Properties

Label 6042.2.a.be.1.9
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 1.9
Root \(-1.13903\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.13903 q^{5} +1.00000 q^{6} +4.65781 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.13903 q^{5} +1.00000 q^{6} +4.65781 q^{7} -1.00000 q^{8} +1.00000 q^{9} -1.13903 q^{10} -4.30320 q^{11} -1.00000 q^{12} -1.92818 q^{13} -4.65781 q^{14} -1.13903 q^{15} +1.00000 q^{16} -6.94646 q^{17} -1.00000 q^{18} -1.00000 q^{19} +1.13903 q^{20} -4.65781 q^{21} +4.30320 q^{22} +7.91164 q^{23} +1.00000 q^{24} -3.70261 q^{25} +1.92818 q^{26} -1.00000 q^{27} +4.65781 q^{28} +9.83797 q^{29} +1.13903 q^{30} +1.28044 q^{31} -1.00000 q^{32} +4.30320 q^{33} +6.94646 q^{34} +5.30540 q^{35} +1.00000 q^{36} +0.818562 q^{37} +1.00000 q^{38} +1.92818 q^{39} -1.13903 q^{40} -8.18877 q^{41} +4.65781 q^{42} -10.6577 q^{43} -4.30320 q^{44} +1.13903 q^{45} -7.91164 q^{46} -6.78781 q^{47} -1.00000 q^{48} +14.6952 q^{49} +3.70261 q^{50} +6.94646 q^{51} -1.92818 q^{52} -1.00000 q^{53} +1.00000 q^{54} -4.90148 q^{55} -4.65781 q^{56} +1.00000 q^{57} -9.83797 q^{58} -6.44178 q^{59} -1.13903 q^{60} +11.1555 q^{61} -1.28044 q^{62} +4.65781 q^{63} +1.00000 q^{64} -2.19626 q^{65} -4.30320 q^{66} +13.1712 q^{67} -6.94646 q^{68} -7.91164 q^{69} -5.30540 q^{70} -0.159757 q^{71} -1.00000 q^{72} -14.3502 q^{73} -0.818562 q^{74} +3.70261 q^{75} -1.00000 q^{76} -20.0435 q^{77} -1.92818 q^{78} +8.88482 q^{79} +1.13903 q^{80} +1.00000 q^{81} +8.18877 q^{82} -10.8297 q^{83} -4.65781 q^{84} -7.91224 q^{85} +10.6577 q^{86} -9.83797 q^{87} +4.30320 q^{88} -7.73787 q^{89} -1.13903 q^{90} -8.98109 q^{91} +7.91164 q^{92} -1.28044 q^{93} +6.78781 q^{94} -1.13903 q^{95} +1.00000 q^{96} -3.07525 q^{97} -14.6952 q^{98} -4.30320 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} - 12 q^{3} + 12 q^{4} - 3 q^{5} + 12 q^{6} - q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) 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
\(5\) 1.13903 0.509390 0.254695 0.967021i \(-0.418025\pi\)
0.254695 + 0.967021i \(0.418025\pi\)
\(6\) 1.00000 0.408248
\(7\) 4.65781 1.76049 0.880244 0.474521i \(-0.157379\pi\)
0.880244 + 0.474521i \(0.157379\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −1.13903 −0.360193
\(11\) −4.30320 −1.29746 −0.648732 0.761017i \(-0.724700\pi\)
−0.648732 + 0.761017i \(0.724700\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.92818 −0.534780 −0.267390 0.963588i \(-0.586161\pi\)
−0.267390 + 0.963588i \(0.586161\pi\)
\(14\) −4.65781 −1.24485
\(15\) −1.13903 −0.294097
\(16\) 1.00000 0.250000
\(17\) −6.94646 −1.68476 −0.842382 0.538881i \(-0.818847\pi\)
−0.842382 + 0.538881i \(0.818847\pi\)
\(18\) −1.00000 −0.235702
\(19\) −1.00000 −0.229416
\(20\) 1.13903 0.254695
\(21\) −4.65781 −1.01642
\(22\) 4.30320 0.917445
\(23\) 7.91164 1.64969 0.824846 0.565357i \(-0.191262\pi\)
0.824846 + 0.565357i \(0.191262\pi\)
\(24\) 1.00000 0.204124
\(25\) −3.70261 −0.740521
\(26\) 1.92818 0.378147
\(27\) −1.00000 −0.192450
\(28\) 4.65781 0.880244
\(29\) 9.83797 1.82686 0.913432 0.406990i \(-0.133422\pi\)
0.913432 + 0.406990i \(0.133422\pi\)
\(30\) 1.13903 0.207958
\(31\) 1.28044 0.229975 0.114987 0.993367i \(-0.463317\pi\)
0.114987 + 0.993367i \(0.463317\pi\)
\(32\) −1.00000 −0.176777
\(33\) 4.30320 0.749091
\(34\) 6.94646 1.19131
\(35\) 5.30540 0.896776
\(36\) 1.00000 0.166667
\(37\) 0.818562 0.134571 0.0672854 0.997734i \(-0.478566\pi\)
0.0672854 + 0.997734i \(0.478566\pi\)
\(38\) 1.00000 0.162221
\(39\) 1.92818 0.308756
\(40\) −1.13903 −0.180097
\(41\) −8.18877 −1.27887 −0.639435 0.768845i \(-0.720832\pi\)
−0.639435 + 0.768845i \(0.720832\pi\)
\(42\) 4.65781 0.718716
\(43\) −10.6577 −1.62528 −0.812641 0.582764i \(-0.801971\pi\)
−0.812641 + 0.582764i \(0.801971\pi\)
\(44\) −4.30320 −0.648732
\(45\) 1.13903 0.169797
\(46\) −7.91164 −1.16651
\(47\) −6.78781 −0.990104 −0.495052 0.868863i \(-0.664851\pi\)
−0.495052 + 0.868863i \(0.664851\pi\)
\(48\) −1.00000 −0.144338
\(49\) 14.6952 2.09932
\(50\) 3.70261 0.523628
\(51\) 6.94646 0.972699
\(52\) −1.92818 −0.267390
\(53\) −1.00000 −0.137361
\(54\) 1.00000 0.136083
\(55\) −4.90148 −0.660916
\(56\) −4.65781 −0.622427
\(57\) 1.00000 0.132453
\(58\) −9.83797 −1.29179
\(59\) −6.44178 −0.838648 −0.419324 0.907837i \(-0.637733\pi\)
−0.419324 + 0.907837i \(0.637733\pi\)
\(60\) −1.13903 −0.147048
\(61\) 11.1555 1.42832 0.714158 0.699985i \(-0.246810\pi\)
0.714158 + 0.699985i \(0.246810\pi\)
\(62\) −1.28044 −0.162617
\(63\) 4.65781 0.586829
\(64\) 1.00000 0.125000
\(65\) −2.19626 −0.272412
\(66\) −4.30320 −0.529687
\(67\) 13.1712 1.60912 0.804558 0.593874i \(-0.202402\pi\)
0.804558 + 0.593874i \(0.202402\pi\)
\(68\) −6.94646 −0.842382
\(69\) −7.91164 −0.952450
\(70\) −5.30540 −0.634116
\(71\) −0.159757 −0.0189597 −0.00947986 0.999955i \(-0.503018\pi\)
−0.00947986 + 0.999955i \(0.503018\pi\)
\(72\) −1.00000 −0.117851
\(73\) −14.3502 −1.67957 −0.839783 0.542922i \(-0.817318\pi\)
−0.839783 + 0.542922i \(0.817318\pi\)
\(74\) −0.818562 −0.0951559
\(75\) 3.70261 0.427540
\(76\) −1.00000 −0.114708
