Properties

Label 6042.2.a.be.1.5
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
Analytic rank $1$
Dimension $12$
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] = [6042,2,Mod(1,6042)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6042, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("6042.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6042 = 2 \cdot 3 \cdot 19 \cdot 53 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6042.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: \(48.2456129013\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 3 x^{11} - 36 x^{10} + 105 x^{9} + 465 x^{8} - 1287 x^{7} - 2668 x^{6} + 6443 x^{5} + 7140 x^{4} - 10858 x^{3} - 10086 x^{2} + 2072 x + 1496 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.5
Root \(2.45965\) of defining polynomial
Character \(\chi\) \(=\) 6042.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.45965 q^{5} +1.00000 q^{6} -4.70163 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.45965 q^{5} +1.00000 q^{6} -4.70163 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.45965 q^{10} +1.91016 q^{11} -1.00000 q^{12} -5.90772 q^{13} +4.70163 q^{14} +2.45965 q^{15} +1.00000 q^{16} -1.14256 q^{17} -1.00000 q^{18} -1.00000 q^{19} -2.45965 q^{20} +4.70163 q^{21} -1.91016 q^{22} +3.30477 q^{23} +1.00000 q^{24} +1.04987 q^{25} +5.90772 q^{26} -1.00000 q^{27} -4.70163 q^{28} +2.41724 q^{29} -2.45965 q^{30} +7.67209 q^{31} -1.00000 q^{32} -1.91016 q^{33} +1.14256 q^{34} +11.5643 q^{35} +1.00000 q^{36} -4.55685 q^{37} +1.00000 q^{38} +5.90772 q^{39} +2.45965 q^{40} +6.19475 q^{41} -4.70163 q^{42} -5.22642 q^{43} +1.91016 q^{44} -2.45965 q^{45} -3.30477 q^{46} +5.78403 q^{47} -1.00000 q^{48} +15.1053 q^{49} -1.04987 q^{50} +1.14256 q^{51} -5.90772 q^{52} -1.00000 q^{53} +1.00000 q^{54} -4.69832 q^{55} +4.70163 q^{56} +1.00000 q^{57} -2.41724 q^{58} -5.66428 q^{59} +2.45965 q^{60} +13.6683 q^{61} -7.67209 q^{62} -4.70163 q^{63} +1.00000 q^{64} +14.5309 q^{65} +1.91016 q^{66} -3.23769 q^{67} -1.14256 q^{68} -3.30477 q^{69} -11.5643 q^{70} +1.22023 q^{71} -1.00000 q^{72} +7.85807 q^{73} +4.55685 q^{74} -1.04987 q^{75} -1.00000 q^{76} -8.98085 q^{77} -5.90772 q^{78} +15.2607 q^{79} -2.45965 q^{80} +1.00000 q^{81} -6.19475 q^{82} -14.6060 q^{83} +4.70163 q^{84} +2.81030 q^{85} +5.22642 q^{86} -2.41724 q^{87} -1.91016 q^{88} -5.90313 q^{89} +2.45965 q^{90} +27.7759 q^{91} +3.30477 q^{92} -7.67209 q^{93} -5.78403 q^{94} +2.45965 q^{95} +1.00000 q^{96} -0.0999559 q^{97} -15.1053 q^{98} +1.91016 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9} + 3 q^{10} - 7 q^{11} - 12 q^{12} + 4 q^{13} + q^{14} + 3 q^{15} + 12 q^{16} - 4 q^{17} - 12 q^{18} - 12 q^{19} - 3 q^{20} + q^{21} + 7 q^{22} - 18 q^{23} + 12 q^{24} + 21 q^{25} - 4 q^{26} - 12 q^{27} - q^{28} + 10 q^{29} - 3 q^{30} + 14 q^{31} - 12 q^{32} + 7 q^{33} + 4 q^{34} - 19 q^{35} + 12 q^{36} + 14 q^{37} + 12 q^{38} - 4 q^{39} + 3 q^{40} - 9 q^{41} - q^{42} - 4 q^{43} - 7 q^{44} - 3 q^{45} + 18 q^{46} - 14 q^{47} - 12 q^{48} + 37 q^{49} - 21 q^{50} + 4 q^{51} + 4 q^{52} - 12 q^{53} + 12 q^{54} - 4 q^{55} + q^{56} + 12 q^{57} - 10 q^{58} - 18 q^{59} + 3 q^{60} - 7 q^{61} - 14 q^{62} - q^{63} + 12 q^{64} + 3 q^{65} - 7 q^{66} + 8 q^{67} - 4 q^{68} + 18 q^{69} + 19 q^{70} - q^{71} - 12 q^{72} - 31 q^{73} - 14 q^{74} - 21 q^{75} - 12 q^{76} - 25 q^{77} + 4 q^{78} - 3 q^{80} + 12 q^{81} + 9 q^{82} - 48 q^{83} + q^{84} + 4 q^{85} + 4 q^{86} - 10 q^{87} + 7 q^{88} + 13 q^{89} + 3 q^{90} + 9 q^{91} - 18 q^{92} - 14 q^{93} + 14 q^{94} + 3 q^{95} + 12 q^{96} + 25 q^{97} - 37 q^{98} - 7 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) −2.45965 −1.09999 −0.549994 0.835169i \(-0.685370\pi\)
−0.549994 + 0.835169i \(0.685370\pi\)
\(6\) 1.00000 0.408248
\(7\) −4.70163 −1.77705 −0.888524 0.458830i \(-0.848269\pi\)
−0.888524 + 0.458830i \(0.848269\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 2.45965 0.777809
\(11\) 1.91016 0.575934 0.287967 0.957640i \(-0.407021\pi\)
0.287967 + 0.957640i \(0.407021\pi\)
\(12\) −1.00000 −0.288675
\(13\) −5.90772 −1.63851 −0.819254 0.573431i \(-0.805612\pi\)
−0.819254 + 0.573431i \(0.805612\pi\)
\(14\) 4.70163 1.25656
\(15\) 2.45965 0.635079
\(16\) 1.00000 0.250000
\(17\) −1.14256 −0.277112 −0.138556 0.990355i \(-0.544246\pi\)
−0.138556 + 0.990355i \(0.544246\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) −2.45965 −0.549994
\(21\) 4.70163 1.02598
\(22\) −1.91016 −0.407247
\(23\) 3.30477 0.689092 0.344546 0.938769i \(-0.388033\pi\)
0.344546 + 0.938769i \(0.388033\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.04987 0.209974
\(26\) 5.90772 1.15860
\(27\) −1.00000 −0.192450
\(28\) −4.70163 −0.888524
\(29\) 2.41724 0.448869 0.224435 0.974489i \(-0.427946\pi\)
0.224435 + 0.974489i \(0.427946\pi\)
\(30\) −2.45965 −0.449068
\(31\) 7.67209 1.37795 0.688974 0.724786i \(-0.258061\pi\)
0.688974 + 0.724786i \(0.258061\pi\)
\(32\) −1.00000 −0.176777
\(33\) −1.91016 −0.332516
\(34\) 1.14256 0.195948
\(35\) 11.5643 1.95473
\(36\) 1.00000 0.166667
\(37\) −4.55685 −0.749142 −0.374571 0.927198i \(-0.622210\pi\)
−0.374571 + 0.927198i \(0.622210\pi\)
\(38\) 1.00000 0.162221
\(39\) 5.90772 0.945993
\(40\) 2.45965 0.388905
\(41\) 6.19475 0.967457 0.483728 0.875218i \(-0.339282\pi\)
