Properties

Label 5577.2.a.n.1.4
Level $5577$
Weight $2$
Character 5577.1
Self dual yes
Analytic conductor $44.533$
Analytic rank $1$
Dimension $5$
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] = [5577,2,Mod(1,5577)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(5577, 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("5577.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 5577 = 3 \cdot 11 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 5577.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: \(44.5325692073\)
Analytic rank: \(1\)
Dimension: \(5\)
Coefficient field: 5.5.233489.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - x^{4} - 6x^{3} + x^{2} + 5x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 429)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Root \(0.881963\) of defining polynomial
Character \(\chi\) \(=\) 5577.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-0.118037 q^{2} +1.00000 q^{3} -1.98607 q^{4} -2.39753 q^{5} -0.118037 q^{6} -2.97027 q^{7} +0.470505 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-0.118037 q^{2} +1.00000 q^{3} -1.98607 q^{4} -2.39753 q^{5} -0.118037 q^{6} -2.97027 q^{7} +0.470505 q^{8} +1.00000 q^{9} +0.282997 q^{10} +1.00000 q^{11} -1.98607 q^{12} +0.350603 q^{14} -2.39753 q^{15} +3.91660 q^{16} +4.58456 q^{17} -0.118037 q^{18} -4.85223 q^{19} +4.76165 q^{20} -2.97027 q^{21} -0.118037 q^{22} +1.75350 q^{23} +0.470505 q^{24} +0.748128 q^{25} +1.00000 q^{27} +5.89916 q^{28} +5.42498 q^{29} +0.282997 q^{30} -1.91984 q^{31} -1.40331 q^{32} +1.00000 q^{33} -0.541149 q^{34} +7.12130 q^{35} -1.98607 q^{36} -3.76743 q^{37} +0.572744 q^{38} -1.12805 q^{40} +5.41472 q^{41} +0.350603 q^{42} -7.01491 q^{43} -1.98607 q^{44} -2.39753 q^{45} -0.206979 q^{46} +12.3378 q^{47} +3.91660 q^{48} +1.82250 q^{49} -0.0883070 q^{50} +4.58456 q^{51} +4.41104 q^{53} -0.118037 q^{54} -2.39753 q^{55} -1.39753 q^{56} -4.85223 q^{57} -0.640350 q^{58} +2.76930 q^{59} +4.76165 q^{60} +5.44077 q^{61} +0.226612 q^{62} -2.97027 q^{63} -7.66755 q^{64} -0.118037 q^{66} +5.70032 q^{67} -9.10525 q^{68} +1.75350 q^{69} -0.840579 q^{70} -14.9505 q^{71} +0.470505 q^{72} +10.5499 q^{73} +0.444698 q^{74} +0.748128 q^{75} +9.63686 q^{76} -2.97027 q^{77} +2.55834 q^{79} -9.39014 q^{80} +1.00000 q^{81} -0.639139 q^{82} -9.41168 q^{83} +5.89916 q^{84} -10.9916 q^{85} +0.828021 q^{86} +5.42498 q^{87} +0.470505 q^{88} +0.0288464 q^{89} +0.282997 q^{90} -3.48257 q^{92} -1.91984 q^{93} -1.45632 q^{94} +11.6334 q^{95} -1.40331 q^{96} -11.0811 q^{97} -0.215123 q^{98} +1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q - 4 q^{2} + 5 q^{3} + 6 q^{4} - 4 q^{5} - 4 q^{6} - 3 q^{7} - 9 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q - 4 q^{2} + 5 q^{3} + 6 q^{4} - 4 q^{5} - 4 q^{6} - 3 q^{7} - 9 q^{8} + 5 q^{9} - 5 q^{10} + 5 q^{11} + 6 q^{12} + 7 q^{14} - 4 q^{15} + 12 q^{16} + 9 q^{17} - 4 q^{18} - 9 q^{19} - 6 q^{20} - 3 q^{21} - 4 q^{22} - 9 q^{23} - 9 q^{24} + q^{25} + 5 q^{27} + q^{28} + 8 q^{29} - 5 q^{30} - 6 q^{31} - 27 q^{32} + 5 q^{33} - 7 q^{34} - 2 q^{35} + 6 q^{36} - 17 q^{37} - q^{38} + 23 q^{40} - 6 q^{41} + 7 q^{42} + 13 q^{43} + 6 q^{44} - 4 q^{45} + 6 q^{46} - 16 q^{47} + 12 q^{48} - 18 q^{49} + 8 q^{50} + 9 q^{51} - 13 q^{53} - 4 q^{54} - 4 q^{55} + q^{56} - 9 q^{57} - 11 q^{58} - 8 q^{59} - 6 q^{60} + 4 q^{61} - 22 q^{62} - 3 q^{63} + 11 q^{64} - 4 q^{66} - 5 q^{67} + 34 q^{68} - 9 q^{69} + 4 q^{70} - 19 q^{71} - 9 q^{72} - 2 q^{73} + 27 q^{74} + q^{75} + 6 q^{76} - 3 q^{77} - 8 q^{79} - 32 q^{80} + 5 q^{81} + 16 q^{82} - 10 q^{83} + q^{84} - 21 q^{85} + 4 q^{86} + 8 q^{87} - 9 q^{88} - 32 q^{89} - 5 q^{90} - 42 q^{92} - 6 q^{93} + 66 q^{94} + 11 q^{95} - 27 q^{96} + 5 q^{97} + 18 q^{98} + 5 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.118037 −0.0834650 −0.0417325 0.999129i \(-0.513288\pi\)
−0.0417325 + 0.999129i \(0.513288\pi\)
\(3\) 1.00000 0.577350
\(4\) −1.98607 −0.993034
\(5\) −2.39753 −1.07221 −0.536103 0.844153i \(-0.680104\pi\)
−0.536103 + 0.844153i \(0.680104\pi\)
\(6\) −0.118037 −0.0481885
\(7\) −2.97027 −1.12266 −0.561328 0.827593i \(-0.689709\pi\)
−0.561328 + 0.827593i \(0.689709\pi\)
\(8\) 0.470505 0.166348
\(9\) 1.00000 0.333333
\(10\) 0.282997 0.0894916
\(11\) 1.00000 0.301511
\(12\) −1.98607 −0.573328
\(13\) 0 0
\(14\) 0.350603 0.0937025
\(15\) −2.39753 −0.619038
\(16\) 3.91660 0.979149
\(17\) 4.58456 1.11192 0.555960 0.831209i \(-0.312351\pi\)
0.555960 + 0.831209i \(0.312351\pi\)
\(18\) −0.118037 −0.0278217
\(19\) −4.85223 −1.11318 −0.556589 0.830788i \(-0.687890\pi\)
−0.556589 + 0.830788i \(0.687890\pi\)
\(20\) 4.76165 1.06474
\(21\) −2.97027 −0.648166
\(22\) −0.118037 −0.0251656
\(23\) 1.75350 0.365630 0.182815 0.983147i \(-0.441479\pi\)
0.182815 + 0.983147i \(0.441479\pi\)
\(24\) 0.470505 0.0960413
\(25\) 0.748128 0.149626
\(26\) 0 0
\(27\) 1.00000 0.192450
\(28\) 5.89916 1.11484
\(29\) 5.42498 1.00739 0.503696 0.863881i \(-0.331973\pi\)
0.503696 + 0.863881i \(0.331973\pi\)
\(30\) 0.282997 0.0516680
\(31\) −1.91984 −0.344813 −0.172407 0.985026i \(-0.555154\pi\)
−0.172407 + 0.985026i \(0.555154\pi\)
\(32\) −1.40331 −0.248073
\(33\) 1.00000 0.174078
\(34\) −0.541149 −0.0928063
\(35\) 7.12130 1.20372
