Properties

Label 731.2.a.e.1.11
Level $731$
Weight $2$
Character 731.1
Self dual yes
Analytic conductor $5.837$
Analytic rank $0$
Dimension $19$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [731,2,Mod(1,731)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(731, 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("731.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 731 = 17 \cdot 43 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 731.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: \(5.83706438776\)
Analytic rank: \(0\)
Dimension: \(19\)
Coefficient field: \(\mathbb{Q}[x]/(x^{19} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{19} - 2 x^{18} - 30 x^{17} + 62 x^{16} + 365 x^{15} - 786 x^{14} - 2295 x^{13} + 5233 x^{12} + 7816 x^{11} - 19517 x^{10} - 13527 x^{9} + 40173 x^{8} + 8942 x^{7} - 41911 x^{6} + \cdots + 64 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.11
Root \(0.614204\) of defining polynomial
Character \(\chi\) \(=\) 731.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+0.614204 q^{2} -2.75618 q^{3} -1.62275 q^{4} -1.26314 q^{5} -1.69286 q^{6} -3.83612 q^{7} -2.22511 q^{8} +4.59652 q^{9} +O(q^{10})\) \(q+0.614204 q^{2} -2.75618 q^{3} -1.62275 q^{4} -1.26314 q^{5} -1.69286 q^{6} -3.83612 q^{7} -2.22511 q^{8} +4.59652 q^{9} -0.775829 q^{10} +1.37582 q^{11} +4.47260 q^{12} -7.02604 q^{13} -2.35616 q^{14} +3.48145 q^{15} +1.87883 q^{16} +1.00000 q^{17} +2.82320 q^{18} -2.98388 q^{19} +2.04977 q^{20} +10.5730 q^{21} +0.845036 q^{22} +6.50242 q^{23} +6.13280 q^{24} -3.40447 q^{25} -4.31543 q^{26} -4.40029 q^{27} +6.22507 q^{28} +9.00351 q^{29} +2.13832 q^{30} -10.0383 q^{31} +5.60421 q^{32} -3.79201 q^{33} +0.614204 q^{34} +4.84557 q^{35} -7.45901 q^{36} +6.10591 q^{37} -1.83271 q^{38} +19.3650 q^{39} +2.81064 q^{40} -1.48370 q^{41} +6.49400 q^{42} -1.00000 q^{43} -2.23262 q^{44} -5.80607 q^{45} +3.99381 q^{46} +0.246738 q^{47} -5.17840 q^{48} +7.71581 q^{49} -2.09104 q^{50} -2.75618 q^{51} +11.4015 q^{52} -3.55498 q^{53} -2.70268 q^{54} -1.73786 q^{55} +8.53579 q^{56} +8.22410 q^{57} +5.52999 q^{58} -2.50988 q^{59} -5.64954 q^{60} +12.9426 q^{61} -6.16559 q^{62} -17.6328 q^{63} -0.315539 q^{64} +8.87491 q^{65} -2.32907 q^{66} -5.60551 q^{67} -1.62275 q^{68} -17.9218 q^{69} +2.97617 q^{70} -8.81728 q^{71} -10.2278 q^{72} -11.2882 q^{73} +3.75028 q^{74} +9.38331 q^{75} +4.84210 q^{76} -5.27782 q^{77} +11.8941 q^{78} -3.35889 q^{79} -2.37324 q^{80} -1.66158 q^{81} -0.911296 q^{82} -5.72716 q^{83} -17.1574 q^{84} -1.26314 q^{85} -0.614204 q^{86} -24.8153 q^{87} -3.06136 q^{88} +12.1270 q^{89} -3.56611 q^{90} +26.9527 q^{91} -10.5518 q^{92} +27.6674 q^{93} +0.151547 q^{94} +3.76907 q^{95} -15.4462 q^{96} +0.613967 q^{97} +4.73908 q^{98} +6.32399 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 19 q + 2 q^{2} + 5 q^{3} + 26 q^{4} + 11 q^{5} + 3 q^{6} + 7 q^{7} - 6 q^{8} + 28 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 19 q + 2 q^{2} + 5 q^{3} + 26 q^{4} + 11 q^{5} + 3 q^{6} + 7 q^{7} - 6 q^{8} + 28 q^{9} - 2 q^{10} + 4 q^{11} + 9 q^{12} + 14 q^{13} + 5 q^{14} - 7 q^{15} + 32 q^{16} + 19 q^{17} + 12 q^{18} + 12 q^{19} + 23 q^{20} + 16 q^{21} + 36 q^{22} - q^{23} - 13 q^{24} + 30 q^{25} - 21 q^{26} + 8 q^{27} + 5 q^{28} + 41 q^{29} - 26 q^{30} - 8 q^{31} - 20 q^{32} - 14 q^{33} + 2 q^{34} + 3 q^{35} - 5 q^{36} + 50 q^{37} - 29 q^{38} + 17 q^{39} - 15 q^{40} + 6 q^{41} - q^{42} - 19 q^{43} + 16 q^{44} + 24 q^{45} + 38 q^{46} - 21 q^{47} - 2 q^{48} + 46 q^{49} - 36 q^{50} + 5 q^{51} + 39 q^{52} - 9 q^{53} + 53 q^{54} + 10 q^{55} - 12 q^{56} - 5 q^{57} - 45 q^{58} - 4 q^{59} - 7 q^{60} + 68 q^{61} - 25 q^{62} + 61 q^{63} - 14 q^{64} + 22 q^{65} - 17 q^{66} + 26 q^{68} - 9 q^{69} - 37 q^{70} + 23 q^{71} - 4 q^{72} - q^{73} - 30 q^{74} - 25 q^{75} + 47 q^{76} - 19 q^{77} + 12 q^{78} + 16 q^{79} + 28 q^{80} - 21 q^{81} - 13 q^{82} - 32 q^{83} - 47 q^{84} + 11 q^{85} - 2 q^{86} - 8 q^{87} + 108 q^{88} + 11 q^{89} + 5 q^{90} + 52 q^{91} - 23 q^{92} - 23 q^{93} + 47 q^{94} - 25 q^{95} - 103 q^{96} + 36 q^{97} - 100 q^{98} - 11 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0.614204 0.434308 0.217154 0.976137i \(-0.430323\pi\)
0.217154 + 0.976137i \(0.430323\pi\)
\(3\) −2.75618 −1.59128 −0.795640 0.605770i \(-0.792865\pi\)
−0.795640 + 0.605770i \(0.792865\pi\)
\(4\) −1.62275 −0.811377
\(5\) −1.26314 −0.564895 −0.282448 0.959283i \(-0.591146\pi\)
−0.282448 + 0.959283i \(0.591146\pi\)
\(6\) −1.69286 −0.691106
\(7\) −3.83612 −1.44992 −0.724958 0.688793i \(-0.758141\pi\)
−0.724958 + 0.688793i \(0.758141\pi\)
\(8\) −2.22511 −0.786695
\(9\) 4.59652 1.53217
\(10\) −0.775829 −0.245339
\(11\) 1.37582 0.414826 0.207413 0.978253i \(-0.433496\pi\)
0.207413 + 0.978253i \(0.433496\pi\)
\(12\) 4.47260 1.29113
\(13\) −7.02604 −1.94867 −0.974337 0.225095i \(-0.927731\pi\)
−0.974337 + 0.225095i \(0.927731\pi\)
\(14\) −2.35616 −0.629710
\(15\) 3.48145 0.898907
\(16\) 1.87883 0.469708
\(17\) 1.00000 0.242536
\(18\) 2.82320 0.665435
\(19\) −2.98388 −0.684549 −0.342274 0.939600i \(-0.611197\pi\)
−0.342274 + 0.939600i \(0.611197\pi\)
\(20\) 2.04977 0.458343
\(21\) 10.5730 2.30722
\(22\) 0.845036 0.180162
\(23\) 6.50242 1.35585 0.677924 0.735132i \(-0.262880\pi\)
0.677924 + 0.735132i \(0.262880\pi\)
\(24\) 6.13280 1.25185
\(25\) −3.40447 −0.680893
\(26\) −4.31543 −0.846325
