Properties

Label 1805.2.a.t.1.3
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
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] = [1805,2,Mod(1,1805)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1805, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 0]))
 
N = Newforms(chi, 2, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("1805.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1805 = 5 \cdot 19^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1805.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: \(14.4129975648\)
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: \( x^{9} - 12x^{7} - 4x^{6} + 48x^{5} + 27x^{4} - 72x^{3} - 51x^{2} + 27x + 19 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 95)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-1.46231\) of defining polynomial
Character \(\chi\) \(=\) 1805.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.46231 q^{2} +1.51036 q^{3} +0.138346 q^{4} -1.00000 q^{5} -2.20860 q^{6} +4.07172 q^{7} +2.72231 q^{8} -0.718827 q^{9} +O(q^{10})\) \(q-1.46231 q^{2} +1.51036 q^{3} +0.138346 q^{4} -1.00000 q^{5} -2.20860 q^{6} +4.07172 q^{7} +2.72231 q^{8} -0.718827 q^{9} +1.46231 q^{10} -0.621006 q^{11} +0.208951 q^{12} -5.09329 q^{13} -5.95412 q^{14} -1.51036 q^{15} -4.25755 q^{16} -0.266483 q^{17} +1.05115 q^{18} -0.138346 q^{20} +6.14975 q^{21} +0.908102 q^{22} -5.84174 q^{23} +4.11166 q^{24} +1.00000 q^{25} +7.44796 q^{26} -5.61675 q^{27} +0.563305 q^{28} +4.07754 q^{29} +2.20860 q^{30} -6.48991 q^{31} +0.781227 q^{32} -0.937939 q^{33} +0.389681 q^{34} -4.07172 q^{35} -0.0994465 q^{36} -8.83927 q^{37} -7.69267 q^{39} -2.72231 q^{40} -4.48083 q^{41} -8.99283 q^{42} -1.80073 q^{43} -0.0859133 q^{44} +0.718827 q^{45} +8.54242 q^{46} +11.5465 q^{47} -6.43041 q^{48} +9.57894 q^{49} -1.46231 q^{50} -0.402485 q^{51} -0.704633 q^{52} -2.52292 q^{53} +8.21342 q^{54} +0.621006 q^{55} +11.0845 q^{56} -5.96263 q^{58} +0.890444 q^{59} -0.208951 q^{60} +2.16991 q^{61} +9.49025 q^{62} -2.92687 q^{63} +7.37271 q^{64} +5.09329 q^{65} +1.37156 q^{66} -14.0788 q^{67} -0.0368668 q^{68} -8.82310 q^{69} +5.95412 q^{70} -1.64668 q^{71} -1.95687 q^{72} +1.79675 q^{73} +12.9257 q^{74} +1.51036 q^{75} -2.52856 q^{77} +11.2491 q^{78} +5.14647 q^{79} +4.25755 q^{80} -6.32681 q^{81} +6.55236 q^{82} +13.7833 q^{83} +0.850790 q^{84} +0.266483 q^{85} +2.63322 q^{86} +6.15854 q^{87} -1.69057 q^{88} -0.000747238 q^{89} -1.05115 q^{90} -20.7385 q^{91} -0.808178 q^{92} -9.80207 q^{93} -16.8845 q^{94} +1.17993 q^{96} -10.4225 q^{97} -14.0074 q^{98} +0.446396 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q - 3 q^{3} + 6 q^{4} - 9 q^{5} - 12 q^{6} + 12 q^{8} + 6 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 9 q - 3 q^{3} + 6 q^{4} - 9 q^{5} - 12 q^{6} + 12 q^{8} + 6 q^{9} - 6 q^{12} + 3 q^{13} - 12 q^{14} + 3 q^{15} - 12 q^{16} - 9 q^{17} - 6 q^{18} - 6 q^{20} - 12 q^{21} - 12 q^{22} + 15 q^{24} + 9 q^{25} + 21 q^{26} - 6 q^{27} - 15 q^{28} - 15 q^{29} + 12 q^{30} - 30 q^{31} + 9 q^{32} - 9 q^{33} - 6 q^{36} - 30 q^{37} + 6 q^{39} - 12 q^{40} - 18 q^{41} + 36 q^{42} - 6 q^{43} - 24 q^{44} - 6 q^{45} - 21 q^{46} + 21 q^{47} - 15 q^{48} + 3 q^{49} - 18 q^{51} + 3 q^{52} + 9 q^{53} - 9 q^{54} - 36 q^{56} + 18 q^{58} - 27 q^{59} + 6 q^{60} + 12 q^{61} - 6 q^{62} - 15 q^{63} + 24 q^{64} - 3 q^{65} + 3 q^{66} - 36 q^{67} + 3 q^{68} - 27 q^{69} + 12 q^{70} + 6 q^{71} - 12 q^{72} - 9 q^{73} - 9 q^{74} - 3 q^{75} + 12 q^{77} - 54 q^{78} - 45 q^{79} + 12 q^{80} - 15 q^{81} - 48 q^{82} + 12 q^{84} + 9 q^{85} + 9 q^{86} + 45 q^{87} - 39 q^{88} + 9 q^{89} + 6 q^{90} - 51 q^{91} - 54 q^{92} + 9 q^{93} - 33 q^{94} - 9 q^{96} - 45 q^{97} - 33 q^{98} + 12 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.46231 −1.03401 −0.517004 0.855983i \(-0.672953\pi\)
−0.517004 + 0.855983i \(0.672953\pi\)
\(3\) 1.51036 0.872004 0.436002 0.899946i \(-0.356394\pi\)
0.436002 + 0.899946i \(0.356394\pi\)
\(4\) 0.138346 0.0691728
\(5\) −1.00000 −0.447214
\(6\) −2.20860 −0.901659
\(7\) 4.07172 1.53897 0.769484 0.638667i \(-0.220514\pi\)
0.769484 + 0.638667i \(0.220514\pi\)
\(8\) 2.72231 0.962483
\(9\) −0.718827 −0.239609
\(10\) 1.46231 0.462422
\(11\) −0.621006 −0.187240 −0.0936201 0.995608i \(-0.529844\pi\)
−0.0936201 + 0.995608i \(0.529844\pi\)
\(12\) 0.208951 0.0603189
\(13\) −5.09329 −1.41262 −0.706312 0.707901i \(-0.749642\pi\)
−0.706312 + 0.707901i \(0.749642\pi\)
\(14\) −5.95412 −1.59130
\(15\) −1.51036 −0.389972
\(16\) −4.25755 −1.06439
\(17\) −0.266483 −0.0646317 −0.0323159 0.999478i \(-0.510288\pi\)
−0.0323159 + 0.999478i \(0.510288\pi\)
\(18\) 1.05115 0.247758
\(19\) 0 0
\(20\) −0.138346 −0.0309350
\(21\) 6.14975 1.34199
\(22\) 0.908102 0.193608
\(23\) −5.84174 −1.21809 −0.609043 0.793137i \(-0.708446\pi\)
−0.609043 + 0.793137i \(0.708446\pi\)
\(24\) 4.11166 0.839289
\(25\) 1.00000 0.200000
\(26\) 7.44796 1.46066
\(27\) −5.61675 −1.08094
\(28\) 0.563305 0.106455
\(29\) 4.07754 0.757181 0.378591 0.925564i \(-0.376409\pi\)
0.378591 + 0.925564i \(0.376409\pi\)
\(30\) 2.20860 0.403234
\(31\) −6.48991 −1.16562 −0.582811 0.812608i \(-0.698047\pi\)
−0.582811 + 0.812608i \(0.698047\pi\)
\(32\) 0.781227 0.138103
\(33\) −0.937939 −0.163274
\(34\) 0.389681 0.0668297
\(35\) −4.07172 −0.688247
\(36\) −0.0994465 −0.0165744
\(37\) −8.83927 −1.45317 −0.726584 0.687078i \(-0.758893\pi\)