\(36\) −1.98607 −0.331011
\(37\) −3.76743 −0.619362 −0.309681 0.950840i \(-0.600222\pi\)
−0.309681 + 0.950840i \(0.600222\pi\)
\(38\) 0.572744 0.0929114
\(39\) 0 0
\(40\) −1.12805 −0.178360
\(41\) 5.41472 0.845637 0.422818 0.906214i \(-0.361041\pi\)
0.422818 + 0.906214i \(0.361041\pi\)
\(42\) 0.350603 0.0540992
\(43\) −7.01491 −1.06976 −0.534882 0.844927i \(-0.679644\pi\)
−0.534882 + 0.844927i \(0.679644\pi\)
\(44\) −1.98607 −0.299411
\(45\) −2.39753 −0.357402
\(46\) −0.206979 −0.0305173
\(47\) 12.3378 1.79966 0.899828 0.436246i \(-0.143692\pi\)
0.899828 + 0.436246i \(0.143692\pi\)
\(48\) 3.91660 0.565312
\(49\) 1.82250 0.260357
\(50\) −0.0883070 −0.0124885
\(51\) 4.58456 0.641967
\(52\) 0 0
\(53\) 4.41104 0.605903 0.302952 0.953006i \(-0.402028\pi\)
0.302952 + 0.953006i \(0.402028\pi\)
\(54\) −0.118037 −0.0160628
\(55\) −2.39753 −0.323282
\(56\) −1.39753 −0.186752
\(57\) −4.85223 −0.642694
\(58\) −0.640350 −0.0840820
\(59\) 2.76930 0.360532 0.180266 0.983618i \(-0.442304\pi\)
0.180266 + 0.983618i \(0.442304\pi\)
\(60\) 4.76165 0.614726
\(61\) 5.44077 0.696620 0.348310 0.937379i \(-0.386756\pi\)
0.348310 + 0.937379i \(0.386756\pi\)
\(62\) 0.226612 0.0287798
\(63\) −2.97027 −0.374219
\(64\) −7.66755 −0.958444
\(65\) 0 0
\(66\) −0.118037 −0.0145294
\(67\) 5.70032 0.696405 0.348203 0.937419i \(-0.386792\pi\)
0.348203 + 0.937419i \(0.386792\pi\)
\(68\) −9.10525 −1.10417
\(69\) 1.75350 0.211097
\(70\) −0.840579 −0.100468
\(71\) −14.9505 −1.77430 −0.887152 0.461478i \(-0.847319\pi\)
−0.887152 + 0.461478i \(0.847319\pi\)
\(72\) 0.470505 0.0554495
\(73\) 10.5499 1.23478 0.617389 0.786658i \(-0.288190\pi\)
0.617389 + 0.786658i \(0.288190\pi\)
\(74\) 0.444698 0.0516951
\(75\) 0.748128 0.0863864
\(76\) 9.63686 1.10542
\(77\) −2.97027 −0.338494
\(78\) 0 0
\(79\) 2.55834 0.287836 0.143918 0.989590i \(-0.454030\pi\)
0.143918 + 0.989590i \(0.454030\pi\)
\(80\) −9.39014 −1.04985
\(81\) 1.00000 0.111111
\(82\) −0.639139 −0.0705810
\(83\) −9.41168 −1.03307 −0.516533 0.856267i \(-0.672778\pi\)
−0.516533 + 0.856267i \(0.672778\pi\)
\(84\) 5.89916 0.643651
\(85\) −10.9916 −1.19221
\(86\) 0.828021 0.0892878
\(87\) 5.42498 0.581619
\(88\) 0.470505 0.0501560
\(89\) 0.0288464 0.00305771 0.00152886 0.999999i \(-0.499513\pi\)
0.00152886 + 0.999999i \(0.499513\pi\)
\(90\) 0.282997 0.0298305
\(91\) 0 0
\(92\) −3.48257 −0.363083
\(93\) −1.91984 −0.199078
\(94\) −1.45632 −0.150208
\(95\) 11.6334 1.19356
\(96\) −1.40331 −0.143225
\(97\) −11.0811 −1.12512 −0.562560 0.826757i \(-0.690183\pi\)
−0.562560 + 0.826757i \(0.690183\pi\)
\(98\) −0.215123 −0.0217307
\(99\) 1.00000 0.100504
\(100\) −1.48583 −0.148583
\(101\) −12.8478 −1.27841 −0.639204 0.769037i \(-0.720736\pi\)
−0.639204 + 0.769037i \(0.720736\pi\)
\(102\) −0.541149 −0.0535818
\(103\) 16.2544 1.60159 0.800797 0.598935i \(-0.204409\pi\)
0.800797 + 0.598935i \(0.204409\pi\)
\(104\) 0 0
\(105\) 7.12130 0.694967
\(106\) −0.520668 −0.0505717
\(107\) −0.848723 −0.0820492 −0.0410246 0.999158i \(-0.513062\pi\)
−0.0410246 + 0.999158i \(0.513062\pi\)
\(108\) −1.98607 −0.191109
\(109\) 13.4908 1.29218 0.646092 0.763260i \(-0.276402\pi\)
0.646092 + 0.763260i \(0.276402\pi\)
\(110\) 0.282997 0.0269827
\(111\) −3.76743 −0.357589
\(112\) −11.6334 −1.09925
\(113\) 8.63196 0.812026 0.406013 0.913867i \(-0.366919\pi\)
0.406013 + 0.913867i \(0.366919\pi\)
\(114\) 0.572744 0.0536424
\(115\) −4.20407 −0.392031
\(116\) −10.7744 −1.00037
\(117\) 0 0
\(118\) −0.326881 −0.0300918
\(119\) −13.6174 −1.24830
\(120\) −1.12805 −0.102976
\(121\) 1.00000 0.0909091
\(122\) −0.642214 −0.0581433
\(123\) 5.41472 0.488229
\(124\) 3.81293 0.342411
\(125\) 10.1940 0.911776
\(126\) 0.350603 0.0312342
\(127\) −17.0045 −1.50890 −0.754452 0.656355i \(-0.772097\pi\)
−0.754452 + 0.656355i \(0.772097\pi\)
\(128\) 3.71168 0.328070
\(129\) −7.01491 −0.617629
\(130\) 0 0
\(131\) −14.4417 −1.26178 −0.630889 0.775873i \(-0.717310\pi\)
−0.630889 + 0.775873i \(0.717310\pi\)
\(132\) −1.98607 −0.172865
\(133\) 14.4124 1.24972
\(134\) −0.672850 −0.0581254
\(135\) −2.39753 −0.206346
\(136\) 2.15706 0.184966
\(137\) −14.1069 −1.20523 −0.602617 0.798031i \(-0.705875\pi\)
−0.602617 + 0.798031i \(0.705875\pi\)
\(138\) −0.206979 −0.0176192
\(139\) 4.87137 0.413185 0.206592 0.978427i \(-0.433763\pi\)
0.206592 + 0.978427i \(0.433763\pi\)
\(140\) −14.1434 −1.19533
\(141\) 12.3378 1.03903
\(142\) 1.76472 0.148092
\(143\) 0 0
\(144\) 3.91660 0.326383
\(145\) −13.0065 −1.08013
\(146\) −1.24529 −0.103061
\(147\) 1.82250 0.150317
\(148\) 7.48238 0.615048
\(149\) −8.78924 −0.720043 −0.360021 0.932944i \(-0.617231\pi\)
−0.360021 + 0.932944i \(0.617231\pi\)
\(150\) −0.0883070 −0.00721024
\(151\) −21.4502 −1.74559 −0.872797 0.488083i \(-0.837696\pi\)
−0.872797 + 0.488083i \(0.837696\pi\)
\(152\) −2.28300 −0.185176
\(153\) 4.58456 0.370640
\(154\) 0.350603 0.0282524
\(155\) 4.60286 0.369711
\(156\) 0 0
\(157\) −22.3349 −1.78252 −0.891259 0.453495i \(-0.850177\pi\)
−0.891259 + 0.453495i \(0.850177\pi\)
\(158\) −0.301980 −0.0240242
\(159\) 4.41104 0.349819
\(160\) 3.36448 0.265985
\(161\) −5.20837 −0.410477
\(162\) −0.118037 −0.00927388
\(163\) −18.7766 −1.47069 −0.735347 0.677691i \(-0.762981\pi\)
−0.735347 + 0.677691i \(0.762981\pi\)
