Properties

Label 8031.2.a.c.1.4
Level $8031$
Weight $2$
Character 8031.1
Self dual yes
Analytic conductor $64.128$
Analytic rank $0$
Dimension $121$
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] = [8031,2,Mod(1,8031)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(8031, 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("8031.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 8031 = 3 \cdot 2677 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 8031.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: \(64.1278578633\)
Analytic rank: \(0\)
Dimension: \(121\)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.4
Character \(\chi\) \(=\) 8031.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-2.64556 q^{2} -1.00000 q^{3} +4.99897 q^{4} -1.40412 q^{5} +2.64556 q^{6} -0.985442 q^{7} -7.93394 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.64556 q^{2} -1.00000 q^{3} +4.99897 q^{4} -1.40412 q^{5} +2.64556 q^{6} -0.985442 q^{7} -7.93394 q^{8} +1.00000 q^{9} +3.71467 q^{10} +2.33194 q^{11} -4.99897 q^{12} -2.71275 q^{13} +2.60704 q^{14} +1.40412 q^{15} +10.9918 q^{16} -2.75070 q^{17} -2.64556 q^{18} -2.99847 q^{19} -7.01913 q^{20} +0.985442 q^{21} -6.16927 q^{22} +7.66768 q^{23} +7.93394 q^{24} -3.02846 q^{25} +7.17672 q^{26} -1.00000 q^{27} -4.92619 q^{28} +5.96396 q^{29} -3.71467 q^{30} +7.67624 q^{31} -13.2114 q^{32} -2.33194 q^{33} +7.27713 q^{34} +1.38367 q^{35} +4.99897 q^{36} -10.2707 q^{37} +7.93263 q^{38} +2.71275 q^{39} +11.1402 q^{40} +8.86891 q^{41} -2.60704 q^{42} +0.458711 q^{43} +11.6573 q^{44} -1.40412 q^{45} -20.2853 q^{46} -9.88478 q^{47} -10.9918 q^{48} -6.02890 q^{49} +8.01196 q^{50} +2.75070 q^{51} -13.5609 q^{52} -5.24831 q^{53} +2.64556 q^{54} -3.27431 q^{55} +7.81844 q^{56} +2.99847 q^{57} -15.7780 q^{58} +14.0479 q^{59} +7.01913 q^{60} -7.61343 q^{61} -20.3079 q^{62} -0.985442 q^{63} +12.9681 q^{64} +3.80901 q^{65} +6.16927 q^{66} +14.2913 q^{67} -13.7507 q^{68} -7.66768 q^{69} -3.66059 q^{70} -1.35018 q^{71} -7.93394 q^{72} -6.43090 q^{73} +27.1716 q^{74} +3.02846 q^{75} -14.9893 q^{76} -2.29799 q^{77} -7.17672 q^{78} +6.95662 q^{79} -15.4337 q^{80} +1.00000 q^{81} -23.4632 q^{82} +9.57486 q^{83} +4.92619 q^{84} +3.86230 q^{85} -1.21354 q^{86} -5.96396 q^{87} -18.5014 q^{88} -10.3413 q^{89} +3.71467 q^{90} +2.67325 q^{91} +38.3305 q^{92} -7.67624 q^{93} +26.1507 q^{94} +4.21020 q^{95} +13.2114 q^{96} +0.279354 q^{97} +15.9498 q^{98} +2.33194 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 121 q + 7 q^{2} - 121 q^{3} + 123 q^{4} + 24 q^{5} - 7 q^{6} - 14 q^{7} + 18 q^{8} + 121 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 121 q + 7 q^{2} - 121 q^{3} + 123 q^{4} + 24 q^{5} - 7 q^{6} - 14 q^{7} + 18 q^{8} + 121 q^{9} + 18 q^{10} + 32 q^{11} - 123 q^{12} + 2 q^{13} + 37 q^{14} - 24 q^{15} + 131 q^{16} + 87 q^{17} + 7 q^{18} - 10 q^{19} + 60 q^{20} + 14 q^{21} - 22 q^{22} + 31 q^{23} - 18 q^{24} + 147 q^{25} + 37 q^{26} - 121 q^{27} - 29 q^{28} + 68 q^{29} - 18 q^{30} + 25 q^{31} + 43 q^{32} - 32 q^{33} + 27 q^{34} + 51 q^{35} + 123 q^{36} - 4 q^{37} + 36 q^{38} - 2 q^{39} + 61 q^{40} + 132 q^{41} - 37 q^{42} - 91 q^{43} + 94 q^{44} + 24 q^{45} + 39 q^{47} - 131 q^{48} + 217 q^{49} + 54 q^{50} - 87 q^{51} - 12 q^{52} + 55 q^{53} - 7 q^{54} + 7 q^{55} + 104 q^{56} + 10 q^{57} - 3 q^{58} + 58 q^{59} - 60 q^{60} + 126 q^{61} + 74 q^{62} - 14 q^{63} + 122 q^{64} + 128 q^{65} + 22 q^{66} - 139 q^{67} + 190 q^{68} - 31 q^{69} - 18 q^{70} + 37 q^{71} + 18 q^{72} + 84 q^{73} + 79 q^{74} - 147 q^{75} + 23 q^{76} + 95 q^{77} - 37 q^{78} - 14 q^{79} + 145 q^{80} + 121 q^{81} + 9 q^{82} + 58 q^{83} + 29 q^{84} + 32 q^{85} + 28 q^{86} - 68 q^{87} - 84 q^{88} + 198 q^{89} + 18 q^{90} + 5 q^{91} + 98 q^{92} - 25 q^{93} + 9 q^{94} + 42 q^{95} - 43 q^{96} + 73 q^{97} + 69 q^{98} + 32 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.64556 −1.87069 −0.935345 0.353736i \(-0.884911\pi\)
−0.935345 + 0.353736i \(0.884911\pi\)
\(3\) −1.00000 −0.577350
\(4\) 4.99897 2.49948
\(5\) −1.40412 −0.627939 −0.313970 0.949433i \(-0.601659\pi\)
−0.313970 + 0.949433i \(0.601659\pi\)
\(6\) 2.64556 1.08004
\(7\) −0.985442 −0.372462 −0.186231 0.982506i \(-0.559627\pi\)
−0.186231 + 0.982506i \(0.559627\pi\)
\(8\) −7.93394 −2.80507
\(9\) 1.00000 0.333333
\(10\) 3.71467 1.17468
\(11\) 2.33194 0.703105 0.351553 0.936168i \(-0.385654\pi\)
0.351553 + 0.936168i \(0.385654\pi\)
\(12\) −4.99897 −1.44308
\(13\) −2.71275 −0.752380 −0.376190 0.926543i \(-0.622766\pi\)
−0.376190 + 0.926543i \(0.622766\pi\)
\(14\) 2.60704 0.696761
\(15\) 1.40412 0.362541
\(16\) 10.9918 2.74794
\(17\) −2.75070 −0.667143 −0.333571 0.942725i \(-0.608254\pi\)
−0.333571 + 0.942725i \(0.608254\pi\)
\(18\) −2.64556 −0.623564
\(19\) −2.99847 −0.687897 −0.343948 0.938989i \(-0.611765\pi\)
−0.343948 + 0.938989i \(0.611765\pi\)
\(20\) −7.01913 −1.56953
\(21\) 0.985442 0.215041
\(22\) −6.16927 −1.31529
\(23\) 7.66768 1.59882 0.799410 0.600785i \(-0.205145\pi\)
0.799410 + 0.600785i \(0.205145\pi\)
\(24\) 7.93394 1.61951
\(25\) −3.02846 −0.605692
\(26\) 7.17672 1.40747
\(27\) −1.00000 −0.192450
\(28\) −4.92619 −0.930963
\(29\) 5.96396 1.10748 0.553739 0.832690i \(-0.313200\pi\)
0.553739 + 0.832690i \(0.313200\pi\)
\(30\) −3.71467 −0.678202
