Properties

Label 6027.2.a.bi.1.10
Level $6027$
Weight $2$
Character 6027.1
Self dual yes
Analytic conductor $48.126$
Analytic rank $1$
Dimension $13$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [6027,2,Mod(1,6027)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6027, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("6027.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6027 = 3 \cdot 7^{2} \cdot 41 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6027.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: \(48.1258372982\)
Analytic rank: \(1\)
Dimension: \(13\)
Coefficient field: \(\mathbb{Q}[x]/(x^{13} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{13} - 4 x^{12} - 11 x^{11} + 56 x^{10} + 26 x^{9} - 263 x^{8} + 50 x^{7} + 478 x^{6} - 174 x^{5} - 311 x^{4} + 84 x^{3} + 69 x^{2} - 12 x - 4 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 861)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(-0.646272\) of defining polynomial
Character \(\chi\) \(=\) 6027.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.646272 q^{2} +1.00000 q^{3} -1.58233 q^{4} +2.20733 q^{5} +0.646272 q^{6} -2.31516 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+0.646272 q^{2} +1.00000 q^{3} -1.58233 q^{4} +2.20733 q^{5} +0.646272 q^{6} -2.31516 q^{8} +1.00000 q^{9} +1.42653 q^{10} -4.98739 q^{11} -1.58233 q^{12} -0.0981362 q^{13} +2.20733 q^{15} +1.66844 q^{16} +6.26794 q^{17} +0.646272 q^{18} -4.27090 q^{19} -3.49272 q^{20} -3.22321 q^{22} +0.961432 q^{23} -2.31516 q^{24} -0.127708 q^{25} -0.0634227 q^{26} +1.00000 q^{27} -8.78221 q^{29} +1.42653 q^{30} +5.62989 q^{31} +5.70859 q^{32} -4.98739 q^{33} +4.05080 q^{34} -1.58233 q^{36} -8.43213 q^{37} -2.76016 q^{38} -0.0981362 q^{39} -5.11032 q^{40} +1.00000 q^{41} -9.42371 q^{43} +7.89171 q^{44} +2.20733 q^{45} +0.621347 q^{46} +8.39723 q^{47} +1.66844 q^{48} -0.0825339 q^{50} +6.26794 q^{51} +0.155284 q^{52} -12.5970 q^{53} +0.646272 q^{54} -11.0088 q^{55} -4.27090 q^{57} -5.67570 q^{58} -1.30381 q^{59} -3.49272 q^{60} +1.74328 q^{61} +3.63844 q^{62} +0.352424 q^{64} -0.216619 q^{65} -3.22321 q^{66} +9.82028 q^{67} -9.91796 q^{68} +0.961432 q^{69} -7.18882 q^{71} -2.31516 q^{72} +0.547746 q^{73} -5.44945 q^{74} -0.127708 q^{75} +6.75797 q^{76} -0.0634227 q^{78} -2.59244 q^{79} +3.68279 q^{80} +1.00000 q^{81} +0.646272 q^{82} -1.01281 q^{83} +13.8354 q^{85} -6.09028 q^{86} -8.78221 q^{87} +11.5466 q^{88} +14.7801 q^{89} +1.42653 q^{90} -1.52130 q^{92} +5.62989 q^{93} +5.42690 q^{94} -9.42726 q^{95} +5.70859 q^{96} -17.6877 q^{97} -4.98739 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 13 q - 4 q^{2} + 13 q^{3} + 12 q^{4} - 8 q^{5} - 4 q^{6} - 12 q^{8} + 13 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 13 q - 4 q^{2} + 13 q^{3} + 12 q^{4} - 8 q^{5} - 4 q^{6} - 12 q^{8} + 13 q^{9} - q^{10} - 10 q^{11} + 12 q^{12} - 16 q^{13} - 8 q^{15} + 26 q^{16} - 12 q^{17} - 4 q^{18} - 11 q^{19} - 6 q^{20} + q^{22} - 15 q^{23} - 12 q^{24} + 15 q^{25} - 18 q^{26} + 13 q^{27} - 8 q^{29} - q^{30} - 9 q^{31} - 23 q^{32} - 10 q^{33} + 7 q^{34} + 12 q^{36} - 2 q^{37} - 20 q^{38} - 16 q^{39} - 49 q^{40} + 13 q^{41} - 7 q^{43} - 22 q^{44} - 8 q^{45} - 4 q^{46} - 26 q^{47} + 26 q^{48} - 15 q^{50} - 12 q^{51} - 24 q^{52} + 4 q^{53} - 4 q^{54} - q^{55} - 11 q^{57} + 39 q^{58} + 3 q^{59} - 6 q^{60} - 28 q^{61} - 7 q^{62} + 2 q^{64} - 20 q^{65} + q^{66} + 7 q^{67} - 55 q^{68} - 15 q^{69} - 40 q^{71} - 12 q^{72} + 2 q^{73} + q^{74} + 15 q^{75} + 26 q^{76} - 18 q^{78} + 13 q^{79} - 22 q^{80} + 13 q^{81} - 4 q^{82} - 14 q^{83} + 48 q^{85} - 49 q^{86} - 8 q^{87} + 20 q^{88} - 35 q^{89} - q^{90} - 105 q^{92} - 9 q^{93} + 2 q^{94} + 7 q^{95} - 23 q^{96} - 64 q^{97} - 10 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.646272 0.456983 0.228492 0.973546i \(-0.426621\pi\)
0.228492 + 0.973546i \(0.426621\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.58233 −0.791166
\(5\) 2.20733 0.987147 0.493573 0.869704i \(-0.335690\pi\)
0.493573 + 0.869704i \(0.335690\pi\)
\(6\) 0.646272 0.263840
\(7\) 0 0
\(8\) −2.31516 −0.818533
\(9\) 1.00000 0.333333
\(10\) 1.42653 0.451110
\(11\) −4.98739 −1.50376 −0.751878 0.659302i \(-0.770852\pi\)
−0.751878 + 0.659302i \(0.770852\pi\)
\(12\) −1.58233 −0.456780
\(13\) −0.0981362 −0.0272181 −0.0136090 0.999907i \(-0.504332\pi\)
−0.0136090 + 0.999907i \(0.504332\pi\)
\(14\) 0 0
\(15\) 2.20733 0.569929
\(16\) 1.66844 0.417110
\(17\) 6.26794 1.52020 0.760099 0.649807i \(-0.225150\pi\)
0.760099 + 0.649807i \(0.225150\pi\)
\(18\) 0.646272 0.152328
\(19\) −4.27090 −0.979811 −0.489905 0.871776i \(-0.662969\pi\)
−0.489905 + 0.871776i \(0.662969\pi\)
\(20\) −3.49272 −0.780997
\(21\) 0 0
\(22\) −3.22321 −0.687192
\(23\) 0.961432 0.200472 0.100236 0.994964i \(-0.468040\pi\)
0.100236 + 0.994964i \(0.468040\pi\)
\(24\) −2.31516 −0.472580
\(25\) −0.127708 −0.0255415
\(26\) −0.0634227 −0.0124382
\(27\) 1.00000 0.192450
\(28\) 0 0
\(29\) −8.78221 −1.63082 −0.815408 0.578887i \(-0.803487\pi\)
−0.815408 + 0.578887i \(0.803487\pi\)
\(30\) 1.42653 0.260448
\(31\) 5.62989 1.01116 0.505579 0.862780i \(-0.331279\pi\)