−0.726584 + 0.687078i \(0.758893\pi\)
\(38\) 0 0
\(39\) −7.69267 −1.23181
\(40\) −2.72231 −0.430435
\(41\) −4.48083 −0.699788 −0.349894 0.936789i \(-0.613782\pi\)
−0.349894 + 0.936789i \(0.613782\pi\)
\(42\) −8.99283 −1.38762
\(43\) −1.80073 −0.274608 −0.137304 0.990529i \(-0.543844\pi\)
−0.137304 + 0.990529i \(0.543844\pi\)
\(44\) −0.0859133 −0.0129519
\(45\) 0.718827 0.107156
\(46\) 8.54242 1.25951
\(47\) 11.5465 1.68423 0.842113 0.539301i \(-0.181312\pi\)
0.842113 + 0.539301i \(0.181312\pi\)
\(48\) −6.43041 −0.928150
\(49\) 9.57894 1.36842
\(50\) −1.46231 −0.206802
\(51\) −0.402485 −0.0563591
\(52\) −0.704633 −0.0977151
\(53\) −2.52292 −0.346550 −0.173275 0.984873i \(-0.555435\pi\)
−0.173275 + 0.984873i \(0.555435\pi\)
\(54\) 8.21342 1.11770
\(55\) 0.621006 0.0837364
\(56\) 11.0845 1.48123
\(57\) 0 0
\(58\) −5.96263 −0.782931
\(59\) 0.890444 0.115926 0.0579630 0.998319i \(-0.481539\pi\)
0.0579630 + 0.998319i \(0.481539\pi\)
\(60\) −0.208951 −0.0269754
\(61\) 2.16991 0.277828 0.138914 0.990304i \(-0.455639\pi\)
0.138914 + 0.990304i \(0.455639\pi\)
\(62\) 9.49025 1.20526
\(63\) −2.92687 −0.368751
\(64\) 7.37271 0.921588
\(65\) 5.09329 0.631744
\(66\) 1.37156 0.168827
\(67\) −14.0788 −1.72000 −0.859999 0.510296i \(-0.829536\pi\)
−0.859999 + 0.510296i \(0.829536\pi\)
\(68\) −0.0368668 −0.00447075
\(69\) −8.82310 −1.06218
\(70\) 5.95412 0.711653
\(71\) −1.64668 −0.195425 −0.0977125 0.995215i \(-0.531153\pi\)
−0.0977125 + 0.995215i \(0.531153\pi\)
\(72\) −1.95687 −0.230620
\(73\) 1.79675 0.210294 0.105147 0.994457i \(-0.466469\pi\)
0.105147 + 0.994457i \(0.466469\pi\)
\(74\) 12.9257 1.50259
\(75\) 1.51036 0.174401
\(76\) 0 0
\(77\) −2.52856 −0.288157
\(78\) 11.2491 1.27371
\(79\) 5.14647 0.579023 0.289511 0.957175i \(-0.406507\pi\)
0.289511 + 0.957175i \(0.406507\pi\)
\(80\) 4.25755 0.476009
\(81\) −6.32681 −0.702978
\(82\) 6.55236 0.723587
\(83\) 13.7833 1.51292 0.756459 0.654041i \(-0.226928\pi\)
0.756459 + 0.654041i \(0.226928\pi\)
\(84\) 0.850790 0.0928288
\(85\) 0.266483 0.0289042
\(86\) 2.63322 0.283947
\(87\) 6.15854 0.660265
\(88\) −1.69057 −0.180216
\(89\) −0.000747238 0 −7.92071e−5 0 −3.96035e−5 1.00000i \(-0.500013\pi\)
−3.96035e−5 1.00000i \(0.500013\pi\)
\(90\) −1.05115 −0.110801
\(91\) −20.7385 −2.17398
\(92\) −0.808178 −0.0842584
\(93\) −9.80207 −1.01643
\(94\) −16.8845 −1.74150
\(95\) 0 0
\(96\) 1.17993 0.120426
\(97\) −10.4225 −1.05825 −0.529123 0.848545i \(-0.677479\pi\)
−0.529123 + 0.848545i \(0.677479\pi\)
\(98\) −14.0074 −1.41496
\(99\) 0.446396 0.0448645
\(100\) 0.138346 0.0138346
\(101\) −12.2197 −1.21591 −0.607955 0.793972i \(-0.708010\pi\)
−0.607955 + 0.793972i \(0.708010\pi\)
\(102\) 0.588557 0.0582758
\(103\) −7.80372 −0.768924 −0.384462 0.923141i \(-0.625613\pi\)
−0.384462 + 0.923141i \(0.625613\pi\)
\(104\) −13.8655 −1.35963
\(105\) −6.14975 −0.600154
\(106\) 3.68929 0.358336
\(107\) 5.70414 0.551440 0.275720 0.961238i \(-0.411084\pi\)
0.275720 + 0.961238i \(0.411084\pi\)
\(108\) −0.777052 −0.0747719
\(109\) −6.73318 −0.644922 −0.322461 0.946583i \(-0.604510\pi\)
−0.322461 + 0.946583i \(0.604510\pi\)
\(110\) −0.908102 −0.0865841
\(111\) −13.3504 −1.26717
\(112\) −17.3356 −1.63806
\(113\) −4.78878 −0.450490 −0.225245 0.974302i \(-0.572318\pi\)
−0.225245 + 0.974302i \(0.572318\pi\)
\(114\) 0 0
\(115\) 5.84174 0.544745
\(116\) 0.564110 0.0523763
\(117\) 3.66119 0.338477
\(118\) −1.30210 −0.119868
\(119\) −1.08505 −0.0994661
\(120\) −4.11166 −0.375341
\(121\) −10.6144 −0.964941
\(122\) −3.17308 −0.287277
\(123\) −6.76765 −0.610218
\(124\) −0.897850 −0.0806293
\(125\) −1.00000 −0.0894427
\(126\) 4.27998 0.381291
\(127\) 18.4967 1.64131 0.820656 0.571422i \(-0.193608\pi\)
0.820656 + 0.571422i \(0.193608\pi\)
\(128\) −12.3436 −1.09103
\(129\) −2.71974 −0.239459
\(130\) −7.44796 −0.653229
\(131\) −19.5844 −1.71110 −0.855549 0.517721i \(-0.826780\pi\)
−0.855549 + 0.517721i \(0.826780\pi\)
\(132\) −0.129760 −0.0112941
\(133\) 0 0
\(134\) 20.5875 1.77849
\(135\) 5.61675 0.483413
\(136\) −0.725451 −0.0622069
\(137\) −2.35667 −0.201343 −0.100672 0.994920i \(-0.532099\pi\)
−0.100672 + 0.994920i \(0.532099\pi\)
\(138\) 12.9021 1.09830
\(139\) 3.83050 0.324899 0.162449 0.986717i \(-0.448061\pi\)
0.162449 + 0.986717i \(0.448061\pi\)
\(140\) −0.563305 −0.0476079
\(141\) 17.4393 1.46865
\(142\) 2.40795 0.202071
\(143\) 3.16296 0.264500
\(144\) 3.06044 0.255037
\(145\) −4.07754 −0.338622
\(146\) −2.62741 −0.217446
\(147\) 14.4676 1.19327
\(148\) −1.22287 −0.100520
\(149\) 18.1856 1.48982 0.744910 0.667165i \(-0.232492\pi\)
0.744910 + 0.667165i \(0.232492\pi\)
\(150\) −2.20860 −0.180332
\(151\) −2.29340 −0.186634 −0.0933170 0.995636i \(-0.529747\pi\)
−0.0933170 + 0.995636i \(0.529747\pi\)
\(152\) 0 0
\(153\) 0.191556 0.0154863
\(154\) 3.69754 0.297956
\(155\) 6.48991 0.521282
\(156\) −1.06425 −0.0852079
\(157\) −20.8445 −1.66357 −0.831787 0.555095i \(-0.812682\pi\)
−0.831787 + 0.555095i \(0.812682\pi\)
\(158\) −7.52572 −0.598714
\(159\) −3.81051 −0.302193
\(160\) −0.781227 −0.0617614
\(161\) −23.7859 −1.87460
\(162\) 9.25174 0.726885
\(163\) −7.91691 −0.620100 −0.310050 0.950720i \(-0.600346\pi\)
−0.310050 + 0.950720i \(0.600346\pi\)
\(164\) −0.619903 −0.0484063
\(165\) 0.937939 0.0730184
\(166\) −20.1555 −1.56437
\(167\) 14.0935 1.09059 0.545293 0.838246i \(-0.316418\pi\)