0.483728 + 0.875218i \(0.339282\pi\)
\(42\) −4.70163 −0.725477
\(43\) −5.22642 −0.797022 −0.398511 0.917164i \(-0.630473\pi\)
−0.398511 + 0.917164i \(0.630473\pi\)
\(44\) 1.91016 0.287967
\(45\) −2.45965 −0.366663
\(46\) −3.30477 −0.487262
\(47\) 5.78403 0.843687 0.421844 0.906669i \(-0.361383\pi\)
0.421844 + 0.906669i \(0.361383\pi\)
\(48\) −1.00000 −0.144338
\(49\) 15.1053 2.15790
\(50\) −1.04987 −0.148474
\(51\) 1.14256 0.159991
\(52\) −5.90772 −0.819254
\(53\) −1.00000 −0.137361
\(54\) 1.00000 0.136083
\(55\) −4.69832 −0.633521
\(56\) 4.70163 0.628281
\(57\) 1.00000 0.132453
\(58\) −2.41724 −0.317399
\(59\) −5.66428 −0.737426 −0.368713 0.929543i \(-0.620202\pi\)
−0.368713 + 0.929543i \(0.620202\pi\)
\(60\) 2.45965 0.317539
\(61\) 13.6683 1.75005 0.875026 0.484076i \(-0.160844\pi\)
0.875026 + 0.484076i \(0.160844\pi\)
\(62\) −7.67209 −0.974356
\(63\) −4.70163 −0.592349
\(64\) 1.00000 0.125000
\(65\) 14.5309 1.80234
\(66\) 1.91016 0.235124
\(67\) −3.23769 −0.395546 −0.197773 0.980248i \(-0.563371\pi\)
−0.197773 + 0.980248i \(0.563371\pi\)
\(68\) −1.14256 −0.138556
\(69\) −3.30477 −0.397848
\(70\) −11.5643 −1.38220
\(71\) 1.22023 0.144815 0.0724076 0.997375i \(-0.476932\pi\)
0.0724076 + 0.997375i \(0.476932\pi\)
\(72\) −1.00000 −0.117851
\(73\) 7.85807 0.919718 0.459859 0.887992i \(-0.347900\pi\)
0.459859 + 0.887992i \(0.347900\pi\)
\(74\) 4.55685 0.529723
\(75\) −1.04987 −0.121229
\(76\) −1.00000 −0.114708
\(77\) −8.98085 −1.02346
\(78\) −5.90772 −0.668918
\(79\) 15.2607 1.71697 0.858483 0.512842i \(-0.171407\pi\)
0.858483 + 0.512842i \(0.171407\pi\)
\(80\) −2.45965 −0.274997
\(81\) 1.00000 0.111111
\(82\) −6.19475 −0.684095
\(83\) −14.6060 −1.60322 −0.801611 0.597846i \(-0.796023\pi\)
−0.801611 + 0.597846i \(0.796023\pi\)
\(84\) 4.70163 0.512990
\(85\) 2.81030 0.304820
\(86\) 5.22642 0.563580
\(87\) −2.41724 −0.259155
\(88\) −1.91016 −0.203624
\(89\) −5.90313 −0.625731 −0.312865 0.949797i \(-0.601289\pi\)
−0.312865 + 0.949797i \(0.601289\pi\)
\(90\) 2.45965 0.259270
\(91\) 27.7759 2.91171
\(92\) 3.30477 0.344546
\(93\) −7.67209 −0.795559
\(94\) −5.78403 −0.596577
\(95\) 2.45965 0.252355
\(96\) 1.00000 0.102062
\(97\) −0.0999559 −0.0101490 −0.00507449 0.999987i \(-0.501615\pi\)
−0.00507449 + 0.999987i \(0.501615\pi\)
\(98\) −15.1053 −1.52587
\(99\) 1.91016 0.191978
\(100\) 1.04987 0.104987
\(101\) −17.7871 −1.76988 −0.884942 0.465701i \(-0.845802\pi\)
−0.884942 + 0.465701i \(0.845802\pi\)
\(102\) −1.14256 −0.113131
\(103\) 14.3083 1.40984 0.704918 0.709289i \(-0.250984\pi\)
0.704918 + 0.709289i \(0.250984\pi\)
\(104\) 5.90772 0.579300
\(105\) −11.5643 −1.12856
\(106\) 1.00000 0.0971286
\(107\) 1.86376 0.180177 0.0900883 0.995934i \(-0.471285\pi\)
0.0900883 + 0.995934i \(0.471285\pi\)
\(108\) −1.00000 −0.0962250
\(109\) 5.31128 0.508728 0.254364 0.967108i \(-0.418134\pi\)
0.254364 + 0.967108i \(0.418134\pi\)
\(110\) 4.69832 0.447967
\(111\) 4.55685 0.432517
\(112\) −4.70163 −0.444262
\(113\) −0.179709 −0.0169056 −0.00845278 0.999964i \(-0.502691\pi\)
−0.00845278 + 0.999964i \(0.502691\pi\)
\(114\) −1.00000 −0.0936586
\(115\) −8.12857 −0.757993
\(116\) 2.41724 0.224435
\(117\) −5.90772 −0.546169
\(118\) 5.66428 0.521439
\(119\) 5.37190 0.492442
\(120\) −2.45965 −0.224534
\(121\) −7.35130 −0.668300
\(122\) −13.6683 −1.23747
\(123\) −6.19475 −0.558561
\(124\) 7.67209 0.688974
\(125\) 9.71593 0.869019
\(126\) 4.70163 0.418854
\(127\) 1.73478 0.153937 0.0769683 0.997034i \(-0.475476\pi\)
0.0769683 + 0.997034i \(0.475476\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 5.22642 0.460161
\(130\) −14.5309 −1.27445
\(131\) −13.5180 −1.18108 −0.590539 0.807009i \(-0.701085\pi\)
−0.590539 + 0.807009i \(0.701085\pi\)
\(132\) −1.91016 −0.166258
\(133\) 4.70163 0.407683
\(134\) 3.23769 0.279694
\(135\) 2.45965 0.211693
\(136\) 1.14256 0.0979740
\(137\) 6.46378 0.552238 0.276119 0.961123i \(-0.410952\pi\)
0.276119 + 0.961123i \(0.410952\pi\)
\(138\) 3.30477 0.281321
\(139\) 6.32384 0.536381 0.268191 0.963366i \(-0.413574\pi\)
0.268191 + 0.963366i \(0.413574\pi\)
\(140\) 11.5643 0.977366
\(141\) −5.78403 −0.487103
\(142\) −1.22023 −0.102400
\(143\) −11.2847 −0.943673
\(144\) 1.00000 0.0833333
\(145\) −5.94555 −0.493751
\(146\) −7.85807 −0.650339
\(147\) −15.1053 −1.24586
\(148\) −4.55685 −0.374571
\(149\) 20.9266 1.71437 0.857185 0.515008i \(-0.172211\pi\)
0.857185 + 0.515008i \(0.172211\pi\)
\(150\) 1.04987 0.0857216
\(151\) 17.4133 1.41707 0.708536 0.705674i \(-0.249356\pi\)
0.708536 + 0.705674i \(0.249356\pi\)
\(152\) 1.00000 0.0811107
\(153\) −1.14256 −0.0923707
\(154\) 8.98085 0.723698
\(155\) −18.8706 −1.51573
\(156\) 5.90772 0.472996
\(157\) −13.2749 −1.05945 −0.529724 0.848170i \(-0.677705\pi\)
−0.529724 + 0.848170i \(0.677705\pi\)
\(158\) −15.2607 −1.21408
\(159\) 1.00000 0.0793052
\(160\) 2.45965 0.194452
\(161\) −15.5378 −1.22455
\(162\) −1.00000 −0.0785674
\(163\) 10.5174 0.823786 0.411893 0.911232i \(-0.364868\pi\)
0.411893 + 0.911232i \(0.364868\pi\)
\(164\) 6.19475 0.483728
\(165\) 4.69832 0.365764
\(166\) 14.6060 1.13365
\(167\) −1.01870 −0.0788297 −0.0394148 0.999223i \(-0.512549\pi\)
−0.0394148 + 0.999223i \(0.512549\pi\)
\(168\) −4.70163 −0.362738
\(169\) 21.9012 1.68471
\(170\) −2.81030 −0.215540