\(77\) −20.0435 −2.28417
\(78\) −1.92818 −0.218323
\(79\) 8.88482 0.999620 0.499810 0.866135i \(-0.333403\pi\)
0.499810 + 0.866135i \(0.333403\pi\)
\(80\) 1.13903 0.127348
\(81\) 1.00000 0.111111
\(82\) 8.18877 0.904298
\(83\) −10.8297 −1.18871 −0.594355 0.804203i \(-0.702592\pi\)
−0.594355 + 0.804203i \(0.702592\pi\)
\(84\) −4.65781 −0.508209
\(85\) −7.91224 −0.858203
\(86\) 10.6577 1.14925
\(87\) −9.83797 −1.05474
\(88\) 4.30320 0.458723
\(89\) −7.73787 −0.820213 −0.410107 0.912038i \(-0.634508\pi\)
−0.410107 + 0.912038i \(0.634508\pi\)
\(90\) −1.13903 −0.120064
\(91\) −8.98109 −0.941474
\(92\) 7.91164 0.824846
\(93\) −1.28044 −0.132776
\(94\) 6.78781 0.700110
\(95\) −1.13903 −0.116862
\(96\) 1.00000 0.102062
\(97\) −3.07525 −0.312244 −0.156122 0.987738i \(-0.549899\pi\)
−0.156122 + 0.987738i \(0.549899\pi\)
\(98\) −14.6952 −1.48444
\(99\) −4.30320 −0.432488
\(100\) −3.70261 −0.370261
\(101\) −10.0253 −0.997556 −0.498778 0.866730i \(-0.666218\pi\)
−0.498778 + 0.866730i \(0.666218\pi\)
\(102\) −6.94646 −0.687802
\(103\) 11.3695 1.12027 0.560137 0.828400i \(-0.310748\pi\)
0.560137 + 0.828400i \(0.310748\pi\)
\(104\) 1.92818 0.189073
\(105\) −5.30540 −0.517754
\(106\) 1.00000 0.0971286
\(107\) −8.55575 −0.827115 −0.413558 0.910478i \(-0.635714\pi\)
−0.413558 + 0.910478i \(0.635714\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −14.9575 −1.43267 −0.716335 0.697756i \(-0.754182\pi\)
−0.716335 + 0.697756i \(0.754182\pi\)
\(110\) 4.90148 0.467338
\(111\) −0.818562 −0.0776945
\(112\) 4.65781 0.440122
\(113\) −2.53394 −0.238373 −0.119186 0.992872i \(-0.538029\pi\)
−0.119186 + 0.992872i \(0.538029\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 9.01161 0.840337
\(116\) 9.83797 0.913432
\(117\) −1.92818 −0.178260
\(118\) 6.44178 0.593014
\(119\) −32.3553 −2.96601
\(120\) 1.13903 0.103979
\(121\) 7.51753 0.683412
\(122\) −11.1555 −1.00997
\(123\) 8.18877 0.738356
\(124\) 1.28044 0.114987
\(125\) −9.91254 −0.886605
\(126\) −4.65781 −0.414951
\(127\) −12.5689 −1.11531 −0.557656 0.830072i \(-0.688299\pi\)
−0.557656 + 0.830072i \(0.688299\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 10.6577 0.938357
\(130\) 2.19626 0.192624
\(131\) −12.4130 −1.08453 −0.542265 0.840208i \(-0.682433\pi\)
−0.542265 + 0.840208i \(0.682433\pi\)
\(132\) 4.30320 0.374545
\(133\) −4.65781 −0.403884
\(134\) −13.1712 −1.13782
\(135\) −1.13903 −0.0980322
\(136\) 6.94646 0.595654
\(137\) 2.04158 0.174424 0.0872119 0.996190i \(-0.472204\pi\)
0.0872119 + 0.996190i \(0.472204\pi\)
\(138\) 7.91164 0.673484
\(139\) −16.4948 −1.39907 −0.699536 0.714597i \(-0.746610\pi\)
−0.699536 + 0.714597i \(0.746610\pi\)
\(140\) 5.30540 0.448388
\(141\) 6.78781 0.571637
\(142\) 0.159757 0.0134065
\(143\) 8.29733 0.693858
\(144\) 1.00000 0.0833333
\(145\) 11.2058 0.930587
\(146\) 14.3502 1.18763
\(147\) −14.6952 −1.21204
\(148\) 0.818562 0.0672854
\(149\) 3.48835 0.285777 0.142888 0.989739i \(-0.454361\pi\)
0.142888 + 0.989739i \(0.454361\pi\)
\(150\) −3.70261 −0.302317
\(151\) 12.2747 0.998904 0.499452 0.866342i \(-0.333535\pi\)
0.499452 + 0.866342i \(0.333535\pi\)
\(152\) 1.00000 0.0811107
\(153\) −6.94646 −0.561588
\(154\) 20.0435 1.61515
\(155\) 1.45847 0.117147
\(156\) 1.92818 0.154378
\(157\) 15.5243 1.23898 0.619488 0.785006i \(-0.287340\pi\)
0.619488 + 0.785006i \(0.287340\pi\)
\(158\) −8.88482 −0.706838
\(159\) 1.00000 0.0793052
\(160\) −1.13903 −0.0900484
\(161\) 36.8510 2.90426
\(162\) −1.00000 −0.0785674
\(163\) 5.08704 0.398447 0.199224 0.979954i \(-0.436158\pi\)
0.199224 + 0.979954i \(0.436158\pi\)
\(164\) −8.18877 −0.639435
\(165\) 4.90148 0.381580
\(166\) 10.8297 0.840545
\(167\) 12.5322 0.969772 0.484886 0.874577i \(-0.338861\pi\)
0.484886 + 0.874577i \(0.338861\pi\)
\(168\) 4.65781 0.359358
\(169\) −9.28213 −0.714010
\(170\) 7.91224 0.606841
\(171\) −1.00000 −0.0764719
\(172\) −10.6577 −0.812641
\(173\) −21.4077 −1.62760 −0.813800 0.581146i \(-0.802605\pi\)
−0.813800 + 0.581146i \(0.802605\pi\)
\(174\) 9.83797 0.745814
\(175\) −17.2461 −1.30368
\(176\) −4.30320 −0.324366
\(177\) 6.44178 0.484194
\(178\) 7.73787 0.579978
\(179\) −0.147626 −0.0110341 −0.00551706 0.999985i \(-0.501756\pi\)
−0.00551706 + 0.999985i \(0.501756\pi\)
\(180\) 1.13903 0.0848984
\(181\) 19.8585 1.47607 0.738034 0.674764i \(-0.235755\pi\)
0.738034 + 0.674764i \(0.235755\pi\)
\(182\) 8.98109 0.665723
\(183\) −11.1555 −0.824638
\(184\) −7.91164 −0.583254
\(185\) 0.932368 0.0685491
\(186\) 1.28044 0.0938867
\(187\) 29.8920 2.18592
\(188\) −6.78781 −0.495052
\(189\) −4.65781 −0.338806
\(190\) 1.13903 0.0826340
\(191\) −1.38890 −0.100497 −0.0502485 0.998737i \(-0.516001\pi\)
−0.0502485 + 0.998737i \(0.516001\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 2.72902 0.196439 0.0982197 0.995165i \(-0.468685\pi\)
0.0982197 + 0.995165i \(0.468685\pi\)
\(194\) 3.07525 0.220790
\(195\) 2.19626 0.157277
\(196\) 14.6952 1.04966
\(197\) 2.54933 0.181632 0.0908162 0.995868i \(-0.471052\pi\)
0.0908162 + 0.995868i \(0.471052\pi\)
\(198\) 4.30320 0.305815