\(27\) −4.40029 −0.846836
\(28\) 6.22507 1.17643
\(29\) 9.00351 1.67191 0.835955 0.548798i \(-0.184914\pi\)
0.835955 + 0.548798i \(0.184914\pi\)
\(30\) 2.13832 0.390402
\(31\) −10.0383 −1.80294 −0.901469 0.432844i \(-0.857510\pi\)
−0.901469 + 0.432844i \(0.857510\pi\)
\(32\) 5.60421 0.990693
\(33\) −3.79201 −0.660104
\(34\) 0.614204 0.105335
\(35\) 4.84557 0.819051
\(36\) −7.45901 −1.24317
\(37\) 6.10591 1.00381 0.501903 0.864924i \(-0.332634\pi\)
0.501903 + 0.864924i \(0.332634\pi\)
\(38\) −1.83271 −0.297305
\(39\) 19.3650 3.10089
\(40\) 2.81064 0.444401
\(41\) −1.48370 −0.231715 −0.115858 0.993266i \(-0.536962\pi\)
−0.115858 + 0.993266i \(0.536962\pi\)
\(42\) 6.49400 1.00205
\(43\) −1.00000 −0.152499
\(44\) −2.23262 −0.336580
\(45\) −5.80607 −0.865517
\(46\) 3.99381 0.588856
\(47\) 0.246738 0.0359904 0.0179952 0.999838i \(-0.494272\pi\)
0.0179952 + 0.999838i \(0.494272\pi\)
\(48\) −5.17840 −0.747438
\(49\) 7.71581 1.10226
\(50\) −2.09104 −0.295717
\(51\) −2.75618 −0.385942
\(52\) 11.4015 1.58111
\(53\) −3.55498 −0.488314 −0.244157 0.969736i \(-0.578511\pi\)
−0.244157 + 0.969736i \(0.578511\pi\)
\(54\) −2.70268 −0.367788
\(55\) −1.73786 −0.234333
\(56\) 8.53579 1.14064
\(57\) 8.22410 1.08931
\(58\) 5.52999 0.726124
\(59\) −2.50988 −0.326759 −0.163379 0.986563i \(-0.552239\pi\)
−0.163379 + 0.986563i \(0.552239\pi\)
\(60\) −5.64954 −0.729352
\(61\) 12.9426 1.65713 0.828565 0.559893i \(-0.189158\pi\)
0.828565 + 0.559893i \(0.189158\pi\)
\(62\) −6.16559 −0.783030
\(63\) −17.6328 −2.22152
\(64\) −0.315539 −0.0394424
\(65\) 8.87491 1.10080
\(66\) −2.32907 −0.286689
\(67\) −5.60551 −0.684822 −0.342411 0.939550i \(-0.611244\pi\)
−0.342411 + 0.939550i \(0.611244\pi\)
\(68\) −1.62275 −0.196788
\(69\) −17.9218 −2.15753
\(70\) 2.97617 0.355721
\(71\) −8.81728 −1.04642 −0.523209 0.852204i \(-0.675265\pi\)
−0.523209 + 0.852204i \(0.675265\pi\)
\(72\) −10.2278 −1.20535
\(73\) −11.2882 −1.32119 −0.660593 0.750744i \(-0.729695\pi\)
−0.660593 + 0.750744i \(0.729695\pi\)
\(74\) 3.75028 0.435961
\(75\) 9.38331 1.08349
\(76\) 4.84210 0.555427
\(77\) −5.27782 −0.601463
\(78\) 11.8941 1.34674
\(79\) −3.35889 −0.377904 −0.188952 0.981986i \(-0.560509\pi\)
−0.188952 + 0.981986i \(0.560509\pi\)
\(80\) −2.37324 −0.265336
\(81\) −1.66158 −0.184620
\(82\) −0.911296 −0.100636
\(83\) −5.72716 −0.628637 −0.314319 0.949318i \(-0.601776\pi\)
−0.314319 + 0.949318i \(0.601776\pi\)
\(84\) −17.1574 −1.87203
\(85\) −1.26314 −0.137007
\(86\) −0.614204 −0.0662313
\(87\) −24.8153 −2.66048
\(88\) −3.06136 −0.326342
\(89\) 12.1270 1.28546 0.642730 0.766093i \(-0.277802\pi\)
0.642730 + 0.766093i \(0.277802\pi\)
\(90\) −3.56611 −0.375901
\(91\) 26.9527 2.82541
\(92\) −10.5518 −1.10010
\(93\) 27.6674 2.86898
\(94\) 0.151547 0.0156309
\(95\) 3.76907 0.386699
\(96\) −15.4462 −1.57647
\(97\) 0.613967 0.0623389 0.0311694 0.999514i \(-0.490077\pi\)
0.0311694 + 0.999514i \(0.490077\pi\)
\(98\) 4.73908 0.478720
\(99\) 6.32399 0.635585
\(100\) 5.52461 0.552461
\(101\) 11.8084 1.17498 0.587491 0.809231i \(-0.300116\pi\)
0.587491 + 0.809231i \(0.300116\pi\)
\(102\) −1.69286 −0.167618
\(103\) −13.9161 −1.37120 −0.685598 0.727981i \(-0.740459\pi\)
−0.685598 + 0.727981i \(0.740459\pi\)
\(104\) 15.6337 1.53301
\(105\) −13.3553 −1.30334
\(106\) −2.18349 −0.212079
\(107\) −0.822787 −0.0795418 −0.0397709 0.999209i \(-0.512663\pi\)
−0.0397709 + 0.999209i \(0.512663\pi\)
\(108\) 7.14058 0.687103
\(109\) 10.8275 1.03709 0.518544 0.855051i \(-0.326474\pi\)
0.518544 + 0.855051i \(0.326474\pi\)
\(110\) −1.06740 −0.101773
\(111\) −16.8290 −1.59734
\(112\) −7.20743 −0.681038
\(113\) 12.4803 1.17405 0.587024 0.809570i \(-0.300300\pi\)
0.587024 + 0.809570i \(0.300300\pi\)
\(114\) 5.05128 0.473096
\(115\) −8.21350 −0.765913
\(116\) −14.6105 −1.35655
\(117\) −32.2953 −2.98570
\(118\) −1.54158 −0.141914
\(119\) −3.83612 −0.351656
\(120\) −7.74661 −0.707166
\(121\) −9.10711 −0.827919
\(122\) 7.94940 0.719705
\(123\) 4.08935 0.368724
\(124\) 16.2897 1.46286
\(125\) 10.6161 0.949529
\(126\) −10.8301 −0.964825
\(127\) 1.03453 0.0917993 0.0458997 0.998946i \(-0.485385\pi\)
0.0458997 + 0.998946i \(0.485385\pi\)
\(128\) −11.4022 −1.00782
\(129\) 2.75618 0.242668
\(130\) 5.45101 0.478085
\(131\) 18.0012 1.57277 0.786387 0.617734i \(-0.211949\pi\)
0.786387 + 0.617734i \(0.211949\pi\)
\(132\) 6.15350 0.535593
\(133\) 11.4465 0.992539
\(134\) −3.44293 −0.297424
\(135\) 5.55820 0.478374
\(136\) −2.22511 −0.190802
\(137\) −11.9203 −1.01842 −0.509211 0.860642i \(-0.670063\pi\)
−0.509211 + 0.860642i \(0.670063\pi\)
\(138\) −11.0077 −0.937035
\(139\) −5.29545 −0.449154 −0.224577 0.974456i \(-0.572100\pi\)
−0.224577 + 0.974456i \(0.572100\pi\)
\(140\) −7.86317 −0.664559
\(141\) −0.680053 −0.0572708
\(142\) −5.41561 −0.454468
\(143\) −9.66659 −0.808361
\(144\) 8.63609 0.719675
\(145\) −11.3727 −0.944454
\(146\) −6.93328 −0.573802
\(147\) −21.2661 −1.75400
\(148\) −9.90839 −0.814464
\(149\) −3.20149 −0.262276 −0.131138 0.991364i \(-0.541863\pi\)
−0.131138 + 0.991364i \(0.541863\pi\)
\(150\) 5.76327 0.470569
\(151\) 6.32381 0.514624 0.257312 0.966328i \(-0.417163\pi\)
0.257312 + 0.966328i \(0.417163\pi\)
\(152\) 6.63946 0.538531
\(153\) 4.59652 0.371606
\(154\) −3.24166 −0.261220
\(155\) 12.6799 1.01847
\(156\) −31.4247 −2.51599
\(157\) 5.79963 0.462860 0.231430 0.972852i \(-0.425659\pi\)
