Properties

Label 538.2.a.e.1.3
Level $538$
Weight $2$
Character 538.1
Self dual yes
Analytic conductor $4.296$
Analytic rank $0$
Dimension $7$
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] = [538,2,Mod(1,538)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(538, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("538.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 538 = 2 \cdot 269 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 538.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: \(4.29595162874\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: \(\mathbb{Q}[x]/(x^{7} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - x^{6} - 15x^{5} + 16x^{4} + 49x^{3} - 53x^{2} - 44x + 48 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(3.22542\) of defining polynomial
Character \(\chi\) \(=\) 538.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -0.383827 q^{3} +1.00000 q^{4} +4.15424 q^{5} -0.383827 q^{6} +4.87818 q^{7} +1.00000 q^{8} -2.85268 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -0.383827 q^{3} +1.00000 q^{4} +4.15424 q^{5} -0.383827 q^{6} +4.87818 q^{7} +1.00000 q^{8} -2.85268 q^{9} +4.15424 q^{10} -1.62910 q^{11} -0.383827 q^{12} -2.16682 q^{13} +4.87818 q^{14} -1.59451 q^{15} +1.00000 q^{16} -6.78448 q^{17} -2.85268 q^{18} -8.61766 q^{19} +4.15424 q^{20} -1.87238 q^{21} -1.62910 q^{22} +2.76765 q^{23} -0.383827 q^{24} +12.2577 q^{25} -2.16682 q^{26} +2.24641 q^{27} +4.87818 q^{28} +2.10510 q^{29} -1.59451 q^{30} -3.62414 q^{31} +1.00000 q^{32} +0.625290 q^{33} -6.78448 q^{34} +20.2651 q^{35} -2.85268 q^{36} +1.19760 q^{37} -8.61766 q^{38} +0.831683 q^{39} +4.15424 q^{40} +5.43715 q^{41} -1.87238 q^{42} -9.63402 q^{43} -1.62910 q^{44} -11.8507 q^{45} +2.76765 q^{46} -7.07613 q^{47} -0.383827 q^{48} +16.7967 q^{49} +12.2577 q^{50} +2.60406 q^{51} -2.16682 q^{52} -3.03890 q^{53} +2.24641 q^{54} -6.76765 q^{55} +4.87818 q^{56} +3.30769 q^{57} +2.10510 q^{58} +4.21827 q^{59} -1.59451 q^{60} +8.74744 q^{61} -3.62414 q^{62} -13.9159 q^{63} +1.00000 q^{64} -9.00148 q^{65} +0.625290 q^{66} -1.50761 q^{67} -6.78448 q^{68} -1.06230 q^{69} +20.2651 q^{70} +11.0270 q^{71} -2.85268 q^{72} +1.37039 q^{73} +1.19760 q^{74} -4.70482 q^{75} -8.61766 q^{76} -7.94703 q^{77} +0.831683 q^{78} -14.9749 q^{79} +4.15424 q^{80} +7.69580 q^{81} +5.43715 q^{82} -8.08276 q^{83} -1.87238 q^{84} -28.1843 q^{85} -9.63402 q^{86} -0.807993 q^{87} -1.62910 q^{88} +3.31299 q^{89} -11.8507 q^{90} -10.5701 q^{91} +2.76765 q^{92} +1.39104 q^{93} -7.07613 q^{94} -35.7998 q^{95} -0.383827 q^{96} +16.8474 q^{97} +16.7967 q^{98} +4.64729 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9} + 7 q^{10} - 3 q^{11} + q^{12} - 9 q^{13} + 6 q^{14} + 8 q^{15} + 7 q^{16} + 8 q^{17} + 12 q^{18} - 11 q^{19} + 7 q^{20} - 6 q^{21} - 3 q^{22} + 12 q^{23} + q^{24} + 22 q^{25} - 9 q^{26} - 14 q^{27} + 6 q^{28} - 5 q^{29} + 8 q^{30} + 14 q^{31} + 7 q^{32} - 4 q^{33} + 8 q^{34} - 4 q^{35} + 12 q^{36} + 13 q^{37} - 11 q^{38} - 18 q^{39} + 7 q^{40} + 12 q^{41} - 6 q^{42} - 11 q^{43} - 3 q^{44} + 3 q^{45} + 12 q^{46} + 2 q^{47} + q^{48} + 15 q^{49} + 22 q^{50} - 26 q^{51} - 9 q^{52} + 19 q^{53} - 14 q^{54} - 40 q^{55} + 6 q^{56} - 12 q^{57} - 5 q^{58} - 9 q^{59} + 8 q^{60} - 3 q^{61} + 14 q^{62} - 26 q^{63} + 7 q^{64} - 10 q^{65} - 4 q^{66} - 33 q^{67} + 8 q^{68} - 64 q^{69} - 4 q^{70} + 28 q^{71} + 12 q^{72} - 14 q^{73} + 13 q^{74} - 45 q^{75} - 11 q^{76} + 10 q^{77} - 18 q^{78} + 2 q^{79} + 7 q^{80} + 15 q^{81} + 12 q^{82} - 7 q^{83} - 6 q^{84} - 16 q^{85} - 11 q^{86} + 16 q^{87} - 3 q^{88} + 18 q^{89} + 3 q^{90} - 26 q^{91} + 12 q^{92} + 6 q^{93} + 2 q^{94} - 34 q^{95} + q^{96} - 4 q^{97} + 15 q^{98} - 37 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.00000 0.707107
\(3\) −0.383827 −0.221602 −0.110801 0.993843i \(-0.535342\pi\)
−0.110801 + 0.993843i \(0.535342\pi\)
\(4\) 1.00000 0.500000
\(5\) 4.15424 1.85783 0.928916 0.370291i \(-0.120742\pi\)
0.928916 + 0.370291i \(0.120742\pi\)
\(6\) −0.383827 −0.156697
\(7\) 4.87818 1.84378 0.921890 0.387452i \(-0.126645\pi\)
0.921890 + 0.387452i \(0.126645\pi\)
\(8\) 1.00000 0.353553
\(9\) −2.85268 −0.950892
\(10\) 4.15424 1.31369
\(11\) −1.62910 −0.491191 −0.245596 0.969372i \(-0.578984\pi\)
−0.245596 + 0.969372i \(0.578984\pi\)
\(12\) −0.383827 −0.110801
\(13\) −2.16682 −0.600968 −0.300484 0.953787i \(-0.597148\pi\)
−0.300484 + 0.953787i \(0.597148\pi\)
\(14\) 4.87818 1.30375
\(15\) −1.59451 −0.411700
\(16\) 1.00000 0.250000
\(17\) −6.78448 −1.64548 −0.822739 0.568420i \(-0.807555\pi\)
−0.822739 + 0.568420i \(0.807555\pi\)
\(18\) −2.85268 −0.672382
\(19\) −8.61766 −1.97703 −0.988513 0.151136i \(-0.951707\pi\)
−0.988513 + 0.151136i \(0.951707\pi\)
\(20\) 4.15424 0.928916
\(21\) −1.87238 −0.408586
\(22\) −1.62910 −0.347325
\(23\) 2.76765 0.577096 0.288548 0.957465i \(-0.406828\pi\)
0.288548 + 0.957465i \(0.406828\pi\)
\(24\) −0.383827 −0.0783483
\(25\) 12.2577 2.45154
\(26\) −2.16682 −0.424948
\(27\) 2.24641 0.432322
\(28\) 4.87818 0.921890
\(29\) 2.10510 0.390907 0.195454 0.980713i \(-0.437382\pi\)
0.195454 + 0.980713i \(0.437382\pi\)
\(30\) −1.59451 −0.291116
\(31\) −3.62414 −0.650914 −0.325457 0.945557i \(-0.605518\pi\)
−0.325457 + 0.945557i \(0.605518\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.625290 0.108849
