Properties

Label 1155.2.a.s.1.1
Level $1155$
Weight $2$
Character 1155.1
Self dual yes
Analytic conductor $9.223$
Analytic rank $0$
Dimension $3$
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] = [1155,2,Mod(1,1155)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(1155, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([0, 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("1155.1");
 
S:= CuspForms(chi, 2);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 1155 = 3 \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1155.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: \(9.22272143346\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.316.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 4x + 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.81361\) of defining polynomial
Character \(\chi\) \(=\) 1155.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-1.81361 q^{2} -1.00000 q^{3} +1.28917 q^{4} +1.00000 q^{5} +1.81361 q^{6} -1.00000 q^{7} +1.28917 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.81361 q^{2} -1.00000 q^{3} +1.28917 q^{4} +1.00000 q^{5} +1.81361 q^{6} -1.00000 q^{7} +1.28917 q^{8} +1.00000 q^{9} -1.81361 q^{10} -1.00000 q^{11} -1.28917 q^{12} +5.10278 q^{13} +1.81361 q^{14} -1.00000 q^{15} -4.91638 q^{16} +4.28917 q^{17} -1.81361 q^{18} -3.33804 q^{19} +1.28917 q^{20} +1.00000 q^{21} +1.81361 q^{22} -3.91638 q^{23} -1.28917 q^{24} +1.00000 q^{25} -9.25443 q^{26} -1.00000 q^{27} -1.28917 q^{28} +9.01916 q^{29} +1.81361 q^{30} -3.68111 q^{31} +6.33804 q^{32} +1.00000 q^{33} -7.77886 q^{34} -1.00000 q^{35} +1.28917 q^{36} -1.10278 q^{37} +6.05390 q^{38} -5.10278 q^{39} +1.28917 q^{40} -6.72999 q^{41} -1.81361 q^{42} -5.39194 q^{43} -1.28917 q^{44} +1.00000 q^{45} +7.10278 q^{46} +8.72999 q^{47} +4.91638 q^{48} +1.00000 q^{49} -1.81361 q^{50} -4.28917 q^{51} +6.57834 q^{52} +8.75971 q^{53} +1.81361 q^{54} -1.00000 q^{55} -1.28917 q^{56} +3.33804 q^{57} -16.3572 q^{58} -5.39194 q^{59} -1.28917 q^{60} -13.7491 q^{61} +6.67609 q^{62} -1.00000 q^{63} -1.66196 q^{64} +5.10278 q^{65} -1.81361 q^{66} +10.6761 q^{67} +5.52946 q^{68} +3.91638 q^{69} +1.81361 q^{70} +10.3572 q^{71} +1.28917 q^{72} +3.62721 q^{73} +2.00000 q^{74} -1.00000 q^{75} -4.30330 q^{76} +1.00000 q^{77} +9.25443 q^{78} -12.1517 q^{79} -4.91638 q^{80} +1.00000 q^{81} +12.2056 q^{82} -4.96526 q^{83} +1.28917 q^{84} +4.28917 q^{85} +9.77886 q^{86} -9.01916 q^{87} -1.28917 q^{88} +15.1169 q^{89} -1.81361 q^{90} -5.10278 q^{91} -5.04888 q^{92} +3.68111 q^{93} -15.8328 q^{94} -3.33804 q^{95} -6.33804 q^{96} +11.8625 q^{97} -1.81361 q^{98} -1.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q + q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 3 q + q^{2} - 3 q^{3} + 3 q^{4} + 3 q^{5} - q^{6} - 3 q^{7} + 3 q^{8} + 3 q^{9} + q^{10} - 3 q^{11} - 3 q^{12} + 8 q^{13} - q^{14} - 3 q^{15} - q^{16} + 12 q^{17} + q^{18} + 2 q^{19} + 3 q^{20} + 3 q^{21} - q^{22} + 2 q^{23} - 3 q^{24} + 3 q^{25} - 2 q^{26} - 3 q^{27} - 3 q^{28} + 6 q^{29} - q^{30} - 2 q^{31} + 7 q^{32} + 3 q^{33} + 8 q^{34} - 3 q^{35} + 3 q^{36} + 4 q^{37} + 22 q^{38} - 8 q^{39} + 3 q^{40} + q^{42} - 8 q^{43} - 3 q^{44} + 3 q^{45} + 14 q^{46} + 6 q^{47} + q^{48} + 3 q^{49} + q^{50} - 12 q^{51} + 18 q^{52} + 16 q^{53} - q^{54} - 3 q^{55} - 3 q^{56} - 2 q^{57} - 16 q^{58} - 8 q^{59} - 3 q^{60} - 4 q^{62} - 3 q^{63} - 17 q^{64} + 8 q^{65} + q^{66} + 8 q^{67} + 26 q^{68} - 2 q^{69} - q^{70} - 2 q^{71} + 3 q^{72} - 2 q^{73} + 6 q^{74} - 3 q^{75} + 24 q^{76} + 3 q^{77} + 2 q^{78} - 18 q^{79} - q^{80} + 3 q^{81} + 22 q^{82} + 10 q^{83} + 3 q^{84} + 12 q^{85} - 2 q^{86} - 6 q^{87} - 3 q^{88} + 2 q^{89} + q^{90} - 8 q^{91} - 4 q^{92} + 2 q^{93} - 20 q^{94} + 2 q^{95} - 7 q^{96} + 18 q^{97} + q^{98} - 3 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.81361 −1.28241 −0.641207 0.767368i \(-0.721566\pi\)
−0.641207 + 0.767368i \(0.721566\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.28917 0.644584
\(5\) 1.00000 0.447214
\(6\) 1.81361 0.740402
\(7\) −1.00000 −0.377964
\(8\) 1.28917 0.455790
\(9\) 1.00000 0.333333
\(10\) −1.81361 −0.573513
\(11\) −1.00000 −0.301511
\(12\) −1.28917 −0.372151
\(13\) 5.10278 1.41526 0.707628 0.706586i \(-0.249765\pi\)
0.707628 + 0.706586i \(0.249765\pi\)
\(14\) 1.81361 0.484707
\(15\) −1.00000 −0.258199
\(16\) −4.91638 −1.22910
\(17\) 4.28917 1.04028 0.520138 0.854082i \(-0.325880\pi\)
0.520138 + 0.854082i \(0.325880\pi\)
\(18\) −1.81361 −0.427471
\(19\) −3.33804 −0.765800 −0.382900 0.923790i \(-0.625075\pi\)
−0.382900 + 0.923790i \(0.625075\pi\)
\(20\) 1.28917 0.288267
\(21\) 1.00000 0.218218
\(22\) 1.81361 0.386662
\(23\) −3.91638 −0.816622 −0.408311 0.912843i \(-0.633882\pi\)
−0.408311 + 0.912843i \(0.633882\pi\)
\(24\) −1.28917 −0.263150
\(25\) 1.00000 0.200000
\(26\) −9.25443 −1.81494
\(27\) −1.00000 −0.192450
\(28\) −1.28917 −0.243630
\(29\) 9.01916 1.67482 0.837408 0.546579i \(-0.184070\pi\)
0.837408 + 0.546579i \(0.184070\pi\)
\(30\) 1.81361 0.331118
\(31\) −3.68111 −0.661147 −0.330574 0.943780i \(-0.607242\pi\)
−0.330574 + 0.943780i \(0.607242\pi\)
\(32\) 6.33804 1.12042
\(33\) 1.00000 0.174078
\(34\) −7.77886 −1.33406
\(35\) −1.00000 −0.169031
\(36\) 1.28917 0.214861
\(37\) −1.10278 −0.181295 −0.0906476 0.995883i \(-0.528894\pi\)
