Properties

Label 6018.2.a.v.1.1
Level $6018$
Weight $2$
Character 6018.1
Self dual yes
Analytic conductor $48.054$
Analytic rank $1$
Dimension $9$
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] = [6018,2,Mod(1,6018)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(6018, 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("6018.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 6018 = 2 \cdot 3 \cdot 17 \cdot 59 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6018.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: \(48.0539719364\)
Analytic rank: \(1\)
Dimension: \(9\)
Coefficient field: \(\mathbb{Q}[x]/(x^{9} - \cdots)\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{9} - 3x^{8} - 21x^{7} + 42x^{6} + 121x^{5} - 127x^{4} - 141x^{3} + 27x^{2} + 26x - 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(4.72920\) of defining polynomial
Character \(\chi\) \(=\) 6018.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.72920 q^{5} +1.00000 q^{6} -3.41012 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -3.72920 q^{5} +1.00000 q^{6} -3.41012 q^{7} -1.00000 q^{8} +1.00000 q^{9} +3.72920 q^{10} +5.39669 q^{11} -1.00000 q^{12} -1.44904 q^{13} +3.41012 q^{14} +3.72920 q^{15} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} -6.42732 q^{19} -3.72920 q^{20} +3.41012 q^{21} -5.39669 q^{22} -6.84877 q^{23} +1.00000 q^{24} +8.90692 q^{25} +1.44904 q^{26} -1.00000 q^{27} -3.41012 q^{28} -5.00277 q^{29} -3.72920 q^{30} +8.63951 q^{31} -1.00000 q^{32} -5.39669 q^{33} +1.00000 q^{34} +12.7170 q^{35} +1.00000 q^{36} +6.89487 q^{37} +6.42732 q^{38} +1.44904 q^{39} +3.72920 q^{40} +9.01653 q^{41} -3.41012 q^{42} -6.75841 q^{43} +5.39669 q^{44} -3.72920 q^{45} +6.84877 q^{46} +2.50567 q^{47} -1.00000 q^{48} +4.62890 q^{49} -8.90692 q^{50} +1.00000 q^{51} -1.44904 q^{52} +8.69169 q^{53} +1.00000 q^{54} -20.1253 q^{55} +3.41012 q^{56} +6.42732 q^{57} +5.00277 q^{58} -1.00000 q^{59} +3.72920 q^{60} +7.26008 q^{61} -8.63951 q^{62} -3.41012 q^{63} +1.00000 q^{64} +5.40376 q^{65} +5.39669 q^{66} -5.37634 q^{67} -1.00000 q^{68} +6.84877 q^{69} -12.7170 q^{70} +0.0579369 q^{71} -1.00000 q^{72} -12.4086 q^{73} -6.89487 q^{74} -8.90692 q^{75} -6.42732 q^{76} -18.4033 q^{77} -1.44904 q^{78} -13.3662 q^{79} -3.72920 q^{80} +1.00000 q^{81} -9.01653 q^{82} +14.1536 q^{83} +3.41012 q^{84} +3.72920 q^{85} +6.75841 q^{86} +5.00277 q^{87} -5.39669 q^{88} +5.41906 q^{89} +3.72920 q^{90} +4.94140 q^{91} -6.84877 q^{92} -8.63951 q^{93} -2.50567 q^{94} +23.9687 q^{95} +1.00000 q^{96} +16.4862 q^{97} -4.62890 q^{98} +5.39669 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 9 q^{2} - 9 q^{3} + 9 q^{4} + 6 q^{5} + 9 q^{6} - 11 q^{7} - 9 q^{8} + 9 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 9 q^{2} - 9 q^{3} + 9 q^{4} + 6 q^{5} + 9 q^{6} - 11 q^{7} - 9 q^{8} + 9 q^{9} - 6 q^{10} + q^{11} - 9 q^{12} + 4 q^{13} + 11 q^{14} - 6 q^{15} + 9 q^{16} - 9 q^{17} - 9 q^{18} - 13 q^{19} + 6 q^{20} + 11 q^{21} - q^{22} - 6 q^{23} + 9 q^{24} + 9 q^{25} - 4 q^{26} - 9 q^{27} - 11 q^{28} + 10 q^{29} + 6 q^{30} + q^{31} - 9 q^{32} - q^{33} + 9 q^{34} + 6 q^{35} + 9 q^{36} - 2 q^{37} + 13 q^{38} - 4 q^{39} - 6 q^{40} + 20 q^{41} - 11 q^{42} - 17 q^{43} + q^{44} + 6 q^{45} + 6 q^{46} + 4 q^{47} - 9 q^{48} + 2 q^{49} - 9 q^{50} + 9 q^{51} + 4 q^{52} + 16 q^{53} + 9 q^{54} - 17 q^{55} + 11 q^{56} + 13 q^{57} - 10 q^{58} - 9 q^{59} - 6 q^{60} - 9 q^{61} - q^{62} - 11 q^{63} + 9 q^{64} + q^{66} - 8 q^{67} - 9 q^{68} + 6 q^{69} - 6 q^{70} - 8 q^{71} - 9 q^{72} - 20 q^{73} + 2 q^{74} - 9 q^{75} - 13 q^{76} - 32 q^{77} + 4 q^{78} - 29 q^{79} + 6 q^{80} + 9 q^{81} - 20 q^{82} - 16 q^{83} + 11 q^{84} - 6 q^{85} + 17 q^{86} - 10 q^{87} - q^{88} + 11 q^{89} - 6 q^{90} - 13 q^{91} - 6 q^{92} - q^{93} - 4 q^{94} + 5 q^{95} + 9 q^{96} + 17 q^{97} - 2 q^{98} + q^{99}+O(q^{100}) \) 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
\(5\) −3.72920 −1.66775 −0.833874 0.551955i \(-0.813882\pi\)
−0.833874 + 0.551955i \(0.813882\pi\)
\(6\) 1.00000 0.408248
\(7\) −3.41012 −1.28890 −0.644452 0.764645i \(-0.722914\pi\)
−0.644452 + 0.764645i \(0.722914\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 3.72920 1.17928
\(11\) 5.39669 1.62716 0.813581 0.581451i \(-0.197515\pi\)
0.813581 + 0.581451i \(0.197515\pi\)
\(12\) −1.00000 −0.288675
\(13\) −1.44904 −0.401891 −0.200946 0.979602i \(-0.564401\pi\)
−0.200946 + 0.979602i \(0.564401\pi\)
\(14\) 3.41012 0.911392
\(15\) 3.72920 0.962875
\(16\) 1.00000 0.250000
\(17\) −1.00000 −0.242536
\(18\) −1.00000 −0.235702
\(19\) −6.42732 −1.47453 −0.737264 0.675605i \(-0.763882\pi\)
−0.737264 + 0.675605i \(0.763882\pi\)
\(20\) −3.72920 −0.833874
\(21\) 3.41012 0.744149
\(22\) −5.39669 −1.15058
\(23\) −6.84877 −1.42807 −0.714033 0.700112i \(-0.753133\pi\)
−0.714033 + 0.700112i \(0.753133\pi\)
\(24\) 1.00000 0.204124
\(25\) 8.90692 1.78138
\(26\) 1.44904 0.284180
\(27\) −1.00000 −0.192450
\(28\) −3.41012 −0.644452
\(29\) −5.00277 −0.928990 −0.464495 0.885576i \(-0.653764\pi\)
−0.464495 + 0.885576i \(0.653764\pi\)
\(30\) −3.72920 −0.680855
\(31\) 8.63951 1.55170 0.775851 0.630916i \(-0.217321\pi\)
0.775851 + 0.630916i \(0.217321\pi\)
\(32\) −1.00000 −0.176777
\(33\) −5.39669 −0.939443
\(34\) 1.00000 0.171499
\(35\) 12.7170 2.14957