0.545293 + 0.838246i \(0.316418\pi\)
\(168\) 16.7415 1.29164
\(169\) 12.9416 0.995505
\(170\) −0.389681 −0.0298872
\(171\) 0 0
\(172\) −0.249122 −0.0189954
\(173\) −10.5128 −0.799273 −0.399636 0.916674i \(-0.630864\pi\)
−0.399636 + 0.916674i \(0.630864\pi\)
\(174\) −9.00569 −0.682719
\(175\) 4.07172 0.307793
\(176\) 2.64396 0.199296
\(177\) 1.34489 0.101088
\(178\) 0.00109269 8.19008e−5 0
\(179\) 5.00329 0.373963 0.186982 0.982363i \(-0.440129\pi\)
0.186982 + 0.982363i \(0.440129\pi\)
\(180\) 0.0994465 0.00741231
\(181\) −26.4916 −1.96911 −0.984553 0.175089i \(-0.943979\pi\)
−0.984553 + 0.175089i \(0.943979\pi\)
\(182\) 30.3260 2.24791
\(183\) 3.27733 0.242267
\(184\) −15.9030 −1.17239
\(185\) 8.83927 0.649876
\(186\) 14.3337 1.05099
\(187\) 0.165488 0.0121017
\(188\) 1.59740 0.116503
\(189\) −22.8699 −1.66354
\(190\) 0 0
\(191\) −0.143815 −0.0104061 −0.00520303 0.999986i \(-0.501656\pi\)
−0.00520303 + 0.999986i \(0.501656\pi\)
\(192\) 11.1354 0.803629
\(193\) 2.06115 0.148365 0.0741823 0.997245i \(-0.476365\pi\)
0.0741823 + 0.997245i \(0.476365\pi\)
\(194\) 15.2409 1.09423
\(195\) 7.69267 0.550884
\(196\) 1.32520 0.0946574
\(197\) 18.2483 1.30014 0.650068 0.759876i \(-0.274741\pi\)
0.650068 + 0.759876i \(0.274741\pi\)
\(198\) −0.652768 −0.0463902
\(199\) 8.00301 0.567318 0.283659 0.958925i \(-0.408452\pi\)
0.283659 + 0.958925i \(0.408452\pi\)
\(200\) 2.72231 0.192497
\(201\) −21.2640 −1.49984
\(202\) 17.8690 1.25726
\(203\) 16.6026 1.16528
\(204\) −0.0556819 −0.00389851
\(205\) 4.48083 0.312955
\(206\) 11.4115 0.795073
\(207\) 4.19920 0.291865
\(208\) 21.6849 1.50358
\(209\) 0 0
\(210\) 8.99283 0.620564
\(211\) −14.0828 −0.969501 −0.484750 0.874653i \(-0.661090\pi\)
−0.484750 + 0.874653i \(0.661090\pi\)
\(212\) −0.349035 −0.0239718
\(213\) −2.48707 −0.170411
\(214\) −8.34121 −0.570193
\(215\) 1.80073 0.122808
\(216\) −15.2906 −1.04039
\(217\) −26.4251 −1.79386
\(218\) 9.84599 0.666855
\(219\) 2.71374 0.183377
\(220\) 0.0859133 0.00579228
\(221\) 1.35728 0.0913003
\(222\) 19.5225 1.31026
\(223\) 4.90610 0.328537 0.164268 0.986416i \(-0.447474\pi\)
0.164268 + 0.986416i \(0.447474\pi\)
\(224\) 3.18094 0.212536
\(225\) −0.718827 −0.0479218
\(226\) 7.00267 0.465811
\(227\) 2.65246 0.176050 0.0880250 0.996118i \(-0.471944\pi\)
0.0880250 + 0.996118i \(0.471944\pi\)
\(228\) 0 0
\(229\) −23.1372 −1.52895 −0.764473 0.644655i \(-0.777001\pi\)
−0.764473 + 0.644655i \(0.777001\pi\)
\(230\) −8.54242 −0.563271
\(231\) −3.81903 −0.251274
\(232\) 11.1004 0.728774
\(233\) −28.9098 −1.89394 −0.946971 0.321319i \(-0.895874\pi\)
−0.946971 + 0.321319i \(0.895874\pi\)
\(234\) −5.35379 −0.349988
\(235\) −11.5465 −0.753209
\(236\) 0.123189 0.00801892
\(237\) 7.77299 0.504910
\(238\) 1.58667 0.102849
\(239\) −20.7473 −1.34203 −0.671016 0.741443i \(-0.734142\pi\)
−0.671016 + 0.741443i \(0.734142\pi\)
\(240\) 6.43041 0.415082
\(241\) 6.19740 0.399209 0.199605 0.979877i \(-0.436034\pi\)
0.199605 + 0.979877i \(0.436034\pi\)
\(242\) 15.5215 0.997757
\(243\) 7.29453 0.467944
\(244\) 0.300197 0.0192182
\(245\) −9.57894 −0.611976
\(246\) 9.89639 0.630971
\(247\) 0 0
\(248\) −17.6676 −1.12189
\(249\) 20.8177 1.31927
\(250\) 1.46231 0.0924845
\(251\) 7.78995 0.491697 0.245849 0.969308i \(-0.420933\pi\)
0.245849 + 0.969308i \(0.420933\pi\)
\(252\) −0.404919 −0.0255075
\(253\) 3.62775 0.228075
\(254\) −27.0478 −1.69713
\(255\) 0.402485 0.0252046
\(256\) 3.30477 0.206548
\(257\) 20.1572 1.25737 0.628686 0.777659i \(-0.283593\pi\)
0.628686 + 0.777659i \(0.283593\pi\)
\(258\) 3.97709 0.247603
\(259\) −35.9911 −2.23638
\(260\) 0.704633 0.0436995
\(261\) −2.93105 −0.181427
\(262\) 28.6385 1.76929
\(263\) 10.6858 0.658914 0.329457 0.944171i \(-0.393134\pi\)
0.329457 + 0.944171i \(0.393134\pi\)
\(264\) −2.55336 −0.157149
\(265\) 2.52292 0.154982
\(266\) 0 0
\(267\) −0.00112860 −6.90689e−5 0
\(268\) −1.94774 −0.118977
\(269\) −23.9260 −1.45879 −0.729396 0.684092i \(-0.760199\pi\)
−0.729396 + 0.684092i \(0.760199\pi\)
\(270\) −8.21342 −0.499853
\(271\) 25.9701 1.57757 0.788787 0.614667i \(-0.210709\pi\)
0.788787 + 0.614667i \(0.210709\pi\)
\(272\) 1.13457 0.0687932
\(273\) −31.3224 −1.89572
\(274\) 3.44617 0.208191
\(275\) −0.621006 −0.0374480
\(276\) −1.22064 −0.0734737
\(277\) −1.83370 −0.110176 −0.0550882 0.998481i \(-0.517544\pi\)
−0.0550882 + 0.998481i \(0.517544\pi\)
\(278\) −5.60137 −0.335948
\(279\) 4.66513 0.279294
\(280\) −11.0845 −0.662426
\(281\) −13.6844 −0.816345 −0.408173 0.912905i \(-0.633834\pi\)
−0.408173 + 0.912905i \(0.633834\pi\)
\(282\) −25.5016 −1.51860
\(283\) 10.5363 0.626319 0.313159 0.949701i \(-0.398613\pi\)
0.313159 + 0.949701i \(0.398613\pi\)
\(284\) −0.227811 −0.0135181
\(285\) 0 0
\(286\) −4.62522 −0.273495
\(287\) −18.2447 −1.07695
\(288\) −0.561567 −0.0330907
\(289\) −16.9290 −0.995823
\(290\) 5.96263 0.350138
\(291\) −15.7417 −0.922794
\(292\) 0.248573 0.0145466
\(293\) 23.9043 1.39650 0.698251 0.715853i \(-0.253962\pi\)
0.698251 + 0.715853i \(0.253962\pi\)
\(294\) −21.1561 −1.23385
\(295\) −0.890444 −0.0518437
\(296\) −24.0633 −1.39865
\(297\) 3.48803 0.202396
\(298\) −26.5929 −1.54049
\(299\) 29.7536 1.72070
\(300\) 0.208951 0.0120638
\(301\) −7.33206 −0.422613
\(302\) 3.35366 0.192981
\(303\) −18.4561 −1.06028
\(304\) 0 0
\(305\) −2.16991 −0.124249
\(306\) −0.280113 −0.0160130
\(307\) 15.6686 0.894256 0.447128 0.894470i \(-0.352447\pi\)