\(171\) −1.00000 −0.0764719
\(172\) −5.22642 −0.398511
\(173\) −13.8755 −1.05494 −0.527468 0.849575i \(-0.676859\pi\)
−0.527468 + 0.849575i \(0.676859\pi\)
\(174\) 2.41724 0.183250
\(175\) −4.93610 −0.373134
\(176\) 1.91016 0.143984
\(177\) 5.66428 0.425753
\(178\) 5.90313 0.442458
\(179\) −1.61613 −0.120795 −0.0603977 0.998174i \(-0.519237\pi\)
−0.0603977 + 0.998174i \(0.519237\pi\)
\(180\) −2.45965 −0.183331
\(181\) −13.5282 −1.00554 −0.502770 0.864420i \(-0.667686\pi\)
−0.502770 + 0.864420i \(0.667686\pi\)
\(182\) −27.7759 −2.05889
\(183\) −13.6683 −1.01039
\(184\) −3.30477 −0.243631
\(185\) 11.2083 0.824048
\(186\) 7.67209 0.562545
\(187\) −2.18248 −0.159598
\(188\) 5.78403 0.421844
\(189\) 4.70163 0.341993
\(190\) −2.45965 −0.178442
\(191\) −11.2029 −0.810614 −0.405307 0.914181i \(-0.632835\pi\)
−0.405307 + 0.914181i \(0.632835\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −20.0363 −1.44224 −0.721121 0.692809i \(-0.756373\pi\)
−0.721121 + 0.692809i \(0.756373\pi\)
\(194\) 0.0999559 0.00717641
\(195\) −14.5309 −1.04058
\(196\) 15.1053 1.07895
\(197\) 21.7120 1.54691 0.773456 0.633849i \(-0.218526\pi\)
0.773456 + 0.633849i \(0.218526\pi\)
\(198\) −1.91016 −0.135749
\(199\) 10.6531 0.755176 0.377588 0.925974i \(-0.376754\pi\)
0.377588 + 0.925974i \(0.376754\pi\)
\(200\) −1.04987 −0.0742371
\(201\) 3.23769 0.228369
\(202\) 17.7871 1.25150
\(203\) −11.3649 −0.797662
\(204\) 1.14256 0.0799954
\(205\) −15.2369 −1.06419
\(206\) −14.3083 −0.996905
\(207\) 3.30477 0.229697
\(208\) −5.90772 −0.409627
\(209\) −1.91016 −0.132128
\(210\) 11.5643 0.798016
\(211\) −4.80428 −0.330740 −0.165370 0.986232i \(-0.552882\pi\)
−0.165370 + 0.986232i \(0.552882\pi\)
\(212\) −1.00000 −0.0686803
\(213\) −1.22023 −0.0836091
\(214\) −1.86376 −0.127404
\(215\) 12.8552 0.876715
\(216\) 1.00000 0.0680414
\(217\) −36.0713 −2.44868
\(218\) −5.31128 −0.359725
\(219\) −7.85807 −0.530999
\(220\) −4.69832 −0.316760
\(221\) 6.74995 0.454051
\(222\) −4.55685 −0.305836
\(223\) 10.6493 0.713128 0.356564 0.934271i \(-0.383948\pi\)
0.356564 + 0.934271i \(0.383948\pi\)
\(224\) 4.70163 0.314141
\(225\) 1.04987 0.0699914
\(226\) 0.179709 0.0119540
\(227\) 6.99635 0.464364 0.232182 0.972672i \(-0.425414\pi\)
0.232182 + 0.972672i \(0.425414\pi\)
\(228\) 1.00000 0.0662266
\(229\) −14.2907 −0.944354 −0.472177 0.881504i \(-0.656532\pi\)
−0.472177 + 0.881504i \(0.656532\pi\)
\(230\) 8.12857 0.535982
\(231\) 8.98085 0.590897
\(232\) −2.41724 −0.158699
\(233\) −0.544364 −0.0356624 −0.0178312 0.999841i \(-0.505676\pi\)
−0.0178312 + 0.999841i \(0.505676\pi\)
\(234\) 5.90772 0.386200
\(235\) −14.2267 −0.928046
\(236\) −5.66428 −0.368713
\(237\) −15.2607 −0.991291
\(238\) −5.37190 −0.348209
\(239\) −7.02584 −0.454464 −0.227232 0.973841i \(-0.572968\pi\)
−0.227232 + 0.973841i \(0.572968\pi\)
\(240\) 2.45965 0.158770
\(241\) −3.00462 −0.193545 −0.0967723 0.995307i \(-0.530852\pi\)
−0.0967723 + 0.995307i \(0.530852\pi\)
\(242\) 7.35130 0.472559
\(243\) −1.00000 −0.0641500
\(244\) 13.6683 0.875026
\(245\) −37.1537 −2.37366
\(246\) 6.19475 0.394963
\(247\) 5.90772 0.375899
\(248\) −7.67209 −0.487178
\(249\) 14.6060 0.925620
\(250\) −9.71593 −0.614489
\(251\) −17.3184 −1.09313 −0.546563 0.837418i \(-0.684064\pi\)
−0.546563 + 0.837418i \(0.684064\pi\)
\(252\) −4.70163 −0.296175
\(253\) 6.31263 0.396872
\(254\) −1.73478 −0.108850
\(255\) −2.81030 −0.175988
\(256\) 1.00000 0.0625000
\(257\) 9.24806 0.576878 0.288439 0.957498i \(-0.406864\pi\)
0.288439 + 0.957498i \(0.406864\pi\)
\(258\) −5.22642 −0.325383
\(259\) 21.4246 1.33126
\(260\) 14.5309 0.901170
\(261\) 2.41724 0.149623
\(262\) 13.5180 0.835148
\(263\) 5.43686 0.335251 0.167626 0.985851i \(-0.446390\pi\)
0.167626 + 0.985851i \(0.446390\pi\)
\(264\) 1.91016 0.117562
\(265\) 2.45965 0.151095
\(266\) −4.70163 −0.288275
\(267\) 5.90313 0.361266
\(268\) −3.23769 −0.197773
\(269\) −22.6746 −1.38250 −0.691248 0.722618i \(-0.742939\pi\)
−0.691248 + 0.722618i \(0.742939\pi\)
\(270\) −2.45965 −0.149689
\(271\) −12.4883 −0.758608 −0.379304 0.925272i \(-0.623837\pi\)
−0.379304 + 0.925272i \(0.623837\pi\)
\(272\) −1.14256 −0.0692781
\(273\) −27.7759 −1.68107
\(274\) −6.46378 −0.390492
\(275\) 2.00542 0.120931
\(276\) −3.30477 −0.198924
\(277\) 6.80678 0.408980 0.204490 0.978869i \(-0.434446\pi\)
0.204490 + 0.978869i \(0.434446\pi\)
\(278\) −6.32384 −0.379279
\(279\) 7.67209 0.459316
\(280\) −11.5643 −0.691102
\(281\) −22.1604 −1.32198 −0.660989 0.750396i \(-0.729863\pi\)
−0.660989 + 0.750396i \(0.729863\pi\)
\(282\) 5.78403 0.344434
\(283\) −2.66711 −0.158543 −0.0792717 0.996853i \(-0.525259\pi\)
−0.0792717 + 0.996853i \(0.525259\pi\)
\(284\) 1.22023 0.0724076
\(285\) −2.45965 −0.145697
\(286\) 11.2847 0.667277
\(287\) −29.1254 −1.71922
\(288\) −1.00000 −0.0589256
\(289\) −15.6945 −0.923209
\(290\) 5.94555 0.349135
\(291\) 0.0999559 0.00585952
\(292\) 7.85807 0.459859
\(293\) 13.2532 0.774261 0.387130 0.922025i \(-0.373466\pi\)
0.387130 + 0.922025i \(0.373466\pi\)
\(294\) 15.1053 0.880959
\(295\) 13.9321 0.811160
\(296\) 4.55685 0.264862
\(297\) −1.91016 −0.110839
\(298\) −20.9266 −1.21224
\(299\) −19.5237 −1.12908
\(300\) −1.04987 −0.0606143
\(301\) 24.5727 1.41635
\(302\) −17.4133 −1.00202
\(303\) 17.7871 1.02184
\(304\) −1.00000 −0.0573539