\(36\) 1.00000 0.166667
\(37\) 6.89487 1.13351 0.566755 0.823887i \(-0.308199\pi\)
0.566755 + 0.823887i \(0.308199\pi\)
\(38\) 6.42732 1.04265
\(39\) 1.44904 0.232032
\(40\) 3.72920 0.589638
\(41\) 9.01653 1.40815 0.704073 0.710128i \(-0.251363\pi\)
0.704073 + 0.710128i \(0.251363\pi\)
\(42\) −3.41012 −0.526193
\(43\) −6.75841 −1.03065 −0.515324 0.856996i \(-0.672328\pi\)
−0.515324 + 0.856996i \(0.672328\pi\)
\(44\) 5.39669 0.813581
\(45\) −3.72920 −0.555916
\(46\) 6.84877 1.00980
\(47\) 2.50567 0.365489 0.182744 0.983160i \(-0.441502\pi\)
0.182744 + 0.983160i \(0.441502\pi\)
\(48\) −1.00000 −0.144338
\(49\) 4.62890 0.661272
\(50\) −8.90692 −1.25963
\(51\) 1.00000 0.140028
\(52\) −1.44904 −0.200946
\(53\) 8.69169 1.19389 0.596947 0.802280i \(-0.296380\pi\)
0.596947 + 0.802280i \(0.296380\pi\)
\(54\) 1.00000 0.136083
\(55\) −20.1253 −2.71370
\(56\) 3.41012 0.455696
\(57\) 6.42732 0.851319
\(58\) 5.00277 0.656895
\(59\) −1.00000 −0.130189
\(60\) 3.72920 0.481437
\(61\) 7.26008 0.929558 0.464779 0.885427i \(-0.346134\pi\)
0.464779 + 0.885427i \(0.346134\pi\)
\(62\) −8.63951 −1.09722
\(63\) −3.41012 −0.429634
\(64\) 1.00000 0.125000
\(65\) 5.40376 0.670253
\(66\) 5.39669 0.664286
\(67\) −5.37634 −0.656825 −0.328412 0.944534i \(-0.606514\pi\)
−0.328412 + 0.944534i \(0.606514\pi\)
\(68\) −1.00000 −0.121268
\(69\) 6.84877 0.824495
\(70\) −12.7170 −1.51997
\(71\) 0.0579369 0.00687584 0.00343792 0.999994i \(-0.498906\pi\)
0.00343792 + 0.999994i \(0.498906\pi\)
\(72\) −1.00000 −0.117851
\(73\) −12.4086 −1.45231 −0.726157 0.687529i \(-0.758696\pi\)
−0.726157 + 0.687529i \(0.758696\pi\)
\(74\) −6.89487 −0.801512
\(75\) −8.90692 −1.02848
\(76\) −6.42732 −0.737264
\(77\) −18.4033 −2.09726
\(78\) −1.44904 −0.164071
\(79\) −13.3662 −1.50381 −0.751905 0.659271i \(-0.770865\pi\)
−0.751905 + 0.659271i \(0.770865\pi\)
\(80\) −3.72920 −0.416937
\(81\) 1.00000 0.111111
\(82\) −9.01653 −0.995709
\(83\) 14.1536 1.55356 0.776782 0.629770i \(-0.216851\pi\)
0.776782 + 0.629770i \(0.216851\pi\)
\(84\) 3.41012 0.372074
\(85\) 3.72920 0.404488
\(86\) 6.75841 0.728778
\(87\) 5.00277 0.536353
\(88\) −5.39669 −0.575289
\(89\) 5.41906 0.574419 0.287210 0.957868i \(-0.407272\pi\)
0.287210 + 0.957868i \(0.407272\pi\)
\(90\) 3.72920 0.393092
\(91\) 4.94140 0.517999
\(92\) −6.84877 −0.714033
\(93\) −8.63951 −0.895876
\(94\) −2.50567 −0.258440
\(95\) 23.9687 2.45914
\(96\) 1.00000 0.102062
\(97\) 16.4862 1.67392 0.836960 0.547265i \(-0.184331\pi\)
0.836960 + 0.547265i \(0.184331\pi\)
\(98\) −4.62890 −0.467590
\(99\) 5.39669 0.542388
\(100\) 8.90692 0.890692
\(101\) 9.67501 0.962699 0.481350 0.876529i \(-0.340147\pi\)
0.481350 + 0.876529i \(0.340147\pi\)
\(102\) −1.00000 −0.0990148
\(103\) 13.3684 1.31722 0.658612 0.752483i \(-0.271144\pi\)
0.658612 + 0.752483i \(0.271144\pi\)
\(104\) 1.44904 0.142090
\(105\) −12.7170 −1.24105
\(106\) −8.69169 −0.844211
\(107\) 9.76675 0.944187 0.472094 0.881548i \(-0.343498\pi\)
0.472094 + 0.881548i \(0.343498\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −9.15105 −0.876511 −0.438256 0.898850i \(-0.644404\pi\)
−0.438256 + 0.898850i \(0.644404\pi\)
\(110\) 20.1253 1.91887
\(111\) −6.89487 −0.654432
\(112\) −3.41012 −0.322226
\(113\) −2.01354 −0.189418 −0.0947090 0.995505i \(-0.530192\pi\)
−0.0947090 + 0.995505i \(0.530192\pi\)
\(114\) −6.42732 −0.601973
\(115\) 25.5404 2.38166
\(116\) −5.00277 −0.464495
\(117\) −1.44904 −0.133964
\(118\) 1.00000 0.0920575
\(119\) 3.41012 0.312605
\(120\) −3.72920 −0.340428
\(121\) 18.1242 1.64766
\(122\) −7.26008 −0.657297
\(123\) −9.01653 −0.812993
\(124\) 8.63951 0.775851
\(125\) −14.5697 −1.30315
\(126\) 3.41012 0.303797
\(127\) 1.18863 0.105474 0.0527371 0.998608i \(-0.483205\pi\)
0.0527371 + 0.998608i \(0.483205\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.75841 0.595044
\(130\) −5.40376 −0.473941
\(131\) 13.5888 1.18726 0.593628 0.804739i \(-0.297695\pi\)
0.593628 + 0.804739i \(0.297695\pi\)
\(132\) −5.39669 −0.469721
\(133\) 21.9179 1.90052
\(134\) 5.37634 0.464445
\(135\) 3.72920 0.320958
\(136\) 1.00000 0.0857493
\(137\) −7.08660 −0.605449 −0.302725 0.953078i \(-0.597896\pi\)
−0.302725 + 0.953078i \(0.597896\pi\)
\(138\) −6.84877 −0.583006
\(139\) 19.8948 1.68746 0.843729 0.536770i \(-0.180356\pi\)
0.843729 + 0.536770i \(0.180356\pi\)
\(140\) 12.7170 1.07478
\(141\) −2.50567 −0.211015
\(142\) −0.0579369 −0.00486195
\(143\) −7.82001 −0.653942
\(144\) 1.00000 0.0833333
\(145\) 18.6563 1.54932
\(146\) 12.4086 1.02694
\(147\) −4.62890 −0.381785
\(148\) 6.89487 0.566755
\(149\) −13.8088 −1.13126 −0.565630 0.824659i \(-0.691367\pi\)
−0.565630 + 0.824659i \(0.691367\pi\)
\(150\) 8.90692 0.727247
\(151\) −7.58245 −0.617051 −0.308526 0.951216i \(-0.599836\pi\)
−0.308526 + 0.951216i \(0.599836\pi\)
\(152\) 6.42732 0.521324
\(153\) −1.00000 −0.0808452
\(154\) 18.4033 1.48298
\(155\) −32.2185 −2.58785
\(156\) 1.44904 0.116016
\(157\) −5.58450 −0.445691 −0.222846 0.974854i \(-0.571535\pi\)
−0.222846 + 0.974854i \(0.571535\pi\)
\(158\) 13.3662 1.06335
\(159\) −8.69169 −0.689296
\(160\) 3.72920 0.294819
\(161\) 23.3551 1.84064
\(162\) −1.00000 −0.0785674
\(163\) −11.5107 −0.901586 −0.450793 0.892628i \(-0.648859\pi\)
−0.450793 + 0.892628i \(0.648859\pi\)
\(164\) 9.01653 0.704073
\(165\) 20.1253 1.56675
