Properties

Label 538.2.a.e.1.2
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 1.2
Root \(1.61949\) 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)
 
\(f(q)\) \(=\) \(q+1.00000 q^{2} -2.60058 q^{3} +1.00000 q^{4} -3.80694 q^{5} -2.60058 q^{6} +0.487379 q^{7} +1.00000 q^{8} +3.76302 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -2.60058 q^{3} +1.00000 q^{4} -3.80694 q^{5} -2.60058 q^{6} +0.487379 q^{7} +1.00000 q^{8} +3.76302 q^{9} -3.80694 q^{10} +2.94230 q^{11} -2.60058 q^{12} +1.21482 q^{13} +0.487379 q^{14} +9.90026 q^{15} +1.00000 q^{16} +3.19066 q^{17} +3.76302 q^{18} -2.02416 q^{19} -3.80694 q^{20} -1.26747 q^{21} +2.94230 q^{22} +7.20116 q^{23} -2.60058 q^{24} +9.49279 q^{25} +1.21482 q^{26} -1.98430 q^{27} +0.487379 q^{28} +3.48092 q^{29} +9.90026 q^{30} -5.14753 q^{31} +1.00000 q^{32} -7.65169 q^{33} +3.19066 q^{34} -1.85542 q^{35} +3.76302 q^{36} +1.03375 q^{37} -2.02416 q^{38} -3.15923 q^{39} -3.80694 q^{40} +8.43673 q^{41} -1.26747 q^{42} -1.24039 q^{43} +2.94230 q^{44} -14.3256 q^{45} +7.20116 q^{46} +4.41272 q^{47} -2.60058 q^{48} -6.76246 q^{49} +9.49279 q^{50} -8.29757 q^{51} +1.21482 q^{52} +1.19619 q^{53} -1.98430 q^{54} -11.2012 q^{55} +0.487379 q^{56} +5.26399 q^{57} +3.48092 q^{58} -12.1673 q^{59} +9.90026 q^{60} +10.2849 q^{61} -5.14753 q^{62} +1.83402 q^{63} +1.00000 q^{64} -4.62474 q^{65} -7.65169 q^{66} -11.3824 q^{67} +3.19066 q^{68} -18.7272 q^{69} -1.85542 q^{70} +8.87514 q^{71} +3.76302 q^{72} +9.05652 q^{73} +1.03375 q^{74} -24.6868 q^{75} -2.02416 q^{76} +1.43402 q^{77} -3.15923 q^{78} -7.41490 q^{79} -3.80694 q^{80} -6.12873 q^{81} +8.43673 q^{82} +6.48221 q^{83} -1.26747 q^{84} -12.1467 q^{85} -1.24039 q^{86} -9.05241 q^{87} +2.94230 q^{88} +16.3474 q^{89} -14.3256 q^{90} +0.592077 q^{91} +7.20116 q^{92} +13.3866 q^{93} +4.41272 q^{94} +7.70585 q^{95} -2.60058 q^{96} -2.51050 q^{97} -6.76246 q^{98} +11.0719 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 7 q^{2} + q^{3} + 7 q^{4} + 7 q^{5} + q^{6} + 6 q^{7} + 7 q^{8} + 12 q^{9}+O(q^{10}) \) 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
\(3\) −2.60058 −1.50145 −0.750723 0.660617i \(-0.770295\pi\)
−0.750723 + 0.660617i \(0.770295\pi\)
\(4\) 1.00000 0.500000
\(5\) −3.80694 −1.70252 −0.851258 0.524748i \(-0.824160\pi\)
−0.851258 + 0.524748i \(0.824160\pi\)
\(6\) −2.60058 −1.06168
\(7\) 0.487379 0.184212 0.0921060 0.995749i \(-0.470640\pi\)
0.0921060 + 0.995749i \(0.470640\pi\)
\(8\) 1.00000 0.353553
\(9\) 3.76302 1.25434
\(10\) −3.80694 −1.20386
\(11\) 2.94230 0.887137 0.443569 0.896240i \(-0.353712\pi\)
0.443569 + 0.896240i \(0.353712\pi\)
\(12\) −2.60058 −0.750723
\(13\) 1.21482 0.336930 0.168465 0.985708i \(-0.446119\pi\)
0.168465 + 0.985708i \(0.446119\pi\)
\(14\) 0.487379 0.130258
\(15\) 9.90026 2.55624
\(16\) 1.00000 0.250000
\(17\) 3.19066 0.773849 0.386924 0.922111i \(-0.373537\pi\)
0.386924 + 0.922111i \(0.373537\pi\)
\(18\) 3.76302 0.886953
\(19\) −2.02416 −0.464374 −0.232187 0.972671i \(-0.574588\pi\)
−0.232187 + 0.972671i \(0.574588\pi\)
\(20\) −3.80694 −0.851258
\(21\) −1.26747 −0.276585
\(22\) 2.94230 0.627301
\(23\) 7.20116 1.50155 0.750773 0.660560i \(-0.229681\pi\)
0.750773 + 0.660560i \(0.229681\pi\)
\(24\) −2.60058 −0.530841
\(25\) 9.49279 1.89856
\(26\) 1.21482 0.238245
\(27\) −1.98430 −0.381879
\(28\) 0.487379 0.0921060
\(29\) 3.48092 0.646390 0.323195 0.946332i \(-0.395243\pi\)
0.323195 + 0.946332i \(0.395243\pi\)
\(30\) 9.90026 1.80753
\(31\) −5.14753 −0.924524 −0.462262 0.886743i \(-0.652962\pi\)
−0.462262 + 0.886743i \(0.652962\pi\)
\(32\) 1.00000 0.176777
\(33\) −7.65169 −1.33199
\(34\) 3.19066 0.547194
\(35\) −1.85542 −0.313624
\(36\) 3.76302 0.627170
\(37\) 1.03375 0.169947 0.0849733 0.996383i \(-0.472919\pi\)
0.0849733 + 0.996383i \(0.472919\pi\)
\(38\) −2.02416 −0.328362
\(39\) −3.15923 −0.505882
\(40\) −3.80694 −0.601930
\(41\) 8.43673 1.31760 0.658798 0.752320i \(-0.271065\pi\)
0.658798 + 0.752320i \(0.271065\pi\)
\(42\) −1.26747 −0.195575
\(43\) −1.24039 −0.189158 −0.0945788 0.995517i \(-0.530150\pi\)
−0.0945788 + 0.995517i \(0.530150\pi\)
\(44\) 2.94230 0.443569
\(45\) −14.3256 −2.13553
\(46\) 7.20116 1.06175
\(47\) 4.41272 0.643661 0.321830 0.946797i \(-0.395702\pi\)
0.321830 + 0.946797i \(0.395702\pi\)
\(48\) −2.60058 −0.375362
\(49\) −6.76246 −0.966066
\(50\) 9.49279 1.34248
\(51\) −8.29757 −1.16189
\(52\) 1.21482 0.168465
\(53\) 1.19619 0.164309 0.0821544 0.996620i \(-0.473820\pi\)
0.0821544 + 0.996620i \(0.473820\pi\)
\(54\) −1.98430 −0.270029
\(55\) −11.2012 −1.51036
\(56\) 0.487379 0.0651288
\(57\) 5.26399 0.697232
\(58\) 3.48092 0.457067
\(59\) −12.1673 −1.58405 −0.792026 0.610487i \(-0.790974\pi\)
−0.792026 + 0.610487i \(0.790974\pi\)
\(60\) 9.90026 1.27812
\(61\) 10.2849 1.31685 0.658423 0.752648i \(-0.271224\pi\)
0.658423 + 0.752648i \(0.271224\pi\)
\(62\) −5.14753 −0.653737
\(63\) 1.83402 0.231065
\(64\) 1.00000 0.125000
\(65\) −4.62474 −0.573628
\(66\) −7.65169 −0.941858
\(67\) −11.3824 −1.39058 −0.695290 0.718730i \(-0.744724\pi\)
−0.695290 + 0.718730i \(0.744724\pi\)
\(68\) 3.19066 0.386924