\(305\) −33.6193 −1.92504
\(306\) 1.14256 0.0653160
\(307\) −3.64574 −0.208073 −0.104037 0.994573i \(-0.533176\pi\)
−0.104037 + 0.994573i \(0.533176\pi\)
\(308\) −8.98085 −0.511731
\(309\) −14.3083 −0.813969
\(310\) 18.8706 1.07178
\(311\) −31.7037 −1.79775 −0.898877 0.438202i \(-0.855615\pi\)
−0.898877 + 0.438202i \(0.855615\pi\)
\(312\) −5.90772 −0.334459
\(313\) 22.2063 1.25518 0.627588 0.778546i \(-0.284042\pi\)
0.627588 + 0.778546i \(0.284042\pi\)
\(314\) 13.2749 0.749143
\(315\) 11.5643 0.651577
\(316\) 15.2607 0.858483
\(317\) 28.9210 1.62437 0.812184 0.583402i \(-0.198279\pi\)
0.812184 + 0.583402i \(0.198279\pi\)
\(318\) −1.00000 −0.0560772
\(319\) 4.61730 0.258519
\(320\) −2.45965 −0.137499
\(321\) −1.86376 −0.104025
\(322\) 15.5378 0.865887
\(323\) 1.14256 0.0635739
\(324\) 1.00000 0.0555556
\(325\) −6.20235 −0.344044
\(326\) −10.5174 −0.582505
\(327\) −5.31128 −0.293715
\(328\) −6.19475 −0.342048
\(329\) −27.1943 −1.49927
\(330\) −4.69832 −0.258634
\(331\) −18.6818 −1.02685 −0.513423 0.858136i \(-0.671623\pi\)
−0.513423 + 0.858136i \(0.671623\pi\)
\(332\) −14.6060 −0.801611
\(333\) −4.55685 −0.249714
\(334\) 1.01870 0.0557410
\(335\) 7.96357 0.435097
\(336\) 4.70163 0.256495
\(337\) 1.11190 0.0605689 0.0302845 0.999541i \(-0.490359\pi\)
0.0302845 + 0.999541i \(0.490359\pi\)
\(338\) −21.9012 −1.19127
\(339\) 0.179709 0.00976043
\(340\) 2.81030 0.152410
\(341\) 14.6549 0.793608
\(342\) 1.00000 0.0540738
\(343\) −38.1081 −2.05764
\(344\) 5.22642 0.281790
\(345\) 8.12857 0.437628
\(346\) 13.8755 0.745953
\(347\) −13.0010 −0.697928 −0.348964 0.937136i \(-0.613466\pi\)
−0.348964 + 0.937136i \(0.613466\pi\)
\(348\) −2.41724 −0.129577
\(349\) 10.6628 0.570769 0.285385 0.958413i \(-0.407879\pi\)
0.285385 + 0.958413i \(0.407879\pi\)
\(350\) 4.93610 0.263846
\(351\) 5.90772 0.315331
\(352\) −1.91016 −0.101812
\(353\) 17.9807 0.957015 0.478507 0.878084i \(-0.341178\pi\)
0.478507 + 0.878084i \(0.341178\pi\)
\(354\) −5.66428 −0.301053
\(355\) −3.00135 −0.159295
\(356\) −5.90313 −0.312865
\(357\) −5.37190 −0.284311
\(358\) 1.61613 0.0854152
\(359\) −25.5705 −1.34956 −0.674781 0.738018i \(-0.735762\pi\)
−0.674781 + 0.738018i \(0.735762\pi\)
\(360\) 2.45965 0.129635
\(361\) 1.00000 0.0526316
\(362\) 13.5282 0.711024
\(363\) 7.35130 0.385843
\(364\) 27.7759 1.45585
\(365\) −19.3281 −1.01168
\(366\) 13.6683 0.714456
\(367\) −4.42367 −0.230914 −0.115457 0.993312i \(-0.536833\pi\)
−0.115457 + 0.993312i \(0.536833\pi\)
\(368\) 3.30477 0.172273
\(369\) 6.19475 0.322486
\(370\) −11.2083 −0.582690
\(371\) 4.70163 0.244096
\(372\) −7.67209 −0.397779
\(373\) 17.4917 0.905686 0.452843 0.891590i \(-0.350410\pi\)
0.452843 + 0.891590i \(0.350410\pi\)
\(374\) 2.18248 0.112853
\(375\) −9.71593 −0.501728
\(376\) −5.78403 −0.298288
\(377\) −14.2804 −0.735476
\(378\) −4.70163 −0.241826
\(379\) 15.5294 0.797689 0.398845 0.917019i \(-0.369411\pi\)
0.398845 + 0.917019i \(0.369411\pi\)
\(380\) 2.45965 0.126177
\(381\) −1.73478 −0.0888753
\(382\) 11.2029 0.573190
\(383\) −23.9671 −1.22466 −0.612331 0.790601i \(-0.709768\pi\)
−0.612331 + 0.790601i \(0.709768\pi\)
\(384\) 1.00000 0.0510310
\(385\) 22.0897 1.12580
\(386\) 20.0363 1.01982
\(387\) −5.22642 −0.265674
\(388\) −0.0999559 −0.00507449
\(389\) 30.1549 1.52892 0.764458 0.644673i \(-0.223007\pi\)
0.764458 + 0.644673i \(0.223007\pi\)
\(390\) 14.5309 0.735802
\(391\) −3.77591 −0.190956
\(392\) −15.1053 −0.762933
\(393\) 13.5180 0.681895
\(394\) −21.7120 −1.09383
\(395\) −37.5360 −1.88864
\(396\) 1.91016 0.0959891
\(397\) 26.0606 1.30795 0.653973 0.756518i \(-0.273101\pi\)
0.653973 + 0.756518i \(0.273101\pi\)
\(398\) −10.6531 −0.533990
\(399\) −4.70163 −0.235376
\(400\) 1.04987 0.0524936
\(401\) 24.5456 1.22575 0.612876 0.790179i \(-0.290013\pi\)
0.612876 + 0.790179i \(0.290013\pi\)
\(402\) −3.23769 −0.161481
\(403\) −45.3246 −2.25778
\(404\) −17.7871 −0.884942
\(405\) −2.45965 −0.122221
\(406\) 11.3649 0.564033
\(407\) −8.70431 −0.431457
\(408\) −1.14256 −0.0565653
\(409\) −24.8079 −1.22667 −0.613335 0.789823i \(-0.710172\pi\)
−0.613335 + 0.789823i \(0.710172\pi\)
\(410\) 15.2369 0.752497
\(411\) −6.46378 −0.318835
\(412\) 14.3083 0.704918
\(413\) 26.6313 1.31044
\(414\) −3.30477 −0.162421
\(415\) 35.9257 1.76352
\(416\) 5.90772 0.289650
\(417\) −6.32384 −0.309680
\(418\) 1.91016 0.0934289
\(419\) 3.30387 0.161405 0.0807023 0.996738i \(-0.474284\pi\)
0.0807023 + 0.996738i \(0.474284\pi\)
\(420\) −11.5643 −0.564282
\(421\) −17.0365 −0.830306 −0.415153 0.909752i \(-0.636272\pi\)
−0.415153 + 0.909752i \(0.636272\pi\)
\(422\) 4.80428 0.233869
\(423\) 5.78403 0.281229
\(424\) 1.00000 0.0485643
\(425\) −1.19954 −0.0581864
\(426\) 1.22023 0.0591206
\(427\) −64.2635 −3.10993
\(428\) 1.86376 0.0900883
\(429\) 11.2847 0.544830
\(430\) −12.8552 −0.619931
\(431\) 14.8752 0.716512 0.358256 0.933623i \(-0.383371\pi\)
0.358256 + 0.933623i \(0.383371\pi\)
\(432\) −1.00000 −0.0481125
\(433\) 9.55069 0.458977 0.229488 0.973311i \(-0.426295\pi\)
0.229488 + 0.973311i \(0.426295\pi\)
\(434\) 36.0713 1.73148
\(435\) 5.94555 0.285067
\(436\) 5.31128 0.254364
\(437\) −3.30477 −0.158089
\(438\) 7.85807 0.375473
\(439\) −15.0945 −0.720422 −0.360211 0.932871i \(-0.617295\pi\)
−0.360211 + 0.932871i \(0.617295\pi\)
\(440\) 4.69832 0.223983
