\) −10.8722 −0.542255
\(403\) 8.15494i 0.406226i
\(404\) 7.57300 0.376771
\(405\) 1.78763i 0.0888279i
\(406\) 0 0
\(407\) −14.2673 + 4.62191i −0.707204 + 0.229099i
\(408\) −0.106095 −0.00525249
\(409\) −20.1022 −0.993992 −0.496996 0.867753i \(-0.665564\pi\)
−0.496996 + 0.867753i \(0.665564\pi\)
\(410\) −0.0574512 −0.00283731
\(411\) 7.58196i 0.373991i
\(412\) 5.23715i 0.258016i
\(413\) 0 0
\(414\) 7.95866i 0.391147i
\(415\) 21.5690i 1.05878i
\(416\) 6.37742i 0.312679i
\(417\) 1.34643i 0.0659349i
\(418\) −4.24349 13.0992i −0.207556 0.640702i
\(419\) 2.63035i 0.128501i −0.997934 0.0642506i \(-0.979534\pi\)
0.997934 0.0642506i \(-0.0204657\pi\)
\(420\) 0 0
\(421\) 24.9232 1.21468 0.607341 0.794441i \(-0.292236\pi\)
0.607341 + 0.794441i \(0.292236\pi\)
\(422\) −25.2654 −1.22990
\(423\) 9.88198i 0.480478i
\(424\) 0.627441i 0.0304712i
\(425\) −0.191437 −0.00928605
\(426\) 8.42785 0.408331
\(427\) 0 0
\(428\) 4.76790i 0.230465i
\(429\) −6.51854 20.1220i −0.314718 0.971499i
\(430\) 12.2939i 0.592866i
\(431\) 13.6456i 0.657287i 0.944454 + 0.328644i \(0.106591\pi\)
−0.944454 + 0.328644i \(0.893409\pi\)
\(432\) 1.00000i 0.0481125i
\(433\) 10.4001i 0.499795i −0.968272 0.249897i \(-0.919603\pi\)
0.968272 0.249897i \(-0.0803969\pi\)
\(434\) 0 0
\(435\) 13.6795i 0.655879i
\(436\) 8.50535i 0.407332i
\(437\) 33.0414 1.58058
\(438\) 0.125008 0.00597313
\(439\) −23.3239 −1.11319 −0.556594 0.830785i \(-0.687892\pi\)
−0.556594 + 0.830785i \(0.687892\pi\)
\(440\) −5.64031 + 1.82718i −0.268891 + 0.0871076i
\(441\) 0 0
\(442\) 0.676613i 0.0321832i
\(443\) 9.83451 0.467252 0.233626 0.972327i \(-0.424941\pi\)
0.233626 + 0.972327i \(0.424941\pi\)
\(444\) 4.52184i 0.214597i
\(445\) 9.58185 0.454223
\(446\) 9.80133 0.464107
\(447\) −0.490157 −0.0231836
\(448\) 0 0
\(449\) 9.26223 0.437112 0.218556 0.975824i \(-0.429865\pi\)
0.218556 + 0.975824i \(0.429865\pi\)
\(450\) 1.80439i 0.0850597i
\(451\) −0.101403 + 0.0328494i −0.00477486 + 0.00154682i
\(452\) −10.3003 −0.484484
\(453\) −3.37323 −0.158488
\(454\) 23.2907i 1.09309i
\(455\) 0 0
\(456\) 4.15162 0.194418
\(457\) 2.55635i 0.119581i −0.998211 0.0597906i \(-0.980957\pi\)
0.998211 0.0597906i \(-0.0190433\pi\)
\(458\) 6.53063 0.305156
\(459\) 0.106095i 0.00495210i
\(460\) 14.2271i 0.663343i
\(461\) 29.5798 1.37767 0.688834 0.724919i \(-0.258123\pi\)
0.688834 + 0.724919i \(0.258123\pi\)
\(462\) 0 0
\(463\) −9.24447 −0.429627 −0.214813 0.976655i \(-0.568914\pi\)
−0.214813 + 0.976655i \(0.568914\pi\)
\(464\) 7.65230i 0.355249i
\(465\) 2.28588i 0.106005i
\(466\) −10.8677 −0.503434
\(467\) 11.6497i 0.539083i −0.962989 0.269541i \(-0.913128\pi\)
0.962989 0.269541i \(-0.0868721\pi\)
\(468\) 6.37742 0.294796
\(469\) 0 0
\(470\) 17.6653i 0.814840i
\(471\) 4.06299 0.187213
\(472\) −12.4549 −0.573285
\(473\) −7.02941 21.6990i −0.323213 0.997721i
\(474\) 10.1691i 0.467084i
\(475\) 7.49114 0.343717
\(476\) 0 0
\(477\) 0.627441 0.0287285