\(69\) −18.7272 −2.25449
\(70\) −1.85542 −0.221766
\(71\) 8.87514 1.05329 0.526643 0.850087i \(-0.323451\pi\)
0.526643 + 0.850087i \(0.323451\pi\)
\(72\) 3.76302 0.443476
\(73\) 9.05652 1.05999 0.529993 0.848002i \(-0.322195\pi\)
0.529993 + 0.848002i \(0.322195\pi\)
\(74\) 1.03375 0.120170
\(75\) −24.6868 −2.85058
\(76\) −2.02416 −0.232187
\(77\) 1.43402 0.163421
\(78\) −3.15923 −0.357713
\(79\) −7.41490 −0.834241 −0.417121 0.908851i \(-0.636961\pi\)
−0.417121 + 0.908851i \(0.636961\pi\)
\(80\) −3.80694 −0.425629
\(81\) −6.12873 −0.680970
\(82\) 8.43673 0.931680
\(83\) 6.48221 0.711514 0.355757 0.934578i \(-0.384223\pi\)
0.355757 + 0.934578i \(0.384223\pi\)
\(84\) −1.26747 −0.138292
\(85\) −12.1467 −1.31749
\(86\) −1.24039 −0.133755
\(87\) −9.05241 −0.970520
\(88\) 2.94230 0.313650
\(89\) 16.3474 1.73282 0.866412 0.499330i \(-0.166420\pi\)
0.866412 + 0.499330i \(0.166420\pi\)
\(90\) −14.3256 −1.51005
\(91\) 0.592077 0.0620666
\(92\) 7.20116 0.750773
\(93\) 13.3866 1.38812
\(94\) 4.41272 0.455137
\(95\) 7.70585 0.790603
\(96\) −2.60058 −0.265421
\(97\) −2.51050 −0.254902 −0.127451 0.991845i \(-0.540680\pi\)
−0.127451 + 0.991845i \(0.540680\pi\)
\(98\) −6.76246 −0.683112
\(99\) 11.0719 1.11277
\(100\) 9.49279 0.949279
\(101\) 10.7868 1.07332 0.536662 0.843797i \(-0.319685\pi\)
0.536662 + 0.843797i \(0.319685\pi\)
\(102\) −8.29757 −0.821582
\(103\) −7.88460 −0.776893 −0.388446 0.921471i \(-0.626988\pi\)
−0.388446 + 0.921471i \(0.626988\pi\)
\(104\) 1.21482 0.119123
\(105\) 4.82518 0.470889
\(106\) 1.19619 0.116184
\(107\) 11.9088 1.15126 0.575632 0.817709i \(-0.304756\pi\)
0.575632 + 0.817709i \(0.304756\pi\)
\(108\) −1.98430 −0.190940
\(109\) 4.27981 0.409931 0.204966 0.978769i \(-0.434292\pi\)
0.204966 + 0.978769i \(0.434292\pi\)
\(110\) −11.2012 −1.06799
\(111\) −2.68834 −0.255166
\(112\) 0.487379 0.0460530
\(113\) 7.96219 0.749020 0.374510 0.927223i \(-0.377811\pi\)
0.374510 + 0.927223i \(0.377811\pi\)
\(114\) 5.26399 0.493017
\(115\) −27.4144 −2.55641
\(116\) 3.48092 0.323195
\(117\) 4.57139 0.422625
\(118\) −12.1673 −1.12009
\(119\) 1.55506 0.142552
\(120\) 9.90026 0.903766
\(121\) −2.34286 −0.212988
\(122\) 10.2849 0.931150
\(123\) −21.9404 −1.97830
\(124\) −5.14753 −0.462262
\(125\) −17.1038 −1.52981
\(126\) 1.83402 0.163387
\(127\) 18.5720 1.64800 0.824001 0.566588i \(-0.191737\pi\)
0.824001 + 0.566588i \(0.191737\pi\)
\(128\) 1.00000 0.0883883
\(129\) 3.22573 0.284010
\(130\) −4.62474 −0.405616
\(131\) −21.8120 −1.90573 −0.952863 0.303402i \(-0.901877\pi\)
−0.952863 + 0.303402i \(0.901877\pi\)
\(132\) −7.65169 −0.665994
\(133\) −0.986533 −0.0855432
\(134\) −11.3824 −0.983288
\(135\) 7.55412 0.650155
\(136\) 3.19066 0.273597
\(137\) −5.11308 −0.436840 −0.218420 0.975855i \(-0.570090\pi\)
−0.218420 + 0.975855i \(0.570090\pi\)
\(138\) −18.7272 −1.59417
\(139\) −11.9663 −1.01497 −0.507483 0.861662i \(-0.669424\pi\)
−0.507483 + 0.861662i \(0.669424\pi\)
\(140\) −1.85542 −0.156812
\(141\) −11.4756 −0.966422
\(142\) 8.87514 0.744785
\(143\) 3.57436 0.298903
\(144\) 3.76302 0.313585
\(145\) −13.2516 −1.10049
\(146\) 9.05652 0.749523
\(147\) 17.5863 1.45050
\(148\) 1.03375 0.0849733
\(149\) −23.7615 −1.94662 −0.973309 0.229497i \(-0.926292\pi\)
−0.973309 + 0.229497i \(0.926292\pi\)
\(150\) −24.6868 −2.01567
\(151\) 6.87628 0.559584 0.279792 0.960061i \(-0.409735\pi\)
0.279792 + 0.960061i \(0.409735\pi\)
\(152\) −2.02416 −0.164181
\(153\) 12.0065 0.970670
\(154\) 1.43402 0.115556
\(155\) 19.5963 1.57402
\(156\) −3.15923 −0.252941
\(157\) 18.7809 1.49888 0.749439 0.662074i \(-0.230323\pi\)
0.749439 + 0.662074i \(0.230323\pi\)
\(158\) −7.41490 −0.589898
\(159\) −3.11078 −0.246701
\(160\) −3.80694 −0.300965
\(161\) 3.50970 0.276603
\(162\) −6.12873 −0.481518
\(163\) 12.4265 0.973317 0.486659 0.873592i \(-0.338216\pi\)
0.486659 + 0.873592i \(0.338216\pi\)
\(164\) 8.43673 0.658798
\(165\) 29.1295 2.26773
\(166\) 6.48221 0.503117
\(167\) 17.6213 1.36358 0.681789 0.731549i \(-0.261202\pi\)
0.681789 + 0.731549i \(0.261202\pi\)
\(168\) −1.26747 −0.0977874
\(169\) −11.5242 −0.886478
\(170\) −12.1467 −0.931606
\(171\) −7.61695 −0.582483
\(172\) −1.24039 −0.0945788
\(173\) 6.38566 0.485493 0.242746 0.970090i \(-0.421952\pi\)
0.242746 + 0.970090i \(0.421952\pi\)
\(174\) −9.05241 −0.686261
\(175\) 4.62659 0.349737
\(176\) 2.94230 0.221784
\(177\) 31.6421 2.37837
\(178\) 16.3474 1.22529
\(179\) −17.6175 −1.31680 −0.658399 0.752669i \(-0.728766\pi\)
−0.658399 + 0.752669i \(0.728766\pi\)
\(180\) −14.3256 −1.06777
\(181\) 23.6018 1.75431 0.877155 0.480208i \(-0.159439\pi\)
0.877155 + 0.480208i \(0.159439\pi\)
\(182\) 0.592077 0.0438877
\(183\) −26.7467 −1.97717
\(184\) 7.20116 0.530877
\(185\) −3.93541 −0.289337
\(186\) 13.3866 0.981551
\(187\) 9.38788 0.686510
\(188\) 4.41272 0.321830
\(189\) −0.967108 −0.0703467
\(190\) 7.70585 0.559041
\(191\) −2.26422 −0.163833 −0.0819165 0.996639i \(-0.526104\pi\)
−0.0819165 + 0.996639i \(0.526104\pi\)
\(192\) −2.60058 −0.187681
\(193\) 9.40014 0.676637 0.338319 0.941032i \(-0.390142\pi\)
0.338319 + 0.941032i \(0.390142\pi\)