\(34\) −6.78448 −1.16353
\(35\) 20.2651 3.42543
\(36\) −2.85268 −0.475446
\(37\) 1.19760 0.196885 0.0984424 0.995143i \(-0.468614\pi\)
0.0984424 + 0.995143i \(0.468614\pi\)
\(38\) −8.61766 −1.39797
\(39\) 0.831683 0.133176
\(40\) 4.15424 0.656843
\(41\) 5.43715 0.849140 0.424570 0.905395i \(-0.360425\pi\)
0.424570 + 0.905395i \(0.360425\pi\)
\(42\) −1.87238 −0.288914
\(43\) −9.63402 −1.46917 −0.734587 0.678514i \(-0.762624\pi\)
−0.734587 + 0.678514i \(0.762624\pi\)
\(44\) −1.62910 −0.245596
\(45\) −11.8507 −1.76660
\(46\) 2.76765 0.408068
\(47\) −7.07613 −1.03216 −0.516080 0.856541i \(-0.672609\pi\)
−0.516080 + 0.856541i \(0.672609\pi\)
\(48\) −0.383827 −0.0554006
\(49\) 16.7967 2.39952
\(50\) 12.2577 1.73350
\(51\) 2.60406 0.364642
\(52\) −2.16682 −0.300484
\(53\) −3.03890 −0.417425 −0.208712 0.977977i \(-0.566927\pi\)
−0.208712 + 0.977977i \(0.566927\pi\)
\(54\) 2.24641 0.305698
\(55\) −6.76765 −0.912550
\(56\) 4.87818 0.651875
\(57\) 3.30769 0.438114
\(58\) 2.10510 0.276413
\(59\) 4.21827 0.549172 0.274586 0.961563i \(-0.411459\pi\)
0.274586 + 0.961563i \(0.411459\pi\)
\(60\) −1.59451 −0.205850
\(61\) 8.74744 1.11999 0.559997 0.828494i \(-0.310802\pi\)
0.559997 + 0.828494i \(0.310802\pi\)
\(62\) −3.62414 −0.460266
\(63\) −13.9159 −1.75324
\(64\) 1.00000 0.125000
\(65\) −9.00148 −1.11650
\(66\) 0.625290 0.0769679
\(67\) −1.50761 −0.184184 −0.0920921 0.995750i \(-0.529355\pi\)
−0.0920921 + 0.995750i \(0.529355\pi\)
\(68\) −6.78448 −0.822739
\(69\) −1.06230 −0.127886
\(70\) 20.2651 2.42215
\(71\) 11.0270 1.30866 0.654331 0.756208i \(-0.272950\pi\)
0.654331 + 0.756208i \(0.272950\pi\)
\(72\) −2.85268 −0.336191
\(73\) 1.37039 0.160391 0.0801957 0.996779i \(-0.474445\pi\)
0.0801957 + 0.996779i \(0.474445\pi\)
\(74\) 1.19760 0.139219
\(75\) −4.70482 −0.543266
\(76\) −8.61766 −0.988513
\(77\) −7.94703 −0.905648
\(78\) 0.831683 0.0941695
\(79\) −14.9749 −1.68480 −0.842401 0.538850i \(-0.818859\pi\)
−0.842401 + 0.538850i \(0.818859\pi\)
\(80\) 4.15424 0.464458
\(81\) 7.69580 0.855089
\(82\) 5.43715 0.600432
\(83\) −8.08276 −0.887198 −0.443599 0.896225i \(-0.646299\pi\)
−0.443599 + 0.896225i \(0.646299\pi\)
\(84\) −1.87238 −0.204293
\(85\) −28.1843 −3.05702
\(86\) −9.63402 −1.03886
\(87\) −0.807993 −0.0866259
\(88\) −1.62910 −0.173662
\(89\) 3.31299 0.351176 0.175588 0.984464i \(-0.443817\pi\)
0.175588 + 0.984464i \(0.443817\pi\)
\(90\) −11.8507 −1.24917
\(91\) −10.5701 −1.10805
\(92\) 2.76765 0.288548
\(93\) 1.39104 0.144244
\(94\) −7.07613 −0.729847
\(95\) −35.7998 −3.67298
\(96\) −0.383827 −0.0391741
\(97\) 16.8474 1.71059 0.855296 0.518139i \(-0.173375\pi\)
0.855296 + 0.518139i \(0.173375\pi\)
\(98\) 16.7967 1.69672
\(99\) 4.64729 0.467070
\(100\) 12.2577 1.22577
\(101\) 2.93034 0.291580 0.145790 0.989316i \(-0.453428\pi\)
0.145790 + 0.989316i \(0.453428\pi\)
\(102\) 2.60406 0.257841
\(103\) 1.25819 0.123973 0.0619867 0.998077i \(-0.480256\pi\)
0.0619867 + 0.998077i \(0.480256\pi\)
\(104\) −2.16682 −0.212474
\(105\) −7.77829 −0.759084
\(106\) −3.03890 −0.295164
\(107\) 9.35946 0.904813 0.452407 0.891812i \(-0.350566\pi\)
0.452407 + 0.891812i \(0.350566\pi\)
\(108\) 2.24641 0.216161
\(109\) 11.4192 1.09376 0.546882 0.837210i \(-0.315815\pi\)
0.546882 + 0.837210i \(0.315815\pi\)
\(110\) −6.76765 −0.645270
\(111\) −0.459672 −0.0436301
\(112\) 4.87818 0.460945
\(113\) 0.316816 0.0298036 0.0149018 0.999889i \(-0.495256\pi\)
0.0149018 + 0.999889i \(0.495256\pi\)
\(114\) 3.30769 0.309793
\(115\) 11.4975 1.07215
\(116\) 2.10510 0.195454
\(117\) 6.18124 0.571456
\(118\) 4.21827 0.388323
\(119\) −33.0959 −3.03390
\(120\) −1.59451 −0.145558
\(121\) −8.34604 −0.758731
\(122\) 8.74744 0.791956
\(123\) −2.08692 −0.188171
\(124\) −3.62414 −0.325457
\(125\) 30.1501 2.69671
\(126\) −13.9159 −1.23973
\(127\) −2.08772 −0.185255 −0.0926276 0.995701i \(-0.529527\pi\)
−0.0926276 + 0.995701i \(0.529527\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.69779 0.325572
\(130\) −9.00148 −0.789482
\(131\) −5.17397 −0.452052 −0.226026 0.974121i \(-0.572573\pi\)
−0.226026 + 0.974121i \(0.572573\pi\)
\(132\) 0.625290 0.0544245
\(133\) −42.0385 −3.64520
\(134\) −1.50761 −0.130238
\(135\) 9.33213 0.803182
\(136\) −6.78448 −0.581764
\(137\) 6.35949 0.543328 0.271664 0.962392i \(-0.412426\pi\)
0.271664 + 0.962392i \(0.412426\pi\)
\(138\) −1.06230 −0.0904289
\(139\) −10.6786 −0.905746 −0.452873 0.891575i \(-0.649601\pi\)
−0.452873 + 0.891575i \(0.649601\pi\)
\(140\) 20.2651 1.71272
\(141\) 2.71601 0.228729
\(142\) 11.0270 0.925364
\(143\) 3.52996 0.295190
\(144\) −2.85268 −0.237723
\(145\) 8.74508 0.726239
\(146\) 1.37039 0.113414
\(147\) −6.44701 −0.531740
\(148\) 1.19760 0.0984424
\(149\) −3.47426 −0.284622 −0.142311 0.989822i \(-0.545453\pi\)
−0.142311 + 0.989822i \(0.545453\pi\)
\(150\) −4.70482 −0.384147
\(151\) 11.2407 0.914752 0.457376 0.889273i \(-0.348789\pi\)
0.457376 + 0.889273i \(0.348789\pi\)
\(152\) −8.61766 −0.698984
\(153\) 19.3539 1.56467
\(154\) −7.94703 −0.640390
\(155\) −15.0555 −1.20929
\(156\) 0.831683 0.0665879
\(157\) 8.59533 0.685982 0.342991 0.939339i \(-0.388560\pi\)
0.342991 + 0.939339i \(0.388560\pi\)
\(158\) −14.9749 −1.19134
\(159\) 1.16641 0.0925024
\(160\) 4.15424 0.328421
\(161\) 13.5011 1.06404
\(162\) 7.69580 0.604639
\(163\) 10.1530 0.795241 0.397621 0.917550i \(-0.369836\pi\)