\(478\) 7.69148 0.351800
\(479\) 10.0552 0.459432 0.229716 0.973258i \(-0.426220\pi\)
0.229716 + 0.973258i \(0.426220\pi\)
\(480\) 1.78763i 0.0815936i
\(481\) 28.8377 1.31489
\(482\) 22.8135i 1.03913i
\(483\) 0 0
\(484\) −8.91051 + 6.45003i −0.405023 + 0.293183i
\(485\) −28.1547 −1.27844
\(486\) −1.00000 −0.0453609
\(487\) 17.3146 0.784601 0.392300 0.919837i \(-0.371679\pi\)
0.392300 + 0.919837i \(0.371679\pi\)
\(488\) 9.95024i 0.450426i
\(489\) 20.7414i 0.937960i
\(490\) 0 0
\(491\) 8.44002i 0.380893i −0.981698 0.190446i \(-0.939006\pi\)
0.981698 0.190446i \(-0.0609935\pi\)
\(492\) 0.0321383i 0.00144891i
\(493\) 0.811871i 0.0365648i
\(494\) 26.4766i 1.19124i
\(495\) 1.82718 + 5.64031i 0.0821259 + 0.253513i
\(496\) 1.27872i 0.0574163i
\(497\) 0 0
\(498\) 12.0657 0.540677
\(499\) 19.0719 0.853778 0.426889 0.904304i \(-0.359610\pi\)
0.426889 + 0.904304i \(0.359610\pi\)
\(500\) 12.1637i 0.543978i
\(501\) 3.42870i 0.153183i
\(502\) 18.1111 0.808340
\(503\) 3.32018 0.148039 0.0740197 0.997257i \(-0.476417\pi\)
0.0740197 + 0.997257i \(0.476417\pi\)
\(504\) 0 0
\(505\) 13.5377i 0.602419i
\(506\) −8.13478 25.1111i −0.361635 1.11633i
\(507\) 27.6715i 1.22893i
\(508\) 3.18422i 0.141277i
\(509\) 4.51879i 0.200292i 0.994973 + 0.100146i \(0.0319309\pi\)
−0.994973 + 0.100146i \(0.968069\pi\)
\(510\) 0.189659i 0.00839822i
\(511\) 0 0
\(512\) 1.00000i 0.0441942i
\(513\) 4.15162i 0.183299i
\(514\) 3.94247 0.173895
\(515\) 9.36207 0.412542
\(516\) 6.87723 0.302753
\(517\) −10.1007 31.1796i −0.444226 1.37128i
\(518\) 0 0
\(519\) 21.1524i 0.928488i
\(520\) 11.4004 0.499943
\(521\) 9.36122i 0.410122i 0.978749 + 0.205061i \(0.0657393\pi\)
−0.978749 + 0.205061i \(0.934261\pi\)
\(522\) 7.65230 0.334932
\(523\) −17.4095 −0.761266 −0.380633 0.924726i \(-0.624294\pi\)
−0.380633 + 0.924726i \(0.624294\pi\)
\(524\) 1.08620 0.0474508
\(525\) 0 0
\(526\) −2.06715 −0.0901320
\(527\) 0.135666i 0.00590971i
\(528\) −1.02213 3.15519i −0.0444824 0.137312i
\(529\) 40.3403 1.75393
\(530\) 1.12163 0.0487205
\(531\) 12.4549i 0.540499i
\(532\) 0 0
\(533\) 0.204959 0.00887777
\(534\) 5.36009i 0.231954i
\(535\) 8.52322 0.368491
\(536\) 10.8722i 0.469606i
\(537\) 14.0866i 0.607881i
\(538\) 2.80480 0.120923
\(539\) 0 0
\(540\) −1.78763 −0.0769272
\(541\) 39.3722i 1.69274i −0.532595 0.846370i \(-0.678783\pi\)
0.532595 0.846370i \(-0.321217\pi\)
\(542\) 9.28092i 0.398650i
\(543\) 12.4579 0.534621
\(544\) 0.106095i 0.00454879i
\(545\) −15.2044 −0.651285
\(546\) 0 0
\(547\) 20.3539i 0.870271i 0.900365 + 0.435135i \(0.143299\pi\)
−0.900365 + 0.435135i \(0.856701\pi\)
\(548\) 7.58196 0.323885
\(549\) 9.95024 0.424666
\(550\) −1.84432 5.69320i −0.0786420 0.242759i
\(551\) 31.7695i 1.35342i
\(552\) 7.95866 0.338743
\(553\) 0 0
\(554\) 5.23096 0.222242
\(555\) −8.08337 −0.343120
\(556\) 1.34643 0.0571013
\(557\) 13.0285i 0.552035i −0.961153 0.276017i \(-0.910985\pi\)
0.961153 0.276017i \(-0.0890148\pi\)