\(199\) −8.95183 −0.634579 −0.317289 0.948329i \(-0.602773\pi\)
−0.317289 + 0.948329i \(0.602773\pi\)
\(200\) 3.70261 0.261814
\(201\) −13.1712 −0.929024
\(202\) 10.0253 0.705378
\(203\) 45.8234 3.21617
\(204\) 6.94646 0.486349
\(205\) −9.32727 −0.651445
\(206\) −11.3695 −0.792154
\(207\) 7.91164 0.549897
\(208\) −1.92818 −0.133695
\(209\) 4.30320 0.297659
\(210\) 5.30540 0.366107
\(211\) 16.0300 1.10355 0.551775 0.833993i \(-0.313951\pi\)
0.551775 + 0.833993i \(0.313951\pi\)
\(212\) −1.00000 −0.0686803
\(213\) 0.159757 0.0109464
\(214\) 8.55575 0.584859
\(215\) −12.1394 −0.827903
\(216\) 1.00000 0.0680414
\(217\) 5.96407 0.404868
\(218\) 14.9575 1.01305
\(219\) 14.3502 0.969698
\(220\) −4.90148 −0.330458
\(221\) 13.3940 0.900978
\(222\) 0.818562 0.0549383
\(223\) 15.7753 1.05639 0.528195 0.849123i \(-0.322869\pi\)
0.528195 + 0.849123i \(0.322869\pi\)
\(224\) −4.65781 −0.311213
\(225\) −3.70261 −0.246840
\(226\) 2.53394 0.168555
\(227\) −4.98013 −0.330543 −0.165271 0.986248i \(-0.552850\pi\)
−0.165271 + 0.986248i \(0.552850\pi\)
\(228\) 1.00000 0.0662266
\(229\) 0.629002 0.0415656 0.0207828 0.999784i \(-0.493384\pi\)
0.0207828 + 0.999784i \(0.493384\pi\)
\(230\) −9.01161 −0.594208
\(231\) 20.0435 1.31877
\(232\) −9.83797 −0.645894
\(233\) 14.0856 0.922778 0.461389 0.887198i \(-0.347351\pi\)
0.461389 + 0.887198i \(0.347351\pi\)
\(234\) 1.92818 0.126049
\(235\) −7.73154 −0.504350
\(236\) −6.44178 −0.419324
\(237\) −8.88482 −0.577131
\(238\) 32.3553 2.09728
\(239\) −22.0582 −1.42682 −0.713412 0.700745i \(-0.752851\pi\)
−0.713412 + 0.700745i \(0.752851\pi\)
\(240\) −1.13903 −0.0735242
\(241\) −9.89881 −0.637638 −0.318819 0.947816i \(-0.603286\pi\)
−0.318819 + 0.947816i \(0.603286\pi\)
\(242\) −7.51753 −0.483245
\(243\) −1.00000 −0.0641500
\(244\) 11.1555 0.714158
\(245\) 16.7383 1.06937
\(246\) −8.18877 −0.522097
\(247\) 1.92818 0.122687
\(248\) −1.28044 −0.0813083
\(249\) 10.8297 0.686302
\(250\) 9.91254 0.626924
\(251\) −11.2298 −0.708818 −0.354409 0.935090i \(-0.615318\pi\)
−0.354409 + 0.935090i \(0.615318\pi\)
\(252\) 4.65781 0.293415
\(253\) −34.0454 −2.14042
\(254\) 12.5689 0.788645
\(255\) 7.91224 0.495484
\(256\) 1.00000 0.0625000
\(257\) 5.75460 0.358962 0.179481 0.983761i \(-0.442558\pi\)
0.179481 + 0.983761i \(0.442558\pi\)
\(258\) −10.6577 −0.663519
\(259\) 3.81271 0.236910
\(260\) −2.19626 −0.136206
\(261\) 9.83797 0.608955
\(262\) 12.4130 0.766878
\(263\) 14.0528 0.866531 0.433265 0.901266i \(-0.357361\pi\)
0.433265 + 0.901266i \(0.357361\pi\)
\(264\) −4.30320 −0.264844
\(265\) −1.13903 −0.0699702
\(266\) 4.65781 0.285589
\(267\) 7.73787 0.473550
\(268\) 13.1712 0.804558
\(269\) −11.9464 −0.728382 −0.364191 0.931324i \(-0.618654\pi\)
−0.364191 + 0.931324i \(0.618654\pi\)
\(270\) 1.13903 0.0693193
\(271\) −17.8176 −1.08234 −0.541171 0.840912i \(-0.682019\pi\)
−0.541171 + 0.840912i \(0.682019\pi\)
\(272\) −6.94646 −0.421191
\(273\) 8.98109 0.543560
\(274\) −2.04158 −0.123336
\(275\) 15.9331 0.960800
\(276\) −7.91164 −0.476225
\(277\) −25.1299 −1.50991 −0.754954 0.655778i \(-0.772341\pi\)
−0.754954 + 0.655778i \(0.772341\pi\)
\(278\) 16.4948 0.989293
\(279\) 1.28044 0.0766582
\(280\) −5.30540 −0.317058
\(281\) 1.06382 0.0634623 0.0317312 0.999496i \(-0.489898\pi\)
0.0317312 + 0.999496i \(0.489898\pi\)
\(282\) −6.78781 −0.404208
\(283\) −7.83023 −0.465459 −0.232730 0.972541i \(-0.574766\pi\)
−0.232730 + 0.972541i \(0.574766\pi\)
\(284\) −0.159757 −0.00947986
\(285\) 1.13903 0.0674704
\(286\) −8.29733 −0.490632
\(287\) −38.1418 −2.25144
\(288\) −1.00000 −0.0589256
\(289\) 31.2533 1.83843
\(290\) −11.2058 −0.658025
\(291\) 3.07525 0.180274
\(292\) −14.3502 −0.839783
\(293\) −11.6142 −0.678506 −0.339253 0.940695i \(-0.610174\pi\)
−0.339253 + 0.940695i \(0.610174\pi\)
\(294\) 14.6952 0.857043
\(295\) −7.33739 −0.427199
\(296\) −0.818562 −0.0475780
\(297\) 4.30320 0.249697
\(298\) −3.48835 −0.202075
\(299\) −15.2551 −0.882223
\(300\) 3.70261 0.213770
\(301\) −49.6415 −2.86129
\(302\) −12.2747 −0.706332
\(303\) 10.0253 0.575939
\(304\) −1.00000 −0.0573539
\(305\) 12.7065 0.727570
\(306\) 6.94646 0.397103
\(307\) −28.7680 −1.64187 −0.820937 0.571018i \(-0.806549\pi\)
−0.820937 + 0.571018i \(0.806549\pi\)
\(308\) −20.0435 −1.14208
\(309\) −11.3695 −0.646791
\(310\) −1.45847 −0.0828353
\(311\) 24.8634 1.40987 0.704937 0.709270i \(-0.250975\pi\)
0.704937 + 0.709270i \(0.250975\pi\)
\(312\) −1.92818 −0.109162
\(313\) −26.8861 −1.51969 −0.759846 0.650104i \(-0.774726\pi\)
−0.759846 + 0.650104i \(0.774726\pi\)
\(314\) −15.5243 −0.876088
\(315\) 5.30540 0.298925
\(316\) 8.88482 0.499810
\(317\) −9.22070 −0.517886 −0.258943 0.965893i \(-0.583374\pi\)
−0.258943 + 0.965893i \(0.583374\pi\)
\(318\) −1.00000 −0.0560772
\(319\) −42.3347 −2.37029
\(320\) 1.13903 0.0636738
\(321\) 8.55575 0.477535
\(322\) −36.8510 −2.05362
\(323\) 6.94646 0.386511
\(324\) 1.00000 0.0555556
\(325\) 7.13928 0.396016
\(326\) −5.08704 −0.281745
\(327\) 14.9575 0.827153
\(328\) 8.18877 0.452149
\(329\) −31.6164 −1.74307
\(330\) −4.90148 −0.269818
\(331\) 8.63527 0.474638 0.237319 0.971432i \(-0.423731\pi\)