−0.0906476 + 0.995883i \(0.528894\pi\)
\(38\) 6.05390 0.982072
\(39\) −5.10278 −0.817098
\(40\) 1.28917 0.203835
\(41\) −6.72999 −1.05105 −0.525524 0.850779i \(-0.676131\pi\)
−0.525524 + 0.850779i \(0.676131\pi\)
\(42\) −1.81361 −0.279846
\(43\) −5.39194 −0.822264 −0.411132 0.911576i \(-0.634866\pi\)
−0.411132 + 0.911576i \(0.634866\pi\)
\(44\) −1.28917 −0.194349
\(45\) 1.00000 0.149071
\(46\) 7.10278 1.04725
\(47\) 8.72999 1.27340 0.636700 0.771112i \(-0.280299\pi\)
0.636700 + 0.771112i \(0.280299\pi\)
\(48\) 4.91638 0.709619
\(49\) 1.00000 0.142857
\(50\) −1.81361 −0.256483
\(51\) −4.28917 −0.600604
\(52\) 6.57834 0.912251
\(53\) 8.75971 1.20324 0.601619 0.798783i \(-0.294522\pi\)
0.601619 + 0.798783i \(0.294522\pi\)
\(54\) 1.81361 0.246801
\(55\) −1.00000 −0.134840
\(56\) −1.28917 −0.172272
\(57\) 3.33804 0.442135
\(58\) −16.3572 −2.14781
\(59\) −5.39194 −0.701971 −0.350986 0.936381i \(-0.614153\pi\)
−0.350986 + 0.936381i \(0.614153\pi\)
\(60\) −1.28917 −0.166431
\(61\) −13.7491 −1.76040 −0.880199 0.474605i \(-0.842591\pi\)
−0.880199 + 0.474605i \(0.842591\pi\)
\(62\) 6.67609 0.847864
\(63\) −1.00000 −0.125988
\(64\) −1.66196 −0.207744
\(65\) 5.10278 0.632921
\(66\) −1.81361 −0.223240
\(67\) 10.6761 1.30429 0.652146 0.758093i \(-0.273869\pi\)
0.652146 + 0.758093i \(0.273869\pi\)
\(68\) 5.52946 0.670546
\(69\) 3.91638 0.471477
\(70\) 1.81361 0.216767
\(71\) 10.3572 1.22917 0.614587 0.788849i \(-0.289323\pi\)
0.614587 + 0.788849i \(0.289323\pi\)
\(72\) 1.28917 0.151930
\(73\) 3.62721 0.424533 0.212267 0.977212i \(-0.431915\pi\)
0.212267 + 0.977212i \(0.431915\pi\)
\(74\) 2.00000 0.232495
\(75\) −1.00000 −0.115470
\(76\) −4.30330 −0.493623
\(77\) 1.00000 0.113961
\(78\) 9.25443 1.04786
\(79\) −12.1517 −1.36717 −0.683584 0.729872i \(-0.739580\pi\)
−0.683584 + 0.729872i \(0.739580\pi\)
\(80\) −4.91638 −0.549668
\(81\) 1.00000 0.111111
\(82\) 12.2056 1.34788
\(83\) −4.96526 −0.545008 −0.272504 0.962155i \(-0.587852\pi\)
−0.272504 + 0.962155i \(0.587852\pi\)
\(84\) 1.28917 0.140660
\(85\) 4.28917 0.465226
\(86\) 9.77886 1.05448
\(87\) −9.01916 −0.966955
\(88\) −1.28917 −0.137426
\(89\) 15.1169 1.60239 0.801195 0.598404i \(-0.204198\pi\)
0.801195 + 0.598404i \(0.204198\pi\)
\(90\) −1.81361 −0.191171
\(91\) −5.10278 −0.534916
\(92\) −5.04888 −0.526382
\(93\) 3.68111 0.381714
\(94\) −15.8328 −1.63302
\(95\) −3.33804 −0.342476
\(96\) −6.33804 −0.646874
\(97\) 11.8625 1.20445 0.602226 0.798325i \(-0.294280\pi\)
0.602226 + 0.798325i \(0.294280\pi\)
\(98\) −1.81361 −0.183202
\(99\) −1.00000 −0.100504
\(100\) 1.28917 0.128917
\(101\) 6.00000 0.597022 0.298511 0.954406i \(-0.403510\pi\)
0.298511 + 0.954406i \(0.403510\pi\)
\(102\) 7.77886 0.770222
\(103\) 15.5975 1.53687 0.768433 0.639930i \(-0.221037\pi\)
0.768433 + 0.639930i \(0.221037\pi\)
\(104\) 6.57834 0.645059
\(105\) 1.00000 0.0975900
\(106\) −15.8867 −1.54305
\(107\) 17.7789 1.71875 0.859374 0.511348i \(-0.170854\pi\)
0.859374 + 0.511348i \(0.170854\pi\)
\(108\) −1.28917 −0.124050
\(109\) 8.35720 0.800475 0.400237 0.916412i \(-0.368928\pi\)
0.400237 + 0.916412i \(0.368928\pi\)
\(110\) 1.81361 0.172921
\(111\) 1.10278 0.104671
\(112\) 4.91638 0.464554
\(113\) 16.7003 1.57103 0.785515 0.618843i \(-0.212398\pi\)
0.785515 + 0.618843i \(0.212398\pi\)
\(114\) −6.05390 −0.567000
\(115\) −3.91638 −0.365204
\(116\) 11.6272 1.07956
\(117\) 5.10278 0.471752
\(118\) 9.77886 0.900217
\(119\) −4.28917 −0.393187
\(120\) −1.28917 −0.117684
\(121\) 1.00000 0.0909091
\(122\) 24.9355 2.25756
\(123\) 6.72999 0.606823
\(124\) −4.74557 −0.426165
\(125\) 1.00000 0.0894427
\(126\) 1.81361 0.161569
\(127\) −12.8136 −1.13702 −0.568512 0.822675i \(-0.692481\pi\)
−0.568512 + 0.822675i \(0.692481\pi\)
\(128\) −9.66196 −0.854004
\(129\) 5.39194 0.474734
\(130\) −9.25443 −0.811667
\(131\) 18.9355 1.65441 0.827203 0.561903i \(-0.189931\pi\)
0.827203 + 0.561903i \(0.189931\pi\)
\(132\) 1.28917 0.112208
\(133\) 3.33804 0.289445
\(134\) −19.3622 −1.67264
\(135\) −1.00000 −0.0860663
\(136\) 5.52946 0.474147
\(137\) −4.03831 −0.345016 −0.172508 0.985008i \(-0.555187\pi\)
−0.172508 + 0.985008i \(0.555187\pi\)
\(138\) −7.10278 −0.604628
\(139\) 20.1361 1.70792 0.853959 0.520340i \(-0.174195\pi\)
0.853959 + 0.520340i \(0.174195\pi\)
\(140\) −1.28917 −0.108955
\(141\) −8.72999 −0.735198
\(142\) −18.7839 −1.57631
\(143\) −5.10278 −0.426715
\(144\) −4.91638 −0.409698
\(145\) 9.01916 0.749000
\(146\) −6.57834 −0.544427
\(147\) −1.00000 −0.0824786
\(148\) −1.42166 −0.116860
\(149\) −20.6167 −1.68898 −0.844491 0.535570i \(-0.820097\pi\)
−0.844491 + 0.535570i \(0.820097\pi\)
\(150\) 1.81361 0.148080
\(151\) 5.73501 0.466709 0.233354 0.972392i \(-0.425030\pi\)
0.233354 + 0.972392i \(0.425030\pi\)
\(152\) −4.30330 −0.349044
\(153\) 4.28917 0.346759
\(154\) −1.81361 −0.146145
\(155\) −3.68111 −0.295674
\(156\) −6.57834 −0.526688
\(157\) −10.0680 −0.803516 −0.401758 0.915746i \(-0.631601\pi\)
−0.401758 + 0.915746i \(0.631601\pi\)
\(158\) 22.0383 1.75327
\(159\) −8.75971 −0.694690
\(160\) 6.33804 0.501066
\(161\) 3.91638 0.308654
\(162\) −1.81361 −0.142490
\(163\) 0.837786 0.0656205 0.0328102 0.999462i \(-0.489554\pi\)
0.0328102 + 0.999462i \(0.489554\pi\)
\(164\) −8.67609 −0.677489
\(165\) 1.00000 0.0778499
\(166\) 9.00502 0.698925
\(167\) 24.1361 1.86770 0.933852 0.357659i \(-0.116425\pi\)
