Properties

Label 381.2.a.d.1.5
Level $381$
Weight $2$
Character 381.1
Self dual yes
Analytic conductor $3.042$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [381,2,Mod(1,381)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(381, 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("381.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 381 = 3 \cdot 127 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 381.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: \(3.04230031701\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.246832.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 7x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{5}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.71457\) of defining polynomial
Character \(\chi\) \(=\) 381.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+2.65432 q^{2} -1.00000 q^{3} +5.04540 q^{4} +0.121872 q^{5} -2.65432 q^{6} -0.0602522 q^{7} +8.08346 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+2.65432 q^{2} -1.00000 q^{3} +5.04540 q^{4} +0.121872 q^{5} -2.65432 q^{6} -0.0602522 q^{7} +8.08346 q^{8} +1.00000 q^{9} +0.323487 q^{10} -0.159928 q^{11} -5.04540 q^{12} -1.06025 q^{13} -0.159928 q^{14} -0.121872 q^{15} +11.3653 q^{16} -3.67288 q^{17} +2.65432 q^{18} -2.41792 q^{19} +0.614893 q^{20} +0.0602522 q^{21} -0.424501 q^{22} +3.62748 q^{23} -8.08346 q^{24} -4.98515 q^{25} -2.81425 q^{26} -1.00000 q^{27} -0.303996 q^{28} +5.54367 q^{29} -0.323487 q^{30} +4.40965 q^{31} +14.0001 q^{32} +0.159928 q^{33} -9.74900 q^{34} -0.00734305 q^{35} +5.04540 q^{36} -7.01586 q^{37} -6.41792 q^{38} +1.06025 q^{39} +0.985147 q^{40} -9.62849 q^{41} +0.159928 q^{42} -7.54875 q^{43} -0.806902 q^{44} +0.121872 q^{45} +9.62849 q^{46} -0.250645 q^{47} -11.3653 q^{48} -6.99637 q^{49} -13.2322 q^{50} +3.67288 q^{51} -5.34940 q^{52} +1.64173 q^{53} -2.65432 q^{54} -0.0194908 q^{55} -0.487046 q^{56} +2.41792 q^{57} +14.7147 q^{58} -3.71828 q^{59} -0.614893 q^{60} +11.6109 q^{61} +11.7046 q^{62} -0.0602522 q^{63} +14.4301 q^{64} -0.129215 q^{65} +0.424501 q^{66} +13.9415 q^{67} -18.5312 q^{68} -3.62748 q^{69} -0.0194908 q^{70} +8.57784 q^{71} +8.08346 q^{72} -11.0944 q^{73} -18.6223 q^{74} +4.98515 q^{75} -12.1994 q^{76} +0.00963603 q^{77} +2.81425 q^{78} +7.67288 q^{79} +1.38511 q^{80} +1.00000 q^{81} -25.5571 q^{82} +7.42087 q^{83} +0.303996 q^{84} -0.447622 q^{85} -20.0368 q^{86} -5.54367 q^{87} -1.29277 q^{88} +11.9941 q^{89} +0.323487 q^{90} +0.0638825 q^{91} +18.3021 q^{92} -4.40965 q^{93} -0.665290 q^{94} -0.294676 q^{95} -14.0001 q^{96} -0.974091 q^{97} -18.5706 q^{98} -0.159928 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5 q + 2 q^{2} - 5 q^{3} + 6 q^{4} + q^{5} - 2 q^{6} + 6 q^{8} + 5 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 5 q + 2 q^{2} - 5 q^{3} + 6 q^{4} + q^{5} - 2 q^{6} + 6 q^{8} + 5 q^{9} - 2 q^{10} + 14 q^{11} - 6 q^{12} - 5 q^{13} + 14 q^{14} - q^{15} + 8 q^{16} + 4 q^{17} + 2 q^{18} + 4 q^{19} + 6 q^{20} - 2 q^{22} + 15 q^{23} - 6 q^{24} - 6 q^{25} + 12 q^{26} - 5 q^{27} - 2 q^{28} + 9 q^{29} + 2 q^{30} + 3 q^{31} + 14 q^{32} - 14 q^{33} + 4 q^{34} + 4 q^{35} + 6 q^{36} - 5 q^{37} - 16 q^{38} + 5 q^{39} - 14 q^{40} + 4 q^{41} - 14 q^{42} + 10 q^{43} + 18 q^{44} + q^{45} - 4 q^{46} - 4 q^{47} - 8 q^{48} - 9 q^{49} - 24 q^{50} - 4 q^{51} - 8 q^{52} + 3 q^{53} - 2 q^{54} + 4 q^{55} - 10 q^{56} - 4 q^{57} + 6 q^{58} + 23 q^{59} - 6 q^{60} - 15 q^{61} - 24 q^{62} + 3 q^{65} + 2 q^{66} + 18 q^{67} - 24 q^{68} - 15 q^{69} + 4 q^{70} + 12 q^{71} + 6 q^{72} - 43 q^{73} - 36 q^{74} + 6 q^{75} - 32 q^{76} + 4 q^{77} - 12 q^{78} + 16 q^{79} + 4 q^{80} + 5 q^{81} - 44 q^{82} + 11 q^{83} + 2 q^{84} - 24 q^{85} - 28 q^{86} - 9 q^{87} - 14 q^{88} + 9 q^{89} - 2 q^{90} + 26 q^{91} + 14 q^{92} - 3 q^{93} - 14 q^{94} + 16 q^{95} - 14 q^{96} - 20 q^{97} - 10 q^{98} + 14 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.65432 1.87689 0.938443 0.345435i \(-0.112268\pi\)
0.938443 + 0.345435i \(0.112268\pi\)
\(3\) −1.00000 −0.577350
\(4\) 5.04540 2.52270
\(5\) 0.121872 0.0545028 0.0272514 0.999629i \(-0.491325\pi\)
0.0272514 + 0.999629i \(0.491325\pi\)
\(6\) −2.65432 −1.08362
\(7\) −0.0602522 −0.0227732 −0.0113866 0.999935i \(-0.503625\pi\)
−0.0113866 + 0.999935i \(0.503625\pi\)
\(8\) 8.08346 2.85793
\(9\) 1.00000 0.333333
\(10\) 0.323487 0.102296
\(11\) −0.159928 −0.0482202 −0.0241101 0.999709i \(-0.507675\pi\)
−0.0241101 + 0.999709i \(0.507675\pi\)
\(12\) −5.04540 −1.45648
\(13\) −1.06025 −0.294061 −0.147031 0.989132i \(-0.546972\pi\)
−0.147031 + 0.989132i \(0.546972\pi\)
\(14\) −0.159928 −0.0427427
\(15\) −0.121872 −0.0314672
\(16\) 11.3653 2.84131
\(17\) −3.67288 −0.890805 −0.445402 0.895330i \(-0.646939\pi\)
−0.445402 + 0.895330i \(0.646939\pi\)
\(18\) 2.65432 0.625629
\(19\) −2.41792 −0.554708 −0.277354 0.960768i \(-0.589457\pi\)
−0.277354 + 0.960768i \(0.589457\pi\)
\(20\) 0.614893 0.137494
\(21\) 0.0602522 0.0131481
\(22\) −0.424501 −0.0905038
\(23\) 3.62748 0.756383 0.378191 0.925727i \(-0.376546\pi\)
0.378191 + 0.925727i \(0.376546\pi\)
\(24\) −8.08346 −1.65003
\(25\) −4.98515 −0.997029
\(26\) −2.81425 −0.551919
\(27\) −1.00000 −0.192450
\(28\) −0.303996 −0.0574499
\(29\) 5.54367 1.02943 0.514717 0.857360i \(-0.327897\pi\)
0.514717 + 0.857360i \(0.327897\pi\)
\(30\) −0.323487 −0.0590604
\(31\) 4.40965 0.791996 0.395998 0.918251i \(-0.370399\pi\)
0.395998 + 0.918251i \(0.370399\pi\)