0.237319 + 0.971432i \(0.423731\pi\)
\(332\) −10.8297 −0.594355
\(333\) 0.818562 0.0448569
\(334\) −12.5322 −0.685733
\(335\) 15.0024 0.819669
\(336\) −4.65781 −0.254105
\(337\) 7.80648 0.425246 0.212623 0.977134i \(-0.431799\pi\)
0.212623 + 0.977134i \(0.431799\pi\)
\(338\) 9.28213 0.504881
\(339\) 2.53394 0.137625
\(340\) −7.91224 −0.429101
\(341\) −5.51001 −0.298384
\(342\) 1.00000 0.0540738
\(343\) 35.8430 1.93534
\(344\) 10.6577 0.574624
\(345\) −9.01161 −0.485169
\(346\) 21.4077 1.15089
\(347\) 21.0070 1.12771 0.563856 0.825873i \(-0.309317\pi\)
0.563856 + 0.825873i \(0.309317\pi\)
\(348\) −9.83797 −0.527370
\(349\) −4.89268 −0.261899 −0.130950 0.991389i \(-0.541803\pi\)
−0.130950 + 0.991389i \(0.541803\pi\)
\(350\) 17.2461 0.921840
\(351\) 1.92818 0.102919
\(352\) 4.30320 0.229361
\(353\) −7.99387 −0.425471 −0.212736 0.977110i \(-0.568237\pi\)
−0.212736 + 0.977110i \(0.568237\pi\)
\(354\) −6.44178 −0.342377
\(355\) −0.181969 −0.00965790
\(356\) −7.73787 −0.410107
\(357\) 32.3553 1.71242
\(358\) 0.147626 0.00780230
\(359\) −8.97136 −0.473490 −0.236745 0.971572i \(-0.576081\pi\)
−0.236745 + 0.971572i \(0.576081\pi\)
\(360\) −1.13903 −0.0600322
\(361\) 1.00000 0.0526316
\(362\) −19.8585 −1.04374
\(363\) −7.51753 −0.394568
\(364\) −8.98109 −0.470737
\(365\) −16.3454 −0.855555
\(366\) 11.1555 0.583107
\(367\) −18.0361 −0.941476 −0.470738 0.882273i \(-0.656012\pi\)
−0.470738 + 0.882273i \(0.656012\pi\)
\(368\) 7.91164 0.412423
\(369\) −8.18877 −0.426290
\(370\) −0.932368 −0.0484715
\(371\) −4.65781 −0.241822
\(372\) −1.28044 −0.0663879
\(373\) 8.01853 0.415184 0.207592 0.978216i \(-0.433437\pi\)
0.207592 + 0.978216i \(0.433437\pi\)
\(374\) −29.8920 −1.54568
\(375\) 9.91254 0.511882
\(376\) 6.78781 0.350055
\(377\) −18.9694 −0.976971
\(378\) 4.65781 0.239572
\(379\) −24.9640 −1.28232 −0.641158 0.767409i \(-0.721546\pi\)
−0.641158 + 0.767409i \(0.721546\pi\)
\(380\) −1.13903 −0.0584311
\(381\) 12.5689 0.643926
\(382\) 1.38890 0.0710621
\(383\) 20.3392 1.03928 0.519642 0.854384i \(-0.326065\pi\)
0.519642 + 0.854384i \(0.326065\pi\)
\(384\) 1.00000 0.0510310
\(385\) −22.8302 −1.16353
\(386\) −2.72902 −0.138904
\(387\) −10.6577 −0.541761
\(388\) −3.07525 −0.156122
\(389\) 0.516054 0.0261650 0.0130825 0.999914i \(-0.495836\pi\)
0.0130825 + 0.999914i \(0.495836\pi\)
\(390\) −2.19626 −0.111212
\(391\) −54.9579 −2.77934
\(392\) −14.6952 −0.742221
\(393\) 12.4130 0.626153
\(394\) −2.54933 −0.128434
\(395\) 10.1201 0.509197
\(396\) −4.30320 −0.216244
\(397\) 9.51624 0.477606 0.238803 0.971068i \(-0.423245\pi\)
0.238803 + 0.971068i \(0.423245\pi\)
\(398\) 8.95183 0.448715
\(399\) 4.65781 0.233182
\(400\) −3.70261 −0.185130
\(401\) −12.6804 −0.633228 −0.316614 0.948554i \(-0.602546\pi\)
−0.316614 + 0.948554i \(0.602546\pi\)
\(402\) 13.1712 0.656919
\(403\) −2.46892 −0.122986
\(404\) −10.0253 −0.498778
\(405\) 1.13903 0.0565989
\(406\) −45.8234 −2.27418
\(407\) −3.52244 −0.174601
\(408\) −6.94646 −0.343901
\(409\) 33.2874 1.64596 0.822978 0.568073i \(-0.192311\pi\)
0.822978 + 0.568073i \(0.192311\pi\)
\(410\) 9.32727 0.460641
\(411\) −2.04158 −0.100704
\(412\) 11.3695 0.560137
\(413\) −30.0046 −1.47643
\(414\) −7.91164 −0.388836
\(415\) −12.3353 −0.605517
\(416\) 1.92818 0.0945367
\(417\) 16.4948 0.807755
\(418\) −4.30320 −0.210476
\(419\) −24.4159 −1.19280 −0.596398 0.802689i \(-0.703402\pi\)
−0.596398 + 0.802689i \(0.703402\pi\)
\(420\) −5.30540 −0.258877
\(421\) 37.3518 1.82042 0.910208 0.414152i \(-0.135922\pi\)
0.910208 + 0.414152i \(0.135922\pi\)
\(422\) −16.0300 −0.780328
\(423\) −6.78781 −0.330035
\(424\) 1.00000 0.0485643
\(425\) 25.7200 1.24760
\(426\) −0.159757 −0.00774027
\(427\) 51.9602 2.51453
\(428\) −8.55575 −0.413558
\(429\) −8.29733 −0.400599
\(430\) 12.1394 0.585416
\(431\) 12.2602 0.590551 0.295276 0.955412i \(-0.404589\pi\)
0.295276 + 0.955412i \(0.404589\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −31.3015 −1.50425 −0.752127 0.659019i \(-0.770972\pi\)
−0.752127 + 0.659019i \(0.770972\pi\)
\(434\) −5.96407 −0.286285
\(435\) −11.2058 −0.537275
\(436\) −14.9575 −0.716335
\(437\) −7.91164 −0.378465
\(438\) −14.3502 −0.685680
\(439\) −25.5266 −1.21832 −0.609159 0.793048i \(-0.708493\pi\)
−0.609159 + 0.793048i \(0.708493\pi\)
\(440\) 4.90148 0.233669
\(441\) 14.6952 0.699773
\(442\) −13.3940 −0.637088
\(443\) 0.0728399 0.00346073 0.00173036 0.999999i \(-0.499449\pi\)
0.00173036 + 0.999999i \(0.499449\pi\)
\(444\) −0.818562 −0.0388473
\(445\) −8.81368 −0.417809
\(446\) −15.7753 −0.746981
\(447\) −3.48835 −0.164993
\(448\) 4.65781 0.220061
\(449\) 14.6323 0.690539 0.345270 0.938504i \(-0.387787\pi\)
0.345270 + 0.938504i \(0.387787\pi\)
\(450\) 3.70261 0.174543
\(451\) 35.2379 1.65929
\(452\) −2.53394 −0.119186
\(453\) −12.2747 −0.576717
\(454\) 4.98013 0.233729
\(455\) −10.2297 −0.479578
\(456\) −1.00000 −0.0468293
\(457\) 20.0024 0.935675 0.467838 0.883814i \(-0.345033\pi\)
0.467838 + 0.883814i \(0.345033\pi\)
\(458\) −0.629002 −0.0293913
\(459\) 6.94646 0.324233
\(460\) 9.01161 0.420169