0.933852 + 0.357659i \(0.116425\pi\)
\(168\) 1.28917 0.0994615
\(169\) 13.0383 1.00295
\(170\) −7.77886 −0.596612
\(171\) −3.33804 −0.255267
\(172\) −6.95112 −0.530018
\(173\) −19.2927 −1.46680 −0.733400 0.679797i \(-0.762068\pi\)
−0.733400 + 0.679797i \(0.762068\pi\)
\(174\) 16.3572 1.24004
\(175\) −1.00000 −0.0755929
\(176\) 4.91638 0.370586
\(177\) 5.39194 0.405283
\(178\) −27.4161 −2.05493
\(179\) 6.35720 0.475160 0.237580 0.971368i \(-0.423646\pi\)
0.237580 + 0.971368i \(0.423646\pi\)
\(180\) 1.28917 0.0960890
\(181\) −3.15667 −0.234634 −0.117317 0.993095i \(-0.537429\pi\)
−0.117317 + 0.993095i \(0.537429\pi\)
\(182\) 9.25443 0.685984
\(183\) 13.7491 1.01637
\(184\) −5.04888 −0.372208
\(185\) −1.10278 −0.0810776
\(186\) −6.67609 −0.489515
\(187\) −4.28917 −0.313655
\(188\) 11.2544 0.820813
\(189\) 1.00000 0.0727393
\(190\) 6.05390 0.439196
\(191\) −11.2544 −0.814342 −0.407171 0.913352i \(-0.633485\pi\)
−0.407171 + 0.913352i \(0.633485\pi\)
\(192\) 1.66196 0.119941
\(193\) −23.4600 −1.68869 −0.844343 0.535803i \(-0.820009\pi\)
−0.844343 + 0.535803i \(0.820009\pi\)
\(194\) −21.5139 −1.54461
\(195\) −5.10278 −0.365417
\(196\) 1.28917 0.0920835
\(197\) 8.78389 0.625826 0.312913 0.949782i \(-0.398695\pi\)
0.312913 + 0.949782i \(0.398695\pi\)
\(198\) 1.81361 0.128887
\(199\) 15.6655 1.11050 0.555250 0.831684i \(-0.312623\pi\)
0.555250 + 0.831684i \(0.312623\pi\)
\(200\) 1.28917 0.0911580
\(201\) −10.6761 −0.753033
\(202\) −10.8816 −0.765629
\(203\) −9.01916 −0.633021
\(204\) −5.52946 −0.387140
\(205\) −6.72999 −0.470043
\(206\) −28.2877 −1.97090
\(207\) −3.91638 −0.272207
\(208\) −25.0872 −1.73948
\(209\) 3.33804 0.230897
\(210\) −1.81361 −0.125151
\(211\) 0.881639 0.0606945 0.0303473 0.999539i \(-0.490339\pi\)
0.0303473 + 0.999539i \(0.490339\pi\)
\(212\) 11.2927 0.775589
\(213\) −10.3572 −0.709664
\(214\) −32.2439 −2.20415
\(215\) −5.39194 −0.367728
\(216\) −1.28917 −0.0877168
\(217\) 3.68111 0.249890
\(218\) −15.1567 −1.02654
\(219\) −3.62721 −0.245104
\(220\) −1.28917 −0.0869157
\(221\) 21.8867 1.47226
\(222\) −2.00000 −0.134231
\(223\) −0.235269 −0.0157548 −0.00787740 0.999969i \(-0.502507\pi\)
−0.00787740 + 0.999969i \(0.502507\pi\)
\(224\) −6.33804 −0.423478
\(225\) 1.00000 0.0666667
\(226\) −30.2877 −2.01471
\(227\) 9.71083 0.644531 0.322265 0.946649i \(-0.395556\pi\)
0.322265 + 0.946649i \(0.395556\pi\)
\(228\) 4.30330 0.284993
\(229\) 18.6705 1.23378 0.616892 0.787048i \(-0.288391\pi\)
0.616892 + 0.787048i \(0.288391\pi\)
\(230\) 7.10278 0.468343
\(231\) −1.00000 −0.0657952
\(232\) 11.6272 0.763364
\(233\) −20.9355 −1.37153 −0.685766 0.727822i \(-0.740533\pi\)
−0.685766 + 0.727822i \(0.740533\pi\)
\(234\) −9.25443 −0.604981
\(235\) 8.72999 0.569482
\(236\) −6.95112 −0.452480
\(237\) 12.1517 0.789335
\(238\) 7.77886 0.504229
\(239\) 28.3119 1.83134 0.915672 0.401926i \(-0.131659\pi\)
0.915672 + 0.401926i \(0.131659\pi\)
\(240\) 4.91638 0.317351
\(241\) 4.26499 0.274732 0.137366 0.990520i \(-0.456136\pi\)
0.137366 + 0.990520i \(0.456136\pi\)
\(242\) −1.81361 −0.116583
\(243\) −1.00000 −0.0641500
\(244\) −17.7250 −1.13472
\(245\) 1.00000 0.0638877
\(246\) −12.2056 −0.778197
\(247\) −17.0333 −1.08380
\(248\) −4.74557 −0.301344
\(249\) 4.96526 0.314660
\(250\) −1.81361 −0.114703
\(251\) −18.7244 −1.18188 −0.590938 0.806717i \(-0.701242\pi\)
−0.590938 + 0.806717i \(0.701242\pi\)
\(252\) −1.28917 −0.0812100
\(253\) 3.91638 0.246221
\(254\) 23.2388 1.45813
\(255\) −4.28917 −0.268598
\(256\) 20.8469 1.30293
\(257\) −4.63224 −0.288951 −0.144475 0.989508i \(-0.546149\pi\)
−0.144475 + 0.989508i \(0.546149\pi\)
\(258\) −9.77886 −0.608805
\(259\) 1.10278 0.0685231
\(260\) 6.57834 0.407971
\(261\) 9.01916 0.558272
\(262\) −34.3416 −2.12163
\(263\) −12.7144 −0.784004 −0.392002 0.919964i \(-0.628217\pi\)
−0.392002 + 0.919964i \(0.628217\pi\)
\(264\) 1.28917 0.0793428
\(265\) 8.75971 0.538105
\(266\) −6.05390 −0.371188
\(267\) −15.1169 −0.925140
\(268\) 13.7633 0.840726
\(269\) −21.3225 −1.30005 −0.650027 0.759911i \(-0.725242\pi\)
−0.650027 + 0.759911i \(0.725242\pi\)
\(270\) 1.81361 0.110373
\(271\) 4.21968 0.256328 0.128164 0.991753i \(-0.459092\pi\)
0.128164 + 0.991753i \(0.459092\pi\)
\(272\) −21.0872 −1.27860
\(273\) 5.10278 0.308834
\(274\) 7.32391 0.442454
\(275\) −1.00000 −0.0603023
\(276\) 5.04888 0.303907
\(277\) 9.08719 0.545996 0.272998 0.962015i \(-0.411985\pi\)
0.272998 + 0.962015i \(0.411985\pi\)
\(278\) −36.5189 −2.19026
\(279\) −3.68111 −0.220382
\(280\) −1.28917 −0.0770426
\(281\) −20.6761 −1.23343 −0.616716 0.787186i \(-0.711537\pi\)
−0.616716 + 0.787186i \(0.711537\pi\)
\(282\) 15.8328 0.942827
\(283\) −7.04334 −0.418683 −0.209341 0.977843i \(-0.567132\pi\)
−0.209341 + 0.977843i \(0.567132\pi\)
\(284\) 13.3522 0.792306
\(285\) 3.33804 0.197729
\(286\) 9.25443 0.547226
\(287\) 6.72999 0.397259
\(288\) 6.33804 0.373473
\(289\) 1.39697 0.0821745
\(290\) −16.3572 −0.960528
\(291\) −11.8625 −0.695391
\(292\) 4.67609 0.273647
\(293\) −4.59247 −0.268295 −0.134147 0.990961i \(-0.542830\pi\)
−0.134147 + 0.990961i \(0.542830\pi\)
\(294\) 1.81361 0.105772
\(295\) −5.39194 −0.313931
\(296\) −1.42166 −0.0826325
\(297\) 1.00000 0.0580259
\(298\) 37.3905 2.16597
\(299\) −19.9844 −1.15573
\(300\) −1.28917 −0.0744302
\(301\) 5.39194 0.310786
\(302\) −10.4011 −0.598513