0.447128 + 0.894470i \(0.352447\pi\)
\(308\) −0.349815 −0.0199326
\(309\) −11.7864 −0.670505
\(310\) −9.49025 −0.539010
\(311\) −3.51210 −0.199153 −0.0995764 0.995030i \(-0.531749\pi\)
−0.0995764 + 0.995030i \(0.531749\pi\)
\(312\) −20.9419 −1.18560
\(313\) 1.17307 0.0663061 0.0331530 0.999450i \(-0.489445\pi\)
0.0331530 + 0.999450i \(0.489445\pi\)
\(314\) 30.4811 1.72015
\(315\) 2.92687 0.164910
\(316\) 0.711991 0.0400526
\(317\) −1.31161 −0.0736673 −0.0368336 0.999321i \(-0.511727\pi\)
−0.0368336 + 0.999321i \(0.511727\pi\)
\(318\) 5.57214 0.312470
\(319\) −2.53218 −0.141775
\(320\) −7.37271 −0.412147
\(321\) 8.61527 0.480858
\(322\) 34.7824 1.93835
\(323\) 0 0
\(324\) −0.875285 −0.0486270
\(325\) −5.09329 −0.282525
\(326\) 11.5770 0.641189
\(327\) −10.1695 −0.562375
\(328\) −12.1982 −0.673534
\(329\) 47.0140 2.59197
\(330\) −1.37156 −0.0755017
\(331\) −12.7249 −0.699423 −0.349712 0.936857i \(-0.613720\pi\)
−0.349712 + 0.936857i \(0.613720\pi\)
\(332\) 1.90686 0.104653
\(333\) 6.35391 0.348192
\(334\) −20.6090 −1.12767
\(335\) 14.0788 0.769206
\(336\) −26.1829 −1.42839
\(337\) 15.2196 0.829064 0.414532 0.910035i \(-0.363945\pi\)
0.414532 + 0.910035i \(0.363945\pi\)
\(338\) −18.9246 −1.02936
\(339\) −7.23276 −0.392829
\(340\) 0.0368668 0.00199938
\(341\) 4.03027 0.218251
\(342\) 0 0
\(343\) 10.5007 0.566987
\(344\) −4.90214 −0.264306
\(345\) 8.82310 0.475020
\(346\) 15.3729 0.826455
\(347\) 20.0179 1.07462 0.537308 0.843386i \(-0.319441\pi\)
0.537308 + 0.843386i \(0.319441\pi\)
\(348\) 0.852006 0.0456723
\(349\) 22.2816 1.19271 0.596353 0.802722i \(-0.296616\pi\)
0.596353 + 0.802722i \(0.296616\pi\)
\(350\) −5.95412 −0.318261
\(351\) 28.6077 1.52697
\(352\) −0.485146 −0.0258584
\(353\) −26.2150 −1.39528 −0.697642 0.716447i \(-0.745767\pi\)
−0.697642 + 0.716447i \(0.745767\pi\)
\(354\) −1.96664 −0.104526
\(355\) 1.64668 0.0873967
\(356\) −0.000103377 0 −5.47897e−6 0
\(357\) −1.63881 −0.0867348
\(358\) −7.31635 −0.386681
\(359\) 28.2782 1.49247 0.746234 0.665683i \(-0.231860\pi\)
0.746234 + 0.665683i \(0.231860\pi\)
\(360\) 1.95687 0.103136
\(361\) 0 0
\(362\) 38.7389 2.03607
\(363\) −16.0314 −0.841432
\(364\) −2.86907 −0.150380
\(365\) −1.79675 −0.0940464
\(366\) −4.79247 −0.250507
\(367\) −9.59488 −0.500848 −0.250424 0.968136i \(-0.580570\pi\)
−0.250424 + 0.968136i \(0.580570\pi\)
\(368\) 24.8715 1.29652
\(369\) 3.22094 0.167676
\(370\) −12.9257 −0.671977
\(371\) −10.2727 −0.533330
\(372\) −1.35607 −0.0703091
\(373\) 13.0401 0.675189 0.337595 0.941292i \(-0.390387\pi\)
0.337595 + 0.941292i \(0.390387\pi\)
\(374\) −0.241994 −0.0125132
\(375\) −1.51036 −0.0779944
\(376\) 31.4331 1.62104
\(377\) −20.7681 −1.06961
\(378\) 33.4428 1.72011
\(379\) 9.93895 0.510530 0.255265 0.966871i \(-0.417837\pi\)
0.255265 + 0.966871i \(0.417837\pi\)
\(380\) 0 0
\(381\) 27.9365 1.43123
\(382\) 0.210301 0.0107599
\(383\) −18.0285 −0.921212 −0.460606 0.887605i \(-0.652368\pi\)
−0.460606 + 0.887605i \(0.652368\pi\)
\(384\) −18.6433 −0.951385
\(385\) 2.52856 0.128868
\(386\) −3.01403 −0.153410
\(387\) 1.29441 0.0657986
\(388\) −1.44191 −0.0732017
\(389\) 24.3511 1.23465 0.617325 0.786708i \(-0.288216\pi\)
0.617325 + 0.786708i \(0.288216\pi\)
\(390\) −11.2491 −0.569618
\(391\) 1.55673 0.0787270
\(392\) 26.0769 1.31708
\(393\) −29.5794 −1.49208
\(394\) −26.6846 −1.34435
\(395\) −5.14647 −0.258947
\(396\) 0.0617568 0.00310340
\(397\) −19.9719 −1.00236 −0.501180 0.865343i \(-0.667100\pi\)
−0.501180 + 0.865343i \(0.667100\pi\)
\(398\) −11.7029 −0.586611
\(399\) 0 0
\(400\) −4.25755 −0.212878
\(401\) −16.4037 −0.819163 −0.409581 0.912274i \(-0.634325\pi\)
−0.409581 + 0.912274i \(0.634325\pi\)
\(402\) 31.0945 1.55085
\(403\) 33.0550 1.64659
\(404\) −1.69055 −0.0841078
\(405\) 6.32681 0.314381
\(406\) −24.2782 −1.20491
\(407\) 5.48924 0.272091
\(408\) −1.09569 −0.0542447
\(409\) 8.33303 0.412042 0.206021 0.978548i \(-0.433948\pi\)
0.206021 + 0.978548i \(0.433948\pi\)
\(410\) −6.55236 −0.323598
\(411\) −3.55940 −0.175572
\(412\) −1.07961 −0.0531886
\(413\) 3.62564 0.178406
\(414\) −6.14053 −0.301790
\(415\) −13.7833 −0.676597
\(416\) −3.97901 −0.195087
\(417\) 5.78542 0.283313
\(418\) 0 0
\(419\) −11.6553 −0.569397 −0.284698 0.958617i \(-0.591893\pi\)
−0.284698 + 0.958617i \(0.591893\pi\)
\(420\) −0.850790 −0.0415143
\(421\) 21.1916 1.03282 0.516408 0.856343i \(-0.327269\pi\)
0.516408 + 0.856343i \(0.327269\pi\)
\(422\) 20.5934 1.00247
\(423\) −8.29992 −0.403556
\(424\) −6.86819 −0.333549
\(425\) −0.266483 −0.0129263
\(426\) 3.63687 0.176207
\(427\) 8.83527 0.427569
\(428\) 0.789142 0.0381446
\(429\) 4.77719 0.230645
\(430\) −2.63322 −0.126985
\(431\) 0.392264 0.0188947 0.00944734 0.999955i \(-0.496993\pi\)
0.00944734 + 0.999955i \(0.496993\pi\)
\(432\) 23.9136 1.15054
\(433\) 12.9711 0.623354 0.311677 0.950188i \(-0.399109\pi\)
0.311677 + 0.950188i \(0.399109\pi\)
\(434\) 38.6417 1.85486
\(435\) −6.15854 −0.295279
\(436\) −0.931506 −0.0446110
\(437\) 0 0
\(438\) −3.96832 −0.189614
\(439\) −5.44800 −0.260019 −0.130009 0.991513i \(-0.541501\pi\)
−0.130009 + 0.991513i \(0.541501\pi\)
\(440\) 1.69057 0.0805948
\(441\) −6.88560 −0.327886
\(442\) −1.98476 −0.0944052
\(443\) 24.7427 1.17556 0.587780 0.809021i \(-0.300002\pi\)
0.587780 + 0.809021i \(0.300002\pi\)
\(444\) −1.84697 −0.0876535
\(445\) 0.000747238 0 3.54225e−5 0