0.397621 + 0.917550i \(0.369836\pi\)
\(164\) 5.43715 0.424570
\(165\) 2.59760 0.202223
\(166\) −8.08276 −0.627344
\(167\) 4.37572 0.338604 0.169302 0.985564i \(-0.445849\pi\)
0.169302 + 0.985564i \(0.445849\pi\)
\(168\) −1.87238 −0.144457
\(169\) −8.30489 −0.638838
\(170\) −28.1843 −2.16164
\(171\) 24.5834 1.87994
\(172\) −9.63402 −0.734587
\(173\) −22.7651 −1.73080 −0.865398 0.501085i \(-0.832934\pi\)
−0.865398 + 0.501085i \(0.832934\pi\)
\(174\) −0.807993 −0.0612538
\(175\) 59.7952 4.52009
\(176\) −1.62910 −0.122798
\(177\) −1.61908 −0.121698
\(178\) 3.31299 0.248319
\(179\) 12.1634 0.909139 0.454569 0.890711i \(-0.349793\pi\)
0.454569 + 0.890711i \(0.349793\pi\)
\(180\) −11.8507 −0.883299
\(181\) −4.61048 −0.342694 −0.171347 0.985211i \(-0.554812\pi\)
−0.171347 + 0.985211i \(0.554812\pi\)
\(182\) −10.5701 −0.783511
\(183\) −3.35750 −0.248193
\(184\) 2.76765 0.204034
\(185\) 4.97513 0.365779
\(186\) 1.39104 0.101996
\(187\) 11.0526 0.808244
\(188\) −7.07613 −0.516080
\(189\) 10.9584 0.797107
\(190\) −35.7998 −2.59719
\(191\) 3.30553 0.239180 0.119590 0.992823i \(-0.461842\pi\)
0.119590 + 0.992823i \(0.461842\pi\)
\(192\) −0.383827 −0.0277003
\(193\) −18.5157 −1.33279 −0.666394 0.745600i \(-0.732163\pi\)
−0.666394 + 0.745600i \(0.732163\pi\)
\(194\) 16.8474 1.20957
\(195\) 3.45501 0.247418
\(196\) 16.7967 1.19976
\(197\) 8.78109 0.625627 0.312814 0.949815i \(-0.398729\pi\)
0.312814 + 0.949815i \(0.398729\pi\)
\(198\) 4.64729 0.330268
\(199\) −12.4370 −0.881634 −0.440817 0.897597i \(-0.645311\pi\)
−0.440817 + 0.897597i \(0.645311\pi\)
\(200\) 12.2577 0.866749
\(201\) 0.578661 0.0408156
\(202\) 2.93034 0.206178
\(203\) 10.2691 0.720746
\(204\) 2.60406 0.182321
\(205\) 22.5872 1.57756
\(206\) 1.25819 0.0876625
\(207\) −7.89522 −0.548756
\(208\) −2.16682 −0.150242
\(209\) 14.0390 0.971098
\(210\) −7.77829 −0.536753
\(211\) −10.9290 −0.752381 −0.376191 0.926542i \(-0.622766\pi\)
−0.376191 + 0.926542i \(0.622766\pi\)
\(212\) −3.03890 −0.208712
\(213\) −4.23245 −0.290003
\(214\) 9.35946 0.639800
\(215\) −40.0220 −2.72948
\(216\) 2.24641 0.152849
\(217\) −17.6792 −1.20014
\(218\) 11.4192 0.773408
\(219\) −0.525990 −0.0355431
\(220\) −6.76765 −0.456275
\(221\) 14.7007 0.988879
\(222\) −0.459672 −0.0308512
\(223\) 8.79372 0.588871 0.294436 0.955671i \(-0.404868\pi\)
0.294436 + 0.955671i \(0.404868\pi\)
\(224\) 4.87818 0.325937
\(225\) −34.9672 −2.33115
\(226\) 0.316816 0.0210743
\(227\) 6.74871 0.447928 0.223964 0.974597i \(-0.428100\pi\)
0.223964 + 0.974597i \(0.428100\pi\)
\(228\) 3.30769 0.219057
\(229\) −23.2606 −1.53711 −0.768553 0.639787i \(-0.779023\pi\)
−0.768553 + 0.639787i \(0.779023\pi\)
\(230\) 11.4975 0.758122
\(231\) 3.05028 0.200694
\(232\) 2.10510 0.138206
\(233\) 11.0721 0.725357 0.362679 0.931914i \(-0.381862\pi\)
0.362679 + 0.931914i \(0.381862\pi\)
\(234\) 6.18124 0.404080
\(235\) −29.3959 −1.91758
\(236\) 4.21827 0.274586
\(237\) 5.74775 0.373356
\(238\) −33.0959 −2.14529
\(239\) −3.99072 −0.258138 −0.129069 0.991636i \(-0.541199\pi\)
−0.129069 + 0.991636i \(0.541199\pi\)
\(240\) −1.59451 −0.102925
\(241\) −16.5408 −1.06549 −0.532745 0.846276i \(-0.678839\pi\)
−0.532745 + 0.846276i \(0.678839\pi\)
\(242\) −8.34604 −0.536504
\(243\) −9.69309 −0.621812
\(244\) 8.74744 0.559997
\(245\) 69.7773 4.45791
\(246\) −2.08692 −0.133057
\(247\) 18.6729 1.18813
\(248\) −3.62414 −0.230133
\(249\) 3.10238 0.196605
\(250\) 30.1501 1.90686
\(251\) 24.0533 1.51823 0.759117 0.650955i \(-0.225631\pi\)
0.759117 + 0.650955i \(0.225631\pi\)
\(252\) −13.9159 −0.876618
\(253\) −4.50877 −0.283464
\(254\) −2.08772 −0.130995
\(255\) 10.8179 0.677443
\(256\) 1.00000 0.0625000
\(257\) −13.6822 −0.853470 −0.426735 0.904377i \(-0.640336\pi\)
−0.426735 + 0.904377i \(0.640336\pi\)
\(258\) 3.69779 0.230214
\(259\) 5.84213 0.363012
\(260\) −9.00148 −0.558248
\(261\) −6.00517 −0.371711
\(262\) −5.17397 −0.319649
\(263\) 11.3942 0.702596 0.351298 0.936264i \(-0.385740\pi\)
0.351298 + 0.936264i \(0.385740\pi\)
\(264\) 0.625290 0.0384840
\(265\) −12.6243 −0.775505
\(266\) −42.0385 −2.57755
\(267\) −1.27161 −0.0778214
\(268\) −1.50761 −0.0920921
\(269\) −1.00000 −0.0609711
\(270\) 9.33213 0.567935
\(271\) −24.6167 −1.49536 −0.747679 0.664061i \(-0.768832\pi\)
−0.747679 + 0.664061i \(0.768832\pi\)
\(272\) −6.78448 −0.411369
\(273\) 4.05710 0.245547
\(274\) 6.35949 0.384191
\(275\) −19.9690 −1.20417
\(276\) −1.06230 −0.0639429
\(277\) 4.70762 0.282853 0.141427 0.989949i \(-0.454831\pi\)
0.141427 + 0.989949i \(0.454831\pi\)
\(278\) −10.6786 −0.640459
\(279\) 10.3385 0.618949
\(280\) 20.2651 1.21107
\(281\) 6.51540 0.388676 0.194338 0.980935i \(-0.437744\pi\)
0.194338 + 0.980935i \(0.437744\pi\)
\(282\) 2.71601 0.161736
\(283\) −8.67417 −0.515626 −0.257813 0.966195i \(-0.583002\pi\)
−0.257813 + 0.966195i \(0.583002\pi\)
\(284\) 11.0270 0.654331
\(285\) 13.7409 0.813941
\(286\) 3.52996 0.208731
\(287\) 26.5234 1.56563
\(288\) −2.85268 −0.168096
\(289\) 29.0291 1.70760
\(290\) 8.74508 0.513529
\(291\) −6.46647 −0.379071
\(292\) 1.37039 0.0801957
\(293\) 20.3798 1.19060 0.595300 0.803504i \(-0.297033\pi\)
0.595300 + 0.803504i \(0.297033\pi\)
\(294\) −6.44701 −0.375997
\(295\) 17.5237 1.02027
\(296\) 1.19760 0.0696093
\(297\) −3.65962 −0.212353
\(298\) −3.47426 −0.201258