\(303\) −6.00000 −0.344691
\(304\) 16.4111 0.941241
\(305\) −13.7491 −0.787274
\(306\) −7.77886 −0.444688
\(307\) 26.0383 1.48609 0.743043 0.669244i \(-0.233382\pi\)
0.743043 + 0.669244i \(0.233382\pi\)
\(308\) 1.28917 0.0734572
\(309\) −15.5975 −0.887310
\(310\) 6.67609 0.379176
\(311\) −16.3033 −0.924475 −0.462238 0.886756i \(-0.652953\pi\)
−0.462238 + 0.886756i \(0.652953\pi\)
\(312\) −6.57834 −0.372425
\(313\) 26.3713 1.49060 0.745298 0.666732i \(-0.232307\pi\)
0.745298 + 0.666732i \(0.232307\pi\)
\(314\) 18.2594 1.03044
\(315\) −1.00000 −0.0563436
\(316\) −15.6655 −0.881255
\(317\) 17.4217 0.978498 0.489249 0.872144i \(-0.337271\pi\)
0.489249 + 0.872144i \(0.337271\pi\)
\(318\) 15.8867 0.890880
\(319\) −9.01916 −0.504976
\(320\) −1.66196 −0.0929061
\(321\) −17.7789 −0.992319
\(322\) −7.10278 −0.395822
\(323\) −14.3174 −0.796643
\(324\) 1.28917 0.0716205
\(325\) 5.10278 0.283051
\(326\) −1.51941 −0.0841526
\(327\) −8.35720 −0.462154
\(328\) −8.67609 −0.479057
\(329\) −8.72999 −0.481300
\(330\) −1.81361 −0.0998358
\(331\) −16.2197 −0.891514 −0.445757 0.895154i \(-0.647066\pi\)
−0.445757 + 0.895154i \(0.647066\pi\)
\(332\) −6.40105 −0.351303
\(333\) −1.10278 −0.0604317
\(334\) −43.7733 −2.39517
\(335\) 10.6761 0.583297
\(336\) −4.91638 −0.268211
\(337\) −6.92140 −0.377033 −0.188516 0.982070i \(-0.560368\pi\)
−0.188516 + 0.982070i \(0.560368\pi\)
\(338\) −23.6464 −1.28619
\(339\) −16.7003 −0.907034
\(340\) 5.52946 0.299877
\(341\) 3.68111 0.199343
\(342\) 6.05390 0.327357
\(343\) −1.00000 −0.0539949
\(344\) −6.95112 −0.374779
\(345\) 3.91638 0.210851
\(346\) 34.9894 1.88104
\(347\) −17.8711 −0.959370 −0.479685 0.877441i \(-0.659249\pi\)
−0.479685 + 0.877441i \(0.659249\pi\)
\(348\) −11.6272 −0.623284
\(349\) 2.72139 0.145673 0.0728364 0.997344i \(-0.476795\pi\)
0.0728364 + 0.997344i \(0.476795\pi\)
\(350\) 1.81361 0.0969413
\(351\) −5.10278 −0.272366
\(352\) −6.33804 −0.337819
\(353\) −18.7144 −0.996067 −0.498034 0.867158i \(-0.665944\pi\)
−0.498034 + 0.867158i \(0.665944\pi\)
\(354\) −9.77886 −0.519741
\(355\) 10.3572 0.549703
\(356\) 19.4882 1.03287
\(357\) 4.28917 0.227007
\(358\) −11.5295 −0.609351
\(359\) 6.30475 0.332752 0.166376 0.986062i \(-0.446793\pi\)
0.166376 + 0.986062i \(0.446793\pi\)
\(360\) 1.28917 0.0679451
\(361\) −7.85746 −0.413550
\(362\) 5.72496 0.300897
\(363\) −1.00000 −0.0524864
\(364\) −6.57834 −0.344799
\(365\) 3.62721 0.189857
\(366\) −24.9355 −1.30340
\(367\) 19.0192 0.992792 0.496396 0.868096i \(-0.334656\pi\)
0.496396 + 0.868096i \(0.334656\pi\)
\(368\) 19.2544 1.00371
\(369\) −6.72999 −0.350349
\(370\) 2.00000 0.103975
\(371\) −8.75971 −0.454781
\(372\) 4.74557 0.246047
\(373\) −14.0086 −0.725337 −0.362669 0.931918i \(-0.618134\pi\)
−0.362669 + 0.931918i \(0.618134\pi\)
\(374\) 7.77886 0.402235
\(375\) −1.00000 −0.0516398
\(376\) 11.2544 0.580403
\(377\) 46.0227 2.37029
\(378\) −1.81361 −0.0932819
\(379\) 13.2686 0.681560 0.340780 0.940143i \(-0.389309\pi\)
0.340780 + 0.940143i \(0.389309\pi\)
\(380\) −4.30330 −0.220755
\(381\) 12.8136 0.656461
\(382\) 20.4111 1.04432
\(383\) 18.6917 0.955100 0.477550 0.878605i \(-0.341525\pi\)
0.477550 + 0.878605i \(0.341525\pi\)
\(384\) 9.66196 0.493060
\(385\) 1.00000 0.0509647
\(386\) 42.5472 2.16559
\(387\) −5.39194 −0.274088
\(388\) 15.2927 0.776371
\(389\) −4.35720 −0.220919 −0.110459 0.993881i \(-0.535232\pi\)
−0.110459 + 0.993881i \(0.535232\pi\)
\(390\) 9.25443 0.468616
\(391\) −16.7980 −0.849512
\(392\) 1.28917 0.0651128
\(393\) −18.9355 −0.955172
\(394\) −15.9305 −0.802568
\(395\) −12.1517 −0.611416
\(396\) −1.28917 −0.0647832
\(397\) −14.5783 −0.731666 −0.365833 0.930681i \(-0.619216\pi\)
−0.365833 + 0.930681i \(0.619216\pi\)
\(398\) −28.4111 −1.42412
\(399\) −3.33804 −0.167111
\(400\) −4.91638 −0.245819
\(401\) 10.0922 0.503981 0.251991 0.967730i \(-0.418915\pi\)
0.251991 + 0.967730i \(0.418915\pi\)
\(402\) 19.3622 0.965700
\(403\) −18.7839 −0.935692
\(404\) 7.73501 0.384831
\(405\) 1.00000 0.0496904
\(406\) 16.3572 0.811794
\(407\) 1.10278 0.0546625
\(408\) −5.52946 −0.273749
\(409\) 32.5572 1.60985 0.804925 0.593376i \(-0.202205\pi\)
0.804925 + 0.593376i \(0.202205\pi\)
\(410\) 12.2056 0.602789
\(411\) 4.03831 0.199195
\(412\) 20.1078 0.990640
\(413\) 5.39194 0.265320
\(414\) 7.10278 0.349082
\(415\) −4.96526 −0.243735
\(416\) 32.3416 1.58568
\(417\) −20.1361 −0.986067
\(418\) −6.05390 −0.296106
\(419\) 1.66698 0.0814372 0.0407186 0.999171i \(-0.487035\pi\)
0.0407186 + 0.999171i \(0.487035\pi\)
\(420\) 1.28917 0.0629050
\(421\) −24.0625 −1.17273 −0.586367 0.810045i \(-0.699442\pi\)
−0.586367 + 0.810045i \(0.699442\pi\)
\(422\) −1.59895 −0.0778355
\(423\) 8.72999 0.424467
\(424\) 11.2927 0.548424
\(425\) 4.28917 0.208055
\(426\) 18.7839 0.910082
\(427\) 13.7491 0.665368
\(428\) 22.9200 1.10788
\(429\) 5.10278 0.246364
\(430\) 9.77886 0.471579
\(431\) −0.637776 −0.0307206 −0.0153603 0.999882i \(-0.504890\pi\)
−0.0153603 + 0.999882i \(0.504890\pi\)
\(432\) 4.91638 0.236540
\(433\) 1.69670 0.0815381 0.0407691 0.999169i \(-0.487019\pi\)
0.0407691 + 0.999169i \(0.487019\pi\)
\(434\) −6.67609 −0.320463
\(435\) −9.01916 −0.432435
\(436\) 10.7738 0.515973
\(437\) 13.0731 0.625369
\(438\) 6.57834 0.314325
\(439\) −15.8569 −0.756811 −0.378405 0.925640i \(-0.623527\pi\)