\(32\) 14.0001 2.47489
\(33\) 0.159928 0.0278400
\(34\) −9.74900 −1.67194
\(35\) −0.00734305 −0.00124120
\(36\) 5.04540 0.840900
\(37\) −7.01586 −1.15340 −0.576700 0.816956i \(-0.695660\pi\)
−0.576700 + 0.816956i \(0.695660\pi\)
\(38\) −6.41792 −1.04112
\(39\) 1.06025 0.169776
\(40\) 0.985147 0.155765
\(41\) −9.62849 −1.50372 −0.751859 0.659324i \(-0.770843\pi\)
−0.751859 + 0.659324i \(0.770843\pi\)
\(42\) 0.159928 0.0246775
\(43\) −7.54875 −1.15117 −0.575587 0.817741i \(-0.695226\pi\)
−0.575587 + 0.817741i \(0.695226\pi\)
\(44\) −0.806902 −0.121645
\(45\) 0.121872 0.0181676
\(46\) 9.62849 1.41964
\(47\) −0.250645 −0.0365603 −0.0182801 0.999833i \(-0.505819\pi\)
−0.0182801 + 0.999833i \(0.505819\pi\)
\(48\) −11.3653 −1.64043
\(49\) −6.99637 −0.999481
\(50\) −13.2322 −1.87131
\(51\) 3.67288 0.514306
\(52\) −5.34940 −0.741828
\(53\) 1.64173 0.225509 0.112754 0.993623i \(-0.464033\pi\)
0.112754 + 0.993623i \(0.464033\pi\)
\(54\) −2.65432 −0.361207
\(55\) −0.0194908 −0.00262814
\(56\) −0.487046 −0.0650842
\(57\) 2.41792 0.320261
\(58\) 14.7147 1.93213
\(59\) −3.71828 −0.484079 −0.242040 0.970266i \(-0.577816\pi\)
−0.242040 + 0.970266i \(0.577816\pi\)
\(60\) −0.614893 −0.0793824
\(61\) 11.6109 1.48662 0.743312 0.668945i \(-0.233254\pi\)
0.743312 + 0.668945i \(0.233254\pi\)
\(62\) 11.7046 1.48649
\(63\) −0.0602522 −0.00759106
\(64\) 14.4301 1.80377
\(65\) −0.129215 −0.0160272
\(66\) 0.424501 0.0522524
\(67\) 13.9415 1.70322 0.851612 0.524173i \(-0.175626\pi\)
0.851612 + 0.524173i \(0.175626\pi\)
\(68\) −18.5312 −2.24723
\(69\) −3.62748 −0.436698
\(70\) −0.0194908 −0.00232960
\(71\) 8.57784 1.01800 0.509001 0.860766i \(-0.330015\pi\)
0.509001 + 0.860766i \(0.330015\pi\)
\(72\) 8.08346 0.952644
\(73\) −11.0944 −1.29850 −0.649252 0.760573i \(-0.724918\pi\)
−0.649252 + 0.760573i \(0.724918\pi\)
\(74\) −18.6223 −2.16480
\(75\) 4.98515 0.575635
\(76\) −12.1994 −1.39936
\(77\) 0.00963603 0.00109813
\(78\) 2.81425 0.318651
\(79\) 7.67288 0.863267 0.431633 0.902049i \(-0.357937\pi\)
0.431633 + 0.902049i \(0.357937\pi\)
\(80\) 1.38511 0.154860
\(81\) 1.00000 0.111111
\(82\) −25.5571 −2.82231
\(83\) 7.42087 0.814546 0.407273 0.913306i \(-0.366480\pi\)
0.407273 + 0.913306i \(0.366480\pi\)
\(84\) 0.303996 0.0331687
\(85\) −0.447622 −0.0485514
\(86\) −20.0368 −2.16062
\(87\) −5.54367 −0.594344
\(88\) −1.29277 −0.137810
\(89\) 11.9941 1.27137 0.635686 0.771947i \(-0.280717\pi\)
0.635686 + 0.771947i \(0.280717\pi\)
\(90\) 0.323487 0.0340985
\(91\) 0.0638825 0.00669670
\(92\) 18.3021 1.90813
\(93\) −4.40965 −0.457259
\(94\) −0.665290 −0.0686195
\(95\) −0.294676 −0.0302332
\(96\) −14.0001 −1.42888
\(97\) −0.974091 −0.0989040 −0.0494520 0.998777i \(-0.515747\pi\)
−0.0494520 + 0.998777i \(0.515747\pi\)
\(98\) −18.5706 −1.87591
\(99\) −0.159928 −0.0160734
\(100\) −25.1521 −2.51521
\(101\) −0.153953 −0.0153189 −0.00765945 0.999971i \(-0.502438\pi\)
−0.00765945 + 0.999971i \(0.502438\pi\)
\(102\) 9.74900 0.965294
\(103\) −14.7525 −1.45360 −0.726801 0.686848i \(-0.758994\pi\)
−0.726801 + 0.686848i \(0.758994\pi\)
\(104\) −8.57050 −0.840407
\(105\) 0.00734305 0.000716609 0
\(106\) 4.35766 0.423254
\(107\) 9.26460 0.895643 0.447821 0.894123i \(-0.352200\pi\)
0.447821 + 0.894123i \(0.352200\pi\)
\(108\) −5.04540 −0.485494
\(109\) −0.135190 −0.0129489 −0.00647445 0.999979i \(-0.502061\pi\)
−0.00647445 + 0.999979i \(0.502061\pi\)
\(110\) −0.0517347 −0.00493271
\(111\) 7.01586 0.665916
\(112\) −0.684781 −0.0647058
\(113\) 10.5609 0.993483 0.496741 0.867899i \(-0.334530\pi\)
0.496741 + 0.867899i \(0.334530\pi\)
\(114\) 6.41792 0.601093
\(115\) 0.442089 0.0412250
\(116\) 27.9700 2.59695
\(117\) −1.06025 −0.0980203
\(118\) −9.86950 −0.908561
\(119\) 0.221299 0.0202865
\(120\) −0.985147 −0.0899312
\(121\) −10.9744 −0.997675
\(122\) 30.8190 2.79022
\(123\) 9.62849 0.868172
\(124\) 22.2484 1.99797
\(125\) −1.21691 −0.108844
\(126\) −0.159928 −0.0142476
\(127\) 1.00000 0.0887357
\(128\) 10.3020 0.910578
\(129\) 7.54875 0.664630
\(130\) −0.342978 −0.0300811
\(131\) 11.3916 0.995293 0.497646 0.867380i \(-0.334198\pi\)
0.497646 + 0.867380i \(0.334198\pi\)
\(132\) 0.806902 0.0702318
\(133\) 0.145685 0.0126325
\(134\) 37.0051 3.19676
\(135\) −0.121872 −0.0104891
\(136\) −29.6896 −2.54586
\(137\) 20.5766 1.75797 0.878987 0.476846i \(-0.158220\pi\)
0.878987 + 0.476846i \(0.158220\pi\)
\(138\) −9.62849 −0.819632
\(139\) 14.8583 1.26026 0.630131 0.776489i \(-0.283001\pi\)
0.630131 + 0.776489i \(0.283001\pi\)
\(140\) −0.0370486 −0.00313118
\(141\) 0.250645 0.0211081
\(142\) 22.7683 1.91067
\(143\) 0.169564 0.0141797
\(144\) 11.3653 0.947105
\(145\) 0.675618 0.0561070
\(146\) −29.4481 −2.43714
\(147\) 6.99637 0.577051
\(148\) −35.3978 −2.90968
\(149\) −2.64424 −0.216624 −0.108312 0.994117i \(-0.534545\pi\)
−0.108312 + 0.994117i \(0.534545\pi\)
\(150\) 13.2322 1.08040
\(151\) −7.40875 −0.602916 −0.301458 0.953479i \(-0.597473\pi\)
−0.301458 + 0.953479i \(0.597473\pi\)
\(152\) −19.5451 −1.58532
\(153\) −3.67288 −0.296935
\(154\) 0.0255771 0.00206106
\(155\) 0.537413 0.0431660
\(156\) 5.34940 0.428294
\(157\) 8.13184 0.648991 0.324496 0.945887i \(-0.394805\pi\)
0.324496 + 0.945887i \(0.394805\pi\)
\(158\) 20.3663 1.62025
\(159\) −1.64173 −0.130197
\(160\) 1.70622 0.134888
\(161\) −0.218564 −0.0172252
\(162\) 2.65432 0.208543
\(163\) −1.99447 −0.156219 −0.0781094 0.996945i \(-0.524888\pi\)
−0.0781094 + 0.996945i \(0.524888\pi\)