0.505579 + 0.862780i \(0.331279\pi\)
\(32\) 5.70859 1.00915
\(33\) −4.98739 −0.868194
\(34\) 4.05080 0.694706
\(35\) 0 0
\(36\) −1.58233 −0.263722
\(37\) −8.43213 −1.38623 −0.693117 0.720825i \(-0.743763\pi\)
−0.693117 + 0.720825i \(0.743763\pi\)
\(38\) −2.76016 −0.447757
\(39\) −0.0981362 −0.0157144
\(40\) −5.11032 −0.808012
\(41\) 1.00000 0.156174
\(42\) 0 0
\(43\) −9.42371 −1.43710 −0.718551 0.695474i \(-0.755194\pi\)
−0.718551 + 0.695474i \(0.755194\pi\)
\(44\) 7.89171 1.18972
\(45\) 2.20733 0.329049
\(46\) 0.621347 0.0916126
\(47\) 8.39723 1.22486 0.612431 0.790524i \(-0.290192\pi\)
0.612431 + 0.790524i \(0.290192\pi\)
\(48\) 1.66844 0.240818
\(49\) 0 0
\(50\) −0.0825339 −0.0116721
\(51\) 6.26794 0.877687
\(52\) 0.155284 0.0215340
\(53\) −12.5970 −1.73033 −0.865167 0.501484i \(-0.832788\pi\)
−0.865167 + 0.501484i \(0.832788\pi\)
\(54\) 0.646272 0.0879465
\(55\) −11.0088 −1.48443
\(56\) 0 0
\(57\) −4.27090 −0.565694
\(58\) −5.67570 −0.745256
\(59\) −1.30381 −0.169741 −0.0848707 0.996392i \(-0.527048\pi\)
−0.0848707 + 0.996392i \(0.527048\pi\)
\(60\) −3.49272 −0.450909
\(61\) 1.74328 0.223204 0.111602 0.993753i \(-0.464402\pi\)
0.111602 + 0.993753i \(0.464402\pi\)
\(62\) 3.63844 0.462082
\(63\) 0 0
\(64\) 0.352424 0.0440531
\(65\) −0.216619 −0.0268682
\(66\) −3.22321 −0.396750
\(67\) 9.82028 1.19974 0.599869 0.800098i \(-0.295219\pi\)
0.599869 + 0.800098i \(0.295219\pi\)
\(68\) −9.91796 −1.20273
\(69\) 0.961432 0.115743
\(70\) 0 0
\(71\) −7.18882 −0.853156 −0.426578 0.904451i \(-0.640281\pi\)
−0.426578 + 0.904451i \(0.640281\pi\)
\(72\) −2.31516 −0.272844
\(73\) 0.547746 0.0641089 0.0320544 0.999486i \(-0.489795\pi\)
0.0320544 + 0.999486i \(0.489795\pi\)
\(74\) −5.44945 −0.633486
\(75\) −0.127708 −0.0147464
\(76\) 6.75797 0.775193
\(77\) 0 0
\(78\) −0.0634227 −0.00718121
\(79\) −2.59244 −0.291672 −0.145836 0.989309i \(-0.546587\pi\)
−0.145836 + 0.989309i \(0.546587\pi\)
\(80\) 3.68279 0.411749
\(81\) 1.00000 0.111111
\(82\) 0.646272 0.0713688
\(83\) −1.01281 −0.111170 −0.0555851 0.998454i \(-0.517702\pi\)
−0.0555851 + 0.998454i \(0.517702\pi\)
\(84\) 0 0
\(85\) 13.8354 1.50066
\(86\) −6.09028 −0.656732
\(87\) −8.78221 −0.941552
\(88\) 11.5466 1.23087
\(89\) 14.7801 1.56668 0.783341 0.621592i \(-0.213514\pi\)
0.783341 + 0.621592i \(0.213514\pi\)
\(90\) 1.42653 0.150370
\(91\) 0 0
\(92\) −1.52130 −0.158607
\(93\) 5.62989 0.583792
\(94\) 5.42690 0.559742
\(95\) −9.42726 −0.967217
\(96\) 5.70859 0.582631
\(97\) −17.6877 −1.79591 −0.897956 0.440085i \(-0.854948\pi\)
−0.897956 + 0.440085i \(0.854948\pi\)
\(98\) 0 0
\(99\) −4.98739 −0.501252
\(100\) 0.202076 0.0202076
\(101\) −9.33195 −0.928564 −0.464282 0.885688i \(-0.653688\pi\)
−0.464282 + 0.885688i \(0.653688\pi\)
\(102\) 4.05080 0.401089
\(103\) −8.42596 −0.830234 −0.415117 0.909768i \(-0.636259\pi\)
−0.415117 + 0.909768i \(0.636259\pi\)
\(104\) 0.227201 0.0222789
\(105\) 0 0
\(106\) −8.14110 −0.790734
\(107\) −10.8948 −1.05324 −0.526618 0.850102i \(-0.676540\pi\)
−0.526618 + 0.850102i \(0.676540\pi\)
\(108\) −1.58233 −0.152260
\(109\) −6.99706 −0.670197 −0.335099 0.942183i \(-0.608770\pi\)
−0.335099 + 0.942183i \(0.608770\pi\)
\(110\) −7.11469 −0.678359
\(111\) −8.43213 −0.800342
\(112\) 0 0
\(113\) −20.1381 −1.89443 −0.947215 0.320600i \(-0.896115\pi\)
−0.947215 + 0.320600i \(0.896115\pi\)
\(114\) −2.76016 −0.258513
\(115\) 2.12219 0.197896
\(116\) 13.8964 1.29025
\(117\) −0.0981362 −0.00907269
\(118\) −0.842615 −0.0775690
\(119\) 0 0
\(120\) −5.11032 −0.466506
\(121\) 13.8741 1.26128
\(122\) 1.12663 0.102000
\(123\) 1.00000 0.0901670
\(124\) −8.90835 −0.799994
\(125\) −11.3185 −1.01236
\(126\) 0 0
\(127\) 13.3189 1.18186 0.590929 0.806723i \(-0.298761\pi\)
0.590929 + 0.806723i \(0.298761\pi\)
\(128\) −11.1894 −0.989014
\(129\) −9.42371 −0.829712
\(130\) −0.139995 −0.0122783
\(131\) −10.8574 −0.948616 −0.474308 0.880359i \(-0.657302\pi\)
−0.474308 + 0.880359i \(0.657302\pi\)
\(132\) 7.89171 0.686885
\(133\) 0 0
\(134\) 6.34657 0.548260
\(135\) 2.20733 0.189976
\(136\) −14.5113 −1.24433
\(137\) 1.94714 0.166356 0.0831779 0.996535i \(-0.473493\pi\)
0.0831779 + 0.996535i \(0.473493\pi\)
\(138\) 0.621347 0.0528925
\(139\) 10.0192 0.849815 0.424907 0.905237i \(-0.360307\pi\)
0.424907 + 0.905237i \(0.360307\pi\)
\(140\) 0 0
\(141\) 8.39723 0.707174
\(142\) −4.64594 −0.389878
\(143\) 0.489444 0.0409293
\(144\) 1.66844 0.139037
\(145\) −19.3852 −1.60985
\(146\) 0.353993 0.0292967
\(147\) 0 0
\(148\) 13.3424 1.09674
\(149\) 19.0408 1.55988 0.779940 0.625854i \(-0.215249\pi\)
0.779940 + 0.625854i \(0.215249\pi\)
\(150\) −0.0825339 −0.00673886
\(151\) −14.8270 −1.20661 −0.603303 0.797512i \(-0.706149\pi\)
−0.603303 + 0.797512i \(0.706149\pi\)
\(152\) 9.88781 0.802008
\(153\) 6.26794 0.506733
\(154\) 0 0
\(155\) 12.4270 0.998161
\(156\) 0.155284 0.0124327
\(157\) −6.00342 −0.479125 −0.239563 0.970881i \(-0.577004\pi\)
−0.239563 + 0.970881i \(0.577004\pi\)
\(158\) −1.67542 −0.133289
\(159\) −12.5970 −0.999009
\(160\) 12.6007 0.996175
\(161\) 0 0
\(162\) 0.646272 0.0507759
\(163\) 12.5669 0.984313 0.492156 0.870507i \(-0.336209\pi\)
