Properties

Label 1339.2.a.d.1.2
Level $1339$
Weight $2$
Character 1339.1
Self dual yes
Analytic conductor $10.692$
Analytic rank $1$
Dimension $19$
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] = [1339,2,Mod(1,1339)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1339, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([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("1339.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1339 = 13 \cdot 103 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1339.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: \(10.6919688306\)
Analytic rank: \(1\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 9 x^{18} + 13 x^{17} + 106 x^{16} - 351 x^{15} - 279 x^{14} + 2337 x^{13} - 1079 x^{12} - 6826 x^{11} + 7199 x^{10} + 9364 x^{9} - 14841 x^{8} - 5183 x^{7} + 14037 x^{6} + \cdots - 63 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.64132\) of defining polynomial
Character \(\chi\) \(=\) 1339.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.64132 q^{2} -0.762312 q^{3} +4.97659 q^{4} +1.72196 q^{5} +2.01351 q^{6} +1.20730 q^{7} -7.86215 q^{8} -2.41888 q^{9} +O(q^{10})\) \(q-2.64132 q^{2} -0.762312 q^{3} +4.97659 q^{4} +1.72196 q^{5} +2.01351 q^{6} +1.20730 q^{7} -7.86215 q^{8} -2.41888 q^{9} -4.54826 q^{10} +3.89418 q^{11} -3.79372 q^{12} +1.00000 q^{13} -3.18888 q^{14} -1.31267 q^{15} +10.8133 q^{16} -0.0584553 q^{17} +6.38905 q^{18} -5.98373 q^{19} +8.56950 q^{20} -0.920343 q^{21} -10.2858 q^{22} -5.61188 q^{23} +5.99341 q^{24} -2.03485 q^{25} -2.64132 q^{26} +4.13088 q^{27} +6.00826 q^{28} -8.70027 q^{29} +3.46719 q^{30} -4.16435 q^{31} -12.8371 q^{32} -2.96858 q^{33} +0.154399 q^{34} +2.07893 q^{35} -12.0378 q^{36} +7.87910 q^{37} +15.8050 q^{38} -0.762312 q^{39} -13.5383 q^{40} -12.3647 q^{41} +2.43092 q^{42} +1.92562 q^{43} +19.3798 q^{44} -4.16522 q^{45} +14.8228 q^{46} -8.52454 q^{47} -8.24311 q^{48} -5.54242 q^{49} +5.37470 q^{50} +0.0445612 q^{51} +4.97659 q^{52} +4.15194 q^{53} -10.9110 q^{54} +6.70563 q^{55} -9.49201 q^{56} +4.56147 q^{57} +22.9802 q^{58} -8.39711 q^{59} -6.53264 q^{60} +14.5547 q^{61} +10.9994 q^{62} -2.92032 q^{63} +12.2804 q^{64} +1.72196 q^{65} +7.84099 q^{66} +2.97482 q^{67} -0.290908 q^{68} +4.27801 q^{69} -5.49113 q^{70} +0.367486 q^{71} +19.0176 q^{72} +11.0083 q^{73} -20.8113 q^{74} +1.55119 q^{75} -29.7786 q^{76} +4.70146 q^{77} +2.01351 q^{78} +12.5782 q^{79} +18.6201 q^{80} +4.10762 q^{81} +32.6591 q^{82} +7.81940 q^{83} -4.58017 q^{84} -0.100658 q^{85} -5.08619 q^{86} +6.63233 q^{87} -30.6166 q^{88} -18.0980 q^{89} +11.0017 q^{90} +1.20730 q^{91} -27.9281 q^{92} +3.17454 q^{93} +22.5161 q^{94} -10.3037 q^{95} +9.78590 q^{96} -16.9843 q^{97} +14.6393 q^{98} -9.41956 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q - 9 q^{2} - 2 q^{3} + 17 q^{4} - 18 q^{5} - 8 q^{6} - 8 q^{7} - 24 q^{8} + 7 q^{9} - q^{11} + 18 q^{12} + 19 q^{13} - 8 q^{14} - 11 q^{15} + 21 q^{16} - 16 q^{17} - 10 q^{19} - 20 q^{20} - 33 q^{21} + 2 q^{22} - 14 q^{23} - 9 q^{24} + 23 q^{25} - 9 q^{26} + q^{27} + 10 q^{28} - 22 q^{29} + 28 q^{30} - 3 q^{31} - 47 q^{32} - 25 q^{33} - 35 q^{34} - 3 q^{35} - 33 q^{36} - 19 q^{37} - 15 q^{38} - 2 q^{39} + 8 q^{40} - 52 q^{41} - 3 q^{42} - 2 q^{43} - 54 q^{44} - 40 q^{45} + 33 q^{46} - 24 q^{47} - 8 q^{48} + 7 q^{49} - 10 q^{50} - 7 q^{51} + 17 q^{52} - 11 q^{53} - 23 q^{54} - 29 q^{55} - 17 q^{56} - 34 q^{57} + 18 q^{58} - 52 q^{59} - 71 q^{60} - 25 q^{61} - 22 q^{62} - 19 q^{63} + 48 q^{64} - 18 q^{65} + 9 q^{66} - 2 q^{67} + 18 q^{68} - 26 q^{69} - 12 q^{70} - 44 q^{71} - 6 q^{72} - 39 q^{73} - 4 q^{74} + 17 q^{75} - 10 q^{76} - 18 q^{77} - 8 q^{78} + q^{79} - 9 q^{80} - 13 q^{81} + 47 q^{82} - 27 q^{83} - 24 q^{84} - 26 q^{85} - q^{86} + 31 q^{87} + 19 q^{88} - 86 q^{89} + 48 q^{90} - 8 q^{91} - 19 q^{92} + 32 q^{93} + 45 q^{94} + 17 q^{95} - 25 q^{96} - 20 q^{97} + 39 q^{98} + 50 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.64132 −1.86770 −0.933849 0.357667i \(-0.883572\pi\)
−0.933849 + 0.357667i \(0.883572\pi\)
\(3\) −0.762312 −0.440121 −0.220061 0.975486i \(-0.570626\pi\)
−0.220061 + 0.975486i \(0.570626\pi\)
\(4\) 4.97659 2.48830
\(5\) 1.72196 0.770084 0.385042 0.922899i \(-0.374187\pi\)
0.385042 + 0.922899i \(0.374187\pi\)
\(6\) 2.01351 0.822014
\(7\) 1.20730 0.456318 0.228159 0.973624i \(-0.426729\pi\)
0.228159 + 0.973624i \(0.426729\pi\)
\(8\) −7.86215 −2.77969
\(9\) −2.41888 −0.806293
\(10\) −4.54826 −1.43829
\(11\) 3.89418 1.17414 0.587070 0.809536i \(-0.300281\pi\)
0.587070 + 0.809536i \(0.300281\pi\)
\(12\) −3.79372 −1.09515
\(13\) 1.00000 0.277350
\(14\) −3.18888 −0.852265
\(15\) −1.31267 −0.338931
\(16\) 10.8133 2.70332
\(17\) −0.0584553 −0.0141775 −0.00708874 0.999975i \(-0.502256\pi\)
−0.00708874 + 0.999975i \(0.502256\pi\)
\(18\) 6.38905 1.50591
\(19\) −5.98373 −1.37276 −0.686380 0.727243i \(-0.740801\pi\)
−0.686380 + 0.727243i \(0.740801\pi\)
\(20\) 8.56950 1.91620
\(21\) −0.920343 −0.200835
\(22\) −10.2858 −2.19294
\(23\) −5.61188 −1.17016 −0.585079 0.810976i \(-0.698937\pi\)
−0.585079 + 0.810976i \(0.698937\pi\)
\(24\) 5.99341 1.22340
\(25\) −2.03485 −0.406970
\(26\) −2.64132 −0.518006
\(27\) 4.13088 0.794988
\(28\) 6.00826 1.13546
\(29\) −8.70027 −1.61560 −0.807800 0.589456i \(-0.799342\pi\)
