Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [27,9,Mod(2,27)] 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("27.2"); S:= CuspForms(chi, 9); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(27, base_ring=CyclotomicField(18)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 9, names="a")
 
Level: \( N \) \(=\) \( 27 = 3^{3} \)
Weight: \( k \) \(=\) \( 9 \)
Character orbit: \([\chi]\) \(=\) 27.f (of order \(18\), degree \(6\), 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: \(10.9992224717\)
Analytic rank: \(0\)
Dimension: \(138\)
Relative dimension: \(23\) over \(\Q(\zeta_{18})\)
Twist minimal: yes
Sato-Tate group: $\mathrm{SU}(2)[C_{18}]$

Embedding invariants

Embedding label 2.16
Character \(\chi\) \(=\) 27.2
Dual form 27.9.f.a.14.16

$q$-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} +(37.9905 - 71.5383i) q^{3} +(-49.7072 - 18.0920i) q^{4} +(-245.194 - 292.211i) q^{5} +(710.230 - 910.016i) q^{6} +(-336.792 + 122.582i) q^{7} +(-3812.44 - 2201.11i) q^{8} +(-3674.45 - 5435.54i) q^{9} +(-2718.13 - 4707.95i) q^{10} +(-2622.22 + 3125.04i) q^{11} +(-3182.67 + 2868.65i) q^{12} +(-689.271 - 3909.05i) q^{13} +(-5030.19 + 886.959i) q^{14} +(-30219.3 + 6439.54i) q^{15} +(-37686.4 - 31622.7i) q^{16} +(54109.2 - 31239.9i) q^{17} +(-38119.0 - 85380.6i) q^{18} +(69562.0 - 120485. i) q^{19} +(6901.26 + 18961.1i) q^{20} +(-4025.56 + 28750.4i) q^{21} +(-44536.3 + 37370.4i) q^{22} +(17740.4 - 48741.3i) q^{23} +(-302300. + 189114. i) q^{24} +(42564.2 - 241393. i) q^{25} -56568.9i q^{26} +(-528444. + 56364.9i) q^{27} +18958.7 q^{28} +(577603. + 101847. i) q^{29} +(-440062. + 15593.6i) q^{30} +(169959. + 61860.0i) q^{31} +(273734. + 326223. i) q^{32} +(123941. + 306311. i) q^{33} +(836727. - 304544. i) q^{34} +(118399. + 68357.9i) q^{35} +(84307.1 + 336664. i) q^{36} +(1.42753e6 + 2.47255e6i) q^{37} +(1.27446e6 - 1.51885e6i) q^{38} +(-305832. - 99197.4i) q^{39} +(291598. + 1.65374e6i) q^{40} +(-337957. + 59591.0i) q^{41} +(-127648. + 393547. i) q^{42} +(1.88422e6 + 1.58105e6i) q^{43} +(186882. - 107896. i) q^{44} +(-687373. + 2.40648e6i) q^{45} +(369606. - 640177. i) q^{46} +(-667331. - 1.83348e6i) q^{47} +(-3.69396e6 + 1.49466e6i) q^{48} +(-4.31769e6 + 3.62297e6i) q^{49} +(1.19477e6 - 3.28260e6i) q^{50} +(-179219. - 5.05770e6i) q^{51} +(-36460.6 + 206778. i) q^{52} -1.43620e7i q^{53} +(-7.55614e6 - 516680. i) q^{54} +1.55613e6 q^{55} +(1.55381e6 + 273979. i) q^{56} +(-5.97659e6 - 9.55362e6i) q^{57} +(7.85455e6 + 2.85882e6i) q^{58} +(-1.52519e7 - 1.81765e7i) q^{59} +(1.61862e6 + 226635. i) q^{60} +(-3.80007e6 + 1.38311e6i) q^{61} +(2.23227e6 + 1.28880e6i) q^{62} +(1.90382e6 + 1.38022e6i) q^{63} +(9.33162e6 + 1.61628e7i) q^{64} +(-973263. + 1.15989e6i) q^{65} +(981458. + 4.60576e6i) q^{66} +(3.89152e6 + 2.20699e7i) q^{67} +(-3.25481e6 + 573911. i) q^{68} +(-2.81291e6 - 3.12082e6i) q^{69} +(1.49255e6 + 1.25240e6i) q^{70} +(2.28067e7 - 1.31675e7i) q^{71} +(2.04437e6 + 2.88105e7i) q^{72} +(7.13836e6 - 1.23640e7i) q^{73} +(1.39163e7 + 3.82347e7i) q^{74} +(-1.56518e7 - 1.22156e7i) q^{75} +(-5.63754e6 + 4.73046e6i) q^{76} +(500068. - 1.37393e6i) q^{77} +(-4.04684e6 - 2.14908e6i) q^{78} +(1.03600e7 - 5.87545e7i) q^{79} +1.87661e7i q^{80} +(-1.60436e7 + 3.99453e7i) q^{81} -4.89067e6 q^{82} +(4.60108e7 + 8.11295e6i) q^{83} +(720251. - 1.35627e6i) q^{84} +(-2.23959e7 - 8.15145e6i) q^{85} +(2.25321e7 + 2.68527e7i) q^{86} +(2.92294e7 - 3.74515e7i) q^{87} +(1.68756e7 - 6.14222e6i) q^{88} +(-3.55252e7 - 2.05105e7i) q^{89} +(-1.56026e7 + 3.20736e7i) q^{90} +(711320. + 1.23204e6i) q^{91} +(-1.76365e6 + 2.10184e6i) q^{92} +(1.08822e7 - 9.80847e6i) q^{93} +(-4.82856e6 - 2.73841e7i) q^{94} +(-5.22633e7 + 9.21542e6i) q^{95} +(3.37367e7 - 7.18908e6i) q^{96} +(-3.64414e7 - 3.05780e7i) q^{97} +(-6.95642e7 + 4.01629e7i) q^{98} +(2.66215e7 + 2.77039e6i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 138 q - 6 q^{2} - 6 q^{3} - 6 q^{4} - 447 q^{5} - 774 q^{6} - 6 q^{7} - 9 q^{8} - 12960 q^{9} - 3 q^{10} + 28668 q^{11} + 77421 q^{12} - 6 q^{13} - 120975 q^{14} + 105507 q^{15} - 774 q^{16} - 9 q^{17}+ \cdots + 228876057 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/27\mathbb{Z}\right)^\times\).