0.492156 + 0.870507i \(0.336209\pi\)
\(164\) −1.58233 −0.123559
\(165\) −11.0088 −0.857035
\(166\) −0.654550 −0.0508029
\(167\) 17.9928 1.39233 0.696163 0.717884i \(-0.254889\pi\)
0.696163 + 0.717884i \(0.254889\pi\)
\(168\) 0 0
\(169\) −12.9904 −0.999259
\(170\) 8.94143 0.685776
\(171\) −4.27090 −0.326604
\(172\) 14.9114 1.13699
\(173\) −6.45076 −0.490442 −0.245221 0.969467i \(-0.578861\pi\)
−0.245221 + 0.969467i \(0.578861\pi\)
\(174\) −5.67570 −0.430274
\(175\) 0 0
\(176\) −8.32116 −0.627231
\(177\) −1.30381 −0.0980003
\(178\) 9.55194 0.715948
\(179\) 3.46574 0.259042 0.129521 0.991577i \(-0.458656\pi\)
0.129521 + 0.991577i \(0.458656\pi\)
\(180\) −3.49272 −0.260332
\(181\) 1.73409 0.128894 0.0644469 0.997921i \(-0.479472\pi\)
0.0644469 + 0.997921i \(0.479472\pi\)
\(182\) 0 0
\(183\) 1.74328 0.128867
\(184\) −2.22587 −0.164093
\(185\) −18.6125 −1.36842
\(186\) 3.63844 0.266783
\(187\) −31.2607 −2.28601
\(188\) −13.2872 −0.969069
\(189\) 0 0
\(190\) −6.09258 −0.442002
\(191\) 9.71393 0.702875 0.351438 0.936211i \(-0.385693\pi\)
0.351438 + 0.936211i \(0.385693\pi\)
\(192\) 0.352424 0.0254340
\(193\) −26.6411 −1.91767 −0.958833 0.283970i \(-0.908348\pi\)
−0.958833 + 0.283970i \(0.908348\pi\)
\(194\) −11.4311 −0.820702
\(195\) −0.216619 −0.0155124
\(196\) 0 0
\(197\) −0.169300 −0.0120621 −0.00603106 0.999982i \(-0.501920\pi\)
−0.00603106 + 0.999982i \(0.501920\pi\)
\(198\) −3.22321 −0.229064
\(199\) −24.2237 −1.71717 −0.858587 0.512668i \(-0.828657\pi\)
−0.858587 + 0.512668i \(0.828657\pi\)
\(200\) 0.295664 0.0209066
\(201\) 9.82028 0.692669
\(202\) −6.03098 −0.424338
\(203\) 0 0
\(204\) −9.91796 −0.694396
\(205\) 2.20733 0.154166
\(206\) −5.44546 −0.379403
\(207\) 0.961432 0.0668241
\(208\) −0.163734 −0.0113529
\(209\) 21.3006 1.47340
\(210\) 0 0
\(211\) −7.40691 −0.509912 −0.254956 0.966953i \(-0.582061\pi\)
−0.254956 + 0.966953i \(0.582061\pi\)
\(212\) 19.9327 1.36898
\(213\) −7.18882 −0.492570
\(214\) −7.04098 −0.481312
\(215\) −20.8012 −1.41863
\(216\) −2.31516 −0.157527
\(217\) 0 0
\(218\) −4.52201 −0.306269
\(219\) 0.547746 0.0370133
\(220\) 17.4196 1.17443
\(221\) −0.615112 −0.0413769
\(222\) −5.44945 −0.365743
\(223\) −28.5175 −1.90967 −0.954837 0.297129i \(-0.903971\pi\)
−0.954837 + 0.297129i \(0.903971\pi\)
\(224\) 0 0
\(225\) −0.127708 −0.00851384
\(226\) −13.0147 −0.865723
\(227\) −15.4763 −1.02720 −0.513601 0.858029i \(-0.671689\pi\)
−0.513601 + 0.858029i \(0.671689\pi\)
\(228\) 6.75797 0.447558
\(229\) 2.59305 0.171353 0.0856766 0.996323i \(-0.472695\pi\)
0.0856766 + 0.996323i \(0.472695\pi\)
\(230\) 1.37152 0.0904350
\(231\) 0 0
\(232\) 20.3322 1.33488
\(233\) 7.83806 0.513488 0.256744 0.966479i \(-0.417350\pi\)
0.256744 + 0.966479i \(0.417350\pi\)
\(234\) −0.0634227 −0.00414607
\(235\) 18.5354 1.20912
\(236\) 2.06306 0.134294
\(237\) −2.59244 −0.168397
\(238\) 0 0
\(239\) −5.90515 −0.381972 −0.190986 0.981593i \(-0.561169\pi\)
−0.190986 + 0.981593i \(0.561169\pi\)
\(240\) 3.68279 0.237723
\(241\) 29.9744 1.93082 0.965410 0.260738i \(-0.0839660\pi\)
0.965410 + 0.260738i \(0.0839660\pi\)
\(242\) 8.96644 0.576385
\(243\) 1.00000 0.0641500
\(244\) −2.75844 −0.176591
\(245\) 0 0
\(246\) 0.646272 0.0412048
\(247\) 0.419129 0.0266686
\(248\) −13.0341 −0.827666
\(249\) −1.01281 −0.0641841
\(250\) −7.31485 −0.462632
\(251\) 1.69710 0.107120 0.0535600 0.998565i \(-0.482943\pi\)
0.0535600 + 0.998565i \(0.482943\pi\)
\(252\) 0 0
\(253\) −4.79504 −0.301461
\(254\) 8.60761 0.540090
\(255\) 13.8354 0.866406
\(256\) −7.93626 −0.496016
\(257\) −2.03614 −0.127011 −0.0635054 0.997981i \(-0.520228\pi\)
−0.0635054 + 0.997981i \(0.520228\pi\)
\(258\) −6.09028 −0.379164
\(259\) 0 0
\(260\) 0.342763 0.0212572
\(261\) −8.78221 −0.543605
\(262\) −7.01684 −0.433502
\(263\) 8.79054 0.542048 0.271024 0.962573i \(-0.412638\pi\)
0.271024 + 0.962573i \(0.412638\pi\)
\(264\) 11.5466 0.710646
\(265\) −27.8057 −1.70809
\(266\) 0 0
\(267\) 14.7801 0.904525
\(268\) −15.5389 −0.949192
\(269\) 4.68163 0.285444 0.142722 0.989763i \(-0.454415\pi\)
0.142722 + 0.989763i \(0.454415\pi\)
\(270\) 1.42653 0.0868161
\(271\) 15.2141 0.924189 0.462095 0.886831i \(-0.347098\pi\)
0.462095 + 0.886831i \(0.347098\pi\)
\(272\) 10.4577 0.634090
\(273\) 0 0
\(274\) 1.25838 0.0760218
\(275\) 0.636928 0.0384082
\(276\) −1.52130 −0.0915718
\(277\) −29.2313 −1.75634 −0.878170 0.478349i \(-0.841235\pi\)
−0.878170 + 0.478349i \(0.841235\pi\)
\(278\) 6.47511 0.388351
\(279\) 5.62989 0.337053
\(280\) 0 0
\(281\) 5.86638 0.349959 0.174980 0.984572i \(-0.444014\pi\)
0.174980 + 0.984572i \(0.444014\pi\)
\(282\) 5.42690 0.323167
\(283\) 16.0598 0.954653 0.477327 0.878726i \(-0.341606\pi\)
0.477327 + 0.878726i \(0.341606\pi\)
\(284\) 11.3751 0.674988
\(285\) −9.42726 −0.558423
\(286\) 0.316314 0.0187040
\(287\) 0 0
\(288\) 5.70859 0.336382
\(289\) 22.2871 1.31100
\(290\) −12.5281 −0.735677
\(291\) −17.6877 −1.03687
\(292\) −0.866717 −0.0507208
\(293\) −12.9726 −0.757870 −0.378935 0.925423i \(-0.623710\pi\)
−0.378935 + 0.925423i \(0.623710\pi\)
\(294\) 0 0
\(295\) −2.87793 −0.167560