\(166\) −14.1536 −1.09854
\(167\) −12.5643 −0.972253 −0.486126 0.873889i \(-0.661591\pi\)
−0.486126 + 0.873889i \(0.661591\pi\)
\(168\) −3.41012 −0.263096
\(169\) −10.9003 −0.838483
\(170\) −3.72920 −0.286016
\(171\) −6.42732 −0.491509
\(172\) −6.75841 −0.515324
\(173\) 12.4630 0.947546 0.473773 0.880647i \(-0.342892\pi\)
0.473773 + 0.880647i \(0.342892\pi\)
\(174\) −5.00277 −0.379259
\(175\) −30.3736 −2.29603
\(176\) 5.39669 0.406791
\(177\) 1.00000 0.0751646
\(178\) −5.41906 −0.406176
\(179\) 21.1067 1.57759 0.788796 0.614655i \(-0.210705\pi\)
0.788796 + 0.614655i \(0.210705\pi\)
\(180\) −3.72920 −0.277958
\(181\) −8.94724 −0.665043 −0.332522 0.943096i \(-0.607899\pi\)
−0.332522 + 0.943096i \(0.607899\pi\)
\(182\) −4.94140 −0.366281
\(183\) −7.26008 −0.536680
\(184\) 6.84877 0.504898
\(185\) −25.7123 −1.89041
\(186\) 8.63951 0.633480
\(187\) −5.39669 −0.394645
\(188\) 2.50567 0.182744
\(189\) 3.41012 0.248050
\(190\) −23.9687 −1.73888
\(191\) 18.0589 1.30670 0.653348 0.757058i \(-0.273364\pi\)
0.653348 + 0.757058i \(0.273364\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −14.5709 −1.04884 −0.524418 0.851461i \(-0.675717\pi\)
−0.524418 + 0.851461i \(0.675717\pi\)
\(194\) −16.4862 −1.18364
\(195\) −5.40376 −0.386971
\(196\) 4.62890 0.330636
\(197\) −4.41973 −0.314893 −0.157446 0.987528i \(-0.550326\pi\)
−0.157446 + 0.987528i \(0.550326\pi\)
\(198\) −5.39669 −0.383526
\(199\) −17.6247 −1.24938 −0.624692 0.780871i \(-0.714775\pi\)
−0.624692 + 0.780871i \(0.714775\pi\)
\(200\) −8.90692 −0.629814
\(201\) 5.37634 0.379218
\(202\) −9.67501 −0.680731
\(203\) 17.0600 1.19738
\(204\) 1.00000 0.0700140
\(205\) −33.6244 −2.34843
\(206\) −13.3684 −0.931418
\(207\) −6.84877 −0.476022
\(208\) −1.44904 −0.100473
\(209\) −34.6862 −2.39930
\(210\) 12.7170 0.877557
\(211\) 5.09679 0.350878 0.175439 0.984490i \(-0.443866\pi\)
0.175439 + 0.984490i \(0.443866\pi\)
\(212\) 8.69169 0.596947
\(213\) −0.0579369 −0.00396977
\(214\) −9.76675 −0.667641
\(215\) 25.2034 1.71886
\(216\) 1.00000 0.0680414
\(217\) −29.4618 −1.99999
\(218\) 9.15105 0.619787
\(219\) 12.4086 0.838494
\(220\) −20.1253 −1.35685
\(221\) 1.44904 0.0974729
\(222\) 6.89487 0.462753
\(223\) −17.1560 −1.14885 −0.574426 0.818556i \(-0.694775\pi\)
−0.574426 + 0.818556i \(0.694775\pi\)
\(224\) 3.41012 0.227848
\(225\) 8.90692 0.593795
\(226\) 2.01354 0.133939
\(227\) −13.4350 −0.891709 −0.445855 0.895105i \(-0.647100\pi\)
−0.445855 + 0.895105i \(0.647100\pi\)
\(228\) 6.42732 0.425659
\(229\) 24.9330 1.64762 0.823810 0.566866i \(-0.191844\pi\)
0.823810 + 0.566866i \(0.191844\pi\)
\(230\) −25.5404 −1.68408
\(231\) 18.4033 1.21085
\(232\) 5.00277 0.328448
\(233\) 8.01856 0.525313 0.262657 0.964889i \(-0.415401\pi\)
0.262657 + 0.964889i \(0.415401\pi\)
\(234\) 1.44904 0.0947267
\(235\) −9.34412 −0.609543
\(236\) −1.00000 −0.0650945
\(237\) 13.3662 0.868226
\(238\) −3.41012 −0.221045
\(239\) −0.912919 −0.0590518 −0.0295259 0.999564i \(-0.509400\pi\)
−0.0295259 + 0.999564i \(0.509400\pi\)
\(240\) 3.72920 0.240719
\(241\) −6.89595 −0.444207 −0.222104 0.975023i \(-0.571292\pi\)
−0.222104 + 0.975023i \(0.571292\pi\)
\(242\) −18.1242 −1.16507
\(243\) −1.00000 −0.0641500
\(244\) 7.26008 0.464779
\(245\) −17.2621 −1.10283
\(246\) 9.01653 0.574873
\(247\) 9.31343 0.592600
\(248\) −8.63951 −0.548610
\(249\) −14.1536 −0.896950
\(250\) 14.5697 0.921468
\(251\) 22.9184 1.44660 0.723298 0.690536i \(-0.242625\pi\)
0.723298 + 0.690536i \(0.242625\pi\)
\(252\) −3.41012 −0.214817
\(253\) −36.9607 −2.32370
\(254\) −1.18863 −0.0745816
\(255\) −3.72920 −0.233531
\(256\) 1.00000 0.0625000
\(257\) 4.80093 0.299473 0.149737 0.988726i \(-0.452157\pi\)
0.149737 + 0.988726i \(0.452157\pi\)
\(258\) −6.75841 −0.420760
\(259\) −23.5123 −1.46098
\(260\) 5.40376 0.335127
\(261\) −5.00277 −0.309663
\(262\) −13.5888 −0.839517
\(263\) 14.1103 0.870081 0.435041 0.900411i \(-0.356734\pi\)
0.435041 + 0.900411i \(0.356734\pi\)
\(264\) 5.39669 0.332143
\(265\) −32.4130 −1.99112
\(266\) −21.9179 −1.34387
\(267\) −5.41906 −0.331641
\(268\) −5.37634 −0.328412
\(269\) −23.2981 −1.42051 −0.710254 0.703945i \(-0.751420\pi\)
−0.710254 + 0.703945i \(0.751420\pi\)
\(270\) −3.72920 −0.226952
\(271\) −0.524590 −0.0318666 −0.0159333 0.999873i \(-0.505072\pi\)
−0.0159333 + 0.999873i \(0.505072\pi\)
\(272\) −1.00000 −0.0606339
\(273\) −4.94140 −0.299067
\(274\) 7.08660 0.428117
\(275\) 48.0679 2.89860
\(276\) 6.84877 0.412247
\(277\) −23.6475 −1.42084 −0.710420 0.703778i \(-0.751495\pi\)
−0.710420 + 0.703778i \(0.751495\pi\)
\(278\) −19.8948 −1.19321
\(279\) 8.63951 0.517234
\(280\) −12.7170 −0.759986
\(281\) 10.7845 0.643352 0.321676 0.946850i \(-0.395754\pi\)
0.321676 + 0.946850i \(0.395754\pi\)
\(282\) 2.50567 0.149210
\(283\) −13.2487 −0.787553 −0.393776 0.919206i \(-0.628832\pi\)
−0.393776 + 0.919206i \(0.628832\pi\)
\(284\) 0.0579369 0.00343792
\(285\) −23.9687 −1.41979
\(286\) 7.82001 0.462407
\(287\) −30.7474 −1.81496
\(288\) −1.00000 −0.0589256
\(289\) 1.00000 0.0588235
\(290\) −18.6563 −1.09554
\(291\) −16.4862 −0.966438
\(292\) −12.4086 −0.726157
\(293\) 7.93575 0.463612 0.231806 0.972762i \(-0.425537\pi\)
0.231806 + 0.972762i \(0.425537\pi\)
\(294\) 4.62890 0.269963
\(295\) 3.72920 0.217122
\(296\) −6.89487 −0.400756
\(297\) −5.39669 −0.313148