\(194\) −2.51050 −0.180243
\(195\) 12.0270 0.861272
\(196\) −6.76246 −0.483033
\(197\) 2.81802 0.200776 0.100388 0.994948i \(-0.467992\pi\)
0.100388 + 0.994948i \(0.467992\pi\)
\(198\) 11.0719 0.786849
\(199\) −14.8803 −1.05483 −0.527417 0.849606i \(-0.676840\pi\)
−0.527417 + 0.849606i \(0.676840\pi\)
\(200\) 9.49279 0.671242
\(201\) 29.6008 2.08788
\(202\) 10.7868 0.758955
\(203\) 1.69653 0.119073
\(204\) −8.29757 −0.580946
\(205\) −32.1181 −2.24323
\(206\) −7.88460 −0.549346
\(207\) 27.0981 1.88345
\(208\) 1.21482 0.0842325
\(209\) −5.95568 −0.411963
\(210\) 4.82518 0.332969
\(211\) −14.3512 −0.987980 −0.493990 0.869468i \(-0.664462\pi\)
−0.493990 + 0.869468i \(0.664462\pi\)
\(212\) 1.19619 0.0821544
\(213\) −23.0805 −1.58145
\(214\) 11.9088 0.814066
\(215\) 4.72209 0.322044
\(216\) −1.98430 −0.135015
\(217\) −2.50880 −0.170308
\(218\) 4.27981 0.289865
\(219\) −23.5522 −1.59151
\(220\) −11.2012 −0.755182
\(221\) 3.87607 0.260733
\(222\) −2.68834 −0.180429
\(223\) −4.41280 −0.295503 −0.147751 0.989025i \(-0.547204\pi\)
−0.147751 + 0.989025i \(0.547204\pi\)
\(224\) 0.487379 0.0325644
\(225\) 35.7216 2.38144
\(226\) 7.96219 0.529637
\(227\) −1.94853 −0.129329 −0.0646643 0.997907i \(-0.520598\pi\)
−0.0646643 + 0.997907i \(0.520598\pi\)
\(228\) 5.26399 0.348616
\(229\) 24.5536 1.62254 0.811272 0.584668i \(-0.198775\pi\)
0.811272 + 0.584668i \(0.198775\pi\)
\(230\) −27.4144 −1.80765
\(231\) −3.72928 −0.245368
\(232\) 3.48092 0.228533
\(233\) −21.7296 −1.42355 −0.711776 0.702406i \(-0.752109\pi\)
−0.711776 + 0.702406i \(0.752109\pi\)
\(234\) 4.57139 0.298841
\(235\) −16.7989 −1.09584
\(236\) −12.1673 −0.792026
\(237\) 19.2830 1.25257
\(238\) 1.55506 0.100800
\(239\) 11.3663 0.735227 0.367613 0.929979i \(-0.380175\pi\)
0.367613 + 0.929979i \(0.380175\pi\)
\(240\) 9.90026 0.639059
\(241\) 2.21590 0.142739 0.0713694 0.997450i \(-0.477263\pi\)
0.0713694 + 0.997450i \(0.477263\pi\)
\(242\) −2.34286 −0.150605
\(243\) 21.8912 1.40432
\(244\) 10.2849 0.658423
\(245\) 25.7443 1.64474
\(246\) −21.9404 −1.39887
\(247\) −2.45898 −0.156461
\(248\) −5.14753 −0.326869
\(249\) −16.8575 −1.06830
\(250\) −17.1038 −1.08174
\(251\) −7.23929 −0.456940 −0.228470 0.973551i \(-0.573372\pi\)
−0.228470 + 0.973551i \(0.573372\pi\)
\(252\) 1.83402 0.115532
\(253\) 21.1880 1.33208
\(254\) 18.5720 1.16531
\(255\) 31.5884 1.97814
\(256\) 1.00000 0.0625000
\(257\) −18.7926 −1.17225 −0.586125 0.810221i \(-0.699347\pi\)
−0.586125 + 0.810221i \(0.699347\pi\)
\(258\) 3.22573 0.200825
\(259\) 0.503826 0.0313062
\(260\) −4.62474 −0.286814
\(261\) 13.0988 0.810794
\(262\) −21.8120 −1.34755
\(263\) −23.0460 −1.42108 −0.710538 0.703658i \(-0.751549\pi\)
−0.710538 + 0.703658i \(0.751549\pi\)
\(264\) −7.65169 −0.470929
\(265\) −4.55381 −0.279738
\(266\) −0.986533 −0.0604882
\(267\) −42.5128 −2.60174
\(268\) −11.3824 −0.695290
\(269\) −1.00000 −0.0609711
\(270\) 7.55412 0.459729
\(271\) −5.87186 −0.356690 −0.178345 0.983968i \(-0.557074\pi\)
−0.178345 + 0.983968i \(0.557074\pi\)
\(272\) 3.19066 0.193462
\(273\) −1.53974 −0.0931896
\(274\) −5.11308 −0.308892
\(275\) 27.9306 1.68428
\(276\) −18.7272 −1.12725
\(277\) −5.00802 −0.300903 −0.150452 0.988617i \(-0.548073\pi\)
−0.150452 + 0.988617i \(0.548073\pi\)
\(278\) −11.9663 −0.717689
\(279\) −19.3703 −1.15967
\(280\) −1.85542 −0.110883
\(281\) 24.4842 1.46061 0.730303 0.683124i \(-0.239379\pi\)
0.730303 + 0.683124i \(0.239379\pi\)
\(282\) −11.4756 −0.683364
\(283\) −4.16006 −0.247290 −0.123645 0.992327i \(-0.539458\pi\)
−0.123645 + 0.992327i \(0.539458\pi\)
\(284\) 8.87514 0.526643
\(285\) −20.0397 −1.18705
\(286\) 3.57436 0.211356
\(287\) 4.11189 0.242717
\(288\) 3.76302 0.221738
\(289\) −6.81969 −0.401158
\(290\) −13.2516 −0.778163
\(291\) 6.52875 0.382722
\(292\) 9.05652 0.529993
\(293\) −20.6328 −1.20538 −0.602691 0.797975i \(-0.705905\pi\)
−0.602691 + 0.797975i \(0.705905\pi\)
\(294\) 17.5863 1.02566
\(295\) 46.3203 2.69687
\(296\) 1.03375 0.0600852
\(297\) −5.83841 −0.338779
\(298\) −23.7615 −1.37647
\(299\) 8.74810 0.505916
\(300\) −24.6868 −1.42529
\(301\) −0.604540 −0.0348451
\(302\) 6.87628 0.395685
\(303\) −28.0519 −1.61154
\(304\) −2.02416 −0.116093
\(305\) −39.1540 −2.24195
\(306\) 12.0065 0.686367
\(307\) 30.3443 1.73184 0.865919 0.500184i \(-0.166734\pi\)
0.865919 + 0.500184i \(0.166734\pi\)
\(308\) 1.43402 0.0817107
\(309\) 20.5045 1.16646
\(310\) 19.5963 1.11300
\(311\) −12.2874 −0.696755 −0.348377 0.937354i \(-0.613267\pi\)
−0.348377 + 0.937354i \(0.613267\pi\)
\(312\) −3.15923 −0.178856
\(313\) −10.8210 −0.611641 −0.305821 0.952089i \(-0.598931\pi\)
−0.305821 + 0.952089i \(0.598931\pi\)
\(314\) 18.7809 1.05987
\(315\) −6.98200 −0.393391
\(316\) −7.41490 −0.417121
\(317\) −1.55660 −0.0874273 −0.0437137 0.999044i \(-0.513919\pi\)
−0.0437137 + 0.999044i \(0.513919\pi\)
\(318\) −3.11078 −0.174444
\(319\) 10.2419 0.573437
\(320\) −3.80694 −0.212814
\(321\) −30.9697 −1.72856
\(322\) 3.50970 0.195588
\(323\) −6.45840 −0.359355
\(324\) −6.12873 −0.340485
\(325\) 11.5320 0.639681
\(326\) 12.4265 0.688239
\(327\) −11.1300 −0.615490