\(164\) −48.5796 −3.79343
\(165\) 0.0194908 0.00151736
\(166\) 19.6973 1.52881
\(167\) 15.9154 1.23157 0.615786 0.787914i \(-0.288839\pi\)
0.615786 + 0.787914i \(0.288839\pi\)
\(168\) 0.487046 0.0375764
\(169\) −11.8759 −0.913528
\(170\) −1.18813 −0.0911254
\(171\) −2.41792 −0.184903
\(172\) −38.0864 −2.90406
\(173\) −8.46195 −0.643350 −0.321675 0.946850i \(-0.604246\pi\)
−0.321675 + 0.946850i \(0.604246\pi\)
\(174\) −14.7147 −1.11551
\(175\) 0.300366 0.0227055
\(176\) −1.81763 −0.137009
\(177\) 3.71828 0.279483
\(178\) 31.8362 2.38622
\(179\) −0.828932 −0.0619573 −0.0309786 0.999520i \(-0.509862\pi\)
−0.0309786 + 0.999520i \(0.509862\pi\)
\(180\) 0.614893 0.0458314
\(181\) −17.0055 −1.26401 −0.632004 0.774965i \(-0.717767\pi\)
−0.632004 + 0.774965i \(0.717767\pi\)
\(182\) 0.169564 0.0125689
\(183\) −11.6109 −0.858303
\(184\) 29.3226 2.16169
\(185\) −0.855037 −0.0628636
\(186\) −11.7046 −0.858223
\(187\) 0.587398 0.0429548
\(188\) −1.26460 −0.0922306
\(189\) 0.0602522 0.00438270
\(190\) −0.782165 −0.0567442
\(191\) 10.0074 0.724112 0.362056 0.932156i \(-0.382075\pi\)
0.362056 + 0.932156i \(0.382075\pi\)
\(192\) −14.4301 −1.04141
\(193\) −19.2597 −1.38634 −0.693172 0.720772i \(-0.743788\pi\)
−0.693172 + 0.720772i \(0.743788\pi\)
\(194\) −2.58555 −0.185631
\(195\) 0.129215 0.00925328
\(196\) −35.2995 −2.52139
\(197\) 7.85481 0.559632 0.279816 0.960054i \(-0.409726\pi\)
0.279816 + 0.960054i \(0.409726\pi\)
\(198\) −0.424501 −0.0301679
\(199\) 16.7433 1.18690 0.593451 0.804870i \(-0.297765\pi\)
0.593451 + 0.804870i \(0.297765\pi\)
\(200\) −40.2972 −2.84944
\(201\) −13.9415 −0.983356
\(202\) −0.408640 −0.0287518
\(203\) −0.334018 −0.0234435
\(204\) 18.5312 1.29744
\(205\) −1.17344 −0.0819569
\(206\) −39.1577 −2.72825
\(207\) 3.62748 0.252128
\(208\) −12.0500 −0.835520
\(209\) 0.386693 0.0267481
\(210\) 0.0194908 0.00134499
\(211\) −6.90368 −0.475269 −0.237634 0.971355i \(-0.576372\pi\)
−0.237634 + 0.971355i \(0.576372\pi\)
\(212\) 8.28317 0.568890
\(213\) −8.57784 −0.587744
\(214\) 24.5912 1.68102
\(215\) −0.919981 −0.0627422
\(216\) −8.08346 −0.550009
\(217\) −0.265691 −0.0180363
\(218\) −0.358838 −0.0243036
\(219\) 11.0944 0.749692
\(220\) −0.0983388 −0.00663000
\(221\) 3.89418 0.261951
\(222\) 18.6223 1.24985
\(223\) −25.0408 −1.67685 −0.838427 0.545014i \(-0.816524\pi\)
−0.838427 + 0.545014i \(0.816524\pi\)
\(224\) −0.843535 −0.0563611
\(225\) −4.98515 −0.332343
\(226\) 28.0319 1.86465
\(227\) −16.5898 −1.10110 −0.550552 0.834801i \(-0.685583\pi\)
−0.550552 + 0.834801i \(0.685583\pi\)
\(228\) 12.1994 0.807922
\(229\) 23.4458 1.54934 0.774672 0.632363i \(-0.217915\pi\)
0.774672 + 0.632363i \(0.217915\pi\)
\(230\) 1.17344 0.0773746
\(231\) −0.00963603 −0.000634004 0
\(232\) 44.8120 2.94205
\(233\) 25.6658 1.68142 0.840710 0.541485i \(-0.182138\pi\)
0.840710 + 0.541485i \(0.182138\pi\)
\(234\) −2.81425 −0.183973
\(235\) −0.0305466 −0.00199264
\(236\) −18.7602 −1.22119
\(237\) −7.67288 −0.498407
\(238\) 0.587398 0.0380754
\(239\) −19.1324 −1.23757 −0.618786 0.785560i \(-0.712375\pi\)
−0.618786 + 0.785560i \(0.712375\pi\)
\(240\) −1.38511 −0.0894083
\(241\) 18.8973 1.21728 0.608640 0.793447i \(-0.291715\pi\)
0.608640 + 0.793447i \(0.291715\pi\)
\(242\) −29.1296 −1.87252
\(243\) −1.00000 −0.0641500
\(244\) 58.5816 3.75031
\(245\) −0.852662 −0.0544746
\(246\) 25.5571 1.62946
\(247\) 2.56360 0.163118
\(248\) 35.6452 2.26347
\(249\) −7.42087 −0.470279
\(250\) −3.23007 −0.204287
\(251\) 24.0758 1.51965 0.759825 0.650128i \(-0.225285\pi\)
0.759825 + 0.650128i \(0.225285\pi\)
\(252\) −0.303996 −0.0191500
\(253\) −0.580137 −0.0364729
\(254\) 2.65432 0.166547
\(255\) 0.447622 0.0280312
\(256\) −1.51547 −0.0947166
\(257\) −21.1973 −1.32225 −0.661127 0.750274i \(-0.729921\pi\)
−0.661127 + 0.750274i \(0.729921\pi\)
\(258\) 20.0368 1.24743
\(259\) 0.422721 0.0262666
\(260\) −0.651942 −0.0404317
\(261\) 5.54367 0.343144
\(262\) 30.2370 1.86805
\(263\) −30.1230 −1.85746 −0.928731 0.370755i \(-0.879099\pi\)
−0.928731 + 0.370755i \(0.879099\pi\)
\(264\) 1.29277 0.0795647
\(265\) 0.200081 0.0122909
\(266\) 0.386693 0.0237097
\(267\) −11.9941 −0.734027
\(268\) 70.3404 4.29672
\(269\) −7.96482 −0.485624 −0.242812 0.970073i \(-0.578070\pi\)
−0.242812 + 0.970073i \(0.578070\pi\)
\(270\) −0.323487 −0.0196868
\(271\) 11.7971 0.716625 0.358312 0.933602i \(-0.383352\pi\)
0.358312 + 0.933602i \(0.383352\pi\)
\(272\) −41.7433 −2.53106
\(273\) −0.0638825 −0.00386634
\(274\) 54.6167 3.29952
\(275\) 0.797266 0.0480770
\(276\) −18.3021 −1.10166
\(277\) −30.6509 −1.84164 −0.920818 0.389992i \(-0.872478\pi\)
−0.920818 + 0.389992i \(0.872478\pi\)
\(278\) 39.4386 2.36537
\(279\) 4.40965 0.263999
\(280\) −0.0593573 −0.00354727
\(281\) −17.6965 −1.05568 −0.527842 0.849343i \(-0.676999\pi\)
−0.527842 + 0.849343i \(0.676999\pi\)
\(282\) 0.665290 0.0396175
\(283\) 16.0146 0.951972 0.475986 0.879453i \(-0.342091\pi\)
0.475986 + 0.879453i \(0.342091\pi\)
\(284\) 43.2787 2.56812
\(285\) 0.294676 0.0174551
\(286\) 0.450078 0.0266136
\(287\) 0.580137 0.0342444
\(288\) 14.0001 0.824963
\(289\) −3.50993 −0.206467
\(290\) 1.79330 0.105306
\(291\) 0.974091 0.0571023
\(292\) −55.9758 −3.27574
\(293\) −19.9488 −1.16542 −0.582710 0.812680i \(-0.698008\pi\)
−0.582710 + 0.812680i \(0.698008\pi\)
\(294\) 18.5706 1.08306
\(295\) −0.453155 −0.0263837
\(296\) −56.7124 −3.29634
\(297\) 0.159928 0.00927998