\(558\) −1.27872 −0.0541326
\(559\) 43.8590i 1.85504i
\(560\) 0 0
\(561\) 0.108443 + 0.334751i 0.00457846 + 0.0141332i
\(562\) −6.87281 −0.289912
\(563\) 33.5633 1.41452 0.707262 0.706951i \(-0.249930\pi\)
0.707262 + 0.706951i \(0.249930\pi\)
\(564\) 9.88198 0.416107
\(565\) 18.4131i 0.774643i
\(566\) 12.0108i 0.504853i
\(567\) 0 0
\(568\) 8.42785i 0.353625i
\(569\) 5.54679i 0.232534i 0.993218 + 0.116267i \(0.0370928\pi\)
−0.993218 + 0.116267i \(0.962907\pi\)
\(570\) 7.42156i 0.310855i
\(571\) 37.0923i 1.55226i −0.630571 0.776132i \(-0.717179\pi\)
0.630571 0.776132i \(-0.282821\pi\)
\(572\) 20.1220 6.51854i 0.841343 0.272554i
\(573\) 27.2293i 1.13752i
\(574\) 0 0
\(575\) 14.3605 0.598875
\(576\) 1.00000 0.0416667
\(577\) 3.08219i 0.128313i −0.997940 0.0641567i \(-0.979564\pi\)
0.997940 0.0641567i \(-0.0204357\pi\)
\(578\) 16.9887i 0.706639i
\(579\) −26.6381 −1.10704
\(580\) 13.6795 0.568008
\(581\) 0 0
\(582\) 15.7498i 0.652849i
\(583\) 1.97970 0.641325i 0.0819907 0.0265610i
\(584\) 0.125008i 0.00517288i
\(585\) 11.4004i 0.471350i
\(586\) 14.1147i 0.583075i
\(587\) 46.9255i 1.93682i −0.249356 0.968412i \(-0.580219\pi\)
0.249356 0.968412i \(-0.419781\pi\)
\(588\) 0 0
\(589\) 5.30877i 0.218744i
\(590\) 22.2648i 0.916627i
\(591\) −11.7764 −0.484415
\(592\) 4.52184 0.185847
\(593\) −5.14958 −0.211468 −0.105734 0.994394i \(-0.533719\pi\)
−0.105734 + 0.994394i \(0.533719\pi\)
\(594\) −3.15519 + 1.02213i −0.129459 + 0.0419384i
\(595\) 0 0
\(596\) 0.490157i 0.0200776i
\(597\) 21.4008 0.875876
\(598\) 50.7557i 2.07556i
\(599\) −35.1462 −1.43603 −0.718017 0.696026i \(-0.754950\pi\)
−0.718017 + 0.696026i \(0.754950\pi\)
\(600\) 1.80439 0.0736639
\(601\) 42.8385 1.74742 0.873710 0.486447i \(-0.161707\pi\)
0.873710 + 0.486447i \(0.161707\pi\)
\(602\) 0 0
\(603\) −10.8722 −0.442749
\(604\) 3.37323i 0.137255i
\(605\) 11.5302 + 15.9287i 0.468771 + 0.647592i
\(606\) 7.57300 0.307632
\(607\) 16.6655 0.676432 0.338216 0.941069i \(-0.390177\pi\)
0.338216 + 0.941069i \(0.390177\pi\)
\(608\) 4.15162i 0.168371i
\(609\) 0 0
\(610\) 17.7873 0.720187
\(611\) 63.0215i 2.54958i
\(612\) −0.106095 −0.00428864
\(613\) 11.8955i 0.480456i 0.970717 + 0.240228i \(0.0772222\pi\)
−0.970717 + 0.240228i \(0.922778\pi\)
\(614\) 21.0753i 0.850531i
\(615\) −0.0574512 −0.00231666
\(616\) 0 0
\(617\) −0.0716192 −0.00288328 −0.00144164 0.999999i \(-0.500459\pi\)
−0.00144164 + 0.999999i \(0.500459\pi\)
\(618\) 5.23715i 0.210669i
\(619\) 29.2187i 1.17440i −0.809442 0.587200i \(-0.800230\pi\)
0.809442 0.587200i \(-0.199770\pi\)
\(620\) −2.28588 −0.0918030
\(621\) 7.95866i 0.319370i
\(622\) −13.9495 −0.559324
\(623\) 0 0
\(624\) 6.37742i 0.255301i
\(625\) −12.7222 −0.508889
\(626\) −3.98407 −0.159235
\(627\) −4.24349 13.0992i −0.169469 0.523131i
\(628\) 4.06299i 0.162131i
\(629\) −0.479746 −0.0191287
\(630\) 0 0
\(631\) −43.3663 −1.72638 −0.863192 0.504876i \(-0.831538\pi\)
−0.863192 + 0.504876i \(0.831538\pi\)
