Properties

Label 6042.2.a.bf.1.12
Level $6042$
Weight $2$
Character 6042.1
Self dual yes
Analytic conductor $48.246$
Analytic rank $0$
Dimension $12$
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] = [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: \(0\)
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: \( x^{12} - 5 x^{11} - 27 x^{10} + 134 x^{9} + 294 x^{8} - 1313 x^{7} - 1685 x^{6} + 5910 x^{5} + 5237 x^{4} - 12206 x^{3} - 7876 x^{2} + 9264 x + 4048 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.12
Root \(4.49493\) 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)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.49493 q^{5} -1.00000 q^{6} +0.415035 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +3.49493 q^{5} -1.00000 q^{6} +0.415035 q^{7} -1.00000 q^{8} +1.00000 q^{9} -3.49493 q^{10} -4.08684 q^{11} +1.00000 q^{12} -4.91017 q^{13} -0.415035 q^{14} +3.49493 q^{15} +1.00000 q^{16} +5.09702 q^{17} -1.00000 q^{18} +1.00000 q^{19} +3.49493 q^{20} +0.415035 q^{21} +4.08684 q^{22} +1.96030 q^{23} -1.00000 q^{24} +7.21452 q^{25} +4.91017 q^{26} +1.00000 q^{27} +0.415035 q^{28} +7.83262 q^{29} -3.49493 q^{30} +4.64876 q^{31} -1.00000 q^{32} -4.08684 q^{33} -5.09702 q^{34} +1.45052 q^{35} +1.00000 q^{36} +3.20548 q^{37} -1.00000 q^{38} -4.91017 q^{39} -3.49493 q^{40} +0.147607 q^{41} -0.415035 q^{42} +0.0396988 q^{43} -4.08684 q^{44} +3.49493 q^{45} -1.96030 q^{46} -0.0503944 q^{47} +1.00000 q^{48} -6.82775 q^{49} -7.21452 q^{50} +5.09702 q^{51} -4.91017 q^{52} +1.00000 q^{53} -1.00000 q^{54} -14.2832 q^{55} -0.415035 q^{56} +1.00000 q^{57} -7.83262 q^{58} -11.4052 q^{59} +3.49493 q^{60} +6.82283 q^{61} -4.64876 q^{62} +0.415035 q^{63} +1.00000 q^{64} -17.1607 q^{65} +4.08684 q^{66} +2.20770 q^{67} +5.09702 q^{68} +1.96030 q^{69} -1.45052 q^{70} +13.8103 q^{71} -1.00000 q^{72} -16.3320 q^{73} -3.20548 q^{74} +7.21452 q^{75} +1.00000 q^{76} -1.69618 q^{77} +4.91017 q^{78} +15.7254 q^{79} +3.49493 q^{80} +1.00000 q^{81} -0.147607 q^{82} +8.45796 q^{83} +0.415035 q^{84} +17.8137 q^{85} -0.0396988 q^{86} +7.83262 q^{87} +4.08684 q^{88} -9.19983 q^{89} -3.49493 q^{90} -2.03789 q^{91} +1.96030 q^{92} +4.64876 q^{93} +0.0503944 q^{94} +3.49493 q^{95} -1.00000 q^{96} -11.2553 q^{97} +6.82775 q^{98} -4.08684 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 12 q - 12 q^{2} + 12 q^{3} + 12 q^{4} - 7 q^{5} - 12 q^{6} + 5 q^{7} - 12 q^{8} + 12 q^{9} + 7 q^{10} - 5 q^{11} + 12 q^{12} + 10 q^{13} - 5 q^{14} - 7 q^{15} + 12 q^{16} + 6 q^{17} - 12 q^{18} + 12 q^{19} - 7 q^{20} + 5 q^{21} + 5 q^{22} - 12 q^{24} + 21 q^{25} - 10 q^{26} + 12 q^{27} + 5 q^{28} + 7 q^{30} + 2 q^{31} - 12 q^{32} - 5 q^{33} - 6 q^{34} - 17 q^{35} + 12 q^{36} + 16 q^{37} - 12 q^{38} + 10 q^{39} + 7 q^{40} + 7 q^{41} - 5 q^{42} + 24 q^{43} - 5 q^{44} - 7 q^{45} + 4 q^{47} + 12 q^{48} + 29 q^{49} - 21 q^{50} + 6 q^{51} + 10 q^{52} + 12 q^{53} - 12 q^{54} - 2 q^{55} - 5 q^{56} + 12 q^{57} + 2 q^{59} - 7 q^{60} + 29 q^{61} - 2 q^{62} + 5 q^{63} + 12 q^{64} - 17 q^{65} + 5 q^{66} + 36 q^{67} + 6 q^{68} + 17 q^{70} - 3 q^{71} - 12 q^{72} + 7 q^{73} - 16 q^{74} + 21 q^{75} + 12 q^{76} - 35 q^{77} - 10 q^{78} + 40 q^{79} - 7 q^{80} + 12 q^{81} - 7 q^{82} + 24 q^{83} + 5 q^{84} - 22 q^{85} - 24 q^{86} + 5 q^{88} - 45 q^{89} + 7 q^{90} + 71 q^{91} + 2 q^{93} - 4 q^{94} - 7 q^{95} - 12 q^{96} + q^{97} - 29 q^{98} - 5 q^{99}+O(q^{100}) \) 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
\(5\) 3.49493 1.56298 0.781490 0.623918i \(-0.214460\pi\)
0.781490 + 0.623918i \(0.214460\pi\)
\(6\) −1.00000 −0.408248
\(7\) 0.415035 0.156869 0.0784343 0.996919i \(-0.475008\pi\)
0.0784343 + 0.996919i \(0.475008\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) −3.49493 −1.10519
\(11\) −4.08684 −1.23223 −0.616114 0.787657i \(-0.711294\pi\)
−0.616114 + 0.787657i \(0.711294\pi\)
\(12\) 1.00000 0.288675
\(13\) −4.91017 −1.36184 −0.680918 0.732360i \(-0.738419\pi\)
−0.680918 + 0.732360i \(0.738419\pi\)
\(14\) −0.415035 −0.110923
\(15\) 3.49493 0.902387
\(16\) 1.00000 0.250000
\(17\) 5.09702 1.23621 0.618105 0.786096i \(-0.287901\pi\)
0.618105 + 0.786096i \(0.287901\pi\)
\(18\) −1.00000 −0.235702
\(19\) 1.00000 0.229416
\(20\) 3.49493 0.781490
\(21\) 0.415035 0.0905681
\(22\) 4.08684 0.871317
\(23\) 1.96030 0.408751 0.204376 0.978893i \(-0.434484\pi\)
0.204376 + 0.978893i \(0.434484\pi\)
\(24\) −1.00000 −0.204124
\(25\) 7.21452 1.44290
\(26\) 4.91017 0.962963
\(27\) 1.00000 0.192450
\(28\) 0.415035 0.0784343
\(29\) 7.83262 1.45448 0.727241 0.686383i \(-0.240802\pi\)
0.727241 + 0.686383i \(0.240802\pi\)
\(30\) −3.49493 −0.638084
\(31\) 4.64876 0.834941 0.417471 0.908690i \(-0.362917\pi\)
0.417471 + 0.908690i \(0.362917\pi\)
\(32\) −1.00000 −0.176777
\(33\) −4.08684 −0.711428
\(34\) −5.09702 −0.874132
\(35\) 1.45052 0.245182
\(36\) 1.00000 0.166667
\(37\) 3.20548 0.526978 0.263489 0.964662i \(-0.415127\pi\)
0.263489 + 0.964662i \(0.415127\pi\)
\(38\) −1.00000 −0.162221
\(39\) −4.91017 −0.786256
\(40\) −3.49493 −0.552597
\(41\) 0.147607 0.0230524 0.0115262 0.999934i \(-0.496331\pi\)
0.0115262 + 0.999934i \(0.496331\pi\)
\(42\) −0.415035 −0.0640413
\(43\) 0.0396988 0.00605401 0.00302700 0.999995i \(-0.499036\pi\)
0.00302700 + 0.999995i \(0.499036\pi\)
