Properties

Label 1805.2.a.u.1.8
Level $1805$
Weight $2$
Character 1805.1
Self dual yes
Analytic conductor $14.413$
Analytic rank $0$
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: \(0\)
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 1.8
Root \(-1.57047\) 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)
 
\(f(q)\) \(=\) \(q+1.57047 q^{2} -2.28502 q^{3} +0.466387 q^{4} -1.00000 q^{5} -3.58856 q^{6} -4.01337 q^{7} -2.40850 q^{8} +2.22131 q^{9} +O(q^{10})\) \(q+1.57047 q^{2} -2.28502 q^{3} +0.466387 q^{4} -1.00000 q^{5} -3.58856 q^{6} -4.01337 q^{7} -2.40850 q^{8} +2.22131 q^{9} -1.57047 q^{10} +2.76620 q^{11} -1.06570 q^{12} -2.78732 q^{13} -6.30289 q^{14} +2.28502 q^{15} -4.71526 q^{16} -3.60673 q^{17} +3.48851 q^{18} -0.466387 q^{20} +9.17062 q^{21} +4.34425 q^{22} +1.36746 q^{23} +5.50347 q^{24} +1.00000 q^{25} -4.37742 q^{26} +1.77932 q^{27} -1.87178 q^{28} +9.50940 q^{29} +3.58856 q^{30} -1.55371 q^{31} -2.58819 q^{32} -6.32083 q^{33} -5.66428 q^{34} +4.01337 q^{35} +1.03599 q^{36} -8.51183 q^{37} +6.36909 q^{39} +2.40850 q^{40} +6.60899 q^{41} +14.4022 q^{42} -6.25476 q^{43} +1.29012 q^{44} -2.22131 q^{45} +2.14756 q^{46} +9.10988 q^{47} +10.7745 q^{48} +9.10711 q^{49} +1.57047 q^{50} +8.24145 q^{51} -1.29997 q^{52} +3.94456 q^{53} +2.79437 q^{54} -2.76620 q^{55} +9.66619 q^{56} +14.9343 q^{58} -3.61183 q^{59} +1.06570 q^{60} -7.48908 q^{61} -2.44005 q^{62} -8.91494 q^{63} +5.36583 q^{64} +2.78732 q^{65} -9.92669 q^{66} +14.5720 q^{67} -1.68213 q^{68} -3.12468 q^{69} +6.30289 q^{70} +3.31888 q^{71} -5.35003 q^{72} -10.0077 q^{73} -13.3676 q^{74} -2.28502 q^{75} -11.1018 q^{77} +10.0025 q^{78} +1.94735 q^{79} +4.71526 q^{80} -10.7297 q^{81} +10.3792 q^{82} +1.61868 q^{83} +4.27705 q^{84} +3.60673 q^{85} -9.82294 q^{86} -21.7292 q^{87} -6.66240 q^{88} +12.2510 q^{89} -3.48851 q^{90} +11.1866 q^{91} +0.637766 q^{92} +3.55025 q^{93} +14.3068 q^{94} +5.91406 q^{96} +11.5912 q^{97} +14.3025 q^{98} +6.14460 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 9 q + 3 q^{3} + 6 q^{4} - 9 q^{5} - 12 q^{6} - 12 q^{8} + 6 q^{9}+O(q^{10}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 1.57047 1.11049 0.555246 0.831686i \(-0.312624\pi\)
0.555246 + 0.831686i \(0.312624\pi\)
\(3\) −2.28502 −1.31926 −0.659628 0.751592i \(-0.729286\pi\)
−0.659628 + 0.751592i \(0.729286\pi\)
\(4\) 0.466387 0.233193
\(5\) −1.00000 −0.447214
\(6\) −3.58856 −1.46502
\(7\) −4.01337 −1.51691 −0.758455 0.651725i \(-0.774045\pi\)
−0.758455 + 0.651725i \(0.774045\pi\)
\(8\) −2.40850 −0.851533
\(9\) 2.22131 0.740437
\(10\) −1.57047 −0.496627
\(11\) 2.76620 0.834041 0.417021 0.908897i \(-0.363074\pi\)
0.417021 + 0.908897i \(0.363074\pi\)
\(12\) −1.06570 −0.307642
\(13\) −2.78732 −0.773065 −0.386532 0.922276i \(-0.626327\pi\)
−0.386532 + 0.922276i \(0.626327\pi\)
\(14\) −6.30289 −1.68452
\(15\) 2.28502 0.589989
\(16\) −4.71526 −1.17881
\(17\) −3.60673 −0.874761 −0.437381 0.899277i \(-0.644094\pi\)
−0.437381 + 0.899277i \(0.644094\pi\)
\(18\) 3.48851 0.822250
\(19\) 0 0
\(20\) −0.466387 −0.104287
\(21\) 9.17062 2.00119
\(22\) 4.34425 0.926197
\(23\) 1.36746 0.285136 0.142568 0.989785i \(-0.454464\pi\)
0.142568 + 0.989785i \(0.454464\pi\)
\(24\) 5.50347 1.12339
\(25\) 1.00000 0.200000
\(26\) −4.37742 −0.858482
\(27\) 1.77932 0.342430
\(28\) −1.87178 −0.353733
\(29\) 9.50940 1.76585 0.882926 0.469512i \(-0.155570\pi\)
0.882926 + 0.469512i \(0.155570\pi\)
\(30\) 3.58856 0.655179
\(31\) −1.55371 −0.279054 −0.139527 0.990218i \(-0.544558\pi\)
−0.139527 + 0.990218i \(0.544558\pi\)
\(32\) −2.58819 −0.457531
\(33\) −6.32083 −1.10031
\(34\) −5.66428 −0.971415
\(35\) 4.01337 0.678383
\(36\) 1.03599 0.172665
\(37\) −8.51183 −1.39934 −0.699668 0.714468i \(-0.746669\pi\)
−0.699668 + 0.714468i \(0.746669\pi\)
\(38\) 0 0
\(39\) 6.36909 1.01987
\(40\) 2.40850 0.380817
\(41\) 6.60899 1.03215 0.516075 0.856543i \(-0.327393\pi\)
0.516075 + 0.856543i \(0.327393\pi\)
\(42\) 14.4022 2.22231
\(43\) −6.25476 −0.953842 −0.476921 0.878946i \(-0.658247\pi\)
−0.476921 + 0.878946i \(0.658247\pi\)
\(44\) 1.29012 0.194493
\(45\) −2.22131 −0.331134
\(46\) 2.14756 0.316641
\(47\) 9.10988 1.32881 0.664406 0.747371i \(-0.268684\pi\)
0.664406 + 0.747371i \(0.268684\pi\)
\(48\) 10.7745 1.55516
\(49\) 9.10711 1.30102
\(50\) 1.57047 0.222098
\(51\) 8.24145 1.15403
\(52\) −1.29997 −0.180274
\(53\) 3.94456 0.541827 0.270913 0.962604i \(-0.412674\pi\)
0.270913 + 0.962604i \(0.412674\pi\)
\(54\) 2.79437 0.380265
\(55\) −2.76620 −0.372995
\(56\) 9.66619 1.29170
\(57\) 0 0
\(58\) 14.9343 1.96096
\(59\) −3.61183 −0.470220 −0.235110 0.971969i \(-0.575545\pi\)
−0.235110 + 0.971969i \(0.575545\pi\)
\(60\) 1.06570 0.137582
\(61\) −7.48908 −0.958879 −0.479439 0.877575i \(-0.659160\pi\)
−0.479439 + 0.877575i \(0.659160\pi\)
\(62\) −2.44005 −0.309887
\(63\) −8.91494 −1.12318
\(64\) 5.36583 0.670729
\(65\) 2.78732 0.345725
\(66\) −9.92669 −1.22189
\(67\) 14.5720 1.78025 0.890125 0.455717i \(-0.150617\pi\)
0.890125 + 0.455717i \(0.150617\pi\)
\(68\) −1.68213 −0.203988
\(69\) −3.12468 −0.376167
\(70\) 6.30289 0.753339