\(328\) 8.43673 0.465840
\(329\) 2.15067 0.118570
\(330\) 29.1295 1.60353
\(331\) 4.76335 0.261818 0.130909 0.991394i \(-0.458210\pi\)
0.130909 + 0.991394i \(0.458210\pi\)
\(332\) 6.48221 0.355757
\(333\) 3.89001 0.213171
\(334\) 17.6213 0.964195
\(335\) 43.3320 2.36748
\(336\) −1.26747 −0.0691461
\(337\) 20.9918 1.14350 0.571749 0.820429i \(-0.306265\pi\)
0.571749 + 0.820429i \(0.306265\pi\)
\(338\) −11.5242 −0.626835
\(339\) −20.7063 −1.12461
\(340\) −12.1467 −0.658745
\(341\) −15.1456 −0.820180
\(342\) −7.61695 −0.411878
\(343\) −6.70754 −0.362173
\(344\) −1.24039 −0.0668773
\(345\) 71.2933 3.83830
\(346\) 6.38566 0.343295
\(347\) 28.6045 1.53557 0.767784 0.640709i \(-0.221359\pi\)
0.767784 + 0.640709i \(0.221359\pi\)
\(348\) −9.05241 −0.485260
\(349\) −13.6612 −0.731268 −0.365634 0.930759i \(-0.619148\pi\)
−0.365634 + 0.930759i \(0.619148\pi\)
\(350\) 4.62659 0.247302
\(351\) −2.41057 −0.128666
\(352\) 2.94230 0.156825
\(353\) 23.8109 1.26733 0.633664 0.773609i \(-0.281550\pi\)
0.633664 + 0.773609i \(0.281550\pi\)
\(354\) 31.6421 1.68176
\(355\) −33.7871 −1.79323
\(356\) 16.3474 0.866412
\(357\) −4.04407 −0.214035
\(358\) −17.6175 −0.931116
\(359\) −11.2449 −0.593483 −0.296742 0.954958i \(-0.595900\pi\)
−0.296742 + 0.954958i \(0.595900\pi\)
\(360\) −14.3256 −0.755025
\(361\) −14.9028 −0.784357
\(362\) 23.6018 1.24048
\(363\) 6.09281 0.319790
\(364\) 0.592077 0.0310333
\(365\) −34.4776 −1.80464
\(366\) −26.7467 −1.39807
\(367\) −21.1011 −1.10147 −0.550735 0.834680i \(-0.685653\pi\)
−0.550735 + 0.834680i \(0.685653\pi\)
\(368\) 7.20116 0.375387
\(369\) 31.7476 1.65271
\(370\) −3.93541 −0.204592
\(371\) 0.582997 0.0302677
\(372\) 13.3866 0.694062
\(373\) −33.9513 −1.75793 −0.878967 0.476883i \(-0.841766\pi\)
−0.878967 + 0.476883i \(0.841766\pi\)
\(374\) 9.38788 0.485436
\(375\) 44.4798 2.29693
\(376\) 4.41272 0.227569
\(377\) 4.22868 0.217788
\(378\) −0.967108 −0.0497427
\(379\) 4.79211 0.246154 0.123077 0.992397i \(-0.460724\pi\)
0.123077 + 0.992397i \(0.460724\pi\)
\(380\) 7.70585 0.395302
\(381\) −48.2981 −2.47439
\(382\) −2.26422 −0.115847
\(383\) 32.6191 1.66676 0.833378 0.552704i \(-0.186404\pi\)
0.833378 + 0.552704i \(0.186404\pi\)
\(384\) −2.60058 −0.132710
\(385\) −5.45922 −0.278227
\(386\) 9.40014 0.478455
\(387\) −4.66761 −0.237268
\(388\) −2.51050 −0.127451
\(389\) 25.3685 1.28623 0.643117 0.765768i \(-0.277641\pi\)
0.643117 + 0.765768i \(0.277641\pi\)
\(390\) 12.0270 0.609011
\(391\) 22.9765 1.16197
\(392\) −6.76246 −0.341556
\(393\) 56.7239 2.86134
\(394\) 2.81802 0.141970
\(395\) 28.2281 1.42031
\(396\) 11.0719 0.556386
\(397\) −28.8866 −1.44978 −0.724889 0.688866i \(-0.758109\pi\)
−0.724889 + 0.688866i \(0.758109\pi\)
\(398\) −14.8803 −0.745881
\(399\) 2.56556 0.128439
\(400\) 9.49279 0.474639
\(401\) −16.8718 −0.842536 −0.421268 0.906936i \(-0.638415\pi\)
−0.421268 + 0.906936i \(0.638415\pi\)
\(402\) 29.6008 1.47635
\(403\) −6.25331 −0.311500
\(404\) 10.7868 0.536662
\(405\) 23.3317 1.15936
\(406\) 1.69653 0.0841973
\(407\) 3.04159 0.150766
\(408\) −8.29757 −0.410791
\(409\) −8.98467 −0.444263 −0.222132 0.975017i \(-0.571301\pi\)
−0.222132 + 0.975017i \(0.571301\pi\)
\(410\) −32.1181 −1.58620
\(411\) 13.2970 0.655891
\(412\) −7.88460 −0.388446
\(413\) −5.93011 −0.291802
\(414\) 27.0981 1.33180
\(415\) −24.6774 −1.21136
\(416\) 1.21482 0.0595614
\(417\) 31.1193 1.52392
\(418\) −5.95568 −0.291302
\(419\) −32.3970 −1.58270 −0.791348 0.611366i \(-0.790620\pi\)
−0.791348 + 0.611366i \(0.790620\pi\)
\(420\) 4.82518 0.235445
\(421\) 6.47692 0.315666 0.157833 0.987466i \(-0.449549\pi\)
0.157833 + 0.987466i \(0.449549\pi\)
\(422\) −14.3512 −0.698608
\(423\) 16.6052 0.807370
\(424\) 1.19619 0.0580919
\(425\) 30.2883 1.46920
\(426\) −23.0805 −1.11826
\(427\) 5.01264 0.242579
\(428\) 11.9088 0.575632
\(429\) −9.29541 −0.448787
\(430\) 4.72209 0.227719
\(431\) 12.7851 0.615836 0.307918 0.951413i \(-0.400368\pi\)
0.307918 + 0.951413i \(0.400368\pi\)
\(432\) −1.98430 −0.0954698
\(433\) −17.4222 −0.837256 −0.418628 0.908158i \(-0.637489\pi\)
−0.418628 + 0.908158i \(0.637489\pi\)
\(434\) −2.50880 −0.120426
\(435\) 34.4620 1.65233
\(436\) 4.27981 0.204966
\(437\) −14.5763 −0.697278
\(438\) −23.5522 −1.12537
\(439\) −4.52030 −0.215742 −0.107871 0.994165i \(-0.534403\pi\)
−0.107871 + 0.994165i \(0.534403\pi\)
\(440\) −11.2012 −0.533994
\(441\) −25.4473 −1.21178
\(442\) 3.87607 0.184366
\(443\) −13.2163 −0.627926 −0.313963 0.949435i \(-0.601657\pi\)
−0.313963 + 0.949435i \(0.601657\pi\)
\(444\) −2.68834 −0.127583
\(445\) −62.2337 −2.95016
\(446\) −4.41280 −0.208952
\(447\) 61.7937 2.92274
\(448\) 0.487379 0.0230265
\(449\) −10.4441 −0.492887 −0.246443 0.969157i \(-0.579262\pi\)
−0.246443 + 0.969157i \(0.579262\pi\)
\(450\) 35.7216 1.68393
\(451\) 24.8234 1.16889
\(452\) 7.96219 0.374510
\(453\) −17.8823 −0.840185
\(454\) −1.94853 −0.0914491
\(455\) −2.25400 −0.105669
\(456\) 5.26399 0.246509
\(457\) 10.7134 0.501153 0.250577 0.968097i \(-0.419380\pi\)
0.250577 + 0.968097i \(0.419380\pi\)
\(458\) 24.5536 1.14731
\(459\) −6.33123 −0.295517