−0.378405 + 0.925640i \(0.623527\pi\)
\(440\) −1.28917 −0.0614587
\(441\) 1.00000 0.0476190
\(442\) −39.6938 −1.88804
\(443\) 40.7628 1.93670 0.968349 0.249602i \(-0.0802996\pi\)
0.968349 + 0.249602i \(0.0802996\pi\)
\(444\) 1.42166 0.0674691
\(445\) 15.1169 0.716610
\(446\) 0.426686 0.0202042
\(447\) 20.6167 0.975134
\(448\) 1.66196 0.0785200
\(449\) 1.78891 0.0844239 0.0422120 0.999109i \(-0.486560\pi\)
0.0422120 + 0.999109i \(0.486560\pi\)
\(450\) −1.81361 −0.0854942
\(451\) 6.72999 0.316903
\(452\) 21.5295 1.01266
\(453\) −5.73501 −0.269454
\(454\) −17.6116 −0.826555
\(455\) −5.10278 −0.239222
\(456\) 4.30330 0.201521
\(457\) 3.39194 0.158668 0.0793342 0.996848i \(-0.474721\pi\)
0.0793342 + 0.996848i \(0.474721\pi\)
\(458\) −33.8610 −1.58222
\(459\) −4.28917 −0.200201
\(460\) −5.04888 −0.235405
\(461\) 33.7733 1.57298 0.786490 0.617603i \(-0.211896\pi\)
0.786490 + 0.617603i \(0.211896\pi\)
\(462\) 1.81361 0.0843766
\(463\) 11.7250 0.544905 0.272453 0.962169i \(-0.412165\pi\)
0.272453 + 0.962169i \(0.412165\pi\)
\(464\) −44.3416 −2.05851
\(465\) 3.68111 0.170707
\(466\) 37.9688 1.75887
\(467\) 29.4600 1.36325 0.681623 0.731704i \(-0.261275\pi\)
0.681623 + 0.731704i \(0.261275\pi\)
\(468\) 6.57834 0.304084
\(469\) −10.6761 −0.492976
\(470\) −15.8328 −0.730311
\(471\) 10.0680 0.463910
\(472\) −6.95112 −0.319951
\(473\) 5.39194 0.247922
\(474\) −22.0383 −1.01225
\(475\) −3.33804 −0.153160
\(476\) −5.52946 −0.253442
\(477\) 8.75971 0.401079
\(478\) −51.3466 −2.34854
\(479\) −17.6711 −0.807412 −0.403706 0.914889i \(-0.632278\pi\)
−0.403706 + 0.914889i \(0.632278\pi\)
\(480\) −6.33804 −0.289291
\(481\) −5.62721 −0.256579
\(482\) −7.73501 −0.352320
\(483\) −3.91638 −0.178202
\(484\) 1.28917 0.0585986
\(485\) 11.8625 0.538648
\(486\) 1.81361 0.0822669
\(487\) −18.7839 −0.851179 −0.425590 0.904916i \(-0.639933\pi\)
−0.425590 + 0.904916i \(0.639933\pi\)
\(488\) −17.7250 −0.802371
\(489\) −0.837786 −0.0378860
\(490\) −1.81361 −0.0819304
\(491\) −5.08864 −0.229647 −0.114824 0.993386i \(-0.536630\pi\)
−0.114824 + 0.993386i \(0.536630\pi\)
\(492\) 8.67609 0.391148
\(493\) 38.6847 1.74227
\(494\) 30.8917 1.38988
\(495\) −1.00000 −0.0449467
\(496\) 18.0978 0.812613
\(497\) −10.3572 −0.464584
\(498\) −9.00502 −0.403525
\(499\) 8.72139 0.390423 0.195212 0.980761i \(-0.437461\pi\)
0.195212 + 0.980761i \(0.437461\pi\)
\(500\) 1.28917 0.0576534
\(501\) −24.1361 −1.07832
\(502\) 33.9588 1.51565
\(503\) −17.4842 −0.779580 −0.389790 0.920904i \(-0.627452\pi\)
−0.389790 + 0.920904i \(0.627452\pi\)
\(504\) −1.28917 −0.0574241
\(505\) 6.00000 0.266996
\(506\) −7.10278 −0.315757
\(507\) −13.0383 −0.579052
\(508\) −16.5189 −0.732908
\(509\) −1.46143 −0.0647767 −0.0323883 0.999475i \(-0.510311\pi\)
−0.0323883 + 0.999475i \(0.510311\pi\)
\(510\) 7.77886 0.344454
\(511\) −3.62721 −0.160458
\(512\) −18.4842 −0.816892
\(513\) 3.33804 0.147378
\(514\) 8.40105 0.370555
\(515\) 15.5975 0.687308
\(516\) 6.95112 0.306006
\(517\) −8.72999 −0.383944
\(518\) −2.00000 −0.0878750
\(519\) 19.2927 0.846857
\(520\) 6.57834 0.288479
\(521\) 12.8036 0.560934 0.280467 0.959864i \(-0.409511\pi\)
0.280467 + 0.959864i \(0.409511\pi\)
\(522\) −16.3572 −0.715935
\(523\) 5.67557 0.248175 0.124088 0.992271i \(-0.460400\pi\)
0.124088 + 0.992271i \(0.460400\pi\)
\(524\) 24.4111 1.06640
\(525\) 1.00000 0.0436436
\(526\) 23.0589 1.00542
\(527\) −15.7889 −0.687776
\(528\) −4.91638 −0.213958
\(529\) −7.66196 −0.333129
\(530\) −15.8867 −0.690073
\(531\) −5.39194 −0.233990
\(532\) 4.30330 0.186572
\(533\) −34.3416 −1.48750
\(534\) 27.4161 1.18641
\(535\) 17.7789 0.768647
\(536\) 13.7633 0.594483
\(537\) −6.35720 −0.274333
\(538\) 38.6705 1.66721
\(539\) −1.00000 −0.0430730
\(540\) −1.28917 −0.0554770
\(541\) 27.2772 1.17274 0.586368 0.810045i \(-0.300557\pi\)
0.586368 + 0.810045i \(0.300557\pi\)
\(542\) −7.65285 −0.328718
\(543\) 3.15667 0.135466
\(544\) 27.1849 1.16554
\(545\) 8.35720 0.357983
\(546\) −9.25443 −0.396053
\(547\) −30.3330 −1.29695 −0.648473 0.761238i \(-0.724592\pi\)
−0.648473 + 0.761238i \(0.724592\pi\)
\(548\) −5.20607 −0.222392
\(549\) −13.7491 −0.586799
\(550\) 1.81361 0.0773324
\(551\) −30.1063 −1.28257
\(552\) 5.04888 0.214894
\(553\) 12.1517 0.516741
\(554\) −16.4806 −0.700193
\(555\) 1.10278 0.0468102
\(556\) 25.9588 1.10090
\(557\) −3.45998 −0.146604 −0.0733019 0.997310i \(-0.523354\pi\)
−0.0733019 + 0.997310i \(0.523354\pi\)
\(558\) 6.67609 0.282621
\(559\) −27.5139 −1.16371
\(560\) 4.91638 0.207755
\(561\) 4.28917 0.181089
\(562\) 37.4983 1.58177
\(563\) 20.9894 0.884599 0.442300 0.896867i \(-0.354163\pi\)
0.442300 + 0.896867i \(0.354163\pi\)
\(564\) −11.2544 −0.473897
\(565\) 16.7003 0.702586
\(566\) 12.7738 0.536925
\(567\) −1.00000 −0.0419961
\(568\) 13.3522 0.560245
\(569\) 19.1763 0.803914 0.401957 0.915658i \(-0.368330\pi\)
0.401957 + 0.915658i \(0.368330\pi\)
\(570\) −6.05390 −0.253570
\(571\) −8.45495 −0.353829 −0.176914 0.984226i \(-0.556612\pi\)
−0.176914 + 0.984226i \(0.556612\pi\)
\(572\) −6.57834 −0.275054
\(573\) 11.2544 0.470160
\(574\) −12.2056 −0.509450
\(575\) −3.91638 −0.163324
\(576\) −1.66196 −0.0692481
\(577\) −12.1744 −0.506826 −0.253413 0.967358i \(-0.581553\pi\)
−0.253413 + 0.967358i \(0.581553\pi\)
\(578\) −2.53355 −0.105382
\(579\) 23.4600 0.974963
\(580\) 11.6272 0.482794
\(581\) 4.96526 0.205994