\(296\) 19.5217 1.13468
\(297\) −4.98739 −0.289398
\(298\) 12.3055 0.712840
\(299\) −0.0943512 −0.00545647
\(300\) 0.202076 0.0116669
\(301\) 0 0
\(302\) −9.58229 −0.551399
\(303\) −9.33195 −0.536106
\(304\) −7.12573 −0.408689
\(305\) 3.84798 0.220335
\(306\) 4.05080 0.231569
\(307\) 24.4190 1.39367 0.696834 0.717233i \(-0.254592\pi\)
0.696834 + 0.717233i \(0.254592\pi\)
\(308\) 0 0
\(309\) −8.42596 −0.479336
\(310\) 8.03123 0.456143
\(311\) 18.2933 1.03732 0.518660 0.854981i \(-0.326431\pi\)
0.518660 + 0.854981i \(0.326431\pi\)
\(312\) 0.227201 0.0128627
\(313\) 9.76655 0.552038 0.276019 0.961152i \(-0.410985\pi\)
0.276019 + 0.961152i \(0.410985\pi\)
\(314\) −3.87984 −0.218952
\(315\) 0 0
\(316\) 4.10209 0.230761
\(317\) 11.2980 0.634560 0.317280 0.948332i \(-0.397230\pi\)
0.317280 + 0.948332i \(0.397230\pi\)
\(318\) −8.14110 −0.456530
\(319\) 43.8004 2.45235
\(320\) 0.777916 0.0434868
\(321\) −10.8948 −0.608086
\(322\) 0 0
\(323\) −26.7697 −1.48951
\(324\) −1.58233 −0.0879073
\(325\) 0.0125327 0.000695191 0
\(326\) 8.12162 0.449815
\(327\) −6.99706 −0.386938
\(328\) −2.31516 −0.127833
\(329\) 0 0
\(330\) −7.11469 −0.391651
\(331\) −2.09392 −0.115092 −0.0575460 0.998343i \(-0.518328\pi\)
−0.0575460 + 0.998343i \(0.518328\pi\)
\(332\) 1.60260 0.0879540
\(333\) −8.43213 −0.462078
\(334\) 11.6283 0.636270
\(335\) 21.6766 1.18432
\(336\) 0 0
\(337\) 3.67269 0.200064 0.100032 0.994984i \(-0.468105\pi\)
0.100032 + 0.994984i \(0.468105\pi\)
\(338\) −8.39532 −0.456645
\(339\) −20.1381 −1.09375
\(340\) −21.8922 −1.18727
\(341\) −28.0785 −1.52053
\(342\) −2.76016 −0.149252
\(343\) 0 0
\(344\) 21.8174 1.17632
\(345\) 2.12219 0.114255
\(346\) −4.16895 −0.224124
\(347\) 27.4690 1.47461 0.737307 0.675558i \(-0.236097\pi\)
0.737307 + 0.675558i \(0.236097\pi\)
\(348\) 13.8964 0.744924
\(349\) −12.9032 −0.690690 −0.345345 0.938476i \(-0.612238\pi\)
−0.345345 + 0.938476i \(0.612238\pi\)
\(350\) 0 0
\(351\) −0.0981362 −0.00523812
\(352\) −28.4710 −1.51751
\(353\) −0.376084 −0.0200169 −0.0100085 0.999950i \(-0.503186\pi\)
−0.0100085 + 0.999950i \(0.503186\pi\)
\(354\) −0.842615 −0.0447845
\(355\) −15.8681 −0.842190
\(356\) −23.3870 −1.23951
\(357\) 0 0
\(358\) 2.23981 0.118378
\(359\) −18.0589 −0.953115 −0.476557 0.879143i \(-0.658116\pi\)
−0.476557 + 0.879143i \(0.658116\pi\)
\(360\) −5.11032 −0.269337
\(361\) −0.759455 −0.0399713
\(362\) 1.12069 0.0589024
\(363\) 13.8741 0.728201
\(364\) 0 0
\(365\) 1.20906 0.0632849
\(366\) 1.12663 0.0588900
\(367\) 15.7573 0.822523 0.411262 0.911517i \(-0.365088\pi\)
0.411262 + 0.911517i \(0.365088\pi\)
\(368\) 1.60409 0.0836190
\(369\) 1.00000 0.0520579
\(370\) −12.0287 −0.625343
\(371\) 0 0
\(372\) −8.90835 −0.461877
\(373\) 26.1027 1.35155 0.675773 0.737110i \(-0.263810\pi\)
0.675773 + 0.737110i \(0.263810\pi\)
\(374\) −20.2029 −1.04467
\(375\) −11.3185 −0.584486
\(376\) −19.4409 −1.00259
\(377\) 0.861853 0.0443877
\(378\) 0 0
\(379\) 6.93660 0.356309 0.178155 0.984003i \(-0.442987\pi\)
0.178155 + 0.984003i \(0.442987\pi\)
\(380\) 14.9171 0.765229
\(381\) 13.3189 0.682346
\(382\) 6.27784 0.321202
\(383\) 24.0548 1.22914 0.614572 0.788861i \(-0.289329\pi\)
0.614572 + 0.788861i \(0.289329\pi\)
\(384\) −11.1894 −0.571008
\(385\) 0 0
\(386\) −17.2174 −0.876342
\(387\) −9.42371 −0.479034
\(388\) 27.9878 1.42086
\(389\) 4.08597 0.207167 0.103584 0.994621i \(-0.466969\pi\)
0.103584 + 0.994621i \(0.466969\pi\)
\(390\) −0.139995 −0.00708890
\(391\) 6.02620 0.304758
\(392\) 0 0
\(393\) −10.8574 −0.547684
\(394\) −0.109414 −0.00551219
\(395\) −5.72235 −0.287923
\(396\) 7.89171 0.396574
\(397\) 0.556877 0.0279489 0.0139744 0.999902i \(-0.495552\pi\)
0.0139744 + 0.999902i \(0.495552\pi\)
\(398\) −15.6551 −0.784720
\(399\) 0 0
\(400\) −0.213072 −0.0106536
\(401\) −30.2668 −1.51145 −0.755726 0.654888i \(-0.772716\pi\)
−0.755726 + 0.654888i \(0.772716\pi\)
\(402\) 6.34657 0.316538
\(403\) −0.552496 −0.0275218
\(404\) 14.7662 0.734648
\(405\) 2.20733 0.109683
\(406\) 0 0
\(407\) 42.0543 2.08456
\(408\) −14.5113 −0.718416
\(409\) 7.57506 0.374562 0.187281 0.982306i \(-0.440032\pi\)
0.187281 + 0.982306i \(0.440032\pi\)
\(410\) 1.42653 0.0704515
\(411\) 1.94714 0.0960455
\(412\) 13.3327 0.656853
\(413\) 0 0
\(414\) 0.621347 0.0305375
\(415\) −2.23560 −0.109741
\(416\) −0.560219 −0.0274670
\(417\) 10.0192 0.490641
\(418\) 13.7660 0.673318
\(419\) −4.76050 −0.232565 −0.116283 0.993216i \(-0.537098\pi\)
−0.116283 + 0.993216i \(0.537098\pi\)
\(420\) 0 0
\(421\) 5.72997 0.279262 0.139631 0.990204i \(-0.455408\pi\)
0.139631 + 0.990204i \(0.455408\pi\)
\(422\) −4.78688 −0.233022
\(423\) 8.39723 0.408287
\(424\) 29.1641 1.41634
\(425\) −0.800463 −0.0388282
\(426\) −4.64594 −0.225096
\(427\) 0 0
\(428\) 17.2391 0.833285
\(429\) 0.489444 0.0236306
\(430\) −13.4432 −0.648291
\(431\) −36.1635 −1.74193 −0.870967 0.491341i \(-0.836507\pi\)
−0.870967 + 0.491341i \(0.836507\pi\)
\(432\) 1.66844 0.0802728
\(433\) −40.4381 −1.94333 −0.971666 0.236358i \(-0.924046\pi\)
−0.971666 + 0.236358i \(0.924046\pi\)
\(434\) 0 0
\(435\) −19.3852 −0.929450
\(436\) 11.0717 0.530237
\(437\) −4.10617 −0.196425