\(446\) −7.17423 −0.339710
\(447\) 27.4667 1.29913
\(448\) 30.0196 1.41829
\(449\) 34.5619 1.63108 0.815538 0.578704i \(-0.196441\pi\)
0.815538 + 0.578704i \(0.196441\pi\)
\(450\) 1.05115 0.0495515
\(451\) 2.78262 0.131029
\(452\) −0.662506 −0.0311617
\(453\) −3.46385 −0.162746
\(454\) −3.87872 −0.182037
\(455\) 20.7385 0.972234
\(456\) 0 0
\(457\) −11.6425 −0.544613 −0.272307 0.962211i \(-0.587786\pi\)
−0.272307 + 0.962211i \(0.587786\pi\)
\(458\) 33.8337 1.58094
\(459\) 1.49677 0.0698633
\(460\) 0.808178 0.0376815
\(461\) 2.98973 0.139246 0.0696228 0.997573i \(-0.477820\pi\)
0.0696228 + 0.997573i \(0.477820\pi\)
\(462\) 5.58460 0.259819
\(463\) −8.48715 −0.394431 −0.197216 0.980360i \(-0.563190\pi\)
−0.197216 + 0.980360i \(0.563190\pi\)
\(464\) −17.3604 −0.805934
\(465\) 9.80207 0.454560
\(466\) 42.2750 1.95835
\(467\) 17.9302 0.829712 0.414856 0.909887i \(-0.363832\pi\)
0.414856 + 0.909887i \(0.363832\pi\)
\(468\) 0.506510 0.0234134
\(469\) −57.3249 −2.64702
\(470\) 16.8845 0.778824
\(471\) −31.4826 −1.45064
\(472\) 2.42407 0.111577
\(473\) 1.11826 0.0514177
\(474\) −11.3665 −0.522081
\(475\) 0 0
\(476\) −0.150111 −0.00688034
\(477\) 1.81355 0.0830366
\(478\) 30.3389 1.38767
\(479\) 3.38955 0.154873 0.0774363 0.996997i \(-0.475327\pi\)
0.0774363 + 0.996997i \(0.475327\pi\)
\(480\) −1.17993 −0.0538562
\(481\) 45.0210 2.05278
\(482\) −9.06250 −0.412786
\(483\) −35.9252 −1.63465
\(484\) −1.46845 −0.0667476
\(485\) 10.4225 0.473262
\(486\) −10.6668 −0.483858
\(487\) −11.1620 −0.505797 −0.252899 0.967493i \(-0.581384\pi\)
−0.252899 + 0.967493i \(0.581384\pi\)
\(488\) 5.90717 0.267405
\(489\) −11.9573 −0.540730
\(490\) 14.0074 0.632788
\(491\) 22.9830 1.03721 0.518604 0.855015i \(-0.326452\pi\)
0.518604 + 0.855015i \(0.326452\pi\)
\(492\) −0.936274 −0.0422105
\(493\) −1.08660 −0.0489379
\(494\) 0 0
\(495\) −0.446396 −0.0200640
\(496\) 27.6311 1.24067
\(497\) −6.70483 −0.300753
\(498\) −30.4419 −1.36414
\(499\) 0.980732 0.0439036 0.0219518 0.999759i \(-0.493012\pi\)
0.0219518 + 0.999759i \(0.493012\pi\)
\(500\) −0.138346 −0.00618700
\(501\) 21.2862 0.950995
\(502\) −11.3913 −0.508419
\(503\) 3.57195 0.159265 0.0796326 0.996824i \(-0.474625\pi\)
0.0796326 + 0.996824i \(0.474625\pi\)
\(504\) −7.96785 −0.354916
\(505\) 12.2197 0.543771
\(506\) −5.30489 −0.235831
\(507\) 19.5464 0.868085
\(508\) 2.55893 0.113534
\(509\) −12.9903 −0.575783 −0.287892 0.957663i \(-0.592954\pi\)
−0.287892 + 0.957663i \(0.592954\pi\)
\(510\) −0.588557 −0.0260617
\(511\) 7.31589 0.323636
\(512\) 19.8547 0.877460
\(513\) 0 0
\(514\) −29.4761 −1.30013
\(515\) 7.80372 0.343873
\(516\) −0.376263 −0.0165641
\(517\) −7.17042 −0.315355
\(518\) 52.6301 2.31243
\(519\) −15.8781 −0.696969
\(520\) 13.8655 0.608043
\(521\) 22.9197 1.00413 0.502064 0.864830i \(-0.332574\pi\)
0.502064 + 0.864830i \(0.332574\pi\)
\(522\) 4.28610 0.187597
\(523\) 1.97236 0.0862455 0.0431227 0.999070i \(-0.486269\pi\)
0.0431227 + 0.999070i \(0.486269\pi\)
\(524\) −2.70942 −0.118361
\(525\) 6.14975 0.268397
\(526\) −15.6259 −0.681323
\(527\) 1.72945 0.0753362
\(528\) 3.99332 0.173787
\(529\) 11.1259 0.483735
\(530\) −3.68929 −0.160253
\(531\) −0.640075 −0.0277769
\(532\) 0 0
\(533\) 22.8222 0.988538
\(534\) 0.00165035 7.14178e−5 0
\(535\) −5.70414 −0.246611
\(536\) −38.3269 −1.65547
\(537\) 7.55674 0.326098
\(538\) 34.9871 1.50840
\(539\) −5.94858 −0.256223
\(540\) 0.777052 0.0334390
\(541\) −27.5270 −1.18348 −0.591739 0.806129i \(-0.701559\pi\)
−0.591739 + 0.806129i \(0.701559\pi\)
\(542\) −37.9764 −1.63122
\(543\) −40.0117 −1.71707
\(544\) −0.208184 −0.00892582
\(545\) 6.73318 0.288418
\(546\) 45.8031 1.96019
\(547\) 8.34646 0.356869 0.178434 0.983952i \(-0.442897\pi\)
0.178434 + 0.983952i \(0.442897\pi\)
\(548\) −0.326034 −0.0139275
\(549\) −1.55979 −0.0665702
\(550\) 0.908102 0.0387216
\(551\) 0 0
\(552\) −24.0192 −1.02233
\(553\) 20.9550 0.891097
\(554\) 2.68144 0.113923
\(555\) 13.3504 0.566695
\(556\) 0.529933 0.0224742
\(557\) 23.8334 1.00985 0.504926 0.863163i \(-0.331520\pi\)
0.504926 + 0.863163i \(0.331520\pi\)
\(558\) −6.82185 −0.288792
\(559\) 9.17161 0.387918
\(560\) 17.3356 0.732562
\(561\) 0.249945 0.0105527
\(562\) 20.0109 0.844108
\(563\) −19.6274 −0.827198 −0.413599 0.910459i \(-0.635728\pi\)
−0.413599 + 0.910459i \(0.635728\pi\)
\(564\) 2.41264 0.101591
\(565\) 4.78878 0.201465
\(566\) −15.4073 −0.647618
\(567\) −25.7610 −1.08186
\(568\) −4.48278 −0.188093
\(569\) 23.7705 0.996510 0.498255 0.867030i \(-0.333974\pi\)
0.498255 + 0.867030i \(0.333974\pi\)
\(570\) 0 0
\(571\) −17.5361 −0.733865 −0.366933 0.930248i \(-0.619592\pi\)
−0.366933 + 0.930248i \(0.619592\pi\)
\(572\) 0.437581 0.0182962
\(573\) −0.217211 −0.00907412
\(574\) 26.6794 1.11358
\(575\) −5.84174 −0.243617
\(576\) −5.29970 −0.220821
\(577\) −25.9280 −1.07940 −0.539699 0.841858i \(-0.681462\pi\)
−0.539699 + 0.841858i \(0.681462\pi\)
\(578\) 24.7554 1.02969
\(579\) 3.11306 0.129375
\(580\) −0.564110 −0.0234234
\(581\) 56.1220 2.32833
\(582\) 23.0192 0.954177
\(583\) 1.56675 0.0648882
\(584\) 4.89133 0.202405
\(585\) −3.66119 −0.151372
\(586\) −34.9554 −1.44399
\(587\) −4.04205 −0.166833 −0.0834166 0.996515i \(-0.526583\pi\)
−0.0834166 + 0.996515i \(0.526583\pi\)
\(588\) 2.00153 0.0825416
\(589\) 0 0
\(590\) 1.30210 0.0536068
\(591\) 27.5614 1.13372
\(592\) 37.6337 1.54673