\(460\) −27.4144 −1.27820
\(461\) −20.5805 −0.958530 −0.479265 0.877670i \(-0.659097\pi\)
−0.479265 + 0.877670i \(0.659097\pi\)
\(462\) −3.72928 −0.173502
\(463\) −20.2894 −0.942930 −0.471465 0.881885i \(-0.656275\pi\)
−0.471465 + 0.881885i \(0.656275\pi\)
\(464\) 3.48092 0.161598
\(465\) −50.9619 −2.36330
\(466\) −21.7296 −1.00660
\(467\) −12.8280 −0.593609 −0.296805 0.954938i \(-0.595921\pi\)
−0.296805 + 0.954938i \(0.595921\pi\)
\(468\) 4.57139 0.211312
\(469\) −5.54754 −0.256162
\(470\) −16.7989 −0.774878
\(471\) −48.8412 −2.25048
\(472\) −12.1673 −0.560047
\(473\) −3.64960 −0.167809
\(474\) 19.2830 0.885699
\(475\) −19.2149 −0.881640
\(476\) 1.55506 0.0712762
\(477\) 4.50128 0.206099
\(478\) 11.3663 0.519884
\(479\) 36.1270 1.65069 0.825343 0.564631i \(-0.190982\pi\)
0.825343 + 0.564631i \(0.190982\pi\)
\(480\) 9.90026 0.451883
\(481\) 1.25581 0.0572601
\(482\) 2.21590 0.100932
\(483\) −9.12726 −0.415304
\(484\) −2.34286 −0.106494
\(485\) 9.55731 0.433975
\(486\) 21.8912 0.993003
\(487\) 11.1794 0.506586 0.253293 0.967390i \(-0.418486\pi\)
0.253293 + 0.967390i \(0.418486\pi\)
\(488\) 10.2849 0.465575
\(489\) −32.3161 −1.46138
\(490\) 25.7443 1.16301
\(491\) 9.49758 0.428620 0.214310 0.976766i \(-0.431250\pi\)
0.214310 + 0.976766i \(0.431250\pi\)
\(492\) −21.9404 −0.989149
\(493\) 11.1064 0.500208
\(494\) −2.45898 −0.110635
\(495\) −42.1502 −1.89451
\(496\) −5.14753 −0.231131
\(497\) 4.32556 0.194028
\(498\) −16.8575 −0.755403
\(499\) 20.4507 0.915498 0.457749 0.889081i \(-0.348656\pi\)
0.457749 + 0.889081i \(0.348656\pi\)
\(500\) −17.1038 −0.764904
\(501\) −45.8257 −2.04734
\(502\) −7.23929 −0.323106
\(503\) −5.31796 −0.237116 −0.118558 0.992947i \(-0.537827\pi\)
−0.118558 + 0.992947i \(0.537827\pi\)
\(504\) 1.83402 0.0816937
\(505\) −41.0646 −1.82735
\(506\) 21.1880 0.941921
\(507\) 29.9697 1.33100
\(508\) 18.5720 0.824001
\(509\) 17.2984 0.766738 0.383369 0.923595i \(-0.374764\pi\)
0.383369 + 0.923595i \(0.374764\pi\)
\(510\) 31.5884 1.39876
\(511\) 4.41396 0.195262
\(512\) 1.00000 0.0441942
\(513\) 4.01654 0.177335
\(514\) −18.7926 −0.828906
\(515\) 30.0162 1.32267
\(516\) 3.22573 0.142005
\(517\) 12.9835 0.571016
\(518\) 0.503826 0.0221368
\(519\) −16.6064 −0.728941
\(520\) −4.62474 −0.202808
\(521\) −38.1750 −1.67248 −0.836239 0.548364i \(-0.815251\pi\)
−0.836239 + 0.548364i \(0.815251\pi\)
\(522\) 13.0988 0.573318
\(523\) −29.2320 −1.27823 −0.639113 0.769113i \(-0.720698\pi\)
−0.639113 + 0.769113i \(0.720698\pi\)
\(524\) −21.8120 −0.952863
\(525\) −12.0318 −0.525112
\(526\) −23.0460 −1.00485
\(527\) −16.4240 −0.715442
\(528\) −7.65169 −0.332997
\(529\) 28.8567 1.25464
\(530\) −4.55381 −0.197805
\(531\) −45.7860 −1.98694
\(532\) −0.986533 −0.0427716
\(533\) 10.2491 0.443937
\(534\) −42.5128 −1.83971
\(535\) −45.3359 −1.96004
\(536\) −11.3824 −0.491644
\(537\) 45.8159 1.97710
\(538\) −1.00000 −0.0431131
\(539\) −19.8972 −0.857033
\(540\) 7.55412 0.325077
\(541\) −33.8660 −1.45601 −0.728007 0.685569i \(-0.759553\pi\)
−0.728007 + 0.685569i \(0.759553\pi\)
\(542\) −5.87186 −0.252218
\(543\) −61.3785 −2.63400
\(544\) 3.19066 0.136798
\(545\) −16.2930 −0.697915
\(546\) −1.53974 −0.0658950
\(547\) −33.7387 −1.44256 −0.721280 0.692643i \(-0.756446\pi\)
−0.721280 + 0.692643i \(0.756446\pi\)
\(548\) −5.11308 −0.218420
\(549\) 38.7023 1.65177
\(550\) 27.9306 1.19097
\(551\) −7.04593 −0.300167
\(552\) −18.7272 −0.797083
\(553\) −3.61387 −0.153677
\(554\) −5.00802 −0.212771
\(555\) 10.2343 0.434424
\(556\) −11.9663 −0.507483
\(557\) −6.11043 −0.258907 −0.129454 0.991585i \(-0.541322\pi\)
−0.129454 + 0.991585i \(0.541322\pi\)
\(558\) −19.3703 −0.820009
\(559\) −1.50685 −0.0637329
\(560\) −1.85542 −0.0784060
\(561\) −24.4140 −1.03076
\(562\) 24.4842 1.03280
\(563\) 36.4999 1.53829 0.769143 0.639077i \(-0.220683\pi\)
0.769143 + 0.639077i \(0.220683\pi\)
\(564\) −11.4756 −0.483211
\(565\) −30.3116 −1.27522
\(566\) −4.16006 −0.174861
\(567\) −2.98702 −0.125443
\(568\) 8.87514 0.372393
\(569\) −12.5604 −0.526558 −0.263279 0.964720i \(-0.584804\pi\)
−0.263279 + 0.964720i \(0.584804\pi\)
\(570\) −20.0397 −0.839370
\(571\) 29.0079 1.21394 0.606972 0.794724i \(-0.292384\pi\)
0.606972 + 0.794724i \(0.292384\pi\)
\(572\) 3.57436 0.149452
\(573\) 5.88828 0.245986
\(574\) 4.11189 0.171627
\(575\) 68.3591 2.85077
\(576\) 3.76302 0.156793
\(577\) −41.1128 −1.71155 −0.855775 0.517348i \(-0.826919\pi\)
−0.855775 + 0.517348i \(0.826919\pi\)
\(578\) −6.81969 −0.283662
\(579\) −24.4458 −1.01593
\(580\) −13.2516 −0.550245
\(581\) 3.15929 0.131070
\(582\) 6.52875 0.270625
\(583\) 3.51954 0.145764
\(584\) 9.05652 0.374762
\(585\) −17.4030 −0.719525
\(586\) −20.6328 −0.852334
\(587\) −29.9889 −1.23777 −0.618887 0.785480i \(-0.712416\pi\)
−0.618887 + 0.785480i \(0.712416\pi\)
\(588\) 17.5863 0.725248
\(589\) 10.4194 0.429325
\(590\) 46.3203 1.90698
\(591\) −7.32849 −0.301454
\(592\) 1.03375 0.0424867
\(593\) −4.55260 −0.186953 −0.0934765 0.995621i \(-0.529798\pi\)
−0.0934765 + 0.995621i \(0.529798\pi\)
\(594\) −5.83841 −0.239553
\(595\) −5.92003 −0.242697
\(596\) −23.7615 −0.973309