\(461\) 17.7467 0.826547 0.413273 0.910607i \(-0.364385\pi\)
0.413273 + 0.910607i \(0.364385\pi\)
\(462\) −20.0435 −0.932508
\(463\) −4.84490 −0.225162 −0.112581 0.993643i \(-0.535912\pi\)
−0.112581 + 0.993643i \(0.535912\pi\)
\(464\) 9.83797 0.456716
\(465\) −1.45847 −0.0676348
\(466\) −14.0856 −0.652503
\(467\) −27.1798 −1.25773 −0.628866 0.777513i \(-0.716481\pi\)
−0.628866 + 0.777513i \(0.716481\pi\)
\(468\) −1.92818 −0.0891300
\(469\) 61.3489 2.83283
\(470\) 7.73154 0.356629
\(471\) −15.5243 −0.715323
\(472\) 6.44178 0.296507
\(473\) 45.8622 2.10874
\(474\) 8.88482 0.408093
\(475\) 3.70261 0.169887
\(476\) −32.3553 −1.48300
\(477\) −1.00000 −0.0457869
\(478\) 22.0582 1.00892
\(479\) −32.4135 −1.48101 −0.740506 0.672049i \(-0.765414\pi\)
−0.740506 + 0.672049i \(0.765414\pi\)
\(480\) 1.13903 0.0519894
\(481\) −1.57833 −0.0719658
\(482\) 9.89881 0.450878
\(483\) −36.8510 −1.67678
\(484\) 7.51753 0.341706
\(485\) −3.50281 −0.159054
\(486\) 1.00000 0.0453609
\(487\) 10.7839 0.488665 0.244332 0.969692i \(-0.421431\pi\)
0.244332 + 0.969692i \(0.421431\pi\)
\(488\) −11.1555 −0.504986
\(489\) −5.08704 −0.230044
\(490\) −16.7383 −0.756161
\(491\) −9.33661 −0.421355 −0.210678 0.977556i \(-0.567567\pi\)
−0.210678 + 0.977556i \(0.567567\pi\)
\(492\) 8.18877 0.369178
\(493\) −68.3390 −3.07784
\(494\) −1.92818 −0.0867528
\(495\) −4.90148 −0.220305
\(496\) 1.28044 0.0574936
\(497\) −0.744120 −0.0333784
\(498\) −10.8297 −0.485289
\(499\) −28.6119 −1.28085 −0.640423 0.768022i \(-0.721241\pi\)
−0.640423 + 0.768022i \(0.721241\pi\)
\(500\) −9.91254 −0.443302
\(501\) −12.5322 −0.559898
\(502\) 11.2298 0.501210
\(503\) −30.7278 −1.37008 −0.685042 0.728504i \(-0.740216\pi\)
−0.685042 + 0.728504i \(0.740216\pi\)
\(504\) −4.65781 −0.207476
\(505\) −11.4191 −0.508145
\(506\) 34.0454 1.51350
\(507\) 9.28213 0.412234
\(508\) −12.5689 −0.557656
\(509\) −3.37076 −0.149406 −0.0747031 0.997206i \(-0.523801\pi\)
−0.0747031 + 0.997206i \(0.523801\pi\)
\(510\) −7.91224 −0.350360
\(511\) −66.8406 −2.95686
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) −5.75460 −0.253825
\(515\) 12.9503 0.570657
\(516\) 10.6577 0.469179
\(517\) 29.2093 1.28462
\(518\) −3.81271 −0.167521
\(519\) 21.4077 0.939695
\(520\) 2.19626 0.0963122
\(521\) 11.1178 0.487079 0.243540 0.969891i \(-0.421691\pi\)
0.243540 + 0.969891i \(0.421691\pi\)
\(522\) −9.83797 −0.430596
\(523\) −35.8578 −1.56795 −0.783977 0.620790i \(-0.786812\pi\)
−0.783977 + 0.620790i \(0.786812\pi\)
\(524\) −12.4130 −0.542265
\(525\) 17.2461 0.752680
\(526\) −14.0528 −0.612730
\(527\) −8.89455 −0.387453
\(528\) 4.30320 0.187273
\(529\) 39.5941 1.72148
\(530\) 1.13903 0.0494764
\(531\) −6.44178 −0.279549
\(532\) −4.65781 −0.201942
\(533\) 15.7894 0.683915
\(534\) −7.73787 −0.334851
\(535\) −9.74527 −0.421325
\(536\) −13.1712 −0.568909
\(537\) 0.147626 0.00637055
\(538\) 11.9464 0.515044
\(539\) −63.2365 −2.72379
\(540\) −1.13903 −0.0490161
\(541\) 23.8571 1.02570 0.512849 0.858479i \(-0.328590\pi\)
0.512849 + 0.858479i \(0.328590\pi\)
\(542\) 17.8176 0.765332
\(543\) −19.8585 −0.852208
\(544\) 6.94646 0.297827
\(545\) −17.0371 −0.729788
\(546\) −8.98109 −0.384355
\(547\) −28.7101 −1.22756 −0.613779 0.789478i \(-0.710351\pi\)
−0.613779 + 0.789478i \(0.710351\pi\)
\(548\) 2.04158 0.0872119
\(549\) 11.1555 0.476105
\(550\) −15.9331 −0.679388
\(551\) −9.83797 −0.419112
\(552\) 7.91164 0.336742
\(553\) 41.3838 1.75982
\(554\) 25.1299 1.06767
\(555\) −0.932368 −0.0395768
\(556\) −16.4948 −0.699536
\(557\) −35.8573 −1.51932 −0.759662 0.650318i \(-0.774635\pi\)
−0.759662 + 0.650318i \(0.774635\pi\)
\(558\) −1.28044 −0.0542055
\(559\) 20.5499 0.869169
\(560\) 5.30540 0.224194
\(561\) −29.8920 −1.26204
\(562\) −1.06382 −0.0448746
\(563\) −34.1720 −1.44018 −0.720090 0.693881i \(-0.755899\pi\)
−0.720090 + 0.693881i \(0.755899\pi\)
\(564\) 6.78781 0.285819
\(565\) −2.88624 −0.121425
\(566\) 7.83023 0.329129
\(567\) 4.65781 0.195610
\(568\) 0.159757 0.00670327
\(569\) −19.2906 −0.808704 −0.404352 0.914603i \(-0.632503\pi\)
−0.404352 + 0.914603i \(0.632503\pi\)
\(570\) −1.13903 −0.0477088
\(571\) −23.1667 −0.969496 −0.484748 0.874654i \(-0.661089\pi\)
−0.484748 + 0.874654i \(0.661089\pi\)
\(572\) 8.29733 0.346929
\(573\) 1.38890 0.0580220
\(574\) 38.1418 1.59201
\(575\) −29.2937 −1.22163
\(576\) 1.00000 0.0416667
\(577\) 6.17964 0.257262 0.128631 0.991693i \(-0.458942\pi\)
0.128631 + 0.991693i \(0.458942\pi\)
\(578\) −31.2533 −1.29997
\(579\) −2.72902 −0.113414
\(580\) 11.2058 0.465294
\(581\) −50.4425 −2.09271
\(582\) −3.07525 −0.127473
\(583\) 4.30320 0.178220
\(584\) 14.3502 0.593816
\(585\) −2.19626 −0.0908040
\(586\) 11.6142 0.479776
\(587\) −34.0495 −1.40538 −0.702688 0.711499i \(-0.748017\pi\)
−0.702688 + 0.711499i \(0.748017\pi\)
\(588\) −14.6952 −0.606021
\(589\) −1.28044 −0.0527598
\(590\) 7.33739 0.302075
\(591\) −2.54933 −0.104866
\(592\) 0.818562 0.0336427
\(593\) 41.7749 1.71549 0.857745 0.514076i \(-0.171865\pi\)
0.857745 + 0.514076i \(0.171865\pi\)
\(594\) −4.30320 −0.176562
\(595\) −36.8537 −1.51086
\(596\) 3.48835 0.142888