\(593\) 18.8561 0.774329 0.387165 0.922011i \(-0.373455\pi\)
0.387165 + 0.922011i \(0.373455\pi\)
\(594\) −5.10058 −0.209279
\(595\) 1.08505 0.0444826
\(596\) 2.51589 0.103055
\(597\) 12.0874 0.494704
\(598\) −43.5090 −1.77922
\(599\) −38.8211 −1.58619 −0.793094 0.609099i \(-0.791531\pi\)
−0.793094 + 0.609099i \(0.791531\pi\)
\(600\) 4.11166 0.167858
\(601\) 0.358419 0.0146202 0.00731011 0.999973i \(-0.497673\pi\)
0.00731011 + 0.999973i \(0.497673\pi\)
\(602\) 10.7217 0.436985
\(603\) 10.1202 0.412127
\(604\) −0.317281 −0.0129100
\(605\) 10.6144 0.431535
\(606\) 26.9886 1.09634
\(607\) 4.56885 0.185444 0.0927219 0.995692i \(-0.470443\pi\)
0.0927219 + 0.995692i \(0.470443\pi\)
\(608\) 0 0
\(609\) 25.0759 1.01613
\(610\) 3.17308 0.128474
\(611\) −58.8095 −2.37918
\(612\) 0.0265008 0.00107123
\(613\) 16.8926 0.682287 0.341143 0.940011i \(-0.389186\pi\)
0.341143 + 0.940011i \(0.389186\pi\)
\(614\) −22.9124 −0.924668
\(615\) 6.76765 0.272898
\(616\) −6.88354 −0.277346
\(617\) 24.3654 0.980914 0.490457 0.871465i \(-0.336830\pi\)
0.490457 + 0.871465i \(0.336830\pi\)
\(618\) 17.2353 0.693307
\(619\) −33.3232 −1.33937 −0.669687 0.742644i \(-0.733572\pi\)
−0.669687 + 0.742644i \(0.733572\pi\)
\(620\) 0.897850 0.0360585
\(621\) 32.8116 1.31668
\(622\) 5.13577 0.205926
\(623\) −0.00304255 −0.000121897 0
\(624\) 32.7519 1.31113
\(625\) 1.00000 0.0400000
\(626\) −1.71540 −0.0685610
\(627\) 0 0
\(628\) −2.88375 −0.115074
\(629\) 2.35552 0.0939207
\(630\) −4.27998 −0.170519
\(631\) 37.5315 1.49411 0.747053 0.664765i \(-0.231468\pi\)
0.747053 + 0.664765i \(0.231468\pi\)
\(632\) 14.0103 0.557300
\(633\) −21.2701 −0.845409
\(634\) 1.91798 0.0761725
\(635\) −18.4967 −0.734017
\(636\) −0.527167 −0.0209035
\(637\) −48.7883 −1.93306
\(638\) 3.70282 0.146596
\(639\) 1.18368 0.0468256
\(640\) 12.3436 0.487925
\(641\) −7.04598 −0.278300 −0.139150 0.990271i \(-0.544437\pi\)
−0.139150 + 0.990271i \(0.544437\pi\)
\(642\) −12.5982 −0.497211
\(643\) 15.2815 0.602645 0.301323 0.953522i \(-0.402572\pi\)
0.301323 + 0.953522i \(0.402572\pi\)
\(644\) −3.29068 −0.129671
\(645\) 2.71974 0.107089
\(646\) 0 0
\(647\) 23.8972 0.939495 0.469748 0.882801i \(-0.344345\pi\)
0.469748 + 0.882801i \(0.344345\pi\)
\(648\) −17.2235 −0.676605
\(649\) −0.552971 −0.0217060
\(650\) 7.44796 0.292133
\(651\) −39.9113 −1.56425
\(652\) −1.09527 −0.0428940
\(653\) −27.1407 −1.06210 −0.531050 0.847341i \(-0.678202\pi\)
−0.531050 + 0.847341i \(0.678202\pi\)
\(654\) 14.8709 0.581500
\(655\) 19.5844 0.765227
\(656\) 19.0774 0.744846
\(657\) −1.29156 −0.0503884
\(658\) −68.7490 −2.68012
\(659\) −45.8304 −1.78530 −0.892650 0.450750i \(-0.851156\pi\)
−0.892650 + 0.450750i \(0.851156\pi\)
\(660\) 0.129760 0.00505089
\(661\) 18.3884 0.715226 0.357613 0.933870i \(-0.383591\pi\)
0.357613 + 0.933870i \(0.383591\pi\)
\(662\) 18.6077 0.723209
\(663\) 2.04997 0.0796142
\(664\) 37.5226 1.45616
\(665\) 0 0
\(666\) −9.29138 −0.360034
\(667\) −23.8199 −0.922312
\(668\) 1.94977 0.0754388
\(669\) 7.40995 0.286485
\(670\) −20.5875 −0.795366
\(671\) −1.34753 −0.0520207
\(672\) 4.80435 0.185332
\(673\) −5.81977 −0.224336 −0.112168 0.993689i \(-0.535779\pi\)
−0.112168 + 0.993689i \(0.535779\pi\)
\(674\) −22.2557 −0.857259
\(675\) −5.61675 −0.216189
\(676\) 1.79041 0.0688619
\(677\) 17.8836 0.687324 0.343662 0.939093i \(-0.388332\pi\)
0.343662 + 0.939093i \(0.388332\pi\)
\(678\) 10.5765 0.406189
\(679\) −42.4376 −1.62861
\(680\) 0.725451 0.0278198
\(681\) 4.00616 0.153516
\(682\) −5.89350 −0.225674
\(683\) −13.4534 −0.514779 −0.257390 0.966308i \(-0.582862\pi\)
−0.257390 + 0.966308i \(0.582862\pi\)
\(684\) 0 0
\(685\) 2.35667 0.0900435
\(686\) −15.3553 −0.586269
\(687\) −34.9453 −1.33325
\(688\) 7.66668 0.292290
\(689\) 12.8500 0.489545
\(690\) −12.9021 −0.491174
\(691\) 2.31515 0.0880724 0.0440362 0.999030i \(-0.485978\pi\)
0.0440362 + 0.999030i \(0.485978\pi\)
\(692\) −1.45440 −0.0552879
\(693\) 1.81760 0.0690449
\(694\) −29.2723 −1.11116
\(695\) −3.83050 −0.145299
\(696\) 16.7655 0.635494
\(697\) 1.19407 0.0452285
\(698\) −32.5825 −1.23327
\(699\) −43.6640 −1.65152
\(700\) 0.563305 0.0212909
\(701\) 3.12614 0.118073 0.0590364 0.998256i \(-0.481197\pi\)
0.0590364 + 0.998256i \(0.481197\pi\)
\(702\) −41.8333 −1.57890
\(703\) 0 0
\(704\) −4.57849 −0.172558
\(705\) −17.4393 −0.656801
\(706\) 38.3344 1.44273
\(707\) −49.7554 −1.87124
\(708\) 0.186059 0.00699253
\(709\) −28.1738 −1.05809 −0.529045 0.848594i \(-0.677450\pi\)
−0.529045 + 0.848594i \(0.677450\pi\)
\(710\) −2.40795 −0.0903689
\(711\) −3.69942 −0.138739
\(712\) −0.00203422 −7.62355e−5 0
\(713\) 37.9124 1.41983
\(714\) 2.39644 0.0896845
\(715\) −3.16296 −0.118288
\(716\) 0.692183 0.0258681
\(717\) −31.3358 −1.17026
\(718\) −41.3515 −1.54322
\(719\) −1.21291 −0.0452338 −0.0226169 0.999744i \(-0.507200\pi\)
−0.0226169 + 0.999744i \(0.507200\pi\)
\(720\) −3.06044 −0.114056
\(721\) −31.7746 −1.18335
\(722\) 0 0
\(723\) 9.36027 0.348112
\(724\) −3.66499 −0.136208
\(725\) 4.07754 0.151436
\(726\) 23.4429 0.870048
\(727\) 35.7719 1.32671 0.663354 0.748306i \(-0.269132\pi\)
0.663354 + 0.748306i \(0.269132\pi\)
\(728\) −56.4566 −2.09242
\(729\) 29.9977 1.11103
\(730\) 2.62741 0.0972447
\(731\) 0.479864 0.0177484
\(732\) 0.453404 0.0167583
\(733\) 37.8471 1.39791 0.698957 0.715164i \(-0.253648\pi\)
0.698957 + 0.715164i \(0.253648\pi\)