\(298\) 13.8088 0.799922
\(299\) 9.92413 0.573927
\(300\) −8.90692 −0.514241
\(301\) 23.0470 1.32840
\(302\) 7.58245 0.436321
\(303\) −9.67501 −0.555815
\(304\) −6.42732 −0.368632
\(305\) −27.0743 −1.55027
\(306\) 1.00000 0.0571662
\(307\) −16.7504 −0.955996 −0.477998 0.878361i \(-0.658637\pi\)
−0.477998 + 0.878361i \(0.658637\pi\)
\(308\) −18.4033 −1.04863
\(309\) −13.3684 −0.760499
\(310\) 32.2185 1.82989
\(311\) −25.3463 −1.43726 −0.718629 0.695394i \(-0.755230\pi\)
−0.718629 + 0.695394i \(0.755230\pi\)
\(312\) −1.44904 −0.0820357
\(313\) −3.98350 −0.225161 −0.112580 0.993643i \(-0.535912\pi\)
−0.112580 + 0.993643i \(0.535912\pi\)
\(314\) 5.58450 0.315151
\(315\) 12.7170 0.716522
\(316\) −13.3662 −0.751905
\(317\) −13.3269 −0.748511 −0.374255 0.927326i \(-0.622102\pi\)
−0.374255 + 0.927326i \(0.622102\pi\)
\(318\) 8.69169 0.487406
\(319\) −26.9984 −1.51162
\(320\) −3.72920 −0.208469
\(321\) −9.76675 −0.545127
\(322\) −23.3551 −1.30153
\(323\) 6.42732 0.357625
\(324\) 1.00000 0.0555556
\(325\) −12.9065 −0.715923
\(326\) 11.5107 0.637518
\(327\) 9.15105 0.506054
\(328\) −9.01653 −0.497855
\(329\) −8.54461 −0.471080
\(330\) −20.1253 −1.10786
\(331\) 1.85247 0.101821 0.0509106 0.998703i \(-0.483788\pi\)
0.0509106 + 0.998703i \(0.483788\pi\)
\(332\) 14.1536 0.776782
\(333\) 6.89487 0.377836
\(334\) 12.5643 0.687486
\(335\) 20.0494 1.09542
\(336\) 3.41012 0.186037
\(337\) 24.5279 1.33612 0.668060 0.744108i \(-0.267125\pi\)
0.668060 + 0.744108i \(0.267125\pi\)
\(338\) 10.9003 0.592897
\(339\) 2.01354 0.109361
\(340\) 3.72920 0.202244
\(341\) 46.6248 2.52487
\(342\) 6.42732 0.347549
\(343\) 8.08572 0.436588
\(344\) 6.75841 0.364389
\(345\) −25.5404 −1.37505
\(346\) −12.4630 −0.670016
\(347\) −23.8090 −1.27813 −0.639066 0.769152i \(-0.720679\pi\)
−0.639066 + 0.769152i \(0.720679\pi\)
\(348\) 5.00277 0.268176
\(349\) −27.5691 −1.47574 −0.737871 0.674942i \(-0.764169\pi\)
−0.737871 + 0.674942i \(0.764169\pi\)
\(350\) 30.3736 1.62354
\(351\) 1.44904 0.0773440
\(352\) −5.39669 −0.287644
\(353\) −15.1210 −0.804812 −0.402406 0.915461i \(-0.631826\pi\)
−0.402406 + 0.915461i \(0.631826\pi\)
\(354\) −1.00000 −0.0531494
\(355\) −0.216058 −0.0114672
\(356\) 5.41906 0.287210
\(357\) −3.41012 −0.180483
\(358\) −21.1067 −1.11553
\(359\) −12.2143 −0.644647 −0.322324 0.946630i \(-0.604464\pi\)
−0.322324 + 0.946630i \(0.604464\pi\)
\(360\) 3.72920 0.196546
\(361\) 22.3104 1.17423
\(362\) 8.94724 0.470257
\(363\) −18.1242 −0.951276
\(364\) 4.94140 0.258999
\(365\) 46.2741 2.42210
\(366\) 7.26008 0.379490
\(367\) −35.7303 −1.86511 −0.932553 0.361033i \(-0.882424\pi\)
−0.932553 + 0.361033i \(0.882424\pi\)
\(368\) −6.84877 −0.357017
\(369\) 9.01653 0.469382
\(370\) 25.7123 1.33672
\(371\) −29.6397 −1.53882
\(372\) −8.63951 −0.447938
\(373\) −5.24218 −0.271430 −0.135715 0.990748i \(-0.543333\pi\)
−0.135715 + 0.990748i \(0.543333\pi\)
\(374\) 5.39669 0.279056
\(375\) 14.5697 0.752375
\(376\) −2.50567 −0.129220
\(377\) 7.24920 0.373353
\(378\) −3.41012 −0.175398
\(379\) −7.59961 −0.390366 −0.195183 0.980767i \(-0.562530\pi\)
−0.195183 + 0.980767i \(0.562530\pi\)
\(380\) 23.9687 1.22957
\(381\) −1.18863 −0.0608956
\(382\) −18.0589 −0.923973
\(383\) 20.2254 1.03347 0.516735 0.856146i \(-0.327147\pi\)
0.516735 + 0.856146i \(0.327147\pi\)
\(384\) 1.00000 0.0510310
\(385\) 68.6297 3.49769
\(386\) 14.5709 0.741639
\(387\) −6.75841 −0.343549
\(388\) 16.4862 0.836960
\(389\) 21.3128 1.08060 0.540300 0.841472i \(-0.318311\pi\)
0.540300 + 0.841472i \(0.318311\pi\)
\(390\) 5.40376 0.273630
\(391\) 6.84877 0.346357
\(392\) −4.62890 −0.233795
\(393\) −13.5888 −0.685463
\(394\) 4.41973 0.222663
\(395\) 49.8451 2.50798
\(396\) 5.39669 0.271194
\(397\) −16.0240 −0.804220 −0.402110 0.915591i \(-0.631723\pi\)
−0.402110 + 0.915591i \(0.631723\pi\)
\(398\) 17.6247 0.883448
\(399\) −21.9179 −1.09727
\(400\) 8.90692 0.445346
\(401\) 21.6412 1.08071 0.540355 0.841437i \(-0.318290\pi\)
0.540355 + 0.841437i \(0.318290\pi\)
\(402\) −5.37634 −0.268148
\(403\) −12.5190 −0.623615
\(404\) 9.67501 0.481350
\(405\) −3.72920 −0.185305
\(406\) −17.0600 −0.846674
\(407\) 37.2095 1.84440
\(408\) −1.00000 −0.0495074
\(409\) 35.6246 1.76153 0.880763 0.473558i \(-0.157031\pi\)
0.880763 + 0.473558i \(0.157031\pi\)
\(410\) 33.6244 1.66059
\(411\) 7.08660 0.349556
\(412\) 13.3684 0.658612
\(413\) 3.41012 0.167801
\(414\) 6.84877 0.336599
\(415\) −52.7817 −2.59095
\(416\) 1.44904 0.0710450
\(417\) −19.8948 −0.974254
\(418\) 34.6862 1.69656
\(419\) −22.2459 −1.08678 −0.543392 0.839479i \(-0.682860\pi\)
−0.543392 + 0.839479i \(0.682860\pi\)
\(420\) −12.7170 −0.620526
\(421\) −16.5179 −0.805031 −0.402515 0.915413i \(-0.631864\pi\)
−0.402515 + 0.915413i \(0.631864\pi\)
\(422\) −5.09679 −0.248108
\(423\) 2.50567 0.121830
\(424\) −8.69169 −0.422106
\(425\) −8.90692 −0.432049
\(426\) 0.0579369 0.00280705
\(427\) −24.7577 −1.19811
\(428\) 9.76675 0.472094
\(429\) 7.82001 0.377554
\(430\) −25.2034 −1.21542
\(431\) −31.4356 −1.51420 −0.757100 0.653300i \(-0.773384\pi\)
−0.757100 + 0.653300i \(0.773384\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −19.6591 −0.944758 −0.472379 0.881395i \(-0.656605\pi\)
−0.472379 + 0.881395i \(0.656605\pi\)
\(434\) 29.4618 1.41421
\(435\) −18.6563 −0.894501
\(436\) −9.15105 −0.438256
\(437\) 44.0192 2.10572