\(597\) 8.95183 0.366374
\(598\) 15.2551 0.623826
\(599\) −4.22921 −0.172801 −0.0864005 0.996260i \(-0.527536\pi\)
−0.0864005 + 0.996260i \(0.527536\pi\)
\(600\) −3.70261 −0.151158
\(601\) −31.3288 −1.27793 −0.638965 0.769236i \(-0.720637\pi\)
−0.638965 + 0.769236i \(0.720637\pi\)
\(602\) 49.6415 2.02324
\(603\) 13.1712 0.536372
\(604\) 12.2747 0.499452
\(605\) 8.56270 0.348123
\(606\) −10.0253 −0.407250
\(607\) 32.6789 1.32639 0.663197 0.748445i \(-0.269199\pi\)
0.663197 + 0.748445i \(0.269199\pi\)
\(608\) 1.00000 0.0405554
\(609\) −45.8234 −1.85686
\(610\) −12.7065 −0.514470
\(611\) 13.0881 0.529488
\(612\) −6.94646 −0.280794
\(613\) 21.6424 0.874128 0.437064 0.899430i \(-0.356018\pi\)
0.437064 + 0.899430i \(0.356018\pi\)
\(614\) 28.7680 1.16098
\(615\) 9.32727 0.376112
\(616\) 20.0435 0.807576
\(617\) 39.0830 1.57342 0.786710 0.617322i \(-0.211783\pi\)
0.786710 + 0.617322i \(0.211783\pi\)
\(618\) 11.3695 0.457350
\(619\) 29.5880 1.18924 0.594620 0.804007i \(-0.297302\pi\)
0.594620 + 0.804007i \(0.297302\pi\)
\(620\) 1.45847 0.0585734
\(621\) −7.91164 −0.317483
\(622\) −24.8634 −0.996931
\(623\) −36.0416 −1.44398
\(624\) 1.92818 0.0771889
\(625\) 7.22233 0.288893
\(626\) 26.8861 1.07458
\(627\) −4.30320 −0.171853
\(628\) 15.5243 0.619488
\(629\) −5.68611 −0.226720
\(630\) −5.30540 −0.211372
\(631\) −20.1558 −0.802390 −0.401195 0.915993i \(-0.631405\pi\)
−0.401195 + 0.915993i \(0.631405\pi\)
\(632\) −8.88482 −0.353419
\(633\) −16.0300 −0.637135
\(634\) 9.22070 0.366201
\(635\) −14.3164 −0.568130
\(636\) 1.00000 0.0396526
\(637\) −28.3350 −1.12267
\(638\) 42.3347 1.67605
\(639\) −0.159757 −0.00631990
\(640\) −1.13903 −0.0450242
\(641\) 13.9222 0.549893 0.274947 0.961460i \(-0.411340\pi\)
0.274947 + 0.961460i \(0.411340\pi\)
\(642\) −8.55575 −0.337668
\(643\) −32.4486 −1.27965 −0.639823 0.768522i \(-0.720992\pi\)
−0.639823 + 0.768522i \(0.720992\pi\)
\(644\) 36.8510 1.45213
\(645\) 12.1394 0.477990
\(646\) −6.94646 −0.273305
\(647\) 23.4019 0.920025 0.460012 0.887913i \(-0.347845\pi\)
0.460012 + 0.887913i \(0.347845\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 27.7203 1.08812
\(650\) −7.13928 −0.280026
\(651\) −5.96407 −0.233750
\(652\) 5.08704 0.199224
\(653\) 8.44705 0.330559 0.165279 0.986247i \(-0.447147\pi\)
0.165279 + 0.986247i \(0.447147\pi\)
\(654\) −14.9575 −0.584885
\(655\) −14.1388 −0.552449
\(656\) −8.18877 −0.319718
\(657\) −14.3502 −0.559855
\(658\) 31.6164 1.23253
\(659\) 36.8682 1.43618 0.718090 0.695950i \(-0.245017\pi\)
0.718090 + 0.695950i \(0.245017\pi\)
\(660\) 4.90148 0.190790
\(661\) −5.28919 −0.205725 −0.102863 0.994696i \(-0.532800\pi\)
−0.102863 + 0.994696i \(0.532800\pi\)
\(662\) −8.63527 −0.335619
\(663\) −13.3940 −0.520180
\(664\) 10.8297 0.420272
\(665\) −5.30540 −0.205734
\(666\) −0.818562 −0.0317186
\(667\) 77.8345 3.01376
\(668\) 12.5322 0.484886
\(669\) −15.7753 −0.609907
\(670\) −15.0024 −0.579593
\(671\) −48.0043 −1.85319
\(672\) 4.65781 0.179679
\(673\) 7.62613 0.293966 0.146983 0.989139i \(-0.453044\pi\)
0.146983 + 0.989139i \(0.453044\pi\)
\(674\) −7.80648 −0.300695
\(675\) 3.70261 0.142513
\(676\) −9.28213 −0.357005
\(677\) −23.6354 −0.908383 −0.454191 0.890904i \(-0.650072\pi\)
−0.454191 + 0.890904i \(0.650072\pi\)
\(678\) −2.53394 −0.0973153
\(679\) −14.3239 −0.549703
\(680\) 7.91224 0.303420
\(681\) 4.98013 0.190839
\(682\) 5.51001 0.210989
\(683\) 45.0734 1.72468 0.862342 0.506326i \(-0.168997\pi\)
0.862342 + 0.506326i \(0.168997\pi\)
\(684\) −1.00000 −0.0382360
\(685\) 2.32542 0.0888498
\(686\) −35.8430 −1.36849
\(687\) −0.629002 −0.0239979
\(688\) −10.6577 −0.406321
\(689\) 1.92818 0.0734577
\(690\) 9.01161 0.343066
\(691\) 0.553888 0.0210709 0.0105355 0.999945i \(-0.496646\pi\)
0.0105355 + 0.999945i \(0.496646\pi\)
\(692\) −21.4077 −0.813800
\(693\) −20.0435 −0.761390
\(694\) −21.0070 −0.797413
\(695\) −18.7881 −0.712674
\(696\) 9.83797 0.372907
\(697\) 56.8829 2.15460
\(698\) 4.89268 0.185191
\(699\) −14.0856 −0.532766
\(700\) −17.2461 −0.651840
\(701\) −46.2402 −1.74647 −0.873233 0.487302i \(-0.837981\pi\)
−0.873233 + 0.487302i \(0.837981\pi\)
\(702\) −1.92818 −0.0727744
\(703\) −0.818562 −0.0308727
\(704\) −4.30320 −0.162183
\(705\) 7.73154 0.291186
\(706\) 7.99387 0.300853
\(707\) −46.6960 −1.75619
\(708\) 6.44178 0.242097
\(709\) 50.6279 1.90137 0.950685 0.310159i \(-0.100382\pi\)
0.950685 + 0.310159i \(0.100382\pi\)
\(710\) 0.181969 0.00682916
\(711\) 8.88482 0.333207
\(712\) 7.73787 0.289989
\(713\) 10.1304 0.379387
\(714\) −32.3553 −1.21087
\(715\) 9.45093 0.353445
\(716\) −0.147626 −0.00551706
\(717\) 22.0582 0.823778
\(718\) 8.97136 0.334808
\(719\) −8.33996 −0.311028 −0.155514 0.987834i \(-0.549703\pi\)
−0.155514 + 0.987834i \(0.549703\pi\)
\(720\) 1.13903 0.0424492
\(721\) 52.9572 1.97223
\(722\) −1.00000 −0.0372161
\(723\) 9.89881 0.368141
\(724\) 19.8585 0.738034
\(725\) −36.4261 −1.35283
\(726\) 7.51753 0.279002
\(727\) 19.3224 0.716627 0.358314 0.933601i \(-0.383352\pi\)
0.358314 + 0.933601i \(0.383352\pi\)
\(728\) 8.98109 0.332861
\(729\) 1.00000 0.0370370
