Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,9,Mod(8,81)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma:// Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("81.8"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 81.f (of order \(18\), degree \(6\), not minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(32.9976674150\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 8.8
Character \(\chi\) \(=\) 81.8
Dual form 81.9.f.a.71.8

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-14.0349 - 2.47473i) q^{2} +(-49.7072 - 18.0920i) q^{4} +(245.194 + 292.211i) q^{5} +(-336.792 + 122.582i) q^{7} +(3812.44 + 2201.11i) q^{8} +(-2718.13 - 4707.95i) q^{10} +(2622.22 - 3125.04i) q^{11} +(-689.271 - 3909.05i) q^{13} +(5030.19 - 886.959i) q^{14} +(-37686.4 - 31622.7i) q^{16} +(-54109.2 + 31239.9i) q^{17} +(69562.0 - 120485. i) q^{19} +(-6901.26 - 18961.1i) q^{20} +(-44536.3 + 37370.4i) q^{22} +(-17740.4 + 48741.3i) q^{23} +(42564.2 - 241393. i) q^{25} +56568.9i q^{26} +18958.7 q^{28} +(-577603. - 101847. i) q^{29} +(169959. + 61860.0i) q^{31} +(-273734. - 326223. i) q^{32} +(836727. - 304544. i) q^{34} +(-118399. - 68357.9i) q^{35} +(1.42753e6 + 2.47255e6i) q^{37} +(-1.27446e6 + 1.51885e6i) q^{38} +(291598. + 1.65374e6i) q^{40} +(337957. - 59591.0i) q^{41} +(1.88422e6 + 1.58105e6i) q^{43} +(-186882. + 107896. i) q^{44} +(369606. - 640177. i) q^{46} +(667331. + 1.83348e6i) q^{47} +(-4.31769e6 + 3.62297e6i) q^{49} +(-1.19477e6 + 3.28260e6i) q^{50} +(-36460.6 + 206778. i) q^{52} +1.43620e7i q^{53} +1.55613e6 q^{55} +(-1.55381e6 - 273979. i) q^{56} +(7.85455e6 + 2.85882e6i) q^{58} +(1.52519e7 + 1.81765e7i) q^{59} +(-3.80007e6 + 1.38311e6i) q^{61} +(-2.23227e6 - 1.28880e6i) q^{62} +(9.33162e6 + 1.61628e7i) q^{64} +(973263. - 1.15989e6i) q^{65} +(3.89152e6 + 2.20699e7i) q^{67} +(3.25481e6 - 573911. i) q^{68} +(1.49255e6 + 1.25240e6i) q^{70} +(-2.28067e7 + 1.31675e7i) q^{71} +(7.13836e6 - 1.23640e7i) q^{73} +(-1.39163e7 - 3.82347e7i) q^{74} +(-5.63754e6 + 4.73046e6i) q^{76} +(-500068. + 1.37393e6i) q^{77} +(1.03600e7 - 5.87545e7i) q^{79} -1.87661e7i q^{80} -4.89067e6 q^{82} +(-4.60108e7 - 8.11295e6i) q^{83} +(-2.23959e7 - 8.15145e6i) q^{85} +(-2.25321e7 - 2.68527e7i) q^{86} +(1.68756e7 - 6.14222e6i) q^{88} +(3.55252e7 + 2.05105e7i) q^{89} +(711320. + 1.23204e6i) q^{91} +(1.76365e6 - 2.10184e6i) q^{92} +(-4.82856e6 - 2.73841e7i) q^{94} +(5.22633e7 - 9.21542e6i) q^{95} +(-3.64414e7 - 3.05780e7i) q^{97} +(6.95642e7 - 4.01629e7i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q + 6 q^{2} - 6 q^{4} + 447 q^{5} - 6 q^{7} + 9 q^{8} - 3 q^{10} - 28668 q^{11} - 6 q^{13} + 120975 q^{14} - 774 q^{16} + 9 q^{17} - 3 q^{19} - 137913 q^{20} - 185478 q^{22} - 68376 q^{23} + 507585 q^{25}+ \cdots - 1293135102 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/81\mathbb{Z}\right)^\times\).