\(441\) 15.1053 0.719300
\(442\) −6.74995 −0.321062
\(443\) 10.8991 0.517830 0.258915 0.965900i \(-0.416635\pi\)
0.258915 + 0.965900i \(0.416635\pi\)
\(444\) 4.55685 0.216259
\(445\) 14.5196 0.688296
\(446\) −10.6493 −0.504258
\(447\) −20.9266 −0.989792
\(448\) −4.70163 −0.222131
\(449\) 13.4361 0.634086 0.317043 0.948411i \(-0.397310\pi\)
0.317043 + 0.948411i \(0.397310\pi\)
\(450\) −1.04987 −0.0494914
\(451\) 11.8329 0.557192
\(452\) −0.179709 −0.00845278
\(453\) −17.4133 −0.818147
\(454\) −6.99635 −0.328355
\(455\) −68.3190 −3.20284
\(456\) −1.00000 −0.0468293
\(457\) 10.4744 0.489973 0.244987 0.969526i \(-0.421216\pi\)
0.244987 + 0.969526i \(0.421216\pi\)
\(458\) 14.2907 0.667759
\(459\) 1.14256 0.0533303
\(460\) −8.12857 −0.378997
\(461\) −11.0887 −0.516452 −0.258226 0.966085i \(-0.583138\pi\)
−0.258226 + 0.966085i \(0.583138\pi\)
\(462\) −8.98085 −0.417827
\(463\) 0.621532 0.0288851 0.0144425 0.999896i \(-0.495403\pi\)
0.0144425 + 0.999896i \(0.495403\pi\)
\(464\) 2.41724 0.112217
\(465\) 18.8706 0.875105
\(466\) 0.544364 0.0252172
\(467\) −30.9060 −1.43016 −0.715080 0.699043i \(-0.753610\pi\)
−0.715080 + 0.699043i \(0.753610\pi\)
\(468\) −5.90772 −0.273085
\(469\) 15.2224 0.702905
\(470\) 14.2267 0.656228
\(471\) 13.2749 0.611673
\(472\) 5.66428 0.260720
\(473\) −9.98330 −0.459032
\(474\) 15.2607 0.700949
\(475\) −1.04987 −0.0481714
\(476\) 5.37190 0.246221
\(477\) −1.00000 −0.0457869
\(478\) 7.02584 0.321355
\(479\) 23.7155 1.08359 0.541795 0.840510i \(-0.317745\pi\)
0.541795 + 0.840510i \(0.317745\pi\)
\(480\) −2.45965 −0.112267
\(481\) 26.9206 1.22748
\(482\) 3.00462 0.136857
\(483\) 15.5378 0.706994
\(484\) −7.35130 −0.334150
\(485\) 0.245856 0.0111638
\(486\) 1.00000 0.0453609
\(487\) −9.42158 −0.426932 −0.213466 0.976950i \(-0.568475\pi\)
−0.213466 + 0.976950i \(0.568475\pi\)
\(488\) −13.6683 −0.618737
\(489\) −10.5174 −0.475613
\(490\) 37.1537 1.67843
\(491\) −13.0936 −0.590906 −0.295453 0.955357i \(-0.595471\pi\)
−0.295453 + 0.955357i \(0.595471\pi\)
\(492\) −6.19475 −0.279281
\(493\) −2.76184 −0.124387
\(494\) −5.90772 −0.265801
\(495\) −4.69832 −0.211174
\(496\) 7.67209 0.344487
\(497\) −5.73709 −0.257344
\(498\) −14.6060 −0.654512
\(499\) −9.16202 −0.410149 −0.205074 0.978746i \(-0.565744\pi\)
−0.205074 + 0.978746i \(0.565744\pi\)
\(500\) 9.71593 0.434510
\(501\) 1.01870 0.0455123
\(502\) 17.3184 0.772957
\(503\) 15.3316 0.683602 0.341801 0.939772i \(-0.388963\pi\)
0.341801 + 0.939772i \(0.388963\pi\)
\(504\) 4.70163 0.209427
\(505\) 43.7501 1.94685
\(506\) −6.31263 −0.280631
\(507\) −21.9012 −0.972666
\(508\) 1.73478 0.0769683
\(509\) −29.0500 −1.28762 −0.643809 0.765187i \(-0.722647\pi\)
−0.643809 + 0.765187i \(0.722647\pi\)
\(510\) 2.81030 0.124442
\(511\) −36.9457 −1.63438
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) −9.24806 −0.407915
\(515\) −35.1933 −1.55080
\(516\) 5.22642 0.230080
\(517\) 11.0484 0.485908
\(518\) −21.4246 −0.941344
\(519\) 13.8755 0.609068
\(520\) −14.5309 −0.637223
\(521\) −0.299447 −0.0131190 −0.00655951 0.999978i \(-0.502088\pi\)
−0.00655951 + 0.999978i \(0.502088\pi\)
\(522\) −2.41724 −0.105800
\(523\) 34.4182 1.50500 0.752502 0.658590i \(-0.228847\pi\)
0.752502 + 0.658590i \(0.228847\pi\)
\(524\) −13.5180 −0.590539
\(525\) 4.93610 0.215429
\(526\) −5.43686 −0.237058
\(527\) −8.76585 −0.381846
\(528\) −1.91016 −0.0831290
\(529\) −12.0785 −0.525152
\(530\) −2.45965 −0.106840
\(531\) −5.66428 −0.245809
\(532\) 4.70163 0.203841
\(533\) −36.5968 −1.58519
\(534\) −5.90313 −0.255453
\(535\) −4.58420 −0.198192
\(536\) 3.23769 0.139847
\(537\) 1.61613 0.0697412
\(538\) 22.6746 0.977572
\(539\) 28.8535 1.24281
\(540\) 2.45965 0.105846
\(541\) 10.7819 0.463549 0.231775 0.972770i \(-0.425547\pi\)
0.231775 + 0.972770i \(0.425547\pi\)
\(542\) 12.4883 0.536417
\(543\) 13.5282 0.580549
\(544\) 1.14256 0.0489870
\(545\) −13.0639 −0.559595
\(546\) 27.7759 1.18870
\(547\) 13.5372 0.578807 0.289403 0.957207i \(-0.406543\pi\)
0.289403 + 0.957207i \(0.406543\pi\)
\(548\) 6.46378 0.276119
\(549\) 13.6683 0.583351
\(550\) −2.00542 −0.0855114
\(551\) −2.41724 −0.102978
\(552\) 3.30477 0.140660
\(553\) −71.7503 −3.05113
\(554\) −6.80678 −0.289193
\(555\) −11.2083 −0.475764
\(556\) 6.32384 0.268191
\(557\) −6.94401 −0.294227 −0.147114 0.989120i \(-0.546998\pi\)
−0.147114 + 0.989120i \(0.546998\pi\)
\(558\) −7.67209 −0.324785
\(559\) 30.8763 1.30593
\(560\) 11.5643 0.488683
\(561\) 2.18248 0.0921442
\(562\) 22.1604 0.934779
\(563\) −15.1283 −0.637580 −0.318790 0.947825i \(-0.603276\pi\)
−0.318790 + 0.947825i \(0.603276\pi\)
\(564\) −5.78403 −0.243552
\(565\) 0.442020 0.0185959
\(566\) 2.66711 0.112107
\(567\) −4.70163 −0.197450
\(568\) −1.22023 −0.0511999
\(569\) −29.6326 −1.24226 −0.621132 0.783706i \(-0.713327\pi\)
−0.621132 + 0.783706i \(0.713327\pi\)
\(570\) 2.45965 0.103023
\(571\) −32.1379 −1.34493 −0.672465 0.740129i \(-0.734764\pi\)
−0.672465 + 0.740129i \(0.734764\pi\)
\(572\) −11.2847 −0.471836
\(573\) 11.2029 0.468008
\(574\) 29.1254 1.21567
\(575\) 3.46958 0.144692
\(576\) 1.00000 0.0416667
\(577\) 13.0365 0.542718 0.271359 0.962478i \(-0.412527\pi\)
0.271359 + 0.962478i \(0.412527\pi\)
\(578\) 15.6945 0.652807
\(579\) 20.0363 0.832679
\(580\) −5.94555 −0.246876
\(581\) 68.6722 2.84900