0.231430 + 0.972852i \(0.425659\pi\)
\(158\) −2.06304 −0.164127
\(159\) 9.79817 0.777045
\(160\) −7.07893 −0.559638
\(161\) −24.9441 −1.96587
\(162\) −1.02055 −0.0801818
\(163\) −8.92023 −0.698687 −0.349343 0.936995i \(-0.613595\pi\)
−0.349343 + 0.936995i \(0.613595\pi\)
\(164\) 2.40768 0.188008
\(165\) 4.78986 0.372890
\(166\) −3.51765 −0.273022
\(167\) −13.6893 −1.05931 −0.529654 0.848214i \(-0.677678\pi\)
−0.529654 + 0.848214i \(0.677678\pi\)
\(168\) −23.5262 −1.81508
\(169\) 36.3653 2.79733
\(170\) −0.775829 −0.0595034
\(171\) −13.7155 −1.04885
\(172\) 1.62275 0.123734
\(173\) 12.1611 0.924590 0.462295 0.886726i \(-0.347026\pi\)
0.462295 + 0.886726i \(0.347026\pi\)
\(174\) −15.2416 −1.15547
\(175\) 13.0599 0.987238
\(176\) 2.58494 0.194847
\(177\) 6.91768 0.519964
\(178\) 7.44845 0.558285
\(179\) 23.1184 1.72795 0.863976 0.503534i \(-0.167967\pi\)
0.863976 + 0.503534i \(0.167967\pi\)
\(180\) 9.42181 0.702261
\(181\) −1.14490 −0.0850996 −0.0425498 0.999094i \(-0.513548\pi\)
−0.0425498 + 0.999094i \(0.513548\pi\)
\(182\) 16.5545 1.22710
\(183\) −35.6721 −2.63696
\(184\) −14.4686 −1.06664
\(185\) −7.71265 −0.567045
\(186\) 16.9935 1.24602
\(187\) 1.37582 0.100610
\(188\) −0.400395 −0.0292018
\(189\) 16.8800 1.22784
\(190\) 2.31498 0.167946
\(191\) 1.64848 0.119280 0.0596398 0.998220i \(-0.481005\pi\)
0.0596398 + 0.998220i \(0.481005\pi\)
\(192\) 0.869682 0.0627639
\(193\) 16.8532 1.21312 0.606561 0.795037i \(-0.292548\pi\)
0.606561 + 0.795037i \(0.292548\pi\)
\(194\) 0.377101 0.0270743
\(195\) −24.4608 −1.75168
\(196\) −12.5209 −0.894347
\(197\) −5.74840 −0.409556 −0.204778 0.978808i \(-0.565647\pi\)
−0.204778 + 0.978808i \(0.565647\pi\)
\(198\) 3.88422 0.276040
\(199\) 7.91781 0.561279 0.280639 0.959813i \(-0.409454\pi\)
0.280639 + 0.959813i \(0.409454\pi\)
\(200\) 7.57531 0.535655
\(201\) 15.4498 1.08974
\(202\) 7.25278 0.510304
\(203\) −34.5385 −2.42413
\(204\) 4.47260 0.313144
\(205\) 1.87413 0.130895
\(206\) −8.54733 −0.595521
\(207\) 29.8885 2.07739
\(208\) −13.2008 −0.915309
\(209\) −4.10529 −0.283969
\(210\) −8.20286 −0.566051
\(211\) −1.98947 −0.136961 −0.0684804 0.997652i \(-0.521815\pi\)
−0.0684804 + 0.997652i \(0.521815\pi\)
\(212\) 5.76886 0.396207
\(213\) 24.3020 1.66515
\(214\) −0.505359 −0.0345457
\(215\) 1.26314 0.0861457
\(216\) 9.79113 0.666202
\(217\) 38.5082 2.61411
\(218\) 6.65030 0.450415
\(219\) 31.1124 2.10238
\(220\) 2.82012 0.190133
\(221\) −7.02604 −0.472623
\(222\) −10.3364 −0.693736
\(223\) 8.24378 0.552044 0.276022 0.961151i \(-0.410984\pi\)
0.276022 + 0.961151i \(0.410984\pi\)
\(224\) −21.4984 −1.43642
\(225\) −15.6487 −1.04325
\(226\) 7.66545 0.509898
\(227\) 11.1989 0.743297 0.371648 0.928374i \(-0.378793\pi\)
0.371648 + 0.928374i \(0.378793\pi\)
\(228\) −13.3457 −0.883840
\(229\) 17.3516 1.14663 0.573314 0.819336i \(-0.305658\pi\)
0.573314 + 0.819336i \(0.305658\pi\)
\(230\) −5.04476 −0.332642
\(231\) 14.5466 0.957096
\(232\) −20.0338 −1.31528
\(233\) −19.3065 −1.26481 −0.632406 0.774637i \(-0.717933\pi\)
−0.632406 + 0.774637i \(0.717933\pi\)
\(234\) −19.8359 −1.29672
\(235\) −0.311666 −0.0203308
\(236\) 4.07292 0.265124
\(237\) 9.25769 0.601352
\(238\) −2.35616 −0.152727
\(239\) −0.131800 −0.00852545 −0.00426273 0.999991i \(-0.501357\pi\)
−0.00426273 + 0.999991i \(0.501357\pi\)
\(240\) 6.54107 0.422224
\(241\) 12.7105 0.818757 0.409379 0.912365i \(-0.365746\pi\)
0.409379 + 0.912365i \(0.365746\pi\)
\(242\) −5.59363 −0.359572
\(243\) 17.7805 1.14062
\(244\) −21.0026 −1.34456
\(245\) −9.74618 −0.622661
\(246\) 2.51169 0.160140
\(247\) 20.9649 1.33396
\(248\) 22.3364 1.41836
\(249\) 15.7851 1.00034
\(250\) 6.52043 0.412388
\(251\) −11.1387 −0.703068 −0.351534 0.936175i \(-0.614340\pi\)
−0.351534 + 0.936175i \(0.614340\pi\)
\(252\) 28.6137 1.80249
\(253\) 8.94617 0.562441
\(254\) 0.635410 0.0398692
\(255\) 3.48145 0.218017
\(256\) −6.37222 −0.398263
\(257\) 24.6214 1.53584 0.767919 0.640547i \(-0.221292\pi\)
0.767919 + 0.640547i \(0.221292\pi\)
\(258\) 1.69286 0.105393
\(259\) −23.4230 −1.45543
\(260\) −14.4018 −0.893161
\(261\) 41.3848 2.56165
\(262\) 11.0564 0.683069
\(263\) 14.7248 0.907970 0.453985 0.891009i \(-0.350002\pi\)
0.453985 + 0.891009i \(0.350002\pi\)
\(264\) 8.43764 0.519301
\(265\) 4.49046 0.275847
\(266\) 7.03050 0.431068
\(267\) −33.4242 −2.04553
\(268\) 9.09636 0.555649
\(269\) 18.0695 1.10172 0.550858 0.834599i \(-0.314300\pi\)
0.550858 + 0.834599i \(0.314300\pi\)
\(270\) 3.41387 0.207762
\(271\) −16.9413 −1.02911 −0.514554 0.857458i \(-0.672042\pi\)
−0.514554 + 0.857458i \(0.672042\pi\)
\(272\) 1.87883 0.113921
\(273\) −74.2865 −4.49603
\(274\) −7.32152 −0.442309
\(275\) −4.68394 −0.282452
\(276\) 29.0827 1.75057
\(277\) −23.7585 −1.42751 −0.713754 0.700396i \(-0.753007\pi\)
−0.713754 + 0.700396i \(0.753007\pi\)
\(278\) −3.25249 −0.195071
\(279\) −46.1414 −2.76241
\(280\) −10.7819 −0.644344
\(281\) −13.8505 −0.826249 −0.413125 0.910674i \(-0.635563\pi\)
−0.413125 + 0.910674i \(0.635563\pi\)
\(282\) −0.417692 −0.0248732
\(283\) −19.2658 −1.14524 −0.572618 0.819823i \(-0.694072\pi\)
−0.572618 + 0.819823i \(0.694072\pi\)
\(284\) 14.3083 0.849040
\(285\) −10.3882 −0.615346
\(286\) −5.93726 −0.351077
\(287\) 5.69166 0.335968
\(288\) 25.7598 1.51791
\(289\) 1.00000 0.0588235
\(290\) −6.98518 −0.410184
\(291\) −1.69220 −0.0991986
\(292\) 18.3180 1.07198