\(734\) 14.0307 0.517881
\(735\) −14.4676 −0.533646
\(736\) −4.56373 −0.168221
\(737\) 8.74301 0.322053
\(738\) −4.71001 −0.173378
\(739\) 7.54675 0.277612 0.138806 0.990320i \(-0.455674\pi\)
0.138806 + 0.990320i \(0.455674\pi\)
\(740\) 1.22287 0.0449537
\(741\) 0 0
\(742\) 15.0218 0.551467
\(743\) −9.17584 −0.336629 −0.168314 0.985733i \(-0.553832\pi\)
−0.168314 + 0.985733i \(0.553832\pi\)
\(744\) −26.6843 −0.978294
\(745\) −18.1856 −0.666268
\(746\) −19.0686 −0.698151
\(747\) −9.90784 −0.362509
\(748\) 0.0228945 0.000837105 0
\(749\) 23.2257 0.848648
\(750\) 2.20860 0.0806468
\(751\) 10.4714 0.382105 0.191053 0.981580i \(-0.438810\pi\)
0.191053 + 0.981580i \(0.438810\pi\)
\(752\) −49.1597 −1.79267
\(753\) 11.7656 0.428762
\(754\) 30.3694 1.10599
\(755\) 2.29340 0.0834653
\(756\) −3.16394 −0.115071
\(757\) −53.0755 −1.92906 −0.964532 0.263967i \(-0.914969\pi\)
−0.964532 + 0.263967i \(0.914969\pi\)
\(758\) −14.5338 −0.527892
\(759\) 5.47919 0.198882
\(760\) 0 0
\(761\) −16.6886 −0.604960 −0.302480 0.953156i \(-0.597815\pi\)
−0.302480 + 0.953156i \(0.597815\pi\)
\(762\) −40.8518 −1.47990
\(763\) −27.4157 −0.992514
\(764\) −0.0198961 −0.000719816 0
\(765\) −0.191556 −0.00692570
\(766\) 26.3632 0.952541
\(767\) −4.53529 −0.163760
\(768\) 4.99138 0.180111
\(769\) −6.17780 −0.222777 −0.111389 0.993777i \(-0.535530\pi\)
−0.111389 + 0.993777i \(0.535530\pi\)
\(770\) −3.69754 −0.133250
\(771\) 30.4445 1.09643
\(772\) 0.285150 0.0102628
\(773\) 28.1558 1.01270 0.506348 0.862330i \(-0.330995\pi\)
0.506348 + 0.862330i \(0.330995\pi\)
\(774\) −1.89283 −0.0680363
\(775\) −6.48991 −0.233125
\(776\) −28.3733 −1.01854
\(777\) −54.3593 −1.95013
\(778\) −35.6088 −1.27664
\(779\) 0 0
\(780\) 1.06425 0.0381061
\(781\) 1.02260 0.0365914
\(782\) −2.27641 −0.0814044
\(783\) −22.9025 −0.818470
\(784\) −40.7828 −1.45653
\(785\) 20.8445 0.743973
\(786\) 43.2542 1.54283
\(787\) 5.46867 0.194937 0.0974685 0.995239i \(-0.468925\pi\)
0.0974685 + 0.995239i \(0.468925\pi\)
\(788\) 2.52457 0.0899339
\(789\) 16.1393 0.574576
\(790\) 7.52572 0.267753
\(791\) −19.4986 −0.693290
\(792\) 1.21523 0.0431813
\(793\) −11.0520 −0.392467
\(794\) 29.2050 1.03645
\(795\) 3.81051 0.135145
\(796\) 1.10718 0.0392429
\(797\) −47.2273 −1.67288 −0.836439 0.548061i \(-0.815366\pi\)
−0.836439 + 0.548061i \(0.815366\pi\)
\(798\) 0 0
\(799\) −3.07694 −0.108854
\(800\) 0.781227 0.0276206
\(801\) 0.000537135 0 1.89787e−5 0
\(802\) 23.9873 0.847021
\(803\) −1.11579 −0.0393755
\(804\) −2.94177 −0.103748
\(805\) 23.7859 0.838345
\(806\) −48.3366 −1.70258
\(807\) −36.1367 −1.27207
\(808\) −33.2659 −1.17029
\(809\) −6.13382 −0.215654 −0.107827 0.994170i \(-0.534389\pi\)
−0.107827 + 0.994170i \(0.534389\pi\)
\(810\) −9.25174 −0.325073
\(811\) 26.9998 0.948093 0.474046 0.880500i \(-0.342793\pi\)
0.474046 + 0.880500i \(0.342793\pi\)
\(812\) 2.29690 0.0806054
\(813\) 39.2241 1.37565
\(814\) −8.02696 −0.281345
\(815\) 7.91691 0.277317
\(816\) 1.71360 0.0599880
\(817\) 0 0
\(818\) −12.1855 −0.426055
\(819\) 14.9074 0.520906
\(820\) 0.619903 0.0216480
\(821\) −45.2474 −1.57914 −0.789572 0.613658i \(-0.789697\pi\)
−0.789572 + 0.613658i \(0.789697\pi\)
\(822\) 5.20494 0.181543
\(823\) 30.1438 1.05075 0.525373 0.850872i \(-0.323926\pi\)
0.525373 + 0.850872i \(0.323926\pi\)
\(824\) −21.2442 −0.740076
\(825\) −0.937939 −0.0326548
\(826\) −5.30181 −0.184473
\(827\) −20.4710 −0.711846 −0.355923 0.934515i \(-0.615833\pi\)
−0.355923 + 0.934515i \(0.615833\pi\)
\(828\) 0.580941 0.0201891
\(829\) −20.9207 −0.726606 −0.363303 0.931671i \(-0.618351\pi\)
−0.363303 + 0.931671i \(0.618351\pi\)
\(830\) 20.1555 0.699607
\(831\) −2.76954 −0.0960743
\(832\) −37.5513 −1.30186
\(833\) −2.55263 −0.0884433
\(834\) −8.46006 −0.292948
\(835\) −14.0935 −0.487725
\(836\) 0 0
\(837\) 36.4522 1.25997
\(838\) 17.0436 0.588761
\(839\) 4.24739 0.146636 0.0733180 0.997309i \(-0.476641\pi\)
0.0733180 + 0.997309i \(0.476641\pi\)
\(840\) −16.7415 −0.577638
\(841\) −12.3736 −0.426677
\(842\) −30.9887 −1.06794
\(843\) −20.6684 −0.711856
\(844\) −1.94829 −0.0670630
\(845\) −12.9416 −0.445204
\(846\) 12.1370 0.417280
\(847\) −43.2187 −1.48501
\(848\) 10.7415 0.368864
\(849\) 15.9136 0.546152
\(850\) 0.389681 0.0133659
\(851\) 51.6367 1.77008
\(852\) −0.344075 −0.0117878
\(853\) −1.66945 −0.0571610 −0.0285805 0.999591i \(-0.509099\pi\)
−0.0285805 + 0.999591i \(0.509099\pi\)
\(854\) −12.9199 −0.442110
\(855\) 0 0
\(856\) 15.5284 0.530751
\(857\) 0.856668 0.0292632 0.0146316 0.999893i \(-0.495342\pi\)
0.0146316 + 0.999893i \(0.495342\pi\)
\(858\) −6.98573 −0.238489
\(859\) −53.1009 −1.81178 −0.905889 0.423515i \(-0.860796\pi\)
−0.905889 + 0.423515i \(0.860796\pi\)
\(860\) 0.249122 0.00849500
\(861\) −27.5560 −0.939106
\(862\) −0.573611 −0.0195373
\(863\) 29.5608 1.00626 0.503130 0.864211i \(-0.332182\pi\)
0.503130 + 0.864211i \(0.332182\pi\)
\(864\) −4.38796 −0.149281
\(865\) 10.5128 0.357446
\(866\) −18.9678 −0.644553
\(867\) −25.5688 −0.868361
\(868\) −3.65580 −0.124086
\(869\) −3.19598 −0.108416
\(870\) 9.00569 0.305321
\(871\) 71.7073 2.42971
\(872\) −18.3298 −0.620727
\(873\) 7.49198 0.253565
\(874\) 0 0
\(875\) −4.07172 −0.137649
\(876\) 0.375433 0.0126847
\(877\) 8.79077 0.296843 0.148422 0.988924i \(-0.452581\pi\)
0.148422 + 0.988924i \(0.452581\pi\)
\(878\) 7.96665 0.268861
\(879\) 36.1039 1.21776