\(438\) −12.4086 −0.592905
\(439\) −23.4092 −1.11726 −0.558631 0.829416i \(-0.688673\pi\)
−0.558631 + 0.829416i \(0.688673\pi\)
\(440\) 20.1253 0.959437
\(441\) 4.62890 0.220424
\(442\) −1.44904 −0.0689238
\(443\) −14.1011 −0.669963 −0.334982 0.942225i \(-0.608730\pi\)
−0.334982 + 0.942225i \(0.608730\pi\)
\(444\) −6.89487 −0.327216
\(445\) −20.2087 −0.957986
\(446\) 17.1560 0.812361
\(447\) 13.8088 0.653133
\(448\) −3.41012 −0.161113
\(449\) −10.3411 −0.488027 −0.244013 0.969772i \(-0.578464\pi\)
−0.244013 + 0.969772i \(0.578464\pi\)
\(450\) −8.90692 −0.419876
\(451\) 48.6594 2.29128
\(452\) −2.01354 −0.0947090
\(453\) 7.58245 0.356255
\(454\) 13.4350 0.630533
\(455\) −18.4274 −0.863892
\(456\) −6.42732 −0.300987
\(457\) 25.8808 1.21065 0.605327 0.795977i \(-0.293042\pi\)
0.605327 + 0.795977i \(0.293042\pi\)
\(458\) −24.9330 −1.16504
\(459\) 1.00000 0.0466760
\(460\) 25.5404 1.19083
\(461\) −0.642862 −0.0299411 −0.0149705 0.999888i \(-0.504765\pi\)
−0.0149705 + 0.999888i \(0.504765\pi\)
\(462\) −18.4033 −0.856201
\(463\) −8.42833 −0.391698 −0.195849 0.980634i \(-0.562746\pi\)
−0.195849 + 0.980634i \(0.562746\pi\)
\(464\) −5.00277 −0.232248
\(465\) 32.2185 1.49409
\(466\) −8.01856 −0.371453
\(467\) 0.311005 0.0143916 0.00719580 0.999974i \(-0.497709\pi\)
0.00719580 + 0.999974i \(0.497709\pi\)
\(468\) −1.44904 −0.0669819
\(469\) 18.3340 0.846583
\(470\) 9.34412 0.431012
\(471\) 5.58450 0.257320
\(472\) 1.00000 0.0460287
\(473\) −36.4730 −1.67703
\(474\) −13.3662 −0.613928
\(475\) −57.2476 −2.62670
\(476\) 3.41012 0.156302
\(477\) 8.69169 0.397965
\(478\) 0.912919 0.0417559
\(479\) −14.3397 −0.655198 −0.327599 0.944817i \(-0.606239\pi\)
−0.327599 + 0.944817i \(0.606239\pi\)
\(480\) −3.72920 −0.170214
\(481\) −9.99094 −0.455547
\(482\) 6.89595 0.314102
\(483\) −23.3551 −1.06269
\(484\) 18.1242 0.823829
\(485\) −61.4803 −2.79168
\(486\) 1.00000 0.0453609
\(487\) −15.1595 −0.686944 −0.343472 0.939163i \(-0.611603\pi\)
−0.343472 + 0.939163i \(0.611603\pi\)
\(488\) −7.26008 −0.328648
\(489\) 11.5107 0.520531
\(490\) 17.2621 0.779822
\(491\) −24.2062 −1.09241 −0.546206 0.837651i \(-0.683928\pi\)
−0.546206 + 0.837651i \(0.683928\pi\)
\(492\) −9.01653 −0.406497
\(493\) 5.00277 0.225313
\(494\) −9.31343 −0.419031
\(495\) −20.1253 −0.904566
\(496\) 8.63951 0.387926
\(497\) −0.197572 −0.00886229
\(498\) 14.1536 0.634240
\(499\) 9.06767 0.405924 0.202962 0.979187i \(-0.434943\pi\)
0.202962 + 0.979187i \(0.434943\pi\)
\(500\) −14.5697 −0.651576
\(501\) 12.5643 0.561330
\(502\) −22.9184 −1.02290
\(503\) −14.1050 −0.628911 −0.314455 0.949272i \(-0.601822\pi\)
−0.314455 + 0.949272i \(0.601822\pi\)
\(504\) 3.41012 0.151899
\(505\) −36.0800 −1.60554
\(506\) 36.9607 1.64310
\(507\) 10.9003 0.484099
\(508\) 1.18863 0.0527371
\(509\) 25.9226 1.14900 0.574499 0.818505i \(-0.305197\pi\)
0.574499 + 0.818505i \(0.305197\pi\)
\(510\) 3.72920 0.165132
\(511\) 42.3147 1.87189
\(512\) −1.00000 −0.0441942
\(513\) 6.42732 0.283773
\(514\) −4.80093 −0.211760
\(515\) −49.8533 −2.19680
\(516\) 6.75841 0.297522
\(517\) 13.5223 0.594710
\(518\) 23.5123 1.03307
\(519\) −12.4630 −0.547066
\(520\) −5.40376 −0.236970
\(521\) −6.64697 −0.291209 −0.145605 0.989343i \(-0.546513\pi\)
−0.145605 + 0.989343i \(0.546513\pi\)
\(522\) 5.00277 0.218965
\(523\) 6.13178 0.268124 0.134062 0.990973i \(-0.457198\pi\)
0.134062 + 0.990973i \(0.457198\pi\)
\(524\) 13.5888 0.593628
\(525\) 30.3736 1.32561
\(526\) −14.1103 −0.615240
\(527\) −8.63951 −0.376343
\(528\) −5.39669 −0.234861
\(529\) 23.9056 1.03937
\(530\) 32.4130 1.40793
\(531\) −1.00000 −0.0433963
\(532\) 21.9179 0.950262
\(533\) −13.0653 −0.565921
\(534\) 5.41906 0.234506
\(535\) −36.4221 −1.57467
\(536\) 5.37634 0.232223
\(537\) −21.1067 −0.910823
\(538\) 23.2981 1.00445
\(539\) 24.9807 1.07600
\(540\) 3.72920 0.160479
\(541\) 8.39640 0.360989 0.180495 0.983576i \(-0.442230\pi\)
0.180495 + 0.983576i \(0.442230\pi\)
\(542\) 0.524590 0.0225331
\(543\) 8.94724 0.383963
\(544\) 1.00000 0.0428746
\(545\) 34.1261 1.46180
\(546\) 4.94140 0.211472
\(547\) 2.56790 0.109795 0.0548977 0.998492i \(-0.482517\pi\)
0.0548977 + 0.998492i \(0.482517\pi\)
\(548\) −7.08660 −0.302725
\(549\) 7.26008 0.309853
\(550\) −48.0679 −2.04962
\(551\) 32.1544 1.36982
\(552\) −6.84877 −0.291503
\(553\) 45.5802 1.93827
\(554\) 23.6475 1.00469
\(555\) 25.7123 1.09143
\(556\) 19.8948 0.843729
\(557\) −13.5198 −0.572851 −0.286425 0.958103i \(-0.592467\pi\)
−0.286425 + 0.958103i \(0.592467\pi\)
\(558\) −8.63951 −0.365740
\(559\) 9.79320 0.414208
\(560\) 12.7170 0.537392
\(561\) 5.39669 0.227848
\(562\) −10.7845 −0.454918
\(563\) −8.20834 −0.345940 −0.172970 0.984927i \(-0.555336\pi\)
−0.172970 + 0.984927i \(0.555336\pi\)
\(564\) −2.50567 −0.105508
\(565\) 7.50889 0.315902
\(566\) 13.2487 0.556884
\(567\) −3.41012 −0.143211
\(568\) −0.0579369 −0.00243098
\(569\) −37.8164 −1.58535 −0.792673 0.609647i \(-0.791311\pi\)
−0.792673 + 0.609647i \(0.791311\pi\)
\(570\) 23.9687 1.00394
\(571\) −0.164607 −0.00688859 −0.00344429 0.999994i \(-0.501096\pi\)
−0.00344429 + 0.999994i \(0.501096\pi\)
\(572\) −7.82001 −0.326971
\(573\) −18.0589 −0.754421
\(574\) 30.7474 1.28337
\(575\) −61.0014 −2.54394
\(576\) 1.00000 0.0416667
\(577\) −6.77447 −0.282025 −0.141012 0.990008i \(-0.545036\pi\)