\(293\) −6.87426 −0.401598 −0.200799 0.979632i \(-0.564354\pi\)
−0.200799 + 0.979632i \(0.564354\pi\)
\(294\) −13.0618 −0.761777
\(295\) 3.17034 0.184584
\(296\) −13.5863 −0.789689
\(297\) −6.05401 −0.351290
\(298\) −1.96637 −0.113909
\(299\) −45.6863 −2.64211
\(300\) −15.2268 −0.879120
\(301\) 3.83612 0.221110
\(302\) 3.88411 0.223505
\(303\) −32.5461 −1.86973
\(304\) −5.60621 −0.321538
\(305\) −16.3484 −0.936105
\(306\) 2.82320 0.161392
\(307\) 23.5940 1.34658 0.673290 0.739379i \(-0.264881\pi\)
0.673290 + 0.739379i \(0.264881\pi\)
\(308\) 8.56460 0.488013
\(309\) 38.3553 2.18196
\(310\) 7.78803 0.442330
\(311\) 34.3961 1.95043 0.975213 0.221270i \(-0.0710201\pi\)
0.975213 + 0.221270i \(0.0710201\pi\)
\(312\) −43.0893 −2.43945
\(313\) −25.4996 −1.44132 −0.720661 0.693288i \(-0.756162\pi\)
−0.720661 + 0.693288i \(0.756162\pi\)
\(314\) 3.56215 0.201024
\(315\) 22.2728 1.25493
\(316\) 5.45065 0.306623
\(317\) 24.1380 1.35572 0.677862 0.735189i \(-0.262907\pi\)
0.677862 + 0.735189i \(0.262907\pi\)
\(318\) 6.01808 0.337477
\(319\) 12.3872 0.693552
\(320\) 0.398572 0.0222808
\(321\) 2.26775 0.126573
\(322\) −15.3207 −0.853792
\(323\) −2.98388 −0.166028
\(324\) 2.69633 0.149796
\(325\) 23.9199 1.32684
\(326\) −5.47885 −0.303445
\(327\) −29.8426 −1.65030
\(328\) 3.30140 0.182289
\(329\) −0.946516 −0.0521831
\(330\) 2.94195 0.161949
\(331\) −27.4047 −1.50630 −0.753149 0.657850i \(-0.771466\pi\)
−0.753149 + 0.657850i \(0.771466\pi\)
\(332\) 9.29377 0.510062
\(333\) 28.0659 1.53800
\(334\) −8.40801 −0.460066
\(335\) 7.08057 0.386853
\(336\) 19.8650 1.08372
\(337\) −10.8198 −0.589391 −0.294696 0.955591i \(-0.595218\pi\)
−0.294696 + 0.955591i \(0.595218\pi\)
\(338\) 22.3357 1.21490
\(339\) −34.3979 −1.86824
\(340\) 2.04977 0.111164
\(341\) −13.8110 −0.747905
\(342\) −8.42409 −0.455523
\(343\) −2.74592 −0.148266
\(344\) 2.22511 0.119970
\(345\) 22.6379 1.21878
\(346\) 7.46939 0.401557
\(347\) −9.49423 −0.509677 −0.254839 0.966984i \(-0.582022\pi\)
−0.254839 + 0.966984i \(0.582022\pi\)
\(348\) 40.2691 2.15865
\(349\) 31.1355 1.66665 0.833323 0.552786i \(-0.186435\pi\)
0.833323 + 0.552786i \(0.186435\pi\)
\(350\) 8.02147 0.428765
\(351\) 30.9166 1.65021
\(352\) 7.71040 0.410965
\(353\) −15.6254 −0.831654 −0.415827 0.909444i \(-0.636508\pi\)
−0.415827 + 0.909444i \(0.636508\pi\)
\(354\) 4.24887 0.225825
\(355\) 11.1375 0.591117
\(356\) −19.6791 −1.04299
\(357\) 10.5730 0.559584
\(358\) 14.1994 0.750463
\(359\) 2.33909 0.123452 0.0617261 0.998093i \(-0.480339\pi\)
0.0617261 + 0.998093i \(0.480339\pi\)
\(360\) 12.9191 0.680898
\(361\) −10.0965 −0.531393
\(362\) −0.703201 −0.0369594
\(363\) 25.1008 1.31745
\(364\) −43.7376 −2.29247
\(365\) 14.2587 0.746332
\(366\) −21.9100 −1.14525
\(367\) 36.4028 1.90021 0.950105 0.311929i \(-0.100975\pi\)
0.950105 + 0.311929i \(0.100975\pi\)
\(368\) 12.2170 0.636853
\(369\) −6.81986 −0.355028
\(370\) −4.73714 −0.246272
\(371\) 13.6373 0.708015
\(372\) −44.8974 −2.32782
\(373\) 10.0402 0.519864 0.259932 0.965627i \(-0.416300\pi\)
0.259932 + 0.965627i \(0.416300\pi\)
\(374\) 0.845036 0.0436958
\(375\) −29.2597 −1.51097
\(376\) −0.549019 −0.0283135
\(377\) −63.2590 −3.25801
\(378\) 10.3678 0.533261
\(379\) −3.95923 −0.203372 −0.101686 0.994817i \(-0.532424\pi\)
−0.101686 + 0.994817i \(0.532424\pi\)
\(380\) −6.11627 −0.313758
\(381\) −2.85134 −0.146078
\(382\) 1.01250 0.0518041
\(383\) 18.6383 0.952372 0.476186 0.879345i \(-0.342019\pi\)
0.476186 + 0.879345i \(0.342019\pi\)
\(384\) 31.4266 1.60373
\(385\) 6.66665 0.339764
\(386\) 10.3513 0.526869
\(387\) −4.59652 −0.233654
\(388\) −0.996317 −0.0505803
\(389\) −17.1284 −0.868445 −0.434223 0.900806i \(-0.642977\pi\)
−0.434223 + 0.900806i \(0.642977\pi\)
\(390\) −15.0239 −0.760767
\(391\) 6.50242 0.328842
\(392\) −17.1685 −0.867141
\(393\) −49.6146 −2.50273
\(394\) −3.53069 −0.177874
\(395\) 4.24276 0.213477
\(396\) −10.2623 −0.515699
\(397\) −17.5299 −0.879801 −0.439901 0.898046i \(-0.644986\pi\)
−0.439901 + 0.898046i \(0.644986\pi\)
\(398\) 4.86315 0.243768
\(399\) −31.5486 −1.57941
\(400\) −6.39643 −0.319821
\(401\) −31.0247 −1.54930 −0.774649 0.632391i \(-0.782073\pi\)
−0.774649 + 0.632391i \(0.782073\pi\)
\(402\) 9.48933 0.473285
\(403\) 70.5297 3.51334
\(404\) −19.1622 −0.953353
\(405\) 2.09881 0.104291
\(406\) −21.2137 −1.05282
\(407\) 8.40065 0.416405
\(408\) 6.13280 0.303619
\(409\) 21.5180 1.06400 0.531998 0.846746i \(-0.321442\pi\)
0.531998 + 0.846746i \(0.321442\pi\)
\(410\) 1.15110 0.0568487
\(411\) 32.8545 1.62060
\(412\) 22.5824 1.11256
\(413\) 9.62820 0.473773
\(414\) 18.3576 0.902229
\(415\) 7.23423 0.355114
\(416\) −39.3754 −1.93054
\(417\) 14.5952 0.714730
\(418\) −2.52149 −0.123330
\(419\) 4.32573 0.211326 0.105663 0.994402i \(-0.466304\pi\)
0.105663 + 0.994402i \(0.466304\pi\)
\(420\) 21.6723 1.05750
\(421\) −14.2478 −0.694396 −0.347198 0.937792i \(-0.612867\pi\)
−0.347198 + 0.937792i \(0.612867\pi\)
\(422\) −1.22194 −0.0594832
\(423\) 1.13414 0.0551435
\(424\) 7.91023 0.384155
\(425\) −3.40447 −0.165141
\(426\) 14.9264 0.723186
\(427\) −49.6493 −2.40270
\(428\) 1.33518 0.0645384
\(429\) 26.6428 1.28633
\(430\) 0.775829 0.0374138
\(431\) 14.6180 0.704125 0.352063 0.935976i \(-0.385480\pi\)
0.352063 + 0.935976i \(0.385480\pi\)
\(432\) −8.26741 −0.397766
\(433\) −29.6884 −1.42673 −0.713367 0.700791i \(-0.752831\pi\)