\(71\) 3.31888 0.393879 0.196940 0.980416i \(-0.436900\pi\)
0.196940 + 0.980416i \(0.436900\pi\)
\(72\) −5.35003 −0.630507
\(73\) −10.0077 −1.17132 −0.585659 0.810558i \(-0.699164\pi\)
−0.585659 + 0.810558i \(0.699164\pi\)
\(74\) −13.3676 −1.55395
\(75\) −2.28502 −0.263851
\(76\) 0 0
\(77\) −11.1018 −1.26517
\(78\) 10.0025 1.13256
\(79\) 1.94735 0.219094 0.109547 0.993982i \(-0.465060\pi\)
0.109547 + 0.993982i \(0.465060\pi\)
\(80\) 4.71526 0.527182
\(81\) −10.7297 −1.19219
\(82\) 10.3792 1.14619
\(83\) 1.61868 0.177673 0.0888365 0.996046i \(-0.471685\pi\)
0.0888365 + 0.996046i \(0.471685\pi\)
\(84\) 4.27705 0.466665
\(85\) 3.60673 0.391205
\(86\) −9.82294 −1.05923
\(87\) −21.7292 −2.32961
\(88\) −6.66240 −0.710214
\(89\) 12.2510 1.29860 0.649300 0.760532i \(-0.275062\pi\)
0.649300 + 0.760532i \(0.275062\pi\)
\(90\) −3.48851 −0.367721
\(91\) 11.1866 1.17267
\(92\) 0.637766 0.0664917
\(93\) 3.55025 0.368143
\(94\) 14.3068 1.47564
\(95\) 0 0
\(96\) 5.91406 0.603601
\(97\) 11.5912 1.17690 0.588452 0.808532i \(-0.299738\pi\)
0.588452 + 0.808532i \(0.299738\pi\)
\(98\) 14.3025 1.44477
\(99\) 6.14460 0.617555
\(100\) 0.466387 0.0466387
\(101\) −6.98240 −0.694774 −0.347387 0.937722i \(-0.612931\pi\)
−0.347387 + 0.937722i \(0.612931\pi\)
\(102\) 12.9430 1.28155
\(103\) −10.1338 −0.998510 −0.499255 0.866455i \(-0.666393\pi\)
−0.499255 + 0.866455i \(0.666393\pi\)
\(104\) 6.71327 0.658290
\(105\) −9.17062 −0.894961
\(106\) 6.19482 0.601694
\(107\) 11.7835 1.13916 0.569578 0.821937i \(-0.307107\pi\)
0.569578 + 0.821937i \(0.307107\pi\)
\(108\) 0.829849 0.0798523
\(109\) 1.60407 0.153642 0.0768212 0.997045i \(-0.475523\pi\)
0.0768212 + 0.997045i \(0.475523\pi\)
\(110\) −4.34425 −0.414208
\(111\) 19.4497 1.84608
\(112\) 18.9241 1.78815
\(113\) −12.9449 −1.21775 −0.608876 0.793266i \(-0.708379\pi\)
−0.608876 + 0.793266i \(0.708379\pi\)
\(114\) 0 0
\(115\) −1.36746 −0.127517
\(116\) 4.43506 0.411785
\(117\) −6.19152 −0.572406
\(118\) −5.67228 −0.522175
\(119\) 14.4751 1.32693
\(120\) −5.50347 −0.502395
\(121\) −3.34812 −0.304375
\(122\) −11.7614 −1.06483
\(123\) −15.1017 −1.36167
\(124\) −0.724627 −0.0650735
\(125\) −1.00000 −0.0894427
\(126\) −14.0007 −1.24728
\(127\) 8.29802 0.736331 0.368165 0.929760i \(-0.379986\pi\)
0.368165 + 0.929760i \(0.379986\pi\)
\(128\) 13.6033 1.20237
\(129\) 14.2923 1.25836
\(130\) 4.37742 0.383925
\(131\) −6.51185 −0.568943 −0.284471 0.958685i \(-0.591818\pi\)
−0.284471 + 0.958685i \(0.591818\pi\)
\(132\) −2.94795 −0.256586
\(133\) 0 0
\(134\) 22.8849 1.97695
\(135\) −1.77932 −0.153139
\(136\) 8.68681 0.744888
\(137\) 14.8686 1.27031 0.635156 0.772384i \(-0.280936\pi\)
0.635156 + 0.772384i \(0.280936\pi\)
\(138\) −4.90722 −0.417731
\(139\) 9.95823 0.844646 0.422323 0.906445i \(-0.361215\pi\)
0.422323 + 0.906445i \(0.361215\pi\)
\(140\) 1.87178 0.158194
\(141\) −20.8163 −1.75304
\(142\) 5.21222 0.437400
\(143\) −7.71030 −0.644768
\(144\) −10.4741 −0.872838
\(145\) −9.50940 −0.789713
\(146\) −15.7169 −1.30074
\(147\) −20.8099 −1.71637
\(148\) −3.96980 −0.326316
\(149\) −1.43162 −0.117283 −0.0586416 0.998279i \(-0.518677\pi\)
−0.0586416 + 0.998279i \(0.518677\pi\)
\(150\) −3.58856 −0.293005
\(151\) 8.18383 0.665990 0.332995 0.942929i \(-0.391941\pi\)
0.332995 + 0.942929i \(0.391941\pi\)
\(152\) 0 0
\(153\) −8.01168 −0.647706
\(154\) −17.4351 −1.40496
\(155\) 1.55371 0.124797
\(156\) 2.97046 0.237827
\(157\) −7.83512 −0.625311 −0.312655 0.949867i \(-0.601218\pi\)
−0.312655 + 0.949867i \(0.601218\pi\)
\(158\) 3.05827 0.243303
\(159\) −9.01339 −0.714808
\(160\) 2.58819 0.204614
\(161\) −5.48813 −0.432525
\(162\) −16.8507 −1.32392
\(163\) −19.2994 −1.51164 −0.755821 0.654778i \(-0.772762\pi\)
−0.755821 + 0.654778i \(0.772762\pi\)
\(164\) 3.08234 0.240690
\(165\) 6.32083 0.492076
\(166\) 2.54209 0.197304
\(167\) −9.38968 −0.726595 −0.363297 0.931673i \(-0.618349\pi\)
−0.363297 + 0.931673i \(0.618349\pi\)
\(168\) −22.0874 −1.70408
\(169\) −5.23082 −0.402371
\(170\) 5.66428 0.434430
\(171\) 0 0
\(172\) −2.91714 −0.222430
\(173\) −8.31084 −0.631861 −0.315931 0.948782i \(-0.602317\pi\)
−0.315931 + 0.948782i \(0.602317\pi\)
\(174\) −34.1251 −2.58702
\(175\) −4.01337 −0.303382
\(176\) −13.0434 −0.983180
\(177\) 8.25309 0.620340
\(178\) 19.2398 1.44209
\(179\) 18.5319 1.38514 0.692568 0.721352i \(-0.256479\pi\)
0.692568 + 0.721352i \(0.256479\pi\)
\(180\) −1.03599 −0.0772182
\(181\) −11.3292 −0.842094 −0.421047 0.907039i \(-0.638337\pi\)
−0.421047 + 0.907039i \(0.638337\pi\)
\(182\) 17.5682 1.30224
\(183\) 17.1127 1.26501
\(184\) −3.29353 −0.242802
\(185\) 8.51183 0.625802
\(186\) 5.57557 0.408820
\(187\) −9.97695 −0.729587
\(188\) 4.24873 0.309870
\(189\) −7.14105 −0.519435
\(190\) 0 0
\(191\) 11.8879 0.860175 0.430088 0.902787i \(-0.358483\pi\)
0.430088 + 0.902787i \(0.358483\pi\)
\(192\) −12.2610 −0.884864
\(193\) 0.957834 0.0689464 0.0344732 0.999406i \(-0.489025\pi\)
0.0344732 + 0.999406i \(0.489025\pi\)
\(194\) 18.2036 1.30694
\(195\) −6.36909 −0.456100
\(196\) 4.24743 0.303388
\(197\) 14.2776 1.01724 0.508619 0.860991i \(-0.330156\pi\)