\(298\) −7.01865 −0.406579
\(299\) −3.84605 −0.222423
\(300\) 25.1521 1.45215
\(301\) 0.454828 0.0262159
\(302\) −19.6652 −1.13160
\(303\) 0.153953 0.00884437
\(304\) −27.4802 −1.57610
\(305\) 1.41504 0.0810252
\(306\) −9.74900 −0.557313
\(307\) 1.74509 0.0995973 0.0497986 0.998759i \(-0.484142\pi\)
0.0497986 + 0.998759i \(0.484142\pi\)
\(308\) 0.0486176 0.00277025
\(309\) 14.7525 0.839238
\(310\) 1.42646 0.0810177
\(311\) 16.3942 0.929632 0.464816 0.885407i \(-0.346120\pi\)
0.464816 + 0.885407i \(0.346120\pi\)
\(312\) 8.57050 0.485209
\(313\) −3.25949 −0.184237 −0.0921187 0.995748i \(-0.529364\pi\)
−0.0921187 + 0.995748i \(0.529364\pi\)
\(314\) 21.5845 1.21808
\(315\) −0.00734305 −0.000413734 0
\(316\) 38.7128 2.17776
\(317\) −3.81807 −0.214444 −0.107222 0.994235i \(-0.534196\pi\)
−0.107222 + 0.994235i \(0.534196\pi\)
\(318\) −4.35766 −0.244366
\(319\) −0.886590 −0.0496395
\(320\) 1.75863 0.0983105
\(321\) −9.26460 −0.517100
\(322\) −0.580137 −0.0323298
\(323\) 8.88072 0.494137
\(324\) 5.04540 0.280300
\(325\) 5.28551 0.293188
\(326\) −5.29395 −0.293205
\(327\) 0.135190 0.00747605
\(328\) −77.8315 −4.29752
\(329\) 0.0151019 0.000832594 0
\(330\) 0.0517347 0.00284790
\(331\) 13.9352 0.765949 0.382975 0.923759i \(-0.374900\pi\)
0.382975 + 0.923759i \(0.374900\pi\)
\(332\) 37.4413 2.05486
\(333\) −7.01586 −0.384467
\(334\) 42.2445 2.31152
\(335\) 1.69908 0.0928305
\(336\) 0.684781 0.0373579
\(337\) −2.48928 −0.135600 −0.0677998 0.997699i \(-0.521598\pi\)
−0.0677998 + 0.997699i \(0.521598\pi\)
\(338\) −31.5223 −1.71459
\(339\) −10.5609 −0.573587
\(340\) −2.25843 −0.122481
\(341\) −0.705228 −0.0381902
\(342\) −6.41792 −0.347041
\(343\) 0.843312 0.0455345
\(344\) −61.0200 −3.28998
\(345\) −0.442089 −0.0238013
\(346\) −22.4607 −1.20749
\(347\) 10.2540 0.550462 0.275231 0.961378i \(-0.411246\pi\)
0.275231 + 0.961378i \(0.411246\pi\)
\(348\) −27.9700 −1.49935
\(349\) −23.4312 −1.25424 −0.627120 0.778922i \(-0.715767\pi\)
−0.627120 + 0.778922i \(0.715767\pi\)
\(350\) 0.797266 0.0426157
\(351\) 1.06025 0.0565921
\(352\) −2.23901 −0.119340
\(353\) −9.26151 −0.492940 −0.246470 0.969150i \(-0.579271\pi\)
−0.246470 + 0.969150i \(0.579271\pi\)
\(354\) 9.86950 0.524558
\(355\) 1.04540 0.0554840
\(356\) 60.5151 3.20729
\(357\) −0.221299 −0.0117124
\(358\) −2.20025 −0.116287
\(359\) −1.52535 −0.0805048 −0.0402524 0.999190i \(-0.512816\pi\)
−0.0402524 + 0.999190i \(0.512816\pi\)
\(360\) 0.985147 0.0519218
\(361\) −13.1537 −0.692299
\(362\) −45.1379 −2.37240
\(363\) 10.9744 0.576008
\(364\) 0.322313 0.0168938
\(365\) −1.35210 −0.0707722
\(366\) −30.8190 −1.61094
\(367\) −28.9807 −1.51278 −0.756389 0.654122i \(-0.773038\pi\)
−0.756389 + 0.654122i \(0.773038\pi\)
\(368\) 41.2273 2.14912
\(369\) −9.62849 −0.501239
\(370\) −2.26954 −0.117988
\(371\) −0.0989176 −0.00513555
\(372\) −22.2484 −1.15353
\(373\) 1.07773 0.0558025 0.0279013 0.999611i \(-0.491118\pi\)
0.0279013 + 0.999611i \(0.491118\pi\)
\(374\) 1.55914 0.0806213
\(375\) 1.21691 0.0628410
\(376\) −2.02608 −0.104487
\(377\) −5.87769 −0.302716
\(378\) 0.159928 0.00822583
\(379\) −0.694276 −0.0356626 −0.0178313 0.999841i \(-0.505676\pi\)
−0.0178313 + 0.999841i \(0.505676\pi\)
\(380\) −1.48676 −0.0762692
\(381\) −1.00000 −0.0512316
\(382\) 26.5629 1.35908
\(383\) −24.2426 −1.23874 −0.619369 0.785100i \(-0.712612\pi\)
−0.619369 + 0.785100i \(0.712612\pi\)
\(384\) −10.3020 −0.525723
\(385\) 0.00117436 5.98511e−5 0
\(386\) −51.1214 −2.60201
\(387\) −7.54875 −0.383724
\(388\) −4.91468 −0.249505
\(389\) 38.4127 1.94760 0.973800 0.227407i \(-0.0730247\pi\)
0.973800 + 0.227407i \(0.0730247\pi\)
\(390\) 0.342978 0.0173674
\(391\) −13.3233 −0.673789
\(392\) −56.5548 −2.85645
\(393\) −11.3916 −0.574633
\(394\) 20.8492 1.05037
\(395\) 0.935110 0.0470505
\(396\) −0.806902 −0.0405484
\(397\) 29.9092 1.50110 0.750549 0.660815i \(-0.229789\pi\)
0.750549 + 0.660815i \(0.229789\pi\)
\(398\) 44.4420 2.22768
\(399\) −0.145685 −0.00729336
\(400\) −56.6575 −2.83287
\(401\) 0.194391 0.00970743 0.00485371 0.999988i \(-0.498455\pi\)
0.00485371 + 0.999988i \(0.498455\pi\)
\(402\) −37.0051 −1.84565
\(403\) −4.67534 −0.232895
\(404\) −0.776754 −0.0386450
\(405\) 0.121872 0.00605587
\(406\) −0.886590 −0.0440007
\(407\) 1.12204 0.0556172
\(408\) 29.6896 1.46985
\(409\) −6.01049 −0.297200 −0.148600 0.988897i \(-0.547477\pi\)
−0.148600 + 0.988897i \(0.547477\pi\)
\(410\) −3.11469 −0.153824
\(411\) −20.5766 −1.01497
\(412\) −74.4320 −3.66700
\(413\) 0.224035 0.0110240
\(414\) 9.62849 0.473215
\(415\) 0.904397 0.0443951
\(416\) −14.8436 −0.727768
\(417\) −14.8583 −0.727613
\(418\) 1.02641 0.0502032
\(419\) −2.47049 −0.120691 −0.0603456 0.998178i \(-0.519220\pi\)
−0.0603456 + 0.998178i \(0.519220\pi\)
\(420\) 0.0370486 0.00180779
\(421\) −1.47555 −0.0719137 −0.0359568 0.999353i \(-0.511448\pi\)
−0.0359568 + 0.999353i \(0.511448\pi\)
\(422\) −18.3245 −0.892025
\(423\) −0.250645 −0.0121868
\(424\) 13.2708 0.644488
\(425\) 18.3099 0.888159
\(426\) −22.7683 −1.10313
\(427\) −0.699582 −0.0338551
\(428\) 46.7436 2.25944
\(429\) −0.169564 −0.00818665
\(430\) −2.44192 −0.117760
\(431\) −24.6705 −1.18834 −0.594168 0.804341i \(-0.702519\pi\)
−0.594168 + 0.804341i \(0.702519\pi\)
\(432\) −11.3653 −0.546811
\(433\) −11.4334 −0.549456 −0.274728 0.961522i \(-0.588588\pi\)
−0.274728 + 0.961522i \(0.588588\pi\)
\(434\) −0.705228 −0.0338520
\(435\) −0.675618 −0.0323934
\(436\) −0.682090 −0.0326662