\(597\) 38.6974 1.58378
\(598\) 8.74810 0.357736
\(599\) −39.5205 −1.61477 −0.807383 0.590028i \(-0.799117\pi\)
−0.807383 + 0.590028i \(0.799117\pi\)
\(600\) −24.6868 −1.00783
\(601\) 32.0934 1.30912 0.654558 0.756011i \(-0.272855\pi\)
0.654558 + 0.756011i \(0.272855\pi\)
\(602\) −0.604540 −0.0246392
\(603\) −42.8322 −1.74426
\(604\) 6.87628 0.279792
\(605\) 8.91915 0.362615
\(606\) −28.0519 −1.13953
\(607\) −4.60057 −0.186732 −0.0933658 0.995632i \(-0.529763\pi\)
−0.0933658 + 0.995632i \(0.529763\pi\)
\(608\) −2.02416 −0.0820904
\(609\) −4.41196 −0.178782
\(610\) −39.1540 −1.58530
\(611\) 5.36065 0.216869
\(612\) 12.0065 0.485335
\(613\) 12.2463 0.494622 0.247311 0.968936i \(-0.420453\pi\)
0.247311 + 0.968936i \(0.420453\pi\)
\(614\) 30.3443 1.22459
\(615\) 83.5257 3.36808
\(616\) 1.43402 0.0577782
\(617\) 46.4794 1.87119 0.935595 0.353076i \(-0.114864\pi\)
0.935595 + 0.353076i \(0.114864\pi\)
\(618\) 20.5045 0.824814
\(619\) −36.5698 −1.46986 −0.734932 0.678141i \(-0.762786\pi\)
−0.734932 + 0.678141i \(0.762786\pi\)
\(620\) 19.5963 0.787008
\(621\) −14.2893 −0.573409
\(622\) −12.2874 −0.492680
\(623\) 7.96740 0.319207
\(624\) −3.15923 −0.126471
\(625\) 17.6491 0.705964
\(626\) −10.8210 −0.432496
\(627\) 15.4882 0.618540
\(628\) 18.7809 0.749439
\(629\) 3.29833 0.131513
\(630\) −6.98200 −0.278170
\(631\) −21.2855 −0.847363 −0.423681 0.905811i \(-0.639262\pi\)
−0.423681 + 0.905811i \(0.639262\pi\)
\(632\) −7.41490 −0.294949
\(633\) 37.3216 1.48340
\(634\) −1.55660 −0.0618205
\(635\) −70.7026 −2.80575
\(636\) −3.11078 −0.123350
\(637\) −8.21516 −0.325496
\(638\) 10.2419 0.405481
\(639\) 33.3974 1.32118
\(640\) −3.80694 −0.150483
\(641\) 0.625521 0.0247066 0.0123533 0.999924i \(-0.496068\pi\)
0.0123533 + 0.999924i \(0.496068\pi\)
\(642\) −30.9697 −1.22228
\(643\) 48.5331 1.91396 0.956980 0.290155i \(-0.0937069\pi\)
0.956980 + 0.290155i \(0.0937069\pi\)
\(644\) 3.50970 0.138301
\(645\) −12.2802 −0.483531
\(646\) −6.45840 −0.254102
\(647\) 18.6403 0.732825 0.366412 0.930453i \(-0.380586\pi\)
0.366412 + 0.930453i \(0.380586\pi\)
\(648\) −6.12873 −0.240759
\(649\) −35.8000 −1.40527
\(650\) 11.5320 0.452323
\(651\) 6.52434 0.255709
\(652\) 12.4265 0.486659
\(653\) 2.06115 0.0806589 0.0403294 0.999186i \(-0.487159\pi\)
0.0403294 + 0.999186i \(0.487159\pi\)
\(654\) −11.1300 −0.435217
\(655\) 83.0370 3.24453
\(656\) 8.43673 0.329399
\(657\) 34.0799 1.32958
\(658\) 2.15067 0.0838417
\(659\) 16.6159 0.647263 0.323632 0.946183i \(-0.395096\pi\)
0.323632 + 0.946183i \(0.395096\pi\)
\(660\) 29.1295 1.13387
\(661\) −21.7435 −0.845725 −0.422863 0.906194i \(-0.638975\pi\)
−0.422863 + 0.906194i \(0.638975\pi\)
\(662\) 4.76335 0.185133
\(663\) −10.0800 −0.391476
\(664\) 6.48221 0.251558
\(665\) 3.75567 0.145639
\(666\) 3.89001 0.150735
\(667\) 25.0667 0.970585
\(668\) 17.6213 0.681789
\(669\) 11.4758 0.443682
\(670\) 43.3320 1.67406
\(671\) 30.2612 1.16822
\(672\) −1.26747 −0.0488937
\(673\) 22.4637 0.865913 0.432956 0.901415i \(-0.357470\pi\)
0.432956 + 0.901415i \(0.357470\pi\)
\(674\) 20.9918 0.808575
\(675\) −18.8366 −0.725020
\(676\) −11.5242 −0.443239
\(677\) −9.44261 −0.362909 −0.181454 0.983399i \(-0.558080\pi\)
−0.181454 + 0.983399i \(0.558080\pi\)
\(678\) −20.7063 −0.795221
\(679\) −1.22356 −0.0469561
\(680\) −12.1467 −0.465803
\(681\) 5.06731 0.194180
\(682\) −15.1456 −0.579954
\(683\) 9.14788 0.350034 0.175017 0.984565i \(-0.444002\pi\)
0.175017 + 0.984565i \(0.444002\pi\)
\(684\) −7.61695 −0.291241
\(685\) 19.4652 0.743726
\(686\) −6.70754 −0.256095
\(687\) −63.8535 −2.43616
\(688\) −1.24039 −0.0472894
\(689\) 1.45315 0.0553606
\(690\) 71.2933 2.71409
\(691\) −6.73151 −0.256079 −0.128039 0.991769i \(-0.540868\pi\)
−0.128039 + 0.991769i \(0.540868\pi\)
\(692\) 6.38566 0.242746
\(693\) 5.39624 0.204986
\(694\) 28.6045 1.08581
\(695\) 45.5549 1.72799
\(696\) −9.05241 −0.343131
\(697\) 26.9187 1.01962
\(698\) −13.6612 −0.517084
\(699\) 56.5095 2.13739
\(700\) 4.62659 0.174869
\(701\) 26.4298 0.998240 0.499120 0.866533i \(-0.333657\pi\)
0.499120 + 0.866533i \(0.333657\pi\)
\(702\) −2.41057 −0.0909809
\(703\) −2.09246 −0.0789187
\(704\) 2.94230 0.110892
\(705\) 43.6870 1.64535
\(706\) 23.8109 0.896136
\(707\) 5.25725 0.197719
\(708\) 31.6421 1.18918
\(709\) 14.0274 0.526810 0.263405 0.964685i \(-0.415154\pi\)
0.263405 + 0.964685i \(0.415154\pi\)
\(710\) −33.7871 −1.26801
\(711\) −27.9024 −1.04642
\(712\) 16.3474 0.612646
\(713\) −37.0682 −1.38822
\(714\) −4.04407 −0.151345
\(715\) −13.6074 −0.508887
\(716\) −17.6175 −0.658399
\(717\) −29.5591 −1.10390
\(718\) −11.2449 −0.419656
\(719\) −27.6353 −1.03062 −0.515312 0.857003i \(-0.672324\pi\)
−0.515312 + 0.857003i \(0.672324\pi\)
\(720\) −14.3256 −0.533884
\(721\) −3.84279 −0.143113
\(722\) −14.9028 −0.554624
\(723\) −5.76263 −0.214315
\(724\) 23.6018 0.877155
\(725\) 33.0436 1.22721
\(726\) 6.09281 0.226125
\(727\) −20.4908 −0.759962 −0.379981 0.924994i \(-0.624070\pi\)
−0.379981 + 0.924994i \(0.624070\pi\)
\(728\) 0.592077 0.0219438
\(729\) −38.5436 −1.42754
\(730\) −34.4776 −1.27607
\(731\) −3.95766 −0.146379