−0.807800 + 0.589456i \(0.799342\pi\)
\(30\) 3.46719 0.633020
\(31\) −4.16435 −0.747940 −0.373970 0.927441i \(-0.622004\pi\)
−0.373970 + 0.927441i \(0.622004\pi\)
\(32\) −12.8371 −2.26931
\(33\) −2.96858 −0.516764
\(34\) 0.154399 0.0264793
\(35\) 2.07893 0.351403
\(36\) −12.0378 −2.00630
\(37\) 7.87910 1.29532 0.647658 0.761931i \(-0.275749\pi\)
0.647658 + 0.761931i \(0.275749\pi\)
\(38\) 15.8050 2.56390
\(39\) −0.762312 −0.122068
\(40\) −13.5383 −2.14060
\(41\) −12.3647 −1.93104 −0.965518 0.260337i \(-0.916166\pi\)
−0.965518 + 0.260337i \(0.916166\pi\)
\(42\) 2.43092 0.375100
\(43\) 1.92562 0.293655 0.146827 0.989162i \(-0.453094\pi\)
0.146827 + 0.989162i \(0.453094\pi\)
\(44\) 19.3798 2.92161
\(45\) −4.16522 −0.620914
\(46\) 14.8228 2.18550
\(47\) −8.52454 −1.24343 −0.621716 0.783243i \(-0.713564\pi\)
−0.621716 + 0.783243i \(0.713564\pi\)
\(48\) −8.24311 −1.18979
\(49\) −5.54242 −0.791774
\(50\) 5.37470 0.760097
\(51\) 0.0445612 0.00623981
\(52\) 4.97659 0.690129
\(53\) 4.15194 0.570313 0.285156 0.958481i \(-0.407954\pi\)
0.285156 + 0.958481i \(0.407954\pi\)
\(54\) −10.9110 −1.48480
\(55\) 6.70563 0.904187
\(56\) −9.49201 −1.26842
\(57\) 4.56147 0.604181
\(58\) 22.9802 3.01745
\(59\) −8.39711 −1.09321 −0.546605 0.837390i \(-0.684080\pi\)
−0.546605 + 0.837390i \(0.684080\pi\)
\(60\) −6.53264 −0.843360
\(61\) 14.5547 1.86354 0.931772 0.363045i \(-0.118263\pi\)
0.931772 + 0.363045i \(0.118263\pi\)
\(62\) 10.9994 1.39693
\(63\) −2.92032 −0.367926
\(64\) 12.2804 1.53505
\(65\) 1.72196 0.213583
\(66\) 7.84099 0.965159
\(67\) 2.97482 0.363432 0.181716 0.983351i \(-0.441835\pi\)
0.181716 + 0.983351i \(0.441835\pi\)
\(68\) −0.290908 −0.0352778
\(69\) 4.27801 0.515012
\(70\) −5.49113 −0.656316
\(71\) 0.367486 0.0436126 0.0218063 0.999762i \(-0.493058\pi\)
0.0218063 + 0.999762i \(0.493058\pi\)
\(72\) 19.0176 2.24125
\(73\) 11.0083 1.28842 0.644211 0.764848i \(-0.277186\pi\)
0.644211 + 0.764848i \(0.277186\pi\)
\(74\) −20.8113 −2.41926
\(75\) 1.55119 0.179116
\(76\) −29.7786 −3.41584
\(77\) 4.70146 0.535781
\(78\) 2.01351 0.227986
\(79\) 12.5782 1.41516 0.707580 0.706633i \(-0.249787\pi\)
0.707580 + 0.706633i \(0.249787\pi\)
\(80\) 18.6201 2.08179
\(81\) 4.10762 0.456402
\(82\) 32.6591 3.60659
\(83\) 7.81940 0.858291 0.429145 0.903235i \(-0.358815\pi\)
0.429145 + 0.903235i \(0.358815\pi\)
\(84\) −4.58017 −0.499738
\(85\) −0.100658 −0.0109179
\(86\) −5.08619 −0.548458
\(87\) 6.63233 0.711060
\(88\) −30.6166 −3.26374
\(89\) −18.0980 −1.91838 −0.959191 0.282758i \(-0.908751\pi\)
−0.959191 + 0.282758i \(0.908751\pi\)
\(90\) 11.0017 1.15968
\(91\) 1.20730 0.126560
\(92\) −27.9281 −2.91170
\(93\) 3.17454 0.329184
\(94\) 22.5161 2.32236
\(95\) −10.3037 −1.05714
\(96\) 9.78590 0.998769
\(97\) −16.9843 −1.72450 −0.862249 0.506485i \(-0.830945\pi\)
−0.862249 + 0.506485i \(0.830945\pi\)
\(98\) 14.6393 1.47879
\(99\) −9.41956 −0.946701
\(100\) −10.1266 −1.01266
\(101\) −11.6080 −1.15504 −0.577520 0.816377i \(-0.695979\pi\)
−0.577520 + 0.816377i \(0.695979\pi\)
\(102\) −0.117701 −0.0116541
\(103\) 1.00000 0.0985329
\(104\) −7.86215 −0.770947
\(105\) −1.58479 −0.154660
\(106\) −10.9666 −1.06517
\(107\) −4.07642 −0.394083 −0.197041 0.980395i \(-0.563133\pi\)
−0.197041 + 0.980395i \(0.563133\pi\)
\(108\) 20.5577 1.97817
\(109\) −6.28688 −0.602174 −0.301087 0.953597i \(-0.597349\pi\)
−0.301087 + 0.953597i \(0.597349\pi\)
\(110\) −17.7117 −1.68875
\(111\) −6.00634 −0.570096
\(112\) 13.0549 1.23358
\(113\) 18.1047 1.70315 0.851573 0.524237i \(-0.175649\pi\)
0.851573 + 0.524237i \(0.175649\pi\)
\(114\) −12.0483 −1.12843
\(115\) −9.66345 −0.901121
\(116\) −43.2977 −4.02009
\(117\) −2.41888 −0.223626
\(118\) 22.1795 2.04179
\(119\) −0.0705733 −0.00646944
\(120\) 10.3204 0.942122
\(121\) 4.16465 0.378605
\(122\) −38.4438 −3.48054
\(123\) 9.42574 0.849890
\(124\) −20.7243 −1.86110
\(125\) −12.1137 −1.08349
\(126\) 7.71352 0.687175
\(127\) −14.7488 −1.30874 −0.654371 0.756174i \(-0.727067\pi\)
−0.654371 + 0.756174i \(0.727067\pi\)
\(128\) −6.76232 −0.597710
\(129\) −1.46793 −0.129244
\(130\) −4.54826 −0.398909
\(131\) −6.88906 −0.601900 −0.300950 0.953640i \(-0.597304\pi\)
−0.300950 + 0.953640i \(0.597304\pi\)
\(132\) −14.7734 −1.28586
\(133\) −7.22418 −0.626416
\(134\) −7.85747 −0.678782
\(135\) 7.11321 0.612208
\(136\) 0.459584 0.0394090
\(137\) −20.3110 −1.73529 −0.867644 0.497186i \(-0.834367\pi\)
−0.867644 + 0.497186i \(0.834367\pi\)
\(138\) −11.2996 −0.961887
\(139\) 6.11416 0.518596 0.259298 0.965797i \(-0.416509\pi\)
0.259298 + 0.965797i \(0.416509\pi\)
\(140\) 10.3460 0.874396
\(141\) 6.49836 0.547261
\(142\) −0.970651 −0.0814552
\(143\) 3.89418 0.325648
\(144\) −26.1561 −2.17967
\(145\) −14.9815 −1.24415
\(146\) −29.0764 −2.40638
\(147\) 4.22505 0.348476
\(148\) 39.2111 3.22313
\(149\) 13.9439 1.14233 0.571164 0.820836i \(-0.306492\pi\)
0.571164 + 0.820836i \(0.306492\pi\)
\(150\) −4.09720 −0.334535
\(151\) −8.10283 −0.659399 −0.329699 0.944086i \(-0.606947\pi\)
−0.329699 + 0.944086i \(0.606947\pi\)
\(152\) 47.0449 3.81585
\(153\) 0.141396 0.0114312
\(154\) −12.4181 −1.00068
\(155\) −7.17085 −0.575977
\(156\) −3.79372 −0.303741
\(157\) −4.61492 −0.368310 −0.184155 0.982897i \(-0.558955\pi\)
−0.184155 + 0.982897i \(0.558955\pi\)
\(158\) −33.2232 −2.64309
\(159\) −3.16508 −0.251007