−0.713367 + 0.700791i \(0.752831\pi\)
\(434\) 23.6519 1.13533
\(435\) 31.3453 1.50289
\(436\) −17.5704 −0.841469
\(437\) −19.4024 −0.928145
\(438\) 19.1093 0.913080
\(439\) 35.7749 1.70744 0.853720 0.520732i \(-0.174341\pi\)
0.853720 + 0.520732i \(0.174341\pi\)
\(440\) 3.86694 0.184349
\(441\) 35.4658 1.68885
\(442\) −4.31543 −0.205264
\(443\) 16.8157 0.798936 0.399468 0.916747i \(-0.369195\pi\)
0.399468 + 0.916747i \(0.369195\pi\)
\(444\) 27.3093 1.29604
\(445\) −15.3182 −0.726150
\(446\) 5.06336 0.239757
\(447\) 8.82386 0.417354
\(448\) 1.21045 0.0571882
\(449\) −3.87481 −0.182863 −0.0914317 0.995811i \(-0.529144\pi\)
−0.0914317 + 0.995811i \(0.529144\pi\)
\(450\) −9.61149 −0.453090
\(451\) −2.04131 −0.0961216
\(452\) −20.2524 −0.952594
\(453\) −17.4295 −0.818912
\(454\) 6.87841 0.322820
\(455\) −34.0452 −1.59606
\(456\) −18.2995 −0.856955
\(457\) 3.89913 0.182394 0.0911969 0.995833i \(-0.470931\pi\)
0.0911969 + 0.995833i \(0.470931\pi\)
\(458\) 10.6574 0.497989
\(459\) −4.40029 −0.205388
\(460\) 13.3285 0.621443
\(461\) 7.77268 0.362010 0.181005 0.983482i \(-0.442065\pi\)
0.181005 + 0.983482i \(0.442065\pi\)
\(462\) 8.93459 0.415675
\(463\) 3.58890 0.166790 0.0833952 0.996517i \(-0.473424\pi\)
0.0833952 + 0.996517i \(0.473424\pi\)
\(464\) 16.9161 0.785310
\(465\) −34.9480 −1.62067
\(466\) −11.8581 −0.549318
\(467\) 28.2210 1.30591 0.652956 0.757396i \(-0.273529\pi\)
0.652956 + 0.757396i \(0.273529\pi\)
\(468\) 52.4073 2.42253
\(469\) 21.5034 0.992935
\(470\) −0.191426 −0.00882984
\(471\) −15.9848 −0.736541
\(472\) 5.58476 0.257059
\(473\) −1.37582 −0.0632604
\(474\) 5.68612 0.261172
\(475\) 10.1585 0.466105
\(476\) 6.22507 0.285326
\(477\) −16.3405 −0.748182
\(478\) −0.0809523 −0.00370267
\(479\) −30.5347 −1.39516 −0.697582 0.716505i \(-0.745741\pi\)
−0.697582 + 0.716505i \(0.745741\pi\)
\(480\) 19.5108 0.890541
\(481\) −42.9004 −1.95609
\(482\) 7.80686 0.355593
\(483\) 68.7503 3.12825
\(484\) 14.7786 0.671754
\(485\) −0.775529 −0.0352150
\(486\) 10.9208 0.495379
\(487\) 13.5108 0.612235 0.306117 0.951994i \(-0.400970\pi\)
0.306117 + 0.951994i \(0.400970\pi\)
\(488\) −28.7987 −1.30366
\(489\) 24.5858 1.11181
\(490\) −5.98615 −0.270427
\(491\) −14.2174 −0.641622 −0.320811 0.947143i \(-0.603955\pi\)
−0.320811 + 0.947143i \(0.603955\pi\)
\(492\) −6.63600 −0.299174
\(493\) 9.00351 0.405498
\(494\) 12.8767 0.579351
\(495\) −7.98812 −0.359039
\(496\) −18.8604 −0.846855
\(497\) 33.8241 1.51722
\(498\) 9.69526 0.434455
\(499\) 1.46685 0.0656653 0.0328326 0.999461i \(-0.489547\pi\)
0.0328326 + 0.999461i \(0.489547\pi\)
\(500\) −17.2272 −0.770425
\(501\) 37.7301 1.68565
\(502\) −6.84143 −0.305348
\(503\) −12.2136 −0.544579 −0.272290 0.962215i \(-0.587781\pi\)
−0.272290 + 0.962215i \(0.587781\pi\)
\(504\) 39.2349 1.74766
\(505\) −14.9157 −0.663742
\(506\) 5.49478 0.244273
\(507\) −100.229 −4.45133
\(508\) −1.67878 −0.0744838
\(509\) −15.6020 −0.691546 −0.345773 0.938318i \(-0.612383\pi\)
−0.345773 + 0.938318i \(0.612383\pi\)
\(510\) 2.13832 0.0946865
\(511\) 43.3030 1.91561
\(512\) 18.8906 0.834855
\(513\) 13.1299 0.579701
\(514\) 15.1225 0.667027
\(515\) 17.5781 0.774582
\(516\) −4.47260 −0.196895
\(517\) 0.339467 0.0149298
\(518\) −14.3865 −0.632107
\(519\) −33.5181 −1.47128
\(520\) −19.7476 −0.865992
\(521\) −43.2179 −1.89341 −0.946705 0.322102i \(-0.895611\pi\)
−0.946705 + 0.322102i \(0.895611\pi\)
\(522\) 25.4187 1.11255
\(523\) −11.1444 −0.487310 −0.243655 0.969862i \(-0.578346\pi\)
−0.243655 + 0.969862i \(0.578346\pi\)
\(524\) −29.2115 −1.27611
\(525\) −35.9955 −1.57097
\(526\) 9.04403 0.394339
\(527\) −10.0383 −0.437277
\(528\) −7.12456 −0.310057
\(529\) 19.2815 0.838325
\(530\) 2.75806 0.119802
\(531\) −11.5367 −0.500650
\(532\) −18.5749 −0.805323
\(533\) 10.4246 0.451538
\(534\) −20.5293 −0.888388
\(535\) 1.03930 0.0449328
\(536\) 12.4729 0.538747
\(537\) −63.7185 −2.74965
\(538\) 11.0984 0.478484
\(539\) 10.6156 0.457245
\(540\) −9.01959 −0.388141
\(541\) −8.93756 −0.384256 −0.192128 0.981370i \(-0.561539\pi\)
−0.192128 + 0.981370i \(0.561539\pi\)
\(542\) −10.4054 −0.446950
\(543\) 3.15554 0.135417
\(544\) 5.60421 0.240278
\(545\) −13.6767 −0.585846
\(546\) −45.6271 −1.95266
\(547\) 30.2438 1.29313 0.646567 0.762857i \(-0.276204\pi\)
0.646567 + 0.762857i \(0.276204\pi\)
\(548\) 19.3437 0.826324
\(549\) 59.4909 2.53901
\(550\) −2.87690 −0.122671
\(551\) −26.8654 −1.14450
\(552\) 39.8780 1.69732
\(553\) 12.8851 0.547930
\(554\) −14.5926 −0.619978
\(555\) 21.2574 0.902328
\(556\) 8.59321 0.364433
\(557\) 13.6108 0.576707 0.288354 0.957524i \(-0.406892\pi\)
0.288354 + 0.957524i \(0.406892\pi\)
\(558\) −28.3402 −1.19974
\(559\) 7.02604 0.297170
\(560\) 9.10403 0.384715
\(561\) −3.79201 −0.160099
\(562\) −8.50701 −0.358847
\(563\) 19.5674 0.824669 0.412335 0.911033i \(-0.364713\pi\)
0.412335 + 0.911033i \(0.364713\pi\)
\(564\) 1.10356 0.0464682
\(565\) −15.7644 −0.663214
\(566\) −11.8332 −0.497385
\(567\) 6.37401 0.267683
\(568\) 19.6194 0.823213
\(569\) 11.1657 0.468090 0.234045 0.972226i \(-0.424804\pi\)
0.234045 + 0.972226i \(0.424804\pi\)
\(570\) −6.38050 −0.267250
\(571\) 5.03739 0.210808 0.105404 0.994429i \(-0.466386\pi\)
0.105404 + 0.994429i \(0.466386\pi\)
\(572\) 15.6865 0.655885
\(573\) −4.54350 −0.189807
\(574\) 3.49584 0.145914
\(575\) −22.1373 −0.923188
\(576\) −1.45038 −0.0604325