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

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 0.594067 − 0.804416i \(-0.297521\pi\)
0.283114 + 0.959086i \(0.408632\pi\)
\(3\) 37.9905 − 71.5383i 0.469018 − 0.883189i
\(4\) −49.7072 − 18.0920i −0.194169 − 0.0706717i
\(5\) −245.194 − 292.211i −0.392311 − 0.467538i 0.533349 − 0.845896i \(-0.320933\pi\)
−0.925660 + 0.378358i \(0.876489\pi\)
\(6\) 710.230 − 910.016i 0.548017 − 0.702173i
\(7\) −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
\(9\) −3674.45 − 5435.54i −0.560044 − 0.828463i
\(10\) −2718.13 − 4707.95i −0.271813 − 0.470795i
\(11\) −2622.22 + 3125.04i −0.179101 + 0.213445i −0.848125 − 0.529797i \(-0.822268\pi\)
0.669023 + 0.743241i \(0.266713\pi\)
\(12\) −3182.67 + 2868.65i −0.153485 + 0.138341i
\(13\) −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
\(15\) −30219.3 + 6439.54i −0.596925 + 0.127201i
\(16\) −37686.4 − 31622.7i −0.575049 − 0.482524i
\(17\) 54109.2 − 31239.9i 0.647851 − 0.374037i −0.139781 − 0.990182i \(-0.544640\pi\)
0.787632 + 0.616145i \(0.211307\pi\)
\(18\) −38119.0 − 85380.6i −0.363121 − 0.813334i
\(19\) 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
\(21\) −4025.56 + 28750.4i −0.0206990 + 0.147832i
\(22\) −44536.3 + 37370.4i −0.190118 + 0.159528i
\(23\) 17740.4 − 48741.3i 0.0633946 − 0.174175i −0.903951 − 0.427636i \(-0.859347\pi\)
0.967346 + 0.253461i \(0.0815688\pi\)
\(24\) −302300. + 189114.i −0.911157 + 0.570005i
\(25\) 42564.2 − 241393.i 0.108964 − 0.617967i
\(26\) − 56568.9i − 0.123790i
\(27\) −528444. + 56364.9i −0.994360 + 0.106061i
\(28\) 18958.7 0.0308445
\(29\) 577603. + 101847.i 0.816652 + 0.143998i 0.566345 − 0.824168i \(-0.308357\pi\)
0.250308 + 0.968166i \(0.419468\pi\)
\(30\) −440062. + 15593.6i −0.543286 + 0.0192513i
\(31\) 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
\(33\) 123941. + 306311.i 0.104510 + 0.258290i
\(34\) 836727. − 304544.i 0.626135 − 0.227895i
\(35\) 118399. + 68357.9i 0.0789000 + 0.0455529i
\(36\) 84307.1 + 336664.i 0.0501943 + 0.200441i
\(37\) 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
\(39\) −305832. − 99197.4i −0.132198 − 0.0428787i
\(40\) 291598. + 1.65374e6i 0.113906 + 0.645991i
\(41\) −337957. + 59591.0i −0.119599 + 0.0210885i −0.233127 − 0.972446i \(-0.574896\pi\)
0.113528 + 0.993535i \(0.463785\pi\)
\(42\) −127648. + 393547.i −0.0410220 + 0.126474i
\(43\) 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
\(45\) −687373. + 2.40648e6i −0.167626 + 0.586857i
\(46\) 369606. − 640177.i 0.0825483 − 0.142978i
\(47\) −667331. − 1.83348e6i −0.136757 − 0.375737i 0.852343 − 0.522984i \(-0.175181\pi\)
−0.989100 + 0.147247i \(0.952959\pi\)
\(48\) −3.69396e6 + 1.49466e6i −0.695868 + 0.281565i
\(49\) −4.31769e6 + 3.62297e6i −0.748975 + 0.628465i
\(50\) 1.19477e6 − 3.28260e6i 0.191163 − 0.525215i
\(51\) −179219. − 5.05770e6i −0.0264913 − 0.747605i
\(52\) −36460.6 + 206778.i −0.00498667 + 0.0282808i