\(632\) −10.1691 −0.404507
\(633\) −25.2654 −1.00421
\(634\) 12.2985i 0.488438i
\(635\) 5.69219 0.225888
\(636\) 0.627441i 0.0248796i
\(637\) 0 0
\(638\) 24.1445 7.82163i 0.955889 0.309661i
\(639\) 8.42785 0.333401
\(640\) 1.78763 0.0706622
\(641\) −5.45490 −0.215456 −0.107728 0.994180i \(-0.534357\pi\)
−0.107728 + 0.994180i \(0.534357\pi\)
\(642\) 4.76790i 0.188174i
\(643\) 4.26488i 0.168190i 0.996458 + 0.0840952i \(0.0268000\pi\)
−0.996458 + 0.0840952i \(0.973200\pi\)
\(644\) 0 0
\(645\) 12.2939i 0.484073i
\(646\) 0.440467i 0.0173299i
\(647\) 23.7320i 0.933003i −0.884520 0.466501i \(-0.845514\pi\)
0.884520 0.466501i \(-0.154486\pi\)
\(648\) 1.00000i 0.0392837i
\(649\) 12.7306 + 39.2978i 0.499718 + 1.54257i
\(650\) 11.5073i 0.451355i
\(651\) 0 0
\(652\) −20.7414 −0.812297
\(653\) 33.6185 1.31559 0.657797 0.753195i \(-0.271489\pi\)
0.657797 + 0.753195i \(0.271489\pi\)
\(654\) 8.50535i 0.332586i
\(655\) 1.94172i 0.0758692i
\(656\) 0.0321383 0.00125479
\(657\) 0.125008 0.00487704
\(658\) 0 0
\(659\) 28.9405i 1.12736i 0.825993 + 0.563681i \(0.190615\pi\)
−0.825993 + 0.563681i \(0.809385\pi\)
\(660\) −5.64031 + 1.82718i −0.219549 + 0.0711231i
\(661\) 0.598886i 0.0232940i 0.999932 + 0.0116470i \(0.00370743\pi\)
−0.999932 + 0.0116470i \(0.996293\pi\)
\(662\) 31.0132i 1.20536i
\(663\) 0.676613i 0.0262775i
\(664\) 12.0657i 0.468240i
\(665\) 0 0
\(666\) 4.52184i 0.175218i
\(667\) 60.9020i 2.35814i
\(668\) 3.42870 0.132660
\(669\) 9.80133 0.378941
\(670\) −19.4354 −0.750855
\(671\) 31.3949 10.1704i 1.21199 0.392625i
\(672\) 0 0
\(673\) 36.3353i 1.40062i −0.713837 0.700312i \(-0.753044\pi\)
0.713837 0.700312i \(-0.246956\pi\)
\(674\) 13.6161 0.524474
\(675\) 1.80439i 0.0694510i
\(676\) −27.6715 −1.06429
\(677\) −19.9914 −0.768332 −0.384166 0.923264i \(-0.625511\pi\)
−0.384166 + 0.923264i \(0.625511\pi\)
\(678\) −10.3003 −0.395580
\(679\) 0 0
\(680\) −0.189659 −0.00727307
\(681\) 23.2907i 0.892503i
\(682\) −4.03461 + 1.30702i −0.154493 + 0.0500483i
\(683\) −22.0325 −0.843049 −0.421524 0.906817i \(-0.638505\pi\)
−0.421524 + 0.906817i \(0.638505\pi\)
\(684\) 4.15162 0.158741
\(685\) 13.5537i 0.517861i
\(686\) 0 0
\(687\) 6.53063 0.249159
\(688\) 6.87723i 0.262192i
\(689\) −4.00145 −0.152443
\(690\) 14.2271i 0.541617i
\(691\) 11.4103i 0.434068i 0.976164 + 0.217034i \(0.0696383\pi\)
−0.976164 + 0.217034i \(0.930362\pi\)
\(692\) −21.1524 −0.804094
\(693\) 0 0
\(694\) −27.9100 −1.05945
\(695\) 2.40691i 0.0912994i
\(696\) 7.65230i 0.290060i
\(697\) −0.00340971 −0.000129152
\(698\) 27.8147i 1.05280i
\(699\) −10.8677 −0.411052
\(700\) 0 0
\(701\) 1.92874i 0.0728474i 0.999336 + 0.0364237i \(0.0115966\pi\)
−0.999336 + 0.0364237i \(0.988403\pi\)
\(702\) 6.37742 0.240700
\(703\) 18.7730 0.708037
\(704\) 3.15519 1.02213i 0.118916 0.0385229i
\(705\) 17.6653i 0.665314i
\(706\) −21.5149 −0.809725
\(707\) 0 0
\(708\) −12.4549 −0.468086
\(709\) 19.0032 0.713681 0.356841 0.934165i \(-0.383854\pi\)