\(577\) −27.1242 −1.12919 −0.564597 0.825366i \(-0.690969\pi\)
−0.564597 + 0.825366i \(0.690969\pi\)
\(578\) 0.614204 0.0255475
\(579\) −46.4505 −1.93042
\(580\) 18.4551 0.766308
\(581\) 21.9701 0.911472
\(582\) −1.03936 −0.0430828
\(583\) −4.89102 −0.202566
\(584\) 25.1175 1.03937
\(585\) 40.7937 1.68661
\(586\) −4.22220 −0.174417
\(587\) −30.4133 −1.25529 −0.627646 0.778499i \(-0.715981\pi\)
−0.627646 + 0.778499i \(0.715981\pi\)
\(588\) 34.5097 1.42316
\(589\) 29.9532 1.23420
\(590\) 1.94724 0.0801665
\(591\) 15.8436 0.651719
\(592\) 11.4720 0.471496
\(593\) 22.6475 0.930021 0.465010 0.885305i \(-0.346051\pi\)
0.465010 + 0.885305i \(0.346051\pi\)
\(594\) −3.71840 −0.152568
\(595\) 4.84557 0.198649
\(596\) 5.19522 0.212805
\(597\) −21.8229 −0.893151
\(598\) −28.0607 −1.14749
\(599\) −27.6329 −1.12905 −0.564525 0.825416i \(-0.690940\pi\)
−0.564525 + 0.825416i \(0.690940\pi\)
\(600\) −20.8789 −0.852378
\(601\) −0.342259 −0.0139610 −0.00698051 0.999976i \(-0.502222\pi\)
−0.00698051 + 0.999976i \(0.502222\pi\)
\(602\) 2.35616 0.0960299
\(603\) −25.7658 −1.04927
\(604\) −10.2620 −0.417554
\(605\) 11.5036 0.467688
\(606\) −19.9900 −0.812037
\(607\) −33.4744 −1.35869 −0.679343 0.733821i \(-0.737735\pi\)
−0.679343 + 0.733821i \(0.737735\pi\)
\(608\) −16.7223 −0.678178
\(609\) 95.1944 3.85747
\(610\) −10.0412 −0.406558
\(611\) −1.73359 −0.0701336
\(612\) −7.45901 −0.301513
\(613\) 19.9080 0.804076 0.402038 0.915623i \(-0.368302\pi\)
0.402038 + 0.915623i \(0.368302\pi\)
\(614\) 14.4915 0.584830
\(615\) −5.16544 −0.208291
\(616\) 11.7437 0.473168
\(617\) 22.8521 0.919989 0.459994 0.887922i \(-0.347851\pi\)
0.459994 + 0.887922i \(0.347851\pi\)
\(618\) 23.5580 0.947641
\(619\) −3.17417 −0.127581 −0.0637904 0.997963i \(-0.520319\pi\)
−0.0637904 + 0.997963i \(0.520319\pi\)
\(620\) −20.5763 −0.826364
\(621\) −28.6125 −1.14818
\(622\) 21.1262 0.847085
\(623\) −46.5206 −1.86381
\(624\) 36.3837 1.45651
\(625\) 3.61271 0.144509
\(626\) −15.6620 −0.625978
\(627\) 11.3149 0.451874
\(628\) −9.41136 −0.375554
\(629\) 6.10591 0.243459
\(630\) 13.6800 0.545025
\(631\) −7.09662 −0.282512 −0.141256 0.989973i \(-0.545114\pi\)
−0.141256 + 0.989973i \(0.545114\pi\)
\(632\) 7.47390 0.297296
\(633\) 5.48334 0.217943
\(634\) 14.8256 0.588802
\(635\) −1.30676 −0.0518570
\(636\) −15.9000 −0.630476
\(637\) −54.2116 −2.14794
\(638\) 7.60829 0.301215
\(639\) −40.5288 −1.60329
\(640\) 14.4027 0.569315
\(641\) 39.4650 1.55877 0.779387 0.626543i \(-0.215531\pi\)
0.779387 + 0.626543i \(0.215531\pi\)
\(642\) 1.39286 0.0549718
\(643\) −29.0359 −1.14506 −0.572532 0.819883i \(-0.694039\pi\)
−0.572532 + 0.819883i \(0.694039\pi\)
\(644\) 40.4780 1.59506
\(645\) −3.48145 −0.137082
\(646\) −1.83271 −0.0721071
\(647\) −8.23501 −0.323752 −0.161876 0.986811i \(-0.551754\pi\)
−0.161876 + 0.986811i \(0.551754\pi\)
\(648\) 3.69719 0.145239
\(649\) −3.45315 −0.135548
\(650\) 14.6917 0.576257
\(651\) −106.136 −4.15978
\(652\) 14.4753 0.566898
\(653\) 16.5418 0.647330 0.323665 0.946172i \(-0.395085\pi\)
0.323665 + 0.946172i \(0.395085\pi\)
\(654\) −18.3294 −0.716737
\(655\) −22.7382 −0.888453
\(656\) −2.78763 −0.108839
\(657\) −51.8865 −2.02429
\(658\) −0.581354 −0.0226635
\(659\) 8.48349 0.330470 0.165235 0.986254i \(-0.447162\pi\)
0.165235 + 0.986254i \(0.447162\pi\)
\(660\) −7.77276 −0.302554
\(661\) 19.7339 0.767559 0.383779 0.923425i \(-0.374622\pi\)
0.383779 + 0.923425i \(0.374622\pi\)
\(662\) −16.8321 −0.654198
\(663\) 19.3650 0.752075
\(664\) 12.7436 0.494546
\(665\) −14.4586 −0.560681
\(666\) 17.2382 0.667967
\(667\) 58.5446 2.26686
\(668\) 22.2143 0.859497
\(669\) −22.7213 −0.878457
\(670\) 4.34892 0.168013
\(671\) 17.8067 0.687421
\(672\) 59.2534 2.28575
\(673\) 19.4335 0.749106 0.374553 0.927206i \(-0.377796\pi\)
0.374553 + 0.927206i \(0.377796\pi\)
\(674\) −6.64556 −0.255977
\(675\) 14.9806 0.576605
\(676\) −59.0119 −2.26969
\(677\) −13.7510 −0.528493 −0.264247 0.964455i \(-0.585123\pi\)
−0.264247 + 0.964455i \(0.585123\pi\)
\(678\) −21.1273 −0.811391
\(679\) −2.35525 −0.0903862
\(680\) 2.81064 0.107783
\(681\) −30.8661 −1.18279
\(682\) −8.48275 −0.324821
\(683\) 40.3806 1.54512 0.772560 0.634942i \(-0.218976\pi\)
0.772560 + 0.634942i \(0.218976\pi\)
\(684\) 22.2568 0.851010
\(685\) 15.0571 0.575302
\(686\) −1.68656 −0.0643931
\(687\) −47.8242 −1.82461
\(688\) −1.87883 −0.0716299
\(689\) 24.9775 0.951566
\(690\) 13.9043 0.529327
\(691\) −38.5069 −1.46487 −0.732435 0.680837i \(-0.761616\pi\)
−0.732435 + 0.680837i \(0.761616\pi\)
\(692\) −19.7344 −0.750191
\(693\) −24.2596 −0.921545
\(694\) −5.83140 −0.221357
\(695\) 6.68892 0.253725
\(696\) 55.2167 2.09298
\(697\) −1.48370 −0.0561992
\(698\) 19.1236 0.723838
\(699\) 53.2122 2.01267
\(700\) −21.1930 −0.801022
\(701\) −49.7827 −1.88027 −0.940134 0.340806i \(-0.889300\pi\)
−0.940134 + 0.340806i \(0.889300\pi\)
\(702\) 18.9891 0.716698
\(703\) −18.2193 −0.687154
\(704\) −0.434126 −0.0163617
\(705\) 0.859006 0.0323520
\(706\) −9.59717 −0.361194
\(707\) −45.2985 −1.70363
\(708\) −11.2257 −0.421887
\(709\) −2.57721 −0.0967892 −0.0483946 0.998828i \(-0.515410\pi\)
−0.0483946 + 0.998828i \(0.515410\pi\)
\(710\) 6.84070 0.256727
\(711\) −15.4392 −0.579015
\(712\) −26.9839 −1.01126
\(713\) −65.2734 −2.44451
\(714\) 6.49400 0.243032
\(715\) 12.2103 0.456639
\(716\) −37.5155 −1.40202
\(717\) 0.363265 0.0135664