\(31\) 7.67624 1.37869 0.689347 0.724431i \(-0.257898\pi\)
0.689347 + 0.724431i \(0.257898\pi\)
\(32\) −13.2114 −2.33547
\(33\) −2.33194 −0.405938
\(34\) 7.27713 1.24802
\(35\) 1.38367 0.233884
\(36\) 4.99897 0.833162
\(37\) −10.2707 −1.68849 −0.844244 0.535959i \(-0.819950\pi\)
−0.844244 + 0.535959i \(0.819950\pi\)
\(38\) 7.93263 1.28684
\(39\) 2.71275 0.434387
\(40\) 11.1402 1.76142
\(41\) 8.86891 1.38509 0.692546 0.721374i \(-0.256489\pi\)
0.692546 + 0.721374i \(0.256489\pi\)
\(42\) −2.60704 −0.402275
\(43\) 0.458711 0.0699527 0.0349764 0.999388i \(-0.488864\pi\)
0.0349764 + 0.999388i \(0.488864\pi\)
\(44\) 11.6573 1.75740
\(45\) −1.40412 −0.209313
\(46\) −20.2853 −2.99090
\(47\) −9.88478 −1.44184 −0.720921 0.693017i \(-0.756281\pi\)
−0.720921 + 0.693017i \(0.756281\pi\)
\(48\) −10.9918 −1.58652
\(49\) −6.02890 −0.861272
\(50\) 8.01196 1.13306
\(51\) 2.75070 0.385175
\(52\) −13.5609 −1.88056
\(53\) −5.24831 −0.720911 −0.360455 0.932776i \(-0.617379\pi\)
−0.360455 + 0.932776i \(0.617379\pi\)
\(54\) 2.64556 0.360015
\(55\) −3.27431 −0.441507
\(56\) 7.81844 1.04478
\(57\) 2.99847 0.397157
\(58\) −15.7780 −2.07175
\(59\) 14.0479 1.82889 0.914443 0.404716i \(-0.132630\pi\)
0.914443 + 0.404716i \(0.132630\pi\)
\(60\) 7.01913 0.906166
\(61\) −7.61343 −0.974800 −0.487400 0.873179i \(-0.662055\pi\)
−0.487400 + 0.873179i \(0.662055\pi\)
\(62\) −20.3079 −2.57911
\(63\) −0.985442 −0.124154
\(64\) 12.9681 1.62101
\(65\) 3.80901 0.472449
\(66\) 6.16927 0.759384
\(67\) 14.2913 1.74596 0.872979 0.487758i \(-0.162185\pi\)
0.872979 + 0.487758i \(0.162185\pi\)
\(68\) −13.7507 −1.66751
\(69\) −7.66768 −0.923080
\(70\) −3.66059 −0.437524
\(71\) −1.35018 −0.160237 −0.0801186 0.996785i \(-0.525530\pi\)
−0.0801186 + 0.996785i \(0.525530\pi\)
\(72\) −7.93394 −0.935024
\(73\) −6.43090 −0.752680 −0.376340 0.926482i \(-0.622817\pi\)
−0.376340 + 0.926482i \(0.622817\pi\)
\(74\) 27.1716 3.15864
\(75\) 3.02846 0.349696
\(76\) −14.9893 −1.71939
\(77\) −2.29799 −0.261880
\(78\) −7.17672 −0.812604
\(79\) 6.95662 0.782681 0.391341 0.920246i \(-0.372011\pi\)
0.391341 + 0.920246i \(0.372011\pi\)
\(80\) −15.4337 −1.72554
\(81\) 1.00000 0.111111
\(82\) −23.4632 −2.59108
\(83\) 9.57486 1.05098 0.525489 0.850801i \(-0.323883\pi\)
0.525489 + 0.850801i \(0.323883\pi\)
\(84\) 4.92619 0.537492
\(85\) 3.86230 0.418925
\(86\) −1.21354 −0.130860
\(87\) −5.96396 −0.639403
\(88\) −18.5014 −1.97226
\(89\) −10.3413 −1.09618 −0.548088 0.836421i \(-0.684644\pi\)
−0.548088 + 0.836421i \(0.684644\pi\)
\(90\) 3.71467 0.391560
\(91\) 2.67325 0.280233
\(92\) 38.3305 3.99623
\(93\) −7.67624 −0.795989
\(94\) 26.1507 2.69724
\(95\) 4.21020 0.431958
\(96\) 13.2114 1.34839
\(97\) 0.279354 0.0283641 0.0141820 0.999899i \(-0.495486\pi\)
0.0141820 + 0.999899i \(0.495486\pi\)
\(98\) 15.9498 1.61117
\(99\) 2.33194 0.234368
\(100\) −15.1392 −1.51392
\(101\) −8.32117 −0.827987 −0.413994 0.910280i \(-0.635866\pi\)
−0.413994 + 0.910280i \(0.635866\pi\)
\(102\) −7.27713 −0.720543
\(103\) −10.8506 −1.06914 −0.534570 0.845124i \(-0.679526\pi\)
−0.534570 + 0.845124i \(0.679526\pi\)
\(104\) 21.5228 2.11048
\(105\) −1.38367 −0.135033
\(106\) 13.8847 1.34860
\(107\) 6.04572 0.584462 0.292231 0.956348i \(-0.405602\pi\)
0.292231 + 0.956348i \(0.405602\pi\)
\(108\) −4.99897 −0.481026
\(109\) −13.4335 −1.28670 −0.643350 0.765572i \(-0.722456\pi\)
−0.643350 + 0.765572i \(0.722456\pi\)
\(110\) 8.66236 0.825924
\(111\) 10.2707 0.974849
\(112\) −10.8317 −1.02350
\(113\) −14.9232 −1.40386 −0.701929 0.712247i \(-0.747677\pi\)
−0.701929 + 0.712247i \(0.747677\pi\)
\(114\) −7.93263 −0.742959
\(115\) −10.7663 −1.00396
\(116\) 29.8136 2.76813
\(117\) −2.71275 −0.250793
\(118\) −37.1646 −3.42128
\(119\) 2.71065 0.248485
\(120\) −11.1402 −1.01695
\(121\) −5.56207 −0.505643
\(122\) 20.1418 1.82355
\(123\) −8.86891 −0.799683
\(124\) 38.3733 3.44603
\(125\) 11.2729 1.00828
\(126\) 2.60704 0.232254
\(127\) 5.83385 0.517671 0.258835 0.965921i \(-0.416661\pi\)
0.258835 + 0.965921i \(0.416661\pi\)
\(128\) −7.88489 −0.696933
\(129\) −0.458711 −0.0403872
\(130\) −10.0769 −0.883806
\(131\) 11.7221 1.02417 0.512084 0.858935i \(-0.328874\pi\)
0.512084 + 0.858935i \(0.328874\pi\)
\(132\) −11.6573 −1.01464
\(133\) 2.95482 0.256215
\(134\) −37.8084 −3.26615
\(135\) 1.40412 0.120847
\(136\) 21.8239 1.87138
\(137\) −8.09665 −0.691744 −0.345872 0.938282i \(-0.612417\pi\)
−0.345872 + 0.938282i \(0.612417\pi\)
\(138\) 20.2853 1.72680
\(139\) 3.25457 0.276049 0.138024 0.990429i \(-0.455925\pi\)
0.138024 + 0.990429i \(0.455925\pi\)
\(140\) 6.91694 0.584588
\(141\) 9.88478 0.832448
\(142\) 3.57199 0.299754
\(143\) −6.32595 −0.529002
\(144\) 10.9918 0.915980
\(145\) −8.37408 −0.695430
\(146\) 17.0133 1.40803
\(147\) 6.02890 0.497256
\(148\) −51.3428 −4.22035
\(149\) 6.34177 0.519538 0.259769 0.965671i \(-0.416354\pi\)
0.259769 + 0.965671i \(0.416354\pi\)
\(150\) −8.01196 −0.654174
\(151\) −7.44976 −0.606253 −0.303127 0.952950i \(-0.598030\pi\)
−0.303127 + 0.952950i \(0.598030\pi\)
\(152\) 23.7897 1.92960
\(153\) −2.75070 −0.222381
\(154\) 6.07945 0.489896
\(155\) −10.7783 −0.865736
\(156\) 13.5609 1.08574
\(157\) 22.0200 1.75738 0.878692 0.477389i \(-0.158417\pi\)
0.878692 + 0.477389i \(0.158417\pi\)
\(158\) −18.4041 −1.46415
\(159\) 5.24831 0.416218
\(160\) 18.5504 1.46654