\(438\) 0.353993 0.0169145
\(439\) −4.51440 −0.215461 −0.107730 0.994180i \(-0.534358\pi\)
−0.107730 + 0.994180i \(0.534358\pi\)
\(440\) 25.4872 1.21505
\(441\) 0 0
\(442\) −0.397530 −0.0189086
\(443\) 2.29267 0.108928 0.0544641 0.998516i \(-0.482655\pi\)
0.0544641 + 0.998516i \(0.482655\pi\)
\(444\) 13.3424 0.633204
\(445\) 32.6244 1.54655
\(446\) −18.4301 −0.872690
\(447\) 19.0408 0.900597
\(448\) 0 0
\(449\) −0.770455 −0.0363600 −0.0181800 0.999835i \(-0.505787\pi\)
−0.0181800 + 0.999835i \(0.505787\pi\)
\(450\) −0.0825339 −0.00389068
\(451\) −4.98739 −0.234847
\(452\) 31.8651 1.49881
\(453\) −14.8270 −0.696634
\(454\) −10.0019 −0.469414
\(455\) 0 0
\(456\) 9.88781 0.463039
\(457\) 9.33921 0.436870 0.218435 0.975851i \(-0.429905\pi\)
0.218435 + 0.975851i \(0.429905\pi\)
\(458\) 1.67581 0.0783056
\(459\) 6.26794 0.292562
\(460\) −3.35802 −0.156568
\(461\) −38.2020 −1.77925 −0.889623 0.456695i \(-0.849033\pi\)
−0.889623 + 0.456695i \(0.849033\pi\)
\(462\) 0 0
\(463\) 12.6784 0.589217 0.294609 0.955618i \(-0.404811\pi\)
0.294609 + 0.955618i \(0.404811\pi\)
\(464\) −14.6526 −0.680229
\(465\) 12.4270 0.576288
\(466\) 5.06552 0.234656
\(467\) 2.95874 0.136914 0.0684571 0.997654i \(-0.478192\pi\)
0.0684571 + 0.997654i \(0.478192\pi\)
\(468\) 0.155284 0.00717801
\(469\) 0 0
\(470\) 11.9789 0.552547
\(471\) −6.00342 −0.276623
\(472\) 3.01853 0.138939
\(473\) 46.9998 2.16105
\(474\) −1.67542 −0.0769546
\(475\) 0.545426 0.0250258
\(476\) 0 0
\(477\) −12.5970 −0.576778
\(478\) −3.81633 −0.174555
\(479\) −13.9672 −0.638179 −0.319090 0.947725i \(-0.603377\pi\)
−0.319090 + 0.947725i \(0.603377\pi\)
\(480\) 12.6007 0.575142
\(481\) 0.827497 0.0377306
\(482\) 19.3716 0.882352
\(483\) 0 0
\(484\) −21.9534 −0.997883
\(485\) −39.0425 −1.77283
\(486\) 0.646272 0.0293155
\(487\) −1.83687 −0.0832364 −0.0416182 0.999134i \(-0.513251\pi\)
−0.0416182 + 0.999134i \(0.513251\pi\)
\(488\) −4.03597 −0.182700
\(489\) 12.5669 0.568293
\(490\) 0 0
\(491\) −15.5775 −0.703003 −0.351502 0.936187i \(-0.614329\pi\)
−0.351502 + 0.936187i \(0.614329\pi\)
\(492\) −1.58233 −0.0713370
\(493\) −55.0464 −2.47916
\(494\) 0.270872 0.0121871
\(495\) −11.0088 −0.494809
\(496\) 9.39313 0.421764
\(497\) 0 0
\(498\) −0.654550 −0.0293311
\(499\) −19.1022 −0.855133 −0.427566 0.903984i \(-0.640629\pi\)
−0.427566 + 0.903984i \(0.640629\pi\)
\(500\) 17.9097 0.800945
\(501\) 17.9928 0.803860
\(502\) 1.09679 0.0489520
\(503\) 24.8976 1.11013 0.555066 0.831807i \(-0.312693\pi\)
0.555066 + 0.831807i \(0.312693\pi\)
\(504\) 0 0
\(505\) −20.5987 −0.916628
\(506\) −3.09890 −0.137763
\(507\) −12.9904 −0.576923
\(508\) −21.0749 −0.935046
\(509\) −18.5591 −0.822617 −0.411309 0.911496i \(-0.634928\pi\)
−0.411309 + 0.911496i \(0.634928\pi\)
\(510\) 8.94143 0.395933
\(511\) 0 0
\(512\) 17.2499 0.762343
\(513\) −4.27090 −0.188565
\(514\) −1.31590 −0.0580419
\(515\) −18.5988 −0.819563
\(516\) 14.9114 0.656440
\(517\) −41.8803 −1.84189
\(518\) 0 0
\(519\) −6.45076 −0.283157
\(520\) 0.501507 0.0219925
\(521\) −11.3049 −0.495275 −0.247638 0.968853i \(-0.579654\pi\)
−0.247638 + 0.968853i \(0.579654\pi\)
\(522\) −5.67570 −0.248419
\(523\) 0.881218 0.0385330 0.0192665 0.999814i \(-0.493867\pi\)
0.0192665 + 0.999814i \(0.493867\pi\)
\(524\) 17.1800 0.750513
\(525\) 0 0
\(526\) 5.68108 0.247707
\(527\) 35.2878 1.53716
\(528\) −8.32116 −0.362132
\(529\) −22.0756 −0.959811
\(530\) −17.9701 −0.780570
\(531\) −1.30381 −0.0565805
\(532\) 0 0
\(533\) −0.0981362 −0.00425075
\(534\) 9.55194 0.413353
\(535\) −24.0483 −1.03970
\(536\) −22.7355 −0.982025
\(537\) 3.46574 0.149558
\(538\) 3.02561 0.130443
\(539\) 0 0
\(540\) −3.49272 −0.150303
\(541\) 18.6630 0.802387 0.401194 0.915993i \(-0.368595\pi\)
0.401194 + 0.915993i \(0.368595\pi\)
\(542\) 9.83244 0.422339
\(543\) 1.73409 0.0744169
\(544\) 35.7811 1.53410
\(545\) −15.4448 −0.661583
\(546\) 0 0
\(547\) −23.5439 −1.00667 −0.503333 0.864093i \(-0.667893\pi\)
−0.503333 + 0.864093i \(0.667893\pi\)
\(548\) −3.08103 −0.131615
\(549\) 1.74328 0.0744012
\(550\) 0.411629 0.0175519
\(551\) 37.5079 1.59789
\(552\) −2.22587 −0.0947393
\(553\) 0 0
\(554\) −18.8914 −0.802618
\(555\) −18.6125 −0.790055
\(556\) −15.8537 −0.672345
\(557\) 9.73807 0.412615 0.206308 0.978487i \(-0.433855\pi\)
0.206308 + 0.978487i \(0.433855\pi\)
\(558\) 3.63844 0.154027
\(559\) 0.924807 0.0391152
\(560\) 0 0
\(561\) −31.2607 −1.31983
\(562\) 3.79128 0.159926
\(563\) 28.0681 1.18293 0.591466 0.806330i \(-0.298550\pi\)
0.591466 + 0.806330i \(0.298550\pi\)
\(564\) −13.2872 −0.559492
\(565\) −44.4513 −1.87008
\(566\) 10.3790 0.436261
\(567\) 0 0
\(568\) 16.6433 0.698337
\(569\) −34.0959 −1.42938 −0.714688 0.699444i \(-0.753431\pi\)
−0.714688 + 0.699444i \(0.753431\pi\)
\(570\) −6.09258 −0.255190
\(571\) 24.5038 1.02545 0.512727 0.858552i \(-0.328635\pi\)
0.512727 + 0.858552i \(0.328635\pi\)
\(572\) −0.774463 −0.0323819
\(573\) 9.71393 0.405805
\(574\) 0 0
\(575\) −0.122782 −0.00512037
\(576\) 0.352424 0.0146844
\(577\) −3.13999 −0.130719 −0.0653597 0.997862i \(-0.520819\pi\)
−0.0653597 + 0.997862i \(0.520819\pi\)
\(578\) 14.4035 0.599107