\(160\) −22.1050 −1.74756
\(161\) −6.77525 −0.533965
\(162\) −10.8496 −0.852422
\(163\) 11.4170 0.894247 0.447124 0.894472i \(-0.352448\pi\)
0.447124 + 0.894472i \(0.352448\pi\)
\(164\) −61.5339 −4.80499
\(165\) −5.11178 −0.397952
\(166\) −20.6536 −1.60303
\(167\) −8.40805 −0.650635 −0.325317 0.945605i \(-0.605471\pi\)
−0.325317 + 0.945605i \(0.605471\pi\)
\(168\) 7.23587 0.558260
\(169\) 1.00000 0.0769231
\(170\) 0.265870 0.0203913
\(171\) 14.4739 1.10685
\(172\) 9.58304 0.730700
\(173\) −3.78023 −0.287405 −0.143703 0.989621i \(-0.545901\pi\)
−0.143703 + 0.989621i \(0.545901\pi\)
\(174\) −17.5181 −1.32805
\(175\) −2.45668 −0.185708
\(176\) 42.1090 3.17408
\(177\) 6.40122 0.481145
\(178\) 47.8026 3.58296
\(179\) 15.4398 1.15402 0.577011 0.816736i \(-0.304219\pi\)
0.577011 + 0.816736i \(0.304219\pi\)
\(180\) −20.7286 −1.54502
\(181\) −1.64581 −0.122332 −0.0611659 0.998128i \(-0.519482\pi\)
−0.0611659 + 0.998128i \(0.519482\pi\)
\(182\) −3.18888 −0.236376
\(183\) −11.0953 −0.820185
\(184\) 44.1215 3.25268
\(185\) 13.5675 0.997503
\(186\) −8.38498 −0.614817
\(187\) −0.227635 −0.0166464
\(188\) −42.4232 −3.09403
\(189\) 4.98723 0.362767
\(190\) 27.2155 1.97442
\(191\) 26.5533 1.92133 0.960665 0.277710i \(-0.0895756\pi\)
0.960665 + 0.277710i \(0.0895756\pi\)
\(192\) −9.36152 −0.675609
\(193\) −15.8380 −1.14004 −0.570021 0.821630i \(-0.693065\pi\)
−0.570021 + 0.821630i \(0.693065\pi\)
\(194\) 44.8611 3.22084
\(195\) −1.31267 −0.0940024
\(196\) −27.5824 −1.97017
\(197\) −0.455598 −0.0324600 −0.0162300 0.999868i \(-0.505166\pi\)
−0.0162300 + 0.999868i \(0.505166\pi\)
\(198\) 24.8801 1.76815
\(199\) −16.2559 −1.15235 −0.576174 0.817327i \(-0.695455\pi\)
−0.576174 + 0.817327i \(0.695455\pi\)
\(200\) 15.9983 1.13125
\(201\) −2.26774 −0.159954
\(202\) 30.6605 2.15727
\(203\) −10.5039 −0.737228
\(204\) 0.221763 0.0155265
\(205\) −21.2915 −1.48706
\(206\) −2.64132 −0.184030
\(207\) 13.5745 0.943491
\(208\) 10.8133 0.749767
\(209\) −23.3017 −1.61181
\(210\) 4.18596 0.288858
\(211\) −8.05535 −0.554553 −0.277277 0.960790i \(-0.589432\pi\)
−0.277277 + 0.960790i \(0.589432\pi\)
\(212\) 20.6625 1.41911
\(213\) −0.280139 −0.0191948
\(214\) 10.7672 0.736027
\(215\) 3.31585 0.226139
\(216\) −32.4776 −2.20982
\(217\) −5.02764 −0.341299
\(218\) 16.6057 1.12468
\(219\) −8.39174 −0.567061
\(220\) 33.3712 2.24989
\(221\) −0.0584553 −0.00393213
\(222\) 15.8647 1.06477
\(223\) −24.3618 −1.63139 −0.815695 0.578482i \(-0.803645\pi\)
−0.815695 + 0.578482i \(0.803645\pi\)
\(224\) −15.4983 −1.03553
\(225\) 4.92206 0.328137
\(226\) −47.8203 −3.18096
\(227\) 20.6110 1.36800 0.684000 0.729482i \(-0.260239\pi\)
0.684000 + 0.729482i \(0.260239\pi\)
\(228\) 22.7006 1.50338
\(229\) 12.5177 0.827192 0.413596 0.910460i \(-0.364273\pi\)
0.413596 + 0.910460i \(0.364273\pi\)
\(230\) 25.5243 1.68302
\(231\) −3.58398 −0.235809
\(232\) 68.4029 4.49087
\(233\) −20.3300 −1.33186 −0.665930 0.746014i \(-0.731965\pi\)
−0.665930 + 0.746014i \(0.731965\pi\)
\(234\) 6.38905 0.417665
\(235\) −14.6789 −0.957548
\(236\) −41.7890 −2.72023
\(237\) −9.58853 −0.622842
\(238\) 0.186407 0.0120830
\(239\) −21.3704 −1.38233 −0.691167 0.722695i \(-0.742903\pi\)
−0.691167 + 0.722695i \(0.742903\pi\)
\(240\) −14.1943 −0.916239
\(241\) 9.51199 0.612721 0.306361 0.951916i \(-0.400889\pi\)
0.306361 + 0.951916i \(0.400889\pi\)
\(242\) −11.0002 −0.707120
\(243\) −15.5239 −0.995860
\(244\) 72.4330 4.63705
\(245\) −9.54382 −0.609733
\(246\) −24.8964 −1.58734
\(247\) −5.98373 −0.380735
\(248\) 32.7408 2.07904
\(249\) −5.96082 −0.377752
\(250\) 31.9963 2.02362
\(251\) −4.46858 −0.282054 −0.141027 0.990006i \(-0.545041\pi\)
−0.141027 + 0.990006i \(0.545041\pi\)
\(252\) −14.5333 −0.915510
\(253\) −21.8537 −1.37393
\(254\) 38.9563 2.44434
\(255\) 0.0767326 0.00480518
\(256\) −6.69936 −0.418710
\(257\) 0.331613 0.0206854 0.0103427 0.999947i \(-0.496708\pi\)
0.0103427 + 0.999947i \(0.496708\pi\)
\(258\) 3.87727 0.241388
\(259\) 9.51247 0.591076
\(260\) 8.56950 0.531458
\(261\) 21.0449 1.30265
\(262\) 18.1962 1.12417
\(263\) 26.9020 1.65885 0.829424 0.558620i \(-0.188669\pi\)
0.829424 + 0.558620i \(0.188669\pi\)
\(264\) 23.3394 1.43644
\(265\) 7.14948 0.439189
\(266\) 19.0814 1.16996
\(267\) 13.7963 0.844321
\(268\) 14.8045 0.904327
\(269\) −10.4968 −0.640004 −0.320002 0.947417i \(-0.603684\pi\)
−0.320002 + 0.947417i \(0.603684\pi\)
\(270\) −18.7883 −1.14342
\(271\) 7.87413 0.478319 0.239160 0.970980i \(-0.423128\pi\)
0.239160 + 0.970980i \(0.423128\pi\)
\(272\) −0.632094 −0.0383263
\(273\) −0.920343 −0.0557017
\(274\) 53.6480 3.24100
\(275\) −7.92408 −0.477840
\(276\) 21.2899 1.28150
\(277\) −2.14339 −0.128784 −0.0643919 0.997925i \(-0.520511\pi\)
−0.0643919 + 0.997925i \(0.520511\pi\)
\(278\) −16.1495 −0.968582
\(279\) 10.0731 0.603059
\(280\) −16.3449 −0.976793
\(281\) −2.84828 −0.169914 −0.0849572 0.996385i \(-0.527075\pi\)
−0.0849572 + 0.996385i \(0.527075\pi\)
\(282\) −17.1643 −1.02212
\(283\) 13.9600 0.829838 0.414919 0.909858i \(-0.363810\pi\)
0.414919 + 0.909858i \(0.363810\pi\)
\(284\) 1.82883 0.108521
\(285\) 7.85467 0.465270
\(286\) −10.2858 −0.608212
\(287\) −14.9279 −0.881167
\(288\) 31.0515 1.82973
\(289\) −16.9966 −0.999799
\(290\) 39.5711 2.32369
\(291\) 12.9474 0.758988
\(292\) 54.7837 3.20597
\(293\) 19.2917 1.12703 0.563517 0.826104i \(-0.309448\pi\)