\(44\) −4.08684 −0.616114
\(45\) 3.49493 0.520993
\(46\) −1.96030 −0.289031
\(47\) −0.0503944 −0.00735077 −0.00367539 0.999993i \(-0.501170\pi\)
−0.00367539 + 0.999993i \(0.501170\pi\)
\(48\) 1.00000 0.144338
\(49\) −6.82775 −0.975392
\(50\) −7.21452 −1.02029
\(51\) 5.09702 0.713726
\(52\) −4.91017 −0.680918
\(53\) 1.00000 0.137361
\(54\) −1.00000 −0.136083
\(55\) −14.2832 −1.92595
\(56\) −0.415035 −0.0554614
\(57\) 1.00000 0.132453
\(58\) −7.83262 −1.02847
\(59\) −11.4052 −1.48483 −0.742417 0.669938i \(-0.766321\pi\)
−0.742417 + 0.669938i \(0.766321\pi\)
\(60\) 3.49493 0.451193
\(61\) 6.82283 0.873574 0.436787 0.899565i \(-0.356116\pi\)
0.436787 + 0.899565i \(0.356116\pi\)
\(62\) −4.64876 −0.590393
\(63\) 0.415035 0.0522895
\(64\) 1.00000 0.125000
\(65\) −17.1607 −2.12852
\(66\) 4.08684 0.503055
\(67\) 2.20770 0.269714 0.134857 0.990865i \(-0.456942\pi\)
0.134857 + 0.990865i \(0.456942\pi\)
\(68\) 5.09702 0.618105
\(69\) 1.96030 0.235993
\(70\) −1.45052 −0.173370
\(71\) 13.8103 1.63899 0.819493 0.573089i \(-0.194255\pi\)
0.819493 + 0.573089i \(0.194255\pi\)
\(72\) −1.00000 −0.117851
\(73\) −16.3320 −1.91151 −0.955757 0.294157i \(-0.904961\pi\)
−0.955757 + 0.294157i \(0.904961\pi\)
\(74\) −3.20548 −0.372630
\(75\) 7.21452 0.833061
\(76\) 1.00000 0.114708
\(77\) −1.69618 −0.193298
\(78\) 4.91017 0.555967
\(79\) 15.7254 1.76925 0.884624 0.466305i \(-0.154415\pi\)
0.884624 + 0.466305i \(0.154415\pi\)
\(80\) 3.49493 0.390745
\(81\) 1.00000 0.111111
\(82\) −0.147607 −0.0163005
\(83\) 8.45796 0.928382 0.464191 0.885735i \(-0.346345\pi\)
0.464191 + 0.885735i \(0.346345\pi\)
\(84\) 0.415035 0.0452841
\(85\) 17.8137 1.93217
\(86\) −0.0396988 −0.00428083
\(87\) 7.83262 0.839745
\(88\) 4.08684 0.435659
\(89\) −9.19983 −0.975180 −0.487590 0.873073i \(-0.662124\pi\)
−0.487590 + 0.873073i \(0.662124\pi\)
\(90\) −3.49493 −0.368398
\(91\) −2.03789 −0.213629
\(92\) 1.96030 0.204376
\(93\) 4.64876 0.482054
\(94\) 0.0503944 0.00519778
\(95\) 3.49493 0.358572
\(96\) −1.00000 −0.102062
\(97\) −11.2553 −1.14280 −0.571400 0.820672i \(-0.693600\pi\)
−0.571400 + 0.820672i \(0.693600\pi\)
\(98\) 6.82775 0.689706
\(99\) −4.08684 −0.410743
\(100\) 7.21452 0.721452
\(101\) −5.72812 −0.569969 −0.284985 0.958532i \(-0.591988\pi\)
−0.284985 + 0.958532i \(0.591988\pi\)
\(102\) −5.09702 −0.504680
\(103\) 16.7849 1.65386 0.826932 0.562302i \(-0.190084\pi\)
0.826932 + 0.562302i \(0.190084\pi\)
\(104\) 4.91017 0.481482
\(105\) 1.45052 0.141556
\(106\) −1.00000 −0.0971286
\(107\) 13.4762 1.30279 0.651396 0.758738i \(-0.274184\pi\)
0.651396 + 0.758738i \(0.274184\pi\)
\(108\) 1.00000 0.0962250
\(109\) −4.89315 −0.468679 −0.234340 0.972155i \(-0.575293\pi\)
−0.234340 + 0.972155i \(0.575293\pi\)
\(110\) 14.2832 1.36185
\(111\) 3.20548 0.304251
\(112\) 0.415035 0.0392171
\(113\) 14.2894 1.34424 0.672119 0.740443i \(-0.265384\pi\)
0.672119 + 0.740443i \(0.265384\pi\)
\(114\) −1.00000 −0.0936586
\(115\) 6.85111 0.638869
\(116\) 7.83262 0.727241
\(117\) −4.91017 −0.453945
\(118\) 11.4052 1.04994
\(119\) 2.11544 0.193922
\(120\) −3.49493 −0.319042
\(121\) 5.70226 0.518388
\(122\) −6.82283 −0.617710
\(123\) 0.147607 0.0133093
\(124\) 4.64876 0.417471
\(125\) 7.73960 0.692251
\(126\) −0.415035 −0.0369743
\(127\) 14.7621 1.30992 0.654962 0.755661i \(-0.272684\pi\)
0.654962 + 0.755661i \(0.272684\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0.0396988 0.00349528
\(130\) 17.1607 1.50509
\(131\) 10.9973 0.960836 0.480418 0.877040i \(-0.340485\pi\)
0.480418 + 0.877040i \(0.340485\pi\)
\(132\) −4.08684 −0.355714
\(133\) 0.415035 0.0359881
\(134\) −2.20770 −0.190717
\(135\) 3.49493 0.300796
\(136\) −5.09702 −0.437066
\(137\) −16.6482 −1.42235 −0.711176 0.703014i \(-0.751837\pi\)
−0.711176 + 0.703014i \(0.751837\pi\)
\(138\) −1.96030 −0.166872
\(139\) 4.76043 0.403775 0.201887 0.979409i \(-0.435293\pi\)
0.201887 + 0.979409i \(0.435293\pi\)
\(140\) 1.45052 0.122591
\(141\) −0.0503944 −0.00424397
\(142\) −13.8103 −1.15894
\(143\) 20.0671 1.67809
\(144\) 1.00000 0.0833333
\(145\) 27.3745 2.27332
\(146\) 16.3320 1.35164
\(147\) −6.82775 −0.563143
\(148\) 3.20548 0.263489
\(149\) −17.0799 −1.39924 −0.699620 0.714515i \(-0.746647\pi\)
−0.699620 + 0.714515i \(0.746647\pi\)
\(150\) −7.21452 −0.589063
\(151\) 14.1138 1.14857 0.574284 0.818656i \(-0.305281\pi\)
0.574284 + 0.818656i \(0.305281\pi\)
\(152\) −1.00000 −0.0811107
\(153\) 5.09702 0.412070
\(154\) 1.69618 0.136682
\(155\) 16.2471 1.30500
\(156\) −4.91017 −0.393128
\(157\) 9.41661 0.751527 0.375764 0.926716i \(-0.377380\pi\)
0.375764 + 0.926716i \(0.377380\pi\)
\(158\) −15.7254 −1.25105
\(159\) 1.00000 0.0793052
\(160\) −3.49493 −0.276298
\(161\) 0.813594 0.0641202
\(162\) −1.00000 −0.0785674
\(163\) −17.2516 −1.35125 −0.675627 0.737244i \(-0.736127\pi\)
−0.675627 + 0.737244i \(0.736127\pi\)
\(164\) 0.147607 0.0115262
\(165\) −14.2832 −1.11195
\(166\) −8.45796 −0.656465
\(167\) −3.67750 −0.284573 −0.142287 0.989825i \(-0.545445\pi\)
−0.142287 + 0.989825i \(0.545445\pi\)
\(168\) −0.415035 −0.0320207
\(169\) 11.1098 0.854596
\(170\) −17.8137 −1.36625
\(171\) 1.00000 0.0764719
\(172\) 0.0396988 0.00302700
\(173\) 21.5612 1.63927 0.819635 0.572886i \(-0.194176\pi\)
0.819635 + 0.572886i \(0.194176\pi\)
\(174\) −7.83262 −0.593790