\(164\) −10.7540 −0.839746
\(165\) −2.39753 −0.186647
\(166\) 1.11093 0.0862248
\(167\) −2.35518 −0.182249 −0.0911247 0.995839i \(-0.529046\pi\)
−0.0911247 + 0.995839i \(0.529046\pi\)
\(168\) −1.39753 −0.107821
\(169\) 0 0
\(170\) 1.29742 0.0995075
\(171\) −4.85223 −0.371059
\(172\) 13.9321 1.06231
\(173\) −1.46817 −0.111623 −0.0558115 0.998441i \(-0.517775\pi\)
−0.0558115 + 0.998441i \(0.517775\pi\)
\(174\) −0.640350 −0.0485448
\(175\) −2.22214 −0.167978
\(176\) 3.91660 0.295225
\(177\) 2.76930 0.208153
\(178\) −0.00340495 −0.000255212 0
\(179\) 8.61218 0.643704 0.321852 0.946790i \(-0.395695\pi\)
0.321852 + 0.946790i \(0.395695\pi\)
\(180\) 4.76165 0.354912
\(181\) −16.8486 −1.25235 −0.626173 0.779684i \(-0.715380\pi\)
−0.626173 + 0.779684i \(0.715380\pi\)
\(182\) 0 0
\(183\) 5.44077 0.402194
\(184\) 0.825031 0.0608221
\(185\) 9.03252 0.664084
\(186\) 0.226612 0.0166160
\(187\) 4.58456 0.335256
\(188\) −24.5037 −1.78712
\(189\) −2.97027 −0.216055
\(190\) −1.37317 −0.0996202
\(191\) −3.66308 −0.265051 −0.132526 0.991180i \(-0.542309\pi\)
−0.132526 + 0.991180i \(0.542309\pi\)
\(192\) −7.66755 −0.553358
\(193\) −3.93409 −0.283182 −0.141591 0.989925i \(-0.545222\pi\)
−0.141591 + 0.989925i \(0.545222\pi\)
\(194\) 1.30799 0.0939081
\(195\) 0 0
\(196\) −3.61961 −0.258544
\(197\) 17.1892 1.22468 0.612339 0.790596i \(-0.290229\pi\)
0.612339 + 0.790596i \(0.290229\pi\)
\(198\) −0.118037 −0.00838854
\(199\) −7.87050 −0.557925 −0.278962 0.960302i \(-0.589990\pi\)
−0.278962 + 0.960302i \(0.589990\pi\)
\(200\) 0.351998 0.0248900
\(201\) 5.70032 0.402070
\(202\) 1.51652 0.106702
\(203\) −16.1136 −1.13096
\(204\) −9.10525 −0.637495
\(205\) −12.9819 −0.906697
\(206\) −1.91863 −0.133677
\(207\) 1.75350 0.121877
\(208\) 0 0
\(209\) −4.85223 −0.335636
\(210\) −0.840579 −0.0580054
\(211\) 11.7612 0.809671 0.404836 0.914389i \(-0.367329\pi\)
0.404836 + 0.914389i \(0.367329\pi\)
\(212\) −8.76063 −0.601683
\(213\) −14.9505 −1.02439
\(214\) 0.100181 0.00684823
\(215\) 16.8184 1.14701
\(216\) 0.470505 0.0320138
\(217\) 5.70244 0.387107
\(218\) −1.59242 −0.107852
\(219\) 10.5499 0.712899
\(220\) 4.76165 0.321030
\(221\) 0 0
\(222\) 0.444698 0.0298462
\(223\) −25.2830 −1.69307 −0.846536 0.532332i \(-0.821316\pi\)
−0.846536 + 0.532332i \(0.821316\pi\)
\(224\) 4.16822 0.278501
\(225\) 0.748128 0.0498752
\(226\) −1.01889 −0.0677757
\(227\) 11.7460 0.779612 0.389806 0.920897i \(-0.372542\pi\)
0.389806 + 0.920897i \(0.372542\pi\)
\(228\) 9.63686 0.638217
\(229\) 9.52159 0.629205 0.314602 0.949224i \(-0.398129\pi\)
0.314602 + 0.949224i \(0.398129\pi\)
\(230\) 0.496236 0.0327209
\(231\) −2.97027 −0.195429
\(232\) 2.55248 0.167578
\(233\) 3.30327 0.216404 0.108202 0.994129i \(-0.465491\pi\)
0.108202 + 0.994129i \(0.465491\pi\)
\(234\) 0 0
\(235\) −29.5802 −1.92960
\(236\) −5.50001 −0.358020
\(237\) 2.55834 0.166182
\(238\) 1.60736 0.104190
\(239\) −9.99602 −0.646589 −0.323294 0.946298i \(-0.604790\pi\)
−0.323294 + 0.946298i \(0.604790\pi\)
\(240\) −9.39014 −0.606131
\(241\) 1.26482 0.0814740 0.0407370 0.999170i \(-0.487029\pi\)
0.0407370 + 0.999170i \(0.487029\pi\)
\(242\) −0.118037 −0.00758772
\(243\) 1.00000 0.0641500
\(244\) −10.8057 −0.691767
\(245\) −4.36950 −0.279157
\(246\) −0.639139 −0.0407500
\(247\) 0 0
\(248\) −0.903292 −0.0573591
\(249\) −9.41168 −0.596441
\(250\) −1.20327 −0.0761014
\(251\) −12.4728 −0.787279 −0.393639 0.919265i \(-0.628784\pi\)
−0.393639 + 0.919265i \(0.628784\pi\)
\(252\) 5.89916 0.371612
\(253\) 1.75350 0.110242
\(254\) 2.00716 0.125941
\(255\) −10.9916 −0.688321
\(256\) 14.8970 0.931062
\(257\) 3.42223 0.213473 0.106736 0.994287i \(-0.465960\pi\)
0.106736 + 0.994287i \(0.465960\pi\)
\(258\) 0.828021 0.0515504
\(259\) 11.1903 0.695331
\(260\) 0 0
\(261\) 5.42498 0.335798
\(262\) 1.70466 0.105314
\(263\) −20.2069 −1.24601 −0.623004 0.782219i \(-0.714088\pi\)
−0.623004 + 0.782219i \(0.714088\pi\)
\(264\) 0.470505 0.0289576
\(265\) −10.5756 −0.649653
\(266\) −1.70121 −0.104308
\(267\) 0.0288464 0.00176537
\(268\) −11.3212 −0.691554
\(269\) −24.7199 −1.50720 −0.753598 0.657335i \(-0.771684\pi\)
−0.753598 + 0.657335i \(0.771684\pi\)
\(270\) 0.282997 0.0172227
\(271\) −8.10763 −0.492504 −0.246252 0.969206i \(-0.579199\pi\)
−0.246252 + 0.969206i \(0.579199\pi\)
\(272\) 17.9559 1.08874
\(273\) 0 0
\(274\) 1.66514 0.100595
\(275\) 0.748128 0.0451138
\(276\) −3.48257 −0.209626
\(277\) 25.7639 1.54800 0.774001 0.633185i \(-0.218253\pi\)
0.774001 + 0.633185i \(0.218253\pi\)
\(278\) −0.575004 −0.0344864
\(279\) −1.91984 −0.114938
\(280\) 3.35060 0.200237
\(281\) −8.10067 −0.483245 −0.241623 0.970370i \(-0.577680\pi\)
−0.241623 + 0.970370i \(0.577680\pi\)
\(282\) −1.45632 −0.0867227
\(283\) −16.8117 −0.999350 −0.499675 0.866213i \(-0.666547\pi\)
−0.499675 + 0.866213i \(0.666547\pi\)
\(284\) 29.6928 1.76194
\(285\) 11.6334 0.689100
\(286\) 0 0
\(287\) −16.0832 −0.949360
\(288\) −1.40331 −0.0826910
\(289\) 4.01822 0.236366
\(290\) 1.53525 0.0901532
\(291\) −11.0811 −0.649588
\(292\) −20.9529 −1.22618
\(293\) −16.2136 −0.947208 −0.473604 0.880738i \(-0.657047\pi\)
−0.473604 + 0.880738i \(0.657047\pi\)
\(294\) −0.215123 −0.0125462
\(295\) −6.63947 −0.386565
\(296\) −1.77260 −0.103030
\(297\) 1.00000 0.0580259
\(298\) 1.03746 0.0600983