\(582\) −0.0999559 −0.00414330
\(583\) −1.91016 −0.0791107
\(584\) −7.85807 −0.325169
\(585\) 14.5309 0.600780
\(586\) −13.2532 −0.547485
\(587\) −2.89693 −0.119569 −0.0597846 0.998211i \(-0.519041\pi\)
−0.0597846 + 0.998211i \(0.519041\pi\)
\(588\) −15.1053 −0.622932
\(589\) −7.67209 −0.316123
\(590\) −13.9321 −0.573577
\(591\) −21.7120 −0.893111
\(592\) −4.55685 −0.187286
\(593\) 14.7541 0.605879 0.302939 0.953010i \(-0.402032\pi\)
0.302939 + 0.953010i \(0.402032\pi\)
\(594\) 1.91016 0.0783747
\(595\) −13.2130 −0.541680
\(596\) 20.9266 0.857185
\(597\) −10.6531 −0.436001
\(598\) 19.5237 0.798382
\(599\) 13.3995 0.547488 0.273744 0.961803i \(-0.411738\pi\)
0.273744 + 0.961803i \(0.411738\pi\)
\(600\) 1.04987 0.0428608
\(601\) 38.9454 1.58862 0.794309 0.607514i \(-0.207833\pi\)
0.794309 + 0.607514i \(0.207833\pi\)
\(602\) −24.5727 −1.00151
\(603\) −3.23769 −0.131849
\(604\) 17.4133 0.708536
\(605\) 18.0816 0.735122
\(606\) −17.7871 −0.722552
\(607\) −38.8800 −1.57809 −0.789045 0.614336i \(-0.789424\pi\)
−0.789045 + 0.614336i \(0.789424\pi\)
\(608\) 1.00000 0.0405554
\(609\) 11.3649 0.460531
\(610\) 33.6193 1.36121
\(611\) −34.1704 −1.38239
\(612\) −1.14256 −0.0461854
\(613\) 1.59254 0.0643220 0.0321610 0.999483i \(-0.489761\pi\)
0.0321610 + 0.999483i \(0.489761\pi\)
\(614\) 3.64574 0.147130
\(615\) 15.2369 0.614411
\(616\) 8.98085 0.361849
\(617\) 26.2045 1.05495 0.527476 0.849570i \(-0.323138\pi\)
0.527476 + 0.849570i \(0.323138\pi\)
\(618\) 14.3083 0.575563
\(619\) 25.5487 1.02689 0.513445 0.858122i \(-0.328369\pi\)
0.513445 + 0.858122i \(0.328369\pi\)
\(620\) −18.8706 −0.757863
\(621\) −3.30477 −0.132616
\(622\) 31.7037 1.27120
\(623\) 27.7543 1.11195
\(624\) 5.90772 0.236498
\(625\) −29.1471 −1.16589
\(626\) −22.2063 −0.887543
\(627\) 1.91016 0.0762844
\(628\) −13.2749 −0.529724
\(629\) 5.20649 0.207596
\(630\) −11.5643 −0.460735
\(631\) −12.1464 −0.483540 −0.241770 0.970334i \(-0.577728\pi\)
−0.241770 + 0.970334i \(0.577728\pi\)
\(632\) −15.2607 −0.607039
\(633\) 4.80428 0.190953
\(634\) −28.9210 −1.14860
\(635\) −4.26694 −0.169328
\(636\) 1.00000 0.0396526
\(637\) −89.2379 −3.53573
\(638\) −4.61730 −0.182801
\(639\) 1.22023 0.0482717
\(640\) 2.45965 0.0972261
\(641\) −14.4032 −0.568892 −0.284446 0.958692i \(-0.591810\pi\)
−0.284446 + 0.958692i \(0.591810\pi\)
\(642\) 1.86376 0.0735568
\(643\) −20.6470 −0.814239 −0.407120 0.913375i \(-0.633467\pi\)
−0.407120 + 0.913375i \(0.633467\pi\)
\(644\) −15.5378 −0.612275
\(645\) −12.8552 −0.506172
\(646\) −1.14256 −0.0449535
\(647\) −13.2938 −0.522635 −0.261318 0.965253i \(-0.584157\pi\)
−0.261318 + 0.965253i \(0.584157\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −10.8197 −0.424709
\(650\) 6.20235 0.243276
\(651\) 36.0713 1.41375
\(652\) 10.5174 0.411893
\(653\) −15.9776 −0.625253 −0.312626 0.949876i \(-0.601209\pi\)
−0.312626 + 0.949876i \(0.601209\pi\)
\(654\) 5.31128 0.207688
\(655\) 33.2497 1.29917
\(656\) 6.19475 0.241864
\(657\) 7.85807 0.306573
\(658\) 27.1943 1.06015
\(659\) −41.8008 −1.62833 −0.814164 0.580634i \(-0.802805\pi\)
−0.814164 + 0.580634i \(0.802805\pi\)
\(660\) 4.69832 0.182882
\(661\) 10.6457 0.414068 0.207034 0.978334i \(-0.433619\pi\)
0.207034 + 0.978334i \(0.433619\pi\)
\(662\) 18.6818 0.726089
\(663\) −6.74995 −0.262146
\(664\) 14.6060 0.566824
\(665\) −11.5643 −0.448446
\(666\) 4.55685 0.176574
\(667\) 7.98841 0.309312
\(668\) −1.01870 −0.0394148
\(669\) −10.6493 −0.411725
\(670\) −7.96357 −0.307660
\(671\) 26.1087 1.00792
\(672\) −4.70163 −0.181369
\(673\) −33.7745 −1.30191 −0.650955 0.759116i \(-0.725631\pi\)
−0.650955 + 0.759116i \(0.725631\pi\)
\(674\) −1.11190 −0.0428287
\(675\) −1.04987 −0.0404096
\(676\) 21.9012 0.842354
\(677\) −38.4563 −1.47800 −0.738999 0.673707i \(-0.764701\pi\)
−0.738999 + 0.673707i \(0.764701\pi\)
\(678\) −0.179709 −0.00690166
\(679\) 0.469955 0.0180352
\(680\) −2.81030 −0.107770
\(681\) −6.99635 −0.268101
\(682\) −14.6549 −0.561165
\(683\) 45.5455 1.74275 0.871375 0.490617i \(-0.163228\pi\)
0.871375 + 0.490617i \(0.163228\pi\)
\(684\) −1.00000 −0.0382360
\(685\) −15.8986 −0.607456
\(686\) 38.1081 1.45497
\(687\) 14.2907 0.545223
\(688\) −5.22642 −0.199256
\(689\) 5.90772 0.225066
\(690\) −8.12857 −0.309449
\(691\) −40.3712 −1.53579 −0.767897 0.640574i \(-0.778697\pi\)
−0.767897 + 0.640574i \(0.778697\pi\)
\(692\) −13.8755 −0.527468
\(693\) −8.98085 −0.341154
\(694\) 13.0010 0.493509
\(695\) −15.5544 −0.590013
\(696\) 2.41724 0.0916251
\(697\) −7.07789 −0.268094
\(698\) −10.6628 −0.403595
\(699\) 0.544364 0.0205897
\(700\) −4.93610 −0.186567
\(701\) 7.09800 0.268088 0.134044 0.990975i \(-0.457204\pi\)
0.134044 + 0.990975i \(0.457204\pi\)
\(702\) −5.90772 −0.222973
\(703\) 4.55685 0.171865
\(704\) 1.91016 0.0719918
\(705\) 14.2267 0.535808
\(706\) −17.9807 −0.676712
\(707\) 83.6284 3.14517
\(708\) 5.66428 0.212877
\(709\) −0.475003 −0.0178391 −0.00891956 0.999960i \(-0.502839\pi\)
−0.00891956 + 0.999960i \(0.502839\pi\)
\(710\) 3.00135 0.112639
\(711\) 15.2607 0.572322
\(712\) 5.90313 0.221229
\(713\) 25.3545 0.949533
\(714\) 5.37190 0.201038
\(715\) 27.7564 1.03803
\(716\) −1.61613 −0.0603977
\(717\) 7.02584 0.262385
\(718\) 25.5705 0.954284
\(719\) −31.9718 −1.19235 −0.596173 0.802856i \(-0.703313\pi\)