\(730\) 16.3454 0.604969
\(731\) 74.0332 2.73822
\(732\) −11.1555 −0.412319
\(733\) 49.0113 1.81027 0.905137 0.425120i \(-0.139768\pi\)
0.905137 + 0.425120i \(0.139768\pi\)
\(734\) 18.0361 0.665724
\(735\) −16.7383 −0.617403
\(736\) −7.91164 −0.291627
\(737\) −56.6782 −2.08777
\(738\) 8.18877 0.301433
\(739\) 26.8178 0.986509 0.493255 0.869885i \(-0.335807\pi\)
0.493255 + 0.869885i \(0.335807\pi\)
\(740\) 0.932368 0.0342745
\(741\) −1.92818 −0.0708334
\(742\) 4.65781 0.170994
\(743\) −9.16505 −0.336233 −0.168117 0.985767i \(-0.553769\pi\)
−0.168117 + 0.985767i \(0.553769\pi\)
\(744\) 1.28044 0.0469434
\(745\) 3.97334 0.145572
\(746\) −8.01853 −0.293579
\(747\) −10.8297 −0.396236
\(748\) 29.8920 1.09296
\(749\) −39.8511 −1.45613
\(750\) −9.91254 −0.361955
\(751\) 1.03554 0.0377872 0.0188936 0.999821i \(-0.493986\pi\)
0.0188936 + 0.999821i \(0.493986\pi\)
\(752\) −6.78781 −0.247526
\(753\) 11.2298 0.409236
\(754\) 18.9694 0.690823
\(755\) 13.9813 0.508832
\(756\) −4.65781 −0.169403
\(757\) −52.8901 −1.92232 −0.961161 0.275989i \(-0.910995\pi\)
−0.961161 + 0.275989i \(0.910995\pi\)
\(758\) 24.9640 0.906734
\(759\) 34.0454 1.23577
\(760\) 1.13903 0.0413170
\(761\) −3.57290 −0.129517 −0.0647587 0.997901i \(-0.520628\pi\)
−0.0647587 + 0.997901i \(0.520628\pi\)
\(762\) −12.5689 −0.455324
\(763\) −69.6693 −2.52220
\(764\) −1.38890 −0.0502485
\(765\) −7.91224 −0.286068
\(766\) −20.3392 −0.734885
\(767\) 12.4209 0.448492
\(768\) −1.00000 −0.0360844
\(769\) −8.02930 −0.289544 −0.144772 0.989465i \(-0.546245\pi\)
−0.144772 + 0.989465i \(0.546245\pi\)
\(770\) 22.8302 0.822743
\(771\) −5.75460 −0.207247
\(772\) 2.72902 0.0982197
\(773\) −29.3074 −1.05411 −0.527057 0.849830i \(-0.676705\pi\)
−0.527057 + 0.849830i \(0.676705\pi\)
\(774\) 10.6577 0.383083
\(775\) −4.74098 −0.170301
\(776\) 3.07525 0.110395
\(777\) −3.81271 −0.136780
\(778\) −0.516054 −0.0185014
\(779\) 8.18877 0.293393
\(780\) 2.19626 0.0786386
\(781\) 0.687468 0.0245995
\(782\) 54.9579 1.96529
\(783\) −9.83797 −0.351580
\(784\) 14.6952 0.524830
\(785\) 17.6827 0.631122
\(786\) −12.4130 −0.442757
\(787\) −39.8830 −1.42168 −0.710838 0.703355i \(-0.751684\pi\)
−0.710838 + 0.703355i \(0.751684\pi\)
\(788\) 2.54933 0.0908162
\(789\) −14.0528 −0.500292
\(790\) −10.1201 −0.360057
\(791\) −11.8026 −0.419653
\(792\) 4.30320 0.152908
\(793\) −21.5098 −0.763835
\(794\) −9.51624 −0.337719
\(795\) 1.13903 0.0403973
\(796\) −8.95183 −0.317289
\(797\) −7.64697 −0.270870 −0.135435 0.990786i \(-0.543243\pi\)
−0.135435 + 0.990786i \(0.543243\pi\)
\(798\) −4.65781 −0.164885
\(799\) 47.1513 1.66809
\(800\) 3.70261 0.130907
\(801\) −7.73787 −0.273404
\(802\) 12.6804 0.447760
\(803\) 61.7518 2.17918
\(804\) −13.1712 −0.464512
\(805\) 41.9744 1.47940
\(806\) 2.46892 0.0869641
\(807\) 11.9464 0.420532
\(808\) 10.0253 0.352689
\(809\) 28.1399 0.989347 0.494673 0.869079i \(-0.335288\pi\)
0.494673 + 0.869079i \(0.335288\pi\)
\(810\) −1.13903 −0.0400215
\(811\) −20.5745 −0.722470 −0.361235 0.932475i \(-0.617645\pi\)
−0.361235 + 0.932475i \(0.617645\pi\)
\(812\) 45.8234 1.60809
\(813\) 17.8176 0.624891
\(814\) 3.52244 0.123461
\(815\) 5.79429 0.202965
\(816\) 6.94646 0.243175
\(817\) 10.6577 0.372865
\(818\) −33.2874 −1.16387
\(819\) −8.98109 −0.313825
\(820\) −9.32727 −0.325722
\(821\) −17.2291 −0.601298 −0.300649 0.953735i \(-0.597203\pi\)
−0.300649 + 0.953735i \(0.597203\pi\)
\(822\) 2.04158 0.0712082
\(823\) 10.4954 0.365848 0.182924 0.983127i \(-0.441444\pi\)
0.182924 + 0.983127i \(0.441444\pi\)
\(824\) −11.3695 −0.396077
\(825\) −15.9331 −0.554718
\(826\) 30.0046 1.04399
\(827\) 17.9759 0.625082 0.312541 0.949904i \(-0.398820\pi\)
0.312541 + 0.949904i \(0.398820\pi\)
\(828\) 7.91164 0.274949
\(829\) 19.3940 0.673582 0.336791 0.941579i \(-0.390658\pi\)
0.336791 + 0.941579i \(0.390658\pi\)
\(830\) 12.3353 0.428165
\(831\) 25.1299 0.871746
\(832\) −1.92818 −0.0668475
\(833\) −102.080 −3.53686
\(834\) −16.4948 −0.571169
\(835\) 14.2746 0.493993
\(836\) 4.30320 0.148829
\(837\) −1.28044 −0.0442586
\(838\) 24.4159 0.843434
\(839\) 22.8320 0.788249 0.394125 0.919057i \(-0.371048\pi\)
0.394125 + 0.919057i \(0.371048\pi\)
\(840\) 5.30540 0.183054
\(841\) 67.7856 2.33744
\(842\) −37.3518 −1.28723
\(843\) −1.06382 −0.0366400
\(844\) 16.0300 0.551775
\(845\) −10.5726 −0.363710
\(846\) 6.78781 0.233370
\(847\) 35.0153 1.20314
\(848\) −1.00000 −0.0343401
\(849\) 7.83023 0.268733
\(850\) −25.7200 −0.882189
\(851\) 6.47617 0.222000
\(852\) 0.159757 0.00547320
\(853\) 6.14514 0.210405 0.105203 0.994451i \(-0.466451\pi\)
0.105203 + 0.994451i \(0.466451\pi\)
\(854\) −51.9602 −1.77804
\(855\) −1.13903 −0.0389541
\(856\) 8.55575 0.292429
\(857\) 34.8887 1.19178 0.595888 0.803068i \(-0.296800\pi\)
0.595888 + 0.803068i \(0.296800\pi\)
\(858\) 8.29733 0.283266
\(859\) −21.8355 −0.745019 −0.372509 0.928028i \(-0.621503\pi\)
−0.372509 + 0.928028i \(0.621503\pi\)
\(860\) −12.1394 −0.413952
\(861\) 38.1418 1.29987
\(862\) −12.2602 −0.417583
\(863\) −42.0704 −1.43209 −0.716046 0.698053i \(-0.754050\pi\)
−0.716046 + 0.698053i \(0.754050\pi\)