\(582\) 21.5139 0.891779
\(583\) −8.75971 −0.362790
\(584\) 4.67609 0.193498
\(585\) 5.10278 0.210974
\(586\) 8.32893 0.344065
\(587\) −23.2983 −0.961623 −0.480811 0.876824i \(-0.659658\pi\)
−0.480811 + 0.876824i \(0.659658\pi\)
\(588\) −1.28917 −0.0531644
\(589\) 12.2877 0.506307
\(590\) 9.77886 0.402589
\(591\) −8.78389 −0.361321
\(592\) 5.42166 0.222829
\(593\) −3.73501 −0.153379 −0.0766893 0.997055i \(-0.524435\pi\)
−0.0766893 + 0.997055i \(0.524435\pi\)
\(594\) −1.81361 −0.0744132
\(595\) −4.28917 −0.175839
\(596\) −26.5783 −1.08869
\(597\) −15.6655 −0.641147
\(598\) 36.2439 1.48212
\(599\) 9.62721 0.393357 0.196679 0.980468i \(-0.436984\pi\)
0.196679 + 0.980468i \(0.436984\pi\)
\(600\) −1.28917 −0.0526301
\(601\) −17.2786 −0.704809 −0.352405 0.935848i \(-0.614636\pi\)
−0.352405 + 0.935848i \(0.614636\pi\)
\(602\) −9.77886 −0.398557
\(603\) 10.6761 0.434764
\(604\) 7.39340 0.300833
\(605\) 1.00000 0.0406558
\(606\) 10.8816 0.442036
\(607\) −47.1950 −1.91559 −0.957793 0.287460i \(-0.907189\pi\)
−0.957793 + 0.287460i \(0.907189\pi\)
\(608\) −21.1567 −0.858016
\(609\) 9.01916 0.365475
\(610\) 24.9355 1.00961
\(611\) 44.5472 1.80219
\(612\) 5.52946 0.223515
\(613\) 27.0177 1.09123 0.545617 0.838034i \(-0.316295\pi\)
0.545617 + 0.838034i \(0.316295\pi\)
\(614\) −47.2233 −1.90578
\(615\) 6.72999 0.271379
\(616\) 1.28917 0.0519421
\(617\) −30.6550 −1.23412 −0.617061 0.786915i \(-0.711677\pi\)
−0.617061 + 0.786915i \(0.711677\pi\)
\(618\) 28.2877 1.13790
\(619\) 5.93051 0.238368 0.119184 0.992872i \(-0.461972\pi\)
0.119184 + 0.992872i \(0.461972\pi\)
\(620\) −4.74557 −0.190587
\(621\) 3.91638 0.157159
\(622\) 29.5678 1.18556
\(623\) −15.1169 −0.605646
\(624\) 25.0872 1.00429
\(625\) 1.00000 0.0400000
\(626\) −47.8272 −1.91156
\(627\) −3.33804 −0.133309
\(628\) −12.9794 −0.517934
\(629\) −4.72999 −0.188597
\(630\) 1.81361 0.0722558
\(631\) −32.1602 −1.28028 −0.640140 0.768259i \(-0.721123\pi\)
−0.640140 + 0.768259i \(0.721123\pi\)
\(632\) −15.6655 −0.623141
\(633\) −0.881639 −0.0350420
\(634\) −31.5960 −1.25484
\(635\) −12.8136 −0.508492
\(636\) −11.2927 −0.447786
\(637\) 5.10278 0.202179
\(638\) 16.3572 0.647588
\(639\) 10.3572 0.409725
\(640\) −9.66196 −0.381922
\(641\) −19.5678 −0.772881 −0.386440 0.922314i \(-0.626295\pi\)
−0.386440 + 0.922314i \(0.626295\pi\)
\(642\) 32.2439 1.27256
\(643\) −16.0086 −0.631317 −0.315659 0.948873i \(-0.602225\pi\)
−0.315659 + 0.948873i \(0.602225\pi\)
\(644\) 5.04888 0.198954
\(645\) 5.39194 0.212308
\(646\) 25.9662 1.02163
\(647\) 1.45998 0.0573976 0.0286988 0.999588i \(-0.490864\pi\)
0.0286988 + 0.999588i \(0.490864\pi\)
\(648\) 1.28917 0.0506433
\(649\) 5.39194 0.211652
\(650\) −9.25443 −0.362988
\(651\) −3.68111 −0.144274
\(652\) 1.08005 0.0422979
\(653\) 23.6514 0.925551 0.462775 0.886476i \(-0.346854\pi\)
0.462775 + 0.886476i \(0.346854\pi\)
\(654\) 15.1567 0.592673
\(655\) 18.9355 0.739873
\(656\) 33.0872 1.29184
\(657\) 3.62721 0.141511
\(658\) 15.8328 0.617225
\(659\) 27.2147 1.06013 0.530066 0.847956i \(-0.322167\pi\)
0.530066 + 0.847956i \(0.322167\pi\)
\(660\) 1.28917 0.0501808
\(661\) 34.6705 1.34853 0.674264 0.738490i \(-0.264461\pi\)
0.674264 + 0.738490i \(0.264461\pi\)
\(662\) 29.4161 1.14329
\(663\) −21.8867 −0.850008
\(664\) −6.40105 −0.248409
\(665\) 3.33804 0.129444
\(666\) 2.00000 0.0774984
\(667\) −35.3225 −1.36769
\(668\) 31.1155 1.20389
\(669\) 0.235269 0.00909604
\(670\) −19.3622 −0.748028
\(671\) 13.7491 0.530780
\(672\) 6.33804 0.244495
\(673\) −27.1169 −1.04528 −0.522640 0.852553i \(-0.675053\pi\)
−0.522640 + 0.852553i \(0.675053\pi\)
\(674\) 12.5527 0.483512
\(675\) −1.00000 −0.0384900
\(676\) 16.8086 0.646484
\(677\) 34.0524 1.30874 0.654371 0.756174i \(-0.272934\pi\)
0.654371 + 0.756174i \(0.272934\pi\)
\(678\) 30.2877 1.16319
\(679\) −11.8625 −0.455240
\(680\) 5.52946 0.212045
\(681\) −9.71083 −0.372120
\(682\) −6.67609 −0.255641
\(683\) 24.1955 0.925815 0.462908 0.886407i \(-0.346806\pi\)
0.462908 + 0.886407i \(0.346806\pi\)
\(684\) −4.30330 −0.164541
\(685\) −4.03831 −0.154296
\(686\) 1.81361 0.0692438
\(687\) −18.6705 −0.712326
\(688\) 26.5089 1.01064
\(689\) 44.6988 1.70289
\(690\) −7.10278 −0.270398
\(691\) 30.8716 1.17441 0.587205 0.809438i \(-0.300228\pi\)
0.587205 + 0.809438i \(0.300228\pi\)
\(692\) −24.8716 −0.945476
\(693\) 1.00000 0.0379869
\(694\) 32.4111 1.23031
\(695\) 20.1361 0.763804
\(696\) −11.6272 −0.440728
\(697\) −28.8661 −1.09338
\(698\) −4.93554 −0.186813
\(699\) 20.9355 0.791855
\(700\) −1.28917 −0.0487260
\(701\) −26.7058 −1.00866 −0.504332 0.863510i \(-0.668261\pi\)
−0.504332 + 0.863510i \(0.668261\pi\)
\(702\) 9.25443 0.349286
\(703\) 3.68111 0.138836
\(704\) 1.66196 0.0626373
\(705\) −8.72999 −0.328790
\(706\) 33.9406 1.27737
\(707\) −6.00000 −0.225653
\(708\) 6.95112 0.261239
\(709\) 4.21254 0.158205 0.0791027 0.996866i \(-0.474795\pi\)
0.0791027 + 0.996866i \(0.474795\pi\)
\(710\) −18.7839 −0.704947
\(711\) −12.1517 −0.455723
\(712\) 19.4882 0.730353
\(713\) 14.4166 0.539907
\(714\) −7.77886 −0.291117
\(715\) −5.10278 −0.190833
\(716\) 8.19550 0.306280
\(717\) −28.3119 −1.05733
\(718\) −11.4343 −0.426726
\(719\) 11.3536 0.423419 0.211709 0.977333i \(-0.432097\pi\)
0.211709 + 0.977333i \(0.432097\pi\)
\(720\) −4.91638 −0.183223
\(721\) −15.5975 −0.580881