−0.596173 + 0.802856i \(0.703313\pi\)
\(720\) −2.45965 −0.0916657
\(721\) −67.2722 −2.50535
\(722\) −1.00000 −0.0372161
\(723\) 3.00462 0.111743
\(724\) −13.5282 −0.502770
\(725\) 2.53779 0.0942510
\(726\) −7.35130 −0.272832
\(727\) −32.2269 −1.19523 −0.597615 0.801783i \(-0.703885\pi\)
−0.597615 + 0.801783i \(0.703885\pi\)
\(728\) −27.7759 −1.02944
\(729\) 1.00000 0.0370370
\(730\) 19.3281 0.715365
\(731\) 5.97152 0.220865
\(732\) −13.6683 −0.505197
\(733\) −5.96157 −0.220196 −0.110098 0.993921i \(-0.535116\pi\)
−0.110098 + 0.993921i \(0.535116\pi\)
\(734\) 4.42367 0.163281
\(735\) 37.1537 1.37044
\(736\) −3.30477 −0.121815
\(737\) −6.18449 −0.227809
\(738\) −6.19475 −0.228032
\(739\) 10.8385 0.398701 0.199350 0.979928i \(-0.436117\pi\)
0.199350 + 0.979928i \(0.436117\pi\)
\(740\) 11.2083 0.412024
\(741\) −5.90772 −0.217026
\(742\) −4.70163 −0.172602
\(743\) 7.57622 0.277945 0.138972 0.990296i \(-0.455620\pi\)
0.138972 + 0.990296i \(0.455620\pi\)
\(744\) 7.67209 0.281272
\(745\) −51.4720 −1.88579
\(746\) −17.4917 −0.640416
\(747\) −14.6060 −0.534407
\(748\) −2.18248 −0.0797992
\(749\) −8.76271 −0.320182
\(750\) 9.71593 0.354776
\(751\) −46.7407 −1.70559 −0.852795 0.522245i \(-0.825095\pi\)
−0.852795 + 0.522245i \(0.825095\pi\)
\(752\) 5.78403 0.210922
\(753\) 17.3184 0.631116
\(754\) 14.2804 0.520060
\(755\) −42.8306 −1.55876
\(756\) 4.70163 0.170997
\(757\) 22.7514 0.826913 0.413457 0.910524i \(-0.364321\pi\)
0.413457 + 0.910524i \(0.364321\pi\)
\(758\) −15.5294 −0.564052
\(759\) −6.31263 −0.229134
\(760\) −2.45965 −0.0892208
\(761\) −24.6727 −0.894384 −0.447192 0.894438i \(-0.647576\pi\)
−0.447192 + 0.894438i \(0.647576\pi\)
\(762\) 1.73478 0.0628444
\(763\) −24.9717 −0.904035
\(764\) −11.2029 −0.405307
\(765\) 2.81030 0.101607
\(766\) 23.9671 0.865967
\(767\) 33.4630 1.20828
\(768\) −1.00000 −0.0360844
\(769\) 15.8286 0.570793 0.285396 0.958410i \(-0.407875\pi\)
0.285396 + 0.958410i \(0.407875\pi\)
\(770\) −22.0897 −0.796059
\(771\) −9.24806 −0.333061
\(772\) −20.0363 −0.721121
\(773\) 20.6158 0.741498 0.370749 0.928733i \(-0.379101\pi\)
0.370749 + 0.928733i \(0.379101\pi\)
\(774\) 5.22642 0.187860
\(775\) 8.05471 0.289334
\(776\) 0.0999559 0.00358821
\(777\) −21.4246 −0.768604
\(778\) −30.1549 −1.08111
\(779\) −6.19475 −0.221950
\(780\) −14.5309 −0.520291
\(781\) 2.33084 0.0834041
\(782\) 3.77591 0.135026
\(783\) −2.41724 −0.0863850
\(784\) 15.1053 0.539475
\(785\) 32.6515 1.16538
\(786\) −13.5180 −0.482173
\(787\) 13.7562 0.490355 0.245178 0.969478i \(-0.421154\pi\)
0.245178 + 0.969478i \(0.421154\pi\)
\(788\) 21.7120 0.773456
\(789\) −5.43686 −0.193557
\(790\) 37.5360 1.33547
\(791\) 0.844922 0.0300420
\(792\) −1.91016 −0.0678745
\(793\) −80.7488 −2.86747
\(794\) −26.0606 −0.924857
\(795\) −2.45965 −0.0872347
\(796\) 10.6531 0.377588
\(797\) 7.10558 0.251693 0.125846 0.992050i \(-0.459835\pi\)
0.125846 + 0.992050i \(0.459835\pi\)
\(798\) 4.70163 0.166436
\(799\) −6.60862 −0.233796
\(800\) −1.04987 −0.0371186
\(801\) −5.90313 −0.208577
\(802\) −24.5456 −0.866737
\(803\) 15.0102 0.529697
\(804\) 3.23769 0.114184
\(805\) 38.2175 1.34699
\(806\) 45.3246 1.59649
\(807\) 22.6746 0.798184
\(808\) 17.7871 0.625749
\(809\) −16.1001 −0.566049 −0.283025 0.959113i \(-0.591338\pi\)
−0.283025 + 0.959113i \(0.591338\pi\)
\(810\) 2.45965 0.0864232
\(811\) 38.9957 1.36932 0.684662 0.728860i \(-0.259950\pi\)
0.684662 + 0.728860i \(0.259950\pi\)
\(812\) −11.3649 −0.398831
\(813\) 12.4883 0.437983
\(814\) 8.70431 0.305086
\(815\) −25.8691 −0.906155
\(816\) 1.14256 0.0399977
\(817\) 5.22642 0.182849
\(818\) 24.8079 0.867387
\(819\) 27.7759 0.970569
\(820\) −15.2369 −0.532096
\(821\) 3.24548 0.113268 0.0566340 0.998395i \(-0.481963\pi\)
0.0566340 + 0.998395i \(0.481963\pi\)
\(822\) 6.46378 0.225450
\(823\) −29.2411 −1.01928 −0.509640 0.860387i \(-0.670222\pi\)
−0.509640 + 0.860387i \(0.670222\pi\)
\(824\) −14.3083 −0.498452
\(825\) −2.00542 −0.0698198
\(826\) −26.6313 −0.926622
\(827\) −47.6510 −1.65699 −0.828494 0.559997i \(-0.810802\pi\)
−0.828494 + 0.559997i \(0.810802\pi\)
\(828\) 3.30477 0.114849
\(829\) 21.6971 0.753572 0.376786 0.926300i \(-0.377029\pi\)
0.376786 + 0.926300i \(0.377029\pi\)
\(830\) −35.9257 −1.24700
\(831\) −6.80678 −0.236125
\(832\) −5.90772 −0.204813
\(833\) −17.2588 −0.597980
\(834\) 6.32384 0.218977
\(835\) 2.50565 0.0867117
\(836\) −1.91016 −0.0660642
\(837\) −7.67209 −0.265186
\(838\) −3.30387 −0.114130
\(839\) −28.8691 −0.996671 −0.498335 0.866984i \(-0.666055\pi\)
−0.498335 + 0.866984i \(0.666055\pi\)
\(840\) 11.5643 0.399008
\(841\) −23.1570 −0.798516
\(842\) 17.0365 0.587115
\(843\) 22.1604 0.763244
\(844\) −4.80428 −0.165370
\(845\) −53.8693 −1.85316
\(846\) −5.78403 −0.198859
\(847\) 34.5631 1.18760
\(848\) −1.00000 −0.0343401
\(849\) 2.66711 0.0915351
\(850\) 1.19954 0.0411440
\(851\) −15.0594 −0.516228
\(852\) −1.22023 −0.0418046
\(853\) −47.6587 −1.63180 −0.815902 0.578191i \(-0.803759\pi\)
−0.815902 + 0.578191i \(0.803759\pi\)
\(854\) 64.2635 2.19905
\(855\) 2.45965 0.0841182
\(856\) −1.86376 −0.0637020
\(857\) 18.8446 0.643718 0.321859 0.946788i \(-0.395692\pi\)
0.321859 + 0.946788i \(0.395692\pi\)
\(858\) −11.2847 −0.385253
\(859\) −6.79165 −0.231728 −0.115864 0.993265i \(-0.536964\pi\)
−0.115864 + 0.993265i \(0.536964\pi\)