−0.141012 + 0.990008i \(0.545036\pi\)
\(578\) −1.00000 −0.0415945
\(579\) 14.5709 0.605546
\(580\) 18.6563 0.774661
\(581\) −48.2656 −2.00239
\(582\) 16.4862 0.683375
\(583\) 46.9063 1.94266
\(584\) 12.4086 0.513471
\(585\) 5.40376 0.223418
\(586\) −7.93575 −0.327823
\(587\) 21.9837 0.907363 0.453682 0.891164i \(-0.350110\pi\)
0.453682 + 0.891164i \(0.350110\pi\)
\(588\) −4.62890 −0.190893
\(589\) −55.5289 −2.28803
\(590\) −3.72920 −0.153529
\(591\) 4.41973 0.181804
\(592\) 6.89487 0.283377
\(593\) 22.4242 0.920853 0.460427 0.887698i \(-0.347696\pi\)
0.460427 + 0.887698i \(0.347696\pi\)
\(594\) 5.39669 0.221429
\(595\) −12.7170 −0.521346
\(596\) −13.8088 −0.565630
\(597\) 17.6247 0.721332
\(598\) −9.92413 −0.405828
\(599\) 34.7479 1.41976 0.709881 0.704322i \(-0.248749\pi\)
0.709881 + 0.704322i \(0.248749\pi\)
\(600\) 8.90692 0.363624
\(601\) −48.0849 −1.96142 −0.980712 0.195460i \(-0.937380\pi\)
−0.980712 + 0.195460i \(0.937380\pi\)
\(602\) −23.0470 −0.939324
\(603\) −5.37634 −0.218942
\(604\) −7.58245 −0.308526
\(605\) −67.5889 −2.74788
\(606\) 9.67501 0.393020
\(607\) 4.10121 0.166463 0.0832315 0.996530i \(-0.473476\pi\)
0.0832315 + 0.996530i \(0.473476\pi\)
\(608\) 6.42732 0.260662
\(609\) −17.0600 −0.691307
\(610\) 27.0743 1.09621
\(611\) −3.63081 −0.146887
\(612\) −1.00000 −0.0404226
\(613\) −5.61823 −0.226918 −0.113459 0.993543i \(-0.536193\pi\)
−0.113459 + 0.993543i \(0.536193\pi\)
\(614\) 16.7504 0.675992
\(615\) 33.6244 1.35587
\(616\) 18.4033 0.741492
\(617\) 9.18586 0.369809 0.184904 0.982757i \(-0.440802\pi\)
0.184904 + 0.982757i \(0.440802\pi\)
\(618\) 13.3684 0.537754
\(619\) 19.2887 0.775280 0.387640 0.921811i \(-0.373290\pi\)
0.387640 + 0.921811i \(0.373290\pi\)
\(620\) −32.2185 −1.29392
\(621\) 6.84877 0.274832
\(622\) 25.3463 1.01629
\(623\) −18.4796 −0.740371
\(624\) 1.44904 0.0580080
\(625\) 9.79864 0.391945
\(626\) 3.98350 0.159213
\(627\) 34.6862 1.38523
\(628\) −5.58450 −0.222846
\(629\) −6.89487 −0.274916
\(630\) −12.7170 −0.506658
\(631\) −8.10629 −0.322706 −0.161353 0.986897i \(-0.551586\pi\)
−0.161353 + 0.986897i \(0.551586\pi\)
\(632\) 13.3662 0.531677
\(633\) −5.09679 −0.202579
\(634\) 13.3269 0.529277
\(635\) −4.43265 −0.175904
\(636\) −8.69169 −0.344648
\(637\) −6.70746 −0.265759
\(638\) 26.9984 1.06888
\(639\) 0.0579369 0.00229195
\(640\) 3.72920 0.147410
\(641\) 37.6291 1.48626 0.743130 0.669147i \(-0.233340\pi\)
0.743130 + 0.669147i \(0.233340\pi\)
\(642\) 9.76675 0.385463
\(643\) 11.5791 0.456634 0.228317 0.973587i \(-0.426678\pi\)
0.228317 + 0.973587i \(0.426678\pi\)
\(644\) 23.3551 0.920320
\(645\) −25.2034 −0.992384
\(646\) −6.42732 −0.252879
\(647\) 22.4611 0.883036 0.441518 0.897252i \(-0.354440\pi\)
0.441518 + 0.897252i \(0.354440\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −5.39669 −0.211839
\(650\) 12.9065 0.506234
\(651\) 29.4618 1.15470
\(652\) −11.5107 −0.450793
\(653\) −3.89084 −0.152260 −0.0761301 0.997098i \(-0.524256\pi\)
−0.0761301 + 0.997098i \(0.524256\pi\)
\(654\) −9.15105 −0.357834
\(655\) −50.6752 −1.98004
\(656\) 9.01653 0.352036
\(657\) −12.4086 −0.484105
\(658\) 8.54461 0.333104
\(659\) −31.4823 −1.22638 −0.613188 0.789937i \(-0.710113\pi\)
−0.613188 + 0.789937i \(0.710113\pi\)
\(660\) 20.1253 0.783377
\(661\) 26.5615 1.03312 0.516562 0.856250i \(-0.327212\pi\)
0.516562 + 0.856250i \(0.327212\pi\)
\(662\) −1.85247 −0.0719985
\(663\) −1.44904 −0.0562760
\(664\) −14.1536 −0.549268
\(665\) −81.7362 −3.16959
\(666\) −6.89487 −0.267171
\(667\) 34.2628 1.32666
\(668\) −12.5643 −0.486126
\(669\) 17.1560 0.663290
\(670\) −20.0494 −0.774578
\(671\) 39.1804 1.51254
\(672\) −3.41012 −0.131548
\(673\) −20.2553 −0.780784 −0.390392 0.920649i \(-0.627661\pi\)
−0.390392 + 0.920649i \(0.627661\pi\)
\(674\) −24.5279 −0.944779
\(675\) −8.90692 −0.342828
\(676\) −10.9003 −0.419242
\(677\) −4.89243 −0.188031 −0.0940156 0.995571i \(-0.529970\pi\)
−0.0940156 + 0.995571i \(0.529970\pi\)
\(678\) −2.01354 −0.0773296
\(679\) −56.2199 −2.15752
\(680\) −3.72920 −0.143008
\(681\) 13.4350 0.514828
\(682\) −46.6248 −1.78535
\(683\) 15.7553 0.602861 0.301431 0.953488i \(-0.402536\pi\)
0.301431 + 0.953488i \(0.402536\pi\)
\(684\) −6.42732 −0.245755
\(685\) 26.4274 1.00974
\(686\) −8.08572 −0.308714
\(687\) −24.9330 −0.951254
\(688\) −6.75841 −0.257662
\(689\) −12.5946 −0.479816
\(690\) 25.5404 0.972307
\(691\) −5.65293 −0.215048 −0.107524 0.994203i \(-0.534292\pi\)
−0.107524 + 0.994203i \(0.534292\pi\)
\(692\) 12.4630 0.473773
\(693\) −18.4033 −0.699085
\(694\) 23.8090 0.903775
\(695\) −74.1918 −2.81425
\(696\) −5.00277 −0.189629
\(697\) −9.01653 −0.341526
\(698\) 27.5691 1.04351
\(699\) −8.01856 −0.303290
\(700\) −30.3736 −1.14802
\(701\) −50.2178 −1.89670 −0.948350 0.317227i \(-0.897248\pi\)
−0.948350 + 0.317227i \(0.897248\pi\)
\(702\) −1.44904 −0.0546905
\(703\) −44.3155 −1.67139
\(704\) 5.39669 0.203395
\(705\) 9.34412 0.351920
\(706\) 15.1210 0.569088
\(707\) −32.9929 −1.24083
\(708\) 1.00000 0.0375823
\(709\) 45.8204 1.72082 0.860410 0.509603i \(-0.170208\pi\)
0.860410 + 0.509603i \(0.170208\pi\)
\(710\) 0.216058 0.00810851
\(711\) −13.3662 −0.501270
\(712\) −5.41906 −0.203088
\(713\) −59.1700 −2.21593
\(714\) 3.41012 0.127620
\(715\) 29.1624 1.09061
\(716\) 21.1067 0.788796
\(717\) 0.912919 0.0340936