\(437\) −8.77095 −0.419571
\(438\) 29.4481 1.40709
\(439\) 20.2215 0.965118 0.482559 0.875863i \(-0.339707\pi\)
0.482559 + 0.875863i \(0.339707\pi\)
\(440\) −0.157553 −0.00751104
\(441\) −6.99637 −0.333160
\(442\) 10.3364 0.491652
\(443\) −12.2600 −0.582490 −0.291245 0.956648i \(-0.594070\pi\)
−0.291245 + 0.956648i \(0.594070\pi\)
\(444\) 35.3978 1.67991
\(445\) 1.46175 0.0692934
\(446\) −66.4661 −3.14726
\(447\) 2.64424 0.125068
\(448\) −0.869448 −0.0410775
\(449\) −23.0518 −1.08788 −0.543941 0.839123i \(-0.683069\pi\)
−0.543941 + 0.839123i \(0.683069\pi\)
\(450\) −13.2322 −0.623770
\(451\) 1.53987 0.0725096
\(452\) 53.2838 2.50626
\(453\) 7.40875 0.348094
\(454\) −44.0346 −2.06665
\(455\) 0.00778549 0.000364989 0
\(456\) 19.5451 0.915284
\(457\) 22.9884 1.07535 0.537676 0.843152i \(-0.319302\pi\)
0.537676 + 0.843152i \(0.319302\pi\)
\(458\) 62.2327 2.90794
\(459\) 3.67288 0.171435
\(460\) 2.23051 0.103998
\(461\) −22.2984 −1.03854 −0.519269 0.854611i \(-0.673796\pi\)
−0.519269 + 0.854611i \(0.673796\pi\)
\(462\) −0.0255771 −0.00118995
\(463\) −4.52005 −0.210065 −0.105032 0.994469i \(-0.533495\pi\)
−0.105032 + 0.994469i \(0.533495\pi\)
\(464\) 63.0052 2.92494
\(465\) −0.537413 −0.0249219
\(466\) 68.1251 3.15583
\(467\) −3.44014 −0.159191 −0.0795954 0.996827i \(-0.525363\pi\)
−0.0795954 + 0.996827i \(0.525363\pi\)
\(468\) −5.34940 −0.247276
\(469\) −0.840005 −0.0387878
\(470\) −0.0810803 −0.00373996
\(471\) −8.13184 −0.374695
\(472\) −30.0566 −1.38347
\(473\) 1.20726 0.0555098
\(474\) −20.3663 −0.935453
\(475\) 12.0537 0.553060
\(476\) 1.11654 0.0511766
\(477\) 1.64173 0.0751695
\(478\) −50.7835 −2.32278
\(479\) −16.4602 −0.752084 −0.376042 0.926603i \(-0.622715\pi\)
−0.376042 + 0.926603i \(0.622715\pi\)
\(480\) −1.70622 −0.0778779
\(481\) 7.43858 0.339170
\(482\) 50.1593 2.28469
\(483\) 0.218564 0.00994499
\(484\) −55.3703 −2.51683
\(485\) −0.118715 −0.00539055
\(486\) −2.65432 −0.120402
\(487\) 28.0787 1.27237 0.636183 0.771538i \(-0.280512\pi\)
0.636183 + 0.771538i \(0.280512\pi\)
\(488\) 93.8562 4.24867
\(489\) 1.99447 0.0901929
\(490\) −2.26323 −0.102243
\(491\) 5.02776 0.226900 0.113450 0.993544i \(-0.463810\pi\)
0.113450 + 0.993544i \(0.463810\pi\)
\(492\) 48.5796 2.19014
\(493\) −20.3612 −0.917024
\(494\) 6.80461 0.306154
\(495\) −0.0194908 −0.000876046 0
\(496\) 50.1168 2.25031
\(497\) −0.516834 −0.0231832
\(498\) −19.6973 −0.882659
\(499\) 23.2126 1.03914 0.519570 0.854428i \(-0.326092\pi\)
0.519570 + 0.854428i \(0.326092\pi\)
\(500\) −6.13980 −0.274580
\(501\) −15.9154 −0.711048
\(502\) 63.9048 2.85221
\(503\) 15.9960 0.713228 0.356614 0.934252i \(-0.383931\pi\)
0.356614 + 0.934252i \(0.383931\pi\)
\(504\) −0.487046 −0.0216947
\(505\) −0.0187626 −0.000834923 0
\(506\) −1.53987 −0.0684555
\(507\) 11.8759 0.527426
\(508\) 5.04540 0.223853
\(509\) 24.2158 1.07335 0.536674 0.843790i \(-0.319681\pi\)
0.536674 + 0.843790i \(0.319681\pi\)
\(510\) 1.18813 0.0526113
\(511\) 0.668463 0.0295711
\(512\) −24.6266 −1.08835
\(513\) 2.41792 0.106754
\(514\) −56.2645 −2.48172
\(515\) −1.79791 −0.0792255
\(516\) 38.0864 1.67666
\(517\) 0.0400852 0.00176294
\(518\) 1.12204 0.0492994
\(519\) 8.46195 0.371438
\(520\) −1.04450 −0.0458046
\(521\) 35.8187 1.56925 0.784624 0.619972i \(-0.212856\pi\)
0.784624 + 0.619972i \(0.212856\pi\)
\(522\) 14.7147 0.644043
\(523\) 7.47258 0.326753 0.163377 0.986564i \(-0.447761\pi\)
0.163377 + 0.986564i \(0.447761\pi\)
\(524\) 57.4754 2.51082
\(525\) −0.300366 −0.0131090
\(526\) −79.9559 −3.48624
\(527\) −16.1961 −0.705514
\(528\) 1.81763 0.0791020
\(529\) −9.84137 −0.427885
\(530\) 0.531077 0.0230685
\(531\) −3.71828 −0.161360
\(532\) 0.735037 0.0318679
\(533\) 10.2086 0.442185
\(534\) −31.8362 −1.37769
\(535\) 1.12910 0.0488151
\(536\) 112.695 4.86770
\(537\) 0.828932 0.0357710
\(538\) −21.1412 −0.911460
\(539\) 1.11892 0.0481952
\(540\) −0.614893 −0.0264608
\(541\) −25.3744 −1.09093 −0.545466 0.838133i \(-0.683647\pi\)
−0.545466 + 0.838133i \(0.683647\pi\)
\(542\) 31.3133 1.34502
\(543\) 17.0055 0.729775
\(544\) −51.4207 −2.20464
\(545\) −0.0164759 −0.000705751 0
\(546\) −0.169564 −0.00725669
\(547\) 19.2589 0.823450 0.411725 0.911308i \(-0.364926\pi\)
0.411725 + 0.911308i \(0.364926\pi\)
\(548\) 103.817 4.43484
\(549\) 11.6109 0.495541
\(550\) 2.11620 0.0902350
\(551\) −13.4041 −0.571035
\(552\) −29.3226 −1.24805
\(553\) −0.462308 −0.0196593
\(554\) −81.3573 −3.45654
\(555\) 0.855037 0.0362943
\(556\) 74.9659 3.17926
\(557\) 31.0604 1.31607 0.658035 0.752987i \(-0.271388\pi\)
0.658035 + 0.752987i \(0.271388\pi\)
\(558\) 11.7046 0.495495
\(559\) 8.00358 0.338515
\(560\) −0.0834557 −0.00352665
\(561\) −0.587398 −0.0248000
\(562\) −46.9721 −1.98140
\(563\) −33.1705 −1.39797 −0.698986 0.715136i \(-0.746365\pi\)
−0.698986 + 0.715136i \(0.746365\pi\)
\(564\) 1.26460 0.0532494
\(565\) 1.28707 0.0541476
\(566\) 42.5079 1.78674
\(567\) −0.0602522 −0.00253035
\(568\) 69.3386 2.90938
\(569\) −34.7023 −1.45479 −0.727397 0.686217i \(-0.759270\pi\)
−0.727397 + 0.686217i \(0.759270\pi\)
\(570\) 0.782165 0.0327613
\(571\) −20.5890 −0.861621 −0.430810 0.902442i \(-0.641772\pi\)
−0.430810 + 0.902442i \(0.641772\pi\)
\(572\) 0.855520 0.0357711
\(573\) −10.0074 −0.418066
\(574\) 1.53987 0.0642729
\(575\) −18.0835 −0.754136
\(576\) 14.4301 0.601256
\(577\) −23.6592 −0.984946 −0.492473 0.870328i \(-0.663907\pi\)
−0.492473 + 0.870328i \(0.663907\pi\)