\(299\) −5.99701 −0.346816
\(300\) −4.70482 −0.271633
\(301\) −46.9965 −2.70883
\(302\) 11.2407 0.646827
\(303\) −1.12474 −0.0646147
\(304\) −8.61766 −0.494256
\(305\) 36.3389 2.08076
\(306\) 19.3539 1.10639
\(307\) −17.5028 −0.998939 −0.499470 0.866331i \(-0.666472\pi\)
−0.499470 + 0.866331i \(0.666472\pi\)
\(308\) −7.94703 −0.452824
\(309\) −0.482928 −0.0274728
\(310\) −15.0555 −0.855096
\(311\) 22.4715 1.27424 0.637120 0.770764i \(-0.280125\pi\)
0.637120 + 0.770764i \(0.280125\pi\)
\(312\) 0.831683 0.0470848
\(313\) −32.6321 −1.84447 −0.922236 0.386626i \(-0.873640\pi\)
−0.922236 + 0.386626i \(0.873640\pi\)
\(314\) 8.59533 0.485063
\(315\) −57.8099 −3.25722
\(316\) −14.9749 −0.842401
\(317\) −19.5532 −1.09822 −0.549110 0.835750i \(-0.685033\pi\)
−0.549110 + 0.835750i \(0.685033\pi\)
\(318\) 1.16641 0.0654090
\(319\) −3.42941 −0.192010
\(320\) 4.15424 0.232229
\(321\) −3.59241 −0.200509
\(322\) 13.5011 0.752388
\(323\) 58.4663 3.25315
\(324\) 7.69580 0.427544
\(325\) −26.5602 −1.47329
\(326\) 10.1530 0.562321
\(327\) −4.38300 −0.242381
\(328\) 5.43715 0.300216
\(329\) −34.5186 −1.90307
\(330\) 2.59760 0.142993
\(331\) −20.1680 −1.10853 −0.554266 0.832340i \(-0.687001\pi\)
−0.554266 + 0.832340i \(0.687001\pi\)
\(332\) −8.08276 −0.443599
\(333\) −3.41638 −0.187216
\(334\) 4.37572 0.239429
\(335\) −6.26298 −0.342183
\(336\) −1.87238 −0.102146
\(337\) 29.1409 1.58741 0.793704 0.608305i \(-0.208150\pi\)
0.793704 + 0.608305i \(0.208150\pi\)
\(338\) −8.30489 −0.451727
\(339\) −0.121603 −0.00660454
\(340\) −28.1843 −1.52851
\(341\) 5.90407 0.319723
\(342\) 24.5834 1.32932
\(343\) 47.7899 2.58041
\(344\) −9.63402 −0.519432
\(345\) −4.41304 −0.237590
\(346\) −22.7651 −1.22386
\(347\) 12.2141 0.655686 0.327843 0.944732i \(-0.393678\pi\)
0.327843 + 0.944732i \(0.393678\pi\)
\(348\) −0.807993 −0.0433130
\(349\) 15.3346 0.820843 0.410421 0.911896i \(-0.365382\pi\)
0.410421 + 0.911896i \(0.365382\pi\)
\(350\) 59.7952 3.19619
\(351\) −4.86757 −0.259812
\(352\) −1.62910 −0.0868311
\(353\) 30.6859 1.63324 0.816622 0.577173i \(-0.195844\pi\)
0.816622 + 0.577173i \(0.195844\pi\)
\(354\) −1.61908 −0.0860533
\(355\) 45.8087 2.43127
\(356\) 3.31299 0.175588
\(357\) 12.7031 0.672319
\(358\) 12.1634 0.642858
\(359\) 0.344824 0.0181991 0.00909956 0.999959i \(-0.497103\pi\)
0.00909956 + 0.999959i \(0.497103\pi\)
\(360\) −11.8507 −0.624587
\(361\) 55.2640 2.90863
\(362\) −4.61048 −0.242322
\(363\) 3.20343 0.168137
\(364\) −10.5701 −0.554026
\(365\) 5.69291 0.297980
\(366\) −3.35750 −0.175499
\(367\) 22.3072 1.16443 0.582214 0.813035i \(-0.302186\pi\)
0.582214 + 0.813035i \(0.302186\pi\)
\(368\) 2.76765 0.144274
\(369\) −15.5104 −0.807440
\(370\) 4.97513 0.258645
\(371\) −14.8243 −0.769640
\(372\) 1.39104 0.0721220
\(373\) −5.92843 −0.306962 −0.153481 0.988152i \(-0.549048\pi\)
−0.153481 + 0.988152i \(0.549048\pi\)
\(374\) 11.0526 0.571515
\(375\) −11.5724 −0.597597
\(376\) −7.07613 −0.364923
\(377\) −4.56137 −0.234922
\(378\) 10.9584 0.563640
\(379\) −2.01119 −0.103308 −0.0516540 0.998665i \(-0.516449\pi\)
−0.0516540 + 0.998665i \(0.516449\pi\)
\(380\) −35.7998 −1.83649
\(381\) 0.801322 0.0410530
\(382\) 3.30553 0.169126
\(383\) 4.21696 0.215477 0.107738 0.994179i \(-0.465639\pi\)
0.107738 + 0.994179i \(0.465639\pi\)
\(384\) −0.383827 −0.0195871
\(385\) −33.0138 −1.68254
\(386\) −18.5157 −0.942423
\(387\) 27.4827 1.39703
\(388\) 16.8474 0.855296
\(389\) 4.54939 0.230663 0.115332 0.993327i \(-0.463207\pi\)
0.115332 + 0.993327i \(0.463207\pi\)
\(390\) 3.45501 0.174951
\(391\) −18.7771 −0.949598
\(392\) 16.7967 0.848360
\(393\) 1.98591 0.100176
\(394\) 8.78109 0.442385
\(395\) −62.2091 −3.13008
\(396\) 4.64729 0.233535
\(397\) −15.2162 −0.763677 −0.381839 0.924229i \(-0.624709\pi\)
−0.381839 + 0.924229i \(0.624709\pi\)
\(398\) −12.4370 −0.623409
\(399\) 16.1355 0.807785
\(400\) 12.2577 0.612884
\(401\) 18.4280 0.920250 0.460125 0.887854i \(-0.347805\pi\)
0.460125 + 0.887854i \(0.347805\pi\)
\(402\) 0.578661 0.0288610
\(403\) 7.85285 0.391178
\(404\) 2.93034 0.145790
\(405\) 31.9702 1.58861
\(406\) 10.2691 0.509645
\(407\) −1.95101 −0.0967081
\(408\) 2.60406 0.128920
\(409\) 36.3293 1.79637 0.898184 0.439619i \(-0.144887\pi\)
0.898184 + 0.439619i \(0.144887\pi\)
\(410\) 22.5872 1.11550
\(411\) −2.44094 −0.120403
\(412\) 1.25819 0.0619867
\(413\) 20.5775 1.01255
\(414\) −7.89522 −0.388029
\(415\) −33.5777 −1.64826
\(416\) −2.16682 −0.106237
\(417\) 4.09872 0.200715
\(418\) 14.0390 0.686670
\(419\) −6.63242 −0.324015 −0.162007 0.986790i \(-0.551797\pi\)
−0.162007 + 0.986790i \(0.551797\pi\)
\(420\) −7.77829 −0.379542
\(421\) −33.1090 −1.61363 −0.806817 0.590801i \(-0.798812\pi\)
−0.806817 + 0.590801i \(0.798812\pi\)
\(422\) −10.9290 −0.532014
\(423\) 20.1859 0.981472
\(424\) −3.03890 −0.147582
\(425\) −83.1620 −4.03395
\(426\) −4.23245 −0.205063
\(427\) 42.6716 2.06502
\(428\) 9.35946 0.452407
\(429\) −1.35489 −0.0654148
\(430\) −40.0220 −1.93003
\(431\) −16.6498 −0.801995 −0.400997 0.916079i \(-0.631336\pi\)
−0.400997 + 0.916079i \(0.631336\pi\)
\(432\) 2.24641 0.108081
\(433\) 5.36131 0.257648 0.128824 0.991667i \(-0.458880\pi\)
0.128824 + 0.991667i \(0.458880\pi\)
\(434\) −17.6792 −0.848629
\(435\) −3.35659 −0.160936
\(436\) 11.4192 0.546882
\(437\) −23.8507 −1.14093
\(438\) −0.525990 −0.0251328