\(718\) 1.43668 0.0536163
\(719\) −20.5368 −0.765892 −0.382946 0.923771i \(-0.625090\pi\)
−0.382946 + 0.923771i \(0.625090\pi\)
\(720\) −10.9086 −0.406541
\(721\) 53.3839 1.98812
\(722\) −6.20129 −0.230788
\(723\) −35.0325 −1.30287
\(724\) 1.85789 0.0690478
\(725\) −30.6521 −1.13839
\(726\) 15.4170 0.572180
\(727\) 1.84717 0.0685079 0.0342539 0.999413i \(-0.489095\pi\)
0.0342539 + 0.999413i \(0.489095\pi\)
\(728\) −59.9728 −2.22274
\(729\) −44.0214 −1.63042
\(730\) 8.75773 0.324138
\(731\) −1.00000 −0.0369863
\(732\) 57.8870 2.13957
\(733\) −42.2608 −1.56094 −0.780469 0.625195i \(-0.785019\pi\)
−0.780469 + 0.625195i \(0.785019\pi\)
\(734\) 22.3588 0.825277
\(735\) 26.8622 0.990828
\(736\) 36.4409 1.34323
\(737\) −7.71219 −0.284082
\(738\) −4.18879 −0.154191
\(739\) −12.8553 −0.472891 −0.236445 0.971645i \(-0.575982\pi\)
−0.236445 + 0.971645i \(0.575982\pi\)
\(740\) 12.5157 0.460087
\(741\) −57.7829 −2.12271
\(742\) 8.37611 0.307497
\(743\) 7.99183 0.293192 0.146596 0.989196i \(-0.453168\pi\)
0.146596 + 0.989196i \(0.453168\pi\)
\(744\) −61.5631 −2.25701
\(745\) 4.04394 0.148158
\(746\) 6.16675 0.225781
\(747\) −26.3250 −0.963181
\(748\) −2.23262 −0.0816327
\(749\) 3.15631 0.115329
\(750\) −17.9715 −0.656225
\(751\) 21.1542 0.771926 0.385963 0.922514i \(-0.373869\pi\)
0.385963 + 0.922514i \(0.373869\pi\)
\(752\) 0.463579 0.0169050
\(753\) 30.7002 1.11878
\(754\) −38.8540 −1.41498
\(755\) −7.98789 −0.290709
\(756\) −27.3921 −0.996242
\(757\) 9.14310 0.332312 0.166156 0.986100i \(-0.446865\pi\)
0.166156 + 0.986100i \(0.446865\pi\)
\(758\) −2.43178 −0.0883261
\(759\) −24.6573 −0.895002
\(760\) −8.38660 −0.304214
\(761\) 1.57450 0.0570757 0.0285378 0.999593i \(-0.490915\pi\)
0.0285378 + 0.999593i \(0.490915\pi\)
\(762\) −1.75130 −0.0634431
\(763\) −41.5356 −1.50369
\(764\) −2.67507 −0.0967807
\(765\) −5.80607 −0.209919
\(766\) 11.4477 0.413623
\(767\) 17.6345 0.636746
\(768\) 17.5630 0.633749
\(769\) −10.0371 −0.361946 −0.180973 0.983488i \(-0.557925\pi\)
−0.180973 + 0.983488i \(0.557925\pi\)
\(770\) 4.09468 0.147562
\(771\) −67.8609 −2.44395
\(772\) −27.3486 −0.984299
\(773\) −6.88424 −0.247609 −0.123804 0.992307i \(-0.539510\pi\)
−0.123804 + 0.992307i \(0.539510\pi\)
\(774\) −2.82320 −0.101478
\(775\) 34.1752 1.22761
\(776\) −1.36614 −0.0490417
\(777\) 64.5580 2.31600
\(778\) −10.5203 −0.377173
\(779\) 4.42719 0.158621
\(780\) 39.6939 1.42127
\(781\) −12.1310 −0.434082
\(782\) 3.99381 0.142819
\(783\) −39.6180 −1.41583
\(784\) 14.4967 0.517740
\(785\) −7.32577 −0.261468
\(786\) −30.4735 −1.08695
\(787\) −17.3622 −0.618894 −0.309447 0.950917i \(-0.600144\pi\)
−0.309447 + 0.950917i \(0.600144\pi\)
\(788\) 9.32823 0.332304
\(789\) −40.5842 −1.44483
\(790\) 2.60592 0.0927146
\(791\) −47.8759 −1.70227
\(792\) −14.0716 −0.500012
\(793\) −90.9352 −3.22920
\(794\) −10.7669 −0.382105
\(795\) −12.3765 −0.438949
\(796\) −12.8486 −0.455408
\(797\) 1.32793 0.0470377 0.0235189 0.999723i \(-0.492513\pi\)
0.0235189 + 0.999723i \(0.492513\pi\)
\(798\) −19.3773 −0.685949
\(799\) 0.246738 0.00872896
\(800\) −19.0793 −0.674556
\(801\) 55.7420 1.96955
\(802\) −19.0555 −0.672872
\(803\) −15.5306 −0.548063
\(804\) −25.0712 −0.884193
\(805\) 31.5080 1.11051
\(806\) 43.3197 1.52587
\(807\) −49.8027 −1.75314
\(808\) −26.2750 −0.924353
\(809\) −18.5672 −0.652789 −0.326394 0.945234i \(-0.605834\pi\)
−0.326394 + 0.945234i \(0.605834\pi\)
\(810\) 1.28910 0.0452943
\(811\) −1.18659 −0.0416670 −0.0208335 0.999783i \(-0.506632\pi\)
−0.0208335 + 0.999783i \(0.506632\pi\)
\(812\) 56.0475 1.96688
\(813\) 46.6931 1.63760
\(814\) 5.15971 0.180848
\(815\) 11.2675 0.394685
\(816\) −5.17840 −0.181280
\(817\) 2.98388 0.104393
\(818\) 13.2164 0.462102
\(819\) 123.889 4.32902
\(820\) −3.04125 −0.106205
\(821\) −48.4440 −1.69071 −0.845354 0.534206i \(-0.820611\pi\)
−0.845354 + 0.534206i \(0.820611\pi\)
\(822\) 20.1794 0.703837
\(823\) 9.73575 0.339367 0.169684 0.985499i \(-0.445725\pi\)
0.169684 + 0.985499i \(0.445725\pi\)
\(824\) 30.9649 1.07871
\(825\) 12.9098 0.449461
\(826\) 5.91368 0.205763
\(827\) 54.8873 1.90862 0.954310 0.298819i \(-0.0965926\pi\)
0.954310 + 0.298819i \(0.0965926\pi\)
\(828\) −48.5016 −1.68555
\(829\) 19.6707 0.683193 0.341597 0.939847i \(-0.389032\pi\)
0.341597 + 0.939847i \(0.389032\pi\)
\(830\) 4.44330 0.154229
\(831\) 65.4826 2.27157
\(832\) 2.21699 0.0768603
\(833\) 7.71581 0.267337
\(834\) 8.96444 0.310413
\(835\) 17.2915 0.598398
\(836\) 6.66187 0.230406
\(837\) 44.1715 1.52679
\(838\) 2.65688 0.0917804
\(839\) −6.89842 −0.238160 −0.119080 0.992885i \(-0.537994\pi\)
−0.119080 + 0.992885i \(0.537994\pi\)
\(840\) 29.7169 1.02533
\(841\) 52.0632 1.79528
\(842\) −8.75106 −0.301582
\(843\) 38.1743 1.31479
\(844\) 3.22842 0.111127
\(845\) −45.9346 −1.58020
\(846\) 0.696591 0.0239493
\(847\) 34.9360 1.20041
\(848\) −6.67922 −0.229365
\(849\) 53.1001 1.82239
\(850\) −2.09104 −0.0717220
\(851\) 39.7032 1.36101
\(852\) −39.4361 −1.35106
\(853\) 22.9675 0.786393 0.393196 0.919455i \(-0.371369\pi\)
0.393196 + 0.919455i \(0.371369\pi\)
\(854\) −30.4948 −1.04351
\(855\) 17.3246 0.592489
\(856\) 1.83079 0.0625752
\(857\) 13.4636 0.459907 0.229953 0.973202i \(-0.426143\pi\)
0.229953 + 0.973202i \(0.426143\pi\)
\(858\) 16.3641 0.558663
\(859\) −3.83384 −0.130809 −0.0654044 0.997859i \(-0.520834\pi\)