\(578\) −9.31648 −0.387514
\(579\) 19.2597 0.800407
\(580\) 3.40876 0.141541
\(581\) −0.447124 −0.0185498
\(582\) 2.58555 0.107174
\(583\) −0.262559 −0.0108741
\(584\) −89.6813 −3.71104
\(585\) −0.129215 −0.00534239
\(586\) −52.9504 −2.18736
\(587\) −31.9623 −1.31922 −0.659612 0.751606i \(-0.729279\pi\)
−0.659612 + 0.751606i \(0.729279\pi\)
\(588\) 35.2995 1.45573
\(589\) −10.6622 −0.439327
\(590\) −1.20282 −0.0495192
\(591\) −7.85481 −0.323104
\(592\) −79.7371 −3.27717
\(593\) 26.0212 1.06856 0.534282 0.845306i \(-0.320582\pi\)
0.534282 + 0.845306i \(0.320582\pi\)
\(594\) 0.424501 0.0174175
\(595\) 0.0269702 0.00110567
\(596\) −13.3412 −0.546478
\(597\) −16.7433 −0.685258
\(598\) −10.2086 −0.417462
\(599\) −14.6966 −0.600485 −0.300243 0.953863i \(-0.597068\pi\)
−0.300243 + 0.953863i \(0.597068\pi\)
\(600\) 40.2972 1.64513
\(601\) −6.64010 −0.270855 −0.135428 0.990787i \(-0.543241\pi\)
−0.135428 + 0.990787i \(0.543241\pi\)
\(602\) 1.20726 0.0492042
\(603\) 13.9415 0.567741
\(604\) −37.3801 −1.52098
\(605\) −1.33748 −0.0543761
\(606\) 0.408640 0.0165999
\(607\) 30.2066 1.22605 0.613023 0.790065i \(-0.289953\pi\)
0.613023 + 0.790065i \(0.289953\pi\)
\(608\) −33.8510 −1.37284
\(609\) 0.334018 0.0135351
\(610\) 3.75598 0.152075
\(611\) 0.265747 0.0107510
\(612\) −18.5312 −0.749078
\(613\) 43.6245 1.76198 0.880989 0.473136i \(-0.156878\pi\)
0.880989 + 0.473136i \(0.156878\pi\)
\(614\) 4.63201 0.186933
\(615\) 1.17344 0.0473178
\(616\) 0.0778924 0.00313838
\(617\) −34.1094 −1.37319 −0.686596 0.727039i \(-0.740896\pi\)
−0.686596 + 0.727039i \(0.740896\pi\)
\(618\) 39.1577 1.57515
\(619\) 5.16120 0.207446 0.103723 0.994606i \(-0.466924\pi\)
0.103723 + 0.994606i \(0.466924\pi\)
\(620\) 2.71146 0.108895
\(621\) −3.62748 −0.145566
\(622\) 43.5155 1.74481
\(623\) −0.722671 −0.0289532
\(624\) 12.0500 0.482388
\(625\) 24.7774 0.991097
\(626\) −8.65172 −0.345792
\(627\) −0.386693 −0.0154430
\(628\) 41.0284 1.63721
\(629\) 25.7684 1.02745
\(630\) −0.0194908 −0.000776532 0
\(631\) 44.1665 1.75824 0.879121 0.476599i \(-0.158131\pi\)
0.879121 + 0.476599i \(0.158131\pi\)
\(632\) 62.0234 2.46716
\(633\) 6.90368 0.274397
\(634\) −10.1344 −0.402487
\(635\) 0.121872 0.00483634
\(636\) −8.28317 −0.328449
\(637\) 7.41792 0.293909
\(638\) −2.35329 −0.0931676
\(639\) 8.57784 0.339334
\(640\) 1.25553 0.0496291
\(641\) 35.9696 1.42071 0.710357 0.703842i \(-0.248534\pi\)
0.710357 + 0.703842i \(0.248534\pi\)
\(642\) −24.5912 −0.970537
\(643\) −46.3358 −1.82731 −0.913654 0.406494i \(-0.866751\pi\)
−0.913654 + 0.406494i \(0.866751\pi\)
\(644\) −1.10274 −0.0434541
\(645\) 0.919981 0.0362242
\(646\) 23.5723 0.927438
\(647\) −37.8776 −1.48912 −0.744561 0.667554i \(-0.767341\pi\)
−0.744561 + 0.667554i \(0.767341\pi\)
\(648\) 8.08346 0.317548
\(649\) 0.594659 0.0233424
\(650\) 14.0294 0.550279
\(651\) 0.265691 0.0104132
\(652\) −10.0629 −0.394093
\(653\) −21.7248 −0.850156 −0.425078 0.905157i \(-0.639753\pi\)
−0.425078 + 0.905157i \(0.639753\pi\)
\(654\) 0.358838 0.0140317
\(655\) 1.38832 0.0542463
\(656\) −109.430 −4.27253
\(657\) −11.0944 −0.432835
\(658\) 0.0400852 0.00156268
\(659\) 1.43232 0.0557954 0.0278977 0.999611i \(-0.491119\pi\)
0.0278977 + 0.999611i \(0.491119\pi\)
\(660\) 0.0983388 0.00382783
\(661\) 38.3355 1.49108 0.745539 0.666462i \(-0.232192\pi\)
0.745539 + 0.666462i \(0.232192\pi\)
\(662\) 36.9885 1.43760
\(663\) −3.89418 −0.151237
\(664\) 59.9863 2.32792
\(665\) 0.0177549 0.000688505 0
\(666\) −18.6223 −0.721600
\(667\) 20.1096 0.778645
\(668\) 80.2996 3.10688
\(669\) 25.0408 0.968132
\(670\) 4.50989 0.174232
\(671\) −1.85691 −0.0716853
\(672\) 0.843535 0.0325401
\(673\) −4.23191 −0.163128 −0.0815640 0.996668i \(-0.525991\pi\)
−0.0815640 + 0.996668i \(0.525991\pi\)
\(674\) −6.60733 −0.254505
\(675\) 4.98515 0.191878
\(676\) −59.9185 −2.30456
\(677\) 3.17305 0.121950 0.0609751 0.998139i \(-0.480579\pi\)
0.0609751 + 0.998139i \(0.480579\pi\)
\(678\) −28.0319 −1.07656
\(679\) 0.0586911 0.00225236
\(680\) −3.61833 −0.138757
\(681\) 16.5898 0.635722
\(682\) −1.87190 −0.0716787
\(683\) 44.4169 1.69956 0.849782 0.527134i \(-0.176734\pi\)
0.849782 + 0.527134i \(0.176734\pi\)
\(684\) −12.1994 −0.466454
\(685\) 2.50771 0.0958146
\(686\) 2.23842 0.0854631
\(687\) −23.4458 −0.894515
\(688\) −85.7935 −3.27084
\(689\) −1.74064 −0.0663133
\(690\) −1.17344 −0.0446722
\(691\) 35.3985 1.34662 0.673312 0.739359i \(-0.264871\pi\)
0.673312 + 0.739359i \(0.264871\pi\)
\(692\) −42.6939 −1.62298
\(693\) 0.00963603 0.000366043 0
\(694\) 27.2173 1.03315
\(695\) 1.81081 0.0686879
\(696\) −44.8120 −1.69859
\(697\) 35.3643 1.33952
\(698\) −62.1937 −2.35407
\(699\) −25.6658 −0.970769
\(700\) 1.51547 0.0572792
\(701\) −8.50547 −0.321247 −0.160624 0.987016i \(-0.551351\pi\)
−0.160624 + 0.987016i \(0.551351\pi\)
\(702\) 2.81425 0.106217
\(703\) 16.9638 0.639800
\(704\) −2.30779 −0.0869781
\(705\) 0.0305466 0.00115045
\(706\) −24.5830 −0.925193
\(707\) 0.00927600 0.000348860 0
\(708\) 18.7602 0.705052
\(709\) −24.4835 −0.919498 −0.459749 0.888049i \(-0.652061\pi\)
−0.459749 + 0.888049i \(0.652061\pi\)
\(710\) 2.77482 0.104137
\(711\) 7.67288 0.287756
\(712\) 96.9538 3.63350
\(713\) 15.9959 0.599052
\(714\) −0.587398 −0.0219828
\(715\) 0.0206652 0.000772833 0
\(716\) −4.18229 −0.156300
\(717\) 19.1324 0.714513
\(718\) −4.04876 −0.151098
\(719\) 42.5741 1.58775 0.793874 0.608083i \(-0.208061\pi\)
0.793874 + 0.608083i \(0.208061\pi\)