\(299\) 0 0
\(300\) −1.48583 −0.0857846
\(301\) 20.8362 1.20098
\(302\) 2.53193 0.145696
\(303\) −12.8478 −0.738089
\(304\) −19.0042 −1.08997
\(305\) −13.0444 −0.746920
\(306\) −0.541149 −0.0309354
\(307\) −3.19713 −0.182470 −0.0912349 0.995829i \(-0.529081\pi\)
−0.0912349 + 0.995829i \(0.529081\pi\)
\(308\) 5.89916 0.336136
\(309\) 16.2544 0.924681
\(310\) −0.543309 −0.0308579
\(311\) −19.9402 −1.13071 −0.565353 0.824849i \(-0.691260\pi\)
−0.565353 + 0.824849i \(0.691260\pi\)
\(312\) 0 0
\(313\) −21.3968 −1.20942 −0.604709 0.796447i \(-0.706711\pi\)
−0.604709 + 0.796447i \(0.706711\pi\)
\(314\) 2.63635 0.148778
\(315\) 7.12130 0.401240
\(316\) −5.08104 −0.285831
\(317\) −13.6654 −0.767526 −0.383763 0.923432i \(-0.625372\pi\)
−0.383763 + 0.923432i \(0.625372\pi\)
\(318\) −0.520668 −0.0291976
\(319\) 5.42498 0.303740
\(320\) 18.3831 1.02765
\(321\) −0.848723 −0.0473711
\(322\) 0.614782 0.0342605
\(323\) −22.2454 −1.23777
\(324\) −1.98607 −0.110337
\(325\) 0 0
\(326\) 2.21633 0.122751
\(327\) 13.4908 0.746043
\(328\) 2.54765 0.140670
\(329\) −36.6466 −2.02039
\(330\) 0.282997 0.0155785
\(331\) 9.56314 0.525638 0.262819 0.964845i \(-0.415348\pi\)
0.262819 + 0.964845i \(0.415348\pi\)
\(332\) 18.6922 1.02587
\(333\) −3.76743 −0.206454
\(334\) 0.277999 0.0152114
\(335\) −13.6667 −0.746690
\(336\) −11.6334 −0.634651
\(337\) 30.5568 1.66453 0.832266 0.554376i \(-0.187043\pi\)
0.832266 + 0.554376i \(0.187043\pi\)
\(338\) 0 0
\(339\) 8.63196 0.468823
\(340\) 21.8301 1.18390
\(341\) −1.91984 −0.103965
\(342\) 0.572744 0.0309705
\(343\) 15.3786 0.830364
\(344\) −3.30055 −0.177954
\(345\) −4.20407 −0.226339
\(346\) 0.173299 0.00931660
\(347\) 7.61080 0.408569 0.204285 0.978912i \(-0.434513\pi\)
0.204285 + 0.978912i \(0.434513\pi\)
\(348\) −10.7744 −0.577567
\(349\) −14.4841 −0.775316 −0.387658 0.921803i \(-0.626716\pi\)
−0.387658 + 0.921803i \(0.626716\pi\)
\(350\) 0.262296 0.0140203
\(351\) 0 0
\(352\) −1.40331 −0.0747969
\(353\) −26.6472 −1.41829 −0.709144 0.705064i \(-0.750918\pi\)
−0.709144 + 0.705064i \(0.750918\pi\)
\(354\) −0.326881 −0.0173735
\(355\) 35.8443 1.90242
\(356\) −0.0572909 −0.00303641
\(357\) −13.6174 −0.720709
\(358\) −1.01656 −0.0537268
\(359\) −26.4230 −1.39455 −0.697277 0.716802i \(-0.745605\pi\)
−0.697277 + 0.716802i \(0.745605\pi\)
\(360\) −1.12805 −0.0594533
\(361\) 4.54416 0.239166
\(362\) 1.98876 0.104527
\(363\) 1.00000 0.0524864
\(364\) 0 0
\(365\) −25.2938 −1.32394
\(366\) −0.642214 −0.0335691
\(367\) −18.7868 −0.980662 −0.490331 0.871536i \(-0.663124\pi\)
−0.490331 + 0.871536i \(0.663124\pi\)
\(368\) 6.86776 0.358007
\(369\) 5.41472 0.281879
\(370\) −1.06617 −0.0554278
\(371\) −13.1020 −0.680221
\(372\) 3.81293 0.197691
\(373\) 22.6389 1.17220 0.586099 0.810240i \(-0.300663\pi\)
0.586099 + 0.810240i \(0.300663\pi\)
\(374\) −0.541149 −0.0279822
\(375\) 10.1940 0.526414
\(376\) 5.80500 0.299370
\(377\) 0 0
\(378\) 0.350603 0.0180331
\(379\) −1.22574 −0.0629621 −0.0314810 0.999504i \(-0.510022\pi\)
−0.0314810 + 0.999504i \(0.510022\pi\)
\(380\) −23.1046 −1.18524
\(381\) −17.0045 −0.871166
\(382\) 0.432380 0.0221225
\(383\) 32.1997 1.64533 0.822664 0.568528i \(-0.192487\pi\)
0.822664 + 0.568528i \(0.192487\pi\)
\(384\) 3.71168 0.189411
\(385\) 7.12130 0.362935
\(386\) 0.464370 0.0236358
\(387\) −7.01491 −0.356588
\(388\) 22.0079 1.11728
\(389\) 37.4190 1.89722 0.948609 0.316451i \(-0.102491\pi\)
0.948609 + 0.316451i \(0.102491\pi\)
\(390\) 0 0
\(391\) 8.03904 0.406552
\(392\) 0.857496 0.0433101
\(393\) −14.4417 −0.728487
\(394\) −2.02896 −0.102218
\(395\) −6.13369 −0.308619
\(396\) −1.98607 −0.0998036
\(397\) −11.1588 −0.560046 −0.280023 0.959993i \(-0.590342\pi\)
−0.280023 + 0.959993i \(0.590342\pi\)
\(398\) 0.929012 0.0465672
\(399\) 14.4124 0.721524
\(400\) 2.93012 0.146506
\(401\) 9.92369 0.495566 0.247783 0.968816i \(-0.420298\pi\)
0.247783 + 0.968816i \(0.420298\pi\)
\(402\) −0.672850 −0.0335587
\(403\) 0 0
\(404\) 25.5167 1.26950
\(405\) −2.39753 −0.119134
\(406\) 1.90201 0.0943952
\(407\) −3.76743 −0.186745
\(408\) 2.15706 0.106790
\(409\) 30.8931 1.52756 0.763782 0.645475i \(-0.223340\pi\)
0.763782 + 0.645475i \(0.223340\pi\)
\(410\) 1.53235 0.0756774
\(411\) −14.1069 −0.695842
\(412\) −32.2824 −1.59044
\(413\) −8.22557 −0.404754
\(414\) −0.206979 −0.0101724
\(415\) 22.5647 1.10766
\(416\) 0 0
\(417\) 4.87137 0.238552
\(418\) 0.572744 0.0280138
\(419\) 24.2611 1.18523 0.592616 0.805485i \(-0.298095\pi\)
0.592616 + 0.805485i \(0.298095\pi\)
\(420\) −14.1434 −0.690126
\(421\) 1.39751 0.0681103 0.0340551 0.999420i \(-0.489158\pi\)
0.0340551 + 0.999420i \(0.489158\pi\)
\(422\) −1.38825 −0.0675792
\(423\) 12.3378 0.599885
\(424\) 2.07542 0.100791
\(425\) 3.42984 0.166372
\(426\) 1.76472 0.0855011
\(427\) −16.1606 −0.782065
\(428\) 1.68562 0.0814776
\(429\) 0 0
\(430\) −1.98520 −0.0957349
\(431\) −25.2996 −1.21864 −0.609320 0.792925i \(-0.708557\pi\)
−0.609320 + 0.792925i \(0.708557\pi\)
\(432\) 3.91660 0.188437
\(433\) 33.9631 1.63216 0.816081 0.577938i \(-0.196142\pi\)
0.816081 + 0.577938i \(0.196142\pi\)
\(434\) −0.673100 −0.0323098
\(435\) −13.0065 −0.623615
\(436\) −26.7936 −1.28318
\(437\) −8.50840 −0.407012
\(438\) −1.24529 −0.0595021