0.508619 + 0.860991i \(0.330156\pi\)
\(198\) 9.64993 0.685791
\(199\) −7.76571 −0.550497 −0.275248 0.961373i \(-0.588760\pi\)
−0.275248 + 0.961373i \(0.588760\pi\)
\(200\) −2.40850 −0.170307
\(201\) −33.2972 −2.34861
\(202\) −10.9657 −0.771542
\(203\) −38.1647 −2.67864
\(204\) 3.84370 0.269113
\(205\) −6.60899 −0.461592
\(206\) −15.9148 −1.10884
\(207\) 3.03756 0.211125
\(208\) 13.1430 0.911300
\(209\) 0 0
\(210\) −14.4022 −0.993847
\(211\) −7.62274 −0.524771 −0.262386 0.964963i \(-0.584509\pi\)
−0.262386 + 0.964963i \(0.584509\pi\)
\(212\) 1.83969 0.126350
\(213\) −7.58371 −0.519627
\(214\) 18.5057 1.26502
\(215\) 6.25476 0.426571
\(216\) −4.28548 −0.291590
\(217\) 6.23559 0.423299
\(218\) 2.51916 0.170619
\(219\) 22.8679 1.54527
\(220\) −1.29012 −0.0869799
\(221\) 10.0531 0.676247
\(222\) 30.5452 2.05006
\(223\) 8.00331 0.535942 0.267971 0.963427i \(-0.413647\pi\)
0.267971 + 0.963427i \(0.413647\pi\)
\(224\) 10.3873 0.694034
\(225\) 2.22131 0.148087
\(226\) −20.3296 −1.35230
\(227\) 17.7409 1.17750 0.588752 0.808314i \(-0.299619\pi\)
0.588752 + 0.808314i \(0.299619\pi\)
\(228\) 0 0
\(229\) −24.2231 −1.60071 −0.800354 0.599528i \(-0.795355\pi\)
−0.800354 + 0.599528i \(0.795355\pi\)
\(230\) −2.14756 −0.141606
\(231\) 25.3678 1.66908
\(232\) −22.9034 −1.50368
\(233\) 13.6491 0.894183 0.447092 0.894488i \(-0.352460\pi\)
0.447092 + 0.894488i \(0.352460\pi\)
\(234\) −9.72361 −0.635653
\(235\) −9.10988 −0.594263
\(236\) −1.68451 −0.109652
\(237\) −4.44974 −0.289042
\(238\) 22.7328 1.47355
\(239\) 3.73764 0.241768 0.120884 0.992667i \(-0.461427\pi\)
0.120884 + 0.992667i \(0.461427\pi\)
\(240\) −10.7745 −0.695488
\(241\) 5.28146 0.340209 0.170104 0.985426i \(-0.445590\pi\)
0.170104 + 0.985426i \(0.445590\pi\)
\(242\) −5.25814 −0.338006
\(243\) 19.1796 1.23037
\(244\) −3.49281 −0.223604
\(245\) −9.10711 −0.581832
\(246\) −23.7168 −1.51212
\(247\) 0 0
\(248\) 3.74210 0.237623
\(249\) −3.69871 −0.234396
\(250\) −1.57047 −0.0993255
\(251\) −12.5445 −0.791801 −0.395900 0.918293i \(-0.629567\pi\)
−0.395900 + 0.918293i \(0.629567\pi\)
\(252\) −4.15781 −0.261917
\(253\) 3.78268 0.237815
\(254\) 13.0318 0.817689
\(255\) −8.24145 −0.516100
\(256\) 10.6319 0.664494
\(257\) −8.59982 −0.536442 −0.268221 0.963357i \(-0.586436\pi\)
−0.268221 + 0.963357i \(0.586436\pi\)
\(258\) 22.4456 1.39740
\(259\) 34.1611 2.12267
\(260\) 1.29997 0.0806208
\(261\) 21.1234 1.30750
\(262\) −10.2267 −0.631806
\(263\) 26.6903 1.64580 0.822898 0.568189i \(-0.192356\pi\)
0.822898 + 0.568189i \(0.192356\pi\)
\(264\) 15.2237 0.936954
\(265\) −3.94456 −0.242312
\(266\) 0 0
\(267\) −27.9937 −1.71319
\(268\) 6.79617 0.415142
\(269\) 3.30320 0.201400 0.100700 0.994917i \(-0.467892\pi\)
0.100700 + 0.994917i \(0.467892\pi\)
\(270\) −2.79437 −0.170060
\(271\) −18.3875 −1.11696 −0.558482 0.829517i \(-0.688616\pi\)
−0.558482 + 0.829517i \(0.688616\pi\)
\(272\) 17.0067 1.03118
\(273\) −25.5615 −1.54705
\(274\) 23.3508 1.41067
\(275\) 2.76620 0.166808
\(276\) −1.45731 −0.0877196
\(277\) 26.7787 1.60898 0.804488 0.593969i \(-0.202440\pi\)
0.804488 + 0.593969i \(0.202440\pi\)
\(278\) 15.6391 0.937973
\(279\) −3.45126 −0.206622
\(280\) −9.66619 −0.577665
\(281\) −21.5462 −1.28534 −0.642669 0.766144i \(-0.722173\pi\)
−0.642669 + 0.766144i \(0.722173\pi\)
\(282\) −32.6914 −1.94674
\(283\) 18.3471 1.09062 0.545311 0.838234i \(-0.316412\pi\)
0.545311 + 0.838234i \(0.316412\pi\)
\(284\) 1.54788 0.0918499
\(285\) 0 0
\(286\) −12.1088 −0.716010
\(287\) −26.5243 −1.56568
\(288\) −5.74917 −0.338773
\(289\) −3.99148 −0.234793
\(290\) −14.9343 −0.876970
\(291\) −26.4860 −1.55264
\(292\) −4.66748 −0.273143
\(293\) 9.28872 0.542653 0.271326 0.962487i \(-0.412538\pi\)
0.271326 + 0.962487i \(0.412538\pi\)
\(294\) −32.6814 −1.90602
\(295\) 3.61183 0.210289
\(296\) 20.5007 1.19158
\(297\) 4.92195 0.285600
\(298\) −2.24833 −0.130242
\(299\) −3.81156 −0.220428
\(300\) −1.06570 −0.0615284
\(301\) 25.1027 1.44689
\(302\) 12.8525 0.739577
\(303\) 15.9549 0.916585
\(304\) 0 0
\(305\) 7.48908 0.428824
\(306\) −12.5821 −0.719272
\(307\) 11.0151 0.628666 0.314333 0.949313i \(-0.398219\pi\)
0.314333 + 0.949313i \(0.398219\pi\)
\(308\) −5.17772 −0.295028
\(309\) 23.1559 1.31729
\(310\) 2.44005 0.138586
\(311\) 25.3821 1.43929 0.719644 0.694343i \(-0.244305\pi\)
0.719644 + 0.694343i \(0.244305\pi\)
\(312\) −15.3399 −0.868453
\(313\) 21.0559 1.19015 0.595074 0.803671i \(-0.297123\pi\)
0.595074 + 0.803671i \(0.297123\pi\)
\(314\) −12.3048 −0.694403
\(315\) 8.91494 0.502300
\(316\) 0.908220 0.0510914
\(317\) −4.00986 −0.225216 −0.112608 0.993639i \(-0.535920\pi\)
−0.112608 + 0.993639i \(0.535920\pi\)
\(318\) −14.1553 −0.793789
\(319\) 26.3049 1.47279
\(320\) −5.36583 −0.299959
\(321\) −26.9256 −1.50284
\(322\) −8.61896 −0.480316
\(323\) 0 0
\(324\) −5.00419 −0.278011
\(325\) −2.78732 −0.154613
\(326\) −30.3091 −1.67867
\(327\) −3.66534 −0.202694
\(328\) −15.9177 −0.878910
\(329\) −36.5613 −2.01569
\(330\) 9.92669 0.546446
\(331\) 32.1360 1.76635 0.883176 0.469042i \(-0.155401\pi\)
0.883176 + 0.469042i \(0.155401\pi\)
\(332\) 0.754929 0.0414321