\(718\) 12.2143 0.455834
\(719\) 27.2899 1.01774 0.508871 0.860843i \(-0.330063\pi\)
0.508871 + 0.860843i \(0.330063\pi\)
\(720\) −3.72920 −0.138979
\(721\) −45.5877 −1.69777
\(722\) −22.3104 −0.830307
\(723\) 6.89595 0.256463
\(724\) −8.94724 −0.332522
\(725\) −44.5592 −1.65489
\(726\) 18.1242 0.672654
\(727\) 7.19232 0.266749 0.133374 0.991066i \(-0.457419\pi\)
0.133374 + 0.991066i \(0.457419\pi\)
\(728\) −4.94140 −0.183140
\(729\) 1.00000 0.0370370
\(730\) −46.2741 −1.71268
\(731\) 6.75841 0.249969
\(732\) −7.26008 −0.268340
\(733\) 24.5020 0.905001 0.452501 0.891764i \(-0.350532\pi\)
0.452501 + 0.891764i \(0.350532\pi\)
\(734\) 35.7303 1.31883
\(735\) 17.2621 0.636722
\(736\) 6.84877 0.252449
\(737\) −29.0144 −1.06876
\(738\) −9.01653 −0.331903
\(739\) 34.7569 1.27855 0.639277 0.768977i \(-0.279234\pi\)
0.639277 + 0.768977i \(0.279234\pi\)
\(740\) −25.7123 −0.945204
\(741\) −9.31343 −0.342138
\(742\) 29.6397 1.08811
\(743\) −38.2351 −1.40271 −0.701355 0.712812i \(-0.747421\pi\)
−0.701355 + 0.712812i \(0.747421\pi\)
\(744\) 8.63951 0.316740
\(745\) 51.4957 1.88666
\(746\) 5.24218 0.191930
\(747\) 14.1536 0.517855
\(748\) −5.39669 −0.197322
\(749\) −33.3058 −1.21697
\(750\) −14.5697 −0.532010
\(751\) −52.1115 −1.90157 −0.950787 0.309844i \(-0.899723\pi\)
−0.950787 + 0.309844i \(0.899723\pi\)
\(752\) 2.50567 0.0913722
\(753\) −22.9184 −0.835193
\(754\) −7.24920 −0.264000
\(755\) 28.2765 1.02909
\(756\) 3.41012 0.124025
\(757\) 46.5082 1.69037 0.845185 0.534474i \(-0.179490\pi\)
0.845185 + 0.534474i \(0.179490\pi\)
\(758\) 7.59961 0.276030
\(759\) 36.9607 1.34159
\(760\) −23.9687 −0.869438
\(761\) 44.8810 1.62693 0.813467 0.581611i \(-0.197577\pi\)
0.813467 + 0.581611i \(0.197577\pi\)
\(762\) 1.18863 0.0430597
\(763\) 31.2061 1.12974
\(764\) 18.0589 0.653348
\(765\) 3.72920 0.134829
\(766\) −20.2254 −0.730773
\(767\) 1.44904 0.0523218
\(768\) −1.00000 −0.0360844
\(769\) −6.92105 −0.249579 −0.124790 0.992183i \(-0.539826\pi\)
−0.124790 + 0.992183i \(0.539826\pi\)
\(770\) −68.6297 −2.47324
\(771\) −4.80093 −0.172901
\(772\) −14.5709 −0.524418
\(773\) −19.9056 −0.715953 −0.357977 0.933731i \(-0.616533\pi\)
−0.357977 + 0.933731i \(0.616533\pi\)
\(774\) 6.75841 0.242926
\(775\) 76.9514 2.76418
\(776\) −16.4862 −0.591820
\(777\) 23.5123 0.843499
\(778\) −21.3128 −0.764100
\(779\) −57.9521 −2.07635
\(780\) −5.40376 −0.193485
\(781\) 0.312667 0.0111881
\(782\) −6.84877 −0.244911
\(783\) 5.00277 0.178784
\(784\) 4.62890 0.165318
\(785\) 20.8257 0.743301
\(786\) 13.5888 0.484695
\(787\) −31.3782 −1.11851 −0.559256 0.828995i \(-0.688913\pi\)
−0.559256 + 0.828995i \(0.688913\pi\)
\(788\) −4.41973 −0.157446
\(789\) −14.1103 −0.502342
\(790\) −49.8451 −1.77341
\(791\) 6.86641 0.244141
\(792\) −5.39669 −0.191763
\(793\) −10.5201 −0.373581
\(794\) 16.0240 0.568669
\(795\) 32.4130 1.14957
\(796\) −17.6247 −0.624692
\(797\) 22.6892 0.803694 0.401847 0.915707i \(-0.368368\pi\)
0.401847 + 0.915707i \(0.368368\pi\)
\(798\) 21.9179 0.775885
\(799\) −2.50567 −0.0886441
\(800\) −8.90692 −0.314907
\(801\) 5.41906 0.191473
\(802\) −21.6412 −0.764178
\(803\) −66.9652 −2.36315
\(804\) 5.37634 0.189609
\(805\) −87.0958 −3.06972
\(806\) 12.5190 0.440963
\(807\) 23.2981 0.820131
\(808\) −9.67501 −0.340366
\(809\) 5.75662 0.202392 0.101196 0.994867i \(-0.467733\pi\)
0.101196 + 0.994867i \(0.467733\pi\)
\(810\) 3.72920 0.131031
\(811\) 53.5935 1.88192 0.940961 0.338516i \(-0.109925\pi\)
0.940961 + 0.338516i \(0.109925\pi\)
\(812\) 17.0600 0.598689
\(813\) 0.524590 0.0183982
\(814\) −37.2095 −1.30419
\(815\) 42.9256 1.50362
\(816\) 1.00000 0.0350070
\(817\) 43.4384 1.51972
\(818\) −35.6246 −1.24559
\(819\) 4.94140 0.172666
\(820\) −33.6244 −1.17422
\(821\) 21.4024 0.746950 0.373475 0.927640i \(-0.378166\pi\)
0.373475 + 0.927640i \(0.378166\pi\)
\(822\) −7.08660 −0.247174
\(823\) −16.6260 −0.579547 −0.289773 0.957095i \(-0.593580\pi\)
−0.289773 + 0.957095i \(0.593580\pi\)
\(824\) −13.3684 −0.465709
\(825\) −48.0679 −1.67351
\(826\) −3.41012 −0.118653
\(827\) 20.0328 0.696610 0.348305 0.937381i \(-0.386757\pi\)
0.348305 + 0.937381i \(0.386757\pi\)
\(828\) −6.84877 −0.238011
\(829\) 20.9181 0.726515 0.363257 0.931689i \(-0.381665\pi\)
0.363257 + 0.931689i \(0.381665\pi\)
\(830\) 52.7817 1.83208
\(831\) 23.6475 0.820323
\(832\) −1.44904 −0.0502364
\(833\) −4.62890 −0.160382
\(834\) 19.8948 0.688902
\(835\) 46.8547 1.62147
\(836\) −34.6862 −1.19965
\(837\) −8.63951 −0.298625
\(838\) 22.2459 0.768473
\(839\) 4.78548 0.165213 0.0826065 0.996582i \(-0.473676\pi\)
0.0826065 + 0.996582i \(0.473676\pi\)
\(840\) 12.7170 0.438778
\(841\) −3.97234 −0.136977
\(842\) 16.5179 0.569243
\(843\) −10.7845 −0.371439
\(844\) 5.09679 0.175439
\(845\) 40.6493 1.39838
\(846\) −2.50567 −0.0861465
\(847\) −61.8058 −2.12367
\(848\) 8.69169 0.298474
\(849\) 13.2487 0.454694
\(850\) 8.90692 0.305505
\(851\) −47.2213 −1.61873
\(852\) −0.0579369 −0.00198488
\(853\) 53.2174 1.82213 0.911064 0.412264i \(-0.135262\pi\)
0.911064 + 0.412264i \(0.135262\pi\)
\(854\) 24.7577 0.847192
\(855\) 23.9687 0.819714
\(856\) −9.76675 −0.333821
\(857\) 6.26206 0.213908 0.106954 0.994264i \(-0.465890\pi\)
0.106954 + 0.994264i \(0.465890\pi\)
\(858\) −7.82001 −0.266971
\(859\) 8.18485 0.279264 0.139632 0.990203i \(-0.455408\pi\)