\(439\) −11.5414 −0.550840 −0.275420 0.961324i \(-0.588817\pi\)
−0.275420 + 0.961324i \(0.588817\pi\)
\(440\) −1.12805 −0.0537775
\(441\) 1.82250 0.0867858
\(442\) 0 0
\(443\) 41.2202 1.95843 0.979216 0.202823i \(-0.0650115\pi\)
0.979216 + 0.202823i \(0.0650115\pi\)
\(444\) 7.48238 0.355098
\(445\) −0.0691600 −0.00327850
\(446\) 2.98433 0.141312
\(447\) −8.78924 −0.415717
\(448\) 22.7747 1.07600
\(449\) 11.3930 0.537671 0.268836 0.963186i \(-0.413361\pi\)
0.268836 + 0.963186i \(0.413361\pi\)
\(450\) −0.0883070 −0.00416283
\(451\) 5.41472 0.254969
\(452\) −17.1436 −0.806369
\(453\) −21.4502 −1.00782
\(454\) −1.38647 −0.0650703
\(455\) 0 0
\(456\) −2.28300 −0.106911
\(457\) 9.69328 0.453433 0.226716 0.973961i \(-0.427201\pi\)
0.226716 + 0.973961i \(0.427201\pi\)
\(458\) −1.12390 −0.0525165
\(459\) 4.58456 0.213989
\(460\) 8.34956 0.389300
\(461\) −11.5806 −0.539362 −0.269681 0.962950i \(-0.586918\pi\)
−0.269681 + 0.962950i \(0.586918\pi\)
\(462\) 0.350603 0.0163115
\(463\) 34.5395 1.60519 0.802593 0.596527i \(-0.203453\pi\)
0.802593 + 0.596527i \(0.203453\pi\)
\(464\) 21.2474 0.986388
\(465\) 4.60286 0.213452
\(466\) −0.389908 −0.0180622
\(467\) 3.73036 0.172621 0.0863103 0.996268i \(-0.472492\pi\)
0.0863103 + 0.996268i \(0.472492\pi\)
\(468\) 0 0
\(469\) −16.9315 −0.781824
\(470\) 3.49157 0.161054
\(471\) −22.3349 −1.02914
\(472\) 1.30297 0.0599740
\(473\) −7.01491 −0.322546
\(474\) −0.301980 −0.0138704
\(475\) −3.63009 −0.166560
\(476\) 27.0450 1.23961
\(477\) 4.41104 0.201968
\(478\) 1.17990 0.0539675
\(479\) −14.1843 −0.648099 −0.324050 0.946040i \(-0.605044\pi\)
−0.324050 + 0.946040i \(0.605044\pi\)
\(480\) 3.36448 0.153567
\(481\) 0 0
\(482\) −0.149296 −0.00680022
\(483\) −5.20837 −0.236989
\(484\) −1.98607 −0.0902758
\(485\) 26.5673 1.20636
\(486\) −0.118037 −0.00535428
\(487\) −26.8742 −1.21778 −0.608892 0.793253i \(-0.708386\pi\)
−0.608892 + 0.793253i \(0.708386\pi\)
\(488\) 2.55991 0.115882
\(489\) −18.7766 −0.849105
\(490\) 0.515763 0.0232998
\(491\) 34.0614 1.53717 0.768585 0.639747i \(-0.220961\pi\)
0.768585 + 0.639747i \(0.220961\pi\)
\(492\) −10.7540 −0.484827
\(493\) 24.8711 1.12014
\(494\) 0 0
\(495\) −2.39753 −0.107761
\(496\) −7.51923 −0.337623
\(497\) 44.4072 1.99193
\(498\) 1.11093 0.0497819
\(499\) −0.146884 −0.00657542 −0.00328771 0.999995i \(-0.501047\pi\)
−0.00328771 + 0.999995i \(0.501047\pi\)
\(500\) −20.2459 −0.905425
\(501\) −2.35518 −0.105222
\(502\) 1.47226 0.0657102
\(503\) 7.30757 0.325829 0.162914 0.986640i \(-0.447911\pi\)
0.162914 + 0.986640i \(0.447911\pi\)
\(504\) −1.39753 −0.0622507
\(505\) 30.8030 1.37072
\(506\) −0.206979 −0.00920132
\(507\) 0 0
\(508\) 33.7721 1.49839
\(509\) 22.1135 0.980163 0.490081 0.871677i \(-0.336967\pi\)
0.490081 + 0.871677i \(0.336967\pi\)
\(510\) 1.29742 0.0574507
\(511\) −31.3362 −1.38623
\(512\) −9.18177 −0.405781
\(513\) −4.85223 −0.214231
\(514\) −0.403951 −0.0178175
\(515\) −38.9704 −1.71724
\(516\) 13.9321 0.613326
\(517\) 12.3378 0.542616
\(518\) −1.32087 −0.0580358
\(519\) −1.46817 −0.0644455
\(520\) 0 0
\(521\) −3.57882 −0.156791 −0.0783954 0.996922i \(-0.524980\pi\)
−0.0783954 + 0.996922i \(0.524980\pi\)
\(522\) −0.640350 −0.0280273
\(523\) 19.2320 0.840955 0.420478 0.907303i \(-0.361862\pi\)
0.420478 + 0.907303i \(0.361862\pi\)
\(524\) 28.6822 1.25299
\(525\) −2.22214 −0.0969822
\(526\) 2.38516 0.103998
\(527\) −8.80162 −0.383404
\(528\) 3.91660 0.170448
\(529\) −19.9252 −0.866314
\(530\) 1.24831 0.0542233
\(531\) 2.76930 0.120177
\(532\) −28.6241 −1.24101
\(533\) 0 0
\(534\) −0.00340495 −0.000147347 0
\(535\) 2.03484 0.0879736
\(536\) 2.68203 0.115846
\(537\) 8.61218 0.371643
\(538\) 2.91787 0.125798
\(539\) 1.82250 0.0785007
\(540\) 4.76165 0.204909
\(541\) −13.5069 −0.580705 −0.290353 0.956920i \(-0.593773\pi\)
−0.290353 + 0.956920i \(0.593773\pi\)
\(542\) 0.957003 0.0411068
\(543\) −16.8486 −0.723043
\(544\) −6.43358 −0.275837
\(545\) −32.3445 −1.38549
\(546\) 0 0
\(547\) −44.5839 −1.90627 −0.953134 0.302548i \(-0.902163\pi\)
−0.953134 + 0.302548i \(0.902163\pi\)
\(548\) 28.0172 1.19684
\(549\) 5.44077 0.232207
\(550\) −0.0883070 −0.00376542
\(551\) −26.3232 −1.12141
\(552\) 0.825031 0.0351156
\(553\) −7.59897 −0.323141
\(554\) −3.04110 −0.129204
\(555\) 9.03252 0.383409
\(556\) −9.67488 −0.410306
\(557\) −18.0166 −0.763389 −0.381694 0.924289i \(-0.624659\pi\)
−0.381694 + 0.924289i \(0.624659\pi\)
\(558\) 0.226612 0.00959327
\(559\) 0 0
\(560\) 27.8913 1.17862
\(561\) 4.58456 0.193560
\(562\) 0.956181 0.0403341
\(563\) −23.2744 −0.980901 −0.490450 0.871469i \(-0.663168\pi\)
−0.490450 + 0.871469i \(0.663168\pi\)
\(564\) −24.5037 −1.03179
\(565\) −20.6953 −0.870659
\(566\) 1.98440 0.0834107
\(567\) −2.97027 −0.124740
\(568\) −7.03430 −0.295153
\(569\) −32.8125 −1.37557 −0.687785 0.725914i \(-0.741417\pi\)
−0.687785 + 0.725914i \(0.741417\pi\)
\(570\) −1.37317 −0.0575157
\(571\) −22.2440 −0.930883 −0.465441 0.885079i \(-0.654104\pi\)
−0.465441 + 0.885079i \(0.654104\pi\)
\(572\) 0 0
\(573\) −3.66308 −0.153027
\(574\) 1.89841 0.0792383
\(575\) 1.31184 0.0547077
\(576\) −7.66755 −0.319481
\(577\) 15.5700 0.648188 0.324094 0.946025i \(-0.394941\pi\)
0.324094 + 0.946025i \(0.394941\pi\)
\(578\) −0.474299 −0.0197283
\(579\) −3.93409 −0.163495