\(720\) 1.38511 0.0516199
\(721\) 0.888868 0.0331032
\(722\) −34.9140 −1.29937
\(723\) −18.8973 −0.702797
\(724\) −85.7994 −3.18871
\(725\) −27.6360 −1.02638
\(726\) 29.1296 1.08110
\(727\) 6.89992 0.255904 0.127952 0.991780i \(-0.459160\pi\)
0.127952 + 0.991780i \(0.459160\pi\)
\(728\) 0.516391 0.0191387
\(729\) 1.00000 0.0370370
\(730\) −3.58890 −0.132831
\(731\) 27.7257 1.02547
\(732\) −58.5816 −2.16524
\(733\) −8.87156 −0.327679 −0.163839 0.986487i \(-0.552388\pi\)
−0.163839 + 0.986487i \(0.552388\pi\)
\(734\) −76.9239 −2.83931
\(735\) 0.852662 0.0314509
\(736\) 50.7851 1.87196
\(737\) −2.22964 −0.0821298
\(738\) −25.5571 −0.940769
\(739\) 20.1269 0.740381 0.370190 0.928956i \(-0.379292\pi\)
0.370190 + 0.928956i \(0.379292\pi\)
\(740\) −4.31400 −0.158586
\(741\) −2.56360 −0.0941762
\(742\) −0.262559 −0.00963883
\(743\) −18.0241 −0.661240 −0.330620 0.943764i \(-0.607258\pi\)
−0.330620 + 0.943764i \(0.607258\pi\)
\(744\) −35.6452 −1.30682
\(745\) −0.322259 −0.0118066
\(746\) 2.86063 0.104735
\(747\) 7.42087 0.271515
\(748\) 2.96366 0.108362
\(749\) −0.558212 −0.0203966
\(750\) 3.23007 0.117945
\(751\) −27.8187 −1.01512 −0.507560 0.861616i \(-0.669452\pi\)
−0.507560 + 0.861616i \(0.669452\pi\)
\(752\) −2.84864 −0.103879
\(753\) −24.0758 −0.877370
\(754\) −15.6012 −0.568164
\(755\) −0.902920 −0.0328606
\(756\) 0.303996 0.0110562
\(757\) 13.2283 0.480792 0.240396 0.970675i \(-0.422723\pi\)
0.240396 + 0.970675i \(0.422723\pi\)
\(758\) −1.84283 −0.0669346
\(759\) 0.580137 0.0210577
\(760\) −2.38200 −0.0864043
\(761\) 13.8717 0.502848 0.251424 0.967877i \(-0.419101\pi\)
0.251424 + 0.967877i \(0.419101\pi\)
\(762\) −2.65432 −0.0961558
\(763\) 0.00814552 0.000294888 0
\(764\) 50.4915 1.82672
\(765\) −0.447622 −0.0161838
\(766\) −64.3475 −2.32497
\(767\) 3.94232 0.142349
\(768\) 1.51547 0.0546847
\(769\) −11.3251 −0.408394 −0.204197 0.978930i \(-0.565458\pi\)
−0.204197 + 0.978930i \(0.565458\pi\)
\(770\) 0.00311713 0.000112334 0
\(771\) 21.1973 0.763404
\(772\) −97.1730 −3.49733
\(773\) 21.9384 0.789068 0.394534 0.918881i \(-0.370906\pi\)
0.394534 + 0.918881i \(0.370906\pi\)
\(774\) −20.0368 −0.720207
\(775\) −21.9827 −0.789643
\(776\) −7.87402 −0.282661
\(777\) −0.422721 −0.0151650
\(778\) 101.959 3.65542
\(779\) 23.2809 0.834124
\(780\) 0.651942 0.0233433
\(781\) −1.37184 −0.0490883
\(782\) −35.3643 −1.26463
\(783\) −5.54367 −0.198115
\(784\) −79.5155 −2.83984
\(785\) 0.991044 0.0353719
\(786\) −30.2370 −1.07852
\(787\) 27.2535 0.971484 0.485742 0.874102i \(-0.338550\pi\)
0.485742 + 0.874102i \(0.338550\pi\)
\(788\) 39.6307 1.41178
\(789\) 30.1230 1.07241
\(790\) 2.48208 0.0883084
\(791\) −0.636315 −0.0226248
\(792\) −1.29277 −0.0459367
\(793\) −12.3105 −0.437158
\(794\) 79.3884 2.81739
\(795\) −0.200081 −0.00709613
\(796\) 84.4767 2.99420
\(797\) 25.8283 0.914887 0.457443 0.889239i \(-0.348765\pi\)
0.457443 + 0.889239i \(0.348765\pi\)
\(798\) −0.386693 −0.0136888
\(799\) 0.920589 0.0325681
\(800\) −69.7925 −2.46754
\(801\) 11.9941 0.423791
\(802\) 0.515975 0.0182197
\(803\) 1.77431 0.0626142
\(804\) −70.3404 −2.48071
\(805\) −0.0266368 −0.000938824 0
\(806\) −12.4098 −0.437118
\(807\) 7.96482 0.280375
\(808\) −1.24447 −0.0437804
\(809\) 35.8182 1.25930 0.629651 0.776878i \(-0.283198\pi\)
0.629651 + 0.776878i \(0.283198\pi\)
\(810\) 0.323487 0.0113662
\(811\) 13.5766 0.476739 0.238370 0.971174i \(-0.423387\pi\)
0.238370 + 0.971174i \(0.423387\pi\)
\(812\) −1.68525 −0.0591408
\(813\) −11.7971 −0.413743
\(814\) 2.97824 0.104387
\(815\) −0.243070 −0.00851436
\(816\) 41.7433 1.46131
\(817\) 18.2522 0.638565
\(818\) −15.9538 −0.557810
\(819\) 0.0638825 0.00223223
\(820\) −5.92049 −0.206753
\(821\) −22.3923 −0.781495 −0.390748 0.920498i \(-0.627783\pi\)
−0.390748 + 0.920498i \(0.627783\pi\)
\(822\) −54.6167 −1.90498
\(823\) −46.2161 −1.61099 −0.805495 0.592602i \(-0.798101\pi\)
−0.805495 + 0.592602i \(0.798101\pi\)
\(824\) −119.251 −4.15430
\(825\) −0.797266 −0.0277573
\(826\) 0.594659 0.0206908
\(827\) −43.3279 −1.50666 −0.753329 0.657644i \(-0.771553\pi\)
−0.753329 + 0.657644i \(0.771553\pi\)
\(828\) 18.3021 0.636042
\(829\) 30.7055 1.06645 0.533223 0.845975i \(-0.320981\pi\)
0.533223 + 0.845975i \(0.320981\pi\)
\(830\) 2.40056 0.0833245
\(831\) 30.6509 1.06327
\(832\) −15.2996 −0.530418
\(833\) 25.6968 0.890343
\(834\) −39.4386 −1.36565
\(835\) 1.93964 0.0671241
\(836\) 1.95102 0.0674775
\(837\) −4.40965 −0.152420
\(838\) −6.55746 −0.226524
\(839\) −1.04914 −0.0362202 −0.0181101 0.999836i \(-0.505765\pi\)
−0.0181101 + 0.999836i \(0.505765\pi\)
\(840\) 0.0593573 0.00204802
\(841\) 1.73225 0.0597328
\(842\) −3.91657 −0.134974
\(843\) 17.6965 0.609499
\(844\) −34.8318 −1.19896
\(845\) −1.44734 −0.0497899
\(846\) −0.665290 −0.0228732
\(847\) 0.661233 0.0227202
\(848\) 18.6586 0.640741
\(849\) −16.0146 −0.549621
\(850\) 48.6002 1.66697
\(851\) −25.4499 −0.872412
\(852\) −43.2787 −1.48270
\(853\) −21.7655 −0.745238 −0.372619 0.927984i \(-0.621540\pi\)
−0.372619 + 0.927984i \(0.621540\pi\)
\(854\) −1.85691 −0.0635422
\(855\) −0.294676 −0.0100777
\(856\) 74.8900 2.55969
\(857\) 1.31119 0.0447895 0.0223948 0.999749i \(-0.492871\pi\)
0.0223948 + 0.999749i \(0.492871\pi\)
\(858\) −0.450078 −0.0153654
\(859\) −22.6717 −0.773549 −0.386774 0.922174i \(-0.626411\pi\)
−0.386774 + 0.922174i \(0.626411\pi\)
\(860\) −4.64167 −0.158280
\(861\) −0.580137 −0.0197710
\(862\) −65.4834 −2.23037