\(579\) −26.6411 −1.10717
\(580\) 30.6739 1.27366
\(581\) 0 0
\(582\) −11.4311 −0.473833
\(583\) 62.8263 2.60200
\(584\) −1.26812 −0.0524753
\(585\) −0.216619 −0.00895608
\(586\) −8.38386 −0.346334
\(587\) −12.3401 −0.509329 −0.254665 0.967029i \(-0.581965\pi\)
−0.254665 + 0.967029i \(0.581965\pi\)
\(588\) 0 0
\(589\) −24.0447 −0.990743
\(590\) −1.85993 −0.0765720
\(591\) −0.169300 −0.00696407
\(592\) −14.0685 −0.578212
\(593\) 28.6742 1.17751 0.588754 0.808312i \(-0.299619\pi\)
0.588754 + 0.808312i \(0.299619\pi\)
\(594\) −3.22321 −0.132250
\(595\) 0 0
\(596\) −30.1288 −1.23412
\(597\) −24.2237 −0.991411
\(598\) −0.0609766 −0.00249352
\(599\) 6.80349 0.277983 0.138991 0.990294i \(-0.455614\pi\)
0.138991 + 0.990294i \(0.455614\pi\)
\(600\) 0.295664 0.0120704
\(601\) −20.4130 −0.832665 −0.416333 0.909212i \(-0.636685\pi\)
−0.416333 + 0.909212i \(0.636685\pi\)
\(602\) 0 0
\(603\) 9.82028 0.399913
\(604\) 23.4613 0.954625
\(605\) 30.6247 1.24507
\(606\) −6.03098 −0.244992
\(607\) 9.15724 0.371681 0.185840 0.982580i \(-0.440499\pi\)
0.185840 + 0.982580i \(0.440499\pi\)
\(608\) −24.3808 −0.988772
\(609\) 0 0
\(610\) 2.48684 0.100689
\(611\) −0.824072 −0.0333384
\(612\) −9.91796 −0.400910
\(613\) −36.1686 −1.46083 −0.730417 0.683001i \(-0.760674\pi\)
−0.730417 + 0.683001i \(0.760674\pi\)
\(614\) 15.7813 0.636883
\(615\) 2.20733 0.0890080
\(616\) 0 0
\(617\) 9.38805 0.377949 0.188974 0.981982i \(-0.439484\pi\)
0.188974 + 0.981982i \(0.439484\pi\)
\(618\) −5.44546 −0.219049
\(619\) −1.65438 −0.0664951 −0.0332476 0.999447i \(-0.510585\pi\)
−0.0332476 + 0.999447i \(0.510585\pi\)
\(620\) −19.6636 −0.789711
\(621\) 0.961432 0.0385809
\(622\) 11.8225 0.474038
\(623\) 0 0
\(624\) −0.163734 −0.00655462
\(625\) −24.3452 −0.973806
\(626\) 6.31185 0.252272
\(627\) 21.3006 0.850665
\(628\) 9.49940 0.379068
\(629\) −52.8521 −2.10735
\(630\) 0 0
\(631\) −3.71318 −0.147819 −0.0739097 0.997265i \(-0.523548\pi\)
−0.0739097 + 0.997265i \(0.523548\pi\)
\(632\) 6.00191 0.238743
\(633\) −7.40691 −0.294398
\(634\) 7.30160 0.289984
\(635\) 29.3991 1.16667
\(636\) 19.9327 0.790382
\(637\) 0 0
\(638\) 28.3070 1.12068
\(639\) −7.18882 −0.284385
\(640\) −24.6987 −0.976302
\(641\) −12.6040 −0.497829 −0.248914 0.968526i \(-0.580074\pi\)
−0.248914 + 0.968526i \(0.580074\pi\)
\(642\) −7.04098 −0.277885
\(643\) 26.4140 1.04167 0.520833 0.853658i \(-0.325621\pi\)
0.520833 + 0.853658i \(0.325621\pi\)
\(644\) 0 0
\(645\) −20.8012 −0.819047
\(646\) −17.3005 −0.680680
\(647\) 27.0862 1.06487 0.532434 0.846471i \(-0.321277\pi\)
0.532434 + 0.846471i \(0.321277\pi\)
\(648\) −2.31516 −0.0909481
\(649\) 6.50261 0.255250
\(650\) 0.00809956 0.000317691 0
\(651\) 0 0
\(652\) −19.8850 −0.778755
\(653\) −8.49488 −0.332431 −0.166215 0.986089i \(-0.553155\pi\)
−0.166215 + 0.986089i \(0.553155\pi\)
\(654\) −4.52201 −0.176825
\(655\) −23.9658 −0.936423
\(656\) 1.66844 0.0651416
\(657\) 0.547746 0.0213696
\(658\) 0 0
\(659\) −2.15227 −0.0838405 −0.0419202 0.999121i \(-0.513348\pi\)
−0.0419202 + 0.999121i \(0.513348\pi\)
\(660\) 17.4196 0.678057
\(661\) −14.3926 −0.559808 −0.279904 0.960028i \(-0.590303\pi\)
−0.279904 + 0.960028i \(0.590303\pi\)
\(662\) −1.35324 −0.0525952
\(663\) −0.615112 −0.0238890
\(664\) 2.34481 0.0909964
\(665\) 0 0
\(666\) −5.44945 −0.211162
\(667\) −8.44350 −0.326934
\(668\) −28.4706 −1.10156
\(669\) −28.5175 −1.10255
\(670\) 14.0090 0.541213
\(671\) −8.69441 −0.335644
\(672\) 0 0
\(673\) 44.6210 1.72001 0.860007 0.510282i \(-0.170459\pi\)
0.860007 + 0.510282i \(0.170459\pi\)
\(674\) 2.37356 0.0914260
\(675\) −0.127708 −0.00491547
\(676\) 20.5551 0.790580
\(677\) −34.6693 −1.33245 −0.666226 0.745750i \(-0.732091\pi\)
−0.666226 + 0.745750i \(0.732091\pi\)
\(678\) −13.0147 −0.499825
\(679\) 0 0
\(680\) −32.0312 −1.22834
\(681\) −15.4763 −0.593055
\(682\) −18.1463 −0.694859
\(683\) 25.2748 0.967113 0.483556 0.875313i \(-0.339345\pi\)
0.483556 + 0.875313i \(0.339345\pi\)
\(684\) 6.75797 0.258398
\(685\) 4.29798 0.164217
\(686\) 0 0
\(687\) 2.59305 0.0989309
\(688\) −15.7229 −0.599430
\(689\) 1.23622 0.0470964
\(690\) 1.37152 0.0522127
\(691\) −8.65506 −0.329254 −0.164627 0.986356i \(-0.552642\pi\)
−0.164627 + 0.986356i \(0.552642\pi\)
\(692\) 10.2072 0.388021
\(693\) 0 0
\(694\) 17.7525 0.673874
\(695\) 22.1156 0.838892
\(696\) 20.3322 0.770692
\(697\) 6.26794 0.237415
\(698\) −8.33896 −0.315634
\(699\) 7.83806 0.296463
\(700\) 0 0
\(701\) −35.7541 −1.35041 −0.675207 0.737628i \(-0.735946\pi\)
−0.675207 + 0.737628i \(0.735946\pi\)
\(702\) −0.0634227 −0.00239374
\(703\) 36.0127 1.35825
\(704\) −1.75768 −0.0662450
\(705\) 18.5354 0.698085
\(706\) −0.243053 −0.00914740
\(707\) 0 0
\(708\) 2.06306 0.0775345
\(709\) 18.7083 0.702606 0.351303 0.936262i \(-0.385739\pi\)
0.351303 + 0.936262i \(0.385739\pi\)
\(710\) −10.2551 −0.384867
\(711\) −2.59244 −0.0972239
\(712\) −34.2182 −1.28238
\(713\) 5.41275 0.202709
\(714\) 0 0
\(715\) 1.08036 0.0404033
\(716\) −5.48395 −0.204945
\(717\) −5.90515 −0.220532
\(718\) −11.6710 −0.435558
\(719\) 19.2531 0.718021 0.359011 0.933334i \(-0.383114\pi\)
