↑

Properties

Label 81.12.e.a.10.32
Level $81$
Weight $12$
Character 81.10
Analytic conductor $62.236$
Analytic rank $0$
Dimension $192$
Inner twists $2$

Related objects

Downloads

Learn more

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [81,12,Mod(10,81)] 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("81.10"); S:= CuspForms(chi, 12); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(81, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([8])) N = Newforms(chi, 12, names="a")
 
Level: \( N \) \(=\) \( 81 = 3^{4} \)
Weight: \( k \) \(=\) \( 12 \)
Character orbit: \([\chi]\) \(=\) 81.e (of order \(9\), 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: \(62.2357976253\)
Analytic rank: \(0\)
Dimension: \(192\)
Relative dimension: \(32\) over \(\Q(\zeta_{9})\)
Twist minimal: no (minimal twist has level 27)
Sato-Tate group: $\mathrm{SU}(2)[C_{9}]$

Embedding invariants

Embedding label 10.32
Character \(\chi\) \(=\) 81.10
Dual form 81.12.e.a.73.32

$q$-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+(14.4291 + 81.8314i) q^{2} +(-4563.68 + 1661.05i) q^{4} +(-352.134 - 295.476i) q^{5} +(13655.9 + 4970.33i) q^{7} +(-116687. - 202108. i) q^{8} +(19098.2 - 33079.1i) q^{10} +(435836. - 365710. i) q^{11} +(138893. - 787699. i) q^{13} +(-209687. + 1.18920e6i) q^{14} +(7.23584e6 - 6.07159e6i) q^{16} +(-412835. + 715050. i) q^{17} +(8.89079e6 + 1.53993e7i) q^{19} +(2.09783e6 + 763547. i) q^{20} +(3.62153e7 + 3.03882e7i) q^{22} +(3.44286e7 - 1.25310e7i) q^{23} +(-8.44222e6 - 4.78782e7i) q^{25} +6.64626e7 q^{26} -7.05771e7 q^{28} +(5.30764e6 + 3.01011e7i) q^{29} +(1.90768e8 - 6.94340e7i) q^{31} +(2.35121e8 + 1.97290e8i) q^{32} +(-6.44704e7 - 2.34653e7i) q^{34} +(-3.34009e6 - 5.78521e6i) q^{35} +(-2.38556e8 + 4.13191e8i) q^{37} +(-1.13186e9 + 9.49743e8i) q^{38} +(-1.86285e7 + 1.05648e8i) q^{40} +(-2.48305e8 + 1.40821e9i) q^{41} +(1.10921e9 - 9.30739e8i) q^{43} +(-1.38156e9 + 2.39293e9i) q^{44} +(1.52220e9 + 2.63653e9i) q^{46} +(-7.21362e8 - 2.62554e8i) q^{47} +(-1.35294e9 - 1.13525e9i) q^{49} +(3.79613e9 - 1.38168e9i) q^{50} +(6.74542e8 + 3.82552e9i) q^{52} +4.89893e8 q^{53} -2.61531e8 q^{55} +(-5.88922e8 - 3.33994e9i) q^{56} +(-2.38663e9 + 8.68662e8i) q^{58} +(-4.15021e9 - 3.48244e9i) q^{59} +(1.07301e10 + 3.90543e9i) q^{61} +(8.43449e9 + 1.46090e10i) q^{62} +(-3.07953e9 + 5.33390e9i) q^{64} +(-2.81655e8 + 2.36337e8i) q^{65} +(1.31047e9 - 7.43205e9i) q^{67} +(6.96316e8 - 3.94900e9i) q^{68} +(4.25217e8 - 3.56799e8i) q^{70} +(-8.45043e9 + 1.46366e10i) q^{71} +(1.24406e10 + 2.15477e10i) q^{73} +(-3.72542e10 - 1.35594e10i) q^{74} +(-6.61537e10 - 5.55095e10i) q^{76} +(7.76942e9 - 2.82784e9i) q^{77} +(-4.77608e9 - 2.70865e10i) q^{79} -4.34200e9 q^{80} -1.18818e11 q^{82} +(-4.25866e9 - 2.41521e10i) q^{83} +(3.56653e8 - 1.29811e8i) q^{85} +(9.21686e10 + 7.73386e10i) q^{86} +(-1.24770e11 - 4.54124e10i) q^{88} +(-1.26135e10 - 2.18473e10i) q^{89} +(5.81183e9 - 1.00664e10i) q^{91} +(-1.36307e11 + 1.14375e11i) q^{92} +(1.10766e10 - 6.28184e10i) q^{94} +(1.41937e9 - 8.04963e9i) q^{95} +(-6.21527e10 + 5.21523e10i) q^{97} +(7.33776e10 - 1.27094e11i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 192 q + 6 q^{2} - 6 q^{4} - 2931 q^{5} - 6 q^{7} + 294915 q^{8} - 3 q^{10} + 1594293 q^{11} - 6 q^{13} - 5263143 q^{14} + 6138 q^{16} + 17038287 q^{17} - 3 q^{19} - 33966483 q^{20} - 6811494 q^{22} + 154612956 q^{23}+ \cdots - 282639387726 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Character values