\(722\) 14.2503 0.530343
\(723\) −4.26499 −0.158617
\(724\) −4.06949 −0.151241
\(725\) 9.01916 0.334963
\(726\) 1.81361 0.0673093
\(727\) 42.2141 1.56564 0.782818 0.622251i \(-0.213782\pi\)
0.782818 + 0.622251i \(0.213782\pi\)
\(728\) −6.57834 −0.243809
\(729\) 1.00000 0.0370370
\(730\) −6.57834 −0.243475
\(731\) −23.1270 −0.855381
\(732\) 17.7250 0.655134
\(733\) 26.9300 0.994682 0.497341 0.867555i \(-0.334310\pi\)
0.497341 + 0.867555i \(0.334310\pi\)
\(734\) −34.4933 −1.27317
\(735\) −1.00000 −0.0368856
\(736\) −24.8222 −0.914958
\(737\) −10.6761 −0.393259
\(738\) 12.2056 0.449293
\(739\) −37.6272 −1.38414 −0.692070 0.721831i \(-0.743301\pi\)
−0.692070 + 0.721831i \(0.743301\pi\)
\(740\) −1.42166 −0.0522614
\(741\) 17.0333 0.625734
\(742\) 15.8867 0.583218
\(743\) −14.0383 −0.515016 −0.257508 0.966276i \(-0.582901\pi\)
−0.257508 + 0.966276i \(0.582901\pi\)
\(744\) 4.74557 0.173981
\(745\) −20.6167 −0.755336
\(746\) 25.4061 0.930183
\(747\) −4.96526 −0.181669
\(748\) −5.52946 −0.202177
\(749\) −17.7789 −0.649626
\(750\) 1.81361 0.0662235
\(751\) −27.2786 −0.995410 −0.497705 0.867346i \(-0.665824\pi\)
−0.497705 + 0.867346i \(0.665824\pi\)
\(752\) −42.9200 −1.56513
\(753\) 18.7244 0.682357
\(754\) −83.4671 −3.03969
\(755\) 5.73501 0.208718
\(756\) 1.28917 0.0468866
\(757\) −15.9900 −0.581165 −0.290582 0.956850i \(-0.593849\pi\)
−0.290582 + 0.956850i \(0.593849\pi\)
\(758\) −24.0639 −0.874042
\(759\) −3.91638 −0.142156
\(760\) −4.30330 −0.156097
\(761\) −0.432226 −0.0156682 −0.00783408 0.999969i \(-0.502494\pi\)
−0.00783408 + 0.999969i \(0.502494\pi\)
\(762\) −23.2388 −0.841854
\(763\) −8.35720 −0.302551
\(764\) −14.5089 −0.524912
\(765\) 4.28917 0.155075
\(766\) −33.8993 −1.22483
\(767\) −27.5139 −0.993468
\(768\) −20.8469 −0.752248
\(769\) 3.46697 0.125022 0.0625110 0.998044i \(-0.480089\pi\)
0.0625110 + 0.998044i \(0.480089\pi\)
\(770\) −1.81361 −0.0653578
\(771\) 4.63224 0.166826
\(772\) −30.2439 −1.08850
\(773\) −12.7355 −0.458065 −0.229033 0.973419i \(-0.573556\pi\)
−0.229033 + 0.973419i \(0.573556\pi\)
\(774\) 9.77886 0.351494
\(775\) −3.68111 −0.132229
\(776\) 15.2927 0.548977
\(777\) −1.10278 −0.0395618
\(778\) 7.90225 0.283309
\(779\) 22.4650 0.804892
\(780\) −6.57834 −0.235542
\(781\) −10.3572 −0.370610
\(782\) 30.4650 1.08943
\(783\) −9.01916 −0.322318
\(784\) −4.91638 −0.175585
\(785\) −10.0680 −0.359343
\(786\) 34.3416 1.22492
\(787\) −10.6277 −0.378838 −0.189419 0.981896i \(-0.560660\pi\)
−0.189419 + 0.981896i \(0.560660\pi\)
\(788\) 11.3239 0.403398
\(789\) 12.7144 0.452645
\(790\) 22.0383 0.784088
\(791\) −16.7003 −0.593793
\(792\) −1.28917 −0.0458086
\(793\) −70.1588 −2.49141
\(794\) 26.4394 0.938298
\(795\) −8.75971 −0.310675
\(796\) 20.1955 0.715811
\(797\) −2.85337 −0.101072 −0.0505358 0.998722i \(-0.516093\pi\)
−0.0505358 + 0.998722i \(0.516093\pi\)
\(798\) 6.05390 0.214306
\(799\) 37.4444 1.32469
\(800\) 6.33804 0.224084
\(801\) 15.1169 0.534130
\(802\) −18.3033 −0.646312
\(803\) −3.62721 −0.128002
\(804\) −13.7633 −0.485393
\(805\) 3.91638 0.138034
\(806\) 34.0666 1.19994
\(807\) 21.3225 0.750586
\(808\) 7.73501 0.272117
\(809\) −36.8122 −1.29425 −0.647123 0.762385i \(-0.724028\pi\)
−0.647123 + 0.762385i \(0.724028\pi\)
\(810\) −1.81361 −0.0637236
\(811\) −14.3133 −0.502610 −0.251305 0.967908i \(-0.580860\pi\)
−0.251305 + 0.967908i \(0.580860\pi\)
\(812\) −11.6272 −0.408035
\(813\) −4.21968 −0.147991
\(814\) −2.00000 −0.0701000
\(815\) 0.837786 0.0293464
\(816\) 21.0872 0.738199
\(817\) 17.9985 0.629689
\(818\) −59.0460 −2.06449
\(819\) −5.10278 −0.178305
\(820\) −8.67609 −0.302982
\(821\) −46.9185 −1.63747 −0.818733 0.574174i \(-0.805323\pi\)
−0.818733 + 0.574174i \(0.805323\pi\)
\(822\) −7.32391 −0.255451
\(823\) 4.31889 0.150547 0.0752735 0.997163i \(-0.476017\pi\)
0.0752735 + 0.997163i \(0.476017\pi\)
\(824\) 20.1078 0.700488
\(825\) 1.00000 0.0348155
\(826\) −9.77886 −0.340250
\(827\) −54.0666 −1.88008 −0.940040 0.341065i \(-0.889212\pi\)
−0.940040 + 0.341065i \(0.889212\pi\)
\(828\) −5.04888 −0.175461
\(829\) 9.89220 0.343570 0.171785 0.985134i \(-0.445047\pi\)
0.171785 + 0.985134i \(0.445047\pi\)
\(830\) 9.00502 0.312569
\(831\) −9.08719 −0.315231
\(832\) −8.48059 −0.294011
\(833\) 4.28917 0.148611
\(834\) 36.5189 1.26455
\(835\) 24.1361 0.835263
\(836\) 4.30330 0.148833
\(837\) 3.68111 0.127238
\(838\) −3.02324 −0.104436
\(839\) −37.3325 −1.28886 −0.644431 0.764663i \(-0.722906\pi\)
−0.644431 + 0.764663i \(0.722906\pi\)
\(840\) 1.28917 0.0444805
\(841\) 52.3452 1.80501
\(842\) 43.6399 1.50393
\(843\) 20.6761 0.712122
\(844\) 1.13658 0.0391227
\(845\) 13.0383 0.448532
\(846\) −15.8328 −0.544342
\(847\) −1.00000 −0.0343604
\(848\) −43.0661 −1.47889
\(849\) 7.04334 0.241727
\(850\) −7.77886 −0.266813
\(851\) 4.31889 0.148050
\(852\) −13.3522 −0.457438
\(853\) −21.7688 −0.745350 −0.372675 0.927962i \(-0.621559\pi\)
−0.372675 + 0.927962i \(0.621559\pi\)
\(854\) −24.9355 −0.853277
\(855\) −3.33804 −0.114159
\(856\) 22.9200 0.783388
\(857\) 46.6832 1.59467 0.797334 0.603538i \(-0.206243\pi\)
0.797334 + 0.603538i \(0.206243\pi\)
\(858\) −9.25443 −0.315941
\(859\) −22.8277 −0.778872 −0.389436 0.921053i \(-0.627330\pi\)
−0.389436 + 0.921053i \(0.627330\pi\)
\(860\) −6.95112 −0.237031
\(861\) −6.72999 −0.229357
\(862\) 1.15667 0.0393965