\(53\) − 1.43620e7i − 1.82016i −0.414429 − 0.910082i \(-0.636019\pi\)
0.414429 − 0.910082i \(-0.363981\pi\)
\(54\) −7.55614e6 − 516680.i −0.888638 − 0.0607640i
\(55\) 1.55613e6 0.170057
\(56\) 1.55381e6 + 273979.i 0.157996 + 0.0278590i
\(57\) −5.97659e6 − 9.55362e6i −0.566179 − 0.905042i
\(58\) 7.85455e6 + 2.85882e6i 0.694080 + 0.252624i
\(59\) −1.52519e7 − 1.81765e7i −1.25868 − 1.50004i −0.785208 − 0.619231i \(-0.787444\pi\)
−0.473474 − 0.880808i \(-0.657000\pi\)
\(60\) 1.61862e6 + 226635.i 0.124894 + 0.0174873i
\(61\) −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
\(63\) 1.90382e6 + 1.38022e6i 0.120855 + 0.0876168i
\(64\) 9.33162e6 + 1.61628e7i 0.556208 + 0.963380i
\(65\) −973263. + 1.15989e6i −0.0545226 + 0.0649775i
\(66\) 981458. + 4.60576e6i 0.0517244 + 0.242731i
\(67\) 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
\(69\) −2.81291e6 − 3.12082e6i −0.124096 − 0.137681i
\(70\) 1.49255e6 + 1.25240e6i 0.0621639 + 0.0521617i
\(71\) 2.28067e7 − 1.31675e7i 0.897489 − 0.518165i 0.0211042 − 0.999777i \(-0.493282\pi\)
0.876385 + 0.481612i \(0.159948\pi\)
\(72\) 2.04437e6 + 2.88105e7i 0.0760727 + 1.07207i
\(73\) 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
\(75\) −1.56518e7 − 1.22156e7i −0.494675 − 0.386074i
\(76\) −5.63754e6 + 4.73046e6i −0.168980 + 0.141791i
\(77\) 500068. − 1.37393e6i 0.0142255 − 0.0390841i
\(78\) −4.04684e6 − 2.14908e6i −0.109330 − 0.0580596i
\(79\) 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
\(81\) −1.60436e7 + 3.99453e7i −0.372701 + 0.927951i
\(82\) −4.89067e6 −0.108172
\(83\) 4.60108e7 + 8.11295e6i 0.969500 + 0.170949i 0.635905 − 0.771767i \(-0.280627\pi\)
0.333595 + 0.942716i \(0.391738\pi\)
\(84\) 720251. − 1.35627e6i 0.0144666 − 0.0272415i
\(85\) −2.23959e7 − 8.15145e6i −0.429036 − 0.156156i
\(86\) 2.25321e7 + 2.68527e7i 0.411916 + 0.490902i
\(87\) 2.92294e7 − 3.74515e7i 0.510202 − 0.653720i
\(88\) 1.68756e7 − 6.14222e6i 0.281403 − 0.102422i
\(89\) −3.55252e7 − 2.05105e7i −0.566208 − 0.326900i 0.189425 − 0.981895i \(-0.439337\pi\)
−0.755633 + 0.654995i \(0.772671\pi\)
\(90\) −1.56026e7 + 3.20736e7i −0.237808 + 0.488853i
\(91\) 711320. + 1.23204e6i 0.0103729 + 0.0179664i
\(92\) −1.76365e6 + 2.10184e6i −0.0246185 + 0.0293392i
\(93\) 1.08822e7 − 9.80847e6i 0.145473 − 0.131120i
\(94\) −4.82856e6 − 2.73841e7i −0.0618452 − 0.350742i
\(95\) −5.22633e7 + 9.21542e6i −0.641656 + 0.113141i
\(96\) 3.37367e7 − 7.18908e6i 0.397208 − 0.0846424i
\(97\) −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
\(99\) 2.66215e7 + 2.77039e6i 0.277136 + 0.0288404i
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 27.9.f.a.2.16 ✓ 138
3.2 odd 2 81.9.f.a.8.8 138
27.13 even 9 81.9.f.a.71.8 138
27.14 odd 18 inner 27.9.f.a.14.16 yes 138
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
27.9.f.a.2.16 ✓ 138 1.1 even 1 trivial
27.9.f.a.14.16 yes 138 27.14 odd 18 inner
81.9.f.a.8.8 138 3.2 odd 2
81.9.f.a.71.8 138 27.13 even 9