\(860\) 12.8552 0.438358
\(861\) 29.1254 0.992590
\(862\) −14.8752 −0.506651
\(863\) 36.1406 1.23024 0.615119 0.788434i \(-0.289108\pi\)
0.615119 + 0.788434i \(0.289108\pi\)
\(864\) 1.00000 0.0340207
\(865\) 34.1289 1.16042
\(866\) −9.55069 −0.324546
\(867\) 15.6945 0.533015
\(868\) −36.0713 −1.22434
\(869\) 29.1504 0.988860
\(870\) −5.94555 −0.201573
\(871\) 19.1274 0.648106
\(872\) −5.31128 −0.179863
\(873\) −0.0999559 −0.00338299
\(874\) 3.30477 0.111786
\(875\) −45.6807 −1.54429
\(876\) −7.85807 −0.265500
\(877\) 33.1313 1.11877 0.559383 0.828909i \(-0.311038\pi\)
0.559383 + 0.828909i \(0.311038\pi\)
\(878\) 15.0945 0.509415
\(879\) −13.2532 −0.447020
\(880\) −4.69832 −0.158380
\(881\) −18.6265 −0.627544 −0.313772 0.949498i \(-0.601593\pi\)
−0.313772 + 0.949498i \(0.601593\pi\)
\(882\) −15.1053 −0.508622
\(883\) 13.9277 0.468703 0.234351 0.972152i \(-0.424703\pi\)
0.234351 + 0.972152i \(0.424703\pi\)
\(884\) 6.74995 0.227025
\(885\) −13.9321 −0.468324
\(886\) −10.8991 −0.366161
\(887\) −57.8038 −1.94086 −0.970430 0.241382i \(-0.922399\pi\)
−0.970430 + 0.241382i \(0.922399\pi\)
\(888\) −4.55685 −0.152918
\(889\) −8.15628 −0.273553
\(890\) −14.5196 −0.486699
\(891\) 1.91016 0.0639927
\(892\) 10.6493 0.356564
\(893\) −5.78403 −0.193555
\(894\) 20.9266 0.699889
\(895\) 3.97512 0.132873
\(896\) 4.70163 0.157070
\(897\) 19.5237 0.651876
\(898\) −13.4361 −0.448367
\(899\) 18.5453 0.618519
\(900\) 1.04987 0.0349957
\(901\) 1.14256 0.0380643
\(902\) −11.8329 −0.393994
\(903\) −24.5727 −0.817728
\(904\) 0.179709 0.00597702
\(905\) 33.2745 1.10608
\(906\) 17.4133 0.578518
\(907\) −11.4415 −0.379908 −0.189954 0.981793i \(-0.560834\pi\)
−0.189954 + 0.981793i \(0.560834\pi\)
\(908\) 6.99635 0.232182
\(909\) −17.7871 −0.589961
\(910\) 68.3190 2.26475
\(911\) 10.1995 0.337925 0.168962 0.985623i \(-0.445958\pi\)
0.168962 + 0.985623i \(0.445958\pi\)
\(912\) 1.00000 0.0331133
\(913\) −27.8998 −0.923350
\(914\) −10.4744 −0.346463
\(915\) 33.6193 1.11142
\(916\) −14.2907 −0.472177
\(917\) 63.5568 2.09883
\(918\) −1.14256 −0.0377102
\(919\) −47.6073 −1.57042 −0.785210 0.619229i \(-0.787445\pi\)
−0.785210 + 0.619229i \(0.787445\pi\)
\(920\) 8.12857 0.267991
\(921\) 3.64574 0.120131
\(922\) 11.0887 0.365187
\(923\) −7.20881 −0.237281
\(924\) 8.98085 0.295448
\(925\) −4.78411 −0.157301
\(926\) −0.621532 −0.0204248
\(927\) 14.3083 0.469945
\(928\) −2.41724 −0.0793496
\(929\) 8.12727 0.266647 0.133324 0.991073i \(-0.457435\pi\)
0.133324 + 0.991073i \(0.457435\pi\)
\(930\) −18.8706 −0.618793
\(931\) −15.1053 −0.495056
\(932\) −0.544364 −0.0178312
\(933\) 31.7037 1.03793
\(934\) 30.9060 1.01128
\(935\) 5.36812 0.175556
\(936\) 5.90772 0.193100
\(937\) 21.3791 0.698426 0.349213 0.937043i \(-0.386449\pi\)
0.349213 + 0.937043i \(0.386449\pi\)
\(938\) −15.2224 −0.497029
\(939\) −22.2063 −0.724676
\(940\) −14.2267 −0.464023
\(941\) −39.8229 −1.29819 −0.649094 0.760708i \(-0.724852\pi\)
−0.649094 + 0.760708i \(0.724852\pi\)
\(942\) −13.2749 −0.432518
\(943\) 20.4722 0.666667
\(944\) −5.66428 −0.184357
\(945\) −11.5643 −0.376188
\(946\) 9.98330 0.324585
\(947\) 4.11934 0.133860 0.0669302 0.997758i \(-0.478680\pi\)
0.0669302 + 0.997758i \(0.478680\pi\)
\(948\) −15.2607 −0.495646
\(949\) −46.4233 −1.50697
\(950\) 1.04987 0.0340623
\(951\) −28.9210 −0.937829
\(952\) −5.37190 −0.174104
\(953\) 1.91115 0.0619081 0.0309541 0.999521i \(-0.490145\pi\)
0.0309541 + 0.999521i \(0.490145\pi\)
\(954\) 1.00000 0.0323762
\(955\) 27.5552 0.891665
\(956\) −7.02584 −0.227232
\(957\) −4.61730 −0.149256
\(958\) −23.7155 −0.766214
\(959\) −30.3903 −0.981354
\(960\) 2.45965 0.0793848
\(961\) 27.8610 0.898741
\(962\) −26.9206 −0.867956
\(963\) 1.86376 0.0600588
\(964\) −3.00462 −0.0967723
\(965\) 49.2822 1.58645
\(966\) −15.5378 −0.499920
\(967\) −22.9112 −0.736775 −0.368387 0.929672i \(-0.620090\pi\)
−0.368387 + 0.929672i \(0.620090\pi\)
\(968\) 7.35130 0.236280
\(969\) −1.14256 −0.0367044
\(970\) −0.245856 −0.00789397
\(971\) 27.9653 0.897448 0.448724 0.893670i \(-0.351879\pi\)
0.448724 + 0.893670i \(0.351879\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −29.7323 −0.953175
\(974\) 9.42158 0.301887
\(975\) 6.20235 0.198634
\(976\) 13.6683 0.437513
\(977\) −36.4664 −1.16666 −0.583331 0.812234i \(-0.698251\pi\)
−0.583331 + 0.812234i \(0.698251\pi\)
\(978\) 10.5174 0.336309
\(979\) −11.2759 −0.360380
\(980\) −37.1537 −1.18683
\(981\) 5.31128 0.169576
\(982\) 13.0936 0.417834
\(983\) 43.7564 1.39561 0.697806 0.716287i \(-0.254160\pi\)
0.697806 + 0.716287i \(0.254160\pi\)
\(984\) 6.19475 0.197481
\(985\) −53.4038 −1.70159
\(986\) 2.76184 0.0879550
\(987\) 27.1943 0.865605
\(988\) 5.90772 0.187950
\(989\) −17.2721 −0.549222
\(990\) 4.69832 0.149322
\(991\) 45.8946 1.45789 0.728945 0.684573i \(-0.240011\pi\)
0.728945 + 0.684573i \(0.240011\pi\)
\(992\) −7.67209 −0.243589
\(993\) 18.6818 0.592849
\(994\) 5.73709 0.181969
\(995\) −26.2028 −0.830685
\(996\) 14.6060 0.462810
\(997\) 5.88414 0.186353 0.0931764 0.995650i \(-0.470298\pi\)
0.0931764 + 0.995650i \(0.470298\pi\)
\(998\) 9.16202 0.290019
\(999\) 4.55685 0.144172
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 6042.2.a.be.1.5 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.be.1.5 12 1.1 even 1 trivial