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{1}{18}\right)\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −14.0349 − 2.47473i −0.877181 − 0.154671i −0.283114 − 0.959086i \(-0.591368\pi\)
−0.594067 + 0.804416i \(0.702479\pi\)
\(3\) 0 0
\(4\) −49.7072 − 18.0920i −0.194169 − 0.0706717i
\(5\) 245.194 + 292.211i 0.392311 + 0.467538i 0.925660 − 0.378358i \(-0.123511\pi\)
−0.533349 + 0.845896i \(0.679067\pi\)
\(6\) 0 0
\(7\) −336.792 + 122.582i −0.140271 + 0.0510546i −0.411202 − 0.911544i \(-0.634891\pi\)
0.270930 + 0.962599i \(0.412669\pi\)
\(8\) 3812.44 + 2201.11i 0.930771 + 0.537381i
\(9\) 0 0
\(10\) −2718.13 − 4707.95i −0.271813 − 0.470795i
\(11\) 2622.22 − 3125.04i 0.179101 − 0.213445i −0.669023 − 0.743241i \(-0.733287\pi\)
0.848125 + 0.529797i \(0.177732\pi\)
\(12\) 0 0
\(13\) −689.271 − 3909.05i −0.0241333 − 0.136867i 0.970361 − 0.241662i \(-0.0776924\pi\)
−0.994494 + 0.104795i \(0.966581\pi\)
\(14\) 5030.19 − 886.959i 0.130940 − 0.0230883i
\(15\) 0 0
\(16\) −37686.4 − 31622.7i −0.575049 − 0.482524i
\(17\) −54109.2 + 31239.9i −0.647851 + 0.374037i −0.787632 − 0.616145i \(-0.788693\pi\)
0.139781 + 0.990182i \(0.455360\pi\)
\(18\) 0 0
\(19\) 69562.0 − 120485.i 0.533774 − 0.924524i −0.465447 − 0.885076i \(-0.654107\pi\)
0.999222 − 0.0394486i \(-0.0125601\pi\)
\(20\) −6901.26 − 18961.1i −0.0431329 − 0.118507i
\(21\) 0 0
\(22\) −44536.3 + 37370.4i −0.190118 + 0.159528i
\(23\) −17740.4 + 48741.3i −0.0633946 + 0.174175i −0.967346 − 0.253461i \(-0.918431\pi\)
0.903951 + 0.427636i \(0.140653\pi\)
\(24\) 0 0
\(25\) 42564.2 − 241393.i 0.108964 − 0.617967i
\(26\) 56568.9i 0.123790i
\(27\) 0 0
\(28\) 18958.7 0.0308445
\(29\) −577603. − 101847.i −0.816652 − 0.143998i −0.250308 − 0.968166i \(-0.580532\pi\)
−0.566345 + 0.824168i \(0.691643\pi\)
\(30\) 0 0
\(31\) 169959. + 61860.0i 0.184034 + 0.0669827i 0.432393 − 0.901685i \(-0.357669\pi\)
−0.248360 + 0.968668i \(0.579891\pi\)
\(32\) −273734. − 326223.i −0.261053 − 0.311111i
\(33\) 0 0
\(34\) 836727. − 304544.i 0.626135 − 0.227895i
\(35\) −118399. − 68357.9i −0.0789000 − 0.0455529i
\(36\) 0 0
\(37\) 1.42753e6 + 2.47255e6i 0.761688 + 1.31928i 0.941980 + 0.335669i \(0.108962\pi\)
−0.180292 + 0.983613i \(0.557704\pi\)
\(38\) −1.27446e6 + 1.51885e6i −0.611214 + 0.728416i
\(39\) 0 0
\(40\) 291598. + 1.65374e6i 0.113906 + 0.645991i
\(41\) 337957. − 59591.0i 0.119599 − 0.0210885i −0.113528 − 0.993535i \(-0.536215\pi\)
0.233127 + 0.972446i \(0.425104\pi\)
\(42\) 0 0
\(43\) 1.88422e6 + 1.58105e6i 0.551134 + 0.462456i 0.875325 − 0.483535i \(-0.160648\pi\)
−0.324191 + 0.945992i \(0.605092\pi\)
\(44\) −186882. + 107896.i −0.0498604 + 0.0287869i
\(45\) 0 0
\(46\) 369606. − 640177.i 0.0825483 − 0.142978i
\(47\) 667331. + 1.83348e6i 0.136757 + 0.375737i 0.989100 − 0.147247i \(-0.0470411\pi\)
−0.852343 + 0.522984i \(0.824819\pi\)
\(48\) 0 0
\(49\) −4.31769e6 + 3.62297e6i −0.748975 + 0.628465i