0.356841 + 0.934165i \(0.383854\pi\)
\(710\) 15.0659 0.565411
\(711\) 10.1691i 0.381373i
\(712\) −5.36009 −0.200878
\(713\) 10.1769i 0.381128i
\(714\) 0 0
\(715\) −11.6527 35.9706i −0.435787 1.34523i
\(716\) 14.0866 0.526441
\(717\) 7.69148 0.287244
\(718\) −20.8942 −0.779766
\(719\) 20.6690i 0.770822i 0.922745 + 0.385411i \(0.125940\pi\)
−0.922745 + 0.385411i \(0.874060\pi\)
\(720\) 1.78763i 0.0666209i
\(721\) 0 0
\(722\) 1.76402i 0.0656500i
\(723\) 22.8135i 0.848443i
\(724\) 12.4579i 0.462996i
\(725\) 13.8077i 0.512806i
\(726\) −8.91051 + 6.45003i −0.330700 + 0.239383i
\(727\) 1.72185i 0.0638600i 0.999490 + 0.0319300i \(0.0101654\pi\)
−0.999490 + 0.0319300i \(0.989835\pi\)
\(728\) 0 0
\(729\) −1.00000 −0.0370370
\(730\) 0.223468 0.00827093
\(731\) 0.729641i 0.0269867i
\(732\) 9.95024i 0.367771i
\(733\) −1.19425 −0.0441106 −0.0220553 0.999757i \(-0.507021\pi\)
−0.0220553 + 0.999757i \(0.507021\pi\)
\(734\) 5.34315 0.197219
\(735\) 0 0
\(736\) 7.95866i 0.293360i
\(737\) −34.3038 + 11.1128i −1.26360 + 0.409344i
\(738\) 0.0321383i 0.00118303i
\(739\) 12.4376i 0.457525i −0.973482 0.228763i \(-0.926532\pi\)
0.973482 0.228763i \(-0.0734680\pi\)
\(740\) 8.08337i 0.297151i
\(741\) 26.4766i 0.972644i
\(742\) 0 0
\(743\) 19.1754i 0.703479i −0.936098 0.351739i \(-0.885590\pi\)
0.936098 0.351739i \(-0.114410\pi\)
\(744\) 1.27872i 0.0468802i
\(745\) −0.876218 −0.0321022
\(746\) −25.3735 −0.928990
\(747\) 12.0657 0.441461
\(748\) −0.334751 + 0.108443i −0.0122397 + 0.00396506i
\(749\) 0 0
\(750\) 12.1637i 0.444156i
\(751\) 45.9739 1.67761 0.838807 0.544430i \(-0.183254\pi\)
0.838807 + 0.544430i \(0.183254\pi\)
\(752\) 9.88198i 0.360359i
\(753\) 18.1111 0.660007
\(754\) −48.8019 −1.77726
\(755\) −6.03007 −0.219457
\(756\) 0 0
\(757\) 32.1348 1.16796 0.583979 0.811768i \(-0.301495\pi\)
0.583979 + 0.811768i \(0.301495\pi\)
\(758\) 22.7532i 0.826433i
\(759\) −8.13478 25.1111i −0.295274 0.911476i
\(760\) 7.42156 0.269208
\(761\) 20.4998 0.743116 0.371558 0.928410i \(-0.378824\pi\)
0.371558 + 0.928410i \(0.378824\pi\)
\(762\) 3.18422i 0.115352i
\(763\) 0 0
\(764\) 27.2293 0.985122
\(765\) 0.189659i 0.00685712i
\(766\) 16.5480 0.597905
\(767\) 79.4304i 2.86807i
\(768\) 1.00000i 0.0360844i
\(769\) −14.3312 −0.516796 −0.258398 0.966039i \(-0.583195\pi\)
−0.258398 + 0.966039i \(0.583195\pi\)
\(770\) 0 0
\(771\) 3.94247 0.141984
\(772\) 26.6381i 0.958724i
\(773\) 29.5740i 1.06370i 0.846838 + 0.531851i \(0.178503\pi\)
−0.846838 + 0.531851i \(0.821497\pi\)
\(774\) 6.87723 0.247197
\(775\) 2.30731i 0.0828810i
\(776\) 15.7498 0.565384
\(777\) 0 0
\(778\) 7.39812i 0.265235i
\(779\) 0.133426 0.00478048
\(780\) 11.4004 0.408201
\(781\) 26.5915 8.61435i 0.951519 0.308246i
\(782\) 0.844376i 0.0301948i
\(783\) 7.65230 0.273471
\(784\) 0 0
\(785\) 7.26311 0.259232
\(786\) 1.08620 0.0387434
\(787\) −26.6975 −0.951662 −0.475831 0.879537i \(-0.657853\pi\)