\(439\) −5.42961 −0.259141 −0.129571 0.991570i \(-0.541360\pi\)
−0.129571 + 0.991570i \(0.541360\pi\)
\(440\) −6.76765 −0.322635
\(441\) −47.9155 −2.28169
\(442\) 14.7007 0.699243
\(443\) 3.66946 0.174341 0.0871706 0.996193i \(-0.472217\pi\)
0.0871706 + 0.996193i \(0.472217\pi\)
\(444\) −0.459672 −0.0218151
\(445\) 13.7629 0.652426
\(446\) 8.79372 0.416395
\(447\) 1.33351 0.0630729
\(448\) 4.87818 0.230472
\(449\) 6.43513 0.303693 0.151846 0.988404i \(-0.451478\pi\)
0.151846 + 0.988404i \(0.451478\pi\)
\(450\) −34.9672 −1.64837
\(451\) −8.85764 −0.417090
\(452\) 0.316816 0.0149018
\(453\) −4.31446 −0.202711
\(454\) 6.74871 0.316733
\(455\) −43.9109 −2.05857
\(456\) 3.30769 0.154897
\(457\) 2.97606 0.139214 0.0696071 0.997574i \(-0.477825\pi\)
0.0696071 + 0.997574i \(0.477825\pi\)
\(458\) −23.2606 −1.08690
\(459\) −15.2407 −0.711377
\(460\) 11.4975 0.536073
\(461\) 9.73958 0.453617 0.226809 0.973939i \(-0.427171\pi\)
0.226809 + 0.973939i \(0.427171\pi\)
\(462\) 3.05028 0.141912
\(463\) −19.9810 −0.928598 −0.464299 0.885679i \(-0.653694\pi\)
−0.464299 + 0.885679i \(0.653694\pi\)
\(464\) 2.10510 0.0977268
\(465\) 5.77871 0.267981
\(466\) 11.0721 0.512905
\(467\) −14.3949 −0.666115 −0.333057 0.942907i \(-0.608080\pi\)
−0.333057 + 0.942907i \(0.608080\pi\)
\(468\) 6.18124 0.285728
\(469\) −7.35441 −0.339595
\(470\) −29.3959 −1.35593
\(471\) −3.29912 −0.152015
\(472\) 4.21827 0.194161
\(473\) 15.6947 0.721645
\(474\) 5.74775 0.264003
\(475\) −105.633 −4.84675
\(476\) −33.0959 −1.51695
\(477\) 8.66900 0.396926
\(478\) −3.99072 −0.182531
\(479\) −22.4679 −1.02659 −0.513293 0.858213i \(-0.671575\pi\)
−0.513293 + 0.858213i \(0.671575\pi\)
\(480\) −1.59451 −0.0727789
\(481\) −2.59499 −0.118321
\(482\) −16.5408 −0.753415
\(483\) −5.18209 −0.235793
\(484\) −8.34604 −0.379366
\(485\) 69.9880 3.17799
\(486\) −9.69309 −0.439687
\(487\) −15.3289 −0.694618 −0.347309 0.937751i \(-0.612904\pi\)
−0.347309 + 0.937751i \(0.612904\pi\)
\(488\) 8.74744 0.395978
\(489\) −3.89698 −0.176227
\(490\) 69.7773 3.15222
\(491\) −23.8945 −1.07834 −0.539172 0.842196i \(-0.681263\pi\)
−0.539172 + 0.842196i \(0.681263\pi\)
\(492\) −2.08692 −0.0940857
\(493\) −14.2820 −0.643229
\(494\) 18.6729 0.840134
\(495\) 19.3059 0.867737
\(496\) −3.62414 −0.162729
\(497\) 53.7917 2.41289
\(498\) 3.10238 0.139021
\(499\) −15.7627 −0.705635 −0.352818 0.935692i \(-0.614776\pi\)
−0.352818 + 0.935692i \(0.614776\pi\)
\(500\) 30.1501 1.34836
\(501\) −1.67952 −0.0750353
\(502\) 24.0533 1.07355
\(503\) 0.635388 0.0283305 0.0141653 0.999900i \(-0.495491\pi\)
0.0141653 + 0.999900i \(0.495491\pi\)
\(504\) −13.9159 −0.619863
\(505\) 12.1733 0.541706
\(506\) −4.50877 −0.200439
\(507\) 3.18764 0.141568
\(508\) −2.08772 −0.0926276
\(509\) 1.74035 0.0771399 0.0385699 0.999256i \(-0.487720\pi\)
0.0385699 + 0.999256i \(0.487720\pi\)
\(510\) 10.8179 0.479024
\(511\) 6.68499 0.295727
\(512\) 1.00000 0.0441942
\(513\) −19.3588 −0.854712
\(514\) −13.6822 −0.603495
\(515\) 5.22683 0.230322
\(516\) 3.69779 0.162786
\(517\) 11.5277 0.506987
\(518\) 5.84213 0.256688
\(519\) 8.73784 0.383548
\(520\) −9.00148 −0.394741
\(521\) 10.5683 0.463004 0.231502 0.972834i \(-0.425636\pi\)
0.231502 + 0.972834i \(0.425636\pi\)
\(522\) −6.00517 −0.262839
\(523\) 10.5329 0.460572 0.230286 0.973123i \(-0.426034\pi\)
0.230286 + 0.973123i \(0.426034\pi\)
\(524\) −5.17397 −0.226026
\(525\) −22.9510 −1.00166
\(526\) 11.3942 0.496810
\(527\) 24.5879 1.07106
\(528\) 0.625290 0.0272123
\(529\) −15.3401 −0.666961
\(530\) −12.6243 −0.548365
\(531\) −12.0334 −0.522203
\(532\) −42.0385 −1.82260
\(533\) −11.7813 −0.510305
\(534\) −1.27161 −0.0550281
\(535\) 38.8814 1.68099
\(536\) −1.50761 −0.0651189
\(537\) −4.66865 −0.201467
\(538\) −1.00000 −0.0431131
\(539\) −27.3634 −1.17862
\(540\) 9.33213 0.401591
\(541\) −0.0839881 −0.00361093 −0.00180546 0.999998i \(-0.500575\pi\)
−0.00180546 + 0.999998i \(0.500575\pi\)
\(542\) −24.6167 −1.05738
\(543\) 1.76963 0.0759419
\(544\) −6.78448 −0.290882
\(545\) 47.4382 2.03203
\(546\) 4.05710 0.173628
\(547\) 1.38614 0.0592669 0.0296334 0.999561i \(-0.490566\pi\)
0.0296334 + 0.999561i \(0.490566\pi\)
\(548\) 6.35949 0.271664
\(549\) −24.9536 −1.06499
\(550\) −19.9690 −0.851479
\(551\) −18.1410 −0.772833
\(552\) −1.06230 −0.0452144
\(553\) −73.0501 −3.10641
\(554\) 4.70762 0.200008
\(555\) −1.90959 −0.0810575
\(556\) −10.6786 −0.452873
\(557\) 17.1840 0.728111 0.364056 0.931377i \(-0.381392\pi\)
0.364056 + 0.931377i \(0.381392\pi\)
\(558\) 10.3385 0.437663
\(559\) 20.8752 0.882926
\(560\) 20.2651 0.856358
\(561\) −4.24227 −0.179109
\(562\) 6.51540 0.274836
\(563\) −12.1968 −0.514033 −0.257017 0.966407i \(-0.582740\pi\)
−0.257017 + 0.966407i \(0.582740\pi\)
\(564\) 2.71601 0.114364
\(565\) 1.31613 0.0553700
\(566\) −8.67417 −0.364602
\(567\) 37.5415 1.57660
\(568\) 11.0270 0.462682
\(569\) 23.4816 0.984401 0.492201 0.870482i \(-0.336193\pi\)
0.492201 + 0.870482i \(0.336193\pi\)
\(570\) 13.7409 0.575543
\(571\) −2.61940 −0.109619 −0.0548093 0.998497i \(-0.517455\pi\)
−0.0548093 + 0.998497i \(0.517455\pi\)
\(572\) 3.52996 0.147595
\(573\) −1.26875 −0.0530028
\(574\) 26.5234 1.10707
\(575\) 33.9250 1.41477
\(576\) −2.85268 −0.118862
\(577\) 12.1035 0.503874 0.251937 0.967744i \(-0.418932\pi\)
0.251937 + 0.967744i \(0.418932\pi\)