0.139632 + 0.990203i \(0.455408\pi\)
\(860\) 25.2034 0.859430
\(861\) 30.7474 1.04787
\(862\) 31.4356 1.07070
\(863\) 18.5631 0.631895 0.315948 0.948777i \(-0.397678\pi\)
0.315948 + 0.948777i \(0.397678\pi\)
\(864\) 1.00000 0.0340207
\(865\) −46.4771 −1.58027
\(866\) 19.6591 0.668045
\(867\) −1.00000 −0.0339618
\(868\) −29.4618 −0.999997
\(869\) −72.1330 −2.44695
\(870\) 18.6563 0.632508
\(871\) 7.79053 0.263972
\(872\) 9.15105 0.309893
\(873\) 16.4862 0.557973
\(874\) −44.0192 −1.48897
\(875\) 49.6843 1.67964
\(876\) 12.4086 0.419247
\(877\) −0.271813 −0.00917847 −0.00458923 0.999989i \(-0.501461\pi\)
−0.00458923 + 0.999989i \(0.501461\pi\)
\(878\) 23.4092 0.790024
\(879\) −7.93575 −0.267666
\(880\) −20.1253 −0.678424
\(881\) 24.5046 0.825582 0.412791 0.910826i \(-0.364554\pi\)
0.412791 + 0.910826i \(0.364554\pi\)
\(882\) −4.62890 −0.155863
\(883\) 34.0358 1.14540 0.572698 0.819766i \(-0.305897\pi\)
0.572698 + 0.819766i \(0.305897\pi\)
\(884\) 1.44904 0.0487365
\(885\) −3.72920 −0.125356
\(886\) 14.1011 0.473736
\(887\) 11.8849 0.399057 0.199528 0.979892i \(-0.436059\pi\)
0.199528 + 0.979892i \(0.436059\pi\)
\(888\) 6.89487 0.231377
\(889\) −4.05338 −0.135946
\(890\) 20.2087 0.677399
\(891\) 5.39669 0.180796
\(892\) −17.1560 −0.574426
\(893\) −16.1047 −0.538923
\(894\) −13.8088 −0.461835
\(895\) −78.7112 −2.63103
\(896\) 3.41012 0.113924
\(897\) −9.92413 −0.331357
\(898\) 10.3411 0.345087
\(899\) −43.2214 −1.44152
\(900\) 8.90692 0.296897
\(901\) −8.69169 −0.289562
\(902\) −48.6594 −1.62018
\(903\) −23.0470 −0.766955
\(904\) 2.01354 0.0669694
\(905\) 33.3660 1.10912
\(906\) −7.58245 −0.251910
\(907\) −3.97541 −0.132001 −0.0660007 0.997820i \(-0.521024\pi\)
−0.0660007 + 0.997820i \(0.521024\pi\)
\(908\) −13.4350 −0.445855
\(909\) 9.67501 0.320900
\(910\) 18.4274 0.610864
\(911\) −46.2699 −1.53299 −0.766496 0.642249i \(-0.778001\pi\)
−0.766496 + 0.642249i \(0.778001\pi\)
\(912\) 6.42732 0.212830
\(913\) 76.3828 2.52790
\(914\) −25.8808 −0.856062
\(915\) 27.0743 0.895048
\(916\) 24.9330 0.823810
\(917\) −46.3393 −1.53026
\(918\) −1.00000 −0.0330049
\(919\) −56.0361 −1.84846 −0.924231 0.381834i \(-0.875293\pi\)
−0.924231 + 0.381834i \(0.875293\pi\)
\(920\) −25.5404 −0.842042
\(921\) 16.7504 0.551945
\(922\) 0.642862 0.0211715
\(923\) −0.0839528 −0.00276334
\(924\) 18.4033 0.605426
\(925\) 61.4120 2.01922
\(926\) 8.42833 0.276972
\(927\) 13.3684 0.439074
\(928\) 5.00277 0.164224
\(929\) 53.2321 1.74649 0.873244 0.487284i \(-0.162012\pi\)
0.873244 + 0.487284i \(0.162012\pi\)
\(930\) −32.2185 −1.05648
\(931\) −29.7514 −0.975064
\(932\) 8.01856 0.262657
\(933\) 25.3463 0.829801
\(934\) −0.311005 −0.0101764
\(935\) 20.1253 0.658168
\(936\) 1.44904 0.0473633
\(937\) 33.3503 1.08951 0.544753 0.838597i \(-0.316623\pi\)
0.544753 + 0.838597i \(0.316623\pi\)
\(938\) −18.3340 −0.598625
\(939\) 3.98350 0.129996
\(940\) −9.34412 −0.304772
\(941\) −48.6087 −1.58460 −0.792299 0.610133i \(-0.791116\pi\)
−0.792299 + 0.610133i \(0.791116\pi\)
\(942\) −5.58450 −0.181953
\(943\) −61.7521 −2.01093
\(944\) −1.00000 −0.0325472
\(945\) −12.7170 −0.413684
\(946\) 36.4730 1.18584
\(947\) −46.3613 −1.50654 −0.753270 0.657711i \(-0.771525\pi\)
−0.753270 + 0.657711i \(0.771525\pi\)
\(948\) 13.3662 0.434113
\(949\) 17.9805 0.583672
\(950\) 57.2476 1.85736
\(951\) 13.3269 0.432153
\(952\) −3.41012 −0.110523
\(953\) 13.6861 0.443335 0.221667 0.975122i \(-0.428850\pi\)
0.221667 + 0.975122i \(0.428850\pi\)
\(954\) −8.69169 −0.281404
\(955\) −67.3452 −2.17924
\(956\) −0.912919 −0.0295259
\(957\) 26.9984 0.872733
\(958\) 14.3397 0.463295
\(959\) 24.1662 0.780366
\(960\) 3.72920 0.120359
\(961\) 43.6412 1.40778
\(962\) 9.99094 0.322121
\(963\) 9.76675 0.314729
\(964\) −6.89595 −0.222104
\(965\) 54.3378 1.74919
\(966\) 23.3551 0.751438
\(967\) −2.60879 −0.0838930 −0.0419465 0.999120i \(-0.513356\pi\)
−0.0419465 + 0.999120i \(0.513356\pi\)
\(968\) −18.1242 −0.582535
\(969\) −6.42732 −0.206475
\(970\) 61.4803 1.97401
\(971\) −51.4859 −1.65226 −0.826131 0.563479i \(-0.809463\pi\)
−0.826131 + 0.563479i \(0.809463\pi\)
\(972\) −1.00000 −0.0320750
\(973\) −67.8437 −2.17497
\(974\) 15.1595 0.485742
\(975\) 12.9065 0.413338
\(976\) 7.26008 0.232389
\(977\) −35.3155 −1.12984 −0.564922 0.825144i \(-0.691094\pi\)
−0.564922 + 0.825144i \(0.691094\pi\)
\(978\) −11.5107 −0.368071
\(979\) 29.2450 0.934673
\(980\) −17.2621 −0.551417
\(981\) −9.15105 −0.292170
\(982\) 24.2062 0.772451
\(983\) 6.75397 0.215418 0.107709 0.994182i \(-0.465648\pi\)
0.107709 + 0.994182i \(0.465648\pi\)
\(984\) 9.01653 0.287437
\(985\) 16.4821 0.525162
\(986\) −5.00277 −0.159320
\(987\) 8.54461 0.271978
\(988\) 9.31343 0.296300
\(989\) 46.2867 1.47183
\(990\) 20.1253 0.639625
\(991\) −51.3017 −1.62965 −0.814827 0.579705i \(-0.803168\pi\)
−0.814827 + 0.579705i \(0.803168\pi\)
\(992\) −8.63951 −0.274305
\(993\) −1.85247 −0.0587865
\(994\) 0.197572 0.00626659
\(995\) 65.7261 2.08366
\(996\) −14.1536 −0.448475
\(997\) −47.0178 −1.48907 −0.744534 0.667585i \(-0.767328\pi\)
−0.744534 + 0.667585i \(0.767328\pi\)
\(998\) −9.06767 −0.287032
\(999\) −6.89487 −0.218144
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 6018.2.a.v.1.1 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
6018.2.a.v.1.1 9 1.1 even 1 trivial