\(161\) −7.55605 −0.595500
\(162\) −2.64556 −0.207855
\(163\) 6.26580 0.490775 0.245388 0.969425i \(-0.421085\pi\)
0.245388 + 0.969425i \(0.421085\pi\)
\(164\) 44.3354 3.46202
\(165\) 3.27431 0.254904
\(166\) −25.3308 −1.96605
\(167\) 16.1336 1.24846 0.624228 0.781242i \(-0.285414\pi\)
0.624228 + 0.781242i \(0.285414\pi\)
\(168\) −7.81844 −0.603206
\(169\) −5.64101 −0.433924
\(170\) −10.2179 −0.783679
\(171\) −2.99847 −0.229299
\(172\) 2.29308 0.174846
\(173\) 0.726527 0.0552368 0.0276184 0.999619i \(-0.491208\pi\)
0.0276184 + 0.999619i \(0.491208\pi\)
\(174\) 15.7780 1.19613
\(175\) 2.98437 0.225597
\(176\) 25.6321 1.93209
\(177\) −14.0479 −1.05591
\(178\) 27.3585 2.05060
\(179\) −20.8854 −1.56105 −0.780524 0.625126i \(-0.785047\pi\)
−0.780524 + 0.625126i \(0.785047\pi\)
\(180\) −7.01913 −0.523175
\(181\) 13.4557 1.00016 0.500079 0.865980i \(-0.333304\pi\)
0.500079 + 0.865980i \(0.333304\pi\)
\(182\) −7.07224 −0.524229
\(183\) 7.61343 0.562801
\(184\) −60.8349 −4.48481
\(185\) 14.4212 1.06027
\(186\) 20.3079 1.48905
\(187\) −6.41445 −0.469071
\(188\) −49.4137 −3.60386
\(189\) 0.985442 0.0716803
\(190\) −11.1383 −0.808059
\(191\) 5.37858 0.389180 0.194590 0.980885i \(-0.437662\pi\)
0.194590 + 0.980885i \(0.437662\pi\)
\(192\) −12.9681 −0.935889
\(193\) 0.0783143 0.00563719 0.00281859 0.999996i \(-0.499103\pi\)
0.00281859 + 0.999996i \(0.499103\pi\)
\(194\) −0.739046 −0.0530604
\(195\) −3.80901 −0.272769
\(196\) −30.1383 −2.15274
\(197\) −12.4020 −0.883603 −0.441801 0.897113i \(-0.645660\pi\)
−0.441801 + 0.897113i \(0.645660\pi\)
\(198\) −6.16927 −0.438431
\(199\) −17.8105 −1.26255 −0.631275 0.775559i \(-0.717468\pi\)
−0.631275 + 0.775559i \(0.717468\pi\)
\(200\) 24.0276 1.69901
\(201\) −14.2913 −1.00803
\(202\) 22.0141 1.54891
\(203\) −5.87713 −0.412494
\(204\) 13.7507 0.962739
\(205\) −12.4530 −0.869754
\(206\) 28.7058 2.00003
\(207\) 7.66768 0.532940
\(208\) −29.8178 −2.06749
\(209\) −6.99225 −0.483664
\(210\) 3.66059 0.252604
\(211\) 0.860963 0.0592712 0.0296356 0.999561i \(-0.490565\pi\)
0.0296356 + 0.999561i \(0.490565\pi\)
\(212\) −26.2361 −1.80191
\(213\) 1.35018 0.0925130
\(214\) −15.9943 −1.09335
\(215\) −0.644083 −0.0439261
\(216\) 7.93394 0.539836
\(217\) −7.56449 −0.513511
\(218\) 35.5392 2.40702
\(219\) 6.43090 0.434560
\(220\) −16.3682 −1.10354
\(221\) 7.46195 0.501945
\(222\) −27.1716 −1.82364
\(223\) −6.15709 −0.412309 −0.206155 0.978519i \(-0.566095\pi\)
−0.206155 + 0.978519i \(0.566095\pi\)
\(224\) 13.0191 0.869875
\(225\) −3.02846 −0.201897
\(226\) 39.4802 2.62618
\(227\) −28.0897 −1.86438 −0.932189 0.361971i \(-0.882104\pi\)
−0.932189 + 0.361971i \(0.882104\pi\)
\(228\) 14.9893 0.992689
\(229\) 1.04031 0.0687455 0.0343727 0.999409i \(-0.489057\pi\)
0.0343727 + 0.999409i \(0.489057\pi\)
\(230\) 28.4829 1.87810
\(231\) 2.29799 0.151196
\(232\) −47.3177 −3.10656
\(233\) 5.12414 0.335693 0.167847 0.985813i \(-0.446319\pi\)
0.167847 + 0.985813i \(0.446319\pi\)
\(234\) 7.17672 0.469157
\(235\) 13.8794 0.905390
\(236\) 70.2252 4.57127
\(237\) −6.95662 −0.451881
\(238\) −7.17119 −0.464839
\(239\) 6.04150 0.390792 0.195396 0.980724i \(-0.437401\pi\)
0.195396 + 0.980724i \(0.437401\pi\)
\(240\) 15.4337 0.996241
\(241\) 3.28202 0.211413 0.105707 0.994397i \(-0.466290\pi\)
0.105707 + 0.994397i \(0.466290\pi\)
\(242\) 14.7148 0.945902
\(243\) −1.00000 −0.0641500
\(244\) −38.0593 −2.43650
\(245\) 8.46528 0.540827
\(246\) 23.4632 1.49596
\(247\) 8.13409 0.517560
\(248\) −60.9029 −3.86734
\(249\) −9.57486 −0.606782
\(250\) −29.8231 −1.88618
\(251\) 24.9522 1.57497 0.787484 0.616335i \(-0.211383\pi\)
0.787484 + 0.616335i \(0.211383\pi\)
\(252\) −4.92619 −0.310321
\(253\) 17.8805 1.12414
\(254\) −15.4338 −0.968402
\(255\) −3.86230 −0.241867
\(256\) −5.07620 −0.317262
\(257\) 9.84196 0.613925 0.306962 0.951722i \(-0.400687\pi\)
0.306962 + 0.951722i \(0.400687\pi\)
\(258\) 1.21354 0.0755520
\(259\) 10.1211 0.628898
\(260\) 19.0411 1.18088
\(261\) 5.96396 0.369160
\(262\) −31.0116 −1.91590
\(263\) −29.9080 −1.84421 −0.922104 0.386942i \(-0.873531\pi\)
−0.922104 + 0.386942i \(0.873531\pi\)
\(264\) 18.5014 1.13869
\(265\) 7.36923 0.452688
\(266\) −7.81714 −0.479300
\(267\) 10.3413 0.632877
\(268\) 71.4417 4.36400
\(269\) 12.0051 0.731963 0.365981 0.930622i \(-0.380733\pi\)
0.365981 + 0.930622i \(0.380733\pi\)
\(270\) −3.71467 −0.226067
\(271\) −11.2519 −0.683506 −0.341753 0.939790i \(-0.611021\pi\)
−0.341753 + 0.939790i \(0.611021\pi\)
\(272\) −30.2350 −1.83327
\(273\) −2.67325 −0.161793
\(274\) 21.4201 1.29404
\(275\) −7.06218 −0.425865
\(276\) −38.3305 −2.30722
\(277\) 14.5854 0.876355 0.438177 0.898889i \(-0.355624\pi\)
0.438177 + 0.898889i \(0.355624\pi\)
\(278\) −8.61014 −0.516402
\(279\) 7.67624 0.459565
\(280\) −10.9780 −0.656060
\(281\) 26.4126 1.57564 0.787822 0.615903i \(-0.211209\pi\)
0.787822 + 0.615903i \(0.211209\pi\)
\(282\) −26.1507 −1.55725
\(283\) 11.5981 0.689438 0.344719 0.938706i \(-0.387974\pi\)
0.344719 + 0.938706i \(0.387974\pi\)
\(284\) −6.74952 −0.400511
\(285\) −4.21020 −0.249391
\(286\) 16.7357 0.989600
\(287\) −8.73980 −0.515894
\(288\) −13.2114 −0.778491
\(289\) −9.43365 −0.554921
\(290\) 22.1541 1.30093
\(291\) −0.279354 −0.0163760
\(292\) −32.1479 −1.88131
\(293\) −16.3897 −0.957499 −0.478749 0.877952i \(-0.658910\pi\)