\(578\) 29.0291 1.20745
\(579\) 7.10681 0.295349
\(580\) 8.74508 0.363120
\(581\) −39.4292 −1.63580
\(582\) −6.46647 −0.268044
\(583\) 4.95066 0.205035
\(584\) 1.37039 0.0567070
\(585\) 25.6783 1.06167
\(586\) 20.3798 0.841881
\(587\) −3.09578 −0.127776 −0.0638882 0.997957i \(-0.520350\pi\)
−0.0638882 + 0.997957i \(0.520350\pi\)
\(588\) −6.44701 −0.265870
\(589\) 31.2316 1.28687
\(590\) 17.5237 0.721439
\(591\) −3.37042 −0.138640
\(592\) 1.19760 0.0492212
\(593\) −24.1730 −0.992666 −0.496333 0.868132i \(-0.665321\pi\)
−0.496333 + 0.868132i \(0.665321\pi\)
\(594\) −3.65962 −0.150156
\(595\) −137.488 −5.63647
\(596\) −3.47426 −0.142311
\(597\) 4.77364 0.195372
\(598\) −5.99701 −0.245236
\(599\) 39.0984 1.59752 0.798759 0.601651i \(-0.205490\pi\)
0.798759 + 0.601651i \(0.205490\pi\)
\(600\) −4.70482 −0.192074
\(601\) −29.5069 −1.20361 −0.601807 0.798642i \(-0.705552\pi\)
−0.601807 + 0.798642i \(0.705552\pi\)
\(602\) −46.9965 −1.91543
\(603\) 4.30073 0.175139
\(604\) 11.2407 0.457376
\(605\) −34.6714 −1.40959
\(606\) −1.12474 −0.0456895
\(607\) 17.2674 0.700861 0.350431 0.936589i \(-0.386035\pi\)
0.350431 + 0.936589i \(0.386035\pi\)
\(608\) −8.61766 −0.349492
\(609\) −3.94154 −0.159719
\(610\) 36.3389 1.47132
\(611\) 15.3327 0.620294
\(612\) 19.3539 0.782336
\(613\) −7.94254 −0.320796 −0.160398 0.987052i \(-0.551278\pi\)
−0.160398 + 0.987052i \(0.551278\pi\)
\(614\) −17.5028 −0.706357
\(615\) −8.66956 −0.349591
\(616\) −7.94703 −0.320195
\(617\) 18.0886 0.728218 0.364109 0.931356i \(-0.381374\pi\)
0.364109 + 0.931356i \(0.381374\pi\)
\(618\) −0.482928 −0.0194262
\(619\) −33.4949 −1.34627 −0.673136 0.739518i \(-0.735053\pi\)
−0.673136 + 0.739518i \(0.735053\pi\)
\(620\) −15.0555 −0.604644
\(621\) 6.21729 0.249491
\(622\) 22.4715 0.901024
\(623\) 16.1614 0.647491
\(624\) 0.831683 0.0332940
\(625\) 63.9624 2.55850
\(626\) −32.6321 −1.30424
\(627\) −5.38854 −0.215197
\(628\) 8.59533 0.342991
\(629\) −8.12512 −0.323970
\(630\) −57.8099 −2.30320
\(631\) 37.3162 1.48554 0.742768 0.669549i \(-0.233512\pi\)
0.742768 + 0.669549i \(0.233512\pi\)
\(632\) −14.9749 −0.595668
\(633\) 4.19483 0.166729
\(634\) −19.5532 −0.776559
\(635\) −8.67288 −0.344173
\(636\) 1.16641 0.0462512
\(637\) −36.3954 −1.44204
\(638\) −3.42941 −0.135772
\(639\) −31.4564 −1.24440
\(640\) 4.15424 0.164211
\(641\) −18.0393 −0.712509 −0.356254 0.934389i \(-0.615946\pi\)
−0.356254 + 0.934389i \(0.615946\pi\)
\(642\) −3.59241 −0.141781
\(643\) −33.0496 −1.30335 −0.651675 0.758498i \(-0.725933\pi\)
−0.651675 + 0.758498i \(0.725933\pi\)
\(644\) 13.5011 0.532019
\(645\) 15.3615 0.604859
\(646\) 58.4663 2.30033
\(647\) −40.3067 −1.58462 −0.792310 0.610118i \(-0.791122\pi\)
−0.792310 + 0.610118i \(0.791122\pi\)
\(648\) 7.69580 0.302320
\(649\) −6.87196 −0.269748
\(650\) −26.5602 −1.04178
\(651\) 6.78575 0.265954
\(652\) 10.1530 0.397621
\(653\) −21.3784 −0.836601 −0.418300 0.908309i \(-0.637374\pi\)
−0.418300 + 0.908309i \(0.637374\pi\)
\(654\) −4.38300 −0.171389
\(655\) −21.4939 −0.839836
\(656\) 5.43715 0.212285
\(657\) −3.90927 −0.152515
\(658\) −34.5186 −1.34568
\(659\) 23.1777 0.902875 0.451438 0.892303i \(-0.350911\pi\)
0.451438 + 0.892303i \(0.350911\pi\)
\(660\) 2.59760 0.101112
\(661\) −3.96791 −0.154334 −0.0771669 0.997018i \(-0.524587\pi\)
−0.0771669 + 0.997018i \(0.524587\pi\)
\(662\) −20.1680 −0.783850
\(663\) −5.64253 −0.219138
\(664\) −8.08276 −0.313672
\(665\) −174.638 −6.77217
\(666\) −3.41638 −0.132382
\(667\) 5.82618 0.225591
\(668\) 4.37572 0.169302
\(669\) −3.37526 −0.130495
\(670\) −6.26298 −0.241960
\(671\) −14.2504 −0.550131
\(672\) −1.87238 −0.0722285
\(673\) 32.9589 1.27047 0.635235 0.772319i \(-0.280903\pi\)
0.635235 + 0.772319i \(0.280903\pi\)
\(674\) 29.1409 1.12247
\(675\) 27.5358 1.05985
\(676\) −8.30489 −0.319419
\(677\) 17.1159 0.657816 0.328908 0.944362i \(-0.393319\pi\)
0.328908 + 0.944362i \(0.393319\pi\)
\(678\) −0.121603 −0.00467012
\(679\) 82.1846 3.15396
\(680\) −28.1843 −1.08082
\(681\) −2.59033 −0.0992618
\(682\) 5.90407 0.226078
\(683\) 33.2813 1.27347 0.636736 0.771082i \(-0.280284\pi\)
0.636736 + 0.771082i \(0.280284\pi\)
\(684\) 24.5834 0.939969
\(685\) 26.4188 1.00941
\(686\) 47.7899 1.82463
\(687\) 8.92804 0.340626
\(688\) −9.63402 −0.367294
\(689\) 6.58475 0.250859
\(690\) −4.41304 −0.168002
\(691\) −21.3236 −0.811189 −0.405594 0.914053i \(-0.632935\pi\)
−0.405594 + 0.914053i \(0.632935\pi\)
\(692\) −22.7651 −0.865398
\(693\) 22.6703 0.861174
\(694\) 12.2141 0.463640
\(695\) −44.3614 −1.68272
\(696\) −0.807993 −0.0306269
\(697\) −36.8882 −1.39724
\(698\) 15.3346 0.580423
\(699\) −4.24976 −0.160741
\(700\) 59.7952 2.26005
\(701\) −28.5333 −1.07769 −0.538844 0.842406i \(-0.681139\pi\)
−0.538844 + 0.842406i \(0.681139\pi\)
\(702\) −4.86757 −0.183715
\(703\) −10.3205 −0.389246
\(704\) −1.62910 −0.0613989
\(705\) 11.2829 0.424940
\(706\) 30.6859 1.15488
\(707\) 14.2947 0.537609
\(708\) −1.61908 −0.0608489
\(709\) 4.38294 0.164605 0.0823024 0.996607i \(-0.473773\pi\)
0.0823024 + 0.996607i \(0.473773\pi\)
\(710\) 45.8087 1.71917
\(711\) 42.7184 1.60207
\(712\) 3.31299 0.124159
\(713\) −10.0304 −0.375640
\(714\) 12.7031 0.475401
\(715\) 14.6643 0.548413
\(716\) 12.1634 0.454569
\(717\) 1.53174 0.0572040
\(718\) 0.344824 0.0128687
\(719\) −36.1805 −1.34930 −0.674652 0.738136i \(-0.735706\pi\)