\(175\) 2.99428 0.226346
\(176\) −4.08684 −0.308057
\(177\) −11.4052 −0.857269
\(178\) 9.19983 0.689556
\(179\) −15.5774 −1.16431 −0.582153 0.813079i \(-0.697790\pi\)
−0.582153 + 0.813079i \(0.697790\pi\)
\(180\) 3.49493 0.260497
\(181\) −5.32170 −0.395559 −0.197779 0.980247i \(-0.563373\pi\)
−0.197779 + 0.980247i \(0.563373\pi\)
\(182\) 2.03789 0.151059
\(183\) 6.82283 0.504358
\(184\) −1.96030 −0.144515
\(185\) 11.2029 0.823656
\(186\) −4.64876 −0.340863
\(187\) −20.8307 −1.52329
\(188\) −0.0503944 −0.00367539
\(189\) 0.415035 0.0301894
\(190\) −3.49493 −0.253549
\(191\) −10.8987 −0.788599 −0.394300 0.918982i \(-0.629013\pi\)
−0.394300 + 0.918982i \(0.629013\pi\)
\(192\) 1.00000 0.0721688
\(193\) 9.27849 0.667880 0.333940 0.942594i \(-0.391622\pi\)
0.333940 + 0.942594i \(0.391622\pi\)
\(194\) 11.2553 0.808082
\(195\) −17.1607 −1.22890
\(196\) −6.82775 −0.487696
\(197\) 0.0491723 0.00350338 0.00175169 0.999998i \(-0.499442\pi\)
0.00175169 + 0.999998i \(0.499442\pi\)
\(198\) 4.08684 0.290439
\(199\) 20.2449 1.43512 0.717562 0.696495i \(-0.245258\pi\)
0.717562 + 0.696495i \(0.245258\pi\)
\(200\) −7.21452 −0.510144
\(201\) 2.20770 0.155719
\(202\) 5.72812 0.403029
\(203\) 3.25081 0.228162
\(204\) 5.09702 0.356863
\(205\) 0.515877 0.0360304
\(206\) −16.7849 −1.16946
\(207\) 1.96030 0.136250
\(208\) −4.91017 −0.340459
\(209\) −4.08684 −0.282693
\(210\) −1.45052 −0.100095
\(211\) 19.2314 1.32394 0.661972 0.749528i \(-0.269720\pi\)
0.661972 + 0.749528i \(0.269720\pi\)
\(212\) 1.00000 0.0686803
\(213\) 13.8103 0.946269
\(214\) −13.4762 −0.921213
\(215\) 0.138744 0.00946229
\(216\) −1.00000 −0.0680414
\(217\) 1.92940 0.130976
\(218\) 4.89315 0.331406
\(219\) −16.3320 −1.10361
\(220\) −14.2832 −0.962974
\(221\) −25.0272 −1.68351
\(222\) −3.20548 −0.215138
\(223\) −8.05779 −0.539589 −0.269795 0.962918i \(-0.586956\pi\)
−0.269795 + 0.962918i \(0.586956\pi\)
\(224\) −0.415035 −0.0277307
\(225\) 7.21452 0.480968
\(226\) −14.2894 −0.950520
\(227\) 16.8358 1.11743 0.558714 0.829360i \(-0.311295\pi\)
0.558714 + 0.829360i \(0.311295\pi\)
\(228\) 1.00000 0.0662266
\(229\) 9.71143 0.641750 0.320875 0.947122i \(-0.396023\pi\)
0.320875 + 0.947122i \(0.396023\pi\)
\(230\) −6.85111 −0.451749
\(231\) −1.69618 −0.111601
\(232\) −7.83262 −0.514237
\(233\) 12.5612 0.822913 0.411457 0.911429i \(-0.365020\pi\)
0.411457 + 0.911429i \(0.365020\pi\)
\(234\) 4.91017 0.320988
\(235\) −0.176125 −0.0114891
\(236\) −11.4052 −0.742417
\(237\) 15.7254 1.02148
\(238\) −2.11544 −0.137124
\(239\) 0.440965 0.0285236 0.0142618 0.999898i \(-0.495460\pi\)
0.0142618 + 0.999898i \(0.495460\pi\)
\(240\) 3.49493 0.225597
\(241\) 5.43948 0.350388 0.175194 0.984534i \(-0.443945\pi\)
0.175194 + 0.984534i \(0.443945\pi\)
\(242\) −5.70226 −0.366555
\(243\) 1.00000 0.0641500
\(244\) 6.82283 0.436787
\(245\) −23.8625 −1.52452
\(246\) −0.147607 −0.00941110
\(247\) −4.91017 −0.312427
\(248\) −4.64876 −0.295196
\(249\) 8.45796 0.536002
\(250\) −7.73960 −0.489495
\(251\) −3.56968 −0.225316 −0.112658 0.993634i \(-0.535936\pi\)
−0.112658 + 0.993634i \(0.535936\pi\)
\(252\) 0.415035 0.0261448
\(253\) −8.01144 −0.503675
\(254\) −14.7621 −0.926257
\(255\) 17.8137 1.11554
\(256\) 1.00000 0.0625000
\(257\) −9.34972 −0.583220 −0.291610 0.956537i \(-0.594191\pi\)
−0.291610 + 0.956537i \(0.594191\pi\)
\(258\) −0.0396988 −0.00247154
\(259\) 1.33039 0.0826664
\(260\) −17.1607 −1.06426
\(261\) 7.83262 0.484827
\(262\) −10.9973 −0.679414
\(263\) 21.4780 1.32439 0.662194 0.749332i \(-0.269625\pi\)
0.662194 + 0.749332i \(0.269625\pi\)
\(264\) 4.08684 0.251528
\(265\) 3.49493 0.214692
\(266\) −0.415035 −0.0254474
\(267\) −9.19983 −0.563020
\(268\) 2.20770 0.134857
\(269\) 4.13801 0.252299 0.126149 0.992011i \(-0.459738\pi\)
0.126149 + 0.992011i \(0.459738\pi\)
\(270\) −3.49493 −0.212695
\(271\) −16.8861 −1.02576 −0.512880 0.858460i \(-0.671421\pi\)
−0.512880 + 0.858460i \(0.671421\pi\)
\(272\) 5.09702 0.309052
\(273\) −2.03789 −0.123339
\(274\) 16.6482 1.00575
\(275\) −29.4846 −1.77799
\(276\) 1.96030 0.117996
\(277\) 3.23480 0.194361 0.0971803 0.995267i \(-0.469018\pi\)
0.0971803 + 0.995267i \(0.469018\pi\)
\(278\) −4.76043 −0.285512
\(279\) 4.64876 0.278314
\(280\) −1.45052 −0.0866851
\(281\) −30.2443 −1.80422 −0.902111 0.431505i \(-0.857983\pi\)
−0.902111 + 0.431505i \(0.857983\pi\)
\(282\) 0.0503944 0.00300094
\(283\) −16.9773 −1.00920 −0.504599 0.863354i \(-0.668359\pi\)
−0.504599 + 0.863354i \(0.668359\pi\)
\(284\) 13.8103 0.819493
\(285\) 3.49493 0.207022
\(286\) −20.0671 −1.18659
\(287\) 0.0612622 0.00361619
\(288\) −1.00000 −0.0589256
\(289\) 8.97963 0.528214
\(290\) −27.3745 −1.60748
\(291\) −11.2553 −0.659796
\(292\) −16.3320 −0.955757
\(293\) −9.13408 −0.533619 −0.266809 0.963749i \(-0.585969\pi\)
−0.266809 + 0.963749i \(0.585969\pi\)
\(294\) 6.82775 0.398202
\(295\) −39.8604 −2.32076
\(296\) −3.20548 −0.186315
\(297\) −4.08684 −0.237143
\(298\) 17.0799 0.989412
\(299\) −9.62541 −0.556652
\(300\) 7.21452 0.416531
\(301\) 0.0164764 0.000949683 0
\(302\) −14.1138 −0.812160
\(303\) −5.72812 −0.329072
\(304\) 1.00000 0.0573539
\(305\) 23.8453 1.36538
\(306\) −5.09702 −0.291377
\(307\) 23.8662 1.36212 0.681059 0.732229i \(-0.261520\pi\)
0.681059 + 0.732229i \(0.261520\pi\)