\(580\) 25.8318 1.07261
\(581\) 27.9552 1.15978
\(582\) 1.30799 0.0542178
\(583\) 4.41104 0.182687
\(584\) 4.96380 0.205403
\(585\) 0 0
\(586\) 1.91381 0.0790586
\(587\) 8.16350 0.336944 0.168472 0.985706i \(-0.446117\pi\)
0.168472 + 0.985706i \(0.446117\pi\)
\(588\) −3.61961 −0.149270
\(589\) 9.31550 0.383838
\(590\) 0.783704 0.0322646
\(591\) 17.1892 0.707068
\(592\) −14.7555 −0.606448
\(593\) 19.9116 0.817673 0.408837 0.912608i \(-0.365935\pi\)
0.408837 + 0.912608i \(0.365935\pi\)
\(594\) −0.118037 −0.00484313
\(595\) 32.6480 1.33844
\(596\) 17.4560 0.715027
\(597\) −7.87050 −0.322118
\(598\) 0 0
\(599\) 29.4044 1.20143 0.600716 0.799463i \(-0.294882\pi\)
0.600716 + 0.799463i \(0.294882\pi\)
\(600\) 0.351998 0.0143702
\(601\) 18.9161 0.771604 0.385802 0.922582i \(-0.373925\pi\)
0.385802 + 0.922582i \(0.373925\pi\)
\(602\) −2.45945 −0.100240
\(603\) 5.70032 0.232135
\(604\) 42.6016 1.73343
\(605\) −2.39753 −0.0974733
\(606\) 1.51652 0.0616046
\(607\) −46.9451 −1.90544 −0.952722 0.303845i \(-0.901730\pi\)
−0.952722 + 0.303845i \(0.901730\pi\)
\(608\) 6.80920 0.276150
\(609\) −16.1136 −0.652958
\(610\) 1.53972 0.0623416
\(611\) 0 0
\(612\) −9.10525 −0.368058
\(613\) 36.4861 1.47366 0.736830 0.676078i \(-0.236322\pi\)
0.736830 + 0.676078i \(0.236322\pi\)
\(614\) 0.377380 0.0152298
\(615\) −12.9819 −0.523482
\(616\) −1.39753 −0.0563079
\(617\) −3.43595 −0.138326 −0.0691632 0.997605i \(-0.522033\pi\)
−0.0691632 + 0.997605i \(0.522033\pi\)
\(618\) −1.91863 −0.0771785
\(619\) 39.5437 1.58939 0.794697 0.607006i \(-0.207630\pi\)
0.794697 + 0.607006i \(0.207630\pi\)
\(620\) −9.14159 −0.367135
\(621\) 1.75350 0.0703656
\(622\) 2.35369 0.0943743
\(623\) −0.0856816 −0.00343276
\(624\) 0 0
\(625\) −28.1809 −1.12724
\(626\) 2.52562 0.100944
\(627\) −4.85223 −0.193779
\(628\) 44.3586 1.77010
\(629\) −17.2720 −0.688681
\(630\) −0.840579 −0.0334895
\(631\) 10.4143 0.414586 0.207293 0.978279i \(-0.433535\pi\)
0.207293 + 0.978279i \(0.433535\pi\)
\(632\) 1.20371 0.0478811
\(633\) 11.7612 0.467464
\(634\) 1.61303 0.0640615
\(635\) 40.7687 1.61786
\(636\) −8.76063 −0.347382
\(637\) 0 0
\(638\) −0.640350 −0.0253517
\(639\) −14.9505 −0.591435
\(640\) −8.89886 −0.351758
\(641\) −49.1700 −1.94210 −0.971049 0.238882i \(-0.923219\pi\)
−0.971049 + 0.238882i \(0.923219\pi\)
\(642\) 0.100181 0.00395383
\(643\) −3.00230 −0.118399 −0.0591995 0.998246i \(-0.518855\pi\)
−0.0591995 + 0.998246i \(0.518855\pi\)
\(644\) 10.3442 0.407618
\(645\) 16.8184 0.662225
\(646\) 2.62578 0.103310
\(647\) 0.306983 0.0120687 0.00603437 0.999982i \(-0.498079\pi\)
0.00603437 + 0.999982i \(0.498079\pi\)
\(648\) 0.470505 0.0184832
\(649\) 2.76930 0.108705
\(650\) 0 0
\(651\) 5.70244 0.223496
\(652\) 37.2915 1.46045
\(653\) −50.4557 −1.97449 −0.987243 0.159223i \(-0.949101\pi\)
−0.987243 + 0.159223i \(0.949101\pi\)
\(654\) −1.59242 −0.0622684
\(655\) 34.6243 1.35288
\(656\) 21.2073 0.828005
\(657\) 10.5499 0.411593
\(658\) 4.32567 0.168632
\(659\) 37.7604 1.47094 0.735468 0.677560i \(-0.236963\pi\)
0.735468 + 0.677560i \(0.236963\pi\)
\(660\) 4.76165 0.185347
\(661\) −3.39981 −0.132237 −0.0661186 0.997812i \(-0.521062\pi\)
−0.0661186 + 0.997812i \(0.521062\pi\)
\(662\) −1.12881 −0.0438723
\(663\) 0 0
\(664\) −4.42824 −0.171849
\(665\) −34.5542 −1.33995
\(666\) 0.444698 0.0172317
\(667\) 9.51271 0.368333
\(668\) 4.67755 0.180980
\(669\) −25.2830 −0.977495
\(670\) 1.61318 0.0623224
\(671\) 5.44077 0.210039
\(672\) 4.16822 0.160793
\(673\) 23.7851 0.916846 0.458423 0.888734i \(-0.348414\pi\)
0.458423 + 0.888734i \(0.348414\pi\)
\(674\) −3.60684 −0.138930
\(675\) 0.748128 0.0287955
\(676\) 0 0
\(677\) −50.2857 −1.93264 −0.966319 0.257349i \(-0.917151\pi\)
−0.966319 + 0.257349i \(0.917151\pi\)
\(678\) −1.01889 −0.0391303
\(679\) 32.9140 1.26312
\(680\) −5.17160 −0.198322
\(681\) 11.7460 0.450109
\(682\) 0.226612 0.00867744
\(683\) −9.68266 −0.370497 −0.185248 0.982692i \(-0.559309\pi\)
−0.185248 + 0.982692i \(0.559309\pi\)
\(684\) 9.63686 0.368475
\(685\) 33.8216 1.29226
\(686\) −1.81524 −0.0693063
\(687\) 9.52159 0.363271
\(688\) −27.4746 −1.04746
\(689\) 0 0
\(690\) 0.496236 0.0188914
\(691\) 7.44899 0.283373 0.141686 0.989912i \(-0.454748\pi\)
0.141686 + 0.989912i \(0.454748\pi\)
\(692\) 2.91588 0.110845
\(693\) −2.97027 −0.112831
\(694\) −0.898359 −0.0341012
\(695\) −11.6792 −0.443019
\(696\) 2.55248 0.0967514
\(697\) 24.8241 0.940280
\(698\) 1.70966 0.0647117
\(699\) 3.30327 0.124941
\(700\) 4.41332 0.166808
\(701\) 33.0152 1.24697 0.623483 0.781837i \(-0.285717\pi\)
0.623483 + 0.781837i \(0.285717\pi\)
\(702\) 0 0
\(703\) 18.2805 0.689461
\(704\) −7.66755 −0.288982
\(705\) −29.5802 −1.11406
\(706\) 3.14536 0.118377
\(707\) 38.1615 1.43521
\(708\) −5.50001 −0.206703
\(709\) 30.1377 1.13185 0.565923 0.824458i \(-0.308520\pi\)
0.565923 + 0.824458i \(0.308520\pi\)
\(710\) −4.23097 −0.158785
\(711\) 2.55834 0.0959453
\(712\) 0.0135724 0.000508646 0
\(713\) −3.36644 −0.126074
\(714\) 1.60736 0.0601539
\(715\) 0 0
\(716\) −17.1044 −0.639220
\(717\) −9.99602 −0.373308
\(718\) 3.11890 0.116396
\(719\) −15.5594 −0.580268 −0.290134 0.956986i \(-0.593700\pi\)
−0.290134 + 0.956986i \(0.593700\pi\)
\(720\) −9.39014 −0.349950
\(721\) −48.2800 −1.79804
\(722\) −0.536380 −0.0199620