\(333\) −18.9074 −1.03612
\(334\) −14.7462 −0.806878
\(335\) −14.5720 −0.796152
\(336\) −43.2418 −2.35903
\(337\) −23.4476 −1.27727 −0.638636 0.769509i \(-0.720501\pi\)
−0.638636 + 0.769509i \(0.720501\pi\)
\(338\) −8.21487 −0.446830
\(339\) 29.5793 1.60653
\(340\) 1.68213 0.0912264
\(341\) −4.29786 −0.232742
\(342\) 0 0
\(343\) −8.45661 −0.456614
\(344\) 15.0646 0.812228
\(345\) 3.12468 0.168227
\(346\) −13.0519 −0.701677
\(347\) 6.49353 0.348591 0.174295 0.984693i \(-0.444235\pi\)
0.174295 + 0.984693i \(0.444235\pi\)
\(348\) −10.1342 −0.543250
\(349\) −27.2008 −1.45603 −0.728013 0.685563i \(-0.759556\pi\)
−0.728013 + 0.685563i \(0.759556\pi\)
\(350\) −6.30289 −0.336903
\(351\) −4.95953 −0.264720
\(352\) −7.15945 −0.381600
\(353\) −6.56125 −0.349220 −0.174610 0.984638i \(-0.555866\pi\)
−0.174610 + 0.984638i \(0.555866\pi\)
\(354\) 12.9613 0.688883
\(355\) −3.31888 −0.176148
\(356\) 5.71369 0.302825
\(357\) −33.0760 −1.75057
\(358\) 29.1038 1.53818
\(359\) −2.73194 −0.144186 −0.0720932 0.997398i \(-0.522968\pi\)
−0.0720932 + 0.997398i \(0.522968\pi\)
\(360\) 5.35003 0.281971
\(361\) 0 0
\(362\) −17.7922 −0.935139
\(363\) 7.65053 0.401549
\(364\) 5.21726 0.273459
\(365\) 10.0077 0.523829
\(366\) 26.8750 1.40478
\(367\) 6.57903 0.343423 0.171711 0.985147i \(-0.445070\pi\)
0.171711 + 0.985147i \(0.445070\pi\)
\(368\) −6.44794 −0.336122
\(369\) 14.6806 0.764243
\(370\) 13.3676 0.694949
\(371\) −15.8310 −0.821902
\(372\) 1.65579 0.0858486
\(373\) 8.55993 0.443216 0.221608 0.975136i \(-0.428869\pi\)
0.221608 + 0.975136i \(0.428869\pi\)
\(374\) −15.6685 −0.810201
\(375\) 2.28502 0.117998
\(376\) −21.9411 −1.13153
\(377\) −26.5058 −1.36512
\(378\) −11.2148 −0.576828
\(379\) −14.2477 −0.731856 −0.365928 0.930643i \(-0.619248\pi\)
−0.365928 + 0.930643i \(0.619248\pi\)
\(380\) 0 0
\(381\) −18.9611 −0.971409
\(382\) 18.6696 0.955218
\(383\) −21.3664 −1.09177 −0.545885 0.837860i \(-0.683807\pi\)
−0.545885 + 0.837860i \(0.683807\pi\)
\(384\) −31.0837 −1.58624
\(385\) 11.1018 0.565799
\(386\) 1.50425 0.0765644
\(387\) −13.8938 −0.706261
\(388\) 5.40596 0.274446
\(389\) 15.2926 0.775367 0.387683 0.921793i \(-0.373275\pi\)
0.387683 + 0.921793i \(0.373275\pi\)
\(390\) −10.0025 −0.506496
\(391\) −4.93207 −0.249426
\(392\) −21.9345 −1.10786
\(393\) 14.8797 0.750581
\(394\) 22.4226 1.12964
\(395\) −1.94735 −0.0979820
\(396\) 2.86576 0.144010
\(397\) 32.6020 1.63625 0.818124 0.575042i \(-0.195014\pi\)
0.818124 + 0.575042i \(0.195014\pi\)
\(398\) −12.1958 −0.611322
\(399\) 0 0
\(400\) −4.71526 −0.235763
\(401\) 17.3119 0.864517 0.432258 0.901750i \(-0.357717\pi\)
0.432258 + 0.901750i \(0.357717\pi\)
\(402\) −52.2924 −2.60811
\(403\) 4.33068 0.215727
\(404\) −3.25650 −0.162017
\(405\) 10.7297 0.533164
\(406\) −59.9367 −2.97461
\(407\) −23.5455 −1.16710
\(408\) −19.8495 −0.982698
\(409\) 10.8272 0.535370 0.267685 0.963506i \(-0.413741\pi\)
0.267685 + 0.963506i \(0.413741\pi\)
\(410\) −10.3792 −0.512594
\(411\) −33.9751 −1.67587
\(412\) −4.72625 −0.232846
\(413\) 14.4956 0.713281
\(414\) 4.77041 0.234453
\(415\) −1.61868 −0.0794577
\(416\) 7.21412 0.353701
\(417\) −22.7547 −1.11430
\(418\) 0 0
\(419\) 33.9763 1.65985 0.829925 0.557876i \(-0.188383\pi\)
0.829925 + 0.557876i \(0.188383\pi\)
\(420\) −4.27705 −0.208699
\(421\) −8.30825 −0.404919 −0.202460 0.979291i \(-0.564894\pi\)
−0.202460 + 0.979291i \(0.564894\pi\)
\(422\) −11.9713 −0.582754
\(423\) 20.2359 0.983903
\(424\) −9.50046 −0.461383
\(425\) −3.60673 −0.174952
\(426\) −11.9100 −0.577042
\(427\) 30.0564 1.45453
\(428\) 5.49568 0.265644
\(429\) 17.6182 0.850614
\(430\) 9.82294 0.473704
\(431\) 12.1298 0.584271 0.292135 0.956377i \(-0.405634\pi\)
0.292135 + 0.956377i \(0.405634\pi\)
\(432\) −8.38993 −0.403661
\(433\) 5.43541 0.261209 0.130605 0.991435i \(-0.458308\pi\)
0.130605 + 0.991435i \(0.458308\pi\)
\(434\) 9.79283 0.470071
\(435\) 21.7292 1.04183
\(436\) 0.748119 0.0358284
\(437\) 0 0
\(438\) 35.9134 1.71601
\(439\) 25.3807 1.21136 0.605678 0.795710i \(-0.292902\pi\)
0.605678 + 0.795710i \(0.292902\pi\)
\(440\) 6.66240 0.317617
\(441\) 20.2297 0.963321
\(442\) 15.7882 0.750967
\(443\) −4.92125 −0.233816 −0.116908 0.993143i \(-0.537298\pi\)
−0.116908 + 0.993143i \(0.537298\pi\)
\(444\) 9.07108 0.430494
\(445\) −12.2510 −0.580752
\(446\) 12.5690 0.595159
\(447\) 3.27129 0.154727
\(448\) −21.5351 −1.01744
\(449\) −4.06649 −0.191910 −0.0959548 0.995386i \(-0.530590\pi\)
−0.0959548 + 0.995386i \(0.530590\pi\)
\(450\) 3.48851 0.164450
\(451\) 18.2818 0.860856
\(452\) −6.03732 −0.283971
\(453\) −18.7002 −0.878612
\(454\) 27.8616 1.30761
\(455\) −11.1866 −0.524434
\(456\) 0 0
\(457\) −13.5517 −0.633922 −0.316961 0.948439i \(-0.602662\pi\)
−0.316961 + 0.948439i \(0.602662\pi\)
\(458\) −38.0418 −1.77757
\(459\) −6.41752 −0.299544
\(460\) −0.637766 −0.0297360
\(461\) 17.8812 0.832812 0.416406 0.909179i \(-0.363290\pi\)
0.416406 + 0.909179i \(0.363290\pi\)
\(462\) 39.8394 1.85350
\(463\) 1.30695 0.0607393 0.0303696 0.999539i \(-0.490332\pi\)
0.0303696 + 0.999539i \(0.490332\pi\)
\(464\) −44.8393 −2.08161
\(465\) −3.55025 −0.164639