\(308\) −1.69618 −0.0966490
\(309\) 16.7849 0.954859
\(310\) −16.2471 −0.922772
\(311\) 6.45660 0.366120 0.183060 0.983102i \(-0.441400\pi\)
0.183060 + 0.983102i \(0.441400\pi\)
\(312\) 4.91017 0.277984
\(313\) −10.2086 −0.577021 −0.288511 0.957477i \(-0.593160\pi\)
−0.288511 + 0.957477i \(0.593160\pi\)
\(314\) −9.41661 −0.531410
\(315\) 1.45052 0.0817275
\(316\) 15.7254 0.884624
\(317\) −15.9024 −0.893168 −0.446584 0.894742i \(-0.647360\pi\)
−0.446584 + 0.894742i \(0.647360\pi\)
\(318\) −1.00000 −0.0560772
\(319\) −32.0107 −1.79225
\(320\) 3.49493 0.195372
\(321\) 13.4762 0.752167
\(322\) −0.813594 −0.0453398
\(323\) 5.09702 0.283606
\(324\) 1.00000 0.0555556
\(325\) −35.4245 −1.96500
\(326\) 17.2516 0.955480
\(327\) −4.89315 −0.270592
\(328\) −0.147607 −0.00815025
\(329\) −0.0209154 −0.00115311
\(330\) 14.2832 0.786265
\(331\) 24.0475 1.32177 0.660886 0.750487i \(-0.270181\pi\)
0.660886 + 0.750487i \(0.270181\pi\)
\(332\) 8.45796 0.464191
\(333\) 3.20548 0.175659
\(334\) 3.67750 0.201224
\(335\) 7.71577 0.421558
\(336\) 0.415035 0.0226420
\(337\) 3.33476 0.181656 0.0908280 0.995867i \(-0.471049\pi\)
0.0908280 + 0.995867i \(0.471049\pi\)
\(338\) −11.1098 −0.604291
\(339\) 14.2894 0.776096
\(340\) 17.8137 0.966085
\(341\) −18.9987 −1.02884
\(342\) −1.00000 −0.0540738
\(343\) −5.73900 −0.309877
\(344\) −0.0396988 −0.00214041
\(345\) 6.85111 0.368851
\(346\) −21.5612 −1.15914
\(347\) −35.8949 −1.92694 −0.963469 0.267819i \(-0.913697\pi\)
−0.963469 + 0.267819i \(0.913697\pi\)
\(348\) 7.83262 0.419873
\(349\) 17.3197 0.927100 0.463550 0.886071i \(-0.346575\pi\)
0.463550 + 0.886071i \(0.346575\pi\)
\(350\) −2.99428 −0.160051
\(351\) −4.91017 −0.262085
\(352\) 4.08684 0.217829
\(353\) −15.9809 −0.850575 −0.425288 0.905058i \(-0.639827\pi\)
−0.425288 + 0.905058i \(0.639827\pi\)
\(354\) 11.4052 0.606181
\(355\) 48.2662 2.56170
\(356\) −9.19983 −0.487590
\(357\) 2.11544 0.111961
\(358\) 15.5774 0.823289
\(359\) −19.6405 −1.03659 −0.518294 0.855203i \(-0.673433\pi\)
−0.518294 + 0.855203i \(0.673433\pi\)
\(360\) −3.49493 −0.184199
\(361\) 1.00000 0.0526316
\(362\) 5.32170 0.279702
\(363\) 5.70226 0.299291
\(364\) −2.03789 −0.106815
\(365\) −57.0791 −2.98766
\(366\) −6.82283 −0.356635
\(367\) −5.75736 −0.300532 −0.150266 0.988646i \(-0.548013\pi\)
−0.150266 + 0.988646i \(0.548013\pi\)
\(368\) 1.96030 0.102188
\(369\) 0.147607 0.00768413
\(370\) −11.2029 −0.582413
\(371\) 0.415035 0.0215476
\(372\) 4.64876 0.241027
\(373\) 27.0308 1.39960 0.699801 0.714338i \(-0.253272\pi\)
0.699801 + 0.714338i \(0.253272\pi\)
\(374\) 20.8307 1.07713
\(375\) 7.73960 0.399671
\(376\) 0.0503944 0.00259889
\(377\) −38.4595 −1.98076
\(378\) −0.415035 −0.0213471
\(379\) 16.1485 0.829494 0.414747 0.909937i \(-0.363870\pi\)
0.414747 + 0.909937i \(0.363870\pi\)
\(380\) 3.49493 0.179286
\(381\) 14.7621 0.756285
\(382\) 10.8987 0.557624
\(383\) 28.1444 1.43811 0.719057 0.694951i \(-0.244574\pi\)
0.719057 + 0.694951i \(0.244574\pi\)
\(384\) −1.00000 −0.0510310
\(385\) −5.92804 −0.302121
\(386\) −9.27849 −0.472263
\(387\) 0.0396988 0.00201800
\(388\) −11.2553 −0.571400
\(389\) 26.2108 1.32894 0.664469 0.747316i \(-0.268658\pi\)
0.664469 + 0.747316i \(0.268658\pi\)
\(390\) 17.1607 0.868965
\(391\) 9.99170 0.505302
\(392\) 6.82775 0.344853
\(393\) 10.9973 0.554739
\(394\) −0.0491723 −0.00247727
\(395\) 54.9592 2.76530
\(396\) −4.08684 −0.205371
\(397\) 10.5870 0.531348 0.265674 0.964063i \(-0.414405\pi\)
0.265674 + 0.964063i \(0.414405\pi\)
\(398\) −20.2449 −1.01479
\(399\) 0.415035 0.0207778
\(400\) 7.21452 0.360726
\(401\) −3.50880 −0.175221 −0.0876105 0.996155i \(-0.527923\pi\)
−0.0876105 + 0.996155i \(0.527923\pi\)
\(402\) −2.20770 −0.110110
\(403\) −22.8262 −1.13705
\(404\) −5.72812 −0.284985
\(405\) 3.49493 0.173664
\(406\) −3.25081 −0.161335
\(407\) −13.1003 −0.649358
\(408\) −5.09702 −0.252340
\(409\) −15.8684 −0.784644 −0.392322 0.919828i \(-0.628328\pi\)
−0.392322 + 0.919828i \(0.628328\pi\)
\(410\) −0.515877 −0.0254773
\(411\) −16.6482 −0.821195
\(412\) 16.7849 0.826932
\(413\) −4.73357 −0.232924
\(414\) −1.96030 −0.0963435
\(415\) 29.5600 1.45104
\(416\) 4.91017 0.240741
\(417\) 4.76043 0.233119
\(418\) 4.08684 0.199894
\(419\) 37.5886 1.83632 0.918162 0.396205i \(-0.129673\pi\)
0.918162 + 0.396205i \(0.129673\pi\)
\(420\) 1.45052 0.0707781
\(421\) 8.13549 0.396500 0.198250 0.980152i \(-0.436474\pi\)
0.198250 + 0.980152i \(0.436474\pi\)
\(422\) −19.2314 −0.936170
\(423\) −0.0503944 −0.00245026
\(424\) −1.00000 −0.0485643
\(425\) 36.7726 1.78373
\(426\) −13.8103 −0.669113
\(427\) 2.83172 0.137036
\(428\) 13.4762 0.651396
\(429\) 20.0671 0.968847
\(430\) −0.138744 −0.00669085
\(431\) −20.9723 −1.01020 −0.505100 0.863061i \(-0.668544\pi\)
−0.505100 + 0.863061i \(0.668544\pi\)
\(432\) 1.00000 0.0481125
\(433\) 24.2854 1.16708 0.583540 0.812084i \(-0.301667\pi\)
0.583540 + 0.812084i \(0.301667\pi\)
\(434\) −1.92940 −0.0926141
\(435\) 27.3745 1.31250
\(436\) −4.89315 −0.234340
\(437\) 1.96030 0.0937739
\(438\) 16.3320 0.780373
\(439\) 10.4349 0.498033 0.249016 0.968499i \(-0.419893\pi\)
0.249016 + 0.968499i \(0.419893\pi\)
\(440\) 14.2832 0.680925
\(441\) −6.82775 −0.325131
\(442\) 25.0272 1.19042
\(443\) −36.6413 −1.74088 −0.870440 0.492275i \(-0.836165\pi\)