0.563517 + 0.826104i \(0.309448\pi\)
\(294\) −11.1597 −0.650849
\(295\) −14.4595 −0.841864
\(296\) −61.9467 −3.60058
\(297\) 16.0864 0.933427
\(298\) −36.8303 −2.13352
\(299\) −5.61188 −0.324544
\(300\) 7.71965 0.445694
\(301\) 2.32481 0.134000
\(302\) 21.4022 1.23156
\(303\) 8.84892 0.508357
\(304\) −64.7038 −3.71102
\(305\) 25.0627 1.43509
\(306\) −0.373473 −0.0213501
\(307\) 13.2691 0.757308 0.378654 0.925538i \(-0.376387\pi\)
0.378654 + 0.925538i \(0.376387\pi\)
\(308\) 23.3973 1.33318
\(309\) −0.762312 −0.0433664
\(310\) 18.9406 1.07575
\(311\) 6.23805 0.353728 0.176864 0.984235i \(-0.443405\pi\)
0.176864 + 0.984235i \(0.443405\pi\)
\(312\) 5.99341 0.339310
\(313\) 9.47102 0.535334 0.267667 0.963512i \(-0.413747\pi\)
0.267667 + 0.963512i \(0.413747\pi\)
\(314\) 12.1895 0.687893
\(315\) −5.02868 −0.283334
\(316\) 62.5967 3.52134
\(317\) −11.5158 −0.646790 −0.323395 0.946264i \(-0.604824\pi\)
−0.323395 + 0.946264i \(0.604824\pi\)
\(318\) 8.35999 0.468805
\(319\) −33.8804 −1.89694
\(320\) 21.1464 1.18212
\(321\) 3.10751 0.173444
\(322\) 17.8956 0.997285
\(323\) 0.349780 0.0194623
\(324\) 20.4420 1.13566
\(325\) −2.03485 −0.112873
\(326\) −30.1560 −1.67018
\(327\) 4.79256 0.265029
\(328\) 97.2128 5.36768
\(329\) −10.2917 −0.567401
\(330\) 13.5019 0.743254
\(331\) −9.87943 −0.543023 −0.271511 0.962435i \(-0.587523\pi\)
−0.271511 + 0.962435i \(0.587523\pi\)
\(332\) 38.9140 2.13568
\(333\) −19.0586 −1.04440
\(334\) 22.2084 1.21519
\(335\) 5.12253 0.279873
\(336\) −9.95194 −0.542923
\(337\) −13.1699 −0.717408 −0.358704 0.933451i \(-0.616781\pi\)
−0.358704 + 0.933451i \(0.616781\pi\)
\(338\) −2.64132 −0.143669
\(339\) −13.8014 −0.749590
\(340\) −0.500932 −0.0271669
\(341\) −16.2168 −0.878186
\(342\) −38.2303 −2.06726
\(343\) −15.1425 −0.817619
\(344\) −15.1395 −0.816269
\(345\) 7.36656 0.396602
\(346\) 9.98480 0.536786
\(347\) 9.72141 0.521872 0.260936 0.965356i \(-0.415969\pi\)
0.260936 + 0.965356i \(0.415969\pi\)
\(348\) 33.0064 1.76933
\(349\) 2.97097 0.159032 0.0795162 0.996834i \(-0.474662\pi\)
0.0795162 + 0.996834i \(0.474662\pi\)
\(350\) 6.48890 0.346846
\(351\) 4.13088 0.220490
\(352\) −49.9901 −2.66448
\(353\) −20.1012 −1.06988 −0.534941 0.844890i \(-0.679666\pi\)
−0.534941 + 0.844890i \(0.679666\pi\)
\(354\) −16.9077 −0.898634
\(355\) 0.632797 0.0335854
\(356\) −90.0663 −4.77350
\(357\) 0.0537989 0.00284734
\(358\) −40.7814 −2.15536
\(359\) 5.94631 0.313834 0.156917 0.987612i \(-0.449844\pi\)
0.156917 + 0.987612i \(0.449844\pi\)
\(360\) 32.7476 1.72595
\(361\) 16.8050 0.884472
\(362\) 4.34711 0.228479
\(363\) −3.17477 −0.166632
\(364\) 6.00826 0.314919
\(365\) 18.9558 0.992193
\(366\) 29.3062 1.53186
\(367\) 24.7077 1.28973 0.644867 0.764295i \(-0.276913\pi\)
0.644867 + 0.764295i \(0.276913\pi\)
\(368\) −60.6830 −3.16332
\(369\) 29.9086 1.55698
\(370\) −35.8362 −1.86303
\(371\) 5.01266 0.260244
\(372\) 15.7984 0.819108
\(373\) 4.30041 0.222667 0.111333 0.993783i \(-0.464488\pi\)
0.111333 + 0.993783i \(0.464488\pi\)
\(374\) 0.601259 0.0310904
\(375\) 9.23445 0.476865
\(376\) 67.0212 3.45636
\(377\) −8.70027 −0.448087
\(378\) −13.1729 −0.677540
\(379\) −36.5270 −1.87627 −0.938134 0.346273i \(-0.887447\pi\)
−0.938134 + 0.346273i \(0.887447\pi\)
\(380\) −51.2775 −2.63048
\(381\) 11.2432 0.576005
\(382\) −70.1359 −3.58846
\(383\) −23.8547 −1.21892 −0.609459 0.792818i \(-0.708613\pi\)
−0.609459 + 0.792818i \(0.708613\pi\)
\(384\) 5.15500 0.263065
\(385\) 8.09574 0.412597
\(386\) 41.8332 2.12926
\(387\) −4.65785 −0.236772
\(388\) −84.5241 −4.29106
\(389\) −22.1914 −1.12515 −0.562575 0.826746i \(-0.690189\pi\)
−0.562575 + 0.826746i \(0.690189\pi\)
\(390\) 3.46719 0.175568
\(391\) 0.328044 0.0165899
\(392\) 43.5753 2.20089
\(393\) 5.25162 0.264909
\(394\) 1.20338 0.0606255
\(395\) 21.6592 1.08979
\(396\) −46.8773 −2.35567
\(397\) −30.2099 −1.51619 −0.758096 0.652143i \(-0.773870\pi\)
−0.758096 + 0.652143i \(0.773870\pi\)
\(398\) 42.9370 2.15224
\(399\) 5.50708 0.275699
\(400\) −22.0034 −1.10017
\(401\) 11.0603 0.552327 0.276163 0.961111i \(-0.410937\pi\)
0.276163 + 0.961111i \(0.410937\pi\)
\(402\) 5.98984 0.298746
\(403\) −4.16435 −0.207441
\(404\) −57.7683 −2.87408
\(405\) 7.07316 0.351468
\(406\) 27.7442 1.37692
\(407\) 30.6827 1.52088
\(408\) −0.350347 −0.0173447
\(409\) −7.17152 −0.354609 −0.177304 0.984156i \(-0.556738\pi\)
−0.177304 + 0.984156i \(0.556738\pi\)
\(410\) 56.2377 2.77738
\(411\) 15.4834 0.763737
\(412\) 4.97659 0.245179
\(413\) −10.1379 −0.498852
\(414\) −35.8546 −1.76216
\(415\) 13.4647 0.660956
\(416\) −12.8371 −0.629392
\(417\) −4.66090 −0.228245
\(418\) 61.5474 3.01038
\(419\) −5.89661 −0.288068 −0.144034 0.989573i \(-0.546008\pi\)
−0.144034 + 0.989573i \(0.546008\pi\)
\(420\) −7.88688 −0.384840
\(421\) 28.3580 1.38209 0.691043 0.722814i \(-0.257152\pi\)
0.691043 + 0.722814i \(0.257152\pi\)
\(422\) 21.2768 1.03574
\(423\) 20.6198 1.00257
\(424\) −32.6432 −1.58529
\(425\) 0.118948 0.00576981
\(426\) 0.739939 0.0358502
\(427\) 17.5720 0.850369
\(428\) −20.2867 −0.980595
\(429\) −2.96858 −0.143325
\(430\) −8.75823 −0.422359
\(431\) 22.8449 1.10040 0.550201 0.835032i \(-0.314551\pi\)
0.550201 + 0.835032i \(0.314551\pi\)
\(432\) 44.6684 2.14911
\(433\) −11.4452 −0.550023 −0.275011 0.961441i \(-0.588682\pi\)
−0.275011 + 0.961441i \(0.588682\pi\)