−0.674652 + 0.738136i \(0.735706\pi\)
\(720\) −11.8507 −0.441649
\(721\) 6.13770 0.228580
\(722\) 55.2640 2.05671
\(723\) 6.34881 0.236115
\(724\) −4.61048 −0.171347
\(725\) 25.8036 0.958323
\(726\) 3.20343 0.118891
\(727\) −23.0348 −0.854312 −0.427156 0.904178i \(-0.640485\pi\)
−0.427156 + 0.904178i \(0.640485\pi\)
\(728\) −10.5701 −0.391756
\(729\) −19.3669 −0.717294
\(730\) 5.69291 0.210704
\(731\) 65.3618 2.41749
\(732\) −3.35750 −0.124097
\(733\) −41.3032 −1.52557 −0.762785 0.646652i \(-0.776169\pi\)
−0.762785 + 0.646652i \(0.776169\pi\)
\(734\) 22.3072 0.823375
\(735\) −26.7824 −0.987884
\(736\) 2.76765 0.102017
\(737\) 2.45605 0.0904696
\(738\) −15.5104 −0.570947
\(739\) −15.2014 −0.559191 −0.279596 0.960118i \(-0.590200\pi\)
−0.279596 + 0.960118i \(0.590200\pi\)
\(740\) 4.97513 0.182889
\(741\) −7.16716 −0.263292
\(742\) −14.8243 −0.544218
\(743\) 51.4130 1.88616 0.943079 0.332568i \(-0.107915\pi\)
0.943079 + 0.332568i \(0.107915\pi\)
\(744\) 1.39104 0.0509980
\(745\) −14.4329 −0.528780
\(746\) −5.92843 −0.217055
\(747\) 23.0575 0.843630
\(748\) 11.0526 0.404122
\(749\) 45.6572 1.66828
\(750\) −11.5724 −0.422565
\(751\) 37.2197 1.35816 0.679082 0.734062i \(-0.262378\pi\)
0.679082 + 0.734062i \(0.262378\pi\)
\(752\) −7.07613 −0.258040
\(753\) −9.23231 −0.336444
\(754\) −4.56137 −0.166115
\(755\) 46.6964 1.69945
\(756\) 10.9584 0.398554
\(757\) 51.7191 1.87976 0.939882 0.341499i \(-0.110935\pi\)
0.939882 + 0.341499i \(0.110935\pi\)
\(758\) −2.01119 −0.0730499
\(759\) 1.73059 0.0628163
\(760\) −35.7998 −1.29859
\(761\) 16.8448 0.610623 0.305312 0.952253i \(-0.401239\pi\)
0.305312 + 0.952253i \(0.401239\pi\)
\(762\) 0.801322 0.0290288
\(763\) 55.7051 2.01666
\(764\) 3.30553 0.119590
\(765\) 80.4008 2.90690
\(766\) 4.21696 0.152365
\(767\) −9.14022 −0.330034
\(768\) −0.383827 −0.0138501
\(769\) 10.2220 0.368614 0.184307 0.982869i \(-0.440996\pi\)
0.184307 + 0.982869i \(0.440996\pi\)
\(770\) −33.0138 −1.18974
\(771\) 5.25158 0.189131
\(772\) −18.5157 −0.666394
\(773\) −31.6879 −1.13973 −0.569867 0.821737i \(-0.693005\pi\)
−0.569867 + 0.821737i \(0.693005\pi\)
\(774\) 27.4827 0.987847
\(775\) −44.4235 −1.59574
\(776\) 16.8474 0.604786
\(777\) −2.24236 −0.0804444
\(778\) 4.54939 0.163103
\(779\) −46.8555 −1.67877
\(780\) 3.45501 0.123709
\(781\) −17.9640 −0.642803
\(782\) −18.7771 −0.671467
\(783\) 4.72892 0.168998
\(784\) 16.7967 0.599881
\(785\) 35.7070 1.27444
\(786\) 1.98591 0.0708350
\(787\) 21.1252 0.753033 0.376516 0.926410i \(-0.377122\pi\)
0.376516 + 0.926410i \(0.377122\pi\)
\(788\) 8.78109 0.312814
\(789\) −4.37339 −0.155697
\(790\) −62.2091 −2.21330
\(791\) 1.54549 0.0549512
\(792\) 4.64729 0.165134
\(793\) −18.9541 −0.673080
\(794\) −15.2162 −0.540001
\(795\) 4.84554 0.171854
\(796\) −12.4370 −0.440817
\(797\) −4.27630 −0.151474 −0.0757371 0.997128i \(-0.524131\pi\)
−0.0757371 + 0.997128i \(0.524131\pi\)
\(798\) 16.1355 0.571190
\(799\) 48.0078 1.69839
\(800\) 12.2577 0.433375
\(801\) −9.45089 −0.333931
\(802\) 18.4280 0.650715
\(803\) −2.23249 −0.0787829
\(804\) 0.578661 0.0204078
\(805\) 56.0868 1.97680
\(806\) 7.85285 0.276605
\(807\) 0.383827 0.0135113
\(808\) 2.93034 0.103089
\(809\) 4.78386 0.168192 0.0840958 0.996458i \(-0.473200\pi\)
0.0840958 + 0.996458i \(0.473200\pi\)
\(810\) 31.9702 1.12332
\(811\) −34.4127 −1.20839 −0.604197 0.796835i \(-0.706506\pi\)
−0.604197 + 0.796835i \(0.706506\pi\)
\(812\) 10.2691 0.360373
\(813\) 9.44854 0.331375
\(814\) −1.95101 −0.0683829
\(815\) 42.1778 1.47742
\(816\) 2.60406 0.0911604
\(817\) 83.0227 2.90460
\(818\) 36.3293 1.27022
\(819\) 30.1532 1.05364
\(820\) 22.5872 0.788779
\(821\) −38.7973 −1.35404 −0.677018 0.735967i \(-0.736728\pi\)
−0.677018 + 0.735967i \(0.736728\pi\)
\(822\) −2.44094 −0.0851375
\(823\) −8.52567 −0.297186 −0.148593 0.988898i \(-0.547474\pi\)
−0.148593 + 0.988898i \(0.547474\pi\)
\(824\) 1.25819 0.0438312
\(825\) 7.66461 0.266848
\(826\) 20.5775 0.715982
\(827\) 3.84932 0.133854 0.0669271 0.997758i \(-0.478681\pi\)
0.0669271 + 0.997758i \(0.478681\pi\)
\(828\) −7.89522 −0.274378
\(829\) 1.76719 0.0613771 0.0306886 0.999529i \(-0.490230\pi\)
0.0306886 + 0.999529i \(0.490230\pi\)
\(830\) −33.5777 −1.16550
\(831\) −1.80691 −0.0626810
\(832\) −2.16682 −0.0751210
\(833\) −113.957 −3.94836
\(834\) 4.09872 0.141927
\(835\) 18.1778 0.629068
\(836\) 14.0390 0.485549
\(837\) −8.14131 −0.281405
\(838\) −6.63242 −0.229113
\(839\) 48.4527 1.67277 0.836386 0.548141i \(-0.184664\pi\)
0.836386 + 0.548141i \(0.184664\pi\)
\(840\) −7.77829 −0.268377
\(841\) −24.5686 −0.847192
\(842\) −33.1090 −1.14101
\(843\) −2.50078 −0.0861315
\(844\) −10.9290 −0.376191
\(845\) −34.5005 −1.18685
\(846\) 20.1859 0.694006
\(847\) −40.7135 −1.39893
\(848\) −3.03890 −0.104356
\(849\) 3.32938 0.114264
\(850\) −83.1620 −2.85243
\(851\) 3.31455 0.113621
\(852\) −4.23245 −0.145001
\(853\) 35.2024 1.20531 0.602654 0.798002i \(-0.294110\pi\)
0.602654 + 0.798002i \(0.294110\pi\)
\(854\) 42.6716 1.46019
\(855\) 102.125 3.49261
\(856\) 9.35946 0.319900
\(857\) 8.30685 0.283757 0.141878 0.989884i \(-0.454686\pi\)
0.141878 + 0.989884i \(0.454686\pi\)
\(858\) −1.35489 −0.0462552
\(859\) 48.5886 1.65782 0.828911 0.559381i \(-0.188961\pi\)
0.828911 + 0.559381i \(0.188961\pi\)
\(860\) −40.0220 −1.36474
\(861\) −10.1804 −0.346947