\(863\) −4.08362 −0.139008 −0.0695040 0.997582i \(-0.522142\pi\)
−0.0695040 + 0.997582i \(0.522142\pi\)
\(864\) −6.33804 −0.215625
\(865\) −19.2927 −0.655973
\(866\) −3.07714 −0.104566
\(867\) −1.39697 −0.0474435
\(868\) 4.74557 0.161075
\(869\) 12.1517 0.412217
\(870\) 16.3572 0.554561
\(871\) 54.4777 1.84591
\(872\) 10.7738 0.364848
\(873\) 11.8625 0.401484
\(874\) −23.7094 −0.801982
\(875\) −1.00000 −0.0338062
\(876\) −4.67609 −0.157990
\(877\) −34.8419 −1.17653 −0.588263 0.808670i \(-0.700188\pi\)
−0.588263 + 0.808670i \(0.700188\pi\)
\(878\) 28.7583 0.970544
\(879\) 4.59247 0.154900
\(880\) 4.91638 0.165731
\(881\) −49.1864 −1.65713 −0.828566 0.559892i \(-0.810843\pi\)
−0.828566 + 0.559892i \(0.810843\pi\)
\(882\) −1.81361 −0.0610673
\(883\) 29.4600 0.991407 0.495704 0.868492i \(-0.334910\pi\)
0.495704 + 0.868492i \(0.334910\pi\)
\(884\) 28.2156 0.948993
\(885\) 5.39194 0.181248
\(886\) −73.9276 −2.48365
\(887\) 7.47411 0.250956 0.125478 0.992096i \(-0.459954\pi\)
0.125478 + 0.992096i \(0.459954\pi\)
\(888\) 1.42166 0.0477079
\(889\) 12.8136 0.429755
\(890\) −27.4161 −0.918991
\(891\) −1.00000 −0.0335013
\(892\) −0.303302 −0.0101553
\(893\) −29.1411 −0.975169
\(894\) −37.3905 −1.25053
\(895\) 6.35720 0.212498
\(896\) 9.66196 0.322783
\(897\) 19.9844 0.667260
\(898\) −3.24438 −0.108266
\(899\) −33.2005 −1.10730
\(900\) 1.28917 0.0429723
\(901\) 37.5719 1.25170
\(902\) −12.2056 −0.406400
\(903\) −5.39194 −0.179433
\(904\) 21.5295 0.716059
\(905\) −3.15667 −0.104931
\(906\) 10.4011 0.345552
\(907\) −43.1200 −1.43177 −0.715887 0.698216i \(-0.753978\pi\)
−0.715887 + 0.698216i \(0.753978\pi\)
\(908\) 12.5189 0.415454
\(909\) 6.00000 0.199007
\(910\) 9.25443 0.306781
\(911\) 31.0278 1.02800 0.513998 0.857792i \(-0.328164\pi\)
0.513998 + 0.857792i \(0.328164\pi\)
\(912\) −16.4111 −0.543426
\(913\) 4.96526 0.164326
\(914\) −6.15165 −0.203479
\(915\) 13.7491 0.454533
\(916\) 24.0695 0.795278
\(917\) −18.9355 −0.625307
\(918\) 7.77886 0.256741
\(919\) −43.3028 −1.42843 −0.714214 0.699928i \(-0.753215\pi\)
−0.714214 + 0.699928i \(0.753215\pi\)
\(920\) −5.04888 −0.166457
\(921\) −26.0383 −0.857992
\(922\) −61.2515 −2.01721
\(923\) 52.8505 1.73959
\(924\) −1.28917 −0.0424105
\(925\) −1.10278 −0.0362590
\(926\) −21.2645 −0.698794
\(927\) 15.5975 0.512289
\(928\) 57.1638 1.87649
\(929\) 14.4595 0.474399 0.237200 0.971461i \(-0.423770\pi\)
0.237200 + 0.971461i \(0.423770\pi\)
\(930\) −6.67609 −0.218918
\(931\) −3.33804 −0.109400
\(932\) −26.9894 −0.884068
\(933\) 16.3033 0.533746
\(934\) −53.4288 −1.74824
\(935\) −4.28917 −0.140271
\(936\) 6.57834 0.215020
\(937\) −16.3088 −0.532787 −0.266393 0.963864i \(-0.585832\pi\)
−0.266393 + 0.963864i \(0.585832\pi\)
\(938\) 19.3622 0.632199
\(939\) −26.3713 −0.860596
\(940\) 11.2544 0.367079
\(941\) 21.1028 0.687931 0.343965 0.938982i \(-0.388230\pi\)
0.343965 + 0.938982i \(0.388230\pi\)
\(942\) −18.2594 −0.594925
\(943\) 26.3572 0.858309
\(944\) 26.5089 0.862790
\(945\) 1.00000 0.0325300
\(946\) −9.77886 −0.317938
\(947\) 25.5436 0.830055 0.415028 0.909809i \(-0.363772\pi\)
0.415028 + 0.909809i \(0.363772\pi\)
\(948\) 15.6655 0.508793
\(949\) 18.5089 0.600823
\(950\) 6.05390 0.196414
\(951\) −17.4217 −0.564936
\(952\) −5.52946 −0.179211
\(953\) 22.0766 0.715132 0.357566 0.933888i \(-0.383607\pi\)
0.357566 + 0.933888i \(0.383607\pi\)
\(954\) −15.8867 −0.514350
\(955\) −11.2544 −0.364185
\(956\) 36.4988 1.18046
\(957\) 9.01916 0.291548
\(958\) 32.0484 1.03544
\(959\) 4.03831 0.130404
\(960\) 1.66196 0.0536394
\(961\) −17.4494 −0.562884
\(962\) 10.2056 0.329040
\(963\) 17.7789 0.572916
\(964\) 5.49829 0.177088
\(965\) −23.4600 −0.755203
\(966\) 7.10278 0.228528
\(967\) −57.9985 −1.86511 −0.932554 0.361031i \(-0.882425\pi\)
−0.932554 + 0.361031i \(0.882425\pi\)
\(968\) 1.28917 0.0414354
\(969\) 14.3174 0.459942
\(970\) −21.5139 −0.690769
\(971\) −10.9114 −0.350162 −0.175081 0.984554i \(-0.556019\pi\)
−0.175081 + 0.984554i \(0.556019\pi\)
\(972\) −1.28917 −0.0413501
\(973\) −20.1361 −0.645533
\(974\) 34.0666 1.09156
\(975\) −5.10278 −0.163420
\(976\) 67.5960 2.16370
\(977\) −20.0625 −0.641856 −0.320928 0.947104i \(-0.603995\pi\)
−0.320928 + 0.947104i \(0.603995\pi\)
\(978\) 1.51941 0.0485855
\(979\) −15.1169 −0.483138
\(980\) 1.28917 0.0411810
\(981\) 8.35720 0.266825
\(982\) 9.22879 0.294503
\(983\) −8.95827 −0.285724 −0.142862 0.989743i \(-0.545631\pi\)
−0.142862 + 0.989743i \(0.545631\pi\)
\(984\) 8.67609 0.276584
\(985\) 8.78389 0.279878
\(986\) −70.1588 −2.23431
\(987\) 8.72999 0.277879
\(988\) −21.9588 −0.698602
\(989\) 21.1169 0.671479
\(990\) 1.81361 0.0576402
\(991\) 22.2297 0.706151 0.353075 0.935595i \(-0.385136\pi\)
0.353075 + 0.935595i \(0.385136\pi\)
\(992\) −23.3311 −0.740762
\(993\) 16.2197 0.514716
\(994\) 18.7839 0.595789
\(995\) 15.6655 0.496631
\(996\) 6.40105 0.202825
\(997\) 1.37330 0.0434930 0.0217465 0.999764i \(-0.493077\pi\)
0.0217465 + 0.999764i \(0.493077\pi\)
\(998\) −15.8172 −0.500684
\(999\) 1.10278 0.0348903
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 1155.2.a.s.1.1 3
3.2 odd 2 3465.2.a.bc.1.3 3
5.4 even 2 5775.2.a.br.1.3 3
7.6 odd 2 8085.2.a.bm.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
1155.2.a.s.1.1 3 1.1 even 1 trivial
3465.2.a.bc.1.3 3 3.2 odd 2
5775.2.a.br.1.3 3 5.4 even 2
8085.2.a.bm.1.1 3 7.6 odd 2