0.359011 + 0.933334i \(0.383114\pi\)
\(720\) 3.68279 0.137250
\(721\) 0 0
\(722\) −0.490815 −0.0182662
\(723\) 29.9744 1.11476
\(724\) −2.74391 −0.101976
\(725\) 1.12156 0.0416535
\(726\) 8.96644 0.332776
\(727\) 23.3415 0.865689 0.432845 0.901469i \(-0.357510\pi\)
0.432845 + 0.901469i \(0.357510\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0.781379 0.0289201
\(731\) −59.0673 −2.18468
\(732\) −2.75844 −0.101955
\(733\) 41.3374 1.52683 0.763416 0.645907i \(-0.223521\pi\)
0.763416 + 0.645907i \(0.223521\pi\)
\(734\) 10.1835 0.375880
\(735\) 0 0
\(736\) 5.48842 0.202306
\(737\) −48.9776 −1.80411
\(738\) 0.646272 0.0237896
\(739\) 18.3470 0.674904 0.337452 0.941343i \(-0.390435\pi\)
0.337452 + 0.941343i \(0.390435\pi\)
\(740\) 29.4511 1.08264
\(741\) 0.419129 0.0153971
\(742\) 0 0
\(743\) 34.1507 1.25287 0.626433 0.779475i \(-0.284514\pi\)
0.626433 + 0.779475i \(0.284514\pi\)
\(744\) −13.0341 −0.477853
\(745\) 42.0292 1.53983
\(746\) 16.8694 0.617634
\(747\) −1.01281 −0.0370567
\(748\) 49.4648 1.80861
\(749\) 0 0
\(750\) −7.31485 −0.267101
\(751\) 25.7182 0.938471 0.469236 0.883073i \(-0.344529\pi\)
0.469236 + 0.883073i \(0.344529\pi\)
\(752\) 14.0103 0.510902
\(753\) 1.69710 0.0618457
\(754\) 0.556992 0.0202844
\(755\) −32.7281 −1.19110
\(756\) 0 0
\(757\) −17.4055 −0.632614 −0.316307 0.948657i \(-0.602443\pi\)
−0.316307 + 0.948657i \(0.602443\pi\)
\(758\) 4.48293 0.162827
\(759\) −4.79504 −0.174049
\(760\) 21.8256 0.791699
\(761\) 17.9597 0.651040 0.325520 0.945535i \(-0.394461\pi\)
0.325520 + 0.945535i \(0.394461\pi\)
\(762\) 8.60761 0.311821
\(763\) 0 0
\(764\) −15.3707 −0.556091
\(765\) 13.8354 0.500220
\(766\) 15.5460 0.561698
\(767\) 0.127951 0.00462004
\(768\) −7.93626 −0.286375
\(769\) 42.5558 1.53460 0.767301 0.641287i \(-0.221599\pi\)
0.767301 + 0.641287i \(0.221599\pi\)
\(770\) 0 0
\(771\) −2.03614 −0.0733297
\(772\) 42.1550 1.51719
\(773\) −30.2158 −1.08679 −0.543393 0.839478i \(-0.682861\pi\)
−0.543393 + 0.839478i \(0.682861\pi\)
\(774\) −6.09028 −0.218911
\(775\) −0.718979 −0.0258265
\(776\) 40.9498 1.47001
\(777\) 0 0
\(778\) 2.64065 0.0946719
\(779\) −4.27090 −0.153021
\(780\) 0.342763 0.0122729
\(781\) 35.8535 1.28294
\(782\) 3.89456 0.139269
\(783\) −8.78221 −0.313851
\(784\) 0 0
\(785\) −13.2515 −0.472967
\(786\) −7.01684 −0.250282
\(787\) 49.5326 1.76565 0.882824 0.469705i \(-0.155640\pi\)
0.882824 + 0.469705i \(0.155640\pi\)
\(788\) 0.267889 0.00954315
\(789\) 8.79054 0.312952
\(790\) −3.69820 −0.131576
\(791\) 0 0
\(792\) 11.5466 0.410291
\(793\) −0.171078 −0.00607518
\(794\) 0.359894 0.0127722
\(795\) −27.8057 −0.986168
\(796\) 38.3300 1.35857
\(797\) −4.84228 −0.171522 −0.0857611 0.996316i \(-0.527332\pi\)
−0.0857611 + 0.996316i \(0.527332\pi\)
\(798\) 0 0
\(799\) 52.6333 1.86203
\(800\) −0.729030 −0.0257751
\(801\) 14.7801 0.522228
\(802\) −19.5606 −0.690708
\(803\) −2.73183 −0.0964041
\(804\) −15.5389 −0.548016
\(805\) 0 0
\(806\) −0.357063 −0.0125770
\(807\) 4.68163 0.164801
\(808\) 21.6050 0.760060
\(809\) 0.0759852 0.00267150 0.00133575 0.999999i \(-0.499575\pi\)
0.00133575 + 0.999999i \(0.499575\pi\)
\(810\) 1.42653 0.0501233
\(811\) 39.6990 1.39402 0.697010 0.717062i \(-0.254513\pi\)
0.697010 + 0.717062i \(0.254513\pi\)
\(812\) 0 0
\(813\) 15.2141 0.533581
\(814\) 27.1786 0.952608
\(815\) 27.7392 0.971661
\(816\) 10.4577 0.366092
\(817\) 40.2477 1.40809
\(818\) 4.89555 0.171169
\(819\) 0 0
\(820\) −3.49272 −0.121971
\(821\) −8.42503 −0.294036 −0.147018 0.989134i \(-0.546967\pi\)
−0.147018 + 0.989134i \(0.546967\pi\)
\(822\) 1.25838 0.0438912
\(823\) −32.0659 −1.11775 −0.558874 0.829253i \(-0.688766\pi\)
−0.558874 + 0.829253i \(0.688766\pi\)
\(824\) 19.5075 0.679574
\(825\) 0.636928 0.0221750
\(826\) 0 0
\(827\) −34.3100 −1.19308 −0.596539 0.802584i \(-0.703458\pi\)
−0.596539 + 0.802584i \(0.703458\pi\)
\(828\) −1.52130 −0.0528690
\(829\) 43.6558 1.51623 0.758114 0.652122i \(-0.226121\pi\)
0.758114 + 0.652122i \(0.226121\pi\)
\(830\) −1.44481 −0.0501499
\(831\) −29.2313 −1.01402
\(832\) −0.0345856 −0.00119904
\(833\) 0 0
\(834\) 6.47511 0.224215
\(835\) 39.7160 1.37443
\(836\) −33.7047 −1.16570
\(837\) 5.62989 0.194597
\(838\) −3.07658 −0.106279
\(839\) 39.1592 1.35192 0.675962 0.736936i \(-0.263728\pi\)
0.675962 + 0.736936i \(0.263728\pi\)
\(840\) 0 0
\(841\) 48.1273 1.65956
\(842\) 3.70312 0.127618
\(843\) 5.86638 0.202049
\(844\) 11.7202 0.403425
\(845\) −28.6740 −0.986415
\(846\) 5.42690 0.186581
\(847\) 0 0
\(848\) −21.0174 −0.721739
\(849\) 16.0598 0.551169
\(850\) −0.517317 −0.0177438
\(851\) −8.10692 −0.277901
\(852\) 11.3751 0.389705
\(853\) 9.54494 0.326812 0.163406 0.986559i \(-0.447752\pi\)
0.163406 + 0.986559i \(0.447752\pi\)
\(854\) 0 0
\(855\) −9.42726 −0.322406
\(856\) 25.2231 0.862109
\(857\) −17.3645 −0.593160 −0.296580 0.955008i \(-0.595846\pi\)
−0.296580 + 0.955008i \(0.595846\pi\)
\(858\) 0.316314 0.0107988
\(859\) 1.22800 0.0418987 0.0209493 0.999781i \(-0.493331\pi\)
0.0209493 + 0.999781i \(0.493331\pi\)
\(860\) 32.9144 1.12237
\(861\) 0 0
\(862\) −23.3715 −0.796035
\(863\) −18.2094 −0.619854 −0.309927 0.950760i \(-0.600305\pi\)