−0.475831 + 0.879537i \(0.657853\pi\)
\(788\) 11.7764i 0.419515i
\(789\) −2.06715 −0.0735925
\(790\) 18.1786i 0.646767i
\(791\) 0 0
\(792\) −1.02213 3.15519i −0.0363198 0.112115i
\(793\) −63.4568 −2.25342
\(794\) −3.86738 −0.137248
\(795\) 1.12163 0.0397801
\(796\) 21.4008i 0.758531i
\(797\) 47.2556i 1.67388i 0.547294 + 0.836940i \(0.315658\pi\)
−0.547294 + 0.836940i \(0.684342\pi\)
\(798\) 0 0
\(799\) 1.04843i 0.0370908i
\(800\) 1.80439i 0.0637948i
\(801\) 5.36009i 0.189390i
\(802\) 3.13361i 0.110652i
\(803\) 0.394426 0.127775i 0.0139190 0.00450907i
\(804\) 10.8722i 0.383432i
\(805\) 0 0
\(806\) 8.15494 0.287245
\(807\) 2.80480 0.0987335
\(808\) 7.57300i 0.266417i
\(809\) 21.0775i 0.741044i 0.928824 + 0.370522i \(0.120821\pi\)
−0.928824 + 0.370522i \(0.879179\pi\)
\(810\) −1.78763 −0.0628108
\(811\) 38.7457 1.36055 0.680273 0.732959i \(-0.261861\pi\)
0.680273 + 0.732959i \(0.261861\pi\)
\(812\) 0 0
\(813\) 9.28092i 0.325496i
\(814\) −4.62191 14.2673i −0.161998 0.500068i
\(815\) 37.0779i 1.29878i
\(816\) 0.106095i 0.00371407i
\(817\) 28.5517i 0.998897i
\(818\) 20.1022i 0.702858i
\(819\) 0 0
\(820\) 0.0574512i 0.00200628i
\(821\) 42.8317i 1.49483i −0.664355 0.747417i \(-0.731294\pi\)
0.664355 0.747417i \(-0.268706\pi\)
\(822\) 7.58196 0.264451
\(823\) 33.8619 1.18035 0.590176 0.807275i \(-0.299059\pi\)
0.590176 + 0.807275i \(0.299059\pi\)
\(824\) −5.23715 −0.182445
\(825\) −1.84432 5.69320i −0.0642109 0.198212i
\(826\) 0 0
\(827\) 15.2572i 0.530544i −0.964174 0.265272i \(-0.914538\pi\)
0.964174 0.265272i \(-0.0854618\pi\)
\(828\) 7.95866 0.276583
\(829\) 17.2483i 0.599058i 0.954087 + 0.299529i \(0.0968295\pi\)
−0.954087 + 0.299529i \(0.903171\pi\)
\(830\) 21.5690 0.748670
\(831\) 5.23096 0.181460
\(832\) −6.37742 −0.221097
\(833\) 0 0
\(834\) 1.34643 0.0466230
\(835\) 6.12924i 0.212111i
\(836\) 13.0992 4.24349i 0.453045 0.146764i
\(837\) −1.27872 −0.0441991
\(838\) 2.63035 0.0908640
\(839\) 9.80607i 0.338543i −0.985569 0.169272i \(-0.945859\pi\)
0.985569 0.169272i \(-0.0541415\pi\)
\(840\) 0 0
\(841\) −29.5576 −1.01923
\(842\) 24.9232i 0.858910i
\(843\) −6.87281 −0.236712
\(844\) 25.2654i 0.869672i
\(845\) 49.4663i 1.70169i
\(846\) 9.88198 0.339750
\(847\) 0 0
\(848\) −0.627441 −0.0215464
\(849\) 12.0108i 0.412211i
\(850\) 0.191437i 0.00656623i
\(851\) 35.9878 1.23365
\(852\) 8.42785i 0.288733i
\(853\) −58.0237 −1.98669 −0.993346 0.115168i \(-0.963259\pi\)
−0.993346 + 0.115168i \(0.963259\pi\)
\(854\) 0 0
\(855\) 7.42156i 0.253812i
\(856\) −4.76790 −0.162963
\(857\) −41.0867 −1.40350 −0.701748 0.712426i \(-0.747597\pi\)
−0.701748 + 0.712426i \(0.747597\pi\)
\(858\) 20.1220 6.51854i 0.686954 0.222539i
\(859\) 26.5613i 0.906261i −0.891444 0.453130i \(-0.850307\pi\)
0.891444 0.453130i \(-0.149693\pi\)
\(860\) 12.2939 0.419219
\(861\) 0 0
\(862\) −13.6456 −0.464772
\(863\) 9.08791 0.309356 0.154678 0.987965i \(-0.450566\pi\)
0.154678 + 0.987965i \(0.450566\pi\)