\(466\) 21.4356 0.992983
\(467\) 27.0266 1.25064 0.625321 0.780368i \(-0.284968\pi\)
0.625321 + 0.780368i \(0.284968\pi\)
\(468\) −2.88764 −0.133481
\(469\) −58.4826 −2.70048
\(470\) −14.3068 −0.659925
\(471\) 17.9034 0.824945
\(472\) 8.69908 0.400408
\(473\) −17.3019 −0.795544
\(474\) −6.98820 −0.320979
\(475\) 0 0
\(476\) 6.75101 0.309432
\(477\) 8.76210 0.401189
\(478\) 5.86987 0.268481
\(479\) 0.289474 0.0132264 0.00661321 0.999978i \(-0.497895\pi\)
0.00661321 + 0.999978i \(0.497895\pi\)
\(480\) −5.91406 −0.269939
\(481\) 23.7252 1.08178
\(482\) 8.29439 0.377799
\(483\) 12.5405 0.570611
\(484\) −1.56152 −0.0709782
\(485\) −11.5912 −0.526327
\(486\) 30.1211 1.36632
\(487\) −15.3711 −0.696530 −0.348265 0.937396i \(-0.613229\pi\)
−0.348265 + 0.937396i \(0.613229\pi\)
\(488\) 18.0374 0.816517
\(489\) 44.0994 1.99424
\(490\) −14.3025 −0.646120
\(491\) 14.5101 0.654834 0.327417 0.944880i \(-0.393822\pi\)
0.327417 + 0.944880i \(0.393822\pi\)
\(492\) −7.04321 −0.317532
\(493\) −34.2979 −1.54470
\(494\) 0 0
\(495\) −6.14460 −0.276179
\(496\) 7.32612 0.328952
\(497\) −13.3199 −0.597479
\(498\) −5.80872 −0.260295
\(499\) −14.7379 −0.659761 −0.329880 0.944023i \(-0.607008\pi\)
−0.329880 + 0.944023i \(0.607008\pi\)
\(500\) −0.466387 −0.0208574
\(501\) 21.4556 0.958565
\(502\) −19.7008 −0.879289
\(503\) 4.35253 0.194070 0.0970348 0.995281i \(-0.469064\pi\)
0.0970348 + 0.995281i \(0.469064\pi\)
\(504\) 21.4716 0.956422
\(505\) 6.98240 0.310713
\(506\) 5.94060 0.264092
\(507\) 11.9525 0.530830
\(508\) 3.87009 0.171707
\(509\) −8.59364 −0.380906 −0.190453 0.981696i \(-0.560996\pi\)
−0.190453 + 0.981696i \(0.560996\pi\)
\(510\) −12.9430 −0.573125
\(511\) 40.1647 1.77678
\(512\) −10.5094 −0.464455
\(513\) 0 0
\(514\) −13.5058 −0.595715
\(515\) 10.1338 0.446547
\(516\) 6.66571 0.293442
\(517\) 25.1998 1.10828
\(518\) 53.6491 2.35721
\(519\) 18.9904 0.833587
\(520\) −6.71327 −0.294396
\(521\) −2.06951 −0.0906668 −0.0453334 0.998972i \(-0.514435\pi\)
−0.0453334 + 0.998972i \(0.514435\pi\)
\(522\) 33.1737 1.45197
\(523\) −14.0120 −0.612702 −0.306351 0.951919i \(-0.599108\pi\)
−0.306351 + 0.951919i \(0.599108\pi\)
\(524\) −3.03704 −0.132674
\(525\) 9.17062 0.400239
\(526\) 41.9164 1.82764
\(527\) 5.60380 0.244105
\(528\) 29.8043 1.29707
\(529\) −21.1300 −0.918698
\(530\) −6.19482 −0.269086
\(531\) −8.02299 −0.348168
\(532\) 0 0
\(533\) −18.4214 −0.797919
\(534\) −43.9634 −1.90248
\(535\) −11.7835 −0.509446
\(536\) −35.0966 −1.51594
\(537\) −42.3457 −1.82735
\(538\) 5.18759 0.223653
\(539\) 25.1921 1.08510
\(540\) −0.829849 −0.0357110
\(541\) −16.6193 −0.714520 −0.357260 0.934005i \(-0.616289\pi\)
−0.357260 + 0.934005i \(0.616289\pi\)
\(542\) −28.8771 −1.24038
\(543\) 25.8875 1.11094
\(544\) 9.33490 0.400230
\(545\) −1.60407 −0.0687110
\(546\) −40.1436 −1.71799
\(547\) 6.03456 0.258019 0.129010 0.991643i \(-0.458820\pi\)
0.129010 + 0.991643i \(0.458820\pi\)
\(548\) 6.93453 0.296228
\(549\) −16.6356 −0.709990
\(550\) 4.34425 0.185239
\(551\) 0 0
\(552\) 7.52578 0.320319
\(553\) −7.81545 −0.332347
\(554\) 42.0552 1.78676
\(555\) −19.4497 −0.825594
\(556\) 4.64439 0.196966
\(557\) −28.1274 −1.19180 −0.595898 0.803060i \(-0.703204\pi\)
−0.595898 + 0.803060i \(0.703204\pi\)
\(558\) −5.42012 −0.229452
\(559\) 17.4341 0.737382
\(560\) −18.9241 −0.799687
\(561\) 22.7975 0.962512
\(562\) −33.8377 −1.42736
\(563\) 1.21155 0.0510609 0.0255305 0.999674i \(-0.491873\pi\)
0.0255305 + 0.999674i \(0.491873\pi\)
\(564\) −9.70842 −0.408798
\(565\) 12.9449 0.544595
\(566\) 28.8136 1.21113
\(567\) 43.0623 1.80844
\(568\) −7.99353 −0.335401
\(569\) 17.5814 0.737049 0.368524 0.929618i \(-0.379863\pi\)
0.368524 + 0.929618i \(0.379863\pi\)
\(570\) 0 0
\(571\) 42.6982 1.78686 0.893432 0.449198i \(-0.148290\pi\)
0.893432 + 0.449198i \(0.148290\pi\)
\(572\) −3.59598 −0.150356
\(573\) −27.1640 −1.13479
\(574\) −41.6557 −1.73867
\(575\) 1.36746 0.0570271
\(576\) 11.9192 0.496633
\(577\) 5.49356 0.228700 0.114350 0.993441i \(-0.463521\pi\)
0.114350 + 0.993441i \(0.463521\pi\)
\(578\) −6.26852 −0.260736
\(579\) −2.18867 −0.0909579
\(580\) −4.43506 −0.184156
\(581\) −6.49635 −0.269514
\(582\) −41.5956 −1.72419
\(583\) 10.9114 0.451906
\(584\) 24.1036 0.997415
\(585\) 6.19152 0.255988
\(586\) 14.5877 0.602611
\(587\) −12.4236 −0.512778 −0.256389 0.966574i \(-0.582533\pi\)
−0.256389 + 0.966574i \(0.582533\pi\)
\(588\) −9.70547 −0.400247
\(589\) 0 0
\(590\) 5.67228 0.233524
\(591\) −32.6247 −1.34200
\(592\) 40.1355 1.64956
\(593\) −1.11776 −0.0459009 −0.0229505 0.999737i \(-0.507306\pi\)
−0.0229505 + 0.999737i \(0.507306\pi\)
\(594\) 7.72979 0.317157
\(595\) −14.4751 −0.593423
\(596\) −0.667690 −0.0273497
\(597\) 17.7448 0.726246
\(598\) −5.98596 −0.244784
\(599\) 27.5096 1.12401 0.562007 0.827133i \(-0.310030\pi\)
0.562007 + 0.827133i \(0.310030\pi\)
\(600\) 5.50347 0.224678
\(601\) −14.0537 −0.573263 −0.286631 0.958041i \(-0.592536\pi\)
−0.286631 + 0.958041i \(0.592536\pi\)
\(602\) 39.4230 1.60676
\(603\) 32.3689 1.31816
\(604\) 3.81683 0.155305
\(605\) 3.34812 0.136121
\(606\) 25.0568 1.01786