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

\(n\) \(2\)
\(\chi(n)\) \(e\left(\frac{4}{9}\right)\)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 14.4291 + 81.8314i 0.318841 + 1.80823i 0.549828 + 0.835278i \(0.314693\pi\)
−0.230987 + 0.972957i \(0.574196\pi\)
\(3\) 0 0
\(4\) −4563.68 + 1661.05i −2.22836 + 0.811057i
\(5\) −352.134 − 295.476i −0.0503934 − 0.0422851i 0.617243 − 0.786772i \(-0.288249\pi\)
−0.667636 + 0.744487i \(0.732694\pi\)
\(6\) 0 0
\(7\) 13655.9 + 4970.33i 0.307100 + 0.111775i 0.490973 − 0.871175i \(-0.336641\pi\)
−0.183873 + 0.982950i \(0.558863\pi\)
\(8\) −116687. − 202108.i −1.25901 − 2.18067i
\(9\) 0 0
\(10\) 19098.2 − 33079.1i 0.0603939 − 0.104605i
\(11\) 435836. − 365710.i 0.815950 − 0.684663i −0.136070 − 0.990699i \(-0.543447\pi\)
0.952020 + 0.306036i \(0.0990028\pi\)
\(12\) 0 0
\(13\) 138893. − 787699.i 0.103751 − 0.588399i −0.887961 − 0.459918i \(-0.847879\pi\)
0.991712 − 0.128481i \(-0.0410101\pi\)
\(14\) −209687. + 1.18920e6i −0.104200 + 0.590948i
\(15\) 0 0
\(16\) 7.23584e6 − 6.07159e6i 1.72516 − 1.44758i
\(17\) −412835. + 715050.i −0.0705191 + 0.122143i −0.899129 − 0.437684i \(-0.855799\pi\)
0.828610 + 0.559827i \(0.189132\pi\)
\(18\) 0 0
\(19\) 8.89079e6 + 1.53993e7i 0.823750 + 1.42678i 0.902871 + 0.429912i \(0.141455\pi\)
−0.0791212 + 0.996865i \(0.525211\pi\)
\(20\) 2.09783e6 + 763547.i 0.146590 + 0.0533545i
\(21\) 0 0
\(22\) 3.62153e7 + 3.03882e7i 1.49819 + 1.25713i
\(23\) 3.44286e7 − 1.25310e7i 1.11536 − 0.405958i 0.282404 − 0.959296i \(-0.408868\pi\)
0.832957 + 0.553337i \(0.186646\pi\)
\(24\) 0 0
\(25\) −8.44222e6 − 4.78782e7i −0.172897 − 0.980546i
\(26\) 6.64626e7 1.09704
\(27\) 0 0
\(28\) −7.05771e7 −0.774987
\(29\) 5.30764e6 + 3.01011e7i 0.0480521 + 0.272517i 0.999362 − 0.0357196i \(-0.0113723\pi\)
−0.951310 + 0.308236i \(0.900261\pi\)
\(30\) 0 0
\(31\) 1.90768e8 − 6.94340e7i 1.19679 − 0.435595i 0.334685 − 0.942330i \(-0.391370\pi\)
0.862102 + 0.506735i \(0.169148\pi\)
\(32\) 2.35121e8 + 1.97290e8i 1.23870 + 1.03940i
\(33\) 0 0
\(34\) −6.44704e7 − 2.34653e7i −0.243347 − 0.0885711i
\(35\) −3.34009e6 − 5.78521e6i −0.0107494 − 0.0186185i
\(36\) 0 0
\(37\) −2.38556e8 + 4.13191e8i −0.565563 + 0.979584i 0.431434 + 0.902145i \(0.358008\pi\)
−0.996997 + 0.0774398i \(0.975325\pi\)
\(38\) −1.13186e9 + 9.49743e8i −2.31730 + 1.94445i
\(39\) 0 0
\(40\) −1.86285e7 + 1.05648e8i −0.0287640 + 0.163129i
\(41\) −2.48305e8 + 1.40821e9i −0.334714 + 1.89826i 0.0953202 + 0.995447i \(0.469612\pi\)
−0.430035 + 0.902812i \(0.641499\pi\)
\(42\) 0 0
\(43\) 1.10921e9 − 9.30739e8i 1.15064 − 0.965498i 0.150902 − 0.988549i \(-0.451782\pi\)
0.999734 + 0.0230506i \(0.00733790\pi\)
\(44\) −1.38156e9 + 2.39293e9i −1.26293 + 2.18746i
\(45\) 0 0
\(46\) 1.52220e9 + 2.63653e9i 1.08969 + 1.88740i
\(47\) −7.21362e8 − 2.62554e8i −0.458791 − 0.166986i 0.102277 − 0.994756i \(-0.467387\pi\)
−0.561068 + 0.827770i \(0.689609\pi\)
\(48\) 0 0
\(49\) −1.35294e9 − 1.13525e9i −0.684228 − 0.574135i