\(50\) −1.19477e6 + 3.28260e6i −0.191163 + 0.525215i
\(51\) 0 0
\(52\) −36460.6 + 206778.i −0.00498667 + 0.0282808i
\(53\) 1.43620e7i 1.82016i 0.414429 + 0.910082i \(0.363981\pi\)
−0.414429 + 0.910082i \(0.636019\pi\)
\(54\) 0 0
\(55\) 1.55613e6 0.170057
\(56\) −1.55381e6 − 273979.i −0.157996 − 0.0278590i
\(57\) 0 0
\(58\) 7.85455e6 + 2.85882e6i 0.694080 + 0.252624i
\(59\) 1.52519e7 + 1.81765e7i 1.25868 + 1.50004i 0.785208 + 0.619231i \(0.212556\pi\)
0.473474 + 0.880808i \(0.343000\pi\)
\(60\) 0 0
\(61\) −3.80007e6 + 1.38311e6i −0.274455 + 0.0998936i −0.475581 − 0.879672i \(-0.657762\pi\)
0.201126 + 0.979565i \(0.435540\pi\)
\(62\) −2.23227e6 − 1.28880e6i −0.151070 − 0.0872206i
\(63\) 0 0
\(64\) 9.33162e6 + 1.61628e7i 0.556208 + 0.963380i
\(65\) 973263. − 1.15989e6i 0.0545226 − 0.0649775i
\(66\) 0 0
\(67\) 3.89152e6 + 2.20699e7i 0.193117 + 1.09522i 0.915075 + 0.403283i \(0.132131\pi\)
−0.721959 + 0.691936i \(0.756758\pi\)
\(68\) 3.25481e6 − 573911.i 0.152226 − 0.0268416i
\(69\) 0 0
\(70\) 1.49255e6 + 1.25240e6i 0.0621639 + 0.0521617i
\(71\) −2.28067e7 + 1.31675e7i −0.897489 + 0.518165i −0.876385 − 0.481612i \(-0.840052\pi\)
−0.0211042 + 0.999777i \(0.506718\pi\)
\(72\) 0 0
\(73\) 7.13836e6 − 1.23640e7i 0.251366 − 0.435379i −0.712536 − 0.701636i \(-0.752453\pi\)
0.963902 + 0.266256i \(0.0857868\pi\)
\(74\) −1.39163e7 − 3.82347e7i −0.464084 − 1.27506i
\(75\) 0 0
\(76\) −5.63754e6 + 4.73046e6i −0.168980 + 0.141791i
\(77\) −500068. + 1.37393e6i −0.0142255 + 0.0390841i
\(78\) 0 0
\(79\) 1.03600e7 − 5.87545e7i 0.265982 − 1.50846i −0.500243 − 0.865885i \(-0.666756\pi\)
0.766225 − 0.642573i \(-0.222133\pi\)
\(80\) − 1.87661e7i − 0.458157i
\(81\) 0 0
\(82\) −4.89067e6 −0.108172
\(83\) −4.60108e7 − 8.11295e6i −0.969500 − 0.170949i −0.333595 − 0.942716i \(-0.608262\pi\)
−0.635905 + 0.771767i \(0.719373\pi\)
\(84\) 0 0
\(85\) −2.23959e7 − 8.15145e6i −0.429036 − 0.156156i
\(86\) −2.25321e7 − 2.68527e7i −0.411916 − 0.490902i
\(87\) 0 0
\(88\) 1.68756e7 − 6.14222e6i 0.281403 − 0.102422i
\(89\) 3.55252e7 + 2.05105e7i 0.566208 + 0.326900i 0.755633 − 0.654995i \(-0.227329\pi\)
−0.189425 + 0.981895i \(0.560663\pi\)
\(90\) 0 0
\(91\) 711320. + 1.23204e6i 0.0103729 + 0.0179664i
\(92\) 1.76365e6 − 2.10184e6i 0.0246185 − 0.0293392i
\(93\) 0 0
\(94\) −4.82856e6 − 2.73841e7i −0.0618452 − 0.350742i
\(95\) 5.22633e7 − 9.21542e6i 0.641656 − 0.113141i
\(96\) 0 0
\(97\) −3.64414e7 − 3.05780e7i −0.411631 − 0.345399i 0.413338 − 0.910578i \(-0.364363\pi\)
−0.824969 + 0.565178i \(0.808807\pi\)
\(98\) 6.95642e7 − 4.01629e7i 0.754192 − 0.435433i
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 81.9.f.a.8.8 138
3.2 odd 2 27.9.f.a.2.16 ✓ 138
27.13 even 9 27.9.f.a.14.16 yes 138
27.14 odd 18 inner 81.9.f.a.71.8 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.16 ✓ 138 3.2 odd 2
27.9.f.a.14.16 yes 138 27.13 even 9
81.9.f.a.8.8 138 1.1 even 1 trivial
81.9.f.a.71.8 138 27.14 odd 18 inner