\(434\) 13.2796 0.637443
\(435\) 11.4206 0.547576
\(436\) −31.2872 −1.49839
\(437\) 33.5800 1.60635
\(438\) 22.1653 1.05910
\(439\) 13.8740 0.662168 0.331084 0.943601i \(-0.392586\pi\)
0.331084 + 0.943601i \(0.392586\pi\)
\(440\) −52.7207 −2.51336
\(441\) 13.4064 0.638402
\(442\) 0.154399 0.00734403
\(443\) 20.9100 0.993463 0.496732 0.867904i \(-0.334533\pi\)
0.496732 + 0.867904i \(0.334533\pi\)
\(444\) −29.8911 −1.41857
\(445\) −31.1640 −1.47732
\(446\) 64.3475 3.04694
\(447\) −10.6296 −0.502763
\(448\) 14.8262 0.700472
\(449\) −8.64121 −0.407804 −0.203902 0.978991i \(-0.565362\pi\)
−0.203902 + 0.978991i \(0.565362\pi\)
\(450\) −13.0008 −0.612861
\(451\) −48.1502 −2.26731
\(452\) 90.0997 4.23793
\(453\) 6.17689 0.290215
\(454\) −54.4403 −2.55501
\(455\) 2.07893 0.0974618
\(456\) −35.8629 −1.67944
\(457\) −22.3853 −1.04714 −0.523569 0.851983i \(-0.675400\pi\)
−0.523569 + 0.851983i \(0.675400\pi\)
\(458\) −33.0633 −1.54495
\(459\) −0.241472 −0.0112709
\(460\) −48.0910 −2.24226
\(461\) −10.3907 −0.483942 −0.241971 0.970284i \(-0.577794\pi\)
−0.241971 + 0.970284i \(0.577794\pi\)
\(462\) 9.46646 0.440420
\(463\) 18.6261 0.865627 0.432814 0.901483i \(-0.357521\pi\)
0.432814 + 0.901483i \(0.357521\pi\)
\(464\) −94.0787 −4.36749
\(465\) 5.46643 0.253500
\(466\) 53.6980 2.48751
\(467\) 28.7158 1.32881 0.664405 0.747373i \(-0.268685\pi\)
0.664405 + 0.747373i \(0.268685\pi\)
\(468\) −12.0378 −0.556447
\(469\) 3.59151 0.165841
\(470\) 38.7718 1.78841
\(471\) 3.51801 0.162101
\(472\) 66.0193 3.03879
\(473\) 7.49873 0.344792
\(474\) 25.3264 1.16328
\(475\) 12.1760 0.558672
\(476\) −0.351215 −0.0160979
\(477\) −10.0430 −0.459839
\(478\) 56.4461 2.58178
\(479\) 4.08885 0.186824 0.0934122 0.995628i \(-0.470223\pi\)
0.0934122 + 0.995628i \(0.470223\pi\)
\(480\) 16.8509 0.769137
\(481\) 7.87910 0.359256
\(482\) −25.1243 −1.14438
\(483\) 5.16486 0.235009
\(484\) 20.7258 0.942081
\(485\) −29.2464 −1.32801
\(486\) 41.0037 1.85997
\(487\) 10.1198 0.458573 0.229287 0.973359i \(-0.426361\pi\)
0.229287 + 0.973359i \(0.426361\pi\)
\(488\) −114.432 −5.18007
\(489\) −8.70331 −0.393577
\(490\) 25.2083 1.13880
\(491\) −1.20805 −0.0545186 −0.0272593 0.999628i \(-0.508678\pi\)
−0.0272593 + 0.999628i \(0.508678\pi\)
\(492\) 46.9081 2.11478
\(493\) 0.508577 0.0229051
\(494\) 15.8050 0.711099
\(495\) −16.2201 −0.729040
\(496\) −45.0304 −2.02192
\(497\) 0.443668 0.0199012
\(498\) 15.7445 0.705527
\(499\) 18.7895 0.841134 0.420567 0.907262i \(-0.361831\pi\)
0.420567 + 0.907262i \(0.361831\pi\)
\(500\) −60.2852 −2.69603
\(501\) 6.40956 0.286358
\(502\) 11.8030 0.526793
\(503\) 34.1846 1.52422 0.762108 0.647450i \(-0.224164\pi\)
0.762108 + 0.647450i \(0.224164\pi\)
\(504\) 22.9600 1.02272
\(505\) −19.9885 −0.889478
\(506\) 57.7227 2.56609
\(507\) −0.762312 −0.0338555
\(508\) −73.3986 −3.25654
\(509\) −5.78374 −0.256360 −0.128180 0.991751i \(-0.540914\pi\)
−0.128180 + 0.991751i \(0.540914\pi\)
\(510\) −0.202676 −0.00897463
\(511\) 13.2903 0.587930
\(512\) 31.2198 1.37973
\(513\) −24.7180 −1.09133
\(514\) −0.875897 −0.0386342
\(515\) 1.72196 0.0758787
\(516\) −7.30527 −0.321597
\(517\) −33.1961 −1.45996
\(518\) −25.1255 −1.10395
\(519\) 2.88171 0.126493
\(520\) −13.5383 −0.593694
\(521\) −32.1466 −1.40837 −0.704184 0.710017i \(-0.748687\pi\)
−0.704184 + 0.710017i \(0.748687\pi\)
\(522\) −55.5865 −2.43295
\(523\) 29.9727 1.31061 0.655306 0.755363i \(-0.272540\pi\)
0.655306 + 0.755363i \(0.272540\pi\)
\(524\) −34.2840 −1.49771
\(525\) 1.87276 0.0817340
\(526\) −71.0569 −3.09823
\(527\) 0.243428 0.0106039
\(528\) −32.1002 −1.39698
\(529\) 8.49324 0.369271
\(530\) −18.8841 −0.820273
\(531\) 20.3116 0.881448
\(532\) −35.9518 −1.55871
\(533\) −12.3647 −0.535573
\(534\) −36.4405 −1.57694
\(535\) −7.01944 −0.303477
\(536\) −23.3885 −1.01023
\(537\) −11.7699 −0.507910
\(538\) 27.7256 1.19533
\(539\) −21.5832 −0.929653
\(540\) 35.3996 1.52336
\(541\) 17.9152 0.770233 0.385116 0.922868i \(-0.374161\pi\)
0.385116 + 0.922868i \(0.374161\pi\)
\(542\) −20.7981 −0.893356
\(543\) 1.25462 0.0538408
\(544\) 0.750398 0.0321730
\(545\) −10.8258 −0.463724
\(546\) 2.43092 0.104034
\(547\) −17.4198 −0.744816 −0.372408 0.928069i \(-0.621468\pi\)
−0.372408 + 0.928069i \(0.621468\pi\)
\(548\) −101.080 −4.31791
\(549\) −35.2062 −1.50256
\(550\) 20.9301 0.892461
\(551\) 52.0600 2.21783
\(552\) −33.6343 −1.43157
\(553\) 15.1857 0.645764
\(554\) 5.66138 0.240529
\(555\) −10.3427 −0.439022
\(556\) 30.4277 1.29042
\(557\) −6.33083 −0.268246 −0.134123 0.990965i \(-0.542822\pi\)
−0.134123 + 0.990965i \(0.542822\pi\)
\(558\) −26.6062 −1.12633
\(559\) 1.92562 0.0814452
\(560\) 22.4801 0.949958
\(561\) 0.173529 0.00732641
\(562\) 7.52324 0.317349
\(563\) 15.9403 0.671803 0.335902 0.941897i \(-0.390959\pi\)
0.335902 + 0.941897i \(0.390959\pi\)
\(564\) 32.3397 1.36175
\(565\) 31.1756 1.31157
\(566\) −36.8730 −1.54989
\(567\) 4.95915 0.208265
\(568\) −2.88923 −0.121230
\(569\) 11.2380 0.471123 0.235561 0.971859i \(-0.424307\pi\)
0.235561 + 0.971859i \(0.424307\pi\)
\(570\) −20.7467 −0.868985
\(571\) 23.4027 0.979372 0.489686 0.871899i \(-0.337111\pi\)
0.489686 + 0.871899i \(0.337111\pi\)
\(572\) 19.3798 0.810309
\(573\) −20.2419 −0.845618
\(574\) 39.4295 1.64575
\(575\) 11.4193 0.476220
\(576\) −29.7049 −1.23770