\(50\) 3.79613e9 − 1.38168e9i 1.71793 − 0.625276i
\(51\) 0 0
\(52\) 6.74542e8 + 3.82552e9i 0.246031 + 1.39531i
\(53\) 4.89893e8 0.160910 0.0804552 − 0.996758i \(-0.474363\pi\)
0.0804552 + 0.996758i \(0.474363\pi\)
\(54\) 0 0
\(55\) −2.61531e8 −0.0700695
\(56\) −5.88922e8 − 3.33994e9i −0.142897 − 0.810410i
\(57\) 0 0
\(58\) −2.38663e9 + 8.68662e8i −0.477454 + 0.173779i
\(59\) −4.15021e9 − 3.48244e9i −0.755760 − 0.634158i 0.181260 − 0.983435i \(-0.441983\pi\)
−0.937019 + 0.349278i \(0.886427\pi\)
\(60\) 0 0
\(61\) 1.07301e10 + 3.90543e9i 1.62663 + 0.592044i 0.984629 − 0.174660i \(-0.0558828\pi\)
0.642000 + 0.766705i \(0.278105\pi\)
\(62\) 8.43449e9 + 1.46090e10i 1.16924 + 2.02519i
\(63\) 0 0
\(64\) −3.07953e9 + 5.33390e9i −0.358504 + 0.620947i
\(65\) −2.81655e8 + 2.36337e8i −0.0301088 + 0.0252643i
\(66\) 0 0
\(67\) 1.31047e9 − 7.43205e9i 0.118581 − 0.672508i −0.866333 − 0.499466i \(-0.833529\pi\)
0.984915 − 0.173041i \(-0.0553594\pi\)
\(68\) 6.96316e8 − 3.94900e9i 0.0580774 − 0.329373i
\(69\) 0 0
\(70\) 4.25217e8 − 3.56799e8i 0.0302393 − 0.0253738i
\(71\) −8.45043e9 + 1.46366e10i −0.555850 + 0.962761i 0.441987 + 0.897022i \(0.354274\pi\)
−0.997837 + 0.0657391i \(0.979059\pi\)
\(72\) 0 0
\(73\) 1.24406e10 + 2.15477e10i 0.702367 + 1.21653i 0.967633 + 0.252360i \(0.0812067\pi\)
−0.265267 + 0.964175i \(0.585460\pi\)
\(74\) −3.72542e10 − 1.35594e10i −1.95164 − 0.710340i
\(75\) 0 0
\(76\) −6.61537e10 − 5.55095e10i −2.99281 − 2.51127i
\(77\) 7.76942e9 − 2.82784e9i 0.327107 − 0.119057i
\(78\) 0 0
\(79\) −4.77608e9 − 2.70865e10i −0.174632 − 0.990385i −0.938568 − 0.345094i \(-0.887847\pi\)
0.763937 − 0.645291i \(-0.223264\pi\)
\(80\) −4.34200e9 −0.148148
\(81\) 0 0
\(82\) −1.18818e11 −3.53922
\(83\) −4.25866e9 − 2.41521e10i −0.118671 − 0.673015i −0.984867 − 0.173311i \(-0.944553\pi\)
0.866196 − 0.499704i \(-0.166558\pi\)
\(84\) 0 0
\(85\) 3.56653e8 − 1.29811e8i 0.00871851 − 0.00317328i
\(86\) 9.21686e10 + 7.73386e10i 2.11272 + 1.77278i
\(87\) 0 0
\(88\) −1.24770e11 − 4.54124e10i −2.52031 − 0.917318i
\(89\) −1.26135e10 − 2.18473e10i −0.239437 − 0.414717i 0.721116 − 0.692815i \(-0.243630\pi\)
−0.960553 + 0.278097i \(0.910296\pi\)
\(90\) 0 0
\(91\) 5.81183e9 − 1.00664e10i 0.0976304 − 0.169101i
\(92\) −1.36307e11 + 1.14375e11i −2.15617 + 1.80924i
\(93\) 0 0
\(94\) 1.10766e10 − 6.28184e10i 0.155669 − 0.882844i
\(95\) 1.41937e9 − 8.04963e9i 0.0188198 − 0.106732i
\(96\) 0 0
\(97\) −6.21527e10 + 5.21523e10i −0.734878 + 0.616636i −0.931457 − 0.363852i \(-0.881461\pi\)
0.196579 + 0.980488i \(0.437017\pi\)
\(98\) 7.33776e10 − 1.27094e11i 0.820012 − 1.42030i
\(99\) 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 81.12.e.a.10.32 192
3.2 odd 2 27.12.e.a.13.1 ✓ 192
27.2 odd 18 27.12.e.a.25.1 yes 192
27.25 even 9 inner 81.12.e.a.73.32 192
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.12.e.a.13.1 ✓ 192 3.2 odd 2
27.12.e.a.25.1 yes 192 27.2 odd 18
81.12.e.a.10.32 192 1.1 even 1 trivial
81.12.e.a.73.32 192 27.25 even 9 inner