\(732\) −26.7467 −0.988586
\(733\) 23.4829 0.867360 0.433680 0.901067i \(-0.357215\pi\)
0.433680 + 0.901067i \(0.357215\pi\)
\(734\) −21.1011 −0.778857
\(735\) −66.9501 −2.46949
\(736\) 7.20116 0.265438
\(737\) −33.4904 −1.23363
\(738\) 31.7476 1.16864
\(739\) −1.54880 −0.0569737 −0.0284868 0.999594i \(-0.509069\pi\)
−0.0284868 + 0.999594i \(0.509069\pi\)
\(740\) −3.93541 −0.144668
\(741\) 6.39479 0.234918
\(742\) 0.582997 0.0214025
\(743\) 6.56645 0.240900 0.120450 0.992719i \(-0.461566\pi\)
0.120450 + 0.992719i \(0.461566\pi\)
\(744\) 13.3866 0.490776
\(745\) 90.4586 3.31415
\(746\) −33.9513 −1.24305
\(747\) 24.3927 0.892482
\(748\) 9.38788 0.343255
\(749\) 5.80408 0.212077
\(750\) 44.4798 1.62417
\(751\) −29.3323 −1.07035 −0.535175 0.844741i \(-0.679754\pi\)
−0.535175 + 0.844741i \(0.679754\pi\)
\(752\) 4.41272 0.160915
\(753\) 18.8264 0.686071
\(754\) 4.22868 0.154000
\(755\) −26.1776 −0.952700
\(756\) −0.967108 −0.0351734
\(757\) 2.63020 0.0955962 0.0477981 0.998857i \(-0.484780\pi\)
0.0477981 + 0.998857i \(0.484780\pi\)
\(758\) 4.79211 0.174057
\(759\) −55.1011 −2.00004
\(760\) 7.70585 0.279520
\(761\) −24.8658 −0.901385 −0.450693 0.892679i \(-0.648823\pi\)
−0.450693 + 0.892679i \(0.648823\pi\)
\(762\) −48.2981 −1.74966
\(763\) 2.08589 0.0755143
\(764\) −2.26422 −0.0819165
\(765\) −45.7081 −1.65258
\(766\) 32.6191 1.17857
\(767\) −14.7811 −0.533714
\(768\) −2.60058 −0.0938404
\(769\) 37.4087 1.34899 0.674496 0.738279i \(-0.264361\pi\)
0.674496 + 0.738279i \(0.264361\pi\)
\(770\) −5.45922 −0.196736
\(771\) 48.8717 1.76007
\(772\) 9.40014 0.338319
\(773\) −7.67556 −0.276071 −0.138035 0.990427i \(-0.544079\pi\)
−0.138035 + 0.990427i \(0.544079\pi\)
\(774\) −4.66761 −0.167774
\(775\) −48.8644 −1.75526
\(776\) −2.51050 −0.0901215
\(777\) −1.31024 −0.0470046
\(778\) 25.3685 0.909505
\(779\) −17.0773 −0.611856
\(780\) 12.0270 0.430636
\(781\) 26.1133 0.934409
\(782\) 22.9765 0.821637
\(783\) −6.90719 −0.246843
\(784\) −6.76246 −0.241516
\(785\) −71.4977 −2.55186
\(786\) 56.7239 2.02328
\(787\) −10.9680 −0.390966 −0.195483 0.980707i \(-0.562627\pi\)
−0.195483 + 0.980707i \(0.562627\pi\)
\(788\) 2.81802 0.100388
\(789\) 59.9330 2.13367
\(790\) 28.2281 1.00431
\(791\) 3.88061 0.137978
\(792\) 11.0719 0.393424
\(793\) 12.4943 0.443685
\(794\) −28.8866 −1.02515
\(795\) 11.8426 0.420012
\(796\) −14.8803 −0.527417
\(797\) 45.7135 1.61926 0.809628 0.586943i \(-0.199669\pi\)
0.809628 + 0.586943i \(0.199669\pi\)
\(798\) 2.56556 0.0908198
\(799\) 14.0795 0.498096
\(800\) 9.49279 0.335621
\(801\) 61.5157 2.17355
\(802\) −16.8718 −0.595763
\(803\) 26.6470 0.940353
\(804\) 29.6008 1.04394
\(805\) −13.3612 −0.470921
\(806\) −6.25331 −0.220264
\(807\) 2.60058 0.0915448
\(808\) 10.7868 0.379478
\(809\) −12.8403 −0.451441 −0.225721 0.974192i \(-0.572474\pi\)
−0.225721 + 0.974192i \(0.572474\pi\)
\(810\) 23.3317 0.819792
\(811\) 44.9655 1.57895 0.789475 0.613782i \(-0.210353\pi\)
0.789475 + 0.613782i \(0.210353\pi\)
\(812\) 1.69653 0.0595365
\(813\) 15.2702 0.535551
\(814\) 3.04159 0.106608
\(815\) −47.3069 −1.65709
\(816\) −8.29757 −0.290473
\(817\) 2.51074 0.0878398
\(818\) −8.98467 −0.314141
\(819\) 2.22800 0.0778526
\(820\) −32.1181 −1.12161
\(821\) 48.4659 1.69147 0.845736 0.533602i \(-0.179162\pi\)
0.845736 + 0.533602i \(0.179162\pi\)
\(822\) 13.2970 0.463785
\(823\) −47.5382 −1.65708 −0.828539 0.559932i \(-0.810827\pi\)
−0.828539 + 0.559932i \(0.810827\pi\)
\(824\) −7.88460 −0.274673
\(825\) −72.6359 −2.52886
\(826\) −5.93011 −0.206335
\(827\) −8.93030 −0.310537 −0.155268 0.987872i \(-0.549624\pi\)
−0.155268 + 0.987872i \(0.549624\pi\)
\(828\) 27.0981 0.941725
\(829\) 34.4302 1.19581 0.597905 0.801567i \(-0.296000\pi\)
0.597905 + 0.801567i \(0.296000\pi\)
\(830\) −24.6774 −0.856564
\(831\) 13.0238 0.451790
\(832\) 1.21482 0.0421162
\(833\) −21.5767 −0.747589
\(834\) 31.1193 1.07757
\(835\) −67.0833 −2.32151
\(836\) −5.95568 −0.205982
\(837\) 10.2143 0.353056
\(838\) −32.3970 −1.11914
\(839\) 54.7089 1.88876 0.944381 0.328853i \(-0.106662\pi\)
0.944381 + 0.328853i \(0.106662\pi\)
\(840\) 4.82518 0.166485
\(841\) −16.8832 −0.582180
\(842\) 6.47692 0.223209
\(843\) −63.6732 −2.19302
\(844\) −14.3512 −0.493990
\(845\) 43.8720 1.50924
\(846\) 16.6052 0.570897
\(847\) −1.14186 −0.0392349
\(848\) 1.19619 0.0410772
\(849\) 10.8186 0.371293
\(850\) 30.2883 1.03888
\(851\) 7.44417 0.255183
\(852\) −23.0805 −0.790726
\(853\) 52.4297 1.79516 0.897579 0.440853i \(-0.145324\pi\)
0.897579 + 0.440853i \(0.145324\pi\)
\(854\) 5.01264 0.171529
\(855\) 28.9973 0.991686
\(856\) 11.9088 0.407033
\(857\) 14.4635 0.494065 0.247033 0.969007i \(-0.420545\pi\)
0.247033 + 0.969007i \(0.420545\pi\)
\(858\) −9.29541 −0.317340
\(859\) −14.5781 −0.497398 −0.248699 0.968581i \(-0.580003\pi\)
−0.248699 + 0.968581i \(0.580003\pi\)
\(860\) 4.72209 0.161022
\(861\) −10.6933 −0.364426
\(862\) 12.7851 0.435462
\(863\) 45.1782 1.53788 0.768941 0.639319i \(-0.220784\pi\)
0.768941 + 0.639319i \(0.220784\pi\)
\(864\) −1.98430 −0.0675073
\(865\) −24.3098 −0.826558
\(866\) −17.4222 −0.592030