−0.478749 + 0.877952i \(0.658910\pi\)
\(294\) −15.9498 −0.930212
\(295\) −19.7249 −1.14843
\(296\) 81.4869 4.73633
\(297\) −2.33194 −0.135313
\(298\) −16.7775 −0.971896
\(299\) −20.8004 −1.20292
\(300\) 15.1392 0.874061
\(301\) −0.452032 −0.0260547
\(302\) 19.7088 1.13411
\(303\) 8.32117 0.478039
\(304\) −32.9585 −1.89030
\(305\) 10.6901 0.612116
\(306\) 7.27713 0.416006
\(307\) −3.05538 −0.174380 −0.0871898 0.996192i \(-0.527789\pi\)
−0.0871898 + 0.996192i \(0.527789\pi\)
\(308\) −11.4876 −0.654565
\(309\) 10.8506 0.617268
\(310\) 28.5147 1.61953
\(311\) 9.26605 0.525430 0.262715 0.964874i \(-0.415382\pi\)
0.262715 + 0.964874i \(0.415382\pi\)
\(312\) −21.5228 −1.21849
\(313\) 2.32324 0.131317 0.0656586 0.997842i \(-0.479085\pi\)
0.0656586 + 0.997842i \(0.479085\pi\)
\(314\) −58.2550 −3.28752
\(315\) 1.38367 0.0779612
\(316\) 34.7760 1.95630
\(317\) −25.0313 −1.40590 −0.702948 0.711241i \(-0.748133\pi\)
−0.702948 + 0.711241i \(0.748133\pi\)
\(318\) −13.8847 −0.778615
\(319\) 13.9076 0.778674
\(320\) −18.2087 −1.01789
\(321\) −6.04572 −0.337439
\(322\) 19.9899 1.11400
\(323\) 8.24790 0.458925
\(324\) 4.99897 0.277721
\(325\) 8.21544 0.455711
\(326\) −16.5765 −0.918089
\(327\) 13.4335 0.742877
\(328\) −70.3655 −3.88528
\(329\) 9.74087 0.537032
\(330\) −8.66236 −0.476847
\(331\) −13.9106 −0.764594 −0.382297 0.924039i \(-0.624867\pi\)
−0.382297 + 0.924039i \(0.624867\pi\)
\(332\) 47.8644 2.62690
\(333\) −10.2707 −0.562829
\(334\) −42.6824 −2.33548
\(335\) −20.0666 −1.09636
\(336\) 10.8317 0.590920
\(337\) 26.4623 1.44149 0.720746 0.693199i \(-0.243799\pi\)
0.720746 + 0.693199i \(0.243799\pi\)
\(338\) 14.9236 0.811738
\(339\) 14.9232 0.810517
\(340\) 19.3075 1.04710
\(341\) 17.9005 0.969367
\(342\) 7.93263 0.428948
\(343\) 12.8392 0.693253
\(344\) −3.63938 −0.196222
\(345\) 10.7663 0.579638
\(346\) −1.92207 −0.103331
\(347\) −34.7535 −1.86566 −0.932832 0.360312i \(-0.882670\pi\)
−0.932832 + 0.360312i \(0.882670\pi\)
\(348\) −29.8136 −1.59818
\(349\) 7.79611 0.417316 0.208658 0.977989i \(-0.433090\pi\)
0.208658 + 0.977989i \(0.433090\pi\)
\(350\) −7.89532 −0.422023
\(351\) 2.71275 0.144796
\(352\) −30.8082 −1.64208
\(353\) 16.9282 0.900998 0.450499 0.892777i \(-0.351246\pi\)
0.450499 + 0.892777i \(0.351246\pi\)
\(354\) 37.1646 1.97528
\(355\) 1.89581 0.100619
\(356\) −51.6958 −2.73987
\(357\) −2.71065 −0.143463
\(358\) 55.2535 2.92024
\(359\) 22.9438 1.21093 0.605463 0.795874i \(-0.292988\pi\)
0.605463 + 0.795874i \(0.292988\pi\)
\(360\) 11.1402 0.587139
\(361\) −10.0092 −0.526798
\(362\) −35.5979 −1.87099
\(363\) 5.56207 0.291933
\(364\) 13.3635 0.700438
\(365\) 9.02972 0.472637
\(366\) −20.1418 −1.05283
\(367\) 23.5888 1.23133 0.615664 0.788009i \(-0.288888\pi\)
0.615664 + 0.788009i \(0.288888\pi\)
\(368\) 84.2812 4.39346
\(369\) 8.86891 0.461697
\(370\) −38.1521 −1.98343
\(371\) 5.17190 0.268512
\(372\) −38.3733 −1.98956
\(373\) 10.1278 0.524396 0.262198 0.965014i \(-0.415553\pi\)
0.262198 + 0.965014i \(0.415553\pi\)
\(374\) 16.9698 0.877488
\(375\) −11.2729 −0.582129
\(376\) 78.4253 4.04447
\(377\) −16.1787 −0.833245
\(378\) −2.60704 −0.134092
\(379\) 5.57389 0.286311 0.143156 0.989700i \(-0.454275\pi\)
0.143156 + 0.989700i \(0.454275\pi\)
\(380\) 21.0467 1.07967
\(381\) −5.83385 −0.298877
\(382\) −14.2293 −0.728036
\(383\) 9.26151 0.473241 0.236621 0.971602i \(-0.423960\pi\)
0.236621 + 0.971602i \(0.423960\pi\)
\(384\) 7.88489 0.402374
\(385\) 3.22664 0.164445
\(386\) −0.207185 −0.0105454
\(387\) 0.458711 0.0233176
\(388\) 1.39648 0.0708955
\(389\) −34.1362 −1.73077 −0.865387 0.501105i \(-0.832927\pi\)
−0.865387 + 0.501105i \(0.832927\pi\)
\(390\) 10.0769 0.510266
\(391\) −21.0915 −1.06664
\(392\) 47.8330 2.41593
\(393\) −11.7221 −0.591304
\(394\) 32.8101 1.65295
\(395\) −9.76790 −0.491477
\(396\) 11.6573 0.585800
\(397\) −5.60042 −0.281077 −0.140538 0.990075i \(-0.544883\pi\)
−0.140538 + 0.990075i \(0.544883\pi\)
\(398\) 47.1186 2.36184
\(399\) −2.95482 −0.147926
\(400\) −33.2881 −1.66440
\(401\) 1.60184 0.0799919 0.0399960 0.999200i \(-0.487265\pi\)
0.0399960 + 0.999200i \(0.487265\pi\)
\(402\) 37.8084 1.88571
\(403\) −20.8237 −1.03730
\(404\) −41.5973 −2.06954
\(405\) −1.40412 −0.0697711
\(406\) 15.5483 0.771648
\(407\) −23.9506 −1.18718
\(408\) −21.8239 −1.08044
\(409\) −23.1867 −1.14651 −0.573255 0.819377i \(-0.694320\pi\)
−0.573255 + 0.819377i \(0.694320\pi\)
\(410\) 32.9451 1.62704
\(411\) 8.09665 0.399378
\(412\) −54.2418 −2.67230
\(413\) −13.8434 −0.681190
\(414\) −20.2853 −0.996967
\(415\) −13.4442 −0.659950
\(416\) 35.8392 1.75716
\(417\) −3.25457 −0.159377
\(418\) 18.4984 0.904786
\(419\) −39.3742 −1.92356 −0.961779 0.273827i \(-0.911710\pi\)
−0.961779 + 0.273827i \(0.911710\pi\)
\(420\) −6.91694 −0.337512
\(421\) −35.4382 −1.72715 −0.863576 0.504219i \(-0.831781\pi\)
−0.863576 + 0.504219i \(0.831781\pi\)
\(422\) −2.27773 −0.110878
\(423\) −9.88478 −0.480614
\(424\) 41.6398 2.02221
\(425\) 8.33038 0.404083
\(426\) −3.57199 −0.173063
\(427\) 7.50259 0.363076
\(428\) 30.2224 1.46085
\(429\) 6.32595 0.305420
\(430\) 1.70396 0.0821721
\(431\) −24.7313 −1.19126 −0.595632 0.803257i \(-0.703099\pi\)
−0.595632 + 0.803257i \(0.703099\pi\)
\(432\) −10.9918 −0.528841
\(433\) 11.2160 0.539006 0.269503 0.963000i \(-0.413141\pi\)