\(880\) −2.64396 −0.0891280
\(881\) 27.2670 0.918647 0.459324 0.888269i \(-0.348092\pi\)
0.459324 + 0.888269i \(0.348092\pi\)
\(882\) 10.0689 0.339037
\(883\) −8.65079 −0.291122 −0.145561 0.989349i \(-0.546499\pi\)
−0.145561 + 0.989349i \(0.546499\pi\)
\(884\) 0.187773 0.00631549
\(885\) −1.34489 −0.0452079
\(886\) −36.1814 −1.21554
\(887\) 39.6495 1.33130 0.665651 0.746263i \(-0.268154\pi\)
0.665651 + 0.746263i \(0.268154\pi\)
\(888\) −36.3441 −1.21963
\(889\) 75.3133 2.52593
\(890\) −0.00109269 −3.66271e−5 0
\(891\) 3.92898 0.131626
\(892\) 0.678737 0.0227258
\(893\) 0 0
\(894\) −40.1648 −1.34331
\(895\) −5.00329 −0.167242
\(896\) −50.2598 −1.67906
\(897\) 44.9386 1.50046
\(898\) −50.5401 −1.68655
\(899\) −26.4629 −0.882587
\(900\) −0.0994465 −0.00331488
\(901\) 0.672317 0.0223981
\(902\) −4.06905 −0.135485
\(903\) −11.0740 −0.368520
\(904\) −13.0366 −0.433589
\(905\) 26.4916 0.880611
\(906\) 5.06521 0.168280
\(907\) −33.3774 −1.10828 −0.554139 0.832424i \(-0.686952\pi\)
−0.554139 + 0.832424i \(0.686952\pi\)
\(908\) 0.366956 0.0121779
\(909\) 8.78388 0.291343
\(910\) −30.3260 −1.00530
\(911\) −46.3977 −1.53722 −0.768611 0.639716i \(-0.779052\pi\)
−0.768611 + 0.639716i \(0.779052\pi\)
\(912\) 0 0
\(913\) −8.55953 −0.283279
\(914\) 17.0249 0.563134
\(915\) −3.27733 −0.108345
\(916\) −3.20092 −0.105761
\(917\) −79.7424 −2.63333
\(918\) −2.18874 −0.0722392
\(919\) 8.51995 0.281047 0.140524 0.990077i \(-0.455121\pi\)
0.140524 + 0.990077i \(0.455121\pi\)
\(920\) 15.9030 0.524308
\(921\) 23.6652 0.779795
\(922\) −4.37191 −0.143981
\(923\) 8.38701 0.276062
\(924\) −0.528345 −0.0173813
\(925\) −8.83927 −0.290634
\(926\) 12.4108 0.407845
\(927\) 5.60953 0.184241
\(928\) 3.18549 0.104569
\(929\) −47.1584 −1.54722 −0.773608 0.633665i \(-0.781550\pi\)
−0.773608 + 0.633665i \(0.781550\pi\)
\(930\) −14.3337 −0.470019
\(931\) 0 0
\(932\) −3.99954 −0.131009
\(933\) −5.30452 −0.173662
\(934\) −26.2195 −0.857929
\(935\) −0.165488 −0.00541203
\(936\) 9.96691 0.325779
\(937\) −29.5004 −0.963735 −0.481867 0.876244i \(-0.660041\pi\)
−0.481867 + 0.876244i \(0.660041\pi\)
\(938\) 83.8267 2.73704
\(939\) 1.77176 0.0578192
\(940\) −1.59740 −0.0521015
\(941\) −9.49895 −0.309657 −0.154829 0.987941i \(-0.549483\pi\)
−0.154829 + 0.987941i \(0.549483\pi\)
\(942\) 46.0373 1.49998
\(943\) 26.1759 0.852403
\(944\) −3.79111 −0.123390
\(945\) 22.8699 0.743957
\(946\) −1.63524 −0.0531663
\(947\) −9.85963 −0.320395 −0.160198 0.987085i \(-0.551213\pi\)
−0.160198 + 0.987085i \(0.551213\pi\)
\(948\) 1.07536 0.0349260
\(949\) −9.15138 −0.297066
\(950\) 0 0
\(951\) −1.98099 −0.0642381
\(952\) −2.95384 −0.0957344
\(953\) 53.6746 1.73869 0.869346 0.494204i \(-0.164540\pi\)
0.869346 + 0.494204i \(0.164540\pi\)
\(954\) −2.65196 −0.0858605
\(955\) 0.143815 0.00465373
\(956\) −2.87030 −0.0928320
\(957\) −3.82449 −0.123628
\(958\) −4.95657 −0.160139
\(959\) −9.59569 −0.309861
\(960\) −11.1354 −0.359394
\(961\) 11.1190 0.358676
\(962\) −65.8345 −2.12259
\(963\) −4.10029 −0.132130
\(964\) 0.857382 0.0276144
\(965\) −2.06115 −0.0663507
\(966\) 52.5338 1.69025
\(967\) 34.2375 1.10100 0.550502 0.834834i \(-0.314436\pi\)
0.550502 + 0.834834i \(0.314436\pi\)
\(968\) −28.8956 −0.928739
\(969\) 0 0
\(970\) −15.2409 −0.489356
\(971\) 14.1698 0.454730 0.227365 0.973810i \(-0.426989\pi\)
0.227365 + 0.973810i \(0.426989\pi\)
\(972\) 1.00916 0.0323690
\(973\) 15.5967 0.500009
\(974\) 16.3222 0.522998
\(975\) −7.69267 −0.246363
\(976\) −9.23850 −0.295717
\(977\) 45.6175 1.45943 0.729717 0.683749i \(-0.239652\pi\)
0.729717 + 0.683749i \(0.239652\pi\)
\(978\) 17.4853 0.559119
\(979\) 0.000464039 0 1.48308e−5 0
\(980\) −1.32520 −0.0423321
\(981\) 4.84000 0.154529
\(982\) −33.6082 −1.07248
\(983\) 20.1860 0.643832 0.321916 0.946768i \(-0.395673\pi\)
0.321916 + 0.946768i \(0.395673\pi\)
\(984\) −18.4237 −0.587325
\(985\) −18.2483 −0.581438
\(986\) 1.58894 0.0506022
\(987\) 71.0079 2.26021
\(988\) 0 0
\(989\) 10.5194 0.334496
\(990\) 0.652768 0.0207463
\(991\) −11.0566 −0.351226 −0.175613 0.984459i \(-0.556191\pi\)
−0.175613 + 0.984459i \(0.556191\pi\)
\(992\) −5.07010 −0.160976
\(993\) −19.2191 −0.609900
\(994\) 9.80452 0.310981
\(995\) −8.00301 −0.253712
\(996\) 2.88004 0.0912576
\(997\) 31.2270 0.988969 0.494485 0.869186i \(-0.335357\pi\)
0.494485 + 0.869186i \(0.335357\pi\)
\(998\) −1.43413 −0.0453967
\(999\) 49.6480 1.57079
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 1805.2.a.t.1.3 9
5.4 even 2 9025.2.a.ce.1.7 9
19.6 even 9 95.2.k.b.36.1 18
19.16 even 9 95.2.k.b.66.1 yes 18
19.18 odd 2 1805.2.a.u.1.7 9
57.35 odd 18 855.2.bs.b.541.3 18
57.44 odd 18 855.2.bs.b.226.3 18
95.44 even 18 475.2.l.b.226.3 18
95.54 even 18 475.2.l.b.351.3 18
95.63 odd 36 475.2.u.c.74.5 36
95.73 odd 36 475.2.u.c.199.2 36
95.82 odd 36 475.2.u.c.74.2 36
95.92 odd 36 475.2.u.c.199.5 36
95.94 odd 2 9025.2.a.cd.1.3 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.k.b.36.1 18 19.6 even 9
95.2.k.b.66.1 yes 18 19.16 even 9
475.2.l.b.226.3 18 95.44 even 18
475.2.l.b.351.3 18 95.54 even 18
475.2.u.c.74.2 36 95.82 odd 36
475.2.u.c.74.5 36 95.63 odd 36
475.2.u.c.199.2 36 95.73 odd 36
475.2.u.c.199.5 36 95.92 odd 36
855.2.bs.b.226.3 18 57.44 odd 18
855.2.bs.b.541.3 18 57.35 odd 18
1805.2.a.t.1.3 9 1.1 even 1 trivial
1805.2.a.u.1.7 9 19.18 odd 2
9025.2.a.cd.1.3 9 95.94 odd 2
9025.2.a.ce.1.7 9 5.4 even 2