\(864\) 1.00000 0.0340207
\(865\) 37.8126i 1.28567i
\(866\) 10.4001 0.353408
\(867\) 16.9887i 0.576968i
\(868\) 0 0
\(869\) 10.3942 + 32.0856i 0.352598 + 1.08843i
\(870\) 13.6795 0.463777
\(871\) 69.3364 2.34937
\(872\) 8.50535 0.288028
\(873\) 15.7498i 0.533049i
\(874\) 33.0414i 1.11764i
\(875\) 0 0
\(876\) 0.125008i 0.00422364i
\(877\) 4.56656i 0.154202i 0.997023 + 0.0771009i \(0.0245664\pi\)
−0.997023 + 0.0771009i \(0.975434\pi\)
\(878\) 23.3239i 0.787143i
\(879\) 14.1147i 0.476078i
\(880\) −1.82718 5.64031i −0.0615944 0.190135i
\(881\) 35.4288i 1.19363i −0.802380 0.596813i \(-0.796433\pi\)
0.802380 0.596813i \(-0.203567\pi\)
\(882\) 0 0
\(883\) 25.2987 0.851368 0.425684 0.904872i \(-0.360033\pi\)
0.425684 + 0.904872i \(0.360033\pi\)
\(884\) 0.676613 0.0227570
\(885\) 22.2648i 0.748423i
\(886\) 9.83451i 0.330397i
\(887\) 44.9662 1.50982 0.754909 0.655830i \(-0.227681\pi\)
0.754909 + 0.655830i \(0.227681\pi\)
\(888\) 4.52184 0.151743
\(889\) 0 0
\(890\) 9.58185i 0.321184i
\(891\) −3.15519 + 1.02213i −0.105703 + 0.0342426i
\(892\) 9.80133i 0.328173i
\(893\) 41.0263i 1.37289i
\(894\) 0.490157i 0.0163933i
\(895\) 25.1816i 0.841727i
\(896\) 0 0
\(897\) 50.7557i 1.69468i
\(898\) 9.26223i 0.309085i
\(899\) 9.78515 0.326353
\(900\) 1.80439 0.0601463
\(901\) 0.0665684 0.00221772
\(902\) −0.0328494 0.101403i −0.00109377 0.00337633i
\(903\) 0 0
\(904\) 10.3003i 0.342582i
\(905\) 22.2702 0.740285
\(906\) 3.37323i 0.112068i
\(907\) −17.2613 −0.573151 −0.286576 0.958058i \(-0.592517\pi\)
−0.286576 + 0.958058i \(0.592517\pi\)
\(908\) −23.2907 −0.772931
\(909\) 7.57300 0.251180
\(910\) 0 0
\(911\) −18.9606 −0.628194 −0.314097 0.949391i \(-0.601702\pi\)
−0.314097 + 0.949391i \(0.601702\pi\)
\(912\) 4.15162i 0.137474i
\(913\) 38.0696 12.3327i 1.25992 0.408153i
\(914\) 2.55635 0.0845566
\(915\) 17.7873 0.588030
\(916\) 6.53063i 0.215778i
\(917\) 0 0
\(918\) −0.106095 −0.00350166
\(919\) 35.7234i 1.17841i 0.807985 + 0.589203i \(0.200558\pi\)
−0.807985 + 0.589203i \(0.799442\pi\)
\(920\) 14.2271 0.469054
\(921\) 21.0753i 0.694456i
\(922\) 29.5798i 0.974158i
\(923\) −53.7479 −1.76913
\(924\) 0 0
\(925\) 8.15916 0.268272
\(926\) 9.24447i 0.303792i
\(927\) 5.23715i 0.172011i
\(928\) −7.65230 −0.251199
\(929\) 40.5119i 1.32915i 0.747220 + 0.664577i \(0.231388\pi\)
−0.747220 + 0.664577i \(0.768612\pi\)
\(930\) −2.28588 −0.0749568
\(931\) 0 0
\(932\) 10.8677i 0.355982i
\(933\) −13.9495 −0.456686
\(934\) 11.6497 0.381189
\(935\) 0.193855 + 0.598410i 0.00633975 + 0.0195701i
\(936\) 6.37742i 0.208452i
\(937\) 36.2513 1.18428 0.592140 0.805835i \(-0.298283\pi\)
0.592140 + 0.805835i \(0.298283\pi\)
\(938\) 0 0
\(939\) −3.98407 −0.130015
\(940\) 17.6653 0.576179
\(941\) 44.7796 1.45977 0.729887 0.683568i \(-0.239573\pi\)
0.729887 + 0.683568i \(0.239573\pi\)
\(942\) 4.06299i 0.132379i
\(943\) 0.255778 0.00832927
\(944\) 12.4549i 0.405374i
\(945\) 0 0
\(946\) 21.6990 7.02941i 0.705495 0.228546i