0.269503 + 0.963000i \(0.413141\pi\)
\(434\) 20.0123 0.960620
\(435\) 8.37408 0.401506
\(436\) −67.1539 −3.21609
\(437\) −22.9913 −1.09982
\(438\) −17.0133 −0.812927
\(439\) −23.0691 −1.10103 −0.550513 0.834827i \(-0.685568\pi\)
−0.550513 + 0.834827i \(0.685568\pi\)
\(440\) 25.9782 1.23846
\(441\) −6.02890 −0.287091
\(442\) −19.7410 −0.938984
\(443\) −9.55290 −0.453872 −0.226936 0.973910i \(-0.572871\pi\)
−0.226936 + 0.973910i \(0.572871\pi\)
\(444\) 51.3428 2.43662
\(445\) 14.5204 0.688332
\(446\) 16.2889 0.771303
\(447\) −6.34177 −0.299956
\(448\) −12.7793 −0.603764
\(449\) 6.33551 0.298991 0.149496 0.988762i \(-0.452235\pi\)
0.149496 + 0.988762i \(0.452235\pi\)
\(450\) 8.01196 0.377688
\(451\) 20.6817 0.973865
\(452\) −74.6006 −3.50892
\(453\) 7.44976 0.350020
\(454\) 74.3129 3.48768
\(455\) −3.75355 −0.175969
\(456\) −23.7897 −1.11406
\(457\) −1.78517 −0.0835070 −0.0417535 0.999128i \(-0.513294\pi\)
−0.0417535 + 0.999128i \(0.513294\pi\)
\(458\) −2.75219 −0.128602
\(459\) 2.75070 0.128392
\(460\) −53.8204 −2.50939
\(461\) 10.4892 0.488532 0.244266 0.969708i \(-0.421453\pi\)
0.244266 + 0.969708i \(0.421453\pi\)
\(462\) −6.07945 −0.282842
\(463\) −13.2197 −0.614372 −0.307186 0.951649i \(-0.599387\pi\)
−0.307186 + 0.951649i \(0.599387\pi\)
\(464\) 65.5543 3.04328
\(465\) 10.7783 0.499833
\(466\) −13.5562 −0.627979
\(467\) 20.9613 0.969974 0.484987 0.874521i \(-0.338824\pi\)
0.484987 + 0.874521i \(0.338824\pi\)
\(468\) −13.5609 −0.626854
\(469\) −14.0832 −0.650303
\(470\) −36.7186 −1.69370
\(471\) −22.0200 −1.01463
\(472\) −111.456 −5.13016
\(473\) 1.06968 0.0491841
\(474\) 18.4041 0.845330
\(475\) 9.08076 0.416654
\(476\) 13.5505 0.621085
\(477\) −5.24831 −0.240304
\(478\) −15.9831 −0.731051
\(479\) −9.23570 −0.421990 −0.210995 0.977487i \(-0.567670\pi\)
−0.210995 + 0.977487i \(0.567670\pi\)
\(480\) −18.5504 −0.846705
\(481\) 27.8617 1.27038
\(482\) −8.68277 −0.395489
\(483\) 7.55605 0.343812
\(484\) −27.8046 −1.26385
\(485\) −0.392245 −0.0178109
\(486\) 2.64556 0.120005
\(487\) 9.66202 0.437828 0.218914 0.975744i \(-0.429749\pi\)
0.218914 + 0.975744i \(0.429749\pi\)
\(488\) 60.4046 2.73439
\(489\) −6.26580 −0.283349
\(490\) −22.3954 −1.01172
\(491\) 32.3129 1.45826 0.729130 0.684375i \(-0.239925\pi\)
0.729130 + 0.684375i \(0.239925\pi\)
\(492\) −44.3354 −1.99880
\(493\) −16.4050 −0.738846
\(494\) −21.5192 −0.968195
\(495\) −3.27431 −0.147169
\(496\) 84.3754 3.78857
\(497\) 1.33053 0.0596823
\(498\) 25.3308 1.13510
\(499\) −32.2772 −1.44493 −0.722463 0.691409i \(-0.756990\pi\)
−0.722463 + 0.691409i \(0.756990\pi\)
\(500\) 56.3528 2.52017
\(501\) −16.1336 −0.720797
\(502\) −66.0124 −2.94628
\(503\) 38.8639 1.73286 0.866428 0.499301i \(-0.166410\pi\)
0.866428 + 0.499301i \(0.166410\pi\)
\(504\) 7.81844 0.348261
\(505\) 11.6839 0.519926
\(506\) −47.3039 −2.10292
\(507\) 5.64101 0.250526
\(508\) 29.1632 1.29391
\(509\) −19.4160 −0.860599 −0.430299 0.902686i \(-0.641592\pi\)
−0.430299 + 0.902686i \(0.641592\pi\)
\(510\) 10.2179 0.452458
\(511\) 6.33727 0.280344
\(512\) 29.1992 1.29043
\(513\) 2.99847 0.132386
\(514\) −26.0375 −1.14846
\(515\) 15.2355 0.671355
\(516\) −2.29308 −0.100947
\(517\) −23.0507 −1.01377
\(518\) −26.7761 −1.17647
\(519\) −0.726527 −0.0318910
\(520\) −30.2204 −1.32525
\(521\) −19.9389 −0.873540 −0.436770 0.899573i \(-0.643878\pi\)
−0.436770 + 0.899573i \(0.643878\pi\)
\(522\) −15.7780 −0.690583
\(523\) −13.8287 −0.604685 −0.302343 0.953199i \(-0.597769\pi\)
−0.302343 + 0.953199i \(0.597769\pi\)
\(524\) 58.5986 2.55989
\(525\) −2.98437 −0.130249
\(526\) 79.1234 3.44994
\(527\) −21.1150 −0.919786
\(528\) −25.6321 −1.11549
\(529\) 35.7932 1.55623
\(530\) −19.4957 −0.846840
\(531\) 14.0479 0.609628
\(532\) 14.7711 0.640406
\(533\) −24.0591 −1.04212
\(534\) −27.3585 −1.18392
\(535\) −8.48889 −0.367007
\(536\) −113.386 −4.89754
\(537\) 20.8854 0.901271
\(538\) −31.7601 −1.36928
\(539\) −14.0590 −0.605565
\(540\) 7.01913 0.302055
\(541\) −20.5690 −0.884331 −0.442165 0.896934i \(-0.645790\pi\)
−0.442165 + 0.896934i \(0.645790\pi\)
\(542\) 29.7676 1.27863
\(543\) −13.4557 −0.577441
\(544\) 36.3407 1.55809
\(545\) 18.8623 0.807970
\(546\) 7.07224 0.302664
\(547\) −30.4875 −1.30355 −0.651775 0.758413i \(-0.725975\pi\)
−0.651775 + 0.758413i \(0.725975\pi\)
\(548\) −40.4749 −1.72900
\(549\) −7.61343 −0.324933
\(550\) 18.6834 0.796662
\(551\) −17.8828 −0.761831
\(552\) 60.8349 2.58931
\(553\) −6.85535 −0.291519
\(554\) −38.5866 −1.63939
\(555\) −14.4212 −0.612146
\(556\) 16.2695 0.689980
\(557\) 6.96523 0.295126 0.147563 0.989053i \(-0.452857\pi\)
0.147563 + 0.989053i \(0.452857\pi\)
\(558\) −20.3079 −0.859704
\(559\) −1.24436 −0.0526310
\(560\) 15.2090 0.642698
\(561\) 6.41445 0.270818
\(562\) −69.8760 −2.94754
\(563\) 3.25826 0.137319 0.0686597 0.997640i \(-0.478128\pi\)
0.0686597 + 0.997640i \(0.478128\pi\)
\(564\) 49.4137 2.08069
\(565\) 20.9539 0.881537
\(566\) −30.6836 −1.28973
\(567\) −0.985442 −0.0413847
\(568\) 10.7123 0.449477
\(569\) 12.6387 0.529843 0.264922 0.964270i \(-0.414654\pi\)
0.264922 + 0.964270i \(0.414654\pi\)
\(570\) 11.1383 0.466533
\(571\) 24.7001 1.03367 0.516834 0.856086i \(-0.327111\pi\)
0.516834 + 0.856086i \(0.327111\pi\)
\(572\) −31.6232 −1.32223
\(573\) −5.37858 −0.224693
\(574\) 23.1216 0.965078
\(575\) −23.2212 −0.968393