−0.870440 + 0.492275i \(0.836165\pi\)
\(444\) 3.20548 0.152126
\(445\) −32.1527 −1.52419
\(446\) 8.05779 0.381547
\(447\) −17.0799 −0.807852
\(448\) 0.415035 0.0196086
\(449\) −23.7926 −1.12284 −0.561421 0.827530i \(-0.689745\pi\)
−0.561421 + 0.827530i \(0.689745\pi\)
\(450\) −7.21452 −0.340096
\(451\) −0.603247 −0.0284058
\(452\) 14.2894 0.672119
\(453\) 14.1138 0.663126
\(454\) −16.8358 −0.790141
\(455\) −7.12229 −0.333898
\(456\) −1.00000 −0.0468293
\(457\) −22.5476 −1.05473 −0.527367 0.849638i \(-0.676821\pi\)
−0.527367 + 0.849638i \(0.676821\pi\)
\(458\) −9.71143 −0.453785
\(459\) 5.09702 0.237909
\(460\) 6.85111 0.319435
\(461\) 9.76411 0.454760 0.227380 0.973806i \(-0.426984\pi\)
0.227380 + 0.973806i \(0.426984\pi\)
\(462\) 1.69618 0.0789136
\(463\) −15.7547 −0.732183 −0.366091 0.930579i \(-0.619304\pi\)
−0.366091 + 0.930579i \(0.619304\pi\)
\(464\) 7.83262 0.363620
\(465\) 16.2471 0.753440
\(466\) −12.5612 −0.581887
\(467\) 2.92445 0.135327 0.0676636 0.997708i \(-0.478446\pi\)
0.0676636 + 0.997708i \(0.478446\pi\)
\(468\) −4.91017 −0.226973
\(469\) 0.916275 0.0423097
\(470\) 0.176125 0.00812403
\(471\) 9.41661 0.433894
\(472\) 11.4052 0.524968
\(473\) −0.162243 −0.00745992
\(474\) −15.7254 −0.722292
\(475\) 7.21452 0.331025
\(476\) 2.11544 0.0969612
\(477\) 1.00000 0.0457869
\(478\) −0.440965 −0.0201693
\(479\) −8.66304 −0.395825 −0.197912 0.980220i \(-0.563416\pi\)
−0.197912 + 0.980220i \(0.563416\pi\)
\(480\) −3.49493 −0.159521
\(481\) −15.7395 −0.717658
\(482\) −5.43948 −0.247762
\(483\) 0.813594 0.0370198
\(484\) 5.70226 0.259194
\(485\) −39.3364 −1.78617
\(486\) −1.00000 −0.0453609
\(487\) −36.4364 −1.65109 −0.825544 0.564337i \(-0.809132\pi\)
−0.825544 + 0.564337i \(0.809132\pi\)
\(488\) −6.82283 −0.308855
\(489\) −17.2516 −0.780146
\(490\) 23.8625 1.07800
\(491\) 19.8059 0.893830 0.446915 0.894577i \(-0.352523\pi\)
0.446915 + 0.894577i \(0.352523\pi\)
\(492\) 0.147607 0.00665465
\(493\) 39.9230 1.79804
\(494\) 4.91017 0.220919
\(495\) −14.2832 −0.641983
\(496\) 4.64876 0.208735
\(497\) 5.73178 0.257105
\(498\) −8.45796 −0.379010
\(499\) 25.2560 1.13061 0.565306 0.824881i \(-0.308758\pi\)
0.565306 + 0.824881i \(0.308758\pi\)
\(500\) 7.73960 0.346125
\(501\) −3.67750 −0.164299
\(502\) 3.56968 0.159322
\(503\) −9.35473 −0.417107 −0.208553 0.978011i \(-0.566875\pi\)
−0.208553 + 0.978011i \(0.566875\pi\)
\(504\) −0.415035 −0.0184871
\(505\) −20.0194 −0.890850
\(506\) 8.01144 0.356152
\(507\) 11.1098 0.493401
\(508\) 14.7621 0.654962
\(509\) −32.3109 −1.43216 −0.716078 0.698021i \(-0.754064\pi\)
−0.716078 + 0.698021i \(0.754064\pi\)
\(510\) −17.8137 −0.788805
\(511\) −6.77835 −0.299857
\(512\) −1.00000 −0.0441942
\(513\) 1.00000 0.0441511
\(514\) 9.34972 0.412398
\(515\) 58.6620 2.58496
\(516\) 0.0396988 0.00174764
\(517\) 0.205954 0.00905783
\(518\) −1.33039 −0.0584539
\(519\) 21.5612 0.946433
\(520\) 17.1607 0.752546
\(521\) 18.3558 0.804184 0.402092 0.915599i \(-0.368283\pi\)
0.402092 + 0.915599i \(0.368283\pi\)
\(522\) −7.83262 −0.342825
\(523\) 2.38320 0.104210 0.0521050 0.998642i \(-0.483407\pi\)
0.0521050 + 0.998642i \(0.483407\pi\)
\(524\) 10.9973 0.480418
\(525\) 2.99428 0.130681
\(526\) −21.4780 −0.936484
\(527\) 23.6948 1.03216
\(528\) −4.08684 −0.177857
\(529\) −19.1572 −0.832923
\(530\) −3.49493 −0.151810
\(531\) −11.4052 −0.494945
\(532\) 0.415035 0.0179941
\(533\) −0.724776 −0.0313936
\(534\) 9.19983 0.398116
\(535\) 47.0983 2.03624
\(536\) −2.20770 −0.0953583
\(537\) −15.5774 −0.672212
\(538\) −4.13801 −0.178402
\(539\) 27.9039 1.20191
\(540\) 3.49493 0.150398
\(541\) 14.3594 0.617357 0.308679 0.951166i \(-0.400113\pi\)
0.308679 + 0.951166i \(0.400113\pi\)
\(542\) 16.8861 0.725322
\(543\) −5.32170 −0.228376
\(544\) −5.09702 −0.218533
\(545\) −17.1012 −0.732536
\(546\) 2.03789 0.0872138
\(547\) 5.70761 0.244040 0.122020 0.992528i \(-0.461063\pi\)
0.122020 + 0.992528i \(0.461063\pi\)
\(548\) −16.6482 −0.711176
\(549\) 6.82283 0.291191
\(550\) 29.4846 1.25723
\(551\) 7.83262 0.333681
\(552\) −1.96030 −0.0834360
\(553\) 6.52660 0.277539
\(554\) −3.23480 −0.137434
\(555\) 11.2029 0.475538
\(556\) 4.76043 0.201887
\(557\) −5.10258 −0.216203 −0.108102 0.994140i \(-0.534477\pi\)
−0.108102 + 0.994140i \(0.534477\pi\)
\(558\) −4.64876 −0.196798
\(559\) −0.194928 −0.00824456
\(560\) 1.45052 0.0612956
\(561\) −20.8307 −0.879473
\(562\) 30.2443 1.27578
\(563\) −13.6236 −0.574168 −0.287084 0.957905i \(-0.592686\pi\)
−0.287084 + 0.957905i \(0.592686\pi\)
\(564\) −0.0503944 −0.00212199
\(565\) 49.9406 2.10102
\(566\) 16.9773 0.713610
\(567\) 0.415035 0.0174298
\(568\) −13.8103 −0.579469
\(569\) −45.5719 −1.91047 −0.955236 0.295844i \(-0.904399\pi\)
−0.955236 + 0.295844i \(0.904399\pi\)
\(570\) −3.49493 −0.146386
\(571\) −31.6212 −1.32331 −0.661653 0.749810i \(-0.730145\pi\)
−0.661653 + 0.749810i \(0.730145\pi\)
\(572\) 20.0671 0.839046
\(573\) −10.8987 −0.455298
\(574\) −0.0612622 −0.00255704
\(575\) 14.1426 0.589789
\(576\) 1.00000 0.0416667
\(577\) 14.6238 0.608797 0.304399 0.952545i \(-0.401545\pi\)
0.304399 + 0.952545i \(0.401545\pi\)
\(578\) −8.97963 −0.373504
\(579\) 9.27849 0.385601
\(580\) 27.3745 1.13666
\(581\) 3.51035 0.145634
\(582\) 11.2553 0.466546
\(583\) −4.08684 −0.169260
\(584\) 16.3320 0.675822