\(577\) −6.07093 −0.252736 −0.126368 0.991983i \(-0.540332\pi\)
−0.126368 + 0.991983i \(0.540332\pi\)
\(578\) 44.8935 1.86732
\(579\) 12.0735 0.501757
\(580\) −74.5570 −3.09581
\(581\) 9.44040 0.391654
\(582\) −34.1982 −1.41756
\(583\) 16.1684 0.669627
\(584\) −86.5487 −3.58141
\(585\) −4.16522 −0.172211
\(586\) −50.9557 −2.10496
\(587\) −35.8925 −1.48144 −0.740720 0.671814i \(-0.765515\pi\)
−0.740720 + 0.671814i \(0.765515\pi\)
\(588\) 21.0264 0.867113
\(589\) 24.9183 1.02674
\(590\) 38.1922 1.57235
\(591\) 0.347308 0.0142863
\(592\) 85.1991 3.50166
\(593\) −40.8481 −1.67743 −0.838715 0.544571i \(-0.816692\pi\)
−0.838715 + 0.544571i \(0.816692\pi\)
\(594\) −42.4894 −1.74336
\(595\) −0.121524 −0.00498202
\(596\) 69.3931 2.84245
\(597\) 12.3920 0.507173
\(598\) 14.8228 0.606150
\(599\) 25.7487 1.05206 0.526031 0.850465i \(-0.323679\pi\)
0.526031 + 0.850465i \(0.323679\pi\)
\(600\) −12.1957 −0.497887
\(601\) 12.9563 0.528500 0.264250 0.964454i \(-0.414876\pi\)
0.264250 + 0.964454i \(0.414876\pi\)
\(602\) −6.14058 −0.250272
\(603\) −7.19573 −0.293033
\(604\) −40.3245 −1.64078
\(605\) 7.17137 0.291558
\(606\) −23.3729 −0.949458
\(607\) 20.1175 0.816543 0.408271 0.912861i \(-0.366132\pi\)
0.408271 + 0.912861i \(0.366132\pi\)
\(608\) 76.8139 3.11521
\(609\) 8.00724 0.324470
\(610\) −66.1987 −2.68031
\(611\) −8.52454 −0.344866
\(612\) 0.703672 0.0284442
\(613\) −4.19807 −0.169558 −0.0847792 0.996400i \(-0.527018\pi\)
−0.0847792 + 0.996400i \(0.527018\pi\)
\(614\) −35.0480 −1.41442
\(615\) 16.2307 0.654487
\(616\) −36.9636 −1.48931
\(617\) 43.3447 1.74499 0.872496 0.488620i \(-0.162500\pi\)
0.872496 + 0.488620i \(0.162500\pi\)
\(618\) 2.01351 0.0809954
\(619\) 0.314554 0.0126430 0.00632149 0.999980i \(-0.497988\pi\)
0.00632149 + 0.999980i \(0.497988\pi\)
\(620\) −35.6864 −1.43320
\(621\) −23.1820 −0.930262
\(622\) −16.4767 −0.660656
\(623\) −21.8498 −0.875393
\(624\) −8.24311 −0.329989
\(625\) −10.6851 −0.427405
\(626\) −25.0160 −0.999842
\(627\) 17.7632 0.709393
\(628\) −22.9666 −0.916466
\(629\) −0.460575 −0.0183643
\(630\) 13.2824 0.529183
\(631\) 14.6013 0.581269 0.290634 0.956834i \(-0.406134\pi\)
0.290634 + 0.956834i \(0.406134\pi\)
\(632\) −98.8919 −3.93371
\(633\) 6.14070 0.244071
\(634\) 30.4169 1.20801
\(635\) −25.3968 −1.00784
\(636\) −15.7513 −0.624579
\(637\) −5.54242 −0.219599
\(638\) 89.4893 3.54291
\(639\) −0.888906 −0.0351646
\(640\) −11.6445 −0.460287
\(641\) −38.5844 −1.52399 −0.761997 0.647581i \(-0.775781\pi\)
−0.761997 + 0.647581i \(0.775781\pi\)
\(642\) −8.20793 −0.323941
\(643\) −13.4610 −0.530851 −0.265426 0.964131i \(-0.585512\pi\)
−0.265426 + 0.964131i \(0.585512\pi\)
\(644\) −33.7177 −1.32866
\(645\) −2.52771 −0.0995285
\(646\) −0.923883 −0.0363497
\(647\) 5.20195 0.204510 0.102255 0.994758i \(-0.467394\pi\)
0.102255 + 0.994758i \(0.467394\pi\)
\(648\) −32.2947 −1.26866
\(649\) −32.6999 −1.28358
\(650\) 5.37470 0.210813
\(651\) 3.83263 0.150213
\(652\) 56.8177 2.22515
\(653\) −29.3086 −1.14693 −0.573466 0.819229i \(-0.694402\pi\)
−0.573466 + 0.819229i \(0.694402\pi\)
\(654\) −12.6587 −0.494995
\(655\) −11.8627 −0.463514
\(656\) −133.703 −5.22022
\(657\) −26.6277 −1.03885
\(658\) 27.1838 1.05973
\(659\) −30.4381 −1.18570 −0.592851 0.805312i \(-0.701998\pi\)
−0.592851 + 0.805312i \(0.701998\pi\)
\(660\) −25.4393 −0.990222
\(661\) −9.51139 −0.369950 −0.184975 0.982743i \(-0.559220\pi\)
−0.184975 + 0.982743i \(0.559220\pi\)
\(662\) 26.0948 1.01420
\(663\) 0.0445612 0.00173061
\(664\) −61.4773 −2.38578
\(665\) −12.4398 −0.482393
\(666\) 50.3399 1.95063
\(667\) 48.8249 1.89051
\(668\) −41.8435 −1.61897
\(669\) 18.5713 0.718009
\(670\) −13.5303 −0.522719
\(671\) 56.6788 2.18806
\(672\) 11.8146 0.455757
\(673\) −15.0736 −0.581043 −0.290521 0.956868i \(-0.593829\pi\)
−0.290521 + 0.956868i \(0.593829\pi\)
\(674\) 34.7859 1.33990
\(675\) −8.40572 −0.323536
\(676\) 4.97659 0.191407
\(677\) 24.9232 0.957875 0.478938 0.877849i \(-0.341022\pi\)
0.478938 + 0.877849i \(0.341022\pi\)
\(678\) 36.4540 1.40001
\(679\) −20.5053 −0.786920
\(680\) 0.791386 0.0303483
\(681\) −15.7120 −0.602086
\(682\) 42.8337 1.64019
\(683\) −24.6663 −0.943831 −0.471916 0.881644i \(-0.656437\pi\)
−0.471916 + 0.881644i \(0.656437\pi\)
\(684\) 72.0308 2.75417
\(685\) −34.9748 −1.33632
\(686\) 39.9963 1.52707
\(687\) −9.54239 −0.364065
\(688\) 20.8223 0.793844
\(689\) 4.15194 0.158176
\(690\) −19.4575 −0.740734
\(691\) −13.5160 −0.514172 −0.257086 0.966389i \(-0.582762\pi\)
−0.257086 + 0.966389i \(0.582762\pi\)
\(692\) −18.8126 −0.715150
\(693\) −11.3723 −0.431997
\(694\) −25.6774 −0.974700
\(695\) 10.5283 0.399363
\(696\) −52.1443 −1.97653
\(697\) 0.722780 0.0273772
\(698\) −7.84729 −0.297024
\(699\) 15.4978 0.586180
\(700\) −12.2259 −0.462096
\(701\) 30.4944 1.15176 0.575879 0.817535i \(-0.304660\pi\)
0.575879 + 0.817535i \(0.304660\pi\)
\(702\) −10.9110 −0.411809
\(703\) −47.1464 −1.77816
\(704\) 47.8222 1.80237
\(705\) 11.1899 0.421437
\(706\) 53.0939 1.99822
\(707\) −14.0144 −0.527066
\(708\) 31.8563 1.19723
\(709\) 17.0796 0.641438 0.320719 0.947174i \(-0.396075\pi\)
0.320719 + 0.947174i \(0.396075\pi\)
\(710\) −1.67142 −0.0627274
\(711\) −30.4252 −1.14103
\(712\) 142.289 5.33251
\(713\) 23.3699 0.875209
\(714\) −0.142100 −0.00531797
\(715\) 6.70563 0.250776
\(716\) 76.8374 2.87155