\(576\) 12.9681 0.540336
\(577\) 22.8234 0.950149 0.475074 0.879946i \(-0.342421\pi\)
0.475074 + 0.879946i \(0.342421\pi\)
\(578\) 24.9573 1.03809
\(579\) −0.0783143 −0.00325463
\(580\) −41.8618 −1.73822
\(581\) −9.43546 −0.391449
\(582\) 0.739046 0.0306344
\(583\) −12.2387 −0.506876
\(584\) 51.0224 2.11132
\(585\) 3.80901 0.157483
\(586\) 43.3600 1.79118
\(587\) 13.7674 0.568241 0.284120 0.958789i \(-0.408298\pi\)
0.284120 + 0.958789i \(0.408298\pi\)
\(588\) 30.1383 1.24288
\(589\) −23.0170 −0.948399
\(590\) 52.1834 2.14836
\(591\) 12.4020 0.510148
\(592\) −112.893 −4.63986
\(593\) −9.62996 −0.395455 −0.197727 0.980257i \(-0.563356\pi\)
−0.197727 + 0.980257i \(0.563356\pi\)
\(594\) 6.16927 0.253128
\(595\) −3.80607 −0.156034
\(596\) 31.7023 1.29858
\(597\) 17.8105 0.728933
\(598\) 55.0288 2.25029
\(599\) 31.7589 1.29763 0.648816 0.760945i \(-0.275264\pi\)
0.648816 + 0.760945i \(0.275264\pi\)
\(600\) −24.0276 −0.980924
\(601\) 41.5128 1.69334 0.846671 0.532117i \(-0.178603\pi\)
0.846671 + 0.532117i \(0.178603\pi\)
\(602\) 1.19588 0.0487403
\(603\) 14.2913 0.581986
\(604\) −37.2411 −1.51532
\(605\) 7.80979 0.317513
\(606\) −22.0141 −0.894263
\(607\) 14.5205 0.589370 0.294685 0.955594i \(-0.404785\pi\)
0.294685 + 0.955594i \(0.404785\pi\)
\(608\) 39.6141 1.60656
\(609\) 5.87713 0.238153
\(610\) −28.2814 −1.14508
\(611\) 26.8149 1.08481
\(612\) −13.7507 −0.555838
\(613\) −33.9679 −1.37195 −0.685976 0.727624i \(-0.740625\pi\)
−0.685976 + 0.727624i \(0.740625\pi\)
\(614\) 8.08317 0.326210
\(615\) 12.4530 0.502153
\(616\) 18.2321 0.734592
\(617\) 38.2551 1.54009 0.770045 0.637989i \(-0.220234\pi\)
0.770045 + 0.637989i \(0.220234\pi\)
\(618\) −28.7058 −1.15472
\(619\) 32.5598 1.30869 0.654345 0.756196i \(-0.272944\pi\)
0.654345 + 0.756196i \(0.272944\pi\)
\(620\) −53.8806 −2.16390
\(621\) −7.66768 −0.307693
\(622\) −24.5139 −0.982916
\(623\) 10.1907 0.408283
\(624\) 29.8178 1.19367
\(625\) −0.686128 −0.0274451
\(626\) −6.14626 −0.245654
\(627\) 6.99225 0.279243
\(628\) 110.077 4.39255
\(629\) 28.2515 1.12646
\(630\) −3.66059 −0.145841
\(631\) 28.3573 1.12889 0.564444 0.825472i \(-0.309091\pi\)
0.564444 + 0.825472i \(0.309091\pi\)
\(632\) −55.1935 −2.19548
\(633\) −0.860963 −0.0342202
\(634\) 66.2216 2.63000
\(635\) −8.19140 −0.325066
\(636\) 26.2361 1.04033
\(637\) 16.3549 0.648004
\(638\) −36.7932 −1.45666
\(639\) −1.35018 −0.0534124
\(640\) 11.0713 0.437632
\(641\) 4.34915 0.171781 0.0858905 0.996305i \(-0.472626\pi\)
0.0858905 + 0.996305i \(0.472626\pi\)
\(642\) 15.9943 0.631245
\(643\) 15.4463 0.609141 0.304571 0.952490i \(-0.401487\pi\)
0.304571 + 0.952490i \(0.401487\pi\)
\(644\) −37.7724 −1.48844
\(645\) 0.644083 0.0253607
\(646\) −21.8203 −0.858507
\(647\) 28.1720 1.10756 0.553778 0.832664i \(-0.313185\pi\)
0.553778 + 0.832664i \(0.313185\pi\)
\(648\) −7.93394 −0.311675
\(649\) 32.7589 1.28590
\(650\) −21.7344 −0.852494
\(651\) 7.56449 0.296476
\(652\) 31.3225 1.22669
\(653\) −1.66654 −0.0652166 −0.0326083 0.999468i \(-0.510381\pi\)
−0.0326083 + 0.999468i \(0.510381\pi\)
\(654\) −35.5392 −1.38969
\(655\) −16.4592 −0.643115
\(656\) 97.4850 3.80615
\(657\) −6.43090 −0.250893
\(658\) −25.7700 −1.00462
\(659\) 34.2933 1.33588 0.667940 0.744215i \(-0.267176\pi\)
0.667940 + 0.744215i \(0.267176\pi\)
\(660\) 16.3682 0.637130
\(661\) −15.2152 −0.591801 −0.295901 0.955219i \(-0.595620\pi\)
−0.295901 + 0.955219i \(0.595620\pi\)
\(662\) 36.8012 1.43032
\(663\) −7.46195 −0.289798
\(664\) −75.9664 −2.94807
\(665\) −4.14891 −0.160888
\(666\) 27.1716 1.05288
\(667\) 45.7297 1.77066
\(668\) 80.6514 3.12050
\(669\) 6.15709 0.238047
\(670\) 53.0873 2.05094
\(671\) −17.7540 −0.685387
\(672\) −13.0191 −0.502222
\(673\) 6.58049 0.253659 0.126830 0.991925i \(-0.459520\pi\)
0.126830 + 0.991925i \(0.459520\pi\)
\(674\) −70.0075 −2.69659
\(675\) 3.02846 0.116565
\(676\) −28.1993 −1.08459
\(677\) 35.6621 1.37061 0.685304 0.728257i \(-0.259669\pi\)
0.685304 + 0.728257i \(0.259669\pi\)
\(678\) −39.4802 −1.51623
\(679\) −0.275287 −0.0105645
\(680\) −30.6433 −1.17512
\(681\) 28.0897 1.07640
\(682\) −47.3568 −1.81339
\(683\) −49.1335 −1.88004 −0.940020 0.341119i \(-0.889194\pi\)
−0.940020 + 0.341119i \(0.889194\pi\)
\(684\) −14.9893 −0.573129
\(685\) 11.3686 0.434373
\(686\) −33.9669 −1.29686
\(687\) −1.04031 −0.0396902
\(688\) 5.04204 0.192226
\(689\) 14.2373 0.542399
\(690\) −28.4829 −1.08432
\(691\) 21.5540 0.819951 0.409976 0.912097i \(-0.365537\pi\)
0.409976 + 0.912097i \(0.365537\pi\)
\(692\) 3.63188 0.138064
\(693\) −2.29799 −0.0872933
\(694\) 91.9422 3.49008
\(695\) −4.56979 −0.173342
\(696\) 47.3177 1.79357
\(697\) −24.3957 −0.924054
\(698\) −20.6251 −0.780670
\(699\) −5.12414 −0.193813
\(700\) 14.9188 0.563877
\(701\) 37.4040 1.41273 0.706366 0.707847i \(-0.250333\pi\)
0.706366 + 0.707847i \(0.250333\pi\)
\(702\) −7.17672 −0.270868
\(703\) 30.7963 1.16151
\(704\) 30.2407 1.13974
\(705\) −13.8794 −0.522727
\(706\) −44.7845 −1.68549
\(707\) 8.20003 0.308394
\(708\) −70.2252 −2.63922
\(709\) −17.9676 −0.674787 −0.337393 0.941364i \(-0.609545\pi\)
−0.337393 + 0.941364i \(0.609545\pi\)
\(710\) −5.01548 −0.188228
\(711\) 6.95662 0.260894
\(712\) 82.0472 3.07485
\(713\) 58.8589 2.20428
\(714\) 7.17119 0.268375
\(715\) 8.88236 0.332181
\(716\) −104.405 −3.90181