\(585\) −17.1607 −0.709507
\(586\) 9.13408 0.377325
\(587\) 40.3285 1.66453 0.832267 0.554375i \(-0.187043\pi\)
0.832267 + 0.554375i \(0.187043\pi\)
\(588\) −6.82775 −0.281571
\(589\) 4.64876 0.191549
\(590\) 39.8604 1.64103
\(591\) 0.0491723 0.00202268
\(592\) 3.20548 0.131745
\(593\) −35.4019 −1.45378 −0.726890 0.686754i \(-0.759035\pi\)
−0.726890 + 0.686754i \(0.759035\pi\)
\(594\) 4.08684 0.167685
\(595\) 7.39332 0.303097
\(596\) −17.0799 −0.699620
\(597\) 20.2449 0.828569
\(598\) 9.62541 0.393612
\(599\) −17.6271 −0.720225 −0.360112 0.932909i \(-0.617262\pi\)
−0.360112 + 0.932909i \(0.617262\pi\)
\(600\) −7.21452 −0.294532
\(601\) 21.5204 0.877837 0.438919 0.898527i \(-0.355362\pi\)
0.438919 + 0.898527i \(0.355362\pi\)
\(602\) −0.0164764 −0.000671528 0
\(603\) 2.20770 0.0899047
\(604\) 14.1138 0.574284
\(605\) 19.9290 0.810229
\(606\) 5.72812 0.232689
\(607\) −34.7096 −1.40882 −0.704410 0.709793i \(-0.748788\pi\)
−0.704410 + 0.709793i \(0.748788\pi\)
\(608\) −1.00000 −0.0405554
\(609\) 3.25081 0.131730
\(610\) −23.8453 −0.965468
\(611\) 0.247445 0.0100105
\(612\) 5.09702 0.206035
\(613\) −0.984912 −0.0397802 −0.0198901 0.999802i \(-0.506332\pi\)
−0.0198901 + 0.999802i \(0.506332\pi\)
\(614\) −23.8662 −0.963162
\(615\) 0.515877 0.0208022
\(616\) 1.69618 0.0683412
\(617\) 15.1661 0.610566 0.305283 0.952262i \(-0.401249\pi\)
0.305283 + 0.952262i \(0.401249\pi\)
\(618\) −16.7849 −0.675187
\(619\) −35.1489 −1.41275 −0.706377 0.707836i \(-0.749671\pi\)
−0.706377 + 0.707836i \(0.749671\pi\)
\(620\) 16.2471 0.652498
\(621\) 1.96030 0.0786642
\(622\) −6.45660 −0.258886
\(623\) −3.81825 −0.152975
\(624\) −4.91017 −0.196564
\(625\) −9.02327 −0.360931
\(626\) 10.2086 0.408016
\(627\) −4.08684 −0.163213
\(628\) 9.41661 0.375764
\(629\) 16.3384 0.651456
\(630\) −1.45052 −0.0577900
\(631\) 0.181817 0.00723803 0.00361902 0.999993i \(-0.498848\pi\)
0.00361902 + 0.999993i \(0.498848\pi\)
\(632\) −15.7254 −0.625523
\(633\) 19.2314 0.764380
\(634\) 15.9024 0.631565
\(635\) 51.5925 2.04739
\(636\) 1.00000 0.0396526
\(637\) 33.5254 1.32832
\(638\) 32.0107 1.26731
\(639\) 13.8103 0.546329
\(640\) −3.49493 −0.138149
\(641\) −9.50673 −0.375493 −0.187747 0.982217i \(-0.560118\pi\)
−0.187747 + 0.982217i \(0.560118\pi\)
\(642\) −13.4762 −0.531862
\(643\) −10.1644 −0.400846 −0.200423 0.979709i \(-0.564232\pi\)
−0.200423 + 0.979709i \(0.564232\pi\)
\(644\) 0.813594 0.0320601
\(645\) 0.138744 0.00546305
\(646\) −5.09702 −0.200540
\(647\) −40.2766 −1.58343 −0.791717 0.610887i \(-0.790813\pi\)
−0.791717 + 0.610887i \(0.790813\pi\)
\(648\) −1.00000 −0.0392837
\(649\) 46.6113 1.82965
\(650\) 35.4245 1.38946
\(651\) 1.92940 0.0756191
\(652\) −17.2516 −0.675627
\(653\) −46.0722 −1.80294 −0.901472 0.432837i \(-0.857513\pi\)
−0.901472 + 0.432837i \(0.857513\pi\)
\(654\) 4.89315 0.191337
\(655\) 38.4347 1.50177
\(656\) 0.147607 0.00576310
\(657\) −16.3320 −0.637171
\(658\) 0.0209154 0.000815369 0
\(659\) −27.5267 −1.07229 −0.536143 0.844127i \(-0.680119\pi\)
−0.536143 + 0.844127i \(0.680119\pi\)
\(660\) −14.2832 −0.555973
\(661\) −40.7418 −1.58467 −0.792336 0.610086i \(-0.791135\pi\)
−0.792336 + 0.610086i \(0.791135\pi\)
\(662\) −24.0475 −0.934633
\(663\) −25.0272 −0.971977
\(664\) −8.45796 −0.328233
\(665\) 1.45052 0.0562487
\(666\) −3.20548 −0.124210
\(667\) 15.3543 0.594521
\(668\) −3.67750 −0.142287
\(669\) −8.05779 −0.311532
\(670\) −7.71577 −0.298086
\(671\) −27.8838 −1.07644
\(672\) −0.415035 −0.0160103
\(673\) 48.9819 1.88811 0.944056 0.329786i \(-0.106977\pi\)
0.944056 + 0.329786i \(0.106977\pi\)
\(674\) −3.33476 −0.128450
\(675\) 7.21452 0.277687
\(676\) 11.1098 0.427298
\(677\) 39.5360 1.51949 0.759745 0.650221i \(-0.225324\pi\)
0.759745 + 0.650221i \(0.225324\pi\)
\(678\) −14.2894 −0.548783
\(679\) −4.67134 −0.179269
\(680\) −17.8137 −0.683125
\(681\) 16.8358 0.645147
\(682\) 18.9987 0.727499
\(683\) 8.82629 0.337729 0.168864 0.985639i \(-0.445990\pi\)
0.168864 + 0.985639i \(0.445990\pi\)
\(684\) 1.00000 0.0382360
\(685\) −58.1843 −2.22311
\(686\) 5.73900 0.219116
\(687\) 9.71143 0.370514
\(688\) 0.0396988 0.00151350
\(689\) −4.91017 −0.187063
\(690\) −6.85111 −0.260817
\(691\) 18.2774 0.695305 0.347652 0.937624i \(-0.386979\pi\)
0.347652 + 0.937624i \(0.386979\pi\)
\(692\) 21.5612 0.819635
\(693\) −1.69618 −0.0644327
\(694\) 35.8949 1.36255
\(695\) 16.6374 0.631091
\(696\) −7.83262 −0.296895
\(697\) 0.752357 0.0284976
\(698\) −17.3197 −0.655559
\(699\) 12.5612 0.475109
\(700\) 2.99428 0.113173
\(701\) −3.92380 −0.148200 −0.0741000 0.997251i \(-0.523608\pi\)
−0.0741000 + 0.997251i \(0.523608\pi\)
\(702\) 4.91017 0.185322
\(703\) 3.20548 0.120897
\(704\) −4.08684 −0.154029
\(705\) −0.176125 −0.00663324
\(706\) 15.9809 0.601447
\(707\) −2.37737 −0.0894103
\(708\) −11.4052 −0.428635
\(709\) 6.36341 0.238983 0.119491 0.992835i \(-0.461874\pi\)
0.119491 + 0.992835i \(0.461874\pi\)
\(710\) −48.2662 −1.81140
\(711\) 15.7254 0.589749
\(712\) 9.19983 0.344778
\(713\) 9.11296 0.341283
\(714\) −2.11544 −0.0791685
\(715\) 70.1330 2.62282
\(716\) −15.5774 −0.582153
\(717\) 0.440965 0.0164681
\(718\) 19.6405 0.732978
\(719\) −47.5181 −1.77212 −0.886062 0.463566i \(-0.846570\pi\)
−0.886062 + 0.463566i \(0.846570\pi\)
\(720\) 3.49493 0.130248
\(721\) 6.96632 0.259439