\(717\) 16.2909 0.608395
\(718\) −15.7061 −0.586148
\(719\) 25.1709 0.938715 0.469358 0.883008i \(-0.344486\pi\)
0.469358 + 0.883008i \(0.344486\pi\)
\(720\) −45.0397 −1.67853
\(721\) 1.20730 0.0449624
\(722\) −44.3874 −1.65193
\(723\) −7.25111 −0.269672
\(724\) −8.19051 −0.304398
\(725\) 17.7038 0.657501
\(726\) 8.38559 0.311218
\(727\) 17.4201 0.646075 0.323038 0.946386i \(-0.395296\pi\)
0.323038 + 0.946386i \(0.395296\pi\)
\(728\) −9.49201 −0.351797
\(729\) −0.488777 −0.0181028
\(730\) −50.0685 −1.85312
\(731\) −0.112563 −0.00416329
\(732\) −55.2166 −2.04086
\(733\) 15.0667 0.556502 0.278251 0.960508i \(-0.410245\pi\)
0.278251 + 0.960508i \(0.410245\pi\)
\(734\) −65.2612 −2.40883
\(735\) 7.27538 0.268356
\(736\) 72.0405 2.65545
\(737\) 11.5845 0.426720
\(738\) −78.9984 −2.90797
\(739\) −23.3677 −0.859593 −0.429797 0.902926i \(-0.641415\pi\)
−0.429797 + 0.902926i \(0.641415\pi\)
\(740\) 67.5200 2.48208
\(741\) 4.56147 0.167570
\(742\) −13.2400 −0.486057
\(743\) −14.8019 −0.543029 −0.271515 0.962434i \(-0.587524\pi\)
−0.271515 + 0.962434i \(0.587524\pi\)
\(744\) −24.9587 −0.915030
\(745\) 24.0108 0.879689
\(746\) −11.3588 −0.415874
\(747\) −18.9142 −0.692034
\(748\) −1.13285 −0.0414211
\(749\) −4.92148 −0.179827
\(750\) −24.3912 −0.890640
\(751\) −31.8584 −1.16253 −0.581265 0.813715i \(-0.697442\pi\)
−0.581265 + 0.813715i \(0.697442\pi\)
\(752\) −92.1784 −3.36140
\(753\) 3.40646 0.124138
\(754\) 22.9802 0.836891
\(755\) −13.9528 −0.507793
\(756\) 24.8194 0.902673
\(757\) 17.6112 0.640088 0.320044 0.947403i \(-0.396302\pi\)
0.320044 + 0.947403i \(0.396302\pi\)
\(758\) 96.4797 3.50430
\(759\) 16.6593 0.604696
\(760\) 81.0096 2.93853
\(761\) −15.3209 −0.555384 −0.277692 0.960670i \(-0.589569\pi\)
−0.277692 + 0.960670i \(0.589569\pi\)
\(762\) −29.6969 −1.07580
\(763\) −7.59017 −0.274783
\(764\) 132.145 4.78084
\(765\) 0.243479 0.00880300
\(766\) 63.0080 2.27657
\(767\) −8.39711 −0.303202
\(768\) 5.10701 0.184283
\(769\) 4.30357 0.155191 0.0775954 0.996985i \(-0.475276\pi\)
0.0775954 + 0.996985i \(0.475276\pi\)
\(770\) −21.3835 −0.770607
\(771\) −0.252793 −0.00910411
\(772\) −78.8192 −2.83677
\(773\) 1.54770 0.0556670 0.0278335 0.999613i \(-0.491139\pi\)
0.0278335 + 0.999613i \(0.491139\pi\)
\(774\) 12.3029 0.442218
\(775\) 8.47384 0.304389
\(776\) 133.533 4.79357
\(777\) −7.25148 −0.260145
\(778\) 58.6148 2.10144
\(779\) 73.9867 2.65085
\(780\) −6.53264 −0.233906
\(781\) 1.43106 0.0512073
\(782\) −0.866471 −0.0309849
\(783\) −35.9398 −1.28438
\(784\) −59.9318 −2.14042
\(785\) −7.94671 −0.283630
\(786\) −13.8712 −0.494770
\(787\) −8.25042 −0.294096 −0.147048 0.989129i \(-0.546977\pi\)
−0.147048 + 0.989129i \(0.546977\pi\)
\(788\) −2.26733 −0.0807702
\(789\) −20.5077 −0.730094
\(790\) −57.2090 −2.03540
\(791\) 21.8579 0.777176
\(792\) 74.0580 2.63154
\(793\) 14.5547 0.516854
\(794\) 79.7942 2.83179
\(795\) −5.45014 −0.193296
\(796\) −80.8988 −2.86738
\(797\) 36.4560 1.29134 0.645669 0.763617i \(-0.276579\pi\)
0.645669 + 0.763617i \(0.276579\pi\)
\(798\) −14.5460 −0.514922
\(799\) 0.498304 0.0176287
\(800\) 26.1216 0.923539
\(801\) 43.7768 1.54678
\(802\) −29.2139 −1.03158
\(803\) 42.8682 1.51279
\(804\) −11.2856 −0.398014
\(805\) −11.6667 −0.411198
\(806\) 10.9994 0.387438
\(807\) 8.00188 0.281679
\(808\) 91.2639 3.21065
\(809\) −34.9356 −1.22827 −0.614135 0.789201i \(-0.710495\pi\)
−0.614135 + 0.789201i \(0.710495\pi\)
\(810\) −18.6825 −0.656436
\(811\) 42.5876 1.49545 0.747727 0.664006i \(-0.231145\pi\)
0.747727 + 0.664006i \(0.231145\pi\)
\(812\) −52.2735 −1.83444
\(813\) −6.00255 −0.210518
\(814\) −81.0428 −2.84055
\(815\) 19.6596 0.688646
\(816\) 0.481853 0.0168682
\(817\) −11.5224 −0.403118
\(818\) 18.9423 0.662302
\(819\) −2.92032 −0.102044
\(820\) −105.959 −3.70025
\(821\) 48.0950 1.67853 0.839264 0.543724i \(-0.182986\pi\)
0.839264 + 0.543724i \(0.182986\pi\)
\(822\) −40.8965 −1.42643
\(823\) −40.1416 −1.39925 −0.699623 0.714512i \(-0.746649\pi\)
−0.699623 + 0.714512i \(0.746649\pi\)
\(824\) −7.86215 −0.273891
\(825\) 6.04062 0.210307
\(826\) 26.7774 0.931705
\(827\) −27.6109 −0.960125 −0.480062 0.877234i \(-0.659386\pi\)
−0.480062 + 0.877234i \(0.659386\pi\)
\(828\) 67.5546 2.34769
\(829\) −26.7196 −0.928011 −0.464005 0.885832i \(-0.653588\pi\)
−0.464005 + 0.885832i \(0.653588\pi\)
\(830\) −35.5646 −1.23447
\(831\) 1.63393 0.0566804
\(832\) 12.2804 0.425747
\(833\) 0.323983 0.0112254
\(834\) 12.3110 0.426293
\(835\) −14.4783 −0.501044
\(836\) −115.963 −4.01067
\(837\) −17.2024 −0.594603
\(838\) 15.5749 0.538025
\(839\) 32.7767 1.13158 0.565789 0.824550i \(-0.308572\pi\)
0.565789 + 0.824550i \(0.308572\pi\)
\(840\) 12.4599 0.429907
\(841\) 46.6948 1.61016
\(842\) −74.9028 −2.58132
\(843\) 2.17128 0.0747830
\(844\) −40.0882 −1.37989
\(845\) 1.72196 0.0592373
\(846\) −54.4637 −1.87250
\(847\) 5.02800 0.172764
\(848\) 44.8962 1.54174
\(849\) −10.6419 −0.365230
\(850\) −0.314179 −0.0107763
\(851\) −44.2166 −1.51573
\(852\) −1.39414 −0.0477625
\(853\) −27.3084 −0.935020 −0.467510 0.883988i \(-0.654849\pi\)
−0.467510 + 0.883988i \(0.654849\pi\)
\(854\) −46.4134 −1.58823
\(855\) 24.9235 0.852366
\(856\) 32.0494 1.09543
\(857\) 30.5265 1.04277 0.521383 0.853323i \(-0.325416\pi\)
0.521383 + 0.853323i \(0.325416\pi\)
\(858\) 7.84099 0.267687