\(864\) 1.00000 0.0340207
\(865\) −24.3841 −0.829083
\(866\) 31.3015 1.06367
\(867\) −31.2533 −1.06142
\(868\) 5.96407 0.202434
\(869\) −38.2332 −1.29697
\(870\) 11.2058 0.379911
\(871\) −25.3964 −0.860524
\(872\) 14.9575 0.506525
\(873\) −3.07525 −0.104081
\(874\) 7.91164 0.267615
\(875\) −46.1708 −1.56086
\(876\) 14.3502 0.484849
\(877\) −38.0029 −1.28327 −0.641634 0.767011i \(-0.721743\pi\)
−0.641634 + 0.767011i \(0.721743\pi\)
\(878\) 25.5266 0.861482
\(879\) 11.6142 0.391736
\(880\) −4.90148 −0.165229
\(881\) 19.0070 0.640362 0.320181 0.947356i \(-0.396256\pi\)
0.320181 + 0.947356i \(0.396256\pi\)
\(882\) −14.6952 −0.494814
\(883\) −58.6118 −1.97244 −0.986222 0.165426i \(-0.947100\pi\)
−0.986222 + 0.165426i \(0.947100\pi\)
\(884\) 13.3940 0.450489
\(885\) 7.33739 0.246644
\(886\) −0.0728399 −0.00244710
\(887\) 51.3140 1.72296 0.861478 0.507796i \(-0.169539\pi\)
0.861478 + 0.507796i \(0.169539\pi\)
\(888\) 0.818562 0.0274692
\(889\) −58.5438 −1.96349
\(890\) 8.81368 0.295435
\(891\) −4.30320 −0.144163
\(892\) 15.7753 0.528195
\(893\) 6.78781 0.227146
\(894\) 3.48835 0.116668
\(895\) −0.168151 −0.00562067
\(896\) −4.65781 −0.155607
\(897\) 15.2551 0.509352
\(898\) −14.6323 −0.488285
\(899\) 12.5970 0.420133
\(900\) −3.70261 −0.123420
\(901\) 6.94646 0.231420
\(902\) −35.2379 −1.17329
\(903\) 49.6415 1.65197
\(904\) 2.53394 0.0842775
\(905\) 22.6194 0.751895
\(906\) 12.2747 0.407801
\(907\) −17.5093 −0.581387 −0.290694 0.956816i \(-0.593886\pi\)
−0.290694 + 0.956816i \(0.593886\pi\)
\(908\) −4.98013 −0.165271
\(909\) −10.0253 −0.332519
\(910\) 10.2297 0.339113
\(911\) −3.07644 −0.101927 −0.0509635 0.998701i \(-0.516229\pi\)
−0.0509635 + 0.998701i \(0.516229\pi\)
\(912\) 1.00000 0.0331133
\(913\) 46.6022 1.54231
\(914\) −20.0024 −0.661622
\(915\) −12.7065 −0.420063
\(916\) 0.629002 0.0207828
\(917\) −57.8175 −1.90930
\(918\) −6.94646 −0.229267
\(919\) −25.4540 −0.839651 −0.419826 0.907605i \(-0.637909\pi\)
−0.419826 + 0.907605i \(0.637909\pi\)
\(920\) −9.01161 −0.297104
\(921\) 28.7680 0.947937
\(922\) −17.7467 −0.584457
\(923\) 0.308041 0.0101393
\(924\) 20.0435 0.659383
\(925\) −3.03081 −0.0996526
\(926\) 4.84490 0.159213
\(927\) 11.3695 0.373425
\(928\) −9.83797 −0.322947
\(929\) 28.6295 0.939304 0.469652 0.882852i \(-0.344379\pi\)
0.469652 + 0.882852i \(0.344379\pi\)
\(930\) 1.45847 0.0478250
\(931\) −14.6952 −0.481617
\(932\) 14.0856 0.461389
\(933\) −24.8634 −0.813991
\(934\) 27.1798 0.889351
\(935\) 34.0479 1.11349
\(936\) 1.92818 0.0630245
\(937\) 26.4693 0.864716 0.432358 0.901702i \(-0.357682\pi\)
0.432358 + 0.901702i \(0.357682\pi\)
\(938\) −61.3489 −2.00311
\(939\) 26.8861 0.877394
\(940\) −7.73154 −0.252175
\(941\) 47.0663 1.53432 0.767159 0.641457i \(-0.221670\pi\)
0.767159 + 0.641457i \(0.221670\pi\)
\(942\) 15.5243 0.505810
\(943\) −64.7866 −2.10974
\(944\) −6.44178 −0.209662
\(945\) −5.30540 −0.172585
\(946\) −45.8622 −1.49111
\(947\) 6.08820 0.197840 0.0989199 0.995095i \(-0.468461\pi\)
0.0989199 + 0.995095i \(0.468461\pi\)
\(948\) −8.88482 −0.288566
\(949\) 27.6698 0.898199
\(950\) −3.70261 −0.120128
\(951\) 9.22070 0.299002
\(952\) 32.3553 1.04864
\(953\) 16.9019 0.547506 0.273753 0.961800i \(-0.411735\pi\)
0.273753 + 0.961800i \(0.411735\pi\)
\(954\) 1.00000 0.0323762
\(955\) −1.58200 −0.0511922
\(956\) −22.0582 −0.713412
\(957\) 42.3347 1.36849
\(958\) 32.4135 1.04723
\(959\) 9.50929 0.307071
\(960\) −1.13903 −0.0367621
\(961\) −29.3605 −0.947112
\(962\) 1.57833 0.0508875
\(963\) −8.55575 −0.275705
\(964\) −9.89881 −0.318819
\(965\) 3.10844 0.100064
\(966\) 36.8510 1.18566
\(967\) −25.6436 −0.824641 −0.412321 0.911039i \(-0.635282\pi\)
−0.412321 + 0.911039i \(0.635282\pi\)
\(968\) −7.51753 −0.241623
\(969\) −6.94646 −0.223152
\(970\) 3.50281 0.112468
\(971\) −56.2777 −1.80604 −0.903018 0.429602i \(-0.858654\pi\)
−0.903018 + 0.429602i \(0.858654\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −76.8298 −2.46305
\(974\) −10.7839 −0.345538
\(975\) −7.13928 −0.228640
\(976\) 11.1555 0.357079
\(977\) 43.3750 1.38769 0.693845 0.720125i \(-0.255915\pi\)
0.693845 + 0.720125i \(0.255915\pi\)
\(978\) 5.08704 0.162665
\(979\) 33.2976 1.06420
\(980\) 16.7383 0.534686
\(981\) −14.9575 −0.477557
\(982\) 9.33661 0.297943
\(983\) 12.3347 0.393415 0.196707 0.980462i \(-0.436975\pi\)
0.196707 + 0.980462i \(0.436975\pi\)
\(984\) −8.18877 −0.261048
\(985\) 2.90377 0.0925219
\(986\) 68.3390 2.17636
\(987\) 31.6164 1.00636
\(988\) 1.92818 0.0613435
\(989\) −84.3199 −2.68122
\(990\) 4.90148 0.155779
\(991\) −28.2911 −0.898697 −0.449349 0.893357i \(-0.648344\pi\)
−0.449349 + 0.893357i \(0.648344\pi\)
\(992\) −1.28044 −0.0406541
\(993\) −8.63527 −0.274032
\(994\) 0.744120 0.0236021
\(995\) −10.1964 −0.323248
\(996\) 10.8297 0.343151
\(997\) 9.80170 0.310423 0.155211 0.987881i \(-0.450394\pi\)
0.155211 + 0.987881i \(0.450394\pi\)
\(998\) 28.6119 0.905695
\(999\) −0.818562 −0.0258982
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.1.9 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.be.1.9 12 1.1 even 1 trivial