\(607\) 16.1942 0.657304 0.328652 0.944451i \(-0.393406\pi\)
0.328652 + 0.944451i \(0.393406\pi\)
\(608\) 0 0
\(609\) 87.2071 3.53381
\(610\) 11.7614 0.476205
\(611\) −25.3922 −1.02726
\(612\) −3.73654 −0.151041
\(613\) 27.6779 1.11790 0.558949 0.829202i \(-0.311205\pi\)
0.558949 + 0.829202i \(0.311205\pi\)
\(614\) 17.2990 0.698129
\(615\) 15.1017 0.608958
\(616\) 26.7386 1.07733
\(617\) 6.09559 0.245399 0.122700 0.992444i \(-0.460845\pi\)
0.122700 + 0.992444i \(0.460845\pi\)
\(618\) 36.3656 1.46284
\(619\) −11.7187 −0.471014 −0.235507 0.971873i \(-0.575675\pi\)
−0.235507 + 0.971873i \(0.575675\pi\)
\(620\) 0.724627 0.0291017
\(621\) 2.43315 0.0976389
\(622\) 39.8619 1.59832
\(623\) −49.1677 −1.96986
\(624\) −30.0319 −1.20224
\(625\) 1.00000 0.0400000
\(626\) 33.0677 1.32165
\(627\) 0 0
\(628\) −3.65419 −0.145818
\(629\) 30.6999 1.22409
\(630\) 14.0007 0.557800
\(631\) 20.9211 0.832856 0.416428 0.909169i \(-0.363282\pi\)
0.416428 + 0.909169i \(0.363282\pi\)
\(632\) −4.69020 −0.186566
\(633\) 17.4181 0.692308
\(634\) −6.29738 −0.250101
\(635\) −8.29802 −0.329297
\(636\) −4.20372 −0.166689
\(637\) −25.3845 −1.00577
\(638\) 41.3112 1.63553
\(639\) 7.37228 0.291643
\(640\) −13.6033 −0.537717
\(641\) 9.48683 0.374707 0.187354 0.982293i \(-0.440009\pi\)
0.187354 + 0.982293i \(0.440009\pi\)
\(642\) −42.2859 −1.66889
\(643\) 12.5393 0.494500 0.247250 0.968952i \(-0.420473\pi\)
0.247250 + 0.968952i \(0.420473\pi\)
\(644\) −2.55959 −0.100862
\(645\) −14.2923 −0.562757
\(646\) 0 0
\(647\) −15.6504 −0.615281 −0.307640 0.951503i \(-0.599539\pi\)
−0.307640 + 0.951503i \(0.599539\pi\)
\(648\) 25.8425 1.01519
\(649\) −9.99104 −0.392183
\(650\) −4.37742 −0.171696
\(651\) −14.2484 −0.558440
\(652\) −9.00096 −0.352505
\(653\) 19.0906 0.747071 0.373536 0.927616i \(-0.378145\pi\)
0.373536 + 0.927616i \(0.378145\pi\)
\(654\) −5.75632 −0.225090
\(655\) 6.51185 0.254439
\(656\) −31.1631 −1.21671
\(657\) −22.2303 −0.867287
\(658\) −57.4185 −2.23841
\(659\) 10.4682 0.407783 0.203891 0.978994i \(-0.434641\pi\)
0.203891 + 0.978994i \(0.434641\pi\)
\(660\) 2.94795 0.114749
\(661\) 29.1420 1.13349 0.566745 0.823893i \(-0.308202\pi\)
0.566745 + 0.823893i \(0.308202\pi\)
\(662\) 50.4687 1.96152
\(663\) −22.9716 −0.892143
\(664\) −3.89858 −0.151294
\(665\) 0 0
\(666\) −29.6936 −1.15060
\(667\) 13.0038 0.503507
\(668\) −4.37922 −0.169437
\(669\) −18.2877 −0.707044
\(670\) −22.8849 −0.884120
\(671\) −20.7163 −0.799745
\(672\) −23.7353 −0.915608
\(673\) 14.5550 0.561052 0.280526 0.959846i \(-0.409491\pi\)
0.280526 + 0.959846i \(0.409491\pi\)
\(674\) −36.8239 −1.41840
\(675\) 1.77932 0.0684859
\(676\) −2.43959 −0.0938302
\(677\) 10.9800 0.421995 0.210998 0.977487i \(-0.432329\pi\)
0.210998 + 0.977487i \(0.432329\pi\)
\(678\) 46.4535 1.78404
\(679\) −46.5196 −1.78526
\(680\) −8.68681 −0.333124
\(681\) −40.5383 −1.55343
\(682\) −6.74968 −0.258459
\(683\) 43.3449 1.65854 0.829272 0.558844i \(-0.188755\pi\)
0.829272 + 0.558844i \(0.188755\pi\)
\(684\) 0 0
\(685\) −14.8686 −0.568101
\(686\) −13.2809 −0.507066
\(687\) 55.3503 2.11174
\(688\) 29.4928 1.12440
\(689\) −10.9948 −0.418867
\(690\) 4.90722 0.186815
\(691\) 30.4057 1.15669 0.578344 0.815793i \(-0.303699\pi\)
0.578344 + 0.815793i \(0.303699\pi\)
\(692\) −3.87606 −0.147346
\(693\) −24.6605 −0.936776
\(694\) 10.1979 0.387107
\(695\) −9.95823 −0.377737
\(696\) 52.3347 1.98374
\(697\) −23.8368 −0.902885
\(698\) −42.7181 −1.61691
\(699\) −31.1885 −1.17966
\(700\) −1.87178 −0.0707466
\(701\) 21.3688 0.807088 0.403544 0.914960i \(-0.367778\pi\)
0.403544 + 0.914960i \(0.367778\pi\)
\(702\) −7.78881 −0.293970
\(703\) 0 0
\(704\) 14.8430 0.559416
\(705\) 20.8163 0.783985
\(706\) −10.3043 −0.387806
\(707\) 28.0229 1.05391
\(708\) 3.84913 0.144659
\(709\) −20.0113 −0.751541 −0.375770 0.926713i \(-0.622622\pi\)
−0.375770 + 0.926713i \(0.622622\pi\)
\(710\) −5.21222 −0.195611
\(711\) 4.32568 0.162226
\(712\) −29.5065 −1.10580
\(713\) −2.12463 −0.0795682
\(714\) −51.9449 −1.94399
\(715\) 7.71030 0.288349
\(716\) 8.64301 0.323005
\(717\) −8.54058 −0.318954
\(718\) −4.29045 −0.160118
\(719\) −4.10144 −0.152958 −0.0764790 0.997071i \(-0.524368\pi\)
−0.0764790 + 0.997071i \(0.524368\pi\)
\(720\) 10.4741 0.390345
\(721\) 40.6705 1.51465
\(722\) 0 0
\(723\) −12.0682 −0.448822
\(724\) −5.28379 −0.196371
\(725\) 9.50940 0.353170
\(726\) 12.0149 0.445917
\(727\) −35.7495 −1.32588 −0.662938 0.748674i \(-0.730691\pi\)
−0.662938 + 0.748674i \(0.730691\pi\)
\(728\) −26.9428 −0.998567
\(729\) −11.6367 −0.430990
\(730\) 15.7169 0.581708
\(731\) 22.5593 0.834384
\(732\) 7.98113 0.294991
\(733\) −31.2944 −1.15589 −0.577943 0.816077i \(-0.696145\pi\)
−0.577943 + 0.816077i \(0.696145\pi\)
\(734\) 10.3322 0.381368
\(735\) 20.8099 0.767586
\(736\) −3.53925 −0.130458
\(737\) 40.3090 1.48480
\(738\) 23.0555 0.848686
\(739\) 32.5127 1.19600 0.597999 0.801497i \(-0.295963\pi\)
0.597999 + 0.801497i \(0.295963\pi\)
\(740\) 3.96980 0.145933
\(741\) 0 0
\(742\) −24.8621 −0.912716
\(743\) −38.8471 −1.42516 −0.712580 0.701590i \(-0.752474\pi\)
−0.712580 + 0.701590i \(0.752474\pi\)