\(867\) 17.7351 0.602317
\(868\) −2.50880 −0.0851542
\(869\) −21.8169 −0.740086
\(870\) 34.4620 1.16837
\(871\) −13.8275 −0.468528
\(872\) 4.27981 0.144933
\(873\) −9.44705 −0.319734
\(874\) −14.5763 −0.493050
\(875\) −8.33603 −0.281809
\(876\) −23.5522 −0.795756
\(877\) 22.1120 0.746668 0.373334 0.927697i \(-0.378214\pi\)
0.373334 + 0.927697i \(0.378214\pi\)
\(878\) −4.52030 −0.152553
\(879\) 53.6573 1.80982
\(880\) −11.2012 −0.377591
\(881\) −23.7418 −0.799882 −0.399941 0.916541i \(-0.630969\pi\)
−0.399941 + 0.916541i \(0.630969\pi\)
\(882\) −25.4473 −0.856855
\(883\) −6.97848 −0.234844 −0.117422 0.993082i \(-0.537463\pi\)
−0.117422 + 0.993082i \(0.537463\pi\)
\(884\) 3.87607 0.130366
\(885\) −120.460 −4.04921
\(886\) −13.2163 −0.444011
\(887\) −34.2177 −1.14892 −0.574459 0.818533i \(-0.694788\pi\)
−0.574459 + 0.818533i \(0.694788\pi\)
\(888\) −2.68834 −0.0902147
\(889\) 9.05163 0.303582
\(890\) −62.2337 −2.08608
\(891\) −18.0326 −0.604114
\(892\) −4.41280 −0.147751
\(893\) −8.93203 −0.298899
\(894\) 61.7937 2.06669
\(895\) 67.0689 2.24187
\(896\) 0.487379 0.0162822
\(897\) −22.7501 −0.759605
\(898\) −10.4441 −0.348523
\(899\) −17.9181 −0.597603
\(900\) 35.7216 1.19072
\(901\) 3.81662 0.127150
\(902\) 24.8234 0.826528
\(903\) 1.57216 0.0523181
\(904\) 7.96219 0.264818
\(905\) −89.8507 −2.98674
\(906\) −17.8823 −0.594100
\(907\) −21.2280 −0.704864 −0.352432 0.935837i \(-0.614645\pi\)
−0.352432 + 0.935837i \(0.614645\pi\)
\(908\) −1.94853 −0.0646643
\(909\) 40.5909 1.34631
\(910\) −2.25400 −0.0747195
\(911\) 0.586051 0.0194167 0.00970836 0.999953i \(-0.496910\pi\)
0.00970836 + 0.999953i \(0.496910\pi\)
\(912\) 5.26399 0.174308
\(913\) 19.0726 0.631211
\(914\) 10.7134 0.354369
\(915\) 101.823 3.36617
\(916\) 24.5536 0.811272
\(917\) −10.6307 −0.351058
\(918\) −6.33123 −0.208962
\(919\) 33.5346 1.10620 0.553102 0.833114i \(-0.313444\pi\)
0.553102 + 0.833114i \(0.313444\pi\)
\(920\) −27.4144 −0.903826
\(921\) −78.9127 −2.60026
\(922\) −20.5805 −0.677783
\(923\) 10.7817 0.354883
\(924\) −3.72928 −0.122684
\(925\) 9.81312 0.322654
\(926\) −20.2894 −0.666752
\(927\) −29.6699 −0.974489
\(928\) 3.48092 0.114267
\(929\) −13.7972 −0.452672 −0.226336 0.974049i \(-0.572675\pi\)
−0.226336 + 0.974049i \(0.572675\pi\)
\(930\) −50.9619 −1.67111
\(931\) 13.6883 0.448616
\(932\) −21.7296 −0.711776
\(933\) 31.9544 1.04614
\(934\) −12.8280 −0.419745
\(935\) −35.7391 −1.16879
\(936\) 4.57139 0.149420
\(937\) −22.6646 −0.740419 −0.370210 0.928948i \(-0.620714\pi\)
−0.370210 + 0.928948i \(0.620714\pi\)
\(938\) −5.54754 −0.181134
\(939\) 28.1410 0.918346
\(940\) −16.7989 −0.547921
\(941\) 26.4988 0.863836 0.431918 0.901913i \(-0.357837\pi\)
0.431918 + 0.901913i \(0.357837\pi\)
\(942\) −48.8412 −1.59133
\(943\) 60.7542 1.97843
\(944\) −12.1673 −0.396013
\(945\) 3.68172 0.119766
\(946\) −3.64960 −0.118659
\(947\) 57.8561 1.88007 0.940035 0.341078i \(-0.110792\pi\)
0.940035 + 0.341078i \(0.110792\pi\)
\(948\) 19.2830 0.626284
\(949\) 11.0020 0.357141
\(950\) −19.2149 −0.623414
\(951\) 4.04806 0.131267
\(952\) 1.55506 0.0503999
\(953\) −46.4902 −1.50597 −0.752983 0.658040i \(-0.771386\pi\)
−0.752983 + 0.658040i \(0.771386\pi\)
\(954\) 4.50128 0.145734
\(955\) 8.61974 0.278928
\(956\) 11.3663 0.367613
\(957\) −26.6349 −0.860985
\(958\) 36.1270 1.16721
\(959\) −2.49201 −0.0804711
\(960\) 9.90026 0.319529
\(961\) −4.50292 −0.145255
\(962\) 1.25581 0.0404890
\(963\) 44.8129 1.44408
\(964\) 2.21590 0.0713694
\(965\) −35.7858 −1.15199
\(966\) −9.12726 −0.293665
\(967\) −5.69625 −0.183179 −0.0915895 0.995797i \(-0.529195\pi\)
−0.0915895 + 0.995797i \(0.529195\pi\)
\(968\) −2.34286 −0.0753025
\(969\) 16.7956 0.539552
\(970\) 9.55731 0.306867
\(971\) 41.2535 1.32389 0.661944 0.749553i \(-0.269732\pi\)
0.661944 + 0.749553i \(0.269732\pi\)
\(972\) 21.8912 0.702159
\(973\) −5.83211 −0.186969
\(974\) 11.1794 0.358210
\(975\) −29.9899 −0.960447
\(976\) 10.2849 0.329211
\(977\) 21.7501 0.695848 0.347924 0.937523i \(-0.386887\pi\)
0.347924 + 0.937523i \(0.386887\pi\)
\(978\) −32.3161 −1.03335
\(979\) 48.0991 1.53725
\(980\) 25.7443 0.822371
\(981\) 16.1050 0.514194
\(982\) 9.49758 0.303080
\(983\) −20.7357 −0.661365 −0.330682 0.943742i \(-0.607279\pi\)
−0.330682 + 0.943742i \(0.607279\pi\)
\(984\) −21.9404 −0.699434
\(985\) −10.7280 −0.341824
\(986\) 11.1064 0.353701
\(987\) −5.59299 −0.178027
\(988\) −2.45898 −0.0782307
\(989\) −8.93225 −0.284029
\(990\) −42.1502 −1.33962
\(991\) 14.2286 0.451987 0.225994 0.974129i \(-0.427437\pi\)
0.225994 + 0.974129i \(0.427437\pi\)
\(992\) −5.14753 −0.163434
\(993\) −12.3875 −0.393105
\(994\) 4.32556 0.137198
\(995\) 56.6483 1.79587
\(996\) −16.8575 −0.534150
\(997\) 34.9594 1.10718 0.553588 0.832791i \(-0.313258\pi\)
0.553588 + 0.832791i \(0.313258\pi\)
\(998\) 20.4507 0.647355
\(999\) −2.05126 −0.0648991
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.1.2 7
3.2 odd 2 4842.2.a.n.1.7 7
4.3 odd 2 4304.2.a.h.1.6 7
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
538.2.a.e.1.2 7 1.1 even 1 trivial
4304.2.a.h.1.6 7 4.3 odd 2
4842.2.a.n.1.7 7 3.2 odd 2