−0.309927 + 0.950760i \(0.600305\pi\)
\(864\) 5.70859 0.194210
\(865\) −14.2389 −0.484139
\(866\) −26.1340 −0.888071
\(867\) 22.2871 0.756909
\(868\) 0 0
\(869\) 12.9295 0.438603
\(870\) −12.5281 −0.424743
\(871\) −0.963725 −0.0326546
\(872\) 16.1993 0.548579
\(873\) −17.6877 −0.598637
\(874\) −2.65371 −0.0897629
\(875\) 0 0
\(876\) −0.866717 −0.0292836
\(877\) 33.3435 1.12593 0.562964 0.826481i \(-0.309661\pi\)
0.562964 + 0.826481i \(0.309661\pi\)
\(878\) −2.91753 −0.0984619
\(879\) −12.9726 −0.437556
\(880\) −18.3675 −0.619169
\(881\) 26.1078 0.879594 0.439797 0.898097i \(-0.355050\pi\)
0.439797 + 0.898097i \(0.355050\pi\)
\(882\) 0 0
\(883\) 30.8036 1.03662 0.518311 0.855192i \(-0.326561\pi\)
0.518311 + 0.855192i \(0.326561\pi\)
\(884\) 0.973311 0.0327360
\(885\) −2.87793 −0.0967406
\(886\) 1.48169 0.0497784
\(887\) −9.34789 −0.313871 −0.156936 0.987609i \(-0.550162\pi\)
−0.156936 + 0.987609i \(0.550162\pi\)
\(888\) 19.5217 0.655107
\(889\) 0 0
\(890\) 21.0843 0.706746
\(891\) −4.98739 −0.167084
\(892\) 45.1242 1.51087
\(893\) −35.8637 −1.20013
\(894\) 12.3055 0.411558
\(895\) 7.65002 0.255712
\(896\) 0 0
\(897\) −0.0943512 −0.00315030
\(898\) −0.497924 −0.0166159
\(899\) −49.4429 −1.64901
\(900\) 0.202076 0.00673586
\(901\) −78.9574 −2.63045
\(902\) −3.22321 −0.107321
\(903\) 0 0
\(904\) 46.6229 1.55065
\(905\) 3.82770 0.127237
\(906\) −9.58229 −0.318350
\(907\) −59.6860 −1.98184 −0.990920 0.134453i \(-0.957072\pi\)
−0.990920 + 0.134453i \(0.957072\pi\)
\(908\) 24.4887 0.812687
\(909\) −9.33195 −0.309521
\(910\) 0 0
\(911\) 23.3828 0.774705 0.387353 0.921932i \(-0.373390\pi\)
0.387353 + 0.921932i \(0.373390\pi\)
\(912\) −7.12573 −0.235956
\(913\) 5.05127 0.167173
\(914\) 6.03568 0.199642
\(915\) 3.84798 0.127210
\(916\) −4.10306 −0.135569
\(917\) 0 0
\(918\) 4.05080 0.133696
\(919\) −29.2948 −0.966347 −0.483174 0.875525i \(-0.660516\pi\)
−0.483174 + 0.875525i \(0.660516\pi\)
\(920\) −4.91322 −0.161984
\(921\) 24.4190 0.804634
\(922\) −24.6889 −0.813086
\(923\) 0.705484 0.0232213
\(924\) 0 0
\(925\) 1.07685 0.0354065
\(926\) 8.19372 0.269262
\(927\) −8.42596 −0.276745
\(928\) −50.1341 −1.64573
\(929\) 30.3704 0.996422 0.498211 0.867056i \(-0.333991\pi\)
0.498211 + 0.867056i \(0.333991\pi\)
\(930\) 8.03123 0.263354
\(931\) 0 0
\(932\) −12.4024 −0.406255
\(933\) 18.2933 0.598897
\(934\) 1.91215 0.0625675
\(935\) −69.0025 −2.25662
\(936\) 0.227201 0.00742630
\(937\) 0.491253 0.0160485 0.00802426 0.999968i \(-0.497446\pi\)
0.00802426 + 0.999968i \(0.497446\pi\)
\(938\) 0 0
\(939\) 9.76655 0.318719
\(940\) −29.3292 −0.956613
\(941\) 30.8534 1.00579 0.502895 0.864347i \(-0.332268\pi\)
0.502895 + 0.864347i \(0.332268\pi\)
\(942\) −3.87984 −0.126412
\(943\) 0.961432 0.0313085
\(944\) −2.17533 −0.0708008
\(945\) 0 0
\(946\) 30.3746 0.987565
\(947\) −48.8549 −1.58757 −0.793786 0.608197i \(-0.791893\pi\)
−0.793786 + 0.608197i \(0.791893\pi\)
\(948\) 4.10209 0.133230
\(949\) −0.0537537 −0.00174492
\(950\) 0.352493 0.0114364
\(951\) 11.2980 0.366364
\(952\) 0 0
\(953\) 24.3510 0.788806 0.394403 0.918938i \(-0.370951\pi\)
0.394403 + 0.918938i \(0.370951\pi\)
\(954\) −8.14110 −0.263578
\(955\) 21.4418 0.693841
\(956\) 9.34390 0.302203
\(957\) 43.8004 1.41586
\(958\) −9.02663 −0.291637
\(959\) 0 0
\(960\) 0.777916 0.0251071
\(961\) 0.695634 0.0224398
\(962\) 0.534788 0.0172423
\(963\) −10.8948 −0.351079
\(964\) −47.4294 −1.52760
\(965\) −58.8056 −1.89302
\(966\) 0 0
\(967\) −21.6517 −0.696273 −0.348136 0.937444i \(-0.613185\pi\)
−0.348136 + 0.937444i \(0.613185\pi\)
\(968\) −32.1208 −1.03240
\(969\) −26.7697 −0.859967
\(970\) −25.2321 −0.810153
\(971\) −15.0904 −0.484274 −0.242137 0.970242i \(-0.577848\pi\)
−0.242137 + 0.970242i \(0.577848\pi\)
\(972\) −1.58233 −0.0507533
\(973\) 0 0
\(974\) −1.18712 −0.0380377
\(975\) 0.0125327 0.000401369 0
\(976\) 2.90855 0.0931004
\(977\) −51.7542 −1.65576 −0.827882 0.560903i \(-0.810454\pi\)
−0.827882 + 0.560903i \(0.810454\pi\)
\(978\) 8.12162 0.259701
\(979\) −73.7139 −2.35591
\(980\) 0 0
\(981\) −6.99706 −0.223399
\(982\) −10.0673 −0.321261
\(983\) −5.13872 −0.163900 −0.0819499 0.996636i \(-0.526115\pi\)
−0.0819499 + 0.996636i \(0.526115\pi\)
\(984\) −2.31516 −0.0738047
\(985\) −0.373701 −0.0119071
\(986\) −35.5750 −1.13294
\(987\) 0 0
\(988\) −0.663202 −0.0210993
\(989\) −9.06025 −0.288099
\(990\) −7.11469 −0.226120
\(991\) 57.3168 1.82073 0.910364 0.413808i \(-0.135801\pi\)
0.910364 + 0.413808i \(0.135801\pi\)
\(992\) 32.1387 1.02041
\(993\) −2.09392 −0.0664485
\(994\) 0 0
\(995\) −53.4697 −1.69510
\(996\) 1.60260 0.0507803
\(997\) 55.0860 1.74459 0.872296 0.488978i \(-0.162630\pi\)
0.872296 + 0.488978i \(0.162630\pi\)
\(998\) −12.3452 −0.390782
\(999\) −8.43213 −0.266781
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 6027.2.a.bi.1.10 13
7.2 even 3 861.2.i.f.739.4 yes 26
7.4 even 3 861.2.i.f.247.4 26
7.6 odd 2 6027.2.a.bh.1.10 13
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
861.2.i.f.247.4 26 7.4 even 3
861.2.i.f.739.4 yes 26 7.2 even 3
6027.2.a.bh.1.10 13 7.6 odd 2
6027.2.a.bi.1.10 13 1.1 even 1 trivial