\(723\) 1.26482 0.0470390
\(724\) 33.4625 1.24362
\(725\) 4.05858 0.150732
\(726\) −0.118037 −0.00438077
\(727\) −41.0629 −1.52294 −0.761469 0.648201i \(-0.775522\pi\)
−0.761469 + 0.648201i \(0.775522\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 2.98561 0.110502
\(731\) −32.1603 −1.18949
\(732\) −10.8057 −0.399392
\(733\) −2.28843 −0.0845251 −0.0422625 0.999107i \(-0.513457\pi\)
−0.0422625 + 0.999107i \(0.513457\pi\)
\(734\) 2.21754 0.0818509
\(735\) −4.36950 −0.161171
\(736\) −2.46071 −0.0907031
\(737\) 5.70032 0.209974
\(738\) −0.639139 −0.0235270
\(739\) 44.6950 1.64413 0.822066 0.569393i \(-0.192822\pi\)
0.822066 + 0.569393i \(0.192822\pi\)
\(740\) −17.9392 −0.659458
\(741\) 0 0
\(742\) 1.54652 0.0567747
\(743\) 20.9315 0.767902 0.383951 0.923354i \(-0.374563\pi\)
0.383951 + 0.923354i \(0.374563\pi\)
\(744\) −0.903292 −0.0331163
\(745\) 21.0724 0.772034
\(746\) −2.67223 −0.0978374
\(747\) −9.41168 −0.344355
\(748\) −9.10525 −0.332921
\(749\) 2.52094 0.0921130
\(750\) −1.20327 −0.0439372
\(751\) −1.37108 −0.0500316 −0.0250158 0.999687i \(-0.507964\pi\)
−0.0250158 + 0.999687i \(0.507964\pi\)
\(752\) 48.3223 1.76213
\(753\) −12.4728 −0.454536
\(754\) 0 0
\(755\) 51.4275 1.87164
\(756\) 5.89916 0.214550
\(757\) −9.16942 −0.333268 −0.166634 0.986019i \(-0.553290\pi\)
−0.166634 + 0.986019i \(0.553290\pi\)
\(758\) 0.144683 0.00525513
\(759\) 1.75350 0.0636481
\(760\) 5.47354 0.198546
\(761\) −29.5359 −1.07067 −0.535337 0.844638i \(-0.679815\pi\)
−0.535337 + 0.844638i \(0.679815\pi\)
\(762\) 2.00716 0.0727119
\(763\) −40.0713 −1.45068
\(764\) 7.27512 0.263205
\(765\) −10.9916 −0.397402
\(766\) −3.80076 −0.137327
\(767\) 0 0
\(768\) 14.8970 0.537549
\(769\) 21.1718 0.763475 0.381737 0.924271i \(-0.375326\pi\)
0.381737 + 0.924271i \(0.375326\pi\)
\(770\) −0.840579 −0.0302923
\(771\) 3.42223 0.123249
\(772\) 7.81338 0.281210
\(773\) 46.3060 1.66551 0.832755 0.553641i \(-0.186762\pi\)
0.832755 + 0.553641i \(0.186762\pi\)
\(774\) 0.828021 0.0297626
\(775\) −1.43628 −0.0515929
\(776\) −5.21373 −0.187162
\(777\) 11.1903 0.401450
\(778\) −4.41683 −0.158351
\(779\) −26.2735 −0.941345
\(780\) 0 0
\(781\) −14.9505 −0.534973
\(782\) −0.948906 −0.0339328
\(783\) 5.42498 0.193873
\(784\) 7.13801 0.254929
\(785\) 53.5484 1.91123
\(786\) 1.70466 0.0608032
\(787\) −9.12758 −0.325363 −0.162681 0.986679i \(-0.552014\pi\)
−0.162681 + 0.986679i \(0.552014\pi\)
\(788\) −34.1388 −1.21615
\(789\) −20.2069 −0.719383
\(790\) 0.724004 0.0257589
\(791\) −25.6392 −0.911626
\(792\) 0.470505 0.0167187
\(793\) 0 0
\(794\) 1.31716 0.0467442
\(795\) −10.5756 −0.375078
\(796\) 15.6313 0.554038
\(797\) −26.9051 −0.953028 −0.476514 0.879167i \(-0.658100\pi\)
−0.476514 + 0.879167i \(0.658100\pi\)
\(798\) −1.70121 −0.0602220
\(799\) 56.5635 2.00107
\(800\) −1.04986 −0.0371181
\(801\) 0.0288464 0.00101924
\(802\) −1.17137 −0.0413624
\(803\) 10.5499 0.372300
\(804\) −11.3212 −0.399269
\(805\) 12.4872 0.440116
\(806\) 0 0
\(807\) −24.7199 −0.870181
\(808\) −6.04497 −0.212661
\(809\) −15.6777 −0.551199 −0.275599 0.961273i \(-0.588876\pi\)
−0.275599 + 0.961273i \(0.588876\pi\)
\(810\) 0.282997 0.00994351
\(811\) 21.5800 0.757775 0.378888 0.925443i \(-0.376307\pi\)
0.378888 + 0.925443i \(0.376307\pi\)
\(812\) 32.0028 1.12308
\(813\) −8.10763 −0.284347
\(814\) 0.444698 0.0155866
\(815\) 45.0173 1.57689
\(816\) 17.9559 0.628582
\(817\) 34.0380 1.19084
\(818\) −3.64653 −0.127498
\(819\) 0 0
\(820\) 25.7830 0.900380
\(821\) −51.1405 −1.78482 −0.892408 0.451229i \(-0.850986\pi\)
−0.892408 + 0.451229i \(0.850986\pi\)
\(822\) 1.66514 0.0580784
\(823\) −46.5261 −1.62180 −0.810900 0.585185i \(-0.801022\pi\)
−0.810900 + 0.585185i \(0.801022\pi\)
\(824\) 7.64777 0.266423
\(825\) 0.748128 0.0260465
\(826\) 0.970923 0.0337827
\(827\) −47.1916 −1.64101 −0.820506 0.571637i \(-0.806308\pi\)
−0.820506 + 0.571637i \(0.806308\pi\)
\(828\) −3.48257 −0.121028
\(829\) −51.5875 −1.79171 −0.895855 0.444347i \(-0.853436\pi\)
−0.895855 + 0.444347i \(0.853436\pi\)
\(830\) −2.66348 −0.0924508
\(831\) 25.7639 0.893739
\(832\) 0 0
\(833\) 8.35538 0.289497
\(834\) −0.575004 −0.0199108
\(835\) 5.64661 0.195409
\(836\) 9.63686 0.333298
\(837\) −1.91984 −0.0663593
\(838\) −2.86371 −0.0989253
\(839\) −38.3814 −1.32507 −0.662536 0.749030i \(-0.730520\pi\)
−0.662536 + 0.749030i \(0.730520\pi\)
\(840\) 3.35060 0.115607
\(841\) 0.430373 0.0148404
\(842\) −0.164958 −0.00568482
\(843\) −8.10067 −0.279002
\(844\) −23.3584 −0.804031
\(845\) 0 0
\(846\) −1.45632 −0.0500694
\(847\) −2.97027 −0.102060
\(848\) 17.2763 0.593270
\(849\) −16.8117 −0.576975
\(850\) −0.404849 −0.0138862
\(851\) −6.60620 −0.226458
\(852\) 29.6928 1.01726
\(853\) 17.7010 0.606070 0.303035 0.952979i \(-0.402000\pi\)
0.303035 + 0.952979i \(0.402000\pi\)
\(854\) 1.90755 0.0652750
\(855\) 11.6334 0.397852
\(856\) −0.399328 −0.0136488
\(857\) −4.15263 −0.141851 −0.0709256 0.997482i \(-0.522595\pi\)
−0.0709256 + 0.997482i \(0.522595\pi\)
\(858\) 0 0
\(859\) 18.5684 0.633547 0.316773 0.948501i \(-0.397401\pi\)
0.316773 + 0.948501i \(0.397401\pi\)
\(860\) −33.4025 −1.13902
\(861\) −16.0832 −0.548113
\(862\) 2.98630 0.101714
\(863\) −2.09641 −0.0713626 −0.0356813 0.999363i \(-0.511360\pi\)