\(722\) −1.00000 −0.0372161
\(723\) 5.43948 0.202297
\(724\) −5.32170 −0.197779
\(725\) 56.5086 2.09868
\(726\) −5.70226 −0.211631
\(727\) −43.0459 −1.59649 −0.798243 0.602336i \(-0.794237\pi\)
−0.798243 + 0.602336i \(0.794237\pi\)
\(728\) 2.03789 0.0755293
\(729\) 1.00000 0.0370370
\(730\) 57.0791 2.11259
\(731\) 0.202346 0.00748402
\(732\) 6.82283 0.252179
\(733\) 51.3459 1.89650 0.948251 0.317521i \(-0.102850\pi\)
0.948251 + 0.317521i \(0.102850\pi\)
\(734\) 5.75736 0.212508
\(735\) −23.8625 −0.880181
\(736\) −1.96030 −0.0722577
\(737\) −9.02254 −0.332349
\(738\) −0.147607 −0.00543350
\(739\) 37.7221 1.38763 0.693815 0.720153i \(-0.255928\pi\)
0.693815 + 0.720153i \(0.255928\pi\)
\(740\) 11.2029 0.411828
\(741\) −4.91017 −0.180380
\(742\) −0.415035 −0.0152364
\(743\) −0.497714 −0.0182594 −0.00912969 0.999958i \(-0.502906\pi\)
−0.00912969 + 0.999958i \(0.502906\pi\)
\(744\) −4.64876 −0.170432
\(745\) −59.6931 −2.18698
\(746\) −27.0308 −0.989668
\(747\) 8.45796 0.309461
\(748\) −20.8307 −0.761646
\(749\) 5.59309 0.204367
\(750\) −7.73960 −0.282610
\(751\) −19.6513 −0.717087 −0.358543 0.933513i \(-0.616726\pi\)
−0.358543 + 0.933513i \(0.616726\pi\)
\(752\) −0.0503944 −0.00183769
\(753\) −3.56968 −0.130086
\(754\) 38.4595 1.40061
\(755\) 49.3268 1.79519
\(756\) 0.415035 0.0150947
\(757\) −21.4267 −0.778766 −0.389383 0.921076i \(-0.627312\pi\)
−0.389383 + 0.921076i \(0.627312\pi\)
\(758\) −16.1485 −0.586541
\(759\) −8.01144 −0.290797
\(760\) −3.49493 −0.126774
\(761\) −34.3057 −1.24358 −0.621790 0.783184i \(-0.713594\pi\)
−0.621790 + 0.783184i \(0.713594\pi\)
\(762\) −14.7621 −0.534775
\(763\) −2.03083 −0.0735210
\(764\) −10.8987 −0.394300
\(765\) 17.8137 0.644057
\(766\) −28.1444 −1.01690
\(767\) 56.0016 2.02210
\(768\) 1.00000 0.0360844
\(769\) −19.5776 −0.705987 −0.352993 0.935626i \(-0.614836\pi\)
−0.352993 + 0.935626i \(0.614836\pi\)
\(770\) 5.92804 0.213632
\(771\) −9.34972 −0.336722
\(772\) 9.27849 0.333940
\(773\) 7.30683 0.262808 0.131404 0.991329i \(-0.458051\pi\)
0.131404 + 0.991329i \(0.458051\pi\)
\(774\) −0.0396988 −0.00142694
\(775\) 33.5386 1.20474
\(776\) 11.2553 0.404041
\(777\) 1.33039 0.0477274
\(778\) −26.2108 −0.939701
\(779\) 0.147607 0.00528858
\(780\) −17.1607 −0.614451
\(781\) −56.4407 −2.01961
\(782\) −9.99170 −0.357302
\(783\) 7.83262 0.279915
\(784\) −6.82775 −0.243848
\(785\) 32.9104 1.17462
\(786\) −10.9973 −0.392260
\(787\) 18.3602 0.654472 0.327236 0.944943i \(-0.393883\pi\)
0.327236 + 0.944943i \(0.393883\pi\)
\(788\) 0.0491723 0.00175169
\(789\) 21.4780 0.764636
\(790\) −54.9592 −1.95536
\(791\) 5.93062 0.210869
\(792\) 4.08684 0.145220
\(793\) −33.5012 −1.18966
\(794\) −10.5870 −0.375720
\(795\) 3.49493 0.123952
\(796\) 20.2449 0.717562
\(797\) −49.4087 −1.75015 −0.875074 0.483990i \(-0.839187\pi\)
−0.875074 + 0.483990i \(0.839187\pi\)
\(798\) −0.415035 −0.0146921
\(799\) −0.256861 −0.00908710
\(800\) −7.21452 −0.255072
\(801\) −9.19983 −0.325060
\(802\) 3.50880 0.123900
\(803\) 66.7462 2.35542
\(804\) 2.20770 0.0778597
\(805\) 2.84345 0.100219
\(806\) 22.8262 0.804018
\(807\) 4.13801 0.145665
\(808\) 5.72812 0.201515
\(809\) 31.0964 1.09329 0.546645 0.837364i \(-0.315905\pi\)
0.546645 + 0.837364i \(0.315905\pi\)
\(810\) −3.49493 −0.122799
\(811\) −35.9571 −1.26262 −0.631312 0.775529i \(-0.717483\pi\)
−0.631312 + 0.775529i \(0.717483\pi\)
\(812\) 3.25081 0.114081
\(813\) −16.8861 −0.592223
\(814\) 13.1003 0.459165
\(815\) −60.2933 −2.11198
\(816\) 5.09702 0.178431
\(817\) 0.0396988 0.00138888
\(818\) 15.8684 0.554827
\(819\) −2.03789 −0.0712097
\(820\) 0.515877 0.0180152
\(821\) 1.89421 0.0661083 0.0330542 0.999454i \(-0.489477\pi\)
0.0330542 + 0.999454i \(0.489477\pi\)
\(822\) 16.6482 0.580673
\(823\) 40.1468 1.39943 0.699715 0.714422i \(-0.253310\pi\)
0.699715 + 0.714422i \(0.253310\pi\)
\(824\) −16.7849 −0.584729
\(825\) −29.4846 −1.02652
\(826\) 4.73357 0.164702
\(827\) −30.1148 −1.04719 −0.523597 0.851966i \(-0.675410\pi\)
−0.523597 + 0.851966i \(0.675410\pi\)
\(828\) 1.96030 0.0681252
\(829\) −37.0113 −1.28546 −0.642728 0.766094i \(-0.722197\pi\)
−0.642728 + 0.766094i \(0.722197\pi\)
\(830\) −29.5600 −1.02604
\(831\) 3.23480 0.112214
\(832\) −4.91017 −0.170229
\(833\) −34.8012 −1.20579
\(834\) −4.76043 −0.164840
\(835\) −12.8526 −0.444782
\(836\) −4.08684 −0.141346
\(837\) 4.64876 0.160685
\(838\) −37.5886 −1.29848
\(839\) 12.9499 0.447078 0.223539 0.974695i \(-0.428239\pi\)
0.223539 + 0.974695i \(0.428239\pi\)
\(840\) −1.45052 −0.0500476
\(841\) 32.3500 1.11552
\(842\) −8.13549 −0.280368
\(843\) −30.2443 −1.04167
\(844\) 19.2314 0.661972
\(845\) 38.8278 1.33572
\(846\) 0.0503944 0.00173259
\(847\) 2.36664 0.0813187
\(848\) 1.00000 0.0343401
\(849\) −16.9773 −0.582660
\(850\) −36.7726 −1.26129
\(851\) 6.28372 0.215403
\(852\) 13.8103 0.473135
\(853\) 35.2445 1.20675 0.603375 0.797458i \(-0.293822\pi\)
0.603375 + 0.797458i \(0.293822\pi\)
\(854\) −2.83172 −0.0968993
\(855\) 3.49493 0.119524
\(856\) −13.4762 −0.460606
\(857\) −15.4683 −0.528385 −0.264193 0.964470i \(-0.585105\pi\)
−0.264193 + 0.964470i \(0.585105\pi\)
\(858\) −20.0671 −0.685079
\(859\) 6.42187 0.219111 0.109556 0.993981i \(-0.465057\pi\)
0.109556 + 0.993981i \(0.465057\pi\)
\(860\) 0.138744 0.00473114
\(861\) 0.0612622 0.00208781