−0.0654044 + 0.997859i \(0.520834\pi\)
\(860\) −2.04977 −0.0698966
\(861\) −15.6872 −0.534619
\(862\) 8.97845 0.305807
\(863\) −28.1275 −0.957470 −0.478735 0.877960i \(-0.658904\pi\)
−0.478735 + 0.877960i \(0.658904\pi\)
\(864\) −24.6601 −0.838955
\(865\) −15.3612 −0.522297
\(866\) −18.2347 −0.619642
\(867\) −2.75618 −0.0936047
\(868\) −62.4894 −2.12103
\(869\) −4.62123 −0.156765
\(870\) 19.2524 0.652718
\(871\) 39.3846 1.33450
\(872\) −24.0924 −0.815872
\(873\) 2.82211 0.0955140
\(874\) −11.9171 −0.403101
\(875\) −40.7245 −1.37674
\(876\) −50.4877 −1.70582
\(877\) 22.6618 0.765235 0.382617 0.923907i \(-0.375023\pi\)
0.382617 + 0.923907i \(0.375023\pi\)
\(878\) 21.9731 0.741555
\(879\) 18.9467 0.639056
\(880\) −3.26516 −0.110068
\(881\) 42.1176 1.41898 0.709488 0.704717i \(-0.248926\pi\)
0.709488 + 0.704717i \(0.248926\pi\)
\(882\) 21.7833 0.733481
\(883\) −14.4855 −0.487476 −0.243738 0.969841i \(-0.578374\pi\)
−0.243738 + 0.969841i \(0.578374\pi\)
\(884\) 11.4015 0.383475
\(885\) −8.73803 −0.293726
\(886\) 10.3283 0.346984
\(887\) 51.5845 1.73204 0.866019 0.500010i \(-0.166670\pi\)
0.866019 + 0.500010i \(0.166670\pi\)
\(888\) 37.4463 1.25662
\(889\) −3.96856 −0.133101
\(890\) −9.40847 −0.315373
\(891\) −2.28603 −0.0765850
\(892\) −13.3776 −0.447916
\(893\) −0.736236 −0.0246372
\(894\) 5.41965 0.181260
\(895\) −29.2019 −0.976112
\(896\) 43.7403 1.46126
\(897\) 125.920 4.20433
\(898\) −2.37992 −0.0794190
\(899\) −90.3802 −3.01435
\(900\) 25.3940 0.846465
\(901\) −3.55498 −0.118434
\(902\) −1.25378 −0.0417464
\(903\) −10.5730 −0.351848
\(904\) −27.7700 −0.923617
\(905\) 1.44617 0.0480724
\(906\) −10.7053 −0.355660
\(907\) −44.1493 −1.46595 −0.732977 0.680254i \(-0.761870\pi\)
−0.732977 + 0.680254i \(0.761870\pi\)
\(908\) −18.1730 −0.603093
\(909\) 54.2776 1.80028
\(910\) −20.9107 −0.693183
\(911\) 34.6750 1.14883 0.574417 0.818563i \(-0.305229\pi\)
0.574417 + 0.818563i \(0.305229\pi\)
\(912\) 15.4517 0.511658
\(913\) −7.87955 −0.260775
\(914\) 2.39486 0.0792151
\(915\) 45.0590 1.48961
\(916\) −28.1574 −0.930346
\(917\) −69.0549 −2.28039
\(918\) −2.70268 −0.0892016
\(919\) −33.2580 −1.09708 −0.548539 0.836125i \(-0.684816\pi\)
−0.548539 + 0.836125i \(0.684816\pi\)
\(920\) 18.2759 0.602540
\(921\) −65.0292 −2.14278
\(922\) 4.77401 0.157224
\(923\) 61.9506 2.03913
\(924\) −23.6056 −0.776566
\(925\) −20.7874 −0.683484
\(926\) 2.20432 0.0724384
\(927\) −63.9657 −2.10091
\(928\) 50.4575 1.65635
\(929\) 14.9942 0.491942 0.245971 0.969277i \(-0.420893\pi\)
0.245971 + 0.969277i \(0.420893\pi\)
\(930\) −21.4652 −0.703871
\(931\) −23.0230 −0.754550
\(932\) 31.3297 1.02624
\(933\) −94.8018 −3.10367
\(934\) 17.3335 0.567168
\(935\) −1.73786 −0.0568342
\(936\) 71.8607 2.34884
\(937\) −7.24245 −0.236601 −0.118300 0.992978i \(-0.537745\pi\)
−0.118300 + 0.992978i \(0.537745\pi\)
\(938\) 13.2075 0.431240
\(939\) 70.2814 2.29355
\(940\) 0.505756 0.0164960
\(941\) 13.2309 0.431316 0.215658 0.976469i \(-0.430810\pi\)
0.215658 + 0.976469i \(0.430810\pi\)
\(942\) −9.81793 −0.319886
\(943\) −9.64766 −0.314171
\(944\) −4.71565 −0.153481
\(945\) −21.3219 −0.693602
\(946\) −0.845036 −0.0274745
\(947\) 26.0266 0.845752 0.422876 0.906188i \(-0.361021\pi\)
0.422876 + 0.906188i \(0.361021\pi\)
\(948\) −15.0230 −0.487923
\(949\) 79.3116 2.57456
\(950\) 6.23940 0.202433
\(951\) −66.5285 −2.15734
\(952\) 8.53579 0.276646
\(953\) 29.6109 0.959192 0.479596 0.877489i \(-0.340783\pi\)
0.479596 + 0.877489i \(0.340783\pi\)
\(954\) −10.0364 −0.324941
\(955\) −2.08227 −0.0673805
\(956\) 0.213879 0.00691735
\(957\) −34.1414 −1.10364
\(958\) −18.7545 −0.605931
\(959\) 45.7278 1.47663
\(960\) −1.09853 −0.0354550
\(961\) 69.7681 2.25058
\(962\) −26.3496 −0.849545
\(963\) −3.78196 −0.121872
\(964\) −20.6261 −0.664320
\(965\) −21.2881 −0.685287
\(966\) 42.2267 1.35862
\(967\) 2.67336 0.0859694 0.0429847 0.999076i \(-0.486313\pi\)
0.0429847 + 0.999076i \(0.486313\pi\)
\(968\) 20.2643 0.651320
\(969\) 8.22410 0.264196
\(970\) −0.476333 −0.0152941
\(971\) 34.4447 1.10538 0.552692 0.833386i \(-0.313601\pi\)
0.552692 + 0.833386i \(0.313601\pi\)
\(972\) −28.8533 −0.925470
\(973\) 20.3140 0.651236
\(974\) 8.29842 0.265898
\(975\) −65.9276 −2.11137
\(976\) 24.3170 0.778368
\(977\) −30.1830 −0.965639 −0.482819 0.875720i \(-0.660387\pi\)
−0.482819 + 0.875720i \(0.660387\pi\)
\(978\) 15.1007 0.482866
\(979\) 16.6846 0.533242
\(980\) 15.8156 0.505212
\(981\) 49.7689 1.58900
\(982\) −8.73238 −0.278662
\(983\) 33.7783 1.07736 0.538680 0.842510i \(-0.318923\pi\)
0.538680 + 0.842510i \(0.318923\pi\)
\(984\) −9.09925 −0.290074
\(985\) 7.26106 0.231356
\(986\) 5.52999 0.176111
\(987\) 2.60877 0.0830379
\(988\) −34.0208 −1.08235
\(989\) −6.50242 −0.206765
\(990\) −4.90634 −0.155934
\(991\) −22.7541 −0.722809 −0.361404 0.932409i \(-0.617703\pi\)
−0.361404 + 0.932409i \(0.617703\pi\)
\(992\) −56.2569 −1.78616
\(993\) 75.5322 2.39694
\(994\) 20.7749 0.658941
\(995\) −10.0013 −0.317064
\(996\) −25.6153 −0.811651
\(997\) −0.573910 −0.0181759 −0.00908795 0.999959i \(-0.502893\pi\)
−0.00908795 + 0.999959i \(0.502893\pi\)
\(998\) 0.900947 0.0285190
\(999\) −26.8678 −0.850058
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 731.2.a.e.1.11 19
3.2 odd 2 6579.2.a.t.1.9 19
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
731.2.a.e.1.11 19 1.1 even 1 trivial
6579.2.a.t.1.9 19 3.2 odd 2