\(947\) −14.7070 −0.477912 −0.238956 0.971030i \(-0.576805\pi\)
−0.238956 + 0.971030i \(0.576805\pi\)
\(948\) −10.1691 −0.330278
\(949\) −0.797230 −0.0258792
\(950\) 7.49114i 0.243045i
\(951\) 12.2985i 0.398808i
\(952\) 0 0
\(953\) 23.4615i 0.759991i −0.924988 0.379996i \(-0.875926\pi\)
0.924988 0.379996i \(-0.124074\pi\)
\(954\) 0.627441i 0.0203141i
\(955\) 48.6759i 1.57511i
\(956\) 7.69148i 0.248760i
\(957\) 24.1445 7.82163i 0.780480 0.252837i
\(958\) 10.0552i 0.324868i
\(959\) 0 0
\(960\) 1.78763 0.0576954
\(961\) 29.3649 0.947254
\(962\) 28.8377i 0.929764i
\(963\) 4.76790i 0.153643i
\(964\) 22.8135 0.734773
\(965\) −47.6189 −1.53291
\(966\) 0 0
\(967\) 57.3317i 1.84366i −0.387589 0.921832i \(-0.626692\pi\)
0.387589 0.921832i \(-0.373308\pi\)
\(968\) −6.45003 8.91051i −0.207312 0.286395i
\(969\) 0.440467i 0.0141498i
\(970\) 28.1547i 0.903993i
\(971\) 48.6951i 1.56270i −0.624092 0.781351i \(-0.714531\pi\)
0.624092 0.781351i \(-0.285469\pi\)
\(972\) 1.00000i 0.0320750i
\(973\) 0 0
\(974\) 17.3146i 0.554797i
\(975\) 11.5073i 0.368530i
\(976\) −9.95024 −0.318499
\(977\) 42.2556 1.35188 0.675939 0.736958i \(-0.263738\pi\)
0.675939 + 0.736958i \(0.263738\pi\)
\(978\) −20.7414 −0.663238
\(979\) 5.47871 + 16.9121i 0.175100 + 0.540514i
\(980\) 0 0
\(981\) 8.50535i 0.271555i
\(982\) 8.44002 0.269332
\(983\) 23.6174i 0.753279i −0.926360 0.376639i \(-0.877080\pi\)
0.926360 0.376639i \(-0.122920\pi\)
\(984\) 0.0321383 0.00102453
\(985\) −21.0517 −0.670764
\(986\) 0.811871 0.0258553
\(987\) 0 0
\(988\) −26.4766 −0.842334
\(989\) 54.7336i 1.74043i
\(990\) −5.64031 + 1.82718i −0.179261 + 0.0580718i
\(991\) 26.9740 0.856857 0.428429 0.903576i \(-0.359067\pi\)
0.428429 + 0.903576i \(0.359067\pi\)
\(992\) 1.27872 0.0405994
\(993\) 31.0132i 0.984173i
\(994\) 0 0
\(995\) 38.2566 1.21282
\(996\) 12.0657i 0.382316i
\(997\) 22.3415 0.707561 0.353781 0.935328i \(-0.384896\pi\)
0.353781 + 0.935328i \(0.384896\pi\)
\(998\) 19.0719i 0.603712i
\(999\) 4.52184i 0.143065i
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 3234.2.e.a.2155.14 16
7.2 even 3 462.2.p.a.241.7 16
7.3 odd 6 462.2.p.b.439.3 yes 16
7.6 odd 2 3234.2.e.b.2155.11 16
11.10 odd 2 3234.2.e.b.2155.6 16
21.2 odd 6 1386.2.bk.a.703.2 16
21.17 even 6 1386.2.bk.b.901.6 16
77.10 even 6 462.2.p.a.439.7 yes 16
77.65 odd 6 462.2.p.b.241.3 yes 16
77.76 even 2 inner 3234.2.e.a.2155.3 16
231.65 even 6 1386.2.bk.b.703.6 16
231.164 odd 6 1386.2.bk.a.901.2 16
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
462.2.p.a.241.7 16 7.2 even 3
462.2.p.a.439.7 yes 16 77.10 even 6
462.2.p.b.241.3 yes 16 77.65 odd 6
462.2.p.b.439.3 yes 16 7.3 odd 6
1386.2.bk.a.703.2 16 21.2 odd 6
1386.2.bk.a.901.2 16 231.164 odd 6
1386.2.bk.b.703.6 16 231.65 even 6
1386.2.bk.b.901.6 16 21.17 even 6
3234.2.e.a.2155.3 16 77.76 even 2 inner
3234.2.e.a.2155.14 16 1.1 even 1 trivial
3234.2.e.b.2155.6 16 11.10 odd 2
3234.2.e.b.2155.11 16 7.6 odd 2