\(862\) 20.9723 0.714320
\(863\) −11.5249 −0.392311 −0.196155 0.980573i \(-0.562846\pi\)
−0.196155 + 0.980573i \(0.562846\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 75.3550 2.56215
\(866\) −24.2854 −0.825251
\(867\) 8.97963 0.304964
\(868\) 1.92940 0.0654880
\(869\) −64.2673 −2.18012
\(870\) −27.3745 −0.928081
\(871\) −10.8402 −0.367306
\(872\) 4.89315 0.165703
\(873\) −11.2553 −0.380933
\(874\) −1.96030 −0.0663082
\(875\) 3.21221 0.108592
\(876\) −16.3320 −0.551807
\(877\) 0.664469 0.0224375 0.0112188 0.999937i \(-0.496429\pi\)
0.0112188 + 0.999937i \(0.496429\pi\)
\(878\) −10.4349 −0.352162
\(879\) −9.13408 −0.308085
\(880\) −14.2832 −0.481487
\(881\) 4.43302 0.149352 0.0746761 0.997208i \(-0.476208\pi\)
0.0746761 + 0.997208i \(0.476208\pi\)
\(882\) 6.82775 0.229902
\(883\) −5.55256 −0.186858 −0.0934292 0.995626i \(-0.529783\pi\)
−0.0934292 + 0.995626i \(0.529783\pi\)
\(884\) −25.0272 −0.841757
\(885\) −39.8604 −1.33989
\(886\) 36.6413 1.23099
\(887\) −20.7613 −0.697097 −0.348549 0.937291i \(-0.613325\pi\)
−0.348549 + 0.937291i \(0.613325\pi\)
\(888\) −3.20548 −0.107569
\(889\) 6.12679 0.205486
\(890\) 32.1527 1.07776
\(891\) −4.08684 −0.136914
\(892\) −8.05779 −0.269795
\(893\) −0.0503944 −0.00168638
\(894\) 17.0799 0.571238
\(895\) −54.4417 −1.81979
\(896\) −0.415035 −0.0138654
\(897\) −9.62541 −0.321383
\(898\) 23.7926 0.793969
\(899\) 36.4120 1.21441
\(900\) 7.21452 0.240484
\(901\) 5.09702 0.169806
\(902\) 0.603247 0.0200859
\(903\) 0.0164764 0.000548300 0
\(904\) −14.2894 −0.475260
\(905\) −18.5990 −0.618250
\(906\) −14.1138 −0.468901
\(907\) −38.1089 −1.26538 −0.632692 0.774404i \(-0.718050\pi\)
−0.632692 + 0.774404i \(0.718050\pi\)
\(908\) 16.8358 0.558714
\(909\) −5.72812 −0.189990
\(910\) 7.12229 0.236102
\(911\) 32.5119 1.07717 0.538583 0.842573i \(-0.318960\pi\)
0.538583 + 0.842573i \(0.318960\pi\)
\(912\) 1.00000 0.0331133
\(913\) −34.5663 −1.14398
\(914\) 22.5476 0.745809
\(915\) 23.8453 0.788301
\(916\) 9.71143 0.320875
\(917\) 4.56425 0.150725
\(918\) −5.09702 −0.168227
\(919\) 46.8658 1.54596 0.772979 0.634431i \(-0.218766\pi\)
0.772979 + 0.634431i \(0.218766\pi\)
\(920\) −6.85111 −0.225874
\(921\) 23.8662 0.786419
\(922\) −9.76411 −0.321564
\(923\) −67.8111 −2.23203
\(924\) −1.69618 −0.0558003
\(925\) 23.1260 0.760380
\(926\) 15.7547 0.517731
\(927\) 16.7849 0.551288
\(928\) −7.83262 −0.257118
\(929\) −29.7878 −0.977306 −0.488653 0.872478i \(-0.662512\pi\)
−0.488653 + 0.872478i \(0.662512\pi\)
\(930\) −16.2471 −0.532762
\(931\) −6.82775 −0.223770
\(932\) 12.5612 0.411457
\(933\) 6.45660 0.211380
\(934\) −2.92445 −0.0956908
\(935\) −72.8019 −2.38088
\(936\) 4.91017 0.160494
\(937\) 22.5070 0.735273 0.367636 0.929970i \(-0.380167\pi\)
0.367636 + 0.929970i \(0.380167\pi\)
\(938\) −0.916275 −0.0299174
\(939\) −10.2086 −0.333143
\(940\) −0.176125 −0.00574455
\(941\) 6.56150 0.213899 0.106949 0.994264i \(-0.465892\pi\)
0.106949 + 0.994264i \(0.465892\pi\)
\(942\) −9.41661 −0.306810
\(943\) 0.289355 0.00942269
\(944\) −11.4052 −0.371208
\(945\) 1.45052 0.0471854
\(946\) 0.162243 0.00527496
\(947\) 2.22644 0.0723495 0.0361748 0.999345i \(-0.488483\pi\)
0.0361748 + 0.999345i \(0.488483\pi\)
\(948\) 15.7254 0.510738
\(949\) 80.1928 2.60317
\(950\) −7.21452 −0.234070
\(951\) −15.9024 −0.515671
\(952\) −2.11544 −0.0685619
\(953\) 2.24347 0.0726733 0.0363366 0.999340i \(-0.488431\pi\)
0.0363366 + 0.999340i \(0.488431\pi\)
\(954\) −1.00000 −0.0323762
\(955\) −38.0900 −1.23256
\(956\) 0.440965 0.0142618
\(957\) −32.0107 −1.03476
\(958\) 8.66304 0.279890
\(959\) −6.90959 −0.223122
\(960\) 3.49493 0.112798
\(961\) −9.38906 −0.302873
\(962\) 15.7395 0.507461
\(963\) 13.4762 0.434264
\(964\) 5.43948 0.175194
\(965\) 32.4277 1.04388
\(966\) −0.813594 −0.0261770
\(967\) 25.0956 0.807020 0.403510 0.914975i \(-0.367790\pi\)
0.403510 + 0.914975i \(0.367790\pi\)
\(968\) −5.70226 −0.183278
\(969\) 5.09702 0.163740
\(970\) 39.3364 1.26302
\(971\) −23.8720 −0.766090 −0.383045 0.923730i \(-0.625125\pi\)
−0.383045 + 0.923730i \(0.625125\pi\)
\(972\) 1.00000 0.0320750
\(973\) 1.97575 0.0633396
\(974\) 36.4364 1.16750
\(975\) −35.4245 −1.13449
\(976\) 6.82283 0.218393
\(977\) 3.52776 0.112863 0.0564315 0.998406i \(-0.482028\pi\)
0.0564315 + 0.998406i \(0.482028\pi\)
\(978\) 17.2516 0.551647
\(979\) 37.5982 1.20164
\(980\) −23.8625 −0.762259
\(981\) −4.89315 −0.156226
\(982\) −19.8059 −0.632033
\(983\) −21.2352 −0.677299 −0.338649 0.940913i \(-0.609970\pi\)
−0.338649 + 0.940913i \(0.609970\pi\)
\(984\) −0.147607 −0.00470555
\(985\) 0.171854 0.00547571
\(986\) −39.9230 −1.27141
\(987\) −0.0209154 −0.000665746 0
\(988\) −4.91017 −0.156213
\(989\) 0.0778216 0.00247458
\(990\) 14.2832 0.453950
\(991\) −22.1328 −0.703073 −0.351537 0.936174i \(-0.614341\pi\)
−0.351537 + 0.936174i \(0.614341\pi\)
\(992\) −4.64876 −0.147598
\(993\) 24.0475 0.763125
\(994\) −5.73178 −0.181801
\(995\) 70.7545 2.24307
\(996\) 8.45796 0.268001
\(997\) 46.0248 1.45762 0.728810 0.684716i \(-0.240074\pi\)
0.728810 + 0.684716i \(0.240074\pi\)
\(998\) −25.2560 −0.799464
\(999\) 3.20548 0.101417
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.bf.1.12 12
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6042.2.a.bf.1.12 12 1.1 even 1 trivial