\(717\) −6.04150 −0.225624
\(718\) −60.6990 −2.26527
\(719\) 4.56041 0.170075 0.0850373 0.996378i \(-0.472899\pi\)
0.0850373 + 0.996378i \(0.472899\pi\)
\(720\) −15.4337 −0.575180
\(721\) 10.6926 0.398214
\(722\) 26.4798 0.985476
\(723\) −3.28202 −0.122060
\(724\) 67.2649 2.49988
\(725\) −18.0616 −0.670791
\(726\) −14.7148 −0.546117
\(727\) 46.3194 1.71789 0.858946 0.512066i \(-0.171120\pi\)
0.858946 + 0.512066i \(0.171120\pi\)
\(728\) −21.2094 −0.786074
\(729\) 1.00000 0.0370370
\(730\) −23.8886 −0.884158
\(731\) −1.26177 −0.0466684
\(732\) 38.0593 1.40671
\(733\) 6.07642 0.224438 0.112219 0.993684i \(-0.464204\pi\)
0.112219 + 0.993684i \(0.464204\pi\)
\(734\) −62.4056 −2.30343
\(735\) −8.46528 −0.312246
\(736\) −101.301 −3.73400
\(737\) 33.3264 1.22759
\(738\) −23.4632 −0.863693
\(739\) −24.3286 −0.894944 −0.447472 0.894298i \(-0.647676\pi\)
−0.447472 + 0.894298i \(0.647676\pi\)
\(740\) 72.0912 2.65012
\(741\) −8.13409 −0.298813
\(742\) −13.6826 −0.502303
\(743\) −2.52505 −0.0926349 −0.0463175 0.998927i \(-0.514749\pi\)
−0.0463175 + 0.998927i \(0.514749\pi\)
\(744\) 60.9029 2.23281
\(745\) −8.90458 −0.326239
\(746\) −26.7936 −0.980984
\(747\) 9.57486 0.350326
\(748\) −32.0657 −1.17244
\(749\) −5.95771 −0.217690
\(750\) 29.8231 1.08898
\(751\) 39.2295 1.43150 0.715752 0.698354i \(-0.246084\pi\)
0.715752 + 0.698354i \(0.246084\pi\)
\(752\) −108.651 −3.96210
\(753\) −24.9522 −0.909309
\(754\) 42.8016 1.55874
\(755\) 10.4603 0.380690
\(756\) 4.92619 0.179164
\(757\) −45.5952 −1.65719 −0.828594 0.559851i \(-0.810858\pi\)
−0.828594 + 0.559851i \(0.810858\pi\)
\(758\) −14.7460 −0.535600
\(759\) −17.8805 −0.649022
\(760\) −33.4035 −1.21167
\(761\) −19.3832 −0.702640 −0.351320 0.936256i \(-0.614267\pi\)
−0.351320 + 0.936256i \(0.614267\pi\)
\(762\) 15.4338 0.559107
\(763\) 13.2380 0.479247
\(764\) 26.8874 0.972751
\(765\) 3.86230 0.139642
\(766\) −24.5019 −0.885288
\(767\) −38.1085 −1.37602
\(768\) 5.07620 0.183171
\(769\) 21.9465 0.791410 0.395705 0.918378i \(-0.370500\pi\)
0.395705 + 0.918378i \(0.370500\pi\)
\(770\) −8.53625 −0.307625
\(771\) −9.84196 −0.354450
\(772\) 0.391491 0.0140901
\(773\) −25.6914 −0.924056 −0.462028 0.886865i \(-0.652878\pi\)
−0.462028 + 0.886865i \(0.652878\pi\)
\(774\) −1.21354 −0.0436200
\(775\) −23.2472 −0.835064
\(776\) −2.21638 −0.0795632
\(777\) −10.1211 −0.363094
\(778\) 90.3092 3.23774
\(779\) −26.5932 −0.952800
\(780\) −19.0411 −0.681781
\(781\) −3.14854 −0.112664
\(782\) 55.7987 1.99536
\(783\) −5.96396 −0.213134
\(784\) −66.2683 −2.36672
\(785\) −30.9186 −1.10353
\(786\) 31.0116 1.10615
\(787\) 53.8180 1.91840 0.959202 0.282723i \(-0.0912377\pi\)
0.959202 + 0.282723i \(0.0912377\pi\)
\(788\) −61.9970 −2.20855
\(789\) 29.9080 1.06475
\(790\) 25.8415 0.919401
\(791\) 14.7059 0.522883
\(792\) −18.5014 −0.657420
\(793\) 20.6533 0.733420
\(794\) 14.8162 0.525808
\(795\) −7.36923 −0.261360
\(796\) −89.0339 −3.15572
\(797\) 32.5335 1.15239 0.576197 0.817311i \(-0.304536\pi\)
0.576197 + 0.817311i \(0.304536\pi\)
\(798\) 7.81714 0.276724
\(799\) 27.1900 0.961915
\(800\) 40.0103 1.41458
\(801\) −10.3413 −0.365392
\(802\) −4.23775 −0.149640
\(803\) −14.9964 −0.529213
\(804\) −71.4417 −2.51955
\(805\) 10.6096 0.373938
\(806\) 55.0903 1.94047
\(807\) −12.0051 −0.422599
\(808\) 66.0197 2.32256
\(809\) −52.1544 −1.83365 −0.916825 0.399289i \(-0.869257\pi\)
−0.916825 + 0.399289i \(0.869257\pi\)
\(810\) 3.71467 0.130520
\(811\) −6.01280 −0.211138 −0.105569 0.994412i \(-0.533666\pi\)
−0.105569 + 0.994412i \(0.533666\pi\)
\(812\) −29.3796 −1.03102
\(813\) 11.2519 0.394622
\(814\) 63.3625 2.22086
\(815\) −8.79791 −0.308177
\(816\) 30.2350 1.05844
\(817\) −1.37543 −0.0481203
\(818\) 61.3418 2.14477
\(819\) 2.67325 0.0934110
\(820\) −62.2521 −2.17394
\(821\) 28.9166 1.00920 0.504599 0.863354i \(-0.331640\pi\)
0.504599 + 0.863354i \(0.331640\pi\)
\(822\) −21.4201 −0.747113
\(823\) 52.7250 1.83788 0.918939 0.394400i \(-0.129048\pi\)
0.918939 + 0.394400i \(0.129048\pi\)
\(824\) 86.0879 2.99902
\(825\) 7.06218 0.245873
\(826\) 36.6235 1.27430
\(827\) −4.87869 −0.169649 −0.0848243 0.996396i \(-0.527033\pi\)
−0.0848243 + 0.996396i \(0.527033\pi\)
\(828\) 38.3305 1.33208
\(829\) 1.95236 0.0678083 0.0339042 0.999425i \(-0.489206\pi\)
0.0339042 + 0.999425i \(0.489206\pi\)
\(830\) 35.5674 1.23456
\(831\) −14.5854 −0.505964
\(832\) −35.1791 −1.21961
\(833\) 16.5837 0.574591
\(834\) 8.61014 0.298145
\(835\) −22.6535 −0.783955
\(836\) −34.9540 −1.20891
\(837\) −7.67624 −0.265330
\(838\) 104.167 3.59838
\(839\) 43.9451 1.51715 0.758577 0.651583i \(-0.225895\pi\)
0.758577 + 0.651583i \(0.225895\pi\)
\(840\) 10.9780 0.378777
\(841\) 6.56876 0.226509
\(842\) 93.7538 3.23097
\(843\) −26.4126 −0.909698
\(844\) 4.30393 0.148147
\(845\) 7.92063 0.272478
\(846\) 26.1507 0.899081
\(847\) 5.48110 0.188333
\(848\) −57.6881 −1.98102
\(849\) −11.5981 −0.398047
\(850\) −22.0385 −0.755914
\(851\) −78.7522 −2.69959
\(852\) 6.74952 0.231235
\(853\) 30.1785 1.03329 0.516646 0.856199i \(-0.327180\pi\)
0.516646 + 0.856199i \(0.327180\pi\)
\(854\) −19.8485 −0.679203
\(855\) 4.21020 0.143986
\(856\) −47.9664 −1.63946
\(857\) 45.0534 1.53900 0.769498 0.638650i \(-0.220507\pi\)
0.769498 + 0.638650i \(0.220507\pi\)
\(858\) −16.7357 −0.571346