\(859\) 15.1596 0.517240 0.258620 0.965979i \(-0.416732\pi\)
0.258620 + 0.965979i \(0.416732\pi\)
\(860\) 16.5016 0.562701
\(861\) 11.3797 0.387820
\(862\) −60.3409 −2.05522
\(863\) 9.53717 0.324649 0.162324 0.986737i \(-0.448101\pi\)
0.162324 + 0.986737i \(0.448101\pi\)
\(864\) −53.0286 −1.80407
\(865\) −6.50940 −0.221326
\(866\) 30.2306 1.02728
\(867\) 12.9567 0.440033
\(868\) −25.0205 −0.849252
\(869\) 48.9819 1.66160
\(870\) −30.1655 −1.02271
\(871\) 2.97482 0.100798
\(872\) 49.4284 1.67386
\(873\) 41.0831 1.39045
\(874\) −88.6956 −3.00017
\(875\) −14.6250 −0.494414
\(876\) −41.7623 −1.41102
\(877\) 26.9268 0.909255 0.454627 0.890682i \(-0.349772\pi\)
0.454627 + 0.890682i \(0.349772\pi\)
\(878\) −36.6456 −1.23673
\(879\) −14.7063 −0.496032
\(880\) 72.5100 2.44431
\(881\) 7.53629 0.253904 0.126952 0.991909i \(-0.459481\pi\)
0.126952 + 0.991909i \(0.459481\pi\)
\(882\) −35.4108 −1.19234
\(883\) 31.2649 1.05215 0.526074 0.850439i \(-0.323664\pi\)
0.526074 + 0.850439i \(0.323664\pi\)
\(884\) −0.290908 −0.00978430
\(885\) 11.0226 0.370522
\(886\) −55.2301 −1.85549
\(887\) −3.40037 −0.114173 −0.0570866 0.998369i \(-0.518181\pi\)
−0.0570866 + 0.998369i \(0.518181\pi\)
\(888\) 47.2227 1.58469
\(889\) −17.8063 −0.597203
\(890\) 82.3143 2.75918
\(891\) 15.9958 0.535880
\(892\) −121.239 −4.05938
\(893\) 51.0085 1.70694
\(894\) 28.0762 0.939009
\(895\) 26.5867 0.888694
\(896\) −8.16418 −0.272746
\(897\) 4.27801 0.142839
\(898\) 22.8242 0.761654
\(899\) 36.2310 1.20837
\(900\) 24.4951 0.816503
\(901\) −0.242703 −0.00808560
\(902\) 127.180 4.23464
\(903\) −1.77223 −0.0589762
\(904\) −142.342 −4.73422
\(905\) −2.83401 −0.0942058
\(906\) −16.3152 −0.542035
\(907\) 35.2903 1.17179 0.585897 0.810385i \(-0.300742\pi\)
0.585897 + 0.810385i \(0.300742\pi\)
\(908\) 102.573 3.40399
\(909\) 28.0784 0.931301
\(910\) −5.49113 −0.182029
\(911\) −36.7498 −1.21758 −0.608788 0.793333i \(-0.708344\pi\)
−0.608788 + 0.793333i \(0.708344\pi\)
\(912\) 49.3245 1.63330
\(913\) 30.4502 1.00775
\(914\) 59.1268 1.95574
\(915\) −19.1056 −0.631612
\(916\) 62.2955 2.05830
\(917\) −8.31719 −0.274658
\(918\) 0.637805 0.0210507
\(919\) −35.9149 −1.18472 −0.592362 0.805672i \(-0.701804\pi\)
−0.592362 + 0.805672i \(0.701804\pi\)
\(920\) 75.9755 2.50484
\(921\) −10.1152 −0.333307
\(922\) 27.4451 0.903857
\(923\) 0.367486 0.0120960
\(924\) −17.8360 −0.586762
\(925\) −16.0328 −0.527155
\(926\) −49.1975 −1.61673
\(927\) −2.41888 −0.0794464
\(928\) 111.687 3.66629
\(929\) −43.1704 −1.41637 −0.708187 0.706025i \(-0.750486\pi\)
−0.708187 + 0.706025i \(0.750486\pi\)
\(930\) −14.4386 −0.473461
\(931\) 33.1643 1.08692
\(932\) −101.174 −3.31406
\(933\) −4.75535 −0.155683
\(934\) −75.8478 −2.48181
\(935\) −0.391979 −0.0128191
\(936\) 19.0176 0.621610
\(937\) −38.6169 −1.26156 −0.630779 0.775962i \(-0.717265\pi\)
−0.630779 + 0.775962i \(0.717265\pi\)
\(938\) −9.48635 −0.309740
\(939\) −7.21987 −0.235612
\(940\) −73.0511 −2.38266
\(941\) −23.1432 −0.754446 −0.377223 0.926122i \(-0.623121\pi\)
−0.377223 + 0.926122i \(0.623121\pi\)
\(942\) −9.29220 −0.302756
\(943\) 69.3891 2.25962
\(944\) −90.8004 −2.95530
\(945\) 8.58781 0.279362
\(946\) −19.8066 −0.643967
\(947\) 56.3926 1.83251 0.916256 0.400594i \(-0.131196\pi\)
0.916256 + 0.400594i \(0.131196\pi\)
\(948\) −47.7182 −1.54982
\(949\) 11.0083 0.357344
\(950\) −32.1607 −1.04343
\(951\) 8.77861 0.284666
\(952\) 0.554858 0.0179830
\(953\) −5.81186 −0.188265 −0.0941323 0.995560i \(-0.530008\pi\)
−0.0941323 + 0.995560i \(0.530008\pi\)
\(954\) 26.5269 0.858841
\(955\) 45.7238 1.47959
\(956\) −106.352 −3.43966
\(957\) 25.8275 0.834884
\(958\) −10.8000 −0.348932
\(959\) −24.5216 −0.791844
\(960\) −16.1202 −0.520276
\(961\) −13.6582 −0.440586
\(962\) −20.8113 −0.670982
\(963\) 9.86038 0.317746
\(964\) 47.3373 1.52463
\(965\) −27.2724 −0.877929
\(966\) −13.6421 −0.438926
\(967\) 38.5738 1.24045 0.620224 0.784425i \(-0.287042\pi\)
0.620224 + 0.784425i \(0.287042\pi\)
\(968\) −32.7431 −1.05240
\(969\) −0.266642 −0.00856577
\(970\) 77.2491 2.48032
\(971\) −30.3351 −0.973501 −0.486750 0.873541i \(-0.661818\pi\)
−0.486750 + 0.873541i \(0.661818\pi\)
\(972\) −77.2563 −2.47800
\(973\) 7.38165 0.236645
\(974\) −26.7298 −0.856477
\(975\) 1.55119 0.0496779
\(976\) 157.385 5.03776
\(977\) −50.9220 −1.62914 −0.814569 0.580067i \(-0.803027\pi\)
−0.814569 + 0.580067i \(0.803027\pi\)
\(978\) 22.9883 0.735083
\(979\) −70.4768 −2.25245
\(980\) −47.4957 −1.51720
\(981\) 15.2072 0.485529
\(982\) 3.19086 0.101824
\(983\) 40.5055 1.29193 0.645963 0.763369i \(-0.276456\pi\)
0.645963 + 0.763369i \(0.276456\pi\)
\(984\) −74.1065 −2.36243
\(985\) −0.784522 −0.0249970
\(986\) −1.34332 −0.0427799
\(987\) 7.84550 0.249725
\(988\) −29.7786 −0.947383
\(989\) −10.8064 −0.343623
\(990\) 42.8426 1.36163
\(991\) −24.7278 −0.785504 −0.392752 0.919644i \(-0.628477\pi\)
−0.392752 + 0.919644i \(0.628477\pi\)
\(992\) 53.4583 1.69730
\(993\) 7.53121 0.238996
\(994\) −1.17187 −0.0371695
\(995\) −27.9920 −0.887405
\(996\) −29.6646 −0.939959
\(997\) −10.1884 −0.322671 −0.161335 0.986900i \(-0.551580\pi\)
−0.161335 + 0.986900i \(0.551580\pi\)
\(998\) −49.6292 −1.57098
\(999\) 32.5476 1.02976
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 1339.2.a.d.1.2 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1339.2.a.d.1.2 19 1.1 even 1 trivial