−0.0356813 + 0.999363i \(0.511360\pi\)
\(864\) −1.40331 −0.0477417
\(865\) 3.51998 0.119683
\(866\) −4.00891 −0.136228
\(867\) 4.01822 0.136466
\(868\) −11.3254 −0.384410
\(869\) 2.55834 0.0867858
\(870\) 1.53525 0.0520500
\(871\) 0 0
\(872\) 6.34748 0.214953
\(873\) −11.0811 −0.375040
\(874\) 1.00431 0.0339712
\(875\) −30.2788 −1.02361
\(876\) −20.9529 −0.707933
\(877\) −7.78390 −0.262843 −0.131422 0.991327i \(-0.541954\pi\)
−0.131422 + 0.991327i \(0.541954\pi\)
\(878\) 1.36231 0.0459759
\(879\) −16.2136 −0.546871
\(880\) −9.39014 −0.316542
\(881\) 54.9122 1.85004 0.925019 0.379922i \(-0.124049\pi\)
0.925019 + 0.379922i \(0.124049\pi\)
\(882\) −0.215123 −0.00724357
\(883\) 12.5693 0.422990 0.211495 0.977379i \(-0.432167\pi\)
0.211495 + 0.977379i \(0.432167\pi\)
\(884\) 0 0
\(885\) −6.63947 −0.223183
\(886\) −4.86552 −0.163460
\(887\) −31.9437 −1.07256 −0.536282 0.844039i \(-0.680172\pi\)
−0.536282 + 0.844039i \(0.680172\pi\)
\(888\) −1.77260 −0.0594844
\(889\) 50.5079 1.69398
\(890\) 0.00816346 0.000273640 0
\(891\) 1.00000 0.0335013
\(892\) 50.2136 1.68128
\(893\) −59.8659 −2.00334
\(894\) 1.03746 0.0346978
\(895\) −20.6479 −0.690184
\(896\) −11.0247 −0.368309
\(897\) 0 0
\(898\) −1.34480 −0.0448767
\(899\) −10.4151 −0.347362
\(900\) −1.48583 −0.0495277
\(901\) 20.2227 0.673716
\(902\) −0.639139 −0.0212810
\(903\) 20.8362 0.693385
\(904\) 4.06137 0.135079
\(905\) 40.3949 1.34277
\(906\) 2.53193 0.0841176
\(907\) −15.1727 −0.503802 −0.251901 0.967753i \(-0.581056\pi\)
−0.251901 + 0.967753i \(0.581056\pi\)
\(908\) −23.3284 −0.774181
\(909\) −12.8478 −0.426136
\(910\) 0 0
\(911\) 39.7790 1.31794 0.658969 0.752170i \(-0.270993\pi\)
0.658969 + 0.752170i \(0.270993\pi\)
\(912\) −19.0042 −0.629293
\(913\) −9.41168 −0.311481
\(914\) −1.14417 −0.0378457
\(915\) −13.0444 −0.431234
\(916\) −18.9105 −0.624821
\(917\) 42.8957 1.41654
\(918\) −0.541149 −0.0178606
\(919\) −29.8226 −0.983757 −0.491878 0.870664i \(-0.663690\pi\)
−0.491878 + 0.870664i \(0.663690\pi\)
\(920\) −1.97803 −0.0652138
\(921\) −3.19713 −0.105349
\(922\) 1.36694 0.0450179
\(923\) 0 0
\(924\) 5.89916 0.194068
\(925\) −2.81852 −0.0926725
\(926\) −4.07695 −0.133977
\(927\) 16.2544 0.533865
\(928\) −7.61294 −0.249907
\(929\) 25.2173 0.827353 0.413676 0.910424i \(-0.364244\pi\)
0.413676 + 0.910424i \(0.364244\pi\)
\(930\) −0.543309 −0.0178158
\(931\) −8.84320 −0.289824
\(932\) −6.56051 −0.214897
\(933\) −19.9402 −0.652813
\(934\) −0.440322 −0.0144078
\(935\) −10.9916 −0.359464
\(936\) 0 0
\(937\) −5.42706 −0.177294 −0.0886471 0.996063i \(-0.528254\pi\)
−0.0886471 + 0.996063i \(0.528254\pi\)
\(938\) 1.99855 0.0652549
\(939\) −21.3968 −0.698258
\(940\) 58.7483 1.91616
\(941\) −25.3338 −0.825859 −0.412929 0.910763i \(-0.635494\pi\)
−0.412929 + 0.910763i \(0.635494\pi\)
\(942\) 2.63635 0.0858969
\(943\) 9.49472 0.309191
\(944\) 10.8462 0.353015
\(945\) 7.12130 0.231656
\(946\) 0.828021 0.0269213
\(947\) −57.5038 −1.86862 −0.934312 0.356457i \(-0.883985\pi\)
−0.934312 + 0.356457i \(0.883985\pi\)
\(948\) −5.08104 −0.165024
\(949\) 0 0
\(950\) 0.428486 0.0139019
\(951\) −13.6654 −0.443131
\(952\) −6.40704 −0.207653
\(953\) 12.5498 0.406529 0.203265 0.979124i \(-0.434845\pi\)
0.203265 + 0.979124i \(0.434845\pi\)
\(954\) −0.520668 −0.0168572
\(955\) 8.78233 0.284189
\(956\) 19.8528 0.642084
\(957\) 5.42498 0.175365
\(958\) 1.67428 0.0540936
\(959\) 41.9013 1.35306
\(960\) 18.3831 0.593314
\(961\) −27.3142 −0.881104
\(962\) 0 0
\(963\) −0.848723 −0.0273497
\(964\) −2.51201 −0.0809064
\(965\) 9.43209 0.303630
\(966\) 0.614782 0.0197803
\(967\) −1.49036 −0.0479266 −0.0239633 0.999713i \(-0.507628\pi\)
−0.0239633 + 0.999713i \(0.507628\pi\)
\(968\) 0.470505 0.0151226
\(969\) −22.2454 −0.714624
\(970\) −3.13593 −0.100689
\(971\) −39.4787 −1.26693 −0.633466 0.773771i \(-0.718368\pi\)
−0.633466 + 0.773771i \(0.718368\pi\)
\(972\) −1.98607 −0.0637031
\(973\) −14.4693 −0.463864
\(974\) 3.17215 0.101642
\(975\) 0 0
\(976\) 21.3093 0.682095
\(977\) −18.7508 −0.599891 −0.299945 0.953956i \(-0.596968\pi\)
−0.299945 + 0.953956i \(0.596968\pi\)
\(978\) 2.21633 0.0708706
\(979\) 0.0288464 0.000921935 0
\(980\) 8.67811 0.277212
\(981\) 13.4908 0.430728
\(982\) −4.02052 −0.128300
\(983\) −23.9822 −0.764914 −0.382457 0.923973i \(-0.624922\pi\)
−0.382457 + 0.923973i \(0.624922\pi\)
\(984\) 2.54765 0.0812161
\(985\) −41.2115 −1.31311
\(986\) −2.93572 −0.0934925
\(987\) −36.6466 −1.16648
\(988\) 0 0
\(989\) −12.3007 −0.391138
\(990\) 0.282997 0.00899425
\(991\) −21.7643 −0.691365 −0.345683 0.938351i \(-0.612353\pi\)
−0.345683 + 0.938351i \(0.612353\pi\)
\(992\) 2.69413 0.0855388
\(993\) 9.56314 0.303477
\(994\) −5.24170 −0.166257
\(995\) 18.8697 0.598210
\(996\) 18.6922 0.592286
\(997\) −34.5397 −1.09388 −0.546941 0.837171i \(-0.684208\pi\)
−0.546941 + 0.837171i \(0.684208\pi\)
\(998\) 0.0173378 0.000548817 0
\(999\) −3.76743 −0.119196
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 5577.2.a.n.1.4 5
13.3 even 3 429.2.i.f.100.2 10
13.9 even 3 429.2.i.f.133.2 yes 10
13.12 even 2 5577.2.a.w.1.2 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
429.2.i.f.100.2 10 13.3 even 3
429.2.i.f.133.2 yes 10 13.9 even 3
5577.2.a.n.1.4 5 1.1 even 1 trivial
5577.2.a.w.1.2 5 13.12 even 2