\(859\) 36.3472 1.24015 0.620075 0.784543i \(-0.287102\pi\)
0.620075 + 0.784543i \(0.287102\pi\)
\(860\) −3.21975 −0.109793
\(861\) 8.73980 0.297851
\(862\) 65.4281 2.22849
\(863\) 40.1227 1.36579 0.682896 0.730516i \(-0.260720\pi\)
0.682896 + 0.730516i \(0.260720\pi\)
\(864\) 13.2114 0.449462
\(865\) −1.02013 −0.0346854
\(866\) −29.6725 −1.00831
\(867\) 9.43365 0.320384
\(868\) −37.8147 −1.28351
\(869\) 16.2224 0.550307
\(870\) −22.1541 −0.751094
\(871\) −38.7686 −1.31362
\(872\) 106.581 3.60929
\(873\) 0.279354 0.00945469
\(874\) 60.8248 2.05743
\(875\) −11.1088 −0.375545
\(876\) 32.1479 1.08618
\(877\) −13.9143 −0.469853 −0.234927 0.972013i \(-0.575485\pi\)
−0.234927 + 0.972013i \(0.575485\pi\)
\(878\) 61.0305 2.05968
\(879\) 16.3897 0.552812
\(880\) −35.9904 −1.21324
\(881\) 29.0714 0.979440 0.489720 0.871880i \(-0.337099\pi\)
0.489720 + 0.871880i \(0.337099\pi\)
\(882\) 15.9498 0.537058
\(883\) −44.1650 −1.48627 −0.743136 0.669141i \(-0.766662\pi\)
−0.743136 + 0.669141i \(0.766662\pi\)
\(884\) 37.3020 1.25460
\(885\) 19.7249 0.663046
\(886\) 25.2727 0.849054
\(887\) 54.8069 1.84024 0.920118 0.391640i \(-0.128092\pi\)
0.920118 + 0.391640i \(0.128092\pi\)
\(888\) −81.4869 −2.73452
\(889\) −5.74892 −0.192813
\(890\) −38.4145 −1.28766
\(891\) 2.33194 0.0781228
\(892\) −30.7791 −1.03056
\(893\) 29.6392 0.991839
\(894\) 16.7775 0.561124
\(895\) 29.3255 0.980243
\(896\) 7.77010 0.259581
\(897\) 20.8004 0.694507
\(898\) −16.7609 −0.559320
\(899\) 45.7808 1.52687
\(900\) −15.1392 −0.504639
\(901\) 14.4365 0.480950
\(902\) −54.7147 −1.82180
\(903\) 0.452032 0.0150427
\(904\) 118.400 3.93792
\(905\) −18.8934 −0.628039
\(906\) −19.7088 −0.654780
\(907\) −10.4974 −0.348561 −0.174281 0.984696i \(-0.555760\pi\)
−0.174281 + 0.984696i \(0.555760\pi\)
\(908\) −140.420 −4.65999
\(909\) −8.32117 −0.275996
\(910\) 9.93024 0.329184
\(911\) 28.1392 0.932294 0.466147 0.884707i \(-0.345642\pi\)
0.466147 + 0.884707i \(0.345642\pi\)
\(912\) 32.9585 1.09136
\(913\) 22.3280 0.738947
\(914\) 4.72278 0.156216
\(915\) −10.6901 −0.353405
\(916\) 5.20047 0.171828
\(917\) −11.5515 −0.381463
\(918\) −7.27713 −0.240181
\(919\) 34.7194 1.14529 0.572643 0.819805i \(-0.305918\pi\)
0.572643 + 0.819805i \(0.305918\pi\)
\(920\) 85.4192 2.81619
\(921\) 3.05538 0.100678
\(922\) −27.7499 −0.913893
\(923\) 3.66270 0.120559
\(924\) 11.4876 0.377913
\(925\) 31.1043 1.02270
\(926\) 34.9735 1.14930
\(927\) −10.8506 −0.356380
\(928\) −78.7924 −2.58649
\(929\) −30.6217 −1.00467 −0.502333 0.864674i \(-0.667525\pi\)
−0.502333 + 0.864674i \(0.667525\pi\)
\(930\) −28.5147 −0.935033
\(931\) 18.0775 0.592466
\(932\) 25.6154 0.839061
\(933\) −9.26605 −0.303357
\(934\) −55.4543 −1.81452
\(935\) 9.00663 0.294548
\(936\) 21.5228 0.703494
\(937\) −12.9862 −0.424241 −0.212121 0.977243i \(-0.568037\pi\)
−0.212121 + 0.977243i \(0.568037\pi\)
\(938\) 37.2580 1.21652
\(939\) −2.32324 −0.0758160
\(940\) 69.3825 2.26301
\(941\) −4.93044 −0.160728 −0.0803638 0.996766i \(-0.525608\pi\)
−0.0803638 + 0.996766i \(0.525608\pi\)
\(942\) 58.2550 1.89805
\(943\) 68.0040 2.21451
\(944\) 154.411 5.02567
\(945\) −1.38367 −0.0450109
\(946\) −2.82991 −0.0920083
\(947\) 22.6290 0.735343 0.367672 0.929956i \(-0.380155\pi\)
0.367672 + 0.929956i \(0.380155\pi\)
\(948\) −34.7760 −1.12947
\(949\) 17.4454 0.566301
\(950\) −24.0237 −0.779430
\(951\) 25.0313 0.811694
\(952\) −21.5062 −0.697019
\(953\) 29.8943 0.968372 0.484186 0.874965i \(-0.339116\pi\)
0.484186 + 0.874965i \(0.339116\pi\)
\(954\) 13.8847 0.449534
\(955\) −7.55215 −0.244382
\(956\) 30.2013 0.976779
\(957\) −13.9076 −0.449568
\(958\) 24.4336 0.789413
\(959\) 7.97878 0.257648
\(960\) 18.2087 0.587682
\(961\) 27.9247 0.900798
\(962\) −73.7097 −2.37650
\(963\) 6.04572 0.194821
\(964\) 16.4067 0.528425
\(965\) −0.109962 −0.00353981
\(966\) −19.9899 −0.643166
\(967\) −0.0665875 −0.00214131 −0.00107065 0.999999i \(-0.500341\pi\)
−0.00107065 + 0.999999i \(0.500341\pi\)
\(968\) 44.1292 1.41837
\(969\) −8.24790 −0.264961
\(970\) 1.03771 0.0333187
\(971\) 60.8438 1.95257 0.976286 0.216486i \(-0.0694596\pi\)
0.976286 + 0.216486i \(0.0694596\pi\)
\(972\) −4.99897 −0.160342
\(973\) −3.20719 −0.102818
\(974\) −25.5614 −0.819041
\(975\) −8.21544 −0.263105
\(976\) −83.6850 −2.67869
\(977\) 8.97093 0.287006 0.143503 0.989650i \(-0.454163\pi\)
0.143503 + 0.989650i \(0.454163\pi\)
\(978\) 16.5765 0.530059
\(979\) −24.1152 −0.770726
\(980\) 42.3177 1.35179
\(981\) −13.4335 −0.428900
\(982\) −85.4856 −2.72795
\(983\) 5.29704 0.168949 0.0844746 0.996426i \(-0.473079\pi\)
0.0844746 + 0.996426i \(0.473079\pi\)
\(984\) 70.3655 2.24317
\(985\) 17.4138 0.554849
\(986\) 43.4005 1.38215
\(987\) −9.74087 −0.310055
\(988\) 40.6621 1.29363
\(989\) 3.51724 0.111842
\(990\) 8.66236 0.275308
\(991\) −11.4066 −0.362343 −0.181172 0.983451i \(-0.557989\pi\)
−0.181172 + 0.983451i \(0.557989\pi\)
\(992\) −101.414 −3.21990
\(993\) 13.9106 0.441439
\(994\) −3.51998 −0.111647
\(995\) 25.0079 0.792805
\(996\) −47.8644 −1.51664
\(997\) −37.4798 −1.18700 −0.593498 0.804836i \(-0.702253\pi\)
−0.593498 + 0.804836i \(0.702253\pi\)
\(998\) 85.3912 2.70301
\(999\) 10.2707 0.324950
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 8031.2.a.c.1.4 121
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
8031.2.a.c.1.4 121 1.1 even 1 trivial