\(863\) −52.8484 −1.79898 −0.899490 0.436941i \(-0.856062\pi\)
−0.899490 + 0.436941i \(0.856062\pi\)
\(864\) −14.0001 −0.476292
\(865\) −1.03127 −0.0350644
\(866\) −30.3480 −1.03127
\(867\) 3.50993 0.119204
\(868\) −1.34052 −0.0455001
\(869\) −1.22711 −0.0416269
\(870\) −1.79330 −0.0607987
\(871\) −14.7815 −0.500852
\(872\) −1.09281 −0.0370071
\(873\) −0.974091 −0.0329680
\(874\) −23.2809 −0.787488
\(875\) 0.0733215 0.00247872
\(876\) 55.9758 1.89125
\(877\) 19.1137 0.645423 0.322712 0.946497i \(-0.395406\pi\)
0.322712 + 0.946497i \(0.395406\pi\)
\(878\) 53.6742 1.81142
\(879\) 19.9488 0.672856
\(880\) −0.221518 −0.00746737
\(881\) −10.8265 −0.364756 −0.182378 0.983229i \(-0.558379\pi\)
−0.182378 + 0.983229i \(0.558379\pi\)
\(882\) −18.5706 −0.625304
\(883\) 28.4711 0.958128 0.479064 0.877780i \(-0.340976\pi\)
0.479064 + 0.877780i \(0.340976\pi\)
\(884\) 19.6477 0.660824
\(885\) 0.453155 0.0152326
\(886\) −32.5419 −1.09327
\(887\) −9.46381 −0.317763 −0.158882 0.987298i \(-0.550789\pi\)
−0.158882 + 0.987298i \(0.550789\pi\)
\(888\) 56.7124 1.90314
\(889\) −0.0602522 −0.00202079
\(890\) 3.87994 0.130056
\(891\) −0.159928 −0.00535780
\(892\) −126.341 −4.23020
\(893\) 0.606038 0.0202803
\(894\) 7.01865 0.234739
\(895\) −0.101024 −0.00337685
\(896\) −0.620719 −0.0207368
\(897\) 3.84605 0.128416
\(898\) −61.1869 −2.04183
\(899\) 24.4456 0.815307
\(900\) −25.1521 −0.838402
\(901\) −6.02987 −0.200884
\(902\) 4.08730 0.136092
\(903\) −0.454828 −0.0151357
\(904\) 85.3683 2.83931
\(905\) −2.07249 −0.0688920
\(906\) 19.6652 0.653332
\(907\) 21.5831 0.716655 0.358327 0.933596i \(-0.383347\pi\)
0.358327 + 0.933596i \(0.383347\pi\)
\(908\) −83.7021 −2.77775
\(909\) −0.153953 −0.00510630
\(910\) 0.0206652 0.000685043 0
\(911\) 32.1087 1.06381 0.531904 0.846805i \(-0.321477\pi\)
0.531904 + 0.846805i \(0.321477\pi\)
\(912\) 27.4802 0.909962
\(913\) −1.18681 −0.0392776
\(914\) 61.0185 2.01831
\(915\) −1.41504 −0.0467799
\(916\) 118.294 3.90853
\(917\) −0.686371 −0.0226660
\(918\) 9.74900 0.321765
\(919\) −34.2000 −1.12815 −0.564077 0.825722i \(-0.690768\pi\)
−0.564077 + 0.825722i \(0.690768\pi\)
\(920\) 3.57361 0.117818
\(921\) −1.74509 −0.0575025
\(922\) −59.1869 −1.94922
\(923\) −9.09468 −0.299355
\(924\) −0.0486176 −0.00159940
\(925\) 34.9751 1.14997
\(926\) −11.9977 −0.394267
\(927\) −14.7525 −0.484534
\(928\) 77.6118 2.54773
\(929\) 29.2758 0.960508 0.480254 0.877129i \(-0.340544\pi\)
0.480254 + 0.877129i \(0.340544\pi\)
\(930\) −1.42646 −0.0467756
\(931\) 16.9166 0.554420
\(932\) 129.494 4.24172
\(933\) −16.3942 −0.536723
\(934\) −9.13123 −0.298783
\(935\) 0.0715874 0.00234116
\(936\) −8.57050 −0.280136
\(937\) 28.7035 0.937704 0.468852 0.883277i \(-0.344668\pi\)
0.468852 + 0.883277i \(0.344668\pi\)
\(938\) −2.22964 −0.0728003
\(939\) 3.25949 0.106369
\(940\) −0.154120 −0.00502683
\(941\) −3.41244 −0.111242 −0.0556211 0.998452i \(-0.517714\pi\)
−0.0556211 + 0.998452i \(0.517714\pi\)
\(942\) −21.5845 −0.703260
\(943\) −34.9272 −1.13739
\(944\) −42.2592 −1.37542
\(945\) 0.00734305 0.000238870 0
\(946\) 3.20445 0.104186
\(947\) 15.1032 0.490789 0.245395 0.969423i \(-0.421082\pi\)
0.245395 + 0.969423i \(0.421082\pi\)
\(948\) −38.7128 −1.25733
\(949\) 11.7629 0.381840
\(950\) 31.9943 1.03803
\(951\) 3.81807 0.123809
\(952\) 1.78886 0.0579773
\(953\) −32.9900 −1.06865 −0.534326 0.845279i \(-0.679434\pi\)
−0.534326 + 0.845279i \(0.679434\pi\)
\(954\) 4.35766 0.141085
\(955\) 1.21963 0.0394661
\(956\) −96.5306 −3.12202
\(957\) 0.886590 0.0286594
\(958\) −43.6905 −1.41158
\(959\) −1.23978 −0.0400347
\(960\) −1.75863 −0.0567596
\(961\) −11.5550 −0.372742
\(962\) 19.7444 0.636584
\(963\) 9.26460 0.298548
\(964\) 95.3442 3.07083
\(965\) −2.34722 −0.0755597
\(966\) 0.580137 0.0186656
\(967\) −31.9734 −1.02819 −0.514097 0.857732i \(-0.671873\pi\)
−0.514097 + 0.857732i \(0.671873\pi\)
\(968\) −88.7113 −2.85129
\(969\) −8.88072 −0.285290
\(970\) −0.315106 −0.0101174
\(971\) 9.23119 0.296243 0.148122 0.988969i \(-0.452677\pi\)
0.148122 + 0.988969i \(0.452677\pi\)
\(972\) −5.04540 −0.161831
\(973\) −0.895243 −0.0287002
\(974\) 74.5297 2.38809
\(975\) −5.28551 −0.169272
\(976\) 131.961 4.22396
\(977\) 9.34029 0.298822 0.149411 0.988775i \(-0.452262\pi\)
0.149411 + 0.988775i \(0.452262\pi\)
\(978\) 5.29395 0.169282
\(979\) −1.91820 −0.0613059
\(980\) −4.30202 −0.137423
\(981\) −0.135190 −0.00431630
\(982\) 13.3453 0.425865
\(983\) 22.9545 0.732135 0.366068 0.930588i \(-0.380704\pi\)
0.366068 + 0.930588i \(0.380704\pi\)
\(984\) 77.8315 2.48118
\(985\) 0.957282 0.0305015
\(986\) −54.0452 −1.72115
\(987\) −0.0151019 −0.000480698 0
\(988\) 12.9344 0.411498
\(989\) −27.3830 −0.870727
\(990\) −0.0517347 −0.00164424
\(991\) 7.38759 0.234674 0.117337 0.993092i \(-0.462564\pi\)
0.117337 + 0.993092i \(0.462564\pi\)
\(992\) 61.7354 1.96010
\(993\) −13.9352 −0.442221
\(994\) −1.37184 −0.0435121
\(995\) 2.04054 0.0646895
\(996\) −37.4413 −1.18637
\(997\) −21.8910 −0.693296 −0.346648 0.937995i \(-0.612680\pi\)
−0.346648 + 0.937995i \(0.612680\pi\)
\(998\) 61.6137 1.95035
\(999\) 7.01586 0.221972
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 381.2.a.d.1.5 5
3.2 odd 2 1143.2.a.g.1.1 5
4.3 odd 2 6096.2.a.bf.1.3 5
5.4 even 2 9525.2.a.j.1.1 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
381.2.a.d.1.5 5 1.1 even 1 trivial
1143.2.a.g.1.1 5 3.2 odd 2
6096.2.a.bf.1.3 5 4.3 odd 2
9525.2.a.j.1.1 5 5.4 even 2