\(744\) −8.55076 −0.313486
\(745\) 1.43162 0.0524506
\(746\) 13.4431 0.492188
\(747\) 3.59559 0.131556
\(748\) −4.65312 −0.170135
\(749\) −47.2916 −1.72800
\(750\) 3.58856 0.131036
\(751\) 1.80781 0.0659681 0.0329840 0.999456i \(-0.489499\pi\)
0.0329840 + 0.999456i \(0.489499\pi\)
\(752\) −42.9554 −1.56642
\(753\) 28.6644 1.04459
\(754\) −41.6266 −1.51595
\(755\) −8.18383 −0.297840
\(756\) −3.33049 −0.121129
\(757\) 22.5632 0.820074 0.410037 0.912069i \(-0.365516\pi\)
0.410037 + 0.912069i \(0.365516\pi\)
\(758\) −22.3757 −0.812721
\(759\) −8.64349 −0.313739
\(760\) 0 0
\(761\) 2.85080 0.103341 0.0516707 0.998664i \(-0.483545\pi\)
0.0516707 + 0.998664i \(0.483545\pi\)
\(762\) −29.7780 −1.07874
\(763\) −6.43774 −0.233062
\(764\) 5.54434 0.200587
\(765\) 8.01168 0.289663
\(766\) −33.5553 −1.21240
\(767\) 10.0673 0.363510
\(768\) −24.2941 −0.876639
\(769\) 6.11973 0.220683 0.110342 0.993894i \(-0.464806\pi\)
0.110342 + 0.993894i \(0.464806\pi\)
\(770\) 17.4351 0.628316
\(771\) 19.6508 0.707705
\(772\) 0.446721 0.0160778
\(773\) −50.1577 −1.80405 −0.902023 0.431688i \(-0.857918\pi\)
−0.902023 + 0.431688i \(0.857918\pi\)
\(774\) −21.8198 −0.784297
\(775\) −1.55371 −0.0558107
\(776\) −27.9173 −1.00217
\(777\) −78.0588 −2.80034
\(778\) 24.0167 0.861039
\(779\) 0 0
\(780\) −2.97046 −0.106359
\(781\) 9.18070 0.328511
\(782\) −7.74569 −0.276985
\(783\) 16.9202 0.604680
\(784\) −42.9424 −1.53366
\(785\) 7.83512 0.279647
\(786\) 23.3682 0.833515
\(787\) 33.0203 1.17705 0.588524 0.808480i \(-0.299709\pi\)
0.588524 + 0.808480i \(0.299709\pi\)
\(788\) 6.65889 0.237213
\(789\) −60.9879 −2.17123
\(790\) −3.05827 −0.108808
\(791\) 51.9525 1.84722
\(792\) −14.7993 −0.525869
\(793\) 20.8745 0.741275
\(794\) 51.2006 1.81704
\(795\) 9.01339 0.319672
\(796\) −3.62182 −0.128372
\(797\) −47.2358 −1.67318 −0.836590 0.547830i \(-0.815454\pi\)
−0.836590 + 0.547830i \(0.815454\pi\)
\(798\) 0 0
\(799\) −32.8569 −1.16239
\(800\) −2.58819 −0.0915062
\(801\) 27.2132 0.961533
\(802\) 27.1879 0.960039
\(803\) −27.6834 −0.976927
\(804\) −15.5294 −0.547679
\(805\) 5.48813 0.193431
\(806\) 6.80122 0.239563
\(807\) −7.54788 −0.265698
\(808\) 16.8171 0.591623
\(809\) 4.38236 0.154076 0.0770378 0.997028i \(-0.475454\pi\)
0.0770378 + 0.997028i \(0.475454\pi\)
\(810\) 16.8507 0.592074
\(811\) 33.9613 1.19254 0.596271 0.802783i \(-0.296649\pi\)
0.596271 + 0.802783i \(0.296649\pi\)
\(812\) −17.7995 −0.624640
\(813\) 42.0159 1.47356
\(814\) −36.9775 −1.29606
\(815\) 19.2994 0.676027
\(816\) −38.8606 −1.36039
\(817\) 0 0
\(818\) 17.0038 0.594525
\(819\) 24.8488 0.868288
\(820\) −3.08234 −0.107640
\(821\) 40.1364 1.40077 0.700385 0.713766i \(-0.253012\pi\)
0.700385 + 0.713766i \(0.253012\pi\)
\(822\) −53.3570 −1.86104
\(823\) 6.31582 0.220156 0.110078 0.993923i \(-0.464890\pi\)
0.110078 + 0.993923i \(0.464890\pi\)
\(824\) 24.4072 0.850264
\(825\) −6.32083 −0.220063
\(826\) 22.7649 0.792093
\(827\) −21.3614 −0.742808 −0.371404 0.928471i \(-0.621123\pi\)
−0.371404 + 0.928471i \(0.621123\pi\)
\(828\) 1.41668 0.0492330
\(829\) −49.0602 −1.70393 −0.851966 0.523597i \(-0.824590\pi\)
−0.851966 + 0.523597i \(0.824590\pi\)
\(830\) −2.54209 −0.0882372
\(831\) −61.1898 −2.12265
\(832\) −14.9563 −0.518517
\(833\) −32.8469 −1.13808
\(834\) −35.7357 −1.23743
\(835\) 9.38968 0.324943
\(836\) 0 0
\(837\) −2.76453 −0.0955562
\(838\) 53.3588 1.84325
\(839\) 7.38997 0.255130 0.127565 0.991830i \(-0.459284\pi\)
0.127565 + 0.991830i \(0.459284\pi\)
\(840\) 22.0874 0.762089
\(841\) 61.4287 2.11823
\(842\) −13.0479 −0.449660
\(843\) 49.2334 1.69569
\(844\) −3.55514 −0.122373
\(845\) 5.23082 0.179946
\(846\) 31.7799 1.09262
\(847\) 13.4372 0.461709
\(848\) −18.5996 −0.638713
\(849\) −41.9235 −1.43881
\(850\) −5.66428 −0.194283
\(851\) −11.6396 −0.399001
\(852\) −3.53694 −0.121174
\(853\) 2.75310 0.0942642 0.0471321 0.998889i \(-0.484992\pi\)
0.0471321 + 0.998889i \(0.484992\pi\)
\(854\) 47.2028 1.61525
\(855\) 0 0
\(856\) −28.3806 −0.970029
\(857\) 44.1261 1.50732 0.753659 0.657266i \(-0.228287\pi\)
0.753659 + 0.657266i \(0.228287\pi\)
\(858\) 27.6689 0.944601
\(859\) −11.7531 −0.401010 −0.200505 0.979693i \(-0.564258\pi\)
−0.200505 + 0.979693i \(0.564258\pi\)
\(860\) 2.91714 0.0994736
\(861\) 60.6085 2.06553
\(862\) 19.0495 0.648828
\(863\) 34.2831 1.16701 0.583504 0.812110i \(-0.301681\pi\)
0.583504 + 0.812110i \(0.301681\pi\)
\(864\) −4.60520 −0.156672
\(865\) 8.31084 0.282577
\(866\) 8.53617 0.290071
\(867\) 9.12061 0.309752
\(868\) 2.90820 0.0987106
\(869\) 5.38678 0.182734
\(870\) 34.1251 1.15695
\(871\) −40.6168 −1.37625
\(872\) −3.86341 −0.130832
\(873\) 25.7476 0.871423
\(874\) 0 0
\(875\) 4.01337 0.135677
\(876\) 10.6653 0.360346
\(877\) 24.2041 0.817313 0.408656 0.912688i \(-0.365997\pi\)
0.408656 + 0.912688i \(0.365997\pi\)
\(878\) 39.8597 1.34520
\(879\) −21.2249 −0.715898
\(880\) 13.0434 0.439691
\(881\) −39.1733 −1.31978 −0.659892 0.751361i \(-0.729398\pi\)
−0.659892 + 0.751361i \(0.729398\pi\)
\(882\) 31.7703 1.06976
\(883\) 32.3695 1.08932 0.544660 0.838657i \(-0.316659\pi\)
0.544660 + 0.838657i \(0.316659\pi\)