\(862\) −16.6498 −0.567096
\(863\) 50.0516 1.70378 0.851888 0.523723i \(-0.175457\pi\)
0.851888 + 0.523723i \(0.175457\pi\)
\(864\) 2.24641 0.0764245
\(865\) −94.5715 −3.21553
\(866\) 5.36131 0.182185
\(867\) −11.1421 −0.378407
\(868\) −17.6792 −0.600071
\(869\) 24.3955 0.827560
\(870\) −3.35659 −0.113799
\(871\) 3.26672 0.110689
\(872\) 11.4192 0.386704
\(873\) −48.0601 −1.62659
\(874\) −23.8507 −0.806761
\(875\) 147.078 4.97214
\(876\) −0.525990 −0.0177716
\(877\) −35.1921 −1.18835 −0.594177 0.804334i \(-0.702522\pi\)
−0.594177 + 0.804334i \(0.702522\pi\)
\(878\) −5.42961 −0.183240
\(879\) −7.82230 −0.263840
\(880\) −6.76765 −0.228138
\(881\) −29.4895 −0.993527 −0.496764 0.867886i \(-0.665478\pi\)
−0.496764 + 0.867886i \(0.665478\pi\)
\(882\) −47.9155 −1.61340
\(883\) −28.1353 −0.946829 −0.473415 0.880840i \(-0.656979\pi\)
−0.473415 + 0.880840i \(0.656979\pi\)
\(884\) 14.7007 0.494439
\(885\) −6.72605 −0.226094
\(886\) 3.66946 0.123278
\(887\) −12.3917 −0.416072 −0.208036 0.978121i \(-0.566707\pi\)
−0.208036 + 0.978121i \(0.566707\pi\)
\(888\) −0.459672 −0.0154256
\(889\) −10.1843 −0.341570
\(890\) 13.7629 0.461335
\(891\) −12.5372 −0.420012
\(892\) 8.79372 0.294436
\(893\) 60.9796 2.04061
\(894\) 1.33351 0.0445993
\(895\) 50.5298 1.68903
\(896\) 4.87818 0.162969
\(897\) 2.30181 0.0768552
\(898\) 6.43513 0.214743
\(899\) −7.62916 −0.254447
\(900\) −34.9672 −1.16557
\(901\) 20.6173 0.686863
\(902\) −8.85764 −0.294927
\(903\) 18.0385 0.600284
\(904\) 0.316816 0.0105372
\(905\) −19.1530 −0.636669
\(906\) −4.31446 −0.143338
\(907\) −14.9432 −0.496181 −0.248090 0.968737i \(-0.579803\pi\)
−0.248090 + 0.968737i \(0.579803\pi\)
\(908\) 6.74871 0.223964
\(909\) −8.35931 −0.277261
\(910\) −43.9109 −1.45563
\(911\) 29.3458 0.972269 0.486134 0.873884i \(-0.338407\pi\)
0.486134 + 0.873884i \(0.338407\pi\)
\(912\) 3.30769 0.109528
\(913\) 13.1676 0.435784
\(914\) 2.97606 0.0984393
\(915\) −13.9478 −0.461101
\(916\) −23.2606 −0.768553
\(917\) −25.2396 −0.833485
\(918\) −15.2407 −0.503019
\(919\) 24.7551 0.816595 0.408297 0.912849i \(-0.366123\pi\)
0.408297 + 0.912849i \(0.366123\pi\)
\(920\) 11.4975 0.379061
\(921\) 6.71805 0.221367
\(922\) 9.73958 0.320756
\(923\) −23.8935 −0.786464
\(924\) 3.05028 0.100347
\(925\) 14.6799 0.482671
\(926\) −19.9810 −0.656618
\(927\) −3.58922 −0.117885
\(928\) 2.10510 0.0691032
\(929\) −27.1303 −0.890117 −0.445059 0.895501i \(-0.646817\pi\)
−0.445059 + 0.895501i \(0.646817\pi\)
\(930\) 5.77871 0.189491
\(931\) −144.748 −4.74392
\(932\) 11.0721 0.362679
\(933\) −8.62515 −0.282375
\(934\) −14.3949 −0.471014
\(935\) 45.9150 1.50158
\(936\) 6.18124 0.202040
\(937\) 44.2931 1.44699 0.723496 0.690328i \(-0.242534\pi\)
0.723496 + 0.690328i \(0.242534\pi\)
\(938\) −7.35441 −0.240130
\(939\) 12.5250 0.408740
\(940\) −29.3959 −0.958789
\(941\) −27.0894 −0.883090 −0.441545 0.897239i \(-0.645569\pi\)
−0.441545 + 0.897239i \(0.645569\pi\)
\(942\) −3.29912 −0.107491
\(943\) 15.0481 0.490035
\(944\) 4.21827 0.137293
\(945\) 45.5238 1.48089
\(946\) 15.6947 0.510280
\(947\) 44.9196 1.45969 0.729845 0.683613i \(-0.239592\pi\)
0.729845 + 0.683613i \(0.239592\pi\)
\(948\) 5.74775 0.186678
\(949\) −2.96938 −0.0963901
\(950\) −105.633 −3.42717
\(951\) 7.50505 0.243368
\(952\) −33.0959 −1.07264
\(953\) −21.0858 −0.683036 −0.341518 0.939875i \(-0.610941\pi\)
−0.341518 + 0.939875i \(0.610941\pi\)
\(954\) 8.66900 0.280669
\(955\) 13.7320 0.444356
\(956\) −3.99072 −0.129069
\(957\) 1.31630 0.0425499
\(958\) −22.4679 −0.725906
\(959\) 31.0227 1.00178
\(960\) −1.59451 −0.0514625
\(961\) −17.8656 −0.576311
\(962\) −2.59499 −0.0836659
\(963\) −26.6995 −0.860380
\(964\) −16.5408 −0.532745
\(965\) −76.9185 −2.47609
\(966\) −5.18209 −0.166731
\(967\) 25.2415 0.811711 0.405855 0.913937i \(-0.366974\pi\)
0.405855 + 0.913937i \(0.366974\pi\)
\(968\) −8.34604 −0.268252
\(969\) −22.4409 −0.720906
\(970\) 69.9880 2.24718
\(971\) −34.5382 −1.10838 −0.554192 0.832389i \(-0.686973\pi\)
−0.554192 + 0.832389i \(0.686973\pi\)
\(972\) −9.69309 −0.310906
\(973\) −52.0921 −1.67000
\(974\) −15.3289 −0.491169
\(975\) 10.1945 0.326486
\(976\) 8.74744 0.279999
\(977\) 61.0213 1.95224 0.976122 0.217225i \(-0.0697004\pi\)
0.976122 + 0.217225i \(0.0697004\pi\)
\(978\) −3.89698 −0.124612
\(979\) −5.39718 −0.172495
\(980\) 69.7773 2.22896
\(981\) −32.5754 −1.04005
\(982\) −23.8945 −0.762504
\(983\) −47.5260 −1.51585 −0.757923 0.652345i \(-0.773786\pi\)
−0.757923 + 0.652345i \(0.773786\pi\)
\(984\) −2.08692 −0.0665286
\(985\) 36.4787 1.16231
\(986\) −14.2820 −0.454831
\(987\) 13.2492 0.421726
\(988\) 18.6729 0.594064
\(989\) −26.6636 −0.847854
\(990\) 19.3059 0.613583
\(991\) −35.6855 −1.13359 −0.566794 0.823860i \(-0.691816\pi\)
−0.566794 + 0.823860i \(0.691816\pi\)
\(992\) −3.62414 −0.115066
\(993\) 7.74100 0.245653
\(994\) 53.7917 1.70617
\(995\) −51.6662 −1.63793
\(996\) 3.10238 0.0983026
\(997\) 12.2019 0.386438 0.193219 0.981156i \(-0.438107\pi\)
0.193219 + 0.981156i \(0.438107\pi\)
\(998\) −15.7627 −0.498960
\(999\) 2.69031 0.0851177
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 538.2.a.e.1.3 7
3.2 odd 2 4842.2.a.n.1.1 7
4.3 odd 2 4304.2.a.h.1.5 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.e.1.3 7 1.1 even 1 trivial
4304.2.a.h.1.5 7 4.3 odd 2
4842.2.a.n.1.1 7 3.2 odd 2