\(884\) 4.68865 0.157696
\(885\) −8.25309 −0.277425
\(886\) −7.72870 −0.259651
\(887\) −0.380819 −0.0127866 −0.00639332 0.999980i \(-0.502035\pi\)
−0.00639332 + 0.999980i \(0.502035\pi\)
\(888\) −46.8446 −1.57200
\(889\) −33.3030 −1.11695
\(890\) −19.2398 −0.644921
\(891\) −29.6805 −0.994336
\(892\) 3.73264 0.124978
\(893\) 0 0
\(894\) 5.13747 0.171823
\(895\) −18.5319 −0.619452
\(896\) −54.5949 −1.82389
\(897\) 8.70949 0.290801
\(898\) −6.38632 −0.213114
\(899\) −14.7748 −0.492767
\(900\) 1.03599 0.0345330
\(901\) −14.2270 −0.473969
\(902\) 28.7111 0.955974
\(903\) −57.3600 −1.90882
\(904\) 31.1777 1.03696
\(905\) 11.3292 0.376596
\(906\) −29.3682 −0.975692
\(907\) 4.92616 0.163571 0.0817853 0.996650i \(-0.473938\pi\)
0.0817853 + 0.996650i \(0.473938\pi\)
\(908\) 8.27411 0.274586
\(909\) −15.5101 −0.514437
\(910\) −17.5682 −0.582380
\(911\) −20.6775 −0.685075 −0.342537 0.939504i \(-0.611286\pi\)
−0.342537 + 0.939504i \(0.611286\pi\)
\(912\) 0 0
\(913\) 4.47759 0.148187
\(914\) −21.2826 −0.703965
\(915\) −17.1127 −0.565728
\(916\) −11.2973 −0.373274
\(917\) 26.1344 0.863035
\(918\) −10.0785 −0.332641
\(919\) −60.5868 −1.99857 −0.999287 0.0377591i \(-0.987978\pi\)
−0.999287 + 0.0377591i \(0.987978\pi\)
\(920\) 3.29353 0.108585
\(921\) −25.1698 −0.829372
\(922\) 28.0820 0.924831
\(923\) −9.25080 −0.304494
\(924\) 11.8312 0.389218
\(925\) −8.51183 −0.279867
\(926\) 2.05254 0.0674505
\(927\) −22.5103 −0.739334
\(928\) −24.6121 −0.807932
\(929\) −0.290620 −0.00953494 −0.00476747 0.999989i \(-0.501518\pi\)
−0.00476747 + 0.999989i \(0.501518\pi\)
\(930\) −5.57557 −0.182830
\(931\) 0 0
\(932\) 6.36576 0.208517
\(933\) −57.9986 −1.89879
\(934\) 42.4446 1.38883
\(935\) 9.97695 0.326281
\(936\) 14.9123 0.487423
\(937\) 5.30585 0.173335 0.0866673 0.996237i \(-0.472378\pi\)
0.0866673 + 0.996237i \(0.472378\pi\)
\(938\) −91.8454 −2.99886
\(939\) −48.1131 −1.57011
\(940\) −4.24873 −0.138578
\(941\) 28.4255 0.926646 0.463323 0.886189i \(-0.346657\pi\)
0.463323 + 0.886189i \(0.346657\pi\)
\(942\) 28.1168 0.916095
\(943\) 9.03754 0.294303
\(944\) 17.0307 0.554302
\(945\) 7.14105 0.232298
\(946\) −27.1722 −0.883446
\(947\) −49.9790 −1.62410 −0.812049 0.583589i \(-0.801648\pi\)
−0.812049 + 0.583589i \(0.801648\pi\)
\(948\) −2.07530 −0.0674026
\(949\) 27.8948 0.905504
\(950\) 0 0
\(951\) 9.16261 0.297118
\(952\) −34.8634 −1.12993
\(953\) 12.1003 0.391966 0.195983 0.980607i \(-0.437210\pi\)
0.195983 + 0.980607i \(0.437210\pi\)
\(954\) 13.7606 0.445517
\(955\) −11.8879 −0.384682
\(956\) 1.74319 0.0563787
\(957\) −60.1073 −1.94299
\(958\) 0.454611 0.0146878
\(959\) −59.6732 −1.92695
\(960\) 12.2610 0.395723
\(961\) −28.5860 −0.922129
\(962\) 37.2599 1.20131
\(963\) 26.1749 0.843474
\(964\) 2.46320 0.0793344
\(965\) −0.957834 −0.0308338
\(966\) 19.6945 0.633660
\(967\) 13.1688 0.423480 0.211740 0.977326i \(-0.432087\pi\)
0.211740 + 0.977326i \(0.432087\pi\)
\(968\) 8.06395 0.259185
\(969\) 0 0
\(970\) −18.2036 −0.584482
\(971\) 17.5135 0.562035 0.281018 0.959703i \(-0.409328\pi\)
0.281018 + 0.959703i \(0.409328\pi\)
\(972\) 8.94513 0.286915
\(973\) −39.9660 −1.28125
\(974\) −24.1399 −0.773491
\(975\) 6.36909 0.203974
\(976\) 35.3129 1.13034
\(977\) 10.5361 0.337081 0.168541 0.985695i \(-0.446095\pi\)
0.168541 + 0.985695i \(0.446095\pi\)
\(978\) 69.2569 2.21459
\(979\) 33.8887 1.08309
\(980\) −4.24743 −0.135679
\(981\) 3.56315 0.113763
\(982\) 22.7878 0.727188
\(983\) −51.9781 −1.65784 −0.828922 0.559364i \(-0.811046\pi\)
−0.828922 + 0.559364i \(0.811046\pi\)
\(984\) 36.3723 1.15951
\(985\) −14.2776 −0.454923
\(986\) −53.8639 −1.71538
\(987\) 83.5433 2.65921
\(988\) 0 0
\(989\) −8.55315 −0.271974
\(990\) −9.64993 −0.306695
\(991\) −6.77628 −0.215256 −0.107628 0.994191i \(-0.534326\pi\)
−0.107628 + 0.994191i \(0.534326\pi\)
\(992\) 4.02128 0.127676
\(993\) −73.4313 −2.33027
\(994\) −20.9185 −0.663496
\(995\) 7.76571 0.246190
\(996\) −1.72503 −0.0546596
\(997\) −40.2693 −1.27534 −0.637671 0.770309i \(-0.720102\pi\)
−0.637671 + 0.770309i \(0.720102\pi\)
\(998\) −23.1455 −0.732659
\(999\) −15.1452 −0.479174
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.u.1.8 9
5.4 even 2 9025.2.a.cd.1.2 9
19.14 odd 18 95.2.k.b.6.1 18
19.15 odd 18 95.2.k.b.16.1 yes 18
19.18 odd 2 1805.2.a.t.1.2 9
57.14 even 18 855.2.bs.b.766.3 18
57.53 even 18 855.2.bs.b.586.3 18
95.14 odd 18 475.2.l.b.101.3 18
95.33 even 36 475.2.u.c.424.5 36
95.34 odd 18 475.2.l.b.301.3 18
95.52 even 36 475.2.u.c.424.2 36
95.53 even 36 475.2.u.c.149.2 36
95.72 even 36 475.2.u.c.149.5 36
95.94 odd 2 9025.2.a.ce.1.8 9
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
95.2.k.b.6.1 18 19.14 odd 18
95.2.k.b.16.1 yes 18 19.15 odd 18
475.2.l.b.101.3 18 95.14 odd 18
475.2.l.b.301.3 18 95.34 odd 18
475.2.u.c.149.2 36 95.53 even 36
475.2.u.c.149.5 36 95.72 even 36
475.2.u.c.424.2 36 95.52 even 36
475.2.u.c.424.5 36 95.33 even 36
855.2.bs.b.586.3 18 57.53 even 18
855.2.bs.b.766.3 18 57.14 even 18
1805.2.a.t.1.2 9 19.18 odd 2
1805.2.a.u.1.8 9 1.1 even 1 trivial
9025.2.a.cd.1.2 9 5.4 even 2
9025.2.a.ce.1.8 9 95.94 odd 2